From 6cfd79d615970e5d902d17343de01f0f22807318 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Thu, 2 Apr 2026 00:30:56 +0100 Subject: [PATCH 001/116] Added new section for cloud-schemes and new doc within that for PC2. --- .../science_guide/cloud_schemes/index.rst | 21 +++++++++++++++++++ .../science_guide/cloud_schemes/pc2.rst | 15 +++++++++++++ 2 files changed, 36 insertions(+) create mode 100644 documentation/source/science_guide/cloud_schemes/index.rst create mode 100644 documentation/source/science_guide/cloud_schemes/pc2.rst diff --git a/documentation/source/science_guide/cloud_schemes/index.rst b/documentation/source/science_guide/cloud_schemes/index.rst new file mode 100644 index 0000000000..817ef3c101 --- /dev/null +++ b/documentation/source/science_guide/cloud_schemes/index.rst @@ -0,0 +1,21 @@ +.. ----------------------------------------------------------------------------- + (c) Crown copyright Met Office. All rights reserved. + The file LICENCE, distributed with this code, contains details of the terms + under which the code may be used. + ----------------------------------------------------------------------------- +.. _cloud_schemes_index: + +Cloud Schemes available in LFRic +================================ + +Contained here are descriptions of the cloud-schemes available in the +LFRic atmosphere model. By "cloud-scheme", we mean a scheme for calculating +the sub-grid cloud-fraction and water-content as a function of the +grid resolved variables. This requires making some assumptions about +the PDF of sub-grid moisture variability. + +.. toctree:: + :maxdepth: 1 + :glob: + + * diff --git a/documentation/source/science_guide/cloud_schemes/pc2.rst b/documentation/source/science_guide/cloud_schemes/pc2.rst new file mode 100644 index 0000000000..5de847ffb6 --- /dev/null +++ b/documentation/source/science_guide/cloud_schemes/pc2.rst @@ -0,0 +1,15 @@ +.. ----------------------------------------------------------------------------- + (c) Crown copyright Met Office. All rights reserved. + The file LICENCE, distributed with this code, contains details of the terms + under which the code may be used. + ----------------------------------------------------------------------------- + +.. _pc2: + +The PC2 cloud-scheme +==================== + +Here we describe the PC2 ("Prognostic Cloud 2") scheme. +As implied by the name, this scheme uses prognostic variables for +the sub-grid cloud-fractions (both liquid and ice) +and condensed water mixing-ratios. From 1b35c04cb23023198ed4fa40ee94d73b183ffaf6 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Thu, 2 Apr 2026 00:43:24 +0100 Subject: [PATCH 002/116] Imported the PC2 cloud-scheme latex source from the UMDPs. --- .../cloud_schemes/UMDP30_PC2CloudScheme.tex | 5734 +++++++++++++++++ 1 file changed, 5734 insertions(+) create mode 100644 documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.tex diff --git a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.tex b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.tex new file mode 100644 index 0000000000..d5173e6ad8 --- /dev/null +++ b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.tex @@ -0,0 +1,5734 @@ +\documentclass{UMDP_article} + +\title{The PC2 Cloud Scheme} +\paperno{030} +\umversion{14.0} +\owner{Cyril Morcrette} +\author{D.~Wilson, A.~Bushell, C.~Morcrette, V.~Varma$^{1}$, M.~Whitall} + +\titlecontent{ + \footnotesize + $^1$ National Institute of Water and Atmospheric Research, + Wellington, New Zealand \\ + } + +\usepackage{textcomp} +\usepackage{amstext,natbib} + +% Packages needed for the UM subroutine tree diagram in um_call_tree.txt: +\input{../029/um_call_tree_preamble} + +\newcommand{\mmax}[1] {\mbox{\footnotesize \sf MAX} \left[#1\right] } + +%%% These are definitions used in the convection documentation +%%% Definitions running across several chapters are defined in PC2_main. +\newcommand{\gbmll}{\ensuremath{\overline{l}_{\rm{l}}}} +\newcommand{\gbmlf}{\ensuremath{\overline{l}_{\rm{f}}}} +\newcommand{\gbmlm}{\ensuremath{\overline{l}_{\rm{m}}}} +\newcommand{\gbmlx}{\ensuremath{\overline{l}_{\rm{x}}}} +\newcommand{\incml}[1]{\ensuremath{\overline{l}_{\rm c}^{\rm #1}}} +\newcommand{\bigqc}[1]{\ensuremath{Q_{\rm C}^{\rm #1}}} +\newcommand{\GpdfB}{\ensuremath{G^{\rm B}}} +\newcommand{\injectcl}{\ensuremath{C_{\rm{l}}^{\ast}}} +\newcommand{\injectcf}{\ensuremath{C_{\rm{f}}^{\ast}}} +% +\newcommand{\lsubsup}[2]{\ensuremath{l_{\rm{#1}}^{\rm{#2}}}} +\newcommand{\csubsup}[2]{\ensuremath{l_{\rm{#1}}^{\rm{#2}}}} +\newcommand{\qsubsup}[2]{\ensuremath{q_{\rm{#1}}^{\rm{#2}}}} +\newcommand{\tsubsup}[2]{\ensuremath{T_{\rm{#1}}^{\rm{#2}}}} +\newcommand{\thsubsup}[2]{\ensuremath{\theta_{\rm{#1}}^{\rm{#2}}}} +\newcommand{\xsubsup}[2]{\ensuremath{{\chi}_{\rm{#1}}^{\rm{#2}}}} +%%% Private definitions (shorthand commands etc.) +\newcommand{\lc}{\left[} +\newcommand{\rc}{\right]} +\newcommand{\ov}{\overline} +\newcommand{\lp}{\left(} +\newcommand{\rp}{\right)} +\newcommand{\dr}{\partial} + +%%% Define a format for partial derivatives +\newcommand{\pardbyd}[2]{\ensuremath{\frac{\partial \, #1}{\partial \, #2}}} + +%%% Define a few of the commonest variables +\newcommand{\Tliq}{\ensuremath{T_{\rm L}}} +\newcommand{\qtot}{\ensuremath{q_{\rm T}}} +\newcommand{\qsat}{\ensuremath{q_{\rm s}}} +\newcommand{\aliq}{\ensuremath{a_{\rm L}}} +\newcommand{\gbmTliq}{\ensuremath{\overline{T}_{\rm L}}} +\newcommand{\gbmqtot}{\ensuremath{\overline{q}_{\rm T}}} +%Add a new command to do a horizontal line +\newcommand{\HRule}{\rule{\linewidth}{1.0mm}} + +\begin{document} % Every document must start with this. +\maketitle + +\tableofcontents + +\newpage +\section{PC2 developers} +We would like to acknowledge those who developed the PC2 cloud scheme: +Damian Wilson, Andrew Bushell, David Gregory, Amanda Kerr-Munslow, +John Edwards, Jeremy Price, Cyril Morcrette, Martin Sharpe, Thomas Mirfield, Ian Boutle. Many +others offered considerable help, advice and analysis, including Roy Kershaw, +Malcolm Brooks, Richard Forbes and Alejandro Bodas-Salcedo, and we would +like to thank them all for their input. + +\section{Introduction} + +This document describes the PC2 \textit{(prognostic cloud, +prognostic condensate)} cloud scheme. It should be seen as +a complete reference source for the scheme's physical assumptions, +numerical techniques, +application to the Unified Model and coding within the Unified Model. It does +not describe results from the scheme, please refer to the various reports and papers +written on this. Except where commented on explicitly, the description applies +to the PC2:66 version of the PC2 scheme, which is the version that will be +available at UM6.5. The version available at 6.4 is PC2:64. + +This paper will first introduce the concepts that underlie cloud schemes, +before developing a study of the theoretical behaviour of the prognostic PC2 scheme +under certain, well-defined, situations. The next sections shows how +the theory can be applied to the physical and dynamical processes +represented in the Unified Model. Finally, we outline the way in which +the PC2 scheme is implemented within the code of the Unified Model. + +%%\subsection{How to use this documentation} + +\subsection{Cloud schemes} + +The basic requirements of any cloud scheme within a large-scale model are to: +\begin{itemize} +\item{calculate the amount of condensation (from water vapour to liquid water or vice-versa) within each gridbox each timestep} +\item{to calculate or update the cloud fractions for use by the radiation and large-scale precipitation schemes (or any other physics scheme).} +\end{itemize} + +Depending on the model involved, cloud schemes may also treat the +deposition / sublimation process from vapour to ice. The problem is +straightforward to solve if one is allowed to assume that there is no +variability of moisture or temperature on a scale of a model gridbox. +In this case the cloud fraction scheme is redundant and only the condensation +part remains, which may be solved diagnostically using the instantaneous +condensation assumption in section \ref{sec:s_dist}. However, the +`no-variability' assumption is poor until +very high resolutions close to, or maybe exceeding, 1 km in the horizontal +are reached. Although we may eventually assume that computer power +will enable such resolutions to be reached globally, for many years we +will need a subgrid-scale cloud scheme to properly account for the +variability in the atmosphere. This is the principal challenge of +cloud parametrization. + +There are several approaches to take to the solution of the problem, +although they are not as independent as often portrayed, since they +nearly all require the same instantaneous condensation assumption +(discussed in section \ref{sec:s_dist}). Hence there are mathematical +links between +all the approaches. \textit{The following are all valid structures +to use in this respect.} +\begin{itemize} +\item{One may diagnose cloud fractions and condensate contents from knowledge of gridbox mean variables. This forms the basis of the \cite{smith90} scheme, which is described in +\citeumdp{029}.} +\item{A mixed scheme, such as \cite{sundqvist1978} uses a prediction of condensate contents, but a diagnostic cloud fraction.} +\item{Alternatively, one may predict cloud fraction and condensate content changes as a result of each modelled process. This forms the basis of the \cite{t93} scheme and the PC2 scheme.} +\item{Hybrid schemes, such as \cite{t02}, will predict various moments of the subgrid-scale variability, and use this knowledge to diagnose the cloud fraction and condensate contents.} +\end{itemize} + +Many years of experience of the results from the \cite{smith90} +scheme have highlighted deficiences in the diagnosis of cloud from +this scheme, which we feel can only be tackled by adding the +memory of cloud history available by using a prognostic based +scheme. We chose to develop a scheme that directly specified +the impacts on observable prognostics (condensates and +cloud fractions, as in \cite{t93}) rather than on moments of a +probability density function (as in \cite{t02}). This is because +we believe it is easier to physically relate (and hence parametrize) +the effect processes to quantities +such as cloud fraction and condensate rather than to the more abstract +quantities of moments of a probability +density function of moisture. However, although the +PC2 scheme is similar to \cite{t93} in its very basic prognostic variable +structure, the assumptions behind the formulation of the prognostic terms +in PC2 are very different and much improved. The PC2 scheme should not be +considered to be merely an extension of \cite{t93}. + +In particular, we wish to use a prognostic formulation in order to link +the detraiment of moisture from convection directly to cloud fraction, and +to break the hard diagnostic link between cloud fraction and condensate. These +major features of the \cite{t93} scheme provide the motivation to develop +the PC2 cloud scheme. + +\subsection{The `s' distribution} +\label{sec:s_dist} + +Most cloud schemes are based on the concept of a distribution +of fluctuations of moisture and temperature in the gridbox. Here we +mathematically formalize this concept, since it is used both in +the PC2 scheme and the \cite{smith90} scheme. + +This method was first formulated by \cite{m77} and \cite{sommeria_deardorff_1977} +for large-eddy simulations. It can also be applied +to larger scale models. It allows us to calculate vapour and liquid +contents and liquid cloud fraction from knowledge only of the combined +vapour+liquid content, $\overline{q_T}$, and the liquid temperature, +$\overline{T_L}$. These variables are unchanged during condensation +processes, so it is useful to write the cloud scheme in terms of these +variables. + +In this derivation we will consider only liquid condensate. +We assume, as above, that, +locally, the water content in a cloud is such as to remove any +supersaturation. This gives the equation + +\begin{equation} +q_{cl} = q_T - q_{sat}(T,p) +\label{eq:basic_qcl} +\end{equation} + +assuming that $q_T > q_{sat}(T,p)$ ($q_{cl}$ will be zero otherwise). +$q_T$ is the local total water +content, equal to the sum of the condensate $(q_{cl})$ plus the vapour +$(q)$, $T$ is the temperature, $p$ is the pressure and $q_{sat}(T,p)$ +is the saturation specific humidity at temperature T and pressure +p \textit{with respect to liquid water}. (Many earlier diagnostic +cloud schemes use a similar instantaneous condensation assumption +for ice, which would mean that $q_{sat}$ must be taken with respect +to ice when $T < 0 ^{\circ} C$, but the Unified Model does not). +We now introduce the liquid temperature ($T_L$), where $T_L$ is given by + +\begin{equation} +T_L = T - \frac{L}{c_p} q_{cl} , +\label{eq:tl} +\end{equation} + +and $L$ is the latent heat of +vaporization and $c_p$ is the heat capacity of air. +Note that $T_L$ is unaffected by changes of phase between vapour and liquid. +We now write (\ref{eq:basic_qcl}) as an \textit{equality} + +\begin{equation} +q_{cl} = q_T - \left( q_{sat}(T_L,p) + \alpha (T - T_L) \right) +\label{eq:alpha_t_tl} +\end{equation} + +where + +\begin{equation} +\alpha = \frac{ q_{sat}(T,p) - q_{sat}(T_L,p) }{T - T_L } . +\label{eq:alpha} +\end{equation} + +Using (\ref{eq:tl}) in (\ref{eq:alpha_t_tl}) gives the expression + +\begin{equation} +q_{cl} = q_T - q_{sat} (T_L) - \alpha \frac{L}{c_p} q_{cl} +\end{equation} + +or + +\begin{equation} +q_{cl} = a_L \left( q_T - q_{sat}(T_L,p) \right) +\label{eq:l_eq_al} +\end{equation} + +where $a_L$ is given by + +\begin{equation} +a_L = \left( 1 + \alpha \frac{L}{c_p} \right) ^{-1} . +\label{eq:a_L} +\end{equation} + +Thus (\ref{eq:basic_qcl}) has been rewritten \textit{exactly} in terms of the conserved +variables, $q_T$ and $T_L$, although the temperature, $T$, does remain in the +definition of $a_L$. +We will need to consider variations across a gridbox for a +parametrization scheme, +so we expand the expression for condensate (\ref{eq:l_eq_al}) into terms +relating to the gridbox mean and variation from the gridbox mean. + +\begin{equation} +q_{cl} = \overline{ a_L \left( q_T - q_{sat}(T_L,p) \right)} ++ [ a_L \left( q_T - q_{sat}(T_L,p) \right) ]' +\label{eq:bar_plus_pri1} +\end{equation} + +where $\overline{\phi}$ represents the mean of a distibution of $\phi$ and +$\phi = \overline{\phi} + {\phi}'$. The expression (\ref{eq:bar_plus_pri1}) +is \textit{exact} when +using the definition of $\alpha$ given in (\ref{eq:alpha}). + +The idea of a PDF scheme is to calculate the first (mean) term, +$\overline{\phi}$, from the known gridbox mean parameters, $q_T$, $T_L$ +and $p$, and to parametrize the distribution of the second, +variable term, ${\phi}'$. Unfortunately, the mean term is difficult +to write in terms of the gridbox mean variables $\overline{q_T}$ +and $\overline{T_L}$ because +$q_{sat}(T_L,p)$ is not a linear function of $T_L$ (or of $p$). In order to +proceed, we will now make an \textit{approximation} that $q_{sat}(T,p)$ +is a linear function of $T_L$ and $p$. +This equivalently implies that $a_L$ and $\alpha$ are approximated as +being constant across the gridbox. The expression +now becomes more tractable, (\ref{eq:bar_plus_pri1}) becoming: + +\begin{equation} +q_{cl} = a_L \left( \overline{q_T} - q_{sat}(\overline{T_L},\overline{p}) +\right) + a_L \left( {q_T}' - \alpha {T_L}' - \beta {p}' \right) +\label{eq:l_eq_bar_plus_pri} +\end{equation} + +where $\beta = {\frac{\partial q_{sat}}{\partial p}}$ at constant +temperature. The first term is connected with the mean properties of +the gridbox, and is written as $Q_c$, the second term is connected with the +deviation of the local conditions from the mean and is written as +$s$. + +\begin{equation} +Q_c = a_L \left( \overline{q_T} - q_{sat}(\overline{T_L},\overline{p}) \right) +\label{eq:qc_eq_qt-qs} +\end{equation} + +\begin{equation} +s = a_L \left( {q_T}' - \alpha {T_L}' - \beta {p}' \right) +\label{eq:s} +\end{equation} + +This gives the equation + +\begin{equation} +q_{cl} = Q_c + s +\label{eq:l_qc_s} +\end{equation} + +with the assumption that $s \ge -Q_c$ (i.e. $q_{cl} \ge 0$). +If $s < -Q_c$ then $q_{cl} = 0$. The term $a_L$ can be calculated +using (\ref{eq:a_L}) +from (\ref{eq:alpha}) with gridbox mean temperatures, i.e. + +\begin{equation} +\alpha = \frac{ q_{sat}(\overline{T},\overline{p}) +- q_{sat}(\overline{T_L},\overline{p}) }{\overline{T} - +\overline{T_L} } . +\label{eq:alpha_mean} +\end{equation} + +This definition of $\alpha$ and $a_L$ will retrieve an \textit{exact} value for +the gridbox mean $\overline{q_{cl}}$ \textit{if} +the distribution is monodispersed. Hence it is the sensible form to use for a +purely diagnostic representation such as \cite{smith90} where we explicitly +consider distributions of $s$. Strictly, the linear approximation +implies that other +approximations for $\alpha$ are valid: PC2 will do this +(see section \ref{sec:homog_num_app}) since we are concerned in PC2 with +the best estimate of the \textit{changes} to $\overline{q_{cl}}$, not the best +estimate of $\overline{q_{cl}}$ itself. + +We now assume that within +any particular gridbox a distribution $G$ of $s$ occurs (with mean, by +definition, of zero). Considering cloud to be where the water content +is greater than zero (i.e. where $s > -Q_c$) gives an expression for the +liquid cloud \textit{volume} fraction, $C_l$, within the gridbox as + +\begin{equation} +C_l = \int_{s=-Q_c}^{\infty} G(s) ds +\label{eq:int_gs_ds} +\end{equation} + +and the expression for mean condensate, ${\overline{q_{cl}}}$, using +(\ref{eq:l_qc_s}) to expand $q_{cl}$, is + +\begin{equation} +\overline{q_{cl}} = \int_{s=-Q_c}^{\infty} (Q_c + s) G(s) ds . +\label{eq:qclbar=int} +\end{equation} + +If we know (parametrize) the PDF given by $G(s)$ then we can solve +for $C_l$ and $\overline{q_{cl}}$. Note that this distribution is in terms of +$s$, there is no need to know the three-dimensional distribution in terms +of three separate variables $q_T$, $T_L$ and $p$. This is the method used +by \cite{smith90}, where a +symmetric triangular distribution function is used. For further +information on the \cite{smith90} scheme, please refer to \citeumdp{029}. Physics and dynamics schemes hence only need to +provide increments to $\overline{q_T}$ and $\overline{T_L}$, +provided that a diagnostic scheme (such as \cite{smith90}) is called at some point in the +timestep to partition $\overline{q_T}$ into $\overline{q}$ and $\overline{q_{cl}}$, +to calculate the dry bulb temperature $\overline{T}$ (from $\overline{T_L}$ +and $\overline{q_{cl}}$) and to calculate the liquid +cloud fraction, $C_l$. The diagnostic scheme effectively allows a calculation of +condensation associated with any physical process. However, its results remain +tied to the distribution of $G(s)$ that is chosen in (\ref{eq:int_gs_ds}) and +(\ref{eq:qclbar=int}) and it is this tie that we seek to break by the use of +a prognostic scheme. + +\subsection{Concept of PC2} + +The PC2 scheme develops prognostic expressions for the rates of change of +cloud fraction and condensate contents as a result of each process that acts in +the model. We consider ice and liquid condensate as two distinct +aspects of clouds, which may or may not overlap +Figure \ref{fig:schematic} provides a schematic summary of the PC2 scheme. +The equations for the five prognostic cloud variables can be written schematically: + +\begin{eqnarray} +\frac{\partial \overline{q_{cl}}}{\partial t} = +\frac{\partial \overline{q_{cl}}}{\partial t} |_{advection} + +\frac{\partial \overline{q_{cl}}}{\partial t} |_{convection} + +\frac{\partial \overline{q_{cl}}}{\partial t} |_{boundary \, layer} + +\frac{\partial \overline{q_{cl}}}{\partial t} |_{precipitation} + ... \nonumber \\ +\frac{\partial \overline{q_{cf}}}{\partial t} = +\frac{\partial \overline{q_{cf}}}{\partial t} |_{advection} + +\frac{\partial \overline{q_{cf}}}{\partial t} |_{convection} + +\frac{\partial \overline{q_{cf}}}{\partial t} |_{boundary \, layer} + +\frac{\partial \overline{q_{cf}}}{\partial t} |_{precipitation} + ... \nonumber \\ +\frac{\partial C_l}{\partial t} = +\frac{\partial C_l}{\partial t} |_{advection} + +\frac{\partial C_l}{\partial t} |_{convection} + +\frac{\partial C_l}{\partial t} |_{boundary \, layer} + +\frac{\partial C_l}{\partial t} |_{precipitation} + ... \nonumber \\ +\frac{\partial C_i}{\partial t} = +\frac{\partial C_i}{\partial t} |_{advection} + +\frac{\partial C_i}{\partial t} |_{convection} + +\frac{\partial C_i}{\partial t} |_{boundary \, layer} + +\frac{\partial C_i}{\partial t} |_{precipitation} + ... \nonumber \\ +\frac{\partial C_t}{\partial t} = +\frac{\partial C_t}{\partial t} |_{advection} + +\frac{\partial C_t}{\partial t} |_{convection} + +\frac{\partial C_t}{\partial t} |_{boundary \, layer} + +\frac{\partial C_t}{\partial t} |_{precipitation} + ... , +\label{eq:dqcldt_and_dcdt} +\end{eqnarray} + +where $\overline{q_{cf}}$ is the ice water specfic humidity, $C_l$ is the liquid +cloud \textit{volume} fraction, $C_i$ is the ice cloud volume fraction, and $C_t$ +is the combined ice or liquid cloud volume fraction. The amount of mixed +phase cloud, $C_{mp}$, can be calculated by the overlap of the ice +and liquid fractions: + +\begin{equation} +C_{mp} = C_i + C_l - C_t. +\label{eq:mp} +\end{equation} + +The idea is to parametrize each of the terms in the above equations. This +approach removes the diagnostic method, hence it will be critical that +we can write expressions for $\frac{\partial \overline{q_{cl}}}{\partial t}$ +and +$\frac{\partial C_l}{\partial t}$ for \textit{each process that alters +$\overline{T}$, $\overline{p}$, +$\overline{q}$, or $\overline{q_{cl}}$ in the model} (and similarly +for the ice terms). In doing so, +we will not lose sight of underlying PDF approach given by +(\ref{eq:int_gs_ds}) and (\ref{eq:qclbar=int}) since we will still use +the concept of +instantaneous condensation for liquid clouds. Equations +\ref{eq:int_gs_ds} and \ref{eq:qclbar=int} will form the +basis of the homogeneous forcing methods discussed in section +\ref{sec:homog}. +We note in particular that the convective cloud fraction, +previously a quantity that is diagnosed separately from the +large-scale cloud fraction calculated by the \cite{smith90} scheme, +may, in PC2, be included as part of the large-scale +cloud fraction. This aspect is similar to the \cite{t93} approach. + +The final aim of PC2 is +that the parametrization of each term in (\ref{eq:dqcldt_and_dcdt}) +is performed by each part of the model that alters $\overline{T}$, +$\overline{p}$, +$\overline{q}$, $\overline{q_{cl}}$ or $\overline{q_{cf}}$ as an +integral part of that physics or dynamics scheme. +However, in this PC2 scheme we acknowledge that this will not be possible, +at least, not to begin with. +Hence we have specifically developed generic approaches +that can be used to calculate expressions for +$\frac{\partial \overline{q_{cl}}}{\partial t}$ and $\frac{\partial C_l} +{\partial t}$ . +These are referred to as Homogeneous forcing (section \ref{sec:homog}), +Injection source (or inhomogeneous forcing, section \ref{sec:inhomog}) +and Width Changing (section \ref{sec:width}). Two additional modules +are available to assist with PC2, liquid cloud initiaion (section +\ref{sec:init}) and the calculation of total cloud fraction changes +(section \ref{sec:ct}). At +the present time, only the large-scale precipitation (section +\ref{sec:precip}) scheme has been rewritten fully to use the PC2 +concept of prognostic cloud fractions. The existing mass-flux +convection scheme has been modified to enable calculation of the detrained +condensate, but direct modification to the cloud fraction is not +included. All other physics schemes use one of the generic +approaches below. + +\subsubsection{A note on convective cloud fraction} + +It was the original intention that PC2 be able to replace the two +separate diagnostic cloud fractions (large-scale and convective) with +a single cloud fraction, as in \cite{t93}. The hypothesis was that +by detraining cloud +directly from the convection scheme we would no longer need a separate +representation of this cloud type. Our experience with PC2 is that +this is not necessarily the case. We suspect that the basic reason is +that we are unable to truely represent the extreme PDF shapes that +result from convective activity. Additionally, we only create cloud +associated with the detrainment part of the convection scheme, assuming +that cloud associated with the active updraughts in convection is small. +This assumption is not necessarily applicable. Similar arguments, +and model results, come from analysis of the \cite{t93} and +\cite{t02} scheme (Ben Johnson, personal communication). We also note +that with two cloud fraction types and two different optical depths +it is possible to have a basic degree of representation of cloud inhomogeneity. + +Hence the code still exists to enable PC2 to be +run with or without a diagnostic convective cloud fraction, although PC2:66 +does not include a diagnostic term. More details are +in section \ref{sec:convec}. + +\section{Physical basis of the PC2 prognostic cloud scheme} + +In this section we will develop the physical models that PC2 uses in order +to calculate its prognostic increment terms. We will also consider the +numerical solution of the models. The way in which these are incorporated +into the Unifed Model will be discussed in section \ref{sec:um} + +\subsection{Instantaneous condensation} + +Liquid clouds in PC2 use the concept of instantaneous condensation. Hence the +`s' distribution methods are fully applicable to the development of the +equations that govern the parametrization of liquid cloud in PC2. We will +start by looking at changes to $\overline{q_{cl}}$ and $C_l$ when a +uniform forcing +is applied to a gridbox, under the assumption of instantaneous condensation. + +\subsection{Homogeneous forcing} +\label{sec:homog} + +We define the expression +\textit{uniform forcing} (or \textit{homogeneous forcing}) to refer to +changes in local values of $T_L$ +and $q_T$ that occur at a rate independent of the part of the +gridbox in which they are located. This implies that $G(s)$ will not alter +due to such a process. +Uniform forcing simply alters $Q_c$ in (\ref{eq:int_gs_ds}) and +(\ref{eq:qclbar=int}). +In the Unified Model, this concept will be applied to several different sets of +physics increments in order to calculate the condensation and cloud fraction +changes associated with each one, where the physics routine does not allow +the explicit calculation of condensation and cloud fraction changes by +another method. +Large-scale ascent may be considered a meteorological example of such +a process. By differentiating (\ref{eq:int_gs_ds}) and (\ref{eq:qclbar=int}) +with respect to time, assuming uniform forcing +(so ${\frac{\partial G}{\partial t}}$ terms are zero), we obtain + +\begin{equation} +{{\frac{\partial C_l}{\partial t}} = G(-Q_c) {\frac{\partial Q_c} +{\partial t} }.} +\label{dcdt} +\end{equation} + +\begin{equation} +{\frac{\partial \overline{q_{cl}}}{\partial t}} = +C_l {\frac{\partial Q_c}{\partial t}} +\label{dqcldt} +\end{equation} + +The quantity $G(-Q_c)$ is the value of the PDF +of $G$ at $s=-Q_c$, which defines the boundary between +the saturated and unsaturated parts of the distribution. + +If we wish to consider a prognostic cloud scheme with equations for +the rate of change of condensate and cloud fraction based upon (\ref{dqcldt}) +and (\ref{dcdt}) then we need to close (\ref{dcdt}) by specifying the +value of $G(-Q_c)$. We will choose to develop a parametrization for this +quantity based upon the quantities $C_l$, $\overline{q_{cl}}$ and the saturation +deficit, $SD$, rather than tie $G(-Q_c)$ to a process. +The saturation deficit is \textit{defined} here in the `s' +framework to be the first moment of the PDF for `s' values less than $-Q_c$. +In this way it is analogous to the liquid water content, $\overline{q_{cl}}$. +Appendix A of \cite{wg03} writes this \textit{definition} as + +\begin{equation} +{SD = - \int_{-\infty}^{-Q_c} {( s+Q_c ) G(s) ds}} +\label{SD} +\end{equation} + +and shows this is equivalent to + +\begin{equation} +{SD = a_L ( q_{sat}({\overline{T}},{\overline{p}}) - {\overline{q}} ) .} +\label{SD2} +\end{equation} + +The basis behind the parametrization for $G(-Q_c)$ is to consider +an underlying form of the distribution $G(s)$ near the $+b_s$ and +$-b_s$ ends. We borrow the notation of \cite{smith90} and refer +to a quantity $b_s$ that is the value of $s$ when a monomodal +distribution $G(s)$ just equals zero. We suppose that the +distribution G can be described as a power law near $s=b_s$. + +\begin{equation} +G(s) ~ \propto ~ {(-s + b_s)}^n +\label{eqn19} +\end{equation} + +provided $s 1$ then the distribution is narrowed. +For the liquid cloud fraction we therefore have + +\begin{equation} +C_l^{[n+1]} = \int_{s=-Q_c}^{\infty} \xi G(\xi s) ds . +\label{eq:c_l_xi} +\end{equation} + +If we transform variables to $s' = \xi s$ we can rewrite this integral as + +\begin{equation} +C_l^{[n+1]} = \int_{s'=-Q_c \xi}^{\infty} G(s') ds' . +\label{eq:c_l_xi2} +\end{equation} + +Hence the expression for $C_l^{[n+1]}$ is equivalent to using the same +distribution function $G(s)$ as for $C_l^{[n]}$ except that the saturation +boundary has been moved from $-Q_c$ to $-Q_c \xi$. The result is the same as applying +a homogeneous forcing (\ref{eq:deltac}) with a modified forcing, + +\begin{equation} +\Delta Q_c \equiv \xi Q_c - Q_c , +\label{eq:deltac_modified} +\end{equation} + +or the continuous version + +\begin{equation} +\frac{\partial Q_c}{\partial t} \equiv +Q_c \frac{\partial}{\partial t}(\xi - 1) . +\label{eq:xi_equiv} +\end{equation} + +We can write $\xi$ in a slightly more +informative way by linking it to the relative change in width of the PDF +$\frac{1}{b_s} \frac{\partial b_s}{\partial t}$. For a PDF that changes +its width, $\xi$ is defined as + +\begin{equation} +\xi = \frac{b_s}{b_s + \delta b_s} = \frac{1}{1 + \frac{\delta b_s}{b_s}}. +\label{eq:xi_equiv1} +\end{equation} + + +For an infintessimal timestep $\delta t$ we therefore have + +\begin{equation} +\xi = \frac{1}{1 + \frac{1}{b_s} \frac{\partial b_s}{\partial t} \delta t } +\label{eq:xi} +\end{equation} + +and hence, by expanding (\ref{eq:xi}) to give $\xi = 1 - \frac{1}{b_s} +\frac{\partial b_s}{\partial t} \delta t$ and using the homogeneous +forcing expression (\ref{dcdt}) with the modified forcing (\ref{eq:xi_equiv}), +we retrieve the continuous form + +\begin{equation} +\frac{\partial C_l}{\partial t} = - G(-Q_c) Q_c \frac{1}{b_s} +\frac{\partial b_s}{\partial t} . +\label{eq:dcdt_width} +\end{equation} + +A similar analysis can be performed for $\frac{\partial \overline{q_{cl}}} +{\partial t}$ from (\ref{dqcldt}) to give + +\begin{equation} +\overline{q_{cl}}^{[n+1]} = \frac{1}{\xi} \int_{s'=-Q_c \xi}^{\infty} +(- \xi Q_c + s') G(s') ds' . +\label{eq:qcl_xi2} +\end{equation} + +Again, this is equivalent to using the homogeneous forcing with the modified +forcing (\ref{eq:xi_equiv}), but it also includes a scaling +term $\frac{1}{\xi}$. In the infinitessimal limit, this scaling gives +a second term that is proportional to the value of the integral (i.e. +$\overline{q_{cl}}$). Hence we obtain the final continuous solution + +\begin{equation} +\frac{\partial \overline{q_{cl}}}{\partial t} = +(- C_l Q_c+\overline{q_{cl}}) \frac{1}{b_s} \frac{\partial b_s}{\partial t} . +\label{eq:dqcldt_width} +\end{equation} + +To close the solution, we need to parametrize $\frac{1}{b_s} +\frac{\partial b_s}{\partial t}$ , +which could be linked to the physics of the process that is occuring. Note +we don't need to calculate $b_s$ separately, just its \textit{fractional} +rate of change. Options for the parameterisation of +$\frac{1}{b_s}\frac{\partial b_s}{\partial t}$ +due to turbulent ``erosion'' are described in section \ref{sec:turb}, +along with the numerical methods used to integrate the equations. + + +\subsection{Initiation of cloud} +\label{sec:init} +In section \ref{sec:homog} we commented that the closure (\ref{eqn22}) for $G(-Qc)$ +is only valid if $C_l$ is not identically 0 or 1. If $C_l$ +is 0 or 1 we know that $G(-Q_c)$ is equal to 0 but we have lost the +information that will tell us when $G(-Q_c + \Delta Q_c)$ starts +differing from 0. Hence the homogeneous forcing equation set +(\ref{dcdt}), (\ref{dqcldt}) and (\ref{eqn22}) is not complete if +we start from a position where $C_l$ is 0 or 1. To complete this set, +we will need to define a width, $b_s$, to the PDF and provide an initiation +increment to $C_l$ and $\overline{q_{cl}}$ when the value of $-Q_c$ crosses +the limit of the distribution. There is more discussion in +\cite{wg03}. + +To initiate new partial cloud-cover (or new partial clear-sky), we essentially +call a diagnostic cloud scheme to initialise the prognostics +$C_l$ and $\overline{q_{cl}}$. +In the UM there is currently a choice of 2 different diagnostic cloud +schemes that can be used for this; either a version of the Smith scheme +(see UMDP 029), or the bimodal scheme (see UMDP 039). +These two options are described below... + +\subsubsection{Initiation using a ``Smith-like'' method} + +This option is selected by setting the UM namelist switch +{\bf i\_pc2\_init\_method = 1} (Smith). + +We will assume the same form of the PDF at its boundaries as is +assumed in the derivation of the $G(-Q_c)$ closure. For the high `$s$' end +of the PDF distribution we integrate the power law description in +(\ref{eqn19}) to obtain the expressions + +\begin{equation} +C_l = \frac{1}{2 b_s^{n+1}} (b_s + Q_c)^{n+1} , +\label{eq:initc} +\end{equation} + +\begin{equation} +\overline{q_{cl}} = \frac{1}{2 b_s^{n+1}} \frac{(b_s + Q_c)^{n+2}}{n+2} . +\label{eq:initqcl} +\end{equation} + +We now need to parametrize the PDF width $b_s$. Unlike the \cite{smith90} +scheme, this is the only +location in the PC2 cloud scheme where the width needs to be defined +for the liquid cloud (although see section \ref{sec:mp_depsub} for +a discussion of an equivalent width in the deposition / sublimation +relationship for ice cloud). We still choose to define $b_s$ in +terms of a critical relative humidity parameter, $RH_{crit}$. Like +the \cite{smith90} scheme (see \citeumdp{029}), +we define the value of $b_s$ as + +\begin{equation} +b_s = a_L (1 - RH_{crit}) q_{sat} (\overline{T_L}) . +\label{eq:bs} +\end{equation} + +Hence, if the parameter $n$ was the same in PC2 as the equivalent +in \cite{smith90}, the +initial creation of liquid cloud would follow precisely that diagnosed +by the \cite{smith90} scheme (assuming that the numerical implementation of +the calculation is the same). +Its subsequent behaviour in PC2, though, would be different, because +the subsequent physical processes that act are parametrized in different +ways. +Note: for some reason, the implementation in the UM uses a fixed value +of $n = 0$ (corresponding to a top-hat distribution) if a constant $RH_{crit}$ +profile is used, but instead sets $n = 1$ (a triangular distribution) +in the PC2 initiation calculation if a TKE-based variable $RH_{crit}$ is used. +In the latter case, $n = 0$ is still hardwired in the PC2 homogeneous forcing +calculations, so it is not handled consistently. + +An equivalent initiation scheme is required if $C_l$ is 1 and $Q_c$ +is being reduced - at some point we need to introduce clear sky into +the solution. Because we make the choice of symmetry (which could be +relaxed if we used different $RH_{crit}$ values for $C_l$ of 1 and $C_l$ of 0), +the problem is entirely equivalent to that of initiating +from $C_l = 0$, with the exception that $\overline{q_{cl}}$ is replaced by $SD$, +$C_l$ is replaced by $(1-C_l)$, and $Q_c$ is replaced by $-Q_c$. +We hence have the solution + +\begin{equation} +1 - C_l = \frac{1}{2 b_s^{n+1}} (b_s - Q_c)^{n+1} , +\label{eq:init1mc} +\end{equation} + +\begin{equation} +SD = \frac{1}{2 b_s^{n+1}} \frac{(b_s - Q_c)^{n+2}}{n+2} . +\label{eq:initSD} +\end{equation} + +The conversion between $SD$ and $\overline{q_{cl}}$ follows (\ref{SD2}). +We will choose, +as we do throughout PC2, to define $\alpha$ (and hence $a_L$) in terms of +$\frac{\partial q_{sat}(\overline{T})}{\partial t}$, although within +this diagnostic calculation of SD it might actually be better to use +the representation (\ref{eq:alpha}) used by the diagnostic \cite{smith90} scheme. + +\subsubsection{Numerical Application of the Smith method} +\label{sec:numapp_init} + +In order to calculate and compare the state of the model to $b_s$, +we first calculate $T_L$, $q_{sat}(\overline{T_L})$ and calculate +the mean relative total humidity, $RH_T$, where + +\begin{equation} +RH_T = \frac{ \overline{q} + \overline{q_{cl}} } {q_{sat}(\overline{T_L}) } . +\label{eq:rht} +\end{equation} + +We then assess whether initiation is required. There are only two +circumstances in which we wish to proceed further: +\begin{itemize} +\item{If the current cloud fraction $C_l$ is 0 and $-Q_c < b_s$. By dividing the second condition by $a_L q_{sat} (\overline{T_L})$ we see, using the definitions (\ref{eq:qc_eq_qt-qs}) and (\ref{eq:bs}), that this second condition is equivalent to $RH_T > RH_{crit}$.} +\item{If the current cloud fraction $C_l$ is 1 and $-Q_c > -b_s$ (or, equivalently, $RH_T < 2 - RH_{crit})$.} +\end{itemize} +Note: in the UM implementation, the actual conditions for when initiation +may occur are more complicated than this, and there are several options +depending on a namelist switch. See section \ref{sec:init2} for details... + +In the second case, we then make the temporary transformation of variables +in order to use the same solution set as in the first case: $C_l'$ takes the +value $(1-C_l)$ and $RH_t'$ takes the value $(2-RH_t)$ (which is equivalent +to the replacing of $Q_c$ by $-Q_c$). In the first case, $C_l'$ and $RH_t$ take +the same values as $C_l$ and $RH_t$ respectively. + +We then solve for the initiated cloud fraction $C_l'$, using the similar +methods as described in \citeumdp{029}, except +that we allow the solution to vary with the PDF shape $n$. We first +write $Q_N$ as + +\begin{equation} +Q_N = \frac{Q_c}{b_s} = \frac{ a_L (\overline{q_T} - q_{sat}(\overline{T_L})) } +{ a_L (1 - RH_{crit}) q_{sat} (\overline{T_L})} = \frac{RH_T - 1}{1-RH_{crit}} +\label{eq:qn_def} +\end{equation} + +and then use $Q_N$ to solve the initiated cloud fraction. We assume a PDF +described by a power law as in (\ref{eqn19}) (and the equivalent for the +other end of the distribution, the two expressions switching at $Q_c=0$), +which is normalized. The solution to (\ref{eq:int_gs_ds}) is hence + +\begin{equation} +C_l^{init'} = \left\{ \begin{array}{ll} + 0, & Q_N \le -1 \\ + \frac{1}{2} {\left( 1 + Q_N \right)}^{n+1}, & -1 < Q_N \le 0 \\ + 1 - \frac{1}{2} {\left( 1 - Q_N \right)}^{n+1}, & 0 < Q_N < 1 \\ + 1, & 1 \le Q_N . + \end{array} \right. +\label{eq:c_qn} +\end{equation} + +where $C_l^{init'}$ is the initiated value of liquid cloud fraction. +If we had performed the variable transformation we then we need to +transform back, so $C_l^{init} = 1 - C_l^{init'}$, otherwise +$C_l^{init} = C_l^{init'}$. + +In practice, it is likely to be only the second of the +options in (\ref{eq:c_qn}) that the scheme uses, since we will be at +that end of the distribution function, unless previous +parts of the model timestep have resulted in large +forcings to $Q_c$. + +The solution for the initiated liquid water, $\overline{q_{cl}}^{init}$ is +more difficult, since it depends on the width of the distribution $b_s$, +hence on $a_L$ and $\alpha$, and $\alpha$ is a function of the dry-bulb +temperature $\overline{T}$, which is not known until we know the +amount of condensation. +Hence we will need to iterate to a solution. + +We first calculate $q_{sat}(\overline{T})$, $\alpha$, $a_L$ and $b_s$, using +(\ref{eq:alpha_exp}), (\ref{eq:a_L}) and (\ref{eq:bs}). We then solve for the liquid +water content: + +\begin{equation} +\frac{\overline{q_{cl}}^{init'}}{b_s} = \left\{ \begin{array}{ll} + 0, & Q_N \le -1 \\ + \frac{1}{2 (n+2)} {\left( 1 + Q_N \right)}^{n+2}, & -1 < Q_N \le 0 \\ + Q_N + \frac{1}{2 (n+2)} {\left( 1 - Q_N \right)}^{n+2}, & 0 < Q_N < 1 \\ + Q_N, & 1 \le Q_N . + \end{array} \right. +\label{eq:l_bar} +\end{equation} + +If we have been working in transformed variables we now transform +back, so the initiated saturation deficit, $SD^{init}$, takes the +value of $\overline{q_{cl}}^{init'}$. We then use (\ref{SD2}) to +estimate $\overline{q_{cl}}^{init}$ using our initial estimates of +$q_{sat}(\overline{T})$ and $a_L$. If we are not in transformed +variables, we have the first estimate +$\overline{q_{cl}}^{init}=\overline{q_{cl}}^{init'}$. + +We now use this estimate of $\overline{q_{cl}}^{init}$ to +calculate a more accurate estimate of $a_L$ etc. by iteration. In order to +achieve a faster convergence of the iteration, we do not use +(\ref{eq:l_bar}) directly in the estimation of $a_L$ etc., but +use a combination of this value and the one from the previous iteration. + +\begin{equation} +\overline{q_{cl}}^{init~[i+1]} = f \overline{q_{cl}}^{init~[i]} + +(1 - f) \overline{q_{cl}}^{init~[i-1]} +\label{eq:iter} +\end{equation} + +where the superscript $[i]$ labels each iteration. We find that 10 +iterations is effective for convergence, with the weighting +$f$ given by $a_L^{[i]}$. + +\subsubsection{Initiation using the bimodal scheme} +\label{sec:bimodal_init} + +This option is selected by setting the UM namelist switch +{\bf i\_pc2\_init\_method = 2} (Bimodal). + +First, the diagnosis of entrainment zones is performed at all grid-points, +as described in UMDP 039. The parameters of the moisture PDF are then +constructed, assuming either a sum of two Gaussian modes from the top +and bottom of an inversion layer (when within an entrainment zone), +or a single symmetric Gaussian mode (when not in an entrainment zone). +The variance of each Gaussian mode is estimated based on the TKE and other +information output by the boundary-layer scheme +(with a minimum limit applied to the PDF width, consistent with +$RH_{crit}$ = 99\%. +Crucially, each Gaussian mode is truncated to zero at plus and minus +3 standard deviations; this sets the overall width of the moisture +PDF at each point. + +The positions of the upper and lower truncated bounds of the moisture PDF +relative to the saturation threshold are expressed in terms of a normalised +$Q_N$ = $Q_c$ over PDF-width (see equation \ref{eq:qn_def}). +In entrainment zones, the sum of the two Gaussian modes can lead to +a highly skewed distribution; hence $Q_N$ can have different values for the +upper and lower bounds, each normalised by the different widths on +either side of the PDF. The upper and lower values of $Q_N$ are then +compared to -1 and 1 respectively, to determine whether the saturation +boundary lies within the PDF bounds. This is the basic condition for +initiation to occur (though there are additional conditions and various +options for these in the soure code; see section \ref{sec:init2}). + +If the initiation conditions are met, the diagnostic bimodal cloud scheme code +is then called (see UMDP 039), and the diagnosed $C_l$ and $q_{cl}$ are used to +set the prognostic $C_l$ and $q_{cl}$. + +\subsection{Injection forcing} +\label{sec:inhomog} + +Injection forcing (sometimes referred to as inhomogeneous forcing) +uses another concept of how the underlying moisture PDF may change in +order to calculate a change in cloud fraction as a result of a known +injection of condensate into a gridbox. The term was developed in +order to be coupled with a modified mass-flux convection scheme, +but is first presented here in its basic form. + +We will assume a physical model whereby saturated air, containing +condensate, randomly replaces already existing air in the gridbox. +(Such a formulation is designed to represent air detrained from convection +replacing pre-existing air when averaged over a large horizontal domain). +\cite{bwg03} discusses the situation in more detail. Briefly, +we consider two parts to the distribution function $G(s)$. One part +represents the background air. This maintains its PDF shape (in terms +of absolute $q_T$ and $T_L$ values) because we assume it is \textit{randomly} +replaced, but will reduce in amplitude as it is replaced by a +second PDF representing the injected air. + +The fractional rate at which existing air +is replaced by the injected source air we will write as +$\frac{\partial{C_S}}{\partial{t}}$. Provided that only the liquid +phase exists (see section \ref{sec:multiple} for the extention to multiple phases), +we then note that the rate +of change of liquid cloud fraction and liquid water content in +the gridbox can be written in two parts: firstly the change +due to the background, and secondly the change due to the source. + +\begin{equation} +\frac{\partial{C_l}}{\partial{t}} = +- C_l \frac{\partial{C_S}}{\partial{t}} + \frac{\partial{C_S}}{\partial{t}} +\label{eq:dcdt_inhom} +\end{equation} + +\begin{equation} +\frac{\partial{\overline{q_{cl}}}}{\partial{t}} = +- \overline{q_{cl}} \frac{\partial{C_S}}{\partial{t}} ++ q_{cl}^S \frac{\partial{C_S}}{\partial{t}} +\label{eq:dqcldt_inhom} +\end{equation} + +where $q_{cl}^S$ is the liquid water content of the injected +air. Eliminating $\frac{\partial{C_S}}{\partial{t}}$ gives the +relationship + +\begin{equation} +\frac{\partial{C_l}}{\partial{t}} = \frac{1 - C_l}{q_{cl}^S +- \overline{q_{cl}}} Q4_l, +\label{eq:dcdt_inhom2} +\end{equation} + +where $Q4_l$ is the net (\textit{including} the liquid water +in the background distribution +that was randomally replaced) injection source change of $\overline{q_{cl}}$: + +\begin{equation} +Q4_l = \frac{\partial{\overline{q_{cl}}}}{\partial{t}} |_{injection \, source}. +\label{eq:q4} +\end{equation} + +We see that we do not need to know anything about the nature of +the two PDFs involved, except the assumption that the injected +PDF contains completely cloudy air. +This equation allows one to calculate the change in $C_l$ associated +with an injection source change of $\overline{q_{cl}}$ for the example of +convection. Modifications +to the mass-flux convection scheme for PC2 (far from trivial and discussed +in depth in section \ref{sec:convec}) +allow $Q4_l$ to be calculated ($q_{cl}^S$ is already available), +and (\ref{eq:dcdt_inhom2}) can then be used +to calculate the equivalent $C_l$ change. We note at this stage +that the denominator in (\ref{eq:dcdt_inhom2}), being the difference +in two terms that may be close to each other, may cause problems +when we attempt to numerically apply this equation. + +It is reasonable to ask what happens to the air in the distribution that +was replaced. In this mathematical representation of a single gridbox we +need not know anything other than that the air is displaced into a neighbouring +gridbox. In practical use with a mass-flux convection scheme we know more that +this air is displaced downwards in the column. We could reasonably calculate +the change in cloud fraction following the same methods as used to calculate +the change in $\overline{q}$ or the change in a tracer and we discuss this +later. + +\subsubsection{Multiple phases in the injection source} +\label{sec:multiple} + +The injection source formulation can be extended to multiple +phases of condensate. In practice, this will simply be the +two phases ice and liquid, although we need to recognize that +they can overlap with each other. \cite{wilson2001} provides the +background to the derivation and it is briefly presented below. + +We firstly rewrite (\ref{eq:dqcldt_inhom}) but use the net +condensate ($\overline{q_c} = \overline{q_{cl}} + \overline{q_{cf}}$) instead of just the +liquid water expression, and the net cloud amount $C_t$, instead +of the liquid cloud amount $C_l$. The same argument as before leads +to the expressions + +\begin{equation} +\frac{\partial{C_t}}{\partial{t}} = +- C_t \frac{\partial{C_S}}{\partial{t}} + \frac{\partial{C_S}}{\partial{t}} +\label{eq:dctdt_inhom} +\end{equation} + +and + +\begin{equation} +\frac{\partial{\overline{q_{c}}}}{\partial{t}} = +- \overline{q_{c}} \frac{\partial{C_S}}{\partial{t}} ++ q_{c}^S \frac{\partial{C_S}}{\partial{t}} . +\label{eq:dqcdt_inhom} +\end{equation} + +The left hand side of (\ref{eq:dqcdt_inhom}) is written as $Q4_c$. +$q_{c}^S$ is the in-cloud +condensate content (ice plus liquid) of the source. + +Hence eliminating $\frac{\partial{C_S}}{\partial{t}}$ we obtain + +\begin{equation} +\frac{\partial{C_t}}{\partial{t}} = \frac{(1-C_t)}{q_{c}^S - +\overline{q_{c}}} Q4_c . +\label{eq:dctdt_q4} +\end{equation} + +We will assume that the proportion of the injected volume that +contains liquid cloud can be written as $g_l$, and the proportion +that contains ice cloud can be written as $g_i$. Note that it +is not necessary to have $g_l + g_i = 1$ if there is mixed +phase cloud injected. We can write the change in \textit{liquid} cloud +fraction equivalently to (\ref{eq:dcdt_inhom}) as + +\begin{equation} +\frac{\partial{C_l}}{\partial{t}} = +- C_l \frac{\partial{C_S}}{\partial{t}} + g_l \frac{\partial{C_S}}{\partial{t}} . +\label{eq:dcldt_inhom} +\end{equation} + +Combining (\ref{eq:dcldt_inhom}) and (\ref{eq:dctdt_inhom}) by eliminating +$\frac{\partial{C_S}}{\partial{t}}$ gives + +\begin{equation} +\frac{\partial{C_l}}{\partial{t}} = \frac{g_l - C_l}{1 - C_t} +\frac{\partial{C_t}}{\partial{t}} +\label{eq:dctdt_dcdt} +\end{equation} + +and hence from (\ref{eq:dctdt_q4}) we have the result + +\begin{equation} +\frac{\partial{C_l}}{\partial{t}} = +\frac{g_l - C_l}{q_c^S - \overline{q_{c}}} Q4_c . +\label{eq:dcltdt_almost_final} +\end{equation} + +An equivalent expression holds for the ice cloud. Hence the change in the +amount of cloud for each phase may be calculated assuming we know +the volume proportions of the source term that contain each of +the phases and the net increase in the amount of condensate, $Q4_c$ +(regardless of phase). This expression is coded for use in +a generically available inhomogeneous forcing module. However, +we can also write this in a slightly more +accessible form by noting the ratio of (\ref{eq:dqcdt_inhom}) and +(\ref{eq:dqcldt_inhom}) with the $Q4$ definitions following (\ref{eq:q4}). + +\begin{equation} +\frac{Q4_c}{q_c^S - \overline{q_c}} = +\frac{Q4_l}{q_{cl}^S - \overline{q_{cl}}} . +\label{eq:q4_ratios} +\end{equation} + +Using (\ref{eq:q4_ratios}) in +(\ref{eq:dcltdt_almost_final}) gives the final expression + +\begin{equation} +\frac{\partial{C_l}}{\partial{t}} = +\frac{g_l - C_l}{q_{cl}^S - \overline{q_{cl}}} Q4_l +\label{eq:dctdt_final} +\end{equation} + +and similarly for the ice. Note that this expression accounts +for the possibility that liquid cloud is displaced from the +gridbox by added ice cloud. We can further write +$q_{cl}^S$ as a fraction of $q_{c}^S$ + +\begin{equation} +q_{cl}^S = h_l q_{c}^S +\label{eq:qcls_qcs} +\end{equation} + +where $h_l$ is the factor between them (i.e. the \textit{mass} fraction +of the injected condensate that is liquid). It is not necessary +in this theory to have $h_l$ equal to $g_l$: if a mixed +phase plume exists $g_l + g_i$ need not equal 1, but since +$h_l$ and its ice equivalent, $h_i$, refer to mass, $h_l + h_i$ +must equal 1. However, if we do not allow a mixed phase injection +(which is the case in the current mass-flux convection +scheme, where only one phase can be injected), $h_l$ and $g_l$ are equal (and either zero or one in the +current mass-flux convection scheme) and we can write (\ref{eq:dctdt_final}) as + +\begin{equation} +\frac{\partial{C_l}}{\partial{t}} = +\frac{ (\delta_{xl} - C_l) }{ \delta_{xl} q_{c}^S - \overline{q_{cl}} } +Q4_l +\label{eq:dctdt_xl} +\end{equation} + +where $\delta_{xl} = h_l = g_l$. Equivalent expressions exist for the +ice cloud fraction and total cloud fraction. + +\begin{equation} +\frac{\partial{C_i}}{\partial{t}} = +\frac{ (\delta_{xi} - C_l) }{ \delta_{xi} q_{c}^S - \overline{q_{cf}} } +Q4_i +\label{eq:dctdt_xi} +\end{equation} + +\begin{equation} +\frac{\partial{C_t}}{\partial{t}} = +\frac{ (1 - C_t) }{ q_{c}^S - \overline{q_{c}} } Q4_c +\label{eq:dctdt_xc} +\end{equation} + +with $\delta_{xi} = h_i = g_i$. These are the expressions that are used +within the convection scheme. It still remains to parametrize $\delta_{xl}$, +which is given by the convection scheme itself. This is discussed in +section \ref{sec:plume_phase}. + +\subsubsection{Numerical application} +\label{sec:multi_numapp} +The numerical application using (\ref{eq:dcltdt_almost_final}) may be performed +with a basic forward timestep. Each of the three cloud fractions can +be incremented, assuming we know $\Delta{\overline{q_{cl}}}$ and +$\Delta{\overline{q_{cf}}}$, as + +\begin{equation} +\Delta{C_t} = \frac{(1 - C_t)} {q_c^S - \overline{q_{cl}} - \overline{q_{cf}}} +( \Delta{\overline{q_{cl}}} + \Delta{\overline{q_{cf}}} ), +\label{eq:cft_ts} +\end{equation} + +\begin{equation} +\Delta{C_l} = \frac{ (g_l - C_l)} +{q_c^S - \overline{q_{cl}} - \overline{q_{cf}}} +( \Delta{\overline{q_{cl}}} + \Delta{\overline{q_{cf}}} ), +\label{eq:cfl_ts} +\end{equation} + +\begin{equation} +\Delta{C_i} = \frac{ (g_i - C_i)} +{q_c^S - \overline{q_{cl}} - \overline{q_{cf}}} +( \Delta{\overline{q_{cl}}} + \Delta{\overline{q_{cf}}} ). +\label{eq:cff_ts} +\end{equation} + +The application from within the convection scheme is slightly different. +We start with (\ref{eq:dctdt_xl}), but enforce two numerical restrictions +to avoid the equation set becoming ill-conditioned. Firstly, we limit +the denominator $q_c^S - \overline{q_{cl}}$ to a minimum value if we +are considering changes of the same phase as the injected source. + +\begin{equation} +\Delta C_l = \frac {\delta_{xl} - C_l} {\delta_{xl} \text{Max}( q_{c}^S +- \overline{q_{cl}} , q_c^{S0} ) + ( 1 - \delta_{xl} ) (-\overline{q_{cl}}) } +Q4_l +\label{eq:delta_cl} +\end{equation} + +where $q_c^{S0}$ is specified as $5 \times 10^{-5} kg \, kg^{-1}$. +The denominator also has an additional check. If its absolute +value is less than a tolerance value of $1 \times 10^{-10} kg \, kg^{-1}$ +then no change in cloud fraction will be considered. A similar +equation is used for the ice cloud and the change in total cloud fraction + + +\begin{equation} +\Delta C_i = \frac {\delta_{xi} - C_i} {\delta_{xi} \text{Max}( q_{c}^S +- \overline{q_{ci}} , q_c^{S0} ) + ( 1 - \delta_{xi} ) (-\overline{q_{cf}}) } +Q4_i +\label{eq:delta_ci} +\end{equation} + +\begin{equation} +\Delta C_t = \frac {1 - C_t} +{ \text{Max}(q_c^S - \overline{q_c} , q_c^{S0} ) } Q4_c . +\label{eq:delta_ct} +\end{equation} + +We now limit the change in cloud fraction to ensure that the cloud +fraction remains within its physical bounds. + +\begin{equation} +C_l^{[n+1]} = ( 0, C_l^{[n]} + \Delta C_l, 1) +\label{eq:delta_cl_conv_final} +\end{equation} + +and similar equations are used for $C_i^{[n+1]}$ and $C_t^{[n+1]}$. + +\subsubsection{A note on the implementation of the cloud fraction change} +\label{sec:conv_imp_note} + +Equation \ref{eq:dcdt_inhom2} has been derived assuming that the +only change in the cloud properties within the gridbox comes from +the detrainment of air from the convective plume (so that the injection +source is an appropriate model). Attention should be drawn to the fact +that this is not the only source of change from the convection scheme. +Two other terms require consideration, namely advection of the environmental +air downwards by compensating subsidence and the condensation resulting +from the adiabatic warming due to this subsidence. +The former is considered correctly in the calculation of +$\frac{\partial \overline{q_{cl}}}{\partial t}$, which corresponds to $Q4$. +However, the calculation of $\frac{\partial C_l}{\partial t}$ +is then performed using (\ref{eq:dcdt_inhom2}) and \textbf{incorrectly} assuming +that all the $\overline{q_{cl}}$ change comes from the detrainment. It is +possible to calculate directly the change in $C_l$ that should occur due to +the detrainment and compensating subsidence treated together, in the same way +that $\Delta \overline{q_{cl}}$ is calculated (see section +\ref{subsect:q4calculation}), and this is the way in which the cloud fraction +change \textbf{should} be done. +It is an unfortunate historical emphasis in the early development of PC2 +on the derivation of (\ref{eq:dcdt_inhom2}) that has led to the treatment +used within the Unified Model for the change in cloud fractions due to convection. + +The change in $\overline{q_{cl}}$ and $C_l$ due to the adiabatic warming +associated with the compensating subsidence is considered explicitly +in the model implementation (see +section \ref{sec:conv_homog}) for both $\overline{q_{cl}}$ and $C_l$ +after the rest of the convective process has been calculated. It is perhaps +arguable that if (\ref{eq:dcdt_inhom2}) is going to be applied then +the value of $Q4$ used in (\ref{eq:dcdt_inhom2}) should include this term. + +Any major future developments of PC2 for a mass-flux convection scheme would be +advised to consider whether it is appropriate to use (\ref{eq:dcdt_inhom2}) at +all. + +\subsection{Ice cloud and mixed phase regions} +\label{sec:ct} + +The homogeneous forcing, initiation and PC2 erosion sections described +above have only considered the generation and dissipation of liquid +clouds. Although the forcing methods will not influence the generation +and dissipation of ice cloud (which is primarily performed in the +large-scale precipitation scheme, section \ref{sec:precip}) we +are still left with the issue of how created or dissipated liquid +cloud overlaps with existing ice cloud in the gridbox. The +opposite situation, where changes in ice cloud are specified +and changes in the overlap with liquid cloud need to be calculated, +is also possible in PC2 (e.g. in the boundary layer, see +section \ref{sec:bl}). + +Here we +need a simple assumption to close the problem. The assumption +that we now choose is that liquid cloud fraction \textit{changes} are +\textit{minimally} overlapped with ice cloud fraction changes. +This choice is based upon observational evidence that mixed +phase cloud is relatively rare, and also on results from earlier PC2 +development that indicated less supercooled liquid water cloud than +is observed from ground-based lidar. + +With this assumption, the equation set becomes straightforward to +write down. We firstly consider that a change in liquid cloud fraction +$\Delta C_l$ is known and we wish to estimate the resulting change in +the total cloud fraction. There is, of course, no change in the ice +cloud fraction $C_i$, since, from our \textit{definitions} in +(\ref{eq:dqcldt_and_dcdt}) and (\ref{eq:mp}), this includes the mixed phase +contribution. Hence we write + +\begin{equation} +\Delta C_i = 0 . +\label{eq:deltaci_eq_0} +\end{equation} + +The change in the total cloud fraction, $C_t$ will depend upon +the sign of the change of the liquid cloud fraction. If +$\Delta C_l > 0$, then $\Delta C_t$ is going to be the same as $\Delta C_l$ +($C_l$ is being added with minimum overlap to $C_i$), unless the +gridbox becomes completely covered in cloud, when there is no +choice but to generate mixed phase cloud. Hence we have + +\begin{equation} +\Delta C_t = \text{Min} ( \Delta C_l , 1 - C_t ). +\label{eq:deltact_min} +\end{equation} + +If $\Delta C_l < 0$, then we still consider minimum overlap +of the \textit{changes} (this is so that the solution is reversible as +much as possible). Hence $\Delta C_t$ is going to be the same as +$\Delta C_l$ unless $C_l$ is reduced below the existing $C_i$, in +which case no more change to $C_t$ is possible. + +\begin{equation} +\Delta C_t = \text{Max} ( \Delta C_l , C_i - C_t ) , +\label{eq:deltact_min2} +\end{equation} + +remembering that both quantities in the maximum expression in +(\ref{eq:deltact_min2}) have negative values. + +We can write similar expressions if a known amount of ice cloud +is added or removed, and we need to calculate the effect on $C_t$. +Similar to the results above we have: + +\begin{equation} +\Delta C_l = 0 . +\label{eq:deltacl_eq_0} +\end{equation} + +and + +\begin{equation} +\Delta C_t = \left\{ \begin{array}{ll} + \text{Max} ( \Delta C_i , C_l - C_t ), & \Delta C_i < 0 \\ + \text{Min} ( \Delta C_i , 1 - C_t ), & \Delta C_i > 0 . + \end{array} \right. +\label{eq:deltact_min_array} +\end{equation} + +For completeness, we also present here the equation set for +random overlap of changes in liquid cloud with existing ice cloud. +We have, as before, + +\begin{equation} +\Delta C_i = 0 . +\label{eq:deltaci_eq_0_2} +\end{equation} + +For $\Delta C_l > 0$ additional liquid cloud is added +randomly to any location outside that of the current liquid +cloud. A proportion $\frac{1-C_t}{1-C_l}$ of this will be additionally +outside that of existing ice cloud. Hence the net change in +total cloud fraction can be written as + +\begin{equation} +\Delta C_t = \Delta C_l \frac{1 - C_t}{1 - C_l} . +\label{eq:deltact_ran1} +\end{equation} + +Similarly, if $\Delta C_l < 0$, the liquid cloud is removed +randomly from the existing liquid cloud. A proportion +$\frac{C_t - C_i}{C_l}$ of this is from liquid cloud that does not +overlap with existing ice cloud. Hence, + +\begin{equation} +\Delta C_t = \Delta C_l \frac{C_t - C_i}{C_l} . +\label{eq:deltact_ran2} +\end{equation} + +Equivalent equations to (\ref{eq:deltact_ran1}) and +(\ref{eq:deltact_ran2}) but with $C_l$ and $C_i$ swapped apply when +we need to estimate changes in $C_t$ from a known $\Delta C_i$, when +assuming random overlap. + +\subsubsection{Numerical Implementation} + +In general, although the situation does not occur within the current +implementation of PC2 , we might have increments to both $C_l$ and +$C_i$ simultaneously. Hence the implementation is to calculate +$\Delta C_t$ from the sum of that predicted by (\ref{eq:deltact_min}) +or (\ref{eq:deltact_min2}), and (\ref{eq:deltact_min_array}). +For the random overlap situation we also need to apply a check on +the denominator in (\ref{eq:deltact_ran1}) and (\ref{eq:deltact_ran2}) before +calculation, with the result set to the limit $\Delta C_t = 0$ +if the denominator is 0. For the minimum overlap situation a final check +is made that $C_t$ lies between 0 and 1, with the value being reset to +0 or 1 if not. + +\subsection{Forced convective cloud} + +Forced convective clouds are clouds that form at the top of a convective boundary layer +but are too shallow to reach their level of free convection (and become fully fledged +cumulus clouds). These clouds currently require special treatment because initiation +in PC2 uses the Smith scheme with a specified value of $RH_{crit}$ while the large $RH$ +variability associated with these clouds implies much lower values than are typically used. + +A profile of ``forced cloud fraction'', $C_{forced}$, is parametrized as +linearly varying with height between a cloud-base value, at the lifting +condensation level (LCL) from the convection diagnosis parcel ascent, and a cloud-top +value of 0.1 at the top of the capping inversion, $z_i^{top}$. The cloud-base +value of $C_{forced}$ varies linearly between 0.1 and 0.3 for cloud depths +between 100 m and 300 m based loosely on SGP ARM site observations \cite{zk13}. The +inversion top is taken to be the boundary layer depth, $z_h$ plus the inversion +thickness, $\Delta z_i$ parametrized following \cite{rb08} as: +\begin{equation} +\Delta z_i = 6.3 \, w_m^2 / \int_{z_h}^{z_h+\Delta z_i} b \, dz +\label{dz_param} +\end{equation} +where $w_m$ is the boundary layer velocity scale ($w_m^3 = u_*^3 + 0.25 w_*^3$) and $b$ is +the parcel buoyancy that is integrated over the depth of the inversion assuming a +piece-wise linear variation between grid-levels. Note that the constant in (\ref{dz_param}) +is the same as in \cite{rb08} because $6.3 = 2.5 * 4^{2/3}$ and $w_m^3$ differs by a factor of 4. + +The in-cloud water content at the top of the inversion is estimated using the water content +from the diagnostic parcel ascent (used to diagnose boundary layer type and trigger convection), +with linear interpolation used between the lifting condensation level and inversion +top. To allow for sub-adiabatic water content (due to lateral mixing or microphysical +processes) the in-cloud water content can be reduced by a factor, forced\_cu\_fac, that has been +set to 0.5 in GA7. + +These cloud fraction and water content profiles are then used as minimum values and +increments to $C$ and $\overline{q_{cl}}$ calculated if necessary. +This methodology can also optionally be applied to cloud layers diagnosed as cumulus, if the +boundary layer option to mix across the lifting condensation level is selected that generates +a cloud base transition zone thickness which is then treated analgously to the inversion +thickness above. + +Also, there is an option to treat the calculated forced cumulus cloud +fraction and water content as diagnostic quantities passed directly to +the radiation scheme as part of the ``convective'' cloud, instead of +using them to modify the prognostic ``large-scale'' cloud variables +$C$ and $\overline{q_{cl}}$. If this option is used, the convective +cloud fraction $CCA$ and water content $CCW$ output by the convection +scheme are updated, by taking the forced cumulus profiles as their +minimum allowed values. Note that only the convective cloud fields +passed to radiation are updated (i.e. the versions of $CCA$ and $CCW$ +that are stored in the model dump / D1 array). The UM code contains +other copies of the convective cloud fields that are only used for +diagnostics; these are {\em not} updated. + +The different options for how to treat forced cumulus cloud are +controlled by the cloud namelist input $forced\_cu$, and are +summarised below: + +\begin{itemize} +\item $forced\_cu = 0$: No treatment of forced cumulus clouds. +\item $forced\_cu = 1$: Forced cumulus cloud applied to +$C$ and $\overline{q_{cl}}$ only in dry-convective boundary-layers. +\item $forced\_cu = 2$: Forced cumulus cloud applied to +$C$ and $\overline{q_{cl}}$ in both dry-convective and +cumulus-capped boundary-layers. +\item $forced\_cu = 3$: Forced cumulus cloud applied to +$CCA$ and $CCW$ in both dry-convective and +cumulus-capped boundary-layers. +\end{itemize} + + +\subsection{Turbulence-driven production of subgrid scale liquid cloud} +\label{sec:turb_qcl_scheme} + +\subsubsection{Introduction}\label{sec:sgt_intro} + +\cite{fhfk14} developed a model for +subgrid liquid water production by turbulent motions. +Their method uses an exactly soluble +stochastic process to describe +subgrid relative humidity (RH) fluctuations. +The probability density function (PDF) of the fluctuations +can be diagnosed in terms of the local turbulent local state +and any pre-existing ice cloud. The +liquid cloud properties (cloud fraction and liquid water content) +can be then be calculated as truncated moments of the PDF. + +\cite{fhfk14} initially used their model to +understand and parametrize the results of Large Eddy Simulations (LES) +of shear-induced, Altostratus clouds. They obtained excellent +agreement between their theoretically predicted predicted mean +cloud properties and the bulk properties of the LES clouds. +Subsequently, their model has been used as the basis of +subgrid cloud initiation method for use in the Unified Model +in conjunction with the PC2 prognostic cloud scheme. In Section \ref{sec:sgt_model_describe} +we outline the model of \cite{fhfk14}. In Section \ref{sec:sgt_model_implement} +we described its implementation in the GCM. + + +\subsubsection{Model description} +\label{sec:sgt_model_describe} + +\cite{fhfk14} started from the equation for the dynamics of +ice supersaturation $S_i=e_v/e_{sat\;ice}-1$: +\begin{equation}\label{eqn:squires_eqn} + \frac{D S_i}{D t} = -b_i B_0 {\cal M}_1 S_i + -\left(\frac{\varepsilon}{L^2}\right)^{1/3}(S_i-S_E) + a_i w, +\end{equation} +where ${\cal M}_1$ is the first moment of ice particle size distribution (PSD), +$\varepsilon$ is the turbulent dissipation rate, $L$ is a prescribed mixing length +for the turbulence, $S_{\rm E}$ is the ice supersaturation of the +environment surrounding the cloud and $b_i,B_0$ and $a_i$ are function of $p$ and $T$ given by +\begin{eqnarray} + b_i &=& \frac{1}{q} + \frac{\epsilon L_s^2}{c_p R T^2}, \\ + B_0 &=& 4\pi C \left( \frac{\epsilon L_s^2}{K_a R T^2} + \frac{R T}{\epsilon e_{si} \psi} \right)^{-1}, \\ + a_i &=& \frac{g}{R T}\left( \frac{\epsilon L_s}{c_p T} - 1 \right), \\ +\end{eqnarray} +The first term on the right hand side of Eq.~\ref{eqn:squires_eqn} is +the sink of vapor due to depositional growth of ice crystals, the second +term models entrainment (mixing) of environmental air into the cloudy +volume and the third term is a source term due to vertical air motions. + +\cite{fhfk14} modeled vertical velocity as a white-noise process +with autocorrelation function: +\begin{equation} + \overline{w(t)w(s)} = \sigma_w^2 \tau_{\rm d} \delta(t-s), +\end{equation} +where $\delta$ is the Dirac distribution and the intensity of the +noise, $\sigma_w^2$, will be called the +standard derivation of the vertical velocity fluctuations (due to the white nature of +noise, a true expectation value $\overline{w^2}$ is not defined) and $\tau_{\rm d}$ +a Lagrangian decorrelation time define here by the relation used by \cite{rodean1997}: +\begin{equation} + \tau_{\rm d} = \frac{2\sigma_w^2}{\varepsilon C_0}, +\label{eqn:taud} +\end{equation} +where $C_0$ is a known constant. + +Because it is linear in $S_i$, Equation \ref{eqn:squires_eqn} can be solved exactly, +for any given realisation of the noise term. By averaging the solutions over the the noise +and taking a steady-state limit (see \cite{fhfk14} for details) it can be shown +that the solution PDF is Gaussian with mean and variance given by: +\begin{eqnarray} + \overline{S_i} &=& + S_{\rm E}\frac{ \left(\varepsilon/L^2\right)^{1/3} }{ b_i B_0 {\cal M}_1 + \left(\varepsilon/L^2\right)^{1/3} }. + \label{eqn:si_avg} \\ + \overline{S_i^2} &=& + \frac{a^2_{\rm i} \sigma^2_w \tau_{\rm d}}{ 2\left(b_i B_0 {\cal M}_1 + \left(\varepsilon/L^2\right)^{1/3}\right)}, + \label{eqn:si_var} +\end{eqnarray} + +Equation \ref{eqn:si_avg} and \ref{eqn:si_var} completely specify the PDF, $F(S_i)$, of +steady-state humidity variations for the subgrid model. The liquid cloud fraction and +liquid water mass mixing ratio are given by +\begin{eqnarray} + C_l^{sgt} &=& \int_{S_{i,wat}}^\infty d S_i F(S_i), \label{eqn:cloud_fraction} \\ + q_{cl}^{sgt} &=& q_{sat\;ice}\int_{S_{i,wat}}^\infty d S_i (S_i -S_{i,wat}) F(S_i) \label{eqn:cloud_liquid}, +\end{eqnarray} +where $S_{i,wat} = e_{sat\;wat}/e_{sat\;ice}-1$ is the value of ice +supersaturation at water saturation. +We use the superscription `$sgt$'(=`{\it s}ub{\it g}rid {\it t}urbulence') to indicate +that $C_l^{sgt}$ and $q_{cl}^{sgt}$ are values of cloud fraction and water content +diagnosed from a parametrization of small-scale turbulent processes. + + +\subsubsection{Model implementation and closure relations} +\label{sec:sgt_model_implement} + +To implement the model of Section \ref{sec:sgt_model_describe} in the +Unified Model, closure relations are needed for the quantities $\sigma_w^2$, +$\varepsilon$, $L$, $\tau_{\rm d}$ and $S_E$, subject to the constraining relationship +given by Eq. \ref{eqn:taud}. +In each model grid box, these parameters specify the subgrid PDF, $F(S_i)$, and +from this the liquid cloud fraction and water content produced by turbulence +can be found using Eqs \ref{eqn:cloud_fraction} and \ref{eqn:cloud_liquid}. + +In addition we need to make some assumptions about how the diagnosed values +$C_l^{sgt}$ and $q_{cl}^{sgt}$ relate to the model prognostic fields, $C_l$ and $q_{cl}$. +Two methods are available for doing this. In the simplest case, the diagnosed +values $C_l^{sgt}$ and $q_{cl}^{sgt}$ are just treated as increments to model prognostics +(option one, in Sec. \ref{sec:sgt_increments} below). +A more complex option (see option two, below) is to increment the +model fields via the PC2 Erosion functionality. + +\subsubsection{Closure relations}\label{sec:sgt_closures} + +The vertical velocity variance, $\sigma_w^2$, is available as a diagnostic from the +Boundary Layer scheme. Because the Boundary Layer scheme is called after the +Microphysics on each model timestep, the diagnostic value is stored in a (non-advected) +model prognostic field. The scheme will operate only where there is diagnosed turbulence, +i.e., non-zero $\sigma_w^2$. + +We take the mixing length scale, $L$, to be proportional to the vertical +grid spacing in each grid box: $L=\beta_{mix} \Delta z$, where $\Delta z$ is +calculated as the height different between the $\rho$-levels adjacent +to the given $\theta$-point. The parameter, $\beta_{mix}$, +is an adjustable constant that the user can define (see Section \ref{sec:sgt_options} below), +however it should be of order one. + +To obtain $\tau_{\rm d}$ we impose an eddy size constraint: +\begin{equation} + \tau_{\rm d} = \frac{L}{\sigma_w} = \beta_{mix} \frac{\Delta z}{\sigma_w} +\label{eqn:eddy_size} +\end{equation} +Eq.~\ref{eqn:taud} then determines the dissipation rate, $\varepsilon$, that is consistent +with the other parameters. The constant $C_0=10$ by default, but can be adjusted by the user. + +The scheme is limited to act only in grid boxes where $\tau_{\rm d}$ is less than a +prescribed value, $\tau_{d}^{max}$. The default is $\tau_d^{max}=1200\;{\rm sec}$, which typically +coincides with a couple of model timesteps. The motivation for this is that a +motion that takes longer than a few timestep to decorrelate will be partially resolved by +the dynamics and therefore cannot be considered as `subgrid' turbulence. + +Finally, where $T$, $p$ and $q$ appear in the expressions for $C_l^{sgt}$ and $q_{cl}^{sgt}$, +these are taken to be the grid box mean values. The first moment of the ice PSD, ${\cal M}_1$, +is found from the parametrization, due to \cite{fhbicc05}, described +in Section 4.1 of UMDP26. + + +\subsubsection{Options for incrementing model prognostics}\label{sec:sgt_increments} + +Using the information in Section \ref{sec:sgt_closures} to obtain closed expressions +for the subgrid PDF of $S_i$-fluctuations allows $C_l^{sgt}$ and $q_{cl}^{sgt}$ to be +calculated. These will be non-zero only where there is turbulence as diagnosed by the +Boundary Layer scheme (and hence non-zero $\sigma_w^2$). To calculate $C_l^{sgt}$ and $q_{cl}^{sgt}$ +the integrals in Eqs \ref{eqn:cloud_fraction} and \ref{eqn:cloud_liquid} are evaluated +numerically using discretisation based on user-specified number of bins. + +Given $C_l^{sgt}$ and $q_{cl}^{sgt}$, two options are available for relating these +to changes in the model prognostics: + +\paragraph{Option one: direct increments} + +The values of $C_l^{sgt}$ and $q_{cl}^{sgt}$ can be added as increments to the +model prognostic fields, $C_l$ and $q_{cl}$. In this case +\begin{eqnarray} + \left( \Delta C_l \right)_{sgt} &=& C_l^{sgt} \\ + \left( \Delta q_{cl} \right)_{sgt} &=& q_{cl}^{sgt}, \\ + \left( \Delta q \right)_{sgt} &=& -\left( \Delta q_{cl} \right)_{sgt}, \\ + \left( \Delta T \right)_{sgt} &=& \frac{L_v}{c_p} \left( \Delta q_{cl} \right)_{sgt}, \\ + \left( \Delta C \right)_{sgt} &=& C_l^{sgt} \\ +\end{eqnarray} +where the left hand sides denote the increments to $C_l$, $q_{cl}$, $T$ and the +total cloud fraction, $C$, due to +the subgrid scheme. Some bounds-checking is then applied to ensure that: +(a) the resultant cloud fractions to not exceed one; (b) the scheme does not +condense out more liquid than there is available moisture. + +\paragraph{Option two: PC2 Erosion method} + +Option one gives a simple method for incrementing the model prognostics, but +it gives rise to a potential inconsistency with the PC2 cloud scheme. This arises because +the subgrid production scheme can elevate cloud fraction to unity in grid boxes +that are subsequently diagnosed by PC2 Initiation to meet the criteria for clear-sky initiation. +PC2 then counteracts the scheme by removing some of the liquid cloud. To try to mitigate +against this issue, cloud fraction increments can be applied using PC2 Erosion. In this case: +\begin{eqnarray} + \left( \Delta q_{cl} \right)_{sgt} &=& q_{cl}^{sgt} - q_{cl}, \\ + \left( \Delta q \right)_{sgt} &=& -\left( \Delta q_{cl} \right)_{sgt}, \\ + \left( \Delta T \right)_{sgt} &=& \frac{L_v}{c_p} \left( \Delta q_{cl} \right)_{sgt}, \\ +\end{eqnarray} +where $q_{cl}$ is the liquid cloud amount prior to calling to the +turbulent production scheme. The cloud fraction increments are calculated +by calling PC2 Erosion with $\left( \Delta q_{cl} \right)_{sgt}$ as input. +See Section \ref{sec:turb} for details on how the PC2 Erosion process works. +This method gives cloud fraction increments that are consistent with +PC2 cloud scheme. + + +\subsubsection{Other user options}\label{sec:sgt_options} + +The following variables and logical switches are optional inputs: +\begin{enumerate} + \item The logical \verb!l_dcfl_by_erosion! provides a switch to + apply cloud fraction increments using PC2 Erosion. Defaults to {\it FALSE}. + \item Setting the logical \verb!l_mixed_phase_t_limit! to {\it TRUE} + allows the user to use the variable \verb!mp_t_limit! to define a temperature limit, $T_{max}$, + above which the scheme is not applied. The default is $T_{max}=0^\circ\;{\rm C}$, so the + scheme is only applied to cold clouds. + \item The input variable \verb!mp_tau_d_lim! defines the + upper limit, $\tau_d^{max}$, on the value of $\tau_d$ above which the scheme is not applied. + The default value is $\tau_d^{max}=1200.0$, so the scheme is not applied in grid boxes + where the decorrelation time scale exceeds $1200$ seconds. + \item \verb!nbins_mp! is the number of bins used in the discretisation of the integrals in + Eqs \ref{eqn:cloud_fraction} and \ref{eqn:cloud_liquid} for $C_l^{sgt}$ and $q_{cl}^{sgt}$. + The default value is $100$ bins. + \item \verb!mp_dz_scal! is the scale parameter, $\beta_{mix}$, in the definition of the mixing length, $L=\beta_{mix}\Delta z$. + \item \verb!mp_czero! defines the constant parameter $C_0$ (defaults to $C_0=10$). +\end{enumerate} + + +\section{Application to the Unified Model} +\label{sec:app_um} + +This section describes the way in which the physical concepts described in the above +section are applied to the sections of the Unified Model, in order to build +up the complete prognostic scheme. Description of the actual subroutines +themselves follow in section \ref{sec:code}. +Note that the large-scale precipitation +and convection schemes have considerable documentation below, since these +schemes have been heavily modified for PC2. The other schemes use generic +forcing scenarios, hence their desciption here is much shorter. Remember, +whenever a signficiant $\overline{T}$ or $\overline{q}$ change occurs, +PC2 must be able to +represent the corresponding condensation and changes in cloud fractions. + +\subsection{Radiation} +\label{sec:rad} + +The shortwave and longwave radiation schemes both alter the temperature +of the atmosphere, hence we need to calculate the corresponding condensation +and cloud fraction changes. For both shortwave and longwave, we use +the homogeneous forcing routines (section \ref{sec:homog}) +for $\overline{q_{cl}}$ and $C_l$, (using eqn. +\ref{eq:deltaqc_exp2} to calculate the $Q_c$ forcing) and then the method in +section \ref{sec:ct} to calculate $C_t$ changes. There is no +$\overline{q_{cf}}$ change associated with this process since the +deposition / sublimation process is performed within the large-scale +precipitation scheme (as it also is in the absence of PC2). + +It is reasonable to question whether homogeneous forcing is a reasonable +model to use when we know that a large proportion of the heating +associated with radiative transfer in the atmosphere comes from the +cloudy air and is not evenly spread across the gridbox. Possible developments +are discussed in section \ref{sec:homog_improve}. + +\subsection{Large-scale precipitation} +\label{sec:precip} + +Precipitation processes have a large effect on cloud fractions. Here we +present the simple physical models that are applied to the transfer +terms included in the large-scale precipitation scheme. They are also +presented within the large-scale precipitation documentation (\citeumdp{026}). + +The basis of the physical model is that microphysical transfer processes can +be calculated separately in different partitions of the model cloud +(i.e. mixed phase cloud, liquid phase cloud, ice phase cloud or clear sky). +However, processes may change the size of these partitions. We consider +here separately each process that is modelled in the large-scale +precipitation scheme. The changes in $\overline{q_{cl}}$, $\overline{q_{cf}}$ +and $\overline{q}$ remain mathematically the same as in the non-PC2 version +of the code (\citeumdp{026}), we only need +to introduce calculations for the changes in cloud fractions. We will see +that many of these +changes can be well modelled by assuming no change to the cloud fractions, +and the others by using simple assumptions. + +Although the model may use two ice prognostic ice categories, only +a single ice cloud fraction is stored, the assumption being that the +two ice categories are completely overlapped with each other. Graupel +is not considered to contribute to the ice cloud fraction. + +\subsubsection{Fall of ice} +\label{sec:lsp_fall} + +The fall of ice is the process that contributes most to the growth of +ice cloud fraction in the model. The model results are therefore sensitive +to the formulation of this process. We will make the basic assumption +that a trail of falling ice does not reduce the horizontal spread of +ice cloud fraction at a particular level (hence $\overline{q_{cf}}$ +that leaves a gridbox does not reduce $C_f$ in that gridbox). The in-cloud +ice content simply reduces due to the fall out of ice - it is the +sublimation term (section \ref{sec:mp_depsub}) that erodes the fall streaks. +However, ice that falls into a clear layer from above may increase +the ice cloud fraction. We parametrize this by considering the +horizontal overlap of ice clouds between two model layers, and the +fall speed of ice between them. We will assume an overlap that +is nearly, but not quite, maximum, the difference being dependent +upon the windshear and the time taken for ice to fall between the +levels. + +\begin{equation} +O^{[k,k+1]} = \text{Max}( C_{i}^{[k+1]} - C_i^{[k]} , 0) ++ w \frac{\Delta z^{[k]}}{v_i^{[k]}} +\label{eq:overhang} +\end{equation} + +where $O^{[k,k+1]}$ is the amount of ice cloud `overhanging' the current +(i.e. $k$'th) layer +from the layer above, $w$ is a parameter that is closely related to the +windshear, $\Delta z^{[k]}$ is the model layer thickness and $v_i^{[k]}$ is +the fallspeed of ice in the layer. $v_i^{[k]}$ is calculated in the microphysics +scheme and, if two ice prognostics are used, is the mass-weighted average fall +speed of the two categories. +The factor $\frac{\Delta z}{v_i}$ is simply the time +taken for the ice to fall through one model layer. Multiplying this +by the windshear would give an estimate to the amount +of overlap between a cloud source and its fall streak in the layer +below (it is an \textit{estimate} since we assume that the cloud +source is continuous and unbroken). Although it is quite possible within +PC2 to do this, to date we have not programmed this link, and we +use a constant, but tunable, value of $1.5 \times 10^{-4} s^{-1}$ for $w$. + +The change in $C_i$ over the timestep is then given by the overlap proportion +multiplied by the how much (in the vertical dimension) of the layer below +can be filled by ice in the timestep: + +\begin{equation} +\Delta C_i = \text{Max}(O^{[k,k+1]} , 1) \text{Min} (v_i \frac{\Delta t}{\Delta z^{[k]}} , 1) +\label{eq:lsp_fall} +\end{equation} + +where $\Delta t$ is the timestep. We now choose to assume a minimum overlap +between the liquid and the ice phases (as in section \ref{sec:ct}). + +\begin{equation} +\Delta C_t = \text{Min} ( \Delta C_i , A_{clear} ) +\label{eq:lsp_fall_ct} +\end{equation} + +where $A_{clear}$ is the proportion of the gridbox that has neither +ice nor liquid cloud present. + +\textbf{An inconsistency has been found in the way that the fall-of-ice term is linked to the globally constant ``wind-shear value'' +when calculting the ice cloud fraction overhang. Consequently, +the option not to use the ``wind shear value'' when calculating the overhang is available in +the UMUI (from version 7.6 onwards).} + +\subsubsection{Homogeneous nucleation} +\label{sec:lsp_homo} +This will freeze all supercooled liquid water when a temperature threshold +is exceeded. Hence we turn all existing liquid and mixed phase cloud to +ice cloud. The cloud fraction changes are: + +\begin{eqnarray} +C_l \leftarrow 0 \nonumber \\ +C_i \leftarrow C_t \nonumber \\ +\Delta C_t = 0. +\label{eq:lsp_homo} +\end{eqnarray} + +\subsubsection{Heterogeneous nucleation} +This process will freeze a small amount of supercooled liquid water, +regardless of the previous presence of ice cloud. This will mean that +previously existing `liquid-only' cloud is converted to mixed phase +cloud. These give the following changes: + +\begin{eqnarray} +\Delta C_l = 0 \nonumber \\ +C_i \leftarrow C_t \nonumber \\ +\Delta C_t = 0. +\label{eq:lsp_het} +\end{eqnarray} + +\subsubsection{Deposition and sublimation} +\label{sec:mp_depsub} +This term exerts one of the most important influences on the ice cloud in +the whole model (this applies to the control as well as for PC2). Contained +in the formulation is a subgrid-scale assumption that causes equivalent +effects to that for a moisture PDF under the `$s$' framework +(section \ref{sec:s_dist}). However, since ${q_{cf}}$ changes +slowly in response to local changes in $q$ and $T$, we cannot base the +$q_{cf}$ response on the same instantaneous condensation framework. It would +be useful to investigate in the future whether the two descriptions of the +moisture variability could be brought together. +Because of its importance, +we describe the method below, although we note it is also described in +\citeumdp{026}. + +We can calculate the local rate of change of $q_{cf}$, given local $T$ and $q$ +etc. using the standard microphysical growth equations (see \citeumdp{026}). +However, it is critical to know the way in which +the moisture is correlated with the ice in the gridbox. We will assume +there exists a distribution of vapour in the gridbox. We know that +the regions where liquid cloud exists must be saturated with respect +to liquid water, hence we need only consider the part of the gridbox +that does not have liquid water present. The average value, $q_a$, +of $q$ within the liquid-free part of the gridbox is thus + +\begin{equation} +q_a = \frac{ \overline{q} - C_l q_{sat \, liq}(\overline{T}) } {1 - C_l} +\label{eq:qa} +\end{equation} + +where we have assumed that the fluctuation of $q_{sat~liq}$ across +the gridbox due to temperature fluctuations is not significant compared +to the fluctuation of $q$ described below. We then parametrize a width, $b_i$, +to the $q$ (not $s$) fluctuations \textit{across the non-liquid cloud part +of the gridbox}, based upon $RH_{crit}$. This is like that for the `$s$' +distribution width, $b_s$ but modified: + +\begin{equation} +b_i = (1 - RH_{crit} ) q_{sat \, liq} ( 1 - \frac{1}{2} +~ \frac{\overline{q_{cf}}} {i q_{sat \, liq}(\overline{T})} ) . +\label{eq:b_i} +\end{equation} + +where the factor $( 1 - \frac{1}{2} +\frac{\overline{q_{cf}}} {i ~ q_{sat \, liq}(\overline{T})})$ should be limited +to a minimum value of zero, but, for numerical reasons, is limited to +a minimum value of 0.001. We note that $b_i$ has a similar form to $b_s$, +except the multiplier $a_L$ and the factor in brackets. If we remember +from (\ref{eq:s}) that the definition of `$s$' includes a factor $a_L$ +we see that the absence of the $a_L$ factor in (\ref{eq:b_i}) is +consistent. The factor in brackets is a \textit{parametrization} of the +effect that, when ice +is present, deposition in the moistier parts and sublimation in the +drier parts of the gridbox must reduce the width of the distribution +of $q$ across the gridbox. It is a simple linear function of +$\frac{\overline{q_{cf}}}{q_{sat~liq}(\overline{T})}$, and is tunable +using the factor $i$, which takes the value of 0.04. + +We note that this formulation isn't totally consistent with the liquid +cloud formulation, which considers an underlying PDF across the whole +gridbox and does not have, in general, its width prescribed. +Remember that we do not calculate on-line the whole of the liquid +- vapour PDF, we only parametrize the single point $G(-Qc)$, +using equation \ref{eqn22}). + +The width is then limited further to be no greater than $\overline{q}$, +to make sure that there are no negative values of $q$ predicted within the +gridbox (possible at low +temperatures where $q_{sat~liq}(\overline{T})$ diverges from +$q_{sat~ice}(\overline{T})$). + +We then calculate the average value of $q$ in the ice-only and clear-sky +partitions of the gridbox. To do this, we make the further assumption +that the ice is correlated with the moistest part of the distribution +(an instantaneous condensation formulation would make the +same assumption). Some algebra retrieves the expressions: + +\begin{eqnarray} +q_{clear} = q_a - b_i A_{ice} ; \\ +q_{ice} = \frac {\overline{q} - C_l q_{sat~liq} - A_{clear} q_{clear} } +{A_{ice}}, +\label{eq:q_clear_and_q_ice} +\end{eqnarray} + +where $A_{ice}$ is the proportion of the gridbox with ice cloud but not +liquid cloud and $A_{clear}$ is the proportion of the gridbox without cloud. +The numerical application will set $q_{clear}$ to $q_a$ if $A_{ice}$ +is zero. We now have a representation of the $q$ values in each of the +gridbox cloud partitions, and can solve the microphysical transfer equation +in each partition. + +The cloud fraction changes now need to be parametrized. We use the +model that deposition will \textit{not} adjust the ice cloud +\textit{fraction} (increases will be done within the fall-of-ice microphysics +section). However, deposition can +decrease the liquid cloud fraction (locally, $q$ can be reduced by +deposition to below $q_{sat~liq}$, hence this is not inconsistent with +the assumptions for the riming term below. This is the principal sink +of supercooled liquid cloud fraction in the model. Sublimation will +be allowed to decrease the ice cloud fraction (since sublimation cannot +act in liquid cloud, there is no impact on the liquid cloud). To solve +for these models, we will need to further split the ice-only partition +to give the proportion of that partition that is above and below ice +saturation. This gives, in general, an area of the gridbox $A_{ice1}$ +that contains ice and is above saturation where + +\begin{equation} + A_{ice1} = \frac{1}{2} A_{ice} + \frac{1}{2} + \frac{ (q_{ice}-q_{sat~ice}(\overline{T})) } {b_i}, +\label{eq:q_ice_above_sat} +\end{equation} + +having assumed that $A_{ice1}$ is between 0 and $A_{ice}$. +If not, it is trivial to partition the gridbox, since the moisture in the ice-only +partition is either completely above or completely below $q_{sat~ice}(\overline{T})$. +The corresponding +area that contains ice and is below saturation is given by +$A_{ice2} = A_{ice} - A_{ice1}$. +We can now parametrize the change in cloud fractions. For deposition, +we shall assume a uniform distribution of local values of $q_{cl}$ about +the local mean. If we assume a uniform removal of local $q_{cl}$ then, with +a little algebra, we can obtain an expression for the change in $C_l$: + +\begin{equation} +\Delta C_l = C_l ( 1 - \frac {\Delta \overline{q_{cl}}} {\overline{q_{cl}}} ) +^{\frac{1}{2}} - C_l +\label{eq:deltacfl_dep} +\end{equation}. + +Since this occurs only in the mixed phase part of the gridbox, we can say +that $\Delta C_t = 0$. We will also note that the change in $\overline{q_{cl}}$ +due to deposition is limited by the amount of $\overline{q_{cl}}$ that is in +the mixed phase partition in the gridbox, hence (\ref{eq:deltacfl_dep}), +although it formally allows removal of $C_l$ from an ice-free partition, will +be unlikely to do so. + +The sublimation forms the main method by which ice cloud is destroyed in PC2, +hence PC2 results are relatively sensitive to its formulation. Here we +make a similar assumption to that used for liquid in the deposition term, +except that we limit the changes only to the region of the gridbox where +ice is subliming. + +\begin{equation} +\Delta C_i = A_{ice2} ( 1 + \frac{\Delta \overline{q_{cf}} } +{ \overline{q_{cf}} ( \frac{A_{ice2}}{C_i} ) } + )^{\frac{1}{2}} - A_{ice2} +\label{eq:deltacfi_sub} +\end{equation}. + +The term $\overline{q_{cf}} ( \frac{A_{ice2}}{C_i} )$ is the amount of +$\overline{q_{cf}}$ that is present in the subliming ice region, hence its +ratio with $\Delta \overline{q_{cf}}$ is the fractional change in that region. The +change in the total cloud fraction must also be equal to the change above, +since sublimation cannot occur in the presence of liquid cloud: + +\begin{equation} +\Delta C_t = \Delta C_i . +\label{eq:deltacft_sub} +\end{equation} + +\subsubsection{Riming} +This process acts only where mixed phase cloud occurs - although, in theory, +it could remove any supercooled liquid totally, the air would remain +saturated with respect to liquid water. Hence any subsequent cooling would +regenerate the same amount of liquid cloud. Hence we choose to model this +process as having \textit{no effect} on the cloud fractions. + +\subsubsection{Capture} +This is the freezing of raindrops onto ice crystals by collision. This does +not alter the ice cloud \textit{fraction} in the gridbox (although it does +alter $\overline{q_{cf}}$, and it has no interaction with the liquid cloud. +Again, we therefore choose to model this process as having \textit{no effect} +on the cloud fractions. + +\subsubsection{Evaporation of melting ice} +Here we simply assume that ice cloud fraction is removed in proportion to the +ice content that is removed. + +\begin{equation} +\Delta C_i = C_i \frac{ \Delta \overline{q_{cf}}}{\overline{q_{cf}}} . +\label{eq:lsp_evapmeltsnow} +\end{equation} + +Because the evaporation cannot occur in the liquid part of the gridbox, +there is no change to $C_t$ (or to $C_l$). + +\subsubsection{Melting} +Again, the change in $C_i$ is calculated using the method +in (\ref{eq:lsp_evapmeltsnow}). + +\begin{equation} +\Delta C_i = C_i \frac{ \Delta \overline{q_{cf}}}{\overline{q_{cf}}} . +\label{eq:lsp_melt} +\end{equation} + +The change in $C_t$ is calculated assuming that there is no correlation +in the gridbox between where the ice melts and the liquid cloud. Hence +we must multiply (\ref{eq:lsp_melt}) by the proportion of ice cloud +fraction that exists without liquid cloud (i.e. $\frac{A_{ice}}{C_i}$). + +\begin{equation} +\Delta C_t = C_i \frac{ \Delta \overline{q_{cf}}}{\overline{q_{cf}}} +\frac{A_{ice}}{C_i} . +\label{eq:lsp_melt2} +\end{equation} + +\subsubsection{Evaporation of rain} +Evaporation of rain will not, \textit{on its own}, +generate liquid cloud, since a +large-scale lifting process will be required in order to condense water +from the moistened air. We cannot, therefore, allow any change in cloud +fractions to occur as a result, subsequent changes are calculated elsewhere +in the model (e.g. by the lifting process, section \ref{sec:pres}). + +\subsubsection{Accretion} +Accretion is the sweep-out of liquid water droplets by rain. We argue +in a similar way to the riming term, that this will not remove any liquid +cloud fraction, since a small amount of lifting will regenerate the same +amount of liquid cloud. Hence we choose to model this process as having +\textit{no effect} on the cloud fractions. The arguments underlying the +formulation of the evaporation of rain and the accretion cloud fraction +changes may appear to be inconsistent in their limiting cases and the +subsequent response to lifting. However, when the limiting case is +not reached the formulations are both correct. For the moment, it is +not considered necessary to increase the complexity of the current, +simple representations. + +\subsubsection{Autoconversion} +As for accretion, the generation of rain directly from collision +and coalescence of liquid water droplets will not alter the cloud fractions. + +\subsubsection{Other microphysics terms} +There are already (i.e. also in the control) +two numerical tidy-up terms at the end of the microphysics +section that remove small rain amounts and provide an additional +melting term for the snow. These do not change the cloud fractions. + +If there are small amounts of ice present at the end of the +microphysics then these are removed at the end of the microphysics +timestep (also in the control). PC2 responds by resetting the +cloud fractions appropriately, so $C_t$ is reset to $C_l$ etc. + +\subsubsection{Numerical implementation} + +Note that after each process has been applied, we do \textit{not} +recalculate the sizes of the ice-only, liquid-only and mixed phase +partitions, but use the values at the start of the microphysics (this +includes the values of $C_i$ used in the calculation of `in-cloud' +water contents above. However, we do update the cloud fractions +themselves sequentially. We also recalculate after each process +the overlaps between the rain fraction (see \citeumdp{026}) and the cloud fractions. + +There is also a final set of checks that $C_l$ and $C_i$ lie +between 0 and 1 and that $C_t$ is bounded between $\text{Max}(C_l, C_i)$ +(maximum overlap of liquid and ice) and $\text{Min}(C_l+C_i,1)$ +(minimum overlap of liquid and ice). + +We should note in particular, that these parametrizations allow a +considerable reduction in $\overline{q_{cl}}$ without a corresponding +large reduction in $C_l$. This is an underlying feature of the PC2 scheme +(discussed in \cite{wg03}), and necessarily implies the +skewing of the underlying moisture PDF. Subsequent parts of +the model (e.g. the width narrowing, section \ref{sec:width}) will, of +course, act on the modified fields to adjust the cloud fractions further, +but remember that these are separate processes and modelled elsewhere in the +timestep. + +\subsection{PC2 erosion} +\label{sec:turb} + +\subsubsection{Original width-narrowing method} +(selected by setting {\bf i\_pc2\_erosion\_method = 1} in the UM namelist). + +In parallel with the homogeneous forcing part of the PC2 response to +convection, we introduce +a new block of code that allows a background change of the PDF width. +At earlier versions of PC2 (PC2:65 and earlier) this block was included as +a separate section of code that was called in as part of the atmphya parallel +timestepping. This was later moved to better numerically balance +increments from +the convection scheme with the cloud fraction erosion term. From VN8.1 onwards, a further option was introduced to implement the erosion prior to the microphysics parametrization. This was primarily to allow PC2 to be run at convection-resolving scales, at which the convection scheme is not called and therefore the erosion is not called. +We have empirically selected a rate of change of width that depends upon the +relative total humidity of the grid box, such that there is more +erosion in drier gridboxes. This promotes more rapid erosion of +shallow convective cloud, which is the main effect that we seek to +include, although the physical implication that dry air is +more turbulent than moist air does not match the way the real atmosphere +works, especially in the +stratosphere. No doubt the link can be improved upon with more +research. The formulation used is: + +\begin{equation} +\frac{1}{b_s} \frac{\partial b_s}{\partial t} = \Upsilon exp ( - \frac{2.01 Q_c}{0.2 a_L q_{sat liq}(T_L)} ) +\label{eq:dbsbydtbs_turb} +\end{equation} + +where the 0.2 factor is chosen to be closely equivalent to $1 - RH_{crit}$ +and the value of 2.01 has been selected through tuning. The code merges the +two numerical values into a single quantity (dbsdtbs-exp), equal to 10.05. +We note that in PC2:64 +(the library 6.4 code, a value of 0.62 is used rather than 2.01). As a guide +to the $RH_T$ dependence, note +that when the value of $RH_T$ is 0.85, the value of +$\frac{1}{b_s} \frac{\partial b_s}{\partial t}$ is close to $1 \times 10^{-4} s^{-1}$. + +Note: the source-code for this erosion method ({\bf pc2\_hom\_conv}, +{\bf pc2\_homog\_plus\_turb}, {\bf pc2\_delta\_hom\_turb}) +includes an additional term ``dbsdtbs1'' +which scales with the rate of homogeneous forcing +$\frac{\partial Q_c}{\partial t}$. However this term is always +set to zero on input to these routines so is never used. + +The width-narrowing formulation of section \ref{sec:width} is used to +calculate increments in $\overline{q_{cl}}$ and $C_l$. Using the liquid - +ice cloud overlap ideas of section \ref{sec:ct} then gives the associated +$C_t$ change. This background narrowing term, $\Upsilon$, is originally based upon work +by \cite{sg03}, although it is a parameter that has been +extensively tuned during PC2 development, a typical value would be $\Upsilon=-2.25 \times 10^{-5} s^{-1}$. + +\subsubsection{Numerical application of the original width-narrowing method} + +Because of the strong link the mathematical expressions for width narrowing +(section \ref{sec:width}) have with the expressions for the +homogeneous forcing (section \ref{sec:homog}), we choose to represent +the timestepping of this process in exactly the same way as for +the homogeneous forcing (in fact, in the Unified Model code we use +the same subroutine, see section \ref{sec:code}). As +before, we use a simple forward timestepping of $C_l$, with +$Q_c$ given by (\ref{eq:qc_eq_qt-qs}) and $a_L$ defined as discussed +in section \ref{sec:homog_num_app} and discretize eq \ref{eq:dcdt_width} as: + +\begin{equation} +\Delta C_l^{[n+1]} = - G(-Q_c) Q_c \frac{1}{b_s} +\frac{\partial b_s}{\partial t} \Delta t. +\label{eq:dcl_turb_final} +\end{equation} + +Similarly to (\ref{eq:c_l^n+1}), we then limit the cloud fraction to 0 and +1 and then apply a mid-point value of $C_l$ to calculate the change in +$\overline{q_{cl}}$ (discretizing eq \ref{eq:dqcldt_width}): + +\begin{equation} +\Delta q_{cl}^{[n+1]} = (q_{cl}^{[n]} - Q_c \frac{1}{2}(C_l^{[n]}+C_l^{[n+1]})) +\frac{1}{b_s} \frac{\partial b_s}{\partial t} \Delta t. +\label{eq:dqcl_turb_final} +\end{equation} + +In this case the value of $\Delta q_{cl}$ \textit{is} limited to ensure that +no more $\overline{q_{cl}}$ is removed than the model has available. This was +chosen to ensure that the erosion process itself contains this physical limit, +not a numerical tidying-up process. + +The option ``l\_fixbug\_pc2\_qcl\_incr'' ensures that qcl is set to zero +if the CFL has reached zero. + + +\subsubsection{Cloud-surface-area hybrid erosion method} +(selected by setting {\bf i\_pc2\_erosion\_method = 3} in the UM namelist). + +\cite{morcrette_petch} showed that changes to the erosion parameter ($\Upsilon$ in Eqn. \ref{eq:dbsbydtbs_turb}) did not have as significant an impact +on the global work done by the erosion process as might be expected. This was due to a feedback process +whereby, reducing the erosion parameter leads to more cloud water, more autoconversion of cloud water to rain, more +fall-out of rain and more drying of the layer, hence increasing the $exp ( - \frac{2.01 Q_c}{0.2 a_L q_{sat liq}(T_L)} )$ part of +Eqn. \ref{eq:dbsbydtbs_turb}. Although the feedback is physically plausible it crucially depends on the formulation of +Eqn. \ref{eq:dbsbydtbs_turb} and the dependence of the rate of narrowing of the PDF on the moisture, a dependence that was developed +from a pragmatic rather than theoretical stand-point. +The option for an alternative way of calculating the erosion was introduced at vn8.0 + +We use equation 30 from \cite{t93} to specify the sink of $q_{cl}$ due to erosion, i.e. +\begin{equation} +\frac{\partial q_{cl}}{\partial t}=-A K(q_{sat}-q_v) +\label{eq:dqcldt_hybrid} +\end{equation} +(note we have changed the sign as we have replaced the evaporation rate $E_2$ +in \cite{t93} with $-\frac{\partial q_{cl}}{\partial t}$ on the left-hand-side). +In the \cite{t93} scheme, $A$ is set to the cloud fraction (i.e. $A=C_l$). +Here we recall that "cloud erosion" is meant to represent the evaporation of cloud water due to the +mixing of clear and cloudy air and that this can only happen on the edges of cloud, where saturated air is exposed to sub-saturated air. +If the cloud fraction is small (e.g. 5$\%$), then there are not many clouds, so there is only a small surface area from which evaporation can occur. +Similarly if the cloud cover is very high (e.g. 95$\%$) then there is again not much surface area exposed to clear sky. +A maximum in exposed surface area is expected when the cloud cover is 50$\%$. + +By imagining that the grid-box is broken up into cubes whose horizontal dimension equal the layer depth it is possible +to work out what the maximum lateral surface area would be, as a function of cloud fraction, for different arrangements of cloudy cubes. +The maximum lateral surface area, is when the clear and cloudy cubes are arranged in a chess-board pattern, and the minimum is when then are all grouped +together into a circular clump. Numerical tests using randomly distributed cloudy cube shows that the variation +in lateral surface area, $S$, as a function of cloud fraction can be expressed as: +\begin{equation} +S= - 2 C_l ^{2} + 2 C_l +\label{eq:S_Cl} \end{equation} +The maximum normalised surface area of 0.5 occurs at a cloud fraction of 0.5. +Using a cloud mask derived from satellite imagery shows that real cloud fields do follow this kind of dependence, +but that the peak surface area is nearer to 0.35, meaning that real clouds are not as randomly distributed as random +ones and that there is some kind of clumping together, which is what we might have expected. +When it comes to implementing such a scheme in the model, there will need to be a tunable parameter to govern the +rate of evaporation. This will not affect the shape of the lateral surface area function. +As a result the details of whether the peak lateral surface area is 0.5 or 0.35 are simply absorbed into the tunable parameter $K$, +which is supplied from the UMUI (using the same text box as was used for supplying $\Upsilon$). + +The exposed surface area associated with the tops and bottom of the clouds is calculated assuming maximum overlap in adjacent layers and is added to the lateral +surface area to give a total surface area, +\begin{equation} +A=max(C_l(k)-C_l(k+1),0.0)+max(C_l(k)-C_l(k-1),0.0)+S +\label{eq:A_top_and_bottom} \end{equation} +it is this value of $A$ which we use in Eqn. \ref{eq:dqcldt_hybrid}. + +Note that the contributions from the top and bottom interfaces of the current +model-level $max(C_l(k)-C_l(k+1),0.0)$ and $max(C_l(k)-C_l(k-1),0.0)$ +may optionally either be included or excluded, depending on the +UM namelist switch \textbf{i\_pc2\_erosion\_method}. +Further note: at present these contributions are hardwired to be excluded, +as they prevented the erosion calculations from being parallelised +in the vertical direction using OpenMP, and no operational model configurations +were using them. + +Having calculated a reduction in $q_{cl}$ using the Tiedtke-surface-area method, we +then work out the relative rate of narrowing that would have given the same sink of $q_{cl}$. This value of $\frac{1}{b_s} \frac{\partial b_s}{\partial t}$ +is then used to calculate the change in $C_l$ using the same moisture PDF assumptions as were used in the original PC2 erosion formulation. +To achieve this, we combine equations \ref{eq:dcdt_width} and +\ref{eq:dqcldt_width} from section \ref{sec:width} to eliminate +$\frac{1}{b_s} \frac{\partial b_s}{\partial t}$ and write +$\frac{\partial C_l}{\partial t}$ as a function of +$\frac{\partial \overline{q_{cl}}}{\partial t}$: + +\begin{equation} +\frac{\partial C_l}{\partial t} + = - \frac{ G(-Q_c) Q_c \frac{\partial \overline{q_{cl}}}{\partial t} } + { (- C_l Q_c+\overline{q_{cl}}) } +\label{eq:dcdt_hybrid} +\end{equation} + +Where the change in liquid water content +$\frac{\partial \overline{q_{cl}}}{\partial t}$ +is given by eq \ref{eq:dqcldt_hybrid} above. + +This combination of a Tiedkte sink term for $q_{cl}$, a PC2 term for $C_l$ and the introduction of some surface area dependence leads to this formulation +being referred to as a ``hybrid'' cloud-surface-area erosion method. + +\subsubsection{Numerical application of the hybrid erosion method} +\label{sec:erosion_numerics} + +Next, we consider how to numerically discretise equations +\ref{eq:dqcldt_hybrid} and \ref{eq:dcdt_hybrid} +to compute cloud increments due to erosion. +The simplest approach is an explicit forwards-in-time discretisation: + +\begin{equation} +\frac{ \Delta {q_{cl}}_{ero}}{\Delta t} = A(C_l^n) K(q_{sat}-q_v) +\label{eq:hybrid_erosion_expl} \end{equation} + +i.e. the increment is calculated by evaluating the term $A$ from equations +\ref{eq:S_Cl} and \ref{eq:A_top_and_bottom} using the value of cloud-fraction +$C_l$ \textit{before} erosion has been applied. + +However, when the environment is significantly subsaturated +(so that the term $(q_{sat}-q_v)$ is large and negative), +and long timesteps $\Delta t$ are used +(e.g. order 1000 s used in global climate simulations), +this discretization can suffer severe numerical overshoot. +i.e. the increment based on $C_l^n$ is large enough to reduce $q_{cl}$ +(and hence also $C_l$) to less than zero within a single timestep. +If the continuous equation were solved analytically this wouldn't happen; +as $C_l$ declines due to the erosion, so will $A(C_l)$ +and hence the erosion rate, so that $q_{cl}$ and $C_l$ smoothly decline +towards zero. + +Three options are available in the code to address this problem, +selected by the UM namelist switch \textbf{i\_pc2\_erosion\_numerics}, +detailed below. +Single-Column Model tests indicate that the 2nd and 3rd options yield much less +timestep sensitivity for detrained cloud in shallow cumulus regimes. + +\begin{enumerate} + +\item \textbf{Retain the explicit discretization, but limit the resulting +erosion increments to ensure $q_{cl}$ and $C_l$ don't go negative. +(i\_pc2\_erosion\_numerics=1)} +Also, to ensure that some cloud remains at end-of-timestep where +shallow cumulus is detraining into dry environments, the erosion +calculation is fed copies of the fields with the current timestep's +convection increments subtracted off. This means any cloud detrained +by convection during the current timestep cannot be eroded until the +following timestep, and so is still present at end-of-timestep. +As discussed in section \ref{sec:timestepping}, +this leads to a problematic timestep sensitivity, +since the amount of cloud not subject to erosion is the convection increment, +which scales with the timestep length. + +Having computed the erosion $q_{cl}$ increment using +\ref{eq:hybrid_erosion_expl}, the consistent $C_l$ increment is computed +by discretising \ref{eq:dcdt_hybrid} as: + +\begin{equation} +\frac{\Delta {C_l}_{ero}}{\Delta t} + = - \frac{ G(-Q_c)^n Q_c^n \frac{\Delta \overline{{q_{cl}}_{ero}}}{\Delta t} } + { (- C_l^n Q_c^n + ( \overline{q_{cl}^n} + + \frac{1}{2} \Delta \overline{{q_{cl}}_{ero}} ) ) } +\label{eq:dcdt_hybrid_discr} +\end{equation} + +i.e. all terms are treated explicitly (using the values before erosion), +except for $\overline{q_{cl}}$ which takes the mid-point interpolated +half-way between its values before and after erosion, +to give some improvement in accuracy. + +\item \textbf{Use an approximate implicit discretisation, +which intrinsically yields a positive solution for $q_{cl}$ and $C_l$. +(i\_pc2\_erosion\_numerics=2)} +The copies of the fields passed to the erosion calculation are +fully updated with the convection increments. +We then write equation \ref{eq:dqcldt_hybrid} in the form: + +\[ +\frac{\partial q_{cl}}{\partial t} = q_{cl} f(q_{cl},C_l,(q_{sat}-q_v)) +\] + +(where the term $f(q_{cl},C_l,(q_{sat}-q_v)) = \frac{A K(q_{sat}-q_v)}{q_{cl}}$ +will be treated explicitly, under the assumption that this ratio will +evolve more slowly while erosion rapidly reduces both the numerator +and the denominator). + +We then take a backwards-in-time implicit discretisation in terms of the +leading factor $q_{cl}$: + +\[ +\frac{ q_{cl}^{n+1} - q_{cl}^{n}}{\Delta t} = q_{cl}^{n+1} f^n +\] + +Now, the problem is somewhat complicated by the fact that in the code, +erosion is calculated in parallel with the homogeneous forcing by convection, +and we need to account for the homogeneous forcing increment +$\Delta q_{cl}^{hom}$ in our implicit solution. We therefore write the above as: + +\[ +\Delta q_{cl}^{ero} = \Delta t + ( q_{cl}^n + \Delta q_{cl}^{hom} + \Delta q_{cl}^{ero} ) f^n +\] + +Rearranging: + +\[ +\Delta q_{cl}^{ero} = \Delta t f^n q_{cl}^n \frac{ q_{cl}^n + \Delta q_{cl}^{hom} } + { q_{cl}^n - \Delta t f^n q_{cl}^n } +\] + +Note that the term $\Delta t f^n q_{cl}^n$ is the erosion increment +we would obtain from the purely explicit discretisation, +$\Delta q_{cl}^{ero\,expl}$. The implicit discretisation is implemented by +first calculating $\Delta q_{cl}^{ero\,expl}$ using equation +\ref{eq:hybrid_erosion_expl} +(as we do for \textbf{i\_pc2\_erosion\_numerics=1}) +but then rescaling it using the above expression, which becomes: + +\begin{equation} +\Delta q_{cl}^{ero} = \Delta q_{cl}^{ero\,expl} \frac{ q_{cl}^n + \Delta q_{cl}^{hom} } + { q_{cl}^n - \Delta q_{cl}^{ero\,expl} } +\label{eq:hybrid_erosion_impl_qcl} \end{equation} + +Provided erosion is acting to reduce cloud-water ($\Delta q_{cl}^{ero\,expl} < 0$), +and homogeneous forcing by convection has not already completely removed +the cloud ($q_{cl}^n + \Delta q_{cl}^{hom} > 0$), \ref{eq:hybrid_erosion_impl_qcl} +is guaranteed to yield a stable, positive solution for $q_{cl}$. + +We also apply exactly the same argument to the equation for the cloud-fraction +increment $C_l$, and obtain: + +\begin{equation} +\Delta C_l^{ero} = \Delta C_l^{ero\,expl} \frac{ C_l^n + \Delta C_l^{hom} } + { C_l^n - \Delta C_l^{ero\,expl} } +\label{eq:hybrid_erosion_impl_Cl} \end{equation} + +Where $\Delta C_l^{ero\,expl}$ is computed using eq \ref{eq:dcdt_hybrid_discr}, +except that the term $\frac{1}{2} \Delta \overline{{q_{cl}}_{ero}}$ +is omitted (interpolating to the mid-point value of $\overline{q_{cl}}$ +in the denominator would be ``double-counting'' if we are already making +an implicit correction to the full increment). + +In the case where the homogeneous forcing increments have already removed +all of the cloud water content or fraction, erosion is not performed, +and $q_{cl}$ and $C_l$ are both set to zero. In the case where erosion is +actually acting to increase cloud-fraction, the code defaults to retaining +the explicit discretisation solution $\Delta q_{cl}^{ero\,expl}$ and +$\Delta C_l^{ero\,expl}$. Otherwise, equations \ref{eq:hybrid_erosion_impl_qcl} +and \ref{eq:hybrid_erosion_impl_Cl} are applied to yield the implicit solution. + +\item \textbf{Use an analytic solution to the integration of the +time-derivatives in (\ref{eq:dqcldt_hybrid}) and (\ref{eq:dcdt_hybrid}) +for greater accuracy. +(i\_pc2\_erosion\_numerics=3)} + +Two problems have been identified with the above implicit numerical method: +\begin{itemize} + +\item The implicit correction is applied completely independently to the +increments for $q_{cl}$ and $C_l$. So as with the explicit method, +differing numerical error in the increments for the two variables +can lead to them becoming inconsistent with eachother. +It was found by experimentation that even with the implicit correction, +it is possible for erosion to reduce $C_l$ by a bigger fraction than $q_{cl}$, +so that the in-cloud water content $\frac{q_{cl}}{C_l}$ is {\em increased}. +Narrowing the PDF should only {\em decrease} the in-cloud water-content; +occasional large increases due to numerical error can lead to spurious +precipitation being produced by the microphysics scheme. + +\item The implicit correction makes it impossible for erosion to reduce +$q_{cl}$, $C_l$ to zero. As we will show below, the analytic solution +to the equations posed does in fact go to zero after a finite time +under grid-mean subsaturation +(although the erosion rate declines with $C_l$ as it approaches zero, +$C_l$ approaches zero more slowly than $q_{cl}$, so that both variables decrease +following power-law curves not exponentials). +When erosion (wrongly) can never entirely remove cloud, this allows +tiny values of $q_{cl}$ and $C_l$ to spuriously spread across the domain +via numerical diffusion from the model's advection scheme. + +\end{itemize} + +Under this option, we attempt to compute an analytic solution to the +simultaneous differential equations \ref{eq:dqcldt_hybrid} and +\ref{eq:dcdt_hybrid} so that $q_{cl}$ and $C_l$ both decrease smoothly and +consistently. +The equations lead to somewhat different behaviour depending on whether +the grid-mean state is subsaturated ($Q_c < 0$), supersaturated ($Q_c > 0$), +or close to saturation ($Q_c$ near-zero). We can employ different +approximations to integrate the equations in each case. +In the code, we first test the value of $Q_c$ and compute the erosion +increments as follows: + +\begin{enumerate} + +\item {\bf Grid-mean subsaturation ($Q_c < 0$):} + +The relation between the erosion tendencies in liquid-cloud-fraction and +liquid water content (\ref{eq:dcdt_hybrid}) can be expressed in terms of +{\em fractional} rates of change +(dividing the top and bottom by $-C_l Q_c$, and dividing both sides by $C_l$): + +\begin{equation} +\frac{1}{C_l} \frac{\partial C_l}{\partial t} + = \frac{ G(-Q_c) \frac{q_{cl}}{C_l^2} }{ 1 - \frac{q_{cl}}{C_l Q_c} } \; + \frac{1}{q_{cl}} \frac{\partial q_{cl}}{\partial t} +\label{eq:dcdt_hybrid_1} +\end{equation} + +Under homogeneous forcing (section \ref{sec:homog}), we defined the PDF height +at the saturation boundary when near the cloudy end of the PDF as +$G(-Q_c) = \frac{n+1}{n+2} \frac{C_l^2}{q_{cl}}$ (eq \ref{eqn20}). +In fact, $G(-Q_c)$ is set to some blend between this and the value near +the clear end of the PDF (eq \ref{eqn21}). But we will assume that +when eroding cloud under grid-mean subsaturated conditions ($Q_c < 0$), +$G(-Q_c)$ follows this scaling with $\frac{C_l^2}{q_{cl}}$ even if its +value differs somewhat from eq \ref{eqn20}. +Therefore the quantity $c_1 = G(-Q_c) \frac{q_{cl}}{C_l^2}$ remains constant +during the erosion process, and eq \ref{eq:dcdt_hybrid_1} becomes: + +\begin{equation} +\frac{1}{C_l} \frac{\partial C_l}{\partial t} + = \frac{ c_1 }{ 1 - \frac{q_{cl}}{C_l Q_c} } \; + \frac{1}{q_{cl}} \frac{\partial q_{cl}}{\partial t} +\label{eq:dcdt_hybrid_2} +\end{equation} + +The term $1 - \frac{q_{cl}}{C_l Q_c}$ (which is $> 1$ since we are considering +grid-mean subsaturation $Q_c < 0$) usually remains close to 1 in practice, +so we can assume its fractional variation over the timestep is small +compared to the other terms, and treat it explicitly. +We can therefore straightforwardly integrate eq \ref{eq:dcdt_hybrid_2} +to obtain the scaling of $C_l$ with $q_{cl}$ as both are reduced by erosion: + +\begin{equation} +\frac{C_l}{{C_l}_0} = \left( \frac{q_{cl}}{{q_{cl}}_0} \right)^{b_1} +\label{eq:cl_qcl_scaling} +\end{equation} + +where ${C_l}_0$, ${q_{cl}}_0$ are the values before erosion is applied, +and the exponent is $b_1 = \frac{ c_1 }{ 1 - \frac{q_{cl}}{C_l Q_c} }$. +When $G(-Q_c)$ takes its value from the cloudy end of the PDF, we have +$c_1 = \frac{n+1}{n+2}$. Since the PDF power $n > 0$ and $Q_c < 0$ +under the considered grid-mean subsaturation, we always have $b_1 < 1$. +This ensures that erosion reduces $C_l$ at a slower fractional rate than +$q_{cl}$, so that in-cloud water content $\frac{q_{cl}}{C_l}$ always decreases. + +Next we derive an integral solution for the decline of $q_{cl}$ with time. +Ignoring the cloud surface-area contributions from the levels above and below +(they are disabled in the code anyway), the erosion liquid water content +tendency is obtained by combining \ref{eq:dqcldt_hybrid} and \ref{eq:S_Cl}: + +\begin{equation} +\frac{\partial q_{cl}}{\partial t} = -K \, 2 C_l (1 - C_l) \, (q_{sat}(T)-q_v) +\label{eq:dqcldt_hybrid_1} +\end{equation} + +From eq \ref{SD2}, $q_{sat}(T)-q_v = \frac{SD}{a_L}$, where $SD$ is the +saturation defecit, and $a_L$ is the dimensionless factor defined in +eq \ref{eq:a_L}. Following the derivation in section +\ref{sec:smooth_initiation} (eq \ref{eq:qc_plus_sd}), +we can write this in terms of the liquid-water content: $SD = q_{cl} - Q_c$ +(where $Q_c$ was defined in eq \ref{eq:qc_eq_qt-qs}, and corresponds to the +grid-mean supersaturation converted to an equivalent liquid water content). +Substituting this into (\ref{eq:dqcldt_hybrid_1}) above, we obtain: + +\begin{equation} +\frac{\partial q_{cl}}{\partial t} = -\frac{K}{a_L} \, 2 C_l (1 - C_l) \, + (q_{cl}-Q_c) +\label{eq:dqcldt_hybrid_2} +\end{equation} + +Substituting eq \ref{eq:cl_qcl_scaling} for the leading factor of $C_l$ +on the right-hand-side and rearranging: + +\[ +\left( \frac{q_{cl}}{{q_{cl}}_0} \right)^{-b_1} \frac{\partial q_{cl}}{\partial t} + = -\frac{K}{a_L} \, 2 {C_l}_0 (1 - C_l) \, (q_{cl}-Q_c) +\] + +In significantly subsaturated conditions the r.h.s. has only weak dependence +on $C_l$ and $q_{cl}$ ($C_l << 1$, $q_{cl} << -Q_c$), so we can treat the +whole r.h.s. explicitly (i.e. neglect its variation during each timestep), +so that the above integrates to: + +\[ +\left[ \frac{{q_{cl}}_0}{1-b_1} \left( \frac{q_{cl}}{{q_{cl}}_0} \right)^{1-b_1} +\right]_{{q_{cl}}_0}^{{q_{cl}}_{\Delta t}} + = -\frac{K}{a_L} \, 2 {C_l}_0 (1 - C_l) \, (q_{cl}-Q_c) \Delta t +\] + +Inserting the limits of the integral on the l.h.s. and rearranging, +we obtain our analytical solution for $q_{cl}$ after time $\Delta t$: + +\begin{equation} +{q_{cl}}_{\Delta t} = {q_{cl}}_0 \left( 1 - \frac{1-b_1}{{q_{cl}}_0} + \frac{K}{a_L} \, 2 {C_l}_0 (1 - C_l) \, (q_{cl}-Q_c) \Delta t + \right)^\frac{1}{1-b_1} +\label{eq:qcl_int_hybrid} +\end{equation} + +Note that $q_{cl}$ falls to zero after a finite time +$\frac{{q_{cl}}_0}{1-b_1} \frac{a_L}{K} + \frac{1}{2 {C_l}_0 (1 - C_l) \, (q_{cl}-Q_c)}$. +If the timestep $\Delta t$ is longer than this time, then erosion +completely removes the cloud during the current timestep. + +We first set ${q_{cl}}_0$ and ${C_l}_0$ to the values already updated +by homogeneous forcing, and then sequentially compute the updated +$q_{cl}$ after erosion using (\ref{eq:qcl_int_hybrid}). +Then we substitute this value into (\ref{eq:cl_qcl_scaling}) to compute +the consistent updated value of $C_l$. +Finally, to improve accuracy, a small number of iterations are performed +to find the solution with the explicitly-treated terms +(the exponent $b_1 = \frac{ c_1 }{ 1 - \frac{q_{cl}}{C_l Q_c} }$ +and the terms $(1 - C_l)$ and $(q_{cl}-Q_c)$ in eq \ref{eq:qcl_int_hybrid}) +adjusted to values linearly-interpolated to half-way between the +start and end of the erosion timestep. + +\item {\bf Grid-mean supersaturation ($Q_c > 0$):} + +In this case, erosion does not act to reduce $C_l$ and $q_{cl}$ towards zero. +Instead, the narrow PDF limit it adjusts towards has +no remaining subsaturated air, so that $C_l = 1$ and $q_{cl} = Q_c$. +In this case, we can repeat the above derivation, but considering the +equations for the clear-fraction $1-C_l$ in place of $C_l$, +and the saturation defecit $SD = q_{cl}-Q_c$ in place of $q_{cl}$. +Assuming that $G(-Qc)$ follows the scaling for the clear end of the PDF +(\ref{eqn21}), this leads to a similar equation to (\ref{eq:cl_qcl_scaling}) +but for the scaling as erosion reduces $1-C_l$ and $SD$ towards zero .: + +\begin{equation} +\frac{1-C_l}{1-{C_l}_0} = \left( \frac{SD}{SD_0} \right)^{b_2} +\label{eq:ca_sd_scaling} +\end{equation} + +with $b_2 = \frac{ c_2 }{ 1 + \frac{SD}{(1-C_l) Q_c} }$ +and $c_2 = G(-Q_c) \frac{SD}{(1-C_l)^2}$ +(note we must have $0 < b_2 < 1$). + +And then the tendency equation for $SD$ is: + +\begin{equation} +\frac{\partial SD}{\partial t} = -\frac{K}{a_L} \, 2 (1 - C_l) C_l \, SD +\label{eq:dsddt_hybrid_2} +\end{equation} + +The one asymmetry between this and the $q_{cl}$ tendency equation +(\ref{eq:dqcldt_hybrid_2}) is that for $SD$ the r.h.s. is directly proportional +to the quantity in the time-derivative, whereas for $q_{cl}$ there is an +additional $Q_c$ term which is constant during erosion. +Substituting (\ref{eq:ca_sd_scaling}) for the leading factor of +$(1 - C_l)$ in (\ref{eq:dsddt_hybrid_2}), integrating over time $\Delta t$ +(neglecting the fractional variation of $C_l$ over the timestep) +and rearranging, we obtain: + +\begin{equation} +{SD}_{\Delta t} = {SD}_0 \left( 1 + b_2 + \frac{K}{a_L} \, 2 (1-{C_l}_0) C_l \, \Delta t + \right)^{-\frac{1}{b_2}} +\label{eq:sd_int_hybrid} +\end{equation} + +Note that the additional power of $SD$ on the r.h.s. of +(\ref{eq:dsddt_hybrid_2}) leads to the integral solution having +a negative exponent. This means that under grid-mean supersaturation, +erosion makes $SD$ and $1-C_l$ approach but never quite reach zero, +which is quite different behaviour to grid-mean subsaturation where +$q_{cl}$ and $C_l$ go to zero over a finite time. +This asymmetry is because the erosion rate is parameterised to be +proportional to $SD$, and this tends to zero as the PDF is narrowed +under supersaturation, but remains finite positive under subsaturation. + +We first set ${SD}_0 = {q_{cl}}_0 - Q_c$ (where as above ${q_{cl}}_0$ is +the value already updated by homogeneous forcing), +then compute the value of $SD$ updated by erosion using +(\ref{eq:sd_int_hybrid}). +Then we substitute this value into (\ref{eq:ca_sd_scaling}) to compute +the consistent updated value of $1-C_l$. +A small number of iterations are then performed +to find the solution with the explicitly-treated terms +(the exponent $b_2 = \frac{ c_2 }{ 1 + \frac{SD}{(1-C_l) Q_c} }$ +and the term $C_l$ in eq \ref{eq:sd_int_hybrid}) +adjusted to values linearly-interpolated to half-way between the +start and end of the erosion timestep. +Then the final values of $SD$ and $1-C_l$ are used to increment +$q_{cl} = Q_c + SD$ and $C_l$, as prognosed by the rest of the model. + +\item {\bf grid-mean saturation ($Q_c$ near-zero):} + +In this case, the PDF is centred on the +saturation boundary, so that narrowing it does not change the cloud-fraction. +In the limit $Q_c = 0$, we have $q_{cl} = SD$, and (\ref{eq:dqcldt_hybrid_2}) +or (\ref{eq:dsddt_hybrid_2}) becomes: + +\begin{equation} +\frac{1}{q_{cl}} \frac{\partial q_{cl}}{\partial t} + = -\frac{K}{a_L} \, 2 C_l (1 - C_l) +\end{equation} + +where everything on the r.h.s. is constant under erosion. +This simply integrates to give exponential decline of $q_{cl}$ +(and $SD$) towards zero: + +\begin{equation} +{q_{cl}}_{\Delta t} = {q_{cl}}_0 e^{ -\frac{K}{a_L} \, 2 C_l (1 - C_l) \Delta t } +\end{equation} + +\end{enumerate} + +\end{enumerate} + + +\subsection{Orographic and Gravity Wave Drag} +The Orographic and Gravity Wave Drag sections do not alter the temperature +or moisture content of the model gridboxes, hence PC2 assumes no change in the +condensate and cloud fractions as a result of these processes. + +\subsection{Advection} +\label{sec:advec} + +The advection of $\overline{q_{cl}}$ and $\overline{q_{cf}}$ are already +performed separately by the semi-Lagrangian advection scheme. +Advection of the three cloud fractions $C_l$, $C_i$ and $C_t$ are all +performed by PC2 in the same way. + +Note that ascent or subsidence by advection entails a pressure change following +each parcel, which will cause an accompanying adiabatic temperature change. +These advective pressure and temperature changes imply a homogeneous forcing, +which yields a change in $\overline{q_{cl}}$ and $C_l$ in addition to their +transport by the winds. This is described in section \ref{sec:pres}. + +If the UM namelist switch \textbf{l\_pc2\_sl\_advection} is turned on, +the PC2 homogeneous forcing response to advection is calculated +straight after the call to Semi-Lagrangian advection. +Otherwise, the pressure change from advection is combined with the +Eulerian pressure change from the dynamics Helmholtz solver, and the resulting +homogeneous forcing of liquid cloud is computed at the end of the timestep. + +\subsection{Boundary Layer} +\label{sec:bl} +At a basic level, the boundary layer scheme works by +mixing $\overline{q_T}$ and $\overline{T_L}$, and tracer mixing +$\overline{q_{cf}}$. The condensation and $C_l$ changes are represented +using the homogeneous forcing representation. The forcing of $Q_c$ can be +written in $\Delta \overline{q_T}$ and $\Delta \overline{T_L}$ terms +using (\ref{eq:deltaqc_exp}). + +$\overline{q_{cf}}$ is already mixed using the tracer mixing scheme. PC2 +will calculate the corresponding $C_i$ change assuming the inhomogeneous +forcing scenario. Although this is not necessarily an appropriate physical +model to use, it is the only generic physical model we have currently +developed in order to convert increments in a condensate to increments in +a cloud fraction. We use a value of the in-cloud water content $q_c^S$ +based upon a linear combination of the current in-cloud ice water +content, $\frac{\overline{q_{cf}}}{C_i}$, and a fixed value. + +\begin{equation} +q_C^S = C_i \frac{\overline{q_{cf}}}{C_i} + ( 1 - C_i) q_{cf0 \, BL} +\label{eq:qcf_ci} +\end{equation} + +where $q_{cf0 \, BL}$ is a specified value of $1 \times 10^{-4} \, kg \, kg^{-1}$. +We then use the inhomogeneous forcing equation based upon +(\ref{eq:dcdt_inhom2}) but for ice water content to write + +\begin{equation} +\Delta C_i = \frac{(1 - C_i)}{q_C^S - \overline{q_{cf}}} Q4_i . +\label{eq:deltaci_bl} +\end{equation} + +Since the physical model will have $C_i$ tend to 1 if the +denominator is small, we will, to avoid numerical problems, set +$C_i$ to 1 if $q_C^S - \overline{q_{cf}} < 1 \times 10^{-10} kg kg^{-1}$. +Note that we do not use the multiple phases injection source +expressions (section \ref{sec:multiple} and equation \ref{eq:cff_ts}). +This is because the liquid water changes are not associated with the plume model. + +Equation \ref{eq:qcf_ci} assumes that the change to the ice water content has led to an increase in ice water content. +However, if the ince water content has reduced, the change to the ice cloud fraction is not consistent. +The option to "Use consistent formulation of ice cloud fraction changes due to boundary-layer processes" ensure that +if the ice water content is reduced, the ice cloud fraction is reduced, in such as way as to maintain +the same in-cloud ice water content. + +The $C_t$ changes are calculated using the minimum overlap method of +section \ref{sec:ct}. + +In \textit{ni-imp-ctl} the control code inhibits the +call to the diagnostic cloud scheme +if there is deep or shallow convection occurring and the model level +is less than \textit{or equal to} the layer immediately above the top of the +boundary layer mixed layer (i.e. level ntml+1). This is in order to +ensure that there is +no large-scale cloud present below the base of the convective cloud, but +additionally performs this calculation at the level above, probably +in order that latent heating from large-scale condensation does not +inhibit the convection. A similar thing is performed for PC2, with +any large-scale cloud being evaporated if the same criteria are met, +\textit{except that it is not performed on the level above the boundary +layer mixed layer}. This choice (i.e. ntml) is seen to give improved +results in PC2, and is arguably a more physical reasonable choice +anyway than using ntml+1. + +\subsection{Convection} +\label{sec:convec} + +This section concentrates specifically upon the PC2 interface to the +convection scheme. In the current formulation of the UM, only a +mass-flux convection scheme exists, and this is what is described +below. Work to interface PC2 to the developing turbulence based +convection scheme is commented upon in section \ref{sec:tbcs}. + +An alternative way of calculating cloud fraction increments is currently under development and +is described in section \ref{sec:conv-simpler}. + +A traditional view of convective parametrization is a scheme that +transports vapour, $q$, +heat, $\theta$, and momentum, $u$ and $v$ winds within a single column. It +does not consider transport sideways to adjoining columns, and (at least +in the Gregory-Rowntree scheme used in the UM) is considered independent of +any resolved scale vertical air motions. This necessitates the view of +compensating subsidence within the column, whereas some conceptual models +of tropical convection would have the bulk of the ascent in the +convective cores and the +associated descent thousands of miles away in the downward branch of the +Hadley circulation. The parametrization schemes traditionally overlook the +existence of condensate in the model column. The non-PC2 version +of the mass-flux convection scheme used in the UM would have the same +large-scale liquid and ice prognostics before and after convection occurs +(apart from a bolt-on evaporation below convective cloud base), with no +regard at all to what happens to it or its effect on the rest of the +convection. Within PC2 we have had to work to more fully incorporate +the condensate into the convection scheme. + +\subsubsection{Introduction to the convective mass flux scheme} + +Within the mass flux scheme the net change in +$\overline{q_{cl}}$ and $C_l$ etc. +comes from two distinct sources. Firstly, the condensate and cloud +fraction injected from the plume (the $Q4$ terms, section \ref{sec:inhomog}); +secondly, the condensation response to the vapour and heat changes associated +with the detrainment and compensating subsidence. Strictly, we will see that the +$Q4$ terms also include the contribution to the condensate transport +by the compensating subsidence - this casts doubt on the validity of +the application of the injection forcing scenario to calculate the +equivalent cloud fraction change, since ideally the cloud fractions +ought to be transported by the compensating subsidence in a similar +way to the condensate transport (which is documented below). + +We therefore split the convective contribution in (\ref{eq:dqcldt_and_dcdt}) +into two parts: + +\begin{equation} +\frac{\partial \overline{q_{cl}}}{\partial t} |_{convection} = +Q4_l + Q_{environment} +\label{eq:inhomg_plus_homog} +\end{equation} + +where $Q_{environment}$ is the condensation associated with changes +in the vapour and temperature from the detrainment and compensating +subsidence. Similar splits are made for the cloud variables, where +the injection forcing, section \ref{sec:inhomog}, is used to calculate +the first term from $Q4_l$. Section \ref{subsect:q4calculation} looks +at the issue of the +calculation of $Q4_l$ etc., and section \ref{sec:conv_homog} looks at +the calculation of $Q_{environment}$, and its associated cloud +fraction change. We first look at the basic transport equations in a +mass flux convection scheme. + +\subsubsection{Basic Equations for a Convective Mass Flux Scheme} +\label{subsect:basmaseqs} + +We first consider a generic mass-flux scheme before its application to PC2. +As discussed by Grant and Stirling (personal communication), +the equations for convective +tendencies are most simply applied to a variable, ${\chi}$, that is conserved +under moist adiabatic processes (e.g. total water content). In this case, +% +\begin{equation} +{\pardbyd{\xsubsup{ }{E}}{t}}_{\rm{conv}} = +- \frac{1}{\ov{\rho}} \, \pardbyd{\ov{\rho w^{'} \xsubsup{ }{E'}}}{z} +\label{eq:chibasic} \end{equation} + +To parametrize \ref{eq:chibasic}, the current UM convection scheme takes a +mass flux approximation +% +\begin{equation} +\lp {\ov{\rho w^{'} \xsubsup{ }{E'}}} \rp_{\rm{conv}} = M^{\rm{P}} \, +\lp { \xsubsup{ }{P} - \xsubsup{ }{E} } \rp +\label{eq:massflux} \end{equation} +% +which can be differentiated to give +% +\begin{equation} +- \frac{1}{\ov{\rho}} \, \pardbyd{\ov{\rho w^{'} \xsubsup{ }{E'}}}{z} = +\pardbyd{\xsubsup{ }{P} \, M^{\rm{P}}}{p} - +\xsubsup{ }{E} \, \pardbyd{M^{\rm{P}}}{p} - +M^{\rm{P}} \, \pardbyd{\xsubsup{ }{E}}{p} +\label{eq:eddyflux} \end{equation} + +The bulk cloud model plume equations for mass and ${\chi}$ are: +% +\begin{eqnarray} +- \pardbyd{M^{\rm{P}}}{p} & = & +\lp { \varepsilon \, M^{\rm{P}} - \mu \, M^{\rm{P}} - \delta \, M^{\rm{P}} } \rp +\label{eq:dbydpmassflux} \\ +- \pardbyd{\xsubsup{ }{P} \, M^{\rm{P}}}{p} & = & \lp { +\varepsilon \, M^{\rm{P}} \, \xsubsup{ }{E} +- \mu \, M^{\rm{P}} \, \xsubsup{ }{R} - \delta \, M^{\rm{P}} \, \xsubsup{ }{P} +} \rp \label{eq:dbydpmfchi} +\end{eqnarray} + +Equations \ref{eq:eddyflux}, \ref{eq:dbydpmassflux} and +\ref{eq:dbydpmfchi} can then be substituted into \ref{eq:chibasic} to give: +% +\begin{equation} +{\pardbyd{\xsubsup{ }{E}}{t}}_{\rm{conv}} = +- M^{\rm{P}} \, \pardbyd{\xsubsup{ }{E}}{p} ++ \mu \, M^{\rm{P}} \, \lp { \xsubsup{ }{R} - \xsubsup{ }{E} } \rp ++ \delta \, M^{\rm{P}} \, \lp { \xsubsup{ }{P} - \xsubsup{ }{E} } \rp +\label{eq:chimassflux} \end{equation} +% +while \xsubsup{}{P} is obtained from the vertical gradient derived by combining +\ref{eq:dbydpmassflux} and \ref{eq:dbydpmfchi} : +% +\begin{equation} +M^{\rm{P}} \, \pardbyd{\xsubsup{ }{P}}{p} = +\varepsilon \, M^{\rm{P}} \, \lp { \xsubsup{ }{P} - \xsubsup{ }{E} } \rp - +\mu \, M^{\rm{P}} \, \lp { \xsubsup{ }{P} - \xsubsup{ }{R} } \rp +\label{eq:gradchipar} \end{equation} + +Within the model, eqn~\ref{eq:chimassflux} would take a discretized form +which actually depends upon whether the model level, k, is above or at the +lowest cloud level (k = cb). Note that the formal cloud base lies at the +half-level below, i.e. on the layer boundary which is also the top of the +turbulent mixed boundary layer. A simple discretized form of +\ref{eq:chimassflux}, setting ${ \mu = 0 }$, is: +% +\begin{eqnarray} +{\pardbyd{\xsubsup{ }{E}}{t}}_{\rm{conv, \, k}} & = & m_{\rm{k+1/2}} \, +\frac{ \lp {\xsubsup{k+1}{E} - \xsubsup{k}{E}} \rp } +{{\Delta z}_{\rm{k \, \rightarrow \, k+1}}} ++ {\delta}_{\rm{k}} \, m_{\rm{k}} \, \lp { \xsubsup{k}{P} - \xsubsup{k}{E} } \rp +\qquad \ldots \; \mbox{for k $>$ cb} \label{eq:chidisck} \\ +{\pardbyd{\xsubsup{ }{E}}{t}}_{\rm{conv, \, cb}} & = & m_{\rm{cb+1/2}} \, +\frac{ \lp {\xsubsup{cb+1}{E} - \xsubsup{cb}{E}} \rp } +{{\Delta z}_{\rm{cb \, \rightarrow \, cb+1}}} +- m_{\rm{cb}} \, +\lp { \xsubsup{i,cb}{P} - \xsubsup{cb}{E} } \rp \label{eq:chidisccb} +\end{eqnarray} +% +where the initial parcel value \xsubsup{i,cb}{P} may be chosen to produce a +fixed increment or place a closure condition on the cloud base flux. In fact, +the convection equations (see \citeumdp{027}) differ from \ref{eq:chidisck} and +\ref{eq:chidisccb} because a different discretization is used, but the +principle is unaltered. + +The model convection variables are NOT conserved under moist adiabatic processes +because precipitation processes deplete the column moisture and condensation +processes affect the temperature, specific humidity and cloud condensate +variables. Surprisingly, however, the form of eqn~\ref{eq:chimassflux} is +retained even though the basic equation \ref{eq:chibasic} acquires additional +terms for temperature and specific humidity: +% +\begin{eqnarray} +{\pardbyd{\tsubsup{ }{E}}{t}}_{\rm{conv}} = Q1 & \equiv & +\lp { \frac{L}{c_{P}} } \rp \, {\ov{Q}}_{\rm{par}} +- \frac{1}{\ov{\rho}} \, \pardbyd{\ov{\rho w^{'} \tsubsup{ }{E'}}}{z} +\label{eq:defineq1} \\ +{\pardbyd{\qsubsup{ }{E}}{t}}_{\rm{conv}} = Q2 & \equiv & - {\ov{Q}}_{\rm{par}} +- \frac{1}{\ov{\rho}} \, \pardbyd{\ov{\rho w^{'} \qsubsup{ }{E'}}}{z} +\label{eq:defineq2} +\end{eqnarray} +% +where ${\ov{Q}}_{\rm{par}}$ is the rate of condensation which occurs in the +ascending plumes. + +The reason that \ref{eq:defineq1} and \ref{eq:defineq2} retain this form +is due to cancellation from the bulk cloud terms equivalent to +\ref{eq:dbydpmfchi} which are modified in the same way as +\ref{eq:defineq1} and \ref{eq:defineq2}. The change is seen in the +vertical gradient equations based upon \ref{eq:gradchipar} +% +\begin{eqnarray} +M^{\rm{P}} \, \pardbyd{\tsubsup{ }{P}}{p} & = & +\varepsilon \, M^{\rm{P}} \, \lp { \tsubsup{ }{P} - \tsubsup{ }{E} } \rp - +\mu \, M^{\rm{P}} \, \lp { \tsubsup{ }{P} - \tsubsup{ }{R} } \rp - +\lp { \frac{L}{c_{P}} } \rp \, {\ov{Q}}_{\rm{par}} \label{eq:gradtpar} \\ +M^{\rm{P}} \, \pardbyd{\qsubsup{ }{P}}{p} & = & +\varepsilon \, M^{\rm{P}} \, \lp { \qsubsup{ }{P} - \qsubsup{ }{E} } \rp - +\mu \, M^{\rm{P}} \, \lp { \qsubsup{ }{P} - \qsubsup{ }{R} } \rp + +{\ov{Q}}_{\rm{par}} \label{eq:gradqpar} \\ +M^{\rm{P}} \, \pardbyd{\lsubsup{ }{P}}{p} & = & +\varepsilon \, M^{\rm{P}} \, \lp { \lsubsup{ }{P} - \lsubsup{ }{E} } \rp +- {\ov{Q}}_{\rm{par}} + PPN \label{eq:gradlpar} +\end{eqnarray} + +The final calculation of rates in the current condensation scheme (\citeumdp{027}, +section 10) assumes a further condensation term, ${\ov{Q}}_{\rm{reset}}$, which +acts to make the net rate of change of condensate equal zero, and a final +assumption is made that the environment values of condensate remain zero (and +also that \lsubsup{ }{R} = \lsubsup{ }{P}). The +result is basic equations +% +\begin{eqnarray} +{\pardbyd{\tsubsup{ }{E}}{t}}_{\rm{conv}} & = & Q1 - +\lp { \frac{L}{c_{P}} } \rp \, {\ov{Q}}_{\rm{reset}} +\label{eq:basictold} \\ +{\pardbyd{\qsubsup{ }{E}}{t}}_{\rm{conv}} & = & Q2 + {\ov{Q}}_{\rm{reset}} +\label{eq:basicqold} \\ +0 \equiv {\pardbyd{\lsubsup{ }{E}}{t}}_{\rm{conv}} & = & {\ov{Q}}_{\rm{par}} - +{\ov{Q}}_{\rm{reset}} - PPN +- \frac{1}{\ov{\rho}} \, \pardbyd{\ov{\rho w^{'} \lsubsup{ }{E'}}}{z} \nonumber \\ +& = & +\mu \, M^{\rm{P}} \, \lsubsup{ }{P} + \delta \, M^{\rm{P}} \, \lsubsup{ }{P} - +{\ov{Q}}_{\rm{reset}} +\label{eq:basiclold} +\end{eqnarray} + +By analogy with equations \ref{eq:defineq1} and \ref{eq:defineq2}, we can +define a $Q4$ from \ref{eq:basiclold} and state that for the control +convection scheme $Q4 = 0$. The PC2 scheme requires a reassessment of these +assumptions because we wish to allow non-zero environment condensate values and +to allow them to change. + +\subsubsection{Calculation of Grid-Box Averaged Condensate Rate (Q4)} +\label{subsect:q4calculation} + +The PC2 condensation scheme allows convection to feed cloud condensate (ice or +liquid) directly into the large scale and to update the cloud amount accordingly. + +Define +% +\begin{eqnarray} +\lp { \pardbyd{\lsubsup{l}{ }}{t} } \rp_{\rm{conv}} = Q4_{\rm{l}} & \equiv & +{\ov{Q}}_{\rm{l, par}} - {\ov{Q}}_{\rm{l, reset}} - RAIN - +\frac{1}{\ov{\rho}} \, \pardbyd{\ov{\rho w^{'} \lsubsup{l}{'}}}{z} +\label{eq:defineq4l} \\ +\lp { \pardbyd{\lsubsup{f}{ }}{t} } \rp_{\rm{conv}} = Q4_{\rm{f}} & \equiv & +{\ov{Q}}_{\rm{f, par}} - {\ov{Q}}_{\rm{f, reset}} - SNOW - +\frac{1}{\ov{\rho}} \, \pardbyd{\ov{\rho w^{'} \lsubsup{f}{'}}}{z} +\label{eq:defineq4f} +\end{eqnarray} +% +where the PC2 assumption thus far has been that ${\ov{Q}}_{\rm{l, reset}} = 0 += {\ov{Q}}_{\rm{f, reset}}$. + +\begin{itemize} +\item{The current convection scheme assumes that parcel +condensate is single phase +(ie. either all liquid or all frozen) and this is seriously hard-wired into the +code. Thus we can treat the precipitation and parcel condensation +processes in $ Q4_{\rm{l}} $ and $ Q4_{\rm{f}} $ separately without worrying +about cross-transfer between the two because at most only one set will ever be +active in a given grid box at one time. However, even for the inactive (zero +parcel condensate) phase, convection mixes environmental air into the +parcel and +can therefore maintain a non-zero $Q4$. Enablement of multiple phase condensate +in the current scheme is a task requiring great caution as the formulations +are extremely sensitive to errors in assignment of condensate phase.} +\end{itemize} + +Based on \ref{eq:gradlpar}, the vertical dependence of condensate is +calculated as +% +\begin{eqnarray} +\pardbyd{\lsubsup{l}{P}}{p} & = & \varepsilon \, +\lp { \lsubsup{l}{P} - \lsubsup{l}{E} } \rp - +\frac{{\ov{Q}}_{\rm{l, par}}}{M^{\rm{P}}} - +\frac{RAIN}{M^{\rm{P}}} \label{eq:vertparl} \\ +\pardbyd{\lsubsup{f}{P}}{p} & = & \varepsilon \, +\lp { \lsubsup{f}{P} - \lsubsup{f}{E} } \rp - +\frac{{\ov{Q}}_{\rm{f, par}}}{M^{\rm{P}}} - +\frac{SNOW}{M^{\rm{P}}}\label{eq:vertparf} +\end{eqnarray} + +Following \citeumdp{027}, equations \ref{eq:dbydpmassflux}, \ref{eq:vertparl} and +\ref{eq:vertparf} are discretized: +% +\begin{eqnarray} +M_{\rm{k} + 1} & = & M_{\rm{k}} \, +\lp { 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp \, +\lp { 1 - \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp \, +EPSS_{\rm{k}} \label{eq:discdmfbydp} \\ +\lsubsup{l \, k + 1}{P} & = & \lp { +\lsubsup{l \, k}{P} + +\varepsilon_{\rm{k} + 1/4} \, \Delta p_{\rm{k} + 1/4} \, \lsubsup{l \, k}{E} + +\varepsilon_{\rm{k} + 3/4} \, \Delta p_{\rm{k} + 3/4} \, +\lc {1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4}} \rc \, +\lsubsup{l \, k + 1}{E} +} \rp \, / \, \lp {EPSS_{\rm{k}}} \rp \nonumber \\ +{ } & { } & + \lp { {\ov{Q}}_{\rm{l} \, \rm{k} + 1} \, / \, M_{\rm{k} + 1}} \rp +- \lp { RAIN_{\rm{k} + 1} \, / \, M_{\rm{k} + 1} } \rp +\label{eq:discvparl} \\ +\lsubsup{f \, k + 1}{P} & = & \lp { +\lsubsup{f \, k}{P} + +\varepsilon_{\rm{k} + 1/4} \, \Delta p_{\rm{k} + 1/4} \, \lsubsup{f \, k}{E} + +\varepsilon_{\rm{k} + 3/4} \, \Delta p_{\rm{k} + 3/4} \, +\lc {1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4}} \rc \, +\lsubsup{f \, k + 1}{E} +} \rp \, / \, \lp {EPSS_{\rm{k}}} \rp \nonumber \\ +{ } & { } & + \lp { {\ov{Q}}_{\rm{f} \, \rm{k} + 1} \, / \, M_{\rm{k} + 1}} \rp +- \lp { SNOW_{\rm{k} + 1} \, / \, M_{\rm{k} + 1} } \rp +\label{eq:discvparf} +\end{eqnarray} +% +where $EPSS_{\rm{k}} = +\lp {1 + \varepsilon_{\rm{k} + 3 / 4} \, \Delta p_{\rm{k} + 3 / 4}} \rp \, +\lp {1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4}} \rp $. + +The condensation and precipitation terms in equations \ref{eq:discdmfbydp}, +\ref{eq:discvparl} and \ref{eq:discvparf} make the equations implicit. +They are therefore solved by starting with an ascent in which condensation and +precipitation terms are suppressed: +% +\begin{eqnarray} +\lsubsup{l \, k + 1}{P} & = & \frac{\lp { +\lsubsup{l \, k}{P} + +\varepsilon_{\rm{k} + 1/4} \, \Delta p_{\rm{k} + 1/4} \, \lsubsup{l \, k}{E} + +\varepsilon_{\rm{k} + 3/4} \, \Delta p_{\rm{k} + 3/4} \, +\lc {1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4}} \rc \, +\lsubsup{l \, k + 1}{E} +} \rp}{EPSS_{\rm{k}}} \label{eq:discvparldry} \\ +\lsubsup{f \, k + 1}{P} & = & \frac{\lp { +\lsubsup{f \, k}{P} + +\varepsilon_{\rm{k} + 1/4} \, \Delta p_{\rm{k} + 1/4} \, \lsubsup{f \, k}{E} + +\varepsilon_{\rm{k} + 3/4} \, \Delta p_{\rm{k} + 3/4} \, +\lc {1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4}} \rc \, +\lsubsup{f \, k + 1}{E} +} \rp}{EPSS_{\rm{k}}} \label{eq:discvparfdry} +\end{eqnarray} +% +At the base of the convective plume (ie. the level immediately above cloud +base), \lsubsup{l \, k}{P} is initialized to \lsubsup{l \, i}{P} and +\lsubsup{f \, k}{P} to \lsubsup{f \, i}{P}, where the initial values are chosen +such that the modified form of \ref{eq:chidisccb} produces zero fluxes at +cloud base: +% +\begin{eqnarray} +Q4_{\rm{l}}(cb) = 0 & = & M_{\rm{cb+1/2}}^{\rm{P}} \, +\pardbyd{\lsubsup{l}{E}}{p} - M_{\rm{cb}}^{\rm{P}}\, +\lp { \lsubsup{l}{P \, i} - \lsubsup{l}{E}(\rm{cb}) } \rp \label{eq:q4lcbi} \\ +Q4_{\rm{f}}(cb) = 0 & = & M_{\rm{cb+1/2}}^{\rm{P}} \, +\pardbyd{\lsubsup{f}{E}}{p} - M_{\rm{cb}}^{\rm{P}}\, +\lp { \lsubsup{f}{P \, i} - \lsubsup{f}{E}(\rm{cb}) } \rp \label{eq:q4fcbi} +\end{eqnarray} + + +As the convection scheme makes the single phase assumption for parcel +condensate, it may be necessary to melt or freeze entrained condensate at this +point and adjust the temperature accordingly. +% +\begin{eqnarray} +\theta_{\rm{k + 1}}^{\rm{P}} = \theta_{\rm{k + 1}}^{\rm{P}} - +\lp \frac{L_{\rm{F}}}{C_{p} \, \Pi_{\rm{k + 1}}} \rp \, \lsubsup{f \, k + 1}{P} +& \; \ldots \; & \mbox{ if \lsubsup{f \, k + 1}{P} is melted } +\label{eqn:meltlf} \\ +\theta_{\rm{k + 1}}^{\rm{P}} = \theta_{\rm{k + 1}}^{\rm{P}} + +\lp \frac{L_{\rm{F}}}{C_{p} \, \Pi_{\rm{k + 1}}} \rp \, \lsubsup{l \, k + 1}{P} +& \; \ldots \; & \mbox{ if \lsubsup{l \, k + 1}{P} is frozen } +\label{eqn:freezell} +\end{eqnarray} + +Once a final value for the condensation term +$ {\ov{Q}}_{\rm{x} \, \rm{k} + 1} \, / \, M_{\rm{k} + 1} $ has been calculated +from the parcel specific humidity equations, it can then be added to the parcel +condensate to give a final pre-precipitation value. + +\begin{itemize} +\item{In practice, the rates $ {\ov{Q}}_{\rm{x} \, \rm{k} + 1}$ and +$ PPN $ are not calculated explicitly in the code. +Instead, their effect is applied directly as increments to the temperature and +moisture fields.} +\end{itemize} + +The precipitation calculation is unaltered. +% +\begin{equation} +P_{\rm{k} + 1} = \lp { \lsubsup{k + 1}{P} - \lsubsup{MIN}{P} } \rp \, +M_{\rm{k} + 1} \, / \, g +\label{eq:precip} \end{equation} +% +where \lsubsup{k + 1}{P} = \lsubsup{l \, k + 1}{P} + \lsubsup{f \, k + 1}{P}. + +\begin{itemize} +\item{Actually, given that the precipitation calculation appears to be +based upon the hydrostatic equation, it is debatable whether it is even suitable +for use with the New Dynamics model and I guess therefore that this needs +revisiting at some point.} +\end{itemize} + +This reduces the parcel condensate to : +% +\begin{eqnarray} +\lsubsup{l \, k + 1}{P} & = & \lp { +\frac{\lsubsup{l \, k + 1}{P}}{\lsubsup{k + 1}{P}} +} \rp \, \lsubsup{MIN}{P} \label{eq:vparlfinal} \\ +\lsubsup{f \, k + 1}{P} & = & \lp { +\frac{\lsubsup{f \, k + 1}{P}}{\lsubsup{k + 1}{P}} +} \rp \, \lsubsup{MIN}{P} \label{eq:vparffinal} +\end{eqnarray} + +The final parcel condensate values are then used in the rate calculation based +upon eqn~\ref{eq:basiclold}: +% +\begin{eqnarray} +Q4_{\rm{l}}(k) & = & M_{\rm{k+1/2}}^{\rm{P}} \, \pardbyd{\lsubsup{l}{E}}{p} + +\lp { {\mu}_{\rm{k}} \, M_{\rm{k}}^{\rm{P}} + +{\delta}_{\rm{k}} \, M_{\rm{k}}^{\rm{P}} } \rp \, +\lp { \lsubsup{l}{P}(\rm{k}) - \lsubsup{l}{E}(\rm{k}) } \rp - +{\ov{Q}}_{\rm{l, reset}} \label{eq:q4lmassf} \\ +Q4_{\rm{f}}(k) & = & M_{\rm{k+1/2}}^{\rm{P}} \, \pardbyd{\lsubsup{f}{E}}{p} + +\lp { {\mu}_{\rm{k}} \, M_{\rm{k}}^{\rm{P}} + +{\delta}_{\rm{k}} \, M_{\rm{k}}^{\rm{P}} } \rp \, +\lp { \lsubsup{f}{P}(\rm{k}) - \lsubsup{f}{E}(\rm{k}) } \rp - +{\ov{Q}}_{\rm{f, reset}} \label{eq:q4fmassf} +\end{eqnarray} + + +Note that, as a side-effect, the \citeumdp{027} environment equations for potential +temperature and specific humidity are also altered because the condensate is no +longer re-evaporated at the end (${\ov{Q}}_{\rm{l, reset}} = 0 += {\ov{Q}}_{\rm{f, reset}}$): +% +\begin{eqnarray} +\frac{\Delta \, \theta_{\rm{k}}^{\rm{E}}}{\Delta \, t} = +\lp \frac{ M_{\rm{k}} }{ \Delta \, p_{\rm{k}} } \rp +\lc { +\lp { 1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4} } \rp +\lp { 1 - \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp +\lp { 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp +\lp { \theta_{\rm{k + 1}}^{\rm{E}} - \theta_{\rm{k}}^{\rm{E}} } \rp +} \right . & + & \nonumber \\ +\lp { \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp +\lp { 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp +\lp { \theta_{\rm{k}}^{\rm{R}} - \theta_{\rm{k}}^{\rm{E}} } \rp +& + & \nonumber \\ +\left . { +\lp { \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp +\lp { \theta_{\rm{k}}^{\rm{P}} - \theta_{\rm{k}}^{\rm{E}} } \rp +} \rc & { } & \label{eq:enviroth} +\end{eqnarray} +% +and +% +\begin{eqnarray} +\frac{\Delta \, q_{\rm{k}}^{\rm{E}}}{\Delta \, t} = +\lp \frac{ M_{\rm{k}} }{ \Delta \, p_{\rm{k}} } \rp +\lc { +\lp { 1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4} } \rp +\lp { 1 - \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp +\lp { 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp +\lp { q_{\rm{k + 1}}^{\rm{E}} - q_{\rm{k}}^{\rm{E}} } \rp +} \right . & + & \nonumber \\ +\lp { \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp +\lp { 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp +\lp { q_{\rm{k}}^{\rm{R}} - q_{\rm{k}}^{\rm{E}} } \rp +& + & \nonumber \\ +\left . { +\lp { \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp +\lp { q_{\rm{k}}^{\rm{P}} - q_{\rm{k}}^{\rm{E}} } \rp +} \rc & { } & \label{eq:enviroq} +\end{eqnarray} + +Similarly, eqns \ref{eq:q4lmassf} and \ref{eq:q4fmassf} have +a discretized form as follows: +% +\begin{eqnarray} +\frac{\Delta \, \lsubsup{l \, k}{E}}{\Delta \, t} = +\lp \frac{ M_{\rm{k}} }{ \Delta \, p_{\rm{k}} } \rp +\lc { +\lp { 1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4} } \rp +\lp { 1 - \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp +\lp { 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp +\lp { \lsubsup{l \, k + 1}{E} - \lsubsup{l \, k}{E} } \rp +} \right . & + & \nonumber \\ +\lp { \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp +\lp { 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp +\lp { \lsubsup{l \, k}{P} - \lsubsup{l \, k}{E} } \rp +& + & \nonumber \\ +\left . { +\lp { \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp +\lp { \lsubsup{l \, k}{P} - \lsubsup{l \, k}{E} } \rp +} \rc & { } & \label{eq:enviroll} +\end{eqnarray} +% +and +% +\begin{eqnarray} +\frac{\Delta \, \lsubsup{f \, k}{E}}{\Delta \, t} = +\lp \frac{ M_{\rm{k}} }{ \Delta \, p_{\rm{k}} } \rp +\lc { +\lp { 1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4} } \rp +\lp { 1 - \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp +\lp { 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp +\lp { \lsubsup{f \, k + 1}{E} - \lsubsup{f \, k}{E} } \rp +} \right . & + & \nonumber \\ +\lp { \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp +\lp { 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp +\lp { \lsubsup{f \, k}{P} - \lsubsup{f \, k}{E} } \rp +& + & \nonumber \\ +\left . { +\lp { \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp +\lp { \lsubsup{f \, k }{P} - \lsubsup{f \, k}{E} } \rp +} \rc & { } & \label{eq:envirolf} +\end{eqnarray} + +\subsubsection{Background condensation} +\label{sec:conv_homog} +The modification to the convective plume will result in the transport, +detrainment and entrainment of condensate, in addition to the +transport of vapour and heat. Although condensation processes within +the plume are treated, it does not treat condensation in the +environment, which is forced by the compensating subsidence. We wish +to relate the environmental increments of vapour and temperature +to a forcing that can be applied in the environment. Because we +know that any detrained air associated with detrained liquid water +from the plume must be saturated with respect to liquid water, we +are able to translate the environmental changes into forcings. + +Here we will consider that the vapour change in the gridbox is as a result of +\textit{saturated with respect to liquid water} air being injected +from the plume and background air being displaced. +We do not consider whether the background air is at saturation yet, for we wish +to derive the expression for the required condensation if this is not the case. +We consider only liquid water clouds, ice clouds have no background condensation +applied as we do not make the instantaneous condensation assumption. + +Hence we can write + +\begin{equation} +\Delta \overline{q} = \Delta C_S ( q_{sat liq}(\overline{T_{s}}) - \overline{q} ) ++ (1 - \Delta C_S) \Delta \overline{q_{background}} +\end{equation} + +where $\Delta C_S$ is the volume of plume air that is detrained into the gridbox, +as discussed by \cite{bwg03}. $T_s$ is the temperature of the air injected into +the gridbox by the plume. The first term is simply the difference +between the value of $q$ in the plume and what was previously in the gridbox, and +the second term is the effect of a background change of $q$ that will be applied +across the part of the gridbox that is not associated with the injected air. We write this as: + +\begin{equation} +(1 - \Delta C_S) \Delta \overline{q_{background}} = \Delta \overline{q} - +\Delta C_S ( q_{sat liq}(\overline{T_{s}}) - \overline{q} ) . +\label{eqn:1mcs} +\end{equation} + +Now we recognise that + +\begin{equation} +\Delta \overline{q} = Q2~ \Delta t +\label{eqn:Q2} +\end{equation} + +where $Q2$ is the rate of moistening of the whole gridbox due to convection. +Remember that, at this stage, we haven't done any condensation outside of the plume. +Hence to calculate the condensation we should apply the background change in $\overline{q}$ +as a uniform forcing for the background air. Hence (\ref{eqn:1mcs}) becomes, using +(\ref{eqn:Q2}), + +\begin{equation} +(1 - \Delta C_S) A_q |_{background} \Delta t = Q2 ~ \Delta t - \Delta C_S +( q_{sat liq}(T_{s}) - \overline{q} ) . +\label{eqn:Aq} +\end{equation} + +where $A_q |_{background}$ is the currently unknown background forcing of +$q$ (see \cite{gwb02}) and $\Delta t$ is the timestep. +We can do the same analysis for the temperature change, and obtain + +\begin{equation} +(1 - \Delta C_S) A_T |_{background} \Delta t = Q1~ \Delta t - +\Delta C_S (T_s - \overline{T} ) +\label{eqn:AT} +\end{equation} + +where Q1 is the rate of warming in the gridbox due to convection and $A_T |_{background}$ is +the currently unknown background forcing of temperature. + +The full change of liquid water content in the gridbox is that injected, +$Q4~\Delta t$, plus the amount of condensation in the background from +the uniform forcings (see +\cite{wg03}). Note that the uniform forcings are only applied across +a proportion $1 - \Delta C_S$ of the gridbox. Hence these two terms give, using the +homogeneous forcing equations (\ref{dqcldt}) and (\ref{eq:deltaqc_exp2}), + +\begin{equation} +\Delta \overline{q_{cl}} |_{convection} = Q4 \Delta t ++ (1 - \Delta C_S) a_L C_l (A_q |_{background} \Delta t +- \alpha A_T |_{background} \Delta t ). +\label{eqn:qclconv} +\end{equation} + +Using (\ref{eqn:Aq}) and (\ref{eqn:AT}) to expand the forcing terms in (\ref{eqn:qclconv}) gives + +\begin{equation} +\Delta \overline{q_{cl}} |_{convection} = Q4 \Delta t ++ \Delta t a_L C_l ( Q2 - \alpha Q1) - \Delta C_S a_L C_l +(q_{sat liq}(T_s)-\overline{q} - \alpha (T_s - \overline{T})) . +\end{equation} + +We now note that + +\begin{equation} +q_{sat} (T_s) - q_{sat liq} (\overline{T}) = \alpha (T_s - \overline{T} ) +\end{equation} + +and hence the final result + +\begin{equation} +\Delta \overline{q_{cl}} |_{convection} = Q4 \Delta t ++ \Delta t ~ a_L C_l ( ( Q2 - \alpha Q1) - \Delta C_S +(q_{sat liq}(\overline{T}) - \overline{q} ) ) . +\label{eqn:dqcl} +\end{equation} + +There is thus an extra term, $-\Delta C_S (q_{sat}(\overline{T})-\overline{q} )$, +which needs to be included in addition to the standard application of the homogeneous +forcing of $Q1$ and $Q2$ (this is represented +by the second term of the expression). This has arisen from the requirement that +the vapour injected by the plume is saturated. We need simply +to retrieve the value of $\Delta C_S$ to complete the parametrization. +This can be straightforwardly obtained +from (\ref{eq:dcldt_inhom}), which links the net change of liquid cloudy volume +due to the injection, $\Delta C_{injection}$, with $\Delta C_S$. + +\begin{equation} +\Delta C_{injection} = (g_l - C_l) \Delta C_S +\end{equation} + +where $g_l$ is 1 if the injected cloud is of liquid phase and 0 if it +is of ice phase. We already know $\Delta C_{injection}$ from the +injection forcing arguments (\ref{eq:dctdt_xl}) above that link it to $Q4$. +We therefore complete the parametrization by calculating $\Delta C_S$ based +on whether $\Delta C$ is positive or negative. If $\Delta C$ is positive, +we assume that the plume must be of liquid phase and hence + +\begin{equation} +\Delta C_S = \frac{\Delta C_{injection}} {1 -C_l} . +\label{eqn:cs1} +\end{equation} + +If $\Delta C_l$ is negative, we assume that the plume must be of ice phase and hence + +\begin{equation} +\Delta C_S = - \frac{\Delta C_{injection}} {C_l} . +\label{eqn:cs2} +\end{equation} + +Here we have still assumed that the vapour content in the detrained plume is equal to +$q_{sat liq}$. A better assumption may be to replace the $q_{sat liq}$ term in +(\ref{eqn:dqcl}) with a $q_{sat}$ expression that depends on the volume fraction +of detrained condensate that is liquid phase, $g_l$. + +If $\Delta C_{injection}$ is zero, we assume that $\Delta C_S$ is 0 also. Equations +(\ref{eqn:dqcl}),(\ref{eqn:cs1}), and (\ref{eqn:cs2}) form the parametrization +for $\Delta \overline{q_{cl}}|_{convection}$. The +representation of $\Delta C_{convection}$ is similar in form to +$\Delta \overline{q_{cl}}|_{convection}$: + +\begin{equation} +\Delta C |_{convection} = \Delta C_{injection} ++ \Delta t ~ a_L G(-Q_c) ( ( Q2 - \alpha Q1) - \Delta C_S (q_{sat}(\overline{T}) +- \overline{q} ) ) . +\end{equation} + +where the specification of $G(-Q_c)$ follows (\ref{eqn22}). Note +that the code includes the numerical limit restriction that +$\Delta C_S$ is between 0 and 1. + +Thus we are able to parametrize the net condensation and cloud changes +associated with the $Q1$ and $Q2$ terms in a physically more consistent way than +using simple homogeneous application of these terms. + +As an aside, we note that in the \cite{t93} scheme the condensation and cloud +fraction change associated with the compensating subsidence is taken out of +the convection term by adding the vertical motion associated with the compensating +subsidence to the large-scale vertical velocity before the +\cite{t93} equivalent of the homogeneous forcing term is applied. By doing +so it ensures that any balance between these two terms (as the tropical circulation +is commonly analysed to show) is removed before the net effect is calculated, +leading to more accurate numerical behaviour. + +\subsubsection{Homogeneous forcing of the environment by +convective-subsidence pressure change} + +To this end, the code includes an option to perform the homogeneous forcing +of liquid cloud by convection using the ``pressure forcing'' from the +convective subsidence, consistent with the pressure forcing by large-scale +advection (see sections \ref{sec:advec} and \ref{sec:pres}). +This approach replaces the above method of homogeneous forcing by convection +if the UM namelist switch \textbf{l\_pc2\_homog\_conv\_pressure} is turned on. +By applying the same homogeneous forcing method for advection and +convectively-forced subsidence, we should get the correct zero net +change in liquid cloud in the common situation where the large-scale ascent +and convective subsidence are in balance (implying no net vertical displacement +of environment parcels). + +Under this option, the increments to $\overline{q_{cl}}$ and $C_l$ produced +by the convection scheme are assumed to already include the effects +of entrainment, detrainment (i.e. injection) and compensating subsidence +(i.e. vertical advection) as expressed by equation \ref{eq:chimassflux}, +but exclude the effects of homogeneous forcing of clouds in the enviroment. +Note that taking equation \ref{eq:chimassflux} with $\chi$ set to +water vapour $q$, detrainment of saturated air into a subsaturated +environment will imply a positive tendency of $\overline{q}$, +but this is \textit{not} a homogeneous forcing, since the increase +in $\overline{q}$ is entirely due to injecting new parcels of saturated +air without altering the existing environment parcels. +Setting $\chi$ to be $\overline{q_{cl}}$ or $C_l$ in equation +\ref{eq:chimassflux}, there is a simply-calculated source of cloud +water and fraction wherever the detrained air is cloudy +($C_l=1$ in the detrained parcel), and we assume +these terms have been calculated this way inside the convection scheme. + +Since entrainment and detrainment do not constitute a homgeneous forcing +and are already accounted for in the convection scheme, + the only component of the convective forcing of liquid cloud +that needs to be done by the PC2 call after convection is the homogeneous +forcing by the subsidence term. This is in essence a vertical advection +(environmental forced descent by updrafts, or forced ascent by downdrafts). +The homogeneous forcing can be calculated from the expected pressure change +(and accompaying adiabatic temperature change) following the environment +as it is vertically displaced. +Conveniently, the UM already holds the convective mass-flux in units +of Pa s$^{-1}$, so it already expresses the pressure vertical velocity forced +by subsidence in the environment: + +\begin{equation} +\Delta p^E = \Delta t \left( M_{up} - M_{dwn} \right) +\label{eq:delta_p_conv} \end{equation} + +where $M_{up}$ is the updraft mass-flux, $M_{dwn}$ is the downdraft mass-flux, +and $\Delta t$ is the model timestep length. +The adiabatic temperature change following an environment parcel +subsided from pressure $p - \Delta p^E$ to $p$ is then given by: + +\begin{equation} +\Delta T^E = \theta^E \left( \left(\frac{p}{p_{ref}}\right)^\kappa + - \left(\frac{p - \Delta p^E}{p_{ref}}\right)^\kappa + \right) +\label{eq:delta_t_conv} \end{equation} + +where $\theta^E$ is the environment potential temperature, +$p_{ref}$ is the reference pressure used to define potential temperature, +and $\kappa = \frac{R_d}{c_p}$ is the ratio of the gas constant for dry air +over its heat capacity at constant pressure. +\ref{eq:delta_p_conv} and \ref{eq:delta_t_conv} are passed into the +PC2 homogeneous forcing routine after convection as the forcings +to be applied (with the forcings to all other variables set to zero). + + +\subsubsection{Convective cloud amount} +It is a debatable point whether the convective cloud fraction should +be set to zero. Although this was one of the original key concepts of +PC2, the cloud that is detrained from the convection scheme is into +the \textit{environment}, and does not represent the tower cloud. However, +it should be able to represent recently detrained cloudy air in a more +accurate way than by simply appealing to a diagnostic large-scale cloud scheme. +There are similar issues associated with the cloud fraction predicted +from the Tiedtke scheme. Probably the most consistent interpretation +is the inclusion of a tower cloud fraction within PC2, but not an +anvil cloud. However, we need to consider carefully any double +counting (or non-counting) implications. In the PC2:64 formulation, +we can represent the large optical depths associated +with new anvils, although we also tend to overestimate the optical +depth of shallow convective clouds. +Hence we choose to apply neither a diagnostic anvil or tower cloud, +so similar to Tiedtke, and let the large-scale cloud fraction represent +the convection completely. + +Strictly speaking these choices are independent of the PC2 scheme, +being simply choices that are available as part of the existing convection +scheme, but they are clearly directly related to the rest of the +cloud scheme formulation. + +\subsubsection{CAPE scaling} +The CAPE scaling option in the mass-flux convection scheme scales its +increments by the calculated values of $\frac{1}{CAPE} \frac{dCAPE}{dt}$. +This applies +also to all the PC2 calculated condensate and cloud fraction increments. +Additionally, in order to achieve reasonable mass flux profiles, +it has proved necessary to adjust the calculation of +$\frac{dCAPE}{dt}$ to use increments of +$\Delta \theta$ (potential +temperature) and $\Delta q$ calculated using a non-PC2 calculation +of these terms. Hence we consider any detrained condensate to have been +evaporated when we calculate $\frac{dCAPE}{dt}$. + +\subsubsection{Convective precipitation} +The amount of condensate detrained from convective plumes, and hence +the amount of moisture in the upper levels of the atmosphere, is very +dependent upon the amount of convective precipitation that is allowed +to fall from the column. The standard parametrization of this is that +any condensate greater than a specified value (dependent on $T$) +is precipitated, leaving the rest to be detrained. + +PC2 incorporates a tuning to this function of temperature by applying +the additional restriction that the limit may not fall to less than +$2 \times 10^{-4}~kg~kg^{-1}$. This implies a difference at temperatures +less than around $-42 ^{\circ} C$, with the tuning allowing less +precipitation and greater detrainment. This change is necessary +in order to produce thick enough anvil clouds. + +\subsubsection{Phase of condensate} +\label{sec:plume_phase} +The phase of the convective condensate \textit{carried in the plume} +is determined by a single phase change temperature TICE, with +condensate entirely in the +ice phase at colder temperatures and condensate entirely in the liquid +phase at warmer temperatures. For PC2:66, this temperature is -10 $^{\circ}$ C. + +\begin{equation} +\delta_{xl} = \left\{ \begin{array}{ll} + 1, & T_{plume} \ge -10 ^{\circ} C \\ + 0, & T_{plume} < -10 ^{\circ} C + \end{array} \right. +\end{equation} + +\begin{equation} +\delta_{xi} = \left\{ \begin{array}{ll} + 0, & T_{plume} \ge -10 ^{\circ} C \\ + 1, & T_{plume} < -10 ^{\circ} C + \end{array} \right. +\end{equation} + +\subsubsection{Tidier way of coupling convection and PC2} +\label{sec:conv-simpler} + +This area is still under development. But in brief, work is udner way to ensure that the +convective plume smoothly transitions from detraining liquid to detraining ice, rather +than using the abrupt change implied by the current formulation of the convection scheme. +Additionally, rather than using inhomogeneous increments to condensate (combining detrainment +and subsidence advection) to calculate cloud fraction increments, an alternative is to use +the detrainment of condensate to simply grow cloud fraction to ensure a specified in-cloud +liquid water content. The cloud fraction are then advected downwards byt he subsidence advection. + The increments to cloud fraction from detrainment and subsidence are then combined. + +\subsubsection{Prognostic dust approach} +A prognostic dust approach is implemented in the micro-physics scheme under +large-sale-precipitation where by the heterogeneous nucleation temperature +can be defined to vary three dimensionally globally as an arc-tangent +function of the mineral dust distribution in the model (documented +in \citeumdp{026}). By default, both liquid and ice are detrained simultaneously at the same +height, and the fraction of condensate that is ice linearly ramps as a function of temperature. +i.e. condensate is assumed to be all-liquid when T is greater than one tuneable threshold; all-ice +when T is less than another tuneable threshold, and vary linearly in-between (the threshold values are +given by starticeTkelvin and alliceTdegC in the UM cloud-scheme namelist. The new heterogeneous +nucleation temperatures calculated in the large-scale-precipitation are passed to the convection +scheme and are used as the above detrainment temperature thresholds by +maintaining a similar linear ramp. For e.g., condensate is +assumed to be all-liquid for T $\geq$ $tnuc_n$ and all-ice for T +$\leq$ $tnuc_n$ - 10.0 + +\subsubsection{Condensation adjustment in the profiles input to the +convection scheme} +\label{sec:conv_input_profs} + +The convection scheme itself is highly sensitive to the input environment +temperature and moisture profiles {\it before} the convection increments +(or PC2 response) are calculated. In particular, the parcel buoyancy +(and hence the CAPE and mass-flux scaling) maybe radically different +depending on whether a ``large-scale'' condensation / evaporation adjustment +is performed before the convection call. + +Where there is large-scale ascent, the profiles after Semi-Lagrangian advection +may have become supersaturated and unrealistically unstable, until the +expected condensation adjustment is performed. If the convection scheme +``sees'' these unrealistic intermediate profiles, it is likely to +predict an excessive, unrealistic mass-flux. + +To address this problem, there are two namelist switches that enable +additional condensation adjustments from PC2 before the convection call: + +\begin{itemize} +\item {\bf l\_pc2\_sl\_advection}: performs homogeneous forcing response +to Semi-Lagrangian advection immediately after the advection calculation, +instead of at the end of the timestep (see section \ref{sec:pres}). +\item {\bf l\_cloud\_call\_b4\_conv}: performs an additional call to +PC2 initiation (and PC2 checks) before the convection scheme +(see section \ref{sec:init2}). +This should catch any instances where large-scale ascent or other processes +have brought the profiles after advection to near or beyond saturation, +in grid-points where there was no liquid cloud already present +(and so no homogeneous forcing response). +\end{itemize} + +\subsection{Response to pressure changes} + \label{sec:pres} + +A pressure change following the parcel during the timestep will result +in an adiabatic temperature change which will force condensation, +hence we must include this temperature change forcing within PC2. +The majority of this pressure change comes from vertical advection +(although not all). +Remember that the advection (section \ref{sec:advec}), on its own, +does not cause condensation, it merely moves the existing cloud field. + + Using the semi-Lagrangian advection in the same way as is performed +for $\overline{q_{cl}}$ etc., the PC2 scheme will obtain the value of the +model prognostic \textit{Exner}, ($\prod$) on the departure points +($\prod_{dep}$). \textit{Exner} is defined as + +\begin{equation} +\label{eq:exner} +\prod = \frac{T}{\theta} = \left( \frac{p}{p_{ref}} \right)^{\kappa} +\end{equation} + +where $\theta$ is the potential temperature, $p_{ref}$ is a reference +pressure set to 1000 hPa, and $\kappa = +\frac{c_p - c_v}{c_p}$ , where $c_v$ is the heat capacity of dry +air at constant volume. The \textit{Exner} quantity is kept as a prognostic +variable in the model (this is unchanged from the control model), and the +value of $\prod$ on the departure points represents the initial value +in the timestep, since there is no update to $\prod$ until the end of +the timestep. After the second physics updates have been performed +(\textit{atmos-physics2}), the +model (including the control) recalculates the value of \textit{Exner} +($\prod^{[n+1]}$). +From $\prod_{dep}$ and $\prod^{[n+1]}$ we can calculate, using the definition +(\ref{eq:exner}), the values of departure pressure and temperature: + +\[ +\overline{p}_{dep} = p_{ref} {\prod_{dep}}^{\frac{1}{\kappa}} +\] + +\[ +\overline{T}_{dep} = \theta \prod_{dep} +\] + +Hence we obtain the net forcing values + +\begin{equation} +\Delta \overline{T} = \overline{T}^{[n+1]} - \overline{T}_{dep} +\label{eq:deltatsl} +\end{equation} + +and + +\begin{equation} +\Delta \overline{p} = \overline{p}^{[n+1]} - \overline{p}_{dep} . +\label{eq:deltapsl} +\end{equation} + +where $\overline{T}^{[n+1]}$ and $\overline{p}^{[n+1]}$ are the temperature +and pressure at the arrival point, after the dynamics call. +(\ref{eq:deltatsl}) and (\ref{eq:deltapsl}) are passed to the homogeneous +forcing routine in order to calculate +the condensation and cloud fraction changes associated with the pressure +change. + +We include this forcing towards the end of the timestep. There are two +reasons for this: +firstly, values of $\prod^{[n+1]}$ are not calculated by the control model +until after the physics is complete; secondly, it makes sense to locate this +process in the timestep in a similar location +to where the large-scale cloud scheme is included in the control (i.e. +after the implicit part of the boundary layer has finished). + +However, there +is a counter argument that says we should include this process immediately +after the dynamics, since we can then apply a forcing on an initial state +that has not already been modified by the dynamics, boundary layer and +convection schemes. This improves the numerics of the problem, since the +homogeneous forcing is designed to take time level n values as inputs. + +These issue are optionally addressed by turning on the UM namelist switch +\textbf{l\_pc2\_sl\_advection}. Under this switch, the PC2 homogeneous +forcing response to pressure change is split: +\begin{enumerate} +\item Forcing by the \textit{Lagrangian} component of pressure change, +performed immediately after the Semil-Lagrangian advection scheme +(before the call to atmos\_physics2). +This calculates the pressure change from the departure point value of +\textit{Exner} described above, to the start-of-timestep value of +\textit{Exner} at the arrival point. +\item Forcing by the \textit{Eulerian} component of pressure change, +performed at the end of the timestep (after the dynamics Helmholtz solver). +This calculates the pressure change from the start-of-timestep \textit{Exner} +at the arrival point, to the end-of-timestep \textit{Exner}. +\end{enumerate} + +Having to calculate the pressure forcing twice obviously adds some +computational cost, but has several advantages: +\begin{itemize} +\item As noted above, the PC2 homogeneous forcing calls can now take +as input the temperature and water-vapour content \textit{before} +the pressure change has been applied, as intended. This should improve +the numerical accuracy. +\item Most of the condensation or evaporation from the dynamics comes from +the \textit{Lagrangian} component of the pressure change, which has now moved +from the end of the timestep to before the dynamics Helmholtz solver. +This means that any latent heating from condensation forced by ascent is now +accounted for by the solver within the same timestep. +This improves the numerical accuracy of the dynamics-physics coupling. +\item If the condensation forced by resolved ascent is only added on at the +end of the timestep, the profiles passed into atmos\_physics2 can contain +out-of-balance thermodynamic states (e.g. if the profile has been lifted +by advection, it maybe supersaturated / unrealistically unstable before +the resulting condensation is added on). This may adversely affect the +convection scheme, which must act upon the profiles passed into +atmos\_physics2. +\end{itemize} + +The splitting of the pressure forcing call under the +\textbf{l\_pc2\_sl\_advection} switch was originally implemented to make the +profiles passed to convection more realistic. + +\subsection{Initiation} +\label{sec:init2} +As discussed in section \ref{sec:init}, there are occasions when +$\overline{q_{cl}}$ and $C_l$ need to be initiated from 0 or 1. +The application of the initiation is given in section \ref{sec:init}. +The initiation forms a new, separate block of PC2 code to perform this +calculation, and is located immediately following the pressure change +response (section \ref{sec:pres}). +Also, if the UM namelist switch {\bf l\_cloud\_call\_b4\_conv} is set to +true, an additional call to PC2 initiation is performed before the +convection scheme, to ensure that the condensation response to +advection and other forcings earlier in the timestep has been accounted for +in the profiles passed to the convection scheme, even if there was no +cloud already present for homogeneous forcing to act upon. +(see section \ref{sec:conv_input_profs}). + +There are currently 3 options for the conditions under-which initiation +may occur. For all of these options, +if using the bimodal cloud scheme to do initiation within PC2, +then the tests on $RH_T$ relative to $RH_{crit}$ are replaced by equivalent +tests for whether the saturation boundary lies within the bounds +of the bimodal scheme's assumed PDF, as described in section +\ref{sec:bimodal_init}. + +\subsubsection{``Original'' initiation logic} + +This option is selected by setting the UM namelist switch +{\bf i\_pc2\_init\_logic = 1} (Original) + +The initiation will be called if the liquid cloud fraction is either +0 or 1 and appropriate $RH$ criteria hold, along with other restrictions. +$C_l$ is initiated away from 0 if + +\begin{itemize} +\item{ $RH_T > RH_{crit} + RH_{crit \, tol}$ \textbf{and} } +\item{ Cumulus convection has {\em not} been diagnosed from the + boundary-layer in the current column \textbf{and} } +\item{ The current level is not below the surface mixed-layer LCL \textbf{and} } +\item{ $C_l = 0$ \textbf{and} } +\item{ $RH_T^{[n+1]} > RH_T^{[n]}$ ,} +\end{itemize} + +where $RH_{crit \, tol}$ is a specified tolerance parameter, of value 0.01, +and $RH_T$ is defined in (\ref{eq:rht}). $RH_T^{[n]}$ is the start of +timestep value of $RH_T$ (i.e. at time level n) and $RH_T^{[n+1]}$ is the +value when initiation is called. +Additionally, there is another possibility for the last of the relations. +This second option also allows initiation when the water +is supercooled: + +\begin{itemize} +\item{ $C_l < 0.05$ \textit{and} $\overline{T} < 0 ^{\circ} C$ .} +\end{itemize} + +Equivalently, $C_l$ is initiated away from 1 if + +\begin{itemize} +\item{ $RH_T < 2 - RH_{crit} - RH_{crit \, tol}$ \textbf{and} } +\item{ $C_l = 1$ \textbf{and}} +\item{ $RH_T^{[n+1]} < RH_T^{[n]}$ .} +\end{itemize} + +\subsubsection{``Simplified'' initiation logic} + +This option is selected by setting the UM namelist switch +{\bf i\_pc2\_init\_logic = 2} (Simplified) + +Under this option, the conditions for initiation are: + +Either: +\begin{itemize} +\item $RH_T > RH_{crit} + RH_{crit \, tol}$ \textbf{and} +\item $C_l < C_{tol}$ \textbf{and} +\item The current level is not below the surface mixed-layer LCL \textbf{and} +\item $RH_T^{[n+1]} > RH_T^{[n]}$ +\end{itemize} +Or: +\begin{itemize} +\item $RH_T < 2 - RH_{crit} - RH_{crit \, tol}$ \textbf{and} +\item $C_l > 1 - C_{tol}$ \textbf{and} +\item $RH_T^{[n+1]} < RH_T^{[n]}$ +\end{itemize} + +where $C_{tol}$ can be set via the UM namelist; its original standard value +is 0.005. Note this threshold is also used to remove small cloud-fractions +after initiation; see section \ref{sec:checks2}. + +This is very similar to the ``Original'' initiation logic described above, +but with the following differences: +\begin{itemize} +\item The condition that the boundary-layer hasn't diagnosed cumulus + convection in the column is removed. + Note that this condition spuriously suppressed initiation in the free + troposphere {\em above} any cumulus cloud produced by the convection + scheme. +\item $C_l$ only needs to be within a numerical tolerance $C_{tol}$ from + 0 or 1, rather than having to be {\it exactly} 0 or 1. +\item The different threshold when initiating super-cooled cloud is removed. +\end{itemize} + +\subsubsection{``Smooth'' initiation logic} +\label{sec:smooth_initiation} + +This option is selected by setting the UM namelist switch +{\bf i\_pc2\_init\_logic = 3} (Smooth) + +There is a fundamental numerical problem with the above options, in that +the initiation process is not permitted to have any effect at all unless +$C_l$ goes to (near) 0 or 1, but can predict values of $C_l$ very different +to 0 or 1 when it does activate. This leads to unphysical sudden noisy jumps +in $C_l$ and $q_{cl}$ when initiation occurs. +For example, if erosion causes $C_l$ to steadily decline, it will continue +to decline (even when the grid-mean $RH_T$ exceeds $RH_{crit}$) until +it reaches the threshold (0 or $C_{tol}$). At this point, initiation suddenly +increases $C_l$ and $q_{cl}$ to the values predicted by the diagnostic cloud +scheme. Erosion may then gradually remove them again, and the cycle repeats. +There is no physical reason for this internal mode of variability in the +scheme. + +Another problem arises if we consider the sensitivity to model resolution. +Suppose we have many adjacent small grid-boxes with similar $RH_T$, +a few containing cloud, the rest containing no cloud. If the whole +region cools to the point where $RH_T > RH_{crit}$, then new cloud +will initiate in the cloud-free grid-boxes, but not in the cloudy grid-boxes. +Now suppose we run a coarse-grained version of the same simulation; +the many small grid-boxes are replaced by a single grid-box containing the +average $C_l$ over the small grid-boxes. Since we now have just one +grid-box already containing partial cloud-cover, initiation of new cloud +can no longer occur anywhere. + +To address these problems, there is an option to use a much simpler / +numerically better-posed initiation method; +always allow the diagnostic cloud scheme to be called +(provided it is expected to predict nonzero cloud water, +i.e. $RH_T > RH_{crit}$ in the case of the Smith scheme). +The $q_{cl}$ predicted by the diagnostic cloud scheme is then taken +as a minimum limit applied to the prognostic $q_{cl}$. +This amounts to taking the diagnostic cloud scheme's assumed PDF as a minimum +allowed width to the actual prognostic moisture PDF. +The prognostic $C_l$ and $q_{cl}$ are incremented as follows: + +\begin{itemize} + +\item If ${q_{cl}}_{diag} > q_{cl}$: + +$\Delta q_{cl} = {q_{cl}}_{diag} - q_{cl} +\quad \refstepcounter{equation}(\theequation)\label{eq:dqcl_init}$ + + + \begin{itemize} + + \item If $Q_C < 0$: + + $\Delta C_{l} = \frac{\Delta q_{cl}}{{q_{cl}}_{diag}} + \left( {C_{l}}_{diag} - C_{l} \right) + \quad \refstepcounter{equation}(\theequation)\label{eq:dcl_init1}$ + + \item If $Q_C > 0$: + + $\Delta C_{l} = \frac{\Delta SD}{{SD}_{diag}} + \left( {C_{l}}_{diag} - C_{l} \right) + \quad \refstepcounter{equation}(\theequation)\label{eq:dcl_init2}$ + + \end{itemize} + +\item Otherwise: + +$\Delta q_{cl} = 0$ + +$\Delta C_{l} = 0$ + +\end{itemize} + +where the subscript $_{diag}$ denotes the liquid cloud water content and +fraction predicted by the diagnostic cloud scheme (either Smith or Bimodal). + +Equation \ref{eq:dcl_init1} simply sets the cloud-fraction to a weighted +mean of the pre-existing and diagnostic-scheme cloud-fractions, in proportion +to the fraction of the water content that was created by initiation +versus that which was already there. +If the pre-existing $q_{cl}$ is zero, \ref{eq:dqcl_init} and \ref{eq:dcl_init1} +simply set $q_{cl}$ and $C_l$ to their new diagnosed values, +as in the previous options. +Crucially, in the limit that the pre-existing $q_{cl}$ approaches +${q_{cl}}_{diag}$, the increments to $q_{cl}$ and $C_l$ smoothly go to zero. +This is important to make the initiation process numerically well-posed, +so that it yields a smooth, continuous solution. + +Note that when we are initiating from $C_l = 1$ instead of $C_l = 0$, +we expect the pre-existing $q_{cl}$ to be nonzero even when there is +no pre-existing sub-grid PDF width. In this case, the completely +uninitiated state will have zero saturation deficit $SD$, rather than +zero $q_{cl}$. Therefore, in this case the increment to $C_l$ is calculated +based on the fractional increase in $SD$ from initiation +(equation \ref{eq:dcl_init2}), instead of the fractional increase in $q_{cl}$. + +Whether to increment $C_l$ based on the increase in $q_{cl}$ or $SD$ is +determined based on the sign of $Q_C$, which is defined as in equation +\ref{eq:qc_eq_qt-qs} (reproduced here for clarity): + +\[ +Q_c = a_L \left( \overline{q_T} - q_{sat}(\overline{T_L}) \right) +\] + +The saturation deficit $SD$ is defined by equation \ref{SD2}: + +\[ +SD = a_L \left( q_{sat}(\overline{T}) - \overline{q} \right) +\] + +Under the reasonable approximation that $q_{sat}$ varies linearly between +$\overline{T}$ and $\overline{T_L}$, so that the values of +$\alpha$ and $a_L$ are the same in +both of these equations, and: + +\[ +q_{sat}(\overline{T_L}) = q_{sat}(\overline{T}) - \alpha \frac{L}{c_p} q_{cl} +\] + +we obtain: + +\begin{equation} +q_{cl} = Q_c + SD +\label{eq:qc_plus_sd} +\end{equation} + +It can be seen that when $Q_C > 0$ (total-water super-saturation), +it represents the value $q_{cl}$ would have if the whole grid-box +were saturated ($SD = 0$, $C_l = 1$). Note that $q_{cl}$ cannot fall below +$Q_C$, since $SD$ cannot be negative. +Since $Q_c$ is invariant under condensation / evaporation, we must have +$\Delta SD = \Delta q_{cl}$ +(hence the implementation of \ref{eq:dcl_init2} in the code simply uses +$q_{cl} - Q_c$ in place of $SD$, and $\Delta q_{cl}$ in place of $\Delta SD$). + +\subsubsection{Additional checks after PC2 initiation} + \label{sec:checks2} + +The initiation is followed immediately by a section of resetting code. +For numerical reasons, it is possible to obtain very low, but non zero, +values of $C_l$ (and equivalently values very close to, but not equal to, +1). The code will reset these clouds to either a fraction of 0 or 1, as +appropriate. We choose to apply these terms here and not in the +Bounds Checking part of the code (section \ref{sec:checks}) because +these are not required to obtain consistency between fields, but are +`tidying up' pieces of code, although they may reasonably also be +applied in the Bounds Checking. Care needs to be taken when choosing +the thresholds, since +we do not wish to reset small values that are genuinely created +by a physics scheme in the model. + +We first calculate $RH_T$ using (\ref{eq:rht}) and compare +this to the critical relative humidity, $RH_{crit}$. The liquid +cloud fraction will be reset to 1 if: +\begin{itemize} +\item{ $RH_T > 2 - RH_{crit}$ and $C_l \ge C_{high}$} +\item{ or $C_l \ge C_{high 2}$ } +\end{itemize} +where $C_{high}$ and $C_{high 2}$ are defined in \ref{eq:chigh-chigh2}. +The evaporation is done by calculating $SD$ using (\ref{SD2}) with (\ref{eq:a_L}) and +(\ref{eq:alpha_exp}) and evaporating the equivalent amount of liquid +into the gridbox to take it to saturation, according to +(\ref{eq:qsdcheck1}) below. + +Similarly, the equivalent check for low values of $RH_T$ is performed. +The liquid +cloud fraction will be reset to 0 if: +\begin{itemize} +\item{ $RH_T < RH_{crit}$ and $C_l \le C_{low}$} +\item{ or $C_l \le C_{low 2}$ .} +\end{itemize} +The remaining $\overline{q_{cl}}$ is evaporated into the gridbox +using (\ref{eq:qclcheck}) below. + +The thresholds $C_{high}$, $C_{high 2}$, $C_{low}$ and $C_{low 2}$ are +set using the parameters $C_{tol}$ and $C_{tol 2}$, according to: + +\begin{eqnarray} +C_{high} = 1 - C_{tol}, \nonumber \\ +C_{high 2} = 1 - C_{tol 2}, \nonumber \\ +C_{low} = C_{tol}, \nonumber \\ +C_{low 2} = C_{tol 2}, +\label{eq:chigh-chigh2} +\end{eqnarray} + +where the parameters $C_{tol}$ and $C_{tol 2}$ can be set via the UM namelist +variables {\bf cloud\_pc2\_tol} and {\bf cloud\_pc2\_tol\_2}. +The original standard values of these parameters are +$C_{tol} = 0.005$ and a lower value $C_{tol 2} = 0.001$. + +Investigations in SCM runs using the comorph convection scheme +(which behaves more smoothly and so typically gives smaller increments +to $C_l$ over a single timestep than other schemes which exhibit intermittent +behaviour) suggested these thresholds are too high to avoid spuriously +resetting physical values of $C_l$ to zero. Detrainment from sparse +shallow cumulus, or advection of cloud into a neighbouring grid-box +under light winds, commonly give increments which increase $C_l$ from zero +to a value less than $0.005$ in one timestep (but would eventually increase +$C_l$ to a significant value over subsequent timesteps if the checks did +not keep resetting $C_l$ to zero). + +Note that if these checks are relaxed by lowering the thresholds +$C_{tol}$ and $C_{tol 2}$ to near-zero, +similar checks are still performed independently by the bounds checking +described in section \ref{sec:checks}, but with a much lower +threshold of $C_{tol 3} = 1 \times 10^{-12}$. + +\subsection{Bounds checking} + \label{sec:checks} +Ideally, model prognostics would never become inconsistent with one another. +However, even although the mathematical solution of the governing equations +may be well behaved, due to numerical inaccuracies values may become +inconsistent. For the cloud and condensate quantities, there are a number +of consistencies that must apply. The bounds checking forms a subroutine +that will, if necessary, adjust $\overline{q}$, $\overline{q_{cl}}$, +$\overline{q_{cf}}$, $C_l$, $C_i$, $C_t$ and, for latent heating, +$\overline{T}$, to ensure consistency between these values. + +The bounds checking is performed three times during the timestep. Firstly, +after the parallel part of the physics (\textit{atmos-physics1}) is complete; +secondly, before the initiation (section \ref{sec:init}) is called; thirdly, +after the initiation is called. + +\subsubsection{} +Firstly, if $C_l > 1 - C_{tol 3}$ then $C_l$ is set to 1. +Accordingly, $C_t$ is set to 1 as well. +$C_{tol 3}$ is a tiny numerical tolerance set to $1 \times 10^{-12}$, +a value intended to be in the realm of floating point rounding error rather +than anything that represents a physical solution. + +\subsubsection{} +The second check is to reset $\overline{C_l}$ to zero. This may be performed +for two reasons. Firstly, if the amount of $\overline{q_{cl}}$ is very small +($\overline{q_{cl}} < q_{c0}$, where $q_{c0} = 1 \times 10^{-10} kg kg^{-1}$), +so we avoid carrying negligible, but non-zero values of $\overline{q_{cl}}$ +and $C_l$. Secondly, if $C_l < C_{tol 3}$ then we reasonably reset $C_l$ to zero. +$C_t$ gets reset, as it must if there is no liquid cloud, to be equal to $C_i$. + +\subsubsection{} +\label{sec:pc2_checks_sd} +The next check complements the first but updates the moisture fields. +We firstly calculate $SD$ using (\ref{SD2}) and +(\ref{eq:alpha_exp}). We then check whether $SD < 0$. +This check catches instances where we have grid-mean supersaturation, +which ought to be impossible (under the instantaneous condensation +assumption made by PC2, condensation should occur to instantly adjust +any supersaturated regions of the gridbox to saturation, so we +{\it must always} have $SD \ge 0$. +When this condition is violated, we condense water vapour to adjust to +grid-mean saturation. $-SD$ corresponds to the amount of vapour that must be +condensed to achieve this, so we have: + +\begin{eqnarray} +\overline{q} \leftarrow \overline{q} + SD \nonumber \\ +\overline{q_{cl}} \leftarrow \overline{q_{cl}} - SD \nonumber \\ +\overline{T} \leftarrow \overline{T} - \frac{L_c}{c_p} SD +\label{eq:qsdcheck1} +\end{eqnarray} + +The original version of this check on $SD$ +(which may increase $\overline{q_{cl}}$), made no accompanying changes to +liquid cloud fraction. However, increases in $\overline{q_{cl}}$ +without any increase in $C_l$ can lead to spurious high in-cloud condensate +which is then converted to rain by the microphysics at the next time-step. +There are currently 4 options for how to treat $C_l$ when increasing +$\overline{q_{cl}}$ under this saturation adjustment, selected by the +UM large-scale cloud namelist switch {\bf i\_pc2\_checks\_cld\_frac\_method}: +\begin{itemize} +\item {\bf i\_pc2\_checks\_cld\_frac\_method = 0} - +Original method; $C_l$ is left unaltered. +\item {\bf i\_pc2\_checks\_cld\_frac\_method = 1} - +Set $C_l$ and $C_t$ to 1. +\item {\bf i\_pc2\_checks\_cld\_frac\_method = 2} - +If $\overline{q_{cl}}$ and $C_l$ were already nonzero before the adjustment, +increase $C_l$ at the same fractional rate as $\overline{q_{cl}}$, so that +the in-cloud water content $\frac{\overline{q_{cl}}}{C_l}$ is conserved. +Otherwise, increase $C_l$ so-as to yield a prescribed in-cloud water +content set to 0.5 g kg$^{-1}$. $C_t$ is then increased by the same +amount as $C_l$, to maintain consistency. +\item {\bf i\_pc2\_checks\_cld\_frac\_method = 3} - +This is the same as option 2 above, except in the case where +$\overline{q_{cl}}$ or $C_l$ was zero before the adjustment. In this case, +$C_l$ is set based on an empirical power-law function of $\overline{q_{cl}}$. +\end{itemize} + +\subsubsection{} + +Next we check whether $SD > 0$, {\it and} $C_l = 1$ +(the first of our checks has ensured that $C_l$ is no greater than 1). +This check catches instances where we have total cloud-cover in a subsaturated +grid-box, which ought to be impossible (if the whole grid-box is full of liquid +cloud, then it must be at grid-mean saturation, i.e. $SD = 0$). +When this happens, we adjust $\overline{q}$ and $\overline{q_{cl}}$ +to take $SD$ to zero, +\textit{provided} that $\overline{q_{cl}} > SD$. Remember that $SD$ +corresponds to the amount of vapour that must be +\textit{evaporated} into the gridbox to give saturation, so we simply +make exactly the same adjustments as we do for removing supersaturated +states above (\ref{eq:qsdcheck1}), except that here $SD$ is positive rather +than negative. + +Our proviso that $\overline{q_{cl}} > SD$ ensures that we do not make +$\overline{q_{cl}}$ negative by this adjustment. +If $\overline{q_{cl}} < SD$, then we cannot bring the gridbox to saturation, +but it is still wrong to allow $C_l = 1$ in a subsaturated gridbox! +This was identified as a bug in the bounds-checking code, which sometimes +caused instances of $C_l = 1$ to spuriously persist in dry environments. +This behaviour is currently controlled by a temporary logical in the +{\bf temp\_fixes} namelist: +\begin{itemize} +\item If {\bf l\_pc2\_checks\_sdfix} is set to false, the code simply does +nothing when it finds instances of $C_l = 1$, $SD > 0$ and +$SD > \overline{q_{cl}}$, allowing such artefacts to persist. +\item If {\bf l\_pc2\_checks\_sdfix} is set to true, in these instances +we simply evaporate all the remaining liquid water, and reset $C_l$ to zero: +\begin{eqnarray} +\overline{q} \leftarrow \overline{q} + \overline{q_{cl}} \nonumber \\ +\overline{T} \leftarrow \overline{T} - \frac{L_c}{c_p} \overline{q_{cl}} +\nonumber \\ +\overline{q_{cl}} \leftarrow 0 \nonumber \\ +C_l \leftarrow 0\nonumber \\ +C_t \leftarrow C_i +\label{eq:qsdcheck2} +\end{eqnarray} +\end{itemize} + +\subsubsection{} +The next check is similar to above but for the $C_l = 0$ situation. + +If $\overline{q_{cl}} < q_{c0}$ or $C_l = 0$ then we evaporate the +small amount of $\overline{q_{cl}}$ that remains in the gridbox: + +\begin{eqnarray} +\overline{q} \leftarrow \overline{q} + \overline{q_{cl}} \nonumber \\ +\overline{q_{cl}} \leftarrow 0 \nonumber \\ +\overline{T} \leftarrow \overline{T} - \frac{L_c}{c_p} \overline{q_{cl}} +\label{eq:qclcheck} +\end{eqnarray} + +\subsubsection{} +Next, if $C_i > 1$ then $C_i$ is set to 1. Accordingly, $C_t$ is set to +1 as well. + +\subsubsection{} +The following check is on the ice water content, $\overline{q_{cf}}$, and +the ice fraction $C_i$. If $\overline{q_{cf}} < q_{c0}$ we simply condense some +vapour to remove the negative quantity. + +\subsubsection{} +However, instead of removing small amounts of +$\overline{q_{cf}}$ when $C_i = 0$ but $\overline{q_{cf}} > 0$, we choose instead to create +some $C_i$ to keep consistency. This is to allow small, but significant, +amounts of $\overline{q_{cf}}$ created by the microphysics scheme to +be maintained. + +\begin{equation} +C_i \leftarrow \frac { \overline{q_{cf}} }{q_{cf0}} +\label{eq:cf_reset} +\end{equation} + +where the `in-cloud' ice content $q_{cf0} = 1 \times 10^{-4} kg kg^{-1}$. + +\subsubsection{} +The next two checks are on the total cloud fraction, $C_t$, to ensure +that it takes on a value that is physically possible, given the values +of $C_l$ and $C_i$. We have, firstly, the maximum overlap situation and +then the minimum overlap situation. + +\begin{eqnarray} +C_t \leftarrow \text{Max}( C_t, C_i, C_l ) \nonumber \\ +C_t \leftarrow \text{Min}( C_t , C_l + C_i, 1) +\label{eq:ctchecks} +\end{eqnarray} + +\subsubsection{} +Finally, there is a homogeneous nucleation term applied, similar +to that in the large-scale precipitation (section \ref{sec:lsp_homo}). This is +a fast microphysics process, and must act to ensure that no liquid cloud +created by the initiation is allowed to persist in this phase if the +temperature is cold enough. Hence, if $\overline{T} < T_{homo}$ then + +\begin{eqnarray} +\overline{q_{cf}} \leftarrow \overline{q_{cf}} + \overline{q_{cl}} \nonumber \\ +\overline{q_{cl}} \leftarrow 0 \nonumber \\ +\overline{T} \leftarrow \overline{T} + \frac{L_f}{c_p} \overline{q_{cl}} \nonumber \\ +C_i \leftarrow C_t \nonumber \\ +C_l \leftarrow 0. +\label{eq:homochecks} +\end{eqnarray} + +\subsubsection{Qpos checks} +\label{sec:qpos} + +The implementation of the PC2 code includes an additional bounds check after +the \textit{atmos-physics-2} part of the model timestep has been completed. This +check is necessary to trap a rare failure, and uses the \textit{Qpos} subroutines +to check that $\overline{q_{cl}}$ is greater or equal to 0. + +During trialling prior to operational implementation, it was found that relying on Q-Pos +to deal with negative condensate values was very expensive, as the Q-Pos routine does a lot of communications between +different processors. It may be preferable to deal with the cause of negative condensate amounts at their source. +The option to ``Ensure consistent sinks of qcl and CFL'' +prevents the QCL increment from +trying to remove too much liquid condensate and hence reduces the models reliance on Q-Pos to +deal with the inconsistencies. + +\subsection{Data Assimilation} +\label{sec:da} + +The data assimilation section in the model will output assimilation increments +that represent changes to $\overline{q}$ and $\overline{T}$ which +\textit{include} the condensation contributions. We hence need to calculate +equivalent increments to $\overline{q_{cl}}$, $C_l$ and $C_t$. We assume +that the assimilation has not calculated these using a different method. +We consider the homogeneous framework and assume that there is a forcing +value of $Q_c$ that exists that will produce the known increment to +$\overline{q}$ and $\overline{T}$. + +Discritising (\ref{dqcldt}) we have, using (\ref{eq:deltaqc_exp}) and +expanding $\Delta T_L$ in terms of $\Delta T$ and $\Delta q_{cl}$, + +\begin{equation} +\Delta \overline{q_{cl}} = C_l ( a_L ( \Delta \overline{q} - +\alpha \Delta \overline{T} - \beta \Delta \overline{p}) + \Delta \overline{q_{cl}} ). +\label{eq:da1} +\end{equation} + +Remember that $Q_c$ (and hence $\Delta Q_c$) is independent of condensation. +Rearranging, we obtain + +\begin{equation} +\Delta \overline{q_{cl}} = \frac{1}{1 - C_l} C_l +a_L ( \Delta \overline{q} - \alpha \Delta \overline{T} - \beta \Delta \overline{p}) +\label{eq:da2} +\end{equation} + +and hence an expression for the condensate increment, +$\Delta \overline{q_{cl}}$, that accompanies the known increments +to $\overline{q}$ and $\overline{T}$. The similar analysis, from +(\ref{dcdt}) and (\ref{eq:da1}) gives + +\begin{equation} +\Delta C_l = \frac{1}{1 - C_l} G(-Q_c) + a_L ( \Delta \overline{q} - \alpha \Delta \overline{T} +- \beta \Delta \overline{p} ) . +\label{eq:da3} +\end{equation} + +Hence the equation set is equivalent to the use of the homogeneous +forcing set, except for the multiplier $\frac{1}{1 - C_l}$. Although this +is a clean solution, we +need to be very careful with the ill-conditioning of this solution +near $C_l = 1$. + +In practice, the ill-conditioning of (\ref{eq:da2}) and (\ref{eq:da3}) becomes too +numerically awkward for us to apply the full solution based on homogeneous +forcing, although, for completeness, we outline it in Appendix +\ref{sec:appendix-da}. Hence we have +chosen to apply a much simpler model. Here we use simply the data assimilation +increments $\Delta \overline{q}$ and $\Delta \overline{T}$ within the standard +homogeneous forcing (section \ref{sec:homog}), even though we are fully +aware that this is inconsistent (because $\Delta \overline{q}$ and $\Delta +\overline{T}$ are not forcings, but are forcings plus the condensation. +This allows us an \textit{estimate} of $\Delta \overline{q_{cl}}$ and $\Delta{C_l}$, +via the homogeneous forcing routine (and $\Delta C_t$ via the standard +updating described in section \ref{sec:ct}). These are the quantities applied +as the equivalent data assimilation increments for $\Delta \overline{q_{cl}}$, +$\Delta{C_l}$ and $\Delta C_t$. The increments $\Delta \overline{q}$ and +$\Delta \overline{T}$ remain those that the data assimilation scheme itself +calculated. + +Appendix \ref{sec:appendix-da} gives, for completeness, the alternative +numerical technique for the solution of (\ref{eq:da2}) and (\ref{eq:da3}). +However, we stress that this technique is not used within the current +PC2 formulation. + +\section{Implementation in the Unified Model} +\label{sec:um} + +This section considers the implementation of PC2 within the Unified Model +code and provides a brief guide to its use. + +In general, we have written PC2 so that the timestepping of the +cloud fraction variables within the \textit{atm\_step\_4a} +subroutine is treated as much as possible in a similar way to +the condensate variables. +Hence, wherever the condensed water variables $q_{cl}$ and $q_{cf}$ +are updated, the cloud fractions need to be updated consistently. + +\subsection{Area cloud fraction} +\label{sec:acf} + +Two area cloud fraction parametrizations are available for use with PC2. + +The area cloud fraction of Cusack (documented in \citeumdp{029}) has been +adapted by \cite{boutle_morcrette10} so it can be used with PC2 (and is available from the UMUI as the +``Cusack'' option from version 7.6 onwards). This method aims to +reproduce some of the detail of the thermodynamic +profile lost due to the coarseness of the grid. The interpolation/extrapolation technique is +used prior to PC2 initiation (which is then called with three times as many levels) +and it is used, along with the homogeneous forcing idea at the start of the timestep to +allow more cloud to be seen by radiation. + +The diagnostic area cloud fraction of \cite{bhi05} +has also been implemented in the model (available from the UMUI at version 6.4 onwards), +and this is used in PC2:64. This method diagnoses the area cloud fraction +given the volume cloud fraction, taking into account the size of the grid +box. The setting of the area cloud fraction is performed at the end of the timestep. + +\subsection{Code Structure} +\label{sec:code} + +A detailed description of the UM's timestep structure, +showing where in the model all the PC2 cloud scheme subroutine calls are made, +is given in the subsections below. + +Note that there are three different subroutines that all do +the PC2 homogeneous forcing, with slightly different details: + +\begin{itemize} +\item {\bf{\it pc2\_delta\_hom\_turb}} outputs increments due to the +condensation or evaporation, but doesn't update the fields themselves. +\item {\bf{\it pc2\_homog\_plus\_turb}} just updates the fields that +are passed in, instead of outputting separate increment arrays. +\item {\bf{\it pc2\_hom\_conv}} outputs increments but includes additional +calculations for various cloud erosion formulations. +\end{itemize} + +Note that code exists in the first two of these routines to do erosion, +but they can only do it via an input fixed rate of narrowing of the +moisture PDF (which is currently set to zero in all instances). +PC2 development has settled on a more complicated treatment of erosion, +which has only been implemented in {\it pc2\_hom\_conv}. +This can either be called after the convection scheme +(within {\it pc2\_from\_conv\_ctl}), +or before the microphysics scheme (within {\it pc2\_turbulence\_ctl}). + +Note there is also an optional call to {\it pc2\_turbulence\_ctl} +after the microphysics scheme, which is used only to estimate the +cloud fraction change consistent with the turbulent production of +liquid cloud (see section \ref{sec:turb_qcl_scheme}). + +Most PC2 code is protected by IF tests on the namelist input +{\it i\_cld\_vn} = 2 (PC2 in the GUI). +However, within the convection scheme, the code is controlled by logicals +{\it l\_calc\_dxek} (which is just set to true if using PC2, and set false +otherwise), and {\it l\_q\_interact}, which controls +whether to allow the interactive detrainment and entrainment of condensate. + +There is also a switch (currently hardwired to .false. in the code) called +{\it l\_pc2\_reset}. Turning this on (not recommended!) does 2 things: + +\begin{itemize} +\item Convective entrainment and detrainment of condensate is disabled, +by setting {\it l\_q\_interact} to false. +\item The prognostic cloud variables are overwritten by a call to +the diagnostic cloud scheme at the end of the timestep, +in subroutine {\it qt\_bal\_cld}. +NOTE: this functionality will no longer work, because inside {\it qt\_bal\_cld} +the call to the diagnostic cloud scheme is now protected by IF tests on +using either the Smith or bimodal cloud schemes. If using PC2, no cloud scheme +is called here, and required output variables are just left unset! +\end{itemize} + +The location of the various cloud scheme routine calls within the UM +is summarised in the list below. + +% The latex source input here contains a colour-coded itemize list +% of the UM subroutine tree, showing the locations of all the cloud-scheme +% routines. To edit this, open the source file source/029/um_call_tree.tex +\input{../029/um_call_tree} + +\subsection{Diagnostics} +\label{sec:diags} + +Nearly all diagnostics retain their meaning when PC2 is run. However, there +are a few that are subtly modified. + +The convective diagnostics that use the convective cloud base and top +calculations remain the same if PC2 is used with a zeroed convective +cloud fraction. These values are not reset by the convection scheme, since +the model is still predicting convection between the diagnosed levels. + +The visibility diagnostics need modifying if the convective cloud +fraction is switched off, since they use the convective cloud fraction +within their calculation. Here we use a value of 0.2 for the convective +cloud amount if there is convective precipitation but the two-dimensional +convective cloud amount is zero. This will be the case if the PC2 +scheme has zeroed the convective cloud amount. + +There are a number of increment diagnostics that are required to +fully diagnose the moisture cycle within PC2. Since most physics +sections can cause condensation, condensate and cloud fraction increment +diagnostics have been written for each of these sections. + +\begin{itemize} +\item{ $\overline{T}$, $\overline{q}$, $\overline{q_{cl}}$, $\overline{C_t}$ and $\overline{C_l}$ increments from SW radiation, $\overline{T}$ increment from SW Radiation without including the condensation: \textbf{Section 1} .} +\item{ $\overline{T}$, $\overline{q}$, $\overline{q_{cl}}$, $\overline{C_t}$ and $\overline{C_l}$ increments from LW radiation, $\overline{T}$ increment from LW Radiation without including the condensation: \textbf{Section 2} .} +\item{ $\overline{T}$, $\overline{q}$, $\overline{q_{cl}}$, $\overline{q_{cf}}$, $\overline{C_t}$, $\overline{C_l}$ and $\overline{C_f}$ increments from Boundary Layer: \textbf{Section 3} .} +\item{ $\overline{T}$, $\overline{q}$, $\overline{q_{cl}}$, $\overline{q_{cf}}$, $\overline{C_t}$, $\overline{C_l}$ and $\overline{C_f}$ increments from Large-scale precipitation: \textbf{Section 4} .} +\item{ $\overline{T}$, $\overline{q}$, $\overline{q_{cl}}$, $\overline{q_{cf}}$, $\overline{C_t}$, $\overline{C_l}$ and $\overline{C_f}$ increments from Convection, $\overline{q}$, $\overline{q_{cl}}$, $\overline{q_{cf}}$, $\overline{C_t}$, $\overline{C_l}$ and $\overline{C_f}$ increments from the inhomogeneous part of the Convection scheme only: \textbf{Section 5} .} +\item{ $\overline{T}$, $\overline{q}$, $\overline{q_{cl}}$, $\overline{q_{cf}}$, $\overline{C_t}$, $\overline{C_l}$ and $\overline{C_f}$ increments from the Advection: \textbf{Section 12} .} +\end{itemize} + +However, there +are a number of parts of PC2 that do not fit into a pre-existing section of +code, and hence the associated increment diagnostics are not easily placed +within the UM framework. These increments were available using a +modification set or branch and a user-STASHmaster file up to version 7.5. From version 7.6 these diagnostics are available as standard. + +\begin{itemize} +\item{ $\overline{T}$, $\overline{q}$, $\overline{q_{cl}}$, $\overline{C_t}$, and $\overline{C_l}$ increments from the PC2 erosion section: \textbf{Section 4} or {\bf Section 5} depending on where the erosion is called.} +\item{ $\overline{T}$, $\overline{q}$, $\overline{q_{cl}}$, $\overline{q_{cf}}$, $\overline{C_t}$, $\overline{C_l}$ and $\overline{C_f}$ increments from the Bounds Checking after atmphya: \textbf{Section 4} .} +\item{ $\overline{T}$, $\overline{q}$, $\overline{q_{cl}}$, $\overline{q_{cf}}$, $\overline{C_t}$, $\overline{C_l}$ and $\overline{C_f}$ increments from the Initiation and Bounds checking at the end of the timestep: \textbf{Section 16} .} +\item{ $\overline{T}$, $\overline{q}$, $\overline{q_{cl}}$, $\overline{q_{cf}}$, $\overline{C_t}$, $\overline{C_l}$ and $\overline{C_f}$ increments from the Pressure Forcing section: \textbf{Section 16} .} +\end{itemize} + +\subsection{Single Column Model} + +The updating in the single column model follows the same timestepping as +that in the full model, but the changes to atm-step are mirrored +within scm\_main. The method used is to store the driving SCM forcing +increments of vapour, liquid and temperature across the forcing subroutine. +The forcing of pressure is +set to zero. After atmos\_physics2 has been called, +a PC2 section of code calls the +homogeneous forcing subroutine with these increments. This therefore +treats the response of PC2 to the prescribed dynamical forcing in the +SCM as homogeneous. Following +this calculation, the initiation scheme is called, as usual. Finally, the area +cloud fraction is set to the bulk cloud fraction and $\overline{\Theta}$ +(potential temperature) +is made consistent with $\overline{T}$ (dry-bulb temperature) which was +changed by the condensation in the PC2 response to the homogeneous forcing. +The rest of the SCM uses the same PC2 code as the full model. + +Note that the change to PC2 homogeneous forcing from advection under the UM +namelist switch \textbf{l\_pc2\_sl\_advection} (see section \ref{sec:pres}) +is also mirrored in the Single-Column Model. If this switch is turned on, +the PC2 homogeneous forcing call using the SCM forcing increments is moved +straight after the call to the forcing routine, so that the condensation +adjustment is performed before the call to atmos\_physics2. +If \textbf{l\_pc2\_sl\_advection} is turned on, the PC2 response to SCM +forcings is also improved as follows... + +The SCM forcings may comprise one or both of the following: +\begin{itemize} +\item (a) Prescribed tendencies or relaxation applied to T,q. +\item (b) Interactive vertical advection applied to T,q. +\end{itemize} +For the latter, we can calculate the pressure change experienced by +vertically-advected parcels, and so calculate the PC2 homogeneous forcing +response in the same way as we do for Semi-Lagrangian advection in the +full model (see section \ref{sec:pres}). +For the former, we don't know if the prescribed T,q tendencies are due to +advection, radiation, or some other process, so we calculate the PC2 +homogeneous forcing response as if the tendencies are applied "in-situ". + +To split the PC2 homogeneous response into these 2 components, the SCM +forcing routine outputs: +\begin{itemize} +\item (a) The forcing increments to T,q excluding the contribution from +interactive vertical advection. +\item (b) The value of exner pressure at departure points, consistent with +the vertical advection. +\end{itemize} +The PC2 homogeneous forcing responses to these 2 forcing components +are then calculated by 2 separate PC2 calls in scm\_main. + +\subsection{Limited Area Boundary Conditions} + +Cloud fractions on the limited area boundaries are fully updateable. +Writing of cloud fraction Limited Area Boundary Conditions (LBCs) +will be automatic if PC2 is selected. +A PC2 LAM may be run from an LBC file with or without cloud fraction LBCs +(this is specified by the logical l-pc2-lbc, which is set in the UMUI). +If there are no cloud fraction lbcs then around the edge of the domain the +checking and initiation routines will be applying significant increments +to the cloud and condensate fields near the boundaries, but this does +not have an adverse effect well away from the boundaries. If there are no +cloud fraction LBCs the cloud fraction fields themselves are not forced to +zero around the edge of the domain but are allowed to freely find their +own value. A PC2 run that outputs lbcs will, by default, always +output cloud fractions as part of the LBCs file. + +\subsection{Parameter values} + +Table \ref{tab:pc2_names} summarizes the values of parameters used in the PC2 +scheme and their location within various comdecks. Those parameters marked +as `Num' are those that are not part of the mathematical equation +set that is being solved, but are required in order to achieve a stable, +realistic, numerical solution. These include, for instance, thresholds for +resetting cloud fractions back to 0 or 1. Those marked 'Phy' are physical +quantities that form an integral part of the equation set that we wish to solve. +Those marked 'Clo' form part of a closure needed to form the equation +set, but are less readily related to physical quantities. +Variables marked 'Diag' form a part of the diagnostic output routines. +\begin{table}[ht] +\begin{center} +\footnotesize +\begin{tabular}{llllll} +\hline +Symbol & Code variable & Description & Value & Location & Notes and ref. \\ \hline +- & init-iterations & Number of iterations in initiation & 10 & pc2-const & Num: \ref{sec:numapp_init} \\ +$C_{tol}$ & cloud-pc2-tol & Bounds checking $C_l$ threshold & 0.005 & UM namelist & Num: \ref{sec:init2} \\ +$C_{tol 2}$ & cloud-pc2-tol-2 & Bounds checking $C_l$ threshold & 0.001 & UM namelist & Num: \ref{sec:init2} \\ +$RH_{tol}$ & rhcrit-tol & $RH_{crit}$ tolerance in initiation & 0.01 & pc2-const & Num: \ref{sec:init2} \\ +$q_{cf0 \, BL}$ & ls-bl0 & Fixed value of BL in-plume $\overline{q_{cf}}$ & $1.0 \times 10^{-4} \, kg \, kg^{-1}$ & imp-ctl & Clo: \ref{sec:bl} \\ +$q_{cf0}$ & one-over-qcf & Fixed in-cloud $\overline{q_{cf}}$ if $C_f$=0 & $1.0 \times 10^{-4} \, kg \, kg^{-1}$ & pc2-chck & Num: \ref{sec:checks} \\ +$m$ & pdf-merge-power & Merging power for $G(-Q_c)$ & 0.5 & pc2-const & Clo: \ref{sec:homog} \\ +$n$ & pdf-power & Shape parameter for $G(-Q_c)$ & 0.0 & pc2-const & Phy: \ref{sec:homog} \\ +$w$ & wind-shear-factor & Wind shear in fallout of ice term & $1.5 \times 10^{-4} \, s^{-1}$ & pc2-const & Phy: \ref{sec:lsp_fall} \\ +$i$ & ice-width & Scaling factor for reduction in $b_i$ & 0.04 & pc2-const & Phy: \ref{sec:mp_depsub} \\ +$a$ & dbsdtbs-turb-0 & Rate of reduction of PDF width & $-2.25 \times 10^{-5} \, s^{-1}$ & UM namelist & Phy: \ref{sec:width} \\ +$b$ & dbsdtbs-turb-1 & Rate of reduction of PDF width & 0 & pc2-const & Phy: \ref{sec:width} \\ + & dbsdtbs-conv & Redn of PDF width in convection & 0 & pc2-const & Phy: \ref{sec:width} \\ + & dbsdtbs-exp & Variation of erosion on RH & 10.05 & pc2-const & Phy: \ref{sec:width} \\ +$RH_{crit}$ & RHCRIT & Critical RH for cloud formation & & UM namelist & Phy: \ref{sec:init}, \ref{sec:mp_depsub} \\ +$q_{c0}$ & condensate-limit& Minimum allowed condensate & $1 \times 10^{-10} \, kg \, kg^{-1}$ & pc2-chck & Num: \ref{sec:checks} \\ +$q_c^{S0}$ & ls0 & Lower limit of plume condensate & $5 \times 10^{-5} \, kg \, kg^{-1} $ & enviro?a & Num: \ref{sec:multi_numapp} \\ + & \textit{Hard-wired} & Conv cloud fraction for visibility& 0.2 & imp-ctl2 & Diag: \ref{sec:diags} \\ + & \textit{Hard-wired} & Limit on width of ice distribution& 0.001 & lspice3d & Num: \ref{sec:mp_depsub} \\ + & \textit{Hard-wired} & $C_l$ limit for init if $T < 0 ^{\circ} C$ & 0.05 & pc2-init & Num: \ref{sec:init2} \\ + & \textit{Hard-wired} & Tolerance on calc. of $q_C^s$ in BL & $1.0 \times 10^{-10} \, kg \, kg^{-1}$ & imp-ctl & Num: \ref{sec:bl} \\ +\hline +\end{tabular} +\end{center} +\caption{PC2 parameter values and locations } +\label{tab:pc2_names} +\end{table} + +PC2 also recommends some tunings of the existing convection +scheme parameters. These cannot be placed in the library code, since they +would interact with non-PC2 simulations, hence would need to be specified +with modification sets. We have included those parameters that have been +investigated throughout testing, although only two are different between +PC2:64 and a non-PC2 run. + +\begin{table}[ht] +\begin{center} +\tiny +\begin{tabular}{llllll} +\hline +Code variable & Description & Value in PC2 & Value in Control & Location & Notes and reference \\ \hline +TICE & Temperature at which plume freezes & $-10 ^{\circ} C$ & $0 ^{\circ} C$* & tice.cdk or UMUI & Phy: \ref{sec:convec} \\ +QSTICE & Approximate qsat(TICE) & $3.5 \times 10^{-3}$ & $3.5 \times 10^{-3}$ & qstice.cdk or UMUI & Phy: \ref{sec:convec} \\ +\textit{Hard-wired} & Limit on conv. cond. after precip & 0.5 $q_{sat}, 2 \times 10^{-4}$ & $0.5 \, q_{sat}$ & cloudw & Phy: \ref{sec:convec} \\ +Anvil factor & Shape parameter for conv. cloud anvil & 0 & 0.3* & UMUI & Phy: \ref{sec:convec} \\ +Tower factor & Shape parameter for conv. cloud tower & 0 & 0.25* & UMUI & Phy: \ref{sec:convec} \\ +\hline +\end{tabular} +\end{center} +\caption{PC2 parameter values and locations relating to the convection. *These values are those used in HadGAM} +\label{tab:pc2_conv_names} +\end{table} + +\subsection{How to run the PC2 scheme} +Running PC2 is straightforward, but you should seek advice as to +modification sets that you need to include to ensure you are +running the most up-to-date version of PC2. +The following is a brief checklist of the options in the UMUI which need +to be selected in order to run PC2. No hand-edits are required. +\begin{itemize} +\item{In the LS cloud panel (atmos-science-section-LScloud) push the button marked 'use the PC2 cloud scheme'.} +\item{If you wish to use PC2 in the diagnostic only mode, also push 'run the PC2 scheme in diagnostic only mode'. If you wish to run PC2 fully then do not push this button} +\item{In the large-scale precipitation section (atmos-science-section-LSprecip) select the 3D large-scale precipitation scheme.} +\item{The specification of the LA boundary conditions can be set in the atmos-InFiles-OtherAncil-LBC panel.} +\item{You will need to select modsets to include update the library code to the PC2 version described here. Seek advice on these.} +\item{You may wish to adjust the convective anvil parameters in atmos-science-section-convec. Again, seek advice.} +\end{itemize} + +\subsection{More information} + +Information on results of the scheme and how to run the PC2 code at +various model versions is available on the PC2 web site. + +\section{Appendix: Alternative PC2 - Data Assimilation formulations} +\label{sec:appendix-da} + +In this alternative method to section \ref{sec:da} we will assume that there +exists a homogeneous forcing, $\Delta Q_c$, +that gives changes, net of condensation, of $\Delta\overline{q}$ and +$\Delta\overline{T}$. If we can recover +what $\Delta Q_c$ is then we can use this to calculate the liquid, +$\overline{q_{cl}}$, and liquid cloud fraction, $C_l$, increments. + +As in section \ref{sec:da}, we start by discretising (\ref{dqcldt}) to give + +\begin{equation} +\Delta \overline{q_{cl}} = C_l \Delta Q_c +\label{eq:dqcldt_discrete} +\end{equation} + +and hence, using the discrete form of $\Delta Q_c$ from +(\ref{eq:deltaqc_exp2}) gives + +\begin{equation} +\Delta \overline{q_{cl}} = C_l ( a_L ( \Delta \overline{q} - \alpha \Delta +\overline{T} ) + \Delta \overline{q_{cl}} ) , +\end{equation} + +which rearranges to + +\begin{equation} +\Delta \overline{q_{cl}} = \frac{1}{1-C_l} C_l a_L ( \Delta \overline{q} +- \alpha \Delta \overline{T} - \beta \Delta \overline{p}) . +\label{eqn:delataqcl} +\end{equation} + +Comparing to (\ref{eq:deltaqc_exp2}) and (\ref{eq:dqcldt_discrete}) we see that +$\Delta \overline{q_{cl}} $ is the same as if we had applied the +homogeneous forcing technique +using $\Delta \overline{q}$, $\Delta \overline{T}$ and $\Delta \overline{p}$ as +forcings, except multiplied by a factor of $\frac{1}{1-C_l}$. + +We can calculate $\Delta C$ in a similar way. From (\ref{eq:deltac}) + +\begin{equation} +\Delta C_l = G(-Q_c) \Delta Q_c +\end{equation} + +and hence, using our value of $\Delta Q_c$ from (\ref{eq:deltaqc_exp2}) +and $\Delta \overline{q_{cl}}$ from (\ref{eqn:delataqcl}) + +\begin{equation} +\Delta C_l = G(-Q_c) (a_L ( \Delta \overline{q} - \alpha \Delta \overline{T} ) ++ \frac{1}{1-C_l} C_l a_L ( \Delta \overline{q} - \alpha \Delta \overline{T} ) ) +\end{equation} + +which rearranges to + +\begin{equation} +\Delta C_l = \frac{1}{1-C_l} G(-Q_c) a_L ( \Delta \overline{q} +- \alpha \Delta \overline{T} -\beta \Delta \overline{p}) . +\label{eqn:c1mc} +\end{equation} + +This is also a factor of $\frac{1}{1-C_l}$ different from using +$\Delta \overline{q}$, +$\Delta \overline{T}$ and $\Delta \overline{p}$ +directly as forcings (the factor must be the same, as we are still +using the homogeneous forcing hypothesis). This equation forms the basis +for the more advanced technique discussed in this section. However, +it is undefined at $C_l=1$ and becomes ill-conditioned near $C_l=1$, +hence there must be care taken when this expression is solved numerically. + +\subsection{Numerical solution} + +The timestepping applied is picked as a result of numerical tests +forcing a single gridbox with uniform increments. Many numerical techniques +were tested, this gives a fast but reasonably well behaved solution. + +Initially, we calculate $G(-Qc)$ and $\Delta Q_c$ from the input fields, +as in the homogeneous forcing +technique (section \ref{sec:homog}) and (\ref{eq:deltaqc_exp2}). + +An initial increment, $\Delta C_l^1$ is estimated directly using the +basic equation + +\begin{equation} +\Delta C_l^1 = \frac{1}{1-C_l^(n)} G(-Q_c) \Delta Q_c . +\end{equation} + +We then recalculate this expression, using a mid-timestep estimate +for $\Delta C_l$; + +\begin{equation} +\Delta C_l = \frac{1}{1-(C_l^{[n]} + \frac{1}{2} \Delta C_l^1)} +G(-Q_c) \Delta Q_c +\end{equation} + +where the term $C_l^{[n]} + \frac{1}{2} \Delta C_l^1$ is limited to be no more +than 0.9999 to avoid divide by zero problems. The final, updated value of +cloud fraction, $C_l^{[n+1]}$, is then + +\begin{equation} +C_l^{[n+1]} = C_l^{[n]} + \Delta C_l +\end{equation} + +and this value is limited to 0 or 1. + +The liquid water term simply uses the final version of $C_l$ in its +calculation. + +\begin{equation} +\Delta \overline{q_{cl}} = \frac{1}{1-C_l^{[n+1]}} C_l^{[n+1]} \Delta Q_c +\end{equation} + +and will be set to 0 if $C^{[n+1]}$ is 0. There is an additional limit, +see below, applied to the liquid +water term, which will prevent the value of $\Delta \overline{q_{cl}}$ +increasing to a large number if $C_l^{[n+1]}$ is very close to 1. + +\subsection{Limit on the liquid water content} + +We will choose a limit on $\overline{q_{cl}}$ to be equal to its value when +the underlying PDF just corresponds to total cloud cover. Therefore, from +(\ref{eq:qclbar=int}) + +\begin{equation} +\overline{q_{cl \, max}} = \int_{s=-b_s}^{\infty} G(s) (b_s + s) ds . +\end{equation} + +We will use the current value of $Q_c$ (which won't in general to be equal to +$b_s$) to split the integral into two ranges of s: + +\begin{equation} +\overline{q_{cl \, max}} = \int_{s=-b_s}^{-Q_c} G(s) (b_s + s) ds ++ \int_{s=-Q_c}^{\infty} G(s) (b_s + s) ds . +\end{equation} + +For the moment we write the first of these integrals as $I1$, and split the +second integral whilst introducing a $(+ Q_c - Q_c)$ term to the integrand: + +\begin{equation} +\overline{q_{cl \, max}} = I1 + \int_{s=-Q_c}^{\infty} G(s) (b_s - Q_c) ds ++ \int_{s=-Q_c}^{\infty} G(s) (s + Q_c) ds . +\end{equation} + +The last of the integrals is now the current liquid water content, +$\overline{q_{cl}}$. +The second integral is proportional to the liquid cloud fraction $C_l$. + +\begin{equation} +\overline{q_{cl \, max}} = I1 + C_l (b_s - Q_c) + \overline{q_{cl}} +\end{equation} + +or + +\begin{equation} +\overline{\Delta q_{cl \, max}} = I1 + C_l (b_s - Q_c) . +\label{eqn:deltaqclmax} +\end{equation} + +Now consider the expression for the saturation deficit, which we have +defined, from (\ref{SD}) as + +\begin{equation} +SD = \int_{-b_s}^{-Q_c} G(s) (-Q_c - s) ds . +\end{equation} + +Splitting and adding the term $(+b_s - b_s)$ to the integrand in a +similar way to above gives + +\begin{eqnarray} +SD = \int_{-b_s}^{-Q_c} G(s) (-Q_c + b_s) ds + \int_{-b_s}^{-Q_c} +G(s) (-s - b_s) ds \nonumber \\ += (-Q_c + b_s) (1 - C_l) - I1 , +\end{eqnarray} + +and hence $I1$ in terms of $SD$. Using this value of $I1$ in +(\ref{eqn:deltaqclmax}) and cancelling the $C_l$ terms gives +$\Delta \overline{q_{cl \, max}}$ as + +\begin{equation} +\Delta \overline{q_{cl \, max}} = (-Q_c + b_s) - SD . +\label{eqn:delta2} +\end{equation} + +This is a general expression, it is not fixed for a particular PDF. To +complete the analysis, we need to estimate $-Q_c+b_s$. To do this, we now +make the \textit{assumption} of a power-law type PDF, as in section +\ref{sec:init}. If we start from the equivalent of +(\ref{eqn19}) but at the $s=-bs$ end of the distribution, equation (B.3) +in \cite{wg03} can be equivalently written for $(1-C_l)$ as: + +\begin{equation} +(1-C_l) = \frac{ A (-Q_c + b_s)^{n+1} }{n+1} . +\label{eqn:1mc} +\end{equation} + +To derive this from (B.3) note that $C_l$ is swapped for $1-C_l$ and +$(b_s - (-Q_c))$ is swapped for $(-Qc - (-b_s))$, as in section +\ref{sec:numapp_init}. Similarly, noting that $\overline{q_{cl}}$ can be +swapped with $SD$, gives the equivalent to (B.4) in \cite{wg03} +as + +\begin{equation} +SD = \frac{ A (-Q_c + b_s)^{n+2} }{(n+1)(n+2)}. +\label{eqn:sd} +\end{equation} + +Using the value $(1-C_l)$ from (\ref{eqn:1mc}) in (\ref{eqn:sd}) gives + +\begin{equation} +\frac{SD}{1-C_l} = \frac {-Q_c + b_s}{n+2} . +\end{equation} + +Finally, we use this expression for $(-Q_c + b_s)$ in (\ref{eqn:delta2}) to +parametrize $\Delta \overline{q_{cl \, max}}$ in terms +of the saturation deficit + +\begin{equation} +\Delta \overline{q_{cl \, max}} = SD ( \frac{n+2}{1-C_l} - 1 ) . +\label{eqn:sdr1mc} +\end{equation} + +This is the expression that is used for the limit on $\overline{q_{cl}}$. +We subsequently apply a second limit, since numerically this expression is +still not well behaved when $C_l$ is close to 1. Here we note that just at +complete cloud cover for a symmetric PDF we have +$\overline{q_{cl}} = b_s$. Hence we estimate $b_s$ as in \cite{smith90}, + +\begin{equation} +b_s = a_L ( 1 - RH_{crit} ) q_{sat}(\overline{T_L}) , +\label{eqn:bs} +\end{equation} + +and take the smaller value for +of (\ref{eqn:sdr1mc}) and (\ref{eqn:bs}) for $\Delta \overline{q_{cl \, max}}$. + + +\subsubsection{Initiation from $C_l=1$} +The equations are not defined when $C_l=1$. (Note that when $C_l=0$ we +will calculate $G(-Q_c)=0$ so there is no change in cloud fraction or liquid water +content in this case). The assimilation is capable of lowering $\overline{q}$ +below $q_{sat}(\overline{T})$ and hence there should be a corresponding +change in cloud fraction and liquid water content. In theory, we can use +the expression for $\Delta \overline{q_{cl \, max}}$ and assume that the +initial liquid water is equal to $b_s$. However, this produces +a tricky set of simulataneous equations, which are not easily solvable +except in the case where $n=0$. We proceed by making this assumption for $n$, +acknowledging that this is not necessarily entirely consistent with the +rest of the model (although it is in the PC2:64 formulation). + +We have (equivalent to B.6 from \cite{wg03}) + +\begin{equation} +\frac{ (1-C_l)^2 }{SD} = G(-Q_c) \frac{n+2}{n+1}. +\end{equation} + +If n=0 (i.e. a `top-hat' function) then $G(-Q_c) = \frac{1}{2 b_s}$ +and we can write + +\begin{equation} +C_l = 1 - \sqrt{ \frac{SD}{b_s} } . +\end{equation} + +We now assume $b_s$ is equal to our current value of $\overline{q_{cl}}$ +and hence + +\begin{equation} +C_l^{[n+1]} = 1 - \sqrt{ \frac{SD^{[n+1]}}{\overline{q_{cl}^{[n]}}} } +\label{eqn:1msqrt} +\end{equation} + +where $C_l^{[n+1]}$ and $SD^{[n+1]}$ are the values of $C_l$ and $SD$ after +this initiation has been applied. +Using our previous expression (\ref{eqn:sdr1mc}) for +$\Delta \overline{q_{cl max}}$ gives (remembering that we are considering +the reverse process, so the sign is opposite), + +\begin{equation} +\Delta \overline{q_{cl}} = - SD^{[n+1]} ( \frac{2}{1-C_l^{[n+1]}} - 1 ) +\end{equation} + +(remembering that $n=0$ is assumed). Hence, replacing $C_l^{[n+1]}$ by +(\ref{eqn:1msqrt}) we have + +\begin{equation} +\Delta \overline{q_{cl}} = SD^{[n+1]} - 2 \sqrt{ SD^{[n+1]} +\overline{q_{cl}}^{[n]} } . +\end{equation} + +This is the expression we use, $SD^{[n+1]}$ is calculated after the +ssimilation increments have been applied, using (\ref{SD2}): + +\begin{equation} +SD^{(n+1)} = a_L^{[n+1]} ( q_{sat}(\overline{T}^{[n+1]}, +\overline{p}^{[n+1]}) - \overline{q}^{[n+1]} ). +\end{equation} + +\subsection{Results} + +Results demonstrate a problem in that there is a distinct asymmetry +between changes when $\Delta Q_c$ is large and positive and when +$\Delta Q_c$ is large and negative, when +cloud fractions start near 1. In the former case, the limit to the amount of +liquid and cloud fraction that can be created means that changes must be +kept relatively small, whereas in the latter case, all the cloud and +liquid water can be removed easily. (The $1/(1-C_l)$ term allows this +to be done relatively quickly). Hence this assimilation +method has a net tendency to remove cloud from the simulation, which, +at the moment, gives poorer results than simply using the homogeneous +forcing method. + +Further work will be required to enable the implementation of +this $\overline{q}$ +and $\overline{T}$ preserving method. + +\section{Appendix: Essentials of PC2 for code developers} +\label{sec:code-development} +This section provides some guidance to code developers on the treatment +of PC2. Code developers are advised to read the relevant part of section +\ref{sec:app_um} to understand the way in which the current PC2 scheme +interacts with their section of code. + +The essence of a prognostic cloud scheme is that each physical part of the +model is able to calculate increments to the cloud fractions and condensate +contents. These form an integral part of each physics scheme and should be +considered by code owners as such, hence any alteration to a scheme +\textit{must} consider also the impact on $q_{cl}$, $q_{cf}$, $C_t$, +$C_l$ and $C_f$, as +well as on the more traditional $T$, $q$ and wind prognostics. Often +there should be no impact, but this cannot be assumed without consideration. +There is no diagnostic cloud fraction and condensation scheme which can be +run in PC2, since this would reset any effect of the cloud prognostics used +elsewhere in the model. (The diagnostic scheme can still be used for model +\textit{diagnostics}, such as visibility and fog fraction, and will be +kept in later versions of the UM). + +Since this places a significant burden on code developers, the PC2 +developers have produced two generic representations which can take increments +to $q$ and $T$ etc. and produce an estimate of the condensation and +cloud fraction changes associated with the increments. These are the +homogeneous forcing and injection forcing (or inhomogeneous forcing) +methods. + +\subsection{Homogeneous forcing} +This is described fully in section \ref{sec:homog}. This assumes that the +distribution of $q_T - q_{sat}(T_L)$ about its gridbox mean is unchanged when +a process acts. (The mean will change of course, but we assume that the +variations in each part of the gridbox from the mean do not). Since this +is equivalent to every part of the gridbox receiving the same $q_T$ and $T_L$ +increment, we call this `Homogeneous Forcing'. We have provided a subroutine +\textit{pc2-homog-plus-turb}, in deck \textit{pc2-homo} in order to +provide the necessary updates. + +\subsection{Injection forcing} +This is described fully in section \ref{sec:inhomog}. We assume that +we already know a condensate increment $q_{cl}$ or $q_{cf}$ and that a +corresponding cloud fraction increment $C_l$ or $C_f$ (and $C_t$) remains +to be estimated. The injection forcing assumes that new cloud randomly +displaces existing cloud in a gridbox, and is designed with detrainment +from deep convection in mind, although it is also used elsewhere. It will +require as an input an estimate of the `in-cloud' water content of +the new cloud that is produced. + +If you consider that both the homogeneous and injection forcing representations +are both poor assumptions for your scheme, you will need to provide +another method for calculating the condensation and cloud fraction changes. +The PC2 team can advise, but you should not expect them to do the work. +You can, of course, replace existing homogeneous and inhomogeneous forcing +calls with new representations of changes to the prognostics if you think +you have improved representations available. This is part of the +development of any prognostic variable representation. + +\subsection{Do I need to modify anything when I change a parametrization scheme?} + +Here we assume that you wish to do the minimum work possible to get +PC2 to work, rather than a full reconsideration of the physics of the PC2 +increment terms. + +If your scheme is currently using the homogeneous forcing +then there is no need to update the cloud part of the scheme, +\textit{provided that +you do not alter values of $T$ and $q$ after the homogeneous forcing +section is called} and that the physical interpretation of your $q$ and $T$ +increments does not change. You need to be careful if you are moving code from +one subroutine to another that you don't inadvertently do this, although +the forcing usually sits at the end of the control subroutine. + +If your scheme is currently using the injection forcing \textit{subroutine}, +which necessitates that condensate +increments are already calculated by the scheme, then there is also no need +to update the cloud part of the scheme. This currently applies to the boundary +layer, where $q_{cf}$ is altered by tracer mixing. Like for the +homogeneous schemes, this +is provided that you \textit{do not alter $T$, $q$ or condensate values after +the injection forcing subroutine is called} and that the physical +interpretation of your $q$ and $T$ increments does not change. + +Changes to winds do \textit{not} need to have a condensation or +cloud fraction increment +associated with them. There may be future scope for developing an +orographic cloud representation (probably diagnostic), but this is not +an essential part of the scheme as it stands. + +If your scheme uses hardwired assumptions about what is happening e.g. +convection or microphysics, then you \textit{do} need to be careful that +$T$, $q$ and condensates +are still calculated correctly after you have performed your changes. +Currently there are many PC2 assumptions hard-wired into the mass-flux +convection scheme: +\begin{itemize} +\item{Any change to the scientific basis by which changes to $T$, $q$, $q_{cl}$ and $q_{cf}$ are calculated requires careful consideration} +\item{Simple changes to convective parameters, such as detrainment rates, should not require a change to the PC2 code} +\item{Be particularly careful if you move code around, \textit{especially the calculation of convective cloud fractions}, since PC2 incorporates a set-to-zero in the code. This will need to be replicated or there is a risk that the diagnostic cloud fraction is no longer set to zero correctly by PC2.} +\end{itemize} +Each microphysics transfer term has been considered individually for PC2 and this +should remain the case. + +Be especially careful when you do anything in the atmphy and atmstep levels of +the code that includes additional changes $T$, $q$, $q_{cl}$ or $q_{cf}$, since +they may need cloud fraction or condensation changes to go along with them. + +In summary, changes to existing increments of $T$, $q$ etc. within the current +UM structure are unlikely to +necessitate a modification for PC2 if their physical interpretation has not +changed. However, new methods of generating $T$ and +$q$ increments will require new code to be added for PC2. + +\subsection{Further PC2 development work} +There are a number of areas in which the PC2:66 formulation can be +developed further, and many of these have been mentioned in the documentation +above. Some +of these are simple sensitivity studies which have not been fully explored in +development, others are more complex alterations. It is fair to say +that the link to the convection has proved the most problematic issue +so far with PC2 development. + +\subsubsection{PC2 cloud erosion} +The cloud erosion is a critical term for the simulation of shallow convective +cloud. A large amount of erosion is required to keep the cloud fractions relatively +low in shallow convection, which is why we have linked the erosion to the relative +humidity. We recognise, however, that this is more an empirical choice than a +physically informed choice. In particular, a low relative humidity (e.g. in the +stratosphere) would imply a very high erosion rate - although the net effect +is to remove any cloud, which is a reasonable thing to do, there is an implication of +the parametrization that mixing within the stratosphere is high, which is +clearly incorrect. We have also seen relatively low cloud fractions in the +mid-levels of deep convection in PC2, and presume that this is influenced +by the erosion formulation. A link to mass flux has also been proposed, but tests +with CRMs do not support a clear link. Perhaps it is more natural to compare the +erosion with the turbulent kinetic energy. This should be available within the +boundary layer and convection schemes, but not outside of these in the current +UM. + +The erosion formulation in PC2:66 is one where the width of the PDF is +always narrowed (developed following \cite{sg03}). +It may be advantageous to think whether there are unmodelled +processes in the atmosphere that result in an increase in width. Clearly +convection is likely to be one, but this is already represented in PC2. +There may be other models entirely for the way in which the PDF changes as a result +of mixing of air within a gridbox or within the column, these may prove +fruitful to explore. + +Another issue is whether width-narrowing (or widening) is really an effective +way of representing the erosion process. CRM evidence suggests that the required +erosion rates to balance convective cloud generation are larger for +liquid cloud fraction than liquid water (by up to a factor of 2), suggesting +that the real atmospheric erosion favours removal of cloud fraction +over liquid water more strongly than the model. + +The in-cloud condensate that is detrained from convective plumes is high. +We might think that the mixing in of environmental air in reality is +likely to lead to more cloud around the plumes and lower condensate within +the plumes. However, the width narrowing scheme is not a good model of mixing +in this situation, always reducing the amount of cloud because it is incorrectly +assumed that much of the detrained plume has condensate contents only just above zero +and that the shape of the moisture PDF remains unchanged. This may have a +bearing on the problem of the lack of mid-level cloud in the model (although I +think there are many reasons for this). A different +mixing method may give significantly different results for the areas around +convective plumes. + +\subsubsection{Narrowing of the moisture PDF} +Most of the parametrized terms in PC2 act to reduce the width of the +moisture PDF. The only terms that can increase the width are the convection, +and the initiation (which can reset the width). This may not be the +best way to describe the way in which the PDF evolves, in particular it +is sensible to ask whether the erosion term should actually increase +the width in the presence of large vertical gradients of moisture. + +\subsubsection{Convective cloud increments in the mass-flux framework} +As discussed in section \ref{sec:conv_imp_note}, it would be useful +to code up the convective cloud fraction changes to link directly to +the mass-flux convection scheme, and not to estimate them from the values +of $Q4$, which can introduce errors. + +\subsubsection{Turbulence based convection scheme} +\label{sec:tbcs} +We will need to properly consider the links between PC2 and the +turbulence based convection scheme. In essence, we can use the diagnosed +cloud fraction and condensate values from the convection scheme to +start off the cloud again when convection has ceased. This has been +tested to some degree but will need proper analysis. The difficult +decision comes in choosing what to do with the condensate and cloud fraction +that is present \textit{before} the convection starts, since we must +ensure conservation of moisture. This is not helped by the traditional +view of convective parametrization that ignores the existence of the condensate +phase in the atmosphere (i.e. it is only concerned with transport of $q$ and +$\theta$, not of $q_{cl}$ and $q_{cf}$) despite the phase changes forming +an integral part of the convection scheme. + +\subsubsection{Detailed convective comparisons with CRM/LEM data} +This work is already underway at the Met Office, in order to properly +evaluate the performance of the convective cloud parametrization +in PC2 against high resolution research models. + +\subsubsection{Choice of PDF parameters} +Work by Dan Tang at Leeds University has highlighted an interesting +and undesirable property of the choice of $m$ and $n$ parameters in the +homogeneous forcing formulation. If a distribution is homogeneously +forced to $C_l = 0$, then we do not necessarily get $\overline{q_{cl}}$ +tending to zero. This is because there is enough influence from the +$\frac{{(1-C_l)}^2}{SD}$ term in the combination (\ref{eqn22}) to +stop the natural convergence of the $\frac{{C_l}^2}{\overline{q_{cl}}}$ term +to $C_l =0$ and $\overline{q_{cl}}=0$. Increasing the power of $m$ should +help. However, we note that the tests that have been done on the chosen +$n$ and $m$ values (0 and 0.5 respectively) do not show particularly +poor behaviour, and we do not pick up substantial evidence of problems +from this in the full model. This remains something to be investigated. + +\subsubsection{Homogeneous forcing section improvements} +\label{sec:homog_improve} +Although the homogeneous forcing provides a convenient method to +calculate increments to $C_l$ and $\overline{q_{cl}}$, it is clearly +not the best representation possible of the processes that use it. +For example, although the clear-sky radiative heating may perhaps +best be considered as a homogeneous process, the part of the +radiative heating influenced by clouds should, ideally, be applied +to the cloudy part of the gridbox and not the clear part. Vertical +advection is likely to be correlated with where there is already +cloud, rather than being uniform throughout the gridbox. There is +no reason that a process that uses homogeneous forcing as its +condensation model should not be looked at with a view to using +something better. This is one of the strengths of the PC2 framework and +is an intention of the project. + +\subsubsection{Overlap of ice and liquid cloud changes} +We have assumed within PC2 that ice and liquid cloud changes are +minimally overlapped with each other (within the same gridbox) in +order to maintain as much supercooled liquid water as possible. Although +there is good observational evidence to say that the two condensate +phases tend not to coexist together in a cloud, it may be possible to +characterise and apply this overlap in a more quantiative way. + +\subsubsection{Parameter tuning} +The sensitivity of some of the parameters in PC2 have not been properly +tested, mainly due to a lack of resources rather than a physical reason. +We have seen that the most effective method of tuning cloud is with the +erosion term, which has been increased to high values in order to remove +enough cloud and is probably as high as we reasonably wish to take it given +the length of the timestep. +\begin{itemize} +\item{The phase change temperature (between liquid and ice) in the +convective plume, TICE, is known to influence the strength of the convection +through the latent heat differences. It also impacts on the amount of +supercooled liquid water in the model. The quantitative impact of altering +this could be explored. We note that CRM simulations of deep convection +suggest that some supercooled liquid water exists within the plumes to +$-40 ^{\circ} C$ and that a representation with partial liquid and partial +ice phase would be more appropriate, based possibly on the current diagnosed +convective cloud phase in the non-PC2 model. Although the theoretical work +has been done to allow partial phases, we repeat the caution that care +must be taken when doing the work and appropriate testing done to ensure +that heat and moisture are properly conserved within the convection scheme.} + +\item{The growth of $C_f$ due to the fall-out of ice term in the microphysics +is parametrized with a dependence on windshear. We have never linked this +directly to the windshear, instead we have used estimated the windshear +as a fixed value. There is no reason why the actual model windshear cannot +be passed into the scheme in order to properly calculate this term.} +\item{$RH_{crit}$ remains a tunable parameter. Although its impact is less +than in a non-PC2 simulation, it is still significant in initiating cloud +and in determining the evolution of the ice cloud. There is also an implicit +overlap assumption regarding the ice cloud fractions, again this might be +improved upon.} +\item{$n$ and $m$ values in the homogeneous forcing have not been +thoroughly investigated for a long time now, and may yield some sensitivities}. +\end{itemize} + +\subsubsection{Cloud inhomogeneities} +A cloud generator approach to cloud inhomogeneities is currently +being developed. However we note two particular issues that relate +to PC2. +\begin{itemize} +\item{The first is that in the diagnostic scheme, the two cloud +fractions (convective and large-scale) allows, to some degree, a +representation of cloud inhomogeneity. This is absent from PC2, +although we note that the convective cloud fraction variable has +not been removed from the radiative transfer code for PC2, it is merely set +to zero, so it is easy to put back.} +\item{The generation of inhomogeneities using a cloud generator +requires some estimate of the variance (and possibly skewness) +of the condensate in the +gridbox. It is possible to back out the full moisture PDF at +each grid point by homogeneous forcing (providing $C_l$ is not equal +to 0 or 1), but this is very expensive and cannot be done on-line. Is +there a quick \textit{estimate} of the variance or skewness that it is +possible to obtain from knowledge only of $\overline{q}$, +$q_{sat}$, $\overline{q_{cl}}$ and $C_l$ etc.?} +\end{itemize} + +\subsubsection{Time-stepping} +\label{sec:timestepping} +A proper analysis of timestep sensitivities of PC2 (as opposed to +microphysics, convection etc) in the full UM +or SCM has not been done for a long time. +In the early development stages much effort was +placed in developing good numerical techniques for each of the terms +in PC2, and to explore the way in which they coupled together. An example +is the homogeneous forcing timestep investigated by \cite{wg03}. +We note that in shallow convection at 30 minutes timestep the erosion +term is trying to remove +most of the cloud that the convective detrainment places into the model. +Since the erosion is limited by the amount of cloud fraction and +condensate present, what ends +up happening is that the `equilibrium' that is achieved is actually one where +the cloud fraction and condensate at the end of the timestep are simply +the values that were detrained by the convection scheme (and hence depend +on the timestep). The CRM suggests a cycling time of around 15 minutes for +liquid water content and just less than half and hour for the cloud fraction, +so we would expect timestep dependency to occur from around a timestep of +15 minutes upwards. We might just about get away with the 30 minute step +of the climate model, but it is not a good situation to try to model. +This is demonstrating the difficulty of modelling shallow convective cloud +by a prognostic scheme, where the physical lifetime of the clouds is +of order the timestep - ideally we wouldn't want to try to model anything +prognostically when the cycling time is less than the timestep. + +As discussed in section \ref{sec:erosion_numerics}, the timestep sensitivity +of cloud amounts in shallow cumulus regimes can be addressed by using +a more accurate numerical method to solve the erosion term. +Several options are available under the UM namelist switch +\textbf{i\_pc2\_erosion\_numerics}. + +In the early development of PC2 we chose to incorporate the PC2 cloud +and condensation increments in the same location where the increments +were calculated (e.g. the microphysics cloud fraction increments +get added along with the microphysics $\overline{T}$ and $\overline{q}$ +increments). This choice was made in order not to confuse the timestepping +method in the UM, which has been carefully developed over a number of +years to achieve numerical accuracy. However, we note that the rapidly +varying nature (in space and time) of variables such as $\overline{q_{cl}}$ +and $C_l$ is very different from the smooth fields of $\overline{q_T}$ and +$\overline{T}$, for which the timestepping was developed, and it may +not be appropriate to implement these in the same locations. In +particular, we might wish to store the increments through the timestep +and update values of $\overline{q_{cl}}$ and $C_l$ etc. at the end +of the timestep, where many of the balances can be cancelled. + +One issue is that we are calculating the increments due to condensation +associated with the adiabatic response to pressure changes after the +Helmholtz solver. Pragmatically, we need to do it here since we do not know +the arrival value of pressure until after the Helmholtz solver has been +used. However, in order to achieve balanced dynamical fields, it is useful +the Helmholtz solver to be called after all the latent heating terms have +been calculated (which not only includes the adiabatic response to +lifting but the cloud initiation term). We have shown that PC2 can run with +the two terms switched over, but this implies that we are missing part of +the pressure change following the parcel (the time changing part +rather than the spatially changing adiabatic part). Although the adiabatic +change is usually likely to dominate, it may be a significant loss. +Under the UM namelist switch \textbf{l\_pc2\_sl\_advection}, +we can call the PC2 response twice, once before the Helmholtz solver and +once afterwards in order to pick up most of the latent heat change before +the solver, but not to have PC2 miss some of the pressure change. +The call for the advective part (before the Helmholtz solver) is +actually done before the call to atmos\_physics2 as well, and so results in +more realistic, saturation-adjusted, profiles being passed to the convection +scheme. + +We have placed the initiation at the end of the timestep, but it is sensible to +ask whether this could ideally be located elsewhere. + +\subsubsection{Initiation formulation} +Ideally this should be a relatively infrequent part of the model +but remains an essential part of the code. It is reasonable to ask whether +the initiation is optimal, particular in the diagnosis of when it is +applied. For example, we note that the initiation is currently +symmetrical, with initiation from $C_l=1$ occuring with the same +$RH_{crit}$ value as from $C_l=0$. However, the \cite{wf00} +observations hint that a higher $RH_{crit}$ might be more appropriate +for initiation from $C_l=1$. + +\subsubsection{70-levels performance} +The performance of PC2:66 in the 70-levels model is not good as +far as shallow convective cloud is concerned (there is far too +much of it in the trade regions). It may be that PC2 is latching onto +a convection sensitivity that is present on going from L38 to L70 but +had little effect in a non-PC2 simulation. It may also be related +to a reduction in timestep from 30 minutes to 20 minutes. +Investigations have not +made much progress in identifying the reasons for the differences, +or producing effective tunings to counter the problem. + +\subsubsection{High horizontal resolution performace} +PC2 has only been tested once at 4 km horizontal resolution. This +produced excessive shallow convective cloud (this may or may not be related +to the 70-levels problem above). Since this simulation the erosion +term has been increased dramatically, which may help. We note that +one of the main advantages of PC2, that of a prognostic link of +cloud to convection, is reduced at high resolution, as convection +becomes more explicit rather than diagnosed. We hence see a +resolution limit beyond which it is no longer appropriate to use +PC2. Results look acceptable at 12 km resolution, but we have not +quantitatively explored this limit. + +\subsubsection{Diagnostic evaluation} +One of the principal areas for future cloud scheme development +work planned in the future is in the area of detailed evaluation against +a number of data sources, such as CloudSat, ground based radar, +or case study campaigns. The quantitative evaluation has been +lacking to a significant degree in the development of the scheme, +as the focus has been on tackling qualitatively poor results. +Hence new sources of evaluation work on PC2 would be very welcome. + +\subsubsection{Moisture distribution within the deposition/sublimation term} +The liquid cloud changes in PC2 (or in a non-PC2 run) are based upon +a moisture PDF, as are the deposition/sublimation changes. However, it is +not the same PDF. It has always been the case with the prognostic ice +microphysics term that its PDF, whether explicit or implicit, has not +been rigorously consistent with the PDF used in the calculation of liquid +water, because it was most easily developed that way and produced reasonable +results. It may be useful to investigate whether the two PDF +representations can be brought together in a rigourous way, both for the PC2 +scheme and the \cite{smith90} scheme. + +We have similarly noted potential inconsistencies in the parametrization +of cloud fraction changes between the evaporation of rain term and the +riming (or accretion) term. Again, it might be possible to bring together +these formulations into a single consistent framework. + +\subsubsection{Area cloud fraction representation} +The current area cloud fraction representation is not used when +convection is taking place (signified by the \textit{cumulus} logical). +This inevitably leads to a potential switching between two different values +of the cloud fields if the convective boundary layer (not whether the +convection is shallow or deep) switches on and off, which is undesirable, +although not as bad as switching cloud on and off completely (as for the +current convective cloud formulation). Additionally, it is reasonable to argue that +having an area cloud fraction for cirrus cloud depend upon whether the boundary +layer is well mixed or has shallow convection occuring is not a reasonable link. + +Work in Australia on a TWP-ICE single column model case study using PC2 +suggests the area cloud fraction scheme over estimates the area cloud coverage +for tropical anvil clouds (which exist long after the convection itself has +ceased). This is perhaps not surprising since the \cite{bhi05} area +cloud fraction scheme was evaluated against mid-latitude cloud and it is +known that tropical clouds have greater vertical coherence. Tuning the +parameters in $large_scale_cloud/ls_acf_brooks.F90$ may be beneficial. + +%%\subsection{Acknowledgements} + +\begin{figure} +\begin{center} +\includegraphics[scale=1.0]{pc2_process_explanation} +\caption{Schematic summary of the PC2 cloud scheme.} +\label{fig:schematic} +\end{center} +\end{figure} + +\begin{figure} +\begin{center} +\includegraphics[scale=0.6]{Timestepping_ctl66.epsi} +\caption{Timestepping diagram for the control (non-PC2) scheme} +\label{fig:tstep_diag} +\end{center} +\end{figure} + +\begin{figure} +\begin{center} +\includegraphics[scale=0.6]{Timestepping_pc266.epsi} +\caption{Timestepping diagram for the PC2 scheme} +\label{fig:tstep_prog} +\end{center} +\end{figure} + +\bibliography{../029/refs} +\bibliographystyle{plainnat} + +\end{document} From a513efb5edd815b654b86aead28e692702c6574b Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Thu, 2 Apr 2026 00:48:26 +0100 Subject: [PATCH 003/116] Imported include files used by the PC2 cloud-scheme latex source. --- .../cloud_schemes/UMDP30_PC2CloudScheme.tex | 4 +- .../cloud_schemes/um_call_tree.tex | 530 ++++++++++++++++++ .../cloud_schemes/um_call_tree_preamble.tex | 25 + 3 files changed, 557 insertions(+), 2 deletions(-) create mode 100644 documentation/source/science_guide/cloud_schemes/um_call_tree.tex create mode 100644 documentation/source/science_guide/cloud_schemes/um_call_tree_preamble.tex diff --git a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.tex b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.tex index d5173e6ad8..8faee4121d 100644 --- a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.tex +++ b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.tex @@ -16,7 +16,7 @@ \usepackage{amstext,natbib} % Packages needed for the UM subroutine tree diagram in um_call_tree.txt: -\input{../029/um_call_tree_preamble} +\input{um_call_tree_preamble} \newcommand{\mmax}[1] {\mbox{\footnotesize \sf MAX} \left[#1\right] } @@ -4738,7 +4738,7 @@ \subsection{Code Structure} % The latex source input here contains a colour-coded itemize list % of the UM subroutine tree, showing the locations of all the cloud-scheme % routines. To edit this, open the source file source/029/um_call_tree.tex -\input{../029/um_call_tree} +\input{um_call_tree} \subsection{Diagnostics} \label{sec:diags} diff --git a/documentation/source/science_guide/cloud_schemes/um_call_tree.tex b/documentation/source/science_guide/cloud_schemes/um_call_tree.tex new file mode 100644 index 0000000000..97d6a73f15 --- /dev/null +++ b/documentation/source/science_guide/cloud_schemes/um_call_tree.tex @@ -0,0 +1,530 @@ + +% Latex source to make a diagram of the UM subroutine call tree, showing the +% locations of all cloud scheme calls. This diagram is included in both +% UMDP 029 (large-scale cloud scheme) and UMDP 030 (PC2). + +% NOTE: any preamble text required for this should be put in the file +% um_call_tree_preamble.tex, which is also inlcuded in both the UMDPs. + +Subroutines only called for the \textcolor{blue}{Smith} scheme are highlighted +in \textcolor{blue}{blue}, those only called for \textcolor{mygreen}{PC2} are +in \textcolor{mygreen}{green}, and those only called for +the \textcolor{purple}{bimodal} scheme are in \textcolor{purple}{purple}. + +\subsubsection{Main Tree from atm\_step\_4a} + +\begin{itemize} + +\item {\bf atm\_step\_4a} \\* +(performs one timestep of the Unified Model...) + \begin{itemize} + + \begin{tcolorbox}[enhanced jigsaw, breakable] + \item {\bf atm\_step\_alloc\_4a} \\* + (does miscellaneous initialisations in atm\_step) + + \begin{itemize} + + \item \textcolor{mygreen}{\bf pc2\_rhtl} \\* + (calculate start-of-timestep Relative Humidity, used by PC2 initiation) + + \end{itemize} + \end{tcolorbox} + + \begin{tcolorbox}[enhanced jigsaw, breakable] + \item {\bf atmos\_physics1} \\* + (calls explicit ``slow'' physics routines...) + \begin{itemize} + + \begin{tcolorbox} + \item {\bf microphys\_ctl} \\* + (interface to microphysics scheme) + \begin{itemize} + + \item \textcolor{mygreen}{\bf pc2\_turbulence\_ctl} \\* + (Perform optional erosion of liquid-cloud; + done here if NOT doing erosion after convection, + e.g. if no convection scheme is used). + \begin{itemize} + + \item \textcolor{mygreen}{\bf pc2\_hom\_conv} \\* + (called here just to do erosion) + + \end{itemize} + + \item \textcolor{blue}{\bf ls\_cld} \\* + (Smith scheme without area cloud fraction calculation, + to set initial cloud fields passed into microphysics) + + \item {\bf ls\_ppn} \\* + (microphysics scheme) + + \item {\bf mphys\_turb\_gen\_mixed\_phase} \\* + (turbulent production of liquid cloud) + + \item \textcolor{mygreen}{\bf pc2\_turbulence\_ctl} \\* + (optionally use the PC2 pdf-width-change code to calculate + the cloud-fraction change from the above turbulent production + of liquid cloud) + + \begin{itemize} + + \item \textcolor{mygreen}{\bf pc2\_hom\_conv} \\* + (called here just to calculate the cloud fraction increment + consistent with the turbulent qcl increment) + + \end{itemize} + + \end{itemize} + \end{tcolorbox} + + \begin{tcolorbox} + \item {\bf rad\_ctl} \\* + (interface to radiation scheme) + \begin{itemize} + + \item {\bf sw\_rad} \\* + (short-wave radiation scheme) + + \item \textcolor{mygreen}{\bf pc2\_homog\_plus\_turb} \\* + (PC2 homogeneous forcing of liquid-cloud by SW radiation heating) + + \item {\bf lw\_rad} \\* + (long-wave radiation scheme) + + \item \textcolor{mygreen}{\bf pc2\_homog\_plus\_turb} \\* + (PC2 homogeneous forcing of liquid-cloud by LW radiation tendency) + + \end{itemize} + \end{tcolorbox} + + \begin{tcolorbox} + \item {\bf atmos\_physics1\_alloc\_pc2} + (wrapper for PC2 self-consistency checks at end of atmos\_physics1) + + \begin{itemize} + + \item Add increments from microphysics + radiation onto + start-of-timestep fields to form updated fields. + + \item \textcolor{mygreen}{\bf pc2\_checks} \\* + (self-consistency checks on cloud fractions and water contents) + + \item Convert corrected updated fields back to increments. + + \end{itemize} + \end{tcolorbox} + + \end{itemize} + \end{tcolorbox} + + Begin loop over solver outer cycles + + \begin{itemize} + + \begin{tcolorbox}[enhanced jigsaw, breakable] + \item {\bf atm\_step\_phys\_reset} \\* + (for PC2, on subsequent solver outer cycles, + reset cloud-fractions to saved values after atmos\_physics1) + \end{tcolorbox} + + \begin{tcolorbox}[enhanced jigsaw, breakable] + \item {\bf eg\_sl\_moisture} \\* + (large-scale advection of cloud water contents and fractions) + \end{tcolorbox} + + \begin{tcolorbox}[enhanced jigsaw, breakable] + \item \textcolor{mygreen}{\bf pc2\_pressure\_forcing\_only} \\* + (Optionally calculate homogeneous forcing of liquid cloud by the + pressure change along the trajectory from departure point to + arrival point). + \begin{itemize} + + \item \textcolor{mygreen}{\bf pc2\_homog\_plus\_turb} \\* + (generic homogeneous forcing routine used here). + + \end{itemize} + \end{tcolorbox} + + \begin{tcolorbox}[enhanced jigsaw, breakable] + \item {\bf atmos\_physics2} \\* + (calls ``fast'' physics routines...) + + \begin{itemize} + + \begin{tcolorbox} + \item {\bf ni\_bl\_ctl} \\* + (interface to explicit boundary-layer and surface scheme calls, + including calculation of TKE and TKE-based $RH_{crit}$) + \end{tcolorbox} + + \begin{tcolorbox} + \item \textcolor{purple}{\bf bm\_calc\_tau} \\* + (calculates turbulence properties used in the bimodal cloud scheme, + based on the boundary-layer scheme TKE and mixing-length) + \end{tcolorbox} + + \begin{tcolorbox} + \item {\bf cloud\_call\_b4\_conv} \\* + (routine for optional cloud-scheme calls before convection) + \begin{itemize} + + \item \textcolor{blue}{\bf ls\_arcld} \\* + (Smith scheme with area cloud fraction; + see \ref{subsubsec:smith_acf} for a drill-down inside this routine) + + \item \textcolor{purple}{\bf bm\_ctl} \\* + (bimodal scheme) + + \item \textcolor{purple}{Set area cloud fraction equal to bulk + cloud fraction} + + \item \textcolor{mygreen}{\bf pc2\_initiation\_ctl} \\* + (interface to PC2 initiation and consistency-checks; + see \ref{subsubsec:pc2_initiation} for a drill-down inside this + routine) + + \end{itemize} + \end{tcolorbox} + + \begin{tcolorbox} + \item {\bf ni\_conv\_ctl} or {\bf other\_conv\_ctl} \\* + (interface routines to various convection schemes...) + \begin{itemize} + + \item {\bf glue\_conv\_5a/6a} \\* + (calls deep, shallow and mid-level convection schemes) + \begin{itemize} + + \item{\bf deep/shallow/mid\_conv} \\* + (convection scheme main routines) + \begin{itemize} + + \item {\bf convec2} + (completes lifting of the convective parcel by one model-level) + \begin{itemize} + + \item {\bf parcel} + (calculates new parcel properties at next level) + + \item {\bf environ} + (calculates grid-mean increments to primary fields; + includes PC2 partitioning of detrained condensate mass + between liquid and ice phases) + + \item \textcolor{mygreen}{\bf pc2\_environ} + (calculates increments to PC2 cloud fractions due to + convective detrainment and subsidence) + + \end{itemize} + + \end{itemize} + + \end{itemize} + + \item \textcolor{mygreen}{\bf pc2\_from\_conv\_ctl} \\* + (PC2 calculations after convection) + \begin{itemize} + + \item \textcolor{mygreen}{\bf pc2\_hom\_conv} \\* + (homogeneous forcing by convection, and erosion of liquid-cloud) + + \end{itemize} + + \end{itemize} + \end{tcolorbox} + + \begin{tcolorbox} + \item {\bf ni\_imp\_ctl} \\* + (interface to boundary-layer implicit solver) + \begin{itemize} + + \item {\bf imp\_solver} \\* + (implicitly solves vertical diffusion to find $T_l$ and $q_T$ + updated by turbulent fluxes). + + \item \textcolor{mygreen}{\bf pc2\_bl\_inhom\_ice} \\* + (inhomogeneous forcing of ice-cloud) + + \item \textcolor{mygreen}{\bf pc2\_delta\_hom\_turb} \\* + (homogeneous forcing of liquid cloud by the turbulent fluxes) + + \item \textcolor{mygreen}{\bf pc2\_bl\_forced\_cu} \\* + (adds diagnosed ``forced cumulus'' cloud fraction and water content + onto the PC2 prognostics) + + \item Calculate area cloud fraction: + + \textcolor{mygreen}{\bf ls\_acf\_brooks} \\* + (for the Brooks epirical method) + + \textcolor{mygreen}{\bf pc2\_hom\_arcld} \\* + (for the Cusack vertical interpolation method) + \begin{itemize} + + \item \textcolor{mygreen}{\bf pc2\_homog\_plus\_turb} \\* + (generic homogeneous forcing routine used to interpolate) + + \end{itemize} + + \item \textcolor{blue}{\bf ls\_arcld} \\* + (interface to diagnostic Smith scheme and area cloud fraction; + see \ref{subsubsec:smith_acf} for a drill-down inside this routine) + + \item \textcolor{purple}{\bf bm\_ctl} \\* + (bimodal cloud scheme) + + \item \textcolor{purple}{Set area cloud fraction equal to bulk + cloud fraction} + + \item {\bf diagnostics\_bl} \\* + (outputs boundary-layer diagnostics to STASH) + \begin{itemize} + + \item {\bf ls\_cld} \\* + (Smith scheme used here to calculate various diagnostics of + near-surface temperature and humidity, by extrapolating pressure, + $T_l$ and $q_t$ down to the desired height and then + re-diagnosing $q_{cl}$. + + \end{itemize} + + \end{itemize} + \end{tcolorbox} + + \end{itemize} + \end{tcolorbox} + + \begin{tcolorbox}[enhanced jigsaw, breakable] + \item {\bf atm\_step\_ac\_assim} \\* + (interface to Data Assimilation analysis increments...) + + \begin{itemize} + + \item {\bf ac\_ctl} + (control routine for Data Assimilation analysis increments...) + \begin{itemize} + + \item{\bf ac} + (main analysis increment routine) + + \item \textcolor{mygreen}{\bf pc2\_assim} \\* + (PC2 reponse to the analysis increments; + see \ref{subsubsec:pc2_assim} for a drill-down inside this routine) + + \item \textcolor{mygreen}{\bf ls\_acf\_brooks} + (calculate area cloud fraction using Brooks empirical method if active) + + \item \textcolor{blue}{\bf ls\_arcld} + (call diagnostic Smith scheme with area cloud fraction again to + account for the analysis increments; + see \ref{subsubsec:smith_acf} for a drill-down inside this routine) + + \end{itemize} + + \end{itemize} + \end{tcolorbox} + + \begin{tcolorbox}[enhanced jigsaw, breakable] + \item {\bf eg\_sl\_helmholtz} \\* + (dynamics pressure solver; updates pressure, and the winds used + to perform advection on the next solver outer cycle) + \end{tcolorbox} + + \end{itemize} + + End loop over solver outer cycles + + \begin{tcolorbox}[enhanced jigsaw, breakable] + \item \textcolor{mygreen}{\bf pc2\_pressure\_forcing} \\* + (interface to miscellaneous PC2 calculations at end-of-timestep) + \begin{itemize} + + \item \textcolor{mygreen}{\bf pc2\_homog\_plus\_turb} \\* + (homogeneous forcing of liquid-cloud by the dynamics pressure change; + optionally either uses total pressure change including the + Lagrangian component following the winds, or only the Eulerian + component from the dynamics solver) + + \item \textcolor{mygreen}{\bf pc2\_initiation\_ctl} \\* + (interface to PC2 initiation and consistency-checks; + see \ref{subsubsec:pc2_initiation} for a drill-down inside this routine) + + \end{itemize} + \end{tcolorbox} + + \begin{tcolorbox}[enhanced jigsaw, breakable] + \item {\bf qt\_bal\_cld} \\* + (calculates end-of-timestep cloud state consistent with final pressure...) + \begin{itemize} + + \item \textcolor{blue}{\bf ls\_arcld} \\* + (interface to diagnostic Smith scheme and area cloud fraction; + see \ref{subsubsec:smith_acf} for a drill-down inside this routine) + + \item \textcolor{purple}{\bf bm\_ctl} \\* + (bimodal cloud scheme) + + \item \textcolor{purple}{Set area cloud fraction equal to bulk + cloud fraction} + + \end{itemize} + \end{tcolorbox} + + \begin{tcolorbox}[enhanced jigsaw, breakable] + \item {\bf iau} \\* + (incremental analysis update; part of data assimilation) + \begin{itemize} + + \item \textcolor{mygreen}{\bf pc2\_assim} \\* + (PC2 reponse to the analysis increments; + see \ref{subsubsec:pc2_assim} for a drill-down inside this routine) + + \item \textcolor{mygreen}{\bf initial\_pc2\_check} \\* + (wrapper for optional self-consistency checks on prognostic cloud variables + if not doing PC2 response to analysis increments) + + \begin{itemize} + + \item \textcolor{mygreen}{\bf pc2\_checks} \\* + (self-consistency checks on cloud fractions and water contents) + + \end{itemize} + + \end{itemize} + \end{tcolorbox} + + \end{itemize} + +\end{itemize} + + +Drill-downs within some routines in the call tree are listed separately below, +to avoid duplication +(since these routines are called in multiple different places in the tree)... + +\subsubsection{Smith scheme with area cloud fraction} +\label{subsubsec:smith_acf} + +\begin{itemize} + +\begin{tcolorbox}[enhanced jigsaw, breakable] +\item \textcolor{blue}{\bf ls\_arcld} \\* +(interface to diagnostic Smith scheme and area cloud fraction) + \begin{itemize} + + \item If no area cloud fraction scheme: + + \textcolor{blue}{\bf ls\_cld} \\* + (just directly call Smith scheme) + + Set area cloud fraction equal to bulk cloud fraction. + + \item If using Cusack vertical interpolation method: + + Interpolate fields onto finer vertical grid + + \textcolor{blue}{\bf ls\_cld} \\* + (call Smith scheme using higher vertical resolution fields) + + Coarse-grain cloud fields back to model grid, but set area cloud + fraction to max of bulk cloud fraction over corresponding fine-grid levels. + + \item If using Brooks empirical area cloud fraction method: + + \textcolor{blue}{\bf ls\_cld} \\* + (just directly call Smith scheme) + + \textcolor{blue}{\bf ls\_acf\_brooks} \\* + (estimate area cloud fraction) + + \end{itemize} +\end{tcolorbox} + +\end{itemize} + + +\subsubsection{PC2 initiation} +\label{subsubsec:pc2_initiation} + +\begin{itemize} + +\begin{tcolorbox}[enhanced jigsaw, breakable] +\item \textcolor{mygreen}{\bf pc2\_initiation\_ctl} \\* +(interface to PC2 initiation and consistency-checks) + \begin{itemize} + + \item \textcolor{mygreen}{\bf pc2\_checks} \\* + (self-consistency checks on cloud fractions and water contents) + + \item PC2 initiation of liquid-cloud: + + \textcolor{mygreen}{\bf pc2\_bm\_initiate} \\* + (using the bimodal cloud scheme) + + \textcolor{mygreen}{\bf pc2\_arcld} \\* + (using the Smith scheme with the Cusack vertical interpolation method) + + \begin{itemize} + + \item \textcolor{mygreen}{\bf pc2\_initiate} \\* + (initiation using the Smith scheme, + called here on a finer vertical grid as per the Cusack method) + + \end{itemize} + + \textcolor{mygreen}{\bf pc2\_initiate} \\* + (using the Smith scheme with no area cloud representation) + + \item \textcolor{mygreen}{\bf pc2\_checks2} \\* + (further self-consistency checks on cloud-fractions) + + \item \textcolor{mygreen}{\bf pc2\_checks} \\* + (repeat the first lot of self-consistency checks again, + just in case we broke something in the mean-time!) + + \item \textcolor{mygreen}{\bf pc2\_hom\_arcld} \\* + (finds area cloud fraction using a version of the Cusack method, + where the cloud fraction on the finer vertical grid is estimated by + applying homogeneous forcing relative to the original grid fields) + + \begin{itemize} + + \item \textcolor{mygreen}{\bf pc2\_homog\_plus\_turb} \\* + (generic homogeneous forcing routine used to interpolate) + + \end{itemize} + + \end{itemize} +\end{tcolorbox} + +\end{itemize} + + +\subsubsection{PC2 Data Assimilation} +\label{subsubsec:pc2_assim} + +\begin{itemize} + +\begin{tcolorbox}[enhanced jigsaw, breakable] +\item \textcolor{mygreen}{\bf pc2\_assim} \\* +(PC2 reponse to the analysis increments) + \begin{itemize} + + \item \textcolor{mygreen}{\bf pc2\_homog\_plus\_turb} \\* + (generic PC2 homogeneous forcing routine used here for liquid-cloud) + + \item Estimate change in ice-cloud fraction from the assimilation + increment to ice-cloud mass. + + \item \textcolor{mygreen}{\bf pc2\_total\_cf} \\* + (update bulk cloud fraction due to change in ice cloud fraction) + + \item \textcolor{mygreen}{\bf pc2\_checks} \\* + (self-consistency checks on prognostic cloud fractions and + water contents) + + \end{itemize} +\end{tcolorbox} + +\end{itemize} diff --git a/documentation/source/science_guide/cloud_schemes/um_call_tree_preamble.tex b/documentation/source/science_guide/cloud_schemes/um_call_tree_preamble.tex new file mode 100644 index 0000000000..009537991a --- /dev/null +++ b/documentation/source/science_guide/cloud_schemes/um_call_tree_preamble.tex @@ -0,0 +1,25 @@ + +% Packages needed for the UM subroutine tree diagram in um_call_tree.txt + +% Used to colour-code things in the subroutine call tree diagram: +\usepackage{xcolor} +% Define a darker green, as in some pdf viewers the standard green +% is too bright to be readable on the grey background. +\definecolor{mygreen}{rgb}{0.0, 0.667, 0.0} + +% Allow more deeply nested lists, for writing the subroutine call tree: +\usepackage{enumitem} +\setlistdepth{20} +\renewlist{itemize}{itemize}{20} +\setlist[itemize]{label=$\cdot$} +\setlist[itemize,1]{label=\textcolor{black}{$\bullet$}} +\setlist[itemize,2]{label=\textcolor{blue}{$\bullet$}} +\setlist[itemize,3]{label=\textcolor{purple}{$\bullet$}} +\setlist[itemize,4]{label=\textcolor{red}{$\bullet$}} +\setlist[itemize,5]{label=\textcolor{orange}{$\bullet$}} +\setlist[itemize,6]{label=\textcolor{yellow}{$\bullet$}} +\setlist[itemize,7]{label=\textcolor{green}{$\bullet$}} +\setlist[itemize,8]{label=\textcolor{cyan}{$\bullet$}} + +% Used to draw boxes around subroutines in the call tree diagram: +\usepackage[most]{tcolorbox} \ No newline at end of file From 39312d83d32d558cfc008bca0dc6b0c9b2b9a46c Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Thu, 2 Apr 2026 00:53:03 +0100 Subject: [PATCH 004/116] Ran pandoc -s -f latex -t rst -o UMDP30_PC2CloudScheme.rst UMDP30_PC2CloudScheme.tex to convert the latex source to .rst format. --- .../cloud_schemes/UMDP30_PC2CloudScheme.rst | 6916 +++++++++++++++++ 1 file changed, 6916 insertions(+) create mode 100644 documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst diff --git a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst new file mode 100644 index 0000000000..c8670fe618 --- /dev/null +++ b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst @@ -0,0 +1,6916 @@ +==================== +The PC2 Cloud Scheme +==================== + +:Author: D. Wilson, A. Bushell, C. Morcrette, V. Varma\ :math:`^{1}`, + M. Whitall + +.. role:: raw-latex(raw) + :format: latex +.. + +PC2 developers +============== + +We would like to acknowledge those who developed the PC2 cloud scheme: +Damian Wilson, Andrew Bushell, David Gregory, Amanda Kerr-Munslow, John +Edwards, Jeremy Price, Cyril Morcrette, Martin Sharpe, Thomas Mirfield, +Ian Boutle. Many others offered considerable help, advice and analysis, +including Roy Kershaw, Malcolm Brooks, Richard Forbes and Alejandro +Bodas-Salcedo, and we would like to thank them all for their input. + +Introduction +============ + +This document describes the PC2 *(prognostic cloud, prognostic +condensate)* cloud scheme. It should be seen as a complete reference +source for the scheme’s physical assumptions, numerical techniques, +application to the Unified Model and coding within the Unified Model. It +does not describe results from the scheme, please refer to the various +reports and papers written on this. Except where commented on +explicitly, the description applies to the PC2:66 version of the PC2 +scheme, which is the version that will be available at UM6.5. The +version available at 6.4 is PC2:64. + +This paper will first introduce the concepts that underlie cloud +schemes, before developing a study of the theoretical behaviour of the +prognostic PC2 scheme under certain, well-defined, situations. The next +sections shows how the theory can be applied to the physical and +dynamical processes represented in the Unified Model. Finally, we +outline the way in which the PC2 scheme is implemented within the code +of the Unified Model. + +Cloud schemes +------------- + +The basic requirements of any cloud scheme within a large-scale model +are to: + +- calculate the amount of condensation (from water vapour to liquid + water or vice-versa) within each gridbox each timestep + +- to calculate or update the cloud fractions for use by the radiation + and large-scale precipitation schemes (or any other physics scheme). + +Depending on the model involved, cloud schemes may also treat the +deposition / sublimation process from vapour to ice. The problem is +straightforward to solve if one is allowed to assume that there is no +variability of moisture or temperature on a scale of a model gridbox. In +this case the cloud fraction scheme is redundant and only the +condensation part remains, which may be solved diagnostically using the +instantaneous condensation assumption in section `2.2 <#sec:s_dist>`__. +However, the ‘no-variability’ assumption is poor until very high +resolutions close to, or maybe exceeding, 1 km in the horizontal are +reached. Although we may eventually assume that computer power will +enable such resolutions to be reached globally, for many years we will +need a subgrid-scale cloud scheme to properly account for the +variability in the atmosphere. This is the principal challenge of cloud +parametrization. + +There are several approaches to take to the solution of the problem, +although they are not as independent as often portrayed, since they +nearly all require the same instantaneous condensation assumption +(discussed in section `2.2 <#sec:s_dist>`__). Hence there are +mathematical links between all the approaches. *The following are all +valid structures to use in this respect.* + +- One may diagnose cloud fractions and condensate contents from + knowledge of gridbox mean variables. This forms the basis of the + :raw-latex:`\cite{smith90}` scheme, which is described in . + +- A mixed scheme, such as :raw-latex:`\cite{sundqvist1978}` uses a + prediction of condensate contents, but a diagnostic cloud fraction. + +- Alternatively, one may predict cloud fraction and condensate content + changes as a result of each modelled process. This forms the basis of + the :raw-latex:`\cite{t93}` scheme and the PC2 scheme. + +- Hybrid schemes, such as :raw-latex:`\cite{t02}`, will predict various + moments of the subgrid-scale variability, and use this knowledge to + diagnose the cloud fraction and condensate contents. + +Many years of experience of the results from the +:raw-latex:`\cite{smith90}` scheme have highlighted deficiences in the +diagnosis of cloud from this scheme, which we feel can only be tackled +by adding the memory of cloud history available by using a prognostic +based scheme. We chose to develop a scheme that directly specified the +impacts on observable prognostics (condensates and cloud fractions, as +in :raw-latex:`\cite{t93}`) rather than on moments of a probability +density function (as in :raw-latex:`\cite{t02}`). This is because we +believe it is easier to physically relate (and hence parametrize) the +effect processes to quantities such as cloud fraction and condensate +rather than to the more abstract quantities of moments of a probability +density function of moisture. However, although the PC2 scheme is +similar to :raw-latex:`\cite{t93}` in its very basic prognostic variable +structure, the assumptions behind the formulation of the prognostic +terms in PC2 are very different and much improved. The PC2 scheme should +not be considered to be merely an extension of :raw-latex:`\cite{t93}`. + +In particular, we wish to use a prognostic formulation in order to link +the detraiment of moisture from convection directly to cloud fraction, +and to break the hard diagnostic link between cloud fraction and +condensate. These major features of the :raw-latex:`\cite{t93}` scheme +provide the motivation to develop the PC2 cloud scheme. + +.. _`sec:s_dist`: + +The ‘s’ distribution +-------------------- + +Most cloud schemes are based on the concept of a distribution of +fluctuations of moisture and temperature in the gridbox. Here we +mathematically formalize this concept, since it is used both in the PC2 +scheme and the :raw-latex:`\cite{smith90}` scheme. + +This method was first formulated by :raw-latex:`\cite{m77}` and +:raw-latex:`\cite{sommeria_deardorff_1977}` for large-eddy simulations. +It can also be applied to larger scale models. It allows us to calculate +vapour and liquid contents and liquid cloud fraction from knowledge only +of the combined vapour+liquid content, :math:`\overline{q_T}`, and the +liquid temperature, :math:`\overline{T_L}`. These variables are +unchanged during condensation processes, so it is useful to write the +cloud scheme in terms of these variables. + +In this derivation we will consider only liquid condensate. We assume, +as above, that, locally, the water content in a cloud is such as to +remove any supersaturation. This gives the equation + +.. math:: + + q_{cl} = q_T - q_{sat}(T,p) + \label{eq:basic_qcl} + +assuming that :math:`q_T > q_{sat}(T,p)` (:math:`q_{cl}` will be zero +otherwise). :math:`q_T` is the local total water content, equal to the +sum of the condensate :math:`(q_{cl})` plus the vapour :math:`(q)`, +:math:`T` is the temperature, :math:`p` is the pressure and +:math:`q_{sat}(T,p)` is the saturation specific humidity at temperature +T and pressure p *with respect to liquid water*. (Many earlier +diagnostic cloud schemes use a similar instantaneous condensation +assumption for ice, which would mean that :math:`q_{sat}` must be taken +with respect to ice when :math:`T < 0 ^{\circ} C`, but the Unified Model +does not). We now introduce the liquid temperature (:math:`T_L`), where +:math:`T_L` is given by + +.. math:: + + T_L = T - \frac{L}{c_p} q_{cl} , + \label{eq:tl} + +and :math:`L` is the latent heat of vaporization and :math:`c_p` is the +heat capacity of air. Note that :math:`T_L` is unaffected by changes of +phase between vapour and liquid. We now write +(`[eq:basic_qcl] <#eq:basic_qcl>`__) as an *equality* + +.. math:: + + q_{cl} = q_T - \left( q_{sat}(T_L,p) + \alpha (T - T_L) \right) + \label{eq:alpha_t_tl} + +where + +.. math:: + + \alpha = \frac{ q_{sat}(T,p) - q_{sat}(T_L,p) }{T - T_L } . + \label{eq:alpha} + +Using (`[eq:tl] <#eq:tl>`__) in (`[eq:alpha_t_tl] <#eq:alpha_t_tl>`__) +gives the expression + +.. math:: q_{cl} = q_T - q_{sat} (T_L) - \alpha \frac{L}{c_p} q_{cl} + +or + +.. math:: + + q_{cl} = a_L \left( q_T - q_{sat}(T_L,p) \right) + \label{eq:l_eq_al} + +where :math:`a_L` is given by + +.. math:: + + a_L = \left( 1 + \alpha \frac{L}{c_p} \right) ^{-1} . + \label{eq:a_L} + +Thus (`[eq:basic_qcl] <#eq:basic_qcl>`__) has been rewritten *exactly* +in terms of the conserved variables, :math:`q_T` and :math:`T_L`, +although the temperature, :math:`T`, does remain in the definition of +:math:`a_L`. We will need to consider variations across a gridbox for a +parametrization scheme, so we expand the expression for condensate +(`[eq:l_eq_al] <#eq:l_eq_al>`__) into terms relating to the gridbox mean +and variation from the gridbox mean. + +.. math:: + + q_{cl} = \overline{ a_L \left( q_T - q_{sat}(T_L,p) \right)} + + [ a_L \left( q_T - q_{sat}(T_L,p) \right) ]' + \label{eq:bar_plus_pri1} + +where :math:`\overline{\phi}` represents the mean of a distibution of +:math:`\phi` and :math:`\phi = \overline{\phi} + {\phi}'`. The +expression (`[eq:bar_plus_pri1] <#eq:bar_plus_pri1>`__) is *exact* when +using the definition of :math:`\alpha` given in +(`[eq:alpha] <#eq:alpha>`__). + +The idea of a PDF scheme is to calculate the first (mean) term, +:math:`\overline{\phi}`, from the known gridbox mean parameters, +:math:`q_T`, :math:`T_L` and :math:`p`, and to parametrize the +distribution of the second, variable term, :math:`{\phi}'`. +Unfortunately, the mean term is difficult to write in terms of the +gridbox mean variables :math:`\overline{q_T}` and :math:`\overline{T_L}` +because :math:`q_{sat}(T_L,p)` is not a linear function of :math:`T_L` +(or of :math:`p`). In order to proceed, we will now make an +*approximation* that :math:`q_{sat}(T,p)` is a linear function of +:math:`T_L` and :math:`p`. This equivalently implies that :math:`a_L` +and :math:`\alpha` are approximated as being constant across the +gridbox. The expression now becomes more tractable, +(`[eq:bar_plus_pri1] <#eq:bar_plus_pri1>`__) becoming: + +.. math:: + + q_{cl} = a_L \left( \overline{q_T} - q_{sat}(\overline{T_L},\overline{p}) + \right) + a_L \left( {q_T}' - \alpha {T_L}' - \beta {p}' \right) + \label{eq:l_eq_bar_plus_pri} + +where :math:`\beta = {\frac{\partial q_{sat}}{\partial p}}` at constant +temperature. The first term is connected with the mean properties of the +gridbox, and is written as :math:`Q_c`, the second term is connected +with the deviation of the local conditions from the mean and is written +as :math:`s`. + +.. math:: + + Q_c = a_L \left( \overline{q_T} - q_{sat}(\overline{T_L},\overline{p}) \right) + \label{eq:qc_eq_qt-qs} + +.. math:: + + s = a_L \left( {q_T}' - \alpha {T_L}' - \beta {p}' \right) + \label{eq:s} + +This gives the equation + +.. math:: + + q_{cl} = Q_c + s + \label{eq:l_qc_s} + +with the assumption that :math:`s \ge -Q_c` (i.e. :math:`q_{cl} \ge 0`). +If :math:`s < -Q_c` then :math:`q_{cl} = 0`. The term :math:`a_L` can be +calculated using (`[eq:a_L] <#eq:a_L>`__) from +(`[eq:alpha] <#eq:alpha>`__) with gridbox mean temperatures, i.e. + +.. math:: + + \alpha = \frac{ q_{sat}(\overline{T},\overline{p}) + - q_{sat}(\overline{T_L},\overline{p}) }{\overline{T} - + \overline{T_L} } . + \label{eq:alpha_mean} + +This definition of :math:`\alpha` and :math:`a_L` will retrieve an +*exact* value for the gridbox mean :math:`\overline{q_{cl}}` *if* the +distribution is monodispersed. Hence it is the sensible form to use for +a purely diagnostic representation such as :raw-latex:`\cite{smith90}` +where we explicitly consider distributions of :math:`s`. Strictly, the +linear approximation implies that other approximations for +:math:`\alpha` are valid: PC2 will do this (see section +`3.2.3 <#sec:homog_num_app>`__) since we are concerned in PC2 with the +best estimate of the *changes* to :math:`\overline{q_{cl}}`, not the +best estimate of :math:`\overline{q_{cl}}` itself. + +We now assume that within any particular gridbox a distribution +:math:`G` of :math:`s` occurs (with mean, by definition, of zero). +Considering cloud to be where the water content is greater than zero +(i.e. where :math:`s > -Q_c`) gives an expression for the liquid cloud +*volume* fraction, :math:`C_l`, within the gridbox as + +.. math:: + + C_l = \int_{s=-Q_c}^{\infty} G(s) ds + \label{eq:int_gs_ds} + +and the expression for mean condensate, :math:`{\overline{q_{cl}}}`, +using (`[eq:l_qc_s] <#eq:l_qc_s>`__) to expand :math:`q_{cl}`, is + +.. math:: + + \overline{q_{cl}} = \int_{s=-Q_c}^{\infty} (Q_c + s) G(s) ds . + \label{eq:qclbar=int} + +If we know (parametrize) the PDF given by :math:`G(s)` then we can solve +for :math:`C_l` and :math:`\overline{q_{cl}}`. Note that this +distribution is in terms of :math:`s`, there is no need to know the +three-dimensional distribution in terms of three separate variables +:math:`q_T`, :math:`T_L` and :math:`p`. This is the method used by +:raw-latex:`\cite{smith90}`, where a symmetric triangular distribution +function is used. For further information on the +:raw-latex:`\cite{smith90}` scheme, please refer to . Physics and +dynamics schemes hence only need to provide increments to +:math:`\overline{q_T}` and :math:`\overline{T_L}`, provided that a +diagnostic scheme (such as :raw-latex:`\cite{smith90}`) is called at +some point in the timestep to partition :math:`\overline{q_T}` into +:math:`\overline{q}` and :math:`\overline{q_{cl}}`, to calculate the dry +bulb temperature :math:`\overline{T}` (from :math:`\overline{T_L}` and +:math:`\overline{q_{cl}}`) and to calculate the liquid cloud fraction, +:math:`C_l`. The diagnostic scheme effectively allows a calculation of +condensation associated with any physical process. However, its results +remain tied to the distribution of :math:`G(s)` that is chosen in +(`[eq:int_gs_ds] <#eq:int_gs_ds>`__) and +(`[eq:qclbar=int] <#eq:qclbar=int>`__) and it is this tie that we seek +to break by the use of a prognostic scheme. + +Concept of PC2 +-------------- + +The PC2 scheme develops prognostic expressions for the rates of change +of cloud fraction and condensate contents as a result of each process +that acts in the model. We consider ice and liquid condensate as two +distinct aspects of clouds, which may or may not overlap Figure +`1 <#fig:schematic>`__ provides a schematic summary of the PC2 scheme. +The equations for the five prognostic cloud variables can be written +schematically: + +.. math:: + + \begin{aligned} + \frac{\partial \overline{q_{cl}}}{\partial t} = + \frac{\partial \overline{q_{cl}}}{\partial t} |_{advection} + + \frac{\partial \overline{q_{cl}}}{\partial t} |_{convection} + + \frac{\partial \overline{q_{cl}}}{\partial t} |_{boundary \, layer} + + \frac{\partial \overline{q_{cl}}}{\partial t} |_{precipitation} + ... \nonumber \\ + \frac{\partial \overline{q_{cf}}}{\partial t} = + \frac{\partial \overline{q_{cf}}}{\partial t} |_{advection} + + \frac{\partial \overline{q_{cf}}}{\partial t} |_{convection} + + \frac{\partial \overline{q_{cf}}}{\partial t} |_{boundary \, layer} + + \frac{\partial \overline{q_{cf}}}{\partial t} |_{precipitation} + ... \nonumber \\ + \frac{\partial C_l}{\partial t} = + \frac{\partial C_l}{\partial t} |_{advection} + + \frac{\partial C_l}{\partial t} |_{convection} + + \frac{\partial C_l}{\partial t} |_{boundary \, layer} + + \frac{\partial C_l}{\partial t} |_{precipitation} + ... \nonumber \\ + \frac{\partial C_i}{\partial t} = + \frac{\partial C_i}{\partial t} |_{advection} + + \frac{\partial C_i}{\partial t} |_{convection} + + \frac{\partial C_i}{\partial t} |_{boundary \, layer} + + \frac{\partial C_i}{\partial t} |_{precipitation} + ... \nonumber \\ + \frac{\partial C_t}{\partial t} = + \frac{\partial C_t}{\partial t} |_{advection} + + \frac{\partial C_t}{\partial t} |_{convection} + + \frac{\partial C_t}{\partial t} |_{boundary \, layer} + + \frac{\partial C_t}{\partial t} |_{precipitation} + ... , + \label{eq:dqcldt_and_dcdt} + \end{aligned} + +where :math:`\overline{q_{cf}}` is the ice water specfic humidity, +:math:`C_l` is the liquid cloud *volume* fraction, :math:`C_i` is the +ice cloud volume fraction, and :math:`C_t` is the combined ice or liquid +cloud volume fraction. The amount of mixed phase cloud, :math:`C_{mp}`, +can be calculated by the overlap of the ice and liquid fractions: + +.. math:: + + C_{mp} = C_i + C_l - C_t. + \label{eq:mp} + +The idea is to parametrize each of the terms in the above equations. +This approach removes the diagnostic method, hence it will be critical +that we can write expressions for +:math:`\frac{\partial \overline{q_{cl}}}{\partial t}` and +:math:`\frac{\partial C_l}{\partial t}` for *each process that alters +:math:`\overline{T}`, :math:`\overline{p}`, :math:`\overline{q}`, or +:math:`\overline{q_{cl}}` in the model* (and similarly for the ice +terms). In doing so, we will not lose sight of underlying PDF approach +given by (`[eq:int_gs_ds] <#eq:int_gs_ds>`__) and +(`[eq:qclbar=int] <#eq:qclbar=int>`__) since we will still use the +concept of instantaneous condensation for liquid clouds. Equations +`[eq:int_gs_ds] <#eq:int_gs_ds>`__ and +`[eq:qclbar=int] <#eq:qclbar=int>`__ will form the basis of the +homogeneous forcing methods discussed in section `3.2 <#sec:homog>`__. +We note in particular that the convective cloud fraction, previously a +quantity that is diagnosed separately from the large-scale cloud +fraction calculated by the :raw-latex:`\cite{smith90}` scheme, may, in +PC2, be included as part of the large-scale cloud fraction. This aspect +is similar to the :raw-latex:`\cite{t93}` approach. + +The final aim of PC2 is that the parametrization of each term in +(`[eq:dqcldt_and_dcdt] <#eq:dqcldt_and_dcdt>`__) is performed by each +part of the model that alters :math:`\overline{T}`, +:math:`\overline{p}`, :math:`\overline{q}`, :math:`\overline{q_{cl}}` or +:math:`\overline{q_{cf}}` as an integral part of that physics or +dynamics scheme. However, in this PC2 scheme we acknowledge that this +will not be possible, at least, not to begin with. Hence we have +specifically developed generic approaches that can be used to calculate +expressions for :math:`\frac{\partial \overline{q_{cl}}}{\partial t}` +and :math:`\frac{\partial C_l} +{\partial t}` . These are referred to as Homogeneous forcing (section +`3.2 <#sec:homog>`__), Injection source (or inhomogeneous forcing, +section `3.5 <#sec:inhomog>`__) and Width Changing (section +`3.3 <#sec:width>`__). Two additional modules are available to assist +with PC2, liquid cloud initiaion (section `3.4 <#sec:init>`__) and the +calculation of total cloud fraction changes (section `3.6 <#sec:ct>`__). +At the present time, only the large-scale precipitation (section +`4.2 <#sec:precip>`__) scheme has been rewritten fully to use the PC2 +concept of prognostic cloud fractions. The existing mass-flux convection +scheme has been modified to enable calculation of the detrained +condensate, but direct modification to the cloud fraction is not +included. All other physics schemes use one of the generic approaches +below. + +A note on convective cloud fraction +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +It was the original intention that PC2 be able to replace the two +separate diagnostic cloud fractions (large-scale and convective) with a +single cloud fraction, as in :raw-latex:`\cite{t93}`. The hypothesis was +that by detraining cloud directly from the convection scheme we would no +longer need a separate representation of this cloud type. Our experience +with PC2 is that this is not necessarily the case. We suspect that the +basic reason is that we are unable to truely represent the extreme PDF +shapes that result from convective activity. Additionally, we only +create cloud associated with the detrainment part of the convection +scheme, assuming that cloud associated with the active updraughts in +convection is small. This assumption is not necessarily applicable. +Similar arguments, and model results, come from analysis of the +:raw-latex:`\cite{t93}` and :raw-latex:`\cite{t02}` scheme (Ben Johnson, +personal communication). We also note that with two cloud fraction types +and two different optical depths it is possible to have a basic degree +of representation of cloud inhomogeneity. + +Hence the code still exists to enable PC2 to be run with or without a +diagnostic convective cloud fraction, although PC2:66 does not include a +diagnostic term. More details are in section `4.7 <#sec:convec>`__. + +Physical basis of the PC2 prognostic cloud scheme +================================================= + +In this section we will develop the physical models that PC2 uses in +order to calculate its prognostic increment terms. We will also consider +the numerical solution of the models. The way in which these are +incorporated into the Unifed Model will be discussed in section +`5 <#sec:um>`__ + +Instantaneous condensation +-------------------------- + +Liquid clouds in PC2 use the concept of instantaneous condensation. +Hence the ‘s’ distribution methods are fully applicable to the +development of the equations that govern the parametrization of liquid +cloud in PC2. We will start by looking at changes to +:math:`\overline{q_{cl}}` and :math:`C_l` when a uniform forcing is +applied to a gridbox, under the assumption of instantaneous +condensation. + +.. _`sec:homog`: + +Homogeneous forcing +------------------- + +We define the expression *uniform forcing* (or *homogeneous forcing*) to +refer to changes in local values of :math:`T_L` and :math:`q_T` that +occur at a rate independent of the part of the gridbox in which they are +located. This implies that :math:`G(s)` will not alter due to such a +process. Uniform forcing simply alters :math:`Q_c` in +(`[eq:int_gs_ds] <#eq:int_gs_ds>`__) and +(`[eq:qclbar=int] <#eq:qclbar=int>`__). In the Unified Model, this +concept will be applied to several different sets of physics increments +in order to calculate the condensation and cloud fraction changes +associated with each one, where the physics routine does not allow the +explicit calculation of condensation and cloud fraction changes by +another method. Large-scale ascent may be considered a meteorological +example of such a process. By differentiating +(`[eq:int_gs_ds] <#eq:int_gs_ds>`__) and +(`[eq:qclbar=int] <#eq:qclbar=int>`__) with respect to time, assuming +uniform forcing (so :math:`{\frac{\partial G}{\partial t}}` terms are +zero), we obtain + +.. math:: + + {{\frac{\partial C_l}{\partial t}} = G(-Q_c) {\frac{\partial Q_c} + {\partial t} }.} + \label{dcdt} + +.. math:: + + {\frac{\partial \overline{q_{cl}}}{\partial t}} = + C_l {\frac{\partial Q_c}{\partial t}} + \label{dqcldt} + +The quantity :math:`G(-Q_c)` is the value of the PDF of :math:`G` at +:math:`s=-Q_c`, which defines the boundary between the saturated and +unsaturated parts of the distribution. + +If we wish to consider a prognostic cloud scheme with equations for the +rate of change of condensate and cloud fraction based upon +(`[dqcldt] <#dqcldt>`__) and (`[dcdt] <#dcdt>`__) then we need to close +(`[dcdt] <#dcdt>`__) by specifying the value of :math:`G(-Q_c)`. We will +choose to develop a parametrization for this quantity based upon the +quantities :math:`C_l`, :math:`\overline{q_{cl}}` and the saturation +deficit, :math:`SD`, rather than tie :math:`G(-Q_c)` to a process. The +saturation deficit is *defined* here in the ‘s’ framework to be the +first moment of the PDF for ‘s’ values less than :math:`-Q_c`. In this +way it is analogous to the liquid water content, +:math:`\overline{q_{cl}}`. Appendix A of :raw-latex:`\cite{wg03}` writes +this *definition* as + +.. math:: + + {SD = - \int_{-\infty}^{-Q_c} {( s+Q_c ) G(s) ds}} + \label{SD} + +and shows this is equivalent to + +.. math:: + + {SD = a_L ( q_{sat}({\overline{T}},{\overline{p}}) - {\overline{q}} ) .} + \label{SD2} + +The basis behind the parametrization for :math:`G(-Q_c)` is to consider +an underlying form of the distribution :math:`G(s)` near the +:math:`+b_s` and :math:`-b_s` ends. We borrow the notation of +:raw-latex:`\cite{smith90}` and refer to a quantity :math:`b_s` that is +the value of :math:`s` when a monomodal distribution :math:`G(s)` just +equals zero. We suppose that the distribution G can be described as a +power law near :math:`s=b_s`. + +.. math:: + + G(s) ~ \propto ~ {(-s + b_s)}^n + \label{eqn19} + +provided :math:`s`__) it can be shown (see appendix +B of :raw-latex:`\cite{wg03}`) that + +.. math:: + + {G_1(-Q_c) = {\frac{(n+1)}{(n+2)}} {\frac{C_l^2}{\overline{q_{cl}}}} .} + \label{eqn20} + +An important feature is that the proportionality between :math:`G(-Q_c)` +and :math:`{\frac{C^2}{\overline{l}} }` holds for any power law +description. Also, this relationship is independent of the value of +:math:`b_s`. The triangular :raw-latex:`\cite{smith90}` scheme obeys +this relationship (for :math:`C_l` less than 0.5) with :math:`n`\ =1, as +does a ‘top hat’ function which is a limiting case of :math:`n` tending +to zero. This invariant functional form can be exploited in deriving a +generalized :math:`G(-Q_c`) closure. If we assume a similar power law +for the other end of the distribution we can write a second estimate of +:math:`G(-Q_c)` as: + +.. math:: + + {G_2(-Q_c) = {\frac{(n+1)}{(n+2)}} {\frac{{(1-C_l)}^2}{SD}} .} + \label{eqn21} + +We note that if n tends to zero then (`[eqn21] <#eqn21>`__) is identical +to the expression used by :raw-latex:`\cite{jgt99}`. This is because +:raw-latex:`\cite{jgt99}` also uses a similar description of a ‘top-hat’ +PDF of fluctuations. + +In order that the closure of :math:`G(-Q_c)` is reversible, we take a +linear combination of :math:`G_1(-Q_c)` and :math:`G_2(-Q_c)`. To close +our parameterisation, we must choose suitable weights to apply to the +two solutions, and there are currently 2 options for the choice of +weights, discussed below. + +The equations (`[dqcldt] <#dqcldt>`__), (`[dcdt] <#dcdt>`__) and either +(`[eqn22] <#eqn22>`__) or (`[eq:gmqc_width] <#eq:gmqc_width>`__) below +form a complete mathematical set for the solution of :math:`C_l` and +:math:`\overline{q_{cl}}` under homogeneous forcing, provided that the +initial value of :math:`C_l` is not identically 0 or 1. If :math:`C_l` +is 0 or 1 then :math:`G(-Q_c)` remains at zero. The equation set then +needs to be initiated in some way. This is discussed further in +:raw-latex:`\cite{wg03}` and in section `3.4 <#sec:init>`__. If +:math:`G(-Q_c)` is defined, then application of the above equation set +may be used to trace out an underlying PDF for any input values of +:math:`C_l`, :math:`\overline{q_{cl}}` and :math:`SD`. Although we never +need to define the complete PDF in PC2 (just the value of +:math:`G(-Q_c)` from equation `[eqn22] <#eqn22>`__), a PDF can be +inferred off-line if required. + +:raw-latex:`\cite{wg03}` analyse the performance of this parametrization +under idealised tests, and shows it performs well against other cloud +parametrizations used in large-scale models. After extensive analysis +including observed PDF shapes and moments from balloon, performance in +the Unified Model, and numerical stability tests, the value of the shape +parameter :math:`n` has been chosen to be 0.0, corresponding to a +top-hat distribution shape. + +Weight as a function of cloud-fraction +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +This option is selected by setting **i_pc2_homog_g_method=1** in the UM +large-scale cloud namelist. + +Under this closure, we choose the weights such that :math:`G_1(-Q_c)` +(`[eqn20] <#eqn20>`__) is used when cloud fractions are small, and +:math:`G_2(-Q_c)` (`[eqn21] <#eqn21>`__) is used when cloud fractions +are large (a small cloud fraction indicates that the saturation boundary +is close to the right-hand end of the PDF, which :math:`G_1(-Q_c)` is +based on). We choose relative weights of :math:`{{(1-C_l)}^{m}}` and +:math:`{C_l^m }` respectively, where :math:`m` is a power currently set +to 0.5. Hence the complete suggested closure of :math:`G(-Q_c)` is: + +.. math:: + + {G(-Q_c) = {\frac{(n+1)}{(n+2)}} {\frac{ ( {( 1-C_l )}^m {\frac{C_l^2} + {\overline{q_{cl}}}} + C_l^m + {\frac{{(1-C_l)}^2}{SD}} ) }{( {(1-C_l)}^m + C_l^m ) }} . } + \label{eqn22} + +A problematic property of equation `[eqn22] <#eqn22>`__ is that it goes +to infinity if either :math:`q_{cl}` or :math:`SD` goes to zero (and +:math:`C_l` is not zero or unity). There are realistic scenarios in +which this limit will be approached; e.g. if heavy rain falls through a +layer of cloud, nearly all of the cloud liquid water content maybe +removed by accretion, without reducing the cloud-fraction. In this +situation, any subsequent homogeneous forcing applied to the cloud will +result in a huge tendency in cloud-fraction in equation +`[dcdt] <#dcdt>`__, due to the term +:math:`\frac{C_l^2}{\overline{q_{cl}}}` in `[eqn22] <#eqn22>`__ becoming +huge. + +Note that for a homogeneous forcing acting to dry the layer / reduce the +cloud, a huge negative tendency is the “right” answer; if there is only +an infinitessimally small amount of liquid water content left within the +cloud, then the cloud fraction should indeed vanish extremely rapidly +under a negative forcing. However, for a positive homogeneous forcing, +the cloud-fraction will very rapidly increase in this scenario, for no +physical reason. This might not be a problem if one exactly integrated +the differential equations `[dcdt] <#dcdt>`__ and +`[dqcldt] <#dqcldt>`__, since :math:`q_{cl}` would immediately increase +away from zero, so that `[eqn22] <#eqn22>`__ immediately becomes +well-defined (an infinitely large tendency maintained for an infinitely +small period of time can yield a finite, sensible increment!) +Unfortunately, PC2 uses an explicit numerical method, and a finite +(often quite large) model timestep, so an instantaneous very large +tendency will yield a very large increment, even if the continuous +differential equations would not have done. + +In practice, the code that implements `[eqn22] <#eqn22>`__ simply sets +:math:`G(-Q_c)` to zero if either :math:`q_{cl}` or :math:`SD` falls +below 1.0E-10 kg kg\ :math:`^{-1}`, to avoid a floating-point error. +This in itself can be problematic, since leaving :math:`C_l` unmodified +under a homogeneous forcing can allow unrealistic states to develop. + +Weight in proportion to PDF width +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +This option is selected by setting **i_pc2_homog_g_method=2** in the UM +large-scale cloud namelist. + +:raw-latex:`\cite{morcrette_2020}` proposed an alternative choice of +weights applied to :math:`G_1(-Q_c)` (`[eqn20] <#eqn20>`__) and +:math:`G_2(-Q_c)` (`[eqn21] <#eqn21>`__), so-as to make each one’s +weight go to zero in the limit that it goes to infinity, reliably +yielding a sensible, finite solution for :math:`G(-Q_c)`. + +We choose the weights to be :math:`\frac{\overline{q_{cl}}}{C_l}` +applied to :math:`G_1(-Q_c)`, and :math:`\frac{SD}{1-C_l}` applied to +:math:`G_2(-Q_c)`, yielding: + +.. math:: + + G(-Q_c) = \frac{(n+1)}{(n+2)} \frac{ + \frac{\overline{q_{cl}}}{C_l} {\frac{C_l^2}{\overline{q_{cl}}}} + + \frac{SD}{1-C_l} {\frac{{(1-C_l)}^2}{SD}} + }{ \frac{\overline{q_{cl}}}{C_l} + \frac{SD}{1-C_l} } + +This is equivalent to weighting the saturated and subsaturated solutions +for :math:`G(-Q_c)` by their respective PDF-widths. Note that everything +cancels-out in the numerator, so this reduces to: + +.. math:: + + G(-Q_c) = \frac{(n+1)}{(n+2)} \frac{ 1 + }{ \frac{\overline{q_{cl}}}{C_l} + \frac{SD}{1-C_l} } + \label{eq:gmqc_width} + +In full UM tests, :raw-latex:`\cite{morcrette_2020}` found that this +closure removed occasional spurious very large cloud-fraction increments +that occur when using `[eqn22] <#eqn22>`__, but did not significantly +impact the performance of the model forecast. + +.. _`sec:homog_num_app`: + +Numerical application +~~~~~~~~~~~~~~~~~~~~~ + +The timestepping methods that are used in the homogeneous forcing were +developed off-line using a single gridbox model, to ensure smooth, +accurate and convergent behaviour of the solution, rather than as a +result of a mathematical analysis of the problem. + +We need to timestep forward (`[dqcldt] <#dqcldt>`__) and +(`[dcdt] <#dcdt>`__) using a timestep of :math:`\Delta{t}`, knowing +values of :math:`\Delta{\overline{T}}`, :math:`\Delta{p}`, +:math:`\Delta{\overline{q}}` and :math:`\Delta{\overline{q_{cl}}}`, +which are provided by an existing physics scheme in the model. We assume +that the physics scheme we are applying this to has not already +calculated its condensation and cloud fraction increments by another +means. + +Firstly, we need to calculate the forcing term :math:`\Delta{Q_c}`. From +(`[eq:qc_eq_qt-qs] <#eq:qc_eq_qt-qs>`__) we can write + +.. math:: + + \Delta{Q_c} = a_L ( \Delta{\overline{q_T}} - \Delta{\overline{q_{sat}(T_L)}} ) + \label{eq:deltaqc} + +assuming that :math:`a_L` does not change (see below). This can be +expanded, using a linear approximation for +:math:`q_{sat}(T+\Delta{T},p+\Delta{p})` in terms of +:math:`q_{sat}(T,p)` as + +.. math:: + + \Delta{Q_c} = a_L ( \Delta{\overline{q}} + \Delta{\overline{q_{cl}}} + - \alpha \Delta{\overline{T_L}} + - \beta \Delta{\overline{p}} ) + \label{eq:deltaqc_exp} + +where :math:`\alpha` is the rate of change of :math:`q_{sat}` with +respect to temperature at constant pressure +(:math:`\frac{\partial q_{sat}}{\partial T}`), and :math:`\beta` is the +rate of change of :math:`q_{sat}` with respect to pressure at constant +temperature (:math:`\frac{\partial q_{sat}}{\partial p}`). Using +(`[eq:tl] <#eq:tl>`__) to expand :math:`T_L` in terms of :math:`T` and +:math:`q_{cl}` and gathering terms together we obtain + +.. math:: + + \Delta{Q_c} = a_L ( \Delta{\overline{q}} - \alpha \Delta{\overline{T}} + - \beta \Delta{\overline{p}} ) + \Delta{\overline{q_{cl}}} . + \label{eq:deltaqc_exp2} + +We must now note the method we use here to calculate :math:`a_L`, +defined in (`[eq:a_L] <#eq:a_L>`__), uses a gradient expansion value of +:math:`\alpha` that corresponds to :math:`\frac{\partial{q_{sat}}} +{\partial{T}}` at constant pressure and not the chord expression in +(`[eq:alpha] <#eq:alpha>`__). This is because we are trying to find the +best estimate of the increments, not the absolute value. We have seen +errors arise in simple numerical tests when the chord expression is +used. We need to define the temperature around which this calculation of +:math:`\frac{\partial{q_{sat}}}{\partial{T}}` is made. Throughout PC2 we +choose the dry-bulb temperature :math:`\overline{T}`, and not the +liquid-temperature (:math:`\overline{T_L}`), since :math:`q_{sat}` +locally is defined by the local dry-bulb temperature (:math:`T`) and we +need to consider *changes* in the condensate. This has been confirmed +using simulations using a single gridbox model. contains a longer +discussion of this issue, but we note here that the +:raw-latex:`\cite{smith90}` scheme performs best when it does not use +:math:`T` to calculate :math:`\alpha` but the gradient of the chord +between :math:`(\overline{T_L}, q_{sat}(\overline{T_L}))` and +:math:`(\overline{T}, q_{sat}(\overline{T}))`, as in +(`[eq:alpha] <#eq:alpha>`__). The best choice is dependent on the method +of implementation. PC2 uses (`[eq:a_L] <#eq:a_L>`__) with the standard +thermodynamic relationships (e.g. :raw-latex:`\cite{ry89}`, chapter 2) + +.. math:: + + \alpha = \frac{ \epsilon L q_{sat}(\overline{T}) } { R \overline{T}^2} , + \label{eq:alpha_exp} + +where :math:`R` is the gas constant for dry air, and + +.. math:: + + \beta = \frac{-q_{sat}(\overline{T})}{\overline{p}} . + \label{eq:beta} + +The right hand side of (`[eq:deltaqc_exp2] <#eq:deltaqc_exp2>`__) now +contains forcing values which we know from the physics scheme we are +applying the homogeneous forcing to. + +We next estimate :math:`G(-Q_c)` from the parametrization +(`[eqn22] <#eqn22>`__) and the expression for the saturation deficit +(`[SD2] <#SD2>`__). The change in :math:`C_l` is then estimated using a +simple forward step of (`[dcdt] <#dcdt>`__) using +(`[eq:deltaqc_exp2] <#eq:deltaqc_exp2>`__): + +.. math:: + + \Delta{C_l} = G(-Q_c) \Delta{Q_c} . + \label{eq:deltac} + +The final value of :math:`C_l` is then limited to lie between 0 and 1. + +.. math:: + + C_l^{[n+1]} = (0, ~ C_l^{[n]} + \Delta{C_l}, ~ 1) + \label{eq:c_l^n+1} + +where :math:`[n]` and :math:`[n+1]` label the timesteps. The +timestepping of :math:`\overline{q_{cl}}` is more involved. We will use +a mid-timestep estimate of :math:`C_l` in the discrete form of +(`[dqcldt] <#dqcldt>`__). + +.. math:: + + \overline{q_{cl}}^{[n+1]} = \overline{q_{cl}}^{[n]} + \frac{1}{2} + (C_l^{[n]} + C_l^{[n+1]}) \Delta{Q_c}. + \label{eq:qcl_l^n+1} + +This completes the homogeneous forcing routine. We note that it is +possible for :math:`\overline{q_{cl}}^{[n+1]}` to be negative if the +forcing is strong enough. Originally, it was not deemed desirable to +prevent homogeneous forcing processes from doing this, in order not to +interfere with the possible cancellation of positive and negative +increments from different physics schemes. A checking routine is +applied, however, in the Unified Model to remove any negative values +that are generated, which is discussed in section +`4.10 <#sec:checks>`__. However, the checking routine (Q-Pos) involves a +lot of communication between processors and can significantly increase +the run-time of the model. The option to “Ensure consistent sinks of qcl +and CFL” performs a check at the end of the homogeneous forcing routines +to ensure that we are not trying to remove more condensate than was +there to start with. + +Note that (`[eq:c_l^n+1] <#eq:c_l^n+1>`__) and +(`[eq:qcl_l^n+1] <#eq:qcl_l^n+1>`__) *include* the contribution of the +forcing itself, it is not just the reactionary condensation. If we wish +to isolate the condensation associated with the forcing, then we must +subtract any liquid forcing from the final solution. The net change in +:math:`\overline{q}` as a result of the homogeneous forcing is simply +the net change in :math:`\overline{q_T}` minus the net change in +:math:`\overline{q_{cl}}`. The net change in :math:`\overline{T}` is +calculated simply to account for the latent heat released due to the +condensation. + +.. _`sec:width`: + +Changing the width of the PDF - PC2 erosion +------------------------------------------- + +Another basic change to the PDF that can be mathematically analysed is +if the PDF shape is kept constant but its width (and therefore height) +is altered. This is, perhaps, the simplest method of representing a +process that changes the shape of the PDF, and we will, in PC2, apply it +to represent mixing of air within a gridbox, although this is a +significant approximation of the process. Its application fulfils the +role of the “cloud erosion” term in the :raw-latex:`\cite{t93}` scheme. +By linking the term to the PDF shape we can place this term on a +stronger mathematical footing than the simple reduction term +parametrized by :raw-latex:`\cite{t93}`. Equivalent arguments enabled +:raw-latex:`\cite{ww99}` to retrieve the same result as presented here. + +We can consider a change in the width of the PDF to alter its form +according to + +.. math:: + + G^{[n+1]}(s) = \xi G^{[n]} (\xi s) + \label{eq:g_xi} + +where :math:`G^{[n+1]}(s)` is the distribution after the change in +width, :math:`G^{[n]} (s)` is the distribution before the change in +width and :math:`\xi` is a scaling factor. If :math:`\xi > 1` then the +distribution is narrowed. For the liquid cloud fraction we therefore +have + +.. math:: + + C_l^{[n+1]} = \int_{s=-Q_c}^{\infty} \xi G(\xi s) ds . + \label{eq:c_l_xi} + +If we transform variables to :math:`s' = \xi s` we can rewrite this +integral as + +.. math:: + + C_l^{[n+1]} = \int_{s'=-Q_c \xi}^{\infty} G(s') ds' . + \label{eq:c_l_xi2} + +Hence the expression for :math:`C_l^{[n+1]}` is equivalent to using the +same distribution function :math:`G(s)` as for :math:`C_l^{[n]}` except +that the saturation boundary has been moved from :math:`-Q_c` to +:math:`-Q_c \xi`. The result is the same as applying a homogeneous +forcing (`[eq:deltac] <#eq:deltac>`__) with a modified forcing, + +.. math:: + + \Delta Q_c \equiv \xi Q_c - Q_c , + \label{eq:deltac_modified} + +or the continuous version + +.. math:: + + \frac{\partial Q_c}{\partial t} \equiv + Q_c \frac{\partial}{\partial t}(\xi - 1) . + \label{eq:xi_equiv} + +We can write :math:`\xi` in a slightly more informative way by linking +it to the relative change in width of the PDF +:math:`\frac{1}{b_s} \frac{\partial b_s}{\partial t}`. For a PDF that +changes its width, :math:`\xi` is defined as + +.. math:: + + \xi = \frac{b_s}{b_s + \delta b_s} = \frac{1}{1 + \frac{\delta b_s}{b_s}}. + \label{eq:xi_equiv1} + +For an infintessimal timestep :math:`\delta t` we therefore have + +.. math:: + + \xi = \frac{1}{1 + \frac{1}{b_s} \frac{\partial b_s}{\partial t} \delta t } + \label{eq:xi} + +and hence, by expanding (`[eq:xi] <#eq:xi>`__) to give +:math:`\xi = 1 - \frac{1}{b_s} +\frac{\partial b_s}{\partial t} \delta t` and using the homogeneous +forcing expression (`[dcdt] <#dcdt>`__) with the modified forcing +(`[eq:xi_equiv] <#eq:xi_equiv>`__), we retrieve the continuous form + +.. math:: + + \frac{\partial C_l}{\partial t} = - G(-Q_c) Q_c \frac{1}{b_s} + \frac{\partial b_s}{\partial t} . + \label{eq:dcdt_width} + +A similar analysis can be performed for +:math:`\frac{\partial \overline{q_{cl}}} +{\partial t}` from (`[dqcldt] <#dqcldt>`__) to give + +.. math:: + + \overline{q_{cl}}^{[n+1]} = \frac{1}{\xi} \int_{s'=-Q_c \xi}^{\infty} + (- \xi Q_c + s') G(s') ds' . + \label{eq:qcl_xi2} + +Again, this is equivalent to using the homogeneous forcing with the +modified forcing (`[eq:xi_equiv] <#eq:xi_equiv>`__), but it also +includes a scaling term :math:`\frac{1}{\xi}`. In the infinitessimal +limit, this scaling gives a second term that is proportional to the +value of the integral (i.e. :math:`\overline{q_{cl}}`). Hence we obtain +the final continuous solution + +.. math:: + + \frac{\partial \overline{q_{cl}}}{\partial t} = + (- C_l Q_c+\overline{q_{cl}}) \frac{1}{b_s} \frac{\partial b_s}{\partial t} . + \label{eq:dqcldt_width} + +To close the solution, we need to parametrize :math:`\frac{1}{b_s} +\frac{\partial b_s}{\partial t}` , which could be linked to the physics +of the process that is occuring. Note we don’t need to calculate +:math:`b_s` separately, just its *fractional* rate of change. Options +for the parameterisation of +:math:`\frac{1}{b_s}\frac{\partial b_s}{\partial t}` due to turbulent +“erosion” are described in section `4.3 <#sec:turb>`__, along with the +numerical methods used to integrate the equations. + +.. _`sec:init`: + +Initiation of cloud +------------------- + +In section `3.2 <#sec:homog>`__ we commented that the closure +(`[eqn22] <#eqn22>`__) for :math:`G(-Qc)` is only valid if :math:`C_l` +is not identically 0 or 1. If :math:`C_l` is 0 or 1 we know that +:math:`G(-Q_c)` is equal to 0 but we have lost the information that will +tell us when :math:`G(-Q_c + \Delta Q_c)` starts differing from 0. Hence +the homogeneous forcing equation set (`[dcdt] <#dcdt>`__), +(`[dqcldt] <#dqcldt>`__) and (`[eqn22] <#eqn22>`__) is not complete if +we start from a position where :math:`C_l` is 0 or 1. To complete this +set, we will need to define a width, :math:`b_s`, to the PDF and provide +an initiation increment to :math:`C_l` and :math:`\overline{q_{cl}}` +when the value of :math:`-Q_c` crosses the limit of the distribution. +There is more discussion in :raw-latex:`\cite{wg03}`. + +To initiate new partial cloud-cover (or new partial clear-sky), we +essentially call a diagnostic cloud scheme to initialise the prognostics +:math:`C_l` and :math:`\overline{q_{cl}}`. In the UM there is currently +a choice of 2 different diagnostic cloud schemes that can be used for +this; either a version of the Smith scheme (see UMDP 029), or the +bimodal scheme (see UMDP 039). These two options are described below... + +Initiation using a “Smith-like” method +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +This option is selected by setting the UM namelist switch +**i_pc2_init_method = 1** (Smith). + +We will assume the same form of the PDF at its boundaries as is assumed +in the derivation of the :math:`G(-Q_c)` closure. For the high +‘:math:`s`’ end of the PDF distribution we integrate the power law +description in (`[eqn19] <#eqn19>`__) to obtain the expressions + +.. math:: + + C_l = \frac{1}{2 b_s^{n+1}} (b_s + Q_c)^{n+1} , + \label{eq:initc} + +.. math:: + + \overline{q_{cl}} = \frac{1}{2 b_s^{n+1}} \frac{(b_s + Q_c)^{n+2}}{n+2} . + \label{eq:initqcl} + +We now need to parametrize the PDF width :math:`b_s`. Unlike the +:raw-latex:`\cite{smith90}` scheme, this is the only location in the PC2 +cloud scheme where the width needs to be defined for the liquid cloud +(although see section `4.2.4 <#sec:mp_depsub>`__ for a discussion of an +equivalent width in the deposition / sublimation relationship for ice +cloud). We still choose to define :math:`b_s` in terms of a critical +relative humidity parameter, :math:`RH_{crit}`. Like the +:raw-latex:`\cite{smith90}` scheme (see ), we define the value of +:math:`b_s` as + +.. math:: + + b_s = a_L (1 - RH_{crit}) q_{sat} (\overline{T_L}) . + \label{eq:bs} + +Hence, if the parameter :math:`n` was the same in PC2 as the equivalent +in :raw-latex:`\cite{smith90}`, the initial creation of liquid cloud +would follow precisely that diagnosed by the :raw-latex:`\cite{smith90}` +scheme (assuming that the numerical implementation of the calculation is +the same). Its subsequent behaviour in PC2, though, would be different, +because the subsequent physical processes that act are parametrized in +different ways. Note: for some reason, the implementation in the UM uses +a fixed value of :math:`n = 0` (corresponding to a top-hat distribution) +if a constant :math:`RH_{crit}` profile is used, but instead sets +:math:`n = 1` (a triangular distribution) in the PC2 initiation +calculation if a TKE-based variable :math:`RH_{crit}` is used. In the +latter case, :math:`n = 0` is still hardwired in the PC2 homogeneous +forcing calculations, so it is not handled consistently. + +An equivalent initiation scheme is required if :math:`C_l` is 1 and +:math:`Q_c` is being reduced - at some point we need to introduce clear +sky into the solution. Because we make the choice of symmetry (which +could be relaxed if we used different :math:`RH_{crit}` values for +:math:`C_l` of 1 and :math:`C_l` of 0), the problem is entirely +equivalent to that of initiating from :math:`C_l = 0`, with the +exception that :math:`\overline{q_{cl}}` is replaced by :math:`SD`, +:math:`C_l` is replaced by :math:`(1-C_l)`, and :math:`Q_c` is replaced +by :math:`-Q_c`. We hence have the solution + +.. math:: + + 1 - C_l = \frac{1}{2 b_s^{n+1}} (b_s - Q_c)^{n+1} , + \label{eq:init1mc} + +.. math:: + + SD = \frac{1}{2 b_s^{n+1}} \frac{(b_s - Q_c)^{n+2}}{n+2} . + \label{eq:initSD} + +The conversion between :math:`SD` and :math:`\overline{q_{cl}}` follows +(`[SD2] <#SD2>`__). We will choose, as we do throughout PC2, to define +:math:`\alpha` (and hence :math:`a_L`) in terms of +:math:`\frac{\partial q_{sat}(\overline{T})}{\partial t}`, although +within this diagnostic calculation of SD it might actually be better to +use the representation (`[eq:alpha] <#eq:alpha>`__) used by the +diagnostic :raw-latex:`\cite{smith90}` scheme. + +.. _`sec:numapp_init`: + +Numerical Application of the Smith method +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +In order to calculate and compare the state of the model to :math:`b_s`, +we first calculate :math:`T_L`, :math:`q_{sat}(\overline{T_L})` and +calculate the mean relative total humidity, :math:`RH_T`, where + +.. math:: + + RH_T = \frac{ \overline{q} + \overline{q_{cl}} } {q_{sat}(\overline{T_L}) } . + \label{eq:rht} + +We then assess whether initiation is required. There are only two +circumstances in which we wish to proceed further: + +- If the current cloud fraction :math:`C_l` is 0 and :math:`-Q_c < b_s`. + By dividing the second condition by + :math:`a_L q_{sat} (\overline{T_L})` we see, using the definitions + (`[eq:qc_eq_qt-qs] <#eq:qc_eq_qt-qs>`__) and (`[eq:bs] <#eq:bs>`__), + that this second condition is equivalent to :math:`RH_T > RH_{crit}`. + +- If the current cloud fraction :math:`C_l` is 1 and :math:`-Q_c > -b_s` + (or, equivalently, :math:`RH_T < 2 - RH_{crit})`. + +Note: in the UM implementation, the actual conditions for when +initiation may occur are more complicated than this, and there are +several options depending on a namelist switch. See section +`4.9 <#sec:init2>`__ for details... + +In the second case, we then make the temporary transformation of +variables in order to use the same solution set as in the first case: +:math:`C_l'` takes the value :math:`(1-C_l)` and :math:`RH_t'` takes the +value :math:`(2-RH_t)` (which is equivalent to the replacing of +:math:`Q_c` by :math:`-Q_c`). In the first case, :math:`C_l'` and +:math:`RH_t` take the same values as :math:`C_l` and :math:`RH_t` +respectively. + +We then solve for the initiated cloud fraction :math:`C_l'`, using the +similar methods as described in , except that we allow the solution to +vary with the PDF shape :math:`n`. We first write :math:`Q_N` as + +.. math:: + + Q_N = \frac{Q_c}{b_s} = \frac{ a_L (\overline{q_T} - q_{sat}(\overline{T_L})) } + { a_L (1 - RH_{crit}) q_{sat} (\overline{T_L})} = \frac{RH_T - 1}{1-RH_{crit}} + \label{eq:qn_def} + +and then use :math:`Q_N` to solve the initiated cloud fraction. We +assume a PDF described by a power law as in (`[eqn19] <#eqn19>`__) (and +the equivalent for the other end of the distribution, the two +expressions switching at :math:`Q_c=0`), which is normalized. The +solution to (`[eq:int_gs_ds] <#eq:int_gs_ds>`__) is hence + +.. math:: + + C_l^{init'} = \left\{ \begin{array}{ll} + 0, & Q_N \le -1 \\ + \frac{1}{2} {\left( 1 + Q_N \right)}^{n+1}, & -1 < Q_N \le 0 \\ + 1 - \frac{1}{2} {\left( 1 - Q_N \right)}^{n+1}, & 0 < Q_N < 1 \\ + 1, & 1 \le Q_N . + \end{array} \right. + \label{eq:c_qn} + +where :math:`C_l^{init'}` is the initiated value of liquid cloud +fraction. If we had performed the variable transformation we then we +need to transform back, so :math:`C_l^{init} = 1 - C_l^{init'}`, +otherwise :math:`C_l^{init} = C_l^{init'}`. + +In practice, it is likely to be only the second of the options in +(`[eq:c_qn] <#eq:c_qn>`__) that the scheme uses, since we will be at +that end of the distribution function, unless previous parts of the +model timestep have resulted in large forcings to :math:`Q_c`. + +The solution for the initiated liquid water, +:math:`\overline{q_{cl}}^{init}` is more difficult, since it depends on +the width of the distribution :math:`b_s`, hence on :math:`a_L` and +:math:`\alpha`, and :math:`\alpha` is a function of the dry-bulb +temperature :math:`\overline{T}`, which is not known until we know the +amount of condensation. Hence we will need to iterate to a solution. + +We first calculate :math:`q_{sat}(\overline{T})`, :math:`\alpha`, +:math:`a_L` and :math:`b_s`, using (`[eq:alpha_exp] <#eq:alpha_exp>`__), +(`[eq:a_L] <#eq:a_L>`__) and (`[eq:bs] <#eq:bs>`__). We then solve for +the liquid water content: + +.. math:: + + \frac{\overline{q_{cl}}^{init'}}{b_s} = \left\{ \begin{array}{ll} + 0, & Q_N \le -1 \\ + \frac{1}{2 (n+2)} {\left( 1 + Q_N \right)}^{n+2}, & -1 < Q_N \le 0 \\ + Q_N + \frac{1}{2 (n+2)} {\left( 1 - Q_N \right)}^{n+2}, & 0 < Q_N < 1 \\ + Q_N, & 1 \le Q_N . + \end{array} \right. + \label{eq:l_bar} + +If we have been working in transformed variables we now transform back, +so the initiated saturation deficit, :math:`SD^{init}`, takes the value +of :math:`\overline{q_{cl}}^{init'}`. We then use (`[SD2] <#SD2>`__) to +estimate :math:`\overline{q_{cl}}^{init}` using our initial estimates of +:math:`q_{sat}(\overline{T})` and :math:`a_L`. If we are not in +transformed variables, we have the first estimate +:math:`\overline{q_{cl}}^{init}=\overline{q_{cl}}^{init'}`. + +We now use this estimate of :math:`\overline{q_{cl}}^{init}` to +calculate a more accurate estimate of :math:`a_L` etc. by iteration. In +order to achieve a faster convergence of the iteration, we do not use +(`[eq:l_bar] <#eq:l_bar>`__) directly in the estimation of :math:`a_L` +etc., but use a combination of this value and the one from the previous +iteration. + +.. math:: + + \overline{q_{cl}}^{init~[i+1]} = f \overline{q_{cl}}^{init~[i]} + + (1 - f) \overline{q_{cl}}^{init~[i-1]} + \label{eq:iter} + +where the superscript :math:`[i]` labels each iteration. We find that 10 +iterations is effective for convergence, with the weighting :math:`f` +given by :math:`a_L^{[i]}`. + +.. _`sec:bimodal_init`: + +Initiation using the bimodal scheme +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +This option is selected by setting the UM namelist switch +**i_pc2_init_method = 2** (Bimodal). + +First, the diagnosis of entrainment zones is performed at all +grid-points, as described in UMDP 039. The parameters of the moisture +PDF are then constructed, assuming either a sum of two Gaussian modes +from the top and bottom of an inversion layer (when within an +entrainment zone), or a single symmetric Gaussian mode (when not in an +entrainment zone). The variance of each Gaussian mode is estimated based +on the TKE and other information output by the boundary-layer scheme +(with a minimum limit applied to the PDF width, consistent with +:math:`RH_{crit}` = 99%. Crucially, each Gaussian mode is truncated to +zero at plus and minus 3 standard deviations; this sets the overall +width of the moisture PDF at each point. + +The positions of the upper and lower truncated bounds of the moisture +PDF relative to the saturation threshold are expressed in terms of a +normalised :math:`Q_N` = :math:`Q_c` over PDF-width (see equation +`[eq:qn_def] <#eq:qn_def>`__). In entrainment zones, the sum of the two +Gaussian modes can lead to a highly skewed distribution; hence +:math:`Q_N` can have different values for the upper and lower bounds, +each normalised by the different widths on either side of the PDF. The +upper and lower values of :math:`Q_N` are then compared to -1 and 1 +respectively, to determine whether the saturation boundary lies within +the PDF bounds. This is the basic condition for initiation to occur +(though there are additional conditions and various options for these in +the soure code; see section `4.9 <#sec:init2>`__). + +If the initiation conditions are met, the diagnostic bimodal cloud +scheme code is then called (see UMDP 039), and the diagnosed :math:`C_l` +and :math:`q_{cl}` are used to set the prognostic :math:`C_l` and +:math:`q_{cl}`. + +.. _`sec:inhomog`: + +Injection forcing +----------------- + +Injection forcing (sometimes referred to as inhomogeneous forcing) uses +another concept of how the underlying moisture PDF may change in order +to calculate a change in cloud fraction as a result of a known injection +of condensate into a gridbox. The term was developed in order to be +coupled with a modified mass-flux convection scheme, but is first +presented here in its basic form. + +We will assume a physical model whereby saturated air, containing +condensate, randomly replaces already existing air in the gridbox. (Such +a formulation is designed to represent air detrained from convection +replacing pre-existing air when averaged over a large horizontal +domain). :raw-latex:`\cite{bwg03}` discusses the situation in more +detail. Briefly, we consider two parts to the distribution function +:math:`G(s)`. One part represents the background air. This maintains its +PDF shape (in terms of absolute :math:`q_T` and :math:`T_L` values) +because we assume it is *randomly* replaced, but will reduce in +amplitude as it is replaced by a second PDF representing the injected +air. + +The fractional rate at which existing air is replaced by the injected +source air we will write as :math:`\frac{\partial{C_S}}{\partial{t}}`. +Provided that only the liquid phase exists (see section +`3.5.1 <#sec:multiple>`__ for the extention to multiple phases), we then +note that the rate of change of liquid cloud fraction and liquid water +content in the gridbox can be written in two parts: firstly the change +due to the background, and secondly the change due to the source. + +.. math:: + + \frac{\partial{C_l}}{\partial{t}} = + - C_l \frac{\partial{C_S}}{\partial{t}} + \frac{\partial{C_S}}{\partial{t}} + \label{eq:dcdt_inhom} + +.. math:: + + \frac{\partial{\overline{q_{cl}}}}{\partial{t}} = + - \overline{q_{cl}} \frac{\partial{C_S}}{\partial{t}} + + q_{cl}^S \frac{\partial{C_S}}{\partial{t}} + \label{eq:dqcldt_inhom} + +where :math:`q_{cl}^S` is the liquid water content of the injected air. +Eliminating :math:`\frac{\partial{C_S}}{\partial{t}}` gives the +relationship + +.. math:: + + \frac{\partial{C_l}}{\partial{t}} = \frac{1 - C_l}{q_{cl}^S + - \overline{q_{cl}}} Q4_l, + \label{eq:dcdt_inhom2} + +where :math:`Q4_l` is the net (*including* the liquid water in the +background distribution that was randomally replaced) injection source +change of :math:`\overline{q_{cl}}`: + +.. math:: + + Q4_l = \frac{\partial{\overline{q_{cl}}}}{\partial{t}} |_{injection \, source}. + \label{eq:q4} + +We see that we do not need to know anything about the nature of the two +PDFs involved, except the assumption that the injected PDF contains +completely cloudy air. This equation allows one to calculate the change +in :math:`C_l` associated with an injection source change of +:math:`\overline{q_{cl}}` for the example of convection. Modifications +to the mass-flux convection scheme for PC2 (far from trivial and +discussed in depth in section `4.7 <#sec:convec>`__) allow :math:`Q4_l` +to be calculated (:math:`q_{cl}^S` is already available), and +(`[eq:dcdt_inhom2] <#eq:dcdt_inhom2>`__) can then be used to calculate +the equivalent :math:`C_l` change. We note at this stage that the +denominator in (`[eq:dcdt_inhom2] <#eq:dcdt_inhom2>`__), being the +difference in two terms that may be close to each other, may cause +problems when we attempt to numerically apply this equation. + +It is reasonable to ask what happens to the air in the distribution that +was replaced. In this mathematical representation of a single gridbox we +need not know anything other than that the air is displaced into a +neighbouring gridbox. In practical use with a mass-flux convection +scheme we know more that this air is displaced downwards in the column. +We could reasonably calculate the change in cloud fraction following the +same methods as used to calculate the change in :math:`\overline{q}` or +the change in a tracer and we discuss this later. + +.. _`sec:multiple`: + +Multiple phases in the injection source +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +The injection source formulation can be extended to multiple phases of +condensate. In practice, this will simply be the two phases ice and +liquid, although we need to recognize that they can overlap with each +other. :raw-latex:`\cite{wilson2001}` provides the background to the +derivation and it is briefly presented below. + +We firstly rewrite (`[eq:dqcldt_inhom] <#eq:dqcldt_inhom>`__) but use +the net condensate +(:math:`\overline{q_c} = \overline{q_{cl}} + \overline{q_{cf}}`) instead +of just the liquid water expression, and the net cloud amount +:math:`C_t`, instead of the liquid cloud amount :math:`C_l`. The same +argument as before leads to the expressions + +.. math:: + + \frac{\partial{C_t}}{\partial{t}} = + - C_t \frac{\partial{C_S}}{\partial{t}} + \frac{\partial{C_S}}{\partial{t}} + \label{eq:dctdt_inhom} + +and + +.. math:: + + \frac{\partial{\overline{q_{c}}}}{\partial{t}} = + - \overline{q_{c}} \frac{\partial{C_S}}{\partial{t}} + + q_{c}^S \frac{\partial{C_S}}{\partial{t}} . + \label{eq:dqcdt_inhom} + +The left hand side of (`[eq:dqcdt_inhom] <#eq:dqcdt_inhom>`__) is +written as :math:`Q4_c`. :math:`q_{c}^S` is the in-cloud condensate +content (ice plus liquid) of the source. + +Hence eliminating :math:`\frac{\partial{C_S}}{\partial{t}}` we obtain + +.. math:: + + \frac{\partial{C_t}}{\partial{t}} = \frac{(1-C_t)}{q_{c}^S - + \overline{q_{c}}} Q4_c . + \label{eq:dctdt_q4} + +We will assume that the proportion of the injected volume that contains +liquid cloud can be written as :math:`g_l`, and the proportion that +contains ice cloud can be written as :math:`g_i`. Note that it is not +necessary to have :math:`g_l + g_i = 1` if there is mixed phase cloud +injected. We can write the change in *liquid* cloud fraction +equivalently to (`[eq:dcdt_inhom] <#eq:dcdt_inhom>`__) as + +.. math:: + + \frac{\partial{C_l}}{\partial{t}} = + - C_l \frac{\partial{C_S}}{\partial{t}} + g_l \frac{\partial{C_S}}{\partial{t}} . + \label{eq:dcldt_inhom} + +Combining (`[eq:dcldt_inhom] <#eq:dcldt_inhom>`__) and +(`[eq:dctdt_inhom] <#eq:dctdt_inhom>`__) by eliminating +:math:`\frac{\partial{C_S}}{\partial{t}}` gives + +.. math:: + + \frac{\partial{C_l}}{\partial{t}} = \frac{g_l - C_l}{1 - C_t} + \frac{\partial{C_t}}{\partial{t}} + \label{eq:dctdt_dcdt} + +and hence from (`[eq:dctdt_q4] <#eq:dctdt_q4>`__) we have the result + +.. math:: + + \frac{\partial{C_l}}{\partial{t}} = + \frac{g_l - C_l}{q_c^S - \overline{q_{c}}} Q4_c . + \label{eq:dcltdt_almost_final} + +An equivalent expression holds for the ice cloud. Hence the change in +the amount of cloud for each phase may be calculated assuming we know +the volume proportions of the source term that contain each of the +phases and the net increase in the amount of condensate, :math:`Q4_c` +(regardless of phase). This expression is coded for use in a generically +available inhomogeneous forcing module. However, we can also write this +in a slightly more accessible form by noting the ratio of +(`[eq:dqcdt_inhom] <#eq:dqcdt_inhom>`__) and +(`[eq:dqcldt_inhom] <#eq:dqcldt_inhom>`__) with the :math:`Q4` +definitions following (`[eq:q4] <#eq:q4>`__). + +.. math:: + + \frac{Q4_c}{q_c^S - \overline{q_c}} = + \frac{Q4_l}{q_{cl}^S - \overline{q_{cl}}} . + \label{eq:q4_ratios} + +Using (`[eq:q4_ratios] <#eq:q4_ratios>`__) in +(`[eq:dcltdt_almost_final] <#eq:dcltdt_almost_final>`__) gives the final +expression + +.. math:: + + \frac{\partial{C_l}}{\partial{t}} = + \frac{g_l - C_l}{q_{cl}^S - \overline{q_{cl}}} Q4_l + \label{eq:dctdt_final} + +and similarly for the ice. Note that this expression accounts for the +possibility that liquid cloud is displaced from the gridbox by added ice +cloud. We can further write :math:`q_{cl}^S` as a fraction of +:math:`q_{c}^S` + +.. math:: + + q_{cl}^S = h_l q_{c}^S + \label{eq:qcls_qcs} + +where :math:`h_l` is the factor between them (i.e. the *mass* fraction +of the injected condensate that is liquid). It is not necessary in this +theory to have :math:`h_l` equal to :math:`g_l`: if a mixed phase plume +exists :math:`g_l + g_i` need not equal 1, but since :math:`h_l` and its +ice equivalent, :math:`h_i`, refer to mass, :math:`h_l + h_i` must equal +1. However, if we do not allow a mixed phase injection (which is the +case in the current mass-flux convection scheme, where only one phase +can be injected), :math:`h_l` and :math:`g_l` are equal (and either zero +or one in the current mass-flux convection scheme) and we can write +(`[eq:dctdt_final] <#eq:dctdt_final>`__) as + +.. math:: + + \frac{\partial{C_l}}{\partial{t}} = + \frac{ (\delta_{xl} - C_l) }{ \delta_{xl} q_{c}^S - \overline{q_{cl}} } + Q4_l + \label{eq:dctdt_xl} + +where :math:`\delta_{xl} = h_l = g_l`. Equivalent expressions exist for +the ice cloud fraction and total cloud fraction. + +.. math:: + + \frac{\partial{C_i}}{\partial{t}} = + \frac{ (\delta_{xi} - C_l) }{ \delta_{xi} q_{c}^S - \overline{q_{cf}} } + Q4_i + \label{eq:dctdt_xi} + +.. math:: + + \frac{\partial{C_t}}{\partial{t}} = + \frac{ (1 - C_t) }{ q_{c}^S - \overline{q_{c}} } Q4_c + \label{eq:dctdt_xc} + +with :math:`\delta_{xi} = h_i = g_i`. These are the expressions that are +used within the convection scheme. It still remains to parametrize +:math:`\delta_{xl}`, which is given by the convection scheme itself. +This is discussed in section `4.7.9 <#sec:plume_phase>`__. + +.. _`sec:multi_numapp`: + +Numerical application +~~~~~~~~~~~~~~~~~~~~~ + +The numerical application using +(`[eq:dcltdt_almost_final] <#eq:dcltdt_almost_final>`__) may be +performed with a basic forward timestep. Each of the three cloud +fractions can be incremented, assuming we know +:math:`\Delta{\overline{q_{cl}}}` and :math:`\Delta{\overline{q_{cf}}}`, +as + +.. math:: + + \Delta{C_t} = \frac{(1 - C_t)} {q_c^S - \overline{q_{cl}} - \overline{q_{cf}}} + ( \Delta{\overline{q_{cl}}} + \Delta{\overline{q_{cf}}} ), + \label{eq:cft_ts} + +.. math:: + + \Delta{C_l} = \frac{ (g_l - C_l)} + {q_c^S - \overline{q_{cl}} - \overline{q_{cf}}} + ( \Delta{\overline{q_{cl}}} + \Delta{\overline{q_{cf}}} ), + \label{eq:cfl_ts} + +.. math:: + + \Delta{C_i} = \frac{ (g_i - C_i)} + {q_c^S - \overline{q_{cl}} - \overline{q_{cf}}} + ( \Delta{\overline{q_{cl}}} + \Delta{\overline{q_{cf}}} ). + \label{eq:cff_ts} + +The application from within the convection scheme is slightly different. +We start with (`[eq:dctdt_xl] <#eq:dctdt_xl>`__), but enforce two +numerical restrictions to avoid the equation set becoming +ill-conditioned. Firstly, we limit the denominator +:math:`q_c^S - \overline{q_{cl}}` to a minimum value if we are +considering changes of the same phase as the injected source. + +.. math:: + + \Delta C_l = \frac {\delta_{xl} - C_l} {\delta_{xl} \text{Max}( q_{c}^S + - \overline{q_{cl}} , q_c^{S0} ) + ( 1 - \delta_{xl} ) (-\overline{q_{cl}}) } + Q4_l + \label{eq:delta_cl} + +where :math:`q_c^{S0}` is specified as +:math:`5 \times 10^{-5} kg \, kg^{-1}`. The denominator also has an +additional check. If its absolute value is less than a tolerance value +of :math:`1 \times 10^{-10} kg \, kg^{-1}` then no change in cloud +fraction will be considered. A similar equation is used for the ice +cloud and the change in total cloud fraction + +.. math:: + + \Delta C_i = \frac {\delta_{xi} - C_i} {\delta_{xi} \text{Max}( q_{c}^S + - \overline{q_{ci}} , q_c^{S0} ) + ( 1 - \delta_{xi} ) (-\overline{q_{cf}}) } + Q4_i + \label{eq:delta_ci} + +.. math:: + + \Delta C_t = \frac {1 - C_t} + { \text{Max}(q_c^S - \overline{q_c} , q_c^{S0} ) } Q4_c . + \label{eq:delta_ct} + +We now limit the change in cloud fraction to ensure that the cloud +fraction remains within its physical bounds. + +.. math:: + + C_l^{[n+1]} = ( 0, C_l^{[n]} + \Delta C_l, 1) + \label{eq:delta_cl_conv_final} + +and similar equations are used for :math:`C_i^{[n+1]}` and +:math:`C_t^{[n+1]}`. + +.. _`sec:conv_imp_note`: + +A note on the implementation of the cloud fraction change +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +Equation `[eq:dcdt_inhom2] <#eq:dcdt_inhom2>`__ has been derived +assuming that the only change in the cloud properties within the gridbox +comes from the detrainment of air from the convective plume (so that the +injection source is an appropriate model). Attention should be drawn to +the fact that this is not the only source of change from the convection +scheme. Two other terms require consideration, namely advection of the +environmental air downwards by compensating subsidence and the +condensation resulting from the adiabatic warming due to this +subsidence. The former is considered correctly in the calculation of +:math:`\frac{\partial \overline{q_{cl}}}{\partial t}`, which corresponds +to :math:`Q4`. However, the calculation of +:math:`\frac{\partial C_l}{\partial t}` is then performed using +(`[eq:dcdt_inhom2] <#eq:dcdt_inhom2>`__) and **incorrectly** assuming +that all the :math:`\overline{q_{cl}}` change comes from the +detrainment. It is possible to calculate directly the change in +:math:`C_l` that should occur due to the detrainment and compensating +subsidence treated together, in the same way that +:math:`\Delta \overline{q_{cl}}` is calculated (see section +`4.7.3 <#subsect:q4calculation>`__), and this is the way in which the +cloud fraction change **should** be done. It is an unfortunate +historical emphasis in the early development of PC2 on the derivation of +(`[eq:dcdt_inhom2] <#eq:dcdt_inhom2>`__) that has led to the treatment +used within the Unified Model for the change in cloud fractions due to +convection. + +The change in :math:`\overline{q_{cl}}` and :math:`C_l` due to the +adiabatic warming associated with the compensating subsidence is +considered explicitly in the model implementation (see section +`4.7.4 <#sec:conv_homog>`__) for both :math:`\overline{q_{cl}}` and +:math:`C_l` after the rest of the convective process has been +calculated. It is perhaps arguable that if +(`[eq:dcdt_inhom2] <#eq:dcdt_inhom2>`__) is going to be applied then the +value of :math:`Q4` used in (`[eq:dcdt_inhom2] <#eq:dcdt_inhom2>`__) +should include this term. + +Any major future developments of PC2 for a mass-flux convection scheme +would be advised to consider whether it is appropriate to use +(`[eq:dcdt_inhom2] <#eq:dcdt_inhom2>`__) at all. + +.. _`sec:ct`: + +Ice cloud and mixed phase regions +--------------------------------- + +The homogeneous forcing, initiation and PC2 erosion sections described +above have only considered the generation and dissipation of liquid +clouds. Although the forcing methods will not influence the generation +and dissipation of ice cloud (which is primarily performed in the +large-scale precipitation scheme, section `4.2 <#sec:precip>`__) we are +still left with the issue of how created or dissipated liquid cloud +overlaps with existing ice cloud in the gridbox. The opposite situation, +where changes in ice cloud are specified and changes in the overlap with +liquid cloud need to be calculated, is also possible in PC2 (e.g. in the +boundary layer, see section `4.6 <#sec:bl>`__). + +Here we need a simple assumption to close the problem. The assumption +that we now choose is that liquid cloud fraction *changes* are +*minimally* overlapped with ice cloud fraction changes. This choice is +based upon observational evidence that mixed phase cloud is relatively +rare, and also on results from earlier PC2 development that indicated +less supercooled liquid water cloud than is observed from ground-based +lidar. + +With this assumption, the equation set becomes straightforward to write +down. We firstly consider that a change in liquid cloud fraction +:math:`\Delta C_l` is known and we wish to estimate the resulting change +in the total cloud fraction. There is, of course, no change in the ice +cloud fraction :math:`C_i`, since, from our *definitions* in +(`[eq:dqcldt_and_dcdt] <#eq:dqcldt_and_dcdt>`__) and +(`[eq:mp] <#eq:mp>`__), this includes the mixed phase contribution. +Hence we write + +.. math:: + + \Delta C_i = 0 . + \label{eq:deltaci_eq_0} + +The change in the total cloud fraction, :math:`C_t` will depend upon the +sign of the change of the liquid cloud fraction. If +:math:`\Delta C_l > 0`, then :math:`\Delta C_t` is going to be the same +as :math:`\Delta C_l` (:math:`C_l` is being added with minimum overlap +to :math:`C_i`), unless the gridbox becomes completely covered in cloud, +when there is no choice but to generate mixed phase cloud. Hence we have + +.. math:: + + \Delta C_t = \text{Min} ( \Delta C_l , 1 - C_t ). + \label{eq:deltact_min} + +If :math:`\Delta C_l < 0`, then we still consider minimum overlap of the +*changes* (this is so that the solution is reversible as much as +possible). Hence :math:`\Delta C_t` is going to be the same as +:math:`\Delta C_l` unless :math:`C_l` is reduced below the existing +:math:`C_i`, in which case no more change to :math:`C_t` is possible. + +.. math:: + + \Delta C_t = \text{Max} ( \Delta C_l , C_i - C_t ) , + \label{eq:deltact_min2} + +remembering that both quantities in the maximum expression in +(`[eq:deltact_min2] <#eq:deltact_min2>`__) have negative values. + +We can write similar expressions if a known amount of ice cloud is added +or removed, and we need to calculate the effect on :math:`C_t`. Similar +to the results above we have: + +.. math:: + + \Delta C_l = 0 . + \label{eq:deltacl_eq_0} + +and + +.. math:: + + \Delta C_t = \left\{ \begin{array}{ll} + \text{Max} ( \Delta C_i , C_l - C_t ), & \Delta C_i < 0 \\ + \text{Min} ( \Delta C_i , 1 - C_t ), & \Delta C_i > 0 . + \end{array} \right. + \label{eq:deltact_min_array} + +For completeness, we also present here the equation set for random +overlap of changes in liquid cloud with existing ice cloud. We have, as +before, + +.. math:: + + \Delta C_i = 0 . + \label{eq:deltaci_eq_0_2} + +For :math:`\Delta C_l > 0` additional liquid cloud is added randomly to +any location outside that of the current liquid cloud. A proportion +:math:`\frac{1-C_t}{1-C_l}` of this will be additionally outside that of +existing ice cloud. Hence the net change in total cloud fraction can be +written as + +.. math:: + + \Delta C_t = \Delta C_l \frac{1 - C_t}{1 - C_l} . + \label{eq:deltact_ran1} + +Similarly, if :math:`\Delta C_l < 0`, the liquid cloud is removed +randomly from the existing liquid cloud. A proportion +:math:`\frac{C_t - C_i}{C_l}` of this is from liquid cloud that does not +overlap with existing ice cloud. Hence, + +.. math:: + + \Delta C_t = \Delta C_l \frac{C_t - C_i}{C_l} . + \label{eq:deltact_ran2} + +Equivalent equations to (`[eq:deltact_ran1] <#eq:deltact_ran1>`__) and +(`[eq:deltact_ran2] <#eq:deltact_ran2>`__) but with :math:`C_l` and +:math:`C_i` swapped apply when we need to estimate changes in +:math:`C_t` from a known :math:`\Delta C_i`, when assuming random +overlap. + +Numerical Implementation +~~~~~~~~~~~~~~~~~~~~~~~~ + +In general, although the situation does not occur within the current +implementation of PC2 , we might have increments to both :math:`C_l` and +:math:`C_i` simultaneously. Hence the implementation is to calculate +:math:`\Delta C_t` from the sum of that predicted by +(`[eq:deltact_min] <#eq:deltact_min>`__) or +(`[eq:deltact_min2] <#eq:deltact_min2>`__), and +(`[eq:deltact_min_array] <#eq:deltact_min_array>`__). For the random +overlap situation we also need to apply a check on the denominator in +(`[eq:deltact_ran1] <#eq:deltact_ran1>`__) and +(`[eq:deltact_ran2] <#eq:deltact_ran2>`__) before calculation, with the +result set to the limit :math:`\Delta C_t = 0` if the denominator is 0. +For the minimum overlap situation a final check is made that :math:`C_t` +lies between 0 and 1, with the value being reset to 0 or 1 if not. + +Forced convective cloud +----------------------- + +Forced convective clouds are clouds that form at the top of a convective +boundary layer but are too shallow to reach their level of free +convection (and become fully fledged cumulus clouds). These clouds +currently require special treatment because initiation in PC2 uses the +Smith scheme with a specified value of :math:`RH_{crit}` while the large +:math:`RH` variability associated with these clouds implies much lower +values than are typically used. + +A profile of “forced cloud fraction”, :math:`C_{forced}`, is +parametrized as linearly varying with height between a cloud-base value, +at the lifting condensation level (LCL) from the convection diagnosis +parcel ascent, and a cloud-top value of 0.1 at the top of the capping +inversion, :math:`z_i^{top}`. The cloud-base value of :math:`C_{forced}` +varies linearly between 0.1 and 0.3 for cloud depths between 100 m and +300 m based loosely on SGP ARM site observations +:raw-latex:`\cite{zk13}`. The inversion top is taken to be the boundary +layer depth, :math:`z_h` plus the inversion thickness, +:math:`\Delta z_i` parametrized following :raw-latex:`\cite{rb08}` as: + +.. math:: + + \Delta z_i = 6.3 \, w_m^2 / \int_{z_h}^{z_h+\Delta z_i} b \, dz + \label{dz_param} + +where :math:`w_m` is the boundary layer velocity scale +(:math:`w_m^3 = u_*^3 + 0.25 w_*^3`) and :math:`b` is the parcel +buoyancy that is integrated over the depth of the inversion assuming a +piece-wise linear variation between grid-levels. Note that the constant +in (`[dz_param] <#dz_param>`__) is the same as in +:raw-latex:`\cite{rb08}` because :math:`6.3 = 2.5 * 4^{2/3}` and +:math:`w_m^3` differs by a factor of 4. + +The in-cloud water content at the top of the inversion is estimated +using the water content from the diagnostic parcel ascent (used to +diagnose boundary layer type and trigger convection), with linear +interpolation used between the lifting condensation level and inversion +top. To allow for sub-adiabatic water content (due to lateral mixing or +microphysical processes) the in-cloud water content can be reduced by a +factor, forced_cu_fac, that has been set to 0.5 in GA7. + +These cloud fraction and water content profiles are then used as minimum +values and increments to :math:`C` and :math:`\overline{q_{cl}}` +calculated if necessary. This methodology can also optionally be applied +to cloud layers diagnosed as cumulus, if the boundary layer option to +mix across the lifting condensation level is selected that generates a +cloud base transition zone thickness which is then treated analgously to +the inversion thickness above. + +Also, there is an option to treat the calculated forced cumulus cloud +fraction and water content as diagnostic quantities passed directly to +the radiation scheme as part of the “convective” cloud, instead of using +them to modify the prognostic “large-scale” cloud variables :math:`C` +and :math:`\overline{q_{cl}}`. If this option is used, the convective +cloud fraction :math:`CCA` and water content :math:`CCW` output by the +convection scheme are updated, by taking the forced cumulus profiles as +their minimum allowed values. Note that only the convective cloud fields +passed to radiation are updated (i.e. the versions of :math:`CCA` and +:math:`CCW` that are stored in the model dump / D1 array). The UM code +contains other copies of the convective cloud fields that are only used +for diagnostics; these are *not* updated. + +The different options for how to treat forced cumulus cloud are +controlled by the cloud namelist input :math:`forced\_cu`, and are +summarised below: + +- :math:`forced\_cu = 0`: No treatment of forced cumulus clouds. + +- :math:`forced\_cu = 1`: Forced cumulus cloud applied to :math:`C` and + :math:`\overline{q_{cl}}` only in dry-convective boundary-layers. + +- :math:`forced\_cu = 2`: Forced cumulus cloud applied to :math:`C` and + :math:`\overline{q_{cl}}` in both dry-convective and cumulus-capped + boundary-layers. + +- :math:`forced\_cu = 3`: Forced cumulus cloud applied to :math:`CCA` + and :math:`CCW` in both dry-convective and cumulus-capped + boundary-layers. + +.. _`sec:turb_qcl_scheme`: + +Turbulence-driven production of subgrid scale liquid cloud +---------------------------------------------------------- + +.. _`sec:sgt_intro`: + +Introduction +~~~~~~~~~~~~ + +:raw-latex:`\cite{fhfk14}` developed a model for subgrid liquid water +production by turbulent motions. Their method uses an exactly soluble +stochastic process to describe subgrid relative humidity (RH) +fluctuations. The probability density function (PDF) of the fluctuations +can be diagnosed in terms of the local turbulent local state and any +pre-existing ice cloud. The liquid cloud properties (cloud fraction and +liquid water content) can be then be calculated as truncated moments of +the PDF. + +:raw-latex:`\cite{fhfk14}` initially used their model to understand and +parametrize the results of Large Eddy Simulations (LES) of +shear-induced, Altostratus clouds. They obtained excellent agreement +between their theoretically predicted predicted mean cloud properties +and the bulk properties of the LES clouds. Subsequently, their model has +been used as the basis of subgrid cloud initiation method for use in the +Unified Model in conjunction with the PC2 prognostic cloud scheme. In +Section `3.8.2 <#sec:sgt_model_describe>`__ we outline the model of +:raw-latex:`\cite{fhfk14}`. In Section +`3.8.3 <#sec:sgt_model_implement>`__ we described its implementation in +the GCM. + +.. _`sec:sgt_model_describe`: + +Model description +~~~~~~~~~~~~~~~~~ + +:raw-latex:`\cite{fhfk14}` started from the equation for the dynamics of +ice supersaturation :math:`S_i=e_v/e_{sat\;ice}-1`: + +.. math:: + + \label{eqn:squires_eqn} + \frac{D S_i}{D t} = -b_i B_0 {\cal M}_1 S_i + -\left(\frac{\varepsilon}{L^2}\right)^{1/3}(S_i-S_E) + a_i w, + +where :math:`{\cal M}_1` is the first moment of ice particle size +distribution (PSD), :math:`\varepsilon` is the turbulent dissipation +rate, :math:`L` is a prescribed mixing length for the turbulence, +:math:`S_{\rm E}` is the ice supersaturation of the environment +surrounding the cloud and :math:`b_i,B_0` and :math:`a_i` are function +of :math:`p` and :math:`T` given by + +.. math:: + + \begin{aligned} + b_i &=& \frac{1}{q} + \frac{\epsilon L_s^2}{c_p R T^2}, \\ + B_0 &=& 4\pi C \left( \frac{\epsilon L_s^2}{K_a R T^2} + \frac{R T}{\epsilon e_{si} \psi} \right)^{-1}, \\ + a_i &=& \frac{g}{R T}\left( \frac{\epsilon L_s}{c_p T} - 1 \right), \\ + \end{aligned} + +The first term on the right hand side of +Eq. `[eqn:squires_eqn] <#eqn:squires_eqn>`__ is the sink of vapor due to +depositional growth of ice crystals, the second term models entrainment +(mixing) of environmental air into the cloudy volume and the third term +is a source term due to vertical air motions. + +:raw-latex:`\cite{fhfk14}` modeled vertical velocity as a white-noise +process with autocorrelation function: + +.. math:: \overline{w(t)w(s)} = \sigma_w^2 \tau_{\rm d} \delta(t-s), + +where :math:`\delta` is the Dirac distribution and the intensity of the +noise, :math:`\sigma_w^2`, will be called the standard derivation of the +vertical velocity fluctuations (due to the white nature of noise, a true +expectation value :math:`\overline{w^2}` is not defined) and +:math:`\tau_{\rm d}` a Lagrangian decorrelation time define here by the +relation used by :raw-latex:`\cite{rodean1997}`: + +.. math:: + + \tau_{\rm d} = \frac{2\sigma_w^2}{\varepsilon C_0}, + \label{eqn:taud} + +where :math:`C_0` is a known constant. + +Because it is linear in :math:`S_i`, Equation +`[eqn:squires_eqn] <#eqn:squires_eqn>`__ can be solved exactly, for any +given realisation of the noise term. By averaging the solutions over the +the noise and taking a steady-state limit (see +:raw-latex:`\cite{fhfk14}` for details) it can be shown that the +solution PDF is Gaussian with mean and variance given by: + +.. math:: + + \begin{aligned} + \overline{S_i} &=& + S_{\rm E}\frac{ \left(\varepsilon/L^2\right)^{1/3} }{ b_i B_0 {\cal M}_1 + \left(\varepsilon/L^2\right)^{1/3} }. + \label{eqn:si_avg} \\ + \overline{S_i^2} &=& + \frac{a^2_{\rm i} \sigma^2_w \tau_{\rm d}}{ 2\left(b_i B_0 {\cal M}_1 + \left(\varepsilon/L^2\right)^{1/3}\right)}, + \label{eqn:si_var} + \end{aligned} + +Equation `[eqn:si_avg] <#eqn:si_avg>`__ and +`[eqn:si_var] <#eqn:si_var>`__ completely specify the PDF, +:math:`F(S_i)`, of steady-state humidity variations for the subgrid +model. The liquid cloud fraction and liquid water mass mixing ratio are +given by + +.. math:: + + \begin{aligned} + C_l^{sgt} &=& \int_{S_{i,wat}}^\infty d S_i F(S_i), \label{eqn:cloud_fraction} \\ + q_{cl}^{sgt} &=& q_{sat\;ice}\int_{S_{i,wat}}^\infty d S_i (S_i -S_{i,wat}) F(S_i) \label{eqn:cloud_liquid}, + \end{aligned} + +where :math:`S_{i,wat} = e_{sat\;wat}/e_{sat\;ice}-1` is the value of +ice supersaturation at water saturation. We use the superscription +‘:math:`sgt`’(=‘*s*\ ub\ *g*\ rid *t*\ urbulence’) to indicate that +:math:`C_l^{sgt}` and :math:`q_{cl}^{sgt}` are values of cloud fraction +and water content diagnosed from a parametrization of small-scale +turbulent processes. + +.. _`sec:sgt_model_implement`: + +Model implementation and closure relations +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +To implement the model of Section `3.8.2 <#sec:sgt_model_describe>`__ in +the Unified Model, closure relations are needed for the quantities +:math:`\sigma_w^2`, :math:`\varepsilon`, :math:`L`, :math:`\tau_{\rm d}` +and :math:`S_E`, subject to the constraining relationship given by Eq. +`[eqn:taud] <#eqn:taud>`__. In each model grid box, these parameters +specify the subgrid PDF, :math:`F(S_i)`, and from this the liquid cloud +fraction and water content produced by turbulence can be found using Eqs +`[eqn:cloud_fraction] <#eqn:cloud_fraction>`__ and +`[eqn:cloud_liquid] <#eqn:cloud_liquid>`__. + +In addition we need to make some assumptions about how the diagnosed +values :math:`C_l^{sgt}` and :math:`q_{cl}^{sgt}` relate to the model +prognostic fields, :math:`C_l` and :math:`q_{cl}`. Two methods are +available for doing this. In the simplest case, the diagnosed values +:math:`C_l^{sgt}` and :math:`q_{cl}^{sgt}` are just treated as +increments to model prognostics (option one, in Sec. +`3.8.5 <#sec:sgt_increments>`__ below). A more complex option (see +option two, below) is to increment the model fields via the PC2 Erosion +functionality. + +.. _`sec:sgt_closures`: + +Closure relations +~~~~~~~~~~~~~~~~~ + +The vertical velocity variance, :math:`\sigma_w^2`, is available as a +diagnostic from the Boundary Layer scheme. Because the Boundary Layer +scheme is called after the Microphysics on each model timestep, the +diagnostic value is stored in a (non-advected) model prognostic field. +The scheme will operate only where there is diagnosed turbulence, i.e., +non-zero :math:`\sigma_w^2`. + +We take the mixing length scale, :math:`L`, to be proportional to the +vertical grid spacing in each grid box: :math:`L=\beta_{mix} \Delta z`, +where :math:`\Delta z` is calculated as the height different between the +:math:`\rho`-levels adjacent to the given :math:`\theta`-point. The +parameter, :math:`\beta_{mix}`, is an adjustable constant that the user +can define (see Section `3.8.6 <#sec:sgt_options>`__ below), however it +should be of order one. + +To obtain :math:`\tau_{\rm d}` we impose an eddy size constraint: + +.. math:: + + \tau_{\rm d} = \frac{L}{\sigma_w} = \beta_{mix} \frac{\Delta z}{\sigma_w} + \label{eqn:eddy_size} + +Eq. `[eqn:taud] <#eqn:taud>`__ then determines the dissipation rate, +:math:`\varepsilon`, that is consistent with the other parameters. The +constant :math:`C_0=10` by default, but can be adjusted by the user. + +The scheme is limited to act only in grid boxes where +:math:`\tau_{\rm d}` is less than a prescribed value, +:math:`\tau_{d}^{max}`. The default is +:math:`\tau_d^{max}=1200\;{\rm sec}`, which typically coincides with a +couple of model timesteps. The motivation for this is that a motion that +takes longer than a few timestep to decorrelate will be partially +resolved by the dynamics and therefore cannot be considered as ‘subgrid’ +turbulence. + +Finally, where :math:`T`, :math:`p` and :math:`q` appear in the +expressions for :math:`C_l^{sgt}` and :math:`q_{cl}^{sgt}`, these are +taken to be the grid box mean values. The first moment of the ice PSD, +:math:`{\cal M}_1`, is found from the parametrization, due to +:raw-latex:`\cite{fhbicc05}`, described in Section 4.1 of UMDP26. + +.. _`sec:sgt_increments`: + +Options for incrementing model prognostics +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +Using the information in Section `3.8.4 <#sec:sgt_closures>`__ to obtain +closed expressions for the subgrid PDF of :math:`S_i`-fluctuations +allows :math:`C_l^{sgt}` and :math:`q_{cl}^{sgt}` to be calculated. +These will be non-zero only where there is turbulence as diagnosed by +the Boundary Layer scheme (and hence non-zero :math:`\sigma_w^2`). To +calculate :math:`C_l^{sgt}` and :math:`q_{cl}^{sgt}` the integrals in +Eqs `[eqn:cloud_fraction] <#eqn:cloud_fraction>`__ and +`[eqn:cloud_liquid] <#eqn:cloud_liquid>`__ are evaluated numerically +using discretisation based on user-specified number of bins. + +Given :math:`C_l^{sgt}` and :math:`q_{cl}^{sgt}`, two options are +available for relating these to changes in the model prognostics: + +Option one: direct increments +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +The values of :math:`C_l^{sgt}` and :math:`q_{cl}^{sgt}` can be added as +increments to the model prognostic fields, :math:`C_l` and +:math:`q_{cl}`. In this case + +.. math:: + + \begin{aligned} + \left( \Delta C_l \right)_{sgt} &=& C_l^{sgt} \\ + \left( \Delta q_{cl} \right)_{sgt} &=& q_{cl}^{sgt}, \\ + \left( \Delta q \right)_{sgt} &=& -\left( \Delta q_{cl} \right)_{sgt}, \\ + \left( \Delta T \right)_{sgt} &=& \frac{L_v}{c_p} \left( \Delta q_{cl} \right)_{sgt}, \\ + \left( \Delta C \right)_{sgt} &=& C_l^{sgt} \\ + \end{aligned} + +where the left hand sides denote the increments to :math:`C_l`, +:math:`q_{cl}`, :math:`T` and the total cloud fraction, :math:`C`, due +to the subgrid scheme. Some bounds-checking is then applied to ensure +that: (a) the resultant cloud fractions to not exceed one; (b) the +scheme does not condense out more liquid than there is available +moisture. + +Option two: PC2 Erosion method +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +Option one gives a simple method for incrementing the model prognostics, +but it gives rise to a potential inconsistency with the PC2 cloud +scheme. This arises because the subgrid production scheme can elevate +cloud fraction to unity in grid boxes that are subsequently diagnosed by +PC2 Initiation to meet the criteria for clear-sky initiation. PC2 then +counteracts the scheme by removing some of the liquid cloud. To try to +mitigate against this issue, cloud fraction increments can be applied +using PC2 Erosion. In this case: + +.. math:: + + \begin{aligned} + \left( \Delta q_{cl} \right)_{sgt} &=& q_{cl}^{sgt} - q_{cl}, \\ + \left( \Delta q \right)_{sgt} &=& -\left( \Delta q_{cl} \right)_{sgt}, \\ + \left( \Delta T \right)_{sgt} &=& \frac{L_v}{c_p} \left( \Delta q_{cl} \right)_{sgt}, \\ + \end{aligned} + +where :math:`q_{cl}` is the liquid cloud amount prior to calling to the +turbulent production scheme. The cloud fraction increments are +calculated by calling PC2 Erosion with +:math:`\left( \Delta q_{cl} \right)_{sgt}` as input. See Section +`4.3 <#sec:turb>`__ for details on how the PC2 Erosion process works. +This method gives cloud fraction increments that are consistent with PC2 +cloud scheme. + +.. _`sec:sgt_options`: + +Other user options +~~~~~~~~~~~~~~~~~~ + +The following variables and logical switches are optional inputs: + +#. The logical ``l_dcfl_by_erosion`` provides a switch to apply cloud + fraction increments using PC2 Erosion. Defaults to *FALSE*. + +#. Setting the logical ``l_mixed_phase_t_limit`` to *TRUE* allows the + user to use the variable ``mp_t_limit`` to define a temperature + limit, :math:`T_{max}`, above which the scheme is not applied. The + default is :math:`T_{max}=0^\circ\;{\rm C}`, so the scheme is only + applied to cold clouds. + +#. The input variable ``mp_tau_d_lim`` defines the upper limit, + :math:`\tau_d^{max}`, on the value of :math:`\tau_d` above which the + scheme is not applied. The default value is + :math:`\tau_d^{max}=1200.0`, so the scheme is not applied in grid + boxes where the decorrelation time scale exceeds :math:`1200` + seconds. + +#. ``nbins_mp`` is the number of bins used in the discretisation of the + integrals in Eqs `[eqn:cloud_fraction] <#eqn:cloud_fraction>`__ and + `[eqn:cloud_liquid] <#eqn:cloud_liquid>`__ for :math:`C_l^{sgt}` and + :math:`q_{cl}^{sgt}`. The default value is :math:`100` bins. + +#. ``mp_dz_scal`` is the scale parameter, :math:`\beta_{mix}`, in the + definition of the mixing length, :math:`L=\beta_{mix}\Delta z`. + +#. ``mp_czero`` defines the constant parameter :math:`C_0` (defaults to + :math:`C_0=10`). + +.. _`sec:app_um`: + +Application to the Unified Model +================================ + +This section describes the way in which the physical concepts described +in the above section are applied to the sections of the Unified Model, +in order to build up the complete prognostic scheme. Description of the +actual subroutines themselves follow in section `5.2 <#sec:code>`__. +Note that the large-scale precipitation and convection schemes have +considerable documentation below, since these schemes have been heavily +modified for PC2. The other schemes use generic forcing scenarios, hence +their desciption here is much shorter. Remember, whenever a signficiant +:math:`\overline{T}` or :math:`\overline{q}` change occurs, PC2 must be +able to represent the corresponding condensation and changes in cloud +fractions. + +.. _`sec:rad`: + +Radiation +--------- + +The shortwave and longwave radiation schemes both alter the temperature +of the atmosphere, hence we need to calculate the corresponding +condensation and cloud fraction changes. For both shortwave and +longwave, we use the homogeneous forcing routines (section +`3.2 <#sec:homog>`__) for :math:`\overline{q_{cl}}` and :math:`C_l`, +(using eqn. `[eq:deltaqc_exp2] <#eq:deltaqc_exp2>`__ to calculate the +:math:`Q_c` forcing) and then the method in section `3.6 <#sec:ct>`__ to +calculate :math:`C_t` changes. There is no :math:`\overline{q_{cf}}` +change associated with this process since the deposition / sublimation +process is performed within the large-scale precipitation scheme (as it +also is in the absence of PC2). + +It is reasonable to question whether homogeneous forcing is a reasonable +model to use when we know that a large proportion of the heating +associated with radiative transfer in the atmosphere comes from the +cloudy air and is not evenly spread across the gridbox. Possible +developments are discussed in section `7.4.7 <#sec:homog_improve>`__. + +.. _`sec:precip`: + +Large-scale precipitation +------------------------- + +Precipitation processes have a large effect on cloud fractions. Here we +present the simple physical models that are applied to the transfer +terms included in the large-scale precipitation scheme. They are also +presented within the large-scale precipitation documentation (). + +The basis of the physical model is that microphysical transfer processes +can be calculated separately in different partitions of the model cloud +(i.e. mixed phase cloud, liquid phase cloud, ice phase cloud or clear +sky). However, processes may change the size of these partitions. We +consider here separately each process that is modelled in the +large-scale precipitation scheme. The changes in +:math:`\overline{q_{cl}}`, :math:`\overline{q_{cf}}` and +:math:`\overline{q}` remain mathematically the same as in the non-PC2 +version of the code (), we only need to introduce calculations for the +changes in cloud fractions. We will see that many of these changes can +be well modelled by assuming no change to the cloud fractions, and the +others by using simple assumptions. + +Although the model may use two ice prognostic ice categories, only a +single ice cloud fraction is stored, the assumption being that the two +ice categories are completely overlapped with each other. Graupel is not +considered to contribute to the ice cloud fraction. + +.. _`sec:lsp_fall`: + +Fall of ice +~~~~~~~~~~~ + +The fall of ice is the process that contributes most to the growth of +ice cloud fraction in the model. The model results are therefore +sensitive to the formulation of this process. We will make the basic +assumption that a trail of falling ice does not reduce the horizontal +spread of ice cloud fraction at a particular level (hence +:math:`\overline{q_{cf}}` that leaves a gridbox does not reduce +:math:`C_f` in that gridbox). The in-cloud ice content simply reduces +due to the fall out of ice - it is the sublimation term (section +`4.2.4 <#sec:mp_depsub>`__) that erodes the fall streaks. However, ice +that falls into a clear layer from above may increase the ice cloud +fraction. We parametrize this by considering the horizontal overlap of +ice clouds between two model layers, and the fall speed of ice between +them. We will assume an overlap that is nearly, but not quite, maximum, +the difference being dependent upon the windshear and the time taken for +ice to fall between the levels. + +.. math:: + + O^{[k,k+1]} = \text{Max}( C_{i}^{[k+1]} - C_i^{[k]} , 0) + + w \frac{\Delta z^{[k]}}{v_i^{[k]}} + \label{eq:overhang} + +where :math:`O^{[k,k+1]}` is the amount of ice cloud ‘overhanging’ the +current (i.e. :math:`k`\ ’th) layer from the layer above, :math:`w` is a +parameter that is closely related to the windshear, +:math:`\Delta z^{[k]}` is the model layer thickness and +:math:`v_i^{[k]}` is the fallspeed of ice in the layer. +:math:`v_i^{[k]}` is calculated in the microphysics scheme and, if two +ice prognostics are used, is the mass-weighted average fall speed of the +two categories. The factor :math:`\frac{\Delta z}{v_i}` is simply the +time taken for the ice to fall through one model layer. Multiplying this +by the windshear would give an estimate to the amount of overlap between +a cloud source and its fall streak in the layer below (it is an +*estimate* since we assume that the cloud source is continuous and +unbroken). Although it is quite possible within PC2 to do this, to date +we have not programmed this link, and we use a constant, but tunable, +value of :math:`1.5 \times 10^{-4} s^{-1}` for :math:`w`. + +The change in :math:`C_i` over the timestep is then given by the overlap +proportion multiplied by the how much (in the vertical dimension) of the +layer below can be filled by ice in the timestep: + +.. math:: + + \Delta C_i = \text{Max}(O^{[k,k+1]} , 1) \text{Min} (v_i \frac{\Delta t}{\Delta z^{[k]}} , 1) + \label{eq:lsp_fall} + +where :math:`\Delta t` is the timestep. We now choose to assume a +minimum overlap between the liquid and the ice phases (as in section +`3.6 <#sec:ct>`__). + +.. math:: + + \Delta C_t = \text{Min} ( \Delta C_i , A_{clear} ) + \label{eq:lsp_fall_ct} + +where :math:`A_{clear}` is the proportion of the gridbox that has +neither ice nor liquid cloud present. + +**An inconsistency has been found in the way that the fall-of-ice term +is linked to the globally constant “wind-shear value” when calculting +the ice cloud fraction overhang. Consequently, the option not to use the +“wind shear value” when calculating the overhang is available in the +UMUI (from version 7.6 onwards).** + +.. _`sec:lsp_homo`: + +Homogeneous nucleation +~~~~~~~~~~~~~~~~~~~~~~ + +This will freeze all supercooled liquid water when a temperature +threshold is exceeded. Hence we turn all existing liquid and mixed phase +cloud to ice cloud. The cloud fraction changes are: + +.. math:: + + \begin{aligned} + C_l \leftarrow 0 \nonumber \\ + C_i \leftarrow C_t \nonumber \\ + \Delta C_t = 0. + \label{eq:lsp_homo} + \end{aligned} + +Heterogeneous nucleation +~~~~~~~~~~~~~~~~~~~~~~~~ + +This process will freeze a small amount of supercooled liquid water, +regardless of the previous presence of ice cloud. This will mean that +previously existing ‘liquid-only’ cloud is converted to mixed phase +cloud. These give the following changes: + +.. math:: + + \begin{aligned} + \Delta C_l = 0 \nonumber \\ + C_i \leftarrow C_t \nonumber \\ + \Delta C_t = 0. + \label{eq:lsp_het} + \end{aligned} + +.. _`sec:mp_depsub`: + +Deposition and sublimation +~~~~~~~~~~~~~~~~~~~~~~~~~~ + +This term exerts one of the most important influences on the ice cloud +in the whole model (this applies to the control as well as for PC2). +Contained in the formulation is a subgrid-scale assumption that causes +equivalent effects to that for a moisture PDF under the ‘:math:`s`’ +framework (section `2.2 <#sec:s_dist>`__). However, since +:math:`{q_{cf}}` changes slowly in response to local changes in +:math:`q` and :math:`T`, we cannot base the :math:`q_{cf}` response on +the same instantaneous condensation framework. It would be useful to +investigate in the future whether the two descriptions of the moisture +variability could be brought together. Because of its importance, we +describe the method below, although we note it is also described in . + +We can calculate the local rate of change of :math:`q_{cf}`, given local +:math:`T` and :math:`q` etc. using the standard microphysical growth +equations (see ). However, it is critical to know the way in which the +moisture is correlated with the ice in the gridbox. We will assume there +exists a distribution of vapour in the gridbox. We know that the regions +where liquid cloud exists must be saturated with respect to liquid +water, hence we need only consider the part of the gridbox that does not +have liquid water present. The average value, :math:`q_a`, of :math:`q` +within the liquid-free part of the gridbox is thus + +.. math:: + + q_a = \frac{ \overline{q} - C_l q_{sat \, liq}(\overline{T}) } {1 - C_l} + \label{eq:qa} + +where we have assumed that the fluctuation of :math:`q_{sat~liq}` across +the gridbox due to temperature fluctuations is not significant compared +to the fluctuation of :math:`q` described below. We then parametrize a +width, :math:`b_i`, to the :math:`q` (not :math:`s`) fluctuations +*across the non-liquid cloud part of the gridbox*, based upon +:math:`RH_{crit}`. This is like that for the ‘:math:`s`’ distribution +width, :math:`b_s` but modified: + +.. math:: + + b_i = (1 - RH_{crit} ) q_{sat \, liq} ( 1 - \frac{1}{2} + ~ \frac{\overline{q_{cf}}} {i q_{sat \, liq}(\overline{T})} ) . + \label{eq:b_i} + +where the factor :math:`( 1 - \frac{1}{2} +\frac{\overline{q_{cf}}} {i ~ q_{sat \, liq}(\overline{T})})` should be +limited to a minimum value of zero, but, for numerical reasons, is +limited to a minimum value of 0.001. We note that :math:`b_i` has a +similar form to :math:`b_s`, except the multiplier :math:`a_L` and the +factor in brackets. If we remember from (`[eq:s] <#eq:s>`__) that the +definition of ‘:math:`s`’ includes a factor :math:`a_L` we see that the +absence of the :math:`a_L` factor in (`[eq:b_i] <#eq:b_i>`__) is +consistent. The factor in brackets is a *parametrization* of the effect +that, when ice is present, deposition in the moistier parts and +sublimation in the drier parts of the gridbox must reduce the width of +the distribution of :math:`q` across the gridbox. It is a simple linear +function of :math:`\frac{\overline{q_{cf}}}{q_{sat~liq}(\overline{T})}`, +and is tunable using the factor :math:`i`, which takes the value of +0.04. + +We note that this formulation isn’t totally consistent with the liquid +cloud formulation, which considers an underlying PDF across the whole +gridbox and does not have, in general, its width prescribed. Remember +that we do not calculate on-line the whole of the liquid - vapour PDF, +we only parametrize the single point :math:`G(-Qc)`, using equation +`[eqn22] <#eqn22>`__). + +The width is then limited further to be no greater than +:math:`\overline{q}`, to make sure that there are no negative values of +:math:`q` predicted within the gridbox (possible at low temperatures +where :math:`q_{sat~liq}(\overline{T})` diverges from +:math:`q_{sat~ice}(\overline{T})`). + +We then calculate the average value of :math:`q` in the ice-only and +clear-sky partitions of the gridbox. To do this, we make the further +assumption that the ice is correlated with the moistest part of the +distribution (an instantaneous condensation formulation would make the +same assumption). Some algebra retrieves the expressions: + +.. math:: + + \begin{aligned} + q_{clear} = q_a - b_i A_{ice} ; \\ + q_{ice} = \frac {\overline{q} - C_l q_{sat~liq} - A_{clear} q_{clear} } + {A_{ice}}, + \label{eq:q_clear_and_q_ice} + \end{aligned} + +where :math:`A_{ice}` is the proportion of the gridbox with ice cloud +but not liquid cloud and :math:`A_{clear}` is the proportion of the +gridbox without cloud. The numerical application will set +:math:`q_{clear}` to :math:`q_a` if :math:`A_{ice}` is zero. We now have +a representation of the :math:`q` values in each of the gridbox cloud +partitions, and can solve the microphysical transfer equation in each +partition. + +The cloud fraction changes now need to be parametrized. We use the model +that deposition will *not* adjust the ice cloud *fraction* (increases +will be done within the fall-of-ice microphysics section). However, +deposition can decrease the liquid cloud fraction (locally, :math:`q` +can be reduced by deposition to below :math:`q_{sat~liq}`, hence this is +not inconsistent with the assumptions for the riming term below. This is +the principal sink of supercooled liquid cloud fraction in the model. +Sublimation will be allowed to decrease the ice cloud fraction (since +sublimation cannot act in liquid cloud, there is no impact on the liquid +cloud). To solve for these models, we will need to further split the +ice-only partition to give the proportion of that partition that is +above and below ice saturation. This gives, in general, an area of the +gridbox :math:`A_{ice1}` that contains ice and is above saturation where + +.. math:: + + A_{ice1} = \frac{1}{2} A_{ice} + \frac{1}{2} + \frac{ (q_{ice}-q_{sat~ice}(\overline{T})) } {b_i}, + \label{eq:q_ice_above_sat} + +having assumed that :math:`A_{ice1}` is between 0 and :math:`A_{ice}`. +If not, it is trivial to partition the gridbox, since the moisture in +the ice-only partition is either completely above or completely below +:math:`q_{sat~ice}(\overline{T})`. The corresponding area that contains +ice and is below saturation is given by +:math:`A_{ice2} = A_{ice} - A_{ice1}`. We can now parametrize the change +in cloud fractions. For deposition, we shall assume a uniform +distribution of local values of :math:`q_{cl}` about the local mean. If +we assume a uniform removal of local :math:`q_{cl}` then, with a little +algebra, we can obtain an expression for the change in :math:`C_l`: + +.. math:: + + \Delta C_l = C_l ( 1 - \frac {\Delta \overline{q_{cl}}} {\overline{q_{cl}}} ) + ^{\frac{1}{2}} - C_l + \label{eq:deltacfl_dep} + +. + +Since this occurs only in the mixed phase part of the gridbox, we can +say that :math:`\Delta C_t = 0`. We will also note that the change in +:math:`\overline{q_{cl}}` due to deposition is limited by the amount of +:math:`\overline{q_{cl}}` that is in the mixed phase partition in the +gridbox, hence (`[eq:deltacfl_dep] <#eq:deltacfl_dep>`__), although it +formally allows removal of :math:`C_l` from an ice-free partition, will +be unlikely to do so. + +The sublimation forms the main method by which ice cloud is destroyed in +PC2, hence PC2 results are relatively sensitive to its formulation. Here +we make a similar assumption to that used for liquid in the deposition +term, except that we limit the changes only to the region of the gridbox +where ice is subliming. + +.. math:: + + \Delta C_i = A_{ice2} ( 1 + \frac{\Delta \overline{q_{cf}} } + { \overline{q_{cf}} ( \frac{A_{ice2}}{C_i} ) } + )^{\frac{1}{2}} - A_{ice2} + \label{eq:deltacfi_sub} + +. + +The term :math:`\overline{q_{cf}} ( \frac{A_{ice2}}{C_i} )` is the +amount of :math:`\overline{q_{cf}}` that is present in the subliming ice +region, hence its ratio with :math:`\Delta \overline{q_{cf}}` is the +fractional change in that region. The change in the total cloud fraction +must also be equal to the change above, since sublimation cannot occur +in the presence of liquid cloud: + +.. math:: + + \Delta C_t = \Delta C_i . + \label{eq:deltacft_sub} + +Riming +~~~~~~ + +This process acts only where mixed phase cloud occurs - although, in +theory, it could remove any supercooled liquid totally, the air would +remain saturated with respect to liquid water. Hence any subsequent +cooling would regenerate the same amount of liquid cloud. Hence we +choose to model this process as having *no effect* on the cloud +fractions. + +Capture +~~~~~~~ + +This is the freezing of raindrops onto ice crystals by collision. This +does not alter the ice cloud *fraction* in the gridbox (although it does +alter :math:`\overline{q_{cf}}`, and it has no interaction with the +liquid cloud. Again, we therefore choose to model this process as having +*no effect* on the cloud fractions. + +Evaporation of melting ice +~~~~~~~~~~~~~~~~~~~~~~~~~~ + +Here we simply assume that ice cloud fraction is removed in proportion +to the ice content that is removed. + +.. math:: + + \Delta C_i = C_i \frac{ \Delta \overline{q_{cf}}}{\overline{q_{cf}}} . + \label{eq:lsp_evapmeltsnow} + +Because the evaporation cannot occur in the liquid part of the gridbox, +there is no change to :math:`C_t` (or to :math:`C_l`). + +Melting +~~~~~~~ + +Again, the change in :math:`C_i` is calculated using the method in +(`[eq:lsp_evapmeltsnow] <#eq:lsp_evapmeltsnow>`__). + +.. math:: + + \Delta C_i = C_i \frac{ \Delta \overline{q_{cf}}}{\overline{q_{cf}}} . + \label{eq:lsp_melt} + +The change in :math:`C_t` is calculated assuming that there is no +correlation in the gridbox between where the ice melts and the liquid +cloud. Hence we must multiply (`[eq:lsp_melt] <#eq:lsp_melt>`__) by the +proportion of ice cloud fraction that exists without liquid cloud (i.e. +:math:`\frac{A_{ice}}{C_i}`). + +.. math:: + + \Delta C_t = C_i \frac{ \Delta \overline{q_{cf}}}{\overline{q_{cf}}} + \frac{A_{ice}}{C_i} . + \label{eq:lsp_melt2} + +Evaporation of rain +~~~~~~~~~~~~~~~~~~~ + +Evaporation of rain will not, *on its own*, generate liquid cloud, since +a large-scale lifting process will be required in order to condense +water from the moistened air. We cannot, therefore, allow any change in +cloud fractions to occur as a result, subsequent changes are calculated +elsewhere in the model (e.g. by the lifting process, section +`4.8 <#sec:pres>`__). + +Accretion +~~~~~~~~~ + +Accretion is the sweep-out of liquid water droplets by rain. We argue in +a similar way to the riming term, that this will not remove any liquid +cloud fraction, since a small amount of lifting will regenerate the same +amount of liquid cloud. Hence we choose to model this process as having +*no effect* on the cloud fractions. The arguments underlying the +formulation of the evaporation of rain and the accretion cloud fraction +changes may appear to be inconsistent in their limiting cases and the +subsequent response to lifting. However, when the limiting case is not +reached the formulations are both correct. For the moment, it is not +considered necessary to increase the complexity of the current, simple +representations. + +Autoconversion +~~~~~~~~~~~~~~ + +As for accretion, the generation of rain directly from collision and +coalescence of liquid water droplets will not alter the cloud fractions. + +Other microphysics terms +~~~~~~~~~~~~~~~~~~~~~~~~ + +There are already (i.e. also in the control) two numerical tidy-up terms +at the end of the microphysics section that remove small rain amounts +and provide an additional melting term for the snow. These do not change +the cloud fractions. + +If there are small amounts of ice present at the end of the microphysics +then these are removed at the end of the microphysics timestep (also in +the control). PC2 responds by resetting the cloud fractions +appropriately, so :math:`C_t` is reset to :math:`C_l` etc. + +.. _numerical-implementation-1: + +Numerical implementation +~~~~~~~~~~~~~~~~~~~~~~~~ + +Note that after each process has been applied, we do *not* recalculate +the sizes of the ice-only, liquid-only and mixed phase partitions, but +use the values at the start of the microphysics (this includes the +values of :math:`C_i` used in the calculation of ‘in-cloud’ water +contents above. However, we do update the cloud fractions themselves +sequentially. We also recalculate after each process the overlaps +between the rain fraction (see ) and the cloud fractions. + +There is also a final set of checks that :math:`C_l` and :math:`C_i` lie +between 0 and 1 and that :math:`C_t` is bounded between +:math:`\text{Max}(C_l, C_i)` (maximum overlap of liquid and ice) and +:math:`\text{Min}(C_l+C_i,1)` (minimum overlap of liquid and ice). + +We should note in particular, that these parametrizations allow a +considerable reduction in :math:`\overline{q_{cl}}` without a +corresponding large reduction in :math:`C_l`. This is an underlying +feature of the PC2 scheme (discussed in :raw-latex:`\cite{wg03}`), and +necessarily implies the skewing of the underlying moisture PDF. +Subsequent parts of the model (e.g. the width narrowing, section +`3.3 <#sec:width>`__) will, of course, act on the modified fields to +adjust the cloud fractions further, but remember that these are separate +processes and modelled elsewhere in the timestep. + +.. _`sec:turb`: + +PC2 erosion +----------- + +Original width-narrowing method +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +(selected by setting **i_pc2_erosion_method = 1** in the UM namelist). + +In parallel with the homogeneous forcing part of the PC2 response to +convection, we introduce a new block of code that allows a background +change of the PDF width. At earlier versions of PC2 (PC2:65 and earlier) +this block was included as a separate section of code that was called in +as part of the atmphya parallel timestepping. This was later moved to +better numerically balance increments from the convection scheme with +the cloud fraction erosion term. From VN8.1 onwards, a further option +was introduced to implement the erosion prior to the microphysics +parametrization. This was primarily to allow PC2 to be run at +convection-resolving scales, at which the convection scheme is not +called and therefore the erosion is not called. We have empirically +selected a rate of change of width that depends upon the relative total +humidity of the grid box, such that there is more erosion in drier +gridboxes. This promotes more rapid erosion of shallow convective cloud, +which is the main effect that we seek to include, although the physical +implication that dry air is more turbulent than moist air does not match +the way the real atmosphere works, especially in the stratosphere. No +doubt the link can be improved upon with more research. The formulation +used is: + +.. math:: + + \frac{1}{b_s} \frac{\partial b_s}{\partial t} = \Upsilon exp ( - \frac{2.01 Q_c}{0.2 a_L q_{sat liq}(T_L)} ) + \label{eq:dbsbydtbs_turb} + +where the 0.2 factor is chosen to be closely equivalent to +:math:`1 - RH_{crit}` and the value of 2.01 has been selected through +tuning. The code merges the two numerical values into a single quantity +(dbsdtbs-exp), equal to 10.05. We note that in PC2:64 (the library 6.4 +code, a value of 0.62 is used rather than 2.01). As a guide to the +:math:`RH_T` dependence, note that when the value of :math:`RH_T` is +0.85, the value of :math:`\frac{1}{b_s} \frac{\partial b_s}{\partial t}` +is close to :math:`1 \times 10^{-4} s^{-1}`. + +Note: the source-code for this erosion method (**pc2_hom_conv**, +**pc2_homog_plus_turb**, **pc2_delta_hom_turb**) includes an additional +term “dbsdtbs1” which scales with the rate of homogeneous forcing +:math:`\frac{\partial Q_c}{\partial t}`. However this term is always set +to zero on input to these routines so is never used. + +The width-narrowing formulation of section `3.3 <#sec:width>`__ is used +to calculate increments in :math:`\overline{q_{cl}}` and :math:`C_l`. +Using the liquid - ice cloud overlap ideas of section `3.6 <#sec:ct>`__ +then gives the associated :math:`C_t` change. This background narrowing +term, :math:`\Upsilon`, is originally based upon work by +:raw-latex:`\cite{sg03}`, although it is a parameter that has been +extensively tuned during PC2 development, a typical value would be +:math:`\Upsilon=-2.25 \times 10^{-5} s^{-1}`. + +Numerical application of the original width-narrowing method +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +Because of the strong link the mathematical expressions for width +narrowing (section `3.3 <#sec:width>`__) have with the expressions for +the homogeneous forcing (section `3.2 <#sec:homog>`__), we choose to +represent the timestepping of this process in exactly the same way as +for the homogeneous forcing (in fact, in the Unified Model code we use +the same subroutine, see section `5.2 <#sec:code>`__). As before, we use +a simple forward timestepping of :math:`C_l`, with :math:`Q_c` given by +(`[eq:qc_eq_qt-qs] <#eq:qc_eq_qt-qs>`__) and :math:`a_L` defined as +discussed in section `3.2.3 <#sec:homog_num_app>`__ and discretize eq +`[eq:dcdt_width] <#eq:dcdt_width>`__ as: + +.. math:: + + \Delta C_l^{[n+1]} = - G(-Q_c) Q_c \frac{1}{b_s} + \frac{\partial b_s}{\partial t} \Delta t. + \label{eq:dcl_turb_final} + +Similarly to (`[eq:c_l^n+1] <#eq:c_l^n+1>`__), we then limit the cloud +fraction to 0 and 1 and then apply a mid-point value of :math:`C_l` to +calculate the change in :math:`\overline{q_{cl}}` (discretizing eq +`[eq:dqcldt_width] <#eq:dqcldt_width>`__): + +.. math:: + + \Delta q_{cl}^{[n+1]} = (q_{cl}^{[n]} - Q_c \frac{1}{2}(C_l^{[n]}+C_l^{[n+1]})) + \frac{1}{b_s} \frac{\partial b_s}{\partial t} \Delta t. + \label{eq:dqcl_turb_final} + +In this case the value of :math:`\Delta q_{cl}` *is* limited to ensure +that no more :math:`\overline{q_{cl}}` is removed than the model has +available. This was chosen to ensure that the erosion process itself +contains this physical limit, not a numerical tidying-up process. + +The option “l_fixbug_pc2_qcl_incr” ensures that qcl is set to zero if +the CFL has reached zero. + +Cloud-surface-area hybrid erosion method +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +(selected by setting **i_pc2_erosion_method = 3** in the UM namelist). + +:raw-latex:`\cite{morcrette_petch}` showed that changes to the erosion +parameter (:math:`\Upsilon` in Eqn. +`[eq:dbsbydtbs_turb] <#eq:dbsbydtbs_turb>`__) did not have as +significant an impact on the global work done by the erosion process as +might be expected. This was due to a feedback process whereby, reducing +the erosion parameter leads to more cloud water, more autoconversion of +cloud water to rain, more fall-out of rain and more drying of the layer, +hence increasing the +:math:`exp ( - \frac{2.01 Q_c}{0.2 a_L q_{sat liq}(T_L)} )` part of Eqn. +`[eq:dbsbydtbs_turb] <#eq:dbsbydtbs_turb>`__. Although the feedback is +physically plausible it crucially depends on the formulation of Eqn. +`[eq:dbsbydtbs_turb] <#eq:dbsbydtbs_turb>`__ and the dependence of the +rate of narrowing of the PDF on the moisture, a dependence that was +developed from a pragmatic rather than theoretical stand-point. The +option for an alternative way of calculating the erosion was introduced +at vn8.0 + +We use equation 30 from :raw-latex:`\cite{t93}` to specify the sink of +:math:`q_{cl}` due to erosion, i.e. + +.. math:: + + \frac{\partial q_{cl}}{\partial t}=-A K(q_{sat}-q_v) + \label{eq:dqcldt_hybrid} + +(note we have changed the sign as we have replaced the evaporation rate +:math:`E_2` in :raw-latex:`\cite{t93}` with +:math:`-\frac{\partial q_{cl}}{\partial t}` on the left-hand-side). In +the :raw-latex:`\cite{t93}` scheme, :math:`A` is set to the cloud +fraction (i.e. :math:`A=C_l`). Here we recall that "cloud erosion" is +meant to represent the evaporation of cloud water due to the mixing of +clear and cloudy air and that this can only happen on the edges of +cloud, where saturated air is exposed to sub-saturated air. If the cloud +fraction is small (e.g. 5\ :math:`\%`), then there are not many clouds, +so there is only a small surface area from which evaporation can occur. +Similarly if the cloud cover is very high (e.g. 95\ :math:`\%`) then +there is again not much surface area exposed to clear sky. A maximum in +exposed surface area is expected when the cloud cover is 50\ :math:`\%`. + +By imagining that the grid-box is broken up into cubes whose horizontal +dimension equal the layer depth it is possible to work out what the +maximum lateral surface area would be, as a function of cloud fraction, +for different arrangements of cloudy cubes. The maximum lateral surface +area, is when the clear and cloudy cubes are arranged in a chess-board +pattern, and the minimum is when then are all grouped together into a +circular clump. Numerical tests using randomly distributed cloudy cube +shows that the variation in lateral surface area, :math:`S`, as a +function of cloud fraction can be expressed as: + +.. math:: + + S= - 2 C_l ^{2} + 2 C_l + \label{eq:S_Cl} + +The maximum normalised surface area of 0.5 occurs at a cloud fraction of +0.5. Using a cloud mask derived from satellite imagery shows that real +cloud fields do follow this kind of dependence, but that the peak +surface area is nearer to 0.35, meaning that real clouds are not as +randomly distributed as random ones and that there is some kind of +clumping together, which is what we might have expected. When it comes +to implementing such a scheme in the model, there will need to be a +tunable parameter to govern the rate of evaporation. This will not +affect the shape of the lateral surface area function. As a result the +details of whether the peak lateral surface area is 0.5 or 0.35 are +simply absorbed into the tunable parameter :math:`K`, which is supplied +from the UMUI (using the same text box as was used for supplying +:math:`\Upsilon`). + +The exposed surface area associated with the tops and bottom of the +clouds is calculated assuming maximum overlap in adjacent layers and is +added to the lateral surface area to give a total surface area, + +.. math:: + + A=max(C_l(k)-C_l(k+1),0.0)+max(C_l(k)-C_l(k-1),0.0)+S + \label{eq:A_top_and_bottom} + +it is this value of :math:`A` which we use in Eqn. +`[eq:dqcldt_hybrid] <#eq:dqcldt_hybrid>`__. + +Note that the contributions from the top and bottom interfaces of the +current model-level :math:`max(C_l(k)-C_l(k+1),0.0)` and +:math:`max(C_l(k)-C_l(k-1),0.0)` may optionally either be included or +excluded, depending on the UM namelist switch **i_pc2_erosion_method**. +Further note: at present these contributions are hardwired to be +excluded, as they prevented the erosion calculations from being +parallelised in the vertical direction using OpenMP, and no operational +model configurations were using them. + +Having calculated a reduction in :math:`q_{cl}` using the +Tiedtke-surface-area method, we then work out the relative rate of +narrowing that would have given the same sink of :math:`q_{cl}`. This +value of :math:`\frac{1}{b_s} \frac{\partial b_s}{\partial t}` is then +used to calculate the change in :math:`C_l` using the same moisture PDF +assumptions as were used in the original PC2 erosion formulation. To +achieve this, we combine equations `[eq:dcdt_width] <#eq:dcdt_width>`__ +and `[eq:dqcldt_width] <#eq:dqcldt_width>`__ from section +`3.3 <#sec:width>`__ to eliminate +:math:`\frac{1}{b_s} \frac{\partial b_s}{\partial t}` and write +:math:`\frac{\partial C_l}{\partial t}` as a function of +:math:`\frac{\partial \overline{q_{cl}}}{\partial t}`: + +.. math:: + + \frac{\partial C_l}{\partial t} + = - \frac{ G(-Q_c) Q_c \frac{\partial \overline{q_{cl}}}{\partial t} } + { (- C_l Q_c+\overline{q_{cl}}) } + \label{eq:dcdt_hybrid} + +Where the change in liquid water content +:math:`\frac{\partial \overline{q_{cl}}}{\partial t}` is given by eq +`[eq:dqcldt_hybrid] <#eq:dqcldt_hybrid>`__ above. + +This combination of a Tiedkte sink term for :math:`q_{cl}`, a PC2 term +for :math:`C_l` and the introduction of some surface area dependence +leads to this formulation being referred to as a “hybrid” +cloud-surface-area erosion method. + +.. _`sec:erosion_numerics`: + +Numerical application of the hybrid erosion method +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +Next, we consider how to numerically discretise equations +`[eq:dqcldt_hybrid] <#eq:dqcldt_hybrid>`__ and +`[eq:dcdt_hybrid] <#eq:dcdt_hybrid>`__ to compute cloud increments due +to erosion. The simplest approach is an explicit forwards-in-time +discretisation: + +.. math:: + + \frac{ \Delta {q_{cl}}_{ero}}{\Delta t} = A(C_l^n) K(q_{sat}-q_v) + \label{eq:hybrid_erosion_expl} + +i.e. the increment is calculated by evaluating the term :math:`A` from +equations `[eq:S_Cl] <#eq:S_Cl>`__ and +`[eq:A_top_and_bottom] <#eq:A_top_and_bottom>`__ using the value of +cloud-fraction :math:`C_l` *before* erosion has been applied. + +However, when the environment is significantly subsaturated (so that the +term :math:`(q_{sat}-q_v)` is large and negative), and long timesteps +:math:`\Delta t` are used (e.g. order 1000 s used in global climate +simulations), this discretization can suffer severe numerical overshoot. +i.e. the increment based on :math:`C_l^n` is large enough to reduce +:math:`q_{cl}` (and hence also :math:`C_l`) to less than zero within a +single timestep. If the continuous equation were solved analytically +this wouldn’t happen; as :math:`C_l` declines due to the erosion, so +will :math:`A(C_l)` and hence the erosion rate, so that :math:`q_{cl}` +and :math:`C_l` smoothly decline towards zero. + +Three options are available in the code to address this problem, +selected by the UM namelist switch **i_pc2_erosion_numerics**, detailed +below. Single-Column Model tests indicate that the 2nd and 3rd options +yield much less timestep sensitivity for detrained cloud in shallow +cumulus regimes. + +#. **Retain the explicit discretization, but limit the resulting erosion + increments to ensure :math:`q_{cl}` and :math:`C_l` don’t go + negative. (i_pc2_erosion_numerics=1)** Also, to ensure that some + cloud remains at end-of-timestep where shallow cumulus is detraining + into dry environments, the erosion calculation is fed copies of the + fields with the current timestep’s convection increments subtracted + off. This means any cloud detrained by convection during the current + timestep cannot be eroded until the following timestep, and so is + still present at end-of-timestep. As discussed in section + `7.4.11 <#sec:timestepping>`__, this leads to a problematic timestep + sensitivity, since the amount of cloud not subject to erosion is the + convection increment, which scales with the timestep length. + + Having computed the erosion :math:`q_{cl}` increment using + `[eq:hybrid_erosion_expl] <#eq:hybrid_erosion_expl>`__, the + consistent :math:`C_l` increment is computed by discretising + `[eq:dcdt_hybrid] <#eq:dcdt_hybrid>`__ as: + + .. math:: + + \frac{\Delta {C_l}_{ero}}{\Delta t} + = - \frac{ G(-Q_c)^n Q_c^n \frac{\Delta \overline{{q_{cl}}_{ero}}}{\Delta t} } + { (- C_l^n Q_c^n + ( \overline{q_{cl}^n} + + \frac{1}{2} \Delta \overline{{q_{cl}}_{ero}} ) ) } + \label{eq:dcdt_hybrid_discr} + + i.e. all terms are treated explicitly (using the values before + erosion), except for :math:`\overline{q_{cl}}` which takes the + mid-point interpolated half-way between its values before and after + erosion, to give some improvement in accuracy. + +#. **Use an approximate implicit discretisation, which intrinsically + yields a positive solution for :math:`q_{cl}` and :math:`C_l`. + (i_pc2_erosion_numerics=2)** The copies of the fields passed to the + erosion calculation are fully updated with the convection increments. + We then write equation `[eq:dqcldt_hybrid] <#eq:dqcldt_hybrid>`__ in + the form: + + .. math:: \frac{\partial q_{cl}}{\partial t} = q_{cl} f(q_{cl},C_l,(q_{sat}-q_v)) + + (where the term + :math:`f(q_{cl},C_l,(q_{sat}-q_v)) = \frac{A K(q_{sat}-q_v)}{q_{cl}}` + will be treated explicitly, under the assumption that this ratio will + evolve more slowly while erosion rapidly reduces both the numerator + and the denominator). + + We then take a backwards-in-time implicit discretisation in terms of + the leading factor :math:`q_{cl}`: + + .. math:: \frac{ q_{cl}^{n+1} - q_{cl}^{n}}{\Delta t} = q_{cl}^{n+1} f^n + + Now, the problem is somewhat complicated by the fact that in the + code, erosion is calculated in parallel with the homogeneous forcing + by convection, and we need to account for the homogeneous forcing + increment :math:`\Delta q_{cl}^{hom}` in our implicit solution. We + therefore write the above as: + + .. math:: + + \Delta q_{cl}^{ero} = \Delta t + ( q_{cl}^n + \Delta q_{cl}^{hom} + \Delta q_{cl}^{ero} ) f^n + + Rearranging: + + .. math:: + + \Delta q_{cl}^{ero} = \Delta t f^n q_{cl}^n \frac{ q_{cl}^n + \Delta q_{cl}^{hom} } + { q_{cl}^n - \Delta t f^n q_{cl}^n } + + Note that the term :math:`\Delta t f^n q_{cl}^n` is the erosion + increment we would obtain from the purely explicit discretisation, + :math:`\Delta q_{cl}^{ero\,expl}`. The implicit discretisation is + implemented by first calculating :math:`\Delta q_{cl}^{ero\,expl}` + using equation `[eq:hybrid_erosion_expl] <#eq:hybrid_erosion_expl>`__ + (as we do for **i_pc2_erosion_numerics=1**) but then rescaling it + using the above expression, which becomes: + + .. math:: + + \Delta q_{cl}^{ero} = \Delta q_{cl}^{ero\,expl} \frac{ q_{cl}^n + \Delta q_{cl}^{hom} } + { q_{cl}^n - \Delta q_{cl}^{ero\,expl} } + \label{eq:hybrid_erosion_impl_qcl} + + Provided erosion is acting to reduce cloud-water + (:math:`\Delta q_{cl}^{ero\,expl} < 0`), and homogeneous forcing by + convection has not already completely removed the cloud + (:math:`q_{cl}^n + \Delta q_{cl}^{hom} > 0`), + `[eq:hybrid_erosion_impl_qcl] <#eq:hybrid_erosion_impl_qcl>`__ is + guaranteed to yield a stable, positive solution for :math:`q_{cl}`. + + We also apply exactly the same argument to the equation for the + cloud-fraction increment :math:`C_l`, and obtain: + + .. math:: + + \Delta C_l^{ero} = \Delta C_l^{ero\,expl} \frac{ C_l^n + \Delta C_l^{hom} } + { C_l^n - \Delta C_l^{ero\,expl} } + \label{eq:hybrid_erosion_impl_Cl} + + Where :math:`\Delta C_l^{ero\,expl}` is computed using eq + `[eq:dcdt_hybrid_discr] <#eq:dcdt_hybrid_discr>`__, except that the + term :math:`\frac{1}{2} \Delta \overline{{q_{cl}}_{ero}}` is omitted + (interpolating to the mid-point value of :math:`\overline{q_{cl}}` in + the denominator would be “double-counting” if we are already making + an implicit correction to the full increment). + + In the case where the homogeneous forcing increments have already + removed all of the cloud water content or fraction, erosion is not + performed, and :math:`q_{cl}` and :math:`C_l` are both set to zero. + In the case where erosion is actually acting to increase + cloud-fraction, the code defaults to retaining the explicit + discretisation solution :math:`\Delta q_{cl}^{ero\,expl}` and + :math:`\Delta C_l^{ero\,expl}`. Otherwise, equations + `[eq:hybrid_erosion_impl_qcl] <#eq:hybrid_erosion_impl_qcl>`__ and + `[eq:hybrid_erosion_impl_Cl] <#eq:hybrid_erosion_impl_Cl>`__ are + applied to yield the implicit solution. + +#. **Use an analytic solution to the integration of the time-derivatives + in (**\ `[eq:dqcldt_hybrid] <#eq:dqcldt_hybrid>`__\ **) and + (**\ `[eq:dcdt_hybrid] <#eq:dcdt_hybrid>`__\ **) for greater + accuracy. (i_pc2_erosion_numerics=3)** + + Two problems have been identified with the above implicit numerical + method: + + - The implicit correction is applied completely independently to the + increments for :math:`q_{cl}` and :math:`C_l`. So as with the + explicit method, differing numerical error in the increments for + the two variables can lead to them becoming inconsistent with + eachother. It was found by experimentation that even with the + implicit correction, it is possible for erosion to reduce + :math:`C_l` by a bigger fraction than :math:`q_{cl}`, so that the + in-cloud water content :math:`\frac{q_{cl}}{C_l}` is *increased*. + Narrowing the PDF should only *decrease* the in-cloud + water-content; occasional large increases due to numerical error + can lead to spurious precipitation being produced by the + microphysics scheme. + + - The implicit correction makes it impossible for erosion to reduce + :math:`q_{cl}`, :math:`C_l` to zero. As we will show below, the + analytic solution to the equations posed does in fact go to zero + after a finite time under grid-mean subsaturation (although the + erosion rate declines with :math:`C_l` as it approaches zero, + :math:`C_l` approaches zero more slowly than :math:`q_{cl}`, so + that both variables decrease following power-law curves not + exponentials). When erosion (wrongly) can never entirely remove + cloud, this allows tiny values of :math:`q_{cl}` and :math:`C_l` to + spuriously spread across the domain via numerical diffusion from + the model’s advection scheme. + + Under this option, we attempt to compute an analytic solution to the + simultaneous differential equations + `[eq:dqcldt_hybrid] <#eq:dqcldt_hybrid>`__ and + `[eq:dcdt_hybrid] <#eq:dcdt_hybrid>`__ so that :math:`q_{cl}` and + :math:`C_l` both decrease smoothly and consistently. The equations + lead to somewhat different behaviour depending on whether the + grid-mean state is subsaturated (:math:`Q_c < 0`), supersaturated + (:math:`Q_c > 0`), or close to saturation (:math:`Q_c` near-zero). We + can employ different approximations to integrate the equations in + each case. In the code, we first test the value of :math:`Q_c` and + compute the erosion increments as follows: + + #. **Grid-mean subsaturation (:math:`Q_c < 0`):** + + The relation between the erosion tendencies in + liquid-cloud-fraction and liquid water content + (`[eq:dcdt_hybrid] <#eq:dcdt_hybrid>`__) can be expressed in terms + of *fractional* rates of change (dividing the top and bottom by + :math:`-C_l Q_c`, and dividing both sides by :math:`C_l`): + + .. math:: + + \frac{1}{C_l} \frac{\partial C_l}{\partial t} + = \frac{ G(-Q_c) \frac{q_{cl}}{C_l^2} }{ 1 - \frac{q_{cl}}{C_l Q_c} } \; + \frac{1}{q_{cl}} \frac{\partial q_{cl}}{\partial t} + \label{eq:dcdt_hybrid_1} + + Under homogeneous forcing (section `3.2 <#sec:homog>`__), we + defined the PDF height at the saturation boundary when near the + cloudy end of the PDF as + :math:`G(-Q_c) = \frac{n+1}{n+2} \frac{C_l^2}{q_{cl}}` (eq + `[eqn20] <#eqn20>`__). In fact, :math:`G(-Q_c)` is set to some + blend between this and the value near the clear end of the PDF (eq + `[eqn21] <#eqn21>`__). But we will assume that when eroding cloud + under grid-mean subsaturated conditions (:math:`Q_c < 0`), + :math:`G(-Q_c)` follows this scaling with + :math:`\frac{C_l^2}{q_{cl}}` even if its value differs somewhat + from eq `[eqn20] <#eqn20>`__. Therefore the quantity + :math:`c_1 = G(-Q_c) \frac{q_{cl}}{C_l^2}` remains constant during + the erosion process, and eq + `[eq:dcdt_hybrid_1] <#eq:dcdt_hybrid_1>`__ becomes: + + .. math:: + + \frac{1}{C_l} \frac{\partial C_l}{\partial t} + = \frac{ c_1 }{ 1 - \frac{q_{cl}}{C_l Q_c} } \; + \frac{1}{q_{cl}} \frac{\partial q_{cl}}{\partial t} + \label{eq:dcdt_hybrid_2} + + The term :math:`1 - \frac{q_{cl}}{C_l Q_c}` (which is :math:`> 1` + since we are considering grid-mean subsaturation :math:`Q_c < 0`) + usually remains close to 1 in practice, so we can assume its + fractional variation over the timestep is small compared to the + other terms, and treat it explicitly. We can therefore + straightforwardly integrate eq + `[eq:dcdt_hybrid_2] <#eq:dcdt_hybrid_2>`__ to obtain the scaling + of :math:`C_l` with :math:`q_{cl}` as both are reduced by erosion: + + .. math:: + + \frac{C_l}{{C_l}_0} = \left( \frac{q_{cl}}{{q_{cl}}_0} \right)^{b_1} + \label{eq:cl_qcl_scaling} + + where :math:`{C_l}_0`, :math:`{q_{cl}}_0` are the values before + erosion is applied, and the exponent is + :math:`b_1 = \frac{ c_1 }{ 1 - \frac{q_{cl}}{C_l Q_c} }`. When + :math:`G(-Q_c)` takes its value from the cloudy end of the PDF, we + have :math:`c_1 = \frac{n+1}{n+2}`. Since the PDF power + :math:`n > 0` and :math:`Q_c < 0` under the considered grid-mean + subsaturation, we always have :math:`b_1 < 1`. This ensures that + erosion reduces :math:`C_l` at a slower fractional rate than + :math:`q_{cl}`, so that in-cloud water content + :math:`\frac{q_{cl}}{C_l}` always decreases. + + Next we derive an integral solution for the decline of + :math:`q_{cl}` with time. Ignoring the cloud surface-area + contributions from the levels above and below (they are disabled + in the code anyway), the erosion liquid water content tendency is + obtained by combining `[eq:dqcldt_hybrid] <#eq:dqcldt_hybrid>`__ + and `[eq:S_Cl] <#eq:S_Cl>`__: + + .. math:: + + \frac{\partial q_{cl}}{\partial t} = -K \, 2 C_l (1 - C_l) \, (q_{sat}(T)-q_v) + \label{eq:dqcldt_hybrid_1} + + From eq `[SD2] <#SD2>`__, :math:`q_{sat}(T)-q_v = \frac{SD}{a_L}`, + where :math:`SD` is the saturation defecit, and :math:`a_L` is the + dimensionless factor defined in eq `[eq:a_L] <#eq:a_L>`__. + Following the derivation in section + `4.9.3 <#sec:smooth_initiation>`__ (eq + `[eq:qc_plus_sd] <#eq:qc_plus_sd>`__), we can write this in terms + of the liquid-water content: :math:`SD = q_{cl} - Q_c` (where + :math:`Q_c` was defined in eq + `[eq:qc_eq_qt-qs] <#eq:qc_eq_qt-qs>`__, and corresponds to the + grid-mean supersaturation converted to an equivalent liquid water + content). Substituting this into + (`[eq:dqcldt_hybrid_1] <#eq:dqcldt_hybrid_1>`__) above, we obtain: + + .. math:: + + \frac{\partial q_{cl}}{\partial t} = -\frac{K}{a_L} \, 2 C_l (1 - C_l) \, + (q_{cl}-Q_c) + \label{eq:dqcldt_hybrid_2} + + Substituting eq `[eq:cl_qcl_scaling] <#eq:cl_qcl_scaling>`__ for + the leading factor of :math:`C_l` on the right-hand-side and + rearranging: + + .. math:: + + \left( \frac{q_{cl}}{{q_{cl}}_0} \right)^{-b_1} \frac{\partial q_{cl}}{\partial t} + = -\frac{K}{a_L} \, 2 {C_l}_0 (1 - C_l) \, (q_{cl}-Q_c) + + In significantly subsaturated conditions the r.h.s. has only weak + dependence on :math:`C_l` and :math:`q_{cl}` (:math:`C_l << 1`, + :math:`q_{cl} << -Q_c`), so we can treat the whole r.h.s. + explicitly (i.e. neglect its variation during each timestep), so + that the above integrates to: + + .. math:: + + \left[ \frac{{q_{cl}}_0}{1-b_1} \left( \frac{q_{cl}}{{q_{cl}}_0} \right)^{1-b_1} + \right]_{{q_{cl}}_0}^{{q_{cl}}_{\Delta t}} + = -\frac{K}{a_L} \, 2 {C_l}_0 (1 - C_l) \, (q_{cl}-Q_c) \Delta t + + Inserting the limits of the integral on the l.h.s. and + rearranging, we obtain our analytical solution for :math:`q_{cl}` + after time :math:`\Delta t`: + + .. math:: + + {q_{cl}}_{\Delta t} = {q_{cl}}_0 \left( 1 - \frac{1-b_1}{{q_{cl}}_0} + \frac{K}{a_L} \, 2 {C_l}_0 (1 - C_l) \, (q_{cl}-Q_c) \Delta t + \right)^\frac{1}{1-b_1} + \label{eq:qcl_int_hybrid} + + Note that :math:`q_{cl}` falls to zero after a finite time + :math:`\frac{{q_{cl}}_0}{1-b_1} \frac{a_L}{K} + \frac{1}{2 {C_l}_0 (1 - C_l) \, (q_{cl}-Q_c)}`. If the timestep + :math:`\Delta t` is longer than this time, then erosion completely + removes the cloud during the current timestep. + + We first set :math:`{q_{cl}}_0` and :math:`{C_l}_0` to the values + already updated by homogeneous forcing, and then sequentially + compute the updated :math:`q_{cl}` after erosion using + (`[eq:qcl_int_hybrid] <#eq:qcl_int_hybrid>`__). Then we substitute + this value into (`[eq:cl_qcl_scaling] <#eq:cl_qcl_scaling>`__) to + compute the consistent updated value of :math:`C_l`. Finally, to + improve accuracy, a small number of iterations are performed to + find the solution with the explicitly-treated terms (the exponent + :math:`b_1 = \frac{ c_1 }{ 1 - \frac{q_{cl}}{C_l Q_c} }` and the + terms :math:`(1 - C_l)` and :math:`(q_{cl}-Q_c)` in eq + `[eq:qcl_int_hybrid] <#eq:qcl_int_hybrid>`__) adjusted to values + linearly-interpolated to half-way between the start and end of the + erosion timestep. + + #. **Grid-mean supersaturation (:math:`Q_c > 0`):** + + In this case, erosion does not act to reduce :math:`C_l` and + :math:`q_{cl}` towards zero. Instead, the narrow PDF limit it + adjusts towards has no remaining subsaturated air, so that + :math:`C_l = 1` and :math:`q_{cl} = Q_c`. In this case, we can + repeat the above derivation, but considering the equations for the + clear-fraction :math:`1-C_l` in place of :math:`C_l`, and the + saturation defecit :math:`SD = q_{cl}-Q_c` in place of + :math:`q_{cl}`. Assuming that :math:`G(-Qc)` follows the scaling + for the clear end of the PDF (`[eqn21] <#eqn21>`__), this leads to + a similar equation to + (`[eq:cl_qcl_scaling] <#eq:cl_qcl_scaling>`__) but for the scaling + as erosion reduces :math:`1-C_l` and :math:`SD` towards zero .: + + .. math:: + + \frac{1-C_l}{1-{C_l}_0} = \left( \frac{SD}{SD_0} \right)^{b_2} + \label{eq:ca_sd_scaling} + + with :math:`b_2 = \frac{ c_2 }{ 1 + \frac{SD}{(1-C_l) Q_c} }` and + :math:`c_2 = G(-Q_c) \frac{SD}{(1-C_l)^2}` (note we must have + :math:`0 < b_2 < 1`). + + And then the tendency equation for :math:`SD` is: + + .. math:: + + \frac{\partial SD}{\partial t} = -\frac{K}{a_L} \, 2 (1 - C_l) C_l \, SD + \label{eq:dsddt_hybrid_2} + + The one asymmetry between this and the :math:`q_{cl}` tendency + equation (`[eq:dqcldt_hybrid_2] <#eq:dqcldt_hybrid_2>`__) is that + for :math:`SD` the r.h.s. is directly proportional to the quantity + in the time-derivative, whereas for :math:`q_{cl}` there is an + additional :math:`Q_c` term which is constant during erosion. + Substituting (`[eq:ca_sd_scaling] <#eq:ca_sd_scaling>`__) for the + leading factor of :math:`(1 - C_l)` in + (`[eq:dsddt_hybrid_2] <#eq:dsddt_hybrid_2>`__), integrating over + time :math:`\Delta t` (neglecting the fractional variation of + :math:`C_l` over the timestep) and rearranging, we obtain: + + .. math:: + + {SD}_{\Delta t} = {SD}_0 \left( 1 + b_2 + \frac{K}{a_L} \, 2 (1-{C_l}_0) C_l \, \Delta t + \right)^{-\frac{1}{b_2}} + \label{eq:sd_int_hybrid} + + Note that the additional power of :math:`SD` on the r.h.s. of + (`[eq:dsddt_hybrid_2] <#eq:dsddt_hybrid_2>`__) leads to the + integral solution having a negative exponent. This means that + under grid-mean supersaturation, erosion makes :math:`SD` and + :math:`1-C_l` approach but never quite reach zero, which is quite + different behaviour to grid-mean subsaturation where + :math:`q_{cl}` and :math:`C_l` go to zero over a finite time. This + asymmetry is because the erosion rate is parameterised to be + proportional to :math:`SD`, and this tends to zero as the PDF is + narrowed under supersaturation, but remains finite positive under + subsaturation. + + We first set :math:`{SD}_0 = {q_{cl}}_0 - Q_c` (where as above + :math:`{q_{cl}}_0` is the value already updated by homogeneous + forcing), then compute the value of :math:`SD` updated by erosion + using (`[eq:sd_int_hybrid] <#eq:sd_int_hybrid>`__). Then we + substitute this value into + (`[eq:ca_sd_scaling] <#eq:ca_sd_scaling>`__) to compute the + consistent updated value of :math:`1-C_l`. A small number of + iterations are then performed to find the solution with the + explicitly-treated terms (the exponent + :math:`b_2 = \frac{ c_2 }{ 1 + \frac{SD}{(1-C_l) Q_c} }` and the + term :math:`C_l` in eq `[eq:sd_int_hybrid] <#eq:sd_int_hybrid>`__) + adjusted to values linearly-interpolated to half-way between the + start and end of the erosion timestep. Then the final values of + :math:`SD` and :math:`1-C_l` are used to increment + :math:`q_{cl} = Q_c + SD` and :math:`C_l`, as prognosed by the + rest of the model. + + #. **grid-mean saturation (:math:`Q_c` near-zero):** + + In this case, the PDF is centred on the saturation boundary, so + that narrowing it does not change the cloud-fraction. In the limit + :math:`Q_c = 0`, we have :math:`q_{cl} = SD`, and + (`[eq:dqcldt_hybrid_2] <#eq:dqcldt_hybrid_2>`__) or + (`[eq:dsddt_hybrid_2] <#eq:dsddt_hybrid_2>`__) becomes: + + .. math:: + + \frac{1}{q_{cl}} \frac{\partial q_{cl}}{\partial t} + = -\frac{K}{a_L} \, 2 C_l (1 - C_l) + + where everything on the r.h.s. is constant under erosion. This + simply integrates to give exponential decline of :math:`q_{cl}` + (and :math:`SD`) towards zero: + + .. math:: {q_{cl}}_{\Delta t} = {q_{cl}}_0 e^{ -\frac{K}{a_L} \, 2 C_l (1 - C_l) \Delta t } + +Orographic and Gravity Wave Drag +-------------------------------- + +The Orographic and Gravity Wave Drag sections do not alter the +temperature or moisture content of the model gridboxes, hence PC2 +assumes no change in the condensate and cloud fractions as a result of +these processes. + +.. _`sec:advec`: + +Advection +--------- + +The advection of :math:`\overline{q_{cl}}` and :math:`\overline{q_{cf}}` +are already performed separately by the semi-Lagrangian advection +scheme. Advection of the three cloud fractions :math:`C_l`, :math:`C_i` +and :math:`C_t` are all performed by PC2 in the same way. + +Note that ascent or subsidence by advection entails a pressure change +following each parcel, which will cause an accompanying adiabatic +temperature change. These advective pressure and temperature changes +imply a homogeneous forcing, which yields a change in +:math:`\overline{q_{cl}}` and :math:`C_l` in addition to their transport +by the winds. This is described in section `4.8 <#sec:pres>`__. + +If the UM namelist switch **l_pc2_sl_advection** is turned on, the PC2 +homogeneous forcing response to advection is calculated straight after +the call to Semi-Lagrangian advection. Otherwise, the pressure change +from advection is combined with the Eulerian pressure change from the +dynamics Helmholtz solver, and the resulting homogeneous forcing of +liquid cloud is computed at the end of the timestep. + +.. _`sec:bl`: + +Boundary Layer +-------------- + +At a basic level, the boundary layer scheme works by mixing +:math:`\overline{q_T}` and :math:`\overline{T_L}`, and tracer mixing +:math:`\overline{q_{cf}}`. The condensation and :math:`C_l` changes are +represented using the homogeneous forcing representation. The forcing of +:math:`Q_c` can be written in :math:`\Delta \overline{q_T}` and +:math:`\Delta \overline{T_L}` terms using +(`[eq:deltaqc_exp] <#eq:deltaqc_exp>`__). + +:math:`\overline{q_{cf}}` is already mixed using the tracer mixing +scheme. PC2 will calculate the corresponding :math:`C_i` change assuming +the inhomogeneous forcing scenario. Although this is not necessarily an +appropriate physical model to use, it is the only generic physical model +we have currently developed in order to convert increments in a +condensate to increments in a cloud fraction. We use a value of the +in-cloud water content :math:`q_c^S` based upon a linear combination of +the current in-cloud ice water content, +:math:`\frac{\overline{q_{cf}}}{C_i}`, and a fixed value. + +.. math:: + + q_C^S = C_i \frac{\overline{q_{cf}}}{C_i} + ( 1 - C_i) q_{cf0 \, BL} + \label{eq:qcf_ci} + +where :math:`q_{cf0 \, BL}` is a specified value of +:math:`1 \times 10^{-4} \, kg \, kg^{-1}`. We then use the inhomogeneous +forcing equation based upon (`[eq:dcdt_inhom2] <#eq:dcdt_inhom2>`__) but +for ice water content to write + +.. math:: + + \Delta C_i = \frac{(1 - C_i)}{q_C^S - \overline{q_{cf}}} Q4_i . + \label{eq:deltaci_bl} + +Since the physical model will have :math:`C_i` tend to 1 if the +denominator is small, we will, to avoid numerical problems, set +:math:`C_i` to 1 if +:math:`q_C^S - \overline{q_{cf}} < 1 \times 10^{-10} kg kg^{-1}`. Note +that we do not use the multiple phases injection source expressions +(section `3.5.1 <#sec:multiple>`__ and equation +`[eq:cff_ts] <#eq:cff_ts>`__). This is because the liquid water changes +are not associated with the plume model. + +Equation `[eq:qcf_ci] <#eq:qcf_ci>`__ assumes that the change to the ice +water content has led to an increase in ice water content. However, if +the ince water content has reduced, the change to the ice cloud fraction +is not consistent. The option to "Use consistent formulation of ice +cloud fraction changes due to boundary-layer processes" ensure that if +the ice water content is reduced, the ice cloud fraction is reduced, in +such as way as to maintain the same in-cloud ice water content. + +The :math:`C_t` changes are calculated using the minimum overlap method +of section `3.6 <#sec:ct>`__. + +In *ni-imp-ctl* the control code inhibits the call to the diagnostic +cloud scheme if there is deep or shallow convection occurring and the +model level is less than *or equal to* the layer immediately above the +top of the boundary layer mixed layer (i.e. level ntml+1). This is in +order to ensure that there is no large-scale cloud present below the +base of the convective cloud, but additionally performs this calculation +at the level above, probably in order that latent heating from +large-scale condensation does not inhibit the convection. A similar +thing is performed for PC2, with any large-scale cloud being evaporated +if the same criteria are met, *except that it is not performed on the +level above the boundary layer mixed layer*. This choice (i.e. ntml) is +seen to give improved results in PC2, and is arguably a more physical +reasonable choice anyway than using ntml+1. + +.. _`sec:convec`: + +Convection +---------- + +This section concentrates specifically upon the PC2 interface to the +convection scheme. In the current formulation of the UM, only a +mass-flux convection scheme exists, and this is what is described below. +Work to interface PC2 to the developing turbulence based convection +scheme is commented upon in section `7.4.4 <#sec:tbcs>`__. + +An alternative way of calculating cloud fraction increments is currently +under development and is described in section +`4.7.10 <#sec:conv-simpler>`__. + +A traditional view of convective parametrization is a scheme that +transports vapour, :math:`q`, heat, :math:`\theta`, and momentum, +:math:`u` and :math:`v` winds within a single column. It does not +consider transport sideways to adjoining columns, and (at least in the +Gregory-Rowntree scheme used in the UM) is considered independent of any +resolved scale vertical air motions. This necessitates the view of +compensating subsidence within the column, whereas some conceptual +models of tropical convection would have the bulk of the ascent in the +convective cores and the associated descent thousands of miles away in +the downward branch of the Hadley circulation. The parametrization +schemes traditionally overlook the existence of condensate in the model +column. The non-PC2 version of the mass-flux convection scheme used in +the UM would have the same large-scale liquid and ice prognostics before +and after convection occurs (apart from a bolt-on evaporation below +convective cloud base), with no regard at all to what happens to it or +its effect on the rest of the convection. Within PC2 we have had to work +to more fully incorporate the condensate into the convection scheme. + +Introduction to the convective mass flux scheme +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +Within the mass flux scheme the net change in :math:`\overline{q_{cl}}` +and :math:`C_l` etc. comes from two distinct sources. Firstly, the +condensate and cloud fraction injected from the plume (the :math:`Q4` +terms, section `3.5 <#sec:inhomog>`__); secondly, the condensation +response to the vapour and heat changes associated with the detrainment +and compensating subsidence. Strictly, we will see that the :math:`Q4` +terms also include the contribution to the condensate transport by the +compensating subsidence - this casts doubt on the validity of the +application of the injection forcing scenario to calculate the +equivalent cloud fraction change, since ideally the cloud fractions +ought to be transported by the compensating subsidence in a similar way +to the condensate transport (which is documented below). + +We therefore split the convective contribution in +(`[eq:dqcldt_and_dcdt] <#eq:dqcldt_and_dcdt>`__) into two parts: + +.. math:: + + \frac{\partial \overline{q_{cl}}}{\partial t} |_{convection} = + Q4_l + Q_{environment} + \label{eq:inhomg_plus_homog} + +where :math:`Q_{environment}` is the condensation associated with +changes in the vapour and temperature from the detrainment and +compensating subsidence. Similar splits are made for the cloud +variables, where the injection forcing, section `3.5 <#sec:inhomog>`__, +is used to calculate the first term from :math:`Q4_l`. Section +`4.7.3 <#subsect:q4calculation>`__ looks at the issue of the calculation +of :math:`Q4_l` etc., and section `4.7.4 <#sec:conv_homog>`__ looks at +the calculation of :math:`Q_{environment}`, and its associated cloud +fraction change. We first look at the basic transport equations in a +mass flux convection scheme. + +.. _`subsect:basmaseqs`: + +Basic Equations for a Convective Mass Flux Scheme +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +We first consider a generic mass-flux scheme before its application to +PC2. As discussed by Grant and Stirling (personal communication), the +equations for convective tendencies are most simply applied to a +variable, :math:`{\chi}`, that is conserved under moist adiabatic +processes (e.g. total water content). In this case, + +.. math:: + + {\ensuremath{\frac{\partial \, \ensuremath{{\chi}_{\rm{ }}^{\rm{E}}}}{\partial \, t}}}_{\rm{conv}} = + - \frac{1}{\overline{\rho}} \, \ensuremath{\frac{\partial \, \overline{\rho w^{'} \ensuremath{{\chi}_{\rm{ }}^{\rm{E'}}}}}{\partial \, z}} + \label{eq:chibasic} + +To parametrize `[eq:chibasic] <#eq:chibasic>`__, the current UM +convection scheme takes a mass flux approximation + +.. math:: + + \left({\overline{\rho w^{'} \ensuremath{{\chi}_{\rm{ }}^{\rm{E'}}}}} \right)_{\rm{conv}} = M^{\rm{P}} \, + \left({ \ensuremath{{\chi}_{\rm{ }}^{\rm{P}}} - \ensuremath{{\chi}_{\rm{ }}^{\rm{E}}} } \right) + \label{eq:massflux} + +which can be differentiated to give + +.. math:: + + - \frac{1}{\overline{\rho}} \, \ensuremath{\frac{\partial \, \overline{\rho w^{'} \ensuremath{{\chi}_{\rm{ }}^{\rm{E'}}}}}{\partial \, z}} = + \ensuremath{\frac{\partial \, \ensuremath{{\chi}_{\rm{ }}^{\rm{P}}} \, M^{\rm{P}}}{\partial \, p}} - + \ensuremath{{\chi}_{\rm{ }}^{\rm{E}}} \, \ensuremath{\frac{\partial \, M^{\rm{P}}}{\partial \, p}} - + M^{\rm{P}} \, \ensuremath{\frac{\partial \, \ensuremath{{\chi}_{\rm{ }}^{\rm{E}}}}{\partial \, p}} + \label{eq:eddyflux} + +The bulk cloud model plume equations for mass and :math:`{\chi}` are: + +.. math:: + + \begin{aligned} + - \ensuremath{\frac{\partial \, M^{\rm{P}}}{\partial \, p}} & = & + \left({ \varepsilon \, M^{\rm{P}} - \mu \, M^{\rm{P}} - \delta \, M^{\rm{P}} } \right) + \label{eq:dbydpmassflux} \\ + - \ensuremath{\frac{\partial \, \ensuremath{{\chi}_{\rm{ }}^{\rm{P}}} \, M^{\rm{P}}}{\partial \, p}} & = & \left({ + \varepsilon \, M^{\rm{P}} \, \ensuremath{{\chi}_{\rm{ }}^{\rm{E}}} + - \mu \, M^{\rm{P}} \, \ensuremath{{\chi}_{\rm{ }}^{\rm{R}}} - \delta \, M^{\rm{P}} \, \ensuremath{{\chi}_{\rm{ }}^{\rm{P}}} + } \right)\label{eq:dbydpmfchi} + \end{aligned} + +Equations `[eq:eddyflux] <#eq:eddyflux>`__, +`[eq:dbydpmassflux] <#eq:dbydpmassflux>`__ and +`[eq:dbydpmfchi] <#eq:dbydpmfchi>`__ can then be substituted into +`[eq:chibasic] <#eq:chibasic>`__ to give: + +.. math:: + + {\ensuremath{\frac{\partial \, \ensuremath{{\chi}_{\rm{ }}^{\rm{E}}}}{\partial \, t}}}_{\rm{conv}} = + - M^{\rm{P}} \, \ensuremath{\frac{\partial \, \ensuremath{{\chi}_{\rm{ }}^{\rm{E}}}}{\partial \, p}} + + \mu \, M^{\rm{P}} \, \left({ \ensuremath{{\chi}_{\rm{ }}^{\rm{R}}} - \ensuremath{{\chi}_{\rm{ }}^{\rm{E}}} } \right) + + \delta \, M^{\rm{P}} \, \left({ \ensuremath{{\chi}_{\rm{ }}^{\rm{P}}} - \ensuremath{{\chi}_{\rm{ }}^{\rm{E}}} } \right) + \label{eq:chimassflux} + +while :math:`{\chi}_{\rm{}}^{\rm{P}}` is obtained from the vertical +gradient derived by combining `[eq:dbydpmassflux] <#eq:dbydpmassflux>`__ +and `[eq:dbydpmfchi] <#eq:dbydpmfchi>`__ : + +.. math:: + + M^{\rm{P}} \, \ensuremath{\frac{\partial \, \ensuremath{{\chi}_{\rm{ }}^{\rm{P}}}}{\partial \, p}} = + \varepsilon \, M^{\rm{P}} \, \left({ \ensuremath{{\chi}_{\rm{ }}^{\rm{P}}} - \ensuremath{{\chi}_{\rm{ }}^{\rm{E}}} } \right)- + \mu \, M^{\rm{P}} \, \left({ \ensuremath{{\chi}_{\rm{ }}^{\rm{P}}} - \ensuremath{{\chi}_{\rm{ }}^{\rm{R}}} } \right) + \label{eq:gradchipar} + +Within the model, eqn `[eq:chimassflux] <#eq:chimassflux>`__ would take +a discretized form which actually depends upon whether the model level, +k, is above or at the lowest cloud level (k = cb). Note that the formal +cloud base lies at the half-level below, i.e. on the layer boundary +which is also the top of the turbulent mixed boundary layer. A simple +discretized form of `[eq:chimassflux] <#eq:chimassflux>`__, setting +:math:`{ \mu = 0 }`, is: + +.. math:: + + \begin{aligned} + {\ensuremath{\frac{\partial \, \ensuremath{{\chi}_{\rm{ }}^{\rm{E}}}}{\partial \, t}}}_{\rm{conv, \, k}} & = & m_{\rm{k+1/2}} \, + \frac{ \left({\ensuremath{{\chi}_{\rm{k+1}}^{\rm{E}}} - \ensuremath{{\chi}_{\rm{k}}^{\rm{E}}}} \right)} + {{\Delta z}_{\rm{k \, \rightarrow \, k+1}}} + + {\delta}_{\rm{k}} \, m_{\rm{k}} \, \left({ \ensuremath{{\chi}_{\rm{k}}^{\rm{P}}} - \ensuremath{{\chi}_{\rm{k}}^{\rm{E}}} } \right) + \qquad \ldots \; \mbox{for k $>$ cb} \label{eq:chidisck} \\ + {\ensuremath{\frac{\partial \, \ensuremath{{\chi}_{\rm{ }}^{\rm{E}}}}{\partial \, t}}}_{\rm{conv, \, cb}} & = & m_{\rm{cb+1/2}} \, + \frac{ \left({\ensuremath{{\chi}_{\rm{cb+1}}^{\rm{E}}} - \ensuremath{{\chi}_{\rm{cb}}^{\rm{E}}}} \right)} + {{\Delta z}_{\rm{cb \, \rightarrow \, cb+1}}} + - m_{\rm{cb}} \, + \left({ \ensuremath{{\chi}_{\rm{i,cb}}^{\rm{P}}} - \ensuremath{{\chi}_{\rm{cb}}^{\rm{E}}} } \right)\label{eq:chidisccb} + \end{aligned} + +where the initial parcel value :math:`{\chi}_{\rm{i,cb}}^{\rm{P}}` may +be chosen to produce a fixed increment or place a closure condition on +the cloud base flux. In fact, the convection equations (see ) differ +from `[eq:chidisck] <#eq:chidisck>`__ and +`[eq:chidisccb] <#eq:chidisccb>`__ because a different discretization is +used, but the principle is unaltered. + +The model convection variables are NOT conserved under moist adiabatic +processes because precipitation processes deplete the column moisture +and condensation processes affect the temperature, specific humidity and +cloud condensate variables. Surprisingly, however, the form of +eqn `[eq:chimassflux] <#eq:chimassflux>`__ is retained even though the +basic equation `[eq:chibasic] <#eq:chibasic>`__ acquires additional +terms for temperature and specific humidity: + +.. math:: + + \begin{aligned} + {\ensuremath{\frac{\partial \, \ensuremath{T_{\rm{ }}^{\rm{E}}}}{\partial \, t}}}_{\rm{conv}} = Q1 & \equiv & + \left({ \frac{L}{c_{P}} } \right)\, {\overline{Q}}_{\rm{par}} + - \frac{1}{\overline{\rho}} \, \ensuremath{\frac{\partial \, \overline{\rho w^{'} \ensuremath{T_{\rm{ }}^{\rm{E'}}}}}{\partial \, z}} + \label{eq:defineq1} \\ + {\ensuremath{\frac{\partial \, \ensuremath{q_{\rm{ }}^{\rm{E}}}}{\partial \, t}}}_{\rm{conv}} = Q2 & \equiv & - {\overline{Q}}_{\rm{par}} + - \frac{1}{\overline{\rho}} \, \ensuremath{\frac{\partial \, \overline{\rho w^{'} \ensuremath{q_{\rm{ }}^{\rm{E'}}}}}{\partial \, z}} + \label{eq:defineq2} + \end{aligned} + +where :math:`{\overline{Q}}_{\rm{par}}` is the rate of condensation +which occurs in the ascending plumes. + +The reason that `[eq:defineq1] <#eq:defineq1>`__ and +`[eq:defineq2] <#eq:defineq2>`__ retain this form is due to cancellation +from the bulk cloud terms equivalent to +`[eq:dbydpmfchi] <#eq:dbydpmfchi>`__ which are modified in the same way +as `[eq:defineq1] <#eq:defineq1>`__ and +`[eq:defineq2] <#eq:defineq2>`__. The change is seen in the vertical +gradient equations based upon `[eq:gradchipar] <#eq:gradchipar>`__ + +.. math:: + + \begin{aligned} + M^{\rm{P}} \, \ensuremath{\frac{\partial \, \ensuremath{T_{\rm{ }}^{\rm{P}}}}{\partial \, p}} & = & + \varepsilon \, M^{\rm{P}} \, \left({ \ensuremath{T_{\rm{ }}^{\rm{P}}} - \ensuremath{T_{\rm{ }}^{\rm{E}}} } \right)- + \mu \, M^{\rm{P}} \, \left({ \ensuremath{T_{\rm{ }}^{\rm{P}}} - \ensuremath{T_{\rm{ }}^{\rm{R}}} } \right)- + \left({ \frac{L}{c_{P}} } \right)\, {\overline{Q}}_{\rm{par}} \label{eq:gradtpar} \\ + M^{\rm{P}} \, \ensuremath{\frac{\partial \, \ensuremath{q_{\rm{ }}^{\rm{P}}}}{\partial \, p}} & = & + \varepsilon \, M^{\rm{P}} \, \left({ \ensuremath{q_{\rm{ }}^{\rm{P}}} - \ensuremath{q_{\rm{ }}^{\rm{E}}} } \right)- + \mu \, M^{\rm{P}} \, \left({ \ensuremath{q_{\rm{ }}^{\rm{P}}} - \ensuremath{q_{\rm{ }}^{\rm{R}}} } \right)+ + {\overline{Q}}_{\rm{par}} \label{eq:gradqpar} \\ + M^{\rm{P}} \, \ensuremath{\frac{\partial \, \ensuremath{l_{\rm{ }}^{\rm{P}}}}{\partial \, p}} & = & + \varepsilon \, M^{\rm{P}} \, \left({ \ensuremath{l_{\rm{ }}^{\rm{P}}} - \ensuremath{l_{\rm{ }}^{\rm{E}}} } \right) + - {\overline{Q}}_{\rm{par}} + PPN \label{eq:gradlpar} + \end{aligned} + +The final calculation of rates in the current condensation scheme (, +section 10) assumes a further condensation term, +:math:`{\overline{Q}}_{\rm{reset}}`, which acts to make the net rate of +change of condensate equal zero, and a final assumption is made that the +environment values of condensate remain zero (and also that +:math:`l_{\rm{ }}^{\rm{R}}` = :math:`l_{\rm{ }}^{\rm{P}}`). The result +is basic equations + +.. math:: + + \begin{aligned} + {\ensuremath{\frac{\partial \, \ensuremath{T_{\rm{ }}^{\rm{E}}}}{\partial \, t}}}_{\rm{conv}} & = & Q1 - + \left({ \frac{L}{c_{P}} } \right)\, {\overline{Q}}_{\rm{reset}} + \label{eq:basictold} \\ + {\ensuremath{\frac{\partial \, \ensuremath{q_{\rm{ }}^{\rm{E}}}}{\partial \, t}}}_{\rm{conv}} & = & Q2 + {\overline{Q}}_{\rm{reset}} + \label{eq:basicqold} \\ + 0 \equiv {\ensuremath{\frac{\partial \, \ensuremath{l_{\rm{ }}^{\rm{E}}}}{\partial \, t}}}_{\rm{conv}} & = & {\overline{Q}}_{\rm{par}} - + {\overline{Q}}_{\rm{reset}} - PPN + - \frac{1}{\overline{\rho}} \, \ensuremath{\frac{\partial \, \overline{\rho w^{'} \ensuremath{l_{\rm{ }}^{\rm{E'}}}}}{\partial \, z}} \nonumber \\ + & = & + \mu \, M^{\rm{P}} \, \ensuremath{l_{\rm{ }}^{\rm{P}}} + \delta \, M^{\rm{P}} \, \ensuremath{l_{\rm{ }}^{\rm{P}}} - + {\overline{Q}}_{\rm{reset}} + \label{eq:basiclold} + \end{aligned} + +By analogy with equations `[eq:defineq1] <#eq:defineq1>`__ and +`[eq:defineq2] <#eq:defineq2>`__, we can define a :math:`Q4` from +`[eq:basiclold] <#eq:basiclold>`__ and state that for the control +convection scheme :math:`Q4 = 0`. The PC2 scheme requires a reassessment +of these assumptions because we wish to allow non-zero environment +condensate values and to allow them to change. + +.. _`subsect:q4calculation`: + +Calculation of Grid-Box Averaged Condensate Rate (Q4) +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +The PC2 condensation scheme allows convection to feed cloud condensate +(ice or liquid) directly into the large scale and to update the cloud +amount accordingly. + +Define + +.. math:: + + \begin{aligned} + \left({ \ensuremath{\frac{\partial \, \ensuremath{l_{\rm{l}}^{\rm{ }}}}{\partial \, t}} } \right)_{\rm{conv}} = Q4_{\rm{l}} & \equiv & + {\overline{Q}}_{\rm{l, par}} - {\overline{Q}}_{\rm{l, reset}} - RAIN - + \frac{1}{\overline{\rho}} \, \ensuremath{\frac{\partial \, \overline{\rho w^{'} \ensuremath{l_{\rm{l}}^{\rm{'}}}}}{\partial \, z}} + \label{eq:defineq4l} \\ + \left({ \ensuremath{\frac{\partial \, \ensuremath{l_{\rm{f}}^{\rm{ }}}}{\partial \, t}} } \right)_{\rm{conv}} = Q4_{\rm{f}} & \equiv & + {\overline{Q}}_{\rm{f, par}} - {\overline{Q}}_{\rm{f, reset}} - SNOW - + \frac{1}{\overline{\rho}} \, \ensuremath{\frac{\partial \, \overline{\rho w^{'} \ensuremath{l_{\rm{f}}^{\rm{'}}}}}{\partial \, z}} + \label{eq:defineq4f} + \end{aligned} + +where the PC2 assumption thus far has been that +:math:`{\overline{Q}}_{\rm{l, reset}} = 0 += {\overline{Q}}_{\rm{f, reset}}`. + +- The current convection scheme assumes that parcel condensate is single + phase (ie. either all liquid or all frozen) and this is seriously + hard-wired into the code. Thus we can treat the precipitation and + parcel condensation processes in :math:`Q4_{\rm{l}}` and + :math:`Q4_{\rm{f}}` separately without worrying about cross-transfer + between the two because at most only one set will ever be active in a + given grid box at one time. However, even for the inactive (zero + parcel condensate) phase, convection mixes environmental air into the + parcel and can therefore maintain a non-zero :math:`Q4`. Enablement of + multiple phase condensate in the current scheme is a task requiring + great caution as the formulations are extremely sensitive to errors in + assignment of condensate phase. + +Based on `[eq:gradlpar] <#eq:gradlpar>`__, the vertical dependence of +condensate is calculated as + +.. math:: + + \begin{aligned} + \ensuremath{\frac{\partial \, \ensuremath{l_{\rm{l}}^{\rm{P}}}}{\partial \, p}} & = & \varepsilon \, + \left({ \ensuremath{l_{\rm{l}}^{\rm{P}}} - \ensuremath{l_{\rm{l}}^{\rm{E}}} } \right)- + \frac{{\overline{Q}}_{\rm{l, par}}}{M^{\rm{P}}} - + \frac{RAIN}{M^{\rm{P}}} \label{eq:vertparl} \\ + \ensuremath{\frac{\partial \, \ensuremath{l_{\rm{f}}^{\rm{P}}}}{\partial \, p}} & = & \varepsilon \, + \left({ \ensuremath{l_{\rm{f}}^{\rm{P}}} - \ensuremath{l_{\rm{f}}^{\rm{E}}} } \right)- + \frac{{\overline{Q}}_{\rm{f, par}}}{M^{\rm{P}}} - + \frac{SNOW}{M^{\rm{P}}}\label{eq:vertparf} + \end{aligned} + +Following , equations `[eq:dbydpmassflux] <#eq:dbydpmassflux>`__, +`[eq:vertparl] <#eq:vertparl>`__ and `[eq:vertparf] <#eq:vertparf>`__ +are discretized: + +.. math:: + + \begin{aligned} + M_{\rm{k} + 1} & = & M_{\rm{k}} \, + \left({ 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right)\, + \left({ 1 - \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right)\, + EPSS_{\rm{k}} \label{eq:discdmfbydp} \\ + \ensuremath{l_{\rm{l \, k + 1}}^{\rm{P}}} & = & \left({ + \ensuremath{l_{\rm{l \, k}}^{\rm{P}}} + + \varepsilon_{\rm{k} + 1/4} \, \Delta p_{\rm{k} + 1/4} \, \ensuremath{l_{\rm{l \, k}}^{\rm{E}}} + + \varepsilon_{\rm{k} + 3/4} \, \Delta p_{\rm{k} + 3/4} \, + \left[{1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4}} \right]\, + \ensuremath{l_{\rm{l \, k + 1}}^{\rm{E}}} + } \right)\, / \, \left({EPSS_{\rm{k}}} \right)\nonumber \\ + { } & { } & + \left({ {\overline{Q}}_{\rm{l} \, \rm{k} + 1} \, / \, M_{\rm{k} + 1}} \right) + - \left({ RAIN_{\rm{k} + 1} \, / \, M_{\rm{k} + 1} } \right) + \label{eq:discvparl} \\ + \ensuremath{l_{\rm{f \, k + 1}}^{\rm{P}}} & = & \left({ + \ensuremath{l_{\rm{f \, k}}^{\rm{P}}} + + \varepsilon_{\rm{k} + 1/4} \, \Delta p_{\rm{k} + 1/4} \, \ensuremath{l_{\rm{f \, k}}^{\rm{E}}} + + \varepsilon_{\rm{k} + 3/4} \, \Delta p_{\rm{k} + 3/4} \, + \left[{1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4}} \right]\, + \ensuremath{l_{\rm{f \, k + 1}}^{\rm{E}}} + } \right)\, / \, \left({EPSS_{\rm{k}}} \right)\nonumber \\ + { } & { } & + \left({ {\overline{Q}}_{\rm{f} \, \rm{k} + 1} \, / \, M_{\rm{k} + 1}} \right) + - \left({ SNOW_{\rm{k} + 1} \, / \, M_{\rm{k} + 1} } \right) + \label{eq:discvparf} + \end{aligned} + +where :math:`EPSS_{\rm{k}} = +\left({1 + \varepsilon_{\rm{k} + 3 / 4} \, \Delta p_{\rm{k} + 3 / 4}} \right)\, +\left({1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4}} \right)`. + +The condensation and precipitation terms in equations +`[eq:discdmfbydp] <#eq:discdmfbydp>`__, +`[eq:discvparl] <#eq:discvparl>`__ and +`[eq:discvparf] <#eq:discvparf>`__ make the equations implicit. They are +therefore solved by starting with an ascent in which condensation and +precipitation terms are suppressed: + +.. math:: + + \begin{aligned} + \ensuremath{l_{\rm{l \, k + 1}}^{\rm{P}}} & = & \frac{\left({ + \ensuremath{l_{\rm{l \, k}}^{\rm{P}}} + + \varepsilon_{\rm{k} + 1/4} \, \Delta p_{\rm{k} + 1/4} \, \ensuremath{l_{\rm{l \, k}}^{\rm{E}}} + + \varepsilon_{\rm{k} + 3/4} \, \Delta p_{\rm{k} + 3/4} \, + \left[{1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4}} \right]\, + \ensuremath{l_{\rm{l \, k + 1}}^{\rm{E}}} + } \right)}{EPSS_{\rm{k}}} \label{eq:discvparldry} \\ + \ensuremath{l_{\rm{f \, k + 1}}^{\rm{P}}} & = & \frac{\left({ + \ensuremath{l_{\rm{f \, k}}^{\rm{P}}} + + \varepsilon_{\rm{k} + 1/4} \, \Delta p_{\rm{k} + 1/4} \, \ensuremath{l_{\rm{f \, k}}^{\rm{E}}} + + \varepsilon_{\rm{k} + 3/4} \, \Delta p_{\rm{k} + 3/4} \, + \left[{1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4}} \right]\, + \ensuremath{l_{\rm{f \, k + 1}}^{\rm{E}}} + } \right)}{EPSS_{\rm{k}}} \label{eq:discvparfdry} + \end{aligned} + +At the base of the convective plume (ie. the level immediately above +cloud base), :math:`l_{\rm{l \, k}}^{\rm{P}}` is initialized to +:math:`l_{\rm{l \, i}}^{\rm{P}}` and :math:`l_{\rm{f \, k}}^{\rm{P}}` to +:math:`l_{\rm{f \, i}}^{\rm{P}}`, where the initial values are chosen +such that the modified form of `[eq:chidisccb] <#eq:chidisccb>`__ +produces zero fluxes at cloud base: + +.. math:: + + \begin{aligned} + Q4_{\rm{l}}(cb) = 0 & = & M_{\rm{cb+1/2}}^{\rm{P}} \, + \ensuremath{\frac{\partial \, \ensuremath{l_{\rm{l}}^{\rm{E}}}}{\partial \, p}} - M_{\rm{cb}}^{\rm{P}}\, + \left({ \ensuremath{l_{\rm{l}}^{\rm{P \, i}}} - \ensuremath{l_{\rm{l}}^{\rm{E}}}(\rm{cb}) } \right)\label{eq:q4lcbi} \\ + Q4_{\rm{f}}(cb) = 0 & = & M_{\rm{cb+1/2}}^{\rm{P}} \, + \ensuremath{\frac{\partial \, \ensuremath{l_{\rm{f}}^{\rm{E}}}}{\partial \, p}} - M_{\rm{cb}}^{\rm{P}}\, + \left({ \ensuremath{l_{\rm{f}}^{\rm{P \, i}}} - \ensuremath{l_{\rm{f}}^{\rm{E}}}(\rm{cb}) } \right)\label{eq:q4fcbi} + \end{aligned} + +As the convection scheme makes the single phase assumption for parcel +condensate, it may be necessary to melt or freeze entrained condensate +at this point and adjust the temperature accordingly. + +.. math:: + + \begin{aligned} + \theta_{\rm{k + 1}}^{\rm{P}} = \theta_{\rm{k + 1}}^{\rm{P}} - + \left(\frac{L_{\rm{F}}}{C_{p} \, \Pi_{\rm{k + 1}}} \right)\, \ensuremath{l_{\rm{f \, k + 1}}^{\rm{P}}} + & \; \ldots \; & \mbox{ if \ensuremath{l_{\rm{f \, k + 1}}^{\rm{P}}} is melted } + \label{eqn:meltlf} \\ + \theta_{\rm{k + 1}}^{\rm{P}} = \theta_{\rm{k + 1}}^{\rm{P}} + + \left(\frac{L_{\rm{F}}}{C_{p} \, \Pi_{\rm{k + 1}}} \right)\, \ensuremath{l_{\rm{l \, k + 1}}^{\rm{P}}} + & \; \ldots \; & \mbox{ if \ensuremath{l_{\rm{l \, k + 1}}^{\rm{P}}} is frozen } + \label{eqn:freezell} + \end{aligned} + +Once a final value for the condensation term +:math:`{\overline{Q}}_{\rm{x} \, \rm{k} + 1} \, / \, M_{\rm{k} + 1}` has +been calculated from the parcel specific humidity equations, it can then +be added to the parcel condensate to give a final pre-precipitation +value. + +- In practice, the rates :math:`{\overline{Q}}_{\rm{x} \, \rm{k} + 1}` + and :math:`PPN` are not calculated explicitly in the code. Instead, + their effect is applied directly as increments to the temperature and + moisture fields. + +The precipitation calculation is unaltered. + +.. math:: + + P_{\rm{k} + 1} = \left({ \ensuremath{l_{\rm{k + 1}}^{\rm{P}}} - \ensuremath{l_{\rm{MIN}}^{\rm{P}}} } \right)\, + M_{\rm{k} + 1} \, / \, g + \label{eq:precip} + +where :math:`l_{\rm{k + 1}}^{\rm{P}}` = +:math:`l_{\rm{l \, k + 1}}^{\rm{P}}` + +:math:`l_{\rm{f \, k + 1}}^{\rm{P}}`. + +- Actually, given that the precipitation calculation appears to be based + upon the hydrostatic equation, it is debatable whether it is even + suitable for use with the New Dynamics model and I guess therefore + that this needs revisiting at some point. + +This reduces the parcel condensate to : + +.. math:: + + \begin{aligned} + \ensuremath{l_{\rm{l \, k + 1}}^{\rm{P}}} & = & \left({ + \frac{\ensuremath{l_{\rm{l \, k + 1}}^{\rm{P}}}}{\ensuremath{l_{\rm{k + 1}}^{\rm{P}}}} + } \right)\, \ensuremath{l_{\rm{MIN}}^{\rm{P}}} \label{eq:vparlfinal} \\ + \ensuremath{l_{\rm{f \, k + 1}}^{\rm{P}}} & = & \left({ + \frac{\ensuremath{l_{\rm{f \, k + 1}}^{\rm{P}}}}{\ensuremath{l_{\rm{k + 1}}^{\rm{P}}}} + } \right)\, \ensuremath{l_{\rm{MIN}}^{\rm{P}}} \label{eq:vparffinal} + \end{aligned} + +The final parcel condensate values are then used in the rate calculation +based upon eqn `[eq:basiclold] <#eq:basiclold>`__: + +.. math:: + + \begin{aligned} + Q4_{\rm{l}}(k) & = & M_{\rm{k+1/2}}^{\rm{P}} \, \ensuremath{\frac{\partial \, \ensuremath{l_{\rm{l}}^{\rm{E}}}}{\partial \, p}} + + \left({ {\mu}_{\rm{k}} \, M_{\rm{k}}^{\rm{P}} + + {\delta}_{\rm{k}} \, M_{\rm{k}}^{\rm{P}} } \right)\, + \left({ \ensuremath{l_{\rm{l}}^{\rm{P}}}(\rm{k}) - \ensuremath{l_{\rm{l}}^{\rm{E}}}(\rm{k}) } \right)- + {\overline{Q}}_{\rm{l, reset}} \label{eq:q4lmassf} \\ + Q4_{\rm{f}}(k) & = & M_{\rm{k+1/2}}^{\rm{P}} \, \ensuremath{\frac{\partial \, \ensuremath{l_{\rm{f}}^{\rm{E}}}}{\partial \, p}} + + \left({ {\mu}_{\rm{k}} \, M_{\rm{k}}^{\rm{P}} + + {\delta}_{\rm{k}} \, M_{\rm{k}}^{\rm{P}} } \right)\, + \left({ \ensuremath{l_{\rm{f}}^{\rm{P}}}(\rm{k}) - \ensuremath{l_{\rm{f}}^{\rm{E}}}(\rm{k}) } \right)- + {\overline{Q}}_{\rm{f, reset}} \label{eq:q4fmassf} + \end{aligned} + +Note that, as a side-effect, the environment equations for potential +temperature and specific humidity are also altered because the +condensate is no longer re-evaporated at the end +(:math:`{\overline{Q}}_{\rm{l, reset}} = 0 += {\overline{Q}}_{\rm{f, reset}}`): + +.. math:: + + \begin{aligned} + \frac{\Delta \, \theta_{\rm{k}}^{\rm{E}}}{\Delta \, t} = + \left(\frac{ M_{\rm{k}} }{ \Delta \, p_{\rm{k}} } \right) + \left[{ + \left({ 1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4} } \right) + \left({ 1 - \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) + \left({ 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) + \left({ \theta_{\rm{k + 1}}^{\rm{E}} - \theta_{\rm{k}}^{\rm{E}} } \right) + } \right . & + & \nonumber \\ + \left({ \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) + \left({ 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) + \left({ \theta_{\rm{k}}^{\rm{R}} - \theta_{\rm{k}}^{\rm{E}} } \right) + & + & \nonumber \\ + \left . { + \left({ \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) + \left({ \theta_{\rm{k}}^{\rm{P}} - \theta_{\rm{k}}^{\rm{E}} } \right) + } \right]& { } & \label{eq:enviroth} + \end{aligned} + +and + +.. math:: + + \begin{aligned} + \frac{\Delta \, q_{\rm{k}}^{\rm{E}}}{\Delta \, t} = + \left(\frac{ M_{\rm{k}} }{ \Delta \, p_{\rm{k}} } \right) + \left[{ + \left({ 1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4} } \right) + \left({ 1 - \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) + \left({ 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) + \left({ q_{\rm{k + 1}}^{\rm{E}} - q_{\rm{k}}^{\rm{E}} } \right) + } \right . & + & \nonumber \\ + \left({ \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) + \left({ 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) + \left({ q_{\rm{k}}^{\rm{R}} - q_{\rm{k}}^{\rm{E}} } \right) + & + & \nonumber \\ + \left . { + \left({ \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) + \left({ q_{\rm{k}}^{\rm{P}} - q_{\rm{k}}^{\rm{E}} } \right) + } \right]& { } & \label{eq:enviroq} + \end{aligned} + +Similarly, eqns `[eq:q4lmassf] <#eq:q4lmassf>`__ and +`[eq:q4fmassf] <#eq:q4fmassf>`__ have a discretized form as follows: + +.. math:: + + \begin{aligned} + \frac{\Delta \, \ensuremath{l_{\rm{l \, k}}^{\rm{E}}}}{\Delta \, t} = + \left(\frac{ M_{\rm{k}} }{ \Delta \, p_{\rm{k}} } \right) + \left[{ + \left({ 1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4} } \right) + \left({ 1 - \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) + \left({ 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) + \left({ \ensuremath{l_{\rm{l \, k + 1}}^{\rm{E}}} - \ensuremath{l_{\rm{l \, k}}^{\rm{E}}} } \right) + } \right . & + & \nonumber \\ + \left({ \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) + \left({ 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) + \left({ \ensuremath{l_{\rm{l \, k}}^{\rm{P}}} - \ensuremath{l_{\rm{l \, k}}^{\rm{E}}} } \right) + & + & \nonumber \\ + \left . { + \left({ \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) + \left({ \ensuremath{l_{\rm{l \, k}}^{\rm{P}}} - \ensuremath{l_{\rm{l \, k}}^{\rm{E}}} } \right) + } \right]& { } & \label{eq:enviroll} + \end{aligned} + +and + +.. math:: + + \begin{aligned} + \frac{\Delta \, \ensuremath{l_{\rm{f \, k}}^{\rm{E}}}}{\Delta \, t} = + \left(\frac{ M_{\rm{k}} }{ \Delta \, p_{\rm{k}} } \right) + \left[{ + \left({ 1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4} } \right) + \left({ 1 - \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) + \left({ 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) + \left({ \ensuremath{l_{\rm{f \, k + 1}}^{\rm{E}}} - \ensuremath{l_{\rm{f \, k}}^{\rm{E}}} } \right) + } \right . & + & \nonumber \\ + \left({ \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) + \left({ 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) + \left({ \ensuremath{l_{\rm{f \, k}}^{\rm{P}}} - \ensuremath{l_{\rm{f \, k}}^{\rm{E}}} } \right) + & + & \nonumber \\ + \left . { + \left({ \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) + \left({ \ensuremath{l_{\rm{f \, k }}^{\rm{P}}} - \ensuremath{l_{\rm{f \, k}}^{\rm{E}}} } \right) + } \right]& { } & \label{eq:envirolf} + \end{aligned} + +.. _`sec:conv_homog`: + +Background condensation +~~~~~~~~~~~~~~~~~~~~~~~ + +The modification to the convective plume will result in the transport, +detrainment and entrainment of condensate, in addition to the transport +of vapour and heat. Although condensation processes within the plume are +treated, it does not treat condensation in the environment, which is +forced by the compensating subsidence. We wish to relate the +environmental increments of vapour and temperature to a forcing that can +be applied in the environment. Because we know that any detrained air +associated with detrained liquid water from the plume must be saturated +with respect to liquid water, we are able to translate the environmental +changes into forcings. + +Here we will consider that the vapour change in the gridbox is as a +result of *saturated with respect to liquid water* air being injected +from the plume and background air being displaced. We do not consider +whether the background air is at saturation yet, for we wish to derive +the expression for the required condensation if this is not the case. We +consider only liquid water clouds, ice clouds have no background +condensation applied as we do not make the instantaneous condensation +assumption. + +Hence we can write + +.. math:: + + \Delta \overline{q} = \Delta C_S ( q_{sat liq}(\overline{T_{s}}) - \overline{q} ) + + (1 - \Delta C_S) \Delta \overline{q_{background}} + +where :math:`\Delta C_S` is the volume of plume air that is detrained +into the gridbox, as discussed by :raw-latex:`\cite{bwg03}`. :math:`T_s` +is the temperature of the air injected into the gridbox by the plume. +The first term is simply the difference between the value of :math:`q` +in the plume and what was previously in the gridbox, and the second term +is the effect of a background change of :math:`q` that will be applied +across the part of the gridbox that is not associated with the injected +air. We write this as: + +.. math:: + + (1 - \Delta C_S) \Delta \overline{q_{background}} = \Delta \overline{q} - + \Delta C_S ( q_{sat liq}(\overline{T_{s}}) - \overline{q} ) . + \label{eqn:1mcs} + +Now we recognise that + +.. math:: + + \Delta \overline{q} = Q2~ \Delta t + \label{eqn:Q2} + +where :math:`Q2` is the rate of moistening of the whole gridbox due to +convection. Remember that, at this stage, we haven’t done any +condensation outside of the plume. Hence to calculate the condensation +we should apply the background change in :math:`\overline{q}` as a +uniform forcing for the background air. Hence +(`[eqn:1mcs] <#eqn:1mcs>`__) becomes, using (`[eqn:Q2] <#eqn:Q2>`__), + +.. math:: + + (1 - \Delta C_S) A_q |_{background} \Delta t = Q2 ~ \Delta t - \Delta C_S + ( q_{sat liq}(T_{s}) - \overline{q} ) . + \label{eqn:Aq} + +where :math:`A_q |_{background}` is the currently unknown background +forcing of :math:`q` (see :raw-latex:`\cite{gwb02}`) and +:math:`\Delta t` is the timestep. We can do the same analysis for the +temperature change, and obtain + +.. math:: + + (1 - \Delta C_S) A_T |_{background} \Delta t = Q1~ \Delta t - + \Delta C_S (T_s - \overline{T} ) + \label{eqn:AT} + +where Q1 is the rate of warming in the gridbox due to convection and +:math:`A_T |_{background}` is the currently unknown background forcing +of temperature. + +The full change of liquid water content in the gridbox is that injected, +:math:`Q4~\Delta t`, plus the amount of condensation in the background +from the uniform forcings (see :raw-latex:`\cite{wg03}`). Note that the +uniform forcings are only applied across a proportion +:math:`1 - \Delta C_S` of the gridbox. Hence these two terms give, using +the homogeneous forcing equations (`[dqcldt] <#dqcldt>`__) and +(`[eq:deltaqc_exp2] <#eq:deltaqc_exp2>`__), + +.. math:: + + \Delta \overline{q_{cl}} |_{convection} = Q4 \Delta t + + (1 - \Delta C_S) a_L C_l (A_q |_{background} \Delta t + - \alpha A_T |_{background} \Delta t ). + \label{eqn:qclconv} + +Using (`[eqn:Aq] <#eqn:Aq>`__) and (`[eqn:AT] <#eqn:AT>`__) to expand +the forcing terms in (`[eqn:qclconv] <#eqn:qclconv>`__) gives + +.. math:: + + \Delta \overline{q_{cl}} |_{convection} = Q4 \Delta t + + \Delta t a_L C_l ( Q2 - \alpha Q1) - \Delta C_S a_L C_l + (q_{sat liq}(T_s)-\overline{q} - \alpha (T_s - \overline{T})) . + +We now note that + +.. math:: q_{sat} (T_s) - q_{sat liq} (\overline{T}) = \alpha (T_s - \overline{T} ) + +and hence the final result + +.. math:: + + \Delta \overline{q_{cl}} |_{convection} = Q4 \Delta t + + \Delta t ~ a_L C_l ( ( Q2 - \alpha Q1) - \Delta C_S + (q_{sat liq}(\overline{T}) - \overline{q} ) ) . + \label{eqn:dqcl} + +There is thus an extra term, +:math:`-\Delta C_S (q_{sat}(\overline{T})-\overline{q} )`, which needs +to be included in addition to the standard application of the +homogeneous forcing of :math:`Q1` and :math:`Q2` (this is represented by +the second term of the expression). This has arisen from the requirement +that the vapour injected by the plume is saturated. We need simply to +retrieve the value of :math:`\Delta C_S` to complete the +parametrization. This can be straightforwardly obtained from +(`[eq:dcldt_inhom] <#eq:dcldt_inhom>`__), which links the net change of +liquid cloudy volume due to the injection, :math:`\Delta C_{injection}`, +with :math:`\Delta C_S`. + +.. math:: \Delta C_{injection} = (g_l - C_l) \Delta C_S + +where :math:`g_l` is 1 if the injected cloud is of liquid phase and 0 if +it is of ice phase. We already know :math:`\Delta C_{injection}` from +the injection forcing arguments (`[eq:dctdt_xl] <#eq:dctdt_xl>`__) above +that link it to :math:`Q4`. We therefore complete the parametrization by +calculating :math:`\Delta C_S` based on whether :math:`\Delta C` is +positive or negative. If :math:`\Delta C` is positive, we assume that +the plume must be of liquid phase and hence + +.. math:: + + \Delta C_S = \frac{\Delta C_{injection}} {1 -C_l} . + \label{eqn:cs1} + +If :math:`\Delta C_l` is negative, we assume that the plume must be of +ice phase and hence + +.. math:: + + \Delta C_S = - \frac{\Delta C_{injection}} {C_l} . + \label{eqn:cs2} + +Here we have still assumed that the vapour content in the detrained +plume is equal to :math:`q_{sat liq}`. A better assumption may be to +replace the :math:`q_{sat liq}` term in (`[eqn:dqcl] <#eqn:dqcl>`__) +with a :math:`q_{sat}` expression that depends on the volume fraction of +detrained condensate that is liquid phase, :math:`g_l`. + +If :math:`\Delta C_{injection}` is zero, we assume that +:math:`\Delta C_S` is 0 also. Equations +(`[eqn:dqcl] <#eqn:dqcl>`__),(`[eqn:cs1] <#eqn:cs1>`__), and +(`[eqn:cs2] <#eqn:cs2>`__) form the parametrization for +:math:`\Delta \overline{q_{cl}}|_{convection}`. The representation of +:math:`\Delta C_{convection}` is similar in form to +:math:`\Delta \overline{q_{cl}}|_{convection}`: + +.. math:: + + \Delta C |_{convection} = \Delta C_{injection} + + \Delta t ~ a_L G(-Q_c) ( ( Q2 - \alpha Q1) - \Delta C_S (q_{sat}(\overline{T}) + - \overline{q} ) ) . + +where the specification of :math:`G(-Q_c)` follows +(`[eqn22] <#eqn22>`__). Note that the code includes the numerical limit +restriction that :math:`\Delta C_S` is between 0 and 1. + +Thus we are able to parametrize the net condensation and cloud changes +associated with the :math:`Q1` and :math:`Q2` terms in a physically more +consistent way than using simple homogeneous application of these terms. + +As an aside, we note that in the :raw-latex:`\cite{t93}` scheme the +condensation and cloud fraction change associated with the compensating +subsidence is taken out of the convection term by adding the vertical +motion associated with the compensating subsidence to the large-scale +vertical velocity before the :raw-latex:`\cite{t93}` equivalent of the +homogeneous forcing term is applied. By doing so it ensures that any +balance between these two terms (as the tropical circulation is commonly +analysed to show) is removed before the net effect is calculated, +leading to more accurate numerical behaviour. + +Homogeneous forcing of the environment by convective-subsidence pressure change +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +To this end, the code includes an option to perform the homogeneous +forcing of liquid cloud by convection using the “pressure forcing” from +the convective subsidence, consistent with the pressure forcing by +large-scale advection (see sections `4.5 <#sec:advec>`__ and +`4.8 <#sec:pres>`__). This approach replaces the above method of +homogeneous forcing by convection if the UM namelist switch +**l_pc2_homog_conv_pressure** is turned on. By applying the same +homogeneous forcing method for advection and convectively-forced +subsidence, we should get the correct zero net change in liquid cloud in +the common situation where the large-scale ascent and convective +subsidence are in balance (implying no net vertical displacement of +environment parcels). + +Under this option, the increments to :math:`\overline{q_{cl}}` and +:math:`C_l` produced by the convection scheme are assumed to already +include the effects of entrainment, detrainment (i.e. injection) and +compensating subsidence (i.e. vertical advection) as expressed by +equation `[eq:chimassflux] <#eq:chimassflux>`__, but exclude the effects +of homogeneous forcing of clouds in the enviroment. Note that taking +equation `[eq:chimassflux] <#eq:chimassflux>`__ with :math:`\chi` set to +water vapour :math:`q`, detrainment of saturated air into a subsaturated +environment will imply a positive tendency of :math:`\overline{q}`, but +this is *not* a homogeneous forcing, since the increase in +:math:`\overline{q}` is entirely due to injecting new parcels of +saturated air without altering the existing environment parcels. Setting +:math:`\chi` to be :math:`\overline{q_{cl}}` or :math:`C_l` in equation +`[eq:chimassflux] <#eq:chimassflux>`__, there is a simply-calculated +source of cloud water and fraction wherever the detrained air is cloudy +(:math:`C_l=1` in the detrained parcel), and we assume these terms have +been calculated this way inside the convection scheme. + +Since entrainment and detrainment do not constitute a homgeneous forcing +and are already accounted for in the convection scheme, the only +component of the convective forcing of liquid cloud that needs to be +done by the PC2 call after convection is the homogeneous forcing by the +subsidence term. This is in essence a vertical advection (environmental +forced descent by updrafts, or forced ascent by downdrafts). The +homogeneous forcing can be calculated from the expected pressure change +(and accompaying adiabatic temperature change) following the environment +as it is vertically displaced. Conveniently, the UM already holds the +convective mass-flux in units of Pa s\ :math:`^{-1}`, so it already +expresses the pressure vertical velocity forced by subsidence in the +environment: + +.. math:: + + \Delta p^E = \Delta t \left( M_{up} - M_{dwn} \right) + \label{eq:delta_p_conv} + +where :math:`M_{up}` is the updraft mass-flux, :math:`M_{dwn}` is the +downdraft mass-flux, and :math:`\Delta t` is the model timestep length. +The adiabatic temperature change following an environment parcel +subsided from pressure :math:`p - \Delta p^E` to :math:`p` is then given +by: + +.. math:: + + \Delta T^E = \theta^E \left( \left(\frac{p}{p_{ref}}\right)^\kappa + - \left(\frac{p - \Delta p^E}{p_{ref}}\right)^\kappa + \right) + \label{eq:delta_t_conv} + +where :math:`\theta^E` is the environment potential temperature, +:math:`p_{ref}` is the reference pressure used to define potential +temperature, and :math:`\kappa = \frac{R_d}{c_p}` is the ratio of the +gas constant for dry air over its heat capacity at constant pressure. +`[eq:delta_p_conv] <#eq:delta_p_conv>`__ and +`[eq:delta_t_conv] <#eq:delta_t_conv>`__ are passed into the PC2 +homogeneous forcing routine after convection as the forcings to be +applied (with the forcings to all other variables set to zero). + +Convective cloud amount +~~~~~~~~~~~~~~~~~~~~~~~ + +It is a debatable point whether the convective cloud fraction should be +set to zero. Although this was one of the original key concepts of PC2, +the cloud that is detrained from the convection scheme is into the +*environment*, and does not represent the tower cloud. However, it +should be able to represent recently detrained cloudy air in a more +accurate way than by simply appealing to a diagnostic large-scale cloud +scheme. There are similar issues associated with the cloud fraction +predicted from the Tiedtke scheme. Probably the most consistent +interpretation is the inclusion of a tower cloud fraction within PC2, +but not an anvil cloud. However, we need to consider carefully any +double counting (or non-counting) implications. In the PC2:64 +formulation, we can represent the large optical depths associated with +new anvils, although we also tend to overestimate the optical depth of +shallow convective clouds. Hence we choose to apply neither a diagnostic +anvil or tower cloud, so similar to Tiedtke, and let the large-scale +cloud fraction represent the convection completely. + +Strictly speaking these choices are independent of the PC2 scheme, being +simply choices that are available as part of the existing convection +scheme, but they are clearly directly related to the rest of the cloud +scheme formulation. + +CAPE scaling +~~~~~~~~~~~~ + +The CAPE scaling option in the mass-flux convection scheme scales its +increments by the calculated values of +:math:`\frac{1}{CAPE} \frac{dCAPE}{dt}`. This applies also to all the +PC2 calculated condensate and cloud fraction increments. Additionally, +in order to achieve reasonable mass flux profiles, it has proved +necessary to adjust the calculation of :math:`\frac{dCAPE}{dt}` to use +increments of :math:`\Delta \theta` (potential temperature) and +:math:`\Delta q` calculated using a non-PC2 calculation of these terms. +Hence we consider any detrained condensate to have been evaporated when +we calculate :math:`\frac{dCAPE}{dt}`. + +Convective precipitation +~~~~~~~~~~~~~~~~~~~~~~~~ + +The amount of condensate detrained from convective plumes, and hence the +amount of moisture in the upper levels of the atmosphere, is very +dependent upon the amount of convective precipitation that is allowed to +fall from the column. The standard parametrization of this is that any +condensate greater than a specified value (dependent on :math:`T`) is +precipitated, leaving the rest to be detrained. + +PC2 incorporates a tuning to this function of temperature by applying +the additional restriction that the limit may not fall to less than +:math:`2 \times 10^{-4}~kg~kg^{-1}`. This implies a difference at +temperatures less than around :math:`-42 ^{\circ} C`, with the tuning +allowing less precipitation and greater detrainment. This change is +necessary in order to produce thick enough anvil clouds. + +.. _`sec:plume_phase`: + +Phase of condensate +~~~~~~~~~~~~~~~~~~~ + +The phase of the convective condensate *carried in the plume* is +determined by a single phase change temperature TICE, with condensate +entirely in the ice phase at colder temperatures and condensate entirely +in the liquid phase at warmer temperatures. For PC2:66, this temperature +is -10 :math:`^{\circ}` C. + +.. math:: + + \delta_{xl} = \left\{ \begin{array}{ll} + 1, & T_{plume} \ge -10 ^{\circ} C \\ + 0, & T_{plume} < -10 ^{\circ} C + \end{array} \right. + +.. math:: + + \delta_{xi} = \left\{ \begin{array}{ll} + 0, & T_{plume} \ge -10 ^{\circ} C \\ + 1, & T_{plume} < -10 ^{\circ} C + \end{array} \right. + +.. _`sec:conv-simpler`: + +Tidier way of coupling convection and PC2 +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +This area is still under development. But in brief, work is udner way to +ensure that the convective plume smoothly transitions from detraining +liquid to detraining ice, rather than using the abrupt change implied by +the current formulation of the convection scheme. Additionally, rather +than using inhomogeneous increments to condensate (combining detrainment +and subsidence advection) to calculate cloud fraction increments, an +alternative is to use the detrainment of condensate to simply grow cloud +fraction to ensure a specified in-cloud liquid water content. The cloud +fraction are then advected downwards byt he subsidence advection. The +increments to cloud fraction from detrainment and subsidence are then +combined. + +Prognostic dust approach +~~~~~~~~~~~~~~~~~~~~~~~~ + +A prognostic dust approach is implemented in the micro-physics scheme +under large-sale-precipitation where by the heterogeneous nucleation +temperature can be defined to vary three dimensionally globally as an +arc-tangent function of the mineral dust distribution in the model +(documented in ). By default, both liquid and ice are detrained +simultaneously at the same height, and the fraction of condensate that +is ice linearly ramps as a function of temperature. i.e. condensate is +assumed to be all-liquid when T is greater than one tuneable threshold; +all-ice when T is less than another tuneable threshold, and vary +linearly in-between (the threshold values are given by starticeTkelvin +and alliceTdegC in the UM cloud-scheme namelist. The new heterogeneous +nucleation temperatures calculated in the large-scale-precipitation are +passed to the convection scheme and are used as the above detrainment +temperature thresholds by maintaining a similar linear ramp. For e.g., +condensate is assumed to be all-liquid for T :math:`\geq` :math:`tnuc_n` +and all-ice for T :math:`\leq` :math:`tnuc_n` - 10.0 + +.. _`sec:conv_input_profs`: + +Condensation adjustment in the profiles input to the convection scheme +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +The convection scheme itself is highly sensitive to the input +environment temperature and moisture profiles *before* the convection +increments (or PC2 response) are calculated. In particular, the parcel +buoyancy (and hence the CAPE and mass-flux scaling) maybe radically +different depending on whether a “large-scale” condensation / +evaporation adjustment is performed before the convection call. + +Where there is large-scale ascent, the profiles after Semi-Lagrangian +advection may have become supersaturated and unrealistically unstable, +until the expected condensation adjustment is performed. If the +convection scheme “sees” these unrealistic intermediate profiles, it is +likely to predict an excessive, unrealistic mass-flux. + +To address this problem, there are two namelist switches that enable +additional condensation adjustments from PC2 before the convection call: + +- **l_pc2_sl_advection**: performs homogeneous forcing response to + Semi-Lagrangian advection immediately after the advection calculation, + instead of at the end of the timestep (see section + `4.8 <#sec:pres>`__). + +- **l_cloud_call_b4_conv**: performs an additional call to PC2 + initiation (and PC2 checks) before the convection scheme (see section + `4.9 <#sec:init2>`__). This should catch any instances where + large-scale ascent or other processes have brought the profiles after + advection to near or beyond saturation, in grid-points where there was + no liquid cloud already present (and so no homogeneous forcing + response). + +.. _`sec:pres`: + +Response to pressure changes +---------------------------- + +A pressure change following the parcel during the timestep will result +in an adiabatic temperature change which will force condensation, hence +we must include this temperature change forcing within PC2. The majority +of this pressure change comes from vertical advection (although not +all). Remember that the advection (section `4.5 <#sec:advec>`__), on its +own, does not cause condensation, it merely moves the existing cloud +field. + +Using the semi-Lagrangian advection in the same way as is performed for +:math:`\overline{q_{cl}}` etc., the PC2 scheme will obtain the value of +the model prognostic *Exner*, (:math:`\prod`) on the departure points +(:math:`\prod_{dep}`). *Exner* is defined as + +.. math:: + + \label{eq:exner} + \prod = \frac{T}{\theta} = \left( \frac{p}{p_{ref}} \right)^{\kappa} + +where :math:`\theta` is the potential temperature, :math:`p_{ref}` is a +reference pressure set to 1000 hPa, and :math:`\kappa = +\frac{c_p - c_v}{c_p}` , where :math:`c_v` is the heat capacity of dry +air at constant volume. The *Exner* quantity is kept as a prognostic +variable in the model (this is unchanged from the control model), and +the value of :math:`\prod` on the departure points represents the +initial value in the timestep, since there is no update to :math:`\prod` +until the end of the timestep. After the second physics updates have +been performed (*atmos-physics2*), the model (including the control) +recalculates the value of *Exner* (:math:`\prod^{[n+1]}`). From +:math:`\prod_{dep}` and :math:`\prod^{[n+1]}` we can calculate, using +the definition (`[eq:exner] <#eq:exner>`__), the values of departure +pressure and temperature: + +.. math:: \overline{p}_{dep} = p_{ref} {\prod_{dep}}^{\frac{1}{\kappa}} + +.. math:: \overline{T}_{dep} = \theta \prod_{dep} + +Hence we obtain the net forcing values + +.. math:: + + \Delta \overline{T} = \overline{T}^{[n+1]} - \overline{T}_{dep} + \label{eq:deltatsl} + +and + +.. math:: + + \Delta \overline{p} = \overline{p}^{[n+1]} - \overline{p}_{dep} . + \label{eq:deltapsl} + +where :math:`\overline{T}^{[n+1]}` and :math:`\overline{p}^{[n+1]}` are +the temperature and pressure at the arrival point, after the dynamics +call. (`[eq:deltatsl] <#eq:deltatsl>`__) and +(`[eq:deltapsl] <#eq:deltapsl>`__) are passed to the homogeneous forcing +routine in order to calculate the condensation and cloud fraction +changes associated with the pressure change. + +We include this forcing towards the end of the timestep. There are two +reasons for this: firstly, values of :math:`\prod^{[n+1]}` are not +calculated by the control model until after the physics is complete; +secondly, it makes sense to locate this process in the timestep in a +similar location to where the large-scale cloud scheme is included in +the control (i.e. after the implicit part of the boundary layer has +finished). + +However, there is a counter argument that says we should include this +process immediately after the dynamics, since we can then apply a +forcing on an initial state that has not already been modified by the +dynamics, boundary layer and convection schemes. This improves the +numerics of the problem, since the homogeneous forcing is designed to +take time level n values as inputs. + +These issue are optionally addressed by turning on the UM namelist +switch **l_pc2_sl_advection**. Under this switch, the PC2 homogeneous +forcing response to pressure change is split: + +#. Forcing by the *Lagrangian* component of pressure change, performed + immediately after the Semil-Lagrangian advection scheme (before the + call to atmos_physics2). This calculates the pressure change from the + departure point value of *Exner* described above, to the + start-of-timestep value of *Exner* at the arrival point. + +#. Forcing by the *Eulerian* component of pressure change, performed at + the end of the timestep (after the dynamics Helmholtz solver). This + calculates the pressure change from the start-of-timestep *Exner* at + the arrival point, to the end-of-timestep *Exner*. + +Having to calculate the pressure forcing twice obviously adds some +computational cost, but has several advantages: + +- As noted above, the PC2 homogeneous forcing calls can now take as + input the temperature and water-vapour content *before* the pressure + change has been applied, as intended. This should improve the + numerical accuracy. + +- Most of the condensation or evaporation from the dynamics comes from + the *Lagrangian* component of the pressure change, which has now moved + from the end of the timestep to before the dynamics Helmholtz solver. + This means that any latent heating from condensation forced by ascent + is now accounted for by the solver within the same timestep. This + improves the numerical accuracy of the dynamics-physics coupling. + +- If the condensation forced by resolved ascent is only added on at the + end of the timestep, the profiles passed into atmos_physics2 can + contain out-of-balance thermodynamic states (e.g. if the profile has + been lifted by advection, it maybe supersaturated / unrealistically + unstable before the resulting condensation is added on). This may + adversely affect the convection scheme, which must act upon the + profiles passed into atmos_physics2. + +The splitting of the pressure forcing call under the +**l_pc2_sl_advection** switch was originally implemented to make the +profiles passed to convection more realistic. + +.. _`sec:init2`: + +Initiation +---------- + +As discussed in section `3.4 <#sec:init>`__, there are occasions when +:math:`\overline{q_{cl}}` and :math:`C_l` need to be initiated from 0 or +1. The application of the initiation is given in section +`3.4 <#sec:init>`__. The initiation forms a new, separate block of PC2 +code to perform this calculation, and is located immediately following +the pressure change response (section `4.8 <#sec:pres>`__). Also, if the +UM namelist switch **l_cloud_call_b4_conv** is set to true, an +additional call to PC2 initiation is performed before the convection +scheme, to ensure that the condensation response to advection and other +forcings earlier in the timestep has been accounted for in the profiles +passed to the convection scheme, even if there was no cloud already +present for homogeneous forcing to act upon. (see section +`4.7.12 <#sec:conv_input_profs>`__). + +There are currently 3 options for the conditions under-which initiation +may occur. For all of these options, if using the bimodal cloud scheme +to do initiation within PC2, then the tests on :math:`RH_T` relative to +:math:`RH_{crit}` are replaced by equivalent tests for whether the +saturation boundary lies within the bounds of the bimodal scheme’s +assumed PDF, as described in section `3.4.3 <#sec:bimodal_init>`__. + +“Original” initiation logic +~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +This option is selected by setting the UM namelist switch +**i_pc2_init_logic = 1** (Original) + +The initiation will be called if the liquid cloud fraction is either 0 +or 1 and appropriate :math:`RH` criteria hold, along with other +restrictions. :math:`C_l` is initiated away from 0 if + +- :math:`RH_T > RH_{crit} + RH_{crit \, tol}` **and** + +- Cumulus convection has *not* been diagnosed from the boundary-layer + in the current column **and** + +- The current level is not below the surface mixed-layer LCL **and** + +- :math:`C_l = 0` **and** + +- :math:`RH_T^{[n+1]} > RH_T^{[n]}` , + +where :math:`RH_{crit \, tol}` is a specified tolerance parameter, of +value 0.01, and :math:`RH_T` is defined in (`[eq:rht] <#eq:rht>`__). +:math:`RH_T^{[n]}` is the start of timestep value of :math:`RH_T` (i.e. +at time level n) and :math:`RH_T^{[n+1]}` is the value when initiation +is called. Additionally, there is another possibility for the last of +the relations. This second option also allows initiation when the water +is supercooled: + +- :math:`C_l < 0.05` *and* :math:`\overline{T} < 0 ^{\circ} C` . + +Equivalently, :math:`C_l` is initiated away from 1 if + +- :math:`RH_T < 2 - RH_{crit} - RH_{crit \, tol}` **and** + +- :math:`C_l = 1` **and** + +- :math:`RH_T^{[n+1]} < RH_T^{[n]}` . + +“Simplified” initiation logic +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +This option is selected by setting the UM namelist switch +**i_pc2_init_logic = 2** (Simplified) + +Under this option, the conditions for initiation are: + +Either: + +- :math:`RH_T > RH_{crit} + RH_{crit \, tol}` **and** + +- :math:`C_l < C_{tol}` **and** + +- The current level is not below the surface mixed-layer LCL **and** + +- :math:`RH_T^{[n+1]} > RH_T^{[n]}` + +Or: + +- :math:`RH_T < 2 - RH_{crit} - RH_{crit \, tol}` **and** + +- :math:`C_l > 1 - C_{tol}` **and** + +- :math:`RH_T^{[n+1]} < RH_T^{[n]}` + +where :math:`C_{tol}` can be set via the UM namelist; its original +standard value is 0.005. Note this threshold is also used to remove +small cloud-fractions after initiation; see section +`4.9.4 <#sec:checks2>`__. + +This is very similar to the “Original” initiation logic described above, +but with the following differences: + +- The condition that the boundary-layer hasn’t diagnosed cumulus + convection in the column is removed. Note that this condition + spuriously suppressed initiation in the free troposphere *above* any + cumulus cloud produced by the convection scheme. + +- :math:`C_l` only needs to be within a numerical tolerance + :math:`C_{tol}` from 0 or 1, rather than having to be *exactly* 0 or + 1. + +- The different threshold when initiating super-cooled cloud is removed. + +.. _`sec:smooth_initiation`: + +“Smooth” initiation logic +~~~~~~~~~~~~~~~~~~~~~~~~~ + +This option is selected by setting the UM namelist switch +**i_pc2_init_logic = 3** (Smooth) + +There is a fundamental numerical problem with the above options, in that +the initiation process is not permitted to have any effect at all unless +:math:`C_l` goes to (near) 0 or 1, but can predict values of :math:`C_l` +very different to 0 or 1 when it does activate. This leads to unphysical +sudden noisy jumps in :math:`C_l` and :math:`q_{cl}` when initiation +occurs. For example, if erosion causes :math:`C_l` to steadily decline, +it will continue to decline (even when the grid-mean :math:`RH_T` +exceeds :math:`RH_{crit}`) until it reaches the threshold (0 or +:math:`C_{tol}`). At this point, initiation suddenly increases +:math:`C_l` and :math:`q_{cl}` to the values predicted by the diagnostic +cloud scheme. Erosion may then gradually remove them again, and the +cycle repeats. There is no physical reason for this internal mode of +variability in the scheme. + +Another problem arises if we consider the sensitivity to model +resolution. Suppose we have many adjacent small grid-boxes with similar +:math:`RH_T`, a few containing cloud, the rest containing no cloud. If +the whole region cools to the point where :math:`RH_T > RH_{crit}`, then +new cloud will initiate in the cloud-free grid-boxes, but not in the +cloudy grid-boxes. Now suppose we run a coarse-grained version of the +same simulation; the many small grid-boxes are replaced by a single +grid-box containing the average :math:`C_l` over the small grid-boxes. +Since we now have just one grid-box already containing partial +cloud-cover, initiation of new cloud can no longer occur anywhere. + +To address these problems, there is an option to use a much simpler / +numerically better-posed initiation method; always allow the diagnostic +cloud scheme to be called (provided it is expected to predict nonzero +cloud water, i.e. :math:`RH_T > RH_{crit}` in the case of the Smith +scheme). The :math:`q_{cl}` predicted by the diagnostic cloud scheme is +then taken as a minimum limit applied to the prognostic :math:`q_{cl}`. +This amounts to taking the diagnostic cloud scheme’s assumed PDF as a +minimum allowed width to the actual prognostic moisture PDF. The +prognostic :math:`C_l` and :math:`q_{cl}` are incremented as follows: + +- If :math:`{q_{cl}}_{diag} > q_{cl}`: + + :math:`\Delta q_{cl} = {q_{cl}}_{diag} - q_{cl} + \quad \refstepcounter{equation}(\theequation)\label{eq:dqcl_init}` + + - If :math:`Q_C < 0`: + + :math:`\Delta C_{l} = \frac{\Delta q_{cl}}{{q_{cl}}_{diag}} + \left( {C_{l}}_{diag} - C_{l} \right) + \quad \refstepcounter{equation}(\theequation)\label{eq:dcl_init1}` + + - If :math:`Q_C > 0`: + + :math:`\Delta C_{l} = \frac{\Delta SD}{{SD}_{diag}} + \left( {C_{l}}_{diag} - C_{l} \right) + \quad \refstepcounter{equation}(\theequation)\label{eq:dcl_init2}` + +- Otherwise: + + :math:`\Delta q_{cl} = 0` + + :math:`\Delta C_{l} = 0` + +where the subscript :math:`_{diag}` denotes the liquid cloud water +content and fraction predicted by the diagnostic cloud scheme (either +Smith or Bimodal). + +Equation `[eq:dcl_init1] <#eq:dcl_init1>`__ simply sets the +cloud-fraction to a weighted mean of the pre-existing and +diagnostic-scheme cloud-fractions, in proportion to the fraction of the +water content that was created by initiation versus that which was +already there. If the pre-existing :math:`q_{cl}` is zero, +`[eq:dqcl_init] <#eq:dqcl_init>`__ and +`[eq:dcl_init1] <#eq:dcl_init1>`__ simply set :math:`q_{cl}` and +:math:`C_l` to their new diagnosed values, as in the previous options. +Crucially, in the limit that the pre-existing :math:`q_{cl}` approaches +:math:`{q_{cl}}_{diag}`, the increments to :math:`q_{cl}` and +:math:`C_l` smoothly go to zero. This is important to make the +initiation process numerically well-posed, so that it yields a smooth, +continuous solution. + +Note that when we are initiating from :math:`C_l = 1` instead of +:math:`C_l = 0`, we expect the pre-existing :math:`q_{cl}` to be nonzero +even when there is no pre-existing sub-grid PDF width. In this case, the +completely uninitiated state will have zero saturation deficit +:math:`SD`, rather than zero :math:`q_{cl}`. Therefore, in this case the +increment to :math:`C_l` is calculated based on the fractional increase +in :math:`SD` from initiation (equation +`[eq:dcl_init2] <#eq:dcl_init2>`__), instead of the fractional increase +in :math:`q_{cl}`. + +Whether to increment :math:`C_l` based on the increase in :math:`q_{cl}` +or :math:`SD` is determined based on the sign of :math:`Q_C`, which is +defined as in equation `[eq:qc_eq_qt-qs] <#eq:qc_eq_qt-qs>`__ +(reproduced here for clarity): + +.. math:: Q_c = a_L \left( \overline{q_T} - q_{sat}(\overline{T_L}) \right) + +The saturation deficit :math:`SD` is defined by equation +`[SD2] <#SD2>`__: + +.. math:: SD = a_L \left( q_{sat}(\overline{T}) - \overline{q} \right) + +Under the reasonable approximation that :math:`q_{sat}` varies linearly +between :math:`\overline{T}` and :math:`\overline{T_L}`, so that the +values of :math:`\alpha` and :math:`a_L` are the same in both of these +equations, and: + +.. math:: q_{sat}(\overline{T_L}) = q_{sat}(\overline{T}) - \alpha \frac{L}{c_p} q_{cl} + +we obtain: + +.. math:: + + q_{cl} = Q_c + SD + \label{eq:qc_plus_sd} + +It can be seen that when :math:`Q_C > 0` (total-water super-saturation), +it represents the value :math:`q_{cl}` would have if the whole grid-box +were saturated (:math:`SD = 0`, :math:`C_l = 1`). Note that +:math:`q_{cl}` cannot fall below :math:`Q_C`, since :math:`SD` cannot be +negative. Since :math:`Q_c` is invariant under condensation / +evaporation, we must have :math:`\Delta SD = \Delta q_{cl}` (hence the +implementation of `[eq:dcl_init2] <#eq:dcl_init2>`__ in the code simply +uses :math:`q_{cl} - Q_c` in place of :math:`SD`, and +:math:`\Delta q_{cl}` in place of :math:`\Delta SD`). + +.. _`sec:checks2`: + +Additional checks after PC2 initiation +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +The initiation is followed immediately by a section of resetting code. +For numerical reasons, it is possible to obtain very low, but non zero, +values of :math:`C_l` (and equivalently values very close to, but not +equal to, 1). The code will reset these clouds to either a fraction of 0 +or 1, as appropriate. We choose to apply these terms here and not in the +Bounds Checking part of the code (section `4.10 <#sec:checks>`__) +because these are not required to obtain consistency between fields, but +are ‘tidying up’ pieces of code, although they may reasonably also be +applied in the Bounds Checking. Care needs to be taken when choosing the +thresholds, since we do not wish to reset small values that are +genuinely created by a physics scheme in the model. + +We first calculate :math:`RH_T` using (`[eq:rht] <#eq:rht>`__) and +compare this to the critical relative humidity, :math:`RH_{crit}`. The +liquid cloud fraction will be reset to 1 if: + +- :math:`RH_T > 2 - RH_{crit}` and :math:`C_l \ge C_{high}` + +- or :math:`C_l \ge C_{high 2}` + +where :math:`C_{high}` and :math:`C_{high 2}` are defined in +`[eq:chigh-chigh2] <#eq:chigh-chigh2>`__. The evaporation is done by +calculating :math:`SD` using (`[SD2] <#SD2>`__) with +(`[eq:a_L] <#eq:a_L>`__) and (`[eq:alpha_exp] <#eq:alpha_exp>`__) and +evaporating the equivalent amount of liquid into the gridbox to take it +to saturation, according to (`[eq:qsdcheck1] <#eq:qsdcheck1>`__) below. + +Similarly, the equivalent check for low values of :math:`RH_T` is +performed. The liquid cloud fraction will be reset to 0 if: + +- :math:`RH_T < RH_{crit}` and :math:`C_l \le C_{low}` + +- or :math:`C_l \le C_{low 2}` . + +The remaining :math:`\overline{q_{cl}}` is evaporated into the gridbox +using (`[eq:qclcheck] <#eq:qclcheck>`__) below. + +The thresholds :math:`C_{high}`, :math:`C_{high 2}`, :math:`C_{low}` and +:math:`C_{low 2}` are set using the parameters :math:`C_{tol}` and +:math:`C_{tol 2}`, according to: + +.. math:: + + \begin{aligned} + C_{high} = 1 - C_{tol}, \nonumber \\ + C_{high 2} = 1 - C_{tol 2}, \nonumber \\ + C_{low} = C_{tol}, \nonumber \\ + C_{low 2} = C_{tol 2}, + \label{eq:chigh-chigh2} + \end{aligned} + +where the parameters :math:`C_{tol}` and :math:`C_{tol 2}` can be set +via the UM namelist variables **cloud_pc2_tol** and **cloud_pc2_tol_2**. +The original standard values of these parameters are +:math:`C_{tol} = 0.005` and a lower value :math:`C_{tol 2} = 0.001`. + +Investigations in SCM runs using the comorph convection scheme (which +behaves more smoothly and so typically gives smaller increments to +:math:`C_l` over a single timestep than other schemes which exhibit +intermittent behaviour) suggested these thresholds are too high to avoid +spuriously resetting physical values of :math:`C_l` to zero. Detrainment +from sparse shallow cumulus, or advection of cloud into a neighbouring +grid-box under light winds, commonly give increments which increase +:math:`C_l` from zero to a value less than :math:`0.005` in one timestep +(but would eventually increase :math:`C_l` to a significant value over +subsequent timesteps if the checks did not keep resetting :math:`C_l` to +zero). + +Note that if these checks are relaxed by lowering the thresholds +:math:`C_{tol}` and :math:`C_{tol 2}` to near-zero, similar checks are +still performed independently by the bounds checking described in +section `4.10 <#sec:checks>`__, but with a much lower threshold of +:math:`C_{tol 3} = 1 \times 10^{-12}`. + +.. _`sec:checks`: + +Bounds checking +--------------- + +Ideally, model prognostics would never become inconsistent with one +another. However, even although the mathematical solution of the +governing equations may be well behaved, due to numerical inaccuracies +values may become inconsistent. For the cloud and condensate quantities, +there are a number of consistencies that must apply. The bounds checking +forms a subroutine that will, if necessary, adjust :math:`\overline{q}`, +:math:`\overline{q_{cl}}`, :math:`\overline{q_{cf}}`, :math:`C_l`, +:math:`C_i`, :math:`C_t` and, for latent heating, :math:`\overline{T}`, +to ensure consistency between these values. + +The bounds checking is performed three times during the timestep. +Firstly, after the parallel part of the physics (*atmos-physics1*) is +complete; secondly, before the initiation (section `3.4 <#sec:init>`__) +is called; thirdly, after the initiation is called. + +Firstly, if :math:`C_l > 1 - C_{tol 3}` then :math:`C_l` is set to 1. +Accordingly, :math:`C_t` is set to 1 as well. :math:`C_{tol 3}` is a +tiny numerical tolerance set to :math:`1 \times 10^{-12}`, a value +intended to be in the realm of floating point rounding error rather than +anything that represents a physical solution. + +.. _section-1: + +The second check is to reset :math:`\overline{C_l}` to zero. This may be +performed for two reasons. Firstly, if the amount of +:math:`\overline{q_{cl}}` is very small +(:math:`\overline{q_{cl}} < q_{c0}`, where +:math:`q_{c0} = 1 \times 10^{-10} kg kg^{-1}`), so we avoid carrying +negligible, but non-zero values of :math:`\overline{q_{cl}}` and +:math:`C_l`. Secondly, if :math:`C_l < C_{tol 3}` then we reasonably +reset :math:`C_l` to zero. :math:`C_t` gets reset, as it must if there +is no liquid cloud, to be equal to :math:`C_i`. + +.. _`sec:pc2_checks_sd`: + +The next check complements the first but updates the moisture fields. We +firstly calculate :math:`SD` using (`[SD2] <#SD2>`__) and +(`[eq:alpha_exp] <#eq:alpha_exp>`__). We then check whether +:math:`SD < 0`. This check catches instances where we have grid-mean +supersaturation, which ought to be impossible (under the instantaneous +condensation assumption made by PC2, condensation should occur to +instantly adjust any supersaturated regions of the gridbox to +saturation, so we *must always* have :math:`SD \ge 0`. When this +condition is violated, we condense water vapour to adjust to grid-mean +saturation. :math:`-SD` corresponds to the amount of vapour that must be +condensed to achieve this, so we have: + +.. math:: + + \begin{aligned} + \overline{q} \leftarrow \overline{q} + SD \nonumber \\ + \overline{q_{cl}} \leftarrow \overline{q_{cl}} - SD \nonumber \\ + \overline{T} \leftarrow \overline{T} - \frac{L_c}{c_p} SD + \label{eq:qsdcheck1} + \end{aligned} + +The original version of this check on :math:`SD` (which may increase +:math:`\overline{q_{cl}}`), made no accompanying changes to liquid cloud +fraction. However, increases in :math:`\overline{q_{cl}}` without any +increase in :math:`C_l` can lead to spurious high in-cloud condensate +which is then converted to rain by the microphysics at the next +time-step. There are currently 4 options for how to treat :math:`C_l` +when increasing :math:`\overline{q_{cl}}` under this saturation +adjustment, selected by the UM large-scale cloud namelist switch +**i_pc2_checks_cld_frac_method**: + +- **i_pc2_checks_cld_frac_method = 0** - Original method; :math:`C_l` is + left unaltered. + +- **i_pc2_checks_cld_frac_method = 1** - Set :math:`C_l` and :math:`C_t` + to 1. + +- **i_pc2_checks_cld_frac_method = 2** - If :math:`\overline{q_{cl}}` + and :math:`C_l` were already nonzero before the adjustment, increase + :math:`C_l` at the same fractional rate as :math:`\overline{q_{cl}}`, + so that the in-cloud water content + :math:`\frac{\overline{q_{cl}}}{C_l}` is conserved. Otherwise, + increase :math:`C_l` so-as to yield a prescribed in-cloud water + content set to 0.5 g kg\ :math:`^{-1}`. :math:`C_t` is then increased + by the same amount as :math:`C_l`, to maintain consistency. + +- **i_pc2_checks_cld_frac_method = 3** - This is the same as option 2 + above, except in the case where :math:`\overline{q_{cl}}` or + :math:`C_l` was zero before the adjustment. In this case, :math:`C_l` + is set based on an empirical power-law function of + :math:`\overline{q_{cl}}`. + +.. _section-2: + +Next we check whether :math:`SD > 0`, *and* :math:`C_l = 1` (the first +of our checks has ensured that :math:`C_l` is no greater than 1). This +check catches instances where we have total cloud-cover in a +subsaturated grid-box, which ought to be impossible (if the whole +grid-box is full of liquid cloud, then it must be at grid-mean +saturation, i.e. :math:`SD = 0`). When this happens, we adjust +:math:`\overline{q}` and :math:`\overline{q_{cl}}` to take :math:`SD` to +zero, *provided* that :math:`\overline{q_{cl}} > SD`. Remember that +:math:`SD` corresponds to the amount of vapour that must be *evaporated* +into the gridbox to give saturation, so we simply make exactly the same +adjustments as we do for removing supersaturated states above +(`[eq:qsdcheck1] <#eq:qsdcheck1>`__), except that here :math:`SD` is +positive rather than negative. + +Our proviso that :math:`\overline{q_{cl}} > SD` ensures that we do not +make :math:`\overline{q_{cl}}` negative by this adjustment. If +:math:`\overline{q_{cl}} < SD`, then we cannot bring the gridbox to +saturation, but it is still wrong to allow :math:`C_l = 1` in a +subsaturated gridbox! This was identified as a bug in the +bounds-checking code, which sometimes caused instances of +:math:`C_l = 1` to spuriously persist in dry environments. This +behaviour is currently controlled by a temporary logical in the +**temp_fixes** namelist: + +- If **l_pc2_checks_sdfix** is set to false, the code simply does + nothing when it finds instances of :math:`C_l = 1`, :math:`SD > 0` and + :math:`SD > \overline{q_{cl}}`, allowing such artefacts to persist. + +- If **l_pc2_checks_sdfix** is set to true, in these instances we simply + evaporate all the remaining liquid water, and reset :math:`C_l` to + zero: + + .. math:: + + \begin{aligned} + \overline{q} \leftarrow \overline{q} + \overline{q_{cl}} \nonumber \\ + \overline{T} \leftarrow \overline{T} - \frac{L_c}{c_p} \overline{q_{cl}} + \nonumber \\ + \overline{q_{cl}} \leftarrow 0 \nonumber \\ + C_l \leftarrow 0\nonumber \\ + C_t \leftarrow C_i + \label{eq:qsdcheck2} + \end{aligned} + +.. _section-3: + +The next check is similar to above but for the :math:`C_l = 0` +situation. + +If :math:`\overline{q_{cl}} < q_{c0}` or :math:`C_l = 0` then we +evaporate the small amount of :math:`\overline{q_{cl}}` that remains in +the gridbox: + +.. math:: + + \begin{aligned} + \overline{q} \leftarrow \overline{q} + \overline{q_{cl}} \nonumber \\ + \overline{q_{cl}} \leftarrow 0 \nonumber \\ + \overline{T} \leftarrow \overline{T} - \frac{L_c}{c_p} \overline{q_{cl}} + \label{eq:qclcheck} + \end{aligned} + +.. _section-4: + +Next, if :math:`C_i > 1` then :math:`C_i` is set to 1. Accordingly, +:math:`C_t` is set to 1 as well. + +.. _section-5: + +The following check is on the ice water content, +:math:`\overline{q_{cf}}`, and the ice fraction :math:`C_i`. If +:math:`\overline{q_{cf}} < q_{c0}` we simply condense some vapour to +remove the negative quantity. + +.. _section-6: + +However, instead of removing small amounts of :math:`\overline{q_{cf}}` +when :math:`C_i = 0` but :math:`\overline{q_{cf}} > 0`, we choose +instead to create some :math:`C_i` to keep consistency. This is to allow +small, but significant, amounts of :math:`\overline{q_{cf}}` created by +the microphysics scheme to be maintained. + +.. math:: + + C_i \leftarrow \frac { \overline{q_{cf}} }{q_{cf0}} + \label{eq:cf_reset} + +where the ‘in-cloud’ ice content +:math:`q_{cf0} = 1 \times 10^{-4} kg kg^{-1}`. + +.. _section-7: + +The next two checks are on the total cloud fraction, :math:`C_t`, to +ensure that it takes on a value that is physically possible, given the +values of :math:`C_l` and :math:`C_i`. We have, firstly, the maximum +overlap situation and then the minimum overlap situation. + +.. math:: + + \begin{aligned} + C_t \leftarrow \text{Max}( C_t, C_i, C_l ) \nonumber \\ + C_t \leftarrow \text{Min}( C_t , C_l + C_i, 1) + \label{eq:ctchecks} + \end{aligned} + +.. _section-8: + +Finally, there is a homogeneous nucleation term applied, similar to that +in the large-scale precipitation (section `4.2.2 <#sec:lsp_homo>`__). +This is a fast microphysics process, and must act to ensure that no +liquid cloud created by the initiation is allowed to persist in this +phase if the temperature is cold enough. Hence, if +:math:`\overline{T} < T_{homo}` then + +.. math:: + + \begin{aligned} + \overline{q_{cf}} \leftarrow \overline{q_{cf}} + \overline{q_{cl}} \nonumber \\ + \overline{q_{cl}} \leftarrow 0 \nonumber \\ + \overline{T} \leftarrow \overline{T} + \frac{L_f}{c_p} \overline{q_{cl}} \nonumber \\ + C_i \leftarrow C_t \nonumber \\ + C_l \leftarrow 0. + \label{eq:homochecks} + \end{aligned} + +.. _`sec:qpos`: + +Qpos checks +~~~~~~~~~~~ + +The implementation of the PC2 code includes an additional bounds check +after the *atmos-physics-2* part of the model timestep has been +completed. This check is necessary to trap a rare failure, and uses the +*Qpos* subroutines to check that :math:`\overline{q_{cl}}` is greater or +equal to 0. + +During trialling prior to operational implementation, it was found that +relying on Q-Pos to deal with negative condensate values was very +expensive, as the Q-Pos routine does a lot of communications between +different processors. It may be preferable to deal with the cause of +negative condensate amounts at their source. The option to “Ensure +consistent sinks of qcl and CFL” prevents the QCL increment from trying +to remove too much liquid condensate and hence reduces the models +reliance on Q-Pos to deal with the inconsistencies. + +.. _`sec:da`: + +Data Assimilation +----------------- + +The data assimilation section in the model will output assimilation +increments that represent changes to :math:`\overline{q}` and +:math:`\overline{T}` which *include* the condensation contributions. We +hence need to calculate equivalent increments to +:math:`\overline{q_{cl}}`, :math:`C_l` and :math:`C_t`. We assume that +the assimilation has not calculated these using a different method. We +consider the homogeneous framework and assume that there is a forcing +value of :math:`Q_c` that exists that will produce the known increment +to :math:`\overline{q}` and :math:`\overline{T}`. + +Discritising (`[dqcldt] <#dqcldt>`__) we have, using +(`[eq:deltaqc_exp] <#eq:deltaqc_exp>`__) and expanding +:math:`\Delta T_L` in terms of :math:`\Delta T` and +:math:`\Delta q_{cl}`, + +.. math:: + + \Delta \overline{q_{cl}} = C_l ( a_L ( \Delta \overline{q} - + \alpha \Delta \overline{T} - \beta \Delta \overline{p}) + \Delta \overline{q_{cl}} ). + \label{eq:da1} + +Remember that :math:`Q_c` (and hence :math:`\Delta Q_c`) is independent +of condensation. Rearranging, we obtain + +.. math:: + + \Delta \overline{q_{cl}} = \frac{1}{1 - C_l} C_l + a_L ( \Delta \overline{q} - \alpha \Delta \overline{T} - \beta \Delta \overline{p}) + \label{eq:da2} + +and hence an expression for the condensate increment, +:math:`\Delta \overline{q_{cl}}`, that accompanies the known increments +to :math:`\overline{q}` and :math:`\overline{T}`. The similar analysis, +from (`[dcdt] <#dcdt>`__) and (`[eq:da1] <#eq:da1>`__) gives + +.. math:: + + \Delta C_l = \frac{1}{1 - C_l} G(-Q_c) + a_L ( \Delta \overline{q} - \alpha \Delta \overline{T} + - \beta \Delta \overline{p} ) . + \label{eq:da3} + +Hence the equation set is equivalent to the use of the homogeneous +forcing set, except for the multiplier :math:`\frac{1}{1 - C_l}`. +Although this is a clean solution, we need to be very careful with the +ill-conditioning of this solution near :math:`C_l = 1`. + +In practice, the ill-conditioning of (`[eq:da2] <#eq:da2>`__) and +(`[eq:da3] <#eq:da3>`__) becomes too numerically awkward for us to apply +the full solution based on homogeneous forcing, although, for +completeness, we outline it in Appendix `6 <#sec:appendix-da>`__. Hence +we have chosen to apply a much simpler model. Here we use simply the +data assimilation increments :math:`\Delta \overline{q}` and +:math:`\Delta \overline{T}` within the standard homogeneous forcing +(section `3.2 <#sec:homog>`__), even though we are fully aware that this +is inconsistent (because :math:`\Delta \overline{q}` and :math:`\Delta +\overline{T}` are not forcings, but are forcings plus the condensation. +This allows us an *estimate* of :math:`\Delta \overline{q_{cl}}` and +:math:`\Delta{C_l}`, via the homogeneous forcing routine (and +:math:`\Delta C_t` via the standard updating described in section +`3.6 <#sec:ct>`__). These are the quantities applied as the equivalent +data assimilation increments for :math:`\Delta \overline{q_{cl}}`, +:math:`\Delta{C_l}` and :math:`\Delta C_t`. The increments +:math:`\Delta \overline{q}` and :math:`\Delta \overline{T}` remain those +that the data assimilation scheme itself calculated. + +Appendix `6 <#sec:appendix-da>`__ gives, for completeness, the +alternative numerical technique for the solution of +(`[eq:da2] <#eq:da2>`__) and (`[eq:da3] <#eq:da3>`__). However, we +stress that this technique is not used within the current PC2 +formulation. + +.. _`sec:um`: + +Implementation in the Unified Model +=================================== + +This section considers the implementation of PC2 within the Unified +Model code and provides a brief guide to its use. + +In general, we have written PC2 so that the timestepping of the cloud +fraction variables within the *atm_step_4a* subroutine is treated as +much as possible in a similar way to the condensate variables. Hence, +wherever the condensed water variables :math:`q_{cl}` and :math:`q_{cf}` +are updated, the cloud fractions need to be updated consistently. + +.. _`sec:acf`: + +Area cloud fraction +------------------- + +Two area cloud fraction parametrizations are available for use with PC2. + +The area cloud fraction of Cusack (documented in ) has been adapted by +:raw-latex:`\cite{boutle_morcrette10}` so it can be used with PC2 (and +is available from the UMUI as the “Cusack” option from version 7.6 +onwards). This method aims to reproduce some of the detail of the +thermodynamic profile lost due to the coarseness of the grid. The +interpolation/extrapolation technique is used prior to PC2 initiation +(which is then called with three times as many levels) and it is used, +along with the homogeneous forcing idea at the start of the timestep to +allow more cloud to be seen by radiation. + +The diagnostic area cloud fraction of :raw-latex:`\cite{bhi05}` has also +been implemented in the model (available from the UMUI at version 6.4 +onwards), and this is used in PC2:64. This method diagnoses the area +cloud fraction given the volume cloud fraction, taking into account the +size of the grid box. The setting of the area cloud fraction is +performed at the end of the timestep. + +.. _`sec:code`: + +Code Structure +-------------- + +A detailed description of the UM’s timestep structure, showing where in +the model all the PC2 cloud scheme subroutine calls are made, is given +in the subsections below. + +Note that there are three different subroutines that all do the PC2 +homogeneous forcing, with slightly different details: + +- ***pc2_delta_hom_turb*** outputs increments due to the condensation or + evaporation, but doesn’t update the fields themselves. + +- ***pc2_homog_plus_turb*** just updates the fields that are passed in, + instead of outputting separate increment arrays. + +- ***pc2_hom_conv*** outputs increments but includes additional + calculations for various cloud erosion formulations. + +Note that code exists in the first two of these routines to do erosion, +but they can only do it via an input fixed rate of narrowing of the +moisture PDF (which is currently set to zero in all instances). PC2 +development has settled on a more complicated treatment of erosion, +which has only been implemented in *pc2_hom_conv*. This can either be +called after the convection scheme (within *pc2_from_conv_ctl*), or +before the microphysics scheme (within *pc2_turbulence_ctl*). + +Note there is also an optional call to *pc2_turbulence_ctl* after the +microphysics scheme, which is used only to estimate the cloud fraction +change consistent with the turbulent production of liquid cloud (see +section `3.8 <#sec:turb_qcl_scheme>`__). + +Most PC2 code is protected by IF tests on the namelist input *i_cld_vn* += 2 (PC2 in the GUI). However, within the convection scheme, the code is +controlled by logicals *l_calc_dxek* (which is just set to true if using +PC2, and set false otherwise), and *l_q_interact*, which controls +whether to allow the interactive detrainment and entrainment of +condensate. + +There is also a switch (currently hardwired to .false. in the code) +called *l_pc2_reset*. Turning this on (not recommended!) does 2 things: + +- Convective entrainment and detrainment of condensate is disabled, by + setting *l_q_interact* to false. + +- The prognostic cloud variables are overwritten by a call to the + diagnostic cloud scheme at the end of the timestep, in subroutine + *qt_bal_cld*. NOTE: this functionality will no longer work, because + inside *qt_bal_cld* the call to the diagnostic cloud scheme is now + protected by IF tests on using either the Smith or bimodal cloud + schemes. If using PC2, no cloud scheme is called here, and required + output variables are just left unset! + +The location of the various cloud scheme routine calls within the UM is +summarised in the list below. + +Subroutines only called for the Smith scheme are highlighted in blue, +those only called for PC2 are in green, and those only called for the +bimodal scheme are in purple. + +Main Tree from atm_step_4a +~~~~~~~~~~~~~~~~~~~~~~~~~~ + +.. container:: itemize + + | **atm_step_4a** + | \* (performs one timestep of the Unified Model...) + + .. container:: itemize + + .. container:: tcolorbox + + | **atm_step_alloc_4a** + | \* (does miscellaneous initialisations in atm_step) + + - | pc2_rhtl + | \* (calculate start-of-timestep Relative Humidity, used by + PC2 initiation) + + .. container:: tcolorbox + + | **atmos_physics1** + | \* (calls explicit “slow” physics routines...) + + .. container:: itemize + + .. container:: tcolorbox + + | **microphys_ctl** + | \* (interface to microphysics scheme) + + - | pc2_turbulence_ctl + | \* (Perform optional erosion of liquid-cloud; done + here if NOT doing erosion after convection, e.g. if + no convection scheme is used). + + - | pc2_hom_conv + | \* (called here just to do erosion) + + - | ls_cld + | \* (Smith scheme without area cloud fraction + calculation, to set initial cloud fields passed into + microphysics) + + - | **ls_ppn** + | \* (microphysics scheme) + + - | **mphys_turb_gen_mixed_phase** + | \* (turbulent production of liquid cloud) + + - | pc2_turbulence_ctl + | \* (optionally use the PC2 pdf-width-change code to + calculate the cloud-fraction change from the above + turbulent production of liquid cloud) + + - | pc2_hom_conv + | \* (called here just to calculate the cloud + fraction increment consistent with the turbulent + qcl increment) + + .. container:: tcolorbox + + | **rad_ctl** + | \* (interface to radiation scheme) + + - | **sw_rad** + | \* (short-wave radiation scheme) + + - | pc2_homog_plus_turb + | \* (PC2 homogeneous forcing of liquid-cloud by SW + radiation heating) + + - | **lw_rad** + | \* (long-wave radiation scheme) + + - | pc2_homog_plus_turb + | \* (PC2 homogeneous forcing of liquid-cloud by LW + radiation tendency) + + .. container:: tcolorbox + + **atmos_physics1_alloc_pc2** (wrapper for PC2 + self-consistency checks at end of atmos_physics1) + + - Add increments from microphysics + radiation onto + start-of-timestep fields to form updated fields. + + - | pc2_checks + | \* (self-consistency checks on cloud fractions and + water contents) + + - Convert corrected updated fields back to increments. + + Begin loop over solver outer cycles + + .. container:: itemize + + .. container:: tcolorbox + + | **atm_step_phys_reset** + | \* (for PC2, on subsequent solver outer cycles, reset + cloud-fractions to saved values after atmos_physics1) + + .. container:: tcolorbox + + | **eg_sl_moisture** + | \* (large-scale advection of cloud water contents and + fractions) + + .. container:: tcolorbox + + | pc2_pressure_forcing_only + | \* (Optionally calculate homogeneous forcing of liquid + cloud by the pressure change along the trajectory from + departure point to arrival point). + + - | pc2_homog_plus_turb + | \* (generic homogeneous forcing routine used here). + + .. container:: tcolorbox + + | **atmos_physics2** + | \* (calls “fast” physics routines...) + + .. container:: itemize + + .. container:: tcolorbox + + | **ni_bl_ctl** + | \* (interface to explicit boundary-layer and surface + scheme calls, including calculation of TKE and + TKE-based :math:`RH_{crit}`) + + .. container:: tcolorbox + + | bm_calc_tau + | \* (calculates turbulence properties used in the + bimodal cloud scheme, based on the boundary-layer + scheme TKE and mixing-length) + + .. container:: tcolorbox + + | **cloud_call_b4_conv** + | \* (routine for optional cloud-scheme calls before + convection) + + - | ls_arcld + | \* (Smith scheme with area cloud fraction; see + `5.2.2 <#subsubsec:smith_acf>`__ for a drill-down + inside this routine) + + - | bm_ctl + | \* (bimodal scheme) + + - Set area cloud fraction equal to bulk cloud fraction + + - | pc2_initiation_ctl + | \* (interface to PC2 initiation and + consistency-checks; see + `5.2.3 <#subsubsec:pc2_initiation>`__ for a + drill-down inside this routine) + + .. container:: tcolorbox + + | **ni_conv_ctl** or **other_conv_ctl** + | \* (interface routines to various convection + schemes...) + + - | **glue_conv_5a/6a** + | \* (calls deep, shallow and mid-level convection + schemes) + + - | **deep/shallow/mid_conv** + | \* (convection scheme main routines) + + - **convec2** (completes lifting of the convective + parcel by one model-level) + + - **parcel** (calculates new parcel properties + at next level) + + - **environ** (calculates grid-mean increments + to primary fields; includes PC2 partitioning + of detrained condensate mass between liquid + and ice phases) + + - pc2_environ (calculates increments to PC2 + cloud fractions due to convective detrainment + and subsidence) + + - | pc2_from_conv_ctl + | \* (PC2 calculations after convection) + + - | pc2_hom_conv + | \* (homogeneous forcing by convection, and + erosion of liquid-cloud) + + .. container:: tcolorbox + + | **ni_imp_ctl** + | \* (interface to boundary-layer implicit solver) + + - | **imp_solver** + | \* (implicitly solves vertical diffusion to find + :math:`T_l` and :math:`q_T` updated by turbulent + fluxes). + + - | pc2_bl_inhom_ice + | \* (inhomogeneous forcing of ice-cloud) + + - | pc2_delta_hom_turb + | \* (homogeneous forcing of liquid cloud by the + turbulent fluxes) + + - | pc2_bl_forced_cu + | \* (adds diagnosed “forced cumulus” cloud fraction + and water content onto the PC2 prognostics) + + - Calculate area cloud fraction: + + | ls_acf_brooks + | \* (for the Brooks epirical method) + + | pc2_hom_arcld + | \* (for the Cusack vertical interpolation method) + + - | pc2_homog_plus_turb + | \* (generic homogeneous forcing routine used to + interpolate) + + - | ls_arcld + | \* (interface to diagnostic Smith scheme and area + cloud fraction; see + `5.2.2 <#subsubsec:smith_acf>`__ for a drill-down + inside this routine) + + - | bm_ctl + | \* (bimodal cloud scheme) + + - Set area cloud fraction equal to bulk cloud fraction + + - | **diagnostics_bl** + | \* (outputs boundary-layer diagnostics to STASH) + + - | **ls_cld** + | \* (Smith scheme used here to calculate various + diagnostics of near-surface temperature and + humidity, by extrapolating pressure, :math:`T_l` + and :math:`q_t` down to the desired height and + then re-diagnosing :math:`q_{cl}`. + + .. container:: tcolorbox + + | **atm_step_ac_assim** + | \* (interface to Data Assimilation analysis increments...) + + - **ac_ctl** (control routine for Data Assimilation analysis + increments...) + + - **ac** (main analysis increment routine) + + - | pc2_assim + | \* (PC2 reponse to the analysis increments; see + `5.2.4 <#subsubsec:pc2_assim>`__ for a drill-down + inside this routine) + + - ls_acf_brooks (calculate area cloud fraction using + Brooks empirical method if active) + + - ls_arcld (call diagnostic Smith scheme with area cloud + fraction again to account for the analysis increments; + see `5.2.2 <#subsubsec:smith_acf>`__ for a drill-down + inside this routine) + + .. container:: tcolorbox + + | **eg_sl_helmholtz** + | \* (dynamics pressure solver; updates pressure, and the + winds used to perform advection on the next solver outer + cycle) + + End loop over solver outer cycles + + .. container:: tcolorbox + + | pc2_pressure_forcing + | \* (interface to miscellaneous PC2 calculations at + end-of-timestep) + + - | pc2_homog_plus_turb + | \* (homogeneous forcing of liquid-cloud by the dynamics + pressure change; optionally either uses total pressure + change including the Lagrangian component following the + winds, or only the Eulerian component from the dynamics + solver) + + - | pc2_initiation_ctl + | \* (interface to PC2 initiation and consistency-checks; see + `5.2.3 <#subsubsec:pc2_initiation>`__ for a drill-down + inside this routine) + + .. container:: tcolorbox + + | **qt_bal_cld** + | \* (calculates end-of-timestep cloud state consistent with + final pressure...) + + - | ls_arcld + | \* (interface to diagnostic Smith scheme and area cloud + fraction; see `5.2.2 <#subsubsec:smith_acf>`__ for a + drill-down inside this routine) + + - | bm_ctl + | \* (bimodal cloud scheme) + + - Set area cloud fraction equal to bulk cloud fraction + + .. container:: tcolorbox + + | **iau** + | \* (incremental analysis update; part of data assimilation) + + - | pc2_assim + | \* (PC2 reponse to the analysis increments; see + `5.2.4 <#subsubsec:pc2_assim>`__ for a drill-down inside + this routine) + + - | initial_pc2_check + | \* (wrapper for optional self-consistency checks on + prognostic cloud variables if not doing PC2 response to + analysis increments) + + - | pc2_checks + | \* (self-consistency checks on cloud fractions and water + contents) + +Drill-downs within some routines in the call tree are listed separately +below, to avoid duplication (since these routines are called in multiple +different places in the tree)... + +.. _`subsubsec:smith_acf`: + +Smith scheme with area cloud fraction +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +.. container:: itemize + + .. container:: tcolorbox + + | ls_arcld + | \* (interface to diagnostic Smith scheme and area cloud + fraction) + + - If no area cloud fraction scheme: + + | ls_cld + | \* (just directly call Smith scheme) + + Set area cloud fraction equal to bulk cloud fraction. + + - If using Cusack vertical interpolation method: + + Interpolate fields onto finer vertical grid + + | ls_cld + | \* (call Smith scheme using higher vertical resolution fields) + + Coarse-grain cloud fields back to model grid, but set area cloud + fraction to max of bulk cloud fraction over corresponding + fine-grid levels. + + - If using Brooks empirical area cloud fraction method: + + | ls_cld + | \* (just directly call Smith scheme) + + | ls_acf_brooks + | \* (estimate area cloud fraction) + +.. _`subsubsec:pc2_initiation`: + +PC2 initiation +~~~~~~~~~~~~~~ + +.. container:: itemize + + .. container:: tcolorbox + + | pc2_initiation_ctl + | \* (interface to PC2 initiation and consistency-checks) + + - | pc2_checks + | \* (self-consistency checks on cloud fractions and water + contents) + + - PC2 initiation of liquid-cloud: + + | pc2_bm_initiate + | \* (using the bimodal cloud scheme) + + | pc2_arcld + | \* (using the Smith scheme with the Cusack vertical + interpolation method) + + - | pc2_initiate + | \* (initiation using the Smith scheme, called here on a + finer vertical grid as per the Cusack method) + + | pc2_initiate + | \* (using the Smith scheme with no area cloud representation) + + - | pc2_checks2 + | \* (further self-consistency checks on cloud-fractions) + + - | pc2_checks + | \* (repeat the first lot of self-consistency checks again, + just in case we broke something in the mean-time!) + + - | pc2_hom_arcld + | \* (finds area cloud fraction using a version of the Cusack + method, where the cloud fraction on the finer vertical grid is + estimated by applying homogeneous forcing relative to the + original grid fields) + + - | pc2_homog_plus_turb + | \* (generic homogeneous forcing routine used to interpolate) + +.. _`subsubsec:pc2_assim`: + +PC2 Data Assimilation +~~~~~~~~~~~~~~~~~~~~~ + +.. container:: itemize + + .. container:: tcolorbox + + | pc2_assim + | \* (PC2 reponse to the analysis increments) + + - | pc2_homog_plus_turb + | \* (generic PC2 homogeneous forcing routine used here for + liquid-cloud) + + - Estimate change in ice-cloud fraction from the assimilation + increment to ice-cloud mass. + + - | pc2_total_cf + | \* (update bulk cloud fraction due to change in ice cloud + fraction) + + - | pc2_checks + | \* (self-consistency checks on prognostic cloud fractions and + water contents) + +.. _`sec:diags`: + +Diagnostics +----------- + +Nearly all diagnostics retain their meaning when PC2 is run. However, +there are a few that are subtly modified. + +The convective diagnostics that use the convective cloud base and top +calculations remain the same if PC2 is used with a zeroed convective +cloud fraction. These values are not reset by the convection scheme, +since the model is still predicting convection between the diagnosed +levels. + +The visibility diagnostics need modifying if the convective cloud +fraction is switched off, since they use the convective cloud fraction +within their calculation. Here we use a value of 0.2 for the convective +cloud amount if there is convective precipitation but the +two-dimensional convective cloud amount is zero. This will be the case +if the PC2 scheme has zeroed the convective cloud amount. + +There are a number of increment diagnostics that are required to fully +diagnose the moisture cycle within PC2. Since most physics sections can +cause condensation, condensate and cloud fraction increment diagnostics +have been written for each of these sections. + +- :math:`\overline{T}`, :math:`\overline{q}`, + :math:`\overline{q_{cl}}`, :math:`\overline{C_t}` and + :math:`\overline{C_l}` increments from SW radiation, + :math:`\overline{T}` increment from SW Radiation without including the + condensation: **Section 1** . + +- :math:`\overline{T}`, :math:`\overline{q}`, + :math:`\overline{q_{cl}}`, :math:`\overline{C_t}` and + :math:`\overline{C_l}` increments from LW radiation, + :math:`\overline{T}` increment from LW Radiation without including the + condensation: **Section 2** . + +- :math:`\overline{T}`, :math:`\overline{q}`, + :math:`\overline{q_{cl}}`, :math:`\overline{q_{cf}}`, + :math:`\overline{C_t}`, :math:`\overline{C_l}` and + :math:`\overline{C_f}` increments from Boundary Layer: **Section 3** . + +- :math:`\overline{T}`, :math:`\overline{q}`, + :math:`\overline{q_{cl}}`, :math:`\overline{q_{cf}}`, + :math:`\overline{C_t}`, :math:`\overline{C_l}` and + :math:`\overline{C_f}` increments from Large-scale precipitation: + **Section 4** . + +- :math:`\overline{T}`, :math:`\overline{q}`, + :math:`\overline{q_{cl}}`, :math:`\overline{q_{cf}}`, + :math:`\overline{C_t}`, :math:`\overline{C_l}` and + :math:`\overline{C_f}` increments from Convection, + :math:`\overline{q}`, :math:`\overline{q_{cl}}`, + :math:`\overline{q_{cf}}`, :math:`\overline{C_t}`, + :math:`\overline{C_l}` and :math:`\overline{C_f}` increments from the + inhomogeneous part of the Convection scheme only: **Section 5** . + +- :math:`\overline{T}`, :math:`\overline{q}`, + :math:`\overline{q_{cl}}`, :math:`\overline{q_{cf}}`, + :math:`\overline{C_t}`, :math:`\overline{C_l}` and + :math:`\overline{C_f}` increments from the Advection: **Section 12** . + +However, there are a number of parts of PC2 that do not fit into a +pre-existing section of code, and hence the associated increment +diagnostics are not easily placed within the UM framework. These +increments were available using a modification set or branch and a +user-STASHmaster file up to version 7.5. From version 7.6 these +diagnostics are available as standard. + +- :math:`\overline{T}`, :math:`\overline{q}`, + :math:`\overline{q_{cl}}`, :math:`\overline{C_t}`, and + :math:`\overline{C_l}` increments from the PC2 erosion section: + **Section 4** or **Section 5** depending on where the erosion is + called. + +- :math:`\overline{T}`, :math:`\overline{q}`, + :math:`\overline{q_{cl}}`, :math:`\overline{q_{cf}}`, + :math:`\overline{C_t}`, :math:`\overline{C_l}` and + :math:`\overline{C_f}` increments from the Bounds Checking after + atmphya: **Section 4** . + +- :math:`\overline{T}`, :math:`\overline{q}`, + :math:`\overline{q_{cl}}`, :math:`\overline{q_{cf}}`, + :math:`\overline{C_t}`, :math:`\overline{C_l}` and + :math:`\overline{C_f}` increments from the Initiation and Bounds + checking at the end of the timestep: **Section 16** . + +- :math:`\overline{T}`, :math:`\overline{q}`, + :math:`\overline{q_{cl}}`, :math:`\overline{q_{cf}}`, + :math:`\overline{C_t}`, :math:`\overline{C_l}` and + :math:`\overline{C_f}` increments from the Pressure Forcing section: + **Section 16** . + +Single Column Model +------------------- + +The updating in the single column model follows the same timestepping as +that in the full model, but the changes to atm-step are mirrored within +scm_main. The method used is to store the driving SCM forcing increments +of vapour, liquid and temperature across the forcing subroutine. The +forcing of pressure is set to zero. After atmos_physics2 has been +called, a PC2 section of code calls the homogeneous forcing subroutine +with these increments. This therefore treats the response of PC2 to the +prescribed dynamical forcing in the SCM as homogeneous. Following this +calculation, the initiation scheme is called, as usual. Finally, the +area cloud fraction is set to the bulk cloud fraction and +:math:`\overline{\Theta}` (potential temperature) is made consistent +with :math:`\overline{T}` (dry-bulb temperature) which was changed by +the condensation in the PC2 response to the homogeneous forcing. The +rest of the SCM uses the same PC2 code as the full model. + +Note that the change to PC2 homogeneous forcing from advection under the +UM namelist switch **l_pc2_sl_advection** (see section +`4.8 <#sec:pres>`__) is also mirrored in the Single-Column Model. If +this switch is turned on, the PC2 homogeneous forcing call using the SCM +forcing increments is moved straight after the call to the forcing +routine, so that the condensation adjustment is performed before the +call to atmos_physics2. If **l_pc2_sl_advection** is turned on, the PC2 +response to SCM forcings is also improved as follows... + +The SCM forcings may comprise one or both of the following: + +- (a) Prescribed tendencies or relaxation applied to T,q. + +- (b) Interactive vertical advection applied to T,q. + +For the latter, we can calculate the pressure change experienced by +vertically-advected parcels, and so calculate the PC2 homogeneous +forcing response in the same way as we do for Semi-Lagrangian advection +in the full model (see section `4.8 <#sec:pres>`__). For the former, we +don’t know if the prescribed T,q tendencies are due to advection, +radiation, or some other process, so we calculate the PC2 homogeneous +forcing response as if the tendencies are applied "in-situ". + +To split the PC2 homogeneous response into these 2 components, the SCM +forcing routine outputs: + +- (a) The forcing increments to T,q excluding the contribution from + interactive vertical advection. + +- (b) The value of exner pressure at departure points, consistent with + the vertical advection. + +The PC2 homogeneous forcing responses to these 2 forcing components are +then calculated by 2 separate PC2 calls in scm_main. + +Limited Area Boundary Conditions +-------------------------------- + +Cloud fractions on the limited area boundaries are fully updateable. +Writing of cloud fraction Limited Area Boundary Conditions (LBCs) will +be automatic if PC2 is selected. A PC2 LAM may be run from an LBC file +with or without cloud fraction LBCs (this is specified by the logical +l-pc2-lbc, which is set in the UMUI). If there are no cloud fraction +lbcs then around the edge of the domain the checking and initiation +routines will be applying significant increments to the cloud and +condensate fields near the boundaries, but this does not have an adverse +effect well away from the boundaries. If there are no cloud fraction +LBCs the cloud fraction fields themselves are not forced to zero around +the edge of the domain but are allowed to freely find their own value. A +PC2 run that outputs lbcs will, by default, always output cloud +fractions as part of the LBCs file. + +Parameter values +---------------- + +Table `1 <#tab:pc2_names>`__ summarizes the values of parameters used in +the PC2 scheme and their location within various comdecks. Those +parameters marked as ‘Num’ are those that are not part of the +mathematical equation set that is being solved, but are required in +order to achieve a stable, realistic, numerical solution. These include, +for instance, thresholds for resetting cloud fractions back to 0 or 1. +Those marked ’Phy’ are physical quantities that form an integral part of +the equation set that we wish to solve. Those marked ’Clo’ form part of +a closure needed to form the equation set, but are less readily related +to physical quantities. Variables marked ’Diag’ form a part of the +diagnostic output routines. + +.. container:: center + + .. container:: + :name: tab:pc2_names + + .. table:: PC2 parameter values and locations + + +----------+----------+----------+----------+----------+----------+ + | Symbol | Code | Des | Value | Location | Notes | + | | variable | cription | | | and ref. | + +==========+==========+==========+==========+==========+==========+ + | - | init-it | Number | 10 | p | Num: | + | | erations | of | | c2-const | `3.4 | + | | | it | | | .2 <#sec | + | | | erations | | | :numapp_ | + | | | in | | | init>`__ | + | | | in | | | | + | | | itiation | | | | + +----------+----------+----------+----------+----------+----------+ + | :math:` | cloud | Bounds | 0.005 | UM | Num: | + | C_{tol}` | -pc2-tol | checking | | namelist | `4.9 | + | | | :ma | | | <#sec:i | + | | | th:`C_l` | | | nit2>`__ | + | | | t | | | | + | | | hreshold | | | | + +----------+----------+----------+----------+----------+----------+ + | : | cloud-p | Bounds | 0.001 | UM | Num: | + | math:`C_ | c2-tol-2 | checking | | namelist | `4.9 | + | {tol 2}` | | :ma | | | <#sec:i | + | | | th:`C_l` | | | nit2>`__ | + | | | t | | | | + | | | hreshold | | | | + +----------+----------+----------+----------+----------+----------+ + | :math:`R | rh | : | 0.01 | p | Num: | + | H_{tol}` | crit-tol | math:`RH | | c2-const | `4.9 | + | | | _{crit}` | | | <#sec:i | + | | | t | | | nit2>`__ | + | | | olerance | | | | + | | | in | | | | + | | | in | | | | + | | | itiation | | | | + +----------+----------+----------+----------+----------+----------+ + | :math | ls-bl0 | Fixed | :ma | imp-ctl | Clo: | + | :`q_{cf0 | | value of | th:`1.0 | | ` | + | \, BL}` | | BL | \times 1 | | 4.6 <#se | + | | | in-plume | 0^{-4} \ | | c:bl>`__ | + | | | : | , kg \, | | | + | | | math:`\o | kg^{-1}` | | | + | | | verline{ | | | | + | | | q_{cf}}` | | | | + +----------+----------+----------+----------+----------+----------+ + | :math:` | one- | Fixed | :ma | pc2-chck | Num: | + | q_{cf0}` | over-qcf | in-cloud | th:`1.0 | | `4.10 | + | | | : | \times 1 | | <#sec:ch | + | | | math:`\o | 0^{-4} \ | | ecks>`__ | + | | | verline{ | , kg \, | | | + | | | q_{cf}}` | kg^{-1}` | | | + | | | if | | | | + | | | :math:` | | | | + | | | C_f`\ =0 | | | | + +----------+----------+----------+----------+----------+----------+ + | : | pdf-mer | Merging | 0.5 | p | Clo: | + | math:`m` | ge-power | power | | c2-const | `3.2 | + | | | for | | | <#sec:h | + | | | :math:` | | | omog>`__ | + | | | G(-Q_c)` | | | | + +----------+----------+----------+----------+----------+----------+ + | : | p | Shape | 0.0 | p | Phy: | + | math:`n` | df-power | p | | c2-const | `3.2 | + | | | arameter | | | <#sec:h | + | | | for | | | omog>`__ | + | | | :math:` | | | | + | | | G(-Q_c)` | | | | + +----------+----------+----------+----------+----------+----------+ + | : | w | Wind | :mat | p | Phy: | + | math:`w` | ind-shea | shear in | h:`1.5 \ | c2-const | ` | + | | r-factor | fallout | times 10 | | 4.2.1 <# | + | | | of ice | ^{-4} \, | | sec:lsp_ | + | | | term | s^{-1}` | | fall>`__ | + +----------+----------+----------+----------+----------+----------+ + | : | i | Scaling | 0.04 | p | Phy: | + | math:`i` | ce-width | factor | | c2-const | `4 | + | | | for | | | .2.4 <#s | + | | | r | | | ec:mp_de | + | | | eduction | | | psub>`__ | + | | | in | | | | + | | | :ma | | | | + | | | th:`b_i` | | | | + +----------+----------+----------+----------+----------+----------+ + | : | dbsdtb | Rate of | :math: | UM | Phy: | + | math:`a` | s-turb-0 | r | `-2.25 \ | namelist | `3.3 | + | | | eduction | times 10 | | <#sec:w | + | | | of PDF | ^{-5} \, | | idth>`__ | + | | | width | s^{-1}` | | | + +----------+----------+----------+----------+----------+----------+ + | : | dbsdtb | Rate of | 0 | p | Phy: | + | math:`b` | s-turb-1 | r | | c2-const | `3.3 | + | | | eduction | | | <#sec:w | + | | | of PDF | | | idth>`__ | + | | | width | | | | + +----------+----------+----------+----------+----------+----------+ + | | dbsd | Redn of | 0 | p | Phy: | + | | tbs-conv | PDF | | c2-const | `3.3 | + | | | width in | | | <#sec:w | + | | | co | | | idth>`__ | + | | | nvection | | | | + +----------+----------+----------+----------+----------+----------+ + | | dbs | V | 10.05 | p | Phy: | + | | dtbs-exp | ariation | | c2-const | `3.3 | + | | | of | | | <#sec:w | + | | | erosion | | | idth>`__ | + | | | on RH | | | | + +----------+----------+----------+----------+----------+----------+ + | : | RHCRIT | Critical | | UM | Phy: | + | math:`RH | | RH for | | namelist | `3.4 | + | _{crit}` | | cloud | | | <#sec:i | + | | | f | | | nit>`__, | + | | | ormation | | | `4 | + | | | | | | .2.4 <#s | + | | | | | | ec:mp_de | + | | | | | | psub>`__ | + +----------+----------+----------+----------+----------+----------+ + | :math: | condensa | Minimum | :m | pc2-chck | Num: | + | `q_{c0}` | te-limit | allowed | ath:`1 \ | | `4.10 | + | | | co | times 10 | | <#sec:ch | + | | | ndensate | ^{-10} \ | | ecks>`__ | + | | | | , kg \, | | | + | | | | kg^{-1}` | | | + +----------+----------+----------+----------+----------+----------+ + | :math:`q | ls0 | Lower | : | enviro?a | Num: | + | _c^{S0}` | | limit of | math:`5 | | `3.5. | + | | | plume | \times 1 | | 2 <#sec: | + | | | co | 0^{-5} \ | | multi_nu | + | | | ndensate | , kg \, | | mapp>`__ | + | | | | kg^{-1}` | | | + +----------+----------+----------+----------+----------+----------+ + | | *Har | Conv | 0.2 | imp-ctl2 | Diag: | + | | d-wired* | cloud | | | `5.3 | + | | | fraction | | | <#sec:d | + | | | for | | | iags>`__ | + | | | vi | | | | + | | | sibility | | | | + +----------+----------+----------+----------+----------+----------+ + | | *Har | Limit on | 0.001 | lspice3d | Num: | + | | d-wired* | width of | | | `4 | + | | | ice | | | .2.4 <#s | + | | | dist | | | ec:mp_de | + | | | ribution | | | psub>`__ | + +----------+----------+----------+----------+----------+----------+ + | | *Har | :ma | 0.05 | pc2-init | Num: | + | | d-wired* | th:`C_l` | | | `4.9 | + | | | limit | | | <#sec:i | + | | | for init | | | nit2>`__ | + | | | if | | | | + | | | :math:`T | | | | + | | | < 0 ^{\ | | | | + | | | circ} C` | | | | + +----------+----------+----------+----------+----------+----------+ + | | *Har | T | :mat | imp-ctl | Num: | + | | d-wired* | olerance | h:`1.0 \ | | ` | + | | | on calc. | times 10 | | 4.6 <#se | + | | | of | ^{-10} \ | | c:bl>`__ | + | | | :math | , kg \, | | | + | | | :`q_C^s` | kg^{-1}` | | | + | | | in BL | | | | + +----------+----------+----------+----------+----------+----------+ + +PC2 also recommends some tunings of the existing convection scheme +parameters. These cannot be placed in the library code, since they would +interact with non-PC2 simulations, hence would need to be specified with +modification sets. We have included those parameters that have been +investigated throughout testing, although only two are different between +PC2:64 and a non-PC2 run. + +.. container:: center + + .. container:: + :name: tab:pc2_conv_names + + .. table:: PC2 parameter values and locations relating to the + convection. \*These values are those used in HadGAM + + +----------+----------+----------+----------+----------+----------+ + | Code | Des | Value in | Value in | Location | Notes | + | variable | cription | PC2 | Control | | and | + | | | | | | r | + | | | | | | eference | + +==========+==========+==========+==========+==========+==========+ + | TICE | Tem | :math: | :math:`0 | tice.cdk | Phy: | + | | perature | `-10 ^{\ | ^{\circ | or UMUI | `4.7 | + | | at which | circ} C` | } C`\ \* | | <#sec:co | + | | plume | | | | nvec>`__ | + | | freezes | | | | | + +----------+----------+----------+----------+----------+----------+ + | QSTICE | App | :m | :m | qs | Phy: | + | | roximate | ath:`3.5 | ath:`3.5 | tice.cdk | `4.7 | + | | qs | \times | \times | or UMUI | <#sec:co | + | | at(TICE) | 10^{-3}` | 10^{-3}` | | nvec>`__ | + +----------+----------+----------+----------+----------+----------+ + | *Har | Limit on | 0.5 | :math: | cloudw | Phy: | + | d-wired* | conv. | : | `0.5 \, | | `4.7 | + | | cond. | math:`q_ | q_{sat}` | | <#sec:co | + | | after | {sat}, 2 | | | nvec>`__ | + | | precip | \times | | | | + | | | 10^{-4}` | | | | + +----------+----------+----------+----------+----------+----------+ + | Anvil | Shape | 0 | 0.3\* | UMUI | Phy: | + | factor | p | | | | `4.7 | + | | arameter | | | | <#sec:co | + | | for | | | | nvec>`__ | + | | conv. | | | | | + | | cloud | | | | | + | | anvil | | | | | + +----------+----------+----------+----------+----------+----------+ + | Tower | Shape | 0 | 0.25\* | UMUI | Phy: | + | factor | p | | | | `4.7 | + | | arameter | | | | <#sec:co | + | | for | | | | nvec>`__ | + | | conv. | | | | | + | | cloud | | | | | + | | tower | | | | | + +----------+----------+----------+----------+----------+----------+ + +How to run the PC2 scheme +------------------------- + +Running PC2 is straightforward, but you should seek advice as to +modification sets that you need to include to ensure you are running the +most up-to-date version of PC2. The following is a brief checklist of +the options in the UMUI which need to be selected in order to run PC2. +No hand-edits are required. + +- In the LS cloud panel (atmos-science-section-LScloud) push the button + marked ’use the PC2 cloud scheme’. + +- If you wish to use PC2 in the diagnostic only mode, also push ’run the + PC2 scheme in diagnostic only mode’. If you wish to run PC2 fully then + do not push this button + +- In the large-scale precipitation section + (atmos-science-section-LSprecip) select the 3D large-scale + precipitation scheme. + +- The specification of the LA boundary conditions can be set in the + atmos-InFiles-OtherAncil-LBC panel. + +- You will need to select modsets to include update the library code to + the PC2 version described here. Seek advice on these. + +- You may wish to adjust the convective anvil parameters in + atmos-science-section-convec. Again, seek advice. + +More information +---------------- + +Information on results of the scheme and how to run the PC2 code at +various model versions is available on the PC2 web site. + +.. _`sec:appendix-da`: + +Appendix: Alternative PC2 - Data Assimilation formulations +========================================================== + +In this alternative method to section `4.11 <#sec:da>`__ we will assume +that there exists a homogeneous forcing, :math:`\Delta Q_c`, that gives +changes, net of condensation, of :math:`\Delta\overline{q}` and +:math:`\Delta\overline{T}`. If we can recover what :math:`\Delta Q_c` is +then we can use this to calculate the liquid, :math:`\overline{q_{cl}}`, +and liquid cloud fraction, :math:`C_l`, increments. + +As in section `4.11 <#sec:da>`__, we start by discretising +(`[dqcldt] <#dqcldt>`__) to give + +.. math:: + + \Delta \overline{q_{cl}} = C_l \Delta Q_c + \label{eq:dqcldt_discrete} + +and hence, using the discrete form of :math:`\Delta Q_c` from +(`[eq:deltaqc_exp2] <#eq:deltaqc_exp2>`__) gives + +.. math:: + + \Delta \overline{q_{cl}} = C_l ( a_L ( \Delta \overline{q} - \alpha \Delta + \overline{T} ) + \Delta \overline{q_{cl}} ) , + +which rearranges to + +.. math:: + + \Delta \overline{q_{cl}} = \frac{1}{1-C_l} C_l a_L ( \Delta \overline{q} + - \alpha \Delta \overline{T} - \beta \Delta \overline{p}) . + \label{eqn:delataqcl} + +Comparing to (`[eq:deltaqc_exp2] <#eq:deltaqc_exp2>`__) and +(`[eq:dqcldt_discrete] <#eq:dqcldt_discrete>`__) we see that +:math:`\Delta \overline{q_{cl}}` is the same as if we had applied the +homogeneous forcing technique using :math:`\Delta \overline{q}`, +:math:`\Delta \overline{T}` and :math:`\Delta \overline{p}` as forcings, +except multiplied by a factor of :math:`\frac{1}{1-C_l}`. + +We can calculate :math:`\Delta C` in a similar way. From +(`[eq:deltac] <#eq:deltac>`__) + +.. math:: \Delta C_l = G(-Q_c) \Delta Q_c + +and hence, using our value of :math:`\Delta Q_c` from +(`[eq:deltaqc_exp2] <#eq:deltaqc_exp2>`__) and +:math:`\Delta \overline{q_{cl}}` from +(`[eqn:delataqcl] <#eqn:delataqcl>`__) + +.. math:: + + \Delta C_l = G(-Q_c) (a_L ( \Delta \overline{q} - \alpha \Delta \overline{T} ) + + \frac{1}{1-C_l} C_l a_L ( \Delta \overline{q} - \alpha \Delta \overline{T} ) ) + +which rearranges to + +.. math:: + + \Delta C_l = \frac{1}{1-C_l} G(-Q_c) a_L ( \Delta \overline{q} + - \alpha \Delta \overline{T} -\beta \Delta \overline{p}) . + \label{eqn:c1mc} + +This is also a factor of :math:`\frac{1}{1-C_l}` different from using +:math:`\Delta \overline{q}`, :math:`\Delta \overline{T}` and +:math:`\Delta \overline{p}` directly as forcings (the factor must be the +same, as we are still using the homogeneous forcing hypothesis). This +equation forms the basis for the more advanced technique discussed in +this section. However, it is undefined at :math:`C_l=1` and becomes +ill-conditioned near :math:`C_l=1`, hence there must be care taken when +this expression is solved numerically. + +Numerical solution +------------------ + +The timestepping applied is picked as a result of numerical tests +forcing a single gridbox with uniform increments. Many numerical +techniques were tested, this gives a fast but reasonably well behaved +solution. + +Initially, we calculate :math:`G(-Qc)` and :math:`\Delta Q_c` from the +input fields, as in the homogeneous forcing technique (section +`3.2 <#sec:homog>`__) and (`[eq:deltaqc_exp2] <#eq:deltaqc_exp2>`__). + +An initial increment, :math:`\Delta C_l^1` is estimated directly using +the basic equation + +.. math:: \Delta C_l^1 = \frac{1}{1-C_l^(n)} G(-Q_c) \Delta Q_c . + +We then recalculate this expression, using a mid-timestep estimate for +:math:`\Delta C_l`; + +.. math:: + + \Delta C_l = \frac{1}{1-(C_l^{[n]} + \frac{1}{2} \Delta C_l^1)} + G(-Q_c) \Delta Q_c + +where the term :math:`C_l^{[n]} + \frac{1}{2} \Delta C_l^1` is limited +to be no more than 0.9999 to avoid divide by zero problems. The final, +updated value of cloud fraction, :math:`C_l^{[n+1]}`, is then + +.. math:: C_l^{[n+1]} = C_l^{[n]} + \Delta C_l + +and this value is limited to 0 or 1. + +The liquid water term simply uses the final version of :math:`C_l` in +its calculation. + +.. math:: \Delta \overline{q_{cl}} = \frac{1}{1-C_l^{[n+1]}} C_l^{[n+1]} \Delta Q_c + +and will be set to 0 if :math:`C^{[n+1]}` is 0. There is an additional +limit, see below, applied to the liquid water term, which will prevent +the value of :math:`\Delta \overline{q_{cl}}` increasing to a large +number if :math:`C_l^{[n+1]}` is very close to 1. + +Limit on the liquid water content +--------------------------------- + +We will choose a limit on :math:`\overline{q_{cl}}` to be equal to its +value when the underlying PDF just corresponds to total cloud cover. +Therefore, from (`[eq:qclbar=int] <#eq:qclbar=int>`__) + +.. math:: \overline{q_{cl \, max}} = \int_{s=-b_s}^{\infty} G(s) (b_s + s) ds . + +We will use the current value of :math:`Q_c` (which won’t in general to +be equal to :math:`b_s`) to split the integral into two ranges of s: + +.. math:: + + \overline{q_{cl \, max}} = \int_{s=-b_s}^{-Q_c} G(s) (b_s + s) ds + + \int_{s=-Q_c}^{\infty} G(s) (b_s + s) ds . + +For the moment we write the first of these integrals as :math:`I1`, and +split the second integral whilst introducing a :math:`(+ Q_c - Q_c)` +term to the integrand: + +.. math:: + + \overline{q_{cl \, max}} = I1 + \int_{s=-Q_c}^{\infty} G(s) (b_s - Q_c) ds + + \int_{s=-Q_c}^{\infty} G(s) (s + Q_c) ds . + +The last of the integrals is now the current liquid water content, +:math:`\overline{q_{cl}}`. The second integral is proportional to the +liquid cloud fraction :math:`C_l`. + +.. math:: \overline{q_{cl \, max}} = I1 + C_l (b_s - Q_c) + \overline{q_{cl}} + +or + +.. math:: + + \overline{\Delta q_{cl \, max}} = I1 + C_l (b_s - Q_c) . + \label{eqn:deltaqclmax} + +Now consider the expression for the saturation deficit, which we have +defined, from (`[SD] <#SD>`__) as + +.. math:: SD = \int_{-b_s}^{-Q_c} G(s) (-Q_c - s) ds . + +Splitting and adding the term :math:`(+b_s - b_s)` to the integrand in a +similar way to above gives + +.. math:: + + \begin{aligned} + SD = \int_{-b_s}^{-Q_c} G(s) (-Q_c + b_s) ds + \int_{-b_s}^{-Q_c} + G(s) (-s - b_s) ds \nonumber \\ + = (-Q_c + b_s) (1 - C_l) - I1 , + \end{aligned} + +and hence :math:`I1` in terms of :math:`SD`. Using this value of +:math:`I1` in (`[eqn:deltaqclmax] <#eqn:deltaqclmax>`__) and cancelling +the :math:`C_l` terms gives :math:`\Delta \overline{q_{cl \, max}}` as + +.. math:: + + \Delta \overline{q_{cl \, max}} = (-Q_c + b_s) - SD . + \label{eqn:delta2} + +This is a general expression, it is not fixed for a particular PDF. To +complete the analysis, we need to estimate :math:`-Q_c+b_s`. To do this, +we now make the *assumption* of a power-law type PDF, as in section +`3.4 <#sec:init>`__. If we start from the equivalent of +(`[eqn19] <#eqn19>`__) but at the :math:`s=-bs` end of the distribution, +equation (B.3) in :raw-latex:`\cite{wg03}` can be equivalently written +for :math:`(1-C_l)` as: + +.. math:: + + (1-C_l) = \frac{ A (-Q_c + b_s)^{n+1} }{n+1} . + \label{eqn:1mc} + +To derive this from (B.3) note that :math:`C_l` is swapped for +:math:`1-C_l` and :math:`(b_s - (-Q_c))` is swapped for +:math:`(-Qc - (-b_s))`, as in section `3.4.2 <#sec:numapp_init>`__. +Similarly, noting that :math:`\overline{q_{cl}}` can be swapped with +:math:`SD`, gives the equivalent to (B.4) in :raw-latex:`\cite{wg03}` as + +.. math:: + + SD = \frac{ A (-Q_c + b_s)^{n+2} }{(n+1)(n+2)}. + \label{eqn:sd} + +Using the value :math:`(1-C_l)` from (`[eqn:1mc] <#eqn:1mc>`__) in +(`[eqn:sd] <#eqn:sd>`__) gives + +.. math:: \frac{SD}{1-C_l} = \frac {-Q_c + b_s}{n+2} . + +Finally, we use this expression for :math:`(-Q_c + b_s)` in +(`[eqn:delta2] <#eqn:delta2>`__) to parametrize +:math:`\Delta \overline{q_{cl \, max}}` in terms of the saturation +deficit + +.. math:: + + \Delta \overline{q_{cl \, max}} = SD ( \frac{n+2}{1-C_l} - 1 ) . + \label{eqn:sdr1mc} + +This is the expression that is used for the limit on +:math:`\overline{q_{cl}}`. We subsequently apply a second limit, since +numerically this expression is still not well behaved when :math:`C_l` +is close to 1. Here we note that just at complete cloud cover for a +symmetric PDF we have :math:`\overline{q_{cl}} = b_s`. Hence we estimate +:math:`b_s` as in :raw-latex:`\cite{smith90}`, + +.. math:: + + b_s = a_L ( 1 - RH_{crit} ) q_{sat}(\overline{T_L}) , + \label{eqn:bs} + +and take the smaller value for of (`[eqn:sdr1mc] <#eqn:sdr1mc>`__) and +(`[eqn:bs] <#eqn:bs>`__) for :math:`\Delta \overline{q_{cl \, max}}`. + +Initiation from :math:`C_l=1` +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +The equations are not defined when :math:`C_l=1`. (Note that when +:math:`C_l=0` we will calculate :math:`G(-Q_c)=0` so there is no change +in cloud fraction or liquid water content in this case). The +assimilation is capable of lowering :math:`\overline{q}` below +:math:`q_{sat}(\overline{T})` and hence there should be a corresponding +change in cloud fraction and liquid water content. In theory, we can use +the expression for :math:`\Delta \overline{q_{cl \, max}}` and assume +that the initial liquid water is equal to :math:`b_s`. However, this +produces a tricky set of simulataneous equations, which are not easily +solvable except in the case where :math:`n=0`. We proceed by making this +assumption for :math:`n`, acknowledging that this is not necessarily +entirely consistent with the rest of the model (although it is in the +PC2:64 formulation). + +We have (equivalent to B.6 from :raw-latex:`\cite{wg03}`) + +.. math:: \frac{ (1-C_l)^2 }{SD} = G(-Q_c) \frac{n+2}{n+1}. + +If n=0 (i.e. a ‘top-hat’ function) then +:math:`G(-Q_c) = \frac{1}{2 b_s}` and we can write + +.. math:: C_l = 1 - \sqrt{ \frac{SD}{b_s} } . + +We now assume :math:`b_s` is equal to our current value of +:math:`\overline{q_{cl}}` and hence + +.. math:: + + C_l^{[n+1]} = 1 - \sqrt{ \frac{SD^{[n+1]}}{\overline{q_{cl}^{[n]}}} } + \label{eqn:1msqrt} + +where :math:`C_l^{[n+1]}` and :math:`SD^{[n+1]}` are the values of +:math:`C_l` and :math:`SD` after this initiation has been applied. Using +our previous expression (`[eqn:sdr1mc] <#eqn:sdr1mc>`__) for +:math:`\Delta \overline{q_{cl max}}` gives (remembering that we are +considering the reverse process, so the sign is opposite), + +.. math:: \Delta \overline{q_{cl}} = - SD^{[n+1]} ( \frac{2}{1-C_l^{[n+1]}} - 1 ) + +(remembering that :math:`n=0` is assumed). Hence, replacing +:math:`C_l^{[n+1]}` by (`[eqn:1msqrt] <#eqn:1msqrt>`__) we have + +.. math:: + + \Delta \overline{q_{cl}} = SD^{[n+1]} - 2 \sqrt{ SD^{[n+1]} + \overline{q_{cl}}^{[n]} } . + +This is the expression we use, :math:`SD^{[n+1]}` is calculated after +the ssimilation increments have been applied, using (`[SD2] <#SD2>`__): + +.. math:: + + SD^{(n+1)} = a_L^{[n+1]} ( q_{sat}(\overline{T}^{[n+1]}, + \overline{p}^{[n+1]}) - \overline{q}^{[n+1]} ). + +Results +------- + +Results demonstrate a problem in that there is a distinct asymmetry +between changes when :math:`\Delta Q_c` is large and positive and when +:math:`\Delta Q_c` is large and negative, when cloud fractions start +near 1. In the former case, the limit to the amount of liquid and cloud +fraction that can be created means that changes must be kept relatively +small, whereas in the latter case, all the cloud and liquid water can be +removed easily. (The :math:`1/(1-C_l)` term allows this to be done +relatively quickly). Hence this assimilation method has a net tendency +to remove cloud from the simulation, which, at the moment, gives poorer +results than simply using the homogeneous forcing method. + +Further work will be required to enable the implementation of this +:math:`\overline{q}` and :math:`\overline{T}` preserving method. + +.. _`sec:code-development`: + +Appendix: Essentials of PC2 for code developers +=============================================== + +This section provides some guidance to code developers on the treatment +of PC2. Code developers are advised to read the relevant part of section +`4 <#sec:app_um>`__ to understand the way in which the current PC2 +scheme interacts with their section of code. + +The essence of a prognostic cloud scheme is that each physical part of +the model is able to calculate increments to the cloud fractions and +condensate contents. These form an integral part of each physics scheme +and should be considered by code owners as such, hence any alteration to +a scheme *must* consider also the impact on :math:`q_{cl}`, +:math:`q_{cf}`, :math:`C_t`, :math:`C_l` and :math:`C_f`, as well as on +the more traditional :math:`T`, :math:`q` and wind prognostics. Often +there should be no impact, but this cannot be assumed without +consideration. There is no diagnostic cloud fraction and condensation +scheme which can be run in PC2, since this would reset any effect of the +cloud prognostics used elsewhere in the model. (The diagnostic scheme +can still be used for model *diagnostics*, such as visibility and fog +fraction, and will be kept in later versions of the UM). + +Since this places a significant burden on code developers, the PC2 +developers have produced two generic representations which can take +increments to :math:`q` and :math:`T` etc. and produce an estimate of +the condensation and cloud fraction changes associated with the +increments. These are the homogeneous forcing and injection forcing (or +inhomogeneous forcing) methods. + +Homogeneous forcing +------------------- + +This is described fully in section `3.2 <#sec:homog>`__. This assumes +that the distribution of :math:`q_T - q_{sat}(T_L)` about its gridbox +mean is unchanged when a process acts. (The mean will change of course, +but we assume that the variations in each part of the gridbox from the +mean do not). Since this is equivalent to every part of the gridbox +receiving the same :math:`q_T` and :math:`T_L` increment, we call this +‘Homogeneous Forcing’. We have provided a subroutine +*pc2-homog-plus-turb*, in deck *pc2-homo* in order to provide the +necessary updates. + +Injection forcing +----------------- + +This is described fully in section `3.5 <#sec:inhomog>`__. We assume +that we already know a condensate increment :math:`q_{cl}` or +:math:`q_{cf}` and that a corresponding cloud fraction increment +:math:`C_l` or :math:`C_f` (and :math:`C_t`) remains to be estimated. +The injection forcing assumes that new cloud randomly displaces existing +cloud in a gridbox, and is designed with detrainment from deep +convection in mind, although it is also used elsewhere. It will require +as an input an estimate of the ‘in-cloud’ water content of the new cloud +that is produced. + +If you consider that both the homogeneous and injection forcing +representations are both poor assumptions for your scheme, you will need +to provide another method for calculating the condensation and cloud +fraction changes. The PC2 team can advise, but you should not expect +them to do the work. You can, of course, replace existing homogeneous +and inhomogeneous forcing calls with new representations of changes to +the prognostics if you think you have improved representations +available. This is part of the development of any prognostic variable +representation. + +Do I need to modify anything when I change a parametrization scheme? +-------------------------------------------------------------------- + +Here we assume that you wish to do the minimum work possible to get PC2 +to work, rather than a full reconsideration of the physics of the PC2 +increment terms. + +If your scheme is currently using the homogeneous forcing then there is +no need to update the cloud part of the scheme, *provided that you do +not alter values of :math:`T` and :math:`q` after the homogeneous +forcing section is called* and that the physical interpretation of your +:math:`q` and :math:`T` increments does not change. You need to be +careful if you are moving code from one subroutine to another that you +don’t inadvertently do this, although the forcing usually sits at the +end of the control subroutine. + +If your scheme is currently using the injection forcing *subroutine*, +which necessitates that condensate increments are already calculated by +the scheme, then there is also no need to update the cloud part of the +scheme. This currently applies to the boundary layer, where +:math:`q_{cf}` is altered by tracer mixing. Like for the homogeneous +schemes, this is provided that you *do not alter :math:`T`, :math:`q` or +condensate values after the injection forcing subroutine is called* and +that the physical interpretation of your :math:`q` and :math:`T` +increments does not change. + +Changes to winds do *not* need to have a condensation or cloud fraction +increment associated with them. There may be future scope for developing +an orographic cloud representation (probably diagnostic), but this is +not an essential part of the scheme as it stands. + +If your scheme uses hardwired assumptions about what is happening e.g. +convection or microphysics, then you *do* need to be careful that +:math:`T`, :math:`q` and condensates are still calculated correctly +after you have performed your changes. Currently there are many PC2 +assumptions hard-wired into the mass-flux convection scheme: + +- Any change to the scientific basis by which changes to :math:`T`, + :math:`q`, :math:`q_{cl}` and :math:`q_{cf}` are calculated requires + careful consideration + +- Simple changes to convective parameters, such as detrainment rates, + should not require a change to the PC2 code + +- Be particularly careful if you move code around, *especially the + calculation of convective cloud fractions*, since PC2 incorporates a + set-to-zero in the code. This will need to be replicated or there is a + risk that the diagnostic cloud fraction is no longer set to zero + correctly by PC2. + +Each microphysics transfer term has been considered individually for PC2 +and this should remain the case. + +Be especially careful when you do anything in the atmphy and atmstep +levels of the code that includes additional changes :math:`T`, +:math:`q`, :math:`q_{cl}` or :math:`q_{cf}`, since they may need cloud +fraction or condensation changes to go along with them. + +In summary, changes to existing increments of :math:`T`, :math:`q` etc. +within the current UM structure are unlikely to necessitate a +modification for PC2 if their physical interpretation has not changed. +However, new methods of generating :math:`T` and :math:`q` increments +will require new code to be added for PC2. + +Further PC2 development work +---------------------------- + +There are a number of areas in which the PC2:66 formulation can be +developed further, and many of these have been mentioned in the +documentation above. Some of these are simple sensitivity studies which +have not been fully explored in development, others are more complex +alterations. It is fair to say that the link to the convection has +proved the most problematic issue so far with PC2 development. + +PC2 cloud erosion +~~~~~~~~~~~~~~~~~ + +The cloud erosion is a critical term for the simulation of shallow +convective cloud. A large amount of erosion is required to keep the +cloud fractions relatively low in shallow convection, which is why we +have linked the erosion to the relative humidity. We recognise, however, +that this is more an empirical choice than a physically informed choice. +In particular, a low relative humidity (e.g. in the stratosphere) would +imply a very high erosion rate - although the net effect is to remove +any cloud, which is a reasonable thing to do, there is an implication of +the parametrization that mixing within the stratosphere is high, which +is clearly incorrect. We have also seen relatively low cloud fractions +in the mid-levels of deep convection in PC2, and presume that this is +influenced by the erosion formulation. A link to mass flux has also been +proposed, but tests with CRMs do not support a clear link. Perhaps it is +more natural to compare the erosion with the turbulent kinetic energy. +This should be available within the boundary layer and convection +schemes, but not outside of these in the current UM. + +The erosion formulation in PC2:66 is one where the width of the PDF is +always narrowed (developed following :raw-latex:`\cite{sg03}`). It may +be advantageous to think whether there are unmodelled processes in the +atmosphere that result in an increase in width. Clearly convection is +likely to be one, but this is already represented in PC2. There may be +other models entirely for the way in which the PDF changes as a result +of mixing of air within a gridbox or within the column, these may prove +fruitful to explore. + +Another issue is whether width-narrowing (or widening) is really an +effective way of representing the erosion process. CRM evidence suggests +that the required erosion rates to balance convective cloud generation +are larger for liquid cloud fraction than liquid water (by up to a +factor of 2), suggesting that the real atmospheric erosion favours +removal of cloud fraction over liquid water more strongly than the +model. + +The in-cloud condensate that is detrained from convective plumes is +high. We might think that the mixing in of environmental air in reality +is likely to lead to more cloud around the plumes and lower condensate +within the plumes. However, the width narrowing scheme is not a good +model of mixing in this situation, always reducing the amount of cloud +because it is incorrectly assumed that much of the detrained plume has +condensate contents only just above zero and that the shape of the +moisture PDF remains unchanged. This may have a bearing on the problem +of the lack of mid-level cloud in the model (although I think there are +many reasons for this). A different mixing method may give significantly +different results for the areas around convective plumes. + +Narrowing of the moisture PDF +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +Most of the parametrized terms in PC2 act to reduce the width of the +moisture PDF. The only terms that can increase the width are the +convection, and the initiation (which can reset the width). This may not +be the best way to describe the way in which the PDF evolves, in +particular it is sensible to ask whether the erosion term should +actually increase the width in the presence of large vertical gradients +of moisture. + +Convective cloud increments in the mass-flux framework +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +As discussed in section `3.5.3 <#sec:conv_imp_note>`__, it would be +useful to code up the convective cloud fraction changes to link directly +to the mass-flux convection scheme, and not to estimate them from the +values of :math:`Q4`, which can introduce errors. + +.. _`sec:tbcs`: + +Turbulence based convection scheme +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +We will need to properly consider the links between PC2 and the +turbulence based convection scheme. In essence, we can use the diagnosed +cloud fraction and condensate values from the convection scheme to start +off the cloud again when convection has ceased. This has been tested to +some degree but will need proper analysis. The difficult decision comes +in choosing what to do with the condensate and cloud fraction that is +present *before* the convection starts, since we must ensure +conservation of moisture. This is not helped by the traditional view of +convective parametrization that ignores the existence of the condensate +phase in the atmosphere (i.e. it is only concerned with transport of +:math:`q` and :math:`\theta`, not of :math:`q_{cl}` and :math:`q_{cf}`) +despite the phase changes forming an integral part of the convection +scheme. + +Detailed convective comparisons with CRM/LEM data +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +This work is already underway at the Met Office, in order to properly +evaluate the performance of the convective cloud parametrization in PC2 +against high resolution research models. + +Choice of PDF parameters +~~~~~~~~~~~~~~~~~~~~~~~~ + +Work by Dan Tang at Leeds University has highlighted an interesting and +undesirable property of the choice of :math:`m` and :math:`n` parameters +in the homogeneous forcing formulation. If a distribution is +homogeneously forced to :math:`C_l = 0`, then we do not necessarily get +:math:`\overline{q_{cl}}` tending to zero. This is because there is +enough influence from the :math:`\frac{{(1-C_l)}^2}{SD}` term in the +combination (`[eqn22] <#eqn22>`__) to stop the natural convergence of +the :math:`\frac{{C_l}^2}{\overline{q_{cl}}}` term to :math:`C_l =0` and +:math:`\overline{q_{cl}}=0`. Increasing the power of :math:`m` should +help. However, we note that the tests that have been done on the chosen +:math:`n` and :math:`m` values (0 and 0.5 respectively) do not show +particularly poor behaviour, and we do not pick up substantial evidence +of problems from this in the full model. This remains something to be +investigated. + +.. _`sec:homog_improve`: + +Homogeneous forcing section improvements +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +Although the homogeneous forcing provides a convenient method to +calculate increments to :math:`C_l` and :math:`\overline{q_{cl}}`, it is +clearly not the best representation possible of the processes that use +it. For example, although the clear-sky radiative heating may perhaps +best be considered as a homogeneous process, the part of the radiative +heating influenced by clouds should, ideally, be applied to the cloudy +part of the gridbox and not the clear part. Vertical advection is likely +to be correlated with where there is already cloud, rather than being +uniform throughout the gridbox. There is no reason that a process that +uses homogeneous forcing as its condensation model should not be looked +at with a view to using something better. This is one of the strengths +of the PC2 framework and is an intention of the project. + +Overlap of ice and liquid cloud changes +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +We have assumed within PC2 that ice and liquid cloud changes are +minimally overlapped with each other (within the same gridbox) in order +to maintain as much supercooled liquid water as possible. Although there +is good observational evidence to say that the two condensate phases +tend not to coexist together in a cloud, it may be possible to +characterise and apply this overlap in a more quantiative way. + +Parameter tuning +~~~~~~~~~~~~~~~~ + +The sensitivity of some of the parameters in PC2 have not been properly +tested, mainly due to a lack of resources rather than a physical reason. +We have seen that the most effective method of tuning cloud is with the +erosion term, which has been increased to high values in order to remove +enough cloud and is probably as high as we reasonably wish to take it +given the length of the timestep. + +- The phase change temperature (between liquid and ice) in the + convective plume, TICE, is known to influence the strength of the + convection through the latent heat differences. It also impacts on the + amount of supercooled liquid water in the model. The quantitative + impact of altering this could be explored. We note that CRM + simulations of deep convection suggest that some supercooled liquid + water exists within the plumes to :math:`-40 ^{\circ} C` and that a + representation with partial liquid and partial ice phase would be more + appropriate, based possibly on the current diagnosed convective cloud + phase in the non-PC2 model. Although the theoretical work has been + done to allow partial phases, we repeat the caution that care must be + taken when doing the work and appropriate testing done to ensure that + heat and moisture are properly conserved within the convection scheme. + +- The growth of :math:`C_f` due to the fall-out of ice term in the + microphysics is parametrized with a dependence on windshear. We have + never linked this directly to the windshear, instead we have used + estimated the windshear as a fixed value. There is no reason why the + actual model windshear cannot be passed into the scheme in order to + properly calculate this term. + +- :math:`RH_{crit}` remains a tunable parameter. Although its impact is + less than in a non-PC2 simulation, it is still significant in + initiating cloud and in determining the evolution of the ice cloud. + There is also an implicit overlap assumption regarding the ice cloud + fractions, again this might be improved upon. + +- :math:`n` and :math:`m` values in the homogeneous forcing have not + been thoroughly investigated for a long time now, and may yield some + sensitivities. + +Cloud inhomogeneities +~~~~~~~~~~~~~~~~~~~~~ + +A cloud generator approach to cloud inhomogeneities is currently being +developed. However we note two particular issues that relate to PC2. + +- The first is that in the diagnostic scheme, the two cloud fractions + (convective and large-scale) allows, to some degree, a representation + of cloud inhomogeneity. This is absent from PC2, although we note that + the convective cloud fraction variable has not been removed from the + radiative transfer code for PC2, it is merely set to zero, so it is + easy to put back. + +- The generation of inhomogeneities using a cloud generator requires + some estimate of the variance (and possibly skewness) of the + condensate in the gridbox. It is possible to back out the full + moisture PDF at each grid point by homogeneous forcing (providing + :math:`C_l` is not equal to 0 or 1), but this is very expensive and + cannot be done on-line. Is there a quick *estimate* of the variance or + skewness that it is possible to obtain from knowledge only of + :math:`\overline{q}`, :math:`q_{sat}`, :math:`\overline{q_{cl}}` and + :math:`C_l` etc.? + +.. _`sec:timestepping`: + +Time-stepping +~~~~~~~~~~~~~ + +A proper analysis of timestep sensitivities of PC2 (as opposed to +microphysics, convection etc) in the full UM or SCM has not been done +for a long time. In the early development stages much effort was placed +in developing good numerical techniques for each of the terms in PC2, +and to explore the way in which they coupled together. An example is the +homogeneous forcing timestep investigated by :raw-latex:`\cite{wg03}`. +We note that in shallow convection at 30 minutes timestep the erosion +term is trying to remove most of the cloud that the convective +detrainment places into the model. Since the erosion is limited by the +amount of cloud fraction and condensate present, what ends up happening +is that the ‘equilibrium’ that is achieved is actually one where the +cloud fraction and condensate at the end of the timestep are simply the +values that were detrained by the convection scheme (and hence depend on +the timestep). The CRM suggests a cycling time of around 15 minutes for +liquid water content and just less than half and hour for the cloud +fraction, so we would expect timestep dependency to occur from around a +timestep of 15 minutes upwards. We might just about get away with the 30 +minute step of the climate model, but it is not a good situation to try +to model. This is demonstrating the difficulty of modelling shallow +convective cloud by a prognostic scheme, where the physical lifetime of +the clouds is of order the timestep - ideally we wouldn’t want to try to +model anything prognostically when the cycling time is less than the +timestep. + +As discussed in section `4.3.4 <#sec:erosion_numerics>`__, the timestep +sensitivity of cloud amounts in shallow cumulus regimes can be addressed +by using a more accurate numerical method to solve the erosion term. +Several options are available under the UM namelist switch +**i_pc2_erosion_numerics**. + +In the early development of PC2 we chose to incorporate the PC2 cloud +and condensation increments in the same location where the increments +were calculated (e.g. the microphysics cloud fraction increments get +added along with the microphysics :math:`\overline{T}` and +:math:`\overline{q}` increments). This choice was made in order not to +confuse the timestepping method in the UM, which has been carefully +developed over a number of years to achieve numerical accuracy. However, +we note that the rapidly varying nature (in space and time) of variables +such as :math:`\overline{q_{cl}}` and :math:`C_l` is very different from +the smooth fields of :math:`\overline{q_T}` and :math:`\overline{T}`, +for which the timestepping was developed, and it may not be appropriate +to implement these in the same locations. In particular, we might wish +to store the increments through the timestep and update values of +:math:`\overline{q_{cl}}` and :math:`C_l` etc. at the end of the +timestep, where many of the balances can be cancelled. + +One issue is that we are calculating the increments due to condensation +associated with the adiabatic response to pressure changes after the +Helmholtz solver. Pragmatically, we need to do it here since we do not +know the arrival value of pressure until after the Helmholtz solver has +been used. However, in order to achieve balanced dynamical fields, it is +useful the Helmholtz solver to be called after all the latent heating +terms have been calculated (which not only includes the adiabatic +response to lifting but the cloud initiation term). We have shown that +PC2 can run with the two terms switched over, but this implies that we +are missing part of the pressure change following the parcel (the time +changing part rather than the spatially changing adiabatic part). +Although the adiabatic change is usually likely to dominate, it may be a +significant loss. Under the UM namelist switch **l_pc2_sl_advection**, +we can call the PC2 response twice, once before the Helmholtz solver and +once afterwards in order to pick up most of the latent heat change +before the solver, but not to have PC2 miss some of the pressure change. +The call for the advective part (before the Helmholtz solver) is +actually done before the call to atmos_physics2 as well, and so results +in more realistic, saturation-adjusted, profiles being passed to the +convection scheme. + +We have placed the initiation at the end of the timestep, but it is +sensible to ask whether this could ideally be located elsewhere. + +Initiation formulation +~~~~~~~~~~~~~~~~~~~~~~ + +Ideally this should be a relatively infrequent part of the model but +remains an essential part of the code. It is reasonable to ask whether +the initiation is optimal, particular in the diagnosis of when it is +applied. For example, we note that the initiation is currently +symmetrical, with initiation from :math:`C_l=1` occuring with the same +:math:`RH_{crit}` value as from :math:`C_l=0`. However, the +:raw-latex:`\cite{wf00}` observations hint that a higher +:math:`RH_{crit}` might be more appropriate for initiation from +:math:`C_l=1`. + +70-levels performance +~~~~~~~~~~~~~~~~~~~~~ + +The performance of PC2:66 in the 70-levels model is not good as far as +shallow convective cloud is concerned (there is far too much of it in +the trade regions). It may be that PC2 is latching onto a convection +sensitivity that is present on going from L38 to L70 but had little +effect in a non-PC2 simulation. It may also be related to a reduction in +timestep from 30 minutes to 20 minutes. Investigations have not made +much progress in identifying the reasons for the differences, or +producing effective tunings to counter the problem. + +High horizontal resolution performace +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +PC2 has only been tested once at 4 km horizontal resolution. This +produced excessive shallow convective cloud (this may or may not be +related to the 70-levels problem above). Since this simulation the +erosion term has been increased dramatically, which may help. We note +that one of the main advantages of PC2, that of a prognostic link of +cloud to convection, is reduced at high resolution, as convection +becomes more explicit rather than diagnosed. We hence see a resolution +limit beyond which it is no longer appropriate to use PC2. Results look +acceptable at 12 km resolution, but we have not quantitatively explored +this limit. + +Diagnostic evaluation +~~~~~~~~~~~~~~~~~~~~~ + +One of the principal areas for future cloud scheme development work +planned in the future is in the area of detailed evaluation against a +number of data sources, such as CloudSat, ground based radar, or case +study campaigns. The quantitative evaluation has been lacking to a +significant degree in the development of the scheme, as the focus has +been on tackling qualitatively poor results. Hence new sources of +evaluation work on PC2 would be very welcome. + +Moisture distribution within the deposition/sublimation term +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +The liquid cloud changes in PC2 (or in a non-PC2 run) are based upon a +moisture PDF, as are the deposition/sublimation changes. However, it is +not the same PDF. It has always been the case with the prognostic ice +microphysics term that its PDF, whether explicit or implicit, has not +been rigorously consistent with the PDF used in the calculation of +liquid water, because it was most easily developed that way and produced +reasonable results. It may be useful to investigate whether the two PDF +representations can be brought together in a rigourous way, both for the +PC2 scheme and the :raw-latex:`\cite{smith90}` scheme. + +We have similarly noted potential inconsistencies in the parametrization +of cloud fraction changes between the evaporation of rain term and the +riming (or accretion) term. Again, it might be possible to bring +together these formulations into a single consistent framework. + +Area cloud fraction representation +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +The current area cloud fraction representation is not used when +convection is taking place (signified by the *cumulus* logical). This +inevitably leads to a potential switching between two different values +of the cloud fields if the convective boundary layer (not whether the +convection is shallow or deep) switches on and off, which is +undesirable, although not as bad as switching cloud on and off +completely (as for the current convective cloud formulation). +Additionally, it is reasonable to argue that having an area cloud +fraction for cirrus cloud depend upon whether the boundary layer is well +mixed or has shallow convection occuring is not a reasonable link. + +Work in Australia on a TWP-ICE single column model case study using PC2 +suggests the area cloud fraction scheme over estimates the area cloud +coverage for tropical anvil clouds (which exist long after the +convection itself has ceased). This is perhaps not surprising since the +:raw-latex:`\cite{bhi05}` area cloud fraction scheme was evaluated +against mid-latitude cloud and it is known that tropical clouds have +greater vertical coherence. Tuning the parameters in +:math:`large_scale_cloud/ls_acf_brooks.F90` may be beneficial. + +.. container:: float + :name: fig:schematic + + .. container:: center + + |image1| + +.. container:: float + :name: fig:tstep_diag + + .. container:: center + + |image2| + +.. container:: float + :name: fig:tstep_prog + + .. container:: center + + |image3| + +.. |image1| image:: pc2_process_explanation +.. |image2| image:: Timestepping_ctl66.epsi +.. |image3| image:: Timestepping_pc266.epsi From 6c02d25fab2a89311690cd5727a265ae90847f74 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Thu, 2 Apr 2026 11:43:21 +0100 Subject: [PATCH 005/116] Attempting to import the figures. --- .../cloud_schemes/Timestepping_ctl66.epsi | 9927 +++++++++++++++ .../cloud_schemes/Timestepping_pc266.epsi | 10230 ++++++++++++++++ .../cloud_schemes/UMDP30_PC2CloudScheme.rst | 2 +- .../cloud_schemes/pc2_process_explanation.eps | 9032 ++++++++++++++ 4 files changed, 29190 insertions(+), 1 deletion(-) create mode 100644 documentation/source/science_guide/cloud_schemes/Timestepping_ctl66.epsi create mode 100644 documentation/source/science_guide/cloud_schemes/Timestepping_pc266.epsi create mode 100644 documentation/source/science_guide/cloud_schemes/pc2_process_explanation.eps diff --git a/documentation/source/science_guide/cloud_schemes/Timestepping_ctl66.epsi b/documentation/source/science_guide/cloud_schemes/Timestepping_ctl66.epsi new file mode 100644 index 0000000000..2b235fbf34 --- /dev/null +++ b/documentation/source/science_guide/cloud_schemes/Timestepping_ctl66.epsi @@ -0,0 +1,9927 @@ +%!PS-Adobe-3.0 EPSF-3.0 +%%Invocation: path/gs -q -sDEVICE=eps2write -sstdout=? -sOutputFile=? -dNOPAUSE -dBATCH -P- -dSAFER -dDEVICEWIDTH=250000 -dDEVICEHEIGHT=250000 ? +%%BoundingBox: 0 0 720 960 +%%HiResBoundingBox: 0.00 0.00 720.00 960.00 +%%Creator: GPL Ghostscript 9540 (eps2write) +%%LanguageLevel: 2 +%%CreationDate: D:20250417013840+01'00' +%%Pages: 1 +%%EndComments +%%BeginProlog +10 dict dup begin +/DSC_OPDFREAD true def +/SetPageSize false def +/EPS2Write true def +end +count 0 ne{ +dup type/dicttype eq{ +dup/EPS2Write known{ +dup/EPS2Write get not +} +{ +true +}ifelse +} +{ +true +}ifelse +} +{ +true +}ifelse +10 dict begin +/this currentdict def +/y 720 def +/ebuf 200 string def +/prnt{ +36//this/y get moveto//ebuf cvs show +//this/y 2 copy get 12 sub put +}bind def +/newline{ +36//this/y get moveto +//this/y 2 copy get 12 sub put +}bind def +{ +errordict/handleerror +{systemdict begin +$error begin +newerror +{(%%[ Error handled by opdfread.ps : )print errorname//ebuf cvs print(; OffendingCommand: ) +print/command load//ebuf cvs print( ]%%)= flush +/newerror false store vmstatus pop pop 0 ne +{grestoreall +}if +errorname(VMerror)ne +{showpage +}if +initgraphics +0 720 moveto +errorname(VMerror)eq +{//this/ehsave known +{clear//this/ehsave get restore 2 vmreclaim +}if +vmstatus exch pop exch pop +} +/Courier 12 selectfont +{ +(ERROR: )//prnt exec errorname//prnt exec +(OFFENDING COMMAND: )//prnt exec +/command load//prnt exec +$error/ostack known{ +(%%[STACK:)= +(STACK:)//prnt exec +$error/ostack get aload length{ +//newline exec +dup mark eq{ +(-mark-)dup = show +}{ +dup type/nametype eq{ +dup xcheck not{ +(/)show +(/)print +}if +}if +dup =//ebuf cvs show +}ifelse +}repeat +}if +}ifelse +(%%]%)= +//systemdict/showpage get exec +quit +}if +end +end +}bind readonly put +}if +end +50 dict begin +count 0 ne{ +dup type/dicttype eq{ +{def}forall +false +} +{ +true +}ifelse +} +{ +true +}ifelse +{ +( *** Warning: global definitions dictionary not found, file may be corrupted.\n)print flush +}if +/DefaultSwitch +{ +dup where{ +pop pop +}{ +false def +}ifelse +}bind def +/=string 256 string def +/=only{ +//=string cvs print +}bind def +/HexDigits(0123456789ABCDEF)readonly def +/PrintHex +{8{ +dup -28 bitshift 15 and//HexDigits exch 1 getinterval//=only exec +4 bitshift +}repeat +pop +}bind def +/PDFR_DEBUG DefaultSwitch +/PDFR_DUMP DefaultSwitch +/PDFR_STREAM DefaultSwitch +/TTFDEBUG DefaultSwitch +/RotatePages DefaultSwitch +/FitPages DefaultSwitch +/CenterPages DefaultSwitch +/SetPageSize DefaultSwitch +/error +{ +counttomark 1 sub -1 0{ +index dup type/arraytype eq{==}{=only}ifelse +}for +()= +cleartomark +....Undefined +}bind def +//SetPageSize{ +//RotatePages//FitPages or//CenterPages or{ +mark(/RotatePages, /FitPages and CenterPages are not allowed with /SetPageSize)//error exec +}if +} +{ +//FitPages//CenterPages and{ +mark(CenterPages is not allowed with /FitPages)//error exec +}if +} +ifelse +/knownget +{ +2 copy known{ +get true +}{ +pop pop false +}ifelse +}bind def +/IsUpper +{dup(A)0 get ge exch(Z)0 get le and +}bind def +/cpa2g{ +dup length array +0 1 2 index length 1 sub{ +dup 3 index exch get cp2g +3 copy put pop pop +}for +exch pop +}bind def +/cpd2g{ +dup length dict exch{ +cp2g 2 index 3 1 roll put +}forall +}bind def +/cps2g{ +dup length string copy +}bind def +/cp2gprocs +<> +def +/cp2g{ +dup gcheck not{ +dup//cp2gprocs 1 index type +2 copy known{ +get currentglobal 3 1 roll true setglobal exec exch setglobal +1 index wcheck not{readonly}if +1 index xcheck{cvx}if +exch pop +}{ +pop pop +}ifelse +}if +}bind def +/BlockBuffer 65535 string def +/PDFReader currentdict def +/ObjectRegistryMaxLength 50000 def +/ObjectRegistry 10 dict def +ObjectRegistry +begin +0 ObjectRegistryMaxLength dict def +end +/CurrentObject null def +/DoneDocumentStructure false def +/GraphicState 20 dict begin +/InitialTextMatrix matrix def +/InitialMatrix matrix currentmatrix def +currentdict end def +/TempMatrix matrix def +/GraphicStateStack 20 array def +/GraphicStateStackPointer 0 def +/InitialTextMatrixStack 20 array def +/InitialTextMatrixStackPointer 0 def +/PDFColorSpaces 50 dict def +/InstalledFonts 50 dict def +/MacRomanEncodingInverse null def +currentglobal false setglobal +userdict/PDFR_InitialGS gstate put +userdict/PDFR_Patterns 50 dict put +userdict/FuncDataReader 10 dict put +setglobal +/InitialExtGState 20 dict begin +/BG2 currentblackgeneration cp2g def +/UCR2 currentundercolorremoval cp2g def +/TR2 currentglobal false setglobal[currentcolortransfer]exch setglobal cp2g def +/HT currenthalftone cp2g def +currentdict end readonly def +/InitialGraphicState 20 dict begin +/FontSize 0 def +/CharacterSpacing 0 def +/TextLeading 0 def +/TextRenderingMode 0 def +/WordSpacing 0 def +currentdict end readonly def +/SimpleColorSpaceNames 15 dict begin +/DeviceGray true def +/DeviceRGB true def +/DeviceCMYK true def +currentdict end readonly def +/1_24_bitshift_1_sub 1 24 bitshift 1 sub def +/ReadFontProcs 10 dict def +/GetObject +{ +dup ObjectRegistryMaxLength idiv +//PDFReader/ObjectRegistry get exch knownget{ +exch knownget +}{ +pop false +}ifelse +}bind def +/PutObject +{ +1 index ObjectRegistryMaxLength idiv +//PDFReader/ObjectRegistry get 1 index knownget{ +exch pop +3 1 roll put +}{ +//PDFReader/ObjectRegistry get dup +begin +1 index ObjectRegistryMaxLength dict def +end +exch get +3 1 roll put +}ifelse +}bind def +/Register +{ +1 index GetObject{ +dup xcheck{ +4 3 roll pop +//PDFR_DEBUG{ +(Have a daemon for )print 2 index == +}if +exec +}{ +dup null ne{ +mark(The object )4 index(is already defined : )4 index//error exec +}{ +pop +}ifelse +3 2 roll +exec +}ifelse +}{ +3 2 roll +exec +}ifelse +PutObject +}bind def +/IsRegistered +{ +GetObject{ +null ne +}{ +false +}ifelse +}bind def +/GetRegistered +{ +dup GetObject not{ +exch mark exch(Object )exch( isn't defined before needed (1).)//error exec +}if +dup xcheck{ +exch mark exch(Object )exch( isn't defined before needed (2).)//error exec +}{ +dup null eq{ +exch mark exch(Object )exch( isn't defined before needed (3).)//error exec +}if +exch pop +}ifelse +}bind def +/StandardFontNames<< +/Times-Roman true +/Helvetica true +/Courier true +/Symbol true +/Times-Bold true +/Helvetica-Bold true +/Courier-Bold true +/ZapfDingbats true +/Times-Italic true +/Helvetica-Oblique true +/Courier-Oblique true +/Times-BoldItalic true +/Helvetica-BoldOblique true +/Courier-BoldOblique true +>>def +/CleanAllResources +{//PDFR_DEBUG{ +(CleanAllResources beg)= +}if +//PDFReader/ObjectRegistry get{ +dup length 0 exch 1 exch 1 sub{ +2 copy get dup xcheck{ +pop pop +}{ +dup null eq{ +pop pop +}{ +dup type/dicttype eq{/.Global known}{pop false}ifelse{ +pop +}{ +//PDFR_DEBUG{ +(Dropping )print dup = +}if +1 index exch/DroppedObject put +}ifelse +}ifelse +}ifelse +}for +pop +}forall +FontDirectory length dict begin +FontDirectory{ +pop +dup//StandardFontNames exch known not{ +dup null def +}if +pop +}forall +currentdict +end{ +pop +//PDFR_DEBUG{ +(Undefining font )print dup = +}if +undefinefont +}forall +//PDFR_DEBUG{ +(CleanAllResources end)= +}if +}bind def +/PrintReference +{ +//PDFR_DEBUG{ +({ )print +dup{ +=only( )print +}forall +( })= +}if +}bind def +/R +{ +0 ne{ +exch mark exch(A referred object generation )exch( isn't 0.)//error exec +}if +[ +exch//GetRegistered/exec load +]cvx +//PrintReference exec +}bind def +/IsObjRef +{ +dup type/arraytype eq{ +dup length 3 eq{ +dup xcheck exch +dup 0 get type/integertype eq 3 2 roll and exch +dup 1 get//GetRegistered eq 3 2 roll and exch +2 get/exec load eq and +}{ +pop false +}ifelse +}{ +pop false +}ifelse +}bind def +/DoNothing +{ +}def +/RunTypeDaemon +{ +dup type/dicttype eq{ +dup/Type//knownget exec{ +//PDFReader/TypeDaemons get exch +//knownget exec{ +exec +}if +}if +}if +}bind def +/obj +{ +//PDFR_DEBUG{ +(Defining )print 1 index =only( )print dup =only( obj)= +}if +0 ne{ +exch mark exch(An object generation )exch( isn't 0.)//error exec +}if +}bind def +/endobj +{ +//PDFR_DEBUG{ +(endobj )= +}if +count 1 eq{ +pop +}{ +dup type/dicttype eq{ +dup/.endobj_daemon//knownget exec{ +//PDFR_DEBUG{(.endobj_daemon for )print 2 index =}if +exec +}if +}if +dup type/dicttype eq{dup/ImmediateExec known}{false}ifelse{ +pop pop +}{ +//PDFR_DEBUG{ +(Storing )print 1 index = +}if +//RunTypeDaemon exec +//DoNothing 3 1 roll//Register exec +}ifelse +}ifelse +}bind def +/StoreBlock +{ +//PDFR_DEBUG{ +(StoreBlock )print//PDFReader/BlockCount get =only(, Length = )print dup length = +}if +dup length string copy +//PDFReader/BlockCount get exch +//PDFReader/CurrentObject get 3 1 roll +put +//PDFReader/BlockCount get 1 add +//PDFReader exch/BlockCount exch put +}bind def +/CheckLength +{dup type/integertype ne{ +mark(Object length isn't an integer.)//error exec +}if +}bind def +/ResolveD +{ +3 copy pop get +dup//IsObjRef exec{ +//PDFR_DEBUG{ +(Resolving )print//PrintReference exec +}if +exec +exch exec +}{ +exch pop +}ifelse +dup 4 1 roll +put +}bind def +/ResolveA +{2 index 2 index get +dup//IsObjRef exec{ +exec +exch exec +3 copy put +}{ +exch pop +}ifelse +exch pop exch pop +}bind def +/StoreStream +{ +dup//PDFReader exch/CurrentObject exch put +//PDFReader/BlockCount 0 put +dup/Length//CheckLength//ResolveD exec +//PDFR_DEBUG{ +(StoreStream Length = )print dup = +}if +currentfile exch()/SubFileDecode filter +{dup//BlockBuffer readstring{ +//StoreBlock exec +}{ +//StoreBlock exec +exit +}ifelse +}loop +pop +//PDFReader/CurrentObject null put +//PDFR_DEBUG{ +(StoreStream end.)= +}if +}bind def +/MakeStreamDumper +{ +//PDFR_DEBUG{ +(MakeStreamDumper beg.)= +}if +currentglobal exch dup gcheck setglobal +[exch +1 dict dup/c 0 put exch +1024 string +{readstring pop +(StreamDumper )print 1 index/c get =string cvs print( )print +dup length =string cvs print( <)print dup print(>\n)print +dup length +3 2 roll +dup/c get +3 2 roll +add/c exch put +}/exec load +] +cvx 0()/SubFileDecode filter +exch setglobal +//PDFR_DEBUG{ +(MakeStreamDumper end.)= +}if +}bind def +/ShortFilterNames 15 dict begin +/AHx/ASCIIHexDecode def +/A85/ASCII85Decode def +/LZW/LZWDecode def +/Fl/FlateDecode def +/RL/RunLengthDecode def +/CCF/CCITTFaxDecode def +/DCT/DCTDecode def +currentdict end readonly def +/AppendFilters +{ +//PDFR_DEBUG{ +(AppendFilters beg.)= +}if +dup 3 1 roll +/Filter//knownget exec{ +dup type/nametype eq{ +dup//ShortFilterNames exch//knownget exec{ +exch pop +}if +2 index/DecodeParms//knownget exec{ +exch +}if +filter +}{ +dup 0 exch 1 exch length 1 sub{ +2 copy get +dup//ShortFilterNames exch//knownget exec{ +exch pop +}if +3 1 roll +4 index/DecodeParms//knownget exec{ +exch get +}{ +pop null +}ifelse +dup null eq{ +pop 3 1 roll filter exch +}{ +3 1 roll +4 1 roll filter exch +}ifelse +}for +pop +}ifelse +//PDFR_DEBUG//PDFR_DUMP and{ +//MakeStreamDumper exec +}if +}if +exch pop +//PDFR_DEBUG{ +(AppendFilters end.)= +}if +}bind def +/ExecuteStream +{ +dup//PDFReader exch/CurrentObject exch put +dup/Length//CheckLength//ResolveD exec +//PDFR_DEBUG{ +(ExecuteStream id = )print 2 index =only( Length = )print dup = +}if +//PDFReader/InitialGraphicState get +//PDFReader/GraphicState get copy pop +//PDFReader/Operators get begin +currentfile exch()/SubFileDecode filter +1 index//AppendFilters exec +cvx mark exch +exec +counttomark 0 ne{ +mark(Data left on ostack after an immediate stream execution.)//error exec +}if +cleartomark +end +//PDFR_DEBUG{ +(ExecuteStream end.)= +}if +//PDFReader/CurrentObject null put +dup/IsPage known{ +dup/Context get/NumCopies//knownget exec{ +1 sub{ +copypage +}repeat +}if +EPS2Write not{showpage}if +pagesave restore +}if +}bind def +/stream +{ +//PDFR_DEBUG{ +1 index =only( stream)= +}if +1 index GetObject{ +dup xcheck{ +exec +1 index null PutObject +}{ +pop +}ifelse +}if +dup/ImmediateExec known{ +dup/GlobalExec//knownget exec{ +currentglobal 4 1 roll +setglobal +//ExecuteStream exec +3 2 roll setglobal +}{ +//ExecuteStream exec +}ifelse +}{ +//StoreStream exec +}ifelse +dup/.CleanResources//knownget exec{ +/All eq{ +//CleanAllResources exec +}if +}if +}bind def +/HookFont +{ +//PDFR_DEBUG{ +(Loaded the font )print dup/FontName get = +}if +{ +dup/FontFileType get dup/Type1 eq exch/MMType1 eq or{ +dup/FontName get +//PDFReader/RemoveFontNamePrefix get exec +findfont +exit +}if +dup/FontFileType get/TrueType eq{ +//PDFReader/MakeType42 get exec +//PDFR_DEBUG{ +(Font dict <<)= +dup{ +1 index/sfnts eq{ +exch pop +(/sfnts [)print +{ +(-string\()print length//=only exec(\)- )= +}forall +(])= +}{ +exch//=only exec( )print == +}ifelse +}forall +(>>)= +}if +dup/FontName get exch definefont +exit +}if +mark(FontHook has no proc for )2 index/FontFileType get//error exec +}loop +/Font exch put +}bind def +/endstream +{ +}bind def +/xref +{ +//PDFR_DEBUG{ +(xref)= +//PDFR_DUMP{ +//PDFReader/ObjectRegistry get == +}if +}if +end +count 0 ne{ +mark(Excessive data on estack at the end of the interpretation.)//error exec +}if +currentfile 1(%%EOF)/SubFileDecode filter +flushfile +cleardictstack +}bind def +/ResolveDict +{dup{ +pop 1 index exch +//DoNothing//ResolveD exec +pop +}forall +pop +}bind def +/SetupPageView +{ +//PDFR_DEBUG{ +(SetupPageView beg)= +}if +//DSC_OPDFREAD not{ +//GraphicState/InitialMatrix get setmatrix +}if +/MediaBox get aload pop +3 index neg 3 index neg translate +3 -1 roll sub 3 1 roll exch sub exch +userdict/.HWMargins//knownget exec{ +aload pop +}{ +currentpagedevice/.HWMargins//knownget exec{ +aload pop +}{ +0 0 0 0 +}ifelse +}ifelse +currentpagedevice/PageSize get aload pop +3 -1 roll sub 3 1 roll exch sub exch +exch 3 index sub exch 3 index sub +//SetPageSize{ +//PDFR_DEBUG{ +(Setting page size to )print 1 index//=only exec( )print dup = +}if +pop pop 3 index 3 index 2 copy +currentglobal false setglobal 3 1 roll +currentpagedevice dup/PageSize known{ +/PageSize get aload pop +}{ +0 0 +}ifelse +round cvi 2 index round cvi eq +exch round cvi 3 index round cvi eq and +{ +//PDFR_DEBUG{(PageSize matches request)== flush}if +pop pop +}{ +/MediaRequested where{ +//PDFR_DEBUG{(MediaRequested is true, check against new request)== flush}if +/MediaRequested get aload pop +round cvi 2 index round cvi eq +exch round cvi 3 index round cvi eq and +{ +//PDFR_DEBUG{(MediaRequested same as current request, ignore)== flush}if +pop pop false +}{ +//PDFR_DEBUG{(MediaRequested different to current request)== flush}if +true +}ifelse +}{ +//PDFR_DEBUG{(No MediaRequested yet)== flush}if +true +}ifelse +{ +//PDFR_DEBUG{(Setting pagesize)== flush}if +2 array astore +dup/MediaRequested exch def +<< exch/PageSize exch >>setpagedevice +}if +}ifelse +userdict/PDFR_InitialGS gstate put +setglobal +}if +//RotatePages{ +2 copy gt 6 index 6 index gt ne{ +1 index 5 index le 1 index 5 index le and not +}{ +false +}ifelse +}{ +false +}ifelse +{//CenterPages{ +//PDFR_DEBUG{ +(Rotating page, and then centering it)== +}if +90 rotate +0 5 index neg translate +5 index 1 index exch sub 2 div +2 index 6 index sub 2 div neg +translate +}{ +//FitPages{ +1 index 5 index div 1 index 7 index div +2 copy gt{ +exch +}if +pop dup scale +}if +90 rotate +0 5 index neg translate +}ifelse +}{ +//CenterPages{ +//PDFR_DEBUG{ +(Ccentering page)== +}if +1 index 6 index sub 2 div +1 index 6 index sub 2 div +translate +}{ +//FitPages{ +1 index 6 index div 1 index 6 index div +2 copy gt{ +exch +}if +pop dup scale +}if +}ifelse +}ifelse +pop pop +translate +pop pop +//PDFR_DEBUG{ +(SetupPageView end)= +}if +}bind def +/PageContentsDaemon +{ +//PDFR_DEBUG{ +(Executing PageContentsDaemon for )print 2 index = +}if +1 index exch/Context exch put +dup/ImmediateExec true put +/pagesave save def +dup/IsPage true put +SetPageSize{dup/Context get//SetupPageView exec}if +}bind def +/FontFileDaemon +{ +//PDFR_DEBUG{ +(Executing FontFileDaemon for )print 2 index = +}if +dup/FontFileType get +2 index exch +dup//ReadFontProcs exch//knownget exec{ +exch pop exec +}{ +mark(FontFile reader for )2 index( isn't implemented yet.)//error exec +}ifelse +//PDFR_DEBUG{ +(FontFileDaemon end)= +}if +pop +}bind def +/FontDescriptorDaemon +{ +//PDFR_DEBUG{ +(Executing FontDescriptorDaemon for )print 2 index = +}if +2 copy/FontResource exch put +/Subtype get 1 index exch/FontFileType exch put +}bind def +/UnPDFEscape{ +dup dup length string cvs +dup(#)search{ +{ +pop +(16#--)2 index 0 2 getinterval +1 index 3 2 getinterval copy pop +cvi +0 exch put +0 +1 index 2 1 index length 2 sub getinterval +3 copy putinterval +length +3 copy exch put +getinterval +(#)search not{ +pop exit +}if +}loop +(\0)search pop exch pop exch pop +cvn +exch pop +}{ +pop pop +}ifelse +}bind def +/TypeDaemons<< +/Page +{//PDFR_DEBUG{ +(Recognized a page.)= +}if +dup/Contents//knownget exec{ +0 get//DoNothing exch +[ +3 index//PageContentsDaemon/exec load +]cvx +//Register exec +}{ +(fixme: page with no Contents won't be printed.)= +}ifelse +}bind +/FontDescriptor +{//PDFR_DEBUG{ +(Recognized a font descriptor.)= +}if +dup/FontName//knownget exec{ +1 index/FontName 3 -1 roll//UnPDFEscape exec put +}if +dup dup/FontFile known{/FontFile}{/FontFile2}ifelse +//knownget exec{ +0 get//DoNothing exch +[ +3 index//FontFileDaemon/exec load +]cvx +//Register exec +}{ +(Font descriptor )print 1 index =only( has no FontFile.)= +}ifelse +}bind +/Font +{//PDFR_DEBUG{ +(Recognized a font resource.)= +}if +dup/BaseFont//knownget exec{ +//UnPDFEscape exec 2 copy/BaseFont exch put +//PDFReader/RemoveFontNamePrefix get exec +currentglobal exch +dup/Font resourcestatus{ +pop pop +//PDFReader/GetInstalledFont get exec pop +}{ +pop +}ifelse +setglobal +}if +dup/FontDescriptor//knownget exec{ +0 get +dup//IsRegistered exec{ +//PDFR_DEBUG{ +(already registered )print dup = +}if +pop +}{ +//DoNothing exch +[ +3 index//FontDescriptorDaemon/exec load +]cvx +//Register exec +}ifelse +}if +}bind +>>def +/MakeStreamReader +{dup +[ +exch +//PDFR_DEBUG{ +(Stream proc ) +/print load +//PDFR_STREAM{ +(<) +/print load +}if +}if +1 dict dup/i -1 put +/dup load +/i +/get load +1 +/add load +/dup load +3 +1 +/roll load +/i +/exch load +/put load +//knownget +/exec load +/not load +{()} +/if load +//PDFR_DEBUG{ +//PDFR_STREAM{ +/dup load +/print load +(>) +/print load +}if +( end of stream proc.\n) +/print load +}if +]cvx +//PDFR_DEBUG{ +(Stream reader )print dup == +}if +0()/SubFileDecode filter +exch//AppendFilters exec +}bind def +/RunDelayedStream +{ +//GraphicState/InitialTextMatrix get +//InitialTextMatrixStack//PDFReader/InitialTextMatrixStackPointer get +2 copy get null eq{ +2 copy currentglobal true setglobal matrix exch setglobal put +}if +get copy pop +//PDFReader/InitialTextMatrixStackPointer 2 copy get 1 add put +//MakeStreamReader exec +mark exch +cvx exec +counttomark 0 ne{ +mark(Data left on ostack after a delayed stream execution.)//error exec +}if +cleartomark +//PDFReader/InitialTextMatrixStackPointer 2 copy get 1 sub put +//InitialTextMatrixStack//PDFReader/InitialTextMatrixStackPointer get get +//GraphicState/InitialTextMatrix get +copy pop +}bind def +//ReadFontProcs begin +/Type1 +{//PDFR_DEBUG{ +(ReadFontProcs.Type1)= +}if +dup/.endobj_daemon[4 index//HookFont/exec load]cvx put +dup/ImmediateExec true put +/GlobalExec true put +}bind def +/MMType1//Type1 def +/TrueType +{//PDFR_DEBUG{ +(ReadFontProcs.TrueType)= +}if +dup/.endobj_daemon[4 index//HookFont/exec load]cvx put +pop +}bind def +end +/.opdloadttfontdict 50 dict def +.opdloadttfontdict begin +/maxstring 65400 def +end +/.InsertionSort +{ +/CompareProc exch def +/Array exch def +1 1 Array length 1 sub +{ +/Ix exch def +/Value1 Array Ix get def +/Jx Ix 1 sub def +{ +Jx 0 lt{ +exit +}if +/Value2 Array Jx get def +Value1 Value2 CompareProc{ +exit +}if +Array Jx 1 add Value2 put +/Jx Jx 1 sub def +}loop +Array Jx 1 add Value1 put +}for +Array +}bind def +/putu16{ +3 copy -8 bitshift put +exch 1 add exch 16#ff and put +}bind def +/putu32{ +3 copy -16 bitshift putu16 +exch 2 add exch 16#ffff and putu16 +}bind def +/.readtable{ +dup dup 1 and add string +dup 0 4 -1 roll getinterval +3 -1 roll exch +dup()ne{readstring}if pop pop +}bind def +/.readbigtable{ +dup maxstring lt{ +.readtable +}{ +currentuserparams/VMReclaim get -2 vmreclaim +[4 2 roll{ +dup maxstring le{exit}if +1 index maxstring string readstring pop 3 1 roll maxstring sub +}loop .readtable] +exch vmreclaim +}ifelse +}bind def +/ReadTTF +{ +.opdloadttfontdict begin +/TTFontFile exch def +/TableDir TTFontFile 12 string readstring pop def +/tables TTFontFile TableDir 4 getu16 16 mul string readstring pop def +/tabarray tables length 16 idiv array def +TableDir 0 4 getinterval(ttcf)eq{ +QUIET not{(Can't handle TrueType font Collections.)=}if +/.loadttfonttables cvx/invalidfont signalerror +}{ +0 16 tables length 1 sub{ +dup +tables exch 16 getinterval +exch 16 div cvi exch +tabarray 3 1 roll put +}for +}ifelse +tabarray{exch 8 getu32 exch 8 getu32 gt}.InsertionSort pop +/Read TableDir length tables length add def +/tabs[ +tabarray{ +dup 8 getu32 +Read sub +dup 0 gt{ +dup string TTFontFile exch readstring pop pop +Read add/Read exch def +}{ +pop +}ifelse +12 getu32 +dup Read add +/Read exch def +TTFontFile exch .readbigtable +}forall +]def +end +}bind def +/GetLocaType +{ +0 1 tabarray length 1 sub{ +dup tabarray exch get +0 4 getinterval(head)eq{ +tabs exch get +50 gets16 +/LocaType exch def +exit +}{ +pop +}ifelse +}for +}bind def +/GetNumGlyphs +{ +0 1 tabarray length 1 sub{ +dup tabarray exch get +0 4 getinterval(maxp)eq{ +tabs exch get +4 getu16 +/NumGlyphs exch def +exit +}{ +pop +}ifelse +}for +}bind def +/StringToLoca +{ +/LocaIndex exch def +/StringOffset 0 def +{ +dup length StringOffset gt{ +dup +LocaType 1 eq{ +StringOffset getu32 +LocaArray LocaIndex 3 -1 roll put +/LocaIndex LocaIndex 1 add def +/StringOffset StringOffset 4 add +def +}{ +StringOffset getu16 2 mul +LocaArray length LocaIndex gt{ +LocaArray LocaIndex 3 -1 roll put +}{ +pop +}ifelse +/LocaIndex LocaIndex 1 add def +/StringOffset StringOffset 2 add +def +}ifelse +}{ +pop +LocaIndex +exit +}ifelse +}loop +}bind def +/GetSortedLoca +{ +NumGlyphs 1 add array/LocaArray exch def +0 1 tabarray length 1 sub{ +dup tabarray exch get +0 4 getinterval(loca)eq{ +tabs exch get +exit +}{ +pop +}ifelse +}for +dup type/stringtype eq{ +0 StringToLoca pop +}{ +0 exch +{ +exch StringToLoca +}forall +pop +}ifelse +LocaArray{gt}.InsertionSort pop +}bind def +/GetWorkingString +{ +WorkString 0 +GlyfArray GlyfStringIndex get +putinterval +/WorkBytes GlyfArray GlyfStringIndex get length def +/GlyfStringIndex GlyfStringIndex 1 add def +}bind def +/GetWorkingBytes +{ +/BytesToRead exch def +WorkString 0 BytesToRead getinterval +dup length string copy +WorkString BytesToRead WorkBytes BytesToRead sub getinterval +dup length string copy +WorkString 0 3 -1 roll putinterval +/WorkBytes WorkBytes BytesToRead sub def +}bind def +/GetGlyfBytes +{ +/ToRead exch def +WorkBytes 0 eq{ +GetWorkingString +}if +WorkBytes ToRead ge{ +ToRead string dup 0 +ToRead GetWorkingBytes putinterval +}{ +ToRead string +dup +0 +WorkString 0 WorkBytes getinterval +putinterval +dup +WorkBytes +ToRead WorkBytes sub +GetWorkingString +GetWorkingBytes +putinterval +}ifelse +}bind def +/SplitGlyf +{ +/GlyfArray exch def +/DestArray GlyfArray length 2 mul array def +/DestArrayIndex 0 def +/LastLoca 0 def +/NextLocaIndex 0 def +/LastLocaIndex 0 def +/GlyfStringIndex 0 def +/WorkString maxstring string def +/WorkBytes 0 def +{ +LocaArray NextLocaIndex get +LastLoca sub maxstring gt +{ +LocaArray LastLocaIndex get LastLoca sub +GetGlyfBytes +DestArray DestArrayIndex 3 -1 roll put +/DestArrayIndex DestArrayIndex 1 add def +LocaArray LastLocaIndex get/LastLoca exch def +}{ +/LastLocaIndex NextLocaIndex def +/NextLocaIndex NextLocaIndex 1 add def +NextLocaIndex NumGlyphs gt +{ +WorkBytes +GlyfStringIndex GlyfArray length lt{ +GlyfArray GlyfStringIndex get length +add string dup +0 +WorkString 0 WorkBytes getinterval +putinterval +dup +WorkBytes +GetWorkingString +WorkString 0 WorkBytes getinterval +putinterval +}{ +pop +WorkString 0 WorkBytes getinterval +}ifelse +dup length string copy +DestArray DestArrayIndex 3 -1 roll put +exit +}if +}ifelse +}loop +DestArray +}bind def +/ProcessTTData +{ +.opdloadttfontdict begin +0 1 tabarray length 1 sub{ +/ix exch def +tabarray ix get +12 getu32 dup maxstring le{ +dup 4 mod 0 ne{ +4 div cvi 1 add 4 mul string/newstring exch def +/oldstring tabs ix get def +newstring 0 oldstring putinterval +0 1 newstring length oldstring length sub 1 sub{ +newstring exch oldstring length add 0 put +}for +tabs ix newstring put +}{ +pop +}ifelse +}{ +dup 4 mod 0 ne{ +dup maxstring idiv maxstring mul sub +4 idiv 1 add 4 mul string/newstring exch def +tabs ix get +dup length 1 sub dup/iy exch def get/oldstring exch def +newstring 0 oldstring putinterval +0 1 newstring length oldstring length sub 1 sub{ +newstring exch oldstring length add 0 put +}for +tabs ix get iy newstring put +}{ +pop +}ifelse +}ifelse +}for +0 1 tabarray length 1 sub{ +dup tabarray exch get +dup 12 getu32 maxstring gt{ +0 4 getinterval dup(glyf)eq{ +pop +GetLocaType +GetNumGlyphs +GetSortedLoca +dup tabs exch get +SplitGlyf +tabs 3 1 roll put +}{ +(Warning, table )print print( > 64Kb\n)print +pop +}ifelse +}{ +pop +pop +}ifelse +}for +end +}bind def +/Makesfnts +{ +.opdloadttfontdict begin +0 +tabs{ +dup type/stringtype eq{ +pop +1 add +}{ +{ +type/stringtype eq{ +1 add +}if +}forall +}ifelse +}forall +1 add +/TTOffset +TableDir length +tabarray length 16 mul add +def +0 +tabarray{ +exch dup 1 add +3 1 roll +dup +tabs exch get +dup type/stringtype eq{ +length +2 index exch +TTOffset +dup 3 1 roll add +/TTOffset exch def +8 exch putu32 +exch tabarray 3 1 roll +put +}{ +0 exch +{ +dup type/stringtype eq{ +length add +}{ +pop +}ifelse +}forall +2 index exch +TTOffset +dup 3 1 roll add +/TTOffset exch def +8 exch putu32 +exch tabarray 3 1 roll +put +}ifelse +}forall +pop +array +dup 0 +TableDir length +tables length add +string +dup 0 TableDir putinterval +dup 12 tables putinterval +put +dup +/ix 1 def +tabs{ +dup type/stringtype eq{ +ix exch +put dup +/ix ix 1 add def +}{ +{ +dup type/stringtype eq{ +ix exch put dup +/ix ix 1 add def +}{ +pop +}ifelse +}forall +}ifelse +}forall +pop +end +}bind def +/MakeType42 +{ +//PDFR_DEBUG{ +(MakeType42 beg)= +}if +10 dict begin +/FontName 1 index/FontName get def +/FontType 42 def +/FontMatrix[1 0 0 1 0 0]def +/FontBBox 1 index/FontBBox get def +dup/FontResource get +dup/Encoding known{ +//PDFReader/ObtainEncoding get exec +/Encoding get +}{ +pop null +}ifelse +/PDFEncoding exch def +/CharStrings 2 index//PDFReader/MakeTTCharStrings get exec def +/sfnts 2 index//MakeStreamReader exec +ReadTTF +ProcessTTData +Makesfnts +def +/Encoding StandardEncoding def +/PaintType 0 def +currentdict end +//PDFR_DEBUG{ +(MakeType42 end)= +}if +}bind def +/GetInstalledFont +{ +dup//InstalledFonts exch knownget{ +exch pop +}{ +dup findfont dup 3 1 roll +//InstalledFonts 3 1 roll put +}ifelse +}bind def +/RemoveFontNamePrefix +{//=string cvs true +0 1 5{ +2 index exch get//IsUpper exec not{ +pop false exit +}if +}for +{(+)search{ +pop pop +}if +}if +cvn +}bind def +/CheckFont +{dup/Type get/Font ne{ +mark(Resource )3 index( must have /Type/Font .)//error exec +}if +}bind def +/CheckEncoding +{dup type/nametype ne{ +dup/Type get/Encoding ne{ +mark(Resource )3 index( must have /Type/Encoding .)//error exec +}if +}if +}bind def +/ObtainEncoding +{dup/Encoding known{ +dup dup/Encoding//CheckEncoding//ResolveD exec +dup type dup/arraytype eq exch/packedarraytype eq or{ +pop pop +}{ +dup type/nametype eq{ +/Encoding findresource +}{ +dup/BaseEncoding//knownget exec not{ +/StandardEncoding +}if +/Encoding findresource +exch +/Differences//knownget exec{ +exch dup length array copy exch +0 exch +{ +dup type/integertype eq{ +exch pop +}{ +3 copy put pop +1 add +}ifelse +}forall +pop +}if +}ifelse +/Encoding exch put +}ifelse +}{ +dup/Encoding/StandardEncoding/Encoding findresource put +}ifelse +}bind def +/ObtainMetrics +{dup/Widths//knownget exec{ +1 index/Encoding get +256 dict +3 index/Subtype get/TrueType eq{ +1000 +}{ +1 +}ifelse +4 index/MissingWidth//knownget exec not{ +0 +}if +5 index/FirstChar//knownget exec not{ +0 +}if +6 5 roll +dup 0 exch 1 exch length 1 sub{ +2 copy get +exch 3 index add +7 index exch get +dup dup null ne exch/.notdef ne and{ +6 index 3 1 roll exch +6 index div +3 copy pop//knownget exec{ +0 eq +}{ +true +}ifelse +{put +}{ +pop pop pop +}ifelse +}{ +pop pop +}ifelse +}for +pop pop pop pop exch pop +1 index exch/Metrics exch put +}{ +dup/MissingWidth//knownget exec{ +256 dict +2 index/Encoding get{ +dup null ne{ +3 copy 3 2 roll put +}if +pop +}forall +exch pop +1 index exch/Metrics exch put +}if +}ifelse +}bind def +/NotDef +{ +FontMatrix aload pop pop pop exch pop exch pop +1 exch div exch +1 exch div exch +1 index 0 setcharwidth +0 setlinewidth +0 0 moveto +2 copy rlineto +1 index 0 rlineto +neg exch neg exch rlineto +closepath stroke +}bind def +/SaveResourcesToStack +{ +[ +//PDFReader/OldResources known{ +//PDFReader/OldResources get +}{ +null +}ifelse +//PDFReader/CurrentObject get/Context get/Resources get +] +//PDFReader/OldResources 3 -1 roll put +}bind def +/RestoreResourcesFromStack +{ +//PDFReader/OldResources get dup +0 get//PDFReader/OldResources 3 -1 roll put +1 get//PDFReader/CurrentObject get/Context get/Resources 3 -1 roll put +}bind def +/BuildChar +{//PDFR_DEBUG{ +(BuildChar )print dup//=only exec( )print +}if +exch begin +Encoding exch get +//PDFR_DEBUG{ +dup = +}if +dup null eq{ +pop//NotDef exec +} +{ +CharProcs exch//knownget exec +{ +currentfont/Font get/Resources//knownget exec{ +exec +SaveResourcesToStack +//PDFReader/CurrentObject get/Context get +/Resources 3 -1 roll put +//RunDelayedStream exec +RestoreResourcesFromStack +}{ +//RunDelayedStream exec +}ifelse +} +{ +//NotDef exec +}ifelse +}ifelse +end +}bind def +/printdict +{(<<)= +{exch = ==}forall +(>>)= +}bind def +/printfont +{ +dup{ +exch dup = +dup/Encoding eq{ +pop = +}{ +dup/FontInfo eq exch/Private eq or{ +//printdict exec +}{ +== +}ifelse +}ifelse +}forall +}bind def +/ScaleMetrics +{1 index{ +2 index div +3 index +3 1 roll put +}forall +pop +}bind def +/ResolveAndSetFontAux +{exch dup +//PDFReader/CurrentObject get/Context get/Resources get +/Font//DoNothing//ResolveD exec +exch//CheckFont//ResolveD exec +dup/Font//knownget exec{ +exch pop exch pop +}{ +{ +dup/Subtype get dup dup/Type1 eq exch/TrueType eq or exch/MMType1 eq or{ +exch pop +dup/BaseFont get +//RemoveFontNamePrefix exec +//PDFR_DEBUG{ +(Font )print dup = +}if +1 index/FontDescriptor known{ +//PDFR_DEBUG{ +(Font from a font descriptor.)= +}if +1 index +/FontDescriptor//DoNothing//ResolveD exec +/Font//knownget exec{ +exch pop +}{ +//PDFR_DEBUG{ +(Font descriptor has no Font resolved.)= +}if +//GetInstalledFont exec +}ifelse +}{ +//GetInstalledFont exec +}ifelse +exch +dup/Encoding known not{ +1 index/Encoding get 1 index exch/Encoding exch put +}if +//ObtainEncoding exec +//ObtainMetrics exec +exch +dup length dict copy +dup 2 index/Encoding get +/Encoding exch put +1 index/Metrics//knownget exec{ +2 index/Subtype get/TrueType ne{ +1 index/FontMatrix get 0 get +dup 0 eq{ +pop +1 index/FontMatrix get 1 get +dup 0 eq{pop 1}if +}if +0.001 div +//ScaleMetrics exec +}{ +1 index/sfnts known not{ +1 index/FontMatrix get 0 get +dup 0 eq{ +pop +1 index/FontMatrix get 1 get +dup 0 eq{pop 1}if +}if +//ScaleMetrics exec +}if +}ifelse +1 index exch/Metrics exch put +}if +1 index/BaseFont get +exch +dup/FID undef +dup/UniqueID undef +definefont +dup 3 1 roll +/Font exch put +exit +}if +dup/Subtype get/Type3 eq{ +//ObtainEncoding exec +2 copy exch/FontName exch put +dup/CharProcs get//ResolveDict exec +dup/FontType 3 put +dup/BuildChar//BuildChar put +dup dup/Font exch put +dup 3 1 roll +definefont +2 copy ne{ +2 copy/Font exch put +}if +exch pop +exit +}if +dup/Subtype get/Type0 eq{ +}if +dup/Subtype get/CIDFontType0 eq{ +}if +dup/Subtype get/CIDFontType2 eq{ +}if +mark(Unknown font type )2 index/Subtype get//error exec +}loop +}ifelse +exch scalefont setfont +}bind def +/ResolveAndSetFont +{ +//ResolveAndSetFontAux exec +}bind def +/.knownget +{2 copy known{ +get true +}{ +pop pop false +}ifelse +}bind def +/.min +{2 copy lt{ +exch +}if +pop +}bind def +/.max +{2 copy gt{ +exch +}if +pop +}bind def +/.dicttomark +{>> +}bind def +/getu16{ +2 copy get 8 bitshift 3 1 roll 1 add get add +}bind def +/gets16{ +getu16 16#8000 xor 16#8000 sub +}bind def +/getu32{ +2 copy getu16 16 bitshift 3 1 roll 2 add getu16 add +}bind def +/gets32{ +2 copy gets16 16 bitshift 3 1 roll 2 add getu16 add +}bind def +/cmapformats mark +0{ +6 256 getinterval{}forall 256 packedarray +}bind +2{ +/sHK_sz 2 def +/sH_sz 8 def +dup 2 getu16/cmapf2_tblen exch def +dup 4 getu16/cmapf2_lang exch def +dup 6 256 sHK_sz mul getinterval/sHKs exch def +0 +0 1 255{ +sHKs exch +2 mul getu16 +1 index +1 index +lt{exch}if pop +}for +/sH_len exch def +dup 6 256 sHK_sz mul add +cmapf2_tblen 1 index sub getinterval +/sH_gIA exch def +/cmapf2_glyph_array 65535 array def +/.cmapf2_putGID{ +/cmapf2_ch cmapf2_ch_hi 8 bitshift cmapf2_ch_lo add def +firstCode cmapf2_ch_lo le +cmapf2_ch_lo firstCode entryCount add lt +and{ +sH_offset idRangeOffset add +cmapf2_ch_lo firstCode sub 2 mul +add 6 add +sH_gIA exch getu16 +dup 0 gt{ +idDelta add +cmapf2_glyph_array exch cmapf2_ch exch put +}{ +pop +}ifelse +}{ +}ifelse +}def +16#00 1 16#ff{ +/cmapf2_ch_hi exch def +sHKs cmapf2_ch_hi sHK_sz mul getu16 +/sH_offset exch def +sH_gIA sH_offset sH_sz getinterval +dup 0 getu16/firstCode exch def +dup 2 getu16/entryCount exch def +dup 4 gets16/idDelta exch def +dup 6 getu16/idRangeOffset exch def +pop +sH_offset 0 eq{ +/cmapf2_ch_lo cmapf2_ch_hi def +/cmapf2_ch_hi 0 def +.cmapf2_putGID +}{ +16#00 1 16#ff{ +/cmapf2_ch_lo exch def +.cmapf2_putGID +}for +}ifelse +}for +pop +0 1 cmapf2_glyph_array length 1 sub{ +dup cmapf2_glyph_array exch get +null eq{cmapf2_glyph_array exch 0 put}{pop}ifelse +}for +cmapf2_glyph_array +}bind +4{ +/etab exch def +/nseg2 etab 6 getu16 def +14/endc etab 2 index nseg2 getinterval def +2 add +nseg2 add/startc etab 2 index nseg2 getinterval def +nseg2 add/iddelta etab 2 index nseg2 getinterval def +nseg2 add/idroff etab 2 index nseg2 getinterval def +pop +/firstcode startc 0 getu16 16#ff00 and dup 16#f000 ne{pop 0}if def +/lastcode firstcode def +/striptopbyte false def +/putglyph{ +glyphs code 3 -1 roll put/code code 1 add def +}bind def +/numcodes 0 def/glyphs 0 0 2 nseg2 3 sub{ +/i2 exch def +/scode startc i2 getu16 def +/ecode endc i2 getu16 def +ecode lastcode gt{ +/lastcode ecode def +}if +}for pop +firstcode 16#f000 ge lastcode firstcode sub 255 le and{ +lastcode 255 and +/striptopbyte true def +}{ +lastcode +}ifelse +1 add +array def +glyphs length 1024 ge{ +.array1024z 0 1024 glyphs length 1023 sub{glyphs exch 2 index putinterval}for +glyphs dup length 1024 sub 3 -1 roll +putinterval +}{ +0 1 glyphs length 1 sub{glyphs exch 0 put}for +}ifelse +/numcodes 0 def/code 0 def +0 2 nseg2 3 sub{ +/i2 exch def +/scode startc i2 getu16 def +/ecode endc i2 getu16 def +numcodes scode firstcode sub +exch sub 0 .max dup/code exch code exch add def +ecode scode sub 1 add add numcodes add/numcodes exch def +/delta iddelta i2 gets16 def +TTFDEBUG{ +(scode=)print scode =only +( ecode=)print ecode =only +( delta=)print delta =only +( droff=)print idroff i2 getu16 = +}if +idroff i2 getu16 dup 0 eq{ +pop scode delta add 65535 and 1 ecode delta add 65535 and +striptopbyte{ +/code scode 255 and def +}{ +/code scode def +}ifelse +{putglyph}for +}{ +/gloff exch 14 nseg2 3 mul add 2 add i2 add add def +striptopbyte{ +/code scode 255 and def +}{ +/code scode def +}ifelse +0 1 ecode scode sub{ +2 mul gloff add etab exch getu16 +dup 0 ne{delta add 65535 and}if putglyph +}for +}ifelse +}for glyphs/glyphs null def +}bind +6{ +dup 6 getu16/firstcode exch def dup 8 getu16/ng exch def +firstcode ng add array +0 1 firstcode 1 sub{2 copy 0 put pop}for +dup firstcode ng getinterval +0 1 ng 1 sub{ +dup 2 mul 10 add 4 index exch getu16 3 copy put pop pop +}for pop exch pop +}bind +.dicttomark readonly def +/cmaparray{ +dup 0 getu16 cmapformats exch .knownget{ +TTFDEBUG{ +(cmap: format )print 1 index 0 getu16 = flush +}if exec +}{ +(Can't handle format )print 0 getu16 = flush +0 1 255{}for 256 packedarray +}ifelse +TTFDEBUG{ +(cmap: length=)print dup length = dup == +}if +}bind def +/postremap mark +/Cdot/Cdotaccent +/Edot/Edotaccent +/Eoverdot/Edotaccent +/Gdot/Gdotaccent +/Ldot/Ldotaccent +/Zdot/Zdotaccent +/cdot/cdotaccent +/edot/edotaccent +/eoverdot/edotaccent +/gdot/gdotaccent +/ldot/ldotaccent +/zdot/zdotaccent +.dicttomark readonly def +/get_from_stringarray +{1 index type/stringtype eq{ +get +}{ +exch{ +2 copy length ge{ +length sub +}{ +exch get exit +}ifelse +}forall +}ifelse +}bind def +/getinterval_from_stringarray +{ +2 index type/stringtype eq{ +getinterval +}{ +string exch 0 +4 3 roll{ +dup length +dup 4 index lt{ +3 index exch sub +exch pop 3 1 roll exch pop +}{ +dup 3 1 roll +4 index sub +5 index length 4 index sub +2 copy gt{exch}if pop +dup 3 1 roll +5 index exch getinterval +5 index 4 index 3 index +getinterval +copy pop +exch pop add exch pop 0 exch +dup 3 index length ge{exit}if +}ifelse +}forall +pop pop +}ifelse +}bind def +/string_array_size +{dup type/stringtype eq{ +length +}{ +0 exch{length add}forall +}ifelse +}bind def +/postformats mark +16#00010000{ +pop MacGlyphEncoding +} +16#00020000{ +dup dup type/arraytype eq{0 get}if length 36 lt{ +TTFDEBUG{(post format 2.0 invalid.)= flush}if +pop[] +}{ +/postglyphs exch def +/post_first postglyphs dup type/arraytype eq{0 get}if def +post_first 32 getu16/numglyphs exch def +/glyphnames numglyphs 2 mul 34 add def +/postpos glyphnames def +/total_length postglyphs//string_array_size exec def +numglyphs array 0 1 numglyphs 1 sub{ +postpos total_length ge{ +1 numglyphs 1 sub{1 index exch/.notdef put}for +exit +}if +postglyphs postpos//get_from_stringarray exec +postglyphs postpos 1 add 2 index//getinterval_from_stringarray exec cvn +exch postpos add 1 add/postpos exch def +2 index 3 1 roll +put +}for +/postnames exch def +numglyphs array 0 1 numglyphs 1 sub{ +dup 2 mul 34 add postglyphs exch 2//getinterval_from_stringarray exec +dup 0 get 8 bitshift exch 1 get add dup 258 lt{ +MacGlyphEncoding exch get +}{ +dup 32768 ge{ +pop/.notdef +}{ +258 sub dup postnames length ge{ +TTFDEBUG{( *** warning: glyph index past end of 'post' table)= flush}if +pop +exit +}if +postnames exch get +postremap 1 index .knownget{exch pop}if +}ifelse +}ifelse +2 index 3 1 roll put +}for +} +ifelse +}bind +16#00030000{ +pop[] +}bind +.dicttomark readonly def +/first_post_string +{ +post dup type/arraytype eq{0 get}if +}bind def +/.getpost{ +/glyphencoding post null eq{ +TTFDEBUG{(post missing)= flush}if[] +}{ +postformats first_post_string 0 getu32 .knownget{ +TTFDEBUG{ +(post: format )print +first_post_string +dup 0 getu16 =only(,)print 2 getu16 = flush +}if +post exch exec +}{ +TTFDEBUG{(post: unknown format )print post 0 getu32 = flush}if[] +}ifelse +}ifelse def +}bind def +/MacRomanEncoding[ +StandardEncoding 0 39 getinterval aload pop +/quotesingle +StandardEncoding 40 56 getinterval aload pop +/grave +StandardEncoding 97 31 getinterval aload pop +/Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute +/agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave +/ecircumflex/edieresis/iacute/igrave +/icircumflex/idieresis/ntilde/oacute +/ograve/ocircumflex/odieresis/otilde +/uacute/ugrave/ucircumflex/udieresis +/dagger/degree/cent/sterling/section/bullet/paragraph/germandbls +/registered/copyright/trademark/acute/dieresis/.notdef/AE/Oslash +/.notdef/plusminus/.notdef/.notdef/yen/mu/.notdef/.notdef +/.notdef/.notdef/.notdef/ordfeminine/ordmasculine/.notdef/ae/oslash +/questiondown/exclamdown/logicalnot/.notdef +/florin/.notdef/.notdef/guillemotleft +/guillemotright/ellipsis/space/Agrave/Atilde/Otilde/OE/oe +/endash/emdash/quotedblleft/quotedblright +/quoteleft/quoteright/divide/.notdef +/ydieresis/Ydieresis/fraction/currency +/guilsinglleft/guilsinglright/fi/fl +/daggerdbl/periodcentered/quotesinglbase/quotedblbase +/perthousand/Acircumflex/Ecircumflex/Aacute +/Edieresis/Egrave/Iacute/Icircumflex +/Idieresis/Igrave/Oacute/Ocircumflex +/.notdef/Ograve/Uacute/Ucircumflex +/Ugrave/dotlessi/circumflex/tilde +/macron/breve/dotaccent/ring/cedilla/hungarumlaut/ogonek/caron +]/Encoding defineresource pop +/TTParser<< +/Pos 0 +/post null +>>def +/readu8 +{read not{ +mark(Insufficient data in the stream.)//error exec +}if +}bind def +/readu16 +{dup//readu8 exec 8 bitshift exch//readu8 exec or +}bind def +/reads16 +{//readu16 exec 16#8000 xor 16#8000 sub +}bind def +/readu32 +{dup//readu16 exec 16 bitshift exch//readu16 exec or +}bind def +/reads32 +{dup//reads16 exec 16 bitshift exch//readu16 exec or +}bind def +/SkipToPosition +{dup//TTParser/Pos get +exch//TTParser exch/Pos exch put +sub +//PDFR_DEBUG{ +(Skipping )print dup//=only exec( bytes.)= +}if +dup 0 eq{ +pop pop +}{ +dup 3 1 roll +()/SubFileDecode filter +exch +{1 index//BlockBuffer readstring pop length +dup 0 eq{pop exch pop exit}if +sub +}loop +0 ne{ +mark(Insufficient data in the stream for SkipToPosition.)//error exec +}if +}ifelse +}bind def +/TagBuffer 4 string def +/ParseTTTableDirectory +{//PDFR_DEBUG{ +(ParseTTTableDirectory beg)= +}if +15 dict begin +dup//readu32 exec 16#00010000 ne{ +mark(Unknown True Type version.)//error exec +}if +dup//readu16 exec/NumTables exch def +dup//readu16 exec/SearchRange exch def +dup//readu16 exec/EntrySelector exch def +dup//readu16 exec/RangeShift exch def +//PDFR_DEBUG{ +(NumTables = )print NumTables = +}if +NumTables{ +dup//TagBuffer readstring not{ +mark(Could not read TT tag.)//error exec +}if +cvn +[2 index//readu32 exec pop +2 index//readu32 exec +3 index//readu32 exec +] +//PDFR_DEBUG{ +2 copy exch//=only exec( )print == +}if +def +}repeat +pop +//TTParser/Pos 12 NumTables 16 mul add put +currentdict end +//PDFR_DEBUG{ +(ParseTTTableDirectory end)= +}if +}bind def +/ParseTTcmap +{//PDFR_DEBUG{ +(ParseTTcmap beg)= +}if +/cmap get aload pop +3 1 roll +7 dict begin +//PDFR_DEBUG{ +(Current position = )print//TTParser/Pos get = +(cmap position = )print dup = +}if +1 index exch//SkipToPosition exec +//TTParser/Pos get/TablePos exch def +dup//readu16 exec pop +dup//readu16 exec/NumEncodings exch def +//PDFR_DEBUG{ +(NumEncodings = )print NumEncodings = +}if +null +NumEncodings{ +1 index//readu32 exec +2 index//readu32 exec +3 array dup 3 2 roll 0 exch put +2 index null ne{ +dup 0 get 3 index 0 get sub +3 index exch 1 exch put +}if +dup 4 3 roll pop 3 1 roll +def +}repeat +dup 0 get +4 3 roll exch sub +1 exch put +//PDFR_DEBUG{ +currentdict{ +exch dup type/integertype eq{ +//PrintHex exec( )print == +}{ +pop pop +}ifelse +}forall +}if +4 NumEncodings 8 mul add/HeaderLength exch def +//TTParser/Pos//TTParser/Pos get HeaderLength add put +0 +NumEncodings{ +16#7FFFFFF null +currentdict{ +1 index type/integertype eq{ +exch pop dup 0 get +dup 5 index gt{ +dup 4 index lt{ +4 1 roll +exch pop exch pop +}{ +pop pop +}ifelse +}{ +pop pop +}ifelse +}{ +pop pop +}ifelse +}forall +//PDFR_DEBUG{ +(Obtaining subtable for )print dup == +}if +3 2 roll pop +3 copy pop +TablePos add//SkipToPosition exec +3 copy exch pop 1 get +//TTParser/Pos//TTParser/Pos get 3 index add put +string +readstring not{ +mark(Can't read a cmap subtable.)//error exec +}if +2 exch put +}repeat +pop pop +currentdict end +//PDFR_DEBUG{ +(ParseTTcmap end)= +}if +}bind def +/GetTTEncoding +{//PDFR_DEBUG{ +(GetTTEncoding beg)= +}if +get +exch pop +2 get +10 dict begin +/TTFDEBUG//PDFR_DEBUG def +//cmaparray exec +end +//PDFR_DEBUG{ +(GetTTEncoding end)= +dup == +}if +}bind def +/InverseEncoding +{ +256 dict begin +dup length 1 sub -1 0{ +2 copy get +exch +1 index currentdict exch//knownget exec{ +dup type/arraytype eq{ +aload length 1 add array astore +}{ +2 array astore +}ifelse +}if +def +}for +pop +currentdict end +}bind def +/GetMacRomanEncodingInverse +{//PDFReader/MacRomanEncodingInverse get +dup null eq{ +pop +MacRomanEncoding//InverseEncoding exec +dup//PDFReader exch/MacRomanEncodingInverse exch put +}if +}bind def +/PutCharStringSingle +{ +dup 3 index length lt{ +2 index exch get +dup 0 ne{ +def +}{ +pop pop +}ifelse +}{ +pop pop +}ifelse +}bind def +/PutCharString +{1 index type/nametype ne{ +mark(Bad charstring name)//error exec +}if +dup type/arraytype eq{ +{ +3 copy//PutCharStringSingle exec +pop pop +}forall +pop +}{ +//PutCharStringSingle exec +}ifelse +}bind def +/ComposeCharStrings +{ +//PDFR_DEBUG{ +(ComposeCharStrings beg)= +}if +1 index length 1 add dict begin +/.notdef 0 def +exch +//TTParser/post get +dup null ne{ +exch +1 index length 1 sub -1 0{ +dup 3 index exch get exch +dup 0 eq 2 index/.notdef eq or{ +pop pop +}{ +def +}ifelse +}for +}if +exch pop exch +{ +//PutCharString exec +}forall +pop +currentdict end +//PDFR_DEBUG{ +(ComposeCharStrings end)= +}if +}bind def +/ParseTTpost +{ +//PDFR_DEBUG{ +(ParseTTpost beg)= +}if +/post get aload pop +3 1 roll +//PDFR_DEBUG{ +(Current position = )print//TTParser/Pos get = +(post position = )print dup = +}if +1 index exch//SkipToPosition exec +//TTParser/Pos//TTParser/Pos get 4 index add put +exch dup 65535 le{ +string +readstring not{ +mark(Insufficient data in the stream for ParseTTpost.)//error exec +}if +}{ +[3 1 roll +dup 16384 div floor cvi +exch 1 index 16384 mul +sub exch +1 sub 0 1 3 -1 roll +{ +1 add index +16384 string readstring not{ +mark(Insufficient data in the stream for ParseTTpost.)//error exec +}if +}for +counttomark -2 roll +string readstring not{ +mark(Insufficient data in the stream for ParseTTpost.)//error exec +}if +] +}ifelse +1 dict begin +/post exch def +//.getpost exec +//TTParser/post glyphencoding put +//PDFR_DEBUG{ +(ParseTTpost end)= +glyphencoding == +}if +end +}bind def +/MakeTTCharStrings +{//MakeStreamReader exec +dup dup//ParseTTTableDirectory exec +//TTParser/post null put +dup/post//knownget exec{ +0 get +1 index/cmap get 0 get +lt{ +2 copy//ParseTTpost exec +//ParseTTcmap exec +}{ +2 copy//ParseTTcmap exec +3 1 roll +//ParseTTpost exec +}ifelse +}{ +//ParseTTcmap exec +}ifelse +{ +dup 16#00030001 known{ +//PDFR_DEBUG{ +(Using the TT cmap encoding for Windows Unicode.)= +}if +16#00030001//GetTTEncoding exec +AdobeGlyphList//ComposeCharStrings exec +exit +}if +dup 16#00010000 known{ +//PDFR_DEBUG{ +(Using the TT cmap encoding for Macintosh Roman.)= +}if +16#00010000//GetTTEncoding exec +PDFEncoding dup null eq{ +pop//GetMacRomanEncodingInverse exec +}{ +//InverseEncoding exec +}ifelse +//ComposeCharStrings exec +exit +}if +dup 16#00030000 known{ +//PDFR_DEBUG{ +(Using the TT cmap encoding 3.0 - not sure why Ghostscript writes it since old versions.)= +}if +16#00030000//GetTTEncoding exec +PDFEncoding dup null eq{ +pop//GetMacRomanEncodingInverse exec +}{ +//InverseEncoding exec +}ifelse +//ComposeCharStrings exec +exit +}if +mark(True Type cmap has no useful encodings.)//error exec +}loop +//PDFR_DEBUG{ +(CharStrings <<)= +dup{ +exch +dup type/nametype eq{ +//=only exec +}{ +== +}ifelse +( )print == +}forall +(>>)= +}if +}bind def +/ScaleVal +{ +aload pop +1 index sub +3 2 roll mul add +}bind def +/ScaleArg +{ +aload pop +1 index sub +3 1 roll +sub exch div +}bind def +/ScaleArgN +{ +dup length 2 sub -2 0{ +2 +2 index 3 1 roll getinterval +3 2 roll +exch//ScaleArg exec +1 index length 2 idiv 1 add 1 roll +}for +pop +}bind def +/ComputeFunction_10 +{ +//PDFR_DEBUG{ +(ComputeFunction_10 beg )print 1 index//=only exec( stack=)print count = +}if +exch +dup 1 eq{ +pop dup length 1 sub get +}{ +1 index length 1 sub mul +dup dup floor sub +dup 0 eq{ +pop cvi get +}{ +3 1 roll floor cvi +2 getinterval +aload pop +2 index mul 3 2 roll 1 exch sub 3 2 roll mul add +}ifelse +}ifelse +//PDFR_DEBUG{ +(ComputeFunction_10 end )print dup//=only exec( stack=)print count = +}if +}bind def +/ComputeFunction_n0 +{ +//PDFR_DEBUG{ +(ComputeFunction_n0 beg N=)print dup//=only exec( stack=)print count = +}if +dup 0 eq{ +pop +}{ +dup 2 add -1 roll +dup 3 index length 1 sub ge{ +pop 1 sub +exch dup length 1 sub get exch +//PDFReader/ComputeFunction_n0 get exec +}{ +dup floor cvi dup +4 index exch get +3 index dup +5 add copy +6 2 roll +pop pop pop pop +1 sub +//PDFReader/ComputeFunction_n0 get exec +3 2 roll pop +exch +4 3 roll exch +4 add 2 roll 1 add +3 2 roll exch get +exch 1 sub +//PDFReader/ComputeFunction_n0 get exec +1 index mul +3 1 roll +1 exch sub mul add +}ifelse +}ifelse +//PDFR_DEBUG{ +(ComputeFunction_n0 end )print dup//=only exec( stack=)print count = +}if +}bind def +/FunctionToProc_x01 +{ +dup/Domain get exch +dup/Data get 0 get exch +/Size get length +[4 1 roll +//PDFR_DEBUG{ +{(function beg, stack =)print count//=only exec(\n)print}/exec load +5 2 roll +}if +dup 1 gt{ +{mark exch +3 add 2 roll +//ScaleArgN exec +counttomark dup +3 add -2 roll +pop exch +//ComputeFunction_n0 exec +}/exec load +}{ +pop +3 1/roll load//ScaleArg/exec load +/exch load +//ComputeFunction_10/exec load +}ifelse +//PDFR_DEBUG{ +(function end, stack =)/print load/count load//=only/exec load(\n)/print load +}if +]cvx +//PDFR_DEBUG{ +(Made a procedure for the 1-result function :)= +dup == +}if +}bind def +/FunctionProcDebugBeg +{(FunctionProcDebugBeg )print count = +}bind def +/FunctionProcDebugEnd +{(FunctionProcDebugEnd )print count = +}bind def +/FunctionToProc_x0n +{ +PDFR_DEBUG{ +(FunctionToProc_x0n beg m=)print dup = +}if +1 index/Size get length exch +dup 7 mul 2 add array +PDFR_DEBUG{ +dup 0//FunctionProcDebugBeg put +}{ +dup 0//DoNothing put +}ifelse +dup 1/exec load put +dup 2 5 index/Domain get put +2 index 1 eq{ +dup 3//ScaleArg put +}{ +dup 3//ScaleArgN put +}ifelse +dup 4/exec load put +1 index 1 sub 0 exch 1 exch{ +dup 7 mul 5 add +1 index 4 index 1 sub ne{ +dup 3 index exch 6 index put 1 add +dup 3 index exch/copy load put 1 add +}if +[ +6 index/Data get 3 index get +6 index 1 eq{ +//ComputeFunction_10/exec load +}{ +6 index +//ComputeFunction_n0/exec load +}ifelse +]cvx +3 index exch 2 index exch put 1 add +2 index 1 index/exec load put 1 add +1 index 4 index 1 sub ne{ +2 index 1 index 6 index 1 add put 1 add +2 index 1 index 1 put 1 add +2 index 1 index/roll load put +}if +pop pop +}for +PDFR_DEBUG{ +dup dup length 2 sub//FunctionProcDebugEnd put +}{ +dup dup length 2 sub//DoNothing put +}ifelse +dup dup length 1 sub/exec load put +cvx exch pop exch pop exch pop +//PDFR_DEBUG{ +(Made a procedure for the n-argument function :)= +dup == +}if +PDFR_DEBUG{ +(FunctionToProc_x0n end)= +}if +}bind def +/MakeTableRec +{ +0 +exec +}bind def +/MakeTable +{//PDFR_DEBUG{ +(MakeTable beg )print count = +}if +1 index/Size get exch +1 sub dup +3 1 roll +get +array +1 index 0 eq{ +exch pop exch pop +}{ +dup length 1 sub -1 0{ +3 index 3 index//MakeTableRec exec +2 index 3 1 roll put +}for +exch pop exch pop +}ifelse +//PDFR_DEBUG{ +(MakeTable end )print count = +}if +}bind def +//MakeTableRec 0//MakeTable put +/StoreSample +{ +1 sub +dup 0 eq{ +pop +}{ +-1 1{ +I exch get get +}for +}ifelse +I 0 get 3 2 roll put +}bind def +/ReadSample32 +{ +4{ +File read not{ +mark(Insufficient data for function.)//error exec +}if +}repeat +pop +3 1 roll exch +256 mul add 256 mul add +//1_24_bitshift_1_sub div +}bind def +/ReadSample +{ +Buffer BitsLeft BitsPerSample +{2 copy ge{ +exit +}if +3 1 roll +8 add 3 1 roll +256 mul File read not{ +mark(Insufficient data for function.)//error exec +}if +add +3 1 roll +}loop +sub dup +2 index exch +neg bitshift +2 copy exch bitshift +4 3 roll exch sub +/Buffer exch def +exch/BitsLeft exch def +Div div +}bind def +/ReadSamplesRec +{0 +exec +}bind def +/ReadSamples +{ +//PDFR_DEBUG{ +(ReadSamples beg )print count = +}if +dup 1 eq{ +pop +0 1 Size 0 get 1 sub{ +I exch 0 exch put +0 1 M 1 sub{ +dup Range exch 2 mul 2 getinterval +//PDFR_DEBUG{ +(Will read a sample ... )print +}if +BitsPerSample 32 eq{//ReadSample32}{//ReadSample}ifelse +exec exch//ScaleVal exec +//PDFR_DEBUG{ +(value=)print dup = +}if +exch Table exch get +Size length//StoreSample exec +}for +}for +}{ +1 sub +dup Size exch get 0 exch 1 exch 1 sub{ +I exch 2 index exch put +dup//ReadSamplesRec exec +}for +pop +}ifelse +//PDFR_DEBUG{ +(ReadSamples end )print count = +}if +}bind def +//ReadSamplesRec 0//ReadSamples put +/StreamToArray +{//PDFR_DEBUG{ +(StreamToArray beg )print count = +}if +userdict/FuncDataReader get begin +dup/BitsPerSample get/BitsPerSample exch def +dup/Size get length/N exch def +dup/Range get length 2 idiv/M exch def +1 BitsPerSample bitshift 1 sub/Div exch def +/BitsLeft 0 def +/Buffer 0 def +dup/Size get/Size exch def +dup/Range get/Range exch def +/File 1 index//MakeStreamReader exec def +/I[N{0}repeat]def +M array +dup length 1 sub -1 0{ +2 index N//MakeTable exec +2 index 3 1 roll put +}for +/Table exch def +N//ReadSamples exec +PDFR_DEBUG{ +(Table = )print Table == +}if +/Data Table put +end +//PDFR_DEBUG{ +(StreamToArray end )print count = +}if +}bind def +/FunctionToProc10 +{ +PDFR_DEBUG{ +(FunctionToProc10 beg, Range = )print dup/Range get == +}if +dup/Order//knownget exec{ +1 ne{ +(Underimplemented function Type 0 Order 3.)= +}if +}if +dup//StreamToArray exec +dup/Range get length dup 2 eq{ +pop//FunctionToProc_x01 exec +}{ +2 idiv//FunctionToProc_x0n exec +}ifelse +PDFR_DEBUG{ +(FunctionToProc10 end)= +}if +}bind def +/FunctionToProc12 +{begin +currentdict/C0//knownget exec{length 1 eq}{true}ifelse{ +N +currentdict/C0//knownget exec{ +0 get +}{ +0 +}ifelse +currentdict/C1//knownget exec{ +0 get +}{ +1 +}ifelse +1 index sub +[4 1 roll +{ +4 2 roll +exp mul add +}aload pop +]cvx +}{ +[ +0 1 C0 length 1 sub{ +N +C0 2 index get +C1 3 index get +4 3 roll pop +1 index sub +[/dup load +5 2 roll +{ +4 2 roll +exp mul add +exch +}aload pop +]cvx +/exec load +}for +/pop load +]cvx +}ifelse +end +//PDFR_DEBUG{ +(FunctionType2Proc : )print dup == +}if +}bind def +/FunctionToProc14 +{//MakeStreamReader exec cvx exec +//PDFR_DEBUG{ +(FunctionType4Proc : )print dup == +}if +}bind def +/FunctionToProc1 +{ +dup/FunctionType get +{dup 0 eq{ +pop//FunctionToProc10 exec exit +}if +dup 2 eq{ +pop//FunctionToProc12 exec exit +}if +dup 4 eq{ +pop//FunctionToProc14 exec exit +}if +mark exch(Function type )exch( isn't implemented yet.)//error exec +}loop +}bind def +/FunctionToProc20 +{ +PDFR_DEBUG{ +(FunctionToProc20, Range = )print dup/Range get == +}if +dup/Order//knownget exec{ +1 ne{ +(Underimplemented function Type 0 Order 3.)= +}if +}if +dup//StreamToArray exec +dup/Range get length dup 2 eq{ +pop//FunctionToProc_x01 exec +}{ +2 idiv//FunctionToProc_x0n exec +}ifelse +}bind def +/FunctionToProc +{//PDFR_DEBUG{ +(FunctionToProc beg )print count = +}if +dup type/dicttype eq{ +dup/Domain get length 2 idiv +{ +dup 1 eq{ +pop//FunctionToProc1 exec exit +}if +dup 2 eq{ +pop//FunctionToProc20 exec exit +}if +mark(Functions with many arguments aren't implemented yet.)//error exec +}loop +}{ +//PDFR_DEBUG{(Not a function dict, assume already a procedure.)print}if +}ifelse +//PDFR_DEBUG{ +(FunctionToProc end )print count = +}if +}bind def +/spotfunctions mark +/Round{ +abs exch abs 2 copy add 1 le{ +dup mul exch dup mul add 1 exch sub +}{ +1 sub dup mul exch 1 sub dup mul add 1 sub +}ifelse +} +/Diamond{ +abs exch abs 2 copy add .75 le{ +dup mul exch dup mul add 1 exch sub +}{ +2 copy add 1.23 le{ +.85 mul add 1 exch sub +}{ +1 sub dup mul exch 1 sub dup mul add 1 sub +}ifelse +}ifelse +} +/Ellipse{ +abs exch abs 2 copy 3 mul exch 4 mul add 3 sub dup 0 lt{ +pop dup mul exch .75 div dup mul add 4 div 1 exch sub +}{ +dup 1 gt{ +pop 1 exch sub dup mul exch 1 exch sub +.75 div dup mul add 4 div 1 sub +}{ +.5 exch sub exch pop exch pop +}ifelse +}ifelse +} +/EllipseA{dup mul .9 mul exch dup mul add 1 exch sub} +/InvertedEllipseA{dup mul .9 mul exch dup mul add 1 sub} +/EllipseB{dup 5 mul 8 div mul exch dup mul exch add sqrt 1 exch sub} +/EllipseC{dup mul .9 mul exch dup mul add 1 exch sub} +/InvertedEllipseC{dup mul .9 mul exch dup mul add 1 sub} +/Line{exch pop abs neg} +/LineX{pop} +/LineY{exch pop} +/Square{abs exch abs 2 copy lt{exch}if pop neg} +/Cross{abs exch abs 2 copy gt{exch}if pop neg} +/Rhomboid{abs exch abs 0.9 mul add 2 div} +/DoubleDot{2{360 mul sin 2 div exch}repeat add} +/InvertedDoubleDot{2{360 mul sin 2 div exch}repeat add neg} +/SimpleDot{dup mul exch dup mul add 1 exch sub} +/InvertedSimpleDot{dup mul exch dup mul add 1 sub} +/CosineDot{180 mul cos exch 180 mul cos add 2 div} +/Double{exch 2 div exch 2{360 mul sin 2 div exch}repeat add} +/InvertedDouble{ +exch 2 div exch 2{360 mul sin 2 div exch}repeat add neg +} +.dicttomark readonly def +/CheckColorSpace +{ +dup type/arraytype ne{ +mark(Resource )3 index( must be an array.)//error exec +}if +}bind def +/SubstitutePDFColorSpaceRec +{0 +exec +}bind def +/SubstitutePDFColorSpace +{ +{ +dup 0 get/Pattern eq{ +dup length 1 gt{ +dup dup 1//CheckColorSpace//ResolveA exec +dup type/nametype ne{ +//SubstitutePDFColorSpaceRec exec +}if +1 exch put +}if +exit +}if +dup 0 get/Indexed eq{ +exit +}if +dup 0 get/Separation eq{ +dup dup 2//CheckColorSpace//ResolveA exec +dup type/nametype ne{ +//SubstitutePDFColorSpaceRec exec +}if +2 exch put +exit +}if +dup 0 get/CalGray eq{ +1 get +dup/Gamma//knownget exec{ +[exch[exch/exp load]cvx dup dup] +1 index exch/DecodeLMN exch put +}if +[exch/CIEBasedA exch] +exit +}if +dup 0 get/CalRGB eq{ +1 get +dup/Matrix//knownget exec{ +1 index exch/MatrixLMN exch put +}if +dup/Gamma//knownget exec{ +aload pop +[exch/exp load]cvx +3 1 roll +[exch/exp load]cvx +3 1 roll +[exch/exp load]cvx +3 1 roll +3 array astore +1 index exch/DecodeLMN exch put +}if +[exch/CIEBasedABC exch] +exit +}if +dup 0 get/Lab eq{ +1 get +begin +currentdict/Range//knownget exec{aload pop}{-100 100 -100 100}ifelse +0 100 6 2 roll 6 array astore +/RangeABC exch def +/DecodeABC[{16 add 116 div}bind{500 div}bind{200 div}bind]def +/MatrixABC[1 1 1 1 0 0 0 0 -1]def +{dup 6 29 div ge{dup dup mul mul}{4 29 div sub 108 841 div mul}ifelse} +/DecodeLMN[ +[3 index aload pop WhitePoint 0 get/mul load]cvx +[4 index aload pop WhitePoint 1 get/mul load]cvx +[5 index aload pop WhitePoint 2 get/mul load]cvx +]def pop +//PDFR_DEBUG{ +(Constructed from Lab <<)= +currentdict{exch = ==}forall +(>>)= +}if +[/CIEBasedABC currentdict] +end +exit +pop +}if +dup 0 get/CIEBasedA eq{exit}if +dup 0 get/CIEBasedABC eq{exit}if +mark exch(Unimplemented color space )exch//error exec +}loop +}bind def +//SubstitutePDFColorSpaceRec 0//SubstitutePDFColorSpace put +/ResolveArrayElement +{2 copy get +dup type dup/arraytype eq exch +/packedarraytype eq or{ +dup length 1 ge exch xcheck and{ +2 copy get +dup 0 get type/integertype eq +1 index 1 get type dup/arraytype +eq exch +/packedarraytype eq or +and{ +exec +2 index 4 1 roll put +}{ +pop pop +}ifelse +}{ +pop +}ifelse +}{ +pop pop +}ifelse +}bind def +/ResolveColorSpaceArrayRec +{0 +exec +}bind def +/SetColorSpaceSafe +{ +PDFR_DEBUG{ +(SetColorSpaceSafe beg)= +}if +currentcolorspace dup type/arraytype eq{ +1 index type/arraytype eq{ +dup length 2 index length eq{ +false exch +dup length 0 exch 1 exch 1 sub{ +dup +4 index exch get exch +2 index exch get +ne{ +exch pop true exch exit +}if +}for +pop +{ +setcolorspace +}{ +pop +}ifelse +}{ +pop setcolorspace +}ifelse +}{ +pop setcolorspace +}ifelse +}{ +pop setcolorspace +}ifelse +PDFR_DEBUG{ +(SetColorSpaceSafe end)= +}if +}bind def +/ResolveColorSpaceArray +{ +//PDFR_DEBUG{ +(ResolveColorSpaceArray beg )print dup == +}if +dup 0 get/Indexed eq{ +1//ResolveArrayElement exec +dup dup 1 get +dup type/arraytype eq{ +//SubstitutePDFColorSpace exec +//ResolveColorSpaceArrayRec exec +1 exch put +}{ +pop pop +}ifelse +}if +dup 0 get/Separation eq{ +dup dup 1 get UnPDFEscape 1 exch put +3//ResolveArrayElement exec +dup 3 get//FunctionToProc exec +2 copy 3 exch put +pop +}if +dup 0 get/Pattern eq{ +dup length 1 gt{ +dup 1 get dup type/arraytype eq{ +ResolveColorSpaceArray +1 index 1 3 -1 roll put +}{ +pop +}ifelse +}if +}if +PDFR_DEBUG{ +(Construcrted color space :)= +dup == +}if +//PDFR_DEBUG{ +(ResolveColorSpaceArray end )print dup == +}if +}bind def +//ResolveColorSpaceArrayRec 0//ResolveColorSpaceArray put +/ResolveColorSpace +{ +//PDFR_DEBUG{ +(ResolveColorSpace beg )print dup = +}if +dup//SimpleColorSpaceNames exch known not{ +dup//PDFColorSpaces exch//knownget exec{ +exch pop +//PDFR_DEBUG{ +(ResolveColorSpace known )= +}if +}{ +dup +//PDFReader/CurrentObject get/Context get/Resources get +/ColorSpace//DoNothing//ResolveD exec +exch//CheckColorSpace//ResolveD exec +dup type/arraytype eq{ +//SubstitutePDFColorSpace exec +//ResolveColorSpaceArray exec +dup//PDFColorSpaces 4 2 roll put +}if +}ifelse +}if +//PDFR_DEBUG{ +(ResolveColorSpace end )print dup == +}if +}bind def +/CheckPattern +{ +dup/PatternType//knownget exec{ +dup 1 ne{ +mark(Resource )4 index( is a shading, which can't be handled at level 2. )//error exec +}if +pop +}if +dup/Type knownget{ +/Pattern ne{ +mark(Resource )4 index( must have /Type/Pattern .)//error exec +}if +}if +}bind def +/PaintProc +{/Context get +//RunDelayedStream exec +}bind def +/ResolvePattern +{ +dup +userdict/PDFR_Patterns get +exch//knownget exec{ +exch pop +}{ +dup +//PDFReader/CurrentObject get/Context get/Resources get +/Pattern//DoNothing//ResolveD exec +exch//CheckPattern//ResolveD exec +dup dup/Context exch put +dup/Resources//DoNothing//ResolveD exec pop +dup/PaintProc//PaintProc put +gsave userdict/PDFR_InitialGS get setgstate +currentglobal exch false setglobal +dup/Matrix get +makepattern +exch setglobal +grestore +dup userdict/PDFR_Patterns get +4 2 roll +put +}ifelse +}bind def +/SetColor +{//PDFR_DEBUG{ +(SetColor beg)= +}if +currentcolorspace dup type/nametype eq{ +pop setcolor +}{ +0 get/Pattern eq{ +//ResolvePattern exec setpattern +}{ +setcolor +}ifelse +}ifelse +//PDFR_DEBUG{ +(SetColor end)= +}if +}bind def +/ImageKeys 15 dict begin +/BPC/BitsPerComponent def +/CS/ColorSpace def +/D/Decode def +/DP/DecodeParms def +/F/Filter def +/H/Height def +/IM/ImageMask def +/I/Interpolate def +/W/Width def +currentdict end readonly def +/ImageValues 15 dict begin +/G/DeviceGray def +/RGB/DeviceRGB def +/CMYK/DeviceCMYK def +/I/Indexed def +/AHx/ASCIIHexDecode def +/A85/ASCII85Decode def +/LZW/LZWDecode def +/Fl/FlateDecode def +/RL/RunLengthDecode def +/CCF/CCITTFaxDecode def +/DCT/DCTDecode def +currentdict end readonly def +/GetColorSpaceRange +{2 index/ColorSpace get +dup type/arraytype eq{ +1 get +}if +exch//knownget exec{ +exch pop +}if +}bind def +/DecodeArrays 15 dict begin +/DeviceGray{[0 1]}def +/DeviceRGB{[0 1 0 1 0 1]}def +/DeviceCMYK{[0 1 0 1 0 1 0 1]}def +/Indexed{ +dup/BitsPerComponent get 1 exch bitshift 1 sub[exch 0 exch] +}def +/Separation{[0 1]}def +/CIEBasedA{[0 1]/RangeA//GetColorSpaceRange exec}def +/CIEBasedABC{[0 1 0 1 0 1]/RangeABC//GetColorSpaceRange exec}def +currentdict end readonly def +/Substitute +{1 index//knownget exec{ +exch pop +}if +}bind def +/DebugImagePrinting +{ +//PDFR_DEBUG{ +(Image :)= +dup{exch//=only exec( )print == +}forall +}if +}bind def +/CompleteImage +{ +dup/ColorSpace known{ +dup/ColorSpace//CheckColorSpace//ResolveD exec pop +}if +dup/Decode known not{ +dup/ColorSpace//knownget exec{ +dup type/arraytype eq{ +0 get +}if +//DecodeArrays exch get exec +}{ +[0 1] +}ifelse +1 index exch/Decode exch put +}if +dup/ImageMatrix[2 index/Width get 0 0 5 index/Height get neg +0 7 index/Height get]put +//DebugImagePrinting exec +}bind def +/CompleteInlineImage +{ +//PDFR_DEBUG{ +(CompleteInlineImage beg)= +}if +dup/ImageType known not{ +dup/ImageType 1 put +}if +dup length dict exch{ +exch//ImageKeys//Substitute exec +dup/Filter eq{ +exch//ImageValues//Substitute exec exch +}if +dup/ColorSpace eq{ +exch +dup//ImageValues exch//knownget exec{ +exch pop +}{ +//ResolveColorSpace exec +}ifelse +exch +}if +exch +2 index 3 1 roll put +}forall +//CompleteImage exec +dup/DataSource 2 copy get +2 index//AppendFilters exec put +//PDFR_DEBUG{ +(CompleteInlineImage end)= +}if +}bind def +/CompleteOutlineImage +{ +currentglobal exch dup gcheck setglobal +//PDFR_DEBUG{ +(CompleteOutlineImage beg)= +}if +dup dup//MakeStreamReader exec/DataSource exch put +dup/ImageType known not{ +//CompleteImage exec +dup/ImageType 1 put +dup/ColorSpace known{ +dup/ColorSpace//CheckColorSpace//ResolveD exec +dup type/arraytype eq{ +//ResolveColorSpaceArray exec +//SubstitutePDFColorSpace exec +1 index exch/ColorSpace exch put +}{ +pop +}ifelse +}if +}if +//PDFR_DEBUG{ +(CompleteOutlineImage end)= +}if +exch setglobal +}bind def +/DoImage +{ +//PDFR_DEBUG{ +(DoImage beg)= +}if +gsave +dup/ColorSpace//knownget exec{setcolorspace}if +dup/ImageMask//knownget exec not{false}if +{imagemask}{image}ifelse +grestore +//PDFR_DEBUG{ +(DoImage end)= +}if +}bind def +/GSave +{ +gsave +//PDFReader/GraphicStateStackPointer get +dup//GraphicStateStack exch get null eq{ +dup//GraphicStateStack exch//InitialGraphicState length dict put +}if +dup//GraphicStateStack exch get +//GraphicState exch copy pop +1 add//PDFReader exch/GraphicStateStackPointer exch put +}bind def +/GRestore +{ +grestore +//PDFReader/GraphicStateStackPointer get +1 sub dup +//PDFReader exch/GraphicStateStackPointer exch put +//GraphicStateStack exch get +//GraphicState copy pop +}bind def +/SetFont +{dup//GraphicState exch/FontSize exch put +//ResolveAndSetFont exec +//GraphicState/FontMatrixNonHV currentfont/FontMatrix get 1 get 0 ne put +}bind def +/ShowText +{ +//GraphicState/TextRenderingMode get dup 0 eq +exch 3 eq not currentfont/FontType get 3 eq and or +{ +//GraphicState/WordSpacing get 0 +32 +//GraphicState/CharacterSpacing get 0 +6 5 roll +//GraphicState/FontMatrixNonHV get{ +[ +7 -2 roll pop +5 -2 roll pop +5 -1 roll +{ +exch +pop +3 index add +exch 2 index eq{3 index add}if +4 1 roll +} +currentfont/FontMatrix get 0 get 0 ne{ +1 1 index length 1 sub getinterval cvx +}if +5 index +cshow +pop pop pop] +xshow +}{ +awidthshow +}ifelse +}{ +//GraphicState/CharacterSpacing get 0 eq +//GraphicState/FontMatrixNonHV get not and +//GraphicState/WordSpacing get 0 eq and{ +true charpath +}{ +{ +exch +pop 0 +currentpoint 5 4 roll +( )dup 0 3 index put true charpath +5 1 roll +moveto rmoveto +//GraphicState/CharacterSpacing get 0 rmoveto +32 eq{ +//GraphicState/WordSpacing get 0 rmoveto +}if +} +//GraphicState/FontMatrixNonHV get dup not exch{ +pop currentfont/FontMatrix get 0 get 0 ne +}if{ +1 1 index length 1 sub getinterval cvx +}if +exch cshow +}ifelse +}ifelse +}bind def +/ShowTextBeg +{ +//GraphicState/TextRenderingMode get dup 0 ne +{ +3 ne +currentfont/FontType get 3 eq not and{ +currentpoint newpath moveto +}if +} +{ +pop +}ifelse +}bind def +/ShowTextEnd +{ +//GraphicState/TextRenderingMode get +currentfont/FontType get 3 eq{ +dup 3 ne{ +pop 0 +}if +}if +{dup 1 eq{ +stroke exit +}if +dup 2 eq{ +gsave fill grestore stroke exit +}if +dup 3 eq{ +currentpoint newpath moveto +}if +dup 4 eq{ +gsave fill grestore clip exit +}if +dup 5 eq{ +gsave stroke grestore clip exit +}if +dup 6 eq{ +gsave fill grestore gsave stroke grestore fill exit +}if +dup 7 eq{ +clip exit +}if +exit +}loop +pop +}bind def +/ShowTextWithGlyphPositioning +{//ShowTextBeg exec +{dup type/stringtype eq{ +//ShowText exec +}{ +neg 1000 div//GraphicState/FontSize get mul 0 rmoveto +}ifelse +}forall +//ShowTextEnd exec +}bind def +/CheckFont +{dup/Type get/ExtGState ne{ +mark(Resource )3 index( must have /Type/ExtGState.)//error exec +}if +}bind def +/SetTransfer +{ +//PDFR_DEBUG{(SetTransfer beg )print count =}if +dup type/arraytype eq 1 index xcheck not and{ +0 4 getinterval aload pop +setcolortransfer +}{ +settransfer +}ifelse +//PDFR_DEBUG{(SetTransfer end )print count =}if +}bind def +/CheckExtGState +{dup/Type get/ExtGState ne{ +mark(Resource )3 index( must have /Type/ExtGState.)//error exec +}if +}bind def +/CheckHalftone +{dup/HalftoneType known not{ +mark(Resource )3 index( must have /HalftoneType.)//error exec +}if +}bind def +/ResolveFunction +{ +//PDFR_DEBUG{(ResolveFunction beg )print dup = count =}if +2 copy get//IsObjRef exec{ +2 copy//DoNothing//ResolveD exec +3 copy put pop +}if +2 copy get dup type/arraytype eq exch xcheck and not{ +2 copy get +dup type/arraytype eq 1 index xcheck not and{ +dup length 1 sub -1 0{ +2 copy//DoNothing ResolveA +dup/Identity eq{ +pop 2 copy{}put +}{ +//FunctionToProc exec +3 copy put pop +}ifelse +pop +}for +}{ +dup/Default eq{ +}{ +dup/Identity eq{ +pop{} +}{dup type/nametype eq{ +//spotfunctions exch get +}{ +//FunctionToProc exec +}ifelse +}ifelse +}ifelse +}ifelse +3 copy put +exch pop +}{ +1 index exch get +}ifelse +//PDFR_DEBUG{(ResolveFunction end )print dup == count =}if +}bind def +/ResolveFunctionSafe +{2 copy known{ +//ResolveFunction exec +}if +pop +}bind def +/CreateHalftoneThresholds +{ +dup/Thresholds known not{ +dup/HalftoneType get 10 eq{ +dup dup//MakeStreamReader exec +/Thresholds exch put +}if +dup/HalftoneType get dup 3 eq exch 6 eq or{ +dup dup//MakeStreamReader exec +//BlockBuffer readstring pop +dup length +dup 0 eq{ +mark(Could not read Thresholds)//error exec +}if +string copy/Thresholds exch put +dup/HalftoneType 3 put +}if +}if +}bind def +/SetExtGState +{ +//PDFReader/CurrentObject get/Context get/Resources get +/ExtGState//DoNothing//ResolveD exec +exch//CheckExtGState//ResolveD exec +dup/LW//knownget exec{ +setlinewidth +}if +dup/LC//knownget exec{ +setlinecap +}if +dup/LJ//knownget exec{ +setlinejoin +}if +dup/ML//knownget exec{ +setmeterlimit +}if +dup/D//knownget exec{ +setdash +}if +dup/RI//knownget exec{ +mark(Unimplemented ExtGState.RI)//error exec +}if +dup/OP//knownget exec{ +setoverprint +}if +dup/op//knownget exec{ +setoverprint +}if +dup/OPM//knownget exec{ +mark(Unimplemented ExtGState.OPM)//error exec +}if +dup/Font//knownget exec{ +mark(Unimplemented ExtGState.Font)//error exec +}if +dup/BG known{ +/BG//ResolveFunction exec +setblackgeneration +}if +dup/BG2 known{ +/BG2//ResolveFunction exec +dup/Default eq{ +//InitialExtGState/BG2 get +}if +setblackgeneration +}if +dup/UCR known{ +/UCR//ResolveFunction exec +setundercolorremoval +}if +dup/UCR2 known{ +/UCR2//ResolveFunction exec +dup/Default eq{ +//InitialExtGState/UCR2 get +}if +setundercolorremoval +}if +dup/TR known{ +/TR//ResolveFunction exec +//SetTransfer exec +}if +dup/TR2 known{ +/TR2//ResolveFunction exec +dup/Default eq{ +pop//InitialExtGState/TR2 get +aload pop setcolortransfer +}{ +//SetTransfer exec +}ifelse +}if +dup/HT//knownget exec{ +dup/Default eq{ +pop//InitialExtGState/HT get +sethalftone +}{ +//PDFR_DEBUG{(Ht beg)=}if +pop dup/HT//CheckHalftone//ResolveD exec +/SpotFunction//ResolveFunctionSafe exec +/TransferFunction//ResolveFunctionSafe exec +null exch +dup/HalftoneType get dup 5 eq exch dup 4 eq exch 2 eq or or{ +dup{ +dup//IsObjRef exec{ +pop +1 index exch//CheckHalftone ResolveD +}if +dup type/dicttype eq{ +dup/SpotFunction//ResolveFunctionSafe exec +/TransferFunction//ResolveFunctionSafe exec +//CreateHalftoneThresholds exec +dup/HalftoneType get 5 gt{ +4 3 roll pop +dup 4 1 roll +}if +}if +pop pop +}forall +}if +//CreateHalftoneThresholds exec +//PDFR_DEBUG{ +(HT:)= +dup{ +1 index/Default eq{ +(Default <<)= +exch pop +{exch = ==}forall +(>>)= +}{ +exch = == +}ifelse +}forall +(HT end)= flush +}if +exch dup null ne{ +(Warning: Ignoring a halftone with a Level 3 component halftone Type )print dup/HalftoneType get = +pop pop +}{ +pop +dup/HalftoneType get 5 gt{ +(Warning: Ignoring a Level 3 halftone Type )print dup/HalftoneType get = +pop +}{ +sethalftone +}ifelse +}ifelse +//PDFR_DEBUG{(HT set)= flush}if +}ifelse +}if +dup/FL//knownget exec{ +setflattness +}if +dup/SM//knownget exec{ +setsmoothness +}if +dup/SA//knownget exec{ +setstrokeadjust +}if +dup/BM//knownget exec{ +mark(Unimplemented ExtGState.BM)//error exec +}if +dup/SMask//knownget exec{ +mark(Unimplemented ExtGState.SMask)//error exec +}if +dup/CA//knownget exec{ +mark(Unimplemented ExtGState.CA)//error exec +}if +dup/ca//knownget exec{ +mark(Unimplemented ExtGState.ca)//error exec +}if +dup/AIS//knownget exec{ +mark(Unimplemented ExtGState.AIS)//error exec +}if +dup/TK//knownget exec{ +mark(Unimplemented ExtGState.TK)//error exec +}if +pop +}bind def +/CheckXObject +{dup/Subtype get dup/Image ne exch dup/Form ne exch/PS ne and and{ +mark(Resource )3 index( must have /Subtype /Image or /Form or /PS.)//error exec +}if +}bind def +/DoXObject +{ +//PDFReader/CurrentObject get/Context get/Resources get +/XObject//DoNothing//ResolveD exec +exch//CheckXObject//ResolveD exec +dup/Subtype get +dup/Image eq{ +pop +//CompleteOutlineImage exec +//DoImage exec +}{ +dup/PS eq{ +PDFR_DEBUG{ +(Executing a PS Xobject)= +}if +pop +//RunDelayedStream exec +}{ +dup/Form eq{ +pop +PDFR_DEBUG{ +(Executing a Form XObject)= +}if +//PDFReader/CurrentObject get exch +dup//PDFReader exch<< exch/Context exch >>/CurrentObject exch put +dup/Matrix get concat +dup/BBox get aload pop exch 3 index sub exch 2 index sub rectclip +//RunDelayedStream exec +//PDFReader exch/CurrentObject exch put +}{ +mark exch(unimplemented XObject type )exch//error exec +}ifelse +}ifelse +}ifelse +}bind def +/Operators 50 dict begin +/q{//GSave exec}bind def +/Q{//GRestore exec}bind def +/cm{//TempMatrix astore concat}bind def +/i{1 .min setflat}bind def +/J/setlinecap load def +/d/setdash load def +/j/setlinejoin load def +/w/setlinewidth load def +/M/setmiterlimit load def +/gs{SetExtGState}bind def +/g/setgray load def +/rg/setrgbcolor load def +/k/setcmykcolor load def +/cs{//ResolveColorSpace exec//SetColorSpaceSafe exec +}bind def +/sc/setcolor load def +/scn{//SetColor exec}bind def +/G/setgray load def +/RG/setrgbcolor load def +/K/setcmykcolor load def +/CS//cs def +/ri{SetColorRenderingIntent}bind def +/SC/setcolor load def +/SCN{//SetColor exec}bind def +/m/moveto load def +/l/lineto load def +/c/curveto load def +/v{currentpoint 6 2 roll curveto}bind def +/y{2 copy curveto}bind def +/re{ +4 2 roll moveto exch dup 0 rlineto 0 3 -1 roll rlineto neg 0 rlineto +closepath +}def +/h/closepath load def +/n/newpath load def +/S/stroke load def +/s{closepath stroke}bind def +/f/fill load def +/f*/eofill load def +/B{gsave fill grestore stroke}bind def +/b{closepath gsave fill grestore stroke}bind def +/B*{gsave eofill grestore stroke}bind def +/b*{closepath gsave eofill grestore stroke}bind def +/W/clip load def +/W*/eoclip load def +/sh{ +ResolveShading +dup/Background known{ +gsave +dup/ColorSpace get setcolorspace +dup/Background get aload pop setcolor +pathbbox +2 index sub exch 3 index sub exch +rectfill +grestore +}if +shfill +}bind def +/Do{//DoXObject exec}bind def +/BI{currentglobal false setglobal<<}bind def +/ID{>> +dup/DataSource currentfile +2 index/F//knownget exec{ +/A85 eq{ +0(~>)/SubFileDecode filter +}if +}if +put +//CompleteInlineImage exec +exch setglobal +//DoImage exec +}bind def +/EI{}bind def +/BT{gsave//GraphicState/InitialTextMatrix get currentmatrix pop}bind def +/ET{grestore}bind def +/Tc{//GraphicState exch/CharacterSpacing exch put}bind def +/TL{//GraphicState exch/TextLeading exch put}bind def +/Tr{//GraphicState exch/TextRenderingMode exch put}bind def +/Ts{ +mark(Unimplemented SetTextRise)//error exec +}bind def +/Tw{//GraphicState exch/WordSpacing exch put}bind def +/Tz{ +mark(Unimplemented SetHorizontalTextScaling)//error exec +}bind def +/Td{translate 0 0 moveto}bind def +/TD{dup neg//TL exec//Td exec}bind def +/Tm{//GraphicState/InitialTextMatrix get setmatrix +//TempMatrix astore concat +0 0 moveto}bind def +/T*{0//GraphicState/TextLeading get neg//Td exec}bind def +/Tj{//ShowTextBeg exec//ShowText exec//ShowTextEnd exec}bind def +/'{//T* exec//ShowText exec//ShowTextEnd exec}bind def +/"{3 2 roll//Tw exec exch//Tc exec//' exec}bind def +/TJ//ShowTextWithGlyphPositioning def +/Tf//SetFont def +/d0/setcharwidth load def +/d1/setcachedevice load def +/BDC{pop pop}bind def +/BMC{pop}bind def +/EMC{}bind def +/BX{BeginCompatibilitySection}bind def +/EX{EndCompatibilitySection}bind def +/DP{DefineMarkedContentPointWithPropertyList}bind def +/MP{DefineMarkedContentPoint}bind def +/PS{cvx exec}bind def +currentdict end def +//PDFR_STREAM{ +//Operators length dict begin +//Operators{ +exch dup +[exch//=only/exec load +( )/print load +8 7 roll +dup type/arraytype eq{ +/exec load +}if +( )/print load +]cvx +def +}forall +currentdict end/Operators exch def +}if +/.registerencoding +{pop pop +}bind def +/.defineencoding +{def +}bind def +/.findencoding +{load +}bind def +/currentglobal where +{pop currentglobal{setglobal}true setglobal} +{{}} +ifelse +/MacRomanEncoding +StandardEncoding 0 39 getinterval aload pop +/quotesingle +StandardEncoding 40 56 getinterval aload pop +/grave +StandardEncoding 97 31 getinterval aload pop +/Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute +/agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave +/ecircumflex/edieresis/iacute/igrave +/icircumflex/idieresis/ntilde/oacute +/ograve/ocircumflex/odieresis/otilde +/uacute/ugrave/ucircumflex/udieresis +/dagger/degree/cent/sterling/section/bullet/paragraph/germandbls +/registered/copyright/trademark/acute/dieresis/.notdef/AE/Oslash +/.notdef/plusminus/.notdef/.notdef/yen/mu/.notdef/.notdef +/.notdef/.notdef/.notdef/ordfeminine/ordmasculine/.notdef/ae/oslash +/questiondown/exclamdown/logicalnot/.notdef +/florin/.notdef/.notdef/guillemotleft +/guillemotright/ellipsis/space/Agrave/Atilde/Otilde/OE/oe +/endash/emdash/quotedblleft/quotedblright +/quoteleft/quoteright/divide/.notdef +/ydieresis/Ydieresis/fraction/currency +/guilsinglleft/guilsinglright/fi/fl +/daggerdbl/periodcentered/quotesinglbase/quotedblbase +/perthousand/Acircumflex/Ecircumflex/Aacute +/Edieresis/Egrave/Iacute/Icircumflex +/Idieresis/Igrave/Oacute/Ocircumflex +/.notdef/Ograve/Uacute/Ucircumflex +/Ugrave/dotlessi/circumflex/tilde +/macron/breve/dotaccent/ring/cedilla/hungarumlaut/ogonek/caron +256 packedarray +5 1 index .registerencoding +.defineencoding +exec +/AdobeGlyphList mark +/A 16#0041 +/AE 16#00c6 +/AEacute 16#01fc +/AEmacron 16#01e2 +/AEsmall 16#f7e6 +/Aacute 16#00c1 +/Aacutesmall 16#f7e1 +/Abreve 16#0102 +/Abreveacute 16#1eae +/Abrevecyrillic 16#04d0 +/Abrevedotbelow 16#1eb6 +/Abrevegrave 16#1eb0 +/Abrevehookabove 16#1eb2 +/Abrevetilde 16#1eb4 +/Acaron 16#01cd +/Acircle 16#24b6 +/Acircumflex 16#00c2 +/Acircumflexacute 16#1ea4 +/Acircumflexdotbelow 16#1eac +/Acircumflexgrave 16#1ea6 +/Acircumflexhookabove 16#1ea8 +/Acircumflexsmall 16#f7e2 +/Acircumflextilde 16#1eaa +/Acute 16#f6c9 +/Acutesmall 16#f7b4 +/Acyrillic 16#0410 +/Adblgrave 16#0200 +/Adieresis 16#00c4 +/Adieresiscyrillic 16#04d2 +/Adieresismacron 16#01de +/Adieresissmall 16#f7e4 +/Adotbelow 16#1ea0 +/Adotmacron 16#01e0 +/Agrave 16#00c0 +/Agravesmall 16#f7e0 +/Ahookabove 16#1ea2 +/Aiecyrillic 16#04d4 +/Ainvertedbreve 16#0202 +/Alpha 16#0391 +/Alphatonos 16#0386 +/Amacron 16#0100 +/Amonospace 16#ff21 +/Aogonek 16#0104 +/Aring 16#00c5 +/Aringacute 16#01fa +/Aringbelow 16#1e00 +/Aringsmall 16#f7e5 +/Asmall 16#f761 +/Atilde 16#00c3 +/Atildesmall 16#f7e3 +/Aybarmenian 16#0531 +/B 16#0042 +/Bcircle 16#24b7 +/Bdotaccent 16#1e02 +/Bdotbelow 16#1e04 +/Becyrillic 16#0411 +/Benarmenian 16#0532 +/Beta 16#0392 +/Bhook 16#0181 +/Blinebelow 16#1e06 +/Bmonospace 16#ff22 +/Brevesmall 16#f6f4 +/Bsmall 16#f762 +/Btopbar 16#0182 +/C 16#0043 +/Caarmenian 16#053e +/Cacute 16#0106 +/Caron 16#f6ca +/Caronsmall 16#f6f5 +/Ccaron 16#010c +/Ccedilla 16#00c7 +/Ccedillaacute 16#1e08 +/Ccedillasmall 16#f7e7 +/Ccircle 16#24b8 +/Ccircumflex 16#0108 +/Cdot 16#010a +/Cdotaccent 16#010a +/Cedillasmall 16#f7b8 +/Chaarmenian 16#0549 +/Cheabkhasiancyrillic 16#04bc +/Checyrillic 16#0427 +/Chedescenderabkhasiancyrillic 16#04be +/Chedescendercyrillic 16#04b6 +/Chedieresiscyrillic 16#04f4 +/Cheharmenian 16#0543 +/Chekhakassiancyrillic 16#04cb +/Cheverticalstrokecyrillic 16#04b8 +/Chi 16#03a7 +/Chook 16#0187 +/Circumflexsmall 16#f6f6 +/Cmonospace 16#ff23 +/Coarmenian 16#0551 +/Csmall 16#f763 +/D 16#0044 +/DZ 16#01f1 +/DZcaron 16#01c4 +/Daarmenian 16#0534 +/Dafrican 16#0189 +/Dcaron 16#010e +/Dcedilla 16#1e10 +/Dcircle 16#24b9 +/Dcircumflexbelow 16#1e12 +/Dcroat 16#0110 +/Ddotaccent 16#1e0a +/Ddotbelow 16#1e0c +/Decyrillic 16#0414 +/Deicoptic 16#03ee +/Delta 16#2206 +/Deltagreek 16#0394 +/Dhook 16#018a +/Dieresis 16#f6cb +/DieresisAcute 16#f6cc +/DieresisGrave 16#f6cd +/Dieresissmall 16#f7a8 +/Digammagreek 16#03dc +/Djecyrillic 16#0402 +/Dlinebelow 16#1e0e +/Dmonospace 16#ff24 +/Dotaccentsmall 16#f6f7 +/Dslash 16#0110 +/Dsmall 16#f764 +/Dtopbar 16#018b +/Dz 16#01f2 +/Dzcaron 16#01c5 +/Dzeabkhasiancyrillic 16#04e0 +/Dzecyrillic 16#0405 +/Dzhecyrillic 16#040f +/E 16#0045 +/Eacute 16#00c9 +/Eacutesmall 16#f7e9 +/Ebreve 16#0114 +/Ecaron 16#011a +/Ecedillabreve 16#1e1c +/Echarmenian 16#0535 +/Ecircle 16#24ba +/Ecircumflex 16#00ca +/Ecircumflexacute 16#1ebe +/Ecircumflexbelow 16#1e18 +/Ecircumflexdotbelow 16#1ec6 +/Ecircumflexgrave 16#1ec0 +/Ecircumflexhookabove 16#1ec2 +/Ecircumflexsmall 16#f7ea +/Ecircumflextilde 16#1ec4 +/Ecyrillic 16#0404 +/Edblgrave 16#0204 +/Edieresis 16#00cb +/Edieresissmall 16#f7eb +/Edot 16#0116 +/Edotaccent 16#0116 +/Edotbelow 16#1eb8 +/Efcyrillic 16#0424 +/Egrave 16#00c8 +/Egravesmall 16#f7e8 +/Eharmenian 16#0537 +/Ehookabove 16#1eba +/Eightroman 16#2167 +/Einvertedbreve 16#0206 +/Eiotifiedcyrillic 16#0464 +/Elcyrillic 16#041b +/Elevenroman 16#216a +/Emacron 16#0112 +/Emacronacute 16#1e16 +/Emacrongrave 16#1e14 +/Emcyrillic 16#041c +/Emonospace 16#ff25 +/Encyrillic 16#041d +/Endescendercyrillic 16#04a2 +/Eng 16#014a +/Enghecyrillic 16#04a4 +/Enhookcyrillic 16#04c7 +/Eogonek 16#0118 +/Eopen 16#0190 +/Epsilon 16#0395 +/Epsilontonos 16#0388 +/Ercyrillic 16#0420 +/Ereversed 16#018e +/Ereversedcyrillic 16#042d +/Escyrillic 16#0421 +/Esdescendercyrillic 16#04aa +/Esh 16#01a9 +/Esmall 16#f765 +/Eta 16#0397 +/Etarmenian 16#0538 +/Etatonos 16#0389 +/Eth 16#00d0 +/Ethsmall 16#f7f0 +/Etilde 16#1ebc +/Etildebelow 16#1e1a +/Euro 16#20ac +/Ezh 16#01b7 +/Ezhcaron 16#01ee +/Ezhreversed 16#01b8 +/F 16#0046 +/Fcircle 16#24bb +/Fdotaccent 16#1e1e +/Feharmenian 16#0556 +/Feicoptic 16#03e4 +/Fhook 16#0191 +/Fitacyrillic 16#0472 +/Fiveroman 16#2164 +/Fmonospace 16#ff26 +/Fourroman 16#2163 +/Fsmall 16#f766 +/G 16#0047 +/GBsquare 16#3387 +/Gacute 16#01f4 +/Gamma 16#0393 +/Gammaafrican 16#0194 +/Gangiacoptic 16#03ea +/Gbreve 16#011e +/Gcaron 16#01e6 +/Gcedilla 16#0122 +/Gcircle 16#24bc +/Gcircumflex 16#011c +/Gcommaaccent 16#0122 +/Gdot 16#0120 +/Gdotaccent 16#0120 +/Gecyrillic 16#0413 +/Ghadarmenian 16#0542 +/Ghemiddlehookcyrillic 16#0494 +/Ghestrokecyrillic 16#0492 +/Gheupturncyrillic 16#0490 +/Ghook 16#0193 +/Gimarmenian 16#0533 +/Gjecyrillic 16#0403 +/Gmacron 16#1e20 +/Gmonospace 16#ff27 +/Grave 16#f6ce +/Gravesmall 16#f760 +/Gsmall 16#f767 +/Gsmallhook 16#029b +/Gstroke 16#01e4 +/H 16#0048 +/H18533 16#25cf +/H18543 16#25aa +/H18551 16#25ab +/H22073 16#25a1 +/HPsquare 16#33cb +/Haabkhasiancyrillic 16#04a8 +/Hadescendercyrillic 16#04b2 +/Hardsigncyrillic 16#042a +/Hbar 16#0126 +/Hbrevebelow 16#1e2a +/Hcedilla 16#1e28 +/Hcircle 16#24bd +/Hcircumflex 16#0124 +/Hdieresis 16#1e26 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+/Lcircle 16#24c1 +/Lcircumflexbelow 16#1e3c +/Lcommaaccent 16#013b +/Ldot 16#013f +/Ldotaccent 16#013f +/Ldotbelow 16#1e36 +/Ldotbelowmacron 16#1e38 +/Liwnarmenian 16#053c +/Lj 16#01c8 +/Ljecyrillic 16#0409 +/Llinebelow 16#1e3a +/Lmonospace 16#ff2c +/Lslash 16#0141 +/Lslashsmall 16#f6f9 +/Lsmall 16#f76c +/M 16#004d +/MBsquare 16#3386 +/Macron 16#f6d0 +/Macronsmall 16#f7af +/Macute 16#1e3e +/Mcircle 16#24c2 +/Mdotaccent 16#1e40 +/Mdotbelow 16#1e42 +/Menarmenian 16#0544 +/Mmonospace 16#ff2d +/Msmall 16#f76d +/Mturned 16#019c +/Mu 16#039c +/N 16#004e +/NJ 16#01ca +/Nacute 16#0143 +/Ncaron 16#0147 +/Ncedilla 16#0145 +/Ncircle 16#24c3 +/Ncircumflexbelow 16#1e4a +/Ncommaaccent 16#0145 +/Ndotaccent 16#1e44 +/Ndotbelow 16#1e46 +/Nhookleft 16#019d +/Nineroman 16#2168 +/Nj 16#01cb +/Njecyrillic 16#040a +/Nlinebelow 16#1e48 +/Nmonospace 16#ff2e +/Nowarmenian 16#0546 +/Nsmall 16#f76e +/Ntilde 16#00d1 +/Ntildesmall 16#f7f1 +/Nu 16#039d +/O 16#004f +/OE 16#0152 +/OEsmall 16#f6fa +/Oacute 16#00d3 +/Oacutesmall 16#f7f3 +/Obarredcyrillic 16#04e8 +/Obarreddieresiscyrillic 16#04ea +/Obreve 16#014e +/Ocaron 16#01d1 +/Ocenteredtilde 16#019f +/Ocircle 16#24c4 +/Ocircumflex 16#00d4 +/Ocircumflexacute 16#1ed0 +/Ocircumflexdotbelow 16#1ed8 +/Ocircumflexgrave 16#1ed2 +/Ocircumflexhookabove 16#1ed4 +/Ocircumflexsmall 16#f7f4 +/Ocircumflextilde 16#1ed6 +/Ocyrillic 16#041e +/Odblacute 16#0150 +/Odblgrave 16#020c +/Odieresis 16#00d6 +/Odieresiscyrillic 16#04e6 +/Odieresissmall 16#f7f6 +/Odotbelow 16#1ecc +/Ogoneksmall 16#f6fb +/Ograve 16#00d2 +/Ogravesmall 16#f7f2 +/Oharmenian 16#0555 +/Ohm 16#2126 +/Ohookabove 16#1ece +/Ohorn 16#01a0 +/Ohornacute 16#1eda +/Ohorndotbelow 16#1ee2 +/Ohorngrave 16#1edc +/Ohornhookabove 16#1ede +/Ohorntilde 16#1ee0 +/Ohungarumlaut 16#0150 +/Oi 16#01a2 +/Oinvertedbreve 16#020e +/Omacron 16#014c +/Omacronacute 16#1e52 +/Omacrongrave 16#1e50 +/Omega 16#2126 +/Omegacyrillic 16#0460 +/Omegagreek 16#03a9 +/Omegaroundcyrillic 16#047a +/Omegatitlocyrillic 16#047c 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16#f6fc +/Rinvertedbreve 16#0212 +/Rlinebelow 16#1e5e +/Rmonospace 16#ff32 +/Rsmall 16#f772 +/Rsmallinverted 16#0281 +/Rsmallinvertedsuperior 16#02b6 +/S 16#0053 +/SF010000 16#250c +/SF020000 16#2514 +/SF030000 16#2510 +/SF040000 16#2518 +/SF050000 16#253c +/SF060000 16#252c +/SF070000 16#2534 +/SF080000 16#251c +/SF090000 16#2524 +/SF100000 16#2500 +/SF110000 16#2502 +/SF190000 16#2561 +/SF200000 16#2562 +/SF210000 16#2556 +/SF220000 16#2555 +/SF230000 16#2563 +/SF240000 16#2551 +/SF250000 16#2557 +/SF260000 16#255d +/SF270000 16#255c +/SF280000 16#255b +/SF360000 16#255e +/SF370000 16#255f +/SF380000 16#255a +/SF390000 16#2554 +/SF400000 16#2569 +/SF410000 16#2566 +/SF420000 16#2560 +/SF430000 16#2550 +/SF440000 16#256c +/SF450000 16#2567 +/SF460000 16#2568 +/SF470000 16#2564 +/SF480000 16#2565 +/SF490000 16#2559 +/SF500000 16#2558 +/SF510000 16#2552 +/SF520000 16#2553 +/SF530000 16#256b +/SF540000 16#256a +/Sacute 16#015a +/Sacutedotaccent 16#1e64 +/Sampigreek 16#03e0 +/Scaron 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b/documentation/source/science_guide/cloud_schemes/Timestepping_pc266.epsi new file mode 100644 index 0000000000..92fceeff6d --- /dev/null +++ b/documentation/source/science_guide/cloud_schemes/Timestepping_pc266.epsi @@ -0,0 +1,10230 @@ +%!PS-Adobe-3.0 EPSF-3.0 +%%Invocation: path/gs -q -sDEVICE=eps2write -sstdout=? -sOutputFile=? -dNOPAUSE -dBATCH -P- -dSAFER -dDEVICEWIDTH=250000 -dDEVICEHEIGHT=250000 ? +%%BoundingBox: 0 0 720 960 +%%HiResBoundingBox: 0.00 0.00 720.00 960.00 +%%Creator: GPL Ghostscript 9540 (eps2write) +%%LanguageLevel: 2 +%%CreationDate: D:20250417013840+01'00' +%%Pages: 1 +%%EndComments +%%BeginProlog +10 dict dup begin +/DSC_OPDFREAD true def +/SetPageSize false def +/EPS2Write true def +end +count 0 ne{ +dup type/dicttype eq{ +dup/EPS2Write known{ +dup/EPS2Write get not +} +{ +true +}ifelse +} +{ +true +}ifelse +} +{ +true +}ifelse +10 dict begin +/this currentdict def +/y 720 def +/ebuf 200 string def +/prnt{ +36//this/y get moveto//ebuf cvs show +//this/y 2 copy get 12 sub put +}bind def +/newline{ +36//this/y get moveto +//this/y 2 copy get 12 sub put +}bind def +{ +errordict/handleerror +{systemdict begin +$error begin +newerror +{(%%[ Error handled by opdfread.ps : )print errorname//ebuf cvs print(; OffendingCommand: ) +print/command load//ebuf cvs print( ]%%)= flush +/newerror false store vmstatus pop pop 0 ne +{grestoreall +}if +errorname(VMerror)ne +{showpage +}if +initgraphics +0 720 moveto +errorname(VMerror)eq +{//this/ehsave known +{clear//this/ehsave get restore 2 vmreclaim +}if +vmstatus exch pop exch pop +} +/Courier 12 selectfont +{ +(ERROR: )//prnt exec errorname//prnt exec +(OFFENDING COMMAND: )//prnt exec +/command load//prnt exec +$error/ostack known{ +(%%[STACK:)= +(STACK:)//prnt exec +$error/ostack get aload length{ +//newline exec +dup mark eq{ +(-mark-)dup = show +}{ +dup type/nametype eq{ +dup xcheck not{ +(/)show +(/)print +}if +}if +dup =//ebuf cvs show +}ifelse +}repeat +}if +}ifelse +(%%]%)= +//systemdict/showpage get exec +quit +}if +end +end +}bind readonly put +}if +end +50 dict begin +count 0 ne{ +dup type/dicttype eq{ +{def}forall +false +} +{ +true +}ifelse +} +{ +true +}ifelse +{ +( *** Warning: global definitions dictionary not found, file may be corrupted.\n)print flush +}if +/DefaultSwitch +{ +dup where{ +pop pop +}{ +false def +}ifelse +}bind def +/=string 256 string def +/=only{ +//=string cvs print +}bind def +/HexDigits(0123456789ABCDEF)readonly def +/PrintHex +{8{ +dup -28 bitshift 15 and//HexDigits exch 1 getinterval//=only exec +4 bitshift +}repeat +pop +}bind def +/PDFR_DEBUG DefaultSwitch +/PDFR_DUMP DefaultSwitch +/PDFR_STREAM DefaultSwitch +/TTFDEBUG DefaultSwitch +/RotatePages DefaultSwitch +/FitPages DefaultSwitch +/CenterPages DefaultSwitch +/SetPageSize DefaultSwitch +/error +{ +counttomark 1 sub -1 0{ +index dup type/arraytype eq{==}{=only}ifelse +}for +()= +cleartomark +....Undefined +}bind def +//SetPageSize{ +//RotatePages//FitPages or//CenterPages or{ +mark(/RotatePages, /FitPages and CenterPages are not allowed with /SetPageSize)//error exec +}if +} +{ +//FitPages//CenterPages and{ +mark(CenterPages is not allowed with /FitPages)//error exec +}if +} +ifelse +/knownget +{ +2 copy known{ +get true +}{ +pop pop false +}ifelse +}bind def +/IsUpper +{dup(A)0 get ge exch(Z)0 get le and +}bind def +/cpa2g{ +dup length array +0 1 2 index length 1 sub{ +dup 3 index exch get cp2g +3 copy put pop pop +}for +exch pop +}bind def +/cpd2g{ +dup length dict exch{ +cp2g 2 index 3 1 roll put +}forall +}bind def +/cps2g{ +dup length string copy +}bind def +/cp2gprocs +<> +def +/cp2g{ +dup gcheck not{ +dup//cp2gprocs 1 index type +2 copy known{ +get currentglobal 3 1 roll true setglobal exec exch setglobal +1 index wcheck not{readonly}if +1 index xcheck{cvx}if +exch pop +}{ +pop pop +}ifelse +}if +}bind def +/BlockBuffer 65535 string def +/PDFReader currentdict def +/ObjectRegistryMaxLength 50000 def +/ObjectRegistry 10 dict def +ObjectRegistry +begin +0 ObjectRegistryMaxLength dict def +end +/CurrentObject null def +/DoneDocumentStructure false def +/GraphicState 20 dict begin +/InitialTextMatrix matrix def +/InitialMatrix matrix currentmatrix def +currentdict end def +/TempMatrix matrix def +/GraphicStateStack 20 array def +/GraphicStateStackPointer 0 def +/InitialTextMatrixStack 20 array def +/InitialTextMatrixStackPointer 0 def +/PDFColorSpaces 50 dict def +/InstalledFonts 50 dict def +/MacRomanEncodingInverse null def +currentglobal false setglobal +userdict/PDFR_InitialGS gstate put +userdict/PDFR_Patterns 50 dict put +userdict/FuncDataReader 10 dict put +setglobal +/InitialExtGState 20 dict begin +/BG2 currentblackgeneration cp2g def +/UCR2 currentundercolorremoval cp2g def +/TR2 currentglobal false setglobal[currentcolortransfer]exch setglobal cp2g def +/HT currenthalftone cp2g def +currentdict end readonly def +/InitialGraphicState 20 dict begin +/FontSize 0 def +/CharacterSpacing 0 def +/TextLeading 0 def +/TextRenderingMode 0 def +/WordSpacing 0 def +currentdict end readonly def +/SimpleColorSpaceNames 15 dict begin +/DeviceGray true def +/DeviceRGB true def +/DeviceCMYK true def +currentdict end readonly def +/1_24_bitshift_1_sub 1 24 bitshift 1 sub def +/ReadFontProcs 10 dict def +/GetObject +{ +dup ObjectRegistryMaxLength idiv +//PDFReader/ObjectRegistry get exch knownget{ +exch knownget +}{ +pop false +}ifelse +}bind def +/PutObject +{ +1 index ObjectRegistryMaxLength idiv +//PDFReader/ObjectRegistry get 1 index knownget{ +exch pop +3 1 roll put +}{ +//PDFReader/ObjectRegistry get dup +begin +1 index ObjectRegistryMaxLength dict def +end +exch get +3 1 roll put +}ifelse +}bind def +/Register +{ +1 index GetObject{ +dup xcheck{ +4 3 roll pop +//PDFR_DEBUG{ +(Have a daemon for )print 2 index == +}if +exec +}{ +dup null ne{ +mark(The object )4 index(is already defined : )4 index//error exec +}{ +pop +}ifelse +3 2 roll +exec +}ifelse +}{ +3 2 roll +exec +}ifelse +PutObject +}bind def +/IsRegistered +{ +GetObject{ +null ne +}{ +false +}ifelse +}bind def +/GetRegistered +{ +dup GetObject not{ +exch mark exch(Object )exch( isn't defined before needed (1).)//error exec +}if +dup xcheck{ +exch mark exch(Object )exch( isn't defined before needed (2).)//error exec +}{ +dup null eq{ +exch mark exch(Object )exch( isn't defined before needed (3).)//error exec +}if +exch pop +}ifelse +}bind def +/StandardFontNames<< +/Times-Roman true +/Helvetica true +/Courier true +/Symbol true +/Times-Bold true +/Helvetica-Bold true +/Courier-Bold true +/ZapfDingbats true +/Times-Italic true +/Helvetica-Oblique true +/Courier-Oblique true +/Times-BoldItalic true +/Helvetica-BoldOblique true +/Courier-BoldOblique true +>>def +/CleanAllResources +{//PDFR_DEBUG{ +(CleanAllResources beg)= +}if +//PDFReader/ObjectRegistry get{ +dup length 0 exch 1 exch 1 sub{ +2 copy get dup xcheck{ +pop pop +}{ +dup null eq{ +pop pop +}{ +dup type/dicttype eq{/.Global known}{pop false}ifelse{ +pop +}{ +//PDFR_DEBUG{ +(Dropping )print dup = +}if +1 index exch/DroppedObject put +}ifelse +}ifelse +}ifelse +}for +pop +}forall +FontDirectory length dict begin +FontDirectory{ +pop +dup//StandardFontNames exch known not{ +dup null def +}if +pop +}forall +currentdict +end{ +pop +//PDFR_DEBUG{ +(Undefining font )print dup = +}if +undefinefont +}forall +//PDFR_DEBUG{ +(CleanAllResources end)= +}if +}bind def +/PrintReference +{ +//PDFR_DEBUG{ +({ )print +dup{ +=only( )print +}forall +( })= +}if +}bind def +/R +{ +0 ne{ +exch mark exch(A referred object generation )exch( isn't 0.)//error exec +}if +[ +exch//GetRegistered/exec load +]cvx +//PrintReference exec +}bind def +/IsObjRef +{ +dup type/arraytype eq{ +dup length 3 eq{ +dup xcheck exch +dup 0 get type/integertype eq 3 2 roll and exch +dup 1 get//GetRegistered eq 3 2 roll and exch +2 get/exec load eq and +}{ +pop false +}ifelse +}{ +pop false +}ifelse +}bind def +/DoNothing +{ +}def +/RunTypeDaemon +{ +dup type/dicttype eq{ +dup/Type//knownget exec{ +//PDFReader/TypeDaemons get exch +//knownget exec{ +exec +}if +}if +}if +}bind def +/obj +{ +//PDFR_DEBUG{ +(Defining )print 1 index =only( )print dup =only( obj)= +}if +0 ne{ +exch mark exch(An object generation )exch( isn't 0.)//error exec +}if +}bind def +/endobj +{ +//PDFR_DEBUG{ +(endobj )= +}if +count 1 eq{ +pop +}{ +dup type/dicttype eq{ +dup/.endobj_daemon//knownget exec{ +//PDFR_DEBUG{(.endobj_daemon for )print 2 index =}if +exec +}if +}if +dup type/dicttype eq{dup/ImmediateExec known}{false}ifelse{ +pop pop +}{ +//PDFR_DEBUG{ +(Storing )print 1 index = +}if +//RunTypeDaemon exec +//DoNothing 3 1 roll//Register exec +}ifelse +}ifelse +}bind def +/StoreBlock +{ +//PDFR_DEBUG{ +(StoreBlock )print//PDFReader/BlockCount get =only(, Length = )print dup length = +}if +dup length string copy +//PDFReader/BlockCount get exch +//PDFReader/CurrentObject get 3 1 roll +put +//PDFReader/BlockCount get 1 add +//PDFReader exch/BlockCount exch put +}bind def +/CheckLength +{dup type/integertype ne{ +mark(Object length isn't an integer.)//error exec +}if +}bind def +/ResolveD +{ +3 copy pop get +dup//IsObjRef exec{ +//PDFR_DEBUG{ +(Resolving )print//PrintReference exec +}if +exec +exch exec +}{ +exch pop +}ifelse +dup 4 1 roll +put +}bind def +/ResolveA +{2 index 2 index get +dup//IsObjRef exec{ +exec +exch exec +3 copy put +}{ +exch pop +}ifelse +exch pop exch pop +}bind def +/StoreStream +{ +dup//PDFReader exch/CurrentObject exch put +//PDFReader/BlockCount 0 put +dup/Length//CheckLength//ResolveD exec +//PDFR_DEBUG{ +(StoreStream Length = )print dup = +}if +currentfile exch()/SubFileDecode filter +{dup//BlockBuffer readstring{ +//StoreBlock exec +}{ +//StoreBlock exec +exit +}ifelse +}loop +pop +//PDFReader/CurrentObject null put +//PDFR_DEBUG{ +(StoreStream end.)= +}if +}bind def +/MakeStreamDumper +{ +//PDFR_DEBUG{ +(MakeStreamDumper beg.)= +}if +currentglobal exch dup gcheck setglobal +[exch +1 dict dup/c 0 put exch +1024 string +{readstring pop +(StreamDumper )print 1 index/c get =string cvs print( )print +dup length =string cvs print( <)print dup print(>\n)print +dup length +3 2 roll +dup/c get +3 2 roll +add/c exch put +}/exec load +] +cvx 0()/SubFileDecode filter +exch setglobal +//PDFR_DEBUG{ +(MakeStreamDumper end.)= +}if +}bind def +/ShortFilterNames 15 dict begin +/AHx/ASCIIHexDecode def +/A85/ASCII85Decode def +/LZW/LZWDecode def +/Fl/FlateDecode def +/RL/RunLengthDecode def +/CCF/CCITTFaxDecode def +/DCT/DCTDecode def +currentdict end readonly def +/AppendFilters +{ +//PDFR_DEBUG{ +(AppendFilters beg.)= +}if +dup 3 1 roll +/Filter//knownget exec{ +dup type/nametype eq{ +dup//ShortFilterNames exch//knownget exec{ +exch pop +}if +2 index/DecodeParms//knownget exec{ +exch +}if +filter +}{ +dup 0 exch 1 exch length 1 sub{ +2 copy get +dup//ShortFilterNames exch//knownget exec{ +exch pop +}if +3 1 roll +4 index/DecodeParms//knownget exec{ +exch get +}{ +pop null +}ifelse +dup null eq{ +pop 3 1 roll filter exch +}{ +3 1 roll +4 1 roll filter exch +}ifelse +}for +pop +}ifelse +//PDFR_DEBUG//PDFR_DUMP and{ +//MakeStreamDumper exec +}if +}if +exch pop +//PDFR_DEBUG{ +(AppendFilters end.)= +}if +}bind def +/ExecuteStream +{ +dup//PDFReader exch/CurrentObject exch put +dup/Length//CheckLength//ResolveD exec +//PDFR_DEBUG{ +(ExecuteStream id = )print 2 index =only( Length = )print dup = +}if +//PDFReader/InitialGraphicState get +//PDFReader/GraphicState get copy pop +//PDFReader/Operators get begin +currentfile exch()/SubFileDecode filter +1 index//AppendFilters exec +cvx mark exch +exec +counttomark 0 ne{ +mark(Data left on ostack after an immediate stream execution.)//error exec +}if +cleartomark +end +//PDFR_DEBUG{ +(ExecuteStream end.)= +}if +//PDFReader/CurrentObject null put +dup/IsPage known{ +dup/Context get/NumCopies//knownget exec{ +1 sub{ +copypage +}repeat +}if +EPS2Write not{showpage}if +pagesave restore +}if +}bind def +/stream +{ +//PDFR_DEBUG{ +1 index =only( stream)= +}if +1 index GetObject{ +dup xcheck{ +exec +1 index null PutObject +}{ +pop +}ifelse +}if +dup/ImmediateExec known{ +dup/GlobalExec//knownget exec{ +currentglobal 4 1 roll +setglobal +//ExecuteStream exec +3 2 roll setglobal +}{ +//ExecuteStream exec +}ifelse +}{ +//StoreStream exec +}ifelse +dup/.CleanResources//knownget exec{ +/All eq{ +//CleanAllResources exec +}if +}if +}bind def +/HookFont +{ +//PDFR_DEBUG{ +(Loaded the font )print dup/FontName get = +}if +{ +dup/FontFileType get dup/Type1 eq exch/MMType1 eq or{ +dup/FontName get +//PDFReader/RemoveFontNamePrefix get exec +findfont +exit +}if +dup/FontFileType get/TrueType eq{ +//PDFReader/MakeType42 get exec +//PDFR_DEBUG{ +(Font dict <<)= +dup{ +1 index/sfnts eq{ +exch pop +(/sfnts [)print +{ +(-string\()print length//=only exec(\)- )= +}forall +(])= +}{ +exch//=only exec( )print == +}ifelse +}forall +(>>)= +}if +dup/FontName get exch definefont +exit +}if +mark(FontHook has no proc for )2 index/FontFileType get//error exec +}loop +/Font exch put +}bind def +/endstream +{ +}bind def +/xref +{ +//PDFR_DEBUG{ +(xref)= +//PDFR_DUMP{ +//PDFReader/ObjectRegistry get == +}if +}if +end +count 0 ne{ +mark(Excessive data on estack at the end of the interpretation.)//error exec +}if +currentfile 1(%%EOF)/SubFileDecode filter +flushfile +cleardictstack +}bind def +/ResolveDict +{dup{ +pop 1 index exch +//DoNothing//ResolveD exec +pop +}forall +pop +}bind def +/SetupPageView +{ +//PDFR_DEBUG{ +(SetupPageView beg)= +}if +//DSC_OPDFREAD not{ +//GraphicState/InitialMatrix get setmatrix +}if +/MediaBox get aload pop +3 index neg 3 index neg translate +3 -1 roll sub 3 1 roll exch sub exch +userdict/.HWMargins//knownget exec{ +aload pop +}{ +currentpagedevice/.HWMargins//knownget exec{ +aload pop +}{ +0 0 0 0 +}ifelse +}ifelse +currentpagedevice/PageSize get aload pop +3 -1 roll sub 3 1 roll exch sub exch +exch 3 index sub exch 3 index sub +//SetPageSize{ +//PDFR_DEBUG{ +(Setting page size to )print 1 index//=only exec( )print dup = +}if +pop pop 3 index 3 index 2 copy +currentglobal false setglobal 3 1 roll +currentpagedevice dup/PageSize known{ +/PageSize get aload pop +}{ +0 0 +}ifelse +round cvi 2 index round cvi eq +exch round cvi 3 index round cvi eq and +{ +//PDFR_DEBUG{(PageSize matches request)== flush}if +pop pop +}{ +/MediaRequested where{ +//PDFR_DEBUG{(MediaRequested is true, check against new request)== flush}if +/MediaRequested get aload pop +round cvi 2 index round cvi eq +exch round cvi 3 index round cvi eq and +{ +//PDFR_DEBUG{(MediaRequested same as current request, ignore)== flush}if +pop pop false +}{ +//PDFR_DEBUG{(MediaRequested different to current request)== flush}if +true +}ifelse +}{ +//PDFR_DEBUG{(No MediaRequested yet)== flush}if +true +}ifelse +{ +//PDFR_DEBUG{(Setting pagesize)== flush}if +2 array astore +dup/MediaRequested exch def +<< exch/PageSize exch >>setpagedevice +}if +}ifelse +userdict/PDFR_InitialGS gstate put +setglobal +}if +//RotatePages{ +2 copy gt 6 index 6 index gt ne{ +1 index 5 index le 1 index 5 index le and not +}{ +false +}ifelse +}{ +false +}ifelse +{//CenterPages{ +//PDFR_DEBUG{ +(Rotating page, and then centering it)== +}if +90 rotate +0 5 index neg translate +5 index 1 index exch sub 2 div +2 index 6 index sub 2 div neg +translate +}{ +//FitPages{ +1 index 5 index div 1 index 7 index div +2 copy gt{ +exch +}if +pop dup scale +}if +90 rotate +0 5 index neg translate +}ifelse +}{ +//CenterPages{ +//PDFR_DEBUG{ +(Ccentering page)== +}if +1 index 6 index sub 2 div +1 index 6 index sub 2 div +translate +}{ +//FitPages{ +1 index 6 index div 1 index 6 index div +2 copy gt{ +exch +}if +pop dup scale +}if +}ifelse +}ifelse +pop pop +translate +pop pop +//PDFR_DEBUG{ +(SetupPageView end)= +}if +}bind def +/PageContentsDaemon +{ +//PDFR_DEBUG{ +(Executing PageContentsDaemon for )print 2 index = +}if +1 index exch/Context exch put +dup/ImmediateExec true put +/pagesave save def +dup/IsPage true put +SetPageSize{dup/Context get//SetupPageView exec}if +}bind def +/FontFileDaemon +{ +//PDFR_DEBUG{ +(Executing FontFileDaemon for )print 2 index = +}if +dup/FontFileType get +2 index exch +dup//ReadFontProcs exch//knownget exec{ +exch pop exec +}{ +mark(FontFile reader for )2 index( isn't implemented yet.)//error exec +}ifelse +//PDFR_DEBUG{ +(FontFileDaemon end)= +}if +pop +}bind def +/FontDescriptorDaemon +{ +//PDFR_DEBUG{ +(Executing FontDescriptorDaemon for )print 2 index = +}if +2 copy/FontResource exch put +/Subtype get 1 index exch/FontFileType exch put +}bind def +/UnPDFEscape{ +dup dup length string cvs +dup(#)search{ +{ +pop +(16#--)2 index 0 2 getinterval +1 index 3 2 getinterval copy pop +cvi +0 exch put +0 +1 index 2 1 index length 2 sub getinterval +3 copy putinterval +length +3 copy exch put +getinterval +(#)search not{ +pop exit +}if +}loop +(\0)search pop exch pop exch pop +cvn +exch pop +}{ +pop pop +}ifelse +}bind def +/TypeDaemons<< +/Page +{//PDFR_DEBUG{ +(Recognized a page.)= +}if +dup/Contents//knownget exec{ +0 get//DoNothing exch +[ +3 index//PageContentsDaemon/exec load +]cvx +//Register exec +}{ +(fixme: page with no Contents won't be printed.)= +}ifelse +}bind +/FontDescriptor +{//PDFR_DEBUG{ +(Recognized a font descriptor.)= +}if +dup/FontName//knownget exec{ +1 index/FontName 3 -1 roll//UnPDFEscape exec put +}if +dup dup/FontFile known{/FontFile}{/FontFile2}ifelse +//knownget exec{ +0 get//DoNothing exch +[ +3 index//FontFileDaemon/exec load +]cvx +//Register exec +}{ +(Font descriptor )print 1 index =only( has no FontFile.)= +}ifelse +}bind +/Font +{//PDFR_DEBUG{ +(Recognized a font resource.)= +}if +dup/BaseFont//knownget exec{ +//UnPDFEscape exec 2 copy/BaseFont exch put +//PDFReader/RemoveFontNamePrefix get exec +currentglobal exch +dup/Font resourcestatus{ +pop pop +//PDFReader/GetInstalledFont get exec pop +}{ +pop +}ifelse +setglobal +}if +dup/FontDescriptor//knownget exec{ +0 get +dup//IsRegistered exec{ +//PDFR_DEBUG{ +(already registered )print dup = +}if +pop +}{ +//DoNothing exch +[ +3 index//FontDescriptorDaemon/exec load +]cvx +//Register exec +}ifelse +}if +}bind +>>def +/MakeStreamReader +{dup +[ +exch +//PDFR_DEBUG{ +(Stream proc ) +/print load +//PDFR_STREAM{ +(<) +/print load +}if +}if +1 dict dup/i -1 put +/dup load +/i +/get load +1 +/add load +/dup load +3 +1 +/roll load +/i +/exch load +/put load +//knownget +/exec load +/not load +{()} +/if load +//PDFR_DEBUG{ +//PDFR_STREAM{ +/dup load +/print load +(>) +/print load +}if +( end of stream proc.\n) +/print load +}if +]cvx +//PDFR_DEBUG{ +(Stream reader )print dup == +}if +0()/SubFileDecode filter +exch//AppendFilters exec +}bind def +/RunDelayedStream +{ +//GraphicState/InitialTextMatrix get +//InitialTextMatrixStack//PDFReader/InitialTextMatrixStackPointer get +2 copy get null eq{ +2 copy currentglobal true setglobal matrix exch setglobal put +}if +get copy pop +//PDFReader/InitialTextMatrixStackPointer 2 copy get 1 add put +//MakeStreamReader exec +mark exch +cvx exec +counttomark 0 ne{ +mark(Data left on ostack after a delayed stream execution.)//error exec +}if +cleartomark +//PDFReader/InitialTextMatrixStackPointer 2 copy get 1 sub put +//InitialTextMatrixStack//PDFReader/InitialTextMatrixStackPointer get get +//GraphicState/InitialTextMatrix get +copy pop +}bind def +//ReadFontProcs begin +/Type1 +{//PDFR_DEBUG{ +(ReadFontProcs.Type1)= +}if +dup/.endobj_daemon[4 index//HookFont/exec load]cvx put +dup/ImmediateExec true put +/GlobalExec true put +}bind def +/MMType1//Type1 def +/TrueType +{//PDFR_DEBUG{ +(ReadFontProcs.TrueType)= +}if +dup/.endobj_daemon[4 index//HookFont/exec load]cvx put +pop +}bind def +end +/.opdloadttfontdict 50 dict def +.opdloadttfontdict begin +/maxstring 65400 def +end +/.InsertionSort +{ +/CompareProc exch def +/Array exch def +1 1 Array length 1 sub +{ +/Ix exch def +/Value1 Array Ix get def +/Jx Ix 1 sub def +{ +Jx 0 lt{ +exit +}if +/Value2 Array Jx get def +Value1 Value2 CompareProc{ +exit +}if +Array Jx 1 add Value2 put +/Jx Jx 1 sub def +}loop +Array Jx 1 add Value1 put +}for +Array +}bind def +/putu16{ +3 copy -8 bitshift put +exch 1 add exch 16#ff and put +}bind def +/putu32{ +3 copy -16 bitshift putu16 +exch 2 add exch 16#ffff and putu16 +}bind def +/.readtable{ +dup dup 1 and add string +dup 0 4 -1 roll getinterval +3 -1 roll exch +dup()ne{readstring}if pop pop +}bind def +/.readbigtable{ +dup maxstring lt{ +.readtable +}{ +currentuserparams/VMReclaim get -2 vmreclaim +[4 2 roll{ +dup maxstring le{exit}if +1 index maxstring string readstring pop 3 1 roll maxstring sub +}loop .readtable] +exch vmreclaim +}ifelse +}bind def +/ReadTTF +{ +.opdloadttfontdict begin +/TTFontFile exch def +/TableDir TTFontFile 12 string readstring pop def +/tables TTFontFile TableDir 4 getu16 16 mul string readstring pop def +/tabarray tables length 16 idiv array def +TableDir 0 4 getinterval(ttcf)eq{ +QUIET not{(Can't handle TrueType font Collections.)=}if +/.loadttfonttables cvx/invalidfont signalerror +}{ +0 16 tables length 1 sub{ +dup +tables exch 16 getinterval +exch 16 div cvi exch +tabarray 3 1 roll put +}for +}ifelse +tabarray{exch 8 getu32 exch 8 getu32 gt}.InsertionSort pop +/Read TableDir length tables length add def +/tabs[ +tabarray{ +dup 8 getu32 +Read sub +dup 0 gt{ +dup string TTFontFile exch readstring pop pop +Read add/Read exch def +}{ +pop +}ifelse +12 getu32 +dup Read add +/Read exch def +TTFontFile exch .readbigtable +}forall +]def +end +}bind def +/GetLocaType +{ +0 1 tabarray length 1 sub{ +dup tabarray exch get +0 4 getinterval(head)eq{ +tabs exch get +50 gets16 +/LocaType exch def +exit +}{ +pop +}ifelse +}for +}bind def +/GetNumGlyphs +{ +0 1 tabarray length 1 sub{ +dup tabarray exch get +0 4 getinterval(maxp)eq{ +tabs exch get +4 getu16 +/NumGlyphs exch def +exit +}{ +pop +}ifelse +}for +}bind def +/StringToLoca +{ +/LocaIndex exch def +/StringOffset 0 def +{ +dup length StringOffset gt{ +dup +LocaType 1 eq{ +StringOffset getu32 +LocaArray LocaIndex 3 -1 roll put +/LocaIndex LocaIndex 1 add def +/StringOffset StringOffset 4 add +def +}{ +StringOffset getu16 2 mul +LocaArray length LocaIndex gt{ +LocaArray LocaIndex 3 -1 roll put +}{ +pop +}ifelse +/LocaIndex LocaIndex 1 add def +/StringOffset StringOffset 2 add +def +}ifelse +}{ +pop +LocaIndex +exit +}ifelse +}loop +}bind def +/GetSortedLoca +{ +NumGlyphs 1 add array/LocaArray exch def +0 1 tabarray length 1 sub{ +dup tabarray exch get +0 4 getinterval(loca)eq{ +tabs exch get +exit +}{ +pop +}ifelse +}for +dup type/stringtype eq{ +0 StringToLoca pop +}{ +0 exch +{ +exch StringToLoca +}forall +pop +}ifelse +LocaArray{gt}.InsertionSort pop +}bind def +/GetWorkingString +{ +WorkString 0 +GlyfArray GlyfStringIndex get +putinterval +/WorkBytes GlyfArray GlyfStringIndex get length def +/GlyfStringIndex GlyfStringIndex 1 add def +}bind def +/GetWorkingBytes +{ +/BytesToRead exch def +WorkString 0 BytesToRead getinterval +dup length string copy +WorkString BytesToRead WorkBytes BytesToRead sub getinterval +dup length string copy +WorkString 0 3 -1 roll putinterval +/WorkBytes WorkBytes BytesToRead sub def +}bind def +/GetGlyfBytes +{ +/ToRead exch def +WorkBytes 0 eq{ +GetWorkingString +}if +WorkBytes ToRead ge{ +ToRead string dup 0 +ToRead GetWorkingBytes putinterval +}{ +ToRead string +dup +0 +WorkString 0 WorkBytes getinterval +putinterval +dup +WorkBytes +ToRead WorkBytes sub +GetWorkingString +GetWorkingBytes +putinterval +}ifelse +}bind def +/SplitGlyf +{ +/GlyfArray exch def +/DestArray GlyfArray length 2 mul array def +/DestArrayIndex 0 def +/LastLoca 0 def +/NextLocaIndex 0 def +/LastLocaIndex 0 def +/GlyfStringIndex 0 def +/WorkString maxstring string def +/WorkBytes 0 def +{ +LocaArray NextLocaIndex get +LastLoca sub maxstring gt +{ +LocaArray LastLocaIndex get LastLoca sub +GetGlyfBytes +DestArray DestArrayIndex 3 -1 roll put +/DestArrayIndex DestArrayIndex 1 add def +LocaArray LastLocaIndex get/LastLoca exch def +}{ +/LastLocaIndex NextLocaIndex def +/NextLocaIndex NextLocaIndex 1 add def +NextLocaIndex NumGlyphs gt +{ +WorkBytes +GlyfStringIndex GlyfArray length lt{ +GlyfArray GlyfStringIndex get length +add string dup +0 +WorkString 0 WorkBytes getinterval +putinterval +dup +WorkBytes +GetWorkingString +WorkString 0 WorkBytes getinterval +putinterval +}{ +pop +WorkString 0 WorkBytes getinterval +}ifelse +dup length string copy +DestArray DestArrayIndex 3 -1 roll put +exit +}if +}ifelse +}loop +DestArray +}bind def +/ProcessTTData +{ +.opdloadttfontdict begin +0 1 tabarray length 1 sub{ +/ix exch def +tabarray ix get +12 getu32 dup maxstring le{ +dup 4 mod 0 ne{ +4 div cvi 1 add 4 mul string/newstring exch def +/oldstring tabs ix get def +newstring 0 oldstring putinterval +0 1 newstring length oldstring length sub 1 sub{ +newstring exch oldstring length add 0 put +}for +tabs ix newstring put +}{ +pop +}ifelse +}{ +dup 4 mod 0 ne{ +dup maxstring idiv maxstring mul sub +4 idiv 1 add 4 mul string/newstring exch def +tabs ix get +dup length 1 sub dup/iy exch def get/oldstring exch def +newstring 0 oldstring putinterval +0 1 newstring length oldstring length sub 1 sub{ +newstring exch oldstring length add 0 put +}for +tabs ix get iy newstring put +}{ +pop +}ifelse +}ifelse +}for +0 1 tabarray length 1 sub{ +dup tabarray exch get +dup 12 getu32 maxstring gt{ +0 4 getinterval dup(glyf)eq{ +pop +GetLocaType +GetNumGlyphs +GetSortedLoca +dup tabs exch get +SplitGlyf +tabs 3 1 roll put +}{ +(Warning, table )print print( > 64Kb\n)print +pop +}ifelse +}{ +pop +pop +}ifelse +}for +end +}bind def +/Makesfnts +{ +.opdloadttfontdict begin +0 +tabs{ +dup type/stringtype eq{ +pop +1 add +}{ +{ +type/stringtype eq{ +1 add +}if +}forall +}ifelse +}forall +1 add +/TTOffset +TableDir length +tabarray length 16 mul add +def +0 +tabarray{ +exch dup 1 add +3 1 roll +dup +tabs exch get +dup type/stringtype eq{ +length +2 index exch +TTOffset +dup 3 1 roll add +/TTOffset exch def +8 exch putu32 +exch tabarray 3 1 roll +put +}{ +0 exch +{ +dup type/stringtype eq{ +length add +}{ +pop +}ifelse +}forall +2 index exch +TTOffset +dup 3 1 roll add +/TTOffset exch def +8 exch putu32 +exch tabarray 3 1 roll +put +}ifelse +}forall +pop +array +dup 0 +TableDir length +tables length add +string +dup 0 TableDir putinterval +dup 12 tables putinterval +put +dup +/ix 1 def +tabs{ +dup type/stringtype eq{ +ix exch +put dup +/ix ix 1 add def +}{ +{ +dup type/stringtype eq{ +ix exch put dup +/ix ix 1 add def +}{ +pop +}ifelse +}forall +}ifelse +}forall +pop +end +}bind def +/MakeType42 +{ +//PDFR_DEBUG{ +(MakeType42 beg)= +}if +10 dict begin +/FontName 1 index/FontName get def +/FontType 42 def +/FontMatrix[1 0 0 1 0 0]def +/FontBBox 1 index/FontBBox get def +dup/FontResource get +dup/Encoding known{ +//PDFReader/ObtainEncoding get exec +/Encoding get +}{ +pop null +}ifelse +/PDFEncoding exch def +/CharStrings 2 index//PDFReader/MakeTTCharStrings get exec def +/sfnts 2 index//MakeStreamReader exec +ReadTTF +ProcessTTData +Makesfnts +def +/Encoding StandardEncoding def +/PaintType 0 def +currentdict end +//PDFR_DEBUG{ +(MakeType42 end)= +}if +}bind def +/GetInstalledFont +{ +dup//InstalledFonts exch knownget{ +exch pop +}{ +dup findfont dup 3 1 roll +//InstalledFonts 3 1 roll put +}ifelse +}bind def +/RemoveFontNamePrefix +{//=string cvs true +0 1 5{ +2 index exch get//IsUpper exec not{ +pop false exit +}if +}for +{(+)search{ +pop pop +}if +}if +cvn +}bind def +/CheckFont +{dup/Type get/Font ne{ +mark(Resource )3 index( must have /Type/Font .)//error exec +}if +}bind def +/CheckEncoding +{dup type/nametype ne{ +dup/Type get/Encoding ne{ +mark(Resource )3 index( must have /Type/Encoding .)//error exec +}if +}if +}bind def +/ObtainEncoding +{dup/Encoding known{ +dup dup/Encoding//CheckEncoding//ResolveD exec +dup type dup/arraytype eq exch/packedarraytype eq or{ +pop pop +}{ +dup type/nametype eq{ +/Encoding findresource +}{ +dup/BaseEncoding//knownget exec not{ +/StandardEncoding +}if +/Encoding findresource +exch +/Differences//knownget exec{ +exch dup length array copy exch +0 exch +{ +dup type/integertype eq{ +exch pop +}{ +3 copy put pop +1 add +}ifelse +}forall +pop +}if +}ifelse +/Encoding exch put +}ifelse +}{ +dup/Encoding/StandardEncoding/Encoding findresource put +}ifelse +}bind def +/ObtainMetrics +{dup/Widths//knownget exec{ +1 index/Encoding get +256 dict +3 index/Subtype get/TrueType eq{ +1000 +}{ +1 +}ifelse +4 index/MissingWidth//knownget exec not{ +0 +}if +5 index/FirstChar//knownget exec not{ +0 +}if +6 5 roll +dup 0 exch 1 exch length 1 sub{ +2 copy get +exch 3 index add +7 index exch get +dup dup null ne exch/.notdef ne and{ +6 index 3 1 roll exch +6 index div +3 copy pop//knownget exec{ +0 eq +}{ +true +}ifelse +{put +}{ +pop pop pop +}ifelse +}{ +pop pop +}ifelse +}for +pop pop pop pop exch pop +1 index exch/Metrics exch put +}{ +dup/MissingWidth//knownget exec{ +256 dict +2 index/Encoding get{ +dup null ne{ +3 copy 3 2 roll put +}if +pop +}forall +exch pop +1 index exch/Metrics exch put +}if +}ifelse +}bind def +/NotDef +{ +FontMatrix aload pop pop pop exch pop exch pop +1 exch div exch +1 exch div exch +1 index 0 setcharwidth +0 setlinewidth +0 0 moveto +2 copy rlineto +1 index 0 rlineto +neg exch neg exch rlineto +closepath stroke +}bind def +/SaveResourcesToStack +{ +[ +//PDFReader/OldResources known{ +//PDFReader/OldResources get +}{ +null +}ifelse +//PDFReader/CurrentObject get/Context get/Resources get +] +//PDFReader/OldResources 3 -1 roll put +}bind def +/RestoreResourcesFromStack +{ +//PDFReader/OldResources get dup +0 get//PDFReader/OldResources 3 -1 roll put +1 get//PDFReader/CurrentObject get/Context get/Resources 3 -1 roll put +}bind def +/BuildChar +{//PDFR_DEBUG{ +(BuildChar )print dup//=only exec( )print +}if +exch begin +Encoding exch get +//PDFR_DEBUG{ +dup = +}if +dup null eq{ +pop//NotDef exec +} +{ +CharProcs exch//knownget exec +{ +currentfont/Font get/Resources//knownget exec{ +exec +SaveResourcesToStack +//PDFReader/CurrentObject get/Context get +/Resources 3 -1 roll put +//RunDelayedStream exec +RestoreResourcesFromStack +}{ +//RunDelayedStream exec +}ifelse +} +{ +//NotDef exec +}ifelse +}ifelse +end +}bind def +/printdict +{(<<)= +{exch = ==}forall +(>>)= +}bind def +/printfont +{ +dup{ +exch dup = +dup/Encoding eq{ +pop = +}{ +dup/FontInfo eq exch/Private eq or{ +//printdict exec +}{ +== +}ifelse +}ifelse +}forall +}bind def +/ScaleMetrics +{1 index{ +2 index div +3 index +3 1 roll put +}forall +pop +}bind def +/ResolveAndSetFontAux +{exch dup +//PDFReader/CurrentObject get/Context get/Resources get +/Font//DoNothing//ResolveD exec +exch//CheckFont//ResolveD exec +dup/Font//knownget exec{ +exch pop exch pop +}{ +{ +dup/Subtype get dup dup/Type1 eq exch/TrueType eq or exch/MMType1 eq or{ +exch pop +dup/BaseFont get +//RemoveFontNamePrefix exec +//PDFR_DEBUG{ +(Font )print dup = +}if +1 index/FontDescriptor known{ +//PDFR_DEBUG{ +(Font from a font descriptor.)= +}if +1 index +/FontDescriptor//DoNothing//ResolveD exec +/Font//knownget exec{ +exch pop +}{ +//PDFR_DEBUG{ +(Font descriptor has no Font resolved.)= +}if +//GetInstalledFont exec +}ifelse +}{ +//GetInstalledFont exec +}ifelse +exch +dup/Encoding known not{ +1 index/Encoding get 1 index exch/Encoding exch put +}if +//ObtainEncoding exec +//ObtainMetrics exec +exch +dup length dict copy +dup 2 index/Encoding get +/Encoding exch put +1 index/Metrics//knownget exec{ +2 index/Subtype get/TrueType ne{ +1 index/FontMatrix get 0 get +dup 0 eq{ +pop +1 index/FontMatrix get 1 get +dup 0 eq{pop 1}if +}if +0.001 div +//ScaleMetrics exec +}{ +1 index/sfnts known not{ +1 index/FontMatrix get 0 get +dup 0 eq{ +pop +1 index/FontMatrix get 1 get +dup 0 eq{pop 1}if +}if +//ScaleMetrics exec +}if +}ifelse +1 index exch/Metrics exch put +}if +1 index/BaseFont get +exch +dup/FID undef +dup/UniqueID undef +definefont +dup 3 1 roll +/Font exch put +exit +}if +dup/Subtype get/Type3 eq{ +//ObtainEncoding exec +2 copy exch/FontName exch put +dup/CharProcs get//ResolveDict exec +dup/FontType 3 put +dup/BuildChar//BuildChar put +dup dup/Font exch put +dup 3 1 roll +definefont +2 copy ne{ +2 copy/Font exch put +}if +exch pop +exit +}if +dup/Subtype get/Type0 eq{ +}if +dup/Subtype get/CIDFontType0 eq{ +}if +dup/Subtype get/CIDFontType2 eq{ +}if +mark(Unknown font type )2 index/Subtype get//error exec +}loop +}ifelse +exch scalefont setfont +}bind def +/ResolveAndSetFont +{ +//ResolveAndSetFontAux exec +}bind def +/.knownget +{2 copy known{ +get true +}{ +pop pop false +}ifelse +}bind def +/.min +{2 copy lt{ +exch +}if +pop +}bind def +/.max +{2 copy gt{ +exch +}if +pop +}bind def +/.dicttomark +{>> +}bind def +/getu16{ +2 copy get 8 bitshift 3 1 roll 1 add get add +}bind def +/gets16{ +getu16 16#8000 xor 16#8000 sub +}bind def +/getu32{ +2 copy getu16 16 bitshift 3 1 roll 2 add getu16 add +}bind def +/gets32{ +2 copy gets16 16 bitshift 3 1 roll 2 add getu16 add +}bind def +/cmapformats mark +0{ +6 256 getinterval{}forall 256 packedarray +}bind +2{ +/sHK_sz 2 def +/sH_sz 8 def +dup 2 getu16/cmapf2_tblen exch def +dup 4 getu16/cmapf2_lang exch def +dup 6 256 sHK_sz mul getinterval/sHKs exch def +0 +0 1 255{ +sHKs exch +2 mul getu16 +1 index +1 index +lt{exch}if pop +}for +/sH_len exch def +dup 6 256 sHK_sz mul add +cmapf2_tblen 1 index sub getinterval +/sH_gIA exch def +/cmapf2_glyph_array 65535 array def +/.cmapf2_putGID{ +/cmapf2_ch cmapf2_ch_hi 8 bitshift cmapf2_ch_lo add def +firstCode cmapf2_ch_lo le +cmapf2_ch_lo firstCode entryCount add lt +and{ +sH_offset idRangeOffset add +cmapf2_ch_lo firstCode sub 2 mul +add 6 add +sH_gIA exch getu16 +dup 0 gt{ +idDelta add +cmapf2_glyph_array exch cmapf2_ch exch put +}{ +pop +}ifelse +}{ +}ifelse +}def +16#00 1 16#ff{ +/cmapf2_ch_hi exch def +sHKs cmapf2_ch_hi sHK_sz mul getu16 +/sH_offset exch def +sH_gIA sH_offset sH_sz getinterval +dup 0 getu16/firstCode exch def +dup 2 getu16/entryCount exch def +dup 4 gets16/idDelta exch def +dup 6 getu16/idRangeOffset exch def +pop +sH_offset 0 eq{ +/cmapf2_ch_lo cmapf2_ch_hi def +/cmapf2_ch_hi 0 def +.cmapf2_putGID +}{ +16#00 1 16#ff{ +/cmapf2_ch_lo exch def +.cmapf2_putGID +}for +}ifelse +}for +pop +0 1 cmapf2_glyph_array length 1 sub{ +dup cmapf2_glyph_array exch get +null eq{cmapf2_glyph_array exch 0 put}{pop}ifelse +}for +cmapf2_glyph_array +}bind +4{ +/etab exch def +/nseg2 etab 6 getu16 def +14/endc etab 2 index nseg2 getinterval def +2 add +nseg2 add/startc etab 2 index nseg2 getinterval def +nseg2 add/iddelta etab 2 index nseg2 getinterval def +nseg2 add/idroff etab 2 index nseg2 getinterval def +pop +/firstcode startc 0 getu16 16#ff00 and dup 16#f000 ne{pop 0}if def +/lastcode firstcode def +/striptopbyte false def +/putglyph{ +glyphs code 3 -1 roll put/code code 1 add def +}bind def +/numcodes 0 def/glyphs 0 0 2 nseg2 3 sub{ +/i2 exch def +/scode startc i2 getu16 def +/ecode endc i2 getu16 def +ecode lastcode gt{ +/lastcode ecode def +}if +}for pop +firstcode 16#f000 ge lastcode firstcode sub 255 le and{ +lastcode 255 and +/striptopbyte true def +}{ +lastcode +}ifelse +1 add +array def +glyphs length 1024 ge{ +.array1024z 0 1024 glyphs length 1023 sub{glyphs exch 2 index putinterval}for +glyphs dup length 1024 sub 3 -1 roll +putinterval +}{ +0 1 glyphs length 1 sub{glyphs exch 0 put}for +}ifelse +/numcodes 0 def/code 0 def +0 2 nseg2 3 sub{ +/i2 exch def +/scode startc i2 getu16 def +/ecode endc i2 getu16 def +numcodes scode firstcode sub +exch sub 0 .max dup/code exch code exch add def +ecode scode sub 1 add add numcodes add/numcodes exch def +/delta iddelta i2 gets16 def +TTFDEBUG{ +(scode=)print scode =only +( ecode=)print ecode =only +( delta=)print delta =only +( droff=)print idroff i2 getu16 = +}if +idroff i2 getu16 dup 0 eq{ +pop scode delta add 65535 and 1 ecode delta add 65535 and +striptopbyte{ +/code scode 255 and def +}{ +/code scode def +}ifelse +{putglyph}for +}{ +/gloff exch 14 nseg2 3 mul add 2 add i2 add add def +striptopbyte{ +/code scode 255 and def +}{ +/code scode def +}ifelse +0 1 ecode scode sub{ +2 mul gloff add etab exch getu16 +dup 0 ne{delta add 65535 and}if putglyph +}for +}ifelse +}for glyphs/glyphs null def +}bind +6{ +dup 6 getu16/firstcode exch def dup 8 getu16/ng exch def +firstcode ng add array +0 1 firstcode 1 sub{2 copy 0 put pop}for +dup firstcode ng getinterval +0 1 ng 1 sub{ +dup 2 mul 10 add 4 index exch getu16 3 copy put pop pop +}for pop exch pop +}bind +.dicttomark readonly def +/cmaparray{ +dup 0 getu16 cmapformats exch .knownget{ +TTFDEBUG{ +(cmap: format )print 1 index 0 getu16 = flush +}if exec +}{ +(Can't handle format )print 0 getu16 = flush +0 1 255{}for 256 packedarray +}ifelse +TTFDEBUG{ +(cmap: length=)print dup length = dup == +}if +}bind def +/postremap mark +/Cdot/Cdotaccent +/Edot/Edotaccent +/Eoverdot/Edotaccent +/Gdot/Gdotaccent +/Ldot/Ldotaccent +/Zdot/Zdotaccent +/cdot/cdotaccent +/edot/edotaccent +/eoverdot/edotaccent +/gdot/gdotaccent +/ldot/ldotaccent +/zdot/zdotaccent +.dicttomark readonly def +/get_from_stringarray +{1 index type/stringtype eq{ +get +}{ +exch{ +2 copy length ge{ +length sub +}{ +exch get exit +}ifelse +}forall +}ifelse +}bind def +/getinterval_from_stringarray +{ +2 index type/stringtype eq{ +getinterval +}{ +string exch 0 +4 3 roll{ +dup length +dup 4 index lt{ +3 index exch sub +exch pop 3 1 roll exch pop +}{ +dup 3 1 roll +4 index sub +5 index length 4 index sub +2 copy gt{exch}if pop +dup 3 1 roll +5 index exch getinterval +5 index 4 index 3 index +getinterval +copy pop +exch pop add exch pop 0 exch +dup 3 index length ge{exit}if +}ifelse +}forall +pop pop +}ifelse +}bind def +/string_array_size +{dup type/stringtype eq{ +length +}{ +0 exch{length add}forall +}ifelse +}bind def +/postformats mark +16#00010000{ +pop MacGlyphEncoding +} +16#00020000{ +dup dup type/arraytype eq{0 get}if length 36 lt{ +TTFDEBUG{(post format 2.0 invalid.)= flush}if +pop[] +}{ +/postglyphs exch def +/post_first postglyphs dup type/arraytype eq{0 get}if def +post_first 32 getu16/numglyphs exch def +/glyphnames numglyphs 2 mul 34 add def +/postpos glyphnames def +/total_length postglyphs//string_array_size exec def +numglyphs array 0 1 numglyphs 1 sub{ +postpos total_length ge{ +1 numglyphs 1 sub{1 index exch/.notdef put}for +exit +}if +postglyphs postpos//get_from_stringarray exec +postglyphs postpos 1 add 2 index//getinterval_from_stringarray exec cvn +exch postpos add 1 add/postpos exch def +2 index 3 1 roll +put +}for +/postnames exch def +numglyphs array 0 1 numglyphs 1 sub{ +dup 2 mul 34 add postglyphs exch 2//getinterval_from_stringarray exec +dup 0 get 8 bitshift exch 1 get add dup 258 lt{ +MacGlyphEncoding exch get +}{ +dup 32768 ge{ +pop/.notdef +}{ +258 sub dup postnames length ge{ +TTFDEBUG{( *** warning: glyph index past end of 'post' table)= flush}if +pop +exit +}if +postnames exch get +postremap 1 index .knownget{exch pop}if +}ifelse +}ifelse +2 index 3 1 roll put +}for +} +ifelse +}bind +16#00030000{ +pop[] +}bind +.dicttomark readonly def +/first_post_string +{ +post dup type/arraytype eq{0 get}if +}bind def +/.getpost{ +/glyphencoding post null eq{ +TTFDEBUG{(post missing)= flush}if[] +}{ +postformats first_post_string 0 getu32 .knownget{ +TTFDEBUG{ +(post: format )print +first_post_string +dup 0 getu16 =only(,)print 2 getu16 = flush +}if +post exch exec +}{ +TTFDEBUG{(post: unknown format )print post 0 getu32 = flush}if[] +}ifelse +}ifelse def +}bind def +/MacRomanEncoding[ +StandardEncoding 0 39 getinterval aload pop +/quotesingle +StandardEncoding 40 56 getinterval aload pop +/grave +StandardEncoding 97 31 getinterval aload pop +/Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute +/agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave +/ecircumflex/edieresis/iacute/igrave +/icircumflex/idieresis/ntilde/oacute +/ograve/ocircumflex/odieresis/otilde +/uacute/ugrave/ucircumflex/udieresis +/dagger/degree/cent/sterling/section/bullet/paragraph/germandbls +/registered/copyright/trademark/acute/dieresis/.notdef/AE/Oslash +/.notdef/plusminus/.notdef/.notdef/yen/mu/.notdef/.notdef +/.notdef/.notdef/.notdef/ordfeminine/ordmasculine/.notdef/ae/oslash +/questiondown/exclamdown/logicalnot/.notdef +/florin/.notdef/.notdef/guillemotleft +/guillemotright/ellipsis/space/Agrave/Atilde/Otilde/OE/oe +/endash/emdash/quotedblleft/quotedblright +/quoteleft/quoteright/divide/.notdef +/ydieresis/Ydieresis/fraction/currency +/guilsinglleft/guilsinglright/fi/fl +/daggerdbl/periodcentered/quotesinglbase/quotedblbase +/perthousand/Acircumflex/Ecircumflex/Aacute +/Edieresis/Egrave/Iacute/Icircumflex +/Idieresis/Igrave/Oacute/Ocircumflex +/.notdef/Ograve/Uacute/Ucircumflex +/Ugrave/dotlessi/circumflex/tilde +/macron/breve/dotaccent/ring/cedilla/hungarumlaut/ogonek/caron +]/Encoding defineresource pop +/TTParser<< +/Pos 0 +/post null +>>def +/readu8 +{read not{ +mark(Insufficient data in the stream.)//error exec +}if +}bind def +/readu16 +{dup//readu8 exec 8 bitshift exch//readu8 exec or +}bind def +/reads16 +{//readu16 exec 16#8000 xor 16#8000 sub +}bind def +/readu32 +{dup//readu16 exec 16 bitshift exch//readu16 exec or +}bind def +/reads32 +{dup//reads16 exec 16 bitshift exch//readu16 exec or +}bind def +/SkipToPosition +{dup//TTParser/Pos get +exch//TTParser exch/Pos exch put +sub +//PDFR_DEBUG{ +(Skipping )print dup//=only exec( bytes.)= +}if +dup 0 eq{ +pop pop +}{ +dup 3 1 roll +()/SubFileDecode filter +exch +{1 index//BlockBuffer readstring pop length +dup 0 eq{pop exch pop exit}if +sub +}loop +0 ne{ +mark(Insufficient data in the stream for SkipToPosition.)//error exec +}if +}ifelse +}bind def +/TagBuffer 4 string def +/ParseTTTableDirectory +{//PDFR_DEBUG{ +(ParseTTTableDirectory beg)= +}if +15 dict begin +dup//readu32 exec 16#00010000 ne{ +mark(Unknown True Type version.)//error exec +}if +dup//readu16 exec/NumTables exch def +dup//readu16 exec/SearchRange exch def +dup//readu16 exec/EntrySelector exch def +dup//readu16 exec/RangeShift exch def +//PDFR_DEBUG{ +(NumTables = )print NumTables = +}if +NumTables{ +dup//TagBuffer readstring not{ +mark(Could not read TT tag.)//error exec +}if +cvn +[2 index//readu32 exec pop +2 index//readu32 exec +3 index//readu32 exec +] +//PDFR_DEBUG{ +2 copy exch//=only exec( )print == +}if +def +}repeat +pop +//TTParser/Pos 12 NumTables 16 mul add put +currentdict end +//PDFR_DEBUG{ +(ParseTTTableDirectory end)= +}if +}bind def +/ParseTTcmap +{//PDFR_DEBUG{ +(ParseTTcmap beg)= +}if +/cmap get aload pop +3 1 roll +7 dict begin +//PDFR_DEBUG{ +(Current position = )print//TTParser/Pos get = +(cmap position = )print dup = +}if +1 index exch//SkipToPosition exec +//TTParser/Pos get/TablePos exch def +dup//readu16 exec pop +dup//readu16 exec/NumEncodings exch def +//PDFR_DEBUG{ +(NumEncodings = )print NumEncodings = +}if +null +NumEncodings{ +1 index//readu32 exec +2 index//readu32 exec +3 array dup 3 2 roll 0 exch put +2 index null ne{ +dup 0 get 3 index 0 get sub +3 index exch 1 exch put +}if +dup 4 3 roll pop 3 1 roll +def +}repeat +dup 0 get +4 3 roll exch sub +1 exch put +//PDFR_DEBUG{ +currentdict{ +exch dup type/integertype eq{ +//PrintHex exec( )print == +}{ +pop pop +}ifelse +}forall +}if +4 NumEncodings 8 mul add/HeaderLength exch def +//TTParser/Pos//TTParser/Pos get HeaderLength add put +0 +NumEncodings{ +16#7FFFFFF null +currentdict{ +1 index type/integertype eq{ +exch pop dup 0 get +dup 5 index gt{ +dup 4 index lt{ +4 1 roll +exch pop exch pop +}{ +pop pop +}ifelse +}{ +pop pop +}ifelse +}{ +pop pop +}ifelse +}forall +//PDFR_DEBUG{ +(Obtaining subtable for )print dup == +}if +3 2 roll pop +3 copy pop +TablePos add//SkipToPosition exec +3 copy exch pop 1 get +//TTParser/Pos//TTParser/Pos get 3 index add put +string +readstring not{ +mark(Can't read a cmap subtable.)//error exec +}if +2 exch put +}repeat +pop pop +currentdict end +//PDFR_DEBUG{ +(ParseTTcmap end)= +}if +}bind def +/GetTTEncoding +{//PDFR_DEBUG{ +(GetTTEncoding beg)= +}if +get +exch pop +2 get +10 dict begin +/TTFDEBUG//PDFR_DEBUG def +//cmaparray exec +end +//PDFR_DEBUG{ +(GetTTEncoding end)= +dup == +}if +}bind def +/InverseEncoding +{ +256 dict begin +dup length 1 sub -1 0{ +2 copy get +exch +1 index currentdict exch//knownget exec{ +dup type/arraytype eq{ +aload length 1 add array astore +}{ +2 array astore +}ifelse +}if +def +}for +pop +currentdict end +}bind def +/GetMacRomanEncodingInverse +{//PDFReader/MacRomanEncodingInverse get +dup null eq{ +pop +MacRomanEncoding//InverseEncoding exec +dup//PDFReader exch/MacRomanEncodingInverse exch put +}if +}bind def +/PutCharStringSingle +{ +dup 3 index length lt{ +2 index exch get +dup 0 ne{ +def +}{ +pop pop +}ifelse +}{ +pop pop +}ifelse +}bind def +/PutCharString +{1 index type/nametype ne{ +mark(Bad charstring name)//error exec +}if +dup type/arraytype eq{ +{ +3 copy//PutCharStringSingle exec +pop pop +}forall +pop +}{ +//PutCharStringSingle exec +}ifelse +}bind def +/ComposeCharStrings +{ +//PDFR_DEBUG{ +(ComposeCharStrings beg)= +}if +1 index length 1 add dict begin +/.notdef 0 def +exch +//TTParser/post get +dup null ne{ +exch +1 index length 1 sub -1 0{ +dup 3 index exch get exch +dup 0 eq 2 index/.notdef eq or{ +pop pop +}{ +def +}ifelse +}for +}if +exch pop exch +{ +//PutCharString exec +}forall +pop +currentdict end +//PDFR_DEBUG{ +(ComposeCharStrings end)= +}if +}bind def +/ParseTTpost +{ +//PDFR_DEBUG{ +(ParseTTpost beg)= +}if +/post get aload pop +3 1 roll +//PDFR_DEBUG{ +(Current position = )print//TTParser/Pos get = +(post position = )print dup = +}if +1 index exch//SkipToPosition exec +//TTParser/Pos//TTParser/Pos get 4 index add put +exch dup 65535 le{ +string +readstring not{ +mark(Insufficient data in the stream for ParseTTpost.)//error exec +}if +}{ +[3 1 roll +dup 16384 div floor cvi +exch 1 index 16384 mul +sub exch +1 sub 0 1 3 -1 roll +{ +1 add index +16384 string readstring not{ +mark(Insufficient data in the stream for ParseTTpost.)//error exec +}if +}for +counttomark -2 roll +string readstring not{ +mark(Insufficient data in the stream for ParseTTpost.)//error exec +}if +] +}ifelse +1 dict begin +/post exch def +//.getpost exec +//TTParser/post glyphencoding put +//PDFR_DEBUG{ +(ParseTTpost end)= +glyphencoding == +}if +end +}bind def +/MakeTTCharStrings +{//MakeStreamReader exec +dup dup//ParseTTTableDirectory exec +//TTParser/post null put +dup/post//knownget exec{ +0 get +1 index/cmap get 0 get +lt{ +2 copy//ParseTTpost exec +//ParseTTcmap exec +}{ +2 copy//ParseTTcmap exec +3 1 roll +//ParseTTpost exec +}ifelse +}{ +//ParseTTcmap exec +}ifelse +{ +dup 16#00030001 known{ +//PDFR_DEBUG{ +(Using the TT cmap encoding for Windows Unicode.)= +}if +16#00030001//GetTTEncoding exec +AdobeGlyphList//ComposeCharStrings exec +exit +}if +dup 16#00010000 known{ +//PDFR_DEBUG{ +(Using the TT cmap encoding for Macintosh Roman.)= +}if +16#00010000//GetTTEncoding exec +PDFEncoding dup null eq{ +pop//GetMacRomanEncodingInverse exec +}{ +//InverseEncoding exec +}ifelse +//ComposeCharStrings exec +exit +}if +dup 16#00030000 known{ +//PDFR_DEBUG{ +(Using the TT cmap encoding 3.0 - not sure why Ghostscript writes it since old versions.)= +}if +16#00030000//GetTTEncoding exec +PDFEncoding dup null eq{ +pop//GetMacRomanEncodingInverse exec +}{ +//InverseEncoding exec +}ifelse +//ComposeCharStrings exec +exit +}if +mark(True Type cmap has no useful encodings.)//error exec +}loop +//PDFR_DEBUG{ +(CharStrings <<)= +dup{ +exch +dup type/nametype eq{ +//=only exec +}{ +== +}ifelse +( )print == +}forall +(>>)= +}if +}bind def +/ScaleVal +{ +aload pop +1 index sub +3 2 roll mul add +}bind def +/ScaleArg +{ +aload pop +1 index sub +3 1 roll +sub exch div +}bind def +/ScaleArgN +{ +dup length 2 sub -2 0{ +2 +2 index 3 1 roll getinterval +3 2 roll +exch//ScaleArg exec +1 index length 2 idiv 1 add 1 roll +}for +pop +}bind def +/ComputeFunction_10 +{ +//PDFR_DEBUG{ +(ComputeFunction_10 beg )print 1 index//=only exec( stack=)print count = +}if +exch +dup 1 eq{ +pop dup length 1 sub get +}{ +1 index length 1 sub mul +dup dup floor sub +dup 0 eq{ +pop cvi get +}{ +3 1 roll floor cvi +2 getinterval +aload pop +2 index mul 3 2 roll 1 exch sub 3 2 roll mul add +}ifelse +}ifelse +//PDFR_DEBUG{ +(ComputeFunction_10 end )print dup//=only exec( stack=)print count = +}if +}bind def +/ComputeFunction_n0 +{ +//PDFR_DEBUG{ +(ComputeFunction_n0 beg N=)print dup//=only exec( stack=)print count = +}if +dup 0 eq{ +pop +}{ +dup 2 add -1 roll +dup 3 index length 1 sub ge{ +pop 1 sub +exch dup length 1 sub get exch +//PDFReader/ComputeFunction_n0 get exec +}{ +dup floor cvi dup +4 index exch get +3 index dup +5 add copy +6 2 roll +pop pop pop pop +1 sub +//PDFReader/ComputeFunction_n0 get exec +3 2 roll pop +exch +4 3 roll exch +4 add 2 roll 1 add +3 2 roll exch get +exch 1 sub +//PDFReader/ComputeFunction_n0 get exec +1 index mul +3 1 roll +1 exch sub mul add +}ifelse +}ifelse +//PDFR_DEBUG{ +(ComputeFunction_n0 end )print dup//=only exec( stack=)print count = +}if +}bind def +/FunctionToProc_x01 +{ +dup/Domain get exch +dup/Data get 0 get exch +/Size get length +[4 1 roll +//PDFR_DEBUG{ +{(function beg, stack =)print count//=only exec(\n)print}/exec load +5 2 roll +}if +dup 1 gt{ +{mark exch +3 add 2 roll +//ScaleArgN exec +counttomark dup +3 add -2 roll +pop exch +//ComputeFunction_n0 exec +}/exec load +}{ +pop +3 1/roll load//ScaleArg/exec load +/exch load +//ComputeFunction_10/exec load +}ifelse +//PDFR_DEBUG{ +(function end, stack =)/print load/count load//=only/exec load(\n)/print load +}if +]cvx +//PDFR_DEBUG{ +(Made a procedure for the 1-result function :)= +dup == +}if +}bind def +/FunctionProcDebugBeg +{(FunctionProcDebugBeg )print count = +}bind def +/FunctionProcDebugEnd +{(FunctionProcDebugEnd )print count = +}bind def +/FunctionToProc_x0n +{ +PDFR_DEBUG{ +(FunctionToProc_x0n beg m=)print dup = +}if +1 index/Size get length exch +dup 7 mul 2 add array +PDFR_DEBUG{ +dup 0//FunctionProcDebugBeg put +}{ +dup 0//DoNothing put +}ifelse +dup 1/exec load put +dup 2 5 index/Domain get put +2 index 1 eq{ +dup 3//ScaleArg put +}{ +dup 3//ScaleArgN put +}ifelse +dup 4/exec load put +1 index 1 sub 0 exch 1 exch{ +dup 7 mul 5 add +1 index 4 index 1 sub ne{ +dup 3 index exch 6 index put 1 add +dup 3 index exch/copy load put 1 add +}if +[ +6 index/Data get 3 index get +6 index 1 eq{ +//ComputeFunction_10/exec load +}{ +6 index +//ComputeFunction_n0/exec load +}ifelse +]cvx +3 index exch 2 index exch put 1 add +2 index 1 index/exec load put 1 add +1 index 4 index 1 sub ne{ +2 index 1 index 6 index 1 add put 1 add +2 index 1 index 1 put 1 add +2 index 1 index/roll load put +}if +pop pop +}for +PDFR_DEBUG{ +dup dup length 2 sub//FunctionProcDebugEnd put +}{ +dup dup length 2 sub//DoNothing put +}ifelse +dup dup length 1 sub/exec load put +cvx exch pop exch pop exch pop +//PDFR_DEBUG{ +(Made a procedure for the n-argument function :)= +dup == +}if +PDFR_DEBUG{ +(FunctionToProc_x0n end)= +}if +}bind def +/MakeTableRec +{ +0 +exec +}bind def +/MakeTable +{//PDFR_DEBUG{ +(MakeTable beg )print count = +}if +1 index/Size get exch +1 sub dup +3 1 roll +get +array +1 index 0 eq{ +exch pop exch pop +}{ +dup length 1 sub -1 0{ +3 index 3 index//MakeTableRec exec +2 index 3 1 roll put +}for +exch pop exch pop +}ifelse +//PDFR_DEBUG{ +(MakeTable end )print count = +}if +}bind def +//MakeTableRec 0//MakeTable put +/StoreSample +{ +1 sub +dup 0 eq{ +pop +}{ +-1 1{ +I exch get get +}for +}ifelse +I 0 get 3 2 roll put +}bind def +/ReadSample32 +{ +4{ +File read not{ +mark(Insufficient data for function.)//error exec +}if +}repeat +pop +3 1 roll exch +256 mul add 256 mul add +//1_24_bitshift_1_sub div +}bind def +/ReadSample +{ +Buffer BitsLeft BitsPerSample +{2 copy ge{ +exit +}if +3 1 roll +8 add 3 1 roll +256 mul File read not{ +mark(Insufficient data for function.)//error exec +}if +add +3 1 roll +}loop +sub dup +2 index exch +neg bitshift +2 copy exch bitshift +4 3 roll exch sub +/Buffer exch def +exch/BitsLeft exch def +Div div +}bind def +/ReadSamplesRec +{0 +exec +}bind def +/ReadSamples +{ +//PDFR_DEBUG{ +(ReadSamples beg )print count = +}if +dup 1 eq{ +pop +0 1 Size 0 get 1 sub{ +I exch 0 exch put +0 1 M 1 sub{ +dup Range exch 2 mul 2 getinterval +//PDFR_DEBUG{ +(Will read a sample ... )print +}if +BitsPerSample 32 eq{//ReadSample32}{//ReadSample}ifelse +exec exch//ScaleVal exec +//PDFR_DEBUG{ +(value=)print dup = +}if +exch Table exch get +Size length//StoreSample exec +}for +}for +}{ +1 sub +dup Size exch get 0 exch 1 exch 1 sub{ +I exch 2 index exch put +dup//ReadSamplesRec exec +}for +pop +}ifelse +//PDFR_DEBUG{ +(ReadSamples end )print count = +}if +}bind def +//ReadSamplesRec 0//ReadSamples put +/StreamToArray +{//PDFR_DEBUG{ +(StreamToArray beg )print count = +}if +userdict/FuncDataReader get begin +dup/BitsPerSample get/BitsPerSample exch def +dup/Size get length/N exch def +dup/Range get length 2 idiv/M exch def +1 BitsPerSample bitshift 1 sub/Div exch def +/BitsLeft 0 def +/Buffer 0 def +dup/Size get/Size exch def +dup/Range get/Range exch def +/File 1 index//MakeStreamReader exec def +/I[N{0}repeat]def +M array +dup length 1 sub -1 0{ +2 index N//MakeTable exec +2 index 3 1 roll put +}for +/Table exch def +N//ReadSamples exec +PDFR_DEBUG{ +(Table = )print Table == +}if +/Data Table put +end +//PDFR_DEBUG{ +(StreamToArray end )print count = +}if +}bind def +/FunctionToProc10 +{ +PDFR_DEBUG{ +(FunctionToProc10 beg, Range = )print dup/Range get == +}if +dup/Order//knownget exec{ +1 ne{ +(Underimplemented function Type 0 Order 3.)= +}if +}if +dup//StreamToArray exec +dup/Range get length dup 2 eq{ +pop//FunctionToProc_x01 exec +}{ +2 idiv//FunctionToProc_x0n exec +}ifelse +PDFR_DEBUG{ +(FunctionToProc10 end)= +}if +}bind def +/FunctionToProc12 +{begin +currentdict/C0//knownget exec{length 1 eq}{true}ifelse{ +N +currentdict/C0//knownget exec{ +0 get +}{ +0 +}ifelse +currentdict/C1//knownget exec{ +0 get +}{ +1 +}ifelse +1 index sub +[4 1 roll +{ +4 2 roll +exp mul add +}aload pop +]cvx +}{ +[ +0 1 C0 length 1 sub{ +N +C0 2 index get +C1 3 index get +4 3 roll pop +1 index sub +[/dup load +5 2 roll +{ +4 2 roll +exp mul add +exch +}aload pop +]cvx +/exec load +}for +/pop load +]cvx +}ifelse +end +//PDFR_DEBUG{ +(FunctionType2Proc : )print dup == +}if +}bind def +/FunctionToProc14 +{//MakeStreamReader exec cvx exec +//PDFR_DEBUG{ +(FunctionType4Proc : )print dup == +}if +}bind def +/FunctionToProc1 +{ +dup/FunctionType get +{dup 0 eq{ +pop//FunctionToProc10 exec exit +}if +dup 2 eq{ +pop//FunctionToProc12 exec exit +}if +dup 4 eq{ +pop//FunctionToProc14 exec exit +}if +mark exch(Function type )exch( isn't implemented yet.)//error exec +}loop +}bind def +/FunctionToProc20 +{ +PDFR_DEBUG{ +(FunctionToProc20, Range = )print dup/Range get == +}if +dup/Order//knownget exec{ +1 ne{ +(Underimplemented function Type 0 Order 3.)= +}if +}if +dup//StreamToArray exec +dup/Range get length dup 2 eq{ +pop//FunctionToProc_x01 exec +}{ +2 idiv//FunctionToProc_x0n exec +}ifelse +}bind def +/FunctionToProc +{//PDFR_DEBUG{ +(FunctionToProc beg )print count = +}if +dup type/dicttype eq{ +dup/Domain get length 2 idiv +{ +dup 1 eq{ +pop//FunctionToProc1 exec exit +}if +dup 2 eq{ +pop//FunctionToProc20 exec exit +}if +mark(Functions with many arguments aren't implemented yet.)//error exec +}loop +}{ +//PDFR_DEBUG{(Not a function dict, assume already a procedure.)print}if +}ifelse +//PDFR_DEBUG{ +(FunctionToProc end )print count = +}if +}bind def +/spotfunctions mark +/Round{ +abs exch abs 2 copy add 1 le{ +dup mul exch dup mul add 1 exch sub +}{ +1 sub dup mul exch 1 sub dup mul add 1 sub +}ifelse +} +/Diamond{ +abs exch abs 2 copy add .75 le{ +dup mul exch dup mul add 1 exch sub +}{ +2 copy add 1.23 le{ +.85 mul add 1 exch sub +}{ +1 sub dup mul exch 1 sub dup mul add 1 sub +}ifelse +}ifelse +} +/Ellipse{ +abs exch abs 2 copy 3 mul exch 4 mul add 3 sub dup 0 lt{ +pop dup mul exch .75 div dup mul add 4 div 1 exch sub +}{ +dup 1 gt{ +pop 1 exch sub dup mul exch 1 exch sub +.75 div dup mul add 4 div 1 sub +}{ +.5 exch sub exch pop exch pop +}ifelse +}ifelse +} +/EllipseA{dup mul .9 mul exch dup mul add 1 exch sub} +/InvertedEllipseA{dup mul .9 mul exch dup mul add 1 sub} +/EllipseB{dup 5 mul 8 div mul exch dup mul exch add sqrt 1 exch sub} +/EllipseC{dup mul .9 mul exch dup mul add 1 exch sub} +/InvertedEllipseC{dup mul .9 mul exch dup mul add 1 sub} +/Line{exch pop abs neg} +/LineX{pop} +/LineY{exch pop} +/Square{abs exch abs 2 copy lt{exch}if pop neg} +/Cross{abs exch abs 2 copy gt{exch}if pop neg} +/Rhomboid{abs exch abs 0.9 mul add 2 div} +/DoubleDot{2{360 mul sin 2 div exch}repeat add} +/InvertedDoubleDot{2{360 mul sin 2 div exch}repeat add neg} +/SimpleDot{dup mul exch dup mul add 1 exch sub} +/InvertedSimpleDot{dup mul exch dup mul add 1 sub} +/CosineDot{180 mul cos exch 180 mul cos add 2 div} +/Double{exch 2 div exch 2{360 mul sin 2 div exch}repeat add} +/InvertedDouble{ +exch 2 div exch 2{360 mul sin 2 div exch}repeat add neg +} +.dicttomark readonly def +/CheckColorSpace +{ +dup type/arraytype ne{ +mark(Resource )3 index( must be an array.)//error exec +}if +}bind def +/SubstitutePDFColorSpaceRec +{0 +exec +}bind def +/SubstitutePDFColorSpace +{ +{ +dup 0 get/Pattern eq{ +dup length 1 gt{ +dup dup 1//CheckColorSpace//ResolveA exec +dup type/nametype ne{ +//SubstitutePDFColorSpaceRec exec +}if +1 exch put +}if +exit +}if +dup 0 get/Indexed eq{ +exit +}if +dup 0 get/Separation eq{ +dup dup 2//CheckColorSpace//ResolveA exec +dup type/nametype ne{ +//SubstitutePDFColorSpaceRec exec +}if +2 exch put +exit +}if +dup 0 get/CalGray eq{ +1 get +dup/Gamma//knownget exec{ +[exch[exch/exp load]cvx dup dup] +1 index exch/DecodeLMN exch put +}if +[exch/CIEBasedA exch] +exit +}if +dup 0 get/CalRGB eq{ +1 get +dup/Matrix//knownget exec{ +1 index exch/MatrixLMN exch put +}if +dup/Gamma//knownget exec{ +aload pop +[exch/exp load]cvx +3 1 roll +[exch/exp load]cvx +3 1 roll +[exch/exp load]cvx +3 1 roll +3 array astore +1 index exch/DecodeLMN exch put +}if +[exch/CIEBasedABC exch] +exit +}if +dup 0 get/Lab eq{ +1 get +begin +currentdict/Range//knownget exec{aload pop}{-100 100 -100 100}ifelse +0 100 6 2 roll 6 array astore +/RangeABC exch def +/DecodeABC[{16 add 116 div}bind{500 div}bind{200 div}bind]def +/MatrixABC[1 1 1 1 0 0 0 0 -1]def +{dup 6 29 div ge{dup dup mul mul}{4 29 div sub 108 841 div mul}ifelse} +/DecodeLMN[ +[3 index aload pop WhitePoint 0 get/mul load]cvx +[4 index aload pop WhitePoint 1 get/mul load]cvx +[5 index aload pop WhitePoint 2 get/mul load]cvx +]def pop +//PDFR_DEBUG{ +(Constructed from Lab <<)= +currentdict{exch = ==}forall +(>>)= +}if +[/CIEBasedABC currentdict] +end +exit +pop +}if +dup 0 get/CIEBasedA eq{exit}if +dup 0 get/CIEBasedABC eq{exit}if +mark exch(Unimplemented color space )exch//error exec +}loop +}bind def +//SubstitutePDFColorSpaceRec 0//SubstitutePDFColorSpace put +/ResolveArrayElement +{2 copy get +dup type dup/arraytype eq exch +/packedarraytype eq or{ +dup length 1 ge exch xcheck and{ +2 copy get +dup 0 get type/integertype eq +1 index 1 get type dup/arraytype +eq exch +/packedarraytype eq or +and{ +exec +2 index 4 1 roll put +}{ +pop pop +}ifelse +}{ +pop +}ifelse +}{ +pop pop +}ifelse +}bind def +/ResolveColorSpaceArrayRec +{0 +exec +}bind def +/SetColorSpaceSafe +{ +PDFR_DEBUG{ +(SetColorSpaceSafe beg)= +}if +currentcolorspace dup type/arraytype eq{ +1 index type/arraytype eq{ +dup length 2 index length eq{ +false exch +dup length 0 exch 1 exch 1 sub{ +dup +4 index exch get exch +2 index exch get +ne{ +exch pop true exch exit +}if +}for +pop +{ +setcolorspace +}{ +pop +}ifelse +}{ +pop setcolorspace +}ifelse +}{ +pop setcolorspace +}ifelse +}{ +pop setcolorspace +}ifelse +PDFR_DEBUG{ +(SetColorSpaceSafe end)= +}if +}bind def +/ResolveColorSpaceArray +{ +//PDFR_DEBUG{ +(ResolveColorSpaceArray beg )print dup == +}if +dup 0 get/Indexed eq{ +1//ResolveArrayElement exec +dup dup 1 get +dup type/arraytype eq{ +//SubstitutePDFColorSpace exec +//ResolveColorSpaceArrayRec exec +1 exch put +}{ +pop pop +}ifelse +}if +dup 0 get/Separation eq{ +dup dup 1 get UnPDFEscape 1 exch put +3//ResolveArrayElement exec +dup 3 get//FunctionToProc exec +2 copy 3 exch put +pop +}if +dup 0 get/Pattern eq{ +dup length 1 gt{ +dup 1 get dup type/arraytype eq{ +ResolveColorSpaceArray +1 index 1 3 -1 roll put +}{ +pop +}ifelse +}if +}if +PDFR_DEBUG{ +(Construcrted color space :)= +dup == +}if +//PDFR_DEBUG{ +(ResolveColorSpaceArray end )print dup == +}if +}bind def +//ResolveColorSpaceArrayRec 0//ResolveColorSpaceArray put +/ResolveColorSpace +{ +//PDFR_DEBUG{ +(ResolveColorSpace beg )print dup = +}if +dup//SimpleColorSpaceNames exch known not{ +dup//PDFColorSpaces exch//knownget exec{ +exch pop +//PDFR_DEBUG{ +(ResolveColorSpace known )= +}if +}{ +dup +//PDFReader/CurrentObject get/Context get/Resources get +/ColorSpace//DoNothing//ResolveD exec +exch//CheckColorSpace//ResolveD exec +dup type/arraytype eq{ +//SubstitutePDFColorSpace exec +//ResolveColorSpaceArray exec +dup//PDFColorSpaces 4 2 roll put +}if +}ifelse +}if +//PDFR_DEBUG{ +(ResolveColorSpace end )print dup == +}if +}bind def +/CheckPattern +{ +dup/PatternType//knownget exec{ +dup 1 ne{ +mark(Resource )4 index( is a shading, which can't be handled at level 2. )//error exec +}if +pop +}if +dup/Type knownget{ +/Pattern ne{ +mark(Resource )4 index( must have /Type/Pattern .)//error exec +}if +}if +}bind def +/PaintProc +{/Context get +//RunDelayedStream exec +}bind def +/ResolvePattern +{ +dup +userdict/PDFR_Patterns get +exch//knownget exec{ +exch pop +}{ +dup +//PDFReader/CurrentObject get/Context get/Resources get +/Pattern//DoNothing//ResolveD exec +exch//CheckPattern//ResolveD exec +dup dup/Context exch put +dup/Resources//DoNothing//ResolveD exec pop +dup/PaintProc//PaintProc put +gsave userdict/PDFR_InitialGS get setgstate +currentglobal exch false setglobal +dup/Matrix get +makepattern +exch setglobal +grestore +dup userdict/PDFR_Patterns get +4 2 roll +put +}ifelse +}bind def +/SetColor +{//PDFR_DEBUG{ +(SetColor beg)= +}if +currentcolorspace dup type/nametype eq{ +pop setcolor +}{ +0 get/Pattern eq{ +//ResolvePattern exec setpattern +}{ +setcolor +}ifelse +}ifelse +//PDFR_DEBUG{ +(SetColor end)= +}if +}bind def +/ImageKeys 15 dict begin +/BPC/BitsPerComponent def +/CS/ColorSpace def +/D/Decode def +/DP/DecodeParms def +/F/Filter def +/H/Height def +/IM/ImageMask def +/I/Interpolate def +/W/Width def +currentdict end readonly def +/ImageValues 15 dict begin +/G/DeviceGray def +/RGB/DeviceRGB def +/CMYK/DeviceCMYK def +/I/Indexed def +/AHx/ASCIIHexDecode def +/A85/ASCII85Decode def +/LZW/LZWDecode def +/Fl/FlateDecode def +/RL/RunLengthDecode def +/CCF/CCITTFaxDecode def +/DCT/DCTDecode def +currentdict end readonly def +/GetColorSpaceRange +{2 index/ColorSpace get +dup type/arraytype eq{ +1 get +}if +exch//knownget exec{ +exch pop +}if +}bind def +/DecodeArrays 15 dict begin +/DeviceGray{[0 1]}def +/DeviceRGB{[0 1 0 1 0 1]}def +/DeviceCMYK{[0 1 0 1 0 1 0 1]}def +/Indexed{ +dup/BitsPerComponent get 1 exch bitshift 1 sub[exch 0 exch] +}def +/Separation{[0 1]}def +/CIEBasedA{[0 1]/RangeA//GetColorSpaceRange exec}def +/CIEBasedABC{[0 1 0 1 0 1]/RangeABC//GetColorSpaceRange exec}def +currentdict end readonly def +/Substitute +{1 index//knownget exec{ +exch pop +}if +}bind def +/DebugImagePrinting +{ +//PDFR_DEBUG{ +(Image :)= +dup{exch//=only exec( )print == +}forall +}if +}bind def +/CompleteImage +{ +dup/ColorSpace known{ +dup/ColorSpace//CheckColorSpace//ResolveD exec pop +}if +dup/Decode known not{ +dup/ColorSpace//knownget exec{ +dup type/arraytype eq{ +0 get +}if +//DecodeArrays exch get exec +}{ +[0 1] +}ifelse +1 index exch/Decode exch put +}if +dup/ImageMatrix[2 index/Width get 0 0 5 index/Height get neg +0 7 index/Height get]put +//DebugImagePrinting exec +}bind def +/CompleteInlineImage +{ +//PDFR_DEBUG{ +(CompleteInlineImage beg)= +}if +dup/ImageType known not{ +dup/ImageType 1 put +}if +dup length dict exch{ +exch//ImageKeys//Substitute exec +dup/Filter eq{ +exch//ImageValues//Substitute exec exch +}if +dup/ColorSpace eq{ +exch +dup//ImageValues exch//knownget exec{ +exch pop +}{ +//ResolveColorSpace exec +}ifelse +exch +}if +exch +2 index 3 1 roll put +}forall +//CompleteImage exec +dup/DataSource 2 copy get +2 index//AppendFilters exec put +//PDFR_DEBUG{ +(CompleteInlineImage end)= +}if +}bind def +/CompleteOutlineImage +{ +currentglobal exch dup gcheck setglobal +//PDFR_DEBUG{ +(CompleteOutlineImage beg)= +}if +dup dup//MakeStreamReader exec/DataSource exch put +dup/ImageType known not{ +//CompleteImage exec +dup/ImageType 1 put +dup/ColorSpace known{ +dup/ColorSpace//CheckColorSpace//ResolveD exec +dup type/arraytype eq{ +//ResolveColorSpaceArray exec +//SubstitutePDFColorSpace exec +1 index exch/ColorSpace exch put +}{ +pop +}ifelse +}if +}if +//PDFR_DEBUG{ +(CompleteOutlineImage end)= +}if +exch setglobal +}bind def +/DoImage +{ +//PDFR_DEBUG{ +(DoImage beg)= +}if +gsave +dup/ColorSpace//knownget exec{setcolorspace}if +dup/ImageMask//knownget exec not{false}if +{imagemask}{image}ifelse +grestore +//PDFR_DEBUG{ +(DoImage end)= +}if +}bind def +/GSave +{ +gsave +//PDFReader/GraphicStateStackPointer get +dup//GraphicStateStack exch get null eq{ +dup//GraphicStateStack exch//InitialGraphicState length dict put +}if +dup//GraphicStateStack exch get +//GraphicState exch copy pop +1 add//PDFReader exch/GraphicStateStackPointer exch put +}bind def +/GRestore +{ +grestore +//PDFReader/GraphicStateStackPointer get +1 sub dup +//PDFReader exch/GraphicStateStackPointer exch put +//GraphicStateStack exch get +//GraphicState copy pop +}bind def +/SetFont +{dup//GraphicState exch/FontSize exch put +//ResolveAndSetFont exec +//GraphicState/FontMatrixNonHV currentfont/FontMatrix get 1 get 0 ne put +}bind def +/ShowText +{ +//GraphicState/TextRenderingMode get dup 0 eq +exch 3 eq not currentfont/FontType get 3 eq and or +{ +//GraphicState/WordSpacing get 0 +32 +//GraphicState/CharacterSpacing get 0 +6 5 roll +//GraphicState/FontMatrixNonHV get{ +[ +7 -2 roll pop +5 -2 roll pop +5 -1 roll +{ +exch +pop +3 index add +exch 2 index eq{3 index add}if +4 1 roll +} +currentfont/FontMatrix get 0 get 0 ne{ +1 1 index length 1 sub getinterval cvx +}if +5 index +cshow +pop pop pop] +xshow +}{ +awidthshow +}ifelse +}{ +//GraphicState/CharacterSpacing get 0 eq +//GraphicState/FontMatrixNonHV get not and +//GraphicState/WordSpacing get 0 eq and{ +true charpath +}{ +{ +exch +pop 0 +currentpoint 5 4 roll +( )dup 0 3 index put true charpath +5 1 roll +moveto rmoveto +//GraphicState/CharacterSpacing get 0 rmoveto +32 eq{ +//GraphicState/WordSpacing get 0 rmoveto +}if +} +//GraphicState/FontMatrixNonHV get dup not exch{ +pop currentfont/FontMatrix get 0 get 0 ne +}if{ +1 1 index length 1 sub getinterval cvx +}if +exch cshow +}ifelse +}ifelse +}bind def +/ShowTextBeg +{ +//GraphicState/TextRenderingMode get dup 0 ne +{ +3 ne +currentfont/FontType get 3 eq not and{ +currentpoint newpath moveto +}if +} +{ +pop +}ifelse +}bind def +/ShowTextEnd +{ +//GraphicState/TextRenderingMode get +currentfont/FontType get 3 eq{ +dup 3 ne{ +pop 0 +}if +}if +{dup 1 eq{ +stroke exit +}if +dup 2 eq{ +gsave fill grestore stroke exit +}if +dup 3 eq{ +currentpoint newpath moveto +}if +dup 4 eq{ +gsave fill grestore clip exit +}if +dup 5 eq{ +gsave stroke grestore clip exit +}if +dup 6 eq{ +gsave fill grestore gsave stroke grestore fill exit +}if +dup 7 eq{ +clip exit +}if +exit +}loop +pop +}bind def +/ShowTextWithGlyphPositioning +{//ShowTextBeg exec +{dup type/stringtype eq{ +//ShowText exec +}{ +neg 1000 div//GraphicState/FontSize get mul 0 rmoveto +}ifelse +}forall +//ShowTextEnd exec +}bind def +/CheckFont +{dup/Type get/ExtGState ne{ +mark(Resource )3 index( must have /Type/ExtGState.)//error exec +}if +}bind def +/SetTransfer +{ +//PDFR_DEBUG{(SetTransfer beg )print count =}if +dup type/arraytype eq 1 index xcheck not and{ +0 4 getinterval aload pop +setcolortransfer +}{ +settransfer +}ifelse +//PDFR_DEBUG{(SetTransfer end )print count =}if +}bind def +/CheckExtGState +{dup/Type get/ExtGState ne{ +mark(Resource )3 index( must have /Type/ExtGState.)//error exec +}if +}bind def +/CheckHalftone +{dup/HalftoneType known not{ +mark(Resource )3 index( must have /HalftoneType.)//error exec +}if +}bind def +/ResolveFunction +{ +//PDFR_DEBUG{(ResolveFunction beg )print dup = count =}if +2 copy get//IsObjRef exec{ +2 copy//DoNothing//ResolveD exec +3 copy put pop +}if +2 copy get dup type/arraytype eq exch xcheck and not{ +2 copy get +dup type/arraytype eq 1 index xcheck not and{ +dup length 1 sub -1 0{ +2 copy//DoNothing ResolveA +dup/Identity eq{ +pop 2 copy{}put +}{ +//FunctionToProc exec +3 copy put pop +}ifelse +pop +}for +}{ +dup/Default eq{ +}{ +dup/Identity eq{ +pop{} +}{dup type/nametype eq{ +//spotfunctions exch get +}{ +//FunctionToProc exec +}ifelse +}ifelse +}ifelse +}ifelse +3 copy put +exch pop +}{ +1 index exch get +}ifelse +//PDFR_DEBUG{(ResolveFunction end )print dup == count =}if +}bind def +/ResolveFunctionSafe +{2 copy known{ +//ResolveFunction exec +}if +pop +}bind def +/CreateHalftoneThresholds +{ +dup/Thresholds known not{ +dup/HalftoneType get 10 eq{ +dup dup//MakeStreamReader exec +/Thresholds exch put +}if +dup/HalftoneType get dup 3 eq exch 6 eq or{ +dup dup//MakeStreamReader exec +//BlockBuffer readstring pop +dup length +dup 0 eq{ +mark(Could not read Thresholds)//error exec +}if +string copy/Thresholds exch put +dup/HalftoneType 3 put +}if +}if +}bind def +/SetExtGState +{ +//PDFReader/CurrentObject get/Context get/Resources get +/ExtGState//DoNothing//ResolveD exec +exch//CheckExtGState//ResolveD exec +dup/LW//knownget exec{ +setlinewidth +}if +dup/LC//knownget exec{ +setlinecap +}if +dup/LJ//knownget exec{ +setlinejoin +}if +dup/ML//knownget exec{ +setmeterlimit +}if +dup/D//knownget exec{ +setdash +}if +dup/RI//knownget exec{ +mark(Unimplemented ExtGState.RI)//error exec +}if +dup/OP//knownget exec{ +setoverprint +}if +dup/op//knownget exec{ +setoverprint +}if +dup/OPM//knownget exec{ +mark(Unimplemented ExtGState.OPM)//error exec +}if +dup/Font//knownget exec{ +mark(Unimplemented ExtGState.Font)//error exec +}if +dup/BG known{ +/BG//ResolveFunction exec +setblackgeneration +}if +dup/BG2 known{ +/BG2//ResolveFunction exec +dup/Default eq{ +//InitialExtGState/BG2 get +}if +setblackgeneration +}if +dup/UCR known{ +/UCR//ResolveFunction exec +setundercolorremoval +}if +dup/UCR2 known{ +/UCR2//ResolveFunction exec +dup/Default eq{ +//InitialExtGState/UCR2 get +}if +setundercolorremoval +}if +dup/TR known{ +/TR//ResolveFunction exec +//SetTransfer exec +}if +dup/TR2 known{ +/TR2//ResolveFunction exec +dup/Default eq{ +pop//InitialExtGState/TR2 get +aload pop setcolortransfer +}{ +//SetTransfer exec +}ifelse +}if +dup/HT//knownget exec{ +dup/Default eq{ +pop//InitialExtGState/HT get +sethalftone +}{ +//PDFR_DEBUG{(Ht beg)=}if +pop dup/HT//CheckHalftone//ResolveD exec +/SpotFunction//ResolveFunctionSafe exec +/TransferFunction//ResolveFunctionSafe exec +null exch +dup/HalftoneType get dup 5 eq exch dup 4 eq exch 2 eq or or{ +dup{ +dup//IsObjRef exec{ +pop +1 index exch//CheckHalftone ResolveD +}if +dup type/dicttype eq{ +dup/SpotFunction//ResolveFunctionSafe exec +/TransferFunction//ResolveFunctionSafe exec +//CreateHalftoneThresholds exec +dup/HalftoneType get 5 gt{ +4 3 roll pop +dup 4 1 roll +}if +}if +pop pop +}forall +}if +//CreateHalftoneThresholds exec +//PDFR_DEBUG{ +(HT:)= +dup{ +1 index/Default eq{ +(Default <<)= +exch pop +{exch = ==}forall +(>>)= +}{ +exch = == +}ifelse +}forall +(HT end)= flush +}if +exch dup null ne{ +(Warning: Ignoring a halftone with a Level 3 component halftone Type )print dup/HalftoneType get = +pop pop +}{ +pop +dup/HalftoneType get 5 gt{ +(Warning: Ignoring a Level 3 halftone Type )print dup/HalftoneType get = +pop +}{ +sethalftone +}ifelse +}ifelse +//PDFR_DEBUG{(HT set)= flush}if +}ifelse +}if +dup/FL//knownget exec{ +setflattness +}if +dup/SM//knownget exec{ +setsmoothness +}if +dup/SA//knownget exec{ +setstrokeadjust +}if +dup/BM//knownget exec{ +mark(Unimplemented ExtGState.BM)//error exec +}if +dup/SMask//knownget exec{ +mark(Unimplemented ExtGState.SMask)//error exec +}if +dup/CA//knownget exec{ +mark(Unimplemented ExtGState.CA)//error exec +}if +dup/ca//knownget exec{ +mark(Unimplemented ExtGState.ca)//error exec +}if +dup/AIS//knownget exec{ +mark(Unimplemented ExtGState.AIS)//error exec +}if +dup/TK//knownget exec{ +mark(Unimplemented ExtGState.TK)//error exec +}if +pop +}bind def +/CheckXObject +{dup/Subtype get dup/Image ne exch dup/Form ne exch/PS ne and and{ +mark(Resource )3 index( must have /Subtype /Image or /Form or /PS.)//error exec +}if +}bind def +/DoXObject +{ +//PDFReader/CurrentObject get/Context get/Resources get +/XObject//DoNothing//ResolveD exec +exch//CheckXObject//ResolveD exec +dup/Subtype get +dup/Image eq{ +pop +//CompleteOutlineImage exec +//DoImage exec +}{ +dup/PS eq{ +PDFR_DEBUG{ +(Executing a PS Xobject)= +}if +pop +//RunDelayedStream exec +}{ +dup/Form eq{ +pop +PDFR_DEBUG{ +(Executing a Form XObject)= +}if +//PDFReader/CurrentObject get exch +dup//PDFReader exch<< exch/Context exch >>/CurrentObject exch put +dup/Matrix get concat +dup/BBox get aload pop exch 3 index sub exch 2 index sub rectclip +//RunDelayedStream exec +//PDFReader exch/CurrentObject exch put +}{ +mark exch(unimplemented XObject type )exch//error exec +}ifelse +}ifelse +}ifelse +}bind def +/Operators 50 dict begin +/q{//GSave exec}bind def +/Q{//GRestore exec}bind def +/cm{//TempMatrix astore concat}bind def +/i{1 .min setflat}bind def +/J/setlinecap load def +/d/setdash load def +/j/setlinejoin load def +/w/setlinewidth load def +/M/setmiterlimit load def +/gs{SetExtGState}bind def +/g/setgray load def +/rg/setrgbcolor load def +/k/setcmykcolor load def +/cs{//ResolveColorSpace exec//SetColorSpaceSafe exec +}bind def +/sc/setcolor load def +/scn{//SetColor exec}bind def +/G/setgray load def +/RG/setrgbcolor load def +/K/setcmykcolor load def +/CS//cs def +/ri{SetColorRenderingIntent}bind def +/SC/setcolor load def +/SCN{//SetColor exec}bind def +/m/moveto load def +/l/lineto load def +/c/curveto load def +/v{currentpoint 6 2 roll curveto}bind def +/y{2 copy curveto}bind def +/re{ +4 2 roll moveto exch dup 0 rlineto 0 3 -1 roll rlineto neg 0 rlineto +closepath +}def +/h/closepath load def +/n/newpath load def +/S/stroke load def +/s{closepath stroke}bind def +/f/fill load def +/f*/eofill load def +/B{gsave fill grestore stroke}bind def +/b{closepath gsave fill grestore stroke}bind def +/B*{gsave eofill grestore stroke}bind def +/b*{closepath gsave eofill grestore stroke}bind def +/W/clip load def +/W*/eoclip load def +/sh{ +ResolveShading +dup/Background known{ +gsave +dup/ColorSpace get setcolorspace +dup/Background get aload pop setcolor +pathbbox +2 index sub exch 3 index sub exch +rectfill +grestore +}if +shfill +}bind def +/Do{//DoXObject exec}bind def +/BI{currentglobal false setglobal<<}bind def +/ID{>> +dup/DataSource currentfile +2 index/F//knownget exec{ +/A85 eq{ +0(~>)/SubFileDecode filter +}if +}if +put +//CompleteInlineImage exec +exch setglobal +//DoImage exec +}bind def +/EI{}bind def +/BT{gsave//GraphicState/InitialTextMatrix get currentmatrix pop}bind def +/ET{grestore}bind def +/Tc{//GraphicState exch/CharacterSpacing exch put}bind def +/TL{//GraphicState exch/TextLeading exch put}bind def +/Tr{//GraphicState exch/TextRenderingMode exch put}bind def +/Ts{ +mark(Unimplemented SetTextRise)//error exec +}bind def +/Tw{//GraphicState exch/WordSpacing exch put}bind def +/Tz{ +mark(Unimplemented SetHorizontalTextScaling)//error exec +}bind def +/Td{translate 0 0 moveto}bind def +/TD{dup neg//TL exec//Td exec}bind def +/Tm{//GraphicState/InitialTextMatrix get setmatrix +//TempMatrix astore concat +0 0 moveto}bind def +/T*{0//GraphicState/TextLeading get neg//Td exec}bind def +/Tj{//ShowTextBeg exec//ShowText exec//ShowTextEnd exec}bind def +/'{//T* exec//ShowText exec//ShowTextEnd exec}bind def +/"{3 2 roll//Tw exec exch//Tc exec//' exec}bind def +/TJ//ShowTextWithGlyphPositioning def +/Tf//SetFont def +/d0/setcharwidth load def +/d1/setcachedevice load def +/BDC{pop pop}bind def +/BMC{pop}bind def +/EMC{}bind def +/BX{BeginCompatibilitySection}bind def +/EX{EndCompatibilitySection}bind def +/DP{DefineMarkedContentPointWithPropertyList}bind def +/MP{DefineMarkedContentPoint}bind def +/PS{cvx exec}bind def +currentdict end def +//PDFR_STREAM{ +//Operators length dict begin +//Operators{ +exch dup +[exch//=only/exec load +( )/print load +8 7 roll +dup type/arraytype eq{ +/exec load +}if +( )/print load +]cvx +def +}forall +currentdict end/Operators exch def +}if +/.registerencoding +{pop pop +}bind def +/.defineencoding +{def +}bind def +/.findencoding +{load +}bind def +/currentglobal where +{pop currentglobal{setglobal}true setglobal} +{{}} +ifelse +/MacRomanEncoding +StandardEncoding 0 39 getinterval aload pop +/quotesingle +StandardEncoding 40 56 getinterval aload pop +/grave +StandardEncoding 97 31 getinterval aload pop +/Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute +/agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave +/ecircumflex/edieresis/iacute/igrave +/icircumflex/idieresis/ntilde/oacute +/ograve/ocircumflex/odieresis/otilde +/uacute/ugrave/ucircumflex/udieresis +/dagger/degree/cent/sterling/section/bullet/paragraph/germandbls +/registered/copyright/trademark/acute/dieresis/.notdef/AE/Oslash +/.notdef/plusminus/.notdef/.notdef/yen/mu/.notdef/.notdef +/.notdef/.notdef/.notdef/ordfeminine/ordmasculine/.notdef/ae/oslash +/questiondown/exclamdown/logicalnot/.notdef +/florin/.notdef/.notdef/guillemotleft +/guillemotright/ellipsis/space/Agrave/Atilde/Otilde/OE/oe +/endash/emdash/quotedblleft/quotedblright +/quoteleft/quoteright/divide/.notdef +/ydieresis/Ydieresis/fraction/currency +/guilsinglleft/guilsinglright/fi/fl +/daggerdbl/periodcentered/quotesinglbase/quotedblbase +/perthousand/Acircumflex/Ecircumflex/Aacute +/Edieresis/Egrave/Iacute/Icircumflex +/Idieresis/Igrave/Oacute/Ocircumflex +/.notdef/Ograve/Uacute/Ucircumflex +/Ugrave/dotlessi/circumflex/tilde +/macron/breve/dotaccent/ring/cedilla/hungarumlaut/ogonek/caron +256 packedarray +5 1 index .registerencoding +.defineencoding +exec +/AdobeGlyphList mark +/A 16#0041 +/AE 16#00c6 +/AEacute 16#01fc +/AEmacron 16#01e2 +/AEsmall 16#f7e6 +/Aacute 16#00c1 +/Aacutesmall 16#f7e1 +/Abreve 16#0102 +/Abreveacute 16#1eae +/Abrevecyrillic 16#04d0 +/Abrevedotbelow 16#1eb6 +/Abrevegrave 16#1eb0 +/Abrevehookabove 16#1eb2 +/Abrevetilde 16#1eb4 +/Acaron 16#01cd +/Acircle 16#24b6 +/Acircumflex 16#00c2 +/Acircumflexacute 16#1ea4 +/Acircumflexdotbelow 16#1eac +/Acircumflexgrave 16#1ea6 +/Acircumflexhookabove 16#1ea8 +/Acircumflexsmall 16#f7e2 +/Acircumflextilde 16#1eaa +/Acute 16#f6c9 +/Acutesmall 16#f7b4 +/Acyrillic 16#0410 +/Adblgrave 16#0200 +/Adieresis 16#00c4 +/Adieresiscyrillic 16#04d2 +/Adieresismacron 16#01de +/Adieresissmall 16#f7e4 +/Adotbelow 16#1ea0 +/Adotmacron 16#01e0 +/Agrave 16#00c0 +/Agravesmall 16#f7e0 +/Ahookabove 16#1ea2 +/Aiecyrillic 16#04d4 +/Ainvertedbreve 16#0202 +/Alpha 16#0391 +/Alphatonos 16#0386 +/Amacron 16#0100 +/Amonospace 16#ff21 +/Aogonek 16#0104 +/Aring 16#00c5 +/Aringacute 16#01fa +/Aringbelow 16#1e00 +/Aringsmall 16#f7e5 +/Asmall 16#f761 +/Atilde 16#00c3 +/Atildesmall 16#f7e3 +/Aybarmenian 16#0531 +/B 16#0042 +/Bcircle 16#24b7 +/Bdotaccent 16#1e02 +/Bdotbelow 16#1e04 +/Becyrillic 16#0411 +/Benarmenian 16#0532 +/Beta 16#0392 +/Bhook 16#0181 +/Blinebelow 16#1e06 +/Bmonospace 16#ff22 +/Brevesmall 16#f6f4 +/Bsmall 16#f762 +/Btopbar 16#0182 +/C 16#0043 +/Caarmenian 16#053e +/Cacute 16#0106 +/Caron 16#f6ca +/Caronsmall 16#f6f5 +/Ccaron 16#010c +/Ccedilla 16#00c7 +/Ccedillaacute 16#1e08 +/Ccedillasmall 16#f7e7 +/Ccircle 16#24b8 +/Ccircumflex 16#0108 +/Cdot 16#010a +/Cdotaccent 16#010a +/Cedillasmall 16#f7b8 +/Chaarmenian 16#0549 +/Cheabkhasiancyrillic 16#04bc +/Checyrillic 16#0427 +/Chedescenderabkhasiancyrillic 16#04be +/Chedescendercyrillic 16#04b6 +/Chedieresiscyrillic 16#04f4 +/Cheharmenian 16#0543 +/Chekhakassiancyrillic 16#04cb +/Cheverticalstrokecyrillic 16#04b8 +/Chi 16#03a7 +/Chook 16#0187 +/Circumflexsmall 16#f6f6 +/Cmonospace 16#ff23 +/Coarmenian 16#0551 +/Csmall 16#f763 +/D 16#0044 +/DZ 16#01f1 +/DZcaron 16#01c4 +/Daarmenian 16#0534 +/Dafrican 16#0189 +/Dcaron 16#010e +/Dcedilla 16#1e10 +/Dcircle 16#24b9 +/Dcircumflexbelow 16#1e12 +/Dcroat 16#0110 +/Ddotaccent 16#1e0a +/Ddotbelow 16#1e0c +/Decyrillic 16#0414 +/Deicoptic 16#03ee +/Delta 16#2206 +/Deltagreek 16#0394 +/Dhook 16#018a +/Dieresis 16#f6cb +/DieresisAcute 16#f6cc +/DieresisGrave 16#f6cd +/Dieresissmall 16#f7a8 +/Digammagreek 16#03dc +/Djecyrillic 16#0402 +/Dlinebelow 16#1e0e +/Dmonospace 16#ff24 +/Dotaccentsmall 16#f6f7 +/Dslash 16#0110 +/Dsmall 16#f764 +/Dtopbar 16#018b +/Dz 16#01f2 +/Dzcaron 16#01c5 +/Dzeabkhasiancyrillic 16#04e0 +/Dzecyrillic 16#0405 +/Dzhecyrillic 16#040f +/E 16#0045 +/Eacute 16#00c9 +/Eacutesmall 16#f7e9 +/Ebreve 16#0114 +/Ecaron 16#011a +/Ecedillabreve 16#1e1c +/Echarmenian 16#0535 +/Ecircle 16#24ba +/Ecircumflex 16#00ca +/Ecircumflexacute 16#1ebe +/Ecircumflexbelow 16#1e18 +/Ecircumflexdotbelow 16#1ec6 +/Ecircumflexgrave 16#1ec0 +/Ecircumflexhookabove 16#1ec2 +/Ecircumflexsmall 16#f7ea +/Ecircumflextilde 16#1ec4 +/Ecyrillic 16#0404 +/Edblgrave 16#0204 +/Edieresis 16#00cb +/Edieresissmall 16#f7eb +/Edot 16#0116 +/Edotaccent 16#0116 +/Edotbelow 16#1eb8 +/Efcyrillic 16#0424 +/Egrave 16#00c8 +/Egravesmall 16#f7e8 +/Eharmenian 16#0537 +/Ehookabove 16#1eba +/Eightroman 16#2167 +/Einvertedbreve 16#0206 +/Eiotifiedcyrillic 16#0464 +/Elcyrillic 16#041b +/Elevenroman 16#216a +/Emacron 16#0112 +/Emacronacute 16#1e16 +/Emacrongrave 16#1e14 +/Emcyrillic 16#041c +/Emonospace 16#ff25 +/Encyrillic 16#041d +/Endescendercyrillic 16#04a2 +/Eng 16#014a +/Enghecyrillic 16#04a4 +/Enhookcyrillic 16#04c7 +/Eogonek 16#0118 +/Eopen 16#0190 +/Epsilon 16#0395 +/Epsilontonos 16#0388 +/Ercyrillic 16#0420 +/Ereversed 16#018e +/Ereversedcyrillic 16#042d +/Escyrillic 16#0421 +/Esdescendercyrillic 16#04aa +/Esh 16#01a9 +/Esmall 16#f765 +/Eta 16#0397 +/Etarmenian 16#0538 +/Etatonos 16#0389 +/Eth 16#00d0 +/Ethsmall 16#f7f0 +/Etilde 16#1ebc +/Etildebelow 16#1e1a +/Euro 16#20ac +/Ezh 16#01b7 +/Ezhcaron 16#01ee +/Ezhreversed 16#01b8 +/F 16#0046 +/Fcircle 16#24bb +/Fdotaccent 16#1e1e +/Feharmenian 16#0556 +/Feicoptic 16#03e4 +/Fhook 16#0191 +/Fitacyrillic 16#0472 +/Fiveroman 16#2164 +/Fmonospace 16#ff26 +/Fourroman 16#2163 +/Fsmall 16#f766 +/G 16#0047 +/GBsquare 16#3387 +/Gacute 16#01f4 +/Gamma 16#0393 +/Gammaafrican 16#0194 +/Gangiacoptic 16#03ea +/Gbreve 16#011e +/Gcaron 16#01e6 +/Gcedilla 16#0122 +/Gcircle 16#24bc +/Gcircumflex 16#011c +/Gcommaaccent 16#0122 +/Gdot 16#0120 +/Gdotaccent 16#0120 +/Gecyrillic 16#0413 +/Ghadarmenian 16#0542 +/Ghemiddlehookcyrillic 16#0494 +/Ghestrokecyrillic 16#0492 +/Gheupturncyrillic 16#0490 +/Ghook 16#0193 +/Gimarmenian 16#0533 +/Gjecyrillic 16#0403 +/Gmacron 16#1e20 +/Gmonospace 16#ff27 +/Grave 16#f6ce +/Gravesmall 16#f760 +/Gsmall 16#f767 +/Gsmallhook 16#029b +/Gstroke 16#01e4 +/H 16#0048 +/H18533 16#25cf +/H18543 16#25aa +/H18551 16#25ab +/H22073 16#25a1 +/HPsquare 16#33cb +/Haabkhasiancyrillic 16#04a8 +/Hadescendercyrillic 16#04b2 +/Hardsigncyrillic 16#042a +/Hbar 16#0126 +/Hbrevebelow 16#1e2a +/Hcedilla 16#1e28 +/Hcircle 16#24bd +/Hcircumflex 16#0124 +/Hdieresis 16#1e26 +/Hdotaccent 16#1e22 +/Hdotbelow 16#1e24 +/Hmonospace 16#ff28 +/Hoarmenian 16#0540 +/Horicoptic 16#03e8 +/Hsmall 16#f768 +/Hungarumlaut 16#f6cf +/Hungarumlautsmall 16#f6f8 +/Hzsquare 16#3390 +/I 16#0049 +/IAcyrillic 16#042f +/IJ 16#0132 +/IUcyrillic 16#042e +/Iacute 16#00cd +/Iacutesmall 16#f7ed +/Ibreve 16#012c +/Icaron 16#01cf +/Icircle 16#24be +/Icircumflex 16#00ce +/Icircumflexsmall 16#f7ee +/Icyrillic 16#0406 +/Idblgrave 16#0208 +/Idieresis 16#00cf +/Idieresisacute 16#1e2e +/Idieresiscyrillic 16#04e4 +/Idieresissmall 16#f7ef +/Idot 16#0130 +/Idotaccent 16#0130 +/Idotbelow 16#1eca +/Iebrevecyrillic 16#04d6 +/Iecyrillic 16#0415 +/Ifraktur 16#2111 +/Igrave 16#00cc +/Igravesmall 16#f7ec +/Ihookabove 16#1ec8 +/Iicyrillic 16#0418 +/Iinvertedbreve 16#020a +/Iishortcyrillic 16#0419 +/Imacron 16#012a +/Imacroncyrillic 16#04e2 +/Imonospace 16#ff29 +/Iniarmenian 16#053b +/Iocyrillic 16#0401 +/Iogonek 16#012e +/Iota 16#0399 +/Iotaafrican 16#0196 +/Iotadieresis 16#03aa +/Iotatonos 16#038a +/Ismall 16#f769 +/Istroke 16#0197 +/Itilde 16#0128 +/Itildebelow 16#1e2c +/Izhitsacyrillic 16#0474 +/Izhitsadblgravecyrillic 16#0476 +/J 16#004a +/Jaarmenian 16#0541 +/Jcircle 16#24bf +/Jcircumflex 16#0134 +/Jecyrillic 16#0408 +/Jheharmenian 16#054b +/Jmonospace 16#ff2a +/Jsmall 16#f76a +/K 16#004b +/KBsquare 16#3385 +/KKsquare 16#33cd +/Kabashkircyrillic 16#04a0 +/Kacute 16#1e30 +/Kacyrillic 16#041a +/Kadescendercyrillic 16#049a +/Kahookcyrillic 16#04c3 +/Kappa 16#039a +/Kastrokecyrillic 16#049e +/Kaverticalstrokecyrillic 16#049c +/Kcaron 16#01e8 +/Kcedilla 16#0136 +/Kcircle 16#24c0 +/Kcommaaccent 16#0136 +/Kdotbelow 16#1e32 +/Keharmenian 16#0554 +/Kenarmenian 16#053f +/Khacyrillic 16#0425 +/Kheicoptic 16#03e6 +/Khook 16#0198 +/Kjecyrillic 16#040c +/Klinebelow 16#1e34 +/Kmonospace 16#ff2b +/Koppacyrillic 16#0480 +/Koppagreek 16#03de +/Ksicyrillic 16#046e +/Ksmall 16#f76b +/L 16#004c +/LJ 16#01c7 +/LL 16#f6bf +/Lacute 16#0139 +/Lambda 16#039b +/Lcaron 16#013d +/Lcedilla 16#013b +/Lcircle 16#24c1 +/Lcircumflexbelow 16#1e3c +/Lcommaaccent 16#013b +/Ldot 16#013f +/Ldotaccent 16#013f +/Ldotbelow 16#1e36 +/Ldotbelowmacron 16#1e38 +/Liwnarmenian 16#053c +/Lj 16#01c8 +/Ljecyrillic 16#0409 +/Llinebelow 16#1e3a 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Tuning the parameters in |image3| -.. |image1| image:: pc2_process_explanation +.. |image1| image:: pc2_process_explanation.eps .. |image2| image:: Timestepping_ctl66.epsi .. |image3| image:: Timestepping_pc266.epsi diff --git a/documentation/source/science_guide/cloud_schemes/pc2_process_explanation.eps b/documentation/source/science_guide/cloud_schemes/pc2_process_explanation.eps new file mode 100644 index 0000000000..b9caaec6e0 --- /dev/null +++ b/documentation/source/science_guide/cloud_schemes/pc2_process_explanation.eps @@ -0,0 +1,9032 @@ +%!PS-Adobe-3.0 EPSF-3.0 +%%Invocation: path/gs -q -sDEVICE=eps2write -sstdout=? -sOutputFile=? -dNOPAUSE -dBATCH -P- -dSAFER -dDEVICEWIDTH=250000 -dDEVICEHEIGHT=250000 ? +%%BoundingBox: 36 73 558 770 +%%HiResBoundingBox: 36.00 73.00 558.00 769.02 +%%Creator: GPL Ghostscript 9540 (eps2write) +%%LanguageLevel: 2 +%%CreationDate: D:20250417013840+01'00' +%%Pages: 1 +%%EndComments +%%BeginProlog +10 dict dup begin +/DSC_OPDFREAD true def +/SetPageSize false def +/EPS2Write true def +end +count 0 ne{ +dup type/dicttype eq{ +dup/EPS2Write known{ +dup/EPS2Write get not +} +{ +true +}ifelse +} +{ +true +}ifelse +} +{ +true +}ifelse +10 dict begin +/this currentdict def +/y 720 def +/ebuf 200 string def +/prnt{ +36//this/y get moveto//ebuf cvs show +//this/y 2 copy get 12 sub put +}bind def +/newline{ +36//this/y get moveto +//this/y 2 copy get 12 sub put +}bind def +{ +errordict/handleerror +{systemdict begin +$error begin +newerror +{(%%[ Error handled by opdfread.ps : )print errorname//ebuf cvs print(; OffendingCommand: ) +print/command load//ebuf cvs print( ]%%)= flush +/newerror false store vmstatus pop pop 0 ne +{grestoreall +}if +errorname(VMerror)ne +{showpage +}if +initgraphics +0 720 moveto +errorname(VMerror)eq +{//this/ehsave known +{clear//this/ehsave get restore 2 vmreclaim +}if +vmstatus exch pop exch pop +} +/Courier 12 selectfont +{ +(ERROR: )//prnt exec errorname//prnt exec +(OFFENDING COMMAND: )//prnt exec +/command load//prnt exec +$error/ostack known{ +(%%[STACK:)= +(STACK:)//prnt exec +$error/ostack get aload length{ +//newline exec +dup mark eq{ +(-mark-)dup = show +}{ +dup type/nametype eq{ +dup xcheck not{ +(/)show +(/)print +}if +}if +dup =//ebuf cvs show +}ifelse +}repeat +}if +}ifelse +(%%]%)= +//systemdict/showpage get exec +quit +}if +end +end +}bind readonly put +}if +end +50 dict begin +count 0 ne{ +dup type/dicttype eq{ +{def}forall +false +} +{ +true +}ifelse +} +{ +true +}ifelse +{ +( *** Warning: global definitions dictionary not found, file may be corrupted.\n)print flush +}if +/DefaultSwitch +{ +dup where{ +pop pop +}{ +false def +}ifelse +}bind def +/=string 256 string def +/=only{ +//=string cvs print +}bind def +/HexDigits(0123456789ABCDEF)readonly def +/PrintHex +{8{ +dup -28 bitshift 15 and//HexDigits exch 1 getinterval//=only exec +4 bitshift +}repeat +pop +}bind def +/PDFR_DEBUG DefaultSwitch +/PDFR_DUMP DefaultSwitch +/PDFR_STREAM DefaultSwitch +/TTFDEBUG DefaultSwitch +/RotatePages DefaultSwitch +/FitPages DefaultSwitch +/CenterPages DefaultSwitch +/SetPageSize DefaultSwitch +/error +{ +counttomark 1 sub -1 0{ +index dup type/arraytype eq{==}{=only}ifelse +}for +()= +cleartomark +....Undefined +}bind def +//SetPageSize{ +//RotatePages//FitPages or//CenterPages or{ +mark(/RotatePages, /FitPages and CenterPages are not allowed with /SetPageSize)//error exec +}if +} +{ +//FitPages//CenterPages and{ +mark(CenterPages is not allowed with /FitPages)//error exec +}if +} +ifelse +/knownget +{ +2 copy known{ +get true +}{ +pop pop false +}ifelse +}bind def +/IsUpper +{dup(A)0 get ge exch(Z)0 get le and +}bind def +/cpa2g{ +dup length array +0 1 2 index length 1 sub{ +dup 3 index exch get cp2g +3 copy put pop pop +}for +exch pop +}bind def +/cpd2g{ +dup length dict exch{ +cp2g 2 index 3 1 roll put +}forall +}bind def +/cps2g{ +dup length string copy +}bind def +/cp2gprocs +<> +def +/cp2g{ +dup gcheck not{ +dup//cp2gprocs 1 index type +2 copy known{ +get currentglobal 3 1 roll true setglobal exec exch setglobal +1 index wcheck not{readonly}if +1 index xcheck{cvx}if +exch pop +}{ +pop pop +}ifelse +}if +}bind def +/BlockBuffer 65535 string def +/PDFReader currentdict def +/ObjectRegistryMaxLength 50000 def +/ObjectRegistry 10 dict def +ObjectRegistry +begin +0 ObjectRegistryMaxLength dict def +end +/CurrentObject null def +/DoneDocumentStructure false def +/GraphicState 20 dict begin +/InitialTextMatrix matrix def +/InitialMatrix matrix currentmatrix def +currentdict end def +/TempMatrix matrix def +/GraphicStateStack 20 array def +/GraphicStateStackPointer 0 def +/InitialTextMatrixStack 20 array def +/InitialTextMatrixStackPointer 0 def +/PDFColorSpaces 50 dict def +/InstalledFonts 50 dict def +/MacRomanEncodingInverse null def +currentglobal false setglobal +userdict/PDFR_InitialGS gstate put +userdict/PDFR_Patterns 50 dict put +userdict/FuncDataReader 10 dict put +setglobal +/InitialExtGState 20 dict begin +/BG2 currentblackgeneration cp2g def +/UCR2 currentundercolorremoval cp2g def +/TR2 currentglobal false setglobal[currentcolortransfer]exch setglobal cp2g def +/HT currenthalftone cp2g def +currentdict end readonly def +/InitialGraphicState 20 dict begin +/FontSize 0 def +/CharacterSpacing 0 def +/TextLeading 0 def +/TextRenderingMode 0 def +/WordSpacing 0 def +currentdict end readonly def +/SimpleColorSpaceNames 15 dict begin +/DeviceGray true def +/DeviceRGB true def +/DeviceCMYK true def +currentdict end readonly def +/1_24_bitshift_1_sub 1 24 bitshift 1 sub def +/ReadFontProcs 10 dict def +/GetObject +{ +dup ObjectRegistryMaxLength idiv +//PDFReader/ObjectRegistry get exch knownget{ +exch knownget +}{ +pop false +}ifelse +}bind def +/PutObject +{ +1 index ObjectRegistryMaxLength idiv +//PDFReader/ObjectRegistry get 1 index knownget{ +exch pop +3 1 roll put +}{ +//PDFReader/ObjectRegistry get dup +begin +1 index ObjectRegistryMaxLength dict def +end +exch get +3 1 roll put +}ifelse +}bind def +/Register +{ +1 index GetObject{ +dup xcheck{ +4 3 roll pop +//PDFR_DEBUG{ +(Have a daemon for )print 2 index == +}if +exec +}{ +dup null ne{ +mark(The object )4 index(is already defined : )4 index//error exec +}{ +pop +}ifelse +3 2 roll +exec +}ifelse +}{ +3 2 roll +exec +}ifelse +PutObject +}bind def +/IsRegistered +{ +GetObject{ +null ne +}{ +false +}ifelse +}bind def +/GetRegistered +{ +dup GetObject not{ +exch mark exch(Object )exch( isn't defined before needed (1).)//error exec +}if +dup xcheck{ +exch mark exch(Object )exch( isn't defined before needed (2).)//error exec +}{ +dup null eq{ +exch mark exch(Object )exch( isn't defined before needed (3).)//error exec +}if +exch pop +}ifelse +}bind def +/StandardFontNames<< +/Times-Roman true +/Helvetica true +/Courier true +/Symbol true +/Times-Bold true +/Helvetica-Bold true +/Courier-Bold true +/ZapfDingbats true +/Times-Italic true +/Helvetica-Oblique true +/Courier-Oblique true +/Times-BoldItalic true +/Helvetica-BoldOblique true +/Courier-BoldOblique true +>>def +/CleanAllResources +{//PDFR_DEBUG{ +(CleanAllResources beg)= +}if +//PDFReader/ObjectRegistry get{ +dup length 0 exch 1 exch 1 sub{ +2 copy get dup xcheck{ +pop pop +}{ +dup null eq{ +pop pop +}{ +dup type/dicttype eq{/.Global known}{pop false}ifelse{ +pop +}{ +//PDFR_DEBUG{ +(Dropping )print dup = +}if +1 index exch/DroppedObject put +}ifelse +}ifelse +}ifelse +}for +pop +}forall +FontDirectory length dict begin +FontDirectory{ +pop +dup//StandardFontNames exch known not{ +dup null def +}if +pop +}forall +currentdict +end{ +pop +//PDFR_DEBUG{ +(Undefining font )print dup = +}if +undefinefont +}forall +//PDFR_DEBUG{ +(CleanAllResources end)= +}if +}bind def +/PrintReference +{ +//PDFR_DEBUG{ +({ )print +dup{ +=only( )print +}forall +( })= +}if +}bind def +/R +{ +0 ne{ +exch mark exch(A referred object generation )exch( isn't 0.)//error exec +}if +[ +exch//GetRegistered/exec load +]cvx +//PrintReference exec +}bind def +/IsObjRef +{ +dup type/arraytype eq{ +dup length 3 eq{ +dup xcheck exch +dup 0 get type/integertype eq 3 2 roll and exch +dup 1 get//GetRegistered eq 3 2 roll and exch +2 get/exec load eq and +}{ +pop false +}ifelse +}{ +pop false +}ifelse +}bind def +/DoNothing +{ +}def +/RunTypeDaemon +{ +dup type/dicttype eq{ +dup/Type//knownget exec{ +//PDFReader/TypeDaemons get exch +//knownget exec{ +exec +}if +}if +}if +}bind def +/obj +{ +//PDFR_DEBUG{ +(Defining )print 1 index =only( )print dup =only( obj)= +}if +0 ne{ +exch mark exch(An object generation )exch( isn't 0.)//error exec +}if +}bind def +/endobj +{ +//PDFR_DEBUG{ +(endobj )= +}if +count 1 eq{ +pop +}{ +dup type/dicttype eq{ +dup/.endobj_daemon//knownget exec{ +//PDFR_DEBUG{(.endobj_daemon for )print 2 index =}if +exec +}if +}if +dup type/dicttype eq{dup/ImmediateExec known}{false}ifelse{ +pop pop +}{ +//PDFR_DEBUG{ +(Storing )print 1 index = +}if +//RunTypeDaemon exec +//DoNothing 3 1 roll//Register exec +}ifelse +}ifelse +}bind def +/StoreBlock +{ +//PDFR_DEBUG{ +(StoreBlock )print//PDFReader/BlockCount get =only(, Length = )print dup length = +}if +dup length string copy +//PDFReader/BlockCount get exch +//PDFReader/CurrentObject get 3 1 roll +put +//PDFReader/BlockCount get 1 add +//PDFReader exch/BlockCount exch put +}bind def +/CheckLength +{dup type/integertype ne{ +mark(Object length isn't an integer.)//error exec +}if +}bind def +/ResolveD +{ +3 copy pop get +dup//IsObjRef exec{ +//PDFR_DEBUG{ +(Resolving )print//PrintReference exec +}if +exec +exch exec +}{ +exch pop +}ifelse +dup 4 1 roll +put +}bind def +/ResolveA +{2 index 2 index get +dup//IsObjRef exec{ +exec +exch exec +3 copy put +}{ +exch pop +}ifelse +exch pop exch pop +}bind def +/StoreStream +{ +dup//PDFReader exch/CurrentObject exch put +//PDFReader/BlockCount 0 put +dup/Length//CheckLength//ResolveD exec +//PDFR_DEBUG{ +(StoreStream Length = )print dup = +}if +currentfile exch()/SubFileDecode filter +{dup//BlockBuffer readstring{ +//StoreBlock exec +}{ +//StoreBlock exec +exit +}ifelse +}loop +pop +//PDFReader/CurrentObject null put +//PDFR_DEBUG{ +(StoreStream end.)= +}if +}bind def +/MakeStreamDumper +{ +//PDFR_DEBUG{ +(MakeStreamDumper beg.)= +}if +currentglobal exch dup gcheck setglobal +[exch +1 dict dup/c 0 put exch +1024 string +{readstring pop +(StreamDumper )print 1 index/c get =string cvs print( )print +dup length =string cvs print( <)print dup print(>\n)print +dup length +3 2 roll +dup/c get +3 2 roll +add/c exch put +}/exec load +] +cvx 0()/SubFileDecode filter +exch setglobal +//PDFR_DEBUG{ +(MakeStreamDumper end.)= +}if +}bind def +/ShortFilterNames 15 dict begin +/AHx/ASCIIHexDecode def +/A85/ASCII85Decode def +/LZW/LZWDecode def +/Fl/FlateDecode def +/RL/RunLengthDecode def +/CCF/CCITTFaxDecode def +/DCT/DCTDecode def +currentdict end readonly def +/AppendFilters +{ +//PDFR_DEBUG{ +(AppendFilters beg.)= +}if +dup 3 1 roll +/Filter//knownget exec{ +dup type/nametype eq{ +dup//ShortFilterNames exch//knownget exec{ +exch pop +}if +2 index/DecodeParms//knownget exec{ +exch +}if +filter +}{ +dup 0 exch 1 exch length 1 sub{ +2 copy get +dup//ShortFilterNames exch//knownget exec{ +exch pop +}if +3 1 roll +4 index/DecodeParms//knownget exec{ +exch get +}{ +pop null +}ifelse +dup null eq{ +pop 3 1 roll filter exch +}{ +3 1 roll +4 1 roll filter exch +}ifelse +}for +pop +}ifelse +//PDFR_DEBUG//PDFR_DUMP and{ +//MakeStreamDumper exec +}if +}if +exch pop +//PDFR_DEBUG{ +(AppendFilters end.)= +}if +}bind def +/ExecuteStream +{ +dup//PDFReader exch/CurrentObject exch put +dup/Length//CheckLength//ResolveD exec +//PDFR_DEBUG{ +(ExecuteStream id = )print 2 index =only( Length = )print dup = +}if +//PDFReader/InitialGraphicState get +//PDFReader/GraphicState get copy pop +//PDFReader/Operators get begin +currentfile exch()/SubFileDecode filter +1 index//AppendFilters exec +cvx mark exch +exec +counttomark 0 ne{ +mark(Data left on ostack after an immediate stream execution.)//error exec +}if +cleartomark +end +//PDFR_DEBUG{ +(ExecuteStream end.)= +}if +//PDFReader/CurrentObject null put +dup/IsPage known{ +dup/Context get/NumCopies//knownget exec{ +1 sub{ +copypage +}repeat +}if +EPS2Write not{showpage}if +pagesave restore +}if +}bind def +/stream +{ +//PDFR_DEBUG{ +1 index =only( stream)= +}if +1 index GetObject{ +dup xcheck{ +exec +1 index null PutObject +}{ +pop +}ifelse +}if +dup/ImmediateExec known{ +dup/GlobalExec//knownget exec{ +currentglobal 4 1 roll +setglobal +//ExecuteStream exec +3 2 roll setglobal +}{ +//ExecuteStream exec +}ifelse +}{ +//StoreStream exec +}ifelse +dup/.CleanResources//knownget exec{ +/All eq{ +//CleanAllResources exec +}if +}if +}bind def +/HookFont +{ +//PDFR_DEBUG{ +(Loaded the font )print dup/FontName get = +}if +{ +dup/FontFileType get dup/Type1 eq exch/MMType1 eq or{ +dup/FontName get +//PDFReader/RemoveFontNamePrefix get exec +findfont +exit +}if +dup/FontFileType get/TrueType eq{ +//PDFReader/MakeType42 get exec +//PDFR_DEBUG{ +(Font dict <<)= +dup{ +1 index/sfnts eq{ +exch pop +(/sfnts [)print +{ +(-string\()print length//=only exec(\)- )= +}forall +(])= +}{ +exch//=only exec( )print == +}ifelse +}forall +(>>)= +}if +dup/FontName get exch definefont +exit +}if +mark(FontHook has no proc for )2 index/FontFileType get//error exec +}loop +/Font exch put +}bind def +/endstream +{ +}bind def +/xref +{ +//PDFR_DEBUG{ +(xref)= +//PDFR_DUMP{ +//PDFReader/ObjectRegistry get == +}if +}if +end +count 0 ne{ +mark(Excessive data on estack at the end of the interpretation.)//error exec +}if +currentfile 1(%%EOF)/SubFileDecode filter +flushfile +cleardictstack +}bind def +/ResolveDict +{dup{ +pop 1 index exch +//DoNothing//ResolveD exec +pop +}forall +pop +}bind def +/SetupPageView +{ +//PDFR_DEBUG{ +(SetupPageView beg)= +}if +//DSC_OPDFREAD not{ +//GraphicState/InitialMatrix get setmatrix +}if +/MediaBox get aload pop +3 index neg 3 index neg translate +3 -1 roll sub 3 1 roll exch sub exch +userdict/.HWMargins//knownget exec{ +aload pop +}{ +currentpagedevice/.HWMargins//knownget exec{ +aload pop +}{ +0 0 0 0 +}ifelse +}ifelse +currentpagedevice/PageSize get aload pop +3 -1 roll sub 3 1 roll exch sub exch +exch 3 index sub exch 3 index sub +//SetPageSize{ +//PDFR_DEBUG{ +(Setting page size to )print 1 index//=only exec( )print dup = +}if +pop pop 3 index 3 index 2 copy +currentglobal false setglobal 3 1 roll +currentpagedevice dup/PageSize known{ +/PageSize get aload pop +}{ +0 0 +}ifelse +round cvi 2 index round cvi eq +exch round cvi 3 index round cvi eq and +{ +//PDFR_DEBUG{(PageSize matches request)== flush}if +pop pop +}{ +/MediaRequested where{ +//PDFR_DEBUG{(MediaRequested is true, check against new request)== flush}if +/MediaRequested get aload pop +round cvi 2 index round cvi eq +exch round cvi 3 index round cvi eq and +{ +//PDFR_DEBUG{(MediaRequested same as current request, ignore)== flush}if +pop pop false +}{ +//PDFR_DEBUG{(MediaRequested different to current request)== flush}if +true +}ifelse +}{ +//PDFR_DEBUG{(No MediaRequested yet)== flush}if +true +}ifelse +{ +//PDFR_DEBUG{(Setting pagesize)== flush}if +2 array astore +dup/MediaRequested exch def +<< exch/PageSize exch >>setpagedevice +}if +}ifelse +userdict/PDFR_InitialGS gstate put +setglobal +}if +//RotatePages{ +2 copy gt 6 index 6 index gt ne{ +1 index 5 index le 1 index 5 index le and not +}{ +false +}ifelse +}{ +false +}ifelse +{//CenterPages{ +//PDFR_DEBUG{ +(Rotating page, and then centering it)== +}if +90 rotate +0 5 index neg translate +5 index 1 index exch sub 2 div +2 index 6 index sub 2 div neg +translate +}{ +//FitPages{ +1 index 5 index div 1 index 7 index div +2 copy gt{ +exch +}if +pop dup scale +}if +90 rotate +0 5 index neg translate +}ifelse +}{ +//CenterPages{ +//PDFR_DEBUG{ +(Ccentering page)== +}if +1 index 6 index sub 2 div +1 index 6 index sub 2 div +translate +}{ +//FitPages{ +1 index 6 index div 1 index 6 index div +2 copy gt{ +exch +}if +pop dup scale +}if +}ifelse +}ifelse +pop pop +translate +pop pop +//PDFR_DEBUG{ +(SetupPageView end)= +}if +}bind def +/PageContentsDaemon +{ +//PDFR_DEBUG{ +(Executing PageContentsDaemon for )print 2 index = +}if +1 index exch/Context exch put +dup/ImmediateExec true put +/pagesave save def +dup/IsPage true put +SetPageSize{dup/Context get//SetupPageView exec}if +}bind def +/FontFileDaemon +{ +//PDFR_DEBUG{ +(Executing FontFileDaemon for )print 2 index = +}if +dup/FontFileType get +2 index exch +dup//ReadFontProcs exch//knownget exec{ +exch pop exec +}{ +mark(FontFile reader for )2 index( isn't implemented yet.)//error exec +}ifelse +//PDFR_DEBUG{ +(FontFileDaemon end)= +}if +pop +}bind def +/FontDescriptorDaemon +{ +//PDFR_DEBUG{ +(Executing FontDescriptorDaemon for )print 2 index = +}if +2 copy/FontResource exch put +/Subtype get 1 index exch/FontFileType exch put +}bind def +/UnPDFEscape{ +dup dup length string cvs +dup(#)search{ +{ +pop +(16#--)2 index 0 2 getinterval +1 index 3 2 getinterval copy pop +cvi +0 exch put +0 +1 index 2 1 index length 2 sub getinterval +3 copy putinterval +length +3 copy exch put +getinterval +(#)search not{ +pop exit +}if +}loop +(\0)search pop exch pop exch pop +cvn +exch pop +}{ +pop pop +}ifelse +}bind def +/TypeDaemons<< +/Page +{//PDFR_DEBUG{ +(Recognized a page.)= +}if +dup/Contents//knownget exec{ +0 get//DoNothing exch +[ +3 index//PageContentsDaemon/exec load +]cvx +//Register exec +}{ +(fixme: page with no Contents won't be printed.)= +}ifelse +}bind +/FontDescriptor +{//PDFR_DEBUG{ +(Recognized a font descriptor.)= +}if +dup/FontName//knownget exec{ +1 index/FontName 3 -1 roll//UnPDFEscape exec put +}if +dup dup/FontFile known{/FontFile}{/FontFile2}ifelse +//knownget exec{ +0 get//DoNothing exch +[ +3 index//FontFileDaemon/exec load +]cvx +//Register exec +}{ +(Font descriptor )print 1 index =only( has no FontFile.)= +}ifelse +}bind +/Font +{//PDFR_DEBUG{ +(Recognized a font resource.)= +}if +dup/BaseFont//knownget exec{ +//UnPDFEscape exec 2 copy/BaseFont exch put +//PDFReader/RemoveFontNamePrefix get exec +currentglobal exch +dup/Font resourcestatus{ +pop pop +//PDFReader/GetInstalledFont get exec pop +}{ +pop +}ifelse +setglobal +}if +dup/FontDescriptor//knownget exec{ +0 get +dup//IsRegistered exec{ +//PDFR_DEBUG{ +(already registered )print dup = +}if +pop +}{ +//DoNothing exch +[ +3 index//FontDescriptorDaemon/exec load +]cvx +//Register exec +}ifelse +}if +}bind +>>def +/MakeStreamReader +{dup +[ +exch +//PDFR_DEBUG{ +(Stream proc ) +/print load +//PDFR_STREAM{ +(<) +/print load +}if +}if +1 dict dup/i -1 put +/dup load +/i +/get load +1 +/add load +/dup load +3 +1 +/roll load +/i +/exch load +/put load +//knownget +/exec load +/not load +{()} +/if load +//PDFR_DEBUG{ +//PDFR_STREAM{ +/dup load +/print load +(>) +/print load +}if +( end of stream proc.\n) +/print load +}if +]cvx +//PDFR_DEBUG{ +(Stream reader )print dup == +}if +0()/SubFileDecode filter +exch//AppendFilters exec +}bind def +/RunDelayedStream +{ +//GraphicState/InitialTextMatrix get +//InitialTextMatrixStack//PDFReader/InitialTextMatrixStackPointer get +2 copy get null eq{ +2 copy currentglobal true setglobal matrix exch setglobal put +}if +get copy pop +//PDFReader/InitialTextMatrixStackPointer 2 copy get 1 add put +//MakeStreamReader exec +mark exch +cvx exec +counttomark 0 ne{ +mark(Data left on ostack after a delayed stream execution.)//error exec +}if +cleartomark +//PDFReader/InitialTextMatrixStackPointer 2 copy get 1 sub put +//InitialTextMatrixStack//PDFReader/InitialTextMatrixStackPointer get get +//GraphicState/InitialTextMatrix get +copy pop +}bind def +//ReadFontProcs begin +/Type1 +{//PDFR_DEBUG{ +(ReadFontProcs.Type1)= +}if +dup/.endobj_daemon[4 index//HookFont/exec load]cvx put +dup/ImmediateExec true put +/GlobalExec true put +}bind def +/MMType1//Type1 def +/TrueType +{//PDFR_DEBUG{ +(ReadFontProcs.TrueType)= +}if +dup/.endobj_daemon[4 index//HookFont/exec load]cvx put +pop +}bind def +end +/.opdloadttfontdict 50 dict def +.opdloadttfontdict begin +/maxstring 65400 def +end +/.InsertionSort +{ +/CompareProc exch def +/Array exch def +1 1 Array length 1 sub +{ +/Ix exch def +/Value1 Array Ix get def +/Jx Ix 1 sub def +{ +Jx 0 lt{ +exit +}if +/Value2 Array Jx get def +Value1 Value2 CompareProc{ +exit +}if +Array Jx 1 add Value2 put +/Jx Jx 1 sub def +}loop +Array Jx 1 add Value1 put +}for +Array +}bind def +/putu16{ +3 copy -8 bitshift put +exch 1 add exch 16#ff and put +}bind def +/putu32{ +3 copy -16 bitshift putu16 +exch 2 add exch 16#ffff and putu16 +}bind def +/.readtable{ +dup dup 1 and add string +dup 0 4 -1 roll getinterval +3 -1 roll exch +dup()ne{readstring}if pop pop +}bind def +/.readbigtable{ +dup maxstring lt{ +.readtable +}{ +currentuserparams/VMReclaim get -2 vmreclaim +[4 2 roll{ +dup maxstring le{exit}if +1 index maxstring string readstring pop 3 1 roll maxstring sub +}loop .readtable] +exch vmreclaim +}ifelse +}bind def +/ReadTTF +{ +.opdloadttfontdict begin +/TTFontFile exch def +/TableDir TTFontFile 12 string readstring pop def +/tables TTFontFile TableDir 4 getu16 16 mul string readstring pop def +/tabarray tables length 16 idiv array def +TableDir 0 4 getinterval(ttcf)eq{ +QUIET not{(Can't handle TrueType font Collections.)=}if +/.loadttfonttables cvx/invalidfont signalerror +}{ +0 16 tables length 1 sub{ +dup +tables exch 16 getinterval +exch 16 div cvi exch +tabarray 3 1 roll put +}for +}ifelse +tabarray{exch 8 getu32 exch 8 getu32 gt}.InsertionSort pop +/Read TableDir length tables length add def +/tabs[ +tabarray{ +dup 8 getu32 +Read sub +dup 0 gt{ +dup string TTFontFile exch readstring pop pop +Read add/Read exch def +}{ +pop +}ifelse +12 getu32 +dup Read add +/Read exch def +TTFontFile exch .readbigtable +}forall +]def +end +}bind def +/GetLocaType +{ +0 1 tabarray length 1 sub{ +dup tabarray exch get +0 4 getinterval(head)eq{ +tabs exch get +50 gets16 +/LocaType exch def +exit +}{ +pop +}ifelse +}for +}bind def +/GetNumGlyphs +{ +0 1 tabarray length 1 sub{ +dup tabarray exch get +0 4 getinterval(maxp)eq{ +tabs exch get +4 getu16 +/NumGlyphs exch def +exit +}{ +pop +}ifelse +}for +}bind def +/StringToLoca +{ +/LocaIndex exch def +/StringOffset 0 def +{ +dup length StringOffset gt{ +dup +LocaType 1 eq{ +StringOffset getu32 +LocaArray LocaIndex 3 -1 roll put +/LocaIndex LocaIndex 1 add def +/StringOffset StringOffset 4 add +def +}{ +StringOffset getu16 2 mul +LocaArray length LocaIndex gt{ +LocaArray LocaIndex 3 -1 roll put +}{ +pop +}ifelse +/LocaIndex LocaIndex 1 add def +/StringOffset StringOffset 2 add +def +}ifelse +}{ +pop +LocaIndex +exit +}ifelse +}loop +}bind def +/GetSortedLoca +{ +NumGlyphs 1 add array/LocaArray exch def +0 1 tabarray length 1 sub{ +dup tabarray exch get +0 4 getinterval(loca)eq{ +tabs exch get +exit +}{ +pop +}ifelse +}for +dup type/stringtype eq{ +0 StringToLoca pop +}{ +0 exch +{ +exch StringToLoca +}forall +pop +}ifelse +LocaArray{gt}.InsertionSort pop +}bind def +/GetWorkingString +{ +WorkString 0 +GlyfArray GlyfStringIndex get +putinterval +/WorkBytes GlyfArray GlyfStringIndex get length def +/GlyfStringIndex GlyfStringIndex 1 add def +}bind def +/GetWorkingBytes +{ +/BytesToRead exch def +WorkString 0 BytesToRead getinterval +dup length string copy +WorkString BytesToRead WorkBytes BytesToRead sub getinterval +dup length string copy +WorkString 0 3 -1 roll putinterval +/WorkBytes WorkBytes BytesToRead sub def +}bind def +/GetGlyfBytes +{ +/ToRead exch def +WorkBytes 0 eq{ +GetWorkingString +}if +WorkBytes ToRead ge{ +ToRead string dup 0 +ToRead GetWorkingBytes putinterval +}{ +ToRead string +dup +0 +WorkString 0 WorkBytes getinterval +putinterval +dup +WorkBytes +ToRead WorkBytes sub +GetWorkingString +GetWorkingBytes +putinterval +}ifelse +}bind def +/SplitGlyf +{ +/GlyfArray exch def +/DestArray GlyfArray length 2 mul array def +/DestArrayIndex 0 def +/LastLoca 0 def +/NextLocaIndex 0 def +/LastLocaIndex 0 def +/GlyfStringIndex 0 def +/WorkString maxstring string def +/WorkBytes 0 def +{ +LocaArray NextLocaIndex get +LastLoca sub maxstring gt +{ +LocaArray LastLocaIndex get LastLoca sub +GetGlyfBytes +DestArray DestArrayIndex 3 -1 roll put +/DestArrayIndex DestArrayIndex 1 add def +LocaArray LastLocaIndex get/LastLoca exch def +}{ +/LastLocaIndex NextLocaIndex def +/NextLocaIndex NextLocaIndex 1 add def +NextLocaIndex NumGlyphs gt +{ +WorkBytes +GlyfStringIndex GlyfArray length lt{ +GlyfArray GlyfStringIndex get length +add string dup +0 +WorkString 0 WorkBytes getinterval +putinterval +dup +WorkBytes +GetWorkingString +WorkString 0 WorkBytes getinterval +putinterval +}{ +pop +WorkString 0 WorkBytes getinterval +}ifelse +dup length string copy +DestArray DestArrayIndex 3 -1 roll put +exit +}if +}ifelse +}loop +DestArray +}bind def +/ProcessTTData +{ +.opdloadttfontdict begin +0 1 tabarray length 1 sub{ +/ix exch def +tabarray ix get +12 getu32 dup maxstring le{ +dup 4 mod 0 ne{ +4 div cvi 1 add 4 mul string/newstring exch def +/oldstring tabs ix get def +newstring 0 oldstring putinterval +0 1 newstring length oldstring length sub 1 sub{ +newstring exch oldstring length add 0 put +}for +tabs ix newstring put +}{ +pop +}ifelse +}{ +dup 4 mod 0 ne{ +dup maxstring idiv maxstring mul sub +4 idiv 1 add 4 mul string/newstring exch def +tabs ix get +dup length 1 sub dup/iy exch def get/oldstring exch def +newstring 0 oldstring putinterval +0 1 newstring length oldstring length sub 1 sub{ +newstring exch oldstring length add 0 put +}for +tabs ix get iy newstring put +}{ +pop +}ifelse +}ifelse +}for +0 1 tabarray length 1 sub{ +dup tabarray exch get +dup 12 getu32 maxstring gt{ +0 4 getinterval dup(glyf)eq{ +pop +GetLocaType +GetNumGlyphs +GetSortedLoca +dup tabs exch get +SplitGlyf +tabs 3 1 roll put +}{ +(Warning, table )print print( > 64Kb\n)print +pop +}ifelse +}{ +pop +pop +}ifelse +}for +end +}bind def +/Makesfnts +{ +.opdloadttfontdict begin +0 +tabs{ +dup type/stringtype eq{ +pop +1 add +}{ +{ +type/stringtype eq{ +1 add +}if +}forall +}ifelse +}forall +1 add +/TTOffset +TableDir length +tabarray length 16 mul add +def +0 +tabarray{ +exch dup 1 add +3 1 roll +dup +tabs exch get +dup type/stringtype eq{ +length +2 index exch +TTOffset +dup 3 1 roll add +/TTOffset exch def +8 exch putu32 +exch tabarray 3 1 roll +put +}{ +0 exch +{ +dup type/stringtype eq{ +length add +}{ +pop +}ifelse +}forall +2 index exch +TTOffset +dup 3 1 roll add +/TTOffset exch def +8 exch putu32 +exch tabarray 3 1 roll +put +}ifelse +}forall +pop +array +dup 0 +TableDir length +tables length add +string +dup 0 TableDir putinterval +dup 12 tables putinterval +put +dup +/ix 1 def +tabs{ +dup type/stringtype eq{ +ix exch +put dup +/ix ix 1 add def +}{ +{ +dup type/stringtype eq{ +ix exch put dup +/ix ix 1 add def +}{ +pop +}ifelse +}forall +}ifelse +}forall +pop +end +}bind def +/MakeType42 +{ +//PDFR_DEBUG{ +(MakeType42 beg)= +}if +10 dict begin +/FontName 1 index/FontName get def +/FontType 42 def +/FontMatrix[1 0 0 1 0 0]def +/FontBBox 1 index/FontBBox get def +dup/FontResource get +dup/Encoding known{ +//PDFReader/ObtainEncoding get exec +/Encoding get +}{ +pop null +}ifelse +/PDFEncoding exch def +/CharStrings 2 index//PDFReader/MakeTTCharStrings get exec def +/sfnts 2 index//MakeStreamReader exec +ReadTTF +ProcessTTData +Makesfnts +def +/Encoding StandardEncoding def +/PaintType 0 def +currentdict end +//PDFR_DEBUG{ +(MakeType42 end)= +}if +}bind def +/GetInstalledFont +{ +dup//InstalledFonts exch knownget{ +exch pop +}{ +dup findfont dup 3 1 roll +//InstalledFonts 3 1 roll put +}ifelse +}bind def +/RemoveFontNamePrefix +{//=string cvs true +0 1 5{ +2 index exch get//IsUpper exec not{ +pop false exit +}if +}for +{(+)search{ +pop pop +}if +}if +cvn +}bind def +/CheckFont +{dup/Type get/Font ne{ +mark(Resource )3 index( must have /Type/Font .)//error exec +}if +}bind def +/CheckEncoding +{dup type/nametype ne{ +dup/Type get/Encoding ne{ +mark(Resource )3 index( must have /Type/Encoding .)//error exec +}if +}if +}bind def +/ObtainEncoding +{dup/Encoding known{ +dup dup/Encoding//CheckEncoding//ResolveD exec +dup type dup/arraytype eq exch/packedarraytype eq or{ +pop pop +}{ +dup type/nametype eq{ +/Encoding findresource +}{ +dup/BaseEncoding//knownget exec not{ +/StandardEncoding +}if +/Encoding findresource +exch +/Differences//knownget exec{ +exch dup length array copy exch +0 exch +{ +dup type/integertype eq{ +exch pop +}{ +3 copy put pop +1 add +}ifelse +}forall +pop +}if +}ifelse +/Encoding exch put +}ifelse +}{ +dup/Encoding/StandardEncoding/Encoding findresource put +}ifelse +}bind def +/ObtainMetrics +{dup/Widths//knownget exec{ +1 index/Encoding get +256 dict +3 index/Subtype get/TrueType eq{ +1000 +}{ +1 +}ifelse +4 index/MissingWidth//knownget exec not{ +0 +}if +5 index/FirstChar//knownget exec not{ +0 +}if +6 5 roll +dup 0 exch 1 exch length 1 sub{ +2 copy get +exch 3 index add +7 index exch get +dup dup null ne exch/.notdef ne and{ +6 index 3 1 roll exch +6 index div +3 copy pop//knownget exec{ +0 eq +}{ +true +}ifelse +{put +}{ +pop pop pop +}ifelse +}{ +pop pop +}ifelse +}for +pop pop pop pop exch pop +1 index exch/Metrics exch put +}{ +dup/MissingWidth//knownget exec{ +256 dict +2 index/Encoding get{ +dup null ne{ +3 copy 3 2 roll put +}if +pop +}forall +exch pop +1 index exch/Metrics exch put +}if +}ifelse +}bind def +/NotDef +{ +FontMatrix aload pop pop pop exch pop exch pop +1 exch div exch +1 exch div exch +1 index 0 setcharwidth +0 setlinewidth +0 0 moveto +2 copy rlineto +1 index 0 rlineto +neg exch neg exch rlineto +closepath stroke +}bind def +/SaveResourcesToStack +{ +[ +//PDFReader/OldResources known{ +//PDFReader/OldResources get +}{ +null +}ifelse +//PDFReader/CurrentObject get/Context get/Resources get +] +//PDFReader/OldResources 3 -1 roll put +}bind def +/RestoreResourcesFromStack +{ +//PDFReader/OldResources get dup +0 get//PDFReader/OldResources 3 -1 roll put +1 get//PDFReader/CurrentObject get/Context get/Resources 3 -1 roll put +}bind def +/BuildChar +{//PDFR_DEBUG{ +(BuildChar )print dup//=only exec( )print +}if +exch begin +Encoding exch get +//PDFR_DEBUG{ +dup = +}if +dup null eq{ +pop//NotDef exec +} +{ +CharProcs exch//knownget exec +{ +currentfont/Font get/Resources//knownget exec{ +exec +SaveResourcesToStack +//PDFReader/CurrentObject get/Context get +/Resources 3 -1 roll put +//RunDelayedStream exec +RestoreResourcesFromStack +}{ +//RunDelayedStream exec +}ifelse +} +{ +//NotDef exec +}ifelse +}ifelse +end +}bind def +/printdict +{(<<)= +{exch = ==}forall +(>>)= +}bind def +/printfont +{ +dup{ +exch dup = +dup/Encoding eq{ +pop = +}{ +dup/FontInfo eq exch/Private eq or{ +//printdict exec +}{ +== +}ifelse +}ifelse +}forall +}bind def +/ScaleMetrics +{1 index{ +2 index div +3 index +3 1 roll put +}forall +pop +}bind def +/ResolveAndSetFontAux +{exch dup +//PDFReader/CurrentObject get/Context get/Resources get +/Font//DoNothing//ResolveD exec +exch//CheckFont//ResolveD exec +dup/Font//knownget exec{ +exch pop exch pop +}{ +{ +dup/Subtype get dup dup/Type1 eq exch/TrueType eq or exch/MMType1 eq or{ +exch pop +dup/BaseFont get +//RemoveFontNamePrefix exec +//PDFR_DEBUG{ +(Font )print dup = +}if +1 index/FontDescriptor known{ +//PDFR_DEBUG{ +(Font from a font descriptor.)= +}if +1 index +/FontDescriptor//DoNothing//ResolveD exec +/Font//knownget exec{ +exch pop +}{ +//PDFR_DEBUG{ +(Font descriptor has no Font resolved.)= +}if +//GetInstalledFont exec +}ifelse +}{ +//GetInstalledFont exec +}ifelse +exch +dup/Encoding known not{ +1 index/Encoding get 1 index exch/Encoding exch put +}if +//ObtainEncoding exec +//ObtainMetrics exec +exch +dup length dict copy +dup 2 index/Encoding get +/Encoding exch put +1 index/Metrics//knownget exec{ +2 index/Subtype get/TrueType ne{ +1 index/FontMatrix get 0 get +dup 0 eq{ +pop +1 index/FontMatrix get 1 get +dup 0 eq{pop 1}if +}if +0.001 div +//ScaleMetrics exec +}{ +1 index/sfnts known not{ +1 index/FontMatrix get 0 get +dup 0 eq{ +pop +1 index/FontMatrix get 1 get +dup 0 eq{pop 1}if +}if +//ScaleMetrics exec +}if +}ifelse +1 index exch/Metrics exch put +}if +1 index/BaseFont get +exch +dup/FID undef +dup/UniqueID undef +definefont +dup 3 1 roll +/Font exch put +exit +}if +dup/Subtype get/Type3 eq{ +//ObtainEncoding exec +2 copy exch/FontName exch put +dup/CharProcs get//ResolveDict exec +dup/FontType 3 put +dup/BuildChar//BuildChar put +dup dup/Font exch put +dup 3 1 roll +definefont +2 copy ne{ +2 copy/Font exch put +}if +exch pop +exit +}if +dup/Subtype get/Type0 eq{ +}if +dup/Subtype get/CIDFontType0 eq{ +}if +dup/Subtype get/CIDFontType2 eq{ +}if +mark(Unknown font type )2 index/Subtype get//error exec +}loop +}ifelse +exch scalefont setfont +}bind def +/ResolveAndSetFont +{ +//ResolveAndSetFontAux exec +}bind def +/.knownget +{2 copy known{ +get true +}{ +pop pop false +}ifelse +}bind def +/.min +{2 copy lt{ +exch +}if +pop +}bind def +/.max +{2 copy gt{ +exch +}if +pop +}bind def +/.dicttomark +{>> +}bind def +/getu16{ +2 copy get 8 bitshift 3 1 roll 1 add get add +}bind def +/gets16{ +getu16 16#8000 xor 16#8000 sub +}bind def +/getu32{ +2 copy getu16 16 bitshift 3 1 roll 2 add getu16 add +}bind def +/gets32{ +2 copy gets16 16 bitshift 3 1 roll 2 add getu16 add +}bind def +/cmapformats mark +0{ +6 256 getinterval{}forall 256 packedarray +}bind +2{ +/sHK_sz 2 def +/sH_sz 8 def +dup 2 getu16/cmapf2_tblen exch def +dup 4 getu16/cmapf2_lang exch def +dup 6 256 sHK_sz mul getinterval/sHKs exch def +0 +0 1 255{ +sHKs exch +2 mul getu16 +1 index +1 index +lt{exch}if pop +}for +/sH_len exch def +dup 6 256 sHK_sz mul add +cmapf2_tblen 1 index sub getinterval +/sH_gIA exch def +/cmapf2_glyph_array 65535 array def +/.cmapf2_putGID{ +/cmapf2_ch cmapf2_ch_hi 8 bitshift cmapf2_ch_lo add def +firstCode cmapf2_ch_lo le +cmapf2_ch_lo firstCode entryCount add lt +and{ +sH_offset idRangeOffset add +cmapf2_ch_lo firstCode sub 2 mul +add 6 add +sH_gIA exch getu16 +dup 0 gt{ +idDelta add +cmapf2_glyph_array exch cmapf2_ch exch put +}{ +pop +}ifelse +}{ +}ifelse +}def +16#00 1 16#ff{ +/cmapf2_ch_hi exch def +sHKs cmapf2_ch_hi sHK_sz mul getu16 +/sH_offset exch def +sH_gIA sH_offset sH_sz getinterval +dup 0 getu16/firstCode exch def +dup 2 getu16/entryCount exch def +dup 4 gets16/idDelta exch def +dup 6 getu16/idRangeOffset exch def +pop +sH_offset 0 eq{ +/cmapf2_ch_lo cmapf2_ch_hi def +/cmapf2_ch_hi 0 def +.cmapf2_putGID +}{ +16#00 1 16#ff{ +/cmapf2_ch_lo exch def +.cmapf2_putGID +}for +}ifelse +}for +pop +0 1 cmapf2_glyph_array length 1 sub{ +dup cmapf2_glyph_array exch get +null eq{cmapf2_glyph_array exch 0 put}{pop}ifelse +}for +cmapf2_glyph_array +}bind +4{ +/etab exch def +/nseg2 etab 6 getu16 def +14/endc etab 2 index nseg2 getinterval def +2 add +nseg2 add/startc etab 2 index nseg2 getinterval def +nseg2 add/iddelta etab 2 index nseg2 getinterval def +nseg2 add/idroff etab 2 index nseg2 getinterval def +pop +/firstcode startc 0 getu16 16#ff00 and dup 16#f000 ne{pop 0}if def +/lastcode firstcode def +/striptopbyte false def +/putglyph{ +glyphs code 3 -1 roll put/code code 1 add def +}bind def +/numcodes 0 def/glyphs 0 0 2 nseg2 3 sub{ +/i2 exch def +/scode startc i2 getu16 def +/ecode endc i2 getu16 def +ecode lastcode gt{ +/lastcode ecode def +}if +}for pop +firstcode 16#f000 ge lastcode firstcode sub 255 le and{ +lastcode 255 and +/striptopbyte true def +}{ +lastcode +}ifelse +1 add +array def +glyphs length 1024 ge{ +.array1024z 0 1024 glyphs length 1023 sub{glyphs exch 2 index putinterval}for +glyphs dup length 1024 sub 3 -1 roll +putinterval +}{ +0 1 glyphs length 1 sub{glyphs exch 0 put}for +}ifelse +/numcodes 0 def/code 0 def +0 2 nseg2 3 sub{ +/i2 exch def +/scode startc i2 getu16 def +/ecode endc i2 getu16 def +numcodes scode firstcode sub +exch sub 0 .max dup/code exch code exch add def +ecode scode sub 1 add add numcodes add/numcodes exch def +/delta iddelta i2 gets16 def +TTFDEBUG{ +(scode=)print scode =only +( ecode=)print ecode =only +( delta=)print delta =only +( droff=)print idroff i2 getu16 = +}if +idroff i2 getu16 dup 0 eq{ +pop scode delta add 65535 and 1 ecode delta add 65535 and +striptopbyte{ +/code scode 255 and def +}{ +/code scode def +}ifelse +{putglyph}for +}{ +/gloff exch 14 nseg2 3 mul add 2 add i2 add add def +striptopbyte{ +/code scode 255 and def +}{ +/code scode def +}ifelse +0 1 ecode scode sub{ +2 mul gloff add etab exch getu16 +dup 0 ne{delta add 65535 and}if putglyph +}for +}ifelse +}for glyphs/glyphs null def +}bind +6{ +dup 6 getu16/firstcode exch def dup 8 getu16/ng exch def +firstcode ng add array +0 1 firstcode 1 sub{2 copy 0 put pop}for +dup firstcode ng getinterval +0 1 ng 1 sub{ +dup 2 mul 10 add 4 index exch getu16 3 copy put pop pop +}for pop exch pop +}bind +.dicttomark readonly def +/cmaparray{ +dup 0 getu16 cmapformats exch .knownget{ +TTFDEBUG{ +(cmap: format )print 1 index 0 getu16 = flush +}if exec +}{ +(Can't handle format )print 0 getu16 = flush +0 1 255{}for 256 packedarray +}ifelse +TTFDEBUG{ +(cmap: length=)print dup length = dup == +}if +}bind def +/postremap mark +/Cdot/Cdotaccent +/Edot/Edotaccent +/Eoverdot/Edotaccent +/Gdot/Gdotaccent +/Ldot/Ldotaccent +/Zdot/Zdotaccent +/cdot/cdotaccent +/edot/edotaccent +/eoverdot/edotaccent +/gdot/gdotaccent +/ldot/ldotaccent +/zdot/zdotaccent +.dicttomark readonly def +/get_from_stringarray +{1 index type/stringtype eq{ +get +}{ +exch{ +2 copy length ge{ +length sub +}{ +exch get exit +}ifelse +}forall +}ifelse +}bind def +/getinterval_from_stringarray +{ +2 index type/stringtype eq{ +getinterval +}{ +string exch 0 +4 3 roll{ +dup length +dup 4 index lt{ +3 index exch sub +exch pop 3 1 roll exch pop +}{ +dup 3 1 roll +4 index sub +5 index length 4 index sub +2 copy gt{exch}if pop +dup 3 1 roll +5 index exch getinterval +5 index 4 index 3 index +getinterval +copy pop +exch pop add exch pop 0 exch +dup 3 index length ge{exit}if +}ifelse +}forall +pop pop +}ifelse +}bind def +/string_array_size +{dup type/stringtype eq{ +length +}{ +0 exch{length add}forall +}ifelse +}bind def +/postformats mark +16#00010000{ +pop MacGlyphEncoding +} +16#00020000{ +dup dup type/arraytype eq{0 get}if length 36 lt{ +TTFDEBUG{(post format 2.0 invalid.)= flush}if +pop[] +}{ +/postglyphs exch def +/post_first postglyphs dup type/arraytype eq{0 get}if def +post_first 32 getu16/numglyphs exch def +/glyphnames numglyphs 2 mul 34 add def +/postpos glyphnames def +/total_length postglyphs//string_array_size exec def +numglyphs array 0 1 numglyphs 1 sub{ +postpos total_length ge{ +1 numglyphs 1 sub{1 index exch/.notdef put}for +exit +}if +postglyphs postpos//get_from_stringarray exec +postglyphs postpos 1 add 2 index//getinterval_from_stringarray exec cvn +exch postpos add 1 add/postpos exch def +2 index 3 1 roll +put +}for +/postnames exch def +numglyphs array 0 1 numglyphs 1 sub{ +dup 2 mul 34 add postglyphs exch 2//getinterval_from_stringarray exec +dup 0 get 8 bitshift exch 1 get add dup 258 lt{ +MacGlyphEncoding exch get +}{ +dup 32768 ge{ +pop/.notdef +}{ +258 sub dup postnames length ge{ +TTFDEBUG{( *** warning: glyph index past end of 'post' table)= flush}if +pop +exit +}if +postnames exch get +postremap 1 index .knownget{exch pop}if +}ifelse +}ifelse +2 index 3 1 roll put +}for +} +ifelse +}bind +16#00030000{ +pop[] +}bind +.dicttomark readonly def +/first_post_string +{ +post dup type/arraytype eq{0 get}if +}bind def +/.getpost{ +/glyphencoding post null eq{ +TTFDEBUG{(post missing)= flush}if[] +}{ +postformats first_post_string 0 getu32 .knownget{ +TTFDEBUG{ +(post: format )print +first_post_string +dup 0 getu16 =only(,)print 2 getu16 = flush +}if +post exch exec +}{ +TTFDEBUG{(post: unknown format )print post 0 getu32 = flush}if[] +}ifelse +}ifelse def +}bind def +/MacRomanEncoding[ +StandardEncoding 0 39 getinterval aload pop +/quotesingle +StandardEncoding 40 56 getinterval aload pop +/grave +StandardEncoding 97 31 getinterval aload pop +/Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute +/agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave +/ecircumflex/edieresis/iacute/igrave +/icircumflex/idieresis/ntilde/oacute +/ograve/ocircumflex/odieresis/otilde +/uacute/ugrave/ucircumflex/udieresis +/dagger/degree/cent/sterling/section/bullet/paragraph/germandbls +/registered/copyright/trademark/acute/dieresis/.notdef/AE/Oslash +/.notdef/plusminus/.notdef/.notdef/yen/mu/.notdef/.notdef +/.notdef/.notdef/.notdef/ordfeminine/ordmasculine/.notdef/ae/oslash +/questiondown/exclamdown/logicalnot/.notdef +/florin/.notdef/.notdef/guillemotleft +/guillemotright/ellipsis/space/Agrave/Atilde/Otilde/OE/oe +/endash/emdash/quotedblleft/quotedblright +/quoteleft/quoteright/divide/.notdef +/ydieresis/Ydieresis/fraction/currency +/guilsinglleft/guilsinglright/fi/fl +/daggerdbl/periodcentered/quotesinglbase/quotedblbase +/perthousand/Acircumflex/Ecircumflex/Aacute +/Edieresis/Egrave/Iacute/Icircumflex +/Idieresis/Igrave/Oacute/Ocircumflex +/.notdef/Ograve/Uacute/Ucircumflex +/Ugrave/dotlessi/circumflex/tilde +/macron/breve/dotaccent/ring/cedilla/hungarumlaut/ogonek/caron +]/Encoding defineresource pop +/TTParser<< +/Pos 0 +/post null +>>def +/readu8 +{read not{ +mark(Insufficient data in the stream.)//error exec +}if +}bind def +/readu16 +{dup//readu8 exec 8 bitshift exch//readu8 exec or +}bind def +/reads16 +{//readu16 exec 16#8000 xor 16#8000 sub +}bind def +/readu32 +{dup//readu16 exec 16 bitshift exch//readu16 exec or +}bind def +/reads32 +{dup//reads16 exec 16 bitshift exch//readu16 exec or +}bind def +/SkipToPosition +{dup//TTParser/Pos get +exch//TTParser exch/Pos exch put +sub +//PDFR_DEBUG{ +(Skipping )print dup//=only exec( bytes.)= +}if +dup 0 eq{ +pop pop +}{ +dup 3 1 roll +()/SubFileDecode filter +exch +{1 index//BlockBuffer readstring pop length +dup 0 eq{pop exch pop exit}if +sub +}loop +0 ne{ +mark(Insufficient data in the stream for SkipToPosition.)//error exec +}if +}ifelse +}bind def +/TagBuffer 4 string def +/ParseTTTableDirectory +{//PDFR_DEBUG{ +(ParseTTTableDirectory beg)= +}if +15 dict begin +dup//readu32 exec 16#00010000 ne{ +mark(Unknown True Type version.)//error exec +}if +dup//readu16 exec/NumTables exch def +dup//readu16 exec/SearchRange exch def +dup//readu16 exec/EntrySelector exch def +dup//readu16 exec/RangeShift exch def +//PDFR_DEBUG{ +(NumTables = )print NumTables = +}if +NumTables{ +dup//TagBuffer readstring not{ +mark(Could not read TT tag.)//error exec +}if +cvn +[2 index//readu32 exec pop +2 index//readu32 exec +3 index//readu32 exec +] +//PDFR_DEBUG{ +2 copy exch//=only exec( )print == +}if +def +}repeat +pop +//TTParser/Pos 12 NumTables 16 mul add put +currentdict end +//PDFR_DEBUG{ +(ParseTTTableDirectory end)= +}if +}bind def +/ParseTTcmap +{//PDFR_DEBUG{ +(ParseTTcmap beg)= +}if +/cmap get aload pop +3 1 roll +7 dict begin +//PDFR_DEBUG{ +(Current position = )print//TTParser/Pos get = +(cmap position = )print dup = +}if +1 index exch//SkipToPosition exec +//TTParser/Pos get/TablePos exch def +dup//readu16 exec pop +dup//readu16 exec/NumEncodings exch def +//PDFR_DEBUG{ +(NumEncodings = )print NumEncodings = +}if +null +NumEncodings{ +1 index//readu32 exec +2 index//readu32 exec +3 array dup 3 2 roll 0 exch put +2 index null ne{ +dup 0 get 3 index 0 get sub +3 index exch 1 exch put +}if +dup 4 3 roll pop 3 1 roll +def +}repeat +dup 0 get +4 3 roll exch sub +1 exch put +//PDFR_DEBUG{ +currentdict{ +exch dup type/integertype eq{ +//PrintHex exec( )print == +}{ +pop pop +}ifelse +}forall +}if +4 NumEncodings 8 mul add/HeaderLength exch def +//TTParser/Pos//TTParser/Pos get HeaderLength add put +0 +NumEncodings{ +16#7FFFFFF null +currentdict{ +1 index type/integertype eq{ +exch pop dup 0 get +dup 5 index gt{ +dup 4 index lt{ +4 1 roll +exch pop exch pop +}{ +pop pop +}ifelse +}{ +pop pop +}ifelse +}{ +pop pop +}ifelse +}forall +//PDFR_DEBUG{ +(Obtaining subtable for )print dup == +}if +3 2 roll pop +3 copy pop +TablePos add//SkipToPosition exec +3 copy exch pop 1 get +//TTParser/Pos//TTParser/Pos get 3 index add put +string +readstring not{ +mark(Can't read a cmap subtable.)//error exec +}if +2 exch put +}repeat +pop pop +currentdict end +//PDFR_DEBUG{ +(ParseTTcmap end)= +}if +}bind def +/GetTTEncoding +{//PDFR_DEBUG{ +(GetTTEncoding beg)= +}if +get +exch pop +2 get +10 dict begin +/TTFDEBUG//PDFR_DEBUG def +//cmaparray exec +end +//PDFR_DEBUG{ +(GetTTEncoding end)= +dup == +}if +}bind def +/InverseEncoding +{ +256 dict begin +dup length 1 sub -1 0{ +2 copy get +exch +1 index currentdict exch//knownget exec{ +dup type/arraytype eq{ +aload length 1 add array astore +}{ +2 array astore +}ifelse +}if +def +}for +pop +currentdict end +}bind def +/GetMacRomanEncodingInverse +{//PDFReader/MacRomanEncodingInverse get +dup null eq{ +pop +MacRomanEncoding//InverseEncoding exec +dup//PDFReader exch/MacRomanEncodingInverse exch put +}if +}bind def +/PutCharStringSingle +{ +dup 3 index length lt{ +2 index exch get +dup 0 ne{ +def +}{ +pop pop +}ifelse +}{ +pop pop +}ifelse +}bind def +/PutCharString +{1 index type/nametype ne{ +mark(Bad charstring name)//error exec +}if +dup type/arraytype eq{ +{ +3 copy//PutCharStringSingle exec +pop pop +}forall +pop +}{ +//PutCharStringSingle exec +}ifelse +}bind def +/ComposeCharStrings +{ +//PDFR_DEBUG{ +(ComposeCharStrings beg)= +}if +1 index length 1 add dict begin +/.notdef 0 def +exch +//TTParser/post get +dup null ne{ +exch +1 index length 1 sub -1 0{ +dup 3 index exch get exch +dup 0 eq 2 index/.notdef eq or{ +pop pop +}{ +def +}ifelse +}for +}if +exch pop exch +{ +//PutCharString exec +}forall +pop +currentdict end +//PDFR_DEBUG{ +(ComposeCharStrings end)= +}if +}bind def +/ParseTTpost +{ +//PDFR_DEBUG{ +(ParseTTpost beg)= +}if +/post get aload pop +3 1 roll +//PDFR_DEBUG{ +(Current position = )print//TTParser/Pos get = +(post position = )print dup = +}if +1 index exch//SkipToPosition exec +//TTParser/Pos//TTParser/Pos get 4 index add put +exch dup 65535 le{ +string +readstring not{ +mark(Insufficient data in the stream for ParseTTpost.)//error exec +}if +}{ +[3 1 roll +dup 16384 div floor cvi +exch 1 index 16384 mul +sub exch +1 sub 0 1 3 -1 roll +{ +1 add index +16384 string readstring not{ +mark(Insufficient data in the stream for ParseTTpost.)//error exec +}if +}for +counttomark -2 roll +string readstring not{ +mark(Insufficient data in the stream for ParseTTpost.)//error exec +}if +] +}ifelse +1 dict begin +/post exch def +//.getpost exec +//TTParser/post glyphencoding put +//PDFR_DEBUG{ +(ParseTTpost end)= +glyphencoding == +}if +end +}bind def +/MakeTTCharStrings +{//MakeStreamReader exec +dup dup//ParseTTTableDirectory exec +//TTParser/post null put +dup/post//knownget exec{ +0 get +1 index/cmap get 0 get +lt{ +2 copy//ParseTTpost exec +//ParseTTcmap exec +}{ +2 copy//ParseTTcmap exec +3 1 roll +//ParseTTpost exec +}ifelse +}{ +//ParseTTcmap exec +}ifelse +{ +dup 16#00030001 known{ +//PDFR_DEBUG{ +(Using the TT cmap encoding for Windows Unicode.)= +}if +16#00030001//GetTTEncoding exec +AdobeGlyphList//ComposeCharStrings exec +exit +}if +dup 16#00010000 known{ +//PDFR_DEBUG{ +(Using the TT cmap encoding for Macintosh Roman.)= +}if +16#00010000//GetTTEncoding exec +PDFEncoding dup null eq{ +pop//GetMacRomanEncodingInverse exec +}{ +//InverseEncoding exec +}ifelse +//ComposeCharStrings exec +exit +}if +dup 16#00030000 known{ +//PDFR_DEBUG{ +(Using the TT cmap encoding 3.0 - not sure why Ghostscript writes it since old versions.)= +}if +16#00030000//GetTTEncoding exec +PDFEncoding dup null eq{ +pop//GetMacRomanEncodingInverse exec +}{ +//InverseEncoding exec +}ifelse +//ComposeCharStrings exec +exit +}if +mark(True Type cmap has no useful encodings.)//error exec +}loop +//PDFR_DEBUG{ +(CharStrings <<)= +dup{ +exch +dup type/nametype eq{ +//=only exec +}{ +== +}ifelse +( )print == +}forall +(>>)= +}if +}bind def +/ScaleVal +{ +aload pop +1 index sub +3 2 roll mul add +}bind def +/ScaleArg +{ +aload pop +1 index sub +3 1 roll +sub exch div +}bind def +/ScaleArgN +{ +dup length 2 sub -2 0{ +2 +2 index 3 1 roll getinterval +3 2 roll +exch//ScaleArg exec +1 index length 2 idiv 1 add 1 roll +}for +pop +}bind def +/ComputeFunction_10 +{ +//PDFR_DEBUG{ +(ComputeFunction_10 beg )print 1 index//=only exec( stack=)print count = +}if +exch +dup 1 eq{ +pop dup length 1 sub get +}{ +1 index length 1 sub mul +dup dup floor sub +dup 0 eq{ +pop cvi get +}{ +3 1 roll floor cvi +2 getinterval +aload pop +2 index mul 3 2 roll 1 exch sub 3 2 roll mul add +}ifelse +}ifelse +//PDFR_DEBUG{ +(ComputeFunction_10 end )print dup//=only exec( stack=)print count = +}if +}bind def +/ComputeFunction_n0 +{ +//PDFR_DEBUG{ +(ComputeFunction_n0 beg N=)print dup//=only exec( stack=)print count = +}if +dup 0 eq{ +pop +}{ +dup 2 add -1 roll +dup 3 index length 1 sub ge{ +pop 1 sub +exch dup length 1 sub get exch +//PDFReader/ComputeFunction_n0 get exec +}{ +dup floor cvi dup +4 index exch get +3 index dup +5 add copy +6 2 roll +pop pop pop pop +1 sub +//PDFReader/ComputeFunction_n0 get exec +3 2 roll pop +exch +4 3 roll exch +4 add 2 roll 1 add +3 2 roll exch get +exch 1 sub +//PDFReader/ComputeFunction_n0 get exec +1 index mul +3 1 roll +1 exch sub mul add +}ifelse +}ifelse +//PDFR_DEBUG{ +(ComputeFunction_n0 end )print dup//=only exec( stack=)print count = +}if +}bind def +/FunctionToProc_x01 +{ +dup/Domain get exch +dup/Data get 0 get exch +/Size get length +[4 1 roll +//PDFR_DEBUG{ +{(function beg, stack =)print count//=only exec(\n)print}/exec load +5 2 roll +}if +dup 1 gt{ +{mark exch +3 add 2 roll +//ScaleArgN exec +counttomark dup +3 add -2 roll +pop exch +//ComputeFunction_n0 exec +}/exec load +}{ +pop +3 1/roll load//ScaleArg/exec load +/exch load +//ComputeFunction_10/exec load +}ifelse +//PDFR_DEBUG{ +(function end, stack =)/print load/count load//=only/exec load(\n)/print load +}if +]cvx +//PDFR_DEBUG{ +(Made a procedure for the 1-result function :)= +dup == +}if +}bind def +/FunctionProcDebugBeg +{(FunctionProcDebugBeg )print count = +}bind def +/FunctionProcDebugEnd +{(FunctionProcDebugEnd )print count = +}bind def +/FunctionToProc_x0n +{ +PDFR_DEBUG{ +(FunctionToProc_x0n beg m=)print dup = +}if +1 index/Size get length exch +dup 7 mul 2 add array +PDFR_DEBUG{ +dup 0//FunctionProcDebugBeg put +}{ +dup 0//DoNothing put +}ifelse +dup 1/exec load put +dup 2 5 index/Domain get put +2 index 1 eq{ +dup 3//ScaleArg put +}{ +dup 3//ScaleArgN put +}ifelse +dup 4/exec load put +1 index 1 sub 0 exch 1 exch{ +dup 7 mul 5 add +1 index 4 index 1 sub ne{ +dup 3 index exch 6 index put 1 add +dup 3 index exch/copy load put 1 add +}if +[ +6 index/Data get 3 index get +6 index 1 eq{ +//ComputeFunction_10/exec load +}{ +6 index +//ComputeFunction_n0/exec load +}ifelse +]cvx +3 index exch 2 index exch put 1 add +2 index 1 index/exec load put 1 add +1 index 4 index 1 sub ne{ +2 index 1 index 6 index 1 add put 1 add +2 index 1 index 1 put 1 add +2 index 1 index/roll load put +}if +pop pop +}for +PDFR_DEBUG{ +dup dup length 2 sub//FunctionProcDebugEnd put +}{ +dup dup length 2 sub//DoNothing put +}ifelse +dup dup length 1 sub/exec load put +cvx exch pop exch pop exch pop +//PDFR_DEBUG{ +(Made a procedure for the n-argument function :)= +dup == +}if +PDFR_DEBUG{ +(FunctionToProc_x0n end)= +}if +}bind def +/MakeTableRec +{ +0 +exec +}bind def +/MakeTable +{//PDFR_DEBUG{ +(MakeTable beg )print count = +}if +1 index/Size get exch +1 sub dup +3 1 roll +get +array +1 index 0 eq{ +exch pop exch pop +}{ +dup length 1 sub -1 0{ +3 index 3 index//MakeTableRec exec +2 index 3 1 roll put +}for +exch pop exch pop +}ifelse +//PDFR_DEBUG{ +(MakeTable end )print count = +}if +}bind def +//MakeTableRec 0//MakeTable put +/StoreSample +{ +1 sub +dup 0 eq{ +pop +}{ +-1 1{ +I exch get get +}for +}ifelse +I 0 get 3 2 roll put +}bind def +/ReadSample32 +{ +4{ +File read not{ +mark(Insufficient data for function.)//error exec +}if +}repeat +pop +3 1 roll exch +256 mul add 256 mul add +//1_24_bitshift_1_sub div +}bind def +/ReadSample +{ +Buffer BitsLeft BitsPerSample +{2 copy ge{ +exit +}if +3 1 roll +8 add 3 1 roll +256 mul File read not{ +mark(Insufficient data for function.)//error exec +}if +add +3 1 roll +}loop +sub dup +2 index exch +neg bitshift +2 copy exch bitshift +4 3 roll exch sub +/Buffer exch def +exch/BitsLeft exch def +Div div +}bind def +/ReadSamplesRec +{0 +exec +}bind def +/ReadSamples +{ +//PDFR_DEBUG{ +(ReadSamples beg )print count = +}if +dup 1 eq{ +pop +0 1 Size 0 get 1 sub{ +I exch 0 exch put +0 1 M 1 sub{ +dup Range exch 2 mul 2 getinterval +//PDFR_DEBUG{ +(Will read a sample ... )print +}if +BitsPerSample 32 eq{//ReadSample32}{//ReadSample}ifelse +exec exch//ScaleVal exec +//PDFR_DEBUG{ +(value=)print dup = +}if +exch Table exch get +Size length//StoreSample exec +}for +}for +}{ +1 sub +dup Size exch get 0 exch 1 exch 1 sub{ +I exch 2 index exch put +dup//ReadSamplesRec exec +}for +pop +}ifelse +//PDFR_DEBUG{ +(ReadSamples end )print count = +}if +}bind def +//ReadSamplesRec 0//ReadSamples put +/StreamToArray +{//PDFR_DEBUG{ +(StreamToArray beg )print count = +}if +userdict/FuncDataReader get begin +dup/BitsPerSample get/BitsPerSample exch def +dup/Size get length/N exch def +dup/Range get length 2 idiv/M exch def +1 BitsPerSample bitshift 1 sub/Div exch def +/BitsLeft 0 def +/Buffer 0 def +dup/Size get/Size exch def +dup/Range get/Range exch def +/File 1 index//MakeStreamReader exec def +/I[N{0}repeat]def +M array +dup length 1 sub -1 0{ +2 index N//MakeTable exec +2 index 3 1 roll put +}for +/Table exch def +N//ReadSamples exec +PDFR_DEBUG{ +(Table = )print Table == +}if +/Data Table put +end +//PDFR_DEBUG{ +(StreamToArray end )print count = +}if +}bind def +/FunctionToProc10 +{ +PDFR_DEBUG{ +(FunctionToProc10 beg, Range = )print dup/Range get == +}if +dup/Order//knownget exec{ +1 ne{ +(Underimplemented function Type 0 Order 3.)= +}if +}if +dup//StreamToArray exec +dup/Range get length dup 2 eq{ +pop//FunctionToProc_x01 exec +}{ +2 idiv//FunctionToProc_x0n exec +}ifelse +PDFR_DEBUG{ +(FunctionToProc10 end)= +}if +}bind def +/FunctionToProc12 +{begin +currentdict/C0//knownget exec{length 1 eq}{true}ifelse{ +N +currentdict/C0//knownget exec{ +0 get +}{ +0 +}ifelse +currentdict/C1//knownget exec{ +0 get +}{ +1 +}ifelse +1 index sub +[4 1 roll +{ +4 2 roll +exp mul add +}aload pop +]cvx +}{ +[ +0 1 C0 length 1 sub{ +N +C0 2 index get +C1 3 index get +4 3 roll pop +1 index sub +[/dup load +5 2 roll +{ +4 2 roll +exp mul add +exch +}aload pop +]cvx +/exec load +}for +/pop load +]cvx +}ifelse +end +//PDFR_DEBUG{ +(FunctionType2Proc : )print dup == +}if +}bind def +/FunctionToProc14 +{//MakeStreamReader exec cvx exec +//PDFR_DEBUG{ +(FunctionType4Proc : )print dup == +}if +}bind def +/FunctionToProc1 +{ +dup/FunctionType get +{dup 0 eq{ +pop//FunctionToProc10 exec exit +}if +dup 2 eq{ +pop//FunctionToProc12 exec exit +}if +dup 4 eq{ +pop//FunctionToProc14 exec exit +}if +mark exch(Function type )exch( isn't implemented yet.)//error exec +}loop +}bind def +/FunctionToProc20 +{ +PDFR_DEBUG{ +(FunctionToProc20, Range = )print dup/Range get == +}if +dup/Order//knownget exec{ +1 ne{ +(Underimplemented function Type 0 Order 3.)= +}if +}if +dup//StreamToArray exec +dup/Range get length dup 2 eq{ +pop//FunctionToProc_x01 exec +}{ +2 idiv//FunctionToProc_x0n exec +}ifelse +}bind def +/FunctionToProc +{//PDFR_DEBUG{ +(FunctionToProc beg )print count = +}if +dup type/dicttype eq{ +dup/Domain get length 2 idiv +{ +dup 1 eq{ +pop//FunctionToProc1 exec exit +}if +dup 2 eq{ +pop//FunctionToProc20 exec exit +}if +mark(Functions with many arguments aren't implemented yet.)//error exec +}loop +}{ +//PDFR_DEBUG{(Not a function dict, assume already a procedure.)print}if +}ifelse +//PDFR_DEBUG{ +(FunctionToProc end )print count = +}if +}bind def +/spotfunctions mark +/Round{ +abs exch abs 2 copy add 1 le{ +dup mul exch dup mul add 1 exch sub +}{ +1 sub dup mul exch 1 sub dup mul add 1 sub +}ifelse +} +/Diamond{ +abs exch abs 2 copy add .75 le{ +dup mul exch dup mul add 1 exch sub +}{ +2 copy add 1.23 le{ +.85 mul add 1 exch sub +}{ +1 sub dup mul exch 1 sub dup mul add 1 sub +}ifelse +}ifelse +} +/Ellipse{ +abs exch abs 2 copy 3 mul exch 4 mul add 3 sub dup 0 lt{ +pop dup mul exch .75 div dup mul add 4 div 1 exch sub +}{ +dup 1 gt{ +pop 1 exch sub dup mul exch 1 exch sub +.75 div dup mul add 4 div 1 sub +}{ +.5 exch sub exch pop exch pop +}ifelse +}ifelse +} +/EllipseA{dup mul .9 mul exch dup mul add 1 exch sub} +/InvertedEllipseA{dup mul .9 mul exch dup mul add 1 sub} +/EllipseB{dup 5 mul 8 div mul exch dup mul exch add sqrt 1 exch sub} +/EllipseC{dup mul .9 mul exch dup mul add 1 exch sub} +/InvertedEllipseC{dup mul .9 mul exch dup mul add 1 sub} +/Line{exch pop abs neg} +/LineX{pop} +/LineY{exch pop} +/Square{abs exch abs 2 copy lt{exch}if pop neg} +/Cross{abs exch abs 2 copy gt{exch}if pop neg} +/Rhomboid{abs exch abs 0.9 mul add 2 div} +/DoubleDot{2{360 mul sin 2 div exch}repeat add} +/InvertedDoubleDot{2{360 mul sin 2 div exch}repeat add neg} +/SimpleDot{dup mul exch dup mul add 1 exch sub} +/InvertedSimpleDot{dup mul exch dup mul add 1 sub} +/CosineDot{180 mul cos exch 180 mul cos add 2 div} +/Double{exch 2 div exch 2{360 mul sin 2 div exch}repeat add} +/InvertedDouble{ +exch 2 div exch 2{360 mul sin 2 div exch}repeat add neg +} +.dicttomark readonly def +/CheckColorSpace +{ +dup type/arraytype ne{ +mark(Resource )3 index( must be an array.)//error exec +}if +}bind def +/SubstitutePDFColorSpaceRec +{0 +exec +}bind def +/SubstitutePDFColorSpace +{ +{ +dup 0 get/Pattern eq{ +dup length 1 gt{ +dup dup 1//CheckColorSpace//ResolveA exec +dup type/nametype ne{ +//SubstitutePDFColorSpaceRec exec +}if +1 exch put +}if +exit +}if +dup 0 get/Indexed eq{ +exit +}if +dup 0 get/Separation eq{ +dup dup 2//CheckColorSpace//ResolveA exec +dup type/nametype ne{ +//SubstitutePDFColorSpaceRec exec +}if +2 exch put +exit +}if +dup 0 get/CalGray eq{ +1 get +dup/Gamma//knownget exec{ +[exch[exch/exp load]cvx dup dup] +1 index exch/DecodeLMN exch put +}if +[exch/CIEBasedA exch] +exit +}if +dup 0 get/CalRGB eq{ +1 get +dup/Matrix//knownget exec{ +1 index exch/MatrixLMN exch put +}if +dup/Gamma//knownget exec{ +aload pop +[exch/exp load]cvx +3 1 roll +[exch/exp load]cvx +3 1 roll +[exch/exp load]cvx +3 1 roll +3 array astore +1 index exch/DecodeLMN exch put +}if +[exch/CIEBasedABC exch] +exit +}if +dup 0 get/Lab eq{ +1 get +begin +currentdict/Range//knownget exec{aload pop}{-100 100 -100 100}ifelse +0 100 6 2 roll 6 array astore +/RangeABC exch def +/DecodeABC[{16 add 116 div}bind{500 div}bind{200 div}bind]def +/MatrixABC[1 1 1 1 0 0 0 0 -1]def +{dup 6 29 div ge{dup dup mul mul}{4 29 div sub 108 841 div mul}ifelse} +/DecodeLMN[ +[3 index aload pop WhitePoint 0 get/mul load]cvx +[4 index aload pop WhitePoint 1 get/mul load]cvx +[5 index aload pop WhitePoint 2 get/mul load]cvx +]def pop +//PDFR_DEBUG{ +(Constructed from Lab <<)= +currentdict{exch = ==}forall +(>>)= +}if +[/CIEBasedABC currentdict] +end +exit +pop +}if +dup 0 get/CIEBasedA eq{exit}if +dup 0 get/CIEBasedABC eq{exit}if +mark exch(Unimplemented color space )exch//error exec +}loop +}bind def +//SubstitutePDFColorSpaceRec 0//SubstitutePDFColorSpace put +/ResolveArrayElement +{2 copy get +dup type dup/arraytype eq exch +/packedarraytype eq or{ +dup length 1 ge exch xcheck and{ +2 copy get +dup 0 get type/integertype eq +1 index 1 get type dup/arraytype +eq exch +/packedarraytype eq or +and{ +exec +2 index 4 1 roll put +}{ +pop pop +}ifelse +}{ +pop +}ifelse +}{ +pop pop +}ifelse +}bind def +/ResolveColorSpaceArrayRec +{0 +exec +}bind def +/SetColorSpaceSafe +{ +PDFR_DEBUG{ +(SetColorSpaceSafe beg)= +}if +currentcolorspace dup type/arraytype eq{ +1 index type/arraytype eq{ +dup length 2 index length eq{ +false exch +dup length 0 exch 1 exch 1 sub{ +dup +4 index exch get exch +2 index exch get +ne{ +exch pop true exch exit +}if +}for +pop +{ +setcolorspace +}{ +pop +}ifelse +}{ +pop setcolorspace +}ifelse +}{ +pop setcolorspace +}ifelse +}{ +pop setcolorspace +}ifelse +PDFR_DEBUG{ +(SetColorSpaceSafe end)= +}if +}bind def +/ResolveColorSpaceArray +{ +//PDFR_DEBUG{ +(ResolveColorSpaceArray beg )print dup == +}if +dup 0 get/Indexed eq{ +1//ResolveArrayElement exec +dup dup 1 get +dup type/arraytype eq{ +//SubstitutePDFColorSpace exec +//ResolveColorSpaceArrayRec exec +1 exch put +}{ +pop pop +}ifelse +}if +dup 0 get/Separation eq{ +dup dup 1 get UnPDFEscape 1 exch put +3//ResolveArrayElement exec +dup 3 get//FunctionToProc exec +2 copy 3 exch put +pop +}if +dup 0 get/Pattern eq{ +dup length 1 gt{ +dup 1 get dup type/arraytype eq{ +ResolveColorSpaceArray +1 index 1 3 -1 roll put +}{ +pop +}ifelse +}if +}if +PDFR_DEBUG{ +(Construcrted color space :)= +dup == +}if +//PDFR_DEBUG{ +(ResolveColorSpaceArray end )print dup == +}if +}bind def +//ResolveColorSpaceArrayRec 0//ResolveColorSpaceArray put +/ResolveColorSpace +{ +//PDFR_DEBUG{ +(ResolveColorSpace beg )print dup = +}if +dup//SimpleColorSpaceNames exch known not{ +dup//PDFColorSpaces exch//knownget exec{ +exch pop +//PDFR_DEBUG{ +(ResolveColorSpace known )= +}if +}{ +dup +//PDFReader/CurrentObject get/Context get/Resources get +/ColorSpace//DoNothing//ResolveD exec +exch//CheckColorSpace//ResolveD exec +dup type/arraytype eq{ +//SubstitutePDFColorSpace exec +//ResolveColorSpaceArray exec +dup//PDFColorSpaces 4 2 roll put +}if +}ifelse +}if +//PDFR_DEBUG{ +(ResolveColorSpace end )print dup == +}if +}bind def +/CheckPattern +{ +dup/PatternType//knownget exec{ +dup 1 ne{ +mark(Resource )4 index( is a shading, which can't be handled at level 2. )//error exec +}if +pop +}if +dup/Type knownget{ +/Pattern ne{ +mark(Resource )4 index( must have /Type/Pattern .)//error exec +}if +}if +}bind def +/PaintProc +{/Context get +//RunDelayedStream exec +}bind def +/ResolvePattern +{ +dup +userdict/PDFR_Patterns get +exch//knownget exec{ +exch pop +}{ +dup +//PDFReader/CurrentObject get/Context get/Resources get +/Pattern//DoNothing//ResolveD exec +exch//CheckPattern//ResolveD exec +dup dup/Context exch put +dup/Resources//DoNothing//ResolveD exec pop +dup/PaintProc//PaintProc put +gsave userdict/PDFR_InitialGS get setgstate +currentglobal exch false setglobal +dup/Matrix get +makepattern +exch setglobal +grestore +dup userdict/PDFR_Patterns get +4 2 roll +put +}ifelse +}bind def +/SetColor +{//PDFR_DEBUG{ +(SetColor beg)= +}if +currentcolorspace dup type/nametype eq{ +pop setcolor +}{ +0 get/Pattern eq{ +//ResolvePattern exec setpattern +}{ +setcolor +}ifelse +}ifelse +//PDFR_DEBUG{ +(SetColor end)= +}if +}bind def +/ImageKeys 15 dict begin +/BPC/BitsPerComponent def +/CS/ColorSpace def +/D/Decode def +/DP/DecodeParms def +/F/Filter def +/H/Height def +/IM/ImageMask def +/I/Interpolate def +/W/Width def +currentdict end readonly def +/ImageValues 15 dict begin +/G/DeviceGray def +/RGB/DeviceRGB def +/CMYK/DeviceCMYK def +/I/Indexed def +/AHx/ASCIIHexDecode def +/A85/ASCII85Decode def +/LZW/LZWDecode def +/Fl/FlateDecode def +/RL/RunLengthDecode def +/CCF/CCITTFaxDecode def +/DCT/DCTDecode def +currentdict end readonly def +/GetColorSpaceRange +{2 index/ColorSpace get +dup type/arraytype eq{ +1 get +}if +exch//knownget exec{ +exch pop +}if +}bind def +/DecodeArrays 15 dict begin +/DeviceGray{[0 1]}def +/DeviceRGB{[0 1 0 1 0 1]}def +/DeviceCMYK{[0 1 0 1 0 1 0 1]}def +/Indexed{ +dup/BitsPerComponent get 1 exch bitshift 1 sub[exch 0 exch] +}def +/Separation{[0 1]}def +/CIEBasedA{[0 1]/RangeA//GetColorSpaceRange exec}def +/CIEBasedABC{[0 1 0 1 0 1]/RangeABC//GetColorSpaceRange exec}def +currentdict end readonly def +/Substitute +{1 index//knownget exec{ +exch pop +}if +}bind def +/DebugImagePrinting +{ +//PDFR_DEBUG{ +(Image :)= +dup{exch//=only exec( )print == +}forall +}if +}bind def +/CompleteImage +{ +dup/ColorSpace known{ +dup/ColorSpace//CheckColorSpace//ResolveD exec pop +}if +dup/Decode known not{ +dup/ColorSpace//knownget exec{ +dup type/arraytype eq{ +0 get +}if +//DecodeArrays exch get exec +}{ +[0 1] +}ifelse +1 index exch/Decode exch put +}if +dup/ImageMatrix[2 index/Width get 0 0 5 index/Height get neg +0 7 index/Height get]put +//DebugImagePrinting exec +}bind def +/CompleteInlineImage +{ +//PDFR_DEBUG{ +(CompleteInlineImage beg)= +}if +dup/ImageType known not{ +dup/ImageType 1 put +}if +dup length dict exch{ +exch//ImageKeys//Substitute exec +dup/Filter eq{ +exch//ImageValues//Substitute exec exch +}if +dup/ColorSpace eq{ +exch +dup//ImageValues exch//knownget exec{ +exch pop +}{ +//ResolveColorSpace exec +}ifelse +exch +}if +exch +2 index 3 1 roll put +}forall +//CompleteImage exec +dup/DataSource 2 copy get +2 index//AppendFilters exec put +//PDFR_DEBUG{ +(CompleteInlineImage end)= +}if +}bind def +/CompleteOutlineImage +{ +currentglobal exch dup gcheck setglobal +//PDFR_DEBUG{ +(CompleteOutlineImage beg)= +}if +dup dup//MakeStreamReader exec/DataSource exch put +dup/ImageType known not{ +//CompleteImage exec +dup/ImageType 1 put +dup/ColorSpace known{ +dup/ColorSpace//CheckColorSpace//ResolveD exec +dup type/arraytype eq{ +//ResolveColorSpaceArray exec +//SubstitutePDFColorSpace exec +1 index exch/ColorSpace exch put +}{ +pop +}ifelse +}if +}if +//PDFR_DEBUG{ +(CompleteOutlineImage end)= +}if +exch setglobal +}bind def +/DoImage +{ +//PDFR_DEBUG{ +(DoImage beg)= +}if +gsave +dup/ColorSpace//knownget exec{setcolorspace}if +dup/ImageMask//knownget exec not{false}if +{imagemask}{image}ifelse +grestore +//PDFR_DEBUG{ +(DoImage end)= +}if +}bind def +/GSave +{ +gsave +//PDFReader/GraphicStateStackPointer get +dup//GraphicStateStack exch get null eq{ +dup//GraphicStateStack exch//InitialGraphicState length dict put +}if +dup//GraphicStateStack exch get +//GraphicState exch copy pop +1 add//PDFReader exch/GraphicStateStackPointer exch put +}bind def +/GRestore +{ +grestore +//PDFReader/GraphicStateStackPointer get +1 sub dup +//PDFReader exch/GraphicStateStackPointer exch put +//GraphicStateStack exch get +//GraphicState copy pop +}bind def +/SetFont +{dup//GraphicState exch/FontSize exch put +//ResolveAndSetFont exec +//GraphicState/FontMatrixNonHV currentfont/FontMatrix get 1 get 0 ne put +}bind def +/ShowText +{ +//GraphicState/TextRenderingMode get dup 0 eq +exch 3 eq not currentfont/FontType get 3 eq and or +{ +//GraphicState/WordSpacing get 0 +32 +//GraphicState/CharacterSpacing get 0 +6 5 roll +//GraphicState/FontMatrixNonHV get{ +[ +7 -2 roll pop +5 -2 roll pop +5 -1 roll +{ +exch +pop +3 index add +exch 2 index eq{3 index add}if +4 1 roll +} +currentfont/FontMatrix get 0 get 0 ne{ +1 1 index length 1 sub getinterval cvx +}if +5 index +cshow +pop pop pop] +xshow +}{ +awidthshow +}ifelse +}{ +//GraphicState/CharacterSpacing get 0 eq +//GraphicState/FontMatrixNonHV get not and +//GraphicState/WordSpacing get 0 eq and{ +true charpath +}{ +{ +exch +pop 0 +currentpoint 5 4 roll +( )dup 0 3 index put true charpath +5 1 roll +moveto rmoveto +//GraphicState/CharacterSpacing get 0 rmoveto +32 eq{ +//GraphicState/WordSpacing get 0 rmoveto +}if +} +//GraphicState/FontMatrixNonHV get dup not exch{ +pop currentfont/FontMatrix get 0 get 0 ne +}if{ +1 1 index length 1 sub getinterval cvx +}if +exch cshow +}ifelse +}ifelse +}bind def +/ShowTextBeg +{ +//GraphicState/TextRenderingMode get dup 0 ne +{ +3 ne +currentfont/FontType get 3 eq not and{ +currentpoint newpath moveto +}if +} +{ +pop +}ifelse +}bind def +/ShowTextEnd +{ +//GraphicState/TextRenderingMode get +currentfont/FontType get 3 eq{ +dup 3 ne{ +pop 0 +}if +}if +{dup 1 eq{ +stroke exit +}if +dup 2 eq{ +gsave fill grestore stroke exit +}if +dup 3 eq{ +currentpoint newpath moveto +}if +dup 4 eq{ +gsave fill grestore clip exit +}if +dup 5 eq{ +gsave stroke grestore clip exit +}if +dup 6 eq{ +gsave fill grestore gsave stroke grestore fill exit +}if +dup 7 eq{ +clip exit +}if +exit +}loop +pop +}bind def +/ShowTextWithGlyphPositioning +{//ShowTextBeg exec +{dup type/stringtype eq{ +//ShowText exec +}{ +neg 1000 div//GraphicState/FontSize get mul 0 rmoveto +}ifelse +}forall +//ShowTextEnd exec +}bind def +/CheckFont +{dup/Type get/ExtGState ne{ +mark(Resource )3 index( must have /Type/ExtGState.)//error exec +}if +}bind def +/SetTransfer +{ +//PDFR_DEBUG{(SetTransfer beg )print count =}if +dup type/arraytype eq 1 index xcheck not and{ +0 4 getinterval aload pop +setcolortransfer +}{ +settransfer +}ifelse +//PDFR_DEBUG{(SetTransfer end )print count =}if +}bind def +/CheckExtGState +{dup/Type get/ExtGState ne{ +mark(Resource )3 index( must have /Type/ExtGState.)//error exec +}if +}bind def +/CheckHalftone +{dup/HalftoneType known not{ +mark(Resource )3 index( must have /HalftoneType.)//error exec +}if +}bind def +/ResolveFunction +{ +//PDFR_DEBUG{(ResolveFunction beg )print dup = count =}if +2 copy get//IsObjRef exec{ +2 copy//DoNothing//ResolveD exec +3 copy put pop +}if +2 copy get dup type/arraytype eq exch xcheck and not{ +2 copy get +dup type/arraytype eq 1 index xcheck not and{ +dup length 1 sub -1 0{ +2 copy//DoNothing ResolveA +dup/Identity eq{ +pop 2 copy{}put +}{ +//FunctionToProc exec +3 copy put pop +}ifelse +pop +}for +}{ +dup/Default eq{ +}{ +dup/Identity eq{ +pop{} +}{dup type/nametype eq{ +//spotfunctions exch get +}{ +//FunctionToProc exec +}ifelse +}ifelse +}ifelse +}ifelse +3 copy put +exch pop +}{ +1 index exch get +}ifelse +//PDFR_DEBUG{(ResolveFunction end )print dup == count =}if +}bind def +/ResolveFunctionSafe +{2 copy known{ +//ResolveFunction exec +}if +pop +}bind def +/CreateHalftoneThresholds +{ +dup/Thresholds known not{ +dup/HalftoneType get 10 eq{ +dup dup//MakeStreamReader exec +/Thresholds exch put +}if +dup/HalftoneType get dup 3 eq exch 6 eq or{ +dup dup//MakeStreamReader exec +//BlockBuffer readstring pop +dup length +dup 0 eq{ +mark(Could not read Thresholds)//error exec +}if +string copy/Thresholds exch put +dup/HalftoneType 3 put +}if +}if +}bind def +/SetExtGState +{ +//PDFReader/CurrentObject get/Context get/Resources get +/ExtGState//DoNothing//ResolveD exec +exch//CheckExtGState//ResolveD exec +dup/LW//knownget exec{ +setlinewidth +}if +dup/LC//knownget exec{ +setlinecap +}if +dup/LJ//knownget exec{ +setlinejoin +}if +dup/ML//knownget exec{ +setmeterlimit +}if +dup/D//knownget exec{ +setdash +}if +dup/RI//knownget exec{ +mark(Unimplemented ExtGState.RI)//error exec +}if +dup/OP//knownget exec{ +setoverprint +}if +dup/op//knownget exec{ +setoverprint +}if +dup/OPM//knownget exec{ +mark(Unimplemented ExtGState.OPM)//error exec +}if +dup/Font//knownget exec{ +mark(Unimplemented ExtGState.Font)//error exec +}if +dup/BG known{ +/BG//ResolveFunction exec +setblackgeneration +}if +dup/BG2 known{ +/BG2//ResolveFunction exec +dup/Default eq{ +//InitialExtGState/BG2 get +}if +setblackgeneration +}if +dup/UCR known{ +/UCR//ResolveFunction exec +setundercolorremoval +}if +dup/UCR2 known{ +/UCR2//ResolveFunction exec +dup/Default eq{ +//InitialExtGState/UCR2 get +}if +setundercolorremoval +}if +dup/TR known{ +/TR//ResolveFunction exec +//SetTransfer exec +}if +dup/TR2 known{ +/TR2//ResolveFunction exec +dup/Default eq{ +pop//InitialExtGState/TR2 get +aload pop setcolortransfer +}{ +//SetTransfer exec +}ifelse +}if +dup/HT//knownget exec{ +dup/Default eq{ +pop//InitialExtGState/HT get +sethalftone +}{ +//PDFR_DEBUG{(Ht beg)=}if +pop dup/HT//CheckHalftone//ResolveD exec +/SpotFunction//ResolveFunctionSafe exec +/TransferFunction//ResolveFunctionSafe exec +null exch +dup/HalftoneType get dup 5 eq exch dup 4 eq exch 2 eq or or{ +dup{ +dup//IsObjRef exec{ +pop +1 index exch//CheckHalftone ResolveD +}if +dup type/dicttype eq{ +dup/SpotFunction//ResolveFunctionSafe exec +/TransferFunction//ResolveFunctionSafe exec +//CreateHalftoneThresholds exec +dup/HalftoneType get 5 gt{ +4 3 roll pop +dup 4 1 roll +}if +}if +pop pop +}forall +}if +//CreateHalftoneThresholds exec +//PDFR_DEBUG{ +(HT:)= +dup{ +1 index/Default eq{ +(Default <<)= +exch pop +{exch = ==}forall +(>>)= +}{ +exch = == +}ifelse +}forall +(HT end)= flush +}if +exch dup null ne{ +(Warning: Ignoring a halftone with a Level 3 component halftone Type )print dup/HalftoneType get = +pop pop +}{ +pop +dup/HalftoneType get 5 gt{ +(Warning: Ignoring a Level 3 halftone Type )print dup/HalftoneType get = +pop +}{ +sethalftone +}ifelse +}ifelse +//PDFR_DEBUG{(HT set)= flush}if +}ifelse +}if +dup/FL//knownget exec{ +setflattness +}if +dup/SM//knownget exec{ +setsmoothness +}if +dup/SA//knownget exec{ +setstrokeadjust +}if +dup/BM//knownget exec{ +mark(Unimplemented ExtGState.BM)//error exec +}if +dup/SMask//knownget exec{ +mark(Unimplemented ExtGState.SMask)//error exec +}if +dup/CA//knownget exec{ +mark(Unimplemented ExtGState.CA)//error exec +}if +dup/ca//knownget exec{ +mark(Unimplemented ExtGState.ca)//error exec +}if +dup/AIS//knownget exec{ +mark(Unimplemented ExtGState.AIS)//error exec +}if +dup/TK//knownget exec{ +mark(Unimplemented ExtGState.TK)//error exec +}if +pop +}bind def +/CheckXObject +{dup/Subtype get dup/Image ne exch dup/Form ne exch/PS ne and and{ +mark(Resource )3 index( must have /Subtype /Image or /Form or /PS.)//error exec +}if +}bind def +/DoXObject +{ +//PDFReader/CurrentObject get/Context get/Resources get +/XObject//DoNothing//ResolveD exec +exch//CheckXObject//ResolveD exec +dup/Subtype get +dup/Image eq{ +pop +//CompleteOutlineImage exec +//DoImage exec +}{ +dup/PS eq{ +PDFR_DEBUG{ +(Executing a PS Xobject)= +}if +pop +//RunDelayedStream exec +}{ +dup/Form eq{ +pop +PDFR_DEBUG{ +(Executing a Form XObject)= +}if +//PDFReader/CurrentObject get exch +dup//PDFReader exch<< exch/Context exch >>/CurrentObject exch put +dup/Matrix get concat +dup/BBox get aload pop exch 3 index sub exch 2 index sub rectclip +//RunDelayedStream exec +//PDFReader exch/CurrentObject exch put +}{ +mark exch(unimplemented XObject type )exch//error exec +}ifelse +}ifelse +}ifelse +}bind def +/Operators 50 dict begin +/q{//GSave exec}bind def +/Q{//GRestore exec}bind def +/cm{//TempMatrix astore concat}bind def +/i{1 .min setflat}bind def +/J/setlinecap load def +/d/setdash load def +/j/setlinejoin load def +/w/setlinewidth load def +/M/setmiterlimit load def +/gs{SetExtGState}bind def +/g/setgray load def +/rg/setrgbcolor load def +/k/setcmykcolor load def +/cs{//ResolveColorSpace exec//SetColorSpaceSafe exec +}bind def +/sc/setcolor load def +/scn{//SetColor exec}bind def +/G/setgray load def +/RG/setrgbcolor load def +/K/setcmykcolor load def +/CS//cs def +/ri{SetColorRenderingIntent}bind def +/SC/setcolor load def +/SCN{//SetColor exec}bind def +/m/moveto load def +/l/lineto load def +/c/curveto load def +/v{currentpoint 6 2 roll curveto}bind def +/y{2 copy curveto}bind def +/re{ +4 2 roll moveto exch dup 0 rlineto 0 3 -1 roll rlineto neg 0 rlineto +closepath +}def +/h/closepath load def +/n/newpath load def +/S/stroke load def +/s{closepath stroke}bind def +/f/fill load def +/f*/eofill load def +/B{gsave fill grestore stroke}bind def +/b{closepath gsave fill grestore stroke}bind def +/B*{gsave eofill grestore stroke}bind def +/b*{closepath gsave eofill grestore stroke}bind def +/W/clip load def +/W*/eoclip load def +/sh{ +ResolveShading +dup/Background known{ +gsave +dup/ColorSpace get setcolorspace +dup/Background get aload pop setcolor +pathbbox +2 index sub exch 3 index sub exch +rectfill +grestore +}if +shfill +}bind def +/Do{//DoXObject exec}bind def +/BI{currentglobal false setglobal<<}bind def +/ID{>> +dup/DataSource currentfile +2 index/F//knownget exec{ +/A85 eq{ +0(~>)/SubFileDecode filter +}if +}if +put +//CompleteInlineImage exec +exch setglobal +//DoImage exec +}bind def +/EI{}bind def +/BT{gsave//GraphicState/InitialTextMatrix get currentmatrix pop}bind def +/ET{grestore}bind def +/Tc{//GraphicState exch/CharacterSpacing exch put}bind def +/TL{//GraphicState exch/TextLeading exch put}bind def +/Tr{//GraphicState exch/TextRenderingMode exch put}bind def +/Ts{ +mark(Unimplemented SetTextRise)//error exec +}bind def +/Tw{//GraphicState exch/WordSpacing exch put}bind def +/Tz{ +mark(Unimplemented SetHorizontalTextScaling)//error exec +}bind def +/Td{translate 0 0 moveto}bind def +/TD{dup neg//TL exec//Td exec}bind def +/Tm{//GraphicState/InitialTextMatrix get setmatrix +//TempMatrix astore concat +0 0 moveto}bind def +/T*{0//GraphicState/TextLeading get neg//Td exec}bind def +/Tj{//ShowTextBeg exec//ShowText exec//ShowTextEnd exec}bind def +/'{//T* exec//ShowText exec//ShowTextEnd exec}bind def +/"{3 2 roll//Tw exec exch//Tc exec//' exec}bind def +/TJ//ShowTextWithGlyphPositioning def +/Tf//SetFont def +/d0/setcharwidth load def +/d1/setcachedevice load def +/BDC{pop pop}bind def +/BMC{pop}bind def +/EMC{}bind def +/BX{BeginCompatibilitySection}bind def +/EX{EndCompatibilitySection}bind def +/DP{DefineMarkedContentPointWithPropertyList}bind def +/MP{DefineMarkedContentPoint}bind def +/PS{cvx exec}bind def +currentdict end def +//PDFR_STREAM{ +//Operators length dict begin +//Operators{ +exch dup +[exch//=only/exec load +( )/print load +8 7 roll +dup type/arraytype eq{ +/exec load +}if +( )/print load +]cvx +def +}forall +currentdict end/Operators exch def +}if +/.registerencoding +{pop pop +}bind def +/.defineencoding +{def +}bind def +/.findencoding +{load +}bind def +/currentglobal where +{pop currentglobal{setglobal}true setglobal} +{{}} +ifelse +/MacRomanEncoding +StandardEncoding 0 39 getinterval aload pop +/quotesingle +StandardEncoding 40 56 getinterval aload pop +/grave +StandardEncoding 97 31 getinterval aload pop +/Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute +/agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave +/ecircumflex/edieresis/iacute/igrave +/icircumflex/idieresis/ntilde/oacute +/ograve/ocircumflex/odieresis/otilde +/uacute/ugrave/ucircumflex/udieresis +/dagger/degree/cent/sterling/section/bullet/paragraph/germandbls +/registered/copyright/trademark/acute/dieresis/.notdef/AE/Oslash +/.notdef/plusminus/.notdef/.notdef/yen/mu/.notdef/.notdef +/.notdef/.notdef/.notdef/ordfeminine/ordmasculine/.notdef/ae/oslash +/questiondown/exclamdown/logicalnot/.notdef +/florin/.notdef/.notdef/guillemotleft +/guillemotright/ellipsis/space/Agrave/Atilde/Otilde/OE/oe +/endash/emdash/quotedblleft/quotedblright +/quoteleft/quoteright/divide/.notdef +/ydieresis/Ydieresis/fraction/currency +/guilsinglleft/guilsinglright/fi/fl +/daggerdbl/periodcentered/quotesinglbase/quotedblbase +/perthousand/Acircumflex/Ecircumflex/Aacute +/Edieresis/Egrave/Iacute/Icircumflex +/Idieresis/Igrave/Oacute/Ocircumflex +/.notdef/Ograve/Uacute/Ucircumflex +/Ugrave/dotlessi/circumflex/tilde +/macron/breve/dotaccent/ring/cedilla/hungarumlaut/ogonek/caron +256 packedarray +5 1 index .registerencoding +.defineencoding +exec +/AdobeGlyphList mark +/A 16#0041 +/AE 16#00c6 +/AEacute 16#01fc +/AEmacron 16#01e2 +/AEsmall 16#f7e6 +/Aacute 16#00c1 +/Aacutesmall 16#f7e1 +/Abreve 16#0102 +/Abreveacute 16#1eae +/Abrevecyrillic 16#04d0 +/Abrevedotbelow 16#1eb6 +/Abrevegrave 16#1eb0 +/Abrevehookabove 16#1eb2 +/Abrevetilde 16#1eb4 +/Acaron 16#01cd +/Acircle 16#24b6 +/Acircumflex 16#00c2 +/Acircumflexacute 16#1ea4 +/Acircumflexdotbelow 16#1eac +/Acircumflexgrave 16#1ea6 +/Acircumflexhookabove 16#1ea8 +/Acircumflexsmall 16#f7e2 +/Acircumflextilde 16#1eaa +/Acute 16#f6c9 +/Acutesmall 16#f7b4 +/Acyrillic 16#0410 +/Adblgrave 16#0200 +/Adieresis 16#00c4 +/Adieresiscyrillic 16#04d2 +/Adieresismacron 16#01de +/Adieresissmall 16#f7e4 +/Adotbelow 16#1ea0 +/Adotmacron 16#01e0 +/Agrave 16#00c0 +/Agravesmall 16#f7e0 +/Ahookabove 16#1ea2 +/Aiecyrillic 16#04d4 +/Ainvertedbreve 16#0202 +/Alpha 16#0391 +/Alphatonos 16#0386 +/Amacron 16#0100 +/Amonospace 16#ff21 +/Aogonek 16#0104 +/Aring 16#00c5 +/Aringacute 16#01fa +/Aringbelow 16#1e00 +/Aringsmall 16#f7e5 +/Asmall 16#f761 +/Atilde 16#00c3 +/Atildesmall 16#f7e3 +/Aybarmenian 16#0531 +/B 16#0042 +/Bcircle 16#24b7 +/Bdotaccent 16#1e02 +/Bdotbelow 16#1e04 +/Becyrillic 16#0411 +/Benarmenian 16#0532 +/Beta 16#0392 +/Bhook 16#0181 +/Blinebelow 16#1e06 +/Bmonospace 16#ff22 +/Brevesmall 16#f6f4 +/Bsmall 16#f762 +/Btopbar 16#0182 +/C 16#0043 +/Caarmenian 16#053e +/Cacute 16#0106 +/Caron 16#f6ca +/Caronsmall 16#f6f5 +/Ccaron 16#010c +/Ccedilla 16#00c7 +/Ccedillaacute 16#1e08 +/Ccedillasmall 16#f7e7 +/Ccircle 16#24b8 +/Ccircumflex 16#0108 +/Cdot 16#010a +/Cdotaccent 16#010a +/Cedillasmall 16#f7b8 +/Chaarmenian 16#0549 +/Cheabkhasiancyrillic 16#04bc +/Checyrillic 16#0427 +/Chedescenderabkhasiancyrillic 16#04be +/Chedescendercyrillic 16#04b6 +/Chedieresiscyrillic 16#04f4 +/Cheharmenian 16#0543 +/Chekhakassiancyrillic 16#04cb +/Cheverticalstrokecyrillic 16#04b8 +/Chi 16#03a7 +/Chook 16#0187 +/Circumflexsmall 16#f6f6 +/Cmonospace 16#ff23 +/Coarmenian 16#0551 +/Csmall 16#f763 +/D 16#0044 +/DZ 16#01f1 +/DZcaron 16#01c4 +/Daarmenian 16#0534 +/Dafrican 16#0189 +/Dcaron 16#010e +/Dcedilla 16#1e10 +/Dcircle 16#24b9 +/Dcircumflexbelow 16#1e12 +/Dcroat 16#0110 +/Ddotaccent 16#1e0a +/Ddotbelow 16#1e0c +/Decyrillic 16#0414 +/Deicoptic 16#03ee +/Delta 16#2206 +/Deltagreek 16#0394 +/Dhook 16#018a +/Dieresis 16#f6cb +/DieresisAcute 16#f6cc +/DieresisGrave 16#f6cd +/Dieresissmall 16#f7a8 +/Digammagreek 16#03dc +/Djecyrillic 16#0402 +/Dlinebelow 16#1e0e +/Dmonospace 16#ff24 +/Dotaccentsmall 16#f6f7 +/Dslash 16#0110 +/Dsmall 16#f764 +/Dtopbar 16#018b +/Dz 16#01f2 +/Dzcaron 16#01c5 +/Dzeabkhasiancyrillic 16#04e0 +/Dzecyrillic 16#0405 +/Dzhecyrillic 16#040f +/E 16#0045 +/Eacute 16#00c9 +/Eacutesmall 16#f7e9 +/Ebreve 16#0114 +/Ecaron 16#011a +/Ecedillabreve 16#1e1c +/Echarmenian 16#0535 +/Ecircle 16#24ba +/Ecircumflex 16#00ca +/Ecircumflexacute 16#1ebe +/Ecircumflexbelow 16#1e18 +/Ecircumflexdotbelow 16#1ec6 +/Ecircumflexgrave 16#1ec0 +/Ecircumflexhookabove 16#1ec2 +/Ecircumflexsmall 16#f7ea +/Ecircumflextilde 16#1ec4 +/Ecyrillic 16#0404 +/Edblgrave 16#0204 +/Edieresis 16#00cb +/Edieresissmall 16#f7eb +/Edot 16#0116 +/Edotaccent 16#0116 +/Edotbelow 16#1eb8 +/Efcyrillic 16#0424 +/Egrave 16#00c8 +/Egravesmall 16#f7e8 +/Eharmenian 16#0537 +/Ehookabove 16#1eba +/Eightroman 16#2167 +/Einvertedbreve 16#0206 +/Eiotifiedcyrillic 16#0464 +/Elcyrillic 16#041b +/Elevenroman 16#216a +/Emacron 16#0112 +/Emacronacute 16#1e16 +/Emacrongrave 16#1e14 +/Emcyrillic 16#041c +/Emonospace 16#ff25 +/Encyrillic 16#041d +/Endescendercyrillic 16#04a2 +/Eng 16#014a +/Enghecyrillic 16#04a4 +/Enhookcyrillic 16#04c7 +/Eogonek 16#0118 +/Eopen 16#0190 +/Epsilon 16#0395 +/Epsilontonos 16#0388 +/Ercyrillic 16#0420 +/Ereversed 16#018e +/Ereversedcyrillic 16#042d +/Escyrillic 16#0421 +/Esdescendercyrillic 16#04aa +/Esh 16#01a9 +/Esmall 16#f765 +/Eta 16#0397 +/Etarmenian 16#0538 +/Etatonos 16#0389 +/Eth 16#00d0 +/Ethsmall 16#f7f0 +/Etilde 16#1ebc +/Etildebelow 16#1e1a +/Euro 16#20ac +/Ezh 16#01b7 +/Ezhcaron 16#01ee +/Ezhreversed 16#01b8 +/F 16#0046 +/Fcircle 16#24bb +/Fdotaccent 16#1e1e +/Feharmenian 16#0556 +/Feicoptic 16#03e4 +/Fhook 16#0191 +/Fitacyrillic 16#0472 +/Fiveroman 16#2164 +/Fmonospace 16#ff26 +/Fourroman 16#2163 +/Fsmall 16#f766 +/G 16#0047 +/GBsquare 16#3387 +/Gacute 16#01f4 +/Gamma 16#0393 +/Gammaafrican 16#0194 +/Gangiacoptic 16#03ea +/Gbreve 16#011e +/Gcaron 16#01e6 +/Gcedilla 16#0122 +/Gcircle 16#24bc +/Gcircumflex 16#011c +/Gcommaaccent 16#0122 +/Gdot 16#0120 +/Gdotaccent 16#0120 +/Gecyrillic 16#0413 +/Ghadarmenian 16#0542 +/Ghemiddlehookcyrillic 16#0494 +/Ghestrokecyrillic 16#0492 +/Gheupturncyrillic 16#0490 +/Ghook 16#0193 +/Gimarmenian 16#0533 +/Gjecyrillic 16#0403 +/Gmacron 16#1e20 +/Gmonospace 16#ff27 +/Grave 16#f6ce +/Gravesmall 16#f760 +/Gsmall 16#f767 +/Gsmallhook 16#029b +/Gstroke 16#01e4 +/H 16#0048 +/H18533 16#25cf +/H18543 16#25aa +/H18551 16#25ab +/H22073 16#25a1 +/HPsquare 16#33cb +/Haabkhasiancyrillic 16#04a8 +/Hadescendercyrillic 16#04b2 +/Hardsigncyrillic 16#042a +/Hbar 16#0126 +/Hbrevebelow 16#1e2a +/Hcedilla 16#1e28 +/Hcircle 16#24bd +/Hcircumflex 16#0124 +/Hdieresis 16#1e26 +/Hdotaccent 16#1e22 +/Hdotbelow 16#1e24 +/Hmonospace 16#ff28 +/Hoarmenian 16#0540 +/Horicoptic 16#03e8 +/Hsmall 16#f768 +/Hungarumlaut 16#f6cf +/Hungarumlautsmall 16#f6f8 +/Hzsquare 16#3390 +/I 16#0049 +/IAcyrillic 16#042f +/IJ 16#0132 +/IUcyrillic 16#042e +/Iacute 16#00cd +/Iacutesmall 16#f7ed +/Ibreve 16#012c +/Icaron 16#01cf +/Icircle 16#24be +/Icircumflex 16#00ce +/Icircumflexsmall 16#f7ee +/Icyrillic 16#0406 +/Idblgrave 16#0208 +/Idieresis 16#00cf +/Idieresisacute 16#1e2e +/Idieresiscyrillic 16#04e4 +/Idieresissmall 16#f7ef +/Idot 16#0130 +/Idotaccent 16#0130 +/Idotbelow 16#1eca +/Iebrevecyrillic 16#04d6 +/Iecyrillic 16#0415 +/Ifraktur 16#2111 +/Igrave 16#00cc +/Igravesmall 16#f7ec +/Ihookabove 16#1ec8 +/Iicyrillic 16#0418 +/Iinvertedbreve 16#020a +/Iishortcyrillic 16#0419 +/Imacron 16#012a +/Imacroncyrillic 16#04e2 +/Imonospace 16#ff29 +/Iniarmenian 16#053b +/Iocyrillic 16#0401 +/Iogonek 16#012e +/Iota 16#0399 +/Iotaafrican 16#0196 +/Iotadieresis 16#03aa +/Iotatonos 16#038a +/Ismall 16#f769 +/Istroke 16#0197 +/Itilde 16#0128 +/Itildebelow 16#1e2c +/Izhitsacyrillic 16#0474 +/Izhitsadblgravecyrillic 16#0476 +/J 16#004a +/Jaarmenian 16#0541 +/Jcircle 16#24bf +/Jcircumflex 16#0134 +/Jecyrillic 16#0408 +/Jheharmenian 16#054b +/Jmonospace 16#ff2a +/Jsmall 16#f76a +/K 16#004b +/KBsquare 16#3385 +/KKsquare 16#33cd +/Kabashkircyrillic 16#04a0 +/Kacute 16#1e30 +/Kacyrillic 16#041a +/Kadescendercyrillic 16#049a +/Kahookcyrillic 16#04c3 +/Kappa 16#039a +/Kastrokecyrillic 16#049e +/Kaverticalstrokecyrillic 16#049c +/Kcaron 16#01e8 +/Kcedilla 16#0136 +/Kcircle 16#24c0 +/Kcommaaccent 16#0136 +/Kdotbelow 16#1e32 +/Keharmenian 16#0554 +/Kenarmenian 16#053f +/Khacyrillic 16#0425 +/Kheicoptic 16#03e6 +/Khook 16#0198 +/Kjecyrillic 16#040c +/Klinebelow 16#1e34 +/Kmonospace 16#ff2b +/Koppacyrillic 16#0480 +/Koppagreek 16#03de +/Ksicyrillic 16#046e +/Ksmall 16#f76b +/L 16#004c +/LJ 16#01c7 +/LL 16#f6bf +/Lacute 16#0139 +/Lambda 16#039b +/Lcaron 16#013d +/Lcedilla 16#013b +/Lcircle 16#24c1 +/Lcircumflexbelow 16#1e3c +/Lcommaaccent 16#013b +/Ldot 16#013f +/Ldotaccent 16#013f +/Ldotbelow 16#1e36 +/Ldotbelowmacron 16#1e38 +/Liwnarmenian 16#053c +/Lj 16#01c8 +/Ljecyrillic 16#0409 +/Llinebelow 16#1e3a +/Lmonospace 16#ff2c +/Lslash 16#0141 +/Lslashsmall 16#f6f9 +/Lsmall 16#f76c +/M 16#004d +/MBsquare 16#3386 +/Macron 16#f6d0 +/Macronsmall 16#f7af +/Macute 16#1e3e +/Mcircle 16#24c2 +/Mdotaccent 16#1e40 +/Mdotbelow 16#1e42 +/Menarmenian 16#0544 +/Mmonospace 16#ff2d +/Msmall 16#f76d +/Mturned 16#019c +/Mu 16#039c +/N 16#004e +/NJ 16#01ca +/Nacute 16#0143 +/Ncaron 16#0147 +/Ncedilla 16#0145 +/Ncircle 16#24c3 +/Ncircumflexbelow 16#1e4a +/Ncommaaccent 16#0145 +/Ndotaccent 16#1e44 +/Ndotbelow 16#1e46 +/Nhookleft 16#019d +/Nineroman 16#2168 +/Nj 16#01cb +/Njecyrillic 16#040a +/Nlinebelow 16#1e48 +/Nmonospace 16#ff2e +/Nowarmenian 16#0546 +/Nsmall 16#f76e +/Ntilde 16#00d1 +/Ntildesmall 16#f7f1 +/Nu 16#039d +/O 16#004f +/OE 16#0152 +/OEsmall 16#f6fa +/Oacute 16#00d3 +/Oacutesmall 16#f7f3 +/Obarredcyrillic 16#04e8 +/Obarreddieresiscyrillic 16#04ea +/Obreve 16#014e +/Ocaron 16#01d1 +/Ocenteredtilde 16#019f +/Ocircle 16#24c4 +/Ocircumflex 16#00d4 +/Ocircumflexacute 16#1ed0 +/Ocircumflexdotbelow 16#1ed8 +/Ocircumflexgrave 16#1ed2 +/Ocircumflexhookabove 16#1ed4 +/Ocircumflexsmall 16#f7f4 +/Ocircumflextilde 16#1ed6 +/Ocyrillic 16#041e +/Odblacute 16#0150 +/Odblgrave 16#020c +/Odieresis 16#00d6 +/Odieresiscyrillic 16#04e6 +/Odieresissmall 16#f7f6 +/Odotbelow 16#1ecc +/Ogoneksmall 16#f6fb +/Ograve 16#00d2 +/Ogravesmall 16#f7f2 +/Oharmenian 16#0555 +/Ohm 16#2126 +/Ohookabove 16#1ece +/Ohorn 16#01a0 +/Ohornacute 16#1eda +/Ohorndotbelow 16#1ee2 +/Ohorngrave 16#1edc +/Ohornhookabove 16#1ede +/Ohorntilde 16#1ee0 +/Ohungarumlaut 16#0150 +/Oi 16#01a2 +/Oinvertedbreve 16#020e +/Omacron 16#014c +/Omacronacute 16#1e52 +/Omacrongrave 16#1e50 +/Omega 16#2126 +/Omegacyrillic 16#0460 +/Omegagreek 16#03a9 +/Omegaroundcyrillic 16#047a +/Omegatitlocyrillic 16#047c +/Omegatonos 16#038f +/Omicron 16#039f +/Omicrontonos 16#038c +/Omonospace 16#ff2f +/Oneroman 16#2160 +/Oogonek 16#01ea +/Oogonekmacron 16#01ec +/Oopen 16#0186 +/Oslash 16#00d8 +/Oslashacute 16#01fe +/Oslashsmall 16#f7f8 +/Osmall 16#f76f +/Ostrokeacute 16#01fe +/Otcyrillic 16#047e +/Otilde 16#00d5 +/Otildeacute 16#1e4c +/Otildedieresis 16#1e4e +/Otildesmall 16#f7f5 +/P 16#0050 +/Pacute 16#1e54 +/Pcircle 16#24c5 +/Pdotaccent 16#1e56 +/Pecyrillic 16#041f +/Peharmenian 16#054a +/Pemiddlehookcyrillic 16#04a6 +/Phi 16#03a6 +/Phook 16#01a4 +/Pi 16#03a0 +/Piwrarmenian 16#0553 +/Pmonospace 16#ff30 +/Psi 16#03a8 +/Psicyrillic 16#0470 +/Psmall 16#f770 +/Q 16#0051 +/Qcircle 16#24c6 +/Qmonospace 16#ff31 +/Qsmall 16#f771 +/R 16#0052 +/Raarmenian 16#054c +/Racute 16#0154 +/Rcaron 16#0158 +/Rcedilla 16#0156 +/Rcircle 16#24c7 +/Rcommaaccent 16#0156 +/Rdblgrave 16#0210 +/Rdotaccent 16#1e58 +/Rdotbelow 16#1e5a +/Rdotbelowmacron 16#1e5c +/Reharmenian 16#0550 +/Rfraktur 16#211c +/Rho 16#03a1 +/Ringsmall 16#f6fc +/Rinvertedbreve 16#0212 +/Rlinebelow 16#1e5e +/Rmonospace 16#ff32 +/Rsmall 16#f772 +/Rsmallinverted 16#0281 +/Rsmallinvertedsuperior 16#02b6 +/S 16#0053 +/SF010000 16#250c +/SF020000 16#2514 +/SF030000 16#2510 +/SF040000 16#2518 +/SF050000 16#253c +/SF060000 16#252c +/SF070000 16#2534 +/SF080000 16#251c +/SF090000 16#2524 +/SF100000 16#2500 +/SF110000 16#2502 +/SF190000 16#2561 +/SF200000 16#2562 +/SF210000 16#2556 +/SF220000 16#2555 +/SF230000 16#2563 +/SF240000 16#2551 +/SF250000 16#2557 +/SF260000 16#255d +/SF270000 16#255c +/SF280000 16#255b +/SF360000 16#255e +/SF370000 16#255f +/SF380000 16#255a +/SF390000 16#2554 +/SF400000 16#2569 +/SF410000 16#2566 +/SF420000 16#2560 +/SF430000 16#2550 +/SF440000 16#256c +/SF450000 16#2567 +/SF460000 16#2568 +/SF470000 16#2564 +/SF480000 16#2565 +/SF490000 16#2559 +/SF500000 16#2558 +/SF510000 16#2552 +/SF520000 16#2553 +/SF530000 16#256b +/SF540000 16#256a +/Sacute 16#015a +/Sacutedotaccent 16#1e64 +/Sampigreek 16#03e0 +/Scaron 16#0160 +/Scarondotaccent 16#1e66 +/Scaronsmall 16#f6fd +/Scedilla 16#015e +/Schwa 16#018f +/Schwacyrillic 16#04d8 +/Schwadieresiscyrillic 16#04da +/Scircle 16#24c8 +/Scircumflex 16#015c +/Scommaaccent 16#0218 +/Sdotaccent 16#1e60 +/Sdotbelow 16#1e62 +/Sdotbelowdotaccent 16#1e68 +/Seharmenian 16#054d +/Sevenroman 16#2166 +/Shaarmenian 16#0547 +/Shacyrillic 16#0428 +/Shchacyrillic 16#0429 +/Sheicoptic 16#03e2 +/Shhacyrillic 16#04ba +/Shimacoptic 16#03ec +/Sigma 16#03a3 +/Sixroman 16#2165 +/Smonospace 16#ff33 +/Softsigncyrillic 16#042c +/Ssmall 16#f773 +/Stigmagreek 16#03da +/T 16#0054 +/Tau 16#03a4 +/Tbar 16#0166 +/Tcaron 16#0164 +/Tcedilla 16#0162 +/Tcircle 16#24c9 +/Tcircumflexbelow 16#1e70 +/Tcommaaccent 16#0162 +/Tdotaccent 16#1e6a +/Tdotbelow 16#1e6c +/Tecyrillic 16#0422 +/Tedescendercyrillic 16#04ac +/Tenroman 16#2169 +/Tetsecyrillic 16#04b4 +/Theta 16#0398 +/Thook 16#01ac +/Thorn 16#00de +/Thornsmall 16#f7fe +/Threeroman 16#2162 +/Tildesmall 16#f6fe +/Tiwnarmenian 16#054f 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+jj)^WR:*RZF.28PeosMm=lZ-CBD9DaR-7^3O/$BCkTLc,\os^EC-%ZYWbT,RYH2M#W$(>=Rb#6X +Cc`EQZ>@-2RKm8WYT:(;]6<)ADEF0I\o,-gn% +EI Q +Q + +endstream +endobj +%%PageTrailer +%%Trailer +end +%%EOF From c2d7729e8128b7bcac2bb47587f3bac60d55facc Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Thu, 2 Apr 2026 11:45:15 +0100 Subject: [PATCH 006/116] Deleted placeholder PC2 description. --- .../source/science_guide/cloud_schemes/pc2.rst | 15 --------------- 1 file changed, 15 deletions(-) delete mode 100644 documentation/source/science_guide/cloud_schemes/pc2.rst diff --git a/documentation/source/science_guide/cloud_schemes/pc2.rst b/documentation/source/science_guide/cloud_schemes/pc2.rst deleted file mode 100644 index 5de847ffb6..0000000000 --- a/documentation/source/science_guide/cloud_schemes/pc2.rst +++ /dev/null @@ -1,15 +0,0 @@ -.. ----------------------------------------------------------------------------- - (c) Crown copyright Met Office. All rights reserved. - The file LICENCE, distributed with this code, contains details of the terms - under which the code may be used. - ----------------------------------------------------------------------------- - -.. _pc2: - -The PC2 cloud-scheme -==================== - -Here we describe the PC2 ("Prognostic Cloud 2") scheme. -As implied by the name, this scheme uses prognostic variables for -the sub-grid cloud-fractions (both liquid and ice) -and condensed water mixing-ratios. From 3aa778f36b792303c363070cf21cb9333f37fc34 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Thu, 2 Apr 2026 11:47:59 +0100 Subject: [PATCH 007/116] Deleted the latex source now we've converted to .rst --- .../cloud_schemes/UMDP30_PC2CloudScheme.tex | 5734 ----------------- .../cloud_schemes/um_call_tree.tex | 530 -- .../cloud_schemes/um_call_tree_preamble.tex | 25 - 3 files changed, 6289 deletions(-) delete mode 100644 documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.tex delete mode 100644 documentation/source/science_guide/cloud_schemes/um_call_tree.tex delete mode 100644 documentation/source/science_guide/cloud_schemes/um_call_tree_preamble.tex diff --git a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.tex b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.tex deleted file mode 100644 index 8faee4121d..0000000000 --- a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.tex +++ /dev/null @@ -1,5734 +0,0 @@ -\documentclass{UMDP_article} - -\title{The PC2 Cloud Scheme} -\paperno{030} -\umversion{14.0} -\owner{Cyril Morcrette} -\author{D.~Wilson, A.~Bushell, C.~Morcrette, V.~Varma$^{1}$, M.~Whitall} - -\titlecontent{ - \footnotesize - $^1$ National Institute of Water and Atmospheric Research, - Wellington, New Zealand \\ - } - -\usepackage{textcomp} -\usepackage{amstext,natbib} - -% Packages needed for the UM subroutine tree diagram in um_call_tree.txt: -\input{um_call_tree_preamble} - -\newcommand{\mmax}[1] {\mbox{\footnotesize \sf MAX} \left[#1\right] } - -%%% These are definitions used in the convection documentation -%%% Definitions running across several chapters are defined in PC2_main. -\newcommand{\gbmll}{\ensuremath{\overline{l}_{\rm{l}}}} -\newcommand{\gbmlf}{\ensuremath{\overline{l}_{\rm{f}}}} -\newcommand{\gbmlm}{\ensuremath{\overline{l}_{\rm{m}}}} -\newcommand{\gbmlx}{\ensuremath{\overline{l}_{\rm{x}}}} -\newcommand{\incml}[1]{\ensuremath{\overline{l}_{\rm c}^{\rm #1}}} -\newcommand{\bigqc}[1]{\ensuremath{Q_{\rm C}^{\rm #1}}} -\newcommand{\GpdfB}{\ensuremath{G^{\rm B}}} -\newcommand{\injectcl}{\ensuremath{C_{\rm{l}}^{\ast}}} -\newcommand{\injectcf}{\ensuremath{C_{\rm{f}}^{\ast}}} -% -\newcommand{\lsubsup}[2]{\ensuremath{l_{\rm{#1}}^{\rm{#2}}}} -\newcommand{\csubsup}[2]{\ensuremath{l_{\rm{#1}}^{\rm{#2}}}} -\newcommand{\qsubsup}[2]{\ensuremath{q_{\rm{#1}}^{\rm{#2}}}} -\newcommand{\tsubsup}[2]{\ensuremath{T_{\rm{#1}}^{\rm{#2}}}} -\newcommand{\thsubsup}[2]{\ensuremath{\theta_{\rm{#1}}^{\rm{#2}}}} -\newcommand{\xsubsup}[2]{\ensuremath{{\chi}_{\rm{#1}}^{\rm{#2}}}} -%%% Private definitions (shorthand commands etc.) -\newcommand{\lc}{\left[} -\newcommand{\rc}{\right]} -\newcommand{\ov}{\overline} -\newcommand{\lp}{\left(} -\newcommand{\rp}{\right)} -\newcommand{\dr}{\partial} - -%%% Define a format for partial derivatives -\newcommand{\pardbyd}[2]{\ensuremath{\frac{\partial \, #1}{\partial \, #2}}} - -%%% Define a few of the commonest variables -\newcommand{\Tliq}{\ensuremath{T_{\rm L}}} -\newcommand{\qtot}{\ensuremath{q_{\rm T}}} -\newcommand{\qsat}{\ensuremath{q_{\rm s}}} -\newcommand{\aliq}{\ensuremath{a_{\rm L}}} -\newcommand{\gbmTliq}{\ensuremath{\overline{T}_{\rm L}}} -\newcommand{\gbmqtot}{\ensuremath{\overline{q}_{\rm T}}} -%Add a new command to do a horizontal line -\newcommand{\HRule}{\rule{\linewidth}{1.0mm}} - -\begin{document} % Every document must start with this. -\maketitle - -\tableofcontents - -\newpage -\section{PC2 developers} -We would like to acknowledge those who developed the PC2 cloud scheme: -Damian Wilson, Andrew Bushell, David Gregory, Amanda Kerr-Munslow, -John Edwards, Jeremy Price, Cyril Morcrette, Martin Sharpe, Thomas Mirfield, Ian Boutle. Many -others offered considerable help, advice and analysis, including Roy Kershaw, -Malcolm Brooks, Richard Forbes and Alejandro Bodas-Salcedo, and we would -like to thank them all for their input. - -\section{Introduction} - -This document describes the PC2 \textit{(prognostic cloud, -prognostic condensate)} cloud scheme. It should be seen as -a complete reference source for the scheme's physical assumptions, -numerical techniques, -application to the Unified Model and coding within the Unified Model. It does -not describe results from the scheme, please refer to the various reports and papers -written on this. Except where commented on explicitly, the description applies -to the PC2:66 version of the PC2 scheme, which is the version that will be -available at UM6.5. The version available at 6.4 is PC2:64. - -This paper will first introduce the concepts that underlie cloud schemes, -before developing a study of the theoretical behaviour of the prognostic PC2 scheme -under certain, well-defined, situations. The next sections shows how -the theory can be applied to the physical and dynamical processes -represented in the Unified Model. Finally, we outline the way in which -the PC2 scheme is implemented within the code of the Unified Model. - -%%\subsection{How to use this documentation} - -\subsection{Cloud schemes} - -The basic requirements of any cloud scheme within a large-scale model are to: -\begin{itemize} -\item{calculate the amount of condensation (from water vapour to liquid water or vice-versa) within each gridbox each timestep} -\item{to calculate or update the cloud fractions for use by the radiation and large-scale precipitation schemes (or any other physics scheme).} -\end{itemize} - -Depending on the model involved, cloud schemes may also treat the -deposition / sublimation process from vapour to ice. The problem is -straightforward to solve if one is allowed to assume that there is no -variability of moisture or temperature on a scale of a model gridbox. -In this case the cloud fraction scheme is redundant and only the condensation -part remains, which may be solved diagnostically using the instantaneous -condensation assumption in section \ref{sec:s_dist}. However, the -`no-variability' assumption is poor until -very high resolutions close to, or maybe exceeding, 1 km in the horizontal -are reached. Although we may eventually assume that computer power -will enable such resolutions to be reached globally, for many years we -will need a subgrid-scale cloud scheme to properly account for the -variability in the atmosphere. This is the principal challenge of -cloud parametrization. - -There are several approaches to take to the solution of the problem, -although they are not as independent as often portrayed, since they -nearly all require the same instantaneous condensation assumption -(discussed in section \ref{sec:s_dist}). Hence there are mathematical -links between -all the approaches. \textit{The following are all valid structures -to use in this respect.} -\begin{itemize} -\item{One may diagnose cloud fractions and condensate contents from knowledge of gridbox mean variables. This forms the basis of the \cite{smith90} scheme, which is described in -\citeumdp{029}.} -\item{A mixed scheme, such as \cite{sundqvist1978} uses a prediction of condensate contents, but a diagnostic cloud fraction.} -\item{Alternatively, one may predict cloud fraction and condensate content changes as a result of each modelled process. This forms the basis of the \cite{t93} scheme and the PC2 scheme.} -\item{Hybrid schemes, such as \cite{t02}, will predict various moments of the subgrid-scale variability, and use this knowledge to diagnose the cloud fraction and condensate contents.} -\end{itemize} - -Many years of experience of the results from the \cite{smith90} -scheme have highlighted deficiences in the diagnosis of cloud from -this scheme, which we feel can only be tackled by adding the -memory of cloud history available by using a prognostic based -scheme. We chose to develop a scheme that directly specified -the impacts on observable prognostics (condensates and -cloud fractions, as in \cite{t93}) rather than on moments of a -probability density function (as in \cite{t02}). This is because -we believe it is easier to physically relate (and hence parametrize) -the effect processes to quantities -such as cloud fraction and condensate rather than to the more abstract -quantities of moments of a probability -density function of moisture. However, although the -PC2 scheme is similar to \cite{t93} in its very basic prognostic variable -structure, the assumptions behind the formulation of the prognostic terms -in PC2 are very different and much improved. The PC2 scheme should not be -considered to be merely an extension of \cite{t93}. - -In particular, we wish to use a prognostic formulation in order to link -the detraiment of moisture from convection directly to cloud fraction, and -to break the hard diagnostic link between cloud fraction and condensate. These -major features of the \cite{t93} scheme provide the motivation to develop -the PC2 cloud scheme. - -\subsection{The `s' distribution} -\label{sec:s_dist} - -Most cloud schemes are based on the concept of a distribution -of fluctuations of moisture and temperature in the gridbox. Here we -mathematically formalize this concept, since it is used both in -the PC2 scheme and the \cite{smith90} scheme. - -This method was first formulated by \cite{m77} and \cite{sommeria_deardorff_1977} -for large-eddy simulations. It can also be applied -to larger scale models. It allows us to calculate vapour and liquid -contents and liquid cloud fraction from knowledge only of the combined -vapour+liquid content, $\overline{q_T}$, and the liquid temperature, -$\overline{T_L}$. These variables are unchanged during condensation -processes, so it is useful to write the cloud scheme in terms of these -variables. - -In this derivation we will consider only liquid condensate. -We assume, as above, that, -locally, the water content in a cloud is such as to remove any -supersaturation. This gives the equation - -\begin{equation} -q_{cl} = q_T - q_{sat}(T,p) -\label{eq:basic_qcl} -\end{equation} - -assuming that $q_T > q_{sat}(T,p)$ ($q_{cl}$ will be zero otherwise). -$q_T$ is the local total water -content, equal to the sum of the condensate $(q_{cl})$ plus the vapour -$(q)$, $T$ is the temperature, $p$ is the pressure and $q_{sat}(T,p)$ -is the saturation specific humidity at temperature T and pressure -p \textit{with respect to liquid water}. (Many earlier diagnostic -cloud schemes use a similar instantaneous condensation assumption -for ice, which would mean that $q_{sat}$ must be taken with respect -to ice when $T < 0 ^{\circ} C$, but the Unified Model does not). -We now introduce the liquid temperature ($T_L$), where $T_L$ is given by - -\begin{equation} -T_L = T - \frac{L}{c_p} q_{cl} , -\label{eq:tl} -\end{equation} - -and $L$ is the latent heat of -vaporization and $c_p$ is the heat capacity of air. -Note that $T_L$ is unaffected by changes of phase between vapour and liquid. -We now write (\ref{eq:basic_qcl}) as an \textit{equality} - -\begin{equation} -q_{cl} = q_T - \left( q_{sat}(T_L,p) + \alpha (T - T_L) \right) -\label{eq:alpha_t_tl} -\end{equation} - -where - -\begin{equation} -\alpha = \frac{ q_{sat}(T,p) - q_{sat}(T_L,p) }{T - T_L } . -\label{eq:alpha} -\end{equation} - -Using (\ref{eq:tl}) in (\ref{eq:alpha_t_tl}) gives the expression - -\begin{equation} -q_{cl} = q_T - q_{sat} (T_L) - \alpha \frac{L}{c_p} q_{cl} -\end{equation} - -or - -\begin{equation} -q_{cl} = a_L \left( q_T - q_{sat}(T_L,p) \right) -\label{eq:l_eq_al} -\end{equation} - -where $a_L$ is given by - -\begin{equation} -a_L = \left( 1 + \alpha \frac{L}{c_p} \right) ^{-1} . -\label{eq:a_L} -\end{equation} - -Thus (\ref{eq:basic_qcl}) has been rewritten \textit{exactly} in terms of the conserved -variables, $q_T$ and $T_L$, although the temperature, $T$, does remain in the -definition of $a_L$. -We will need to consider variations across a gridbox for a -parametrization scheme, -so we expand the expression for condensate (\ref{eq:l_eq_al}) into terms -relating to the gridbox mean and variation from the gridbox mean. - -\begin{equation} -q_{cl} = \overline{ a_L \left( q_T - q_{sat}(T_L,p) \right)} -+ [ a_L \left( q_T - q_{sat}(T_L,p) \right) ]' -\label{eq:bar_plus_pri1} -\end{equation} - -where $\overline{\phi}$ represents the mean of a distibution of $\phi$ and -$\phi = \overline{\phi} + {\phi}'$. The expression (\ref{eq:bar_plus_pri1}) -is \textit{exact} when -using the definition of $\alpha$ given in (\ref{eq:alpha}). - -The idea of a PDF scheme is to calculate the first (mean) term, -$\overline{\phi}$, from the known gridbox mean parameters, $q_T$, $T_L$ -and $p$, and to parametrize the distribution of the second, -variable term, ${\phi}'$. Unfortunately, the mean term is difficult -to write in terms of the gridbox mean variables $\overline{q_T}$ -and $\overline{T_L}$ because -$q_{sat}(T_L,p)$ is not a linear function of $T_L$ (or of $p$). In order to -proceed, we will now make an \textit{approximation} that $q_{sat}(T,p)$ -is a linear function of $T_L$ and $p$. -This equivalently implies that $a_L$ and $\alpha$ are approximated as -being constant across the gridbox. The expression -now becomes more tractable, (\ref{eq:bar_plus_pri1}) becoming: - -\begin{equation} -q_{cl} = a_L \left( \overline{q_T} - q_{sat}(\overline{T_L},\overline{p}) -\right) + a_L \left( {q_T}' - \alpha {T_L}' - \beta {p}' \right) -\label{eq:l_eq_bar_plus_pri} -\end{equation} - -where $\beta = {\frac{\partial q_{sat}}{\partial p}}$ at constant -temperature. The first term is connected with the mean properties of -the gridbox, and is written as $Q_c$, the second term is connected with the -deviation of the local conditions from the mean and is written as -$s$. - -\begin{equation} -Q_c = a_L \left( \overline{q_T} - q_{sat}(\overline{T_L},\overline{p}) \right) -\label{eq:qc_eq_qt-qs} -\end{equation} - -\begin{equation} -s = a_L \left( {q_T}' - \alpha {T_L}' - \beta {p}' \right) -\label{eq:s} -\end{equation} - -This gives the equation - -\begin{equation} -q_{cl} = Q_c + s -\label{eq:l_qc_s} -\end{equation} - -with the assumption that $s \ge -Q_c$ (i.e. $q_{cl} \ge 0$). -If $s < -Q_c$ then $q_{cl} = 0$. The term $a_L$ can be calculated -using (\ref{eq:a_L}) -from (\ref{eq:alpha}) with gridbox mean temperatures, i.e. - -\begin{equation} -\alpha = \frac{ q_{sat}(\overline{T},\overline{p}) -- q_{sat}(\overline{T_L},\overline{p}) }{\overline{T} - -\overline{T_L} } . -\label{eq:alpha_mean} -\end{equation} - -This definition of $\alpha$ and $a_L$ will retrieve an \textit{exact} value for -the gridbox mean $\overline{q_{cl}}$ \textit{if} -the distribution is monodispersed. Hence it is the sensible form to use for a -purely diagnostic representation such as \cite{smith90} where we explicitly -consider distributions of $s$. Strictly, the linear approximation -implies that other -approximations for $\alpha$ are valid: PC2 will do this -(see section \ref{sec:homog_num_app}) since we are concerned in PC2 with -the best estimate of the \textit{changes} to $\overline{q_{cl}}$, not the best -estimate of $\overline{q_{cl}}$ itself. - -We now assume that within -any particular gridbox a distribution $G$ of $s$ occurs (with mean, by -definition, of zero). Considering cloud to be where the water content -is greater than zero (i.e. where $s > -Q_c$) gives an expression for the -liquid cloud \textit{volume} fraction, $C_l$, within the gridbox as - -\begin{equation} -C_l = \int_{s=-Q_c}^{\infty} G(s) ds -\label{eq:int_gs_ds} -\end{equation} - -and the expression for mean condensate, ${\overline{q_{cl}}}$, using -(\ref{eq:l_qc_s}) to expand $q_{cl}$, is - -\begin{equation} -\overline{q_{cl}} = \int_{s=-Q_c}^{\infty} (Q_c + s) G(s) ds . -\label{eq:qclbar=int} -\end{equation} - -If we know (parametrize) the PDF given by $G(s)$ then we can solve -for $C_l$ and $\overline{q_{cl}}$. Note that this distribution is in terms of -$s$, there is no need to know the three-dimensional distribution in terms -of three separate variables $q_T$, $T_L$ and $p$. This is the method used -by \cite{smith90}, where a -symmetric triangular distribution function is used. For further -information on the \cite{smith90} scheme, please refer to \citeumdp{029}. Physics and dynamics schemes hence only need to -provide increments to $\overline{q_T}$ and $\overline{T_L}$, -provided that a diagnostic scheme (such as \cite{smith90}) is called at some point in the -timestep to partition $\overline{q_T}$ into $\overline{q}$ and $\overline{q_{cl}}$, -to calculate the dry bulb temperature $\overline{T}$ (from $\overline{T_L}$ -and $\overline{q_{cl}}$) and to calculate the liquid -cloud fraction, $C_l$. The diagnostic scheme effectively allows a calculation of -condensation associated with any physical process. However, its results remain -tied to the distribution of $G(s)$ that is chosen in (\ref{eq:int_gs_ds}) and -(\ref{eq:qclbar=int}) and it is this tie that we seek to break by the use of -a prognostic scheme. - -\subsection{Concept of PC2} - -The PC2 scheme develops prognostic expressions for the rates of change of -cloud fraction and condensate contents as a result of each process that acts in -the model. We consider ice and liquid condensate as two distinct -aspects of clouds, which may or may not overlap -Figure \ref{fig:schematic} provides a schematic summary of the PC2 scheme. -The equations for the five prognostic cloud variables can be written schematically: - -\begin{eqnarray} -\frac{\partial \overline{q_{cl}}}{\partial t} = -\frac{\partial \overline{q_{cl}}}{\partial t} |_{advection} + -\frac{\partial \overline{q_{cl}}}{\partial t} |_{convection} + -\frac{\partial \overline{q_{cl}}}{\partial t} |_{boundary \, layer} + -\frac{\partial \overline{q_{cl}}}{\partial t} |_{precipitation} + ... \nonumber \\ -\frac{\partial \overline{q_{cf}}}{\partial t} = -\frac{\partial \overline{q_{cf}}}{\partial t} |_{advection} + -\frac{\partial \overline{q_{cf}}}{\partial t} |_{convection} + -\frac{\partial \overline{q_{cf}}}{\partial t} |_{boundary \, layer} + -\frac{\partial \overline{q_{cf}}}{\partial t} |_{precipitation} + ... \nonumber \\ -\frac{\partial C_l}{\partial t} = -\frac{\partial C_l}{\partial t} |_{advection} + -\frac{\partial C_l}{\partial t} |_{convection} + -\frac{\partial C_l}{\partial t} |_{boundary \, layer} + -\frac{\partial C_l}{\partial t} |_{precipitation} + ... \nonumber \\ -\frac{\partial C_i}{\partial t} = -\frac{\partial C_i}{\partial t} |_{advection} + -\frac{\partial C_i}{\partial t} |_{convection} + -\frac{\partial C_i}{\partial t} |_{boundary \, layer} + -\frac{\partial C_i}{\partial t} |_{precipitation} + ... \nonumber \\ -\frac{\partial C_t}{\partial t} = -\frac{\partial C_t}{\partial t} |_{advection} + -\frac{\partial C_t}{\partial t} |_{convection} + -\frac{\partial C_t}{\partial t} |_{boundary \, layer} + -\frac{\partial C_t}{\partial t} |_{precipitation} + ... , -\label{eq:dqcldt_and_dcdt} -\end{eqnarray} - -where $\overline{q_{cf}}$ is the ice water specfic humidity, $C_l$ is the liquid -cloud \textit{volume} fraction, $C_i$ is the ice cloud volume fraction, and $C_t$ -is the combined ice or liquid cloud volume fraction. The amount of mixed -phase cloud, $C_{mp}$, can be calculated by the overlap of the ice -and liquid fractions: - -\begin{equation} -C_{mp} = C_i + C_l - C_t. -\label{eq:mp} -\end{equation} - -The idea is to parametrize each of the terms in the above equations. This -approach removes the diagnostic method, hence it will be critical that -we can write expressions for $\frac{\partial \overline{q_{cl}}}{\partial t}$ -and -$\frac{\partial C_l}{\partial t}$ for \textit{each process that alters -$\overline{T}$, $\overline{p}$, -$\overline{q}$, or $\overline{q_{cl}}$ in the model} (and similarly -for the ice terms). In doing so, -we will not lose sight of underlying PDF approach given by -(\ref{eq:int_gs_ds}) and (\ref{eq:qclbar=int}) since we will still use -the concept of -instantaneous condensation for liquid clouds. Equations -\ref{eq:int_gs_ds} and \ref{eq:qclbar=int} will form the -basis of the homogeneous forcing methods discussed in section -\ref{sec:homog}. -We note in particular that the convective cloud fraction, -previously a quantity that is diagnosed separately from the -large-scale cloud fraction calculated by the \cite{smith90} scheme, -may, in PC2, be included as part of the large-scale -cloud fraction. This aspect is similar to the \cite{t93} approach. - -The final aim of PC2 is -that the parametrization of each term in (\ref{eq:dqcldt_and_dcdt}) -is performed by each part of the model that alters $\overline{T}$, -$\overline{p}$, -$\overline{q}$, $\overline{q_{cl}}$ or $\overline{q_{cf}}$ as an -integral part of that physics or dynamics scheme. -However, in this PC2 scheme we acknowledge that this will not be possible, -at least, not to begin with. -Hence we have specifically developed generic approaches -that can be used to calculate expressions for -$\frac{\partial \overline{q_{cl}}}{\partial t}$ and $\frac{\partial C_l} -{\partial t}$ . -These are referred to as Homogeneous forcing (section \ref{sec:homog}), -Injection source (or inhomogeneous forcing, section \ref{sec:inhomog}) -and Width Changing (section \ref{sec:width}). Two additional modules -are available to assist with PC2, liquid cloud initiaion (section -\ref{sec:init}) and the calculation of total cloud fraction changes -(section \ref{sec:ct}). At -the present time, only the large-scale precipitation (section -\ref{sec:precip}) scheme has been rewritten fully to use the PC2 -concept of prognostic cloud fractions. The existing mass-flux -convection scheme has been modified to enable calculation of the detrained -condensate, but direct modification to the cloud fraction is not -included. All other physics schemes use one of the generic -approaches below. - -\subsubsection{A note on convective cloud fraction} - -It was the original intention that PC2 be able to replace the two -separate diagnostic cloud fractions (large-scale and convective) with -a single cloud fraction, as in \cite{t93}. The hypothesis was that -by detraining cloud -directly from the convection scheme we would no longer need a separate -representation of this cloud type. Our experience with PC2 is that -this is not necessarily the case. We suspect that the basic reason is -that we are unable to truely represent the extreme PDF shapes that -result from convective activity. Additionally, we only create cloud -associated with the detrainment part of the convection scheme, assuming -that cloud associated with the active updraughts in convection is small. -This assumption is not necessarily applicable. Similar arguments, -and model results, come from analysis of the \cite{t93} and -\cite{t02} scheme (Ben Johnson, personal communication). We also note -that with two cloud fraction types and two different optical depths -it is possible to have a basic degree of representation of cloud inhomogeneity. - -Hence the code still exists to enable PC2 to be -run with or without a diagnostic convective cloud fraction, although PC2:66 -does not include a diagnostic term. More details are -in section \ref{sec:convec}. - -\section{Physical basis of the PC2 prognostic cloud scheme} - -In this section we will develop the physical models that PC2 uses in order -to calculate its prognostic increment terms. We will also consider the -numerical solution of the models. The way in which these are incorporated -into the Unifed Model will be discussed in section \ref{sec:um} - -\subsection{Instantaneous condensation} - -Liquid clouds in PC2 use the concept of instantaneous condensation. Hence the -`s' distribution methods are fully applicable to the development of the -equations that govern the parametrization of liquid cloud in PC2. We will -start by looking at changes to $\overline{q_{cl}}$ and $C_l$ when a -uniform forcing -is applied to a gridbox, under the assumption of instantaneous condensation. - -\subsection{Homogeneous forcing} -\label{sec:homog} - -We define the expression -\textit{uniform forcing} (or \textit{homogeneous forcing}) to refer to -changes in local values of $T_L$ -and $q_T$ that occur at a rate independent of the part of the -gridbox in which they are located. This implies that $G(s)$ will not alter -due to such a process. -Uniform forcing simply alters $Q_c$ in (\ref{eq:int_gs_ds}) and -(\ref{eq:qclbar=int}). -In the Unified Model, this concept will be applied to several different sets of -physics increments in order to calculate the condensation and cloud fraction -changes associated with each one, where the physics routine does not allow -the explicit calculation of condensation and cloud fraction changes by -another method. -Large-scale ascent may be considered a meteorological example of such -a process. By differentiating (\ref{eq:int_gs_ds}) and (\ref{eq:qclbar=int}) -with respect to time, assuming uniform forcing -(so ${\frac{\partial G}{\partial t}}$ terms are zero), we obtain - -\begin{equation} -{{\frac{\partial C_l}{\partial t}} = G(-Q_c) {\frac{\partial Q_c} -{\partial t} }.} -\label{dcdt} -\end{equation} - -\begin{equation} -{\frac{\partial \overline{q_{cl}}}{\partial t}} = -C_l {\frac{\partial Q_c}{\partial t}} -\label{dqcldt} -\end{equation} - -The quantity $G(-Q_c)$ is the value of the PDF -of $G$ at $s=-Q_c$, which defines the boundary between -the saturated and unsaturated parts of the distribution. - -If we wish to consider a prognostic cloud scheme with equations for -the rate of change of condensate and cloud fraction based upon (\ref{dqcldt}) -and (\ref{dcdt}) then we need to close (\ref{dcdt}) by specifying the -value of $G(-Q_c)$. We will choose to develop a parametrization for this -quantity based upon the quantities $C_l$, $\overline{q_{cl}}$ and the saturation -deficit, $SD$, rather than tie $G(-Q_c)$ to a process. -The saturation deficit is \textit{defined} here in the `s' -framework to be the first moment of the PDF for `s' values less than $-Q_c$. -In this way it is analogous to the liquid water content, $\overline{q_{cl}}$. -Appendix A of \cite{wg03} writes this \textit{definition} as - -\begin{equation} -{SD = - \int_{-\infty}^{-Q_c} {( s+Q_c ) G(s) ds}} -\label{SD} -\end{equation} - -and shows this is equivalent to - -\begin{equation} -{SD = a_L ( q_{sat}({\overline{T}},{\overline{p}}) - {\overline{q}} ) .} -\label{SD2} -\end{equation} - -The basis behind the parametrization for $G(-Q_c)$ is to consider -an underlying form of the distribution $G(s)$ near the $+b_s$ and -$-b_s$ ends. We borrow the notation of \cite{smith90} and refer -to a quantity $b_s$ that is the value of $s$ when a monomodal -distribution $G(s)$ just equals zero. We suppose that the -distribution G can be described as a power law near $s=b_s$. - -\begin{equation} -G(s) ~ \propto ~ {(-s + b_s)}^n -\label{eqn19} -\end{equation} - -provided $s 1$ then the distribution is narrowed. -For the liquid cloud fraction we therefore have - -\begin{equation} -C_l^{[n+1]} = \int_{s=-Q_c}^{\infty} \xi G(\xi s) ds . -\label{eq:c_l_xi} -\end{equation} - -If we transform variables to $s' = \xi s$ we can rewrite this integral as - -\begin{equation} -C_l^{[n+1]} = \int_{s'=-Q_c \xi}^{\infty} G(s') ds' . -\label{eq:c_l_xi2} -\end{equation} - -Hence the expression for $C_l^{[n+1]}$ is equivalent to using the same -distribution function $G(s)$ as for $C_l^{[n]}$ except that the saturation -boundary has been moved from $-Q_c$ to $-Q_c \xi$. The result is the same as applying -a homogeneous forcing (\ref{eq:deltac}) with a modified forcing, - -\begin{equation} -\Delta Q_c \equiv \xi Q_c - Q_c , -\label{eq:deltac_modified} -\end{equation} - -or the continuous version - -\begin{equation} -\frac{\partial Q_c}{\partial t} \equiv -Q_c \frac{\partial}{\partial t}(\xi - 1) . -\label{eq:xi_equiv} -\end{equation} - -We can write $\xi$ in a slightly more -informative way by linking it to the relative change in width of the PDF -$\frac{1}{b_s} \frac{\partial b_s}{\partial t}$. For a PDF that changes -its width, $\xi$ is defined as - -\begin{equation} -\xi = \frac{b_s}{b_s + \delta b_s} = \frac{1}{1 + \frac{\delta b_s}{b_s}}. -\label{eq:xi_equiv1} -\end{equation} - - -For an infintessimal timestep $\delta t$ we therefore have - -\begin{equation} -\xi = \frac{1}{1 + \frac{1}{b_s} \frac{\partial b_s}{\partial t} \delta t } -\label{eq:xi} -\end{equation} - -and hence, by expanding (\ref{eq:xi}) to give $\xi = 1 - \frac{1}{b_s} -\frac{\partial b_s}{\partial t} \delta t$ and using the homogeneous -forcing expression (\ref{dcdt}) with the modified forcing (\ref{eq:xi_equiv}), -we retrieve the continuous form - -\begin{equation} -\frac{\partial C_l}{\partial t} = - G(-Q_c) Q_c \frac{1}{b_s} -\frac{\partial b_s}{\partial t} . -\label{eq:dcdt_width} -\end{equation} - -A similar analysis can be performed for $\frac{\partial \overline{q_{cl}}} -{\partial t}$ from (\ref{dqcldt}) to give - -\begin{equation} -\overline{q_{cl}}^{[n+1]} = \frac{1}{\xi} \int_{s'=-Q_c \xi}^{\infty} -(- \xi Q_c + s') G(s') ds' . -\label{eq:qcl_xi2} -\end{equation} - -Again, this is equivalent to using the homogeneous forcing with the modified -forcing (\ref{eq:xi_equiv}), but it also includes a scaling -term $\frac{1}{\xi}$. In the infinitessimal limit, this scaling gives -a second term that is proportional to the value of the integral (i.e. -$\overline{q_{cl}}$). Hence we obtain the final continuous solution - -\begin{equation} -\frac{\partial \overline{q_{cl}}}{\partial t} = -(- C_l Q_c+\overline{q_{cl}}) \frac{1}{b_s} \frac{\partial b_s}{\partial t} . -\label{eq:dqcldt_width} -\end{equation} - -To close the solution, we need to parametrize $\frac{1}{b_s} -\frac{\partial b_s}{\partial t}$ , -which could be linked to the physics of the process that is occuring. Note -we don't need to calculate $b_s$ separately, just its \textit{fractional} -rate of change. Options for the parameterisation of -$\frac{1}{b_s}\frac{\partial b_s}{\partial t}$ -due to turbulent ``erosion'' are described in section \ref{sec:turb}, -along with the numerical methods used to integrate the equations. - - -\subsection{Initiation of cloud} -\label{sec:init} -In section \ref{sec:homog} we commented that the closure (\ref{eqn22}) for $G(-Qc)$ -is only valid if $C_l$ is not identically 0 or 1. If $C_l$ -is 0 or 1 we know that $G(-Q_c)$ is equal to 0 but we have lost the -information that will tell us when $G(-Q_c + \Delta Q_c)$ starts -differing from 0. Hence the homogeneous forcing equation set -(\ref{dcdt}), (\ref{dqcldt}) and (\ref{eqn22}) is not complete if -we start from a position where $C_l$ is 0 or 1. To complete this set, -we will need to define a width, $b_s$, to the PDF and provide an initiation -increment to $C_l$ and $\overline{q_{cl}}$ when the value of $-Q_c$ crosses -the limit of the distribution. There is more discussion in -\cite{wg03}. - -To initiate new partial cloud-cover (or new partial clear-sky), we essentially -call a diagnostic cloud scheme to initialise the prognostics -$C_l$ and $\overline{q_{cl}}$. -In the UM there is currently a choice of 2 different diagnostic cloud -schemes that can be used for this; either a version of the Smith scheme -(see UMDP 029), or the bimodal scheme (see UMDP 039). -These two options are described below... - -\subsubsection{Initiation using a ``Smith-like'' method} - -This option is selected by setting the UM namelist switch -{\bf i\_pc2\_init\_method = 1} (Smith). - -We will assume the same form of the PDF at its boundaries as is -assumed in the derivation of the $G(-Q_c)$ closure. For the high `$s$' end -of the PDF distribution we integrate the power law description in -(\ref{eqn19}) to obtain the expressions - -\begin{equation} -C_l = \frac{1}{2 b_s^{n+1}} (b_s + Q_c)^{n+1} , -\label{eq:initc} -\end{equation} - -\begin{equation} -\overline{q_{cl}} = \frac{1}{2 b_s^{n+1}} \frac{(b_s + Q_c)^{n+2}}{n+2} . -\label{eq:initqcl} -\end{equation} - -We now need to parametrize the PDF width $b_s$. Unlike the \cite{smith90} -scheme, this is the only -location in the PC2 cloud scheme where the width needs to be defined -for the liquid cloud (although see section \ref{sec:mp_depsub} for -a discussion of an equivalent width in the deposition / sublimation -relationship for ice cloud). We still choose to define $b_s$ in -terms of a critical relative humidity parameter, $RH_{crit}$. Like -the \cite{smith90} scheme (see \citeumdp{029}), -we define the value of $b_s$ as - -\begin{equation} -b_s = a_L (1 - RH_{crit}) q_{sat} (\overline{T_L}) . -\label{eq:bs} -\end{equation} - -Hence, if the parameter $n$ was the same in PC2 as the equivalent -in \cite{smith90}, the -initial creation of liquid cloud would follow precisely that diagnosed -by the \cite{smith90} scheme (assuming that the numerical implementation of -the calculation is the same). -Its subsequent behaviour in PC2, though, would be different, because -the subsequent physical processes that act are parametrized in different -ways. -Note: for some reason, the implementation in the UM uses a fixed value -of $n = 0$ (corresponding to a top-hat distribution) if a constant $RH_{crit}$ -profile is used, but instead sets $n = 1$ (a triangular distribution) -in the PC2 initiation calculation if a TKE-based variable $RH_{crit}$ is used. -In the latter case, $n = 0$ is still hardwired in the PC2 homogeneous forcing -calculations, so it is not handled consistently. - -An equivalent initiation scheme is required if $C_l$ is 1 and $Q_c$ -is being reduced - at some point we need to introduce clear sky into -the solution. Because we make the choice of symmetry (which could be -relaxed if we used different $RH_{crit}$ values for $C_l$ of 1 and $C_l$ of 0), -the problem is entirely equivalent to that of initiating -from $C_l = 0$, with the exception that $\overline{q_{cl}}$ is replaced by $SD$, -$C_l$ is replaced by $(1-C_l)$, and $Q_c$ is replaced by $-Q_c$. -We hence have the solution - -\begin{equation} -1 - C_l = \frac{1}{2 b_s^{n+1}} (b_s - Q_c)^{n+1} , -\label{eq:init1mc} -\end{equation} - -\begin{equation} -SD = \frac{1}{2 b_s^{n+1}} \frac{(b_s - Q_c)^{n+2}}{n+2} . -\label{eq:initSD} -\end{equation} - -The conversion between $SD$ and $\overline{q_{cl}}$ follows (\ref{SD2}). -We will choose, -as we do throughout PC2, to define $\alpha$ (and hence $a_L$) in terms of -$\frac{\partial q_{sat}(\overline{T})}{\partial t}$, although within -this diagnostic calculation of SD it might actually be better to use -the representation (\ref{eq:alpha}) used by the diagnostic \cite{smith90} scheme. - -\subsubsection{Numerical Application of the Smith method} -\label{sec:numapp_init} - -In order to calculate and compare the state of the model to $b_s$, -we first calculate $T_L$, $q_{sat}(\overline{T_L})$ and calculate -the mean relative total humidity, $RH_T$, where - -\begin{equation} -RH_T = \frac{ \overline{q} + \overline{q_{cl}} } {q_{sat}(\overline{T_L}) } . -\label{eq:rht} -\end{equation} - -We then assess whether initiation is required. There are only two -circumstances in which we wish to proceed further: -\begin{itemize} -\item{If the current cloud fraction $C_l$ is 0 and $-Q_c < b_s$. By dividing the second condition by $a_L q_{sat} (\overline{T_L})$ we see, using the definitions (\ref{eq:qc_eq_qt-qs}) and (\ref{eq:bs}), that this second condition is equivalent to $RH_T > RH_{crit}$.} -\item{If the current cloud fraction $C_l$ is 1 and $-Q_c > -b_s$ (or, equivalently, $RH_T < 2 - RH_{crit})$.} -\end{itemize} -Note: in the UM implementation, the actual conditions for when initiation -may occur are more complicated than this, and there are several options -depending on a namelist switch. See section \ref{sec:init2} for details... - -In the second case, we then make the temporary transformation of variables -in order to use the same solution set as in the first case: $C_l'$ takes the -value $(1-C_l)$ and $RH_t'$ takes the value $(2-RH_t)$ (which is equivalent -to the replacing of $Q_c$ by $-Q_c$). In the first case, $C_l'$ and $RH_t$ take -the same values as $C_l$ and $RH_t$ respectively. - -We then solve for the initiated cloud fraction $C_l'$, using the similar -methods as described in \citeumdp{029}, except -that we allow the solution to vary with the PDF shape $n$. We first -write $Q_N$ as - -\begin{equation} -Q_N = \frac{Q_c}{b_s} = \frac{ a_L (\overline{q_T} - q_{sat}(\overline{T_L})) } -{ a_L (1 - RH_{crit}) q_{sat} (\overline{T_L})} = \frac{RH_T - 1}{1-RH_{crit}} -\label{eq:qn_def} -\end{equation} - -and then use $Q_N$ to solve the initiated cloud fraction. We assume a PDF -described by a power law as in (\ref{eqn19}) (and the equivalent for the -other end of the distribution, the two expressions switching at $Q_c=0$), -which is normalized. The solution to (\ref{eq:int_gs_ds}) is hence - -\begin{equation} -C_l^{init'} = \left\{ \begin{array}{ll} - 0, & Q_N \le -1 \\ - \frac{1}{2} {\left( 1 + Q_N \right)}^{n+1}, & -1 < Q_N \le 0 \\ - 1 - \frac{1}{2} {\left( 1 - Q_N \right)}^{n+1}, & 0 < Q_N < 1 \\ - 1, & 1 \le Q_N . - \end{array} \right. -\label{eq:c_qn} -\end{equation} - -where $C_l^{init'}$ is the initiated value of liquid cloud fraction. -If we had performed the variable transformation we then we need to -transform back, so $C_l^{init} = 1 - C_l^{init'}$, otherwise -$C_l^{init} = C_l^{init'}$. - -In practice, it is likely to be only the second of the -options in (\ref{eq:c_qn}) that the scheme uses, since we will be at -that end of the distribution function, unless previous -parts of the model timestep have resulted in large -forcings to $Q_c$. - -The solution for the initiated liquid water, $\overline{q_{cl}}^{init}$ is -more difficult, since it depends on the width of the distribution $b_s$, -hence on $a_L$ and $\alpha$, and $\alpha$ is a function of the dry-bulb -temperature $\overline{T}$, which is not known until we know the -amount of condensation. -Hence we will need to iterate to a solution. - -We first calculate $q_{sat}(\overline{T})$, $\alpha$, $a_L$ and $b_s$, using -(\ref{eq:alpha_exp}), (\ref{eq:a_L}) and (\ref{eq:bs}). We then solve for the liquid -water content: - -\begin{equation} -\frac{\overline{q_{cl}}^{init'}}{b_s} = \left\{ \begin{array}{ll} - 0, & Q_N \le -1 \\ - \frac{1}{2 (n+2)} {\left( 1 + Q_N \right)}^{n+2}, & -1 < Q_N \le 0 \\ - Q_N + \frac{1}{2 (n+2)} {\left( 1 - Q_N \right)}^{n+2}, & 0 < Q_N < 1 \\ - Q_N, & 1 \le Q_N . - \end{array} \right. -\label{eq:l_bar} -\end{equation} - -If we have been working in transformed variables we now transform -back, so the initiated saturation deficit, $SD^{init}$, takes the -value of $\overline{q_{cl}}^{init'}$. We then use (\ref{SD2}) to -estimate $\overline{q_{cl}}^{init}$ using our initial estimates of -$q_{sat}(\overline{T})$ and $a_L$. If we are not in transformed -variables, we have the first estimate -$\overline{q_{cl}}^{init}=\overline{q_{cl}}^{init'}$. - -We now use this estimate of $\overline{q_{cl}}^{init}$ to -calculate a more accurate estimate of $a_L$ etc. by iteration. In order to -achieve a faster convergence of the iteration, we do not use -(\ref{eq:l_bar}) directly in the estimation of $a_L$ etc., but -use a combination of this value and the one from the previous iteration. - -\begin{equation} -\overline{q_{cl}}^{init~[i+1]} = f \overline{q_{cl}}^{init~[i]} + -(1 - f) \overline{q_{cl}}^{init~[i-1]} -\label{eq:iter} -\end{equation} - -where the superscript $[i]$ labels each iteration. We find that 10 -iterations is effective for convergence, with the weighting -$f$ given by $a_L^{[i]}$. - -\subsubsection{Initiation using the bimodal scheme} -\label{sec:bimodal_init} - -This option is selected by setting the UM namelist switch -{\bf i\_pc2\_init\_method = 2} (Bimodal). - -First, the diagnosis of entrainment zones is performed at all grid-points, -as described in UMDP 039. The parameters of the moisture PDF are then -constructed, assuming either a sum of two Gaussian modes from the top -and bottom of an inversion layer (when within an entrainment zone), -or a single symmetric Gaussian mode (when not in an entrainment zone). -The variance of each Gaussian mode is estimated based on the TKE and other -information output by the boundary-layer scheme -(with a minimum limit applied to the PDF width, consistent with -$RH_{crit}$ = 99\%. -Crucially, each Gaussian mode is truncated to zero at plus and minus -3 standard deviations; this sets the overall width of the moisture -PDF at each point. - -The positions of the upper and lower truncated bounds of the moisture PDF -relative to the saturation threshold are expressed in terms of a normalised -$Q_N$ = $Q_c$ over PDF-width (see equation \ref{eq:qn_def}). -In entrainment zones, the sum of the two Gaussian modes can lead to -a highly skewed distribution; hence $Q_N$ can have different values for the -upper and lower bounds, each normalised by the different widths on -either side of the PDF. The upper and lower values of $Q_N$ are then -compared to -1 and 1 respectively, to determine whether the saturation -boundary lies within the PDF bounds. This is the basic condition for -initiation to occur (though there are additional conditions and various -options for these in the soure code; see section \ref{sec:init2}). - -If the initiation conditions are met, the diagnostic bimodal cloud scheme code -is then called (see UMDP 039), and the diagnosed $C_l$ and $q_{cl}$ are used to -set the prognostic $C_l$ and $q_{cl}$. - -\subsection{Injection forcing} -\label{sec:inhomog} - -Injection forcing (sometimes referred to as inhomogeneous forcing) -uses another concept of how the underlying moisture PDF may change in -order to calculate a change in cloud fraction as a result of a known -injection of condensate into a gridbox. The term was developed in -order to be coupled with a modified mass-flux convection scheme, -but is first presented here in its basic form. - -We will assume a physical model whereby saturated air, containing -condensate, randomly replaces already existing air in the gridbox. -(Such a formulation is designed to represent air detrained from convection -replacing pre-existing air when averaged over a large horizontal domain). -\cite{bwg03} discusses the situation in more detail. Briefly, -we consider two parts to the distribution function $G(s)$. One part -represents the background air. This maintains its PDF shape (in terms -of absolute $q_T$ and $T_L$ values) because we assume it is \textit{randomly} -replaced, but will reduce in amplitude as it is replaced by a -second PDF representing the injected air. - -The fractional rate at which existing air -is replaced by the injected source air we will write as -$\frac{\partial{C_S}}{\partial{t}}$. Provided that only the liquid -phase exists (see section \ref{sec:multiple} for the extention to multiple phases), -we then note that the rate -of change of liquid cloud fraction and liquid water content in -the gridbox can be written in two parts: firstly the change -due to the background, and secondly the change due to the source. - -\begin{equation} -\frac{\partial{C_l}}{\partial{t}} = -- C_l \frac{\partial{C_S}}{\partial{t}} + \frac{\partial{C_S}}{\partial{t}} -\label{eq:dcdt_inhom} -\end{equation} - -\begin{equation} -\frac{\partial{\overline{q_{cl}}}}{\partial{t}} = -- \overline{q_{cl}} \frac{\partial{C_S}}{\partial{t}} -+ q_{cl}^S \frac{\partial{C_S}}{\partial{t}} -\label{eq:dqcldt_inhom} -\end{equation} - -where $q_{cl}^S$ is the liquid water content of the injected -air. Eliminating $\frac{\partial{C_S}}{\partial{t}}$ gives the -relationship - -\begin{equation} -\frac{\partial{C_l}}{\partial{t}} = \frac{1 - C_l}{q_{cl}^S -- \overline{q_{cl}}} Q4_l, -\label{eq:dcdt_inhom2} -\end{equation} - -where $Q4_l$ is the net (\textit{including} the liquid water -in the background distribution -that was randomally replaced) injection source change of $\overline{q_{cl}}$: - -\begin{equation} -Q4_l = \frac{\partial{\overline{q_{cl}}}}{\partial{t}} |_{injection \, source}. -\label{eq:q4} -\end{equation} - -We see that we do not need to know anything about the nature of -the two PDFs involved, except the assumption that the injected -PDF contains completely cloudy air. -This equation allows one to calculate the change in $C_l$ associated -with an injection source change of $\overline{q_{cl}}$ for the example of -convection. Modifications -to the mass-flux convection scheme for PC2 (far from trivial and discussed -in depth in section \ref{sec:convec}) -allow $Q4_l$ to be calculated ($q_{cl}^S$ is already available), -and (\ref{eq:dcdt_inhom2}) can then be used -to calculate the equivalent $C_l$ change. We note at this stage -that the denominator in (\ref{eq:dcdt_inhom2}), being the difference -in two terms that may be close to each other, may cause problems -when we attempt to numerically apply this equation. - -It is reasonable to ask what happens to the air in the distribution that -was replaced. In this mathematical representation of a single gridbox we -need not know anything other than that the air is displaced into a neighbouring -gridbox. In practical use with a mass-flux convection scheme we know more that -this air is displaced downwards in the column. We could reasonably calculate -the change in cloud fraction following the same methods as used to calculate -the change in $\overline{q}$ or the change in a tracer and we discuss this -later. - -\subsubsection{Multiple phases in the injection source} -\label{sec:multiple} - -The injection source formulation can be extended to multiple -phases of condensate. In practice, this will simply be the -two phases ice and liquid, although we need to recognize that -they can overlap with each other. \cite{wilson2001} provides the -background to the derivation and it is briefly presented below. - -We firstly rewrite (\ref{eq:dqcldt_inhom}) but use the net -condensate ($\overline{q_c} = \overline{q_{cl}} + \overline{q_{cf}}$) instead of just the -liquid water expression, and the net cloud amount $C_t$, instead -of the liquid cloud amount $C_l$. The same argument as before leads -to the expressions - -\begin{equation} -\frac{\partial{C_t}}{\partial{t}} = -- C_t \frac{\partial{C_S}}{\partial{t}} + \frac{\partial{C_S}}{\partial{t}} -\label{eq:dctdt_inhom} -\end{equation} - -and - -\begin{equation} -\frac{\partial{\overline{q_{c}}}}{\partial{t}} = -- \overline{q_{c}} \frac{\partial{C_S}}{\partial{t}} -+ q_{c}^S \frac{\partial{C_S}}{\partial{t}} . -\label{eq:dqcdt_inhom} -\end{equation} - -The left hand side of (\ref{eq:dqcdt_inhom}) is written as $Q4_c$. -$q_{c}^S$ is the in-cloud -condensate content (ice plus liquid) of the source. - -Hence eliminating $\frac{\partial{C_S}}{\partial{t}}$ we obtain - -\begin{equation} -\frac{\partial{C_t}}{\partial{t}} = \frac{(1-C_t)}{q_{c}^S - -\overline{q_{c}}} Q4_c . -\label{eq:dctdt_q4} -\end{equation} - -We will assume that the proportion of the injected volume that -contains liquid cloud can be written as $g_l$, and the proportion -that contains ice cloud can be written as $g_i$. Note that it -is not necessary to have $g_l + g_i = 1$ if there is mixed -phase cloud injected. We can write the change in \textit{liquid} cloud -fraction equivalently to (\ref{eq:dcdt_inhom}) as - -\begin{equation} -\frac{\partial{C_l}}{\partial{t}} = -- C_l \frac{\partial{C_S}}{\partial{t}} + g_l \frac{\partial{C_S}}{\partial{t}} . -\label{eq:dcldt_inhom} -\end{equation} - -Combining (\ref{eq:dcldt_inhom}) and (\ref{eq:dctdt_inhom}) by eliminating -$\frac{\partial{C_S}}{\partial{t}}$ gives - -\begin{equation} -\frac{\partial{C_l}}{\partial{t}} = \frac{g_l - C_l}{1 - C_t} -\frac{\partial{C_t}}{\partial{t}} -\label{eq:dctdt_dcdt} -\end{equation} - -and hence from (\ref{eq:dctdt_q4}) we have the result - -\begin{equation} -\frac{\partial{C_l}}{\partial{t}} = -\frac{g_l - C_l}{q_c^S - \overline{q_{c}}} Q4_c . -\label{eq:dcltdt_almost_final} -\end{equation} - -An equivalent expression holds for the ice cloud. Hence the change in the -amount of cloud for each phase may be calculated assuming we know -the volume proportions of the source term that contain each of -the phases and the net increase in the amount of condensate, $Q4_c$ -(regardless of phase). This expression is coded for use in -a generically available inhomogeneous forcing module. However, -we can also write this in a slightly more -accessible form by noting the ratio of (\ref{eq:dqcdt_inhom}) and -(\ref{eq:dqcldt_inhom}) with the $Q4$ definitions following (\ref{eq:q4}). - -\begin{equation} -\frac{Q4_c}{q_c^S - \overline{q_c}} = -\frac{Q4_l}{q_{cl}^S - \overline{q_{cl}}} . -\label{eq:q4_ratios} -\end{equation} - -Using (\ref{eq:q4_ratios}) in -(\ref{eq:dcltdt_almost_final}) gives the final expression - -\begin{equation} -\frac{\partial{C_l}}{\partial{t}} = -\frac{g_l - C_l}{q_{cl}^S - \overline{q_{cl}}} Q4_l -\label{eq:dctdt_final} -\end{equation} - -and similarly for the ice. Note that this expression accounts -for the possibility that liquid cloud is displaced from the -gridbox by added ice cloud. We can further write -$q_{cl}^S$ as a fraction of $q_{c}^S$ - -\begin{equation} -q_{cl}^S = h_l q_{c}^S -\label{eq:qcls_qcs} -\end{equation} - -where $h_l$ is the factor between them (i.e. the \textit{mass} fraction -of the injected condensate that is liquid). It is not necessary -in this theory to have $h_l$ equal to $g_l$: if a mixed -phase plume exists $g_l + g_i$ need not equal 1, but since -$h_l$ and its ice equivalent, $h_i$, refer to mass, $h_l + h_i$ -must equal 1. However, if we do not allow a mixed phase injection -(which is the case in the current mass-flux convection -scheme, where only one phase can be injected), $h_l$ and $g_l$ are equal (and either zero or one in the -current mass-flux convection scheme) and we can write (\ref{eq:dctdt_final}) as - -\begin{equation} -\frac{\partial{C_l}}{\partial{t}} = -\frac{ (\delta_{xl} - C_l) }{ \delta_{xl} q_{c}^S - \overline{q_{cl}} } -Q4_l -\label{eq:dctdt_xl} -\end{equation} - -where $\delta_{xl} = h_l = g_l$. Equivalent expressions exist for the -ice cloud fraction and total cloud fraction. - -\begin{equation} -\frac{\partial{C_i}}{\partial{t}} = -\frac{ (\delta_{xi} - C_l) }{ \delta_{xi} q_{c}^S - \overline{q_{cf}} } -Q4_i -\label{eq:dctdt_xi} -\end{equation} - -\begin{equation} -\frac{\partial{C_t}}{\partial{t}} = -\frac{ (1 - C_t) }{ q_{c}^S - \overline{q_{c}} } Q4_c -\label{eq:dctdt_xc} -\end{equation} - -with $\delta_{xi} = h_i = g_i$. These are the expressions that are used -within the convection scheme. It still remains to parametrize $\delta_{xl}$, -which is given by the convection scheme itself. This is discussed in -section \ref{sec:plume_phase}. - -\subsubsection{Numerical application} -\label{sec:multi_numapp} -The numerical application using (\ref{eq:dcltdt_almost_final}) may be performed -with a basic forward timestep. Each of the three cloud fractions can -be incremented, assuming we know $\Delta{\overline{q_{cl}}}$ and -$\Delta{\overline{q_{cf}}}$, as - -\begin{equation} -\Delta{C_t} = \frac{(1 - C_t)} {q_c^S - \overline{q_{cl}} - \overline{q_{cf}}} -( \Delta{\overline{q_{cl}}} + \Delta{\overline{q_{cf}}} ), -\label{eq:cft_ts} -\end{equation} - -\begin{equation} -\Delta{C_l} = \frac{ (g_l - C_l)} -{q_c^S - \overline{q_{cl}} - \overline{q_{cf}}} -( \Delta{\overline{q_{cl}}} + \Delta{\overline{q_{cf}}} ), -\label{eq:cfl_ts} -\end{equation} - -\begin{equation} -\Delta{C_i} = \frac{ (g_i - C_i)} -{q_c^S - \overline{q_{cl}} - \overline{q_{cf}}} -( \Delta{\overline{q_{cl}}} + \Delta{\overline{q_{cf}}} ). -\label{eq:cff_ts} -\end{equation} - -The application from within the convection scheme is slightly different. -We start with (\ref{eq:dctdt_xl}), but enforce two numerical restrictions -to avoid the equation set becoming ill-conditioned. Firstly, we limit -the denominator $q_c^S - \overline{q_{cl}}$ to a minimum value if we -are considering changes of the same phase as the injected source. - -\begin{equation} -\Delta C_l = \frac {\delta_{xl} - C_l} {\delta_{xl} \text{Max}( q_{c}^S -- \overline{q_{cl}} , q_c^{S0} ) + ( 1 - \delta_{xl} ) (-\overline{q_{cl}}) } -Q4_l -\label{eq:delta_cl} -\end{equation} - -where $q_c^{S0}$ is specified as $5 \times 10^{-5} kg \, kg^{-1}$. -The denominator also has an additional check. If its absolute -value is less than a tolerance value of $1 \times 10^{-10} kg \, kg^{-1}$ -then no change in cloud fraction will be considered. A similar -equation is used for the ice cloud and the change in total cloud fraction - - -\begin{equation} -\Delta C_i = \frac {\delta_{xi} - C_i} {\delta_{xi} \text{Max}( q_{c}^S -- \overline{q_{ci}} , q_c^{S0} ) + ( 1 - \delta_{xi} ) (-\overline{q_{cf}}) } -Q4_i -\label{eq:delta_ci} -\end{equation} - -\begin{equation} -\Delta C_t = \frac {1 - C_t} -{ \text{Max}(q_c^S - \overline{q_c} , q_c^{S0} ) } Q4_c . -\label{eq:delta_ct} -\end{equation} - -We now limit the change in cloud fraction to ensure that the cloud -fraction remains within its physical bounds. - -\begin{equation} -C_l^{[n+1]} = ( 0, C_l^{[n]} + \Delta C_l, 1) -\label{eq:delta_cl_conv_final} -\end{equation} - -and similar equations are used for $C_i^{[n+1]}$ and $C_t^{[n+1]}$. - -\subsubsection{A note on the implementation of the cloud fraction change} -\label{sec:conv_imp_note} - -Equation \ref{eq:dcdt_inhom2} has been derived assuming that the -only change in the cloud properties within the gridbox comes from -the detrainment of air from the convective plume (so that the injection -source is an appropriate model). Attention should be drawn to the fact -that this is not the only source of change from the convection scheme. -Two other terms require consideration, namely advection of the environmental -air downwards by compensating subsidence and the condensation resulting -from the adiabatic warming due to this subsidence. -The former is considered correctly in the calculation of -$\frac{\partial \overline{q_{cl}}}{\partial t}$, which corresponds to $Q4$. -However, the calculation of $\frac{\partial C_l}{\partial t}$ -is then performed using (\ref{eq:dcdt_inhom2}) and \textbf{incorrectly} assuming -that all the $\overline{q_{cl}}$ change comes from the detrainment. It is -possible to calculate directly the change in $C_l$ that should occur due to -the detrainment and compensating subsidence treated together, in the same way -that $\Delta \overline{q_{cl}}$ is calculated (see section -\ref{subsect:q4calculation}), and this is the way in which the cloud fraction -change \textbf{should} be done. -It is an unfortunate historical emphasis in the early development of PC2 -on the derivation of (\ref{eq:dcdt_inhom2}) that has led to the treatment -used within the Unified Model for the change in cloud fractions due to convection. - -The change in $\overline{q_{cl}}$ and $C_l$ due to the adiabatic warming -associated with the compensating subsidence is considered explicitly -in the model implementation (see -section \ref{sec:conv_homog}) for both $\overline{q_{cl}}$ and $C_l$ -after the rest of the convective process has been calculated. It is perhaps -arguable that if (\ref{eq:dcdt_inhom2}) is going to be applied then -the value of $Q4$ used in (\ref{eq:dcdt_inhom2}) should include this term. - -Any major future developments of PC2 for a mass-flux convection scheme would be -advised to consider whether it is appropriate to use (\ref{eq:dcdt_inhom2}) at -all. - -\subsection{Ice cloud and mixed phase regions} -\label{sec:ct} - -The homogeneous forcing, initiation and PC2 erosion sections described -above have only considered the generation and dissipation of liquid -clouds. Although the forcing methods will not influence the generation -and dissipation of ice cloud (which is primarily performed in the -large-scale precipitation scheme, section \ref{sec:precip}) we -are still left with the issue of how created or dissipated liquid -cloud overlaps with existing ice cloud in the gridbox. The -opposite situation, where changes in ice cloud are specified -and changes in the overlap with liquid cloud need to be calculated, -is also possible in PC2 (e.g. in the boundary layer, see -section \ref{sec:bl}). - -Here we -need a simple assumption to close the problem. The assumption -that we now choose is that liquid cloud fraction \textit{changes} are -\textit{minimally} overlapped with ice cloud fraction changes. -This choice is based upon observational evidence that mixed -phase cloud is relatively rare, and also on results from earlier PC2 -development that indicated less supercooled liquid water cloud than -is observed from ground-based lidar. - -With this assumption, the equation set becomes straightforward to -write down. We firstly consider that a change in liquid cloud fraction -$\Delta C_l$ is known and we wish to estimate the resulting change in -the total cloud fraction. There is, of course, no change in the ice -cloud fraction $C_i$, since, from our \textit{definitions} in -(\ref{eq:dqcldt_and_dcdt}) and (\ref{eq:mp}), this includes the mixed phase -contribution. Hence we write - -\begin{equation} -\Delta C_i = 0 . -\label{eq:deltaci_eq_0} -\end{equation} - -The change in the total cloud fraction, $C_t$ will depend upon -the sign of the change of the liquid cloud fraction. If -$\Delta C_l > 0$, then $\Delta C_t$ is going to be the same as $\Delta C_l$ -($C_l$ is being added with minimum overlap to $C_i$), unless the -gridbox becomes completely covered in cloud, when there is no -choice but to generate mixed phase cloud. Hence we have - -\begin{equation} -\Delta C_t = \text{Min} ( \Delta C_l , 1 - C_t ). -\label{eq:deltact_min} -\end{equation} - -If $\Delta C_l < 0$, then we still consider minimum overlap -of the \textit{changes} (this is so that the solution is reversible as -much as possible). Hence $\Delta C_t$ is going to be the same as -$\Delta C_l$ unless $C_l$ is reduced below the existing $C_i$, in -which case no more change to $C_t$ is possible. - -\begin{equation} -\Delta C_t = \text{Max} ( \Delta C_l , C_i - C_t ) , -\label{eq:deltact_min2} -\end{equation} - -remembering that both quantities in the maximum expression in -(\ref{eq:deltact_min2}) have negative values. - -We can write similar expressions if a known amount of ice cloud -is added or removed, and we need to calculate the effect on $C_t$. -Similar to the results above we have: - -\begin{equation} -\Delta C_l = 0 . -\label{eq:deltacl_eq_0} -\end{equation} - -and - -\begin{equation} -\Delta C_t = \left\{ \begin{array}{ll} - \text{Max} ( \Delta C_i , C_l - C_t ), & \Delta C_i < 0 \\ - \text{Min} ( \Delta C_i , 1 - C_t ), & \Delta C_i > 0 . - \end{array} \right. -\label{eq:deltact_min_array} -\end{equation} - -For completeness, we also present here the equation set for -random overlap of changes in liquid cloud with existing ice cloud. -We have, as before, - -\begin{equation} -\Delta C_i = 0 . -\label{eq:deltaci_eq_0_2} -\end{equation} - -For $\Delta C_l > 0$ additional liquid cloud is added -randomly to any location outside that of the current liquid -cloud. A proportion $\frac{1-C_t}{1-C_l}$ of this will be additionally -outside that of existing ice cloud. Hence the net change in -total cloud fraction can be written as - -\begin{equation} -\Delta C_t = \Delta C_l \frac{1 - C_t}{1 - C_l} . -\label{eq:deltact_ran1} -\end{equation} - -Similarly, if $\Delta C_l < 0$, the liquid cloud is removed -randomly from the existing liquid cloud. A proportion -$\frac{C_t - C_i}{C_l}$ of this is from liquid cloud that does not -overlap with existing ice cloud. Hence, - -\begin{equation} -\Delta C_t = \Delta C_l \frac{C_t - C_i}{C_l} . -\label{eq:deltact_ran2} -\end{equation} - -Equivalent equations to (\ref{eq:deltact_ran1}) and -(\ref{eq:deltact_ran2}) but with $C_l$ and $C_i$ swapped apply when -we need to estimate changes in $C_t$ from a known $\Delta C_i$, when -assuming random overlap. - -\subsubsection{Numerical Implementation} - -In general, although the situation does not occur within the current -implementation of PC2 , we might have increments to both $C_l$ and -$C_i$ simultaneously. Hence the implementation is to calculate -$\Delta C_t$ from the sum of that predicted by (\ref{eq:deltact_min}) -or (\ref{eq:deltact_min2}), and (\ref{eq:deltact_min_array}). -For the random overlap situation we also need to apply a check on -the denominator in (\ref{eq:deltact_ran1}) and (\ref{eq:deltact_ran2}) before -calculation, with the result set to the limit $\Delta C_t = 0$ -if the denominator is 0. For the minimum overlap situation a final check -is made that $C_t$ lies between 0 and 1, with the value being reset to -0 or 1 if not. - -\subsection{Forced convective cloud} - -Forced convective clouds are clouds that form at the top of a convective boundary layer -but are too shallow to reach their level of free convection (and become fully fledged -cumulus clouds). These clouds currently require special treatment because initiation -in PC2 uses the Smith scheme with a specified value of $RH_{crit}$ while the large $RH$ -variability associated with these clouds implies much lower values than are typically used. - -A profile of ``forced cloud fraction'', $C_{forced}$, is parametrized as -linearly varying with height between a cloud-base value, at the lifting -condensation level (LCL) from the convection diagnosis parcel ascent, and a cloud-top -value of 0.1 at the top of the capping inversion, $z_i^{top}$. The cloud-base -value of $C_{forced}$ varies linearly between 0.1 and 0.3 for cloud depths -between 100 m and 300 m based loosely on SGP ARM site observations \cite{zk13}. The -inversion top is taken to be the boundary layer depth, $z_h$ plus the inversion -thickness, $\Delta z_i$ parametrized following \cite{rb08} as: -\begin{equation} -\Delta z_i = 6.3 \, w_m^2 / \int_{z_h}^{z_h+\Delta z_i} b \, dz -\label{dz_param} -\end{equation} -where $w_m$ is the boundary layer velocity scale ($w_m^3 = u_*^3 + 0.25 w_*^3$) and $b$ is -the parcel buoyancy that is integrated over the depth of the inversion assuming a -piece-wise linear variation between grid-levels. Note that the constant in (\ref{dz_param}) -is the same as in \cite{rb08} because $6.3 = 2.5 * 4^{2/3}$ and $w_m^3$ differs by a factor of 4. - -The in-cloud water content at the top of the inversion is estimated using the water content -from the diagnostic parcel ascent (used to diagnose boundary layer type and trigger convection), -with linear interpolation used between the lifting condensation level and inversion -top. To allow for sub-adiabatic water content (due to lateral mixing or microphysical -processes) the in-cloud water content can be reduced by a factor, forced\_cu\_fac, that has been -set to 0.5 in GA7. - -These cloud fraction and water content profiles are then used as minimum values and -increments to $C$ and $\overline{q_{cl}}$ calculated if necessary. -This methodology can also optionally be applied to cloud layers diagnosed as cumulus, if the -boundary layer option to mix across the lifting condensation level is selected that generates -a cloud base transition zone thickness which is then treated analgously to the inversion -thickness above. - -Also, there is an option to treat the calculated forced cumulus cloud -fraction and water content as diagnostic quantities passed directly to -the radiation scheme as part of the ``convective'' cloud, instead of -using them to modify the prognostic ``large-scale'' cloud variables -$C$ and $\overline{q_{cl}}$. If this option is used, the convective -cloud fraction $CCA$ and water content $CCW$ output by the convection -scheme are updated, by taking the forced cumulus profiles as their -minimum allowed values. Note that only the convective cloud fields -passed to radiation are updated (i.e. the versions of $CCA$ and $CCW$ -that are stored in the model dump / D1 array). The UM code contains -other copies of the convective cloud fields that are only used for -diagnostics; these are {\em not} updated. - -The different options for how to treat forced cumulus cloud are -controlled by the cloud namelist input $forced\_cu$, and are -summarised below: - -\begin{itemize} -\item $forced\_cu = 0$: No treatment of forced cumulus clouds. -\item $forced\_cu = 1$: Forced cumulus cloud applied to -$C$ and $\overline{q_{cl}}$ only in dry-convective boundary-layers. -\item $forced\_cu = 2$: Forced cumulus cloud applied to -$C$ and $\overline{q_{cl}}$ in both dry-convective and -cumulus-capped boundary-layers. -\item $forced\_cu = 3$: Forced cumulus cloud applied to -$CCA$ and $CCW$ in both dry-convective and -cumulus-capped boundary-layers. -\end{itemize} - - -\subsection{Turbulence-driven production of subgrid scale liquid cloud} -\label{sec:turb_qcl_scheme} - -\subsubsection{Introduction}\label{sec:sgt_intro} - -\cite{fhfk14} developed a model for -subgrid liquid water production by turbulent motions. -Their method uses an exactly soluble -stochastic process to describe -subgrid relative humidity (RH) fluctuations. -The probability density function (PDF) of the fluctuations -can be diagnosed in terms of the local turbulent local state -and any pre-existing ice cloud. The -liquid cloud properties (cloud fraction and liquid water content) -can be then be calculated as truncated moments of the PDF. - -\cite{fhfk14} initially used their model to -understand and parametrize the results of Large Eddy Simulations (LES) -of shear-induced, Altostratus clouds. They obtained excellent -agreement between their theoretically predicted predicted mean -cloud properties and the bulk properties of the LES clouds. -Subsequently, their model has been used as the basis of -subgrid cloud initiation method for use in the Unified Model -in conjunction with the PC2 prognostic cloud scheme. In Section \ref{sec:sgt_model_describe} -we outline the model of \cite{fhfk14}. In Section \ref{sec:sgt_model_implement} -we described its implementation in the GCM. - - -\subsubsection{Model description} -\label{sec:sgt_model_describe} - -\cite{fhfk14} started from the equation for the dynamics of -ice supersaturation $S_i=e_v/e_{sat\;ice}-1$: -\begin{equation}\label{eqn:squires_eqn} - \frac{D S_i}{D t} = -b_i B_0 {\cal M}_1 S_i - -\left(\frac{\varepsilon}{L^2}\right)^{1/3}(S_i-S_E) + a_i w, -\end{equation} -where ${\cal M}_1$ is the first moment of ice particle size distribution (PSD), -$\varepsilon$ is the turbulent dissipation rate, $L$ is a prescribed mixing length -for the turbulence, $S_{\rm E}$ is the ice supersaturation of the -environment surrounding the cloud and $b_i,B_0$ and $a_i$ are function of $p$ and $T$ given by -\begin{eqnarray} - b_i &=& \frac{1}{q} + \frac{\epsilon L_s^2}{c_p R T^2}, \\ - B_0 &=& 4\pi C \left( \frac{\epsilon L_s^2}{K_a R T^2} + \frac{R T}{\epsilon e_{si} \psi} \right)^{-1}, \\ - a_i &=& \frac{g}{R T}\left( \frac{\epsilon L_s}{c_p T} - 1 \right), \\ -\end{eqnarray} -The first term on the right hand side of Eq.~\ref{eqn:squires_eqn} is -the sink of vapor due to depositional growth of ice crystals, the second -term models entrainment (mixing) of environmental air into the cloudy -volume and the third term is a source term due to vertical air motions. - -\cite{fhfk14} modeled vertical velocity as a white-noise process -with autocorrelation function: -\begin{equation} - \overline{w(t)w(s)} = \sigma_w^2 \tau_{\rm d} \delta(t-s), -\end{equation} -where $\delta$ is the Dirac distribution and the intensity of the -noise, $\sigma_w^2$, will be called the -standard derivation of the vertical velocity fluctuations (due to the white nature of -noise, a true expectation value $\overline{w^2}$ is not defined) and $\tau_{\rm d}$ -a Lagrangian decorrelation time define here by the relation used by \cite{rodean1997}: -\begin{equation} - \tau_{\rm d} = \frac{2\sigma_w^2}{\varepsilon C_0}, -\label{eqn:taud} -\end{equation} -where $C_0$ is a known constant. - -Because it is linear in $S_i$, Equation \ref{eqn:squires_eqn} can be solved exactly, -for any given realisation of the noise term. By averaging the solutions over the the noise -and taking a steady-state limit (see \cite{fhfk14} for details) it can be shown -that the solution PDF is Gaussian with mean and variance given by: -\begin{eqnarray} - \overline{S_i} &=& - S_{\rm E}\frac{ \left(\varepsilon/L^2\right)^{1/3} }{ b_i B_0 {\cal M}_1 + \left(\varepsilon/L^2\right)^{1/3} }. - \label{eqn:si_avg} \\ - \overline{S_i^2} &=& - \frac{a^2_{\rm i} \sigma^2_w \tau_{\rm d}}{ 2\left(b_i B_0 {\cal M}_1 + \left(\varepsilon/L^2\right)^{1/3}\right)}, - \label{eqn:si_var} -\end{eqnarray} - -Equation \ref{eqn:si_avg} and \ref{eqn:si_var} completely specify the PDF, $F(S_i)$, of -steady-state humidity variations for the subgrid model. The liquid cloud fraction and -liquid water mass mixing ratio are given by -\begin{eqnarray} - C_l^{sgt} &=& \int_{S_{i,wat}}^\infty d S_i F(S_i), \label{eqn:cloud_fraction} \\ - q_{cl}^{sgt} &=& q_{sat\;ice}\int_{S_{i,wat}}^\infty d S_i (S_i -S_{i,wat}) F(S_i) \label{eqn:cloud_liquid}, -\end{eqnarray} -where $S_{i,wat} = e_{sat\;wat}/e_{sat\;ice}-1$ is the value of ice -supersaturation at water saturation. -We use the superscription `$sgt$'(=`{\it s}ub{\it g}rid {\it t}urbulence') to indicate -that $C_l^{sgt}$ and $q_{cl}^{sgt}$ are values of cloud fraction and water content -diagnosed from a parametrization of small-scale turbulent processes. - - -\subsubsection{Model implementation and closure relations} -\label{sec:sgt_model_implement} - -To implement the model of Section \ref{sec:sgt_model_describe} in the -Unified Model, closure relations are needed for the quantities $\sigma_w^2$, -$\varepsilon$, $L$, $\tau_{\rm d}$ and $S_E$, subject to the constraining relationship -given by Eq. \ref{eqn:taud}. -In each model grid box, these parameters specify the subgrid PDF, $F(S_i)$, and -from this the liquid cloud fraction and water content produced by turbulence -can be found using Eqs \ref{eqn:cloud_fraction} and \ref{eqn:cloud_liquid}. - -In addition we need to make some assumptions about how the diagnosed values -$C_l^{sgt}$ and $q_{cl}^{sgt}$ relate to the model prognostic fields, $C_l$ and $q_{cl}$. -Two methods are available for doing this. In the simplest case, the diagnosed -values $C_l^{sgt}$ and $q_{cl}^{sgt}$ are just treated as increments to model prognostics -(option one, in Sec. \ref{sec:sgt_increments} below). -A more complex option (see option two, below) is to increment the -model fields via the PC2 Erosion functionality. - -\subsubsection{Closure relations}\label{sec:sgt_closures} - -The vertical velocity variance, $\sigma_w^2$, is available as a diagnostic from the -Boundary Layer scheme. Because the Boundary Layer scheme is called after the -Microphysics on each model timestep, the diagnostic value is stored in a (non-advected) -model prognostic field. The scheme will operate only where there is diagnosed turbulence, -i.e., non-zero $\sigma_w^2$. - -We take the mixing length scale, $L$, to be proportional to the vertical -grid spacing in each grid box: $L=\beta_{mix} \Delta z$, where $\Delta z$ is -calculated as the height different between the $\rho$-levels adjacent -to the given $\theta$-point. The parameter, $\beta_{mix}$, -is an adjustable constant that the user can define (see Section \ref{sec:sgt_options} below), -however it should be of order one. - -To obtain $\tau_{\rm d}$ we impose an eddy size constraint: -\begin{equation} - \tau_{\rm d} = \frac{L}{\sigma_w} = \beta_{mix} \frac{\Delta z}{\sigma_w} -\label{eqn:eddy_size} -\end{equation} -Eq.~\ref{eqn:taud} then determines the dissipation rate, $\varepsilon$, that is consistent -with the other parameters. The constant $C_0=10$ by default, but can be adjusted by the user. - -The scheme is limited to act only in grid boxes where $\tau_{\rm d}$ is less than a -prescribed value, $\tau_{d}^{max}$. The default is $\tau_d^{max}=1200\;{\rm sec}$, which typically -coincides with a couple of model timesteps. The motivation for this is that a -motion that takes longer than a few timestep to decorrelate will be partially resolved by -the dynamics and therefore cannot be considered as `subgrid' turbulence. - -Finally, where $T$, $p$ and $q$ appear in the expressions for $C_l^{sgt}$ and $q_{cl}^{sgt}$, -these are taken to be the grid box mean values. The first moment of the ice PSD, ${\cal M}_1$, -is found from the parametrization, due to \cite{fhbicc05}, described -in Section 4.1 of UMDP26. - - -\subsubsection{Options for incrementing model prognostics}\label{sec:sgt_increments} - -Using the information in Section \ref{sec:sgt_closures} to obtain closed expressions -for the subgrid PDF of $S_i$-fluctuations allows $C_l^{sgt}$ and $q_{cl}^{sgt}$ to be -calculated. These will be non-zero only where there is turbulence as diagnosed by the -Boundary Layer scheme (and hence non-zero $\sigma_w^2$). To calculate $C_l^{sgt}$ and $q_{cl}^{sgt}$ -the integrals in Eqs \ref{eqn:cloud_fraction} and \ref{eqn:cloud_liquid} are evaluated -numerically using discretisation based on user-specified number of bins. - -Given $C_l^{sgt}$ and $q_{cl}^{sgt}$, two options are available for relating these -to changes in the model prognostics: - -\paragraph{Option one: direct increments} - -The values of $C_l^{sgt}$ and $q_{cl}^{sgt}$ can be added as increments to the -model prognostic fields, $C_l$ and $q_{cl}$. In this case -\begin{eqnarray} - \left( \Delta C_l \right)_{sgt} &=& C_l^{sgt} \\ - \left( \Delta q_{cl} \right)_{sgt} &=& q_{cl}^{sgt}, \\ - \left( \Delta q \right)_{sgt} &=& -\left( \Delta q_{cl} \right)_{sgt}, \\ - \left( \Delta T \right)_{sgt} &=& \frac{L_v}{c_p} \left( \Delta q_{cl} \right)_{sgt}, \\ - \left( \Delta C \right)_{sgt} &=& C_l^{sgt} \\ -\end{eqnarray} -where the left hand sides denote the increments to $C_l$, $q_{cl}$, $T$ and the -total cloud fraction, $C$, due to -the subgrid scheme. Some bounds-checking is then applied to ensure that: -(a) the resultant cloud fractions to not exceed one; (b) the scheme does not -condense out more liquid than there is available moisture. - -\paragraph{Option two: PC2 Erosion method} - -Option one gives a simple method for incrementing the model prognostics, but -it gives rise to a potential inconsistency with the PC2 cloud scheme. This arises because -the subgrid production scheme can elevate cloud fraction to unity in grid boxes -that are subsequently diagnosed by PC2 Initiation to meet the criteria for clear-sky initiation. -PC2 then counteracts the scheme by removing some of the liquid cloud. To try to mitigate -against this issue, cloud fraction increments can be applied using PC2 Erosion. In this case: -\begin{eqnarray} - \left( \Delta q_{cl} \right)_{sgt} &=& q_{cl}^{sgt} - q_{cl}, \\ - \left( \Delta q \right)_{sgt} &=& -\left( \Delta q_{cl} \right)_{sgt}, \\ - \left( \Delta T \right)_{sgt} &=& \frac{L_v}{c_p} \left( \Delta q_{cl} \right)_{sgt}, \\ -\end{eqnarray} -where $q_{cl}$ is the liquid cloud amount prior to calling to the -turbulent production scheme. The cloud fraction increments are calculated -by calling PC2 Erosion with $\left( \Delta q_{cl} \right)_{sgt}$ as input. -See Section \ref{sec:turb} for details on how the PC2 Erosion process works. -This method gives cloud fraction increments that are consistent with -PC2 cloud scheme. - - -\subsubsection{Other user options}\label{sec:sgt_options} - -The following variables and logical switches are optional inputs: -\begin{enumerate} - \item The logical \verb!l_dcfl_by_erosion! provides a switch to - apply cloud fraction increments using PC2 Erosion. Defaults to {\it FALSE}. - \item Setting the logical \verb!l_mixed_phase_t_limit! to {\it TRUE} - allows the user to use the variable \verb!mp_t_limit! to define a temperature limit, $T_{max}$, - above which the scheme is not applied. The default is $T_{max}=0^\circ\;{\rm C}$, so the - scheme is only applied to cold clouds. - \item The input variable \verb!mp_tau_d_lim! defines the - upper limit, $\tau_d^{max}$, on the value of $\tau_d$ above which the scheme is not applied. - The default value is $\tau_d^{max}=1200.0$, so the scheme is not applied in grid boxes - where the decorrelation time scale exceeds $1200$ seconds. - \item \verb!nbins_mp! is the number of bins used in the discretisation of the integrals in - Eqs \ref{eqn:cloud_fraction} and \ref{eqn:cloud_liquid} for $C_l^{sgt}$ and $q_{cl}^{sgt}$. - The default value is $100$ bins. - \item \verb!mp_dz_scal! is the scale parameter, $\beta_{mix}$, in the definition of the mixing length, $L=\beta_{mix}\Delta z$. - \item \verb!mp_czero! defines the constant parameter $C_0$ (defaults to $C_0=10$). -\end{enumerate} - - -\section{Application to the Unified Model} -\label{sec:app_um} - -This section describes the way in which the physical concepts described in the above -section are applied to the sections of the Unified Model, in order to build -up the complete prognostic scheme. Description of the actual subroutines -themselves follow in section \ref{sec:code}. -Note that the large-scale precipitation -and convection schemes have considerable documentation below, since these -schemes have been heavily modified for PC2. The other schemes use generic -forcing scenarios, hence their desciption here is much shorter. Remember, -whenever a signficiant $\overline{T}$ or $\overline{q}$ change occurs, -PC2 must be able to -represent the corresponding condensation and changes in cloud fractions. - -\subsection{Radiation} -\label{sec:rad} - -The shortwave and longwave radiation schemes both alter the temperature -of the atmosphere, hence we need to calculate the corresponding condensation -and cloud fraction changes. For both shortwave and longwave, we use -the homogeneous forcing routines (section \ref{sec:homog}) -for $\overline{q_{cl}}$ and $C_l$, (using eqn. -\ref{eq:deltaqc_exp2} to calculate the $Q_c$ forcing) and then the method in -section \ref{sec:ct} to calculate $C_t$ changes. There is no -$\overline{q_{cf}}$ change associated with this process since the -deposition / sublimation process is performed within the large-scale -precipitation scheme (as it also is in the absence of PC2). - -It is reasonable to question whether homogeneous forcing is a reasonable -model to use when we know that a large proportion of the heating -associated with radiative transfer in the atmosphere comes from the -cloudy air and is not evenly spread across the gridbox. Possible developments -are discussed in section \ref{sec:homog_improve}. - -\subsection{Large-scale precipitation} -\label{sec:precip} - -Precipitation processes have a large effect on cloud fractions. Here we -present the simple physical models that are applied to the transfer -terms included in the large-scale precipitation scheme. They are also -presented within the large-scale precipitation documentation (\citeumdp{026}). - -The basis of the physical model is that microphysical transfer processes can -be calculated separately in different partitions of the model cloud -(i.e. mixed phase cloud, liquid phase cloud, ice phase cloud or clear sky). -However, processes may change the size of these partitions. We consider -here separately each process that is modelled in the large-scale -precipitation scheme. The changes in $\overline{q_{cl}}$, $\overline{q_{cf}}$ -and $\overline{q}$ remain mathematically the same as in the non-PC2 version -of the code (\citeumdp{026}), we only need -to introduce calculations for the changes in cloud fractions. We will see -that many of these -changes can be well modelled by assuming no change to the cloud fractions, -and the others by using simple assumptions. - -Although the model may use two ice prognostic ice categories, only -a single ice cloud fraction is stored, the assumption being that the -two ice categories are completely overlapped with each other. Graupel -is not considered to contribute to the ice cloud fraction. - -\subsubsection{Fall of ice} -\label{sec:lsp_fall} - -The fall of ice is the process that contributes most to the growth of -ice cloud fraction in the model. The model results are therefore sensitive -to the formulation of this process. We will make the basic assumption -that a trail of falling ice does not reduce the horizontal spread of -ice cloud fraction at a particular level (hence $\overline{q_{cf}}$ -that leaves a gridbox does not reduce $C_f$ in that gridbox). The in-cloud -ice content simply reduces due to the fall out of ice - it is the -sublimation term (section \ref{sec:mp_depsub}) that erodes the fall streaks. -However, ice that falls into a clear layer from above may increase -the ice cloud fraction. We parametrize this by considering the -horizontal overlap of ice clouds between two model layers, and the -fall speed of ice between them. We will assume an overlap that -is nearly, but not quite, maximum, the difference being dependent -upon the windshear and the time taken for ice to fall between the -levels. - -\begin{equation} -O^{[k,k+1]} = \text{Max}( C_{i}^{[k+1]} - C_i^{[k]} , 0) -+ w \frac{\Delta z^{[k]}}{v_i^{[k]}} -\label{eq:overhang} -\end{equation} - -where $O^{[k,k+1]}$ is the amount of ice cloud `overhanging' the current -(i.e. $k$'th) layer -from the layer above, $w$ is a parameter that is closely related to the -windshear, $\Delta z^{[k]}$ is the model layer thickness and $v_i^{[k]}$ is -the fallspeed of ice in the layer. $v_i^{[k]}$ is calculated in the microphysics -scheme and, if two ice prognostics are used, is the mass-weighted average fall -speed of the two categories. -The factor $\frac{\Delta z}{v_i}$ is simply the time -taken for the ice to fall through one model layer. Multiplying this -by the windshear would give an estimate to the amount -of overlap between a cloud source and its fall streak in the layer -below (it is an \textit{estimate} since we assume that the cloud -source is continuous and unbroken). Although it is quite possible within -PC2 to do this, to date we have not programmed this link, and we -use a constant, but tunable, value of $1.5 \times 10^{-4} s^{-1}$ for $w$. - -The change in $C_i$ over the timestep is then given by the overlap proportion -multiplied by the how much (in the vertical dimension) of the layer below -can be filled by ice in the timestep: - -\begin{equation} -\Delta C_i = \text{Max}(O^{[k,k+1]} , 1) \text{Min} (v_i \frac{\Delta t}{\Delta z^{[k]}} , 1) -\label{eq:lsp_fall} -\end{equation} - -where $\Delta t$ is the timestep. We now choose to assume a minimum overlap -between the liquid and the ice phases (as in section \ref{sec:ct}). - -\begin{equation} -\Delta C_t = \text{Min} ( \Delta C_i , A_{clear} ) -\label{eq:lsp_fall_ct} -\end{equation} - -where $A_{clear}$ is the proportion of the gridbox that has neither -ice nor liquid cloud present. - -\textbf{An inconsistency has been found in the way that the fall-of-ice term is linked to the globally constant ``wind-shear value'' -when calculting the ice cloud fraction overhang. Consequently, -the option not to use the ``wind shear value'' when calculating the overhang is available in -the UMUI (from version 7.6 onwards).} - -\subsubsection{Homogeneous nucleation} -\label{sec:lsp_homo} -This will freeze all supercooled liquid water when a temperature threshold -is exceeded. Hence we turn all existing liquid and mixed phase cloud to -ice cloud. The cloud fraction changes are: - -\begin{eqnarray} -C_l \leftarrow 0 \nonumber \\ -C_i \leftarrow C_t \nonumber \\ -\Delta C_t = 0. -\label{eq:lsp_homo} -\end{eqnarray} - -\subsubsection{Heterogeneous nucleation} -This process will freeze a small amount of supercooled liquid water, -regardless of the previous presence of ice cloud. This will mean that -previously existing `liquid-only' cloud is converted to mixed phase -cloud. These give the following changes: - -\begin{eqnarray} -\Delta C_l = 0 \nonumber \\ -C_i \leftarrow C_t \nonumber \\ -\Delta C_t = 0. -\label{eq:lsp_het} -\end{eqnarray} - -\subsubsection{Deposition and sublimation} -\label{sec:mp_depsub} -This term exerts one of the most important influences on the ice cloud in -the whole model (this applies to the control as well as for PC2). Contained -in the formulation is a subgrid-scale assumption that causes equivalent -effects to that for a moisture PDF under the `$s$' framework -(section \ref{sec:s_dist}). However, since ${q_{cf}}$ changes -slowly in response to local changes in $q$ and $T$, we cannot base the -$q_{cf}$ response on the same instantaneous condensation framework. It would -be useful to investigate in the future whether the two descriptions of the -moisture variability could be brought together. -Because of its importance, -we describe the method below, although we note it is also described in -\citeumdp{026}. - -We can calculate the local rate of change of $q_{cf}$, given local $T$ and $q$ -etc. using the standard microphysical growth equations (see \citeumdp{026}). -However, it is critical to know the way in which -the moisture is correlated with the ice in the gridbox. We will assume -there exists a distribution of vapour in the gridbox. We know that -the regions where liquid cloud exists must be saturated with respect -to liquid water, hence we need only consider the part of the gridbox -that does not have liquid water present. The average value, $q_a$, -of $q$ within the liquid-free part of the gridbox is thus - -\begin{equation} -q_a = \frac{ \overline{q} - C_l q_{sat \, liq}(\overline{T}) } {1 - C_l} -\label{eq:qa} -\end{equation} - -where we have assumed that the fluctuation of $q_{sat~liq}$ across -the gridbox due to temperature fluctuations is not significant compared -to the fluctuation of $q$ described below. We then parametrize a width, $b_i$, -to the $q$ (not $s$) fluctuations \textit{across the non-liquid cloud part -of the gridbox}, based upon $RH_{crit}$. This is like that for the `$s$' -distribution width, $b_s$ but modified: - -\begin{equation} -b_i = (1 - RH_{crit} ) q_{sat \, liq} ( 1 - \frac{1}{2} -~ \frac{\overline{q_{cf}}} {i q_{sat \, liq}(\overline{T})} ) . -\label{eq:b_i} -\end{equation} - -where the factor $( 1 - \frac{1}{2} -\frac{\overline{q_{cf}}} {i ~ q_{sat \, liq}(\overline{T})})$ should be limited -to a minimum value of zero, but, for numerical reasons, is limited to -a minimum value of 0.001. We note that $b_i$ has a similar form to $b_s$, -except the multiplier $a_L$ and the factor in brackets. If we remember -from (\ref{eq:s}) that the definition of `$s$' includes a factor $a_L$ -we see that the absence of the $a_L$ factor in (\ref{eq:b_i}) is -consistent. The factor in brackets is a \textit{parametrization} of the -effect that, when ice -is present, deposition in the moistier parts and sublimation in the -drier parts of the gridbox must reduce the width of the distribution -of $q$ across the gridbox. It is a simple linear function of -$\frac{\overline{q_{cf}}}{q_{sat~liq}(\overline{T})}$, and is tunable -using the factor $i$, which takes the value of 0.04. - -We note that this formulation isn't totally consistent with the liquid -cloud formulation, which considers an underlying PDF across the whole -gridbox and does not have, in general, its width prescribed. -Remember that we do not calculate on-line the whole of the liquid -- vapour PDF, we only parametrize the single point $G(-Qc)$, -using equation \ref{eqn22}). - -The width is then limited further to be no greater than $\overline{q}$, -to make sure that there are no negative values of $q$ predicted within the -gridbox (possible at low -temperatures where $q_{sat~liq}(\overline{T})$ diverges from -$q_{sat~ice}(\overline{T})$). - -We then calculate the average value of $q$ in the ice-only and clear-sky -partitions of the gridbox. To do this, we make the further assumption -that the ice is correlated with the moistest part of the distribution -(an instantaneous condensation formulation would make the -same assumption). Some algebra retrieves the expressions: - -\begin{eqnarray} -q_{clear} = q_a - b_i A_{ice} ; \\ -q_{ice} = \frac {\overline{q} - C_l q_{sat~liq} - A_{clear} q_{clear} } -{A_{ice}}, -\label{eq:q_clear_and_q_ice} -\end{eqnarray} - -where $A_{ice}$ is the proportion of the gridbox with ice cloud but not -liquid cloud and $A_{clear}$ is the proportion of the gridbox without cloud. -The numerical application will set $q_{clear}$ to $q_a$ if $A_{ice}$ -is zero. We now have a representation of the $q$ values in each of the -gridbox cloud partitions, and can solve the microphysical transfer equation -in each partition. - -The cloud fraction changes now need to be parametrized. We use the -model that deposition will \textit{not} adjust the ice cloud -\textit{fraction} (increases will be done within the fall-of-ice microphysics -section). However, deposition can -decrease the liquid cloud fraction (locally, $q$ can be reduced by -deposition to below $q_{sat~liq}$, hence this is not inconsistent with -the assumptions for the riming term below. This is the principal sink -of supercooled liquid cloud fraction in the model. Sublimation will -be allowed to decrease the ice cloud fraction (since sublimation cannot -act in liquid cloud, there is no impact on the liquid cloud). To solve -for these models, we will need to further split the ice-only partition -to give the proportion of that partition that is above and below ice -saturation. This gives, in general, an area of the gridbox $A_{ice1}$ -that contains ice and is above saturation where - -\begin{equation} - A_{ice1} = \frac{1}{2} A_{ice} + \frac{1}{2} - \frac{ (q_{ice}-q_{sat~ice}(\overline{T})) } {b_i}, -\label{eq:q_ice_above_sat} -\end{equation} - -having assumed that $A_{ice1}$ is between 0 and $A_{ice}$. -If not, it is trivial to partition the gridbox, since the moisture in the ice-only -partition is either completely above or completely below $q_{sat~ice}(\overline{T})$. -The corresponding -area that contains ice and is below saturation is given by -$A_{ice2} = A_{ice} - A_{ice1}$. -We can now parametrize the change in cloud fractions. For deposition, -we shall assume a uniform distribution of local values of $q_{cl}$ about -the local mean. If we assume a uniform removal of local $q_{cl}$ then, with -a little algebra, we can obtain an expression for the change in $C_l$: - -\begin{equation} -\Delta C_l = C_l ( 1 - \frac {\Delta \overline{q_{cl}}} {\overline{q_{cl}}} ) -^{\frac{1}{2}} - C_l -\label{eq:deltacfl_dep} -\end{equation}. - -Since this occurs only in the mixed phase part of the gridbox, we can say -that $\Delta C_t = 0$. We will also note that the change in $\overline{q_{cl}}$ -due to deposition is limited by the amount of $\overline{q_{cl}}$ that is in -the mixed phase partition in the gridbox, hence (\ref{eq:deltacfl_dep}), -although it formally allows removal of $C_l$ from an ice-free partition, will -be unlikely to do so. - -The sublimation forms the main method by which ice cloud is destroyed in PC2, -hence PC2 results are relatively sensitive to its formulation. Here we -make a similar assumption to that used for liquid in the deposition term, -except that we limit the changes only to the region of the gridbox where -ice is subliming. - -\begin{equation} -\Delta C_i = A_{ice2} ( 1 + \frac{\Delta \overline{q_{cf}} } -{ \overline{q_{cf}} ( \frac{A_{ice2}}{C_i} ) } - )^{\frac{1}{2}} - A_{ice2} -\label{eq:deltacfi_sub} -\end{equation}. - -The term $\overline{q_{cf}} ( \frac{A_{ice2}}{C_i} )$ is the amount of -$\overline{q_{cf}}$ that is present in the subliming ice region, hence its -ratio with $\Delta \overline{q_{cf}}$ is the fractional change in that region. The -change in the total cloud fraction must also be equal to the change above, -since sublimation cannot occur in the presence of liquid cloud: - -\begin{equation} -\Delta C_t = \Delta C_i . -\label{eq:deltacft_sub} -\end{equation} - -\subsubsection{Riming} -This process acts only where mixed phase cloud occurs - although, in theory, -it could remove any supercooled liquid totally, the air would remain -saturated with respect to liquid water. Hence any subsequent cooling would -regenerate the same amount of liquid cloud. Hence we choose to model this -process as having \textit{no effect} on the cloud fractions. - -\subsubsection{Capture} -This is the freezing of raindrops onto ice crystals by collision. This does -not alter the ice cloud \textit{fraction} in the gridbox (although it does -alter $\overline{q_{cf}}$, and it has no interaction with the liquid cloud. -Again, we therefore choose to model this process as having \textit{no effect} -on the cloud fractions. - -\subsubsection{Evaporation of melting ice} -Here we simply assume that ice cloud fraction is removed in proportion to the -ice content that is removed. - -\begin{equation} -\Delta C_i = C_i \frac{ \Delta \overline{q_{cf}}}{\overline{q_{cf}}} . -\label{eq:lsp_evapmeltsnow} -\end{equation} - -Because the evaporation cannot occur in the liquid part of the gridbox, -there is no change to $C_t$ (or to $C_l$). - -\subsubsection{Melting} -Again, the change in $C_i$ is calculated using the method -in (\ref{eq:lsp_evapmeltsnow}). - -\begin{equation} -\Delta C_i = C_i \frac{ \Delta \overline{q_{cf}}}{\overline{q_{cf}}} . -\label{eq:lsp_melt} -\end{equation} - -The change in $C_t$ is calculated assuming that there is no correlation -in the gridbox between where the ice melts and the liquid cloud. Hence -we must multiply (\ref{eq:lsp_melt}) by the proportion of ice cloud -fraction that exists without liquid cloud (i.e. $\frac{A_{ice}}{C_i}$). - -\begin{equation} -\Delta C_t = C_i \frac{ \Delta \overline{q_{cf}}}{\overline{q_{cf}}} -\frac{A_{ice}}{C_i} . -\label{eq:lsp_melt2} -\end{equation} - -\subsubsection{Evaporation of rain} -Evaporation of rain will not, \textit{on its own}, -generate liquid cloud, since a -large-scale lifting process will be required in order to condense water -from the moistened air. We cannot, therefore, allow any change in cloud -fractions to occur as a result, subsequent changes are calculated elsewhere -in the model (e.g. by the lifting process, section \ref{sec:pres}). - -\subsubsection{Accretion} -Accretion is the sweep-out of liquid water droplets by rain. We argue -in a similar way to the riming term, that this will not remove any liquid -cloud fraction, since a small amount of lifting will regenerate the same -amount of liquid cloud. Hence we choose to model this process as having -\textit{no effect} on the cloud fractions. The arguments underlying the -formulation of the evaporation of rain and the accretion cloud fraction -changes may appear to be inconsistent in their limiting cases and the -subsequent response to lifting. However, when the limiting case is -not reached the formulations are both correct. For the moment, it is -not considered necessary to increase the complexity of the current, -simple representations. - -\subsubsection{Autoconversion} -As for accretion, the generation of rain directly from collision -and coalescence of liquid water droplets will not alter the cloud fractions. - -\subsubsection{Other microphysics terms} -There are already (i.e. also in the control) -two numerical tidy-up terms at the end of the microphysics -section that remove small rain amounts and provide an additional -melting term for the snow. These do not change the cloud fractions. - -If there are small amounts of ice present at the end of the -microphysics then these are removed at the end of the microphysics -timestep (also in the control). PC2 responds by resetting the -cloud fractions appropriately, so $C_t$ is reset to $C_l$ etc. - -\subsubsection{Numerical implementation} - -Note that after each process has been applied, we do \textit{not} -recalculate the sizes of the ice-only, liquid-only and mixed phase -partitions, but use the values at the start of the microphysics (this -includes the values of $C_i$ used in the calculation of `in-cloud' -water contents above. However, we do update the cloud fractions -themselves sequentially. We also recalculate after each process -the overlaps between the rain fraction (see \citeumdp{026}) and the cloud fractions. - -There is also a final set of checks that $C_l$ and $C_i$ lie -between 0 and 1 and that $C_t$ is bounded between $\text{Max}(C_l, C_i)$ -(maximum overlap of liquid and ice) and $\text{Min}(C_l+C_i,1)$ -(minimum overlap of liquid and ice). - -We should note in particular, that these parametrizations allow a -considerable reduction in $\overline{q_{cl}}$ without a corresponding -large reduction in $C_l$. This is an underlying feature of the PC2 scheme -(discussed in \cite{wg03}), and necessarily implies the -skewing of the underlying moisture PDF. Subsequent parts of -the model (e.g. the width narrowing, section \ref{sec:width}) will, of -course, act on the modified fields to adjust the cloud fractions further, -but remember that these are separate processes and modelled elsewhere in the -timestep. - -\subsection{PC2 erosion} -\label{sec:turb} - -\subsubsection{Original width-narrowing method} -(selected by setting {\bf i\_pc2\_erosion\_method = 1} in the UM namelist). - -In parallel with the homogeneous forcing part of the PC2 response to -convection, we introduce -a new block of code that allows a background change of the PDF width. -At earlier versions of PC2 (PC2:65 and earlier) this block was included as -a separate section of code that was called in as part of the atmphya parallel -timestepping. This was later moved to better numerically balance -increments from -the convection scheme with the cloud fraction erosion term. From VN8.1 onwards, a further option was introduced to implement the erosion prior to the microphysics parametrization. This was primarily to allow PC2 to be run at convection-resolving scales, at which the convection scheme is not called and therefore the erosion is not called. -We have empirically selected a rate of change of width that depends upon the -relative total humidity of the grid box, such that there is more -erosion in drier gridboxes. This promotes more rapid erosion of -shallow convective cloud, which is the main effect that we seek to -include, although the physical implication that dry air is -more turbulent than moist air does not match the way the real atmosphere -works, especially in the -stratosphere. No doubt the link can be improved upon with more -research. The formulation used is: - -\begin{equation} -\frac{1}{b_s} \frac{\partial b_s}{\partial t} = \Upsilon exp ( - \frac{2.01 Q_c}{0.2 a_L q_{sat liq}(T_L)} ) -\label{eq:dbsbydtbs_turb} -\end{equation} - -where the 0.2 factor is chosen to be closely equivalent to $1 - RH_{crit}$ -and the value of 2.01 has been selected through tuning. The code merges the -two numerical values into a single quantity (dbsdtbs-exp), equal to 10.05. -We note that in PC2:64 -(the library 6.4 code, a value of 0.62 is used rather than 2.01). As a guide -to the $RH_T$ dependence, note -that when the value of $RH_T$ is 0.85, the value of -$\frac{1}{b_s} \frac{\partial b_s}{\partial t}$ is close to $1 \times 10^{-4} s^{-1}$. - -Note: the source-code for this erosion method ({\bf pc2\_hom\_conv}, -{\bf pc2\_homog\_plus\_turb}, {\bf pc2\_delta\_hom\_turb}) -includes an additional term ``dbsdtbs1'' -which scales with the rate of homogeneous forcing -$\frac{\partial Q_c}{\partial t}$. However this term is always -set to zero on input to these routines so is never used. - -The width-narrowing formulation of section \ref{sec:width} is used to -calculate increments in $\overline{q_{cl}}$ and $C_l$. Using the liquid - -ice cloud overlap ideas of section \ref{sec:ct} then gives the associated -$C_t$ change. This background narrowing term, $\Upsilon$, is originally based upon work -by \cite{sg03}, although it is a parameter that has been -extensively tuned during PC2 development, a typical value would be $\Upsilon=-2.25 \times 10^{-5} s^{-1}$. - -\subsubsection{Numerical application of the original width-narrowing method} - -Because of the strong link the mathematical expressions for width narrowing -(section \ref{sec:width}) have with the expressions for the -homogeneous forcing (section \ref{sec:homog}), we choose to represent -the timestepping of this process in exactly the same way as for -the homogeneous forcing (in fact, in the Unified Model code we use -the same subroutine, see section \ref{sec:code}). As -before, we use a simple forward timestepping of $C_l$, with -$Q_c$ given by (\ref{eq:qc_eq_qt-qs}) and $a_L$ defined as discussed -in section \ref{sec:homog_num_app} and discretize eq \ref{eq:dcdt_width} as: - -\begin{equation} -\Delta C_l^{[n+1]} = - G(-Q_c) Q_c \frac{1}{b_s} -\frac{\partial b_s}{\partial t} \Delta t. -\label{eq:dcl_turb_final} -\end{equation} - -Similarly to (\ref{eq:c_l^n+1}), we then limit the cloud fraction to 0 and -1 and then apply a mid-point value of $C_l$ to calculate the change in -$\overline{q_{cl}}$ (discretizing eq \ref{eq:dqcldt_width}): - -\begin{equation} -\Delta q_{cl}^{[n+1]} = (q_{cl}^{[n]} - Q_c \frac{1}{2}(C_l^{[n]}+C_l^{[n+1]})) -\frac{1}{b_s} \frac{\partial b_s}{\partial t} \Delta t. -\label{eq:dqcl_turb_final} -\end{equation} - -In this case the value of $\Delta q_{cl}$ \textit{is} limited to ensure that -no more $\overline{q_{cl}}$ is removed than the model has available. This was -chosen to ensure that the erosion process itself contains this physical limit, -not a numerical tidying-up process. - -The option ``l\_fixbug\_pc2\_qcl\_incr'' ensures that qcl is set to zero -if the CFL has reached zero. - - -\subsubsection{Cloud-surface-area hybrid erosion method} -(selected by setting {\bf i\_pc2\_erosion\_method = 3} in the UM namelist). - -\cite{morcrette_petch} showed that changes to the erosion parameter ($\Upsilon$ in Eqn. \ref{eq:dbsbydtbs_turb}) did not have as significant an impact -on the global work done by the erosion process as might be expected. This was due to a feedback process -whereby, reducing the erosion parameter leads to more cloud water, more autoconversion of cloud water to rain, more -fall-out of rain and more drying of the layer, hence increasing the $exp ( - \frac{2.01 Q_c}{0.2 a_L q_{sat liq}(T_L)} )$ part of -Eqn. \ref{eq:dbsbydtbs_turb}. Although the feedback is physically plausible it crucially depends on the formulation of -Eqn. \ref{eq:dbsbydtbs_turb} and the dependence of the rate of narrowing of the PDF on the moisture, a dependence that was developed -from a pragmatic rather than theoretical stand-point. -The option for an alternative way of calculating the erosion was introduced at vn8.0 - -We use equation 30 from \cite{t93} to specify the sink of $q_{cl}$ due to erosion, i.e. -\begin{equation} -\frac{\partial q_{cl}}{\partial t}=-A K(q_{sat}-q_v) -\label{eq:dqcldt_hybrid} -\end{equation} -(note we have changed the sign as we have replaced the evaporation rate $E_2$ -in \cite{t93} with $-\frac{\partial q_{cl}}{\partial t}$ on the left-hand-side). -In the \cite{t93} scheme, $A$ is set to the cloud fraction (i.e. $A=C_l$). -Here we recall that "cloud erosion" is meant to represent the evaporation of cloud water due to the -mixing of clear and cloudy air and that this can only happen on the edges of cloud, where saturated air is exposed to sub-saturated air. -If the cloud fraction is small (e.g. 5$\%$), then there are not many clouds, so there is only a small surface area from which evaporation can occur. -Similarly if the cloud cover is very high (e.g. 95$\%$) then there is again not much surface area exposed to clear sky. -A maximum in exposed surface area is expected when the cloud cover is 50$\%$. - -By imagining that the grid-box is broken up into cubes whose horizontal dimension equal the layer depth it is possible -to work out what the maximum lateral surface area would be, as a function of cloud fraction, for different arrangements of cloudy cubes. -The maximum lateral surface area, is when the clear and cloudy cubes are arranged in a chess-board pattern, and the minimum is when then are all grouped -together into a circular clump. Numerical tests using randomly distributed cloudy cube shows that the variation -in lateral surface area, $S$, as a function of cloud fraction can be expressed as: -\begin{equation} -S= - 2 C_l ^{2} + 2 C_l -\label{eq:S_Cl} \end{equation} -The maximum normalised surface area of 0.5 occurs at a cloud fraction of 0.5. -Using a cloud mask derived from satellite imagery shows that real cloud fields do follow this kind of dependence, -but that the peak surface area is nearer to 0.35, meaning that real clouds are not as randomly distributed as random -ones and that there is some kind of clumping together, which is what we might have expected. -When it comes to implementing such a scheme in the model, there will need to be a tunable parameter to govern the -rate of evaporation. This will not affect the shape of the lateral surface area function. -As a result the details of whether the peak lateral surface area is 0.5 or 0.35 are simply absorbed into the tunable parameter $K$, -which is supplied from the UMUI (using the same text box as was used for supplying $\Upsilon$). - -The exposed surface area associated with the tops and bottom of the clouds is calculated assuming maximum overlap in adjacent layers and is added to the lateral -surface area to give a total surface area, -\begin{equation} -A=max(C_l(k)-C_l(k+1),0.0)+max(C_l(k)-C_l(k-1),0.0)+S -\label{eq:A_top_and_bottom} \end{equation} -it is this value of $A$ which we use in Eqn. \ref{eq:dqcldt_hybrid}. - -Note that the contributions from the top and bottom interfaces of the current -model-level $max(C_l(k)-C_l(k+1),0.0)$ and $max(C_l(k)-C_l(k-1),0.0)$ -may optionally either be included or excluded, depending on the -UM namelist switch \textbf{i\_pc2\_erosion\_method}. -Further note: at present these contributions are hardwired to be excluded, -as they prevented the erosion calculations from being parallelised -in the vertical direction using OpenMP, and no operational model configurations -were using them. - -Having calculated a reduction in $q_{cl}$ using the Tiedtke-surface-area method, we -then work out the relative rate of narrowing that would have given the same sink of $q_{cl}$. This value of $\frac{1}{b_s} \frac{\partial b_s}{\partial t}$ -is then used to calculate the change in $C_l$ using the same moisture PDF assumptions as were used in the original PC2 erosion formulation. -To achieve this, we combine equations \ref{eq:dcdt_width} and -\ref{eq:dqcldt_width} from section \ref{sec:width} to eliminate -$\frac{1}{b_s} \frac{\partial b_s}{\partial t}$ and write -$\frac{\partial C_l}{\partial t}$ as a function of -$\frac{\partial \overline{q_{cl}}}{\partial t}$: - -\begin{equation} -\frac{\partial C_l}{\partial t} - = - \frac{ G(-Q_c) Q_c \frac{\partial \overline{q_{cl}}}{\partial t} } - { (- C_l Q_c+\overline{q_{cl}}) } -\label{eq:dcdt_hybrid} -\end{equation} - -Where the change in liquid water content -$\frac{\partial \overline{q_{cl}}}{\partial t}$ -is given by eq \ref{eq:dqcldt_hybrid} above. - -This combination of a Tiedkte sink term for $q_{cl}$, a PC2 term for $C_l$ and the introduction of some surface area dependence leads to this formulation -being referred to as a ``hybrid'' cloud-surface-area erosion method. - -\subsubsection{Numerical application of the hybrid erosion method} -\label{sec:erosion_numerics} - -Next, we consider how to numerically discretise equations -\ref{eq:dqcldt_hybrid} and \ref{eq:dcdt_hybrid} -to compute cloud increments due to erosion. -The simplest approach is an explicit forwards-in-time discretisation: - -\begin{equation} -\frac{ \Delta {q_{cl}}_{ero}}{\Delta t} = A(C_l^n) K(q_{sat}-q_v) -\label{eq:hybrid_erosion_expl} \end{equation} - -i.e. the increment is calculated by evaluating the term $A$ from equations -\ref{eq:S_Cl} and \ref{eq:A_top_and_bottom} using the value of cloud-fraction -$C_l$ \textit{before} erosion has been applied. - -However, when the environment is significantly subsaturated -(so that the term $(q_{sat}-q_v)$ is large and negative), -and long timesteps $\Delta t$ are used -(e.g. order 1000 s used in global climate simulations), -this discretization can suffer severe numerical overshoot. -i.e. the increment based on $C_l^n$ is large enough to reduce $q_{cl}$ -(and hence also $C_l$) to less than zero within a single timestep. -If the continuous equation were solved analytically this wouldn't happen; -as $C_l$ declines due to the erosion, so will $A(C_l)$ -and hence the erosion rate, so that $q_{cl}$ and $C_l$ smoothly decline -towards zero. - -Three options are available in the code to address this problem, -selected by the UM namelist switch \textbf{i\_pc2\_erosion\_numerics}, -detailed below. -Single-Column Model tests indicate that the 2nd and 3rd options yield much less -timestep sensitivity for detrained cloud in shallow cumulus regimes. - -\begin{enumerate} - -\item \textbf{Retain the explicit discretization, but limit the resulting -erosion increments to ensure $q_{cl}$ and $C_l$ don't go negative. -(i\_pc2\_erosion\_numerics=1)} -Also, to ensure that some cloud remains at end-of-timestep where -shallow cumulus is detraining into dry environments, the erosion -calculation is fed copies of the fields with the current timestep's -convection increments subtracted off. This means any cloud detrained -by convection during the current timestep cannot be eroded until the -following timestep, and so is still present at end-of-timestep. -As discussed in section \ref{sec:timestepping}, -this leads to a problematic timestep sensitivity, -since the amount of cloud not subject to erosion is the convection increment, -which scales with the timestep length. - -Having computed the erosion $q_{cl}$ increment using -\ref{eq:hybrid_erosion_expl}, the consistent $C_l$ increment is computed -by discretising \ref{eq:dcdt_hybrid} as: - -\begin{equation} -\frac{\Delta {C_l}_{ero}}{\Delta t} - = - \frac{ G(-Q_c)^n Q_c^n \frac{\Delta \overline{{q_{cl}}_{ero}}}{\Delta t} } - { (- C_l^n Q_c^n + ( \overline{q_{cl}^n} - + \frac{1}{2} \Delta \overline{{q_{cl}}_{ero}} ) ) } -\label{eq:dcdt_hybrid_discr} -\end{equation} - -i.e. all terms are treated explicitly (using the values before erosion), -except for $\overline{q_{cl}}$ which takes the mid-point interpolated -half-way between its values before and after erosion, -to give some improvement in accuracy. - -\item \textbf{Use an approximate implicit discretisation, -which intrinsically yields a positive solution for $q_{cl}$ and $C_l$. -(i\_pc2\_erosion\_numerics=2)} -The copies of the fields passed to the erosion calculation are -fully updated with the convection increments. -We then write equation \ref{eq:dqcldt_hybrid} in the form: - -\[ -\frac{\partial q_{cl}}{\partial t} = q_{cl} f(q_{cl},C_l,(q_{sat}-q_v)) -\] - -(where the term $f(q_{cl},C_l,(q_{sat}-q_v)) = \frac{A K(q_{sat}-q_v)}{q_{cl}}$ -will be treated explicitly, under the assumption that this ratio will -evolve more slowly while erosion rapidly reduces both the numerator -and the denominator). - -We then take a backwards-in-time implicit discretisation in terms of the -leading factor $q_{cl}$: - -\[ -\frac{ q_{cl}^{n+1} - q_{cl}^{n}}{\Delta t} = q_{cl}^{n+1} f^n -\] - -Now, the problem is somewhat complicated by the fact that in the code, -erosion is calculated in parallel with the homogeneous forcing by convection, -and we need to account for the homogeneous forcing increment -$\Delta q_{cl}^{hom}$ in our implicit solution. We therefore write the above as: - -\[ -\Delta q_{cl}^{ero} = \Delta t - ( q_{cl}^n + \Delta q_{cl}^{hom} + \Delta q_{cl}^{ero} ) f^n -\] - -Rearranging: - -\[ -\Delta q_{cl}^{ero} = \Delta t f^n q_{cl}^n \frac{ q_{cl}^n + \Delta q_{cl}^{hom} } - { q_{cl}^n - \Delta t f^n q_{cl}^n } -\] - -Note that the term $\Delta t f^n q_{cl}^n$ is the erosion increment -we would obtain from the purely explicit discretisation, -$\Delta q_{cl}^{ero\,expl}$. The implicit discretisation is implemented by -first calculating $\Delta q_{cl}^{ero\,expl}$ using equation -\ref{eq:hybrid_erosion_expl} -(as we do for \textbf{i\_pc2\_erosion\_numerics=1}) -but then rescaling it using the above expression, which becomes: - -\begin{equation} -\Delta q_{cl}^{ero} = \Delta q_{cl}^{ero\,expl} \frac{ q_{cl}^n + \Delta q_{cl}^{hom} } - { q_{cl}^n - \Delta q_{cl}^{ero\,expl} } -\label{eq:hybrid_erosion_impl_qcl} \end{equation} - -Provided erosion is acting to reduce cloud-water ($\Delta q_{cl}^{ero\,expl} < 0$), -and homogeneous forcing by convection has not already completely removed -the cloud ($q_{cl}^n + \Delta q_{cl}^{hom} > 0$), \ref{eq:hybrid_erosion_impl_qcl} -is guaranteed to yield a stable, positive solution for $q_{cl}$. - -We also apply exactly the same argument to the equation for the cloud-fraction -increment $C_l$, and obtain: - -\begin{equation} -\Delta C_l^{ero} = \Delta C_l^{ero\,expl} \frac{ C_l^n + \Delta C_l^{hom} } - { C_l^n - \Delta C_l^{ero\,expl} } -\label{eq:hybrid_erosion_impl_Cl} \end{equation} - -Where $\Delta C_l^{ero\,expl}$ is computed using eq \ref{eq:dcdt_hybrid_discr}, -except that the term $\frac{1}{2} \Delta \overline{{q_{cl}}_{ero}}$ -is omitted (interpolating to the mid-point value of $\overline{q_{cl}}$ -in the denominator would be ``double-counting'' if we are already making -an implicit correction to the full increment). - -In the case where the homogeneous forcing increments have already removed -all of the cloud water content or fraction, erosion is not performed, -and $q_{cl}$ and $C_l$ are both set to zero. In the case where erosion is -actually acting to increase cloud-fraction, the code defaults to retaining -the explicit discretisation solution $\Delta q_{cl}^{ero\,expl}$ and -$\Delta C_l^{ero\,expl}$. Otherwise, equations \ref{eq:hybrid_erosion_impl_qcl} -and \ref{eq:hybrid_erosion_impl_Cl} are applied to yield the implicit solution. - -\item \textbf{Use an analytic solution to the integration of the -time-derivatives in (\ref{eq:dqcldt_hybrid}) and (\ref{eq:dcdt_hybrid}) -for greater accuracy. -(i\_pc2\_erosion\_numerics=3)} - -Two problems have been identified with the above implicit numerical method: -\begin{itemize} - -\item The implicit correction is applied completely independently to the -increments for $q_{cl}$ and $C_l$. So as with the explicit method, -differing numerical error in the increments for the two variables -can lead to them becoming inconsistent with eachother. -It was found by experimentation that even with the implicit correction, -it is possible for erosion to reduce $C_l$ by a bigger fraction than $q_{cl}$, -so that the in-cloud water content $\frac{q_{cl}}{C_l}$ is {\em increased}. -Narrowing the PDF should only {\em decrease} the in-cloud water-content; -occasional large increases due to numerical error can lead to spurious -precipitation being produced by the microphysics scheme. - -\item The implicit correction makes it impossible for erosion to reduce -$q_{cl}$, $C_l$ to zero. As we will show below, the analytic solution -to the equations posed does in fact go to zero after a finite time -under grid-mean subsaturation -(although the erosion rate declines with $C_l$ as it approaches zero, -$C_l$ approaches zero more slowly than $q_{cl}$, so that both variables decrease -following power-law curves not exponentials). -When erosion (wrongly) can never entirely remove cloud, this allows -tiny values of $q_{cl}$ and $C_l$ to spuriously spread across the domain -via numerical diffusion from the model's advection scheme. - -\end{itemize} - -Under this option, we attempt to compute an analytic solution to the -simultaneous differential equations \ref{eq:dqcldt_hybrid} and -\ref{eq:dcdt_hybrid} so that $q_{cl}$ and $C_l$ both decrease smoothly and -consistently. -The equations lead to somewhat different behaviour depending on whether -the grid-mean state is subsaturated ($Q_c < 0$), supersaturated ($Q_c > 0$), -or close to saturation ($Q_c$ near-zero). We can employ different -approximations to integrate the equations in each case. -In the code, we first test the value of $Q_c$ and compute the erosion -increments as follows: - -\begin{enumerate} - -\item {\bf Grid-mean subsaturation ($Q_c < 0$):} - -The relation between the erosion tendencies in liquid-cloud-fraction and -liquid water content (\ref{eq:dcdt_hybrid}) can be expressed in terms of -{\em fractional} rates of change -(dividing the top and bottom by $-C_l Q_c$, and dividing both sides by $C_l$): - -\begin{equation} -\frac{1}{C_l} \frac{\partial C_l}{\partial t} - = \frac{ G(-Q_c) \frac{q_{cl}}{C_l^2} }{ 1 - \frac{q_{cl}}{C_l Q_c} } \; - \frac{1}{q_{cl}} \frac{\partial q_{cl}}{\partial t} -\label{eq:dcdt_hybrid_1} -\end{equation} - -Under homogeneous forcing (section \ref{sec:homog}), we defined the PDF height -at the saturation boundary when near the cloudy end of the PDF as -$G(-Q_c) = \frac{n+1}{n+2} \frac{C_l^2}{q_{cl}}$ (eq \ref{eqn20}). -In fact, $G(-Q_c)$ is set to some blend between this and the value near -the clear end of the PDF (eq \ref{eqn21}). But we will assume that -when eroding cloud under grid-mean subsaturated conditions ($Q_c < 0$), -$G(-Q_c)$ follows this scaling with $\frac{C_l^2}{q_{cl}}$ even if its -value differs somewhat from eq \ref{eqn20}. -Therefore the quantity $c_1 = G(-Q_c) \frac{q_{cl}}{C_l^2}$ remains constant -during the erosion process, and eq \ref{eq:dcdt_hybrid_1} becomes: - -\begin{equation} -\frac{1}{C_l} \frac{\partial C_l}{\partial t} - = \frac{ c_1 }{ 1 - \frac{q_{cl}}{C_l Q_c} } \; - \frac{1}{q_{cl}} \frac{\partial q_{cl}}{\partial t} -\label{eq:dcdt_hybrid_2} -\end{equation} - -The term $1 - \frac{q_{cl}}{C_l Q_c}$ (which is $> 1$ since we are considering -grid-mean subsaturation $Q_c < 0$) usually remains close to 1 in practice, -so we can assume its fractional variation over the timestep is small -compared to the other terms, and treat it explicitly. -We can therefore straightforwardly integrate eq \ref{eq:dcdt_hybrid_2} -to obtain the scaling of $C_l$ with $q_{cl}$ as both are reduced by erosion: - -\begin{equation} -\frac{C_l}{{C_l}_0} = \left( \frac{q_{cl}}{{q_{cl}}_0} \right)^{b_1} -\label{eq:cl_qcl_scaling} -\end{equation} - -where ${C_l}_0$, ${q_{cl}}_0$ are the values before erosion is applied, -and the exponent is $b_1 = \frac{ c_1 }{ 1 - \frac{q_{cl}}{C_l Q_c} }$. -When $G(-Q_c)$ takes its value from the cloudy end of the PDF, we have -$c_1 = \frac{n+1}{n+2}$. Since the PDF power $n > 0$ and $Q_c < 0$ -under the considered grid-mean subsaturation, we always have $b_1 < 1$. -This ensures that erosion reduces $C_l$ at a slower fractional rate than -$q_{cl}$, so that in-cloud water content $\frac{q_{cl}}{C_l}$ always decreases. - -Next we derive an integral solution for the decline of $q_{cl}$ with time. -Ignoring the cloud surface-area contributions from the levels above and below -(they are disabled in the code anyway), the erosion liquid water content -tendency is obtained by combining \ref{eq:dqcldt_hybrid} and \ref{eq:S_Cl}: - -\begin{equation} -\frac{\partial q_{cl}}{\partial t} = -K \, 2 C_l (1 - C_l) \, (q_{sat}(T)-q_v) -\label{eq:dqcldt_hybrid_1} -\end{equation} - -From eq \ref{SD2}, $q_{sat}(T)-q_v = \frac{SD}{a_L}$, where $SD$ is the -saturation defecit, and $a_L$ is the dimensionless factor defined in -eq \ref{eq:a_L}. Following the derivation in section -\ref{sec:smooth_initiation} (eq \ref{eq:qc_plus_sd}), -we can write this in terms of the liquid-water content: $SD = q_{cl} - Q_c$ -(where $Q_c$ was defined in eq \ref{eq:qc_eq_qt-qs}, and corresponds to the -grid-mean supersaturation converted to an equivalent liquid water content). -Substituting this into (\ref{eq:dqcldt_hybrid_1}) above, we obtain: - -\begin{equation} -\frac{\partial q_{cl}}{\partial t} = -\frac{K}{a_L} \, 2 C_l (1 - C_l) \, - (q_{cl}-Q_c) -\label{eq:dqcldt_hybrid_2} -\end{equation} - -Substituting eq \ref{eq:cl_qcl_scaling} for the leading factor of $C_l$ -on the right-hand-side and rearranging: - -\[ -\left( \frac{q_{cl}}{{q_{cl}}_0} \right)^{-b_1} \frac{\partial q_{cl}}{\partial t} - = -\frac{K}{a_L} \, 2 {C_l}_0 (1 - C_l) \, (q_{cl}-Q_c) -\] - -In significantly subsaturated conditions the r.h.s. has only weak dependence -on $C_l$ and $q_{cl}$ ($C_l << 1$, $q_{cl} << -Q_c$), so we can treat the -whole r.h.s. explicitly (i.e. neglect its variation during each timestep), -so that the above integrates to: - -\[ -\left[ \frac{{q_{cl}}_0}{1-b_1} \left( \frac{q_{cl}}{{q_{cl}}_0} \right)^{1-b_1} -\right]_{{q_{cl}}_0}^{{q_{cl}}_{\Delta t}} - = -\frac{K}{a_L} \, 2 {C_l}_0 (1 - C_l) \, (q_{cl}-Q_c) \Delta t -\] - -Inserting the limits of the integral on the l.h.s. and rearranging, -we obtain our analytical solution for $q_{cl}$ after time $\Delta t$: - -\begin{equation} -{q_{cl}}_{\Delta t} = {q_{cl}}_0 \left( 1 - \frac{1-b_1}{{q_{cl}}_0} - \frac{K}{a_L} \, 2 {C_l}_0 (1 - C_l) \, (q_{cl}-Q_c) \Delta t - \right)^\frac{1}{1-b_1} -\label{eq:qcl_int_hybrid} -\end{equation} - -Note that $q_{cl}$ falls to zero after a finite time -$\frac{{q_{cl}}_0}{1-b_1} \frac{a_L}{K} - \frac{1}{2 {C_l}_0 (1 - C_l) \, (q_{cl}-Q_c)}$. -If the timestep $\Delta t$ is longer than this time, then erosion -completely removes the cloud during the current timestep. - -We first set ${q_{cl}}_0$ and ${C_l}_0$ to the values already updated -by homogeneous forcing, and then sequentially compute the updated -$q_{cl}$ after erosion using (\ref{eq:qcl_int_hybrid}). -Then we substitute this value into (\ref{eq:cl_qcl_scaling}) to compute -the consistent updated value of $C_l$. -Finally, to improve accuracy, a small number of iterations are performed -to find the solution with the explicitly-treated terms -(the exponent $b_1 = \frac{ c_1 }{ 1 - \frac{q_{cl}}{C_l Q_c} }$ -and the terms $(1 - C_l)$ and $(q_{cl}-Q_c)$ in eq \ref{eq:qcl_int_hybrid}) -adjusted to values linearly-interpolated to half-way between the -start and end of the erosion timestep. - -\item {\bf Grid-mean supersaturation ($Q_c > 0$):} - -In this case, erosion does not act to reduce $C_l$ and $q_{cl}$ towards zero. -Instead, the narrow PDF limit it adjusts towards has -no remaining subsaturated air, so that $C_l = 1$ and $q_{cl} = Q_c$. -In this case, we can repeat the above derivation, but considering the -equations for the clear-fraction $1-C_l$ in place of $C_l$, -and the saturation defecit $SD = q_{cl}-Q_c$ in place of $q_{cl}$. -Assuming that $G(-Qc)$ follows the scaling for the clear end of the PDF -(\ref{eqn21}), this leads to a similar equation to (\ref{eq:cl_qcl_scaling}) -but for the scaling as erosion reduces $1-C_l$ and $SD$ towards zero .: - -\begin{equation} -\frac{1-C_l}{1-{C_l}_0} = \left( \frac{SD}{SD_0} \right)^{b_2} -\label{eq:ca_sd_scaling} -\end{equation} - -with $b_2 = \frac{ c_2 }{ 1 + \frac{SD}{(1-C_l) Q_c} }$ -and $c_2 = G(-Q_c) \frac{SD}{(1-C_l)^2}$ -(note we must have $0 < b_2 < 1$). - -And then the tendency equation for $SD$ is: - -\begin{equation} -\frac{\partial SD}{\partial t} = -\frac{K}{a_L} \, 2 (1 - C_l) C_l \, SD -\label{eq:dsddt_hybrid_2} -\end{equation} - -The one asymmetry between this and the $q_{cl}$ tendency equation -(\ref{eq:dqcldt_hybrid_2}) is that for $SD$ the r.h.s. is directly proportional -to the quantity in the time-derivative, whereas for $q_{cl}$ there is an -additional $Q_c$ term which is constant during erosion. -Substituting (\ref{eq:ca_sd_scaling}) for the leading factor of -$(1 - C_l)$ in (\ref{eq:dsddt_hybrid_2}), integrating over time $\Delta t$ -(neglecting the fractional variation of $C_l$ over the timestep) -and rearranging, we obtain: - -\begin{equation} -{SD}_{\Delta t} = {SD}_0 \left( 1 + b_2 - \frac{K}{a_L} \, 2 (1-{C_l}_0) C_l \, \Delta t - \right)^{-\frac{1}{b_2}} -\label{eq:sd_int_hybrid} -\end{equation} - -Note that the additional power of $SD$ on the r.h.s. of -(\ref{eq:dsddt_hybrid_2}) leads to the integral solution having -a negative exponent. This means that under grid-mean supersaturation, -erosion makes $SD$ and $1-C_l$ approach but never quite reach zero, -which is quite different behaviour to grid-mean subsaturation where -$q_{cl}$ and $C_l$ go to zero over a finite time. -This asymmetry is because the erosion rate is parameterised to be -proportional to $SD$, and this tends to zero as the PDF is narrowed -under supersaturation, but remains finite positive under subsaturation. - -We first set ${SD}_0 = {q_{cl}}_0 - Q_c$ (where as above ${q_{cl}}_0$ is -the value already updated by homogeneous forcing), -then compute the value of $SD$ updated by erosion using -(\ref{eq:sd_int_hybrid}). -Then we substitute this value into (\ref{eq:ca_sd_scaling}) to compute -the consistent updated value of $1-C_l$. -A small number of iterations are then performed -to find the solution with the explicitly-treated terms -(the exponent $b_2 = \frac{ c_2 }{ 1 + \frac{SD}{(1-C_l) Q_c} }$ -and the term $C_l$ in eq \ref{eq:sd_int_hybrid}) -adjusted to values linearly-interpolated to half-way between the -start and end of the erosion timestep. -Then the final values of $SD$ and $1-C_l$ are used to increment -$q_{cl} = Q_c + SD$ and $C_l$, as prognosed by the rest of the model. - -\item {\bf grid-mean saturation ($Q_c$ near-zero):} - -In this case, the PDF is centred on the -saturation boundary, so that narrowing it does not change the cloud-fraction. -In the limit $Q_c = 0$, we have $q_{cl} = SD$, and (\ref{eq:dqcldt_hybrid_2}) -or (\ref{eq:dsddt_hybrid_2}) becomes: - -\begin{equation} -\frac{1}{q_{cl}} \frac{\partial q_{cl}}{\partial t} - = -\frac{K}{a_L} \, 2 C_l (1 - C_l) -\end{equation} - -where everything on the r.h.s. is constant under erosion. -This simply integrates to give exponential decline of $q_{cl}$ -(and $SD$) towards zero: - -\begin{equation} -{q_{cl}}_{\Delta t} = {q_{cl}}_0 e^{ -\frac{K}{a_L} \, 2 C_l (1 - C_l) \Delta t } -\end{equation} - -\end{enumerate} - -\end{enumerate} - - -\subsection{Orographic and Gravity Wave Drag} -The Orographic and Gravity Wave Drag sections do not alter the temperature -or moisture content of the model gridboxes, hence PC2 assumes no change in the -condensate and cloud fractions as a result of these processes. - -\subsection{Advection} -\label{sec:advec} - -The advection of $\overline{q_{cl}}$ and $\overline{q_{cf}}$ are already -performed separately by the semi-Lagrangian advection scheme. -Advection of the three cloud fractions $C_l$, $C_i$ and $C_t$ are all -performed by PC2 in the same way. - -Note that ascent or subsidence by advection entails a pressure change following -each parcel, which will cause an accompanying adiabatic temperature change. -These advective pressure and temperature changes imply a homogeneous forcing, -which yields a change in $\overline{q_{cl}}$ and $C_l$ in addition to their -transport by the winds. This is described in section \ref{sec:pres}. - -If the UM namelist switch \textbf{l\_pc2\_sl\_advection} is turned on, -the PC2 homogeneous forcing response to advection is calculated -straight after the call to Semi-Lagrangian advection. -Otherwise, the pressure change from advection is combined with the -Eulerian pressure change from the dynamics Helmholtz solver, and the resulting -homogeneous forcing of liquid cloud is computed at the end of the timestep. - -\subsection{Boundary Layer} -\label{sec:bl} -At a basic level, the boundary layer scheme works by -mixing $\overline{q_T}$ and $\overline{T_L}$, and tracer mixing -$\overline{q_{cf}}$. The condensation and $C_l$ changes are represented -using the homogeneous forcing representation. The forcing of $Q_c$ can be -written in $\Delta \overline{q_T}$ and $\Delta \overline{T_L}$ terms -using (\ref{eq:deltaqc_exp}). - -$\overline{q_{cf}}$ is already mixed using the tracer mixing scheme. PC2 -will calculate the corresponding $C_i$ change assuming the inhomogeneous -forcing scenario. Although this is not necessarily an appropriate physical -model to use, it is the only generic physical model we have currently -developed in order to convert increments in a condensate to increments in -a cloud fraction. We use a value of the in-cloud water content $q_c^S$ -based upon a linear combination of the current in-cloud ice water -content, $\frac{\overline{q_{cf}}}{C_i}$, and a fixed value. - -\begin{equation} -q_C^S = C_i \frac{\overline{q_{cf}}}{C_i} + ( 1 - C_i) q_{cf0 \, BL} -\label{eq:qcf_ci} -\end{equation} - -where $q_{cf0 \, BL}$ is a specified value of $1 \times 10^{-4} \, kg \, kg^{-1}$. -We then use the inhomogeneous forcing equation based upon -(\ref{eq:dcdt_inhom2}) but for ice water content to write - -\begin{equation} -\Delta C_i = \frac{(1 - C_i)}{q_C^S - \overline{q_{cf}}} Q4_i . -\label{eq:deltaci_bl} -\end{equation} - -Since the physical model will have $C_i$ tend to 1 if the -denominator is small, we will, to avoid numerical problems, set -$C_i$ to 1 if $q_C^S - \overline{q_{cf}} < 1 \times 10^{-10} kg kg^{-1}$. -Note that we do not use the multiple phases injection source -expressions (section \ref{sec:multiple} and equation \ref{eq:cff_ts}). -This is because the liquid water changes are not associated with the plume model. - -Equation \ref{eq:qcf_ci} assumes that the change to the ice water content has led to an increase in ice water content. -However, if the ince water content has reduced, the change to the ice cloud fraction is not consistent. -The option to "Use consistent formulation of ice cloud fraction changes due to boundary-layer processes" ensure that -if the ice water content is reduced, the ice cloud fraction is reduced, in such as way as to maintain -the same in-cloud ice water content. - -The $C_t$ changes are calculated using the minimum overlap method of -section \ref{sec:ct}. - -In \textit{ni-imp-ctl} the control code inhibits the -call to the diagnostic cloud scheme -if there is deep or shallow convection occurring and the model level -is less than \textit{or equal to} the layer immediately above the top of the -boundary layer mixed layer (i.e. level ntml+1). This is in order to -ensure that there is -no large-scale cloud present below the base of the convective cloud, but -additionally performs this calculation at the level above, probably -in order that latent heating from large-scale condensation does not -inhibit the convection. A similar thing is performed for PC2, with -any large-scale cloud being evaporated if the same criteria are met, -\textit{except that it is not performed on the level above the boundary -layer mixed layer}. This choice (i.e. ntml) is seen to give improved -results in PC2, and is arguably a more physical reasonable choice -anyway than using ntml+1. - -\subsection{Convection} -\label{sec:convec} - -This section concentrates specifically upon the PC2 interface to the -convection scheme. In the current formulation of the UM, only a -mass-flux convection scheme exists, and this is what is described -below. Work to interface PC2 to the developing turbulence based -convection scheme is commented upon in section \ref{sec:tbcs}. - -An alternative way of calculating cloud fraction increments is currently under development and -is described in section \ref{sec:conv-simpler}. - -A traditional view of convective parametrization is a scheme that -transports vapour, $q$, -heat, $\theta$, and momentum, $u$ and $v$ winds within a single column. It -does not consider transport sideways to adjoining columns, and (at least -in the Gregory-Rowntree scheme used in the UM) is considered independent of -any resolved scale vertical air motions. This necessitates the view of -compensating subsidence within the column, whereas some conceptual models -of tropical convection would have the bulk of the ascent in the -convective cores and the -associated descent thousands of miles away in the downward branch of the -Hadley circulation. The parametrization schemes traditionally overlook the -existence of condensate in the model column. The non-PC2 version -of the mass-flux convection scheme used in the UM would have the same -large-scale liquid and ice prognostics before and after convection occurs -(apart from a bolt-on evaporation below convective cloud base), with no -regard at all to what happens to it or its effect on the rest of the -convection. Within PC2 we have had to work to more fully incorporate -the condensate into the convection scheme. - -\subsubsection{Introduction to the convective mass flux scheme} - -Within the mass flux scheme the net change in -$\overline{q_{cl}}$ and $C_l$ etc. -comes from two distinct sources. Firstly, the condensate and cloud -fraction injected from the plume (the $Q4$ terms, section \ref{sec:inhomog}); -secondly, the condensation response to the vapour and heat changes associated -with the detrainment and compensating subsidence. Strictly, we will see that the -$Q4$ terms also include the contribution to the condensate transport -by the compensating subsidence - this casts doubt on the validity of -the application of the injection forcing scenario to calculate the -equivalent cloud fraction change, since ideally the cloud fractions -ought to be transported by the compensating subsidence in a similar -way to the condensate transport (which is documented below). - -We therefore split the convective contribution in (\ref{eq:dqcldt_and_dcdt}) -into two parts: - -\begin{equation} -\frac{\partial \overline{q_{cl}}}{\partial t} |_{convection} = -Q4_l + Q_{environment} -\label{eq:inhomg_plus_homog} -\end{equation} - -where $Q_{environment}$ is the condensation associated with changes -in the vapour and temperature from the detrainment and compensating -subsidence. Similar splits are made for the cloud variables, where -the injection forcing, section \ref{sec:inhomog}, is used to calculate -the first term from $Q4_l$. Section \ref{subsect:q4calculation} looks -at the issue of the -calculation of $Q4_l$ etc., and section \ref{sec:conv_homog} looks at -the calculation of $Q_{environment}$, and its associated cloud -fraction change. We first look at the basic transport equations in a -mass flux convection scheme. - -\subsubsection{Basic Equations for a Convective Mass Flux Scheme} -\label{subsect:basmaseqs} - -We first consider a generic mass-flux scheme before its application to PC2. -As discussed by Grant and Stirling (personal communication), -the equations for convective -tendencies are most simply applied to a variable, ${\chi}$, that is conserved -under moist adiabatic processes (e.g. total water content). In this case, -% -\begin{equation} -{\pardbyd{\xsubsup{ }{E}}{t}}_{\rm{conv}} = -- \frac{1}{\ov{\rho}} \, \pardbyd{\ov{\rho w^{'} \xsubsup{ }{E'}}}{z} -\label{eq:chibasic} \end{equation} - -To parametrize \ref{eq:chibasic}, the current UM convection scheme takes a -mass flux approximation -% -\begin{equation} -\lp {\ov{\rho w^{'} \xsubsup{ }{E'}}} \rp_{\rm{conv}} = M^{\rm{P}} \, -\lp { \xsubsup{ }{P} - \xsubsup{ }{E} } \rp -\label{eq:massflux} \end{equation} -% -which can be differentiated to give -% -\begin{equation} -- \frac{1}{\ov{\rho}} \, \pardbyd{\ov{\rho w^{'} \xsubsup{ }{E'}}}{z} = -\pardbyd{\xsubsup{ }{P} \, M^{\rm{P}}}{p} - -\xsubsup{ }{E} \, \pardbyd{M^{\rm{P}}}{p} - -M^{\rm{P}} \, \pardbyd{\xsubsup{ }{E}}{p} -\label{eq:eddyflux} \end{equation} - -The bulk cloud model plume equations for mass and ${\chi}$ are: -% -\begin{eqnarray} -- \pardbyd{M^{\rm{P}}}{p} & = & -\lp { \varepsilon \, M^{\rm{P}} - \mu \, M^{\rm{P}} - \delta \, M^{\rm{P}} } \rp -\label{eq:dbydpmassflux} \\ -- \pardbyd{\xsubsup{ }{P} \, M^{\rm{P}}}{p} & = & \lp { -\varepsilon \, M^{\rm{P}} \, \xsubsup{ }{E} -- \mu \, M^{\rm{P}} \, \xsubsup{ }{R} - \delta \, M^{\rm{P}} \, \xsubsup{ }{P} -} \rp \label{eq:dbydpmfchi} -\end{eqnarray} - -Equations \ref{eq:eddyflux}, \ref{eq:dbydpmassflux} and -\ref{eq:dbydpmfchi} can then be substituted into \ref{eq:chibasic} to give: -% -\begin{equation} -{\pardbyd{\xsubsup{ }{E}}{t}}_{\rm{conv}} = -- M^{\rm{P}} \, \pardbyd{\xsubsup{ }{E}}{p} -+ \mu \, M^{\rm{P}} \, \lp { \xsubsup{ }{R} - \xsubsup{ }{E} } \rp -+ \delta \, M^{\rm{P}} \, \lp { \xsubsup{ }{P} - \xsubsup{ }{E} } \rp -\label{eq:chimassflux} \end{equation} -% -while \xsubsup{}{P} is obtained from the vertical gradient derived by combining -\ref{eq:dbydpmassflux} and \ref{eq:dbydpmfchi} : -% -\begin{equation} -M^{\rm{P}} \, \pardbyd{\xsubsup{ }{P}}{p} = -\varepsilon \, M^{\rm{P}} \, \lp { \xsubsup{ }{P} - \xsubsup{ }{E} } \rp - -\mu \, M^{\rm{P}} \, \lp { \xsubsup{ }{P} - \xsubsup{ }{R} } \rp -\label{eq:gradchipar} \end{equation} - -Within the model, eqn~\ref{eq:chimassflux} would take a discretized form -which actually depends upon whether the model level, k, is above or at the -lowest cloud level (k = cb). Note that the formal cloud base lies at the -half-level below, i.e. on the layer boundary which is also the top of the -turbulent mixed boundary layer. A simple discretized form of -\ref{eq:chimassflux}, setting ${ \mu = 0 }$, is: -% -\begin{eqnarray} -{\pardbyd{\xsubsup{ }{E}}{t}}_{\rm{conv, \, k}} & = & m_{\rm{k+1/2}} \, -\frac{ \lp {\xsubsup{k+1}{E} - \xsubsup{k}{E}} \rp } -{{\Delta z}_{\rm{k \, \rightarrow \, k+1}}} -+ {\delta}_{\rm{k}} \, m_{\rm{k}} \, \lp { \xsubsup{k}{P} - \xsubsup{k}{E} } \rp -\qquad \ldots \; \mbox{for k $>$ cb} \label{eq:chidisck} \\ -{\pardbyd{\xsubsup{ }{E}}{t}}_{\rm{conv, \, cb}} & = & m_{\rm{cb+1/2}} \, -\frac{ \lp {\xsubsup{cb+1}{E} - \xsubsup{cb}{E}} \rp } -{{\Delta z}_{\rm{cb \, \rightarrow \, cb+1}}} -- m_{\rm{cb}} \, -\lp { \xsubsup{i,cb}{P} - \xsubsup{cb}{E} } \rp \label{eq:chidisccb} -\end{eqnarray} -% -where the initial parcel value \xsubsup{i,cb}{P} may be chosen to produce a -fixed increment or place a closure condition on the cloud base flux. In fact, -the convection equations (see \citeumdp{027}) differ from \ref{eq:chidisck} and -\ref{eq:chidisccb} because a different discretization is used, but the -principle is unaltered. - -The model convection variables are NOT conserved under moist adiabatic processes -because precipitation processes deplete the column moisture and condensation -processes affect the temperature, specific humidity and cloud condensate -variables. Surprisingly, however, the form of eqn~\ref{eq:chimassflux} is -retained even though the basic equation \ref{eq:chibasic} acquires additional -terms for temperature and specific humidity: -% -\begin{eqnarray} -{\pardbyd{\tsubsup{ }{E}}{t}}_{\rm{conv}} = Q1 & \equiv & -\lp { \frac{L}{c_{P}} } \rp \, {\ov{Q}}_{\rm{par}} -- \frac{1}{\ov{\rho}} \, \pardbyd{\ov{\rho w^{'} \tsubsup{ }{E'}}}{z} -\label{eq:defineq1} \\ -{\pardbyd{\qsubsup{ }{E}}{t}}_{\rm{conv}} = Q2 & \equiv & - {\ov{Q}}_{\rm{par}} -- \frac{1}{\ov{\rho}} \, \pardbyd{\ov{\rho w^{'} \qsubsup{ }{E'}}}{z} -\label{eq:defineq2} -\end{eqnarray} -% -where ${\ov{Q}}_{\rm{par}}$ is the rate of condensation which occurs in the -ascending plumes. - -The reason that \ref{eq:defineq1} and \ref{eq:defineq2} retain this form -is due to cancellation from the bulk cloud terms equivalent to -\ref{eq:dbydpmfchi} which are modified in the same way as -\ref{eq:defineq1} and \ref{eq:defineq2}. The change is seen in the -vertical gradient equations based upon \ref{eq:gradchipar} -% -\begin{eqnarray} -M^{\rm{P}} \, \pardbyd{\tsubsup{ }{P}}{p} & = & -\varepsilon \, M^{\rm{P}} \, \lp { \tsubsup{ }{P} - \tsubsup{ }{E} } \rp - -\mu \, M^{\rm{P}} \, \lp { \tsubsup{ }{P} - \tsubsup{ }{R} } \rp - -\lp { \frac{L}{c_{P}} } \rp \, {\ov{Q}}_{\rm{par}} \label{eq:gradtpar} \\ -M^{\rm{P}} \, \pardbyd{\qsubsup{ }{P}}{p} & = & -\varepsilon \, M^{\rm{P}} \, \lp { \qsubsup{ }{P} - \qsubsup{ }{E} } \rp - -\mu \, M^{\rm{P}} \, \lp { \qsubsup{ }{P} - \qsubsup{ }{R} } \rp + -{\ov{Q}}_{\rm{par}} \label{eq:gradqpar} \\ -M^{\rm{P}} \, \pardbyd{\lsubsup{ }{P}}{p} & = & -\varepsilon \, M^{\rm{P}} \, \lp { \lsubsup{ }{P} - \lsubsup{ }{E} } \rp -- {\ov{Q}}_{\rm{par}} + PPN \label{eq:gradlpar} -\end{eqnarray} - -The final calculation of rates in the current condensation scheme (\citeumdp{027}, -section 10) assumes a further condensation term, ${\ov{Q}}_{\rm{reset}}$, which -acts to make the net rate of change of condensate equal zero, and a final -assumption is made that the environment values of condensate remain zero (and -also that \lsubsup{ }{R} = \lsubsup{ }{P}). The -result is basic equations -% -\begin{eqnarray} -{\pardbyd{\tsubsup{ }{E}}{t}}_{\rm{conv}} & = & Q1 - -\lp { \frac{L}{c_{P}} } \rp \, {\ov{Q}}_{\rm{reset}} -\label{eq:basictold} \\ -{\pardbyd{\qsubsup{ }{E}}{t}}_{\rm{conv}} & = & Q2 + {\ov{Q}}_{\rm{reset}} -\label{eq:basicqold} \\ -0 \equiv {\pardbyd{\lsubsup{ }{E}}{t}}_{\rm{conv}} & = & {\ov{Q}}_{\rm{par}} - -{\ov{Q}}_{\rm{reset}} - PPN -- \frac{1}{\ov{\rho}} \, \pardbyd{\ov{\rho w^{'} \lsubsup{ }{E'}}}{z} \nonumber \\ -& = & -\mu \, M^{\rm{P}} \, \lsubsup{ }{P} + \delta \, M^{\rm{P}} \, \lsubsup{ }{P} - -{\ov{Q}}_{\rm{reset}} -\label{eq:basiclold} -\end{eqnarray} - -By analogy with equations \ref{eq:defineq1} and \ref{eq:defineq2}, we can -define a $Q4$ from \ref{eq:basiclold} and state that for the control -convection scheme $Q4 = 0$. The PC2 scheme requires a reassessment of these -assumptions because we wish to allow non-zero environment condensate values and -to allow them to change. - -\subsubsection{Calculation of Grid-Box Averaged Condensate Rate (Q4)} -\label{subsect:q4calculation} - -The PC2 condensation scheme allows convection to feed cloud condensate (ice or -liquid) directly into the large scale and to update the cloud amount accordingly. - -Define -% -\begin{eqnarray} -\lp { \pardbyd{\lsubsup{l}{ }}{t} } \rp_{\rm{conv}} = Q4_{\rm{l}} & \equiv & -{\ov{Q}}_{\rm{l, par}} - {\ov{Q}}_{\rm{l, reset}} - RAIN - -\frac{1}{\ov{\rho}} \, \pardbyd{\ov{\rho w^{'} \lsubsup{l}{'}}}{z} -\label{eq:defineq4l} \\ -\lp { \pardbyd{\lsubsup{f}{ }}{t} } \rp_{\rm{conv}} = Q4_{\rm{f}} & \equiv & -{\ov{Q}}_{\rm{f, par}} - {\ov{Q}}_{\rm{f, reset}} - SNOW - -\frac{1}{\ov{\rho}} \, \pardbyd{\ov{\rho w^{'} \lsubsup{f}{'}}}{z} -\label{eq:defineq4f} -\end{eqnarray} -% -where the PC2 assumption thus far has been that ${\ov{Q}}_{\rm{l, reset}} = 0 -= {\ov{Q}}_{\rm{f, reset}}$. - -\begin{itemize} -\item{The current convection scheme assumes that parcel -condensate is single phase -(ie. either all liquid or all frozen) and this is seriously hard-wired into the -code. Thus we can treat the precipitation and parcel condensation -processes in $ Q4_{\rm{l}} $ and $ Q4_{\rm{f}} $ separately without worrying -about cross-transfer between the two because at most only one set will ever be -active in a given grid box at one time. However, even for the inactive (zero -parcel condensate) phase, convection mixes environmental air into the -parcel and -can therefore maintain a non-zero $Q4$. Enablement of multiple phase condensate -in the current scheme is a task requiring great caution as the formulations -are extremely sensitive to errors in assignment of condensate phase.} -\end{itemize} - -Based on \ref{eq:gradlpar}, the vertical dependence of condensate is -calculated as -% -\begin{eqnarray} -\pardbyd{\lsubsup{l}{P}}{p} & = & \varepsilon \, -\lp { \lsubsup{l}{P} - \lsubsup{l}{E} } \rp - -\frac{{\ov{Q}}_{\rm{l, par}}}{M^{\rm{P}}} - -\frac{RAIN}{M^{\rm{P}}} \label{eq:vertparl} \\ -\pardbyd{\lsubsup{f}{P}}{p} & = & \varepsilon \, -\lp { \lsubsup{f}{P} - \lsubsup{f}{E} } \rp - -\frac{{\ov{Q}}_{\rm{f, par}}}{M^{\rm{P}}} - -\frac{SNOW}{M^{\rm{P}}}\label{eq:vertparf} -\end{eqnarray} - -Following \citeumdp{027}, equations \ref{eq:dbydpmassflux}, \ref{eq:vertparl} and -\ref{eq:vertparf} are discretized: -% -\begin{eqnarray} -M_{\rm{k} + 1} & = & M_{\rm{k}} \, -\lp { 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp \, -\lp { 1 - \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp \, -EPSS_{\rm{k}} \label{eq:discdmfbydp} \\ -\lsubsup{l \, k + 1}{P} & = & \lp { -\lsubsup{l \, k}{P} + -\varepsilon_{\rm{k} + 1/4} \, \Delta p_{\rm{k} + 1/4} \, \lsubsup{l \, k}{E} + -\varepsilon_{\rm{k} + 3/4} \, \Delta p_{\rm{k} + 3/4} \, -\lc {1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4}} \rc \, -\lsubsup{l \, k + 1}{E} -} \rp \, / \, \lp {EPSS_{\rm{k}}} \rp \nonumber \\ -{ } & { } & + \lp { {\ov{Q}}_{\rm{l} \, \rm{k} + 1} \, / \, M_{\rm{k} + 1}} \rp -- \lp { RAIN_{\rm{k} + 1} \, / \, M_{\rm{k} + 1} } \rp -\label{eq:discvparl} \\ -\lsubsup{f \, k + 1}{P} & = & \lp { -\lsubsup{f \, k}{P} + -\varepsilon_{\rm{k} + 1/4} \, \Delta p_{\rm{k} + 1/4} \, \lsubsup{f \, k}{E} + -\varepsilon_{\rm{k} + 3/4} \, \Delta p_{\rm{k} + 3/4} \, -\lc {1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4}} \rc \, -\lsubsup{f \, k + 1}{E} -} \rp \, / \, \lp {EPSS_{\rm{k}}} \rp \nonumber \\ -{ } & { } & + \lp { {\ov{Q}}_{\rm{f} \, \rm{k} + 1} \, / \, M_{\rm{k} + 1}} \rp -- \lp { SNOW_{\rm{k} + 1} \, / \, M_{\rm{k} + 1} } \rp -\label{eq:discvparf} -\end{eqnarray} -% -where $EPSS_{\rm{k}} = -\lp {1 + \varepsilon_{\rm{k} + 3 / 4} \, \Delta p_{\rm{k} + 3 / 4}} \rp \, -\lp {1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4}} \rp $. - -The condensation and precipitation terms in equations \ref{eq:discdmfbydp}, -\ref{eq:discvparl} and \ref{eq:discvparf} make the equations implicit. -They are therefore solved by starting with an ascent in which condensation and -precipitation terms are suppressed: -% -\begin{eqnarray} -\lsubsup{l \, k + 1}{P} & = & \frac{\lp { -\lsubsup{l \, k}{P} + -\varepsilon_{\rm{k} + 1/4} \, \Delta p_{\rm{k} + 1/4} \, \lsubsup{l \, k}{E} + -\varepsilon_{\rm{k} + 3/4} \, \Delta p_{\rm{k} + 3/4} \, -\lc {1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4}} \rc \, -\lsubsup{l \, k + 1}{E} -} \rp}{EPSS_{\rm{k}}} \label{eq:discvparldry} \\ -\lsubsup{f \, k + 1}{P} & = & \frac{\lp { -\lsubsup{f \, k}{P} + -\varepsilon_{\rm{k} + 1/4} \, \Delta p_{\rm{k} + 1/4} \, \lsubsup{f \, k}{E} + -\varepsilon_{\rm{k} + 3/4} \, \Delta p_{\rm{k} + 3/4} \, -\lc {1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4}} \rc \, -\lsubsup{f \, k + 1}{E} -} \rp}{EPSS_{\rm{k}}} \label{eq:discvparfdry} -\end{eqnarray} -% -At the base of the convective plume (ie. the level immediately above cloud -base), \lsubsup{l \, k}{P} is initialized to \lsubsup{l \, i}{P} and -\lsubsup{f \, k}{P} to \lsubsup{f \, i}{P}, where the initial values are chosen -such that the modified form of \ref{eq:chidisccb} produces zero fluxes at -cloud base: -% -\begin{eqnarray} -Q4_{\rm{l}}(cb) = 0 & = & M_{\rm{cb+1/2}}^{\rm{P}} \, -\pardbyd{\lsubsup{l}{E}}{p} - M_{\rm{cb}}^{\rm{P}}\, -\lp { \lsubsup{l}{P \, i} - \lsubsup{l}{E}(\rm{cb}) } \rp \label{eq:q4lcbi} \\ -Q4_{\rm{f}}(cb) = 0 & = & M_{\rm{cb+1/2}}^{\rm{P}} \, -\pardbyd{\lsubsup{f}{E}}{p} - M_{\rm{cb}}^{\rm{P}}\, -\lp { \lsubsup{f}{P \, i} - \lsubsup{f}{E}(\rm{cb}) } \rp \label{eq:q4fcbi} -\end{eqnarray} - - -As the convection scheme makes the single phase assumption for parcel -condensate, it may be necessary to melt or freeze entrained condensate at this -point and adjust the temperature accordingly. -% -\begin{eqnarray} -\theta_{\rm{k + 1}}^{\rm{P}} = \theta_{\rm{k + 1}}^{\rm{P}} - -\lp \frac{L_{\rm{F}}}{C_{p} \, \Pi_{\rm{k + 1}}} \rp \, \lsubsup{f \, k + 1}{P} -& \; \ldots \; & \mbox{ if \lsubsup{f \, k + 1}{P} is melted } -\label{eqn:meltlf} \\ -\theta_{\rm{k + 1}}^{\rm{P}} = \theta_{\rm{k + 1}}^{\rm{P}} + -\lp \frac{L_{\rm{F}}}{C_{p} \, \Pi_{\rm{k + 1}}} \rp \, \lsubsup{l \, k + 1}{P} -& \; \ldots \; & \mbox{ if \lsubsup{l \, k + 1}{P} is frozen } -\label{eqn:freezell} -\end{eqnarray} - -Once a final value for the condensation term -$ {\ov{Q}}_{\rm{x} \, \rm{k} + 1} \, / \, M_{\rm{k} + 1} $ has been calculated -from the parcel specific humidity equations, it can then be added to the parcel -condensate to give a final pre-precipitation value. - -\begin{itemize} -\item{In practice, the rates $ {\ov{Q}}_{\rm{x} \, \rm{k} + 1}$ and -$ PPN $ are not calculated explicitly in the code. -Instead, their effect is applied directly as increments to the temperature and -moisture fields.} -\end{itemize} - -The precipitation calculation is unaltered. -% -\begin{equation} -P_{\rm{k} + 1} = \lp { \lsubsup{k + 1}{P} - \lsubsup{MIN}{P} } \rp \, -M_{\rm{k} + 1} \, / \, g -\label{eq:precip} \end{equation} -% -where \lsubsup{k + 1}{P} = \lsubsup{l \, k + 1}{P} + \lsubsup{f \, k + 1}{P}. - -\begin{itemize} -\item{Actually, given that the precipitation calculation appears to be -based upon the hydrostatic equation, it is debatable whether it is even suitable -for use with the New Dynamics model and I guess therefore that this needs -revisiting at some point.} -\end{itemize} - -This reduces the parcel condensate to : -% -\begin{eqnarray} -\lsubsup{l \, k + 1}{P} & = & \lp { -\frac{\lsubsup{l \, k + 1}{P}}{\lsubsup{k + 1}{P}} -} \rp \, \lsubsup{MIN}{P} \label{eq:vparlfinal} \\ -\lsubsup{f \, k + 1}{P} & = & \lp { -\frac{\lsubsup{f \, k + 1}{P}}{\lsubsup{k + 1}{P}} -} \rp \, \lsubsup{MIN}{P} \label{eq:vparffinal} -\end{eqnarray} - -The final parcel condensate values are then used in the rate calculation based -upon eqn~\ref{eq:basiclold}: -% -\begin{eqnarray} -Q4_{\rm{l}}(k) & = & M_{\rm{k+1/2}}^{\rm{P}} \, \pardbyd{\lsubsup{l}{E}}{p} + -\lp { {\mu}_{\rm{k}} \, M_{\rm{k}}^{\rm{P}} + -{\delta}_{\rm{k}} \, M_{\rm{k}}^{\rm{P}} } \rp \, -\lp { \lsubsup{l}{P}(\rm{k}) - \lsubsup{l}{E}(\rm{k}) } \rp - -{\ov{Q}}_{\rm{l, reset}} \label{eq:q4lmassf} \\ -Q4_{\rm{f}}(k) & = & M_{\rm{k+1/2}}^{\rm{P}} \, \pardbyd{\lsubsup{f}{E}}{p} + -\lp { {\mu}_{\rm{k}} \, M_{\rm{k}}^{\rm{P}} + -{\delta}_{\rm{k}} \, M_{\rm{k}}^{\rm{P}} } \rp \, -\lp { \lsubsup{f}{P}(\rm{k}) - \lsubsup{f}{E}(\rm{k}) } \rp - -{\ov{Q}}_{\rm{f, reset}} \label{eq:q4fmassf} -\end{eqnarray} - - -Note that, as a side-effect, the \citeumdp{027} environment equations for potential -temperature and specific humidity are also altered because the condensate is no -longer re-evaporated at the end (${\ov{Q}}_{\rm{l, reset}} = 0 -= {\ov{Q}}_{\rm{f, reset}}$): -% -\begin{eqnarray} -\frac{\Delta \, \theta_{\rm{k}}^{\rm{E}}}{\Delta \, t} = -\lp \frac{ M_{\rm{k}} }{ \Delta \, p_{\rm{k}} } \rp -\lc { -\lp { 1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4} } \rp -\lp { 1 - \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp -\lp { 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp -\lp { \theta_{\rm{k + 1}}^{\rm{E}} - \theta_{\rm{k}}^{\rm{E}} } \rp -} \right . & + & \nonumber \\ -\lp { \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp -\lp { 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp -\lp { \theta_{\rm{k}}^{\rm{R}} - \theta_{\rm{k}}^{\rm{E}} } \rp -& + & \nonumber \\ -\left . { -\lp { \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp -\lp { \theta_{\rm{k}}^{\rm{P}} - \theta_{\rm{k}}^{\rm{E}} } \rp -} \rc & { } & \label{eq:enviroth} -\end{eqnarray} -% -and -% -\begin{eqnarray} -\frac{\Delta \, q_{\rm{k}}^{\rm{E}}}{\Delta \, t} = -\lp \frac{ M_{\rm{k}} }{ \Delta \, p_{\rm{k}} } \rp -\lc { -\lp { 1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4} } \rp -\lp { 1 - \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp -\lp { 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp -\lp { q_{\rm{k + 1}}^{\rm{E}} - q_{\rm{k}}^{\rm{E}} } \rp -} \right . & + & \nonumber \\ -\lp { \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp -\lp { 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp -\lp { q_{\rm{k}}^{\rm{R}} - q_{\rm{k}}^{\rm{E}} } \rp -& + & \nonumber \\ -\left . { -\lp { \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp -\lp { q_{\rm{k}}^{\rm{P}} - q_{\rm{k}}^{\rm{E}} } \rp -} \rc & { } & \label{eq:enviroq} -\end{eqnarray} - -Similarly, eqns \ref{eq:q4lmassf} and \ref{eq:q4fmassf} have -a discretized form as follows: -% -\begin{eqnarray} -\frac{\Delta \, \lsubsup{l \, k}{E}}{\Delta \, t} = -\lp \frac{ M_{\rm{k}} }{ \Delta \, p_{\rm{k}} } \rp -\lc { -\lp { 1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4} } \rp -\lp { 1 - \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp -\lp { 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp -\lp { \lsubsup{l \, k + 1}{E} - \lsubsup{l \, k}{E} } \rp -} \right . & + & \nonumber \\ -\lp { \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp -\lp { 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp -\lp { \lsubsup{l \, k}{P} - \lsubsup{l \, k}{E} } \rp -& + & \nonumber \\ -\left . { -\lp { \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp -\lp { \lsubsup{l \, k}{P} - \lsubsup{l \, k}{E} } \rp -} \rc & { } & \label{eq:enviroll} -\end{eqnarray} -% -and -% -\begin{eqnarray} -\frac{\Delta \, \lsubsup{f \, k}{E}}{\Delta \, t} = -\lp \frac{ M_{\rm{k}} }{ \Delta \, p_{\rm{k}} } \rp -\lc { -\lp { 1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4} } \rp -\lp { 1 - \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp -\lp { 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp -\lp { \lsubsup{f \, k + 1}{E} - \lsubsup{f \, k}{E} } \rp -} \right . & + & \nonumber \\ -\lp { \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp -\lp { 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp -\lp { \lsubsup{f \, k}{P} - \lsubsup{f \, k}{E} } \rp -& + & \nonumber \\ -\left . { -\lp { \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp -\lp { \lsubsup{f \, k }{P} - \lsubsup{f \, k}{E} } \rp -} \rc & { } & \label{eq:envirolf} -\end{eqnarray} - -\subsubsection{Background condensation} -\label{sec:conv_homog} -The modification to the convective plume will result in the transport, -detrainment and entrainment of condensate, in addition to the -transport of vapour and heat. Although condensation processes within -the plume are treated, it does not treat condensation in the -environment, which is forced by the compensating subsidence. We wish -to relate the environmental increments of vapour and temperature -to a forcing that can be applied in the environment. Because we -know that any detrained air associated with detrained liquid water -from the plume must be saturated with respect to liquid water, we -are able to translate the environmental changes into forcings. - -Here we will consider that the vapour change in the gridbox is as a result of -\textit{saturated with respect to liquid water} air being injected -from the plume and background air being displaced. -We do not consider whether the background air is at saturation yet, for we wish -to derive the expression for the required condensation if this is not the case. -We consider only liquid water clouds, ice clouds have no background condensation -applied as we do not make the instantaneous condensation assumption. - -Hence we can write - -\begin{equation} -\Delta \overline{q} = \Delta C_S ( q_{sat liq}(\overline{T_{s}}) - \overline{q} ) -+ (1 - \Delta C_S) \Delta \overline{q_{background}} -\end{equation} - -where $\Delta C_S$ is the volume of plume air that is detrained into the gridbox, -as discussed by \cite{bwg03}. $T_s$ is the temperature of the air injected into -the gridbox by the plume. The first term is simply the difference -between the value of $q$ in the plume and what was previously in the gridbox, and -the second term is the effect of a background change of $q$ that will be applied -across the part of the gridbox that is not associated with the injected air. We write this as: - -\begin{equation} -(1 - \Delta C_S) \Delta \overline{q_{background}} = \Delta \overline{q} - -\Delta C_S ( q_{sat liq}(\overline{T_{s}}) - \overline{q} ) . -\label{eqn:1mcs} -\end{equation} - -Now we recognise that - -\begin{equation} -\Delta \overline{q} = Q2~ \Delta t -\label{eqn:Q2} -\end{equation} - -where $Q2$ is the rate of moistening of the whole gridbox due to convection. -Remember that, at this stage, we haven't done any condensation outside of the plume. -Hence to calculate the condensation we should apply the background change in $\overline{q}$ -as a uniform forcing for the background air. Hence (\ref{eqn:1mcs}) becomes, using -(\ref{eqn:Q2}), - -\begin{equation} -(1 - \Delta C_S) A_q |_{background} \Delta t = Q2 ~ \Delta t - \Delta C_S -( q_{sat liq}(T_{s}) - \overline{q} ) . -\label{eqn:Aq} -\end{equation} - -where $A_q |_{background}$ is the currently unknown background forcing of -$q$ (see \cite{gwb02}) and $\Delta t$ is the timestep. -We can do the same analysis for the temperature change, and obtain - -\begin{equation} -(1 - \Delta C_S) A_T |_{background} \Delta t = Q1~ \Delta t - -\Delta C_S (T_s - \overline{T} ) -\label{eqn:AT} -\end{equation} - -where Q1 is the rate of warming in the gridbox due to convection and $A_T |_{background}$ is -the currently unknown background forcing of temperature. - -The full change of liquid water content in the gridbox is that injected, -$Q4~\Delta t$, plus the amount of condensation in the background from -the uniform forcings (see -\cite{wg03}). Note that the uniform forcings are only applied across -a proportion $1 - \Delta C_S$ of the gridbox. Hence these two terms give, using the -homogeneous forcing equations (\ref{dqcldt}) and (\ref{eq:deltaqc_exp2}), - -\begin{equation} -\Delta \overline{q_{cl}} |_{convection} = Q4 \Delta t -+ (1 - \Delta C_S) a_L C_l (A_q |_{background} \Delta t -- \alpha A_T |_{background} \Delta t ). -\label{eqn:qclconv} -\end{equation} - -Using (\ref{eqn:Aq}) and (\ref{eqn:AT}) to expand the forcing terms in (\ref{eqn:qclconv}) gives - -\begin{equation} -\Delta \overline{q_{cl}} |_{convection} = Q4 \Delta t -+ \Delta t a_L C_l ( Q2 - \alpha Q1) - \Delta C_S a_L C_l -(q_{sat liq}(T_s)-\overline{q} - \alpha (T_s - \overline{T})) . -\end{equation} - -We now note that - -\begin{equation} -q_{sat} (T_s) - q_{sat liq} (\overline{T}) = \alpha (T_s - \overline{T} ) -\end{equation} - -and hence the final result - -\begin{equation} -\Delta \overline{q_{cl}} |_{convection} = Q4 \Delta t -+ \Delta t ~ a_L C_l ( ( Q2 - \alpha Q1) - \Delta C_S -(q_{sat liq}(\overline{T}) - \overline{q} ) ) . -\label{eqn:dqcl} -\end{equation} - -There is thus an extra term, $-\Delta C_S (q_{sat}(\overline{T})-\overline{q} )$, -which needs to be included in addition to the standard application of the homogeneous -forcing of $Q1$ and $Q2$ (this is represented -by the second term of the expression). This has arisen from the requirement that -the vapour injected by the plume is saturated. We need simply -to retrieve the value of $\Delta C_S$ to complete the parametrization. -This can be straightforwardly obtained -from (\ref{eq:dcldt_inhom}), which links the net change of liquid cloudy volume -due to the injection, $\Delta C_{injection}$, with $\Delta C_S$. - -\begin{equation} -\Delta C_{injection} = (g_l - C_l) \Delta C_S -\end{equation} - -where $g_l$ is 1 if the injected cloud is of liquid phase and 0 if it -is of ice phase. We already know $\Delta C_{injection}$ from the -injection forcing arguments (\ref{eq:dctdt_xl}) above that link it to $Q4$. -We therefore complete the parametrization by calculating $\Delta C_S$ based -on whether $\Delta C$ is positive or negative. If $\Delta C$ is positive, -we assume that the plume must be of liquid phase and hence - -\begin{equation} -\Delta C_S = \frac{\Delta C_{injection}} {1 -C_l} . -\label{eqn:cs1} -\end{equation} - -If $\Delta C_l$ is negative, we assume that the plume must be of ice phase and hence - -\begin{equation} -\Delta C_S = - \frac{\Delta C_{injection}} {C_l} . -\label{eqn:cs2} -\end{equation} - -Here we have still assumed that the vapour content in the detrained plume is equal to -$q_{sat liq}$. A better assumption may be to replace the $q_{sat liq}$ term in -(\ref{eqn:dqcl}) with a $q_{sat}$ expression that depends on the volume fraction -of detrained condensate that is liquid phase, $g_l$. - -If $\Delta C_{injection}$ is zero, we assume that $\Delta C_S$ is 0 also. Equations -(\ref{eqn:dqcl}),(\ref{eqn:cs1}), and (\ref{eqn:cs2}) form the parametrization -for $\Delta \overline{q_{cl}}|_{convection}$. The -representation of $\Delta C_{convection}$ is similar in form to -$\Delta \overline{q_{cl}}|_{convection}$: - -\begin{equation} -\Delta C |_{convection} = \Delta C_{injection} -+ \Delta t ~ a_L G(-Q_c) ( ( Q2 - \alpha Q1) - \Delta C_S (q_{sat}(\overline{T}) -- \overline{q} ) ) . -\end{equation} - -where the specification of $G(-Q_c)$ follows (\ref{eqn22}). Note -that the code includes the numerical limit restriction that -$\Delta C_S$ is between 0 and 1. - -Thus we are able to parametrize the net condensation and cloud changes -associated with the $Q1$ and $Q2$ terms in a physically more consistent way than -using simple homogeneous application of these terms. - -As an aside, we note that in the \cite{t93} scheme the condensation and cloud -fraction change associated with the compensating subsidence is taken out of -the convection term by adding the vertical motion associated with the compensating -subsidence to the large-scale vertical velocity before the -\cite{t93} equivalent of the homogeneous forcing term is applied. By doing -so it ensures that any balance between these two terms (as the tropical circulation -is commonly analysed to show) is removed before the net effect is calculated, -leading to more accurate numerical behaviour. - -\subsubsection{Homogeneous forcing of the environment by -convective-subsidence pressure change} - -To this end, the code includes an option to perform the homogeneous forcing -of liquid cloud by convection using the ``pressure forcing'' from the -convective subsidence, consistent with the pressure forcing by large-scale -advection (see sections \ref{sec:advec} and \ref{sec:pres}). -This approach replaces the above method of homogeneous forcing by convection -if the UM namelist switch \textbf{l\_pc2\_homog\_conv\_pressure} is turned on. -By applying the same homogeneous forcing method for advection and -convectively-forced subsidence, we should get the correct zero net -change in liquid cloud in the common situation where the large-scale ascent -and convective subsidence are in balance (implying no net vertical displacement -of environment parcels). - -Under this option, the increments to $\overline{q_{cl}}$ and $C_l$ produced -by the convection scheme are assumed to already include the effects -of entrainment, detrainment (i.e. injection) and compensating subsidence -(i.e. vertical advection) as expressed by equation \ref{eq:chimassflux}, -but exclude the effects of homogeneous forcing of clouds in the enviroment. -Note that taking equation \ref{eq:chimassflux} with $\chi$ set to -water vapour $q$, detrainment of saturated air into a subsaturated -environment will imply a positive tendency of $\overline{q}$, -but this is \textit{not} a homogeneous forcing, since the increase -in $\overline{q}$ is entirely due to injecting new parcels of saturated -air without altering the existing environment parcels. -Setting $\chi$ to be $\overline{q_{cl}}$ or $C_l$ in equation -\ref{eq:chimassflux}, there is a simply-calculated source of cloud -water and fraction wherever the detrained air is cloudy -($C_l=1$ in the detrained parcel), and we assume -these terms have been calculated this way inside the convection scheme. - -Since entrainment and detrainment do not constitute a homgeneous forcing -and are already accounted for in the convection scheme, - the only component of the convective forcing of liquid cloud -that needs to be done by the PC2 call after convection is the homogeneous -forcing by the subsidence term. This is in essence a vertical advection -(environmental forced descent by updrafts, or forced ascent by downdrafts). -The homogeneous forcing can be calculated from the expected pressure change -(and accompaying adiabatic temperature change) following the environment -as it is vertically displaced. -Conveniently, the UM already holds the convective mass-flux in units -of Pa s$^{-1}$, so it already expresses the pressure vertical velocity forced -by subsidence in the environment: - -\begin{equation} -\Delta p^E = \Delta t \left( M_{up} - M_{dwn} \right) -\label{eq:delta_p_conv} \end{equation} - -where $M_{up}$ is the updraft mass-flux, $M_{dwn}$ is the downdraft mass-flux, -and $\Delta t$ is the model timestep length. -The adiabatic temperature change following an environment parcel -subsided from pressure $p - \Delta p^E$ to $p$ is then given by: - -\begin{equation} -\Delta T^E = \theta^E \left( \left(\frac{p}{p_{ref}}\right)^\kappa - - \left(\frac{p - \Delta p^E}{p_{ref}}\right)^\kappa - \right) -\label{eq:delta_t_conv} \end{equation} - -where $\theta^E$ is the environment potential temperature, -$p_{ref}$ is the reference pressure used to define potential temperature, -and $\kappa = \frac{R_d}{c_p}$ is the ratio of the gas constant for dry air -over its heat capacity at constant pressure. -\ref{eq:delta_p_conv} and \ref{eq:delta_t_conv} are passed into the -PC2 homogeneous forcing routine after convection as the forcings -to be applied (with the forcings to all other variables set to zero). - - -\subsubsection{Convective cloud amount} -It is a debatable point whether the convective cloud fraction should -be set to zero. Although this was one of the original key concepts of -PC2, the cloud that is detrained from the convection scheme is into -the \textit{environment}, and does not represent the tower cloud. However, -it should be able to represent recently detrained cloudy air in a more -accurate way than by simply appealing to a diagnostic large-scale cloud scheme. -There are similar issues associated with the cloud fraction predicted -from the Tiedtke scheme. Probably the most consistent interpretation -is the inclusion of a tower cloud fraction within PC2, but not an -anvil cloud. However, we need to consider carefully any double -counting (or non-counting) implications. In the PC2:64 formulation, -we can represent the large optical depths associated -with new anvils, although we also tend to overestimate the optical -depth of shallow convective clouds. -Hence we choose to apply neither a diagnostic anvil or tower cloud, -so similar to Tiedtke, and let the large-scale cloud fraction represent -the convection completely. - -Strictly speaking these choices are independent of the PC2 scheme, -being simply choices that are available as part of the existing convection -scheme, but they are clearly directly related to the rest of the -cloud scheme formulation. - -\subsubsection{CAPE scaling} -The CAPE scaling option in the mass-flux convection scheme scales its -increments by the calculated values of $\frac{1}{CAPE} \frac{dCAPE}{dt}$. -This applies -also to all the PC2 calculated condensate and cloud fraction increments. -Additionally, in order to achieve reasonable mass flux profiles, -it has proved necessary to adjust the calculation of -$\frac{dCAPE}{dt}$ to use increments of -$\Delta \theta$ (potential -temperature) and $\Delta q$ calculated using a non-PC2 calculation -of these terms. Hence we consider any detrained condensate to have been -evaporated when we calculate $\frac{dCAPE}{dt}$. - -\subsubsection{Convective precipitation} -The amount of condensate detrained from convective plumes, and hence -the amount of moisture in the upper levels of the atmosphere, is very -dependent upon the amount of convective precipitation that is allowed -to fall from the column. The standard parametrization of this is that -any condensate greater than a specified value (dependent on $T$) -is precipitated, leaving the rest to be detrained. - -PC2 incorporates a tuning to this function of temperature by applying -the additional restriction that the limit may not fall to less than -$2 \times 10^{-4}~kg~kg^{-1}$. This implies a difference at temperatures -less than around $-42 ^{\circ} C$, with the tuning allowing less -precipitation and greater detrainment. This change is necessary -in order to produce thick enough anvil clouds. - -\subsubsection{Phase of condensate} -\label{sec:plume_phase} -The phase of the convective condensate \textit{carried in the plume} -is determined by a single phase change temperature TICE, with -condensate entirely in the -ice phase at colder temperatures and condensate entirely in the liquid -phase at warmer temperatures. For PC2:66, this temperature is -10 $^{\circ}$ C. - -\begin{equation} -\delta_{xl} = \left\{ \begin{array}{ll} - 1, & T_{plume} \ge -10 ^{\circ} C \\ - 0, & T_{plume} < -10 ^{\circ} C - \end{array} \right. -\end{equation} - -\begin{equation} -\delta_{xi} = \left\{ \begin{array}{ll} - 0, & T_{plume} \ge -10 ^{\circ} C \\ - 1, & T_{plume} < -10 ^{\circ} C - \end{array} \right. -\end{equation} - -\subsubsection{Tidier way of coupling convection and PC2} -\label{sec:conv-simpler} - -This area is still under development. But in brief, work is udner way to ensure that the -convective plume smoothly transitions from detraining liquid to detraining ice, rather -than using the abrupt change implied by the current formulation of the convection scheme. -Additionally, rather than using inhomogeneous increments to condensate (combining detrainment -and subsidence advection) to calculate cloud fraction increments, an alternative is to use -the detrainment of condensate to simply grow cloud fraction to ensure a specified in-cloud -liquid water content. The cloud fraction are then advected downwards byt he subsidence advection. - The increments to cloud fraction from detrainment and subsidence are then combined. - -\subsubsection{Prognostic dust approach} -A prognostic dust approach is implemented in the micro-physics scheme under -large-sale-precipitation where by the heterogeneous nucleation temperature -can be defined to vary three dimensionally globally as an arc-tangent -function of the mineral dust distribution in the model (documented -in \citeumdp{026}). By default, both liquid and ice are detrained simultaneously at the same -height, and the fraction of condensate that is ice linearly ramps as a function of temperature. -i.e. condensate is assumed to be all-liquid when T is greater than one tuneable threshold; all-ice -when T is less than another tuneable threshold, and vary linearly in-between (the threshold values are -given by starticeTkelvin and alliceTdegC in the UM cloud-scheme namelist. The new heterogeneous -nucleation temperatures calculated in the large-scale-precipitation are passed to the convection -scheme and are used as the above detrainment temperature thresholds by -maintaining a similar linear ramp. For e.g., condensate is -assumed to be all-liquid for T $\geq$ $tnuc_n$ and all-ice for T -$\leq$ $tnuc_n$ - 10.0 - -\subsubsection{Condensation adjustment in the profiles input to the -convection scheme} -\label{sec:conv_input_profs} - -The convection scheme itself is highly sensitive to the input environment -temperature and moisture profiles {\it before} the convection increments -(or PC2 response) are calculated. In particular, the parcel buoyancy -(and hence the CAPE and mass-flux scaling) maybe radically different -depending on whether a ``large-scale'' condensation / evaporation adjustment -is performed before the convection call. - -Where there is large-scale ascent, the profiles after Semi-Lagrangian advection -may have become supersaturated and unrealistically unstable, until the -expected condensation adjustment is performed. If the convection scheme -``sees'' these unrealistic intermediate profiles, it is likely to -predict an excessive, unrealistic mass-flux. - -To address this problem, there are two namelist switches that enable -additional condensation adjustments from PC2 before the convection call: - -\begin{itemize} -\item {\bf l\_pc2\_sl\_advection}: performs homogeneous forcing response -to Semi-Lagrangian advection immediately after the advection calculation, -instead of at the end of the timestep (see section \ref{sec:pres}). -\item {\bf l\_cloud\_call\_b4\_conv}: performs an additional call to -PC2 initiation (and PC2 checks) before the convection scheme -(see section \ref{sec:init2}). -This should catch any instances where large-scale ascent or other processes -have brought the profiles after advection to near or beyond saturation, -in grid-points where there was no liquid cloud already present -(and so no homogeneous forcing response). -\end{itemize} - -\subsection{Response to pressure changes} - \label{sec:pres} - -A pressure change following the parcel during the timestep will result -in an adiabatic temperature change which will force condensation, -hence we must include this temperature change forcing within PC2. -The majority of this pressure change comes from vertical advection -(although not all). -Remember that the advection (section \ref{sec:advec}), on its own, -does not cause condensation, it merely moves the existing cloud field. - - Using the semi-Lagrangian advection in the same way as is performed -for $\overline{q_{cl}}$ etc., the PC2 scheme will obtain the value of the -model prognostic \textit{Exner}, ($\prod$) on the departure points -($\prod_{dep}$). \textit{Exner} is defined as - -\begin{equation} -\label{eq:exner} -\prod = \frac{T}{\theta} = \left( \frac{p}{p_{ref}} \right)^{\kappa} -\end{equation} - -where $\theta$ is the potential temperature, $p_{ref}$ is a reference -pressure set to 1000 hPa, and $\kappa = -\frac{c_p - c_v}{c_p}$ , where $c_v$ is the heat capacity of dry -air at constant volume. The \textit{Exner} quantity is kept as a prognostic -variable in the model (this is unchanged from the control model), and the -value of $\prod$ on the departure points represents the initial value -in the timestep, since there is no update to $\prod$ until the end of -the timestep. After the second physics updates have been performed -(\textit{atmos-physics2}), the -model (including the control) recalculates the value of \textit{Exner} -($\prod^{[n+1]}$). -From $\prod_{dep}$ and $\prod^{[n+1]}$ we can calculate, using the definition -(\ref{eq:exner}), the values of departure pressure and temperature: - -\[ -\overline{p}_{dep} = p_{ref} {\prod_{dep}}^{\frac{1}{\kappa}} -\] - -\[ -\overline{T}_{dep} = \theta \prod_{dep} -\] - -Hence we obtain the net forcing values - -\begin{equation} -\Delta \overline{T} = \overline{T}^{[n+1]} - \overline{T}_{dep} -\label{eq:deltatsl} -\end{equation} - -and - -\begin{equation} -\Delta \overline{p} = \overline{p}^{[n+1]} - \overline{p}_{dep} . -\label{eq:deltapsl} -\end{equation} - -where $\overline{T}^{[n+1]}$ and $\overline{p}^{[n+1]}$ are the temperature -and pressure at the arrival point, after the dynamics call. -(\ref{eq:deltatsl}) and (\ref{eq:deltapsl}) are passed to the homogeneous -forcing routine in order to calculate -the condensation and cloud fraction changes associated with the pressure -change. - -We include this forcing towards the end of the timestep. There are two -reasons for this: -firstly, values of $\prod^{[n+1]}$ are not calculated by the control model -until after the physics is complete; secondly, it makes sense to locate this -process in the timestep in a similar location -to where the large-scale cloud scheme is included in the control (i.e. -after the implicit part of the boundary layer has finished). - -However, there -is a counter argument that says we should include this process immediately -after the dynamics, since we can then apply a forcing on an initial state -that has not already been modified by the dynamics, boundary layer and -convection schemes. This improves the numerics of the problem, since the -homogeneous forcing is designed to take time level n values as inputs. - -These issue are optionally addressed by turning on the UM namelist switch -\textbf{l\_pc2\_sl\_advection}. Under this switch, the PC2 homogeneous -forcing response to pressure change is split: -\begin{enumerate} -\item Forcing by the \textit{Lagrangian} component of pressure change, -performed immediately after the Semil-Lagrangian advection scheme -(before the call to atmos\_physics2). -This calculates the pressure change from the departure point value of -\textit{Exner} described above, to the start-of-timestep value of -\textit{Exner} at the arrival point. -\item Forcing by the \textit{Eulerian} component of pressure change, -performed at the end of the timestep (after the dynamics Helmholtz solver). -This calculates the pressure change from the start-of-timestep \textit{Exner} -at the arrival point, to the end-of-timestep \textit{Exner}. -\end{enumerate} - -Having to calculate the pressure forcing twice obviously adds some -computational cost, but has several advantages: -\begin{itemize} -\item As noted above, the PC2 homogeneous forcing calls can now take -as input the temperature and water-vapour content \textit{before} -the pressure change has been applied, as intended. This should improve -the numerical accuracy. -\item Most of the condensation or evaporation from the dynamics comes from -the \textit{Lagrangian} component of the pressure change, which has now moved -from the end of the timestep to before the dynamics Helmholtz solver. -This means that any latent heating from condensation forced by ascent is now -accounted for by the solver within the same timestep. -This improves the numerical accuracy of the dynamics-physics coupling. -\item If the condensation forced by resolved ascent is only added on at the -end of the timestep, the profiles passed into atmos\_physics2 can contain -out-of-balance thermodynamic states (e.g. if the profile has been lifted -by advection, it maybe supersaturated / unrealistically unstable before -the resulting condensation is added on). This may adversely affect the -convection scheme, which must act upon the profiles passed into -atmos\_physics2. -\end{itemize} - -The splitting of the pressure forcing call under the -\textbf{l\_pc2\_sl\_advection} switch was originally implemented to make the -profiles passed to convection more realistic. - -\subsection{Initiation} -\label{sec:init2} -As discussed in section \ref{sec:init}, there are occasions when -$\overline{q_{cl}}$ and $C_l$ need to be initiated from 0 or 1. -The application of the initiation is given in section \ref{sec:init}. -The initiation forms a new, separate block of PC2 code to perform this -calculation, and is located immediately following the pressure change -response (section \ref{sec:pres}). -Also, if the UM namelist switch {\bf l\_cloud\_call\_b4\_conv} is set to -true, an additional call to PC2 initiation is performed before the -convection scheme, to ensure that the condensation response to -advection and other forcings earlier in the timestep has been accounted for -in the profiles passed to the convection scheme, even if there was no -cloud already present for homogeneous forcing to act upon. -(see section \ref{sec:conv_input_profs}). - -There are currently 3 options for the conditions under-which initiation -may occur. For all of these options, -if using the bimodal cloud scheme to do initiation within PC2, -then the tests on $RH_T$ relative to $RH_{crit}$ are replaced by equivalent -tests for whether the saturation boundary lies within the bounds -of the bimodal scheme's assumed PDF, as described in section -\ref{sec:bimodal_init}. - -\subsubsection{``Original'' initiation logic} - -This option is selected by setting the UM namelist switch -{\bf i\_pc2\_init\_logic = 1} (Original) - -The initiation will be called if the liquid cloud fraction is either -0 or 1 and appropriate $RH$ criteria hold, along with other restrictions. -$C_l$ is initiated away from 0 if - -\begin{itemize} -\item{ $RH_T > RH_{crit} + RH_{crit \, tol}$ \textbf{and} } -\item{ Cumulus convection has {\em not} been diagnosed from the - boundary-layer in the current column \textbf{and} } -\item{ The current level is not below the surface mixed-layer LCL \textbf{and} } -\item{ $C_l = 0$ \textbf{and} } -\item{ $RH_T^{[n+1]} > RH_T^{[n]}$ ,} -\end{itemize} - -where $RH_{crit \, tol}$ is a specified tolerance parameter, of value 0.01, -and $RH_T$ is defined in (\ref{eq:rht}). $RH_T^{[n]}$ is the start of -timestep value of $RH_T$ (i.e. at time level n) and $RH_T^{[n+1]}$ is the -value when initiation is called. -Additionally, there is another possibility for the last of the relations. -This second option also allows initiation when the water -is supercooled: - -\begin{itemize} -\item{ $C_l < 0.05$ \textit{and} $\overline{T} < 0 ^{\circ} C$ .} -\end{itemize} - -Equivalently, $C_l$ is initiated away from 1 if - -\begin{itemize} -\item{ $RH_T < 2 - RH_{crit} - RH_{crit \, tol}$ \textbf{and} } -\item{ $C_l = 1$ \textbf{and}} -\item{ $RH_T^{[n+1]} < RH_T^{[n]}$ .} -\end{itemize} - -\subsubsection{``Simplified'' initiation logic} - -This option is selected by setting the UM namelist switch -{\bf i\_pc2\_init\_logic = 2} (Simplified) - -Under this option, the conditions for initiation are: - -Either: -\begin{itemize} -\item $RH_T > RH_{crit} + RH_{crit \, tol}$ \textbf{and} -\item $C_l < C_{tol}$ \textbf{and} -\item The current level is not below the surface mixed-layer LCL \textbf{and} -\item $RH_T^{[n+1]} > RH_T^{[n]}$ -\end{itemize} -Or: -\begin{itemize} -\item $RH_T < 2 - RH_{crit} - RH_{crit \, tol}$ \textbf{and} -\item $C_l > 1 - C_{tol}$ \textbf{and} -\item $RH_T^{[n+1]} < RH_T^{[n]}$ -\end{itemize} - -where $C_{tol}$ can be set via the UM namelist; its original standard value -is 0.005. Note this threshold is also used to remove small cloud-fractions -after initiation; see section \ref{sec:checks2}. - -This is very similar to the ``Original'' initiation logic described above, -but with the following differences: -\begin{itemize} -\item The condition that the boundary-layer hasn't diagnosed cumulus - convection in the column is removed. - Note that this condition spuriously suppressed initiation in the free - troposphere {\em above} any cumulus cloud produced by the convection - scheme. -\item $C_l$ only needs to be within a numerical tolerance $C_{tol}$ from - 0 or 1, rather than having to be {\it exactly} 0 or 1. -\item The different threshold when initiating super-cooled cloud is removed. -\end{itemize} - -\subsubsection{``Smooth'' initiation logic} -\label{sec:smooth_initiation} - -This option is selected by setting the UM namelist switch -{\bf i\_pc2\_init\_logic = 3} (Smooth) - -There is a fundamental numerical problem with the above options, in that -the initiation process is not permitted to have any effect at all unless -$C_l$ goes to (near) 0 or 1, but can predict values of $C_l$ very different -to 0 or 1 when it does activate. This leads to unphysical sudden noisy jumps -in $C_l$ and $q_{cl}$ when initiation occurs. -For example, if erosion causes $C_l$ to steadily decline, it will continue -to decline (even when the grid-mean $RH_T$ exceeds $RH_{crit}$) until -it reaches the threshold (0 or $C_{tol}$). At this point, initiation suddenly -increases $C_l$ and $q_{cl}$ to the values predicted by the diagnostic cloud -scheme. Erosion may then gradually remove them again, and the cycle repeats. -There is no physical reason for this internal mode of variability in the -scheme. - -Another problem arises if we consider the sensitivity to model resolution. -Suppose we have many adjacent small grid-boxes with similar $RH_T$, -a few containing cloud, the rest containing no cloud. If the whole -region cools to the point where $RH_T > RH_{crit}$, then new cloud -will initiate in the cloud-free grid-boxes, but not in the cloudy grid-boxes. -Now suppose we run a coarse-grained version of the same simulation; -the many small grid-boxes are replaced by a single grid-box containing the -average $C_l$ over the small grid-boxes. Since we now have just one -grid-box already containing partial cloud-cover, initiation of new cloud -can no longer occur anywhere. - -To address these problems, there is an option to use a much simpler / -numerically better-posed initiation method; -always allow the diagnostic cloud scheme to be called -(provided it is expected to predict nonzero cloud water, -i.e. $RH_T > RH_{crit}$ in the case of the Smith scheme). -The $q_{cl}$ predicted by the diagnostic cloud scheme is then taken -as a minimum limit applied to the prognostic $q_{cl}$. -This amounts to taking the diagnostic cloud scheme's assumed PDF as a minimum -allowed width to the actual prognostic moisture PDF. -The prognostic $C_l$ and $q_{cl}$ are incremented as follows: - -\begin{itemize} - -\item If ${q_{cl}}_{diag} > q_{cl}$: - -$\Delta q_{cl} = {q_{cl}}_{diag} - q_{cl} -\quad \refstepcounter{equation}(\theequation)\label{eq:dqcl_init}$ - - - \begin{itemize} - - \item If $Q_C < 0$: - - $\Delta C_{l} = \frac{\Delta q_{cl}}{{q_{cl}}_{diag}} - \left( {C_{l}}_{diag} - C_{l} \right) - \quad \refstepcounter{equation}(\theequation)\label{eq:dcl_init1}$ - - \item If $Q_C > 0$: - - $\Delta C_{l} = \frac{\Delta SD}{{SD}_{diag}} - \left( {C_{l}}_{diag} - C_{l} \right) - \quad \refstepcounter{equation}(\theequation)\label{eq:dcl_init2}$ - - \end{itemize} - -\item Otherwise: - -$\Delta q_{cl} = 0$ - -$\Delta C_{l} = 0$ - -\end{itemize} - -where the subscript $_{diag}$ denotes the liquid cloud water content and -fraction predicted by the diagnostic cloud scheme (either Smith or Bimodal). - -Equation \ref{eq:dcl_init1} simply sets the cloud-fraction to a weighted -mean of the pre-existing and diagnostic-scheme cloud-fractions, in proportion -to the fraction of the water content that was created by initiation -versus that which was already there. -If the pre-existing $q_{cl}$ is zero, \ref{eq:dqcl_init} and \ref{eq:dcl_init1} -simply set $q_{cl}$ and $C_l$ to their new diagnosed values, -as in the previous options. -Crucially, in the limit that the pre-existing $q_{cl}$ approaches -${q_{cl}}_{diag}$, the increments to $q_{cl}$ and $C_l$ smoothly go to zero. -This is important to make the initiation process numerically well-posed, -so that it yields a smooth, continuous solution. - -Note that when we are initiating from $C_l = 1$ instead of $C_l = 0$, -we expect the pre-existing $q_{cl}$ to be nonzero even when there is -no pre-existing sub-grid PDF width. In this case, the completely -uninitiated state will have zero saturation deficit $SD$, rather than -zero $q_{cl}$. Therefore, in this case the increment to $C_l$ is calculated -based on the fractional increase in $SD$ from initiation -(equation \ref{eq:dcl_init2}), instead of the fractional increase in $q_{cl}$. - -Whether to increment $C_l$ based on the increase in $q_{cl}$ or $SD$ is -determined based on the sign of $Q_C$, which is defined as in equation -\ref{eq:qc_eq_qt-qs} (reproduced here for clarity): - -\[ -Q_c = a_L \left( \overline{q_T} - q_{sat}(\overline{T_L}) \right) -\] - -The saturation deficit $SD$ is defined by equation \ref{SD2}: - -\[ -SD = a_L \left( q_{sat}(\overline{T}) - \overline{q} \right) -\] - -Under the reasonable approximation that $q_{sat}$ varies linearly between -$\overline{T}$ and $\overline{T_L}$, so that the values of -$\alpha$ and $a_L$ are the same in -both of these equations, and: - -\[ -q_{sat}(\overline{T_L}) = q_{sat}(\overline{T}) - \alpha \frac{L}{c_p} q_{cl} -\] - -we obtain: - -\begin{equation} -q_{cl} = Q_c + SD -\label{eq:qc_plus_sd} -\end{equation} - -It can be seen that when $Q_C > 0$ (total-water super-saturation), -it represents the value $q_{cl}$ would have if the whole grid-box -were saturated ($SD = 0$, $C_l = 1$). Note that $q_{cl}$ cannot fall below -$Q_C$, since $SD$ cannot be negative. -Since $Q_c$ is invariant under condensation / evaporation, we must have -$\Delta SD = \Delta q_{cl}$ -(hence the implementation of \ref{eq:dcl_init2} in the code simply uses -$q_{cl} - Q_c$ in place of $SD$, and $\Delta q_{cl}$ in place of $\Delta SD$). - -\subsubsection{Additional checks after PC2 initiation} - \label{sec:checks2} - -The initiation is followed immediately by a section of resetting code. -For numerical reasons, it is possible to obtain very low, but non zero, -values of $C_l$ (and equivalently values very close to, but not equal to, -1). The code will reset these clouds to either a fraction of 0 or 1, as -appropriate. We choose to apply these terms here and not in the -Bounds Checking part of the code (section \ref{sec:checks}) because -these are not required to obtain consistency between fields, but are -`tidying up' pieces of code, although they may reasonably also be -applied in the Bounds Checking. Care needs to be taken when choosing -the thresholds, since -we do not wish to reset small values that are genuinely created -by a physics scheme in the model. - -We first calculate $RH_T$ using (\ref{eq:rht}) and compare -this to the critical relative humidity, $RH_{crit}$. The liquid -cloud fraction will be reset to 1 if: -\begin{itemize} -\item{ $RH_T > 2 - RH_{crit}$ and $C_l \ge C_{high}$} -\item{ or $C_l \ge C_{high 2}$ } -\end{itemize} -where $C_{high}$ and $C_{high 2}$ are defined in \ref{eq:chigh-chigh2}. -The evaporation is done by calculating $SD$ using (\ref{SD2}) with (\ref{eq:a_L}) and -(\ref{eq:alpha_exp}) and evaporating the equivalent amount of liquid -into the gridbox to take it to saturation, according to -(\ref{eq:qsdcheck1}) below. - -Similarly, the equivalent check for low values of $RH_T$ is performed. -The liquid -cloud fraction will be reset to 0 if: -\begin{itemize} -\item{ $RH_T < RH_{crit}$ and $C_l \le C_{low}$} -\item{ or $C_l \le C_{low 2}$ .} -\end{itemize} -The remaining $\overline{q_{cl}}$ is evaporated into the gridbox -using (\ref{eq:qclcheck}) below. - -The thresholds $C_{high}$, $C_{high 2}$, $C_{low}$ and $C_{low 2}$ are -set using the parameters $C_{tol}$ and $C_{tol 2}$, according to: - -\begin{eqnarray} -C_{high} = 1 - C_{tol}, \nonumber \\ -C_{high 2} = 1 - C_{tol 2}, \nonumber \\ -C_{low} = C_{tol}, \nonumber \\ -C_{low 2} = C_{tol 2}, -\label{eq:chigh-chigh2} -\end{eqnarray} - -where the parameters $C_{tol}$ and $C_{tol 2}$ can be set via the UM namelist -variables {\bf cloud\_pc2\_tol} and {\bf cloud\_pc2\_tol\_2}. -The original standard values of these parameters are -$C_{tol} = 0.005$ and a lower value $C_{tol 2} = 0.001$. - -Investigations in SCM runs using the comorph convection scheme -(which behaves more smoothly and so typically gives smaller increments -to $C_l$ over a single timestep than other schemes which exhibit intermittent -behaviour) suggested these thresholds are too high to avoid spuriously -resetting physical values of $C_l$ to zero. Detrainment from sparse -shallow cumulus, or advection of cloud into a neighbouring grid-box -under light winds, commonly give increments which increase $C_l$ from zero -to a value less than $0.005$ in one timestep (but would eventually increase -$C_l$ to a significant value over subsequent timesteps if the checks did -not keep resetting $C_l$ to zero). - -Note that if these checks are relaxed by lowering the thresholds -$C_{tol}$ and $C_{tol 2}$ to near-zero, -similar checks are still performed independently by the bounds checking -described in section \ref{sec:checks}, but with a much lower -threshold of $C_{tol 3} = 1 \times 10^{-12}$. - -\subsection{Bounds checking} - \label{sec:checks} -Ideally, model prognostics would never become inconsistent with one another. -However, even although the mathematical solution of the governing equations -may be well behaved, due to numerical inaccuracies values may become -inconsistent. For the cloud and condensate quantities, there are a number -of consistencies that must apply. The bounds checking forms a subroutine -that will, if necessary, adjust $\overline{q}$, $\overline{q_{cl}}$, -$\overline{q_{cf}}$, $C_l$, $C_i$, $C_t$ and, for latent heating, -$\overline{T}$, to ensure consistency between these values. - -The bounds checking is performed three times during the timestep. Firstly, -after the parallel part of the physics (\textit{atmos-physics1}) is complete; -secondly, before the initiation (section \ref{sec:init}) is called; thirdly, -after the initiation is called. - -\subsubsection{} -Firstly, if $C_l > 1 - C_{tol 3}$ then $C_l$ is set to 1. -Accordingly, $C_t$ is set to 1 as well. -$C_{tol 3}$ is a tiny numerical tolerance set to $1 \times 10^{-12}$, -a value intended to be in the realm of floating point rounding error rather -than anything that represents a physical solution. - -\subsubsection{} -The second check is to reset $\overline{C_l}$ to zero. This may be performed -for two reasons. Firstly, if the amount of $\overline{q_{cl}}$ is very small -($\overline{q_{cl}} < q_{c0}$, where $q_{c0} = 1 \times 10^{-10} kg kg^{-1}$), -so we avoid carrying negligible, but non-zero values of $\overline{q_{cl}}$ -and $C_l$. Secondly, if $C_l < C_{tol 3}$ then we reasonably reset $C_l$ to zero. -$C_t$ gets reset, as it must if there is no liquid cloud, to be equal to $C_i$. - -\subsubsection{} -\label{sec:pc2_checks_sd} -The next check complements the first but updates the moisture fields. -We firstly calculate $SD$ using (\ref{SD2}) and -(\ref{eq:alpha_exp}). We then check whether $SD < 0$. -This check catches instances where we have grid-mean supersaturation, -which ought to be impossible (under the instantaneous condensation -assumption made by PC2, condensation should occur to instantly adjust -any supersaturated regions of the gridbox to saturation, so we -{\it must always} have $SD \ge 0$. -When this condition is violated, we condense water vapour to adjust to -grid-mean saturation. $-SD$ corresponds to the amount of vapour that must be -condensed to achieve this, so we have: - -\begin{eqnarray} -\overline{q} \leftarrow \overline{q} + SD \nonumber \\ -\overline{q_{cl}} \leftarrow \overline{q_{cl}} - SD \nonumber \\ -\overline{T} \leftarrow \overline{T} - \frac{L_c}{c_p} SD -\label{eq:qsdcheck1} -\end{eqnarray} - -The original version of this check on $SD$ -(which may increase $\overline{q_{cl}}$), made no accompanying changes to -liquid cloud fraction. However, increases in $\overline{q_{cl}}$ -without any increase in $C_l$ can lead to spurious high in-cloud condensate -which is then converted to rain by the microphysics at the next time-step. -There are currently 4 options for how to treat $C_l$ when increasing -$\overline{q_{cl}}$ under this saturation adjustment, selected by the -UM large-scale cloud namelist switch {\bf i\_pc2\_checks\_cld\_frac\_method}: -\begin{itemize} -\item {\bf i\_pc2\_checks\_cld\_frac\_method = 0} - -Original method; $C_l$ is left unaltered. -\item {\bf i\_pc2\_checks\_cld\_frac\_method = 1} - -Set $C_l$ and $C_t$ to 1. -\item {\bf i\_pc2\_checks\_cld\_frac\_method = 2} - -If $\overline{q_{cl}}$ and $C_l$ were already nonzero before the adjustment, -increase $C_l$ at the same fractional rate as $\overline{q_{cl}}$, so that -the in-cloud water content $\frac{\overline{q_{cl}}}{C_l}$ is conserved. -Otherwise, increase $C_l$ so-as to yield a prescribed in-cloud water -content set to 0.5 g kg$^{-1}$. $C_t$ is then increased by the same -amount as $C_l$, to maintain consistency. -\item {\bf i\_pc2\_checks\_cld\_frac\_method = 3} - -This is the same as option 2 above, except in the case where -$\overline{q_{cl}}$ or $C_l$ was zero before the adjustment. In this case, -$C_l$ is set based on an empirical power-law function of $\overline{q_{cl}}$. -\end{itemize} - -\subsubsection{} - -Next we check whether $SD > 0$, {\it and} $C_l = 1$ -(the first of our checks has ensured that $C_l$ is no greater than 1). -This check catches instances where we have total cloud-cover in a subsaturated -grid-box, which ought to be impossible (if the whole grid-box is full of liquid -cloud, then it must be at grid-mean saturation, i.e. $SD = 0$). -When this happens, we adjust $\overline{q}$ and $\overline{q_{cl}}$ -to take $SD$ to zero, -\textit{provided} that $\overline{q_{cl}} > SD$. Remember that $SD$ -corresponds to the amount of vapour that must be -\textit{evaporated} into the gridbox to give saturation, so we simply -make exactly the same adjustments as we do for removing supersaturated -states above (\ref{eq:qsdcheck1}), except that here $SD$ is positive rather -than negative. - -Our proviso that $\overline{q_{cl}} > SD$ ensures that we do not make -$\overline{q_{cl}}$ negative by this adjustment. -If $\overline{q_{cl}} < SD$, then we cannot bring the gridbox to saturation, -but it is still wrong to allow $C_l = 1$ in a subsaturated gridbox! -This was identified as a bug in the bounds-checking code, which sometimes -caused instances of $C_l = 1$ to spuriously persist in dry environments. -This behaviour is currently controlled by a temporary logical in the -{\bf temp\_fixes} namelist: -\begin{itemize} -\item If {\bf l\_pc2\_checks\_sdfix} is set to false, the code simply does -nothing when it finds instances of $C_l = 1$, $SD > 0$ and -$SD > \overline{q_{cl}}$, allowing such artefacts to persist. -\item If {\bf l\_pc2\_checks\_sdfix} is set to true, in these instances -we simply evaporate all the remaining liquid water, and reset $C_l$ to zero: -\begin{eqnarray} -\overline{q} \leftarrow \overline{q} + \overline{q_{cl}} \nonumber \\ -\overline{T} \leftarrow \overline{T} - \frac{L_c}{c_p} \overline{q_{cl}} -\nonumber \\ -\overline{q_{cl}} \leftarrow 0 \nonumber \\ -C_l \leftarrow 0\nonumber \\ -C_t \leftarrow C_i -\label{eq:qsdcheck2} -\end{eqnarray} -\end{itemize} - -\subsubsection{} -The next check is similar to above but for the $C_l = 0$ situation. - -If $\overline{q_{cl}} < q_{c0}$ or $C_l = 0$ then we evaporate the -small amount of $\overline{q_{cl}}$ that remains in the gridbox: - -\begin{eqnarray} -\overline{q} \leftarrow \overline{q} + \overline{q_{cl}} \nonumber \\ -\overline{q_{cl}} \leftarrow 0 \nonumber \\ -\overline{T} \leftarrow \overline{T} - \frac{L_c}{c_p} \overline{q_{cl}} -\label{eq:qclcheck} -\end{eqnarray} - -\subsubsection{} -Next, if $C_i > 1$ then $C_i$ is set to 1. Accordingly, $C_t$ is set to -1 as well. - -\subsubsection{} -The following check is on the ice water content, $\overline{q_{cf}}$, and -the ice fraction $C_i$. If $\overline{q_{cf}} < q_{c0}$ we simply condense some -vapour to remove the negative quantity. - -\subsubsection{} -However, instead of removing small amounts of -$\overline{q_{cf}}$ when $C_i = 0$ but $\overline{q_{cf}} > 0$, we choose instead to create -some $C_i$ to keep consistency. This is to allow small, but significant, -amounts of $\overline{q_{cf}}$ created by the microphysics scheme to -be maintained. - -\begin{equation} -C_i \leftarrow \frac { \overline{q_{cf}} }{q_{cf0}} -\label{eq:cf_reset} -\end{equation} - -where the `in-cloud' ice content $q_{cf0} = 1 \times 10^{-4} kg kg^{-1}$. - -\subsubsection{} -The next two checks are on the total cloud fraction, $C_t$, to ensure -that it takes on a value that is physically possible, given the values -of $C_l$ and $C_i$. We have, firstly, the maximum overlap situation and -then the minimum overlap situation. - -\begin{eqnarray} -C_t \leftarrow \text{Max}( C_t, C_i, C_l ) \nonumber \\ -C_t \leftarrow \text{Min}( C_t , C_l + C_i, 1) -\label{eq:ctchecks} -\end{eqnarray} - -\subsubsection{} -Finally, there is a homogeneous nucleation term applied, similar -to that in the large-scale precipitation (section \ref{sec:lsp_homo}). This is -a fast microphysics process, and must act to ensure that no liquid cloud -created by the initiation is allowed to persist in this phase if the -temperature is cold enough. Hence, if $\overline{T} < T_{homo}$ then - -\begin{eqnarray} -\overline{q_{cf}} \leftarrow \overline{q_{cf}} + \overline{q_{cl}} \nonumber \\ -\overline{q_{cl}} \leftarrow 0 \nonumber \\ -\overline{T} \leftarrow \overline{T} + \frac{L_f}{c_p} \overline{q_{cl}} \nonumber \\ -C_i \leftarrow C_t \nonumber \\ -C_l \leftarrow 0. -\label{eq:homochecks} -\end{eqnarray} - -\subsubsection{Qpos checks} -\label{sec:qpos} - -The implementation of the PC2 code includes an additional bounds check after -the \textit{atmos-physics-2} part of the model timestep has been completed. This -check is necessary to trap a rare failure, and uses the \textit{Qpos} subroutines -to check that $\overline{q_{cl}}$ is greater or equal to 0. - -During trialling prior to operational implementation, it was found that relying on Q-Pos -to deal with negative condensate values was very expensive, as the Q-Pos routine does a lot of communications between -different processors. It may be preferable to deal with the cause of negative condensate amounts at their source. -The option to ``Ensure consistent sinks of qcl and CFL'' -prevents the QCL increment from -trying to remove too much liquid condensate and hence reduces the models reliance on Q-Pos to -deal with the inconsistencies. - -\subsection{Data Assimilation} -\label{sec:da} - -The data assimilation section in the model will output assimilation increments -that represent changes to $\overline{q}$ and $\overline{T}$ which -\textit{include} the condensation contributions. We hence need to calculate -equivalent increments to $\overline{q_{cl}}$, $C_l$ and $C_t$. We assume -that the assimilation has not calculated these using a different method. -We consider the homogeneous framework and assume that there is a forcing -value of $Q_c$ that exists that will produce the known increment to -$\overline{q}$ and $\overline{T}$. - -Discritising (\ref{dqcldt}) we have, using (\ref{eq:deltaqc_exp}) and -expanding $\Delta T_L$ in terms of $\Delta T$ and $\Delta q_{cl}$, - -\begin{equation} -\Delta \overline{q_{cl}} = C_l ( a_L ( \Delta \overline{q} - -\alpha \Delta \overline{T} - \beta \Delta \overline{p}) + \Delta \overline{q_{cl}} ). -\label{eq:da1} -\end{equation} - -Remember that $Q_c$ (and hence $\Delta Q_c$) is independent of condensation. -Rearranging, we obtain - -\begin{equation} -\Delta \overline{q_{cl}} = \frac{1}{1 - C_l} C_l -a_L ( \Delta \overline{q} - \alpha \Delta \overline{T} - \beta \Delta \overline{p}) -\label{eq:da2} -\end{equation} - -and hence an expression for the condensate increment, -$\Delta \overline{q_{cl}}$, that accompanies the known increments -to $\overline{q}$ and $\overline{T}$. The similar analysis, from -(\ref{dcdt}) and (\ref{eq:da1}) gives - -\begin{equation} -\Delta C_l = \frac{1}{1 - C_l} G(-Q_c) - a_L ( \Delta \overline{q} - \alpha \Delta \overline{T} -- \beta \Delta \overline{p} ) . -\label{eq:da3} -\end{equation} - -Hence the equation set is equivalent to the use of the homogeneous -forcing set, except for the multiplier $\frac{1}{1 - C_l}$. Although this -is a clean solution, we -need to be very careful with the ill-conditioning of this solution -near $C_l = 1$. - -In practice, the ill-conditioning of (\ref{eq:da2}) and (\ref{eq:da3}) becomes too -numerically awkward for us to apply the full solution based on homogeneous -forcing, although, for completeness, we outline it in Appendix -\ref{sec:appendix-da}. Hence we have -chosen to apply a much simpler model. Here we use simply the data assimilation -increments $\Delta \overline{q}$ and $\Delta \overline{T}$ within the standard -homogeneous forcing (section \ref{sec:homog}), even though we are fully -aware that this is inconsistent (because $\Delta \overline{q}$ and $\Delta -\overline{T}$ are not forcings, but are forcings plus the condensation. -This allows us an \textit{estimate} of $\Delta \overline{q_{cl}}$ and $\Delta{C_l}$, -via the homogeneous forcing routine (and $\Delta C_t$ via the standard -updating described in section \ref{sec:ct}). These are the quantities applied -as the equivalent data assimilation increments for $\Delta \overline{q_{cl}}$, -$\Delta{C_l}$ and $\Delta C_t$. The increments $\Delta \overline{q}$ and -$\Delta \overline{T}$ remain those that the data assimilation scheme itself -calculated. - -Appendix \ref{sec:appendix-da} gives, for completeness, the alternative -numerical technique for the solution of (\ref{eq:da2}) and (\ref{eq:da3}). -However, we stress that this technique is not used within the current -PC2 formulation. - -\section{Implementation in the Unified Model} -\label{sec:um} - -This section considers the implementation of PC2 within the Unified Model -code and provides a brief guide to its use. - -In general, we have written PC2 so that the timestepping of the -cloud fraction variables within the \textit{atm\_step\_4a} -subroutine is treated as much as possible in a similar way to -the condensate variables. -Hence, wherever the condensed water variables $q_{cl}$ and $q_{cf}$ -are updated, the cloud fractions need to be updated consistently. - -\subsection{Area cloud fraction} -\label{sec:acf} - -Two area cloud fraction parametrizations are available for use with PC2. - -The area cloud fraction of Cusack (documented in \citeumdp{029}) has been -adapted by \cite{boutle_morcrette10} so it can be used with PC2 (and is available from the UMUI as the -``Cusack'' option from version 7.6 onwards). This method aims to -reproduce some of the detail of the thermodynamic -profile lost due to the coarseness of the grid. The interpolation/extrapolation technique is -used prior to PC2 initiation (which is then called with three times as many levels) -and it is used, along with the homogeneous forcing idea at the start of the timestep to -allow more cloud to be seen by radiation. - -The diagnostic area cloud fraction of \cite{bhi05} -has also been implemented in the model (available from the UMUI at version 6.4 onwards), -and this is used in PC2:64. This method diagnoses the area cloud fraction -given the volume cloud fraction, taking into account the size of the grid -box. The setting of the area cloud fraction is performed at the end of the timestep. - -\subsection{Code Structure} -\label{sec:code} - -A detailed description of the UM's timestep structure, -showing where in the model all the PC2 cloud scheme subroutine calls are made, -is given in the subsections below. - -Note that there are three different subroutines that all do -the PC2 homogeneous forcing, with slightly different details: - -\begin{itemize} -\item {\bf{\it pc2\_delta\_hom\_turb}} outputs increments due to the -condensation or evaporation, but doesn't update the fields themselves. -\item {\bf{\it pc2\_homog\_plus\_turb}} just updates the fields that -are passed in, instead of outputting separate increment arrays. -\item {\bf{\it pc2\_hom\_conv}} outputs increments but includes additional -calculations for various cloud erosion formulations. -\end{itemize} - -Note that code exists in the first two of these routines to do erosion, -but they can only do it via an input fixed rate of narrowing of the -moisture PDF (which is currently set to zero in all instances). -PC2 development has settled on a more complicated treatment of erosion, -which has only been implemented in {\it pc2\_hom\_conv}. -This can either be called after the convection scheme -(within {\it pc2\_from\_conv\_ctl}), -or before the microphysics scheme (within {\it pc2\_turbulence\_ctl}). - -Note there is also an optional call to {\it pc2\_turbulence\_ctl} -after the microphysics scheme, which is used only to estimate the -cloud fraction change consistent with the turbulent production of -liquid cloud (see section \ref{sec:turb_qcl_scheme}). - -Most PC2 code is protected by IF tests on the namelist input -{\it i\_cld\_vn} = 2 (PC2 in the GUI). -However, within the convection scheme, the code is controlled by logicals -{\it l\_calc\_dxek} (which is just set to true if using PC2, and set false -otherwise), and {\it l\_q\_interact}, which controls -whether to allow the interactive detrainment and entrainment of condensate. - -There is also a switch (currently hardwired to .false. in the code) called -{\it l\_pc2\_reset}. Turning this on (not recommended!) does 2 things: - -\begin{itemize} -\item Convective entrainment and detrainment of condensate is disabled, -by setting {\it l\_q\_interact} to false. -\item The prognostic cloud variables are overwritten by a call to -the diagnostic cloud scheme at the end of the timestep, -in subroutine {\it qt\_bal\_cld}. -NOTE: this functionality will no longer work, because inside {\it qt\_bal\_cld} -the call to the diagnostic cloud scheme is now protected by IF tests on -using either the Smith or bimodal cloud schemes. If using PC2, no cloud scheme -is called here, and required output variables are just left unset! -\end{itemize} - -The location of the various cloud scheme routine calls within the UM -is summarised in the list below. - -% The latex source input here contains a colour-coded itemize list -% of the UM subroutine tree, showing the locations of all the cloud-scheme -% routines. To edit this, open the source file source/029/um_call_tree.tex -\input{um_call_tree} - -\subsection{Diagnostics} -\label{sec:diags} - -Nearly all diagnostics retain their meaning when PC2 is run. However, there -are a few that are subtly modified. - -The convective diagnostics that use the convective cloud base and top -calculations remain the same if PC2 is used with a zeroed convective -cloud fraction. These values are not reset by the convection scheme, since -the model is still predicting convection between the diagnosed levels. - -The visibility diagnostics need modifying if the convective cloud -fraction is switched off, since they use the convective cloud fraction -within their calculation. Here we use a value of 0.2 for the convective -cloud amount if there is convective precipitation but the two-dimensional -convective cloud amount is zero. This will be the case if the PC2 -scheme has zeroed the convective cloud amount. - -There are a number of increment diagnostics that are required to -fully diagnose the moisture cycle within PC2. Since most physics -sections can cause condensation, condensate and cloud fraction increment -diagnostics have been written for each of these sections. - -\begin{itemize} -\item{ $\overline{T}$, $\overline{q}$, $\overline{q_{cl}}$, $\overline{C_t}$ and $\overline{C_l}$ increments from SW radiation, $\overline{T}$ increment from SW Radiation without including the condensation: \textbf{Section 1} .} -\item{ $\overline{T}$, $\overline{q}$, $\overline{q_{cl}}$, $\overline{C_t}$ and $\overline{C_l}$ increments from LW radiation, $\overline{T}$ increment from LW Radiation without including the condensation: \textbf{Section 2} .} -\item{ $\overline{T}$, $\overline{q}$, $\overline{q_{cl}}$, $\overline{q_{cf}}$, $\overline{C_t}$, $\overline{C_l}$ and $\overline{C_f}$ increments from Boundary Layer: \textbf{Section 3} .} -\item{ $\overline{T}$, $\overline{q}$, $\overline{q_{cl}}$, $\overline{q_{cf}}$, $\overline{C_t}$, $\overline{C_l}$ and $\overline{C_f}$ increments from Large-scale precipitation: \textbf{Section 4} .} -\item{ $\overline{T}$, $\overline{q}$, $\overline{q_{cl}}$, $\overline{q_{cf}}$, $\overline{C_t}$, $\overline{C_l}$ and $\overline{C_f}$ increments from Convection, $\overline{q}$, $\overline{q_{cl}}$, $\overline{q_{cf}}$, $\overline{C_t}$, $\overline{C_l}$ and $\overline{C_f}$ increments from the inhomogeneous part of the Convection scheme only: \textbf{Section 5} .} -\item{ $\overline{T}$, $\overline{q}$, $\overline{q_{cl}}$, $\overline{q_{cf}}$, $\overline{C_t}$, $\overline{C_l}$ and $\overline{C_f}$ increments from the Advection: \textbf{Section 12} .} -\end{itemize} - -However, there -are a number of parts of PC2 that do not fit into a pre-existing section of -code, and hence the associated increment diagnostics are not easily placed -within the UM framework. These increments were available using a -modification set or branch and a user-STASHmaster file up to version 7.5. From version 7.6 these diagnostics are available as standard. - -\begin{itemize} -\item{ $\overline{T}$, $\overline{q}$, $\overline{q_{cl}}$, $\overline{C_t}$, and $\overline{C_l}$ increments from the PC2 erosion section: \textbf{Section 4} or {\bf Section 5} depending on where the erosion is called.} -\item{ $\overline{T}$, $\overline{q}$, $\overline{q_{cl}}$, $\overline{q_{cf}}$, $\overline{C_t}$, $\overline{C_l}$ and $\overline{C_f}$ increments from the Bounds Checking after atmphya: \textbf{Section 4} .} -\item{ $\overline{T}$, $\overline{q}$, $\overline{q_{cl}}$, $\overline{q_{cf}}$, $\overline{C_t}$, $\overline{C_l}$ and $\overline{C_f}$ increments from the Initiation and Bounds checking at the end of the timestep: \textbf{Section 16} .} -\item{ $\overline{T}$, $\overline{q}$, $\overline{q_{cl}}$, $\overline{q_{cf}}$, $\overline{C_t}$, $\overline{C_l}$ and $\overline{C_f}$ increments from the Pressure Forcing section: \textbf{Section 16} .} -\end{itemize} - -\subsection{Single Column Model} - -The updating in the single column model follows the same timestepping as -that in the full model, but the changes to atm-step are mirrored -within scm\_main. The method used is to store the driving SCM forcing -increments of vapour, liquid and temperature across the forcing subroutine. -The forcing of pressure is -set to zero. After atmos\_physics2 has been called, -a PC2 section of code calls the -homogeneous forcing subroutine with these increments. This therefore -treats the response of PC2 to the prescribed dynamical forcing in the -SCM as homogeneous. Following -this calculation, the initiation scheme is called, as usual. Finally, the area -cloud fraction is set to the bulk cloud fraction and $\overline{\Theta}$ -(potential temperature) -is made consistent with $\overline{T}$ (dry-bulb temperature) which was -changed by the condensation in the PC2 response to the homogeneous forcing. -The rest of the SCM uses the same PC2 code as the full model. - -Note that the change to PC2 homogeneous forcing from advection under the UM -namelist switch \textbf{l\_pc2\_sl\_advection} (see section \ref{sec:pres}) -is also mirrored in the Single-Column Model. If this switch is turned on, -the PC2 homogeneous forcing call using the SCM forcing increments is moved -straight after the call to the forcing routine, so that the condensation -adjustment is performed before the call to atmos\_physics2. -If \textbf{l\_pc2\_sl\_advection} is turned on, the PC2 response to SCM -forcings is also improved as follows... - -The SCM forcings may comprise one or both of the following: -\begin{itemize} -\item (a) Prescribed tendencies or relaxation applied to T,q. -\item (b) Interactive vertical advection applied to T,q. -\end{itemize} -For the latter, we can calculate the pressure change experienced by -vertically-advected parcels, and so calculate the PC2 homogeneous forcing -response in the same way as we do for Semi-Lagrangian advection in the -full model (see section \ref{sec:pres}). -For the former, we don't know if the prescribed T,q tendencies are due to -advection, radiation, or some other process, so we calculate the PC2 -homogeneous forcing response as if the tendencies are applied "in-situ". - -To split the PC2 homogeneous response into these 2 components, the SCM -forcing routine outputs: -\begin{itemize} -\item (a) The forcing increments to T,q excluding the contribution from -interactive vertical advection. -\item (b) The value of exner pressure at departure points, consistent with -the vertical advection. -\end{itemize} -The PC2 homogeneous forcing responses to these 2 forcing components -are then calculated by 2 separate PC2 calls in scm\_main. - -\subsection{Limited Area Boundary Conditions} - -Cloud fractions on the limited area boundaries are fully updateable. -Writing of cloud fraction Limited Area Boundary Conditions (LBCs) -will be automatic if PC2 is selected. -A PC2 LAM may be run from an LBC file with or without cloud fraction LBCs -(this is specified by the logical l-pc2-lbc, which is set in the UMUI). -If there are no cloud fraction lbcs then around the edge of the domain the -checking and initiation routines will be applying significant increments -to the cloud and condensate fields near the boundaries, but this does -not have an adverse effect well away from the boundaries. If there are no -cloud fraction LBCs the cloud fraction fields themselves are not forced to -zero around the edge of the domain but are allowed to freely find their -own value. A PC2 run that outputs lbcs will, by default, always -output cloud fractions as part of the LBCs file. - -\subsection{Parameter values} - -Table \ref{tab:pc2_names} summarizes the values of parameters used in the PC2 -scheme and their location within various comdecks. Those parameters marked -as `Num' are those that are not part of the mathematical equation -set that is being solved, but are required in order to achieve a stable, -realistic, numerical solution. These include, for instance, thresholds for -resetting cloud fractions back to 0 or 1. Those marked 'Phy' are physical -quantities that form an integral part of the equation set that we wish to solve. -Those marked 'Clo' form part of a closure needed to form the equation -set, but are less readily related to physical quantities. -Variables marked 'Diag' form a part of the diagnostic output routines. -\begin{table}[ht] -\begin{center} -\footnotesize -\begin{tabular}{llllll} -\hline -Symbol & Code variable & Description & Value & Location & Notes and ref. \\ \hline -- & init-iterations & Number of iterations in initiation & 10 & pc2-const & Num: \ref{sec:numapp_init} \\ -$C_{tol}$ & cloud-pc2-tol & Bounds checking $C_l$ threshold & 0.005 & UM namelist & Num: \ref{sec:init2} \\ -$C_{tol 2}$ & cloud-pc2-tol-2 & Bounds checking $C_l$ threshold & 0.001 & UM namelist & Num: \ref{sec:init2} \\ -$RH_{tol}$ & rhcrit-tol & $RH_{crit}$ tolerance in initiation & 0.01 & pc2-const & Num: \ref{sec:init2} \\ -$q_{cf0 \, BL}$ & ls-bl0 & Fixed value of BL in-plume $\overline{q_{cf}}$ & $1.0 \times 10^{-4} \, kg \, kg^{-1}$ & imp-ctl & Clo: \ref{sec:bl} \\ -$q_{cf0}$ & one-over-qcf & Fixed in-cloud $\overline{q_{cf}}$ if $C_f$=0 & $1.0 \times 10^{-4} \, kg \, kg^{-1}$ & pc2-chck & Num: \ref{sec:checks} \\ -$m$ & pdf-merge-power & Merging power for $G(-Q_c)$ & 0.5 & pc2-const & Clo: \ref{sec:homog} \\ -$n$ & pdf-power & Shape parameter for $G(-Q_c)$ & 0.0 & pc2-const & Phy: \ref{sec:homog} \\ -$w$ & wind-shear-factor & Wind shear in fallout of ice term & $1.5 \times 10^{-4} \, s^{-1}$ & pc2-const & Phy: \ref{sec:lsp_fall} \\ -$i$ & ice-width & Scaling factor for reduction in $b_i$ & 0.04 & pc2-const & Phy: \ref{sec:mp_depsub} \\ -$a$ & dbsdtbs-turb-0 & Rate of reduction of PDF width & $-2.25 \times 10^{-5} \, s^{-1}$ & UM namelist & Phy: \ref{sec:width} \\ -$b$ & dbsdtbs-turb-1 & Rate of reduction of PDF width & 0 & pc2-const & Phy: \ref{sec:width} \\ - & dbsdtbs-conv & Redn of PDF width in convection & 0 & pc2-const & Phy: \ref{sec:width} \\ - & dbsdtbs-exp & Variation of erosion on RH & 10.05 & pc2-const & Phy: \ref{sec:width} \\ -$RH_{crit}$ & RHCRIT & Critical RH for cloud formation & & UM namelist & Phy: \ref{sec:init}, \ref{sec:mp_depsub} \\ -$q_{c0}$ & condensate-limit& Minimum allowed condensate & $1 \times 10^{-10} \, kg \, kg^{-1}$ & pc2-chck & Num: \ref{sec:checks} \\ -$q_c^{S0}$ & ls0 & Lower limit of plume condensate & $5 \times 10^{-5} \, kg \, kg^{-1} $ & enviro?a & Num: \ref{sec:multi_numapp} \\ - & \textit{Hard-wired} & Conv cloud fraction for visibility& 0.2 & imp-ctl2 & Diag: \ref{sec:diags} \\ - & \textit{Hard-wired} & Limit on width of ice distribution& 0.001 & lspice3d & Num: \ref{sec:mp_depsub} \\ - & \textit{Hard-wired} & $C_l$ limit for init if $T < 0 ^{\circ} C$ & 0.05 & pc2-init & Num: \ref{sec:init2} \\ - & \textit{Hard-wired} & Tolerance on calc. of $q_C^s$ in BL & $1.0 \times 10^{-10} \, kg \, kg^{-1}$ & imp-ctl & Num: \ref{sec:bl} \\ -\hline -\end{tabular} -\end{center} -\caption{PC2 parameter values and locations } -\label{tab:pc2_names} -\end{table} - -PC2 also recommends some tunings of the existing convection -scheme parameters. These cannot be placed in the library code, since they -would interact with non-PC2 simulations, hence would need to be specified -with modification sets. We have included those parameters that have been -investigated throughout testing, although only two are different between -PC2:64 and a non-PC2 run. - -\begin{table}[ht] -\begin{center} -\tiny -\begin{tabular}{llllll} -\hline -Code variable & Description & Value in PC2 & Value in Control & Location & Notes and reference \\ \hline -TICE & Temperature at which plume freezes & $-10 ^{\circ} C$ & $0 ^{\circ} C$* & tice.cdk or UMUI & Phy: \ref{sec:convec} \\ -QSTICE & Approximate qsat(TICE) & $3.5 \times 10^{-3}$ & $3.5 \times 10^{-3}$ & qstice.cdk or UMUI & Phy: \ref{sec:convec} \\ -\textit{Hard-wired} & Limit on conv. cond. after precip & 0.5 $q_{sat}, 2 \times 10^{-4}$ & $0.5 \, q_{sat}$ & cloudw & Phy: \ref{sec:convec} \\ -Anvil factor & Shape parameter for conv. cloud anvil & 0 & 0.3* & UMUI & Phy: \ref{sec:convec} \\ -Tower factor & Shape parameter for conv. cloud tower & 0 & 0.25* & UMUI & Phy: \ref{sec:convec} \\ -\hline -\end{tabular} -\end{center} -\caption{PC2 parameter values and locations relating to the convection. *These values are those used in HadGAM} -\label{tab:pc2_conv_names} -\end{table} - -\subsection{How to run the PC2 scheme} -Running PC2 is straightforward, but you should seek advice as to -modification sets that you need to include to ensure you are -running the most up-to-date version of PC2. -The following is a brief checklist of the options in the UMUI which need -to be selected in order to run PC2. No hand-edits are required. -\begin{itemize} -\item{In the LS cloud panel (atmos-science-section-LScloud) push the button marked 'use the PC2 cloud scheme'.} -\item{If you wish to use PC2 in the diagnostic only mode, also push 'run the PC2 scheme in diagnostic only mode'. If you wish to run PC2 fully then do not push this button} -\item{In the large-scale precipitation section (atmos-science-section-LSprecip) select the 3D large-scale precipitation scheme.} -\item{The specification of the LA boundary conditions can be set in the atmos-InFiles-OtherAncil-LBC panel.} -\item{You will need to select modsets to include update the library code to the PC2 version described here. Seek advice on these.} -\item{You may wish to adjust the convective anvil parameters in atmos-science-section-convec. Again, seek advice.} -\end{itemize} - -\subsection{More information} - -Information on results of the scheme and how to run the PC2 code at -various model versions is available on the PC2 web site. - -\section{Appendix: Alternative PC2 - Data Assimilation formulations} -\label{sec:appendix-da} - -In this alternative method to section \ref{sec:da} we will assume that there -exists a homogeneous forcing, $\Delta Q_c$, -that gives changes, net of condensation, of $\Delta\overline{q}$ and -$\Delta\overline{T}$. If we can recover -what $\Delta Q_c$ is then we can use this to calculate the liquid, -$\overline{q_{cl}}$, and liquid cloud fraction, $C_l$, increments. - -As in section \ref{sec:da}, we start by discretising (\ref{dqcldt}) to give - -\begin{equation} -\Delta \overline{q_{cl}} = C_l \Delta Q_c -\label{eq:dqcldt_discrete} -\end{equation} - -and hence, using the discrete form of $\Delta Q_c$ from -(\ref{eq:deltaqc_exp2}) gives - -\begin{equation} -\Delta \overline{q_{cl}} = C_l ( a_L ( \Delta \overline{q} - \alpha \Delta -\overline{T} ) + \Delta \overline{q_{cl}} ) , -\end{equation} - -which rearranges to - -\begin{equation} -\Delta \overline{q_{cl}} = \frac{1}{1-C_l} C_l a_L ( \Delta \overline{q} -- \alpha \Delta \overline{T} - \beta \Delta \overline{p}) . -\label{eqn:delataqcl} -\end{equation} - -Comparing to (\ref{eq:deltaqc_exp2}) and (\ref{eq:dqcldt_discrete}) we see that -$\Delta \overline{q_{cl}} $ is the same as if we had applied the -homogeneous forcing technique -using $\Delta \overline{q}$, $\Delta \overline{T}$ and $\Delta \overline{p}$ as -forcings, except multiplied by a factor of $\frac{1}{1-C_l}$. - -We can calculate $\Delta C$ in a similar way. From (\ref{eq:deltac}) - -\begin{equation} -\Delta C_l = G(-Q_c) \Delta Q_c -\end{equation} - -and hence, using our value of $\Delta Q_c$ from (\ref{eq:deltaqc_exp2}) -and $\Delta \overline{q_{cl}}$ from (\ref{eqn:delataqcl}) - -\begin{equation} -\Delta C_l = G(-Q_c) (a_L ( \Delta \overline{q} - \alpha \Delta \overline{T} ) -+ \frac{1}{1-C_l} C_l a_L ( \Delta \overline{q} - \alpha \Delta \overline{T} ) ) -\end{equation} - -which rearranges to - -\begin{equation} -\Delta C_l = \frac{1}{1-C_l} G(-Q_c) a_L ( \Delta \overline{q} -- \alpha \Delta \overline{T} -\beta \Delta \overline{p}) . -\label{eqn:c1mc} -\end{equation} - -This is also a factor of $\frac{1}{1-C_l}$ different from using -$\Delta \overline{q}$, -$\Delta \overline{T}$ and $\Delta \overline{p}$ -directly as forcings (the factor must be the same, as we are still -using the homogeneous forcing hypothesis). This equation forms the basis -for the more advanced technique discussed in this section. However, -it is undefined at $C_l=1$ and becomes ill-conditioned near $C_l=1$, -hence there must be care taken when this expression is solved numerically. - -\subsection{Numerical solution} - -The timestepping applied is picked as a result of numerical tests -forcing a single gridbox with uniform increments. Many numerical techniques -were tested, this gives a fast but reasonably well behaved solution. - -Initially, we calculate $G(-Qc)$ and $\Delta Q_c$ from the input fields, -as in the homogeneous forcing -technique (section \ref{sec:homog}) and (\ref{eq:deltaqc_exp2}). - -An initial increment, $\Delta C_l^1$ is estimated directly using the -basic equation - -\begin{equation} -\Delta C_l^1 = \frac{1}{1-C_l^(n)} G(-Q_c) \Delta Q_c . -\end{equation} - -We then recalculate this expression, using a mid-timestep estimate -for $\Delta C_l$; - -\begin{equation} -\Delta C_l = \frac{1}{1-(C_l^{[n]} + \frac{1}{2} \Delta C_l^1)} -G(-Q_c) \Delta Q_c -\end{equation} - -where the term $C_l^{[n]} + \frac{1}{2} \Delta C_l^1$ is limited to be no more -than 0.9999 to avoid divide by zero problems. The final, updated value of -cloud fraction, $C_l^{[n+1]}$, is then - -\begin{equation} -C_l^{[n+1]} = C_l^{[n]} + \Delta C_l -\end{equation} - -and this value is limited to 0 or 1. - -The liquid water term simply uses the final version of $C_l$ in its -calculation. - -\begin{equation} -\Delta \overline{q_{cl}} = \frac{1}{1-C_l^{[n+1]}} C_l^{[n+1]} \Delta Q_c -\end{equation} - -and will be set to 0 if $C^{[n+1]}$ is 0. There is an additional limit, -see below, applied to the liquid -water term, which will prevent the value of $\Delta \overline{q_{cl}}$ -increasing to a large number if $C_l^{[n+1]}$ is very close to 1. - -\subsection{Limit on the liquid water content} - -We will choose a limit on $\overline{q_{cl}}$ to be equal to its value when -the underlying PDF just corresponds to total cloud cover. Therefore, from -(\ref{eq:qclbar=int}) - -\begin{equation} -\overline{q_{cl \, max}} = \int_{s=-b_s}^{\infty} G(s) (b_s + s) ds . -\end{equation} - -We will use the current value of $Q_c$ (which won't in general to be equal to -$b_s$) to split the integral into two ranges of s: - -\begin{equation} -\overline{q_{cl \, max}} = \int_{s=-b_s}^{-Q_c} G(s) (b_s + s) ds -+ \int_{s=-Q_c}^{\infty} G(s) (b_s + s) ds . -\end{equation} - -For the moment we write the first of these integrals as $I1$, and split the -second integral whilst introducing a $(+ Q_c - Q_c)$ term to the integrand: - -\begin{equation} -\overline{q_{cl \, max}} = I1 + \int_{s=-Q_c}^{\infty} G(s) (b_s - Q_c) ds -+ \int_{s=-Q_c}^{\infty} G(s) (s + Q_c) ds . -\end{equation} - -The last of the integrals is now the current liquid water content, -$\overline{q_{cl}}$. -The second integral is proportional to the liquid cloud fraction $C_l$. - -\begin{equation} -\overline{q_{cl \, max}} = I1 + C_l (b_s - Q_c) + \overline{q_{cl}} -\end{equation} - -or - -\begin{equation} -\overline{\Delta q_{cl \, max}} = I1 + C_l (b_s - Q_c) . -\label{eqn:deltaqclmax} -\end{equation} - -Now consider the expression for the saturation deficit, which we have -defined, from (\ref{SD}) as - -\begin{equation} -SD = \int_{-b_s}^{-Q_c} G(s) (-Q_c - s) ds . -\end{equation} - -Splitting and adding the term $(+b_s - b_s)$ to the integrand in a -similar way to above gives - -\begin{eqnarray} -SD = \int_{-b_s}^{-Q_c} G(s) (-Q_c + b_s) ds + \int_{-b_s}^{-Q_c} -G(s) (-s - b_s) ds \nonumber \\ -= (-Q_c + b_s) (1 - C_l) - I1 , -\end{eqnarray} - -and hence $I1$ in terms of $SD$. Using this value of $I1$ in -(\ref{eqn:deltaqclmax}) and cancelling the $C_l$ terms gives -$\Delta \overline{q_{cl \, max}}$ as - -\begin{equation} -\Delta \overline{q_{cl \, max}} = (-Q_c + b_s) - SD . -\label{eqn:delta2} -\end{equation} - -This is a general expression, it is not fixed for a particular PDF. To -complete the analysis, we need to estimate $-Q_c+b_s$. To do this, we now -make the \textit{assumption} of a power-law type PDF, as in section -\ref{sec:init}. If we start from the equivalent of -(\ref{eqn19}) but at the $s=-bs$ end of the distribution, equation (B.3) -in \cite{wg03} can be equivalently written for $(1-C_l)$ as: - -\begin{equation} -(1-C_l) = \frac{ A (-Q_c + b_s)^{n+1} }{n+1} . -\label{eqn:1mc} -\end{equation} - -To derive this from (B.3) note that $C_l$ is swapped for $1-C_l$ and -$(b_s - (-Q_c))$ is swapped for $(-Qc - (-b_s))$, as in section -\ref{sec:numapp_init}. Similarly, noting that $\overline{q_{cl}}$ can be -swapped with $SD$, gives the equivalent to (B.4) in \cite{wg03} -as - -\begin{equation} -SD = \frac{ A (-Q_c + b_s)^{n+2} }{(n+1)(n+2)}. -\label{eqn:sd} -\end{equation} - -Using the value $(1-C_l)$ from (\ref{eqn:1mc}) in (\ref{eqn:sd}) gives - -\begin{equation} -\frac{SD}{1-C_l} = \frac {-Q_c + b_s}{n+2} . -\end{equation} - -Finally, we use this expression for $(-Q_c + b_s)$ in (\ref{eqn:delta2}) to -parametrize $\Delta \overline{q_{cl \, max}}$ in terms -of the saturation deficit - -\begin{equation} -\Delta \overline{q_{cl \, max}} = SD ( \frac{n+2}{1-C_l} - 1 ) . -\label{eqn:sdr1mc} -\end{equation} - -This is the expression that is used for the limit on $\overline{q_{cl}}$. -We subsequently apply a second limit, since numerically this expression is -still not well behaved when $C_l$ is close to 1. Here we note that just at -complete cloud cover for a symmetric PDF we have -$\overline{q_{cl}} = b_s$. Hence we estimate $b_s$ as in \cite{smith90}, - -\begin{equation} -b_s = a_L ( 1 - RH_{crit} ) q_{sat}(\overline{T_L}) , -\label{eqn:bs} -\end{equation} - -and take the smaller value for -of (\ref{eqn:sdr1mc}) and (\ref{eqn:bs}) for $\Delta \overline{q_{cl \, max}}$. - - -\subsubsection{Initiation from $C_l=1$} -The equations are not defined when $C_l=1$. (Note that when $C_l=0$ we -will calculate $G(-Q_c)=0$ so there is no change in cloud fraction or liquid water -content in this case). The assimilation is capable of lowering $\overline{q}$ -below $q_{sat}(\overline{T})$ and hence there should be a corresponding -change in cloud fraction and liquid water content. In theory, we can use -the expression for $\Delta \overline{q_{cl \, max}}$ and assume that the -initial liquid water is equal to $b_s$. However, this produces -a tricky set of simulataneous equations, which are not easily solvable -except in the case where $n=0$. We proceed by making this assumption for $n$, -acknowledging that this is not necessarily entirely consistent with the -rest of the model (although it is in the PC2:64 formulation). - -We have (equivalent to B.6 from \cite{wg03}) - -\begin{equation} -\frac{ (1-C_l)^2 }{SD} = G(-Q_c) \frac{n+2}{n+1}. -\end{equation} - -If n=0 (i.e. a `top-hat' function) then $G(-Q_c) = \frac{1}{2 b_s}$ -and we can write - -\begin{equation} -C_l = 1 - \sqrt{ \frac{SD}{b_s} } . -\end{equation} - -We now assume $b_s$ is equal to our current value of $\overline{q_{cl}}$ -and hence - -\begin{equation} -C_l^{[n+1]} = 1 - \sqrt{ \frac{SD^{[n+1]}}{\overline{q_{cl}^{[n]}}} } -\label{eqn:1msqrt} -\end{equation} - -where $C_l^{[n+1]}$ and $SD^{[n+1]}$ are the values of $C_l$ and $SD$ after -this initiation has been applied. -Using our previous expression (\ref{eqn:sdr1mc}) for -$\Delta \overline{q_{cl max}}$ gives (remembering that we are considering -the reverse process, so the sign is opposite), - -\begin{equation} -\Delta \overline{q_{cl}} = - SD^{[n+1]} ( \frac{2}{1-C_l^{[n+1]}} - 1 ) -\end{equation} - -(remembering that $n=0$ is assumed). Hence, replacing $C_l^{[n+1]}$ by -(\ref{eqn:1msqrt}) we have - -\begin{equation} -\Delta \overline{q_{cl}} = SD^{[n+1]} - 2 \sqrt{ SD^{[n+1]} -\overline{q_{cl}}^{[n]} } . -\end{equation} - -This is the expression we use, $SD^{[n+1]}$ is calculated after the -ssimilation increments have been applied, using (\ref{SD2}): - -\begin{equation} -SD^{(n+1)} = a_L^{[n+1]} ( q_{sat}(\overline{T}^{[n+1]}, -\overline{p}^{[n+1]}) - \overline{q}^{[n+1]} ). -\end{equation} - -\subsection{Results} - -Results demonstrate a problem in that there is a distinct asymmetry -between changes when $\Delta Q_c$ is large and positive and when -$\Delta Q_c$ is large and negative, when -cloud fractions start near 1. In the former case, the limit to the amount of -liquid and cloud fraction that can be created means that changes must be -kept relatively small, whereas in the latter case, all the cloud and -liquid water can be removed easily. (The $1/(1-C_l)$ term allows this -to be done relatively quickly). Hence this assimilation -method has a net tendency to remove cloud from the simulation, which, -at the moment, gives poorer results than simply using the homogeneous -forcing method. - -Further work will be required to enable the implementation of -this $\overline{q}$ -and $\overline{T}$ preserving method. - -\section{Appendix: Essentials of PC2 for code developers} -\label{sec:code-development} -This section provides some guidance to code developers on the treatment -of PC2. Code developers are advised to read the relevant part of section -\ref{sec:app_um} to understand the way in which the current PC2 scheme -interacts with their section of code. - -The essence of a prognostic cloud scheme is that each physical part of the -model is able to calculate increments to the cloud fractions and condensate -contents. These form an integral part of each physics scheme and should be -considered by code owners as such, hence any alteration to a scheme -\textit{must} consider also the impact on $q_{cl}$, $q_{cf}$, $C_t$, -$C_l$ and $C_f$, as -well as on the more traditional $T$, $q$ and wind prognostics. Often -there should be no impact, but this cannot be assumed without consideration. -There is no diagnostic cloud fraction and condensation scheme which can be -run in PC2, since this would reset any effect of the cloud prognostics used -elsewhere in the model. (The diagnostic scheme can still be used for model -\textit{diagnostics}, such as visibility and fog fraction, and will be -kept in later versions of the UM). - -Since this places a significant burden on code developers, the PC2 -developers have produced two generic representations which can take increments -to $q$ and $T$ etc. and produce an estimate of the condensation and -cloud fraction changes associated with the increments. These are the -homogeneous forcing and injection forcing (or inhomogeneous forcing) -methods. - -\subsection{Homogeneous forcing} -This is described fully in section \ref{sec:homog}. This assumes that the -distribution of $q_T - q_{sat}(T_L)$ about its gridbox mean is unchanged when -a process acts. (The mean will change of course, but we assume that the -variations in each part of the gridbox from the mean do not). Since this -is equivalent to every part of the gridbox receiving the same $q_T$ and $T_L$ -increment, we call this `Homogeneous Forcing'. We have provided a subroutine -\textit{pc2-homog-plus-turb}, in deck \textit{pc2-homo} in order to -provide the necessary updates. - -\subsection{Injection forcing} -This is described fully in section \ref{sec:inhomog}. We assume that -we already know a condensate increment $q_{cl}$ or $q_{cf}$ and that a -corresponding cloud fraction increment $C_l$ or $C_f$ (and $C_t$) remains -to be estimated. The injection forcing assumes that new cloud randomly -displaces existing cloud in a gridbox, and is designed with detrainment -from deep convection in mind, although it is also used elsewhere. It will -require as an input an estimate of the `in-cloud' water content of -the new cloud that is produced. - -If you consider that both the homogeneous and injection forcing representations -are both poor assumptions for your scheme, you will need to provide -another method for calculating the condensation and cloud fraction changes. -The PC2 team can advise, but you should not expect them to do the work. -You can, of course, replace existing homogeneous and inhomogeneous forcing -calls with new representations of changes to the prognostics if you think -you have improved representations available. This is part of the -development of any prognostic variable representation. - -\subsection{Do I need to modify anything when I change a parametrization scheme?} - -Here we assume that you wish to do the minimum work possible to get -PC2 to work, rather than a full reconsideration of the physics of the PC2 -increment terms. - -If your scheme is currently using the homogeneous forcing -then there is no need to update the cloud part of the scheme, -\textit{provided that -you do not alter values of $T$ and $q$ after the homogeneous forcing -section is called} and that the physical interpretation of your $q$ and $T$ -increments does not change. You need to be careful if you are moving code from -one subroutine to another that you don't inadvertently do this, although -the forcing usually sits at the end of the control subroutine. - -If your scheme is currently using the injection forcing \textit{subroutine}, -which necessitates that condensate -increments are already calculated by the scheme, then there is also no need -to update the cloud part of the scheme. This currently applies to the boundary -layer, where $q_{cf}$ is altered by tracer mixing. Like for the -homogeneous schemes, this -is provided that you \textit{do not alter $T$, $q$ or condensate values after -the injection forcing subroutine is called} and that the physical -interpretation of your $q$ and $T$ increments does not change. - -Changes to winds do \textit{not} need to have a condensation or -cloud fraction increment -associated with them. There may be future scope for developing an -orographic cloud representation (probably diagnostic), but this is not -an essential part of the scheme as it stands. - -If your scheme uses hardwired assumptions about what is happening e.g. -convection or microphysics, then you \textit{do} need to be careful that -$T$, $q$ and condensates -are still calculated correctly after you have performed your changes. -Currently there are many PC2 assumptions hard-wired into the mass-flux -convection scheme: -\begin{itemize} -\item{Any change to the scientific basis by which changes to $T$, $q$, $q_{cl}$ and $q_{cf}$ are calculated requires careful consideration} -\item{Simple changes to convective parameters, such as detrainment rates, should not require a change to the PC2 code} -\item{Be particularly careful if you move code around, \textit{especially the calculation of convective cloud fractions}, since PC2 incorporates a set-to-zero in the code. This will need to be replicated or there is a risk that the diagnostic cloud fraction is no longer set to zero correctly by PC2.} -\end{itemize} -Each microphysics transfer term has been considered individually for PC2 and this -should remain the case. - -Be especially careful when you do anything in the atmphy and atmstep levels of -the code that includes additional changes $T$, $q$, $q_{cl}$ or $q_{cf}$, since -they may need cloud fraction or condensation changes to go along with them. - -In summary, changes to existing increments of $T$, $q$ etc. within the current -UM structure are unlikely to -necessitate a modification for PC2 if their physical interpretation has not -changed. However, new methods of generating $T$ and -$q$ increments will require new code to be added for PC2. - -\subsection{Further PC2 development work} -There are a number of areas in which the PC2:66 formulation can be -developed further, and many of these have been mentioned in the documentation -above. Some -of these are simple sensitivity studies which have not been fully explored in -development, others are more complex alterations. It is fair to say -that the link to the convection has proved the most problematic issue -so far with PC2 development. - -\subsubsection{PC2 cloud erosion} -The cloud erosion is a critical term for the simulation of shallow convective -cloud. A large amount of erosion is required to keep the cloud fractions relatively -low in shallow convection, which is why we have linked the erosion to the relative -humidity. We recognise, however, that this is more an empirical choice than a -physically informed choice. In particular, a low relative humidity (e.g. in the -stratosphere) would imply a very high erosion rate - although the net effect -is to remove any cloud, which is a reasonable thing to do, there is an implication of -the parametrization that mixing within the stratosphere is high, which is -clearly incorrect. We have also seen relatively low cloud fractions in the -mid-levels of deep convection in PC2, and presume that this is influenced -by the erosion formulation. A link to mass flux has also been proposed, but tests -with CRMs do not support a clear link. Perhaps it is more natural to compare the -erosion with the turbulent kinetic energy. This should be available within the -boundary layer and convection schemes, but not outside of these in the current -UM. - -The erosion formulation in PC2:66 is one where the width of the PDF is -always narrowed (developed following \cite{sg03}). -It may be advantageous to think whether there are unmodelled -processes in the atmosphere that result in an increase in width. Clearly -convection is likely to be one, but this is already represented in PC2. -There may be other models entirely for the way in which the PDF changes as a result -of mixing of air within a gridbox or within the column, these may prove -fruitful to explore. - -Another issue is whether width-narrowing (or widening) is really an effective -way of representing the erosion process. CRM evidence suggests that the required -erosion rates to balance convective cloud generation are larger for -liquid cloud fraction than liquid water (by up to a factor of 2), suggesting -that the real atmospheric erosion favours removal of cloud fraction -over liquid water more strongly than the model. - -The in-cloud condensate that is detrained from convective plumes is high. -We might think that the mixing in of environmental air in reality is -likely to lead to more cloud around the plumes and lower condensate within -the plumes. However, the width narrowing scheme is not a good model of mixing -in this situation, always reducing the amount of cloud because it is incorrectly -assumed that much of the detrained plume has condensate contents only just above zero -and that the shape of the moisture PDF remains unchanged. This may have a -bearing on the problem of the lack of mid-level cloud in the model (although I -think there are many reasons for this). A different -mixing method may give significantly different results for the areas around -convective plumes. - -\subsubsection{Narrowing of the moisture PDF} -Most of the parametrized terms in PC2 act to reduce the width of the -moisture PDF. The only terms that can increase the width are the convection, -and the initiation (which can reset the width). This may not be the -best way to describe the way in which the PDF evolves, in particular it -is sensible to ask whether the erosion term should actually increase -the width in the presence of large vertical gradients of moisture. - -\subsubsection{Convective cloud increments in the mass-flux framework} -As discussed in section \ref{sec:conv_imp_note}, it would be useful -to code up the convective cloud fraction changes to link directly to -the mass-flux convection scheme, and not to estimate them from the values -of $Q4$, which can introduce errors. - -\subsubsection{Turbulence based convection scheme} -\label{sec:tbcs} -We will need to properly consider the links between PC2 and the -turbulence based convection scheme. In essence, we can use the diagnosed -cloud fraction and condensate values from the convection scheme to -start off the cloud again when convection has ceased. This has been -tested to some degree but will need proper analysis. The difficult -decision comes in choosing what to do with the condensate and cloud fraction -that is present \textit{before} the convection starts, since we must -ensure conservation of moisture. This is not helped by the traditional -view of convective parametrization that ignores the existence of the condensate -phase in the atmosphere (i.e. it is only concerned with transport of $q$ and -$\theta$, not of $q_{cl}$ and $q_{cf}$) despite the phase changes forming -an integral part of the convection scheme. - -\subsubsection{Detailed convective comparisons with CRM/LEM data} -This work is already underway at the Met Office, in order to properly -evaluate the performance of the convective cloud parametrization -in PC2 against high resolution research models. - -\subsubsection{Choice of PDF parameters} -Work by Dan Tang at Leeds University has highlighted an interesting -and undesirable property of the choice of $m$ and $n$ parameters in the -homogeneous forcing formulation. If a distribution is homogeneously -forced to $C_l = 0$, then we do not necessarily get $\overline{q_{cl}}$ -tending to zero. This is because there is enough influence from the -$\frac{{(1-C_l)}^2}{SD}$ term in the combination (\ref{eqn22}) to -stop the natural convergence of the $\frac{{C_l}^2}{\overline{q_{cl}}}$ term -to $C_l =0$ and $\overline{q_{cl}}=0$. Increasing the power of $m$ should -help. However, we note that the tests that have been done on the chosen -$n$ and $m$ values (0 and 0.5 respectively) do not show particularly -poor behaviour, and we do not pick up substantial evidence of problems -from this in the full model. This remains something to be investigated. - -\subsubsection{Homogeneous forcing section improvements} -\label{sec:homog_improve} -Although the homogeneous forcing provides a convenient method to -calculate increments to $C_l$ and $\overline{q_{cl}}$, it is clearly -not the best representation possible of the processes that use it. -For example, although the clear-sky radiative heating may perhaps -best be considered as a homogeneous process, the part of the -radiative heating influenced by clouds should, ideally, be applied -to the cloudy part of the gridbox and not the clear part. Vertical -advection is likely to be correlated with where there is already -cloud, rather than being uniform throughout the gridbox. There is -no reason that a process that uses homogeneous forcing as its -condensation model should not be looked at with a view to using -something better. This is one of the strengths of the PC2 framework and -is an intention of the project. - -\subsubsection{Overlap of ice and liquid cloud changes} -We have assumed within PC2 that ice and liquid cloud changes are -minimally overlapped with each other (within the same gridbox) in -order to maintain as much supercooled liquid water as possible. Although -there is good observational evidence to say that the two condensate -phases tend not to coexist together in a cloud, it may be possible to -characterise and apply this overlap in a more quantiative way. - -\subsubsection{Parameter tuning} -The sensitivity of some of the parameters in PC2 have not been properly -tested, mainly due to a lack of resources rather than a physical reason. -We have seen that the most effective method of tuning cloud is with the -erosion term, which has been increased to high values in order to remove -enough cloud and is probably as high as we reasonably wish to take it given -the length of the timestep. -\begin{itemize} -\item{The phase change temperature (between liquid and ice) in the -convective plume, TICE, is known to influence the strength of the convection -through the latent heat differences. It also impacts on the amount of -supercooled liquid water in the model. The quantitative impact of altering -this could be explored. We note that CRM simulations of deep convection -suggest that some supercooled liquid water exists within the plumes to -$-40 ^{\circ} C$ and that a representation with partial liquid and partial -ice phase would be more appropriate, based possibly on the current diagnosed -convective cloud phase in the non-PC2 model. Although the theoretical work -has been done to allow partial phases, we repeat the caution that care -must be taken when doing the work and appropriate testing done to ensure -that heat and moisture are properly conserved within the convection scheme.} - -\item{The growth of $C_f$ due to the fall-out of ice term in the microphysics -is parametrized with a dependence on windshear. We have never linked this -directly to the windshear, instead we have used estimated the windshear -as a fixed value. There is no reason why the actual model windshear cannot -be passed into the scheme in order to properly calculate this term.} -\item{$RH_{crit}$ remains a tunable parameter. Although its impact is less -than in a non-PC2 simulation, it is still significant in initiating cloud -and in determining the evolution of the ice cloud. There is also an implicit -overlap assumption regarding the ice cloud fractions, again this might be -improved upon.} -\item{$n$ and $m$ values in the homogeneous forcing have not been -thoroughly investigated for a long time now, and may yield some sensitivities}. -\end{itemize} - -\subsubsection{Cloud inhomogeneities} -A cloud generator approach to cloud inhomogeneities is currently -being developed. However we note two particular issues that relate -to PC2. -\begin{itemize} -\item{The first is that in the diagnostic scheme, the two cloud -fractions (convective and large-scale) allows, to some degree, a -representation of cloud inhomogeneity. This is absent from PC2, -although we note that the convective cloud fraction variable has -not been removed from the radiative transfer code for PC2, it is merely set -to zero, so it is easy to put back.} -\item{The generation of inhomogeneities using a cloud generator -requires some estimate of the variance (and possibly skewness) -of the condensate in the -gridbox. It is possible to back out the full moisture PDF at -each grid point by homogeneous forcing (providing $C_l$ is not equal -to 0 or 1), but this is very expensive and cannot be done on-line. Is -there a quick \textit{estimate} of the variance or skewness that it is -possible to obtain from knowledge only of $\overline{q}$, -$q_{sat}$, $\overline{q_{cl}}$ and $C_l$ etc.?} -\end{itemize} - -\subsubsection{Time-stepping} -\label{sec:timestepping} -A proper analysis of timestep sensitivities of PC2 (as opposed to -microphysics, convection etc) in the full UM -or SCM has not been done for a long time. -In the early development stages much effort was -placed in developing good numerical techniques for each of the terms -in PC2, and to explore the way in which they coupled together. An example -is the homogeneous forcing timestep investigated by \cite{wg03}. -We note that in shallow convection at 30 minutes timestep the erosion -term is trying to remove -most of the cloud that the convective detrainment places into the model. -Since the erosion is limited by the amount of cloud fraction and -condensate present, what ends -up happening is that the `equilibrium' that is achieved is actually one where -the cloud fraction and condensate at the end of the timestep are simply -the values that were detrained by the convection scheme (and hence depend -on the timestep). The CRM suggests a cycling time of around 15 minutes for -liquid water content and just less than half and hour for the cloud fraction, -so we would expect timestep dependency to occur from around a timestep of -15 minutes upwards. We might just about get away with the 30 minute step -of the climate model, but it is not a good situation to try to model. -This is demonstrating the difficulty of modelling shallow convective cloud -by a prognostic scheme, where the physical lifetime of the clouds is -of order the timestep - ideally we wouldn't want to try to model anything -prognostically when the cycling time is less than the timestep. - -As discussed in section \ref{sec:erosion_numerics}, the timestep sensitivity -of cloud amounts in shallow cumulus regimes can be addressed by using -a more accurate numerical method to solve the erosion term. -Several options are available under the UM namelist switch -\textbf{i\_pc2\_erosion\_numerics}. - -In the early development of PC2 we chose to incorporate the PC2 cloud -and condensation increments in the same location where the increments -were calculated (e.g. the microphysics cloud fraction increments -get added along with the microphysics $\overline{T}$ and $\overline{q}$ -increments). This choice was made in order not to confuse the timestepping -method in the UM, which has been carefully developed over a number of -years to achieve numerical accuracy. However, we note that the rapidly -varying nature (in space and time) of variables such as $\overline{q_{cl}}$ -and $C_l$ is very different from the smooth fields of $\overline{q_T}$ and -$\overline{T}$, for which the timestepping was developed, and it may -not be appropriate to implement these in the same locations. In -particular, we might wish to store the increments through the timestep -and update values of $\overline{q_{cl}}$ and $C_l$ etc. at the end -of the timestep, where many of the balances can be cancelled. - -One issue is that we are calculating the increments due to condensation -associated with the adiabatic response to pressure changes after the -Helmholtz solver. Pragmatically, we need to do it here since we do not know -the arrival value of pressure until after the Helmholtz solver has been -used. However, in order to achieve balanced dynamical fields, it is useful -the Helmholtz solver to be called after all the latent heating terms have -been calculated (which not only includes the adiabatic response to -lifting but the cloud initiation term). We have shown that PC2 can run with -the two terms switched over, but this implies that we are missing part of -the pressure change following the parcel (the time changing part -rather than the spatially changing adiabatic part). Although the adiabatic -change is usually likely to dominate, it may be a significant loss. -Under the UM namelist switch \textbf{l\_pc2\_sl\_advection}, -we can call the PC2 response twice, once before the Helmholtz solver and -once afterwards in order to pick up most of the latent heat change before -the solver, but not to have PC2 miss some of the pressure change. -The call for the advective part (before the Helmholtz solver) is -actually done before the call to atmos\_physics2 as well, and so results in -more realistic, saturation-adjusted, profiles being passed to the convection -scheme. - -We have placed the initiation at the end of the timestep, but it is sensible to -ask whether this could ideally be located elsewhere. - -\subsubsection{Initiation formulation} -Ideally this should be a relatively infrequent part of the model -but remains an essential part of the code. It is reasonable to ask whether -the initiation is optimal, particular in the diagnosis of when it is -applied. For example, we note that the initiation is currently -symmetrical, with initiation from $C_l=1$ occuring with the same -$RH_{crit}$ value as from $C_l=0$. However, the \cite{wf00} -observations hint that a higher $RH_{crit}$ might be more appropriate -for initiation from $C_l=1$. - -\subsubsection{70-levels performance} -The performance of PC2:66 in the 70-levels model is not good as -far as shallow convective cloud is concerned (there is far too -much of it in the trade regions). It may be that PC2 is latching onto -a convection sensitivity that is present on going from L38 to L70 but -had little effect in a non-PC2 simulation. It may also be related -to a reduction in timestep from 30 minutes to 20 minutes. -Investigations have not -made much progress in identifying the reasons for the differences, -or producing effective tunings to counter the problem. - -\subsubsection{High horizontal resolution performace} -PC2 has only been tested once at 4 km horizontal resolution. This -produced excessive shallow convective cloud (this may or may not be related -to the 70-levels problem above). Since this simulation the erosion -term has been increased dramatically, which may help. We note that -one of the main advantages of PC2, that of a prognostic link of -cloud to convection, is reduced at high resolution, as convection -becomes more explicit rather than diagnosed. We hence see a -resolution limit beyond which it is no longer appropriate to use -PC2. Results look acceptable at 12 km resolution, but we have not -quantitatively explored this limit. - -\subsubsection{Diagnostic evaluation} -One of the principal areas for future cloud scheme development -work planned in the future is in the area of detailed evaluation against -a number of data sources, such as CloudSat, ground based radar, -or case study campaigns. The quantitative evaluation has been -lacking to a significant degree in the development of the scheme, -as the focus has been on tackling qualitatively poor results. -Hence new sources of evaluation work on PC2 would be very welcome. - -\subsubsection{Moisture distribution within the deposition/sublimation term} -The liquid cloud changes in PC2 (or in a non-PC2 run) are based upon -a moisture PDF, as are the deposition/sublimation changes. However, it is -not the same PDF. It has always been the case with the prognostic ice -microphysics term that its PDF, whether explicit or implicit, has not -been rigorously consistent with the PDF used in the calculation of liquid -water, because it was most easily developed that way and produced reasonable -results. It may be useful to investigate whether the two PDF -representations can be brought together in a rigourous way, both for the PC2 -scheme and the \cite{smith90} scheme. - -We have similarly noted potential inconsistencies in the parametrization -of cloud fraction changes between the evaporation of rain term and the -riming (or accretion) term. Again, it might be possible to bring together -these formulations into a single consistent framework. - -\subsubsection{Area cloud fraction representation} -The current area cloud fraction representation is not used when -convection is taking place (signified by the \textit{cumulus} logical). -This inevitably leads to a potential switching between two different values -of the cloud fields if the convective boundary layer (not whether the -convection is shallow or deep) switches on and off, which is undesirable, -although not as bad as switching cloud on and off completely (as for the -current convective cloud formulation). Additionally, it is reasonable to argue that -having an area cloud fraction for cirrus cloud depend upon whether the boundary -layer is well mixed or has shallow convection occuring is not a reasonable link. - -Work in Australia on a TWP-ICE single column model case study using PC2 -suggests the area cloud fraction scheme over estimates the area cloud coverage -for tropical anvil clouds (which exist long after the convection itself has -ceased). This is perhaps not surprising since the \cite{bhi05} area -cloud fraction scheme was evaluated against mid-latitude cloud and it is -known that tropical clouds have greater vertical coherence. Tuning the -parameters in $large_scale_cloud/ls_acf_brooks.F90$ may be beneficial. - -%%\subsection{Acknowledgements} - -\begin{figure} -\begin{center} -\includegraphics[scale=1.0]{pc2_process_explanation} -\caption{Schematic summary of the PC2 cloud scheme.} -\label{fig:schematic} -\end{center} -\end{figure} - -\begin{figure} -\begin{center} -\includegraphics[scale=0.6]{Timestepping_ctl66.epsi} -\caption{Timestepping diagram for the control (non-PC2) scheme} -\label{fig:tstep_diag} -\end{center} -\end{figure} - -\begin{figure} -\begin{center} -\includegraphics[scale=0.6]{Timestepping_pc266.epsi} -\caption{Timestepping diagram for the PC2 scheme} -\label{fig:tstep_prog} -\end{center} -\end{figure} - -\bibliography{../029/refs} -\bibliographystyle{plainnat} - -\end{document} diff --git a/documentation/source/science_guide/cloud_schemes/um_call_tree.tex b/documentation/source/science_guide/cloud_schemes/um_call_tree.tex deleted file mode 100644 index 97d6a73f15..0000000000 --- a/documentation/source/science_guide/cloud_schemes/um_call_tree.tex +++ /dev/null @@ -1,530 +0,0 @@ - -% Latex source to make a diagram of the UM subroutine call tree, showing the -% locations of all cloud scheme calls. This diagram is included in both -% UMDP 029 (large-scale cloud scheme) and UMDP 030 (PC2). - -% NOTE: any preamble text required for this should be put in the file -% um_call_tree_preamble.tex, which is also inlcuded in both the UMDPs. - -Subroutines only called for the \textcolor{blue}{Smith} scheme are highlighted -in \textcolor{blue}{blue}, those only called for \textcolor{mygreen}{PC2} are -in \textcolor{mygreen}{green}, and those only called for -the \textcolor{purple}{bimodal} scheme are in \textcolor{purple}{purple}. - -\subsubsection{Main Tree from atm\_step\_4a} - -\begin{itemize} - -\item {\bf atm\_step\_4a} \\* -(performs one timestep of the Unified Model...) - \begin{itemize} - - \begin{tcolorbox}[enhanced jigsaw, breakable] - \item {\bf atm\_step\_alloc\_4a} \\* - (does miscellaneous initialisations in atm\_step) - - \begin{itemize} - - \item \textcolor{mygreen}{\bf pc2\_rhtl} \\* - (calculate start-of-timestep Relative Humidity, used by PC2 initiation) - - \end{itemize} - \end{tcolorbox} - - \begin{tcolorbox}[enhanced jigsaw, breakable] - \item {\bf atmos\_physics1} \\* - (calls explicit ``slow'' physics routines...) - \begin{itemize} - - \begin{tcolorbox} - \item {\bf microphys\_ctl} \\* - (interface to microphysics scheme) - \begin{itemize} - - \item \textcolor{mygreen}{\bf pc2\_turbulence\_ctl} \\* - (Perform optional erosion of liquid-cloud; - done here if NOT doing erosion after convection, - e.g. if no convection scheme is used). - \begin{itemize} - - \item \textcolor{mygreen}{\bf pc2\_hom\_conv} \\* - (called here just to do erosion) - - \end{itemize} - - \item \textcolor{blue}{\bf ls\_cld} \\* - (Smith scheme without area cloud fraction calculation, - to set initial cloud fields passed into microphysics) - - \item {\bf ls\_ppn} \\* - (microphysics scheme) - - \item {\bf mphys\_turb\_gen\_mixed\_phase} \\* - (turbulent production of liquid cloud) - - \item \textcolor{mygreen}{\bf pc2\_turbulence\_ctl} \\* - (optionally use the PC2 pdf-width-change code to calculate - the cloud-fraction change from the above turbulent production - of liquid cloud) - - \begin{itemize} - - \item \textcolor{mygreen}{\bf pc2\_hom\_conv} \\* - (called here just to calculate the cloud fraction increment - consistent with the turbulent qcl increment) - - \end{itemize} - - \end{itemize} - \end{tcolorbox} - - \begin{tcolorbox} - \item {\bf rad\_ctl} \\* - (interface to radiation scheme) - \begin{itemize} - - \item {\bf sw\_rad} \\* - (short-wave radiation scheme) - - \item \textcolor{mygreen}{\bf pc2\_homog\_plus\_turb} \\* - (PC2 homogeneous forcing of liquid-cloud by SW radiation heating) - - \item {\bf lw\_rad} \\* - (long-wave radiation scheme) - - \item \textcolor{mygreen}{\bf pc2\_homog\_plus\_turb} \\* - (PC2 homogeneous forcing of liquid-cloud by LW radiation tendency) - - \end{itemize} - \end{tcolorbox} - - \begin{tcolorbox} - \item {\bf atmos\_physics1\_alloc\_pc2} - (wrapper for PC2 self-consistency checks at end of atmos\_physics1) - - \begin{itemize} - - \item Add increments from microphysics + radiation onto - start-of-timestep fields to form updated fields. - - \item \textcolor{mygreen}{\bf pc2\_checks} \\* - (self-consistency checks on cloud fractions and water contents) - - \item Convert corrected updated fields back to increments. - - \end{itemize} - \end{tcolorbox} - - \end{itemize} - \end{tcolorbox} - - Begin loop over solver outer cycles - - \begin{itemize} - - \begin{tcolorbox}[enhanced jigsaw, breakable] - \item {\bf atm\_step\_phys\_reset} \\* - (for PC2, on subsequent solver outer cycles, - reset cloud-fractions to saved values after atmos\_physics1) - \end{tcolorbox} - - \begin{tcolorbox}[enhanced jigsaw, breakable] - \item {\bf eg\_sl\_moisture} \\* - (large-scale advection of cloud water contents and fractions) - \end{tcolorbox} - - \begin{tcolorbox}[enhanced jigsaw, breakable] - \item \textcolor{mygreen}{\bf pc2\_pressure\_forcing\_only} \\* - (Optionally calculate homogeneous forcing of liquid cloud by the - pressure change along the trajectory from departure point to - arrival point). - \begin{itemize} - - \item \textcolor{mygreen}{\bf pc2\_homog\_plus\_turb} \\* - (generic homogeneous forcing routine used here). - - \end{itemize} - \end{tcolorbox} - - \begin{tcolorbox}[enhanced jigsaw, breakable] - \item {\bf atmos\_physics2} \\* - (calls ``fast'' physics routines...) - - \begin{itemize} - - \begin{tcolorbox} - \item {\bf ni\_bl\_ctl} \\* - (interface to explicit boundary-layer and surface scheme calls, - including calculation of TKE and TKE-based $RH_{crit}$) - \end{tcolorbox} - - \begin{tcolorbox} - \item \textcolor{purple}{\bf bm\_calc\_tau} \\* - (calculates turbulence properties used in the bimodal cloud scheme, - based on the boundary-layer scheme TKE and mixing-length) - \end{tcolorbox} - - \begin{tcolorbox} - \item {\bf cloud\_call\_b4\_conv} \\* - (routine for optional cloud-scheme calls before convection) - \begin{itemize} - - \item \textcolor{blue}{\bf ls\_arcld} \\* - (Smith scheme with area cloud fraction; - see \ref{subsubsec:smith_acf} for a drill-down inside this routine) - - \item \textcolor{purple}{\bf bm\_ctl} \\* - (bimodal scheme) - - \item \textcolor{purple}{Set area cloud fraction equal to bulk - cloud fraction} - - \item \textcolor{mygreen}{\bf pc2\_initiation\_ctl} \\* - (interface to PC2 initiation and consistency-checks; - see \ref{subsubsec:pc2_initiation} for a drill-down inside this - routine) - - \end{itemize} - \end{tcolorbox} - - \begin{tcolorbox} - \item {\bf ni\_conv\_ctl} or {\bf other\_conv\_ctl} \\* - (interface routines to various convection schemes...) - \begin{itemize} - - \item {\bf glue\_conv\_5a/6a} \\* - (calls deep, shallow and mid-level convection schemes) - \begin{itemize} - - \item{\bf deep/shallow/mid\_conv} \\* - (convection scheme main routines) - \begin{itemize} - - \item {\bf convec2} - (completes lifting of the convective parcel by one model-level) - \begin{itemize} - - \item {\bf parcel} - (calculates new parcel properties at next level) - - \item {\bf environ} - (calculates grid-mean increments to primary fields; - includes PC2 partitioning of detrained condensate mass - between liquid and ice phases) - - \item \textcolor{mygreen}{\bf pc2\_environ} - (calculates increments to PC2 cloud fractions due to - convective detrainment and subsidence) - - \end{itemize} - - \end{itemize} - - \end{itemize} - - \item \textcolor{mygreen}{\bf pc2\_from\_conv\_ctl} \\* - (PC2 calculations after convection) - \begin{itemize} - - \item \textcolor{mygreen}{\bf pc2\_hom\_conv} \\* - (homogeneous forcing by convection, and erosion of liquid-cloud) - - \end{itemize} - - \end{itemize} - \end{tcolorbox} - - \begin{tcolorbox} - \item {\bf ni\_imp\_ctl} \\* - (interface to boundary-layer implicit solver) - \begin{itemize} - - \item {\bf imp\_solver} \\* - (implicitly solves vertical diffusion to find $T_l$ and $q_T$ - updated by turbulent fluxes). - - \item \textcolor{mygreen}{\bf pc2\_bl\_inhom\_ice} \\* - (inhomogeneous forcing of ice-cloud) - - \item \textcolor{mygreen}{\bf pc2\_delta\_hom\_turb} \\* - (homogeneous forcing of liquid cloud by the turbulent fluxes) - - \item \textcolor{mygreen}{\bf pc2\_bl\_forced\_cu} \\* - (adds diagnosed ``forced cumulus'' cloud fraction and water content - onto the PC2 prognostics) - - \item Calculate area cloud fraction: - - \textcolor{mygreen}{\bf ls\_acf\_brooks} \\* - (for the Brooks epirical method) - - \textcolor{mygreen}{\bf pc2\_hom\_arcld} \\* - (for the Cusack vertical interpolation method) - \begin{itemize} - - \item \textcolor{mygreen}{\bf pc2\_homog\_plus\_turb} \\* - (generic homogeneous forcing routine used to interpolate) - - \end{itemize} - - \item \textcolor{blue}{\bf ls\_arcld} \\* - (interface to diagnostic Smith scheme and area cloud fraction; - see \ref{subsubsec:smith_acf} for a drill-down inside this routine) - - \item \textcolor{purple}{\bf bm\_ctl} \\* - (bimodal cloud scheme) - - \item \textcolor{purple}{Set area cloud fraction equal to bulk - cloud fraction} - - \item {\bf diagnostics\_bl} \\* - (outputs boundary-layer diagnostics to STASH) - \begin{itemize} - - \item {\bf ls\_cld} \\* - (Smith scheme used here to calculate various diagnostics of - near-surface temperature and humidity, by extrapolating pressure, - $T_l$ and $q_t$ down to the desired height and then - re-diagnosing $q_{cl}$. - - \end{itemize} - - \end{itemize} - \end{tcolorbox} - - \end{itemize} - \end{tcolorbox} - - \begin{tcolorbox}[enhanced jigsaw, breakable] - \item {\bf atm\_step\_ac\_assim} \\* - (interface to Data Assimilation analysis increments...) - - \begin{itemize} - - \item {\bf ac\_ctl} - (control routine for Data Assimilation analysis increments...) - \begin{itemize} - - \item{\bf ac} - (main analysis increment routine) - - \item \textcolor{mygreen}{\bf pc2\_assim} \\* - (PC2 reponse to the analysis increments; - see \ref{subsubsec:pc2_assim} for a drill-down inside this routine) - - \item \textcolor{mygreen}{\bf ls\_acf\_brooks} - (calculate area cloud fraction using Brooks empirical method if active) - - \item \textcolor{blue}{\bf ls\_arcld} - (call diagnostic Smith scheme with area cloud fraction again to - account for the analysis increments; - see \ref{subsubsec:smith_acf} for a drill-down inside this routine) - - \end{itemize} - - \end{itemize} - \end{tcolorbox} - - \begin{tcolorbox}[enhanced jigsaw, breakable] - \item {\bf eg\_sl\_helmholtz} \\* - (dynamics pressure solver; updates pressure, and the winds used - to perform advection on the next solver outer cycle) - \end{tcolorbox} - - \end{itemize} - - End loop over solver outer cycles - - \begin{tcolorbox}[enhanced jigsaw, breakable] - \item \textcolor{mygreen}{\bf pc2\_pressure\_forcing} \\* - (interface to miscellaneous PC2 calculations at end-of-timestep) - \begin{itemize} - - \item \textcolor{mygreen}{\bf pc2\_homog\_plus\_turb} \\* - (homogeneous forcing of liquid-cloud by the dynamics pressure change; - optionally either uses total pressure change including the - Lagrangian component following the winds, or only the Eulerian - component from the dynamics solver) - - \item \textcolor{mygreen}{\bf pc2\_initiation\_ctl} \\* - (interface to PC2 initiation and consistency-checks; - see \ref{subsubsec:pc2_initiation} for a drill-down inside this routine) - - \end{itemize} - \end{tcolorbox} - - \begin{tcolorbox}[enhanced jigsaw, breakable] - \item {\bf qt\_bal\_cld} \\* - (calculates end-of-timestep cloud state consistent with final pressure...) - \begin{itemize} - - \item \textcolor{blue}{\bf ls\_arcld} \\* - (interface to diagnostic Smith scheme and area cloud fraction; - see \ref{subsubsec:smith_acf} for a drill-down inside this routine) - - \item \textcolor{purple}{\bf bm\_ctl} \\* - (bimodal cloud scheme) - - \item \textcolor{purple}{Set area cloud fraction equal to bulk - cloud fraction} - - \end{itemize} - \end{tcolorbox} - - \begin{tcolorbox}[enhanced jigsaw, breakable] - \item {\bf iau} \\* - (incremental analysis update; part of data assimilation) - \begin{itemize} - - \item \textcolor{mygreen}{\bf pc2\_assim} \\* - (PC2 reponse to the analysis increments; - see \ref{subsubsec:pc2_assim} for a drill-down inside this routine) - - \item \textcolor{mygreen}{\bf initial\_pc2\_check} \\* - (wrapper for optional self-consistency checks on prognostic cloud variables - if not doing PC2 response to analysis increments) - - \begin{itemize} - - \item \textcolor{mygreen}{\bf pc2\_checks} \\* - (self-consistency checks on cloud fractions and water contents) - - \end{itemize} - - \end{itemize} - \end{tcolorbox} - - \end{itemize} - -\end{itemize} - - -Drill-downs within some routines in the call tree are listed separately below, -to avoid duplication -(since these routines are called in multiple different places in the tree)... - -\subsubsection{Smith scheme with area cloud fraction} -\label{subsubsec:smith_acf} - -\begin{itemize} - -\begin{tcolorbox}[enhanced jigsaw, breakable] -\item \textcolor{blue}{\bf ls\_arcld} \\* -(interface to diagnostic Smith scheme and area cloud fraction) - \begin{itemize} - - \item If no area cloud fraction scheme: - - \textcolor{blue}{\bf ls\_cld} \\* - (just directly call Smith scheme) - - Set area cloud fraction equal to bulk cloud fraction. - - \item If using Cusack vertical interpolation method: - - Interpolate fields onto finer vertical grid - - \textcolor{blue}{\bf ls\_cld} \\* - (call Smith scheme using higher vertical resolution fields) - - Coarse-grain cloud fields back to model grid, but set area cloud - fraction to max of bulk cloud fraction over corresponding fine-grid levels. - - \item If using Brooks empirical area cloud fraction method: - - \textcolor{blue}{\bf ls\_cld} \\* - (just directly call Smith scheme) - - \textcolor{blue}{\bf ls\_acf\_brooks} \\* - (estimate area cloud fraction) - - \end{itemize} -\end{tcolorbox} - -\end{itemize} - - -\subsubsection{PC2 initiation} -\label{subsubsec:pc2_initiation} - -\begin{itemize} - -\begin{tcolorbox}[enhanced jigsaw, breakable] -\item \textcolor{mygreen}{\bf pc2\_initiation\_ctl} \\* -(interface to PC2 initiation and consistency-checks) - \begin{itemize} - - \item \textcolor{mygreen}{\bf pc2\_checks} \\* - (self-consistency checks on cloud fractions and water contents) - - \item PC2 initiation of liquid-cloud: - - \textcolor{mygreen}{\bf pc2\_bm\_initiate} \\* - (using the bimodal cloud scheme) - - \textcolor{mygreen}{\bf pc2\_arcld} \\* - (using the Smith scheme with the Cusack vertical interpolation method) - - \begin{itemize} - - \item \textcolor{mygreen}{\bf pc2\_initiate} \\* - (initiation using the Smith scheme, - called here on a finer vertical grid as per the Cusack method) - - \end{itemize} - - \textcolor{mygreen}{\bf pc2\_initiate} \\* - (using the Smith scheme with no area cloud representation) - - \item \textcolor{mygreen}{\bf pc2\_checks2} \\* - (further self-consistency checks on cloud-fractions) - - \item \textcolor{mygreen}{\bf pc2\_checks} \\* - (repeat the first lot of self-consistency checks again, - just in case we broke something in the mean-time!) - - \item \textcolor{mygreen}{\bf pc2\_hom\_arcld} \\* - (finds area cloud fraction using a version of the Cusack method, - where the cloud fraction on the finer vertical grid is estimated by - applying homogeneous forcing relative to the original grid fields) - - \begin{itemize} - - \item \textcolor{mygreen}{\bf pc2\_homog\_plus\_turb} \\* - (generic homogeneous forcing routine used to interpolate) - - \end{itemize} - - \end{itemize} -\end{tcolorbox} - -\end{itemize} - - -\subsubsection{PC2 Data Assimilation} -\label{subsubsec:pc2_assim} - -\begin{itemize} - -\begin{tcolorbox}[enhanced jigsaw, breakable] -\item \textcolor{mygreen}{\bf pc2\_assim} \\* -(PC2 reponse to the analysis increments) - \begin{itemize} - - \item \textcolor{mygreen}{\bf pc2\_homog\_plus\_turb} \\* - (generic PC2 homogeneous forcing routine used here for liquid-cloud) - - \item Estimate change in ice-cloud fraction from the assimilation - increment to ice-cloud mass. - - \item \textcolor{mygreen}{\bf pc2\_total\_cf} \\* - (update bulk cloud fraction due to change in ice cloud fraction) - - \item \textcolor{mygreen}{\bf pc2\_checks} \\* - (self-consistency checks on prognostic cloud fractions and - water contents) - - \end{itemize} -\end{tcolorbox} - -\end{itemize} diff --git a/documentation/source/science_guide/cloud_schemes/um_call_tree_preamble.tex b/documentation/source/science_guide/cloud_schemes/um_call_tree_preamble.tex deleted file mode 100644 index 009537991a..0000000000 --- a/documentation/source/science_guide/cloud_schemes/um_call_tree_preamble.tex +++ /dev/null @@ -1,25 +0,0 @@ - -% Packages needed for the UM subroutine tree diagram in um_call_tree.txt - -% Used to colour-code things in the subroutine call tree diagram: -\usepackage{xcolor} -% Define a darker green, as in some pdf viewers the standard green -% is too bright to be readable on the grey background. -\definecolor{mygreen}{rgb}{0.0, 0.667, 0.0} - -% Allow more deeply nested lists, for writing the subroutine call tree: -\usepackage{enumitem} -\setlistdepth{20} -\renewlist{itemize}{itemize}{20} -\setlist[itemize]{label=$\cdot$} -\setlist[itemize,1]{label=\textcolor{black}{$\bullet$}} -\setlist[itemize,2]{label=\textcolor{blue}{$\bullet$}} -\setlist[itemize,3]{label=\textcolor{purple}{$\bullet$}} -\setlist[itemize,4]{label=\textcolor{red}{$\bullet$}} -\setlist[itemize,5]{label=\textcolor{orange}{$\bullet$}} -\setlist[itemize,6]{label=\textcolor{yellow}{$\bullet$}} -\setlist[itemize,7]{label=\textcolor{green}{$\bullet$}} -\setlist[itemize,8]{label=\textcolor{cyan}{$\bullet$}} - -% Used to draw boxes around subroutines in the call tree diagram: -\usepackage[most]{tcolorbox} \ No newline at end of file From 04ed58afb3c344e56ac15769d8ee1ddda43546dd Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Thu, 2 Apr 2026 13:46:11 +0100 Subject: [PATCH 008/116] Convert the figures to .svg and import them correctly. --- .../cloud_schemes/Timestepping_ctl66.svg | 2624 ++++ .../cloud_schemes/Timestepping_pc266.svg | 3374 +++++ .../cloud_schemes/UMDP30_PC2CloudScheme.rst | 33 +- .../cloud_schemes/pc2_process_explanation.svg | 12060 ++++++++++++++++ 4 files changed, 18073 insertions(+), 18 deletions(-) create mode 100644 documentation/source/science_guide/cloud_schemes/Timestepping_ctl66.svg create mode 100644 documentation/source/science_guide/cloud_schemes/Timestepping_pc266.svg create mode 100644 documentation/source/science_guide/cloud_schemes/pc2_process_explanation.svg diff --git a/documentation/source/science_guide/cloud_schemes/Timestepping_ctl66.svg b/documentation/source/science_guide/cloud_schemes/Timestepping_ctl66.svg new file mode 100644 index 0000000000..425fccc35d --- /dev/null +++ b/documentation/source/science_guide/cloud_schemes/Timestepping_ctl66.svg @@ -0,0 +1,2624 @@ + + + + +Created by potrace 1.16, written by Peter Selinger 2001-2019 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/documentation/source/science_guide/cloud_schemes/Timestepping_pc266.svg b/documentation/source/science_guide/cloud_schemes/Timestepping_pc266.svg new file mode 100644 index 0000000000..8b64eb4691 --- /dev/null +++ b/documentation/source/science_guide/cloud_schemes/Timestepping_pc266.svg @@ -0,0 +1,3374 @@ + + + + +Created by potrace 1.16, written by Peter Selinger 2001-2019 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst index e34f0dbc2e..c2791c1bbe 100644 --- a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst +++ b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst @@ -326,8 +326,9 @@ Concept of PC2 The PC2 scheme develops prognostic expressions for the rates of change of cloud fraction and condensate contents as a result of each process that acts in the model. We consider ice and liquid condensate as two -distinct aspects of clouds, which may or may not overlap Figure -`1 <#fig:schematic>`__ provides a schematic summary of the PC2 scheme. +distinct aspects of clouds, which may or may not overlap. +:numref:`Figure %s ` +provides a schematic summary of the PC2 scheme. The equations for the five prognostic cloud variables can be written schematically: @@ -6890,27 +6891,23 @@ against mid-latitude cloud and it is known that tropical clouds have greater vertical coherence. Tuning the parameters in :math:`large_scale_cloud/ls_acf_brooks.F90` may be beneficial. -.. container:: float +.. figure:: pc2_process_explanation.svg :name: fig:schematic + :alt: Schematic summary of the PC2 cloud scheme + :width: 100% - .. container:: center + Schematic summary of the PC2 cloud scheme. - |image1| - -.. container:: float +.. figure:: Timestepping_ctl66.svg :name: fig:tstep_diag + :alt: Timestepping diagram for the control (non-PC2) scheme + :width: 100% - .. container:: center - - |image2| + Timestepping diagram for the control (non-PC2) scheme -.. container:: float +.. figure:: Timestepping_pc266.svg :name: fig:tstep_prog + :alt: Timestepping diagram for the PC2 scheme + :width: 100% - .. container:: center - - |image3| - -.. |image1| image:: pc2_process_explanation.eps -.. |image2| image:: Timestepping_ctl66.epsi -.. |image3| image:: Timestepping_pc266.epsi + Timestepping diagram for the PC2 scheme diff --git a/documentation/source/science_guide/cloud_schemes/pc2_process_explanation.svg b/documentation/source/science_guide/cloud_schemes/pc2_process_explanation.svg new file mode 100644 index 0000000000..2da9790371 --- /dev/null +++ b/documentation/source/science_guide/cloud_schemes/pc2_process_explanation.svg @@ -0,0 +1,12060 @@ + + + + +Created by potrace 1.16, written by Peter Selinger 2001-2019 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + 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+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + From c2d44a5436fbce9bfa034a70c70edbb1166f4b23 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Thu, 2 Apr 2026 14:34:36 +0100 Subject: [PATCH 009/116] Trying to get section numbering and cross-referencing to work (not working yet). --- documentation/source/conf.py | 6 +++++- .../science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst | 7 +++---- 2 files changed, 8 insertions(+), 5 deletions(-) diff --git a/documentation/source/conf.py b/documentation/source/conf.py index 9d6ae59a1c..4331773a1d 100644 --- a/documentation/source/conf.py +++ b/documentation/source/conf.py @@ -20,7 +20,8 @@ extensions = [ 'sphinx_sitemap', 'sphinx_design', - 'sphinx.ext.intersphinx' + 'sphinx.ext.intersphinx', + 'sphinx.ext.autosectionlabel' ] # Add any paths that contain templates here, relative to this directory. @@ -95,6 +96,9 @@ # Enable numbered references to e.g. figures. # numfig = True +html_use_modindex = True +autosectionlabel_prefix_document = True +autosectionlabel_maxdepth = 3 # Exclude files from Sphinx processing exclude_patterns = ['common_links.rst'] diff --git a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst index c2791c1bbe..b16230116c 100644 --- a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst +++ b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst @@ -58,7 +58,7 @@ straightforward to solve if one is allowed to assume that there is no variability of moisture or temperature on a scale of a model gridbox. In this case the cloud fraction scheme is redundant and only the condensation part remains, which may be solved diagnostically using the -instantaneous condensation assumption in section `2.2 <#sec:s_dist>`__. +instantaneous condensation assumption in section :numref:`the-s-distribution`. However, the ‘no-variability’ assumption is poor until very high resolutions close to, or maybe exceeding, 1 km in the horizontal are reached. Although we may eventually assume that computer power will @@ -70,7 +70,7 @@ parametrization. There are several approaches to take to the solution of the problem, although they are not as independent as often portrayed, since they nearly all require the same instantaneous condensation assumption -(discussed in section `2.2 <#sec:s_dist>`__). Hence there are +(discussed in section :numref:`the-s-distribution`). Hence there are mathematical links between all the approaches. *The following are all valid structures to use in this respect.* @@ -112,7 +112,6 @@ and to break the hard diagnostic link between cloud fraction and condensate. These major features of the :raw-latex:`\cite{t93}` scheme provide the motivation to develop the PC2 cloud scheme. -.. _`sec:s_dist`: The ‘s’ distribution -------------------- @@ -2285,7 +2284,7 @@ This term exerts one of the most important influences on the ice cloud in the whole model (this applies to the control as well as for PC2). Contained in the formulation is a subgrid-scale assumption that causes equivalent effects to that for a moisture PDF under the ‘:math:`s`’ -framework (section `2.2 <#sec:s_dist>`__). However, since +framework (section :numref:`the-s-distribution`). However, since :math:`{q_{cf}}` changes slowly in response to local changes in :math:`q` and :math:`T`, we cannot base the :math:`q_{cf}` response on the same instantaneous condensation framework. It would be useful to From ceadd31aa87d5c3e16e394c873f70768b9251888 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Thu, 2 Apr 2026 14:48:14 +0100 Subject: [PATCH 010/116] Further attempt to cross-reference section. --- .../science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst index b16230116c..40dc136719 100644 --- a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst +++ b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst @@ -58,7 +58,7 @@ straightforward to solve if one is allowed to assume that there is no variability of moisture or temperature on a scale of a model gridbox. In this case the cloud fraction scheme is redundant and only the condensation part remains, which may be solved diagnostically using the -instantaneous condensation assumption in section :numref:`the-s-distribution`. +instantaneous condensation assumption in section :ref:`the-s-distribution`. However, the ‘no-variability’ assumption is poor until very high resolutions close to, or maybe exceeding, 1 km in the horizontal are reached. Although we may eventually assume that computer power will @@ -70,7 +70,7 @@ parametrization. There are several approaches to take to the solution of the problem, although they are not as independent as often portrayed, since they nearly all require the same instantaneous condensation assumption -(discussed in section :numref:`the-s-distribution`). Hence there are +(discussed in section :ref:`the-s-distribution`). Hence there are mathematical links between all the approaches. *The following are all valid structures to use in this respect.* @@ -2284,7 +2284,7 @@ This term exerts one of the most important influences on the ice cloud in the whole model (this applies to the control as well as for PC2). Contained in the formulation is a subgrid-scale assumption that causes equivalent effects to that for a moisture PDF under the ‘:math:`s`’ -framework (section :numref:`the-s-distribution`). However, since +framework (section :ref:`the-s-distribution`). However, since :math:`{q_{cf}}` changes slowly in response to local changes in :math:`q` and :math:`T`, we cannot base the :math:`q_{cf}` response on the same instantaneous condensation framework. It would be useful to From 86c8a8ebeffa341e8eadff5a93dad8165c8d3a22 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Thu, 2 Apr 2026 15:32:35 +0100 Subject: [PATCH 011/116] Still fiddling with cross-referencing a section. --- .../cloud_schemes/UMDP30_PC2CloudScheme.rst | 9 ++++++--- 1 file changed, 6 insertions(+), 3 deletions(-) diff --git a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst index 40dc136719..7ac88de9a3 100644 --- a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst +++ b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst @@ -58,7 +58,8 @@ straightforward to solve if one is allowed to assume that there is no variability of moisture or temperature on a scale of a model gridbox. In this case the cloud fraction scheme is redundant and only the condensation part remains, which may be solved diagnostically using the -instantaneous condensation assumption in section :ref:`the-s-distribution`. +instantaneous condensation assumption in section +:ref:`The ‘s’ distribution `:. However, the ‘no-variability’ assumption is poor until very high resolutions close to, or maybe exceeding, 1 km in the horizontal are reached. Although we may eventually assume that computer power will @@ -70,7 +71,8 @@ parametrization. There are several approaches to take to the solution of the problem, although they are not as independent as often portrayed, since they nearly all require the same instantaneous condensation assumption -(discussed in section :ref:`the-s-distribution`). Hence there are +(discussed in section :ref:`The ‘s’ distribution `:). +Hence there are mathematical links between all the approaches. *The following are all valid structures to use in this respect.* @@ -112,6 +114,7 @@ and to break the hard diagnostic link between cloud fraction and condensate. These major features of the :raw-latex:`\cite{t93}` scheme provide the motivation to develop the PC2 cloud scheme. +.. _sec:s_dist: The ‘s’ distribution -------------------- @@ -2284,7 +2287,7 @@ This term exerts one of the most important influences on the ice cloud in the whole model (this applies to the control as well as for PC2). Contained in the formulation is a subgrid-scale assumption that causes equivalent effects to that for a moisture PDF under the ‘:math:`s`’ -framework (section :ref:`the-s-distribution`). However, since +framework (section :ref:`The ‘s’ distribution `:). However, since :math:`{q_{cf}}` changes slowly in response to local changes in :math:`q` and :math:`T`, we cannot base the :math:`q_{cf}` response on the same instantaneous condensation framework. It would be useful to From 340afbdb73e7b4fa960384a819d5fb80d0bcb14e Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Thu, 2 Apr 2026 15:36:40 +0100 Subject: [PATCH 012/116] Just reference sections by title (giving up on tryyying to number them). --- .../cloud_schemes/UMDP30_PC2CloudScheme.rst | 10 ++++------ 1 file changed, 4 insertions(+), 6 deletions(-) diff --git a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst index 7ac88de9a3..86c469a33b 100644 --- a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst +++ b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst @@ -58,8 +58,7 @@ straightforward to solve if one is allowed to assume that there is no variability of moisture or temperature on a scale of a model gridbox. In this case the cloud fraction scheme is redundant and only the condensation part remains, which may be solved diagnostically using the -instantaneous condensation assumption in section -:ref:`The ‘s’ distribution `:. +instantaneous condensation assumption in section :ref:`The ‘s’ distribution`. However, the ‘no-variability’ assumption is poor until very high resolutions close to, or maybe exceeding, 1 km in the horizontal are reached. Although we may eventually assume that computer power will @@ -71,8 +70,7 @@ parametrization. There are several approaches to take to the solution of the problem, although they are not as independent as often portrayed, since they nearly all require the same instantaneous condensation assumption -(discussed in section :ref:`The ‘s’ distribution `:). -Hence there are +(discussed in section :ref:`The ‘s’ distribution`). Hence there are mathematical links between all the approaches. *The following are all valid structures to use in this respect.* @@ -114,7 +112,7 @@ and to break the hard diagnostic link between cloud fraction and condensate. These major features of the :raw-latex:`\cite{t93}` scheme provide the motivation to develop the PC2 cloud scheme. -.. _sec:s_dist: +.. _The ‘s’ distribution: The ‘s’ distribution -------------------- @@ -2287,7 +2285,7 @@ This term exerts one of the most important influences on the ice cloud in the whole model (this applies to the control as well as for PC2). Contained in the formulation is a subgrid-scale assumption that causes equivalent effects to that for a moisture PDF under the ‘:math:`s`’ -framework (section :ref:`The ‘s’ distribution `:). However, since +framework (section :ref:`The ‘s’ distribution`). However, since :math:`{q_{cf}}` changes slowly in response to local changes in :math:`q` and :math:`T`, we cannot base the :math:`q_{cf}` response on the same instantaneous condensation framework. It would be useful to From 1d3d134eb30528bff6ec1f61d109d378f84c744a Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Thu, 2 Apr 2026 15:39:59 +0100 Subject: [PATCH 013/116] Reverted unneeded changes to conf.py --- documentation/source/conf.py | 6 +----- 1 file changed, 1 insertion(+), 5 deletions(-) diff --git a/documentation/source/conf.py b/documentation/source/conf.py index 4331773a1d..9d6ae59a1c 100644 --- a/documentation/source/conf.py +++ b/documentation/source/conf.py @@ -20,8 +20,7 @@ extensions = [ 'sphinx_sitemap', 'sphinx_design', - 'sphinx.ext.intersphinx', - 'sphinx.ext.autosectionlabel' + 'sphinx.ext.intersphinx' ] # Add any paths that contain templates here, relative to this directory. @@ -96,9 +95,6 @@ # Enable numbered references to e.g. figures. # numfig = True -html_use_modindex = True -autosectionlabel_prefix_document = True -autosectionlabel_maxdepth = 3 # Exclude files from Sphinx processing exclude_patterns = ['common_links.rst'] From f55bc3110a61773240799a00d5618aa96043aacb Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Thu, 2 Apr 2026 17:14:00 +0100 Subject: [PATCH 014/116] Rolled-out fix for cross-referencing across all sections. --- .../cloud_schemes/UMDP30_PC2CloudScheme.rst | 342 +++++++++--------- 1 file changed, 177 insertions(+), 165 deletions(-) diff --git a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst index 86c469a33b..17f77308fc 100644 --- a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst +++ b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst @@ -58,7 +58,7 @@ straightforward to solve if one is allowed to assume that there is no variability of moisture or temperature on a scale of a model gridbox. In this case the cloud fraction scheme is redundant and only the condensation part remains, which may be solved diagnostically using the -instantaneous condensation assumption in section :ref:`The ‘s’ distribution`. +instantaneous condensation assumption in section :ref:`The 's' distribution`. However, the ‘no-variability’ assumption is poor until very high resolutions close to, or maybe exceeding, 1 km in the horizontal are reached. Although we may eventually assume that computer power will @@ -70,7 +70,7 @@ parametrization. There are several approaches to take to the solution of the problem, although they are not as independent as often portrayed, since they nearly all require the same instantaneous condensation assumption -(discussed in section :ref:`The ‘s’ distribution`). Hence there are +(discussed in section :ref:`The 's' distribution`). Hence there are mathematical links between all the approaches. *The following are all valid structures to use in this respect.* @@ -112,9 +112,9 @@ and to break the hard diagnostic link between cloud fraction and condensate. These major features of the :raw-latex:`\cite{t93}` scheme provide the motivation to develop the PC2 cloud scheme. -.. _The ‘s’ distribution: +.. _The 's' distribution: -The ‘s’ distribution +The 's' distribution -------------------- Most cloud schemes are based on the concept of a distribution of @@ -275,7 +275,7 @@ a purely diagnostic representation such as :raw-latex:`\cite{smith90}` where we explicitly consider distributions of :math:`s`. Strictly, the linear approximation implies that other approximations for :math:`\alpha` are valid: PC2 will do this (see section -`3.2.3 <#sec:homog_num_app>`__) since we are concerned in PC2 with the +:ref:`Numerical application`) since we are concerned in PC2 with the best estimate of the *changes* to :math:`\overline{q_{cl}}`, not the best estimate of :math:`\overline{q_{cl}}` itself. @@ -387,7 +387,7 @@ given by (`[eq:int_gs_ds] <#eq:int_gs_ds>`__) and concept of instantaneous condensation for liquid clouds. Equations `[eq:int_gs_ds] <#eq:int_gs_ds>`__ and `[eq:qclbar=int] <#eq:qclbar=int>`__ will form the basis of the -homogeneous forcing methods discussed in section `3.2 <#sec:homog>`__. +homogeneous forcing methods discussed in section :ref:`Homogeneous forcing`. We note in particular that the convective cloud fraction, previously a quantity that is diagnosed separately from the large-scale cloud fraction calculated by the :raw-latex:`\cite{smith90}` scheme, may, in @@ -405,13 +405,15 @@ specifically developed generic approaches that can be used to calculate expressions for :math:`\frac{\partial \overline{q_{cl}}}{\partial t}` and :math:`\frac{\partial C_l} {\partial t}` . These are referred to as Homogeneous forcing (section -`3.2 <#sec:homog>`__), Injection source (or inhomogeneous forcing, -section `3.5 <#sec:inhomog>`__) and Width Changing (section -`3.3 <#sec:width>`__). Two additional modules are available to assist -with PC2, liquid cloud initiaion (section `3.4 <#sec:init>`__) and the -calculation of total cloud fraction changes (section `3.6 <#sec:ct>`__). +:ref:`Homogeneous forcing`), Injection source (or inhomogeneous forcing, +section :ref:`Injection forcing`) and Width Changing (section +:ref:`Changing the width of the PDF - PC2 erosion`). +Two additional modules are available to assist +with PC2, liquid cloud initiaion (section :ref:`Initiation of cloud`) and the +calculation of total cloud fraction changes +(section :ref:`Ice cloud and mixed phase regions`). At the present time, only the large-scale precipitation (section -`4.2 <#sec:precip>`__) scheme has been rewritten fully to use the PC2 +:ref:`Large-scale precipitation`) scheme has been rewritten fully to use the PC2 concept of prognostic cloud fractions. The existing mass-flux convection scheme has been modified to enable calculation of the detrained condensate, but direct modification to the cloud fraction is not @@ -440,7 +442,7 @@ of representation of cloud inhomogeneity. Hence the code still exists to enable PC2 to be run with or without a diagnostic convective cloud fraction, although PC2:66 does not include a -diagnostic term. More details are in section `4.7 <#sec:convec>`__. +diagnostic term. More details are in section :ref:`Convection`. Physical basis of the PC2 prognostic cloud scheme ================================================= @@ -449,20 +451,20 @@ In this section we will develop the physical models that PC2 uses in order to calculate its prognostic increment terms. We will also consider the numerical solution of the models. The way in which these are incorporated into the Unifed Model will be discussed in section -`5 <#sec:um>`__ +:ref:`Implementation in the Unified Model` Instantaneous condensation -------------------------- Liquid clouds in PC2 use the concept of instantaneous condensation. -Hence the ‘s’ distribution methods are fully applicable to the +Hence the 's' distribution methods are fully applicable to the development of the equations that govern the parametrization of liquid cloud in PC2. We will start by looking at changes to :math:`\overline{q_{cl}}` and :math:`C_l` when a uniform forcing is applied to a gridbox, under the assumption of instantaneous condensation. -.. _`sec:homog`: +.. _Homogeneous forcing: Homogeneous forcing ------------------- @@ -508,8 +510,8 @@ rate of change of condensate and cloud fraction based upon choose to develop a parametrization for this quantity based upon the quantities :math:`C_l`, :math:`\overline{q_{cl}}` and the saturation deficit, :math:`SD`, rather than tie :math:`G(-Q_c)` to a process. The -saturation deficit is *defined* here in the ‘s’ framework to be the -first moment of the PDF for ‘s’ values less than :math:`-Q_c`. In this +saturation deficit is *defined* here in the 's' framework to be the +first moment of the PDF for 's' values less than :math:`-Q_c`. In this way it is analogous to the liquid water content, :math:`\overline{q_{cl}}`. Appendix A of :raw-latex:`\cite{wg03}` writes this *definition* as @@ -583,7 +585,7 @@ form a complete mathematical set for the solution of :math:`C_l` and initial value of :math:`C_l` is not identically 0 or 1. If :math:`C_l` is 0 or 1 then :math:`G(-Q_c)` remains at zero. The equation set then needs to be initiated in some way. This is discussed further in -:raw-latex:`\cite{wg03}` and in section `3.4 <#sec:init>`__. If +:raw-latex:`\cite{wg03}` and in section :ref:`Initiation of cloud`. If :math:`G(-Q_c)` is defined, then application of the above equation set may be used to trace out an underlying PDF for any input values of :math:`C_l`, :math:`\overline{q_{cl}}` and :math:`SD`. Although we never @@ -694,7 +696,7 @@ closure removed occasional spurious very large cloud-fraction increments that occur when using `[eqn22] <#eqn22>`__, but did not significantly impact the performance of the model forecast. -.. _`sec:homog_num_app`: +.. _Numerical application: Numerical application ~~~~~~~~~~~~~~~~~~~~~ @@ -823,7 +825,7 @@ interfere with the possible cancellation of positive and negative increments from different physics schemes. A checking routine is applied, however, in the Unified Model to remove any negative values that are generated, which is discussed in section -`4.10 <#sec:checks>`__. However, the checking routine (Q-Pos) involves a +:ref:`Bounds checking`. However, the checking routine (Q-Pos) involves a lot of communication between processors and can significantly increase the run-time of the model. The option to “Ensure consistent sinks of qcl and CFL” performs a check at the end of the homogeneous forcing routines @@ -841,7 +843,7 @@ the net change in :math:`\overline{q_T}` minus the net change in calculated simply to account for the latent heat released due to the condensation. -.. _`sec:width`: +.. _Changing the width of the PDF - PC2 erosion: Changing the width of the PDF - PC2 erosion ------------------------------------------- @@ -962,15 +964,15 @@ of the process that is occuring. Note we don’t need to calculate :math:`b_s` separately, just its *fractional* rate of change. Options for the parameterisation of :math:`\frac{1}{b_s}\frac{\partial b_s}{\partial t}` due to turbulent -“erosion” are described in section `4.3 <#sec:turb>`__, along with the +“erosion” are described in section :ref:`PC2 erosion`, along with the numerical methods used to integrate the equations. -.. _`sec:init`: +.. _Initiation of cloud: Initiation of cloud ------------------- -In section `3.2 <#sec:homog>`__ we commented that the closure +In section :ref:`Homogeneous forcing` we commented that the closure (`[eqn22] <#eqn22>`__) for :math:`G(-Qc)` is only valid if :math:`C_l` is not identically 0 or 1. If :math:`C_l` is 0 or 1 we know that :math:`G(-Q_c)` is equal to 0 but we have lost the information that will @@ -1014,7 +1016,7 @@ description in (`[eqn19] <#eqn19>`__) to obtain the expressions We now need to parametrize the PDF width :math:`b_s`. Unlike the :raw-latex:`\cite{smith90}` scheme, this is the only location in the PC2 cloud scheme where the width needs to be defined for the liquid cloud -(although see section `4.2.4 <#sec:mp_depsub>`__ for a discussion of an +(although see section :ref:`Deposition and sublimation` for a discussion of an equivalent width in the deposition / sublimation relationship for ice cloud). We still choose to define :math:`b_s` in terms of a critical relative humidity parameter, :math:`RH_{crit}`. Like the @@ -1068,7 +1070,7 @@ within this diagnostic calculation of SD it might actually be better to use the representation (`[eq:alpha] <#eq:alpha>`__) used by the diagnostic :raw-latex:`\cite{smith90}` scheme. -.. _`sec:numapp_init`: +.. _Numerical Application of the Smith method: Numerical Application of the Smith method ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ @@ -1097,7 +1099,7 @@ circumstances in which we wish to proceed further: Note: in the UM implementation, the actual conditions for when initiation may occur are more complicated than this, and there are several options depending on a namelist switch. See section -`4.9 <#sec:init2>`__ for details... +:ref:`Initiation` for details... In the second case, we then make the temporary transformation of variables in order to use the same solution set as in the first case: @@ -1190,7 +1192,7 @@ where the superscript :math:`[i]` labels each iteration. We find that 10 iterations is effective for convergence, with the weighting :math:`f` given by :math:`a_L^{[i]}`. -.. _`sec:bimodal_init`: +.. _Initiation using the bimodal scheme: Initiation using the bimodal scheme ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ @@ -1221,14 +1223,14 @@ upper and lower values of :math:`Q_N` are then compared to -1 and 1 respectively, to determine whether the saturation boundary lies within the PDF bounds. This is the basic condition for initiation to occur (though there are additional conditions and various options for these in -the soure code; see section `4.9 <#sec:init2>`__). +the soure code; see section :ref:`Initiation`). If the initiation conditions are met, the diagnostic bimodal cloud scheme code is then called (see UMDP 039), and the diagnosed :math:`C_l` and :math:`q_{cl}` are used to set the prognostic :math:`C_l` and :math:`q_{cl}`. -.. _`sec:inhomog`: +.. _Injection forcing: Injection forcing ----------------- @@ -1255,7 +1257,7 @@ air. The fractional rate at which existing air is replaced by the injected source air we will write as :math:`\frac{\partial{C_S}}{\partial{t}}`. Provided that only the liquid phase exists (see section -`3.5.1 <#sec:multiple>`__ for the extention to multiple phases), we then +:ref:`Multiple phases in the injection source` for the extention to multiple phases), we then note that the rate of change of liquid cloud fraction and liquid water content in the gridbox can be written in two parts: firstly the change due to the background, and secondly the change due to the source. @@ -1298,7 +1300,7 @@ completely cloudy air. This equation allows one to calculate the change in :math:`C_l` associated with an injection source change of :math:`\overline{q_{cl}}` for the example of convection. Modifications to the mass-flux convection scheme for PC2 (far from trivial and -discussed in depth in section `4.7 <#sec:convec>`__) allow :math:`Q4_l` +discussed in depth in section :ref:`Convection`) allow :math:`Q4_l` to be calculated (:math:`q_{cl}^S` is already available), and (`[eq:dcdt_inhom2] <#eq:dcdt_inhom2>`__) can then be used to calculate the equivalent :math:`C_l` change. We note at this stage that the @@ -1315,7 +1317,7 @@ We could reasonably calculate the change in cloud fraction following the same methods as used to calculate the change in :math:`\overline{q}` or the change in a tracer and we discuss this later. -.. _`sec:multiple`: +.. _Multiple phases in the injection source: Multiple phases in the injection source ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ @@ -1465,9 +1467,8 @@ the ice cloud fraction and total cloud fraction. with :math:`\delta_{xi} = h_i = g_i`. These are the expressions that are used within the convection scheme. It still remains to parametrize :math:`\delta_{xl}`, which is given by the convection scheme itself. -This is discussed in section `4.7.9 <#sec:plume_phase>`__. +This is discussed in section :ref:`Phase of condensate`. -.. _`sec:multi_numapp`: Numerical application ~~~~~~~~~~~~~~~~~~~~~ @@ -1544,7 +1545,7 @@ fraction remains within its physical bounds. and similar equations are used for :math:`C_i^{[n+1]}` and :math:`C_t^{[n+1]}`. -.. _`sec:conv_imp_note`: +.. _A note on the implementation of the cloud fraction change: A note on the implementation of the cloud fraction change ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ @@ -1577,7 +1578,7 @@ convection. The change in :math:`\overline{q_{cl}}` and :math:`C_l` due to the adiabatic warming associated with the compensating subsidence is considered explicitly in the model implementation (see section -`4.7.4 <#sec:conv_homog>`__) for both :math:`\overline{q_{cl}}` and +:ref:`Background condensation`) for both :math:`\overline{q_{cl}}` and :math:`C_l` after the rest of the convective process has been calculated. It is perhaps arguable that if (`[eq:dcdt_inhom2] <#eq:dcdt_inhom2>`__) is going to be applied then the @@ -1588,7 +1589,7 @@ Any major future developments of PC2 for a mass-flux convection scheme would be advised to consider whether it is appropriate to use (`[eq:dcdt_inhom2] <#eq:dcdt_inhom2>`__) at all. -.. _`sec:ct`: +.. _Ice cloud and mixed phase regions: Ice cloud and mixed phase regions --------------------------------- @@ -1597,12 +1598,12 @@ The homogeneous forcing, initiation and PC2 erosion sections described above have only considered the generation and dissipation of liquid clouds. Although the forcing methods will not influence the generation and dissipation of ice cloud (which is primarily performed in the -large-scale precipitation scheme, section `4.2 <#sec:precip>`__) we are -still left with the issue of how created or dissipated liquid cloud +large-scale precipitation scheme, section :ref:`Large-scale precipitation`) +we are still left with the issue of how created or dissipated liquid cloud overlaps with existing ice cloud in the gridbox. The opposite situation, where changes in ice cloud are specified and changes in the overlap with liquid cloud need to be calculated, is also possible in PC2 (e.g. in the -boundary layer, see section `4.6 <#sec:bl>`__). +boundary layer, see section :ref:`Boundary Layer`). Here we need a simple assumption to close the problem. The assumption that we now choose is that liquid cloud fraction *changes* are @@ -1805,13 +1806,11 @@ summarised below: and :math:`CCW` in both dry-convective and cumulus-capped boundary-layers. -.. _`sec:turb_qcl_scheme`: +.. _Turbulence-driven production of subgrid scale liquid cloud: Turbulence-driven production of subgrid scale liquid cloud ---------------------------------------------------------- -.. _`sec:sgt_intro`: - Introduction ~~~~~~~~~~~~ @@ -1831,12 +1830,12 @@ between their theoretically predicted predicted mean cloud properties and the bulk properties of the LES clouds. Subsequently, their model has been used as the basis of subgrid cloud initiation method for use in the Unified Model in conjunction with the PC2 prognostic cloud scheme. In -Section `3.8.2 <#sec:sgt_model_describe>`__ we outline the model of +Section :ref:`Model description` we outline the model of :raw-latex:`\cite{fhfk14}`. In Section -`3.8.3 <#sec:sgt_model_implement>`__ we described its implementation in -the GCM. +:ref:`Model implementation and closure relations` +we described its implementation in the GCM. -.. _`sec:sgt_model_describe`: +.. _Model description: Model description ~~~~~~~~~~~~~~~~~ @@ -1928,12 +1927,12 @@ ice supersaturation at water saturation. We use the superscription and water content diagnosed from a parametrization of small-scale turbulent processes. -.. _`sec:sgt_model_implement`: +.. _Model implementation and closure relations: Model implementation and closure relations ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -To implement the model of Section `3.8.2 <#sec:sgt_model_describe>`__ in +To implement the model of Section :ref:`Model description` in the Unified Model, closure relations are needed for the quantities :math:`\sigma_w^2`, :math:`\varepsilon`, :math:`L`, :math:`\tau_{\rm d}` and :math:`S_E`, subject to the constraining relationship given by Eq. @@ -1949,11 +1948,12 @@ prognostic fields, :math:`C_l` and :math:`q_{cl}`. Two methods are available for doing this. In the simplest case, the diagnosed values :math:`C_l^{sgt}` and :math:`q_{cl}^{sgt}` are just treated as increments to model prognostics (option one, in Sec. -`3.8.5 <#sec:sgt_increments>`__ below). A more complex option (see +:ref:`Options for incrementing model prognostics` below). +A more complex option (see option two, below) is to increment the model fields via the PC2 Erosion functionality. -.. _`sec:sgt_closures`: +.. _Closure relations: Closure relations ~~~~~~~~~~~~~~~~~ @@ -1970,7 +1970,7 @@ vertical grid spacing in each grid box: :math:`L=\beta_{mix} \Delta z`, where :math:`\Delta z` is calculated as the height different between the :math:`\rho`-levels adjacent to the given :math:`\theta`-point. The parameter, :math:`\beta_{mix}`, is an adjustable constant that the user -can define (see Section `3.8.6 <#sec:sgt_options>`__ below), however it +can define (see Section :ref:`Other user options` below), however it should be of order one. To obtain :math:`\tau_{\rm d}` we impose an eddy size constraint: @@ -1999,12 +1999,12 @@ taken to be the grid box mean values. The first moment of the ice PSD, :math:`{\cal M}_1`, is found from the parametrization, due to :raw-latex:`\cite{fhbicc05}`, described in Section 4.1 of UMDP26. -.. _`sec:sgt_increments`: +.. _Options for incrementing model prognostics: Options for incrementing model prognostics ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -Using the information in Section `3.8.4 <#sec:sgt_closures>`__ to obtain +Using the information in Section :ref:`Closure relations` to obtain closed expressions for the subgrid PDF of :math:`S_i`-fluctuations allows :math:`C_l^{sgt}` and :math:`q_{cl}^{sgt}` to be calculated. These will be non-zero only where there is turbulence as diagnosed by @@ -2065,11 +2065,11 @@ where :math:`q_{cl}` is the liquid cloud amount prior to calling to the turbulent production scheme. The cloud fraction increments are calculated by calling PC2 Erosion with :math:`\left( \Delta q_{cl} \right)_{sgt}` as input. See Section -`4.3 <#sec:turb>`__ for details on how the PC2 Erosion process works. +:ref:`PC2 erosion` for details on how the PC2 Erosion process works. This method gives cloud fraction increments that are consistent with PC2 cloud scheme. -.. _`sec:sgt_options`: +.. _Other user options: Other user options ~~~~~~~~~~~~~~~~~~ @@ -2103,7 +2103,7 @@ The following variables and logical switches are optional inputs: #. ``mp_czero`` defines the constant parameter :math:`C_0` (defaults to :math:`C_0=10`). -.. _`sec:app_um`: +.. _Application to the Unified Model: Application to the Unified Model ================================ @@ -2111,7 +2111,7 @@ Application to the Unified Model This section describes the way in which the physical concepts described in the above section are applied to the sections of the Unified Model, in order to build up the complete prognostic scheme. Description of the -actual subroutines themselves follow in section `5.2 <#sec:code>`__. +actual subroutines themselves follow in section :ref:`Code Structure`. Note that the large-scale precipitation and convection schemes have considerable documentation below, since these schemes have been heavily modified for PC2. The other schemes use generic forcing scenarios, hence @@ -2120,7 +2120,6 @@ their desciption here is much shorter. Remember, whenever a signficiant able to represent the corresponding condensation and changes in cloud fractions. -.. _`sec:rad`: Radiation --------- @@ -2129,9 +2128,10 @@ The shortwave and longwave radiation schemes both alter the temperature of the atmosphere, hence we need to calculate the corresponding condensation and cloud fraction changes. For both shortwave and longwave, we use the homogeneous forcing routines (section -`3.2 <#sec:homog>`__) for :math:`\overline{q_{cl}}` and :math:`C_l`, +:ref:`Homogeneous forcing`) for :math:`\overline{q_{cl}}` and :math:`C_l`, (using eqn. `[eq:deltaqc_exp2] <#eq:deltaqc_exp2>`__ to calculate the -:math:`Q_c` forcing) and then the method in section `3.6 <#sec:ct>`__ to +:math:`Q_c` forcing) and then the method in +section :ref:`Ice cloud and mixed phase regions` to calculate :math:`C_t` changes. There is no :math:`\overline{q_{cf}}` change associated with this process since the deposition / sublimation process is performed within the large-scale precipitation scheme (as it @@ -2141,9 +2141,10 @@ It is reasonable to question whether homogeneous forcing is a reasonable model to use when we know that a large proportion of the heating associated with radiative transfer in the atmosphere comes from the cloudy air and is not evenly spread across the gridbox. Possible -developments are discussed in section `7.4.7 <#sec:homog_improve>`__. +developments are discussed in section +:ref:`Homogeneous forcing section improvements`. -.. _`sec:precip`: +.. _Large-scale precipitation: Large-scale precipitation ------------------------- @@ -2171,7 +2172,6 @@ single ice cloud fraction is stored, the assumption being that the two ice categories are completely overlapped with each other. Graupel is not considered to contribute to the ice cloud fraction. -.. _`sec:lsp_fall`: Fall of ice ~~~~~~~~~~~ @@ -2184,7 +2184,7 @@ spread of ice cloud fraction at a particular level (hence :math:`\overline{q_{cf}}` that leaves a gridbox does not reduce :math:`C_f` in that gridbox). The in-cloud ice content simply reduces due to the fall out of ice - it is the sublimation term (section -`4.2.4 <#sec:mp_depsub>`__) that erodes the fall streaks. However, ice +:ref:`Deposition and sublimation`) that erodes the fall streaks. However, ice that falls into a clear layer from above may increase the ice cloud fraction. We parametrize this by considering the horizontal overlap of ice clouds between two model layers, and the fall speed of ice between @@ -2225,7 +2225,7 @@ layer below can be filled by ice in the timestep: where :math:`\Delta t` is the timestep. We now choose to assume a minimum overlap between the liquid and the ice phases (as in section -`3.6 <#sec:ct>`__). +:ref:`Ice cloud and mixed phase regions`). .. math:: @@ -2241,7 +2241,7 @@ the ice cloud fraction overhang. Consequently, the option not to use the “wind shear value” when calculating the overhang is available in the UMUI (from version 7.6 onwards).** -.. _`sec:lsp_homo`: +.. _Homogeneous nucleation: Homogeneous nucleation ~~~~~~~~~~~~~~~~~~~~~~ @@ -2276,7 +2276,7 @@ cloud. These give the following changes: \label{eq:lsp_het} \end{aligned} -.. _`sec:mp_depsub`: +.. _Deposition and sublimation: Deposition and sublimation ~~~~~~~~~~~~~~~~~~~~~~~~~~ @@ -2285,7 +2285,7 @@ This term exerts one of the most important influences on the ice cloud in the whole model (this applies to the control as well as for PC2). Contained in the formulation is a subgrid-scale assumption that causes equivalent effects to that for a moisture PDF under the ‘:math:`s`’ -framework (section :ref:`The ‘s’ distribution`). However, since +framework (section :ref:`The 's' distribution`). However, since :math:`{q_{cf}}` changes slowly in response to local changes in :math:`q` and :math:`T`, we cannot base the :math:`q_{cf}` response on the same instantaneous condensation framework. It would be useful to @@ -2512,7 +2512,7 @@ a large-scale lifting process will be required in order to condense water from the moistened air. We cannot, therefore, allow any change in cloud fractions to occur as a result, subsequent changes are calculated elsewhere in the model (e.g. by the lifting process, section -`4.8 <#sec:pres>`__). +:ref:`Response to pressure changes`). Accretion ~~~~~~~~~ @@ -2572,11 +2572,12 @@ corresponding large reduction in :math:`C_l`. This is an underlying feature of the PC2 scheme (discussed in :raw-latex:`\cite{wg03}`), and necessarily implies the skewing of the underlying moisture PDF. Subsequent parts of the model (e.g. the width narrowing, section -`3.3 <#sec:width>`__) will, of course, act on the modified fields to +:ref:`Changing the width of the PDF - PC2 erosion`) will, of course, +act on the modified fields to adjust the cloud fractions further, but remember that these are separate processes and modelled elsewhere in the timestep. -.. _`sec:turb`: +.. _PC2 erosion: PC2 erosion ----------- @@ -2626,9 +2627,11 @@ term “dbsdtbs1” which scales with the rate of homogeneous forcing :math:`\frac{\partial Q_c}{\partial t}`. However this term is always set to zero on input to these routines so is never used. -The width-narrowing formulation of section `3.3 <#sec:width>`__ is used +The width-narrowing formulation of section +:ref:`Changing the width of the PDF - PC2 erosion` is used to calculate increments in :math:`\overline{q_{cl}}` and :math:`C_l`. -Using the liquid - ice cloud overlap ideas of section `3.6 <#sec:ct>`__ +Using the liquid - ice cloud overlap ideas of +section :ref:`Ice cloud and mixed phase regions` then gives the associated :math:`C_t` change. This background narrowing term, :math:`\Upsilon`, is originally based upon work by :raw-latex:`\cite{sg03}`, although it is a parameter that has been @@ -2639,14 +2642,15 @@ Numerical application of the original width-narrowing method ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Because of the strong link the mathematical expressions for width -narrowing (section `3.3 <#sec:width>`__) have with the expressions for -the homogeneous forcing (section `3.2 <#sec:homog>`__), we choose to +narrowing (section :ref:`Changing the width of the PDF - PC2 erosion`) +have with the expressions for +the homogeneous forcing (section :ref:`Homogeneous forcing`), we choose to represent the timestepping of this process in exactly the same way as for the homogeneous forcing (in fact, in the Unified Model code we use -the same subroutine, see section `5.2 <#sec:code>`__). As before, we use +the same subroutine, see section :ref:`Code Structure`). As before, we use a simple forward timestepping of :math:`C_l`, with :math:`Q_c` given by (`[eq:qc_eq_qt-qs] <#eq:qc_eq_qt-qs>`__) and :math:`a_L` defined as -discussed in section `3.2.3 <#sec:homog_num_app>`__ and discretize eq +discussed in section :ref:`Numerical application` and discretize eq `[eq:dcdt_width] <#eq:dcdt_width>`__ as: .. math:: @@ -2776,7 +2780,7 @@ used to calculate the change in :math:`C_l` using the same moisture PDF assumptions as were used in the original PC2 erosion formulation. To achieve this, we combine equations `[eq:dcdt_width] <#eq:dcdt_width>`__ and `[eq:dqcldt_width] <#eq:dqcldt_width>`__ from section -`3.3 <#sec:width>`__ to eliminate +:ref:`Changing the width of the PDF - PC2 erosion` to eliminate :math:`\frac{1}{b_s} \frac{\partial b_s}{\partial t}` and write :math:`\frac{\partial C_l}{\partial t}` as a function of :math:`\frac{\partial \overline{q_{cl}}}{\partial t}`: @@ -2797,7 +2801,7 @@ for :math:`C_l` and the introduction of some surface area dependence leads to this formulation being referred to as a “hybrid” cloud-surface-area erosion method. -.. _`sec:erosion_numerics`: +.. _Numerical application of the hybrid erosion method: Numerical application of the hybrid erosion method ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ @@ -2844,7 +2848,7 @@ cumulus regimes. off. This means any cloud detrained by convection during the current timestep cannot be eroded until the following timestep, and so is still present at end-of-timestep. As discussed in section - `7.4.11 <#sec:timestepping>`__, this leads to a problematic timestep + :ref:`Time-stepping`, this leads to a problematic timestep sensitivity, since the amount of cloud not subject to erosion is the convection increment, which scales with the timestep length. @@ -3012,7 +3016,7 @@ cumulus regimes. \frac{1}{q_{cl}} \frac{\partial q_{cl}}{\partial t} \label{eq:dcdt_hybrid_1} - Under homogeneous forcing (section `3.2 <#sec:homog>`__), we + Under homogeneous forcing (section :ref:`Homogeneous forcing`), we defined the PDF height at the saturation boundary when near the cloudy end of the PDF as :math:`G(-Q_c) = \frac{n+1}{n+2} \frac{C_l^2}{q_{cl}}` (eq @@ -3075,7 +3079,7 @@ cumulus regimes. where :math:`SD` is the saturation defecit, and :math:`a_L` is the dimensionless factor defined in eq `[eq:a_L] <#eq:a_L>`__. Following the derivation in section - `4.9.3 <#sec:smooth_initiation>`__ (eq + :ref:`“Smooth” initiation logic` (eq `[eq:qc_plus_sd] <#eq:qc_plus_sd>`__), we can write this in terms of the liquid-water content: :math:`SD = q_{cl} - Q_c` (where :math:`Q_c` was defined in eq @@ -3247,7 +3251,7 @@ temperature or moisture content of the model gridboxes, hence PC2 assumes no change in the condensate and cloud fractions as a result of these processes. -.. _`sec:advec`: +.. _Advection: Advection --------- @@ -3262,7 +3266,7 @@ following each parcel, which will cause an accompanying adiabatic temperature change. These advective pressure and temperature changes imply a homogeneous forcing, which yields a change in :math:`\overline{q_{cl}}` and :math:`C_l` in addition to their transport -by the winds. This is described in section `4.8 <#sec:pres>`__. +by the winds. This is described in section :ref:`Response to pressure changes`. If the UM namelist switch **l_pc2_sl_advection** is turned on, the PC2 homogeneous forcing response to advection is calculated straight after @@ -3271,7 +3275,7 @@ from advection is combined with the Eulerian pressure change from the dynamics Helmholtz solver, and the resulting homogeneous forcing of liquid cloud is computed at the end of the timestep. -.. _`sec:bl`: +.. _Boundary Layer: Boundary Layer -------------- @@ -3314,7 +3318,7 @@ denominator is small, we will, to avoid numerical problems, set :math:`C_i` to 1 if :math:`q_C^S - \overline{q_{cf}} < 1 \times 10^{-10} kg kg^{-1}`. Note that we do not use the multiple phases injection source expressions -(section `3.5.1 <#sec:multiple>`__ and equation +(section :ref:`Multiple phases in the injection source` and equation `[eq:cff_ts] <#eq:cff_ts>`__). This is because the liquid water changes are not associated with the plume model. @@ -3327,7 +3331,7 @@ the ice water content is reduced, the ice cloud fraction is reduced, in such as way as to maintain the same in-cloud ice water content. The :math:`C_t` changes are calculated using the minimum overlap method -of section `3.6 <#sec:ct>`__. +of section :ref:`Ice cloud and mixed phase regions`. In *ni-imp-ctl* the control code inhibits the call to the diagnostic cloud scheme if there is deep or shallow convection occurring and the @@ -3343,7 +3347,7 @@ level above the boundary layer mixed layer*. This choice (i.e. ntml) is seen to give improved results in PC2, and is arguably a more physical reasonable choice anyway than using ntml+1. -.. _`sec:convec`: +.. _Convection: Convection ---------- @@ -3352,11 +3356,11 @@ This section concentrates specifically upon the PC2 interface to the convection scheme. In the current formulation of the UM, only a mass-flux convection scheme exists, and this is what is described below. Work to interface PC2 to the developing turbulence based convection -scheme is commented upon in section `7.4.4 <#sec:tbcs>`__. +scheme is commented upon in section :ref:`Turbulence based convection scheme`. An alternative way of calculating cloud fraction increments is currently under development and is described in section -`4.7.10 <#sec:conv-simpler>`__. +:ref:`Tidier way of coupling convection and PC2`. A traditional view of convective parametrization is a scheme that transports vapour, :math:`q`, heat, :math:`\theta`, and momentum, @@ -3382,7 +3386,7 @@ Introduction to the convective mass flux scheme Within the mass flux scheme the net change in :math:`\overline{q_{cl}}` and :math:`C_l` etc. comes from two distinct sources. Firstly, the condensate and cloud fraction injected from the plume (the :math:`Q4` -terms, section `3.5 <#sec:inhomog>`__); secondly, the condensation +terms, section :ref:`Injection forcing`); secondly, the condensation response to the vapour and heat changes associated with the detrainment and compensating subsidence. Strictly, we will see that the :math:`Q4` terms also include the contribution to the condensate transport by the @@ -3404,10 +3408,10 @@ We therefore split the convective contribution in where :math:`Q_{environment}` is the condensation associated with changes in the vapour and temperature from the detrainment and compensating subsidence. Similar splits are made for the cloud -variables, where the injection forcing, section `3.5 <#sec:inhomog>`__, +variables, where the injection forcing, section :ref:`Injection forcing`, is used to calculate the first term from :math:`Q4_l`. Section `4.7.3 <#subsect:q4calculation>`__ looks at the issue of the calculation -of :math:`Q4_l` etc., and section `4.7.4 <#sec:conv_homog>`__ looks at +of :math:`Q4_l` etc., and section :ref:`Background condensation` looks at the calculation of :math:`Q_{environment}`, and its associated cloud fraction change. We first look at the basic transport equations in a mass flux convection scheme. @@ -3906,7 +3910,7 @@ and } \right]& { } & \label{eq:envirolf} \end{aligned} -.. _`sec:conv_homog`: +.. _Background condensation: Background condensation ~~~~~~~~~~~~~~~~~~~~~~~ @@ -4104,8 +4108,9 @@ Homogeneous forcing of the environment by convective-subsidence pressure change To this end, the code includes an option to perform the homogeneous forcing of liquid cloud by convection using the “pressure forcing” from the convective subsidence, consistent with the pressure forcing by -large-scale advection (see sections `4.5 <#sec:advec>`__ and -`4.8 <#sec:pres>`__). This approach replaces the above method of +large-scale advection (see sections :ref:`Advection` and +:ref:`Response to pressure changes`). +This approach replaces the above method of homogeneous forcing by convection if the UM namelist switch **l_pc2_homog_conv_pressure** is turned on. By applying the same homogeneous forcing method for advection and convectively-forced @@ -4228,7 +4233,7 @@ temperatures less than around :math:`-42 ^{\circ} C`, with the tuning allowing less precipitation and greater detrainment. This change is necessary in order to produce thick enough anvil clouds. -.. _`sec:plume_phase`: +.. _Phase of condensate: Phase of condensate ~~~~~~~~~~~~~~~~~~~ @@ -4253,7 +4258,7 @@ is -10 :math:`^{\circ}` C. 1, & T_{plume} < -10 ^{\circ} C \end{array} \right. -.. _`sec:conv-simpler`: +.. _Tidier way of coupling convection and PC2: Tidier way of coupling convection and PC2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ @@ -4290,7 +4295,7 @@ temperature thresholds by maintaining a similar linear ramp. For e.g., condensate is assumed to be all-liquid for T :math:`\geq` :math:`tnuc_n` and all-ice for T :math:`\leq` :math:`tnuc_n` - 10.0 -.. _`sec:conv_input_profs`: +.. _Condensation adjustment in the profiles input to the convection scheme: Condensation adjustment in the profiles input to the convection scheme ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ @@ -4314,17 +4319,17 @@ additional condensation adjustments from PC2 before the convection call: - **l_pc2_sl_advection**: performs homogeneous forcing response to Semi-Lagrangian advection immediately after the advection calculation, instead of at the end of the timestep (see section - `4.8 <#sec:pres>`__). + :ref:`Response to pressure changes`). - **l_cloud_call_b4_conv**: performs an additional call to PC2 initiation (and PC2 checks) before the convection scheme (see section - `4.9 <#sec:init2>`__). This should catch any instances where + :ref:`Initiation`). This should catch any instances where large-scale ascent or other processes have brought the profiles after advection to near or beyond saturation, in grid-points where there was no liquid cloud already present (and so no homogeneous forcing response). -.. _`sec:pres`: +.. _Response to pressure changes: Response to pressure changes ---------------------------- @@ -4333,7 +4338,7 @@ A pressure change following the parcel during the timestep will result in an adiabatic temperature change which will force condensation, hence we must include this temperature change forcing within PC2. The majority of this pressure change comes from vertical advection (although not -all). Remember that the advection (section `4.5 <#sec:advec>`__), on its +all). Remember that the advection (section :ref:`Advection`), on its own, does not cause condensation, it merely moves the existing cloud field. @@ -4443,31 +4448,32 @@ The splitting of the pressure forcing call under the **l_pc2_sl_advection** switch was originally implemented to make the profiles passed to convection more realistic. -.. _`sec:init2`: +.. _Initiation: Initiation ---------- -As discussed in section `3.4 <#sec:init>`__, there are occasions when +As discussed in section :ref:`Initiation of cloud`, there are occasions when :math:`\overline{q_{cl}}` and :math:`C_l` need to be initiated from 0 or 1. The application of the initiation is given in section -`3.4 <#sec:init>`__. The initiation forms a new, separate block of PC2 +:ref:`Initiation of cloud`. The initiation forms a new, separate block of PC2 code to perform this calculation, and is located immediately following -the pressure change response (section `4.8 <#sec:pres>`__). Also, if the -UM namelist switch **l_cloud_call_b4_conv** is set to true, an +the pressure change response (section :ref:`Response to pressure changes`). +Also, if the UM namelist switch **l_cloud_call_b4_conv** is set to true, an additional call to PC2 initiation is performed before the convection scheme, to ensure that the condensation response to advection and other forcings earlier in the timestep has been accounted for in the profiles passed to the convection scheme, even if there was no cloud already present for homogeneous forcing to act upon. (see section -`4.7.12 <#sec:conv_input_profs>`__). +:ref:`Condensation adjustment in the profiles input to the convection scheme`). There are currently 3 options for the conditions under-which initiation may occur. For all of these options, if using the bimodal cloud scheme to do initiation within PC2, then the tests on :math:`RH_T` relative to :math:`RH_{crit}` are replaced by equivalent tests for whether the saturation boundary lies within the bounds of the bimodal scheme’s -assumed PDF, as described in section `3.4.3 <#sec:bimodal_init>`__. +assumed PDF, as described in section +:ref:`Initiation using the bimodal scheme`. “Original” initiation logic ~~~~~~~~~~~~~~~~~~~~~~~~~~~ @@ -4537,7 +4543,7 @@ Or: where :math:`C_{tol}` can be set via the UM namelist; its original standard value is 0.005. Note this threshold is also used to remove small cloud-fractions after initiation; see section -`4.9.4 <#sec:checks2>`__. +:ref:`Additional checks after PC2 initiation`. This is very similar to the “Original” initiation logic described above, but with the following differences: @@ -4553,7 +4559,7 @@ but with the following differences: - The different threshold when initiating super-cooled cloud is removed. -.. _`sec:smooth_initiation`: +.. _“Smooth” initiation logic: “Smooth” initiation logic ~~~~~~~~~~~~~~~~~~~~~~~~~ @@ -4683,7 +4689,7 @@ implementation of `[eq:dcl_init2] <#eq:dcl_init2>`__ in the code simply uses :math:`q_{cl} - Q_c` in place of :math:`SD`, and :math:`\Delta q_{cl}` in place of :math:`\Delta SD`). -.. _`sec:checks2`: +.. _Additional checks after PC2 initiation: Additional checks after PC2 initiation ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ @@ -4693,7 +4699,7 @@ For numerical reasons, it is possible to obtain very low, but non zero, values of :math:`C_l` (and equivalently values very close to, but not equal to, 1). The code will reset these clouds to either a fraction of 0 or 1, as appropriate. We choose to apply these terms here and not in the -Bounds Checking part of the code (section `4.10 <#sec:checks>`__) +Bounds Checking part of the code (section :ref:`Bounds checking`) because these are not required to obtain consistency between fields, but are ‘tidying up’ pieces of code, although they may reasonably also be applied in the Bounds Checking. Care needs to be taken when choosing the @@ -4759,10 +4765,10 @@ zero). Note that if these checks are relaxed by lowering the thresholds :math:`C_{tol}` and :math:`C_{tol 2}` to near-zero, similar checks are still performed independently by the bounds checking described in -section `4.10 <#sec:checks>`__, but with a much lower threshold of +section :ref:`Bounds checking`, but with a much lower threshold of :math:`C_{tol 3} = 1 \times 10^{-12}`. -.. _`sec:checks`: +.. _Bounds checking: Bounds checking --------------- @@ -4779,7 +4785,7 @@ to ensure consistency between these values. The bounds checking is performed three times during the timestep. Firstly, after the parallel part of the physics (*atmos-physics1*) is -complete; secondly, before the initiation (section `3.4 <#sec:init>`__) +complete; secondly, before the initiation (section :ref:`Initiation of cloud`) is called; thirdly, after the initiation is called. Firstly, if :math:`C_l > 1 - C_{tol 3}` then :math:`C_l` is set to 1. @@ -4964,7 +4970,7 @@ overlap situation and then the minimum overlap situation. .. _section-8: Finally, there is a homogeneous nucleation term applied, similar to that -in the large-scale precipitation (section `4.2.2 <#sec:lsp_homo>`__). +in the large-scale precipitation (section :ref:`Homogeneous nucleation`). This is a fast microphysics process, and must act to ensure that no liquid cloud created by the initiation is allowed to persist in this phase if the temperature is cold enough. Hence, if @@ -5001,7 +5007,7 @@ consistent sinks of qcl and CFL” prevents the QCL increment from trying to remove too much liquid condensate and hence reduces the models reliance on Q-Pos to deal with the inconsistencies. -.. _`sec:da`: +.. _Data Assimilation: Data Assimilation ----------------- @@ -5056,29 +5062,32 @@ ill-conditioning of this solution near :math:`C_l = 1`. In practice, the ill-conditioning of (`[eq:da2] <#eq:da2>`__) and (`[eq:da3] <#eq:da3>`__) becomes too numerically awkward for us to apply the full solution based on homogeneous forcing, although, for -completeness, we outline it in Appendix `6 <#sec:appendix-da>`__. Hence +completeness, we outline it in Appendix +:ref:`Appendix; Alternative PC2 - Data Assimilation formulations`. Hence we have chosen to apply a much simpler model. Here we use simply the data assimilation increments :math:`\Delta \overline{q}` and :math:`\Delta \overline{T}` within the standard homogeneous forcing -(section `3.2 <#sec:homog>`__), even though we are fully aware that this +(section :ref:`Homogeneous forcing`), even though we are fully aware that this is inconsistent (because :math:`\Delta \overline{q}` and :math:`\Delta \overline{T}` are not forcings, but are forcings plus the condensation. This allows us an *estimate* of :math:`\Delta \overline{q_{cl}}` and :math:`\Delta{C_l}`, via the homogeneous forcing routine (and :math:`\Delta C_t` via the standard updating described in section -`3.6 <#sec:ct>`__). These are the quantities applied as the equivalent +:ref:`Ice cloud and mixed phase regions`). +These are the quantities applied as the equivalent data assimilation increments for :math:`\Delta \overline{q_{cl}}`, :math:`\Delta{C_l}` and :math:`\Delta C_t`. The increments :math:`\Delta \overline{q}` and :math:`\Delta \overline{T}` remain those that the data assimilation scheme itself calculated. -Appendix `6 <#sec:appendix-da>`__ gives, for completeness, the +Appendix :ref:`Appendix; Alternative PC2 - Data Assimilation formulations` +gives, for completeness, the alternative numerical technique for the solution of (`[eq:da2] <#eq:da2>`__) and (`[eq:da3] <#eq:da3>`__). However, we stress that this technique is not used within the current PC2 formulation. -.. _`sec:um`: +.. _Implementation in the Unified Model: Implementation in the Unified Model =================================== @@ -5092,7 +5101,6 @@ much as possible in a similar way to the condensate variables. Hence, wherever the condensed water variables :math:`q_{cl}` and :math:`q_{cf}` are updated, the cloud fractions need to be updated consistently. -.. _`sec:acf`: Area cloud fraction ------------------- @@ -5116,7 +5124,7 @@ cloud fraction given the volume cloud fraction, taking into account the size of the grid box. The setting of the area cloud fraction is performed at the end of the timestep. -.. _`sec:code`: +.. _Code Structure: Code Structure -------------- @@ -5148,7 +5156,7 @@ before the microphysics scheme (within *pc2_turbulence_ctl*). Note there is also an optional call to *pc2_turbulence_ctl* after the microphysics scheme, which is used only to estimate the cloud fraction change consistent with the turbulent production of liquid cloud (see -section `3.8 <#sec:turb_qcl_scheme>`__). +section :ref:`Turbulence-driven production of subgrid scale liquid cloud`). Most PC2 code is protected by IF tests on the namelist input *i_cld_vn* = 2 (PC2 in the GUI). However, within the convection scheme, the code is @@ -5326,8 +5334,8 @@ Main Tree from atm_step_4a - | ls_arcld | \* (Smith scheme with area cloud fraction; see - `5.2.2 <#subsubsec:smith_acf>`__ for a drill-down - inside this routine) + :ref:`Smith scheme with area cloud fraction` + for a drill-down inside this routine) - | bm_ctl | \* (bimodal scheme) @@ -5337,7 +5345,7 @@ Main Tree from atm_step_4a - | pc2_initiation_ctl | \* (interface to PC2 initiation and consistency-checks; see - `5.2.3 <#subsubsec:pc2_initiation>`__ for a + :ref:`PC2 initiation` for a drill-down inside this routine) .. container:: tcolorbox @@ -5411,8 +5419,8 @@ Main Tree from atm_step_4a - | ls_arcld | \* (interface to diagnostic Smith scheme and area cloud fraction; see - `5.2.2 <#subsubsec:smith_acf>`__ for a drill-down - inside this routine) + :ref:`Smith scheme with area cloud fraction` + for a drill-down inside this routine) - | bm_ctl | \* (bimodal cloud scheme) @@ -5441,7 +5449,7 @@ Main Tree from atm_step_4a - | pc2_assim | \* (PC2 reponse to the analysis increments; see - `5.2.4 <#subsubsec:pc2_assim>`__ for a drill-down + :ref:`PC2 Data Assimilation` for a drill-down inside this routine) - ls_acf_brooks (calculate area cloud fraction using @@ -5449,8 +5457,8 @@ Main Tree from atm_step_4a - ls_arcld (call diagnostic Smith scheme with area cloud fraction again to account for the analysis increments; - see `5.2.2 <#subsubsec:smith_acf>`__ for a drill-down - inside this routine) + see :ref:`Smith scheme with area cloud fraction` + for a drill-down inside this routine) .. container:: tcolorbox @@ -5476,7 +5484,7 @@ Main Tree from atm_step_4a - | pc2_initiation_ctl | \* (interface to PC2 initiation and consistency-checks; see - `5.2.3 <#subsubsec:pc2_initiation>`__ for a drill-down + :ref:`PC2 initiation` for a drill-down inside this routine) .. container:: tcolorbox @@ -5487,8 +5495,8 @@ Main Tree from atm_step_4a - | ls_arcld | \* (interface to diagnostic Smith scheme and area cloud - fraction; see `5.2.2 <#subsubsec:smith_acf>`__ for a - drill-down inside this routine) + fraction; see :ref:`Smith scheme with area cloud fraction` + for a drill-down inside this routine) - | bm_ctl | \* (bimodal cloud scheme) @@ -5502,7 +5510,7 @@ Main Tree from atm_step_4a - | pc2_assim | \* (PC2 reponse to the analysis increments; see - `5.2.4 <#subsubsec:pc2_assim>`__ for a drill-down inside + :ref:`PC2 Data Assimilation` for a drill-down inside this routine) - | initial_pc2_check @@ -5518,7 +5526,7 @@ Drill-downs within some routines in the call tree are listed separately below, to avoid duplication (since these routines are called in multiple different places in the tree)... -.. _`subsubsec:smith_acf`: +.. _Smith scheme with area cloud fraction: Smith scheme with area cloud fraction ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ @@ -5557,7 +5565,7 @@ Smith scheme with area cloud fraction | ls_acf_brooks | \* (estimate area cloud fraction) -.. _`subsubsec:pc2_initiation`: +.. _PC2 initiation: PC2 initiation ~~~~~~~~~~~~~~ @@ -5605,7 +5613,7 @@ PC2 initiation - | pc2_homog_plus_turb | \* (generic homogeneous forcing routine used to interpolate) -.. _`subsubsec:pc2_assim`: +.. _PC2 Data Assimilation: PC2 Data Assimilation ~~~~~~~~~~~~~~~~~~~~~ @@ -5632,7 +5640,6 @@ PC2 Data Assimilation | \* (self-consistency checks on prognostic cloud fractions and water contents) -.. _`sec:diags`: Diagnostics ----------- @@ -5746,8 +5753,9 @@ rest of the SCM uses the same PC2 code as the full model. Note that the change to PC2 homogeneous forcing from advection under the UM namelist switch **l_pc2_sl_advection** (see section -`4.8 <#sec:pres>`__) is also mirrored in the Single-Column Model. If -this switch is turned on, the PC2 homogeneous forcing call using the SCM +:ref:`Response to pressure changes`) is also mirrored in the +Single-Column Model. +If this switch is turned on, the PC2 homogeneous forcing call using the SCM forcing increments is moved straight after the call to the forcing routine, so that the condensation adjustment is performed before the call to atmos_physics2. If **l_pc2_sl_advection** is turned on, the PC2 @@ -5762,7 +5770,8 @@ The SCM forcings may comprise one or both of the following: For the latter, we can calculate the pressure change experienced by vertically-advected parcels, and so calculate the PC2 homogeneous forcing response in the same way as we do for Semi-Lagrangian advection -in the full model (see section `4.8 <#sec:pres>`__). For the former, we +in the full model (see section :ref:`Response to pressure changes`). +For the former, we don’t know if the prescribed T,q tendencies are due to advection, radiation, or some other process, so we calculate the PC2 homogeneous forcing response as if the tendencies are applied "in-situ". @@ -6069,19 +6078,19 @@ More information Information on results of the scheme and how to run the PC2 code at various model versions is available on the PC2 web site. -.. _`sec:appendix-da`: +.. _Appendix; Alternative PC2 - Data Assimilation formulations: -Appendix: Alternative PC2 - Data Assimilation formulations +Appendix; Alternative PC2 - Data Assimilation formulations ========================================================== -In this alternative method to section `4.11 <#sec:da>`__ we will assume +In this alternative method to section :ref:`Data Assimilation` we will assume that there exists a homogeneous forcing, :math:`\Delta Q_c`, that gives changes, net of condensation, of :math:`\Delta\overline{q}` and :math:`\Delta\overline{T}`. If we can recover what :math:`\Delta Q_c` is then we can use this to calculate the liquid, :math:`\overline{q_{cl}}`, and liquid cloud fraction, :math:`C_l`, increments. -As in section `4.11 <#sec:da>`__, we start by discretising +As in section :ref:`Data Assimilation`, we start by discretising (`[dqcldt] <#dqcldt>`__) to give .. math:: @@ -6154,7 +6163,7 @@ solution. Initially, we calculate :math:`G(-Qc)` and :math:`\Delta Q_c` from the input fields, as in the homogeneous forcing technique (section -`3.2 <#sec:homog>`__) and (`[eq:deltaqc_exp2] <#eq:deltaqc_exp2>`__). +:ref:`Homogeneous forcing`) and (`[eq:deltaqc_exp2] <#eq:deltaqc_exp2>`__). An initial increment, :math:`\Delta C_l^1` is estimated directly using the basic equation @@ -6254,7 +6263,7 @@ the :math:`C_l` terms gives :math:`\Delta \overline{q_{cl \, max}}` as This is a general expression, it is not fixed for a particular PDF. To complete the analysis, we need to estimate :math:`-Q_c+b_s`. To do this, we now make the *assumption* of a power-law type PDF, as in section -`3.4 <#sec:init>`__. If we start from the equivalent of +:ref:`Initiation of cloud`. If we start from the equivalent of (`[eqn19] <#eqn19>`__) but at the :math:`s=-bs` end of the distribution, equation (B.3) in :raw-latex:`\cite{wg03}` can be equivalently written for :math:`(1-C_l)` as: @@ -6266,7 +6275,8 @@ for :math:`(1-C_l)` as: To derive this from (B.3) note that :math:`C_l` is swapped for :math:`1-C_l` and :math:`(b_s - (-Q_c))` is swapped for -:math:`(-Qc - (-b_s))`, as in section `3.4.2 <#sec:numapp_init>`__. +:math:`(-Qc - (-b_s))`, as in +section :ref:`Numerical Application of the Smith method`. Similarly, noting that :math:`\overline{q_{cl}}` can be swapped with :math:`SD`, gives the equivalent to (B.4) in :raw-latex:`\cite{wg03}` as @@ -6380,14 +6390,14 @@ results than simply using the homogeneous forcing method. Further work will be required to enable the implementation of this :math:`\overline{q}` and :math:`\overline{T}` preserving method. -.. _`sec:code-development`: Appendix: Essentials of PC2 for code developers =============================================== This section provides some guidance to code developers on the treatment of PC2. Code developers are advised to read the relevant part of section -`4 <#sec:app_um>`__ to understand the way in which the current PC2 +:ref:`Application to the Unified Model` +to understand the way in which the current PC2 scheme interacts with their section of code. The essence of a prognostic cloud scheme is that each physical part of @@ -6414,7 +6424,7 @@ inhomogeneous forcing) methods. Homogeneous forcing ------------------- -This is described fully in section `3.2 <#sec:homog>`__. This assumes +This is described fully in section :ref:`Homogeneous forcing`. This assumes that the distribution of :math:`q_T - q_{sat}(T_L)` about its gridbox mean is unchanged when a process acts. (The mean will change of course, but we assume that the variations in each part of the gridbox from the @@ -6427,7 +6437,7 @@ necessary updates. Injection forcing ----------------- -This is described fully in section `3.5 <#sec:inhomog>`__. We assume +This is described fully in section :ref:`Injection forcing`. We assume that we already know a condensate increment :math:`q_{cl}` or :math:`q_{cf}` and that a corresponding cloud fraction increment :math:`C_l` or :math:`C_f` (and :math:`C_t`) remains to be estimated. @@ -6584,12 +6594,13 @@ of moisture. Convective cloud increments in the mass-flux framework ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -As discussed in section `3.5.3 <#sec:conv_imp_note>`__, it would be +As discussed in section +:ref:`A note on the implementation of the cloud fraction change`, it would be useful to code up the convective cloud fraction changes to link directly to the mass-flux convection scheme, and not to estimate them from the values of :math:`Q4`, which can introduce errors. -.. _`sec:tbcs`: +.. _Turbulence based convection scheme: Turbulence based convection scheme ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ @@ -6633,7 +6644,7 @@ particularly poor behaviour, and we do not pick up substantial evidence of problems from this in the full model. This remains something to be investigated. -.. _`sec:homog_improve`: +.. _Homogeneous forcing section improvements: Homogeneous forcing section improvements ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ @@ -6725,7 +6736,7 @@ developed. However we note two particular issues that relate to PC2. :math:`\overline{q}`, :math:`q_{sat}`, :math:`\overline{q_{cl}}` and :math:`C_l` etc.? -.. _`sec:timestepping`: +.. _Time-stepping: Time-stepping ~~~~~~~~~~~~~ @@ -6754,7 +6765,8 @@ the clouds is of order the timestep - ideally we wouldn’t want to try to model anything prognostically when the cycling time is less than the timestep. -As discussed in section `4.3.4 <#sec:erosion_numerics>`__, the timestep +As discussed in section +:ref:`Numerical application of the hybrid erosion method`, the timestep sensitivity of cloud amounts in shallow cumulus regimes can be addressed by using a more accurate numerical method to solve the erosion term. Several options are available under the UM namelist switch From 06435421efab1fc1c1f529296c6f0c05bccc9450 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Thu, 2 Apr 2026 17:55:31 +0100 Subject: [PATCH 015/116] First attempt to get an equation reference to work. --- documentation/source/conf.py | 7 ++++++- .../science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst | 2 +- 2 files changed, 7 insertions(+), 2 deletions(-) diff --git a/documentation/source/conf.py b/documentation/source/conf.py index 9d6ae59a1c..40ce0c646e 100644 --- a/documentation/source/conf.py +++ b/documentation/source/conf.py @@ -20,7 +20,8 @@ extensions = [ 'sphinx_sitemap', 'sphinx_design', - 'sphinx.ext.intersphinx' + 'sphinx.ext.intersphinx', + 'sphinx.ext.mathjax' ] # Add any paths that contain templates here, relative to this directory. @@ -96,6 +97,10 @@ # numfig = True +# enable \label/\eqref numbering +# load amsmath functionality +mathjax3_config = { "tex": { "tags": "ams", "packages": {"[+]": ["ams"]} } } + # Exclude files from Sphinx processing exclude_patterns = ['common_links.rst'] diff --git a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst index 17f77308fc..8a5240675d 100644 --- a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst +++ b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst @@ -160,7 +160,7 @@ does not). We now introduce the liquid temperature (:math:`T_L`), where and :math:`L` is the latent heat of vaporization and :math:`c_p` is the heat capacity of air. Note that :math:`T_L` is unaffected by changes of phase between vapour and liquid. We now write -(`[eq:basic_qcl] <#eq:basic_qcl>`__) as an *equality* +:math:`\eqref{eq:basic_qcl}` as an *equality* .. math:: From 965e59a8322ffae44b03e1f8d518aca99c9fb176 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Fri, 3 Apr 2026 00:05:47 +0100 Subject: [PATCH 016/116] Finally got equation numbering and referencing to work using math_jax extension! Updated syntax to use this for the first few equations. --- documentation/source/conf.py | 9 +++--- .../cloud_schemes/UMDP30_PC2CloudScheme.rst | 30 ++++++++----------- 2 files changed, 17 insertions(+), 22 deletions(-) diff --git a/documentation/source/conf.py b/documentation/source/conf.py index 40ce0c646e..e201673aed 100644 --- a/documentation/source/conf.py +++ b/documentation/source/conf.py @@ -21,9 +21,12 @@ 'sphinx_sitemap', 'sphinx_design', 'sphinx.ext.intersphinx', - 'sphinx.ext.mathjax' + 'sphinx.ext.mathjax', ] +# Enable equation referencing and cross-referencing +mathjax3_config = { "tex": { "tags": "ams", "packages": {"[+]": ["ams"]}, } } + # Add any paths that contain templates here, relative to this directory. templates_path = ['_templates'] @@ -97,10 +100,6 @@ # numfig = True -# enable \label/\eqref numbering -# load amsmath functionality -mathjax3_config = { "tex": { "tags": "ams", "packages": {"[+]": ["ams"]} } } - # Exclude files from Sphinx processing exclude_patterns = ['common_links.rst'] diff --git a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst index 8a5240675d..6680765698 100644 --- a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst +++ b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst @@ -135,10 +135,9 @@ In this derivation we will consider only liquid condensate. We assume, as above, that, locally, the water content in a cloud is such as to remove any supersaturation. This gives the equation -.. math:: +.. math:: :label: eq:basic_qcl q_{cl} = q_T - q_{sat}(T,p) - \label{eq:basic_qcl} assuming that :math:`q_T > q_{sat}(T,p)` (:math:`q_{cl}` will be zero otherwise). :math:`q_T` is the local total water content, equal to the @@ -152,29 +151,26 @@ with respect to ice when :math:`T < 0 ^{\circ} C`, but the Unified Model does not). We now introduce the liquid temperature (:math:`T_L`), where :math:`T_L` is given by -.. math:: +.. math:: :label: eq:tl T_L = T - \frac{L}{c_p} q_{cl} , - \label{eq:tl} and :math:`L` is the latent heat of vaporization and :math:`c_p` is the heat capacity of air. Note that :math:`T_L` is unaffected by changes of phase between vapour and liquid. We now write -:math:`\eqref{eq:basic_qcl}` as an *equality* +:eq:`eq:basic_qcl` as an *equality* -.. math:: +.. math:: :label: eq:alpha_t_tl q_{cl} = q_T - \left( q_{sat}(T_L,p) + \alpha (T - T_L) \right) - \label{eq:alpha_t_tl} where -.. math:: +.. math:: :label: eq:alpha \alpha = \frac{ q_{sat}(T,p) - q_{sat}(T_L,p) }{T - T_L } . - \label{eq:alpha} -Using (`[eq:tl] <#eq:tl>`__) in (`[eq:alpha_t_tl] <#eq:alpha_t_tl>`__) +Using :eq:`eq:tl` in :eq:`eq:alpha_t_tl` gives the expression .. math:: q_{cl} = q_T - q_{sat} (T_L) - \alpha \frac{L}{c_p} q_{cl} @@ -193,7 +189,7 @@ where :math:`a_L` is given by a_L = \left( 1 + \alpha \frac{L}{c_p} \right) ^{-1} . \label{eq:a_L} -Thus (`[eq:basic_qcl] <#eq:basic_qcl>`__) has been rewritten *exactly* +Thus :eq:`eq:basic_qcl` has been rewritten *exactly* in terms of the conserved variables, :math:`q_T` and :math:`T_L`, although the temperature, :math:`T`, does remain in the definition of :math:`a_L`. We will need to consider variations across a gridbox for a @@ -211,7 +207,7 @@ where :math:`\overline{\phi}` represents the mean of a distibution of :math:`\phi` and :math:`\phi = \overline{\phi} + {\phi}'`. The expression (`[eq:bar_plus_pri1] <#eq:bar_plus_pri1>`__) is *exact* when using the definition of :math:`\alpha` given in -(`[eq:alpha] <#eq:alpha>`__). +:eq:`eq:alpha`. The idea of a PDF scheme is to calculate the first (mean) term, :math:`\overline{\phi}`, from the known gridbox mean parameters, @@ -259,7 +255,7 @@ This gives the equation with the assumption that :math:`s \ge -Q_c` (i.e. :math:`q_{cl} \ge 0`). If :math:`s < -Q_c` then :math:`q_{cl} = 0`. The term :math:`a_L` can be calculated using (`[eq:a_L] <#eq:a_L>`__) from -(`[eq:alpha] <#eq:alpha>`__) with gridbox mean temperatures, i.e. +:eq:`eq:alpha` with gridbox mean temperatures, i.e. .. math:: @@ -740,7 +736,7 @@ respect to temperature at constant pressure (:math:`\frac{\partial q_{sat}}{\partial T}`), and :math:`\beta` is the rate of change of :math:`q_{sat}` with respect to pressure at constant temperature (:math:`\frac{\partial q_{sat}}{\partial p}`). Using -(`[eq:tl] <#eq:tl>`__) to expand :math:`T_L` in terms of :math:`T` and +:eq:`eq:tl` to expand :math:`T_L` in terms of :math:`T` and :math:`q_{cl}` and gathering terms together we obtain .. math:: @@ -753,7 +749,7 @@ We must now note the method we use here to calculate :math:`a_L`, defined in (`[eq:a_L] <#eq:a_L>`__), uses a gradient expansion value of :math:`\alpha` that corresponds to :math:`\frac{\partial{q_{sat}}} {\partial{T}}` at constant pressure and not the chord expression in -(`[eq:alpha] <#eq:alpha>`__). This is because we are trying to find the +:eq:`eq:alpha`. This is because we are trying to find the best estimate of the increments, not the absolute value. We have seen errors arise in simple numerical tests when the chord expression is used. We need to define the temperature around which this calculation of @@ -768,7 +764,7 @@ discussion of this issue, but we note here that the :math:`T` to calculate :math:`\alpha` but the gradient of the chord between :math:`(\overline{T_L}, q_{sat}(\overline{T_L}))` and :math:`(\overline{T}, q_{sat}(\overline{T}))`, as in -(`[eq:alpha] <#eq:alpha>`__). The best choice is dependent on the method +:eq:`eq:alpha`. The best choice is dependent on the method of implementation. PC2 uses (`[eq:a_L] <#eq:a_L>`__) with the standard thermodynamic relationships (e.g. :raw-latex:`\cite{ry89}`, chapter 2) @@ -1067,7 +1063,7 @@ The conversion between :math:`SD` and :math:`\overline{q_{cl}}` follows :math:`\alpha` (and hence :math:`a_L`) in terms of :math:`\frac{\partial q_{sat}(\overline{T})}{\partial t}`, although within this diagnostic calculation of SD it might actually be better to -use the representation (`[eq:alpha] <#eq:alpha>`__) used by the +use the representation :eq:`eq:alpha` used by the diagnostic :raw-latex:`\cite{smith90}` scheme. .. _Numerical Application of the Smith method: From 2a5b5b04e5a72ac885c0c77ac64cddc6c9053059 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Tue, 7 Apr 2026 12:10:07 +0100 Subject: [PATCH 017/116] Corrected equation labelling and referencing across the doc using python script written by copilot. --- .../cloud_schemes/UMDP30_PC2CloudScheme.rst | 1000 +++++++---------- 1 file changed, 425 insertions(+), 575 deletions(-) diff --git a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst index 6680765698..6be957df1b 100644 --- a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst +++ b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst @@ -177,35 +177,32 @@ gives the expression or -.. math:: +.. math:: :label: eq:l_eq_al q_{cl} = a_L \left( q_T - q_{sat}(T_L,p) \right) - \label{eq:l_eq_al} where :math:`a_L` is given by -.. math:: +.. math:: :label: eq:a_L a_L = \left( 1 + \alpha \frac{L}{c_p} \right) ^{-1} . - \label{eq:a_L} Thus :eq:`eq:basic_qcl` has been rewritten *exactly* in terms of the conserved variables, :math:`q_T` and :math:`T_L`, although the temperature, :math:`T`, does remain in the definition of :math:`a_L`. We will need to consider variations across a gridbox for a parametrization scheme, so we expand the expression for condensate -(`[eq:l_eq_al] <#eq:l_eq_al>`__) into terms relating to the gridbox mean +:eq:`eq:l_eq_al` into terms relating to the gridbox mean and variation from the gridbox mean. -.. math:: +.. math:: :label: eq:bar_plus_pri1 q_{cl} = \overline{ a_L \left( q_T - q_{sat}(T_L,p) \right)} + [ a_L \left( q_T - q_{sat}(T_L,p) \right) ]' - \label{eq:bar_plus_pri1} where :math:`\overline{\phi}` represents the mean of a distibution of :math:`\phi` and :math:`\phi = \overline{\phi} + {\phi}'`. The -expression (`[eq:bar_plus_pri1] <#eq:bar_plus_pri1>`__) is *exact* when +expression :eq:`eq:bar_plus_pri1` is *exact* when using the definition of :math:`\alpha` given in :eq:`eq:alpha`. @@ -221,13 +218,12 @@ because :math:`q_{sat}(T_L,p)` is not a linear function of :math:`T_L` :math:`T_L` and :math:`p`. This equivalently implies that :math:`a_L` and :math:`\alpha` are approximated as being constant across the gridbox. The expression now becomes more tractable, -(`[eq:bar_plus_pri1] <#eq:bar_plus_pri1>`__) becoming: +:eq:`eq:bar_plus_pri1` becoming: -.. math:: +.. math:: :label: eq:l_eq_bar_plus_pri q_{cl} = a_L \left( \overline{q_T} - q_{sat}(\overline{T_L},\overline{p}) \right) + a_L \left( {q_T}' - \alpha {T_L}' - \beta {p}' \right) - \label{eq:l_eq_bar_plus_pri} where :math:`\beta = {\frac{\partial q_{sat}}{\partial p}}` at constant temperature. The first term is connected with the mean properties of the @@ -235,34 +231,30 @@ gridbox, and is written as :math:`Q_c`, the second term is connected with the deviation of the local conditions from the mean and is written as :math:`s`. -.. math:: +.. math:: :label: eq:qc_eq_qt-qs Q_c = a_L \left( \overline{q_T} - q_{sat}(\overline{T_L},\overline{p}) \right) - \label{eq:qc_eq_qt-qs} -.. math:: +.. math:: :label: eq:s s = a_L \left( {q_T}' - \alpha {T_L}' - \beta {p}' \right) - \label{eq:s} This gives the equation -.. math:: +.. math:: :label: eq:l_qc_s q_{cl} = Q_c + s - \label{eq:l_qc_s} with the assumption that :math:`s \ge -Q_c` (i.e. :math:`q_{cl} \ge 0`). If :math:`s < -Q_c` then :math:`q_{cl} = 0`. The term :math:`a_L` can be -calculated using (`[eq:a_L] <#eq:a_L>`__) from +calculated using :eq:`eq:a_L` from :eq:`eq:alpha` with gridbox mean temperatures, i.e. -.. math:: +.. math:: :label: eq:alpha_mean \alpha = \frac{ q_{sat}(\overline{T},\overline{p}) - q_{sat}(\overline{T_L},\overline{p}) }{\overline{T} - \overline{T_L} } . - \label{eq:alpha_mean} This definition of :math:`\alpha` and :math:`a_L` will retrieve an *exact* value for the gridbox mean :math:`\overline{q_{cl}}` *if* the @@ -281,18 +273,16 @@ Considering cloud to be where the water content is greater than zero (i.e. where :math:`s > -Q_c`) gives an expression for the liquid cloud *volume* fraction, :math:`C_l`, within the gridbox as -.. math:: +.. math:: :label: eq:int_gs_ds C_l = \int_{s=-Q_c}^{\infty} G(s) ds - \label{eq:int_gs_ds} and the expression for mean condensate, :math:`{\overline{q_{cl}}}`, -using (`[eq:l_qc_s] <#eq:l_qc_s>`__) to expand :math:`q_{cl}`, is +using :eq:`eq:l_qc_s` to expand :math:`q_{cl}`, is -.. math:: +.. math:: :label: eq:qclbar=int \overline{q_{cl}} = \int_{s=-Q_c}^{\infty} (Q_c + s) G(s) ds . - \label{eq:qclbar=int} If we know (parametrize) the PDF given by :math:`G(s)` then we can solve for :math:`C_l` and :math:`\overline{q_{cl}}`. Note that this @@ -312,8 +302,8 @@ bulb temperature :math:`\overline{T}` (from :math:`\overline{T_L}` and :math:`C_l`. The diagnostic scheme effectively allows a calculation of condensation associated with any physical process. However, its results remain tied to the distribution of :math:`G(s)` that is chosen in -(`[eq:int_gs_ds] <#eq:int_gs_ds>`__) and -(`[eq:qclbar=int] <#eq:qclbar=int>`__) and it is this tie that we seek +:eq:`eq:int_gs_ds` and +:eq:`eq:qclbar=int` and it is this tie that we seek to break by the use of a prognostic scheme. Concept of PC2 @@ -365,10 +355,9 @@ ice cloud volume fraction, and :math:`C_t` is the combined ice or liquid cloud volume fraction. The amount of mixed phase cloud, :math:`C_{mp}`, can be calculated by the overlap of the ice and liquid fractions: -.. math:: +.. math:: :label: eq:mp C_{mp} = C_i + C_l - C_t. - \label{eq:mp} The idea is to parametrize each of the terms in the above equations. This approach removes the diagnostic method, hence it will be critical @@ -378,11 +367,11 @@ that we can write expressions for :math:`\overline{T}`, :math:`\overline{p}`, :math:`\overline{q}`, or :math:`\overline{q_{cl}}` in the model* (and similarly for the ice terms). In doing so, we will not lose sight of underlying PDF approach -given by (`[eq:int_gs_ds] <#eq:int_gs_ds>`__) and -(`[eq:qclbar=int] <#eq:qclbar=int>`__) since we will still use the +given by :eq:`eq:int_gs_ds` and +:eq:`eq:qclbar=int` since we will still use the concept of instantaneous condensation for liquid clouds. Equations -`[eq:int_gs_ds] <#eq:int_gs_ds>`__ and -`[eq:qclbar=int] <#eq:qclbar=int>`__ will form the basis of the +:eq:`eq:int_gs_ds` and +:eq:`eq:qclbar=int` will form the basis of the homogeneous forcing methods discussed in section :ref:`Homogeneous forcing`. We note in particular that the convective cloud fraction, previously a quantity that is diagnosed separately from the large-scale cloud @@ -391,7 +380,7 @@ PC2, be included as part of the large-scale cloud fraction. This aspect is similar to the :raw-latex:`\cite{t93}` approach. The final aim of PC2 is that the parametrization of each term in -(`[eq:dqcldt_and_dcdt] <#eq:dqcldt_and_dcdt>`__) is performed by each +:eq:`eq:dqcldt_and_dcdt` is performed by each part of the model that alters :math:`\overline{T}`, :math:`\overline{p}`, :math:`\overline{q}`, :math:`\overline{q_{cl}}` or :math:`\overline{q_{cf}}` as an integral part of that physics or @@ -470,30 +459,28 @@ refer to changes in local values of :math:`T_L` and :math:`q_T` that occur at a rate independent of the part of the gridbox in which they are located. This implies that :math:`G(s)` will not alter due to such a process. Uniform forcing simply alters :math:`Q_c` in -(`[eq:int_gs_ds] <#eq:int_gs_ds>`__) and -(`[eq:qclbar=int] <#eq:qclbar=int>`__). In the Unified Model, this +:eq:`eq:int_gs_ds` and +:eq:`eq:qclbar=int`. In the Unified Model, this concept will be applied to several different sets of physics increments in order to calculate the condensation and cloud fraction changes associated with each one, where the physics routine does not allow the explicit calculation of condensation and cloud fraction changes by another method. Large-scale ascent may be considered a meteorological example of such a process. By differentiating -(`[eq:int_gs_ds] <#eq:int_gs_ds>`__) and -(`[eq:qclbar=int] <#eq:qclbar=int>`__) with respect to time, assuming +:eq:`eq:int_gs_ds` and +:eq:`eq:qclbar=int` with respect to time, assuming uniform forcing (so :math:`{\frac{\partial G}{\partial t}}` terms are zero), we obtain -.. math:: +.. math:: :label: dcdt {{\frac{\partial C_l}{\partial t}} = G(-Q_c) {\frac{\partial Q_c} {\partial t} }.} - \label{dcdt} -.. math:: +.. math:: :label: dqcldt {\frac{\partial \overline{q_{cl}}}{\partial t}} = C_l {\frac{\partial Q_c}{\partial t}} - \label{dqcldt} The quantity :math:`G(-Q_c)` is the value of the PDF of :math:`G` at :math:`s=-Q_c`, which defines the boundary between the saturated and @@ -501,8 +488,8 @@ unsaturated parts of the distribution. If we wish to consider a prognostic cloud scheme with equations for the rate of change of condensate and cloud fraction based upon -(`[dqcldt] <#dqcldt>`__) and (`[dcdt] <#dcdt>`__) then we need to close -(`[dcdt] <#dcdt>`__) by specifying the value of :math:`G(-Q_c)`. We will +:eq:`dqcldt` and :eq:`dcdt` then we need to close +:eq:`dcdt` by specifying the value of :math:`G(-Q_c)`. We will choose to develop a parametrization for this quantity based upon the quantities :math:`C_l`, :math:`\overline{q_{cl}}` and the saturation deficit, :math:`SD`, rather than tie :math:`G(-Q_c)` to a process. The @@ -512,17 +499,15 @@ way it is analogous to the liquid water content, :math:`\overline{q_{cl}}`. Appendix A of :raw-latex:`\cite{wg03}` writes this *definition* as -.. math:: +.. math:: :label: SD {SD = - \int_{-\infty}^{-Q_c} {( s+Q_c ) G(s) ds}} - \label{SD} and shows this is equivalent to -.. math:: +.. math:: :label: SD2 {SD = a_L ( q_{sat}({\overline{T}},{\overline{p}}) - {\overline{q}} ) .} - \label{SD2} The basis behind the parametrization for :math:`G(-Q_c)` is to consider an underlying form of the distribution :math:`G(s)` near the @@ -532,20 +517,18 @@ the value of :math:`s` when a monomodal distribution :math:`G(s)` just equals zero. We suppose that the distribution G can be described as a power law near :math:`s=b_s`. -.. math:: +.. math:: :label: eqn19 G(s) ~ \propto ~ {(-s + b_s)}^n - \label{eqn19} provided :math:`s`__) it can be shown (see appendix +n is a power. From :eq:`eqn19` it can be shown (see appendix B of :raw-latex:`\cite{wg03}`) that -.. math:: +.. math:: :label: eqn20 {G_1(-Q_c) = {\frac{(n+1)}{(n+2)}} {\frac{C_l^2}{\overline{q_{cl}}}} .} - \label{eqn20} An important feature is that the proportionality between :math:`G(-Q_c)` and :math:`{\frac{C^2}{\overline{l}} }` holds for any power law @@ -558,12 +541,11 @@ generalized :math:`G(-Q_c`) closure. If we assume a similar power law for the other end of the distribution we can write a second estimate of :math:`G(-Q_c)` as: -.. math:: +.. math:: :label: eqn21 {G_2(-Q_c) = {\frac{(n+1)}{(n+2)}} {\frac{{(1-C_l)}^2}{SD}} .} - \label{eqn21} -We note that if n tends to zero then (`[eqn21] <#eqn21>`__) is identical +We note that if n tends to zero then :eq:`eqn21` is identical to the expression used by :raw-latex:`\cite{jgt99}`. This is because :raw-latex:`\cite{jgt99}` also uses a similar description of a ‘top-hat’ PDF of fluctuations. @@ -574,8 +556,8 @@ our parameterisation, we must choose suitable weights to apply to the two solutions, and there are currently 2 options for the choice of weights, discussed below. -The equations (`[dqcldt] <#dqcldt>`__), (`[dcdt] <#dcdt>`__) and either -(`[eqn22] <#eqn22>`__) or (`[eq:gmqc_width] <#eq:gmqc_width>`__) below +The equations :eq:`dqcldt`, :eq:`dcdt` and either +:eq:`eqn22` or :eq:`eq:gmqc_width` below form a complete mathematical set for the solution of :math:`C_l` and :math:`\overline{q_{cl}}` under homogeneous forcing, provided that the initial value of :math:`C_l` is not identically 0 or 1. If :math:`C_l` @@ -586,7 +568,7 @@ needs to be initiated in some way. This is discussed further in may be used to trace out an underlying PDF for any input values of :math:`C_l`, :math:`\overline{q_{cl}}` and :math:`SD`. Although we never need to define the complete PDF in PC2 (just the value of -:math:`G(-Q_c)` from equation `[eqn22] <#eqn22>`__), a PDF can be +:math:`G(-Q_c)` from equation :eq:`eqn22`, a PDF can be inferred off-line if required. :raw-latex:`\cite{wg03}` analyse the performance of this parametrization @@ -604,22 +586,21 @@ This option is selected by setting **i_pc2_homog_g_method=1** in the UM large-scale cloud namelist. Under this closure, we choose the weights such that :math:`G_1(-Q_c)` -(`[eqn20] <#eqn20>`__) is used when cloud fractions are small, and -:math:`G_2(-Q_c)` (`[eqn21] <#eqn21>`__) is used when cloud fractions +:eq:`eqn20` is used when cloud fractions are small, and +:math:`G_2(-Q_c)` :eq:`eqn21` is used when cloud fractions are large (a small cloud fraction indicates that the saturation boundary is close to the right-hand end of the PDF, which :math:`G_1(-Q_c)` is based on). We choose relative weights of :math:`{{(1-C_l)}^{m}}` and :math:`{C_l^m }` respectively, where :math:`m` is a power currently set to 0.5. Hence the complete suggested closure of :math:`G(-Q_c)` is: -.. math:: +.. math:: :label: eqn22 {G(-Q_c) = {\frac{(n+1)}{(n+2)}} {\frac{ ( {( 1-C_l )}^m {\frac{C_l^2} {\overline{q_{cl}}}} + C_l^m {\frac{{(1-C_l)}^2}{SD}} ) }{( {(1-C_l)}^m + C_l^m ) }} . } - \label{eqn22} -A problematic property of equation `[eqn22] <#eqn22>`__ is that it goes +A problematic property of equation :eq:`eqn22` is that it goes to infinity if either :math:`q_{cl}` or :math:`SD` goes to zero (and :math:`C_l` is not zero or unity). There are realistic scenarios in which this limit will be approached; e.g. if heavy rain falls through a @@ -627,8 +608,8 @@ layer of cloud, nearly all of the cloud liquid water content maybe removed by accretion, without reducing the cloud-fraction. In this situation, any subsequent homogeneous forcing applied to the cloud will result in a huge tendency in cloud-fraction in equation -`[dcdt] <#dcdt>`__, due to the term -:math:`\frac{C_l^2}{\overline{q_{cl}}}` in `[eqn22] <#eqn22>`__ becoming +:eq:`dcdt`, due to the term +:math:`\frac{C_l^2}{\overline{q_{cl}}}` in :eq:`eqn22` becoming huge. Note that for a homogeneous forcing acting to dry the layer / reduce the @@ -638,9 +619,9 @@ cloud, then the cloud fraction should indeed vanish extremely rapidly under a negative forcing. However, for a positive homogeneous forcing, the cloud-fraction will very rapidly increase in this scenario, for no physical reason. This might not be a problem if one exactly integrated -the differential equations `[dcdt] <#dcdt>`__ and -`[dqcldt] <#dqcldt>`__, since :math:`q_{cl}` would immediately increase -away from zero, so that `[eqn22] <#eqn22>`__ immediately becomes +the differential equations :eq:`dcdt` and +:eq:`dqcldt`, since :math:`q_{cl}` would immediately increase +away from zero, so that :eq:`eqn22` immediately becomes well-defined (an infinitely large tendency maintained for an infinitely small period of time can yield a finite, sensible increment!) Unfortunately, PC2 uses an explicit numerical method, and a finite @@ -648,7 +629,7 @@ Unfortunately, PC2 uses an explicit numerical method, and a finite tendency will yield a very large increment, even if the continuous differential equations would not have done. -In practice, the code that implements `[eqn22] <#eqn22>`__ simply sets +In practice, the code that implements :eq:`eqn22` simply sets :math:`G(-Q_c)` to zero if either :math:`q_{cl}` or :math:`SD` falls below 1.0E-10 kg kg\ :math:`^{-1}`, to avoid a floating-point error. This in itself can be problematic, since leaving :math:`C_l` unmodified @@ -661,8 +642,8 @@ This option is selected by setting **i_pc2_homog_g_method=2** in the UM large-scale cloud namelist. :raw-latex:`\cite{morcrette_2020}` proposed an alternative choice of -weights applied to :math:`G_1(-Q_c)` (`[eqn20] <#eqn20>`__) and -:math:`G_2(-Q_c)` (`[eqn21] <#eqn21>`__), so-as to make each one’s +weights applied to :math:`G_1(-Q_c)` :eq:`eqn20` and +:math:`G_2(-Q_c)` :eq:`eqn21`, so-as to make each one’s weight go to zero in the limit that it goes to infinity, reliably yielding a sensible, finite solution for :math:`G(-Q_c)`. @@ -681,15 +662,14 @@ This is equivalent to weighting the saturated and subsaturated solutions for :math:`G(-Q_c)` by their respective PDF-widths. Note that everything cancels-out in the numerator, so this reduces to: -.. math:: +.. math:: :label: eq:gmqc_width G(-Q_c) = \frac{(n+1)}{(n+2)} \frac{ 1 }{ \frac{\overline{q_{cl}}}{C_l} + \frac{SD}{1-C_l} } - \label{eq:gmqc_width} In full UM tests, :raw-latex:`\cite{morcrette_2020}` found that this closure removed occasional spurious very large cloud-fraction increments -that occur when using `[eqn22] <#eqn22>`__, but did not significantly +that occur when using :eq:`eqn22`, but did not significantly impact the performance of the model forecast. .. _Numerical application: @@ -702,8 +682,8 @@ developed off-line using a single gridbox model, to ensure smooth, accurate and convergent behaviour of the solution, rather than as a result of a mathematical analysis of the problem. -We need to timestep forward (`[dqcldt] <#dqcldt>`__) and -(`[dcdt] <#dcdt>`__) using a timestep of :math:`\Delta{t}`, knowing +We need to timestep forward :eq:`dqcldt` and +:eq:`dcdt` using a timestep of :math:`\Delta{t}`, knowing values of :math:`\Delta{\overline{T}}`, :math:`\Delta{p}`, :math:`\Delta{\overline{q}}` and :math:`\Delta{\overline{q_{cl}}}`, which are provided by an existing physics scheme in the model. We assume @@ -712,24 +692,22 @@ calculated its condensation and cloud fraction increments by another means. Firstly, we need to calculate the forcing term :math:`\Delta{Q_c}`. From -(`[eq:qc_eq_qt-qs] <#eq:qc_eq_qt-qs>`__) we can write +:eq:`eq:qc_eq_qt-qs` we can write -.. math:: +.. math:: :label: eq:deltaqc \Delta{Q_c} = a_L ( \Delta{\overline{q_T}} - \Delta{\overline{q_{sat}(T_L)}} ) - \label{eq:deltaqc} assuming that :math:`a_L` does not change (see below). This can be expanded, using a linear approximation for :math:`q_{sat}(T+\Delta{T},p+\Delta{p})` in terms of :math:`q_{sat}(T,p)` as -.. math:: +.. math:: :label: eq:deltaqc_exp \Delta{Q_c} = a_L ( \Delta{\overline{q}} + \Delta{\overline{q_{cl}}} - \alpha \Delta{\overline{T_L}} - \beta \Delta{\overline{p}} ) - \label{eq:deltaqc_exp} where :math:`\alpha` is the rate of change of :math:`q_{sat}` with respect to temperature at constant pressure @@ -739,14 +717,13 @@ temperature (:math:`\frac{\partial q_{sat}}{\partial p}`). Using :eq:`eq:tl` to expand :math:`T_L` in terms of :math:`T` and :math:`q_{cl}` and gathering terms together we obtain -.. math:: +.. math:: :label: eq:deltaqc_exp2 \Delta{Q_c} = a_L ( \Delta{\overline{q}} - \alpha \Delta{\overline{T}} - \beta \Delta{\overline{p}} ) + \Delta{\overline{q_{cl}}} . - \label{eq:deltaqc_exp2} We must now note the method we use here to calculate :math:`a_L`, -defined in (`[eq:a_L] <#eq:a_L>`__), uses a gradient expansion value of +defined in :eq:`eq:a_L`, uses a gradient expansion value of :math:`\alpha` that corresponds to :math:`\frac{\partial{q_{sat}}} {\partial{T}}` at constant pressure and not the chord expression in :eq:`eq:alpha`. This is because we are trying to find the @@ -765,53 +742,48 @@ discussion of this issue, but we note here that the between :math:`(\overline{T_L}, q_{sat}(\overline{T_L}))` and :math:`(\overline{T}, q_{sat}(\overline{T}))`, as in :eq:`eq:alpha`. The best choice is dependent on the method -of implementation. PC2 uses (`[eq:a_L] <#eq:a_L>`__) with the standard +of implementation. PC2 uses :eq:`eq:a_L` with the standard thermodynamic relationships (e.g. :raw-latex:`\cite{ry89}`, chapter 2) -.. math:: +.. math:: :label: eq:alpha_exp \alpha = \frac{ \epsilon L q_{sat}(\overline{T}) } { R \overline{T}^2} , - \label{eq:alpha_exp} where :math:`R` is the gas constant for dry air, and -.. math:: +.. math:: :label: eq:beta \beta = \frac{-q_{sat}(\overline{T})}{\overline{p}} . - \label{eq:beta} -The right hand side of (`[eq:deltaqc_exp2] <#eq:deltaqc_exp2>`__) now +The right hand side of :eq:`eq:deltaqc_exp2` now contains forcing values which we know from the physics scheme we are applying the homogeneous forcing to. We next estimate :math:`G(-Q_c)` from the parametrization -(`[eqn22] <#eqn22>`__) and the expression for the saturation deficit -(`[SD2] <#SD2>`__). The change in :math:`C_l` is then estimated using a -simple forward step of (`[dcdt] <#dcdt>`__) using -(`[eq:deltaqc_exp2] <#eq:deltaqc_exp2>`__): +:eq:`eqn22` and the expression for the saturation deficit +:eq:`SD2`. The change in :math:`C_l` is then estimated using a +simple forward step of :eq:`dcdt` using +:eq:`eq:deltaqc_exp2`: -.. math:: +.. math:: :label: eq:deltac \Delta{C_l} = G(-Q_c) \Delta{Q_c} . - \label{eq:deltac} The final value of :math:`C_l` is then limited to lie between 0 and 1. -.. math:: +.. math:: :label: eq:c_l^n+1 C_l^{[n+1]} = (0, ~ C_l^{[n]} + \Delta{C_l}, ~ 1) - \label{eq:c_l^n+1} where :math:`[n]` and :math:`[n+1]` label the timesteps. The timestepping of :math:`\overline{q_{cl}}` is more involved. We will use a mid-timestep estimate of :math:`C_l` in the discrete form of -(`[dqcldt] <#dqcldt>`__). +:eq:`dqcldt`. -.. math:: +.. math:: :label: eq:qcl_l^n+1 \overline{q_{cl}}^{[n+1]} = \overline{q_{cl}}^{[n]} + \frac{1}{2} (C_l^{[n]} + C_l^{[n+1]}) \Delta{Q_c}. - \label{eq:qcl_l^n+1} This completes the homogeneous forcing routine. We note that it is possible for :math:`\overline{q_{cl}}^{[n+1]}` to be negative if the @@ -828,8 +800,8 @@ and CFL” performs a check at the end of the homogeneous forcing routines to ensure that we are not trying to remove more condensate than was there to start with. -Note that (`[eq:c_l^n+1] <#eq:c_l^n+1>`__) and -(`[eq:qcl_l^n+1] <#eq:qcl_l^n+1>`__) *include* the contribution of the +Note that :eq:`eq:c_l^n+1` and +:eq:`eq:qcl_l^n+1` *include* the contribution of the forcing itself, it is not just the reactionary condensation. If we wish to isolate the condensation associated with the forcing, then we must subtract any liquid forcing from the final solution. The net change in @@ -859,10 +831,9 @@ parametrized by :raw-latex:`\cite{t93}`. Equivalent arguments enabled We can consider a change in the width of the PDF to alter its form according to -.. math:: +.. math:: :label: eq:g_xi G^{[n+1]}(s) = \xi G^{[n]} (\xi s) - \label{eq:g_xi} where :math:`G^{[n+1]}(s)` is the distribution after the change in width, :math:`G^{[n]} (s)` is the distribution before the change in @@ -870,89 +841,80 @@ width and :math:`\xi` is a scaling factor. If :math:`\xi > 1` then the distribution is narrowed. For the liquid cloud fraction we therefore have -.. math:: +.. math:: :label: eq:c_l_xi C_l^{[n+1]} = \int_{s=-Q_c}^{\infty} \xi G(\xi s) ds . - \label{eq:c_l_xi} If we transform variables to :math:`s' = \xi s` we can rewrite this integral as -.. math:: +.. math:: :label: eq:c_l_xi2 C_l^{[n+1]} = \int_{s'=-Q_c \xi}^{\infty} G(s') ds' . - \label{eq:c_l_xi2} Hence the expression for :math:`C_l^{[n+1]}` is equivalent to using the same distribution function :math:`G(s)` as for :math:`C_l^{[n]}` except that the saturation boundary has been moved from :math:`-Q_c` to :math:`-Q_c \xi`. The result is the same as applying a homogeneous -forcing (`[eq:deltac] <#eq:deltac>`__) with a modified forcing, +forcing :eq:`eq:deltac` with a modified forcing, -.. math:: +.. math:: :label: eq:deltac_modified \Delta Q_c \equiv \xi Q_c - Q_c , - \label{eq:deltac_modified} or the continuous version -.. math:: +.. math:: :label: eq:xi_equiv \frac{\partial Q_c}{\partial t} \equiv Q_c \frac{\partial}{\partial t}(\xi - 1) . - \label{eq:xi_equiv} We can write :math:`\xi` in a slightly more informative way by linking it to the relative change in width of the PDF :math:`\frac{1}{b_s} \frac{\partial b_s}{\partial t}`. For a PDF that changes its width, :math:`\xi` is defined as -.. math:: +.. math:: :label: eq:xi_equiv1 \xi = \frac{b_s}{b_s + \delta b_s} = \frac{1}{1 + \frac{\delta b_s}{b_s}}. - \label{eq:xi_equiv1} For an infintessimal timestep :math:`\delta t` we therefore have -.. math:: +.. math:: :label: eq:xi \xi = \frac{1}{1 + \frac{1}{b_s} \frac{\partial b_s}{\partial t} \delta t } - \label{eq:xi} -and hence, by expanding (`[eq:xi] <#eq:xi>`__) to give +and hence, by expanding :eq:`eq:xi` to give :math:`\xi = 1 - \frac{1}{b_s} \frac{\partial b_s}{\partial t} \delta t` and using the homogeneous -forcing expression (`[dcdt] <#dcdt>`__) with the modified forcing -(`[eq:xi_equiv] <#eq:xi_equiv>`__), we retrieve the continuous form +forcing expression :eq:`dcdt` with the modified forcing +:eq:`eq:xi_equiv`, we retrieve the continuous form -.. math:: +.. math:: :label: eq:dcdt_width \frac{\partial C_l}{\partial t} = - G(-Q_c) Q_c \frac{1}{b_s} \frac{\partial b_s}{\partial t} . - \label{eq:dcdt_width} A similar analysis can be performed for :math:`\frac{\partial \overline{q_{cl}}} -{\partial t}` from (`[dqcldt] <#dqcldt>`__) to give +{\partial t}` from :eq:`dqcldt` to give -.. math:: +.. math:: :label: eq:qcl_xi2 \overline{q_{cl}}^{[n+1]} = \frac{1}{\xi} \int_{s'=-Q_c \xi}^{\infty} (- \xi Q_c + s') G(s') ds' . - \label{eq:qcl_xi2} Again, this is equivalent to using the homogeneous forcing with the -modified forcing (`[eq:xi_equiv] <#eq:xi_equiv>`__), but it also +modified forcing :eq:`eq:xi_equiv`, but it also includes a scaling term :math:`\frac{1}{\xi}`. In the infinitessimal limit, this scaling gives a second term that is proportional to the value of the integral (i.e. :math:`\overline{q_{cl}}`). Hence we obtain the final continuous solution -.. math:: +.. math:: :label: eq:dqcldt_width \frac{\partial \overline{q_{cl}}}{\partial t} = (- C_l Q_c+\overline{q_{cl}}) \frac{1}{b_s} \frac{\partial b_s}{\partial t} . - \label{eq:dqcldt_width} To close the solution, we need to parametrize :math:`\frac{1}{b_s} \frac{\partial b_s}{\partial t}` , which could be linked to the physics @@ -969,12 +931,12 @@ Initiation of cloud ------------------- In section :ref:`Homogeneous forcing` we commented that the closure -(`[eqn22] <#eqn22>`__) for :math:`G(-Qc)` is only valid if :math:`C_l` +:eq:`eqn22` for :math:`G(-Qc)` is only valid if :math:`C_l` is not identically 0 or 1. If :math:`C_l` is 0 or 1 we know that :math:`G(-Q_c)` is equal to 0 but we have lost the information that will tell us when :math:`G(-Q_c + \Delta Q_c)` starts differing from 0. Hence -the homogeneous forcing equation set (`[dcdt] <#dcdt>`__), -(`[dqcldt] <#dqcldt>`__) and (`[eqn22] <#eqn22>`__) is not complete if +the homogeneous forcing equation set :eq:`dcdt`, +:eq:`dqcldt` and :eq:`eqn22` is not complete if we start from a position where :math:`C_l` is 0 or 1. To complete this set, we will need to define a width, :math:`b_s`, to the PDF and provide an initiation increment to :math:`C_l` and :math:`\overline{q_{cl}}` @@ -997,17 +959,15 @@ This option is selected by setting the UM namelist switch We will assume the same form of the PDF at its boundaries as is assumed in the derivation of the :math:`G(-Q_c)` closure. For the high ‘:math:`s`’ end of the PDF distribution we integrate the power law -description in (`[eqn19] <#eqn19>`__) to obtain the expressions +description in :eq:`eqn19` to obtain the expressions -.. math:: +.. math:: :label: eq:initc C_l = \frac{1}{2 b_s^{n+1}} (b_s + Q_c)^{n+1} , - \label{eq:initc} -.. math:: +.. math:: :label: eq:initqcl \overline{q_{cl}} = \frac{1}{2 b_s^{n+1}} \frac{(b_s + Q_c)^{n+2}}{n+2} . - \label{eq:initqcl} We now need to parametrize the PDF width :math:`b_s`. Unlike the :raw-latex:`\cite{smith90}` scheme, this is the only location in the PC2 @@ -1019,10 +979,9 @@ relative humidity parameter, :math:`RH_{crit}`. Like the :raw-latex:`\cite{smith90}` scheme (see ), we define the value of :math:`b_s` as -.. math:: +.. math:: :label: eq:bs b_s = a_L (1 - RH_{crit}) q_{sat} (\overline{T_L}) . - \label{eq:bs} Hence, if the parameter :math:`n` was the same in PC2 as the equivalent in :raw-latex:`\cite{smith90}`, the initial creation of liquid cloud @@ -1048,18 +1007,16 @@ exception that :math:`\overline{q_{cl}}` is replaced by :math:`SD`, :math:`C_l` is replaced by :math:`(1-C_l)`, and :math:`Q_c` is replaced by :math:`-Q_c`. We hence have the solution -.. math:: +.. math:: :label: eq:init1mc 1 - C_l = \frac{1}{2 b_s^{n+1}} (b_s - Q_c)^{n+1} , - \label{eq:init1mc} -.. math:: +.. math:: :label: eq:initSD SD = \frac{1}{2 b_s^{n+1}} \frac{(b_s - Q_c)^{n+2}}{n+2} . - \label{eq:initSD} The conversion between :math:`SD` and :math:`\overline{q_{cl}}` follows -(`[SD2] <#SD2>`__). We will choose, as we do throughout PC2, to define +:eq:`SD2`. We will choose, as we do throughout PC2, to define :math:`\alpha` (and hence :math:`a_L`) in terms of :math:`\frac{\partial q_{sat}(\overline{T})}{\partial t}`, although within this diagnostic calculation of SD it might actually be better to @@ -1075,10 +1032,9 @@ In order to calculate and compare the state of the model to :math:`b_s`, we first calculate :math:`T_L`, :math:`q_{sat}(\overline{T_L})` and calculate the mean relative total humidity, :math:`RH_T`, where -.. math:: +.. math:: :label: eq:rht RH_T = \frac{ \overline{q} + \overline{q_{cl}} } {q_{sat}(\overline{T_L}) } . - \label{eq:rht} We then assess whether initiation is required. There are only two circumstances in which we wish to proceed further: @@ -1086,7 +1042,7 @@ circumstances in which we wish to proceed further: - If the current cloud fraction :math:`C_l` is 0 and :math:`-Q_c < b_s`. By dividing the second condition by :math:`a_L q_{sat} (\overline{T_L})` we see, using the definitions - (`[eq:qc_eq_qt-qs] <#eq:qc_eq_qt-qs>`__) and (`[eq:bs] <#eq:bs>`__), + :eq:`eq:qc_eq_qt-qs` and :eq:`eq:bs`, that this second condition is equivalent to :math:`RH_T > RH_{crit}`. - If the current cloud fraction :math:`C_l` is 1 and :math:`-Q_c > -b_s` @@ -1109,19 +1065,18 @@ We then solve for the initiated cloud fraction :math:`C_l'`, using the similar methods as described in , except that we allow the solution to vary with the PDF shape :math:`n`. We first write :math:`Q_N` as -.. math:: +.. math:: :label: eq:qn_def Q_N = \frac{Q_c}{b_s} = \frac{ a_L (\overline{q_T} - q_{sat}(\overline{T_L})) } { a_L (1 - RH_{crit}) q_{sat} (\overline{T_L})} = \frac{RH_T - 1}{1-RH_{crit}} - \label{eq:qn_def} and then use :math:`Q_N` to solve the initiated cloud fraction. We -assume a PDF described by a power law as in (`[eqn19] <#eqn19>`__) (and +assume a PDF described by a power law as in :eq:`eqn19` (and the equivalent for the other end of the distribution, the two expressions switching at :math:`Q_c=0`), which is normalized. The -solution to (`[eq:int_gs_ds] <#eq:int_gs_ds>`__) is hence +solution to :eq:`eq:int_gs_ds` is hence -.. math:: +.. math:: :label: eq:c_qn C_l^{init'} = \left\{ \begin{array}{ll} 0, & Q_N \le -1 \\ @@ -1129,7 +1084,6 @@ solution to (`[eq:int_gs_ds] <#eq:int_gs_ds>`__) is hence 1 - \frac{1}{2} {\left( 1 - Q_N \right)}^{n+1}, & 0 < Q_N < 1 \\ 1, & 1 \le Q_N . \end{array} \right. - \label{eq:c_qn} where :math:`C_l^{init'}` is the initiated value of liquid cloud fraction. If we had performed the variable transformation we then we @@ -1137,7 +1091,7 @@ need to transform back, so :math:`C_l^{init} = 1 - C_l^{init'}`, otherwise :math:`C_l^{init} = C_l^{init'}`. In practice, it is likely to be only the second of the options in -(`[eq:c_qn] <#eq:c_qn>`__) that the scheme uses, since we will be at +:eq:`eq:c_qn` that the scheme uses, since we will be at that end of the distribution function, unless previous parts of the model timestep have resulted in large forcings to :math:`Q_c`. @@ -1149,11 +1103,11 @@ temperature :math:`\overline{T}`, which is not known until we know the amount of condensation. Hence we will need to iterate to a solution. We first calculate :math:`q_{sat}(\overline{T})`, :math:`\alpha`, -:math:`a_L` and :math:`b_s`, using (`[eq:alpha_exp] <#eq:alpha_exp>`__), -(`[eq:a_L] <#eq:a_L>`__) and (`[eq:bs] <#eq:bs>`__). We then solve for +:math:`a_L` and :math:`b_s`, using :eq:`eq:alpha_exp`, +:eq:`eq:a_L` and :eq:`eq:bs`. We then solve for the liquid water content: -.. math:: +.. math:: :label: eq:l_bar \frac{\overline{q_{cl}}^{init'}}{b_s} = \left\{ \begin{array}{ll} 0, & Q_N \le -1 \\ @@ -1161,11 +1115,10 @@ the liquid water content: Q_N + \frac{1}{2 (n+2)} {\left( 1 - Q_N \right)}^{n+2}, & 0 < Q_N < 1 \\ Q_N, & 1 \le Q_N . \end{array} \right. - \label{eq:l_bar} If we have been working in transformed variables we now transform back, so the initiated saturation deficit, :math:`SD^{init}`, takes the value -of :math:`\overline{q_{cl}}^{init'}`. We then use (`[SD2] <#SD2>`__) to +of :math:`\overline{q_{cl}}^{init'}`. We then use :eq:`SD2` to estimate :math:`\overline{q_{cl}}^{init}` using our initial estimates of :math:`q_{sat}(\overline{T})` and :math:`a_L`. If we are not in transformed variables, we have the first estimate @@ -1174,15 +1127,14 @@ transformed variables, we have the first estimate We now use this estimate of :math:`\overline{q_{cl}}^{init}` to calculate a more accurate estimate of :math:`a_L` etc. by iteration. In order to achieve a faster convergence of the iteration, we do not use -(`[eq:l_bar] <#eq:l_bar>`__) directly in the estimation of :math:`a_L` +:eq:`eq:l_bar` directly in the estimation of :math:`a_L` etc., but use a combination of this value and the one from the previous iteration. -.. math:: +.. math:: :label: eq:iter \overline{q_{cl}}^{init~[i+1]} = f \overline{q_{cl}}^{init~[i]} + (1 - f) \overline{q_{cl}}^{init~[i-1]} - \label{eq:iter} where the superscript :math:`[i]` labels each iteration. We find that 10 iterations is effective for convergence, with the weighting :math:`f` @@ -1211,7 +1163,7 @@ width of the moisture PDF at each point. The positions of the upper and lower truncated bounds of the moisture PDF relative to the saturation threshold are expressed in terms of a normalised :math:`Q_N` = :math:`Q_c` over PDF-width (see equation -`[eq:qn_def] <#eq:qn_def>`__). In entrainment zones, the sum of the two +:eq:`eq:qn_def`. In entrainment zones, the sum of the two Gaussian modes can lead to a highly skewed distribution; hence :math:`Q_N` can have different values for the upper and lower bounds, each normalised by the different widths on either side of the PDF. The @@ -1258,37 +1210,33 @@ note that the rate of change of liquid cloud fraction and liquid water content in the gridbox can be written in two parts: firstly the change due to the background, and secondly the change due to the source. -.. math:: +.. math:: :label: eq:dcdt_inhom \frac{\partial{C_l}}{\partial{t}} = - C_l \frac{\partial{C_S}}{\partial{t}} + \frac{\partial{C_S}}{\partial{t}} - \label{eq:dcdt_inhom} -.. math:: +.. math:: :label: eq:dqcldt_inhom \frac{\partial{\overline{q_{cl}}}}{\partial{t}} = - \overline{q_{cl}} \frac{\partial{C_S}}{\partial{t}} + q_{cl}^S \frac{\partial{C_S}}{\partial{t}} - \label{eq:dqcldt_inhom} where :math:`q_{cl}^S` is the liquid water content of the injected air. Eliminating :math:`\frac{\partial{C_S}}{\partial{t}}` gives the relationship -.. math:: +.. math:: :label: eq:dcdt_inhom2 \frac{\partial{C_l}}{\partial{t}} = \frac{1 - C_l}{q_{cl}^S - \overline{q_{cl}}} Q4_l, - \label{eq:dcdt_inhom2} where :math:`Q4_l` is the net (*including* the liquid water in the background distribution that was randomally replaced) injection source change of :math:`\overline{q_{cl}}`: -.. math:: +.. math:: :label: eq:q4 Q4_l = \frac{\partial{\overline{q_{cl}}}}{\partial{t}} |_{injection \, source}. - \label{eq:q4} We see that we do not need to know anything about the nature of the two PDFs involved, except the assumption that the injected PDF contains @@ -1298,9 +1246,9 @@ in :math:`C_l` associated with an injection source change of to the mass-flux convection scheme for PC2 (far from trivial and discussed in depth in section :ref:`Convection`) allow :math:`Q4_l` to be calculated (:math:`q_{cl}^S` is already available), and -(`[eq:dcdt_inhom2] <#eq:dcdt_inhom2>`__) can then be used to calculate +:eq:`eq:dcdt_inhom2` can then be used to calculate the equivalent :math:`C_l` change. We note at this stage that the -denominator in (`[eq:dcdt_inhom2] <#eq:dcdt_inhom2>`__), being the +denominator in :eq:`eq:dcdt_inhom2`, being the difference in two terms that may be close to each other, may cause problems when we attempt to numerically apply this equation. @@ -1324,70 +1272,64 @@ liquid, although we need to recognize that they can overlap with each other. :raw-latex:`\cite{wilson2001}` provides the background to the derivation and it is briefly presented below. -We firstly rewrite (`[eq:dqcldt_inhom] <#eq:dqcldt_inhom>`__) but use +We firstly rewrite :eq:`eq:dqcldt_inhom` but use the net condensate (:math:`\overline{q_c} = \overline{q_{cl}} + \overline{q_{cf}}`) instead of just the liquid water expression, and the net cloud amount :math:`C_t`, instead of the liquid cloud amount :math:`C_l`. The same argument as before leads to the expressions -.. math:: +.. math:: :label: eq:dctdt_inhom \frac{\partial{C_t}}{\partial{t}} = - C_t \frac{\partial{C_S}}{\partial{t}} + \frac{\partial{C_S}}{\partial{t}} - \label{eq:dctdt_inhom} and -.. math:: +.. math:: :label: eq:dqcdt_inhom \frac{\partial{\overline{q_{c}}}}{\partial{t}} = - \overline{q_{c}} \frac{\partial{C_S}}{\partial{t}} + q_{c}^S \frac{\partial{C_S}}{\partial{t}} . - \label{eq:dqcdt_inhom} -The left hand side of (`[eq:dqcdt_inhom] <#eq:dqcdt_inhom>`__) is +The left hand side of :eq:`eq:dqcdt_inhom` is written as :math:`Q4_c`. :math:`q_{c}^S` is the in-cloud condensate content (ice plus liquid) of the source. Hence eliminating :math:`\frac{\partial{C_S}}{\partial{t}}` we obtain -.. math:: +.. math:: :label: eq:dctdt_q4 \frac{\partial{C_t}}{\partial{t}} = \frac{(1-C_t)}{q_{c}^S - \overline{q_{c}}} Q4_c . - \label{eq:dctdt_q4} We will assume that the proportion of the injected volume that contains liquid cloud can be written as :math:`g_l`, and the proportion that contains ice cloud can be written as :math:`g_i`. Note that it is not necessary to have :math:`g_l + g_i = 1` if there is mixed phase cloud injected. We can write the change in *liquid* cloud fraction -equivalently to (`[eq:dcdt_inhom] <#eq:dcdt_inhom>`__) as +equivalently to :eq:`eq:dcdt_inhom` as -.. math:: +.. math:: :label: eq:dcldt_inhom \frac{\partial{C_l}}{\partial{t}} = - C_l \frac{\partial{C_S}}{\partial{t}} + g_l \frac{\partial{C_S}}{\partial{t}} . - \label{eq:dcldt_inhom} -Combining (`[eq:dcldt_inhom] <#eq:dcldt_inhom>`__) and -(`[eq:dctdt_inhom] <#eq:dctdt_inhom>`__) by eliminating +Combining :eq:`eq:dcldt_inhom` and +:eq:`eq:dctdt_inhom` by eliminating :math:`\frac{\partial{C_S}}{\partial{t}}` gives -.. math:: +.. math:: :label: eq:dctdt_dcdt \frac{\partial{C_l}}{\partial{t}} = \frac{g_l - C_l}{1 - C_t} \frac{\partial{C_t}}{\partial{t}} - \label{eq:dctdt_dcdt} -and hence from (`[eq:dctdt_q4] <#eq:dctdt_q4>`__) we have the result +and hence from :eq:`eq:dctdt_q4` we have the result -.. math:: +.. math:: :label: eq:dcltdt_almost_final \frac{\partial{C_l}}{\partial{t}} = \frac{g_l - C_l}{q_c^S - \overline{q_{c}}} Q4_c . - \label{eq:dcltdt_almost_final} An equivalent expression holds for the ice cloud. Hence the change in the amount of cloud for each phase may be calculated assuming we know @@ -1396,35 +1338,32 @@ phases and the net increase in the amount of condensate, :math:`Q4_c` (regardless of phase). This expression is coded for use in a generically available inhomogeneous forcing module. However, we can also write this in a slightly more accessible form by noting the ratio of -(`[eq:dqcdt_inhom] <#eq:dqcdt_inhom>`__) and -(`[eq:dqcldt_inhom] <#eq:dqcldt_inhom>`__) with the :math:`Q4` -definitions following (`[eq:q4] <#eq:q4>`__). +:eq:`eq:dqcdt_inhom` and +:eq:`eq:dqcldt_inhom` with the :math:`Q4` +definitions following :eq:`eq:q4`. -.. math:: +.. math:: :label: eq:q4_ratios \frac{Q4_c}{q_c^S - \overline{q_c}} = \frac{Q4_l}{q_{cl}^S - \overline{q_{cl}}} . - \label{eq:q4_ratios} -Using (`[eq:q4_ratios] <#eq:q4_ratios>`__) in -(`[eq:dcltdt_almost_final] <#eq:dcltdt_almost_final>`__) gives the final +Using :eq:`eq:q4_ratios` in +:eq:`eq:dcltdt_almost_final` gives the final expression -.. math:: +.. math:: :label: eq:dctdt_final \frac{\partial{C_l}}{\partial{t}} = \frac{g_l - C_l}{q_{cl}^S - \overline{q_{cl}}} Q4_l - \label{eq:dctdt_final} and similarly for the ice. Note that this expression accounts for the possibility that liquid cloud is displaced from the gridbox by added ice cloud. We can further write :math:`q_{cl}^S` as a fraction of :math:`q_{c}^S` -.. math:: +.. math:: :label: eq:qcls_qcs q_{cl}^S = h_l q_{c}^S - \label{eq:qcls_qcs} where :math:`h_l` is the factor between them (i.e. the *mass* fraction of the injected condensate that is liquid). It is not necessary in this @@ -1435,30 +1374,27 @@ ice equivalent, :math:`h_i`, refer to mass, :math:`h_l + h_i` must equal case in the current mass-flux convection scheme, where only one phase can be injected), :math:`h_l` and :math:`g_l` are equal (and either zero or one in the current mass-flux convection scheme) and we can write -(`[eq:dctdt_final] <#eq:dctdt_final>`__) as +:eq:`eq:dctdt_final` as -.. math:: +.. math:: :label: eq:dctdt_xl \frac{\partial{C_l}}{\partial{t}} = \frac{ (\delta_{xl} - C_l) }{ \delta_{xl} q_{c}^S - \overline{q_{cl}} } Q4_l - \label{eq:dctdt_xl} where :math:`\delta_{xl} = h_l = g_l`. Equivalent expressions exist for the ice cloud fraction and total cloud fraction. -.. math:: +.. math:: :label: eq:dctdt_xi \frac{\partial{C_i}}{\partial{t}} = \frac{ (\delta_{xi} - C_l) }{ \delta_{xi} q_{c}^S - \overline{q_{cf}} } Q4_i - \label{eq:dctdt_xi} -.. math:: +.. math:: :label: eq:dctdt_xc \frac{\partial{C_t}}{\partial{t}} = \frac{ (1 - C_t) }{ q_{c}^S - \overline{q_{c}} } Q4_c - \label{eq:dctdt_xc} with :math:`\delta_{xi} = h_i = g_i`. These are the expressions that are used within the convection scheme. It still remains to parametrize @@ -1470,45 +1406,41 @@ Numerical application ~~~~~~~~~~~~~~~~~~~~~ The numerical application using -(`[eq:dcltdt_almost_final] <#eq:dcltdt_almost_final>`__) may be +:eq:`eq:dcltdt_almost_final` may be performed with a basic forward timestep. Each of the three cloud fractions can be incremented, assuming we know :math:`\Delta{\overline{q_{cl}}}` and :math:`\Delta{\overline{q_{cf}}}`, as -.. math:: +.. math:: :label: eq:cft_ts \Delta{C_t} = \frac{(1 - C_t)} {q_c^S - \overline{q_{cl}} - \overline{q_{cf}}} ( \Delta{\overline{q_{cl}}} + \Delta{\overline{q_{cf}}} ), - \label{eq:cft_ts} -.. math:: +.. math:: :label: eq:cfl_ts \Delta{C_l} = \frac{ (g_l - C_l)} {q_c^S - \overline{q_{cl}} - \overline{q_{cf}}} ( \Delta{\overline{q_{cl}}} + \Delta{\overline{q_{cf}}} ), - \label{eq:cfl_ts} -.. math:: +.. math:: :label: eq:cff_ts \Delta{C_i} = \frac{ (g_i - C_i)} {q_c^S - \overline{q_{cl}} - \overline{q_{cf}}} ( \Delta{\overline{q_{cl}}} + \Delta{\overline{q_{cf}}} ). - \label{eq:cff_ts} The application from within the convection scheme is slightly different. -We start with (`[eq:dctdt_xl] <#eq:dctdt_xl>`__), but enforce two +We start with :eq:`eq:dctdt_xl`, but enforce two numerical restrictions to avoid the equation set becoming ill-conditioned. Firstly, we limit the denominator :math:`q_c^S - \overline{q_{cl}}` to a minimum value if we are considering changes of the same phase as the injected source. -.. math:: +.. math:: :label: eq:delta_cl \Delta C_l = \frac {\delta_{xl} - C_l} {\delta_{xl} \text{Max}( q_{c}^S - \overline{q_{cl}} , q_c^{S0} ) + ( 1 - \delta_{xl} ) (-\overline{q_{cl}}) } Q4_l - \label{eq:delta_cl} where :math:`q_c^{S0}` is specified as :math:`5 \times 10^{-5} kg \, kg^{-1}`. The denominator also has an @@ -1517,26 +1449,23 @@ of :math:`1 \times 10^{-10} kg \, kg^{-1}` then no change in cloud fraction will be considered. A similar equation is used for the ice cloud and the change in total cloud fraction -.. math:: +.. math:: :label: eq:delta_ci \Delta C_i = \frac {\delta_{xi} - C_i} {\delta_{xi} \text{Max}( q_{c}^S - \overline{q_{ci}} , q_c^{S0} ) + ( 1 - \delta_{xi} ) (-\overline{q_{cf}}) } Q4_i - \label{eq:delta_ci} -.. math:: +.. math:: :label: eq:delta_ct \Delta C_t = \frac {1 - C_t} { \text{Max}(q_c^S - \overline{q_c} , q_c^{S0} ) } Q4_c . - \label{eq:delta_ct} We now limit the change in cloud fraction to ensure that the cloud fraction remains within its physical bounds. -.. math:: +.. math:: :label: eq:delta_cl_conv_final C_l^{[n+1]} = ( 0, C_l^{[n]} + \Delta C_l, 1) - \label{eq:delta_cl_conv_final} and similar equations are used for :math:`C_i^{[n+1]}` and :math:`C_t^{[n+1]}`. @@ -1546,7 +1475,7 @@ and similar equations are used for :math:`C_i^{[n+1]}` and A note on the implementation of the cloud fraction change ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -Equation `[eq:dcdt_inhom2] <#eq:dcdt_inhom2>`__ has been derived +Equation :eq:`eq:dcdt_inhom2` has been derived assuming that the only change in the cloud properties within the gridbox comes from the detrainment of air from the convective plume (so that the injection source is an appropriate model). Attention should be drawn to @@ -1558,7 +1487,7 @@ subsidence. The former is considered correctly in the calculation of :math:`\frac{\partial \overline{q_{cl}}}{\partial t}`, which corresponds to :math:`Q4`. However, the calculation of :math:`\frac{\partial C_l}{\partial t}` is then performed using -(`[eq:dcdt_inhom2] <#eq:dcdt_inhom2>`__) and **incorrectly** assuming +:eq:`eq:dcdt_inhom2` and **incorrectly** assuming that all the :math:`\overline{q_{cl}}` change comes from the detrainment. It is possible to calculate directly the change in :math:`C_l` that should occur due to the detrainment and compensating @@ -1567,7 +1496,7 @@ subsidence treated together, in the same way that `4.7.3 <#subsect:q4calculation>`__), and this is the way in which the cloud fraction change **should** be done. It is an unfortunate historical emphasis in the early development of PC2 on the derivation of -(`[eq:dcdt_inhom2] <#eq:dcdt_inhom2>`__) that has led to the treatment +:eq:`eq:dcdt_inhom2` that has led to the treatment used within the Unified Model for the change in cloud fractions due to convection. @@ -1577,13 +1506,13 @@ considered explicitly in the model implementation (see section :ref:`Background condensation`) for both :math:`\overline{q_{cl}}` and :math:`C_l` after the rest of the convective process has been calculated. It is perhaps arguable that if -(`[eq:dcdt_inhom2] <#eq:dcdt_inhom2>`__) is going to be applied then the -value of :math:`Q4` used in (`[eq:dcdt_inhom2] <#eq:dcdt_inhom2>`__) +:eq:`eq:dcdt_inhom2` is going to be applied then the +value of :math:`Q4` used in :eq:`eq:dcdt_inhom2` should include this term. Any major future developments of PC2 for a mass-flux convection scheme would be advised to consider whether it is appropriate to use -(`[eq:dcdt_inhom2] <#eq:dcdt_inhom2>`__) at all. +:eq:`eq:dcdt_inhom2` at all. .. _Ice cloud and mixed phase regions: @@ -1614,14 +1543,13 @@ down. We firstly consider that a change in liquid cloud fraction :math:`\Delta C_l` is known and we wish to estimate the resulting change in the total cloud fraction. There is, of course, no change in the ice cloud fraction :math:`C_i`, since, from our *definitions* in -(`[eq:dqcldt_and_dcdt] <#eq:dqcldt_and_dcdt>`__) and -(`[eq:mp] <#eq:mp>`__), this includes the mixed phase contribution. +:eq:`eq:dqcldt_and_dcdt` and +:eq:`eq:mp`, this includes the mixed phase contribution. Hence we write -.. math:: +.. math:: :label: eq:deltaci_eq_0 \Delta C_i = 0 . - \label{eq:deltaci_eq_0} The change in the total cloud fraction, :math:`C_t` will depend upon the sign of the change of the liquid cloud fraction. If @@ -1630,10 +1558,9 @@ as :math:`\Delta C_l` (:math:`C_l` is being added with minimum overlap to :math:`C_i`), unless the gridbox becomes completely covered in cloud, when there is no choice but to generate mixed phase cloud. Hence we have -.. math:: +.. math:: :label: eq:deltact_min \Delta C_t = \text{Min} ( \Delta C_l , 1 - C_t ). - \label{eq:deltact_min} If :math:`\Delta C_l < 0`, then we still consider minimum overlap of the *changes* (this is so that the solution is reversible as much as @@ -1641,41 +1568,37 @@ possible). Hence :math:`\Delta C_t` is going to be the same as :math:`\Delta C_l` unless :math:`C_l` is reduced below the existing :math:`C_i`, in which case no more change to :math:`C_t` is possible. -.. math:: +.. math:: :label: eq:deltact_min2 \Delta C_t = \text{Max} ( \Delta C_l , C_i - C_t ) , - \label{eq:deltact_min2} remembering that both quantities in the maximum expression in -(`[eq:deltact_min2] <#eq:deltact_min2>`__) have negative values. +:eq:`eq:deltact_min2` have negative values. We can write similar expressions if a known amount of ice cloud is added or removed, and we need to calculate the effect on :math:`C_t`. Similar to the results above we have: -.. math:: +.. math:: :label: eq:deltacl_eq_0 \Delta C_l = 0 . - \label{eq:deltacl_eq_0} and -.. math:: +.. math:: :label: eq:deltact_min_array \Delta C_t = \left\{ \begin{array}{ll} \text{Max} ( \Delta C_i , C_l - C_t ), & \Delta C_i < 0 \\ \text{Min} ( \Delta C_i , 1 - C_t ), & \Delta C_i > 0 . \end{array} \right. - \label{eq:deltact_min_array} For completeness, we also present here the equation set for random overlap of changes in liquid cloud with existing ice cloud. We have, as before, -.. math:: +.. math:: :label: eq:deltaci_eq_0_2 \Delta C_i = 0 . - \label{eq:deltaci_eq_0_2} For :math:`\Delta C_l > 0` additional liquid cloud is added randomly to any location outside that of the current liquid cloud. A proportion @@ -1683,23 +1606,21 @@ any location outside that of the current liquid cloud. A proportion existing ice cloud. Hence the net change in total cloud fraction can be written as -.. math:: +.. math:: :label: eq:deltact_ran1 \Delta C_t = \Delta C_l \frac{1 - C_t}{1 - C_l} . - \label{eq:deltact_ran1} Similarly, if :math:`\Delta C_l < 0`, the liquid cloud is removed randomly from the existing liquid cloud. A proportion :math:`\frac{C_t - C_i}{C_l}` of this is from liquid cloud that does not overlap with existing ice cloud. Hence, -.. math:: +.. math:: :label: eq:deltact_ran2 \Delta C_t = \Delta C_l \frac{C_t - C_i}{C_l} . - \label{eq:deltact_ran2} -Equivalent equations to (`[eq:deltact_ran1] <#eq:deltact_ran1>`__) and -(`[eq:deltact_ran2] <#eq:deltact_ran2>`__) but with :math:`C_l` and +Equivalent equations to :eq:`eq:deltact_ran1` and +:eq:`eq:deltact_ran2` but with :math:`C_l` and :math:`C_i` swapped apply when we need to estimate changes in :math:`C_t` from a known :math:`\Delta C_i`, when assuming random overlap. @@ -1711,12 +1632,12 @@ In general, although the situation does not occur within the current implementation of PC2 , we might have increments to both :math:`C_l` and :math:`C_i` simultaneously. Hence the implementation is to calculate :math:`\Delta C_t` from the sum of that predicted by -(`[eq:deltact_min] <#eq:deltact_min>`__) or -(`[eq:deltact_min2] <#eq:deltact_min2>`__), and -(`[eq:deltact_min_array] <#eq:deltact_min_array>`__). For the random +:eq:`eq:deltact_min` or +:eq:`eq:deltact_min2`, and +:eq:`eq:deltact_min_array`. For the random overlap situation we also need to apply a check on the denominator in -(`[eq:deltact_ran1] <#eq:deltact_ran1>`__) and -(`[eq:deltact_ran2] <#eq:deltact_ran2>`__) before calculation, with the +:eq:`eq:deltact_ran1` and +:eq:`eq:deltact_ran2` before calculation, with the result set to the limit :math:`\Delta C_t = 0` if the denominator is 0. For the minimum overlap situation a final check is made that :math:`C_t` lies between 0 and 1, with the value being reset to 0 or 1 if not. @@ -1743,16 +1664,15 @@ varies linearly between 0.1 and 0.3 for cloud depths between 100 m and layer depth, :math:`z_h` plus the inversion thickness, :math:`\Delta z_i` parametrized following :raw-latex:`\cite{rb08}` as: -.. math:: +.. math:: :label: dz_param \Delta z_i = 6.3 \, w_m^2 / \int_{z_h}^{z_h+\Delta z_i} b \, dz - \label{dz_param} where :math:`w_m` is the boundary layer velocity scale (:math:`w_m^3 = u_*^3 + 0.25 w_*^3`) and :math:`b` is the parcel buoyancy that is integrated over the depth of the inversion assuming a piece-wise linear variation between grid-levels. Note that the constant -in (`[dz_param] <#dz_param>`__) is the same as in +in :eq:`dz_param` is the same as in :raw-latex:`\cite{rb08}` because :math:`6.3 = 2.5 * 4^{2/3}` and :math:`w_m^3` differs by a factor of 4. @@ -1861,7 +1781,7 @@ of :math:`p` and :math:`T` given by \end{aligned} The first term on the right hand side of -Eq. `[eqn:squires_eqn] <#eqn:squires_eqn>`__ is the sink of vapor due to +Eq. :eq:`eqn:squires_eqn` is the sink of vapor due to depositional growth of ice crystals, the second term models entrainment (mixing) of environmental air into the cloudy volume and the third term is a source term due to vertical air motions. @@ -1878,15 +1798,14 @@ expectation value :math:`\overline{w^2}` is not defined) and :math:`\tau_{\rm d}` a Lagrangian decorrelation time define here by the relation used by :raw-latex:`\cite{rodean1997}`: -.. math:: +.. math:: :label: eqn:taud \tau_{\rm d} = \frac{2\sigma_w^2}{\varepsilon C_0}, - \label{eqn:taud} where :math:`C_0` is a known constant. Because it is linear in :math:`S_i`, Equation -`[eqn:squires_eqn] <#eqn:squires_eqn>`__ can be solved exactly, for any +:eq:`eqn:squires_eqn` can be solved exactly, for any given realisation of the noise term. By averaging the solutions over the the noise and taking a steady-state limit (see :raw-latex:`\cite{fhfk14}` for details) it can be shown that the @@ -1903,8 +1822,8 @@ solution PDF is Gaussian with mean and variance given by: \label{eqn:si_var} \end{aligned} -Equation `[eqn:si_avg] <#eqn:si_avg>`__ and -`[eqn:si_var] <#eqn:si_var>`__ completely specify the PDF, +Equation :eq:`eqn:si_avg` and +:eq:`eqn:si_var` completely specify the PDF, :math:`F(S_i)`, of steady-state humidity variations for the subgrid model. The liquid cloud fraction and liquid water mass mixing ratio are given by @@ -1932,11 +1851,11 @@ To implement the model of Section :ref:`Model description` in the Unified Model, closure relations are needed for the quantities :math:`\sigma_w^2`, :math:`\varepsilon`, :math:`L`, :math:`\tau_{\rm d}` and :math:`S_E`, subject to the constraining relationship given by Eq. -`[eqn:taud] <#eqn:taud>`__. In each model grid box, these parameters +:eq:`eqn:taud`. In each model grid box, these parameters specify the subgrid PDF, :math:`F(S_i)`, and from this the liquid cloud fraction and water content produced by turbulence can be found using Eqs -`[eqn:cloud_fraction] <#eqn:cloud_fraction>`__ and -`[eqn:cloud_liquid] <#eqn:cloud_liquid>`__. +:eq:`eqn:cloud_fraction` and +:eq:`eqn:cloud_liquid`. In addition we need to make some assumptions about how the diagnosed values :math:`C_l^{sgt}` and :math:`q_{cl}^{sgt}` relate to the model @@ -1971,12 +1890,11 @@ should be of order one. To obtain :math:`\tau_{\rm d}` we impose an eddy size constraint: -.. math:: +.. math:: :label: eqn:eddy_size \tau_{\rm d} = \frac{L}{\sigma_w} = \beta_{mix} \frac{\Delta z}{\sigma_w} - \label{eqn:eddy_size} -Eq. `[eqn:taud] <#eqn:taud>`__ then determines the dissipation rate, +Eq. :eq:`eqn:taud` then determines the dissipation rate, :math:`\varepsilon`, that is consistent with the other parameters. The constant :math:`C_0=10` by default, but can be adjusted by the user. @@ -2006,8 +1924,8 @@ allows :math:`C_l^{sgt}` and :math:`q_{cl}^{sgt}` to be calculated. These will be non-zero only where there is turbulence as diagnosed by the Boundary Layer scheme (and hence non-zero :math:`\sigma_w^2`). To calculate :math:`C_l^{sgt}` and :math:`q_{cl}^{sgt}` the integrals in -Eqs `[eqn:cloud_fraction] <#eqn:cloud_fraction>`__ and -`[eqn:cloud_liquid] <#eqn:cloud_liquid>`__ are evaluated numerically +Eqs :eq:`eqn:cloud_fraction` and +:eq:`eqn:cloud_liquid` are evaluated numerically using discretisation based on user-specified number of bins. Given :math:`C_l^{sgt}` and :math:`q_{cl}^{sgt}`, two options are @@ -2089,8 +2007,8 @@ The following variables and logical switches are optional inputs: seconds. #. ``nbins_mp`` is the number of bins used in the discretisation of the - integrals in Eqs `[eqn:cloud_fraction] <#eqn:cloud_fraction>`__ and - `[eqn:cloud_liquid] <#eqn:cloud_liquid>`__ for :math:`C_l^{sgt}` and + integrals in Eqs :eq:`eqn:cloud_fraction` and + :eq:`eqn:cloud_liquid` for :math:`C_l^{sgt}` and :math:`q_{cl}^{sgt}`. The default value is :math:`100` bins. #. ``mp_dz_scal`` is the scale parameter, :math:`\beta_{mix}`, in the @@ -2125,7 +2043,7 @@ of the atmosphere, hence we need to calculate the corresponding condensation and cloud fraction changes. For both shortwave and longwave, we use the homogeneous forcing routines (section :ref:`Homogeneous forcing`) for :math:`\overline{q_{cl}}` and :math:`C_l`, -(using eqn. `[eq:deltaqc_exp2] <#eq:deltaqc_exp2>`__ to calculate the +(using eqn. :eq:`eq:deltaqc_exp2` to calculate the :math:`Q_c` forcing) and then the method in section :ref:`Ice cloud and mixed phase regions` to calculate :math:`C_t` changes. There is no :math:`\overline{q_{cf}}` @@ -2188,11 +2106,10 @@ them. We will assume an overlap that is nearly, but not quite, maximum, the difference being dependent upon the windshear and the time taken for ice to fall between the levels. -.. math:: +.. math:: :label: eq:overhang O^{[k,k+1]} = \text{Max}( C_{i}^{[k+1]} - C_i^{[k]} , 0) + w \frac{\Delta z^{[k]}}{v_i^{[k]}} - \label{eq:overhang} where :math:`O^{[k,k+1]}` is the amount of ice cloud ‘overhanging’ the current (i.e. :math:`k`\ ’th) layer from the layer above, :math:`w` is a @@ -2214,19 +2131,17 @@ The change in :math:`C_i` over the timestep is then given by the overlap proportion multiplied by the how much (in the vertical dimension) of the layer below can be filled by ice in the timestep: -.. math:: +.. math:: :label: eq:lsp_fall \Delta C_i = \text{Max}(O^{[k,k+1]} , 1) \text{Min} (v_i \frac{\Delta t}{\Delta z^{[k]}} , 1) - \label{eq:lsp_fall} where :math:`\Delta t` is the timestep. We now choose to assume a minimum overlap between the liquid and the ice phases (as in section :ref:`Ice cloud and mixed phase regions`). -.. math:: +.. math:: :label: eq:lsp_fall_ct \Delta C_t = \text{Min} ( \Delta C_i , A_{clear} ) - \label{eq:lsp_fall_ct} where :math:`A_{clear}` is the proportion of the gridbox that has neither ice nor liquid cloud present. @@ -2299,10 +2214,9 @@ water, hence we need only consider the part of the gridbox that does not have liquid water present. The average value, :math:`q_a`, of :math:`q` within the liquid-free part of the gridbox is thus -.. math:: +.. math:: :label: eq:qa q_a = \frac{ \overline{q} - C_l q_{sat \, liq}(\overline{T}) } {1 - C_l} - \label{eq:qa} where we have assumed that the fluctuation of :math:`q_{sat~liq}` across the gridbox due to temperature fluctuations is not significant compared @@ -2312,20 +2226,19 @@ width, :math:`b_i`, to the :math:`q` (not :math:`s`) fluctuations :math:`RH_{crit}`. This is like that for the ‘:math:`s`’ distribution width, :math:`b_s` but modified: -.. math:: +.. math:: :label: eq:b_i b_i = (1 - RH_{crit} ) q_{sat \, liq} ( 1 - \frac{1}{2} ~ \frac{\overline{q_{cf}}} {i q_{sat \, liq}(\overline{T})} ) . - \label{eq:b_i} where the factor :math:`( 1 - \frac{1}{2} \frac{\overline{q_{cf}}} {i ~ q_{sat \, liq}(\overline{T})})` should be limited to a minimum value of zero, but, for numerical reasons, is limited to a minimum value of 0.001. We note that :math:`b_i` has a similar form to :math:`b_s`, except the multiplier :math:`a_L` and the -factor in brackets. If we remember from (`[eq:s] <#eq:s>`__) that the +factor in brackets. If we remember from :eq:`eq:s` that the definition of ‘:math:`s`’ includes a factor :math:`a_L` we see that the -absence of the :math:`a_L` factor in (`[eq:b_i] <#eq:b_i>`__) is +absence of the :math:`a_L` factor in :eq:`eq:b_i` is consistent. The factor in brackets is a *parametrization* of the effect that, when ice is present, deposition in the moistier parts and sublimation in the drier parts of the gridbox must reduce the width of @@ -2339,7 +2252,7 @@ cloud formulation, which considers an underlying PDF across the whole gridbox and does not have, in general, its width prescribed. Remember that we do not calculate on-line the whole of the liquid - vapour PDF, we only parametrize the single point :math:`G(-Qc)`, using equation -`[eqn22] <#eqn22>`__). +:eq:`eqn22`. The width is then limited further to be no greater than :math:`\overline{q}`, to make sure that there are no negative values of @@ -2384,11 +2297,10 @@ ice-only partition to give the proportion of that partition that is above and below ice saturation. This gives, in general, an area of the gridbox :math:`A_{ice1}` that contains ice and is above saturation where -.. math:: +.. math:: :label: eq:q_ice_above_sat A_{ice1} = \frac{1}{2} A_{ice} + \frac{1}{2} \frac{ (q_{ice}-q_{sat~ice}(\overline{T})) } {b_i}, - \label{eq:q_ice_above_sat} having assumed that :math:`A_{ice1}` is between 0 and :math:`A_{ice}`. If not, it is trivial to partition the gridbox, since the moisture in @@ -2401,11 +2313,10 @@ distribution of local values of :math:`q_{cl}` about the local mean. If we assume a uniform removal of local :math:`q_{cl}` then, with a little algebra, we can obtain an expression for the change in :math:`C_l`: -.. math:: +.. math:: :label: eq:deltacfl_dep \Delta C_l = C_l ( 1 - \frac {\Delta \overline{q_{cl}}} {\overline{q_{cl}}} ) ^{\frac{1}{2}} - C_l - \label{eq:deltacfl_dep} . @@ -2413,7 +2324,7 @@ Since this occurs only in the mixed phase part of the gridbox, we can say that :math:`\Delta C_t = 0`. We will also note that the change in :math:`\overline{q_{cl}}` due to deposition is limited by the amount of :math:`\overline{q_{cl}}` that is in the mixed phase partition in the -gridbox, hence (`[eq:deltacfl_dep] <#eq:deltacfl_dep>`__), although it +gridbox, hence :eq:`eq:deltacfl_dep`, although it formally allows removal of :math:`C_l` from an ice-free partition, will be unlikely to do so. @@ -2423,12 +2334,11 @@ we make a similar assumption to that used for liquid in the deposition term, except that we limit the changes only to the region of the gridbox where ice is subliming. -.. math:: +.. math:: :label: eq:deltacfi_sub \Delta C_i = A_{ice2} ( 1 + \frac{\Delta \overline{q_{cf}} } { \overline{q_{cf}} ( \frac{A_{ice2}}{C_i} ) } )^{\frac{1}{2}} - A_{ice2} - \label{eq:deltacfi_sub} . @@ -2439,10 +2349,9 @@ fractional change in that region. The change in the total cloud fraction must also be equal to the change above, since sublimation cannot occur in the presence of liquid cloud: -.. math:: +.. math:: :label: eq:deltacft_sub \Delta C_t = \Delta C_i . - \label{eq:deltacft_sub} Riming ~~~~~~ @@ -2469,10 +2378,9 @@ Evaporation of melting ice Here we simply assume that ice cloud fraction is removed in proportion to the ice content that is removed. -.. math:: +.. math:: :label: eq:lsp_evapmeltsnow \Delta C_i = C_i \frac{ \Delta \overline{q_{cf}}}{\overline{q_{cf}}} . - \label{eq:lsp_evapmeltsnow} Because the evaporation cannot occur in the liquid part of the gridbox, there is no change to :math:`C_t` (or to :math:`C_l`). @@ -2481,24 +2389,22 @@ Melting ~~~~~~~ Again, the change in :math:`C_i` is calculated using the method in -(`[eq:lsp_evapmeltsnow] <#eq:lsp_evapmeltsnow>`__). +:eq:`eq:lsp_evapmeltsnow`. -.. math:: +.. math:: :label: eq:lsp_melt \Delta C_i = C_i \frac{ \Delta \overline{q_{cf}}}{\overline{q_{cf}}} . - \label{eq:lsp_melt} The change in :math:`C_t` is calculated assuming that there is no correlation in the gridbox between where the ice melts and the liquid -cloud. Hence we must multiply (`[eq:lsp_melt] <#eq:lsp_melt>`__) by the +cloud. Hence we must multiply :eq:`eq:lsp_melt` by the proportion of ice cloud fraction that exists without liquid cloud (i.e. :math:`\frac{A_{ice}}{C_i}`). -.. math:: +.. math:: :label: eq:lsp_melt2 \Delta C_t = C_i \frac{ \Delta \overline{q_{cf}}}{\overline{q_{cf}}} \frac{A_{ice}}{C_i} . - \label{eq:lsp_melt2} Evaporation of rain ~~~~~~~~~~~~~~~~~~~ @@ -2603,10 +2509,9 @@ the way the real atmosphere works, especially in the stratosphere. No doubt the link can be improved upon with more research. The formulation used is: -.. math:: +.. math:: :label: eq:dbsbydtbs_turb \frac{1}{b_s} \frac{\partial b_s}{\partial t} = \Upsilon exp ( - \frac{2.01 Q_c}{0.2 a_L q_{sat liq}(T_L)} ) - \label{eq:dbsbydtbs_turb} where the 0.2 factor is chosen to be closely equivalent to :math:`1 - RH_{crit}` and the value of 2.01 has been selected through @@ -2645,26 +2550,24 @@ represent the timestepping of this process in exactly the same way as for the homogeneous forcing (in fact, in the Unified Model code we use the same subroutine, see section :ref:`Code Structure`). As before, we use a simple forward timestepping of :math:`C_l`, with :math:`Q_c` given by -(`[eq:qc_eq_qt-qs] <#eq:qc_eq_qt-qs>`__) and :math:`a_L` defined as +:eq:`eq:qc_eq_qt-qs` and :math:`a_L` defined as discussed in section :ref:`Numerical application` and discretize eq -`[eq:dcdt_width] <#eq:dcdt_width>`__ as: +:eq:`eq:dcdt_width` as: -.. math:: +.. math:: :label: eq:dcl_turb_final \Delta C_l^{[n+1]} = - G(-Q_c) Q_c \frac{1}{b_s} \frac{\partial b_s}{\partial t} \Delta t. - \label{eq:dcl_turb_final} -Similarly to (`[eq:c_l^n+1] <#eq:c_l^n+1>`__), we then limit the cloud +Similarly to :eq:`eq:c_l^n+1`, we then limit the cloud fraction to 0 and 1 and then apply a mid-point value of :math:`C_l` to calculate the change in :math:`\overline{q_{cl}}` (discretizing eq -`[eq:dqcldt_width] <#eq:dqcldt_width>`__): +:eq:`eq:dqcldt_width`: -.. math:: +.. math:: :label: eq:dqcl_turb_final \Delta q_{cl}^{[n+1]} = (q_{cl}^{[n]} - Q_c \frac{1}{2}(C_l^{[n]}+C_l^{[n+1]})) \frac{1}{b_s} \frac{\partial b_s}{\partial t} \Delta t. - \label{eq:dqcl_turb_final} In this case the value of :math:`\Delta q_{cl}` *is* limited to ensure that no more :math:`\overline{q_{cl}}` is removed than the model has @@ -2681,16 +2584,16 @@ Cloud-surface-area hybrid erosion method :raw-latex:`\cite{morcrette_petch}` showed that changes to the erosion parameter (:math:`\Upsilon` in Eqn. -`[eq:dbsbydtbs_turb] <#eq:dbsbydtbs_turb>`__) did not have as +:eq:`eq:dbsbydtbs_turb` did not have as significant an impact on the global work done by the erosion process as might be expected. This was due to a feedback process whereby, reducing the erosion parameter leads to more cloud water, more autoconversion of cloud water to rain, more fall-out of rain and more drying of the layer, hence increasing the :math:`exp ( - \frac{2.01 Q_c}{0.2 a_L q_{sat liq}(T_L)} )` part of Eqn. -`[eq:dbsbydtbs_turb] <#eq:dbsbydtbs_turb>`__. Although the feedback is +:eq:`eq:dbsbydtbs_turb`. Although the feedback is physically plausible it crucially depends on the formulation of Eqn. -`[eq:dbsbydtbs_turb] <#eq:dbsbydtbs_turb>`__ and the dependence of the +:eq:`eq:dbsbydtbs_turb` and the dependence of the rate of narrowing of the PDF on the moisture, a dependence that was developed from a pragmatic rather than theoretical stand-point. The option for an alternative way of calculating the erosion was introduced @@ -2699,10 +2602,9 @@ at vn8.0 We use equation 30 from :raw-latex:`\cite{t93}` to specify the sink of :math:`q_{cl}` due to erosion, i.e. -.. math:: +.. math:: :label: eq:dqcldt_hybrid \frac{\partial q_{cl}}{\partial t}=-A K(q_{sat}-q_v) - \label{eq:dqcldt_hybrid} (note we have changed the sign as we have replaced the evaporation rate :math:`E_2` in :raw-latex:`\cite{t93}` with @@ -2728,10 +2630,9 @@ circular clump. Numerical tests using randomly distributed cloudy cube shows that the variation in lateral surface area, :math:`S`, as a function of cloud fraction can be expressed as: -.. math:: +.. math:: :label: eq:S_Cl S= - 2 C_l ^{2} + 2 C_l - \label{eq:S_Cl} The maximum normalised surface area of 0.5 occurs at a cloud fraction of 0.5. Using a cloud mask derived from satellite imagery shows that real @@ -2751,13 +2652,12 @@ The exposed surface area associated with the tops and bottom of the clouds is calculated assuming maximum overlap in adjacent layers and is added to the lateral surface area to give a total surface area, -.. math:: +.. math:: :label: eq:A_top_and_bottom A=max(C_l(k)-C_l(k+1),0.0)+max(C_l(k)-C_l(k-1),0.0)+S - \label{eq:A_top_and_bottom} it is this value of :math:`A` which we use in Eqn. -`[eq:dqcldt_hybrid] <#eq:dqcldt_hybrid>`__. +:eq:`eq:dqcldt_hybrid`. Note that the contributions from the top and bottom interfaces of the current model-level :math:`max(C_l(k)-C_l(k+1),0.0)` and @@ -2774,23 +2674,22 @@ narrowing that would have given the same sink of :math:`q_{cl}`. This value of :math:`\frac{1}{b_s} \frac{\partial b_s}{\partial t}` is then used to calculate the change in :math:`C_l` using the same moisture PDF assumptions as were used in the original PC2 erosion formulation. To -achieve this, we combine equations `[eq:dcdt_width] <#eq:dcdt_width>`__ -and `[eq:dqcldt_width] <#eq:dqcldt_width>`__ from section +achieve this, we combine equations :eq:`eq:dcdt_width` +and :eq:`eq:dqcldt_width` from section :ref:`Changing the width of the PDF - PC2 erosion` to eliminate :math:`\frac{1}{b_s} \frac{\partial b_s}{\partial t}` and write :math:`\frac{\partial C_l}{\partial t}` as a function of :math:`\frac{\partial \overline{q_{cl}}}{\partial t}`: -.. math:: +.. math:: :label: eq:dcdt_hybrid \frac{\partial C_l}{\partial t} = - \frac{ G(-Q_c) Q_c \frac{\partial \overline{q_{cl}}}{\partial t} } { (- C_l Q_c+\overline{q_{cl}}) } - \label{eq:dcdt_hybrid} Where the change in liquid water content :math:`\frac{\partial \overline{q_{cl}}}{\partial t}` is given by eq -`[eq:dqcldt_hybrid] <#eq:dqcldt_hybrid>`__ above. +:eq:`eq:dqcldt_hybrid` above. This combination of a Tiedkte sink term for :math:`q_{cl}`, a PC2 term for :math:`C_l` and the introduction of some surface area dependence @@ -2803,19 +2702,18 @@ Numerical application of the hybrid erosion method ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Next, we consider how to numerically discretise equations -`[eq:dqcldt_hybrid] <#eq:dqcldt_hybrid>`__ and -`[eq:dcdt_hybrid] <#eq:dcdt_hybrid>`__ to compute cloud increments due +:eq:`eq:dqcldt_hybrid` and +:eq:`eq:dcdt_hybrid` to compute cloud increments due to erosion. The simplest approach is an explicit forwards-in-time discretisation: -.. math:: +.. math:: :label: eq:hybrid_erosion_expl \frac{ \Delta {q_{cl}}_{ero}}{\Delta t} = A(C_l^n) K(q_{sat}-q_v) - \label{eq:hybrid_erosion_expl} i.e. the increment is calculated by evaluating the term :math:`A` from -equations `[eq:S_Cl] <#eq:S_Cl>`__ and -`[eq:A_top_and_bottom] <#eq:A_top_and_bottom>`__ using the value of +equations :eq:`eq:S_Cl` and +:eq:`eq:A_top_and_bottom` using the value of cloud-fraction :math:`C_l` *before* erosion has been applied. However, when the environment is significantly subsaturated (so that the @@ -2849,17 +2747,16 @@ cumulus regimes. convection increment, which scales with the timestep length. Having computed the erosion :math:`q_{cl}` increment using - `[eq:hybrid_erosion_expl] <#eq:hybrid_erosion_expl>`__, the + :eq:`eq:hybrid_erosion_expl`, the consistent :math:`C_l` increment is computed by discretising - `[eq:dcdt_hybrid] <#eq:dcdt_hybrid>`__ as: + :eq:`eq:dcdt_hybrid` as: - .. math:: + .. math:: :label: eq:dcdt_hybrid_discr \frac{\Delta {C_l}_{ero}}{\Delta t} = - \frac{ G(-Q_c)^n Q_c^n \frac{\Delta \overline{{q_{cl}}_{ero}}}{\Delta t} } { (- C_l^n Q_c^n + ( \overline{q_{cl}^n} + \frac{1}{2} \Delta \overline{{q_{cl}}_{ero}} ) ) } - \label{eq:dcdt_hybrid_discr} i.e. all terms are treated explicitly (using the values before erosion), except for :math:`\overline{q_{cl}}` which takes the @@ -2870,7 +2767,7 @@ cumulus regimes. yields a positive solution for :math:`q_{cl}` and :math:`C_l`. (i_pc2_erosion_numerics=2)** The copies of the fields passed to the erosion calculation are fully updated with the convection increments. - We then write equation `[eq:dqcldt_hybrid] <#eq:dqcldt_hybrid>`__ in + We then write equation :eq:`eq:dqcldt_hybrid` in the form: .. math:: \frac{\partial q_{cl}}{\partial t} = q_{cl} f(q_{cl},C_l,(q_{sat}-q_v)) @@ -2908,34 +2805,32 @@ cumulus regimes. increment we would obtain from the purely explicit discretisation, :math:`\Delta q_{cl}^{ero\,expl}`. The implicit discretisation is implemented by first calculating :math:`\Delta q_{cl}^{ero\,expl}` - using equation `[eq:hybrid_erosion_expl] <#eq:hybrid_erosion_expl>`__ + using equation :eq:`eq:hybrid_erosion_expl` (as we do for **i_pc2_erosion_numerics=1**) but then rescaling it using the above expression, which becomes: - .. math:: + .. math:: :label: eq:hybrid_erosion_impl_qcl \Delta q_{cl}^{ero} = \Delta q_{cl}^{ero\,expl} \frac{ q_{cl}^n + \Delta q_{cl}^{hom} } { q_{cl}^n - \Delta q_{cl}^{ero\,expl} } - \label{eq:hybrid_erosion_impl_qcl} Provided erosion is acting to reduce cloud-water (:math:`\Delta q_{cl}^{ero\,expl} < 0`), and homogeneous forcing by convection has not already completely removed the cloud (:math:`q_{cl}^n + \Delta q_{cl}^{hom} > 0`), - `[eq:hybrid_erosion_impl_qcl] <#eq:hybrid_erosion_impl_qcl>`__ is + :eq:`eq:hybrid_erosion_impl_qcl` is guaranteed to yield a stable, positive solution for :math:`q_{cl}`. We also apply exactly the same argument to the equation for the cloud-fraction increment :math:`C_l`, and obtain: - .. math:: + .. math:: :label: eq:hybrid_erosion_impl_Cl \Delta C_l^{ero} = \Delta C_l^{ero\,expl} \frac{ C_l^n + \Delta C_l^{hom} } { C_l^n - \Delta C_l^{ero\,expl} } - \label{eq:hybrid_erosion_impl_Cl} Where :math:`\Delta C_l^{ero\,expl}` is computed using eq - `[eq:dcdt_hybrid_discr] <#eq:dcdt_hybrid_discr>`__, except that the + :eq:`eq:dcdt_hybrid_discr`, except that the term :math:`\frac{1}{2} \Delta \overline{{q_{cl}}_{ero}}` is omitted (interpolating to the mid-point value of :math:`\overline{q_{cl}}` in the denominator would be “double-counting” if we are already making @@ -2948,13 +2843,13 @@ cumulus regimes. cloud-fraction, the code defaults to retaining the explicit discretisation solution :math:`\Delta q_{cl}^{ero\,expl}` and :math:`\Delta C_l^{ero\,expl}`. Otherwise, equations - `[eq:hybrid_erosion_impl_qcl] <#eq:hybrid_erosion_impl_qcl>`__ and - `[eq:hybrid_erosion_impl_Cl] <#eq:hybrid_erosion_impl_Cl>`__ are + :eq:`eq:hybrid_erosion_impl_qcl` and + :eq:`eq:hybrid_erosion_impl_Cl` are applied to yield the implicit solution. #. **Use an analytic solution to the integration of the time-derivatives - in (**\ `[eq:dqcldt_hybrid] <#eq:dqcldt_hybrid>`__\ **) and - (**\ `[eq:dcdt_hybrid] <#eq:dcdt_hybrid>`__\ **) for greater + in (**\ :eq:`eq:dqcldt_hybrid`\ **) and + (**\ :eq:`eq:dcdt_hybrid`\ **) for greater accuracy. (i_pc2_erosion_numerics=3)** Two problems have been identified with the above implicit numerical @@ -2987,8 +2882,8 @@ cumulus regimes. Under this option, we attempt to compute an analytic solution to the simultaneous differential equations - `[eq:dqcldt_hybrid] <#eq:dqcldt_hybrid>`__ and - `[eq:dcdt_hybrid] <#eq:dcdt_hybrid>`__ so that :math:`q_{cl}` and + :eq:`eq:dqcldt_hybrid` and + :eq:`eq:dcdt_hybrid` so that :math:`q_{cl}` and :math:`C_l` both decrease smoothly and consistently. The equations lead to somewhat different behaviour depending on whether the grid-mean state is subsaturated (:math:`Q_c < 0`), supersaturated @@ -3001,38 +2896,36 @@ cumulus regimes. The relation between the erosion tendencies in liquid-cloud-fraction and liquid water content - (`[eq:dcdt_hybrid] <#eq:dcdt_hybrid>`__) can be expressed in terms + :eq:`eq:dcdt_hybrid` can be expressed in terms of *fractional* rates of change (dividing the top and bottom by :math:`-C_l Q_c`, and dividing both sides by :math:`C_l`): - .. math:: + .. math:: :label: eq:dcdt_hybrid_1 \frac{1}{C_l} \frac{\partial C_l}{\partial t} = \frac{ G(-Q_c) \frac{q_{cl}}{C_l^2} }{ 1 - \frac{q_{cl}}{C_l Q_c} } \; \frac{1}{q_{cl}} \frac{\partial q_{cl}}{\partial t} - \label{eq:dcdt_hybrid_1} Under homogeneous forcing (section :ref:`Homogeneous forcing`), we defined the PDF height at the saturation boundary when near the cloudy end of the PDF as :math:`G(-Q_c) = \frac{n+1}{n+2} \frac{C_l^2}{q_{cl}}` (eq - `[eqn20] <#eqn20>`__). In fact, :math:`G(-Q_c)` is set to some + :eq:`eqn20`. In fact, :math:`G(-Q_c)` is set to some blend between this and the value near the clear end of the PDF (eq - `[eqn21] <#eqn21>`__). But we will assume that when eroding cloud + :eq:`eqn21`. But we will assume that when eroding cloud under grid-mean subsaturated conditions (:math:`Q_c < 0`), :math:`G(-Q_c)` follows this scaling with :math:`\frac{C_l^2}{q_{cl}}` even if its value differs somewhat - from eq `[eqn20] <#eqn20>`__. Therefore the quantity + from eq :eq:`eqn20`. Therefore the quantity :math:`c_1 = G(-Q_c) \frac{q_{cl}}{C_l^2}` remains constant during the erosion process, and eq - `[eq:dcdt_hybrid_1] <#eq:dcdt_hybrid_1>`__ becomes: + :eq:`eq:dcdt_hybrid_1` becomes: - .. math:: + .. math:: :label: eq:dcdt_hybrid_2 \frac{1}{C_l} \frac{\partial C_l}{\partial t} = \frac{ c_1 }{ 1 - \frac{q_{cl}}{C_l Q_c} } \; \frac{1}{q_{cl}} \frac{\partial q_{cl}}{\partial t} - \label{eq:dcdt_hybrid_2} The term :math:`1 - \frac{q_{cl}}{C_l Q_c}` (which is :math:`> 1` since we are considering grid-mean subsaturation :math:`Q_c < 0`) @@ -3040,13 +2933,12 @@ cumulus regimes. fractional variation over the timestep is small compared to the other terms, and treat it explicitly. We can therefore straightforwardly integrate eq - `[eq:dcdt_hybrid_2] <#eq:dcdt_hybrid_2>`__ to obtain the scaling + :eq:`eq:dcdt_hybrid_2` to obtain the scaling of :math:`C_l` with :math:`q_{cl}` as both are reduced by erosion: - .. math:: + .. math:: :label: eq:cl_qcl_scaling \frac{C_l}{{C_l}_0} = \left( \frac{q_{cl}}{{q_{cl}}_0} \right)^{b_1} - \label{eq:cl_qcl_scaling} where :math:`{C_l}_0`, :math:`{q_{cl}}_0` are the values before erosion is applied, and the exponent is @@ -3063,34 +2955,32 @@ cumulus regimes. :math:`q_{cl}` with time. Ignoring the cloud surface-area contributions from the levels above and below (they are disabled in the code anyway), the erosion liquid water content tendency is - obtained by combining `[eq:dqcldt_hybrid] <#eq:dqcldt_hybrid>`__ - and `[eq:S_Cl] <#eq:S_Cl>`__: + obtained by combining :eq:`eq:dqcldt_hybrid` + and :eq:`eq:S_Cl`: - .. math:: + .. math:: :label: eq:dqcldt_hybrid_1 \frac{\partial q_{cl}}{\partial t} = -K \, 2 C_l (1 - C_l) \, (q_{sat}(T)-q_v) - \label{eq:dqcldt_hybrid_1} - From eq `[SD2] <#SD2>`__, :math:`q_{sat}(T)-q_v = \frac{SD}{a_L}`, + From eq :eq:`SD2`, :math:`q_{sat}(T)-q_v = \frac{SD}{a_L}`, where :math:`SD` is the saturation defecit, and :math:`a_L` is the - dimensionless factor defined in eq `[eq:a_L] <#eq:a_L>`__. + dimensionless factor defined in eq :eq:`eq:a_L`. Following the derivation in section :ref:`“Smooth” initiation logic` (eq - `[eq:qc_plus_sd] <#eq:qc_plus_sd>`__), we can write this in terms + :eq:`eq:qc_plus_sd`, we can write this in terms of the liquid-water content: :math:`SD = q_{cl} - Q_c` (where :math:`Q_c` was defined in eq - `[eq:qc_eq_qt-qs] <#eq:qc_eq_qt-qs>`__, and corresponds to the + :eq:`eq:qc_eq_qt-qs`, and corresponds to the grid-mean supersaturation converted to an equivalent liquid water content). Substituting this into - (`[eq:dqcldt_hybrid_1] <#eq:dqcldt_hybrid_1>`__) above, we obtain: + :eq:`eq:dqcldt_hybrid_1` above, we obtain: - .. math:: + .. math:: :label: eq:dqcldt_hybrid_2 \frac{\partial q_{cl}}{\partial t} = -\frac{K}{a_L} \, 2 C_l (1 - C_l) \, (q_{cl}-Q_c) - \label{eq:dqcldt_hybrid_2} - Substituting eq `[eq:cl_qcl_scaling] <#eq:cl_qcl_scaling>`__ for + Substituting eq :eq:`eq:cl_qcl_scaling` for the leading factor of :math:`C_l` on the right-hand-side and rearranging: @@ -3115,12 +3005,11 @@ cumulus regimes. rearranging, we obtain our analytical solution for :math:`q_{cl}` after time :math:`\Delta t`: - .. math:: + .. math:: :label: eq:qcl_int_hybrid {q_{cl}}_{\Delta t} = {q_{cl}}_0 \left( 1 - \frac{1-b_1}{{q_{cl}}_0} \frac{K}{a_L} \, 2 {C_l}_0 (1 - C_l) \, (q_{cl}-Q_c) \Delta t \right)^\frac{1}{1-b_1} - \label{eq:qcl_int_hybrid} Note that :math:`q_{cl}` falls to zero after a finite time :math:`\frac{{q_{cl}}_0}{1-b_1} \frac{a_L}{K} @@ -3131,14 +3020,14 @@ cumulus regimes. We first set :math:`{q_{cl}}_0` and :math:`{C_l}_0` to the values already updated by homogeneous forcing, and then sequentially compute the updated :math:`q_{cl}` after erosion using - (`[eq:qcl_int_hybrid] <#eq:qcl_int_hybrid>`__). Then we substitute - this value into (`[eq:cl_qcl_scaling] <#eq:cl_qcl_scaling>`__) to + :eq:`eq:qcl_int_hybrid`. Then we substitute + this value into :eq:`eq:cl_qcl_scaling` to compute the consistent updated value of :math:`C_l`. Finally, to improve accuracy, a small number of iterations are performed to find the solution with the explicitly-treated terms (the exponent :math:`b_1 = \frac{ c_1 }{ 1 - \frac{q_{cl}}{C_l Q_c} }` and the terms :math:`(1 - C_l)` and :math:`(q_{cl}-Q_c)` in eq - `[eq:qcl_int_hybrid] <#eq:qcl_int_hybrid>`__) adjusted to values + :eq:`eq:qcl_int_hybrid` adjusted to values linearly-interpolated to half-way between the start and end of the erosion timestep. @@ -3152,15 +3041,14 @@ cumulus regimes. clear-fraction :math:`1-C_l` in place of :math:`C_l`, and the saturation defecit :math:`SD = q_{cl}-Q_c` in place of :math:`q_{cl}`. Assuming that :math:`G(-Qc)` follows the scaling - for the clear end of the PDF (`[eqn21] <#eqn21>`__), this leads to + for the clear end of the PDF :eq:`eqn21`, this leads to a similar equation to - (`[eq:cl_qcl_scaling] <#eq:cl_qcl_scaling>`__) but for the scaling + :eq:`eq:cl_qcl_scaling` but for the scaling as erosion reduces :math:`1-C_l` and :math:`SD` towards zero .: - .. math:: + .. math:: :label: eq:ca_sd_scaling \frac{1-C_l}{1-{C_l}_0} = \left( \frac{SD}{SD_0} \right)^{b_2} - \label{eq:ca_sd_scaling} with :math:`b_2 = \frac{ c_2 }{ 1 + \frac{SD}{(1-C_l) Q_c} }` and :math:`c_2 = G(-Q_c) \frac{SD}{(1-C_l)^2}` (note we must have @@ -3168,31 +3056,29 @@ cumulus regimes. And then the tendency equation for :math:`SD` is: - .. math:: + .. math:: :label: eq:dsddt_hybrid_2 \frac{\partial SD}{\partial t} = -\frac{K}{a_L} \, 2 (1 - C_l) C_l \, SD - \label{eq:dsddt_hybrid_2} The one asymmetry between this and the :math:`q_{cl}` tendency - equation (`[eq:dqcldt_hybrid_2] <#eq:dqcldt_hybrid_2>`__) is that + equation :eq:`eq:dqcldt_hybrid_2` is that for :math:`SD` the r.h.s. is directly proportional to the quantity in the time-derivative, whereas for :math:`q_{cl}` there is an additional :math:`Q_c` term which is constant during erosion. - Substituting (`[eq:ca_sd_scaling] <#eq:ca_sd_scaling>`__) for the + Substituting :eq:`eq:ca_sd_scaling` for the leading factor of :math:`(1 - C_l)` in - (`[eq:dsddt_hybrid_2] <#eq:dsddt_hybrid_2>`__), integrating over + :eq:`eq:dsddt_hybrid_2`, integrating over time :math:`\Delta t` (neglecting the fractional variation of :math:`C_l` over the timestep) and rearranging, we obtain: - .. math:: + .. math:: :label: eq:sd_int_hybrid {SD}_{\Delta t} = {SD}_0 \left( 1 + b_2 \frac{K}{a_L} \, 2 (1-{C_l}_0) C_l \, \Delta t \right)^{-\frac{1}{b_2}} - \label{eq:sd_int_hybrid} Note that the additional power of :math:`SD` on the r.h.s. of - (`[eq:dsddt_hybrid_2] <#eq:dsddt_hybrid_2>`__) leads to the + :eq:`eq:dsddt_hybrid_2` leads to the integral solution having a negative exponent. This means that under grid-mean supersaturation, erosion makes :math:`SD` and :math:`1-C_l` approach but never quite reach zero, which is quite @@ -3206,14 +3092,14 @@ cumulus regimes. We first set :math:`{SD}_0 = {q_{cl}}_0 - Q_c` (where as above :math:`{q_{cl}}_0` is the value already updated by homogeneous forcing), then compute the value of :math:`SD` updated by erosion - using (`[eq:sd_int_hybrid] <#eq:sd_int_hybrid>`__). Then we + using :eq:`eq:sd_int_hybrid`. Then we substitute this value into - (`[eq:ca_sd_scaling] <#eq:ca_sd_scaling>`__) to compute the + :eq:`eq:ca_sd_scaling` to compute the consistent updated value of :math:`1-C_l`. A small number of iterations are then performed to find the solution with the explicitly-treated terms (the exponent :math:`b_2 = \frac{ c_2 }{ 1 + \frac{SD}{(1-C_l) Q_c} }` and the - term :math:`C_l` in eq `[eq:sd_int_hybrid] <#eq:sd_int_hybrid>`__) + term :math:`C_l` in eq :eq:`eq:sd_int_hybrid` adjusted to values linearly-interpolated to half-way between the start and end of the erosion timestep. Then the final values of :math:`SD` and :math:`1-C_l` are used to increment @@ -3225,8 +3111,8 @@ cumulus regimes. In this case, the PDF is centred on the saturation boundary, so that narrowing it does not change the cloud-fraction. In the limit :math:`Q_c = 0`, we have :math:`q_{cl} = SD`, and - (`[eq:dqcldt_hybrid_2] <#eq:dqcldt_hybrid_2>`__) or - (`[eq:dsddt_hybrid_2] <#eq:dsddt_hybrid_2>`__) becomes: + :eq:`eq:dqcldt_hybrid_2` or + :eq:`eq:dsddt_hybrid_2` becomes: .. math:: @@ -3282,7 +3168,7 @@ At a basic level, the boundary layer scheme works by mixing represented using the homogeneous forcing representation. The forcing of :math:`Q_c` can be written in :math:`\Delta \overline{q_T}` and :math:`\Delta \overline{T_L}` terms using -(`[eq:deltaqc_exp] <#eq:deltaqc_exp>`__). +:eq:`eq:deltaqc_exp`. :math:`\overline{q_{cf}}` is already mixed using the tracer mixing scheme. PC2 will calculate the corresponding :math:`C_i` change assuming @@ -3294,20 +3180,18 @@ in-cloud water content :math:`q_c^S` based upon a linear combination of the current in-cloud ice water content, :math:`\frac{\overline{q_{cf}}}{C_i}`, and a fixed value. -.. math:: +.. math:: :label: eq:qcf_ci q_C^S = C_i \frac{\overline{q_{cf}}}{C_i} + ( 1 - C_i) q_{cf0 \, BL} - \label{eq:qcf_ci} where :math:`q_{cf0 \, BL}` is a specified value of :math:`1 \times 10^{-4} \, kg \, kg^{-1}`. We then use the inhomogeneous -forcing equation based upon (`[eq:dcdt_inhom2] <#eq:dcdt_inhom2>`__) but +forcing equation based upon :eq:`eq:dcdt_inhom2` but for ice water content to write -.. math:: +.. math:: :label: eq:deltaci_bl \Delta C_i = \frac{(1 - C_i)}{q_C^S - \overline{q_{cf}}} Q4_i . - \label{eq:deltaci_bl} Since the physical model will have :math:`C_i` tend to 1 if the denominator is small, we will, to avoid numerical problems, set @@ -3315,10 +3199,10 @@ denominator is small, we will, to avoid numerical problems, set :math:`q_C^S - \overline{q_{cf}} < 1 \times 10^{-10} kg kg^{-1}`. Note that we do not use the multiple phases injection source expressions (section :ref:`Multiple phases in the injection source` and equation -`[eq:cff_ts] <#eq:cff_ts>`__). This is because the liquid water changes +:eq:`eq:cff_ts`. This is because the liquid water changes are not associated with the plume model. -Equation `[eq:qcf_ci] <#eq:qcf_ci>`__ assumes that the change to the ice +Equation :eq:`eq:qcf_ci` assumes that the change to the ice water content has led to an increase in ice water content. However, if the ince water content has reduced, the change to the ice cloud fraction is not consistent. The option to "Use consistent formulation of ice @@ -3393,13 +3277,12 @@ ought to be transported by the compensating subsidence in a similar way to the condensate transport (which is documented below). We therefore split the convective contribution in -(`[eq:dqcldt_and_dcdt] <#eq:dqcldt_and_dcdt>`__) into two parts: +:eq:`eq:dqcldt_and_dcdt` into two parts: -.. math:: +.. math:: :label: eq:inhomg_plus_homog \frac{\partial \overline{q_{cl}}}{\partial t} |_{convection} = Q4_l + Q_{environment} - \label{eq:inhomg_plus_homog} where :math:`Q_{environment}` is the condensation associated with changes in the vapour and temperature from the detrainment and @@ -3423,30 +3306,27 @@ equations for convective tendencies are most simply applied to a variable, :math:`{\chi}`, that is conserved under moist adiabatic processes (e.g. total water content). In this case, -.. math:: +.. math:: :label: eq:chibasic {\ensuremath{\frac{\partial \, \ensuremath{{\chi}_{\rm{ }}^{\rm{E}}}}{\partial \, t}}}_{\rm{conv}} = - \frac{1}{\overline{\rho}} \, \ensuremath{\frac{\partial \, \overline{\rho w^{'} \ensuremath{{\chi}_{\rm{ }}^{\rm{E'}}}}}{\partial \, z}} - \label{eq:chibasic} -To parametrize `[eq:chibasic] <#eq:chibasic>`__, the current UM +To parametrize :eq:`eq:chibasic`, the current UM convection scheme takes a mass flux approximation -.. math:: +.. math:: :label: eq:massflux \left({\overline{\rho w^{'} \ensuremath{{\chi}_{\rm{ }}^{\rm{E'}}}}} \right)_{\rm{conv}} = M^{\rm{P}} \, \left({ \ensuremath{{\chi}_{\rm{ }}^{\rm{P}}} - \ensuremath{{\chi}_{\rm{ }}^{\rm{E}}} } \right) - \label{eq:massflux} which can be differentiated to give -.. math:: +.. math:: :label: eq:eddyflux - \frac{1}{\overline{\rho}} \, \ensuremath{\frac{\partial \, \overline{\rho w^{'} \ensuremath{{\chi}_{\rm{ }}^{\rm{E'}}}}}{\partial \, z}} = \ensuremath{\frac{\partial \, \ensuremath{{\chi}_{\rm{ }}^{\rm{P}}} \, M^{\rm{P}}}{\partial \, p}} - \ensuremath{{\chi}_{\rm{ }}^{\rm{E}}} \, \ensuremath{\frac{\partial \, M^{\rm{P}}}{\partial \, p}} - M^{\rm{P}} \, \ensuremath{\frac{\partial \, \ensuremath{{\chi}_{\rm{ }}^{\rm{E}}}}{\partial \, p}} - \label{eq:eddyflux} The bulk cloud model plume equations for mass and :math:`{\chi}` are: @@ -3462,36 +3342,34 @@ The bulk cloud model plume equations for mass and :math:`{\chi}` are: } \right)\label{eq:dbydpmfchi} \end{aligned} -Equations `[eq:eddyflux] <#eq:eddyflux>`__, -`[eq:dbydpmassflux] <#eq:dbydpmassflux>`__ and -`[eq:dbydpmfchi] <#eq:dbydpmfchi>`__ can then be substituted into -`[eq:chibasic] <#eq:chibasic>`__ to give: +Equations :eq:`eq:eddyflux`, +:eq:`eq:dbydpmassflux` and +:eq:`eq:dbydpmfchi` can then be substituted into +:eq:`eq:chibasic` to give: -.. math:: +.. math:: :label: eq:chimassflux {\ensuremath{\frac{\partial \, \ensuremath{{\chi}_{\rm{ }}^{\rm{E}}}}{\partial \, t}}}_{\rm{conv}} = - M^{\rm{P}} \, \ensuremath{\frac{\partial \, \ensuremath{{\chi}_{\rm{ }}^{\rm{E}}}}{\partial \, p}} + \mu \, M^{\rm{P}} \, \left({ \ensuremath{{\chi}_{\rm{ }}^{\rm{R}}} - \ensuremath{{\chi}_{\rm{ }}^{\rm{E}}} } \right) + \delta \, M^{\rm{P}} \, \left({ \ensuremath{{\chi}_{\rm{ }}^{\rm{P}}} - \ensuremath{{\chi}_{\rm{ }}^{\rm{E}}} } \right) - \label{eq:chimassflux} while :math:`{\chi}_{\rm{}}^{\rm{P}}` is obtained from the vertical -gradient derived by combining `[eq:dbydpmassflux] <#eq:dbydpmassflux>`__ -and `[eq:dbydpmfchi] <#eq:dbydpmfchi>`__ : +gradient derived by combining :eq:`eq:dbydpmassflux` +and :eq:`eq:dbydpmfchi` : -.. math:: +.. math:: :label: eq:gradchipar M^{\rm{P}} \, \ensuremath{\frac{\partial \, \ensuremath{{\chi}_{\rm{ }}^{\rm{P}}}}{\partial \, p}} = \varepsilon \, M^{\rm{P}} \, \left({ \ensuremath{{\chi}_{\rm{ }}^{\rm{P}}} - \ensuremath{{\chi}_{\rm{ }}^{\rm{E}}} } \right)- \mu \, M^{\rm{P}} \, \left({ \ensuremath{{\chi}_{\rm{ }}^{\rm{P}}} - \ensuremath{{\chi}_{\rm{ }}^{\rm{R}}} } \right) - \label{eq:gradchipar} -Within the model, eqn `[eq:chimassflux] <#eq:chimassflux>`__ would take +Within the model, eqn :eq:`eq:chimassflux` would take a discretized form which actually depends upon whether the model level, k, is above or at the lowest cloud level (k = cb). Note that the formal cloud base lies at the half-level below, i.e. on the layer boundary which is also the top of the turbulent mixed boundary layer. A simple -discretized form of `[eq:chimassflux] <#eq:chimassflux>`__, setting +discretized form of :eq:`eq:chimassflux`, setting :math:`{ \mu = 0 }`, is: .. math:: @@ -3512,16 +3390,16 @@ discretized form of `[eq:chimassflux] <#eq:chimassflux>`__, setting where the initial parcel value :math:`{\chi}_{\rm{i,cb}}^{\rm{P}}` may be chosen to produce a fixed increment or place a closure condition on the cloud base flux. In fact, the convection equations (see ) differ -from `[eq:chidisck] <#eq:chidisck>`__ and -`[eq:chidisccb] <#eq:chidisccb>`__ because a different discretization is +from :eq:`eq:chidisck` and +:eq:`eq:chidisccb` because a different discretization is used, but the principle is unaltered. The model convection variables are NOT conserved under moist adiabatic processes because precipitation processes deplete the column moisture and condensation processes affect the temperature, specific humidity and cloud condensate variables. Surprisingly, however, the form of -eqn `[eq:chimassflux] <#eq:chimassflux>`__ is retained even though the -basic equation `[eq:chibasic] <#eq:chibasic>`__ acquires additional +eqn :eq:`eq:chimassflux` is retained even though the +basic equation :eq:`eq:chibasic` acquires additional terms for temperature and specific humidity: .. math:: @@ -3539,13 +3417,13 @@ terms for temperature and specific humidity: where :math:`{\overline{Q}}_{\rm{par}}` is the rate of condensation which occurs in the ascending plumes. -The reason that `[eq:defineq1] <#eq:defineq1>`__ and -`[eq:defineq2] <#eq:defineq2>`__ retain this form is due to cancellation +The reason that :eq:`eq:defineq1` and +:eq:`eq:defineq2` retain this form is due to cancellation from the bulk cloud terms equivalent to -`[eq:dbydpmfchi] <#eq:dbydpmfchi>`__ which are modified in the same way -as `[eq:defineq1] <#eq:defineq1>`__ and -`[eq:defineq2] <#eq:defineq2>`__. The change is seen in the vertical -gradient equations based upon `[eq:gradchipar] <#eq:gradchipar>`__ +:eq:`eq:dbydpmfchi` which are modified in the same way +as :eq:`eq:defineq1` and +:eq:`eq:defineq2`. The change is seen in the vertical +gradient equations based upon :eq:`eq:gradchipar` .. math:: @@ -3588,9 +3466,9 @@ is basic equations \label{eq:basiclold} \end{aligned} -By analogy with equations `[eq:defineq1] <#eq:defineq1>`__ and -`[eq:defineq2] <#eq:defineq2>`__, we can define a :math:`Q4` from -`[eq:basiclold] <#eq:basiclold>`__ and state that for the control +By analogy with equations :eq:`eq:defineq1` and +:eq:`eq:defineq2`, we can define a :math:`Q4` from +:eq:`eq:basiclold` and state that for the control convection scheme :math:`Q4 = 0`. The PC2 scheme requires a reassessment of these assumptions because we wish to allow non-zero environment condensate values and to allow them to change. @@ -3636,7 +3514,7 @@ where the PC2 assumption thus far has been that great caution as the formulations are extremely sensitive to errors in assignment of condensate phase. -Based on `[eq:gradlpar] <#eq:gradlpar>`__, the vertical dependence of +Based on :eq:`eq:gradlpar`, the vertical dependence of condensate is calculated as .. math:: @@ -3652,8 +3530,8 @@ condensate is calculated as \frac{SNOW}{M^{\rm{P}}}\label{eq:vertparf} \end{aligned} -Following , equations `[eq:dbydpmassflux] <#eq:dbydpmassflux>`__, -`[eq:vertparl] <#eq:vertparl>`__ and `[eq:vertparf] <#eq:vertparf>`__ +Following , equations :eq:`eq:dbydpmassflux`, +:eq:`eq:vertparl` and :eq:`eq:vertparf` are discretized: .. math:: @@ -3690,9 +3568,9 @@ where :math:`EPSS_{\rm{k}} = \left({1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4}} \right)`. The condensation and precipitation terms in equations -`[eq:discdmfbydp] <#eq:discdmfbydp>`__, -`[eq:discvparl] <#eq:discvparl>`__ and -`[eq:discvparf] <#eq:discvparf>`__ make the equations implicit. They are +:eq:`eq:discdmfbydp`, +:eq:`eq:discvparl` and +:eq:`eq:discvparf` make the equations implicit. They are therefore solved by starting with an ascent in which condensation and precipitation terms are suppressed: @@ -3719,7 +3597,7 @@ At the base of the convective plume (ie. the level immediately above cloud base), :math:`l_{\rm{l \, k}}^{\rm{P}}` is initialized to :math:`l_{\rm{l \, i}}^{\rm{P}}` and :math:`l_{\rm{f \, k}}^{\rm{P}}` to :math:`l_{\rm{f \, i}}^{\rm{P}}`, where the initial values are chosen -such that the modified form of `[eq:chidisccb] <#eq:chidisccb>`__ +such that the modified form of :eq:`eq:chidisccb` produces zero fluxes at cloud base: .. math:: @@ -3763,11 +3641,10 @@ value. The precipitation calculation is unaltered. -.. math:: +.. math:: :label: eq:precip P_{\rm{k} + 1} = \left({ \ensuremath{l_{\rm{k + 1}}^{\rm{P}}} - \ensuremath{l_{\rm{MIN}}^{\rm{P}}} } \right)\, M_{\rm{k} + 1} \, / \, g - \label{eq:precip} where :math:`l_{\rm{k + 1}}^{\rm{P}}` = :math:`l_{\rm{l \, k + 1}}^{\rm{P}}` + @@ -3792,7 +3669,7 @@ This reduces the parcel condensate to : \end{aligned} The final parcel condensate values are then used in the rate calculation -based upon eqn `[eq:basiclold] <#eq:basiclold>`__: +based upon eqn :eq:`eq:basiclold`: .. math:: @@ -3859,8 +3736,8 @@ and } \right]& { } & \label{eq:enviroq} \end{aligned} -Similarly, eqns `[eq:q4lmassf] <#eq:q4lmassf>`__ and -`[eq:q4fmassf] <#eq:q4fmassf>`__ have a discretized form as follows: +Similarly, eqns :eq:`eq:q4lmassf` and +:eq:`eq:q4fmassf` have a discretized form as follows: .. math:: @@ -3947,42 +3824,38 @@ is the effect of a background change of :math:`q` that will be applied across the part of the gridbox that is not associated with the injected air. We write this as: -.. math:: +.. math:: :label: eqn:1mcs (1 - \Delta C_S) \Delta \overline{q_{background}} = \Delta \overline{q} - \Delta C_S ( q_{sat liq}(\overline{T_{s}}) - \overline{q} ) . - \label{eqn:1mcs} Now we recognise that -.. math:: +.. math:: :label: eqn:Q2 \Delta \overline{q} = Q2~ \Delta t - \label{eqn:Q2} where :math:`Q2` is the rate of moistening of the whole gridbox due to convection. Remember that, at this stage, we haven’t done any condensation outside of the plume. Hence to calculate the condensation we should apply the background change in :math:`\overline{q}` as a uniform forcing for the background air. Hence -(`[eqn:1mcs] <#eqn:1mcs>`__) becomes, using (`[eqn:Q2] <#eqn:Q2>`__), +:eq:`eqn:1mcs` becomes, using :eq:`eqn:Q2`, -.. math:: +.. math:: :label: eqn:Aq (1 - \Delta C_S) A_q |_{background} \Delta t = Q2 ~ \Delta t - \Delta C_S ( q_{sat liq}(T_{s}) - \overline{q} ) . - \label{eqn:Aq} where :math:`A_q |_{background}` is the currently unknown background forcing of :math:`q` (see :raw-latex:`\cite{gwb02}`) and :math:`\Delta t` is the timestep. We can do the same analysis for the temperature change, and obtain -.. math:: +.. math:: :label: eqn:AT (1 - \Delta C_S) A_T |_{background} \Delta t = Q1~ \Delta t - \Delta C_S (T_s - \overline{T} ) - \label{eqn:AT} where Q1 is the rate of warming in the gridbox due to convection and :math:`A_T |_{background}` is the currently unknown background forcing @@ -3993,18 +3866,17 @@ The full change of liquid water content in the gridbox is that injected, from the uniform forcings (see :raw-latex:`\cite{wg03}`). Note that the uniform forcings are only applied across a proportion :math:`1 - \Delta C_S` of the gridbox. Hence these two terms give, using -the homogeneous forcing equations (`[dqcldt] <#dqcldt>`__) and -(`[eq:deltaqc_exp2] <#eq:deltaqc_exp2>`__), +the homogeneous forcing equations :eq:`dqcldt` and +:eq:`eq:deltaqc_exp2`, -.. math:: +.. math:: :label: eqn:qclconv \Delta \overline{q_{cl}} |_{convection} = Q4 \Delta t + (1 - \Delta C_S) a_L C_l (A_q |_{background} \Delta t - \alpha A_T |_{background} \Delta t ). - \label{eqn:qclconv} -Using (`[eqn:Aq] <#eqn:Aq>`__) and (`[eqn:AT] <#eqn:AT>`__) to expand -the forcing terms in (`[eqn:qclconv] <#eqn:qclconv>`__) gives +Using :eq:`eqn:Aq` and :eq:`eqn:AT` to expand +the forcing terms in :eq:`eqn:qclconv` gives .. math:: @@ -4018,12 +3890,11 @@ We now note that and hence the final result -.. math:: +.. math:: :label: eqn:dqcl \Delta \overline{q_{cl}} |_{convection} = Q4 \Delta t + \Delta t ~ a_L C_l ( ( Q2 - \alpha Q1) - \Delta C_S (q_{sat liq}(\overline{T}) - \overline{q} ) ) . - \label{eqn:dqcl} There is thus an extra term, :math:`-\Delta C_S (q_{sat}(\overline{T})-\overline{q} )`, which needs @@ -4033,7 +3904,7 @@ the second term of the expression). This has arisen from the requirement that the vapour injected by the plume is saturated. We need simply to retrieve the value of :math:`\Delta C_S` to complete the parametrization. This can be straightforwardly obtained from -(`[eq:dcldt_inhom] <#eq:dcldt_inhom>`__), which links the net change of +:eq:`eq:dcldt_inhom`, which links the net change of liquid cloudy volume due to the injection, :math:`\Delta C_{injection}`, with :math:`\Delta C_S`. @@ -4041,35 +3912,33 @@ with :math:`\Delta C_S`. where :math:`g_l` is 1 if the injected cloud is of liquid phase and 0 if it is of ice phase. We already know :math:`\Delta C_{injection}` from -the injection forcing arguments (`[eq:dctdt_xl] <#eq:dctdt_xl>`__) above +the injection forcing arguments :eq:`eq:dctdt_xl` above that link it to :math:`Q4`. We therefore complete the parametrization by calculating :math:`\Delta C_S` based on whether :math:`\Delta C` is positive or negative. If :math:`\Delta C` is positive, we assume that the plume must be of liquid phase and hence -.. math:: +.. math:: :label: eqn:cs1 \Delta C_S = \frac{\Delta C_{injection}} {1 -C_l} . - \label{eqn:cs1} If :math:`\Delta C_l` is negative, we assume that the plume must be of ice phase and hence -.. math:: +.. math:: :label: eqn:cs2 \Delta C_S = - \frac{\Delta C_{injection}} {C_l} . - \label{eqn:cs2} Here we have still assumed that the vapour content in the detrained plume is equal to :math:`q_{sat liq}`. A better assumption may be to -replace the :math:`q_{sat liq}` term in (`[eqn:dqcl] <#eqn:dqcl>`__) +replace the :math:`q_{sat liq}` term in :eq:`eqn:dqcl` with a :math:`q_{sat}` expression that depends on the volume fraction of detrained condensate that is liquid phase, :math:`g_l`. If :math:`\Delta C_{injection}` is zero, we assume that :math:`\Delta C_S` is 0 also. Equations -(`[eqn:dqcl] <#eqn:dqcl>`__),(`[eqn:cs1] <#eqn:cs1>`__), and -(`[eqn:cs2] <#eqn:cs2>`__) form the parametrization for +:eq:`eqn:dqcl`,:eq:`eqn:cs1`, and +:eq:`eqn:cs2` form the parametrization for :math:`\Delta \overline{q_{cl}}|_{convection}`. The representation of :math:`\Delta C_{convection}` is similar in form to :math:`\Delta \overline{q_{cl}}|_{convection}`: @@ -4081,7 +3950,7 @@ If :math:`\Delta C_{injection}` is zero, we assume that - \overline{q} ) ) . where the specification of :math:`G(-Q_c)` follows -(`[eqn22] <#eqn22>`__). Note that the code includes the numerical limit +:eq:`eqn22`. Note that the code includes the numerical limit restriction that :math:`\Delta C_S` is between 0 and 1. Thus we are able to parametrize the net condensation and cloud changes @@ -4119,16 +3988,16 @@ Under this option, the increments to :math:`\overline{q_{cl}}` and :math:`C_l` produced by the convection scheme are assumed to already include the effects of entrainment, detrainment (i.e. injection) and compensating subsidence (i.e. vertical advection) as expressed by -equation `[eq:chimassflux] <#eq:chimassflux>`__, but exclude the effects +equation :eq:`eq:chimassflux`, but exclude the effects of homogeneous forcing of clouds in the enviroment. Note that taking -equation `[eq:chimassflux] <#eq:chimassflux>`__ with :math:`\chi` set to +equation :eq:`eq:chimassflux` with :math:`\chi` set to water vapour :math:`q`, detrainment of saturated air into a subsaturated environment will imply a positive tendency of :math:`\overline{q}`, but this is *not* a homogeneous forcing, since the increase in :math:`\overline{q}` is entirely due to injecting new parcels of saturated air without altering the existing environment parcels. Setting :math:`\chi` to be :math:`\overline{q_{cl}}` or :math:`C_l` in equation -`[eq:chimassflux] <#eq:chimassflux>`__, there is a simply-calculated +:eq:`eq:chimassflux`, there is a simply-calculated source of cloud water and fraction wherever the detrained air is cloudy (:math:`C_l=1` in the detrained parcel), and we assume these terms have been calculated this way inside the convection scheme. @@ -4146,10 +4015,9 @@ convective mass-flux in units of Pa s\ :math:`^{-1}`, so it already expresses the pressure vertical velocity forced by subsidence in the environment: -.. math:: +.. math:: :label: eq:delta_p_conv \Delta p^E = \Delta t \left( M_{up} - M_{dwn} \right) - \label{eq:delta_p_conv} where :math:`M_{up}` is the updraft mass-flux, :math:`M_{dwn}` is the downdraft mass-flux, and :math:`\Delta t` is the model timestep length. @@ -4157,19 +4025,18 @@ The adiabatic temperature change following an environment parcel subsided from pressure :math:`p - \Delta p^E` to :math:`p` is then given by: -.. math:: +.. math:: :label: eq:delta_t_conv \Delta T^E = \theta^E \left( \left(\frac{p}{p_{ref}}\right)^\kappa - \left(\frac{p - \Delta p^E}{p_{ref}}\right)^\kappa \right) - \label{eq:delta_t_conv} where :math:`\theta^E` is the environment potential temperature, :math:`p_{ref}` is the reference pressure used to define potential temperature, and :math:`\kappa = \frac{R_d}{c_p}` is the ratio of the gas constant for dry air over its heat capacity at constant pressure. -`[eq:delta_p_conv] <#eq:delta_p_conv>`__ and -`[eq:delta_t_conv] <#eq:delta_t_conv>`__ are passed into the PC2 +:eq:`eq:delta_p_conv` and +:eq:`eq:delta_t_conv` are passed into the PC2 homogeneous forcing routine after convection as the forcings to be applied (with the forcings to all other variables set to zero). @@ -4359,7 +4226,7 @@ until the end of the timestep. After the second physics updates have been performed (*atmos-physics2*), the model (including the control) recalculates the value of *Exner* (:math:`\prod^{[n+1]}`). From :math:`\prod_{dep}` and :math:`\prod^{[n+1]}` we can calculate, using -the definition (`[eq:exner] <#eq:exner>`__), the values of departure +the definition :eq:`eq:exner`, the values of departure pressure and temperature: .. math:: \overline{p}_{dep} = p_{ref} {\prod_{dep}}^{\frac{1}{\kappa}} @@ -4368,22 +4235,20 @@ pressure and temperature: Hence we obtain the net forcing values -.. math:: +.. math:: :label: eq:deltatsl \Delta \overline{T} = \overline{T}^{[n+1]} - \overline{T}_{dep} - \label{eq:deltatsl} and -.. math:: +.. math:: :label: eq:deltapsl \Delta \overline{p} = \overline{p}^{[n+1]} - \overline{p}_{dep} . - \label{eq:deltapsl} where :math:`\overline{T}^{[n+1]}` and :math:`\overline{p}^{[n+1]}` are the temperature and pressure at the arrival point, after the dynamics -call. (`[eq:deltatsl] <#eq:deltatsl>`__) and -(`[eq:deltapsl] <#eq:deltapsl>`__) are passed to the homogeneous forcing +call. :eq:`eq:deltatsl` and +:eq:`eq:deltapsl` are passed to the homogeneous forcing routine in order to calculate the condensation and cloud fraction changes associated with the pressure change. @@ -4493,7 +4358,7 @@ restrictions. :math:`C_l` is initiated away from 0 if - :math:`RH_T^{[n+1]} > RH_T^{[n]}` , where :math:`RH_{crit \, tol}` is a specified tolerance parameter, of -value 0.01, and :math:`RH_T` is defined in (`[eq:rht] <#eq:rht>`__). +value 0.01, and :math:`RH_T` is defined in :eq:`eq:rht`. :math:`RH_T^{[n]}` is the start of timestep value of :math:`RH_T` (i.e. at time level n) and :math:`RH_T^{[n+1]}` is the value when initiation is called. Additionally, there is another possibility for the last of @@ -4625,13 +4490,13 @@ where the subscript :math:`_{diag}` denotes the liquid cloud water content and fraction predicted by the diagnostic cloud scheme (either Smith or Bimodal). -Equation `[eq:dcl_init1] <#eq:dcl_init1>`__ simply sets the +Equation :eq:`eq:dcl_init1` simply sets the cloud-fraction to a weighted mean of the pre-existing and diagnostic-scheme cloud-fractions, in proportion to the fraction of the water content that was created by initiation versus that which was already there. If the pre-existing :math:`q_{cl}` is zero, -`[eq:dqcl_init] <#eq:dqcl_init>`__ and -`[eq:dcl_init1] <#eq:dcl_init1>`__ simply set :math:`q_{cl}` and +:eq:`eq:dqcl_init` and +:eq:`eq:dcl_init1` simply set :math:`q_{cl}` and :math:`C_l` to their new diagnosed values, as in the previous options. Crucially, in the limit that the pre-existing :math:`q_{cl}` approaches :math:`{q_{cl}}_{diag}`, the increments to :math:`q_{cl}` and @@ -4646,18 +4511,18 @@ completely uninitiated state will have zero saturation deficit :math:`SD`, rather than zero :math:`q_{cl}`. Therefore, in this case the increment to :math:`C_l` is calculated based on the fractional increase in :math:`SD` from initiation (equation -`[eq:dcl_init2] <#eq:dcl_init2>`__), instead of the fractional increase +:eq:`eq:dcl_init2`, instead of the fractional increase in :math:`q_{cl}`. Whether to increment :math:`C_l` based on the increase in :math:`q_{cl}` or :math:`SD` is determined based on the sign of :math:`Q_C`, which is -defined as in equation `[eq:qc_eq_qt-qs] <#eq:qc_eq_qt-qs>`__ +defined as in equation :eq:`eq:qc_eq_qt-qs` (reproduced here for clarity): .. math:: Q_c = a_L \left( \overline{q_T} - q_{sat}(\overline{T_L}) \right) The saturation deficit :math:`SD` is defined by equation -`[SD2] <#SD2>`__: +:eq:`SD2`: .. math:: SD = a_L \left( q_{sat}(\overline{T}) - \overline{q} \right) @@ -4670,10 +4535,9 @@ equations, and: we obtain: -.. math:: +.. math:: :label: eq:qc_plus_sd q_{cl} = Q_c + SD - \label{eq:qc_plus_sd} It can be seen that when :math:`Q_C > 0` (total-water super-saturation), it represents the value :math:`q_{cl}` would have if the whole grid-box @@ -4681,7 +4545,7 @@ were saturated (:math:`SD = 0`, :math:`C_l = 1`). Note that :math:`q_{cl}` cannot fall below :math:`Q_C`, since :math:`SD` cannot be negative. Since :math:`Q_c` is invariant under condensation / evaporation, we must have :math:`\Delta SD = \Delta q_{cl}` (hence the -implementation of `[eq:dcl_init2] <#eq:dcl_init2>`__ in the code simply +implementation of :eq:`eq:dcl_init2` in the code simply uses :math:`q_{cl} - Q_c` in place of :math:`SD`, and :math:`\Delta q_{cl}` in place of :math:`\Delta SD`). @@ -4702,7 +4566,7 @@ applied in the Bounds Checking. Care needs to be taken when choosing the thresholds, since we do not wish to reset small values that are genuinely created by a physics scheme in the model. -We first calculate :math:`RH_T` using (`[eq:rht] <#eq:rht>`__) and +We first calculate :math:`RH_T` using :eq:`eq:rht` and compare this to the critical relative humidity, :math:`RH_{crit}`. The liquid cloud fraction will be reset to 1 if: @@ -4711,11 +4575,11 @@ liquid cloud fraction will be reset to 1 if: - or :math:`C_l \ge C_{high 2}` where :math:`C_{high}` and :math:`C_{high 2}` are defined in -`[eq:chigh-chigh2] <#eq:chigh-chigh2>`__. The evaporation is done by -calculating :math:`SD` using (`[SD2] <#SD2>`__) with -(`[eq:a_L] <#eq:a_L>`__) and (`[eq:alpha_exp] <#eq:alpha_exp>`__) and +:eq:`eq:chigh-chigh2`. The evaporation is done by +calculating :math:`SD` using :eq:`SD2` with +:eq:`eq:a_L` and :eq:`eq:alpha_exp` and evaporating the equivalent amount of liquid into the gridbox to take it -to saturation, according to (`[eq:qsdcheck1] <#eq:qsdcheck1>`__) below. +to saturation, according to :eq:`eq:qsdcheck1` below. Similarly, the equivalent check for low values of :math:`RH_T` is performed. The liquid cloud fraction will be reset to 0 if: @@ -4725,7 +4589,7 @@ performed. The liquid cloud fraction will be reset to 0 if: - or :math:`C_l \le C_{low 2}` . The remaining :math:`\overline{q_{cl}}` is evaporated into the gridbox -using (`[eq:qclcheck] <#eq:qclcheck>`__) below. +using :eq:`eq:qclcheck` below. The thresholds :math:`C_{high}`, :math:`C_{high 2}`, :math:`C_{low}` and :math:`C_{low 2}` are set using the parameters :math:`C_{tol}` and @@ -4805,8 +4669,8 @@ is no liquid cloud, to be equal to :math:`C_i`. .. _`sec:pc2_checks_sd`: The next check complements the first but updates the moisture fields. We -firstly calculate :math:`SD` using (`[SD2] <#SD2>`__) and -(`[eq:alpha_exp] <#eq:alpha_exp>`__). We then check whether +firstly calculate :math:`SD` using :eq:`SD2` and +:eq:`eq:alpha_exp`. We then check whether :math:`SD < 0`. This check catches instances where we have grid-mean supersaturation, which ought to be impossible (under the instantaneous condensation assumption made by PC2, condensation should occur to @@ -4869,7 +4733,7 @@ zero, *provided* that :math:`\overline{q_{cl}} > SD`. Remember that :math:`SD` corresponds to the amount of vapour that must be *evaporated* into the gridbox to give saturation, so we simply make exactly the same adjustments as we do for removing supersaturated states above -(`[eq:qsdcheck1] <#eq:qsdcheck1>`__), except that here :math:`SD` is +:eq:`eq:qsdcheck1`, except that here :math:`SD` is positive rather than negative. Our proviso that :math:`\overline{q_{cl}} > SD` ensures that we do not @@ -4940,10 +4804,9 @@ instead to create some :math:`C_i` to keep consistency. This is to allow small, but significant, amounts of :math:`\overline{q_{cf}}` created by the microphysics scheme to be maintained. -.. math:: +.. math:: :label: eq:cf_reset C_i \leftarrow \frac { \overline{q_{cf}} }{q_{cf0}} - \label{eq:cf_reset} where the ‘in-cloud’ ice content :math:`q_{cf0} = 1 \times 10^{-4} kg kg^{-1}`. @@ -5018,45 +4881,42 @@ consider the homogeneous framework and assume that there is a forcing value of :math:`Q_c` that exists that will produce the known increment to :math:`\overline{q}` and :math:`\overline{T}`. -Discritising (`[dqcldt] <#dqcldt>`__) we have, using -(`[eq:deltaqc_exp] <#eq:deltaqc_exp>`__) and expanding +Discritising :eq:`dqcldt` we have, using +:eq:`eq:deltaqc_exp` and expanding :math:`\Delta T_L` in terms of :math:`\Delta T` and :math:`\Delta q_{cl}`, -.. math:: +.. math:: :label: eq:da1 \Delta \overline{q_{cl}} = C_l ( a_L ( \Delta \overline{q} - \alpha \Delta \overline{T} - \beta \Delta \overline{p}) + \Delta \overline{q_{cl}} ). - \label{eq:da1} Remember that :math:`Q_c` (and hence :math:`\Delta Q_c`) is independent of condensation. Rearranging, we obtain -.. math:: +.. math:: :label: eq:da2 \Delta \overline{q_{cl}} = \frac{1}{1 - C_l} C_l a_L ( \Delta \overline{q} - \alpha \Delta \overline{T} - \beta \Delta \overline{p}) - \label{eq:da2} and hence an expression for the condensate increment, :math:`\Delta \overline{q_{cl}}`, that accompanies the known increments to :math:`\overline{q}` and :math:`\overline{T}`. The similar analysis, -from (`[dcdt] <#dcdt>`__) and (`[eq:da1] <#eq:da1>`__) gives +from :eq:`dcdt` and :eq:`eq:da1` gives -.. math:: +.. math:: :label: eq:da3 \Delta C_l = \frac{1}{1 - C_l} G(-Q_c) a_L ( \Delta \overline{q} - \alpha \Delta \overline{T} - \beta \Delta \overline{p} ) . - \label{eq:da3} Hence the equation set is equivalent to the use of the homogeneous forcing set, except for the multiplier :math:`\frac{1}{1 - C_l}`. Although this is a clean solution, we need to be very careful with the ill-conditioning of this solution near :math:`C_l = 1`. -In practice, the ill-conditioning of (`[eq:da2] <#eq:da2>`__) and -(`[eq:da3] <#eq:da3>`__) becomes too numerically awkward for us to apply +In practice, the ill-conditioning of :eq:`eq:da2` and +:eq:`eq:da3` becomes too numerically awkward for us to apply the full solution based on homogeneous forcing, although, for completeness, we outline it in Appendix :ref:`Appendix; Alternative PC2 - Data Assimilation formulations`. Hence @@ -5079,7 +4939,7 @@ that the data assimilation scheme itself calculated. Appendix :ref:`Appendix; Alternative PC2 - Data Assimilation formulations` gives, for completeness, the alternative numerical technique for the solution of -(`[eq:da2] <#eq:da2>`__) and (`[eq:da3] <#eq:da3>`__). However, we +:eq:`eq:da2` and :eq:`eq:da3`. However, we stress that this technique is not used within the current PC2 formulation. @@ -6087,15 +5947,14 @@ then we can use this to calculate the liquid, :math:`\overline{q_{cl}}`, and liquid cloud fraction, :math:`C_l`, increments. As in section :ref:`Data Assimilation`, we start by discretising -(`[dqcldt] <#dqcldt>`__) to give +:eq:`dqcldt` to give -.. math:: +.. math:: :label: eq:dqcldt_discrete \Delta \overline{q_{cl}} = C_l \Delta Q_c - \label{eq:dqcldt_discrete} and hence, using the discrete form of :math:`\Delta Q_c` from -(`[eq:deltaqc_exp2] <#eq:deltaqc_exp2>`__) gives +:eq:`eq:deltaqc_exp2` gives .. math:: @@ -6104,28 +5963,27 @@ and hence, using the discrete form of :math:`\Delta Q_c` from which rearranges to -.. math:: +.. math:: :label: eqn:delataqcl \Delta \overline{q_{cl}} = \frac{1}{1-C_l} C_l a_L ( \Delta \overline{q} - \alpha \Delta \overline{T} - \beta \Delta \overline{p}) . - \label{eqn:delataqcl} -Comparing to (`[eq:deltaqc_exp2] <#eq:deltaqc_exp2>`__) and -(`[eq:dqcldt_discrete] <#eq:dqcldt_discrete>`__) we see that +Comparing to :eq:`eq:deltaqc_exp2` and +:eq:`eq:dqcldt_discrete` we see that :math:`\Delta \overline{q_{cl}}` is the same as if we had applied the homogeneous forcing technique using :math:`\Delta \overline{q}`, :math:`\Delta \overline{T}` and :math:`\Delta \overline{p}` as forcings, except multiplied by a factor of :math:`\frac{1}{1-C_l}`. We can calculate :math:`\Delta C` in a similar way. From -(`[eq:deltac] <#eq:deltac>`__) +:eq:`eq:deltac` .. math:: \Delta C_l = G(-Q_c) \Delta Q_c and hence, using our value of :math:`\Delta Q_c` from -(`[eq:deltaqc_exp2] <#eq:deltaqc_exp2>`__) and +:eq:`eq:deltaqc_exp2` and :math:`\Delta \overline{q_{cl}}` from -(`[eqn:delataqcl] <#eqn:delataqcl>`__) +:eq:`eqn:delataqcl` .. math:: @@ -6134,11 +5992,10 @@ and hence, using our value of :math:`\Delta Q_c` from which rearranges to -.. math:: +.. math:: :label: eqn:c1mc \Delta C_l = \frac{1}{1-C_l} G(-Q_c) a_L ( \Delta \overline{q} - \alpha \Delta \overline{T} -\beta \Delta \overline{p}) . - \label{eqn:c1mc} This is also a factor of :math:`\frac{1}{1-C_l}` different from using :math:`\Delta \overline{q}`, :math:`\Delta \overline{T}` and @@ -6159,7 +6016,7 @@ solution. Initially, we calculate :math:`G(-Qc)` and :math:`\Delta Q_c` from the input fields, as in the homogeneous forcing technique (section -:ref:`Homogeneous forcing`) and (`[eq:deltaqc_exp2] <#eq:deltaqc_exp2>`__). +:ref:`Homogeneous forcing`) and :eq:`eq:deltaqc_exp2`. An initial increment, :math:`\Delta C_l^1` is estimated directly using the basic equation @@ -6197,7 +6054,7 @@ Limit on the liquid water content We will choose a limit on :math:`\overline{q_{cl}}` to be equal to its value when the underlying PDF just corresponds to total cloud cover. -Therefore, from (`[eq:qclbar=int] <#eq:qclbar=int>`__) +Therefore, from :eq:`eq:qclbar=int` .. math:: \overline{q_{cl \, max}} = \int_{s=-b_s}^{\infty} G(s) (b_s + s) ds . @@ -6226,13 +6083,12 @@ liquid cloud fraction :math:`C_l`. or -.. math:: +.. math:: :label: eqn:deltaqclmax \overline{\Delta q_{cl \, max}} = I1 + C_l (b_s - Q_c) . - \label{eqn:deltaqclmax} Now consider the expression for the saturation deficit, which we have -defined, from (`[SD] <#SD>`__) as +defined, from :eq:`SD` as .. math:: SD = \int_{-b_s}^{-Q_c} G(s) (-Q_c - s) ds . @@ -6248,26 +6104,24 @@ similar way to above gives \end{aligned} and hence :math:`I1` in terms of :math:`SD`. Using this value of -:math:`I1` in (`[eqn:deltaqclmax] <#eqn:deltaqclmax>`__) and cancelling +:math:`I1` in :eq:`eqn:deltaqclmax` and cancelling the :math:`C_l` terms gives :math:`\Delta \overline{q_{cl \, max}}` as -.. math:: +.. math:: :label: eqn:delta2 \Delta \overline{q_{cl \, max}} = (-Q_c + b_s) - SD . - \label{eqn:delta2} This is a general expression, it is not fixed for a particular PDF. To complete the analysis, we need to estimate :math:`-Q_c+b_s`. To do this, we now make the *assumption* of a power-law type PDF, as in section :ref:`Initiation of cloud`. If we start from the equivalent of -(`[eqn19] <#eqn19>`__) but at the :math:`s=-bs` end of the distribution, +:eq:`eqn19` but at the :math:`s=-bs` end of the distribution, equation (B.3) in :raw-latex:`\cite{wg03}` can be equivalently written for :math:`(1-C_l)` as: -.. math:: +.. math:: :label: eqn:1mc (1-C_l) = \frac{ A (-Q_c + b_s)^{n+1} }{n+1} . - \label{eqn:1mc} To derive this from (B.3) note that :math:`C_l` is swapped for :math:`1-C_l` and :math:`(b_s - (-Q_c))` is swapped for @@ -6276,25 +6130,23 @@ section :ref:`Numerical Application of the Smith method`. Similarly, noting that :math:`\overline{q_{cl}}` can be swapped with :math:`SD`, gives the equivalent to (B.4) in :raw-latex:`\cite{wg03}` as -.. math:: +.. math:: :label: eqn:sd SD = \frac{ A (-Q_c + b_s)^{n+2} }{(n+1)(n+2)}. - \label{eqn:sd} -Using the value :math:`(1-C_l)` from (`[eqn:1mc] <#eqn:1mc>`__) in -(`[eqn:sd] <#eqn:sd>`__) gives +Using the value :math:`(1-C_l)` from :eq:`eqn:1mc` in +:eq:`eqn:sd` gives .. math:: \frac{SD}{1-C_l} = \frac {-Q_c + b_s}{n+2} . Finally, we use this expression for :math:`(-Q_c + b_s)` in -(`[eqn:delta2] <#eqn:delta2>`__) to parametrize +:eq:`eqn:delta2` to parametrize :math:`\Delta \overline{q_{cl \, max}}` in terms of the saturation deficit -.. math:: +.. math:: :label: eqn:sdr1mc \Delta \overline{q_{cl \, max}} = SD ( \frac{n+2}{1-C_l} - 1 ) . - \label{eqn:sdr1mc} This is the expression that is used for the limit on :math:`\overline{q_{cl}}`. We subsequently apply a second limit, since @@ -6303,13 +6155,12 @@ is close to 1. Here we note that just at complete cloud cover for a symmetric PDF we have :math:`\overline{q_{cl}} = b_s`. Hence we estimate :math:`b_s` as in :raw-latex:`\cite{smith90}`, -.. math:: +.. math:: :label: eqn:bs b_s = a_L ( 1 - RH_{crit} ) q_{sat}(\overline{T_L}) , - \label{eqn:bs} -and take the smaller value for of (`[eqn:sdr1mc] <#eqn:sdr1mc>`__) and -(`[eqn:bs] <#eqn:bs>`__) for :math:`\Delta \overline{q_{cl \, max}}`. +and take the smaller value for of :eq:`eqn:sdr1mc` and +:eq:`eqn:bs` for :math:`\Delta \overline{q_{cl \, max}}`. Initiation from :math:`C_l=1` ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ @@ -6340,21 +6191,20 @@ If n=0 (i.e. a ‘top-hat’ function) then We now assume :math:`b_s` is equal to our current value of :math:`\overline{q_{cl}}` and hence -.. math:: +.. math:: :label: eqn:1msqrt C_l^{[n+1]} = 1 - \sqrt{ \frac{SD^{[n+1]}}{\overline{q_{cl}^{[n]}}} } - \label{eqn:1msqrt} where :math:`C_l^{[n+1]}` and :math:`SD^{[n+1]}` are the values of :math:`C_l` and :math:`SD` after this initiation has been applied. Using -our previous expression (`[eqn:sdr1mc] <#eqn:sdr1mc>`__) for +our previous expression :eq:`eqn:sdr1mc` for :math:`\Delta \overline{q_{cl max}}` gives (remembering that we are considering the reverse process, so the sign is opposite), .. math:: \Delta \overline{q_{cl}} = - SD^{[n+1]} ( \frac{2}{1-C_l^{[n+1]}} - 1 ) (remembering that :math:`n=0` is assumed). Hence, replacing -:math:`C_l^{[n+1]}` by (`[eqn:1msqrt] <#eqn:1msqrt>`__) we have +:math:`C_l^{[n+1]}` by :eq:`eqn:1msqrt` we have .. math:: @@ -6362,7 +6212,7 @@ considering the reverse process, so the sign is opposite), \overline{q_{cl}}^{[n]} } . This is the expression we use, :math:`SD^{[n+1]}` is calculated after -the ssimilation increments have been applied, using (`[SD2] <#SD2>`__): +the ssimilation increments have been applied, using :eq:`SD2`: .. math:: @@ -6631,7 +6481,7 @@ in the homogeneous forcing formulation. If a distribution is homogeneously forced to :math:`C_l = 0`, then we do not necessarily get :math:`\overline{q_{cl}}` tending to zero. This is because there is enough influence from the :math:`\frac{{(1-C_l)}^2}{SD}` term in the -combination (`[eqn22] <#eqn22>`__) to stop the natural convergence of +combination :eq:`eqn22` to stop the natural convergence of the :math:`\frac{{C_l}^2}{\overline{q_{cl}}}` term to :math:`C_l =0` and :math:`\overline{q_{cl}}=0`. Increasing the power of :math:`m` should help. However, we note that the tests that have been done on the chosen From 2910393d1fa7e3c3d10a395c3d5b69c3ab7ba9f3 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Tue, 7 Apr 2026 13:04:19 +0100 Subject: [PATCH 018/116] Went back to the doc as it was before I corrected the section cross-referecing, and re-applied the corrections using an automatic script (most things stay the same, but my tweaks to reduce line-lengths have been reveretd). --- .../cloud_schemes/UMDP30_PC2CloudScheme.rst | 128 ++++++++---------- 1 file changed, 58 insertions(+), 70 deletions(-) diff --git a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst index 6be957df1b..44fc02f6a8 100644 --- a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst +++ b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst @@ -58,7 +58,7 @@ straightforward to solve if one is allowed to assume that there is no variability of moisture or temperature on a scale of a model gridbox. In this case the cloud fraction scheme is redundant and only the condensation part remains, which may be solved diagnostically using the -instantaneous condensation assumption in section :ref:`The 's' distribution`. +instantaneous condensation assumption in section :ref:`The ‘s’ distribution`. However, the ‘no-variability’ assumption is poor until very high resolutions close to, or maybe exceeding, 1 km in the horizontal are reached. Although we may eventually assume that computer power will @@ -70,7 +70,7 @@ parametrization. There are several approaches to take to the solution of the problem, although they are not as independent as often portrayed, since they nearly all require the same instantaneous condensation assumption -(discussed in section :ref:`The 's' distribution`). Hence there are +(discussed in section :ref:`The ‘s’ distribution`). Hence there are mathematical links between all the approaches. *The following are all valid structures to use in this respect.* @@ -112,9 +112,9 @@ and to break the hard diagnostic link between cloud fraction and condensate. These major features of the :raw-latex:`\cite{t93}` scheme provide the motivation to develop the PC2 cloud scheme. -.. _The 's' distribution: +.. _The ‘s’ distribution: -The 's' distribution +The ‘s’ distribution -------------------- Most cloud schemes are based on the concept of a distribution of @@ -392,11 +392,9 @@ and :math:`\frac{\partial C_l} {\partial t}` . These are referred to as Homogeneous forcing (section :ref:`Homogeneous forcing`), Injection source (or inhomogeneous forcing, section :ref:`Injection forcing`) and Width Changing (section -:ref:`Changing the width of the PDF - PC2 erosion`). -Two additional modules are available to assist +:ref:`Changing the width of the PDF - PC2 erosion`). Two additional modules are available to assist with PC2, liquid cloud initiaion (section :ref:`Initiation of cloud`) and the -calculation of total cloud fraction changes -(section :ref:`Ice cloud and mixed phase regions`). +calculation of total cloud fraction changes (section :ref:`Ice cloud and mixed phase regions`). At the present time, only the large-scale precipitation (section :ref:`Large-scale precipitation`) scheme has been rewritten fully to use the PC2 concept of prognostic cloud fractions. The existing mass-flux convection @@ -442,7 +440,7 @@ Instantaneous condensation -------------------------- Liquid clouds in PC2 use the concept of instantaneous condensation. -Hence the 's' distribution methods are fully applicable to the +Hence the ‘s’ distribution methods are fully applicable to the development of the equations that govern the parametrization of liquid cloud in PC2. We will start by looking at changes to :math:`\overline{q_{cl}}` and :math:`C_l` when a uniform forcing is @@ -493,8 +491,8 @@ rate of change of condensate and cloud fraction based upon choose to develop a parametrization for this quantity based upon the quantities :math:`C_l`, :math:`\overline{q_{cl}}` and the saturation deficit, :math:`SD`, rather than tie :math:`G(-Q_c)` to a process. The -saturation deficit is *defined* here in the 's' framework to be the -first moment of the PDF for 's' values less than :math:`-Q_c`. In this +saturation deficit is *defined* here in the ‘s’ framework to be the +first moment of the PDF for ‘s’ values less than :math:`-Q_c`. In this way it is analogous to the liquid water content, :math:`\overline{q_{cl}}`. Appendix A of :raw-latex:`\cite{wg03}` writes this *definition* as @@ -1401,6 +1399,7 @@ used within the convection scheme. It still remains to parametrize :math:`\delta_{xl}`, which is given by the convection scheme itself. This is discussed in section :ref:`Phase of condensate`. +.. _Numerical application: Numerical application ~~~~~~~~~~~~~~~~~~~~~ @@ -1493,7 +1492,7 @@ detrainment. It is possible to calculate directly the change in :math:`C_l` that should occur due to the detrainment and compensating subsidence treated together, in the same way that :math:`\Delta \overline{q_{cl}}` is calculated (see section -`4.7.3 <#subsect:q4calculation>`__), and this is the way in which the +:ref:`Calculation of Grid-Box Averaged Condensate Rate (Q4)`), and this is the way in which the cloud fraction change **should** be done. It is an unfortunate historical emphasis in the early development of PC2 on the derivation of :eq:`eq:dcdt_inhom2` that has led to the treatment @@ -1523,8 +1522,8 @@ The homogeneous forcing, initiation and PC2 erosion sections described above have only considered the generation and dissipation of liquid clouds. Although the forcing methods will not influence the generation and dissipation of ice cloud (which is primarily performed in the -large-scale precipitation scheme, section :ref:`Large-scale precipitation`) -we are still left with the issue of how created or dissipated liquid cloud +large-scale precipitation scheme, section :ref:`Large-scale precipitation`) we are +still left with the issue of how created or dissipated liquid cloud overlaps with existing ice cloud in the gridbox. The opposite situation, where changes in ice cloud are specified and changes in the overlap with liquid cloud need to be calculated, is also possible in PC2 (e.g. in the @@ -1727,6 +1726,8 @@ summarised below: Turbulence-driven production of subgrid scale liquid cloud ---------------------------------------------------------- +.. _Introduction: + Introduction ~~~~~~~~~~~~ @@ -1748,8 +1749,8 @@ been used as the basis of subgrid cloud initiation method for use in the Unified Model in conjunction with the PC2 prognostic cloud scheme. In Section :ref:`Model description` we outline the model of :raw-latex:`\cite{fhfk14}`. In Section -:ref:`Model implementation and closure relations` -we described its implementation in the GCM. +:ref:`Model implementation and closure relations` we described its implementation in +the GCM. .. _Model description: @@ -1863,8 +1864,7 @@ prognostic fields, :math:`C_l` and :math:`q_{cl}`. Two methods are available for doing this. In the simplest case, the diagnosed values :math:`C_l^{sgt}` and :math:`q_{cl}^{sgt}` are just treated as increments to model prognostics (option one, in Sec. -:ref:`Options for incrementing model prognostics` below). -A more complex option (see +:ref:`Options for incrementing model prognostics` below). A more complex option (see option two, below) is to increment the model fields via the PC2 Erosion functionality. @@ -2034,6 +2034,7 @@ their desciption here is much shorter. Remember, whenever a signficiant able to represent the corresponding condensation and changes in cloud fractions. +.. _Radiation: Radiation --------- @@ -2044,8 +2045,7 @@ condensation and cloud fraction changes. For both shortwave and longwave, we use the homogeneous forcing routines (section :ref:`Homogeneous forcing`) for :math:`\overline{q_{cl}}` and :math:`C_l`, (using eqn. :eq:`eq:deltaqc_exp2` to calculate the -:math:`Q_c` forcing) and then the method in -section :ref:`Ice cloud and mixed phase regions` to +:math:`Q_c` forcing) and then the method in section :ref:`Ice cloud and mixed phase regions` to calculate :math:`C_t` changes. There is no :math:`\overline{q_{cf}}` change associated with this process since the deposition / sublimation process is performed within the large-scale precipitation scheme (as it @@ -2055,8 +2055,7 @@ It is reasonable to question whether homogeneous forcing is a reasonable model to use when we know that a large proportion of the heating associated with radiative transfer in the atmosphere comes from the cloudy air and is not evenly spread across the gridbox. Possible -developments are discussed in section -:ref:`Homogeneous forcing section improvements`. +developments are discussed in section :ref:`Homogeneous forcing section improvements`. .. _Large-scale precipitation: @@ -2086,6 +2085,7 @@ single ice cloud fraction is stored, the assumption being that the two ice categories are completely overlapped with each other. Graupel is not considered to contribute to the ice cloud fraction. +.. _Fall of ice: Fall of ice ~~~~~~~~~~~ @@ -2196,7 +2196,7 @@ This term exerts one of the most important influences on the ice cloud in the whole model (this applies to the control as well as for PC2). Contained in the formulation is a subgrid-scale assumption that causes equivalent effects to that for a moisture PDF under the ‘:math:`s`’ -framework (section :ref:`The 's' distribution`). However, since +framework (section :ref:`The ‘s’ distribution`). However, since :math:`{q_{cf}}` changes slowly in response to local changes in :math:`q` and :math:`T`, we cannot base the :math:`q_{cf}` response on the same instantaneous condensation framework. It would be useful to @@ -2474,8 +2474,7 @@ corresponding large reduction in :math:`C_l`. This is an underlying feature of the PC2 scheme (discussed in :raw-latex:`\cite{wg03}`), and necessarily implies the skewing of the underlying moisture PDF. Subsequent parts of the model (e.g. the width narrowing, section -:ref:`Changing the width of the PDF - PC2 erosion`) will, of course, -act on the modified fields to +:ref:`Changing the width of the PDF - PC2 erosion`) will, of course, act on the modified fields to adjust the cloud fractions further, but remember that these are separate processes and modelled elsewhere in the timestep. @@ -2528,11 +2527,9 @@ term “dbsdtbs1” which scales with the rate of homogeneous forcing :math:`\frac{\partial Q_c}{\partial t}`. However this term is always set to zero on input to these routines so is never used. -The width-narrowing formulation of section -:ref:`Changing the width of the PDF - PC2 erosion` is used +The width-narrowing formulation of section :ref:`Changing the width of the PDF - PC2 erosion` is used to calculate increments in :math:`\overline{q_{cl}}` and :math:`C_l`. -Using the liquid - ice cloud overlap ideas of -section :ref:`Ice cloud and mixed phase regions` +Using the liquid - ice cloud overlap ideas of section :ref:`Ice cloud and mixed phase regions` then gives the associated :math:`C_t` change. This background narrowing term, :math:`\Upsilon`, is originally based upon work by :raw-latex:`\cite{sg03}`, although it is a parameter that has been @@ -2543,8 +2540,7 @@ Numerical application of the original width-narrowing method ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Because of the strong link the mathematical expressions for width -narrowing (section :ref:`Changing the width of the PDF - PC2 erosion`) -have with the expressions for +narrowing (section :ref:`Changing the width of the PDF - PC2 erosion`) have with the expressions for the homogeneous forcing (section :ref:`Homogeneous forcing`), we choose to represent the timestepping of this process in exactly the same way as for the homogeneous forcing (in fact, in the Unified Model code we use @@ -3289,13 +3285,13 @@ changes in the vapour and temperature from the detrainment and compensating subsidence. Similar splits are made for the cloud variables, where the injection forcing, section :ref:`Injection forcing`, is used to calculate the first term from :math:`Q4_l`. Section -`4.7.3 <#subsect:q4calculation>`__ looks at the issue of the calculation +:ref:`Calculation of Grid-Box Averaged Condensate Rate (Q4)` looks at the issue of the calculation of :math:`Q4_l` etc., and section :ref:`Background condensation` looks at the calculation of :math:`Q_{environment}`, and its associated cloud fraction change. We first look at the basic transport equations in a mass flux convection scheme. -.. _`subsect:basmaseqs`: +.. _Basic Equations for a Convective Mass Flux Scheme: Basic Equations for a Convective Mass Flux Scheme ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ @@ -3473,7 +3469,7 @@ convection scheme :math:`Q4 = 0`. The PC2 scheme requires a reassessment of these assumptions because we wish to allow non-zero environment condensate values and to allow them to change. -.. _`subsect:q4calculation`: +.. _Calculation of Grid-Box Averaged Condensate Rate (Q4): Calculation of Grid-Box Averaged Condensate Rate (Q4) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ @@ -3974,8 +3970,7 @@ To this end, the code includes an option to perform the homogeneous forcing of liquid cloud by convection using the “pressure forcing” from the convective subsidence, consistent with the pressure forcing by large-scale advection (see sections :ref:`Advection` and -:ref:`Response to pressure changes`). -This approach replaces the above method of +:ref:`Response to pressure changes`). This approach replaces the above method of homogeneous forcing by convection if the UM namelist switch **l_pc2_homog_conv_pressure** is turned on. By applying the same homogeneous forcing method for advection and convectively-forced @@ -4319,8 +4314,8 @@ As discussed in section :ref:`Initiation of cloud`, there are occasions when 1. The application of the initiation is given in section :ref:`Initiation of cloud`. The initiation forms a new, separate block of PC2 code to perform this calculation, and is located immediately following -the pressure change response (section :ref:`Response to pressure changes`). -Also, if the UM namelist switch **l_cloud_call_b4_conv** is set to true, an +the pressure change response (section :ref:`Response to pressure changes`). Also, if the +UM namelist switch **l_cloud_call_b4_conv** is set to true, an additional call to PC2 initiation is performed before the convection scheme, to ensure that the condensation response to advection and other forcings earlier in the timestep has been accounted for in the profiles @@ -4333,8 +4328,7 @@ may occur. For all of these options, if using the bimodal cloud scheme to do initiation within PC2, then the tests on :math:`RH_T` relative to :math:`RH_{crit}` are replaced by equivalent tests for whether the saturation boundary lies within the bounds of the bimodal scheme’s -assumed PDF, as described in section -:ref:`Initiation using the bimodal scheme`. +assumed PDF, as described in section :ref:`Initiation using the bimodal scheme`. “Original” initiation logic ~~~~~~~~~~~~~~~~~~~~~~~~~~~ @@ -4846,7 +4840,7 @@ phase if the temperature is cold enough. Hence, if \label{eq:homochecks} \end{aligned} -.. _`sec:qpos`: +.. _Qpos checks: Qpos checks ~~~~~~~~~~~ @@ -4918,8 +4912,7 @@ ill-conditioning of this solution near :math:`C_l = 1`. In practice, the ill-conditioning of :eq:`eq:da2` and :eq:`eq:da3` becomes too numerically awkward for us to apply the full solution based on homogeneous forcing, although, for -completeness, we outline it in Appendix -:ref:`Appendix; Alternative PC2 - Data Assimilation formulations`. Hence +completeness, we outline it in Appendix :ref:`Appendix: Alternative PC2 - Data Assimilation formulations`. Hence we have chosen to apply a much simpler model. Here we use simply the data assimilation increments :math:`\Delta \overline{q}` and :math:`\Delta \overline{T}` within the standard homogeneous forcing @@ -4929,15 +4922,13 @@ is inconsistent (because :math:`\Delta \overline{q}` and :math:`\Delta This allows us an *estimate* of :math:`\Delta \overline{q_{cl}}` and :math:`\Delta{C_l}`, via the homogeneous forcing routine (and :math:`\Delta C_t` via the standard updating described in section -:ref:`Ice cloud and mixed phase regions`). -These are the quantities applied as the equivalent +:ref:`Ice cloud and mixed phase regions`). These are the quantities applied as the equivalent data assimilation increments for :math:`\Delta \overline{q_{cl}}`, :math:`\Delta{C_l}` and :math:`\Delta C_t`. The increments :math:`\Delta \overline{q}` and :math:`\Delta \overline{T}` remain those that the data assimilation scheme itself calculated. -Appendix :ref:`Appendix; Alternative PC2 - Data Assimilation formulations` -gives, for completeness, the +Appendix :ref:`Appendix: Alternative PC2 - Data Assimilation formulations` gives, for completeness, the alternative numerical technique for the solution of :eq:`eq:da2` and :eq:`eq:da3`. However, we stress that this technique is not used within the current PC2 @@ -4957,6 +4948,7 @@ much as possible in a similar way to the condensate variables. Hence, wherever the condensed water variables :math:`q_{cl}` and :math:`q_{cf}` are updated, the cloud fractions need to be updated consistently. +.. _Area cloud fraction: Area cloud fraction ------------------- @@ -5190,8 +5182,8 @@ Main Tree from atm_step_4a - | ls_arcld | \* (Smith scheme with area cloud fraction; see - :ref:`Smith scheme with area cloud fraction` - for a drill-down inside this routine) + :ref:`Smith scheme with area cloud fraction` for a drill-down + inside this routine) - | bm_ctl | \* (bimodal scheme) @@ -5275,8 +5267,8 @@ Main Tree from atm_step_4a - | ls_arcld | \* (interface to diagnostic Smith scheme and area cloud fraction; see - :ref:`Smith scheme with area cloud fraction` - for a drill-down inside this routine) + :ref:`Smith scheme with area cloud fraction` for a drill-down + inside this routine) - | bm_ctl | \* (bimodal cloud scheme) @@ -5313,8 +5305,8 @@ Main Tree from atm_step_4a - ls_arcld (call diagnostic Smith scheme with area cloud fraction again to account for the analysis increments; - see :ref:`Smith scheme with area cloud fraction` - for a drill-down inside this routine) + see :ref:`Smith scheme with area cloud fraction` for a drill-down + inside this routine) .. container:: tcolorbox @@ -5351,8 +5343,8 @@ Main Tree from atm_step_4a - | ls_arcld | \* (interface to diagnostic Smith scheme and area cloud - fraction; see :ref:`Smith scheme with area cloud fraction` - for a drill-down inside this routine) + fraction; see :ref:`Smith scheme with area cloud fraction` for a + drill-down inside this routine) - | bm_ctl | \* (bimodal cloud scheme) @@ -5496,6 +5488,7 @@ PC2 Data Assimilation | \* (self-consistency checks on prognostic cloud fractions and water contents) +.. _Diagnostics: Diagnostics ----------- @@ -5609,9 +5602,8 @@ rest of the SCM uses the same PC2 code as the full model. Note that the change to PC2 homogeneous forcing from advection under the UM namelist switch **l_pc2_sl_advection** (see section -:ref:`Response to pressure changes`) is also mirrored in the -Single-Column Model. -If this switch is turned on, the PC2 homogeneous forcing call using the SCM +:ref:`Response to pressure changes`) is also mirrored in the Single-Column Model. If +this switch is turned on, the PC2 homogeneous forcing call using the SCM forcing increments is moved straight after the call to the forcing routine, so that the condensation adjustment is performed before the call to atmos_physics2. If **l_pc2_sl_advection** is turned on, the PC2 @@ -5626,8 +5618,7 @@ The SCM forcings may comprise one or both of the following: For the latter, we can calculate the pressure change experienced by vertically-advected parcels, and so calculate the PC2 homogeneous forcing response in the same way as we do for Semi-Lagrangian advection -in the full model (see section :ref:`Response to pressure changes`). -For the former, we +in the full model (see section :ref:`Response to pressure changes`). For the former, we don’t know if the prescribed T,q tendencies are due to advection, radiation, or some other process, so we calculate the PC2 homogeneous forcing response as if the tendencies are applied "in-situ". @@ -5934,9 +5925,9 @@ More information Information on results of the scheme and how to run the PC2 code at various model versions is available on the PC2 web site. -.. _Appendix; Alternative PC2 - Data Assimilation formulations: +.. _Appendix: Alternative PC2 - Data Assimilation formulations: -Appendix; Alternative PC2 - Data Assimilation formulations +Appendix: Alternative PC2 - Data Assimilation formulations ========================================================== In this alternative method to section :ref:`Data Assimilation` we will assume @@ -6125,8 +6116,7 @@ for :math:`(1-C_l)` as: To derive this from (B.3) note that :math:`C_l` is swapped for :math:`1-C_l` and :math:`(b_s - (-Q_c))` is swapped for -:math:`(-Qc - (-b_s))`, as in -section :ref:`Numerical Application of the Smith method`. +:math:`(-Qc - (-b_s))`, as in section :ref:`Numerical Application of the Smith method`. Similarly, noting that :math:`\overline{q_{cl}}` can be swapped with :math:`SD`, gives the equivalent to (B.4) in :raw-latex:`\cite{wg03}` as @@ -6236,14 +6226,14 @@ results than simply using the homogeneous forcing method. Further work will be required to enable the implementation of this :math:`\overline{q}` and :math:`\overline{T}` preserving method. +.. _Appendix: Essentials of PC2 for code developers: Appendix: Essentials of PC2 for code developers =============================================== This section provides some guidance to code developers on the treatment of PC2. Code developers are advised to read the relevant part of section -:ref:`Application to the Unified Model` -to understand the way in which the current PC2 +:ref:`Application to the Unified Model` to understand the way in which the current PC2 scheme interacts with their section of code. The essence of a prognostic cloud scheme is that each physical part of @@ -6440,8 +6430,7 @@ of moisture. Convective cloud increments in the mass-flux framework ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -As discussed in section -:ref:`A note on the implementation of the cloud fraction change`, it would be +As discussed in section :ref:`A note on the implementation of the cloud fraction change`, it would be useful to code up the convective cloud fraction changes to link directly to the mass-flux convection scheme, and not to estimate them from the values of :math:`Q4`, which can introduce errors. @@ -6611,8 +6600,7 @@ the clouds is of order the timestep - ideally we wouldn’t want to try to model anything prognostically when the cycling time is less than the timestep. -As discussed in section -:ref:`Numerical application of the hybrid erosion method`, the timestep +As discussed in section :ref:`Numerical application of the hybrid erosion method`, the timestep sensitivity of cloud amounts in shallow cumulus regimes can be addressed by using a more accurate numerical method to solve the erosion term. Several options are available under the UM namelist switch From 1ddac0370b09e99d6cfda27e965f3ce914fd70c7 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Tue, 7 Apr 2026 15:20:11 +0100 Subject: [PATCH 019/116] Split-up latex aligned regions into separate math blocks so that it is possible to label and reference the individual equations in sphynx. --- .../cloud_schemes/UMDP30_PC2CloudScheme.rst | 607 +++++++++++------- 1 file changed, 369 insertions(+), 238 deletions(-) diff --git a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst index 44fc02f6a8..95983e7670 100644 --- a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst +++ b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst @@ -320,34 +320,44 @@ schematically: .. math:: - \begin{aligned} \frac{\partial \overline{q_{cl}}}{\partial t} = \frac{\partial \overline{q_{cl}}}{\partial t} |_{advection} + \frac{\partial \overline{q_{cl}}}{\partial t} |_{convection} + \frac{\partial \overline{q_{cl}}}{\partial t} |_{boundary \, layer} + - \frac{\partial \overline{q_{cl}}}{\partial t} |_{precipitation} + ... \nonumber \\ + \frac{\partial \overline{q_{cl}}}{\partial t} |_{precipitation} + ... + +.. math:: + \frac{\partial \overline{q_{cf}}}{\partial t} = \frac{\partial \overline{q_{cf}}}{\partial t} |_{advection} + \frac{\partial \overline{q_{cf}}}{\partial t} |_{convection} + \frac{\partial \overline{q_{cf}}}{\partial t} |_{boundary \, layer} + - \frac{\partial \overline{q_{cf}}}{\partial t} |_{precipitation} + ... \nonumber \\ + \frac{\partial \overline{q_{cf}}}{\partial t} |_{precipitation} + ... + +.. math:: + \frac{\partial C_l}{\partial t} = \frac{\partial C_l}{\partial t} |_{advection} + \frac{\partial C_l}{\partial t} |_{convection} + \frac{\partial C_l}{\partial t} |_{boundary \, layer} + - \frac{\partial C_l}{\partial t} |_{precipitation} + ... \nonumber \\ + \frac{\partial C_l}{\partial t} |_{precipitation} + ... + +.. math:: + \frac{\partial C_i}{\partial t} = \frac{\partial C_i}{\partial t} |_{advection} + \frac{\partial C_i}{\partial t} |_{convection} + \frac{\partial C_i}{\partial t} |_{boundary \, layer} + - \frac{\partial C_i}{\partial t} |_{precipitation} + ... \nonumber \\ + \frac{\partial C_i}{\partial t} |_{precipitation} + ... + +.. math:: :label: eq:dqcldt_and_dcdt + \frac{\partial C_t}{\partial t} = \frac{\partial C_t}{\partial t} |_{advection} + \frac{\partial C_t}{\partial t} |_{convection} + \frac{\partial C_t}{\partial t} |_{boundary \, layer} + \frac{\partial C_t}{\partial t} |_{precipitation} + ... , - \label{eq:dqcldt_and_dcdt} - \end{aligned} + where :math:`\overline{q_{cf}}` is the ice water specfic humidity, :math:`C_l` is the liquid cloud *volume* fraction, :math:`C_i` is the @@ -1775,11 +1785,16 @@ of :math:`p` and :math:`T` given by .. math:: - \begin{aligned} - b_i &=& \frac{1}{q} + \frac{\epsilon L_s^2}{c_p R T^2}, \\ - B_0 &=& 4\pi C \left( \frac{\epsilon L_s^2}{K_a R T^2} + \frac{R T}{\epsilon e_{si} \psi} \right)^{-1}, \\ - a_i &=& \frac{g}{R T}\left( \frac{\epsilon L_s}{c_p T} - 1 \right), \\ - \end{aligned} + b_i = \frac{1}{q} + \frac{\epsilon L_s^2}{c_p R T^2}, + +.. math:: + + B_0 = 4\pi C \left( \frac{\epsilon L_s^2}{K_a R T^2} + \frac{R T}{\epsilon e_{si} \psi} \right)^{-1}, + +.. math:: + + a_i = \frac{g}{R T}\left( \frac{\epsilon L_s}{c_p T} - 1 \right), + The first term on the right hand side of Eq. :eq:`eqn:squires_eqn` is the sink of vapor due to @@ -1812,16 +1827,16 @@ the noise and taking a steady-state limit (see :raw-latex:`\cite{fhfk14}` for details) it can be shown that the solution PDF is Gaussian with mean and variance given by: -.. math:: +.. math:: :label: eqn:si_avg + + \overline{S_i} = + S_{\rm E}\frac{ \left(\varepsilon/L^2\right)^{1/3} }{ b_i B_0 {\cal M}_1 + \left(\varepsilon/L^2\right)^{1/3} }. + +.. math:: :label: eqn:si_var + + \overline{S_i^2} = + \frac{a^2_{\rm i} \sigma^2_w \tau_{\rm d}}{ 2\left(b_i B_0 {\cal M}_1 + \left(\varepsilon/L^2\right)^{1/3}\right)}, - \begin{aligned} - \overline{S_i} &=& - S_{\rm E}\frac{ \left(\varepsilon/L^2\right)^{1/3} }{ b_i B_0 {\cal M}_1 + \left(\varepsilon/L^2\right)^{1/3} }. - \label{eqn:si_avg} \\ - \overline{S_i^2} &=& - \frac{a^2_{\rm i} \sigma^2_w \tau_{\rm d}}{ 2\left(b_i B_0 {\cal M}_1 + \left(\varepsilon/L^2\right)^{1/3}\right)}, - \label{eqn:si_var} - \end{aligned} Equation :eq:`eqn:si_avg` and :eq:`eqn:si_var` completely specify the PDF, @@ -1831,10 +1846,12 @@ given by .. math:: - \begin{aligned} - C_l^{sgt} &=& \int_{S_{i,wat}}^\infty d S_i F(S_i), \label{eqn:cloud_fraction} \\ - q_{cl}^{sgt} &=& q_{sat\;ice}\int_{S_{i,wat}}^\infty d S_i (S_i -S_{i,wat}) F(S_i) \label{eqn:cloud_liquid}, - \end{aligned} + C_l^{sgt} = \int_{S_{i,wat}}^\infty d S_i F(S_i), \label{eqn:cloud_fraction} + +.. math:: + + q_{cl}^{sgt} = q_{sat\;ice}\int_{S_{i,wat}}^\infty d S_i (S_i -S_{i,wat}) F(S_i) \label{eqn:cloud_liquid}, + where :math:`S_{i,wat} = e_{sat\;wat}/e_{sat\;ice}-1` is the value of ice supersaturation at water saturation. We use the superscription @@ -1940,13 +1957,24 @@ increments to the model prognostic fields, :math:`C_l` and .. math:: - \begin{aligned} - \left( \Delta C_l \right)_{sgt} &=& C_l^{sgt} \\ - \left( \Delta q_{cl} \right)_{sgt} &=& q_{cl}^{sgt}, \\ - \left( \Delta q \right)_{sgt} &=& -\left( \Delta q_{cl} \right)_{sgt}, \\ - \left( \Delta T \right)_{sgt} &=& \frac{L_v}{c_p} \left( \Delta q_{cl} \right)_{sgt}, \\ - \left( \Delta C \right)_{sgt} &=& C_l^{sgt} \\ - \end{aligned} + \left( \Delta C_l \right)_{sgt} = C_l^{sgt} + +.. math:: + + \left( \Delta q_{cl} \right)_{sgt} = q_{cl}^{sgt}, + +.. math:: + + \left( \Delta q \right)_{sgt} = -\left( \Delta q_{cl} \right)_{sgt}, + +.. math:: + + \left( \Delta T \right)_{sgt} = \frac{L_v}{c_p} \left( \Delta q_{cl} \right)_{sgt}, + +.. math:: + + \left( \Delta C \right)_{sgt} = C_l^{sgt} + where the left hand sides denote the increments to :math:`C_l`, :math:`q_{cl}`, :math:`T` and the total cloud fraction, :math:`C`, due @@ -1969,11 +1997,16 @@ using PC2 Erosion. In this case: .. math:: - \begin{aligned} - \left( \Delta q_{cl} \right)_{sgt} &=& q_{cl}^{sgt} - q_{cl}, \\ - \left( \Delta q \right)_{sgt} &=& -\left( \Delta q_{cl} \right)_{sgt}, \\ - \left( \Delta T \right)_{sgt} &=& \frac{L_v}{c_p} \left( \Delta q_{cl} \right)_{sgt}, \\ - \end{aligned} + \left( \Delta q_{cl} \right)_{sgt} = q_{cl}^{sgt} - q_{cl}, + +.. math:: + + \left( \Delta q \right)_{sgt} = -\left( \Delta q_{cl} \right)_{sgt}, + +.. math:: + + \left( \Delta T \right)_{sgt} = \frac{L_v}{c_p} \left( \Delta q_{cl} \right)_{sgt}, + where :math:`q_{cl}` is the liquid cloud amount prior to calling to the turbulent production scheme. The cloud fraction increments are @@ -2163,12 +2196,16 @@ cloud to ice cloud. The cloud fraction changes are: .. math:: - \begin{aligned} - C_l \leftarrow 0 \nonumber \\ - C_i \leftarrow C_t \nonumber \\ + C_l \leftarrow 0 + +.. math:: + + C_i \leftarrow C_t + +.. math:: :label: eq:lsp_homo + \Delta C_t = 0. - \label{eq:lsp_homo} - \end{aligned} + Heterogeneous nucleation ~~~~~~~~~~~~~~~~~~~~~~~~ @@ -2180,12 +2217,16 @@ cloud. These give the following changes: .. math:: - \begin{aligned} - \Delta C_l = 0 \nonumber \\ - C_i \leftarrow C_t \nonumber \\ + \Delta C_l = 0 + +.. math:: + + C_i \leftarrow C_t + +.. math:: :label: eq:lsp_het + \Delta C_t = 0. - \label{eq:lsp_het} - \end{aligned} + .. _Deposition and sublimation: @@ -2268,12 +2309,13 @@ same assumption). Some algebra retrieves the expressions: .. math:: - \begin{aligned} - q_{clear} = q_a - b_i A_{ice} ; \\ + q_{clear} = q_a - b_i A_{ice} ; + +.. math:: :label: eq:q_clear_and_q_ice + q_{ice} = \frac {\overline{q} - C_l q_{sat~liq} - A_{clear} q_{clear} } {A_{ice}}, - \label{eq:q_clear_and_q_ice} - \end{aligned} + where :math:`A_{ice}` is the proportion of the gridbox with ice cloud but not liquid cloud and :math:`A_{clear}` is the proportion of the @@ -3326,17 +3368,18 @@ which can be differentiated to give The bulk cloud model plume equations for mass and :math:`{\chi}` are: -.. math:: +.. math:: :label: eq:dbydpmassflux - \begin{aligned} - - \ensuremath{\frac{\partial \, M^{\rm{P}}}{\partial \, p}} & = & + - \ensuremath{\frac{\partial \, M^{\rm{P}}}{\partial \, p}} = \left({ \varepsilon \, M^{\rm{P}} - \mu \, M^{\rm{P}} - \delta \, M^{\rm{P}} } \right) - \label{eq:dbydpmassflux} \\ - - \ensuremath{\frac{\partial \, \ensuremath{{\chi}_{\rm{ }}^{\rm{P}}} \, M^{\rm{P}}}{\partial \, p}} & = & \left({ + +.. math:: + + - \ensuremath{\frac{\partial \, \ensuremath{{\chi}_{\rm{ }}^{\rm{P}}} \, M^{\rm{P}}}{\partial \, p}} = \left({ \varepsilon \, M^{\rm{P}} \, \ensuremath{{\chi}_{\rm{ }}^{\rm{E}}} - - \mu \, M^{\rm{P}} \, \ensuremath{{\chi}_{\rm{ }}^{\rm{R}}} - \delta \, M^{\rm{P}} \, \ensuremath{{\chi}_{\rm{ }}^{\rm{P}}} - } \right)\label{eq:dbydpmfchi} - \end{aligned} + - \mu \, M^{\rm{P}} \, \ensuremath{{\chi}_{\rm{ }}^{\rm{R}}} - \delta \, M^{\rm{P}} \, \ensuremath{{\chi}_{\rm{ }}^{\rm{P}}} + } \right)\label{eq:dbydpmfchi} + Equations :eq:`eq:eddyflux`, :eq:`eq:dbydpmassflux` and @@ -3370,18 +3413,20 @@ discretized form of :eq:`eq:chimassflux`, setting .. math:: - \begin{aligned} - {\ensuremath{\frac{\partial \, \ensuremath{{\chi}_{\rm{ }}^{\rm{E}}}}{\partial \, t}}}_{\rm{conv, \, k}} & = & m_{\rm{k+1/2}} \, + {\ensuremath{\frac{\partial \, \ensuremath{{\chi}_{\rm{ }}^{\rm{E}}}}{\partial \, t}}}_{\rm{conv, \, k}} = m_{\rm{k+1/2}} \, \frac{ \left({\ensuremath{{\chi}_{\rm{k+1}}^{\rm{E}}} - \ensuremath{{\chi}_{\rm{k}}^{\rm{E}}}} \right)} - {{\Delta z}_{\rm{k \, \rightarrow \, k+1}}} + {{\Delta z}_{\rm{k \, \rightarrow \, k+1}}} + {\delta}_{\rm{k}} \, m_{\rm{k}} \, \left({ \ensuremath{{\chi}_{\rm{k}}^{\rm{P}}} - \ensuremath{{\chi}_{\rm{k}}^{\rm{E}}} } \right) - \qquad \ldots \; \mbox{for k $>$ cb} \label{eq:chidisck} \\ - {\ensuremath{\frac{\partial \, \ensuremath{{\chi}_{\rm{ }}^{\rm{E}}}}{\partial \, t}}}_{\rm{conv, \, cb}} & = & m_{\rm{cb+1/2}} \, + \qquad \ldots \; \mbox{for k $>$ cb} \label{eq:chidisck} + +.. math:: + + {\ensuremath{\frac{\partial \, \ensuremath{{\chi}_{\rm{ }}^{\rm{E}}}}{\partial \, t}}}_{\rm{conv, \, cb}} = m_{\rm{cb+1/2}} \, \frac{ \left({\ensuremath{{\chi}_{\rm{cb+1}}^{\rm{E}}} - \ensuremath{{\chi}_{\rm{cb}}^{\rm{E}}}} \right)} - {{\Delta z}_{\rm{cb \, \rightarrow \, cb+1}}} - - m_{\rm{cb}} \, + {{\Delta z}_{\rm{cb \, \rightarrow \, cb+1}}} + - m_{\rm{cb}} \, \left({ \ensuremath{{\chi}_{\rm{i,cb}}^{\rm{P}}} - \ensuremath{{\chi}_{\rm{cb}}^{\rm{E}}} } \right)\label{eq:chidisccb} - \end{aligned} + where the initial parcel value :math:`{\chi}_{\rm{i,cb}}^{\rm{P}}` may be chosen to produce a fixed increment or place a closure condition on @@ -3398,17 +3443,17 @@ eqn :eq:`eq:chimassflux` is retained even though the basic equation :eq:`eq:chibasic` acquires additional terms for temperature and specific humidity: -.. math:: +.. math:: :label: eq:defineq1 - \begin{aligned} - {\ensuremath{\frac{\partial \, \ensuremath{T_{\rm{ }}^{\rm{E}}}}{\partial \, t}}}_{\rm{conv}} = Q1 & \equiv & + {\ensuremath{\frac{\partial \, \ensuremath{T_{\rm{ }}^{\rm{E}}}}{\partial \, t}}}_{\rm{conv}} = Q1 \equiv \left({ \frac{L}{c_{P}} } \right)\, {\overline{Q}}_{\rm{par}} - - \frac{1}{\overline{\rho}} \, \ensuremath{\frac{\partial \, \overline{\rho w^{'} \ensuremath{T_{\rm{ }}^{\rm{E'}}}}}{\partial \, z}} - \label{eq:defineq1} \\ - {\ensuremath{\frac{\partial \, \ensuremath{q_{\rm{ }}^{\rm{E}}}}{\partial \, t}}}_{\rm{conv}} = Q2 & \equiv & - {\overline{Q}}_{\rm{par}} - - \frac{1}{\overline{\rho}} \, \ensuremath{\frac{\partial \, \overline{\rho w^{'} \ensuremath{q_{\rm{ }}^{\rm{E'}}}}}{\partial \, z}} - \label{eq:defineq2} - \end{aligned} + - \frac{1}{\overline{\rho}} \, \ensuremath{\frac{\partial \, \overline{\rho w^{'} \ensuremath{T_{\rm{ }}^{\rm{E'}}}}}{\partial \, z}} + +.. math:: :label: eq:defineq2 + + {\ensuremath{\frac{\partial \, \ensuremath{q_{\rm{ }}^{\rm{E}}}}{\partial \, t}}}_{\rm{conv}} = Q2 \equiv - {\overline{Q}}_{\rm{par}} + - \frac{1}{\overline{\rho}} \, \ensuremath{\frac{\partial \, \overline{\rho w^{'} \ensuremath{q_{\rm{ }}^{\rm{E'}}}}}{\partial \, z}} + where :math:`{\overline{Q}}_{\rm{par}}` is the rate of condensation which occurs in the ascending plumes. @@ -3423,19 +3468,24 @@ gradient equations based upon :eq:`eq:gradchipar` .. math:: - \begin{aligned} - M^{\rm{P}} \, \ensuremath{\frac{\partial \, \ensuremath{T_{\rm{ }}^{\rm{P}}}}{\partial \, p}} & = & - \varepsilon \, M^{\rm{P}} \, \left({ \ensuremath{T_{\rm{ }}^{\rm{P}}} - \ensuremath{T_{\rm{ }}^{\rm{E}}} } \right)- + M^{\rm{P}} \, \ensuremath{\frac{\partial \, \ensuremath{T_{\rm{ }}^{\rm{P}}}}{\partial \, p}} = + \varepsilon \, M^{\rm{P}} \, \left({ \ensuremath{T_{\rm{ }}^{\rm{P}}} - \ensuremath{T_{\rm{ }}^{\rm{E}}} } \right)- \mu \, M^{\rm{P}} \, \left({ \ensuremath{T_{\rm{ }}^{\rm{P}}} - \ensuremath{T_{\rm{ }}^{\rm{R}}} } \right)- - \left({ \frac{L}{c_{P}} } \right)\, {\overline{Q}}_{\rm{par}} \label{eq:gradtpar} \\ - M^{\rm{P}} \, \ensuremath{\frac{\partial \, \ensuremath{q_{\rm{ }}^{\rm{P}}}}{\partial \, p}} & = & - \varepsilon \, M^{\rm{P}} \, \left({ \ensuremath{q_{\rm{ }}^{\rm{P}}} - \ensuremath{q_{\rm{ }}^{\rm{E}}} } \right)- + \left({ \frac{L}{c_{P}} } \right)\, {\overline{Q}}_{\rm{par}} \label{eq:gradtpar} + +.. math:: + + M^{\rm{P}} \, \ensuremath{\frac{\partial \, \ensuremath{q_{\rm{ }}^{\rm{P}}}}{\partial \, p}} = + \varepsilon \, M^{\rm{P}} \, \left({ \ensuremath{q_{\rm{ }}^{\rm{P}}} - \ensuremath{q_{\rm{ }}^{\rm{E}}} } \right)- \mu \, M^{\rm{P}} \, \left({ \ensuremath{q_{\rm{ }}^{\rm{P}}} - \ensuremath{q_{\rm{ }}^{\rm{R}}} } \right)+ - {\overline{Q}}_{\rm{par}} \label{eq:gradqpar} \\ - M^{\rm{P}} \, \ensuremath{\frac{\partial \, \ensuremath{l_{\rm{ }}^{\rm{P}}}}{\partial \, p}} & = & + {\overline{Q}}_{\rm{par}} \label{eq:gradqpar} + +.. math:: + + M^{\rm{P}} \, \ensuremath{\frac{\partial \, \ensuremath{l_{\rm{ }}^{\rm{P}}}}{\partial \, p}} = \varepsilon \, M^{\rm{P}} \, \left({ \ensuremath{l_{\rm{ }}^{\rm{P}}} - \ensuremath{l_{\rm{ }}^{\rm{E}}} } \right) - - {\overline{Q}}_{\rm{par}} + PPN \label{eq:gradlpar} - \end{aligned} + - {\overline{Q}}_{\rm{par}} + PPN \label{eq:gradlpar} + The final calculation of rates in the current condensation scheme (, section 10) assumes a further condensation term, @@ -3445,22 +3495,27 @@ environment values of condensate remain zero (and also that :math:`l_{\rm{ }}^{\rm{R}}` = :math:`l_{\rm{ }}^{\rm{P}}`). The result is basic equations -.. math:: +.. math:: :label: eq:basictold - \begin{aligned} - {\ensuremath{\frac{\partial \, \ensuremath{T_{\rm{ }}^{\rm{E}}}}{\partial \, t}}}_{\rm{conv}} & = & Q1 - + {\ensuremath{\frac{\partial \, \ensuremath{T_{\rm{ }}^{\rm{E}}}}{\partial \, t}}}_{\rm{conv}} = Q1 - \left({ \frac{L}{c_{P}} } \right)\, {\overline{Q}}_{\rm{reset}} - \label{eq:basictold} \\ - {\ensuremath{\frac{\partial \, \ensuremath{q_{\rm{ }}^{\rm{E}}}}{\partial \, t}}}_{\rm{conv}} & = & Q2 + {\overline{Q}}_{\rm{reset}} - \label{eq:basicqold} \\ - 0 \equiv {\ensuremath{\frac{\partial \, \ensuremath{l_{\rm{ }}^{\rm{E}}}}{\partial \, t}}}_{\rm{conv}} & = & {\overline{Q}}_{\rm{par}} - + +.. math:: :label: eq:basicqold + + {\ensuremath{\frac{\partial \, \ensuremath{q_{\rm{ }}^{\rm{E}}}}{\partial \, t}}}_{\rm{conv}} = Q2 + {\overline{Q}}_{\rm{reset}} + +.. math:: + + 0 \equiv {\ensuremath{\frac{\partial \, \ensuremath{l_{\rm{ }}^{\rm{E}}}}{\partial \, t}}}_{\rm{conv}} = {\overline{Q}}_{\rm{par}} - {\overline{Q}}_{\rm{reset}} - PPN - - \frac{1}{\overline{\rho}} \, \ensuremath{\frac{\partial \, \overline{\rho w^{'} \ensuremath{l_{\rm{ }}^{\rm{E'}}}}}{\partial \, z}} \nonumber \\ - & = & + - \frac{1}{\overline{\rho}} \, \ensuremath{\frac{\partial \, \overline{\rho w^{'} \ensuremath{l_{\rm{ }}^{\rm{E'}}}}}{\partial \, z}} + +.. math:: :label: eq:basiclold + + = \mu \, M^{\rm{P}} \, \ensuremath{l_{\rm{ }}^{\rm{P}}} + \delta \, M^{\rm{P}} \, \ensuremath{l_{\rm{ }}^{\rm{P}}} - - {\overline{Q}}_{\rm{reset}} - \label{eq:basiclold} - \end{aligned} + {\overline{Q}}_{\rm{reset}} + By analogy with equations :eq:`eq:defineq1` and :eq:`eq:defineq2`, we can define a :math:`Q4` from @@ -3480,18 +3535,18 @@ amount accordingly. Define -.. math:: +.. math:: :label: eq:defineq4l - \begin{aligned} - \left({ \ensuremath{\frac{\partial \, \ensuremath{l_{\rm{l}}^{\rm{ }}}}{\partial \, t}} } \right)_{\rm{conv}} = Q4_{\rm{l}} & \equiv & - {\overline{Q}}_{\rm{l, par}} - {\overline{Q}}_{\rm{l, reset}} - RAIN - + \left({ \ensuremath{\frac{\partial \, \ensuremath{l_{\rm{l}}^{\rm{ }}}}{\partial \, t}} } \right)_{\rm{conv}} = Q4_{\rm{l}} \equiv + {\overline{Q}}_{\rm{l, par}} - {\overline{Q}}_{\rm{l, reset}} - RAIN - \frac{1}{\overline{\rho}} \, \ensuremath{\frac{\partial \, \overline{\rho w^{'} \ensuremath{l_{\rm{l}}^{\rm{'}}}}}{\partial \, z}} - \label{eq:defineq4l} \\ - \left({ \ensuremath{\frac{\partial \, \ensuremath{l_{\rm{f}}^{\rm{ }}}}{\partial \, t}} } \right)_{\rm{conv}} = Q4_{\rm{f}} & \equiv & - {\overline{Q}}_{\rm{f, par}} - {\overline{Q}}_{\rm{f, reset}} - SNOW - + +.. math:: :label: eq:defineq4f + + \left({ \ensuremath{\frac{\partial \, \ensuremath{l_{\rm{f}}^{\rm{ }}}}{\partial \, t}} } \right)_{\rm{conv}} = Q4_{\rm{f}} \equiv + {\overline{Q}}_{\rm{f, par}} - {\overline{Q}}_{\rm{f, reset}} - SNOW - \frac{1}{\overline{\rho}} \, \ensuremath{\frac{\partial \, \overline{\rho w^{'} \ensuremath{l_{\rm{f}}^{\rm{'}}}}}{\partial \, z}} - \label{eq:defineq4f} - \end{aligned} + where the PC2 assumption thus far has been that :math:`{\overline{Q}}_{\rm{l, reset}} = 0 @@ -3515,16 +3570,18 @@ condensate is calculated as .. math:: - \begin{aligned} - \ensuremath{\frac{\partial \, \ensuremath{l_{\rm{l}}^{\rm{P}}}}{\partial \, p}} & = & \varepsilon \, - \left({ \ensuremath{l_{\rm{l}}^{\rm{P}}} - \ensuremath{l_{\rm{l}}^{\rm{E}}} } \right)- - \frac{{\overline{Q}}_{\rm{l, par}}}{M^{\rm{P}}} - - \frac{RAIN}{M^{\rm{P}}} \label{eq:vertparl} \\ - \ensuremath{\frac{\partial \, \ensuremath{l_{\rm{f}}^{\rm{P}}}}{\partial \, p}} & = & \varepsilon \, - \left({ \ensuremath{l_{\rm{f}}^{\rm{P}}} - \ensuremath{l_{\rm{f}}^{\rm{E}}} } \right)- - \frac{{\overline{Q}}_{\rm{f, par}}}{M^{\rm{P}}} - - \frac{SNOW}{M^{\rm{P}}}\label{eq:vertparf} - \end{aligned} + \ensuremath{\frac{\partial \, \ensuremath{l_{\rm{l}}^{\rm{P}}}}{\partial \, p}} = \varepsilon \, + \left({ \ensuremath{l_{\rm{l}}^{\rm{P}}} - \ensuremath{l_{\rm{l}}^{\rm{E}}} } \right)- + \frac{{\overline{Q}}_{\rm{l, par}}}{M^{\rm{P}}} - + \frac{RAIN}{M^{\rm{P}}} \label{eq:vertparl} + +.. math:: + + \ensuremath{\frac{\partial \, \ensuremath{l_{\rm{f}}^{\rm{P}}}}{\partial \, p}} = \varepsilon \, + \left({ \ensuremath{l_{\rm{f}}^{\rm{P}}} - \ensuremath{l_{\rm{f}}^{\rm{E}}} } \right)- + \frac{{\overline{Q}}_{\rm{f, par}}}{M^{\rm{P}}} - + \frac{SNOW}{M^{\rm{P}}}\label{eq:vertparf} + Following , equations :eq:`eq:dbydpmassflux`, :eq:`eq:vertparl` and :eq:`eq:vertparf` @@ -3532,32 +3589,41 @@ are discretized: .. math:: - \begin{aligned} - M_{\rm{k} + 1} & = & M_{\rm{k}} \, + M_{\rm{k} + 1} = M_{\rm{k}} \, \left({ 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right)\, \left({ 1 - \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right)\, - EPSS_{\rm{k}} \label{eq:discdmfbydp} \\ - \ensuremath{l_{\rm{l \, k + 1}}^{\rm{P}}} & = & \left({ - \ensuremath{l_{\rm{l \, k}}^{\rm{P}}} + + EPSS_{\rm{k}} \label{eq:discdmfbydp} + +.. math:: + + \ensuremath{l_{\rm{l \, k + 1}}^{\rm{P}}} = \left({ + \ensuremath{l_{\rm{l \, k}}^{\rm{P}}} + \varepsilon_{\rm{k} + 1/4} \, \Delta p_{\rm{k} + 1/4} \, \ensuremath{l_{\rm{l \, k}}^{\rm{E}}} + - \varepsilon_{\rm{k} + 3/4} \, \Delta p_{\rm{k} + 3/4} \, + \varepsilon_{\rm{k} + 3/4} \, \Delta p_{\rm{k} + 3/4} \, \left[{1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4}} \right]\, \ensuremath{l_{\rm{l \, k + 1}}^{\rm{E}}} - } \right)\, / \, \left({EPSS_{\rm{k}}} \right)\nonumber \\ - { } & { } & + \left({ {\overline{Q}}_{\rm{l} \, \rm{k} + 1} \, / \, M_{\rm{k} + 1}} \right) + } \right)\, / \, \left({EPSS_{\rm{k}}} \right) + +.. math:: :label: eq:discvparl + + { } { } + \left({ {\overline{Q}}_{\rm{l} \, \rm{k} + 1} \, / \, M_{\rm{k} + 1}} \right) - \left({ RAIN_{\rm{k} + 1} \, / \, M_{\rm{k} + 1} } \right) - \label{eq:discvparl} \\ - \ensuremath{l_{\rm{f \, k + 1}}^{\rm{P}}} & = & \left({ - \ensuremath{l_{\rm{f \, k}}^{\rm{P}}} + + +.. math:: + + \ensuremath{l_{\rm{f \, k + 1}}^{\rm{P}}} = \left({ + \ensuremath{l_{\rm{f \, k}}^{\rm{P}}} + \varepsilon_{\rm{k} + 1/4} \, \Delta p_{\rm{k} + 1/4} \, \ensuremath{l_{\rm{f \, k}}^{\rm{E}}} + - \varepsilon_{\rm{k} + 3/4} \, \Delta p_{\rm{k} + 3/4} \, + \varepsilon_{\rm{k} + 3/4} \, \Delta p_{\rm{k} + 3/4} \, \left[{1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4}} \right]\, \ensuremath{l_{\rm{f \, k + 1}}^{\rm{E}}} - } \right)\, / \, \left({EPSS_{\rm{k}}} \right)\nonumber \\ - { } & { } & + \left({ {\overline{Q}}_{\rm{f} \, \rm{k} + 1} \, / \, M_{\rm{k} + 1}} \right) + } \right)\, / \, \left({EPSS_{\rm{k}}} \right) + +.. math:: :label: eq:discvparf + + { } { } + \left({ {\overline{Q}}_{\rm{f} \, \rm{k} + 1} \, / \, M_{\rm{k} + 1}} \right) - \left({ SNOW_{\rm{k} + 1} \, / \, M_{\rm{k} + 1} } \right) - \label{eq:discvparf} - \end{aligned} + where :math:`EPSS_{\rm{k}} = \left({1 + \varepsilon_{\rm{k} + 3 / 4} \, \Delta p_{\rm{k} + 3 / 4}} \right)\, @@ -3572,22 +3638,24 @@ precipitation terms are suppressed: .. math:: - \begin{aligned} - \ensuremath{l_{\rm{l \, k + 1}}^{\rm{P}}} & = & \frac{\left({ - \ensuremath{l_{\rm{l \, k}}^{\rm{P}}} + + \ensuremath{l_{\rm{l \, k + 1}}^{\rm{P}}} = \frac{\left({ + \ensuremath{l_{\rm{l \, k}}^{\rm{P}}} + \varepsilon_{\rm{k} + 1/4} \, \Delta p_{\rm{k} + 1/4} \, \ensuremath{l_{\rm{l \, k}}^{\rm{E}}} + - \varepsilon_{\rm{k} + 3/4} \, \Delta p_{\rm{k} + 3/4} \, + \varepsilon_{\rm{k} + 3/4} \, \Delta p_{\rm{k} + 3/4} \, \left[{1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4}} \right]\, \ensuremath{l_{\rm{l \, k + 1}}^{\rm{E}}} - } \right)}{EPSS_{\rm{k}}} \label{eq:discvparldry} \\ - \ensuremath{l_{\rm{f \, k + 1}}^{\rm{P}}} & = & \frac{\left({ - \ensuremath{l_{\rm{f \, k}}^{\rm{P}}} + + } \right)}{EPSS_{\rm{k}}} \label{eq:discvparldry} + +.. math:: + + \ensuremath{l_{\rm{f \, k + 1}}^{\rm{P}}} = \frac{\left({ + \ensuremath{l_{\rm{f \, k}}^{\rm{P}}} + \varepsilon_{\rm{k} + 1/4} \, \Delta p_{\rm{k} + 1/4} \, \ensuremath{l_{\rm{f \, k}}^{\rm{E}}} + - \varepsilon_{\rm{k} + 3/4} \, \Delta p_{\rm{k} + 3/4} \, + \varepsilon_{\rm{k} + 3/4} \, \Delta p_{\rm{k} + 3/4} \, \left[{1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4}} \right]\, \ensuremath{l_{\rm{f \, k + 1}}^{\rm{E}}} } \right)}{EPSS_{\rm{k}}} \label{eq:discvparfdry} - \end{aligned} + At the base of the convective plume (ie. the level immediately above cloud base), :math:`l_{\rm{l \, k}}^{\rm{P}}` is initialized to @@ -3598,31 +3666,33 @@ produces zero fluxes at cloud base: .. math:: - \begin{aligned} - Q4_{\rm{l}}(cb) = 0 & = & M_{\rm{cb+1/2}}^{\rm{P}} \, - \ensuremath{\frac{\partial \, \ensuremath{l_{\rm{l}}^{\rm{E}}}}{\partial \, p}} - M_{\rm{cb}}^{\rm{P}}\, - \left({ \ensuremath{l_{\rm{l}}^{\rm{P \, i}}} - \ensuremath{l_{\rm{l}}^{\rm{E}}}(\rm{cb}) } \right)\label{eq:q4lcbi} \\ - Q4_{\rm{f}}(cb) = 0 & = & M_{\rm{cb+1/2}}^{\rm{P}} \, - \ensuremath{\frac{\partial \, \ensuremath{l_{\rm{f}}^{\rm{E}}}}{\partial \, p}} - M_{\rm{cb}}^{\rm{P}}\, - \left({ \ensuremath{l_{\rm{f}}^{\rm{P \, i}}} - \ensuremath{l_{\rm{f}}^{\rm{E}}}(\rm{cb}) } \right)\label{eq:q4fcbi} - \end{aligned} + Q4_{\rm{l}}(cb) = 0 = M_{\rm{cb+1/2}}^{\rm{P}} \, + \ensuremath{\frac{\partial \, \ensuremath{l_{\rm{l}}^{\rm{E}}}}{\partial \, p}} - M_{\rm{cb}}^{\rm{P}}\, + \left({ \ensuremath{l_{\rm{l}}^{\rm{P \, i}}} - \ensuremath{l_{\rm{l}}^{\rm{E}}}(\rm{cb}) } \right)\label{eq:q4lcbi} + +.. math:: + + Q4_{\rm{f}}(cb) = 0 = M_{\rm{cb+1/2}}^{\rm{P}} \, + \ensuremath{\frac{\partial \, \ensuremath{l_{\rm{f}}^{\rm{E}}}}{\partial \, p}} - M_{\rm{cb}}^{\rm{P}}\, + \left({ \ensuremath{l_{\rm{f}}^{\rm{P \, i}}} - \ensuremath{l_{\rm{f}}^{\rm{E}}}(\rm{cb}) } \right)\label{eq:q4fcbi} + As the convection scheme makes the single phase assumption for parcel condensate, it may be necessary to melt or freeze entrained condensate at this point and adjust the temperature accordingly. -.. math:: +.. math:: :label: eqn:meltlf + + \theta_{\rm{k + 1}}^{\rm{P}} = \theta_{\rm{k + 1}}^{\rm{P}} - + \left(\frac{L_{\rm{F}}}{C_{p} \, \Pi_{\rm{k + 1}}} \right)\, \ensuremath{l_{\rm{f \, k + 1}}^{\rm{P}}} + \; \ldots \; \mbox{ if \ensuremath{l_{\rm{f \, k + 1}}^{\rm{P}}} is melted } + +.. math:: :label: eqn:freezell + + \theta_{\rm{k + 1}}^{\rm{P}} = \theta_{\rm{k + 1}}^{\rm{P}} + + \left(\frac{L_{\rm{F}}}{C_{p} \, \Pi_{\rm{k + 1}}} \right)\, \ensuremath{l_{\rm{l \, k + 1}}^{\rm{P}}} + \; \ldots \; \mbox{ if \ensuremath{l_{\rm{l \, k + 1}}^{\rm{P}}} is frozen } - \begin{aligned} - \theta_{\rm{k + 1}}^{\rm{P}} = \theta_{\rm{k + 1}}^{\rm{P}} - - \left(\frac{L_{\rm{F}}}{C_{p} \, \Pi_{\rm{k + 1}}} \right)\, \ensuremath{l_{\rm{f \, k + 1}}^{\rm{P}}} - & \; \ldots \; & \mbox{ if \ensuremath{l_{\rm{f \, k + 1}}^{\rm{P}}} is melted } - \label{eqn:meltlf} \\ - \theta_{\rm{k + 1}}^{\rm{P}} = \theta_{\rm{k + 1}}^{\rm{P}} + - \left(\frac{L_{\rm{F}}}{C_{p} \, \Pi_{\rm{k + 1}}} \right)\, \ensuremath{l_{\rm{l \, k + 1}}^{\rm{P}}} - & \; \ldots \; & \mbox{ if \ensuremath{l_{\rm{l \, k + 1}}^{\rm{P}}} is frozen } - \label{eqn:freezell} - \end{aligned} Once a final value for the condensation term :math:`{\overline{Q}}_{\rm{x} \, \rm{k} + 1} \, / \, M_{\rm{k} + 1}` has @@ -3655,32 +3725,36 @@ This reduces the parcel condensate to : .. math:: - \begin{aligned} - \ensuremath{l_{\rm{l \, k + 1}}^{\rm{P}}} & = & \left({ + \ensuremath{l_{\rm{l \, k + 1}}^{\rm{P}}} = \left({ \frac{\ensuremath{l_{\rm{l \, k + 1}}^{\rm{P}}}}{\ensuremath{l_{\rm{k + 1}}^{\rm{P}}}} - } \right)\, \ensuremath{l_{\rm{MIN}}^{\rm{P}}} \label{eq:vparlfinal} \\ - \ensuremath{l_{\rm{f \, k + 1}}^{\rm{P}}} & = & \left({ + } \right)\, \ensuremath{l_{\rm{MIN}}^{\rm{P}}} \label{eq:vparlfinal} + +.. math:: + + \ensuremath{l_{\rm{f \, k + 1}}^{\rm{P}}} = \left({ \frac{\ensuremath{l_{\rm{f \, k + 1}}^{\rm{P}}}}{\ensuremath{l_{\rm{k + 1}}^{\rm{P}}}} - } \right)\, \ensuremath{l_{\rm{MIN}}^{\rm{P}}} \label{eq:vparffinal} - \end{aligned} + } \right)\, \ensuremath{l_{\rm{MIN}}^{\rm{P}}} \label{eq:vparffinal} + The final parcel condensate values are then used in the rate calculation based upon eqn :eq:`eq:basiclold`: .. math:: - \begin{aligned} - Q4_{\rm{l}}(k) & = & M_{\rm{k+1/2}}^{\rm{P}} \, \ensuremath{\frac{\partial \, \ensuremath{l_{\rm{l}}^{\rm{E}}}}{\partial \, p}} + - \left({ {\mu}_{\rm{k}} \, M_{\rm{k}}^{\rm{P}} + - {\delta}_{\rm{k}} \, M_{\rm{k}}^{\rm{P}} } \right)\, + Q4_{\rm{l}}(k) = M_{\rm{k+1/2}}^{\rm{P}} \, \ensuremath{\frac{\partial \, \ensuremath{l_{\rm{l}}^{\rm{E}}}}{\partial \, p}} + + \left({ {\mu}_{\rm{k}} \, M_{\rm{k}}^{\rm{P}} + + {\delta}_{\rm{k}} \, M_{\rm{k}}^{\rm{P}} } \right)\, \left({ \ensuremath{l_{\rm{l}}^{\rm{P}}}(\rm{k}) - \ensuremath{l_{\rm{l}}^{\rm{E}}}(\rm{k}) } \right)- - {\overline{Q}}_{\rm{l, reset}} \label{eq:q4lmassf} \\ - Q4_{\rm{f}}(k) & = & M_{\rm{k+1/2}}^{\rm{P}} \, \ensuremath{\frac{\partial \, \ensuremath{l_{\rm{f}}^{\rm{E}}}}{\partial \, p}} + - \left({ {\mu}_{\rm{k}} \, M_{\rm{k}}^{\rm{P}} + - {\delta}_{\rm{k}} \, M_{\rm{k}}^{\rm{P}} } \right)\, - \left({ \ensuremath{l_{\rm{f}}^{\rm{P}}}(\rm{k}) - \ensuremath{l_{\rm{f}}^{\rm{E}}}(\rm{k}) } \right)- + {\overline{Q}}_{\rm{l, reset}} \label{eq:q4lmassf} + +.. math:: + + Q4_{\rm{f}}(k) = M_{\rm{k+1/2}}^{\rm{P}} \, \ensuremath{\frac{\partial \, \ensuremath{l_{\rm{f}}^{\rm{E}}}}{\partial \, p}} + + \left({ {\mu}_{\rm{k}} \, M_{\rm{k}}^{\rm{P}} + + {\delta}_{\rm{k}} \, M_{\rm{k}}^{\rm{P}} } \right)\, + \left({ \ensuremath{l_{\rm{f}}^{\rm{P}}}(\rm{k}) - \ensuremath{l_{\rm{f}}^{\rm{E}}}(\rm{k}) } \right)- {\overline{Q}}_{\rm{f, reset}} \label{eq:q4fmassf} - \end{aligned} + Note that, as a side-effect, the environment equations for potential temperature and specific humidity are also altered because the @@ -3690,94 +3764,114 @@ condensate is no longer re-evaporated at the end .. math:: - \begin{aligned} - \frac{\Delta \, \theta_{\rm{k}}^{\rm{E}}}{\Delta \, t} = + \frac{\Delta \, \theta_{\rm{k}}^{\rm{E}}}{\Delta \, t} = \left(\frac{ M_{\rm{k}} }{ \Delta \, p_{\rm{k}} } \right) \left[{ \left({ 1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4} } \right) \left({ 1 - \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) \left({ 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) \left({ \theta_{\rm{k + 1}}^{\rm{E}} - \theta_{\rm{k}}^{\rm{E}} } \right) - } \right . & + & \nonumber \\ + } \right . + + +.. math:: + \left({ \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) \left({ 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) \left({ \theta_{\rm{k}}^{\rm{R}} - \theta_{\rm{k}}^{\rm{E}} } \right) - & + & \nonumber \\ + + + +.. math:: + \left . { \left({ \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) \left({ \theta_{\rm{k}}^{\rm{P}} - \theta_{\rm{k}}^{\rm{E}} } \right) - } \right]& { } & \label{eq:enviroth} - \end{aligned} + } \right] { } \label{eq:enviroth} + and .. math:: - \begin{aligned} - \frac{\Delta \, q_{\rm{k}}^{\rm{E}}}{\Delta \, t} = + \frac{\Delta \, q_{\rm{k}}^{\rm{E}}}{\Delta \, t} = \left(\frac{ M_{\rm{k}} }{ \Delta \, p_{\rm{k}} } \right) \left[{ \left({ 1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4} } \right) \left({ 1 - \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) \left({ 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) \left({ q_{\rm{k + 1}}^{\rm{E}} - q_{\rm{k}}^{\rm{E}} } \right) - } \right . & + & \nonumber \\ + } \right . + + +.. math:: + \left({ \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) \left({ 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) \left({ q_{\rm{k}}^{\rm{R}} - q_{\rm{k}}^{\rm{E}} } \right) - & + & \nonumber \\ + + + +.. math:: + \left . { \left({ \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) \left({ q_{\rm{k}}^{\rm{P}} - q_{\rm{k}}^{\rm{E}} } \right) - } \right]& { } & \label{eq:enviroq} - \end{aligned} + } \right] { } \label{eq:enviroq} + Similarly, eqns :eq:`eq:q4lmassf` and :eq:`eq:q4fmassf` have a discretized form as follows: .. math:: - \begin{aligned} - \frac{\Delta \, \ensuremath{l_{\rm{l \, k}}^{\rm{E}}}}{\Delta \, t} = + \frac{\Delta \, \ensuremath{l_{\rm{l \, k}}^{\rm{E}}}}{\Delta \, t} = \left(\frac{ M_{\rm{k}} }{ \Delta \, p_{\rm{k}} } \right) \left[{ \left({ 1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4} } \right) \left({ 1 - \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) \left({ 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) \left({ \ensuremath{l_{\rm{l \, k + 1}}^{\rm{E}}} - \ensuremath{l_{\rm{l \, k}}^{\rm{E}}} } \right) - } \right . & + & \nonumber \\ + } \right . + + +.. math:: + \left({ \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) \left({ 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) \left({ \ensuremath{l_{\rm{l \, k}}^{\rm{P}}} - \ensuremath{l_{\rm{l \, k}}^{\rm{E}}} } \right) - & + & \nonumber \\ + + + +.. math:: + \left . { \left({ \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) \left({ \ensuremath{l_{\rm{l \, k}}^{\rm{P}}} - \ensuremath{l_{\rm{l \, k}}^{\rm{E}}} } \right) - } \right]& { } & \label{eq:enviroll} - \end{aligned} + } \right] { } \label{eq:enviroll} + and .. math:: - \begin{aligned} - \frac{\Delta \, \ensuremath{l_{\rm{f \, k}}^{\rm{E}}}}{\Delta \, t} = + \frac{\Delta \, \ensuremath{l_{\rm{f \, k}}^{\rm{E}}}}{\Delta \, t} = \left(\frac{ M_{\rm{k}} }{ \Delta \, p_{\rm{k}} } \right) \left[{ \left({ 1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4} } \right) \left({ 1 - \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) \left({ 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) \left({ \ensuremath{l_{\rm{f \, k + 1}}^{\rm{E}}} - \ensuremath{l_{\rm{f \, k}}^{\rm{E}}} } \right) - } \right . & + & \nonumber \\ + } \right . + + +.. math:: + \left({ \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) \left({ 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) \left({ \ensuremath{l_{\rm{f \, k}}^{\rm{P}}} - \ensuremath{l_{\rm{f \, k}}^{\rm{E}}} } \right) - & + & \nonumber \\ + + + +.. math:: + \left . { \left({ \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) \left({ \ensuremath{l_{\rm{f \, k }}^{\rm{P}}} - \ensuremath{l_{\rm{f \, k}}^{\rm{E}}} } \right) - } \right]& { } & \label{eq:envirolf} - \end{aligned} + } \right] { } \label{eq:envirolf} + .. _Background condensation: @@ -4591,13 +4685,20 @@ The thresholds :math:`C_{high}`, :math:`C_{high 2}`, :math:`C_{low}` and .. math:: - \begin{aligned} - C_{high} = 1 - C_{tol}, \nonumber \\ - C_{high 2} = 1 - C_{tol 2}, \nonumber \\ - C_{low} = C_{tol}, \nonumber \\ + C_{high} = 1 - C_{tol}, + +.. math:: + + C_{high 2} = 1 - C_{tol 2}, + +.. math:: + + C_{low} = C_{tol}, + +.. math:: :label: eq:chigh-chigh2 + C_{low 2} = C_{tol 2}, - \label{eq:chigh-chigh2} - \end{aligned} + where the parameters :math:`C_{tol}` and :math:`C_{tol 2}` can be set via the UM namelist variables **cloud_pc2_tol** and **cloud_pc2_tol_2**. @@ -4676,12 +4777,16 @@ condensed to achieve this, so we have: .. math:: - \begin{aligned} - \overline{q} \leftarrow \overline{q} + SD \nonumber \\ - \overline{q_{cl}} \leftarrow \overline{q_{cl}} - SD \nonumber \\ + \overline{q} \leftarrow \overline{q} + SD + +.. math:: + + \overline{q_{cl}} \leftarrow \overline{q_{cl}} - SD + +.. math:: :label: eq:qsdcheck1 + \overline{T} \leftarrow \overline{T} - \frac{L_c}{c_p} SD - \label{eq:qsdcheck1} - \end{aligned} + The original version of this check on :math:`SD` (which may increase :math:`\overline{q_{cl}}`), made no accompanying changes to liquid cloud @@ -4750,15 +4855,24 @@ behaviour is currently controlled by a temporary logical in the .. math:: - \begin{aligned} - \overline{q} \leftarrow \overline{q} + \overline{q_{cl}} \nonumber \\ + \overline{q} \leftarrow \overline{q} + \overline{q_{cl}} + + .. math:: + \overline{T} \leftarrow \overline{T} - \frac{L_c}{c_p} \overline{q_{cl}} - \nonumber \\ - \overline{q_{cl}} \leftarrow 0 \nonumber \\ - C_l \leftarrow 0\nonumber \\ + + .. math:: + + \overline{q_{cl}} \leftarrow 0 + + .. math:: + + C_l \leftarrow 0 + + .. math:: :label: eq:qsdcheck2 + C_t \leftarrow C_i - \label{eq:qsdcheck2} - \end{aligned} + .. _section-3: @@ -4771,12 +4885,16 @@ the gridbox: .. math:: - \begin{aligned} - \overline{q} \leftarrow \overline{q} + \overline{q_{cl}} \nonumber \\ - \overline{q_{cl}} \leftarrow 0 \nonumber \\ + \overline{q} \leftarrow \overline{q} + \overline{q_{cl}} + +.. math:: + + \overline{q_{cl}} \leftarrow 0 + +.. math:: :label: eq:qclcheck + \overline{T} \leftarrow \overline{T} - \frac{L_c}{c_p} \overline{q_{cl}} - \label{eq:qclcheck} - \end{aligned} + .. _section-4: @@ -4814,11 +4932,12 @@ overlap situation and then the minimum overlap situation. .. math:: - \begin{aligned} - C_t \leftarrow \text{Max}( C_t, C_i, C_l ) \nonumber \\ + C_t \leftarrow \text{Max}( C_t, C_i, C_l ) + +.. math:: :label: eq:ctchecks + C_t \leftarrow \text{Min}( C_t , C_l + C_i, 1) - \label{eq:ctchecks} - \end{aligned} + .. _section-8: @@ -4831,14 +4950,24 @@ phase if the temperature is cold enough. Hence, if .. math:: - \begin{aligned} - \overline{q_{cf}} \leftarrow \overline{q_{cf}} + \overline{q_{cl}} \nonumber \\ - \overline{q_{cl}} \leftarrow 0 \nonumber \\ - \overline{T} \leftarrow \overline{T} + \frac{L_f}{c_p} \overline{q_{cl}} \nonumber \\ - C_i \leftarrow C_t \nonumber \\ + \overline{q_{cf}} \leftarrow \overline{q_{cf}} + \overline{q_{cl}} + +.. math:: + + \overline{q_{cl}} \leftarrow 0 + +.. math:: + + \overline{T} \leftarrow \overline{T} + \frac{L_f}{c_p} \overline{q_{cl}} + +.. math:: + + C_i \leftarrow C_t + +.. math:: :label: eq:homochecks + C_l \leftarrow 0. - \label{eq:homochecks} - \end{aligned} + .. _Qpos checks: @@ -6088,11 +6217,13 @@ similar way to above gives .. math:: - \begin{aligned} SD = \int_{-b_s}^{-Q_c} G(s) (-Q_c + b_s) ds + \int_{-b_s}^{-Q_c} - G(s) (-s - b_s) ds \nonumber \\ + G(s) (-s - b_s) ds + +.. math:: + = (-Q_c + b_s) (1 - C_l) - I1 , - \end{aligned} + and hence :math:`I1` in terms of :math:`SD`. Using this value of :math:`I1` in :eq:`eqn:deltaqclmax` and cancelling From f6d57561c24b609f4524acb925b42566dd9dfdae Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Tue, 7 Apr 2026 15:41:29 +0100 Subject: [PATCH 020/116] Now fixing equation labels when they don't appear at the end of the math block! --- .../cloud_schemes/UMDP30_PC2CloudScheme.rst | 98 +++++++++---------- 1 file changed, 48 insertions(+), 50 deletions(-) diff --git a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst index 95983e7670..9799d7603a 100644 --- a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst +++ b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst @@ -1770,9 +1770,8 @@ Model description :raw-latex:`\cite{fhfk14}` started from the equation for the dynamics of ice supersaturation :math:`S_i=e_v/e_{sat\;ice}-1`: -.. math:: +.. math:: :label: eqn:squires_eqn - \label{eqn:squires_eqn} \frac{D S_i}{D t} = -b_i B_0 {\cal M}_1 S_i -\left(\frac{\varepsilon}{L^2}\right)^{1/3}(S_i-S_E) + a_i w, @@ -1844,13 +1843,13 @@ Equation :eq:`eqn:si_avg` and model. The liquid cloud fraction and liquid water mass mixing ratio are given by -.. math:: +.. math:: :label: eqn:cloud_fraction - C_l^{sgt} = \int_{S_{i,wat}}^\infty d S_i F(S_i), \label{eqn:cloud_fraction} + C_l^{sgt} = \int_{S_{i,wat}}^\infty d S_i F(S_i), -.. math:: +.. math:: :label: eqn:cloud_liquid - q_{cl}^{sgt} = q_{sat\;ice}\int_{S_{i,wat}}^\infty d S_i (S_i -S_{i,wat}) F(S_i) \label{eqn:cloud_liquid}, + q_{cl}^{sgt} = q_{sat\;ice}\int_{S_{i,wat}}^\infty d S_i (S_i -S_{i,wat}) F(S_i) , where :math:`S_{i,wat} = e_{sat\;wat}/e_{sat\;ice}-1` is the value of @@ -3373,12 +3372,12 @@ The bulk cloud model plume equations for mass and :math:`{\chi}` are: - \ensuremath{\frac{\partial \, M^{\rm{P}}}{\partial \, p}} = \left({ \varepsilon \, M^{\rm{P}} - \mu \, M^{\rm{P}} - \delta \, M^{\rm{P}} } \right) -.. math:: +.. math:: :label: eq:dbydpmfchi - \ensuremath{\frac{\partial \, \ensuremath{{\chi}_{\rm{ }}^{\rm{P}}} \, M^{\rm{P}}}{\partial \, p}} = \left({ \varepsilon \, M^{\rm{P}} \, \ensuremath{{\chi}_{\rm{ }}^{\rm{E}}} - \mu \, M^{\rm{P}} \, \ensuremath{{\chi}_{\rm{ }}^{\rm{R}}} - \delta \, M^{\rm{P}} \, \ensuremath{{\chi}_{\rm{ }}^{\rm{P}}} - } \right)\label{eq:dbydpmfchi} + } \right) Equations :eq:`eq:eddyflux`, @@ -3411,21 +3410,21 @@ which is also the top of the turbulent mixed boundary layer. A simple discretized form of :eq:`eq:chimassflux`, setting :math:`{ \mu = 0 }`, is: -.. math:: +.. math:: :label: eq:chidisck {\ensuremath{\frac{\partial \, \ensuremath{{\chi}_{\rm{ }}^{\rm{E}}}}{\partial \, t}}}_{\rm{conv, \, k}} = m_{\rm{k+1/2}} \, \frac{ \left({\ensuremath{{\chi}_{\rm{k+1}}^{\rm{E}}} - \ensuremath{{\chi}_{\rm{k}}^{\rm{E}}}} \right)} {{\Delta z}_{\rm{k \, \rightarrow \, k+1}}} + {\delta}_{\rm{k}} \, m_{\rm{k}} \, \left({ \ensuremath{{\chi}_{\rm{k}}^{\rm{P}}} - \ensuremath{{\chi}_{\rm{k}}^{\rm{E}}} } \right) - \qquad \ldots \; \mbox{for k $>$ cb} \label{eq:chidisck} + \qquad \ldots \; \mbox{for k $>$ cb} -.. math:: +.. math:: :label: eq:chidisccb {\ensuremath{\frac{\partial \, \ensuremath{{\chi}_{\rm{ }}^{\rm{E}}}}{\partial \, t}}}_{\rm{conv, \, cb}} = m_{\rm{cb+1/2}} \, \frac{ \left({\ensuremath{{\chi}_{\rm{cb+1}}^{\rm{E}}} - \ensuremath{{\chi}_{\rm{cb}}^{\rm{E}}}} \right)} {{\Delta z}_{\rm{cb \, \rightarrow \, cb+1}}} - m_{\rm{cb}} \, - \left({ \ensuremath{{\chi}_{\rm{i,cb}}^{\rm{P}}} - \ensuremath{{\chi}_{\rm{cb}}^{\rm{E}}} } \right)\label{eq:chidisccb} + \left({ \ensuremath{{\chi}_{\rm{i,cb}}^{\rm{P}}} - \ensuremath{{\chi}_{\rm{cb}}^{\rm{E}}} } \right) where the initial parcel value :math:`{\chi}_{\rm{i,cb}}^{\rm{P}}` may @@ -3466,25 +3465,25 @@ as :eq:`eq:defineq1` and :eq:`eq:defineq2`. The change is seen in the vertical gradient equations based upon :eq:`eq:gradchipar` -.. math:: +.. math:: :label: eq:gradtpar M^{\rm{P}} \, \ensuremath{\frac{\partial \, \ensuremath{T_{\rm{ }}^{\rm{P}}}}{\partial \, p}} = \varepsilon \, M^{\rm{P}} \, \left({ \ensuremath{T_{\rm{ }}^{\rm{P}}} - \ensuremath{T_{\rm{ }}^{\rm{E}}} } \right)- \mu \, M^{\rm{P}} \, \left({ \ensuremath{T_{\rm{ }}^{\rm{P}}} - \ensuremath{T_{\rm{ }}^{\rm{R}}} } \right)- - \left({ \frac{L}{c_{P}} } \right)\, {\overline{Q}}_{\rm{par}} \label{eq:gradtpar} + \left({ \frac{L}{c_{P}} } \right)\, {\overline{Q}}_{\rm{par}} -.. math:: +.. math:: :label: eq:gradqpar M^{\rm{P}} \, \ensuremath{\frac{\partial \, \ensuremath{q_{\rm{ }}^{\rm{P}}}}{\partial \, p}} = \varepsilon \, M^{\rm{P}} \, \left({ \ensuremath{q_{\rm{ }}^{\rm{P}}} - \ensuremath{q_{\rm{ }}^{\rm{E}}} } \right)- \mu \, M^{\rm{P}} \, \left({ \ensuremath{q_{\rm{ }}^{\rm{P}}} - \ensuremath{q_{\rm{ }}^{\rm{R}}} } \right)+ - {\overline{Q}}_{\rm{par}} \label{eq:gradqpar} + {\overline{Q}}_{\rm{par}} -.. math:: +.. math:: :label: eq:gradlpar M^{\rm{P}} \, \ensuremath{\frac{\partial \, \ensuremath{l_{\rm{ }}^{\rm{P}}}}{\partial \, p}} = \varepsilon \, M^{\rm{P}} \, \left({ \ensuremath{l_{\rm{ }}^{\rm{P}}} - \ensuremath{l_{\rm{ }}^{\rm{E}}} } \right) - - {\overline{Q}}_{\rm{par}} + PPN \label{eq:gradlpar} + - {\overline{Q}}_{\rm{par}} + PPN The final calculation of rates in the current condensation scheme (, @@ -3568,31 +3567,31 @@ where the PC2 assumption thus far has been that Based on :eq:`eq:gradlpar`, the vertical dependence of condensate is calculated as -.. math:: +.. math:: :label: eq:vertparl \ensuremath{\frac{\partial \, \ensuremath{l_{\rm{l}}^{\rm{P}}}}{\partial \, p}} = \varepsilon \, \left({ \ensuremath{l_{\rm{l}}^{\rm{P}}} - \ensuremath{l_{\rm{l}}^{\rm{E}}} } \right)- \frac{{\overline{Q}}_{\rm{l, par}}}{M^{\rm{P}}} - - \frac{RAIN}{M^{\rm{P}}} \label{eq:vertparl} + \frac{RAIN}{M^{\rm{P}}} -.. math:: +.. math:: :label: eq:vertparf \ensuremath{\frac{\partial \, \ensuremath{l_{\rm{f}}^{\rm{P}}}}{\partial \, p}} = \varepsilon \, \left({ \ensuremath{l_{\rm{f}}^{\rm{P}}} - \ensuremath{l_{\rm{f}}^{\rm{E}}} } \right)- \frac{{\overline{Q}}_{\rm{f, par}}}{M^{\rm{P}}} - - \frac{SNOW}{M^{\rm{P}}}\label{eq:vertparf} + \frac{SNOW}{M^{\rm{P}}} Following , equations :eq:`eq:dbydpmassflux`, :eq:`eq:vertparl` and :eq:`eq:vertparf` are discretized: -.. math:: +.. math:: :label: eq:discdmfbydp M_{\rm{k} + 1} = M_{\rm{k}} \, \left({ 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right)\, \left({ 1 - \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right)\, - EPSS_{\rm{k}} \label{eq:discdmfbydp} + EPSS_{\rm{k}} .. math:: @@ -3636,7 +3635,7 @@ The condensation and precipitation terms in equations therefore solved by starting with an ascent in which condensation and precipitation terms are suppressed: -.. math:: +.. math:: :label: eq:discvparldry \ensuremath{l_{\rm{l \, k + 1}}^{\rm{P}}} = \frac{\left({ \ensuremath{l_{\rm{l \, k}}^{\rm{P}}} + @@ -3644,9 +3643,9 @@ precipitation terms are suppressed: \varepsilon_{\rm{k} + 3/4} \, \Delta p_{\rm{k} + 3/4} \, \left[{1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4}} \right]\, \ensuremath{l_{\rm{l \, k + 1}}^{\rm{E}}} - } \right)}{EPSS_{\rm{k}}} \label{eq:discvparldry} + } \right)}{EPSS_{\rm{k}}} -.. math:: +.. math:: :label: eq:discvparfdry \ensuremath{l_{\rm{f \, k + 1}}^{\rm{P}}} = \frac{\left({ \ensuremath{l_{\rm{f \, k}}^{\rm{P}}} + @@ -3654,7 +3653,7 @@ precipitation terms are suppressed: \varepsilon_{\rm{k} + 3/4} \, \Delta p_{\rm{k} + 3/4} \, \left[{1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4}} \right]\, \ensuremath{l_{\rm{f \, k + 1}}^{\rm{E}}} - } \right)}{EPSS_{\rm{k}}} \label{eq:discvparfdry} + } \right)}{EPSS_{\rm{k}}} At the base of the convective plume (ie. the level immediately above @@ -3664,17 +3663,17 @@ cloud base), :math:`l_{\rm{l \, k}}^{\rm{P}}` is initialized to such that the modified form of :eq:`eq:chidisccb` produces zero fluxes at cloud base: -.. math:: +.. math:: :label: eq:q4lcbi Q4_{\rm{l}}(cb) = 0 = M_{\rm{cb+1/2}}^{\rm{P}} \, \ensuremath{\frac{\partial \, \ensuremath{l_{\rm{l}}^{\rm{E}}}}{\partial \, p}} - M_{\rm{cb}}^{\rm{P}}\, - \left({ \ensuremath{l_{\rm{l}}^{\rm{P \, i}}} - \ensuremath{l_{\rm{l}}^{\rm{E}}}(\rm{cb}) } \right)\label{eq:q4lcbi} + \left({ \ensuremath{l_{\rm{l}}^{\rm{P \, i}}} - \ensuremath{l_{\rm{l}}^{\rm{E}}}(\rm{cb}) } \right) -.. math:: +.. math:: :label: eq:q4fcbi Q4_{\rm{f}}(cb) = 0 = M_{\rm{cb+1/2}}^{\rm{P}} \, \ensuremath{\frac{\partial \, \ensuremath{l_{\rm{f}}^{\rm{E}}}}{\partial \, p}} - M_{\rm{cb}}^{\rm{P}}\, - \left({ \ensuremath{l_{\rm{f}}^{\rm{P \, i}}} - \ensuremath{l_{\rm{f}}^{\rm{E}}}(\rm{cb}) } \right)\label{eq:q4fcbi} + \left({ \ensuremath{l_{\rm{f}}^{\rm{P \, i}}} - \ensuremath{l_{\rm{f}}^{\rm{E}}}(\rm{cb}) } \right) As the convection scheme makes the single phase assumption for parcel @@ -3723,37 +3722,37 @@ where :math:`l_{\rm{k + 1}}^{\rm{P}}` = This reduces the parcel condensate to : -.. math:: +.. math:: :label: eq:vparlfinal \ensuremath{l_{\rm{l \, k + 1}}^{\rm{P}}} = \left({ \frac{\ensuremath{l_{\rm{l \, k + 1}}^{\rm{P}}}}{\ensuremath{l_{\rm{k + 1}}^{\rm{P}}}} - } \right)\, \ensuremath{l_{\rm{MIN}}^{\rm{P}}} \label{eq:vparlfinal} + } \right)\, \ensuremath{l_{\rm{MIN}}^{\rm{P}}} -.. math:: +.. math:: :label: eq:vparffinal \ensuremath{l_{\rm{f \, k + 1}}^{\rm{P}}} = \left({ \frac{\ensuremath{l_{\rm{f \, k + 1}}^{\rm{P}}}}{\ensuremath{l_{\rm{k + 1}}^{\rm{P}}}} - } \right)\, \ensuremath{l_{\rm{MIN}}^{\rm{P}}} \label{eq:vparffinal} + } \right)\, \ensuremath{l_{\rm{MIN}}^{\rm{P}}} The final parcel condensate values are then used in the rate calculation based upon eqn :eq:`eq:basiclold`: -.. math:: +.. math:: :label: eq:q4lmassf Q4_{\rm{l}}(k) = M_{\rm{k+1/2}}^{\rm{P}} \, \ensuremath{\frac{\partial \, \ensuremath{l_{\rm{l}}^{\rm{E}}}}{\partial \, p}} + \left({ {\mu}_{\rm{k}} \, M_{\rm{k}}^{\rm{P}} + {\delta}_{\rm{k}} \, M_{\rm{k}}^{\rm{P}} } \right)\, \left({ \ensuremath{l_{\rm{l}}^{\rm{P}}}(\rm{k}) - \ensuremath{l_{\rm{l}}^{\rm{E}}}(\rm{k}) } \right)- - {\overline{Q}}_{\rm{l, reset}} \label{eq:q4lmassf} + {\overline{Q}}_{\rm{l, reset}} -.. math:: +.. math:: :label: eq:q4fmassf Q4_{\rm{f}}(k) = M_{\rm{k+1/2}}^{\rm{P}} \, \ensuremath{\frac{\partial \, \ensuremath{l_{\rm{f}}^{\rm{E}}}}{\partial \, p}} + \left({ {\mu}_{\rm{k}} \, M_{\rm{k}}^{\rm{P}} + {\delta}_{\rm{k}} \, M_{\rm{k}}^{\rm{P}} } \right)\, \left({ \ensuremath{l_{\rm{f}}^{\rm{P}}}(\rm{k}) - \ensuremath{l_{\rm{f}}^{\rm{E}}}(\rm{k}) } \right)- - {\overline{Q}}_{\rm{f, reset}} \label{eq:q4fmassf} + {\overline{Q}}_{\rm{f, reset}} Note that, as a side-effect, the environment equations for potential @@ -3780,12 +3779,12 @@ condensate is no longer re-evaporated at the end \left({ \theta_{\rm{k}}^{\rm{R}} - \theta_{\rm{k}}^{\rm{E}} } \right) + -.. math:: +.. math:: :label: eq:enviroth \left . { \left({ \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) \left({ \theta_{\rm{k}}^{\rm{P}} - \theta_{\rm{k}}^{\rm{E}} } \right) - } \right] { } \label{eq:enviroth} + } \right] { } and @@ -3808,12 +3807,12 @@ and \left({ q_{\rm{k}}^{\rm{R}} - q_{\rm{k}}^{\rm{E}} } \right) + -.. math:: +.. math:: :label: eq:enviroq \left . { \left({ \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) \left({ q_{\rm{k}}^{\rm{P}} - q_{\rm{k}}^{\rm{E}} } \right) - } \right] { } \label{eq:enviroq} + } \right] { } Similarly, eqns :eq:`eq:q4lmassf` and @@ -3837,12 +3836,12 @@ Similarly, eqns :eq:`eq:q4lmassf` and \left({ \ensuremath{l_{\rm{l \, k}}^{\rm{P}}} - \ensuremath{l_{\rm{l \, k}}^{\rm{E}}} } \right) + -.. math:: +.. math:: :label: eq:enviroll \left . { \left({ \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) \left({ \ensuremath{l_{\rm{l \, k}}^{\rm{P}}} - \ensuremath{l_{\rm{l \, k}}^{\rm{E}}} } \right) - } \right] { } \label{eq:enviroll} + } \right] { } and @@ -3865,12 +3864,12 @@ and \left({ \ensuremath{l_{\rm{f \, k}}^{\rm{P}}} - \ensuremath{l_{\rm{f \, k}}^{\rm{E}}} } \right) + -.. math:: +.. math:: :label: eq:envirolf \left . { \left({ \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) \left({ \ensuremath{l_{\rm{f \, k }}^{\rm{P}}} - \ensuremath{l_{\rm{f \, k}}^{\rm{E}}} } \right) - } \right] { } \label{eq:envirolf} + } \right] { } .. _Background condensation: @@ -4299,9 +4298,8 @@ Using the semi-Lagrangian advection in the same way as is performed for the model prognostic *Exner*, (:math:`\prod`) on the departure points (:math:`\prod_{dep}`). *Exner* is defined as -.. math:: +.. math:: :label: eq:exner - \label{eq:exner} \prod = \frac{T}{\theta} = \left( \frac{p}{p_{ref}} \right)^{\kappa} where :math:`\theta` is the potential temperature, :math:`p_{ref}` is a From fc60e4285c527d42a048adf6bc4a3aeb05b525c1 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Tue, 7 Apr 2026 23:17:39 +0100 Subject: [PATCH 021/116] Fixed problems due to wrong indent in bullet-point lists. --- .../cloud_schemes/UMDP30_PC2CloudScheme.rst | 46 +++++++++---------- 1 file changed, 23 insertions(+), 23 deletions(-) diff --git a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst index 9799d7603a..0ae4551659 100644 --- a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst +++ b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst @@ -4432,16 +4432,16 @@ The initiation will be called if the liquid cloud fraction is either 0 or 1 and appropriate :math:`RH` criteria hold, along with other restrictions. :math:`C_l` is initiated away from 0 if -- :math:`RH_T > RH_{crit} + RH_{crit \, tol}` **and** +- :math:`RH_T > RH_{crit} + RH_{crit \, tol}` **and** -- Cumulus convection has *not* been diagnosed from the boundary-layer +- Cumulus convection has *not* been diagnosed from the boundary-layer in the current column **and** -- The current level is not below the surface mixed-layer LCL **and** +- The current level is not below the surface mixed-layer LCL **and** -- :math:`C_l = 0` **and** +- :math:`C_l = 0` **and** -- :math:`RH_T^{[n+1]} > RH_T^{[n]}` , +- :math:`RH_T^{[n+1]} > RH_T^{[n]}` , where :math:`RH_{crit \, tol}` is a specified tolerance parameter, of value 0.01, and :math:`RH_T` is defined in :eq:`eq:rht`. @@ -4451,15 +4451,15 @@ is called. Additionally, there is another possibility for the last of the relations. This second option also allows initiation when the water is supercooled: -- :math:`C_l < 0.05` *and* :math:`\overline{T} < 0 ^{\circ} C` . +- :math:`C_l < 0.05` *and* :math:`\overline{T} < 0 ^{\circ} C` . Equivalently, :math:`C_l` is initiated away from 1 if -- :math:`RH_T < 2 - RH_{crit} - RH_{crit \, tol}` **and** +- :math:`RH_T < 2 - RH_{crit} - RH_{crit \, tol}` **and** -- :math:`C_l = 1` **and** +- :math:`C_l = 1` **and** -- :math:`RH_T^{[n+1]} < RH_T^{[n]}` . +- :math:`RH_T^{[n+1]} < RH_T^{[n]}` . “Simplified” initiation logic ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ @@ -4656,9 +4656,9 @@ We first calculate :math:`RH_T` using :eq:`eq:rht` and compare this to the critical relative humidity, :math:`RH_{crit}`. The liquid cloud fraction will be reset to 1 if: -- :math:`RH_T > 2 - RH_{crit}` and :math:`C_l \ge C_{high}` +- :math:`RH_T > 2 - RH_{crit}` and :math:`C_l \ge C_{high}` -- or :math:`C_l \ge C_{high 2}` +- or :math:`C_l \ge C_{high 2}` where :math:`C_{high}` and :math:`C_{high 2}` are defined in :eq:`eq:chigh-chigh2`. The evaporation is done by @@ -4670,9 +4670,9 @@ to saturation, according to :eq:`eq:qsdcheck1` below. Similarly, the equivalent check for low values of :math:`RH_T` is performed. The liquid cloud fraction will be reset to 0 if: -- :math:`RH_T < RH_{crit}` and :math:`C_l \le C_{low}` +- :math:`RH_T < RH_{crit}` and :math:`C_l \le C_{low}` -- or :math:`C_l \le C_{low 2}` . +- or :math:`C_l \le C_{low 2}` . The remaining :math:`\overline{q_{cl}}` is evaporated into the gridbox using :eq:`eq:qclcheck` below. @@ -5641,30 +5641,30 @@ diagnose the moisture cycle within PC2. Since most physics sections can cause condensation, condensate and cloud fraction increment diagnostics have been written for each of these sections. -- :math:`\overline{T}`, :math:`\overline{q}`, +- :math:`\overline{T}`, :math:`\overline{q}`, :math:`\overline{q_{cl}}`, :math:`\overline{C_t}` and :math:`\overline{C_l}` increments from SW radiation, :math:`\overline{T}` increment from SW Radiation without including the condensation: **Section 1** . -- :math:`\overline{T}`, :math:`\overline{q}`, +- :math:`\overline{T}`, :math:`\overline{q}`, :math:`\overline{q_{cl}}`, :math:`\overline{C_t}` and :math:`\overline{C_l}` increments from LW radiation, :math:`\overline{T}` increment from LW Radiation without including the condensation: **Section 2** . -- :math:`\overline{T}`, :math:`\overline{q}`, +- :math:`\overline{T}`, :math:`\overline{q}`, :math:`\overline{q_{cl}}`, :math:`\overline{q_{cf}}`, :math:`\overline{C_t}`, :math:`\overline{C_l}` and :math:`\overline{C_f}` increments from Boundary Layer: **Section 3** . -- :math:`\overline{T}`, :math:`\overline{q}`, +- :math:`\overline{T}`, :math:`\overline{q}`, :math:`\overline{q_{cl}}`, :math:`\overline{q_{cf}}`, :math:`\overline{C_t}`, :math:`\overline{C_l}` and :math:`\overline{C_f}` increments from Large-scale precipitation: **Section 4** . -- :math:`\overline{T}`, :math:`\overline{q}`, +- :math:`\overline{T}`, :math:`\overline{q}`, :math:`\overline{q_{cl}}`, :math:`\overline{q_{cf}}`, :math:`\overline{C_t}`, :math:`\overline{C_l}` and :math:`\overline{C_f}` increments from Convection, @@ -5673,7 +5673,7 @@ have been written for each of these sections. :math:`\overline{C_l}` and :math:`\overline{C_f}` increments from the inhomogeneous part of the Convection scheme only: **Section 5** . -- :math:`\overline{T}`, :math:`\overline{q}`, +- :math:`\overline{T}`, :math:`\overline{q}`, :math:`\overline{q_{cl}}`, :math:`\overline{q_{cf}}`, :math:`\overline{C_t}`, :math:`\overline{C_l}` and :math:`\overline{C_f}` increments from the Advection: **Section 12** . @@ -5685,25 +5685,25 @@ increments were available using a modification set or branch and a user-STASHmaster file up to version 7.5. From version 7.6 these diagnostics are available as standard. -- :math:`\overline{T}`, :math:`\overline{q}`, +- :math:`\overline{T}`, :math:`\overline{q}`, :math:`\overline{q_{cl}}`, :math:`\overline{C_t}`, and :math:`\overline{C_l}` increments from the PC2 erosion section: **Section 4** or **Section 5** depending on where the erosion is called. -- :math:`\overline{T}`, :math:`\overline{q}`, +- :math:`\overline{T}`, :math:`\overline{q}`, :math:`\overline{q_{cl}}`, :math:`\overline{q_{cf}}`, :math:`\overline{C_t}`, :math:`\overline{C_l}` and :math:`\overline{C_f}` increments from the Bounds Checking after atmphya: **Section 4** . -- :math:`\overline{T}`, :math:`\overline{q}`, +- :math:`\overline{T}`, :math:`\overline{q}`, :math:`\overline{q_{cl}}`, :math:`\overline{q_{cf}}`, :math:`\overline{C_t}`, :math:`\overline{C_l}` and :math:`\overline{C_f}` increments from the Initiation and Bounds checking at the end of the timestep: **Section 16** . -- :math:`\overline{T}`, :math:`\overline{q}`, +- :math:`\overline{T}`, :math:`\overline{q}`, :math:`\overline{q_{cl}}`, :math:`\overline{q_{cf}}`, :math:`\overline{C_t}`, :math:`\overline{C_l}` and :math:`\overline{C_f}` increments from the Pressure Forcing section: From 2756298890672be4083f960aaab3806162b0bb52 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Wed, 8 Apr 2026 00:27:43 +0100 Subject: [PATCH 022/116] Replaced very non-standard labelling of in-line equations (which hadn't converted to .rst correctly) with correctly-formatted display-mode equations. --- .../cloud_schemes/UMDP30_PC2CloudScheme.rst | 23 +++++++++++-------- 1 file changed, 13 insertions(+), 10 deletions(-) diff --git a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst index 0ae4551659..b532e9acab 100644 --- a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst +++ b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst @@ -4551,26 +4551,29 @@ prognostic :math:`C_l` and :math:`q_{cl}` are incremented as follows: - If :math:`{q_{cl}}_{diag} > q_{cl}`: - :math:`\Delta q_{cl} = {q_{cl}}_{diag} - q_{cl} - \quad \refstepcounter{equation}(\theequation)\label{eq:dqcl_init}` + .. math:: :label: eq:dqcl_init + + \Delta q_{cl} = {q_{cl}}_{diag} - q_{cl} - If :math:`Q_C < 0`: - :math:`\Delta C_{l} = \frac{\Delta q_{cl}}{{q_{cl}}_{diag}} - \left( {C_{l}}_{diag} - C_{l} \right) - \quad \refstepcounter{equation}(\theequation)\label{eq:dcl_init1}` + .. math:: :label: eq:dcl_init1 + + \Delta C_{l} = \frac{\Delta q_{cl}}{{q_{cl}}_{diag}} + \left( {C_{l}}_{diag} - C_{l} \right) - If :math:`Q_C > 0`: - :math:`\Delta C_{l} = \frac{\Delta SD}{{SD}_{diag}} - \left( {C_{l}}_{diag} - C_{l} \right) - \quad \refstepcounter{equation}(\theequation)\label{eq:dcl_init2}` + .. math:: :label: eq:dcl_init2 + + \Delta C_{l} = \frac{\Delta SD}{{SD}_{diag}} + \left( {C_{l}}_{diag} - C_{l} \right) - Otherwise: - :math:`\Delta q_{cl} = 0` + .. math:: \Delta q_{cl} = 0 - :math:`\Delta C_{l} = 0` + .. math:: \Delta C_{l} = 0 where the subscript :math:`_{diag}` denotes the liquid cloud water content and fraction predicted by the diagnostic cloud scheme (either From facce9336c743f3e3dd371da6d9ef56bf36b9eb5 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Wed, 8 Apr 2026 00:50:25 +0100 Subject: [PATCH 023/116] Miscellaneous fixes: (a) Avoid duplicate section title and label 'numerical application', (b) Corrected a stray wrong indent, (c) Remove colon from appendix section titles (messes-up when used in the label). --- .../cloud_schemes/UMDP30_PC2CloudScheme.rst | 20 +++++++++---------- 1 file changed, 10 insertions(+), 10 deletions(-) diff --git a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst index b532e9acab..601af85b0a 100644 --- a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst +++ b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst @@ -1409,10 +1409,10 @@ used within the convection scheme. It still remains to parametrize :math:`\delta_{xl}`, which is given by the convection scheme itself. This is discussed in section :ref:`Phase of condensate`. -.. _Numerical application: +.. _Numerical application of injection forcing: -Numerical application -~~~~~~~~~~~~~~~~~~~~~ +Numerical application of injection forcing +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The numerical application using :eq:`eq:dcltdt_almost_final` may be @@ -3050,7 +3050,7 @@ cumulus regimes. Note that :math:`q_{cl}` falls to zero after a finite time :math:`\frac{{q_{cl}}_0}{1-b_1} \frac{a_L}{K} - \frac{1}{2 {C_l}_0 (1 - C_l) \, (q_{cl}-Q_c)}`. If the timestep + \frac{1}{2 {C_l}_0 (1 - C_l) \, (q_{cl}-Q_c)}`. If the timestep :math:`\Delta t` is longer than this time, then erosion completely removes the cloud during the current timestep. @@ -5042,7 +5042,7 @@ ill-conditioning of this solution near :math:`C_l = 1`. In practice, the ill-conditioning of :eq:`eq:da2` and :eq:`eq:da3` becomes too numerically awkward for us to apply the full solution based on homogeneous forcing, although, for -completeness, we outline it in Appendix :ref:`Appendix: Alternative PC2 - Data Assimilation formulations`. Hence +completeness, we outline it in Appendix :ref:`Appendix; Alternative PC2 - Data Assimilation formulations`. Hence we have chosen to apply a much simpler model. Here we use simply the data assimilation increments :math:`\Delta \overline{q}` and :math:`\Delta \overline{T}` within the standard homogeneous forcing @@ -5058,7 +5058,7 @@ data assimilation increments for :math:`\Delta \overline{q_{cl}}`, :math:`\Delta \overline{q}` and :math:`\Delta \overline{T}` remain those that the data assimilation scheme itself calculated. -Appendix :ref:`Appendix: Alternative PC2 - Data Assimilation formulations` gives, for completeness, the +Appendix :ref:`Appendix; Alternative PC2 - Data Assimilation formulations` gives, for completeness, the alternative numerical technique for the solution of :eq:`eq:da2` and :eq:`eq:da3`. However, we stress that this technique is not used within the current PC2 @@ -6055,9 +6055,9 @@ More information Information on results of the scheme and how to run the PC2 code at various model versions is available on the PC2 web site. -.. _Appendix: Alternative PC2 - Data Assimilation formulations: +.. _Appendix; Alternative PC2 - Data Assimilation formulations: -Appendix: Alternative PC2 - Data Assimilation formulations +Appendix; Alternative PC2 - Data Assimilation formulations ========================================================== In this alternative method to section :ref:`Data Assimilation` we will assume @@ -6358,9 +6358,9 @@ results than simply using the homogeneous forcing method. Further work will be required to enable the implementation of this :math:`\overline{q}` and :math:`\overline{T}` preserving method. -.. _Appendix: Essentials of PC2 for code developers: +.. _Appendix; Essentials of PC2 for code developers: -Appendix: Essentials of PC2 for code developers +Appendix; Essentials of PC2 for code developers =============================================== This section provides some guidance to code developers on the treatment From bcc316cfdeef5ce60af21ce6596fcce9c02a0ae6 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Fri, 10 Apr 2026 13:52:45 +0100 Subject: [PATCH 024/116] Deleted old .eps copies of the figures. --- .../cloud_schemes/Timestepping_ctl66.epsi | 9927 --------------- .../cloud_schemes/Timestepping_pc266.epsi | 10230 ---------------- .../cloud_schemes/pc2_process_explanation.eps | 9032 -------------- 3 files changed, 29189 deletions(-) delete mode 100644 documentation/source/science_guide/cloud_schemes/Timestepping_ctl66.epsi delete mode 100644 documentation/source/science_guide/cloud_schemes/Timestepping_pc266.epsi delete mode 100644 documentation/source/science_guide/cloud_schemes/pc2_process_explanation.eps diff --git a/documentation/source/science_guide/cloud_schemes/Timestepping_ctl66.epsi b/documentation/source/science_guide/cloud_schemes/Timestepping_ctl66.epsi deleted file mode 100644 index 2b235fbf34..0000000000 --- a/documentation/source/science_guide/cloud_schemes/Timestepping_ctl66.epsi +++ /dev/null @@ -1,9927 +0,0 @@ -%!PS-Adobe-3.0 EPSF-3.0 -%%Invocation: path/gs -q -sDEVICE=eps2write -sstdout=? -sOutputFile=? -dNOPAUSE -dBATCH -P- -dSAFER -dDEVICEWIDTH=250000 -dDEVICEHEIGHT=250000 ? -%%BoundingBox: 0 0 720 960 -%%HiResBoundingBox: 0.00 0.00 720.00 960.00 -%%Creator: GPL Ghostscript 9540 (eps2write) -%%LanguageLevel: 2 -%%CreationDate: D:20250417013840+01'00' -%%Pages: 1 -%%EndComments -%%BeginProlog -10 dict dup begin -/DSC_OPDFREAD true def -/SetPageSize false def -/EPS2Write true def -end -count 0 ne{ -dup type/dicttype eq{ -dup/EPS2Write known{ -dup/EPS2Write get not -} -{ -true -}ifelse -} -{ -true -}ifelse -} -{ -true -}ifelse -10 dict begin -/this currentdict def -/y 720 def -/ebuf 200 string def -/prnt{ -36//this/y get moveto//ebuf cvs show -//this/y 2 copy get 12 sub put -}bind def -/newline{ -36//this/y get moveto -//this/y 2 copy get 12 sub put -}bind def -{ -errordict/handleerror -{systemdict begin -$error begin -newerror -{(%%[ Error handled by opdfread.ps : )print errorname//ebuf cvs print(; OffendingCommand: ) -print/command load//ebuf cvs print( ]%%)= flush -/newerror false store vmstatus pop pop 0 ne -{grestoreall -}if -errorname(VMerror)ne -{showpage -}if -initgraphics -0 720 moveto -errorname(VMerror)eq -{//this/ehsave known -{clear//this/ehsave get restore 2 vmreclaim -}if -vmstatus exch pop exch pop -} -/Courier 12 selectfont -{ -(ERROR: )//prnt exec errorname//prnt exec -(OFFENDING COMMAND: )//prnt exec -/command load//prnt exec -$error/ostack known{ -(%%[STACK:)= -(STACK:)//prnt exec -$error/ostack get aload length{ -//newline exec -dup mark eq{ -(-mark-)dup = show -}{ -dup type/nametype eq{ -dup xcheck not{ -(/)show -(/)print -}if -}if -dup =//ebuf cvs show -}ifelse -}repeat -}if -}ifelse -(%%]%)= -//systemdict/showpage get exec -quit -}if -end -end -}bind readonly put -}if -end -50 dict begin -count 0 ne{ -dup type/dicttype eq{ -{def}forall -false -} -{ -true -}ifelse -} -{ -true -}ifelse -{ -( *** Warning: global definitions dictionary not found, file may be corrupted.\n)print flush -}if -/DefaultSwitch -{ -dup where{ -pop pop -}{ -false def -}ifelse -}bind def -/=string 256 string def -/=only{ -//=string cvs print -}bind def -/HexDigits(0123456789ABCDEF)readonly def -/PrintHex -{8{ -dup -28 bitshift 15 and//HexDigits exch 1 getinterval//=only exec -4 bitshift -}repeat -pop -}bind def -/PDFR_DEBUG DefaultSwitch -/PDFR_DUMP DefaultSwitch -/PDFR_STREAM DefaultSwitch -/TTFDEBUG DefaultSwitch -/RotatePages DefaultSwitch -/FitPages DefaultSwitch -/CenterPages DefaultSwitch -/SetPageSize DefaultSwitch -/error -{ -counttomark 1 sub -1 0{ -index dup type/arraytype eq{==}{=only}ifelse -}for -()= -cleartomark -....Undefined -}bind def -//SetPageSize{ -//RotatePages//FitPages or//CenterPages or{ -mark(/RotatePages, /FitPages and CenterPages are not allowed with /SetPageSize)//error exec -}if -} -{ -//FitPages//CenterPages and{ -mark(CenterPages is not allowed with /FitPages)//error exec -}if -} -ifelse -/knownget -{ -2 copy known{ -get true -}{ -pop pop false -}ifelse -}bind def -/IsUpper -{dup(A)0 get ge exch(Z)0 get le and -}bind def -/cpa2g{ -dup length array -0 1 2 index length 1 sub{ -dup 3 index exch get cp2g -3 copy put pop pop -}for -exch pop -}bind def -/cpd2g{ -dup length dict exch{ -cp2g 2 index 3 1 roll put -}forall -}bind def -/cps2g{ -dup length string copy -}bind def -/cp2gprocs -<> -def -/cp2g{ -dup gcheck not{ -dup//cp2gprocs 1 index type -2 copy known{ -get currentglobal 3 1 roll true setglobal exec exch setglobal -1 index wcheck not{readonly}if -1 index xcheck{cvx}if -exch pop -}{ -pop pop -}ifelse -}if -}bind def -/BlockBuffer 65535 string def -/PDFReader currentdict def -/ObjectRegistryMaxLength 50000 def -/ObjectRegistry 10 dict def -ObjectRegistry -begin -0 ObjectRegistryMaxLength dict def -end -/CurrentObject null def -/DoneDocumentStructure false def -/GraphicState 20 dict begin -/InitialTextMatrix matrix def -/InitialMatrix matrix currentmatrix def -currentdict end def -/TempMatrix matrix def -/GraphicStateStack 20 array def -/GraphicStateStackPointer 0 def -/InitialTextMatrixStack 20 array def -/InitialTextMatrixStackPointer 0 def -/PDFColorSpaces 50 dict def -/InstalledFonts 50 dict def -/MacRomanEncodingInverse null def -currentglobal false setglobal -userdict/PDFR_InitialGS gstate put -userdict/PDFR_Patterns 50 dict put -userdict/FuncDataReader 10 dict put -setglobal -/InitialExtGState 20 dict begin -/BG2 currentblackgeneration cp2g def -/UCR2 currentundercolorremoval cp2g def -/TR2 currentglobal false setglobal[currentcolortransfer]exch setglobal cp2g def -/HT currenthalftone cp2g def -currentdict end readonly def -/InitialGraphicState 20 dict begin -/FontSize 0 def -/CharacterSpacing 0 def -/TextLeading 0 def -/TextRenderingMode 0 def -/WordSpacing 0 def -currentdict end readonly def -/SimpleColorSpaceNames 15 dict begin -/DeviceGray true def -/DeviceRGB true def -/DeviceCMYK true def -currentdict end readonly def -/1_24_bitshift_1_sub 1 24 bitshift 1 sub def -/ReadFontProcs 10 dict def -/GetObject -{ -dup ObjectRegistryMaxLength idiv -//PDFReader/ObjectRegistry get exch knownget{ -exch knownget -}{ -pop false -}ifelse -}bind def -/PutObject -{ -1 index ObjectRegistryMaxLength idiv -//PDFReader/ObjectRegistry get 1 index knownget{ -exch pop -3 1 roll put -}{ -//PDFReader/ObjectRegistry get dup -begin -1 index ObjectRegistryMaxLength dict def -end -exch get -3 1 roll put -}ifelse -}bind def -/Register -{ -1 index GetObject{ -dup xcheck{ -4 3 roll pop -//PDFR_DEBUG{ -(Have a daemon for )print 2 index == -}if -exec -}{ -dup null ne{ -mark(The object )4 index(is already defined : )4 index//error exec -}{ -pop -}ifelse -3 2 roll -exec -}ifelse -}{ -3 2 roll -exec -}ifelse -PutObject -}bind def -/IsRegistered -{ -GetObject{ -null ne -}{ -false -}ifelse -}bind def -/GetRegistered -{ -dup GetObject not{ -exch mark exch(Object )exch( isn't defined before needed (1).)//error exec -}if -dup xcheck{ -exch mark exch(Object )exch( isn't defined before needed (2).)//error exec -}{ -dup null eq{ -exch mark exch(Object )exch( isn't defined before needed (3).)//error exec -}if -exch pop -}ifelse -}bind def -/StandardFontNames<< -/Times-Roman true -/Helvetica true -/Courier true -/Symbol true -/Times-Bold true -/Helvetica-Bold true -/Courier-Bold true -/ZapfDingbats true -/Times-Italic true -/Helvetica-Oblique true -/Courier-Oblique true -/Times-BoldItalic true -/Helvetica-BoldOblique true -/Courier-BoldOblique true ->>def -/CleanAllResources -{//PDFR_DEBUG{ -(CleanAllResources beg)= -}if -//PDFReader/ObjectRegistry get{ -dup length 0 exch 1 exch 1 sub{ -2 copy get dup xcheck{ -pop pop -}{ -dup null eq{ -pop pop -}{ -dup type/dicttype eq{/.Global known}{pop false}ifelse{ -pop -}{ -//PDFR_DEBUG{ -(Dropping )print dup = -}if -1 index exch/DroppedObject put -}ifelse -}ifelse -}ifelse -}for -pop -}forall -FontDirectory length dict begin -FontDirectory{ -pop -dup//StandardFontNames exch known not{ -dup null def -}if -pop -}forall -currentdict -end{ -pop -//PDFR_DEBUG{ -(Undefining font )print dup = -}if -undefinefont -}forall -//PDFR_DEBUG{ -(CleanAllResources end)= -}if -}bind def -/PrintReference -{ -//PDFR_DEBUG{ -({ )print -dup{ -=only( )print -}forall -( })= -}if -}bind def -/R -{ -0 ne{ -exch mark exch(A referred object generation )exch( isn't 0.)//error exec -}if -[ -exch//GetRegistered/exec load -]cvx -//PrintReference exec -}bind def -/IsObjRef -{ -dup type/arraytype eq{ -dup length 3 eq{ -dup xcheck exch -dup 0 get type/integertype eq 3 2 roll and exch -dup 1 get//GetRegistered eq 3 2 roll and exch -2 get/exec load eq and -}{ -pop false -}ifelse -}{ -pop false -}ifelse -}bind def -/DoNothing -{ -}def -/RunTypeDaemon -{ -dup type/dicttype eq{ -dup/Type//knownget exec{ -//PDFReader/TypeDaemons get exch -//knownget exec{ -exec -}if -}if -}if -}bind def -/obj -{ -//PDFR_DEBUG{ -(Defining )print 1 index =only( )print dup =only( obj)= -}if -0 ne{ -exch mark exch(An object generation )exch( isn't 0.)//error exec -}if -}bind def -/endobj -{ -//PDFR_DEBUG{ -(endobj )= -}if -count 1 eq{ -pop -}{ -dup type/dicttype eq{ -dup/.endobj_daemon//knownget exec{ -//PDFR_DEBUG{(.endobj_daemon for )print 2 index =}if -exec -}if -}if -dup type/dicttype eq{dup/ImmediateExec known}{false}ifelse{ -pop pop -}{ -//PDFR_DEBUG{ -(Storing )print 1 index = -}if -//RunTypeDaemon exec -//DoNothing 3 1 roll//Register exec -}ifelse -}ifelse -}bind def -/StoreBlock -{ -//PDFR_DEBUG{ -(StoreBlock )print//PDFReader/BlockCount get =only(, Length = )print dup length = -}if -dup length string copy -//PDFReader/BlockCount get exch -//PDFReader/CurrentObject get 3 1 roll -put -//PDFReader/BlockCount get 1 add -//PDFReader exch/BlockCount exch put -}bind def -/CheckLength -{dup type/integertype ne{ -mark(Object length isn't an integer.)//error exec -}if -}bind def -/ResolveD -{ -3 copy pop get -dup//IsObjRef exec{ -//PDFR_DEBUG{ -(Resolving )print//PrintReference exec -}if -exec -exch exec -}{ -exch pop -}ifelse -dup 4 1 roll -put -}bind def -/ResolveA -{2 index 2 index get -dup//IsObjRef exec{ -exec -exch exec -3 copy put -}{ -exch pop -}ifelse -exch pop exch pop -}bind def -/StoreStream -{ -dup//PDFReader exch/CurrentObject exch put -//PDFReader/BlockCount 0 put -dup/Length//CheckLength//ResolveD exec -//PDFR_DEBUG{ -(StoreStream Length = )print dup = -}if -currentfile exch()/SubFileDecode filter -{dup//BlockBuffer readstring{ -//StoreBlock exec -}{ -//StoreBlock exec -exit -}ifelse -}loop -pop -//PDFReader/CurrentObject null put -//PDFR_DEBUG{ -(StoreStream end.)= -}if -}bind def -/MakeStreamDumper -{ -//PDFR_DEBUG{ -(MakeStreamDumper beg.)= -}if -currentglobal exch dup gcheck setglobal -[exch -1 dict dup/c 0 put exch -1024 string -{readstring pop -(StreamDumper )print 1 index/c get =string cvs print( )print -dup length =string cvs print( <)print dup print(>\n)print -dup length -3 2 roll -dup/c get -3 2 roll -add/c exch put -}/exec load -] -cvx 0()/SubFileDecode filter -exch setglobal -//PDFR_DEBUG{ -(MakeStreamDumper end.)= -}if -}bind def -/ShortFilterNames 15 dict begin -/AHx/ASCIIHexDecode def -/A85/ASCII85Decode def -/LZW/LZWDecode def -/Fl/FlateDecode def -/RL/RunLengthDecode def -/CCF/CCITTFaxDecode def -/DCT/DCTDecode def -currentdict end readonly def -/AppendFilters -{ -//PDFR_DEBUG{ -(AppendFilters beg.)= -}if -dup 3 1 roll -/Filter//knownget exec{ -dup type/nametype eq{ -dup//ShortFilterNames exch//knownget exec{ -exch pop -}if -2 index/DecodeParms//knownget exec{ -exch -}if -filter -}{ -dup 0 exch 1 exch length 1 sub{ -2 copy get -dup//ShortFilterNames exch//knownget exec{ -exch pop -}if -3 1 roll -4 index/DecodeParms//knownget exec{ -exch get -}{ -pop null -}ifelse -dup null eq{ -pop 3 1 roll filter exch -}{ -3 1 roll -4 1 roll filter exch -}ifelse -}for -pop -}ifelse -//PDFR_DEBUG//PDFR_DUMP and{ -//MakeStreamDumper exec -}if -}if -exch pop -//PDFR_DEBUG{ -(AppendFilters end.)= -}if -}bind def -/ExecuteStream -{ -dup//PDFReader exch/CurrentObject exch put -dup/Length//CheckLength//ResolveD exec -//PDFR_DEBUG{ -(ExecuteStream id = )print 2 index =only( Length = )print dup = -}if -//PDFReader/InitialGraphicState get -//PDFReader/GraphicState get copy pop -//PDFReader/Operators get begin -currentfile exch()/SubFileDecode filter -1 index//AppendFilters exec -cvx mark exch -exec -counttomark 0 ne{ -mark(Data left on ostack after an immediate stream execution.)//error exec -}if -cleartomark -end -//PDFR_DEBUG{ -(ExecuteStream end.)= -}if -//PDFReader/CurrentObject null put -dup/IsPage known{ -dup/Context get/NumCopies//knownget exec{ -1 sub{ -copypage -}repeat -}if -EPS2Write not{showpage}if -pagesave restore -}if -}bind def -/stream -{ -//PDFR_DEBUG{ -1 index =only( stream)= -}if -1 index GetObject{ -dup xcheck{ -exec -1 index null PutObject -}{ -pop -}ifelse -}if -dup/ImmediateExec known{ -dup/GlobalExec//knownget exec{ -currentglobal 4 1 roll -setglobal -//ExecuteStream exec -3 2 roll setglobal -}{ -//ExecuteStream exec -}ifelse -}{ -//StoreStream exec -}ifelse -dup/.CleanResources//knownget exec{ -/All eq{ -//CleanAllResources exec -}if -}if -}bind def -/HookFont -{ -//PDFR_DEBUG{ -(Loaded the font )print dup/FontName get = -}if -{ -dup/FontFileType get dup/Type1 eq exch/MMType1 eq or{ -dup/FontName get -//PDFReader/RemoveFontNamePrefix get exec -findfont -exit -}if -dup/FontFileType get/TrueType eq{ -//PDFReader/MakeType42 get exec -//PDFR_DEBUG{ -(Font dict <<)= -dup{ -1 index/sfnts eq{ -exch pop -(/sfnts [)print -{ -(-string\()print length//=only exec(\)- )= -}forall -(])= -}{ -exch//=only exec( )print == -}ifelse -}forall -(>>)= -}if -dup/FontName get exch definefont -exit -}if -mark(FontHook has no proc for )2 index/FontFileType get//error exec -}loop -/Font exch put -}bind def -/endstream -{ -}bind def -/xref -{ -//PDFR_DEBUG{ -(xref)= -//PDFR_DUMP{ -//PDFReader/ObjectRegistry get == -}if -}if -end -count 0 ne{ -mark(Excessive data on estack at the end of the interpretation.)//error exec -}if -currentfile 1(%%EOF)/SubFileDecode filter -flushfile -cleardictstack -}bind def -/ResolveDict -{dup{ -pop 1 index exch -//DoNothing//ResolveD exec -pop -}forall -pop -}bind def -/SetupPageView -{ -//PDFR_DEBUG{ -(SetupPageView beg)= -}if -//DSC_OPDFREAD not{ -//GraphicState/InitialMatrix get setmatrix -}if -/MediaBox get aload pop -3 index neg 3 index neg translate -3 -1 roll sub 3 1 roll exch sub exch -userdict/.HWMargins//knownget exec{ -aload pop -}{ -currentpagedevice/.HWMargins//knownget exec{ -aload pop -}{ -0 0 0 0 -}ifelse -}ifelse -currentpagedevice/PageSize get aload pop -3 -1 roll sub 3 1 roll exch sub exch -exch 3 index sub exch 3 index sub -//SetPageSize{ -//PDFR_DEBUG{ -(Setting page size to )print 1 index//=only exec( )print dup = -}if -pop pop 3 index 3 index 2 copy -currentglobal false setglobal 3 1 roll -currentpagedevice dup/PageSize known{ -/PageSize get aload pop -}{ -0 0 -}ifelse -round cvi 2 index round cvi eq -exch round cvi 3 index round cvi eq and -{ -//PDFR_DEBUG{(PageSize matches request)== flush}if -pop pop -}{ -/MediaRequested where{ -//PDFR_DEBUG{(MediaRequested is true, check against new request)== flush}if -/MediaRequested get aload pop -round cvi 2 index round cvi eq -exch round cvi 3 index round cvi eq and -{ -//PDFR_DEBUG{(MediaRequested same as current request, ignore)== flush}if -pop pop false -}{ -//PDFR_DEBUG{(MediaRequested different to current request)== flush}if -true -}ifelse -}{ -//PDFR_DEBUG{(No MediaRequested yet)== flush}if -true -}ifelse -{ -//PDFR_DEBUG{(Setting pagesize)== flush}if -2 array astore -dup/MediaRequested exch def -<< exch/PageSize exch >>setpagedevice -}if -}ifelse -userdict/PDFR_InitialGS gstate put -setglobal -}if -//RotatePages{ -2 copy gt 6 index 6 index gt ne{ -1 index 5 index le 1 index 5 index le and not -}{ -false -}ifelse -}{ -false -}ifelse -{//CenterPages{ -//PDFR_DEBUG{ -(Rotating page, and then centering it)== -}if -90 rotate -0 5 index neg translate -5 index 1 index exch sub 2 div -2 index 6 index sub 2 div neg -translate -}{ -//FitPages{ -1 index 5 index div 1 index 7 index div -2 copy gt{ -exch -}if -pop dup scale -}if -90 rotate -0 5 index neg translate -}ifelse -}{ -//CenterPages{ -//PDFR_DEBUG{ -(Ccentering page)== -}if -1 index 6 index sub 2 div -1 index 6 index sub 2 div -translate -}{ -//FitPages{ -1 index 6 index div 1 index 6 index div -2 copy gt{ -exch -}if -pop dup scale -}if -}ifelse -}ifelse -pop pop -translate -pop pop -//PDFR_DEBUG{ -(SetupPageView end)= -}if -}bind def -/PageContentsDaemon -{ -//PDFR_DEBUG{ -(Executing PageContentsDaemon for )print 2 index = -}if -1 index exch/Context exch put -dup/ImmediateExec true put -/pagesave save def -dup/IsPage true put -SetPageSize{dup/Context get//SetupPageView exec}if -}bind def -/FontFileDaemon -{ -//PDFR_DEBUG{ -(Executing FontFileDaemon for )print 2 index = -}if -dup/FontFileType get -2 index exch -dup//ReadFontProcs exch//knownget exec{ -exch pop exec -}{ -mark(FontFile reader for )2 index( isn't implemented yet.)//error exec -}ifelse -//PDFR_DEBUG{ -(FontFileDaemon end)= -}if -pop -}bind def -/FontDescriptorDaemon -{ -//PDFR_DEBUG{ -(Executing FontDescriptorDaemon for )print 2 index = -}if -2 copy/FontResource exch put -/Subtype get 1 index exch/FontFileType exch put -}bind def -/UnPDFEscape{ -dup dup length string cvs -dup(#)search{ -{ -pop -(16#--)2 index 0 2 getinterval -1 index 3 2 getinterval copy pop -cvi -0 exch put -0 -1 index 2 1 index length 2 sub getinterval -3 copy putinterval -length -3 copy exch put -getinterval -(#)search not{ -pop exit -}if -}loop -(\0)search pop exch pop exch pop -cvn -exch pop -}{ -pop pop -}ifelse -}bind def -/TypeDaemons<< -/Page -{//PDFR_DEBUG{ -(Recognized a page.)= -}if -dup/Contents//knownget exec{ -0 get//DoNothing exch -[ -3 index//PageContentsDaemon/exec load -]cvx -//Register exec -}{ -(fixme: page with no Contents won't be printed.)= -}ifelse -}bind -/FontDescriptor -{//PDFR_DEBUG{ -(Recognized a font descriptor.)= -}if -dup/FontName//knownget exec{ -1 index/FontName 3 -1 roll//UnPDFEscape exec put -}if -dup dup/FontFile known{/FontFile}{/FontFile2}ifelse -//knownget exec{ -0 get//DoNothing exch -[ -3 index//FontFileDaemon/exec load -]cvx -//Register exec -}{ -(Font descriptor )print 1 index =only( has no FontFile.)= -}ifelse -}bind -/Font -{//PDFR_DEBUG{ -(Recognized a font resource.)= -}if -dup/BaseFont//knownget exec{ -//UnPDFEscape exec 2 copy/BaseFont exch put -//PDFReader/RemoveFontNamePrefix get exec -currentglobal exch -dup/Font resourcestatus{ -pop pop -//PDFReader/GetInstalledFont get exec pop -}{ -pop -}ifelse -setglobal -}if -dup/FontDescriptor//knownget exec{ -0 get -dup//IsRegistered exec{ -//PDFR_DEBUG{ -(already registered )print dup = -}if -pop -}{ -//DoNothing exch -[ -3 index//FontDescriptorDaemon/exec load -]cvx -//Register exec -}ifelse -}if -}bind ->>def -/MakeStreamReader -{dup -[ -exch -//PDFR_DEBUG{ -(Stream proc ) -/print load -//PDFR_STREAM{ -(<) -/print load -}if -}if -1 dict dup/i -1 put -/dup load -/i -/get load -1 -/add load -/dup load -3 -1 -/roll load -/i -/exch load -/put load -//knownget -/exec load -/not load -{()} -/if load -//PDFR_DEBUG{ -//PDFR_STREAM{ -/dup load -/print load -(>) -/print load -}if -( end of stream proc.\n) -/print load -}if -]cvx -//PDFR_DEBUG{ -(Stream reader )print dup == -}if -0()/SubFileDecode filter -exch//AppendFilters exec -}bind def -/RunDelayedStream -{ -//GraphicState/InitialTextMatrix get -//InitialTextMatrixStack//PDFReader/InitialTextMatrixStackPointer get -2 copy get null eq{ -2 copy currentglobal true setglobal matrix exch setglobal put -}if -get copy pop -//PDFReader/InitialTextMatrixStackPointer 2 copy get 1 add put -//MakeStreamReader exec -mark exch -cvx exec -counttomark 0 ne{ -mark(Data left on ostack after a delayed stream execution.)//error exec -}if -cleartomark -//PDFReader/InitialTextMatrixStackPointer 2 copy get 1 sub put -//InitialTextMatrixStack//PDFReader/InitialTextMatrixStackPointer get get -//GraphicState/InitialTextMatrix get -copy pop -}bind def -//ReadFontProcs begin -/Type1 -{//PDFR_DEBUG{ -(ReadFontProcs.Type1)= -}if -dup/.endobj_daemon[4 index//HookFont/exec load]cvx put -dup/ImmediateExec true put -/GlobalExec true put -}bind def -/MMType1//Type1 def -/TrueType -{//PDFR_DEBUG{ -(ReadFontProcs.TrueType)= -}if -dup/.endobj_daemon[4 index//HookFont/exec load]cvx put -pop -}bind def -end -/.opdloadttfontdict 50 dict def -.opdloadttfontdict begin -/maxstring 65400 def -end -/.InsertionSort -{ -/CompareProc exch def -/Array exch def -1 1 Array length 1 sub -{ -/Ix exch def -/Value1 Array Ix get def -/Jx Ix 1 sub def -{ -Jx 0 lt{ -exit -}if -/Value2 Array Jx get def -Value1 Value2 CompareProc{ -exit -}if -Array Jx 1 add Value2 put -/Jx Jx 1 sub def -}loop -Array Jx 1 add Value1 put -}for -Array -}bind def -/putu16{ -3 copy -8 bitshift put -exch 1 add exch 16#ff and put -}bind def -/putu32{ -3 copy -16 bitshift putu16 -exch 2 add exch 16#ffff and putu16 -}bind def -/.readtable{ -dup dup 1 and add string -dup 0 4 -1 roll getinterval -3 -1 roll exch -dup()ne{readstring}if pop pop -}bind def -/.readbigtable{ -dup maxstring lt{ -.readtable -}{ -currentuserparams/VMReclaim get -2 vmreclaim -[4 2 roll{ -dup maxstring le{exit}if -1 index maxstring string readstring pop 3 1 roll maxstring sub -}loop .readtable] -exch vmreclaim -}ifelse -}bind def -/ReadTTF -{ -.opdloadttfontdict begin -/TTFontFile exch def -/TableDir TTFontFile 12 string readstring pop def -/tables TTFontFile TableDir 4 getu16 16 mul string readstring pop def -/tabarray tables length 16 idiv array def -TableDir 0 4 getinterval(ttcf)eq{ -QUIET not{(Can't handle TrueType font Collections.)=}if -/.loadttfonttables cvx/invalidfont signalerror -}{ -0 16 tables length 1 sub{ -dup -tables exch 16 getinterval -exch 16 div cvi exch -tabarray 3 1 roll put -}for -}ifelse -tabarray{exch 8 getu32 exch 8 getu32 gt}.InsertionSort pop -/Read TableDir length tables length add def -/tabs[ -tabarray{ -dup 8 getu32 -Read sub -dup 0 gt{ -dup string TTFontFile exch readstring pop pop -Read add/Read exch def -}{ -pop -}ifelse -12 getu32 -dup Read add -/Read exch def -TTFontFile exch .readbigtable -}forall -]def -end -}bind def -/GetLocaType -{ -0 1 tabarray length 1 sub{ -dup tabarray exch get -0 4 getinterval(head)eq{ -tabs exch get -50 gets16 -/LocaType exch def -exit -}{ -pop -}ifelse -}for -}bind def -/GetNumGlyphs -{ -0 1 tabarray length 1 sub{ -dup tabarray exch get -0 4 getinterval(maxp)eq{ -tabs exch get -4 getu16 -/NumGlyphs exch def -exit -}{ -pop -}ifelse -}for -}bind def -/StringToLoca -{ -/LocaIndex exch def -/StringOffset 0 def -{ -dup length StringOffset gt{ -dup -LocaType 1 eq{ -StringOffset getu32 -LocaArray LocaIndex 3 -1 roll put -/LocaIndex LocaIndex 1 add def -/StringOffset StringOffset 4 add -def -}{ -StringOffset getu16 2 mul -LocaArray length LocaIndex gt{ -LocaArray LocaIndex 3 -1 roll put -}{ -pop -}ifelse -/LocaIndex LocaIndex 1 add def -/StringOffset StringOffset 2 add -def -}ifelse -}{ -pop -LocaIndex -exit -}ifelse -}loop -}bind def -/GetSortedLoca -{ -NumGlyphs 1 add array/LocaArray exch def -0 1 tabarray length 1 sub{ -dup tabarray exch get -0 4 getinterval(loca)eq{ -tabs exch get -exit -}{ -pop -}ifelse -}for -dup type/stringtype eq{ -0 StringToLoca pop -}{ -0 exch -{ -exch StringToLoca -}forall -pop -}ifelse -LocaArray{gt}.InsertionSort pop -}bind def -/GetWorkingString -{ -WorkString 0 -GlyfArray GlyfStringIndex get -putinterval -/WorkBytes GlyfArray GlyfStringIndex get length def -/GlyfStringIndex GlyfStringIndex 1 add def -}bind def -/GetWorkingBytes -{ -/BytesToRead exch def -WorkString 0 BytesToRead getinterval -dup length string copy -WorkString BytesToRead WorkBytes BytesToRead sub getinterval -dup length string copy -WorkString 0 3 -1 roll putinterval -/WorkBytes WorkBytes BytesToRead sub def -}bind def -/GetGlyfBytes -{ -/ToRead exch def -WorkBytes 0 eq{ -GetWorkingString -}if -WorkBytes ToRead ge{ -ToRead string dup 0 -ToRead GetWorkingBytes putinterval -}{ -ToRead string -dup -0 -WorkString 0 WorkBytes getinterval -putinterval -dup -WorkBytes -ToRead WorkBytes sub -GetWorkingString -GetWorkingBytes -putinterval -}ifelse -}bind def -/SplitGlyf -{ -/GlyfArray exch def -/DestArray GlyfArray length 2 mul array def -/DestArrayIndex 0 def -/LastLoca 0 def -/NextLocaIndex 0 def -/LastLocaIndex 0 def -/GlyfStringIndex 0 def -/WorkString maxstring string def -/WorkBytes 0 def -{ -LocaArray NextLocaIndex get -LastLoca sub maxstring gt -{ -LocaArray LastLocaIndex get LastLoca sub -GetGlyfBytes -DestArray DestArrayIndex 3 -1 roll put -/DestArrayIndex DestArrayIndex 1 add def -LocaArray LastLocaIndex get/LastLoca exch def -}{ -/LastLocaIndex NextLocaIndex def -/NextLocaIndex NextLocaIndex 1 add def -NextLocaIndex NumGlyphs gt -{ -WorkBytes -GlyfStringIndex GlyfArray length lt{ -GlyfArray GlyfStringIndex get length -add string dup -0 -WorkString 0 WorkBytes getinterval -putinterval -dup -WorkBytes -GetWorkingString -WorkString 0 WorkBytes getinterval -putinterval -}{ -pop -WorkString 0 WorkBytes getinterval -}ifelse -dup length string copy -DestArray DestArrayIndex 3 -1 roll put -exit -}if -}ifelse -}loop -DestArray -}bind def -/ProcessTTData -{ -.opdloadttfontdict begin -0 1 tabarray length 1 sub{ -/ix exch def -tabarray ix get -12 getu32 dup maxstring le{ -dup 4 mod 0 ne{ -4 div cvi 1 add 4 mul string/newstring exch def -/oldstring tabs ix get def -newstring 0 oldstring putinterval -0 1 newstring length oldstring length sub 1 sub{ -newstring exch oldstring length add 0 put -}for -tabs ix newstring put -}{ -pop -}ifelse -}{ -dup 4 mod 0 ne{ -dup maxstring idiv maxstring mul sub -4 idiv 1 add 4 mul string/newstring exch def -tabs ix get -dup length 1 sub dup/iy exch def get/oldstring exch def -newstring 0 oldstring putinterval -0 1 newstring length oldstring length sub 1 sub{ -newstring exch oldstring length add 0 put -}for -tabs ix get iy newstring put -}{ -pop -}ifelse -}ifelse -}for -0 1 tabarray length 1 sub{ -dup tabarray exch get -dup 12 getu32 maxstring gt{ -0 4 getinterval dup(glyf)eq{ -pop -GetLocaType -GetNumGlyphs -GetSortedLoca -dup tabs exch get -SplitGlyf -tabs 3 1 roll put -}{ -(Warning, table )print print( > 64Kb\n)print -pop -}ifelse -}{ -pop -pop -}ifelse -}for -end -}bind def -/Makesfnts -{ -.opdloadttfontdict begin -0 -tabs{ -dup type/stringtype eq{ -pop -1 add -}{ -{ -type/stringtype eq{ -1 add -}if -}forall -}ifelse -}forall -1 add -/TTOffset -TableDir length -tabarray length 16 mul add -def -0 -tabarray{ -exch dup 1 add -3 1 roll -dup -tabs exch get -dup type/stringtype eq{ -length -2 index exch -TTOffset -dup 3 1 roll add -/TTOffset exch def -8 exch putu32 -exch tabarray 3 1 roll -put -}{ -0 exch -{ -dup type/stringtype eq{ -length add -}{ -pop -}ifelse -}forall -2 index exch -TTOffset -dup 3 1 roll add -/TTOffset exch def -8 exch putu32 -exch tabarray 3 1 roll -put -}ifelse -}forall -pop -array -dup 0 -TableDir length -tables length add -string -dup 0 TableDir putinterval -dup 12 tables putinterval -put -dup -/ix 1 def -tabs{ -dup type/stringtype eq{ -ix exch -put dup -/ix ix 1 add def -}{ -{ -dup type/stringtype eq{ -ix exch put dup -/ix ix 1 add def -}{ -pop -}ifelse -}forall -}ifelse -}forall -pop -end -}bind def -/MakeType42 -{ -//PDFR_DEBUG{ -(MakeType42 beg)= -}if -10 dict begin -/FontName 1 index/FontName get def -/FontType 42 def -/FontMatrix[1 0 0 1 0 0]def -/FontBBox 1 index/FontBBox get def -dup/FontResource get -dup/Encoding known{ -//PDFReader/ObtainEncoding get exec -/Encoding get -}{ -pop null -}ifelse -/PDFEncoding exch def -/CharStrings 2 index//PDFReader/MakeTTCharStrings get exec def -/sfnts 2 index//MakeStreamReader exec -ReadTTF -ProcessTTData -Makesfnts -def -/Encoding StandardEncoding def -/PaintType 0 def -currentdict end -//PDFR_DEBUG{ -(MakeType42 end)= -}if -}bind def -/GetInstalledFont -{ -dup//InstalledFonts exch knownget{ -exch pop -}{ -dup findfont dup 3 1 roll -//InstalledFonts 3 1 roll put -}ifelse -}bind def -/RemoveFontNamePrefix -{//=string cvs true -0 1 5{ -2 index exch get//IsUpper exec not{ -pop false exit -}if -}for -{(+)search{ -pop pop -}if -}if -cvn -}bind def -/CheckFont -{dup/Type get/Font ne{ -mark(Resource )3 index( must have /Type/Font .)//error exec -}if -}bind def -/CheckEncoding -{dup type/nametype ne{ -dup/Type get/Encoding ne{ -mark(Resource )3 index( must have /Type/Encoding .)//error exec -}if -}if -}bind def -/ObtainEncoding -{dup/Encoding known{ -dup dup/Encoding//CheckEncoding//ResolveD exec -dup type dup/arraytype eq exch/packedarraytype eq or{ -pop pop -}{ -dup type/nametype eq{ -/Encoding findresource -}{ -dup/BaseEncoding//knownget exec not{ -/StandardEncoding -}if -/Encoding findresource -exch -/Differences//knownget exec{ -exch dup length array copy exch -0 exch -{ -dup type/integertype eq{ -exch pop -}{ -3 copy put pop -1 add -}ifelse -}forall -pop -}if -}ifelse -/Encoding exch put -}ifelse -}{ -dup/Encoding/StandardEncoding/Encoding findresource put -}ifelse -}bind def -/ObtainMetrics -{dup/Widths//knownget exec{ -1 index/Encoding get -256 dict -3 index/Subtype get/TrueType eq{ -1000 -}{ -1 -}ifelse -4 index/MissingWidth//knownget exec not{ -0 -}if -5 index/FirstChar//knownget exec not{ -0 -}if -6 5 roll -dup 0 exch 1 exch length 1 sub{ -2 copy get -exch 3 index add -7 index exch get -dup dup null ne exch/.notdef ne and{ -6 index 3 1 roll exch -6 index div -3 copy pop//knownget exec{ -0 eq -}{ -true -}ifelse -{put -}{ -pop pop pop -}ifelse -}{ -pop pop -}ifelse -}for -pop pop pop pop exch pop -1 index exch/Metrics exch put -}{ -dup/MissingWidth//knownget exec{ -256 dict -2 index/Encoding get{ -dup null ne{ -3 copy 3 2 roll put -}if -pop -}forall -exch pop -1 index exch/Metrics exch put -}if -}ifelse -}bind def -/NotDef -{ -FontMatrix aload pop pop pop exch pop exch pop -1 exch div exch -1 exch div exch -1 index 0 setcharwidth -0 setlinewidth -0 0 moveto -2 copy rlineto -1 index 0 rlineto -neg exch neg exch rlineto -closepath stroke -}bind def -/SaveResourcesToStack -{ -[ -//PDFReader/OldResources known{ -//PDFReader/OldResources get -}{ -null -}ifelse -//PDFReader/CurrentObject get/Context get/Resources get -] -//PDFReader/OldResources 3 -1 roll put -}bind def -/RestoreResourcesFromStack -{ -//PDFReader/OldResources get dup -0 get//PDFReader/OldResources 3 -1 roll put -1 get//PDFReader/CurrentObject get/Context get/Resources 3 -1 roll put -}bind def -/BuildChar -{//PDFR_DEBUG{ -(BuildChar )print dup//=only exec( )print -}if -exch begin -Encoding exch get -//PDFR_DEBUG{ -dup = -}if -dup null eq{ -pop//NotDef exec -} -{ -CharProcs exch//knownget exec -{ -currentfont/Font get/Resources//knownget exec{ -exec -SaveResourcesToStack -//PDFReader/CurrentObject get/Context get -/Resources 3 -1 roll put -//RunDelayedStream exec -RestoreResourcesFromStack -}{ -//RunDelayedStream exec -}ifelse -} -{ -//NotDef exec -}ifelse -}ifelse -end -}bind def -/printdict -{(<<)= -{exch = ==}forall -(>>)= -}bind def -/printfont -{ -dup{ -exch dup = -dup/Encoding eq{ -pop = -}{ -dup/FontInfo eq exch/Private eq or{ -//printdict exec -}{ -== -}ifelse -}ifelse -}forall -}bind def -/ScaleMetrics -{1 index{ -2 index div -3 index -3 1 roll put -}forall -pop -}bind def -/ResolveAndSetFontAux -{exch dup -//PDFReader/CurrentObject get/Context get/Resources get -/Font//DoNothing//ResolveD exec -exch//CheckFont//ResolveD exec -dup/Font//knownget exec{ -exch pop exch pop -}{ -{ -dup/Subtype get dup dup/Type1 eq exch/TrueType eq or exch/MMType1 eq or{ -exch pop -dup/BaseFont get -//RemoveFontNamePrefix exec -//PDFR_DEBUG{ -(Font )print dup = -}if -1 index/FontDescriptor known{ -//PDFR_DEBUG{ -(Font from a font descriptor.)= -}if -1 index -/FontDescriptor//DoNothing//ResolveD exec -/Font//knownget exec{ -exch pop -}{ -//PDFR_DEBUG{ -(Font descriptor has no Font resolved.)= -}if -//GetInstalledFont exec -}ifelse -}{ -//GetInstalledFont exec -}ifelse -exch -dup/Encoding known not{ -1 index/Encoding get 1 index exch/Encoding exch put -}if -//ObtainEncoding exec -//ObtainMetrics exec -exch -dup length dict copy -dup 2 index/Encoding get -/Encoding exch put -1 index/Metrics//knownget exec{ -2 index/Subtype get/TrueType ne{ -1 index/FontMatrix get 0 get -dup 0 eq{ -pop -1 index/FontMatrix get 1 get -dup 0 eq{pop 1}if -}if -0.001 div -//ScaleMetrics exec -}{ -1 index/sfnts known not{ -1 index/FontMatrix get 0 get -dup 0 eq{ -pop -1 index/FontMatrix get 1 get -dup 0 eq{pop 1}if -}if -//ScaleMetrics exec -}if -}ifelse -1 index exch/Metrics exch put -}if -1 index/BaseFont get -exch -dup/FID undef -dup/UniqueID undef -definefont -dup 3 1 roll -/Font exch put -exit -}if -dup/Subtype get/Type3 eq{ -//ObtainEncoding exec -2 copy exch/FontName exch put -dup/CharProcs get//ResolveDict exec -dup/FontType 3 put -dup/BuildChar//BuildChar put -dup dup/Font exch put -dup 3 1 roll -definefont -2 copy ne{ -2 copy/Font exch put -}if -exch pop -exit -}if -dup/Subtype get/Type0 eq{ -}if -dup/Subtype get/CIDFontType0 eq{ -}if -dup/Subtype get/CIDFontType2 eq{ -}if -mark(Unknown font type )2 index/Subtype get//error exec -}loop -}ifelse -exch scalefont setfont -}bind def -/ResolveAndSetFont -{ -//ResolveAndSetFontAux exec -}bind def -/.knownget -{2 copy known{ -get true -}{ -pop pop false -}ifelse -}bind def -/.min -{2 copy lt{ -exch -}if -pop -}bind def -/.max -{2 copy gt{ -exch -}if -pop -}bind def -/.dicttomark -{>> -}bind def -/getu16{ -2 copy get 8 bitshift 3 1 roll 1 add get add -}bind def -/gets16{ -getu16 16#8000 xor 16#8000 sub -}bind def -/getu32{ -2 copy getu16 16 bitshift 3 1 roll 2 add getu16 add -}bind def -/gets32{ -2 copy gets16 16 bitshift 3 1 roll 2 add getu16 add -}bind def -/cmapformats mark -0{ -6 256 getinterval{}forall 256 packedarray -}bind -2{ -/sHK_sz 2 def -/sH_sz 8 def -dup 2 getu16/cmapf2_tblen exch def -dup 4 getu16/cmapf2_lang exch def -dup 6 256 sHK_sz mul getinterval/sHKs exch def -0 -0 1 255{ -sHKs exch -2 mul getu16 -1 index -1 index -lt{exch}if pop -}for -/sH_len exch def -dup 6 256 sHK_sz mul add -cmapf2_tblen 1 index sub getinterval -/sH_gIA exch def -/cmapf2_glyph_array 65535 array def -/.cmapf2_putGID{ -/cmapf2_ch cmapf2_ch_hi 8 bitshift cmapf2_ch_lo add def -firstCode cmapf2_ch_lo le -cmapf2_ch_lo firstCode entryCount add lt -and{ -sH_offset idRangeOffset add -cmapf2_ch_lo firstCode sub 2 mul -add 6 add -sH_gIA exch getu16 -dup 0 gt{ -idDelta add -cmapf2_glyph_array exch cmapf2_ch exch put -}{ -pop -}ifelse -}{ -}ifelse -}def -16#00 1 16#ff{ -/cmapf2_ch_hi exch def -sHKs cmapf2_ch_hi sHK_sz mul getu16 -/sH_offset exch def -sH_gIA sH_offset sH_sz getinterval -dup 0 getu16/firstCode exch def -dup 2 getu16/entryCount exch def -dup 4 gets16/idDelta exch def -dup 6 getu16/idRangeOffset exch def -pop -sH_offset 0 eq{ -/cmapf2_ch_lo cmapf2_ch_hi def -/cmapf2_ch_hi 0 def -.cmapf2_putGID -}{ -16#00 1 16#ff{ -/cmapf2_ch_lo exch def -.cmapf2_putGID -}for -}ifelse -}for -pop -0 1 cmapf2_glyph_array length 1 sub{ -dup cmapf2_glyph_array exch get -null eq{cmapf2_glyph_array exch 0 put}{pop}ifelse -}for -cmapf2_glyph_array -}bind -4{ -/etab exch def -/nseg2 etab 6 getu16 def -14/endc etab 2 index nseg2 getinterval def -2 add -nseg2 add/startc etab 2 index nseg2 getinterval def -nseg2 add/iddelta etab 2 index nseg2 getinterval def -nseg2 add/idroff etab 2 index nseg2 getinterval def -pop -/firstcode startc 0 getu16 16#ff00 and dup 16#f000 ne{pop 0}if def -/lastcode firstcode def -/striptopbyte false def -/putglyph{ -glyphs code 3 -1 roll put/code code 1 add def -}bind def -/numcodes 0 def/glyphs 0 0 2 nseg2 3 sub{ -/i2 exch def -/scode startc i2 getu16 def -/ecode endc i2 getu16 def -ecode lastcode gt{ -/lastcode ecode def -}if -}for pop -firstcode 16#f000 ge lastcode firstcode sub 255 le and{ -lastcode 255 and -/striptopbyte true def -}{ -lastcode -}ifelse -1 add -array def -glyphs length 1024 ge{ -.array1024z 0 1024 glyphs length 1023 sub{glyphs exch 2 index putinterval}for -glyphs dup length 1024 sub 3 -1 roll -putinterval -}{ -0 1 glyphs length 1 sub{glyphs exch 0 put}for -}ifelse -/numcodes 0 def/code 0 def -0 2 nseg2 3 sub{ -/i2 exch def -/scode startc i2 getu16 def -/ecode endc i2 getu16 def -numcodes scode firstcode sub -exch sub 0 .max dup/code exch code exch add def -ecode scode sub 1 add add numcodes add/numcodes exch def -/delta iddelta i2 gets16 def -TTFDEBUG{ -(scode=)print scode =only -( ecode=)print ecode =only -( delta=)print delta =only -( droff=)print idroff i2 getu16 = -}if -idroff i2 getu16 dup 0 eq{ -pop scode delta add 65535 and 1 ecode delta add 65535 and -striptopbyte{ -/code scode 255 and def -}{ -/code scode def -}ifelse -{putglyph}for -}{ -/gloff exch 14 nseg2 3 mul add 2 add i2 add add def -striptopbyte{ -/code scode 255 and def -}{ -/code scode def -}ifelse -0 1 ecode scode sub{ -2 mul gloff add etab exch getu16 -dup 0 ne{delta add 65535 and}if putglyph -}for -}ifelse -}for glyphs/glyphs null def -}bind -6{ -dup 6 getu16/firstcode exch def dup 8 getu16/ng exch def -firstcode ng add array -0 1 firstcode 1 sub{2 copy 0 put pop}for -dup firstcode ng getinterval -0 1 ng 1 sub{ -dup 2 mul 10 add 4 index exch getu16 3 copy put pop pop -}for pop exch pop -}bind -.dicttomark readonly def -/cmaparray{ -dup 0 getu16 cmapformats exch .knownget{ -TTFDEBUG{ -(cmap: format )print 1 index 0 getu16 = flush -}if exec -}{ -(Can't handle format )print 0 getu16 = flush -0 1 255{}for 256 packedarray -}ifelse -TTFDEBUG{ -(cmap: length=)print dup length = dup == -}if -}bind def -/postremap mark -/Cdot/Cdotaccent -/Edot/Edotaccent -/Eoverdot/Edotaccent -/Gdot/Gdotaccent -/Ldot/Ldotaccent -/Zdot/Zdotaccent -/cdot/cdotaccent -/edot/edotaccent -/eoverdot/edotaccent -/gdot/gdotaccent -/ldot/ldotaccent -/zdot/zdotaccent -.dicttomark readonly def -/get_from_stringarray -{1 index type/stringtype eq{ -get -}{ -exch{ -2 copy length ge{ -length sub -}{ -exch get exit -}ifelse -}forall -}ifelse -}bind def -/getinterval_from_stringarray -{ -2 index type/stringtype eq{ -getinterval -}{ -string exch 0 -4 3 roll{ -dup length -dup 4 index lt{ -3 index exch sub -exch pop 3 1 roll exch pop -}{ -dup 3 1 roll -4 index sub -5 index length 4 index sub -2 copy gt{exch}if pop -dup 3 1 roll -5 index exch getinterval -5 index 4 index 3 index -getinterval -copy pop -exch pop add exch pop 0 exch -dup 3 index length ge{exit}if -}ifelse -}forall -pop pop -}ifelse -}bind def -/string_array_size -{dup type/stringtype eq{ -length -}{ -0 exch{length add}forall -}ifelse -}bind def -/postformats mark -16#00010000{ -pop MacGlyphEncoding -} -16#00020000{ -dup dup type/arraytype eq{0 get}if length 36 lt{ -TTFDEBUG{(post format 2.0 invalid.)= flush}if -pop[] -}{ -/postglyphs exch def -/post_first postglyphs dup type/arraytype eq{0 get}if def -post_first 32 getu16/numglyphs exch def -/glyphnames numglyphs 2 mul 34 add def -/postpos glyphnames def -/total_length postglyphs//string_array_size exec def -numglyphs array 0 1 numglyphs 1 sub{ -postpos total_length ge{ -1 numglyphs 1 sub{1 index exch/.notdef put}for -exit -}if -postglyphs postpos//get_from_stringarray exec -postglyphs postpos 1 add 2 index//getinterval_from_stringarray exec cvn -exch postpos add 1 add/postpos exch def -2 index 3 1 roll -put -}for -/postnames exch def -numglyphs array 0 1 numglyphs 1 sub{ -dup 2 mul 34 add postglyphs exch 2//getinterval_from_stringarray exec -dup 0 get 8 bitshift exch 1 get add dup 258 lt{ -MacGlyphEncoding exch get -}{ -dup 32768 ge{ -pop/.notdef -}{ -258 sub dup postnames length ge{ -TTFDEBUG{( *** warning: glyph index past end of 'post' table)= flush}if -pop -exit -}if -postnames exch get -postremap 1 index .knownget{exch pop}if -}ifelse -}ifelse -2 index 3 1 roll put -}for -} -ifelse -}bind -16#00030000{ -pop[] -}bind -.dicttomark readonly def -/first_post_string -{ -post dup type/arraytype eq{0 get}if -}bind def -/.getpost{ -/glyphencoding post null eq{ -TTFDEBUG{(post missing)= flush}if[] -}{ -postformats first_post_string 0 getu32 .knownget{ -TTFDEBUG{ -(post: format )print -first_post_string -dup 0 getu16 =only(,)print 2 getu16 = flush -}if -post exch exec -}{ -TTFDEBUG{(post: unknown format )print post 0 getu32 = flush}if[] -}ifelse -}ifelse def -}bind def -/MacRomanEncoding[ -StandardEncoding 0 39 getinterval aload pop -/quotesingle -StandardEncoding 40 56 getinterval aload pop -/grave -StandardEncoding 97 31 getinterval aload pop -/Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute -/agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave -/ecircumflex/edieresis/iacute/igrave -/icircumflex/idieresis/ntilde/oacute -/ograve/ocircumflex/odieresis/otilde -/uacute/ugrave/ucircumflex/udieresis -/dagger/degree/cent/sterling/section/bullet/paragraph/germandbls -/registered/copyright/trademark/acute/dieresis/.notdef/AE/Oslash -/.notdef/plusminus/.notdef/.notdef/yen/mu/.notdef/.notdef -/.notdef/.notdef/.notdef/ordfeminine/ordmasculine/.notdef/ae/oslash -/questiondown/exclamdown/logicalnot/.notdef -/florin/.notdef/.notdef/guillemotleft -/guillemotright/ellipsis/space/Agrave/Atilde/Otilde/OE/oe -/endash/emdash/quotedblleft/quotedblright -/quoteleft/quoteright/divide/.notdef -/ydieresis/Ydieresis/fraction/currency -/guilsinglleft/guilsinglright/fi/fl -/daggerdbl/periodcentered/quotesinglbase/quotedblbase -/perthousand/Acircumflex/Ecircumflex/Aacute -/Edieresis/Egrave/Iacute/Icircumflex -/Idieresis/Igrave/Oacute/Ocircumflex -/.notdef/Ograve/Uacute/Ucircumflex -/Ugrave/dotlessi/circumflex/tilde -/macron/breve/dotaccent/ring/cedilla/hungarumlaut/ogonek/caron -]/Encoding defineresource pop -/TTParser<< -/Pos 0 -/post null ->>def -/readu8 -{read not{ -mark(Insufficient data in the stream.)//error exec -}if -}bind def -/readu16 -{dup//readu8 exec 8 bitshift exch//readu8 exec or -}bind def -/reads16 -{//readu16 exec 16#8000 xor 16#8000 sub -}bind def -/readu32 -{dup//readu16 exec 16 bitshift exch//readu16 exec or -}bind def -/reads32 -{dup//reads16 exec 16 bitshift exch//readu16 exec or -}bind def -/SkipToPosition -{dup//TTParser/Pos get -exch//TTParser exch/Pos exch put -sub -//PDFR_DEBUG{ -(Skipping )print dup//=only exec( bytes.)= -}if -dup 0 eq{ -pop pop -}{ -dup 3 1 roll -()/SubFileDecode filter -exch -{1 index//BlockBuffer readstring pop length -dup 0 eq{pop exch pop exit}if -sub -}loop -0 ne{ -mark(Insufficient data in the stream for SkipToPosition.)//error exec -}if -}ifelse -}bind def -/TagBuffer 4 string def -/ParseTTTableDirectory -{//PDFR_DEBUG{ -(ParseTTTableDirectory beg)= -}if -15 dict begin -dup//readu32 exec 16#00010000 ne{ -mark(Unknown True Type version.)//error exec -}if -dup//readu16 exec/NumTables exch def -dup//readu16 exec/SearchRange exch def -dup//readu16 exec/EntrySelector exch def -dup//readu16 exec/RangeShift exch def -//PDFR_DEBUG{ -(NumTables = )print NumTables = -}if -NumTables{ -dup//TagBuffer readstring not{ -mark(Could not read TT tag.)//error exec -}if -cvn -[2 index//readu32 exec pop -2 index//readu32 exec -3 index//readu32 exec -] -//PDFR_DEBUG{ -2 copy exch//=only exec( )print == -}if -def -}repeat -pop -//TTParser/Pos 12 NumTables 16 mul add put -currentdict end -//PDFR_DEBUG{ -(ParseTTTableDirectory end)= -}if -}bind def -/ParseTTcmap -{//PDFR_DEBUG{ -(ParseTTcmap beg)= -}if -/cmap get aload pop -3 1 roll -7 dict begin -//PDFR_DEBUG{ -(Current position = )print//TTParser/Pos get = -(cmap position = )print dup = -}if -1 index exch//SkipToPosition exec -//TTParser/Pos get/TablePos exch def -dup//readu16 exec pop -dup//readu16 exec/NumEncodings exch def -//PDFR_DEBUG{ -(NumEncodings = )print NumEncodings = -}if -null -NumEncodings{ -1 index//readu32 exec -2 index//readu32 exec -3 array dup 3 2 roll 0 exch put -2 index null ne{ -dup 0 get 3 index 0 get sub -3 index exch 1 exch put -}if -dup 4 3 roll pop 3 1 roll -def -}repeat -dup 0 get -4 3 roll exch sub -1 exch put -//PDFR_DEBUG{ -currentdict{ -exch dup type/integertype eq{ -//PrintHex exec( )print == -}{ -pop pop -}ifelse -}forall -}if -4 NumEncodings 8 mul add/HeaderLength exch def -//TTParser/Pos//TTParser/Pos get HeaderLength add put -0 -NumEncodings{ -16#7FFFFFF null -currentdict{ -1 index type/integertype eq{ -exch pop dup 0 get -dup 5 index gt{ -dup 4 index lt{ -4 1 roll -exch pop exch pop -}{ -pop pop -}ifelse -}{ -pop pop -}ifelse -}{ -pop pop -}ifelse -}forall -//PDFR_DEBUG{ -(Obtaining subtable for )print dup == -}if -3 2 roll pop -3 copy pop -TablePos add//SkipToPosition exec -3 copy exch pop 1 get -//TTParser/Pos//TTParser/Pos get 3 index add put -string -readstring not{ -mark(Can't read a cmap subtable.)//error exec -}if -2 exch put -}repeat -pop pop -currentdict end -//PDFR_DEBUG{ -(ParseTTcmap end)= -}if -}bind def -/GetTTEncoding -{//PDFR_DEBUG{ -(GetTTEncoding beg)= -}if -get -exch pop -2 get -10 dict begin -/TTFDEBUG//PDFR_DEBUG def -//cmaparray exec -end -//PDFR_DEBUG{ -(GetTTEncoding end)= -dup == -}if -}bind def -/InverseEncoding -{ -256 dict begin -dup length 1 sub -1 0{ -2 copy get -exch -1 index currentdict exch//knownget exec{ -dup type/arraytype eq{ -aload length 1 add array astore -}{ -2 array astore -}ifelse -}if -def -}for -pop -currentdict end -}bind def -/GetMacRomanEncodingInverse -{//PDFReader/MacRomanEncodingInverse get -dup null eq{ -pop -MacRomanEncoding//InverseEncoding exec -dup//PDFReader exch/MacRomanEncodingInverse exch put -}if -}bind def -/PutCharStringSingle -{ -dup 3 index length lt{ -2 index exch get -dup 0 ne{ -def -}{ -pop pop -}ifelse -}{ -pop pop -}ifelse -}bind def -/PutCharString -{1 index type/nametype ne{ -mark(Bad charstring name)//error exec -}if -dup type/arraytype eq{ -{ -3 copy//PutCharStringSingle exec -pop pop -}forall -pop -}{ -//PutCharStringSingle exec -}ifelse -}bind def -/ComposeCharStrings -{ -//PDFR_DEBUG{ -(ComposeCharStrings beg)= -}if -1 index length 1 add dict begin -/.notdef 0 def -exch -//TTParser/post get -dup null ne{ -exch -1 index length 1 sub -1 0{ -dup 3 index exch get exch -dup 0 eq 2 index/.notdef eq or{ -pop pop -}{ -def -}ifelse -}for -}if -exch pop exch -{ -//PutCharString exec -}forall -pop -currentdict end -//PDFR_DEBUG{ -(ComposeCharStrings end)= -}if -}bind def -/ParseTTpost -{ -//PDFR_DEBUG{ -(ParseTTpost beg)= -}if -/post get aload pop -3 1 roll -//PDFR_DEBUG{ -(Current position = )print//TTParser/Pos get = -(post position = )print dup = -}if -1 index exch//SkipToPosition exec -//TTParser/Pos//TTParser/Pos get 4 index add put -exch dup 65535 le{ -string -readstring not{ -mark(Insufficient data in the stream for ParseTTpost.)//error exec -}if -}{ -[3 1 roll -dup 16384 div floor cvi -exch 1 index 16384 mul -sub exch -1 sub 0 1 3 -1 roll -{ -1 add index -16384 string readstring not{ -mark(Insufficient data in the stream for ParseTTpost.)//error exec -}if -}for -counttomark -2 roll -string readstring not{ -mark(Insufficient data in the stream for ParseTTpost.)//error exec -}if -] -}ifelse -1 dict begin -/post exch def -//.getpost exec -//TTParser/post glyphencoding put -//PDFR_DEBUG{ -(ParseTTpost end)= -glyphencoding == -}if -end -}bind def -/MakeTTCharStrings -{//MakeStreamReader exec -dup dup//ParseTTTableDirectory exec -//TTParser/post null put -dup/post//knownget exec{ -0 get -1 index/cmap get 0 get -lt{ -2 copy//ParseTTpost exec -//ParseTTcmap exec -}{ -2 copy//ParseTTcmap exec -3 1 roll -//ParseTTpost exec -}ifelse -}{ -//ParseTTcmap exec -}ifelse -{ -dup 16#00030001 known{ -//PDFR_DEBUG{ -(Using the TT cmap encoding for Windows Unicode.)= -}if -16#00030001//GetTTEncoding exec -AdobeGlyphList//ComposeCharStrings exec -exit -}if -dup 16#00010000 known{ -//PDFR_DEBUG{ -(Using the TT cmap encoding for Macintosh Roman.)= -}if -16#00010000//GetTTEncoding exec -PDFEncoding dup null eq{ -pop//GetMacRomanEncodingInverse exec -}{ -//InverseEncoding exec -}ifelse -//ComposeCharStrings exec -exit -}if -dup 16#00030000 known{ -//PDFR_DEBUG{ -(Using the TT cmap encoding 3.0 - not sure why Ghostscript writes it since old versions.)= -}if -16#00030000//GetTTEncoding exec -PDFEncoding dup null eq{ -pop//GetMacRomanEncodingInverse exec -}{ -//InverseEncoding exec -}ifelse -//ComposeCharStrings exec -exit -}if -mark(True Type cmap has no useful encodings.)//error exec -}loop -//PDFR_DEBUG{ -(CharStrings <<)= -dup{ -exch -dup type/nametype eq{ -//=only exec -}{ -== -}ifelse -( )print == -}forall -(>>)= -}if -}bind def -/ScaleVal -{ -aload pop -1 index sub -3 2 roll mul add -}bind def -/ScaleArg -{ -aload pop -1 index sub -3 1 roll -sub exch div -}bind def -/ScaleArgN -{ -dup length 2 sub -2 0{ -2 -2 index 3 1 roll getinterval -3 2 roll -exch//ScaleArg exec -1 index length 2 idiv 1 add 1 roll -}for -pop -}bind def -/ComputeFunction_10 -{ -//PDFR_DEBUG{ -(ComputeFunction_10 beg )print 1 index//=only exec( stack=)print count = -}if -exch -dup 1 eq{ -pop dup length 1 sub get -}{ -1 index length 1 sub mul -dup dup floor sub -dup 0 eq{ -pop cvi get -}{ -3 1 roll floor cvi -2 getinterval -aload pop -2 index mul 3 2 roll 1 exch sub 3 2 roll mul add -}ifelse -}ifelse -//PDFR_DEBUG{ -(ComputeFunction_10 end )print dup//=only exec( stack=)print count = -}if -}bind def -/ComputeFunction_n0 -{ -//PDFR_DEBUG{ -(ComputeFunction_n0 beg N=)print dup//=only exec( stack=)print count = -}if -dup 0 eq{ -pop -}{ -dup 2 add -1 roll -dup 3 index length 1 sub ge{ -pop 1 sub -exch dup length 1 sub get exch -//PDFReader/ComputeFunction_n0 get exec -}{ -dup floor cvi dup -4 index exch get -3 index dup -5 add copy -6 2 roll -pop pop pop pop -1 sub -//PDFReader/ComputeFunction_n0 get exec -3 2 roll pop -exch -4 3 roll exch -4 add 2 roll 1 add -3 2 roll exch get -exch 1 sub -//PDFReader/ComputeFunction_n0 get exec -1 index mul -3 1 roll -1 exch sub mul add -}ifelse -}ifelse -//PDFR_DEBUG{ -(ComputeFunction_n0 end )print dup//=only exec( stack=)print count = -}if -}bind def -/FunctionToProc_x01 -{ -dup/Domain get exch -dup/Data get 0 get exch -/Size get length -[4 1 roll -//PDFR_DEBUG{ -{(function beg, stack =)print count//=only exec(\n)print}/exec load -5 2 roll -}if -dup 1 gt{ -{mark exch -3 add 2 roll -//ScaleArgN exec -counttomark dup -3 add -2 roll -pop exch -//ComputeFunction_n0 exec -}/exec load -}{ -pop -3 1/roll load//ScaleArg/exec load -/exch load -//ComputeFunction_10/exec load -}ifelse -//PDFR_DEBUG{ -(function end, stack =)/print load/count load//=only/exec load(\n)/print load -}if -]cvx -//PDFR_DEBUG{ -(Made a procedure for the 1-result function :)= -dup == -}if -}bind def -/FunctionProcDebugBeg -{(FunctionProcDebugBeg )print count = -}bind def -/FunctionProcDebugEnd -{(FunctionProcDebugEnd )print count = -}bind def -/FunctionToProc_x0n -{ -PDFR_DEBUG{ -(FunctionToProc_x0n beg m=)print dup = -}if -1 index/Size get length exch -dup 7 mul 2 add array -PDFR_DEBUG{ -dup 0//FunctionProcDebugBeg put -}{ -dup 0//DoNothing put -}ifelse -dup 1/exec load put -dup 2 5 index/Domain get put -2 index 1 eq{ -dup 3//ScaleArg put -}{ -dup 3//ScaleArgN put -}ifelse -dup 4/exec load put -1 index 1 sub 0 exch 1 exch{ -dup 7 mul 5 add -1 index 4 index 1 sub ne{ -dup 3 index exch 6 index put 1 add -dup 3 index exch/copy load put 1 add -}if -[ -6 index/Data get 3 index get -6 index 1 eq{ -//ComputeFunction_10/exec load -}{ -6 index -//ComputeFunction_n0/exec load -}ifelse -]cvx -3 index exch 2 index exch put 1 add -2 index 1 index/exec load put 1 add -1 index 4 index 1 sub ne{ -2 index 1 index 6 index 1 add put 1 add -2 index 1 index 1 put 1 add -2 index 1 index/roll load put -}if -pop pop -}for -PDFR_DEBUG{ -dup dup length 2 sub//FunctionProcDebugEnd put -}{ -dup dup length 2 sub//DoNothing put -}ifelse -dup dup length 1 sub/exec load put -cvx exch pop exch pop exch pop -//PDFR_DEBUG{ -(Made a procedure for the n-argument function :)= -dup == -}if -PDFR_DEBUG{ -(FunctionToProc_x0n end)= -}if -}bind def -/MakeTableRec -{ -0 -exec -}bind def -/MakeTable -{//PDFR_DEBUG{ -(MakeTable beg )print count = -}if -1 index/Size get exch -1 sub dup -3 1 roll -get -array -1 index 0 eq{ -exch pop exch pop -}{ -dup length 1 sub -1 0{ -3 index 3 index//MakeTableRec exec -2 index 3 1 roll put -}for -exch pop exch pop -}ifelse -//PDFR_DEBUG{ -(MakeTable end )print count = -}if -}bind def -//MakeTableRec 0//MakeTable put -/StoreSample -{ -1 sub -dup 0 eq{ -pop -}{ --1 1{ -I exch get get -}for -}ifelse -I 0 get 3 2 roll put -}bind def -/ReadSample32 -{ -4{ -File read not{ -mark(Insufficient data for function.)//error exec -}if -}repeat -pop -3 1 roll exch -256 mul add 256 mul add -//1_24_bitshift_1_sub div -}bind def -/ReadSample -{ -Buffer BitsLeft BitsPerSample -{2 copy ge{ -exit -}if -3 1 roll -8 add 3 1 roll -256 mul File read not{ -mark(Insufficient data for function.)//error exec -}if -add -3 1 roll -}loop -sub dup -2 index exch -neg bitshift -2 copy exch bitshift -4 3 roll exch sub -/Buffer exch def -exch/BitsLeft exch def -Div div -}bind def -/ReadSamplesRec -{0 -exec -}bind def -/ReadSamples -{ -//PDFR_DEBUG{ -(ReadSamples beg )print count = -}if -dup 1 eq{ -pop -0 1 Size 0 get 1 sub{ -I exch 0 exch put -0 1 M 1 sub{ -dup Range exch 2 mul 2 getinterval -//PDFR_DEBUG{ -(Will read a sample ... )print -}if -BitsPerSample 32 eq{//ReadSample32}{//ReadSample}ifelse -exec exch//ScaleVal exec -//PDFR_DEBUG{ -(value=)print dup = -}if -exch Table exch get -Size length//StoreSample exec -}for -}for -}{ -1 sub -dup Size exch get 0 exch 1 exch 1 sub{ -I exch 2 index exch put -dup//ReadSamplesRec exec -}for -pop -}ifelse -//PDFR_DEBUG{ -(ReadSamples end )print count = -}if -}bind def -//ReadSamplesRec 0//ReadSamples put -/StreamToArray -{//PDFR_DEBUG{ -(StreamToArray beg )print count = -}if -userdict/FuncDataReader get begin -dup/BitsPerSample get/BitsPerSample exch def -dup/Size get length/N exch def -dup/Range get length 2 idiv/M exch def -1 BitsPerSample bitshift 1 sub/Div exch def -/BitsLeft 0 def -/Buffer 0 def -dup/Size get/Size exch def -dup/Range get/Range exch def -/File 1 index//MakeStreamReader exec def -/I[N{0}repeat]def -M array -dup length 1 sub -1 0{ -2 index N//MakeTable exec -2 index 3 1 roll put -}for -/Table exch def -N//ReadSamples exec -PDFR_DEBUG{ -(Table = )print Table == -}if -/Data Table put -end -//PDFR_DEBUG{ -(StreamToArray end )print count = -}if -}bind def -/FunctionToProc10 -{ -PDFR_DEBUG{ -(FunctionToProc10 beg, Range = )print dup/Range get == -}if -dup/Order//knownget exec{ -1 ne{ -(Underimplemented function Type 0 Order 3.)= -}if -}if -dup//StreamToArray exec -dup/Range get length dup 2 eq{ -pop//FunctionToProc_x01 exec -}{ -2 idiv//FunctionToProc_x0n exec -}ifelse -PDFR_DEBUG{ -(FunctionToProc10 end)= -}if -}bind def -/FunctionToProc12 -{begin -currentdict/C0//knownget exec{length 1 eq}{true}ifelse{ -N -currentdict/C0//knownget exec{ -0 get -}{ -0 -}ifelse -currentdict/C1//knownget exec{ -0 get -}{ -1 -}ifelse -1 index sub -[4 1 roll -{ -4 2 roll -exp mul add -}aload pop -]cvx -}{ -[ -0 1 C0 length 1 sub{ -N -C0 2 index get -C1 3 index get -4 3 roll pop -1 index sub -[/dup load -5 2 roll -{ -4 2 roll -exp mul add -exch -}aload pop -]cvx -/exec load -}for -/pop load -]cvx -}ifelse -end -//PDFR_DEBUG{ -(FunctionType2Proc : )print dup == -}if -}bind def -/FunctionToProc14 -{//MakeStreamReader exec cvx exec -//PDFR_DEBUG{ -(FunctionType4Proc : )print dup == -}if -}bind def -/FunctionToProc1 -{ -dup/FunctionType get -{dup 0 eq{ -pop//FunctionToProc10 exec exit -}if -dup 2 eq{ -pop//FunctionToProc12 exec exit -}if -dup 4 eq{ -pop//FunctionToProc14 exec exit -}if -mark exch(Function type )exch( isn't implemented yet.)//error exec -}loop -}bind def -/FunctionToProc20 -{ -PDFR_DEBUG{ -(FunctionToProc20, Range = )print dup/Range get == -}if -dup/Order//knownget exec{ -1 ne{ -(Underimplemented function Type 0 Order 3.)= -}if -}if -dup//StreamToArray exec -dup/Range get length dup 2 eq{ -pop//FunctionToProc_x01 exec -}{ -2 idiv//FunctionToProc_x0n exec -}ifelse -}bind def -/FunctionToProc -{//PDFR_DEBUG{ -(FunctionToProc beg )print count = -}if -dup type/dicttype eq{ -dup/Domain get length 2 idiv -{ -dup 1 eq{ -pop//FunctionToProc1 exec exit -}if -dup 2 eq{ -pop//FunctionToProc20 exec exit -}if -mark(Functions with many arguments aren't implemented yet.)//error exec -}loop -}{ -//PDFR_DEBUG{(Not a function dict, assume already a procedure.)print}if -}ifelse -//PDFR_DEBUG{ -(FunctionToProc end )print count = -}if -}bind def -/spotfunctions mark -/Round{ -abs exch abs 2 copy add 1 le{ -dup mul exch dup mul add 1 exch sub -}{ -1 sub dup mul exch 1 sub dup mul add 1 sub -}ifelse -} -/Diamond{ -abs exch abs 2 copy add .75 le{ -dup mul exch dup mul add 1 exch sub -}{ -2 copy add 1.23 le{ -.85 mul add 1 exch sub -}{ -1 sub dup mul exch 1 sub dup mul add 1 sub -}ifelse -}ifelse -} -/Ellipse{ -abs exch abs 2 copy 3 mul exch 4 mul add 3 sub dup 0 lt{ -pop dup mul exch .75 div dup mul add 4 div 1 exch sub -}{ -dup 1 gt{ -pop 1 exch sub dup mul exch 1 exch sub -.75 div dup mul add 4 div 1 sub -}{ -.5 exch sub exch pop exch pop -}ifelse -}ifelse -} -/EllipseA{dup mul .9 mul exch dup mul add 1 exch sub} -/InvertedEllipseA{dup mul .9 mul exch dup mul add 1 sub} -/EllipseB{dup 5 mul 8 div mul exch dup mul exch add sqrt 1 exch sub} -/EllipseC{dup mul .9 mul exch dup mul add 1 exch sub} -/InvertedEllipseC{dup mul .9 mul exch dup mul add 1 sub} -/Line{exch pop abs neg} -/LineX{pop} -/LineY{exch pop} -/Square{abs exch abs 2 copy lt{exch}if pop neg} -/Cross{abs exch abs 2 copy gt{exch}if pop neg} -/Rhomboid{abs exch abs 0.9 mul add 2 div} -/DoubleDot{2{360 mul sin 2 div exch}repeat add} -/InvertedDoubleDot{2{360 mul sin 2 div exch}repeat add neg} -/SimpleDot{dup mul exch dup mul add 1 exch sub} -/InvertedSimpleDot{dup mul exch dup mul add 1 sub} -/CosineDot{180 mul cos exch 180 mul cos add 2 div} -/Double{exch 2 div exch 2{360 mul sin 2 div exch}repeat add} -/InvertedDouble{ -exch 2 div exch 2{360 mul sin 2 div exch}repeat add neg -} -.dicttomark readonly def -/CheckColorSpace -{ -dup type/arraytype ne{ -mark(Resource )3 index( must be an array.)//error exec -}if -}bind def -/SubstitutePDFColorSpaceRec -{0 -exec -}bind def -/SubstitutePDFColorSpace -{ -{ -dup 0 get/Pattern eq{ -dup length 1 gt{ -dup dup 1//CheckColorSpace//ResolveA exec -dup type/nametype ne{ -//SubstitutePDFColorSpaceRec exec -}if -1 exch put -}if -exit -}if -dup 0 get/Indexed eq{ -exit -}if -dup 0 get/Separation eq{ -dup dup 2//CheckColorSpace//ResolveA exec -dup type/nametype ne{ -//SubstitutePDFColorSpaceRec exec -}if -2 exch put -exit -}if -dup 0 get/CalGray eq{ -1 get -dup/Gamma//knownget exec{ -[exch[exch/exp load]cvx dup dup] -1 index exch/DecodeLMN exch put -}if -[exch/CIEBasedA exch] -exit -}if -dup 0 get/CalRGB eq{ -1 get -dup/Matrix//knownget exec{ -1 index exch/MatrixLMN exch put -}if -dup/Gamma//knownget exec{ -aload pop -[exch/exp load]cvx -3 1 roll -[exch/exp load]cvx -3 1 roll -[exch/exp load]cvx -3 1 roll -3 array astore -1 index exch/DecodeLMN exch put -}if -[exch/CIEBasedABC exch] -exit -}if -dup 0 get/Lab eq{ -1 get -begin -currentdict/Range//knownget exec{aload pop}{-100 100 -100 100}ifelse -0 100 6 2 roll 6 array astore -/RangeABC exch def -/DecodeABC[{16 add 116 div}bind{500 div}bind{200 div}bind]def -/MatrixABC[1 1 1 1 0 0 0 0 -1]def -{dup 6 29 div ge{dup dup mul mul}{4 29 div sub 108 841 div mul}ifelse} -/DecodeLMN[ -[3 index aload pop WhitePoint 0 get/mul load]cvx -[4 index aload pop WhitePoint 1 get/mul load]cvx -[5 index aload pop WhitePoint 2 get/mul load]cvx -]def pop -//PDFR_DEBUG{ -(Constructed from Lab <<)= -currentdict{exch = ==}forall -(>>)= -}if -[/CIEBasedABC currentdict] -end -exit -pop -}if -dup 0 get/CIEBasedA eq{exit}if -dup 0 get/CIEBasedABC eq{exit}if -mark exch(Unimplemented color space )exch//error exec -}loop -}bind def -//SubstitutePDFColorSpaceRec 0//SubstitutePDFColorSpace put -/ResolveArrayElement -{2 copy get -dup type dup/arraytype eq exch -/packedarraytype eq or{ -dup length 1 ge exch xcheck and{ -2 copy get -dup 0 get type/integertype eq -1 index 1 get type dup/arraytype -eq exch -/packedarraytype eq or -and{ -exec -2 index 4 1 roll put -}{ -pop pop -}ifelse -}{ -pop -}ifelse -}{ -pop pop -}ifelse -}bind def -/ResolveColorSpaceArrayRec -{0 -exec -}bind def -/SetColorSpaceSafe -{ -PDFR_DEBUG{ -(SetColorSpaceSafe beg)= -}if -currentcolorspace dup type/arraytype eq{ -1 index type/arraytype eq{ -dup length 2 index length eq{ -false exch -dup length 0 exch 1 exch 1 sub{ -dup -4 index exch get exch -2 index exch get -ne{ -exch pop true exch exit -}if -}for -pop -{ -setcolorspace -}{ -pop -}ifelse -}{ -pop setcolorspace -}ifelse -}{ -pop setcolorspace -}ifelse -}{ -pop setcolorspace -}ifelse -PDFR_DEBUG{ -(SetColorSpaceSafe end)= -}if -}bind def -/ResolveColorSpaceArray -{ -//PDFR_DEBUG{ -(ResolveColorSpaceArray beg )print dup == -}if -dup 0 get/Indexed eq{ -1//ResolveArrayElement exec -dup dup 1 get -dup type/arraytype eq{ -//SubstitutePDFColorSpace exec -//ResolveColorSpaceArrayRec exec -1 exch put -}{ -pop pop -}ifelse -}if -dup 0 get/Separation eq{ -dup dup 1 get UnPDFEscape 1 exch put -3//ResolveArrayElement exec -dup 3 get//FunctionToProc exec -2 copy 3 exch put -pop -}if -dup 0 get/Pattern eq{ -dup length 1 gt{ -dup 1 get dup type/arraytype eq{ -ResolveColorSpaceArray -1 index 1 3 -1 roll put -}{ -pop -}ifelse -}if -}if -PDFR_DEBUG{ -(Construcrted color space :)= -dup == -}if -//PDFR_DEBUG{ -(ResolveColorSpaceArray end )print dup == -}if -}bind def -//ResolveColorSpaceArrayRec 0//ResolveColorSpaceArray put -/ResolveColorSpace -{ -//PDFR_DEBUG{ -(ResolveColorSpace beg )print dup = -}if -dup//SimpleColorSpaceNames exch known not{ -dup//PDFColorSpaces exch//knownget exec{ -exch pop -//PDFR_DEBUG{ -(ResolveColorSpace known )= -}if -}{ -dup -//PDFReader/CurrentObject get/Context get/Resources get -/ColorSpace//DoNothing//ResolveD exec -exch//CheckColorSpace//ResolveD exec -dup type/arraytype eq{ -//SubstitutePDFColorSpace exec -//ResolveColorSpaceArray exec -dup//PDFColorSpaces 4 2 roll put -}if -}ifelse -}if -//PDFR_DEBUG{ -(ResolveColorSpace end )print dup == -}if -}bind def -/CheckPattern -{ -dup/PatternType//knownget exec{ -dup 1 ne{ -mark(Resource )4 index( is a shading, which can't be handled at level 2. )//error exec -}if -pop -}if -dup/Type knownget{ -/Pattern ne{ -mark(Resource )4 index( must have /Type/Pattern .)//error exec -}if -}if -}bind def -/PaintProc -{/Context get -//RunDelayedStream exec -}bind def -/ResolvePattern -{ -dup -userdict/PDFR_Patterns get -exch//knownget exec{ -exch pop -}{ -dup -//PDFReader/CurrentObject get/Context get/Resources get -/Pattern//DoNothing//ResolveD exec -exch//CheckPattern//ResolveD exec -dup dup/Context exch put -dup/Resources//DoNothing//ResolveD exec pop -dup/PaintProc//PaintProc put -gsave userdict/PDFR_InitialGS get setgstate -currentglobal exch false setglobal -dup/Matrix get -makepattern -exch setglobal -grestore -dup userdict/PDFR_Patterns get -4 2 roll -put -}ifelse -}bind def -/SetColor -{//PDFR_DEBUG{ -(SetColor beg)= -}if -currentcolorspace dup type/nametype eq{ -pop setcolor -}{ -0 get/Pattern eq{ -//ResolvePattern exec setpattern -}{ -setcolor -}ifelse -}ifelse -//PDFR_DEBUG{ -(SetColor end)= -}if -}bind def -/ImageKeys 15 dict begin -/BPC/BitsPerComponent def -/CS/ColorSpace def -/D/Decode def -/DP/DecodeParms def -/F/Filter def -/H/Height def -/IM/ImageMask def -/I/Interpolate def -/W/Width def -currentdict end readonly def -/ImageValues 15 dict begin -/G/DeviceGray def -/RGB/DeviceRGB def -/CMYK/DeviceCMYK def -/I/Indexed def -/AHx/ASCIIHexDecode def -/A85/ASCII85Decode def -/LZW/LZWDecode def -/Fl/FlateDecode def -/RL/RunLengthDecode def -/CCF/CCITTFaxDecode def -/DCT/DCTDecode def -currentdict end readonly def -/GetColorSpaceRange -{2 index/ColorSpace get -dup type/arraytype eq{ -1 get -}if -exch//knownget exec{ -exch pop -}if -}bind def -/DecodeArrays 15 dict begin -/DeviceGray{[0 1]}def -/DeviceRGB{[0 1 0 1 0 1]}def -/DeviceCMYK{[0 1 0 1 0 1 0 1]}def -/Indexed{ -dup/BitsPerComponent get 1 exch bitshift 1 sub[exch 0 exch] -}def -/Separation{[0 1]}def -/CIEBasedA{[0 1]/RangeA//GetColorSpaceRange exec}def -/CIEBasedABC{[0 1 0 1 0 1]/RangeABC//GetColorSpaceRange exec}def -currentdict end readonly def -/Substitute -{1 index//knownget exec{ -exch pop -}if -}bind def -/DebugImagePrinting -{ -//PDFR_DEBUG{ -(Image :)= -dup{exch//=only exec( )print == -}forall -}if -}bind def -/CompleteImage -{ -dup/ColorSpace known{ -dup/ColorSpace//CheckColorSpace//ResolveD exec pop -}if -dup/Decode known not{ -dup/ColorSpace//knownget exec{ -dup type/arraytype eq{ -0 get -}if -//DecodeArrays exch get exec -}{ -[0 1] -}ifelse -1 index exch/Decode exch put -}if -dup/ImageMatrix[2 index/Width get 0 0 5 index/Height get neg -0 7 index/Height get]put -//DebugImagePrinting exec -}bind def -/CompleteInlineImage -{ -//PDFR_DEBUG{ -(CompleteInlineImage beg)= -}if -dup/ImageType known not{ -dup/ImageType 1 put -}if -dup length dict exch{ -exch//ImageKeys//Substitute exec -dup/Filter eq{ -exch//ImageValues//Substitute exec exch -}if -dup/ColorSpace eq{ -exch -dup//ImageValues exch//knownget exec{ -exch pop -}{ -//ResolveColorSpace exec -}ifelse -exch -}if -exch -2 index 3 1 roll put -}forall -//CompleteImage exec -dup/DataSource 2 copy get -2 index//AppendFilters exec put -//PDFR_DEBUG{ -(CompleteInlineImage end)= -}if -}bind def -/CompleteOutlineImage -{ -currentglobal exch dup gcheck setglobal -//PDFR_DEBUG{ -(CompleteOutlineImage beg)= -}if -dup dup//MakeStreamReader exec/DataSource exch put -dup/ImageType known not{ -//CompleteImage exec -dup/ImageType 1 put -dup/ColorSpace known{ -dup/ColorSpace//CheckColorSpace//ResolveD exec -dup type/arraytype eq{ -//ResolveColorSpaceArray exec -//SubstitutePDFColorSpace exec -1 index exch/ColorSpace exch put -}{ -pop -}ifelse -}if -}if -//PDFR_DEBUG{ -(CompleteOutlineImage end)= -}if -exch setglobal -}bind def -/DoImage -{ -//PDFR_DEBUG{ -(DoImage beg)= -}if -gsave -dup/ColorSpace//knownget exec{setcolorspace}if -dup/ImageMask//knownget exec not{false}if -{imagemask}{image}ifelse -grestore -//PDFR_DEBUG{ -(DoImage end)= -}if -}bind def -/GSave -{ -gsave -//PDFReader/GraphicStateStackPointer get -dup//GraphicStateStack exch get null eq{ -dup//GraphicStateStack exch//InitialGraphicState length dict put -}if -dup//GraphicStateStack exch get -//GraphicState exch copy pop -1 add//PDFReader exch/GraphicStateStackPointer exch put -}bind def -/GRestore -{ -grestore -//PDFReader/GraphicStateStackPointer get -1 sub dup -//PDFReader exch/GraphicStateStackPointer exch put -//GraphicStateStack exch get -//GraphicState copy pop -}bind def -/SetFont -{dup//GraphicState exch/FontSize exch put -//ResolveAndSetFont exec -//GraphicState/FontMatrixNonHV currentfont/FontMatrix get 1 get 0 ne put -}bind def -/ShowText -{ -//GraphicState/TextRenderingMode get dup 0 eq -exch 3 eq not currentfont/FontType get 3 eq and or -{ -//GraphicState/WordSpacing get 0 -32 -//GraphicState/CharacterSpacing get 0 -6 5 roll -//GraphicState/FontMatrixNonHV get{ -[ -7 -2 roll pop -5 -2 roll pop -5 -1 roll -{ -exch -pop -3 index add -exch 2 index eq{3 index add}if -4 1 roll -} -currentfont/FontMatrix get 0 get 0 ne{ -1 1 index length 1 sub getinterval cvx -}if -5 index -cshow -pop pop pop] -xshow -}{ -awidthshow -}ifelse -}{ -//GraphicState/CharacterSpacing get 0 eq -//GraphicState/FontMatrixNonHV get not and -//GraphicState/WordSpacing get 0 eq and{ -true charpath -}{ -{ -exch -pop 0 -currentpoint 5 4 roll -( )dup 0 3 index put true charpath -5 1 roll -moveto rmoveto -//GraphicState/CharacterSpacing get 0 rmoveto -32 eq{ -//GraphicState/WordSpacing get 0 rmoveto -}if -} -//GraphicState/FontMatrixNonHV get dup not exch{ -pop currentfont/FontMatrix get 0 get 0 ne -}if{ -1 1 index length 1 sub getinterval cvx -}if -exch cshow -}ifelse -}ifelse -}bind def -/ShowTextBeg -{ -//GraphicState/TextRenderingMode get dup 0 ne -{ -3 ne -currentfont/FontType get 3 eq not and{ -currentpoint newpath moveto -}if -} -{ -pop -}ifelse -}bind def -/ShowTextEnd -{ -//GraphicState/TextRenderingMode get -currentfont/FontType get 3 eq{ -dup 3 ne{ -pop 0 -}if -}if -{dup 1 eq{ -stroke exit -}if -dup 2 eq{ -gsave fill grestore stroke exit -}if -dup 3 eq{ -currentpoint newpath moveto -}if -dup 4 eq{ -gsave fill grestore clip exit -}if -dup 5 eq{ -gsave stroke grestore clip exit -}if -dup 6 eq{ -gsave fill grestore gsave stroke grestore fill exit -}if -dup 7 eq{ -clip exit -}if -exit -}loop -pop -}bind def -/ShowTextWithGlyphPositioning -{//ShowTextBeg exec -{dup type/stringtype eq{ -//ShowText exec -}{ -neg 1000 div//GraphicState/FontSize get mul 0 rmoveto -}ifelse -}forall -//ShowTextEnd exec -}bind def -/CheckFont -{dup/Type get/ExtGState ne{ -mark(Resource )3 index( must have /Type/ExtGState.)//error exec -}if -}bind def -/SetTransfer -{ -//PDFR_DEBUG{(SetTransfer beg )print count =}if -dup type/arraytype eq 1 index xcheck not and{ -0 4 getinterval aload pop -setcolortransfer -}{ -settransfer -}ifelse -//PDFR_DEBUG{(SetTransfer end )print count =}if -}bind def -/CheckExtGState -{dup/Type get/ExtGState ne{ -mark(Resource )3 index( must have /Type/ExtGState.)//error exec -}if -}bind def -/CheckHalftone -{dup/HalftoneType known not{ -mark(Resource )3 index( must have /HalftoneType.)//error exec -}if -}bind def -/ResolveFunction -{ -//PDFR_DEBUG{(ResolveFunction beg )print dup = count =}if -2 copy get//IsObjRef exec{ -2 copy//DoNothing//ResolveD exec -3 copy put pop -}if -2 copy get dup type/arraytype eq exch xcheck and not{ -2 copy get -dup type/arraytype eq 1 index xcheck not and{ -dup length 1 sub -1 0{ -2 copy//DoNothing ResolveA -dup/Identity eq{ -pop 2 copy{}put -}{ -//FunctionToProc exec -3 copy put pop -}ifelse -pop -}for -}{ -dup/Default eq{ -}{ -dup/Identity eq{ -pop{} -}{dup type/nametype eq{ -//spotfunctions exch get -}{ -//FunctionToProc exec -}ifelse -}ifelse -}ifelse -}ifelse -3 copy put -exch pop -}{ -1 index exch get -}ifelse -//PDFR_DEBUG{(ResolveFunction end )print dup == count =}if -}bind def -/ResolveFunctionSafe -{2 copy known{ -//ResolveFunction exec -}if -pop -}bind def -/CreateHalftoneThresholds -{ -dup/Thresholds known not{ -dup/HalftoneType get 10 eq{ -dup dup//MakeStreamReader exec -/Thresholds exch put -}if -dup/HalftoneType get dup 3 eq exch 6 eq or{ -dup dup//MakeStreamReader exec -//BlockBuffer readstring pop -dup length -dup 0 eq{ -mark(Could not read Thresholds)//error exec -}if -string copy/Thresholds exch put -dup/HalftoneType 3 put -}if -}if -}bind def -/SetExtGState -{ -//PDFReader/CurrentObject get/Context get/Resources get -/ExtGState//DoNothing//ResolveD exec -exch//CheckExtGState//ResolveD exec -dup/LW//knownget exec{ -setlinewidth -}if -dup/LC//knownget exec{ -setlinecap -}if -dup/LJ//knownget exec{ -setlinejoin -}if -dup/ML//knownget exec{ -setmeterlimit -}if -dup/D//knownget exec{ -setdash -}if -dup/RI//knownget exec{ -mark(Unimplemented ExtGState.RI)//error exec -}if -dup/OP//knownget exec{ -setoverprint -}if -dup/op//knownget exec{ -setoverprint -}if -dup/OPM//knownget exec{ -mark(Unimplemented ExtGState.OPM)//error exec -}if -dup/Font//knownget exec{ -mark(Unimplemented ExtGState.Font)//error exec -}if -dup/BG known{ -/BG//ResolveFunction exec -setblackgeneration -}if -dup/BG2 known{ -/BG2//ResolveFunction exec -dup/Default eq{ -//InitialExtGState/BG2 get -}if -setblackgeneration -}if -dup/UCR known{ -/UCR//ResolveFunction exec -setundercolorremoval -}if -dup/UCR2 known{ -/UCR2//ResolveFunction exec -dup/Default eq{ -//InitialExtGState/UCR2 get -}if -setundercolorremoval -}if -dup/TR known{ -/TR//ResolveFunction exec -//SetTransfer exec -}if -dup/TR2 known{ -/TR2//ResolveFunction exec -dup/Default eq{ -pop//InitialExtGState/TR2 get -aload pop setcolortransfer -}{ -//SetTransfer exec -}ifelse -}if -dup/HT//knownget exec{ -dup/Default eq{ -pop//InitialExtGState/HT get -sethalftone -}{ -//PDFR_DEBUG{(Ht beg)=}if -pop dup/HT//CheckHalftone//ResolveD exec -/SpotFunction//ResolveFunctionSafe exec -/TransferFunction//ResolveFunctionSafe exec -null exch -dup/HalftoneType get dup 5 eq exch dup 4 eq exch 2 eq or or{ -dup{ -dup//IsObjRef exec{ -pop -1 index exch//CheckHalftone ResolveD -}if -dup type/dicttype eq{ -dup/SpotFunction//ResolveFunctionSafe exec -/TransferFunction//ResolveFunctionSafe exec -//CreateHalftoneThresholds exec -dup/HalftoneType get 5 gt{ -4 3 roll pop -dup 4 1 roll -}if -}if -pop pop -}forall -}if -//CreateHalftoneThresholds exec -//PDFR_DEBUG{ -(HT:)= -dup{ -1 index/Default eq{ -(Default <<)= -exch pop -{exch = ==}forall -(>>)= -}{ -exch = == -}ifelse -}forall -(HT end)= flush -}if -exch dup null ne{ -(Warning: Ignoring a halftone with a Level 3 component halftone Type )print dup/HalftoneType get = -pop pop -}{ -pop -dup/HalftoneType get 5 gt{ -(Warning: Ignoring a Level 3 halftone Type )print dup/HalftoneType get = -pop -}{ -sethalftone -}ifelse -}ifelse -//PDFR_DEBUG{(HT set)= flush}if -}ifelse -}if -dup/FL//knownget exec{ -setflattness -}if -dup/SM//knownget exec{ -setsmoothness -}if -dup/SA//knownget exec{ -setstrokeadjust -}if -dup/BM//knownget exec{ -mark(Unimplemented ExtGState.BM)//error exec -}if -dup/SMask//knownget exec{ -mark(Unimplemented ExtGState.SMask)//error exec -}if -dup/CA//knownget exec{ -mark(Unimplemented ExtGState.CA)//error exec -}if -dup/ca//knownget exec{ -mark(Unimplemented ExtGState.ca)//error exec -}if -dup/AIS//knownget exec{ -mark(Unimplemented ExtGState.AIS)//error exec -}if -dup/TK//knownget exec{ -mark(Unimplemented ExtGState.TK)//error exec -}if -pop -}bind def -/CheckXObject -{dup/Subtype get dup/Image ne exch dup/Form ne exch/PS ne and and{ -mark(Resource )3 index( must have /Subtype /Image or /Form or /PS.)//error exec -}if -}bind def -/DoXObject -{ -//PDFReader/CurrentObject get/Context get/Resources get -/XObject//DoNothing//ResolveD exec -exch//CheckXObject//ResolveD exec -dup/Subtype get -dup/Image eq{ -pop -//CompleteOutlineImage exec -//DoImage exec -}{ -dup/PS eq{ -PDFR_DEBUG{ -(Executing a PS Xobject)= -}if -pop -//RunDelayedStream exec -}{ -dup/Form eq{ -pop -PDFR_DEBUG{ -(Executing a Form XObject)= -}if -//PDFReader/CurrentObject get exch -dup//PDFReader exch<< exch/Context exch >>/CurrentObject exch put -dup/Matrix get concat -dup/BBox get aload pop exch 3 index sub exch 2 index sub rectclip -//RunDelayedStream exec -//PDFReader exch/CurrentObject exch put -}{ -mark exch(unimplemented XObject type )exch//error exec -}ifelse -}ifelse -}ifelse -}bind def -/Operators 50 dict begin -/q{//GSave exec}bind def -/Q{//GRestore exec}bind def -/cm{//TempMatrix astore concat}bind def -/i{1 .min setflat}bind def -/J/setlinecap load def -/d/setdash load def -/j/setlinejoin load def -/w/setlinewidth load def -/M/setmiterlimit load def -/gs{SetExtGState}bind def -/g/setgray load def -/rg/setrgbcolor load def -/k/setcmykcolor load def -/cs{//ResolveColorSpace exec//SetColorSpaceSafe exec -}bind def -/sc/setcolor load def -/scn{//SetColor exec}bind def -/G/setgray load def -/RG/setrgbcolor load def -/K/setcmykcolor load def -/CS//cs def -/ri{SetColorRenderingIntent}bind def -/SC/setcolor load def -/SCN{//SetColor exec}bind def -/m/moveto load def -/l/lineto load def -/c/curveto load def -/v{currentpoint 6 2 roll curveto}bind def -/y{2 copy curveto}bind def -/re{ -4 2 roll moveto exch dup 0 rlineto 0 3 -1 roll rlineto neg 0 rlineto -closepath -}def -/h/closepath load def -/n/newpath load def -/S/stroke load def -/s{closepath stroke}bind def -/f/fill load def -/f*/eofill load def -/B{gsave fill grestore stroke}bind def -/b{closepath gsave fill grestore stroke}bind def -/B*{gsave eofill grestore stroke}bind def -/b*{closepath gsave eofill grestore stroke}bind def -/W/clip load def -/W*/eoclip load def -/sh{ -ResolveShading -dup/Background known{ -gsave -dup/ColorSpace get setcolorspace -dup/Background get aload pop setcolor -pathbbox -2 index sub exch 3 index sub exch -rectfill -grestore -}if -shfill -}bind def -/Do{//DoXObject exec}bind def -/BI{currentglobal false setglobal<<}bind def -/ID{>> -dup/DataSource currentfile -2 index/F//knownget exec{ -/A85 eq{ -0(~>)/SubFileDecode filter -}if -}if -put -//CompleteInlineImage exec -exch setglobal -//DoImage exec -}bind def -/EI{}bind def -/BT{gsave//GraphicState/InitialTextMatrix get currentmatrix pop}bind def -/ET{grestore}bind def -/Tc{//GraphicState exch/CharacterSpacing exch put}bind def -/TL{//GraphicState exch/TextLeading exch put}bind def -/Tr{//GraphicState exch/TextRenderingMode exch put}bind def -/Ts{ -mark(Unimplemented SetTextRise)//error exec -}bind def -/Tw{//GraphicState exch/WordSpacing exch put}bind def -/Tz{ -mark(Unimplemented SetHorizontalTextScaling)//error exec -}bind def -/Td{translate 0 0 moveto}bind def -/TD{dup neg//TL exec//Td exec}bind def -/Tm{//GraphicState/InitialTextMatrix get setmatrix -//TempMatrix astore concat -0 0 moveto}bind def -/T*{0//GraphicState/TextLeading get neg//Td exec}bind def -/Tj{//ShowTextBeg exec//ShowText exec//ShowTextEnd exec}bind def -/'{//T* exec//ShowText exec//ShowTextEnd exec}bind def -/"{3 2 roll//Tw exec exch//Tc exec//' exec}bind def -/TJ//ShowTextWithGlyphPositioning def -/Tf//SetFont def -/d0/setcharwidth load def -/d1/setcachedevice load def -/BDC{pop pop}bind def -/BMC{pop}bind def -/EMC{}bind def -/BX{BeginCompatibilitySection}bind def -/EX{EndCompatibilitySection}bind def -/DP{DefineMarkedContentPointWithPropertyList}bind def -/MP{DefineMarkedContentPoint}bind def -/PS{cvx exec}bind def -currentdict end def -//PDFR_STREAM{ -//Operators length dict begin -//Operators{ -exch dup -[exch//=only/exec load -( )/print load -8 7 roll -dup type/arraytype eq{ -/exec load -}if -( )/print load -]cvx -def -}forall -currentdict end/Operators exch def -}if -/.registerencoding -{pop pop -}bind def -/.defineencoding -{def -}bind def -/.findencoding -{load -}bind def -/currentglobal where -{pop currentglobal{setglobal}true setglobal} -{{}} -ifelse -/MacRomanEncoding -StandardEncoding 0 39 getinterval aload pop -/quotesingle -StandardEncoding 40 56 getinterval aload pop -/grave -StandardEncoding 97 31 getinterval aload pop -/Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute -/agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave -/ecircumflex/edieresis/iacute/igrave -/icircumflex/idieresis/ntilde/oacute -/ograve/ocircumflex/odieresis/otilde -/uacute/ugrave/ucircumflex/udieresis -/dagger/degree/cent/sterling/section/bullet/paragraph/germandbls -/registered/copyright/trademark/acute/dieresis/.notdef/AE/Oslash -/.notdef/plusminus/.notdef/.notdef/yen/mu/.notdef/.notdef -/.notdef/.notdef/.notdef/ordfeminine/ordmasculine/.notdef/ae/oslash -/questiondown/exclamdown/logicalnot/.notdef -/florin/.notdef/.notdef/guillemotleft -/guillemotright/ellipsis/space/Agrave/Atilde/Otilde/OE/oe -/endash/emdash/quotedblleft/quotedblright -/quoteleft/quoteright/divide/.notdef -/ydieresis/Ydieresis/fraction/currency -/guilsinglleft/guilsinglright/fi/fl -/daggerdbl/periodcentered/quotesinglbase/quotedblbase -/perthousand/Acircumflex/Ecircumflex/Aacute -/Edieresis/Egrave/Iacute/Icircumflex -/Idieresis/Igrave/Oacute/Ocircumflex -/.notdef/Ograve/Uacute/Ucircumflex -/Ugrave/dotlessi/circumflex/tilde -/macron/breve/dotaccent/ring/cedilla/hungarumlaut/ogonek/caron -256 packedarray -5 1 index .registerencoding -.defineencoding -exec -/AdobeGlyphList mark -/A 16#0041 -/AE 16#00c6 -/AEacute 16#01fc -/AEmacron 16#01e2 -/AEsmall 16#f7e6 -/Aacute 16#00c1 -/Aacutesmall 16#f7e1 -/Abreve 16#0102 -/Abreveacute 16#1eae -/Abrevecyrillic 16#04d0 -/Abrevedotbelow 16#1eb6 -/Abrevegrave 16#1eb0 -/Abrevehookabove 16#1eb2 -/Abrevetilde 16#1eb4 -/Acaron 16#01cd -/Acircle 16#24b6 -/Acircumflex 16#00c2 -/Acircumflexacute 16#1ea4 -/Acircumflexdotbelow 16#1eac -/Acircumflexgrave 16#1ea6 -/Acircumflexhookabove 16#1ea8 -/Acircumflexsmall 16#f7e2 -/Acircumflextilde 16#1eaa -/Acute 16#f6c9 -/Acutesmall 16#f7b4 -/Acyrillic 16#0410 -/Adblgrave 16#0200 -/Adieresis 16#00c4 -/Adieresiscyrillic 16#04d2 -/Adieresismacron 16#01de -/Adieresissmall 16#f7e4 -/Adotbelow 16#1ea0 -/Adotmacron 16#01e0 -/Agrave 16#00c0 -/Agravesmall 16#f7e0 -/Ahookabove 16#1ea2 -/Aiecyrillic 16#04d4 -/Ainvertedbreve 16#0202 -/Alpha 16#0391 -/Alphatonos 16#0386 -/Amacron 16#0100 -/Amonospace 16#ff21 -/Aogonek 16#0104 -/Aring 16#00c5 -/Aringacute 16#01fa -/Aringbelow 16#1e00 -/Aringsmall 16#f7e5 -/Asmall 16#f761 -/Atilde 16#00c3 -/Atildesmall 16#f7e3 -/Aybarmenian 16#0531 -/B 16#0042 -/Bcircle 16#24b7 -/Bdotaccent 16#1e02 -/Bdotbelow 16#1e04 -/Becyrillic 16#0411 -/Benarmenian 16#0532 -/Beta 16#0392 -/Bhook 16#0181 -/Blinebelow 16#1e06 -/Bmonospace 16#ff22 -/Brevesmall 16#f6f4 -/Bsmall 16#f762 -/Btopbar 16#0182 -/C 16#0043 -/Caarmenian 16#053e -/Cacute 16#0106 -/Caron 16#f6ca -/Caronsmall 16#f6f5 -/Ccaron 16#010c -/Ccedilla 16#00c7 -/Ccedillaacute 16#1e08 -/Ccedillasmall 16#f7e7 -/Ccircle 16#24b8 -/Ccircumflex 16#0108 -/Cdot 16#010a -/Cdotaccent 16#010a -/Cedillasmall 16#f7b8 -/Chaarmenian 16#0549 -/Cheabkhasiancyrillic 16#04bc -/Checyrillic 16#0427 -/Chedescenderabkhasiancyrillic 16#04be -/Chedescendercyrillic 16#04b6 -/Chedieresiscyrillic 16#04f4 -/Cheharmenian 16#0543 -/Chekhakassiancyrillic 16#04cb -/Cheverticalstrokecyrillic 16#04b8 -/Chi 16#03a7 -/Chook 16#0187 -/Circumflexsmall 16#f6f6 -/Cmonospace 16#ff23 -/Coarmenian 16#0551 -/Csmall 16#f763 -/D 16#0044 -/DZ 16#01f1 -/DZcaron 16#01c4 -/Daarmenian 16#0534 -/Dafrican 16#0189 -/Dcaron 16#010e -/Dcedilla 16#1e10 -/Dcircle 16#24b9 -/Dcircumflexbelow 16#1e12 -/Dcroat 16#0110 -/Ddotaccent 16#1e0a -/Ddotbelow 16#1e0c -/Decyrillic 16#0414 -/Deicoptic 16#03ee -/Delta 16#2206 -/Deltagreek 16#0394 -/Dhook 16#018a -/Dieresis 16#f6cb -/DieresisAcute 16#f6cc -/DieresisGrave 16#f6cd -/Dieresissmall 16#f7a8 -/Digammagreek 16#03dc -/Djecyrillic 16#0402 -/Dlinebelow 16#1e0e -/Dmonospace 16#ff24 -/Dotaccentsmall 16#f6f7 -/Dslash 16#0110 -/Dsmall 16#f764 -/Dtopbar 16#018b -/Dz 16#01f2 -/Dzcaron 16#01c5 -/Dzeabkhasiancyrillic 16#04e0 -/Dzecyrillic 16#0405 -/Dzhecyrillic 16#040f -/E 16#0045 -/Eacute 16#00c9 -/Eacutesmall 16#f7e9 -/Ebreve 16#0114 -/Ecaron 16#011a -/Ecedillabreve 16#1e1c -/Echarmenian 16#0535 -/Ecircle 16#24ba -/Ecircumflex 16#00ca -/Ecircumflexacute 16#1ebe -/Ecircumflexbelow 16#1e18 -/Ecircumflexdotbelow 16#1ec6 -/Ecircumflexgrave 16#1ec0 -/Ecircumflexhookabove 16#1ec2 -/Ecircumflexsmall 16#f7ea -/Ecircumflextilde 16#1ec4 -/Ecyrillic 16#0404 -/Edblgrave 16#0204 -/Edieresis 16#00cb -/Edieresissmall 16#f7eb -/Edot 16#0116 -/Edotaccent 16#0116 -/Edotbelow 16#1eb8 -/Efcyrillic 16#0424 -/Egrave 16#00c8 -/Egravesmall 16#f7e8 -/Eharmenian 16#0537 -/Ehookabove 16#1eba -/Eightroman 16#2167 -/Einvertedbreve 16#0206 -/Eiotifiedcyrillic 16#0464 -/Elcyrillic 16#041b -/Elevenroman 16#216a -/Emacron 16#0112 -/Emacronacute 16#1e16 -/Emacrongrave 16#1e14 -/Emcyrillic 16#041c -/Emonospace 16#ff25 -/Encyrillic 16#041d -/Endescendercyrillic 16#04a2 -/Eng 16#014a -/Enghecyrillic 16#04a4 -/Enhookcyrillic 16#04c7 -/Eogonek 16#0118 -/Eopen 16#0190 -/Epsilon 16#0395 -/Epsilontonos 16#0388 -/Ercyrillic 16#0420 -/Ereversed 16#018e -/Ereversedcyrillic 16#042d -/Escyrillic 16#0421 -/Esdescendercyrillic 16#04aa -/Esh 16#01a9 -/Esmall 16#f765 -/Eta 16#0397 -/Etarmenian 16#0538 -/Etatonos 16#0389 -/Eth 16#00d0 -/Ethsmall 16#f7f0 -/Etilde 16#1ebc -/Etildebelow 16#1e1a -/Euro 16#20ac -/Ezh 16#01b7 -/Ezhcaron 16#01ee -/Ezhreversed 16#01b8 -/F 16#0046 -/Fcircle 16#24bb -/Fdotaccent 16#1e1e -/Feharmenian 16#0556 -/Feicoptic 16#03e4 -/Fhook 16#0191 -/Fitacyrillic 16#0472 -/Fiveroman 16#2164 -/Fmonospace 16#ff26 -/Fourroman 16#2163 -/Fsmall 16#f766 -/G 16#0047 -/GBsquare 16#3387 -/Gacute 16#01f4 -/Gamma 16#0393 -/Gammaafrican 16#0194 -/Gangiacoptic 16#03ea -/Gbreve 16#011e -/Gcaron 16#01e6 -/Gcedilla 16#0122 -/Gcircle 16#24bc -/Gcircumflex 16#011c -/Gcommaaccent 16#0122 -/Gdot 16#0120 -/Gdotaccent 16#0120 -/Gecyrillic 16#0413 -/Ghadarmenian 16#0542 -/Ghemiddlehookcyrillic 16#0494 -/Ghestrokecyrillic 16#0492 -/Gheupturncyrillic 16#0490 -/Ghook 16#0193 -/Gimarmenian 16#0533 -/Gjecyrillic 16#0403 -/Gmacron 16#1e20 -/Gmonospace 16#ff27 -/Grave 16#f6ce -/Gravesmall 16#f760 -/Gsmall 16#f767 -/Gsmallhook 16#029b -/Gstroke 16#01e4 -/H 16#0048 -/H18533 16#25cf -/H18543 16#25aa -/H18551 16#25ab -/H22073 16#25a1 -/HPsquare 16#33cb -/Haabkhasiancyrillic 16#04a8 -/Hadescendercyrillic 16#04b2 -/Hardsigncyrillic 16#042a -/Hbar 16#0126 -/Hbrevebelow 16#1e2a -/Hcedilla 16#1e28 -/Hcircle 16#24bd -/Hcircumflex 16#0124 -/Hdieresis 16#1e26 -/Hdotaccent 16#1e22 -/Hdotbelow 16#1e24 -/Hmonospace 16#ff28 -/Hoarmenian 16#0540 -/Horicoptic 16#03e8 -/Hsmall 16#f768 -/Hungarumlaut 16#f6cf -/Hungarumlautsmall 16#f6f8 -/Hzsquare 16#3390 -/I 16#0049 -/IAcyrillic 16#042f -/IJ 16#0132 -/IUcyrillic 16#042e -/Iacute 16#00cd -/Iacutesmall 16#f7ed -/Ibreve 16#012c -/Icaron 16#01cf -/Icircle 16#24be -/Icircumflex 16#00ce -/Icircumflexsmall 16#f7ee -/Icyrillic 16#0406 -/Idblgrave 16#0208 -/Idieresis 16#00cf -/Idieresisacute 16#1e2e -/Idieresiscyrillic 16#04e4 -/Idieresissmall 16#f7ef -/Idot 16#0130 -/Idotaccent 16#0130 -/Idotbelow 16#1eca -/Iebrevecyrillic 16#04d6 -/Iecyrillic 16#0415 -/Ifraktur 16#2111 -/Igrave 16#00cc -/Igravesmall 16#f7ec -/Ihookabove 16#1ec8 -/Iicyrillic 16#0418 -/Iinvertedbreve 16#020a -/Iishortcyrillic 16#0419 -/Imacron 16#012a -/Imacroncyrillic 16#04e2 -/Imonospace 16#ff29 -/Iniarmenian 16#053b -/Iocyrillic 16#0401 -/Iogonek 16#012e -/Iota 16#0399 -/Iotaafrican 16#0196 -/Iotadieresis 16#03aa -/Iotatonos 16#038a -/Ismall 16#f769 -/Istroke 16#0197 -/Itilde 16#0128 -/Itildebelow 16#1e2c -/Izhitsacyrillic 16#0474 -/Izhitsadblgravecyrillic 16#0476 -/J 16#004a -/Jaarmenian 16#0541 -/Jcircle 16#24bf -/Jcircumflex 16#0134 -/Jecyrillic 16#0408 -/Jheharmenian 16#054b -/Jmonospace 16#ff2a -/Jsmall 16#f76a -/K 16#004b -/KBsquare 16#3385 -/KKsquare 16#33cd -/Kabashkircyrillic 16#04a0 -/Kacute 16#1e30 -/Kacyrillic 16#041a -/Kadescendercyrillic 16#049a -/Kahookcyrillic 16#04c3 -/Kappa 16#039a -/Kastrokecyrillic 16#049e -/Kaverticalstrokecyrillic 16#049c -/Kcaron 16#01e8 -/Kcedilla 16#0136 -/Kcircle 16#24c0 -/Kcommaaccent 16#0136 -/Kdotbelow 16#1e32 -/Keharmenian 16#0554 -/Kenarmenian 16#053f -/Khacyrillic 16#0425 -/Kheicoptic 16#03e6 -/Khook 16#0198 -/Kjecyrillic 16#040c -/Klinebelow 16#1e34 -/Kmonospace 16#ff2b -/Koppacyrillic 16#0480 -/Koppagreek 16#03de -/Ksicyrillic 16#046e -/Ksmall 16#f76b -/L 16#004c -/LJ 16#01c7 -/LL 16#f6bf -/Lacute 16#0139 -/Lambda 16#039b -/Lcaron 16#013d -/Lcedilla 16#013b -/Lcircle 16#24c1 -/Lcircumflexbelow 16#1e3c -/Lcommaaccent 16#013b -/Ldot 16#013f -/Ldotaccent 16#013f -/Ldotbelow 16#1e36 -/Ldotbelowmacron 16#1e38 -/Liwnarmenian 16#053c -/Lj 16#01c8 -/Ljecyrillic 16#0409 -/Llinebelow 16#1e3a -/Lmonospace 16#ff2c -/Lslash 16#0141 -/Lslashsmall 16#f6f9 -/Lsmall 16#f76c -/M 16#004d -/MBsquare 16#3386 -/Macron 16#f6d0 -/Macronsmall 16#f7af -/Macute 16#1e3e -/Mcircle 16#24c2 -/Mdotaccent 16#1e40 -/Mdotbelow 16#1e42 -/Menarmenian 16#0544 -/Mmonospace 16#ff2d -/Msmall 16#f76d -/Mturned 16#019c -/Mu 16#039c -/N 16#004e -/NJ 16#01ca -/Nacute 16#0143 -/Ncaron 16#0147 -/Ncedilla 16#0145 -/Ncircle 16#24c3 -/Ncircumflexbelow 16#1e4a -/Ncommaaccent 16#0145 -/Ndotaccent 16#1e44 -/Ndotbelow 16#1e46 -/Nhookleft 16#019d -/Nineroman 16#2168 -/Nj 16#01cb -/Njecyrillic 16#040a -/Nlinebelow 16#1e48 -/Nmonospace 16#ff2e -/Nowarmenian 16#0546 -/Nsmall 16#f76e -/Ntilde 16#00d1 -/Ntildesmall 16#f7f1 -/Nu 16#039d -/O 16#004f -/OE 16#0152 -/OEsmall 16#f6fa -/Oacute 16#00d3 -/Oacutesmall 16#f7f3 -/Obarredcyrillic 16#04e8 -/Obarreddieresiscyrillic 16#04ea -/Obreve 16#014e -/Ocaron 16#01d1 -/Ocenteredtilde 16#019f -/Ocircle 16#24c4 -/Ocircumflex 16#00d4 -/Ocircumflexacute 16#1ed0 -/Ocircumflexdotbelow 16#1ed8 -/Ocircumflexgrave 16#1ed2 -/Ocircumflexhookabove 16#1ed4 -/Ocircumflexsmall 16#f7f4 -/Ocircumflextilde 16#1ed6 -/Ocyrillic 16#041e -/Odblacute 16#0150 -/Odblgrave 16#020c -/Odieresis 16#00d6 -/Odieresiscyrillic 16#04e6 -/Odieresissmall 16#f7f6 -/Odotbelow 16#1ecc -/Ogoneksmall 16#f6fb -/Ograve 16#00d2 -/Ogravesmall 16#f7f2 -/Oharmenian 16#0555 -/Ohm 16#2126 -/Ohookabove 16#1ece -/Ohorn 16#01a0 -/Ohornacute 16#1eda -/Ohorndotbelow 16#1ee2 -/Ohorngrave 16#1edc -/Ohornhookabove 16#1ede -/Ohorntilde 16#1ee0 -/Ohungarumlaut 16#0150 -/Oi 16#01a2 -/Oinvertedbreve 16#020e -/Omacron 16#014c -/Omacronacute 16#1e52 -/Omacrongrave 16#1e50 -/Omega 16#2126 -/Omegacyrillic 16#0460 -/Omegagreek 16#03a9 -/Omegaroundcyrillic 16#047a -/Omegatitlocyrillic 16#047c -/Omegatonos 16#038f -/Omicron 16#039f -/Omicrontonos 16#038c -/Omonospace 16#ff2f -/Oneroman 16#2160 -/Oogonek 16#01ea -/Oogonekmacron 16#01ec -/Oopen 16#0186 -/Oslash 16#00d8 -/Oslashacute 16#01fe -/Oslashsmall 16#f7f8 -/Osmall 16#f76f -/Ostrokeacute 16#01fe -/Otcyrillic 16#047e -/Otilde 16#00d5 -/Otildeacute 16#1e4c -/Otildedieresis 16#1e4e -/Otildesmall 16#f7f5 -/P 16#0050 -/Pacute 16#1e54 -/Pcircle 16#24c5 -/Pdotaccent 16#1e56 -/Pecyrillic 16#041f -/Peharmenian 16#054a -/Pemiddlehookcyrillic 16#04a6 -/Phi 16#03a6 -/Phook 16#01a4 -/Pi 16#03a0 -/Piwrarmenian 16#0553 -/Pmonospace 16#ff30 -/Psi 16#03a8 -/Psicyrillic 16#0470 -/Psmall 16#f770 -/Q 16#0051 -/Qcircle 16#24c6 -/Qmonospace 16#ff31 -/Qsmall 16#f771 -/R 16#0052 -/Raarmenian 16#054c -/Racute 16#0154 -/Rcaron 16#0158 -/Rcedilla 16#0156 -/Rcircle 16#24c7 -/Rcommaaccent 16#0156 -/Rdblgrave 16#0210 -/Rdotaccent 16#1e58 -/Rdotbelow 16#1e5a -/Rdotbelowmacron 16#1e5c -/Reharmenian 16#0550 -/Rfraktur 16#211c -/Rho 16#03a1 -/Ringsmall 16#f6fc -/Rinvertedbreve 16#0212 -/Rlinebelow 16#1e5e -/Rmonospace 16#ff32 -/Rsmall 16#f772 -/Rsmallinverted 16#0281 -/Rsmallinvertedsuperior 16#02b6 -/S 16#0053 -/SF010000 16#250c -/SF020000 16#2514 -/SF030000 16#2510 -/SF040000 16#2518 -/SF050000 16#253c -/SF060000 16#252c -/SF070000 16#2534 -/SF080000 16#251c -/SF090000 16#2524 -/SF100000 16#2500 -/SF110000 16#2502 -/SF190000 16#2561 -/SF200000 16#2562 -/SF210000 16#2556 -/SF220000 16#2555 -/SF230000 16#2563 -/SF240000 16#2551 -/SF250000 16#2557 -/SF260000 16#255d -/SF270000 16#255c -/SF280000 16#255b -/SF360000 16#255e -/SF370000 16#255f -/SF380000 16#255a -/SF390000 16#2554 -/SF400000 16#2569 -/SF410000 16#2566 -/SF420000 16#2560 -/SF430000 16#2550 -/SF440000 16#256c -/SF450000 16#2567 -/SF460000 16#2568 -/SF470000 16#2564 -/SF480000 16#2565 -/SF490000 16#2559 -/SF500000 16#2558 -/SF510000 16#2552 -/SF520000 16#2553 -/SF530000 16#256b -/SF540000 16#256a -/Sacute 16#015a -/Sacutedotaccent 16#1e64 -/Sampigreek 16#03e0 -/Scaron 16#0160 -/Scarondotaccent 16#1e66 -/Scaronsmall 16#f6fd -/Scedilla 16#015e -/Schwa 16#018f -/Schwacyrillic 16#04d8 -/Schwadieresiscyrillic 16#04da -/Scircle 16#24c8 -/Scircumflex 16#015c -/Scommaaccent 16#0218 -/Sdotaccent 16#1e60 -/Sdotbelow 16#1e62 -/Sdotbelowdotaccent 16#1e68 -/Seharmenian 16#054d -/Sevenroman 16#2166 -/Shaarmenian 16#0547 -/Shacyrillic 16#0428 -/Shchacyrillic 16#0429 -/Sheicoptic 16#03e2 -/Shhacyrillic 16#04ba -/Shimacoptic 16#03ec -/Sigma 16#03a3 -/Sixroman 16#2165 -/Smonospace 16#ff33 -/Softsigncyrillic 16#042c -/Ssmall 16#f773 -/Stigmagreek 16#03da -/T 16#0054 -/Tau 16#03a4 -/Tbar 16#0166 -/Tcaron 16#0164 -/Tcedilla 16#0162 -/Tcircle 16#24c9 -/Tcircumflexbelow 16#1e70 -/Tcommaaccent 16#0162 -/Tdotaccent 16#1e6a -/Tdotbelow 16#1e6c -/Tecyrillic 16#0422 -/Tedescendercyrillic 16#04ac -/Tenroman 16#2169 -/Tetsecyrillic 16#04b4 -/Theta 16#0398 -/Thook 16#01ac -/Thorn 16#00de -/Thornsmall 16#f7fe -/Threeroman 16#2162 -/Tildesmall 16#f6fe -/Tiwnarmenian 16#054f -/Tlinebelow 16#1e6e -/Tmonospace 16#ff34 -/Toarmenian 16#0539 -/Tonefive 16#01bc -/Tonesix 16#0184 -/Tonetwo 16#01a7 -/Tretroflexhook 16#01ae -/Tsecyrillic 16#0426 -/Tshecyrillic 16#040b -/Tsmall 16#f774 -/Twelveroman 16#216b -/Tworoman 16#2161 -/U 16#0055 -/Uacute 16#00da -/Uacutesmall 16#f7fa -/Ubreve 16#016c -/Ucaron 16#01d3 -/Ucircle 16#24ca -/Ucircumflex 16#00db -/Ucircumflexbelow 16#1e76 -/Ucircumflexsmall 16#f7fb -/Ucyrillic 16#0423 -/Udblacute 16#0170 -/Udblgrave 16#0214 -/Udieresis 16#00dc -/Udieresisacute 16#01d7 -/Udieresisbelow 16#1e72 -/Udieresiscaron 16#01d9 -/Udieresiscyrillic 16#04f0 -/Udieresisgrave 16#01db -/Udieresismacron 16#01d5 -/Udieresissmall 16#f7fc -/Udotbelow 16#1ee4 -/Ugrave 16#00d9 -/Ugravesmall 16#f7f9 -/Uhookabove 16#1ee6 -/Uhorn 16#01af -/Uhornacute 16#1ee8 -/Uhorndotbelow 16#1ef0 -/Uhorngrave 16#1eea -/Uhornhookabove 16#1eec -/Uhorntilde 16#1eee -/Uhungarumlaut 16#0170 -/Uhungarumlautcyrillic 16#04f2 -/Uinvertedbreve 16#0216 -/Ukcyrillic 16#0478 -/Umacron 16#016a -/Umacroncyrillic 16#04ee -/Umacrondieresis 16#1e7a -/Umonospace 16#ff35 -/Uogonek 16#0172 -/Upsilon 16#03a5 -/Upsilon1 16#03d2 -/Upsilonacutehooksymbolgreek 16#03d3 -/Upsilonafrican 16#01b1 -/Upsilondieresis 16#03ab -/Upsilondieresishooksymbolgreek 16#03d4 -/Upsilonhooksymbol 16#03d2 -/Upsilontonos 16#038e -/Uring 16#016e -/Ushortcyrillic 16#040e -/Usmall 16#f775 -/Ustraightcyrillic 16#04ae -/Ustraightstrokecyrillic 16#04b0 -/Utilde 16#0168 -/Utildeacute 16#1e78 -/Utildebelow 16#1e74 -/V 16#0056 -/Vcircle 16#24cb -/Vdotbelow 16#1e7e -/Vecyrillic 16#0412 -/Vewarmenian 16#054e -/Vhook 16#01b2 -/Vmonospace 16#ff36 -/Voarmenian 16#0548 -/Vsmall 16#f776 -/Vtilde 16#1e7c -/W 16#0057 -/Wacute 16#1e82 -/Wcircle 16#24cc -/Wcircumflex 16#0174 -/Wdieresis 16#1e84 -/Wdotaccent 16#1e86 -/Wdotbelow 16#1e88 -/Wgrave 16#1e80 -/Wmonospace 16#ff37 -/Wsmall 16#f777 -/X 16#0058 -/Xcircle 16#24cd -/Xdieresis 16#1e8c -/Xdotaccent 16#1e8a -/Xeharmenian 16#053d -/Xi 16#039e -/Xmonospace 16#ff38 -/Xsmall 16#f778 -/Y 16#0059 -/Yacute 16#00dd -/Yacutesmall 16#f7fd -/Yatcyrillic 16#0462 -/Ycircle 16#24ce -/Ycircumflex 16#0176 -/Ydieresis 16#0178 -/Ydieresissmall 16#f7ff -/Ydotaccent 16#1e8e -/Ydotbelow 16#1ef4 -/Yericyrillic 16#042b -/Yerudieresiscyrillic 16#04f8 -/Ygrave 16#1ef2 -/Yhook 16#01b3 -/Yhookabove 16#1ef6 -/Yiarmenian 16#0545 -/Yicyrillic 16#0407 -/Yiwnarmenian 16#0552 -/Ymonospace 16#ff39 -/Ysmall 16#f779 -/Ytilde 16#1ef8 -/Yusbigcyrillic 16#046a -/Yusbigiotifiedcyrillic 16#046c -/Yuslittlecyrillic 16#0466 -/Yuslittleiotifiedcyrillic 16#0468 -/Z 16#005a -/Zaarmenian 16#0536 -/Zacute 16#0179 -/Zcaron 16#017d -/Zcaronsmall 16#f6ff -/Zcircle 16#24cf -/Zcircumflex 16#1e90 -/Zdot 16#017b -/Zdotaccent 16#017b -/Zdotbelow 16#1e92 -/Zecyrillic 16#0417 -/Zedescendercyrillic 16#0498 -/Zedieresiscyrillic 16#04de -/Zeta 16#0396 -/Zhearmenian 16#053a -/Zhebrevecyrillic 16#04c1 -/Zhecyrillic 16#0416 -/Zhedescendercyrillic 16#0496 -/Zhedieresiscyrillic 16#04dc -/Zlinebelow 16#1e94 -/Zmonospace 16#ff3a -/Zsmall 16#f77a -/Zstroke 16#01b5 -/a 16#0061 -/aabengali 16#0986 -/aacute 16#00e1 -/aadeva 16#0906 -/aagujarati 16#0a86 -/aagurmukhi 16#0a06 -/aamatragurmukhi 16#0a3e -/aarusquare 16#3303 -/aavowelsignbengali 16#09be -/aavowelsigndeva 16#093e -/aavowelsigngujarati 16#0abe -/abbreviationmarkarmenian 16#055f -/abbreviationsigndeva 16#0970 -/abengali 16#0985 -/abopomofo 16#311a -/abreve 16#0103 -/abreveacute 16#1eaf -/abrevecyrillic 16#04d1 -/abrevedotbelow 16#1eb7 -/abrevegrave 16#1eb1 -/abrevehookabove 16#1eb3 -/abrevetilde 16#1eb5 -/acaron 16#01ce -/acircle 16#24d0 -/acircumflex 16#00e2 -/acircumflexacute 16#1ea5 -/acircumflexdotbelow 16#1ead -/acircumflexgrave 16#1ea7 -/acircumflexhookabove 16#1ea9 -/acircumflextilde 16#1eab -/acute 16#00b4 -/acutebelowcmb 16#0317 -/acutecmb 16#0301 -/acutecomb 16#0301 -/acutedeva 16#0954 -/acutelowmod 16#02cf -/acutetonecmb 16#0341 -/acyrillic 16#0430 -/adblgrave 16#0201 -/addakgurmukhi 16#0a71 -/adeva 16#0905 -/adieresis 16#00e4 -/adieresiscyrillic 16#04d3 -/adieresismacron 16#01df -/adotbelow 16#1ea1 -/adotmacron 16#01e1 -/ae 16#00e6 -/aeacute 16#01fd -/aekorean 16#3150 -/aemacron 16#01e3 -/afii00208 16#2015 -/afii08941 16#20a4 -/afii10017 16#0410 -/afii10018 16#0411 -/afii10019 16#0412 -/afii10020 16#0413 -/afii10021 16#0414 -/afii10022 16#0415 -/afii10023 16#0401 -/afii10024 16#0416 -/afii10025 16#0417 -/afii10026 16#0418 -/afii10027 16#0419 -/afii10028 16#041a -/afii10029 16#041b -/afii10030 16#041c -/afii10031 16#041d -/afii10032 16#041e -/afii10033 16#041f -/afii10034 16#0420 -/afii10035 16#0421 -/afii10036 16#0422 -/afii10037 16#0423 -/afii10038 16#0424 -/afii10039 16#0425 -/afii10040 16#0426 -/afii10041 16#0427 -/afii10042 16#0428 -/afii10043 16#0429 -/afii10044 16#042a -/afii10045 16#042b -/afii10046 16#042c -/afii10047 16#042d -/afii10048 16#042e -/afii10049 16#042f -/afii10050 16#0490 -/afii10051 16#0402 -/afii10052 16#0403 -/afii10053 16#0404 -/afii10054 16#0405 -/afii10055 16#0406 -/afii10056 16#0407 -/afii10057 16#0408 -/afii10058 16#0409 -/afii10059 16#040a -/afii10060 16#040b -/afii10061 16#040c -/afii10062 16#040e -/afii10063 16#f6c4 -/afii10064 16#f6c5 -/afii10065 16#0430 -/afii10066 16#0431 -/afii10067 16#0432 -/afii10068 16#0433 -/afii10069 16#0434 -/afii10070 16#0435 -/afii10071 16#0451 -/afii10072 16#0436 -/afii10073 16#0437 -/afii10074 16#0438 -/afii10075 16#0439 -/afii10076 16#043a -/afii10077 16#043b -/afii10078 16#043c -/afii10079 16#043d -/afii10080 16#043e -/afii10081 16#043f -/afii10082 16#0440 -/afii10083 16#0441 -/afii10084 16#0442 -/afii10085 16#0443 -/afii10086 16#0444 -/afii10087 16#0445 -/afii10088 16#0446 -/afii10089 16#0447 -/afii10090 16#0448 -/afii10091 16#0449 -/afii10092 16#044a -/afii10093 16#044b -/afii10094 16#044c -/afii10095 16#044d -/afii10096 16#044e -/afii10097 16#044f -/afii10098 16#0491 -/afii10099 16#0452 -/afii10100 16#0453 -/afii10101 16#0454 -/afii10102 16#0455 -/afii10103 16#0456 -/afii10104 16#0457 -/afii10105 16#0458 -/afii10106 16#0459 -/afii10107 16#045a -/afii10108 16#045b -/afii10109 16#045c -/afii10110 16#045e -/afii10145 16#040f -/afii10146 16#0462 -/afii10147 16#0472 -/afii10148 16#0474 -/afii10192 16#f6c6 -/afii10193 16#045f -/afii10194 16#0463 -/afii10195 16#0473 -/afii10196 16#0475 -/afii10831 16#f6c7 -/afii10832 16#f6c8 -/afii10846 16#04d9 -/afii299 16#200e -/afii300 16#200f -/afii301 16#200d -/afii57381 16#066a -/afii57388 16#060c -/afii57392 16#0660 -/afii57393 16#0661 -/afii57394 16#0662 -/afii57395 16#0663 -/afii57396 16#0664 -/afii57397 16#0665 -/afii57398 16#0666 -/afii57399 16#0667 -/afii57400 16#0668 -/afii57401 16#0669 -/afii57403 16#061b -/afii57407 16#061f -/afii57409 16#0621 -/afii57410 16#0622 -/afii57411 16#0623 -/afii57412 16#0624 -/afii57413 16#0625 -/afii57414 16#0626 -/afii57415 16#0627 -/afii57416 16#0628 -/afii57417 16#0629 -/afii57418 16#062a -/afii57419 16#062b -/afii57420 16#062c -/afii57421 16#062d -/afii57422 16#062e -/afii57423 16#062f -/afii57424 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b/documentation/source/science_guide/cloud_schemes/Timestepping_pc266.epsi deleted file mode 100644 index 92fceeff6d..0000000000 --- a/documentation/source/science_guide/cloud_schemes/Timestepping_pc266.epsi +++ /dev/null @@ -1,10230 +0,0 @@ -%!PS-Adobe-3.0 EPSF-3.0 -%%Invocation: path/gs -q -sDEVICE=eps2write -sstdout=? -sOutputFile=? -dNOPAUSE -dBATCH -P- -dSAFER -dDEVICEWIDTH=250000 -dDEVICEHEIGHT=250000 ? -%%BoundingBox: 0 0 720 960 -%%HiResBoundingBox: 0.00 0.00 720.00 960.00 -%%Creator: GPL Ghostscript 9540 (eps2write) -%%LanguageLevel: 2 -%%CreationDate: D:20250417013840+01'00' -%%Pages: 1 -%%EndComments -%%BeginProlog -10 dict dup begin -/DSC_OPDFREAD true def -/SetPageSize false def -/EPS2Write true def -end -count 0 ne{ -dup type/dicttype eq{ -dup/EPS2Write known{ -dup/EPS2Write get not -} -{ -true -}ifelse -} -{ -true -}ifelse -} -{ -true -}ifelse -10 dict begin -/this currentdict def -/y 720 def -/ebuf 200 string def -/prnt{ -36//this/y get moveto//ebuf cvs show -//this/y 2 copy get 12 sub put -}bind def -/newline{ -36//this/y get moveto -//this/y 2 copy get 12 sub put -}bind def -{ -errordict/handleerror -{systemdict begin -$error begin -newerror -{(%%[ Error handled by opdfread.ps : )print errorname//ebuf cvs print(; OffendingCommand: ) -print/command load//ebuf cvs print( ]%%)= flush -/newerror false store vmstatus pop pop 0 ne -{grestoreall -}if -errorname(VMerror)ne -{showpage -}if -initgraphics -0 720 moveto -errorname(VMerror)eq -{//this/ehsave known -{clear//this/ehsave get restore 2 vmreclaim -}if -vmstatus exch pop exch pop -} -/Courier 12 selectfont -{ -(ERROR: )//prnt exec errorname//prnt exec -(OFFENDING COMMAND: )//prnt exec -/command load//prnt exec -$error/ostack known{ -(%%[STACK:)= -(STACK:)//prnt exec -$error/ostack get aload length{ -//newline exec -dup mark eq{ -(-mark-)dup = show -}{ -dup type/nametype eq{ -dup xcheck not{ -(/)show -(/)print -}if -}if -dup =//ebuf cvs show -}ifelse -}repeat -}if -}ifelse -(%%]%)= -//systemdict/showpage get exec -quit -}if -end -end -}bind readonly put -}if -end -50 dict begin -count 0 ne{ -dup type/dicttype eq{ -{def}forall -false -} -{ -true -}ifelse -} -{ -true -}ifelse -{ -( *** Warning: global definitions dictionary not found, file may be corrupted.\n)print flush -}if -/DefaultSwitch -{ -dup where{ -pop pop -}{ -false def -}ifelse -}bind def -/=string 256 string def -/=only{ -//=string cvs print -}bind def -/HexDigits(0123456789ABCDEF)readonly def -/PrintHex -{8{ -dup -28 bitshift 15 and//HexDigits exch 1 getinterval//=only exec -4 bitshift -}repeat -pop -}bind def -/PDFR_DEBUG DefaultSwitch -/PDFR_DUMP DefaultSwitch -/PDFR_STREAM DefaultSwitch -/TTFDEBUG DefaultSwitch -/RotatePages DefaultSwitch -/FitPages DefaultSwitch -/CenterPages DefaultSwitch -/SetPageSize DefaultSwitch -/error -{ -counttomark 1 sub -1 0{ -index dup type/arraytype eq{==}{=only}ifelse -}for -()= -cleartomark -....Undefined -}bind def -//SetPageSize{ -//RotatePages//FitPages or//CenterPages or{ -mark(/RotatePages, /FitPages and CenterPages are not allowed with /SetPageSize)//error exec -}if -} -{ -//FitPages//CenterPages and{ -mark(CenterPages is not allowed with /FitPages)//error exec -}if -} -ifelse -/knownget -{ -2 copy known{ -get true -}{ -pop pop false -}ifelse -}bind def -/IsUpper -{dup(A)0 get ge exch(Z)0 get le and -}bind def -/cpa2g{ -dup length array -0 1 2 index length 1 sub{ -dup 3 index exch get cp2g -3 copy put pop pop -}for -exch pop -}bind def -/cpd2g{ -dup length dict exch{ -cp2g 2 index 3 1 roll put -}forall -}bind def -/cps2g{ -dup length string copy -}bind def -/cp2gprocs -<> -def -/cp2g{ -dup gcheck not{ -dup//cp2gprocs 1 index type -2 copy known{ -get currentglobal 3 1 roll true setglobal exec exch setglobal -1 index wcheck not{readonly}if -1 index xcheck{cvx}if -exch pop -}{ -pop pop -}ifelse -}if -}bind def -/BlockBuffer 65535 string def -/PDFReader currentdict def -/ObjectRegistryMaxLength 50000 def -/ObjectRegistry 10 dict def -ObjectRegistry -begin -0 ObjectRegistryMaxLength dict def -end -/CurrentObject null def -/DoneDocumentStructure false def -/GraphicState 20 dict begin -/InitialTextMatrix matrix def -/InitialMatrix matrix currentmatrix def -currentdict end def -/TempMatrix matrix def -/GraphicStateStack 20 array def -/GraphicStateStackPointer 0 def -/InitialTextMatrixStack 20 array def -/InitialTextMatrixStackPointer 0 def -/PDFColorSpaces 50 dict def -/InstalledFonts 50 dict def -/MacRomanEncodingInverse null def -currentglobal false setglobal -userdict/PDFR_InitialGS gstate put -userdict/PDFR_Patterns 50 dict put -userdict/FuncDataReader 10 dict put -setglobal -/InitialExtGState 20 dict begin -/BG2 currentblackgeneration cp2g def -/UCR2 currentundercolorremoval cp2g def -/TR2 currentglobal false setglobal[currentcolortransfer]exch setglobal cp2g def -/HT currenthalftone cp2g def -currentdict end readonly def -/InitialGraphicState 20 dict begin -/FontSize 0 def -/CharacterSpacing 0 def -/TextLeading 0 def -/TextRenderingMode 0 def -/WordSpacing 0 def -currentdict end readonly def -/SimpleColorSpaceNames 15 dict begin -/DeviceGray true def -/DeviceRGB true def -/DeviceCMYK true def -currentdict end readonly def -/1_24_bitshift_1_sub 1 24 bitshift 1 sub def -/ReadFontProcs 10 dict def -/GetObject -{ -dup ObjectRegistryMaxLength idiv -//PDFReader/ObjectRegistry get exch knownget{ -exch knownget -}{ -pop false -}ifelse -}bind def -/PutObject -{ -1 index ObjectRegistryMaxLength idiv -//PDFReader/ObjectRegistry get 1 index knownget{ -exch pop -3 1 roll put -}{ -//PDFReader/ObjectRegistry get dup -begin -1 index ObjectRegistryMaxLength dict def -end -exch get -3 1 roll put -}ifelse -}bind def -/Register -{ -1 index GetObject{ -dup xcheck{ -4 3 roll pop -//PDFR_DEBUG{ -(Have a daemon for )print 2 index == -}if -exec -}{ -dup null ne{ -mark(The object )4 index(is already defined : )4 index//error exec -}{ -pop -}ifelse -3 2 roll -exec -}ifelse -}{ -3 2 roll -exec -}ifelse -PutObject -}bind def -/IsRegistered -{ -GetObject{ -null ne -}{ -false -}ifelse -}bind def -/GetRegistered -{ -dup GetObject not{ -exch mark exch(Object )exch( isn't defined before needed (1).)//error exec -}if -dup xcheck{ -exch mark exch(Object )exch( isn't defined before needed (2).)//error exec -}{ -dup null eq{ -exch mark exch(Object )exch( isn't defined before needed (3).)//error exec -}if -exch pop -}ifelse -}bind def -/StandardFontNames<< -/Times-Roman true -/Helvetica true -/Courier true -/Symbol true -/Times-Bold true -/Helvetica-Bold true -/Courier-Bold true -/ZapfDingbats true -/Times-Italic true -/Helvetica-Oblique true -/Courier-Oblique true -/Times-BoldItalic true -/Helvetica-BoldOblique true -/Courier-BoldOblique true ->>def -/CleanAllResources -{//PDFR_DEBUG{ -(CleanAllResources beg)= -}if -//PDFReader/ObjectRegistry get{ -dup length 0 exch 1 exch 1 sub{ -2 copy get dup xcheck{ -pop pop -}{ -dup null eq{ -pop pop -}{ -dup type/dicttype eq{/.Global known}{pop false}ifelse{ -pop -}{ -//PDFR_DEBUG{ -(Dropping )print dup = -}if -1 index exch/DroppedObject put -}ifelse -}ifelse -}ifelse -}for -pop -}forall -FontDirectory length dict begin -FontDirectory{ -pop -dup//StandardFontNames exch known not{ -dup null def -}if -pop -}forall -currentdict -end{ -pop -//PDFR_DEBUG{ -(Undefining font )print dup = -}if -undefinefont -}forall -//PDFR_DEBUG{ -(CleanAllResources end)= -}if -}bind def -/PrintReference -{ -//PDFR_DEBUG{ -({ )print -dup{ -=only( )print -}forall -( })= -}if -}bind def -/R -{ -0 ne{ -exch mark exch(A referred object generation )exch( isn't 0.)//error exec -}if -[ -exch//GetRegistered/exec load -]cvx -//PrintReference exec -}bind def -/IsObjRef -{ -dup type/arraytype eq{ -dup length 3 eq{ -dup xcheck exch -dup 0 get type/integertype eq 3 2 roll and exch -dup 1 get//GetRegistered eq 3 2 roll and exch -2 get/exec load eq and -}{ -pop false -}ifelse -}{ -pop false -}ifelse -}bind def -/DoNothing -{ -}def -/RunTypeDaemon -{ -dup type/dicttype eq{ -dup/Type//knownget exec{ -//PDFReader/TypeDaemons get exch -//knownget exec{ -exec -}if -}if -}if -}bind def -/obj -{ -//PDFR_DEBUG{ -(Defining )print 1 index =only( )print dup =only( obj)= -}if -0 ne{ -exch mark exch(An object generation )exch( isn't 0.)//error exec -}if -}bind def -/endobj -{ -//PDFR_DEBUG{ -(endobj )= -}if -count 1 eq{ -pop -}{ -dup type/dicttype eq{ -dup/.endobj_daemon//knownget exec{ -//PDFR_DEBUG{(.endobj_daemon for )print 2 index =}if -exec -}if -}if -dup type/dicttype eq{dup/ImmediateExec known}{false}ifelse{ -pop pop -}{ -//PDFR_DEBUG{ -(Storing )print 1 index = -}if -//RunTypeDaemon exec -//DoNothing 3 1 roll//Register exec -}ifelse -}ifelse -}bind def -/StoreBlock -{ -//PDFR_DEBUG{ -(StoreBlock )print//PDFReader/BlockCount get =only(, Length = )print dup length = -}if -dup length string copy -//PDFReader/BlockCount get exch -//PDFReader/CurrentObject get 3 1 roll -put -//PDFReader/BlockCount get 1 add -//PDFReader exch/BlockCount exch put -}bind def -/CheckLength -{dup type/integertype ne{ -mark(Object length isn't an integer.)//error exec -}if -}bind def -/ResolveD -{ -3 copy pop get -dup//IsObjRef exec{ -//PDFR_DEBUG{ -(Resolving )print//PrintReference exec -}if -exec -exch exec -}{ -exch pop -}ifelse -dup 4 1 roll -put -}bind def -/ResolveA -{2 index 2 index get -dup//IsObjRef exec{ -exec -exch exec -3 copy put -}{ -exch pop -}ifelse -exch pop exch pop -}bind def -/StoreStream -{ -dup//PDFReader exch/CurrentObject exch put -//PDFReader/BlockCount 0 put -dup/Length//CheckLength//ResolveD exec -//PDFR_DEBUG{ -(StoreStream Length = )print dup = -}if -currentfile exch()/SubFileDecode filter -{dup//BlockBuffer readstring{ -//StoreBlock exec -}{ -//StoreBlock exec -exit -}ifelse -}loop -pop -//PDFReader/CurrentObject null put -//PDFR_DEBUG{ -(StoreStream end.)= -}if -}bind def -/MakeStreamDumper -{ -//PDFR_DEBUG{ -(MakeStreamDumper beg.)= -}if -currentglobal exch dup gcheck setglobal -[exch -1 dict dup/c 0 put exch -1024 string -{readstring pop -(StreamDumper )print 1 index/c get =string cvs print( )print -dup length =string cvs print( <)print dup print(>\n)print -dup length -3 2 roll -dup/c get -3 2 roll -add/c exch put -}/exec load -] -cvx 0()/SubFileDecode filter -exch setglobal -//PDFR_DEBUG{ -(MakeStreamDumper end.)= -}if -}bind def -/ShortFilterNames 15 dict begin -/AHx/ASCIIHexDecode def -/A85/ASCII85Decode def -/LZW/LZWDecode def -/Fl/FlateDecode def -/RL/RunLengthDecode def -/CCF/CCITTFaxDecode def -/DCT/DCTDecode def -currentdict end readonly def -/AppendFilters -{ -//PDFR_DEBUG{ -(AppendFilters beg.)= -}if -dup 3 1 roll -/Filter//knownget exec{ -dup type/nametype eq{ -dup//ShortFilterNames exch//knownget exec{ -exch pop -}if -2 index/DecodeParms//knownget exec{ -exch -}if -filter -}{ -dup 0 exch 1 exch length 1 sub{ -2 copy get -dup//ShortFilterNames exch//knownget exec{ -exch pop -}if -3 1 roll -4 index/DecodeParms//knownget exec{ -exch get -}{ -pop null -}ifelse -dup null eq{ -pop 3 1 roll filter exch -}{ -3 1 roll -4 1 roll filter exch -}ifelse -}for -pop -}ifelse -//PDFR_DEBUG//PDFR_DUMP and{ -//MakeStreamDumper exec -}if -}if -exch pop -//PDFR_DEBUG{ -(AppendFilters end.)= -}if -}bind def -/ExecuteStream -{ -dup//PDFReader exch/CurrentObject exch put -dup/Length//CheckLength//ResolveD exec -//PDFR_DEBUG{ -(ExecuteStream id = )print 2 index =only( Length = )print dup = -}if -//PDFReader/InitialGraphicState get -//PDFReader/GraphicState get copy pop -//PDFReader/Operators get begin -currentfile exch()/SubFileDecode filter -1 index//AppendFilters exec -cvx mark exch -exec -counttomark 0 ne{ -mark(Data left on ostack after an immediate stream execution.)//error exec -}if -cleartomark -end -//PDFR_DEBUG{ -(ExecuteStream end.)= -}if -//PDFReader/CurrentObject null put -dup/IsPage known{ -dup/Context get/NumCopies//knownget exec{ -1 sub{ -copypage -}repeat -}if -EPS2Write not{showpage}if -pagesave restore -}if -}bind def -/stream -{ -//PDFR_DEBUG{ -1 index =only( stream)= -}if -1 index GetObject{ -dup xcheck{ -exec -1 index null PutObject -}{ -pop -}ifelse -}if -dup/ImmediateExec known{ -dup/GlobalExec//knownget exec{ -currentglobal 4 1 roll -setglobal -//ExecuteStream exec -3 2 roll setglobal -}{ -//ExecuteStream exec -}ifelse -}{ -//StoreStream exec -}ifelse -dup/.CleanResources//knownget exec{ -/All eq{ -//CleanAllResources exec -}if -}if -}bind def -/HookFont -{ -//PDFR_DEBUG{ -(Loaded the font )print dup/FontName get = -}if -{ -dup/FontFileType get dup/Type1 eq exch/MMType1 eq or{ -dup/FontName get -//PDFReader/RemoveFontNamePrefix get exec -findfont -exit -}if -dup/FontFileType get/TrueType eq{ -//PDFReader/MakeType42 get exec -//PDFR_DEBUG{ -(Font dict <<)= -dup{ -1 index/sfnts eq{ -exch pop -(/sfnts [)print -{ -(-string\()print length//=only exec(\)- )= -}forall -(])= -}{ -exch//=only exec( )print == -}ifelse -}forall -(>>)= -}if -dup/FontName get exch definefont -exit -}if -mark(FontHook has no proc for )2 index/FontFileType get//error exec -}loop -/Font exch put -}bind def -/endstream -{ -}bind def -/xref -{ -//PDFR_DEBUG{ -(xref)= -//PDFR_DUMP{ -//PDFReader/ObjectRegistry get == -}if -}if -end -count 0 ne{ -mark(Excessive data on estack at the end of the interpretation.)//error exec -}if -currentfile 1(%%EOF)/SubFileDecode filter -flushfile -cleardictstack -}bind def -/ResolveDict -{dup{ -pop 1 index exch -//DoNothing//ResolveD exec -pop -}forall -pop -}bind def -/SetupPageView -{ -//PDFR_DEBUG{ -(SetupPageView beg)= -}if -//DSC_OPDFREAD not{ -//GraphicState/InitialMatrix get setmatrix -}if -/MediaBox get aload pop -3 index neg 3 index neg translate -3 -1 roll sub 3 1 roll exch sub exch -userdict/.HWMargins//knownget exec{ -aload pop -}{ -currentpagedevice/.HWMargins//knownget exec{ -aload pop -}{ -0 0 0 0 -}ifelse -}ifelse -currentpagedevice/PageSize get aload pop -3 -1 roll sub 3 1 roll exch sub exch -exch 3 index sub exch 3 index sub -//SetPageSize{ -//PDFR_DEBUG{ -(Setting page size to )print 1 index//=only exec( )print dup = -}if -pop pop 3 index 3 index 2 copy -currentglobal false setglobal 3 1 roll -currentpagedevice dup/PageSize known{ -/PageSize get aload pop -}{ -0 0 -}ifelse -round cvi 2 index round cvi eq -exch round cvi 3 index round cvi eq and -{ -//PDFR_DEBUG{(PageSize matches request)== flush}if -pop pop -}{ -/MediaRequested where{ -//PDFR_DEBUG{(MediaRequested is true, check against new request)== flush}if -/MediaRequested get aload pop -round cvi 2 index round cvi eq -exch round cvi 3 index round cvi eq and -{ -//PDFR_DEBUG{(MediaRequested same as current request, ignore)== flush}if -pop pop false -}{ -//PDFR_DEBUG{(MediaRequested different to current request)== flush}if -true -}ifelse -}{ -//PDFR_DEBUG{(No MediaRequested yet)== flush}if -true -}ifelse -{ -//PDFR_DEBUG{(Setting pagesize)== flush}if -2 array astore -dup/MediaRequested exch def -<< exch/PageSize exch >>setpagedevice -}if -}ifelse -userdict/PDFR_InitialGS gstate put -setglobal -}if -//RotatePages{ -2 copy gt 6 index 6 index gt ne{ -1 index 5 index le 1 index 5 index le and not -}{ -false -}ifelse -}{ -false -}ifelse -{//CenterPages{ -//PDFR_DEBUG{ -(Rotating page, and then centering it)== -}if -90 rotate -0 5 index neg translate -5 index 1 index exch sub 2 div -2 index 6 index sub 2 div neg -translate -}{ -//FitPages{ -1 index 5 index div 1 index 7 index div -2 copy gt{ -exch -}if -pop dup scale -}if -90 rotate -0 5 index neg translate -}ifelse -}{ -//CenterPages{ -//PDFR_DEBUG{ -(Ccentering page)== -}if -1 index 6 index sub 2 div -1 index 6 index sub 2 div -translate -}{ -//FitPages{ -1 index 6 index div 1 index 6 index div -2 copy gt{ -exch -}if -pop dup scale -}if -}ifelse -}ifelse -pop pop -translate -pop pop -//PDFR_DEBUG{ -(SetupPageView end)= -}if -}bind def -/PageContentsDaemon -{ -//PDFR_DEBUG{ -(Executing PageContentsDaemon for )print 2 index = -}if -1 index exch/Context exch put -dup/ImmediateExec true put -/pagesave save def -dup/IsPage true put -SetPageSize{dup/Context get//SetupPageView exec}if -}bind def -/FontFileDaemon -{ -//PDFR_DEBUG{ -(Executing FontFileDaemon for )print 2 index = -}if -dup/FontFileType get -2 index exch -dup//ReadFontProcs exch//knownget exec{ -exch pop exec -}{ -mark(FontFile reader for )2 index( isn't implemented yet.)//error exec -}ifelse -//PDFR_DEBUG{ -(FontFileDaemon end)= -}if -pop -}bind def -/FontDescriptorDaemon -{ -//PDFR_DEBUG{ -(Executing FontDescriptorDaemon for )print 2 index = -}if -2 copy/FontResource exch put -/Subtype get 1 index exch/FontFileType exch put -}bind def -/UnPDFEscape{ -dup dup length string cvs -dup(#)search{ -{ -pop -(16#--)2 index 0 2 getinterval -1 index 3 2 getinterval copy pop -cvi -0 exch put -0 -1 index 2 1 index length 2 sub getinterval -3 copy putinterval -length -3 copy exch put -getinterval -(#)search not{ -pop exit -}if -}loop -(\0)search pop exch pop exch pop -cvn -exch pop -}{ -pop pop -}ifelse -}bind def -/TypeDaemons<< -/Page -{//PDFR_DEBUG{ -(Recognized a page.)= -}if -dup/Contents//knownget exec{ -0 get//DoNothing exch -[ -3 index//PageContentsDaemon/exec load -]cvx -//Register exec -}{ -(fixme: page with no Contents won't be printed.)= -}ifelse -}bind -/FontDescriptor -{//PDFR_DEBUG{ -(Recognized a font descriptor.)= -}if -dup/FontName//knownget exec{ -1 index/FontName 3 -1 roll//UnPDFEscape exec put -}if -dup dup/FontFile known{/FontFile}{/FontFile2}ifelse -//knownget exec{ -0 get//DoNothing exch -[ -3 index//FontFileDaemon/exec load -]cvx -//Register exec -}{ -(Font descriptor )print 1 index =only( has no FontFile.)= -}ifelse -}bind -/Font -{//PDFR_DEBUG{ -(Recognized a font resource.)= -}if -dup/BaseFont//knownget exec{ -//UnPDFEscape exec 2 copy/BaseFont exch put -//PDFReader/RemoveFontNamePrefix get exec -currentglobal exch -dup/Font resourcestatus{ -pop pop -//PDFReader/GetInstalledFont get exec pop -}{ -pop -}ifelse -setglobal -}if -dup/FontDescriptor//knownget exec{ -0 get -dup//IsRegistered exec{ -//PDFR_DEBUG{ -(already registered )print dup = -}if -pop -}{ -//DoNothing exch -[ -3 index//FontDescriptorDaemon/exec load -]cvx -//Register exec -}ifelse -}if -}bind ->>def -/MakeStreamReader -{dup -[ -exch -//PDFR_DEBUG{ -(Stream proc ) -/print load -//PDFR_STREAM{ -(<) -/print load -}if -}if -1 dict dup/i -1 put -/dup load -/i -/get load -1 -/add load -/dup load -3 -1 -/roll load -/i -/exch load -/put load -//knownget -/exec load -/not load -{()} -/if load -//PDFR_DEBUG{ -//PDFR_STREAM{ -/dup load -/print load -(>) -/print load -}if -( end of stream proc.\n) -/print load -}if -]cvx -//PDFR_DEBUG{ -(Stream reader )print dup == -}if -0()/SubFileDecode filter -exch//AppendFilters exec -}bind def -/RunDelayedStream -{ -//GraphicState/InitialTextMatrix get -//InitialTextMatrixStack//PDFReader/InitialTextMatrixStackPointer get -2 copy get null eq{ -2 copy currentglobal true setglobal matrix exch setglobal put -}if -get copy pop -//PDFReader/InitialTextMatrixStackPointer 2 copy get 1 add put -//MakeStreamReader exec -mark exch -cvx exec -counttomark 0 ne{ -mark(Data left on ostack after a delayed stream execution.)//error exec -}if -cleartomark -//PDFReader/InitialTextMatrixStackPointer 2 copy get 1 sub put -//InitialTextMatrixStack//PDFReader/InitialTextMatrixStackPointer get get -//GraphicState/InitialTextMatrix get -copy pop -}bind def -//ReadFontProcs begin -/Type1 -{//PDFR_DEBUG{ -(ReadFontProcs.Type1)= -}if -dup/.endobj_daemon[4 index//HookFont/exec load]cvx put -dup/ImmediateExec true put -/GlobalExec true put -}bind def -/MMType1//Type1 def -/TrueType -{//PDFR_DEBUG{ -(ReadFontProcs.TrueType)= -}if -dup/.endobj_daemon[4 index//HookFont/exec load]cvx put -pop -}bind def -end -/.opdloadttfontdict 50 dict def -.opdloadttfontdict begin -/maxstring 65400 def -end -/.InsertionSort -{ -/CompareProc exch def -/Array exch def -1 1 Array length 1 sub -{ -/Ix exch def -/Value1 Array Ix get def -/Jx Ix 1 sub def -{ -Jx 0 lt{ -exit -}if -/Value2 Array Jx get def -Value1 Value2 CompareProc{ -exit -}if -Array Jx 1 add Value2 put -/Jx Jx 1 sub def -}loop -Array Jx 1 add Value1 put -}for -Array -}bind def -/putu16{ -3 copy -8 bitshift put -exch 1 add exch 16#ff and put -}bind def -/putu32{ -3 copy -16 bitshift putu16 -exch 2 add exch 16#ffff and putu16 -}bind def -/.readtable{ -dup dup 1 and add string -dup 0 4 -1 roll getinterval -3 -1 roll exch -dup()ne{readstring}if pop pop -}bind def -/.readbigtable{ -dup maxstring lt{ -.readtable -}{ -currentuserparams/VMReclaim get -2 vmreclaim -[4 2 roll{ -dup maxstring le{exit}if -1 index maxstring string readstring pop 3 1 roll maxstring sub -}loop .readtable] -exch vmreclaim -}ifelse -}bind def -/ReadTTF -{ -.opdloadttfontdict begin -/TTFontFile exch def -/TableDir TTFontFile 12 string readstring pop def -/tables TTFontFile TableDir 4 getu16 16 mul string readstring pop def -/tabarray tables length 16 idiv array def -TableDir 0 4 getinterval(ttcf)eq{ -QUIET not{(Can't handle TrueType font Collections.)=}if -/.loadttfonttables cvx/invalidfont signalerror -}{ -0 16 tables length 1 sub{ -dup -tables exch 16 getinterval -exch 16 div cvi exch -tabarray 3 1 roll put -}for -}ifelse -tabarray{exch 8 getu32 exch 8 getu32 gt}.InsertionSort pop -/Read TableDir length tables length add def -/tabs[ -tabarray{ -dup 8 getu32 -Read sub -dup 0 gt{ -dup string TTFontFile exch readstring pop pop -Read add/Read exch def -}{ -pop -}ifelse -12 getu32 -dup Read add -/Read exch def -TTFontFile exch .readbigtable -}forall -]def -end -}bind def -/GetLocaType -{ -0 1 tabarray length 1 sub{ -dup tabarray exch get -0 4 getinterval(head)eq{ -tabs exch get -50 gets16 -/LocaType exch def -exit -}{ -pop -}ifelse -}for -}bind def -/GetNumGlyphs -{ -0 1 tabarray length 1 sub{ -dup tabarray exch get -0 4 getinterval(maxp)eq{ -tabs exch get -4 getu16 -/NumGlyphs exch def -exit -}{ -pop -}ifelse -}for -}bind def -/StringToLoca -{ -/LocaIndex exch def -/StringOffset 0 def -{ -dup length StringOffset gt{ -dup -LocaType 1 eq{ -StringOffset getu32 -LocaArray LocaIndex 3 -1 roll put -/LocaIndex LocaIndex 1 add def -/StringOffset StringOffset 4 add -def -}{ -StringOffset getu16 2 mul -LocaArray length LocaIndex gt{ -LocaArray LocaIndex 3 -1 roll put -}{ -pop -}ifelse -/LocaIndex LocaIndex 1 add def -/StringOffset StringOffset 2 add -def -}ifelse -}{ -pop -LocaIndex -exit -}ifelse -}loop -}bind def -/GetSortedLoca -{ -NumGlyphs 1 add array/LocaArray exch def -0 1 tabarray length 1 sub{ -dup tabarray exch get -0 4 getinterval(loca)eq{ -tabs exch get -exit -}{ -pop -}ifelse -}for -dup type/stringtype eq{ -0 StringToLoca pop -}{ -0 exch -{ -exch StringToLoca -}forall -pop -}ifelse -LocaArray{gt}.InsertionSort pop -}bind def -/GetWorkingString -{ -WorkString 0 -GlyfArray GlyfStringIndex get -putinterval -/WorkBytes GlyfArray GlyfStringIndex get length def -/GlyfStringIndex GlyfStringIndex 1 add def -}bind def -/GetWorkingBytes -{ -/BytesToRead exch def -WorkString 0 BytesToRead getinterval -dup length string copy -WorkString BytesToRead WorkBytes BytesToRead sub getinterval -dup length string copy -WorkString 0 3 -1 roll putinterval -/WorkBytes WorkBytes BytesToRead sub def -}bind def -/GetGlyfBytes -{ -/ToRead exch def -WorkBytes 0 eq{ -GetWorkingString -}if -WorkBytes ToRead ge{ -ToRead string dup 0 -ToRead GetWorkingBytes putinterval -}{ -ToRead string -dup -0 -WorkString 0 WorkBytes getinterval -putinterval -dup -WorkBytes -ToRead WorkBytes sub -GetWorkingString -GetWorkingBytes -putinterval -}ifelse -}bind def -/SplitGlyf -{ -/GlyfArray exch def -/DestArray GlyfArray length 2 mul array def -/DestArrayIndex 0 def -/LastLoca 0 def -/NextLocaIndex 0 def -/LastLocaIndex 0 def -/GlyfStringIndex 0 def -/WorkString maxstring string def -/WorkBytes 0 def -{ -LocaArray NextLocaIndex get -LastLoca sub maxstring gt -{ -LocaArray LastLocaIndex get LastLoca sub -GetGlyfBytes -DestArray DestArrayIndex 3 -1 roll put -/DestArrayIndex DestArrayIndex 1 add def -LocaArray LastLocaIndex get/LastLoca exch def -}{ -/LastLocaIndex NextLocaIndex def -/NextLocaIndex NextLocaIndex 1 add def -NextLocaIndex NumGlyphs gt -{ -WorkBytes -GlyfStringIndex GlyfArray length lt{ -GlyfArray GlyfStringIndex get length -add string dup -0 -WorkString 0 WorkBytes getinterval -putinterval -dup -WorkBytes -GetWorkingString -WorkString 0 WorkBytes getinterval -putinterval -}{ -pop -WorkString 0 WorkBytes getinterval -}ifelse -dup length string copy -DestArray DestArrayIndex 3 -1 roll put -exit -}if -}ifelse -}loop -DestArray -}bind def -/ProcessTTData -{ -.opdloadttfontdict begin -0 1 tabarray length 1 sub{ -/ix exch def -tabarray ix get -12 getu32 dup maxstring le{ -dup 4 mod 0 ne{ -4 div cvi 1 add 4 mul string/newstring exch def -/oldstring tabs ix get def -newstring 0 oldstring putinterval -0 1 newstring length oldstring length sub 1 sub{ -newstring exch oldstring length add 0 put -}for -tabs ix newstring put -}{ -pop -}ifelse -}{ -dup 4 mod 0 ne{ -dup maxstring idiv maxstring mul sub -4 idiv 1 add 4 mul string/newstring exch def -tabs ix get -dup length 1 sub dup/iy exch def get/oldstring exch def -newstring 0 oldstring putinterval -0 1 newstring length oldstring length sub 1 sub{ -newstring exch oldstring length add 0 put -}for -tabs ix get iy newstring put -}{ -pop -}ifelse -}ifelse -}for -0 1 tabarray length 1 sub{ -dup tabarray exch get -dup 12 getu32 maxstring gt{ -0 4 getinterval dup(glyf)eq{ -pop -GetLocaType -GetNumGlyphs -GetSortedLoca -dup tabs exch get -SplitGlyf -tabs 3 1 roll put -}{ -(Warning, table )print print( > 64Kb\n)print -pop -}ifelse -}{ -pop -pop -}ifelse -}for -end -}bind def -/Makesfnts -{ -.opdloadttfontdict begin -0 -tabs{ -dup type/stringtype eq{ -pop -1 add -}{ -{ -type/stringtype eq{ -1 add -}if -}forall -}ifelse -}forall -1 add -/TTOffset -TableDir length -tabarray length 16 mul add -def -0 -tabarray{ -exch dup 1 add -3 1 roll -dup -tabs exch get -dup type/stringtype eq{ -length -2 index exch -TTOffset -dup 3 1 roll add -/TTOffset exch def -8 exch putu32 -exch tabarray 3 1 roll -put -}{ -0 exch -{ -dup type/stringtype eq{ -length add -}{ -pop -}ifelse -}forall -2 index exch -TTOffset -dup 3 1 roll add -/TTOffset exch def -8 exch putu32 -exch tabarray 3 1 roll -put -}ifelse -}forall -pop -array -dup 0 -TableDir length -tables length add -string -dup 0 TableDir putinterval -dup 12 tables putinterval -put -dup -/ix 1 def -tabs{ -dup type/stringtype eq{ -ix exch -put dup -/ix ix 1 add def -}{ -{ -dup type/stringtype eq{ -ix exch put dup -/ix ix 1 add def -}{ -pop -}ifelse -}forall -}ifelse -}forall -pop -end -}bind def -/MakeType42 -{ -//PDFR_DEBUG{ -(MakeType42 beg)= -}if -10 dict begin -/FontName 1 index/FontName get def -/FontType 42 def -/FontMatrix[1 0 0 1 0 0]def -/FontBBox 1 index/FontBBox get def -dup/FontResource get -dup/Encoding known{ -//PDFReader/ObtainEncoding get exec -/Encoding get -}{ -pop null -}ifelse -/PDFEncoding exch def -/CharStrings 2 index//PDFReader/MakeTTCharStrings get exec def -/sfnts 2 index//MakeStreamReader exec -ReadTTF -ProcessTTData -Makesfnts -def -/Encoding StandardEncoding def -/PaintType 0 def -currentdict end -//PDFR_DEBUG{ -(MakeType42 end)= -}if -}bind def -/GetInstalledFont -{ -dup//InstalledFonts exch knownget{ -exch pop -}{ -dup findfont dup 3 1 roll -//InstalledFonts 3 1 roll put -}ifelse -}bind def -/RemoveFontNamePrefix -{//=string cvs true -0 1 5{ -2 index exch get//IsUpper exec not{ -pop false exit -}if -}for -{(+)search{ -pop pop -}if -}if -cvn -}bind def -/CheckFont -{dup/Type get/Font ne{ -mark(Resource )3 index( must have /Type/Font .)//error exec -}if -}bind def -/CheckEncoding -{dup type/nametype ne{ -dup/Type get/Encoding ne{ -mark(Resource )3 index( must have /Type/Encoding .)//error exec -}if -}if -}bind def -/ObtainEncoding -{dup/Encoding known{ -dup dup/Encoding//CheckEncoding//ResolveD exec -dup type dup/arraytype eq exch/packedarraytype eq or{ -pop pop -}{ -dup type/nametype eq{ -/Encoding findresource -}{ -dup/BaseEncoding//knownget exec not{ -/StandardEncoding -}if -/Encoding findresource -exch -/Differences//knownget exec{ -exch dup length array copy exch -0 exch -{ -dup type/integertype eq{ -exch pop -}{ -3 copy put pop -1 add -}ifelse -}forall -pop -}if -}ifelse -/Encoding exch put -}ifelse -}{ -dup/Encoding/StandardEncoding/Encoding findresource put -}ifelse -}bind def -/ObtainMetrics -{dup/Widths//knownget exec{ -1 index/Encoding get -256 dict -3 index/Subtype get/TrueType eq{ -1000 -}{ -1 -}ifelse -4 index/MissingWidth//knownget exec not{ -0 -}if -5 index/FirstChar//knownget exec not{ -0 -}if -6 5 roll -dup 0 exch 1 exch length 1 sub{ -2 copy get -exch 3 index add -7 index exch get -dup dup null ne exch/.notdef ne and{ -6 index 3 1 roll exch -6 index div -3 copy pop//knownget exec{ -0 eq -}{ -true -}ifelse -{put -}{ -pop pop pop -}ifelse -}{ -pop pop -}ifelse -}for -pop pop pop pop exch pop -1 index exch/Metrics exch put -}{ -dup/MissingWidth//knownget exec{ -256 dict -2 index/Encoding get{ -dup null ne{ -3 copy 3 2 roll put -}if -pop -}forall -exch pop -1 index exch/Metrics exch put -}if -}ifelse -}bind def -/NotDef -{ -FontMatrix aload pop pop pop exch pop exch pop -1 exch div exch -1 exch div exch -1 index 0 setcharwidth -0 setlinewidth -0 0 moveto -2 copy rlineto -1 index 0 rlineto -neg exch neg exch rlineto -closepath stroke -}bind def -/SaveResourcesToStack -{ -[ -//PDFReader/OldResources known{ -//PDFReader/OldResources get -}{ -null -}ifelse -//PDFReader/CurrentObject get/Context get/Resources get -] -//PDFReader/OldResources 3 -1 roll put -}bind def -/RestoreResourcesFromStack -{ -//PDFReader/OldResources get dup -0 get//PDFReader/OldResources 3 -1 roll put -1 get//PDFReader/CurrentObject get/Context get/Resources 3 -1 roll put -}bind def -/BuildChar -{//PDFR_DEBUG{ -(BuildChar )print dup//=only exec( )print -}if -exch begin -Encoding exch get -//PDFR_DEBUG{ -dup = -}if -dup null eq{ -pop//NotDef exec -} -{ -CharProcs exch//knownget exec -{ -currentfont/Font get/Resources//knownget exec{ -exec -SaveResourcesToStack -//PDFReader/CurrentObject get/Context get -/Resources 3 -1 roll put -//RunDelayedStream exec -RestoreResourcesFromStack -}{ -//RunDelayedStream exec -}ifelse -} -{ -//NotDef exec -}ifelse -}ifelse -end -}bind def -/printdict -{(<<)= -{exch = ==}forall -(>>)= -}bind def -/printfont -{ -dup{ -exch dup = -dup/Encoding eq{ -pop = -}{ -dup/FontInfo eq exch/Private eq or{ -//printdict exec -}{ -== -}ifelse -}ifelse -}forall -}bind def -/ScaleMetrics -{1 index{ -2 index div -3 index -3 1 roll put -}forall -pop -}bind def -/ResolveAndSetFontAux -{exch dup -//PDFReader/CurrentObject get/Context get/Resources get -/Font//DoNothing//ResolveD exec -exch//CheckFont//ResolveD exec -dup/Font//knownget exec{ -exch pop exch pop -}{ -{ -dup/Subtype get dup dup/Type1 eq exch/TrueType eq or exch/MMType1 eq or{ -exch pop -dup/BaseFont get -//RemoveFontNamePrefix exec -//PDFR_DEBUG{ -(Font )print dup = -}if -1 index/FontDescriptor known{ -//PDFR_DEBUG{ -(Font from a font descriptor.)= -}if -1 index -/FontDescriptor//DoNothing//ResolveD exec -/Font//knownget exec{ -exch pop -}{ -//PDFR_DEBUG{ -(Font descriptor has no Font resolved.)= -}if -//GetInstalledFont exec -}ifelse -}{ -//GetInstalledFont exec -}ifelse -exch -dup/Encoding known not{ -1 index/Encoding get 1 index exch/Encoding exch put -}if -//ObtainEncoding exec -//ObtainMetrics exec -exch -dup length dict copy -dup 2 index/Encoding get -/Encoding exch put -1 index/Metrics//knownget exec{ -2 index/Subtype get/TrueType ne{ -1 index/FontMatrix get 0 get -dup 0 eq{ -pop -1 index/FontMatrix get 1 get -dup 0 eq{pop 1}if -}if -0.001 div -//ScaleMetrics exec -}{ -1 index/sfnts known not{ -1 index/FontMatrix get 0 get -dup 0 eq{ -pop -1 index/FontMatrix get 1 get -dup 0 eq{pop 1}if -}if -//ScaleMetrics exec -}if -}ifelse -1 index exch/Metrics exch put -}if -1 index/BaseFont get -exch -dup/FID undef -dup/UniqueID undef -definefont -dup 3 1 roll -/Font exch put -exit -}if -dup/Subtype get/Type3 eq{ -//ObtainEncoding exec -2 copy exch/FontName exch put -dup/CharProcs get//ResolveDict exec -dup/FontType 3 put -dup/BuildChar//BuildChar put -dup dup/Font exch put -dup 3 1 roll -definefont -2 copy ne{ -2 copy/Font exch put -}if -exch pop -exit -}if -dup/Subtype get/Type0 eq{ -}if -dup/Subtype get/CIDFontType0 eq{ -}if -dup/Subtype get/CIDFontType2 eq{ -}if -mark(Unknown font type )2 index/Subtype get//error exec -}loop -}ifelse -exch scalefont setfont -}bind def -/ResolveAndSetFont -{ -//ResolveAndSetFontAux exec -}bind def -/.knownget -{2 copy known{ -get true -}{ -pop pop false -}ifelse -}bind def -/.min -{2 copy lt{ -exch -}if -pop -}bind def -/.max -{2 copy gt{ -exch -}if -pop -}bind def -/.dicttomark -{>> -}bind def -/getu16{ -2 copy get 8 bitshift 3 1 roll 1 add get add -}bind def -/gets16{ -getu16 16#8000 xor 16#8000 sub -}bind def -/getu32{ -2 copy getu16 16 bitshift 3 1 roll 2 add getu16 add -}bind def -/gets32{ -2 copy gets16 16 bitshift 3 1 roll 2 add getu16 add -}bind def -/cmapformats mark -0{ -6 256 getinterval{}forall 256 packedarray -}bind -2{ -/sHK_sz 2 def -/sH_sz 8 def -dup 2 getu16/cmapf2_tblen exch def -dup 4 getu16/cmapf2_lang exch def -dup 6 256 sHK_sz mul getinterval/sHKs exch def -0 -0 1 255{ -sHKs exch -2 mul getu16 -1 index -1 index -lt{exch}if pop -}for -/sH_len exch def -dup 6 256 sHK_sz mul add -cmapf2_tblen 1 index sub getinterval -/sH_gIA exch def -/cmapf2_glyph_array 65535 array def -/.cmapf2_putGID{ -/cmapf2_ch cmapf2_ch_hi 8 bitshift cmapf2_ch_lo add def -firstCode cmapf2_ch_lo le -cmapf2_ch_lo firstCode entryCount add lt -and{ -sH_offset idRangeOffset add -cmapf2_ch_lo firstCode sub 2 mul -add 6 add -sH_gIA exch getu16 -dup 0 gt{ -idDelta add -cmapf2_glyph_array exch cmapf2_ch exch put -}{ -pop -}ifelse -}{ -}ifelse -}def -16#00 1 16#ff{ -/cmapf2_ch_hi exch def -sHKs cmapf2_ch_hi sHK_sz mul getu16 -/sH_offset exch def -sH_gIA sH_offset sH_sz getinterval -dup 0 getu16/firstCode exch def -dup 2 getu16/entryCount exch def -dup 4 gets16/idDelta exch def -dup 6 getu16/idRangeOffset exch def -pop -sH_offset 0 eq{ -/cmapf2_ch_lo cmapf2_ch_hi def -/cmapf2_ch_hi 0 def -.cmapf2_putGID -}{ -16#00 1 16#ff{ -/cmapf2_ch_lo exch def -.cmapf2_putGID -}for -}ifelse -}for -pop -0 1 cmapf2_glyph_array length 1 sub{ -dup cmapf2_glyph_array exch get -null eq{cmapf2_glyph_array exch 0 put}{pop}ifelse -}for -cmapf2_glyph_array -}bind -4{ -/etab exch def -/nseg2 etab 6 getu16 def -14/endc etab 2 index nseg2 getinterval def -2 add -nseg2 add/startc etab 2 index nseg2 getinterval def -nseg2 add/iddelta etab 2 index nseg2 getinterval def -nseg2 add/idroff etab 2 index nseg2 getinterval def -pop -/firstcode startc 0 getu16 16#ff00 and dup 16#f000 ne{pop 0}if def -/lastcode firstcode def -/striptopbyte false def -/putglyph{ -glyphs code 3 -1 roll put/code code 1 add def -}bind def -/numcodes 0 def/glyphs 0 0 2 nseg2 3 sub{ -/i2 exch def -/scode startc i2 getu16 def -/ecode endc i2 getu16 def -ecode lastcode gt{ -/lastcode ecode def -}if -}for pop -firstcode 16#f000 ge lastcode firstcode sub 255 le and{ -lastcode 255 and -/striptopbyte true def -}{ -lastcode -}ifelse -1 add -array def -glyphs length 1024 ge{ -.array1024z 0 1024 glyphs length 1023 sub{glyphs exch 2 index putinterval}for -glyphs dup length 1024 sub 3 -1 roll -putinterval -}{ -0 1 glyphs length 1 sub{glyphs exch 0 put}for -}ifelse -/numcodes 0 def/code 0 def -0 2 nseg2 3 sub{ -/i2 exch def -/scode startc i2 getu16 def -/ecode endc i2 getu16 def -numcodes scode firstcode sub -exch sub 0 .max dup/code exch code exch add def -ecode scode sub 1 add add numcodes add/numcodes exch def -/delta iddelta i2 gets16 def -TTFDEBUG{ -(scode=)print scode =only -( ecode=)print ecode =only -( delta=)print delta =only -( droff=)print idroff i2 getu16 = -}if -idroff i2 getu16 dup 0 eq{ -pop scode delta add 65535 and 1 ecode delta add 65535 and -striptopbyte{ -/code scode 255 and def -}{ -/code scode def -}ifelse -{putglyph}for -}{ -/gloff exch 14 nseg2 3 mul add 2 add i2 add add def -striptopbyte{ -/code scode 255 and def -}{ -/code scode def -}ifelse -0 1 ecode scode sub{ -2 mul gloff add etab exch getu16 -dup 0 ne{delta add 65535 and}if putglyph -}for -}ifelse -}for glyphs/glyphs null def -}bind -6{ -dup 6 getu16/firstcode exch def dup 8 getu16/ng exch def -firstcode ng add array -0 1 firstcode 1 sub{2 copy 0 put pop}for -dup firstcode ng getinterval -0 1 ng 1 sub{ -dup 2 mul 10 add 4 index exch getu16 3 copy put pop pop -}for pop exch pop -}bind -.dicttomark readonly def -/cmaparray{ -dup 0 getu16 cmapformats exch .knownget{ -TTFDEBUG{ -(cmap: format )print 1 index 0 getu16 = flush -}if exec -}{ -(Can't handle format )print 0 getu16 = flush -0 1 255{}for 256 packedarray -}ifelse -TTFDEBUG{ -(cmap: length=)print dup length = dup == -}if -}bind def -/postremap mark -/Cdot/Cdotaccent -/Edot/Edotaccent -/Eoverdot/Edotaccent -/Gdot/Gdotaccent -/Ldot/Ldotaccent -/Zdot/Zdotaccent -/cdot/cdotaccent -/edot/edotaccent -/eoverdot/edotaccent -/gdot/gdotaccent -/ldot/ldotaccent -/zdot/zdotaccent -.dicttomark readonly def -/get_from_stringarray -{1 index type/stringtype eq{ -get -}{ -exch{ -2 copy length ge{ -length sub -}{ -exch get exit -}ifelse -}forall -}ifelse -}bind def -/getinterval_from_stringarray -{ -2 index type/stringtype eq{ -getinterval -}{ -string exch 0 -4 3 roll{ -dup length -dup 4 index lt{ -3 index exch sub -exch pop 3 1 roll exch pop -}{ -dup 3 1 roll -4 index sub -5 index length 4 index sub -2 copy gt{exch}if pop -dup 3 1 roll -5 index exch getinterval -5 index 4 index 3 index -getinterval -copy pop -exch pop add exch pop 0 exch -dup 3 index length ge{exit}if -}ifelse -}forall -pop pop -}ifelse -}bind def -/string_array_size -{dup type/stringtype eq{ -length -}{ -0 exch{length add}forall -}ifelse -}bind def -/postformats mark -16#00010000{ -pop MacGlyphEncoding -} -16#00020000{ -dup dup type/arraytype eq{0 get}if length 36 lt{ -TTFDEBUG{(post format 2.0 invalid.)= flush}if -pop[] -}{ -/postglyphs exch def -/post_first postglyphs dup type/arraytype eq{0 get}if def -post_first 32 getu16/numglyphs exch def -/glyphnames numglyphs 2 mul 34 add def -/postpos glyphnames def -/total_length postglyphs//string_array_size exec def -numglyphs array 0 1 numglyphs 1 sub{ -postpos total_length ge{ -1 numglyphs 1 sub{1 index exch/.notdef put}for -exit -}if -postglyphs postpos//get_from_stringarray exec -postglyphs postpos 1 add 2 index//getinterval_from_stringarray exec cvn -exch postpos add 1 add/postpos exch def -2 index 3 1 roll -put -}for -/postnames exch def -numglyphs array 0 1 numglyphs 1 sub{ -dup 2 mul 34 add postglyphs exch 2//getinterval_from_stringarray exec -dup 0 get 8 bitshift exch 1 get add dup 258 lt{ -MacGlyphEncoding exch get -}{ -dup 32768 ge{ -pop/.notdef -}{ -258 sub dup postnames length ge{ -TTFDEBUG{( *** warning: glyph index past end of 'post' table)= flush}if -pop -exit -}if -postnames exch get -postremap 1 index .knownget{exch pop}if -}ifelse -}ifelse -2 index 3 1 roll put -}for -} -ifelse -}bind -16#00030000{ -pop[] -}bind -.dicttomark readonly def -/first_post_string -{ -post dup type/arraytype eq{0 get}if -}bind def -/.getpost{ -/glyphencoding post null eq{ -TTFDEBUG{(post missing)= flush}if[] -}{ -postformats first_post_string 0 getu32 .knownget{ -TTFDEBUG{ -(post: format )print -first_post_string -dup 0 getu16 =only(,)print 2 getu16 = flush -}if -post exch exec -}{ -TTFDEBUG{(post: unknown format )print post 0 getu32 = flush}if[] -}ifelse -}ifelse def -}bind def -/MacRomanEncoding[ -StandardEncoding 0 39 getinterval aload pop -/quotesingle -StandardEncoding 40 56 getinterval aload pop -/grave -StandardEncoding 97 31 getinterval aload pop -/Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute -/agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave -/ecircumflex/edieresis/iacute/igrave -/icircumflex/idieresis/ntilde/oacute -/ograve/ocircumflex/odieresis/otilde -/uacute/ugrave/ucircumflex/udieresis -/dagger/degree/cent/sterling/section/bullet/paragraph/germandbls -/registered/copyright/trademark/acute/dieresis/.notdef/AE/Oslash -/.notdef/plusminus/.notdef/.notdef/yen/mu/.notdef/.notdef -/.notdef/.notdef/.notdef/ordfeminine/ordmasculine/.notdef/ae/oslash -/questiondown/exclamdown/logicalnot/.notdef -/florin/.notdef/.notdef/guillemotleft -/guillemotright/ellipsis/space/Agrave/Atilde/Otilde/OE/oe -/endash/emdash/quotedblleft/quotedblright -/quoteleft/quoteright/divide/.notdef -/ydieresis/Ydieresis/fraction/currency -/guilsinglleft/guilsinglright/fi/fl -/daggerdbl/periodcentered/quotesinglbase/quotedblbase -/perthousand/Acircumflex/Ecircumflex/Aacute -/Edieresis/Egrave/Iacute/Icircumflex -/Idieresis/Igrave/Oacute/Ocircumflex -/.notdef/Ograve/Uacute/Ucircumflex -/Ugrave/dotlessi/circumflex/tilde -/macron/breve/dotaccent/ring/cedilla/hungarumlaut/ogonek/caron -]/Encoding defineresource pop -/TTParser<< -/Pos 0 -/post null ->>def -/readu8 -{read not{ -mark(Insufficient data in the stream.)//error exec -}if -}bind def -/readu16 -{dup//readu8 exec 8 bitshift exch//readu8 exec or -}bind def -/reads16 -{//readu16 exec 16#8000 xor 16#8000 sub -}bind def -/readu32 -{dup//readu16 exec 16 bitshift exch//readu16 exec or -}bind def -/reads32 -{dup//reads16 exec 16 bitshift exch//readu16 exec or -}bind def -/SkipToPosition -{dup//TTParser/Pos get -exch//TTParser exch/Pos exch put -sub -//PDFR_DEBUG{ -(Skipping )print dup//=only exec( bytes.)= -}if -dup 0 eq{ -pop pop -}{ -dup 3 1 roll -()/SubFileDecode filter -exch -{1 index//BlockBuffer readstring pop length -dup 0 eq{pop exch pop exit}if -sub -}loop -0 ne{ -mark(Insufficient data in the stream for SkipToPosition.)//error exec -}if -}ifelse -}bind def -/TagBuffer 4 string def -/ParseTTTableDirectory -{//PDFR_DEBUG{ -(ParseTTTableDirectory beg)= -}if -15 dict begin -dup//readu32 exec 16#00010000 ne{ -mark(Unknown True Type version.)//error exec -}if -dup//readu16 exec/NumTables exch def -dup//readu16 exec/SearchRange exch def -dup//readu16 exec/EntrySelector exch def -dup//readu16 exec/RangeShift exch def -//PDFR_DEBUG{ -(NumTables = )print NumTables = -}if -NumTables{ -dup//TagBuffer readstring not{ -mark(Could not read TT tag.)//error exec -}if -cvn -[2 index//readu32 exec pop -2 index//readu32 exec -3 index//readu32 exec -] -//PDFR_DEBUG{ -2 copy exch//=only exec( )print == -}if -def -}repeat -pop -//TTParser/Pos 12 NumTables 16 mul add put -currentdict end -//PDFR_DEBUG{ -(ParseTTTableDirectory end)= -}if -}bind def -/ParseTTcmap -{//PDFR_DEBUG{ -(ParseTTcmap beg)= -}if -/cmap get aload pop -3 1 roll -7 dict begin -//PDFR_DEBUG{ -(Current position = )print//TTParser/Pos get = -(cmap position = )print dup = -}if -1 index exch//SkipToPosition exec -//TTParser/Pos get/TablePos exch def -dup//readu16 exec pop -dup//readu16 exec/NumEncodings exch def -//PDFR_DEBUG{ -(NumEncodings = )print NumEncodings = -}if -null -NumEncodings{ -1 index//readu32 exec -2 index//readu32 exec -3 array dup 3 2 roll 0 exch put -2 index null ne{ -dup 0 get 3 index 0 get sub -3 index exch 1 exch put -}if -dup 4 3 roll pop 3 1 roll -def -}repeat -dup 0 get -4 3 roll exch sub -1 exch put -//PDFR_DEBUG{ -currentdict{ -exch dup type/integertype eq{ -//PrintHex exec( )print == -}{ -pop pop -}ifelse -}forall -}if -4 NumEncodings 8 mul add/HeaderLength exch def -//TTParser/Pos//TTParser/Pos get HeaderLength add put -0 -NumEncodings{ -16#7FFFFFF null -currentdict{ -1 index type/integertype eq{ -exch pop dup 0 get -dup 5 index gt{ -dup 4 index lt{ -4 1 roll -exch pop exch pop -}{ -pop pop -}ifelse -}{ -pop pop -}ifelse -}{ -pop pop -}ifelse -}forall -//PDFR_DEBUG{ -(Obtaining subtable for )print dup == -}if -3 2 roll pop -3 copy pop -TablePos add//SkipToPosition exec -3 copy exch pop 1 get -//TTParser/Pos//TTParser/Pos get 3 index add put -string -readstring not{ -mark(Can't read a cmap subtable.)//error exec -}if -2 exch put -}repeat -pop pop -currentdict end -//PDFR_DEBUG{ -(ParseTTcmap end)= -}if -}bind def -/GetTTEncoding -{//PDFR_DEBUG{ -(GetTTEncoding beg)= -}if -get -exch pop -2 get -10 dict begin -/TTFDEBUG//PDFR_DEBUG def -//cmaparray exec -end -//PDFR_DEBUG{ -(GetTTEncoding end)= -dup == -}if -}bind def -/InverseEncoding -{ -256 dict begin -dup length 1 sub -1 0{ -2 copy get -exch -1 index currentdict exch//knownget exec{ -dup type/arraytype eq{ -aload length 1 add array astore -}{ -2 array astore -}ifelse -}if -def -}for -pop -currentdict end -}bind def -/GetMacRomanEncodingInverse -{//PDFReader/MacRomanEncodingInverse get -dup null eq{ -pop -MacRomanEncoding//InverseEncoding exec -dup//PDFReader exch/MacRomanEncodingInverse exch put -}if -}bind def -/PutCharStringSingle -{ -dup 3 index length lt{ -2 index exch get -dup 0 ne{ -def -}{ -pop pop -}ifelse -}{ -pop pop -}ifelse -}bind def -/PutCharString -{1 index type/nametype ne{ -mark(Bad charstring name)//error exec -}if -dup type/arraytype eq{ -{ -3 copy//PutCharStringSingle exec -pop pop -}forall -pop -}{ -//PutCharStringSingle exec -}ifelse -}bind def -/ComposeCharStrings -{ -//PDFR_DEBUG{ -(ComposeCharStrings beg)= -}if -1 index length 1 add dict begin -/.notdef 0 def -exch -//TTParser/post get -dup null ne{ -exch -1 index length 1 sub -1 0{ -dup 3 index exch get exch -dup 0 eq 2 index/.notdef eq or{ -pop pop -}{ -def -}ifelse -}for -}if -exch pop exch -{ -//PutCharString exec -}forall -pop -currentdict end -//PDFR_DEBUG{ -(ComposeCharStrings end)= -}if -}bind def -/ParseTTpost -{ -//PDFR_DEBUG{ -(ParseTTpost beg)= -}if -/post get aload pop -3 1 roll -//PDFR_DEBUG{ -(Current position = )print//TTParser/Pos get = -(post position = )print dup = -}if -1 index exch//SkipToPosition exec -//TTParser/Pos//TTParser/Pos get 4 index add put -exch dup 65535 le{ -string -readstring not{ -mark(Insufficient data in the stream for ParseTTpost.)//error exec -}if -}{ -[3 1 roll -dup 16384 div floor cvi -exch 1 index 16384 mul -sub exch -1 sub 0 1 3 -1 roll -{ -1 add index -16384 string readstring not{ -mark(Insufficient data in the stream for ParseTTpost.)//error exec -}if -}for -counttomark -2 roll -string readstring not{ -mark(Insufficient data in the stream for ParseTTpost.)//error exec -}if -] -}ifelse -1 dict begin -/post exch def -//.getpost exec -//TTParser/post glyphencoding put -//PDFR_DEBUG{ -(ParseTTpost end)= -glyphencoding == -}if -end -}bind def -/MakeTTCharStrings -{//MakeStreamReader exec -dup dup//ParseTTTableDirectory exec -//TTParser/post null put -dup/post//knownget exec{ -0 get -1 index/cmap get 0 get -lt{ -2 copy//ParseTTpost exec -//ParseTTcmap exec -}{ -2 copy//ParseTTcmap exec -3 1 roll -//ParseTTpost exec -}ifelse -}{ -//ParseTTcmap exec -}ifelse -{ -dup 16#00030001 known{ -//PDFR_DEBUG{ -(Using the TT cmap encoding for Windows Unicode.)= -}if -16#00030001//GetTTEncoding exec -AdobeGlyphList//ComposeCharStrings exec -exit -}if -dup 16#00010000 known{ -//PDFR_DEBUG{ -(Using the TT cmap encoding for Macintosh Roman.)= -}if -16#00010000//GetTTEncoding exec -PDFEncoding dup null eq{ -pop//GetMacRomanEncodingInverse exec -}{ -//InverseEncoding exec -}ifelse -//ComposeCharStrings exec -exit -}if -dup 16#00030000 known{ -//PDFR_DEBUG{ -(Using the TT cmap encoding 3.0 - not sure why Ghostscript writes it since old versions.)= -}if -16#00030000//GetTTEncoding exec -PDFEncoding dup null eq{ -pop//GetMacRomanEncodingInverse exec -}{ -//InverseEncoding exec -}ifelse -//ComposeCharStrings exec -exit -}if -mark(True Type cmap has no useful encodings.)//error exec -}loop -//PDFR_DEBUG{ -(CharStrings <<)= -dup{ -exch -dup type/nametype eq{ -//=only exec -}{ -== -}ifelse -( )print == -}forall -(>>)= -}if -}bind def -/ScaleVal -{ -aload pop -1 index sub -3 2 roll mul add -}bind def -/ScaleArg -{ -aload pop -1 index sub -3 1 roll -sub exch div -}bind def -/ScaleArgN -{ -dup length 2 sub -2 0{ -2 -2 index 3 1 roll getinterval -3 2 roll -exch//ScaleArg exec -1 index length 2 idiv 1 add 1 roll -}for -pop -}bind def -/ComputeFunction_10 -{ -//PDFR_DEBUG{ -(ComputeFunction_10 beg )print 1 index//=only exec( stack=)print count = -}if -exch -dup 1 eq{ -pop dup length 1 sub get -}{ -1 index length 1 sub mul -dup dup floor sub -dup 0 eq{ -pop cvi get -}{ -3 1 roll floor cvi -2 getinterval -aload pop -2 index mul 3 2 roll 1 exch sub 3 2 roll mul add -}ifelse -}ifelse -//PDFR_DEBUG{ -(ComputeFunction_10 end )print dup//=only exec( stack=)print count = -}if -}bind def -/ComputeFunction_n0 -{ -//PDFR_DEBUG{ -(ComputeFunction_n0 beg N=)print dup//=only exec( stack=)print count = -}if -dup 0 eq{ -pop -}{ -dup 2 add -1 roll -dup 3 index length 1 sub ge{ -pop 1 sub -exch dup length 1 sub get exch -//PDFReader/ComputeFunction_n0 get exec -}{ -dup floor cvi dup -4 index exch get -3 index dup -5 add copy -6 2 roll -pop pop pop pop -1 sub -//PDFReader/ComputeFunction_n0 get exec -3 2 roll pop -exch -4 3 roll exch -4 add 2 roll 1 add -3 2 roll exch get -exch 1 sub -//PDFReader/ComputeFunction_n0 get exec -1 index mul -3 1 roll -1 exch sub mul add -}ifelse -}ifelse -//PDFR_DEBUG{ -(ComputeFunction_n0 end )print dup//=only exec( stack=)print count = -}if -}bind def -/FunctionToProc_x01 -{ -dup/Domain get exch -dup/Data get 0 get exch -/Size get length -[4 1 roll -//PDFR_DEBUG{ -{(function beg, stack =)print count//=only exec(\n)print}/exec load -5 2 roll -}if -dup 1 gt{ -{mark exch -3 add 2 roll -//ScaleArgN exec -counttomark dup -3 add -2 roll -pop exch -//ComputeFunction_n0 exec -}/exec load -}{ -pop -3 1/roll load//ScaleArg/exec load -/exch load -//ComputeFunction_10/exec load -}ifelse -//PDFR_DEBUG{ -(function end, stack =)/print load/count load//=only/exec load(\n)/print load -}if -]cvx -//PDFR_DEBUG{ -(Made a procedure for the 1-result function :)= -dup == -}if -}bind def -/FunctionProcDebugBeg -{(FunctionProcDebugBeg )print count = -}bind def -/FunctionProcDebugEnd -{(FunctionProcDebugEnd )print count = -}bind def -/FunctionToProc_x0n -{ -PDFR_DEBUG{ -(FunctionToProc_x0n beg m=)print dup = -}if -1 index/Size get length exch -dup 7 mul 2 add array -PDFR_DEBUG{ -dup 0//FunctionProcDebugBeg put -}{ -dup 0//DoNothing put -}ifelse -dup 1/exec load put -dup 2 5 index/Domain get put -2 index 1 eq{ -dup 3//ScaleArg put -}{ -dup 3//ScaleArgN put -}ifelse -dup 4/exec load put -1 index 1 sub 0 exch 1 exch{ -dup 7 mul 5 add -1 index 4 index 1 sub ne{ -dup 3 index exch 6 index put 1 add -dup 3 index exch/copy load put 1 add -}if -[ -6 index/Data get 3 index get -6 index 1 eq{ -//ComputeFunction_10/exec load -}{ -6 index -//ComputeFunction_n0/exec load -}ifelse -]cvx -3 index exch 2 index exch put 1 add -2 index 1 index/exec load put 1 add -1 index 4 index 1 sub ne{ -2 index 1 index 6 index 1 add put 1 add -2 index 1 index 1 put 1 add -2 index 1 index/roll load put -}if -pop pop -}for -PDFR_DEBUG{ -dup dup length 2 sub//FunctionProcDebugEnd put -}{ -dup dup length 2 sub//DoNothing put -}ifelse -dup dup length 1 sub/exec load put -cvx exch pop exch pop exch pop -//PDFR_DEBUG{ -(Made a procedure for the n-argument function :)= -dup == -}if -PDFR_DEBUG{ -(FunctionToProc_x0n end)= -}if -}bind def -/MakeTableRec -{ -0 -exec -}bind def -/MakeTable -{//PDFR_DEBUG{ -(MakeTable beg )print count = -}if -1 index/Size get exch -1 sub dup -3 1 roll -get -array -1 index 0 eq{ -exch pop exch pop -}{ -dup length 1 sub -1 0{ -3 index 3 index//MakeTableRec exec -2 index 3 1 roll put -}for -exch pop exch pop -}ifelse -//PDFR_DEBUG{ -(MakeTable end )print count = -}if -}bind def -//MakeTableRec 0//MakeTable put -/StoreSample -{ -1 sub -dup 0 eq{ -pop -}{ --1 1{ -I exch get get -}for -}ifelse -I 0 get 3 2 roll put -}bind def -/ReadSample32 -{ -4{ -File read not{ -mark(Insufficient data for function.)//error exec -}if -}repeat -pop -3 1 roll exch -256 mul add 256 mul add -//1_24_bitshift_1_sub div -}bind def -/ReadSample -{ -Buffer BitsLeft BitsPerSample -{2 copy ge{ -exit -}if -3 1 roll -8 add 3 1 roll -256 mul File read not{ -mark(Insufficient data for function.)//error exec -}if -add -3 1 roll -}loop -sub dup -2 index exch -neg bitshift -2 copy exch bitshift -4 3 roll exch sub -/Buffer exch def -exch/BitsLeft exch def -Div div -}bind def -/ReadSamplesRec -{0 -exec -}bind def -/ReadSamples -{ -//PDFR_DEBUG{ -(ReadSamples beg )print count = -}if -dup 1 eq{ -pop -0 1 Size 0 get 1 sub{ -I exch 0 exch put -0 1 M 1 sub{ -dup Range exch 2 mul 2 getinterval -//PDFR_DEBUG{ -(Will read a sample ... )print -}if -BitsPerSample 32 eq{//ReadSample32}{//ReadSample}ifelse -exec exch//ScaleVal exec -//PDFR_DEBUG{ -(value=)print dup = -}if -exch Table exch get -Size length//StoreSample exec -}for -}for -}{ -1 sub -dup Size exch get 0 exch 1 exch 1 sub{ -I exch 2 index exch put -dup//ReadSamplesRec exec -}for -pop -}ifelse -//PDFR_DEBUG{ -(ReadSamples end )print count = -}if -}bind def -//ReadSamplesRec 0//ReadSamples put -/StreamToArray -{//PDFR_DEBUG{ -(StreamToArray beg )print count = -}if -userdict/FuncDataReader get begin -dup/BitsPerSample get/BitsPerSample exch def -dup/Size get length/N exch def -dup/Range get length 2 idiv/M exch def -1 BitsPerSample bitshift 1 sub/Div exch def -/BitsLeft 0 def -/Buffer 0 def -dup/Size get/Size exch def -dup/Range get/Range exch def -/File 1 index//MakeStreamReader exec def -/I[N{0}repeat]def -M array -dup length 1 sub -1 0{ -2 index N//MakeTable exec -2 index 3 1 roll put -}for -/Table exch def -N//ReadSamples exec -PDFR_DEBUG{ -(Table = )print Table == -}if -/Data Table put -end -//PDFR_DEBUG{ -(StreamToArray end )print count = -}if -}bind def -/FunctionToProc10 -{ -PDFR_DEBUG{ -(FunctionToProc10 beg, Range = )print dup/Range get == -}if -dup/Order//knownget exec{ -1 ne{ -(Underimplemented function Type 0 Order 3.)= -}if -}if -dup//StreamToArray exec -dup/Range get length dup 2 eq{ -pop//FunctionToProc_x01 exec -}{ -2 idiv//FunctionToProc_x0n exec -}ifelse -PDFR_DEBUG{ -(FunctionToProc10 end)= -}if -}bind def -/FunctionToProc12 -{begin -currentdict/C0//knownget exec{length 1 eq}{true}ifelse{ -N -currentdict/C0//knownget exec{ -0 get -}{ -0 -}ifelse -currentdict/C1//knownget exec{ -0 get -}{ -1 -}ifelse -1 index sub -[4 1 roll -{ -4 2 roll -exp mul add -}aload pop -]cvx -}{ -[ -0 1 C0 length 1 sub{ -N -C0 2 index get -C1 3 index get -4 3 roll pop -1 index sub -[/dup load -5 2 roll -{ -4 2 roll -exp mul add -exch -}aload pop -]cvx -/exec load -}for -/pop load -]cvx -}ifelse -end -//PDFR_DEBUG{ -(FunctionType2Proc : )print dup == -}if -}bind def -/FunctionToProc14 -{//MakeStreamReader exec cvx exec -//PDFR_DEBUG{ -(FunctionType4Proc : )print dup == -}if -}bind def -/FunctionToProc1 -{ -dup/FunctionType get -{dup 0 eq{ -pop//FunctionToProc10 exec exit -}if -dup 2 eq{ -pop//FunctionToProc12 exec exit -}if -dup 4 eq{ -pop//FunctionToProc14 exec exit -}if -mark exch(Function type )exch( isn't implemented yet.)//error exec -}loop -}bind def -/FunctionToProc20 -{ -PDFR_DEBUG{ -(FunctionToProc20, Range = )print dup/Range get == -}if -dup/Order//knownget exec{ -1 ne{ -(Underimplemented function Type 0 Order 3.)= -}if -}if -dup//StreamToArray exec -dup/Range get length dup 2 eq{ -pop//FunctionToProc_x01 exec -}{ -2 idiv//FunctionToProc_x0n exec -}ifelse -}bind def -/FunctionToProc -{//PDFR_DEBUG{ -(FunctionToProc beg )print count = -}if -dup type/dicttype eq{ -dup/Domain get length 2 idiv -{ -dup 1 eq{ -pop//FunctionToProc1 exec exit -}if -dup 2 eq{ -pop//FunctionToProc20 exec exit -}if -mark(Functions with many arguments aren't implemented yet.)//error exec -}loop -}{ -//PDFR_DEBUG{(Not a function dict, assume already a procedure.)print}if -}ifelse -//PDFR_DEBUG{ -(FunctionToProc end )print count = -}if -}bind def -/spotfunctions mark -/Round{ -abs exch abs 2 copy add 1 le{ -dup mul exch dup mul add 1 exch sub -}{ -1 sub dup mul exch 1 sub dup mul add 1 sub -}ifelse -} -/Diamond{ -abs exch abs 2 copy add .75 le{ -dup mul exch dup mul add 1 exch sub -}{ -2 copy add 1.23 le{ -.85 mul add 1 exch sub -}{ -1 sub dup mul exch 1 sub dup mul add 1 sub -}ifelse -}ifelse -} -/Ellipse{ -abs exch abs 2 copy 3 mul exch 4 mul add 3 sub dup 0 lt{ -pop dup mul exch .75 div dup mul add 4 div 1 exch sub -}{ -dup 1 gt{ -pop 1 exch sub dup mul exch 1 exch sub -.75 div dup mul add 4 div 1 sub -}{ -.5 exch sub exch pop exch pop -}ifelse -}ifelse -} -/EllipseA{dup mul .9 mul exch dup mul add 1 exch sub} -/InvertedEllipseA{dup mul .9 mul exch dup mul add 1 sub} -/EllipseB{dup 5 mul 8 div mul exch dup mul exch add sqrt 1 exch sub} -/EllipseC{dup mul .9 mul exch dup mul add 1 exch sub} -/InvertedEllipseC{dup mul .9 mul exch dup mul add 1 sub} -/Line{exch pop abs neg} -/LineX{pop} -/LineY{exch pop} -/Square{abs exch abs 2 copy lt{exch}if pop neg} -/Cross{abs exch abs 2 copy gt{exch}if pop neg} -/Rhomboid{abs exch abs 0.9 mul add 2 div} -/DoubleDot{2{360 mul sin 2 div exch}repeat add} -/InvertedDoubleDot{2{360 mul sin 2 div exch}repeat add neg} -/SimpleDot{dup mul exch dup mul add 1 exch sub} -/InvertedSimpleDot{dup mul exch dup mul add 1 sub} -/CosineDot{180 mul cos exch 180 mul cos add 2 div} -/Double{exch 2 div exch 2{360 mul sin 2 div exch}repeat add} -/InvertedDouble{ -exch 2 div exch 2{360 mul sin 2 div exch}repeat add neg -} -.dicttomark readonly def -/CheckColorSpace -{ -dup type/arraytype ne{ -mark(Resource )3 index( must be an array.)//error exec -}if -}bind def -/SubstitutePDFColorSpaceRec -{0 -exec -}bind def -/SubstitutePDFColorSpace -{ -{ -dup 0 get/Pattern eq{ -dup length 1 gt{ -dup dup 1//CheckColorSpace//ResolveA exec -dup type/nametype ne{ -//SubstitutePDFColorSpaceRec exec -}if -1 exch put -}if -exit -}if -dup 0 get/Indexed eq{ -exit -}if -dup 0 get/Separation eq{ -dup dup 2//CheckColorSpace//ResolveA exec -dup type/nametype ne{ -//SubstitutePDFColorSpaceRec exec -}if -2 exch put -exit -}if -dup 0 get/CalGray eq{ -1 get -dup/Gamma//knownget exec{ -[exch[exch/exp load]cvx dup dup] -1 index exch/DecodeLMN exch put -}if -[exch/CIEBasedA exch] -exit -}if -dup 0 get/CalRGB eq{ -1 get -dup/Matrix//knownget exec{ -1 index exch/MatrixLMN exch put -}if -dup/Gamma//knownget exec{ -aload pop -[exch/exp load]cvx -3 1 roll -[exch/exp load]cvx -3 1 roll -[exch/exp load]cvx -3 1 roll -3 array astore -1 index exch/DecodeLMN exch put -}if -[exch/CIEBasedABC exch] -exit -}if -dup 0 get/Lab eq{ -1 get -begin -currentdict/Range//knownget exec{aload pop}{-100 100 -100 100}ifelse -0 100 6 2 roll 6 array astore -/RangeABC exch def -/DecodeABC[{16 add 116 div}bind{500 div}bind{200 div}bind]def -/MatrixABC[1 1 1 1 0 0 0 0 -1]def -{dup 6 29 div ge{dup dup mul mul}{4 29 div sub 108 841 div mul}ifelse} -/DecodeLMN[ -[3 index aload pop WhitePoint 0 get/mul load]cvx -[4 index aload pop WhitePoint 1 get/mul load]cvx -[5 index aload pop WhitePoint 2 get/mul load]cvx -]def pop -//PDFR_DEBUG{ -(Constructed from Lab <<)= -currentdict{exch = ==}forall -(>>)= -}if -[/CIEBasedABC currentdict] -end -exit -pop -}if -dup 0 get/CIEBasedA eq{exit}if -dup 0 get/CIEBasedABC eq{exit}if -mark exch(Unimplemented color space )exch//error exec -}loop -}bind def -//SubstitutePDFColorSpaceRec 0//SubstitutePDFColorSpace put -/ResolveArrayElement -{2 copy get -dup type dup/arraytype eq exch -/packedarraytype eq or{ -dup length 1 ge exch xcheck and{ -2 copy get -dup 0 get type/integertype eq -1 index 1 get type dup/arraytype -eq exch -/packedarraytype eq or -and{ -exec -2 index 4 1 roll put -}{ -pop pop -}ifelse -}{ -pop -}ifelse -}{ -pop pop -}ifelse -}bind def -/ResolveColorSpaceArrayRec -{0 -exec -}bind def -/SetColorSpaceSafe -{ -PDFR_DEBUG{ -(SetColorSpaceSafe beg)= -}if -currentcolorspace dup type/arraytype eq{ -1 index type/arraytype eq{ -dup length 2 index length eq{ -false exch -dup length 0 exch 1 exch 1 sub{ -dup -4 index exch get exch -2 index exch get -ne{ -exch pop true exch exit -}if -}for -pop -{ -setcolorspace -}{ -pop -}ifelse -}{ -pop setcolorspace -}ifelse -}{ -pop setcolorspace -}ifelse -}{ -pop setcolorspace -}ifelse -PDFR_DEBUG{ -(SetColorSpaceSafe end)= -}if -}bind def -/ResolveColorSpaceArray -{ -//PDFR_DEBUG{ -(ResolveColorSpaceArray beg )print dup == -}if -dup 0 get/Indexed eq{ -1//ResolveArrayElement exec -dup dup 1 get -dup type/arraytype eq{ -//SubstitutePDFColorSpace exec -//ResolveColorSpaceArrayRec exec -1 exch put -}{ -pop pop -}ifelse -}if -dup 0 get/Separation eq{ -dup dup 1 get UnPDFEscape 1 exch put -3//ResolveArrayElement exec -dup 3 get//FunctionToProc exec -2 copy 3 exch put -pop -}if -dup 0 get/Pattern eq{ -dup length 1 gt{ -dup 1 get dup type/arraytype eq{ -ResolveColorSpaceArray -1 index 1 3 -1 roll put -}{ -pop -}ifelse -}if -}if -PDFR_DEBUG{ -(Construcrted color space :)= -dup == -}if -//PDFR_DEBUG{ -(ResolveColorSpaceArray end )print dup == -}if -}bind def -//ResolveColorSpaceArrayRec 0//ResolveColorSpaceArray put -/ResolveColorSpace -{ -//PDFR_DEBUG{ -(ResolveColorSpace beg )print dup = -}if -dup//SimpleColorSpaceNames exch known not{ -dup//PDFColorSpaces exch//knownget exec{ -exch pop -//PDFR_DEBUG{ -(ResolveColorSpace known )= -}if -}{ -dup -//PDFReader/CurrentObject get/Context get/Resources get -/ColorSpace//DoNothing//ResolveD exec -exch//CheckColorSpace//ResolveD exec -dup type/arraytype eq{ -//SubstitutePDFColorSpace exec -//ResolveColorSpaceArray exec -dup//PDFColorSpaces 4 2 roll put -}if -}ifelse -}if -//PDFR_DEBUG{ -(ResolveColorSpace end )print dup == -}if -}bind def -/CheckPattern -{ -dup/PatternType//knownget exec{ -dup 1 ne{ -mark(Resource )4 index( is a shading, which can't be handled at level 2. )//error exec -}if -pop -}if -dup/Type knownget{ -/Pattern ne{ -mark(Resource )4 index( must have /Type/Pattern .)//error exec -}if -}if -}bind def -/PaintProc -{/Context get -//RunDelayedStream exec -}bind def -/ResolvePattern -{ -dup -userdict/PDFR_Patterns get -exch//knownget exec{ -exch pop -}{ -dup -//PDFReader/CurrentObject get/Context get/Resources get -/Pattern//DoNothing//ResolveD exec -exch//CheckPattern//ResolveD exec -dup dup/Context exch put -dup/Resources//DoNothing//ResolveD exec pop -dup/PaintProc//PaintProc put -gsave userdict/PDFR_InitialGS get setgstate -currentglobal exch false setglobal -dup/Matrix get -makepattern -exch setglobal -grestore -dup userdict/PDFR_Patterns get -4 2 roll -put -}ifelse -}bind def -/SetColor -{//PDFR_DEBUG{ -(SetColor beg)= -}if -currentcolorspace dup type/nametype eq{ -pop setcolor -}{ -0 get/Pattern eq{ -//ResolvePattern exec setpattern -}{ -setcolor -}ifelse -}ifelse -//PDFR_DEBUG{ -(SetColor end)= -}if -}bind def -/ImageKeys 15 dict begin -/BPC/BitsPerComponent def -/CS/ColorSpace def -/D/Decode def -/DP/DecodeParms def -/F/Filter def -/H/Height def -/IM/ImageMask def -/I/Interpolate def -/W/Width def -currentdict end readonly def -/ImageValues 15 dict begin -/G/DeviceGray def -/RGB/DeviceRGB def -/CMYK/DeviceCMYK def -/I/Indexed def -/AHx/ASCIIHexDecode def -/A85/ASCII85Decode def -/LZW/LZWDecode def -/Fl/FlateDecode def -/RL/RunLengthDecode def -/CCF/CCITTFaxDecode def -/DCT/DCTDecode def -currentdict end readonly def -/GetColorSpaceRange -{2 index/ColorSpace get -dup type/arraytype eq{ -1 get -}if -exch//knownget exec{ -exch pop -}if -}bind def -/DecodeArrays 15 dict begin -/DeviceGray{[0 1]}def -/DeviceRGB{[0 1 0 1 0 1]}def -/DeviceCMYK{[0 1 0 1 0 1 0 1]}def -/Indexed{ -dup/BitsPerComponent get 1 exch bitshift 1 sub[exch 0 exch] -}def -/Separation{[0 1]}def -/CIEBasedA{[0 1]/RangeA//GetColorSpaceRange exec}def -/CIEBasedABC{[0 1 0 1 0 1]/RangeABC//GetColorSpaceRange exec}def -currentdict end readonly def -/Substitute -{1 index//knownget exec{ -exch pop -}if -}bind def -/DebugImagePrinting -{ -//PDFR_DEBUG{ -(Image :)= -dup{exch//=only exec( )print == -}forall -}if -}bind def -/CompleteImage -{ -dup/ColorSpace known{ -dup/ColorSpace//CheckColorSpace//ResolveD exec pop -}if -dup/Decode known not{ -dup/ColorSpace//knownget exec{ -dup type/arraytype eq{ -0 get -}if -//DecodeArrays exch get exec -}{ -[0 1] -}ifelse -1 index exch/Decode exch put -}if -dup/ImageMatrix[2 index/Width get 0 0 5 index/Height get neg -0 7 index/Height get]put -//DebugImagePrinting exec -}bind def -/CompleteInlineImage -{ -//PDFR_DEBUG{ -(CompleteInlineImage beg)= -}if -dup/ImageType known not{ -dup/ImageType 1 put -}if -dup length dict exch{ -exch//ImageKeys//Substitute exec -dup/Filter eq{ -exch//ImageValues//Substitute exec exch -}if -dup/ColorSpace eq{ -exch -dup//ImageValues exch//knownget exec{ -exch pop -}{ -//ResolveColorSpace exec -}ifelse -exch -}if -exch -2 index 3 1 roll put -}forall -//CompleteImage exec -dup/DataSource 2 copy get -2 index//AppendFilters exec put -//PDFR_DEBUG{ -(CompleteInlineImage end)= -}if -}bind def -/CompleteOutlineImage -{ -currentglobal exch dup gcheck setglobal -//PDFR_DEBUG{ -(CompleteOutlineImage beg)= -}if -dup dup//MakeStreamReader exec/DataSource exch put -dup/ImageType known not{ -//CompleteImage exec -dup/ImageType 1 put -dup/ColorSpace known{ -dup/ColorSpace//CheckColorSpace//ResolveD exec -dup type/arraytype eq{ -//ResolveColorSpaceArray exec -//SubstitutePDFColorSpace exec -1 index exch/ColorSpace exch put -}{ -pop -}ifelse -}if -}if -//PDFR_DEBUG{ -(CompleteOutlineImage end)= -}if -exch setglobal -}bind def -/DoImage -{ -//PDFR_DEBUG{ -(DoImage beg)= -}if -gsave -dup/ColorSpace//knownget exec{setcolorspace}if -dup/ImageMask//knownget exec not{false}if -{imagemask}{image}ifelse -grestore -//PDFR_DEBUG{ -(DoImage end)= -}if -}bind def -/GSave -{ -gsave -//PDFReader/GraphicStateStackPointer get -dup//GraphicStateStack exch get null eq{ -dup//GraphicStateStack exch//InitialGraphicState length dict put -}if -dup//GraphicStateStack exch get -//GraphicState exch copy pop -1 add//PDFReader exch/GraphicStateStackPointer exch put -}bind def -/GRestore -{ -grestore -//PDFReader/GraphicStateStackPointer get -1 sub dup -//PDFReader exch/GraphicStateStackPointer exch put -//GraphicStateStack exch get -//GraphicState copy pop -}bind def -/SetFont -{dup//GraphicState exch/FontSize exch put -//ResolveAndSetFont exec -//GraphicState/FontMatrixNonHV currentfont/FontMatrix get 1 get 0 ne put -}bind def -/ShowText -{ -//GraphicState/TextRenderingMode get dup 0 eq -exch 3 eq not currentfont/FontType get 3 eq and or -{ -//GraphicState/WordSpacing get 0 -32 -//GraphicState/CharacterSpacing get 0 -6 5 roll -//GraphicState/FontMatrixNonHV get{ -[ -7 -2 roll pop -5 -2 roll pop -5 -1 roll -{ -exch -pop -3 index add -exch 2 index eq{3 index add}if -4 1 roll -} -currentfont/FontMatrix get 0 get 0 ne{ -1 1 index length 1 sub getinterval cvx -}if -5 index -cshow -pop pop pop] -xshow -}{ -awidthshow -}ifelse -}{ -//GraphicState/CharacterSpacing get 0 eq -//GraphicState/FontMatrixNonHV get not and -//GraphicState/WordSpacing get 0 eq and{ -true charpath -}{ -{ -exch -pop 0 -currentpoint 5 4 roll -( )dup 0 3 index put true charpath -5 1 roll -moveto rmoveto -//GraphicState/CharacterSpacing get 0 rmoveto -32 eq{ -//GraphicState/WordSpacing get 0 rmoveto -}if -} -//GraphicState/FontMatrixNonHV get dup not exch{ -pop currentfont/FontMatrix get 0 get 0 ne -}if{ -1 1 index length 1 sub getinterval cvx -}if -exch cshow -}ifelse -}ifelse -}bind def -/ShowTextBeg -{ -//GraphicState/TextRenderingMode get dup 0 ne -{ -3 ne -currentfont/FontType get 3 eq not and{ -currentpoint newpath moveto -}if -} -{ -pop -}ifelse -}bind def -/ShowTextEnd -{ -//GraphicState/TextRenderingMode get -currentfont/FontType get 3 eq{ -dup 3 ne{ -pop 0 -}if -}if -{dup 1 eq{ -stroke exit -}if -dup 2 eq{ -gsave fill grestore stroke exit -}if -dup 3 eq{ -currentpoint newpath moveto -}if -dup 4 eq{ -gsave fill grestore clip exit -}if -dup 5 eq{ -gsave stroke grestore clip exit -}if -dup 6 eq{ -gsave fill grestore gsave stroke grestore fill exit -}if -dup 7 eq{ -clip exit -}if -exit -}loop -pop -}bind def -/ShowTextWithGlyphPositioning -{//ShowTextBeg exec -{dup type/stringtype eq{ -//ShowText exec -}{ -neg 1000 div//GraphicState/FontSize get mul 0 rmoveto -}ifelse -}forall -//ShowTextEnd exec -}bind def -/CheckFont -{dup/Type get/ExtGState ne{ -mark(Resource )3 index( must have /Type/ExtGState.)//error exec -}if -}bind def -/SetTransfer -{ -//PDFR_DEBUG{(SetTransfer beg )print count =}if -dup type/arraytype eq 1 index xcheck not and{ -0 4 getinterval aload pop -setcolortransfer -}{ -settransfer -}ifelse -//PDFR_DEBUG{(SetTransfer end )print count =}if -}bind def -/CheckExtGState -{dup/Type get/ExtGState ne{ -mark(Resource )3 index( must have /Type/ExtGState.)//error exec -}if -}bind def -/CheckHalftone -{dup/HalftoneType known not{ -mark(Resource )3 index( must have /HalftoneType.)//error exec -}if -}bind def -/ResolveFunction -{ -//PDFR_DEBUG{(ResolveFunction beg )print dup = count =}if -2 copy get//IsObjRef exec{ -2 copy//DoNothing//ResolveD exec -3 copy put pop -}if -2 copy get dup type/arraytype eq exch xcheck and not{ -2 copy get -dup type/arraytype eq 1 index xcheck not and{ -dup length 1 sub -1 0{ -2 copy//DoNothing ResolveA -dup/Identity eq{ -pop 2 copy{}put -}{ -//FunctionToProc exec -3 copy put pop -}ifelse -pop -}for -}{ -dup/Default eq{ -}{ -dup/Identity eq{ -pop{} -}{dup type/nametype eq{ -//spotfunctions exch get -}{ -//FunctionToProc exec -}ifelse -}ifelse -}ifelse -}ifelse -3 copy put -exch pop -}{ -1 index exch get -}ifelse -//PDFR_DEBUG{(ResolveFunction end )print dup == count =}if -}bind def -/ResolveFunctionSafe -{2 copy known{ -//ResolveFunction exec -}if -pop -}bind def -/CreateHalftoneThresholds -{ -dup/Thresholds known not{ -dup/HalftoneType get 10 eq{ -dup dup//MakeStreamReader exec -/Thresholds exch put -}if -dup/HalftoneType get dup 3 eq exch 6 eq or{ -dup dup//MakeStreamReader exec -//BlockBuffer readstring pop -dup length -dup 0 eq{ -mark(Could not read Thresholds)//error exec -}if -string copy/Thresholds exch put -dup/HalftoneType 3 put -}if -}if -}bind def -/SetExtGState -{ -//PDFReader/CurrentObject get/Context get/Resources get -/ExtGState//DoNothing//ResolveD exec -exch//CheckExtGState//ResolveD exec -dup/LW//knownget exec{ -setlinewidth -}if -dup/LC//knownget exec{ -setlinecap -}if -dup/LJ//knownget exec{ -setlinejoin -}if -dup/ML//knownget exec{ -setmeterlimit -}if -dup/D//knownget exec{ -setdash -}if -dup/RI//knownget exec{ -mark(Unimplemented ExtGState.RI)//error exec -}if -dup/OP//knownget exec{ -setoverprint -}if -dup/op//knownget exec{ -setoverprint -}if -dup/OPM//knownget exec{ -mark(Unimplemented ExtGState.OPM)//error exec -}if -dup/Font//knownget exec{ -mark(Unimplemented ExtGState.Font)//error exec -}if -dup/BG known{ -/BG//ResolveFunction exec -setblackgeneration -}if -dup/BG2 known{ -/BG2//ResolveFunction exec -dup/Default eq{ -//InitialExtGState/BG2 get -}if -setblackgeneration -}if -dup/UCR known{ -/UCR//ResolveFunction exec -setundercolorremoval -}if -dup/UCR2 known{ -/UCR2//ResolveFunction exec -dup/Default eq{ -//InitialExtGState/UCR2 get -}if -setundercolorremoval -}if -dup/TR known{ -/TR//ResolveFunction exec -//SetTransfer exec -}if -dup/TR2 known{ -/TR2//ResolveFunction exec -dup/Default eq{ -pop//InitialExtGState/TR2 get -aload pop setcolortransfer -}{ -//SetTransfer exec -}ifelse -}if -dup/HT//knownget exec{ -dup/Default eq{ -pop//InitialExtGState/HT get -sethalftone -}{ -//PDFR_DEBUG{(Ht beg)=}if -pop dup/HT//CheckHalftone//ResolveD exec -/SpotFunction//ResolveFunctionSafe exec -/TransferFunction//ResolveFunctionSafe exec -null exch -dup/HalftoneType get dup 5 eq exch dup 4 eq exch 2 eq or or{ -dup{ -dup//IsObjRef exec{ -pop -1 index exch//CheckHalftone ResolveD -}if -dup type/dicttype eq{ -dup/SpotFunction//ResolveFunctionSafe exec -/TransferFunction//ResolveFunctionSafe exec -//CreateHalftoneThresholds exec -dup/HalftoneType get 5 gt{ -4 3 roll pop -dup 4 1 roll -}if -}if -pop pop -}forall -}if -//CreateHalftoneThresholds exec -//PDFR_DEBUG{ -(HT:)= -dup{ -1 index/Default eq{ -(Default <<)= -exch pop -{exch = ==}forall -(>>)= -}{ -exch = == -}ifelse -}forall -(HT end)= flush -}if -exch dup null ne{ -(Warning: Ignoring a halftone with a Level 3 component halftone Type )print dup/HalftoneType get = -pop pop -}{ -pop -dup/HalftoneType get 5 gt{ -(Warning: Ignoring a Level 3 halftone Type )print dup/HalftoneType get = -pop -}{ -sethalftone -}ifelse -}ifelse -//PDFR_DEBUG{(HT set)= flush}if -}ifelse -}if -dup/FL//knownget exec{ -setflattness -}if -dup/SM//knownget exec{ -setsmoothness -}if -dup/SA//knownget exec{ -setstrokeadjust -}if -dup/BM//knownget exec{ -mark(Unimplemented ExtGState.BM)//error exec -}if -dup/SMask//knownget exec{ -mark(Unimplemented ExtGState.SMask)//error exec -}if -dup/CA//knownget exec{ -mark(Unimplemented ExtGState.CA)//error exec -}if -dup/ca//knownget exec{ -mark(Unimplemented ExtGState.ca)//error exec -}if -dup/AIS//knownget exec{ -mark(Unimplemented ExtGState.AIS)//error exec -}if -dup/TK//knownget exec{ -mark(Unimplemented ExtGState.TK)//error exec -}if -pop -}bind def -/CheckXObject -{dup/Subtype get dup/Image ne exch dup/Form ne exch/PS ne and and{ -mark(Resource )3 index( must have /Subtype /Image or /Form or /PS.)//error exec -}if -}bind def -/DoXObject -{ -//PDFReader/CurrentObject get/Context get/Resources get -/XObject//DoNothing//ResolveD exec -exch//CheckXObject//ResolveD exec -dup/Subtype get -dup/Image eq{ -pop -//CompleteOutlineImage exec -//DoImage exec -}{ -dup/PS eq{ -PDFR_DEBUG{ -(Executing a PS Xobject)= -}if -pop -//RunDelayedStream exec -}{ -dup/Form eq{ -pop -PDFR_DEBUG{ -(Executing a Form XObject)= -}if -//PDFReader/CurrentObject get exch -dup//PDFReader exch<< exch/Context exch >>/CurrentObject exch put -dup/Matrix get concat -dup/BBox get aload pop exch 3 index sub exch 2 index sub rectclip -//RunDelayedStream exec -//PDFReader exch/CurrentObject exch put -}{ -mark exch(unimplemented XObject type )exch//error exec -}ifelse -}ifelse -}ifelse -}bind def -/Operators 50 dict begin -/q{//GSave exec}bind def -/Q{//GRestore exec}bind def -/cm{//TempMatrix astore concat}bind def -/i{1 .min setflat}bind def -/J/setlinecap load def -/d/setdash load def -/j/setlinejoin load def -/w/setlinewidth load def -/M/setmiterlimit load def -/gs{SetExtGState}bind def -/g/setgray load def -/rg/setrgbcolor load def -/k/setcmykcolor load def -/cs{//ResolveColorSpace exec//SetColorSpaceSafe exec -}bind def -/sc/setcolor load def -/scn{//SetColor exec}bind def -/G/setgray load def -/RG/setrgbcolor load def -/K/setcmykcolor load def -/CS//cs def -/ri{SetColorRenderingIntent}bind def -/SC/setcolor load def -/SCN{//SetColor exec}bind def -/m/moveto load def -/l/lineto load def -/c/curveto load def -/v{currentpoint 6 2 roll curveto}bind def -/y{2 copy curveto}bind def -/re{ -4 2 roll moveto exch dup 0 rlineto 0 3 -1 roll rlineto neg 0 rlineto -closepath -}def -/h/closepath load def -/n/newpath load def -/S/stroke load def -/s{closepath stroke}bind def -/f/fill load def -/f*/eofill load def -/B{gsave fill grestore stroke}bind def -/b{closepath gsave fill grestore stroke}bind def -/B*{gsave eofill grestore stroke}bind def -/b*{closepath gsave eofill grestore stroke}bind def -/W/clip load def -/W*/eoclip load def -/sh{ -ResolveShading -dup/Background known{ -gsave -dup/ColorSpace get setcolorspace -dup/Background get aload pop setcolor -pathbbox -2 index sub exch 3 index sub exch -rectfill -grestore -}if -shfill -}bind def -/Do{//DoXObject exec}bind def -/BI{currentglobal false setglobal<<}bind def -/ID{>> -dup/DataSource currentfile -2 index/F//knownget exec{ -/A85 eq{ -0(~>)/SubFileDecode filter -}if -}if -put -//CompleteInlineImage exec -exch setglobal -//DoImage exec -}bind def -/EI{}bind def -/BT{gsave//GraphicState/InitialTextMatrix get currentmatrix pop}bind def -/ET{grestore}bind def -/Tc{//GraphicState exch/CharacterSpacing exch put}bind def -/TL{//GraphicState exch/TextLeading exch put}bind def -/Tr{//GraphicState exch/TextRenderingMode exch put}bind def -/Ts{ -mark(Unimplemented SetTextRise)//error exec -}bind def -/Tw{//GraphicState exch/WordSpacing exch put}bind def -/Tz{ -mark(Unimplemented SetHorizontalTextScaling)//error exec -}bind def -/Td{translate 0 0 moveto}bind def -/TD{dup neg//TL exec//Td exec}bind def -/Tm{//GraphicState/InitialTextMatrix get setmatrix -//TempMatrix astore concat -0 0 moveto}bind def -/T*{0//GraphicState/TextLeading get neg//Td exec}bind def -/Tj{//ShowTextBeg exec//ShowText exec//ShowTextEnd exec}bind def -/'{//T* exec//ShowText exec//ShowTextEnd exec}bind def -/"{3 2 roll//Tw exec exch//Tc exec//' exec}bind def -/TJ//ShowTextWithGlyphPositioning def -/Tf//SetFont def -/d0/setcharwidth load def -/d1/setcachedevice load def -/BDC{pop pop}bind def -/BMC{pop}bind def -/EMC{}bind def -/BX{BeginCompatibilitySection}bind def -/EX{EndCompatibilitySection}bind def -/DP{DefineMarkedContentPointWithPropertyList}bind def -/MP{DefineMarkedContentPoint}bind def -/PS{cvx exec}bind def -currentdict end def -//PDFR_STREAM{ -//Operators length dict begin -//Operators{ -exch dup -[exch//=only/exec load -( )/print load -8 7 roll -dup type/arraytype eq{ -/exec load -}if -( )/print load -]cvx -def -}forall -currentdict end/Operators exch def -}if -/.registerencoding -{pop pop -}bind def -/.defineencoding -{def -}bind def -/.findencoding -{load -}bind def -/currentglobal where -{pop currentglobal{setglobal}true setglobal} -{{}} -ifelse -/MacRomanEncoding -StandardEncoding 0 39 getinterval aload pop -/quotesingle -StandardEncoding 40 56 getinterval aload pop -/grave -StandardEncoding 97 31 getinterval aload pop -/Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute -/agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave -/ecircumflex/edieresis/iacute/igrave -/icircumflex/idieresis/ntilde/oacute -/ograve/ocircumflex/odieresis/otilde -/uacute/ugrave/ucircumflex/udieresis -/dagger/degree/cent/sterling/section/bullet/paragraph/germandbls -/registered/copyright/trademark/acute/dieresis/.notdef/AE/Oslash -/.notdef/plusminus/.notdef/.notdef/yen/mu/.notdef/.notdef -/.notdef/.notdef/.notdef/ordfeminine/ordmasculine/.notdef/ae/oslash -/questiondown/exclamdown/logicalnot/.notdef -/florin/.notdef/.notdef/guillemotleft -/guillemotright/ellipsis/space/Agrave/Atilde/Otilde/OE/oe -/endash/emdash/quotedblleft/quotedblright -/quoteleft/quoteright/divide/.notdef -/ydieresis/Ydieresis/fraction/currency -/guilsinglleft/guilsinglright/fi/fl -/daggerdbl/periodcentered/quotesinglbase/quotedblbase -/perthousand/Acircumflex/Ecircumflex/Aacute -/Edieresis/Egrave/Iacute/Icircumflex -/Idieresis/Igrave/Oacute/Ocircumflex -/.notdef/Ograve/Uacute/Ucircumflex -/Ugrave/dotlessi/circumflex/tilde -/macron/breve/dotaccent/ring/cedilla/hungarumlaut/ogonek/caron -256 packedarray -5 1 index .registerencoding -.defineencoding -exec -/AdobeGlyphList mark -/A 16#0041 -/AE 16#00c6 -/AEacute 16#01fc -/AEmacron 16#01e2 -/AEsmall 16#f7e6 -/Aacute 16#00c1 -/Aacutesmall 16#f7e1 -/Abreve 16#0102 -/Abreveacute 16#1eae -/Abrevecyrillic 16#04d0 -/Abrevedotbelow 16#1eb6 -/Abrevegrave 16#1eb0 -/Abrevehookabove 16#1eb2 -/Abrevetilde 16#1eb4 -/Acaron 16#01cd -/Acircle 16#24b6 -/Acircumflex 16#00c2 -/Acircumflexacute 16#1ea4 -/Acircumflexdotbelow 16#1eac -/Acircumflexgrave 16#1ea6 -/Acircumflexhookabove 16#1ea8 -/Acircumflexsmall 16#f7e2 -/Acircumflextilde 16#1eaa -/Acute 16#f6c9 -/Acutesmall 16#f7b4 -/Acyrillic 16#0410 -/Adblgrave 16#0200 -/Adieresis 16#00c4 -/Adieresiscyrillic 16#04d2 -/Adieresismacron 16#01de -/Adieresissmall 16#f7e4 -/Adotbelow 16#1ea0 -/Adotmacron 16#01e0 -/Agrave 16#00c0 -/Agravesmall 16#f7e0 -/Ahookabove 16#1ea2 -/Aiecyrillic 16#04d4 -/Ainvertedbreve 16#0202 -/Alpha 16#0391 -/Alphatonos 16#0386 -/Amacron 16#0100 -/Amonospace 16#ff21 -/Aogonek 16#0104 -/Aring 16#00c5 -/Aringacute 16#01fa -/Aringbelow 16#1e00 -/Aringsmall 16#f7e5 -/Asmall 16#f761 -/Atilde 16#00c3 -/Atildesmall 16#f7e3 -/Aybarmenian 16#0531 -/B 16#0042 -/Bcircle 16#24b7 -/Bdotaccent 16#1e02 -/Bdotbelow 16#1e04 -/Becyrillic 16#0411 -/Benarmenian 16#0532 -/Beta 16#0392 -/Bhook 16#0181 -/Blinebelow 16#1e06 -/Bmonospace 16#ff22 -/Brevesmall 16#f6f4 -/Bsmall 16#f762 -/Btopbar 16#0182 -/C 16#0043 -/Caarmenian 16#053e -/Cacute 16#0106 -/Caron 16#f6ca -/Caronsmall 16#f6f5 -/Ccaron 16#010c -/Ccedilla 16#00c7 -/Ccedillaacute 16#1e08 -/Ccedillasmall 16#f7e7 -/Ccircle 16#24b8 -/Ccircumflex 16#0108 -/Cdot 16#010a -/Cdotaccent 16#010a -/Cedillasmall 16#f7b8 -/Chaarmenian 16#0549 -/Cheabkhasiancyrillic 16#04bc -/Checyrillic 16#0427 -/Chedescenderabkhasiancyrillic 16#04be -/Chedescendercyrillic 16#04b6 -/Chedieresiscyrillic 16#04f4 -/Cheharmenian 16#0543 -/Chekhakassiancyrillic 16#04cb -/Cheverticalstrokecyrillic 16#04b8 -/Chi 16#03a7 -/Chook 16#0187 -/Circumflexsmall 16#f6f6 -/Cmonospace 16#ff23 -/Coarmenian 16#0551 -/Csmall 16#f763 -/D 16#0044 -/DZ 16#01f1 -/DZcaron 16#01c4 -/Daarmenian 16#0534 -/Dafrican 16#0189 -/Dcaron 16#010e -/Dcedilla 16#1e10 -/Dcircle 16#24b9 -/Dcircumflexbelow 16#1e12 -/Dcroat 16#0110 -/Ddotaccent 16#1e0a -/Ddotbelow 16#1e0c -/Decyrillic 16#0414 -/Deicoptic 16#03ee -/Delta 16#2206 -/Deltagreek 16#0394 -/Dhook 16#018a -/Dieresis 16#f6cb -/DieresisAcute 16#f6cc -/DieresisGrave 16#f6cd -/Dieresissmall 16#f7a8 -/Digammagreek 16#03dc -/Djecyrillic 16#0402 -/Dlinebelow 16#1e0e -/Dmonospace 16#ff24 -/Dotaccentsmall 16#f6f7 -/Dslash 16#0110 -/Dsmall 16#f764 -/Dtopbar 16#018b -/Dz 16#01f2 -/Dzcaron 16#01c5 -/Dzeabkhasiancyrillic 16#04e0 -/Dzecyrillic 16#0405 -/Dzhecyrillic 16#040f -/E 16#0045 -/Eacute 16#00c9 -/Eacutesmall 16#f7e9 -/Ebreve 16#0114 -/Ecaron 16#011a -/Ecedillabreve 16#1e1c -/Echarmenian 16#0535 -/Ecircle 16#24ba -/Ecircumflex 16#00ca -/Ecircumflexacute 16#1ebe -/Ecircumflexbelow 16#1e18 -/Ecircumflexdotbelow 16#1ec6 -/Ecircumflexgrave 16#1ec0 -/Ecircumflexhookabove 16#1ec2 -/Ecircumflexsmall 16#f7ea -/Ecircumflextilde 16#1ec4 -/Ecyrillic 16#0404 -/Edblgrave 16#0204 -/Edieresis 16#00cb -/Edieresissmall 16#f7eb -/Edot 16#0116 -/Edotaccent 16#0116 -/Edotbelow 16#1eb8 -/Efcyrillic 16#0424 -/Egrave 16#00c8 -/Egravesmall 16#f7e8 -/Eharmenian 16#0537 -/Ehookabove 16#1eba -/Eightroman 16#2167 -/Einvertedbreve 16#0206 -/Eiotifiedcyrillic 16#0464 -/Elcyrillic 16#041b -/Elevenroman 16#216a -/Emacron 16#0112 -/Emacronacute 16#1e16 -/Emacrongrave 16#1e14 -/Emcyrillic 16#041c -/Emonospace 16#ff25 -/Encyrillic 16#041d -/Endescendercyrillic 16#04a2 -/Eng 16#014a -/Enghecyrillic 16#04a4 -/Enhookcyrillic 16#04c7 -/Eogonek 16#0118 -/Eopen 16#0190 -/Epsilon 16#0395 -/Epsilontonos 16#0388 -/Ercyrillic 16#0420 -/Ereversed 16#018e -/Ereversedcyrillic 16#042d -/Escyrillic 16#0421 -/Esdescendercyrillic 16#04aa -/Esh 16#01a9 -/Esmall 16#f765 -/Eta 16#0397 -/Etarmenian 16#0538 -/Etatonos 16#0389 -/Eth 16#00d0 -/Ethsmall 16#f7f0 -/Etilde 16#1ebc -/Etildebelow 16#1e1a -/Euro 16#20ac -/Ezh 16#01b7 -/Ezhcaron 16#01ee -/Ezhreversed 16#01b8 -/F 16#0046 -/Fcircle 16#24bb -/Fdotaccent 16#1e1e -/Feharmenian 16#0556 -/Feicoptic 16#03e4 -/Fhook 16#0191 -/Fitacyrillic 16#0472 -/Fiveroman 16#2164 -/Fmonospace 16#ff26 -/Fourroman 16#2163 -/Fsmall 16#f766 -/G 16#0047 -/GBsquare 16#3387 -/Gacute 16#01f4 -/Gamma 16#0393 -/Gammaafrican 16#0194 -/Gangiacoptic 16#03ea -/Gbreve 16#011e -/Gcaron 16#01e6 -/Gcedilla 16#0122 -/Gcircle 16#24bc -/Gcircumflex 16#011c -/Gcommaaccent 16#0122 -/Gdot 16#0120 -/Gdotaccent 16#0120 -/Gecyrillic 16#0413 -/Ghadarmenian 16#0542 -/Ghemiddlehookcyrillic 16#0494 -/Ghestrokecyrillic 16#0492 -/Gheupturncyrillic 16#0490 -/Ghook 16#0193 -/Gimarmenian 16#0533 -/Gjecyrillic 16#0403 -/Gmacron 16#1e20 -/Gmonospace 16#ff27 -/Grave 16#f6ce -/Gravesmall 16#f760 -/Gsmall 16#f767 -/Gsmallhook 16#029b -/Gstroke 16#01e4 -/H 16#0048 -/H18533 16#25cf -/H18543 16#25aa -/H18551 16#25ab -/H22073 16#25a1 -/HPsquare 16#33cb -/Haabkhasiancyrillic 16#04a8 -/Hadescendercyrillic 16#04b2 -/Hardsigncyrillic 16#042a -/Hbar 16#0126 -/Hbrevebelow 16#1e2a -/Hcedilla 16#1e28 -/Hcircle 16#24bd -/Hcircumflex 16#0124 -/Hdieresis 16#1e26 -/Hdotaccent 16#1e22 -/Hdotbelow 16#1e24 -/Hmonospace 16#ff28 -/Hoarmenian 16#0540 -/Horicoptic 16#03e8 -/Hsmall 16#f768 -/Hungarumlaut 16#f6cf -/Hungarumlautsmall 16#f6f8 -/Hzsquare 16#3390 -/I 16#0049 -/IAcyrillic 16#042f -/IJ 16#0132 -/IUcyrillic 16#042e -/Iacute 16#00cd -/Iacutesmall 16#f7ed -/Ibreve 16#012c -/Icaron 16#01cf -/Icircle 16#24be -/Icircumflex 16#00ce -/Icircumflexsmall 16#f7ee -/Icyrillic 16#0406 -/Idblgrave 16#0208 -/Idieresis 16#00cf -/Idieresisacute 16#1e2e -/Idieresiscyrillic 16#04e4 -/Idieresissmall 16#f7ef -/Idot 16#0130 -/Idotaccent 16#0130 -/Idotbelow 16#1eca -/Iebrevecyrillic 16#04d6 -/Iecyrillic 16#0415 -/Ifraktur 16#2111 -/Igrave 16#00cc -/Igravesmall 16#f7ec -/Ihookabove 16#1ec8 -/Iicyrillic 16#0418 -/Iinvertedbreve 16#020a -/Iishortcyrillic 16#0419 -/Imacron 16#012a -/Imacroncyrillic 16#04e2 -/Imonospace 16#ff29 -/Iniarmenian 16#053b -/Iocyrillic 16#0401 -/Iogonek 16#012e -/Iota 16#0399 -/Iotaafrican 16#0196 -/Iotadieresis 16#03aa -/Iotatonos 16#038a -/Ismall 16#f769 -/Istroke 16#0197 -/Itilde 16#0128 -/Itildebelow 16#1e2c -/Izhitsacyrillic 16#0474 -/Izhitsadblgravecyrillic 16#0476 -/J 16#004a -/Jaarmenian 16#0541 -/Jcircle 16#24bf -/Jcircumflex 16#0134 -/Jecyrillic 16#0408 -/Jheharmenian 16#054b -/Jmonospace 16#ff2a -/Jsmall 16#f76a -/K 16#004b -/KBsquare 16#3385 -/KKsquare 16#33cd -/Kabashkircyrillic 16#04a0 -/Kacute 16#1e30 -/Kacyrillic 16#041a -/Kadescendercyrillic 16#049a -/Kahookcyrillic 16#04c3 -/Kappa 16#039a -/Kastrokecyrillic 16#049e -/Kaverticalstrokecyrillic 16#049c -/Kcaron 16#01e8 -/Kcedilla 16#0136 -/Kcircle 16#24c0 -/Kcommaaccent 16#0136 -/Kdotbelow 16#1e32 -/Keharmenian 16#0554 -/Kenarmenian 16#053f -/Khacyrillic 16#0425 -/Kheicoptic 16#03e6 -/Khook 16#0198 -/Kjecyrillic 16#040c -/Klinebelow 16#1e34 -/Kmonospace 16#ff2b -/Koppacyrillic 16#0480 -/Koppagreek 16#03de -/Ksicyrillic 16#046e -/Ksmall 16#f76b -/L 16#004c -/LJ 16#01c7 -/LL 16#f6bf -/Lacute 16#0139 -/Lambda 16#039b -/Lcaron 16#013d -/Lcedilla 16#013b -/Lcircle 16#24c1 -/Lcircumflexbelow 16#1e3c -/Lcommaaccent 16#013b -/Ldot 16#013f -/Ldotaccent 16#013f -/Ldotbelow 16#1e36 -/Ldotbelowmacron 16#1e38 -/Liwnarmenian 16#053c -/Lj 16#01c8 -/Ljecyrillic 16#0409 -/Llinebelow 16#1e3a -/Lmonospace 16#ff2c -/Lslash 16#0141 -/Lslashsmall 16#f6f9 -/Lsmall 16#f76c -/M 16#004d -/MBsquare 16#3386 -/Macron 16#f6d0 -/Macronsmall 16#f7af -/Macute 16#1e3e -/Mcircle 16#24c2 -/Mdotaccent 16#1e40 -/Mdotbelow 16#1e42 -/Menarmenian 16#0544 -/Mmonospace 16#ff2d -/Msmall 16#f76d -/Mturned 16#019c -/Mu 16#039c -/N 16#004e -/NJ 16#01ca -/Nacute 16#0143 -/Ncaron 16#0147 -/Ncedilla 16#0145 -/Ncircle 16#24c3 -/Ncircumflexbelow 16#1e4a -/Ncommaaccent 16#0145 -/Ndotaccent 16#1e44 -/Ndotbelow 16#1e46 -/Nhookleft 16#019d -/Nineroman 16#2168 -/Nj 16#01cb -/Njecyrillic 16#040a -/Nlinebelow 16#1e48 -/Nmonospace 16#ff2e -/Nowarmenian 16#0546 -/Nsmall 16#f76e -/Ntilde 16#00d1 -/Ntildesmall 16#f7f1 -/Nu 16#039d -/O 16#004f -/OE 16#0152 -/OEsmall 16#f6fa -/Oacute 16#00d3 -/Oacutesmall 16#f7f3 -/Obarredcyrillic 16#04e8 -/Obarreddieresiscyrillic 16#04ea -/Obreve 16#014e -/Ocaron 16#01d1 -/Ocenteredtilde 16#019f -/Ocircle 16#24c4 -/Ocircumflex 16#00d4 -/Ocircumflexacute 16#1ed0 -/Ocircumflexdotbelow 16#1ed8 -/Ocircumflexgrave 16#1ed2 -/Ocircumflexhookabove 16#1ed4 -/Ocircumflexsmall 16#f7f4 -/Ocircumflextilde 16#1ed6 -/Ocyrillic 16#041e -/Odblacute 16#0150 -/Odblgrave 16#020c -/Odieresis 16#00d6 -/Odieresiscyrillic 16#04e6 -/Odieresissmall 16#f7f6 -/Odotbelow 16#1ecc -/Ogoneksmall 16#f6fb -/Ograve 16#00d2 -/Ogravesmall 16#f7f2 -/Oharmenian 16#0555 -/Ohm 16#2126 -/Ohookabove 16#1ece -/Ohorn 16#01a0 -/Ohornacute 16#1eda -/Ohorndotbelow 16#1ee2 -/Ohorngrave 16#1edc -/Ohornhookabove 16#1ede -/Ohorntilde 16#1ee0 -/Ohungarumlaut 16#0150 -/Oi 16#01a2 -/Oinvertedbreve 16#020e -/Omacron 16#014c -/Omacronacute 16#1e52 -/Omacrongrave 16#1e50 -/Omega 16#2126 -/Omegacyrillic 16#0460 -/Omegagreek 16#03a9 -/Omegaroundcyrillic 16#047a -/Omegatitlocyrillic 16#047c -/Omegatonos 16#038f -/Omicron 16#039f -/Omicrontonos 16#038c -/Omonospace 16#ff2f -/Oneroman 16#2160 -/Oogonek 16#01ea -/Oogonekmacron 16#01ec -/Oopen 16#0186 -/Oslash 16#00d8 -/Oslashacute 16#01fe -/Oslashsmall 16#f7f8 -/Osmall 16#f76f -/Ostrokeacute 16#01fe -/Otcyrillic 16#047e -/Otilde 16#00d5 -/Otildeacute 16#1e4c -/Otildedieresis 16#1e4e -/Otildesmall 16#f7f5 -/P 16#0050 -/Pacute 16#1e54 -/Pcircle 16#24c5 -/Pdotaccent 16#1e56 -/Pecyrillic 16#041f -/Peharmenian 16#054a -/Pemiddlehookcyrillic 16#04a6 -/Phi 16#03a6 -/Phook 16#01a4 -/Pi 16#03a0 -/Piwrarmenian 16#0553 -/Pmonospace 16#ff30 -/Psi 16#03a8 -/Psicyrillic 16#0470 -/Psmall 16#f770 -/Q 16#0051 -/Qcircle 16#24c6 -/Qmonospace 16#ff31 -/Qsmall 16#f771 -/R 16#0052 -/Raarmenian 16#054c -/Racute 16#0154 -/Rcaron 16#0158 -/Rcedilla 16#0156 -/Rcircle 16#24c7 -/Rcommaaccent 16#0156 -/Rdblgrave 16#0210 -/Rdotaccent 16#1e58 -/Rdotbelow 16#1e5a -/Rdotbelowmacron 16#1e5c -/Reharmenian 16#0550 -/Rfraktur 16#211c -/Rho 16#03a1 -/Ringsmall 16#f6fc -/Rinvertedbreve 16#0212 -/Rlinebelow 16#1e5e -/Rmonospace 16#ff32 -/Rsmall 16#f772 -/Rsmallinverted 16#0281 -/Rsmallinvertedsuperior 16#02b6 -/S 16#0053 -/SF010000 16#250c -/SF020000 16#2514 -/SF030000 16#2510 -/SF040000 16#2518 -/SF050000 16#253c -/SF060000 16#252c -/SF070000 16#2534 -/SF080000 16#251c -/SF090000 16#2524 -/SF100000 16#2500 -/SF110000 16#2502 -/SF190000 16#2561 -/SF200000 16#2562 -/SF210000 16#2556 -/SF220000 16#2555 -/SF230000 16#2563 -/SF240000 16#2551 -/SF250000 16#2557 -/SF260000 16#255d -/SF270000 16#255c -/SF280000 16#255b -/SF360000 16#255e -/SF370000 16#255f -/SF380000 16#255a -/SF390000 16#2554 -/SF400000 16#2569 -/SF410000 16#2566 -/SF420000 16#2560 -/SF430000 16#2550 -/SF440000 16#256c -/SF450000 16#2567 -/SF460000 16#2568 -/SF470000 16#2564 -/SF480000 16#2565 -/SF490000 16#2559 -/SF500000 16#2558 -/SF510000 16#2552 -/SF520000 16#2553 -/SF530000 16#256b -/SF540000 16#256a -/Sacute 16#015a -/Sacutedotaccent 16#1e64 -/Sampigreek 16#03e0 -/Scaron 16#0160 -/Scarondotaccent 16#1e66 -/Scaronsmall 16#f6fd -/Scedilla 16#015e -/Schwa 16#018f -/Schwacyrillic 16#04d8 -/Schwadieresiscyrillic 16#04da -/Scircle 16#24c8 -/Scircumflex 16#015c -/Scommaaccent 16#0218 -/Sdotaccent 16#1e60 -/Sdotbelow 16#1e62 -/Sdotbelowdotaccent 16#1e68 -/Seharmenian 16#054d -/Sevenroman 16#2166 -/Shaarmenian 16#0547 -/Shacyrillic 16#0428 -/Shchacyrillic 16#0429 -/Sheicoptic 16#03e2 -/Shhacyrillic 16#04ba -/Shimacoptic 16#03ec -/Sigma 16#03a3 -/Sixroman 16#2165 -/Smonospace 16#ff33 -/Softsigncyrillic 16#042c -/Ssmall 16#f773 -/Stigmagreek 16#03da -/T 16#0054 -/Tau 16#03a4 -/Tbar 16#0166 -/Tcaron 16#0164 -/Tcedilla 16#0162 -/Tcircle 16#24c9 -/Tcircumflexbelow 16#1e70 -/Tcommaaccent 16#0162 -/Tdotaccent 16#1e6a -/Tdotbelow 16#1e6c -/Tecyrillic 16#0422 -/Tedescendercyrillic 16#04ac -/Tenroman 16#2169 -/Tetsecyrillic 16#04b4 -/Theta 16#0398 -/Thook 16#01ac -/Thorn 16#00de -/Thornsmall 16#f7fe -/Threeroman 16#2162 -/Tildesmall 16#f6fe -/Tiwnarmenian 16#054f -/Tlinebelow 16#1e6e -/Tmonospace 16#ff34 -/Toarmenian 16#0539 -/Tonefive 16#01bc -/Tonesix 16#0184 -/Tonetwo 16#01a7 -/Tretroflexhook 16#01ae -/Tsecyrillic 16#0426 -/Tshecyrillic 16#040b -/Tsmall 16#f774 -/Twelveroman 16#216b -/Tworoman 16#2161 -/U 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]%%)= flush -/newerror false store vmstatus pop pop 0 ne -{grestoreall -}if -errorname(VMerror)ne -{showpage -}if -initgraphics -0 720 moveto -errorname(VMerror)eq -{//this/ehsave known -{clear//this/ehsave get restore 2 vmreclaim -}if -vmstatus exch pop exch pop -} -/Courier 12 selectfont -{ -(ERROR: )//prnt exec errorname//prnt exec -(OFFENDING COMMAND: )//prnt exec -/command load//prnt exec -$error/ostack known{ -(%%[STACK:)= -(STACK:)//prnt exec -$error/ostack get aload length{ -//newline exec -dup mark eq{ -(-mark-)dup = show -}{ -dup type/nametype eq{ -dup xcheck not{ -(/)show -(/)print -}if -}if -dup =//ebuf cvs show -}ifelse -}repeat -}if -}ifelse -(%%]%)= -//systemdict/showpage get exec -quit -}if -end -end -}bind readonly put -}if -end -50 dict begin -count 0 ne{ -dup type/dicttype eq{ -{def}forall -false -} -{ -true -}ifelse -} -{ -true -}ifelse -{ -( *** Warning: global definitions dictionary not found, file may be corrupted.\n)print flush -}if -/DefaultSwitch -{ -dup where{ -pop pop -}{ -false def -}ifelse -}bind def -/=string 256 string def -/=only{ -//=string cvs print -}bind def -/HexDigits(0123456789ABCDEF)readonly def -/PrintHex -{8{ -dup -28 bitshift 15 and//HexDigits exch 1 getinterval//=only exec -4 bitshift -}repeat -pop -}bind def -/PDFR_DEBUG DefaultSwitch -/PDFR_DUMP DefaultSwitch -/PDFR_STREAM DefaultSwitch -/TTFDEBUG DefaultSwitch -/RotatePages DefaultSwitch -/FitPages DefaultSwitch -/CenterPages DefaultSwitch -/SetPageSize DefaultSwitch -/error -{ -counttomark 1 sub -1 0{ -index dup type/arraytype eq{==}{=only}ifelse -}for -()= -cleartomark -....Undefined -}bind def -//SetPageSize{ -//RotatePages//FitPages or//CenterPages or{ -mark(/RotatePages, /FitPages and CenterPages are not allowed with /SetPageSize)//error exec -}if -} -{ -//FitPages//CenterPages and{ -mark(CenterPages is not allowed with /FitPages)//error exec -}if -} -ifelse -/knownget -{ -2 copy known{ -get true -}{ -pop pop false -}ifelse -}bind def -/IsUpper -{dup(A)0 get ge exch(Z)0 get le and -}bind def -/cpa2g{ -dup length array -0 1 2 index length 1 sub{ -dup 3 index exch get cp2g -3 copy put pop pop -}for -exch pop -}bind def -/cpd2g{ -dup length dict exch{ -cp2g 2 index 3 1 roll put -}forall -}bind def -/cps2g{ -dup length string copy -}bind def -/cp2gprocs -<> -def -/cp2g{ -dup gcheck not{ -dup//cp2gprocs 1 index type -2 copy known{ -get currentglobal 3 1 roll true setglobal exec exch setglobal -1 index wcheck not{readonly}if -1 index xcheck{cvx}if -exch pop -}{ -pop pop -}ifelse -}if -}bind def -/BlockBuffer 65535 string def -/PDFReader currentdict def -/ObjectRegistryMaxLength 50000 def -/ObjectRegistry 10 dict def -ObjectRegistry -begin -0 ObjectRegistryMaxLength dict def -end -/CurrentObject null def -/DoneDocumentStructure false def -/GraphicState 20 dict begin -/InitialTextMatrix matrix def -/InitialMatrix matrix currentmatrix def -currentdict end def -/TempMatrix matrix def -/GraphicStateStack 20 array def -/GraphicStateStackPointer 0 def -/InitialTextMatrixStack 20 array def -/InitialTextMatrixStackPointer 0 def -/PDFColorSpaces 50 dict def -/InstalledFonts 50 dict def -/MacRomanEncodingInverse null def -currentglobal false setglobal -userdict/PDFR_InitialGS gstate put -userdict/PDFR_Patterns 50 dict put -userdict/FuncDataReader 10 dict put -setglobal -/InitialExtGState 20 dict begin -/BG2 currentblackgeneration cp2g def -/UCR2 currentundercolorremoval cp2g def -/TR2 currentglobal false setglobal[currentcolortransfer]exch setglobal cp2g def -/HT currenthalftone cp2g def -currentdict end readonly def -/InitialGraphicState 20 dict begin -/FontSize 0 def -/CharacterSpacing 0 def -/TextLeading 0 def -/TextRenderingMode 0 def -/WordSpacing 0 def -currentdict end readonly def -/SimpleColorSpaceNames 15 dict begin -/DeviceGray true def -/DeviceRGB true def -/DeviceCMYK true def -currentdict end readonly def -/1_24_bitshift_1_sub 1 24 bitshift 1 sub def -/ReadFontProcs 10 dict def -/GetObject -{ -dup ObjectRegistryMaxLength idiv -//PDFReader/ObjectRegistry get exch knownget{ -exch knownget -}{ -pop false -}ifelse -}bind def -/PutObject -{ -1 index ObjectRegistryMaxLength idiv -//PDFReader/ObjectRegistry get 1 index knownget{ -exch pop -3 1 roll put -}{ -//PDFReader/ObjectRegistry get dup -begin -1 index ObjectRegistryMaxLength dict def -end -exch get -3 1 roll put -}ifelse -}bind def -/Register -{ -1 index GetObject{ -dup xcheck{ -4 3 roll pop -//PDFR_DEBUG{ -(Have a daemon for )print 2 index == -}if -exec -}{ -dup null ne{ -mark(The object )4 index(is already defined : )4 index//error exec -}{ -pop -}ifelse -3 2 roll -exec -}ifelse -}{ -3 2 roll -exec -}ifelse -PutObject -}bind def -/IsRegistered -{ -GetObject{ -null ne -}{ -false -}ifelse -}bind def -/GetRegistered -{ -dup GetObject not{ -exch mark exch(Object )exch( isn't defined before needed (1).)//error exec -}if -dup xcheck{ -exch mark exch(Object )exch( isn't defined before needed (2).)//error exec -}{ -dup null eq{ -exch mark exch(Object )exch( isn't defined before needed (3).)//error exec -}if -exch pop -}ifelse -}bind def -/StandardFontNames<< -/Times-Roman true -/Helvetica true -/Courier true -/Symbol true -/Times-Bold true -/Helvetica-Bold true -/Courier-Bold true -/ZapfDingbats true -/Times-Italic true -/Helvetica-Oblique true -/Courier-Oblique true -/Times-BoldItalic true -/Helvetica-BoldOblique true -/Courier-BoldOblique true ->>def -/CleanAllResources -{//PDFR_DEBUG{ -(CleanAllResources beg)= -}if -//PDFReader/ObjectRegistry get{ -dup length 0 exch 1 exch 1 sub{ -2 copy get dup xcheck{ -pop pop -}{ -dup null eq{ -pop pop -}{ -dup type/dicttype eq{/.Global known}{pop false}ifelse{ -pop -}{ -//PDFR_DEBUG{ -(Dropping )print dup = -}if -1 index exch/DroppedObject put -}ifelse -}ifelse -}ifelse -}for -pop -}forall -FontDirectory length dict begin -FontDirectory{ -pop -dup//StandardFontNames exch known not{ -dup null def -}if -pop -}forall -currentdict -end{ -pop -//PDFR_DEBUG{ -(Undefining font )print dup = -}if -undefinefont -}forall -//PDFR_DEBUG{ -(CleanAllResources end)= -}if -}bind def -/PrintReference -{ -//PDFR_DEBUG{ -({ )print -dup{ -=only( )print -}forall -( })= -}if -}bind def -/R -{ -0 ne{ -exch mark exch(A referred object generation )exch( isn't 0.)//error exec -}if -[ -exch//GetRegistered/exec load -]cvx -//PrintReference exec -}bind def -/IsObjRef -{ -dup type/arraytype eq{ -dup length 3 eq{ -dup xcheck exch -dup 0 get type/integertype eq 3 2 roll and exch -dup 1 get//GetRegistered eq 3 2 roll and exch -2 get/exec load eq and -}{ -pop false -}ifelse -}{ -pop false -}ifelse -}bind def -/DoNothing -{ -}def -/RunTypeDaemon -{ -dup type/dicttype eq{ -dup/Type//knownget exec{ -//PDFReader/TypeDaemons get exch -//knownget exec{ -exec -}if -}if -}if -}bind def -/obj -{ -//PDFR_DEBUG{ -(Defining )print 1 index =only( )print dup =only( obj)= -}if -0 ne{ -exch mark exch(An object generation )exch( isn't 0.)//error exec -}if -}bind def -/endobj -{ -//PDFR_DEBUG{ -(endobj )= -}if -count 1 eq{ -pop -}{ -dup type/dicttype eq{ -dup/.endobj_daemon//knownget exec{ -//PDFR_DEBUG{(.endobj_daemon for )print 2 index =}if -exec -}if -}if -dup type/dicttype eq{dup/ImmediateExec known}{false}ifelse{ -pop pop -}{ -//PDFR_DEBUG{ -(Storing )print 1 index = -}if -//RunTypeDaemon exec -//DoNothing 3 1 roll//Register exec -}ifelse -}ifelse -}bind def -/StoreBlock -{ -//PDFR_DEBUG{ -(StoreBlock )print//PDFReader/BlockCount get =only(, Length = )print dup length = -}if -dup length string copy -//PDFReader/BlockCount get exch -//PDFReader/CurrentObject get 3 1 roll -put -//PDFReader/BlockCount get 1 add -//PDFReader exch/BlockCount exch put -}bind def -/CheckLength -{dup type/integertype ne{ -mark(Object length isn't an integer.)//error exec -}if -}bind def -/ResolveD -{ -3 copy pop get -dup//IsObjRef exec{ -//PDFR_DEBUG{ -(Resolving )print//PrintReference exec -}if -exec -exch exec -}{ -exch pop -}ifelse -dup 4 1 roll -put -}bind def -/ResolveA -{2 index 2 index get -dup//IsObjRef exec{ -exec -exch exec -3 copy put -}{ -exch pop -}ifelse -exch pop exch pop -}bind def -/StoreStream -{ -dup//PDFReader exch/CurrentObject exch put -//PDFReader/BlockCount 0 put -dup/Length//CheckLength//ResolveD exec -//PDFR_DEBUG{ -(StoreStream Length = )print dup = -}if -currentfile exch()/SubFileDecode filter -{dup//BlockBuffer readstring{ -//StoreBlock exec -}{ -//StoreBlock exec -exit -}ifelse -}loop -pop -//PDFReader/CurrentObject null put -//PDFR_DEBUG{ -(StoreStream end.)= -}if -}bind def -/MakeStreamDumper -{ -//PDFR_DEBUG{ -(MakeStreamDumper beg.)= -}if -currentglobal exch dup gcheck setglobal -[exch -1 dict dup/c 0 put exch -1024 string -{readstring pop -(StreamDumper )print 1 index/c get =string cvs print( )print -dup length =string cvs print( <)print dup print(>\n)print -dup length -3 2 roll -dup/c get -3 2 roll -add/c exch put -}/exec load -] -cvx 0()/SubFileDecode filter -exch setglobal -//PDFR_DEBUG{ -(MakeStreamDumper end.)= -}if -}bind def -/ShortFilterNames 15 dict begin -/AHx/ASCIIHexDecode def -/A85/ASCII85Decode def -/LZW/LZWDecode def -/Fl/FlateDecode def -/RL/RunLengthDecode def -/CCF/CCITTFaxDecode def -/DCT/DCTDecode def -currentdict end readonly def -/AppendFilters -{ -//PDFR_DEBUG{ -(AppendFilters beg.)= -}if -dup 3 1 roll -/Filter//knownget exec{ -dup type/nametype eq{ -dup//ShortFilterNames exch//knownget exec{ -exch pop -}if -2 index/DecodeParms//knownget exec{ -exch -}if -filter -}{ -dup 0 exch 1 exch length 1 sub{ -2 copy get -dup//ShortFilterNames exch//knownget exec{ -exch pop -}if -3 1 roll -4 index/DecodeParms//knownget exec{ -exch get -}{ -pop null -}ifelse -dup null eq{ -pop 3 1 roll filter exch -}{ -3 1 roll -4 1 roll filter exch -}ifelse -}for -pop -}ifelse -//PDFR_DEBUG//PDFR_DUMP and{ -//MakeStreamDumper exec -}if -}if -exch pop -//PDFR_DEBUG{ -(AppendFilters end.)= -}if -}bind def -/ExecuteStream -{ -dup//PDFReader exch/CurrentObject exch put -dup/Length//CheckLength//ResolveD exec -//PDFR_DEBUG{ -(ExecuteStream id = )print 2 index =only( Length = )print dup = -}if -//PDFReader/InitialGraphicState get -//PDFReader/GraphicState get copy pop -//PDFReader/Operators get begin -currentfile exch()/SubFileDecode filter -1 index//AppendFilters exec -cvx mark exch -exec -counttomark 0 ne{ -mark(Data left on ostack after an immediate stream execution.)//error exec -}if -cleartomark -end -//PDFR_DEBUG{ -(ExecuteStream end.)= -}if -//PDFReader/CurrentObject null put -dup/IsPage known{ -dup/Context get/NumCopies//knownget exec{ -1 sub{ -copypage -}repeat -}if -EPS2Write not{showpage}if -pagesave restore -}if -}bind def -/stream -{ -//PDFR_DEBUG{ -1 index =only( stream)= -}if -1 index GetObject{ -dup xcheck{ -exec -1 index null PutObject -}{ -pop -}ifelse -}if -dup/ImmediateExec known{ -dup/GlobalExec//knownget exec{ -currentglobal 4 1 roll -setglobal -//ExecuteStream exec -3 2 roll setglobal -}{ -//ExecuteStream exec -}ifelse -}{ -//StoreStream exec -}ifelse -dup/.CleanResources//knownget exec{ -/All eq{ -//CleanAllResources exec -}if -}if -}bind def -/HookFont -{ -//PDFR_DEBUG{ -(Loaded the font )print dup/FontName get = -}if -{ -dup/FontFileType get dup/Type1 eq exch/MMType1 eq or{ -dup/FontName get -//PDFReader/RemoveFontNamePrefix get exec -findfont -exit -}if -dup/FontFileType get/TrueType eq{ -//PDFReader/MakeType42 get exec -//PDFR_DEBUG{ -(Font dict <<)= -dup{ -1 index/sfnts eq{ -exch pop -(/sfnts [)print -{ -(-string\()print length//=only exec(\)- )= -}forall -(])= -}{ -exch//=only exec( )print == -}ifelse -}forall -(>>)= -}if -dup/FontName get exch definefont -exit -}if -mark(FontHook has no proc for )2 index/FontFileType get//error exec -}loop -/Font exch put -}bind def -/endstream -{ -}bind def -/xref -{ -//PDFR_DEBUG{ -(xref)= -//PDFR_DUMP{ -//PDFReader/ObjectRegistry get == -}if -}if -end -count 0 ne{ -mark(Excessive data on estack at the end of the interpretation.)//error exec -}if -currentfile 1(%%EOF)/SubFileDecode filter -flushfile -cleardictstack -}bind def -/ResolveDict -{dup{ -pop 1 index exch -//DoNothing//ResolveD exec -pop -}forall -pop -}bind def -/SetupPageView -{ -//PDFR_DEBUG{ -(SetupPageView beg)= -}if -//DSC_OPDFREAD not{ -//GraphicState/InitialMatrix get setmatrix -}if -/MediaBox get aload pop -3 index neg 3 index neg translate -3 -1 roll sub 3 1 roll exch sub exch -userdict/.HWMargins//knownget exec{ -aload pop -}{ -currentpagedevice/.HWMargins//knownget exec{ -aload pop -}{ -0 0 0 0 -}ifelse -}ifelse -currentpagedevice/PageSize get aload pop -3 -1 roll sub 3 1 roll exch sub exch -exch 3 index sub exch 3 index sub -//SetPageSize{ -//PDFR_DEBUG{ -(Setting page size to )print 1 index//=only exec( )print dup = -}if -pop pop 3 index 3 index 2 copy -currentglobal false setglobal 3 1 roll -currentpagedevice dup/PageSize known{ -/PageSize get aload pop -}{ -0 0 -}ifelse -round cvi 2 index round cvi eq -exch round cvi 3 index round cvi eq and -{ -//PDFR_DEBUG{(PageSize matches request)== flush}if -pop pop -}{ -/MediaRequested where{ -//PDFR_DEBUG{(MediaRequested is true, check against new request)== flush}if -/MediaRequested get aload pop -round cvi 2 index round cvi eq -exch round cvi 3 index round cvi eq and -{ -//PDFR_DEBUG{(MediaRequested same as current request, ignore)== flush}if -pop pop false -}{ -//PDFR_DEBUG{(MediaRequested different to current request)== flush}if -true -}ifelse -}{ -//PDFR_DEBUG{(No MediaRequested yet)== flush}if -true -}ifelse -{ -//PDFR_DEBUG{(Setting pagesize)== flush}if -2 array astore -dup/MediaRequested exch def -<< exch/PageSize exch >>setpagedevice -}if -}ifelse -userdict/PDFR_InitialGS gstate put -setglobal -}if -//RotatePages{ -2 copy gt 6 index 6 index gt ne{ -1 index 5 index le 1 index 5 index le and not -}{ -false -}ifelse -}{ -false -}ifelse -{//CenterPages{ -//PDFR_DEBUG{ -(Rotating page, and then centering it)== -}if -90 rotate -0 5 index neg translate -5 index 1 index exch sub 2 div -2 index 6 index sub 2 div neg -translate -}{ -//FitPages{ -1 index 5 index div 1 index 7 index div -2 copy gt{ -exch -}if -pop dup scale -}if -90 rotate -0 5 index neg translate -}ifelse -}{ -//CenterPages{ -//PDFR_DEBUG{ -(Ccentering page)== -}if -1 index 6 index sub 2 div -1 index 6 index sub 2 div -translate -}{ -//FitPages{ -1 index 6 index div 1 index 6 index div -2 copy gt{ -exch -}if -pop dup scale -}if -}ifelse -}ifelse -pop pop -translate -pop pop -//PDFR_DEBUG{ -(SetupPageView end)= -}if -}bind def -/PageContentsDaemon -{ -//PDFR_DEBUG{ -(Executing PageContentsDaemon for )print 2 index = -}if -1 index exch/Context exch put -dup/ImmediateExec true put -/pagesave save def -dup/IsPage true put -SetPageSize{dup/Context get//SetupPageView exec}if -}bind def -/FontFileDaemon -{ -//PDFR_DEBUG{ -(Executing FontFileDaemon for )print 2 index = -}if -dup/FontFileType get -2 index exch -dup//ReadFontProcs exch//knownget exec{ -exch pop exec -}{ -mark(FontFile reader for )2 index( isn't implemented yet.)//error exec -}ifelse -//PDFR_DEBUG{ -(FontFileDaemon end)= -}if -pop -}bind def -/FontDescriptorDaemon -{ -//PDFR_DEBUG{ -(Executing FontDescriptorDaemon for )print 2 index = -}if -2 copy/FontResource exch put -/Subtype get 1 index exch/FontFileType exch put -}bind def -/UnPDFEscape{ -dup dup length string cvs -dup(#)search{ -{ -pop -(16#--)2 index 0 2 getinterval -1 index 3 2 getinterval copy pop -cvi -0 exch put -0 -1 index 2 1 index length 2 sub getinterval -3 copy putinterval -length -3 copy exch put -getinterval -(#)search not{ -pop exit -}if -}loop -(\0)search pop exch pop exch pop -cvn -exch pop -}{ -pop pop -}ifelse -}bind def -/TypeDaemons<< -/Page -{//PDFR_DEBUG{ -(Recognized a page.)= -}if -dup/Contents//knownget exec{ -0 get//DoNothing exch -[ -3 index//PageContentsDaemon/exec load -]cvx -//Register exec -}{ -(fixme: page with no Contents won't be printed.)= -}ifelse -}bind -/FontDescriptor -{//PDFR_DEBUG{ -(Recognized a font descriptor.)= -}if -dup/FontName//knownget exec{ -1 index/FontName 3 -1 roll//UnPDFEscape exec put -}if -dup dup/FontFile known{/FontFile}{/FontFile2}ifelse -//knownget exec{ -0 get//DoNothing exch -[ -3 index//FontFileDaemon/exec load -]cvx -//Register exec -}{ -(Font descriptor )print 1 index =only( has no FontFile.)= -}ifelse -}bind -/Font -{//PDFR_DEBUG{ -(Recognized a font resource.)= -}if -dup/BaseFont//knownget exec{ -//UnPDFEscape exec 2 copy/BaseFont exch put -//PDFReader/RemoveFontNamePrefix get exec -currentglobal exch -dup/Font resourcestatus{ -pop pop -//PDFReader/GetInstalledFont get exec pop -}{ -pop -}ifelse -setglobal -}if -dup/FontDescriptor//knownget exec{ -0 get -dup//IsRegistered exec{ -//PDFR_DEBUG{ -(already registered )print dup = -}if -pop -}{ -//DoNothing exch -[ -3 index//FontDescriptorDaemon/exec load -]cvx -//Register exec -}ifelse -}if -}bind ->>def -/MakeStreamReader -{dup -[ -exch -//PDFR_DEBUG{ -(Stream proc ) -/print load -//PDFR_STREAM{ -(<) -/print load -}if -}if -1 dict dup/i -1 put -/dup load -/i -/get load -1 -/add load -/dup load -3 -1 -/roll load -/i -/exch load -/put load -//knownget -/exec load -/not load -{()} -/if load -//PDFR_DEBUG{ -//PDFR_STREAM{ -/dup load -/print load -(>) -/print load -}if -( end of stream proc.\n) -/print load -}if -]cvx -//PDFR_DEBUG{ -(Stream reader )print dup == -}if -0()/SubFileDecode filter -exch//AppendFilters exec -}bind def -/RunDelayedStream -{ -//GraphicState/InitialTextMatrix get -//InitialTextMatrixStack//PDFReader/InitialTextMatrixStackPointer get -2 copy get null eq{ -2 copy currentglobal true setglobal matrix exch setglobal put -}if -get copy pop -//PDFReader/InitialTextMatrixStackPointer 2 copy get 1 add put -//MakeStreamReader exec -mark exch -cvx exec -counttomark 0 ne{ -mark(Data left on ostack after a delayed stream execution.)//error exec -}if -cleartomark -//PDFReader/InitialTextMatrixStackPointer 2 copy get 1 sub put -//InitialTextMatrixStack//PDFReader/InitialTextMatrixStackPointer get get -//GraphicState/InitialTextMatrix get -copy pop -}bind def -//ReadFontProcs begin -/Type1 -{//PDFR_DEBUG{ -(ReadFontProcs.Type1)= -}if -dup/.endobj_daemon[4 index//HookFont/exec load]cvx put -dup/ImmediateExec true put -/GlobalExec true put -}bind def -/MMType1//Type1 def -/TrueType -{//PDFR_DEBUG{ -(ReadFontProcs.TrueType)= -}if -dup/.endobj_daemon[4 index//HookFont/exec load]cvx put -pop -}bind def -end -/.opdloadttfontdict 50 dict def -.opdloadttfontdict begin -/maxstring 65400 def -end -/.InsertionSort -{ -/CompareProc exch def -/Array exch def -1 1 Array length 1 sub -{ -/Ix exch def -/Value1 Array Ix get def -/Jx Ix 1 sub def -{ -Jx 0 lt{ -exit -}if -/Value2 Array Jx get def -Value1 Value2 CompareProc{ -exit -}if -Array Jx 1 add Value2 put -/Jx Jx 1 sub def -}loop -Array Jx 1 add Value1 put -}for -Array -}bind def -/putu16{ -3 copy -8 bitshift put -exch 1 add exch 16#ff and put -}bind def -/putu32{ -3 copy -16 bitshift putu16 -exch 2 add exch 16#ffff and putu16 -}bind def -/.readtable{ -dup dup 1 and add string -dup 0 4 -1 roll getinterval -3 -1 roll exch -dup()ne{readstring}if pop pop -}bind def -/.readbigtable{ -dup maxstring lt{ -.readtable -}{ -currentuserparams/VMReclaim get -2 vmreclaim -[4 2 roll{ -dup maxstring le{exit}if -1 index maxstring string readstring pop 3 1 roll maxstring sub -}loop .readtable] -exch vmreclaim -}ifelse -}bind def -/ReadTTF -{ -.opdloadttfontdict begin -/TTFontFile exch def -/TableDir TTFontFile 12 string readstring pop def -/tables TTFontFile TableDir 4 getu16 16 mul string readstring pop def -/tabarray tables length 16 idiv array def -TableDir 0 4 getinterval(ttcf)eq{ -QUIET not{(Can't handle TrueType font Collections.)=}if -/.loadttfonttables cvx/invalidfont signalerror -}{ -0 16 tables length 1 sub{ -dup -tables exch 16 getinterval -exch 16 div cvi exch -tabarray 3 1 roll put -}for -}ifelse -tabarray{exch 8 getu32 exch 8 getu32 gt}.InsertionSort pop -/Read TableDir length tables length add def -/tabs[ -tabarray{ -dup 8 getu32 -Read sub -dup 0 gt{ -dup string TTFontFile exch readstring pop pop -Read add/Read exch def -}{ -pop -}ifelse -12 getu32 -dup Read add -/Read exch def -TTFontFile exch .readbigtable -}forall -]def -end -}bind def -/GetLocaType -{ -0 1 tabarray length 1 sub{ -dup tabarray exch get -0 4 getinterval(head)eq{ -tabs exch get -50 gets16 -/LocaType exch def -exit -}{ -pop -}ifelse -}for -}bind def -/GetNumGlyphs -{ -0 1 tabarray length 1 sub{ -dup tabarray exch get -0 4 getinterval(maxp)eq{ -tabs exch get -4 getu16 -/NumGlyphs exch def -exit -}{ -pop -}ifelse -}for -}bind def -/StringToLoca -{ -/LocaIndex exch def -/StringOffset 0 def -{ -dup length StringOffset gt{ -dup -LocaType 1 eq{ -StringOffset getu32 -LocaArray LocaIndex 3 -1 roll put -/LocaIndex LocaIndex 1 add def -/StringOffset StringOffset 4 add -def -}{ -StringOffset getu16 2 mul -LocaArray length LocaIndex gt{ -LocaArray LocaIndex 3 -1 roll put -}{ -pop -}ifelse -/LocaIndex LocaIndex 1 add def -/StringOffset StringOffset 2 add -def -}ifelse -}{ -pop -LocaIndex -exit -}ifelse -}loop -}bind def -/GetSortedLoca -{ -NumGlyphs 1 add array/LocaArray exch def -0 1 tabarray length 1 sub{ -dup tabarray exch get -0 4 getinterval(loca)eq{ -tabs exch get -exit -}{ -pop -}ifelse -}for -dup type/stringtype eq{ -0 StringToLoca pop -}{ -0 exch -{ -exch StringToLoca -}forall -pop -}ifelse -LocaArray{gt}.InsertionSort pop -}bind def -/GetWorkingString -{ -WorkString 0 -GlyfArray GlyfStringIndex get -putinterval -/WorkBytes GlyfArray GlyfStringIndex get length def -/GlyfStringIndex GlyfStringIndex 1 add def -}bind def -/GetWorkingBytes -{ -/BytesToRead exch def -WorkString 0 BytesToRead getinterval -dup length string copy -WorkString BytesToRead WorkBytes BytesToRead sub getinterval -dup length string copy -WorkString 0 3 -1 roll putinterval -/WorkBytes WorkBytes BytesToRead sub def -}bind def -/GetGlyfBytes -{ -/ToRead exch def -WorkBytes 0 eq{ -GetWorkingString -}if -WorkBytes ToRead ge{ -ToRead string dup 0 -ToRead GetWorkingBytes putinterval -}{ -ToRead string -dup -0 -WorkString 0 WorkBytes getinterval -putinterval -dup -WorkBytes -ToRead WorkBytes sub -GetWorkingString -GetWorkingBytes -putinterval -}ifelse -}bind def -/SplitGlyf -{ -/GlyfArray exch def -/DestArray GlyfArray length 2 mul array def -/DestArrayIndex 0 def -/LastLoca 0 def -/NextLocaIndex 0 def -/LastLocaIndex 0 def -/GlyfStringIndex 0 def -/WorkString maxstring string def -/WorkBytes 0 def -{ -LocaArray NextLocaIndex get -LastLoca sub maxstring gt -{ -LocaArray LastLocaIndex get LastLoca sub -GetGlyfBytes -DestArray DestArrayIndex 3 -1 roll put -/DestArrayIndex DestArrayIndex 1 add def -LocaArray LastLocaIndex get/LastLoca exch def -}{ -/LastLocaIndex NextLocaIndex def -/NextLocaIndex NextLocaIndex 1 add def -NextLocaIndex NumGlyphs gt -{ -WorkBytes -GlyfStringIndex GlyfArray length lt{ -GlyfArray GlyfStringIndex get length -add string dup -0 -WorkString 0 WorkBytes getinterval -putinterval -dup -WorkBytes -GetWorkingString -WorkString 0 WorkBytes getinterval -putinterval -}{ -pop -WorkString 0 WorkBytes getinterval -}ifelse -dup length string copy -DestArray DestArrayIndex 3 -1 roll put -exit -}if -}ifelse -}loop -DestArray -}bind def -/ProcessTTData -{ -.opdloadttfontdict begin -0 1 tabarray length 1 sub{ -/ix exch def -tabarray ix get -12 getu32 dup maxstring le{ -dup 4 mod 0 ne{ -4 div cvi 1 add 4 mul string/newstring exch def -/oldstring tabs ix get def -newstring 0 oldstring putinterval -0 1 newstring length oldstring length sub 1 sub{ -newstring exch oldstring length add 0 put -}for -tabs ix newstring put -}{ -pop -}ifelse -}{ -dup 4 mod 0 ne{ -dup maxstring idiv maxstring mul sub -4 idiv 1 add 4 mul string/newstring exch def -tabs ix get -dup length 1 sub dup/iy exch def get/oldstring exch def -newstring 0 oldstring putinterval -0 1 newstring length oldstring length sub 1 sub{ -newstring exch oldstring length add 0 put -}for -tabs ix get iy newstring put -}{ -pop -}ifelse -}ifelse -}for -0 1 tabarray length 1 sub{ -dup tabarray exch get -dup 12 getu32 maxstring gt{ -0 4 getinterval dup(glyf)eq{ -pop -GetLocaType -GetNumGlyphs -GetSortedLoca -dup tabs exch get -SplitGlyf -tabs 3 1 roll put -}{ -(Warning, table )print print( > 64Kb\n)print -pop -}ifelse -}{ -pop -pop -}ifelse -}for -end -}bind def -/Makesfnts -{ -.opdloadttfontdict begin -0 -tabs{ -dup type/stringtype eq{ -pop -1 add -}{ -{ -type/stringtype eq{ -1 add -}if -}forall -}ifelse -}forall -1 add -/TTOffset -TableDir length -tabarray length 16 mul add -def -0 -tabarray{ -exch dup 1 add -3 1 roll -dup -tabs exch get -dup type/stringtype eq{ -length -2 index exch -TTOffset -dup 3 1 roll add -/TTOffset exch def -8 exch putu32 -exch tabarray 3 1 roll -put -}{ -0 exch -{ -dup type/stringtype eq{ -length add -}{ -pop -}ifelse -}forall -2 index exch -TTOffset -dup 3 1 roll add -/TTOffset exch def -8 exch putu32 -exch tabarray 3 1 roll -put -}ifelse -}forall -pop -array -dup 0 -TableDir length -tables length add -string -dup 0 TableDir putinterval -dup 12 tables putinterval -put -dup -/ix 1 def -tabs{ -dup type/stringtype eq{ -ix exch -put dup -/ix ix 1 add def -}{ -{ -dup type/stringtype eq{ -ix exch put dup -/ix ix 1 add def -}{ -pop -}ifelse -}forall -}ifelse -}forall -pop -end -}bind def -/MakeType42 -{ -//PDFR_DEBUG{ -(MakeType42 beg)= -}if -10 dict begin -/FontName 1 index/FontName get def -/FontType 42 def -/FontMatrix[1 0 0 1 0 0]def -/FontBBox 1 index/FontBBox get def -dup/FontResource get -dup/Encoding known{ -//PDFReader/ObtainEncoding get exec -/Encoding get -}{ -pop null -}ifelse -/PDFEncoding exch def -/CharStrings 2 index//PDFReader/MakeTTCharStrings get exec def -/sfnts 2 index//MakeStreamReader exec -ReadTTF -ProcessTTData -Makesfnts -def -/Encoding StandardEncoding def -/PaintType 0 def -currentdict end -//PDFR_DEBUG{ -(MakeType42 end)= -}if -}bind def -/GetInstalledFont -{ -dup//InstalledFonts exch knownget{ -exch pop -}{ -dup findfont dup 3 1 roll -//InstalledFonts 3 1 roll put -}ifelse -}bind def -/RemoveFontNamePrefix -{//=string cvs true -0 1 5{ -2 index exch get//IsUpper exec not{ -pop false exit -}if -}for -{(+)search{ -pop pop -}if -}if -cvn -}bind def -/CheckFont -{dup/Type get/Font ne{ -mark(Resource )3 index( must have /Type/Font .)//error exec -}if -}bind def -/CheckEncoding -{dup type/nametype ne{ -dup/Type get/Encoding ne{ -mark(Resource )3 index( must have /Type/Encoding .)//error exec -}if -}if -}bind def -/ObtainEncoding -{dup/Encoding known{ -dup dup/Encoding//CheckEncoding//ResolveD exec -dup type dup/arraytype eq exch/packedarraytype eq or{ -pop pop -}{ -dup type/nametype eq{ -/Encoding findresource -}{ -dup/BaseEncoding//knownget exec not{ -/StandardEncoding -}if -/Encoding findresource -exch -/Differences//knownget exec{ -exch dup length array copy exch -0 exch -{ -dup type/integertype eq{ -exch pop -}{ -3 copy put pop -1 add -}ifelse -}forall -pop -}if -}ifelse -/Encoding exch put -}ifelse -}{ -dup/Encoding/StandardEncoding/Encoding findresource put -}ifelse -}bind def -/ObtainMetrics -{dup/Widths//knownget exec{ -1 index/Encoding get -256 dict -3 index/Subtype get/TrueType eq{ -1000 -}{ -1 -}ifelse -4 index/MissingWidth//knownget exec not{ -0 -}if -5 index/FirstChar//knownget exec not{ -0 -}if -6 5 roll -dup 0 exch 1 exch length 1 sub{ -2 copy get -exch 3 index add -7 index exch get -dup dup null ne exch/.notdef ne and{ -6 index 3 1 roll exch -6 index div -3 copy pop//knownget exec{ -0 eq -}{ -true -}ifelse -{put -}{ -pop pop pop -}ifelse -}{ -pop pop -}ifelse -}for -pop pop pop pop exch pop -1 index exch/Metrics exch put -}{ -dup/MissingWidth//knownget exec{ -256 dict -2 index/Encoding get{ -dup null ne{ -3 copy 3 2 roll put -}if -pop -}forall -exch pop -1 index exch/Metrics exch put -}if -}ifelse -}bind def -/NotDef -{ -FontMatrix aload pop pop pop exch pop exch pop -1 exch div exch -1 exch div exch -1 index 0 setcharwidth -0 setlinewidth -0 0 moveto -2 copy rlineto -1 index 0 rlineto -neg exch neg exch rlineto -closepath stroke -}bind def -/SaveResourcesToStack -{ -[ -//PDFReader/OldResources known{ -//PDFReader/OldResources get -}{ -null -}ifelse -//PDFReader/CurrentObject get/Context get/Resources get -] -//PDFReader/OldResources 3 -1 roll put -}bind def -/RestoreResourcesFromStack -{ -//PDFReader/OldResources get dup -0 get//PDFReader/OldResources 3 -1 roll put -1 get//PDFReader/CurrentObject get/Context get/Resources 3 -1 roll put -}bind def -/BuildChar -{//PDFR_DEBUG{ -(BuildChar )print dup//=only exec( )print -}if -exch begin -Encoding exch get -//PDFR_DEBUG{ -dup = -}if -dup null eq{ -pop//NotDef exec -} -{ -CharProcs exch//knownget exec -{ -currentfont/Font get/Resources//knownget exec{ -exec -SaveResourcesToStack -//PDFReader/CurrentObject get/Context get -/Resources 3 -1 roll put -//RunDelayedStream exec -RestoreResourcesFromStack -}{ -//RunDelayedStream exec -}ifelse -} -{ -//NotDef exec -}ifelse -}ifelse -end -}bind def -/printdict -{(<<)= -{exch = ==}forall -(>>)= -}bind def -/printfont -{ -dup{ -exch dup = -dup/Encoding eq{ -pop = -}{ -dup/FontInfo eq exch/Private eq or{ -//printdict exec -}{ -== -}ifelse -}ifelse -}forall -}bind def -/ScaleMetrics -{1 index{ -2 index div -3 index -3 1 roll put -}forall -pop -}bind def -/ResolveAndSetFontAux -{exch dup -//PDFReader/CurrentObject get/Context get/Resources get -/Font//DoNothing//ResolveD exec -exch//CheckFont//ResolveD exec -dup/Font//knownget exec{ -exch pop exch pop -}{ -{ -dup/Subtype get dup dup/Type1 eq exch/TrueType eq or exch/MMType1 eq or{ -exch pop -dup/BaseFont get -//RemoveFontNamePrefix exec -//PDFR_DEBUG{ -(Font )print dup = -}if -1 index/FontDescriptor known{ -//PDFR_DEBUG{ -(Font from a font descriptor.)= -}if -1 index -/FontDescriptor//DoNothing//ResolveD exec -/Font//knownget exec{ -exch pop -}{ -//PDFR_DEBUG{ -(Font descriptor has no Font resolved.)= -}if -//GetInstalledFont exec -}ifelse -}{ -//GetInstalledFont exec -}ifelse -exch -dup/Encoding known not{ -1 index/Encoding get 1 index exch/Encoding exch put -}if -//ObtainEncoding exec -//ObtainMetrics exec -exch -dup length dict copy -dup 2 index/Encoding get -/Encoding exch put -1 index/Metrics//knownget exec{ -2 index/Subtype get/TrueType ne{ -1 index/FontMatrix get 0 get -dup 0 eq{ -pop -1 index/FontMatrix get 1 get -dup 0 eq{pop 1}if -}if -0.001 div -//ScaleMetrics exec -}{ -1 index/sfnts known not{ -1 index/FontMatrix get 0 get -dup 0 eq{ -pop -1 index/FontMatrix get 1 get -dup 0 eq{pop 1}if -}if -//ScaleMetrics exec -}if -}ifelse -1 index exch/Metrics exch put -}if -1 index/BaseFont get -exch -dup/FID undef -dup/UniqueID undef -definefont -dup 3 1 roll -/Font exch put -exit -}if -dup/Subtype get/Type3 eq{ -//ObtainEncoding exec -2 copy exch/FontName exch put -dup/CharProcs get//ResolveDict exec -dup/FontType 3 put -dup/BuildChar//BuildChar put -dup dup/Font exch put -dup 3 1 roll -definefont -2 copy ne{ -2 copy/Font exch put -}if -exch pop -exit -}if -dup/Subtype get/Type0 eq{ -}if -dup/Subtype get/CIDFontType0 eq{ -}if -dup/Subtype get/CIDFontType2 eq{ -}if -mark(Unknown font type )2 index/Subtype get//error exec -}loop -}ifelse -exch scalefont setfont -}bind def -/ResolveAndSetFont -{ -//ResolveAndSetFontAux exec -}bind def -/.knownget -{2 copy known{ -get true -}{ -pop pop false -}ifelse -}bind def -/.min -{2 copy lt{ -exch -}if -pop -}bind def -/.max -{2 copy gt{ -exch -}if -pop -}bind def -/.dicttomark -{>> -}bind def -/getu16{ -2 copy get 8 bitshift 3 1 roll 1 add get add -}bind def -/gets16{ -getu16 16#8000 xor 16#8000 sub -}bind def -/getu32{ -2 copy getu16 16 bitshift 3 1 roll 2 add getu16 add -}bind def -/gets32{ -2 copy gets16 16 bitshift 3 1 roll 2 add getu16 add -}bind def -/cmapformats mark -0{ -6 256 getinterval{}forall 256 packedarray -}bind -2{ -/sHK_sz 2 def -/sH_sz 8 def -dup 2 getu16/cmapf2_tblen exch def -dup 4 getu16/cmapf2_lang exch def -dup 6 256 sHK_sz mul getinterval/sHKs exch def -0 -0 1 255{ -sHKs exch -2 mul getu16 -1 index -1 index -lt{exch}if pop -}for -/sH_len exch def -dup 6 256 sHK_sz mul add -cmapf2_tblen 1 index sub getinterval -/sH_gIA exch def -/cmapf2_glyph_array 65535 array def -/.cmapf2_putGID{ -/cmapf2_ch cmapf2_ch_hi 8 bitshift cmapf2_ch_lo add def -firstCode cmapf2_ch_lo le -cmapf2_ch_lo firstCode entryCount add lt -and{ -sH_offset idRangeOffset add -cmapf2_ch_lo firstCode sub 2 mul -add 6 add -sH_gIA exch getu16 -dup 0 gt{ -idDelta add -cmapf2_glyph_array exch cmapf2_ch exch put -}{ -pop -}ifelse -}{ -}ifelse -}def -16#00 1 16#ff{ -/cmapf2_ch_hi exch def -sHKs cmapf2_ch_hi sHK_sz mul getu16 -/sH_offset exch def -sH_gIA sH_offset sH_sz getinterval -dup 0 getu16/firstCode exch def -dup 2 getu16/entryCount exch def -dup 4 gets16/idDelta exch def -dup 6 getu16/idRangeOffset exch def -pop -sH_offset 0 eq{ -/cmapf2_ch_lo cmapf2_ch_hi def -/cmapf2_ch_hi 0 def -.cmapf2_putGID -}{ -16#00 1 16#ff{ -/cmapf2_ch_lo exch def -.cmapf2_putGID -}for -}ifelse -}for -pop -0 1 cmapf2_glyph_array length 1 sub{ -dup cmapf2_glyph_array exch get -null eq{cmapf2_glyph_array exch 0 put}{pop}ifelse -}for -cmapf2_glyph_array -}bind -4{ -/etab exch def -/nseg2 etab 6 getu16 def -14/endc etab 2 index nseg2 getinterval def -2 add -nseg2 add/startc etab 2 index nseg2 getinterval def -nseg2 add/iddelta etab 2 index nseg2 getinterval def -nseg2 add/idroff etab 2 index nseg2 getinterval def -pop -/firstcode startc 0 getu16 16#ff00 and dup 16#f000 ne{pop 0}if def -/lastcode firstcode def -/striptopbyte false def -/putglyph{ -glyphs code 3 -1 roll put/code code 1 add def -}bind def -/numcodes 0 def/glyphs 0 0 2 nseg2 3 sub{ -/i2 exch def -/scode startc i2 getu16 def -/ecode endc i2 getu16 def -ecode lastcode gt{ -/lastcode ecode def -}if -}for pop -firstcode 16#f000 ge lastcode firstcode sub 255 le and{ -lastcode 255 and -/striptopbyte true def -}{ -lastcode -}ifelse -1 add -array def -glyphs length 1024 ge{ -.array1024z 0 1024 glyphs length 1023 sub{glyphs exch 2 index putinterval}for -glyphs dup length 1024 sub 3 -1 roll -putinterval -}{ -0 1 glyphs length 1 sub{glyphs exch 0 put}for -}ifelse -/numcodes 0 def/code 0 def -0 2 nseg2 3 sub{ -/i2 exch def -/scode startc i2 getu16 def -/ecode endc i2 getu16 def -numcodes scode firstcode sub -exch sub 0 .max dup/code exch code exch add def -ecode scode sub 1 add add numcodes add/numcodes exch def -/delta iddelta i2 gets16 def -TTFDEBUG{ -(scode=)print scode =only -( ecode=)print ecode =only -( delta=)print delta =only -( droff=)print idroff i2 getu16 = -}if -idroff i2 getu16 dup 0 eq{ -pop scode delta add 65535 and 1 ecode delta add 65535 and -striptopbyte{ -/code scode 255 and def -}{ -/code scode def -}ifelse -{putglyph}for -}{ -/gloff exch 14 nseg2 3 mul add 2 add i2 add add def -striptopbyte{ -/code scode 255 and def -}{ -/code scode def -}ifelse -0 1 ecode scode sub{ -2 mul gloff add etab exch getu16 -dup 0 ne{delta add 65535 and}if putglyph -}for -}ifelse -}for glyphs/glyphs null def -}bind -6{ -dup 6 getu16/firstcode exch def dup 8 getu16/ng exch def -firstcode ng add array -0 1 firstcode 1 sub{2 copy 0 put pop}for -dup firstcode ng getinterval -0 1 ng 1 sub{ -dup 2 mul 10 add 4 index exch getu16 3 copy put pop pop -}for pop exch pop -}bind -.dicttomark readonly def -/cmaparray{ -dup 0 getu16 cmapformats exch .knownget{ -TTFDEBUG{ -(cmap: format )print 1 index 0 getu16 = flush -}if exec -}{ -(Can't handle format )print 0 getu16 = flush -0 1 255{}for 256 packedarray -}ifelse -TTFDEBUG{ -(cmap: length=)print dup length = dup == -}if -}bind def -/postremap mark -/Cdot/Cdotaccent -/Edot/Edotaccent -/Eoverdot/Edotaccent -/Gdot/Gdotaccent -/Ldot/Ldotaccent -/Zdot/Zdotaccent -/cdot/cdotaccent -/edot/edotaccent -/eoverdot/edotaccent -/gdot/gdotaccent -/ldot/ldotaccent -/zdot/zdotaccent -.dicttomark readonly def -/get_from_stringarray -{1 index type/stringtype eq{ -get -}{ -exch{ -2 copy length ge{ -length sub -}{ -exch get exit -}ifelse -}forall -}ifelse -}bind def -/getinterval_from_stringarray -{ -2 index type/stringtype eq{ -getinterval -}{ -string exch 0 -4 3 roll{ -dup length -dup 4 index lt{ -3 index exch sub -exch pop 3 1 roll exch pop -}{ -dup 3 1 roll -4 index sub -5 index length 4 index sub -2 copy gt{exch}if pop -dup 3 1 roll -5 index exch getinterval -5 index 4 index 3 index -getinterval -copy pop -exch pop add exch pop 0 exch -dup 3 index length ge{exit}if -}ifelse -}forall -pop pop -}ifelse -}bind def -/string_array_size -{dup type/stringtype eq{ -length -}{ -0 exch{length add}forall -}ifelse -}bind def -/postformats mark -16#00010000{ -pop MacGlyphEncoding -} -16#00020000{ -dup dup type/arraytype eq{0 get}if length 36 lt{ -TTFDEBUG{(post format 2.0 invalid.)= flush}if -pop[] -}{ -/postglyphs exch def -/post_first postglyphs dup type/arraytype eq{0 get}if def -post_first 32 getu16/numglyphs exch def -/glyphnames numglyphs 2 mul 34 add def -/postpos glyphnames def -/total_length postglyphs//string_array_size exec def -numglyphs array 0 1 numglyphs 1 sub{ -postpos total_length ge{ -1 numglyphs 1 sub{1 index exch/.notdef put}for -exit -}if -postglyphs postpos//get_from_stringarray exec -postglyphs postpos 1 add 2 index//getinterval_from_stringarray exec cvn -exch postpos add 1 add/postpos exch def -2 index 3 1 roll -put -}for -/postnames exch def -numglyphs array 0 1 numglyphs 1 sub{ -dup 2 mul 34 add postglyphs exch 2//getinterval_from_stringarray exec -dup 0 get 8 bitshift exch 1 get add dup 258 lt{ -MacGlyphEncoding exch get -}{ -dup 32768 ge{ -pop/.notdef -}{ -258 sub dup postnames length ge{ -TTFDEBUG{( *** warning: glyph index past end of 'post' table)= flush}if -pop -exit -}if -postnames exch get -postremap 1 index .knownget{exch pop}if -}ifelse -}ifelse -2 index 3 1 roll put -}for -} -ifelse -}bind -16#00030000{ -pop[] -}bind -.dicttomark readonly def -/first_post_string -{ -post dup type/arraytype eq{0 get}if -}bind def -/.getpost{ -/glyphencoding post null eq{ -TTFDEBUG{(post missing)= flush}if[] -}{ -postformats first_post_string 0 getu32 .knownget{ -TTFDEBUG{ -(post: format )print -first_post_string -dup 0 getu16 =only(,)print 2 getu16 = flush -}if -post exch exec -}{ -TTFDEBUG{(post: unknown format )print post 0 getu32 = flush}if[] -}ifelse -}ifelse def -}bind def -/MacRomanEncoding[ -StandardEncoding 0 39 getinterval aload pop -/quotesingle -StandardEncoding 40 56 getinterval aload pop -/grave -StandardEncoding 97 31 getinterval aload pop -/Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute -/agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave -/ecircumflex/edieresis/iacute/igrave -/icircumflex/idieresis/ntilde/oacute -/ograve/ocircumflex/odieresis/otilde -/uacute/ugrave/ucircumflex/udieresis -/dagger/degree/cent/sterling/section/bullet/paragraph/germandbls -/registered/copyright/trademark/acute/dieresis/.notdef/AE/Oslash -/.notdef/plusminus/.notdef/.notdef/yen/mu/.notdef/.notdef -/.notdef/.notdef/.notdef/ordfeminine/ordmasculine/.notdef/ae/oslash -/questiondown/exclamdown/logicalnot/.notdef -/florin/.notdef/.notdef/guillemotleft -/guillemotright/ellipsis/space/Agrave/Atilde/Otilde/OE/oe -/endash/emdash/quotedblleft/quotedblright -/quoteleft/quoteright/divide/.notdef -/ydieresis/Ydieresis/fraction/currency -/guilsinglleft/guilsinglright/fi/fl -/daggerdbl/periodcentered/quotesinglbase/quotedblbase -/perthousand/Acircumflex/Ecircumflex/Aacute -/Edieresis/Egrave/Iacute/Icircumflex -/Idieresis/Igrave/Oacute/Ocircumflex -/.notdef/Ograve/Uacute/Ucircumflex -/Ugrave/dotlessi/circumflex/tilde -/macron/breve/dotaccent/ring/cedilla/hungarumlaut/ogonek/caron -]/Encoding defineresource pop -/TTParser<< -/Pos 0 -/post null ->>def -/readu8 -{read not{ -mark(Insufficient data in the stream.)//error exec -}if -}bind def -/readu16 -{dup//readu8 exec 8 bitshift exch//readu8 exec or -}bind def -/reads16 -{//readu16 exec 16#8000 xor 16#8000 sub -}bind def -/readu32 -{dup//readu16 exec 16 bitshift exch//readu16 exec or -}bind def -/reads32 -{dup//reads16 exec 16 bitshift exch//readu16 exec or -}bind def -/SkipToPosition -{dup//TTParser/Pos get -exch//TTParser exch/Pos exch put -sub -//PDFR_DEBUG{ -(Skipping )print dup//=only exec( bytes.)= -}if -dup 0 eq{ -pop pop -}{ -dup 3 1 roll -()/SubFileDecode filter -exch -{1 index//BlockBuffer readstring pop length -dup 0 eq{pop exch pop exit}if -sub -}loop -0 ne{ -mark(Insufficient data in the stream for SkipToPosition.)//error exec -}if -}ifelse -}bind def -/TagBuffer 4 string def -/ParseTTTableDirectory -{//PDFR_DEBUG{ -(ParseTTTableDirectory beg)= -}if -15 dict begin -dup//readu32 exec 16#00010000 ne{ -mark(Unknown True Type version.)//error exec -}if -dup//readu16 exec/NumTables exch def -dup//readu16 exec/SearchRange exch def -dup//readu16 exec/EntrySelector exch def -dup//readu16 exec/RangeShift exch def -//PDFR_DEBUG{ -(NumTables = )print NumTables = -}if -NumTables{ -dup//TagBuffer readstring not{ -mark(Could not read TT tag.)//error exec -}if -cvn -[2 index//readu32 exec pop -2 index//readu32 exec -3 index//readu32 exec -] -//PDFR_DEBUG{ -2 copy exch//=only exec( )print == -}if -def -}repeat -pop -//TTParser/Pos 12 NumTables 16 mul add put -currentdict end -//PDFR_DEBUG{ -(ParseTTTableDirectory end)= -}if -}bind def -/ParseTTcmap -{//PDFR_DEBUG{ -(ParseTTcmap beg)= -}if -/cmap get aload pop -3 1 roll -7 dict begin -//PDFR_DEBUG{ -(Current position = )print//TTParser/Pos get = -(cmap position = )print dup = -}if -1 index exch//SkipToPosition exec -//TTParser/Pos get/TablePos exch def -dup//readu16 exec pop -dup//readu16 exec/NumEncodings exch def -//PDFR_DEBUG{ -(NumEncodings = )print NumEncodings = -}if -null -NumEncodings{ -1 index//readu32 exec -2 index//readu32 exec -3 array dup 3 2 roll 0 exch put -2 index null ne{ -dup 0 get 3 index 0 get sub -3 index exch 1 exch put -}if -dup 4 3 roll pop 3 1 roll -def -}repeat -dup 0 get -4 3 roll exch sub -1 exch put -//PDFR_DEBUG{ -currentdict{ -exch dup type/integertype eq{ -//PrintHex exec( )print == -}{ -pop pop -}ifelse -}forall -}if -4 NumEncodings 8 mul add/HeaderLength exch def -//TTParser/Pos//TTParser/Pos get HeaderLength add put -0 -NumEncodings{ -16#7FFFFFF null -currentdict{ -1 index type/integertype eq{ -exch pop dup 0 get -dup 5 index gt{ -dup 4 index lt{ -4 1 roll -exch pop exch pop -}{ -pop pop -}ifelse -}{ -pop pop -}ifelse -}{ -pop pop -}ifelse -}forall -//PDFR_DEBUG{ -(Obtaining subtable for )print dup == -}if -3 2 roll pop -3 copy pop -TablePos add//SkipToPosition exec -3 copy exch pop 1 get -//TTParser/Pos//TTParser/Pos get 3 index add put -string -readstring not{ -mark(Can't read a cmap subtable.)//error exec -}if -2 exch put -}repeat -pop pop -currentdict end -//PDFR_DEBUG{ -(ParseTTcmap end)= -}if -}bind def -/GetTTEncoding -{//PDFR_DEBUG{ -(GetTTEncoding beg)= -}if -get -exch pop -2 get -10 dict begin -/TTFDEBUG//PDFR_DEBUG def -//cmaparray exec -end -//PDFR_DEBUG{ -(GetTTEncoding end)= -dup == -}if -}bind def -/InverseEncoding -{ -256 dict begin -dup length 1 sub -1 0{ -2 copy get -exch -1 index currentdict exch//knownget exec{ -dup type/arraytype eq{ -aload length 1 add array astore -}{ -2 array astore -}ifelse -}if -def -}for -pop -currentdict end -}bind def -/GetMacRomanEncodingInverse -{//PDFReader/MacRomanEncodingInverse get -dup null eq{ -pop -MacRomanEncoding//InverseEncoding exec -dup//PDFReader exch/MacRomanEncodingInverse exch put -}if -}bind def -/PutCharStringSingle -{ -dup 3 index length lt{ -2 index exch get -dup 0 ne{ -def -}{ -pop pop -}ifelse -}{ -pop pop -}ifelse -}bind def -/PutCharString -{1 index type/nametype ne{ -mark(Bad charstring name)//error exec -}if -dup type/arraytype eq{ -{ -3 copy//PutCharStringSingle exec -pop pop -}forall -pop -}{ -//PutCharStringSingle exec -}ifelse -}bind def -/ComposeCharStrings -{ -//PDFR_DEBUG{ -(ComposeCharStrings beg)= -}if -1 index length 1 add dict begin -/.notdef 0 def -exch -//TTParser/post get -dup null ne{ -exch -1 index length 1 sub -1 0{ -dup 3 index exch get exch -dup 0 eq 2 index/.notdef eq or{ -pop pop -}{ -def -}ifelse -}for -}if -exch pop exch -{ -//PutCharString exec -}forall -pop -currentdict end -//PDFR_DEBUG{ -(ComposeCharStrings end)= -}if -}bind def -/ParseTTpost -{ -//PDFR_DEBUG{ -(ParseTTpost beg)= -}if -/post get aload pop -3 1 roll -//PDFR_DEBUG{ -(Current position = )print//TTParser/Pos get = -(post position = )print dup = -}if -1 index exch//SkipToPosition exec -//TTParser/Pos//TTParser/Pos get 4 index add put -exch dup 65535 le{ -string -readstring not{ -mark(Insufficient data in the stream for ParseTTpost.)//error exec -}if -}{ -[3 1 roll -dup 16384 div floor cvi -exch 1 index 16384 mul -sub exch -1 sub 0 1 3 -1 roll -{ -1 add index -16384 string readstring not{ -mark(Insufficient data in the stream for ParseTTpost.)//error exec -}if -}for -counttomark -2 roll -string readstring not{ -mark(Insufficient data in the stream for ParseTTpost.)//error exec -}if -] -}ifelse -1 dict begin -/post exch def -//.getpost exec -//TTParser/post glyphencoding put -//PDFR_DEBUG{ -(ParseTTpost end)= -glyphencoding == -}if -end -}bind def -/MakeTTCharStrings -{//MakeStreamReader exec -dup dup//ParseTTTableDirectory exec -//TTParser/post null put -dup/post//knownget exec{ -0 get -1 index/cmap get 0 get -lt{ -2 copy//ParseTTpost exec -//ParseTTcmap exec -}{ -2 copy//ParseTTcmap exec -3 1 roll -//ParseTTpost exec -}ifelse -}{ -//ParseTTcmap exec -}ifelse -{ -dup 16#00030001 known{ -//PDFR_DEBUG{ -(Using the TT cmap encoding for Windows Unicode.)= -}if -16#00030001//GetTTEncoding exec -AdobeGlyphList//ComposeCharStrings exec -exit -}if -dup 16#00010000 known{ -//PDFR_DEBUG{ -(Using the TT cmap encoding for Macintosh Roman.)= -}if -16#00010000//GetTTEncoding exec -PDFEncoding dup null eq{ -pop//GetMacRomanEncodingInverse exec -}{ -//InverseEncoding exec -}ifelse -//ComposeCharStrings exec -exit -}if -dup 16#00030000 known{ -//PDFR_DEBUG{ -(Using the TT cmap encoding 3.0 - not sure why Ghostscript writes it since old versions.)= -}if -16#00030000//GetTTEncoding exec -PDFEncoding dup null eq{ -pop//GetMacRomanEncodingInverse exec -}{ -//InverseEncoding exec -}ifelse -//ComposeCharStrings exec -exit -}if -mark(True Type cmap has no useful encodings.)//error exec -}loop -//PDFR_DEBUG{ -(CharStrings <<)= -dup{ -exch -dup type/nametype eq{ -//=only exec -}{ -== -}ifelse -( )print == -}forall -(>>)= -}if -}bind def -/ScaleVal -{ -aload pop -1 index sub -3 2 roll mul add -}bind def -/ScaleArg -{ -aload pop -1 index sub -3 1 roll -sub exch div -}bind def -/ScaleArgN -{ -dup length 2 sub -2 0{ -2 -2 index 3 1 roll getinterval -3 2 roll -exch//ScaleArg exec -1 index length 2 idiv 1 add 1 roll -}for -pop -}bind def -/ComputeFunction_10 -{ -//PDFR_DEBUG{ -(ComputeFunction_10 beg )print 1 index//=only exec( stack=)print count = -}if -exch -dup 1 eq{ -pop dup length 1 sub get -}{ -1 index length 1 sub mul -dup dup floor sub -dup 0 eq{ -pop cvi get -}{ -3 1 roll floor cvi -2 getinterval -aload pop -2 index mul 3 2 roll 1 exch sub 3 2 roll mul add -}ifelse -}ifelse -//PDFR_DEBUG{ -(ComputeFunction_10 end )print dup//=only exec( stack=)print count = -}if -}bind def -/ComputeFunction_n0 -{ -//PDFR_DEBUG{ -(ComputeFunction_n0 beg N=)print dup//=only exec( stack=)print count = -}if -dup 0 eq{ -pop -}{ -dup 2 add -1 roll -dup 3 index length 1 sub ge{ -pop 1 sub -exch dup length 1 sub get exch -//PDFReader/ComputeFunction_n0 get exec -}{ -dup floor cvi dup -4 index exch get -3 index dup -5 add copy -6 2 roll -pop pop pop pop -1 sub -//PDFReader/ComputeFunction_n0 get exec -3 2 roll pop -exch -4 3 roll exch -4 add 2 roll 1 add -3 2 roll exch get -exch 1 sub -//PDFReader/ComputeFunction_n0 get exec -1 index mul -3 1 roll -1 exch sub mul add -}ifelse -}ifelse -//PDFR_DEBUG{ -(ComputeFunction_n0 end )print dup//=only exec( stack=)print count = -}if -}bind def -/FunctionToProc_x01 -{ -dup/Domain get exch -dup/Data get 0 get exch -/Size get length -[4 1 roll -//PDFR_DEBUG{ -{(function beg, stack =)print count//=only exec(\n)print}/exec load -5 2 roll -}if -dup 1 gt{ -{mark exch -3 add 2 roll -//ScaleArgN exec -counttomark dup -3 add -2 roll -pop exch -//ComputeFunction_n0 exec -}/exec load -}{ -pop -3 1/roll load//ScaleArg/exec load -/exch load -//ComputeFunction_10/exec load -}ifelse -//PDFR_DEBUG{ -(function end, stack =)/print load/count load//=only/exec load(\n)/print load -}if -]cvx -//PDFR_DEBUG{ -(Made a procedure for the 1-result function :)= -dup == -}if -}bind def -/FunctionProcDebugBeg -{(FunctionProcDebugBeg )print count = -}bind def -/FunctionProcDebugEnd -{(FunctionProcDebugEnd )print count = -}bind def -/FunctionToProc_x0n -{ -PDFR_DEBUG{ -(FunctionToProc_x0n beg m=)print dup = -}if -1 index/Size get length exch -dup 7 mul 2 add array -PDFR_DEBUG{ -dup 0//FunctionProcDebugBeg put -}{ -dup 0//DoNothing put -}ifelse -dup 1/exec load put -dup 2 5 index/Domain get put -2 index 1 eq{ -dup 3//ScaleArg put -}{ -dup 3//ScaleArgN put -}ifelse -dup 4/exec load put -1 index 1 sub 0 exch 1 exch{ -dup 7 mul 5 add -1 index 4 index 1 sub ne{ -dup 3 index exch 6 index put 1 add -dup 3 index exch/copy load put 1 add -}if -[ -6 index/Data get 3 index get -6 index 1 eq{ -//ComputeFunction_10/exec load -}{ -6 index -//ComputeFunction_n0/exec load -}ifelse -]cvx -3 index exch 2 index exch put 1 add -2 index 1 index/exec load put 1 add -1 index 4 index 1 sub ne{ -2 index 1 index 6 index 1 add put 1 add -2 index 1 index 1 put 1 add -2 index 1 index/roll load put -}if -pop pop -}for -PDFR_DEBUG{ -dup dup length 2 sub//FunctionProcDebugEnd put -}{ -dup dup length 2 sub//DoNothing put -}ifelse -dup dup length 1 sub/exec load put -cvx exch pop exch pop exch pop -//PDFR_DEBUG{ -(Made a procedure for the n-argument function :)= -dup == -}if -PDFR_DEBUG{ -(FunctionToProc_x0n end)= -}if -}bind def -/MakeTableRec -{ -0 -exec -}bind def -/MakeTable -{//PDFR_DEBUG{ -(MakeTable beg )print count = -}if -1 index/Size get exch -1 sub dup -3 1 roll -get -array -1 index 0 eq{ -exch pop exch pop -}{ -dup length 1 sub -1 0{ -3 index 3 index//MakeTableRec exec -2 index 3 1 roll put -}for -exch pop exch pop -}ifelse -//PDFR_DEBUG{ -(MakeTable end )print count = -}if -}bind def -//MakeTableRec 0//MakeTable put -/StoreSample -{ -1 sub -dup 0 eq{ -pop -}{ --1 1{ -I exch get get -}for -}ifelse -I 0 get 3 2 roll put -}bind def -/ReadSample32 -{ -4{ -File read not{ -mark(Insufficient data for function.)//error exec -}if -}repeat -pop -3 1 roll exch -256 mul add 256 mul add -//1_24_bitshift_1_sub div -}bind def -/ReadSample -{ -Buffer BitsLeft BitsPerSample -{2 copy ge{ -exit -}if -3 1 roll -8 add 3 1 roll -256 mul File read not{ -mark(Insufficient data for function.)//error exec -}if -add -3 1 roll -}loop -sub dup -2 index exch -neg bitshift -2 copy exch bitshift -4 3 roll exch sub -/Buffer exch def -exch/BitsLeft exch def -Div div -}bind def -/ReadSamplesRec -{0 -exec -}bind def -/ReadSamples -{ -//PDFR_DEBUG{ -(ReadSamples beg )print count = -}if -dup 1 eq{ -pop -0 1 Size 0 get 1 sub{ -I exch 0 exch put -0 1 M 1 sub{ -dup Range exch 2 mul 2 getinterval -//PDFR_DEBUG{ -(Will read a sample ... )print -}if -BitsPerSample 32 eq{//ReadSample32}{//ReadSample}ifelse -exec exch//ScaleVal exec -//PDFR_DEBUG{ -(value=)print dup = -}if -exch Table exch get -Size length//StoreSample exec -}for -}for -}{ -1 sub -dup Size exch get 0 exch 1 exch 1 sub{ -I exch 2 index exch put -dup//ReadSamplesRec exec -}for -pop -}ifelse -//PDFR_DEBUG{ -(ReadSamples end )print count = -}if -}bind def -//ReadSamplesRec 0//ReadSamples put -/StreamToArray -{//PDFR_DEBUG{ -(StreamToArray beg )print count = -}if -userdict/FuncDataReader get begin -dup/BitsPerSample get/BitsPerSample exch def -dup/Size get length/N exch def -dup/Range get length 2 idiv/M exch def -1 BitsPerSample bitshift 1 sub/Div exch def -/BitsLeft 0 def -/Buffer 0 def -dup/Size get/Size exch def -dup/Range get/Range exch def -/File 1 index//MakeStreamReader exec def -/I[N{0}repeat]def -M array -dup length 1 sub -1 0{ -2 index N//MakeTable exec -2 index 3 1 roll put -}for -/Table exch def -N//ReadSamples exec -PDFR_DEBUG{ -(Table = )print Table == -}if -/Data Table put -end -//PDFR_DEBUG{ -(StreamToArray end )print count = -}if -}bind def -/FunctionToProc10 -{ -PDFR_DEBUG{ -(FunctionToProc10 beg, Range = )print dup/Range get == -}if -dup/Order//knownget exec{ -1 ne{ -(Underimplemented function Type 0 Order 3.)= -}if -}if -dup//StreamToArray exec -dup/Range get length dup 2 eq{ -pop//FunctionToProc_x01 exec -}{ -2 idiv//FunctionToProc_x0n exec -}ifelse -PDFR_DEBUG{ -(FunctionToProc10 end)= -}if -}bind def -/FunctionToProc12 -{begin -currentdict/C0//knownget exec{length 1 eq}{true}ifelse{ -N -currentdict/C0//knownget exec{ -0 get -}{ -0 -}ifelse -currentdict/C1//knownget exec{ -0 get -}{ -1 -}ifelse -1 index sub -[4 1 roll -{ -4 2 roll -exp mul add -}aload pop -]cvx -}{ -[ -0 1 C0 length 1 sub{ -N -C0 2 index get -C1 3 index get -4 3 roll pop -1 index sub -[/dup load -5 2 roll -{ -4 2 roll -exp mul add -exch -}aload pop -]cvx -/exec load -}for -/pop load -]cvx -}ifelse -end -//PDFR_DEBUG{ -(FunctionType2Proc : )print dup == -}if -}bind def -/FunctionToProc14 -{//MakeStreamReader exec cvx exec -//PDFR_DEBUG{ -(FunctionType4Proc : )print dup == -}if -}bind def -/FunctionToProc1 -{ -dup/FunctionType get -{dup 0 eq{ -pop//FunctionToProc10 exec exit -}if -dup 2 eq{ -pop//FunctionToProc12 exec exit -}if -dup 4 eq{ -pop//FunctionToProc14 exec exit -}if -mark exch(Function type )exch( isn't implemented yet.)//error exec -}loop -}bind def -/FunctionToProc20 -{ -PDFR_DEBUG{ -(FunctionToProc20, Range = )print dup/Range get == -}if -dup/Order//knownget exec{ -1 ne{ -(Underimplemented function Type 0 Order 3.)= -}if -}if -dup//StreamToArray exec -dup/Range get length dup 2 eq{ -pop//FunctionToProc_x01 exec -}{ -2 idiv//FunctionToProc_x0n exec -}ifelse -}bind def -/FunctionToProc -{//PDFR_DEBUG{ -(FunctionToProc beg )print count = -}if -dup type/dicttype eq{ -dup/Domain get length 2 idiv -{ -dup 1 eq{ -pop//FunctionToProc1 exec exit -}if -dup 2 eq{ -pop//FunctionToProc20 exec exit -}if -mark(Functions with many arguments aren't implemented yet.)//error exec -}loop -}{ -//PDFR_DEBUG{(Not a function dict, assume already a procedure.)print}if -}ifelse -//PDFR_DEBUG{ -(FunctionToProc end )print count = -}if -}bind def -/spotfunctions mark -/Round{ -abs exch abs 2 copy add 1 le{ -dup mul exch dup mul add 1 exch sub -}{ -1 sub dup mul exch 1 sub dup mul add 1 sub -}ifelse -} -/Diamond{ -abs exch abs 2 copy add .75 le{ -dup mul exch dup mul add 1 exch sub -}{ -2 copy add 1.23 le{ -.85 mul add 1 exch sub -}{ -1 sub dup mul exch 1 sub dup mul add 1 sub -}ifelse -}ifelse -} -/Ellipse{ -abs exch abs 2 copy 3 mul exch 4 mul add 3 sub dup 0 lt{ -pop dup mul exch .75 div dup mul add 4 div 1 exch sub -}{ -dup 1 gt{ -pop 1 exch sub dup mul exch 1 exch sub -.75 div dup mul add 4 div 1 sub -}{ -.5 exch sub exch pop exch pop -}ifelse -}ifelse -} -/EllipseA{dup mul .9 mul exch dup mul add 1 exch sub} -/InvertedEllipseA{dup mul .9 mul exch dup mul add 1 sub} -/EllipseB{dup 5 mul 8 div mul exch dup mul exch add sqrt 1 exch sub} -/EllipseC{dup mul .9 mul exch dup mul add 1 exch sub} -/InvertedEllipseC{dup mul .9 mul exch dup mul add 1 sub} -/Line{exch pop abs neg} -/LineX{pop} -/LineY{exch pop} -/Square{abs exch abs 2 copy lt{exch}if pop neg} -/Cross{abs exch abs 2 copy gt{exch}if pop neg} -/Rhomboid{abs exch abs 0.9 mul add 2 div} -/DoubleDot{2{360 mul sin 2 div exch}repeat add} -/InvertedDoubleDot{2{360 mul sin 2 div exch}repeat add neg} -/SimpleDot{dup mul exch dup mul add 1 exch sub} -/InvertedSimpleDot{dup mul exch dup mul add 1 sub} -/CosineDot{180 mul cos exch 180 mul cos add 2 div} -/Double{exch 2 div exch 2{360 mul sin 2 div exch}repeat add} -/InvertedDouble{ -exch 2 div exch 2{360 mul sin 2 div exch}repeat add neg -} -.dicttomark readonly def -/CheckColorSpace -{ -dup type/arraytype ne{ -mark(Resource )3 index( must be an array.)//error exec -}if -}bind def -/SubstitutePDFColorSpaceRec -{0 -exec -}bind def -/SubstitutePDFColorSpace -{ -{ -dup 0 get/Pattern eq{ -dup length 1 gt{ -dup dup 1//CheckColorSpace//ResolveA exec -dup type/nametype ne{ -//SubstitutePDFColorSpaceRec exec -}if -1 exch put -}if -exit -}if -dup 0 get/Indexed eq{ -exit -}if -dup 0 get/Separation eq{ -dup dup 2//CheckColorSpace//ResolveA exec -dup type/nametype ne{ -//SubstitutePDFColorSpaceRec exec -}if -2 exch put -exit -}if -dup 0 get/CalGray eq{ -1 get -dup/Gamma//knownget exec{ -[exch[exch/exp load]cvx dup dup] -1 index exch/DecodeLMN exch put -}if -[exch/CIEBasedA exch] -exit -}if -dup 0 get/CalRGB eq{ -1 get -dup/Matrix//knownget exec{ -1 index exch/MatrixLMN exch put -}if -dup/Gamma//knownget exec{ -aload pop -[exch/exp load]cvx -3 1 roll -[exch/exp load]cvx -3 1 roll -[exch/exp load]cvx -3 1 roll -3 array astore -1 index exch/DecodeLMN exch put -}if -[exch/CIEBasedABC exch] -exit -}if -dup 0 get/Lab eq{ -1 get -begin -currentdict/Range//knownget exec{aload pop}{-100 100 -100 100}ifelse -0 100 6 2 roll 6 array astore -/RangeABC exch def -/DecodeABC[{16 add 116 div}bind{500 div}bind{200 div}bind]def -/MatrixABC[1 1 1 1 0 0 0 0 -1]def -{dup 6 29 div ge{dup dup mul mul}{4 29 div sub 108 841 div mul}ifelse} -/DecodeLMN[ -[3 index aload pop WhitePoint 0 get/mul load]cvx -[4 index aload pop WhitePoint 1 get/mul load]cvx -[5 index aload pop WhitePoint 2 get/mul load]cvx -]def pop -//PDFR_DEBUG{ -(Constructed from Lab <<)= -currentdict{exch = ==}forall -(>>)= -}if -[/CIEBasedABC currentdict] -end -exit -pop -}if -dup 0 get/CIEBasedA eq{exit}if -dup 0 get/CIEBasedABC eq{exit}if -mark exch(Unimplemented color space )exch//error exec -}loop -}bind def -//SubstitutePDFColorSpaceRec 0//SubstitutePDFColorSpace put -/ResolveArrayElement -{2 copy get -dup type dup/arraytype eq exch -/packedarraytype eq or{ -dup length 1 ge exch xcheck and{ -2 copy get -dup 0 get type/integertype eq -1 index 1 get type dup/arraytype -eq exch -/packedarraytype eq or -and{ -exec -2 index 4 1 roll put -}{ -pop pop -}ifelse -}{ -pop -}ifelse -}{ -pop pop -}ifelse -}bind def -/ResolveColorSpaceArrayRec -{0 -exec -}bind def -/SetColorSpaceSafe -{ -PDFR_DEBUG{ -(SetColorSpaceSafe beg)= -}if -currentcolorspace dup type/arraytype eq{ -1 index type/arraytype eq{ -dup length 2 index length eq{ -false exch -dup length 0 exch 1 exch 1 sub{ -dup -4 index exch get exch -2 index exch get -ne{ -exch pop true exch exit -}if -}for -pop -{ -setcolorspace -}{ -pop -}ifelse -}{ -pop setcolorspace -}ifelse -}{ -pop setcolorspace -}ifelse -}{ -pop setcolorspace -}ifelse -PDFR_DEBUG{ -(SetColorSpaceSafe end)= -}if -}bind def -/ResolveColorSpaceArray -{ -//PDFR_DEBUG{ -(ResolveColorSpaceArray beg )print dup == -}if -dup 0 get/Indexed eq{ -1//ResolveArrayElement exec -dup dup 1 get -dup type/arraytype eq{ -//SubstitutePDFColorSpace exec -//ResolveColorSpaceArrayRec exec -1 exch put -}{ -pop pop -}ifelse -}if -dup 0 get/Separation eq{ -dup dup 1 get UnPDFEscape 1 exch put -3//ResolveArrayElement exec -dup 3 get//FunctionToProc exec -2 copy 3 exch put -pop -}if -dup 0 get/Pattern eq{ -dup length 1 gt{ -dup 1 get dup type/arraytype eq{ -ResolveColorSpaceArray -1 index 1 3 -1 roll put -}{ -pop -}ifelse -}if -}if -PDFR_DEBUG{ -(Construcrted color space :)= -dup == -}if -//PDFR_DEBUG{ -(ResolveColorSpaceArray end )print dup == -}if -}bind def -//ResolveColorSpaceArrayRec 0//ResolveColorSpaceArray put -/ResolveColorSpace -{ -//PDFR_DEBUG{ -(ResolveColorSpace beg )print dup = -}if -dup//SimpleColorSpaceNames exch known not{ -dup//PDFColorSpaces exch//knownget exec{ -exch pop -//PDFR_DEBUG{ -(ResolveColorSpace known )= -}if -}{ -dup -//PDFReader/CurrentObject get/Context get/Resources get -/ColorSpace//DoNothing//ResolveD exec -exch//CheckColorSpace//ResolveD exec -dup type/arraytype eq{ -//SubstitutePDFColorSpace exec -//ResolveColorSpaceArray exec -dup//PDFColorSpaces 4 2 roll put -}if -}ifelse -}if -//PDFR_DEBUG{ -(ResolveColorSpace end )print dup == -}if -}bind def -/CheckPattern -{ -dup/PatternType//knownget exec{ -dup 1 ne{ -mark(Resource )4 index( is a shading, which can't be handled at level 2. )//error exec -}if -pop -}if -dup/Type knownget{ -/Pattern ne{ -mark(Resource )4 index( must have /Type/Pattern .)//error exec -}if -}if -}bind def -/PaintProc -{/Context get -//RunDelayedStream exec -}bind def -/ResolvePattern -{ -dup -userdict/PDFR_Patterns get -exch//knownget exec{ -exch pop -}{ -dup -//PDFReader/CurrentObject get/Context get/Resources get -/Pattern//DoNothing//ResolveD exec -exch//CheckPattern//ResolveD exec -dup dup/Context exch put -dup/Resources//DoNothing//ResolveD exec pop -dup/PaintProc//PaintProc put -gsave userdict/PDFR_InitialGS get setgstate -currentglobal exch false setglobal -dup/Matrix get -makepattern -exch setglobal -grestore -dup userdict/PDFR_Patterns get -4 2 roll -put -}ifelse -}bind def -/SetColor -{//PDFR_DEBUG{ -(SetColor beg)= -}if -currentcolorspace dup type/nametype eq{ -pop setcolor -}{ -0 get/Pattern eq{ -//ResolvePattern exec setpattern -}{ -setcolor -}ifelse -}ifelse -//PDFR_DEBUG{ -(SetColor end)= -}if -}bind def -/ImageKeys 15 dict begin -/BPC/BitsPerComponent def -/CS/ColorSpace def -/D/Decode def -/DP/DecodeParms def -/F/Filter def -/H/Height def -/IM/ImageMask def -/I/Interpolate def -/W/Width def -currentdict end readonly def -/ImageValues 15 dict begin -/G/DeviceGray def -/RGB/DeviceRGB def -/CMYK/DeviceCMYK def -/I/Indexed def -/AHx/ASCIIHexDecode def -/A85/ASCII85Decode def -/LZW/LZWDecode def -/Fl/FlateDecode def -/RL/RunLengthDecode def -/CCF/CCITTFaxDecode def -/DCT/DCTDecode def -currentdict end readonly def -/GetColorSpaceRange -{2 index/ColorSpace get -dup type/arraytype eq{ -1 get -}if -exch//knownget exec{ -exch pop -}if -}bind def -/DecodeArrays 15 dict begin -/DeviceGray{[0 1]}def -/DeviceRGB{[0 1 0 1 0 1]}def -/DeviceCMYK{[0 1 0 1 0 1 0 1]}def -/Indexed{ -dup/BitsPerComponent get 1 exch bitshift 1 sub[exch 0 exch] -}def -/Separation{[0 1]}def -/CIEBasedA{[0 1]/RangeA//GetColorSpaceRange exec}def -/CIEBasedABC{[0 1 0 1 0 1]/RangeABC//GetColorSpaceRange exec}def -currentdict end readonly def -/Substitute -{1 index//knownget exec{ -exch pop -}if -}bind def -/DebugImagePrinting -{ -//PDFR_DEBUG{ -(Image :)= -dup{exch//=only exec( )print == -}forall -}if -}bind def -/CompleteImage -{ -dup/ColorSpace known{ -dup/ColorSpace//CheckColorSpace//ResolveD exec pop -}if -dup/Decode known not{ -dup/ColorSpace//knownget exec{ -dup type/arraytype eq{ -0 get -}if -//DecodeArrays exch get exec -}{ -[0 1] -}ifelse -1 index exch/Decode exch put -}if -dup/ImageMatrix[2 index/Width get 0 0 5 index/Height get neg -0 7 index/Height get]put -//DebugImagePrinting exec -}bind def -/CompleteInlineImage -{ -//PDFR_DEBUG{ -(CompleteInlineImage beg)= -}if -dup/ImageType known not{ -dup/ImageType 1 put -}if -dup length dict exch{ -exch//ImageKeys//Substitute exec -dup/Filter eq{ -exch//ImageValues//Substitute exec exch -}if -dup/ColorSpace eq{ -exch -dup//ImageValues exch//knownget exec{ -exch pop -}{ -//ResolveColorSpace exec -}ifelse -exch -}if -exch -2 index 3 1 roll put -}forall -//CompleteImage exec -dup/DataSource 2 copy get -2 index//AppendFilters exec put -//PDFR_DEBUG{ -(CompleteInlineImage end)= -}if -}bind def -/CompleteOutlineImage -{ -currentglobal exch dup gcheck setglobal -//PDFR_DEBUG{ -(CompleteOutlineImage beg)= -}if -dup dup//MakeStreamReader exec/DataSource exch put -dup/ImageType known not{ -//CompleteImage exec -dup/ImageType 1 put -dup/ColorSpace known{ -dup/ColorSpace//CheckColorSpace//ResolveD exec -dup type/arraytype eq{ -//ResolveColorSpaceArray exec -//SubstitutePDFColorSpace exec -1 index exch/ColorSpace exch put -}{ -pop -}ifelse -}if -}if -//PDFR_DEBUG{ -(CompleteOutlineImage end)= -}if -exch setglobal -}bind def -/DoImage -{ -//PDFR_DEBUG{ -(DoImage beg)= -}if -gsave -dup/ColorSpace//knownget exec{setcolorspace}if -dup/ImageMask//knownget exec not{false}if -{imagemask}{image}ifelse -grestore -//PDFR_DEBUG{ -(DoImage end)= -}if -}bind def -/GSave -{ -gsave -//PDFReader/GraphicStateStackPointer get -dup//GraphicStateStack exch get null eq{ -dup//GraphicStateStack exch//InitialGraphicState length dict put -}if -dup//GraphicStateStack exch get -//GraphicState exch copy pop -1 add//PDFReader exch/GraphicStateStackPointer exch put -}bind def -/GRestore -{ -grestore -//PDFReader/GraphicStateStackPointer get -1 sub dup -//PDFReader exch/GraphicStateStackPointer exch put -//GraphicStateStack exch get -//GraphicState copy pop -}bind def -/SetFont -{dup//GraphicState exch/FontSize exch put -//ResolveAndSetFont exec -//GraphicState/FontMatrixNonHV currentfont/FontMatrix get 1 get 0 ne put -}bind def -/ShowText -{ -//GraphicState/TextRenderingMode get dup 0 eq -exch 3 eq not currentfont/FontType get 3 eq and or -{ -//GraphicState/WordSpacing get 0 -32 -//GraphicState/CharacterSpacing get 0 -6 5 roll -//GraphicState/FontMatrixNonHV get{ -[ -7 -2 roll pop -5 -2 roll pop -5 -1 roll -{ -exch -pop -3 index add -exch 2 index eq{3 index add}if -4 1 roll -} -currentfont/FontMatrix get 0 get 0 ne{ -1 1 index length 1 sub getinterval cvx -}if -5 index -cshow -pop pop pop] -xshow -}{ -awidthshow -}ifelse -}{ -//GraphicState/CharacterSpacing get 0 eq -//GraphicState/FontMatrixNonHV get not and -//GraphicState/WordSpacing get 0 eq and{ -true charpath -}{ -{ -exch -pop 0 -currentpoint 5 4 roll -( )dup 0 3 index put true charpath -5 1 roll -moveto rmoveto -//GraphicState/CharacterSpacing get 0 rmoveto -32 eq{ -//GraphicState/WordSpacing get 0 rmoveto -}if -} -//GraphicState/FontMatrixNonHV get dup not exch{ -pop currentfont/FontMatrix get 0 get 0 ne -}if{ -1 1 index length 1 sub getinterval cvx -}if -exch cshow -}ifelse -}ifelse -}bind def -/ShowTextBeg -{ -//GraphicState/TextRenderingMode get dup 0 ne -{ -3 ne -currentfont/FontType get 3 eq not and{ -currentpoint newpath moveto -}if -} -{ -pop -}ifelse -}bind def -/ShowTextEnd -{ -//GraphicState/TextRenderingMode get -currentfont/FontType get 3 eq{ -dup 3 ne{ -pop 0 -}if -}if -{dup 1 eq{ -stroke exit -}if -dup 2 eq{ -gsave fill grestore stroke exit -}if -dup 3 eq{ -currentpoint newpath moveto -}if -dup 4 eq{ -gsave fill grestore clip exit -}if -dup 5 eq{ -gsave stroke grestore clip exit -}if -dup 6 eq{ -gsave fill grestore gsave stroke grestore fill exit -}if -dup 7 eq{ -clip exit -}if -exit -}loop -pop -}bind def -/ShowTextWithGlyphPositioning -{//ShowTextBeg exec -{dup type/stringtype eq{ -//ShowText exec -}{ -neg 1000 div//GraphicState/FontSize get mul 0 rmoveto -}ifelse -}forall -//ShowTextEnd exec -}bind def -/CheckFont -{dup/Type get/ExtGState ne{ -mark(Resource )3 index( must have /Type/ExtGState.)//error exec -}if -}bind def -/SetTransfer -{ -//PDFR_DEBUG{(SetTransfer beg )print count =}if -dup type/arraytype eq 1 index xcheck not and{ -0 4 getinterval aload pop -setcolortransfer -}{ -settransfer -}ifelse -//PDFR_DEBUG{(SetTransfer end )print count =}if -}bind def -/CheckExtGState -{dup/Type get/ExtGState ne{ -mark(Resource )3 index( must have /Type/ExtGState.)//error exec -}if -}bind def -/CheckHalftone -{dup/HalftoneType known not{ -mark(Resource )3 index( must have /HalftoneType.)//error exec -}if -}bind def -/ResolveFunction -{ -//PDFR_DEBUG{(ResolveFunction beg )print dup = count =}if -2 copy get//IsObjRef exec{ -2 copy//DoNothing//ResolveD exec -3 copy put pop -}if -2 copy get dup type/arraytype eq exch xcheck and not{ -2 copy get -dup type/arraytype eq 1 index xcheck not and{ -dup length 1 sub -1 0{ -2 copy//DoNothing ResolveA -dup/Identity eq{ -pop 2 copy{}put -}{ -//FunctionToProc exec -3 copy put pop -}ifelse -pop -}for -}{ -dup/Default eq{ -}{ -dup/Identity eq{ -pop{} -}{dup type/nametype eq{ -//spotfunctions exch get -}{ -//FunctionToProc exec -}ifelse -}ifelse -}ifelse -}ifelse -3 copy put -exch pop -}{ -1 index exch get -}ifelse -//PDFR_DEBUG{(ResolveFunction end )print dup == count =}if -}bind def -/ResolveFunctionSafe -{2 copy known{ -//ResolveFunction exec -}if -pop -}bind def -/CreateHalftoneThresholds -{ -dup/Thresholds known not{ -dup/HalftoneType get 10 eq{ -dup dup//MakeStreamReader exec -/Thresholds exch put -}if -dup/HalftoneType get dup 3 eq exch 6 eq or{ -dup dup//MakeStreamReader exec -//BlockBuffer readstring pop -dup length -dup 0 eq{ -mark(Could not read Thresholds)//error exec -}if -string copy/Thresholds exch put -dup/HalftoneType 3 put -}if -}if -}bind def -/SetExtGState -{ -//PDFReader/CurrentObject get/Context get/Resources get -/ExtGState//DoNothing//ResolveD exec -exch//CheckExtGState//ResolveD exec -dup/LW//knownget exec{ -setlinewidth -}if -dup/LC//knownget exec{ -setlinecap -}if -dup/LJ//knownget exec{ -setlinejoin -}if -dup/ML//knownget exec{ -setmeterlimit -}if -dup/D//knownget exec{ -setdash -}if -dup/RI//knownget exec{ -mark(Unimplemented ExtGState.RI)//error exec -}if -dup/OP//knownget exec{ -setoverprint -}if -dup/op//knownget exec{ -setoverprint -}if -dup/OPM//knownget exec{ -mark(Unimplemented ExtGState.OPM)//error exec -}if -dup/Font//knownget exec{ -mark(Unimplemented ExtGState.Font)//error exec -}if -dup/BG known{ -/BG//ResolveFunction exec -setblackgeneration -}if -dup/BG2 known{ -/BG2//ResolveFunction exec -dup/Default eq{ -//InitialExtGState/BG2 get -}if -setblackgeneration -}if -dup/UCR known{ -/UCR//ResolveFunction exec -setundercolorremoval -}if -dup/UCR2 known{ -/UCR2//ResolveFunction exec -dup/Default eq{ -//InitialExtGState/UCR2 get -}if -setundercolorremoval -}if -dup/TR known{ -/TR//ResolveFunction exec -//SetTransfer exec -}if -dup/TR2 known{ -/TR2//ResolveFunction exec -dup/Default eq{ -pop//InitialExtGState/TR2 get -aload pop setcolortransfer -}{ -//SetTransfer exec -}ifelse -}if -dup/HT//knownget exec{ -dup/Default eq{ -pop//InitialExtGState/HT get -sethalftone -}{ -//PDFR_DEBUG{(Ht beg)=}if -pop dup/HT//CheckHalftone//ResolveD exec -/SpotFunction//ResolveFunctionSafe exec -/TransferFunction//ResolveFunctionSafe exec -null exch -dup/HalftoneType get dup 5 eq exch dup 4 eq exch 2 eq or or{ -dup{ -dup//IsObjRef exec{ -pop -1 index exch//CheckHalftone ResolveD -}if -dup type/dicttype eq{ -dup/SpotFunction//ResolveFunctionSafe exec -/TransferFunction//ResolveFunctionSafe exec -//CreateHalftoneThresholds exec -dup/HalftoneType get 5 gt{ -4 3 roll pop -dup 4 1 roll -}if -}if -pop pop -}forall -}if -//CreateHalftoneThresholds exec -//PDFR_DEBUG{ -(HT:)= -dup{ -1 index/Default eq{ -(Default <<)= -exch pop -{exch = ==}forall -(>>)= -}{ -exch = == -}ifelse -}forall -(HT end)= flush -}if -exch dup null ne{ -(Warning: Ignoring a halftone with a Level 3 component halftone Type )print dup/HalftoneType get = -pop pop -}{ -pop -dup/HalftoneType get 5 gt{ -(Warning: Ignoring a Level 3 halftone Type )print dup/HalftoneType get = -pop -}{ -sethalftone -}ifelse -}ifelse -//PDFR_DEBUG{(HT set)= flush}if -}ifelse -}if -dup/FL//knownget exec{ -setflattness -}if -dup/SM//knownget exec{ -setsmoothness -}if -dup/SA//knownget exec{ -setstrokeadjust -}if -dup/BM//knownget exec{ -mark(Unimplemented ExtGState.BM)//error exec -}if -dup/SMask//knownget exec{ -mark(Unimplemented ExtGState.SMask)//error exec -}if -dup/CA//knownget exec{ -mark(Unimplemented ExtGState.CA)//error exec -}if -dup/ca//knownget exec{ -mark(Unimplemented ExtGState.ca)//error exec -}if -dup/AIS//knownget exec{ -mark(Unimplemented ExtGState.AIS)//error exec -}if -dup/TK//knownget exec{ -mark(Unimplemented ExtGState.TK)//error exec -}if -pop -}bind def -/CheckXObject -{dup/Subtype get dup/Image ne exch dup/Form ne exch/PS ne and and{ -mark(Resource )3 index( must have /Subtype /Image or /Form or /PS.)//error exec -}if -}bind def -/DoXObject -{ -//PDFReader/CurrentObject get/Context get/Resources get -/XObject//DoNothing//ResolveD exec -exch//CheckXObject//ResolveD exec -dup/Subtype get -dup/Image eq{ -pop -//CompleteOutlineImage exec -//DoImage exec -}{ -dup/PS eq{ -PDFR_DEBUG{ -(Executing a PS Xobject)= -}if -pop -//RunDelayedStream exec -}{ -dup/Form eq{ -pop -PDFR_DEBUG{ -(Executing a Form XObject)= -}if -//PDFReader/CurrentObject get exch -dup//PDFReader exch<< exch/Context exch >>/CurrentObject exch put -dup/Matrix get concat -dup/BBox get aload pop exch 3 index sub exch 2 index sub rectclip -//RunDelayedStream exec -//PDFReader exch/CurrentObject exch put -}{ -mark exch(unimplemented XObject type )exch//error exec -}ifelse -}ifelse -}ifelse -}bind def -/Operators 50 dict begin -/q{//GSave exec}bind def -/Q{//GRestore exec}bind def -/cm{//TempMatrix astore concat}bind def -/i{1 .min setflat}bind def -/J/setlinecap load def -/d/setdash load def -/j/setlinejoin load def -/w/setlinewidth load def -/M/setmiterlimit load def -/gs{SetExtGState}bind def -/g/setgray load def -/rg/setrgbcolor load def -/k/setcmykcolor load def -/cs{//ResolveColorSpace exec//SetColorSpaceSafe exec -}bind def -/sc/setcolor load def -/scn{//SetColor exec}bind def -/G/setgray load def -/RG/setrgbcolor load def -/K/setcmykcolor load def -/CS//cs def -/ri{SetColorRenderingIntent}bind def -/SC/setcolor load def -/SCN{//SetColor exec}bind def -/m/moveto load def -/l/lineto load def -/c/curveto load def -/v{currentpoint 6 2 roll curveto}bind def -/y{2 copy curveto}bind def -/re{ -4 2 roll moveto exch dup 0 rlineto 0 3 -1 roll rlineto neg 0 rlineto -closepath -}def -/h/closepath load def -/n/newpath load def -/S/stroke load def -/s{closepath stroke}bind def -/f/fill load def -/f*/eofill load def -/B{gsave fill grestore stroke}bind def -/b{closepath gsave fill grestore stroke}bind def -/B*{gsave eofill grestore stroke}bind def -/b*{closepath gsave eofill grestore stroke}bind def -/W/clip load def -/W*/eoclip load def -/sh{ -ResolveShading -dup/Background known{ -gsave -dup/ColorSpace get setcolorspace -dup/Background get aload pop setcolor -pathbbox -2 index sub exch 3 index sub exch -rectfill -grestore -}if -shfill -}bind def -/Do{//DoXObject exec}bind def -/BI{currentglobal false setglobal<<}bind def -/ID{>> -dup/DataSource currentfile -2 index/F//knownget exec{ -/A85 eq{ -0(~>)/SubFileDecode filter -}if -}if -put -//CompleteInlineImage exec -exch setglobal -//DoImage exec -}bind def -/EI{}bind def -/BT{gsave//GraphicState/InitialTextMatrix get currentmatrix pop}bind def -/ET{grestore}bind def -/Tc{//GraphicState exch/CharacterSpacing exch put}bind def -/TL{//GraphicState exch/TextLeading exch put}bind def -/Tr{//GraphicState exch/TextRenderingMode exch put}bind def -/Ts{ -mark(Unimplemented SetTextRise)//error exec -}bind def -/Tw{//GraphicState exch/WordSpacing exch put}bind def -/Tz{ -mark(Unimplemented SetHorizontalTextScaling)//error exec -}bind def -/Td{translate 0 0 moveto}bind def -/TD{dup neg//TL exec//Td exec}bind def -/Tm{//GraphicState/InitialTextMatrix get setmatrix -//TempMatrix astore concat -0 0 moveto}bind def -/T*{0//GraphicState/TextLeading get neg//Td exec}bind def -/Tj{//ShowTextBeg exec//ShowText exec//ShowTextEnd exec}bind def -/'{//T* exec//ShowText exec//ShowTextEnd exec}bind def -/"{3 2 roll//Tw exec exch//Tc exec//' exec}bind def -/TJ//ShowTextWithGlyphPositioning def -/Tf//SetFont def -/d0/setcharwidth load def -/d1/setcachedevice load def -/BDC{pop pop}bind def -/BMC{pop}bind def -/EMC{}bind def -/BX{BeginCompatibilitySection}bind def -/EX{EndCompatibilitySection}bind def -/DP{DefineMarkedContentPointWithPropertyList}bind def -/MP{DefineMarkedContentPoint}bind def -/PS{cvx exec}bind def -currentdict end def -//PDFR_STREAM{ -//Operators length dict begin -//Operators{ -exch dup -[exch//=only/exec load -( )/print load -8 7 roll -dup type/arraytype eq{ -/exec load -}if -( )/print load -]cvx -def -}forall -currentdict end/Operators exch def -}if -/.registerencoding -{pop pop -}bind def -/.defineencoding -{def -}bind def -/.findencoding -{load -}bind def -/currentglobal where -{pop currentglobal{setglobal}true setglobal} -{{}} -ifelse -/MacRomanEncoding -StandardEncoding 0 39 getinterval aload pop -/quotesingle -StandardEncoding 40 56 getinterval aload pop -/grave -StandardEncoding 97 31 getinterval aload pop -/Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute -/agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave -/ecircumflex/edieresis/iacute/igrave -/icircumflex/idieresis/ntilde/oacute -/ograve/ocircumflex/odieresis/otilde -/uacute/ugrave/ucircumflex/udieresis -/dagger/degree/cent/sterling/section/bullet/paragraph/germandbls -/registered/copyright/trademark/acute/dieresis/.notdef/AE/Oslash -/.notdef/plusminus/.notdef/.notdef/yen/mu/.notdef/.notdef -/.notdef/.notdef/.notdef/ordfeminine/ordmasculine/.notdef/ae/oslash -/questiondown/exclamdown/logicalnot/.notdef -/florin/.notdef/.notdef/guillemotleft -/guillemotright/ellipsis/space/Agrave/Atilde/Otilde/OE/oe -/endash/emdash/quotedblleft/quotedblright -/quoteleft/quoteright/divide/.notdef -/ydieresis/Ydieresis/fraction/currency -/guilsinglleft/guilsinglright/fi/fl -/daggerdbl/periodcentered/quotesinglbase/quotedblbase -/perthousand/Acircumflex/Ecircumflex/Aacute -/Edieresis/Egrave/Iacute/Icircumflex -/Idieresis/Igrave/Oacute/Ocircumflex -/.notdef/Ograve/Uacute/Ucircumflex -/Ugrave/dotlessi/circumflex/tilde -/macron/breve/dotaccent/ring/cedilla/hungarumlaut/ogonek/caron -256 packedarray -5 1 index .registerencoding -.defineencoding -exec -/AdobeGlyphList mark -/A 16#0041 -/AE 16#00c6 -/AEacute 16#01fc -/AEmacron 16#01e2 -/AEsmall 16#f7e6 -/Aacute 16#00c1 -/Aacutesmall 16#f7e1 -/Abreve 16#0102 -/Abreveacute 16#1eae -/Abrevecyrillic 16#04d0 -/Abrevedotbelow 16#1eb6 -/Abrevegrave 16#1eb0 -/Abrevehookabove 16#1eb2 -/Abrevetilde 16#1eb4 -/Acaron 16#01cd -/Acircle 16#24b6 -/Acircumflex 16#00c2 -/Acircumflexacute 16#1ea4 -/Acircumflexdotbelow 16#1eac -/Acircumflexgrave 16#1ea6 -/Acircumflexhookabove 16#1ea8 -/Acircumflexsmall 16#f7e2 -/Acircumflextilde 16#1eaa -/Acute 16#f6c9 -/Acutesmall 16#f7b4 -/Acyrillic 16#0410 -/Adblgrave 16#0200 -/Adieresis 16#00c4 -/Adieresiscyrillic 16#04d2 -/Adieresismacron 16#01de -/Adieresissmall 16#f7e4 -/Adotbelow 16#1ea0 -/Adotmacron 16#01e0 -/Agrave 16#00c0 -/Agravesmall 16#f7e0 -/Ahookabove 16#1ea2 -/Aiecyrillic 16#04d4 -/Ainvertedbreve 16#0202 -/Alpha 16#0391 -/Alphatonos 16#0386 -/Amacron 16#0100 -/Amonospace 16#ff21 -/Aogonek 16#0104 -/Aring 16#00c5 -/Aringacute 16#01fa -/Aringbelow 16#1e00 -/Aringsmall 16#f7e5 -/Asmall 16#f761 -/Atilde 16#00c3 -/Atildesmall 16#f7e3 -/Aybarmenian 16#0531 -/B 16#0042 -/Bcircle 16#24b7 -/Bdotaccent 16#1e02 -/Bdotbelow 16#1e04 -/Becyrillic 16#0411 -/Benarmenian 16#0532 -/Beta 16#0392 -/Bhook 16#0181 -/Blinebelow 16#1e06 -/Bmonospace 16#ff22 -/Brevesmall 16#f6f4 -/Bsmall 16#f762 -/Btopbar 16#0182 -/C 16#0043 -/Caarmenian 16#053e -/Cacute 16#0106 -/Caron 16#f6ca -/Caronsmall 16#f6f5 -/Ccaron 16#010c -/Ccedilla 16#00c7 -/Ccedillaacute 16#1e08 -/Ccedillasmall 16#f7e7 -/Ccircle 16#24b8 -/Ccircumflex 16#0108 -/Cdot 16#010a -/Cdotaccent 16#010a -/Cedillasmall 16#f7b8 -/Chaarmenian 16#0549 -/Cheabkhasiancyrillic 16#04bc -/Checyrillic 16#0427 -/Chedescenderabkhasiancyrillic 16#04be -/Chedescendercyrillic 16#04b6 -/Chedieresiscyrillic 16#04f4 -/Cheharmenian 16#0543 -/Chekhakassiancyrillic 16#04cb -/Cheverticalstrokecyrillic 16#04b8 -/Chi 16#03a7 -/Chook 16#0187 -/Circumflexsmall 16#f6f6 -/Cmonospace 16#ff23 -/Coarmenian 16#0551 -/Csmall 16#f763 -/D 16#0044 -/DZ 16#01f1 -/DZcaron 16#01c4 -/Daarmenian 16#0534 -/Dafrican 16#0189 -/Dcaron 16#010e -/Dcedilla 16#1e10 -/Dcircle 16#24b9 -/Dcircumflexbelow 16#1e12 -/Dcroat 16#0110 -/Ddotaccent 16#1e0a -/Ddotbelow 16#1e0c -/Decyrillic 16#0414 -/Deicoptic 16#03ee -/Delta 16#2206 -/Deltagreek 16#0394 -/Dhook 16#018a -/Dieresis 16#f6cb -/DieresisAcute 16#f6cc -/DieresisGrave 16#f6cd -/Dieresissmall 16#f7a8 -/Digammagreek 16#03dc -/Djecyrillic 16#0402 -/Dlinebelow 16#1e0e -/Dmonospace 16#ff24 -/Dotaccentsmall 16#f6f7 -/Dslash 16#0110 -/Dsmall 16#f764 -/Dtopbar 16#018b -/Dz 16#01f2 -/Dzcaron 16#01c5 -/Dzeabkhasiancyrillic 16#04e0 -/Dzecyrillic 16#0405 -/Dzhecyrillic 16#040f -/E 16#0045 -/Eacute 16#00c9 -/Eacutesmall 16#f7e9 -/Ebreve 16#0114 -/Ecaron 16#011a -/Ecedillabreve 16#1e1c -/Echarmenian 16#0535 -/Ecircle 16#24ba -/Ecircumflex 16#00ca -/Ecircumflexacute 16#1ebe -/Ecircumflexbelow 16#1e18 -/Ecircumflexdotbelow 16#1ec6 -/Ecircumflexgrave 16#1ec0 -/Ecircumflexhookabove 16#1ec2 -/Ecircumflexsmall 16#f7ea -/Ecircumflextilde 16#1ec4 -/Ecyrillic 16#0404 -/Edblgrave 16#0204 -/Edieresis 16#00cb -/Edieresissmall 16#f7eb -/Edot 16#0116 -/Edotaccent 16#0116 -/Edotbelow 16#1eb8 -/Efcyrillic 16#0424 -/Egrave 16#00c8 -/Egravesmall 16#f7e8 -/Eharmenian 16#0537 -/Ehookabove 16#1eba -/Eightroman 16#2167 -/Einvertedbreve 16#0206 -/Eiotifiedcyrillic 16#0464 -/Elcyrillic 16#041b -/Elevenroman 16#216a -/Emacron 16#0112 -/Emacronacute 16#1e16 -/Emacrongrave 16#1e14 -/Emcyrillic 16#041c -/Emonospace 16#ff25 -/Encyrillic 16#041d -/Endescendercyrillic 16#04a2 -/Eng 16#014a -/Enghecyrillic 16#04a4 -/Enhookcyrillic 16#04c7 -/Eogonek 16#0118 -/Eopen 16#0190 -/Epsilon 16#0395 -/Epsilontonos 16#0388 -/Ercyrillic 16#0420 -/Ereversed 16#018e -/Ereversedcyrillic 16#042d -/Escyrillic 16#0421 -/Esdescendercyrillic 16#04aa -/Esh 16#01a9 -/Esmall 16#f765 -/Eta 16#0397 -/Etarmenian 16#0538 -/Etatonos 16#0389 -/Eth 16#00d0 -/Ethsmall 16#f7f0 -/Etilde 16#1ebc -/Etildebelow 16#1e1a -/Euro 16#20ac -/Ezh 16#01b7 -/Ezhcaron 16#01ee -/Ezhreversed 16#01b8 -/F 16#0046 -/Fcircle 16#24bb -/Fdotaccent 16#1e1e -/Feharmenian 16#0556 -/Feicoptic 16#03e4 -/Fhook 16#0191 -/Fitacyrillic 16#0472 -/Fiveroman 16#2164 -/Fmonospace 16#ff26 -/Fourroman 16#2163 -/Fsmall 16#f766 -/G 16#0047 -/GBsquare 16#3387 -/Gacute 16#01f4 -/Gamma 16#0393 -/Gammaafrican 16#0194 -/Gangiacoptic 16#03ea -/Gbreve 16#011e -/Gcaron 16#01e6 -/Gcedilla 16#0122 -/Gcircle 16#24bc -/Gcircumflex 16#011c -/Gcommaaccent 16#0122 -/Gdot 16#0120 -/Gdotaccent 16#0120 -/Gecyrillic 16#0413 -/Ghadarmenian 16#0542 -/Ghemiddlehookcyrillic 16#0494 -/Ghestrokecyrillic 16#0492 -/Gheupturncyrillic 16#0490 -/Ghook 16#0193 -/Gimarmenian 16#0533 -/Gjecyrillic 16#0403 -/Gmacron 16#1e20 -/Gmonospace 16#ff27 -/Grave 16#f6ce -/Gravesmall 16#f760 -/Gsmall 16#f767 -/Gsmallhook 16#029b -/Gstroke 16#01e4 -/H 16#0048 -/H18533 16#25cf -/H18543 16#25aa -/H18551 16#25ab -/H22073 16#25a1 -/HPsquare 16#33cb -/Haabkhasiancyrillic 16#04a8 -/Hadescendercyrillic 16#04b2 -/Hardsigncyrillic 16#042a -/Hbar 16#0126 -/Hbrevebelow 16#1e2a -/Hcedilla 16#1e28 -/Hcircle 16#24bd -/Hcircumflex 16#0124 -/Hdieresis 16#1e26 -/Hdotaccent 16#1e22 -/Hdotbelow 16#1e24 -/Hmonospace 16#ff28 -/Hoarmenian 16#0540 -/Horicoptic 16#03e8 -/Hsmall 16#f768 -/Hungarumlaut 16#f6cf -/Hungarumlautsmall 16#f6f8 -/Hzsquare 16#3390 -/I 16#0049 -/IAcyrillic 16#042f -/IJ 16#0132 -/IUcyrillic 16#042e -/Iacute 16#00cd -/Iacutesmall 16#f7ed -/Ibreve 16#012c -/Icaron 16#01cf -/Icircle 16#24be -/Icircumflex 16#00ce -/Icircumflexsmall 16#f7ee -/Icyrillic 16#0406 -/Idblgrave 16#0208 -/Idieresis 16#00cf -/Idieresisacute 16#1e2e -/Idieresiscyrillic 16#04e4 -/Idieresissmall 16#f7ef -/Idot 16#0130 -/Idotaccent 16#0130 -/Idotbelow 16#1eca -/Iebrevecyrillic 16#04d6 -/Iecyrillic 16#0415 -/Ifraktur 16#2111 -/Igrave 16#00cc -/Igravesmall 16#f7ec -/Ihookabove 16#1ec8 -/Iicyrillic 16#0418 -/Iinvertedbreve 16#020a -/Iishortcyrillic 16#0419 -/Imacron 16#012a -/Imacroncyrillic 16#04e2 -/Imonospace 16#ff29 -/Iniarmenian 16#053b -/Iocyrillic 16#0401 -/Iogonek 16#012e -/Iota 16#0399 -/Iotaafrican 16#0196 -/Iotadieresis 16#03aa -/Iotatonos 16#038a -/Ismall 16#f769 -/Istroke 16#0197 -/Itilde 16#0128 -/Itildebelow 16#1e2c -/Izhitsacyrillic 16#0474 -/Izhitsadblgravecyrillic 16#0476 -/J 16#004a -/Jaarmenian 16#0541 -/Jcircle 16#24bf -/Jcircumflex 16#0134 -/Jecyrillic 16#0408 -/Jheharmenian 16#054b -/Jmonospace 16#ff2a -/Jsmall 16#f76a -/K 16#004b -/KBsquare 16#3385 -/KKsquare 16#33cd -/Kabashkircyrillic 16#04a0 -/Kacute 16#1e30 -/Kacyrillic 16#041a -/Kadescendercyrillic 16#049a -/Kahookcyrillic 16#04c3 -/Kappa 16#039a -/Kastrokecyrillic 16#049e -/Kaverticalstrokecyrillic 16#049c -/Kcaron 16#01e8 -/Kcedilla 16#0136 -/Kcircle 16#24c0 -/Kcommaaccent 16#0136 -/Kdotbelow 16#1e32 -/Keharmenian 16#0554 -/Kenarmenian 16#053f -/Khacyrillic 16#0425 -/Kheicoptic 16#03e6 -/Khook 16#0198 -/Kjecyrillic 16#040c -/Klinebelow 16#1e34 -/Kmonospace 16#ff2b -/Koppacyrillic 16#0480 -/Koppagreek 16#03de -/Ksicyrillic 16#046e -/Ksmall 16#f76b -/L 16#004c -/LJ 16#01c7 -/LL 16#f6bf -/Lacute 16#0139 -/Lambda 16#039b -/Lcaron 16#013d -/Lcedilla 16#013b -/Lcircle 16#24c1 -/Lcircumflexbelow 16#1e3c -/Lcommaaccent 16#013b -/Ldot 16#013f -/Ldotaccent 16#013f -/Ldotbelow 16#1e36 -/Ldotbelowmacron 16#1e38 -/Liwnarmenian 16#053c -/Lj 16#01c8 -/Ljecyrillic 16#0409 -/Llinebelow 16#1e3a -/Lmonospace 16#ff2c -/Lslash 16#0141 -/Lslashsmall 16#f6f9 -/Lsmall 16#f76c -/M 16#004d -/MBsquare 16#3386 -/Macron 16#f6d0 -/Macronsmall 16#f7af -/Macute 16#1e3e -/Mcircle 16#24c2 -/Mdotaccent 16#1e40 -/Mdotbelow 16#1e42 -/Menarmenian 16#0544 -/Mmonospace 16#ff2d -/Msmall 16#f76d -/Mturned 16#019c -/Mu 16#039c -/N 16#004e -/NJ 16#01ca -/Nacute 16#0143 -/Ncaron 16#0147 -/Ncedilla 16#0145 -/Ncircle 16#24c3 -/Ncircumflexbelow 16#1e4a -/Ncommaaccent 16#0145 -/Ndotaccent 16#1e44 -/Ndotbelow 16#1e46 -/Nhookleft 16#019d -/Nineroman 16#2168 -/Nj 16#01cb -/Njecyrillic 16#040a -/Nlinebelow 16#1e48 -/Nmonospace 16#ff2e -/Nowarmenian 16#0546 -/Nsmall 16#f76e -/Ntilde 16#00d1 -/Ntildesmall 16#f7f1 -/Nu 16#039d -/O 16#004f -/OE 16#0152 -/OEsmall 16#f6fa -/Oacute 16#00d3 -/Oacutesmall 16#f7f3 -/Obarredcyrillic 16#04e8 -/Obarreddieresiscyrillic 16#04ea -/Obreve 16#014e -/Ocaron 16#01d1 -/Ocenteredtilde 16#019f -/Ocircle 16#24c4 -/Ocircumflex 16#00d4 -/Ocircumflexacute 16#1ed0 -/Ocircumflexdotbelow 16#1ed8 -/Ocircumflexgrave 16#1ed2 -/Ocircumflexhookabove 16#1ed4 -/Ocircumflexsmall 16#f7f4 -/Ocircumflextilde 16#1ed6 -/Ocyrillic 16#041e -/Odblacute 16#0150 -/Odblgrave 16#020c -/Odieresis 16#00d6 -/Odieresiscyrillic 16#04e6 -/Odieresissmall 16#f7f6 -/Odotbelow 16#1ecc -/Ogoneksmall 16#f6fb -/Ograve 16#00d2 -/Ogravesmall 16#f7f2 -/Oharmenian 16#0555 -/Ohm 16#2126 -/Ohookabove 16#1ece -/Ohorn 16#01a0 -/Ohornacute 16#1eda -/Ohorndotbelow 16#1ee2 -/Ohorngrave 16#1edc -/Ohornhookabove 16#1ede -/Ohorntilde 16#1ee0 -/Ohungarumlaut 16#0150 -/Oi 16#01a2 -/Oinvertedbreve 16#020e -/Omacron 16#014c -/Omacronacute 16#1e52 -/Omacrongrave 16#1e50 -/Omega 16#2126 -/Omegacyrillic 16#0460 -/Omegagreek 16#03a9 -/Omegaroundcyrillic 16#047a -/Omegatitlocyrillic 16#047c -/Omegatonos 16#038f -/Omicron 16#039f -/Omicrontonos 16#038c -/Omonospace 16#ff2f -/Oneroman 16#2160 -/Oogonek 16#01ea -/Oogonekmacron 16#01ec -/Oopen 16#0186 -/Oslash 16#00d8 -/Oslashacute 16#01fe -/Oslashsmall 16#f7f8 -/Osmall 16#f76f -/Ostrokeacute 16#01fe -/Otcyrillic 16#047e -/Otilde 16#00d5 -/Otildeacute 16#1e4c -/Otildedieresis 16#1e4e -/Otildesmall 16#f7f5 -/P 16#0050 -/Pacute 16#1e54 -/Pcircle 16#24c5 -/Pdotaccent 16#1e56 -/Pecyrillic 16#041f -/Peharmenian 16#054a -/Pemiddlehookcyrillic 16#04a6 -/Phi 16#03a6 -/Phook 16#01a4 -/Pi 16#03a0 -/Piwrarmenian 16#0553 -/Pmonospace 16#ff30 -/Psi 16#03a8 -/Psicyrillic 16#0470 -/Psmall 16#f770 -/Q 16#0051 -/Qcircle 16#24c6 -/Qmonospace 16#ff31 -/Qsmall 16#f771 -/R 16#0052 -/Raarmenian 16#054c -/Racute 16#0154 -/Rcaron 16#0158 -/Rcedilla 16#0156 -/Rcircle 16#24c7 -/Rcommaaccent 16#0156 -/Rdblgrave 16#0210 -/Rdotaccent 16#1e58 -/Rdotbelow 16#1e5a -/Rdotbelowmacron 16#1e5c -/Reharmenian 16#0550 -/Rfraktur 16#211c -/Rho 16#03a1 -/Ringsmall 16#f6fc -/Rinvertedbreve 16#0212 -/Rlinebelow 16#1e5e -/Rmonospace 16#ff32 -/Rsmall 16#f772 -/Rsmallinverted 16#0281 -/Rsmallinvertedsuperior 16#02b6 -/S 16#0053 -/SF010000 16#250c -/SF020000 16#2514 -/SF030000 16#2510 -/SF040000 16#2518 -/SF050000 16#253c -/SF060000 16#252c -/SF070000 16#2534 -/SF080000 16#251c -/SF090000 16#2524 -/SF100000 16#2500 -/SF110000 16#2502 -/SF190000 16#2561 -/SF200000 16#2562 -/SF210000 16#2556 -/SF220000 16#2555 -/SF230000 16#2563 -/SF240000 16#2551 -/SF250000 16#2557 -/SF260000 16#255d -/SF270000 16#255c -/SF280000 16#255b -/SF360000 16#255e -/SF370000 16#255f -/SF380000 16#255a -/SF390000 16#2554 -/SF400000 16#2569 -/SF410000 16#2566 -/SF420000 16#2560 -/SF430000 16#2550 -/SF440000 16#256c -/SF450000 16#2567 -/SF460000 16#2568 -/SF470000 16#2564 -/SF480000 16#2565 -/SF490000 16#2559 -/SF500000 16#2558 -/SF510000 16#2552 -/SF520000 16#2553 -/SF530000 16#256b -/SF540000 16#256a -/Sacute 16#015a -/Sacutedotaccent 16#1e64 -/Sampigreek 16#03e0 -/Scaron 16#0160 -/Scarondotaccent 16#1e66 -/Scaronsmall 16#f6fd -/Scedilla 16#015e -/Schwa 16#018f -/Schwacyrillic 16#04d8 -/Schwadieresiscyrillic 16#04da -/Scircle 16#24c8 -/Scircumflex 16#015c -/Scommaaccent 16#0218 -/Sdotaccent 16#1e60 -/Sdotbelow 16#1e62 -/Sdotbelowdotaccent 16#1e68 -/Seharmenian 16#054d -/Sevenroman 16#2166 -/Shaarmenian 16#0547 -/Shacyrillic 16#0428 -/Shchacyrillic 16#0429 -/Sheicoptic 16#03e2 -/Shhacyrillic 16#04ba -/Shimacoptic 16#03ec -/Sigma 16#03a3 -/Sixroman 16#2165 -/Smonospace 16#ff33 -/Softsigncyrillic 16#042c -/Ssmall 16#f773 -/Stigmagreek 16#03da -/T 16#0054 -/Tau 16#03a4 -/Tbar 16#0166 -/Tcaron 16#0164 -/Tcedilla 16#0162 -/Tcircle 16#24c9 -/Tcircumflexbelow 16#1e70 -/Tcommaaccent 16#0162 -/Tdotaccent 16#1e6a -/Tdotbelow 16#1e6c -/Tecyrillic 16#0422 -/Tedescendercyrillic 16#04ac -/Tenroman 16#2169 -/Tetsecyrillic 16#04b4 -/Theta 16#0398 -/Thook 16#01ac -/Thorn 16#00de -/Thornsmall 16#f7fe -/Threeroman 16#2162 -/Tildesmall 16#f6fe -/Tiwnarmenian 16#054f -/Tlinebelow 16#1e6e -/Tmonospace 16#ff34 -/Toarmenian 16#0539 -/Tonefive 16#01bc -/Tonesix 16#0184 -/Tonetwo 16#01a7 -/Tretroflexhook 16#01ae -/Tsecyrillic 16#0426 -/Tshecyrillic 16#040b -/Tsmall 16#f774 -/Twelveroman 16#216b -/Tworoman 16#2161 -/U 16#0055 -/Uacute 16#00da -/Uacutesmall 16#f7fa -/Ubreve 16#016c -/Ucaron 16#01d3 -/Ucircle 16#24ca -/Ucircumflex 16#00db -/Ucircumflexbelow 16#1e76 -/Ucircumflexsmall 16#f7fb -/Ucyrillic 16#0423 -/Udblacute 16#0170 -/Udblgrave 16#0214 -/Udieresis 16#00dc -/Udieresisacute 16#01d7 -/Udieresisbelow 16#1e72 -/Udieresiscaron 16#01d9 -/Udieresiscyrillic 16#04f0 -/Udieresisgrave 16#01db -/Udieresismacron 16#01d5 -/Udieresissmall 16#f7fc -/Udotbelow 16#1ee4 -/Ugrave 16#00d9 -/Ugravesmall 16#f7f9 -/Uhookabove 16#1ee6 -/Uhorn 16#01af -/Uhornacute 16#1ee8 -/Uhorndotbelow 16#1ef0 -/Uhorngrave 16#1eea -/Uhornhookabove 16#1eec -/Uhorntilde 16#1eee -/Uhungarumlaut 16#0170 -/Uhungarumlautcyrillic 16#04f2 -/Uinvertedbreve 16#0216 -/Ukcyrillic 16#0478 -/Umacron 16#016a -/Umacroncyrillic 16#04ee -/Umacrondieresis 16#1e7a -/Umonospace 16#ff35 -/Uogonek 16#0172 -/Upsilon 16#03a5 -/Upsilon1 16#03d2 -/Upsilonacutehooksymbolgreek 16#03d3 -/Upsilonafrican 16#01b1 -/Upsilondieresis 16#03ab -/Upsilondieresishooksymbolgreek 16#03d4 -/Upsilonhooksymbol 16#03d2 -/Upsilontonos 16#038e -/Uring 16#016e -/Ushortcyrillic 16#040e -/Usmall 16#f775 -/Ustraightcyrillic 16#04ae -/Ustraightstrokecyrillic 16#04b0 -/Utilde 16#0168 -/Utildeacute 16#1e78 -/Utildebelow 16#1e74 -/V 16#0056 -/Vcircle 16#24cb -/Vdotbelow 16#1e7e -/Vecyrillic 16#0412 -/Vewarmenian 16#054e -/Vhook 16#01b2 -/Vmonospace 16#ff36 -/Voarmenian 16#0548 -/Vsmall 16#f776 -/Vtilde 16#1e7c -/W 16#0057 -/Wacute 16#1e82 -/Wcircle 16#24cc -/Wcircumflex 16#0174 -/Wdieresis 16#1e84 -/Wdotaccent 16#1e86 -/Wdotbelow 16#1e88 -/Wgrave 16#1e80 -/Wmonospace 16#ff37 -/Wsmall 16#f777 -/X 16#0058 -/Xcircle 16#24cd -/Xdieresis 16#1e8c -/Xdotaccent 16#1e8a -/Xeharmenian 16#053d -/Xi 16#039e -/Xmonospace 16#ff38 -/Xsmall 16#f778 -/Y 16#0059 -/Yacute 16#00dd -/Yacutesmall 16#f7fd -/Yatcyrillic 16#0462 -/Ycircle 16#24ce -/Ycircumflex 16#0176 -/Ydieresis 16#0178 -/Ydieresissmall 16#f7ff -/Ydotaccent 16#1e8e -/Ydotbelow 16#1ef4 -/Yericyrillic 16#042b -/Yerudieresiscyrillic 16#04f8 -/Ygrave 16#1ef2 -/Yhook 16#01b3 -/Yhookabove 16#1ef6 -/Yiarmenian 16#0545 -/Yicyrillic 16#0407 -/Yiwnarmenian 16#0552 -/Ymonospace 16#ff39 -/Ysmall 16#f779 -/Ytilde 16#1ef8 -/Yusbigcyrillic 16#046a -/Yusbigiotifiedcyrillic 16#046c -/Yuslittlecyrillic 16#0466 -/Yuslittleiotifiedcyrillic 16#0468 -/Z 16#005a -/Zaarmenian 16#0536 -/Zacute 16#0179 -/Zcaron 16#017d -/Zcaronsmall 16#f6ff -/Zcircle 16#24cf -/Zcircumflex 16#1e90 -/Zdot 16#017b -/Zdotaccent 16#017b -/Zdotbelow 16#1e92 -/Zecyrillic 16#0417 -/Zedescendercyrillic 16#0498 -/Zedieresiscyrillic 16#04de -/Zeta 16#0396 -/Zhearmenian 16#053a -/Zhebrevecyrillic 16#04c1 -/Zhecyrillic 16#0416 -/Zhedescendercyrillic 16#0496 -/Zhedieresiscyrillic 16#04dc -/Zlinebelow 16#1e94 -/Zmonospace 16#ff3a -/Zsmall 16#f77a -/Zstroke 16#01b5 -/a 16#0061 -/aabengali 16#0986 -/aacute 16#00e1 -/aadeva 16#0906 -/aagujarati 16#0a86 -/aagurmukhi 16#0a06 -/aamatragurmukhi 16#0a3e -/aarusquare 16#3303 -/aavowelsignbengali 16#09be -/aavowelsigndeva 16#093e -/aavowelsigngujarati 16#0abe -/abbreviationmarkarmenian 16#055f -/abbreviationsigndeva 16#0970 -/abengali 16#0985 -/abopomofo 16#311a -/abreve 16#0103 -/abreveacute 16#1eaf -/abrevecyrillic 16#04d1 -/abrevedotbelow 16#1eb7 -/abrevegrave 16#1eb1 -/abrevehookabove 16#1eb3 -/abrevetilde 16#1eb5 -/acaron 16#01ce -/acircle 16#24d0 -/acircumflex 16#00e2 -/acircumflexacute 16#1ea5 -/acircumflexdotbelow 16#1ead -/acircumflexgrave 16#1ea7 -/acircumflexhookabove 16#1ea9 -/acircumflextilde 16#1eab -/acute 16#00b4 -/acutebelowcmb 16#0317 -/acutecmb 16#0301 -/acutecomb 16#0301 -/acutedeva 16#0954 -/acutelowmod 16#02cf -/acutetonecmb 16#0341 -/acyrillic 16#0430 -/adblgrave 16#0201 -/addakgurmukhi 16#0a71 -/adeva 16#0905 -/adieresis 16#00e4 -/adieresiscyrillic 16#04d3 -/adieresismacron 16#01df -/adotbelow 16#1ea1 -/adotmacron 16#01e1 -/ae 16#00e6 -/aeacute 16#01fd -/aekorean 16#3150 -/aemacron 16#01e3 -/afii00208 16#2015 -/afii08941 16#20a4 -/afii10017 16#0410 -/afii10018 16#0411 -/afii10019 16#0412 -/afii10020 16#0413 -/afii10021 16#0414 -/afii10022 16#0415 -/afii10023 16#0401 -/afii10024 16#0416 -/afii10025 16#0417 -/afii10026 16#0418 -/afii10027 16#0419 -/afii10028 16#041a -/afii10029 16#041b -/afii10030 16#041c -/afii10031 16#041d -/afii10032 16#041e -/afii10033 16#041f -/afii10034 16#0420 -/afii10035 16#0421 -/afii10036 16#0422 -/afii10037 16#0423 -/afii10038 16#0424 -/afii10039 16#0425 -/afii10040 16#0426 -/afii10041 16#0427 -/afii10042 16#0428 -/afii10043 16#0429 -/afii10044 16#042a -/afii10045 16#042b -/afii10046 16#042c -/afii10047 16#042d -/afii10048 16#042e -/afii10049 16#042f -/afii10050 16#0490 -/afii10051 16#0402 -/afii10052 16#0403 -/afii10053 16#0404 -/afii10054 16#0405 -/afii10055 16#0406 -/afii10056 16#0407 -/afii10057 16#0408 -/afii10058 16#0409 -/afii10059 16#040a -/afii10060 16#040b -/afii10061 16#040c -/afii10062 16#040e -/afii10063 16#f6c4 -/afii10064 16#f6c5 -/afii10065 16#0430 -/afii10066 16#0431 -/afii10067 16#0432 -/afii10068 16#0433 -/afii10069 16#0434 -/afii10070 16#0435 -/afii10071 16#0451 -/afii10072 16#0436 -/afii10073 16#0437 -/afii10074 16#0438 -/afii10075 16#0439 -/afii10076 16#043a -/afii10077 16#043b -/afii10078 16#043c -/afii10079 16#043d -/afii10080 16#043e -/afii10081 16#043f -/afii10082 16#0440 -/afii10083 16#0441 -/afii10084 16#0442 -/afii10085 16#0443 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-^(23s5LT/",u.@'4[0^k;esi]:Xo%<7TQT!6Ub&+dN(h%o"B6T.^d/7/uY$-6K(Y)9oV'N2'&'p -;bb/ZA[E2+T%gf@.u)tF[R4mL;Up]EC!o^sMfVZq/Y;o7 -jj)^WR:*RZF.28PeosMm=lZ-CBD9DaR-7^3O/$BCkTLc,\os^EC-%ZYWbT,RYH2M#W$(>=Rb#6X -Cc`EQZ>@-2RKm8WYT:(;]6<)ADEF0I\o,-gn% -EI Q -Q - -endstream -endobj -%%PageTrailer -%%Trailer -end -%%EOF From e6f29751386d1be6a900fae3a114087f65039b01 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Sat, 11 Apr 2026 01:57:57 +0100 Subject: [PATCH 025/116] Fixed tables using script to convert to list-table. --- .../cloud_schemes/UMDP30_PC2CloudScheme.rst | 416 +++++++++--------- 1 file changed, 202 insertions(+), 214 deletions(-) diff --git a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst index 601af85b0a..73ef3b60ea 100644 --- a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst +++ b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst @@ -5797,172 +5797,163 @@ a closure needed to form the equation set, but are less readily related to physical quantities. Variables marked ’Diag’ form a part of the diagnostic output routines. -.. container:: center - - .. container:: - :name: tab:pc2_names - - .. table:: PC2 parameter values and locations - - +----------+----------+----------+----------+----------+----------+ - | Symbol | Code | Des | Value | Location | Notes | - | | variable | cription | | | and ref. | - +==========+==========+==========+==========+==========+==========+ - | - | init-it | Number | 10 | p | Num: | - | | erations | of | | c2-const | `3.4 | - | | | it | | | .2 <#sec | - | | | erations | | | :numapp_ | - | | | in | | | init>`__ | - | | | in | | | | - | | | itiation | | | | - +----------+----------+----------+----------+----------+----------+ - | :math:` | cloud | Bounds | 0.005 | UM | Num: | - | C_{tol}` | -pc2-tol | checking | | namelist | `4.9 | - | | | :ma | | | <#sec:i | - | | | th:`C_l` | | | nit2>`__ | - | | | t | | | | - | | | hreshold | | | | - +----------+----------+----------+----------+----------+----------+ - | : | cloud-p | Bounds | 0.001 | UM | Num: | - | math:`C_ | c2-tol-2 | checking | | namelist | `4.9 | - | {tol 2}` | | :ma | | | <#sec:i | - | | | th:`C_l` | | | nit2>`__ | - | | | t | | | | - | | | hreshold | | | | - +----------+----------+----------+----------+----------+----------+ - | :math:`R | rh | : | 0.01 | p | Num: | - | H_{tol}` | crit-tol | math:`RH | | c2-const | `4.9 | - | | | _{crit}` | | | <#sec:i | - | | | t | | | nit2>`__ | - | | | olerance | | | | - | | | in | | | | - | | | in | | | | - | | | itiation | | | | - +----------+----------+----------+----------+----------+----------+ - | :math | ls-bl0 | Fixed | :ma | imp-ctl | Clo: | - | :`q_{cf0 | | value of | th:`1.0 | | ` | - | \, BL}` | | BL | \times 1 | | 4.6 <#se | - | | | in-plume | 0^{-4} \ | | c:bl>`__ | - | | | : | , kg \, | | | - | | | math:`\o | kg^{-1}` | | | - | | | verline{ | | | | - | | | q_{cf}}` | | | | - +----------+----------+----------+----------+----------+----------+ - | :math:` | one- | Fixed | :ma | pc2-chck | Num: | - | q_{cf0}` | over-qcf | in-cloud | th:`1.0 | | `4.10 | - | | | : | \times 1 | | <#sec:ch | - | | | math:`\o | 0^{-4} \ | | ecks>`__ | - | | | verline{ | , kg \, | | | - | | | q_{cf}}` | kg^{-1}` | | | - | | | if | | | | - | | | :math:` | | | | - | | | C_f`\ =0 | | | | - +----------+----------+----------+----------+----------+----------+ - | : | pdf-mer | Merging | 0.5 | p | Clo: | - | math:`m` | ge-power | power | | c2-const | `3.2 | - | | | for | | | <#sec:h | - | | | :math:` | | | omog>`__ | - | | | G(-Q_c)` | | | | - +----------+----------+----------+----------+----------+----------+ - | : | p | Shape | 0.0 | p | Phy: | - | math:`n` | df-power | p | | c2-const | `3.2 | - | | | arameter | | | <#sec:h | - | | | for | | | omog>`__ | - | | | :math:` | | | | - | | | G(-Q_c)` | | | | - +----------+----------+----------+----------+----------+----------+ - | : | w | Wind | :mat | p | Phy: | - | math:`w` | ind-shea | shear in | h:`1.5 \ | c2-const | ` | - | | r-factor | fallout | times 10 | | 4.2.1 <# | - | | | of ice | ^{-4} \, | | sec:lsp_ | - | | | term | s^{-1}` | | fall>`__ | - +----------+----------+----------+----------+----------+----------+ - | : | i | Scaling | 0.04 | p | Phy: | - | math:`i` | ce-width | factor | | c2-const | `4 | - | | | for | | | .2.4 <#s | - | | | r | | | ec:mp_de | - | | | eduction | | | psub>`__ | - | | | in | | | | - | | | :ma | | | | - | | | th:`b_i` | | | | - +----------+----------+----------+----------+----------+----------+ - | : | dbsdtb | Rate of | :math: | UM | Phy: | - | math:`a` | s-turb-0 | r | `-2.25 \ | namelist | `3.3 | - | | | eduction | times 10 | | <#sec:w | - | | | of PDF | ^{-5} \, | | idth>`__ | - | | | width | s^{-1}` | | | - +----------+----------+----------+----------+----------+----------+ - | : | dbsdtb | Rate of | 0 | p | Phy: | - | math:`b` | s-turb-1 | r | | c2-const | `3.3 | - | | | eduction | | | <#sec:w | - | | | of PDF | | | idth>`__ | - | | | width | | | | - +----------+----------+----------+----------+----------+----------+ - | | dbsd | Redn of | 0 | p | Phy: | - | | tbs-conv | PDF | | c2-const | `3.3 | - | | | width in | | | <#sec:w | - | | | co | | | idth>`__ | - | | | nvection | | | | - +----------+----------+----------+----------+----------+----------+ - | | dbs | V | 10.05 | p | Phy: | - | | dtbs-exp | ariation | | c2-const | `3.3 | - | | | of | | | <#sec:w | - | | | erosion | | | idth>`__ | - | | | on RH | | | | - +----------+----------+----------+----------+----------+----------+ - | : | RHCRIT | Critical | | UM | Phy: | - | math:`RH | | RH for | | namelist | `3.4 | - | _{crit}` | | cloud | | | <#sec:i | - | | | f | | | nit>`__, | - | | | ormation | | | `4 | - | | | | | | .2.4 <#s | - | | | | | | ec:mp_de | - | | | | | | psub>`__ | - +----------+----------+----------+----------+----------+----------+ - | :math: | condensa | Minimum | :m | pc2-chck | Num: | - | `q_{c0}` | te-limit | allowed | ath:`1 \ | | `4.10 | - | | | co | times 10 | | <#sec:ch | - | | | ndensate | ^{-10} \ | | ecks>`__ | - | | | | , kg \, | | | - | | | | kg^{-1}` | | | - +----------+----------+----------+----------+----------+----------+ - | :math:`q | ls0 | Lower | : | enviro?a | Num: | - | _c^{S0}` | | limit of | math:`5 | | `3.5. | - | | | plume | \times 1 | | 2 <#sec: | - | | | co | 0^{-5} \ | | multi_nu | - | | | ndensate | , kg \, | | mapp>`__ | - | | | | kg^{-1}` | | | - +----------+----------+----------+----------+----------+----------+ - | | *Har | Conv | 0.2 | imp-ctl2 | Diag: | - | | d-wired* | cloud | | | `5.3 | - | | | fraction | | | <#sec:d | - | | | for | | | iags>`__ | - | | | vi | | | | - | | | sibility | | | | - +----------+----------+----------+----------+----------+----------+ - | | *Har | Limit on | 0.001 | lspice3d | Num: | - | | d-wired* | width of | | | `4 | - | | | ice | | | .2.4 <#s | - | | | dist | | | ec:mp_de | - | | | ribution | | | psub>`__ | - +----------+----------+----------+----------+----------+----------+ - | | *Har | :ma | 0.05 | pc2-init | Num: | - | | d-wired* | th:`C_l` | | | `4.9 | - | | | limit | | | <#sec:i | - | | | for init | | | nit2>`__ | - | | | if | | | | - | | | :math:`T | | | | - | | | < 0 ^{\ | | | | - | | | circ} C` | | | | - +----------+----------+----------+----------+----------+----------+ - | | *Har | T | :mat | imp-ctl | Num: | - | | d-wired* | olerance | h:`1.0 \ | | ` | - | | | on calc. | times 10 | | 4.6 <#se | - | | | of | ^{-10} \ | | c:bl>`__ | - | | | :math | , kg \, | | | - | | | :`q_C^s` | kg^{-1}` | | | - | | | in BL | | | | - +----------+----------+----------+----------+----------+----------+ +.. list-table:: PC2 parameter values and locations + :name: tab:pc2_names + :header-rows: 1 + + * - Symbol + - Code variable + - Des cription + - Value + - Location + - Notes and ref. + + * - - + - init-it erations + - Number of it erations in in itiation + - 10 + - p c2-const + - Num: `3.4.2 <#sec:numapp_init>`__ + + * - :math:`C_{tol}` + - cloud -pc2-tol + - Bounds checking :math:`C_l` t hreshold + - 0.005 + - UM namelist + - Num: `4.9 <#sec:init2>`__ + + * - :math:`C_{tol 2}` + - cloud-p c2-tol-2 + - Bounds checking :math:`C_l` t hreshold + - 0.001 + - UM namelist + - Num: `4.9 <#sec:init2>`__ + + * - :math:`RH_{tol}` + - rh crit-tol + - :math:`RH_{crit}` t olerance in in itiation + - 0.01 + - p c2-const + - Num: `4.9 <#sec:init2>`__ + + * - :math:`q_{cf0\, BL}` + - ls-bl0 + - Fixed value of BL in-plume :math:`\overline{q_{cf}}` + - :math:`1.0\times 10^{-4} \, kg \,kg^{-1}` + - imp-ctl + - Clo: `4.6 <#sec:bl>`__ + + * - :math:`q_{cf0}` + - one- over-qcf + - Fixed in-cloud :math:`\overline{q_{cf}}` if :math:`C_f`\ =0 + - :math:`1.0\times 10^{-4} \, kg \,kg^{-1}` + - pc2-chck + - Num: `4.10 <#sec:checks>`__ + + * - :math:`m` + - pdf-mer ge-power + - Merging power for :math:`G(-Q_c)` + - 0.5 + - p c2-const + - Clo: `3.2 <#sec:homog>`__ + + * - :math:`n` + - p df-power + - Shape p arameter for :math:`G(-Q_c)` + - 0.0 + - p c2-const + - Phy: `3.2 <#sec:homog>`__ + + * - :math:`w` + - w ind-shea r-factor + - Wind shear in fallout of ice term + - :math:`1.5 \times 10^{-4} \,s^{-1}` + - p c2-const + - Phy: `4.2.1 <#sec:lsp_fall>`__ + + * - :math:`i` + - i ce-width + - Scaling factor for r eduction in :math:`b_i` + - 0.04 + - p c2-const + - Phy: `4.2.4 <#sec:mp_depsub>`__ + + * - :math:`a` + - dbsdtb s-turb-0 + - Rate of r eduction of PDF width + - :math:`-2.25 \times 10^{-5} \,s^{-1}` + - UM namelist + - Phy: `3.3 <#sec:width>`__ + + * - :math:`b` + - dbsdtb s-turb-1 + - Rate of r eduction of PDF width + - 0 + - p c2-const + - Phy: `3.3 <#sec:width>`__ + + * - + - dbsd tbs-conv + - Redn of PDF width in co nvection + - 0 + - p c2-const + - Phy: `3.3 <#sec:width>`__ + + * - + - dbs dtbs-exp + - V ariation of erosion on RH + - 10.05 + - p c2-const + - Phy: `3.3 <#sec:width>`__ + + * - :math:`RH_{crit}` + - RHCRIT + - Critical RH for cloud f ormation + - + - UM namelist + - Phy: `3.4 <#sec:init>`__, `4.2.4 <#sec:mp_depsub>`__ + + * - :math:`q_{c0}` + - condensa te-limit + - Minimum allowed co ndensate + - :math:`1 \times 10^{-10} \, kg \,kg^{-1}` + - pc2-chck + - Num: `4.10 <#sec:checks>`__ + + * - :math:`q_c^{S0}` + - ls0 + - Lower limit of plume co ndensate + - :math:`5\times 10^{-5} \, kg \,kg^{-1}` + - enviro?a + - Num: `3.5.2 <#sec:multi_numapp>`__ + + * - + - *Har d-wired* + - Conv cloud fraction for vi sibility + - 0.2 + - imp-ctl2 + - Diag: `5.3 <#sec:diags>`__ + + * - + - *Har d-wired* + - Limit on width of ice dist ribution + - 0.001 + - lspice3d + - Num: `4.2.4 <#sec:mp_depsub>`__ + + * - + - *Har d-wired* + - :math:`C_l` limit for init if :math:`T< 0 ^{\circ} C` + - 0.05 + - pc2-init + - Num: `4.9 <#sec:init2>`__ + + * - + - *Har d-wired* + - T olerance on calc. of :math:`q_C^s` in BL + - :math:`1.0 \times 10^{-10} \, kg \,kg^{-1}` + - imp-ctl + - Num: `4.6 <#sec:bl>`__ PC2 also recommends some tunings of the existing convection scheme parameters. These cannot be placed in the library code, since they would @@ -5971,54 +5962,51 @@ modification sets. We have included those parameters that have been investigated throughout testing, although only two are different between PC2:64 and a non-PC2 run. -.. container:: center - - .. container:: - :name: tab:pc2_conv_names - - .. table:: PC2 parameter values and locations relating to the - convection. \*These values are those used in HadGAM - - +----------+----------+----------+----------+----------+----------+ - | Code | Des | Value in | Value in | Location | Notes | - | variable | cription | PC2 | Control | | and | - | | | | | | r | - | | | | | | eference | - +==========+==========+==========+==========+==========+==========+ - | TICE | Tem | :math: | :math:`0 | tice.cdk | Phy: | - | | perature | `-10 ^{\ | ^{\circ | or UMUI | `4.7 | - | | at which | circ} C` | } C`\ \* | | <#sec:co | - | | plume | | | | nvec>`__ | - | | freezes | | | | | - +----------+----------+----------+----------+----------+----------+ - | QSTICE | App | :m | :m | qs | Phy: | - | | roximate | ath:`3.5 | ath:`3.5 | tice.cdk | `4.7 | - | | qs | \times | \times | or UMUI | <#sec:co | - | | at(TICE) | 10^{-3}` | 10^{-3}` | | nvec>`__ | - +----------+----------+----------+----------+----------+----------+ - | *Har | Limit on | 0.5 | :math: | cloudw | Phy: | - | d-wired* | conv. | : | `0.5 \, | | `4.7 | - | | cond. | math:`q_ | q_{sat}` | | <#sec:co | - | | after | {sat}, 2 | | | nvec>`__ | - | | precip | \times | | | | - | | | 10^{-4}` | | | | - +----------+----------+----------+----------+----------+----------+ - | Anvil | Shape | 0 | 0.3\* | UMUI | Phy: | - | factor | p | | | | `4.7 | - | | arameter | | | | <#sec:co | - | | for | | | | nvec>`__ | - | | conv. | | | | | - | | cloud | | | | | - | | anvil | | | | | - +----------+----------+----------+----------+----------+----------+ - | Tower | Shape | 0 | 0.25\* | UMUI | Phy: | - | factor | p | | | | `4.7 | - | | arameter | | | | <#sec:co | - | | for | | | | nvec>`__ | - | | conv. | | | | | - | | cloud | | | | | - | | tower | | | | | - +----------+----------+----------+----------+----------+----------+ +.. list-table:: PC2 parameter values and locations relating to the + :name: tab:pc2_conv_names + :header-rows: 1 + + * - Code variable + - Des cription + - Value in PC2 + - Value in Control + - Location + - Notes and r eference + + * - TICE + - Tem perature at which plume freezes + - :math:`-10 ^{\circ} C` + - :math:`0^{\circ} C`\ \* + - tice.cdk or UMUI + - Phy: `4.7 <#sec:convec>`__ + + * - QSTICE + - App roximate qs at(TICE) + - :math:`3.5\times10^{-3}` + - :math:`3.5\times10^{-3}` + - qs tice.cdk or UMUI + - Phy: `4.7 <#sec:convec>`__ + + * - *Har d-wired* + - Limit on conv. cond. after precip + - 0.5 :math:`q_{sat}, 2\times10^{-4}` + - :math:`0.5 \,q_{sat}` + - cloudw + - Phy: `4.7 <#sec:convec>`__ + + * - Anvil factor + - Shape p arameter for conv. cloud anvil + - 0 + - 0.3\* + - UMUI + - Phy: `4.7 <#sec:convec>`__ + + * - Tower factor + - Shape p arameter for conv. cloud tower + - 0 + - 0.25\* + - UMUI + - Phy: `4.7 <#sec:convec>`__ How to run the PC2 scheme ------------------------- From c88b6957d21b04e8551fc6493fa54fda804719b9 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Sat, 11 Apr 2026 02:15:24 +0100 Subject: [PATCH 026/116] Re-checked-out from earlier version and reran all the scripts to re-apply automated changes from scratch. Needed to apply corrections to section references in tables. --- .../cloud_schemes/UMDP30_PC2CloudScheme.rst | 52 +++++++++---------- 1 file changed, 26 insertions(+), 26 deletions(-) diff --git a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst index 73ef3b60ea..f065962114 100644 --- a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst +++ b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst @@ -5813,147 +5813,147 @@ diagnostic output routines. - Number of it erations in in itiation - 10 - p c2-const - - Num: `3.4.2 <#sec:numapp_init>`__ + - Num: :ref:`Numerical Application of the Smith method` * - :math:`C_{tol}` - cloud -pc2-tol - Bounds checking :math:`C_l` t hreshold - 0.005 - UM namelist - - Num: `4.9 <#sec:init2>`__ + - Num: :ref:`Initiation` * - :math:`C_{tol 2}` - cloud-p c2-tol-2 - Bounds checking :math:`C_l` t hreshold - 0.001 - UM namelist - - Num: `4.9 <#sec:init2>`__ + - Num: :ref:`Initiation` * - :math:`RH_{tol}` - rh crit-tol - :math:`RH_{crit}` t olerance in in itiation - 0.01 - p c2-const - - Num: `4.9 <#sec:init2>`__ + - Num: :ref:`Initiation` * - :math:`q_{cf0\, BL}` - ls-bl0 - Fixed value of BL in-plume :math:`\overline{q_{cf}}` - :math:`1.0\times 10^{-4} \, kg \,kg^{-1}` - imp-ctl - - Clo: `4.6 <#sec:bl>`__ + - Clo: :ref:`Boundary Layer` * - :math:`q_{cf0}` - one- over-qcf - Fixed in-cloud :math:`\overline{q_{cf}}` if :math:`C_f`\ =0 - :math:`1.0\times 10^{-4} \, kg \,kg^{-1}` - pc2-chck - - Num: `4.10 <#sec:checks>`__ + - Num: :ref:`Bounds checking` * - :math:`m` - pdf-mer ge-power - Merging power for :math:`G(-Q_c)` - 0.5 - p c2-const - - Clo: `3.2 <#sec:homog>`__ + - Clo: :ref:`Homogeneous forcing` * - :math:`n` - p df-power - Shape p arameter for :math:`G(-Q_c)` - 0.0 - p c2-const - - Phy: `3.2 <#sec:homog>`__ + - Phy: :ref:`Homogeneous forcing` * - :math:`w` - w ind-shea r-factor - Wind shear in fallout of ice term - :math:`1.5 \times 10^{-4} \,s^{-1}` - p c2-const - - Phy: `4.2.1 <#sec:lsp_fall>`__ + - Phy: :ref:`Fall of ice` * - :math:`i` - i ce-width - Scaling factor for r eduction in :math:`b_i` - 0.04 - p c2-const - - Phy: `4.2.4 <#sec:mp_depsub>`__ + - Phy: :ref:`Deposition and sublimation` * - :math:`a` - dbsdtb s-turb-0 - Rate of r eduction of PDF width - :math:`-2.25 \times 10^{-5} \,s^{-1}` - UM namelist - - Phy: `3.3 <#sec:width>`__ + - Phy: :ref:`Changing the width of the PDF - PC2 erosion` * - :math:`b` - dbsdtb s-turb-1 - Rate of r eduction of PDF width - 0 - p c2-const - - Phy: `3.3 <#sec:width>`__ + - Phy: :ref:`Changing the width of the PDF - PC2 erosion` * - - dbsd tbs-conv - Redn of PDF width in co nvection - 0 - p c2-const - - Phy: `3.3 <#sec:width>`__ + - Phy: :ref:`Changing the width of the PDF - PC2 erosion` * - - dbs dtbs-exp - V ariation of erosion on RH - 10.05 - p c2-const - - Phy: `3.3 <#sec:width>`__ + - Phy: :ref:`Changing the width of the PDF - PC2 erosion` * - :math:`RH_{crit}` - RHCRIT - Critical RH for cloud f ormation - - UM namelist - - Phy: `3.4 <#sec:init>`__, `4.2.4 <#sec:mp_depsub>`__ + - Phy: :ref:`Initiation of cloud`, :ref:`Deposition and sublimation` * - :math:`q_{c0}` - condensa te-limit - Minimum allowed co ndensate - :math:`1 \times 10^{-10} \, kg \,kg^{-1}` - pc2-chck - - Num: `4.10 <#sec:checks>`__ + - Num: :ref:`Bounds checking` * - :math:`q_c^{S0}` - ls0 - Lower limit of plume co ndensate - :math:`5\times 10^{-5} \, kg \,kg^{-1}` - enviro?a - - Num: `3.5.2 <#sec:multi_numapp>`__ + - Num: :ref:`Numerical application` * - - *Har d-wired* - Conv cloud fraction for vi sibility - 0.2 - imp-ctl2 - - Diag: `5.3 <#sec:diags>`__ + - Diag: :ref:`Diagnostics` * - - *Har d-wired* - Limit on width of ice dist ribution - 0.001 - lspice3d - - Num: `4.2.4 <#sec:mp_depsub>`__ + - Num: :ref:`Deposition and sublimation` * - - *Har d-wired* - :math:`C_l` limit for init if :math:`T< 0 ^{\circ} C` - 0.05 - pc2-init - - Num: `4.9 <#sec:init2>`__ + - Num: :ref:`Initiation` * - - *Har d-wired* - T olerance on calc. of :math:`q_C^s` in BL - :math:`1.0 \times 10^{-10} \, kg \,kg^{-1}` - imp-ctl - - Num: `4.6 <#sec:bl>`__ + - Num: :ref:`Boundary Layer` PC2 also recommends some tunings of the existing convection scheme parameters. These cannot be placed in the library code, since they would @@ -5978,35 +5978,35 @@ PC2:64 and a non-PC2 run. - :math:`-10 ^{\circ} C` - :math:`0^{\circ} C`\ \* - tice.cdk or UMUI - - Phy: `4.7 <#sec:convec>`__ + - Phy: :ref:`Convection` * - QSTICE - App roximate qs at(TICE) - :math:`3.5\times10^{-3}` - :math:`3.5\times10^{-3}` - qs tice.cdk or UMUI - - Phy: `4.7 <#sec:convec>`__ + - Phy: :ref:`Convection` * - *Har d-wired* - Limit on conv. cond. after precip - 0.5 :math:`q_{sat}, 2\times10^{-4}` - :math:`0.5 \,q_{sat}` - cloudw - - Phy: `4.7 <#sec:convec>`__ + - Phy: :ref:`Convection` * - Anvil factor - Shape p arameter for conv. cloud anvil - 0 - 0.3\* - UMUI - - Phy: `4.7 <#sec:convec>`__ + - Phy: :ref:`Convection` * - Tower factor - Shape p arameter for conv. cloud tower - 0 - 0.25\* - UMUI - - Phy: `4.7 <#sec:convec>`__ + - Phy: :ref:`Convection` How to run the PC2 scheme ------------------------- From 9b61e4e6b1d51a39885c3eec3b00317396d362be Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Mon, 20 Apr 2026 11:47:42 +0100 Subject: [PATCH 027/116] Re-import the latex file, this time with the .bib file so we can try and get the literature citations to work... --- .../cloud_schemes/UMDP30_PC2CloudScheme.tex | 5734 +++++++++++++++++ .../science_guide/cloud_schemes/refs.bib | 371 ++ 2 files changed, 6105 insertions(+) create mode 100644 documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.tex create mode 100644 documentation/source/science_guide/cloud_schemes/refs.bib diff --git a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.tex b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.tex new file mode 100644 index 0000000000..d5173e6ad8 --- /dev/null +++ b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.tex @@ -0,0 +1,5734 @@ +\documentclass{UMDP_article} + +\title{The PC2 Cloud Scheme} +\paperno{030} +\umversion{14.0} +\owner{Cyril Morcrette} +\author{D.~Wilson, A.~Bushell, C.~Morcrette, V.~Varma$^{1}$, M.~Whitall} + +\titlecontent{ + \footnotesize + $^1$ National Institute of Water and Atmospheric Research, + Wellington, New Zealand \\ + } + +\usepackage{textcomp} +\usepackage{amstext,natbib} + +% Packages needed for the UM subroutine tree diagram in um_call_tree.txt: +\input{../029/um_call_tree_preamble} + +\newcommand{\mmax}[1] {\mbox{\footnotesize \sf MAX} \left[#1\right] } + +%%% These are definitions used in the convection documentation +%%% Definitions running across several chapters are defined in PC2_main. +\newcommand{\gbmll}{\ensuremath{\overline{l}_{\rm{l}}}} +\newcommand{\gbmlf}{\ensuremath{\overline{l}_{\rm{f}}}} +\newcommand{\gbmlm}{\ensuremath{\overline{l}_{\rm{m}}}} +\newcommand{\gbmlx}{\ensuremath{\overline{l}_{\rm{x}}}} +\newcommand{\incml}[1]{\ensuremath{\overline{l}_{\rm c}^{\rm #1}}} +\newcommand{\bigqc}[1]{\ensuremath{Q_{\rm C}^{\rm #1}}} +\newcommand{\GpdfB}{\ensuremath{G^{\rm B}}} +\newcommand{\injectcl}{\ensuremath{C_{\rm{l}}^{\ast}}} +\newcommand{\injectcf}{\ensuremath{C_{\rm{f}}^{\ast}}} +% +\newcommand{\lsubsup}[2]{\ensuremath{l_{\rm{#1}}^{\rm{#2}}}} +\newcommand{\csubsup}[2]{\ensuremath{l_{\rm{#1}}^{\rm{#2}}}} +\newcommand{\qsubsup}[2]{\ensuremath{q_{\rm{#1}}^{\rm{#2}}}} +\newcommand{\tsubsup}[2]{\ensuremath{T_{\rm{#1}}^{\rm{#2}}}} +\newcommand{\thsubsup}[2]{\ensuremath{\theta_{\rm{#1}}^{\rm{#2}}}} +\newcommand{\xsubsup}[2]{\ensuremath{{\chi}_{\rm{#1}}^{\rm{#2}}}} +%%% Private definitions (shorthand commands etc.) +\newcommand{\lc}{\left[} +\newcommand{\rc}{\right]} +\newcommand{\ov}{\overline} +\newcommand{\lp}{\left(} +\newcommand{\rp}{\right)} +\newcommand{\dr}{\partial} + +%%% Define a format for partial derivatives +\newcommand{\pardbyd}[2]{\ensuremath{\frac{\partial \, #1}{\partial \, #2}}} + +%%% Define a few of the commonest variables +\newcommand{\Tliq}{\ensuremath{T_{\rm L}}} +\newcommand{\qtot}{\ensuremath{q_{\rm T}}} +\newcommand{\qsat}{\ensuremath{q_{\rm s}}} +\newcommand{\aliq}{\ensuremath{a_{\rm L}}} +\newcommand{\gbmTliq}{\ensuremath{\overline{T}_{\rm L}}} +\newcommand{\gbmqtot}{\ensuremath{\overline{q}_{\rm T}}} +%Add a new command to do a horizontal line +\newcommand{\HRule}{\rule{\linewidth}{1.0mm}} + +\begin{document} % Every document must start with this. +\maketitle + +\tableofcontents + +\newpage +\section{PC2 developers} +We would like to acknowledge those who developed the PC2 cloud scheme: +Damian Wilson, Andrew Bushell, David Gregory, Amanda Kerr-Munslow, +John Edwards, Jeremy Price, Cyril Morcrette, Martin Sharpe, Thomas Mirfield, Ian Boutle. Many +others offered considerable help, advice and analysis, including Roy Kershaw, +Malcolm Brooks, Richard Forbes and Alejandro Bodas-Salcedo, and we would +like to thank them all for their input. + +\section{Introduction} + +This document describes the PC2 \textit{(prognostic cloud, +prognostic condensate)} cloud scheme. It should be seen as +a complete reference source for the scheme's physical assumptions, +numerical techniques, +application to the Unified Model and coding within the Unified Model. It does +not describe results from the scheme, please refer to the various reports and papers +written on this. Except where commented on explicitly, the description applies +to the PC2:66 version of the PC2 scheme, which is the version that will be +available at UM6.5. The version available at 6.4 is PC2:64. + +This paper will first introduce the concepts that underlie cloud schemes, +before developing a study of the theoretical behaviour of the prognostic PC2 scheme +under certain, well-defined, situations. The next sections shows how +the theory can be applied to the physical and dynamical processes +represented in the Unified Model. Finally, we outline the way in which +the PC2 scheme is implemented within the code of the Unified Model. + +%%\subsection{How to use this documentation} + +\subsection{Cloud schemes} + +The basic requirements of any cloud scheme within a large-scale model are to: +\begin{itemize} +\item{calculate the amount of condensation (from water vapour to liquid water or vice-versa) within each gridbox each timestep} +\item{to calculate or update the cloud fractions for use by the radiation and large-scale precipitation schemes (or any other physics scheme).} +\end{itemize} + +Depending on the model involved, cloud schemes may also treat the +deposition / sublimation process from vapour to ice. The problem is +straightforward to solve if one is allowed to assume that there is no +variability of moisture or temperature on a scale of a model gridbox. +In this case the cloud fraction scheme is redundant and only the condensation +part remains, which may be solved diagnostically using the instantaneous +condensation assumption in section \ref{sec:s_dist}. However, the +`no-variability' assumption is poor until +very high resolutions close to, or maybe exceeding, 1 km in the horizontal +are reached. Although we may eventually assume that computer power +will enable such resolutions to be reached globally, for many years we +will need a subgrid-scale cloud scheme to properly account for the +variability in the atmosphere. This is the principal challenge of +cloud parametrization. + +There are several approaches to take to the solution of the problem, +although they are not as independent as often portrayed, since they +nearly all require the same instantaneous condensation assumption +(discussed in section \ref{sec:s_dist}). Hence there are mathematical +links between +all the approaches. \textit{The following are all valid structures +to use in this respect.} +\begin{itemize} +\item{One may diagnose cloud fractions and condensate contents from knowledge of gridbox mean variables. This forms the basis of the \cite{smith90} scheme, which is described in +\citeumdp{029}.} +\item{A mixed scheme, such as \cite{sundqvist1978} uses a prediction of condensate contents, but a diagnostic cloud fraction.} +\item{Alternatively, one may predict cloud fraction and condensate content changes as a result of each modelled process. This forms the basis of the \cite{t93} scheme and the PC2 scheme.} +\item{Hybrid schemes, such as \cite{t02}, will predict various moments of the subgrid-scale variability, and use this knowledge to diagnose the cloud fraction and condensate contents.} +\end{itemize} + +Many years of experience of the results from the \cite{smith90} +scheme have highlighted deficiences in the diagnosis of cloud from +this scheme, which we feel can only be tackled by adding the +memory of cloud history available by using a prognostic based +scheme. We chose to develop a scheme that directly specified +the impacts on observable prognostics (condensates and +cloud fractions, as in \cite{t93}) rather than on moments of a +probability density function (as in \cite{t02}). This is because +we believe it is easier to physically relate (and hence parametrize) +the effect processes to quantities +such as cloud fraction and condensate rather than to the more abstract +quantities of moments of a probability +density function of moisture. However, although the +PC2 scheme is similar to \cite{t93} in its very basic prognostic variable +structure, the assumptions behind the formulation of the prognostic terms +in PC2 are very different and much improved. The PC2 scheme should not be +considered to be merely an extension of \cite{t93}. + +In particular, we wish to use a prognostic formulation in order to link +the detraiment of moisture from convection directly to cloud fraction, and +to break the hard diagnostic link between cloud fraction and condensate. These +major features of the \cite{t93} scheme provide the motivation to develop +the PC2 cloud scheme. + +\subsection{The `s' distribution} +\label{sec:s_dist} + +Most cloud schemes are based on the concept of a distribution +of fluctuations of moisture and temperature in the gridbox. Here we +mathematically formalize this concept, since it is used both in +the PC2 scheme and the \cite{smith90} scheme. + +This method was first formulated by \cite{m77} and \cite{sommeria_deardorff_1977} +for large-eddy simulations. It can also be applied +to larger scale models. It allows us to calculate vapour and liquid +contents and liquid cloud fraction from knowledge only of the combined +vapour+liquid content, $\overline{q_T}$, and the liquid temperature, +$\overline{T_L}$. These variables are unchanged during condensation +processes, so it is useful to write the cloud scheme in terms of these +variables. + +In this derivation we will consider only liquid condensate. +We assume, as above, that, +locally, the water content in a cloud is such as to remove any +supersaturation. This gives the equation + +\begin{equation} +q_{cl} = q_T - q_{sat}(T,p) +\label{eq:basic_qcl} +\end{equation} + +assuming that $q_T > q_{sat}(T,p)$ ($q_{cl}$ will be zero otherwise). +$q_T$ is the local total water +content, equal to the sum of the condensate $(q_{cl})$ plus the vapour +$(q)$, $T$ is the temperature, $p$ is the pressure and $q_{sat}(T,p)$ +is the saturation specific humidity at temperature T and pressure +p \textit{with respect to liquid water}. (Many earlier diagnostic +cloud schemes use a similar instantaneous condensation assumption +for ice, which would mean that $q_{sat}$ must be taken with respect +to ice when $T < 0 ^{\circ} C$, but the Unified Model does not). +We now introduce the liquid temperature ($T_L$), where $T_L$ is given by + +\begin{equation} +T_L = T - \frac{L}{c_p} q_{cl} , +\label{eq:tl} +\end{equation} + +and $L$ is the latent heat of +vaporization and $c_p$ is the heat capacity of air. +Note that $T_L$ is unaffected by changes of phase between vapour and liquid. +We now write (\ref{eq:basic_qcl}) as an \textit{equality} + +\begin{equation} +q_{cl} = q_T - \left( q_{sat}(T_L,p) + \alpha (T - T_L) \right) +\label{eq:alpha_t_tl} +\end{equation} + +where + +\begin{equation} +\alpha = \frac{ q_{sat}(T,p) - q_{sat}(T_L,p) }{T - T_L } . +\label{eq:alpha} +\end{equation} + +Using (\ref{eq:tl}) in (\ref{eq:alpha_t_tl}) gives the expression + +\begin{equation} +q_{cl} = q_T - q_{sat} (T_L) - \alpha \frac{L}{c_p} q_{cl} +\end{equation} + +or + +\begin{equation} +q_{cl} = a_L \left( q_T - q_{sat}(T_L,p) \right) +\label{eq:l_eq_al} +\end{equation} + +where $a_L$ is given by + +\begin{equation} +a_L = \left( 1 + \alpha \frac{L}{c_p} \right) ^{-1} . +\label{eq:a_L} +\end{equation} + +Thus (\ref{eq:basic_qcl}) has been rewritten \textit{exactly} in terms of the conserved +variables, $q_T$ and $T_L$, although the temperature, $T$, does remain in the +definition of $a_L$. +We will need to consider variations across a gridbox for a +parametrization scheme, +so we expand the expression for condensate (\ref{eq:l_eq_al}) into terms +relating to the gridbox mean and variation from the gridbox mean. + +\begin{equation} +q_{cl} = \overline{ a_L \left( q_T - q_{sat}(T_L,p) \right)} ++ [ a_L \left( q_T - q_{sat}(T_L,p) \right) ]' +\label{eq:bar_plus_pri1} +\end{equation} + +where $\overline{\phi}$ represents the mean of a distibution of $\phi$ and +$\phi = \overline{\phi} + {\phi}'$. The expression (\ref{eq:bar_plus_pri1}) +is \textit{exact} when +using the definition of $\alpha$ given in (\ref{eq:alpha}). + +The idea of a PDF scheme is to calculate the first (mean) term, +$\overline{\phi}$, from the known gridbox mean parameters, $q_T$, $T_L$ +and $p$, and to parametrize the distribution of the second, +variable term, ${\phi}'$. Unfortunately, the mean term is difficult +to write in terms of the gridbox mean variables $\overline{q_T}$ +and $\overline{T_L}$ because +$q_{sat}(T_L,p)$ is not a linear function of $T_L$ (or of $p$). In order to +proceed, we will now make an \textit{approximation} that $q_{sat}(T,p)$ +is a linear function of $T_L$ and $p$. +This equivalently implies that $a_L$ and $\alpha$ are approximated as +being constant across the gridbox. The expression +now becomes more tractable, (\ref{eq:bar_plus_pri1}) becoming: + +\begin{equation} +q_{cl} = a_L \left( \overline{q_T} - q_{sat}(\overline{T_L},\overline{p}) +\right) + a_L \left( {q_T}' - \alpha {T_L}' - \beta {p}' \right) +\label{eq:l_eq_bar_plus_pri} +\end{equation} + +where $\beta = {\frac{\partial q_{sat}}{\partial p}}$ at constant +temperature. The first term is connected with the mean properties of +the gridbox, and is written as $Q_c$, the second term is connected with the +deviation of the local conditions from the mean and is written as +$s$. + +\begin{equation} +Q_c = a_L \left( \overline{q_T} - q_{sat}(\overline{T_L},\overline{p}) \right) +\label{eq:qc_eq_qt-qs} +\end{equation} + +\begin{equation} +s = a_L \left( {q_T}' - \alpha {T_L}' - \beta {p}' \right) +\label{eq:s} +\end{equation} + +This gives the equation + +\begin{equation} +q_{cl} = Q_c + s +\label{eq:l_qc_s} +\end{equation} + +with the assumption that $s \ge -Q_c$ (i.e. $q_{cl} \ge 0$). +If $s < -Q_c$ then $q_{cl} = 0$. The term $a_L$ can be calculated +using (\ref{eq:a_L}) +from (\ref{eq:alpha}) with gridbox mean temperatures, i.e. + +\begin{equation} +\alpha = \frac{ q_{sat}(\overline{T},\overline{p}) +- q_{sat}(\overline{T_L},\overline{p}) }{\overline{T} - +\overline{T_L} } . +\label{eq:alpha_mean} +\end{equation} + +This definition of $\alpha$ and $a_L$ will retrieve an \textit{exact} value for +the gridbox mean $\overline{q_{cl}}$ \textit{if} +the distribution is monodispersed. Hence it is the sensible form to use for a +purely diagnostic representation such as \cite{smith90} where we explicitly +consider distributions of $s$. Strictly, the linear approximation +implies that other +approximations for $\alpha$ are valid: PC2 will do this +(see section \ref{sec:homog_num_app}) since we are concerned in PC2 with +the best estimate of the \textit{changes} to $\overline{q_{cl}}$, not the best +estimate of $\overline{q_{cl}}$ itself. + +We now assume that within +any particular gridbox a distribution $G$ of $s$ occurs (with mean, by +definition, of zero). Considering cloud to be where the water content +is greater than zero (i.e. where $s > -Q_c$) gives an expression for the +liquid cloud \textit{volume} fraction, $C_l$, within the gridbox as + +\begin{equation} +C_l = \int_{s=-Q_c}^{\infty} G(s) ds +\label{eq:int_gs_ds} +\end{equation} + +and the expression for mean condensate, ${\overline{q_{cl}}}$, using +(\ref{eq:l_qc_s}) to expand $q_{cl}$, is + +\begin{equation} +\overline{q_{cl}} = \int_{s=-Q_c}^{\infty} (Q_c + s) G(s) ds . +\label{eq:qclbar=int} +\end{equation} + +If we know (parametrize) the PDF given by $G(s)$ then we can solve +for $C_l$ and $\overline{q_{cl}}$. Note that this distribution is in terms of +$s$, there is no need to know the three-dimensional distribution in terms +of three separate variables $q_T$, $T_L$ and $p$. This is the method used +by \cite{smith90}, where a +symmetric triangular distribution function is used. For further +information on the \cite{smith90} scheme, please refer to \citeumdp{029}. Physics and dynamics schemes hence only need to +provide increments to $\overline{q_T}$ and $\overline{T_L}$, +provided that a diagnostic scheme (such as \cite{smith90}) is called at some point in the +timestep to partition $\overline{q_T}$ into $\overline{q}$ and $\overline{q_{cl}}$, +to calculate the dry bulb temperature $\overline{T}$ (from $\overline{T_L}$ +and $\overline{q_{cl}}$) and to calculate the liquid +cloud fraction, $C_l$. The diagnostic scheme effectively allows a calculation of +condensation associated with any physical process. However, its results remain +tied to the distribution of $G(s)$ that is chosen in (\ref{eq:int_gs_ds}) and +(\ref{eq:qclbar=int}) and it is this tie that we seek to break by the use of +a prognostic scheme. + +\subsection{Concept of PC2} + +The PC2 scheme develops prognostic expressions for the rates of change of +cloud fraction and condensate contents as a result of each process that acts in +the model. We consider ice and liquid condensate as two distinct +aspects of clouds, which may or may not overlap +Figure \ref{fig:schematic} provides a schematic summary of the PC2 scheme. +The equations for the five prognostic cloud variables can be written schematically: + +\begin{eqnarray} +\frac{\partial \overline{q_{cl}}}{\partial t} = +\frac{\partial \overline{q_{cl}}}{\partial t} |_{advection} + +\frac{\partial \overline{q_{cl}}}{\partial t} |_{convection} + +\frac{\partial \overline{q_{cl}}}{\partial t} |_{boundary \, layer} + +\frac{\partial \overline{q_{cl}}}{\partial t} |_{precipitation} + ... \nonumber \\ +\frac{\partial \overline{q_{cf}}}{\partial t} = +\frac{\partial \overline{q_{cf}}}{\partial t} |_{advection} + +\frac{\partial \overline{q_{cf}}}{\partial t} |_{convection} + +\frac{\partial \overline{q_{cf}}}{\partial t} |_{boundary \, layer} + +\frac{\partial \overline{q_{cf}}}{\partial t} |_{precipitation} + ... \nonumber \\ +\frac{\partial C_l}{\partial t} = +\frac{\partial C_l}{\partial t} |_{advection} + +\frac{\partial C_l}{\partial t} |_{convection} + +\frac{\partial C_l}{\partial t} |_{boundary \, layer} + +\frac{\partial C_l}{\partial t} |_{precipitation} + ... \nonumber \\ +\frac{\partial C_i}{\partial t} = +\frac{\partial C_i}{\partial t} |_{advection} + +\frac{\partial C_i}{\partial t} |_{convection} + +\frac{\partial C_i}{\partial t} |_{boundary \, layer} + +\frac{\partial C_i}{\partial t} |_{precipitation} + ... \nonumber \\ +\frac{\partial C_t}{\partial t} = +\frac{\partial C_t}{\partial t} |_{advection} + +\frac{\partial C_t}{\partial t} |_{convection} + +\frac{\partial C_t}{\partial t} |_{boundary \, layer} + +\frac{\partial C_t}{\partial t} |_{precipitation} + ... , +\label{eq:dqcldt_and_dcdt} +\end{eqnarray} + +where $\overline{q_{cf}}$ is the ice water specfic humidity, $C_l$ is the liquid +cloud \textit{volume} fraction, $C_i$ is the ice cloud volume fraction, and $C_t$ +is the combined ice or liquid cloud volume fraction. The amount of mixed +phase cloud, $C_{mp}$, can be calculated by the overlap of the ice +and liquid fractions: + +\begin{equation} +C_{mp} = C_i + C_l - C_t. +\label{eq:mp} +\end{equation} + +The idea is to parametrize each of the terms in the above equations. This +approach removes the diagnostic method, hence it will be critical that +we can write expressions for $\frac{\partial \overline{q_{cl}}}{\partial t}$ +and +$\frac{\partial C_l}{\partial t}$ for \textit{each process that alters +$\overline{T}$, $\overline{p}$, +$\overline{q}$, or $\overline{q_{cl}}$ in the model} (and similarly +for the ice terms). In doing so, +we will not lose sight of underlying PDF approach given by +(\ref{eq:int_gs_ds}) and (\ref{eq:qclbar=int}) since we will still use +the concept of +instantaneous condensation for liquid clouds. Equations +\ref{eq:int_gs_ds} and \ref{eq:qclbar=int} will form the +basis of the homogeneous forcing methods discussed in section +\ref{sec:homog}. +We note in particular that the convective cloud fraction, +previously a quantity that is diagnosed separately from the +large-scale cloud fraction calculated by the \cite{smith90} scheme, +may, in PC2, be included as part of the large-scale +cloud fraction. This aspect is similar to the \cite{t93} approach. + +The final aim of PC2 is +that the parametrization of each term in (\ref{eq:dqcldt_and_dcdt}) +is performed by each part of the model that alters $\overline{T}$, +$\overline{p}$, +$\overline{q}$, $\overline{q_{cl}}$ or $\overline{q_{cf}}$ as an +integral part of that physics or dynamics scheme. +However, in this PC2 scheme we acknowledge that this will not be possible, +at least, not to begin with. +Hence we have specifically developed generic approaches +that can be used to calculate expressions for +$\frac{\partial \overline{q_{cl}}}{\partial t}$ and $\frac{\partial C_l} +{\partial t}$ . +These are referred to as Homogeneous forcing (section \ref{sec:homog}), +Injection source (or inhomogeneous forcing, section \ref{sec:inhomog}) +and Width Changing (section \ref{sec:width}). Two additional modules +are available to assist with PC2, liquid cloud initiaion (section +\ref{sec:init}) and the calculation of total cloud fraction changes +(section \ref{sec:ct}). At +the present time, only the large-scale precipitation (section +\ref{sec:precip}) scheme has been rewritten fully to use the PC2 +concept of prognostic cloud fractions. The existing mass-flux +convection scheme has been modified to enable calculation of the detrained +condensate, but direct modification to the cloud fraction is not +included. All other physics schemes use one of the generic +approaches below. + +\subsubsection{A note on convective cloud fraction} + +It was the original intention that PC2 be able to replace the two +separate diagnostic cloud fractions (large-scale and convective) with +a single cloud fraction, as in \cite{t93}. The hypothesis was that +by detraining cloud +directly from the convection scheme we would no longer need a separate +representation of this cloud type. Our experience with PC2 is that +this is not necessarily the case. We suspect that the basic reason is +that we are unable to truely represent the extreme PDF shapes that +result from convective activity. Additionally, we only create cloud +associated with the detrainment part of the convection scheme, assuming +that cloud associated with the active updraughts in convection is small. +This assumption is not necessarily applicable. Similar arguments, +and model results, come from analysis of the \cite{t93} and +\cite{t02} scheme (Ben Johnson, personal communication). We also note +that with two cloud fraction types and two different optical depths +it is possible to have a basic degree of representation of cloud inhomogeneity. + +Hence the code still exists to enable PC2 to be +run with or without a diagnostic convective cloud fraction, although PC2:66 +does not include a diagnostic term. More details are +in section \ref{sec:convec}. + +\section{Physical basis of the PC2 prognostic cloud scheme} + +In this section we will develop the physical models that PC2 uses in order +to calculate its prognostic increment terms. We will also consider the +numerical solution of the models. The way in which these are incorporated +into the Unifed Model will be discussed in section \ref{sec:um} + +\subsection{Instantaneous condensation} + +Liquid clouds in PC2 use the concept of instantaneous condensation. Hence the +`s' distribution methods are fully applicable to the development of the +equations that govern the parametrization of liquid cloud in PC2. We will +start by looking at changes to $\overline{q_{cl}}$ and $C_l$ when a +uniform forcing +is applied to a gridbox, under the assumption of instantaneous condensation. + +\subsection{Homogeneous forcing} +\label{sec:homog} + +We define the expression +\textit{uniform forcing} (or \textit{homogeneous forcing}) to refer to +changes in local values of $T_L$ +and $q_T$ that occur at a rate independent of the part of the +gridbox in which they are located. This implies that $G(s)$ will not alter +due to such a process. +Uniform forcing simply alters $Q_c$ in (\ref{eq:int_gs_ds}) and +(\ref{eq:qclbar=int}). +In the Unified Model, this concept will be applied to several different sets of +physics increments in order to calculate the condensation and cloud fraction +changes associated with each one, where the physics routine does not allow +the explicit calculation of condensation and cloud fraction changes by +another method. +Large-scale ascent may be considered a meteorological example of such +a process. By differentiating (\ref{eq:int_gs_ds}) and (\ref{eq:qclbar=int}) +with respect to time, assuming uniform forcing +(so ${\frac{\partial G}{\partial t}}$ terms are zero), we obtain + +\begin{equation} +{{\frac{\partial C_l}{\partial t}} = G(-Q_c) {\frac{\partial Q_c} +{\partial t} }.} +\label{dcdt} +\end{equation} + +\begin{equation} +{\frac{\partial \overline{q_{cl}}}{\partial t}} = +C_l {\frac{\partial Q_c}{\partial t}} +\label{dqcldt} +\end{equation} + +The quantity $G(-Q_c)$ is the value of the PDF +of $G$ at $s=-Q_c$, which defines the boundary between +the saturated and unsaturated parts of the distribution. + +If we wish to consider a prognostic cloud scheme with equations for +the rate of change of condensate and cloud fraction based upon (\ref{dqcldt}) +and (\ref{dcdt}) then we need to close (\ref{dcdt}) by specifying the +value of $G(-Q_c)$. We will choose to develop a parametrization for this +quantity based upon the quantities $C_l$, $\overline{q_{cl}}$ and the saturation +deficit, $SD$, rather than tie $G(-Q_c)$ to a process. +The saturation deficit is \textit{defined} here in the `s' +framework to be the first moment of the PDF for `s' values less than $-Q_c$. +In this way it is analogous to the liquid water content, $\overline{q_{cl}}$. +Appendix A of \cite{wg03} writes this \textit{definition} as + +\begin{equation} +{SD = - \int_{-\infty}^{-Q_c} {( s+Q_c ) G(s) ds}} +\label{SD} +\end{equation} + +and shows this is equivalent to + +\begin{equation} +{SD = a_L ( q_{sat}({\overline{T}},{\overline{p}}) - {\overline{q}} ) .} +\label{SD2} +\end{equation} + +The basis behind the parametrization for $G(-Q_c)$ is to consider +an underlying form of the distribution $G(s)$ near the $+b_s$ and +$-b_s$ ends. We borrow the notation of \cite{smith90} and refer +to a quantity $b_s$ that is the value of $s$ when a monomodal +distribution $G(s)$ just equals zero. We suppose that the +distribution G can be described as a power law near $s=b_s$. + +\begin{equation} +G(s) ~ \propto ~ {(-s + b_s)}^n +\label{eqn19} +\end{equation} + +provided $s 1$ then the distribution is narrowed. +For the liquid cloud fraction we therefore have + +\begin{equation} +C_l^{[n+1]} = \int_{s=-Q_c}^{\infty} \xi G(\xi s) ds . +\label{eq:c_l_xi} +\end{equation} + +If we transform variables to $s' = \xi s$ we can rewrite this integral as + +\begin{equation} +C_l^{[n+1]} = \int_{s'=-Q_c \xi}^{\infty} G(s') ds' . +\label{eq:c_l_xi2} +\end{equation} + +Hence the expression for $C_l^{[n+1]}$ is equivalent to using the same +distribution function $G(s)$ as for $C_l^{[n]}$ except that the saturation +boundary has been moved from $-Q_c$ to $-Q_c \xi$. The result is the same as applying +a homogeneous forcing (\ref{eq:deltac}) with a modified forcing, + +\begin{equation} +\Delta Q_c \equiv \xi Q_c - Q_c , +\label{eq:deltac_modified} +\end{equation} + +or the continuous version + +\begin{equation} +\frac{\partial Q_c}{\partial t} \equiv +Q_c \frac{\partial}{\partial t}(\xi - 1) . +\label{eq:xi_equiv} +\end{equation} + +We can write $\xi$ in a slightly more +informative way by linking it to the relative change in width of the PDF +$\frac{1}{b_s} \frac{\partial b_s}{\partial t}$. For a PDF that changes +its width, $\xi$ is defined as + +\begin{equation} +\xi = \frac{b_s}{b_s + \delta b_s} = \frac{1}{1 + \frac{\delta b_s}{b_s}}. +\label{eq:xi_equiv1} +\end{equation} + + +For an infintessimal timestep $\delta t$ we therefore have + +\begin{equation} +\xi = \frac{1}{1 + \frac{1}{b_s} \frac{\partial b_s}{\partial t} \delta t } +\label{eq:xi} +\end{equation} + +and hence, by expanding (\ref{eq:xi}) to give $\xi = 1 - \frac{1}{b_s} +\frac{\partial b_s}{\partial t} \delta t$ and using the homogeneous +forcing expression (\ref{dcdt}) with the modified forcing (\ref{eq:xi_equiv}), +we retrieve the continuous form + +\begin{equation} +\frac{\partial C_l}{\partial t} = - G(-Q_c) Q_c \frac{1}{b_s} +\frac{\partial b_s}{\partial t} . +\label{eq:dcdt_width} +\end{equation} + +A similar analysis can be performed for $\frac{\partial \overline{q_{cl}}} +{\partial t}$ from (\ref{dqcldt}) to give + +\begin{equation} +\overline{q_{cl}}^{[n+1]} = \frac{1}{\xi} \int_{s'=-Q_c \xi}^{\infty} +(- \xi Q_c + s') G(s') ds' . +\label{eq:qcl_xi2} +\end{equation} + +Again, this is equivalent to using the homogeneous forcing with the modified +forcing (\ref{eq:xi_equiv}), but it also includes a scaling +term $\frac{1}{\xi}$. In the infinitessimal limit, this scaling gives +a second term that is proportional to the value of the integral (i.e. +$\overline{q_{cl}}$). Hence we obtain the final continuous solution + +\begin{equation} +\frac{\partial \overline{q_{cl}}}{\partial t} = +(- C_l Q_c+\overline{q_{cl}}) \frac{1}{b_s} \frac{\partial b_s}{\partial t} . +\label{eq:dqcldt_width} +\end{equation} + +To close the solution, we need to parametrize $\frac{1}{b_s} +\frac{\partial b_s}{\partial t}$ , +which could be linked to the physics of the process that is occuring. Note +we don't need to calculate $b_s$ separately, just its \textit{fractional} +rate of change. Options for the parameterisation of +$\frac{1}{b_s}\frac{\partial b_s}{\partial t}$ +due to turbulent ``erosion'' are described in section \ref{sec:turb}, +along with the numerical methods used to integrate the equations. + + +\subsection{Initiation of cloud} +\label{sec:init} +In section \ref{sec:homog} we commented that the closure (\ref{eqn22}) for $G(-Qc)$ +is only valid if $C_l$ is not identically 0 or 1. If $C_l$ +is 0 or 1 we know that $G(-Q_c)$ is equal to 0 but we have lost the +information that will tell us when $G(-Q_c + \Delta Q_c)$ starts +differing from 0. Hence the homogeneous forcing equation set +(\ref{dcdt}), (\ref{dqcldt}) and (\ref{eqn22}) is not complete if +we start from a position where $C_l$ is 0 or 1. To complete this set, +we will need to define a width, $b_s$, to the PDF and provide an initiation +increment to $C_l$ and $\overline{q_{cl}}$ when the value of $-Q_c$ crosses +the limit of the distribution. There is more discussion in +\cite{wg03}. + +To initiate new partial cloud-cover (or new partial clear-sky), we essentially +call a diagnostic cloud scheme to initialise the prognostics +$C_l$ and $\overline{q_{cl}}$. +In the UM there is currently a choice of 2 different diagnostic cloud +schemes that can be used for this; either a version of the Smith scheme +(see UMDP 029), or the bimodal scheme (see UMDP 039). +These two options are described below... + +\subsubsection{Initiation using a ``Smith-like'' method} + +This option is selected by setting the UM namelist switch +{\bf i\_pc2\_init\_method = 1} (Smith). + +We will assume the same form of the PDF at its boundaries as is +assumed in the derivation of the $G(-Q_c)$ closure. For the high `$s$' end +of the PDF distribution we integrate the power law description in +(\ref{eqn19}) to obtain the expressions + +\begin{equation} +C_l = \frac{1}{2 b_s^{n+1}} (b_s + Q_c)^{n+1} , +\label{eq:initc} +\end{equation} + +\begin{equation} +\overline{q_{cl}} = \frac{1}{2 b_s^{n+1}} \frac{(b_s + Q_c)^{n+2}}{n+2} . +\label{eq:initqcl} +\end{equation} + +We now need to parametrize the PDF width $b_s$. Unlike the \cite{smith90} +scheme, this is the only +location in the PC2 cloud scheme where the width needs to be defined +for the liquid cloud (although see section \ref{sec:mp_depsub} for +a discussion of an equivalent width in the deposition / sublimation +relationship for ice cloud). We still choose to define $b_s$ in +terms of a critical relative humidity parameter, $RH_{crit}$. Like +the \cite{smith90} scheme (see \citeumdp{029}), +we define the value of $b_s$ as + +\begin{equation} +b_s = a_L (1 - RH_{crit}) q_{sat} (\overline{T_L}) . +\label{eq:bs} +\end{equation} + +Hence, if the parameter $n$ was the same in PC2 as the equivalent +in \cite{smith90}, the +initial creation of liquid cloud would follow precisely that diagnosed +by the \cite{smith90} scheme (assuming that the numerical implementation of +the calculation is the same). +Its subsequent behaviour in PC2, though, would be different, because +the subsequent physical processes that act are parametrized in different +ways. +Note: for some reason, the implementation in the UM uses a fixed value +of $n = 0$ (corresponding to a top-hat distribution) if a constant $RH_{crit}$ +profile is used, but instead sets $n = 1$ (a triangular distribution) +in the PC2 initiation calculation if a TKE-based variable $RH_{crit}$ is used. +In the latter case, $n = 0$ is still hardwired in the PC2 homogeneous forcing +calculations, so it is not handled consistently. + +An equivalent initiation scheme is required if $C_l$ is 1 and $Q_c$ +is being reduced - at some point we need to introduce clear sky into +the solution. Because we make the choice of symmetry (which could be +relaxed if we used different $RH_{crit}$ values for $C_l$ of 1 and $C_l$ of 0), +the problem is entirely equivalent to that of initiating +from $C_l = 0$, with the exception that $\overline{q_{cl}}$ is replaced by $SD$, +$C_l$ is replaced by $(1-C_l)$, and $Q_c$ is replaced by $-Q_c$. +We hence have the solution + +\begin{equation} +1 - C_l = \frac{1}{2 b_s^{n+1}} (b_s - Q_c)^{n+1} , +\label{eq:init1mc} +\end{equation} + +\begin{equation} +SD = \frac{1}{2 b_s^{n+1}} \frac{(b_s - Q_c)^{n+2}}{n+2} . +\label{eq:initSD} +\end{equation} + +The conversion between $SD$ and $\overline{q_{cl}}$ follows (\ref{SD2}). +We will choose, +as we do throughout PC2, to define $\alpha$ (and hence $a_L$) in terms of +$\frac{\partial q_{sat}(\overline{T})}{\partial t}$, although within +this diagnostic calculation of SD it might actually be better to use +the representation (\ref{eq:alpha}) used by the diagnostic \cite{smith90} scheme. + +\subsubsection{Numerical Application of the Smith method} +\label{sec:numapp_init} + +In order to calculate and compare the state of the model to $b_s$, +we first calculate $T_L$, $q_{sat}(\overline{T_L})$ and calculate +the mean relative total humidity, $RH_T$, where + +\begin{equation} +RH_T = \frac{ \overline{q} + \overline{q_{cl}} } {q_{sat}(\overline{T_L}) } . +\label{eq:rht} +\end{equation} + +We then assess whether initiation is required. There are only two +circumstances in which we wish to proceed further: +\begin{itemize} +\item{If the current cloud fraction $C_l$ is 0 and $-Q_c < b_s$. By dividing the second condition by $a_L q_{sat} (\overline{T_L})$ we see, using the definitions (\ref{eq:qc_eq_qt-qs}) and (\ref{eq:bs}), that this second condition is equivalent to $RH_T > RH_{crit}$.} +\item{If the current cloud fraction $C_l$ is 1 and $-Q_c > -b_s$ (or, equivalently, $RH_T < 2 - RH_{crit})$.} +\end{itemize} +Note: in the UM implementation, the actual conditions for when initiation +may occur are more complicated than this, and there are several options +depending on a namelist switch. See section \ref{sec:init2} for details... + +In the second case, we then make the temporary transformation of variables +in order to use the same solution set as in the first case: $C_l'$ takes the +value $(1-C_l)$ and $RH_t'$ takes the value $(2-RH_t)$ (which is equivalent +to the replacing of $Q_c$ by $-Q_c$). In the first case, $C_l'$ and $RH_t$ take +the same values as $C_l$ and $RH_t$ respectively. + +We then solve for the initiated cloud fraction $C_l'$, using the similar +methods as described in \citeumdp{029}, except +that we allow the solution to vary with the PDF shape $n$. We first +write $Q_N$ as + +\begin{equation} +Q_N = \frac{Q_c}{b_s} = \frac{ a_L (\overline{q_T} - q_{sat}(\overline{T_L})) } +{ a_L (1 - RH_{crit}) q_{sat} (\overline{T_L})} = \frac{RH_T - 1}{1-RH_{crit}} +\label{eq:qn_def} +\end{equation} + +and then use $Q_N$ to solve the initiated cloud fraction. We assume a PDF +described by a power law as in (\ref{eqn19}) (and the equivalent for the +other end of the distribution, the two expressions switching at $Q_c=0$), +which is normalized. The solution to (\ref{eq:int_gs_ds}) is hence + +\begin{equation} +C_l^{init'} = \left\{ \begin{array}{ll} + 0, & Q_N \le -1 \\ + \frac{1}{2} {\left( 1 + Q_N \right)}^{n+1}, & -1 < Q_N \le 0 \\ + 1 - \frac{1}{2} {\left( 1 - Q_N \right)}^{n+1}, & 0 < Q_N < 1 \\ + 1, & 1 \le Q_N . + \end{array} \right. +\label{eq:c_qn} +\end{equation} + +where $C_l^{init'}$ is the initiated value of liquid cloud fraction. +If we had performed the variable transformation we then we need to +transform back, so $C_l^{init} = 1 - C_l^{init'}$, otherwise +$C_l^{init} = C_l^{init'}$. + +In practice, it is likely to be only the second of the +options in (\ref{eq:c_qn}) that the scheme uses, since we will be at +that end of the distribution function, unless previous +parts of the model timestep have resulted in large +forcings to $Q_c$. + +The solution for the initiated liquid water, $\overline{q_{cl}}^{init}$ is +more difficult, since it depends on the width of the distribution $b_s$, +hence on $a_L$ and $\alpha$, and $\alpha$ is a function of the dry-bulb +temperature $\overline{T}$, which is not known until we know the +amount of condensation. +Hence we will need to iterate to a solution. + +We first calculate $q_{sat}(\overline{T})$, $\alpha$, $a_L$ and $b_s$, using +(\ref{eq:alpha_exp}), (\ref{eq:a_L}) and (\ref{eq:bs}). We then solve for the liquid +water content: + +\begin{equation} +\frac{\overline{q_{cl}}^{init'}}{b_s} = \left\{ \begin{array}{ll} + 0, & Q_N \le -1 \\ + \frac{1}{2 (n+2)} {\left( 1 + Q_N \right)}^{n+2}, & -1 < Q_N \le 0 \\ + Q_N + \frac{1}{2 (n+2)} {\left( 1 - Q_N \right)}^{n+2}, & 0 < Q_N < 1 \\ + Q_N, & 1 \le Q_N . + \end{array} \right. +\label{eq:l_bar} +\end{equation} + +If we have been working in transformed variables we now transform +back, so the initiated saturation deficit, $SD^{init}$, takes the +value of $\overline{q_{cl}}^{init'}$. We then use (\ref{SD2}) to +estimate $\overline{q_{cl}}^{init}$ using our initial estimates of +$q_{sat}(\overline{T})$ and $a_L$. If we are not in transformed +variables, we have the first estimate +$\overline{q_{cl}}^{init}=\overline{q_{cl}}^{init'}$. + +We now use this estimate of $\overline{q_{cl}}^{init}$ to +calculate a more accurate estimate of $a_L$ etc. by iteration. In order to +achieve a faster convergence of the iteration, we do not use +(\ref{eq:l_bar}) directly in the estimation of $a_L$ etc., but +use a combination of this value and the one from the previous iteration. + +\begin{equation} +\overline{q_{cl}}^{init~[i+1]} = f \overline{q_{cl}}^{init~[i]} + +(1 - f) \overline{q_{cl}}^{init~[i-1]} +\label{eq:iter} +\end{equation} + +where the superscript $[i]$ labels each iteration. We find that 10 +iterations is effective for convergence, with the weighting +$f$ given by $a_L^{[i]}$. + +\subsubsection{Initiation using the bimodal scheme} +\label{sec:bimodal_init} + +This option is selected by setting the UM namelist switch +{\bf i\_pc2\_init\_method = 2} (Bimodal). + +First, the diagnosis of entrainment zones is performed at all grid-points, +as described in UMDP 039. The parameters of the moisture PDF are then +constructed, assuming either a sum of two Gaussian modes from the top +and bottom of an inversion layer (when within an entrainment zone), +or a single symmetric Gaussian mode (when not in an entrainment zone). +The variance of each Gaussian mode is estimated based on the TKE and other +information output by the boundary-layer scheme +(with a minimum limit applied to the PDF width, consistent with +$RH_{crit}$ = 99\%. +Crucially, each Gaussian mode is truncated to zero at plus and minus +3 standard deviations; this sets the overall width of the moisture +PDF at each point. + +The positions of the upper and lower truncated bounds of the moisture PDF +relative to the saturation threshold are expressed in terms of a normalised +$Q_N$ = $Q_c$ over PDF-width (see equation \ref{eq:qn_def}). +In entrainment zones, the sum of the two Gaussian modes can lead to +a highly skewed distribution; hence $Q_N$ can have different values for the +upper and lower bounds, each normalised by the different widths on +either side of the PDF. The upper and lower values of $Q_N$ are then +compared to -1 and 1 respectively, to determine whether the saturation +boundary lies within the PDF bounds. This is the basic condition for +initiation to occur (though there are additional conditions and various +options for these in the soure code; see section \ref{sec:init2}). + +If the initiation conditions are met, the diagnostic bimodal cloud scheme code +is then called (see UMDP 039), and the diagnosed $C_l$ and $q_{cl}$ are used to +set the prognostic $C_l$ and $q_{cl}$. + +\subsection{Injection forcing} +\label{sec:inhomog} + +Injection forcing (sometimes referred to as inhomogeneous forcing) +uses another concept of how the underlying moisture PDF may change in +order to calculate a change in cloud fraction as a result of a known +injection of condensate into a gridbox. The term was developed in +order to be coupled with a modified mass-flux convection scheme, +but is first presented here in its basic form. + +We will assume a physical model whereby saturated air, containing +condensate, randomly replaces already existing air in the gridbox. +(Such a formulation is designed to represent air detrained from convection +replacing pre-existing air when averaged over a large horizontal domain). +\cite{bwg03} discusses the situation in more detail. Briefly, +we consider two parts to the distribution function $G(s)$. One part +represents the background air. This maintains its PDF shape (in terms +of absolute $q_T$ and $T_L$ values) because we assume it is \textit{randomly} +replaced, but will reduce in amplitude as it is replaced by a +second PDF representing the injected air. + +The fractional rate at which existing air +is replaced by the injected source air we will write as +$\frac{\partial{C_S}}{\partial{t}}$. Provided that only the liquid +phase exists (see section \ref{sec:multiple} for the extention to multiple phases), +we then note that the rate +of change of liquid cloud fraction and liquid water content in +the gridbox can be written in two parts: firstly the change +due to the background, and secondly the change due to the source. + +\begin{equation} +\frac{\partial{C_l}}{\partial{t}} = +- C_l \frac{\partial{C_S}}{\partial{t}} + \frac{\partial{C_S}}{\partial{t}} +\label{eq:dcdt_inhom} +\end{equation} + +\begin{equation} +\frac{\partial{\overline{q_{cl}}}}{\partial{t}} = +- \overline{q_{cl}} \frac{\partial{C_S}}{\partial{t}} ++ q_{cl}^S \frac{\partial{C_S}}{\partial{t}} +\label{eq:dqcldt_inhom} +\end{equation} + +where $q_{cl}^S$ is the liquid water content of the injected +air. Eliminating $\frac{\partial{C_S}}{\partial{t}}$ gives the +relationship + +\begin{equation} +\frac{\partial{C_l}}{\partial{t}} = \frac{1 - C_l}{q_{cl}^S +- \overline{q_{cl}}} Q4_l, +\label{eq:dcdt_inhom2} +\end{equation} + +where $Q4_l$ is the net (\textit{including} the liquid water +in the background distribution +that was randomally replaced) injection source change of $\overline{q_{cl}}$: + +\begin{equation} +Q4_l = \frac{\partial{\overline{q_{cl}}}}{\partial{t}} |_{injection \, source}. +\label{eq:q4} +\end{equation} + +We see that we do not need to know anything about the nature of +the two PDFs involved, except the assumption that the injected +PDF contains completely cloudy air. +This equation allows one to calculate the change in $C_l$ associated +with an injection source change of $\overline{q_{cl}}$ for the example of +convection. Modifications +to the mass-flux convection scheme for PC2 (far from trivial and discussed +in depth in section \ref{sec:convec}) +allow $Q4_l$ to be calculated ($q_{cl}^S$ is already available), +and (\ref{eq:dcdt_inhom2}) can then be used +to calculate the equivalent $C_l$ change. We note at this stage +that the denominator in (\ref{eq:dcdt_inhom2}), being the difference +in two terms that may be close to each other, may cause problems +when we attempt to numerically apply this equation. + +It is reasonable to ask what happens to the air in the distribution that +was replaced. In this mathematical representation of a single gridbox we +need not know anything other than that the air is displaced into a neighbouring +gridbox. In practical use with a mass-flux convection scheme we know more that +this air is displaced downwards in the column. We could reasonably calculate +the change in cloud fraction following the same methods as used to calculate +the change in $\overline{q}$ or the change in a tracer and we discuss this +later. + +\subsubsection{Multiple phases in the injection source} +\label{sec:multiple} + +The injection source formulation can be extended to multiple +phases of condensate. In practice, this will simply be the +two phases ice and liquid, although we need to recognize that +they can overlap with each other. \cite{wilson2001} provides the +background to the derivation and it is briefly presented below. + +We firstly rewrite (\ref{eq:dqcldt_inhom}) but use the net +condensate ($\overline{q_c} = \overline{q_{cl}} + \overline{q_{cf}}$) instead of just the +liquid water expression, and the net cloud amount $C_t$, instead +of the liquid cloud amount $C_l$. The same argument as before leads +to the expressions + +\begin{equation} +\frac{\partial{C_t}}{\partial{t}} = +- C_t \frac{\partial{C_S}}{\partial{t}} + \frac{\partial{C_S}}{\partial{t}} +\label{eq:dctdt_inhom} +\end{equation} + +and + +\begin{equation} +\frac{\partial{\overline{q_{c}}}}{\partial{t}} = +- \overline{q_{c}} \frac{\partial{C_S}}{\partial{t}} ++ q_{c}^S \frac{\partial{C_S}}{\partial{t}} . +\label{eq:dqcdt_inhom} +\end{equation} + +The left hand side of (\ref{eq:dqcdt_inhom}) is written as $Q4_c$. +$q_{c}^S$ is the in-cloud +condensate content (ice plus liquid) of the source. + +Hence eliminating $\frac{\partial{C_S}}{\partial{t}}$ we obtain + +\begin{equation} +\frac{\partial{C_t}}{\partial{t}} = \frac{(1-C_t)}{q_{c}^S - +\overline{q_{c}}} Q4_c . +\label{eq:dctdt_q4} +\end{equation} + +We will assume that the proportion of the injected volume that +contains liquid cloud can be written as $g_l$, and the proportion +that contains ice cloud can be written as $g_i$. Note that it +is not necessary to have $g_l + g_i = 1$ if there is mixed +phase cloud injected. We can write the change in \textit{liquid} cloud +fraction equivalently to (\ref{eq:dcdt_inhom}) as + +\begin{equation} +\frac{\partial{C_l}}{\partial{t}} = +- C_l \frac{\partial{C_S}}{\partial{t}} + g_l \frac{\partial{C_S}}{\partial{t}} . +\label{eq:dcldt_inhom} +\end{equation} + +Combining (\ref{eq:dcldt_inhom}) and (\ref{eq:dctdt_inhom}) by eliminating +$\frac{\partial{C_S}}{\partial{t}}$ gives + +\begin{equation} +\frac{\partial{C_l}}{\partial{t}} = \frac{g_l - C_l}{1 - C_t} +\frac{\partial{C_t}}{\partial{t}} +\label{eq:dctdt_dcdt} +\end{equation} + +and hence from (\ref{eq:dctdt_q4}) we have the result + +\begin{equation} +\frac{\partial{C_l}}{\partial{t}} = +\frac{g_l - C_l}{q_c^S - \overline{q_{c}}} Q4_c . +\label{eq:dcltdt_almost_final} +\end{equation} + +An equivalent expression holds for the ice cloud. Hence the change in the +amount of cloud for each phase may be calculated assuming we know +the volume proportions of the source term that contain each of +the phases and the net increase in the amount of condensate, $Q4_c$ +(regardless of phase). This expression is coded for use in +a generically available inhomogeneous forcing module. However, +we can also write this in a slightly more +accessible form by noting the ratio of (\ref{eq:dqcdt_inhom}) and +(\ref{eq:dqcldt_inhom}) with the $Q4$ definitions following (\ref{eq:q4}). + +\begin{equation} +\frac{Q4_c}{q_c^S - \overline{q_c}} = +\frac{Q4_l}{q_{cl}^S - \overline{q_{cl}}} . +\label{eq:q4_ratios} +\end{equation} + +Using (\ref{eq:q4_ratios}) in +(\ref{eq:dcltdt_almost_final}) gives the final expression + +\begin{equation} +\frac{\partial{C_l}}{\partial{t}} = +\frac{g_l - C_l}{q_{cl}^S - \overline{q_{cl}}} Q4_l +\label{eq:dctdt_final} +\end{equation} + +and similarly for the ice. Note that this expression accounts +for the possibility that liquid cloud is displaced from the +gridbox by added ice cloud. We can further write +$q_{cl}^S$ as a fraction of $q_{c}^S$ + +\begin{equation} +q_{cl}^S = h_l q_{c}^S +\label{eq:qcls_qcs} +\end{equation} + +where $h_l$ is the factor between them (i.e. the \textit{mass} fraction +of the injected condensate that is liquid). It is not necessary +in this theory to have $h_l$ equal to $g_l$: if a mixed +phase plume exists $g_l + g_i$ need not equal 1, but since +$h_l$ and its ice equivalent, $h_i$, refer to mass, $h_l + h_i$ +must equal 1. However, if we do not allow a mixed phase injection +(which is the case in the current mass-flux convection +scheme, where only one phase can be injected), $h_l$ and $g_l$ are equal (and either zero or one in the +current mass-flux convection scheme) and we can write (\ref{eq:dctdt_final}) as + +\begin{equation} +\frac{\partial{C_l}}{\partial{t}} = +\frac{ (\delta_{xl} - C_l) }{ \delta_{xl} q_{c}^S - \overline{q_{cl}} } +Q4_l +\label{eq:dctdt_xl} +\end{equation} + +where $\delta_{xl} = h_l = g_l$. Equivalent expressions exist for the +ice cloud fraction and total cloud fraction. + +\begin{equation} +\frac{\partial{C_i}}{\partial{t}} = +\frac{ (\delta_{xi} - C_l) }{ \delta_{xi} q_{c}^S - \overline{q_{cf}} } +Q4_i +\label{eq:dctdt_xi} +\end{equation} + +\begin{equation} +\frac{\partial{C_t}}{\partial{t}} = +\frac{ (1 - C_t) }{ q_{c}^S - \overline{q_{c}} } Q4_c +\label{eq:dctdt_xc} +\end{equation} + +with $\delta_{xi} = h_i = g_i$. These are the expressions that are used +within the convection scheme. It still remains to parametrize $\delta_{xl}$, +which is given by the convection scheme itself. This is discussed in +section \ref{sec:plume_phase}. + +\subsubsection{Numerical application} +\label{sec:multi_numapp} +The numerical application using (\ref{eq:dcltdt_almost_final}) may be performed +with a basic forward timestep. Each of the three cloud fractions can +be incremented, assuming we know $\Delta{\overline{q_{cl}}}$ and +$\Delta{\overline{q_{cf}}}$, as + +\begin{equation} +\Delta{C_t} = \frac{(1 - C_t)} {q_c^S - \overline{q_{cl}} - \overline{q_{cf}}} +( \Delta{\overline{q_{cl}}} + \Delta{\overline{q_{cf}}} ), +\label{eq:cft_ts} +\end{equation} + +\begin{equation} +\Delta{C_l} = \frac{ (g_l - C_l)} +{q_c^S - \overline{q_{cl}} - \overline{q_{cf}}} +( \Delta{\overline{q_{cl}}} + \Delta{\overline{q_{cf}}} ), +\label{eq:cfl_ts} +\end{equation} + +\begin{equation} +\Delta{C_i} = \frac{ (g_i - C_i)} +{q_c^S - \overline{q_{cl}} - \overline{q_{cf}}} +( \Delta{\overline{q_{cl}}} + \Delta{\overline{q_{cf}}} ). +\label{eq:cff_ts} +\end{equation} + +The application from within the convection scheme is slightly different. +We start with (\ref{eq:dctdt_xl}), but enforce two numerical restrictions +to avoid the equation set becoming ill-conditioned. Firstly, we limit +the denominator $q_c^S - \overline{q_{cl}}$ to a minimum value if we +are considering changes of the same phase as the injected source. + +\begin{equation} +\Delta C_l = \frac {\delta_{xl} - C_l} {\delta_{xl} \text{Max}( q_{c}^S +- \overline{q_{cl}} , q_c^{S0} ) + ( 1 - \delta_{xl} ) (-\overline{q_{cl}}) } +Q4_l +\label{eq:delta_cl} +\end{equation} + +where $q_c^{S0}$ is specified as $5 \times 10^{-5} kg \, kg^{-1}$. +The denominator also has an additional check. If its absolute +value is less than a tolerance value of $1 \times 10^{-10} kg \, kg^{-1}$ +then no change in cloud fraction will be considered. A similar +equation is used for the ice cloud and the change in total cloud fraction + + +\begin{equation} +\Delta C_i = \frac {\delta_{xi} - C_i} {\delta_{xi} \text{Max}( q_{c}^S +- \overline{q_{ci}} , q_c^{S0} ) + ( 1 - \delta_{xi} ) (-\overline{q_{cf}}) } +Q4_i +\label{eq:delta_ci} +\end{equation} + +\begin{equation} +\Delta C_t = \frac {1 - C_t} +{ \text{Max}(q_c^S - \overline{q_c} , q_c^{S0} ) } Q4_c . +\label{eq:delta_ct} +\end{equation} + +We now limit the change in cloud fraction to ensure that the cloud +fraction remains within its physical bounds. + +\begin{equation} +C_l^{[n+1]} = ( 0, C_l^{[n]} + \Delta C_l, 1) +\label{eq:delta_cl_conv_final} +\end{equation} + +and similar equations are used for $C_i^{[n+1]}$ and $C_t^{[n+1]}$. + +\subsubsection{A note on the implementation of the cloud fraction change} +\label{sec:conv_imp_note} + +Equation \ref{eq:dcdt_inhom2} has been derived assuming that the +only change in the cloud properties within the gridbox comes from +the detrainment of air from the convective plume (so that the injection +source is an appropriate model). Attention should be drawn to the fact +that this is not the only source of change from the convection scheme. +Two other terms require consideration, namely advection of the environmental +air downwards by compensating subsidence and the condensation resulting +from the adiabatic warming due to this subsidence. +The former is considered correctly in the calculation of +$\frac{\partial \overline{q_{cl}}}{\partial t}$, which corresponds to $Q4$. +However, the calculation of $\frac{\partial C_l}{\partial t}$ +is then performed using (\ref{eq:dcdt_inhom2}) and \textbf{incorrectly} assuming +that all the $\overline{q_{cl}}$ change comes from the detrainment. It is +possible to calculate directly the change in $C_l$ that should occur due to +the detrainment and compensating subsidence treated together, in the same way +that $\Delta \overline{q_{cl}}$ is calculated (see section +\ref{subsect:q4calculation}), and this is the way in which the cloud fraction +change \textbf{should} be done. +It is an unfortunate historical emphasis in the early development of PC2 +on the derivation of (\ref{eq:dcdt_inhom2}) that has led to the treatment +used within the Unified Model for the change in cloud fractions due to convection. + +The change in $\overline{q_{cl}}$ and $C_l$ due to the adiabatic warming +associated with the compensating subsidence is considered explicitly +in the model implementation (see +section \ref{sec:conv_homog}) for both $\overline{q_{cl}}$ and $C_l$ +after the rest of the convective process has been calculated. It is perhaps +arguable that if (\ref{eq:dcdt_inhom2}) is going to be applied then +the value of $Q4$ used in (\ref{eq:dcdt_inhom2}) should include this term. + +Any major future developments of PC2 for a mass-flux convection scheme would be +advised to consider whether it is appropriate to use (\ref{eq:dcdt_inhom2}) at +all. + +\subsection{Ice cloud and mixed phase regions} +\label{sec:ct} + +The homogeneous forcing, initiation and PC2 erosion sections described +above have only considered the generation and dissipation of liquid +clouds. Although the forcing methods will not influence the generation +and dissipation of ice cloud (which is primarily performed in the +large-scale precipitation scheme, section \ref{sec:precip}) we +are still left with the issue of how created or dissipated liquid +cloud overlaps with existing ice cloud in the gridbox. The +opposite situation, where changes in ice cloud are specified +and changes in the overlap with liquid cloud need to be calculated, +is also possible in PC2 (e.g. in the boundary layer, see +section \ref{sec:bl}). + +Here we +need a simple assumption to close the problem. The assumption +that we now choose is that liquid cloud fraction \textit{changes} are +\textit{minimally} overlapped with ice cloud fraction changes. +This choice is based upon observational evidence that mixed +phase cloud is relatively rare, and also on results from earlier PC2 +development that indicated less supercooled liquid water cloud than +is observed from ground-based lidar. + +With this assumption, the equation set becomes straightforward to +write down. We firstly consider that a change in liquid cloud fraction +$\Delta C_l$ is known and we wish to estimate the resulting change in +the total cloud fraction. There is, of course, no change in the ice +cloud fraction $C_i$, since, from our \textit{definitions} in +(\ref{eq:dqcldt_and_dcdt}) and (\ref{eq:mp}), this includes the mixed phase +contribution. Hence we write + +\begin{equation} +\Delta C_i = 0 . +\label{eq:deltaci_eq_0} +\end{equation} + +The change in the total cloud fraction, $C_t$ will depend upon +the sign of the change of the liquid cloud fraction. If +$\Delta C_l > 0$, then $\Delta C_t$ is going to be the same as $\Delta C_l$ +($C_l$ is being added with minimum overlap to $C_i$), unless the +gridbox becomes completely covered in cloud, when there is no +choice but to generate mixed phase cloud. Hence we have + +\begin{equation} +\Delta C_t = \text{Min} ( \Delta C_l , 1 - C_t ). +\label{eq:deltact_min} +\end{equation} + +If $\Delta C_l < 0$, then we still consider minimum overlap +of the \textit{changes} (this is so that the solution is reversible as +much as possible). Hence $\Delta C_t$ is going to be the same as +$\Delta C_l$ unless $C_l$ is reduced below the existing $C_i$, in +which case no more change to $C_t$ is possible. + +\begin{equation} +\Delta C_t = \text{Max} ( \Delta C_l , C_i - C_t ) , +\label{eq:deltact_min2} +\end{equation} + +remembering that both quantities in the maximum expression in +(\ref{eq:deltact_min2}) have negative values. + +We can write similar expressions if a known amount of ice cloud +is added or removed, and we need to calculate the effect on $C_t$. +Similar to the results above we have: + +\begin{equation} +\Delta C_l = 0 . +\label{eq:deltacl_eq_0} +\end{equation} + +and + +\begin{equation} +\Delta C_t = \left\{ \begin{array}{ll} + \text{Max} ( \Delta C_i , C_l - C_t ), & \Delta C_i < 0 \\ + \text{Min} ( \Delta C_i , 1 - C_t ), & \Delta C_i > 0 . + \end{array} \right. +\label{eq:deltact_min_array} +\end{equation} + +For completeness, we also present here the equation set for +random overlap of changes in liquid cloud with existing ice cloud. +We have, as before, + +\begin{equation} +\Delta C_i = 0 . +\label{eq:deltaci_eq_0_2} +\end{equation} + +For $\Delta C_l > 0$ additional liquid cloud is added +randomly to any location outside that of the current liquid +cloud. A proportion $\frac{1-C_t}{1-C_l}$ of this will be additionally +outside that of existing ice cloud. Hence the net change in +total cloud fraction can be written as + +\begin{equation} +\Delta C_t = \Delta C_l \frac{1 - C_t}{1 - C_l} . +\label{eq:deltact_ran1} +\end{equation} + +Similarly, if $\Delta C_l < 0$, the liquid cloud is removed +randomly from the existing liquid cloud. A proportion +$\frac{C_t - C_i}{C_l}$ of this is from liquid cloud that does not +overlap with existing ice cloud. Hence, + +\begin{equation} +\Delta C_t = \Delta C_l \frac{C_t - C_i}{C_l} . +\label{eq:deltact_ran2} +\end{equation} + +Equivalent equations to (\ref{eq:deltact_ran1}) and +(\ref{eq:deltact_ran2}) but with $C_l$ and $C_i$ swapped apply when +we need to estimate changes in $C_t$ from a known $\Delta C_i$, when +assuming random overlap. + +\subsubsection{Numerical Implementation} + +In general, although the situation does not occur within the current +implementation of PC2 , we might have increments to both $C_l$ and +$C_i$ simultaneously. Hence the implementation is to calculate +$\Delta C_t$ from the sum of that predicted by (\ref{eq:deltact_min}) +or (\ref{eq:deltact_min2}), and (\ref{eq:deltact_min_array}). +For the random overlap situation we also need to apply a check on +the denominator in (\ref{eq:deltact_ran1}) and (\ref{eq:deltact_ran2}) before +calculation, with the result set to the limit $\Delta C_t = 0$ +if the denominator is 0. For the minimum overlap situation a final check +is made that $C_t$ lies between 0 and 1, with the value being reset to +0 or 1 if not. + +\subsection{Forced convective cloud} + +Forced convective clouds are clouds that form at the top of a convective boundary layer +but are too shallow to reach their level of free convection (and become fully fledged +cumulus clouds). These clouds currently require special treatment because initiation +in PC2 uses the Smith scheme with a specified value of $RH_{crit}$ while the large $RH$ +variability associated with these clouds implies much lower values than are typically used. + +A profile of ``forced cloud fraction'', $C_{forced}$, is parametrized as +linearly varying with height between a cloud-base value, at the lifting +condensation level (LCL) from the convection diagnosis parcel ascent, and a cloud-top +value of 0.1 at the top of the capping inversion, $z_i^{top}$. The cloud-base +value of $C_{forced}$ varies linearly between 0.1 and 0.3 for cloud depths +between 100 m and 300 m based loosely on SGP ARM site observations \cite{zk13}. The +inversion top is taken to be the boundary layer depth, $z_h$ plus the inversion +thickness, $\Delta z_i$ parametrized following \cite{rb08} as: +\begin{equation} +\Delta z_i = 6.3 \, w_m^2 / \int_{z_h}^{z_h+\Delta z_i} b \, dz +\label{dz_param} +\end{equation} +where $w_m$ is the boundary layer velocity scale ($w_m^3 = u_*^3 + 0.25 w_*^3$) and $b$ is +the parcel buoyancy that is integrated over the depth of the inversion assuming a +piece-wise linear variation between grid-levels. Note that the constant in (\ref{dz_param}) +is the same as in \cite{rb08} because $6.3 = 2.5 * 4^{2/3}$ and $w_m^3$ differs by a factor of 4. + +The in-cloud water content at the top of the inversion is estimated using the water content +from the diagnostic parcel ascent (used to diagnose boundary layer type and trigger convection), +with linear interpolation used between the lifting condensation level and inversion +top. To allow for sub-adiabatic water content (due to lateral mixing or microphysical +processes) the in-cloud water content can be reduced by a factor, forced\_cu\_fac, that has been +set to 0.5 in GA7. + +These cloud fraction and water content profiles are then used as minimum values and +increments to $C$ and $\overline{q_{cl}}$ calculated if necessary. +This methodology can also optionally be applied to cloud layers diagnosed as cumulus, if the +boundary layer option to mix across the lifting condensation level is selected that generates +a cloud base transition zone thickness which is then treated analgously to the inversion +thickness above. + +Also, there is an option to treat the calculated forced cumulus cloud +fraction and water content as diagnostic quantities passed directly to +the radiation scheme as part of the ``convective'' cloud, instead of +using them to modify the prognostic ``large-scale'' cloud variables +$C$ and $\overline{q_{cl}}$. If this option is used, the convective +cloud fraction $CCA$ and water content $CCW$ output by the convection +scheme are updated, by taking the forced cumulus profiles as their +minimum allowed values. Note that only the convective cloud fields +passed to radiation are updated (i.e. the versions of $CCA$ and $CCW$ +that are stored in the model dump / D1 array). The UM code contains +other copies of the convective cloud fields that are only used for +diagnostics; these are {\em not} updated. + +The different options for how to treat forced cumulus cloud are +controlled by the cloud namelist input $forced\_cu$, and are +summarised below: + +\begin{itemize} +\item $forced\_cu = 0$: No treatment of forced cumulus clouds. +\item $forced\_cu = 1$: Forced cumulus cloud applied to +$C$ and $\overline{q_{cl}}$ only in dry-convective boundary-layers. +\item $forced\_cu = 2$: Forced cumulus cloud applied to +$C$ and $\overline{q_{cl}}$ in both dry-convective and +cumulus-capped boundary-layers. +\item $forced\_cu = 3$: Forced cumulus cloud applied to +$CCA$ and $CCW$ in both dry-convective and +cumulus-capped boundary-layers. +\end{itemize} + + +\subsection{Turbulence-driven production of subgrid scale liquid cloud} +\label{sec:turb_qcl_scheme} + +\subsubsection{Introduction}\label{sec:sgt_intro} + +\cite{fhfk14} developed a model for +subgrid liquid water production by turbulent motions. +Their method uses an exactly soluble +stochastic process to describe +subgrid relative humidity (RH) fluctuations. +The probability density function (PDF) of the fluctuations +can be diagnosed in terms of the local turbulent local state +and any pre-existing ice cloud. The +liquid cloud properties (cloud fraction and liquid water content) +can be then be calculated as truncated moments of the PDF. + +\cite{fhfk14} initially used their model to +understand and parametrize the results of Large Eddy Simulations (LES) +of shear-induced, Altostratus clouds. They obtained excellent +agreement between their theoretically predicted predicted mean +cloud properties and the bulk properties of the LES clouds. +Subsequently, their model has been used as the basis of +subgrid cloud initiation method for use in the Unified Model +in conjunction with the PC2 prognostic cloud scheme. In Section \ref{sec:sgt_model_describe} +we outline the model of \cite{fhfk14}. In Section \ref{sec:sgt_model_implement} +we described its implementation in the GCM. + + +\subsubsection{Model description} +\label{sec:sgt_model_describe} + +\cite{fhfk14} started from the equation for the dynamics of +ice supersaturation $S_i=e_v/e_{sat\;ice}-1$: +\begin{equation}\label{eqn:squires_eqn} + \frac{D S_i}{D t} = -b_i B_0 {\cal M}_1 S_i + -\left(\frac{\varepsilon}{L^2}\right)^{1/3}(S_i-S_E) + a_i w, +\end{equation} +where ${\cal M}_1$ is the first moment of ice particle size distribution (PSD), +$\varepsilon$ is the turbulent dissipation rate, $L$ is a prescribed mixing length +for the turbulence, $S_{\rm E}$ is the ice supersaturation of the +environment surrounding the cloud and $b_i,B_0$ and $a_i$ are function of $p$ and $T$ given by +\begin{eqnarray} + b_i &=& \frac{1}{q} + \frac{\epsilon L_s^2}{c_p R T^2}, \\ + B_0 &=& 4\pi C \left( \frac{\epsilon L_s^2}{K_a R T^2} + \frac{R T}{\epsilon e_{si} \psi} \right)^{-1}, \\ + a_i &=& \frac{g}{R T}\left( \frac{\epsilon L_s}{c_p T} - 1 \right), \\ +\end{eqnarray} +The first term on the right hand side of Eq.~\ref{eqn:squires_eqn} is +the sink of vapor due to depositional growth of ice crystals, the second +term models entrainment (mixing) of environmental air into the cloudy +volume and the third term is a source term due to vertical air motions. + +\cite{fhfk14} modeled vertical velocity as a white-noise process +with autocorrelation function: +\begin{equation} + \overline{w(t)w(s)} = \sigma_w^2 \tau_{\rm d} \delta(t-s), +\end{equation} +where $\delta$ is the Dirac distribution and the intensity of the +noise, $\sigma_w^2$, will be called the +standard derivation of the vertical velocity fluctuations (due to the white nature of +noise, a true expectation value $\overline{w^2}$ is not defined) and $\tau_{\rm d}$ +a Lagrangian decorrelation time define here by the relation used by \cite{rodean1997}: +\begin{equation} + \tau_{\rm d} = \frac{2\sigma_w^2}{\varepsilon C_0}, +\label{eqn:taud} +\end{equation} +where $C_0$ is a known constant. + +Because it is linear in $S_i$, Equation \ref{eqn:squires_eqn} can be solved exactly, +for any given realisation of the noise term. By averaging the solutions over the the noise +and taking a steady-state limit (see \cite{fhfk14} for details) it can be shown +that the solution PDF is Gaussian with mean and variance given by: +\begin{eqnarray} + \overline{S_i} &=& + S_{\rm E}\frac{ \left(\varepsilon/L^2\right)^{1/3} }{ b_i B_0 {\cal M}_1 + \left(\varepsilon/L^2\right)^{1/3} }. + \label{eqn:si_avg} \\ + \overline{S_i^2} &=& + \frac{a^2_{\rm i} \sigma^2_w \tau_{\rm d}}{ 2\left(b_i B_0 {\cal M}_1 + \left(\varepsilon/L^2\right)^{1/3}\right)}, + \label{eqn:si_var} +\end{eqnarray} + +Equation \ref{eqn:si_avg} and \ref{eqn:si_var} completely specify the PDF, $F(S_i)$, of +steady-state humidity variations for the subgrid model. The liquid cloud fraction and +liquid water mass mixing ratio are given by +\begin{eqnarray} + C_l^{sgt} &=& \int_{S_{i,wat}}^\infty d S_i F(S_i), \label{eqn:cloud_fraction} \\ + q_{cl}^{sgt} &=& q_{sat\;ice}\int_{S_{i,wat}}^\infty d S_i (S_i -S_{i,wat}) F(S_i) \label{eqn:cloud_liquid}, +\end{eqnarray} +where $S_{i,wat} = e_{sat\;wat}/e_{sat\;ice}-1$ is the value of ice +supersaturation at water saturation. +We use the superscription `$sgt$'(=`{\it s}ub{\it g}rid {\it t}urbulence') to indicate +that $C_l^{sgt}$ and $q_{cl}^{sgt}$ are values of cloud fraction and water content +diagnosed from a parametrization of small-scale turbulent processes. + + +\subsubsection{Model implementation and closure relations} +\label{sec:sgt_model_implement} + +To implement the model of Section \ref{sec:sgt_model_describe} in the +Unified Model, closure relations are needed for the quantities $\sigma_w^2$, +$\varepsilon$, $L$, $\tau_{\rm d}$ and $S_E$, subject to the constraining relationship +given by Eq. \ref{eqn:taud}. +In each model grid box, these parameters specify the subgrid PDF, $F(S_i)$, and +from this the liquid cloud fraction and water content produced by turbulence +can be found using Eqs \ref{eqn:cloud_fraction} and \ref{eqn:cloud_liquid}. + +In addition we need to make some assumptions about how the diagnosed values +$C_l^{sgt}$ and $q_{cl}^{sgt}$ relate to the model prognostic fields, $C_l$ and $q_{cl}$. +Two methods are available for doing this. In the simplest case, the diagnosed +values $C_l^{sgt}$ and $q_{cl}^{sgt}$ are just treated as increments to model prognostics +(option one, in Sec. \ref{sec:sgt_increments} below). +A more complex option (see option two, below) is to increment the +model fields via the PC2 Erosion functionality. + +\subsubsection{Closure relations}\label{sec:sgt_closures} + +The vertical velocity variance, $\sigma_w^2$, is available as a diagnostic from the +Boundary Layer scheme. Because the Boundary Layer scheme is called after the +Microphysics on each model timestep, the diagnostic value is stored in a (non-advected) +model prognostic field. The scheme will operate only where there is diagnosed turbulence, +i.e., non-zero $\sigma_w^2$. + +We take the mixing length scale, $L$, to be proportional to the vertical +grid spacing in each grid box: $L=\beta_{mix} \Delta z$, where $\Delta z$ is +calculated as the height different between the $\rho$-levels adjacent +to the given $\theta$-point. The parameter, $\beta_{mix}$, +is an adjustable constant that the user can define (see Section \ref{sec:sgt_options} below), +however it should be of order one. + +To obtain $\tau_{\rm d}$ we impose an eddy size constraint: +\begin{equation} + \tau_{\rm d} = \frac{L}{\sigma_w} = \beta_{mix} \frac{\Delta z}{\sigma_w} +\label{eqn:eddy_size} +\end{equation} +Eq.~\ref{eqn:taud} then determines the dissipation rate, $\varepsilon$, that is consistent +with the other parameters. The constant $C_0=10$ by default, but can be adjusted by the user. + +The scheme is limited to act only in grid boxes where $\tau_{\rm d}$ is less than a +prescribed value, $\tau_{d}^{max}$. The default is $\tau_d^{max}=1200\;{\rm sec}$, which typically +coincides with a couple of model timesteps. The motivation for this is that a +motion that takes longer than a few timestep to decorrelate will be partially resolved by +the dynamics and therefore cannot be considered as `subgrid' turbulence. + +Finally, where $T$, $p$ and $q$ appear in the expressions for $C_l^{sgt}$ and $q_{cl}^{sgt}$, +these are taken to be the grid box mean values. The first moment of the ice PSD, ${\cal M}_1$, +is found from the parametrization, due to \cite{fhbicc05}, described +in Section 4.1 of UMDP26. + + +\subsubsection{Options for incrementing model prognostics}\label{sec:sgt_increments} + +Using the information in Section \ref{sec:sgt_closures} to obtain closed expressions +for the subgrid PDF of $S_i$-fluctuations allows $C_l^{sgt}$ and $q_{cl}^{sgt}$ to be +calculated. These will be non-zero only where there is turbulence as diagnosed by the +Boundary Layer scheme (and hence non-zero $\sigma_w^2$). To calculate $C_l^{sgt}$ and $q_{cl}^{sgt}$ +the integrals in Eqs \ref{eqn:cloud_fraction} and \ref{eqn:cloud_liquid} are evaluated +numerically using discretisation based on user-specified number of bins. + +Given $C_l^{sgt}$ and $q_{cl}^{sgt}$, two options are available for relating these +to changes in the model prognostics: + +\paragraph{Option one: direct increments} + +The values of $C_l^{sgt}$ and $q_{cl}^{sgt}$ can be added as increments to the +model prognostic fields, $C_l$ and $q_{cl}$. In this case +\begin{eqnarray} + \left( \Delta C_l \right)_{sgt} &=& C_l^{sgt} \\ + \left( \Delta q_{cl} \right)_{sgt} &=& q_{cl}^{sgt}, \\ + \left( \Delta q \right)_{sgt} &=& -\left( \Delta q_{cl} \right)_{sgt}, \\ + \left( \Delta T \right)_{sgt} &=& \frac{L_v}{c_p} \left( \Delta q_{cl} \right)_{sgt}, \\ + \left( \Delta C \right)_{sgt} &=& C_l^{sgt} \\ +\end{eqnarray} +where the left hand sides denote the increments to $C_l$, $q_{cl}$, $T$ and the +total cloud fraction, $C$, due to +the subgrid scheme. Some bounds-checking is then applied to ensure that: +(a) the resultant cloud fractions to not exceed one; (b) the scheme does not +condense out more liquid than there is available moisture. + +\paragraph{Option two: PC2 Erosion method} + +Option one gives a simple method for incrementing the model prognostics, but +it gives rise to a potential inconsistency with the PC2 cloud scheme. This arises because +the subgrid production scheme can elevate cloud fraction to unity in grid boxes +that are subsequently diagnosed by PC2 Initiation to meet the criteria for clear-sky initiation. +PC2 then counteracts the scheme by removing some of the liquid cloud. To try to mitigate +against this issue, cloud fraction increments can be applied using PC2 Erosion. In this case: +\begin{eqnarray} + \left( \Delta q_{cl} \right)_{sgt} &=& q_{cl}^{sgt} - q_{cl}, \\ + \left( \Delta q \right)_{sgt} &=& -\left( \Delta q_{cl} \right)_{sgt}, \\ + \left( \Delta T \right)_{sgt} &=& \frac{L_v}{c_p} \left( \Delta q_{cl} \right)_{sgt}, \\ +\end{eqnarray} +where $q_{cl}$ is the liquid cloud amount prior to calling to the +turbulent production scheme. The cloud fraction increments are calculated +by calling PC2 Erosion with $\left( \Delta q_{cl} \right)_{sgt}$ as input. +See Section \ref{sec:turb} for details on how the PC2 Erosion process works. +This method gives cloud fraction increments that are consistent with +PC2 cloud scheme. + + +\subsubsection{Other user options}\label{sec:sgt_options} + +The following variables and logical switches are optional inputs: +\begin{enumerate} + \item The logical \verb!l_dcfl_by_erosion! provides a switch to + apply cloud fraction increments using PC2 Erosion. Defaults to {\it FALSE}. + \item Setting the logical \verb!l_mixed_phase_t_limit! to {\it TRUE} + allows the user to use the variable \verb!mp_t_limit! to define a temperature limit, $T_{max}$, + above which the scheme is not applied. The default is $T_{max}=0^\circ\;{\rm C}$, so the + scheme is only applied to cold clouds. + \item The input variable \verb!mp_tau_d_lim! defines the + upper limit, $\tau_d^{max}$, on the value of $\tau_d$ above which the scheme is not applied. + The default value is $\tau_d^{max}=1200.0$, so the scheme is not applied in grid boxes + where the decorrelation time scale exceeds $1200$ seconds. + \item \verb!nbins_mp! is the number of bins used in the discretisation of the integrals in + Eqs \ref{eqn:cloud_fraction} and \ref{eqn:cloud_liquid} for $C_l^{sgt}$ and $q_{cl}^{sgt}$. + The default value is $100$ bins. + \item \verb!mp_dz_scal! is the scale parameter, $\beta_{mix}$, in the definition of the mixing length, $L=\beta_{mix}\Delta z$. + \item \verb!mp_czero! defines the constant parameter $C_0$ (defaults to $C_0=10$). +\end{enumerate} + + +\section{Application to the Unified Model} +\label{sec:app_um} + +This section describes the way in which the physical concepts described in the above +section are applied to the sections of the Unified Model, in order to build +up the complete prognostic scheme. Description of the actual subroutines +themselves follow in section \ref{sec:code}. +Note that the large-scale precipitation +and convection schemes have considerable documentation below, since these +schemes have been heavily modified for PC2. The other schemes use generic +forcing scenarios, hence their desciption here is much shorter. Remember, +whenever a signficiant $\overline{T}$ or $\overline{q}$ change occurs, +PC2 must be able to +represent the corresponding condensation and changes in cloud fractions. + +\subsection{Radiation} +\label{sec:rad} + +The shortwave and longwave radiation schemes both alter the temperature +of the atmosphere, hence we need to calculate the corresponding condensation +and cloud fraction changes. For both shortwave and longwave, we use +the homogeneous forcing routines (section \ref{sec:homog}) +for $\overline{q_{cl}}$ and $C_l$, (using eqn. +\ref{eq:deltaqc_exp2} to calculate the $Q_c$ forcing) and then the method in +section \ref{sec:ct} to calculate $C_t$ changes. There is no +$\overline{q_{cf}}$ change associated with this process since the +deposition / sublimation process is performed within the large-scale +precipitation scheme (as it also is in the absence of PC2). + +It is reasonable to question whether homogeneous forcing is a reasonable +model to use when we know that a large proportion of the heating +associated with radiative transfer in the atmosphere comes from the +cloudy air and is not evenly spread across the gridbox. Possible developments +are discussed in section \ref{sec:homog_improve}. + +\subsection{Large-scale precipitation} +\label{sec:precip} + +Precipitation processes have a large effect on cloud fractions. Here we +present the simple physical models that are applied to the transfer +terms included in the large-scale precipitation scheme. They are also +presented within the large-scale precipitation documentation (\citeumdp{026}). + +The basis of the physical model is that microphysical transfer processes can +be calculated separately in different partitions of the model cloud +(i.e. mixed phase cloud, liquid phase cloud, ice phase cloud or clear sky). +However, processes may change the size of these partitions. We consider +here separately each process that is modelled in the large-scale +precipitation scheme. The changes in $\overline{q_{cl}}$, $\overline{q_{cf}}$ +and $\overline{q}$ remain mathematically the same as in the non-PC2 version +of the code (\citeumdp{026}), we only need +to introduce calculations for the changes in cloud fractions. We will see +that many of these +changes can be well modelled by assuming no change to the cloud fractions, +and the others by using simple assumptions. + +Although the model may use two ice prognostic ice categories, only +a single ice cloud fraction is stored, the assumption being that the +two ice categories are completely overlapped with each other. Graupel +is not considered to contribute to the ice cloud fraction. + +\subsubsection{Fall of ice} +\label{sec:lsp_fall} + +The fall of ice is the process that contributes most to the growth of +ice cloud fraction in the model. The model results are therefore sensitive +to the formulation of this process. We will make the basic assumption +that a trail of falling ice does not reduce the horizontal spread of +ice cloud fraction at a particular level (hence $\overline{q_{cf}}$ +that leaves a gridbox does not reduce $C_f$ in that gridbox). The in-cloud +ice content simply reduces due to the fall out of ice - it is the +sublimation term (section \ref{sec:mp_depsub}) that erodes the fall streaks. +However, ice that falls into a clear layer from above may increase +the ice cloud fraction. We parametrize this by considering the +horizontal overlap of ice clouds between two model layers, and the +fall speed of ice between them. We will assume an overlap that +is nearly, but not quite, maximum, the difference being dependent +upon the windshear and the time taken for ice to fall between the +levels. + +\begin{equation} +O^{[k,k+1]} = \text{Max}( C_{i}^{[k+1]} - C_i^{[k]} , 0) ++ w \frac{\Delta z^{[k]}}{v_i^{[k]}} +\label{eq:overhang} +\end{equation} + +where $O^{[k,k+1]}$ is the amount of ice cloud `overhanging' the current +(i.e. $k$'th) layer +from the layer above, $w$ is a parameter that is closely related to the +windshear, $\Delta z^{[k]}$ is the model layer thickness and $v_i^{[k]}$ is +the fallspeed of ice in the layer. $v_i^{[k]}$ is calculated in the microphysics +scheme and, if two ice prognostics are used, is the mass-weighted average fall +speed of the two categories. +The factor $\frac{\Delta z}{v_i}$ is simply the time +taken for the ice to fall through one model layer. Multiplying this +by the windshear would give an estimate to the amount +of overlap between a cloud source and its fall streak in the layer +below (it is an \textit{estimate} since we assume that the cloud +source is continuous and unbroken). Although it is quite possible within +PC2 to do this, to date we have not programmed this link, and we +use a constant, but tunable, value of $1.5 \times 10^{-4} s^{-1}$ for $w$. + +The change in $C_i$ over the timestep is then given by the overlap proportion +multiplied by the how much (in the vertical dimension) of the layer below +can be filled by ice in the timestep: + +\begin{equation} +\Delta C_i = \text{Max}(O^{[k,k+1]} , 1) \text{Min} (v_i \frac{\Delta t}{\Delta z^{[k]}} , 1) +\label{eq:lsp_fall} +\end{equation} + +where $\Delta t$ is the timestep. We now choose to assume a minimum overlap +between the liquid and the ice phases (as in section \ref{sec:ct}). + +\begin{equation} +\Delta C_t = \text{Min} ( \Delta C_i , A_{clear} ) +\label{eq:lsp_fall_ct} +\end{equation} + +where $A_{clear}$ is the proportion of the gridbox that has neither +ice nor liquid cloud present. + +\textbf{An inconsistency has been found in the way that the fall-of-ice term is linked to the globally constant ``wind-shear value'' +when calculting the ice cloud fraction overhang. Consequently, +the option not to use the ``wind shear value'' when calculating the overhang is available in +the UMUI (from version 7.6 onwards).} + +\subsubsection{Homogeneous nucleation} +\label{sec:lsp_homo} +This will freeze all supercooled liquid water when a temperature threshold +is exceeded. Hence we turn all existing liquid and mixed phase cloud to +ice cloud. The cloud fraction changes are: + +\begin{eqnarray} +C_l \leftarrow 0 \nonumber \\ +C_i \leftarrow C_t \nonumber \\ +\Delta C_t = 0. +\label{eq:lsp_homo} +\end{eqnarray} + +\subsubsection{Heterogeneous nucleation} +This process will freeze a small amount of supercooled liquid water, +regardless of the previous presence of ice cloud. This will mean that +previously existing `liquid-only' cloud is converted to mixed phase +cloud. These give the following changes: + +\begin{eqnarray} +\Delta C_l = 0 \nonumber \\ +C_i \leftarrow C_t \nonumber \\ +\Delta C_t = 0. +\label{eq:lsp_het} +\end{eqnarray} + +\subsubsection{Deposition and sublimation} +\label{sec:mp_depsub} +This term exerts one of the most important influences on the ice cloud in +the whole model (this applies to the control as well as for PC2). Contained +in the formulation is a subgrid-scale assumption that causes equivalent +effects to that for a moisture PDF under the `$s$' framework +(section \ref{sec:s_dist}). However, since ${q_{cf}}$ changes +slowly in response to local changes in $q$ and $T$, we cannot base the +$q_{cf}$ response on the same instantaneous condensation framework. It would +be useful to investigate in the future whether the two descriptions of the +moisture variability could be brought together. +Because of its importance, +we describe the method below, although we note it is also described in +\citeumdp{026}. + +We can calculate the local rate of change of $q_{cf}$, given local $T$ and $q$ +etc. using the standard microphysical growth equations (see \citeumdp{026}). +However, it is critical to know the way in which +the moisture is correlated with the ice in the gridbox. We will assume +there exists a distribution of vapour in the gridbox. We know that +the regions where liquid cloud exists must be saturated with respect +to liquid water, hence we need only consider the part of the gridbox +that does not have liquid water present. The average value, $q_a$, +of $q$ within the liquid-free part of the gridbox is thus + +\begin{equation} +q_a = \frac{ \overline{q} - C_l q_{sat \, liq}(\overline{T}) } {1 - C_l} +\label{eq:qa} +\end{equation} + +where we have assumed that the fluctuation of $q_{sat~liq}$ across +the gridbox due to temperature fluctuations is not significant compared +to the fluctuation of $q$ described below. We then parametrize a width, $b_i$, +to the $q$ (not $s$) fluctuations \textit{across the non-liquid cloud part +of the gridbox}, based upon $RH_{crit}$. This is like that for the `$s$' +distribution width, $b_s$ but modified: + +\begin{equation} +b_i = (1 - RH_{crit} ) q_{sat \, liq} ( 1 - \frac{1}{2} +~ \frac{\overline{q_{cf}}} {i q_{sat \, liq}(\overline{T})} ) . +\label{eq:b_i} +\end{equation} + +where the factor $( 1 - \frac{1}{2} +\frac{\overline{q_{cf}}} {i ~ q_{sat \, liq}(\overline{T})})$ should be limited +to a minimum value of zero, but, for numerical reasons, is limited to +a minimum value of 0.001. We note that $b_i$ has a similar form to $b_s$, +except the multiplier $a_L$ and the factor in brackets. If we remember +from (\ref{eq:s}) that the definition of `$s$' includes a factor $a_L$ +we see that the absence of the $a_L$ factor in (\ref{eq:b_i}) is +consistent. The factor in brackets is a \textit{parametrization} of the +effect that, when ice +is present, deposition in the moistier parts and sublimation in the +drier parts of the gridbox must reduce the width of the distribution +of $q$ across the gridbox. It is a simple linear function of +$\frac{\overline{q_{cf}}}{q_{sat~liq}(\overline{T})}$, and is tunable +using the factor $i$, which takes the value of 0.04. + +We note that this formulation isn't totally consistent with the liquid +cloud formulation, which considers an underlying PDF across the whole +gridbox and does not have, in general, its width prescribed. +Remember that we do not calculate on-line the whole of the liquid +- vapour PDF, we only parametrize the single point $G(-Qc)$, +using equation \ref{eqn22}). + +The width is then limited further to be no greater than $\overline{q}$, +to make sure that there are no negative values of $q$ predicted within the +gridbox (possible at low +temperatures where $q_{sat~liq}(\overline{T})$ diverges from +$q_{sat~ice}(\overline{T})$). + +We then calculate the average value of $q$ in the ice-only and clear-sky +partitions of the gridbox. To do this, we make the further assumption +that the ice is correlated with the moistest part of the distribution +(an instantaneous condensation formulation would make the +same assumption). Some algebra retrieves the expressions: + +\begin{eqnarray} +q_{clear} = q_a - b_i A_{ice} ; \\ +q_{ice} = \frac {\overline{q} - C_l q_{sat~liq} - A_{clear} q_{clear} } +{A_{ice}}, +\label{eq:q_clear_and_q_ice} +\end{eqnarray} + +where $A_{ice}$ is the proportion of the gridbox with ice cloud but not +liquid cloud and $A_{clear}$ is the proportion of the gridbox without cloud. +The numerical application will set $q_{clear}$ to $q_a$ if $A_{ice}$ +is zero. We now have a representation of the $q$ values in each of the +gridbox cloud partitions, and can solve the microphysical transfer equation +in each partition. + +The cloud fraction changes now need to be parametrized. We use the +model that deposition will \textit{not} adjust the ice cloud +\textit{fraction} (increases will be done within the fall-of-ice microphysics +section). However, deposition can +decrease the liquid cloud fraction (locally, $q$ can be reduced by +deposition to below $q_{sat~liq}$, hence this is not inconsistent with +the assumptions for the riming term below. This is the principal sink +of supercooled liquid cloud fraction in the model. Sublimation will +be allowed to decrease the ice cloud fraction (since sublimation cannot +act in liquid cloud, there is no impact on the liquid cloud). To solve +for these models, we will need to further split the ice-only partition +to give the proportion of that partition that is above and below ice +saturation. This gives, in general, an area of the gridbox $A_{ice1}$ +that contains ice and is above saturation where + +\begin{equation} + A_{ice1} = \frac{1}{2} A_{ice} + \frac{1}{2} + \frac{ (q_{ice}-q_{sat~ice}(\overline{T})) } {b_i}, +\label{eq:q_ice_above_sat} +\end{equation} + +having assumed that $A_{ice1}$ is between 0 and $A_{ice}$. +If not, it is trivial to partition the gridbox, since the moisture in the ice-only +partition is either completely above or completely below $q_{sat~ice}(\overline{T})$. +The corresponding +area that contains ice and is below saturation is given by +$A_{ice2} = A_{ice} - A_{ice1}$. +We can now parametrize the change in cloud fractions. For deposition, +we shall assume a uniform distribution of local values of $q_{cl}$ about +the local mean. If we assume a uniform removal of local $q_{cl}$ then, with +a little algebra, we can obtain an expression for the change in $C_l$: + +\begin{equation} +\Delta C_l = C_l ( 1 - \frac {\Delta \overline{q_{cl}}} {\overline{q_{cl}}} ) +^{\frac{1}{2}} - C_l +\label{eq:deltacfl_dep} +\end{equation}. + +Since this occurs only in the mixed phase part of the gridbox, we can say +that $\Delta C_t = 0$. We will also note that the change in $\overline{q_{cl}}$ +due to deposition is limited by the amount of $\overline{q_{cl}}$ that is in +the mixed phase partition in the gridbox, hence (\ref{eq:deltacfl_dep}), +although it formally allows removal of $C_l$ from an ice-free partition, will +be unlikely to do so. + +The sublimation forms the main method by which ice cloud is destroyed in PC2, +hence PC2 results are relatively sensitive to its formulation. Here we +make a similar assumption to that used for liquid in the deposition term, +except that we limit the changes only to the region of the gridbox where +ice is subliming. + +\begin{equation} +\Delta C_i = A_{ice2} ( 1 + \frac{\Delta \overline{q_{cf}} } +{ \overline{q_{cf}} ( \frac{A_{ice2}}{C_i} ) } + )^{\frac{1}{2}} - A_{ice2} +\label{eq:deltacfi_sub} +\end{equation}. + +The term $\overline{q_{cf}} ( \frac{A_{ice2}}{C_i} )$ is the amount of +$\overline{q_{cf}}$ that is present in the subliming ice region, hence its +ratio with $\Delta \overline{q_{cf}}$ is the fractional change in that region. The +change in the total cloud fraction must also be equal to the change above, +since sublimation cannot occur in the presence of liquid cloud: + +\begin{equation} +\Delta C_t = \Delta C_i . +\label{eq:deltacft_sub} +\end{equation} + +\subsubsection{Riming} +This process acts only where mixed phase cloud occurs - although, in theory, +it could remove any supercooled liquid totally, the air would remain +saturated with respect to liquid water. Hence any subsequent cooling would +regenerate the same amount of liquid cloud. Hence we choose to model this +process as having \textit{no effect} on the cloud fractions. + +\subsubsection{Capture} +This is the freezing of raindrops onto ice crystals by collision. This does +not alter the ice cloud \textit{fraction} in the gridbox (although it does +alter $\overline{q_{cf}}$, and it has no interaction with the liquid cloud. +Again, we therefore choose to model this process as having \textit{no effect} +on the cloud fractions. + +\subsubsection{Evaporation of melting ice} +Here we simply assume that ice cloud fraction is removed in proportion to the +ice content that is removed. + +\begin{equation} +\Delta C_i = C_i \frac{ \Delta \overline{q_{cf}}}{\overline{q_{cf}}} . +\label{eq:lsp_evapmeltsnow} +\end{equation} + +Because the evaporation cannot occur in the liquid part of the gridbox, +there is no change to $C_t$ (or to $C_l$). + +\subsubsection{Melting} +Again, the change in $C_i$ is calculated using the method +in (\ref{eq:lsp_evapmeltsnow}). + +\begin{equation} +\Delta C_i = C_i \frac{ \Delta \overline{q_{cf}}}{\overline{q_{cf}}} . +\label{eq:lsp_melt} +\end{equation} + +The change in $C_t$ is calculated assuming that there is no correlation +in the gridbox between where the ice melts and the liquid cloud. Hence +we must multiply (\ref{eq:lsp_melt}) by the proportion of ice cloud +fraction that exists without liquid cloud (i.e. $\frac{A_{ice}}{C_i}$). + +\begin{equation} +\Delta C_t = C_i \frac{ \Delta \overline{q_{cf}}}{\overline{q_{cf}}} +\frac{A_{ice}}{C_i} . +\label{eq:lsp_melt2} +\end{equation} + +\subsubsection{Evaporation of rain} +Evaporation of rain will not, \textit{on its own}, +generate liquid cloud, since a +large-scale lifting process will be required in order to condense water +from the moistened air. We cannot, therefore, allow any change in cloud +fractions to occur as a result, subsequent changes are calculated elsewhere +in the model (e.g. by the lifting process, section \ref{sec:pres}). + +\subsubsection{Accretion} +Accretion is the sweep-out of liquid water droplets by rain. We argue +in a similar way to the riming term, that this will not remove any liquid +cloud fraction, since a small amount of lifting will regenerate the same +amount of liquid cloud. Hence we choose to model this process as having +\textit{no effect} on the cloud fractions. The arguments underlying the +formulation of the evaporation of rain and the accretion cloud fraction +changes may appear to be inconsistent in their limiting cases and the +subsequent response to lifting. However, when the limiting case is +not reached the formulations are both correct. For the moment, it is +not considered necessary to increase the complexity of the current, +simple representations. + +\subsubsection{Autoconversion} +As for accretion, the generation of rain directly from collision +and coalescence of liquid water droplets will not alter the cloud fractions. + +\subsubsection{Other microphysics terms} +There are already (i.e. also in the control) +two numerical tidy-up terms at the end of the microphysics +section that remove small rain amounts and provide an additional +melting term for the snow. These do not change the cloud fractions. + +If there are small amounts of ice present at the end of the +microphysics then these are removed at the end of the microphysics +timestep (also in the control). PC2 responds by resetting the +cloud fractions appropriately, so $C_t$ is reset to $C_l$ etc. + +\subsubsection{Numerical implementation} + +Note that after each process has been applied, we do \textit{not} +recalculate the sizes of the ice-only, liquid-only and mixed phase +partitions, but use the values at the start of the microphysics (this +includes the values of $C_i$ used in the calculation of `in-cloud' +water contents above. However, we do update the cloud fractions +themselves sequentially. We also recalculate after each process +the overlaps between the rain fraction (see \citeumdp{026}) and the cloud fractions. + +There is also a final set of checks that $C_l$ and $C_i$ lie +between 0 and 1 and that $C_t$ is bounded between $\text{Max}(C_l, C_i)$ +(maximum overlap of liquid and ice) and $\text{Min}(C_l+C_i,1)$ +(minimum overlap of liquid and ice). + +We should note in particular, that these parametrizations allow a +considerable reduction in $\overline{q_{cl}}$ without a corresponding +large reduction in $C_l$. This is an underlying feature of the PC2 scheme +(discussed in \cite{wg03}), and necessarily implies the +skewing of the underlying moisture PDF. Subsequent parts of +the model (e.g. the width narrowing, section \ref{sec:width}) will, of +course, act on the modified fields to adjust the cloud fractions further, +but remember that these are separate processes and modelled elsewhere in the +timestep. + +\subsection{PC2 erosion} +\label{sec:turb} + +\subsubsection{Original width-narrowing method} +(selected by setting {\bf i\_pc2\_erosion\_method = 1} in the UM namelist). + +In parallel with the homogeneous forcing part of the PC2 response to +convection, we introduce +a new block of code that allows a background change of the PDF width. +At earlier versions of PC2 (PC2:65 and earlier) this block was included as +a separate section of code that was called in as part of the atmphya parallel +timestepping. This was later moved to better numerically balance +increments from +the convection scheme with the cloud fraction erosion term. From VN8.1 onwards, a further option was introduced to implement the erosion prior to the microphysics parametrization. This was primarily to allow PC2 to be run at convection-resolving scales, at which the convection scheme is not called and therefore the erosion is not called. +We have empirically selected a rate of change of width that depends upon the +relative total humidity of the grid box, such that there is more +erosion in drier gridboxes. This promotes more rapid erosion of +shallow convective cloud, which is the main effect that we seek to +include, although the physical implication that dry air is +more turbulent than moist air does not match the way the real atmosphere +works, especially in the +stratosphere. No doubt the link can be improved upon with more +research. The formulation used is: + +\begin{equation} +\frac{1}{b_s} \frac{\partial b_s}{\partial t} = \Upsilon exp ( - \frac{2.01 Q_c}{0.2 a_L q_{sat liq}(T_L)} ) +\label{eq:dbsbydtbs_turb} +\end{equation} + +where the 0.2 factor is chosen to be closely equivalent to $1 - RH_{crit}$ +and the value of 2.01 has been selected through tuning. The code merges the +two numerical values into a single quantity (dbsdtbs-exp), equal to 10.05. +We note that in PC2:64 +(the library 6.4 code, a value of 0.62 is used rather than 2.01). As a guide +to the $RH_T$ dependence, note +that when the value of $RH_T$ is 0.85, the value of +$\frac{1}{b_s} \frac{\partial b_s}{\partial t}$ is close to $1 \times 10^{-4} s^{-1}$. + +Note: the source-code for this erosion method ({\bf pc2\_hom\_conv}, +{\bf pc2\_homog\_plus\_turb}, {\bf pc2\_delta\_hom\_turb}) +includes an additional term ``dbsdtbs1'' +which scales with the rate of homogeneous forcing +$\frac{\partial Q_c}{\partial t}$. However this term is always +set to zero on input to these routines so is never used. + +The width-narrowing formulation of section \ref{sec:width} is used to +calculate increments in $\overline{q_{cl}}$ and $C_l$. Using the liquid - +ice cloud overlap ideas of section \ref{sec:ct} then gives the associated +$C_t$ change. This background narrowing term, $\Upsilon$, is originally based upon work +by \cite{sg03}, although it is a parameter that has been +extensively tuned during PC2 development, a typical value would be $\Upsilon=-2.25 \times 10^{-5} s^{-1}$. + +\subsubsection{Numerical application of the original width-narrowing method} + +Because of the strong link the mathematical expressions for width narrowing +(section \ref{sec:width}) have with the expressions for the +homogeneous forcing (section \ref{sec:homog}), we choose to represent +the timestepping of this process in exactly the same way as for +the homogeneous forcing (in fact, in the Unified Model code we use +the same subroutine, see section \ref{sec:code}). As +before, we use a simple forward timestepping of $C_l$, with +$Q_c$ given by (\ref{eq:qc_eq_qt-qs}) and $a_L$ defined as discussed +in section \ref{sec:homog_num_app} and discretize eq \ref{eq:dcdt_width} as: + +\begin{equation} +\Delta C_l^{[n+1]} = - G(-Q_c) Q_c \frac{1}{b_s} +\frac{\partial b_s}{\partial t} \Delta t. +\label{eq:dcl_turb_final} +\end{equation} + +Similarly to (\ref{eq:c_l^n+1}), we then limit the cloud fraction to 0 and +1 and then apply a mid-point value of $C_l$ to calculate the change in +$\overline{q_{cl}}$ (discretizing eq \ref{eq:dqcldt_width}): + +\begin{equation} +\Delta q_{cl}^{[n+1]} = (q_{cl}^{[n]} - Q_c \frac{1}{2}(C_l^{[n]}+C_l^{[n+1]})) +\frac{1}{b_s} \frac{\partial b_s}{\partial t} \Delta t. +\label{eq:dqcl_turb_final} +\end{equation} + +In this case the value of $\Delta q_{cl}$ \textit{is} limited to ensure that +no more $\overline{q_{cl}}$ is removed than the model has available. This was +chosen to ensure that the erosion process itself contains this physical limit, +not a numerical tidying-up process. + +The option ``l\_fixbug\_pc2\_qcl\_incr'' ensures that qcl is set to zero +if the CFL has reached zero. + + +\subsubsection{Cloud-surface-area hybrid erosion method} +(selected by setting {\bf i\_pc2\_erosion\_method = 3} in the UM namelist). + +\cite{morcrette_petch} showed that changes to the erosion parameter ($\Upsilon$ in Eqn. \ref{eq:dbsbydtbs_turb}) did not have as significant an impact +on the global work done by the erosion process as might be expected. This was due to a feedback process +whereby, reducing the erosion parameter leads to more cloud water, more autoconversion of cloud water to rain, more +fall-out of rain and more drying of the layer, hence increasing the $exp ( - \frac{2.01 Q_c}{0.2 a_L q_{sat liq}(T_L)} )$ part of +Eqn. \ref{eq:dbsbydtbs_turb}. Although the feedback is physically plausible it crucially depends on the formulation of +Eqn. \ref{eq:dbsbydtbs_turb} and the dependence of the rate of narrowing of the PDF on the moisture, a dependence that was developed +from a pragmatic rather than theoretical stand-point. +The option for an alternative way of calculating the erosion was introduced at vn8.0 + +We use equation 30 from \cite{t93} to specify the sink of $q_{cl}$ due to erosion, i.e. +\begin{equation} +\frac{\partial q_{cl}}{\partial t}=-A K(q_{sat}-q_v) +\label{eq:dqcldt_hybrid} +\end{equation} +(note we have changed the sign as we have replaced the evaporation rate $E_2$ +in \cite{t93} with $-\frac{\partial q_{cl}}{\partial t}$ on the left-hand-side). +In the \cite{t93} scheme, $A$ is set to the cloud fraction (i.e. $A=C_l$). +Here we recall that "cloud erosion" is meant to represent the evaporation of cloud water due to the +mixing of clear and cloudy air and that this can only happen on the edges of cloud, where saturated air is exposed to sub-saturated air. +If the cloud fraction is small (e.g. 5$\%$), then there are not many clouds, so there is only a small surface area from which evaporation can occur. +Similarly if the cloud cover is very high (e.g. 95$\%$) then there is again not much surface area exposed to clear sky. +A maximum in exposed surface area is expected when the cloud cover is 50$\%$. + +By imagining that the grid-box is broken up into cubes whose horizontal dimension equal the layer depth it is possible +to work out what the maximum lateral surface area would be, as a function of cloud fraction, for different arrangements of cloudy cubes. +The maximum lateral surface area, is when the clear and cloudy cubes are arranged in a chess-board pattern, and the minimum is when then are all grouped +together into a circular clump. Numerical tests using randomly distributed cloudy cube shows that the variation +in lateral surface area, $S$, as a function of cloud fraction can be expressed as: +\begin{equation} +S= - 2 C_l ^{2} + 2 C_l +\label{eq:S_Cl} \end{equation} +The maximum normalised surface area of 0.5 occurs at a cloud fraction of 0.5. +Using a cloud mask derived from satellite imagery shows that real cloud fields do follow this kind of dependence, +but that the peak surface area is nearer to 0.35, meaning that real clouds are not as randomly distributed as random +ones and that there is some kind of clumping together, which is what we might have expected. +When it comes to implementing such a scheme in the model, there will need to be a tunable parameter to govern the +rate of evaporation. This will not affect the shape of the lateral surface area function. +As a result the details of whether the peak lateral surface area is 0.5 or 0.35 are simply absorbed into the tunable parameter $K$, +which is supplied from the UMUI (using the same text box as was used for supplying $\Upsilon$). + +The exposed surface area associated with the tops and bottom of the clouds is calculated assuming maximum overlap in adjacent layers and is added to the lateral +surface area to give a total surface area, +\begin{equation} +A=max(C_l(k)-C_l(k+1),0.0)+max(C_l(k)-C_l(k-1),0.0)+S +\label{eq:A_top_and_bottom} \end{equation} +it is this value of $A$ which we use in Eqn. \ref{eq:dqcldt_hybrid}. + +Note that the contributions from the top and bottom interfaces of the current +model-level $max(C_l(k)-C_l(k+1),0.0)$ and $max(C_l(k)-C_l(k-1),0.0)$ +may optionally either be included or excluded, depending on the +UM namelist switch \textbf{i\_pc2\_erosion\_method}. +Further note: at present these contributions are hardwired to be excluded, +as they prevented the erosion calculations from being parallelised +in the vertical direction using OpenMP, and no operational model configurations +were using them. + +Having calculated a reduction in $q_{cl}$ using the Tiedtke-surface-area method, we +then work out the relative rate of narrowing that would have given the same sink of $q_{cl}$. This value of $\frac{1}{b_s} \frac{\partial b_s}{\partial t}$ +is then used to calculate the change in $C_l$ using the same moisture PDF assumptions as were used in the original PC2 erosion formulation. +To achieve this, we combine equations \ref{eq:dcdt_width} and +\ref{eq:dqcldt_width} from section \ref{sec:width} to eliminate +$\frac{1}{b_s} \frac{\partial b_s}{\partial t}$ and write +$\frac{\partial C_l}{\partial t}$ as a function of +$\frac{\partial \overline{q_{cl}}}{\partial t}$: + +\begin{equation} +\frac{\partial C_l}{\partial t} + = - \frac{ G(-Q_c) Q_c \frac{\partial \overline{q_{cl}}}{\partial t} } + { (- C_l Q_c+\overline{q_{cl}}) } +\label{eq:dcdt_hybrid} +\end{equation} + +Where the change in liquid water content +$\frac{\partial \overline{q_{cl}}}{\partial t}$ +is given by eq \ref{eq:dqcldt_hybrid} above. + +This combination of a Tiedkte sink term for $q_{cl}$, a PC2 term for $C_l$ and the introduction of some surface area dependence leads to this formulation +being referred to as a ``hybrid'' cloud-surface-area erosion method. + +\subsubsection{Numerical application of the hybrid erosion method} +\label{sec:erosion_numerics} + +Next, we consider how to numerically discretise equations +\ref{eq:dqcldt_hybrid} and \ref{eq:dcdt_hybrid} +to compute cloud increments due to erosion. +The simplest approach is an explicit forwards-in-time discretisation: + +\begin{equation} +\frac{ \Delta {q_{cl}}_{ero}}{\Delta t} = A(C_l^n) K(q_{sat}-q_v) +\label{eq:hybrid_erosion_expl} \end{equation} + +i.e. the increment is calculated by evaluating the term $A$ from equations +\ref{eq:S_Cl} and \ref{eq:A_top_and_bottom} using the value of cloud-fraction +$C_l$ \textit{before} erosion has been applied. + +However, when the environment is significantly subsaturated +(so that the term $(q_{sat}-q_v)$ is large and negative), +and long timesteps $\Delta t$ are used +(e.g. order 1000 s used in global climate simulations), +this discretization can suffer severe numerical overshoot. +i.e. the increment based on $C_l^n$ is large enough to reduce $q_{cl}$ +(and hence also $C_l$) to less than zero within a single timestep. +If the continuous equation were solved analytically this wouldn't happen; +as $C_l$ declines due to the erosion, so will $A(C_l)$ +and hence the erosion rate, so that $q_{cl}$ and $C_l$ smoothly decline +towards zero. + +Three options are available in the code to address this problem, +selected by the UM namelist switch \textbf{i\_pc2\_erosion\_numerics}, +detailed below. +Single-Column Model tests indicate that the 2nd and 3rd options yield much less +timestep sensitivity for detrained cloud in shallow cumulus regimes. + +\begin{enumerate} + +\item \textbf{Retain the explicit discretization, but limit the resulting +erosion increments to ensure $q_{cl}$ and $C_l$ don't go negative. +(i\_pc2\_erosion\_numerics=1)} +Also, to ensure that some cloud remains at end-of-timestep where +shallow cumulus is detraining into dry environments, the erosion +calculation is fed copies of the fields with the current timestep's +convection increments subtracted off. This means any cloud detrained +by convection during the current timestep cannot be eroded until the +following timestep, and so is still present at end-of-timestep. +As discussed in section \ref{sec:timestepping}, +this leads to a problematic timestep sensitivity, +since the amount of cloud not subject to erosion is the convection increment, +which scales with the timestep length. + +Having computed the erosion $q_{cl}$ increment using +\ref{eq:hybrid_erosion_expl}, the consistent $C_l$ increment is computed +by discretising \ref{eq:dcdt_hybrid} as: + +\begin{equation} +\frac{\Delta {C_l}_{ero}}{\Delta t} + = - \frac{ G(-Q_c)^n Q_c^n \frac{\Delta \overline{{q_{cl}}_{ero}}}{\Delta t} } + { (- C_l^n Q_c^n + ( \overline{q_{cl}^n} + + \frac{1}{2} \Delta \overline{{q_{cl}}_{ero}} ) ) } +\label{eq:dcdt_hybrid_discr} +\end{equation} + +i.e. all terms are treated explicitly (using the values before erosion), +except for $\overline{q_{cl}}$ which takes the mid-point interpolated +half-way between its values before and after erosion, +to give some improvement in accuracy. + +\item \textbf{Use an approximate implicit discretisation, +which intrinsically yields a positive solution for $q_{cl}$ and $C_l$. +(i\_pc2\_erosion\_numerics=2)} +The copies of the fields passed to the erosion calculation are +fully updated with the convection increments. +We then write equation \ref{eq:dqcldt_hybrid} in the form: + +\[ +\frac{\partial q_{cl}}{\partial t} = q_{cl} f(q_{cl},C_l,(q_{sat}-q_v)) +\] + +(where the term $f(q_{cl},C_l,(q_{sat}-q_v)) = \frac{A K(q_{sat}-q_v)}{q_{cl}}$ +will be treated explicitly, under the assumption that this ratio will +evolve more slowly while erosion rapidly reduces both the numerator +and the denominator). + +We then take a backwards-in-time implicit discretisation in terms of the +leading factor $q_{cl}$: + +\[ +\frac{ q_{cl}^{n+1} - q_{cl}^{n}}{\Delta t} = q_{cl}^{n+1} f^n +\] + +Now, the problem is somewhat complicated by the fact that in the code, +erosion is calculated in parallel with the homogeneous forcing by convection, +and we need to account for the homogeneous forcing increment +$\Delta q_{cl}^{hom}$ in our implicit solution. We therefore write the above as: + +\[ +\Delta q_{cl}^{ero} = \Delta t + ( q_{cl}^n + \Delta q_{cl}^{hom} + \Delta q_{cl}^{ero} ) f^n +\] + +Rearranging: + +\[ +\Delta q_{cl}^{ero} = \Delta t f^n q_{cl}^n \frac{ q_{cl}^n + \Delta q_{cl}^{hom} } + { q_{cl}^n - \Delta t f^n q_{cl}^n } +\] + +Note that the term $\Delta t f^n q_{cl}^n$ is the erosion increment +we would obtain from the purely explicit discretisation, +$\Delta q_{cl}^{ero\,expl}$. The implicit discretisation is implemented by +first calculating $\Delta q_{cl}^{ero\,expl}$ using equation +\ref{eq:hybrid_erosion_expl} +(as we do for \textbf{i\_pc2\_erosion\_numerics=1}) +but then rescaling it using the above expression, which becomes: + +\begin{equation} +\Delta q_{cl}^{ero} = \Delta q_{cl}^{ero\,expl} \frac{ q_{cl}^n + \Delta q_{cl}^{hom} } + { q_{cl}^n - \Delta q_{cl}^{ero\,expl} } +\label{eq:hybrid_erosion_impl_qcl} \end{equation} + +Provided erosion is acting to reduce cloud-water ($\Delta q_{cl}^{ero\,expl} < 0$), +and homogeneous forcing by convection has not already completely removed +the cloud ($q_{cl}^n + \Delta q_{cl}^{hom} > 0$), \ref{eq:hybrid_erosion_impl_qcl} +is guaranteed to yield a stable, positive solution for $q_{cl}$. + +We also apply exactly the same argument to the equation for the cloud-fraction +increment $C_l$, and obtain: + +\begin{equation} +\Delta C_l^{ero} = \Delta C_l^{ero\,expl} \frac{ C_l^n + \Delta C_l^{hom} } + { C_l^n - \Delta C_l^{ero\,expl} } +\label{eq:hybrid_erosion_impl_Cl} \end{equation} + +Where $\Delta C_l^{ero\,expl}$ is computed using eq \ref{eq:dcdt_hybrid_discr}, +except that the term $\frac{1}{2} \Delta \overline{{q_{cl}}_{ero}}$ +is omitted (interpolating to the mid-point value of $\overline{q_{cl}}$ +in the denominator would be ``double-counting'' if we are already making +an implicit correction to the full increment). + +In the case where the homogeneous forcing increments have already removed +all of the cloud water content or fraction, erosion is not performed, +and $q_{cl}$ and $C_l$ are both set to zero. In the case where erosion is +actually acting to increase cloud-fraction, the code defaults to retaining +the explicit discretisation solution $\Delta q_{cl}^{ero\,expl}$ and +$\Delta C_l^{ero\,expl}$. Otherwise, equations \ref{eq:hybrid_erosion_impl_qcl} +and \ref{eq:hybrid_erosion_impl_Cl} are applied to yield the implicit solution. + +\item \textbf{Use an analytic solution to the integration of the +time-derivatives in (\ref{eq:dqcldt_hybrid}) and (\ref{eq:dcdt_hybrid}) +for greater accuracy. +(i\_pc2\_erosion\_numerics=3)} + +Two problems have been identified with the above implicit numerical method: +\begin{itemize} + +\item The implicit correction is applied completely independently to the +increments for $q_{cl}$ and $C_l$. So as with the explicit method, +differing numerical error in the increments for the two variables +can lead to them becoming inconsistent with eachother. +It was found by experimentation that even with the implicit correction, +it is possible for erosion to reduce $C_l$ by a bigger fraction than $q_{cl}$, +so that the in-cloud water content $\frac{q_{cl}}{C_l}$ is {\em increased}. +Narrowing the PDF should only {\em decrease} the in-cloud water-content; +occasional large increases due to numerical error can lead to spurious +precipitation being produced by the microphysics scheme. + +\item The implicit correction makes it impossible for erosion to reduce +$q_{cl}$, $C_l$ to zero. As we will show below, the analytic solution +to the equations posed does in fact go to zero after a finite time +under grid-mean subsaturation +(although the erosion rate declines with $C_l$ as it approaches zero, +$C_l$ approaches zero more slowly than $q_{cl}$, so that both variables decrease +following power-law curves not exponentials). +When erosion (wrongly) can never entirely remove cloud, this allows +tiny values of $q_{cl}$ and $C_l$ to spuriously spread across the domain +via numerical diffusion from the model's advection scheme. + +\end{itemize} + +Under this option, we attempt to compute an analytic solution to the +simultaneous differential equations \ref{eq:dqcldt_hybrid} and +\ref{eq:dcdt_hybrid} so that $q_{cl}$ and $C_l$ both decrease smoothly and +consistently. +The equations lead to somewhat different behaviour depending on whether +the grid-mean state is subsaturated ($Q_c < 0$), supersaturated ($Q_c > 0$), +or close to saturation ($Q_c$ near-zero). We can employ different +approximations to integrate the equations in each case. +In the code, we first test the value of $Q_c$ and compute the erosion +increments as follows: + +\begin{enumerate} + +\item {\bf Grid-mean subsaturation ($Q_c < 0$):} + +The relation between the erosion tendencies in liquid-cloud-fraction and +liquid water content (\ref{eq:dcdt_hybrid}) can be expressed in terms of +{\em fractional} rates of change +(dividing the top and bottom by $-C_l Q_c$, and dividing both sides by $C_l$): + +\begin{equation} +\frac{1}{C_l} \frac{\partial C_l}{\partial t} + = \frac{ G(-Q_c) \frac{q_{cl}}{C_l^2} }{ 1 - \frac{q_{cl}}{C_l Q_c} } \; + \frac{1}{q_{cl}} \frac{\partial q_{cl}}{\partial t} +\label{eq:dcdt_hybrid_1} +\end{equation} + +Under homogeneous forcing (section \ref{sec:homog}), we defined the PDF height +at the saturation boundary when near the cloudy end of the PDF as +$G(-Q_c) = \frac{n+1}{n+2} \frac{C_l^2}{q_{cl}}$ (eq \ref{eqn20}). +In fact, $G(-Q_c)$ is set to some blend between this and the value near +the clear end of the PDF (eq \ref{eqn21}). But we will assume that +when eroding cloud under grid-mean subsaturated conditions ($Q_c < 0$), +$G(-Q_c)$ follows this scaling with $\frac{C_l^2}{q_{cl}}$ even if its +value differs somewhat from eq \ref{eqn20}. +Therefore the quantity $c_1 = G(-Q_c) \frac{q_{cl}}{C_l^2}$ remains constant +during the erosion process, and eq \ref{eq:dcdt_hybrid_1} becomes: + +\begin{equation} +\frac{1}{C_l} \frac{\partial C_l}{\partial t} + = \frac{ c_1 }{ 1 - \frac{q_{cl}}{C_l Q_c} } \; + \frac{1}{q_{cl}} \frac{\partial q_{cl}}{\partial t} +\label{eq:dcdt_hybrid_2} +\end{equation} + +The term $1 - \frac{q_{cl}}{C_l Q_c}$ (which is $> 1$ since we are considering +grid-mean subsaturation $Q_c < 0$) usually remains close to 1 in practice, +so we can assume its fractional variation over the timestep is small +compared to the other terms, and treat it explicitly. +We can therefore straightforwardly integrate eq \ref{eq:dcdt_hybrid_2} +to obtain the scaling of $C_l$ with $q_{cl}$ as both are reduced by erosion: + +\begin{equation} +\frac{C_l}{{C_l}_0} = \left( \frac{q_{cl}}{{q_{cl}}_0} \right)^{b_1} +\label{eq:cl_qcl_scaling} +\end{equation} + +where ${C_l}_0$, ${q_{cl}}_0$ are the values before erosion is applied, +and the exponent is $b_1 = \frac{ c_1 }{ 1 - \frac{q_{cl}}{C_l Q_c} }$. +When $G(-Q_c)$ takes its value from the cloudy end of the PDF, we have +$c_1 = \frac{n+1}{n+2}$. Since the PDF power $n > 0$ and $Q_c < 0$ +under the considered grid-mean subsaturation, we always have $b_1 < 1$. +This ensures that erosion reduces $C_l$ at a slower fractional rate than +$q_{cl}$, so that in-cloud water content $\frac{q_{cl}}{C_l}$ always decreases. + +Next we derive an integral solution for the decline of $q_{cl}$ with time. +Ignoring the cloud surface-area contributions from the levels above and below +(they are disabled in the code anyway), the erosion liquid water content +tendency is obtained by combining \ref{eq:dqcldt_hybrid} and \ref{eq:S_Cl}: + +\begin{equation} +\frac{\partial q_{cl}}{\partial t} = -K \, 2 C_l (1 - C_l) \, (q_{sat}(T)-q_v) +\label{eq:dqcldt_hybrid_1} +\end{equation} + +From eq \ref{SD2}, $q_{sat}(T)-q_v = \frac{SD}{a_L}$, where $SD$ is the +saturation defecit, and $a_L$ is the dimensionless factor defined in +eq \ref{eq:a_L}. Following the derivation in section +\ref{sec:smooth_initiation} (eq \ref{eq:qc_plus_sd}), +we can write this in terms of the liquid-water content: $SD = q_{cl} - Q_c$ +(where $Q_c$ was defined in eq \ref{eq:qc_eq_qt-qs}, and corresponds to the +grid-mean supersaturation converted to an equivalent liquid water content). +Substituting this into (\ref{eq:dqcldt_hybrid_1}) above, we obtain: + +\begin{equation} +\frac{\partial q_{cl}}{\partial t} = -\frac{K}{a_L} \, 2 C_l (1 - C_l) \, + (q_{cl}-Q_c) +\label{eq:dqcldt_hybrid_2} +\end{equation} + +Substituting eq \ref{eq:cl_qcl_scaling} for the leading factor of $C_l$ +on the right-hand-side and rearranging: + +\[ +\left( \frac{q_{cl}}{{q_{cl}}_0} \right)^{-b_1} \frac{\partial q_{cl}}{\partial t} + = -\frac{K}{a_L} \, 2 {C_l}_0 (1 - C_l) \, (q_{cl}-Q_c) +\] + +In significantly subsaturated conditions the r.h.s. has only weak dependence +on $C_l$ and $q_{cl}$ ($C_l << 1$, $q_{cl} << -Q_c$), so we can treat the +whole r.h.s. explicitly (i.e. neglect its variation during each timestep), +so that the above integrates to: + +\[ +\left[ \frac{{q_{cl}}_0}{1-b_1} \left( \frac{q_{cl}}{{q_{cl}}_0} \right)^{1-b_1} +\right]_{{q_{cl}}_0}^{{q_{cl}}_{\Delta t}} + = -\frac{K}{a_L} \, 2 {C_l}_0 (1 - C_l) \, (q_{cl}-Q_c) \Delta t +\] + +Inserting the limits of the integral on the l.h.s. and rearranging, +we obtain our analytical solution for $q_{cl}$ after time $\Delta t$: + +\begin{equation} +{q_{cl}}_{\Delta t} = {q_{cl}}_0 \left( 1 - \frac{1-b_1}{{q_{cl}}_0} + \frac{K}{a_L} \, 2 {C_l}_0 (1 - C_l) \, (q_{cl}-Q_c) \Delta t + \right)^\frac{1}{1-b_1} +\label{eq:qcl_int_hybrid} +\end{equation} + +Note that $q_{cl}$ falls to zero after a finite time +$\frac{{q_{cl}}_0}{1-b_1} \frac{a_L}{K} + \frac{1}{2 {C_l}_0 (1 - C_l) \, (q_{cl}-Q_c)}$. +If the timestep $\Delta t$ is longer than this time, then erosion +completely removes the cloud during the current timestep. + +We first set ${q_{cl}}_0$ and ${C_l}_0$ to the values already updated +by homogeneous forcing, and then sequentially compute the updated +$q_{cl}$ after erosion using (\ref{eq:qcl_int_hybrid}). +Then we substitute this value into (\ref{eq:cl_qcl_scaling}) to compute +the consistent updated value of $C_l$. +Finally, to improve accuracy, a small number of iterations are performed +to find the solution with the explicitly-treated terms +(the exponent $b_1 = \frac{ c_1 }{ 1 - \frac{q_{cl}}{C_l Q_c} }$ +and the terms $(1 - C_l)$ and $(q_{cl}-Q_c)$ in eq \ref{eq:qcl_int_hybrid}) +adjusted to values linearly-interpolated to half-way between the +start and end of the erosion timestep. + +\item {\bf Grid-mean supersaturation ($Q_c > 0$):} + +In this case, erosion does not act to reduce $C_l$ and $q_{cl}$ towards zero. +Instead, the narrow PDF limit it adjusts towards has +no remaining subsaturated air, so that $C_l = 1$ and $q_{cl} = Q_c$. +In this case, we can repeat the above derivation, but considering the +equations for the clear-fraction $1-C_l$ in place of $C_l$, +and the saturation defecit $SD = q_{cl}-Q_c$ in place of $q_{cl}$. +Assuming that $G(-Qc)$ follows the scaling for the clear end of the PDF +(\ref{eqn21}), this leads to a similar equation to (\ref{eq:cl_qcl_scaling}) +but for the scaling as erosion reduces $1-C_l$ and $SD$ towards zero .: + +\begin{equation} +\frac{1-C_l}{1-{C_l}_0} = \left( \frac{SD}{SD_0} \right)^{b_2} +\label{eq:ca_sd_scaling} +\end{equation} + +with $b_2 = \frac{ c_2 }{ 1 + \frac{SD}{(1-C_l) Q_c} }$ +and $c_2 = G(-Q_c) \frac{SD}{(1-C_l)^2}$ +(note we must have $0 < b_2 < 1$). + +And then the tendency equation for $SD$ is: + +\begin{equation} +\frac{\partial SD}{\partial t} = -\frac{K}{a_L} \, 2 (1 - C_l) C_l \, SD +\label{eq:dsddt_hybrid_2} +\end{equation} + +The one asymmetry between this and the $q_{cl}$ tendency equation +(\ref{eq:dqcldt_hybrid_2}) is that for $SD$ the r.h.s. is directly proportional +to the quantity in the time-derivative, whereas for $q_{cl}$ there is an +additional $Q_c$ term which is constant during erosion. +Substituting (\ref{eq:ca_sd_scaling}) for the leading factor of +$(1 - C_l)$ in (\ref{eq:dsddt_hybrid_2}), integrating over time $\Delta t$ +(neglecting the fractional variation of $C_l$ over the timestep) +and rearranging, we obtain: + +\begin{equation} +{SD}_{\Delta t} = {SD}_0 \left( 1 + b_2 + \frac{K}{a_L} \, 2 (1-{C_l}_0) C_l \, \Delta t + \right)^{-\frac{1}{b_2}} +\label{eq:sd_int_hybrid} +\end{equation} + +Note that the additional power of $SD$ on the r.h.s. of +(\ref{eq:dsddt_hybrid_2}) leads to the integral solution having +a negative exponent. This means that under grid-mean supersaturation, +erosion makes $SD$ and $1-C_l$ approach but never quite reach zero, +which is quite different behaviour to grid-mean subsaturation where +$q_{cl}$ and $C_l$ go to zero over a finite time. +This asymmetry is because the erosion rate is parameterised to be +proportional to $SD$, and this tends to zero as the PDF is narrowed +under supersaturation, but remains finite positive under subsaturation. + +We first set ${SD}_0 = {q_{cl}}_0 - Q_c$ (where as above ${q_{cl}}_0$ is +the value already updated by homogeneous forcing), +then compute the value of $SD$ updated by erosion using +(\ref{eq:sd_int_hybrid}). +Then we substitute this value into (\ref{eq:ca_sd_scaling}) to compute +the consistent updated value of $1-C_l$. +A small number of iterations are then performed +to find the solution with the explicitly-treated terms +(the exponent $b_2 = \frac{ c_2 }{ 1 + \frac{SD}{(1-C_l) Q_c} }$ +and the term $C_l$ in eq \ref{eq:sd_int_hybrid}) +adjusted to values linearly-interpolated to half-way between the +start and end of the erosion timestep. +Then the final values of $SD$ and $1-C_l$ are used to increment +$q_{cl} = Q_c + SD$ and $C_l$, as prognosed by the rest of the model. + +\item {\bf grid-mean saturation ($Q_c$ near-zero):} + +In this case, the PDF is centred on the +saturation boundary, so that narrowing it does not change the cloud-fraction. +In the limit $Q_c = 0$, we have $q_{cl} = SD$, and (\ref{eq:dqcldt_hybrid_2}) +or (\ref{eq:dsddt_hybrid_2}) becomes: + +\begin{equation} +\frac{1}{q_{cl}} \frac{\partial q_{cl}}{\partial t} + = -\frac{K}{a_L} \, 2 C_l (1 - C_l) +\end{equation} + +where everything on the r.h.s. is constant under erosion. +This simply integrates to give exponential decline of $q_{cl}$ +(and $SD$) towards zero: + +\begin{equation} +{q_{cl}}_{\Delta t} = {q_{cl}}_0 e^{ -\frac{K}{a_L} \, 2 C_l (1 - C_l) \Delta t } +\end{equation} + +\end{enumerate} + +\end{enumerate} + + +\subsection{Orographic and Gravity Wave Drag} +The Orographic and Gravity Wave Drag sections do not alter the temperature +or moisture content of the model gridboxes, hence PC2 assumes no change in the +condensate and cloud fractions as a result of these processes. + +\subsection{Advection} +\label{sec:advec} + +The advection of $\overline{q_{cl}}$ and $\overline{q_{cf}}$ are already +performed separately by the semi-Lagrangian advection scheme. +Advection of the three cloud fractions $C_l$, $C_i$ and $C_t$ are all +performed by PC2 in the same way. + +Note that ascent or subsidence by advection entails a pressure change following +each parcel, which will cause an accompanying adiabatic temperature change. +These advective pressure and temperature changes imply a homogeneous forcing, +which yields a change in $\overline{q_{cl}}$ and $C_l$ in addition to their +transport by the winds. This is described in section \ref{sec:pres}. + +If the UM namelist switch \textbf{l\_pc2\_sl\_advection} is turned on, +the PC2 homogeneous forcing response to advection is calculated +straight after the call to Semi-Lagrangian advection. +Otherwise, the pressure change from advection is combined with the +Eulerian pressure change from the dynamics Helmholtz solver, and the resulting +homogeneous forcing of liquid cloud is computed at the end of the timestep. + +\subsection{Boundary Layer} +\label{sec:bl} +At a basic level, the boundary layer scheme works by +mixing $\overline{q_T}$ and $\overline{T_L}$, and tracer mixing +$\overline{q_{cf}}$. The condensation and $C_l$ changes are represented +using the homogeneous forcing representation. The forcing of $Q_c$ can be +written in $\Delta \overline{q_T}$ and $\Delta \overline{T_L}$ terms +using (\ref{eq:deltaqc_exp}). + +$\overline{q_{cf}}$ is already mixed using the tracer mixing scheme. PC2 +will calculate the corresponding $C_i$ change assuming the inhomogeneous +forcing scenario. Although this is not necessarily an appropriate physical +model to use, it is the only generic physical model we have currently +developed in order to convert increments in a condensate to increments in +a cloud fraction. We use a value of the in-cloud water content $q_c^S$ +based upon a linear combination of the current in-cloud ice water +content, $\frac{\overline{q_{cf}}}{C_i}$, and a fixed value. + +\begin{equation} +q_C^S = C_i \frac{\overline{q_{cf}}}{C_i} + ( 1 - C_i) q_{cf0 \, BL} +\label{eq:qcf_ci} +\end{equation} + +where $q_{cf0 \, BL}$ is a specified value of $1 \times 10^{-4} \, kg \, kg^{-1}$. +We then use the inhomogeneous forcing equation based upon +(\ref{eq:dcdt_inhom2}) but for ice water content to write + +\begin{equation} +\Delta C_i = \frac{(1 - C_i)}{q_C^S - \overline{q_{cf}}} Q4_i . +\label{eq:deltaci_bl} +\end{equation} + +Since the physical model will have $C_i$ tend to 1 if the +denominator is small, we will, to avoid numerical problems, set +$C_i$ to 1 if $q_C^S - \overline{q_{cf}} < 1 \times 10^{-10} kg kg^{-1}$. +Note that we do not use the multiple phases injection source +expressions (section \ref{sec:multiple} and equation \ref{eq:cff_ts}). +This is because the liquid water changes are not associated with the plume model. + +Equation \ref{eq:qcf_ci} assumes that the change to the ice water content has led to an increase in ice water content. +However, if the ince water content has reduced, the change to the ice cloud fraction is not consistent. +The option to "Use consistent formulation of ice cloud fraction changes due to boundary-layer processes" ensure that +if the ice water content is reduced, the ice cloud fraction is reduced, in such as way as to maintain +the same in-cloud ice water content. + +The $C_t$ changes are calculated using the minimum overlap method of +section \ref{sec:ct}. + +In \textit{ni-imp-ctl} the control code inhibits the +call to the diagnostic cloud scheme +if there is deep or shallow convection occurring and the model level +is less than \textit{or equal to} the layer immediately above the top of the +boundary layer mixed layer (i.e. level ntml+1). This is in order to +ensure that there is +no large-scale cloud present below the base of the convective cloud, but +additionally performs this calculation at the level above, probably +in order that latent heating from large-scale condensation does not +inhibit the convection. A similar thing is performed for PC2, with +any large-scale cloud being evaporated if the same criteria are met, +\textit{except that it is not performed on the level above the boundary +layer mixed layer}. This choice (i.e. ntml) is seen to give improved +results in PC2, and is arguably a more physical reasonable choice +anyway than using ntml+1. + +\subsection{Convection} +\label{sec:convec} + +This section concentrates specifically upon the PC2 interface to the +convection scheme. In the current formulation of the UM, only a +mass-flux convection scheme exists, and this is what is described +below. Work to interface PC2 to the developing turbulence based +convection scheme is commented upon in section \ref{sec:tbcs}. + +An alternative way of calculating cloud fraction increments is currently under development and +is described in section \ref{sec:conv-simpler}. + +A traditional view of convective parametrization is a scheme that +transports vapour, $q$, +heat, $\theta$, and momentum, $u$ and $v$ winds within a single column. It +does not consider transport sideways to adjoining columns, and (at least +in the Gregory-Rowntree scheme used in the UM) is considered independent of +any resolved scale vertical air motions. This necessitates the view of +compensating subsidence within the column, whereas some conceptual models +of tropical convection would have the bulk of the ascent in the +convective cores and the +associated descent thousands of miles away in the downward branch of the +Hadley circulation. The parametrization schemes traditionally overlook the +existence of condensate in the model column. The non-PC2 version +of the mass-flux convection scheme used in the UM would have the same +large-scale liquid and ice prognostics before and after convection occurs +(apart from a bolt-on evaporation below convective cloud base), with no +regard at all to what happens to it or its effect on the rest of the +convection. Within PC2 we have had to work to more fully incorporate +the condensate into the convection scheme. + +\subsubsection{Introduction to the convective mass flux scheme} + +Within the mass flux scheme the net change in +$\overline{q_{cl}}$ and $C_l$ etc. +comes from two distinct sources. Firstly, the condensate and cloud +fraction injected from the plume (the $Q4$ terms, section \ref{sec:inhomog}); +secondly, the condensation response to the vapour and heat changes associated +with the detrainment and compensating subsidence. Strictly, we will see that the +$Q4$ terms also include the contribution to the condensate transport +by the compensating subsidence - this casts doubt on the validity of +the application of the injection forcing scenario to calculate the +equivalent cloud fraction change, since ideally the cloud fractions +ought to be transported by the compensating subsidence in a similar +way to the condensate transport (which is documented below). + +We therefore split the convective contribution in (\ref{eq:dqcldt_and_dcdt}) +into two parts: + +\begin{equation} +\frac{\partial \overline{q_{cl}}}{\partial t} |_{convection} = +Q4_l + Q_{environment} +\label{eq:inhomg_plus_homog} +\end{equation} + +where $Q_{environment}$ is the condensation associated with changes +in the vapour and temperature from the detrainment and compensating +subsidence. Similar splits are made for the cloud variables, where +the injection forcing, section \ref{sec:inhomog}, is used to calculate +the first term from $Q4_l$. Section \ref{subsect:q4calculation} looks +at the issue of the +calculation of $Q4_l$ etc., and section \ref{sec:conv_homog} looks at +the calculation of $Q_{environment}$, and its associated cloud +fraction change. We first look at the basic transport equations in a +mass flux convection scheme. + +\subsubsection{Basic Equations for a Convective Mass Flux Scheme} +\label{subsect:basmaseqs} + +We first consider a generic mass-flux scheme before its application to PC2. +As discussed by Grant and Stirling (personal communication), +the equations for convective +tendencies are most simply applied to a variable, ${\chi}$, that is conserved +under moist adiabatic processes (e.g. total water content). In this case, +% +\begin{equation} +{\pardbyd{\xsubsup{ }{E}}{t}}_{\rm{conv}} = +- \frac{1}{\ov{\rho}} \, \pardbyd{\ov{\rho w^{'} \xsubsup{ }{E'}}}{z} +\label{eq:chibasic} \end{equation} + +To parametrize \ref{eq:chibasic}, the current UM convection scheme takes a +mass flux approximation +% +\begin{equation} +\lp {\ov{\rho w^{'} \xsubsup{ }{E'}}} \rp_{\rm{conv}} = M^{\rm{P}} \, +\lp { \xsubsup{ }{P} - \xsubsup{ }{E} } \rp +\label{eq:massflux} \end{equation} +% +which can be differentiated to give +% +\begin{equation} +- \frac{1}{\ov{\rho}} \, \pardbyd{\ov{\rho w^{'} \xsubsup{ }{E'}}}{z} = +\pardbyd{\xsubsup{ }{P} \, M^{\rm{P}}}{p} - +\xsubsup{ }{E} \, \pardbyd{M^{\rm{P}}}{p} - +M^{\rm{P}} \, \pardbyd{\xsubsup{ }{E}}{p} +\label{eq:eddyflux} \end{equation} + +The bulk cloud model plume equations for mass and ${\chi}$ are: +% +\begin{eqnarray} +- \pardbyd{M^{\rm{P}}}{p} & = & +\lp { \varepsilon \, M^{\rm{P}} - \mu \, M^{\rm{P}} - \delta \, M^{\rm{P}} } \rp +\label{eq:dbydpmassflux} \\ +- \pardbyd{\xsubsup{ }{P} \, M^{\rm{P}}}{p} & = & \lp { +\varepsilon \, M^{\rm{P}} \, \xsubsup{ }{E} +- \mu \, M^{\rm{P}} \, \xsubsup{ }{R} - \delta \, M^{\rm{P}} \, \xsubsup{ }{P} +} \rp \label{eq:dbydpmfchi} +\end{eqnarray} + +Equations \ref{eq:eddyflux}, \ref{eq:dbydpmassflux} and +\ref{eq:dbydpmfchi} can then be substituted into \ref{eq:chibasic} to give: +% +\begin{equation} +{\pardbyd{\xsubsup{ }{E}}{t}}_{\rm{conv}} = +- M^{\rm{P}} \, \pardbyd{\xsubsup{ }{E}}{p} ++ \mu \, M^{\rm{P}} \, \lp { \xsubsup{ }{R} - \xsubsup{ }{E} } \rp ++ \delta \, M^{\rm{P}} \, \lp { \xsubsup{ }{P} - \xsubsup{ }{E} } \rp +\label{eq:chimassflux} \end{equation} +% +while \xsubsup{}{P} is obtained from the vertical gradient derived by combining +\ref{eq:dbydpmassflux} and \ref{eq:dbydpmfchi} : +% +\begin{equation} +M^{\rm{P}} \, \pardbyd{\xsubsup{ }{P}}{p} = +\varepsilon \, M^{\rm{P}} \, \lp { \xsubsup{ }{P} - \xsubsup{ }{E} } \rp - +\mu \, M^{\rm{P}} \, \lp { \xsubsup{ }{P} - \xsubsup{ }{R} } \rp +\label{eq:gradchipar} \end{equation} + +Within the model, eqn~\ref{eq:chimassflux} would take a discretized form +which actually depends upon whether the model level, k, is above or at the +lowest cloud level (k = cb). Note that the formal cloud base lies at the +half-level below, i.e. on the layer boundary which is also the top of the +turbulent mixed boundary layer. A simple discretized form of +\ref{eq:chimassflux}, setting ${ \mu = 0 }$, is: +% +\begin{eqnarray} +{\pardbyd{\xsubsup{ }{E}}{t}}_{\rm{conv, \, k}} & = & m_{\rm{k+1/2}} \, +\frac{ \lp {\xsubsup{k+1}{E} - \xsubsup{k}{E}} \rp } +{{\Delta z}_{\rm{k \, \rightarrow \, k+1}}} ++ {\delta}_{\rm{k}} \, m_{\rm{k}} \, \lp { \xsubsup{k}{P} - \xsubsup{k}{E} } \rp +\qquad \ldots \; \mbox{for k $>$ cb} \label{eq:chidisck} \\ +{\pardbyd{\xsubsup{ }{E}}{t}}_{\rm{conv, \, cb}} & = & m_{\rm{cb+1/2}} \, +\frac{ \lp {\xsubsup{cb+1}{E} - \xsubsup{cb}{E}} \rp } +{{\Delta z}_{\rm{cb \, \rightarrow \, cb+1}}} +- m_{\rm{cb}} \, +\lp { \xsubsup{i,cb}{P} - \xsubsup{cb}{E} } \rp \label{eq:chidisccb} +\end{eqnarray} +% +where the initial parcel value \xsubsup{i,cb}{P} may be chosen to produce a +fixed increment or place a closure condition on the cloud base flux. In fact, +the convection equations (see \citeumdp{027}) differ from \ref{eq:chidisck} and +\ref{eq:chidisccb} because a different discretization is used, but the +principle is unaltered. + +The model convection variables are NOT conserved under moist adiabatic processes +because precipitation processes deplete the column moisture and condensation +processes affect the temperature, specific humidity and cloud condensate +variables. Surprisingly, however, the form of eqn~\ref{eq:chimassflux} is +retained even though the basic equation \ref{eq:chibasic} acquires additional +terms for temperature and specific humidity: +% +\begin{eqnarray} +{\pardbyd{\tsubsup{ }{E}}{t}}_{\rm{conv}} = Q1 & \equiv & +\lp { \frac{L}{c_{P}} } \rp \, {\ov{Q}}_{\rm{par}} +- \frac{1}{\ov{\rho}} \, \pardbyd{\ov{\rho w^{'} \tsubsup{ }{E'}}}{z} +\label{eq:defineq1} \\ +{\pardbyd{\qsubsup{ }{E}}{t}}_{\rm{conv}} = Q2 & \equiv & - {\ov{Q}}_{\rm{par}} +- \frac{1}{\ov{\rho}} \, \pardbyd{\ov{\rho w^{'} \qsubsup{ }{E'}}}{z} +\label{eq:defineq2} +\end{eqnarray} +% +where ${\ov{Q}}_{\rm{par}}$ is the rate of condensation which occurs in the +ascending plumes. + +The reason that \ref{eq:defineq1} and \ref{eq:defineq2} retain this form +is due to cancellation from the bulk cloud terms equivalent to +\ref{eq:dbydpmfchi} which are modified in the same way as +\ref{eq:defineq1} and \ref{eq:defineq2}. The change is seen in the +vertical gradient equations based upon \ref{eq:gradchipar} +% +\begin{eqnarray} +M^{\rm{P}} \, \pardbyd{\tsubsup{ }{P}}{p} & = & +\varepsilon \, M^{\rm{P}} \, \lp { \tsubsup{ }{P} - \tsubsup{ }{E} } \rp - +\mu \, M^{\rm{P}} \, \lp { \tsubsup{ }{P} - \tsubsup{ }{R} } \rp - +\lp { \frac{L}{c_{P}} } \rp \, {\ov{Q}}_{\rm{par}} \label{eq:gradtpar} \\ +M^{\rm{P}} \, \pardbyd{\qsubsup{ }{P}}{p} & = & +\varepsilon \, M^{\rm{P}} \, \lp { \qsubsup{ }{P} - \qsubsup{ }{E} } \rp - +\mu \, M^{\rm{P}} \, \lp { \qsubsup{ }{P} - \qsubsup{ }{R} } \rp + +{\ov{Q}}_{\rm{par}} \label{eq:gradqpar} \\ +M^{\rm{P}} \, \pardbyd{\lsubsup{ }{P}}{p} & = & +\varepsilon \, M^{\rm{P}} \, \lp { \lsubsup{ }{P} - \lsubsup{ }{E} } \rp +- {\ov{Q}}_{\rm{par}} + PPN \label{eq:gradlpar} +\end{eqnarray} + +The final calculation of rates in the current condensation scheme (\citeumdp{027}, +section 10) assumes a further condensation term, ${\ov{Q}}_{\rm{reset}}$, which +acts to make the net rate of change of condensate equal zero, and a final +assumption is made that the environment values of condensate remain zero (and +also that \lsubsup{ }{R} = \lsubsup{ }{P}). The +result is basic equations +% +\begin{eqnarray} +{\pardbyd{\tsubsup{ }{E}}{t}}_{\rm{conv}} & = & Q1 - +\lp { \frac{L}{c_{P}} } \rp \, {\ov{Q}}_{\rm{reset}} +\label{eq:basictold} \\ +{\pardbyd{\qsubsup{ }{E}}{t}}_{\rm{conv}} & = & Q2 + {\ov{Q}}_{\rm{reset}} +\label{eq:basicqold} \\ +0 \equiv {\pardbyd{\lsubsup{ }{E}}{t}}_{\rm{conv}} & = & {\ov{Q}}_{\rm{par}} - +{\ov{Q}}_{\rm{reset}} - PPN +- \frac{1}{\ov{\rho}} \, \pardbyd{\ov{\rho w^{'} \lsubsup{ }{E'}}}{z} \nonumber \\ +& = & +\mu \, M^{\rm{P}} \, \lsubsup{ }{P} + \delta \, M^{\rm{P}} \, \lsubsup{ }{P} - +{\ov{Q}}_{\rm{reset}} +\label{eq:basiclold} +\end{eqnarray} + +By analogy with equations \ref{eq:defineq1} and \ref{eq:defineq2}, we can +define a $Q4$ from \ref{eq:basiclold} and state that for the control +convection scheme $Q4 = 0$. The PC2 scheme requires a reassessment of these +assumptions because we wish to allow non-zero environment condensate values and +to allow them to change. + +\subsubsection{Calculation of Grid-Box Averaged Condensate Rate (Q4)} +\label{subsect:q4calculation} + +The PC2 condensation scheme allows convection to feed cloud condensate (ice or +liquid) directly into the large scale and to update the cloud amount accordingly. + +Define +% +\begin{eqnarray} +\lp { \pardbyd{\lsubsup{l}{ }}{t} } \rp_{\rm{conv}} = Q4_{\rm{l}} & \equiv & +{\ov{Q}}_{\rm{l, par}} - {\ov{Q}}_{\rm{l, reset}} - RAIN - +\frac{1}{\ov{\rho}} \, \pardbyd{\ov{\rho w^{'} \lsubsup{l}{'}}}{z} +\label{eq:defineq4l} \\ +\lp { \pardbyd{\lsubsup{f}{ }}{t} } \rp_{\rm{conv}} = Q4_{\rm{f}} & \equiv & +{\ov{Q}}_{\rm{f, par}} - {\ov{Q}}_{\rm{f, reset}} - SNOW - +\frac{1}{\ov{\rho}} \, \pardbyd{\ov{\rho w^{'} \lsubsup{f}{'}}}{z} +\label{eq:defineq4f} +\end{eqnarray} +% +where the PC2 assumption thus far has been that ${\ov{Q}}_{\rm{l, reset}} = 0 += {\ov{Q}}_{\rm{f, reset}}$. + +\begin{itemize} +\item{The current convection scheme assumes that parcel +condensate is single phase +(ie. either all liquid or all frozen) and this is seriously hard-wired into the +code. Thus we can treat the precipitation and parcel condensation +processes in $ Q4_{\rm{l}} $ and $ Q4_{\rm{f}} $ separately without worrying +about cross-transfer between the two because at most only one set will ever be +active in a given grid box at one time. However, even for the inactive (zero +parcel condensate) phase, convection mixes environmental air into the +parcel and +can therefore maintain a non-zero $Q4$. Enablement of multiple phase condensate +in the current scheme is a task requiring great caution as the formulations +are extremely sensitive to errors in assignment of condensate phase.} +\end{itemize} + +Based on \ref{eq:gradlpar}, the vertical dependence of condensate is +calculated as +% +\begin{eqnarray} +\pardbyd{\lsubsup{l}{P}}{p} & = & \varepsilon \, +\lp { \lsubsup{l}{P} - \lsubsup{l}{E} } \rp - +\frac{{\ov{Q}}_{\rm{l, par}}}{M^{\rm{P}}} - +\frac{RAIN}{M^{\rm{P}}} \label{eq:vertparl} \\ +\pardbyd{\lsubsup{f}{P}}{p} & = & \varepsilon \, +\lp { \lsubsup{f}{P} - \lsubsup{f}{E} } \rp - +\frac{{\ov{Q}}_{\rm{f, par}}}{M^{\rm{P}}} - +\frac{SNOW}{M^{\rm{P}}}\label{eq:vertparf} +\end{eqnarray} + +Following \citeumdp{027}, equations \ref{eq:dbydpmassflux}, \ref{eq:vertparl} and +\ref{eq:vertparf} are discretized: +% +\begin{eqnarray} +M_{\rm{k} + 1} & = & M_{\rm{k}} \, +\lp { 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp \, +\lp { 1 - \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp \, +EPSS_{\rm{k}} \label{eq:discdmfbydp} \\ +\lsubsup{l \, k + 1}{P} & = & \lp { +\lsubsup{l \, k}{P} + +\varepsilon_{\rm{k} + 1/4} \, \Delta p_{\rm{k} + 1/4} \, \lsubsup{l \, k}{E} + +\varepsilon_{\rm{k} + 3/4} \, \Delta p_{\rm{k} + 3/4} \, +\lc {1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4}} \rc \, +\lsubsup{l \, k + 1}{E} +} \rp \, / \, \lp {EPSS_{\rm{k}}} \rp \nonumber \\ +{ } & { } & + \lp { {\ov{Q}}_{\rm{l} \, \rm{k} + 1} \, / \, M_{\rm{k} + 1}} \rp +- \lp { RAIN_{\rm{k} + 1} \, / \, M_{\rm{k} + 1} } \rp +\label{eq:discvparl} \\ +\lsubsup{f \, k + 1}{P} & = & \lp { +\lsubsup{f \, k}{P} + +\varepsilon_{\rm{k} + 1/4} \, \Delta p_{\rm{k} + 1/4} \, \lsubsup{f \, k}{E} + +\varepsilon_{\rm{k} + 3/4} \, \Delta p_{\rm{k} + 3/4} \, +\lc {1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4}} \rc \, +\lsubsup{f \, k + 1}{E} +} \rp \, / \, \lp {EPSS_{\rm{k}}} \rp \nonumber \\ +{ } & { } & + \lp { {\ov{Q}}_{\rm{f} \, \rm{k} + 1} \, / \, M_{\rm{k} + 1}} \rp +- \lp { SNOW_{\rm{k} + 1} \, / \, M_{\rm{k} + 1} } \rp +\label{eq:discvparf} +\end{eqnarray} +% +where $EPSS_{\rm{k}} = +\lp {1 + \varepsilon_{\rm{k} + 3 / 4} \, \Delta p_{\rm{k} + 3 / 4}} \rp \, +\lp {1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4}} \rp $. + +The condensation and precipitation terms in equations \ref{eq:discdmfbydp}, +\ref{eq:discvparl} and \ref{eq:discvparf} make the equations implicit. +They are therefore solved by starting with an ascent in which condensation and +precipitation terms are suppressed: +% +\begin{eqnarray} +\lsubsup{l \, k + 1}{P} & = & \frac{\lp { +\lsubsup{l \, k}{P} + +\varepsilon_{\rm{k} + 1/4} \, \Delta p_{\rm{k} + 1/4} \, \lsubsup{l \, k}{E} + +\varepsilon_{\rm{k} + 3/4} \, \Delta p_{\rm{k} + 3/4} \, +\lc {1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4}} \rc \, +\lsubsup{l \, k + 1}{E} +} \rp}{EPSS_{\rm{k}}} \label{eq:discvparldry} \\ +\lsubsup{f \, k + 1}{P} & = & \frac{\lp { +\lsubsup{f \, k}{P} + +\varepsilon_{\rm{k} + 1/4} \, \Delta p_{\rm{k} + 1/4} \, \lsubsup{f \, k}{E} + +\varepsilon_{\rm{k} + 3/4} \, \Delta p_{\rm{k} + 3/4} \, +\lc {1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4}} \rc \, +\lsubsup{f \, k + 1}{E} +} \rp}{EPSS_{\rm{k}}} \label{eq:discvparfdry} +\end{eqnarray} +% +At the base of the convective plume (ie. the level immediately above cloud +base), \lsubsup{l \, k}{P} is initialized to \lsubsup{l \, i}{P} and +\lsubsup{f \, k}{P} to \lsubsup{f \, i}{P}, where the initial values are chosen +such that the modified form of \ref{eq:chidisccb} produces zero fluxes at +cloud base: +% +\begin{eqnarray} +Q4_{\rm{l}}(cb) = 0 & = & M_{\rm{cb+1/2}}^{\rm{P}} \, +\pardbyd{\lsubsup{l}{E}}{p} - M_{\rm{cb}}^{\rm{P}}\, +\lp { \lsubsup{l}{P \, i} - \lsubsup{l}{E}(\rm{cb}) } \rp \label{eq:q4lcbi} \\ +Q4_{\rm{f}}(cb) = 0 & = & M_{\rm{cb+1/2}}^{\rm{P}} \, +\pardbyd{\lsubsup{f}{E}}{p} - M_{\rm{cb}}^{\rm{P}}\, +\lp { \lsubsup{f}{P \, i} - \lsubsup{f}{E}(\rm{cb}) } \rp \label{eq:q4fcbi} +\end{eqnarray} + + +As the convection scheme makes the single phase assumption for parcel +condensate, it may be necessary to melt or freeze entrained condensate at this +point and adjust the temperature accordingly. +% +\begin{eqnarray} +\theta_{\rm{k + 1}}^{\rm{P}} = \theta_{\rm{k + 1}}^{\rm{P}} - +\lp \frac{L_{\rm{F}}}{C_{p} \, \Pi_{\rm{k + 1}}} \rp \, \lsubsup{f \, k + 1}{P} +& \; \ldots \; & \mbox{ if \lsubsup{f \, k + 1}{P} is melted } +\label{eqn:meltlf} \\ +\theta_{\rm{k + 1}}^{\rm{P}} = \theta_{\rm{k + 1}}^{\rm{P}} + +\lp \frac{L_{\rm{F}}}{C_{p} \, \Pi_{\rm{k + 1}}} \rp \, \lsubsup{l \, k + 1}{P} +& \; \ldots \; & \mbox{ if \lsubsup{l \, k + 1}{P} is frozen } +\label{eqn:freezell} +\end{eqnarray} + +Once a final value for the condensation term +$ {\ov{Q}}_{\rm{x} \, \rm{k} + 1} \, / \, M_{\rm{k} + 1} $ has been calculated +from the parcel specific humidity equations, it can then be added to the parcel +condensate to give a final pre-precipitation value. + +\begin{itemize} +\item{In practice, the rates $ {\ov{Q}}_{\rm{x} \, \rm{k} + 1}$ and +$ PPN $ are not calculated explicitly in the code. +Instead, their effect is applied directly as increments to the temperature and +moisture fields.} +\end{itemize} + +The precipitation calculation is unaltered. +% +\begin{equation} +P_{\rm{k} + 1} = \lp { \lsubsup{k + 1}{P} - \lsubsup{MIN}{P} } \rp \, +M_{\rm{k} + 1} \, / \, g +\label{eq:precip} \end{equation} +% +where \lsubsup{k + 1}{P} = \lsubsup{l \, k + 1}{P} + \lsubsup{f \, k + 1}{P}. + +\begin{itemize} +\item{Actually, given that the precipitation calculation appears to be +based upon the hydrostatic equation, it is debatable whether it is even suitable +for use with the New Dynamics model and I guess therefore that this needs +revisiting at some point.} +\end{itemize} + +This reduces the parcel condensate to : +% +\begin{eqnarray} +\lsubsup{l \, k + 1}{P} & = & \lp { +\frac{\lsubsup{l \, k + 1}{P}}{\lsubsup{k + 1}{P}} +} \rp \, \lsubsup{MIN}{P} \label{eq:vparlfinal} \\ +\lsubsup{f \, k + 1}{P} & = & \lp { +\frac{\lsubsup{f \, k + 1}{P}}{\lsubsup{k + 1}{P}} +} \rp \, \lsubsup{MIN}{P} \label{eq:vparffinal} +\end{eqnarray} + +The final parcel condensate values are then used in the rate calculation based +upon eqn~\ref{eq:basiclold}: +% +\begin{eqnarray} +Q4_{\rm{l}}(k) & = & M_{\rm{k+1/2}}^{\rm{P}} \, \pardbyd{\lsubsup{l}{E}}{p} + +\lp { {\mu}_{\rm{k}} \, M_{\rm{k}}^{\rm{P}} + +{\delta}_{\rm{k}} \, M_{\rm{k}}^{\rm{P}} } \rp \, +\lp { \lsubsup{l}{P}(\rm{k}) - \lsubsup{l}{E}(\rm{k}) } \rp - +{\ov{Q}}_{\rm{l, reset}} \label{eq:q4lmassf} \\ +Q4_{\rm{f}}(k) & = & M_{\rm{k+1/2}}^{\rm{P}} \, \pardbyd{\lsubsup{f}{E}}{p} + +\lp { {\mu}_{\rm{k}} \, M_{\rm{k}}^{\rm{P}} + +{\delta}_{\rm{k}} \, M_{\rm{k}}^{\rm{P}} } \rp \, +\lp { \lsubsup{f}{P}(\rm{k}) - \lsubsup{f}{E}(\rm{k}) } \rp - +{\ov{Q}}_{\rm{f, reset}} \label{eq:q4fmassf} +\end{eqnarray} + + +Note that, as a side-effect, the \citeumdp{027} environment equations for potential +temperature and specific humidity are also altered because the condensate is no +longer re-evaporated at the end (${\ov{Q}}_{\rm{l, reset}} = 0 += {\ov{Q}}_{\rm{f, reset}}$): +% +\begin{eqnarray} +\frac{\Delta \, \theta_{\rm{k}}^{\rm{E}}}{\Delta \, t} = +\lp \frac{ M_{\rm{k}} }{ \Delta \, p_{\rm{k}} } \rp +\lc { +\lp { 1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4} } \rp +\lp { 1 - \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp +\lp { 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp +\lp { \theta_{\rm{k + 1}}^{\rm{E}} - \theta_{\rm{k}}^{\rm{E}} } \rp +} \right . & + & \nonumber \\ +\lp { \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp +\lp { 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp +\lp { \theta_{\rm{k}}^{\rm{R}} - \theta_{\rm{k}}^{\rm{E}} } \rp +& + & \nonumber \\ +\left . { +\lp { \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp +\lp { \theta_{\rm{k}}^{\rm{P}} - \theta_{\rm{k}}^{\rm{E}} } \rp +} \rc & { } & \label{eq:enviroth} +\end{eqnarray} +% +and +% +\begin{eqnarray} +\frac{\Delta \, q_{\rm{k}}^{\rm{E}}}{\Delta \, t} = +\lp \frac{ M_{\rm{k}} }{ \Delta \, p_{\rm{k}} } \rp +\lc { +\lp { 1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4} } \rp +\lp { 1 - \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp +\lp { 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp +\lp { q_{\rm{k + 1}}^{\rm{E}} - q_{\rm{k}}^{\rm{E}} } \rp +} \right . & + & \nonumber \\ +\lp { \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp +\lp { 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp +\lp { q_{\rm{k}}^{\rm{R}} - q_{\rm{k}}^{\rm{E}} } \rp +& + & \nonumber \\ +\left . { +\lp { \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp +\lp { q_{\rm{k}}^{\rm{P}} - q_{\rm{k}}^{\rm{E}} } \rp +} \rc & { } & \label{eq:enviroq} +\end{eqnarray} + +Similarly, eqns \ref{eq:q4lmassf} and \ref{eq:q4fmassf} have +a discretized form as follows: +% +\begin{eqnarray} +\frac{\Delta \, \lsubsup{l \, k}{E}}{\Delta \, t} = +\lp \frac{ M_{\rm{k}} }{ \Delta \, p_{\rm{k}} } \rp +\lc { +\lp { 1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4} } \rp +\lp { 1 - \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp +\lp { 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp +\lp { \lsubsup{l \, k + 1}{E} - \lsubsup{l \, k}{E} } \rp +} \right . & + & \nonumber \\ +\lp { \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp +\lp { 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp +\lp { \lsubsup{l \, k}{P} - \lsubsup{l \, k}{E} } \rp +& + & \nonumber \\ +\left . { +\lp { \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp +\lp { \lsubsup{l \, k}{P} - \lsubsup{l \, k}{E} } \rp +} \rc & { } & \label{eq:enviroll} +\end{eqnarray} +% +and +% +\begin{eqnarray} +\frac{\Delta \, \lsubsup{f \, k}{E}}{\Delta \, t} = +\lp \frac{ M_{\rm{k}} }{ \Delta \, p_{\rm{k}} } \rp +\lc { +\lp { 1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4} } \rp +\lp { 1 - \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp +\lp { 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp +\lp { \lsubsup{f \, k + 1}{E} - \lsubsup{f \, k}{E} } \rp +} \right . & + & \nonumber \\ +\lp { \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp +\lp { 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp +\lp { \lsubsup{f \, k}{P} - \lsubsup{f \, k}{E} } \rp +& + & \nonumber \\ +\left . { +\lp { \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp +\lp { \lsubsup{f \, k }{P} - \lsubsup{f \, k}{E} } \rp +} \rc & { } & \label{eq:envirolf} +\end{eqnarray} + +\subsubsection{Background condensation} +\label{sec:conv_homog} +The modification to the convective plume will result in the transport, +detrainment and entrainment of condensate, in addition to the +transport of vapour and heat. Although condensation processes within +the plume are treated, it does not treat condensation in the +environment, which is forced by the compensating subsidence. We wish +to relate the environmental increments of vapour and temperature +to a forcing that can be applied in the environment. Because we +know that any detrained air associated with detrained liquid water +from the plume must be saturated with respect to liquid water, we +are able to translate the environmental changes into forcings. + +Here we will consider that the vapour change in the gridbox is as a result of +\textit{saturated with respect to liquid water} air being injected +from the plume and background air being displaced. +We do not consider whether the background air is at saturation yet, for we wish +to derive the expression for the required condensation if this is not the case. +We consider only liquid water clouds, ice clouds have no background condensation +applied as we do not make the instantaneous condensation assumption. + +Hence we can write + +\begin{equation} +\Delta \overline{q} = \Delta C_S ( q_{sat liq}(\overline{T_{s}}) - \overline{q} ) ++ (1 - \Delta C_S) \Delta \overline{q_{background}} +\end{equation} + +where $\Delta C_S$ is the volume of plume air that is detrained into the gridbox, +as discussed by \cite{bwg03}. $T_s$ is the temperature of the air injected into +the gridbox by the plume. The first term is simply the difference +between the value of $q$ in the plume and what was previously in the gridbox, and +the second term is the effect of a background change of $q$ that will be applied +across the part of the gridbox that is not associated with the injected air. We write this as: + +\begin{equation} +(1 - \Delta C_S) \Delta \overline{q_{background}} = \Delta \overline{q} - +\Delta C_S ( q_{sat liq}(\overline{T_{s}}) - \overline{q} ) . +\label{eqn:1mcs} +\end{equation} + +Now we recognise that + +\begin{equation} +\Delta \overline{q} = Q2~ \Delta t +\label{eqn:Q2} +\end{equation} + +where $Q2$ is the rate of moistening of the whole gridbox due to convection. +Remember that, at this stage, we haven't done any condensation outside of the plume. +Hence to calculate the condensation we should apply the background change in $\overline{q}$ +as a uniform forcing for the background air. Hence (\ref{eqn:1mcs}) becomes, using +(\ref{eqn:Q2}), + +\begin{equation} +(1 - \Delta C_S) A_q |_{background} \Delta t = Q2 ~ \Delta t - \Delta C_S +( q_{sat liq}(T_{s}) - \overline{q} ) . +\label{eqn:Aq} +\end{equation} + +where $A_q |_{background}$ is the currently unknown background forcing of +$q$ (see \cite{gwb02}) and $\Delta t$ is the timestep. +We can do the same analysis for the temperature change, and obtain + +\begin{equation} +(1 - \Delta C_S) A_T |_{background} \Delta t = Q1~ \Delta t - +\Delta C_S (T_s - \overline{T} ) +\label{eqn:AT} +\end{equation} + +where Q1 is the rate of warming in the gridbox due to convection and $A_T |_{background}$ is +the currently unknown background forcing of temperature. + +The full change of liquid water content in the gridbox is that injected, +$Q4~\Delta t$, plus the amount of condensation in the background from +the uniform forcings (see +\cite{wg03}). Note that the uniform forcings are only applied across +a proportion $1 - \Delta C_S$ of the gridbox. Hence these two terms give, using the +homogeneous forcing equations (\ref{dqcldt}) and (\ref{eq:deltaqc_exp2}), + +\begin{equation} +\Delta \overline{q_{cl}} |_{convection} = Q4 \Delta t ++ (1 - \Delta C_S) a_L C_l (A_q |_{background} \Delta t +- \alpha A_T |_{background} \Delta t ). +\label{eqn:qclconv} +\end{equation} + +Using (\ref{eqn:Aq}) and (\ref{eqn:AT}) to expand the forcing terms in (\ref{eqn:qclconv}) gives + +\begin{equation} +\Delta \overline{q_{cl}} |_{convection} = Q4 \Delta t ++ \Delta t a_L C_l ( Q2 - \alpha Q1) - \Delta C_S a_L C_l +(q_{sat liq}(T_s)-\overline{q} - \alpha (T_s - \overline{T})) . +\end{equation} + +We now note that + +\begin{equation} +q_{sat} (T_s) - q_{sat liq} (\overline{T}) = \alpha (T_s - \overline{T} ) +\end{equation} + +and hence the final result + +\begin{equation} +\Delta \overline{q_{cl}} |_{convection} = Q4 \Delta t ++ \Delta t ~ a_L C_l ( ( Q2 - \alpha Q1) - \Delta C_S +(q_{sat liq}(\overline{T}) - \overline{q} ) ) . +\label{eqn:dqcl} +\end{equation} + +There is thus an extra term, $-\Delta C_S (q_{sat}(\overline{T})-\overline{q} )$, +which needs to be included in addition to the standard application of the homogeneous +forcing of $Q1$ and $Q2$ (this is represented +by the second term of the expression). This has arisen from the requirement that +the vapour injected by the plume is saturated. We need simply +to retrieve the value of $\Delta C_S$ to complete the parametrization. +This can be straightforwardly obtained +from (\ref{eq:dcldt_inhom}), which links the net change of liquid cloudy volume +due to the injection, $\Delta C_{injection}$, with $\Delta C_S$. + +\begin{equation} +\Delta C_{injection} = (g_l - C_l) \Delta C_S +\end{equation} + +where $g_l$ is 1 if the injected cloud is of liquid phase and 0 if it +is of ice phase. We already know $\Delta C_{injection}$ from the +injection forcing arguments (\ref{eq:dctdt_xl}) above that link it to $Q4$. +We therefore complete the parametrization by calculating $\Delta C_S$ based +on whether $\Delta C$ is positive or negative. If $\Delta C$ is positive, +we assume that the plume must be of liquid phase and hence + +\begin{equation} +\Delta C_S = \frac{\Delta C_{injection}} {1 -C_l} . +\label{eqn:cs1} +\end{equation} + +If $\Delta C_l$ is negative, we assume that the plume must be of ice phase and hence + +\begin{equation} +\Delta C_S = - \frac{\Delta C_{injection}} {C_l} . +\label{eqn:cs2} +\end{equation} + +Here we have still assumed that the vapour content in the detrained plume is equal to +$q_{sat liq}$. A better assumption may be to replace the $q_{sat liq}$ term in +(\ref{eqn:dqcl}) with a $q_{sat}$ expression that depends on the volume fraction +of detrained condensate that is liquid phase, $g_l$. + +If $\Delta C_{injection}$ is zero, we assume that $\Delta C_S$ is 0 also. Equations +(\ref{eqn:dqcl}),(\ref{eqn:cs1}), and (\ref{eqn:cs2}) form the parametrization +for $\Delta \overline{q_{cl}}|_{convection}$. The +representation of $\Delta C_{convection}$ is similar in form to +$\Delta \overline{q_{cl}}|_{convection}$: + +\begin{equation} +\Delta C |_{convection} = \Delta C_{injection} ++ \Delta t ~ a_L G(-Q_c) ( ( Q2 - \alpha Q1) - \Delta C_S (q_{sat}(\overline{T}) +- \overline{q} ) ) . +\end{equation} + +where the specification of $G(-Q_c)$ follows (\ref{eqn22}). Note +that the code includes the numerical limit restriction that +$\Delta C_S$ is between 0 and 1. + +Thus we are able to parametrize the net condensation and cloud changes +associated with the $Q1$ and $Q2$ terms in a physically more consistent way than +using simple homogeneous application of these terms. + +As an aside, we note that in the \cite{t93} scheme the condensation and cloud +fraction change associated with the compensating subsidence is taken out of +the convection term by adding the vertical motion associated with the compensating +subsidence to the large-scale vertical velocity before the +\cite{t93} equivalent of the homogeneous forcing term is applied. By doing +so it ensures that any balance between these two terms (as the tropical circulation +is commonly analysed to show) is removed before the net effect is calculated, +leading to more accurate numerical behaviour. + +\subsubsection{Homogeneous forcing of the environment by +convective-subsidence pressure change} + +To this end, the code includes an option to perform the homogeneous forcing +of liquid cloud by convection using the ``pressure forcing'' from the +convective subsidence, consistent with the pressure forcing by large-scale +advection (see sections \ref{sec:advec} and \ref{sec:pres}). +This approach replaces the above method of homogeneous forcing by convection +if the UM namelist switch \textbf{l\_pc2\_homog\_conv\_pressure} is turned on. +By applying the same homogeneous forcing method for advection and +convectively-forced subsidence, we should get the correct zero net +change in liquid cloud in the common situation where the large-scale ascent +and convective subsidence are in balance (implying no net vertical displacement +of environment parcels). + +Under this option, the increments to $\overline{q_{cl}}$ and $C_l$ produced +by the convection scheme are assumed to already include the effects +of entrainment, detrainment (i.e. injection) and compensating subsidence +(i.e. vertical advection) as expressed by equation \ref{eq:chimassflux}, +but exclude the effects of homogeneous forcing of clouds in the enviroment. +Note that taking equation \ref{eq:chimassflux} with $\chi$ set to +water vapour $q$, detrainment of saturated air into a subsaturated +environment will imply a positive tendency of $\overline{q}$, +but this is \textit{not} a homogeneous forcing, since the increase +in $\overline{q}$ is entirely due to injecting new parcels of saturated +air without altering the existing environment parcels. +Setting $\chi$ to be $\overline{q_{cl}}$ or $C_l$ in equation +\ref{eq:chimassflux}, there is a simply-calculated source of cloud +water and fraction wherever the detrained air is cloudy +($C_l=1$ in the detrained parcel), and we assume +these terms have been calculated this way inside the convection scheme. + +Since entrainment and detrainment do not constitute a homgeneous forcing +and are already accounted for in the convection scheme, + the only component of the convective forcing of liquid cloud +that needs to be done by the PC2 call after convection is the homogeneous +forcing by the subsidence term. This is in essence a vertical advection +(environmental forced descent by updrafts, or forced ascent by downdrafts). +The homogeneous forcing can be calculated from the expected pressure change +(and accompaying adiabatic temperature change) following the environment +as it is vertically displaced. +Conveniently, the UM already holds the convective mass-flux in units +of Pa s$^{-1}$, so it already expresses the pressure vertical velocity forced +by subsidence in the environment: + +\begin{equation} +\Delta p^E = \Delta t \left( M_{up} - M_{dwn} \right) +\label{eq:delta_p_conv} \end{equation} + +where $M_{up}$ is the updraft mass-flux, $M_{dwn}$ is the downdraft mass-flux, +and $\Delta t$ is the model timestep length. +The adiabatic temperature change following an environment parcel +subsided from pressure $p - \Delta p^E$ to $p$ is then given by: + +\begin{equation} +\Delta T^E = \theta^E \left( \left(\frac{p}{p_{ref}}\right)^\kappa + - \left(\frac{p - \Delta p^E}{p_{ref}}\right)^\kappa + \right) +\label{eq:delta_t_conv} \end{equation} + +where $\theta^E$ is the environment potential temperature, +$p_{ref}$ is the reference pressure used to define potential temperature, +and $\kappa = \frac{R_d}{c_p}$ is the ratio of the gas constant for dry air +over its heat capacity at constant pressure. +\ref{eq:delta_p_conv} and \ref{eq:delta_t_conv} are passed into the +PC2 homogeneous forcing routine after convection as the forcings +to be applied (with the forcings to all other variables set to zero). + + +\subsubsection{Convective cloud amount} +It is a debatable point whether the convective cloud fraction should +be set to zero. Although this was one of the original key concepts of +PC2, the cloud that is detrained from the convection scheme is into +the \textit{environment}, and does not represent the tower cloud. However, +it should be able to represent recently detrained cloudy air in a more +accurate way than by simply appealing to a diagnostic large-scale cloud scheme. +There are similar issues associated with the cloud fraction predicted +from the Tiedtke scheme. Probably the most consistent interpretation +is the inclusion of a tower cloud fraction within PC2, but not an +anvil cloud. However, we need to consider carefully any double +counting (or non-counting) implications. In the PC2:64 formulation, +we can represent the large optical depths associated +with new anvils, although we also tend to overestimate the optical +depth of shallow convective clouds. +Hence we choose to apply neither a diagnostic anvil or tower cloud, +so similar to Tiedtke, and let the large-scale cloud fraction represent +the convection completely. + +Strictly speaking these choices are independent of the PC2 scheme, +being simply choices that are available as part of the existing convection +scheme, but they are clearly directly related to the rest of the +cloud scheme formulation. + +\subsubsection{CAPE scaling} +The CAPE scaling option in the mass-flux convection scheme scales its +increments by the calculated values of $\frac{1}{CAPE} \frac{dCAPE}{dt}$. +This applies +also to all the PC2 calculated condensate and cloud fraction increments. +Additionally, in order to achieve reasonable mass flux profiles, +it has proved necessary to adjust the calculation of +$\frac{dCAPE}{dt}$ to use increments of +$\Delta \theta$ (potential +temperature) and $\Delta q$ calculated using a non-PC2 calculation +of these terms. Hence we consider any detrained condensate to have been +evaporated when we calculate $\frac{dCAPE}{dt}$. + +\subsubsection{Convective precipitation} +The amount of condensate detrained from convective plumes, and hence +the amount of moisture in the upper levels of the atmosphere, is very +dependent upon the amount of convective precipitation that is allowed +to fall from the column. The standard parametrization of this is that +any condensate greater than a specified value (dependent on $T$) +is precipitated, leaving the rest to be detrained. + +PC2 incorporates a tuning to this function of temperature by applying +the additional restriction that the limit may not fall to less than +$2 \times 10^{-4}~kg~kg^{-1}$. This implies a difference at temperatures +less than around $-42 ^{\circ} C$, with the tuning allowing less +precipitation and greater detrainment. This change is necessary +in order to produce thick enough anvil clouds. + +\subsubsection{Phase of condensate} +\label{sec:plume_phase} +The phase of the convective condensate \textit{carried in the plume} +is determined by a single phase change temperature TICE, with +condensate entirely in the +ice phase at colder temperatures and condensate entirely in the liquid +phase at warmer temperatures. For PC2:66, this temperature is -10 $^{\circ}$ C. + +\begin{equation} +\delta_{xl} = \left\{ \begin{array}{ll} + 1, & T_{plume} \ge -10 ^{\circ} C \\ + 0, & T_{plume} < -10 ^{\circ} C + \end{array} \right. +\end{equation} + +\begin{equation} +\delta_{xi} = \left\{ \begin{array}{ll} + 0, & T_{plume} \ge -10 ^{\circ} C \\ + 1, & T_{plume} < -10 ^{\circ} C + \end{array} \right. +\end{equation} + +\subsubsection{Tidier way of coupling convection and PC2} +\label{sec:conv-simpler} + +This area is still under development. But in brief, work is udner way to ensure that the +convective plume smoothly transitions from detraining liquid to detraining ice, rather +than using the abrupt change implied by the current formulation of the convection scheme. +Additionally, rather than using inhomogeneous increments to condensate (combining detrainment +and subsidence advection) to calculate cloud fraction increments, an alternative is to use +the detrainment of condensate to simply grow cloud fraction to ensure a specified in-cloud +liquid water content. The cloud fraction are then advected downwards byt he subsidence advection. + The increments to cloud fraction from detrainment and subsidence are then combined. + +\subsubsection{Prognostic dust approach} +A prognostic dust approach is implemented in the micro-physics scheme under +large-sale-precipitation where by the heterogeneous nucleation temperature +can be defined to vary three dimensionally globally as an arc-tangent +function of the mineral dust distribution in the model (documented +in \citeumdp{026}). By default, both liquid and ice are detrained simultaneously at the same +height, and the fraction of condensate that is ice linearly ramps as a function of temperature. +i.e. condensate is assumed to be all-liquid when T is greater than one tuneable threshold; all-ice +when T is less than another tuneable threshold, and vary linearly in-between (the threshold values are +given by starticeTkelvin and alliceTdegC in the UM cloud-scheme namelist. The new heterogeneous +nucleation temperatures calculated in the large-scale-precipitation are passed to the convection +scheme and are used as the above detrainment temperature thresholds by +maintaining a similar linear ramp. For e.g., condensate is +assumed to be all-liquid for T $\geq$ $tnuc_n$ and all-ice for T +$\leq$ $tnuc_n$ - 10.0 + +\subsubsection{Condensation adjustment in the profiles input to the +convection scheme} +\label{sec:conv_input_profs} + +The convection scheme itself is highly sensitive to the input environment +temperature and moisture profiles {\it before} the convection increments +(or PC2 response) are calculated. In particular, the parcel buoyancy +(and hence the CAPE and mass-flux scaling) maybe radically different +depending on whether a ``large-scale'' condensation / evaporation adjustment +is performed before the convection call. + +Where there is large-scale ascent, the profiles after Semi-Lagrangian advection +may have become supersaturated and unrealistically unstable, until the +expected condensation adjustment is performed. If the convection scheme +``sees'' these unrealistic intermediate profiles, it is likely to +predict an excessive, unrealistic mass-flux. + +To address this problem, there are two namelist switches that enable +additional condensation adjustments from PC2 before the convection call: + +\begin{itemize} +\item {\bf l\_pc2\_sl\_advection}: performs homogeneous forcing response +to Semi-Lagrangian advection immediately after the advection calculation, +instead of at the end of the timestep (see section \ref{sec:pres}). +\item {\bf l\_cloud\_call\_b4\_conv}: performs an additional call to +PC2 initiation (and PC2 checks) before the convection scheme +(see section \ref{sec:init2}). +This should catch any instances where large-scale ascent or other processes +have brought the profiles after advection to near or beyond saturation, +in grid-points where there was no liquid cloud already present +(and so no homogeneous forcing response). +\end{itemize} + +\subsection{Response to pressure changes} + \label{sec:pres} + +A pressure change following the parcel during the timestep will result +in an adiabatic temperature change which will force condensation, +hence we must include this temperature change forcing within PC2. +The majority of this pressure change comes from vertical advection +(although not all). +Remember that the advection (section \ref{sec:advec}), on its own, +does not cause condensation, it merely moves the existing cloud field. + + Using the semi-Lagrangian advection in the same way as is performed +for $\overline{q_{cl}}$ etc., the PC2 scheme will obtain the value of the +model prognostic \textit{Exner}, ($\prod$) on the departure points +($\prod_{dep}$). \textit{Exner} is defined as + +\begin{equation} +\label{eq:exner} +\prod = \frac{T}{\theta} = \left( \frac{p}{p_{ref}} \right)^{\kappa} +\end{equation} + +where $\theta$ is the potential temperature, $p_{ref}$ is a reference +pressure set to 1000 hPa, and $\kappa = +\frac{c_p - c_v}{c_p}$ , where $c_v$ is the heat capacity of dry +air at constant volume. The \textit{Exner} quantity is kept as a prognostic +variable in the model (this is unchanged from the control model), and the +value of $\prod$ on the departure points represents the initial value +in the timestep, since there is no update to $\prod$ until the end of +the timestep. After the second physics updates have been performed +(\textit{atmos-physics2}), the +model (including the control) recalculates the value of \textit{Exner} +($\prod^{[n+1]}$). +From $\prod_{dep}$ and $\prod^{[n+1]}$ we can calculate, using the definition +(\ref{eq:exner}), the values of departure pressure and temperature: + +\[ +\overline{p}_{dep} = p_{ref} {\prod_{dep}}^{\frac{1}{\kappa}} +\] + +\[ +\overline{T}_{dep} = \theta \prod_{dep} +\] + +Hence we obtain the net forcing values + +\begin{equation} +\Delta \overline{T} = \overline{T}^{[n+1]} - \overline{T}_{dep} +\label{eq:deltatsl} +\end{equation} + +and + +\begin{equation} +\Delta \overline{p} = \overline{p}^{[n+1]} - \overline{p}_{dep} . +\label{eq:deltapsl} +\end{equation} + +where $\overline{T}^{[n+1]}$ and $\overline{p}^{[n+1]}$ are the temperature +and pressure at the arrival point, after the dynamics call. +(\ref{eq:deltatsl}) and (\ref{eq:deltapsl}) are passed to the homogeneous +forcing routine in order to calculate +the condensation and cloud fraction changes associated with the pressure +change. + +We include this forcing towards the end of the timestep. There are two +reasons for this: +firstly, values of $\prod^{[n+1]}$ are not calculated by the control model +until after the physics is complete; secondly, it makes sense to locate this +process in the timestep in a similar location +to where the large-scale cloud scheme is included in the control (i.e. +after the implicit part of the boundary layer has finished). + +However, there +is a counter argument that says we should include this process immediately +after the dynamics, since we can then apply a forcing on an initial state +that has not already been modified by the dynamics, boundary layer and +convection schemes. This improves the numerics of the problem, since the +homogeneous forcing is designed to take time level n values as inputs. + +These issue are optionally addressed by turning on the UM namelist switch +\textbf{l\_pc2\_sl\_advection}. Under this switch, the PC2 homogeneous +forcing response to pressure change is split: +\begin{enumerate} +\item Forcing by the \textit{Lagrangian} component of pressure change, +performed immediately after the Semil-Lagrangian advection scheme +(before the call to atmos\_physics2). +This calculates the pressure change from the departure point value of +\textit{Exner} described above, to the start-of-timestep value of +\textit{Exner} at the arrival point. +\item Forcing by the \textit{Eulerian} component of pressure change, +performed at the end of the timestep (after the dynamics Helmholtz solver). +This calculates the pressure change from the start-of-timestep \textit{Exner} +at the arrival point, to the end-of-timestep \textit{Exner}. +\end{enumerate} + +Having to calculate the pressure forcing twice obviously adds some +computational cost, but has several advantages: +\begin{itemize} +\item As noted above, the PC2 homogeneous forcing calls can now take +as input the temperature and water-vapour content \textit{before} +the pressure change has been applied, as intended. This should improve +the numerical accuracy. +\item Most of the condensation or evaporation from the dynamics comes from +the \textit{Lagrangian} component of the pressure change, which has now moved +from the end of the timestep to before the dynamics Helmholtz solver. +This means that any latent heating from condensation forced by ascent is now +accounted for by the solver within the same timestep. +This improves the numerical accuracy of the dynamics-physics coupling. +\item If the condensation forced by resolved ascent is only added on at the +end of the timestep, the profiles passed into atmos\_physics2 can contain +out-of-balance thermodynamic states (e.g. if the profile has been lifted +by advection, it maybe supersaturated / unrealistically unstable before +the resulting condensation is added on). This may adversely affect the +convection scheme, which must act upon the profiles passed into +atmos\_physics2. +\end{itemize} + +The splitting of the pressure forcing call under the +\textbf{l\_pc2\_sl\_advection} switch was originally implemented to make the +profiles passed to convection more realistic. + +\subsection{Initiation} +\label{sec:init2} +As discussed in section \ref{sec:init}, there are occasions when +$\overline{q_{cl}}$ and $C_l$ need to be initiated from 0 or 1. +The application of the initiation is given in section \ref{sec:init}. +The initiation forms a new, separate block of PC2 code to perform this +calculation, and is located immediately following the pressure change +response (section \ref{sec:pres}). +Also, if the UM namelist switch {\bf l\_cloud\_call\_b4\_conv} is set to +true, an additional call to PC2 initiation is performed before the +convection scheme, to ensure that the condensation response to +advection and other forcings earlier in the timestep has been accounted for +in the profiles passed to the convection scheme, even if there was no +cloud already present for homogeneous forcing to act upon. +(see section \ref{sec:conv_input_profs}). + +There are currently 3 options for the conditions under-which initiation +may occur. For all of these options, +if using the bimodal cloud scheme to do initiation within PC2, +then the tests on $RH_T$ relative to $RH_{crit}$ are replaced by equivalent +tests for whether the saturation boundary lies within the bounds +of the bimodal scheme's assumed PDF, as described in section +\ref{sec:bimodal_init}. + +\subsubsection{``Original'' initiation logic} + +This option is selected by setting the UM namelist switch +{\bf i\_pc2\_init\_logic = 1} (Original) + +The initiation will be called if the liquid cloud fraction is either +0 or 1 and appropriate $RH$ criteria hold, along with other restrictions. +$C_l$ is initiated away from 0 if + +\begin{itemize} +\item{ $RH_T > RH_{crit} + RH_{crit \, tol}$ \textbf{and} } +\item{ Cumulus convection has {\em not} been diagnosed from the + boundary-layer in the current column \textbf{and} } +\item{ The current level is not below the surface mixed-layer LCL \textbf{and} } +\item{ $C_l = 0$ \textbf{and} } +\item{ $RH_T^{[n+1]} > RH_T^{[n]}$ ,} +\end{itemize} + +where $RH_{crit \, tol}$ is a specified tolerance parameter, of value 0.01, +and $RH_T$ is defined in (\ref{eq:rht}). $RH_T^{[n]}$ is the start of +timestep value of $RH_T$ (i.e. at time level n) and $RH_T^{[n+1]}$ is the +value when initiation is called. +Additionally, there is another possibility for the last of the relations. +This second option also allows initiation when the water +is supercooled: + +\begin{itemize} +\item{ $C_l < 0.05$ \textit{and} $\overline{T} < 0 ^{\circ} C$ .} +\end{itemize} + +Equivalently, $C_l$ is initiated away from 1 if + +\begin{itemize} +\item{ $RH_T < 2 - RH_{crit} - RH_{crit \, tol}$ \textbf{and} } +\item{ $C_l = 1$ \textbf{and}} +\item{ $RH_T^{[n+1]} < RH_T^{[n]}$ .} +\end{itemize} + +\subsubsection{``Simplified'' initiation logic} + +This option is selected by setting the UM namelist switch +{\bf i\_pc2\_init\_logic = 2} (Simplified) + +Under this option, the conditions for initiation are: + +Either: +\begin{itemize} +\item $RH_T > RH_{crit} + RH_{crit \, tol}$ \textbf{and} +\item $C_l < C_{tol}$ \textbf{and} +\item The current level is not below the surface mixed-layer LCL \textbf{and} +\item $RH_T^{[n+1]} > RH_T^{[n]}$ +\end{itemize} +Or: +\begin{itemize} +\item $RH_T < 2 - RH_{crit} - RH_{crit \, tol}$ \textbf{and} +\item $C_l > 1 - C_{tol}$ \textbf{and} +\item $RH_T^{[n+1]} < RH_T^{[n]}$ +\end{itemize} + +where $C_{tol}$ can be set via the UM namelist; its original standard value +is 0.005. Note this threshold is also used to remove small cloud-fractions +after initiation; see section \ref{sec:checks2}. + +This is very similar to the ``Original'' initiation logic described above, +but with the following differences: +\begin{itemize} +\item The condition that the boundary-layer hasn't diagnosed cumulus + convection in the column is removed. + Note that this condition spuriously suppressed initiation in the free + troposphere {\em above} any cumulus cloud produced by the convection + scheme. +\item $C_l$ only needs to be within a numerical tolerance $C_{tol}$ from + 0 or 1, rather than having to be {\it exactly} 0 or 1. +\item The different threshold when initiating super-cooled cloud is removed. +\end{itemize} + +\subsubsection{``Smooth'' initiation logic} +\label{sec:smooth_initiation} + +This option is selected by setting the UM namelist switch +{\bf i\_pc2\_init\_logic = 3} (Smooth) + +There is a fundamental numerical problem with the above options, in that +the initiation process is not permitted to have any effect at all unless +$C_l$ goes to (near) 0 or 1, but can predict values of $C_l$ very different +to 0 or 1 when it does activate. This leads to unphysical sudden noisy jumps +in $C_l$ and $q_{cl}$ when initiation occurs. +For example, if erosion causes $C_l$ to steadily decline, it will continue +to decline (even when the grid-mean $RH_T$ exceeds $RH_{crit}$) until +it reaches the threshold (0 or $C_{tol}$). At this point, initiation suddenly +increases $C_l$ and $q_{cl}$ to the values predicted by the diagnostic cloud +scheme. Erosion may then gradually remove them again, and the cycle repeats. +There is no physical reason for this internal mode of variability in the +scheme. + +Another problem arises if we consider the sensitivity to model resolution. +Suppose we have many adjacent small grid-boxes with similar $RH_T$, +a few containing cloud, the rest containing no cloud. If the whole +region cools to the point where $RH_T > RH_{crit}$, then new cloud +will initiate in the cloud-free grid-boxes, but not in the cloudy grid-boxes. +Now suppose we run a coarse-grained version of the same simulation; +the many small grid-boxes are replaced by a single grid-box containing the +average $C_l$ over the small grid-boxes. Since we now have just one +grid-box already containing partial cloud-cover, initiation of new cloud +can no longer occur anywhere. + +To address these problems, there is an option to use a much simpler / +numerically better-posed initiation method; +always allow the diagnostic cloud scheme to be called +(provided it is expected to predict nonzero cloud water, +i.e. $RH_T > RH_{crit}$ in the case of the Smith scheme). +The $q_{cl}$ predicted by the diagnostic cloud scheme is then taken +as a minimum limit applied to the prognostic $q_{cl}$. +This amounts to taking the diagnostic cloud scheme's assumed PDF as a minimum +allowed width to the actual prognostic moisture PDF. +The prognostic $C_l$ and $q_{cl}$ are incremented as follows: + +\begin{itemize} + +\item If ${q_{cl}}_{diag} > q_{cl}$: + +$\Delta q_{cl} = {q_{cl}}_{diag} - q_{cl} +\quad \refstepcounter{equation}(\theequation)\label{eq:dqcl_init}$ + + + \begin{itemize} + + \item If $Q_C < 0$: + + $\Delta C_{l} = \frac{\Delta q_{cl}}{{q_{cl}}_{diag}} + \left( {C_{l}}_{diag} - C_{l} \right) + \quad \refstepcounter{equation}(\theequation)\label{eq:dcl_init1}$ + + \item If $Q_C > 0$: + + $\Delta C_{l} = \frac{\Delta SD}{{SD}_{diag}} + \left( {C_{l}}_{diag} - C_{l} \right) + \quad \refstepcounter{equation}(\theequation)\label{eq:dcl_init2}$ + + \end{itemize} + +\item Otherwise: + +$\Delta q_{cl} = 0$ + +$\Delta C_{l} = 0$ + +\end{itemize} + +where the subscript $_{diag}$ denotes the liquid cloud water content and +fraction predicted by the diagnostic cloud scheme (either Smith or Bimodal). + +Equation \ref{eq:dcl_init1} simply sets the cloud-fraction to a weighted +mean of the pre-existing and diagnostic-scheme cloud-fractions, in proportion +to the fraction of the water content that was created by initiation +versus that which was already there. +If the pre-existing $q_{cl}$ is zero, \ref{eq:dqcl_init} and \ref{eq:dcl_init1} +simply set $q_{cl}$ and $C_l$ to their new diagnosed values, +as in the previous options. +Crucially, in the limit that the pre-existing $q_{cl}$ approaches +${q_{cl}}_{diag}$, the increments to $q_{cl}$ and $C_l$ smoothly go to zero. +This is important to make the initiation process numerically well-posed, +so that it yields a smooth, continuous solution. + +Note that when we are initiating from $C_l = 1$ instead of $C_l = 0$, +we expect the pre-existing $q_{cl}$ to be nonzero even when there is +no pre-existing sub-grid PDF width. In this case, the completely +uninitiated state will have zero saturation deficit $SD$, rather than +zero $q_{cl}$. Therefore, in this case the increment to $C_l$ is calculated +based on the fractional increase in $SD$ from initiation +(equation \ref{eq:dcl_init2}), instead of the fractional increase in $q_{cl}$. + +Whether to increment $C_l$ based on the increase in $q_{cl}$ or $SD$ is +determined based on the sign of $Q_C$, which is defined as in equation +\ref{eq:qc_eq_qt-qs} (reproduced here for clarity): + +\[ +Q_c = a_L \left( \overline{q_T} - q_{sat}(\overline{T_L}) \right) +\] + +The saturation deficit $SD$ is defined by equation \ref{SD2}: + +\[ +SD = a_L \left( q_{sat}(\overline{T}) - \overline{q} \right) +\] + +Under the reasonable approximation that $q_{sat}$ varies linearly between +$\overline{T}$ and $\overline{T_L}$, so that the values of +$\alpha$ and $a_L$ are the same in +both of these equations, and: + +\[ +q_{sat}(\overline{T_L}) = q_{sat}(\overline{T}) - \alpha \frac{L}{c_p} q_{cl} +\] + +we obtain: + +\begin{equation} +q_{cl} = Q_c + SD +\label{eq:qc_plus_sd} +\end{equation} + +It can be seen that when $Q_C > 0$ (total-water super-saturation), +it represents the value $q_{cl}$ would have if the whole grid-box +were saturated ($SD = 0$, $C_l = 1$). Note that $q_{cl}$ cannot fall below +$Q_C$, since $SD$ cannot be negative. +Since $Q_c$ is invariant under condensation / evaporation, we must have +$\Delta SD = \Delta q_{cl}$ +(hence the implementation of \ref{eq:dcl_init2} in the code simply uses +$q_{cl} - Q_c$ in place of $SD$, and $\Delta q_{cl}$ in place of $\Delta SD$). + +\subsubsection{Additional checks after PC2 initiation} + \label{sec:checks2} + +The initiation is followed immediately by a section of resetting code. +For numerical reasons, it is possible to obtain very low, but non zero, +values of $C_l$ (and equivalently values very close to, but not equal to, +1). The code will reset these clouds to either a fraction of 0 or 1, as +appropriate. We choose to apply these terms here and not in the +Bounds Checking part of the code (section \ref{sec:checks}) because +these are not required to obtain consistency between fields, but are +`tidying up' pieces of code, although they may reasonably also be +applied in the Bounds Checking. Care needs to be taken when choosing +the thresholds, since +we do not wish to reset small values that are genuinely created +by a physics scheme in the model. + +We first calculate $RH_T$ using (\ref{eq:rht}) and compare +this to the critical relative humidity, $RH_{crit}$. The liquid +cloud fraction will be reset to 1 if: +\begin{itemize} +\item{ $RH_T > 2 - RH_{crit}$ and $C_l \ge C_{high}$} +\item{ or $C_l \ge C_{high 2}$ } +\end{itemize} +where $C_{high}$ and $C_{high 2}$ are defined in \ref{eq:chigh-chigh2}. +The evaporation is done by calculating $SD$ using (\ref{SD2}) with (\ref{eq:a_L}) and +(\ref{eq:alpha_exp}) and evaporating the equivalent amount of liquid +into the gridbox to take it to saturation, according to +(\ref{eq:qsdcheck1}) below. + +Similarly, the equivalent check for low values of $RH_T$ is performed. +The liquid +cloud fraction will be reset to 0 if: +\begin{itemize} +\item{ $RH_T < RH_{crit}$ and $C_l \le C_{low}$} +\item{ or $C_l \le C_{low 2}$ .} +\end{itemize} +The remaining $\overline{q_{cl}}$ is evaporated into the gridbox +using (\ref{eq:qclcheck}) below. + +The thresholds $C_{high}$, $C_{high 2}$, $C_{low}$ and $C_{low 2}$ are +set using the parameters $C_{tol}$ and $C_{tol 2}$, according to: + +\begin{eqnarray} +C_{high} = 1 - C_{tol}, \nonumber \\ +C_{high 2} = 1 - C_{tol 2}, \nonumber \\ +C_{low} = C_{tol}, \nonumber \\ +C_{low 2} = C_{tol 2}, +\label{eq:chigh-chigh2} +\end{eqnarray} + +where the parameters $C_{tol}$ and $C_{tol 2}$ can be set via the UM namelist +variables {\bf cloud\_pc2\_tol} and {\bf cloud\_pc2\_tol\_2}. +The original standard values of these parameters are +$C_{tol} = 0.005$ and a lower value $C_{tol 2} = 0.001$. + +Investigations in SCM runs using the comorph convection scheme +(which behaves more smoothly and so typically gives smaller increments +to $C_l$ over a single timestep than other schemes which exhibit intermittent +behaviour) suggested these thresholds are too high to avoid spuriously +resetting physical values of $C_l$ to zero. Detrainment from sparse +shallow cumulus, or advection of cloud into a neighbouring grid-box +under light winds, commonly give increments which increase $C_l$ from zero +to a value less than $0.005$ in one timestep (but would eventually increase +$C_l$ to a significant value over subsequent timesteps if the checks did +not keep resetting $C_l$ to zero). + +Note that if these checks are relaxed by lowering the thresholds +$C_{tol}$ and $C_{tol 2}$ to near-zero, +similar checks are still performed independently by the bounds checking +described in section \ref{sec:checks}, but with a much lower +threshold of $C_{tol 3} = 1 \times 10^{-12}$. + +\subsection{Bounds checking} + \label{sec:checks} +Ideally, model prognostics would never become inconsistent with one another. +However, even although the mathematical solution of the governing equations +may be well behaved, due to numerical inaccuracies values may become +inconsistent. For the cloud and condensate quantities, there are a number +of consistencies that must apply. The bounds checking forms a subroutine +that will, if necessary, adjust $\overline{q}$, $\overline{q_{cl}}$, +$\overline{q_{cf}}$, $C_l$, $C_i$, $C_t$ and, for latent heating, +$\overline{T}$, to ensure consistency between these values. + +The bounds checking is performed three times during the timestep. Firstly, +after the parallel part of the physics (\textit{atmos-physics1}) is complete; +secondly, before the initiation (section \ref{sec:init}) is called; thirdly, +after the initiation is called. + +\subsubsection{} +Firstly, if $C_l > 1 - C_{tol 3}$ then $C_l$ is set to 1. +Accordingly, $C_t$ is set to 1 as well. +$C_{tol 3}$ is a tiny numerical tolerance set to $1 \times 10^{-12}$, +a value intended to be in the realm of floating point rounding error rather +than anything that represents a physical solution. + +\subsubsection{} +The second check is to reset $\overline{C_l}$ to zero. This may be performed +for two reasons. Firstly, if the amount of $\overline{q_{cl}}$ is very small +($\overline{q_{cl}} < q_{c0}$, where $q_{c0} = 1 \times 10^{-10} kg kg^{-1}$), +so we avoid carrying negligible, but non-zero values of $\overline{q_{cl}}$ +and $C_l$. Secondly, if $C_l < C_{tol 3}$ then we reasonably reset $C_l$ to zero. +$C_t$ gets reset, as it must if there is no liquid cloud, to be equal to $C_i$. + +\subsubsection{} +\label{sec:pc2_checks_sd} +The next check complements the first but updates the moisture fields. +We firstly calculate $SD$ using (\ref{SD2}) and +(\ref{eq:alpha_exp}). We then check whether $SD < 0$. +This check catches instances where we have grid-mean supersaturation, +which ought to be impossible (under the instantaneous condensation +assumption made by PC2, condensation should occur to instantly adjust +any supersaturated regions of the gridbox to saturation, so we +{\it must always} have $SD \ge 0$. +When this condition is violated, we condense water vapour to adjust to +grid-mean saturation. $-SD$ corresponds to the amount of vapour that must be +condensed to achieve this, so we have: + +\begin{eqnarray} +\overline{q} \leftarrow \overline{q} + SD \nonumber \\ +\overline{q_{cl}} \leftarrow \overline{q_{cl}} - SD \nonumber \\ +\overline{T} \leftarrow \overline{T} - \frac{L_c}{c_p} SD +\label{eq:qsdcheck1} +\end{eqnarray} + +The original version of this check on $SD$ +(which may increase $\overline{q_{cl}}$), made no accompanying changes to +liquid cloud fraction. However, increases in $\overline{q_{cl}}$ +without any increase in $C_l$ can lead to spurious high in-cloud condensate +which is then converted to rain by the microphysics at the next time-step. +There are currently 4 options for how to treat $C_l$ when increasing +$\overline{q_{cl}}$ under this saturation adjustment, selected by the +UM large-scale cloud namelist switch {\bf i\_pc2\_checks\_cld\_frac\_method}: +\begin{itemize} +\item {\bf i\_pc2\_checks\_cld\_frac\_method = 0} - +Original method; $C_l$ is left unaltered. +\item {\bf i\_pc2\_checks\_cld\_frac\_method = 1} - +Set $C_l$ and $C_t$ to 1. +\item {\bf i\_pc2\_checks\_cld\_frac\_method = 2} - +If $\overline{q_{cl}}$ and $C_l$ were already nonzero before the adjustment, +increase $C_l$ at the same fractional rate as $\overline{q_{cl}}$, so that +the in-cloud water content $\frac{\overline{q_{cl}}}{C_l}$ is conserved. +Otherwise, increase $C_l$ so-as to yield a prescribed in-cloud water +content set to 0.5 g kg$^{-1}$. $C_t$ is then increased by the same +amount as $C_l$, to maintain consistency. +\item {\bf i\_pc2\_checks\_cld\_frac\_method = 3} - +This is the same as option 2 above, except in the case where +$\overline{q_{cl}}$ or $C_l$ was zero before the adjustment. In this case, +$C_l$ is set based on an empirical power-law function of $\overline{q_{cl}}$. +\end{itemize} + +\subsubsection{} + +Next we check whether $SD > 0$, {\it and} $C_l = 1$ +(the first of our checks has ensured that $C_l$ is no greater than 1). +This check catches instances where we have total cloud-cover in a subsaturated +grid-box, which ought to be impossible (if the whole grid-box is full of liquid +cloud, then it must be at grid-mean saturation, i.e. $SD = 0$). +When this happens, we adjust $\overline{q}$ and $\overline{q_{cl}}$ +to take $SD$ to zero, +\textit{provided} that $\overline{q_{cl}} > SD$. Remember that $SD$ +corresponds to the amount of vapour that must be +\textit{evaporated} into the gridbox to give saturation, so we simply +make exactly the same adjustments as we do for removing supersaturated +states above (\ref{eq:qsdcheck1}), except that here $SD$ is positive rather +than negative. + +Our proviso that $\overline{q_{cl}} > SD$ ensures that we do not make +$\overline{q_{cl}}$ negative by this adjustment. +If $\overline{q_{cl}} < SD$, then we cannot bring the gridbox to saturation, +but it is still wrong to allow $C_l = 1$ in a subsaturated gridbox! +This was identified as a bug in the bounds-checking code, which sometimes +caused instances of $C_l = 1$ to spuriously persist in dry environments. +This behaviour is currently controlled by a temporary logical in the +{\bf temp\_fixes} namelist: +\begin{itemize} +\item If {\bf l\_pc2\_checks\_sdfix} is set to false, the code simply does +nothing when it finds instances of $C_l = 1$, $SD > 0$ and +$SD > \overline{q_{cl}}$, allowing such artefacts to persist. +\item If {\bf l\_pc2\_checks\_sdfix} is set to true, in these instances +we simply evaporate all the remaining liquid water, and reset $C_l$ to zero: +\begin{eqnarray} +\overline{q} \leftarrow \overline{q} + \overline{q_{cl}} \nonumber \\ +\overline{T} \leftarrow \overline{T} - \frac{L_c}{c_p} \overline{q_{cl}} +\nonumber \\ +\overline{q_{cl}} \leftarrow 0 \nonumber \\ +C_l \leftarrow 0\nonumber \\ +C_t \leftarrow C_i +\label{eq:qsdcheck2} +\end{eqnarray} +\end{itemize} + +\subsubsection{} +The next check is similar to above but for the $C_l = 0$ situation. + +If $\overline{q_{cl}} < q_{c0}$ or $C_l = 0$ then we evaporate the +small amount of $\overline{q_{cl}}$ that remains in the gridbox: + +\begin{eqnarray} +\overline{q} \leftarrow \overline{q} + \overline{q_{cl}} \nonumber \\ +\overline{q_{cl}} \leftarrow 0 \nonumber \\ +\overline{T} \leftarrow \overline{T} - \frac{L_c}{c_p} \overline{q_{cl}} +\label{eq:qclcheck} +\end{eqnarray} + +\subsubsection{} +Next, if $C_i > 1$ then $C_i$ is set to 1. Accordingly, $C_t$ is set to +1 as well. + +\subsubsection{} +The following check is on the ice water content, $\overline{q_{cf}}$, and +the ice fraction $C_i$. If $\overline{q_{cf}} < q_{c0}$ we simply condense some +vapour to remove the negative quantity. + +\subsubsection{} +However, instead of removing small amounts of +$\overline{q_{cf}}$ when $C_i = 0$ but $\overline{q_{cf}} > 0$, we choose instead to create +some $C_i$ to keep consistency. This is to allow small, but significant, +amounts of $\overline{q_{cf}}$ created by the microphysics scheme to +be maintained. + +\begin{equation} +C_i \leftarrow \frac { \overline{q_{cf}} }{q_{cf0}} +\label{eq:cf_reset} +\end{equation} + +where the `in-cloud' ice content $q_{cf0} = 1 \times 10^{-4} kg kg^{-1}$. + +\subsubsection{} +The next two checks are on the total cloud fraction, $C_t$, to ensure +that it takes on a value that is physically possible, given the values +of $C_l$ and $C_i$. We have, firstly, the maximum overlap situation and +then the minimum overlap situation. + +\begin{eqnarray} +C_t \leftarrow \text{Max}( C_t, C_i, C_l ) \nonumber \\ +C_t \leftarrow \text{Min}( C_t , C_l + C_i, 1) +\label{eq:ctchecks} +\end{eqnarray} + +\subsubsection{} +Finally, there is a homogeneous nucleation term applied, similar +to that in the large-scale precipitation (section \ref{sec:lsp_homo}). This is +a fast microphysics process, and must act to ensure that no liquid cloud +created by the initiation is allowed to persist in this phase if the +temperature is cold enough. Hence, if $\overline{T} < T_{homo}$ then + +\begin{eqnarray} +\overline{q_{cf}} \leftarrow \overline{q_{cf}} + \overline{q_{cl}} \nonumber \\ +\overline{q_{cl}} \leftarrow 0 \nonumber \\ +\overline{T} \leftarrow \overline{T} + \frac{L_f}{c_p} \overline{q_{cl}} \nonumber \\ +C_i \leftarrow C_t \nonumber \\ +C_l \leftarrow 0. +\label{eq:homochecks} +\end{eqnarray} + +\subsubsection{Qpos checks} +\label{sec:qpos} + +The implementation of the PC2 code includes an additional bounds check after +the \textit{atmos-physics-2} part of the model timestep has been completed. This +check is necessary to trap a rare failure, and uses the \textit{Qpos} subroutines +to check that $\overline{q_{cl}}$ is greater or equal to 0. + +During trialling prior to operational implementation, it was found that relying on Q-Pos +to deal with negative condensate values was very expensive, as the Q-Pos routine does a lot of communications between +different processors. It may be preferable to deal with the cause of negative condensate amounts at their source. +The option to ``Ensure consistent sinks of qcl and CFL'' +prevents the QCL increment from +trying to remove too much liquid condensate and hence reduces the models reliance on Q-Pos to +deal with the inconsistencies. + +\subsection{Data Assimilation} +\label{sec:da} + +The data assimilation section in the model will output assimilation increments +that represent changes to $\overline{q}$ and $\overline{T}$ which +\textit{include} the condensation contributions. We hence need to calculate +equivalent increments to $\overline{q_{cl}}$, $C_l$ and $C_t$. We assume +that the assimilation has not calculated these using a different method. +We consider the homogeneous framework and assume that there is a forcing +value of $Q_c$ that exists that will produce the known increment to +$\overline{q}$ and $\overline{T}$. + +Discritising (\ref{dqcldt}) we have, using (\ref{eq:deltaqc_exp}) and +expanding $\Delta T_L$ in terms of $\Delta T$ and $\Delta q_{cl}$, + +\begin{equation} +\Delta \overline{q_{cl}} = C_l ( a_L ( \Delta \overline{q} - +\alpha \Delta \overline{T} - \beta \Delta \overline{p}) + \Delta \overline{q_{cl}} ). +\label{eq:da1} +\end{equation} + +Remember that $Q_c$ (and hence $\Delta Q_c$) is independent of condensation. +Rearranging, we obtain + +\begin{equation} +\Delta \overline{q_{cl}} = \frac{1}{1 - C_l} C_l +a_L ( \Delta \overline{q} - \alpha \Delta \overline{T} - \beta \Delta \overline{p}) +\label{eq:da2} +\end{equation} + +and hence an expression for the condensate increment, +$\Delta \overline{q_{cl}}$, that accompanies the known increments +to $\overline{q}$ and $\overline{T}$. The similar analysis, from +(\ref{dcdt}) and (\ref{eq:da1}) gives + +\begin{equation} +\Delta C_l = \frac{1}{1 - C_l} G(-Q_c) + a_L ( \Delta \overline{q} - \alpha \Delta \overline{T} +- \beta \Delta \overline{p} ) . +\label{eq:da3} +\end{equation} + +Hence the equation set is equivalent to the use of the homogeneous +forcing set, except for the multiplier $\frac{1}{1 - C_l}$. Although this +is a clean solution, we +need to be very careful with the ill-conditioning of this solution +near $C_l = 1$. + +In practice, the ill-conditioning of (\ref{eq:da2}) and (\ref{eq:da3}) becomes too +numerically awkward for us to apply the full solution based on homogeneous +forcing, although, for completeness, we outline it in Appendix +\ref{sec:appendix-da}. Hence we have +chosen to apply a much simpler model. Here we use simply the data assimilation +increments $\Delta \overline{q}$ and $\Delta \overline{T}$ within the standard +homogeneous forcing (section \ref{sec:homog}), even though we are fully +aware that this is inconsistent (because $\Delta \overline{q}$ and $\Delta +\overline{T}$ are not forcings, but are forcings plus the condensation. +This allows us an \textit{estimate} of $\Delta \overline{q_{cl}}$ and $\Delta{C_l}$, +via the homogeneous forcing routine (and $\Delta C_t$ via the standard +updating described in section \ref{sec:ct}). These are the quantities applied +as the equivalent data assimilation increments for $\Delta \overline{q_{cl}}$, +$\Delta{C_l}$ and $\Delta C_t$. The increments $\Delta \overline{q}$ and +$\Delta \overline{T}$ remain those that the data assimilation scheme itself +calculated. + +Appendix \ref{sec:appendix-da} gives, for completeness, the alternative +numerical technique for the solution of (\ref{eq:da2}) and (\ref{eq:da3}). +However, we stress that this technique is not used within the current +PC2 formulation. + +\section{Implementation in the Unified Model} +\label{sec:um} + +This section considers the implementation of PC2 within the Unified Model +code and provides a brief guide to its use. + +In general, we have written PC2 so that the timestepping of the +cloud fraction variables within the \textit{atm\_step\_4a} +subroutine is treated as much as possible in a similar way to +the condensate variables. +Hence, wherever the condensed water variables $q_{cl}$ and $q_{cf}$ +are updated, the cloud fractions need to be updated consistently. + +\subsection{Area cloud fraction} +\label{sec:acf} + +Two area cloud fraction parametrizations are available for use with PC2. + +The area cloud fraction of Cusack (documented in \citeumdp{029}) has been +adapted by \cite{boutle_morcrette10} so it can be used with PC2 (and is available from the UMUI as the +``Cusack'' option from version 7.6 onwards). This method aims to +reproduce some of the detail of the thermodynamic +profile lost due to the coarseness of the grid. The interpolation/extrapolation technique is +used prior to PC2 initiation (which is then called with three times as many levels) +and it is used, along with the homogeneous forcing idea at the start of the timestep to +allow more cloud to be seen by radiation. + +The diagnostic area cloud fraction of \cite{bhi05} +has also been implemented in the model (available from the UMUI at version 6.4 onwards), +and this is used in PC2:64. This method diagnoses the area cloud fraction +given the volume cloud fraction, taking into account the size of the grid +box. The setting of the area cloud fraction is performed at the end of the timestep. + +\subsection{Code Structure} +\label{sec:code} + +A detailed description of the UM's timestep structure, +showing where in the model all the PC2 cloud scheme subroutine calls are made, +is given in the subsections below. + +Note that there are three different subroutines that all do +the PC2 homogeneous forcing, with slightly different details: + +\begin{itemize} +\item {\bf{\it pc2\_delta\_hom\_turb}} outputs increments due to the +condensation or evaporation, but doesn't update the fields themselves. +\item {\bf{\it pc2\_homog\_plus\_turb}} just updates the fields that +are passed in, instead of outputting separate increment arrays. +\item {\bf{\it pc2\_hom\_conv}} outputs increments but includes additional +calculations for various cloud erosion formulations. +\end{itemize} + +Note that code exists in the first two of these routines to do erosion, +but they can only do it via an input fixed rate of narrowing of the +moisture PDF (which is currently set to zero in all instances). +PC2 development has settled on a more complicated treatment of erosion, +which has only been implemented in {\it pc2\_hom\_conv}. +This can either be called after the convection scheme +(within {\it pc2\_from\_conv\_ctl}), +or before the microphysics scheme (within {\it pc2\_turbulence\_ctl}). + +Note there is also an optional call to {\it pc2\_turbulence\_ctl} +after the microphysics scheme, which is used only to estimate the +cloud fraction change consistent with the turbulent production of +liquid cloud (see section \ref{sec:turb_qcl_scheme}). + +Most PC2 code is protected by IF tests on the namelist input +{\it i\_cld\_vn} = 2 (PC2 in the GUI). +However, within the convection scheme, the code is controlled by logicals +{\it l\_calc\_dxek} (which is just set to true if using PC2, and set false +otherwise), and {\it l\_q\_interact}, which controls +whether to allow the interactive detrainment and entrainment of condensate. + +There is also a switch (currently hardwired to .false. in the code) called +{\it l\_pc2\_reset}. Turning this on (not recommended!) does 2 things: + +\begin{itemize} +\item Convective entrainment and detrainment of condensate is disabled, +by setting {\it l\_q\_interact} to false. +\item The prognostic cloud variables are overwritten by a call to +the diagnostic cloud scheme at the end of the timestep, +in subroutine {\it qt\_bal\_cld}. +NOTE: this functionality will no longer work, because inside {\it qt\_bal\_cld} +the call to the diagnostic cloud scheme is now protected by IF tests on +using either the Smith or bimodal cloud schemes. If using PC2, no cloud scheme +is called here, and required output variables are just left unset! +\end{itemize} + +The location of the various cloud scheme routine calls within the UM +is summarised in the list below. + +% The latex source input here contains a colour-coded itemize list +% of the UM subroutine tree, showing the locations of all the cloud-scheme +% routines. To edit this, open the source file source/029/um_call_tree.tex +\input{../029/um_call_tree} + +\subsection{Diagnostics} +\label{sec:diags} + +Nearly all diagnostics retain their meaning when PC2 is run. However, there +are a few that are subtly modified. + +The convective diagnostics that use the convective cloud base and top +calculations remain the same if PC2 is used with a zeroed convective +cloud fraction. These values are not reset by the convection scheme, since +the model is still predicting convection between the diagnosed levels. + +The visibility diagnostics need modifying if the convective cloud +fraction is switched off, since they use the convective cloud fraction +within their calculation. Here we use a value of 0.2 for the convective +cloud amount if there is convective precipitation but the two-dimensional +convective cloud amount is zero. This will be the case if the PC2 +scheme has zeroed the convective cloud amount. + +There are a number of increment diagnostics that are required to +fully diagnose the moisture cycle within PC2. Since most physics +sections can cause condensation, condensate and cloud fraction increment +diagnostics have been written for each of these sections. + +\begin{itemize} +\item{ $\overline{T}$, $\overline{q}$, $\overline{q_{cl}}$, $\overline{C_t}$ and $\overline{C_l}$ increments from SW radiation, $\overline{T}$ increment from SW Radiation without including the condensation: \textbf{Section 1} .} +\item{ $\overline{T}$, $\overline{q}$, $\overline{q_{cl}}$, $\overline{C_t}$ and $\overline{C_l}$ increments from LW radiation, $\overline{T}$ increment from LW Radiation without including the condensation: \textbf{Section 2} .} +\item{ $\overline{T}$, $\overline{q}$, $\overline{q_{cl}}$, $\overline{q_{cf}}$, $\overline{C_t}$, $\overline{C_l}$ and $\overline{C_f}$ increments from Boundary Layer: \textbf{Section 3} .} +\item{ $\overline{T}$, $\overline{q}$, $\overline{q_{cl}}$, $\overline{q_{cf}}$, $\overline{C_t}$, $\overline{C_l}$ and $\overline{C_f}$ increments from Large-scale precipitation: \textbf{Section 4} .} +\item{ $\overline{T}$, $\overline{q}$, $\overline{q_{cl}}$, $\overline{q_{cf}}$, $\overline{C_t}$, $\overline{C_l}$ and $\overline{C_f}$ increments from Convection, $\overline{q}$, $\overline{q_{cl}}$, $\overline{q_{cf}}$, $\overline{C_t}$, $\overline{C_l}$ and $\overline{C_f}$ increments from the inhomogeneous part of the Convection scheme only: \textbf{Section 5} .} +\item{ $\overline{T}$, $\overline{q}$, $\overline{q_{cl}}$, $\overline{q_{cf}}$, $\overline{C_t}$, $\overline{C_l}$ and $\overline{C_f}$ increments from the Advection: \textbf{Section 12} .} +\end{itemize} + +However, there +are a number of parts of PC2 that do not fit into a pre-existing section of +code, and hence the associated increment diagnostics are not easily placed +within the UM framework. These increments were available using a +modification set or branch and a user-STASHmaster file up to version 7.5. From version 7.6 these diagnostics are available as standard. + +\begin{itemize} +\item{ $\overline{T}$, $\overline{q}$, $\overline{q_{cl}}$, $\overline{C_t}$, and $\overline{C_l}$ increments from the PC2 erosion section: \textbf{Section 4} or {\bf Section 5} depending on where the erosion is called.} +\item{ $\overline{T}$, $\overline{q}$, $\overline{q_{cl}}$, $\overline{q_{cf}}$, $\overline{C_t}$, $\overline{C_l}$ and $\overline{C_f}$ increments from the Bounds Checking after atmphya: \textbf{Section 4} .} +\item{ $\overline{T}$, $\overline{q}$, $\overline{q_{cl}}$, $\overline{q_{cf}}$, $\overline{C_t}$, $\overline{C_l}$ and $\overline{C_f}$ increments from the Initiation and Bounds checking at the end of the timestep: \textbf{Section 16} .} +\item{ $\overline{T}$, $\overline{q}$, $\overline{q_{cl}}$, $\overline{q_{cf}}$, $\overline{C_t}$, $\overline{C_l}$ and $\overline{C_f}$ increments from the Pressure Forcing section: \textbf{Section 16} .} +\end{itemize} + +\subsection{Single Column Model} + +The updating in the single column model follows the same timestepping as +that in the full model, but the changes to atm-step are mirrored +within scm\_main. The method used is to store the driving SCM forcing +increments of vapour, liquid and temperature across the forcing subroutine. +The forcing of pressure is +set to zero. After atmos\_physics2 has been called, +a PC2 section of code calls the +homogeneous forcing subroutine with these increments. This therefore +treats the response of PC2 to the prescribed dynamical forcing in the +SCM as homogeneous. Following +this calculation, the initiation scheme is called, as usual. Finally, the area +cloud fraction is set to the bulk cloud fraction and $\overline{\Theta}$ +(potential temperature) +is made consistent with $\overline{T}$ (dry-bulb temperature) which was +changed by the condensation in the PC2 response to the homogeneous forcing. +The rest of the SCM uses the same PC2 code as the full model. + +Note that the change to PC2 homogeneous forcing from advection under the UM +namelist switch \textbf{l\_pc2\_sl\_advection} (see section \ref{sec:pres}) +is also mirrored in the Single-Column Model. If this switch is turned on, +the PC2 homogeneous forcing call using the SCM forcing increments is moved +straight after the call to the forcing routine, so that the condensation +adjustment is performed before the call to atmos\_physics2. +If \textbf{l\_pc2\_sl\_advection} is turned on, the PC2 response to SCM +forcings is also improved as follows... + +The SCM forcings may comprise one or both of the following: +\begin{itemize} +\item (a) Prescribed tendencies or relaxation applied to T,q. +\item (b) Interactive vertical advection applied to T,q. +\end{itemize} +For the latter, we can calculate the pressure change experienced by +vertically-advected parcels, and so calculate the PC2 homogeneous forcing +response in the same way as we do for Semi-Lagrangian advection in the +full model (see section \ref{sec:pres}). +For the former, we don't know if the prescribed T,q tendencies are due to +advection, radiation, or some other process, so we calculate the PC2 +homogeneous forcing response as if the tendencies are applied "in-situ". + +To split the PC2 homogeneous response into these 2 components, the SCM +forcing routine outputs: +\begin{itemize} +\item (a) The forcing increments to T,q excluding the contribution from +interactive vertical advection. +\item (b) The value of exner pressure at departure points, consistent with +the vertical advection. +\end{itemize} +The PC2 homogeneous forcing responses to these 2 forcing components +are then calculated by 2 separate PC2 calls in scm\_main. + +\subsection{Limited Area Boundary Conditions} + +Cloud fractions on the limited area boundaries are fully updateable. +Writing of cloud fraction Limited Area Boundary Conditions (LBCs) +will be automatic if PC2 is selected. +A PC2 LAM may be run from an LBC file with or without cloud fraction LBCs +(this is specified by the logical l-pc2-lbc, which is set in the UMUI). +If there are no cloud fraction lbcs then around the edge of the domain the +checking and initiation routines will be applying significant increments +to the cloud and condensate fields near the boundaries, but this does +not have an adverse effect well away from the boundaries. If there are no +cloud fraction LBCs the cloud fraction fields themselves are not forced to +zero around the edge of the domain but are allowed to freely find their +own value. A PC2 run that outputs lbcs will, by default, always +output cloud fractions as part of the LBCs file. + +\subsection{Parameter values} + +Table \ref{tab:pc2_names} summarizes the values of parameters used in the PC2 +scheme and their location within various comdecks. Those parameters marked +as `Num' are those that are not part of the mathematical equation +set that is being solved, but are required in order to achieve a stable, +realistic, numerical solution. These include, for instance, thresholds for +resetting cloud fractions back to 0 or 1. Those marked 'Phy' are physical +quantities that form an integral part of the equation set that we wish to solve. +Those marked 'Clo' form part of a closure needed to form the equation +set, but are less readily related to physical quantities. +Variables marked 'Diag' form a part of the diagnostic output routines. +\begin{table}[ht] +\begin{center} +\footnotesize +\begin{tabular}{llllll} +\hline +Symbol & Code variable & Description & Value & Location & Notes and ref. \\ \hline +- & init-iterations & Number of iterations in initiation & 10 & pc2-const & Num: \ref{sec:numapp_init} \\ +$C_{tol}$ & cloud-pc2-tol & Bounds checking $C_l$ threshold & 0.005 & UM namelist & Num: \ref{sec:init2} \\ +$C_{tol 2}$ & cloud-pc2-tol-2 & Bounds checking $C_l$ threshold & 0.001 & UM namelist & Num: \ref{sec:init2} \\ +$RH_{tol}$ & rhcrit-tol & $RH_{crit}$ tolerance in initiation & 0.01 & pc2-const & Num: \ref{sec:init2} \\ +$q_{cf0 \, BL}$ & ls-bl0 & Fixed value of BL in-plume $\overline{q_{cf}}$ & $1.0 \times 10^{-4} \, kg \, kg^{-1}$ & imp-ctl & Clo: \ref{sec:bl} \\ +$q_{cf0}$ & one-over-qcf & Fixed in-cloud $\overline{q_{cf}}$ if $C_f$=0 & $1.0 \times 10^{-4} \, kg \, kg^{-1}$ & pc2-chck & Num: \ref{sec:checks} \\ +$m$ & pdf-merge-power & Merging power for $G(-Q_c)$ & 0.5 & pc2-const & Clo: \ref{sec:homog} \\ +$n$ & pdf-power & Shape parameter for $G(-Q_c)$ & 0.0 & pc2-const & Phy: \ref{sec:homog} \\ +$w$ & wind-shear-factor & Wind shear in fallout of ice term & $1.5 \times 10^{-4} \, s^{-1}$ & pc2-const & Phy: \ref{sec:lsp_fall} \\ +$i$ & ice-width & Scaling factor for reduction in $b_i$ & 0.04 & pc2-const & Phy: \ref{sec:mp_depsub} \\ +$a$ & dbsdtbs-turb-0 & Rate of reduction of PDF width & $-2.25 \times 10^{-5} \, s^{-1}$ & UM namelist & Phy: \ref{sec:width} \\ +$b$ & dbsdtbs-turb-1 & Rate of reduction of PDF width & 0 & pc2-const & Phy: \ref{sec:width} \\ + & dbsdtbs-conv & Redn of PDF width in convection & 0 & pc2-const & Phy: \ref{sec:width} \\ + & dbsdtbs-exp & Variation of erosion on RH & 10.05 & pc2-const & Phy: \ref{sec:width} \\ +$RH_{crit}$ & RHCRIT & Critical RH for cloud formation & & UM namelist & Phy: \ref{sec:init}, \ref{sec:mp_depsub} \\ +$q_{c0}$ & condensate-limit& Minimum allowed condensate & $1 \times 10^{-10} \, kg \, kg^{-1}$ & pc2-chck & Num: \ref{sec:checks} \\ +$q_c^{S0}$ & ls0 & Lower limit of plume condensate & $5 \times 10^{-5} \, kg \, kg^{-1} $ & enviro?a & Num: \ref{sec:multi_numapp} \\ + & \textit{Hard-wired} & Conv cloud fraction for visibility& 0.2 & imp-ctl2 & Diag: \ref{sec:diags} \\ + & \textit{Hard-wired} & Limit on width of ice distribution& 0.001 & lspice3d & Num: \ref{sec:mp_depsub} \\ + & \textit{Hard-wired} & $C_l$ limit for init if $T < 0 ^{\circ} C$ & 0.05 & pc2-init & Num: \ref{sec:init2} \\ + & \textit{Hard-wired} & Tolerance on calc. of $q_C^s$ in BL & $1.0 \times 10^{-10} \, kg \, kg^{-1}$ & imp-ctl & Num: \ref{sec:bl} \\ +\hline +\end{tabular} +\end{center} +\caption{PC2 parameter values and locations } +\label{tab:pc2_names} +\end{table} + +PC2 also recommends some tunings of the existing convection +scheme parameters. These cannot be placed in the library code, since they +would interact with non-PC2 simulations, hence would need to be specified +with modification sets. We have included those parameters that have been +investigated throughout testing, although only two are different between +PC2:64 and a non-PC2 run. + +\begin{table}[ht] +\begin{center} +\tiny +\begin{tabular}{llllll} +\hline +Code variable & Description & Value in PC2 & Value in Control & Location & Notes and reference \\ \hline +TICE & Temperature at which plume freezes & $-10 ^{\circ} C$ & $0 ^{\circ} C$* & tice.cdk or UMUI & Phy: \ref{sec:convec} \\ +QSTICE & Approximate qsat(TICE) & $3.5 \times 10^{-3}$ & $3.5 \times 10^{-3}$ & qstice.cdk or UMUI & Phy: \ref{sec:convec} \\ +\textit{Hard-wired} & Limit on conv. cond. after precip & 0.5 $q_{sat}, 2 \times 10^{-4}$ & $0.5 \, q_{sat}$ & cloudw & Phy: \ref{sec:convec} \\ +Anvil factor & Shape parameter for conv. cloud anvil & 0 & 0.3* & UMUI & Phy: \ref{sec:convec} \\ +Tower factor & Shape parameter for conv. cloud tower & 0 & 0.25* & UMUI & Phy: \ref{sec:convec} \\ +\hline +\end{tabular} +\end{center} +\caption{PC2 parameter values and locations relating to the convection. *These values are those used in HadGAM} +\label{tab:pc2_conv_names} +\end{table} + +\subsection{How to run the PC2 scheme} +Running PC2 is straightforward, but you should seek advice as to +modification sets that you need to include to ensure you are +running the most up-to-date version of PC2. +The following is a brief checklist of the options in the UMUI which need +to be selected in order to run PC2. No hand-edits are required. +\begin{itemize} +\item{In the LS cloud panel (atmos-science-section-LScloud) push the button marked 'use the PC2 cloud scheme'.} +\item{If you wish to use PC2 in the diagnostic only mode, also push 'run the PC2 scheme in diagnostic only mode'. If you wish to run PC2 fully then do not push this button} +\item{In the large-scale precipitation section (atmos-science-section-LSprecip) select the 3D large-scale precipitation scheme.} +\item{The specification of the LA boundary conditions can be set in the atmos-InFiles-OtherAncil-LBC panel.} +\item{You will need to select modsets to include update the library code to the PC2 version described here. Seek advice on these.} +\item{You may wish to adjust the convective anvil parameters in atmos-science-section-convec. Again, seek advice.} +\end{itemize} + +\subsection{More information} + +Information on results of the scheme and how to run the PC2 code at +various model versions is available on the PC2 web site. + +\section{Appendix: Alternative PC2 - Data Assimilation formulations} +\label{sec:appendix-da} + +In this alternative method to section \ref{sec:da} we will assume that there +exists a homogeneous forcing, $\Delta Q_c$, +that gives changes, net of condensation, of $\Delta\overline{q}$ and +$\Delta\overline{T}$. If we can recover +what $\Delta Q_c$ is then we can use this to calculate the liquid, +$\overline{q_{cl}}$, and liquid cloud fraction, $C_l$, increments. + +As in section \ref{sec:da}, we start by discretising (\ref{dqcldt}) to give + +\begin{equation} +\Delta \overline{q_{cl}} = C_l \Delta Q_c +\label{eq:dqcldt_discrete} +\end{equation} + +and hence, using the discrete form of $\Delta Q_c$ from +(\ref{eq:deltaqc_exp2}) gives + +\begin{equation} +\Delta \overline{q_{cl}} = C_l ( a_L ( \Delta \overline{q} - \alpha \Delta +\overline{T} ) + \Delta \overline{q_{cl}} ) , +\end{equation} + +which rearranges to + +\begin{equation} +\Delta \overline{q_{cl}} = \frac{1}{1-C_l} C_l a_L ( \Delta \overline{q} +- \alpha \Delta \overline{T} - \beta \Delta \overline{p}) . +\label{eqn:delataqcl} +\end{equation} + +Comparing to (\ref{eq:deltaqc_exp2}) and (\ref{eq:dqcldt_discrete}) we see that +$\Delta \overline{q_{cl}} $ is the same as if we had applied the +homogeneous forcing technique +using $\Delta \overline{q}$, $\Delta \overline{T}$ and $\Delta \overline{p}$ as +forcings, except multiplied by a factor of $\frac{1}{1-C_l}$. + +We can calculate $\Delta C$ in a similar way. From (\ref{eq:deltac}) + +\begin{equation} +\Delta C_l = G(-Q_c) \Delta Q_c +\end{equation} + +and hence, using our value of $\Delta Q_c$ from (\ref{eq:deltaqc_exp2}) +and $\Delta \overline{q_{cl}}$ from (\ref{eqn:delataqcl}) + +\begin{equation} +\Delta C_l = G(-Q_c) (a_L ( \Delta \overline{q} - \alpha \Delta \overline{T} ) ++ \frac{1}{1-C_l} C_l a_L ( \Delta \overline{q} - \alpha \Delta \overline{T} ) ) +\end{equation} + +which rearranges to + +\begin{equation} +\Delta C_l = \frac{1}{1-C_l} G(-Q_c) a_L ( \Delta \overline{q} +- \alpha \Delta \overline{T} -\beta \Delta \overline{p}) . +\label{eqn:c1mc} +\end{equation} + +This is also a factor of $\frac{1}{1-C_l}$ different from using +$\Delta \overline{q}$, +$\Delta \overline{T}$ and $\Delta \overline{p}$ +directly as forcings (the factor must be the same, as we are still +using the homogeneous forcing hypothesis). This equation forms the basis +for the more advanced technique discussed in this section. However, +it is undefined at $C_l=1$ and becomes ill-conditioned near $C_l=1$, +hence there must be care taken when this expression is solved numerically. + +\subsection{Numerical solution} + +The timestepping applied is picked as a result of numerical tests +forcing a single gridbox with uniform increments. Many numerical techniques +were tested, this gives a fast but reasonably well behaved solution. + +Initially, we calculate $G(-Qc)$ and $\Delta Q_c$ from the input fields, +as in the homogeneous forcing +technique (section \ref{sec:homog}) and (\ref{eq:deltaqc_exp2}). + +An initial increment, $\Delta C_l^1$ is estimated directly using the +basic equation + +\begin{equation} +\Delta C_l^1 = \frac{1}{1-C_l^(n)} G(-Q_c) \Delta Q_c . +\end{equation} + +We then recalculate this expression, using a mid-timestep estimate +for $\Delta C_l$; + +\begin{equation} +\Delta C_l = \frac{1}{1-(C_l^{[n]} + \frac{1}{2} \Delta C_l^1)} +G(-Q_c) \Delta Q_c +\end{equation} + +where the term $C_l^{[n]} + \frac{1}{2} \Delta C_l^1$ is limited to be no more +than 0.9999 to avoid divide by zero problems. The final, updated value of +cloud fraction, $C_l^{[n+1]}$, is then + +\begin{equation} +C_l^{[n+1]} = C_l^{[n]} + \Delta C_l +\end{equation} + +and this value is limited to 0 or 1. + +The liquid water term simply uses the final version of $C_l$ in its +calculation. + +\begin{equation} +\Delta \overline{q_{cl}} = \frac{1}{1-C_l^{[n+1]}} C_l^{[n+1]} \Delta Q_c +\end{equation} + +and will be set to 0 if $C^{[n+1]}$ is 0. There is an additional limit, +see below, applied to the liquid +water term, which will prevent the value of $\Delta \overline{q_{cl}}$ +increasing to a large number if $C_l^{[n+1]}$ is very close to 1. + +\subsection{Limit on the liquid water content} + +We will choose a limit on $\overline{q_{cl}}$ to be equal to its value when +the underlying PDF just corresponds to total cloud cover. Therefore, from +(\ref{eq:qclbar=int}) + +\begin{equation} +\overline{q_{cl \, max}} = \int_{s=-b_s}^{\infty} G(s) (b_s + s) ds . +\end{equation} + +We will use the current value of $Q_c$ (which won't in general to be equal to +$b_s$) to split the integral into two ranges of s: + +\begin{equation} +\overline{q_{cl \, max}} = \int_{s=-b_s}^{-Q_c} G(s) (b_s + s) ds ++ \int_{s=-Q_c}^{\infty} G(s) (b_s + s) ds . +\end{equation} + +For the moment we write the first of these integrals as $I1$, and split the +second integral whilst introducing a $(+ Q_c - Q_c)$ term to the integrand: + +\begin{equation} +\overline{q_{cl \, max}} = I1 + \int_{s=-Q_c}^{\infty} G(s) (b_s - Q_c) ds ++ \int_{s=-Q_c}^{\infty} G(s) (s + Q_c) ds . +\end{equation} + +The last of the integrals is now the current liquid water content, +$\overline{q_{cl}}$. +The second integral is proportional to the liquid cloud fraction $C_l$. + +\begin{equation} +\overline{q_{cl \, max}} = I1 + C_l (b_s - Q_c) + \overline{q_{cl}} +\end{equation} + +or + +\begin{equation} +\overline{\Delta q_{cl \, max}} = I1 + C_l (b_s - Q_c) . +\label{eqn:deltaqclmax} +\end{equation} + +Now consider the expression for the saturation deficit, which we have +defined, from (\ref{SD}) as + +\begin{equation} +SD = \int_{-b_s}^{-Q_c} G(s) (-Q_c - s) ds . +\end{equation} + +Splitting and adding the term $(+b_s - b_s)$ to the integrand in a +similar way to above gives + +\begin{eqnarray} +SD = \int_{-b_s}^{-Q_c} G(s) (-Q_c + b_s) ds + \int_{-b_s}^{-Q_c} +G(s) (-s - b_s) ds \nonumber \\ += (-Q_c + b_s) (1 - C_l) - I1 , +\end{eqnarray} + +and hence $I1$ in terms of $SD$. Using this value of $I1$ in +(\ref{eqn:deltaqclmax}) and cancelling the $C_l$ terms gives +$\Delta \overline{q_{cl \, max}}$ as + +\begin{equation} +\Delta \overline{q_{cl \, max}} = (-Q_c + b_s) - SD . +\label{eqn:delta2} +\end{equation} + +This is a general expression, it is not fixed for a particular PDF. To +complete the analysis, we need to estimate $-Q_c+b_s$. To do this, we now +make the \textit{assumption} of a power-law type PDF, as in section +\ref{sec:init}. If we start from the equivalent of +(\ref{eqn19}) but at the $s=-bs$ end of the distribution, equation (B.3) +in \cite{wg03} can be equivalently written for $(1-C_l)$ as: + +\begin{equation} +(1-C_l) = \frac{ A (-Q_c + b_s)^{n+1} }{n+1} . +\label{eqn:1mc} +\end{equation} + +To derive this from (B.3) note that $C_l$ is swapped for $1-C_l$ and +$(b_s - (-Q_c))$ is swapped for $(-Qc - (-b_s))$, as in section +\ref{sec:numapp_init}. Similarly, noting that $\overline{q_{cl}}$ can be +swapped with $SD$, gives the equivalent to (B.4) in \cite{wg03} +as + +\begin{equation} +SD = \frac{ A (-Q_c + b_s)^{n+2} }{(n+1)(n+2)}. +\label{eqn:sd} +\end{equation} + +Using the value $(1-C_l)$ from (\ref{eqn:1mc}) in (\ref{eqn:sd}) gives + +\begin{equation} +\frac{SD}{1-C_l} = \frac {-Q_c + b_s}{n+2} . +\end{equation} + +Finally, we use this expression for $(-Q_c + b_s)$ in (\ref{eqn:delta2}) to +parametrize $\Delta \overline{q_{cl \, max}}$ in terms +of the saturation deficit + +\begin{equation} +\Delta \overline{q_{cl \, max}} = SD ( \frac{n+2}{1-C_l} - 1 ) . +\label{eqn:sdr1mc} +\end{equation} + +This is the expression that is used for the limit on $\overline{q_{cl}}$. +We subsequently apply a second limit, since numerically this expression is +still not well behaved when $C_l$ is close to 1. Here we note that just at +complete cloud cover for a symmetric PDF we have +$\overline{q_{cl}} = b_s$. Hence we estimate $b_s$ as in \cite{smith90}, + +\begin{equation} +b_s = a_L ( 1 - RH_{crit} ) q_{sat}(\overline{T_L}) , +\label{eqn:bs} +\end{equation} + +and take the smaller value for +of (\ref{eqn:sdr1mc}) and (\ref{eqn:bs}) for $\Delta \overline{q_{cl \, max}}$. + + +\subsubsection{Initiation from $C_l=1$} +The equations are not defined when $C_l=1$. (Note that when $C_l=0$ we +will calculate $G(-Q_c)=0$ so there is no change in cloud fraction or liquid water +content in this case). The assimilation is capable of lowering $\overline{q}$ +below $q_{sat}(\overline{T})$ and hence there should be a corresponding +change in cloud fraction and liquid water content. In theory, we can use +the expression for $\Delta \overline{q_{cl \, max}}$ and assume that the +initial liquid water is equal to $b_s$. However, this produces +a tricky set of simulataneous equations, which are not easily solvable +except in the case where $n=0$. We proceed by making this assumption for $n$, +acknowledging that this is not necessarily entirely consistent with the +rest of the model (although it is in the PC2:64 formulation). + +We have (equivalent to B.6 from \cite{wg03}) + +\begin{equation} +\frac{ (1-C_l)^2 }{SD} = G(-Q_c) \frac{n+2}{n+1}. +\end{equation} + +If n=0 (i.e. a `top-hat' function) then $G(-Q_c) = \frac{1}{2 b_s}$ +and we can write + +\begin{equation} +C_l = 1 - \sqrt{ \frac{SD}{b_s} } . +\end{equation} + +We now assume $b_s$ is equal to our current value of $\overline{q_{cl}}$ +and hence + +\begin{equation} +C_l^{[n+1]} = 1 - \sqrt{ \frac{SD^{[n+1]}}{\overline{q_{cl}^{[n]}}} } +\label{eqn:1msqrt} +\end{equation} + +where $C_l^{[n+1]}$ and $SD^{[n+1]}$ are the values of $C_l$ and $SD$ after +this initiation has been applied. +Using our previous expression (\ref{eqn:sdr1mc}) for +$\Delta \overline{q_{cl max}}$ gives (remembering that we are considering +the reverse process, so the sign is opposite), + +\begin{equation} +\Delta \overline{q_{cl}} = - SD^{[n+1]} ( \frac{2}{1-C_l^{[n+1]}} - 1 ) +\end{equation} + +(remembering that $n=0$ is assumed). Hence, replacing $C_l^{[n+1]}$ by +(\ref{eqn:1msqrt}) we have + +\begin{equation} +\Delta \overline{q_{cl}} = SD^{[n+1]} - 2 \sqrt{ SD^{[n+1]} +\overline{q_{cl}}^{[n]} } . +\end{equation} + +This is the expression we use, $SD^{[n+1]}$ is calculated after the +ssimilation increments have been applied, using (\ref{SD2}): + +\begin{equation} +SD^{(n+1)} = a_L^{[n+1]} ( q_{sat}(\overline{T}^{[n+1]}, +\overline{p}^{[n+1]}) - \overline{q}^{[n+1]} ). +\end{equation} + +\subsection{Results} + +Results demonstrate a problem in that there is a distinct asymmetry +between changes when $\Delta Q_c$ is large and positive and when +$\Delta Q_c$ is large and negative, when +cloud fractions start near 1. In the former case, the limit to the amount of +liquid and cloud fraction that can be created means that changes must be +kept relatively small, whereas in the latter case, all the cloud and +liquid water can be removed easily. (The $1/(1-C_l)$ term allows this +to be done relatively quickly). Hence this assimilation +method has a net tendency to remove cloud from the simulation, which, +at the moment, gives poorer results than simply using the homogeneous +forcing method. + +Further work will be required to enable the implementation of +this $\overline{q}$ +and $\overline{T}$ preserving method. + +\section{Appendix: Essentials of PC2 for code developers} +\label{sec:code-development} +This section provides some guidance to code developers on the treatment +of PC2. Code developers are advised to read the relevant part of section +\ref{sec:app_um} to understand the way in which the current PC2 scheme +interacts with their section of code. + +The essence of a prognostic cloud scheme is that each physical part of the +model is able to calculate increments to the cloud fractions and condensate +contents. These form an integral part of each physics scheme and should be +considered by code owners as such, hence any alteration to a scheme +\textit{must} consider also the impact on $q_{cl}$, $q_{cf}$, $C_t$, +$C_l$ and $C_f$, as +well as on the more traditional $T$, $q$ and wind prognostics. Often +there should be no impact, but this cannot be assumed without consideration. +There is no diagnostic cloud fraction and condensation scheme which can be +run in PC2, since this would reset any effect of the cloud prognostics used +elsewhere in the model. (The diagnostic scheme can still be used for model +\textit{diagnostics}, such as visibility and fog fraction, and will be +kept in later versions of the UM). + +Since this places a significant burden on code developers, the PC2 +developers have produced two generic representations which can take increments +to $q$ and $T$ etc. and produce an estimate of the condensation and +cloud fraction changes associated with the increments. These are the +homogeneous forcing and injection forcing (or inhomogeneous forcing) +methods. + +\subsection{Homogeneous forcing} +This is described fully in section \ref{sec:homog}. This assumes that the +distribution of $q_T - q_{sat}(T_L)$ about its gridbox mean is unchanged when +a process acts. (The mean will change of course, but we assume that the +variations in each part of the gridbox from the mean do not). Since this +is equivalent to every part of the gridbox receiving the same $q_T$ and $T_L$ +increment, we call this `Homogeneous Forcing'. We have provided a subroutine +\textit{pc2-homog-plus-turb}, in deck \textit{pc2-homo} in order to +provide the necessary updates. + +\subsection{Injection forcing} +This is described fully in section \ref{sec:inhomog}. We assume that +we already know a condensate increment $q_{cl}$ or $q_{cf}$ and that a +corresponding cloud fraction increment $C_l$ or $C_f$ (and $C_t$) remains +to be estimated. The injection forcing assumes that new cloud randomly +displaces existing cloud in a gridbox, and is designed with detrainment +from deep convection in mind, although it is also used elsewhere. It will +require as an input an estimate of the `in-cloud' water content of +the new cloud that is produced. + +If you consider that both the homogeneous and injection forcing representations +are both poor assumptions for your scheme, you will need to provide +another method for calculating the condensation and cloud fraction changes. +The PC2 team can advise, but you should not expect them to do the work. +You can, of course, replace existing homogeneous and inhomogeneous forcing +calls with new representations of changes to the prognostics if you think +you have improved representations available. This is part of the +development of any prognostic variable representation. + +\subsection{Do I need to modify anything when I change a parametrization scheme?} + +Here we assume that you wish to do the minimum work possible to get +PC2 to work, rather than a full reconsideration of the physics of the PC2 +increment terms. + +If your scheme is currently using the homogeneous forcing +then there is no need to update the cloud part of the scheme, +\textit{provided that +you do not alter values of $T$ and $q$ after the homogeneous forcing +section is called} and that the physical interpretation of your $q$ and $T$ +increments does not change. You need to be careful if you are moving code from +one subroutine to another that you don't inadvertently do this, although +the forcing usually sits at the end of the control subroutine. + +If your scheme is currently using the injection forcing \textit{subroutine}, +which necessitates that condensate +increments are already calculated by the scheme, then there is also no need +to update the cloud part of the scheme. This currently applies to the boundary +layer, where $q_{cf}$ is altered by tracer mixing. Like for the +homogeneous schemes, this +is provided that you \textit{do not alter $T$, $q$ or condensate values after +the injection forcing subroutine is called} and that the physical +interpretation of your $q$ and $T$ increments does not change. + +Changes to winds do \textit{not} need to have a condensation or +cloud fraction increment +associated with them. There may be future scope for developing an +orographic cloud representation (probably diagnostic), but this is not +an essential part of the scheme as it stands. + +If your scheme uses hardwired assumptions about what is happening e.g. +convection or microphysics, then you \textit{do} need to be careful that +$T$, $q$ and condensates +are still calculated correctly after you have performed your changes. +Currently there are many PC2 assumptions hard-wired into the mass-flux +convection scheme: +\begin{itemize} +\item{Any change to the scientific basis by which changes to $T$, $q$, $q_{cl}$ and $q_{cf}$ are calculated requires careful consideration} +\item{Simple changes to convective parameters, such as detrainment rates, should not require a change to the PC2 code} +\item{Be particularly careful if you move code around, \textit{especially the calculation of convective cloud fractions}, since PC2 incorporates a set-to-zero in the code. This will need to be replicated or there is a risk that the diagnostic cloud fraction is no longer set to zero correctly by PC2.} +\end{itemize} +Each microphysics transfer term has been considered individually for PC2 and this +should remain the case. + +Be especially careful when you do anything in the atmphy and atmstep levels of +the code that includes additional changes $T$, $q$, $q_{cl}$ or $q_{cf}$, since +they may need cloud fraction or condensation changes to go along with them. + +In summary, changes to existing increments of $T$, $q$ etc. within the current +UM structure are unlikely to +necessitate a modification for PC2 if their physical interpretation has not +changed. However, new methods of generating $T$ and +$q$ increments will require new code to be added for PC2. + +\subsection{Further PC2 development work} +There are a number of areas in which the PC2:66 formulation can be +developed further, and many of these have been mentioned in the documentation +above. Some +of these are simple sensitivity studies which have not been fully explored in +development, others are more complex alterations. It is fair to say +that the link to the convection has proved the most problematic issue +so far with PC2 development. + +\subsubsection{PC2 cloud erosion} +The cloud erosion is a critical term for the simulation of shallow convective +cloud. A large amount of erosion is required to keep the cloud fractions relatively +low in shallow convection, which is why we have linked the erosion to the relative +humidity. We recognise, however, that this is more an empirical choice than a +physically informed choice. In particular, a low relative humidity (e.g. in the +stratosphere) would imply a very high erosion rate - although the net effect +is to remove any cloud, which is a reasonable thing to do, there is an implication of +the parametrization that mixing within the stratosphere is high, which is +clearly incorrect. We have also seen relatively low cloud fractions in the +mid-levels of deep convection in PC2, and presume that this is influenced +by the erosion formulation. A link to mass flux has also been proposed, but tests +with CRMs do not support a clear link. Perhaps it is more natural to compare the +erosion with the turbulent kinetic energy. This should be available within the +boundary layer and convection schemes, but not outside of these in the current +UM. + +The erosion formulation in PC2:66 is one where the width of the PDF is +always narrowed (developed following \cite{sg03}). +It may be advantageous to think whether there are unmodelled +processes in the atmosphere that result in an increase in width. Clearly +convection is likely to be one, but this is already represented in PC2. +There may be other models entirely for the way in which the PDF changes as a result +of mixing of air within a gridbox or within the column, these may prove +fruitful to explore. + +Another issue is whether width-narrowing (or widening) is really an effective +way of representing the erosion process. CRM evidence suggests that the required +erosion rates to balance convective cloud generation are larger for +liquid cloud fraction than liquid water (by up to a factor of 2), suggesting +that the real atmospheric erosion favours removal of cloud fraction +over liquid water more strongly than the model. + +The in-cloud condensate that is detrained from convective plumes is high. +We might think that the mixing in of environmental air in reality is +likely to lead to more cloud around the plumes and lower condensate within +the plumes. However, the width narrowing scheme is not a good model of mixing +in this situation, always reducing the amount of cloud because it is incorrectly +assumed that much of the detrained plume has condensate contents only just above zero +and that the shape of the moisture PDF remains unchanged. This may have a +bearing on the problem of the lack of mid-level cloud in the model (although I +think there are many reasons for this). A different +mixing method may give significantly different results for the areas around +convective plumes. + +\subsubsection{Narrowing of the moisture PDF} +Most of the parametrized terms in PC2 act to reduce the width of the +moisture PDF. The only terms that can increase the width are the convection, +and the initiation (which can reset the width). This may not be the +best way to describe the way in which the PDF evolves, in particular it +is sensible to ask whether the erosion term should actually increase +the width in the presence of large vertical gradients of moisture. + +\subsubsection{Convective cloud increments in the mass-flux framework} +As discussed in section \ref{sec:conv_imp_note}, it would be useful +to code up the convective cloud fraction changes to link directly to +the mass-flux convection scheme, and not to estimate them from the values +of $Q4$, which can introduce errors. + +\subsubsection{Turbulence based convection scheme} +\label{sec:tbcs} +We will need to properly consider the links between PC2 and the +turbulence based convection scheme. In essence, we can use the diagnosed +cloud fraction and condensate values from the convection scheme to +start off the cloud again when convection has ceased. This has been +tested to some degree but will need proper analysis. The difficult +decision comes in choosing what to do with the condensate and cloud fraction +that is present \textit{before} the convection starts, since we must +ensure conservation of moisture. This is not helped by the traditional +view of convective parametrization that ignores the existence of the condensate +phase in the atmosphere (i.e. it is only concerned with transport of $q$ and +$\theta$, not of $q_{cl}$ and $q_{cf}$) despite the phase changes forming +an integral part of the convection scheme. + +\subsubsection{Detailed convective comparisons with CRM/LEM data} +This work is already underway at the Met Office, in order to properly +evaluate the performance of the convective cloud parametrization +in PC2 against high resolution research models. + +\subsubsection{Choice of PDF parameters} +Work by Dan Tang at Leeds University has highlighted an interesting +and undesirable property of the choice of $m$ and $n$ parameters in the +homogeneous forcing formulation. If a distribution is homogeneously +forced to $C_l = 0$, then we do not necessarily get $\overline{q_{cl}}$ +tending to zero. This is because there is enough influence from the +$\frac{{(1-C_l)}^2}{SD}$ term in the combination (\ref{eqn22}) to +stop the natural convergence of the $\frac{{C_l}^2}{\overline{q_{cl}}}$ term +to $C_l =0$ and $\overline{q_{cl}}=0$. Increasing the power of $m$ should +help. However, we note that the tests that have been done on the chosen +$n$ and $m$ values (0 and 0.5 respectively) do not show particularly +poor behaviour, and we do not pick up substantial evidence of problems +from this in the full model. This remains something to be investigated. + +\subsubsection{Homogeneous forcing section improvements} +\label{sec:homog_improve} +Although the homogeneous forcing provides a convenient method to +calculate increments to $C_l$ and $\overline{q_{cl}}$, it is clearly +not the best representation possible of the processes that use it. +For example, although the clear-sky radiative heating may perhaps +best be considered as a homogeneous process, the part of the +radiative heating influenced by clouds should, ideally, be applied +to the cloudy part of the gridbox and not the clear part. Vertical +advection is likely to be correlated with where there is already +cloud, rather than being uniform throughout the gridbox. There is +no reason that a process that uses homogeneous forcing as its +condensation model should not be looked at with a view to using +something better. This is one of the strengths of the PC2 framework and +is an intention of the project. + +\subsubsection{Overlap of ice and liquid cloud changes} +We have assumed within PC2 that ice and liquid cloud changes are +minimally overlapped with each other (within the same gridbox) in +order to maintain as much supercooled liquid water as possible. Although +there is good observational evidence to say that the two condensate +phases tend not to coexist together in a cloud, it may be possible to +characterise and apply this overlap in a more quantiative way. + +\subsubsection{Parameter tuning} +The sensitivity of some of the parameters in PC2 have not been properly +tested, mainly due to a lack of resources rather than a physical reason. +We have seen that the most effective method of tuning cloud is with the +erosion term, which has been increased to high values in order to remove +enough cloud and is probably as high as we reasonably wish to take it given +the length of the timestep. +\begin{itemize} +\item{The phase change temperature (between liquid and ice) in the +convective plume, TICE, is known to influence the strength of the convection +through the latent heat differences. It also impacts on the amount of +supercooled liquid water in the model. The quantitative impact of altering +this could be explored. We note that CRM simulations of deep convection +suggest that some supercooled liquid water exists within the plumes to +$-40 ^{\circ} C$ and that a representation with partial liquid and partial +ice phase would be more appropriate, based possibly on the current diagnosed +convective cloud phase in the non-PC2 model. Although the theoretical work +has been done to allow partial phases, we repeat the caution that care +must be taken when doing the work and appropriate testing done to ensure +that heat and moisture are properly conserved within the convection scheme.} + +\item{The growth of $C_f$ due to the fall-out of ice term in the microphysics +is parametrized with a dependence on windshear. We have never linked this +directly to the windshear, instead we have used estimated the windshear +as a fixed value. There is no reason why the actual model windshear cannot +be passed into the scheme in order to properly calculate this term.} +\item{$RH_{crit}$ remains a tunable parameter. Although its impact is less +than in a non-PC2 simulation, it is still significant in initiating cloud +and in determining the evolution of the ice cloud. There is also an implicit +overlap assumption regarding the ice cloud fractions, again this might be +improved upon.} +\item{$n$ and $m$ values in the homogeneous forcing have not been +thoroughly investigated for a long time now, and may yield some sensitivities}. +\end{itemize} + +\subsubsection{Cloud inhomogeneities} +A cloud generator approach to cloud inhomogeneities is currently +being developed. However we note two particular issues that relate +to PC2. +\begin{itemize} +\item{The first is that in the diagnostic scheme, the two cloud +fractions (convective and large-scale) allows, to some degree, a +representation of cloud inhomogeneity. This is absent from PC2, +although we note that the convective cloud fraction variable has +not been removed from the radiative transfer code for PC2, it is merely set +to zero, so it is easy to put back.} +\item{The generation of inhomogeneities using a cloud generator +requires some estimate of the variance (and possibly skewness) +of the condensate in the +gridbox. It is possible to back out the full moisture PDF at +each grid point by homogeneous forcing (providing $C_l$ is not equal +to 0 or 1), but this is very expensive and cannot be done on-line. Is +there a quick \textit{estimate} of the variance or skewness that it is +possible to obtain from knowledge only of $\overline{q}$, +$q_{sat}$, $\overline{q_{cl}}$ and $C_l$ etc.?} +\end{itemize} + +\subsubsection{Time-stepping} +\label{sec:timestepping} +A proper analysis of timestep sensitivities of PC2 (as opposed to +microphysics, convection etc) in the full UM +or SCM has not been done for a long time. +In the early development stages much effort was +placed in developing good numerical techniques for each of the terms +in PC2, and to explore the way in which they coupled together. An example +is the homogeneous forcing timestep investigated by \cite{wg03}. +We note that in shallow convection at 30 minutes timestep the erosion +term is trying to remove +most of the cloud that the convective detrainment places into the model. +Since the erosion is limited by the amount of cloud fraction and +condensate present, what ends +up happening is that the `equilibrium' that is achieved is actually one where +the cloud fraction and condensate at the end of the timestep are simply +the values that were detrained by the convection scheme (and hence depend +on the timestep). The CRM suggests a cycling time of around 15 minutes for +liquid water content and just less than half and hour for the cloud fraction, +so we would expect timestep dependency to occur from around a timestep of +15 minutes upwards. We might just about get away with the 30 minute step +of the climate model, but it is not a good situation to try to model. +This is demonstrating the difficulty of modelling shallow convective cloud +by a prognostic scheme, where the physical lifetime of the clouds is +of order the timestep - ideally we wouldn't want to try to model anything +prognostically when the cycling time is less than the timestep. + +As discussed in section \ref{sec:erosion_numerics}, the timestep sensitivity +of cloud amounts in shallow cumulus regimes can be addressed by using +a more accurate numerical method to solve the erosion term. +Several options are available under the UM namelist switch +\textbf{i\_pc2\_erosion\_numerics}. + +In the early development of PC2 we chose to incorporate the PC2 cloud +and condensation increments in the same location where the increments +were calculated (e.g. the microphysics cloud fraction increments +get added along with the microphysics $\overline{T}$ and $\overline{q}$ +increments). This choice was made in order not to confuse the timestepping +method in the UM, which has been carefully developed over a number of +years to achieve numerical accuracy. However, we note that the rapidly +varying nature (in space and time) of variables such as $\overline{q_{cl}}$ +and $C_l$ is very different from the smooth fields of $\overline{q_T}$ and +$\overline{T}$, for which the timestepping was developed, and it may +not be appropriate to implement these in the same locations. In +particular, we might wish to store the increments through the timestep +and update values of $\overline{q_{cl}}$ and $C_l$ etc. at the end +of the timestep, where many of the balances can be cancelled. + +One issue is that we are calculating the increments due to condensation +associated with the adiabatic response to pressure changes after the +Helmholtz solver. Pragmatically, we need to do it here since we do not know +the arrival value of pressure until after the Helmholtz solver has been +used. However, in order to achieve balanced dynamical fields, it is useful +the Helmholtz solver to be called after all the latent heating terms have +been calculated (which not only includes the adiabatic response to +lifting but the cloud initiation term). We have shown that PC2 can run with +the two terms switched over, but this implies that we are missing part of +the pressure change following the parcel (the time changing part +rather than the spatially changing adiabatic part). Although the adiabatic +change is usually likely to dominate, it may be a significant loss. +Under the UM namelist switch \textbf{l\_pc2\_sl\_advection}, +we can call the PC2 response twice, once before the Helmholtz solver and +once afterwards in order to pick up most of the latent heat change before +the solver, but not to have PC2 miss some of the pressure change. +The call for the advective part (before the Helmholtz solver) is +actually done before the call to atmos\_physics2 as well, and so results in +more realistic, saturation-adjusted, profiles being passed to the convection +scheme. + +We have placed the initiation at the end of the timestep, but it is sensible to +ask whether this could ideally be located elsewhere. + +\subsubsection{Initiation formulation} +Ideally this should be a relatively infrequent part of the model +but remains an essential part of the code. It is reasonable to ask whether +the initiation is optimal, particular in the diagnosis of when it is +applied. For example, we note that the initiation is currently +symmetrical, with initiation from $C_l=1$ occuring with the same +$RH_{crit}$ value as from $C_l=0$. However, the \cite{wf00} +observations hint that a higher $RH_{crit}$ might be more appropriate +for initiation from $C_l=1$. + +\subsubsection{70-levels performance} +The performance of PC2:66 in the 70-levels model is not good as +far as shallow convective cloud is concerned (there is far too +much of it in the trade regions). It may be that PC2 is latching onto +a convection sensitivity that is present on going from L38 to L70 but +had little effect in a non-PC2 simulation. It may also be related +to a reduction in timestep from 30 minutes to 20 minutes. +Investigations have not +made much progress in identifying the reasons for the differences, +or producing effective tunings to counter the problem. + +\subsubsection{High horizontal resolution performace} +PC2 has only been tested once at 4 km horizontal resolution. This +produced excessive shallow convective cloud (this may or may not be related +to the 70-levels problem above). Since this simulation the erosion +term has been increased dramatically, which may help. We note that +one of the main advantages of PC2, that of a prognostic link of +cloud to convection, is reduced at high resolution, as convection +becomes more explicit rather than diagnosed. We hence see a +resolution limit beyond which it is no longer appropriate to use +PC2. Results look acceptable at 12 km resolution, but we have not +quantitatively explored this limit. + +\subsubsection{Diagnostic evaluation} +One of the principal areas for future cloud scheme development +work planned in the future is in the area of detailed evaluation against +a number of data sources, such as CloudSat, ground based radar, +or case study campaigns. The quantitative evaluation has been +lacking to a significant degree in the development of the scheme, +as the focus has been on tackling qualitatively poor results. +Hence new sources of evaluation work on PC2 would be very welcome. + +\subsubsection{Moisture distribution within the deposition/sublimation term} +The liquid cloud changes in PC2 (or in a non-PC2 run) are based upon +a moisture PDF, as are the deposition/sublimation changes. However, it is +not the same PDF. It has always been the case with the prognostic ice +microphysics term that its PDF, whether explicit or implicit, has not +been rigorously consistent with the PDF used in the calculation of liquid +water, because it was most easily developed that way and produced reasonable +results. It may be useful to investigate whether the two PDF +representations can be brought together in a rigourous way, both for the PC2 +scheme and the \cite{smith90} scheme. + +We have similarly noted potential inconsistencies in the parametrization +of cloud fraction changes between the evaporation of rain term and the +riming (or accretion) term. Again, it might be possible to bring together +these formulations into a single consistent framework. + +\subsubsection{Area cloud fraction representation} +The current area cloud fraction representation is not used when +convection is taking place (signified by the \textit{cumulus} logical). +This inevitably leads to a potential switching between two different values +of the cloud fields if the convective boundary layer (not whether the +convection is shallow or deep) switches on and off, which is undesirable, +although not as bad as switching cloud on and off completely (as for the +current convective cloud formulation). Additionally, it is reasonable to argue that +having an area cloud fraction for cirrus cloud depend upon whether the boundary +layer is well mixed or has shallow convection occuring is not a reasonable link. + +Work in Australia on a TWP-ICE single column model case study using PC2 +suggests the area cloud fraction scheme over estimates the area cloud coverage +for tropical anvil clouds (which exist long after the convection itself has +ceased). This is perhaps not surprising since the \cite{bhi05} area +cloud fraction scheme was evaluated against mid-latitude cloud and it is +known that tropical clouds have greater vertical coherence. Tuning the +parameters in $large_scale_cloud/ls_acf_brooks.F90$ may be beneficial. + +%%\subsection{Acknowledgements} + +\begin{figure} +\begin{center} +\includegraphics[scale=1.0]{pc2_process_explanation} +\caption{Schematic summary of the PC2 cloud scheme.} +\label{fig:schematic} +\end{center} +\end{figure} + +\begin{figure} +\begin{center} +\includegraphics[scale=0.6]{Timestepping_ctl66.epsi} +\caption{Timestepping diagram for the control (non-PC2) scheme} +\label{fig:tstep_diag} +\end{center} +\end{figure} + +\begin{figure} +\begin{center} +\includegraphics[scale=0.6]{Timestepping_pc266.epsi} +\caption{Timestepping diagram for the PC2 scheme} +\label{fig:tstep_prog} +\end{center} +\end{figure} + +\bibliography{../029/refs} +\bibliographystyle{plainnat} + +\end{document} diff --git a/documentation/source/science_guide/cloud_schemes/refs.bib b/documentation/source/science_guide/cloud_schemes/refs.bib new file mode 100644 index 0000000000..136ad7991e --- /dev/null +++ b/documentation/source/science_guide/cloud_schemes/refs.bib @@ -0,0 +1,371 @@ +%@string{qj="Q. 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From Development of a New Cloud Scheme for the +Unified Model}, + institution = {Met Office internal note}, + year = {2001}, +} + +@article{abel_etal_2017, +author = {S. J. Abel and I. A. Boutle and K. Waite and S. Fox and P. R. A. Brown and R. Cotton and G. Lloyd and T. W. Choularton and K. N. Bower}, +title = {The Role of Precipitation in Controlling the Transition from Stratocumulus to Cumulus Clouds in a Northern Hemisphere Cold-Air Outbreak}, +journal = jas, +volume = {74}, +number = {7}, +pages = {2293-2314}, +year = {2017}, +doi = {10.1175/JAS-D-16-0362.1}, +} + +@Article{morcrette_etal_2019, + author = {C. Morcrette and K. Brown and R. Bowyer and P. Gill and D. Suri}, + title = {Development and evaluation of in-flight icing index for aviation }, + journal = {Weather and Forecasting}, + year = {2019}, + volume = {34}, + pages = {731-750}, + doi = {10.1175/WAF-D-18-0177.1}, +} + +@Article{morcrette_2020, + author = {C. J. Morcrette}, + title = {Modification of the thermodynamic variability closure in the Met Office Unified Model prognostic cloud scheme}, + journal = {Atmospheric Science Letters}, + year = {2020}, + volume = {}, + pages = {}, + doi = {10.1002/asl.1021}, +} + +@Article{vanweverberg2020, + author = {Van Weverberg, K. and C.J. Morcrette and I.A. Boutle}, + title = {Bi-modal diagnostic cloud fraction parameterization. Part I: Motivating analysis and scheme description}, + journal = mwr, + year = {2020}, + volume = {}, + pages = {-}, +} + From 640dc90223df5665b26082d2e493a99e1fd6b241 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Mon, 20 Apr 2026 11:52:07 +0100 Subject: [PATCH 028/116] Re-importing include files used by the latex doc. --- .../cloud_schemes/um_call_tree.tex | 530 ++++++++++++++++++ .../cloud_schemes/um_call_tree_preamble.tex | 25 + 2 files changed, 555 insertions(+) create mode 100644 documentation/source/science_guide/cloud_schemes/um_call_tree.tex create mode 100644 documentation/source/science_guide/cloud_schemes/um_call_tree_preamble.tex diff --git a/documentation/source/science_guide/cloud_schemes/um_call_tree.tex b/documentation/source/science_guide/cloud_schemes/um_call_tree.tex new file mode 100644 index 0000000000..97d6a73f15 --- /dev/null +++ b/documentation/source/science_guide/cloud_schemes/um_call_tree.tex @@ -0,0 +1,530 @@ + +% Latex source to make a diagram of the UM subroutine call tree, showing the +% locations of all cloud scheme calls. This diagram is included in both +% UMDP 029 (large-scale cloud scheme) and UMDP 030 (PC2). + +% NOTE: any preamble text required for this should be put in the file +% um_call_tree_preamble.tex, which is also inlcuded in both the UMDPs. + +Subroutines only called for the \textcolor{blue}{Smith} scheme are highlighted +in \textcolor{blue}{blue}, those only called for \textcolor{mygreen}{PC2} are +in \textcolor{mygreen}{green}, and those only called for +the \textcolor{purple}{bimodal} scheme are in \textcolor{purple}{purple}. + +\subsubsection{Main Tree from atm\_step\_4a} + +\begin{itemize} + +\item {\bf atm\_step\_4a} \\* +(performs one timestep of the Unified Model...) + \begin{itemize} + + \begin{tcolorbox}[enhanced jigsaw, breakable] + \item {\bf atm\_step\_alloc\_4a} \\* + (does miscellaneous initialisations in atm\_step) + + \begin{itemize} + + \item \textcolor{mygreen}{\bf pc2\_rhtl} \\* + (calculate start-of-timestep Relative Humidity, used by PC2 initiation) + + \end{itemize} + \end{tcolorbox} + + \begin{tcolorbox}[enhanced jigsaw, breakable] + \item {\bf atmos\_physics1} \\* + (calls explicit ``slow'' physics routines...) + \begin{itemize} + + \begin{tcolorbox} + \item {\bf microphys\_ctl} \\* + (interface to microphysics scheme) + \begin{itemize} + + \item \textcolor{mygreen}{\bf pc2\_turbulence\_ctl} \\* + (Perform optional erosion of liquid-cloud; + done here if NOT doing erosion after convection, + e.g. if no convection scheme is used). + \begin{itemize} + + \item \textcolor{mygreen}{\bf pc2\_hom\_conv} \\* + (called here just to do erosion) + + \end{itemize} + + \item \textcolor{blue}{\bf ls\_cld} \\* + (Smith scheme without area cloud fraction calculation, + to set initial cloud fields passed into microphysics) + + \item {\bf ls\_ppn} \\* + (microphysics scheme) + + \item {\bf mphys\_turb\_gen\_mixed\_phase} \\* + (turbulent production of liquid cloud) + + \item \textcolor{mygreen}{\bf pc2\_turbulence\_ctl} \\* + (optionally use the PC2 pdf-width-change code to calculate + the cloud-fraction change from the above turbulent production + of liquid cloud) + + \begin{itemize} + + \item \textcolor{mygreen}{\bf pc2\_hom\_conv} \\* + (called here just to calculate the cloud fraction increment + consistent with the turbulent qcl increment) + + \end{itemize} + + \end{itemize} + \end{tcolorbox} + + \begin{tcolorbox} + \item {\bf rad\_ctl} \\* + (interface to radiation scheme) + \begin{itemize} + + \item {\bf sw\_rad} \\* + (short-wave radiation scheme) + + \item \textcolor{mygreen}{\bf pc2\_homog\_plus\_turb} \\* + (PC2 homogeneous forcing of liquid-cloud by SW radiation heating) + + \item {\bf lw\_rad} \\* + (long-wave radiation scheme) + + \item \textcolor{mygreen}{\bf pc2\_homog\_plus\_turb} \\* + (PC2 homogeneous forcing of liquid-cloud by LW radiation tendency) + + \end{itemize} + \end{tcolorbox} + + \begin{tcolorbox} + \item {\bf atmos\_physics1\_alloc\_pc2} + (wrapper for PC2 self-consistency checks at end of atmos\_physics1) + + \begin{itemize} + + \item Add increments from microphysics + radiation onto + start-of-timestep fields to form updated fields. + + \item \textcolor{mygreen}{\bf pc2\_checks} \\* + (self-consistency checks on cloud fractions and water contents) + + \item Convert corrected updated fields back to increments. + + \end{itemize} + \end{tcolorbox} + + \end{itemize} + \end{tcolorbox} + + Begin loop over solver outer cycles + + \begin{itemize} + + \begin{tcolorbox}[enhanced jigsaw, breakable] + \item {\bf atm\_step\_phys\_reset} \\* + (for PC2, on subsequent solver outer cycles, + reset cloud-fractions to saved values after atmos\_physics1) + \end{tcolorbox} + + \begin{tcolorbox}[enhanced jigsaw, breakable] + \item {\bf eg\_sl\_moisture} \\* + (large-scale advection of cloud water contents and fractions) + \end{tcolorbox} + + \begin{tcolorbox}[enhanced jigsaw, breakable] + \item \textcolor{mygreen}{\bf pc2\_pressure\_forcing\_only} \\* + (Optionally calculate homogeneous forcing of liquid cloud by the + pressure change along the trajectory from departure point to + arrival point). + \begin{itemize} + + \item \textcolor{mygreen}{\bf pc2\_homog\_plus\_turb} \\* + (generic homogeneous forcing routine used here). + + \end{itemize} + \end{tcolorbox} + + \begin{tcolorbox}[enhanced jigsaw, breakable] + \item {\bf atmos\_physics2} \\* + (calls ``fast'' physics routines...) + + \begin{itemize} + + \begin{tcolorbox} + \item {\bf ni\_bl\_ctl} \\* + (interface to explicit boundary-layer and surface scheme calls, + including calculation of TKE and TKE-based $RH_{crit}$) + \end{tcolorbox} + + \begin{tcolorbox} + \item \textcolor{purple}{\bf bm\_calc\_tau} \\* + (calculates turbulence properties used in the bimodal cloud scheme, + based on the boundary-layer scheme TKE and mixing-length) + \end{tcolorbox} + + \begin{tcolorbox} + \item {\bf cloud\_call\_b4\_conv} \\* + (routine for optional cloud-scheme calls before convection) + \begin{itemize} + + \item \textcolor{blue}{\bf ls\_arcld} \\* + (Smith scheme with area cloud fraction; + see \ref{subsubsec:smith_acf} for a drill-down inside this routine) + + \item \textcolor{purple}{\bf bm\_ctl} \\* + (bimodal scheme) + + \item \textcolor{purple}{Set area cloud fraction equal to bulk + cloud fraction} + + \item \textcolor{mygreen}{\bf pc2\_initiation\_ctl} \\* + (interface to PC2 initiation and consistency-checks; + see \ref{subsubsec:pc2_initiation} for a drill-down inside this + routine) + + \end{itemize} + \end{tcolorbox} + + \begin{tcolorbox} + \item {\bf ni\_conv\_ctl} or {\bf other\_conv\_ctl} \\* + (interface routines to various convection schemes...) + \begin{itemize} + + \item {\bf glue\_conv\_5a/6a} \\* + (calls deep, shallow and mid-level convection schemes) + \begin{itemize} + + \item{\bf deep/shallow/mid\_conv} \\* + (convection scheme main routines) + \begin{itemize} + + \item {\bf convec2} + (completes lifting of the convective parcel by one model-level) + \begin{itemize} + + \item {\bf parcel} + (calculates new parcel properties at next level) + + \item {\bf environ} + (calculates grid-mean increments to primary fields; + includes PC2 partitioning of detrained condensate mass + between liquid and ice phases) + + \item \textcolor{mygreen}{\bf pc2\_environ} + (calculates increments to PC2 cloud fractions due to + convective detrainment and subsidence) + + \end{itemize} + + \end{itemize} + + \end{itemize} + + \item \textcolor{mygreen}{\bf pc2\_from\_conv\_ctl} \\* + (PC2 calculations after convection) + \begin{itemize} + + \item \textcolor{mygreen}{\bf pc2\_hom\_conv} \\* + (homogeneous forcing by convection, and erosion of liquid-cloud) + + \end{itemize} + + \end{itemize} + \end{tcolorbox} + + \begin{tcolorbox} + \item {\bf ni\_imp\_ctl} \\* + (interface to boundary-layer implicit solver) + \begin{itemize} + + \item {\bf imp\_solver} \\* + (implicitly solves vertical diffusion to find $T_l$ and $q_T$ + updated by turbulent fluxes). + + \item \textcolor{mygreen}{\bf pc2\_bl\_inhom\_ice} \\* + (inhomogeneous forcing of ice-cloud) + + \item \textcolor{mygreen}{\bf pc2\_delta\_hom\_turb} \\* + (homogeneous forcing of liquid cloud by the turbulent fluxes) + + \item \textcolor{mygreen}{\bf pc2\_bl\_forced\_cu} \\* + (adds diagnosed ``forced cumulus'' cloud fraction and water content + onto the PC2 prognostics) + + \item Calculate area cloud fraction: + + \textcolor{mygreen}{\bf ls\_acf\_brooks} \\* + (for the Brooks epirical method) + + \textcolor{mygreen}{\bf pc2\_hom\_arcld} \\* + (for the Cusack vertical interpolation method) + \begin{itemize} + + \item \textcolor{mygreen}{\bf pc2\_homog\_plus\_turb} \\* + (generic homogeneous forcing routine used to interpolate) + + \end{itemize} + + \item \textcolor{blue}{\bf ls\_arcld} \\* + (interface to diagnostic Smith scheme and area cloud fraction; + see \ref{subsubsec:smith_acf} for a drill-down inside this routine) + + \item \textcolor{purple}{\bf bm\_ctl} \\* + (bimodal cloud scheme) + + \item \textcolor{purple}{Set area cloud fraction equal to bulk + cloud fraction} + + \item {\bf diagnostics\_bl} \\* + (outputs boundary-layer diagnostics to STASH) + \begin{itemize} + + \item {\bf ls\_cld} \\* + (Smith scheme used here to calculate various diagnostics of + near-surface temperature and humidity, by extrapolating pressure, + $T_l$ and $q_t$ down to the desired height and then + re-diagnosing $q_{cl}$. + + \end{itemize} + + \end{itemize} + \end{tcolorbox} + + \end{itemize} + \end{tcolorbox} + + \begin{tcolorbox}[enhanced jigsaw, breakable] + \item {\bf atm\_step\_ac\_assim} \\* + (interface to Data Assimilation analysis increments...) + + \begin{itemize} + + \item {\bf ac\_ctl} + (control routine for Data Assimilation analysis increments...) + \begin{itemize} + + \item{\bf ac} + (main analysis increment routine) + + \item \textcolor{mygreen}{\bf pc2\_assim} \\* + (PC2 reponse to the analysis increments; + see \ref{subsubsec:pc2_assim} for a drill-down inside this routine) + + \item \textcolor{mygreen}{\bf ls\_acf\_brooks} + (calculate area cloud fraction using Brooks empirical method if active) + + \item \textcolor{blue}{\bf ls\_arcld} + (call diagnostic Smith scheme with area cloud fraction again to + account for the analysis increments; + see \ref{subsubsec:smith_acf} for a drill-down inside this routine) + + \end{itemize} + + \end{itemize} + \end{tcolorbox} + + \begin{tcolorbox}[enhanced jigsaw, breakable] + \item {\bf eg\_sl\_helmholtz} \\* + (dynamics pressure solver; updates pressure, and the winds used + to perform advection on the next solver outer cycle) + \end{tcolorbox} + + \end{itemize} + + End loop over solver outer cycles + + \begin{tcolorbox}[enhanced jigsaw, breakable] + \item \textcolor{mygreen}{\bf pc2\_pressure\_forcing} \\* + (interface to miscellaneous PC2 calculations at end-of-timestep) + \begin{itemize} + + \item \textcolor{mygreen}{\bf pc2\_homog\_plus\_turb} \\* + (homogeneous forcing of liquid-cloud by the dynamics pressure change; + optionally either uses total pressure change including the + Lagrangian component following the winds, or only the Eulerian + component from the dynamics solver) + + \item \textcolor{mygreen}{\bf pc2\_initiation\_ctl} \\* + (interface to PC2 initiation and consistency-checks; + see \ref{subsubsec:pc2_initiation} for a drill-down inside this routine) + + \end{itemize} + \end{tcolorbox} + + \begin{tcolorbox}[enhanced jigsaw, breakable] + \item {\bf qt\_bal\_cld} \\* + (calculates end-of-timestep cloud state consistent with final pressure...) + \begin{itemize} + + \item \textcolor{blue}{\bf ls\_arcld} \\* + (interface to diagnostic Smith scheme and area cloud fraction; + see \ref{subsubsec:smith_acf} for a drill-down inside this routine) + + \item \textcolor{purple}{\bf bm\_ctl} \\* + (bimodal cloud scheme) + + \item \textcolor{purple}{Set area cloud fraction equal to bulk + cloud fraction} + + \end{itemize} + \end{tcolorbox} + + \begin{tcolorbox}[enhanced jigsaw, breakable] + \item {\bf iau} \\* + (incremental analysis update; part of data assimilation) + \begin{itemize} + + \item \textcolor{mygreen}{\bf pc2\_assim} \\* + (PC2 reponse to the analysis increments; + see \ref{subsubsec:pc2_assim} for a drill-down inside this routine) + + \item \textcolor{mygreen}{\bf initial\_pc2\_check} \\* + (wrapper for optional self-consistency checks on prognostic cloud variables + if not doing PC2 response to analysis increments) + + \begin{itemize} + + \item \textcolor{mygreen}{\bf pc2\_checks} \\* + (self-consistency checks on cloud fractions and water contents) + + \end{itemize} + + \end{itemize} + \end{tcolorbox} + + \end{itemize} + +\end{itemize} + + +Drill-downs within some routines in the call tree are listed separately below, +to avoid duplication +(since these routines are called in multiple different places in the tree)... + +\subsubsection{Smith scheme with area cloud fraction} +\label{subsubsec:smith_acf} + +\begin{itemize} + +\begin{tcolorbox}[enhanced jigsaw, breakable] +\item \textcolor{blue}{\bf ls\_arcld} \\* +(interface to diagnostic Smith scheme and area cloud fraction) + \begin{itemize} + + \item If no area cloud fraction scheme: + + \textcolor{blue}{\bf ls\_cld} \\* + (just directly call Smith scheme) + + Set area cloud fraction equal to bulk cloud fraction. + + \item If using Cusack vertical interpolation method: + + Interpolate fields onto finer vertical grid + + \textcolor{blue}{\bf ls\_cld} \\* + (call Smith scheme using higher vertical resolution fields) + + Coarse-grain cloud fields back to model grid, but set area cloud + fraction to max of bulk cloud fraction over corresponding fine-grid levels. + + \item If using Brooks empirical area cloud fraction method: + + \textcolor{blue}{\bf ls\_cld} \\* + (just directly call Smith scheme) + + \textcolor{blue}{\bf ls\_acf\_brooks} \\* + (estimate area cloud fraction) + + \end{itemize} +\end{tcolorbox} + +\end{itemize} + + +\subsubsection{PC2 initiation} +\label{subsubsec:pc2_initiation} + +\begin{itemize} + +\begin{tcolorbox}[enhanced jigsaw, breakable] +\item \textcolor{mygreen}{\bf pc2\_initiation\_ctl} \\* +(interface to PC2 initiation and consistency-checks) + \begin{itemize} + + \item \textcolor{mygreen}{\bf pc2\_checks} \\* + (self-consistency checks on cloud fractions and water contents) + + \item PC2 initiation of liquid-cloud: + + \textcolor{mygreen}{\bf pc2\_bm\_initiate} \\* + (using the bimodal cloud scheme) + + \textcolor{mygreen}{\bf pc2\_arcld} \\* + (using the Smith scheme with the Cusack vertical interpolation method) + + \begin{itemize} + + \item \textcolor{mygreen}{\bf pc2\_initiate} \\* + (initiation using the Smith scheme, + called here on a finer vertical grid as per the Cusack method) + + \end{itemize} + + \textcolor{mygreen}{\bf pc2\_initiate} \\* + (using the Smith scheme with no area cloud representation) + + \item \textcolor{mygreen}{\bf pc2\_checks2} \\* + (further self-consistency checks on cloud-fractions) + + \item \textcolor{mygreen}{\bf pc2\_checks} \\* + (repeat the first lot of self-consistency checks again, + just in case we broke something in the mean-time!) + + \item \textcolor{mygreen}{\bf pc2\_hom\_arcld} \\* + (finds area cloud fraction using a version of the Cusack method, + where the cloud fraction on the finer vertical grid is estimated by + applying homogeneous forcing relative to the original grid fields) + + \begin{itemize} + + \item \textcolor{mygreen}{\bf pc2\_homog\_plus\_turb} \\* + (generic homogeneous forcing routine used to interpolate) + + \end{itemize} + + \end{itemize} +\end{tcolorbox} + +\end{itemize} + + +\subsubsection{PC2 Data Assimilation} +\label{subsubsec:pc2_assim} + +\begin{itemize} + +\begin{tcolorbox}[enhanced jigsaw, breakable] +\item \textcolor{mygreen}{\bf pc2\_assim} \\* +(PC2 reponse to the analysis increments) + \begin{itemize} + + \item \textcolor{mygreen}{\bf pc2\_homog\_plus\_turb} \\* + (generic PC2 homogeneous forcing routine used here for liquid-cloud) + + \item Estimate change in ice-cloud fraction from the assimilation + increment to ice-cloud mass. + + \item \textcolor{mygreen}{\bf pc2\_total\_cf} \\* + (update bulk cloud fraction due to change in ice cloud fraction) + + \item \textcolor{mygreen}{\bf pc2\_checks} \\* + (self-consistency checks on prognostic cloud fractions and + water contents) + + \end{itemize} +\end{tcolorbox} + +\end{itemize} diff --git a/documentation/source/science_guide/cloud_schemes/um_call_tree_preamble.tex b/documentation/source/science_guide/cloud_schemes/um_call_tree_preamble.tex new file mode 100644 index 0000000000..009537991a --- /dev/null +++ b/documentation/source/science_guide/cloud_schemes/um_call_tree_preamble.tex @@ -0,0 +1,25 @@ + +% Packages needed for the UM subroutine tree diagram in um_call_tree.txt + +% Used to colour-code things in the subroutine call tree diagram: +\usepackage{xcolor} +% Define a darker green, as in some pdf viewers the standard green +% is too bright to be readable on the grey background. +\definecolor{mygreen}{rgb}{0.0, 0.667, 0.0} + +% Allow more deeply nested lists, for writing the subroutine call tree: +\usepackage{enumitem} +\setlistdepth{20} +\renewlist{itemize}{itemize}{20} +\setlist[itemize]{label=$\cdot$} +\setlist[itemize,1]{label=\textcolor{black}{$\bullet$}} +\setlist[itemize,2]{label=\textcolor{blue}{$\bullet$}} +\setlist[itemize,3]{label=\textcolor{purple}{$\bullet$}} +\setlist[itemize,4]{label=\textcolor{red}{$\bullet$}} +\setlist[itemize,5]{label=\textcolor{orange}{$\bullet$}} +\setlist[itemize,6]{label=\textcolor{yellow}{$\bullet$}} +\setlist[itemize,7]{label=\textcolor{green}{$\bullet$}} +\setlist[itemize,8]{label=\textcolor{cyan}{$\bullet$}} + +% Used to draw boxes around subroutines in the call tree diagram: +\usepackage[most]{tcolorbox} \ No newline at end of file From 5fac9fb262df248071d12a42f2a859dcdb2b327b Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Mon, 20 Apr 2026 11:54:38 +0100 Subject: [PATCH 029/116] Point to included tex and bib files in the current directory. --- .../science_guide/cloud_schemes/UMDP30_PC2CloudScheme.tex | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.tex b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.tex index d5173e6ad8..f04d3aa29a 100644 --- a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.tex +++ b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.tex @@ -16,7 +16,7 @@ \usepackage{amstext,natbib} % Packages needed for the UM subroutine tree diagram in um_call_tree.txt: -\input{../029/um_call_tree_preamble} +\input{um_call_tree_preamble} \newcommand{\mmax}[1] {\mbox{\footnotesize \sf MAX} \left[#1\right] } @@ -4738,7 +4738,7 @@ \subsection{Code Structure} % The latex source input here contains a colour-coded itemize list % of the UM subroutine tree, showing the locations of all the cloud-scheme % routines. To edit this, open the source file source/029/um_call_tree.tex -\input{../029/um_call_tree} +\input{um_call_tree} \subsection{Diagnostics} \label{sec:diags} @@ -5728,7 +5728,7 @@ \subsubsection{Area cloud fraction representation} \end{center} \end{figure} -\bibliography{../029/refs} +\bibliography{refs} \bibliographystyle{plainnat} \end{document} From b29b65ff9d4e1402764fbd7cd83ec7f883aa660c Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Mon, 20 Apr 2026 15:16:55 +0100 Subject: [PATCH 030/116] Re-ran pandoc then re-applied corrections script from scratch, now automating the corrections to figures and figure cross-referencing (then re-applied the manual fixes from 8 April 2026 using git apply). --- .../cloud_schemes/UMDP30_PC2CloudScheme.rst | 14 ++++++-------- 1 file changed, 6 insertions(+), 8 deletions(-) diff --git a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst index f065962114..c6e01fa8ec 100644 --- a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst +++ b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst @@ -312,9 +312,7 @@ Concept of PC2 The PC2 scheme develops prognostic expressions for the rates of change of cloud fraction and condensate contents as a result of each process that acts in the model. We consider ice and liquid condensate as two -distinct aspects of clouds, which may or may not overlap. -:numref:`Figure %s ` -provides a schematic summary of the PC2 scheme. +distinct aspects of clouds, which may or may not overlap :numref:`Figure %s ` provides a schematic summary of the PC2 scheme. The equations for the five prognostic cloud variables can be written schematically: @@ -6859,21 +6857,21 @@ greater vertical coherence. Tuning the parameters in .. figure:: pc2_process_explanation.svg :name: fig:schematic - :alt: Schematic summary of the PC2 cloud scheme + :alt: Schematic summary of the PC2 cloud scheme. :width: 100% Schematic summary of the PC2 cloud scheme. -.. figure:: Timestepping_ctl66.svg +.. figure:: Timestepping_ctl66.epsi.svg :name: fig:tstep_diag :alt: Timestepping diagram for the control (non-PC2) scheme - :width: 100% + :width: 60% Timestepping diagram for the control (non-PC2) scheme -.. figure:: Timestepping_pc266.svg +.. figure:: Timestepping_pc266.epsi.svg :name: fig:tstep_prog :alt: Timestepping diagram for the PC2 scheme - :width: 100% + :width: 60% Timestepping diagram for the PC2 scheme From 769b468cbe3e4dfe3405395c35334a9f22984716 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Mon, 20 Apr 2026 15:21:11 +0100 Subject: [PATCH 031/116] Reran pandoc and then applied only automated corrections. --- .../cloud_schemes/UMDP30_PC2CloudScheme.rst | 43 +++++++++---------- 1 file changed, 20 insertions(+), 23 deletions(-) diff --git a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst index c6e01fa8ec..ab8428194e 100644 --- a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst +++ b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst @@ -1407,10 +1407,10 @@ used within the convection scheme. It still remains to parametrize :math:`\delta_{xl}`, which is given by the convection scheme itself. This is discussed in section :ref:`Phase of condensate`. -.. _Numerical application of injection forcing: +.. _Numerical application: -Numerical application of injection forcing -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +Numerical application +~~~~~~~~~~~~~~~~~~~~~ The numerical application using :eq:`eq:dcltdt_almost_final` may be @@ -3048,7 +3048,7 @@ cumulus regimes. Note that :math:`q_{cl}` falls to zero after a finite time :math:`\frac{{q_{cl}}_0}{1-b_1} \frac{a_L}{K} - \frac{1}{2 {C_l}_0 (1 - C_l) \, (q_{cl}-Q_c)}`. If the timestep + \frac{1}{2 {C_l}_0 (1 - C_l) \, (q_{cl}-Q_c)}`. If the timestep :math:`\Delta t` is longer than this time, then erosion completely removes the cloud during the current timestep. @@ -4549,29 +4549,26 @@ prognostic :math:`C_l` and :math:`q_{cl}` are incremented as follows: - If :math:`{q_{cl}}_{diag} > q_{cl}`: - .. math:: :label: eq:dqcl_init - - \Delta q_{cl} = {q_{cl}}_{diag} - q_{cl} + :math:`\Delta q_{cl} = {q_{cl}}_{diag} - q_{cl} + \quad \refstepcounter{equation}(\theequation)\label{eq:dqcl_init}` - If :math:`Q_C < 0`: - .. math:: :label: eq:dcl_init1 - - \Delta C_{l} = \frac{\Delta q_{cl}}{{q_{cl}}_{diag}} - \left( {C_{l}}_{diag} - C_{l} \right) + :math:`\Delta C_{l} = \frac{\Delta q_{cl}}{{q_{cl}}_{diag}} + \left( {C_{l}}_{diag} - C_{l} \right) + \quad \refstepcounter{equation}(\theequation)\label{eq:dcl_init1}` - If :math:`Q_C > 0`: - .. math:: :label: eq:dcl_init2 - - \Delta C_{l} = \frac{\Delta SD}{{SD}_{diag}} - \left( {C_{l}}_{diag} - C_{l} \right) + :math:`\Delta C_{l} = \frac{\Delta SD}{{SD}_{diag}} + \left( {C_{l}}_{diag} - C_{l} \right) + \quad \refstepcounter{equation}(\theequation)\label{eq:dcl_init2}` - Otherwise: - .. math:: \Delta q_{cl} = 0 + :math:`\Delta q_{cl} = 0` - .. math:: \Delta C_{l} = 0 + :math:`\Delta C_{l} = 0` where the subscript :math:`_{diag}` denotes the liquid cloud water content and fraction predicted by the diagnostic cloud scheme (either @@ -5040,7 +5037,7 @@ ill-conditioning of this solution near :math:`C_l = 1`. In practice, the ill-conditioning of :eq:`eq:da2` and :eq:`eq:da3` becomes too numerically awkward for us to apply the full solution based on homogeneous forcing, although, for -completeness, we outline it in Appendix :ref:`Appendix; Alternative PC2 - Data Assimilation formulations`. Hence +completeness, we outline it in Appendix :ref:`Appendix: Alternative PC2 - Data Assimilation formulations`. Hence we have chosen to apply a much simpler model. Here we use simply the data assimilation increments :math:`\Delta \overline{q}` and :math:`\Delta \overline{T}` within the standard homogeneous forcing @@ -5056,7 +5053,7 @@ data assimilation increments for :math:`\Delta \overline{q_{cl}}`, :math:`\Delta \overline{q}` and :math:`\Delta \overline{T}` remain those that the data assimilation scheme itself calculated. -Appendix :ref:`Appendix; Alternative PC2 - Data Assimilation formulations` gives, for completeness, the +Appendix :ref:`Appendix: Alternative PC2 - Data Assimilation formulations` gives, for completeness, the alternative numerical technique for the solution of :eq:`eq:da2` and :eq:`eq:da3`. However, we stress that this technique is not used within the current PC2 @@ -6041,9 +6038,9 @@ More information Information on results of the scheme and how to run the PC2 code at various model versions is available on the PC2 web site. -.. _Appendix; Alternative PC2 - Data Assimilation formulations: +.. _Appendix: Alternative PC2 - Data Assimilation formulations: -Appendix; Alternative PC2 - Data Assimilation formulations +Appendix: Alternative PC2 - Data Assimilation formulations ========================================================== In this alternative method to section :ref:`Data Assimilation` we will assume @@ -6344,9 +6341,9 @@ results than simply using the homogeneous forcing method. Further work will be required to enable the implementation of this :math:`\overline{q}` and :math:`\overline{T}` preserving method. -.. _Appendix; Essentials of PC2 for code developers: +.. _Appendix: Essentials of PC2 for code developers: -Appendix; Essentials of PC2 for code developers +Appendix: Essentials of PC2 for code developers =============================================== This section provides some guidance to code developers on the treatment From 3894d6f5884c23a7596b57f4fdb1fd364006117e Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Mon, 20 Apr 2026 15:25:36 +0100 Subject: [PATCH 032/116] Applied all manual corrections in 1 commit, for convenience of re-applying without missing something. --- .../cloud_schemes/UMDP30_PC2CloudScheme.rst | 51 ++++++++++--------- 1 file changed, 28 insertions(+), 23 deletions(-) diff --git a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst index ab8428194e..40b0107800 100644 --- a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst +++ b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst @@ -312,7 +312,9 @@ Concept of PC2 The PC2 scheme develops prognostic expressions for the rates of change of cloud fraction and condensate contents as a result of each process that acts in the model. We consider ice and liquid condensate as two -distinct aspects of clouds, which may or may not overlap :numref:`Figure %s ` provides a schematic summary of the PC2 scheme. +distinct aspects of clouds, which may or may not overlap. +:numref:`Figure %s ` +provides a schematic summary of the PC2 scheme. The equations for the five prognostic cloud variables can be written schematically: @@ -1407,10 +1409,10 @@ used within the convection scheme. It still remains to parametrize :math:`\delta_{xl}`, which is given by the convection scheme itself. This is discussed in section :ref:`Phase of condensate`. -.. _Numerical application: +.. _Numerical application of injection forcing: -Numerical application -~~~~~~~~~~~~~~~~~~~~~ +Numerical application of injection forcing +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The numerical application using :eq:`eq:dcltdt_almost_final` may be @@ -3048,7 +3050,7 @@ cumulus regimes. Note that :math:`q_{cl}` falls to zero after a finite time :math:`\frac{{q_{cl}}_0}{1-b_1} \frac{a_L}{K} - \frac{1}{2 {C_l}_0 (1 - C_l) \, (q_{cl}-Q_c)}`. If the timestep + \frac{1}{2 {C_l}_0 (1 - C_l) \, (q_{cl}-Q_c)}`. If the timestep :math:`\Delta t` is longer than this time, then erosion completely removes the cloud during the current timestep. @@ -4549,26 +4551,29 @@ prognostic :math:`C_l` and :math:`q_{cl}` are incremented as follows: - If :math:`{q_{cl}}_{diag} > q_{cl}`: - :math:`\Delta q_{cl} = {q_{cl}}_{diag} - q_{cl} - \quad \refstepcounter{equation}(\theequation)\label{eq:dqcl_init}` + .. math:: :label: eq:dqcl_init + + \Delta q_{cl} = {q_{cl}}_{diag} - q_{cl} - If :math:`Q_C < 0`: - :math:`\Delta C_{l} = \frac{\Delta q_{cl}}{{q_{cl}}_{diag}} - \left( {C_{l}}_{diag} - C_{l} \right) - \quad \refstepcounter{equation}(\theequation)\label{eq:dcl_init1}` + .. math:: :label: eq:dcl_init1 + + \Delta C_{l} = \frac{\Delta q_{cl}}{{q_{cl}}_{diag}} + \left( {C_{l}}_{diag} - C_{l} \right) - If :math:`Q_C > 0`: - :math:`\Delta C_{l} = \frac{\Delta SD}{{SD}_{diag}} - \left( {C_{l}}_{diag} - C_{l} \right) - \quad \refstepcounter{equation}(\theequation)\label{eq:dcl_init2}` + .. math:: :label: eq:dcl_init2 + + \Delta C_{l} = \frac{\Delta SD}{{SD}_{diag}} + \left( {C_{l}}_{diag} - C_{l} \right) - Otherwise: - :math:`\Delta q_{cl} = 0` + .. math:: \Delta q_{cl} = 0 - :math:`\Delta C_{l} = 0` + .. math:: \Delta C_{l} = 0 where the subscript :math:`_{diag}` denotes the liquid cloud water content and fraction predicted by the diagnostic cloud scheme (either @@ -5037,7 +5042,7 @@ ill-conditioning of this solution near :math:`C_l = 1`. In practice, the ill-conditioning of :eq:`eq:da2` and :eq:`eq:da3` becomes too numerically awkward for us to apply the full solution based on homogeneous forcing, although, for -completeness, we outline it in Appendix :ref:`Appendix: Alternative PC2 - Data Assimilation formulations`. Hence +completeness, we outline it in Appendix :ref:`Appendix; Alternative PC2 - Data Assimilation formulations`. Hence we have chosen to apply a much simpler model. Here we use simply the data assimilation increments :math:`\Delta \overline{q}` and :math:`\Delta \overline{T}` within the standard homogeneous forcing @@ -5053,7 +5058,7 @@ data assimilation increments for :math:`\Delta \overline{q_{cl}}`, :math:`\Delta \overline{q}` and :math:`\Delta \overline{T}` remain those that the data assimilation scheme itself calculated. -Appendix :ref:`Appendix: Alternative PC2 - Data Assimilation formulations` gives, for completeness, the +Appendix :ref:`Appendix; Alternative PC2 - Data Assimilation formulations` gives, for completeness, the alternative numerical technique for the solution of :eq:`eq:da2` and :eq:`eq:da3`. However, we stress that this technique is not used within the current PC2 @@ -6038,9 +6043,9 @@ More information Information on results of the scheme and how to run the PC2 code at various model versions is available on the PC2 web site. -.. _Appendix: Alternative PC2 - Data Assimilation formulations: +.. _Appendix; Alternative PC2 - Data Assimilation formulations: -Appendix: Alternative PC2 - Data Assimilation formulations +Appendix; Alternative PC2 - Data Assimilation formulations ========================================================== In this alternative method to section :ref:`Data Assimilation` we will assume @@ -6341,9 +6346,9 @@ results than simply using the homogeneous forcing method. Further work will be required to enable the implementation of this :math:`\overline{q}` and :math:`\overline{T}` preserving method. -.. _Appendix: Essentials of PC2 for code developers: +.. _Appendix; Essentials of PC2 for code developers: -Appendix: Essentials of PC2 for code developers +Appendix; Essentials of PC2 for code developers =============================================== This section provides some guidance to code developers on the treatment @@ -6859,14 +6864,14 @@ greater vertical coherence. Tuning the parameters in Schematic summary of the PC2 cloud scheme. -.. figure:: Timestepping_ctl66.epsi.svg +.. figure:: Timestepping_ctl66.svg :name: fig:tstep_diag :alt: Timestepping diagram for the control (non-PC2) scheme :width: 60% Timestepping diagram for the control (non-PC2) scheme -.. figure:: Timestepping_pc266.epsi.svg +.. figure:: Timestepping_pc266.svg :name: fig:tstep_prog :alt: Timestepping diagram for the PC2 scheme :width: 60% From a5db077afc301661d5cb13324d5fe81c10937716 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Tue, 21 Apr 2026 01:30:42 +0100 Subject: [PATCH 033/116] Fixed bibliography and references. --- .../cloud_schemes/UMDP30_PC2CloudScheme.rst | 318 +++++++++++++----- 1 file changed, 241 insertions(+), 77 deletions(-) diff --git a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst index 40b0107800..e755c764ae 100644 --- a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst +++ b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst @@ -76,40 +76,40 @@ valid structures to use in this respect.* - One may diagnose cloud fractions and condensate contents from knowledge of gridbox mean variables. This forms the basis of the - :raw-latex:`\cite{smith90}` scheme, which is described in . + `Smith (1990)`_ scheme, which is described in . -- A mixed scheme, such as :raw-latex:`\cite{sundqvist1978}` uses a +- A mixed scheme, such as `Sundqvist (1978)`_ uses a prediction of condensate contents, but a diagnostic cloud fraction. - Alternatively, one may predict cloud fraction and condensate content changes as a result of each modelled process. This forms the basis of - the :raw-latex:`\cite{t93}` scheme and the PC2 scheme. + the `Tiedtke (1993)`_ scheme and the PC2 scheme. -- Hybrid schemes, such as :raw-latex:`\cite{t02}`, will predict various +- Hybrid schemes, such as `Tompkins (2002)`_, will predict various moments of the subgrid-scale variability, and use this knowledge to diagnose the cloud fraction and condensate contents. Many years of experience of the results from the -:raw-latex:`\cite{smith90}` scheme have highlighted deficiences in the +`Smith (1990)`_ scheme have highlighted deficiences in the diagnosis of cloud from this scheme, which we feel can only be tackled by adding the memory of cloud history available by using a prognostic based scheme. We chose to develop a scheme that directly specified the impacts on observable prognostics (condensates and cloud fractions, as -in :raw-latex:`\cite{t93}`) rather than on moments of a probability -density function (as in :raw-latex:`\cite{t02}`). This is because we +in `Tiedtke (1993)`_) rather than on moments of a probability +density function (as in `Tompkins (2002)`_). This is because we believe it is easier to physically relate (and hence parametrize) the effect processes to quantities such as cloud fraction and condensate rather than to the more abstract quantities of moments of a probability density function of moisture. However, although the PC2 scheme is -similar to :raw-latex:`\cite{t93}` in its very basic prognostic variable +similar to `Tiedtke (1993)`_ in its very basic prognostic variable structure, the assumptions behind the formulation of the prognostic terms in PC2 are very different and much improved. The PC2 scheme should -not be considered to be merely an extension of :raw-latex:`\cite{t93}`. +not be considered to be merely an extension of `Tiedtke (1993)`_. In particular, we wish to use a prognostic formulation in order to link the detraiment of moisture from convection directly to cloud fraction, and to break the hard diagnostic link between cloud fraction and -condensate. These major features of the :raw-latex:`\cite{t93}` scheme +condensate. These major features of the `Tiedtke (1993)`_ scheme provide the motivation to develop the PC2 cloud scheme. .. _The ‘s’ distribution: @@ -120,10 +120,10 @@ The ‘s’ distribution Most cloud schemes are based on the concept of a distribution of fluctuations of moisture and temperature in the gridbox. Here we mathematically formalize this concept, since it is used both in the PC2 -scheme and the :raw-latex:`\cite{smith90}` scheme. +scheme and the `Smith (1990)`_ scheme. -This method was first formulated by :raw-latex:`\cite{m77}` and -:raw-latex:`\cite{sommeria_deardorff_1977}` for large-eddy simulations. +This method was first formulated by `Mellor (1977)`_ and +`Sommeria and Deardorff (1977)`_ for large-eddy simulations. It can also be applied to larger scale models. It allows us to calculate vapour and liquid contents and liquid cloud fraction from knowledge only of the combined vapour+liquid content, :math:`\overline{q_T}`, and the @@ -259,7 +259,7 @@ calculated using :eq:`eq:a_L` from This definition of :math:`\alpha` and :math:`a_L` will retrieve an *exact* value for the gridbox mean :math:`\overline{q_{cl}}` *if* the distribution is monodispersed. Hence it is the sensible form to use for -a purely diagnostic representation such as :raw-latex:`\cite{smith90}` +a purely diagnostic representation such as `Smith (1990)`_ where we explicitly consider distributions of :math:`s`. Strictly, the linear approximation implies that other approximations for :math:`\alpha` are valid: PC2 will do this (see section @@ -289,12 +289,12 @@ for :math:`C_l` and :math:`\overline{q_{cl}}`. Note that this distribution is in terms of :math:`s`, there is no need to know the three-dimensional distribution in terms of three separate variables :math:`q_T`, :math:`T_L` and :math:`p`. This is the method used by -:raw-latex:`\cite{smith90}`, where a symmetric triangular distribution +`Smith (1990)`_, where a symmetric triangular distribution function is used. For further information on the -:raw-latex:`\cite{smith90}` scheme, please refer to . Physics and +`Smith (1990)`_ scheme, please refer to . Physics and dynamics schemes hence only need to provide increments to :math:`\overline{q_T}` and :math:`\overline{T_L}`, provided that a -diagnostic scheme (such as :raw-latex:`\cite{smith90}`) is called at +diagnostic scheme (such as `Smith (1990)`_) is called at some point in the timestep to partition :math:`\overline{q_T}` into :math:`\overline{q}` and :math:`\overline{q_{cl}}`, to calculate the dry bulb temperature :math:`\overline{T}` (from :math:`\overline{T_L}` and @@ -385,9 +385,9 @@ concept of instantaneous condensation for liquid clouds. Equations homogeneous forcing methods discussed in section :ref:`Homogeneous forcing`. We note in particular that the convective cloud fraction, previously a quantity that is diagnosed separately from the large-scale cloud -fraction calculated by the :raw-latex:`\cite{smith90}` scheme, may, in +fraction calculated by the `Smith (1990)`_ scheme, may, in PC2, be included as part of the large-scale cloud fraction. This aspect -is similar to the :raw-latex:`\cite{t93}` approach. +is similar to the `Tiedtke (1993)`_ approach. The final aim of PC2 is that the parametrization of each term in :eq:`eq:dqcldt_and_dcdt` is performed by each @@ -418,7 +418,7 @@ A note on convective cloud fraction It was the original intention that PC2 be able to replace the two separate diagnostic cloud fractions (large-scale and convective) with a -single cloud fraction, as in :raw-latex:`\cite{t93}`. The hypothesis was +single cloud fraction, as in `Tiedtke (1993)`_. The hypothesis was that by detraining cloud directly from the convection scheme we would no longer need a separate representation of this cloud type. Our experience with PC2 is that this is not necessarily the case. We suspect that the @@ -428,7 +428,7 @@ create cloud associated with the detrainment part of the convection scheme, assuming that cloud associated with the active updraughts in convection is small. This assumption is not necessarily applicable. Similar arguments, and model results, come from analysis of the -:raw-latex:`\cite{t93}` and :raw-latex:`\cite{t02}` scheme (Ben Johnson, +`Tiedtke (1993)`_ and `Tompkins (2002)`_ scheme (Ben Johnson, personal communication). We also note that with two cloud fraction types and two different optical depths it is possible to have a basic degree of representation of cloud inhomogeneity. @@ -504,7 +504,7 @@ deficit, :math:`SD`, rather than tie :math:`G(-Q_c)` to a process. The saturation deficit is *defined* here in the ‘s’ framework to be the first moment of the PDF for ‘s’ values less than :math:`-Q_c`. In this way it is analogous to the liquid water content, -:math:`\overline{q_{cl}}`. Appendix A of :raw-latex:`\cite{wg03}` writes +:math:`\overline{q_{cl}}`. Appendix A of `Wilson and Gregory (2003)`_ writes this *definition* as .. math:: :label: SD @@ -520,7 +520,7 @@ and shows this is equivalent to The basis behind the parametrization for :math:`G(-Q_c)` is to consider an underlying form of the distribution :math:`G(s)` near the :math:`+b_s` and :math:`-b_s` ends. We borrow the notation of -:raw-latex:`\cite{smith90}` and refer to a quantity :math:`b_s` that is +`Smith (1990)`_ and refer to a quantity :math:`b_s` that is the value of :math:`s` when a monomodal distribution :math:`G(s)` just equals zero. We suppose that the distribution G can be described as a power law near :math:`s=b_s`. @@ -532,7 +532,7 @@ power law near :math:`s=b_s`. provided :math:`s Date: Wed, 22 Apr 2026 16:54:32 +0100 Subject: [PATCH 034/116] Removed spurious ensuremath directives in equations. --- .../cloud_schemes/UMDP30_PC2CloudScheme.rst | 186 +++++++++--------- 1 file changed, 93 insertions(+), 93 deletions(-) diff --git a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst index e755c764ae..54ea2d886c 100644 --- a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst +++ b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst @@ -3345,38 +3345,38 @@ processes (e.g. total water content). In this case, .. math:: :label: eq:chibasic - {\ensuremath{\frac{\partial \, \ensuremath{{\chi}_{\rm{ }}^{\rm{E}}}}{\partial \, t}}}_{\rm{conv}} = - - \frac{1}{\overline{\rho}} \, \ensuremath{\frac{\partial \, \overline{\rho w^{'} \ensuremath{{\chi}_{\rm{ }}^{\rm{E'}}}}}{\partial \, z}} + {\frac{\partial \, {\chi}_{\rm{ }}^{\rm{E}}}{\partial \, t}}_{\rm{conv}} = + - \frac{1}{\overline{\rho}} \, \frac{\partial \, \overline{\rho w^{'} {\chi}_{\rm{ }}^{\rm{E'}}}}{\partial \, z} To parametrize :eq:`eq:chibasic`, the current UM convection scheme takes a mass flux approximation .. math:: :label: eq:massflux - \left({\overline{\rho w^{'} \ensuremath{{\chi}_{\rm{ }}^{\rm{E'}}}}} \right)_{\rm{conv}} = M^{\rm{P}} \, - \left({ \ensuremath{{\chi}_{\rm{ }}^{\rm{P}}} - \ensuremath{{\chi}_{\rm{ }}^{\rm{E}}} } \right) + \left({\overline{\rho w^{'} {\chi}_{\rm{ }}^{\rm{E'}}}} \right)_{\rm{conv}} = M^{\rm{P}} \, + \left({ {\chi}_{\rm{ }}^{\rm{P}} - {\chi}_{\rm{ }}^{\rm{E}} } \right) which can be differentiated to give .. math:: :label: eq:eddyflux - - \frac{1}{\overline{\rho}} \, \ensuremath{\frac{\partial \, \overline{\rho w^{'} \ensuremath{{\chi}_{\rm{ }}^{\rm{E'}}}}}{\partial \, z}} = - \ensuremath{\frac{\partial \, \ensuremath{{\chi}_{\rm{ }}^{\rm{P}}} \, M^{\rm{P}}}{\partial \, p}} - - \ensuremath{{\chi}_{\rm{ }}^{\rm{E}}} \, \ensuremath{\frac{\partial \, M^{\rm{P}}}{\partial \, p}} - - M^{\rm{P}} \, \ensuremath{\frac{\partial \, \ensuremath{{\chi}_{\rm{ }}^{\rm{E}}}}{\partial \, p}} + - \frac{1}{\overline{\rho}} \, \frac{\partial \, \overline{\rho w^{'} {\chi}_{\rm{ }}^{\rm{E'}}}}{\partial \, z} = + \frac{\partial \, {\chi}_{\rm{ }}^{\rm{P}} \, M^{\rm{P}}}{\partial \, p} - + {\chi}_{\rm{ }}^{\rm{E}} \, \frac{\partial \, M^{\rm{P}}}{\partial \, p} - + M^{\rm{P}} \, \frac{\partial \, {\chi}_{\rm{ }}^{\rm{E}}}{\partial \, p} The bulk cloud model plume equations for mass and :math:`{\chi}` are: .. math:: :label: eq:dbydpmassflux - - \ensuremath{\frac{\partial \, M^{\rm{P}}}{\partial \, p}} = + - \frac{\partial \, M^{\rm{P}}}{\partial \, p} = \left({ \varepsilon \, M^{\rm{P}} - \mu \, M^{\rm{P}} - \delta \, M^{\rm{P}} } \right) .. math:: :label: eq:dbydpmfchi - - \ensuremath{\frac{\partial \, \ensuremath{{\chi}_{\rm{ }}^{\rm{P}}} \, M^{\rm{P}}}{\partial \, p}} = \left({ - \varepsilon \, M^{\rm{P}} \, \ensuremath{{\chi}_{\rm{ }}^{\rm{E}}} - - \mu \, M^{\rm{P}} \, \ensuremath{{\chi}_{\rm{ }}^{\rm{R}}} - \delta \, M^{\rm{P}} \, \ensuremath{{\chi}_{\rm{ }}^{\rm{P}}} + - \frac{\partial \, {\chi}_{\rm{ }}^{\rm{P}} \, M^{\rm{P}}}{\partial \, p} = \left({ + \varepsilon \, M^{\rm{P}} \, {\chi}_{\rm{ }}^{\rm{E}} + - \mu \, M^{\rm{P}} \, {\chi}_{\rm{ }}^{\rm{R}} - \delta \, M^{\rm{P}} \, {\chi}_{\rm{ }}^{\rm{P}} } \right) @@ -3387,10 +3387,10 @@ Equations :eq:`eq:eddyflux`, .. math:: :label: eq:chimassflux - {\ensuremath{\frac{\partial \, \ensuremath{{\chi}_{\rm{ }}^{\rm{E}}}}{\partial \, t}}}_{\rm{conv}} = - - M^{\rm{P}} \, \ensuremath{\frac{\partial \, \ensuremath{{\chi}_{\rm{ }}^{\rm{E}}}}{\partial \, p}} - + \mu \, M^{\rm{P}} \, \left({ \ensuremath{{\chi}_{\rm{ }}^{\rm{R}}} - \ensuremath{{\chi}_{\rm{ }}^{\rm{E}}} } \right) - + \delta \, M^{\rm{P}} \, \left({ \ensuremath{{\chi}_{\rm{ }}^{\rm{P}}} - \ensuremath{{\chi}_{\rm{ }}^{\rm{E}}} } \right) + {\frac{\partial \, {\chi}_{\rm{ }}^{\rm{E}}}{\partial \, t}}_{\rm{conv}} = + - M^{\rm{P}} \, \frac{\partial \, {\chi}_{\rm{ }}^{\rm{E}}}{\partial \, p} + + \mu \, M^{\rm{P}} \, \left({ {\chi}_{\rm{ }}^{\rm{R}} - {\chi}_{\rm{ }}^{\rm{E}} } \right) + + \delta \, M^{\rm{P}} \, \left({ {\chi}_{\rm{ }}^{\rm{P}} - {\chi}_{\rm{ }}^{\rm{E}} } \right) while :math:`{\chi}_{\rm{}}^{\rm{P}}` is obtained from the vertical gradient derived by combining :eq:`eq:dbydpmassflux` @@ -3398,9 +3398,9 @@ and :eq:`eq:dbydpmfchi` : .. math:: :label: eq:gradchipar - M^{\rm{P}} \, \ensuremath{\frac{\partial \, \ensuremath{{\chi}_{\rm{ }}^{\rm{P}}}}{\partial \, p}} = - \varepsilon \, M^{\rm{P}} \, \left({ \ensuremath{{\chi}_{\rm{ }}^{\rm{P}}} - \ensuremath{{\chi}_{\rm{ }}^{\rm{E}}} } \right)- - \mu \, M^{\rm{P}} \, \left({ \ensuremath{{\chi}_{\rm{ }}^{\rm{P}}} - \ensuremath{{\chi}_{\rm{ }}^{\rm{R}}} } \right) + M^{\rm{P}} \, \frac{\partial \, {\chi}_{\rm{ }}^{\rm{P}}}{\partial \, p} = + \varepsilon \, M^{\rm{P}} \, \left({ {\chi}_{\rm{ }}^{\rm{P}} - {\chi}_{\rm{ }}^{\rm{E}} } \right)- + \mu \, M^{\rm{P}} \, \left({ {\chi}_{\rm{ }}^{\rm{P}} - {\chi}_{\rm{ }}^{\rm{R}} } \right) Within the model, eqn :eq:`eq:chimassflux` would take a discretized form which actually depends upon whether the model level, @@ -3412,19 +3412,19 @@ discretized form of :eq:`eq:chimassflux`, setting .. math:: :label: eq:chidisck - {\ensuremath{\frac{\partial \, \ensuremath{{\chi}_{\rm{ }}^{\rm{E}}}}{\partial \, t}}}_{\rm{conv, \, k}} = m_{\rm{k+1/2}} \, - \frac{ \left({\ensuremath{{\chi}_{\rm{k+1}}^{\rm{E}}} - \ensuremath{{\chi}_{\rm{k}}^{\rm{E}}}} \right)} + {\frac{\partial \, {\chi}_{\rm{ }}^{\rm{E}}}{\partial \, t}}_{\rm{conv, \, k}} = m_{\rm{k+1/2}} \, + \frac{ \left({{\chi}_{\rm{k+1}}^{\rm{E}} - {\chi}_{\rm{k}}^{\rm{E}}} \right)} {{\Delta z}_{\rm{k \, \rightarrow \, k+1}}} - + {\delta}_{\rm{k}} \, m_{\rm{k}} \, \left({ \ensuremath{{\chi}_{\rm{k}}^{\rm{P}}} - \ensuremath{{\chi}_{\rm{k}}^{\rm{E}}} } \right) + + {\delta}_{\rm{k}} \, m_{\rm{k}} \, \left({ {\chi}_{\rm{k}}^{\rm{P}} - {\chi}_{\rm{k}}^{\rm{E}} } \right) \qquad \ldots \; \mbox{for k $>$ cb} .. math:: :label: eq:chidisccb - {\ensuremath{\frac{\partial \, \ensuremath{{\chi}_{\rm{ }}^{\rm{E}}}}{\partial \, t}}}_{\rm{conv, \, cb}} = m_{\rm{cb+1/2}} \, - \frac{ \left({\ensuremath{{\chi}_{\rm{cb+1}}^{\rm{E}}} - \ensuremath{{\chi}_{\rm{cb}}^{\rm{E}}}} \right)} + {\frac{\partial \, {\chi}_{\rm{ }}^{\rm{E}}}{\partial \, t}}_{\rm{conv, \, cb}} = m_{\rm{cb+1/2}} \, + \frac{ \left({{\chi}_{\rm{cb+1}}^{\rm{E}} - {\chi}_{\rm{cb}}^{\rm{E}}} \right)} {{\Delta z}_{\rm{cb \, \rightarrow \, cb+1}}} - m_{\rm{cb}} \, - \left({ \ensuremath{{\chi}_{\rm{i,cb}}^{\rm{P}}} - \ensuremath{{\chi}_{\rm{cb}}^{\rm{E}}} } \right) + \left({ {\chi}_{\rm{i,cb}}^{\rm{P}} - {\chi}_{\rm{cb}}^{\rm{E}} } \right) where the initial parcel value :math:`{\chi}_{\rm{i,cb}}^{\rm{P}}` may @@ -3444,14 +3444,14 @@ terms for temperature and specific humidity: .. math:: :label: eq:defineq1 - {\ensuremath{\frac{\partial \, \ensuremath{T_{\rm{ }}^{\rm{E}}}}{\partial \, t}}}_{\rm{conv}} = Q1 \equiv + {\frac{\partial \, T_{\rm{ }}^{\rm{E}}}{\partial \, t}}_{\rm{conv}} = Q1 \equiv \left({ \frac{L}{c_{P}} } \right)\, {\overline{Q}}_{\rm{par}} - - \frac{1}{\overline{\rho}} \, \ensuremath{\frac{\partial \, \overline{\rho w^{'} \ensuremath{T_{\rm{ }}^{\rm{E'}}}}}{\partial \, z}} + - \frac{1}{\overline{\rho}} \, \frac{\partial \, \overline{\rho w^{'} T_{\rm{ }}^{\rm{E'}}}}{\partial \, z} .. math:: :label: eq:defineq2 - {\ensuremath{\frac{\partial \, \ensuremath{q_{\rm{ }}^{\rm{E}}}}{\partial \, t}}}_{\rm{conv}} = Q2 \equiv - {\overline{Q}}_{\rm{par}} - - \frac{1}{\overline{\rho}} \, \ensuremath{\frac{\partial \, \overline{\rho w^{'} \ensuremath{q_{\rm{ }}^{\rm{E'}}}}}{\partial \, z}} + {\frac{\partial \, q_{\rm{ }}^{\rm{E}}}{\partial \, t}}_{\rm{conv}} = Q2 \equiv - {\overline{Q}}_{\rm{par}} + - \frac{1}{\overline{\rho}} \, \frac{\partial \, \overline{\rho w^{'} q_{\rm{ }}^{\rm{E'}}}}{\partial \, z} where :math:`{\overline{Q}}_{\rm{par}}` is the rate of condensation @@ -3467,22 +3467,22 @@ gradient equations based upon :eq:`eq:gradchipar` .. math:: :label: eq:gradtpar - M^{\rm{P}} \, \ensuremath{\frac{\partial \, \ensuremath{T_{\rm{ }}^{\rm{P}}}}{\partial \, p}} = - \varepsilon \, M^{\rm{P}} \, \left({ \ensuremath{T_{\rm{ }}^{\rm{P}}} - \ensuremath{T_{\rm{ }}^{\rm{E}}} } \right)- - \mu \, M^{\rm{P}} \, \left({ \ensuremath{T_{\rm{ }}^{\rm{P}}} - \ensuremath{T_{\rm{ }}^{\rm{R}}} } \right)- + M^{\rm{P}} \, \frac{\partial \, T_{\rm{ }}^{\rm{P}}}{\partial \, p} = + \varepsilon \, M^{\rm{P}} \, \left({ T_{\rm{ }}^{\rm{P}} - T_{\rm{ }}^{\rm{E}} } \right)- + \mu \, M^{\rm{P}} \, \left({ T_{\rm{ }}^{\rm{P}} - T_{\rm{ }}^{\rm{R}} } \right)- \left({ \frac{L}{c_{P}} } \right)\, {\overline{Q}}_{\rm{par}} .. math:: :label: eq:gradqpar - M^{\rm{P}} \, \ensuremath{\frac{\partial \, \ensuremath{q_{\rm{ }}^{\rm{P}}}}{\partial \, p}} = - \varepsilon \, M^{\rm{P}} \, \left({ \ensuremath{q_{\rm{ }}^{\rm{P}}} - \ensuremath{q_{\rm{ }}^{\rm{E}}} } \right)- - \mu \, M^{\rm{P}} \, \left({ \ensuremath{q_{\rm{ }}^{\rm{P}}} - \ensuremath{q_{\rm{ }}^{\rm{R}}} } \right)+ + M^{\rm{P}} \, \frac{\partial \, q_{\rm{ }}^{\rm{P}}}{\partial \, p} = + \varepsilon \, M^{\rm{P}} \, \left({ q_{\rm{ }}^{\rm{P}} - q_{\rm{ }}^{\rm{E}} } \right)- + \mu \, M^{\rm{P}} \, \left({ q_{\rm{ }}^{\rm{P}} - q_{\rm{ }}^{\rm{R}} } \right)+ {\overline{Q}}_{\rm{par}} .. math:: :label: eq:gradlpar - M^{\rm{P}} \, \ensuremath{\frac{\partial \, \ensuremath{l_{\rm{ }}^{\rm{P}}}}{\partial \, p}} = - \varepsilon \, M^{\rm{P}} \, \left({ \ensuremath{l_{\rm{ }}^{\rm{P}}} - \ensuremath{l_{\rm{ }}^{\rm{E}}} } \right) + M^{\rm{P}} \, \frac{\partial \, l_{\rm{ }}^{\rm{P}}}{\partial \, p} = + \varepsilon \, M^{\rm{P}} \, \left({ l_{\rm{ }}^{\rm{P}} - l_{\rm{ }}^{\rm{E}} } \right) - {\overline{Q}}_{\rm{par}} + PPN @@ -3496,23 +3496,23 @@ is basic equations .. math:: :label: eq:basictold - {\ensuremath{\frac{\partial \, \ensuremath{T_{\rm{ }}^{\rm{E}}}}{\partial \, t}}}_{\rm{conv}} = Q1 - + {\frac{\partial \, T_{\rm{ }}^{\rm{E}}}{\partial \, t}}_{\rm{conv}} = Q1 - \left({ \frac{L}{c_{P}} } \right)\, {\overline{Q}}_{\rm{reset}} .. math:: :label: eq:basicqold - {\ensuremath{\frac{\partial \, \ensuremath{q_{\rm{ }}^{\rm{E}}}}{\partial \, t}}}_{\rm{conv}} = Q2 + {\overline{Q}}_{\rm{reset}} + {\frac{\partial \, q_{\rm{ }}^{\rm{E}}}{\partial \, t}}_{\rm{conv}} = Q2 + {\overline{Q}}_{\rm{reset}} .. math:: - 0 \equiv {\ensuremath{\frac{\partial \, \ensuremath{l_{\rm{ }}^{\rm{E}}}}{\partial \, t}}}_{\rm{conv}} = {\overline{Q}}_{\rm{par}} - + 0 \equiv {\frac{\partial \, l_{\rm{ }}^{\rm{E}}}{\partial \, t}}_{\rm{conv}} = {\overline{Q}}_{\rm{par}} - {\overline{Q}}_{\rm{reset}} - PPN - - \frac{1}{\overline{\rho}} \, \ensuremath{\frac{\partial \, \overline{\rho w^{'} \ensuremath{l_{\rm{ }}^{\rm{E'}}}}}{\partial \, z}} + - \frac{1}{\overline{\rho}} \, \frac{\partial \, \overline{\rho w^{'} l_{\rm{ }}^{\rm{E'}}}}{\partial \, z} .. math:: :label: eq:basiclold = - \mu \, M^{\rm{P}} \, \ensuremath{l_{\rm{ }}^{\rm{P}}} + \delta \, M^{\rm{P}} \, \ensuremath{l_{\rm{ }}^{\rm{P}}} - + \mu \, M^{\rm{P}} \, l_{\rm{ }}^{\rm{P}} + \delta \, M^{\rm{P}} \, l_{\rm{ }}^{\rm{P}} - {\overline{Q}}_{\rm{reset}} @@ -3536,15 +3536,15 @@ Define .. math:: :label: eq:defineq4l - \left({ \ensuremath{\frac{\partial \, \ensuremath{l_{\rm{l}}^{\rm{ }}}}{\partial \, t}} } \right)_{\rm{conv}} = Q4_{\rm{l}} \equiv + \left({ \frac{\partial \, l_{\rm{l}}^{\rm{ }}}{\partial \, t} } \right)_{\rm{conv}} = Q4_{\rm{l}} \equiv {\overline{Q}}_{\rm{l, par}} - {\overline{Q}}_{\rm{l, reset}} - RAIN - - \frac{1}{\overline{\rho}} \, \ensuremath{\frac{\partial \, \overline{\rho w^{'} \ensuremath{l_{\rm{l}}^{\rm{'}}}}}{\partial \, z}} + \frac{1}{\overline{\rho}} \, \frac{\partial \, \overline{\rho w^{'} l_{\rm{l}}^{\rm{'}}}}{\partial \, z} .. math:: :label: eq:defineq4f - \left({ \ensuremath{\frac{\partial \, \ensuremath{l_{\rm{f}}^{\rm{ }}}}{\partial \, t}} } \right)_{\rm{conv}} = Q4_{\rm{f}} \equiv + \left({ \frac{\partial \, l_{\rm{f}}^{\rm{ }}}{\partial \, t} } \right)_{\rm{conv}} = Q4_{\rm{f}} \equiv {\overline{Q}}_{\rm{f, par}} - {\overline{Q}}_{\rm{f, reset}} - SNOW - - \frac{1}{\overline{\rho}} \, \ensuremath{\frac{\partial \, \overline{\rho w^{'} \ensuremath{l_{\rm{f}}^{\rm{'}}}}}{\partial \, z}} + \frac{1}{\overline{\rho}} \, \frac{\partial \, \overline{\rho w^{'} l_{\rm{f}}^{\rm{'}}}}{\partial \, z} where the PC2 assumption thus far has been that @@ -3569,15 +3569,15 @@ condensate is calculated as .. math:: :label: eq:vertparl - \ensuremath{\frac{\partial \, \ensuremath{l_{\rm{l}}^{\rm{P}}}}{\partial \, p}} = \varepsilon \, - \left({ \ensuremath{l_{\rm{l}}^{\rm{P}}} - \ensuremath{l_{\rm{l}}^{\rm{E}}} } \right)- + \frac{\partial \, l_{\rm{l}}^{\rm{P}}}{\partial \, p} = \varepsilon \, + \left({ l_{\rm{l}}^{\rm{P}} - l_{\rm{l}}^{\rm{E}} } \right)- \frac{{\overline{Q}}_{\rm{l, par}}}{M^{\rm{P}}} - \frac{RAIN}{M^{\rm{P}}} .. math:: :label: eq:vertparf - \ensuremath{\frac{\partial \, \ensuremath{l_{\rm{f}}^{\rm{P}}}}{\partial \, p}} = \varepsilon \, - \left({ \ensuremath{l_{\rm{f}}^{\rm{P}}} - \ensuremath{l_{\rm{f}}^{\rm{E}}} } \right)- + \frac{\partial \, l_{\rm{f}}^{\rm{P}}}{\partial \, p} = \varepsilon \, + \left({ l_{\rm{f}}^{\rm{P}} - l_{\rm{f}}^{\rm{E}} } \right)- \frac{{\overline{Q}}_{\rm{f, par}}}{M^{\rm{P}}} - \frac{SNOW}{M^{\rm{P}}} @@ -3595,12 +3595,12 @@ are discretized: .. math:: - \ensuremath{l_{\rm{l \, k + 1}}^{\rm{P}}} = \left({ - \ensuremath{l_{\rm{l \, k}}^{\rm{P}}} + - \varepsilon_{\rm{k} + 1/4} \, \Delta p_{\rm{k} + 1/4} \, \ensuremath{l_{\rm{l \, k}}^{\rm{E}}} + + l_{\rm{l \, k + 1}}^{\rm{P}} = \left({ + l_{\rm{l \, k}}^{\rm{P}} + + \varepsilon_{\rm{k} + 1/4} \, \Delta p_{\rm{k} + 1/4} \, l_{\rm{l \, k}}^{\rm{E}} + \varepsilon_{\rm{k} + 3/4} \, \Delta p_{\rm{k} + 3/4} \, \left[{1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4}} \right]\, - \ensuremath{l_{\rm{l \, k + 1}}^{\rm{E}}} + l_{\rm{l \, k + 1}}^{\rm{E}} } \right)\, / \, \left({EPSS_{\rm{k}}} \right) .. math:: :label: eq:discvparl @@ -3610,12 +3610,12 @@ are discretized: .. math:: - \ensuremath{l_{\rm{f \, k + 1}}^{\rm{P}}} = \left({ - \ensuremath{l_{\rm{f \, k}}^{\rm{P}}} + - \varepsilon_{\rm{k} + 1/4} \, \Delta p_{\rm{k} + 1/4} \, \ensuremath{l_{\rm{f \, k}}^{\rm{E}}} + + l_{\rm{f \, k + 1}}^{\rm{P}} = \left({ + l_{\rm{f \, k}}^{\rm{P}} + + \varepsilon_{\rm{k} + 1/4} \, \Delta p_{\rm{k} + 1/4} \, l_{\rm{f \, k}}^{\rm{E}} + \varepsilon_{\rm{k} + 3/4} \, \Delta p_{\rm{k} + 3/4} \, \left[{1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4}} \right]\, - \ensuremath{l_{\rm{f \, k + 1}}^{\rm{E}}} + l_{\rm{f \, k + 1}}^{\rm{E}} } \right)\, / \, \left({EPSS_{\rm{k}}} \right) .. math:: :label: eq:discvparf @@ -3637,22 +3637,22 @@ precipitation terms are suppressed: .. math:: :label: eq:discvparldry - \ensuremath{l_{\rm{l \, k + 1}}^{\rm{P}}} = \frac{\left({ - \ensuremath{l_{\rm{l \, k}}^{\rm{P}}} + - \varepsilon_{\rm{k} + 1/4} \, \Delta p_{\rm{k} + 1/4} \, \ensuremath{l_{\rm{l \, k}}^{\rm{E}}} + + l_{\rm{l \, k + 1}}^{\rm{P}} = \frac{\left({ + l_{\rm{l \, k}}^{\rm{P}} + + \varepsilon_{\rm{k} + 1/4} \, \Delta p_{\rm{k} + 1/4} \, l_{\rm{l \, k}}^{\rm{E}} + \varepsilon_{\rm{k} + 3/4} \, \Delta p_{\rm{k} + 3/4} \, \left[{1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4}} \right]\, - \ensuremath{l_{\rm{l \, k + 1}}^{\rm{E}}} + l_{\rm{l \, k + 1}}^{\rm{E}} } \right)}{EPSS_{\rm{k}}} .. math:: :label: eq:discvparfdry - \ensuremath{l_{\rm{f \, k + 1}}^{\rm{P}}} = \frac{\left({ - \ensuremath{l_{\rm{f \, k}}^{\rm{P}}} + - \varepsilon_{\rm{k} + 1/4} \, \Delta p_{\rm{k} + 1/4} \, \ensuremath{l_{\rm{f \, k}}^{\rm{E}}} + + l_{\rm{f \, k + 1}}^{\rm{P}} = \frac{\left({ + l_{\rm{f \, k}}^{\rm{P}} + + \varepsilon_{\rm{k} + 1/4} \, \Delta p_{\rm{k} + 1/4} \, l_{\rm{f \, k}}^{\rm{E}} + \varepsilon_{\rm{k} + 3/4} \, \Delta p_{\rm{k} + 3/4} \, \left[{1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4}} \right]\, - \ensuremath{l_{\rm{f \, k + 1}}^{\rm{E}}} + l_{\rm{f \, k + 1}}^{\rm{E}} } \right)}{EPSS_{\rm{k}}} @@ -3666,14 +3666,14 @@ produces zero fluxes at cloud base: .. math:: :label: eq:q4lcbi Q4_{\rm{l}}(cb) = 0 = M_{\rm{cb+1/2}}^{\rm{P}} \, - \ensuremath{\frac{\partial \, \ensuremath{l_{\rm{l}}^{\rm{E}}}}{\partial \, p}} - M_{\rm{cb}}^{\rm{P}}\, - \left({ \ensuremath{l_{\rm{l}}^{\rm{P \, i}}} - \ensuremath{l_{\rm{l}}^{\rm{E}}}(\rm{cb}) } \right) + \frac{\partial \, l_{\rm{l}}^{\rm{E}}}{\partial \, p} - M_{\rm{cb}}^{\rm{P}}\, + \left({ l_{\rm{l}}^{\rm{P \, i}} - l_{\rm{l}}^{\rm{E}}(\rm{cb}) } \right) .. math:: :label: eq:q4fcbi Q4_{\rm{f}}(cb) = 0 = M_{\rm{cb+1/2}}^{\rm{P}} \, - \ensuremath{\frac{\partial \, \ensuremath{l_{\rm{f}}^{\rm{E}}}}{\partial \, p}} - M_{\rm{cb}}^{\rm{P}}\, - \left({ \ensuremath{l_{\rm{f}}^{\rm{P \, i}}} - \ensuremath{l_{\rm{f}}^{\rm{E}}}(\rm{cb}) } \right) + \frac{\partial \, l_{\rm{f}}^{\rm{E}}}{\partial \, p} - M_{\rm{cb}}^{\rm{P}}\, + \left({ l_{\rm{f}}^{\rm{P \, i}} - l_{\rm{f}}^{\rm{E}}(\rm{cb}) } \right) As the convection scheme makes the single phase assumption for parcel @@ -3683,14 +3683,14 @@ at this point and adjust the temperature accordingly. .. math:: :label: eqn:meltlf \theta_{\rm{k + 1}}^{\rm{P}} = \theta_{\rm{k + 1}}^{\rm{P}} - - \left(\frac{L_{\rm{F}}}{C_{p} \, \Pi_{\rm{k + 1}}} \right)\, \ensuremath{l_{\rm{f \, k + 1}}^{\rm{P}}} - \; \ldots \; \mbox{ if \ensuremath{l_{\rm{f \, k + 1}}^{\rm{P}}} is melted } + \left(\frac{L_{\rm{F}}}{C_{p} \, \Pi_{\rm{k + 1}}} \right)\, l_{\rm{f \, k + 1}}^{\rm{P}} + \; \ldots \; \mbox{ if l_{\rm{f \, k + 1}}^{\rm{P}} is melted } .. math:: :label: eqn:freezell \theta_{\rm{k + 1}}^{\rm{P}} = \theta_{\rm{k + 1}}^{\rm{P}} + - \left(\frac{L_{\rm{F}}}{C_{p} \, \Pi_{\rm{k + 1}}} \right)\, \ensuremath{l_{\rm{l \, k + 1}}^{\rm{P}}} - \; \ldots \; \mbox{ if \ensuremath{l_{\rm{l \, k + 1}}^{\rm{P}}} is frozen } + \left(\frac{L_{\rm{F}}}{C_{p} \, \Pi_{\rm{k + 1}}} \right)\, l_{\rm{l \, k + 1}}^{\rm{P}} + \; \ldots \; \mbox{ if l_{\rm{l \, k + 1}}^{\rm{P}} is frozen } Once a final value for the condensation term @@ -3708,7 +3708,7 @@ The precipitation calculation is unaltered. .. math:: :label: eq:precip - P_{\rm{k} + 1} = \left({ \ensuremath{l_{\rm{k + 1}}^{\rm{P}}} - \ensuremath{l_{\rm{MIN}}^{\rm{P}}} } \right)\, + P_{\rm{k} + 1} = \left({ l_{\rm{k + 1}}^{\rm{P}} - l_{\rm{MIN}}^{\rm{P}} } \right)\, M_{\rm{k} + 1} \, / \, g where :math:`l_{\rm{k + 1}}^{\rm{P}}` = @@ -3724,15 +3724,15 @@ This reduces the parcel condensate to : .. math:: :label: eq:vparlfinal - \ensuremath{l_{\rm{l \, k + 1}}^{\rm{P}}} = \left({ - \frac{\ensuremath{l_{\rm{l \, k + 1}}^{\rm{P}}}}{\ensuremath{l_{\rm{k + 1}}^{\rm{P}}}} - } \right)\, \ensuremath{l_{\rm{MIN}}^{\rm{P}}} + l_{\rm{l \, k + 1}}^{\rm{P}} = \left({ + \frac{l_{\rm{l \, k + 1}}^{\rm{P}}}{l_{\rm{k + 1}}^{\rm{P}}} + } \right)\, l_{\rm{MIN}}^{\rm{P}} .. math:: :label: eq:vparffinal - \ensuremath{l_{\rm{f \, k + 1}}^{\rm{P}}} = \left({ - \frac{\ensuremath{l_{\rm{f \, k + 1}}^{\rm{P}}}}{\ensuremath{l_{\rm{k + 1}}^{\rm{P}}}} - } \right)\, \ensuremath{l_{\rm{MIN}}^{\rm{P}}} + l_{\rm{f \, k + 1}}^{\rm{P}} = \left({ + \frac{l_{\rm{f \, k + 1}}^{\rm{P}}}{l_{\rm{k + 1}}^{\rm{P}}} + } \right)\, l_{\rm{MIN}}^{\rm{P}} The final parcel condensate values are then used in the rate calculation @@ -3740,18 +3740,18 @@ based upon eqn :eq:`eq:basiclold`: .. math:: :label: eq:q4lmassf - Q4_{\rm{l}}(k) = M_{\rm{k+1/2}}^{\rm{P}} \, \ensuremath{\frac{\partial \, \ensuremath{l_{\rm{l}}^{\rm{E}}}}{\partial \, p}} + + Q4_{\rm{l}}(k) = M_{\rm{k+1/2}}^{\rm{P}} \, \frac{\partial \, l_{\rm{l}}^{\rm{E}}}{\partial \, p} + \left({ {\mu}_{\rm{k}} \, M_{\rm{k}}^{\rm{P}} + {\delta}_{\rm{k}} \, M_{\rm{k}}^{\rm{P}} } \right)\, - \left({ \ensuremath{l_{\rm{l}}^{\rm{P}}}(\rm{k}) - \ensuremath{l_{\rm{l}}^{\rm{E}}}(\rm{k}) } \right)- + \left({ l_{\rm{l}}^{\rm{P}}(\rm{k}) - l_{\rm{l}}^{\rm{E}}(\rm{k}) } \right)- {\overline{Q}}_{\rm{l, reset}} .. math:: :label: eq:q4fmassf - Q4_{\rm{f}}(k) = M_{\rm{k+1/2}}^{\rm{P}} \, \ensuremath{\frac{\partial \, \ensuremath{l_{\rm{f}}^{\rm{E}}}}{\partial \, p}} + + Q4_{\rm{f}}(k) = M_{\rm{k+1/2}}^{\rm{P}} \, \frac{\partial \, l_{\rm{f}}^{\rm{E}}}{\partial \, p} + \left({ {\mu}_{\rm{k}} \, M_{\rm{k}}^{\rm{P}} + {\delta}_{\rm{k}} \, M_{\rm{k}}^{\rm{P}} } \right)\, - \left({ \ensuremath{l_{\rm{f}}^{\rm{P}}}(\rm{k}) - \ensuremath{l_{\rm{f}}^{\rm{E}}}(\rm{k}) } \right)- + \left({ l_{\rm{f}}^{\rm{P}}(\rm{k}) - l_{\rm{f}}^{\rm{E}}(\rm{k}) } \right)- {\overline{Q}}_{\rm{f, reset}} @@ -3820,27 +3820,27 @@ Similarly, eqns :eq:`eq:q4lmassf` and .. math:: - \frac{\Delta \, \ensuremath{l_{\rm{l \, k}}^{\rm{E}}}}{\Delta \, t} = + \frac{\Delta \, l_{\rm{l \, k}}^{\rm{E}}}{\Delta \, t} = \left(\frac{ M_{\rm{k}} }{ \Delta \, p_{\rm{k}} } \right) \left[{ \left({ 1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4} } \right) \left({ 1 - \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) \left({ 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) - \left({ \ensuremath{l_{\rm{l \, k + 1}}^{\rm{E}}} - \ensuremath{l_{\rm{l \, k}}^{\rm{E}}} } \right) + \left({ l_{\rm{l \, k + 1}}^{\rm{E}} - l_{\rm{l \, k}}^{\rm{E}} } \right) } \right . + .. math:: \left({ \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) \left({ 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) - \left({ \ensuremath{l_{\rm{l \, k}}^{\rm{P}}} - \ensuremath{l_{\rm{l \, k}}^{\rm{E}}} } \right) + \left({ l_{\rm{l \, k}}^{\rm{P}} - l_{\rm{l \, k}}^{\rm{E}} } \right) + .. math:: :label: eq:enviroll \left . { \left({ \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) - \left({ \ensuremath{l_{\rm{l \, k}}^{\rm{P}}} - \ensuremath{l_{\rm{l \, k}}^{\rm{E}}} } \right) + \left({ l_{\rm{l \, k}}^{\rm{P}} - l_{\rm{l \, k}}^{\rm{E}} } \right) } \right] { } @@ -3848,27 +3848,27 @@ and .. math:: - \frac{\Delta \, \ensuremath{l_{\rm{f \, k}}^{\rm{E}}}}{\Delta \, t} = + \frac{\Delta \, l_{\rm{f \, k}}^{\rm{E}}}{\Delta \, t} = \left(\frac{ M_{\rm{k}} }{ \Delta \, p_{\rm{k}} } \right) \left[{ \left({ 1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4} } \right) \left({ 1 - \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) \left({ 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) - \left({ \ensuremath{l_{\rm{f \, k + 1}}^{\rm{E}}} - \ensuremath{l_{\rm{f \, k}}^{\rm{E}}} } \right) + \left({ l_{\rm{f \, k + 1}}^{\rm{E}} - l_{\rm{f \, k}}^{\rm{E}} } \right) } \right . + .. math:: \left({ \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) \left({ 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) - \left({ \ensuremath{l_{\rm{f \, k}}^{\rm{P}}} - \ensuremath{l_{\rm{f \, k}}^{\rm{E}}} } \right) + \left({ l_{\rm{f \, k}}^{\rm{P}} - l_{\rm{f \, k}}^{\rm{E}} } \right) + .. math:: :label: eq:envirolf \left . { \left({ \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) - \left({ \ensuremath{l_{\rm{f \, k }}^{\rm{P}}} - \ensuremath{l_{\rm{f \, k}}^{\rm{E}}} } \right) + \left({ l_{\rm{f \, k }}^{\rm{P}} - l_{\rm{f \, k}}^{\rm{E}} } \right) } \right] { } From 3e481bc046e02133b08b4e42b37cb9967668358a Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Wed, 22 Apr 2026 17:28:53 +0100 Subject: [PATCH 035/116] Added code-owner for the ported PC2 documentation. --- .github/CODEOWNERS | 3 +++ 1 file changed, 3 insertions(+) diff --git a/.github/CODEOWNERS b/.github/CODEOWNERS index 979887d626..0eeff37239 100644 --- a/.github/CODEOWNERS +++ b/.github/CODEOWNERS @@ -78,3 +78,6 @@ documentation # @MetOffice/ssdteam .github/ @MetOffice/ssdteam LICENSE @yaswant README.md @MetOffice/ssdteam + +# Owners of specific documentation sections +documentation/source/science_guide/cloud_schemes @paul-barrett From b9b31bde613c93f4bb25c0ad8623314405b6a855 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Wed, 22 Apr 2026 22:06:02 +0100 Subject: [PATCH 036/116] Added LFRic copyright notice. --- .../science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst | 6 ++++++ 1 file changed, 6 insertions(+) diff --git a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst index 54ea2d886c..801adfd09a 100644 --- a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst +++ b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst @@ -1,3 +1,9 @@ +.. ----------------------------------------------------------------------------- + (c) Crown copyright Met Office. All rights reserved. + The file LICENCE, distributed with this code, contains details of the terms + under which the code may be used. + ----------------------------------------------------------------------------- + ==================== The PC2 Cloud Scheme ==================== From 9f2c1a7951fdfbb4c49f913516f5655b834a90f0 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Thu, 23 Apr 2026 10:18:14 +0100 Subject: [PATCH 037/116] Split-up lines longer than 80 characters, as-per the style guide. --- .../cloud_schemes/UMDP30_PC2CloudScheme.rst | 383 ++++++++++++------ 1 file changed, 255 insertions(+), 128 deletions(-) diff --git a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst index 801adfd09a..aed2a30ddf 100644 --- a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst +++ b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst @@ -239,7 +239,8 @@ as :math:`s`. .. math:: :label: eq:qc_eq_qt-qs - Q_c = a_L \left( \overline{q_T} - q_{sat}(\overline{T_L},\overline{p}) \right) + Q_c = a_L \left( \overline{q_T} - q_{sat}(\overline{T_L},\overline{p}) + \right) .. math:: :label: eq:s @@ -408,9 +409,11 @@ and :math:`\frac{\partial C_l} {\partial t}` . These are referred to as Homogeneous forcing (section :ref:`Homogeneous forcing`), Injection source (or inhomogeneous forcing, section :ref:`Injection forcing`) and Width Changing (section -:ref:`Changing the width of the PDF - PC2 erosion`). Two additional modules are available to assist +:ref:`Changing the width of the PDF - PC2 erosion`). Two additional modules are +available to assist with PC2, liquid cloud initiaion (section :ref:`Initiation of cloud`) and the -calculation of total cloud fraction changes (section :ref:`Ice cloud and mixed phase regions`). +calculation of total cloud fraction changes (section :ref:`Ice cloud and mixed +phase regions`). At the present time, only the large-scale precipitation (section :ref:`Large-scale precipitation`) scheme has been rewritten fully to use the PC2 concept of prognostic cloud fractions. The existing mass-flux convection @@ -710,7 +713,8 @@ Firstly, we need to calculate the forcing term :math:`\Delta{Q_c}`. From .. math:: :label: eq:deltaqc - \Delta{Q_c} = a_L ( \Delta{\overline{q_T}} - \Delta{\overline{q_{sat}(T_L)}} ) + \Delta{Q_c} = a_L ( \Delta{\overline{q_T}} - \Delta{\overline{q_{sat}(T_L)}} + ) assuming that :math:`a_L` does not change (see below). This can be expanded, using a linear approximation for @@ -1081,8 +1085,10 @@ vary with the PDF shape :math:`n`. We first write :math:`Q_N` as .. math:: :label: eq:qn_def - Q_N = \frac{Q_c}{b_s} = \frac{ a_L (\overline{q_T} - q_{sat}(\overline{T_L})) } - { a_L (1 - RH_{crit}) q_{sat} (\overline{T_L})} = \frac{RH_T - 1}{1-RH_{crit}} + Q_N = \frac{Q_c}{b_s} = \frac{ a_L (\overline{q_T} - + q_{sat}(\overline{T_L})) } + { a_L (1 - RH_{crit}) q_{sat} (\overline{T_L})} = \frac{RH_T - + 1}{1-RH_{crit}} and then use :math:`Q_N` to solve the initiated cloud fraction. We assume a PDF described by a power law as in :eq:`eqn19` (and @@ -1093,9 +1099,12 @@ solution to :eq:`eq:int_gs_ds` is hence .. math:: :label: eq:c_qn C_l^{init'} = \left\{ \begin{array}{ll} - 0, & Q_N \le -1 \\ - \frac{1}{2} {\left( 1 + Q_N \right)}^{n+1}, & -1 < Q_N \le 0 \\ - 1 - \frac{1}{2} {\left( 1 - Q_N \right)}^{n+1}, & 0 < Q_N < 1 \\ + 0, & Q_N \le -1 + \\ + \frac{1}{2} {\left( 1 + Q_N \right)}^{n+1}, & -1 < Q_N + \le 0 \\ + 1 - \frac{1}{2} {\left( 1 - Q_N \right)}^{n+1}, & 0 < Q_N < + 1 \\ 1, & 1 \le Q_N . \end{array} \right. @@ -1124,10 +1133,14 @@ the liquid water content: .. math:: :label: eq:l_bar \frac{\overline{q_{cl}}^{init'}}{b_s} = \left\{ \begin{array}{ll} - 0, & Q_N \le -1 \\ - \frac{1}{2 (n+2)} {\left( 1 + Q_N \right)}^{n+2}, & -1 < Q_N \le 0 \\ - Q_N + \frac{1}{2 (n+2)} {\left( 1 - Q_N \right)}^{n+2}, & 0 < Q_N < 1 \\ - Q_N, & 1 \le Q_N . + 0, & Q_N \le + -1 \\ + \frac{1}{2 (n+2)} {\left( 1 + Q_N \right)}^{n+2}, & + -1 < Q_N \le 0 \\ + Q_N + \frac{1}{2 (n+2)} {\left( 1 - Q_N \right)}^{n+2}, & + 0 < Q_N < 1 \\ + Q_N, & 1 \le Q_N + . \end{array} \right. If we have been working in transformed variables we now transform back, @@ -1219,7 +1232,8 @@ air. The fractional rate at which existing air is replaced by the injected source air we will write as :math:`\frac{\partial{C_S}}{\partial{t}}`. Provided that only the liquid phase exists (see section -:ref:`Multiple phases in the injection source` for the extention to multiple phases), we then +:ref:`Multiple phases in the injection source` for the extention to multiple +phases), we then note that the rate of change of liquid cloud fraction and liquid water content in the gridbox can be written in two parts: firstly the change due to the background, and secondly the change due to the source. @@ -1250,7 +1264,8 @@ change of :math:`\overline{q_{cl}}`: .. math:: :label: eq:q4 - Q4_l = \frac{\partial{\overline{q_{cl}}}}{\partial{t}} |_{injection \, source}. + Q4_l = \frac{\partial{\overline{q_{cl}}}}{\partial{t}} |_{injection \, + source}. We see that we do not need to know anything about the nature of the two PDFs involved, except the assumption that the injected PDF contains @@ -1327,7 +1342,8 @@ equivalently to :eq:`eq:dcdt_inhom` as .. math:: :label: eq:dcldt_inhom \frac{\partial{C_l}}{\partial{t}} = - - C_l \frac{\partial{C_S}}{\partial{t}} + g_l \frac{\partial{C_S}}{\partial{t}} . + - C_l \frac{\partial{C_S}}{\partial{t}} + g_l + \frac{\partial{C_S}}{\partial{t}} . Combining :eq:`eq:dcldt_inhom` and :eq:`eq:dctdt_inhom` by eliminating @@ -1429,7 +1445,8 @@ as .. math:: :label: eq:cft_ts - \Delta{C_t} = \frac{(1 - C_t)} {q_c^S - \overline{q_{cl}} - \overline{q_{cf}}} + \Delta{C_t} = \frac{(1 - C_t)} {q_c^S - \overline{q_{cl}} - + \overline{q_{cf}}} ( \Delta{\overline{q_{cl}}} + \Delta{\overline{q_{cf}}} ), .. math:: :label: eq:cfl_ts @@ -1508,7 +1525,8 @@ detrainment. It is possible to calculate directly the change in :math:`C_l` that should occur due to the detrainment and compensating subsidence treated together, in the same way that :math:`\Delta \overline{q_{cl}}` is calculated (see section -:ref:`Calculation of Grid-Box Averaged Condensate Rate (Q4)`), and this is the way in which the +:ref:`Calculation of Grid-Box Averaged Condensate Rate (Q4)`), and this is the +way in which the cloud fraction change **should** be done. It is an unfortunate historical emphasis in the early development of PC2 on the derivation of :eq:`eq:dcdt_inhom2` that has led to the treatment @@ -1538,7 +1556,8 @@ The homogeneous forcing, initiation and PC2 erosion sections described above have only considered the generation and dissipation of liquid clouds. Although the forcing methods will not influence the generation and dissipation of ice cloud (which is primarily performed in the -large-scale precipitation scheme, section :ref:`Large-scale precipitation`) we are +large-scale precipitation scheme, section :ref:`Large-scale precipitation`) we +are still left with the issue of how created or dissipated liquid cloud overlaps with existing ice cloud in the gridbox. The opposite situation, where changes in ice cloud are specified and changes in the overlap with @@ -1765,7 +1784,8 @@ been used as the basis of subgrid cloud initiation method for use in the Unified Model in conjunction with the PC2 prognostic cloud scheme. In Section :ref:`Model description` we outline the model of `Field et al. (2014)`_. In Section -:ref:`Model implementation and closure relations` we described its implementation in +:ref:`Model implementation and closure relations` we described its +implementation in the GCM. .. _Model description: @@ -1794,7 +1814,8 @@ of :math:`p` and :math:`T` given by .. math:: - B_0 = 4\pi C \left( \frac{\epsilon L_s^2}{K_a R T^2} + \frac{R T}{\epsilon e_{si} \psi} \right)^{-1}, + B_0 = 4\pi C \left( \frac{\epsilon L_s^2}{K_a R T^2} + \frac{R T}{\epsilon + e_{si} \psi} \right)^{-1}, .. math:: @@ -1835,12 +1856,14 @@ solution PDF is Gaussian with mean and variance given by: .. math:: :label: eqn:si_avg \overline{S_i} = - S_{\rm E}\frac{ \left(\varepsilon/L^2\right)^{1/3} }{ b_i B_0 {\cal M}_1 + \left(\varepsilon/L^2\right)^{1/3} }. + S_{\rm E}\frac{ \left(\varepsilon/L^2\right)^{1/3} }{ b_i B_0 {\cal M}_1 + + \left(\varepsilon/L^2\right)^{1/3} }. .. math:: :label: eqn:si_var \overline{S_i^2} = - \frac{a^2_{\rm i} \sigma^2_w \tau_{\rm d}}{ 2\left(b_i B_0 {\cal M}_1 + \left(\varepsilon/L^2\right)^{1/3}\right)}, + \frac{a^2_{\rm i} \sigma^2_w \tau_{\rm d}}{ 2\left(b_i B_0 {\cal M}_1 + + \left(\varepsilon/L^2\right)^{1/3}\right)}, Equation :eq:`eqn:si_avg` and @@ -1855,7 +1878,8 @@ given by .. math:: :label: eqn:cloud_liquid - q_{cl}^{sgt} = q_{sat\;ice}\int_{S_{i,wat}}^\infty d S_i (S_i -S_{i,wat}) F(S_i) , + q_{cl}^{sgt} = q_{sat\;ice}\int_{S_{i,wat}}^\infty d S_i (S_i -S_{i,wat}) + F(S_i) , where :math:`S_{i,wat} = e_{sat\;wat}/e_{sat\;ice}-1` is the value of @@ -1886,7 +1910,8 @@ prognostic fields, :math:`C_l` and :math:`q_{cl}`. Two methods are available for doing this. In the simplest case, the diagnosed values :math:`C_l^{sgt}` and :math:`q_{cl}^{sgt}` are just treated as increments to model prognostics (option one, in Sec. -:ref:`Options for incrementing model prognostics` below). A more complex option (see +:ref:`Options for incrementing model prognostics` below). A more complex option +(see option two, below) is to increment the model fields via the PC2 Erosion functionality. @@ -1974,7 +1999,8 @@ increments to the model prognostic fields, :math:`C_l` and .. math:: - \left( \Delta T \right)_{sgt} = \frac{L_v}{c_p} \left( \Delta q_{cl} \right)_{sgt}, + \left( \Delta T \right)_{sgt} = \frac{L_v}{c_p} \left( \Delta q_{cl} + \right)_{sgt}, .. math:: @@ -2010,7 +2036,8 @@ using PC2 Erosion. In this case: .. math:: - \left( \Delta T \right)_{sgt} = \frac{L_v}{c_p} \left( \Delta q_{cl} \right)_{sgt}, + \left( \Delta T \right)_{sgt} = \frac{L_v}{c_p} \left( \Delta q_{cl} + \right)_{sgt}, where :math:`q_{cl}` is the liquid cloud amount prior to calling to the @@ -2083,7 +2110,8 @@ condensation and cloud fraction changes. For both shortwave and longwave, we use the homogeneous forcing routines (section :ref:`Homogeneous forcing`) for :math:`\overline{q_{cl}}` and :math:`C_l`, (using eqn. :eq:`eq:deltaqc_exp2` to calculate the -:math:`Q_c` forcing) and then the method in section :ref:`Ice cloud and mixed phase regions` to +:math:`Q_c` forcing) and then the method in section :ref:`Ice cloud and mixed +phase regions` to calculate :math:`C_t` changes. There is no :math:`\overline{q_{cf}}` change associated with this process since the deposition / sublimation process is performed within the large-scale precipitation scheme (as it @@ -2093,7 +2121,8 @@ It is reasonable to question whether homogeneous forcing is a reasonable model to use when we know that a large proportion of the heating associated with radiative transfer in the atmosphere comes from the cloudy air and is not evenly spread across the gridbox. Possible -developments are discussed in section :ref:`Homogeneous forcing section improvements`. +developments are discussed in section :ref:`Homogeneous forcing section +improvements`. .. _Large-scale precipitation: @@ -2171,7 +2200,8 @@ layer below can be filled by ice in the timestep: .. math:: :label: eq:lsp_fall - \Delta C_i = \text{Max}(O^{[k,k+1]} , 1) \text{Min} (v_i \frac{\Delta t}{\Delta z^{[k]}} , 1) + \Delta C_i = \text{Max}(O^{[k,k+1]} , 1) \text{Min} (v_i \frac{\Delta + t}{\Delta z^{[k]}} , 1) where :math:`\Delta t` is the timestep. We now choose to assume a minimum overlap between the liquid and the ice phases (as in section @@ -2521,7 +2551,8 @@ corresponding large reduction in :math:`C_l`. This is an underlying feature of the PC2 scheme (discussed in `Wilson and Gregory (2003)`_), and necessarily implies the skewing of the underlying moisture PDF. Subsequent parts of the model (e.g. the width narrowing, section -:ref:`Changing the width of the PDF - PC2 erosion`) will, of course, act on the modified fields to +:ref:`Changing the width of the PDF - PC2 erosion`) will, of course, act on the +modified fields to adjust the cloud fractions further, but remember that these are separate processes and modelled elsewhere in the timestep. @@ -2557,7 +2588,8 @@ used is: .. math:: :label: eq:dbsbydtbs_turb - \frac{1}{b_s} \frac{\partial b_s}{\partial t} = \Upsilon exp ( - \frac{2.01 Q_c}{0.2 a_L q_{sat liq}(T_L)} ) + \frac{1}{b_s} \frac{\partial b_s}{\partial t} = \Upsilon exp ( - \frac{2.01 + Q_c}{0.2 a_L q_{sat liq}(T_L)} ) where the 0.2 factor is chosen to be closely equivalent to :math:`1 - RH_{crit}` and the value of 2.01 has been selected through @@ -2574,9 +2606,11 @@ term “dbsdtbs1” which scales with the rate of homogeneous forcing :math:`\frac{\partial Q_c}{\partial t}`. However this term is always set to zero on input to these routines so is never used. -The width-narrowing formulation of section :ref:`Changing the width of the PDF - PC2 erosion` is used +The width-narrowing formulation of section :ref:`Changing the width of the PDF +- PC2 erosion` is used to calculate increments in :math:`\overline{q_{cl}}` and :math:`C_l`. -Using the liquid - ice cloud overlap ideas of section :ref:`Ice cloud and mixed phase regions` +Using the liquid - ice cloud overlap ideas of section :ref:`Ice cloud and mixed +phase regions` then gives the associated :math:`C_t` change. This background narrowing term, :math:`\Upsilon`, is originally based upon work by `Stiller and Gregory (2003)`_, although it is a parameter that has been @@ -2587,7 +2621,8 @@ Numerical application of the original width-narrowing method ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Because of the strong link the mathematical expressions for width -narrowing (section :ref:`Changing the width of the PDF - PC2 erosion`) have with the expressions for +narrowing (section :ref:`Changing the width of the PDF - PC2 erosion`) have +with the expressions for the homogeneous forcing (section :ref:`Homogeneous forcing`), we choose to represent the timestepping of this process in exactly the same way as for the homogeneous forcing (in fact, in the Unified Model code we use @@ -2609,7 +2644,8 @@ calculate the change in :math:`\overline{q_{cl}}` (discretizing eq .. math:: :label: eq:dqcl_turb_final - \Delta q_{cl}^{[n+1]} = (q_{cl}^{[n]} - Q_c \frac{1}{2}(C_l^{[n]}+C_l^{[n+1]})) + \Delta q_{cl}^{[n+1]} = (q_{cl}^{[n]} - Q_c + \frac{1}{2}(C_l^{[n]}+C_l^{[n+1]})) \frac{1}{b_s} \frac{\partial b_s}{\partial t} \Delta t. In this case the value of :math:`\Delta q_{cl}` *is* limited to ensure @@ -2797,7 +2833,8 @@ cumulus regimes. .. math:: :label: eq:dcdt_hybrid_discr \frac{\Delta {C_l}_{ero}}{\Delta t} - = - \frac{ G(-Q_c)^n Q_c^n \frac{\Delta \overline{{q_{cl}}_{ero}}}{\Delta t} } + = - \frac{ G(-Q_c)^n Q_c^n \frac{\Delta + \overline{{q_{cl}}_{ero}}}{\Delta t} } { (- C_l^n Q_c^n + ( \overline{q_{cl}^n} + \frac{1}{2} \Delta \overline{{q_{cl}}_{ero}} ) ) } @@ -2813,7 +2850,8 @@ cumulus regimes. We then write equation :eq:`eq:dqcldt_hybrid` in the form: - .. math:: \frac{\partial q_{cl}}{\partial t} = q_{cl} f(q_{cl},C_l,(q_{sat}-q_v)) + .. math:: \frac{\partial q_{cl}}{\partial t} = q_{cl} + f(q_{cl},C_l,(q_{sat}-q_v)) (where the term :math:`f(q_{cl},C_l,(q_{sat}-q_v)) = \frac{A K(q_{sat}-q_v)}{q_{cl}}` @@ -2835,14 +2873,17 @@ cumulus regimes. .. math:: \Delta q_{cl}^{ero} = \Delta t - ( q_{cl}^n + \Delta q_{cl}^{hom} + \Delta q_{cl}^{ero} ) f^n + ( q_{cl}^n + \Delta q_{cl}^{hom} + \Delta q_{cl}^{ero} + ) f^n Rearranging: .. math:: - \Delta q_{cl}^{ero} = \Delta t f^n q_{cl}^n \frac{ q_{cl}^n + \Delta q_{cl}^{hom} } - { q_{cl}^n - \Delta t f^n q_{cl}^n } + \Delta q_{cl}^{ero} = \Delta t f^n q_{cl}^n \frac{ q_{cl}^n + \Delta + q_{cl}^{hom} } + { q_{cl}^n - \Delta t f^n + q_{cl}^n } Note that the term :math:`\Delta t f^n q_{cl}^n` is the erosion increment we would obtain from the purely explicit discretisation, @@ -2854,8 +2895,10 @@ cumulus regimes. .. math:: :label: eq:hybrid_erosion_impl_qcl - \Delta q_{cl}^{ero} = \Delta q_{cl}^{ero\,expl} \frac{ q_{cl}^n + \Delta q_{cl}^{hom} } - { q_{cl}^n - \Delta q_{cl}^{ero\,expl} } + \Delta q_{cl}^{ero} = \Delta q_{cl}^{ero\,expl} \frac{ q_{cl}^n + \Delta + q_{cl}^{hom} } + { q_{cl}^n - \Delta + q_{cl}^{ero\,expl} } Provided erosion is acting to reduce cloud-water (:math:`\Delta q_{cl}^{ero\,expl} < 0`), and homogeneous forcing by @@ -2869,8 +2912,10 @@ cumulus regimes. .. math:: :label: eq:hybrid_erosion_impl_Cl - \Delta C_l^{ero} = \Delta C_l^{ero\,expl} \frac{ C_l^n + \Delta C_l^{hom} } - { C_l^n - \Delta C_l^{ero\,expl} } + \Delta C_l^{ero} = \Delta C_l^{ero\,expl} \frac{ C_l^n + \Delta C_l^{hom} + } + { C_l^n - \Delta + C_l^{ero\,expl} } Where :math:`\Delta C_l^{ero\,expl}` is computed using eq :eq:`eq:dcdt_hybrid_discr`, except that the @@ -2946,7 +2991,8 @@ cumulus regimes. .. math:: :label: eq:dcdt_hybrid_1 \frac{1}{C_l} \frac{\partial C_l}{\partial t} - = \frac{ G(-Q_c) \frac{q_{cl}}{C_l^2} }{ 1 - \frac{q_{cl}}{C_l Q_c} } \; + = \frac{ G(-Q_c) \frac{q_{cl}}{C_l^2} }{ 1 - \frac{q_{cl}}{C_l Q_c} } + \; \frac{1}{q_{cl}} \frac{\partial q_{cl}}{\partial t} Under homogeneous forcing (section :ref:`Homogeneous forcing`), we @@ -3003,7 +3049,8 @@ cumulus regimes. .. math:: :label: eq:dqcldt_hybrid_1 - \frac{\partial q_{cl}}{\partial t} = -K \, 2 C_l (1 - C_l) \, (q_{sat}(T)-q_v) + \frac{\partial q_{cl}}{\partial t} = -K \, 2 C_l (1 - C_l) \, + (q_{sat}(T)-q_v) From eq :eq:`SD2`, :math:`q_{sat}(T)-q_v = \frac{SD}{a_L}`, where :math:`SD` is the saturation defecit, and :math:`a_L` is the @@ -3020,7 +3067,8 @@ cumulus regimes. .. math:: :label: eq:dqcldt_hybrid_2 - \frac{\partial q_{cl}}{\partial t} = -\frac{K}{a_L} \, 2 C_l (1 - C_l) \, + \frac{\partial q_{cl}}{\partial t} = -\frac{K}{a_L} \, 2 C_l (1 - C_l) + \, (q_{cl}-Q_c) Substituting eq :eq:`eq:cl_qcl_scaling` for @@ -3029,7 +3077,8 @@ cumulus regimes. .. math:: - \left( \frac{q_{cl}}{{q_{cl}}_0} \right)^{-b_1} \frac{\partial q_{cl}}{\partial t} + \left( \frac{q_{cl}}{{q_{cl}}_0} \right)^{-b_1} \frac{\partial + q_{cl}}{\partial t} = -\frac{K}{a_L} \, 2 {C_l}_0 (1 - C_l) \, (q_{cl}-Q_c) In significantly subsaturated conditions the r.h.s. has only weak @@ -3040,7 +3089,8 @@ cumulus regimes. .. math:: - \left[ \frac{{q_{cl}}_0}{1-b_1} \left( \frac{q_{cl}}{{q_{cl}}_0} \right)^{1-b_1} + \left[ \frac{{q_{cl}}_0}{1-b_1} \left( \frac{q_{cl}}{{q_{cl}}_0} + \right)^{1-b_1} \right]_{{q_{cl}}_0}^{{q_{cl}}_{\Delta t}} = -\frac{K}{a_L} \, 2 {C_l}_0 (1 - C_l) \, (q_{cl}-Q_c) \Delta t @@ -3101,7 +3151,8 @@ cumulus regimes. .. math:: :label: eq:dsddt_hybrid_2 - \frac{\partial SD}{\partial t} = -\frac{K}{a_L} \, 2 (1 - C_l) C_l \, SD + \frac{\partial SD}{\partial t} = -\frac{K}{a_L} \, 2 (1 - C_l) C_l \, + SD The one asymmetry between this and the :math:`q_{cl}` tendency equation :eq:`eq:dqcldt_hybrid_2` is that @@ -3166,7 +3217,8 @@ cumulus regimes. simply integrates to give exponential decline of :math:`q_{cl}` (and :math:`SD`) towards zero: - .. math:: {q_{cl}}_{\Delta t} = {q_{cl}}_0 e^{ -\frac{K}{a_L} \, 2 C_l (1 - C_l) \Delta t } + .. math:: {q_{cl}}_{\Delta t} = {q_{cl}}_0 e^{ -\frac{K}{a_L} \, 2 C_l (1 + - C_l) \Delta t } Orographic and Gravity Wave Drag -------------------------------- @@ -3332,7 +3384,8 @@ changes in the vapour and temperature from the detrainment and compensating subsidence. Similar splits are made for the cloud variables, where the injection forcing, section :ref:`Injection forcing`, is used to calculate the first term from :math:`Q4_l`. Section -:ref:`Calculation of Grid-Box Averaged Condensate Rate (Q4)` looks at the issue of the calculation +:ref:`Calculation of Grid-Box Averaged Condensate Rate (Q4)` looks at the issue +of the calculation of :math:`Q4_l` etc., and section :ref:`Background condensation` looks at the calculation of :math:`Q_{environment}`, and its associated cloud fraction change. We first look at the basic transport equations in a @@ -3352,23 +3405,27 @@ processes (e.g. total water content). In this case, .. math:: :label: eq:chibasic {\frac{\partial \, {\chi}_{\rm{ }}^{\rm{E}}}{\partial \, t}}_{\rm{conv}} = - - \frac{1}{\overline{\rho}} \, \frac{\partial \, \overline{\rho w^{'} {\chi}_{\rm{ }}^{\rm{E'}}}}{\partial \, z} + - \frac{1}{\overline{\rho}} \, \frac{\partial \, \overline{\rho w^{'} + {\chi}_{\rm{ }}^{\rm{E'}}}}{\partial \, z} To parametrize :eq:`eq:chibasic`, the current UM convection scheme takes a mass flux approximation .. math:: :label: eq:massflux - \left({\overline{\rho w^{'} {\chi}_{\rm{ }}^{\rm{E'}}}} \right)_{\rm{conv}} = M^{\rm{P}} \, + \left({\overline{\rho w^{'} {\chi}_{\rm{ }}^{\rm{E'}}}} \right)_{\rm{conv}} + = M^{\rm{P}} \, \left({ {\chi}_{\rm{ }}^{\rm{P}} - {\chi}_{\rm{ }}^{\rm{E}} } \right) which can be differentiated to give .. math:: :label: eq:eddyflux - - \frac{1}{\overline{\rho}} \, \frac{\partial \, \overline{\rho w^{'} {\chi}_{\rm{ }}^{\rm{E'}}}}{\partial \, z} = + - \frac{1}{\overline{\rho}} \, \frac{\partial \, \overline{\rho w^{'} + {\chi}_{\rm{ }}^{\rm{E'}}}}{\partial \, z} = \frac{\partial \, {\chi}_{\rm{ }}^{\rm{P}} \, M^{\rm{P}}}{\partial \, p} - - {\chi}_{\rm{ }}^{\rm{E}} \, \frac{\partial \, M^{\rm{P}}}{\partial \, p} - + {\chi}_{\rm{ }}^{\rm{E}} \, \frac{\partial \, M^{\rm{P}}}{\partial \, p} - + M^{\rm{P}} \, \frac{\partial \, {\chi}_{\rm{ }}^{\rm{E}}}{\partial \, p} The bulk cloud model plume equations for mass and :math:`{\chi}` are: @@ -3376,13 +3433,16 @@ The bulk cloud model plume equations for mass and :math:`{\chi}` are: .. math:: :label: eq:dbydpmassflux - \frac{\partial \, M^{\rm{P}}}{\partial \, p} = - \left({ \varepsilon \, M^{\rm{P}} - \mu \, M^{\rm{P}} - \delta \, M^{\rm{P}} } \right) + \left({ \varepsilon \, M^{\rm{P}} - \mu \, M^{\rm{P}} - \delta \, M^{\rm{P}} + } \right) .. math:: :label: eq:dbydpmfchi - - \frac{\partial \, {\chi}_{\rm{ }}^{\rm{P}} \, M^{\rm{P}}}{\partial \, p} = \left({ + - \frac{\partial \, {\chi}_{\rm{ }}^{\rm{P}} \, M^{\rm{P}}}{\partial \, p} + = \left({ \varepsilon \, M^{\rm{P}} \, {\chi}_{\rm{ }}^{\rm{E}} - - \mu \, M^{\rm{P}} \, {\chi}_{\rm{ }}^{\rm{R}} - \delta \, M^{\rm{P}} \, {\chi}_{\rm{ }}^{\rm{P}} + - \mu \, M^{\rm{P}} \, {\chi}_{\rm{ }}^{\rm{R}} - \delta \, M^{\rm{P}} \, + {\chi}_{\rm{ }}^{\rm{P}} } \right) @@ -3405,8 +3465,10 @@ and :eq:`eq:dbydpmfchi` : .. math:: :label: eq:gradchipar M^{\rm{P}} \, \frac{\partial \, {\chi}_{\rm{ }}^{\rm{P}}}{\partial \, p} = - \varepsilon \, M^{\rm{P}} \, \left({ {\chi}_{\rm{ }}^{\rm{P}} - {\chi}_{\rm{ }}^{\rm{E}} } \right)- - \mu \, M^{\rm{P}} \, \left({ {\chi}_{\rm{ }}^{\rm{P}} - {\chi}_{\rm{ }}^{\rm{R}} } \right) + \varepsilon \, M^{\rm{P}} \, \left({ {\chi}_{\rm{ }}^{\rm{P}} - {\chi}_{\rm{ + }}^{\rm{E}} } \right)- + \mu \, M^{\rm{P}} \, \left({ {\chi}_{\rm{ }}^{\rm{P}} - {\chi}_{\rm{ + }}^{\rm{R}} } \right) Within the model, eqn :eq:`eq:chimassflux` would take a discretized form which actually depends upon whether the model level, @@ -3418,7 +3480,8 @@ discretized form of :eq:`eq:chimassflux`, setting .. math:: :label: eq:chidisck - {\frac{\partial \, {\chi}_{\rm{ }}^{\rm{E}}}{\partial \, t}}_{\rm{conv, \, k}} = m_{\rm{k+1/2}} \, + {\frac{\partial \, {\chi}_{\rm{ }}^{\rm{E}}}{\partial \, t}}_{\rm{conv, \, + k}} = m_{\rm{k+1/2}} \, \frac{ \left({{\chi}_{\rm{k+1}}^{\rm{E}} - {\chi}_{\rm{k}}^{\rm{E}}} \right)} {{\Delta z}_{\rm{k \, \rightarrow \, k+1}}} + {\delta}_{\rm{k}} \, m_{\rm{k}} \, \left({ {\chi}_{\rm{k}}^{\rm{P}} - {\chi}_{\rm{k}}^{\rm{E}} } \right) @@ -3426,8 +3489,10 @@ discretized form of :eq:`eq:chimassflux`, setting .. math:: :label: eq:chidisccb - {\frac{\partial \, {\chi}_{\rm{ }}^{\rm{E}}}{\partial \, t}}_{\rm{conv, \, cb}} = m_{\rm{cb+1/2}} \, - \frac{ \left({{\chi}_{\rm{cb+1}}^{\rm{E}} - {\chi}_{\rm{cb}}^{\rm{E}}} \right)} + {\frac{\partial \, {\chi}_{\rm{ }}^{\rm{E}}}{\partial \, t}}_{\rm{conv, \, + cb}} = m_{\rm{cb+1/2}} \, + \frac{ \left({{\chi}_{\rm{cb+1}}^{\rm{E}} - {\chi}_{\rm{cb}}^{\rm{E}}} + \right)} {{\Delta z}_{\rm{cb \, \rightarrow \, cb+1}}} - m_{\rm{cb}} \, \left({ {\chi}_{\rm{i,cb}}^{\rm{P}} - {\chi}_{\rm{cb}}^{\rm{E}} } \right) @@ -3450,14 +3515,18 @@ terms for temperature and specific humidity: .. math:: :label: eq:defineq1 - {\frac{\partial \, T_{\rm{ }}^{\rm{E}}}{\partial \, t}}_{\rm{conv}} = Q1 \equiv + {\frac{\partial \, T_{\rm{ }}^{\rm{E}}}{\partial \, t}}_{\rm{conv}} = Q1 + \equiv \left({ \frac{L}{c_{P}} } \right)\, {\overline{Q}}_{\rm{par}} - - \frac{1}{\overline{\rho}} \, \frac{\partial \, \overline{\rho w^{'} T_{\rm{ }}^{\rm{E'}}}}{\partial \, z} + - \frac{1}{\overline{\rho}} \, \frac{\partial \, \overline{\rho w^{'} + T_{\rm{ }}^{\rm{E'}}}}{\partial \, z} .. math:: :label: eq:defineq2 - {\frac{\partial \, q_{\rm{ }}^{\rm{E}}}{\partial \, t}}_{\rm{conv}} = Q2 \equiv - {\overline{Q}}_{\rm{par}} - - \frac{1}{\overline{\rho}} \, \frac{\partial \, \overline{\rho w^{'} q_{\rm{ }}^{\rm{E'}}}}{\partial \, z} + {\frac{\partial \, q_{\rm{ }}^{\rm{E}}}{\partial \, t}}_{\rm{conv}} = Q2 + \equiv - {\overline{Q}}_{\rm{par}} + - \frac{1}{\overline{\rho}} \, \frac{\partial \, \overline{\rho w^{'} + q_{\rm{ }}^{\rm{E'}}}}{\partial \, z} where :math:`{\overline{Q}}_{\rm{par}}` is the rate of condensation @@ -3474,21 +3543,26 @@ gradient equations based upon :eq:`eq:gradchipar` .. math:: :label: eq:gradtpar M^{\rm{P}} \, \frac{\partial \, T_{\rm{ }}^{\rm{P}}}{\partial \, p} = - \varepsilon \, M^{\rm{P}} \, \left({ T_{\rm{ }}^{\rm{P}} - T_{\rm{ }}^{\rm{E}} } \right)- - \mu \, M^{\rm{P}} \, \left({ T_{\rm{ }}^{\rm{P}} - T_{\rm{ }}^{\rm{R}} } \right)- + \varepsilon \, M^{\rm{P}} \, \left({ T_{\rm{ }}^{\rm{P}} - T_{\rm{ + }}^{\rm{E}} } \right)- + \mu \, M^{\rm{P}} \, \left({ T_{\rm{ }}^{\rm{P}} - T_{\rm{ + }}^{\rm{R}} } \right)- \left({ \frac{L}{c_{P}} } \right)\, {\overline{Q}}_{\rm{par}} .. math:: :label: eq:gradqpar M^{\rm{P}} \, \frac{\partial \, q_{\rm{ }}^{\rm{P}}}{\partial \, p} = - \varepsilon \, M^{\rm{P}} \, \left({ q_{\rm{ }}^{\rm{P}} - q_{\rm{ }}^{\rm{E}} } \right)- - \mu \, M^{\rm{P}} \, \left({ q_{\rm{ }}^{\rm{P}} - q_{\rm{ }}^{\rm{R}} } \right)+ + \varepsilon \, M^{\rm{P}} \, \left({ q_{\rm{ }}^{\rm{P}} - q_{\rm{ + }}^{\rm{E}} } \right)- + \mu \, M^{\rm{P}} \, \left({ q_{\rm{ }}^{\rm{P}} - q_{\rm{ + }}^{\rm{R}} } \right)+ {\overline{Q}}_{\rm{par}} .. math:: :label: eq:gradlpar M^{\rm{P}} \, \frac{\partial \, l_{\rm{ }}^{\rm{P}}}{\partial \, p} = - \varepsilon \, M^{\rm{P}} \, \left({ l_{\rm{ }}^{\rm{P}} - l_{\rm{ }}^{\rm{E}} } \right) + \varepsilon \, M^{\rm{P}} \, \left({ l_{\rm{ }}^{\rm{P}} - l_{\rm{ + }}^{\rm{E}} } \right) - {\overline{Q}}_{\rm{par}} + PPN @@ -3507,18 +3581,22 @@ is basic equations .. math:: :label: eq:basicqold - {\frac{\partial \, q_{\rm{ }}^{\rm{E}}}{\partial \, t}}_{\rm{conv}} = Q2 + {\overline{Q}}_{\rm{reset}} + {\frac{\partial \, q_{\rm{ }}^{\rm{E}}}{\partial \, t}}_{\rm{conv}} = Q2 + + {\overline{Q}}_{\rm{reset}} .. math:: - 0 \equiv {\frac{\partial \, l_{\rm{ }}^{\rm{E}}}{\partial \, t}}_{\rm{conv}} = {\overline{Q}}_{\rm{par}} - + 0 \equiv {\frac{\partial \, l_{\rm{ }}^{\rm{E}}}{\partial \, t}}_{\rm{conv}} + = {\overline{Q}}_{\rm{par}} - {\overline{Q}}_{\rm{reset}} - PPN - - \frac{1}{\overline{\rho}} \, \frac{\partial \, \overline{\rho w^{'} l_{\rm{ }}^{\rm{E'}}}}{\partial \, z} + - \frac{1}{\overline{\rho}} \, \frac{\partial \, \overline{\rho w^{'} + l_{\rm{ }}^{\rm{E'}}}}{\partial \, z} .. math:: :label: eq:basiclold = - \mu \, M^{\rm{P}} \, l_{\rm{ }}^{\rm{P}} + \delta \, M^{\rm{P}} \, l_{\rm{ }}^{\rm{P}} - + \mu \, M^{\rm{P}} \, l_{\rm{ }}^{\rm{P}} + \delta \, M^{\rm{P}} \, l_{\rm{ + }}^{\rm{P}} - {\overline{Q}}_{\rm{reset}} @@ -3542,15 +3620,19 @@ Define .. math:: :label: eq:defineq4l - \left({ \frac{\partial \, l_{\rm{l}}^{\rm{ }}}{\partial \, t} } \right)_{\rm{conv}} = Q4_{\rm{l}} \equiv + \left({ \frac{\partial \, l_{\rm{l}}^{\rm{ }}}{\partial \, t} } + \right)_{\rm{conv}} = Q4_{\rm{l}} \equiv {\overline{Q}}_{\rm{l, par}} - {\overline{Q}}_{\rm{l, reset}} - RAIN - - \frac{1}{\overline{\rho}} \, \frac{\partial \, \overline{\rho w^{'} l_{\rm{l}}^{\rm{'}}}}{\partial \, z} + \frac{1}{\overline{\rho}} \, \frac{\partial \, \overline{\rho w^{'} + l_{\rm{l}}^{\rm{'}}}}{\partial \, z} .. math:: :label: eq:defineq4f - \left({ \frac{\partial \, l_{\rm{f}}^{\rm{ }}}{\partial \, t} } \right)_{\rm{conv}} = Q4_{\rm{f}} \equiv + \left({ \frac{\partial \, l_{\rm{f}}^{\rm{ }}}{\partial \, t} } + \right)_{\rm{conv}} = Q4_{\rm{f}} \equiv {\overline{Q}}_{\rm{f, par}} - {\overline{Q}}_{\rm{f, reset}} - SNOW - - \frac{1}{\overline{\rho}} \, \frac{\partial \, \overline{\rho w^{'} l_{\rm{f}}^{\rm{'}}}}{\partial \, z} + \frac{1}{\overline{\rho}} \, \frac{\partial \, \overline{\rho w^{'} + l_{\rm{f}}^{\rm{'}}}}{\partial \, z} where the PC2 assumption thus far has been that @@ -3603,30 +3685,36 @@ are discretized: l_{\rm{l \, k + 1}}^{\rm{P}} = \left({ l_{\rm{l \, k}}^{\rm{P}} + - \varepsilon_{\rm{k} + 1/4} \, \Delta p_{\rm{k} + 1/4} \, l_{\rm{l \, k}}^{\rm{E}} + + \varepsilon_{\rm{k} + 1/4} \, \Delta p_{\rm{k} + 1/4} \, l_{\rm{l \, + k}}^{\rm{E}} + \varepsilon_{\rm{k} + 3/4} \, \Delta p_{\rm{k} + 3/4} \, - \left[{1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4}} \right]\, + \left[{1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4}} + \right]\, l_{\rm{l \, k + 1}}^{\rm{E}} } \right)\, / \, \left({EPSS_{\rm{k}}} \right) .. math:: :label: eq:discvparl - { } { } + \left({ {\overline{Q}}_{\rm{l} \, \rm{k} + 1} \, / \, M_{\rm{k} + 1}} \right) + { } { } + \left({ {\overline{Q}}_{\rm{l} \, \rm{k} + 1} \, / \, M_{\rm{k} + + 1}} \right) - \left({ RAIN_{\rm{k} + 1} \, / \, M_{\rm{k} + 1} } \right) .. math:: l_{\rm{f \, k + 1}}^{\rm{P}} = \left({ l_{\rm{f \, k}}^{\rm{P}} + - \varepsilon_{\rm{k} + 1/4} \, \Delta p_{\rm{k} + 1/4} \, l_{\rm{f \, k}}^{\rm{E}} + + \varepsilon_{\rm{k} + 1/4} \, \Delta p_{\rm{k} + 1/4} \, l_{\rm{f \, + k}}^{\rm{E}} + \varepsilon_{\rm{k} + 3/4} \, \Delta p_{\rm{k} + 3/4} \, - \left[{1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4}} \right]\, + \left[{1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4}} + \right]\, l_{\rm{f \, k + 1}}^{\rm{E}} } \right)\, / \, \left({EPSS_{\rm{k}}} \right) .. math:: :label: eq:discvparf - { } { } + \left({ {\overline{Q}}_{\rm{f} \, \rm{k} + 1} \, / \, M_{\rm{k} + 1}} \right) + { } { } + \left({ {\overline{Q}}_{\rm{f} \, \rm{k} + 1} \, / \, M_{\rm{k} + + 1}} \right) - \left({ SNOW_{\rm{k} + 1} \, / \, M_{\rm{k} + 1} } \right) @@ -3645,9 +3733,11 @@ precipitation terms are suppressed: l_{\rm{l \, k + 1}}^{\rm{P}} = \frac{\left({ l_{\rm{l \, k}}^{\rm{P}} + - \varepsilon_{\rm{k} + 1/4} \, \Delta p_{\rm{k} + 1/4} \, l_{\rm{l \, k}}^{\rm{E}} + + \varepsilon_{\rm{k} + 1/4} \, \Delta p_{\rm{k} + 1/4} \, l_{\rm{l \, + k}}^{\rm{E}} + \varepsilon_{\rm{k} + 3/4} \, \Delta p_{\rm{k} + 3/4} \, - \left[{1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4}} \right]\, + \left[{1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4}} + \right]\, l_{\rm{l \, k + 1}}^{\rm{E}} } \right)}{EPSS_{\rm{k}}} @@ -3655,9 +3745,11 @@ precipitation terms are suppressed: l_{\rm{f \, k + 1}}^{\rm{P}} = \frac{\left({ l_{\rm{f \, k}}^{\rm{P}} + - \varepsilon_{\rm{k} + 1/4} \, \Delta p_{\rm{k} + 1/4} \, l_{\rm{f \, k}}^{\rm{E}} + + \varepsilon_{\rm{k} + 1/4} \, \Delta p_{\rm{k} + 1/4} \, l_{\rm{f \, + k}}^{\rm{E}} + \varepsilon_{\rm{k} + 3/4} \, \Delta p_{\rm{k} + 3/4} \, - \left[{1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4}} \right]\, + \left[{1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4}} + \right]\, l_{\rm{f \, k + 1}}^{\rm{E}} } \right)}{EPSS_{\rm{k}}} @@ -3672,13 +3764,15 @@ produces zero fluxes at cloud base: .. math:: :label: eq:q4lcbi Q4_{\rm{l}}(cb) = 0 = M_{\rm{cb+1/2}}^{\rm{P}} \, - \frac{\partial \, l_{\rm{l}}^{\rm{E}}}{\partial \, p} - M_{\rm{cb}}^{\rm{P}}\, + \frac{\partial \, l_{\rm{l}}^{\rm{E}}}{\partial \, p} - + M_{\rm{cb}}^{\rm{P}}\, \left({ l_{\rm{l}}^{\rm{P \, i}} - l_{\rm{l}}^{\rm{E}}(\rm{cb}) } \right) .. math:: :label: eq:q4fcbi Q4_{\rm{f}}(cb) = 0 = M_{\rm{cb+1/2}}^{\rm{P}} \, - \frac{\partial \, l_{\rm{f}}^{\rm{E}}}{\partial \, p} - M_{\rm{cb}}^{\rm{P}}\, + \frac{\partial \, l_{\rm{f}}^{\rm{E}}}{\partial \, p} - + M_{\rm{cb}}^{\rm{P}}\, \left({ l_{\rm{f}}^{\rm{P \, i}} - l_{\rm{f}}^{\rm{E}}(\rm{cb}) } \right) @@ -3689,13 +3783,15 @@ at this point and adjust the temperature accordingly. .. math:: :label: eqn:meltlf \theta_{\rm{k + 1}}^{\rm{P}} = \theta_{\rm{k + 1}}^{\rm{P}} - - \left(\frac{L_{\rm{F}}}{C_{p} \, \Pi_{\rm{k + 1}}} \right)\, l_{\rm{f \, k + 1}}^{\rm{P}} + \left(\frac{L_{\rm{F}}}{C_{p} \, \Pi_{\rm{k + 1}}} \right)\, l_{\rm{f \, k + + 1}}^{\rm{P}} \; \ldots \; \mbox{ if l_{\rm{f \, k + 1}}^{\rm{P}} is melted } .. math:: :label: eqn:freezell \theta_{\rm{k + 1}}^{\rm{P}} = \theta_{\rm{k + 1}}^{\rm{P}} + - \left(\frac{L_{\rm{F}}}{C_{p} \, \Pi_{\rm{k + 1}}} \right)\, l_{\rm{l \, k + 1}}^{\rm{P}} + \left(\frac{L_{\rm{F}}}{C_{p} \, \Pi_{\rm{k + 1}}} \right)\, l_{\rm{l \, k + + 1}}^{\rm{P}} \; \ldots \; \mbox{ if l_{\rm{l \, k + 1}}^{\rm{P}} is frozen } @@ -3714,7 +3810,8 @@ The precipitation calculation is unaltered. .. math:: :label: eq:precip - P_{\rm{k} + 1} = \left({ l_{\rm{k + 1}}^{\rm{P}} - l_{\rm{MIN}}^{\rm{P}} } \right)\, + P_{\rm{k} + 1} = \left({ l_{\rm{k + 1}}^{\rm{P}} - l_{\rm{MIN}}^{\rm{P}} } + \right)\, M_{\rm{k} + 1} \, / \, g where :math:`l_{\rm{k + 1}}^{\rm{P}}` = @@ -3746,7 +3843,8 @@ based upon eqn :eq:`eq:basiclold`: .. math:: :label: eq:q4lmassf - Q4_{\rm{l}}(k) = M_{\rm{k+1/2}}^{\rm{P}} \, \frac{\partial \, l_{\rm{l}}^{\rm{E}}}{\partial \, p} + + Q4_{\rm{l}}(k) = M_{\rm{k+1/2}}^{\rm{P}} \, \frac{\partial \, + l_{\rm{l}}^{\rm{E}}}{\partial \, p} + \left({ {\mu}_{\rm{k}} \, M_{\rm{k}}^{\rm{P}} + {\delta}_{\rm{k}} \, M_{\rm{k}}^{\rm{P}} } \right)\, \left({ l_{\rm{l}}^{\rm{P}}(\rm{k}) - l_{\rm{l}}^{\rm{E}}(\rm{k}) } \right)- @@ -3754,7 +3852,8 @@ based upon eqn :eq:`eq:basiclold`: .. math:: :label: eq:q4fmassf - Q4_{\rm{f}}(k) = M_{\rm{k+1/2}}^{\rm{P}} \, \frac{\partial \, l_{\rm{f}}^{\rm{E}}}{\partial \, p} + + Q4_{\rm{f}}(k) = M_{\rm{k+1/2}}^{\rm{P}} \, \frac{\partial \, + l_{\rm{f}}^{\rm{E}}}{\partial \, p} + \left({ {\mu}_{\rm{k}} \, M_{\rm{k}}^{\rm{P}} + {\delta}_{\rm{k}} \, M_{\rm{k}}^{\rm{P}} } \right)\, \left({ l_{\rm{f}}^{\rm{P}}(\rm{k}) - l_{\rm{f}}^{\rm{E}}(\rm{k}) } \right)- @@ -3772,7 +3871,8 @@ condensate is no longer re-evaporated at the end \frac{\Delta \, \theta_{\rm{k}}^{\rm{E}}}{\Delta \, t} = \left(\frac{ M_{\rm{k}} }{ \Delta \, p_{\rm{k}} } \right) \left[{ - \left({ 1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4} } \right) + \left({ 1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4} } + \right) \left({ 1 - \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) \left({ 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) \left({ \theta_{\rm{k + 1}}^{\rm{E}} - \theta_{\rm{k}}^{\rm{E}} } \right) @@ -3800,7 +3900,8 @@ and \frac{\Delta \, q_{\rm{k}}^{\rm{E}}}{\Delta \, t} = \left(\frac{ M_{\rm{k}} }{ \Delta \, p_{\rm{k}} } \right) \left[{ - \left({ 1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4} } \right) + \left({ 1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4} } + \right) \left({ 1 - \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) \left({ 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) \left({ q_{\rm{k + 1}}^{\rm{E}} - q_{\rm{k}}^{\rm{E}} } \right) @@ -3829,7 +3930,8 @@ Similarly, eqns :eq:`eq:q4lmassf` and \frac{\Delta \, l_{\rm{l \, k}}^{\rm{E}}}{\Delta \, t} = \left(\frac{ M_{\rm{k}} }{ \Delta \, p_{\rm{k}} } \right) \left[{ - \left({ 1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4} } \right) + \left({ 1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4} } + \right) \left({ 1 - \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) \left({ 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) \left({ l_{\rm{l \, k + 1}}^{\rm{E}} - l_{\rm{l \, k}}^{\rm{E}} } \right) @@ -3857,7 +3959,8 @@ and \frac{\Delta \, l_{\rm{f \, k}}^{\rm{E}}}{\Delta \, t} = \left(\frac{ M_{\rm{k}} }{ \Delta \, p_{\rm{k}} } \right) \left[{ - \left({ 1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4} } \right) + \left({ 1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4} } + \right) \left({ 1 - \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) \left({ 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) \left({ l_{\rm{f \, k + 1}}^{\rm{E}} - l_{\rm{f \, k}}^{\rm{E}} } \right) @@ -3907,7 +4010,8 @@ Hence we can write .. math:: - \Delta \overline{q} = \Delta C_S ( q_{sat liq}(\overline{T_{s}}) - \overline{q} ) + \Delta \overline{q} = \Delta C_S ( q_{sat liq}(\overline{T_{s}}) - + \overline{q} ) + (1 - \Delta C_S) \Delta \overline{q_{background}} where :math:`\Delta C_S` is the volume of plume air that is detrained @@ -3981,7 +4085,8 @@ the forcing terms in :eq:`eqn:qclconv` gives We now note that -.. math:: q_{sat} (T_s) - q_{sat liq} (\overline{T}) = \alpha (T_s - \overline{T} ) +.. math:: q_{sat} (T_s) - q_{sat liq} (\overline{T}) = \alpha (T_s - + \overline{T} ) and hence the final result @@ -4122,7 +4227,8 @@ by: .. math:: :label: eq:delta_t_conv \Delta T^E = \theta^E \left( \left(\frac{p}{p_{ref}}\right)^\kappa - - \left(\frac{p - \Delta p^E}{p_{ref}}\right)^\kappa + - \left(\frac{p - \Delta + p^E}{p_{ref}}\right)^\kappa \right) where :math:`\theta^E` is the environment potential temperature, @@ -4204,15 +4310,19 @@ is -10 :math:`^{\circ}` C. .. math:: \delta_{xl} = \left\{ \begin{array}{ll} - 1, & T_{plume} \ge -10 ^{\circ} C \\ - 0, & T_{plume} < -10 ^{\circ} C + 1, & T_{plume} \ge -10 + ^{\circ} C \\ + 0, & T_{plume} < -10 + ^{\circ} C \end{array} \right. .. math:: \delta_{xi} = \left\{ \begin{array}{ll} - 0, & T_{plume} \ge -10 ^{\circ} C \\ - 1, & T_{plume} < -10 ^{\circ} C + 0, & T_{plume} \ge -10 + ^{\circ} C \\ + 1, & T_{plume} < -10 + ^{\circ} C \end{array} \right. .. _Tidier way of coupling convection and PC2: @@ -4412,7 +4522,8 @@ As discussed in section :ref:`Initiation of cloud`, there are occasions when 1. The application of the initiation is given in section :ref:`Initiation of cloud`. The initiation forms a new, separate block of PC2 code to perform this calculation, and is located immediately following -the pressure change response (section :ref:`Response to pressure changes`). Also, if the +the pressure change response (section :ref:`Response to pressure changes`). +Also, if the UM namelist switch **l_cloud_call_b4_conv** is set to true, an additional call to PC2 initiation is performed before the convection scheme, to ensure that the condensation response to advection and other @@ -4626,7 +4737,8 @@ between :math:`\overline{T}` and :math:`\overline{T_L}`, so that the values of :math:`\alpha` and :math:`a_L` are the same in both of these equations, and: -.. math:: q_{sat}(\overline{T_L}) = q_{sat}(\overline{T}) - \alpha \frac{L}{c_p} q_{cl} +.. math:: q_{sat}(\overline{T_L}) = q_{sat}(\overline{T}) - \alpha + \frac{L}{c_p} q_{cl} we obtain: @@ -5019,7 +5131,8 @@ Discritising :eq:`dqcldt` we have, using .. math:: :label: eq:da1 \Delta \overline{q_{cl}} = C_l ( a_L ( \Delta \overline{q} - - \alpha \Delta \overline{T} - \beta \Delta \overline{p}) + \Delta \overline{q_{cl}} ). + \alpha \Delta \overline{T} - \beta \Delta \overline{p}) + \Delta + \overline{q_{cl}} ). Remember that :math:`Q_c` (and hence :math:`\Delta Q_c`) is independent of condensation. Rearranging, we obtain @@ -5027,7 +5140,8 @@ of condensation. Rearranging, we obtain .. math:: :label: eq:da2 \Delta \overline{q_{cl}} = \frac{1}{1 - C_l} C_l - a_L ( \Delta \overline{q} - \alpha \Delta \overline{T} - \beta \Delta \overline{p}) + a_L ( \Delta \overline{q} - \alpha \Delta \overline{T} - \beta \Delta + \overline{p}) and hence an expression for the condensate increment, :math:`\Delta \overline{q_{cl}}`, that accompanies the known increments @@ -5048,7 +5162,8 @@ ill-conditioning of this solution near :math:`C_l = 1`. In practice, the ill-conditioning of :eq:`eq:da2` and :eq:`eq:da3` becomes too numerically awkward for us to apply the full solution based on homogeneous forcing, although, for -completeness, we outline it in Appendix :ref:`Appendix; Alternative PC2 - Data Assimilation formulations`. Hence +completeness, we outline it in Appendix :ref:`Appendix; Alternative PC2 - Data +Assimilation formulations`. Hence we have chosen to apply a much simpler model. Here we use simply the data assimilation increments :math:`\Delta \overline{q}` and :math:`\Delta \overline{T}` within the standard homogeneous forcing @@ -5058,13 +5173,15 @@ is inconsistent (because :math:`\Delta \overline{q}` and :math:`\Delta This allows us an *estimate* of :math:`\Delta \overline{q_{cl}}` and :math:`\Delta{C_l}`, via the homogeneous forcing routine (and :math:`\Delta C_t` via the standard updating described in section -:ref:`Ice cloud and mixed phase regions`). These are the quantities applied as the equivalent +:ref:`Ice cloud and mixed phase regions`). These are the quantities applied as +the equivalent data assimilation increments for :math:`\Delta \overline{q_{cl}}`, :math:`\Delta{C_l}` and :math:`\Delta C_t`. The increments :math:`\Delta \overline{q}` and :math:`\Delta \overline{T}` remain those that the data assimilation scheme itself calculated. -Appendix :ref:`Appendix; Alternative PC2 - Data Assimilation formulations` gives, for completeness, the +Appendix :ref:`Appendix; Alternative PC2 - Data Assimilation formulations` +gives, for completeness, the alternative numerical technique for the solution of :eq:`eq:da2` and :eq:`eq:da3`. However, we stress that this technique is not used within the current PC2 @@ -5441,7 +5558,8 @@ Main Tree from atm_step_4a - ls_arcld (call diagnostic Smith scheme with area cloud fraction again to account for the analysis increments; - see :ref:`Smith scheme with area cloud fraction` for a drill-down + see :ref:`Smith scheme with area cloud fraction` for a + drill-down inside this routine) .. container:: tcolorbox @@ -5738,7 +5856,8 @@ rest of the SCM uses the same PC2 code as the full model. Note that the change to PC2 homogeneous forcing from advection under the UM namelist switch **l_pc2_sl_advection** (see section -:ref:`Response to pressure changes`) is also mirrored in the Single-Column Model. If +:ref:`Response to pressure changes`) is also mirrored in the Single-Column +Model. If this switch is turned on, the PC2 homogeneous forcing call using the SCM forcing increments is moved straight after the call to the forcing routine, so that the condensation adjustment is performed before the @@ -5754,7 +5873,8 @@ The SCM forcings may comprise one or both of the following: For the latter, we can calculate the pressure change experienced by vertically-advected parcels, and so calculate the PC2 homogeneous forcing response in the same way as we do for Semi-Lagrangian advection -in the full model (see section :ref:`Response to pressure changes`). For the former, we +in the full model (see section :ref:`Response to pressure changes`). For the +former, we don’t know if the prescribed T,q tendencies are due to advection, radiation, or some other process, so we calculate the PC2 homogeneous forcing response as if the tendencies are applied "in-situ". @@ -6102,7 +6222,8 @@ and hence, using our value of :math:`\Delta Q_c` from .. math:: - \Delta C_l = G(-Q_c) (a_L ( \Delta \overline{q} - \alpha \Delta \overline{T} ) + \Delta C_l = G(-Q_c) (a_L ( \Delta \overline{q} - \alpha \Delta \overline{T} + ) + \frac{1}{1-C_l} C_l a_L ( \Delta \overline{q} - \alpha \Delta \overline{T} ) ) which rearranges to @@ -6157,7 +6278,8 @@ and this value is limited to 0 or 1. The liquid water term simply uses the final version of :math:`C_l` in its calculation. -.. math:: \Delta \overline{q_{cl}} = \frac{1}{1-C_l^{[n+1]}} C_l^{[n+1]} \Delta Q_c +.. math:: \Delta \overline{q_{cl}} = \frac{1}{1-C_l^{[n+1]}} C_l^{[n+1]} \Delta + Q_c and will be set to 0 if :math:`C^{[n+1]}` is 0. There is an additional limit, see below, applied to the liquid water term, which will prevent @@ -6242,7 +6364,8 @@ for :math:`(1-C_l)` as: To derive this from (B.3) note that :math:`C_l` is swapped for :math:`1-C_l` and :math:`(b_s - (-Q_c))` is swapped for -:math:`(-Qc - (-b_s))`, as in section :ref:`Numerical Application of the Smith method`. +:math:`(-Qc - (-b_s))`, as in section :ref:`Numerical Application of the Smith +method`. Similarly, noting that :math:`\overline{q_{cl}}` can be swapped with :math:`SD`, gives the equivalent to (B.4) in `Wilson and Gregory (2003)`_ as @@ -6317,7 +6440,8 @@ our previous expression :eq:`eqn:sdr1mc` for :math:`\Delta \overline{q_{cl max}}` gives (remembering that we are considering the reverse process, so the sign is opposite), -.. math:: \Delta \overline{q_{cl}} = - SD^{[n+1]} ( \frac{2}{1-C_l^{[n+1]}} - 1 ) +.. math:: \Delta \overline{q_{cl}} = - SD^{[n+1]} ( \frac{2}{1-C_l^{[n+1]}} - + 1 ) (remembering that :math:`n=0` is assumed). Hence, replacing :math:`C_l^{[n+1]}` by :eq:`eqn:1msqrt` we have @@ -6359,7 +6483,8 @@ Appendix; Essentials of PC2 for code developers This section provides some guidance to code developers on the treatment of PC2. Code developers are advised to read the relevant part of section -:ref:`Application to the Unified Model` to understand the way in which the current PC2 +:ref:`Application to the Unified Model` to understand the way in which the +current PC2 scheme interacts with their section of code. The essence of a prognostic cloud scheme is that each physical part of @@ -6556,7 +6681,8 @@ of moisture. Convective cloud increments in the mass-flux framework ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -As discussed in section :ref:`A note on the implementation of the cloud fraction change`, it would be +As discussed in section :ref:`A note on the implementation of the cloud +fraction change`, it would be useful to code up the convective cloud fraction changes to link directly to the mass-flux convection scheme, and not to estimate them from the values of :math:`Q4`, which can introduce errors. @@ -6726,7 +6852,8 @@ the clouds is of order the timestep - ideally we wouldn’t want to try to model anything prognostically when the cycling time is less than the timestep. -As discussed in section :ref:`Numerical application of the hybrid erosion method`, the timestep +As discussed in section :ref:`Numerical application of the hybrid erosion +method`, the timestep sensitivity of cloud amounts in shallow cumulus regimes can be addressed by using a more accurate numerical method to solve the erosion term. Several options are available under the UM namelist switch @@ -7046,4 +7173,4 @@ References *Modification of the thermodynamic variability closure in the Met Office Unified Model prognostic cloud scheme*. Atmospheric Science Letters. - https://doi.org/10.1002/asl.1021 + https://doi.org/10.1002/asl.1021 \ No newline at end of file From b34e69ac7e56cbe09899ec6a2fa759c289120576 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Thu, 23 Apr 2026 10:40:22 +0100 Subject: [PATCH 038/116] Changed subsubsection header underline format as-per the style guide. --- .../cloud_schemes/UMDP30_PC2CloudScheme.rst | 146 +++++++++--------- 1 file changed, 73 insertions(+), 73 deletions(-) diff --git a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst index aed2a30ddf..5d630f081e 100644 --- a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst +++ b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst @@ -423,7 +423,7 @@ included. All other physics schemes use one of the generic approaches below. A note on convective cloud fraction -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ It was the original intention that PC2 be able to replace the two separate diagnostic cloud fractions (large-scale and convective) with a @@ -597,7 +597,7 @@ parameter :math:`n` has been chosen to be 0.0, corresponding to a top-hat distribution shape. Weight as a function of cloud-fraction -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ This option is selected by setting **i_pc2_homog_g_method=1** in the UM large-scale cloud namelist. @@ -653,7 +653,7 @@ This in itself can be problematic, since leaving :math:`C_l` unmodified under a homogeneous forcing can allow unrealistic states to develop. Weight in proportion to PDF width -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ This option is selected by setting **i_pc2_homog_g_method=2** in the UM large-scale cloud namelist. @@ -692,7 +692,7 @@ impact the performance of the model forecast. .. _Numerical application: Numerical application -~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^ The timestepping methods that are used in the homogeneous forcing were developed off-line using a single gridbox model, to ensure smooth, @@ -969,7 +969,7 @@ this; either a version of the Smith scheme (see UMDP 029), or the bimodal scheme (see UMDP 039). These two options are described below... Initiation using a “Smith-like” method -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ This option is selected by setting the UM namelist switch **i_pc2_init_method = 1** (Smith). @@ -1044,7 +1044,7 @@ diagnostic `Smith (1990)`_ scheme. .. _Numerical Application of the Smith method: Numerical Application of the Smith method -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ In order to calculate and compare the state of the model to :math:`b_s`, we first calculate :math:`T_L`, :math:`q_{sat}(\overline{T_L})` and @@ -1170,7 +1170,7 @@ given by :math:`a_L^{[i]}`. .. _Initiation using the bimodal scheme: Initiation using the bimodal scheme -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ This option is selected by setting the UM namelist switch **i_pc2_init_method = 2** (Bimodal). @@ -1293,7 +1293,7 @@ the change in a tracer and we discuss this later. .. _Multiple phases in the injection source: Multiple phases in the injection source -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ The injection source formulation can be extended to multiple phases of condensate. In practice, this will simply be the two phases ice and @@ -1434,7 +1434,7 @@ This is discussed in section :ref:`Phase of condensate`. .. _Numerical application of injection forcing: Numerical application of injection forcing -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ The numerical application using :eq:`eq:dcltdt_almost_final` may be @@ -1505,7 +1505,7 @@ and similar equations are used for :math:`C_i^{[n+1]}` and .. _A note on the implementation of the cloud fraction change: A note on the implementation of the cloud fraction change -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Equation :eq:`eq:dcdt_inhom2` has been derived assuming that the only change in the cloud properties within the gridbox @@ -1660,7 +1660,7 @@ Equivalent equations to :eq:`eq:deltact_ran1` and overlap. Numerical Implementation -~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^ In general, although the situation does not occur within the current implementation of PC2 , we might have increments to both :math:`C_l` and @@ -1764,7 +1764,7 @@ Turbulence-driven production of subgrid scale liquid cloud .. _Introduction: Introduction -~~~~~~~~~~~~ +^^^^^^^^^^^^ `Field et al. (2014)`_ developed a model for subgrid liquid water production by turbulent motions. Their method uses an exactly soluble @@ -1791,7 +1791,7 @@ the GCM. .. _Model description: Model description -~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^ `Field et al. (2014)`_ started from the equation for the dynamics of ice supersaturation :math:`S_i=e_v/e_{sat\;ice}-1`: @@ -1892,7 +1892,7 @@ turbulent processes. .. _Model implementation and closure relations: Model implementation and closure relations -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ To implement the model of Section :ref:`Model description` in the Unified Model, closure relations are needed for the quantities @@ -1918,7 +1918,7 @@ functionality. .. _Closure relations: Closure relations -~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^ The vertical velocity variance, :math:`\sigma_w^2`, is available as a diagnostic from the Boundary Layer scheme. Because the Boundary Layer @@ -1963,7 +1963,7 @@ taken to be the grid box mean values. The first moment of the ice PSD, .. _Options for incrementing model prognostics: Options for incrementing model prognostics -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Using the information in Section :ref:`Closure relations` to obtain closed expressions for the subgrid PDF of :math:`S_i`-fluctuations @@ -2051,7 +2051,7 @@ cloud scheme. .. _Other user options: Other user options -~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^ The following variables and logical switches are optional inputs: @@ -2155,7 +2155,7 @@ considered to contribute to the ice cloud fraction. .. _Fall of ice: Fall of ice -~~~~~~~~~~~ +^^^^^^^^^^^ The fall of ice is the process that contributes most to the growth of ice cloud fraction in the model. The model results are therefore @@ -2223,7 +2223,7 @@ UMUI (from version 7.6 onwards).** .. _Homogeneous nucleation: Homogeneous nucleation -~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^ This will freeze all supercooled liquid water when a temperature threshold is exceeded. Hence we turn all existing liquid and mixed phase @@ -2243,7 +2243,7 @@ cloud to ice cloud. The cloud fraction changes are: Heterogeneous nucleation -~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^ This process will freeze a small amount of supercooled liquid water, regardless of the previous presence of ice cloud. This will mean that @@ -2266,7 +2266,7 @@ cloud. These give the following changes: .. _Deposition and sublimation: Deposition and sublimation -~~~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^^^ This term exerts one of the most important influences on the ice cloud in the whole model (this applies to the control as well as for PC2). @@ -2431,7 +2431,7 @@ in the presence of liquid cloud: \Delta C_t = \Delta C_i . Riming -~~~~~~ +^^^^^^ This process acts only where mixed phase cloud occurs - although, in theory, it could remove any supercooled liquid totally, the air would @@ -2441,7 +2441,7 @@ choose to model this process as having *no effect* on the cloud fractions. Capture -~~~~~~~ +^^^^^^^ This is the freezing of raindrops onto ice crystals by collision. This does not alter the ice cloud *fraction* in the gridbox (although it does @@ -2450,7 +2450,7 @@ liquid cloud. Again, we therefore choose to model this process as having *no effect* on the cloud fractions. Evaporation of melting ice -~~~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^^^ Here we simply assume that ice cloud fraction is removed in proportion to the ice content that is removed. @@ -2463,7 +2463,7 @@ Because the evaporation cannot occur in the liquid part of the gridbox, there is no change to :math:`C_t` (or to :math:`C_l`). Melting -~~~~~~~ +^^^^^^^ Again, the change in :math:`C_i` is calculated using the method in :eq:`eq:lsp_evapmeltsnow`. @@ -2484,7 +2484,7 @@ proportion of ice cloud fraction that exists without liquid cloud (i.e. \frac{A_{ice}}{C_i} . Evaporation of rain -~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^ Evaporation of rain will not, *on its own*, generate liquid cloud, since a large-scale lifting process will be required in order to condense @@ -2494,7 +2494,7 @@ elsewhere in the model (e.g. by the lifting process, section :ref:`Response to pressure changes`). Accretion -~~~~~~~~~ +^^^^^^^^^ Accretion is the sweep-out of liquid water droplets by rain. We argue in a similar way to the riming term, that this will not remove any liquid @@ -2509,13 +2509,13 @@ considered necessary to increase the complexity of the current, simple representations. Autoconversion -~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^ As for accretion, the generation of rain directly from collision and coalescence of liquid water droplets will not alter the cloud fractions. Other microphysics terms -~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^ There are already (i.e. also in the control) two numerical tidy-up terms at the end of the microphysics section that remove small rain amounts @@ -2530,7 +2530,7 @@ appropriately, so :math:`C_t` is reset to :math:`C_l` etc. .. _numerical-implementation-1: Numerical implementation -~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^ Note that after each process has been applied, we do *not* recalculate the sizes of the ice-only, liquid-only and mixed phase partitions, but @@ -2562,7 +2562,7 @@ PC2 erosion ----------- Original width-narrowing method -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ (selected by setting **i_pc2_erosion_method = 1** in the UM namelist). @@ -2618,7 +2618,7 @@ extensively tuned during PC2 development, a typical value would be :math:`\Upsilon=-2.25 \times 10^{-5} s^{-1}`. Numerical application of the original width-narrowing method -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Because of the strong link the mathematical expressions for width narrowing (section :ref:`Changing the width of the PDF - PC2 erosion`) have @@ -2657,7 +2657,7 @@ The option “l_fixbug_pc2_qcl_incr” ensures that qcl is set to zero if the CFL has reached zero. Cloud-surface-area hybrid erosion method -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ (selected by setting **i_pc2_erosion_method = 3** in the UM namelist). @@ -2778,7 +2778,7 @@ cloud-surface-area erosion method. .. _Numerical application of the hybrid erosion method: Numerical application of the hybrid erosion method -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Next, we consider how to numerically discretise equations :eq:`eq:dqcldt_hybrid` and @@ -3356,7 +3356,7 @@ its effect on the rest of the convection. Within PC2 we have had to work to more fully incorporate the condensate into the convection scheme. Introduction to the convective mass flux scheme -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Within the mass flux scheme the net change in :math:`\overline{q_{cl}}` and :math:`C_l` etc. comes from two distinct sources. Firstly, the @@ -3394,7 +3394,7 @@ mass flux convection scheme. .. _Basic Equations for a Convective Mass Flux Scheme: Basic Equations for a Convective Mass Flux Scheme -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ We first consider a generic mass-flux scheme before its application to PC2. As discussed by Grant and Stirling (personal communication), the @@ -3610,7 +3610,7 @@ condensate values and to allow them to change. .. _Calculation of Grid-Box Averaged Condensate Rate (Q4): Calculation of Grid-Box Averaged Condensate Rate (Q4) -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ The PC2 condensation scheme allows convection to feed cloud condensate (ice or liquid) directly into the large scale and to update the cloud @@ -3984,7 +3984,7 @@ and .. _Background condensation: Background condensation -~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^ The modification to the convective plume will result in the transport, detrainment and entrainment of condensate, in addition to the transport @@ -4168,7 +4168,7 @@ analysed to show) is removed before the net effect is calculated, leading to more accurate numerical behaviour. Homogeneous forcing of the environment by convective-subsidence pressure change -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ To this end, the code includes an option to perform the homogeneous forcing of liquid cloud by convection using the “pressure forcing” from @@ -4241,7 +4241,7 @@ homogeneous forcing routine after convection as the forcings to be applied (with the forcings to all other variables set to zero). Convective cloud amount -~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^ It is a debatable point whether the convective cloud fraction should be set to zero. Although this was one of the original key concepts of PC2, @@ -4266,7 +4266,7 @@ scheme, but they are clearly directly related to the rest of the cloud scheme formulation. CAPE scaling -~~~~~~~~~~~~ +^^^^^^^^^^^^ The CAPE scaling option in the mass-flux convection scheme scales its increments by the calculated values of @@ -4280,7 +4280,7 @@ Hence we consider any detrained condensate to have been evaporated when we calculate :math:`\frac{dCAPE}{dt}`. Convective precipitation -~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^ The amount of condensate detrained from convective plumes, and hence the amount of moisture in the upper levels of the atmosphere, is very @@ -4299,7 +4299,7 @@ necessary in order to produce thick enough anvil clouds. .. _Phase of condensate: Phase of condensate -~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^ The phase of the convective condensate *carried in the plume* is determined by a single phase change temperature TICE, with condensate @@ -4328,7 +4328,7 @@ is -10 :math:`^{\circ}` C. .. _Tidier way of coupling convection and PC2: Tidier way of coupling convection and PC2 -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ This area is still under development. But in brief, work is udner way to ensure that the convective plume smoothly transitions from detraining @@ -4343,7 +4343,7 @@ increments to cloud fraction from detrainment and subsidence are then combined. Prognostic dust approach -~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^ A prognostic dust approach is implemented in the micro-physics scheme under large-sale-precipitation where by the heterogeneous nucleation @@ -4365,7 +4365,7 @@ and all-ice for T :math:`\leq` :math:`tnuc_n` - 10.0 .. _Condensation adjustment in the profiles input to the convection scheme: Condensation adjustment in the profiles input to the convection scheme -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ The convection scheme itself is highly sensitive to the input environment temperature and moisture profiles *before* the convection @@ -4540,7 +4540,7 @@ saturation boundary lies within the bounds of the bimodal scheme’s assumed PDF, as described in section :ref:`Initiation using the bimodal scheme`. “Original” initiation logic -~~~~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^^^^ This option is selected by setting the UM namelist switch **i_pc2_init_logic = 1** (Original) @@ -4579,7 +4579,7 @@ Equivalently, :math:`C_l` is initiated away from 1 if - :math:`RH_T^{[n+1]} < RH_T^{[n]}` . “Simplified” initiation logic -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ This option is selected by setting the UM namelist switch **i_pc2_init_logic = 2** (Simplified) @@ -4626,7 +4626,7 @@ but with the following differences: .. _“Smooth” initiation logic: “Smooth” initiation logic -~~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^^ This option is selected by setting the UM namelist switch **i_pc2_init_logic = 3** (Smooth) @@ -4759,7 +4759,7 @@ uses :math:`q_{cl} - Q_c` in place of :math:`SD`, and .. _Additional checks after PC2 initiation: Additional checks after PC2 initiation -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ The initiation is followed immediately by a section of resetting code. For numerical reasons, it is possible to obtain very low, but non zero, @@ -5091,7 +5091,7 @@ phase if the temperature is cold enough. Hence, if .. _Qpos checks: Qpos checks -~~~~~~~~~~~ +^^^^^^^^^^^ The implementation of the PC2 code includes an additional bounds check after the *atmos-physics-2* part of the model timestep has been @@ -5288,7 +5288,7 @@ those only called for PC2 are in green, and those only called for the bimodal scheme are in purple. Main Tree from atm_step_4a -~~~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^^^ .. container:: itemize @@ -5631,7 +5631,7 @@ different places in the tree)... .. _Smith scheme with area cloud fraction: Smith scheme with area cloud fraction -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ .. container:: itemize @@ -5670,7 +5670,7 @@ Smith scheme with area cloud fraction .. _PC2 initiation: PC2 initiation -~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^ .. container:: itemize @@ -5718,7 +5718,7 @@ PC2 initiation .. _PC2 Data Assimilation: PC2 Data Assimilation -~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^ .. container:: itemize @@ -6402,7 +6402,7 @@ and take the smaller value for of :eq:`eqn:sdr1mc` and :eq:`eqn:bs` for :math:`\Delta \overline{q_{cl \, max}}`. Initiation from :math:`C_l=1` -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ The equations are not defined when :math:`C_l=1`. (Note that when :math:`C_l=0` we will calculate :math:`G(-Q_c)=0` so there is no change @@ -6619,7 +6619,7 @@ alterations. It is fair to say that the link to the convection has proved the most problematic issue so far with PC2 development. PC2 cloud erosion -~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^ The cloud erosion is a critical term for the simulation of shallow convective cloud. A large amount of erosion is required to keep the @@ -6668,7 +6668,7 @@ many reasons for this). A different mixing method may give significantly different results for the areas around convective plumes. Narrowing of the moisture PDF -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Most of the parametrized terms in PC2 act to reduce the width of the moisture PDF. The only terms that can increase the width are the @@ -6679,7 +6679,7 @@ actually increase the width in the presence of large vertical gradients of moisture. Convective cloud increments in the mass-flux framework -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ As discussed in section :ref:`A note on the implementation of the cloud fraction change`, it would be @@ -6690,7 +6690,7 @@ values of :math:`Q4`, which can introduce errors. .. _Turbulence based convection scheme: Turbulence based convection scheme -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ We will need to properly consider the links between PC2 and the turbulence based convection scheme. In essence, we can use the diagnosed @@ -6707,14 +6707,14 @@ despite the phase changes forming an integral part of the convection scheme. Detailed convective comparisons with CRM/LEM data -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ This work is already underway at the Met Office, in order to properly evaluate the performance of the convective cloud parametrization in PC2 against high resolution research models. Choice of PDF parameters -~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^ Work by Dan Tang at Leeds University has highlighted an interesting and undesirable property of the choice of :math:`m` and :math:`n` parameters @@ -6734,7 +6734,7 @@ investigated. .. _Homogeneous forcing section improvements: Homogeneous forcing section improvements -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Although the homogeneous forcing provides a convenient method to calculate increments to :math:`C_l` and :math:`\overline{q_{cl}}`, it is @@ -6750,7 +6750,7 @@ at with a view to using something better. This is one of the strengths of the PC2 framework and is an intention of the project. Overlap of ice and liquid cloud changes -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ We have assumed within PC2 that ice and liquid cloud changes are minimally overlapped with each other (within the same gridbox) in order @@ -6760,7 +6760,7 @@ tend not to coexist together in a cloud, it may be possible to characterise and apply this overlap in a more quantiative way. Parameter tuning -~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^ The sensitivity of some of the parameters in PC2 have not been properly tested, mainly due to a lack of resources rather than a physical reason. @@ -6801,7 +6801,7 @@ given the length of the timestep. sensitivities. Cloud inhomogeneities -~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^ A cloud generator approach to cloud inhomogeneities is currently being developed. However we note two particular issues that relate to PC2. @@ -6826,7 +6826,7 @@ developed. However we note two particular issues that relate to PC2. .. _Time-stepping: Time-stepping -~~~~~~~~~~~~~ +^^^^^^^^^^^^^ A proper analysis of timestep sensitivities of PC2 (as opposed to microphysics, convection etc) in the full UM or SCM has not been done @@ -6900,7 +6900,7 @@ We have placed the initiation at the end of the timestep, but it is sensible to ask whether this could ideally be located elsewhere. Initiation formulation -~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^ Ideally this should be a relatively infrequent part of the model but remains an essential part of the code. It is reasonable to ask whether @@ -6913,7 +6913,7 @@ symmetrical, with initiation from :math:`C_l=1` occuring with the same :math:`C_l=1`. 70-levels performance -~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^ The performance of PC2:66 in the 70-levels model is not good as far as shallow convective cloud is concerned (there is far too much of it in @@ -6925,7 +6925,7 @@ much progress in identifying the reasons for the differences, or producing effective tunings to counter the problem. High horizontal resolution performace -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ PC2 has only been tested once at 4 km horizontal resolution. This produced excessive shallow convective cloud (this may or may not be @@ -6939,7 +6939,7 @@ acceptable at 12 km resolution, but we have not quantitatively explored this limit. Diagnostic evaluation -~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^ One of the principal areas for future cloud scheme development work planned in the future is in the area of detailed evaluation against a @@ -6950,7 +6950,7 @@ been on tackling qualitatively poor results. Hence new sources of evaluation work on PC2 would be very welcome. Moisture distribution within the deposition/sublimation term -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ The liquid cloud changes in PC2 (or in a non-PC2 run) are based upon a moisture PDF, as are the deposition/sublimation changes. However, it is @@ -6968,7 +6968,7 @@ riming (or accretion) term. Again, it might be possible to bring together these formulations into a single consistent framework. Area cloud fraction representation -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ The current area cloud fraction representation is not used when convection is taking place (signified by the *cumulus* logical). This From c6a46656b056e02b870a458646e75a9964520a9f Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Thu, 23 Apr 2026 12:48:15 +0100 Subject: [PATCH 039/116] Preparatory changes for adding boundary-layer / turbulence documentation. --- .github/CODEOWNERS | 3 +++ documentation/source/conf.py | 6 +++++- .../turbulence_schemes/index.rst | 21 +++++++++++++++++++ 3 files changed, 29 insertions(+), 1 deletion(-) create mode 100644 documentation/source/science_guide/turbulence_schemes/index.rst diff --git a/.github/CODEOWNERS b/.github/CODEOWNERS index 979887d626..ca7a6db0ae 100644 --- a/.github/CODEOWNERS +++ b/.github/CODEOWNERS @@ -78,3 +78,6 @@ documentation # @MetOffice/ssdteam .github/ @MetOffice/ssdteam LICENSE @yaswant README.md @MetOffice/ssdteam + +# Owners of specific documentation sections +documentation/source/science_guide/turbulence_schemes @Adrian-Lock diff --git a/documentation/source/conf.py b/documentation/source/conf.py index 9d6ae59a1c..e201673aed 100644 --- a/documentation/source/conf.py +++ b/documentation/source/conf.py @@ -20,9 +20,13 @@ extensions = [ 'sphinx_sitemap', 'sphinx_design', - 'sphinx.ext.intersphinx' + 'sphinx.ext.intersphinx', + 'sphinx.ext.mathjax', ] +# Enable equation referencing and cross-referencing +mathjax3_config = { "tex": { "tags": "ams", "packages": {"[+]": ["ams"]}, } } + # Add any paths that contain templates here, relative to this directory. templates_path = ['_templates'] diff --git a/documentation/source/science_guide/turbulence_schemes/index.rst b/documentation/source/science_guide/turbulence_schemes/index.rst new file mode 100644 index 0000000000..625d6a9e87 --- /dev/null +++ b/documentation/source/science_guide/turbulence_schemes/index.rst @@ -0,0 +1,21 @@ +.. ----------------------------------------------------------------------------- + (c) Crown copyright Met Office. All rights reserved. + The file LICENCE, distributed with this code, contains details of the terms + under which the code may be used. + ----------------------------------------------------------------------------- +.. _turbulence_schemes_index: + +Turbulence Schemes available in LFRic +===================================== + +Contained here are descriptions of the boundary-layer / turbulence schemes +available in the LFRic atmosphere model. These schemes represent +sub-grid-scale mixing of momentum and scalars in the vertical, +and also in the horizontal for some schemes (usually only needed +at km-scale or higher model-resolution). + +.. toctree:: + :maxdepth: 1 + :glob: + + * From f03116d3c3d4b9f5c4d1fcc5aa88363e6cd92a09 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Thu, 23 Apr 2026 13:33:40 +0100 Subject: [PATCH 040/116] Copied in the UMDP 024 source, including figures and include files. --- .../turbulence_schemes/bldoc.tex | 5637 ++ .../turbulence_schemes/div_r071.eps | 410 + .../turbulence_schemes/div_r080.eps | 402 + .../turbulence_schemes/honnert_vs_tanh.eps | 258 + .../turbulence_schemes/ideal_invinteg.eps | 265 + .../turbulence_schemes/ideal_revflux.eps | 517 + .../turbulence_schemes/nbldoc_zidiag.eps | 183 + .../turbulence_schemes/new_ktop_shape.eps | 288 + .../turbulence_schemes/newcommand.tex | 111 + .../turbulence_schemes/stab_dep.eps | 1398 + .../turbulence_schemes/subsent_fig7.eps | 323 + .../turbulence_schemes/wcrp_bltypes1.eps | 51115 +++++++++++++ .../turbulence_schemes/wcrp_bltypes2.eps | 63242 ++++++++++++++++ .../turbulence_schemes/zturb_schem.eps | 127 + 14 files changed, 124276 insertions(+) create mode 100644 documentation/source/science_guide/turbulence_schemes/bldoc.tex create mode 100644 documentation/source/science_guide/turbulence_schemes/div_r071.eps create mode 100644 documentation/source/science_guide/turbulence_schemes/div_r080.eps create mode 100644 documentation/source/science_guide/turbulence_schemes/honnert_vs_tanh.eps create mode 100644 documentation/source/science_guide/turbulence_schemes/ideal_invinteg.eps create mode 100644 documentation/source/science_guide/turbulence_schemes/ideal_revflux.eps create mode 100644 documentation/source/science_guide/turbulence_schemes/nbldoc_zidiag.eps create mode 100644 documentation/source/science_guide/turbulence_schemes/new_ktop_shape.eps create mode 100644 documentation/source/science_guide/turbulence_schemes/newcommand.tex create mode 100644 documentation/source/science_guide/turbulence_schemes/stab_dep.eps create mode 100644 documentation/source/science_guide/turbulence_schemes/subsent_fig7.eps create mode 100644 documentation/source/science_guide/turbulence_schemes/wcrp_bltypes1.eps create mode 100644 documentation/source/science_guide/turbulence_schemes/wcrp_bltypes2.eps create mode 100644 documentation/source/science_guide/turbulence_schemes/zturb_schem.eps diff --git a/documentation/source/science_guide/turbulence_schemes/bldoc.tex b/documentation/source/science_guide/turbulence_schemes/bldoc.tex new file mode 100644 index 0000000000..482de14dbb --- /dev/null +++ b/documentation/source/science_guide/turbulence_schemes/bldoc.tex @@ -0,0 +1,5637 @@ +\documentclass{UMDP_article} + +\title{The Parametrization of Boundary Layer Processes} +\paperno{024} +\umversion{13.8} +\owner{Adrian Lock} +\author{A.~Lock, J.~Edwards and I.~Boutle} + +\usepackage{amsmath,amssymb} +\usepackage{algorithmic} +\usepackage{natbib} +\usepackage{indentfirst} + +\input{newcommand} +\def\gtapp{\raisebox{-.4ex}{$\ \stackrel{>}{{\scriptstyle \sim}} \ $}} + +%\def\captionarb#1{\caption{#1}} +\def\captionarb#1{ + \protect\footnotesize + \caption{\protect\footnotesize#1} + \protect\normalsize\protect\vspace{0.7cm}} +\def\comment#1{({\em #1})} +%\def\comment#1{} % suppress comments +\def\nocomment#1{} + +\begin{document} +\maketitle + +\tableofcontents + +\newpage + +\section{Introduction and code versions} + +This is the documentation for the ``boundary layer'' parametrization +of vertical turbulent transports of heat, moisture and horizontal +momentum. It includes surface exchange but \emph{not} the +parametrization of the surface itself. This is covered within the +surface (JULES) documentation. Although commonly referred to as the +``boundary layer'' parametrization, it includes a free-tropospheric +component. Turbulent fluxes are calculated up to ``BL\_LEVELS'' which +is currently set so that the entire troposphere is included. + +Generally speaking, only version 9C of the boundary layer +parametrization will be documented here as it is now the only +supported version. However, the documentation still makes occasional +references to previous versions of the scheme (8A, 9B) as it is useful +to retain the history of how we have got to where we are. Version 9 +interfaces to the JULES surface code, which is now the only supported +surface code within the UM. + +Several options for higher order closures are available in the 1A +version of the UM boundary layer scheme and these are documented +separately in \citeumdp{025}. + +\section{Model variables and turbulence closure} +\label{sec:closure} + +Given source terms, ${\cal S}$ say, from processes other than boundary +layer turbulence, Reynolds' averaging gives the following equation for +conserved scalar variables, $\chi$, and the two horizontal components +of momentum, ${\bf u}$ on a sphere gives: +\begin{eqnarray} + \frac{\partial \chi}{\partial t} + &=& - \frac{1}{r^2 \rho} \, \frac{\partial }{\partial z} \left( r^2 \rho \overline{w'\chi'} \right) + + {\cal S} + \label{cons_eqn_scal} \\ + \frac{\partial {\bf u}}{\partial t} + &=& \frac{1}{r^2 \rho} \, \frac{\partial }{\partial z} \left( r^2 {\bf \tau} \right) + + {\cal S} + \label{cons_eqn_uv} +\end{eqnarray} +where $\overline{w'\chi'}$ and ${\bf \tau}$ are the vertical turbulent +fluxes to be parametrized, $r$ is the height from the centre of the +planet and $\rho$ is density. The scalar variables treated by +(\ref{cons_eqn_scal}), which are approximately conserved under moist +adiabatic ascent, are: +\begin{eqnarray} + \thetal &=& T_L + \frac{g}{c_p} z = T - \frac{L}{c_p} \ql + - \frac{L_s}{c_p} q_f + \frac{g}{c_p} z \label{thetal} \\ + q_t &=& q_v + \ql + q_f \label{qt} +\end{eqnarray} +where $T$ is temperature, $q_v$ is specific humidity, $\ql$ and $q_f$ +the specific liquid and frozen water contents respectively, and +$L_s=L+L_f$ is the latent heat of sublimation. Note that $\thetal$ is +based on `liquid/frozen water static energy' ($= c_p T + g z - L \ql - +L_s q_f$) rather than potential temperature, $\theta$. Note also that +the option to use mixing ratios in the boundary layer code instead of +specific quantities is also available and the details of the necessary +changes are documented in appendix~\ref{app:mixratio}. Ultimately +turbulent motions are dissipated as heat and so the source term ${\cal + S}$ in (\ref{cons_eqn_scal}) can include an approximation for that +frictional heating, as described in appendix~\ref{app:fricheat}. +Finally, the ice cloud contributions in (\ref{thetal}) and (\ref{qt}) can +optionally be ignored (l\_noice\_in\_turb), which will be more appropriate if +the time scales for ice melting or sublimation are longer than those of the +turbulence (and so may be more appropriate if the ice itself is not being +mixed by parametrized turbulence). In this instance the saturation humidity +will be calculated with respect to liquid water at all temperatures and, for +consistency, only liquid cloud fractions with be considered. + +An additional variable used for diagnostic purposes is +\begin{equation} + \thetavl = \thetal (1 + c_v q_t) + \label{thetavl} +\end{equation} +where $c_v=(1/\epsilon) -1$ and $\epsilon$ is the ratio of the +molecular weights of water vapour and dry air (\mbox{i.e.}, $\epsilon += M_v/M_a \approx 0.62198$). Thus $\thetavl$ is a conserved variable +that is equal to virtual potential temperature ($\theta_v$) in +cloud-free air and so is used as a simplified measure of buoyancy. + +A `first-order' closure is used to parametrize the turbulent fluxes, +although non-local terms are also included. Under the 9C scheme, an +alternative methodology is optionally available, see section +\ref{sec:rev_flux_grad}. The standard closures are: +\begin{eqnarray} + \overline{w'\chi'} &=& - K_h \frac{\partial \chi}{\partial z} + \khsurf \gamma_{\chi} + \label{scal_closure} \\ + {\bf \tau} &=& K_m \frac{\partial {\bf u}}{\partial z} + {\bf \tau}^{nl} + \label{uv_closure} +\end{eqnarray} + +Separate eddy-diffusivities are calculated for momentum, $K_m$, and +for scalar variables, $K_h$. The second term on the right hand side +represents a non-local flux in unstable boundary layers. Currently it +is only applied for transport arising from surface-driven turbulence +($\khsurf$) and is non-zero only for $\chi=\thetal$, as described in +section~\ref{sec:gradadj}. + +Thus, the parametrization reduces to determining $K_h$, $K_m$ and +$\gamma_{\chi}$ and ${\bf \tau}^{nl}$. Two methods are used to +determine $K_h$ and $K_m$ and how they are combined for +(\ref{scal_closure}) and (\ref{uv_closure}) is described in +section~\ref{sec:shear}. The first method is a local Richardson +number ($Ri$) based scheme. It is calculated for all regimes (but +will be responsible for all mixing in stable conditions), over all +levels up to the specified BL\_LEVELS and is described in +section~\ref{sec:local}. The second method is a non-locally specified +profile scheme. This is exclusively for unstable boundary layers, is +calculated up to level NL\_BL\_LEVELS (typically around 6km AMSL) and +is described in more detail in section~\ref{sec:nonlocal}. In this +regime, mixing is assumed to occur in (or lead rapidly to the +formation of) well-mixed layers (in which conserved variables are +approximately uniform with height) that are capped by an inversion. +Mixing is assumed to be driven either from the surface in a `surface +mixed layer' (SML, by a positive surface buoyancy flux and by surface +stresses) or by cloud-top buoyancy sources (radiative and evaporative +cooling, see appendix~\ref{app:vscales}). As described in section +\ref{sec:nonlocal}, separate $K$-profiles are used for these two +turbulence sources. If the cloud-top sources generate mixing +throughout the SML the layer is said to be `coupled' but if the +$K$-profile representing surface-driven mixing does not extend up to +cloud-top, the layer is referred to as being `decoupled'. As +decoupled layers are restricted to being buoyancy driven and typically +below 6km, they are referred to as decoupled stratocumulus (DSC) +layers. The calculation of $\gamma_{\chi}$ is described in +section~\ref{sec:gradadj} and, finally, fluxes across the top of both +SML and DSC layers (the entrainment fluxes) are specified explicitly +through a separate entrainment parametrization, as described in +section \ref{sec:entr}. + +The strategy used to determine precisely where and when the resulting +eddy-diffusivities should be applied is described in +section~\ref{sec:types}. The buoyancy parameters, finite difference +and other notation used here are defined in appendices~\ref{app:buoyp} +and \ref{app:not}. Further papers describing this scheme and its +performance are \cite{lock00} (noting the corrigendum in +\cite{locketal01_corr}), \cite{martin00:_new_bound_layer_mixin_schem}, +\cite{lock01}, \cite{bushetal1999} and +\cite{brown08:_upgrad_bound_layer_schem_met}. + +%------------------------------------------------------------------------ +% DIAGNOSIS OF TYPE AND DEPTHS +%------------------------------------------------------------------------ +\section{Diagnosis of boundary layer depth and type} +\label{sec:types} + +The non-locally specified $K$-profiles require the height of the base +and top of the layer to be diagnosed (see section~\ref{sec:nonlocal}). +Furthermore, as stated in section~\ref{sec:closure}, the mixing +generated by the non-local $K$ profiles is assumed to occur in (or +lead rapidly to the formation of) well-mixed layers capped by an +inversion. Thus, the accurate diagnosis of their vertical extent is +crucial. How to make this diagnosis is dependent on the boundary +layer mixing regime which has been categorised into 7 distinct +`boundary layer types': +\begin{itemize} +\item{\bf Type I}: Stable boundary layer (with or without cloud) --- + turbulent diffusivities are calculated by the `local' scheme + (section~\ref{sec:local}) +\item{\bf Type II}: Boundary layer with stratocumulus over a stable + near-surface layer --- as Type I but with a turbulently mixed cloud + layer driven from its top (a DSC layer, diagnosis described in + section \ref{sec:decouple}) +\item{\bf Type III}: Well mixed boundary layer --- the classic single + mixed layer which may be cloud-topped or clear but is predominantly + buoyancy-driven (\mbox{c.f.} a possible type VII below) --- + diagnosis described in section~\ref{sec:adiapar}) +\item{\bf Type IV}: Unstable boundary layer with a DSC layer not over + cumulus (see section~\ref{sec:decouple}) --- the surface-based and + cloud-top-driven non-local $K$ profiles may or may not overlap and + cloud-top entrainment can still include the surface forcing (see + section~\ref{sec:entr}) +\item{\bf Type V}: Boundary layer with a DSC layer over cumulus --- + the cumulus (treated by the model's mass-flux convection scheme) + provides coupling with the SML (cumulus diagnosis described in + section \ref{sec:adiapar}) +\item{\bf Type VI}: Cumulus-capped boundary layer --- no turbulent + diffusivities are allowed\footnote{unless the option to mix across + the LCL is selected, see section~\ref{sec:lclmixing}} at or above + the LCL as the mass-flux convection scheme operates here (cumulus + diagnosis described in section~\ref{sec:adiapar}) +\item{\bf Type VII}: Shear-dominated unstable layer --- potentially + wind-shear might allow deeper turbulent mixing in unstable boundary + layers than is apparent purely from the thermodynamic profiles + (sufficient even to inhibit the formation of cumulus); the + possibilities are discussed in section~\ref{sec:shear}. +\end{itemize} +Types I to VI are shown schematically in Fig.~\ref{fig:bltypes}. + +\begin{figure}[tbh] + \centering + \scalebox{0.46}{\includegraphics{wcrp_bltypes1}} + \scalebox{0.46}{\includegraphics{wcrp_bltypes2}} + \captionarb{Schematic representation of boundary layer types I to + VI. The top of the upward arrows indicate the height \zhpar while + the top of their solid line portions indicate \zh.} + \label{fig:bltypes} +\end{figure} + +\subsection{The diagnostic parcel ascent and cumulus diagnosis} +\label{sec:adiapar} + +{\bf Summary}: the depth of the non-local $K$-profiles for +surface-driven turbulence (with NTML grid-levels in the mixed layer +and top at height \zh, as required for (\ref{kmsurf})) is determined +from: +\begin{enumerate} +\item a diagnostic moist parcel ascent; top at grid-level NTPAR, + height \zhpar$=z_{\ntpar+\frac{1}{2}}$. Typically this is an + adiabatic parcel but entraining options are available. +\item a diagnosis of cumulus-capped layers (if cumulus-capped then + NTML and \zh are set to the LCL\footnote{unless the option to mix + across the LCL is selected, see section~\ref{sec:lclmixing}}, if + not then to the parcel top) +\end{enumerate} +Note that this process is only performed for unstable boundary layers +(defined by a positive surface buoyancy flux, \mbox{i.e.}, $\wbs>0$). + +{\bf Step 1:} the method assumes that the height to which turbulent +mixing driven by surface processes can extend in unstable boundary +layers (and therefore the vertical extent of the $K$ profile for +surface-driven turbulence) can be determined solely from the +properties of the thermodynamic profiles. In more detail, the first +step in calculating \zh is to lift a parcel, with properties from the +first grid-level ($k=k_s$) above the top of the surface layer, upwards +allowing for latent heat release. The top of the surface layer is +taken to be at the lower of $z=0.1$\zh (this is then consistent with +the $K$-profiles, see section~\ref{sec:nlsurf}; \zh is taken from the +previous timestep) and the grid-level above which $\thetavl$ starts to +increase with height. The ascent is stopped at the grid-level NTPAR +(height \zhpar$=z_{\ntpar+\frac{1}{2}}$) above which the parcel +becomes more negatively buoyant than a given threshold, $\theta_v'$. +Note that the parcel properties themselves are not perturbed in order +to preserve the height of the mixed-layer's lifting condensation level +(LCL). The calculation of the parcel's buoyancy excess is described +in section~\ref{sec:parxs}. Currently, +\begin{equation} + \theta_v' = \mbox{max} \left[A_{plume}, + \, \mbox{min} \left[ B_{plume} \sigma_{Tv1}, + \, G_{max}\zhe \right] \right] + \label{parcel_pert} +\end{equation} +where $A_{plume}=0.2$, $B_{plume}=3.26$, $G_{max}=10^{-3}$Km$^{-1}$, +$\sigma_{Tv1} = 1.93\, \wthvs/w_m$ and $w_m^3=u_*^3+0.25\,\zhe \wbs$. +Following \cite{holtslag93:_local_versus_nonloc_bound_layer}, +$\theta_v'$ is related to the magnitude of the gradient adjustment, +$\gamma_{\thetal}$ (see section \ref{sec:gradadj}). Thus, +$B_{plume}=A_{ga}$, although somewhat arbitrary limits have been +placed on the magnitude of $\theta_v'$ for numerical security (the +upper limit being consistent with that applied to $\gamma_{\thetal}$ +in (\ref{gradadj}) ). Within limits, then, $\theta_v'$ represents a +typical buoyancy excess of boundary layer plumes. + +The pressure at the LCL, $P_{LCL}=P_{k_s} +(T_{LCL}/T_{k_s})^{(1/\kappa)}$, where $\kappa=R/c_p$. The +temperature at the LCL, $T_{LCL}$, is calculated using approximations +in \cite{Bolton1980} as +\begin{equation*} + T_{LCL} = 55 + \frac{2840}{3.5 \log(T_{k_s}) - log(e_{k_s}) - 4.805} +\end{equation*} +where the vapour pressure of air in grid-level $k_s$, $e_{k_s} = +q_{k_s} P_{k_s}/(100 \, \epsilon)$. The full-level below that +containing the LCL is labelled NLCL and \zlcl$=z_{\nlcl+\frac{1}{2}}$. +If the parcel rises above the top of the LCL transition zone (defined +as 1.1\zlcl, its ascent can also be stopped at the grid-level at which +it has maximum buoyancy excess over the environment. This is +identified as the grid-level above which +\begin{equation*} + \frac{d\theta_v}{dz}|_{\rm env} > + \Gamma_{\rm inv}\, \frac{d\theta_v}{dz}|_{\rm par} +\end{equation*} +where currently the tolerance for identifying inversions by this +method, $\Gamma_{\rm inv}=1.1$. This use of the height of maximum +excess (if lower than that given by the straight buoyancy threshold, +$\theta_v'$) is typically of little consequence in stratocumulus +regions (which tend to be well-mixed beneath large inversions), but +can be necessary in order to identify the capping inversion in cumulus +cases (e.g.~in the trade wind regions). + +{\bf Step 2:} having established \zhpar, a crucial additional test is +to determine whether this layer is well-mixed (\mbox{i.e.}, +stratocumulus-capped) or cumulus-capped. The parcel ascent can rise +to cloud-top in both cases but cumulus cloud layers are observed not +to be as well-mixed as stratocumulus layers. Recall that application +of the $K$ profiles is expected to form or maintain well-mixed layers +and so their current formulation is inappropriate for cumulus cloud +layers. Specifically, a logical flag (CUMULUS) is set to true if +\begin{equation*} + \left| \frac{ \Delta_{\rm cld} q_t}{\Delta_{\rm cld} z} \right| > + C_t \, \left| \frac{ \Delta_{\rm sub} q_t}{\Delta_{\rm sub} z} \right| +\end{equation*} +where the cloud-layer gradient, $ \Delta_{\rm cld} $, is taken between +both NTPAR and NTPAR-1 (to allow for the possibility that a Sc layer +has just deepened by a grid-level) and NLCL and the sub-cloud layer +gradient, $\Delta_{\rm sub}$, between grid-levels NLCL and $k_s$. +Currently the threshold factor, $C_t = 1.1$. If cumulus is diagnosed, +the top of the surface-based mixed layer (\zh) is set to \zlcl (rather +than to \zhpar, as illustrated in Fig.~\ref{fig:bltypes} for types V +and VI). There is then an option to diagnose the thickness of the LCL +transition zone, see section~\ref{sec:lclmixing}. Otherwise, the +boundary layer surface-driven mixing is capped at \zlcl so that mixing +into the cumulus cloud layer is only carried out by the model's +mass-flux convection scheme and not by the eddy viscosity based +boundary layer scheme. Note that basing the CUMULUS diagnosis on +cloud and sub-cloud layer gradients limits the model only to being +able to resolve cumulus with cloud and sub-cloud layers at least 2 +grid-levels (and optionally 400m) thick. Otherwise the layer is +considered well-mixed to \zhpar with an option to include a +representation of fluxes into the capping inversion (see +section~\ref{sec:dzi}). + +If the parcel ascent fails to find an inversion below 3km (or +BL\_LEVELS) but the LCL is below BL\_LEVELS, then the layer is assumed +to be cumulus-capped. If the LCL is above BL\_LEVELS, then again +cumulus is diagnosed with NTML$=\mbox{min}[$NLCL, BL\_LEVELS$-1]$, in +the hope that the mass-flux convection scheme (in its moist or dry +mode) will transport the surface fluxes higher! Clearly this +restriction on the boundary layer scheme is not desirable and so a +value of BL\_LEVELS above the tropopause is recommended. + +Note that if cumulus is not diagnosed then a further, subgrid +estimation of the height of the capping inversion is attempted for \zh +(as described in section~\ref{sec:sginv}). + +\subsubsection{Calculation of parcel buoyancy excess} +\label{sec:parxs} + +As described in appendix~\ref{app:buoyp}, virtual temperature, $T_v = +T(1 + c_v q_v - \ql - q_f)$, is used as the measure of buoyancy. The +condensed water in the parcel at a grid-level $k$ ($\qlf^p$, the +superscript $^p$ indicating parcel properties) is estimated using a +Taylor expansion of $q_s$ about the environment at that grid-level +($q_s^p \approx {q_s}_k + \alpha_L (T^p-T_k) $). Assuming that +$\qlf^p=q_t^p - q_s^p$ gives +\begin{equation} + \qlf^p = \mbox{max}\left[ 0.0, \, a_L \left( q_t^p - {q_s}_k + - \alpha_L (\thetal^p - (g z_k/c_p)-T_k)\right) + \right] + \label{qlpar} +\end{equation} +where the buoyancy parameters $a_L$ and $\alpha_L$ are defined in +appendix \ref{app:buoyp}. Recall that the parcel has $q_t$ and +$\thetal$ taken from grid-level $k_s$ which are conserved during its +ascent. Note that (\ref{qlpar}) will not give condensation until the +parcel becomes saturated. In the environment the cloud scheme will +allow some condensation (and therefore warming and stabilisation of +the environment profile) to take place before the grid-level becomes +saturated in the mean. To allow for this in the parcel (without +applying the cloud scheme), (\ref{qlpar}) is also calculated at each +grid-level but using the environment grid-box mean $q_t$ and $\thetal$ +to give $\qlf^e$. The difference in the environment's condensed water +as determined by the UM cloud scheme (\mbox{i.e.}, $\ql+q_f$) and by +(\ref{qlpar}) (\mbox{i.e.}, $\qlf^e$) is then added to $q^p_{lf}$. + +Given $\qlf^p$, (\ref{thetal}) implies $T^p = \thetal^p - (g z_k/c_p) ++ (L \qlf^p/c_p)$ (using $L_s$ if $T_k$ is below the melting point) +and (\ref{qt}) implies $q_v^p = q_t^p - \qlf^p $ and thus $T_v^p$ can +be calculated. Recall that the diagnosis of the parcel's maximum +buoyancy excess over the environment (described in +section~\ref{sec:adiapar}) required $\theta_v$. This is approximated +as $\theta_v = T_v + (g z_k/c_p)$. + +\subsection{Diagnosis of the vertical extent of the K-profiles} +\label{sec:decouple} + +The diagnosis of mixed layers with turbulence driven from cloud-top +has been separated in to three stages. These are: +\begin{enumerate} +\item diagnose the existence of a decoupled stratocumulus (DSC) layer + with approximately uniform $\thetavl$ (label the top grid-level in + the mixed-layer NTDSC and diagnose the subgrid height of its capping + inversion, \zhsc, see section~\ref{sec:sginv}) +\item diagnose an approximate depth of the DSC layer, $\zml$, in order + to be able to calculate the representative turbulent velocity scales + (see appendix \ref{app:vscales}). +\item calculate the depth of the $K$ profiles (see + section~\ref{sec:nonlocal}) in both SML and DSC layers using + constraints on the TKE budget of the layer. This includes the + diagnosis of recoupling of DSC layers and decoupling of SMLs +\end{enumerate} + +{\bf Step 1}: the diagnosis of DSC layers depends on whether cumulus +convection has been diagnosed. If a cumulus-capped layer under an +inversion within BL\_LEVELS has been diagnosed, grid-levels NTPAR and +NTPAR+1 are tested to see if they contain significant layer cloud +($C_F>$ SC\_CFTOL). This threshold for identifying potentially +turbulently-mixed cloud layers is currently SC\_CFTOL$=0.1$. If there +is significant cloud, NTDSC is set to NTPAR. + +Alternatively, if a well-mixed surface-driven boundary layer was +diagnosed, then from grid-level NTML$+2$ upwards, a cloud-top +grid-level ($k_{ct}$) is sought such that ${C_F}_{k_{ct}}>$ SC\_CFTOL +and ${C_F}_{k_{ct}+1}<$ SC\_CFTOL. If $\Delta_{k_{ct}} \thetavl / +\Delta_{k_{ct}} z < 10^{-3}$Km$^{-1}$ (\mbox{i.e.}, $\thetavl$ is +approximately well-mixed over at least two grid-levels), then NTDSC is +set to $k_{ct}$. If grid-levels $k_{ct}$ and $k_{ct}-1$ are not +well-mixed, grid-levels $k_{ct}-1$ and $k_{ct}-2$ are tested using the +same criterion. If they are not well-mixed either, the cloud-layer is +ignored for the purposes of turbulent mixing. If grid-levels +$k_{ct}-1$ and $k_{ct}-2$ are identified as well-mixed a further test +is applied to determine whether the $\theta_v$ (rather than +$\thetavl$) gradient across grid-levels $k_{ct}$ and $k_{ct}-1$ is +greater than adiabatic (\mbox{i.e.}, whether grid-levels $k_{ct}$ and +$k_{ct}-1$ actually form part of an inversion --- note that by +ignoring the $\ql$ contribution to buoyancy, $\thetavl$ is not a good +variable to use to measure the strength of cloud-capping inversions). +To do this, the $\theta_v$ gradient between grid-levels $k_{ct}$ and +$k_{ct}-1$ is compared with that for a parcel lifted adiabatically +from grid-level $k_{ct}-1$, in exactly the same way as for the SML +parcel ascent (see section \ref{sec:parxs}). If $d\theta_v/dz|_{\rm + env} > \Gamma_{\rm inv} d\theta_v/dz|_{\rm par} $ between +grid-levels $k_{ct}$ and $k_{ct}-1$ then NTDSC is set to $k_{ct}-1$; +if not then NTDSC is set to $k_{ct}$ (recall that grid-levels +$k_{ct}-1$ and $k_{ct}-2$ have already been identified as well-mixed). + +{\bf Step 2} is to diagnose an approximate depth of the DSC layer, +$\zml$. The bottom grid-level of the mixed-layer (NBDSC) is diagnosed +as the lowest grid-level, descending from NTDSC, where +${\thetavl}_{\ntdsc} + \thetavl'$ is less than $\thetavl$ of the +environment. The parcel perturbation is given by +\begin{equation} + \thetavl' = - \, \frac{ \tau_{rc} \Delta_\radf }{z_{rc}} + \label{dscd_pert} +\end{equation} +where $\Delta_\radf$ (Kms$^{-1}$) is the magnitude of the cloud-top +radiative divergence (see appendix~\ref{app:vscales}), $\tau_{rc}$ is +a timescale for the exposure of boundary layer eddies to the cloud-top +radiative cooling (taken to be 200s) and $z_{rc}$ is a depth-scale for +the radiatively cooled layer (taken to be 50m). These values of +$\tau_{rc}$ and $z_{rc}$ are only estimates (and will in reality vary +from one cloud to another) but they are consistent with, for example, +the observations of \cite{nicholls1986}. If the parcel failed to fall +(\mbox{i.e.}, NBDSC equals NTDSC) in a DSC layer {\em not} overlying +cumulus, then the layer is assumed not to be well-mixed. At the top +of a cumulus layer, the DSC layer is given a minimum depth of +$\Delta_{\ntdsc+\frac{1}{2}} z$. Otherwise, the layer depth, $\zml$, +is measured from the top of layer NTDSC to the base of layer NBDSC. + +{\bf Step 3}: the step 2 calculation of $\zml$ is used to calculate +the representative velocity scales for the DSC layer but its +calculation is only crude. Here, the vertical extent of the +$K$-profiles is determined more accurately by ensuring that the +magnitude of the integrated buoyancy consumption of TKE within the +mixed layer is less than or equal to a fraction, $D_t$, of the +buoyancy production, following \cite{turton1987}. + +Following appendix~\ref{app:buoyp} the grid-box mean buoyancy flux can +be written as: +\begin{equation} + \wb = g \left[ (1-C_F) \left(\beta_T \overline{w'\thetal'} + \beta_q \wqt \right) + + C_F \left( \tilde{\beta_T} \overline{w'\thetal'} + \tilde{\beta_q} \wqt \right) + \right] + \label{eq:wb_cont} +\end{equation} + +As standard, the fluxes in (\ref{eq:wb_cont}) are then expanded using +the first-order closure in (\ref{scal_closure}) as: +\begin{eqnarray} + \wthl_k &=& -\khsurf \,\frac{\widetilde{\Delta_k \thetal}}{\Delta_k z} + -\khtop \,\frac{\Delta_k \thetal}{\Delta_k z} \nonumber \\ + \wqt_k &=& -\left(\khsurf + \khtop \right) \, + \,\frac{\Delta_k q_t}{\Delta_k z} + \label{eq:wx_std} +\end{eqnarray} +where $\widetilde{\Delta_k \thetal} = \Delta_k \thetal - +\gamma_{\thetal} \Delta_k z$ in order to include the non-local (or +gradient adjustment) term. If the alternative flux-gradient option is +used, see section \ref{sec:rev_flux_grad}, then additional terms are +needed. + +Large-eddy simulations have demonstrated that the crucial region in +determining when decoupling of stratocumulus will occur (\mbox{i.e.}, +when the $K$ profiles no longer span the entire layer from cloud-top +to the surface) is in a thin layer of unsaturated air just below +cloud-base, where $\wb$ first becomes negative. Thus, in the above +calculation, it is crucial both to have an accurate measure of +cloud-base height (which will have to be subgrid) and to include +successfully this thin unsaturated layer in the buoyancy consumption +integral. Thus, the $\wb$ integration is performed over the cloud and +sub-cloud layers separately and the cloud-fraction is taken to be +uniform within the cloud layer (and zero below cloud-base). The +height of cloud-base is given by (\ref{zc_calc}). + +An iterative method is then used to find the vertical extent of mixing +(within certain bounds, as described below) such that the magnitude of +buoyancy consumption of TKE within the mixed layer equals a fraction, +$D_t$, of the buoyancy production, \mbox{i.e.}, +\begin{equation} + \sum_{z_{k-\frac{1}{2}} > z_i-\zml}^{z_{k-\frac{1}{2}} < z_i} + \left|\left[ \wb|_{z_{k-\frac{1}{2}}}<0 \right]\right| \, \Delta_k z \, + \leq \, D_t \, + \sum_{z_{k-\frac{1}{2}} > z_i-\zml}^{z_{k-\frac{1}{2}} < z_i} + \left[ \wb|_{z_{k-\frac{1}{2}}}>0 \right] \, \Delta_k z + \label{deccrit} +\end{equation} +Note that, for simplicity, the ${\cal E}_h$ factors are not included +in $\khsurf$ or $\khtop$ when calculating (\ref{eq:wx_std}) under the +assumption that they will be small. This process is applied to all +unstable mixed layers. For stratocumulus layers, observations and LES +suggest a value of $D_t=0.1$. A separate value of $D_t$ can be used for +the sub-cloud layer in cumulus capped boundary layers if this method is +used to determine the LCL transition zone thickness, see +section \ref{sec:lclmixing}. For cloud-free mixed layers, $D_t=1$ is +used, purely to keep negative buoyancy fluxes down to a reasonably +realistic level (for example, if the parcel top diagnostic returned +too high a boundary layer depth). + +The first step is to test for whether a well-mixed layer is possible +(either decoupling what has so far been diagnosed as a well-mixed +layer or, if one exists, recoupling a decoupled stratocumulus layer), +\mbox{i.e.}, to test whether (\ref{deccrit}) is satisfied with both +$\khsurf$ and $\khtop$ extending from the surface to the cloud-top. +If recoupling is possible then the various flags identifying the DSC +layer are reset ({\em this includes setting the cumulus diagnosis to + false}), any surface-driven entrainment originally applied at \zh is +added to the entrainment at \zhsc (after rescaling for the inversion +strength at \zhsc) and \zbase is set to 0.1\zh (for the reason +discussed above). If decoupling is diagnosed, \zhsc is set to the +original \zh (inversion height), although the entrainment across this +inversion is not recalculated (and so keeps any surface-driven +component --- the COUPLED flag is therefore set to true, see +section~\ref{sec:entr}). + +If a decoupled layer is diagnosed, then an iteration is performed to +find the highest \zh (so top of the $\khsurf$ profile) that still +satisfies (\ref{deccrit}), but with $\khtop=0$ in (\ref{eq:wx_std}). +The iteration proceeds with \zh stepping from its lowest permissible +height to its highest (currently 3 steps are used). If at any stage +(\ref{deccrit}) is violated, then the step below (therefore containing +the height that would give equality in (\ref{deccrit})) is divided by +4 and 3 of those steps are taken downwards. If (\ref{deccrit}) is met +the step above is again reduced by a factor of 4 and 3 steps taken +upwards. A total of 3 sweeps are possible, each with a smaller step +so that \zh approaches the height that gives equality in +(\ref{deccrit}). The accuracy with which this is achieved will be the +difference in the maximum and minimum permissible heights of \zh +divided by $2\times4\times4 = 32$, which will typically be less than +30m. The top grid-level of the SML, NTML, is defined as the highest +grid-level such that $\khtop$ is non-zero at the half-level above. + +The above process is then repeated to find the appropriate \zbase for +$\khtop$, \mbox{i.e.}, for the base of top-driven mixing. Some +constraints are placed on \zbase, namely that it should never go below +$0.1$\zh (to avoid affecting the continuity of the $K$ profiles at the +top of the surface layer, see (\ref{ws_defn})). If cumulus convection +has been diagnosed then \zbase is not allowed to go below +$z_{\ntml+\frac{1}{2}}$ (unless the layer is diagnosed to recouple +completely). Finally, \zbase must always be at or below +$z_{\ntdsc-1}$, so that mixing in decoupled layers is always resolved, +and at least $\Delta z_{rad}$ (the cloud-top radiative cooling depth +defined in section \ref{sec:wbint_inv}) below the t inversion. The +base grid-level of the DSC layer, NBDSC, is defined (analogously to +NTDSC) as the lowest grid-level such that $\khtop$ is non-zero at the +half-level below. + +A possible extension to this diagnosis would be to include the shear +contribution to the TKE budget in (\ref{deccrit}) and so allow +shear-driven mixing to help maintain well-mixed layers. + +\subsubsection{Surface layer $\wb$ integration} + +In the surface layer, below $z_i/10$, the $K$ profiles have a +different functional form from the rest of the mixed layer. Rather +than include this additional complexity in the $\wb$ integration, the +surface layer is treated separately. In place of the +finite-difference form of $\wb$, see (\ref{eq:wb_cont}) and +(\ref{eq:wx_std}) above, $\wb$ is assumed to be linear between $\wbs$ +at the surface and zero at a level which must be estimated. The +surface layer integration is then from the surface up to $z_{{\rm + K_{SURF}}}$, where $\theta$-level K\_SURF is the first above +$z_i/10$. The level where $\wb$ is zero is found by linear +interpolation across the grid-levels where the diagnosed cloud-free +buoyancy flux would become negative. This is where $\beta_T +\widetilde{\Delta_k \thetal} + \beta_q \Delta_k q_t$ becomes positive +and so where the cloud-free part of $\wb$ (\mbox{i.e.}, that part +below cloud-base which is important for decoupling) becomes negative. + + +\subsubsection{Integration of $\wb$ close to the inversion} +\label{sec:wbint_inv} + +Because of the large gradients often seen in fluxes close to the +inversion (in particular, in the LW radiative flux), simple finite +difference flux calculations, (\ref{eq:wx_std}), can be significantly +inaccurate in this region. An example is shown in +Fig.~\ref{fig:inv_integ}. Calculating $\wthl_{\ntml+\frac{1}{2}}$ +from (\ref{eq:wx_std}) gives a negative value, largely because +$\Delta_{\ntml+1} \thetal$ is positive and so the local flux is large +and negative. In reality, $\wthl$ becomes positive only a short +distance below cloud-top such that the integral here will tend also to +be positive. + +The solution adopted is to integrate $\wb$ analytically across the +region just below the inversion, labelled $\Delta z_{rad}$ in +Fig,~\ref{fig:inv_integ}. Since $\Delta_{\ntml} \thetal$ can also be +significantly positive (when the grid-level inversion is rising or +falling, for example), the base of this region is taken to be the +lower of the first $\theta$-level below $z_h-100$ m (a physically +reasonable depth over which cloud-top radiative cooling might be +expected to occur) and $z_{\ntml-1}$. +\begin{figure}[tb] + \begin{center} + \scalebox{1}{\includegraphics{ideal_invinteg}} + \caption{Subgrid (lines) and model (symbols) fluxes of $\thetal$: + turbulent flux (dash-dotted, crosses), radiative flux (dashed, + triangles) and total flux (solid). The shaded area illustrates + the integrated turbulent flux that would be obtained were + (\protect\mbox{\protect\ref{eq:wx_std}}) used.} + \label{fig:inv_integ} + \end{center} +\end{figure} +Then, +\begin{eqnarray} + \int_{z_h-\Delta z_{rad}}^{z_h} \, \wthl \, dz & = & + \int_{z_h-\Delta z_{rad}}^{z_h} \, F_{\thetal}^{Tot} - + F_{\thetal}^{NT}\, dz + \nonumber \\ + & = & I^{Tot} - I^{rad} - I^{ppn} + \label{wthl_int} +\end{eqnarray} +For the radiative flux, it could be assumed that the subgrid flux +distribution is exponentially dependent on the grid-level LWP, for +example. This would give: +\begin{equation} + I^{rad} = \frac{\Delta z_{rad}} + {\ln(F^{rad}|_{z_h}/F^{rad}|_{z_h-\Delta z_{rad}} ) } + \lb F^{rad}|_{z_h}-F^{rad}|_{z_h-\Delta z_{rad}} \rb + \label{irad} +\end{equation} +However, off-line tests indicated this could give a strong and +spurious sensitivity to $F^{rad}|_{z_h-\Delta z_{rad}}$. Furthermore, +for most realistic scenarios, the logarithmic factor in (\ref{irad}) +tends to be close to 3. Consequently, we approximate $I^{rad} = +\Delta z_{rad} ( F^{rad}|_{z_h}-F^{rad}|_{z_h-\Delta z_{rad}} ) /3 $. +In addition, $F^{rad}|_{z_h}-F^{rad}|_{z_h-\Delta z_{rad}} $ is +approximated as $\Delta \radf$, the radiative flux change across +cloud-top used in the calculation of $\vtopo$ (\ref{ctraddiv}). The +precipitation flux is assumed to vary linearly across this region, as +does the total flux, and so its contribution to $I^{Tot}$ cancels with +$I^{ppn}$ in (\ref{wthl_int}). Finally, for simplicity, the total +flux is taken to be constant and equal to the inversion value, such +that $I^{Tot} = \Delta z_{rad} F^{Tot}|_{z_h}$. With these +approximations, (\ref{wthl_int}) becomes +\begin{eqnarray*} + \int_{z_h-\Delta z_{rad}}^{z_h} \, \wthl \, dz & = & + \Delta z_{rad} \lb -w_e \Delta \thetal + \Delta \radf \rb + - \Delta z_{rad} \Delta \radf / 3 \\ + & = & \Delta z_{rad} \lb -w_e \Delta \thetal + \frac{2}{3} \Delta \radf \rb +\end{eqnarray*} +For the integral of $\wqt$ across this cloud-top region, $\wqt$ is +also taken to be constant so that: +\begin{equation} + \int_{z_h-\Delta z_{rad}}^{z_h} \, \wqt \, dz = + - \Delta z_{rad} w_e \Delta q_t +\end{equation} +The integrated buoyancy flux is then found from (\ref{eq:wb_cont}) +using the mixed layer cloud fraction and buoyancy coefficients +evaluated at the grid-level above $ z_h -\Delta z_{rad} $. + +\subsection{Diagnosis of inversion thickness} +\label{sec:dzi} + +Terminating the diagnostic parcel ascent at its level of neutral +buoyancy ignores any overshooting through the parcel's own inertia as +it enters the inversion region. This overshooting region effectively +defines the depth of the inversion over which the negative entrainment +heat fluxes are seen. Typically this will be small relative to the +model vertical grid but at higher vertical resolution or when a +strongly surface-heated boundary layer is capped by weak stability +inversions could be resolved. Following \cite{beare2008}, a simple +energetic argument gives a realistic prediction of the top of the +inversion, $z_{top}$, in LES from +\begin{equation} + 6.3 \, w_m^2 = \int_{z_{nb}}^{z_{top}} \, b \, dz + \label{dz_param} +\end{equation} +where $z_{nb}$ is the level of neutral buoyancy (found by linear +interpolation between grid-levels), $w_m$ is the boundary layer +velocity scale defined in section \ref{sec:nlsurf} and $b$ is the +parcel buoyancy. Note that the constant in (\ref{dz_param}) is the +same as in \cite{beare2008} because $6.3 = 2.5 * 4^{2/3}$ and $w_m^3$ +differs by a factor of 4. The buoyancy integration in +(\ref{dz_param}), that is itself dependent on $z_{top}$, is performed +working upwards from \zhpar assuming piece-wise linear variation of +$b$ between grid-levels. Note that the standard definition of the +boundary layer top in the UM is the height of the first flux level +below the level of neutral buoyancy, so +\zhpar$=z_{\ntpar+\frac{1}{2}}$. The inversion thickness is then +defined as +\begin{equation} + \Delta z_i = z_{top} -\zhpare + \label{dz_definition} +\end{equation} + +\subsection{Diagnosis of the LCL transition zone thickness} +\label{sec:lclmixing} + +As described in section~\ref{sec:adiapar}, when cumulus convection has +been diagnosed surface-driven mixing was originally capped at \zlcl so +that mixing into the cumulus cloud layer was only carried out by the +model's mass-flux convection scheme. This was seen to lead to errors +in the mean profiles across the LCL, with superadiabats being the most +extreme manifestation. Using the boundary layer parametrization to +couple cloud and sub-cloud layers would have the numerical advantage +of being implicit. There is also observational and LES evidence that +appropriately-scaled buoyancy fluxes up to the LCL are +indistinguishable from those in cloud-free convective boundary layers +and so the non-local surface-driven mixed layer K-profiles remain +accurate up to this level. To diagnose the depth to which these +profiles should penetrate above the LCL, the algorithm given in +section~\ref{sec:decouple} to diagnose the extent of the K-profiles in +decoupled boundary layers can be used (using the switch kprof\_cu). +This ensures that the magnitude of the integrated buoyancy consumption +of TKE within the mixed layer is less than or equal to a fraction, $D_t$, +of the buoyancy production. In cumulus layers, cloudy thermals will +generate positive buoyancy fluxes (and are handled by the convection +scheme) but it is assumed that there will also be cloud-free thermals +within the grid box that may penetrate above the grid-box mean LCL +(but are too dry to reach their own LCL). Thus their buoyancy flux is +given by (\ref{eq:wb_cont}) with $C_F=0$. Restricting the negative +integral of this buoyancy flux then gives a new definition for \zh +that is then used in the calculation of the surface-driven K-profiles +in section~\ref{sec:nlsurf} --- the larger the value of $D_t$, the +higher \zh will be. Typically $D_t=0.1$ for decoupled stratocumulus +layers while idealised clear-sky convective boundary layers (where +the magnitude of the entrainment buoyancy flux is a fraction, $A_1$, +of the surface flux) would have $D_t = A_1^2 \sim 0.05$. For GA7 $D_t$ +has been set to 0.05 for this cumulus transition zone calculation. +Because the Gregory-Rowntree convection scheme triggers from the LCL, that +is used as a minimum constraint on the boundary layer mixing depth (so that +the massflux and turbulence schemes remain coupled). With other convection +schemes this may not be appropriate and so this minimum constraint can be +relaxed, which is achieved by setting it to half the height of the LCL (the +factor of a half is arbitrary, with no sensitivity to this choice given that +the diagnosis parcel reached the LCL, but ensures the iteration starts well +below the LCL). + +%------------------------------------------------------------------------ +% LOCAL SCHEME +%------------------------------------------------------------------------ +\section{The local scheme} +\label{sec:local} + +A first order `mixing length' closure is used: +\begin{eqnarray} + K_m &=& {\cal L}_m^2 \, (S+S_d) \, f_m(Ri) \label{kmlocal}\\ + K_h &=& {\cal L}_h \, {\cal L}_m \, + (S+S_d) \, f_h(Ri) \label{khlocal} +\end{eqnarray} +where ${\cal L}_m$ and ${\cal L}_h$ are the neutral mixing lengths and +$S$ is the resolved vertical shear of the horizontal wind components, +$S = \left| \partial {\bf u}/\partial z \right|$. A representation of +the wind shear, $S_d$, generated by drainage flows in complex terrain +can also be included, as described below. Near the surface simple +finite difference calculations for the vertical gradients can become +inaccurate because of the quasi-logarithmic profiles of variables +\cite[]{Ayra1991}. Currently this is ignored above grid-level 2 and +the neutral mixing lengths are given by +\begin{eqnarray*} + {\cal L}_m &=& \frac{k(z+z_{0m})}{1+k(z+z_{0m})/\lambda_m} \\ + {\cal L}_h &=& \frac{k(z+z_{0m})}{1+k(z+z_{0m})/\lambda_h} +\end{eqnarray*} +where $z_{0m}$ includes the orographic component. For the lowest +interior grid-level ($k=1$) they are calculated, incorporating this +log profile correction, as +\begin{equation*} + \tilde{{\cal L}}_{X,k-1/2} = \frac{k \Delta_{k-1/2} z}{ + ln\left( \frac{z_k + z_{0m}}{z_{k-1} + z_{0m}} \right) + + \frac{k \Delta_{k-1/2} z}{\lambda_X} } +\end{equation*} +If near-surface resolution is increased this logarithmic correction +should be considered over more levels. + +The asymptotic mixing lengths are given by +\begin{eqnarray} + \lambda_m &=&\mbox{max}\left[\lambda_0,\, 0.15 \zloce, 2 h_B \right] \nonumber\\ + \lambda_h &=&\mbox{max}\left[\lambda_0,\, 0.15 \zloce \right] + \label{asymp_ml} +\end{eqnarray} +where $\lambda_0$ is a minimum mixing length read in from the namelist +and \zloc is defined below. The orographic blending height, $h_B$ (only +used within the boundary layer, as defined below), is given by +\begin{equation*} + h_B = {\rm max}\left[z_1+(z_{0m})_{\mbox{veg}}, 2^{1/2} \sigma_h \right] +\end{equation*} +where $\sigma_h$ is the standard deviation of the height of the +subgrid orography and $(z_{0m})_{\mbox{veg}}$ is the vegetative part +of the roughness length. The constants in (\ref{asymp_ml}) can be +considered `tuned' (see, in particular, the operational modifications +described in appendix~\ref{app:opmods}). + +The Richardson number, $Ri$, that is used as a local measure of +stability is given by +\begin{equation} + Ri = \frac{\Delta B / \Delta z}{(S+S_d)^2} + \label{ridefn} +\end{equation} + +The measure of buoyancy used in $Ri$ is +\begin{equation} + \Delta B = g\left( \overline{\beta_T} \Delta \thetal + + \overline{\beta_q} \Delta q_t \right) + \label{Bdefn} +\end{equation} +where $\overline{\beta_T}$ and $\overline{\beta_q}$ are the grid-box +mean (\mbox{i.e.}, cloud weighted) buoyancy coefficients, that can be +defined in two different ways, see appendix~\ref{app:buoyp} and +section~\ref{sec:fd_ri}. Note that (\ref{Bdefn}) reduces +to a virtual temperature approximation of buoyancy in cloud-free air +and that neutral buoyancy (in cloudy as well as cloud-free air) is +implied by vertically uniform $\thetal$ and $q_t$. This is then +entirely consistent with the assumption that $\thetal$ and $q_t$ are +conserved variables within the boundary layer scheme. + +As described in \cite{lock2012}, the wind shear generated by drainage +flows in complex terrain is thought to lead to additional vertical +mixing. This wind shear can be approximated as +\begin{equation*} + S_d = \frac{\Delta B }{ \Delta z} \, \alpha_d \, t_d \, {\cal Z}_d +\end{equation*} +The representative slope of the local terrain, $\alpha_d$, is given by +\begin{equation*} + \alpha_d^2 = \frac{1.0}{ 25.0 + (l_h/\sigma_h)^2} +\end{equation*} +with $l_h$ a specified horizontal scale for the terrain, currently +taken to be 1500 m (empirically derived for Scottish orography in the +UKV), and $\sigma_h$ the standard deviation of the full subgrid +orographic height. $\sigma_h$ should also be taken as the average +over the surrounding area of each grid box (typically 6 to 8 grid +lengths), in order to be representative of the local area over which +such flows will be underresolved. The above formula is used so that +$\alpha_d \sim \sigma_h/l_h $ for small $\sigma_h$ but only tends to +0.2 for large values. To limit the vertical extent of $S_d$ to be +below approximately $z=\sigma_h$, a height-dependent factor is +included, ${\cal Z}_d = 0.5( 1 - {\rm tanh}\left[ 4 ((z/\sigma_h)-1) +\right])$. The timescale, $t_d$, takes a fixed value of 30 minutes, +for simplicity. + +Initially, the lowest half-level at which $Ri>Ri_{crit}$ is taken to +be a measure of the boundary layer top (\zloc) and the full-level +below is designated NTLOC. In general $Ri_{crit}=1$ but a value of +0.25 is recommended for use with the 'SHARPEST' stability functions, +see below. If the boundary layer was diagnosed as cumulus-capped by +the non-local scheme (see section~\ref{sec:types}) then \zloc is +lowered to \zlcl (and $K_h$ and $K_m$ are set to zero from the base of +grid-level NLCL upwards) so that transports into and within the +cumulus cloud layer can be performed solely by the mass-flux +convection scheme. Depending on the switch local\_fa, above NTLOC +turbulently-mixed layers (where $Ri 0$), several forms for the stability +functions are available. The `long-tailed' functions are +\begin{equation*} + f_{\rm stable} = \frac{1}{1+g_0 Ri} +\end{equation*} +Alternative functions, which decrease as $1/Ri^2$ with increasing +stability are, from \cite{louis1979}: +\begin{equation*} + f_{\rm stable} = \frac{1}{(1+ 5 Ri)^2} +\end{equation*} +and the family of ``sharp'' functions can be written in terms of a transitional +Richardson number, $Ri_{t}$, as: +\begin{equation*} + f_{\rm stable} = +\begin{cases} + (1 - 5Ri)^2 & {\rm for}\ 0Ri_{t} +\end{cases} +\end{equation*} +where +\begin{eqnarray} +A_{Ri} & = & \left(1-g_0 Ri_{t}\right)/\left(1- g_0 Ri_{t}/2\right)^2 \nonumber\\ +B_{Ri} & = & (g_0/2) /\left(1 - g_0 Ri_{t}/2\right)^2 +\end{eqnarray} +For the `SHARPEST' function of \cite{derbyshire1997}, $Ri_{t}=0.1$, while +larger values give even sharper reduction of turbulence with increasing $Ri$. +An additional option, used operationally in some configurations (originally +in the Mesoscale Model, hence called 'MES tails'), is to blend linearly +from Louis functions at the surface to SHARPEST by 200m. + +A stability dependent Prandtl number ($Pr=f_m/f_h$) is generally used +following \cite{MailhotLock2004} with: +\begin{equation*} + Pr=\min \left( Pr_{\rm max}, \, Pr_N(1+2Ri) \, \right). +\end{equation*} +The maximum permitted Prandtl number, $Pr_{\rm max}$, is currently set +to $5$ for model stability reasons. The stability functions for $Ri>0$ are +then given by: +\begin{eqnarray*} + f_m & =& \frac{Pr}{Pr_N} \, f_{\rm stable} \\ + f_h & =& \frac{1}{Pr_N} \, f_{\rm stable} +\end{eqnarray*} +Note that writing the functions in this way ensures that $f_m=1$ under +neutral conditions and the effect of the variation in $Pr$ is for $f_m$ +to decrease slower with increasing $Ri$ than $f_{\rm stable}$, which can +be explained through increasing gravity-wave activity. + +Finally, the LEM stable functions are also available which cut off all +turbulence beyond a critical Richardson number, $Ri_c=0.25$: +\begin{eqnarray*} + f_m & =& \left( 1 - \frac{Ri}{Ri_c} \right)^4 \\ + f_h & =& \frac{1}{Pr_N} \left( 1 - \frac{Ri}{Ri_c} \right)^4 (1 - g_{LEM} Ri) +\end{eqnarray*} +\label{stable_stab_lem} +with $g_{LEM}=1.2$. + +\subsection{Finite difference calculations} +\label{sec:fd_ri} + +The Charney-Phillips vertical grid staggering used in the UM stores +the horizontal wind components ($u$, $v$) on grid-levels, +$\rho$-levels, that are staggered relative to scalar variables (such +as $\thetal$ and $q_t$) and vertical velocity, $w$. While much of the +boundary layer scheme is grid-independent, this has serious +implications for the calculation of $Ri$. There are two obvious +possibilities, to calculate $Ri$ (and thence $K(Ri)$) on either +$\theta$-levels or $\rho$-levels and then interpolate either $K_h$ or +$K_m$ to be able to calculate the required fluxes. To do the former +requires averaging the buoyancy gradient in the numerator (and is +referred to by \cite{cullen1994} as the `$\theta$-bar' method), the +latter the wind shear in the denominator (referred to as the +`$\rho$-bar' method). Single-column model and other tests +demonstrated that the `$\rho$-bar' method could readily generate +instabilities just above the top of the boundary layer because +averaging the wind shear into this stable air tended to reduce $Ri$ +and so promote mixing. Fortunately, the `$\theta$-bar' method tended +to increase $Ri$ above inversions and so damp mixing. Thus, $Ri$ is +calculated on $\theta$-levels as +\begin{equation*} + Ri_k = \frac{DBDZ_k} + {(\Delta_{k+\frac{1}{2}} {\bf u}/\Delta_{k+\frac{1}{2}} z)^2} +\end{equation*} +The buoyancy gradient on $\theta$-level $k$ can be calculated in two different ways, +depending on the switch {\rm i\_interp\_local}. The long-standing method is given by +\begin{equation*} + DBDZ_k = g\left( \overline{\beta_T}_{k} (D\thetal DZ)_k + + \overline{\beta_q}_{k} (Dq_t DZ)_k \right) +\end{equation*} +where $\overline{\beta_T}$ and $\overline{\beta_q}$ are the grid-box +mean (\mbox{i.e.}, cloud-fraction weighted) buoyancy coefficients, +defined in appendix~\ref{app:buoyp}. Note that because this is +defined on $\theta$-levels, no vertical interpolation of cloud +variables (fractional area and water contents), to which the buoyancy +coefficients are very sensitive, is required. The volume-weighted +gradients of $\thetal$ and $q_t$ are calculated as +\begin{equation} + (D\chi DZ)_k =\left( (z_{k}-z_{k-\frac{1}{2}}) \, \frac{\Delta_{k+1} \chi}{\Delta_{k+1} z} + + (z_{k+\frac{1}{2}}-z_{k}) \, \frac{\Delta_{k} \chi}{\Delta_{k} z} + \right) / \Delta_{k+\frac{1}{2}} z +\label{gradient_interp} +\end{equation} +as long as $\chi_{k-1}$ is defined on an atmospheric model level. To calculate +$DBDZ_1$, between the surface and the lowest $\theta$-level, either the buoyancy gradient +from level 1 to 2 can be extrapolated, \mbox{i.e.}, +\begin{equation*} + (D\chi DZ)_1 = \frac{\Delta_{2} \chi}{\Delta_{2} z} +\end{equation*} +or surface properties can be used. Over sea, the sea-surface temperature +and $q_{sat}$ can be used. Over a heterogeneous (tiled) land surface the +appropriate moisture variable varies between tiles. For the orographic form drag +(\ref{section_2}), an average $Ri_{SL}$ of the surface layer is calculated but, as +discussed above, subsequent vertical averaging of $Ri$ would potentially be +numerically unstable. In principle, the tile-average of $(Dq_t DZ)_1$ could +be calculated but for now, over land, the grid-box average surface temperature is +used to calculate $(D\thetal DZ)_1$ and $(Dq_t DZ)_1$ is extrapolated from +above (\mbox{i.e.}, $(Dq_t DZ)_1 = (Dq_t DZ)_2)$). + +As noted above, a feature of the previous option is that applying the cloudy +buoyancy coefficients at a cloud top level, $k$ say, to strong gradients +interpolated between $k-1$ and $k+1$, can yield an unstable $DBDZ_k$ despite +strong static stability, especially when the upper level is very dry. This can be +related to cloud-top entrainment instability but this process is intended to be +represented within the non-local scheme. Hence, the alternative method is to +calculate the buoyancy gradient directly on $\rho$-levels and then interpolate this +vertically to give $DBDZ_k$, using (\ref{gradient_interp}). This then +requires a cloud fraction on $\rho$-levels. The difficulty comes where there is +a change in cloud fraction between levels. For this ``edge'' fraction, $f_{edge}$ +(the fraction of the grid-box that is cloudy in one level but not in the other), the +change in supersaturation ($ s = q_t-q_{sat}$) between +levels is used to estimate the vertical fraction likely to contain cloud, $f_{lev}$. +For example, where $C_F$ decreases with height, $f_{lev}={q_c}_{k-1}/(s_{k-1}-s_{k})$, +where $q_c$ is the total condensate, and $f_{lev}$ also constrained to be less +than unity. The total cloud volume fraction +is then given by $f_{tot} = {\rm min}[{C_F}_{k-1},{C_F}_k] + f_{edge}f_{lev}$ and this +is used to weight the saturated contribution to the buoyancy parameters on +$rho$-levels, \mbox{e.g.}, +$\overline{\beta_T}_{k-1/2} = f_{tot} \tilde{\beta_T}_{k-1/2} + (1-f_{tot}){\beta_T}_{k-1/2})$, +where the saturated and unsaturated buoyancy parameters are also intepolated to +$\rho$-levels using (\ref{gradient_interp}). + +Having calculated $Ri$ on $\theta$-levels, ${K_m}_{k}$ and ${K_h}_{k}$ +are calculated, still on $\theta$-levels, as in (\ref{kmlocal}) and +(\ref{khlocal}). Finally, $K_h$ must be interpolated to +$\rho$-levels: +\begin{equation*} + {K_h}_{k+\frac{1}{2}} = \left( + (z_{k+\frac{1}{2}}-z_{k}) {K_h}_{k+1} + + (z_{k+1}-z_{k+\frac{1}{2}}) {K_h}_{k} \right) / \Delta_{k} z +\end{equation*} +Note that in the code the convention is for fluxes to be held on the +half-level below the variable itself. Consequently, RHOKM(K), and +therefore RI(K), are held on the `half-level' below $\rho$-level K, +which is $\theta$-level K-1. + +In addition to the above, the log profile correction applied to ${\cal + L}_h$ (to give $\tilde{{\cal L}}_h$) must be applied {\em after} +interpolation of $K_h$ to level $k+\frac{1}{2}$ in order that the +correct cancellation with the finite difference scalar gradient in the +flux calculation can occur. In the unstable stability functions +(\ref{unstable_stab}), however, $\tilde{{\cal L}}_h$ must be +calculated on $\theta$-levels (\mbox{i.e.}, the same as $\tilde{{\cal + L}}_m$ and $Ri$) in order to maintain the same stability +dependence. + +\subsection{Shear-driven mixing and interaction between the local and + non-local schemes} +\label{sec:shear} +The general approach is to take $K_{\chi}$ in (\ref{scal_closure}) and +(\ref{uv_closure}) as +\begin{equation} + K_{\chi} = \mbox{max} \left[ (K_{\chi}^{\rm surf}+K_{\chi}^{\rm Sc}), + K_{\chi}(Ri) \right] + \label{klnl} +\end{equation} +As noted in section~\ref{sec:closure}, this implies that mixing in +stable boundary layers is determined exclusively by the local scheme, +$K_{\chi}(Ri)$. Continuing to calculate $K_{\chi}(Ri)$ in unstable +boundary layers and using (\ref{klnl}) is seen as the simplest way of +achieving a relatively smooth transition between stable and unstable +boundary layers. + +At the top of unstable mixed layers, great care is taken to ensure the +parametrized entrainment mixing is implemented faithfully, see +section~\ref{sec:entr}). Consequently, if a subgrid inversion has +been diagnosed capping a mixed layer (see section~\ref{sec:sginv}), +then $K_{\chi}(Ri)$ is set to zero at the interfaces either side of +the inversion grid-level. There are also options (using the switch +Keep\_Ri\_FA) to set $K_{\chi}(Ri)$ to zero entirely above unstable +boundary layers or across the LCL in cumulus-capped layers. + +However, the mixed-layer depths were only diagnosed from thermodynamic +constraints. In near neutral boundary layers, shear generation of +turbulence might be expected to allow mixing to extend into regions of +weak static stability (and potentially to inhibit the formation of +cumulus). Currently, therefore, if NTLOC$>$NTML+1 (in layers that are +not cumulus-capped) $K_{\chi}(Ri)$ is left unconstrained by the SML +part of the non-local scheme (and so not set to zero from the SML +inversion upwards) and similarly if NTLOC$>$NTDSC+1. It is realised +that this does not cover the case of shear-driven mixing into cloud +layers that have been diagnosed as cumulus-capped (which would be +poorly represented by the current convection scheme). Several methods +have been introduced that attempt to alleviate this problem, giving +rise to the diagnosis of a ``shear-dominated boundary layer'' type +(type VII), discussed in section~\ref{sec:types}. The first (the +``shear-dominated boundary layer fix'') simply sets the CUMULUS flag +to false if NTLOC $>$ NTPAR. This then ensures that the +locally-determined $K$ are not set to zero above the LCL. Several +more rigorous options are available that incorporate a ``dynamic +criteria'' in the diagnosis of boundary layer type. The first of +these prohibits the diagnosis of cumulus boundary layers when the bulk +measure of stability, $-z_i/L$, is small (currently less than 1.6). +Here $z_i$ is taken as the top of the diagnosis parcel ascent (or at +most 3km) and $L$ is the surface Obukhov length. This test also resets +the depth of the surface-based mixed layer to level 1 since the top of +the parcel ascent may not be suitable (having previously +been diagnosed as cumulus cloud top). The second method +effectively increases the importance of the Richardson number +diagnosis and has been developed from analysis of cold-air outbreaks +\cite[]{bodas-salcedo2012}. Because of the strong surface buoyancy +generation of turbulence in these regimes, a calculation of $Ri$ is +made that allows for the gradient adjustment by the non-local scheme, +\mbox{i.e.}, using $\widetilde{\Delta_k \thetal}$ (see +(\ref{eq:wx_std})). The height, \zloc, where $Ri>Ri_{crit}=0.25$ is +found. It is then hypothesised that this level of turbulent +instability (that incorporates the effects of shear) only needs extend +some fractional distance into the cloud layer to disrupt the formation +of cumulus elements. Thus, if $\zloce > \zlcle + f_{\rm sh} +\left(\zhpare-\zlcle\right) $, where $f_{\rm sh}$ is a tunable +parameter ($0 +Ri_{crit}$, \zhsc is the top of any stratocumulus layer and \zh is the +top of surface-based mixed layer, found by adiabatic parcel ascent but +reset to the LCL in cumulus capped layers. Another diagnostic is +available, the ``boundary layer depth'' (STASH 25), that is set to +$=\mbox{max}[\zhe, \zloce]$ and so represents the depth of the stable +boundary layer or ``surface'' mixed layer. Also available are three +diagnostics that represent the calculated value of each of the +individual terms in STASH 3,304: 3,356 is set to \zh; 3,357 is \zhsc +and 3,358 is \zloc. + +%------------------------------------------------------------------------ +% NON-LOCAL SCHEME +%------------------------------------------------------------------------ +\section{The non-local scheme} +\label{sec:nonlocal} + +This method of calculating $K$ values for unstable conditions is +non-local in the sense that, at a given height within the boundary +layer, $K$ is determined not by any local properties of the mean +profiles at that height but solely by the magnitude of the turbulence +forcing applied to the layer (as measured by the representative +velocity scales described in appendix~\ref{app:vscales}) and the +height within the layer. The non-local scheme is therefore +particularly robust but care must be taken where the profiles are +applied. The calculation of the vertical position and extent of the +$K$ profiles is described in section~\ref{sec:types}. + +\subsection{Surface-driven turbulence} +\label{sec:nlsurf} + +For turbulence sources at the surface (namely surface drag with +velocity scale $u_*$, and positive surface buoyancy fluxes with +velocity scale $w_*$) in a layer with top at $z=$\zh, base at $z=0$ we +set +\begin{equation} + \kmsurf = k \ \zhe \ w_m \ \frac{z}{\zhe} + \left( 1 - {\cal E}_m^{\rm surf} \frac{z}{\zhe} \right)^2 + \label{kmsurf} +\end{equation} +where $w_m^3 = u_*^3 + w_s^3$, $u_*$ is the friction velocity +(including the orographic roughness component) and $w_s$ is defined +below. For the 9C version of the scheme, \zh is the +diagnosed subgrid inversion height (see section~\ref{sec:sginv}) for +both $\khsurf$ and $\kmsurf$. In the 8A version, $\kmsurf$ uses +\zh$=z_{\ntml+\frac{1}{2}}$. The factor ${\cal E}_m^{\rm surf}$ is +chosen so that $\kmsurf$ will tend to $ K_m|_{\ntml+\frac{1}{2}}$ as +$z$ tends to \zh, where $K_m|_{\ntml+\frac{1}{2}}$ is the entrainment +eddy-diffusivity (given by (\ref{khent}), although, in order to avoid +altering the shape function too much, ${\cal E}_m^{\rm surf}$ is not +allowed to fall below $0.7$). A similar factor, ${\cal E}_h^{\rm + surf}$, is used in the $\khsurf$ profile even though the entrainment +fluxes of the thermodynamic variables will usually be specified +explicitly rather than through an eddy-diffusivity (see +section~\ref{sec:entr}). + +The form of $w_s$ differs between the surface layer ($ z < 0.1 $\zh) +and the rest of the mixed-layer: +\begin{equation} + w_s^3 = + \begin{cases} + 2.5 \, \frac{z}{\zhe} w_*^3 & {\rm surface\ layer} \\ + 0.25 \, w_*^3 & {\rm mixed\ layer} \\ + \end{cases} + \label{ws_defn} +\end{equation} +and $w_*^3=\zhe \wbs$ using \zh from the current timestep (note that +the use of $w_*$ here will be inconsistent with the use of $\vheato$ +in the entrainment parametrization in cloudy boundary layers). Note +that $w_s$ is continuous across $0.1$\zh and constant with height in +the mixed layer. This form for $w_s$ is motivated by a desire to +match the model's surface transfer formulation within the surface +layer (as described further in section~\ref{sec:hbcomp}) and to use a +cubic sum of velocity scales within the mixed layer (consistent with +dimensional analysis of the TKE equation, see +\cite{holtslag93:_local_versus_nonloc_bound_layer}). + +The formula for $\khsurf$ is identical to (\ref{kmsurf}) but with +$w_m$ replaced by $w_h=w_m/Pr$, where the turbulent Prandtl number is +given by: +\begin{equation} + Pr = 0.75 \frac{u_*^4 + (4/25)w_s^3 w_m}{u_*^4 + (8/25)w_s^3 w_m} + \label{prandtl_nl} +\end{equation} +Thus $Pr$ varies from 0.75 in neutral conditions to 0.375 in +convective. The origin of the functional form of (\ref{prandtl_nl}) +is unknown. + +\subsubsection[Comparison with Holtslag and Boville (1993)]{Comparison with + \cite{holtslag93:_local_versus_nonloc_bound_layer}} +\label{sec:hbcomp} +The surface-driven $K$ profiles are the same as those in +\cite{holtslag93:_local_versus_nonloc_bound_layer}, HB93, except for +(\ref{ws_defn}) and (\ref{prandtl_nl}) and the inclusion of the ${\cal + E}_m^{\rm surf}$ terms. For the latter, HB93 effectively set ${\cal + E}_m^{\rm surf} =1$. To generate entrainment, however, they simply +use $\kmsurf|_{\ntml+\frac{1}{2}}$, as evaluated from (\ref{kmsurf}) +with a subgrid calculation of \zh$>z_{\ntml+\frac{1}{2}}$, rather than +using a separate entrainment parametrization. + +The difference in (\ref{ws_defn}) arises from the surface layer, where +HB93 match $w_m$ to their surface exchange functions (\mbox{i.e.}, +$w_m = u_* / \phi_m $) which results in proportionality constants of 6 +and 0.6 for $w_s$ in the surface and mixed layers respectively. This +matching is greatly simplified because their non-dimensional shear +$\phi_m = ( 1 + 15 k (z/z_i) w_*^3 / u_*^3 )^{-(1/3)}$. To match +$w_m$, through (\ref{ws_defn}), to the UM function, $\phi_m = ( 1 + 16 +k (z/z_i) w_*^3 / u_*^3 )^{-(1/4)}$, would require a complex function +of $u_*$ and $w_*$ in place of the constant and so this is not +attempted. + +The formula for the Prandtl number used in the interior in HB93 is +also matched to that used in the surface exchange functions ($Pr_{\rm + surf}$, say). For the UM, +\begin{equation*} + Pr_{\rm surf} = \frac{\Phi_h}{\Phi_m} + = \left( 1 + 16 \, k \frac{z}{\zhe} \, \frac{w_*^3}{u_*^3} + \right)^{-1/4} +\end{equation*} +giving $Pr_{\rm surf} = 1$ in the neutral limit (compared to 0.75 from +(\ref{prandtl_nl})). In the convective limit, $Pr_{\rm surf}|_{0.1\, + \zhe} \rightarrow 0.9 (w_*/u_*)^{-3/4} = 0.9 \beta^{3/4} = 0.14 $ +(compared to 0.375 from (\ref{prandtl_nl})). Thus, the Prandtl +numbers do not match between the surface layer and interior +formulations in the UM. + +The formulation in HB93 gives $Pr$ varying from 1 to 0.6 (for $-z/L$ +varying from 0 to 10). In convective conditions ($-z/L=10$), HB93 +have $w_m = 0.85 w_*$ and $w_h=1.4 w_*$ while the UM has $w_m = 0.65 +w_*$ and $w_h = 1.7 w_*$. The implications of these differences from +HB93 are unknown. The convective LES in \cite{lock99} suggest $ w_h +\approx w_*$; I don't know where the larger proportionality constants +come from. + +Another difference between the UM and HB93 is that HB93 only apply +gradient adjustment above the surface layer (and this is allowed for +in their mixed layer definition of $Pr$). Simulations in +\cite{brown1996}, however, suggest that this may lead to a cold bias +at the top of the surface layer. It is attempted to alleviate this in +the UM by the application of gradient adjustment down to the surface +(although this will then lead to a dependence on the height of the +lowest grid-level). The implications of this for matching the Prandtl +number between the surface and interior in the UM is not known. + +\subsection{Cloud-top-driven turbulence} + +For cloud-top-driven turbulence over a layer of depth $\zml$ (with top +at \zh or \zhsc and base at \zbase, determined as in +section~\ref{sec:decouple}), +\begin{equation} + \kmtop = 0.63 \ k \ \zml \ \vtopo \left( \frac{z'}{\zml} \right)^2 + \left( 1 - {\cal E}_m^{\rm Sc} \frac{z'}{\zml} \right)^{0.8} + \label{kmtop} +\end{equation} +where $\vtopc = \vrad+\vbr$ (see appendix~\ref{app:vscales}) and $z'$ +is height above \zbase. Then $K_h = K_m / \mbox{Pr}$, where +$\mbox{Pr}=0.75$. The resulting $K_h$ profile was derived against +convective cloudy LES, as described in \cite{lock99_proceedings}. The +appropriate Prandtl number (and therefore $\kmtop$) is unknown, 0.75 +being chosen simply as a number in the middle of the range usually +quoted for turbulent mixing in general. As with (\ref{kmsurf}), \zh +(or \zhsc) are given by the subgrid diagnosis (see section +\ref{sec:sginv}) except for $\kmtop$ in the 8A scheme which uses the +height of the half-level below ($z_{\ntml+\frac{1}{2}}$ or +$z_{\ntdsc+\frac{1}{2}}$). Again following (\ref{kmsurf}), the +factors ${\cal E}_m^{\rm Sc}$ and ${\cal E}_h^{\rm Sc}$ are included +in (\ref{kmtop}) so that $\kmtop$ will tend to +$K_m|_{\ntml+\frac{1}{2}}$ (and $\khtop$ to +$K_h|_{\ntml+\frac{1}{2}}$), given by (\ref{khent}), as $z$ tends to +\zh (and here no restriction is made on the magnitude of either ${\cal + E}_m^{\rm Sc}$ or ${\cal E}_h^{\rm Sc}$). + +\subsection{Gradient adjustment} +\label{sec:gradadj} +Recall that for $\thetal$ only we use +\begin{equation} + \wthl = - K_h \frac{\partial \thetal}{\partial z} + \khsurf \gamma_{\thetal} +\label{wthl} +\end{equation} +where +\begin{equation} + \gamma_{\thetal} = + \mbox{min}\left[ A_{ga} \frac{\sigma_{T1}}{\zhe}, G_{max} \right] + \label{gradadj} +\end{equation} +$A_{ga}=3.26$, $G_{max}=10^{-3}$Km$^{-1}$ and $\sigma_{T1} = 1.93 \, +\wthls/w_m$, where for this calculation of $w_m$ (given by +$w_m^3=u_*^3+0.25\,\zhe \wbs$) \zh is taken from the previous +timestep. The form of (\ref{gradadj}) is similar to that used in HB93 +and the magnitude of $\gamma_{\thetal}$ is the same as in HB93 in the +convective limit --- the difference in $A_{ga}$ exactly allows for the +different constants in (\ref{ws_defn}). + +Consistent with the mixed layer assumptions underlying the non-local +scheme, the flux profile produced by the scheme is assumed to be +essentially determined by the specified surface and entrainment +values. Thus, the effect of including this non-local term ($ \khsurf +\gamma_{\thetal} $) is to allow the model to maintain more well-mixed +$\thetal$ profiles (\mbox{i.e.}, with $\partial \thetal / \partial z$ +less negative or even positive in a cloud-free surface-heated boundary +layer, for example), subject to an arbitrary upper limit included for +numerical safety. Hence the term `gradient adjustment' rather than +non-local flux. When estimating the buoyancy flux, then (as in +(\ref{eq:wx_std})), it is simplest to allow for the non-local term by +adjusting the $\thetal$ gradient. + +The equivalent term for $q_t$ (\mbox{i.e.}, $\gamma_{q_t}$) is set to +zero in order to represent crudely the effects on the mixed-layer +$q_t$ profile of entrainment drying at the mixed-layer top which tend +to make $q_t$ profiles less well mixed than those of $\thetal$ +\cite[]{mahrt1976}. From UM version 5.5, there is the option to +implement the non-gradient stress parametrization of +\cite{brown97:_non}, as described in section~\ref{sec:ngstress}. + +\subsection{Non-gradient stress parametrization} +\label{sec:ngstress} + +There is an option that is operational in the UM to include an +additional non-gradient (or non-local) stress parametrization, ${\bf + \tau}^{nl}$ in (\ref{uv_closure}), as proposed by +\cite{brown97:_non}. They showed that with only a down-gradient +stress parametrization, a one-dimensional model produced wind profiles +in the convective boundary layer that were less well-mixed than +predicted by LES, and underestimated the near surface wind. +Furthermore, \cite{brownetal2006} showed that the operational +verification statistics indicate a slow bias in the 10~m wind over +land by day, especially in spring and summer. + +The non-gradient stress parametrization in the UM is very similar to +that proposed by \cite{brown97:_non}, written +\begin{equation} + (\tau_x^{nl},\tau_y^{nl})= \left[ + \frac{2.7w_*^3}{(u_*^3+0.6w_*^3)}\right] \left[ \left( \frac{z'}{\zhe'} + \right) \left( 1- \frac{z'}{\zhe'} \right)^2 \right] + (\tau_x^{s},\tau_y^{s}) + \label{tau_nl} +\end{equation} +Here $w_*$ is the convective velocity scale, $u_*$ is the friction +velocity, and $(\tau_x^{s},\tau_y^{s})$ are the surface stresses. +Note that the surface stresses here have to be diagnosed explicitly +(from time-level n fields) but experience has shown these can become +unrealistically large when the near-surface wind is significantly out +of balance with the surface characteristics. As a safety measure, +these surface stresses can be limited such that the implied stress +gradient across the boundary layer is always less than a parameter, +MAX\_STRESS\_GRAD, currently set to 0.05 ms$^{-2}$ (which, for example, +gives a maximum $u_*$ of 7 ms$^{-1}$ in a boundary layer 1km deep). The +term involving $u_*$ and $w_*$ is as proposed by \cite{brown97:_non} +(although note that their Table 3 contains a typo), and ensures that +the non-gradient stress is zero in neutral conditions but asymptotes +to a stability-independent fraction of surface stress in convective +conditions. The primed variables in the shape function allow the +non-local stress profile to either be applied across the whole boundary +layer (using $z'=z$ and $\zhe'=\zhe$), as in \cite{brown97:_non}, +or only above the surface layer (using $z'=z-0.1\zhe$, $\zhe'=\zhe-0.1\zhe$). +The motivation for applying the non-local stress above the surface layer +was to ensure that the match to surface layer similarity was maintained below +$0.1\zhe$ (although separate tests suggested that the impact of this +change is small). + +\subsection{The revised scalar flux-gradient formulation} +\label{sec:rev_flux_grad} + +Following detailed analysis of many large-eddy simulations, including +both surface-heated and cloud-top cooled, a revised flux-gradient +relationship has been developed. In this section, the previous +version will be referred to as the standard one. The formulation is +given in terms of the total flux, +\begin{equation*} + F_{\chi}^{Tot}=\wx + F_{\chi}^{NT} +\end{equation*} +where the non-turbulent flux, $F_{\chi}^{NT}$, is the sum of the +radiative (for $\thetal$), microphysics and subsidence fluxes. This +is a crucial difference from the old formulation: the mean profiles in +LES are found to respond to the total flux profile (which is linear in +a mixed layer) rather than to the individual components of the flux. +Hence any flux-gradient relationship can never be generic to both +$\wthl$ and $\wqt$ since, for example, the shape of the $\thetal$ +profile is determined through interactions with radiation while the +$q_t$ profile is not. Physically, this suggests that while processes +like radiation must {\em locally} generate regions of cold (negatively +buoyant) air at cloud-top, subsequent mixing by turbulent eddies +results in a more-or-less uniformly well-mixed {\em mean} $\thetal$ +profile (presumably because these eddies bring locally warm air back +up to the cloud-top region). + +So, the new formulation is written: +\begin{equation} + F_{\chi}^{Tot} = F_{\chi}^{NT}|_{\zbaseq} + -\lb \khsurf + \khtop \rb \f{\p \ol{\chi}}{\p z} + + \wxngs + \wxngt + + f_2 \lb F_{\chi}|_{z_h} - F_{\chi}^{NT}|_{\zbaseq} \rb +\label{fg_new} +\end{equation} +where $z_h$ and $\zbaseq$ are the heights of the top and base of the +mixed layer, respectively. It can be seen that (\ref{fg_new}) is +composed of a local down-gradient component, two non-gradient flux +terms (one generated by surface-driven turbulence and the other by +cloud-top) and a non-local entrainment flux profile. The turbulent +fluxes are then obtained by subtracting off the non-turbulent +component: +\begin{equation*} + \wx = F_{\chi}^{Tot} - F_{\chi}^{NT} +\end{equation*} + +The components of (\ref{fg_new}) are: +\begin{itemize} +\item{$\khmsurf = k z_h w_{h,m} \zonzi \lb 1-\zonzi \rb^2$} +\item{$\khtop = 3.6 k \vtopo z_{ml} \lb \zonzml \rb^{3}\lb 1-\zonzml + \rb^{2}$} +\item{$\wxngs=\khsurf \gamma_{\chi}$ with + $\gamma_{\chi}=A_{ga}\f{\wxs}{w_h z_h}$ and $A_{ga}=10$} +\item{$\wxngt = f^{Sc} \lb F_{\chi}|_{z_h}- F_{\chi}^{NT}|_{\zbaseq} + \rb $ with $f^{Sc}=3.5 \, k \, \f{\vtopo}{\vsumo} + \lb\zonzi\rb^{3}\lb1-\zonzi \rb$} +\item{$f_2 = 0.5 \, \zonzi \, 2^{(z/z_h)^4}$} +\end{itemize} +In the above equations $k$ is von Karman's constant, $z'$ +($=z-\zbaseq$) is height above the mixed layer base, $z_{ml}$ +($=z_h-\zbaseq$) is the mixed layer depth, $u_*$ is the friction +velocity, and $w_*$ and $\vtopo$ are the velocity scales for surface +and cloud-top buoyancy-driven turbulence. + +Although the structure of the surface-driven non-gradient terms is the +same as for the standard flux-gradient formulation, +(\ref{scal_closure}), note that they are now applied to $q_t$ as well +as $\thetal$ and also the empirical coefficients in the velocity +scales have been revised: +\begin{itemize} +\item{$w_h = (u_*^3 + C_{ws} w_*^3)^{\f{1}{3}} / Pr_{\rm neut} $ with + $C_{ws}=0.42$ for $\zonzi \geq 0.1$ and $C_{ws}=4.2 \zonzi$ for + $\zonzi<0.1$ } +\item{$w_m = w_h Pr $} +\end{itemize} +The functional form of the Prandtl number, $Pr$, is unchanged except +that $w_m$ is replaced by its neutral value: +\begin{equation*} + Pr = Pr_{\rm neut} + \frac{u_*^4 + w_*^3 {w_m}^{\rm neut} / 25} + {u_*^4 + w_*^3 {w_m}^{\rm neut} Pr_{\rm neut}/ (25 Pr_{\rm conv} )} +\end{equation*} +and the range is now $ Pr_{\rm neut} = 0.75$ to $ Pr_{\rm conv} = +0.6$. As with the standard scheme, a constant Prandtl number of 0.75 +is used to calculate $\kmtop$. + +\subsubsection{Discussion of some of the revisions} + +\begin{table}[h] +\begin{center} +{\begin{tabular}{c|cc|cc} +Formulation & \multicolumn{2}{c}{Convective limit} & \multicolumn{2}{c}{Neutral limit} \\ + & $w_m$ & $w_h$ & $w_m$ & $w_h$ \\ +\hline +HB & $0.84 \,w_*$ & $1.4 \,w_*$ & $u_*$ & $u_*$ \\ +UM standard & $0.63 \,w_*$ & $1.7 \,w_*$ & $u_*$ & $1.3 \,u_*$ \\ +UM revised & $0.6 \,w_*$ & $ w_*$ & $u_*$ & $1.3 \,u_*$ \\ +\end{tabular}} +\end{center} +\caption{Convective and Neutral limits for velocity scales} +\label{tab:vscales} +\end{table} + +\begin{figure}[p] + \centering + \scalebox{0.8}{\includegraphics{stab_dep}} + \caption{Stability dependence of the surface velocity scales, + Prandtl number (although I hope something is wrong with my coding + of HB here!) and $d$. Solid lines are from HB, dotted from the + standard UM and the dashed from the revised formulation. The + dash-dotted line for $d$ is a potential modification, as described + in the text. } + \label{fig:stab_dep} +\end{figure} + +It is useful to compare the velocity scales in the revised scheme with +those in the standard version, as well as those in +\cite{holtslag93:_local_versus_nonloc_bound_layer}, hereafter HB, on +which the parametrization was originally based. Recall that HB and +the standard UM set $w_m = (u_*^3 + C_{ws} w_*^3)^{\f{1}{3}}$ and $w_h += w_m/Pr$, with $C_{ws} = 0.6$ and 0.25, respectively, above the +surface layer. The convective and neutral limits for $w_h$ and $w_m$ +are given in Table~\ref{tab:vscales} and the stability dependencies of +several parameters are shown in Fig.~\ref{fig:stab_dep}. The +parameter $d$ in Fig.~\ref{fig:stab_dep} contains the stability +dependence of the gradient adjustment parameter: +\begin{equation} + \gamma_{\chi}= d \f{\wxs}{w_* z_h} + \hspace{0.5cm} {\rm with} \hspace{0.5cm} + d^{HB} = 7.2 w_*^2/w_m^2, \hspace{0.2cm} + d^{std} = 6.3 w_*/w_m, \hspace{0.2cm} + d^{rev} = 10 w_*/w_h + \label{grad_adj} +\end{equation} +The inclusion of an extra $w_*/w_m$ factor in $\gamma_{\chi}$ was a +deliberate change by HB from the original +\cite{troen86:_simpl_model_atmos_bound_layer} formulation on which the +UM was based. This seems an appealing feature (HB's $\gamma_{\chi}$ +will tend to zero as $w_* \rightarrow 0$) and probably should be +considered for the revised scheme (the dash-dotted line in +Fig.~\ref{fig:stab_dep} sets $d^{std} = 10 w_*^2/w_h^2$). Similarly, +$f_2$ might benefit from an additional factor of the form $(\vsurf + +\vtopc )/ \vsum $ so that it too tends to zero in the neutral limit. +Further analysis of LES and SCM tests will be required to verify this. + +Note that the most significant change from the standard UM scheme is +the change to $w_h$ in the convective limit. Since $\gamma_{\chi}$ +remains unchanged in the convective limit, this reduction in $w_h$ +will result in a significantly smaller $\wxngs$ for the revised scheme +which gives better agreement against LES. + +Compared to the standard scheme, it appears that the revised $\khtop$ +is very different. However, Fig.\ref{fig:new_ksc} shows that this +actually amounts to a small adjustment in the shape. In addition, +note that the factors $\varepsilon_h^{surf}$ and $\varepsilon_h^{Sc}$ +have been removed since the entrainment flux is now carried via the +explicit $f_2$ term. + +\begin{figure}[tbh] + \begin{center} + \includegraphics{new_ktop_shape} + \caption{Standard UM $\khtop$ (solid) and revised (dotted), both + scaled by $k z_h \vtopo$. An upside-down version of $\khsurf$ + is also shown (dashed) for comparison.} + \label{fig:new_ksc} + \end{center} +\end{figure} + +%------------------------------------------------------------------------ +% The blended scheme +%------------------------------------------------------------------------ +\section{The blended scheme} +\label{sec:blend} + +For high resolution simulations, the UM has a Smagorinsky-type +subgrid turbulence scheme, described in \citeumdp{028}. However, this scheme +is only truly applicable for horizontal grid-lengths of order $10$~m, +and any real-world simulation run at lower resolution than this will +inevitably have unresolved scales somewhere in the domain. Rather than +force the user to make an ad-hoc decision about the scales they are +interested in, and thus grid-length at which to switch from using the +boundary-layer parametrization (1D BL) to the subgrid turbulence +scheme (3D Smag), a method for blending the two parametrizations has +been developed. This blend is regime and scale dependent, allowing a +single parametrization to be used across resolutions, including the +completely unresolved/resolved extremes. This blending process is +described in \cite{Boutleetal2014}, which gives some examples of its +use and comparison to simulations using either the 1D BL or 3D Smag +schemes only. Updated technical details from \cite{Boutleetal2014} are +reproduced below. Several options are available that are selected using +the switch {\tt blending\_option}. These all follow the same principles +but differ in their choice of what should constitute the boundary layer and +how to treat non-turbulent layers of the atmosphere. + +As shown in \cite{Honnertetal2011}, the rate at which turbulent +structures become resolved appears to be different for different +aspects of the flow. For example, moisture fluxes are on a larger +scale than heat or momentum fluxes, and so transition to being +sub-grid at lower resolution. This is just one of many challenges when +creating a truly accurate grey-zone parametrization, and so our aim +here is to start from the simplest possible approach which allows the +model to transition from unresolved to resolved turbulence in a +plausible way, without the user having to decide at which grid-length +to switch from a 1D, non-local, to a 3D, local sub-grid scheme. + +Given some function, $W_{1D}$, which tells us how poorly resolved the +turbulence is ($=1$ if unresolved, $=0$ if well resolved), we can use +this to blend between the 1D BL and 3D Smag schemes. Both schemes have +a local Richardson number formulation: +\begin{equation}\label{eq-kri} + K_\chi(Ri) = l^2 S f_\chi(Ri), +\end{equation} +where $K_\chi$ is the eddy diffusivity, $l$ is the mixing length, $S$ +is the wind shear, $f_\chi(Ri)$ is the stability function and $\chi$ +represents conserved heat and moisture variables, or momentum. Both +schemes use the same stability function, and both schemes can use the +full 3D shear for $S$. Therefore the only difference is in the mixing +length, which is calculated as +\begin{equation}\label{eq-lblend} + l_{\rm blend} = W_{1D}l_{\rm bl}+(1-W_{1D})l_{\rm smag}, +\end{equation} +where $l_{\rm bl}^{-1} = (\kappa z)^{-1} + \lambda_0^{-1}$ and $l_{\rm + smag}^{-2} = (\kappa z)^{-2} + (c_s \Delta x)^{-2}$, $\kappa$ is the +von Karman constant and $c_s$ is the Smagorinsky constant. Near the +surface $l_{\rm bl}$ and $l_{\rm smag}$ are identical, but the +asymptotic values are different and this method weights the asymptotic +value according to the weighting of the two schemes. For example, at +$\Delta x=1$~km, $c_s\Delta x=200$~m (for $c_s=0.2$), whereas +$\lambda_0=\max(40\ {\rm m}, 0.15z_h)$, which allows for a small +mixing length in shallow unresolved boundary layers (e.g.~stable +ones). + +The \cite{lock00} scheme also contains a non-local component to the +turbulent flux, and this is simply down-weighted by $W_{1D}$ to ensure +that it becomes less significant as the turbulence becomes better +resolved. Therefore the full eddy diffusivity is given by +\begin{equation} + K_\chi = \max\left[W_{1D}K_\chi^{\rm NL}, K_\chi(Ri)\right], +\end{equation} +where $K_\chi^{\rm NL}$ is the non-local diffusivity and $l$ in +Eq.~\ref{eq-kri} is given by $l_{\rm blend}$ in +Eq.~\ref{eq-lblend}. The turbulent flux is then calculated as +\begin{equation} + F_\chi=-K_\chi\frac{\partial \chi}{\partial z} + W_{1D}F_\chi^{\rm NL}, +\end{equation} +where $F_\chi^{\rm NL}$ is the non-local flux. Therefore when +$W_{1D}=1$, the scheme of \cite{lock00} is recovered, whilst with +$W_{1D}=0$ the Smagorinsky-type scheme is recovered. + +Now we need to define the function $W_{1D}$ to blend the schemes. Within the +boundary layer this is based on the turbulent kinetic energy partitioning +given by \cite{Honnertetal2011}. We choose the TKE +partitioning because it is most closely linked to the eddy diffusivity +we are trying to parametrize (for example a TKE based scheme would +calculate the eddy diffusivity from the TKE), and simplify the +function slightly, using +\begin{equation}\label{eq-tanh} + W_{1D} = 1 - \tanh\left(\beta\frac{z_{\rm turb}}{\Delta x}\right)\max\left[0,\min\left[1,r_f\left(l_0-\frac{\Delta x}{z_{\rm turb}}\right)\right] \right], +\end{equation} +where $z_{\rm turb}$ is the appropriate lengthscale of the turbulence, +$\beta$ is a scaling parameter which controls the speed of the +transition from unresolved to resolved +turbulence, $r_f=\frac{1}{l_0-l_1}$, $l_0=4$ and $l_1=0.25$ (N.~B.~this +formula is slightly modified from that given in \cite[]{Boutleetal2014}). +\cite{Malavelleetal2014} demonstrated that this scaling +method was applicable to any type of unstable boundary layer given an +appropriate choice of $z_{\rm turb}$. In \cite{Boutleetal2014} this +functional form was applied everywhere, adjusting the values of +$z_{\rm turb}$ and $\beta$ depending on the regime. +The max function is present to force the lowest resolution simulations +to just use the 1D mixing scheme. An alternative approach that differs +above the boundary layer is described below. + +The simplest case is for a well-mixed boundary layer, where the +appropriate lengthscale is the boundary-layer depth (inversion +height). Therefore we set $z_{\rm turb}=z_h$, which is broadly +consistent with \cite{Malavelleetal2014}, and choose $\beta=\beta_{\rm + bl}=0.15$ to give the best match of our function to that of +\cite{Honnertetal2011}. These functions are shown in +Figure~\ref{fig-blend}(a) and are only dissimilar for small $\Delta +x$, where Eq.~\ref{eq-tanh} tends to zero faster. This is by choice, +to force the highest resolution simulations to use the 3D turbulence +scheme. +\begin{figure}[tbh] + \centering + \noindent\includegraphics[width=0.49\columnwidth]{honnert_vs_tanh.eps} + \noindent\includegraphics[width=0.49\columnwidth]{zturb_schem.eps} + \caption{(a) Weighting for the 1D boundary-layer scheme as a + function of $\Delta x/z_{\rm turb}$, showing the function of + Equation~\ref{eq-tanh} (blue solid), the equation in + \cite{Boutleetal2014} (black solid) and the TKE partitioning of + \cite{Honnertetal2011} (mean thick dashed, 5th/95th percentiles + thin dashed). (b) Schematic showing the calculation of $z_{\rm + turb}$ used in Eq.~\ref{eq-tanh} for a well-mixed layer (black + dotted) and a decoupled cloud layer (black solid).} + \label{fig-blend} +\end{figure} + +One of the key benefits of the \cite{lock00} scheme is its ability to +represent decoupled stratocumulus layers, and this is a feature which +needs to be maintained in the blended scheme. Physically they are +similar to well-mixed surface driven boundary layers, and the +\cite{lock00} scheme parametrizes them as such. The appropriate length +scale is now the decoupled cloud mixed layer depth, $z_{\rm sc}$ +\cite[i.e.~the depth through which a negatively buoyant parcel +released at cloud top would descend,][]{lock01}. In this case, below +the decoupled cloud top we set +\begin{equation}\label{zturb_dsc} + z_{\rm turb}=\min\left[\max\left(z,z_{\rm sml}\right),\max\left(z_{\rm sc},z_h-z\right)\right], +\end{equation} +where $z_{\rm sml}$ is the depth of the surface-based mixed layer +\cite[i.e.~the depth through which a positively buoyant parcel +released at the surface would ascend,][]{lock00}. This is shown +schematically in Figure~\ref{fig-blend}(b), and ensures that $W_{1D}$ +has a high value in the poorly resolved surface mixed layer and cloud +layer, and a lower value in between those layers. Again, this choice +of $z_{\rm turb}$ is broadly consistent with the analysis of decoupled +stratocumulus LES presented by \cite{Malavelleetal2014}. Finally, +\cite{Honnertetal2011} also included shallow cumulus simulations and +showed that the relevent length scale there was the cloud top height. Most +of the {\tt blending\_option} choices apply this to all regimes diagnosed +as cumulus-capped (see section~\ref{sec:types}) but alternatively +({\tt blending\_option}$=$4) this can be restricted to strictly shallow +cumulus clouds, defined as contiguously cloudy levels (cloud fraction +greater than SC\_CFTOL) with cloud top height below input parameter +{\tt shallow\_cu\_maxtop}. Note that the diagnosis of shallow cumulus +from the diagnosis parcel ascent (that was used to identify a cumulus regime) +was found frequently to indicate deep convection even when the resolved +clouds were shallow because the diagnosis parcel, being undilute, would +penetrate to the tropopause. However, having decided the regime is shallow +convection, we do still set $z_{\rm turb}$ to the diagnosis parcel top height +because, for current km-scale configurations (without a cumulus convection +parametrization), it was found that the resulting stronger parametrized +vertical mixing was beneficial for the development of the convection, and +that without this a widespread stratiform cloud layer could develop instead. + +Above the boundary layer top, \cite{Boutleetal2014} aimed for any free +atmospheric mixing to be done by the 3D Smagorinsky scheme. Therefore, above +the boundary layer top they use $z$ as the appropriate length scale, and in +general take $z_{\rm turb}$ in Eq.~\ref{eq-tanh} as the greater of that defined +by (\ref{zturb_dsc}) and $z$. However, this did not give a particularly +fast transition using the value of $\beta_{\rm bl}$, therefore they used +$\beta_{\rm fa}=1$ at a height well above the boundary layer +($z_{\rm fa}=z_h+1$~km), and transitioned between these regimes linearly using +\begin{equation} + \beta = \beta_{\rm bl}\frac{z_{\rm fa}-z}{z_{\rm fa}-z_h} + + \beta_{\rm fa}\frac{z-z_h}{z_{\rm fa}-z_h} +\end{equation} +However, because the above method still uses (\ref{eq-tanh}), which depends on +$z_{\rm turb}/\Delta x$, the rate of transition to 3D Smagorinsky with height +above the boundary layer varies in an undesirable way with grid size. It +might be considered more logical to think of non-turbulent regions of the free +troposphere as unresolved turbulence and so revert to the 1D mixing scheme +there. An alternative treatment({\tt blending\_option}$=$3 or 4), then, is to +increase $W_{1D}$ above the boundary layer top smoothly, to reach unity by +some physical height $z_{\rm fa}$, to be independent of both horizontal and +vertical grid sizes. For $z_{\rm turb} < z < z_{\rm fa}$, then, we set +\begin{equation} + W_{1D} = 1 + \frac{1}{2} \left( W_{1D}|_{z=z_{\rm turb}} - 1 \right) + \left[ 1 + {\rm cos}\left( \pi \, \frac{z-z_{\rm turb}}{z_{\rm fa}-z_{\rm turb}} + \right) \right] +\end{equation} +The cosine term in square brackets transitions smoothly from 2 at +$z=z_{\rm turb}$ to zero at $z_{fa}$ where, although somewhat +arbitrary, $z_{\rm fa} = {\rm min}(2 z_{\rm turb}, z_{\rm turb}+1 {\rm km})$. +The former term ensures the transition is well above any shallow boundary +layers while the latter that it does not drift far into the free atmosphere. +In addition, within any layers identified as turbulent, through having +subcritical $Ri$, $z_{\rm turb}$ is set to the layer depth, in the same way +as is done for decoupled stratocumulus in (\ref{zturb_dsc}). + +For current operational convection-permitting model grid sizes (1.5 km +in the UKV), the representation of cumulus convection remains a +challenge. One option is to include a grey-zone convection +parametrization, described in the documentation of that scheme +(see \citeumdp{027}). Tests in the UKV, though, showed some +detriment to the spin-up of resolved scale convection (as well as +somewhat poor discrimination of precipitating versus non-precipitating +parametrized convection) that led to the development of an alternative +strategy, namely to abandon the blended turbulence scheme when pure +cumulus convection was diagnosed and leave the representation of +cumulus entirely to the resolved scales. This option +({\tt blending\_option}$=$2) is also now discouraged. + +%------------------------------------------------------------------------ +% ENTRAINMENT +%------------------------------------------------------------------------ +\section{Entrainment fluxes} +\label{sec:entr} + +{\bf Summary}: parametrized entrainment fluxes (at the top of mixed +layers) are specified for momentum through an eddy-diffusivity, as +described in section~\ref{sec:ent_K}. For scalar variables, if the +inversion is sufficiently sharp so as to be unresolved, the ideal is +to specify the entrainment fluxes explicitly, as described in +section~\ref{sec:ent_flux}, based on the subgrid inversion diagnosis +described in section~\ref{sec:sginv}. Further details can be found in +\cite{lock01}. If the profiles are such that the inversion is sharp +but a subgrid inversion cannot be diagnosed, an eddy-diffusivity +similar to that for momentum is used (see section~\ref{sec:ent_K}). +If the inversion is thick enough to be resolved then an eddy +diffusivity profile is constructed across the inversion (see +section~\ref{sec:entr_prof}) for both scalars and momentum fields. +For tracer variables (scalars other than $\thetal$ and $q_t$), the +entrainment fluxes are specified using an equivalent eddy-diffusivity, +as described in section~\ref{sec:ent_K_flux}. Note that, as indicated +below, several aspects of the implementation of entrainment fluxes +were revised at the 9C scheme and these are documented separately. + +The parametrization of the entrainment rate, $w_e$ (given, in the +absence of subsidence, by the rate of rise of the inversion), can be +written (using the notation given in appendix~\ref{app:vscales}) +\begin{equation} + w_e = \frac{ A_1 \, \vsum / \zml + g \tilde{\beta_T} \tilde{\alpha_t} + \Delta_\radf } + {\Delta b + c_T \vsumo^2/\zml } + \label{we_parm} +\end{equation} +where $ \vsum = \vheat + \vrad + \vbr + A_2 u_*^3 $. The constant +$A_1$ is given a value 0.23, as in \cite{lock98}, and $A_1*A_2=5$, as +in \cite{driedonks1982}. To allow for weak inversions, the +\cite{zilitinkevich1975} correction is included in (\ref{we_parm}) +with the constant, $c_T=1$. A further parametrization for $\alpha_t$, +which is the fraction of the cloud-top radiative divergence +($\Delta_\radf $, in Kms$^{-1}$) that occurs across the +horizontally-averaged inversion in the LES, can be written +\begin{equation*} + \alpha_t = 1 - \exp{ \left\{-\Delta z_i / (2 L_{rad})\right\} } +\end{equation*} +where the thickness of the inversion is parametrized as $\Delta z_i = +\mbox{min}[\vsumo^2/\Delta b, 100]$ and $L_{rad}$ is a depth-scale for +the radiatively-cooled layer (taken to be 15 $\times +\,\mbox{max}[200/z_c, 1]$, where $z_c$ is the cloud depth). To allow +for a feedback with forcing of entrainment by buoyancy reversal (see +appendix~\ref{app:vscales}), $\tilde{\alpha_t} = \alpha_t+ Br +(1-\alpha_t) $. following \cite{lock98} and \cite{lock09:_factor}. +The calculation of the other quantities required for (\ref{we_parm}) +is described in appendix \ref{app:vscales}. At some point during the +transition to a decoupled boundary layer the surface-driven +entrainment terms (the terms in (\ref{we_parm}) proportional to +$\vheat$ and $u_*$) will no longer contribute to entrainment at cloud +top, because the two layers will have become entirely decoupled. If +the {\tt entr\_smooth\_dec} switch is on then the surface contribution +to the parametrized entrainment at \zhsc is decreased linearly as the +$\thetavl$ difference between NTDSC and NTML increases from 0.5 to 1K. +The flag, COUPLED, is set to true and \zhsc is used as the mixed-layer +depth in (\ref{we_parm}) as long as any surface-driven entrainment +remains. If the {\tt entr\_smooth\_dec} switch is off then this +transition is discontinuous at a $\thetavl$ difference of 0.5K. + +It should be noted that (\ref{we_parm}) takes no account of wind shear +anywhere other than at the surface. How to quantify the shear +generation of turbulence in DSC layers is not known. The direct +impact of shear across the inversion is thought to be simply to +diffuse the inversion in the vertical --- this wind shear will +contribute little to the mixed layer TKE and so can not contribute to +the full process of mixing across the inversion and down into the +mixed layer that is entrainment. However, important interactions +between wind shear across inversions and cloud-top radiative cooling +have been observed that are not yet accounted for in the UM. + +The least well-determined part of (\ref{we_parm}) is the constant +$A_2$ --- the constant in the Zilitinkevich correction, $c_T$, is also +approximate but is included to limit the growth of layers capped by +weak inversions and for numerical safety. A further limit is applied +to the value of $w_e$ determined by (\ref{we_parm}) such that the +inversion cannot rise by more than one grid-level in a timestep. With +current vertical resolutions and timesteps this is not a serious +restriction. The constants $A_1$ and $A_{\rm br}$ appeared to be +determined within 10-20 \% in \cite{lock98}, although only solid cloud +sheets were simulated (as discussed further in +appendix~\ref{app:vscales}). Similarly the parametrizations of +$\alpha_t$ and $\Delta z_i$ were found to be accurate but the +parameter $L_{rad}$ is currently only crudely represented in the UM. + +\subsection{Specification of entrainment fluxes in the 9B scheme} +\label{sec:ent_flux} +Note that the 9C scheme (see next section) differs by generalising the +approach to include all processes operating in the inversion +grid-level, rather than just radiation. + +If it is assumed that the turbulent fluxes reduce from their extremum +at $z=z_i$ (the `entrainment' fluxes) to zero at $z=h$ a small +distance above, then $ \wthlzi = - w_e \Delta \thetal + \radf|_h - +\radf|_{z_i}$, so that +\begin{eqnarray} + {\cal H}|_{z_i} & =& - w_e \Delta \thetal + \radfnet|_h \nonumber\\ + \wqtzi & =& - w_e \Delta q_t +\end{eqnarray} +\label{discinv} +where the total heat flux ${\cal H} = \wthl + \radfnet$ and $\radfnet += \radf - \radf|_{\zbaseq}$. The net radiative flux relative to the +base of the mixed layers is simply calculated as +\begin{equation*} + \radfnet|_{z_{k+\frac{1}{2}}} = \sum_{k=\nbdsc}^{k} \mbox{max}\left[ + - \Delta_{k+\frac{1}{2}} z \, {\cal S}_\radf(k), \,0 \right] +\end{equation*} +where NBDSC$=1$ in SMLs, ${\cal S}_\radf$ are the temperature +increments (in Ks$^{-1}$) from the radiation scheme and $\radfnet|_h$ +is estimated by extrapolating down from $\radf|_{z_{\ntml+\frac{3}{2}} +}$ using the flux-divergence in grid-level NTML$+2$ (and similarly for +DSC layers). + +The thermodynamic variables' entrainment fluxes, then, are imposed +nominally at the subgrid inversion height ($z_i=$ \zh and/or \zhsc), +diagnosed as described in section~\ref{sec:sginv}. The required +grid-level fluxes (at $z_{\ntdsc+\frac{1}{2}}$, for example) are then +estimated using linear interpolation of ${\cal H}$ and $\wqt$ between +\zhsc and the base of the mixed layer: +\begin{eqnarray} + \wthl|_{ z_{\ntdsc+\frac{1}{2}} } & =& \wthl|_{\zbaseq} + - \frac{ z'_{\ntdsc+\frac{1}{2}} }{\zml} + \left( \tilde{w_e} \Delta \thetal + \wthl|_{\zbaseq} - \radfnet|_{h} \right) + - \radfnet|_{ z_{\ntdsc+\frac{1}{2}} } \nonumber\\ + \wqt|_{ z_{\ntdsc+\frac{1}{2}} } & =& \wqt|_{\zbaseq} + - \frac{ z'_{\ntdsc+\frac{1}{2}} }{\zml} + \left( \tilde{w_e} \Delta q_t + \wqt|_{\zbaseq} \right) +\end{eqnarray} +\label{fluxinterp} +where $z' = z-\zbaseq$, and similarly for the SML entrainment fluxes +(at $z=z_{\ntml+\frac{1}{2}}$). The turbulent fluxes at the base of +the mixed layer are assumed zero except for the SML where the surface +fluxes are used. This interpolation is illustrated for a SML in +Fig.~\ref{fig:fluxinterp}. +\begin{figure}[tbh] + \centering + \scalebox{1.0}{\includegraphics{subsent_fig7}} + \captionarb{Idealised profiles of (a) $\wqt$ (dash-dotted line) and + (b) ${\cal H}$ (dotted line), $\wthl$ (dash-dotted) and $F$ + (dashed). The continuous lines are the turbulent fluxes on the + model grid indicated by the dashed horizontal lines.} + \label{fig:fluxinterp} +\end{figure} +Note that, because (\ref{fluxinterp}) includes an explicit balance +between the turbulent and radiative fluxes for $\wthl$, it is not +possible to parametrize the entrainment fluxes through a single $K_h$ +for both $\wthl$ and $\wqt$. Furthermore, the radiative forcing of +turbulence in the mixed layer is fixed through the timestep and so it +is consistent to assume the entrainment fluxes (at $z_i$) are also +fixed. Hence (\ref{fluxinterp}) are implemented explicitly, rather +than via an eddy-diffusivity. This is discussed further, with +reference to tracer fluxes, in section~\ref{sec:ent_K_flux}. + +In order to allow for the long timesteps used in NWP and to facilitate +movement of the subgrid inversion across grid-levels within a +timestep, the parametrization of $w_e$ and the model's subsidence +velocity, $w_S|_{z_i}$, are used to calculate $z_i$ at the next +time-level ($z_i^{n+1}$). Currently, the latter is found by linear +interpolation to $z_i$ and both are assumed constant in time. If +$z_i^{n+1} < z_{\ntdsc +\frac{1}{2}}$, then the entrainment fluxes +there (given by (\ref{fluxinterp})) are multiplied by the fraction of +the timestep that $z_i$ was above this grid-level, namely +$(z_i-z_{\ntdsc +\frac{1}{2}})/(z_i - z_i^{n+1}) $. The full +entrainment flux at grid-level NTDSC$-\frac{1}{2}$ must then also be +specified, given by (\ref{fluxinterp}) with $z_{\ntdsc + \frac{1}{2}}$ +replaced by $z_{\ntdsc - \frac{1}{2}}$. If $z_i$ rises above +$z_{\ntdsc+\frac{3}{2}}$, the entrainment flux is specified only at +this higher grid-level (multiplied by the fraction of the timestep +that $z_i$ is above this half-level) and the values of the mixed-layer +$K$ profiles are used in half-level NTDSC$+\frac{1}{2}$ (these will be +non-zero because $z_i>z_{\ntdsc + \frac{1}{2}}$). Wherever the +entrainment fluxes are specified explicitly, the eddy-diffusivities +(both non-local and local) are set to zero. Also, the mean value of +$z_i$ during the timestep is used in (\ref{fluxinterp}) in order best +to approximate the mean flux gradient across the mixed layer. + +Finally, the entrainment flux is adjusted to allow for numerical +entrainment arising from the model's resolved vertical advection (as +discussed in \cite{lock01}). This is performed at whichever +grid-level the entrainment fluxes are specified, to allow for any +entrainment implied by a $\thetal$ subsidence increment, $\Theta^{\rm + S}$ (Ks$^{-1}$), at the model grid-level below. The subsidence +increments could be obtained directly in the SCM but in the full 3D UM +advection increments are dominated by the horizontal component. The +subsidence increments are calculated, therefore, from the vertical +velocity field using first order upwind advection (it would clearly be +preferable to use the model's actual vertical advection algorithm in +the GCM although the errors incurred in this diagnostic calculation +should not be very significant). The interpolated entrainment fluxes +given by (\ref{fluxinterp}) are therefore calculated not using $w_e$ +but using an entrainment velocity, $\tilde{w_e}$, that is reduced to +allow for any subsidence increments applied to the grid-level below +the entrainment flux. To take the case of $z_{\ntdsc+\frac{1}{2}} < +z_i^{n+1} < z_{\ntdsc+\frac{3}{2}}$ as an example, this reduced +entrainment velocity is given by +\begin{equation*} + \tilde{w_e} = w_e + \tilde{w_S} +\end{equation*} +with $\tilde{w_e}$ constrained to lie between 0 and $w_e$ and +\begin{equation} + \tilde{w_S} = - \, \frac{ \Theta^{\rm S}_{\ntml} + ( \Delta_{\ntml+\frac{1}{2}} z ) } + { \Delta \thetal } + \label{we_num} +\end{equation} + +\subsubsection{Diagnosis of a sub-grid inversion} +\label{sec:sginv} + +The profile of $\thetavl$ is used to diagnose the height of a +discontinuous inversion because it is approximately conserved under +adiabatic vertical motion and is equal to the virtual potential +temperature, $\theta_v$, in the absence of cloud. This should ensure +it is monotonically increasing with height in the statically stable +free-troposphere of a GCM. If $\thetavl$ does not increase +monotonically between grid-levels NTML and NTML+2 (or NTDSC and +NTDSC+2), then entrainment fluxes are simply specified via +(\ref{khent}) and none of the coupling with subsidence or radiation +described above is attempted (the local scheme is also currently not +set to zero above NTML or NTDSC when this occurs to allow it to +diffuse out this static instability). + +\begin{figure}[tbh] + \centering + \scalebox{1.0}{\includegraphics{nbldoc_zidiag}} + \captionarb{Schematic illustrating the assumptions behind the + subgrid diagnosis of $z_i$.} + \label{zi_diag} +\end{figure} +Having identified the model grid-level at the top of the well-mixed +layer (either level NTML from the parcel ascent, as described in +section~\ref{sec:adiapar}, or NTDSC for DSC layers, see section +\ref{sec:decouple}--- the analysis is the same for both), the +grid-level above is designated the inversion level within which the +diagnosis of a subgrid $z_i$ will be made. It is assumed that +$\thetavl$ in grid-level NTML$+1$ represents a cell-average value. +Thus, $z_i$ can be calculated by assuming that the integral of +$\thetavl$ over grid-level NTML$+1$ for the model and for a profile +with a discontinuous inversion at $z_i$ are equal, as illustrated by +the hatched areas in Fig.~\ref{zi_diag}. To calculate the integral of +the discontinuous profile, the lapse rate of $\thetavl$ between +grid-levels NTML$-1$ and $NTML$, $\gammaml$, is extended up to $z_i$, +while the stable lapse in the free atmosphere, between grid-levels +NTML$+2$ and NTML$+3$, $\gammafa$, is extrapolated down. Equating +these areas gives a quadratic equation in $\Delta z_{disc} = +z_{\ntml+\frac{3}{2}} - z_i$ which can be written +\begin{equation} + a (\Delta z_{disc})^2 + b \ \Delta z_{disc} +c =0 +\label{zi_interp} +\end{equation} +The coefficients are given by +\begin{eqnarray*} + a & =& 0.5 (\gammafa - \gammaml) \\ + b & =& - \left( {\thetavl}_{\ntml+2} + - \gammafa (z_{\ntml+2}-z_{\ntml+\frac{3}{2}}) \right) + + \left( {\thetavl}_{\ntml} + + \gammaml (z_{\ntml+\frac{3}{2}}-z_{\ntml}) \right) \\ + c & =& (z_{\ntml+\frac{3}{2}}-z_{\ntml+\frac{1}{2}}) + \left( {\thetavl}_{\ntml+1} - + \left( {\thetavl}_{\ntml} + + \gammaml \left( + \frac{1}{2}(z_{\ntml+\frac{1}{2}}+z_{\ntml+\frac{3}{2}}) + -z_{\ntml} \right) \right) + \right) \\ +\end{eqnarray*} + +Clearly, care must be taken to ensure that $z_i$ is not only +well-defined but also sensible (for example, as a rising inversion +encounters more or less stable regions above). If $b>0$ this suggests +the estimated lapse rates are inappropriate and these are therefore +set to zero and (\ref{zi_interp}) is recalculated. The case $c<0$ +suggests the grid-level designated as the inversion level should have +been considered as part of the mixed layer and so $z_i$ is set to be +fractionally below $z_{\ntml+\frac{3}{2}}$ (\mbox{i.e.}, as high as +possible without attempting to diagnose a subgrid $z_i$ in grid-level +NTML$+2$). If $b^2-4ac<0$ the quadratic equation has no real roots. +In this instance $z_i$ is set to fractionally below +$z_{\ntml+\frac{1}{2}}$ and NTML (and therefore the eddy-diffusivity +profiles) is lowered by a grid-level. In all other circumstances, the +required root is then $\Delta z_{disc} = (-b - (b^2-4ac)^{1/2} +)/(2a)$; the other root will either be larger or negative (if $a<0$). + +In addition, from variations seen in $z_i$ during single-column model +simulations, the error in $\Delta z_{disc}$ is estimated to be around +10\% of the vertical resolution, $\Delta_{\ntml+\frac{3}{2}} z$. +Accordingly, if $z_i$ is diagnosed as being less than +$z_{\ntml+\frac{1}{2}} + 0.1 \, \Delta_{\ntml+\frac{3}{2}} z$, NTML is +lowered a grid-level and $z_i$ is set fractionally below +$z_{\ntml+\frac{1}{2}}$. This small distance below the grid-level is +taken to be $(\Delta t/2) \times 10^{-4}$ so that, were a small rate +of rise of $z_i$ (of $10^{-4}$ ms$^{-1}$, say) to be diagnosed, then +$z_i$ would spend at least half the timestep (of length $\Delta t$) in +the next grid-level up. The specified fluxes would then contribute +significantly to that grid-level's evolution. Conversely, if $z_i$ is +subsiding, this technique allows the inversion to drop down a +grid-level without requiring this to be detected by the initial parcel +ascent. + +Having calculated $z_i$, the discontinuous jumps in $\thetal$ and +$q_t$ that are used in the entrainment calculation are calculated from +similar integral assumptions: +\begin{equation} + \Delta \chi = \left( {\chi}_{\ntml+1} - {\chi}_{\ntml} \right) \, + \frac{ z_{\ntml+\frac{3}{2}} - z_{\ntml+\frac{1}{2}} } + { z_{\ntml+\frac{3}{2}} - z_i } +\label{dqt_disc} +\end{equation} +with $\chi = \thetal$ and $q_t$. Note that the lapse rate above the +inversion has been ignored as there is no guarantee of monotonicity in +$q_t$ in the atmosphere above the inversion. In addition, +(\ref{dqt_disc}) will become increasingly inaccurate as $z_i$ tends to +$z_{\ntml+\frac{3}{2}}$ (and so ${\chi}_{\ntml+1}$ approaches +${\chi}_{\ntml}$). Consequently, if the fraction on the right hand +side of (\ref{dqt_disc}) is greater than 10, double grid-level jumps +are used (\mbox{i.e.}, $\Delta \chi = {\chi}_{\ntml+2} - +{\chi}_{\ntml} $). Finally, note that (\ref{dqt_disc}) implicitly +assumes the structure of the $\thetal$ and $q_t$ profiles across the +inversion grid-level are consistent with the diagnosed $z_i$. This is +very unlikely to be the case, for example, when running from an +analysis so the 9C scheme uses what has been found to be a more robust +algorithm, see the separate documentation. + +It would clearly be advantageous to pass knowledge of this subgrid +inversion structure to other parametrizations in the UM, particularly +the cloud scheme as currently the cloud fraction in level NTML$+1$ is +essentially meaningless (being diagnosed from a mixture of cloudy +boundary layer air and typically very dry free tropospheric air). + +\subsection{Specification of entrainment fluxes across sharp inversions in the 9C scheme} +\label{sec:ent_flux_9c} + +As described in section~\ref{sec:ent_flux}, when the capping inversion +is thinner than the model vertical grid it is important for the +entrainment flux implementation that the subsidence increments are +realistically and consistently distributed between the inversion +grid-level and the mixed layer. Rather than work with the increments +themselves, though, a more robust solution is to couple the subsidence +and turbulent fluxes across the inversion, exactly analogously to the +coupling of turbulent and radiative fluxes. This allows the total +tendency of the inversion grid-level to be linked to whether the +inversion should be rising or falling (determined from the balance +between the parametrized entrainment rate, $w_e$, and the large-scale +vertical velocity evaluated at the inversion, $w|_{z_h}$). + +\begin{figure}[p] + \centering + \scalebox{1}{\includegraphics{ideal_revflux}} + \caption{Subgrid (lines) and model (symbols) profiles and fluxes of, + top row, $q_t$ and, bottom row, $\thetal$: turbulent fluxes + (dash-dotted, crosses), subsidence fluxes (dotted, diamonds), + radiative flux (dashed, triangles) and total flux (solid, + squares).} + \label{fig:rev_fluxes} +\end{figure} +An idealised subgrid total flux profile is constructed from the +parametrized entrainment flux and the increments from radiation, +precipitation and subsidence, assuming a well-mixed boundary layer +capped by a diagnosed subgrid inversion. The crucial step is to +ensure that the total flux on the model entrainment grid-level equals +the idealised total flux profile interpolated to that level. Consider +the example illustrated in Fig.~\ref{fig:rev_fluxes} of a well-mixed +boundary layer up to $\theta$-level $\ntml$. For the subgrid $q_t$ +profiles, the turbulent flux divergence generates a moistening across +the inversion while subsidence generates drying. For this example it +has been assumed the entrainment rate is slightly larger than the +subsidence velocity at the inversion and so overall there is a weak +moistening relative to the mixed layer (the total flux gradient is +more negative across the inversion than in the mixed layer), +consistent with the rising tendency of the inversion. For the model, +the subsidence flux-divergence associated with the inversion is split +across levels $\ntml$ and $\ntml+1$. To keep the {\em net} moistening +of the model's boundary layer and inversion consistent with the total +subgrid flux profile, the model's entrainment flux at $\ntml+1/2$ +(shown by the cross in Fig.~\ref{fig:rev_fluxes}) must be found by +subtracting the subsidence flux at $\ntml+1/2$ (diamond) from the +total flux interpolated to $\ntml+1/2$ (square). Exactly the same +arguments follow for the $\thetal$ fluxes except that the situation is +complicated by the addition of the radiative flux. + +The above arguments can be generalised as follows. Writing $\fxtot$ +as the total flux of a conserved variable $\chi$ ($=q_t$ or $\thetal$) +and $\fxntp$ as the flux from physics sources other than turbulence +(\mbox{i.e.}, radiation, $\fx^{rad}$, in the above examples, but +including precipitation fluxes, $\fx^{ppn}$, in the full model) and +$\fx^{subs}$ as the flux from resolved scale subsidence, the total +flux at the subgrid inversion height is given by: +\begin{equation} + \fxtot|_{z_h} = - w_e \Delta \chi + \fxntp|_{z_t} + \fx^{subs}|_{z_h} + \label{fxtot_zi} +\end{equation} +As in section~\ref{sec:ent_flux}, (\ref{fxtot_zi}) is derived by +integrating the conservation equation for $\chi$ over an inversion in +which jumps occur over a thin layer with base at a height $z_h$ and +top at $z_t$ (in the UM, the inversion is assumed to be +infinitesimally thin so that $z_t=z_h$). This integration gives $ - +w_e \Delta \chi = \wx|_{z_h} -(\fxntp|_{z_t}-\fxntp|_{z_h})$. +\cite{lock99} related the non-turbulent flux divergence, +$\fxntp|_{z_t}-\fxntp|_{z_h}$, to radiative cooling occurring within +undulations of the cloudy boundary layer top. Similar considerations +need to be borne in mind when calculating all the non-turbulent fluxes +in (\ref{fxtot_zi}). First, the radiative flux is extrapolated down +from $\ntml+\frac{3}{2}$ to $z=z_t$ using the divergence in the +grid-level above the inversion as representative of the +free-atmospheric divergence. Second, since the microphysical flux is +generated within the cloud, $\fx^{ppn}|_{z_t} = +{\fx}^{ppn}_{\ntml+\frac{3}{2}}$. Finally, the subsidence +flux-divergence across level $\ntml$ and $\ntml+1$ is assumed to be +associated with the inversion so ${\fx}^{Subs}|_{z_h} = +{\fx}^{Subs}_{\ntml-\frac{1}{2}}$. Thus, the finite-difference form +of (\ref{fxtot_zi}) becomes \beqn \fxtot|_{z_h} = - w_e \Delta \chi + +\fx^{rad}|_{z_t} + {\fx}^{ppn}_{\ntml+\frac{3}{2}} + +{\fx}^{subs}_{\ntml-\frac{1}{2}} +\label{fxtot_zi_fd} +\eeqn + +Then, assuming a linear profile of $\fxtot$ in the mixed layer, +interpolating the total flux to the inversion flux grid-level gives +\begin{equation} + \fxtot|_{ \ntml+\frac{1}{2} } = \fxtot|_{\zbaseq} + + \frac{ z'_{\ntml+\frac{1}{2}} }{\zml} + \left( \fxtot|_{z_h} - \fxtot|_{\zbaseq} \right) + \label{fxtot_interp} +\end{equation} +where $z'$ ($=z-\zbaseq$) is height above the base of the mixed layer +at $z=\zbaseq$. Finally, the grid-level turbulent entrainment flux is +given by: +\begin{equation} + \wx|_{ \ntml+\frac{1}{2} } = \fxtot|_{ \ntml+\frac{1}{2} } + - \fxnt|_{ \ntml+\frac{1}{2} } + \label{rev_entflux} +\end{equation} +This revised algorithm has several advantages over the previous. +Firstly, the fluxes for $q_t$ and $\thetal$ are coupled independently, +whereas in the 9B version the coupling with subsidence was estimated +using only the $\thetal$ increments in order to calculate +$\tilde{w_e}$ in \ref{we_num}. Secondly, this method makes it much +simpler to include all processes, and precipitation in particular, in +a consistent manner. Thirdly, since the total grid-level flux, +$\fxtot|_{\ntml+\frac{1}{2}}$ in (\ref{fxtot_interp}), is used to +calculate the entrainment fluxes, it is straightforward to ensure that +the net budget of the inversion grid-level, namely $- ( +\fxtot|_{\ntml+\frac{3}{2}} - \fxtot|_{\ntml+\frac{1}{2}})/\Delta z $, +is consistent with the entrainment/subsidence balance. In other +words, to use $\thetal$ as an example, if the inversion is rising +(falling) then $\fxtot|_{\ntml+\frac{1}{2}}$ is limited to ensure that +the inversion grid-level will cool (warm). Finally, if the inversion +is rising we don't want the inversion grid-level $\thetal$ to cool to +less than $\thetal$ of the mixed layer by the end of the timestep. In +other words, for $\chi=\thetal$, given +\begin{eqnarray*} + \chi_{\ntml+1}^{n+1} & =& \chi_{\ntml+1}^{n} + - \frac{\Delta t}{\Delta z} \left( + \fxtot|_{ \ntml+\frac{3}{2} } - \fxtot|_{ \ntml+\frac{1}{2} } + \right) \\ + \chi_{\ntml}^{n+1} & =& \chi_{\ntml}^{n} + - \frac{\Delta t}{z_{\ntml+\frac{1}{2}}} \left( + \fxtot|_{ \ntml+\frac{1}{2} } - \fxtot|_{\zbaseq} + \right) \\ +\end{eqnarray*} +where the superscripts $n$ and $n+1$ refer to the model timestep, +although strictly speaking $n+1$ refers to fields after the boundary +layer implicit solver. Requiring that +$\chi_{\ntml+1}^{n+1}\geq\chi_{\ntml}^{n+1}$ implies +\begin{equation} + \fxtot|_{ \ntml+\frac{1}{2} } + \left( 1+ \frac{\Delta z}{z_{ml}}\right) + \geq \fxtot|_{ \ntml+\frac{3}{2} } + \Delta z \left( + \frac{\chi_{\ntml}^{n}-\chi_{\ntml+1}^{n}}{\Delta t} + + \frac{\fxtot|_{\zbaseq}}{z_{ml}} \right) +\end{equation} +The same arguments apply for $q_t$, noting that the free atmosphere +can be drier or moister than the mixed layer and so these cases must +be treated separately. If $\thetal$ of the free atmosphere is colder +than the mixed layer then no subgrid inversion treatment is attempted +and entrainment is modelled using a straightforward eddy diffusivity. + +\subsubsection{Calculation of the inversion jumps in the 9C scheme} + +In the 9B scheme, the discontinuous jumps in $\thetal$ and $q_t$ that +are used in the entrainment calculation were calculated from integral +assumptions similar to those used to diagnose the subgrid inversion +height, $z_h$, and were given by (\ref{dqt_disc}). Note that +${\chi}_{\ntml+2}$ does not appear in (\ref{dqt_disc}) and so no +direct information from the free atmosphere is used. Only if the +budgets of $\thetal$ and $q_t$ in level $\ntml+1$ are entirely +consistent with the rise and fall of the subgrid inversion will +(\ref{dqt_disc}) give accurate results. This will not be the case +during an assimilation cycle, for example, neither is it likely to be +the case if the convection scheme is detraining into level $\ntml+1$. + +Instead, a more robust algorithm is used in the 9C scheme and the +subgrid inversion calculation is only attempted where both $\thetal$ +and $\thetavl$ are monotonically increasing and $q_t$ is simply +monotonic across the inversion. The formula used is: +\begin{equation} + \Delta \chi = {\chi}_{\ntml+2} - {\chi}_{\ntml} + - \gamma_{\chi} \left( z_{\ntml+2} - z_h \right) +\label{dqt_disc_9c} +\end{equation} +subject to the constraint that the lapse rate adjustment should not +reduce the two grid-length difference by more than half. The +free-atmospheric lapse rates are given by + +\begin{eqnarray*} + \gamma_{\thetal} & =& {\rm max}\left[ \, 0, \, \f{ {\thetal}_{\ntml+3}-{\thetal}_{\ntml+2} } + { z_{\ntml+3} - z_{\ntml+2} } + \right] \\ + \gamma_{q_t} & =& {\rm min}\left[ \, 0, \, \f{ {q_t}_{\ntml+3}-{q_t}_{\ntml+2} } + { z_{\ntml+3} - z_{\ntml+2} } + \right] +\end{eqnarray*} + +\subsection{Calculation of the subsidence flux} +\label{sec:subs_calc} + +The vertical advection or subsidence flux, ${\fx}^{subs}$, is +calculated by integrating estimates of the vertical advection +increments. These estimates are made at 9B from the model's vertical +velocity field, $w$, using first order upwind advection. As described +above, however, the coupling between different flux profiles is +performed on the model grid and, over land, these coordinate surfaces +follow the underlying terrain. To correct this, the 9C scheme +calculates the subsidence flux in grid-point, rather than physical +space, by using $\dot{\eta}$ (where $\eta$ is the model's vertical +coordinate) rather than $w$. + +The following two examples illustrate why this represents an +improvement. First, consider a boundary layer capped by a horizontal +inversion in a horizontal flow over a rising land surface. Here $w$ +will be zero and yet the model will be generating a vertical advection +flux across the inversion grid-levels, because $\dot{\eta}$ is +negative. Conversely, consider the same boundary layer but in a flow +that follows the coordinate surfaces, going up and over a hill. Now +there will be no vertical advection flux across the model's inversion +grid-level because $\dot{\eta}$ is zero and yet $w$ will be negative +on the down-slope thus giving a spurious subsidence source to the 9B +scheme. + + +\subsection{Specification of entrainment eddy diffusivity} + +As discussed above it is considered beneficial to specify the +thermodynamic entrainment fluxes explicitly under the assumption that +both the turbulence forcing and the inversion jumps change slowly +compared to the timestep. Under the circumstance that no subgrid +inversion can be diagnosed, not only is an alternative derivation of +the entrainment fluxes required, but it is also deemed likely that +these assumptions may be violated and so the entrainment fluxes are +specified via an entrainment eddy diffusivity. Currently, this is +also the case for momentum and tracer variables. + +\subsubsection{For momentum (and scalars if no subgrid inversion)} +\label{sec:ent_K} + +For momentum, and scalars if a subgrid inversion cannot be diagnosed, +see section~\ref{sec:sginv}, fluxes at the mixed layer top are +specified through an eddy diffusivity which is given by +\begin{eqnarray} + K_h|_{\ntml+\frac{1}{2}} & =& w_e \Delta_{\ntml+1} z \nonumber\\ + K_m|_{\ntml} & =& Pr \, w_e \Delta_{\ntml+\frac{1}{2}} z +\end{eqnarray} +\label{khent} +noting the Charney-Philips grid implying stresses are staggered from +scalar fluxes. The Prandtl number, $Pr$, takes the same form as for +the non-local $K$ profiles, see section~\ref{sec:nonlocal}. + +Substituting (\ref{khent}) in (\ref{scal_closure}) gives, for example, +$\wthl|_{\ntml+\frac{1}{2}} = - w_e \Delta_{\ntml+1} \thetal$. Note +that this gives entrainment buoyancy fluxes identical to +(\ref{discinv}) as long as there is no buoyancy reversal generation of +turbulence (\mbox{i.e.},$\vbro=0$) and if variations in the grid-level +jumps across the timestep are ignored. The former is because the +other terms in (\ref{we_parm}) are inversely proportional to $\Delta +b$. The latter will never actually be true and can give rise to large +errors if the inversion is rising quickly. Therefore, the +thermodynamic entrainment fluxes are specified explicitly where +possible. + +The advantages of diagnosing the subgrid inversion are that it allows +consistency between the turbulent and radiative fluxes and large-scale +vertical advection, it reduces grid-resolution errors arising from the +mixed layer depth calculation and it allows a more accurate +calculation of $\vbro$ and $\alpha_t$. For momentum, because the +jumps across inversions are typically small and variable, it seems +unwise numerically to attempt to specify the inversion stresses +explicitly and so (\ref{khent}) is always used. For the 9C +version, the entrainment $K_m$ given by (\ref{khent}) is imposed at +the height of the temperature inversion \zh (either subgrid or at +$z_{\ntml+\frac{1}{2}}$) and $K_m|_{\ntml+\frac{1}{2}}$ is calculated +from (\ref{kmsurf}) and (\ref{kmtop}), noting the use of the ${\cal + E}$ factors. + +\subsubsection{Resolved inversions} +\label{sec:entr_prof} +An inversion is defined as being resolved when it extends above the +flux-level above the usual entrainment interface level (see +section~\ref{sec:dzi}), \mbox{i.e.} when +\begin{equation*} + z_{\ntml+\frac{1}{2}} + \Delta z_i > z_{\ntml+\frac{3}{2}} +\end{equation*} +When this happens, there is no subgrid inversion diagnosis and the +entrainment parametrization follows the methodology given in +section~\ref{sec:ent_K} to give $K_h|_{\ntml+\frac{1}{2}}$. The +diffusion coefficient profile within the inversion is then calculated +assuming the $\thetavl$ flux profile within the inversion decreases +following a cosine shape from the standard parametrized entrainment +flux at the inversion base to zero at the inversion top, \mbox{i.e.}: +\begin{equation} + \overline{w'\thetavl'} = \overline{w'\thetavl'}|_{\ntml+\frac{1}{2}} + cos\left(\pi \frac{z'}{2} \right) + \label{ent_svl} +\end{equation} +where $z'=(z-\zhe)/\Delta z_i$ is scaled height within the inversion. +This flux profile is then converted into a diffusion coefficient +profile by inverting the standard flux parametrization: +\begin{equation*} + K_h|_{k+\frac{1}{2}}= - \, \frac{\overline{w'\thetavl'} } + { ({\thetavl}_{k+1}-{\thetavl}_{k})/(z_{k+1}-z_k) } +\end{equation*} +The diffusion coefficient for momentum entrainment is calculated in +the same way, allowing for the staggered grid, with the same $Pr$ as +in (\ref{khent}). + +\subsubsection{For tracers, when there is a subgrid inversion} +\label{sec:ent_K_flux} +Here `tracers' refers to scalar variables other than $\thetal$ and +$q_t$: aerosols, $q_f$, etc. Ideally, tracer entrainment fluxes would +be specified explicitly in the same way as for $\thetal$ and $q_t$. +However, specifying the entrainment flux effectively specifies the net +change in mixed-layer tracer concentration across the timestep. Thus, +if the mixed-layer tracer concentration is small at the start of a +timestep and the entrainment flux is larger than the surface flux, the +mixed-layer concentration could go negative (and tests indicated that +this did indeed happen). Specifying the entrainment flux assumes that +both the turbulence forcing and the inversion jump change slowly +compared to the timestep. Whilst this is true for atmospheric +$\thetal$ and $q_t$, the latter is not true for tracers with a small +boundary layer concentration. Consequently, for a tracer field +$\chi$, the parametrized entrainment fluxes $\overline{w'\chi'}_{ + z_{\ntml+\frac{1}{2}} }$ are calculated from (\ref{fluxinterp}) but +are implemented through an equivalent entrainment eddy-diffusivity +given by: +\begin{equation} + K_{\chi}|_{\ntml+\frac{1}{2}} = - \overline{w'\chi'}_{ z_{\ntml+\frac{1}{2}} } + \frac{\Delta_{\ntml+1} z}{\Delta_{\ntml+1} \chi} + \label{K_ent_tracer} +\end{equation} +Note from (\ref{scal_closure}) that (\ref{K_ent_tracer}) gives the +parametrized flux if $\Delta_{\ntml+1} \chi$ does not change across +the timestep (see section~\ref{sec:implicit} for a description of the +implicit numerical solution of (\ref{cons_eqn_scal})). As +(\ref{K_ent_tracer}) involves the potentially numerically dangerous +calculation of $\Delta \chi/\Delta_{\ntml+1} \chi$ (where $\Delta +\chi$ is the subgrid inversion jump, given by (\ref{dqt_disc})), the +following constraints are also ensured: +\begin{equation*} + 0 \leq K_{\chi}|_{\ntml+\frac{1}{2}} + \leq 10 \,K_{\chi}|_{\ntml-\frac{1}{2}} +\end{equation*} + +%------------------------------------------------------------------------ +\section{Surface Exchange} + +Note that the surface scheme itself is documented under the JULES +documentation. + +\subsection{The theoretical basis.}\label{section_1} +Making the assumption that \textbf{Monin-Obukhov similarity theory} +for the surface layer is valid the gradients of model variables in the +surface layer are related to the surface fluxes by: +\begin{eqnarray} + \frac{\partial T}{\partial z} + \frac{g}{ c_P }&=&-\frac{ H_0 }{ c_P \rho _0 v_\ast } \frac{ \phi _h (z/L)}{kz}\label{1.1.1}\\ + \frac{\partial q}{\partial z}&=&-\frac{ E_0 }{ \rho _0 v_\ast } \frac{ \phi _h (z/L)}{kz}\label{1.1.2}\\ + \frac{\partial {\rm {\bf v}}}{\partial z}&=&\frac{ {\rm {\bf \tau }}_{0} }{ \rho _0 v_\ast } \frac{ \phi _m (z/L)}{kz},\label{1.1.3} +\end{eqnarray} +where subscript 0 represents a surface value and subscript * +represents a surface layer scaling quantity. $\phi _{m}$ and $\phi +_{h}$ are the Monin-Obukhov stability functions (for the form of these +see section~\ref{section_1.3} below). $L$ is the Monin-Obukhov length +scale defined by +\begin{equation} + L = \frac{- { v_\ast }^3 }{k F_{B0} / \rho _0 }, + \label{1.1.4} +\end{equation} +where F$_{B0}$ is the surface buoyancy flux defined by +\begin{equation} + F_{B0} = \frac{ g }{ c_P } \beta _{T1} H_0 + g \beta _{q1} E_0. + \label{1.1.5} +\end{equation} +The buoyancy coefficients in equation~(\ref{1.1.5}) are given in +appendix~\ref{app:buoyp} with the subscript 1 denoting a value at the +lowest level in the atmosphere model. + +Equations~(\ref{1.1.1})--(\ref{1.1.3}) can be integrated from the +``surface'', i.e. the roughness height where the surface variables are +defined, to a reference height in the surface layer, for modelling +applications, the height, z$_{1}$, of the bottom model layer above the +surface. The resulting expressions for the surface turbulent fluxes +are: +\begin{eqnarray} + \frac{ H_0 }{ c_P \rho _0 }&=&-\frac{ c_H }{ c_D^{1/2} } v_\ast \left( {\Delta T + \frac{g}{c_p }( z_1 + z_{0m} - z_{0h} )} \right)\label{1.1.7}\\ + \frac{ E_0 }{ \rho _0 }&=&-\frac{ c_H }{ c_D^{1/2} } v_\ast \Delta q\label{1.1.8}\\ + \frac{ {\bf \tau }_{0} }{ \rho _{0} }&=& c_D^{1/2} v_\ast \Delta {\rm {\bf v}},\label{1.1.9} +\end{eqnarray} +where $\Delta $X=X$_{1}$-X$_{0}$. From~(\ref{1.1.7}) and~(\ref{1.1.8}) +the surface buoyancy flux in definition~(\ref{1.1.4}) is +\begin{equation} + \frac{ F_{B0} }{ \rho _0 } = -\frac{ c_H }{ c_D^{1/2} } v_\ast \Delta B, + \label{1.1.10} +\end{equation} +\begin{equation} + \Delta B = g \beta _{T1} \left( {\Delta T + \frac{g}{ c_P }( z_1 + z_{0m} - z_{0h} )} \right) + + g \beta _{q1} \Delta q + \label{1.1.11} +\end{equation} +The \textbf{surface exchange coefficients }in +equations~(\ref{1.1.7})--(\ref{1.1.9}), c$_{D}$ and c$_{H}$, are given +by +\begin{eqnarray} + c_D^{1/2}&=&\frac{k}{ \Phi _m (L , z_1 + z_{0m} , z_{0m} )} + \label{1.1.12}\\ + \frac{ c_H }{ c_D^{1/2} }&=&\frac{k}{ \Phi _h (L , z_1 + z_{0m} , z_{0h} )}, + \label{1.1.13} +\end{eqnarray} +where +\begin{eqnarray} + \Phi _m (L , z_1 + z_{0m} , z_{0m} )&=& \int \limits_{ z_{0m} /L}^{( z_1 + z_{0m} )/L} \frac{ \phi _m (\zeta )}{\zeta } d\zeta\label{1.1.14}\\ + \Phi _h (L , z_1 + z_{0m} , z_{0h} )&=& \int \limits_{ z_{0h} /L}^{( z_1 + z_{0m} )/L} \frac{ \phi _h (\zeta )}{\zeta } d\zeta\label{1.1.15}, +\end{eqnarray} +z$_{0m}$ and z$_{0h}$ are the \textbf{surface roughness lengths }for +momentum and scalars respectively. + +The equations for the \textbf{surface turbulent fluxes}, +(\ref{1.1.7})--(\ref{1.1.9}), can be written in the forms +\begin{eqnarray} + \frac{ H_0 }{ c_P \rho _0 }&=&{-c}_H V \left( {\Delta T + \frac{g}{c_p }( z_1 + z_{0m} - z_{0h} )} \right)\nonumber\\ + &=& - C_H \left( {\Delta T + \frac{g}{c_p } ( z_1 + z_{0m} - z_{0h} )} \right)\label{1.1.16}\\ + \frac{ E_0 }{ \rho _0 }&=&- c_H V \Delta q = - C_H \Delta q\label{1.1.17}\\ + \frac{ {\rm {\bf \tau }}_{0} }{ \rho _{0} }&=& c_D V \Delta {\rm {\bf v}}{ }= C_D \Delta {\rm {\bf v}},\label{1.1.18} +\end{eqnarray} +where the effective wind speed for surface turbulent exchanges, $V$, +is defined by +\begin{equation} + V = \frac{ v_\ast }{ c_D^{1/2} } = \frac{ v_\ast ^2 }{ C_D }\label{1.1.19} +\end{equation} +and the \textbf{surface conductances} for scalars and momentum are +respectively +\begin{eqnarray} + C_H&=&\frac{k}{ \Phi _h } v_\ast = c_H V\label{1.1.20}\\ + C_D&=&\frac{k}{ \Phi _m } v_\ast = c_D V\label{1.1.21}. +\end{eqnarray} + +The surface exchange coefficients can then be written in any of the +following forms: +\begin{eqnarray} + c_H&=&\frac{ C_H }{V} = \frac{ C_H C_D }{ v_\ast ^2 } = \frac{ k^2 }{ \Phi _h \Phi _m }\label{1.1.22}\\ + c_D&=&\frac{ C_D }{V} = \frac{ C_D^2 }{ v_\ast ^2 } = \frac{ k^2 }{ \Phi _m^2 }. +\label{1.1.23} +\end{eqnarray} +In order to close the system the surface scaling velocity, v$_{\ast +}$, needs to be specified. If +\begin{equation} + v_\ast = u_\ast \equiv \left| { {\rm {\bf \tau }}_{0} {/} \rho _{0} } \right|^{1/2}\label{1.1.24} +\end{equation} +we have the standard Monin-Obukhov theory and it is easy to deduce +that with this definition $v_{\ast } = c_{D}^{1/2} \Delta $\textbf{v} +and V=$\Delta $\textbf{v}. To allow for the effect of +\textbf{turbulent and cloud-scale gusts }on the surface turbulent +fluxes the surface scaling velocity, v$_{\ast }$, can be defined as +\begin{equation} + v_\ast ^2 = u_\ast ^2 + \gamma _t^2 w_\ast ^2 + \gamma _c^2 w_c^2. + \label{1.1.25} +\end{equation} +The second term represents the effects of turbulent eddy-scale +convective gusts and w$_{\ast }$ is the turbulent convective scaling +velocity defined by +\begin{equation} + w_\ast = {\left( { z_i \frac{ F_{B0} }{ \rho _0 }} \right)}^{1/3} + \label{1.1.26} +\end{equation} +for F$_{B0} >$ 0 and zero otherwise. z$_{i}$ is the height of the top +of the surface-based turbulent mixing layer. $\gamma _{t}$ is a +dimensionless constant which can be determined empirically or tuned +within empirical limits. The third term represents the effects of +deep convective cloud-scale gusts; the inclusion of this term is +optional. The form implemented is taken from +\cite{redelsperger00:_param_mesos_enhan_surfac_fluxes}, in which the +velocity scale, w$_{c}$, is a function of the convective downdraught +mass-flux at cloud base. (Note that the published expression is given +as an adjustment of the 10-m wind and has been scaled to make it +consistent with $v_\ast$.) A further term $\gamma +_{m}^{2}$w$_{m}^{2}$ could be included in low resolution models to +represent the effects of mesoscale gusts but this is not done in the +Unified Model. The\textbf{ low wind speed limit}, i.e. as $\Delta +$\textbf{v} $\to $ 0, for unstable conditions (with w$_{c }$= 0) can +be seen to be +\begin{equation} + v_\ast \sim \gamma _t w_\ast \sim \gamma _t^{3/2} {\left( {\frac{ c_H }{ c_D^{1/2} }} \right)}^{1/2} z_i^{1/2} (-\Delta B )^{1/2} + \label{1.1.27} +\end{equation} +which implies that +\begin{equation} + L \sim -( \gamma _t^3 /k) z_i + \label{1.1.28} +\end{equation} + +Thus the low wind speed limits for the sensible and latent heat fluxes +are obtained by substituting~(\ref{1.1.27}) into (\ref{1.1.7}) and +(\ref{1.1.8}) with the surface transfer coefficients evaluated with L +given by (\ref{1.1.28}). The finite limit for L implies that the form +of the stability functions, $\phi $, for very large and negative +$\zeta $ is unimportant. However, the value of $\Phi_{h}$ for L given +by (\ref{1.1.28}) is needed if the value of $\gamma _{t}$ is +determined from measurements of say the latent heat flux in very low +mean wind conditions. + +\subsubsection[Comparison with Godfrey and Beljaars (1991) formulation for gustiness] +{Comparison with the \cite{godfrey1991} formulation for gustiness} +\label{section_1.2} +We can define the \textbf{mean gust speed} at height z$_{1}$ by +\begin{equation} + v_g = ( V^2 - \left| {\Delta {{\rm {\bf v}}}} \right|^2 {)}^{{1/2}}. + \label{1.2.1} +\end{equation} +Using the definitions of $V$ (\ref{1.1.19}) and $v_{\ast }$ +(\ref{1.1.25}) it can be deduced that +\begin{equation} + v_g^2 = W_g^2 ( z_1 ) + \frac{1}{2}\left| {\Delta {{\rm {\bf v}}}} \right|{ }\left[ {\left( {{ } {\left| {\Delta {{\rm {\bf v}}}} \right|}^2 - W_g^2 ( z_1 )} \right)^{1/2} - \left| {\Delta {{\rm {\bf v}}}} \right|} \right] + \label{1.2.2} +\end{equation} +where +\begin{equation} + W_g (z) = \frac{1}{ c_D^{1/2} } {\left( { \gamma _t^2 w_\ast ^2 + \gamma _c^2 w_c^2 } \right)}^{1/2} = \frac{ \Phi _m (L , z + z_{0m} , z_{0m} )}{k} {\left( { \gamma _t^2 w_\ast ^2 + \gamma _c^2 w_c^2 } \right)}^{1/2}. + \label{1.2.3} +\end{equation} +Thus in this formulation the mean gust speed is a function of height +above the surface through the same factor, $\Phi _{m}$(z), which +determines the profile of the mean wind \textbf{v} in the surface +layer (see Eq.~(\ref{1.1.9})). The values of $\Delta $\textbf{v}, +v$_{g}$ and $V$ thus tend to zero as z $\to $ 0. Note that v$_{g} \to +$ W$_{g}$ as $\Delta $\textbf{v} $\to $ 0 and that v$_{g} \to $ 0 as +the convective gustiness scaling velocities tend to zero. + +Equation~(\ref{1.2.1}) can be rewritten as +\begin{equation} + V^2 = \left| {\Delta {{\rm {\bf v}}}} \right|^2 + v_g^2, + \label{1.2.4} +\end{equation} +which is exactly the form of \cite{godfrey1991}. However +\cite{godfrey1991} define the mean gust speed as $\beta $w$_{\ast +}$. Thus they directly modify the mean surface to air wind difference, +$\Delta $\textbf{v}, with the gustiness or turbulent convective +scaling velocity, w$_{\ast }$, combining a grid dependent quantity +with a constant scaling speed. The two formulations are similar in +that they introduce a mean gust speed but differ in their assumption +about whether this has a non-constant profile in the surface +layer. Although transitory wind gusts may not be as close to +equilibrium with the surface characteristics as the mean wind they +should have a profile in the surface layer which approaches zero at +the surface (strictly at the roughness height z$_{0m})$. + +The two formulations can be made equivalent by assuming that +\cite{godfrey1991} $\beta $ is not constant but is given by $\gamma +_{t}(\Phi _{m}$(z)/k) which tends to zero as the surface is +approached. However, over sea points where the roughness length is +small (of order 10$^{-4}$ m) and for which \cite{godfrey1991} derived +their formulation, $\Phi_{m}$(z) varies at most by about 15{\%} +between 10 m and 50 m. Assuming a constant $\beta $ does not lead to +much inaccuracy in these circumstances. If gustiness is included over +land, as is the case in the Unified Model, the higher roughness +lengths lead to a greater variation in $\Phi _{m}$(z) in the region +where models generally have their lowest level placed. + +If $\gamma _{t}$ = 0.08 then in the low wind speed limit of an +unstable tropical maritime surface layer with a virtual temperature +lapse of 1.5 K, a specific humidity lapse of 7x10$^{-3}$ kg/kg, +SST=303.16 K and a boundary layer depth of 800 m we obtain a latent +heat flux of 37.08 W/m$^{2}$ assuming the form for the stability +functions given below. + + +\subsection{Making surface exchange consistent with flux differencing} + +The boundary layer scheme increments conserved quantities using +differences in fluxes across a layer. This is strictly consistent only +if the conserved quantity is a mass-weighted mean across the layer, +rather than a representative value, such as the value at the middle of +the layer. If the profile of the conserved quantity is linear the mean +of the quantity is the same as the point-value in the middle of the +layer, but this is not so if the profile is not linear. Near the +surface, the profiles will be logarithmic in neutral conditions and so +the values in the middle of the layer will be larger than the mean +values. Surface similarity in the UM is applied treating taking the +wind and temperature in the bottom layer as point values in the +calculation of surface fluxes and is therefore not absolutely +consistent with flux differencing. Whilst the effect of this +difference is not large, it is desirable to have the option of +correcting it, which is done by enabling the option to ``make surface +exchange consistent with flux differencing.'' The following +discussion explains how this is done. + +In effect, the UM takes the displacement height for momentum as +$-z_{0m}$, where $z_{0m}$ is the momentum roughness length. The +profile of wind is therefore determined by Monin-Obukhov theory as +\begin{equation} + \frac{\partial u}{\partial z} = \frac{u_*}{k(z+z_{0m})} \phi_m((z+z_{0m})/L), +\end{equation} +with $u_*$ being the friction velocity, $L$ the surface Obukhov length +and $\phi_m$ the similarity function. It is common practice to +introduce a new function $\psi_m$ such that +$\phi_m(\zeta)=1-\zeta \partial \psi_m / \partial \zeta.$ Redefining +the vertical coordinate as $\zeta=z/L$, we have +\begin{eqnarray} + u(\zeta) &=& \frac{u_*}{k} \int_{0}^{\zeta} \frac{1}{(\zeta'+\zeta_{0m})} + \phi_m(\zeta'+\zeta_{0m}) \, d\zeta' = + \frac{u_*}{k} \int_{\zeta_{0m}}^{\zeta'+\zeta_{0m}} + \frac{1}{\zeta'} \phi_m(\zeta') \, d\zeta' \\ + &=& \frac{u_*}{k} \int_{\zeta_{0m}}^{\zeta'+\zeta_{0m}} \left ( \frac{1}{\zeta'} - + \frac{d\psi_m}{d\zeta'} \right ) \, d\zeta' \nonumber \\ + &=& \frac{u_*}{k} \left \{ \ln \left ( \frac{\zeta+\zeta_{0m}}{\zeta_{0m}} + \right ) - \psi_m(\zeta+\zeta_{0m}) + \psi_m (\zeta_{0m}) \right \}. + \nonumber +\end{eqnarray} +This is also frequently written as +\begin{equation} + u(\zeta) = \frac{u_*}{k} \Phi_m(\zeta). +\end{equation} +The mean over the lowest layer, of depth $z_1$ (or $\zeta_1$ in the +rescaled coordinate), is therefore +\begin{eqnarray} + \bar u &=& \frac{u_*}{k\zeta_1} \int_0^{\zeta_1} u(\zeta) \, d \zeta \\ + &=& \frac{u_*}{k\zeta_1} \int_0^{\zeta_1} + \ln \left ( \frac{\zeta+\zeta_{0m}}{\zeta_{0m}} \right ) + - \psi_m(\zeta+\zeta_{0m}) + \psi_m (\zeta_{0m}) \, d \zeta. + \nonumber +\end{eqnarray} +We consider the three terms within the integral separately. For the +first, +\begin{eqnarray} + \int_0^{\zeta_1} \ln \left ( \frac{\zeta+\zeta_{0m}}{\zeta_{0m}} \right ) \, d \zeta + &=& \zeta_{0m} \int_1^{1+\zeta_1/\zeta_{0m}} \ln(x) \, dx \\ + &=& \zeta_{0m} \left [ \left ( 1+ \frac{\zeta_1}{\zeta_{0m}} \right ) \ln + \left ( 1+ \frac{\zeta_1}{\zeta_{0m}} \right ) - + \left ( 1+ \frac{\zeta_1}{\zeta_{0m}} \right ) +1 \right ] . \nonumber +\end{eqnarray} +For the second, +\begin{eqnarray} + \int_0^{\zeta_1} \psi_m(\zeta+\zeta_{0m}) \, d\zeta &=& + \int_{\zeta_{0m}}^{\zeta_1+\zeta_{0m}} \psi_m(\zeta) \, d\zeta \\ + &=& \left [ \zeta \psi_m + \right ]_{\zeta_{0m}}^{\zeta_1+\zeta_{0m}} + - \int_{\zeta_{0m}}^{\zeta_1+\zeta_{0m}} \zeta \frac{d\psi_m}{d\zeta} + d\zeta \nonumber \\ + &=& (\zeta_1+\zeta_{0m}) \psi_m(\zeta_1+\zeta_{0m}) - \zeta_{0m} + \psi_m(\zeta_{0m}) \nonumber \\ &-& \int_{\zeta_{0m}}^{\zeta_1+\zeta_{0m}} + (1-\phi_m) d\zeta \nonumber \\ + &=& (\zeta_1+\zeta_{0m}) \psi_m(\zeta_1+\zeta_{0m}) - \zeta_{0m} + \psi_m(\zeta_{0m}) \nonumber \\ &+& \int_{\zeta_{0m}}^{\zeta_1+\zeta_{0m}} + (\phi_m -1) \, d\zeta . \nonumber +\end{eqnarray} +$\phi_m-1$ is retained in the last integral since this will prove +convenient in later algebra. The third integral is trivial. Hence, +\begin{eqnarray} + \bar u &=& \frac{u_*}{k} \left \{ + \left ( 1+ \frac{\zeta_{0m}}{\zeta_1} \right ) \left [ + \ln \left ( 1+ \frac{\zeta_1}{\zeta_{0m}} \right ) \right . \right . \\ + &-& \left . \left . \psi_m(\zeta_1+\zeta_{0m}) + \psi_m(\zeta_{0m}) \right ] -1 + - \frac{1}{\zeta_1} \int_{\zeta_{0m}}^{\zeta_1+\zeta_{0m}} + (\phi_m -1) \, d\zeta \right \} \nonumber \\ + &=& \frac{u_*}{k} \left \{ \left ( 1+ \frac{\zeta_{0m}}{\zeta_1} \right ) + \Phi_m(\zeta_1) - \frac{1}{\zeta_1} \int_{\zeta_{0m}}^{\zeta_1+\zeta_{0m}} + \phi_m \, d\zeta \right \} . \nonumber +\end{eqnarray} +Thus, in practical terms, the standard function $\Phi_m$ is evaluated +at the top of the layer, scaled by $1+\zeta_{0m}/\zeta_1$ and reduced +by the mean value of $\phi_m$. For standard Monin-Obukhov functions, +this last integral is easy to perform. In the limit $L \rightarrow +\infty$ it becomes 1 and to avoid a numerical singularity it is set +equal to 1 in this (nearly neutral) limit. The adjustment of the +thermal Monin-Obukhov function is exactly equivalent. + +Algorithmically, the existing routine {\tt PHI\_M\_H} is replaced by +{\tt PHI\_M\_H\_VOL} which takes the same inputs except that the +heights are the top of the layers. This is done when the surface +fluxes, or turbulence scales $u_*$ and $\theta_*$ are +calculated. Monin-Obukhov functions are also used to calculate winds +and temperatures at observed levels. In this case a value at a +particular height is required, so the existing Monin-Obukhov routine +should be used. + +\subsection{The form of the stability functions.} +\label{section_1.3} +For \textbf{stable conditions}, i.e. $\Delta $B $\ge $ 0, the +stability functions are given by \cite{Beljaars1991}: +\begin{eqnarray} + \Phi _m&=&\ln \left( {\frac{ z_1 + z_{0m} }{ z_{0m} }} \right) - \Psi _m ( \zeta _1 ) + \Psi _m ( \zeta _{0m} ) + \label{1.3.11}\\ + \Phi _h&=&\ln \left( {\frac{ z_1 + z_{0m} }{ z_{0h} }} \right) - \Psi _h ( \zeta _1 ) + \Psi _h ( \zeta _{0h} ) + \label{1.3.12} +\end{eqnarray} +where $\zeta _{1}$ = (z$_{1}$ + z$_{0m})$/L, $\zeta _{0m}$ = +z$_{0m}$/L, $\zeta _{0h}$ = z$_{0h}$/L and +\begin{eqnarray} + - \Psi _h (\zeta )&=&\left[ { {\left( {1 + \frac{2}{3}a\zeta } \right)}^{3/2} - 1 } \right] + b\left( {\zeta - \frac{c}{d}} \right)\exp (-d\zeta ) + \frac{bc}{d} + \label{1.3.13}\\ + - \Psi _m (\zeta )&=&a\zeta + b\left( {\zeta - \frac{c}{d}} \right)\exp (-d\zeta ) + \frac{bc}{d}, + \label{1.3.14} +\end{eqnarray} +with $a = 1$, $b =2/3$, $c = 5$, $d = 0.35$. + +Note that the bulk flux Richardson number for the surface layer is +given by +\begin{equation} {Ri}_{fB} = \frac{ c_D^{1/2} }{k} \frac{ z_1 }{L} = + \frac{ z_1 /L}{ \Phi _m } + \label{1.3.8} +\end{equation} +so the \cite{Beljaars1991} functions imply Ri$_{f B} \to $ 1/a = 1 as +z$_{1}$/L $\to \infty $. + +For \textbf{unstable conditions}, i.e. $\Delta $B $<$ 0, the Dyer and +Hicks forms \cite[]{dyer1974} are used: +\begin{eqnarray} + \phi _m&=&(1 - 16\zeta )^{-1/4} + \label{1.3.15} \\ + \phi _h&=&(1 - 16\zeta )^{-1/2} + \label{1.3.16} +\end{eqnarray} +(Note that $\phi _{h}\prime $ is discontinuous at 0.) These are only +empirically verified for $\zeta \ge $ -1. Evaluating the integrals +(\ref{1.1.14}) and (\ref{1.1.15}) we obtain: +\begin{equation} + \Phi _m = \ln \left( {\frac{ z_1 + z_{0m} }{ z_{0m} }} \right) - 2 \ln \left( {\frac{1 + X_1 }{1 + X_0 }} \right) - \ln \left( {\frac{1 + X_1^2 }{1 + X_0^2 }} \right)+ 2 \left( { {\tan }^{-1} X_1 - {\tan }^{-1} X_0 } \right) + \label{1.3.17} +\end{equation} +where +\begin{equation} + X_1 = (1 - 16 \zeta _1 )^{1/4} , X_0 = (1 - 16 \zeta _{0m} )^{1/4} + \label{1.3.18} +\end{equation} +and +\begin{equation} + \Phi _h = \ln \left( {\frac{ z_1 + z_{0m} }{ z_{0h} }} \right) - 2 \ln \left( {\frac{1 + Y_1 }{1 + Y_0 }} \right) + \label{1.3.19} +\end{equation} +where +\begin{equation} + Y_1 = (1 - 16 \zeta _1 )^{1/2} , Y_0 = (1 - 16 \zeta _{0h} )^{1/2}. + \label{1.3.20} +\end{equation} + +\subsection{The iterative algorithm for calculating the surface + exchange coefficients}\label{section_1.4} + +For conditions that are stable, i.e. $\Delta $B $\ge$ 0, or near-neutral (taken as +$\Delta $\textbf{v} $\ge$ 2 ms$^{-1}$), then start the iteration from the neutral limit, so +\begin{eqnarray} + \Phi _m^{(0)}&=&\ln \left( {\frac{ z_1 + z_{0m} }{ z_{0m} }} \right) + \label{1.4.5} \\ + \Phi _h^{(0)}&=&\ln \left( {\frac{ z_1 + z_{0m} }{ z_{0h} }} \right) + \label{1.4.6} \\ + v_\ast ^{(0)}&=& {\left( {\frac{k}{ \Phi _m^{(0)} }} \right)} \left| {\Delta {{{\rm {\bf v}}}}} \right| + \label{1.4.7} +\end{eqnarray} +Otherwise (if $\Delta $B $<$ 0 and $\Delta $\textbf{v} $<$ 2 ms$^{-1}$ ) start from the greater of the neutral and convective limits for $v_\ast^{(0)}$, so +\begin{eqnarray} + \frac{1}{ L^{(0)} }&=&\frac{-k}{ \gamma _t^3 z_i } + \label{1.4.1}\\ + \Phi _m^{(0)}&=& \Phi _m ( L^{(0)} , z_1 + z_{0m} , z_{0m} ) + \label{1.4.2}\\ + \Phi _h^{(0)}&=& \Phi _h ( L^{(0)} , z_1 + z_{0m} , z_{0h} ) + \label{1.4.3}\\ + v_\ast ^{(0)}&= & MAX{\left[ {\left( {\frac{k}{ \Phi _m^{(0)} }} \right)} \left| {\Delta {{{\rm {\bf v}}}}} \right|, \, + {\left[ { \gamma _t^3 \left( {\frac{k}{ \Phi _h^{(0)} }} \right) z_i \left| {-\Delta B} \right| } \right]}^{ 1/2} \right]} + \label{1.4.4} +\end{eqnarray} + +Then calculate +\begin{eqnarray} + C_D^{(0)}&=&\frac{k}{ \Phi _m^{(0)} } v_\ast ^{(0)} + \label{1.4.8} \\ + C_H^{(0)}&=&\frac{k}{ \Phi _h^{(0)} } v_\ast ^{(0)} + \label{1.4.9} +\end{eqnarray} + +Having set up initial values the iteration loop can be entered (this +is the original method used but contains an inconsistency in the +treatment of boundary-layer convective gustiness, as described in +section~\ref{mo_iter_corrn}): + +DO n = 1 to N +\begin{eqnarray} + u_\ast ^{(n)2}&=& C_D^{(n-1)} \left| {\Delta {{\rm {\bf v}}}} \right| + \label{1.4.10} \\ + {\left( {\frac{ F_{B0} }{ \rho _0 }} \right)}^{(n)}&=& { {-C}_H }^{(n-1)} \Delta B + \label{1.4.11}\\ + w_\ast ^{(n)}&=& {\left[ { z_i {\left( {\frac{ F_{B0} }{ \rho _0 }} \right)}^{(n)} } \right]}^{ 1/3} + \label{1.4.12}\\ + v_\ast ^{(n)2}&=& u_\ast ^{(n)2} + \gamma _t^2 w_\ast ^{(n)2} + \gamma _c^2 w_c^2 + \label{1.4.13}\\ + \frac{1}{ L^{(n)} } = \frac{-k( F_{B0} / \rho _0 )^{(n)} }{ v_\ast ^{(n)3} } + \label{1.4.14}\\ + \Phi _m^{(n)}&=& \Phi _m ( L^{(n)} , z_1 + z_{0m} , z_{0m} ) + \label{1.4.15} \\ + \Phi _h^{(n)}&=& \Phi _h ( L^{(n)} , z_1 + z_{0m} , z_{0h} ) + \label{1.4.16} \\ + C_D^{(n)}&=&\frac{k}{ \Phi _m^{(n)} } v_\ast ^{(n)} + \label{1.4.17} \\ + C_H^{(n)}&=&\frac{k}{ \Phi _h^{(n)} } v_\ast ^{(n)} + \label{1.4.18} +\end{eqnarray} + +END DO. + +For neutral and stable conditions ($\Delta $B $\ge $ 0) start the +iteration from the neutral values and set w$_{\ast }$=0 in the above +iteration loop. + +Use the final (N) values of C$_{H}$ and C$_{D}$ to calculate the +surface sensible and latent heat fluxes and surface stress: + +\begin{eqnarray} + H_0&=& {-c}_P \rho _0 C_H^{(N)} \left( {\Delta T + \frac{g}{ c_P }( z_1 + z_{0m} - z_{0h} )} \right) + \label{1.4.19} \\ + E_0&=& {-\rho }_0 C_H^{(N)} \Delta q + \label{1.4.20} \\ + {\rm {\bf \tau }}_{0} &=& \rho _0 C_D^{(N)} \Delta {\rm {\bf v}} + \label{1.4.21} +\end{eqnarray} +N is the last iteration value. N = 5 is currently used. + +For sea points the momentum roughness length and the wind mixing +energy flux are calculated from v$_{\ast }^{(N)}$ using the formulae +in subsection~\ref{section_1.6} below. + +\subsubsection{Correction to the iterative algorithm}\label{mo_iter_corrn} +The above implementation of boundary-layer convective gustiness in the +Monin-Obukhov iteration contains an inconsistency. The overall effect +turns out to be small, but it is desirable to use the corrected form, +which is derived as follows. + +We decompose the wind as ${\bf u}=\bar{\bf u} + {\bf u}_g + {\bf u}'$, +representing, respectively, the large-scale, gust and small-scale +turbulent contributions to the velocity. Locally, Monin-Obukhov theory +then gives +\begin{equation} + |\bar{\bf u} + {\bf u}_g({\bf x}) | = \frac{u_*({\bf x})}{k} \Phi_m({\bf x}). +\end{equation} +We ignore the spatial variation of $\Phi_m$, expecting that the +principal effect of locally stronger winds is to increase the local +stress -- this is exactly true in nearly neutral flow. The local +stress is aligned with the wind so +\begin{equation} {\bf \tau}({\bf x}) = \rho u_*^2({\bf x}) + \frac{\bar{\bf u} + {\bf u}_g({\bf x})} {|\bar{\bf u} + {\bf + u}_g({\bf x})|}. +\end{equation} +With the assumption that $\Phi_m$ does not vary spatially, +\begin{equation} {\bf \tau}({\bf x}) = \rho \frac{k^2}{\Phi_m^2} + |\bar{\bf u} + {\bf u}_g({\bf x})| (\bar{\bf u} + {\bf u}_g({\bf + x})) \equiv \rho C_D |\bar{\bf u} + {\bf u}_g({\bf x})| (\bar{\bf + u} + {\bf u}_g({\bf x})), +\end{equation} +where $C_D$ is the standard drag coefficient, $\frac{k^2}{\Phi_m^2}$. +The grid-box mean effect is +\begin{equation} + \langle{\bf \tau}\rangle = \rho C_D \langle |\bar{\bf u} + {\bf u}_g({\bf x})| + (\bar{\bf u} + {\bf u}_g({\bf x})) \rangle \approx \rho C_D \langle + |\bar{\bf u} + {\bf u}_g({\bf x})| \rangle \bar{\bf u}, +\end{equation} +which is the product of the enhanced wind speed (including gusts) and +the mean velocity. The magnitude of the stress is then +$\langle\tau\rangle = \rho C_D S U$, with the last two symbols +representing the mean wind speed, including gusts, and the mean +background velocity. + +In the model, we want to write this as an effective drag coefficient, +$C_{De}$, so that $\langle\tau\rangle = \rho C_{De}U^2$. Now define +$\tilde u_* = \frac{k}{\Phi_m}U$, the friction velocity due to +large-scale flow, and $\hat u_* = \frac{k}{\Phi_m}S$, the friction +velocity due to the total flow, including gusts (again implicitly +assuming that $\Phi$ is unaffected by the gusts). The representation +is $\hat u_*^2 = \tilde u_*^2 +\gamma_t^2 w_*^2$. Then, +\begin{equation} + C_{De}=\frac{C_DS}{U} = \frac{k^2}{\Phi_m^2} \frac{\hat u_*}{\tilde u_*} + = \frac{k}{\Phi_m}\frac{\hat u_*}{U} +\end{equation} +The variable {\tt CDV} in the routine {\tt FCDCH} within the +Monin-Obukhov iteration will then be $\frac{k}{\Phi_m}{\hat u_*}$ +(note that it is divided by $U$ at the end of the routine). This is +indeed coded at the end of the loop; but in the original code {\tt + CDV} is used to calculate $\tilde u_*^2$ at the beginning of the +routine. In fact, $\tilde u_*^2 = (k/\Phi_m)\tilde u_* U$, which +differs from the coded result by a factor of $\tilde u_*/\hat +u_*$. After this step, the purported $\tilde u_*$ is augmented by the +gust contribution to get $\hat u_*$. This has the consequence of +overestimating $C_{De}$ and also means that what is described as {\tt + U\_S} in this loop is not $\tilde u_*$, but $\sqrt(\tilde u_* \hat +u_*)$. + +If we let $v=\sqrt(\tilde u_* \hat u_*)$, then we get +\begin{equation} + \hat u_*^2 = \frac{1}{2} \left \{ \gamma_t^2 w_*^2 + \sqrt { \gamma_t^4 w_*^4 + + 4 v^2 } \right \}. +\end{equation} +In the corrected version this expression is used to calculate {\tt + V\_S}, namely $\hat u_*$ within the iteration. + + +\subsection{The interpolation of surface layer variables to standard + observation heights}\label{section_1.5} +Integrating~(\ref{1.1.3}) between the roughness height, z$_{0m}$, and +the observation height z$_{ob}$ we obtain +\begin{equation} + {\rm {\bf v}}_{ob} { = } {\rm {\bf v}}_{0} { + + }\frac{ {\rm {\bf \tau }}_{0} }{ \rho _0 v_\ast k} \Phi _m (L, + z_{ob} + z_{0m} , z_{0m} ) + \label{1.5.1} +\end{equation} +Using the expression for the surface turbulent stress this gives the +interpolation formula +\begin{equation} + {\rm {\bf v}}_{ob} { = } {\rm {\bf v}}_{0} { + + }\frac{ C_D }{k v_\ast } \Phi _m (L, z_{ob} + z_{0m} , z_{0m} ) ( + {\rm {\bf v}}_{1} { - } {\rm {\bf v}}_{0} {)} + \label{1.5.2} +\end{equation} +For wind z$_{ob}$ is set to 10m and the last iteration (N) values of +C$_{D}$, L and $v_{\ast }$ are used. Integrating~(\ref{1.1.1}) and +(\ref{1.1.2}) between the roughness height, z$_{0h}$, and the +observation height z$_{ob, }$ we obtain for the scalar $X$ +($=T+(g/c_{P})z$ , $q$ or tracer amount) +\begin{equation} + X_{ob} = X_0 + \frac{ F_{X0} }{ \rho _0 v_\ast k} \Phi _h (L, z_{ob} + z_{0h} , z_{0h} ) + \label{1.5.3} +\end{equation} +and using the expression for the surface flux $F_{X0}$ of the scalar +quantity $X$ this gives the interpolation formula +\begin{equation} + X_{ob} = X_0 + \frac{ C_H }{k v_\ast } \Phi _h (L, z_{ob} + z_{0h} , z_{0h} ) ( X_1 - X_0 ) + \label{1.5.4} +\end{equation} +For temperature and humidity z$_{ob}$ is set to the screen height (1.5 +m) and the last iteration (N) values of C$_{H}$, L and $v_{\ast }$ are +used. + +\subsubsection{The parametrization of decoupling} + +In the foregoing analysis it is tacitly assumed that the surface +layer, up to the model's lowest grid level, is in equilibrium with the +surface and lies within the constant flux layer. In light winds, and +when the surface temperature falls quickly, these assumptions are +invalid; equation~\ref{1.5.4} then yields temperatures at the height +of observation that are too closely tied to the surface +temperature. Observed temperatures may be significantly warmer: this +may be termed decoupling. Two parametrizations of this effect are +available. Both involve the idea that as the wind becomes very light +radiative cooling comes to determine the temperature profile. + +The first parametrization simply sets the interpolation coefficient +between the surface temperature and that on the model's lowest level +according to the radiative equilibrium profile when the Richardson +number exceeds 0.25 (a typical criterion for high stability). + +The second more elaborate scheme is directed at the evening +transition, when the surface temperature is falling rapidly: it is +under such conditions that the impact of decoupling on the air +temperature is greatest. For a couple of hours after the surface +temperature drops below the air temperature, radiative cooling +directly to the surface largely determines the atmospheric cooling +rate when the wind is light and this may easily be calculated, +provided that a parametrization of the transmission between the air +and the surface is available: this parametrization depends on the +absorbing properties of atmospheric trace gases. Explicitly, we have +\begin{equation} + \dot T_{ob, \mbox{\tiny rad, surf}} = \frac{4\sigma T_s^3}{c_P} + {\cal K}(z_{ob}) (T_s-T_{ob}), +\end{equation} +where $\dot T_{ob, \mbox{\tiny rad, surf}}$ is the cooling rate of the +air at the height of observation due to direct radiative exchanges +with the surface, $T_{ob}$ is the air temperature at that height, +$T_s$ is the temperature of the surface and ${\cal K}(z_{ob})$ depends +on the concentration of trace gases in the atmosphere, in practice +water vapour and carbon dioxide, and their spectroscopic +properties. The contributions of water vapour and carbon dioxide are, +to a good approximation, additive, so we may write +\begin{equation} + {\cal K}(z_{ob}) = \left [ q_w {\cal C}_w(\mu_w, + T_{ob}) + q_c {\cal C}_c(\mu_c, T_{ob}) \right ], +\end{equation} +where $q_w$ and $q_c$ are the specific concentrations of water vapour +and carbon dioxide and $\mu_w$ and $\mu_c$ are the respective +pathlengths between the observation height and the surface. The +functions ${\cal C}_w$ and ${\cal C}_c$ are parametrized and explicit +functional forms are included in the code. + +In stronger winds turbulent cooling will be more important, so the +scheme must approach the standard procedure in that limit. Within the +context of local scaling, it can be shown that the depth of the +atmosphere which feels the impact of surface cooling must scale on +${\cal L}=(u_*^3/ (g/T_s)\dot T_s)^{1/2}$, where $u_*$ is the surface +friction velocity, $\dot T_s$ is the surface cooling rate. This +parameter is used as a measure of the strength of turbulence to define +the relaxation back to the strong-wind limit. + +To implement the scheme, the liquid-frozen potential temperature at +the height of observation, $\theta_{ob}=T_{ob}+(g/c_P)z_{ob}-(L/c_P) +q_{cl} - ((L+L_f)/c_P) q_{cf}$ is made a prognostic. Whenever the +surface buoyancy flux changes sign and becomes stable, this prognostic +is initialized using standard theory. On subsequent timesteps, it is +updated to allow for radiative cooling to the surface, giving a +provisional value $\theta_{ob}'$, and then relaxed back towards the +result that would be obtained from standard similarity theory, +$\theta_{ob, \mbox{\tiny sim}}$, as in the last section: +\begin{eqnarray} + \theta_{ob}'(t+\delta t) &\leftarrow & \theta_{ob}(t)+ + \delta t \, \dot T_{ob,\mbox{\tiny rad,surf}} \\ + \theta_{ob}(t+\delta t) &\leftarrow & W \theta_{ob}'(t+\delta t) + +(1-W) \theta_{ob, \mbox{\tiny sim}}. +\end{eqnarray} +By tuning against an idealized highly vertically resolved model based +on local scaling we set, +\begin{equation} + W = \exp(-(0.4 f)^2 \delta t \, t_{\mbox{\tiny trans}})) / + (1+X({\cal L})\delta t), +\end{equation} +with +\begin{equation} + X({\cal L})= \min \left ( 0.000283 \left \{\frac{{\cal L}}{z_{ob}} + \log \left (1+\frac{z_{ob}}{z_0}\right ) \right \}^2, + \; \frac{0.2 u_* }{z_{ob}} \right ). +\end{equation} +The explicit dependence of $W$ on the timestep, $\delta t$, ensures +that the results converge as $\delta t \rightarrow 0$. The exponential +factor involving the Coriolis parameter, $f$, is intended to represent +the recoupling of the surface and atmosphere as growing directional +shear at the top of the incipient stable boundary layer generates +turbulence. This factor is somewhat exaggerated relative to the +results of the model against which it is tuned, as a cautionary +measure to ensure that decoupling is not allowed to persist too long +after the transition. This is the purpose of the inclusion of the +factor $0.4 f t_{\mbox{\tiny trans}}$, where $t_{\mbox{\tiny trans}}$ +is the time since the transition. + +It must be stressed that these schemes are heuristic and that the +precise behaviour in weak turbulence is not fully understood. The +second scheme appears to work well during the evening transition, but +for reasons of caution decoupling is suppressed somewhat too +rapidly. The simpler first scheme underestimates decoupling during the +transition, but allows it to persist longer, although tending to +overestimate it on these timescales. Overall, the second scheme is to +be preferred. + +\subsection{The surface fluxes for sea and sea-ice + gridboxes}\label{section_1.6} +For gridboxes with sea-ice (i.e. where sea-ice fraction, f$_{I} >$ 0) +sensible and latent heat fluxes are calculated separately for the sea +and ice parts of the gridbox and combined with appropriate weighting +to obtain the total fluxes into the atmosphere. This is done because +the two surfaces can have very different temperatures and also differ +in their roughness. + +The sea and sea-ice surface fluxes of sensible heat, moisture and +momentum are calculated using gridbox mean surface transfer +coefficients, $<$C$_{H}>$ and $<$C$_{D}>$. These are linear +combinations of the corresponding coefficients calculated for the +ice-free sea (L), typical Marginal Ice Zone broken sea-ice (MIZ) and +complete ice cover (I). + +The surface sensible heat and moisture fluxes are calculated +separately for the ice-free (leads) and ice-covered parts of the +gridbox. Although the fluxes over the two surfaces are calculated from +gridbox mean surface transfer coefficients, the different surface +temperatures give different fluxes. The ice surface temperature is +predicted from a surface energy balance and ice heat conduction model +(see the documentation for the land and ice surface processes +component of the Unified Model). In current versions of the model the +sea surface temperature (SST) is assumed to be 271.35 K, the freezing +point of sea water, whenever the ice fraction is greater than +zero. This is unrealistic except for genuine leads (i.e. large ice +fraction) and a future version of the model will allow the SST to be +larger than 271.35 K for gridboxes with sea-ice. + +The wind mixing energy flux, F$_{WME}$ , is the rate of production of +turbulent kinetic energy per unit area in the sea surface layer by the +wind stress at the air-sea interface. In atmosphere-only +configurations of the Unified Model this quantity is a useful +diagnostic. When the atmosphere model is coupled to an ocean model the +wind mixing energy flux is accumulated over an ocean model timestep +and then used in the calculation of the mixing in the upper layers of +the ocean. The gridbox mean \textbf{wind mixing energy flux} is given +by + +\begin{equation} + F_{WME} = ( 1- f_I ) \frac{ \rho _0^{3/2} v_\ast ^3 }{ \rho _{(sea)}^{1/2} } + \label{1.6.9} +\end{equation} +where $v_{\ast }$ is calculated using the drag coefficient for the +leads part of the gridbox, c$_{D(L)}$, rather than the gridbox mean +value, $<$c$_{D}>$, when there is partial ice cover. + +\subsubsection{Roughness Lengths over the Sea} + +The roughness lengths for momentum and scalars depend on both the +atmospheric flow and the wave state. The dependence on wave state is +not fully understood and is still a subject of active research. In any +case, it could only fully be represented in a coupled wave-atmosphere +model. Simpler more empirical schemes are therefore currently used in +the Unified Model. + +In all schemes available here the momentum roughness length is given by +\begin{equation} + z_{0m(sea)} = \frac{1.54\times {10}^{-6} }{ v_\ast } + + \frac{\alpha}{g} v_\ast ^2 + \label{eq:z0msea} +\end{equation} +which is a generalisation of Charnock's formula to include low-wind +conditions \cite[]{Smith88}. $\alpha$ is Charnock's coefficient, which +is determined from field measurements. It is often taken as a +constant, but more elaborate schemes include a dependence on wind +speed. In practice the difference between different parametrizations +of the momentum roughness length therefore comes down to the +specification of Charnock's coefficient. + +There is greater uncertainty in the roughness lengths for scalars and +the dependencies are described separately for each scheme. Note that +whilst the full versions of some schemes prescribe different roughness +lengths for heat and moisture, in the Unified Model we have only a +single roughness for all scalars. + +Schemes are selected by setting the variable {\sl iseasurfalg}, as now +described. +\begin{enumerate} +\item Option {\sl iseasurfalg=0}. The original and most basic scheme + comprises a fixed value of Charnock's coefficient and a fixed scalar + roughness length. Typical values of Charnock's coefficient lie in + the range 0.011--0.018 and a typical value of the thermal roughness + length is $z_{0h(sea)}$ = 4x10$^{-5}$ m. + +\item Option {\sl iseasurfalg=1}. The use of a fixed thermal + roughness length, as above, leads to a rapid increase in the + exchange coefficient for moisture as the wind speed increases that + is at variance with observational evidence. A parametrization of the + scalar roughness length was developed from surface divergence theory + \cite[]{csanady2001}, as described by \cite{edwards2007}. This + involves an inverse dependence of $z_{0h}$ on the friction velocity + in the aerodynamically smooth limit and an inverse dependence of + $z_{0h}$ on $z_{0m}$ at higher wind speeds that reduces the increase + in the exchange coefficient with wind speed. + + During iteration of the equations of surface transfer to calculate + the Obukhov length, the friction velocity changes, so implicitly + changing the roughness lengths. Historically, in the algorithm + adopted in the Unified Model,roughness lengths have not been + modified within this iteration, with values from the previous + timestep being used. The scheme was therefore originally implemented + in a form that was based on conditions at the previous timestep, but + did not require adding $z_{0h}$ to the dump. To cope with conditions + of light winds, this required an iterative calculation of $v_\ast$ + from $z_{0m}$ from the previous timestep before the calculation of + the Obukhov length and $v_\ast$. Note that although there is not a + 1-1 relationship between $v_\ast$ and $z_{0m}$, the ambiguity is in + practice removed by the consideration that the inversion is only of + relevance in conditions of light winds. + + With this scheme a fixed value of Charnock's coefficient must be + specified as above. + +\item Option {\sl iseasurfalg=2}. An alternative version of the + foregoing scheme has been developed that includes full iteration of + the roughness lengths within the iteration for the Obukhov length. + +\item Option {\sl iseasurfalg=3}. This option provides various forms + of the COARE algorithm. The COARE algorithm exists in various forms + and continues to be developed. Version 3.0 \cite[]{fairall2003} has + been extensively used, while version 3.5 \cite[]{edson2013} has + recently been released. Whilst the full COARE algorithm provides a + complete description of surface transfer at the sea surface, here we + use only the expressions for the roughness lengths. + + In current versions of the scheme Charnock's coefficient is + specified using a linear relationship between the 10-m wind speed, + valid over a certain range of wind speeds, with fixed values outside + the range: + \begin{equation} + \alpha = a U_{10} +b + \label{eq:charn} + \end{equation} + for $U_{10,min} < U_{10} < U_{10,max}$. The constants $a$, $b$, + $U_{10,min}$ and $U_{10,max}$ differ between different versions of + the algorithm and are specified through namelist + parameters. Strictly, $U_{10}$ here should be the neutral 10-m wind + speed, but over the ocean the difference between the neutral and + stability-adjusted wind speeds is typically small, so the + distinction is often ignored. (Current practice in data assimilation + (2014) is to ignore the distinction). A logical switch is therefore + provided to enable the user to apply the formula using the true + neutral wind or the stability-adjusted wind, as preferred. + + The COARE algorithm does distinguish roughness lengths for heat and + moisture, but this is not currently feasible in the Unified Model, + so, since latent heat fluxes are dominant over the ocean, the scalar + roughness length is set using the expression for the moisture + roughness, + \begin{equation} + z_{0h} = \min(1.15\times 10^{-4}, 5.5\times 10^{-5}/Re_*^{0.6}), + \label{eq:z0h_coare} + \end{equation} + where $Re_*$ is the roughness Reynolds number. + + This scheme has been implemented is a form that allows the roughness + lengths to evolve during iteration to obtain the Obukhov length. + +\item Option {\sl iseasurfalg=4}. Equivalent to option {\sl iseasurfalg=1} + for a variable Charnock parameter. A fixed value of Charnock's coefficient + does not need to be provided. On the other hand, a Charnock field needs + to be provided via wave coupling or initialization. + +\item Option {\sl iseasurfalg=5}. Equivalent to option {\sl iseasurfalg=2} + for a variable Charnock parameter. A fixed value of Charnock's coefficient + does not need to be provided. On the other hand, a Charnock field needs + to be provided via wave coupling or initialization. + +\end{enumerate} + +The observations upon which these schemes are based do not extend to +10-m (neutral) wind speeds much above 20~ms${}^{-1}$ and there is +some uncertainty +over the behaviour of the drag at the wind speeds encountered in +tropical cyclones: indeed, there is considerable evidence that it +does not continue to increase in the manner predicted by schemes +like those described above and may even decrease. \cite{Donelan2004} +presents some measurements suggesting that the drag coefficient should +not be permitted to increase for 10-m neutral winds above about +33~ms${}^{-1}$, when the drag coefficient is about 0.0024. Whilst it is +likely that further work will be required on this topic, the possibility +of limiting the drag coefficient has been allowed for by introducing +the option {\tt i\_high\_wind\_drag} with the options +\begin{enumerate} +\item Option {\sl i\_high\_wind\_drag=0}. This is the default option +of making no modification to the standard scheme at high winds. +\item Option {\sl i\_high\_wind\_drag=1}. This option allows the +user to specify a maximum value of the (neutral) drag coefficient, +{\tt cdn\_max\_sea} (called {\tt cd\_limit\_sea} at versions below 11.5). +\item Option {\sl i\_high\_wind\_drag=2}. Like the previous option, +this allows the user to specify a maximum value of the neutral drag +coefficient, {\tt cdn\_max\_sea}, but at higher wind speeds the drag +coefficient is reduced and attains a limiting value, {\tt cdn\_hw\_sea}. +The reduction is linear in the wind speed between {\tt u\_cdn\_max} and +{\tt u\_cdn\_hw}. This reflects current understanding of the behaviour +of the sea surface at high wind speeds, with the neutral drag coefficient +saturating at around 35 ms${}^{-1}$ and declining at higher wind speeds. +Suggested values of these coefficients are based on \cite{donelan2018} +and \cite{hsu2017}. +\end{enumerate} +It might be thought more logical to subsume the treatment of high winds +under {\tt iseasurfalg}, but given that standard schemes for surface +exchange at lower wind speeds do not explicitly account for this range +of speeds and that the treatment of high wind speeds is less certain, +it is useful to consider the treatment of high wind speeds as a seperate +option. + +\subsubsection{Surface exchange over sea ice} + +As explained above, when sea ice is present, surface exchange involves +exchanges between the atmosphere and the open sea (L), the marginal +ice zone (MIZ) and the zone of pack ice. More mechanistically, one may +consider the interfacial exchanges over the sea and ice surfaces and the +contribution of form drag on the ice freeboard in the marginal ice zone. + +Two approaches are available in uncoupled configurations of the model. + +\begin{enumerate} + +\item +{The Original Scheme} +The exchange coefficients over the sea and sea ice regions of the +gridbox are interpolated between values representative of pack ice, +the marginal ice zone and open sea, using the ice +fraction, f$_{I}$. For 0 $\le $ f$_{I} <$ 0.7 +\begin{eqnarray} + < C_H >&=&( f_I C_{H(MIZ)} + ( 0.7 - f_I ) C_{H(L)} ) / 0.7 + \label{1.6.1} \\ + < C_D >&=&( f_I C_{D(MIZ)} + ( 0.7 - f_I ) C_{D(L)} ) / 0.7 + \label{1.6.2} +\end{eqnarray} +and for 0.7 $\le $ f$_{I} \le $ 1 +\begin{eqnarray} + < C_H >&=&( ( 1 - f_I ) C_{H(MIZ)} + ( f_I - 0.7 ) ) C_{H(I)} ) / 0.3 + \label{1.6.3} \\ + < C_D >&=&( ( 1 - f_I ) C_{D(MIZ)} + ( f_I - 0.7 ) ) C_{D(I)} ) / 0.3 + \label{1.6.4} +\end{eqnarray} +where +\begin{eqnarray} + C_{H(L)}&= &C_H ( L_{(L)} , z_{0m(sea)} , z_{0h(sea)} ) + \label{1.6.5} \\ + C_{H(MIZ)}&=& C_H ( L_{(I)} , z_{0m(MIZ)} , z_{0h(MIZ)} ) + \label{1.6.6} \\ + C_{H(I)}&=& C_H ( L_{(I)} , z_{0m(sea-ice)} , z_{0h(sea-ice)} ) + \label{1.6.7} +\end{eqnarray} +and similarly for the drag coefficient C$_{D}$. + +The roughness lengths over open sea are calculated as above, but those for +ice are prescribed and set using the gui or namelists. The typical +roughness length for pack ice, z$_{0m(sea-ice)}$ = 5x10$^{-4}$ m. +Historically, z$_{0h(sea-ice)}$ was set equal to z$_{0m(sea-ice)}$, +but more recently it has been set equal to one fifth of z$_{0m(sea-ice)}$, +based on \cite{andreas2010}. The setting for marginal ice is more +problematic. Whilst z$_{0m(MIZ)}$ should be larger than z$_{0m(sea-ice)}$, +good simulations of mean sea-level pressure are obtained only if +z$_{0m(MIZ)}$ is substantially greater than z$_{0m(sea-ice)}$ and a +value of 0.1m is typically used. The ratio of z$_{0h(MIZ)}$ to +z$_{0m(MIZ)}$ is standardly set to 0.2 in this case, but smaller values +might be more realistic. However, a better approach in the longer term +is to use an explicit representation of ice form drag. + +\item +{Explicit Treatment of Ice Form Drag} + +\cite{lupkes2012} have suggested a simple parametrization of the +form drag coefficient of marginal ice that has been found to +perform well in comparison to aircraft measurements (\cite{elvidge2016}). +\cite{lupkes2015} have extended the parametrization to include the effects +of stability. When coupled to CICE, it is intended that a more elaborate +scheme will be used, but this scheme is useful for application in +atmosphere-only simulations and its implementation is now described. + +The fundamental quantity involved in representing the drag is the +pressure force on the ice free-board in the up-stream flow, +\begin{equation} +F_p =\int_{z_0}^{h_f} \frac{\rho}{2} [u(z)]^2 \, dz. +\end{equation} +$u(z)$ will in general exhibit a mixed character, but it may be taken as +the developed flow over open sea, as in \cite{lupkes2012}, or may be +interpolated between the developed flows over open sea or pack ice, depending +on the ice fraction, as in \cite{lupkes2015}. +In principle, it will be subject to the effects of stability, but +since the free-board does not much exceed 0.5m, these effects are small +(\cite{lupkes2015}) and the flow may be taken as neutral up to $h_f$. +Hence, +\begin{equation} +F_p \approx \frac{h_f}{2k^2} \rho u_*^2 \left [ (\log(h_f/z_0) -1)^2 +1 +\right ] = +\frac{h_f}{2k^2} \rho C_d U_1^2 \left [(\log(h_f/z_0) -1)^2 +1 \right ]. +\label{eq:int_u2} +\end{equation} +where $C_d$ is the upstream drag coefficient and $U_1$ is the wind +on the model's lowest atmospheric level. Because this will be +significantly above $h_f$, the stability dependence of $C_d$ should +be considered here (again see \cite{lupkes2015}). $U_1$ may be interpreted +as the wind at a specific height, or, consistenly with the flux-difference +form of the momentum equation, as the layer-averaged velocity. This +distinction affects the numerical value of $C_d$, but does not otherwise +affect the foregoing equation. If using the original version of the +scheme (\cite{lupkes2012}), $C_d$ must be taken as the neutral drag +coefficient. Note also that various approximations may be made in +Equation~\ref{eq:int_u2}. \cite{lupkes2012} approximate +$(\log(h_f/z_0) -1)^2 +1$ as $(\log(h_f/z_0) )^2 $; while \cite{lupkes2015} +approximate it as $(\log(h_f/z_0) -1)^2 $. Here we retain the full expression. + +If, in a unit area, there are $N$ floes, each of crosswind dimension $D_i$, +the total drag will be +\begin{equation} +F_d = N c_w S_c^2 D_i F_p, +\label{eq:fd_fp} +\end{equation} +where $c_w$ is a coefficient and $S_c$ is a sheltering coefficient. The +fractional coverage of sea ice within this unit area is $ND_i^2 c_s$, +where $c_s$ is related to the shape of the floe. Overall, the drag per +unit area of {\em ice} is +\begin{equation} +f_d = \frac{h_f}{2k^2} c_e \rho C_d U_1^2 S_c^2 \frac{A}{D_i} +\left [(\log(h_f/z_0) -1)^2 +1 \right ], +\end{equation} +where $c_e=c_w/c_s$. Assuming that $U_1$ is blended, it follows that the +form drag coefficient is +\begin{equation} +C_{df} = \frac{h_f}{2k^2} c_e S_c^2 \frac{1}{D_i} +\left [(\log(h_f/z_0) -1)^2 +1 \right ]. +\end{equation} +Defining, $L=(\log(h_f/z_0) -1)^2 +1$ and interpolating in the ice fraction, +\begin{equation} +C_{df} = \frac{c_e}{2}\frac{h_f}{D_i}\frac{S_c^2}{k^2} \left [ +(1-A) C_{ds} L_s + A C_{di} L_i \right ], +\end{equation} +where the sheltering factor is taken to be the same over ice and water. +\cite{lupkes2012} provides parametrizations for quantities such as +$h_f$, while \cite{elvidge2016} provide suggested values for the +constants in the scheme, based on observations. In using these values +in the Unified Model, $c_e$ should be increased by about 30\% to represent +the effect of differing approximations of the logarithmic wind profile. + +For scalar transfer \cite{lupkes2015} suggest adding a contribution to +the sensible heat flux to represent the impact of form drag; however, +the mechanistic physical basis of the scheme they propose is unclear. +Moreover, when combined with the interfacial drag, this suggests scalar +transfer much larger than observed by \cite{schroder2003}. Consequently, +no enhancement of the scalar transfer coefficient by form drag is +included. + +The overall drag coefficients are now set by interpolation in the ice fraction: +\begin{eqnarray} + < C_D >&=& (1 - f_I) C_{D(L)} + f_I (C_{D(I)} + C_{D(FRM)}) + \label{eq:cdice_int} \\ + < C_H >&=& (1 - f_I) C_{H(L)} + f_I C_{H(I)} + \label{eq:chice_int} +\end{eqnarray} + +\end{enumerate} + + +\subsubsection{Surface exchange in coastal grid-boxes} +\label{sec:coast} + +In coupled ocean-atmosphere modelling the ocean requires appropriate +surface stresses and fluxes over all ocean points. In coastal regions +this means providing sea-surface fluxes from atmospheric grid-boxes +that are partly sea and partly land. This is achieved through coastal +tiling, where the ocean part of the grid box is effectively treated in +the same way as other land surface tiles. However, because of the +very different roughness characteristics of land and sea, this can +lead to serious biases especially in the ocean surface fluxes. One +particular problem that has been identified is that, compared to a +neighbouring sea-only point, coastal points tend to have slower +near-surface wind speeds (because of the rough land surface fraction) +but the ocean surface exchange will still use a typical very small +roughness length. Thus the diagnosed ocean surface stress and +sensible and latent heat fluxes are all significantly smaller than a +neighbouring sea point. Although truly coastal winds are notoriously +complex, in a typical climate model grid box (of 100km or more) the +vast majority of the sea area will be unaffected by the land. Thus a +partial solution to this problem is to take the wind speed over the +sea part of coastal points as the average of that over the +neighbouring sea points. The wind speed over the land component is +then slowed (by up to a factor of 5) to maintain the grid box mean +wind speed. + +\subsection{Surface roughness lengths and resistances to evaporation + over land}\label{section_1.7} +For land points the roughness lengths excluding orographic effects are +specified from land use datasets. The vegetative roughness length for +scalars is assumed to be 0.1 of that for momentum. This is a simple +approximation; in reality the factor depends on the land cover type +and the degree of heterogeneity. [Future versions of the Unified Model +will treat surface heterogeneity explicitly by the ``tiling'' method.] + +The surface moisture flux given by~(\ref{1.1.8}) or (\ref{1.1.17}) +involves a surface humidity value, q$_{0}$. Prior to UM6.3, for +evaporation from all of ocean, sea-ice, lake and snow-covered surfaces +as well as from water on vegetative canopies this surface value is +taken to be the saturated specific humidity at the surface (skin) +temperature and pressure, q$_{sat}$(T$_{0}$,p$_{0})$. [Saturation is +respect to liquid water or ice depending on which the surface is.] +The saturation vapour pressure of a liquid, though, is lowered by +dissolved ionic substances. For typical sea salinities the saturated +vapour pressure is only about 90\% of the value over pure water. From +UM6.3, therefore, there is the option to include this effect, so that +the parametrization of the surface moisture flux over the sea becomes +\begin{equation*} + E_0 = - \rho_0 c_H V (q_1 - 0.98 q_{sat}(T_0,p_0) ) +\end{equation*} +Evapotranspiration through vegetation receives a special treatment +because it is controlled by the physiology of the plants. The +formulation is described in full in the documentation for the land and +ice surface processes component of the Unified Model. The +evapotranspiration for the surface is given by +\begin{equation} + E_t = - \rho _0 \frac{ q_1 - q_{sat} ( T_0 , p_0 )}{( r_a + r_s )} + \label{1.7.1} +\end{equation} +where the aerodynamic resistance, r$_{a}$ , is given by +\begin{equation} + r_a = \frac{1}{ C_H } = \frac{1}{ c_H V} + \label{1.7.2} +\end{equation} +and r$_{s}$ is the surface or stomatal resistance to +evaporation. r$_{s}$ is a function of the available soil moisture, +near surface atmospheric conditions and the radiation impinging on the +plants. [For the formulation see the documentation for the land and +ice surface processes component of the Unified Model.] A similar +formula to~(\ref{1.7.1}) is used for the evaporation from the very +near surface soil layer. Equation~(\ref{1.7.1}) can be written as +\begin{equation} + E_t = - \rho _0 C_E ( q_1 - q_{sat} ( T_0 , p_0 ) ) + \label{1.7.3} +\end{equation} +where +\begin{equation} + C_E = \frac{ C_H }{\left( {1 + \frac{ r_s }{ r_a }} \right)} + \label{1.7.4} +\end{equation} + +\subsection{The modifications needed to incorporate orographic form + drag.} +\label{section_2} +\subsubsection{Effective roughness lengths}\label{section_2.1} +Form drag is included in the surface turbulent flux formulation via +effective roughness lengths for momentum \cite[]{wood93} and for +scalar quantities \cite[]{hewer1998}. The formulae of +section~\ref{section_1} are interpreted as relationships between +gridbox mean quantities and fluxes with the roughness lengths replaced +by effective values, z$_{0m(eff)}$ and z$_{0h(eff)}$. + +When form drag is included via effective roughness lengths equations +(\ref{1.1.7})-(\ref{1.1.9}) become: +\begin{eqnarray} + \frac{ H_{0(eff)} }{ c_P \rho _0 }&=&\frac{-k}{ \Phi _h (L , z_1 + z_{0m(eff)} , z_{0h(eff)} )} v_{\ast (eff)}\nonumber\\ + && \left( {\Delta T + \frac{g}{c_p }( z_1 + z_{0m(eff)} - z_{0h(eff)} )} \right) + \label{2.1.1} \\ + \frac{ E_{0(eff)} }{ \rho _0 }&=&\frac{-k}{ \Phi _h (L , z_1 + z_{0m(eff)} , z_{0h(eff)} )} v_{\ast (eff)} \Delta q + \label{2.1.2} \\ + \frac{ {\rm {\bf \tau }}_{{0(eff)}} }{ \rho _{0} }&=&\frac{k}{ \Phi _m (L , z_1 + z_{0m(eff)} , z_{0m(eff)} )} v_{\ast (eff)} \Delta {\rm {\bf v}} + \label{2.1.3} +\end{eqnarray} +The effective surface scaling velocity, v$_{\ast (eff)}$ , is given by +(cf. (\ref{1.1.25})) +\begin{equation} + v_{\ast (eff)}^2 = u_{\ast (eff)}^2 + \gamma _t^2 w_\ast ^2 + \gamma _c^2 w_c^2 + \label{2.1.4} +\end{equation} +where +\begin{equation} + u_{\ast (eff)}^2 = \left| { {\rm {\bf \tau }}_{{0(eff)}} {/} \rho _{0} } \right| + \label{2.1.5} +\end{equation} +The effective roughness for momentum is derived by setting the total +effective surface stress, \textbf{$\tau $}$_{0(eff)}$, to the sum of +the surface stress over a flat surface with the same vegetative +roughness, \textbf{$\tau $}$_{0(f)}$, and the orographic pressure drag +force at the surface, \textbf{$\tau $}$_{0(p)}$. The stresses are +evaluated in terms of the velocity at height z$_{c}$ above the +surface. z$_{c}$ is currently set to 2$^{1/2}\sigma _{h}$ where +$\sigma _{h}$ is the standard deviation of the unresolved orographic +height. Thus +\begin{equation} + \frac{ {\rm {\bf \tau }}_{{0(eff)}} }{ \rho _{0} }{ = }\frac{{k } {v}_{{\ast (eff)}} }{ \Phi _{m} {(L , } {z}_{c} { , } {z}_{{0m(eff)}} {)}}{ }{\rm {\bf v}}{(} {z}_{c} {)} + \label{2.1.6} +\end{equation} +and +\begin{equation} + \frac{ {\rm {\bf \tau }}_{{0(f)}} }{ \rho _{0} }{ = }\frac{{k } {v}_{{\ast (f)}} }{ \Phi _{m} {(L , } {z}_{c} { , } {z}_{{0m}} {)}}{ }{\rm {\bf v}}{(} {z}_{c} {)} + \label{2.1.7} +\end{equation} +where the scaling velocity based on the stress over a flat surface, v$_{\ast +(f)}$ , is given by +\begin{equation} + v_{\ast (f)}^2 = u_{\ast (f)}^2 + \gamma _t^2 w_\ast ^2 + \gamma _c^2 w_c^2 + \label{2.1.8} +\end{equation} +with +\begin{equation} + u_{\ast (f)}^2 = \left| { {\rm {\bf \tau }}_{{0(f)}} {/} \rho _{0} } \right| + \label{2.1.9} +\end{equation} +[The scaling velocity which appears in the expression~(\ref{1.1.4}) for the +Monin-Obukhov length is chosen to be v$_{\ast (eff)}$ rather than the flat +surface value.] + +The orographic stress is given by +\begin{equation} + \frac{ {\rm {\bf \tau }}_{{0(p)}} }{ \rho _{0} }{ = }\frac{{1}}{{2}}{ } {c}_{{D(orog)}} { } {f}_{D} {(} {{Ri}}_{B} {)}\frac{{A}}{{S}}{ }\left| {{\rm {\bf v}}{(} {z}_{c} {)}} \right|{ }{\rm {\bf v}}{(} {z}_{c} {)} + \label{2.1.10} +\end{equation} +where $A/S$ is the total silhouette area of orography in a gridbox +over the flat surface area of the gridbox taken as an average over all +directions. The function f$_{D}$ is a function of the bulk Richardson +number of the surface layer and is set to 1 for Ri$_{SL} <$ 0 and +decreases linearly to zero at Ri$_{SL(crit)}$ = 0.5. The orographic +drag coefficient c$_{D(orog)}$ is set to the constant value (typically +0.3, \cite{mason1986}). + +If the function $\Phi _{m}$ and v$_{\ast }$ are approximated by their +neutral values in~(\ref{2.1.6}) and~(\ref{2.1.7}) then the equation +for calculating the effective momentum roughness is derived +\begin{equation} + \frac{\ln ( z_c / z_{0m(eff)} )}{\ln ( z_c / z_{0m} )} = {\left( {1 + \frac{1}{2} c_{D(orog)} f_D \frac{A}{S} {\left( {\frac{\ln ( z_c / z_{0m} )}{k}} \right)}^2 } \right)}^{-1/2} + \label{2.1.12} +\end{equation} +The stress for the flat surface is related to the total stress by +\begin{equation} + {\rm {\bf \tau }}_{{0(f)}} { = } {\rm {\bf \tau + }}_{{0(eff)}} { } {\left( {{1 + }\frac{{1}}{{2}}{ } + {c}_{{D(orog)}} { } {f}_{D} { }\frac{{A}}{{S}}{ } {\left( + {\frac{\ln {(} {z}_{c} { / } {z}_{{0m}} {)}}{{k}}} + \right)}^{2} } \right)}^{{-1}} + \label{2.1.13} +\end{equation} +which is derived from equations~(\ref{2.1.6}), (\ref{2.1.7}) and +(\ref{2.1.10}). Equation~(\ref{2.1.13}) implies that +\begin{equation} + C_{D(f)} = C_{D(eff)} {\left( {1 + \frac{1}{2} c_{D(orog)} f_D \frac{A}{S} {\left( {\frac{\ln ( z_c / z_{0m} )}{k}} \right)}^2 } \right)}^{-1} + \label{2.1.14} +\end{equation} + + +\subsubsection{Parametrized orographic drag coefficient} + +\cite{wood93} find that orographic drag coefficient c$_{D(orog)}$ +depends on A/S via the equation +\begin{equation} + c_{D(orog)} = 2\alpha \beta \pi ^2 \frac{A}{S} \frac{ u_{\ast (f)}^2 }{ v^2 ( z_c )} + \label{2.1.11} +\end{equation} +where $\alpha $ and $\beta $ are constants ($\alpha $=12 and $\beta $=1). + +If (\ref{2.1.11}) is used, the formula for the effective roughness length +for momentum becomes +\begin{equation} + \frac{\ln ( z_c / z_{0m(eff)} )}{\ln ( z_c / z_{0m} )} = {\left( {1 + \alpha \beta \pi ^2 f_D {\left( {\frac{A}{S}} \right)}^2 } \right)}^{-1/2} + \label{2.1.15} +\end{equation} +and~(\ref{2.1.13}) and (\ref{2.1.14}) become +\begin{eqnarray} + {\rm {\bf \tau }}_{{0(f)}} &=& {\rm {\bf \tau + }}_{{0(eff)}} {\left( {{1 + }\alpha \beta \pi ^{2} { } {f}_{D} { } + {\left( {\frac{{A}}{{S}}} \right)}^{2} { }} \right)}^{-1} + \label{2.1.16} \\ + C_{D(f)}&= &C_{D(eff)} {\left( {1 + \alpha \beta \pi ^2 f_D {\left( {\frac{A}{S}} \right)}^2 } \right)}^{-1} + \label{2.1.17} +\end{eqnarray} +The effective surface flux of scalar X evaluated in terms of values at +z$_{c}$ is +\begin{equation} + \frac{ F_{X0(eff)} }{ \rho _0 } = \frac{k v_{\ast (eff)} }{ \Phi _h (L , z_c , z_{0h(eff)} )} (X( z_c ) - X_0 ) + \label{2.1.18} +\end{equation} +and the surface flux for the flat surface is given by +\begin{equation} + \frac{ F_{X0(f)} }{ \rho _0 } = \frac{k v_{\ast (f)} }{ \Phi _h (L , z_c , z_{0h} )} (X( z_c ) - X_0 ) + \label{2.1.19} +\end{equation} + +\cite{hewer1998} find that the scalar transport is enhanced when there +is orographic form drag such that +\begin{equation} + F_{X0(eff)} = F_{X0(f)} {\left( {1 - 2.2 f_D \frac{A}{S}} \right)}^{-1} + \label{2.1.20} +\end{equation} +Combining~(\ref{2.1.18})--(\ref{2.1.20}) and using the neutral values +of the stability functions the expression for the effective scalar +roughness length is derived as +\begin{equation} + \frac{\ln ( z_c / z_{0h(eff)} )}{\ln ( z_c / z_{0h} )} = {\left( {1 + \frac{1}{2} c_{D(orog)} f_D \frac{A}{S} {\left( {\frac{\ln ( z_c / z_{0m} )}{k}} \right)}^2 } \right)}^{1/2} \left( {1 - 2.2 f_D \frac{A}{S}} \right) + \label{2.1.21)} +\end{equation} +which becomes +\begin{equation} + \frac{\ln ( z_c / z_{0h(eff)} )}{\ln ( z_c / z_{0h} )} = {\left( {1 + \alpha \beta \pi ^2 f_D {\left( {\frac{A}{S}} \right)}^2 } \right)}^{1/2} \left( {1 - 2.2 f_D \frac{A}{S}} \right) + \label{2.1.22} +\end{equation} +if the \cite{wood93} formulation is used. + +\subsubsection{The iterative algorithm for calculating the effective + surface exchange coefficients}\label{section_2.2} + +For unstable conditions, i.e. $\Delta $B $<$ 0 : + +IF $\Delta $\textbf{v} $<$ 2 ms$^{-1}$ then start the iteration from +the convective limit, so +\begin{eqnarray} + \frac{1}{ L^{(0)} }&=&\frac{-k}{ \gamma _t^3 z_i } + \label{2.2.1} \\ + \Phi _m^{(0)}&=& \Phi _m ( L^{(0)} , z_1 + z_{0m(eff)} , z_{0m(eff)} ) + \label{2.2.2} \\ + \Phi _h^{(0)}&=& \Phi _h ( L^{(0)} , z_1 + z_{0m(eff)} , z_{0h} ) + \label{2.2.3} \\ + v_{\ast (eff)}^{(0)}&=& v_{\ast (f)}^{(0)} = {\left[ { \gamma _t^3 \left( {\frac{k}{ \Phi _h^{(0)} }} \right) z_i \left| {-\Delta B} \right| + \gamma _c^2 w_c^2 } \right]}^{ 1/2} + \label{(2.2.4} +\end{eqnarray} + +ELSE IF ($\Delta $\textbf{v} $\ge $ 2 ms$^{-1}$ ) start iteration from +the neutral end, so +\begin{eqnarray} + \Phi _m^{(0)}&=&\ln \left( {\frac{ z_1 + z_{0m(eff)} }{ z_{0m(eff)} }} \right) + \label{2.2.5} \\ + \Phi _h^{(0)}&=&\ln \left( {\frac{ z_1 + z_{0m(eff)} }{ z_{0h} }} \right) + \label{2.2.6} \\ + u_{\ast (eff)}^{(0)}&=&\frac{k}{ \Phi _m^{(0)} } \left| {\Delta {{{v}}}} \right| + \label{2.2.7} \\ + v_{\ast (eff)}^{(0)}&=& {\left( { u_{\ast (eff)}^{(0) 2} + \gamma _c^2 w_c^2 } \right)}^{ 1/2} + \label{2.2.8} \\ + u_{\ast (f)}&=& u_{\ast (eff)} \frac{\ln ( z_c / z_{0m(eff)} )}{\ln ( z_c / z_{0m} )} + \label{2.2.9} \\ + v_{\ast (f)}^{(0)}&=& {\left( { u_{\ast (f)}^{(0) 2} + \gamma _c^2 w_c^2 } \right)}^{ 1/2} + \label{2.2.10} +\end{eqnarray} + +END IF. + +Then calculate: +\begin{eqnarray} + C_{D(eff)}^{(0)}&=&\frac{k}{ \Phi _m^{(0)} } v_{\ast (eff)}^{(0)} + \label{(2.2.11} \\ + C_{H(eff)}^{(0)}&=&\frac{k}{ \Phi _h^{(0)} } v_{\ast (eff)}^{(0)} + \label{(2.2.12} \\ + C_{D(f)}^{(0)}&=& C_{D(eff)}^{(0)} {\left( {1 + \frac{1}{2} c_{D(orog)} f_D \frac{A}{S} {\left( {\frac{\ln ( z_c / z_{0m} )}{k}} \right)}^2 } \right)}^{-1} + \label{(2.2.13} \\ + C_{H(f)}^{(0)}&=& C_{H(eff)}^{(0)} \left( {1 - 2.2 f_D \frac{A}{S}} \right) + \label{(2.2.14} +\end{eqnarray} + +Having set up initial values the iteration loop can be entered: + +DO n = 1 to N +\begin{eqnarray} + u_{\ast (eff)}^{(n)2}&=& C_{D(eff)}^{(n-1)} \left| {\Delta {{\rm {\bf v}}}} \right| + \label{(2.2.15} \\ + u_{\ast (f)}^{(n)2}&=& C_{D(f)}^{(n-1)} \left| {\Delta {{\rm {\bf v}}}} \right| + \label{(2.2.16} \\ + {\left( {\frac{ F_{B0} }{ \rho _0 }} \right)}^{(n)}&=&- { C_{H(eff)} }^{(n-1)} \Delta B + \label{(2.2.17} \\ + w_\ast ^{(n)}&=& {\left[ { z_i {\left( {\frac{ F_{B0} }{ \rho _0 }} \right)}^{(n)} } \right]}^{ 1/3} + \label{(2.2.18} \\ + v_{\ast (eff)}^{(n)2}&=& u_{\ast (eff)}^{(n)2} + \gamma _t^2 w_\ast ^{(n)2} + \gamma _c^2 w_c^2 + \label{(2.2.19} \\ + v_{\ast (f)}^{(n)2}&= &u_{\ast (f)}^{(n)2} + \gamma _t^2 w_\ast ^{(n)2} + \gamma _c^2 w_c^2 + \label{(2.2.20} \\ + \frac{1}{ L^{(n)} }&=&\frac{-k( F_{B0} / \rho _0 )^{(n)} }{ v_{\ast (eff)}^{(n)3} } + \label{(2.2.21} \\ + \Phi _m^{(n)}&=& \Phi _m ( L^{(n)} , z_1 + z_{0m(eff)} , z_{0m(eff)} ) + \label{(2.2.22} \\ + \Phi _h^{(n)}&=& \Phi _h ( L^{(n)} , z_1 + z_{0m(eff)} , z_{0h} ) + \label{(2.2.23} \\ + C_{D(eff)}^{(n)}&=&\frac{k}{ \Phi _m^{(n)} } v_{\ast (eff)}^{(n)} + \label{(2.2.24} \\ + C_{H(eff)}^{(n)}&=&\frac{k}{ \Phi _h^{(n)} } v_{\ast (eff)}^{(n)} + \label{(2.2.25} \\ + C_{D(f)}^{(n)}&=& C_{D(eff)}^{(n)} {\left( {1 + \frac{1}{2} c_{D(orog)} f_D \frac{A}{S} {\left( {\frac{\ln ( z_c / z_{0m} )}{k}} \right)}^2 } \right)}^{-1} + \label{(2.2.26} \\ + C_{H(f)}^{(n)}&=& C_{H(eff)}^{(n)} \left( {1 - 2.2 f_D \frac{A}{S}} \right) + \label{(2.2.27} +\end{eqnarray} + +END DO. + +For neutral and stable conditions ($\Delta $B $\ge $ 0) start the +iteration from the neutral values and set w$_{\ast }$=0 in the above +iteration loop. Use the final (N) values of C$_{H(eff)}$ and +C$_{D(eff)}$ to calculate the surface sensible and latent heat fluxes +and surface stress: +\begin{eqnarray} + H_{0(eff)}&=&- c_P \rho _0 C_{H(eff)}^{(N)} \left( {\Delta T + \frac{g}{c_p }( z_1 + z_{0m(eff)} - z_{0h} )} \right) + \label{2.2.28} \\ + E_{0(eff)}&=& {-\rho }_0 C_{H(eff)}^{(N)} \Delta q + \label{2.2.29} \\ + {\rm {\bf \tau }}_{{0(eff)}} &=& \rho _0 C_{D(eff)}^{(N)} \Delta {\rm {\bf v}} + \label{2.2.30} +\end{eqnarray} +The stress for a flat surface, if required for output, is calculated from +\begin{equation} + {\rm {\bf \tau }}_{{0(f)}} { = } \rho _0 + C_{D(f)}^{(N)} \Delta {\rm {\bf v}} + \label{2.2.31} +\end{equation} + +\subsubsection{Interpolation of surface layer variables to standard + observation heights}\label{section_2.3} +If the observation height wind is assumed to lie on the profile +defined by the effective roughness length and surface scaling velocity +then (c.f. equation~(\ref{1.5.1})) +\begin{equation} + {\rm {\bf v}}_{{ob}} { = } {\rm {\bf v}}_{0} { + + }\frac{ {\rm {\bf \tau }}_{{0(eff)}} }{ \rho _0 v_{\ast (eff)} k} + \Phi _m (L, z_{ob} + z_{0m(eff)} , z_{0m(eff)} + \label{2.3.1} +\end{equation} +Using the expression for the surface turbulent stress this becomes +\begin{equation} + {\rm {\bf v}}_{{ob}} { = } {\rm {\bf v}}_{0} { + + }\frac{ C_{D(eff)} }{ {kv}_{\ast (eff)} } \Phi _m (L, z_{ob} + + z_{0m(eff)} , z_{0m(eff)} ) ( {\rm {\bf v}}_1 - {\rm {\bf v}}_{0} + {)} + \label{2.3.2} +\end{equation} +For wind z$_{ob}$ is set to 10m and the last iteration (N) values of +C$_{D(eff)}$, L and v$_{\ast (eff)}$ are used. Alternatively if the +observation height wind is assumed to lie on a profile defined by the +flat surface roughness length and scaling velocity then +\begin{equation} + {\rm {\bf v}}_{{ob}} { = } {\rm {\bf v}}_{0} { + + }\frac{ {\rm {\bf \tau }}_{{0(f)}} }{ \rho _0 v_{\ast (f)} k} \Phi + _m (L, z_{ob} + z_{0m} , z_{0m} ) + \label{2.3.3} +\end{equation} +and substituting for the surface stress this becomes +\begin{equation} + {\rm {\bf v}}_{{ob}} { = } {\rm {\bf v}}_{0} { + + }\frac{ C_{D(f)} }{k v_{\ast (f)} } \Phi _m (L, z_{ob} + z_{0m} , + z_{0m} ) ( {\rm {\bf v}}_1 - {\rm {\bf v}}_{0} {)} + \label{2.3.4} +\end{equation} +Most configurations of the Unified Model currently use the latter +assumption with the last iteration value of C$_{D(f)}$, L and $v_{\ast + (f)}$ used in the interpolation formula. + +If the observation height scalar quantities are assumed to lie on the +mean profile defined by the effective roughness length and scaling +quantities then (c.f. equation~(\ref{1.5.3}) we obtain for the generic +scalar $X$ ($T+(g/c_{P})z$, $q$, tracer amount) +\begin{equation} + X_{ob} = X_0 + \frac{ F_{X0(eff)} }{ \rho _0 v_{\ast (eff)} k} \Phi _h (L, z_{ob} + z_{0h(eff)} , z_{0h(eff)} ) + \label{2.3.5} +\end{equation} +and using the expression for the surface flux of the scalar quantity +$X$ this becomes +\begin{equation} + X_{ob} = X_0 + \frac{ C_{H(eff)} }{k v_{\ast (eff)} } \Phi _h (L, z_{ob} + z_{0h(eff)} , z_{0h(eff)} ) ( X_1 - X_0 ). + \label{2.3.6} +\end{equation} +Alternatively if the observation height scalar quantities are assumed +to lie on a profile defined by the flat surface roughness length and +flux then +\begin{equation} + X_{ob} = X_0 + \frac{ C_{H(f)} }{k v_{\ast (f)} } \Phi _h (L, z_{ob} + z_{0h} , z_{0h} ) ( X_1 - X_0 ) + \label{2.3.7} +\end{equation} +For temperature and humidity z$_{ob}$ is set to the screen height (1.5 +m) and the last iteration (N) values of C$_{H}$, L and v$_{\ast }$ are +used. + +\subsection{Distributed form drag -- an alternative to the effective + roughness length parametrization}\label{section_2.4} +An alternative representation of the turbulent form drag due to +sub-grid hills is the explicit orographic stress parametrization +proposed by \cite{wood01:_param}. In this representation the drag is +represented via an orographic stress term, applied directly to the +horizontal momentum equations. The roughness lengths remain at the +vegetative values and no adjustment to the roughness lengths for +scalar quantities is made. + +The turbulent form drag is represented by the term +\begin{equation} + {\bf f}=\frac{1}{\rho}\frac{\partial}{\partial z}{\bf\tau}_{\rm orog} + \label{eq:drag} +\end{equation} +on the right-hand side of the horizontal momentum equation, where +${\bf\tau}_{\rm orog}$ is the horizontal vector containing the extra +stress imparted on the flow by the sub-grid orography This term is +included in the Unified Model as an additional explicit (in terms of +time discretisation) stress. Following \cite{wood01:_param} we define +${\bf\tau}_{\rm orog}$ to be +\begin{equation} + {\bf\tau}_{\rm orog}(z)=\left({F_p}_x,{F_p}_y\right)e^{-z/\ell}, +\end{equation} +where ${\bf F_p}=({F_p}_x,{F_p}_y)$, ${F_p}_x$ and ${F_p}_y$ are the +grid-box average $x$ and $y$ components of the pressure force on the +sub-grid orography, and $\ell$ is a decay scale. We define $\ell$ such +that +\begin{equation} + \ell={\rm min}\left(\lambda,\frac{z_h}{3}\right), + \label{eq:l} +\end{equation} +where $z_h$ is the boundary-layer depth and $\lambda$, a somewhat ill +defined quantity, is related to the horizontal scales of the sub-grid +hills (and set to 300 m). Note that the value of $\ell$ obtained from +Eq.~(\ref{eq:l}) is further constrained to be at least 100 m. + +If the steep-hill expression is to be used, the surface stress applied is +almost identical to that used in the effective roughness parametrization +(Eq.~\ref{2.1.10}), namely: +\begin{equation} + \frac{\bf F_p}{\rho_0}=\frac{1}{2}c_{D(orog)} f_D (Ri_{B}) \frac{A}{S} + \left\vert{\rm{\bf v}}(\ell) \right\vert{\rm{\bf v}}(\ell), + \label{eq:dragsteep} +\end{equation} +the main difference being the dependence on the height scale $\ell$ rather +than $z_c$. Similarly, if the \cite{wood93} low-hill expression is used, the +surface stress is given by the equivalent of (Eq.~\ref{2.1.16}), namely: +\begin{equation} + \frac{\bf F_p}{\rho_0} = {\left( {\frac{\kappa}{\zeta_m}} \right)}^{2} + \alpha \beta \pi ^{2} {f}_{D} (Ri_{B}) + {\left( {\frac{A}{S}} \right)}^{2} + \left\vert{\rm{\bf v}}(\ell) \right\vert{\rm{\bf v}}(\ell), + \label{eq:draglow} +\end{equation} +where $\zeta_m = {\rm log}(\ell/z_{0m})$. There is also an option to use the +low-hill stress (\ref{eq:draglow}) but capped by that from the steep hill +expression (\ref{eq:dragsteep}), to avoid generating huge stresses at large $A/S$. + +There is also a choice for the Richardson number, $Ri_{B}$, that appears +in the stability dependence, $f_D$, which can either use $Ri_{B}=Ri_{SL}$ +(as with the effective roughness length version) or $Ri_{B \ell}$, a bulk +Richardson number between the surface and the scale height, $\ell$: +\begin{equation*} + Ri_{B \ell} = \frac{ \ell \left( g \left( + \overline{\beta_T}_{k\ell} ({\thetal}_{k\ell}-{\thetal}_{1}) + + \overline{\beta_q}_{k\ell} ({q_t}_{k\ell}-{q_t}_{1}) \right) + + \Delta b_{SL} \right) }{U^2(\ell)} +\end{equation*} +where the stability of the atmosphere between the surface and the bottom +model level is included via $\Delta b_{SL}$, which is the numerator of +$Ri_{SL}$, $U$ is the wind speed and the subscript $k \ell$ indicates +the $\theta$-level containing $\ell$. + +%------------------------------------------------------------------------ +% Section: implicit solver +%------------------------------------------------------------------------ +\section{Implicit solution of the diffusion equation} +\label{sec:implicit} + +\subsection{Unconditionally stable implicit solver} + +This is the vertical diffusion scheme of \cite{woodetal2007} which has +the advantages of (i) unconditional stability and non-oscillatory +behaviour for practical NWP cases and (ii) monotonic damping for +suitable choices of a free parameter $P$ which represents the degree +of nonlinearity of the diffusion problem to be solved. If the chosen +value of $P$ is equal to the real value of $P$ then the scheme is +second order accurate. In practical simulations $P$ may vary from +timestep to timestep and from column to column. + +\subsubsection{Algorithmic description} + +Consider the non-linear damping equation: +\begin{equation} + \frac{dX}{dt}=-\left(KX^{P}\right)X+S + \label{eq:damp1} +\end{equation} +Here $S$ is a constant forcing, or source, term and $KX^{P}$ is the +diffusion coefficient, with $K$ constant. $P$ is assumed to be +positive. The new scheme is written +\begin{equation} + \frac{X^{*}-X^{n}}{\Delta t}=-\Ical_{1}\left[K\left(X^{n}\right)^{P}\right] + X^{*}+\Ecal_{1}\left[K\left(X^{n}\right)^{P}\right] + X^{n}+\left(\Ical_{1}-\Ecal_{1}\right)S,\label{eq:sppf1} +\end{equation} +\begin{equation} + \frac{X^{n+1}-X^{*}}{\Delta + t}=-\Ical_{2}\left[K\left(X^{n}\right)^{P}\right]X^{n+1} + + \Ecal_{2}\left[K\left(X^{n}\right)^{P}\right]X^{*} + + \left(\Ical_{2}-\Ecal_{2}\right)S,\label{eq:sppf2} +\end{equation} +where +\begin{equation} + \Ecal_{1}=\left(1+\frac{1}{\sqrt{2}}\right) + \left[P+\frac{1}{\sqrt{2}}\pm\sqrt{P + \left(\sqrt{2}-1\right)+\frac{1}{2}}\right] + \label{eq:E1coeff} +\end{equation} +\begin{equation} + \Ecal_{2}=\left(1+\frac{1}{\sqrt{2}}\right) + \left[P+\frac{1}{\sqrt{2}}\mp\sqrt{P\left(\sqrt{2}-1\right)+\frac{1}{2}}\right] + \label{eq:E2coeff} +\end{equation} +\begin{equation} + \Ical_{1}=\Ical_{2}=\left(1+\frac{1}{\sqrt{2}}\right)\left(1+P\right) + \label{eq:Icoeff} +\end{equation} +Consider the one-dimensional ``forced'' boundary layer diffusion +equation +\begin{equation} + \frac{\partial X}{\partial t}=\frac{\partial F}{\partial z}+S, + \qquad F=K_{X}\frac{\partial X}{\partial z} + \label{eq:vdiff1} +\end{equation} +where $X$ is the scalar variable being diffused, $F$ is the flux of +$X$, $t$ is the time, $z$ is the height from the earth's surface, and +$K$ is the diffusion coefficient which is often non-constant and +depends on $X$ (i.e. the PDE is non-linear) and $S$ is a forcing term +from other processes preceding the boundary layer. In the UM these +processes are: microphysics, gravity wave drag, radiation, dynamics +and optionally (using the switch i\_impsolve\_loc) convection\footnote{If +i\_impsolve\_loc = 1, the boundary-layer implicit solver is +performed before the convection call so that $S$ excludes the convection +increments. If i\_impsolve\_loc = 2, it is performed after the convection +call. There are pros and cons to each option. Calling the implicit solver +before convection reduces the accuracy of the final mixed-layer profile, +since convection may alter the mixed-layer gradients afterwards. +On the other hand, allowing convection to act on a state which includes +the heating and moistening by surface-fluxes over the current timestep +may improve the accuracy of the convective closure. +Also the non-turbulent fluxes used to construct the budgets at entrainment +grid-levels (section \ref{sec:rev_flux_grad}) do not include contributions +from convection, so arguably excluding them from $S$ is consistent. +In the presence of convective subsidence, the top grid-level of the +sub-cloud mixed layer gets warmed and dried by the subsidence +(consistent with a lowering of the mixed-layer top). However if the +implicit solver is called after convection it does not account for this +lowering, so that all the subsided air is forced to be entrained into the +mixed-layer.}. $S$ represents the total tendency from these processes. +Equations (\ref{eq:sppf1}), (\ref{eq:sppf2}) applied to +(\ref{eq:vdiff1}) becomes +\begin{eqnarray} + \frac{X^{*}-X^{n}}{\Delta t} & = & \Ical_{1}\frac{\partial F}{\partial z}^{*}-\Ecal_{1}\frac{\partial F}{\partial z}^{n}+\left(\Ical_{1}-\Ecal_{1}\right)S\label{eq:sppf_bl1}\\ + \frac{X^{n+1}-X^{*}}{\Delta t} & = & \Ical_{2}\frac{\partial + F}{\partial z}^{n+1}-\Ecal_{2}\frac{\partial F}{\partial + z}^{*}+\left(\Ical_{2}-\Ecal_{2}\right)S\label{eq:sppf_bl2} +\end{eqnarray} +where, +\begin{equation*} + F^{n}=K_{X}\frac{\partial X}{\partial z}^{n},\; + F^{*}=K_{X}\frac{\partial X}{\partial z}^{*},\; + F^{n+1}=K_{X}\frac{\partial X}{\partial z}^{n+1},\; K_{X}\equiv + K(X^{n}) +\end{equation*} +i.e. only one evaluation of the exchange coefficient is +required per timestep. Furthermore, the condition +$I_{1}+I_{2}-(\Ecal_{1}+\Ecal_{2})=1$ ensures that if the intermediate +``starred'' quantities are eliminated and the scheme is reduced into a +single equation then the forcing term will be multiplied by $1$. + +Recall from section~\ref{sec:closure} that the boundary layer solver +computes the increment of $X$, where $X=u,\; v,\;\theta_{L},\; q_{w}$. +Let $\delta X^{*}=X^{*}-X^{n}$, $\delta +X^{n+1}=X^{n+1}-X^{*}$. Then, +\begin{equation*} + F^{*}=F^{n}+K_{X}\frac{\partial\delta X}{\partial z}^{*},\qquad + F^{n+1}=F^{*}+K_{X}\frac{\partial\delta X}{\partial z}^{n+1}. +\end{equation*} +Writing equations (\ref{eq:sppf_bl1}), +(\ref{eq:sppf_bl2}) in terms of these increments: +\begin{eqnarray} + \frac{\delta X}{\Delta t}^{*} & = & (\Ical_{1}-\Ecal_{1})\left(\frac{\partial F}{\partial z}^{n}+S\right)+\Ical_{1}\frac{\partial}{\partial z}\left(K_{X}\frac{\partial\delta X}{\partial z}^{*}\right)\label{eq:sppf_inc1}\\ + \frac{\delta X}{\Delta t}^{n+1} & = & (\Ical_{2}-\Ecal_{2})\left(\frac{\partial F}{\partial z}^{*}+S\right)+\Ical_{2}\frac{\partial}{\partial z}\left(K_{X}\frac{\partial\delta X}{\partial z}^{n+1}\right)\label{eq:sppf_inc2}\\ + X^{n+1} & = & X^{n}+\delta X^{*}+\delta X^{n+1}\label{eq:sppf_inc3} +\end{eqnarray} + +\subsubsection{Discrete equations and boundary conditions} +\label{sec:impsolve} + +To derive the boundary conditions for the horizontal wind components +we adapt the technique used in the original scheme. + +\noindent +{\bf Vertical diffusion solver for momentum variables } + +\noindent +Consider the following equivalent form of (\ref{eq:sppf_bl1}): +\begin{equation} + \frac{\delta u^{*}}{\Delta t}=\frac{\partial\bar{\tau}_{x}^{*}}{\partial z}+\left(\Ical_{1}-\Ecal_{1}\right)S\label{eq:du_star} +\end{equation} +where $\tau_{x}$ is the $u$ wind component stress (defined in the same +way as the flux in (\ref{eq:sppf_inc1}) and $\bar{\tau}_{x}^{*}$ its +time-average: +\begin{equation} + \bar{\tau}_{x}^{*}=\Ical_{1}\tau_{x}^{*}-\Ecal_{1}\tau_{x}^{n},\qquad\tau_{x}^{*}=\tau_{x}^{n}+K_{u}\frac{\partial\delta + u^{*}}{\partial z}.\label{eq:tau_star} +\end{equation} +Substituting (\ref{eq:tau_star}) into (\ref{eq:du_star}) the following +is obtained: +\begin{equation*} + \frac{\delta u^{*}}{\Delta + t}=(\Ical_{1}-\Ecal_{1})\left(\frac{\partial\tau_{x}^{n}}{\partial + z}+S\right)+\Ical_{1}\frac{\partial}{\partial + z}\left(K_{u}\frac{\partial\delta u^{*}}{\partial z}\right) +\end{equation*} +which is identical to (\ref{eq:sppf_inc1}) for $X\equiv u,\; +F_{X}\equiv\tau_{x}$. This equivalent derivation is used here as it +presents a more convenient form to express the boundary +conditions. Given that the wind components are defined on +$\rho$-levels (half levels), discretizing the previous equation in $z$ +on all $L$ half-levels except at the bottom and the top one we obtain +\begin{eqnarray*} + \delta u_{k+1/2}^{*} & = & (\Ical_{1}-\Ecal_{1})\Delta t\left(\frac{\tau_{x}^{n}\Big|_{k+1}-\tau_{x}^{n}\Big|_{k}}{z_{k+1}-z_{k}}+S_{k+1/2}\right)\\ + & & +\Ical_{1}\frac{\Delta + t}{z_{k+1}-z_{k}}\left[\left(K_{u}\Big|_{k+1}\frac{\delta + u_{k+3/2}^{*}-\delta + u_{k+1/2}^{*}}{z_{k+3/2}-z_{k+1/2}}\right)-\left(K_{u}\Big|_{k}\frac{\delta + u_{k+1/2}^{*}-\delta + u_{k-1/2}^{*}}{z_{k+1/2}-z_{k-1/2}}\right)\right] +\end{eqnarray*} +or, rearranging +\begin{equation} + A_{k}\delta u_{k+3/2}^{*}+B_{k}\delta + u_{k+1/2}^{*}+C_{k}\delta u_{k-1/2}^{*}=\Delta + t(\Ical_{1}-\Ecal_{1})\left(\frac{\tau_{x}^{n}\Big|_{k+1}-\tau_{x}^{n}\Big|_{k}}{z_{k+1}-z_{k}}+S_{k+1/2}\right),\label{eq:tridiag} +\end{equation} +where $k=1,2,\ldots,L-2$, +\begin{equation*} + A_{k}=-\Ical_{1}\frac{\Delta t\noindent + K_{u}\Big|_{k+1}}{(z_{k+1}-z_{k})(z_{k+3/2}-z_{k+1/2})},\; + C_{k}=-\Ical_{1}\frac{\Delta + tK_{u}\Big|_{k}}{(z_{k+1}-z_{k})(z_{k+1/2}-z_{k-1/2})},\; + B_{k}=1-A_{k}-C_{k}. +\end{equation*} +(Note that the surface is level $0$). + +For the top $\rho$-level, $k=L-1$, the $z$-discretization of +(\ref{eq:du_star}) is: +\begin{equation} + B_{L}\delta u_{L-1/2}^{*}+C_{L}\delta u_{L-3/2}^{*}=\Delta t(\Ical_{1}-\Ecal_{1})\left(\frac{\tau_{x}^{n}\Big|_{L}-\tau_{x}^{n}\Big|_{L-1}}{z_{L}-z_{L-1}}+S_{L-1/2}\right),\label{eq:tridiag_top} +\end{equation} +where $B_{L}$, $C_{L}$ are derived as before setting $A_{L}=0$. + +For the bottom $\rho$-level, $k=0$, the $z$-discretization of +(\ref{eq:du_star}) is: +\begin{eqnarray} + \delta u_{1/2}^{*} & = & \frac{\Delta t}{z_{1}-0}\left(\bar{\tau}_{x}^{*}\Big|_{1}-\bar{\tau}_{x}^{*}\Big|_{0}\right)+\Delta t\left(\Ical_{1}-\Ecal_{1}\right)S_{1/2}\label{eq:u_bc_1} +\end{eqnarray} +where, from (\ref{eq:tau_star}), +\begin{equation} + \bar{\tau}_{x}^{*}\Big|_{1}=\left(\Ical_{1}-\Ecal_{1}\right)\tau_{x}^{n}\Big|_{1}+\Ical_{1}\left(K_{u}\frac{\partial\delta u^{*}}{\partial z}\right)\Big|_{1}.\label{eq:u_bc_2} +\end{equation} +Combining (\ref{eq:u_bc_1}), (\ref{eq:u_bc_2}) the bottom row +discretization is obtained: +\begin{equation} + A_{0}\delta u_{3/2}^{*}+B_{0}\delta u_{1/2}^{*}=\Delta t\left(\Ical_{1}-\Ecal_{1}\right)\left(\frac{\tau_{x}^{n}\Big|_{1}}{z_{1}}+S_{1/2}\right)-\frac{\Delta t}{z_{1}}\bar{\tau}_{x}^{*}\Big|_{0}\label{eq:u_bc_3} +\end{equation} +where +\begin{equation*} + A_{0}=-\Ical_{1}\frac{\Delta tK_{u}\Big|_{1}}{z_{1}(z_{3/2}-z_{1/2})},\quad B_{0}=1-A_{0}. +\end{equation*} + +Equations (\ref{eq:tridiag}), (\ref{eq:tridiag_top}) and +(\ref{eq:u_bc_3}) form a tridiagonal system of linear equations. When +the elimination procedure takes place (\ref{eq:u_bc_3}) becomes +\begin{equation} + \delta u_{1/2}^{*}=\delta u_{1/2}^{'}-\beta\bar{\tau}_{x}^{*}\Big|_{0}\label{eq:du_half} +\end{equation} +where $\delta u_{1/2}^{'}$, $\beta$ are available +quantities. Furthermore, +\begin{equation*} + \bar{\tau}_{x}^{*}\Big|_{0}=\left(\Ical_{1}-\Ecal_{1}\right)\tau_{x}^{n}\Big|_{0}+\Ical_{1}\left(K_{u}\frac{\partial\delta u^{*}}{\partial z}\right)\Big|_{0} +\end{equation*} +Approximating $\left(\frac{\partial\delta u^{*}}{\partial + z}\right)\Big|_{0}\approx\frac{\delta u_{1/2}^{*}-\delta + u_{0}^{*}}{z_{1/2}}$, and assuming that $u_{0}=0$ the previous +equation becomes +\begin{equation} + \bar{\tau}_{x}^{*}\Big|_{0}=\left(\Ical_{1}-\Ecal_{1}\right)\tau_{x}^{n}\Big|_{0}+\Ical_{1}K_{u}\Big|_{0}\frac{\delta u_{1/2}^{*}}{z_{1/2}}.\label{eq:tau_zero} +\end{equation} +From (\ref{eq:du_half}), (\ref{eq:tau_zero}) the following expression +for the implicit surface stress is obtained +\begin{equation} + \bar{\tau}_{x}^{*}\Big|_{0}=\frac{\left(\Ical_{1}-\Ecal_{1}\right)\tau_{x}^{n}\Big|_{0}+\Ical_{1}(K_{u}\Big|_{0}/z_{1/2})\delta + u_{1/2}^{'}}{1+\Ical_{1}(K_{u}\Big|_{0}/z_{1/2})\beta}.\label{eq:imp_tau} +\end{equation} +Then, $\delta u_{1/2}^{*}$ can be computed from (\ref{eq:imp_tau}) and +(\ref{eq:du_half}). Similarly the implicit surface stress for $u$ +which corresponds to the 2nd stage (\ref{eq:sppf_inc2}) will be +\begin{equation} + \bar{\tau}_{x}^{n+1}\Big|_{0}=\frac{\left(\Ical_{2}-\Ecal_{2}\right)\tau_{x}^{*}\Big|_{0}+\Ical_{2}(K_{u}\Big|_{0}/z_{1/2})\delta + u_{1/2}^{'}}{1+\Ical_{2}(K_{u}\Big|_{0}/z_{1/2})\beta}.\label{eq:imp_tau2} +\end{equation} +In the same way $\bar{\tau}_{y}^{*}\Big|_{0}$, +$\bar{\tau}_{y}^{n+1}\Big|_{0}$ can be derived. + +\noindent +{\bf Vertical diffusion solver for scalar variables} + +\noindent +Derivation of boundary conditions for the scalar variables (static +energy and total water content flux) is more difficult: the new scheme +in comparison with the original scheme is more complex and the +procedure for deriving the scalar fluxes described in section 3 of +\cite{esseryetal2001} is also complex. Currently, an alternative +treatment for the boundary conditions has been coded which works well +in practice. The boundary conditions for the scalar variables, +i.e. the surface scalar fluxes for the new scheme are obtained using +the original implicit surface exchange calculation. This is applied as +follows. Consider the equivalent discrete form of (\ref{eq:sppf_bl1}) +for the thermodynamic variable $X$: +\begin{equation} + \frac{\delta X^{*}}{\Delta t} + =\frac{\partial\overline{F}^{*}}{\partial z}+\left(\Ical_{1}-\Ecal_{1}\right)S\label{eq:dX_star} +\end{equation} +Considering that, +\begin{equation} + \overline{F}^{*}=\Ical_{1}F^{*}-\Ecal_{1}F^{n},\qquad + F^{*}=F^{n}+K_{X}\frac{\partial\delta X^{*}}{\partial z}\label{eq:dX_star2} +\end{equation} (\ref{eq:dX_star}) +would re-produce (\ref{eq:sppf_inc1}), which is re-written below, +\begin{equation*} + \frac{\delta X^{*}}{\Delta t}=(\Ical_{1}-\Ecal_{1})\left(\frac{\partial F^{n}}{\partial z}+S\right)+\Ical_{1}\frac{\partial}{\partial z}\left(K_{X}\frac{\partial\delta X^{*}}{\partial z}\right) +\end{equation*} +and thus the following discretization is obtained, on $\theta$-levels: +\begin{eqnarray*} + \frac{\delta X_{k}^{*}}{\Delta t} & = & \left(\Ical_{1}-\Ecal_{1}\right)\left(\frac{F_{k+1/2}^{n}-F_{k-1/2}^{n}}{z_{k+1/2}-z_{k-1/2}}+S_{k}\right)\\ + & +&\frac{\Ical_{1}}{z_{k+1/2}-z_{k-1/2}}\left[K_{X}\Big|_{k+1/2}\left(\frac{\delta X_{k+1}^{*}-\delta X_{k}^{*}}{z_{k+1}-z_{k}}\right)-K_{X}\Big|_{k-1/2}\left(\frac{\delta X_{k}^{*}-\delta X_{k-1}^{*}}{z_{k}-z_{k-1}}\right)\right],\; k=2,\ldots,L-1.\end{eqnarray*} +or, +\begin{equation} + A_{k}\delta X_{k+1}^{*}+B_{k}\delta X_{k}^{*}+C_{k}\delta X_{k-1}^{*}=\left(\Ical_{1}-\Ecal_{1}\right)\left(\frac{F_{k+1/2}^{n}-F_{k-1/2}^{n}}{z_{k+1/2}-z_{k-1/2}}+S_{k}\right),\quad k=2,\ldots,L-1\label{eq:dX_disc} +\end{equation} +where, +\begin{equation*} + A_{k}=-\Ical_{1}\frac{\Delta tK_{X}\Big|_{k+1/2}}{(z_{k+1}-z_{k})(z_{k+1/2}-z_{k-1/2})},\; C_{k}=-\Ical_{1}\frac{\Delta tK_{X}\Big|_{k-1/2}}{(z_{k+1/2}-z_{k-1/2})(z_{k}-z_{k-1})},\quad B_{k}=1-A_{k}-C_{k}. +\end{equation*} +The discrete equation for the top level, $k=L$, will be: +\begin{equation} + B_{L}\delta X_{L}^{*}+C_{L}\delta X_{L-1}^{*}=\left(\Ical_{1}-\Ecal_{1}\right)\left(\frac{-F_{L-1/2}^{n}}{z_{L+1/2}-z_{L-1/2}}+S_{L}\right),\label{eq:dX_disc_top} +\end{equation} +where $B_{L}$, $C_{L}$ are derived as before setting $A_{L}=0$. + +From (\ref{eq:dX_star}), a bottom interior level ($k=1$) +discretization is +\begin{equation} + \delta X_{1}^{*}=\frac{\Delta t}{z_{3/2}-0}\left(\overline{F_{3/2}}^{*}-\overline{F_{0}}^{*}\right)+\Delta t\left(\Ical_{1}-\Ecal_{1}\right)S_{1}\label{eq:dX1_star} +\end{equation} +$F_{0}$ is used instead of $F_{1/2}$. The former is computed by the +implicit surface scheme. This flux gradient is defined in the same way +in the original solver as well. Using (\ref{eq:dX_star2}), +(\ref{eq:dX1_star}) becomes +\begin{equation} + \delta + X_{1}^{*}=\frac{\Delta t}{z_{3/2}} + \left[\left(\Ical_{1}-\Ecal_{1}\right)F_{3/2}^{n}-\overline{F_{0}}^{*}\right] + +\Delta t\left(\Ical_{1}-\Ecal_{1}\right) + S_{1}+\Delta t\Ical_{1}\frac{1}{z_{3/2}} + \left(K_{X}\frac{\partial\delta X^{*}}{\partial z}\right)_{3/2} + \label{eq:dX1_star2} +\end{equation} +where $\overline{F}_{0}^{*}$ can be approximated as +\begin{equation} + \overline{F}_{0}^{*}=\Ical_{1}F_{0}^{*}-\Ecal_{1}F_{0}^{n}\approx\left(\Ical_{1}-\Ecal_{1}\right)F_{JULES}\label{eq:F0_star} +\end{equation} +where, $F_{JULES}$ is the implicit flux calculated by the +\textit{implicit surface scheme using the original implicit + algorithm}. Finalising, the discrete equations for the bottom level +will be +\begin{equation*} + \delta X_{1}^{*}=\Delta t\left(\Ical_{1}-\Ecal_{1}\right) + \left(\frac{F_{3/2}^{n}-F_{JULES}}{z_{3/2}}+S_{1}\right) + +\Delta t\Ical_{1}\frac{1}{z_{3/2}}K_{X}\Big|_{3/2} + \left(\frac{\delta X_{2}^{*}-\delta X_{1}^{*}}{z_{2}-z_{1}}\right) +\end{equation*} +or, +\begin{equation} + A_{1}\delta X_{2}^{*}+B_{1}\delta X_{1}^{*}=\Delta t + \left(\Ical_{1}-\Ecal_{1}\right)\left(\frac{F_{3/2}^{n}-F_{JULES}}{z_{3/2}}+S_{1}\right)\label{eq:dX_bottom} +\end{equation} +where, +\begin{equation*} + A_{1}=-\Ical_{1}\Delta t\frac{K_{X}\Big|_{3/2}}{z_{3/2}(z_{2}-z_{1})}, + \quad B_{1}=1-A_{1}. +\end{equation*} +Equations (\ref{eq:dX_disc}), (\ref{eq:dX_disc_top}) and +(\ref{eq:dX_bottom}) define a tridiagonal system of equations for +$\delta X^{*}$. + +Similarly the corresponding discrete equations for +(\ref{eq:sppf_inc2}) will be: +\begin{equation} + B_{L}^{'}\delta X_{L}^{n+1}+C_{L}^{'}\delta X_{L-1}^{n+1}=\left(\Ical_{2}-\Ecal_{2}\right)\left(\frac{-F_{L-1/2}^{*}}{z_{L+1/2}-z_{L-1/2}}+S_{L}\right),\label{eq:dXtop_np1} +\end{equation} +\begin{equation} + A_{k}^{'}\delta X_{k+1}^{n+1}+B_{k}^{'}\delta X_{k}^{n+1}+C_{k}^{'}\delta X_{k-1}^{n+1}=\left(\Ical_{2}-\Ecal_{2}\right)\left(\frac{F_{k+1/2}^{*}-F_{k-1/2}^{*}}{z_{k+1/2}-z_{k-1/2}}+S_{k}\right),\quad k=L-1,\ldots,2\label{eq:dXk_np1} +\end{equation} +\begin{equation} + A_{1}^{'}\delta X_{2}^{n+1}+B_{1}^{'}\delta X_{1}^{n+1}=\left(\Ical_{2}-\Ecal_{2}\right)\left(\frac{F_{3/2}^{*}-F_{JULES}}{z_{3/2}}+S_{1}\right)\label{eq:dX1_np1} +\end{equation} +where, +\begin{equation*} + A_{k}^{'}=-\Ical_{2}\frac{\Delta tK_{X}\Big|_{k+1/2}}{(z_{k+1}-z_{k})(z_{k+1/2}-z_{k-1/2})},\; C_{k}^{'}=-\Ical_{2}\frac{\Delta tK_{X}\Big|_{k-1/2}}{(z_{k+1/2}-z_{k-1/2})(z_{k}-z_{k-1})},\quad B_{k}^{'}=1-A_{k}^{'}-C_{k}^{'}, +\end{equation*} +for $k=L,\ldots,2,\quad A_{L}=0$. +\begin{equation*} + A_{1}^{'}=-\Ical_{2}\Delta t\frac{K_{X}\Big|_{3/2}}{z_{3/2}(z_{2}-z_{1})},\quad B_{1}^{'}=1-A_{1}^{'} +\end{equation*} +and the approximation +\begin{equation} + \overline{F}_{0}^{n+1}=\Ical_{2}F_{0}^{n+1}-\Ecal_{2}F_{0}^{n}\approx\left(\Ical_{2}-\Ecal_{2}\right)F_{JULES}\label{eq:F0_np1} +\end{equation} +has taken place. The same flux $F_{JULES}$ will be used for both +(\ref{eq:F0_star}) and (\ref{eq:F0_np1}) and therefore needs to be +computed only once, when the 1st or predictor stage is computed, +i.e. $X^{*}$. Briefly the following calculations take place for the +scalar variables: +\begin{center} +{\begin{tabular}{ll} +CALL bdy\_impl3(): & set up coefficients for (\ref{eq:dX_disc_top}), (\ref{eq:dX_disc}) and do a downward sweep; \\ + & do a downward sweep using the original implicit scheme to \\ + & compute information required by the surface implicit solver; \\ +CALL sf\_impl2(): & CALL im\_sf\_pt2(): compute $F_{JULES}$ (scalar implicit fluxes), \\ + & using original surface implicit solver; \\ +CALL bdy\_impl4(): & set up (\ref{eq:dX_bottom}) and complete downward sweep; \\ + & back substitute to compute implicit correction $\delta X^{*}$; \\ +CALL bdy\_impl3(): & compute explicit flux +$F^*=F^n+K_X\frac{\partial \delta X^*}{\partial z}$; \\ + & set up coefficients for (\ref{eq:dXtop_np1}), (\ref{eq:dXk_np1}), (\ref{eq:dX1_np1}) and \\ + & do a downward sweep; \\ +CALL sf\_impl2(): & only momentum variables are affected - no change in scalars; \\ +CALL bdy\_impl4(): & back substitute to compute final implicit correction $\delta X^{n+1}$ \\ +\end{tabular}} +\end{center} +NB: for CABLE compatibility, sf\_impl2 is now called by an +intermediate routine surf\_couple\_implicit. + +\subsubsection{Flux diagnostic formulae} + +The original boundary layer implicit solver computes the total stress +by time averaging the stresses at $t^{n}$ and $t^{n+1}$, where +$[t^{n},t^{n+1}]$ denotes the time integration interval for the +vertical diffusion equation being solved. The averaging which takes +place for the zonal wind component stress is: +\begin{equation} + \overline{\tau_{x}}^{n+1}\equiv(1-\gamma)\tau_{x}^{n}+\gamma\tau_{x}^{n+1} + =\tau_{x}^{n}+\gamma + K_{u}\frac{\partial\delta u^{n+1}}{\partial z}\label{eq:taux_tot} +\end{equation} +where $\delta u^{n+1}=u^{n+1}-u^{n}$. Likewise, $\tau_{y}$ and the +scalar fluxes are derived. + +For the new scheme the total zonal wind component stress is defined +as: +\begin{equation*} + \overline{\tau_{x}}^{n+1}\equiv\overline{\tau_{x}}^{[n,*]}+\overline{\tau_{x}}^{[*,n+1]} +\end{equation*} +where, $\overline{\tau_{x}}^{[n,*]}$ denotes the total stress for the +1st stage of the scheme, i.e. the time averaged stress from $t^{n}$ to +the pseudo-timelevel $t^{*}$ and similarly, +$\overline{\tau_{x}}^{[*,n+1]}$ the total stress for the 2nd stage of +the scheme (corrector). The total stress for the predictor and the +corrector are defined as: +\begin{eqnarray*} + \overline{\tau_{x}}^{[n,*]}\equiv\Ical_{1}\tau_{x}^{*}-\Ecal_{1}\tau_{x}^{n} & = & \left(\Ical_{1}-\Ecal_{1}\right)\tau_{x}^{n}+\Ical_{1}K_{u}\frac{\partial\delta u^{*}}{\partial z}\\ + \overline{\tau_{x}}^{[*,n+1]}\equiv\Ical_{2}\tau_{x}^{n+1}-\Ecal_{2}\tau_{x}^{*} & = & \left(\Ical_{2}-\Ecal_{2}\right)\tau_{x}^{*}+\Ical_{2}K_{u}\frac{\partial\delta u^{n+1}}{\partial z} +\end{eqnarray*} +where, $\delta u^{*}=u^{*}-u^{n},\;\delta u^{n+1}=u^{n+1}-u^{*}$. The +meridional stress $\tau_{y}$ and the scalar fluxes can be derived in a +similar way. These formulae have been validated in SCM experiments. + +\subsubsection{Implicit surface flux and future upgrades} + +This scheme should be incorporated in the calculation of the scalar +implicit fluxes. This would be preferable to the current technique for +calculating the scalar implicit fluxes (use of original implicit +scheme for these). This has been attempted but not yet successfully +completed. In the tested code, 2 calls to the modified im\_sf\_pt +subroutine are done, one per scheme stage (step), i.e. one for the +stage that ${\delta X}_{1}^{*}$ is computed and one for $\delta +X_{1}^{n+1}$ where the subscript denotes level number. At each call, a +modified version of the flux formulae (78), (79) of +\cite{esseryetal2001} is used: + +\noindent \textbf{1st sweep:} +\begin{eqnarray} + \frac{\overline{H^{*}}}{c_{p}} & = & \frac{(1+\beta B_{2})[\gamma_{2}F_{T}^{n}+A_{1}\delta{T'}_{1}+A_{2}\delta{Q'}_{1}]-\beta A_{2}[\gamma_{2}F_{Q}^{n}+B_{1}\delta{T'}_{1}+B_{2}\delta{Q'}_{1}]}{(1+\beta A_{1})(1+\beta B_{2})-\beta^{2}A_{2}B_{1}}\label{eq:FTLstar} +\end{eqnarray} +\begin{eqnarray} + \overline{E^{*}} & = & \frac{(1+\beta A_{1})[\gamma_{2}F_{Q}^{n}+B_{1}\delta{T'}_{1}+B_{2}\delta{Q'}_{1}]-\beta B_{1}[\gamma_{2}F_{T}^{n}+A_{1}\delta{T'}_{1}+A_{2}\delta{Q'}_{1}]}{(1+\beta A_{1})(1+\beta B_{2})-\beta^{2}A_{2}B_{1}}\label{eq:FQWstar} +\end{eqnarray} +where, $F_{T}^{n}$, $F_{Q}^{n}$ denote the surface explicit fluxes, +$\gamma_{2}=\Ical_{1}-\Ecal_{1}$ and the coefficients +$A_{1},\; A_{2},B_{1},\; B_{2}$ are given by +\begin{subequations} + \begin{equation} + A_1=-\gamma_1\sum_j \nu_j RK_{PMj} + [LD_j\psi_jRK_H(1)_j+A_{*j}], + \end{equation} + \begin{equation} + A_2=\gamma_1\sum_j \nu_j RK_{PMj} + L\psi_jRK_H(1)_j, + \end{equation} + \begin{equation} + B_1=\gamma_1c_p\sum_j \nu_j RK_{PMj} + D_j\psi_jRK_H(1)_j + \end{equation} + \begin{equation} + B_2=-\gamma_1\sum_j \nu_j RK_{PMj} + \psi_j[c_pRK_H(1)_j+A_{*j}]. + \end{equation} + \label{ab_coeffs} +\end{subequations} +but with $\gamma_{1}=\Ical_{1}$. Here $RK_H(1) =\rho C_H U_1$, +$RK_{PM}={RK_H(1)\over(c_p+LD\psi)RK_H(1)+A_*}$, +\begin{equation*} + A_* = (1 - f_r){2\lambda\over\Delta z_s} + {C_c\over\Delta t} + + 4(1 + f_r)\sigma T_s^3, +\end{equation*} +\begin{equation*} + D={q_{\rm sat}(T_*^{(n)},p_*)-q_{\rm sat}(T_1^{(n)},p_*) \over + T_*^{(n)}-T_1^{(n)}}, +\end{equation*} +\begin{equation*} + \psi=f_a+(1-f_a){g_s\over g_s+C_HU_1}, +\end{equation*} +and $\nu_j$ represents the fraction of surface tile type $j$. $f_r$ is +the radiative canopy fraction, $\lambda$ is the soil conductivity, +$\Delta z_s$ and $T_s$ are the thickness and temperature of the +surface soil layer, $C_c$ is the canopy heat capacity, $f_a$ is the +saturated fraction of the tile and $g_s$ is the surface +conductance. To derive (\ref{eq:FTLstar}), (\ref{eq:FQWstar}), the +time-weighted level 1 $T$ and $Q$ consistent with the discrete +equations of the new scheme is written as follows: +\begin{equation*} + \overline{T_{1}^{*}}=\gamma_{2}T^{n}+\gamma_{1}\delta T_{1}^{*},\quad\overline{Q_{1}^{*}}=\gamma_{2}Q^{n}+\gamma_{1}\delta Q_{1}^{*},\qquad\delta T_{1}^{*}=T^{*}-T^{n},\quad\delta Q_{1}^{*}=Q^{*}-Q^{n} +\end{equation*} +and the original derivation is followed. From these expressions the +tile flux for $H$ is derived: +\begin{eqnarray*} + \frac{H_{j}^{*}}{c_{p}} & = & \gamma_{2}\frac{H_{j}^{(n)}}{c_{p}}-\gamma_{1}RK_{PMj}[LD_{j}\psi_{j}RK_{H}(1)_{j}+A_{*j}][c_{p}\delta{T'}_{1}-\beta\overline{H^{*}}]\\ + & & \qquad\quad+\gamma_{1}RK_{PMj}L\psi_{j}RK_{H}(1)_{j}[\delta{Q'}_{1}-\beta\overline{E^{*}}] +\end{eqnarray*} +and similarly $E_{j}^{*}$. From these, the tile flux equations +(\ref{eq:FTLstar}), (\ref{eq:FQWstar}) can be obtained. + +\noindent \textbf{2nd sweep:} +\begin{eqnarray} + \frac{\overline{H^{n+1}}}{c_{p}} & = & \frac{(1+\beta B_{2})[\xi_{2}F_{T}^{*}+A_{1}\delta{T'}_{1}+A_{2}\delta{Q'}_{1}]-\beta A_{2}[\xi_{2}F_{Q}^{*}+B_{1}\delta{T'}_{1}+B_{2}\delta{Q'}_{1}]}{(1+\beta A_{1})(1+\beta B_{2})-\beta^{2}A_{2}B_{1}}\label{eq:FTLnp1} +\end{eqnarray} +\begin{eqnarray} + \overline{E^{n+1}} & = & \frac{(1+\beta A_{1})[\xi_{2}F_{Q}^{*}+B_{1}\delta{T'}_{1}+B_{2}\delta{Q'}_{1}]-\beta B_{1}[\xi_{2}F_{T}^{*}+A_{1}\delta{T'}_{1}+A_{2}\delta{Q'}_{1}]}{(1+\beta A_{1})(1+\beta B_{2})-\beta^{2}A_{2}B_{1}}\label{eq:FQWnp1} +\end{eqnarray} +where, the coefficients $A_{1},\; A_{2},B_{1},\; B_{2}$ are given by +(\ref{ab_coeffs}) but with $\gamma_{1}=\Ical_{2}$. The above +formulae are derived as explained earlier. The definitions +\begin{equation*} + \overline{T_{1}^{n+1}}=\gamma_{2}T^{*}+\gamma_{1}\delta T_{1}^{n+1},\quad\overline{Q_{1}^{*}}=\gamma_{2}Q^{*}+\gamma_{1}\delta Q_{1}^{n+1},\qquad\delta T_{1}^{n+1}=T^{n+1}-T^{*},\quad\delta Q_{1}^{n+1}=Q^{n+1}-Q^{*} +\end{equation*} +are used here. The surface fluxes $F_{T}^{*}\equiv{\displaystyle + \frac{H^{*}}{c_{p}}}$, $F_{Q}^{*}\equiv E^{*}$ are computed at model +state ${(T}_{1}^{*},Q_{1}^{*})$. They are the equivalent of the +explicit fluxes $F_{T}^{n}\equiv{\displaystyle \frac{H^{n}}{c_{p}}}$, +$F_{Q}^{n}\equiv E^{n}$. However, they are not equal to the left +hand-side of (\ref{eq:FTLstar}), (\ref{eq:FQWstar}). Both are +connected by a linear relationship, which is simply the definition of +the time-weighted averaging consistent with the new scheme: +\begin{equation*} + \overline{H^{*}}\equiv\Ical_{1}H^{*}-\Ecal_{1}H^{n},\qquad\overline{E^{*}}\equiv\Ical_{1}E^{*}-\Ecal_{1}E^{n}. +\end{equation*} +Therefore, once $\overline{H^{*}},\overline{E^{*}}$ have been +computed, $H^{*}$, $E^{*}$ can be computed as follows: +\begin{equation*} + H^{*}=\frac{\overline{H^{*}}+(\gamma_{1}-\gamma_{2})H^{n}}{\gamma_{1}},\qquad + E^{*}=\frac{\overline{E^{*}}+(\gamma_{1}-\gamma_{2})E^{n}}{\gamma_{1}}. +\end{equation*} +Note that without the previous transformation the model fails within a +few timesteps. + +The calculation of the surface temperature, evaporation and melting +takes place only in the second sweep. The flux increment obtained from +evaporation and melting is added on $\overline{H^{n+1}}$, +$\overline{E^{n+1}}$. The same relationship is used for the surface +temperature: +\begin{equation*} + T_{*}=T_{s}+\frac{1}{A_{*}} + \left[R_{s}-H-LE+\frac{C_{c}}{\Delta t}\left(T_{*}^{n}-T_{s}\right)\right], +\end{equation*} +however, the total averaged flux from $t^{n}$ to $t^{n+1}$ is used: +\begin{equation*} + H=\overline{H^{*}}+\overline{H^{n+1}},\qquad E=\overline{E^{*}}+\overline{E^{n+1}}. +\end{equation*} +The total flux is also kept by the corresponding STASH +diagnostic. This has to be adjusted if evaporation exhausts any of the +moisture stores during the timestep or if the tile has a melting +snowcover. + +\paragraph{Limited evaporation} + +Downward surface moisture fluxes are added to canopy moisture or, if +the surface temperature is below freezing, snowcover. + +For an upward total moisture flux $E$, the rates of evaporation from +the canopy and soil moisture stores are +\begin{equation} + E_c = f_a{E\over\psi} +\end{equation} +and +\begin{equation} + E_s = (1 - f_a)\psi_s {E\over\psi} +\end{equation} +where +\begin{equation} + \psi_s = {g_s\over g_s+C_HU_1}. +\end{equation} +If the predicted canopy evaporation would exhaust the canopy moisture +store $C$ during a timestep, the soil evaporation is recalculated as +\begin{equation} + E_s =\psi_s\left(1 - {f_aC\over E_c\Delta t}\right){E\over\psi} +\end{equation} +and $E_c$ is reset to $C/\Delta t$. If $E_s$ would then exhaust the +available soil moisture $m$, it is limited to $m/\Delta t$. + +For an adjustment $\Delta(LE)$ in the latent heat flux, repartitioning +the surface energy balance gives adjustments +\begin{equation} + \Delta H = - \left[1 + {A_*\over c_pRK_H(1)}\right]^{-1}\Delta(LE) +\end{equation} +and +\begin{equation} + \Delta T_* = - {\Delta H+\Delta(LE)\over A_*} +\end{equation} +in the surface sensible heat flux and temperature. + +Evaporation from a lake tile (or the lake fraction of an aggregated +surface) is not limited and does not draw on the conserved moisture +stores. + +\paragraph{Snowmelt} + +Classical surface energy balance neglects snowmelt heat fluxes. If +$T_*>T_m$ for a snow-covered tile and sufficient snow is available, +$T_*$ is reset to $T_m$ by adding an increment +\begin{equation} + \Delta T_*=T_m-T_*, +\end{equation} +corresponding to a snowmelt heat flux +\begin{equation} + S_m = - [(c_p + L_sD)RK_H(1) + A_*]{\Delta T_*\over L_f}. + \label{eq:Sm} +\end{equation} +The maximum melt rate that can be sustained over a timestep $\Delta +t$, however, is $S/\Delta t-E$, giving +\begin{equation} + \Delta T_*={L_f(S/\Delta t-E) \over + (c_p+L_cD)RK_H(1) + A_*}. + \label{eq:dTmax} +\end{equation} +$\Delta T_*$ is set to the smaller of the values given by Equations +(\ref{eq:Sm}) and (\ref{eq:dTmax}), and the surface energy balance is +repartitioned by adding increments +\begin{equation} + \Delta H = c_pRK_H(1)\Delta T_* +\end{equation} +and +\begin{equation} + \Delta E = DRK_H(1)\Delta T_* +\end{equation} +to the tile heat and moisture fluxes. + +The model with the above changes coded seems to work stably but the +surface fluxes (and therefore the boundary layer increments) are only +qualitatively correct. They seem to be overestimated by the above +scheme. {[}\textit{Could it be that coefficients $D_{j}$, $\psi_{j}$ + need to be modified in the second sweep?}] + +\subsubsection{Blending height coupling} + +The same method is used as in section~\ref{sec:impsolve} to form two +independent tridiagonal systems of linear equations that relate the +increments to momentum, temperature and humidity to the surface fluxes. +The `downward sweep' elimination procedure still takes place to obtain +equation (\ref{eq:du_half}) and a corresponding equation for the +increments to the scalar variables at the bottom model level + +\begin{equation} + \delta X_{1/2}^{*}=\delta X_{1/2}^{'}-\beta_X\frac{\bar{H}_\star}{C_p} +\end{equation} + +where $\delta X_{1/2}^{'}$ and $\beta_X$ are known. An `upward sweep' of +this tridiagonal matrix (i.e. back subsitution) then takes place to +obtain equations for the increment to momentum and scalar variables at a +given level $\kbl$ in terms of the surface fluxes + +\begin{equation} + \delta u_{\kbl}^{*}=\delta u_{\kbl}^{'}+(-1)^{\kbl}\beta\bar{\tau}_{x}^{*}\Big|_{0} \prod^{j=2}_{\kbl} {C_{u}}_{j}^{'} + \sum^{\kbl-1}_{i=1} \left[(-1)^{\kbl+i}\delta u_{\kbl}^{'} \prod^{j=i+1}_{\kbl} {C_{u}}_{j}^{'}\right] +\end{equation} + +\begin{equation} + \delta X_{\kbl}^{*}=\delta X_{\kbl}^{'}+(-1)^{\kbl}\beta_X\frac{\bar{H}_\star}{C_p} \prod^{j=2}_{\kbl} {C_{X}}_{j}^{'} + \sum^{\kbl-1}_{i=1} \left[(-1)^{\kbl+i}\delta X_{\kbl}^{'} \prod^{j=i+1}_{\kbl} {C_{X}}_{j}^{'}\right] +\end{equation} + +where $\delta u_{\kbl}^{*}$ and $\delta X_{\kbl}^{*}$ are the only unknowns. +The coefficients for these equations are passed to the surface implicit +solver so that the surface fluxes are calculated using level $\kbl$ +(which corresponds to a user-specified blending height) rather than +the bottom model level. The rest of the implicit solver continues in the +same way as for coupling at the bottom model level. + +By default, this option is switched off as further work is needed +(there is currently a problem with the input data such that this +blending height option crashes on the first time step). + + +\section{Derived diagnostics} + +\subsection{Boundary layer thermal speed: stash 3,355} + +This diagnostic is intended for use in quantifying the strength of +convective thermals for aviation applications. Updraught velocities +in convective boundary layers will scale with the convective velocity +scale, $w_*$, given by $ w_*^3 = \zhe \wbs $. In addition to the +basic convective velocity scale, the strength of thermals should also +depend on the surface stability --- it would be possible to have +significant heat flux and boundary layer depth in windy conditions +that should not lead to a strong thermal forecast. This sensitivity of +boundary layer turbulence is already included in the parametrization +of non-local momentum fluxes (see section \ref{sec:ngstress}) through +the stability dependence in (\ref{tau_nl}) that can be written as +\begin{equation} + f_{stab} = - \frac{a_{stab} \zhe/L }{1 - a_{stab} \zhe/L} +\end{equation} +for the Obhukov length, (\ref{1.1.4}), $<0$ (\mbox{i.e.}, unstable +boundary layers) and the empirical constant $a_{stab} = 1.5$. This +function tends to unity as $L$ decreases in magnitude (\mbox{i.e.}, +surface heating increases and wind stress decreases). The final +thermal speed, in units of ms$^{-1}$, is given simply by +\begin{equation} + {\rm Thermal} \, {\rm Speed} = f_{stab} \, w_* +\end{equation} + +\subsection{Wind gust: stash 3,463 and 3,515 (scale-dependent)} + +WMO define the wind gust strength as the maximum of the wind averaged +over 3 second intervals. In the boundary layer the strength of gusts +is proportional to the standard deviation of the horizontal wind, +$\sigma_u$, so that +\begin{equation} + U_{gust} = U_{10m} + W_{1D} \, \sigma_u \, \frac{1}{k} \, + {\rm log}\left( \frac{5 \, e^{k \, c_{\rm ugn}} + z_{0m(eff)} } + {5 + z_{0m(eff)}} \right) +\label{windgust} +\end{equation} +The factor $W_{1D}$ is included only in the scale-dependent version of +the diagnostic (stash 3,515) to allow for the larger scales of boundary +layer turbulence that are resolved (and so are already included in +$U_{10m}$). The lowest grid-level value of $W_{1D}$, from (\ref{eq-tanh}), +is used, noting that $W_{1D}$ is constant within the boundary layer. The +constant $c_{\rm ugn}$ in (\ref{windgust}) is determined from universal +turbulence spectra for a 25\% exceeding probability of the three-second +wind gust (\cite{beljaars1987}). It is included through a function that +includes the effective roughness length, $z_{0m(eff)}$, in order to +take into account the very high effective $u_*$ values that occur over +mountainous terrain (due to the orographic form drag parametrization) +and so avoid unrealistic high gust values. Currently the UM takes +$c_{\rm ugn}=4$ which was reduced from the value used at ECMWF based +on evaluation of the wind gust performance. The stability dependence +of $\sigma_u$ is estimated on the basis of the similarity relation +from \cite{panofsky1977} +\begin{equation*} + \sigma_u = +\begin{cases} + A_{gust} u_* (1.0 - \zhe / (24 L) )^{1/3} & {\rm for}\ L<0 \\ + A_{gust} u_* & {\rm for}\ L>0 +\end{cases} +\end{equation*} +with $A_{gust}=2.29$. For $L$ close to zero the wind gust diagnostic +is undefined and so we set $U_{gust} = U_{10m}$. Note that the +friction velocity, $u_*$, must use the implicitly calculated surface +stress components because the explicitly calculated $u_*$ can be +erratic, particularly over mountainous regions. Then, for consistency +with the implicit $u_*$, $L$ must also be calculated implicitly and, to +avoid potential numerical problems in very light winds, the unstable +($L<0$) case is rewritten as: +\begin{equation*} + \sigma_u = A_{gust} (u_*^3 + k w_*^3 / 24 )^{1/3} +\end{equation*} + +\subsection{TKE: stash 3,473} + +A substantial part of the turbulent flux is parametrized in both the +UM's first order closure and closures involving TKE, $e$, through a +simple down-gradient diffusion term. An estimate of subgrid TKE can +then be made by equating the UM's diffusion coefficient, +(\ref{klnl}), with that from a typical TKE-closure, \mbox{i.e.} +\begin{equation} + K_m = l \sqrt{e} + \label{tke_closure} +\end{equation} +where $l$ is a length scale. Initially it was thought to diagnose $e$ +by approximating $l$ as the mixing length in (\ref{kmlocal}) but +closer inspection reveals that many TKE closures have diagnostic +relationships for $l$ that involve the TKE itself! A common one for +stable boundary layers is $l_{st} \sim \sqrt{e} / N$, where $N$ is the +Brunt-Vaisala frequency. \cite{Suselj2012}, for example, also take +$l_{un} = \tau_{un} \sqrt{e} $ in unstable boundary layers, where +$\tau_{un}$ is a turbulence timescale that they take as a constant 400 +seconds. These they combine through $l^{-1}= l_{un}^{-1} + +l_{st}^{-1} = e^{-1/2}( \tau_{un}^{-1} + \tau_{st}^{-1}) \equiv +e^{-1/2}\tau_{turb}^{-1}$ and (\ref{tke_closure}) becomes: +\begin{equation*} + K_m = \tau_{turb} e +\end{equation*} +To derive a TKE diagnostic then requires a parametrization of the +turbulence timescale, $\tau_{turb}$. + +Basic boundary layer scaling (\mbox{e.g.}, Figure 4 of +\cite{holtslag91:_eddy_diffus_count_trans_convec}) shows that +$\ol{w'^2}$ from a variety of convective boundary layer LES and +observations nicely follows the relationship +\begin{equation} + \ol{w'^2} = c_{w2} w_*^2 f(z') + \label{w2_scaling} +\end{equation} +where $w_*$ is the convective velocity scale and $f$ is a shape +function within the boundary layer ($z'=z/\zhe$). The shape of this +function is very similar to that used in the UM for $\kmsurf$ in +(\ref{kmsurf}). We now assume we can generalise (\ref{w2_scaling}) by +replacing $w_*$ with $w_m$ (this really ought to be checked against +neutral boundary layer LES but hasn't yet been). Setting $ f(z')=z' +(1-z')^2$ in (\ref{w2_scaling}) and comparing with Fig.4 of +\cite{holtslag91:_eddy_diffus_count_trans_convec} gives $c_{w2}= 2.66 +/ C_{ws}^{2/3}$ (\mbox{i.e.}, a constant of 2.66 gives the maximum in +$\ol{w'^2}/w_*^2$ at around the observed value of 0.4) so that we can +generalise (\ref{w2_scaling}) to +\begin{equation} + \ol{w'^2} = \frac{2.66 }{C_{ws}^{2/3}} \, w_m^2\, f(z') + \label{gen_w2_scaling} +\end{equation} +where the mixed layer expression for $w_m$ is used. + +\cite{holtslag91:_eddy_diffus_count_trans_convec} also show from +analysis of the scalar flux budget that +\begin{equation*} + \ol{w'\theta'} = - \frac{\tau_{turb}}{2} \, \ol{w'^2} \frac{d \theta}{dz} +\end{equation*} +where $\tau_{turb}$ is a return to isotropy timescale. Ignoring the +non-gradient parametrization in the UM, it follows that +\begin{equation} + \khsurf = \frac{\tau_{turb}}{2} \, \ol{w'^2} + \label{bl_scaling} +\end{equation} +Combining (\ref{bl_scaling}) with (\ref{gen_w2_scaling}) and +(\ref{kmsurf}), and subsuming the Prandtl number into the other +constants, for surface-driven boundary layer mixing we can write: +\begin{equation*} + \kmsurf = \frac{\tau_{\rm surf}}{2} \, \ol{w'^2} = \frac{\tau_{turb}}{2} \, \frac{2.66 }{C_{ws}^{2/3}} w_m^2 \, f(z') = k \zhe w_m f(z') +\end{equation*} +which then gives $\tau_{\rm surf} = C_{ws}^{2/3} k \zhe / (1.33 w_m) +$. An analogous timescale can be derived for top-driven mixing in +decoupled stratocumulus layers, $\tau_{\rm Sc} = g_1 k \zml / (1.33 \, +\vtopo) $. + +There are two options to derive a TKE diagnosis from the Ri-based scheme +and then combine with the non-local TKE (selected via var\_diags\_opt). One +is to assume $\tau_{\rm SBL}=0.7/N$ as the timescale for +stable boundary layers and combine all these timescales following +\cite{Suselj2012}) to give: +\begin{equation} + e = K_m \tau_{turb}^{-1} + \label{tke_diag} +\end{equation} +where $\tau_{turb}^{-1} = MAX[ \tau_{\rm surf}^{-1},\tau_{\rm Sc}^{-1}] + \tau_{\rm SBL}^{-1}$. +Note that (\ref{tke_diag}) gives $\ol{w'^2}$, rather than TKE. As a simple fix +to improve the near-surface TKE in convective boundary layers, where the horizontal +wind variability often dominates, the value of $e$ given by (\ref{tke_diag}) at +the level of the maximum in $\kmsurf$ is copied to all levels below that height. + +The second method diagnoses TKE for the local scheme following the Met Office +LEM and MONC, simplifying and parametrizing the terms in the TKE budget to give +\begin{equation} + e_{loc}^{3/2} = \lambda S^2 K_m (1-Ri/Pr)/C_{e} + \label{tke_diag_loc} +\end{equation} +where $C_{e}=A_{2N}^{3/2}$. Initial tests found that the MONC value of +$A_{2N}=0.23$ gave rather large values of $e_{loc}$ and so $C_e=0.41$ is used. Note +that this is an optional value used in the higher order closure scheme (see section +2.8.5 of \citeumdp{025}). It could also be worth testing the suggested parametrization in +(2.300) there, of $C_e=0.19+0.74 \lambda/\Delta z$ but this has not yet been attempted. +The total non-local TKE is computed by adding the TKE from each non-local component, +as is done for the diffusion coefficients, \mbox{i.e.}, +\begin{equation} + e_{nl} = \frac{3}{2} \left( \frac{\kmsurf}{\tau_{\rm surf}} + + \frac{\kmtop}{\tau_{\rm Sc}} \right) + \label{tke_diag_nl} +\end{equation} +The factor of $3/2$ in (\ref{tke_diag_nl}) arises because we are really diagnosing +$\ol{w'^2}$ and so here we make the assumption of isotropic turbulence to extend +this to TKE. As before, we do also make the simple fix to improve the near-surface +TKE in convective boundary layers, but here we copy only the value of $e_{nl}$ at the +level of the maximum in $\kmsurf$ to all levels of $e_{nl}$ below that height. The +final TKE is then the greater of $e_{nl}$ and $e_{loc}$ (as is done to combine the +diffusion coefficients). + +The methods above will still underestimate the value of subgrid TKE in +regions of parametrized convection, because the value of $K_m$ will be +small (or zero) here as parametrized mixing is assumed to be done via +the convection scheme. Therefore an option {\sl l\_conv\_tke} is +provided to include an estimate of the TKE due to parametrized +convection within the diagnostic. This is given by: +\begin{equation} +e_{\rm conv} = \left(\frac{M}{g\rho \times CCA}\right)^2 +\end{equation} +where $M$ is the convective updraft mass flux (Pa~s$^{-1}$) and CCA is +the convective cloud area. The final diagnostic is then given as the +maximum of $e$ and $e_{\rm conv}$. + +If selected, this option also reduces the minimum value used in UKCA +by an order of magnitude. This is possible, because the minimum is no +longer having to provide a realistic estimate of TKE in convective +cloud regions, and is now a genuine numerical minimum. Selecting this +option also corrects a bug in the level indexing of this diagnostic +when passed to UKCA. + +Note that additional diagnostics of the scalar variances are also made and those +are documented in \citeumdp{029}. + +\subsection{Diagnostics of Neutral Winds and Stresses: stash 3,365 to 3,371} +\label{app:neutwind} + +Conditions near the ocean's surface are often described using 10-m +neutral wind and quantities derived from the neutral winds. Such +diagnostics are therefore potentially very useful for evaluation of +the model and have been added to the scheme. + +The equivalent neutral 10-m wind is the wind that would be observed, +given the surface friction velocity and roughness length, if the +stratification were neutral. As such, it is more simply related to the +surface stress than the true stability-dependent wind. Scatterometers +ultimately respond to backscatter from surface capillary waves, which +are driven by the surface stress, so observations from scatterometers +are typically reported as equivalent neutral winds. + +The pseudostress is the product of the wind speed and the vector wind +at a given height (in practice 10m). The kinematic surface stress is +therefore equal to the product of the pseudostress and the drag +coefficient. Pseudostress is sometimes used in observational products, +notably the Cross-Calibrated Multi-Platform (CCMP) surface wind vector +analysis \cite[]{atlas2011}. + +%------------------------------------------------------------------------ +% APPENDIX: VELOCITY SCALES +%------------------------------------------------------------------------ +%\setcounter{section}{0} +%\renewcommand{\thesection}{\Alph{section}} +%\renewcommand{\theequation}{\Alph{section}.\arabic{equation}} +%\setcounter{equation}{0} + +\section{Appendix: Definitions of the velocity scales} +\label{app:vscales} + +As described in \cite{lock00}, the parametrization of the entrainment +rate in convective boundary layers is based on four velocity scales, +each representative of a turbulence-generating process ($\vheato$ for +surface heating, $u_*$ for surface shear generation, $\vrado$ for +cloud-top radiative cooling and $\vbro$ for buoyancy reversal). The +velocity scales can be written +\begin{eqnarray} + \vheat &=& \zml \! \left( (2-\zeta_s)\zeta_s \wbs + (1-\zeta_s)^2 \wbsat \right) + \label{vsurf} \\ + \vrad &=& \zml \Delta_\radf \, g \, + \left( \beta_T \zeta_r^2 + \tilde{\beta_T} (1-\zeta_r^2) \right) + \label{vrad} \\ + \vbr &=& \abr \chi_s^2 \, \mbox{max}\left[0,-\delta b\right] \, \Delta b ^{1/2} + \, z_c^{3/2} \, C_{fac} \label{vbr} +\end{eqnarray} +Here, $\wbsat = g ( \tilde{\beta_T} \wthls + \tilde{\beta_q}\wqts )$, +where the subscript $_S$ indicates the surface flux; $\Delta_\radf$ is +the divergence of the net radiative flux, $\radf$ (in Kms$^{-1}$), +associated with cloud-top, for which the calculation is described in +section~\ref{app:deltaf}. + +Various depth parameters are given by $\zeta_s = +(\zml-\tilde{z_c})/\zml $, $\zeta = (\zml-z_c)/\zml $ and $ \zeta_r = +\zeta + Br (1-\zeta) $. $\zml$ is the mixed-layer depth, $z_c$ is the +cloud depth and $\tilde{z_c}$ is the cloud-fraction weighted cloud +depth. The former is used in the calculation of $\vrado$ as it is +assumed the radiative cooling will occur predominantly in cloudy air. +To allow for a feedback in the presence of buoyancy reversal, the +parameter $Br$ is included in $ \zeta_r $ and $\tilde{\alpha_t} $ (in +(\ref{we_parm})). It is given in terms of the \cite{siems1990} +parameter, $D = \chi_s \delta b/\Delta b $ and constrained by $0< Br = +10 D < 1$. This gives a linear ramp for this feedback between regimes +where there is no buoyancy reversal ($D \leq 0$) and the feedback seen +in LES of stratocumulus \cite[]{lock98} with significant buoyancy +reversal ($D \gtapp 0.1$). Furthermore, the LES of +\cite{lock09:_factor} indicated the presence of cumulus penetrating up +into stratocumulus could be sufficient to enhance the feedback for +small $D$. Thus the option exists to enhance $Br$ for $0$ SC\_CFTOL in grid-levels NTML or NTML$+1$ or the layer is a +decoupled layer). If no subgrid inversion has been diagnosed and +$C_F(NTML+1)>$ SC\_CFTOL, then $z_c$ is increased by the full depth of +layer $NTML+1$ if $C_F(NTML)>$ SC\_CFTOL and using (\ref{zc_calc}) +otherwise (and similarly for DSC layers). The same ideas are used in +the 9C version, extrapolating using the adiabatic +water gradient, except that the grid-level from which this +extrapolation is made is now not the lowest grid-level with $C_F > $ +SC\_CFTOL but rather the lowest level with $C_F=1$ (or the grid-level +with the maximum $C_F$). Most observations of stratocumulus (i.e., +mixed layer clouds) find $\ql$ to be close to adiabatic over the whole +layer and accurately given by the supersaturation of the well-mixed +$q_t$. If $C_F=1$ then the Smith cloud scheme makes $\ql$ equal to the +supersaturation and so will be reasonably accurate. Extrapolating this +grid-level $\ql$ to zero should then give a reasonably accurate +measure of cloud-base. + +The formula for $\vbro$ was derived using dimensional arguments and +comparison with LES data: $\chi_s = -\qlmax (1+(L/c_p)\alpha_L) / ( +\Delta q_t - \alpha_L \Delta \thetal)$, where $\qlmax$ is the +cloud-top liquid water mixing ratio, $L$ is the latent heat of +vaporisation of water, $c_p$ the specific heat at constant pressure +and T is the temperature; $\delta b = g(\tilde{\beta_T} \Delta \thetal ++ \tilde{\beta_q} \Delta q_t)$ and the buoyancy jump across the +inversion is given by +\begin{equation} + \Delta b = g \, \left( \beta_T \Delta \thetal + \beta_q \Delta q_t + + \left( \beta_T \frac{L}{c_p} - + \frac{1+c_v}{c_v}\beta_q \right)\Delta \ql + + \left( \beta_T \frac{L_s}{c_p} - + \frac{1+c_v}{c_v}\beta_q \right)\Delta q_f \right) + \label{dbinv} +\end{equation} +The empirical constant $\abr = 0.24$. The calculation of $\Delta +\thetal$ and $ \Delta q_t$ is described for a subgrid inversion in +section~\ref{sec:sginv} or, if one is not diagnosed, they are taken +simply as $\Delta_{\ntml+1}$. For $\Delta \ql$, $\Delta q_f$ and +$\qlmax$, in-cloud values extrapolated to $z_i$ (either subgrid or +$z_{\ntml+\frac{1}{2}}$) from above and below using the adiabatic +lapse rates are calculated as: +\begin{eqnarray*} + \qlmax & =& \frac{{\ql}_{\ntml}}{{C_F}^l_{\ntml}} + + (z_i-z_{\ntml}) \gamma_{\ql}\\ + \ql^+ & =& \mbox{max}\left[ 0, \, + \frac{{\ql}_{\ntml+2}}{{C_F}^l_{\ntml+2}} + - (z_{\ntml+2}-z_i) \gamma_{\ql} \right] +\end{eqnarray*} +and similarly for $q_f$ (noting that currently $\gamma_{q_f}=0$) and +for DSC layers. Then, +\begin{eqnarray*} + \Delta \ql & =& {C_F^l}_{\ntml+2}\, \ql^+ - {C_F^l}_{\ntml}\, \qlmax \\ + \Delta q_f & =& {C_F^f}_{\ntml+2}\, q_f^+ - {C_F^f}_{\ntml}\, \qfmax +\end{eqnarray*} + +The only other explicit account of variable cloud fraction is in +(\ref{vbr}) for which it is assumed that buoyancy reversal can only +occur for cloudy air underlying cloud-free air (assuming maximum +overlap). Thus, the cloud fraction factor, $C_{fac} = \mbox{max}[ +0.0, -\Delta C_F ] $, where $\Delta C_F = {C_F}_{\ntml+2} - +{C_F}_{\ntml} $ if a subgrid inversion is diagnosed (because +${C_F}_{\ntml+1}$ is currently meaningless) and $\Delta C_F = +{C_F}_{\ntml+1} - {C_F}_{\ntml} $ if not. A more complete +decomposition is not possible given a cloud scheme in the model +\cite[]{smith90} which does not allow discrete identification of +in-cloud and out-of-cloud profiles. The cloud-fraction dependence of +the radiative generation of turbulence is implicitly treated in +(\ref{vrad}) simply by assuming the grid-box mean radiative flux +divergence, $\Delta_\radf$, occurs solely in the cloudy air. + +The assumption behind the current cloud fraction dependence is that +the fraction of the boundary layer that is cloud-capped entrains as +though it were an infinite solid cloud sheet (as in the LES used to +derive the parametrization). This essentially assumes the cloud +within the grid-box is continuous. An additional explicit dependence +of entrainment on cloud fraction was implemented in version 4.5 which +reduced the cloud-top source terms of entrainment in partially cloudy +boundary layers by a factor +$\exp{\left\{-(0.9-{C_F}_{\ntml})^3/0.075\right\}}$ for $ +{C_F}_{\ntml} < 0.9$. It was argued that partial cloudiness on the +scale of the mixed-layer eddies might reduce the entrainment +efficiency of the cloud-top processes. By reducing the parametrized +entrainment warming and drying in partially cloudy boundary layers it +was hoped that the climatological cloudiness of the sub-tropical +marine stratocumulus might be improved. Only marginal success was +observed, though, as more weakly entraining layers became shallower +and their cloud-top therefore warmer. In addition, timeseries of +cloud fraction from New Dynamics climate simulations suggested these +boundary layers either had high total cloud amount or zero. Coupled +with the generally realistic cloud amounts obtained in the New +Dynamics this arbitrary term has currently been dropped from version 5 +onwards. The issue of how the entrainment rate should be parametrized +in partially cloudy boundary layers, however, remains. + +\subsection{Calculation of $\Delta_\radf$} +\label{app:deltaf} + +An important term in the entrainment parametrization and $\khtop$ is +the velocity scale $\vrado$, the cube of which is proportional to the +net radiative flux difference associated with cloud-top, $\Delta_F$. +Because radiation tends not to be called every timestep, the +calculation of $\Delta_\radf$ is done somewhat independently from the +cloud-top height. + +In the 9B version, $\Delta_\radf$ is calculated as: +\begin{equation} + \Delta_\radf = \sum_{k=k_m-1}^{k_m+1} \mbox{max}\left[ + - \Delta_{k+\frac{1}{2}} z \, {\cal S}_\radf(k), \,0 \right] + \label{ctraddiv} +\end{equation} +where $k_m$ is the grid-level with the greatest radiative cooling +increment, ${\cal S}_\radf$, within 2 grid-levels of cloud-top. In +the 8A scheme, ${\cal S}_\radf$ is simply the net (SW+LW) cooling +increment. During the day, though, the net divergence is partly +reduced from the nocturnal (LW) value due to SW warming of the +cloud-layer. In general, the SW warming is more diffuse than the LW +cooling (which occurs mostly within O(50)m of cloud-top). Using +grid-level net radiative increments at the current coarse vertical +resolution used in the UM ($\sim 200$m) means that the cancellation +between LW and SW during the day is excessive. + +A slightly more accurate estimate of the net divergence can be +obtained by assuming the SW and LW radiative fluxes at a given height, +$z$, have an exponential shape, dependent on the LWP above $z$, +\mbox{i.e.}: +\begin{equation} + F_{LW}(z) = \Delta_\radf^{LW} \exp^{ - \kappa_{LW} \mbox{LWP}(z) } + \label{eq:explw} +\end{equation} +Then the net divergence can be approximated given the SW flux at the +height where $F_{LW}$ becomes some small fraction, $A$, of $ +\Delta_\radf^{LW} $ (which implies $ \mbox{LWP} =-ln(A) / \kappa_{LW} +$). Then +\begin{equation*} + \Delta_\radf \approx \Delta_\radf^{LW} + + (1-\exp^{ln(A)\kappa_{SW}/\kappa_{LW}}) \Delta_\radf^{SW} +\end{equation*} +Empirically, see Fig.~\ref{fig:dradts}, a reasonable fit to LEM data +is obtained with: +\begin{equation} + \Delta_\radf \approx \Delta_\radf^{LW} + 0.35 \Delta_\radf^{SW} + \label{eq:deltaf_emp} +\end{equation} +Note that in the 9B scheme $\Delta_\radf^{LW} $ and $\Delta_\radf^{SW} +$ are calculated as in (\ref{ctraddiv}) but with the LW and SW +increments separately. + +This change in the calculation of $\Delta_\radf$ is illustrated in +Fig.~(\ref{fig:dradts}) from LES of the diurnal cycle of marine +stratocumulus. The top panel is from a simulation which used the code +specified for the EUROCS LES intercomparison, the lower panel used the +Edwards-Slingo radiation scheme in the LES. There are clearly some +differences in the distribution of the SW absorption within the cloud +between these two schemes but the 9B parametrization is clearly an +improvement on the 8A which gives $\Delta_\radf=0$ around midday (and +therefore zero entrainment and turbulent mixing). + +\begin{figure}[tbh] + \begin{centering} + \includegraphics[width=3.5in]{div_r080} + \includegraphics[width=3.5in]{div_r071} + \end{centering} + \captionarb{Time series from LES of $\Delta_\radf$ (solid), + $\Delta_\radf^{LW} $ (dotted), $-\Delta_\radf^{SW} $ (dashed) and + the 8A (dash-dot) and 9B (dash-dot-dot-dot) parametrizations of + $\Delta_\radf$.} + \label{fig:dradts} +\end{figure} + +The 9C version attempted to remove the grid-dependence implied by the +summation over 3 grid-levels in (\ref{ctraddiv}) as follows: +\begin{enumerate} +\item the search for the level with maximum LW radiative + cooling, $k_m$, is restricted to the top half of the mixed layer and + no higher than level NTML$+1$ +\item to allow for the case where the radiative cooling is distributed + roughly equally over two grid-levels, if the LW flux divergence in + level $k_m-1$ is greater than half that in level $k_m$, then $k_m$ + is lowered one grid-level. +\item then, the cloud-top radiative flux change is initially + calculated for LW and SW fluxes separately as: + \begin{equation} + \Delta_\radf = \radf_{k_m+1} - \radf_{k_{rb}} + \label{ctraddiv_9c} + \end{equation} + where the base grid-level for the calculation, $k_{rb}$, is taken to + be the higher of the base of the LW radiatively cooled layer and + $z_h/2$, since cooling can only generate turbulence if it occurs in + the upper part of the mixed layer. For decoupled stratocumulus + layers, $k_{rb}$ is further restricted to be above the top of the + surface mixed layer. +\item finally, the flux divergence across grid-level $k_m+1$ is + separated into cloudy and free-atmospheric contributions by + extrapolating the free-atmospheric flux-gradient downwards. The + cloudy contribution is then included in $\Delta \radf$: + \begin{equation} + \Delta \radf = \Delta \radf + \Delta_{k_m+\f{3}{2}} \radf + - \Delta_{k_m+\f{5}{2}} \f{\Delta_{k_m+\f{3}{2}} z}{k_m+\f{1}{2}} \radf + \label{ctraddiv_9c_inv} + \end{equation} +\item As at 9B above, the calculations in (\ref{ctraddiv_9c}) and + (\ref{ctraddiv_9c_inv}) are performed separately for LW and SW + radiation before the two are combined using the empirical + relationship in (\ref{eq:deltaf_emp}) +\end{enumerate} + +Further single column model tests with fine vertical resolution have shown +the above calculations can still fail to accurately measure the radiative +flux jump across the top of the cloud. The methodology now recommended is +to identify where the LW radiative cooling profile transitions from +free-tropospheric rates above the cloud to stronger rates within it. It +entails only relatively minor changes to the first two steps of the +algorithm above which become: +\begin{enumerate} +\item the search for the level with maximum LW radiative cooling, $k_m$, is +restricted to the top half of the mixed layer and below $1.2 \, \zhe$ (rather +than level NTML$+1$ used above) +\item if the LW flux divergence in level $k_m+1$ is relatively weak (less than +double that in level $k_m+2$), we assume that level $k_m+1$ is actually typical +of the free-troposphere and that $k_m$ must therefore be the inversion +grid-level (despite having the strongest LW cooling). Hence we lower +$k_m$ by one so that it now marks the top of the mixed layer --- note that +LW cooling within the inversion grid-level will be included in step 4 above +(which is unchanged) +\end{enumerate} +These small changes were found sufficient to give a robust measure of +$\Delta \radf$ for grids varying down to 20m spacing where the radiative +flux profile is well resolved. + +%------------------------------------------------------------------------ +% APPENDIX: BUOYANCY PARAMETERS +%------------------------------------------------------------------------ +\newpage +\section{Appendix: Derivation and definitions of the buoyancy parameters} +\label{app:buoyp} + +Buoyancy is measured by the virtual temperature +\begin{equation} + T_v = T(1 + c_v q_v - \ql - q_f) = T V_{fac} + \label{Tv} +\end{equation} +where $c_v=(1/\epsilon) -1$ and $\epsilon$ is the ratio of the +molecular weights of water vapour and dry air (\mbox{i.e.}, $\epsilon += M_v/M_a \approx 0.62198$). The buoyancy flux is then given by +\begin{equation*} + \wb = \frac{g}{T_v}\, \overline{w'T_v'} +\end{equation*} +Linearising gives +\begin{equation*} + \wb = g \left( \beta_T \overline{w'T_L'} + \beta_q \wqt + + \left( \beta_T \frac{L}{c_p} - \frac{1+c_v}{c_v} \beta_q \right) + \overline{w'\ql'} \right) +\end{equation*} +where the buoyancy parameters are given by + +\begin{equation*} + \beta_T = \frac{1}{T}, \qquad \beta_q = \frac{c_v}{V_{fac}} +\end{equation*} + +In saturated cloudy air (see, for example, \cite{stage1981}), the +Clausius-Clapeyron equation can be used to calculate $ +\overline{w'\ql'} $ (via $\ql' = q_t' - q_s' = q_t' - \alpha_L T'$) as +\begin{equation*} + \overline{w'\ql'} = a_L( \wqt - \alpha_L \overline{w'T_L'} ) +\end{equation*} +where +\begin{equation*} + \alpha_L = \frac{\partial q_s}{\partial T} = \frac{\epsilon L q_s(T,p) }{R T^2}, \qquad + a_L = \frac{1}{1+L\alpha_L/c_p} +\end{equation*} +where R is the gas constant ($=287.05$). Thus the buoyancy flux can +be written +\begin{equation*} + \wb = +\begin{cases} + g \left( \beta_T \overline{w'T_L'} + \beta_q \wqt \right) + & {\rm in\ unsaturated\ air} \\ + g \left( \tilde{\beta_T} \overline{w'T_L'} + \tilde{\beta_q} \wqt + \right) & {\rm in\ saturated\ air} +\end{cases} +\end{equation*} +where +\begin{eqnarray*} + \tilde{\beta_T} = \beta_T - \alpha_L \beta_c, & + \tilde{\beta_q} & = \beta_q + \beta_c \\ + {\rm and} & + \beta_c & = a_L \left( \frac{L}{c_p} \beta_T + - \frac{1+c_v}{c_v} \beta_q \right) +\end{eqnarray*} + +Note that here $\tilde{\beta_T}$ and $\tilde{\beta_q}$ are strictly +{\em in}-cloud parameters, while their definitions in boundary layer +code prior to 8A were grid-box mean. Thus, here, any necessary +$C_F$-weighting must be included explicitly, as in (\ref{eq:wb_cont}). + +In all the above, if $T$ is less than the melting point of ice then +the latent heat of sublimation, $L_s = L + L_f$, is used in place of +$L$. + +%------------------------------------------------------------------------ +% APPENDIX: MIXING RATIOS +%------------------------------------------------------------------------ +\newpage +\section{Appendix: changing between specific humidities and mixing ratios} +\label{app:mixratio} + +Denote wet density by +\begin{equation*} + \rho = \rho_y + \rho_v + \rhol + \rho_{f} +\end{equation*} +where $\rho_y$ is the density of dry air and the other $\rho$ are +vapour, liquid and frozen water respectively. Mixing ratios and +specific humidities are then defined as +\begin{equation*} + m_v = \frac{\rho_v}{\rho_y} q_v = \frac{\rho_v}{\rho} +\end{equation*} + +When specific quantities are mixed, the turbulent diffusion equations, +(\ref{cons_eqn_scal}) and (\ref{cons_eqn_uv}), have $\rho$ as the wet +density. This is then consistent with the conservation of globally +integrated quantities such as moisture. For example, neglecting +spherical geometry for simplicity: +\begin{equation} + \int (\rho_v + \rhol + \rho_{f})\, d\underline{x} = + \int \rho (q_v + \ql + q_{f}) \,d\underline{x} = \int \rho q_t \, d\underline{x} +\label{moisture_cons} +\end{equation} + +When mixing ratios are used, the momentum equations, +(\ref{cons_eqn_uv}), remain unchanged and the wet density still +appears. This makes the reasonable assumption that all moisture +components should be included in the momentum budget. For moisture +conservation, (\ref{moisture_cons}) can be rewritten in terms of +mixing ratios as +\begin{equation*} + \int (\rho_v + \rhol + \rho_{f})\, d\underline{x} = + \int \rho_y (m_v + \ml + m_{f}) \,d\underline{x} = \int \rho_y m_t \, d\underline{x} +\end{equation*} +Thus $\rho$ in (\ref{cons_eqn_scal}) is replaced with $\rho_y$ for +$\chi = m_t$ and $\thetal$ when mixing ratios are passed into the +boundary layer code. + +For surface exchange, JULES initially approximates the surface air density as +$\rho_* = p_S/(R T_S)$ (where the subscript $S$ denotes the surface values and $R$ +the gas constant for dry air, 287 JK$^{-1}$kg$^{-1}$). If a more accurate calculation +of surface air density is requested, following Eqs (1) to (6) of +\cite{Webbetal1980} we then calculate the wet or dry surface air densities (to be +used when the atmospheric humidity is specific or mixing ratio, respectively) as: +\begin{eqnarray*} + \rho_{0} & =& \rho_*/(1+(1/\epsilon-1)q_S) \\ + \rho_{y0} & =& \rho_*/(1+(1/\epsilon)m_{vS}) +\end{eqnarray*} +where, in each case, the surface humidity is taken as the surface saturated humidity +over open sea but over land and ice surfaces this is likely to be inappropriate +and so the driving level humidity is used (typically the lowest model level). + +Note that $\thetal$ itself is defined in terms +of mixing ratios as: +\begin{equation} + \thetal = T - \frac{L_c}{c_{pd}} \ml + - \frac{L_c+L_f}{c_{pd}} m_f + \frac{g}{c_{pd}} z + \label{sl_defn} +\end{equation} + +For saturation calculations a version of QSAT is used that is +switchable between input specific and mixing ratio variables. The +rate of change of $q_s$ with temperature is also used in the boundary +layer code (see e.g. appendix~\ref{app:buoyp}): +\begin{equation*} + \frac{ d q_{sat} }{ dT } = \frac{\epsilon L q_{sat} }{RT^2} +\end{equation*} +In fact this expression should really be converted to work for +specific quantities and so simply changing to mixing ratios will +improve the accuracy of this calculation. + +Finally, in appendix~\ref{app:buoyp} virtual temperature is defined in +terms of specific variables as +\begin{equation*} + T_v = T ( 1 + c_v q_v - q_l ) +\end{equation*} +In terms of mixing ratios this becomes +\begin{equation*} + T_v = \frac{T ( 1 + m_v/\epsilon )}{ 1 + m_v + m_l } +\end{equation*} +Linearising, however, gives +\begin{equation*} + T_v = T ( 1 + c_v m_v - m_l ) +\end{equation*} +Thus, the same level of approximation as is currently used is +maintained simply by changing specific variables to mixing ratios. + +Additional points to note are: +\begin{enumerate} +\item the diagnostic of screen humidity, $q$1.5m, will remain as a + specific humidity. Similarly RH1.5m will remain defined in terms of + specific quantities, \mbox{i.e.}, $q$1.5m$/q_s$1.5m +\item all other moisture diagnostics (\mbox{e.g.}, latent heat fluxes + and increments) will simply switch to being mixing ratios if mixing + ratios are selected - no conversion will be made. +\item it is important to note that RHOKM will be wet density times + $K_m$ while RHOKH will be dry density times $K_h$ +\end{enumerate} + +%------------------------------------------------------------------------ +% APPENDIX: frictional heating +%------------------------------------------------------------------------ +\section{Appendix: including the heating from turbulence dissipation} +\label{app:fricheat} + +An estimate of the true molecular dissipation rate, $\epsilon_{mol}$, +can be obtained by assuming local equilibrium in the budget of subgrid +TKE (SKE). Then the sum of the inputs from resolved kinetic energy, +plus that from subgrid buoyancy effects, must equal the dissipation. +The SKE budget is +\begin{equation} + d SKE/dt = S + T + B + \epsilon_{mol} +\label{ske_budg} +\end{equation} +where the shear production, S, is essentially the resolved KE +dissipation term. Note that the buoyancy term B appears in +(\ref{ske_budg}) which indicates that some of the energy from resolved +scale dissipation (i.e. S) should be consumed in doing work against +buoyancy (at least in stable BLs) thus leaving less energy to be +finally dissipated as heat. Note though that CBLs will generate +additional dissipation (and hence heating) through the buoyancy term. +Note that from an atmospheric budget viewpoint the transport term, T, +can be neglected as it will integrate vertically to zero. Locally, +vertical variations could be important but it will be ignored because +finally the heating source is implemented through an integral over the +BL. + +So, this estimate of the molecular dissipation rate should appear as +an additional heating source term, (following \cite{zhang1999}): +\begin{equation*} + \frac{\partial \thetal}{\partial t} = + \left[\frac{du}{dz}\tau_x + \frac{dv}{dz}\tau_y + B \right] / (\rho c_p) +\end{equation*} +Tests in the SCM showed the heating rate gradients can be very large +near the surface. Hence to avoid stability problems (since this +heating increment must be added after the implicit calculation of the +stress (and heat flux) profiles) the increments are summed over the +levels within the BL (i.e. up to \zh) and then that total heating is +applied as a linear decrease from the surface to zero over \zh. + +%------------------------------------------------------------------------ +% APPENDIX: OPERATIONAL FUDGES +%------------------------------------------------------------------------ +\section{Appendix: Operational modifications} +\label{app:opmods} + +The operational global forecast model has been found to give improved +performance on NWP Index parameters when the following modifications +to its local $Ri$-based scheme are used. In (\ref{asymp_ml}), the +definition of $\lambda_m$ only is altered to +\begin{equation*} + \lambda_m = \mbox{max}\left[40,\, 0.3 \zloce, 2 h_B \right] +\end{equation*} +and both $\lambda_m$ and $\lambda_h$ are not reduced (to 40m) above +the boundary layer top. It is possible these modifications point to +problems with the definition of the boundary layer depth and the use +of a Prandtl number of unity with no stability dependence in the local +scheme. Both these issues are under further investigation. + +\newpage +%------------------------------------------------------------------------ +% APPENDIX: UKCA INPUTS +%------------------------------------------------------------------------ +\section{Appendix: Inputs to UKCA} + +The UKCA chemistry and aerosols sub-model takes a number of boundary +layer diagnostics as input. For a list of these and a brief +explanation of how they are used see the table below. If any changes +modify the results for these variables it will prevent UKCA jobs from +regressing. If the changes are significant it would be prudent to +discuss them with the UKCA code owner before lodging the change. + +\begin{table}[htp] + \begin{center} + \begin{tabular}{|p{0.5cm}|p{0.7cm}|p{6cm}|p{6cm}|} + \hline + \multicolumn{4}{|c|}{Boundary layer inputs to UKCA} \\ + \hline + Sec & Item & Description & Use in UKCA \\ + \hline + 0 & 24 & SURFACE TEMPERATURE AFTER TIMESTEP & dry deposition \\ + 0 & 25 & BOUNDARY LAYER DEPTH AFTER TIMESTEP & dry deposition, bl nucleation and call to tr\_mix \\ + 0 & 26 & ROUGHNESS LENGTH AFTER TIMESTEP & dry deposition \\ + 0 & 233 & SURFACE TEMPERATURE ON TILES K & dry deposition \\ + 0 & 234 & ROUGHNESS LENGTH ON TILES m & dry deposition \\ + 3 & 60 & RHOKH\_MIX & call to tr\_mix \\ + 3 & 64 & DTRDZ\_CHARNEY\_GRID & call to tr\_mix \\ + 3 & 65 & GRID-LEVEL OF SML INVERSION (kent) & call to tr\_mix \\ + 3 & 66 & Rho * entrainment rate (we\_lim) & call to tr\_mix \\ + 3 & 67 & Fraction of the timestep (t\_frac) & call to tr\_mix \\ + 3 & 68 & zrzi & call to tr\_mix \\ + 3 & 69 & GRID-LEVEL OF DSC INVERSION (kent) & call to tr\_mix \\ + 3 & 70 & Rho * entrainment rate dsc & call to tr\_mix \\ + 3 & 71 & Fraction of the timestep dsc & call to tr\_mix \\ + 3 & 72 & zrzi dsc & call to tr\_mix \\ + 3 & 73 & ZHSC Top of decoupled layer & call to tr\_mix \\ + 3 & 217 & SURFACE HEAT FLUX W/M2 & dry deposition \\ + 3 & 230 & 10 METRE WIND SPEED ON C-GRID & calculate sea salt emissions \\ + 3 & 401 & Dust Emissions div 1 & GLOMAP dust scheme \\ + 3 & 402 & Dust Emissions div 2 & GLOMAP dust scheme \\ + 3 & 403 & Dust Emissions div 3 & GLOMAP dust scheme \\ + 3 & 404 & Dust Emissions div 4 & GLOMAP dust scheme \\ + 3 & 405 & Dust Emissions div 5 & GLOMAP dust scheme \\ + 3 & 406 & Dust Emissions div 6 & GLOMAP dust scheme \\ + 3 & 430 & Dust Friction velocity (U*) on tiles & dry deposition \\ + 3 & 462 & STOMATAL CONDUCTANCE ON PFTS (M/S) & dry deposition \\ + 3 & 465 & FRICTION VELOCITY & dry deposition \\ + 3 & 473 & TURBULENT KINETIC ENERGY & ACTIVATE cloud scheme \\ + \hline + \end{tabular} + \end{center} +\end{table} + +%------------------------------------------------------------------------ +% APPENDIX: NOTATION +%------------------------------------------------------------------------ +\section{Appendix: Notation} +\label{app:not} + +\begin{flushleft} + \begin{tabular}{|l|l|} + \hline + \multicolumn{2}{|c|}{Finite difference notation} \\ + \hline + $z_k$ & height of the $\theta$-level $k$ \\ + $z_{k+\frac{1}{2}}$ & height of half-level above $\theta$-level $k$ \\ + $\Delta_k$ & indicates a finite difference between $\theta$-levels $k$ and + $k-1$ \\ + $\Delta_{k+\frac{1}{2}}$ & indicates a finite difference between + half-levels $k+\frac{1}{2}$ and $k-\frac{1}{2}$ \\ + $\Delta$ & note: real change (\mbox{i.e.}, not necessarily + finite-difference) in a parameter \\ + & across the capping inversion (see (\ref{dbinv}) and following text) \\ + \hline + \end{tabular} +\end{flushleft} + +\begin{flushleft} + \begin{tabular}{|l|l|} + \hline + \multicolumn{2}{|c|}{Model variables} \\ + \hline + $\theta_l$, $\thetavl$ & thermodynamic variables defined by (\ref{thetal}) + and (\ref{thetavl}) \\ + $T_v$, $\theta_v$ & virtual temperature and potential temperature, \\ + & defined by (\ref{Tv}) and in section (\ref{sec:parxs}) \\ + $b$ & buoyancy ($=g T_v'/T_v$) \\ + $q_t$, $q_v$, $q_s$, $\ql$, $q_f$ & specific humidities: \\ + & total, vapour, saturated, liquid and frozen water, respectively \\ + $C_F$, $C_F^l$, $C_F^f$ & cloud fraction and the liquid and frozen water + parts, respectively \\ + ${\cal H}$ & total heat flux (net radiative plus turbulent, Kms$^{-1}$) \\ + \hline + \end{tabular} +\end{flushleft} + +\begin{flushleft} + \begin{tabular}{|l|l|} + \hline + \multicolumn{2}{|c|}{Thresholds} \\ + \hline + $C_t$ & ($=1.1$) threshold for ratio of layer $q_t$-gradients in cumulus + diagnosis \\ + $\Gamma_{\rm inv}$ & ($=1.1$) threshold on ratio of environment to parcel + $\theta_v$ gradients \\ + & for identifying capping inversions above the LCL \\ + SC\_CFTOL & ($=0.1$) $C_F$ threshold for recognising the presence of Sc \\ + $\Delta_{k_{ct}} \thetavl / \Delta_{k_{ct}} z < 10^{-3}$ & threshold + (in Km$^{-1}$) for initial diagnosis of {\em well-mixed} DSC layers \\ + $D_t$ & ($=0.1$) threshold for the ratio of buoyancy consumption to + production \\ + & before decoupling occurs \\ + \hline + \end{tabular} +\end{flushleft} + +\begin{flushleft} + \begin{tabular}{|l|l|} + \hline + \multicolumn{2}{|c|}{Layer definitions and parameters} \\ + \hline + SML & surface-based mixed layer \\ + NTML & top $\theta$-level within SML \\ + NTPAR & top $\theta$-level reached by parcel ascent \\ + DSC & decoupled stratocumulus (mixed layer) \\ + NTDSC & top $\theta$-level within DSC layer \\ + NBDSC & bottom $\theta$-level within DSC layer \\ + NTLOC & top $\theta$-level below which $Ri<1$ \\ + \zh & height of top of SML (potentially subgrid) \\ + \zhsc & height of top of DSC layer (potentially subgrid) \\ + \zbase & height of base of DSC layer (subgrid) \\ + \zhpar & height of half-level at top of parcel ascent \\ + \zloc & height of half-level marking `top' of local $Ri$-based mixing \\ + & (where $Ri>1$) \\ + $z_i$ & generic inversion height \\ + $z_c$ & cloud depth \\ + $\zml$ & mixed layer depth \\ + $\kmsurf$, $\khsurf$ & $K$ profiles for surface-driven turbulence (in SML) \\ + $\kmtop$, $\khtop$ & $K$ profiles for cloud-top-driven turbulence \\ + & (calculated for both DSC and SML) \\ + LCL & lifting condensation level \\ + \hline + \end{tabular} +\end{flushleft} + +\begin{flushleft} + \begin{tabular}{|l|l|} + \hline + \multicolumn{2}{|c|}{Other parameters} \\ + \hline + $\gamma_{\thetal}$ & gradient adjustment term, given by (\ref{gradadj}) \\ + $w_m$ & scaling velocity for momentum mixing in the SML \\ + & (used in $\kmsurf$, $\gamma_{\thetal}$ + and the SML parcel perturbation, $\theta_v'$) \\ + $w_*$ & `standard' convective velocity scale for a cloud-free convective \\ + & boundary layer, $ w_*^3 = \zhe \wbs $ \\ + $u_*$ & friction velocity (here includes the orographic component) \\ + $w_e$, $\tilde{w_e}$ & entrainment velocity and compensated to allow for + subsidence (ms$^{-1}$) \\ + $w_S$ & subsidence velocity (ms$^{-1}$) \\ + $\Delta_\radf $ & cloud-top net radiative divergence, calculation given in + (\ref{ctraddiv}) \\ + $\alpha_t$ & parameter in entrainment parametrization, (\ref{we_parm}) \\ + $\tau_{rc}$, $z_{rc}$ & parameters in perturbation calculation, + (\ref{dscd_pert}), \\ + & for initial identification of and $\zml$ calculation for DSC layers \\ + $a_L$, $\alpha_L$, $\beta_T$, $\beta_q$, $\tilde{\beta_T}$, $\tilde{\beta_q}$ + & buoyancy parameters, defined in appendix~\ref{app:buoyp} \\ + \hline + \end{tabular} +\end{flushleft} + +\newpage + +\bibliographystyle{plainnat} +\bibliography{../025/papers_db} + +\end{document} % Every document must end with this. + + + + + + diff --git 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332.5 +%%Title: Graphics produced by WAVE +%%For: lock@visual +%%Creator: WAVE Version 7.00 (Linux i386) +%%CreationDate: Tue Aug 20 09:24:05 2002 +%%EndComments +% EPSF created by ps2eps 1.68 +%%BeginProlog +save +countdictstack +mark +newpath +/showpage {} def +/setpagedevice {pop} def +%%EndProlog +%%Page 1 1 +%+ wave_prolog.ps -- Prolog for PV-WAVE CL generated PostScript files +%+ $Id: wave_prolog.ps,v 1.5 1998/06/05 17:44:06 thaux Exp $ +%+ Copyright (c) 1989-1992 Precision Visuals, Inc. All Rights Reserved. +%v 1 +save /$WAVE_DICT 60 dict def $WAVE_DICT begin /bdef { bind def } bind def /C +{currentpoint newpath moveto} bdef /D {currentpoint stroke moveto} bdef /F +{closepath fill} bdef /K { setgray } bdef /L {translate} bdef /M {moveto} +bdef /N {rmoveto} bdef /O {show} bdef /P {lineto} bdef /R {rlineto} bdef /S +{gsave show grestore} bdef /Z {gsave currentpoint lineto 20 setlinewidth 1 +setlinecap stroke grestore} bdef /CC {concat} bdef /CD {currentmatrix def} +bdef /CP {currentpoint pop def} bdef /FM {findfont exch makefont} bdef /FS +{findfont exch scalefont} bdef /GS {get setfont} bdef /MD {matrix def} bdef +/MN {mul neg def} bdef /RS {roll put setfont} bdef /SL {setlinewidth} bdef +/SM {setmatrix} bdef /L0 {[] 0 setdash} bdef /L1 {[40 100] 0 setdash} bdef +/L2 {[200 200] 0 setdash} bdef /L3 {[200 100 50 100] 0 setdash} bdef /L4 +{[300 100 50 100 50 100 50 100] 0 setdash} bdef /L5 {[400 200] 0 setdash} +bdef /IO { gsave 0.0 setgray /val exch def moveto /str 20 string def val str +cvs show grestore } bdef /UL { gsave currentpoint /cury exch def /curx exch +def /curmtx matrix currentmatrix def matrix identmatrix setmatrix newpath 0 +0 moveto true charpath flattenpath pathbbox curmtx setmatrix /height exch +def /width exch def /ybase cury height .1 mul sub def pop pop newpath curx +ybase moveto width 0 rlineto closepath stroke grestore } bdef /BG { gsave +currentpoint /cury exch def /curx exch def /curmtx matrix currentmatrix def +matrix identmatrix setmatrix newpath 0 0 moveto true charpath flattenpath +pathbbox curmtx setmatrix /height exch def /width exch def pop pop newpath 0 +0 moveto (M) true charpath flattenpath pathbbox /xdiff exch .1 mul def +/ydiff exch .1 mul def pop pop newpath curx xdiff 2 div sub cury ydiff 2 div +sub moveto width xdiff add 0 rlineto 0 height ydiff add rlineto width neg +xdiff sub 0 rlineto closepath fill grestore } bdef /$T_DICT 20 dict def /T { +$T_DICT begin /align exch def /orien exch def /size exch def /thestring exch +def gsave moveto /chsize 1.0 def /xsize 0.0 def /SUPER 8#330 def /SUBS 8#331 +def /NORM 8#332 def /SCRIPTWID 0.7 def /OFFSET 0.6 def orien rotate size dup +scale /orien false def thestring { /charcode exch def charcode SUPER eq +charcode SUBS eq or { /chsize SCRIPTWID def /orien true def } { charcode +NORM eq { /chsize 1.0 def /orien true def } { ( ) dup 0 charcode put +stringwidth pop chsize mul xsize add /xsize exch def } ifelse } ifelse } +forall xsize align mul neg 0 rmoveto orien { /regularfont currentfont def +/fractionfont currentfont [ SCRIPTWID 0 0 SCRIPTWID 0 0 ] makefont def gsave +newpath 0 0 moveto (1) true charpath flattenpath pathbbox /height exch def +pop pop pop grestore } if /xsize 0 def thestring { /charcode exch def +charcode SUPER eq { 0 OFFSET height mul dup /xsize exch def rmoveto +fractionfont setfont } { charcode SUBS eq { 0 OFFSET height mul neg dup +/xsize exch def rmoveto fractionfont setfont } { charcode NORM eq { 0 xsize +neg rmoveto regularfont setfont } { ( ) dup 0 charcode put show } ifelse } +ifelse } ifelse } forall grestore end } bdef /IsChar { exch /CharStrings get +exch known } bdef /MapCh { 3 -1 roll /Encoding get 3 1 roll put } bdef +/MapDegree { dup 16#b0 exch /degree IsChar { /degree } { /ring } ifelse +MapCh } bdef /MapBB { dup 16#a6 exch /brokenbar IsChar { /brokenbar } { /bar +} ifelse MapCh } bdef /ReEncode { dup findfont begin currentdict dup length +dict begin { 1 index /FID ne {def} {pop pop} ifelse } forall /FontName exch +def dup length 0 ne {/Encoding Encoding 256 array copy def 0 exch { dup type +/nametype eq { Encoding 2 index 2 index put pop 1 add } { exch pop } ifelse +} forall } if pop currentdict dup end end /FontName get exch definefont dup +MapDegree MapBB } bdef /RF { ISOLatin1Encoding exch ReEncode } bdef /Courier +RF /Courier-Bold RF /Courier-Oblique RF /Courier-BoldOblique RF /Helvetica +RF /Helvetica-Bold RF /Helvetica-Oblique RF /Helvetica-BoldOblique RF +/Helvetica-Narrow RF /Helvetica-Narrow-Bold RF /Helvetica-Narrow-Oblique RF +/Helvetica-Narrow-BoldOblique RF /AvantGarde-Book RF /AvantGarde-DemiOblique +RF /AvantGarde-Demi RF /AvantGarde-DemiOblique RF /Bookman-Demi RF +/Bookman-DemiItalic RF /Bookman-Light RF /Bookman-LightItalic RF +/ZapfChancery-MediumItalic RF /NewCenturySchlbk-Roman RF +/NewCenturySchlbk-Bold RF /NewCenturySchlbk-Italic RF +/NewCenturySchlbk-BoldItalic RF /Palatino-Roman RF /Palatino-Bold RF +/Palatino-Italic RF /Palatino-BoldItalic RF /Times-Roman RF /Times-Bold RF +/Times-Italic RF /Times-BoldItalic RF end +%%EndProlog +%%Page: 0 1 +%%BeginPageSetup +save $WAVE_DICT begin 28 56 L 0.028346 dup scale +%%PageBoundingBox: 28 56 509 339 +%%EndPageSetup +/psFontCache 500 array def /psStack 8 array def /psStackInd 0 def +/psTextWidth 0 def /psCurBase 0 def /ASW { stringwidth pop psCurBase add +/psCurBase exch def psCurBase psTextWidth gt +{ /psTextWidth psCurBase def } if } def /PSS { +psStack psStackInd psCurBase put /psStackInd psStackInd 1 add def } def +/PPS { /psStackInd psStackInd 1 sub def +/psCurBase psStack psStackInd get def } def 10.000000 SL L0 0.000 K +7480 9734 M 0 -306 R D 7495 9734 M 0 -306 R D 7436 9734 M 176 0 R 43 -14 R +15 -15 R 15 -29 R 0 -29 R -15 -30 R -15 -14 R -43 -15 R -117 0 R D +7612 9734 M 29 -14 R 14 -15 R 15 -29 R 0 -29 R -15 -30 R -14 -14 R -29 -15 R +D 7436 9428 M 103 0 R D 7568 9588 M 29 -14 R 15 -15 R 43 -102 R 15 -15 R +15 0 R 14 15 R D 7597 9574 M 15 -30 R 29 -102 R 14 -14 R 30 0 R 14 29 R +0 14 R D 7860 9734 M -44 -14 R -29 -44 R -15 -73 R 0 -44 R 15 -73 R 29 -44 R +44 -14 R 29 0 R 44 14 R 29 44 R 15 73 R 0 44 R -15 73 R -29 44 R -44 14 R +-29 0 R -29 -14 R -15 -15 R -15 -29 R -14 -73 R 0 -44 R 14 -73 R 15 -29 R +15 -15 R 29 -14 R D 7889 9428 M 29 14 R 15 15 R 15 29 R 14 73 R 0 44 R +-14 73 R -15 29 R -15 15 R -29 14 R D 8137 9734 M -43 -14 R -15 -30 R +0 -43 R 15 -30 R 43 -14 R 59 0 R 44 14 R 14 30 R 0 43 R -14 30 R -44 14 R +-59 0 R -29 -14 R -14 -30 R 0 -43 R 14 -30 R 29 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30 R D 10898 9734 M -29 -14 R +-30 -30 R -14 -29 R -15 -44 R 0 -73 R 15 -43 R 14 -30 R 30 -29 R 29 -14 R D +11292 9690 M 15 44 R 0 -87 R -15 43 R -29 30 R -44 14 R -44 0 R -44 -14 R +-29 -30 R 0 -29 R 15 -29 R 14 -15 R 30 -14 R 87 -29 R 30 -15 R 29 -29 R D +11102 9661 M 29 -29 R 30 -15 R 87 -29 R 30 -14 R 14 -15 R 15 -29 R 0 -59 R +-29 -29 R -44 -14 R -44 0 R -44 14 R -29 29 R -15 44 R 0 -87 R 15 43 R D +2220 1408 M 14114 0 R D 2220 1408 M 0 157 R D 2196 1195 M -35 -11 R +-23 -35 R -12 -59 R 0 -35 R 12 -58 R 23 -35 R 35 -12 R 24 0 R 35 12 R +23 35 R 12 58 R 0 35 R -12 59 R -23 35 R -35 11 R -24 0 R -23 -11 R +-12 -12 R -12 -23 R -11 -59 R 0 -35 R 11 -58 R 12 -24 R 12 -11 R 23 -12 R D +2220 950 M 23 12 R 12 11 R 11 24 R 12 58 R 0 35 R -12 59 R -11 23 R -12 12 R +-23 11 R D 7260 1408 M 0 157 R D 7190 1195 M -23 -117 R 23 24 R 35 12 R +35 0 R 35 -12 R 24 -24 R 11 -35 R 0 -23 R -11 -35 R -24 -23 R -35 -12 R +-35 0 R -35 12 R -11 11 R -12 24 R 0 11 R 12 12 R 11 -12 R -11 -11 R D +7260 1114 M 24 -12 R 23 -24 R 12 -35 R 0 -23 R -12 -35 R -23 -23 R -24 -12 R +D 7190 1195 M 117 0 R D 7190 1184 M 59 0 R 58 11 R D 12301 1408 M 0 157 R D +12126 1149 M 23 11 R 35 35 R 0 -245 R D 12173 1184 M 0 -234 R D 12126 950 M +105 0 R D 12395 1195 M -36 -11 R -23 -35 R -12 -59 R 0 -35 R 12 -58 R +23 -35 R 36 -12 R 23 0 R 35 12 R 23 35 R 12 58 R 0 35 R -12 59 R -23 35 R +-35 11 R -23 0 R -24 -11 R -12 -12 R -11 -23 R -12 -59 R 0 -35 R 12 -58 R +11 -24 R 12 -11 R 24 -12 R D 12418 950 M 23 12 R 12 11 R 12 24 R 11 58 R +0 35 R -11 59 R -12 23 R -12 12 R -23 11 R D 3228 1408 M 0 78 R D +4236 1408 M 0 78 R D 5244 1408 M 0 78 R D 6252 1408 M 0 78 R D 8268 1408 M +0 78 R D 9277 1408 M 0 78 R D 10285 1408 M 0 78 R D 11293 1408 M 0 78 R D +13309 1408 M 0 78 R D 14317 1408 M 0 78 R D 15325 1408 M 0 78 R D 8032 738 M +0 -246 R D 8044 738 M 0 -246 R D 7962 738 M -12 -70 R 0 70 R 176 0 R 0 -70 R +-12 70 R D 7997 492 M 82 0 R D 8207 738 M -11 -12 R 11 -12 R 12 12 R +-12 12 R D 8207 656 M 0 -164 R D 8219 656 M 0 -164 R D 8172 656 M 47 0 R D +8172 492 M 82 0 R D 8336 656 M 0 -164 R D 8348 656 M 0 -164 R D 8348 621 M +23 23 R 35 12 R 23 0 R 36 -12 R 11 -23 R 0 -129 R D 8429 656 M 24 -12 R +12 -23 R 0 -129 R D 8476 621 M 24 23 R 35 12 R 23 0 R 35 -12 R 12 -23 R +0 -129 R D 8558 656 M 23 -12 R 12 -23 R 0 -129 R D 8301 656 M 47 0 R D +8301 492 M 82 0 R D 8429 492 M 82 0 R D 8558 492 M 82 0 R D 8710 586 M +140 0 R 0 23 R -12 24 R -11 11 R -24 12 R -35 0 R -35 -12 R -23 -23 R +-12 -35 R 0 -24 R 12 -35 R 23 -23 R 35 -12 R 24 0 R 35 12 R 23 23 R D +8838 586 M 0 35 R -11 23 R D 8768 656 M -23 -12 R -23 -23 R -12 -35 R +0 -24 R 12 -35 R 23 -23 R 23 -12 R D 9131 738 M -12 -12 R 12 -12 R 11 12 R +-11 12 R D 9131 656 M 0 -164 R D 9142 656 M 0 -164 R D 9095 656 M 47 0 R D +9095 492 M 82 0 R D 9259 656 M 0 -164 R D 9271 656 M 0 -164 R D 9271 621 M +23 23 R 35 12 R 24 0 R 35 -12 R 11 -23 R 0 -129 R D 9353 656 M 23 -12 R +12 -23 R 0 -129 R D 9224 656 M 47 0 R D 9224 492 M 82 0 R D 9353 492 M +81 0 R D 9703 738 M 0 -246 R D 9715 738 M 0 -246 R D 9715 621 M 23 23 R +35 12 R 24 0 R 35 -12 R 11 -23 R 0 -129 R D 9797 656 M 23 -12 R 12 -23 R +0 -129 R D 9668 738 M 47 0 R D 9668 492 M 82 0 R D 9797 492 M 81 0 R D +10007 656 M -35 -12 R -24 -23 R -11 -35 R 0 -24 R 11 -35 R 24 -23 R 35 -12 R +23 0 R 35 12 R 24 23 R 11 35 R 0 24 R -11 35 R -24 23 R -35 12 R -23 0 R +-24 -12 R -23 -23 R -12 -35 R 0 -24 R 12 -35 R 23 -23 R 24 -12 R D +10030 492 M 24 12 R 23 23 R 12 35 R 0 24 R -12 35 R -23 23 R -24 12 R D +10194 656 M 0 -129 R 11 -23 R 36 -12 R 23 0 R 35 12 R 23 23 R D 10205 656 M +0 -129 R 12 -23 R 24 -12 R D 10322 656 M 0 -164 R D 10334 656 M 0 -164 R D +10159 656 M 46 0 R D 10287 656 M 47 0 R D 10322 492 M 47 0 R D 10451 656 M +0 -164 R D 10463 656 M 0 -164 R D 10463 586 M 11 35 R 24 23 R 23 12 R 35 0 R +12 -12 R 0 -11 R -12 -12 R -12 12 R 12 11 R D 10416 656 M 47 0 R D +10416 492 M 82 0 R D 10743 633 M 12 23 R 0 -47 R -12 24 R -12 11 R -23 12 R +-47 0 R -23 -12 R -12 -11 R 0 -24 R 12 -11 R 23 -12 R 59 -24 R 23 -11 R +12 -12 R D 10626 621 M 12 -12 R 23 -11 R 59 -24 R 23 -12 R 12 -11 R 0 -35 R +-12 -12 R -23 -12 R -47 0 R -24 12 R -11 12 R -12 23 R 0 -47 R 12 24 R D +2220 9296 M 14114 0 R D 2220 9296 M 0 -158 R D 7260 9296 M 0 -158 R D +12301 9296 M 0 -158 R D 3228 9296 M 0 -79 R D 4236 9296 M 0 -79 R D +5244 9296 M 0 -79 R D 6252 9296 M 0 -79 R D 8268 9296 M 0 -79 R D +9277 9296 M 0 -79 R D 10285 9296 M 0 -79 R D 11293 9296 M 0 -79 R D +13309 9296 M 0 -79 R D 14317 9296 M 0 -79 R D 15325 9296 M 0 -79 R D +2220 1408 M 0 7888 R D 2220 1408 M 282 0 R D 1968 1653 M -35 -12 R -23 -35 R +-12 -58 R 0 -35 R 12 -59 R 23 -35 R 35 -11 R 24 0 R 35 11 R 23 35 R 12 59 R +0 35 R -12 58 R -23 35 R -35 12 R -24 0 R -23 -12 R -12 -11 R -11 -24 R +-12 -58 R 0 -35 R 12 -59 R 11 -23 R 12 -12 R 23 -11 R D 1992 1408 M 23 11 R +12 12 R 11 23 R 12 59 R 0 35 R -12 58 R -11 24 R -12 11 R -23 12 R D +2220 2985 M 282 0 R D 1676 3043 M 12 -12 R -12 -11 R -11 11 R 0 12 R 11 23 R +12 12 R 35 12 R 47 0 R 35 -12 R 11 -12 R 12 -23 R 0 -23 R -12 -24 R +-35 -23 R -58 -23 R -23 -12 R -24 -24 R -11 -35 R 0 -35 R D 1770 3090 M +23 -12 R 12 -12 R 11 -23 R 0 -23 R -11 -24 R -35 -23 R -47 -23 R D +1665 2868 M 11 11 R 24 0 R 58 -23 R 35 0 R 23 12 R 12 11 R D 1700 2879 M +58 -35 R 47 0 R 11 12 R 12 23 R 0 24 R D 1968 3090 M -35 -12 R -23 -35 R +-12 -58 R 0 -35 R 12 -59 R 23 -35 R 35 -12 R 24 0 R 35 12 R 23 35 R 12 59 R +0 35 R -12 58 R -23 35 R -35 12 R -24 0 R -23 -12 R -12 -12 R -11 -23 R +-12 -58 R 0 -35 R 12 -59 R 11 -23 R 12 -12 R 23 -12 R D 1992 2844 M 23 12 R +12 12 R 11 23 R 12 59 R 0 35 R -12 58 R -11 23 R -12 12 R -23 12 R D +2220 4563 M 282 0 R D 1770 4644 M 0 -222 R D 1781 4667 M 0 -245 R D +1781 4667 M -128 -175 R 187 0 R D 1735 4422 M 81 0 R D 1968 4667 M -35 -11 R +-23 -35 R -12 -59 R 0 -35 R 12 -58 R 23 -35 R 35 -12 R 24 0 R 35 12 R +23 35 R 12 58 R 0 35 R -12 59 R -23 35 R -35 11 R -24 0 R -23 -11 R +-12 -12 R -11 -23 R -12 -59 R 0 -35 R 12 -58 R 11 -24 R 12 -11 R 23 -12 R D +1992 4422 M 23 12 R 12 11 R 11 24 R 12 58 R 0 35 R -12 59 R -11 23 R +-12 12 R -23 11 R D 2220 6140 M 282 0 R D 1805 6210 M -12 -12 R 12 -11 R +11 11 R 0 12 R -11 23 R -24 12 R -35 0 R -35 -12 R -23 -23 R -12 -23 R +-11 -47 R 0 -70 R 11 -35 R 24 -24 R 35 -11 R 23 0 R 35 11 R 23 24 R 12 35 R +0 11 R -12 35 R -23 24 R -35 11 R -12 0 R -35 -11 R -23 -24 R -12 -35 R D +1746 6245 M -23 -12 R -23 -23 R -12 -23 R -12 -47 R 0 -70 R 12 -35 R +23 -24 R 24 -11 R D 1758 6000 M 23 11 R 24 24 R 11 35 R 0 11 R -11 35 R +-24 24 R -23 11 R D 1968 6245 M -35 -12 R -23 -35 R -12 -58 R 0 -35 R +12 -59 R 23 -35 R 35 -11 R 24 0 R 35 11 R 23 35 R 12 59 R 0 35 R -12 58 R +-23 35 R -35 12 R -24 0 R -23 -12 R -12 -11 R -11 -24 R -12 -58 R 0 -35 R +12 -59 R 11 -23 R 12 -12 R 23 -11 R D 1992 6000 M 23 11 R 12 12 R 11 23 R +12 59 R 0 35 R -12 58 R -11 24 R -12 11 R -23 12 R D 2220 7718 M 282 0 R D +1723 7823 M -35 -12 R -12 -24 R 0 -35 R 12 -23 R 35 -12 R 47 0 R 35 12 R +11 23 R 0 35 R -11 24 R -35 12 R -47 0 R -23 -12 R -12 -24 R 0 -35 R +12 -23 R 23 -12 R D 1770 7717 M 23 12 R 12 23 R 0 35 R -12 24 R -23 12 R D +1723 7717 M -35 -11 R -12 -12 R -11 -23 R 0 -47 R 11 -23 R 12 -12 R 35 -12 R +47 0 R 35 12 R 11 12 R 12 23 R 0 47 R -12 23 R -11 12 R -35 11 R D +1723 7717 M -23 -11 R -12 -12 R -12 -23 R 0 -47 R 12 -23 R 12 -12 R 23 -12 R +D 1770 7577 M 23 12 R 12 12 R 11 23 R 0 47 R -11 23 R -12 12 R -23 11 R D +1968 7823 M -35 -12 R -23 -35 R -12 -59 R 0 -35 R 12 -58 R 23 -35 R 35 -12 R +24 0 R 35 12 R 23 35 R 12 58 R 0 35 R -12 59 R -23 35 R -35 12 R -24 0 R +-23 -12 R -12 -12 R -11 -23 R -12 -59 R 0 -35 R 12 -58 R 11 -23 R 12 -12 R +23 -12 R D 1992 7577 M 23 12 R 12 12 R 11 23 R 12 58 R 0 35 R -12 59 R +-11 23 R -12 12 R -23 12 R D 2220 9296 M 282 0 R D 1466 9213 M 23 11 R +35 35 R 0 -245 R D 1513 9248 M 0 -234 R D 1466 9014 M 105 0 R D 1735 9259 M +-35 -11 R -24 -35 R -11 -59 R 0 -35 R 11 -58 R 24 -35 R 35 -12 R 23 0 R +35 12 R 23 35 R 12 58 R 0 35 R -12 59 R -23 35 R -35 11 R -23 0 R -24 -11 R +-11 -12 R -12 -23 R -12 -59 R 0 -35 R 12 -58 R 12 -24 R 11 -11 R 24 -12 R D +1758 9014 M 23 12 R 12 11 R 12 24 R 11 58 R 0 35 R -11 59 R -12 23 R +-12 12 R -23 11 R D 1968 9259 M -35 -11 R -23 -35 R -12 -59 R 0 -35 R +12 -58 R 23 -35 R 35 -12 R 24 0 R 35 12 R 23 35 R 12 58 R 0 35 R -12 59 R +-23 35 R -35 11 R -24 0 R -23 -11 R -12 -12 R -11 -23 R -12 -59 R 0 -35 R +12 -58 R 11 -24 R 12 -11 R 23 -12 R D 1992 9014 M 23 12 R 12 11 R 11 24 R +12 58 R 0 35 R -12 59 R -11 23 R -12 12 R -23 11 R D 2220 1802 M 141 0 R D +2220 2196 M 141 0 R D 2220 2591 M 141 0 R D 2220 3380 M 141 0 R D +2220 3774 M 141 0 R D 2220 4168 M 141 0 R D 2220 4957 M 141 0 R D +2220 5352 M 141 0 R D 2220 5746 M 141 0 R D 2220 6535 M 141 0 R D +2220 6929 M 141 0 R D 2220 7324 M 141 0 R D 2220 8112 M 141 0 R D +2220 8507 M 141 0 R D 2220 8901 M 141 0 R D 829 3655 M 246 0 R D 829 3666 M +246 0 R D 829 3620 M 0 116 R 12 35 R 23 24 R 24 11 R 35 12 R 58 0 R 35 -12 R +23 -11 R 24 -24 R 12 -35 R 0 -116 R D 829 3736 M 12 24 R 23 23 R 24 12 R +35 11 R 58 0 R 35 -11 R 23 -12 R 24 -23 R 12 -24 R D 829 3912 M 12 -12 R +12 12 R -12 11 R -12 -11 R D 911 3912 M 164 0 R D 911 3923 M 164 0 R D +911 3877 M 0 46 R D 1075 3877 M 0 81 R D 911 4017 M 164 70 R D 911 4028 M +140 59 R D 911 4157 M 164 -70 R D 911 3993 M 0 71 R D 911 4110 M 0 70 R D +981 4239 M 0 140 R -23 0 R -24 -12 R -11 -11 R -12 -24 R 0 -35 R 12 -35 R +23 -23 R 35 -12 R 23 0 R 35 12 R 24 23 R 12 35 R 0 24 R -12 35 R -24 23 R D +981 4367 M -35 0 R -23 -11 R D 911 4297 M 12 -23 R 23 -24 R 35 -11 R 23 0 R +35 11 R 24 24 R 12 23 R D 911 4472 M 164 0 R D 911 4484 M 164 0 R D +981 4484 M -35 12 R -23 23 R -12 24 R 0 35 R 12 11 R 11 0 R 12 -11 R +-12 -12 R -11 12 R D 911 4437 M 0 47 R D 1075 4437 M 0 82 R D 911 4706 M +12 -23 R 11 -12 R 24 -12 R 23 0 R 23 12 R 12 12 R 12 23 R 0 24 R -12 23 R +-12 12 R -23 11 R -23 0 R -24 -11 R -11 -12 R -12 -23 R 0 -24 R D 923 4683 M +23 -12 R 47 0 R 23 12 R D 1016 4753 M -23 12 R -47 0 R -23 -12 R D +934 4765 M -11 11 R -12 24 R 12 0 R 0 -24 R D 1004 4671 M 12 -12 R 23 -11 R +12 0 R 24 11 R 23 35 R 0 59 R 12 35 R 11 12 R D 1051 4648 M 12 11 R 12 35 R +0 59 R 23 35 R 23 12 R 12 0 R 23 -12 R 12 -35 R 0 -70 R -12 -35 R -23 -12 R +-12 0 R -23 12 R -23 35 R D 981 4881 M 0 141 R -23 0 R -24 -12 R -11 -12 R +-12 -23 R 0 -35 R 12 -35 R 23 -24 R 35 -11 R 23 0 R 35 11 R 24 24 R 12 35 R +0 23 R -12 35 R -24 24 R D 981 5010 M -35 0 R -23 -12 R D 911 4940 M +12 -24 R 23 -23 R 35 -12 R 23 0 R 35 12 R 24 23 R 12 24 R D 911 5115 M +164 0 R D 911 5127 M 164 0 R D 946 5127 M -23 23 R -12 35 R 0 24 R 12 35 R +23 11 R 129 0 R D 911 5209 M 12 23 R 23 12 R 129 0 R D 911 5080 M 0 47 R D +1075 5080 M 0 82 R D 1075 5209 M 0 81 R D 946 5489 M 12 -12 R 11 12 R +-11 12 R -12 0 R -23 -24 R -12 -23 R 0 -35 R 12 -35 R 23 -24 R 35 -11 R +23 0 R 35 11 R 24 24 R 12 35 R 0 23 R -12 35 R -24 24 R D 911 5419 M +12 -23 R 23 -24 R 35 -12 R 23 0 R 35 12 R 24 24 R 12 23 R D 981 5582 M +0 141 R -23 0 R -24 -12 R -11 -12 R -12 -23 R 0 -35 R 12 -35 R 23 -24 R +35 -11 R 23 0 R 35 11 R 24 24 R 12 35 R 0 23 R -12 35 R -24 24 R D +981 5711 M -35 0 R -23 -12 R D 911 5641 M 12 -23 R 23 -24 R 35 -12 R 23 0 R +35 12 R 24 24 R 12 23 R D 782 6073 M 24 -23 R 35 -24 R 47 -23 R 58 -12 R +47 0 R 58 12 R 59 23 R 35 24 R 23 23 R D 806 6050 M 47 -24 R 35 -11 R +58 -12 R 47 0 R 58 12 R 47 11 R 47 24 R D 829 6155 M 246 47 R D 829 6167 M +187 35 R D 829 6248 M 246 -46 R D 829 6248 M 246 47 R D 829 6260 M 187 35 R +D 829 6342 M 246 -47 R D 829 6120 M 0 82 R D 829 6307 M 0 70 R D 911 6447 M +164 0 R D 911 6459 M 164 0 R D 946 6459 M -23 23 R -12 35 R 0 24 R 12 35 R +23 11 R 129 0 R D 911 6541 M 12 23 R 23 12 R 129 0 R D 946 6587 M -23 24 R +-12 35 R 0 23 R 12 35 R 23 12 R 129 0 R D 911 6669 M 12 23 R 23 12 R 129 0 R +D 911 6412 M 0 47 R D 1075 6412 M 0 82 R D 1075 6541 M 0 81 R D 1075 6669 M +0 82 R D 832 6808 M 0 130 R D 774 6996 M 7 7 R 7 -7 R -7 -8 R -7 0 R -14 8 R +-8 7 R -7 22 R 0 29 R 7 21 R 8 7 R 14 8 R 14 0 R 15 -8 R 14 -21 R 15 -36 R +7 -15 R 15 -14 R 21 -8 R 22 0 R D 745 7054 M 7 14 R 8 7 R 14 7 R 14 0 R +15 -7 R 14 -21 R 15 -29 R D 882 6988 M -7 8 R 0 14 R 15 36 R 0 22 R -8 14 R +-7 8 R D 875 7010 M 22 36 R 0 29 R -7 7 R -15 8 R -14 0 R D 782 7142 M +24 23 R 35 24 R 47 23 R 58 12 R 47 0 R 58 -12 R 59 -23 R 35 -24 R 23 -23 R D +806 7165 M 47 24 R 35 11 R 58 12 R 47 0 R 58 -12 R 47 -11 R 47 -24 R D +16334 1408 M 0 7888 R D 16334 1408 M -283 0 R D 16334 2985 M -283 0 R D +16334 4563 M -283 0 R D 16334 6140 M -283 0 R D 16334 7718 M -283 0 R D +16334 9296 M -283 0 R D 16334 1802 M -142 0 R D 16334 2196 M -142 0 R D +16334 2591 M -142 0 R D 16334 3380 M -142 0 R D 16334 3774 M -142 0 R D +16334 4168 M -142 0 R D 16334 4957 M -142 0 R D 16334 5352 M -142 0 R D +16334 5746 M -142 0 R D 16334 6535 M -142 0 R D 16334 6929 M -142 0 R D +16334 7324 M -142 0 R D 16334 8112 M -142 0 R D 16334 8507 M -142 0 R D +16334 8901 M -142 0 R D L1 2220 6894 M 1008 18 R 1008 0 R 1008 0 R 1008 -3 R +1008 9 R 1008 -14 R 1009 -5 R 1008 6 R 1008 -16 R 1008 -15 R 1008 22 R +1008 -8 R 1008 13 R 1009 0 R D L2 2220 1408 M 1008 0 R 1008 391 R 1008 975 R +1008 1236 R 1008 1295 R 1008 1120 R 1009 711 R 1008 155 R 1008 -379 R +1008 -702 R 1008 -754 R 1008 -852 R 1008 -1017 R 1009 -1045 R D L3 +2220 6894 M 1008 18 R 1008 -392 R 1008 -975 R 1008 -1239 R 1008 -1286 R +1008 -1133 R 674 -479 R D 11328 1408 M 973 664 R 1008 776 R 1008 843 R +1008 1031 R 1009 1044 R D L0 2220 6894 M 1008 17 R 1008 -191 R 1008 -370 R +1008 -383 R 1008 -301 R 1008 -182 R 1009 -308 R 1008 -207 R 1008 -77 R +1008 21 R 1008 153 R 1008 273 R 1008 402 R 1009 495 R D L4 2220 6894 M +1008 18 R 1008 -137 R 1008 -341 R 1008 -436 R 1008 -444 R 1008 -406 R +1009 -254 R 1008 -49 R 1008 117 R 1008 231 R 1008 286 R 1008 290 R +1008 370 R 1009 365 R D L0 3161 8901 M 940 0 R D 4242 9068 M 0 -246 R D +4254 9068 M 140 -222 R D 4254 9044 M 140 -222 R 0 246 R D 4207 9068 M 47 0 R +D 4359 9068 M 70 0 R D 4207 8822 M 70 0 R D 4511 9068 M 0 -246 R D +4523 9068 M 0 -246 R D 4593 8998 M 0 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-246 R D +4873 8279 M 35 -187 R D 4955 8279 M -47 -246 R D 4733 8279 M 82 0 R D +4920 8279 M 70 0 R D L3 3161 7718 M 940 0 R D L0 4359 7849 M -12 -11 R +12 -12 R 12 12 R 0 11 R -12 24 R -23 11 R -35 0 R -35 -11 R -24 -24 R +-11 -23 R -12 -47 R 0 -70 R 12 -35 R 23 -23 R 35 -12 R 23 0 R 35 12 R +24 23 R 11 35 R 0 12 R -11 35 R -24 23 R -35 12 R -11 0 R -35 -12 R +-24 -23 R -11 -35 R D 4301 7884 M -24 -11 R -23 -24 R -12 -23 R -11 -47 R +0 -70 R 11 -35 R 24 -23 R 23 -12 R D 4312 7639 M 24 12 R 23 23 R 12 35 R +0 12 R -12 35 R -23 23 R -24 12 R D 4534 7884 M -81 -245 R D 4534 7884 M +82 -245 R D 4534 7849 M 70 -210 R D 4476 7709 M 105 0 R D 4429 7639 M 70 0 R +D 4569 7639 M 71 0 R D 4698 7779 M 210 0 R D 4698 7709 M 210 0 R D +5013 7884 M 0 -245 R D 5025 7884 M 0 -245 R D 4978 7884 M 82 0 R D +4978 7639 M 176 0 R 0 70 R -12 -70 R D 5212 7884 M 47 -245 R D 5224 7884 M +35 -187 R D 5306 7884 M -47 -245 R D 5306 7884 M 46 -245 R D 5317 7884 M +35 -187 R D 5399 7884 M -47 -245 R D 5177 7884 M 82 0 R D 5364 7884 M 70 0 R +D 5598 7849 M 0 -210 R D 5492 7744 M 211 0 R D 5936 7849 M 12 35 R 0 -70 R +-12 35 R -23 24 R -35 11 R -35 0 R -35 -11 R -23 -24 R 0 -23 R 11 -23 R +12 -12 R 23 -12 R 70 -23 R 24 -12 R 23 -23 R D 5785 7826 M 23 -23 R 23 -12 R +70 -23 R 24 -12 R 11 -12 R 12 -23 R 0 -47 R -23 -23 R -35 -12 R -35 0 R +-35 12 R -24 23 R -11 35 R 0 -70 R 11 35 R D 6030 7884 M 47 -245 R D +6042 7884 M 35 -187 R D 6123 7884 M -46 -245 R D 6123 7884 M 47 -245 R D +6135 7884 M 35 -187 R D 6217 7884 M -47 -245 R D 5995 7884 M 82 0 R D +6182 7884 M 70 0 R D L4 3161 7324 M 940 0 R D L0 4359 7455 M -12 -12 R +12 -11 R 12 11 R 0 12 R -12 23 R -23 12 R -35 0 R -35 -12 R -24 -23 R +-11 -23 R -12 -47 R 0 -70 R 12 -35 R 23 -24 R 35 -11 R 23 0 R 35 11 R +24 24 R 11 35 R 0 11 R -11 36 R -24 23 R -35 12 R -11 0 R -35 -12 R +-24 -23 R -11 -36 R D 4301 7490 M -24 -12 R -23 -23 R -12 -23 R -11 -47 R +0 -70 R 11 -35 R 24 -24 R 23 -11 R D 4312 7245 M 24 11 R 23 24 R 12 35 R +0 11 R -12 36 R -23 23 R -24 12 R D 4476 7490 M 0 -245 R D 4488 7490 M +0 -245 R D 4441 7490 M 140 0 R 35 -12 R 12 -11 R 12 -24 R 0 -23 R -12 -23 R +-12 -12 R -35 -12 R D 4581 7490 M 23 -12 R 12 -11 R 12 -24 R 0 -23 R +-12 -23 R -12 -12 R -23 -12 R -93 0 R D 4581 7373 M 35 -11 R 12 -12 R +12 -24 R 0 -35 R -12 -23 R -12 -12 R -35 -11 R -140 0 R D 4581 7373 M +23 -11 R 12 -12 R 12 -24 R 0 -35 R -12 -23 R -12 -12 R -23 -11 R D +4803 7537 M -23 -24 R -24 -35 R -23 -46 R -12 -59 R 0 -47 R 12 -58 R +23 -58 R 24 -35 R 23 -24 R D 4780 7513 M -24 -46 R -11 -35 R -12 -59 R +0 -47 R 12 -58 R 11 -47 R 24 -46 R D 4943 7490 M -35 -12 R -23 -35 R +-12 -58 R 0 -35 R 12 -59 R 23 -35 R 35 -11 R 24 0 R 35 11 R 23 35 R 12 59 R +0 35 R -12 58 R -23 35 R -35 12 R -24 0 R -23 -12 R -12 -11 R -11 -24 R +-12 -58 R 0 -35 R 12 -59 R 11 -23 R 12 -12 R 23 -11 R D 4967 7245 M 23 11 R +12 12 R 11 23 R 12 59 R 0 35 R -12 58 R -11 24 R -12 11 R -23 12 R D +5130 7268 M -11 -12 R 11 -11 R 12 11 R -12 12 R D 5235 7443 M 12 -11 R +-12 -12 R -11 12 R 0 11 R 11 24 R 12 11 R 35 12 R 47 0 R 35 -12 R 12 -23 R +0 -35 R -12 -23 R -35 -12 R -35 0 R D 5329 7490 M 23 -12 R 12 -23 R 0 -35 R +-12 -23 R -23 -12 R 23 -12 R 24 -23 R 11 -24 R 0 -35 R -11 -23 R -12 -12 R +-35 -11 R -47 0 R -35 11 R -12 12 R -11 23 R 0 12 R 11 12 R 12 -12 R +-12 -12 R D 5364 7362 M 12 -36 R 0 -35 R -12 -23 R -12 -12 R -23 -11 R D +5481 7490 M -24 -117 R 24 24 R 35 11 R 35 0 R 35 -11 R 23 -24 R 12 -35 R +0 -23 R -12 -35 R -23 -24 R -35 -11 R -35 0 R -35 11 R -12 12 R -12 23 R +0 12 R 12 12 R 12 -12 R -12 -12 R D 5551 7408 M 23 -11 R 24 -24 R 11 -35 R +0 -23 R -11 -35 R -24 -24 R -23 -11 R D 5481 7490 M 117 0 R D 5481 7478 M +58 0 R 59 12 R D 5691 7537 M 23 -24 R 24 -35 R 23 -46 R 12 -59 R 0 -47 R +-12 -58 R -23 -58 R -24 -35 R -23 -24 R D 5714 7513 M 24 -46 R 12 -35 R +11 -59 R 0 -47 R -11 -58 R -12 -47 R -24 -46 R D +end restore +showpage +%%Trailer +restore +%%Pages: 1 +%%Trailer +cleartomark +countdictstack +exch sub { end } repeat +restore +%%EOF diff --git a/documentation/source/science_guide/turbulence_schemes/honnert_vs_tanh.eps b/documentation/source/science_guide/turbulence_schemes/honnert_vs_tanh.eps new file mode 100644 index 0000000000..1066d61d43 --- /dev/null +++ b/documentation/source/science_guide/turbulence_schemes/honnert_vs_tanh.eps @@ -0,0 +1,258 @@ +%!PS-Adobe-3.0 +%%BoundingBox: 45 68 611 634 +%%Title: Graphics produced by IDL +%%For: frib@eld638, /net/home/h04/frib/idl +%%Creator: IDL Version 8.2 (linux x86_64 m64) +%%CreationDate: Thu Feb 5 14:07:43 2015 +%%DocumentData: Clean7bit +%%Requirements: color +%%LanguageLevel: 1 +%%PageOrder: Ascend +%%Pages: (atend) +%%DocumentNeededResources: (atend) +%%EndComments +%%BeginProlog +save +%+ prolog.ps -- Prolog for IDL generated PostScript files +%+ Copyright (c) 1988-2012 Exelis Visual Information Solutions, Inc. All Rights Reserved. +%v 5 +/$IDL_DICT 40 dict def $IDL_DICT begin /bdef { bind def } bind def /C +{currentpoint newpath moveto} bdef /CP {currentpoint} bdef /D {currentpoint +stroke moveto} bdef /F {closepath fill} bdef /K { setgray } bdef /M {moveto} +bdef /N {rmoveto} bdef /P {lineto} bdef /R {rlineto} bdef /S {gsave show +grestore} bdef /X {currentpoint pop} bdef /Z {gsave currentpoint lineto 20 +setlinewidth 1 setlinecap stroke grestore} bdef /L0 {[] 0 setdash} bdef /L1 +{[40 100] 0 setdash} bdef /L2 {[200 200] 0 setdash} bdef /L3 {[200 100 50 +100] 0 setdash} bdef /L4 {[300 100 50 100 50 100 50 100] 0 setdash} bdef /L5 +{[400 200] 0 setdash} bdef /STDFONT { findfont exch scalefont setfont } bdef +/ISOFONT { findfont dup length dict begin { 1 index /FID ne {def} {pop pop} +ifelse } forall /Encoding ISOLatin1Encoding def currentdict end /idltmpfont +exch definefont exch scalefont setfont } bdef /ISOBULLET { gsave /Helvetica +findfont exch scalefont setfont (\267) show currentpoint grestore moveto} +bdef /MITERLIMIT { 2.5 setmiterlimit } bdef end +%%EndProlog +%%Page: 0 1 +%%PageRequirements: color +%%PageResources: (atend) +%%PageOrientation: Portrait +%%PageBoundingBox: 45 68 611 634 +%%BeginPageSetup +save +$IDL_DICT begin 45 68 translate 0.0283465 dup scale MITERLIMIT +%%IncludeResource: font Helvetica +423.333 /Helvetica STDFONT +%%EndPageSetup +10 setlinewidth L0 0.000 0.000 0.000 setrgbcolor 3330 2112 M 15671 0 R D +5380 2112 M 0 337 R D gsave 5380 1320 translate 0 0 M 1.5 dup scale +-294.217 0 N +(0.1) show grestore 12191 2112 M 0 337 R D gsave 12191 1320 translate 0 0 M +1.5 dup scale -294.217 0 N +(1.0) show grestore 19001 2112 M 0 337 R D gsave 19001 1320 translate 0 0 M +1.5 dup scale -411.903 0 N +(10.0) show grestore 3330 2112 M 0 168 R D 3869 2112 M 0 168 R D 4325 2112 M +0 168 R D 4720 2112 M 0 168 R D 5069 2112 M 0 168 R D 7430 2112 M 0 168 R D +8630 2112 M 0 168 R D 9481 2112 M 0 168 R D 10141 2112 M 0 168 R D +10680 2112 M 0 168 R D 11136 2112 M 0 168 R D 11531 2112 M 0 168 R D +11879 2112 M 0 168 R D 14241 2112 M 0 168 R D 15440 2112 M 0 168 R D +16291 2112 M 0 168 R D 16951 2112 M 0 168 R D 17490 2112 M 0 168 R D +17946 2112 M 0 168 R D 18341 2112 M 0 168 R D 18689 2112 M 0 168 R D +gsave 11166 634 translate 0 0 M 1.5 dup scale -626.165 0 N +%%IncludeResource: font Symbol +423.333 /Symbol STDFONT +(D) show +%%IncludeResource: font Helvetica +423.333 /Helvetica STDFONT +(x/z) show X -101.051 M +%%IncludeResource: font Helvetica +262.467 /Helvetica STDFONT +(turb) show grestore +%%IncludeResource: font Helvetica +423.333 /Helvetica STDFONT 3330 18944 M 15671 0 R D 5380 18944 M 0 -337 R D +12191 18944 M 0 -337 R D 19001 18944 M 0 -337 R D 3330 18944 M 0 -168 R D +3869 18944 M 0 -168 R D 4325 18944 M 0 -168 R D 4720 18944 M 0 -168 R D +5069 18944 M 0 -168 R D 7430 18944 M 0 -168 R D 8630 18944 M 0 -168 R D +9481 18944 M 0 -168 R D 10141 18944 M 0 -168 R D 10680 18944 M 0 -168 R D +11136 18944 M 0 -168 R D 11531 18944 M 0 -168 R D 11879 18944 M 0 -168 R D +14241 18944 M 0 -168 R D 15440 18944 M 0 -168 R D 16291 18944 M 0 -168 R D +16951 18944 M 0 -168 R D 17490 18944 M 0 -168 R D 17946 18944 M 0 -168 R D +18341 18944 M 0 -168 R D 18689 18944 M 0 -168 R D 3330 2112 M 0 16832 R D +3330 2112 M 314 0 R D gsave 3164 2112 translate 0 0 M 1.5 dup scale +-588.433 0 N +(0.0) show grestore 3330 5479 M 314 0 R D gsave 3164 5267 translate 0 0 M +1.5 dup scale -588.433 0 N +(0.2) show grestore 3330 8845 M 314 0 R D gsave 3164 8634 translate 0 0 M +1.5 dup scale -588.433 0 N +(0.4) show grestore 3330 12211 M 314 0 R D gsave 3164 12000 translate 0 0 M +1.5 dup scale -588.433 0 N +(0.6) show grestore 3330 15578 M 314 0 R D gsave 3164 15367 translate 0 0 M +1.5 dup scale -588.433 0 N +(0.8) show grestore 3330 18944 M 314 0 R D gsave 3164 18575 translate 0 0 M +1.5 dup scale -588.433 0 N +(1.0) show grestore 3330 2954 M 157 0 R D 3330 3795 M 157 0 R D 3330 4637 M +157 0 R D 3330 6320 M 157 0 R D 3330 7162 M 157 0 R D 3330 8003 M 157 0 R D +3330 9687 M 157 0 R D 3330 10528 M 157 0 R D 3330 11370 M 157 0 R D +3330 13053 M 157 0 R D 3330 13895 M 157 0 R D 3330 14736 M 157 0 R D +3330 16419 M 157 0 R D 3330 17261 M 157 0 R D 3330 18103 M 157 0 R D +gsave 1782 10528 translate 0 0 M 90 rotate 1.5 dup scale -367.53 0 N +(W) show X -101.051 M +%%IncludeResource: font Helvetica +262.467 /Helvetica STDFONT +(1D) show grestore +%%IncludeResource: font Helvetica +423.333 /Helvetica STDFONT 19001 2112 M 0 16832 R D 19001 2112 M -313 0 R D +19001 5479 M -313 0 R D 19001 8845 M -313 0 R D 19001 12211 M -313 0 R D +19001 15578 M -313 0 R D 19001 18944 M -313 0 R D 19001 2954 M -157 0 R D +19001 3795 M -157 0 R D 19001 4637 M -157 0 R D 19001 6320 M -157 0 R D +19001 7162 M -157 0 R D 19001 8003 M -157 0 R D 19001 9687 M -157 0 R D +19001 10528 M -157 0 R D 19001 11370 M -157 0 R D 19001 13053 M -157 0 R D +19001 13895 M -157 0 R D 19001 14736 M -157 0 R D 19001 16419 M -157 0 R D +19001 17261 M -157 0 R D 19001 18103 M -157 0 R D 60 setlinewidth +3330 2405 M 59 14 R 57 15 R 56 16 R 56 17 R 54 18 R 53 18 R 53 20 R 51 20 R +51 21 R 49 23 R 49 22 R 48 24 R 48 25 R 46 26 R 46 26 R 45 28 R 45 28 R +44 29 R 43 30 R 42 31 R 42 31 R 42 33 R 40 33 R 41 34 R 39 35 R 40 35 R +38 37 R 38 37 R 38 37 R 37 39 R 37 39 R 36 40 R 36 40 R 36 41 R 35 41 R +34 42 R 34 43 R 34 43 R 34 44 R 33 44 R 32 45 R 33 45 R 32 46 R 31 46 R +32 46 R 31 47 R 30 47 R 30 48 R 31 47 R 29 49 R 282 495 R 258 509 R +236 512 R 219 506 R 205 494 R 190 476 R 180 458 R 169 437 R 160 416 R +151 395 R 145 374 R 137 354 R 132 336 R 126 317 R 120 300 R 116 285 R +112 269 R 108 255 R 103 242 R 101 229 R 97 218 R 94 207 R 91 197 R 88 187 R +86 179 R 83 170 R 81 163 R 79 155 R 77 148 R 75 142 R 73 136 R 71 131 R +69 124 R 68 120 R 67 115 R 65 111 R 64 106 R 62 102 R 61 99 R 60 95 R +58 91 R 58 88 R 56 85 R 55 83 R 54 79 R 54 77 R 52 74 R 52 72 R 50 70 R +50 67 R 49 65 R 48 64 R 47 61 R 47 60 R 46 58 R 45 56 R 44 54 R 44 53 R +43 52 R 43 50 R 42 49 R 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+(vn10.1) show grestore +%%PageTrailer +end +restore +showpage +%%PageResources: font Helvetica +%%+ font Symbol +%%Trailer +restore +%%Pages: 1 +%%DocumentNeededResources: font Helvetica +%%+ font Symbol +%%EOF diff --git a/documentation/source/science_guide/turbulence_schemes/ideal_invinteg.eps b/documentation/source/science_guide/turbulence_schemes/ideal_invinteg.eps new file mode 100644 index 0000000000..443becd7aa --- /dev/null +++ b/documentation/source/science_guide/turbulence_schemes/ideal_invinteg.eps @@ -0,0 +1,265 @@ +%!PS-Adobe-2.0 EPSF-2.0 +%%BoundingBox: 56 40 254 236 +%%HiResBoundingBox: 57 41 253.5 235.5 +%%Title: Graphics produced by WAVE +%%For: frlk@eld206 +%%Creator: WAVE Version 8.00 (Linux i386) +%%CreationDate: Thu Mar 30 16:29:43 2006 +%%EndComments +% EPSF created by ps2eps 1.68 +%%BeginProlog +save +countdictstack +mark +newpath +/showpage {} def +/setpagedevice {pop} def +%%EndProlog +%%Page 1 1 +%+ wave_prolog.ps -- Prolog for PV-WAVE CL generated PostScript files +%+ $Id: wave_prolog.ps,v 1.5 1998/06/05 17:44:06 thaux Exp $ +%+ Copyright (c) 1989-1992 Precision Visuals, Inc. All Rights Reserved. +%v 1 +save /$WAVE_DICT 60 dict def $WAVE_DICT begin /bdef { bind def } bind def /C +{currentpoint newpath moveto} bdef /D {currentpoint stroke moveto} bdef /F +{closepath fill} bdef /K { setgray } bdef /L {translate} bdef /M {moveto} +bdef /N {rmoveto} bdef /O {show} bdef /P {lineto} bdef /R {rlineto} bdef /S +{gsave show grestore} bdef /Z {gsave currentpoint lineto 20 setlinewidth 1 +setlinecap stroke grestore} bdef /CC {concat} bdef /CD {currentmatrix def} +bdef /CP {currentpoint pop def} bdef /FM {findfont exch makefont} bdef /FS +{findfont exch scalefont} bdef /GS {get setfont} bdef /MD {matrix def} bdef +/MN {mul neg def} bdef /RS {roll put setfont} bdef /SL {setlinewidth} bdef +/SM {setmatrix} bdef /L0 {[] 0 setdash} bdef /L1 {[40 100] 0 setdash} bdef +/L2 {[200 200] 0 setdash} bdef /L3 {[200 100 50 100] 0 setdash} bdef /L4 +{[300 100 50 100 50 100 50 100] 0 setdash} bdef /L5 {[400 200] 0 setdash} +bdef /IO { gsave 0.0 setgray /val exch def moveto /str 20 string def val str +cvs show grestore } bdef /UL { gsave currentpoint /cury exch def /curx exch +def /curmtx matrix currentmatrix def matrix identmatrix setmatrix newpath 0 +0 moveto true charpath flattenpath pathbbox curmtx setmatrix /height exch +def /width exch def /ybase cury height .1 mul sub def pop pop newpath curx +ybase moveto width 0 rlineto closepath stroke grestore } bdef /BG { gsave +currentpoint /cury exch def /curx exch def /curmtx matrix currentmatrix def +matrix identmatrix setmatrix newpath 0 0 moveto true charpath flattenpath +pathbbox curmtx setmatrix /height exch def /width exch def pop pop newpath 0 +0 moveto (M) true charpath flattenpath pathbbox /xdiff exch .1 mul def +/ydiff exch .1 mul def pop pop newpath curx xdiff 2 div sub cury ydiff 2 div +sub moveto width xdiff add 0 rlineto 0 height ydiff add rlineto width neg +xdiff sub 0 rlineto closepath fill grestore } bdef /$T_DICT 20 dict def /T { +$T_DICT begin /align exch def /orien exch def /size exch def /thestring exch +def gsave moveto /chsize 1.0 def /xsize 0.0 def /SUPER 8#330 def /SUBS 8#331 +def /NORM 8#332 def /SCRIPTWID 0.7 def /OFFSET 0.6 def orien rotate size dup +scale /orien false def thestring { /charcode exch def charcode SUPER eq +charcode SUBS eq or { /chsize SCRIPTWID def /orien true def } { charcode +NORM eq { /chsize 1.0 def /orien true def } { ( ) dup 0 charcode put +stringwidth pop chsize mul xsize add /xsize exch def } ifelse } ifelse } +forall xsize align mul neg 0 rmoveto orien { /regularfont currentfont def +/fractionfont currentfont [ SCRIPTWID 0 0 SCRIPTWID 0 0 ] makefont def gsave +newpath 0 0 moveto (1) true charpath flattenpath pathbbox /height exch def +pop pop pop grestore } if /xsize 0 def thestring { /charcode exch def +charcode SUPER eq { 0 OFFSET height mul dup /xsize exch def rmoveto +fractionfont setfont } { charcode SUBS eq { 0 OFFSET height mul neg dup +/xsize exch def rmoveto fractionfont setfont } { charcode NORM eq { 0 xsize +neg rmoveto regularfont setfont } { ( ) dup 0 charcode put show } ifelse } +ifelse } ifelse } forall grestore end } bdef /IsChar { exch /CharStrings get +exch known } bdef /MapCh { 3 -1 roll /Encoding get 3 1 roll put } bdef +/MapDegree { dup 16#b0 exch /degree IsChar { /degree } { /ring } ifelse +MapCh } bdef /MapBB { dup 16#a6 exch /brokenbar IsChar { /brokenbar } { /bar +} ifelse MapCh } bdef /ReEncode { dup findfont begin currentdict dup length +dict begin { 1 index /FID ne {def} {pop pop} ifelse } forall /FontName exch +def dup length 0 ne {/Encoding Encoding 256 array copy def 0 exch { dup type +/nametype eq { Encoding 2 index 2 index put pop 1 add } { exch pop } ifelse +} forall } if pop currentdict dup end end /FontName get exch definefont dup +MapDegree MapBB } bdef /RF { ISOLatin1Encoding exch ReEncode } bdef /Courier +RF /Courier-Bold RF /Courier-Oblique RF /Courier-BoldOblique RF /Helvetica +RF /Helvetica-Bold RF /Helvetica-Oblique RF /Helvetica-BoldOblique RF +/Helvetica-Narrow RF /Helvetica-Narrow-Bold RF /Helvetica-Narrow-Oblique RF +/Helvetica-Narrow-BoldOblique RF /AvantGarde-Book RF /AvantGarde-DemiOblique +RF /AvantGarde-Demi RF /AvantGarde-DemiOblique RF /Bookman-Demi RF +/Bookman-DemiItalic RF /Bookman-Light RF /Bookman-LightItalic RF +/ZapfChancery-MediumItalic RF /NewCenturySchlbk-Roman RF +/NewCenturySchlbk-Bold RF /NewCenturySchlbk-Italic RF +/NewCenturySchlbk-BoldItalic RF /Palatino-Roman RF /Palatino-Bold RF +/Palatino-Italic RF /Palatino-BoldItalic RF /Times-Roman RF /Times-Bold RF +/Times-Italic RF /Times-BoldItalic RF end +%%EndProlog +%%Page: 0 1 +%%BeginPageSetup +save $WAVE_DICT begin 28 28 L 0.028346 dup scale +%%PageBoundingBox: 28 28 254 254 +%%EndPageSetup +/psFontCache 500 array def /psStack 8 array def /psStackInd 0 def +/psTextWidth 0 def /psCurBase 0 def /ASW { stringwidth pop psCurBase add +/psCurBase exch def psCurBase psTextWidth gt +{ /psTextWidth psCurBase def } if } def /PSS { +psStack psStackInd psCurBase put /psStackInd 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7779 5746 M 6 0 R 0 -111 R D +7727 5694 M 110 0 R 0 -7 R D 7727 5694 M 0 -7 R 110 0 R D 7902 5739 M 13 7 R +20 19 R 0 -136 R D 7902 5739 M 0 -6 R 13 6 R 13 13 R 0 -123 R 7 0 R D +7025 4344 M 10 -32 R 11 -10 R 21 -11 R 21 0 R 31 11 R 21 31 R 11 32 R 0 31 R +-11 21 R -10 11 R -21 10 R -21 0 R -32 -10 R -21 -32 R -10 -31 R -53 -158 R +D 7088 4291 M 21 11 R 21 31 R 10 32 R 0 42 R -10 21 R D 7088 4438 M +-21 -10 R -21 -32 R -11 -31 R -42 -158 R D 7206 4344 M 0 -137 R D +7212 4325 M 0 -118 R -6 0 R D 7212 4325 M 85 -118 R D 7206 4344 M 85 -117 R +0 117 R 6 0 R 0 -137 R D 7375 4338 M 0 -131 R D 7382 4338 M 0 -131 R -7 0 R +D 7336 4344 M 85 0 R 0 -6 R D 7336 4344 M 0 -6 R 85 0 R D 7460 4344 M +0 -137 R D 7466 4312 M 0 -105 R -6 0 R D 7466 4312 M 46 -105 R D 7460 4344 M +52 -117 R 52 117 R D 7557 4312 M -45 -105 R D 7557 4312 M 0 -105 R 7 0 R +0 137 R D 7616 4344 M 0 -137 R D 7616 4344 M 7 0 R 0 -130 R 71 0 R 0 -7 R +-78 0 R D L1 2220 5875 M 3409 0 R D 2220 3560 M 5114 0 R D L0 1737 6091 M +-12 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R D 1240 3654 M 84 -117 R 0 117 R +7 0 R 0 -137 R D 1409 3647 M 0 -130 R D 1415 3647 M 0 -130 R -6 0 R D +1370 3654 M 84 0 R 0 -7 R D 1370 3654 M 0 -7 R 84 0 R D 1493 3654 M 0 -137 R +D 1500 3621 M 0 -104 R -7 0 R D 1500 3621 M 46 -104 R D 1493 3654 M +53 -117 R 52 117 R D 1591 3621 M -45 -104 R D 1591 3621 M 0 -104 R 7 0 R +0 137 R D 1650 3654 M 0 -137 R D 1650 3654 M 6 0 R 0 -130 R 72 0 R 0 -7 R +-78 0 R D 1760 3582 M 111 0 R 0 -6 R D 1760 3582 M 0 -6 R 111 0 R D +1936 3628 M 13 6 R 20 20 R 0 -137 R D 1936 3628 M 0 -7 R 13 7 R 13 13 R +0 -124 R 7 0 R D 1984 5105 M -93 -245 R D 1984 5105 M 94 -245 R D +1984 5070 M 82 -210 R D 1902 4871 M 164 0 R D 1891 4860 M 187 0 R D +2311 5035 M -11 -23 R -24 -24 R -93 -70 R -23 -23 R -12 -24 R D 2300 5012 M +-105 0 R -24 -12 R -11 -24 R D 2276 5012 M -46 11 R -35 0 R -12 -11 R D +2276 5012 M -46 23 R -35 0 R -24 -23 R -11 -36 R D 2160 4895 M 105 0 R +23 11 R 12 24 R D 2183 4895 M 47 -12 R 35 0 R 11 12 R D 2183 4895 M 47 -24 R +35 0 R 23 24 R 12 35 R D 2375 4868 M 0 -101 R D 2383 4868 M 0 -101 R D +2383 4824 M 7 22 R 14 15 R 15 7 R 21 0 R 8 -7 R 0 -8 R -8 -7 R -7 7 R 7 8 R +D 2354 4868 M 29 0 R D 2354 4767 M 50 0 R D 2498 4853 M 0 -7 R -7 0 R 0 7 R +7 8 R 15 7 R 29 0 R 14 -7 R 7 -8 R 8 -14 R 0 -51 R 7 -14 R 7 -7 R D +2563 4853 M 0 -65 R 8 -14 R 14 -7 R 7 0 R D 2563 4839 M -7 -7 R -43 -8 R +-22 -7 R -7 -14 R 0 -15 R 7 -14 R 22 -7 R 22 0 R 14 7 R 14 14 R D +2513 4824 M -15 -7 R -7 -14 R 0 -15 R 7 -14 R 15 -7 R D 2715 4918 M 0 -151 R +D 2723 4918 M 0 -151 R D 2715 4846 M -14 15 R -15 7 R -14 0 R -22 -7 R +-14 -15 R -7 -22 R 0 -14 R 7 -22 R 14 -14 R 22 -7 R 14 0 R 15 7 R 14 14 R D +2672 4868 M -15 -7 R -14 -15 R -7 -22 R 0 -14 R 7 -22 R 14 -14 R 15 -7 R D +2694 4918 M 29 0 R D 2715 4767 M 29 0 R D 20.000000 SL 2288 3560 M 0 1178 R +D 2288 5266 M 0 609 R -68 -122 R D 2288 5875 M 68 -122 R D 2220 3682 M +68 -122 R 68 122 R D 10.000000 SL 4718 1519 M -111 -111 R -111 111 R +111 -111 R D 4823 2189 M -222 -222 R 111 111 R -111 111 R 222 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41 R 0 40 R 0 41 R 0 41 R 0 40 R 0 41 R 0 40 R 0 41 R +0 41 R 0 40 R 0 41 R 0 40 R 0 41 R 0 41 R 0 40 R 0 41 R 0 40 R 1 41 R 0 41 R +1 40 R 2 41 R 2 40 R 5 41 R 6 41 R 10 40 R 14 41 R 20 40 R 28 41 R 39 41 R +51 40 R 68 41 R 87 41 R 108 40 R 131 41 R 155 40 R 180 41 R 201 41 R +218 40 R 231 41 R 234 40 R 228 41 R 213 41 R 186 40 R 149 41 R 105 40 R +54 41 R 10 41 R 11 40 R 10 41 R 10 40 R 10 41 R 11 41 R 10 40 R 10 41 R +10 40 R 10 41 R 11 41 R 10 40 R 10 41 R 10 41 R 11 40 R 10 41 R 10 40 R +10 41 R 10 41 R 11 40 R 10 41 R 10 40 R 10 41 R 11 41 R 10 40 R 10 41 R +10 40 R 11 41 R 10 41 R 10 40 R 10 41 R 10 40 R 11 41 R D L3 4607 1408 M +6 41 R 6 40 R 6 41 R 6 40 R 7 41 R 6 41 R 6 40 R 6 41 R 6 40 R 7 41 R 6 41 R +6 40 R 6 41 R 6 41 R 7 40 R 6 41 R 6 40 R 6 41 R 6 41 R 7 40 R 6 41 R 6 40 R +6 41 R 6 41 R 7 40 R 6 41 R 6 40 R 6 41 R 6 41 R 7 40 R 6 41 R 6 40 R 6 41 R +6 41 R 7 40 R 6 41 R 6 40 R 6 41 R 6 41 R 7 40 R 6 41 R 6 41 R 6 40 R 6 41 R +7 40 R 6 41 R 6 41 R 6 40 R 6 41 R 7 40 R 6 41 R 6 41 R 6 40 R 6 41 R 6 40 R +7 41 R 6 41 R 6 40 R 6 41 R 6 40 R 7 41 R 6 41 R 6 40 R 6 41 R 6 40 R 7 41 R +6 41 R 6 40 R 6 41 R 6 41 R 7 40 R 6 41 R 6 40 R 6 41 R 6 41 R 7 40 R 6 41 R +6 40 R 6 41 R 6 41 R 7 40 R 6 41 R 6 40 R 5 41 R 6 41 R 5 40 R 5 41 R 3 40 R +2 41 R 0 41 R -3 40 R -8 41 R -14 40 R -22 41 R -32 41 R -46 40 R -62 41 R +-80 41 R -102 40 R -125 41 R -149 40 R -173 41 R -195 41 R -212 40 R +-225 41 R -227 40 R -223 41 R -206 41 R -180 40 R -143 41 R 577 40 R +628 41 R 0 41 R 0 40 R 0 41 R 0 40 R 0 41 R 0 41 R 0 40 R 0 41 R 0 40 R +0 41 R 0 41 R 0 40 R 0 41 R 0 41 R 0 40 R 0 41 R 0 40 R 0 41 R 0 41 R 0 40 R +0 41 R 0 40 R 0 41 R 0 41 R 0 40 R 0 41 R 0 40 R 0 41 R 0 41 R 0 40 R 0 41 R +0 40 R 0 41 R D L0 4607 1408 M 6 41 R 6 40 R 6 41 R 6 40 R 7 41 R 6 41 R +6 40 R 6 41 R 6 40 R 7 41 R 6 41 R 6 40 R 6 41 R 6 41 R 7 40 R 6 41 R 6 40 R +6 41 R 6 41 R 7 40 R 6 41 R 6 40 R 6 41 R 6 41 R 7 40 R 6 41 R 6 40 R 6 41 R +6 41 R 7 40 R 6 41 R 6 40 R 6 41 R 6 41 R 7 40 R 6 41 R 6 40 R 6 41 R 6 41 R +7 40 R 6 41 R 6 41 R 6 40 R 6 41 R 7 40 R 6 41 R 6 41 R 6 40 R 6 41 R 7 40 R +6 41 R 6 41 R 6 40 R 6 41 R 6 40 R 7 41 R 6 41 R 6 40 R 6 41 R 6 40 R 7 41 R +6 41 R 6 40 R 6 41 R 6 40 R 7 41 R 6 41 R 6 40 R 6 41 R 6 41 R 7 40 R 6 41 R +6 40 R 6 41 R 6 41 R 7 40 R 6 41 R 6 40 R 6 41 R 6 41 R 7 40 R 6 41 R 6 40 R +6 41 R 6 41 R 7 40 R 6 41 R 6 40 R 6 41 R 6 41 R 7 40 R 6 41 R 6 40 R 6 41 R +6 41 R 7 40 R 6 41 R 6 41 R 6 40 R 6 41 R 7 40 R 6 41 R 6 41 R 6 40 R 6 41 R +7 40 R 6 41 R 6 41 R 6 40 R 6 41 R 682 40 R 682 41 R 10 41 R 11 40 R 10 41 R +10 40 R 10 41 R 11 41 R 10 40 R 10 41 R 10 40 R 10 41 R 11 41 R 10 40 R +10 41 R 10 41 R 11 40 R 10 41 R 10 40 R 10 41 R 10 41 R 11 40 R 10 41 R +10 40 R 10 41 R 11 41 R 10 40 R 10 41 R 10 40 R 11 41 R 10 41 R 10 40 R +10 41 R 10 40 R 11 41 R D +end restore +showpage +%%Trailer +restore +%%Pages: 1 +%%Trailer +cleartomark +countdictstack +exch sub { end } repeat +restore +%%EOF diff --git a/documentation/source/science_guide/turbulence_schemes/ideal_revflux.eps b/documentation/source/science_guide/turbulence_schemes/ideal_revflux.eps new file mode 100644 index 0000000000..b3e8cec02e --- /dev/null +++ b/documentation/source/science_guide/turbulence_schemes/ideal_revflux.eps @@ -0,0 +1,517 @@ +%!PS-Adobe-2.0 EPSF-2.0 +%%BoundingBox: 60 38 464 520 +%%HiResBoundingBox: 60.5 39 463.5 519 +%%Title: Graphics produced by WAVE +%%For: frlk@eld206 +%%Creator: WAVE Version 8.00 (Linux i386) +%%CreationDate: Thu Mar 30 11:10:47 2006 +%%EndComments +% EPSF created by ps2eps 1.68 +%%BeginProlog +save +countdictstack +mark +newpath +/showpage {} def +/setpagedevice {pop} def +%%EndProlog +%%Page 1 1 +%+ wave_prolog.ps -- Prolog for PV-WAVE CL generated PostScript files +%+ $Id: wave_prolog.ps,v 1.5 1998/06/05 17:44:06 thaux Exp $ +%+ Copyright (c) 1989-1992 Precision Visuals, Inc. All Rights Reserved. +%v 1 +save /$WAVE_DICT 60 dict def $WAVE_DICT begin /bdef { bind def } bind def /C +{currentpoint newpath moveto} bdef /D {currentpoint stroke moveto} bdef /F +{closepath fill} bdef /K { setgray } bdef /L {translate} bdef /M {moveto} +bdef /N {rmoveto} bdef /O {show} bdef /P {lineto} bdef /R {rlineto} bdef /S +{gsave show grestore} bdef /Z {gsave currentpoint lineto 20 setlinewidth 1 +setlinecap stroke grestore} bdef /CC {concat} bdef /CD {currentmatrix def} +bdef /CP {currentpoint pop def} bdef /FM {findfont exch makefont} bdef /FS +{findfont exch scalefont} bdef /GS {get setfont} bdef /MD {matrix def} bdef +/MN {mul neg def} bdef /RS {roll put setfont} bdef /SL {setlinewidth} bdef +/SM {setmatrix} bdef /L0 {[] 0 setdash} bdef /L1 {[40 100] 0 setdash} bdef +/L2 {[200 200] 0 setdash} bdef /L3 {[200 100 50 100] 0 setdash} bdef /L4 +{[300 100 50 100 50 100 50 100] 0 setdash} bdef /L5 {[400 200] 0 setdash} +bdef /IO { gsave 0.0 setgray /val exch def moveto /str 20 string def val str +cvs show grestore } bdef /UL { gsave currentpoint /cury exch def /curx exch +def /curmtx matrix currentmatrix def matrix identmatrix setmatrix newpath 0 +0 moveto true charpath flattenpath pathbbox curmtx setmatrix /height exch +def /width exch def /ybase cury height .1 mul sub def pop pop newpath curx +ybase moveto width 0 rlineto closepath stroke grestore } bdef /BG { gsave +currentpoint /cury exch def /curx exch def /curmtx matrix currentmatrix def +matrix identmatrix setmatrix newpath 0 0 moveto true charpath flattenpath +pathbbox curmtx setmatrix /height exch def /width exch def pop pop newpath 0 +0 moveto (M) true charpath flattenpath pathbbox /xdiff exch .1 mul def +/ydiff exch .1 mul def pop pop newpath curx xdiff 2 div sub cury ydiff 2 div +sub moveto width xdiff add 0 rlineto 0 height ydiff add rlineto width neg +xdiff sub 0 rlineto closepath fill grestore } bdef /$T_DICT 20 dict def /T { +$T_DICT begin /align exch def /orien exch def /size exch def /thestring exch +def gsave moveto /chsize 1.0 def /xsize 0.0 def /SUPER 8#330 def /SUBS 8#331 +def /NORM 8#332 def /SCRIPTWID 0.7 def /OFFSET 0.6 def orien rotate size dup +scale /orien false def thestring { /charcode exch def charcode SUPER eq +charcode SUBS eq or { /chsize SCRIPTWID def /orien true def } { charcode +NORM eq { /chsize 1.0 def /orien true def } { ( ) dup 0 charcode put +stringwidth pop chsize mul xsize add /xsize exch def } ifelse } ifelse } +forall xsize align mul neg 0 rmoveto orien { /regularfont currentfont def +/fractionfont currentfont [ SCRIPTWID 0 0 SCRIPTWID 0 0 ] makefont def gsave +newpath 0 0 moveto (1) true charpath flattenpath pathbbox /height exch def +pop pop pop grestore } if /xsize 0 def thestring { /charcode exch def +charcode SUPER eq { 0 OFFSET height mul dup /xsize exch def rmoveto +fractionfont setfont } { charcode SUBS eq { 0 OFFSET height mul neg dup +/xsize exch def rmoveto fractionfont setfont } { charcode NORM eq { 0 xsize +neg rmoveto regularfont setfont } { ( ) dup 0 charcode put show } ifelse } +ifelse } ifelse } forall grestore end } bdef /IsChar { exch /CharStrings get +exch known } bdef /MapCh { 3 -1 roll /Encoding get 3 1 roll put } bdef +/MapDegree { dup 16#b0 exch /degree IsChar { /degree } { /ring } ifelse +MapCh } bdef /MapBB { dup 16#a6 exch /brokenbar IsChar { /brokenbar } { /bar +} ifelse MapCh } bdef /ReEncode { dup findfont begin currentdict dup length +dict begin { 1 index /FID ne {def} {pop pop} ifelse } forall /FontName exch +def dup length 0 ne {/Encoding Encoding 256 array copy def 0 exch { dup type +/nametype eq { Encoding 2 index 2 index put pop 1 add } { exch pop } ifelse +} forall } if pop currentdict dup end end /FontName get exch definefont dup +MapDegree MapBB } bdef /RF { ISOLatin1Encoding exch ReEncode } bdef /Courier +RF /Courier-Bold RF /Courier-Oblique RF /Courier-BoldOblique RF /Helvetica +RF /Helvetica-Bold RF /Helvetica-Oblique RF /Helvetica-BoldOblique RF +/Helvetica-Narrow RF /Helvetica-Narrow-Bold RF /Helvetica-Narrow-Oblique RF +/Helvetica-Narrow-BoldOblique RF /AvantGarde-Book RF /AvantGarde-DemiOblique +RF /AvantGarde-Demi RF /AvantGarde-DemiOblique RF /Bookman-Demi RF +/Bookman-DemiItalic RF /Bookman-Light RF /Bookman-LightItalic RF +/ZapfChancery-MediumItalic RF /NewCenturySchlbk-Roman RF +/NewCenturySchlbk-Bold RF /NewCenturySchlbk-Italic RF +/NewCenturySchlbk-BoldItalic RF /Palatino-Roman RF /Palatino-Bold RF +/Palatino-Italic RF /Palatino-BoldItalic RF /Times-Roman RF /Times-Bold RF +/Times-Italic RF /Times-BoldItalic RF end +%%EndProlog +%%Page: 0 1 +%%BeginPageSetup +save $WAVE_DICT begin 28 28 L 0.028346 dup scale +%%PageBoundingBox: 28 28 481 538 +%%EndPageSetup +/psFontCache 500 array def /psStack 8 array def /psStackInd 0 def +/psTextWidth 0 def /psCurBase 0 def /ASW { stringwidth pop psCurBase add +/psCurBase exch def psCurBase psTextWidth gt +{ /psTextWidth psCurBase def } if } def /PSS { +psStack psStackInd psCurBase put /psStackInd psStackInd 1 add def } def +/PPS { /psStackInd psStackInd 1 sub def +/psCurBase psStack psStackInd get def } def 10.000000 SL L0 0.000 K +2220 10408 M 5114 0 R D 2220 10408 M 0 138 R D 2197 10196 M -35 -12 R +-24 -35 R -11 -58 R 0 -35 R 11 -59 R 24 -35 R 35 -12 R 23 0 R 35 12 R +24 35 R 11 59 R 0 35 R -11 58 R -24 35 R -35 12 R -23 0 R -24 -12 R +-11 -12 R -12 -23 R -12 -58 R 0 -35 R 12 -59 R 12 -23 R 11 -12 R 24 -12 R D +2220 9950 M 23 12 R 12 12 R 12 23 R 12 59 R 0 35 R -12 58 R -12 23 R +-12 12 R -23 12 R D 15005 10408 M 0 138 R D 4697 9668 M -70 -245 R D +4709 9668 M -70 -245 R D 4697 9668 M 24 0 R -70 -245 R D 4674 9586 M 0 35 R +-12 35 R -23 12 R -24 0 R -35 -12 R -23 -35 R -12 -35 R 0 -23 R 12 -35 R +12 -12 R 23 -11 R 23 0 R 24 11 R 12 12 R 11 23 R 12 35 R D 4580 9645 M +-11 -24 R -12 -35 R 0 -35 R 12 -23 R D 4615 9668 M -23 -23 R -12 -24 R +-11 -35 R 0 -35 R 11 -35 R 12 -11 R D 4592 9423 M 94 0 R D 4639 9434 M +-35 -11 R D 4639 9446 M -24 -23 R D 4651 9446 M 11 -23 R D 4639 9434 M +35 -11 R D 4821 9559 M -22 -80 R -7 -29 R 0 -21 R 7 -15 R 7 -7 R 15 0 R +14 15 R 7 14 R D 4828 9559 M -22 -80 R -7 -29 R 0 -36 R D 4821 9559 M 14 0 R +-29 -101 R -7 -29 R D 4777 9508 M 73 0 R D 2220 10408 M 0 6888 R D +2220 10408 M 102 0 R D 1969 10653 M -35 -11 R -24 -35 R -11 -59 R 0 -35 R +11 -58 R 24 -35 R 35 -12 R 23 0 R 35 12 R 24 35 R 11 58 R 0 35 R -11 59 R +-24 35 R -35 11 R -23 0 R -23 -11 R -12 -12 R -12 -23 R -12 -59 R 0 -35 R +12 -58 R 12 -24 R 12 -11 R 23 -12 R D 1992 10408 M 24 12 R 11 11 R 12 24 R +12 58 R 0 35 R -12 59 R -12 23 R -11 12 R -24 11 R D 2220 14303 M 102 0 R D +1915 14373 M -12 -23 R -23 -24 R -94 -70 R -23 -23 R -12 -24 R D +1903 14350 M -105 0 R -23 -12 R -12 -23 R D 1880 14350 M -47 11 R -35 0 R +-12 -11 R D 1880 14350 M -47 23 R -35 0 R -23 -23 R -12 -35 R D 1763 14233 M +105 0 R 24 12 R 11 23 R D 1786 14233 M 47 -12 R 35 0 R 12 12 R D +1786 14233 M 47 -24 R 35 0 R 24 24 R 11 35 R D 2008 14264 M -44 -152 R +15 0 R D 2015 14264 M -43 -152 R D 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-12 R +-24 -23 R -11 -35 R 0 -24 R 11 -35 R 24 -23 R 35 -12 R 23 0 R 35 12 R +24 23 R D 5072 3782 M 0 35 R -12 23 R D 5002 3852 M -24 -12 R -23 -23 R +-12 -35 R 0 -24 R 12 -35 R 23 -23 R 24 -12 R D 5177 3934 M 0 -246 R D +5189 3934 M 0 -246 R D 5142 3934 M 47 0 R D 5142 3688 M 82 0 R D 5493 3934 M +0 -246 R D 5504 3934 M 140 -222 R D 5504 3910 M 140 -222 R 0 246 R D +5457 3934 M 47 0 R D 5609 3934 M 70 0 R D 5457 3688 M 71 0 R D 5808 3934 M +0 -246 R D 5820 3934 M 0 -246 R D 5738 3934 M -12 -70 R 0 70 R 175 0 R +0 -70 R -11 70 R D 5773 3688 M 82 0 R D 5983 3934 M 0 -246 R D 5995 3934 M +70 -211 R D 5983 3934 M 82 -246 R 82 246 R 0 -246 R D 6159 3934 M 0 -246 R D +5948 3934 M 47 0 R D 6147 3934 M 47 0 R D 5948 3688 M 70 0 R D 6112 3688 M +82 0 R D 6275 3934 M 0 -246 R D 6287 3934 M 0 -246 R D 6240 3934 M 82 0 R D +6240 3688 M 176 0 R 0 70 R -12 -70 R D 10220 1408 M 5114 0 R D 11925 1408 M +0 138 R D 11901 1196 M -35 -12 R -23 -35 R -12 -58 R 0 -35 R 12 -59 R +23 -35 R 35 -12 R 24 0 R 35 12 R 23 35 R 12 59 R 0 35 R -12 58 R -23 35 R +-35 12 R -24 0 R -23 -12 R -12 -12 R -11 -23 R -12 -58 R 0 -35 R 12 -59 R +11 -23 R 12 -12 R 23 -12 R D 11925 950 M 23 12 R 12 12 R 11 23 R 12 59 R +0 35 R -12 58 R -11 23 R -12 12 R -23 12 R D 12607 1408 M 0 138 R D +11772 738 M 0 -245 R D 11784 738 M 0 -245 R D 11924 738 M 0 -245 R D +11936 738 M 0 -245 R D 11737 738 M 82 0 R D 11889 738 M 82 0 R D 11784 621 M +140 0 R D 11737 493 M 82 0 R D 11889 493 M 82 0 R D 12041 586 M 140 0 R +0 24 R -11 23 R -12 12 R -24 11 R -35 0 R -35 -11 R -23 -24 R -12 -35 R +0 -23 R 12 -35 R 23 -23 R 35 -12 R 24 0 R 35 12 R 23 23 R D 12170 586 M +0 35 R -12 24 R D 12099 656 M -23 -11 R -23 -24 R -12 -35 R 0 -23 R 12 -35 R +23 -23 R 23 -12 R D 12275 633 M 0 -12 R -12 0 R 0 12 R 12 12 R 23 11 R +47 0 R 23 -11 R 12 -12 R 12 -23 R 0 -82 R 11 -23 R 12 -12 R D 12380 633 M +0 -105 R 12 -23 R 23 -12 R 12 0 R D 12380 610 M -12 -12 R -70 -12 R +-35 -11 R -12 -24 R 0 -23 R 12 -23 R 35 -12 R 35 0 R 23 12 R 24 23 R D +12298 586 M -23 -11 R -12 -24 R 0 -23 R 12 -23 R 23 -12 R D 12508 738 M +0 -198 R 12 -35 R 23 -12 R 24 0 R 23 12 R 12 23 R D 12520 738 M 0 -198 R +12 -35 R 11 -12 R D 12473 656 M 94 0 R D 12929 727 M -12 -12 R 12 -12 R +12 12 R 0 12 R -12 11 R -23 0 R -24 -11 R -11 -24 R 0 -210 R D 12906 738 M +-12 -11 R -12 -24 R 0 -210 R D 12836 656 M 93 0 R D 12836 493 M 81 0 R D +13022 738 M 0 -245 R D 13034 738 M 0 -245 R D 12987 738 M 47 0 R D +12987 493 M 82 0 R D 13151 656 M 0 -128 R 12 -23 R 35 -12 R 23 0 R 35 12 R +24 23 R D 13163 656 M 0 -128 R 11 -23 R 24 -12 R D 13280 656 M 0 -163 R D +13291 656 M 0 -163 R D 13116 656 M 47 0 R D 13244 656 M 47 0 R D 13280 493 M +46 0 R D 13396 656 M 129 -163 R D 13408 656 M 129 -163 R D 13537 656 M +-141 -163 R D 13373 656 M 70 0 R D 13490 656 M 70 0 R D 13373 493 M 70 0 R D +13490 493 M 70 0 R D 13630 586 M 140 0 R 0 24 R -11 23 R -12 12 R -23 11 R +-36 0 R -35 -11 R -23 -24 R -12 -35 R 0 -23 R 12 -35 R 23 -23 R 35 -12 R +24 0 R 35 12 R 23 23 R D 13759 586 M 0 35 R -12 24 R D 13688 656 M -23 -11 R +-23 -24 R -12 -35 R 0 -23 R 12 -35 R 23 -23 R 23 -12 R D 13957 633 M 12 23 R +0 -46 R -12 23 R -11 12 R -24 11 R -47 0 R -23 -11 R -12 -12 R 0 -23 R +12 -12 R 23 -12 R 59 -23 R 23 -12 R 12 -11 R D 13840 621 M 12 -11 R 23 -12 R +59 -23 R 23 -12 R 12 -12 R 0 -35 R -12 -11 R -23 -12 R -47 0 R -23 12 R +-12 11 R -12 24 R 0 -47 R 12 23 R D L2 11925 1408 M 0 48 R 0 47 R 0 48 R +0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R +1 48 R 0 47 R 0 48 R 0 47 R 1 48 R 0 47 R 1 48 R 0 47 R 1 48 R 1 47 R 1 48 R +2 47 R 1 48 R 2 47 R 2 48 R 3 47 R 3 48 R 3 47 R 5 48 R 4 47 R 6 48 R 6 47 R +8 48 R 8 47 R 10 48 R 11 47 R 12 48 R 14 47 R 16 48 R 18 47 R 19 48 R +22 47 R 25 48 R 27 47 R 29 48 R 33 47 R 36 48 R 39 47 R 42 48 R 45 47 R +50 48 R 52 47 R 57 48 R 60 47 R 63 48 R 68 47 R 70 48 R 74 47 R 78 48 R +80 47 R 82 48 R 86 47 R 87 48 R 88 47 R 90 48 R 91 47 R 90 48 R 90 47 R +90 48 R 87 47 R 86 48 R 82 47 R 79 48 R 75 47 R 71 48 R 65 47 R 59 48 R +54 47 R 46 48 R 40 47 R 32 48 R 24 47 R 16 48 R 8 47 R 10 48 R 11 47 R +10 48 R 10 47 R 10 48 R 11 47 R 10 48 R 10 47 R 10 48 R 10 47 R 11 48 R +10 47 R 10 48 R 10 47 R 11 48 R 10 47 R 10 48 R 10 47 R 11 48 R 10 47 R +10 48 R 10 47 R 10 48 R 11 47 R 10 48 R 10 47 R 10 48 R 11 47 R 10 48 R +10 47 R 10 48 R 10 47 R 11 48 R 10 47 R 10 48 R 10 47 R 11 48 R 10 48 R +10 47 R 10 48 R 11 47 R 10 48 R 10 47 R 10 48 R 10 47 R 11 48 R 10 47 R +10 48 R 10 47 R 11 48 R 10 47 R 10 48 R 10 47 R 11 48 R 10 47 R 10 48 R +10 47 R D L3 12607 1408 M 8 48 R 8 47 R 8 48 R 9 47 R 8 48 R 8 47 R 9 48 R +8 47 R 8 48 R 8 47 R 9 48 R 8 47 R 8 48 R 8 47 R 9 48 R 8 47 R 8 48 R 8 47 R +8 48 R 8 47 R 7 48 R 8 47 R 7 48 R 8 47 R 7 48 R 7 47 R 7 48 R 6 47 R 6 48 R +6 47 R 5 48 R 5 47 R 4 48 R 4 47 R 2 48 R 2 47 R 1 48 R 0 47 R -1 48 R +-3 47 R -4 48 R -6 47 R -7 48 R -9 47 R -12 48 R -14 47 R -16 48 R -18 47 R +-22 48 R -24 47 R -27 48 R -31 47 R -34 48 R -37 47 R -41 48 R -44 47 R +-48 48 R -52 47 R -55 48 R -59 47 R -63 48 R -65 47 R -69 48 R -72 47 R +-75 48 R -76 47 R -79 48 R -81 47 R -81 48 R -82 47 R -83 48 R -81 47 R +-81 48 R -80 47 R -77 48 R -74 47 R -71 48 R -66 47 R -63 48 R -56 47 R +-52 48 R -44 47 R 175 48 R 184 47 R 194 48 R 204 47 R 215 48 R 225 47 R +0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R +0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R +0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R +0 47 R 0 48 R 0 47 R 0 48 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R +0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R +0 48 R 0 47 R D L1 11925 1408 M 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R +0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R +0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R +0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R +0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R +0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R +0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R +0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R +-189 48 R -191 47 R -193 48 R -196 47 R -197 48 R -200 47 R -5 48 R -4 47 R +-5 48 R -5 47 R -5 48 R -4 47 R -5 48 R -5 47 R -5 48 R -5 47 R -5 48 R +-5 47 R -5 48 R -6 47 R -5 48 R -5 47 R -6 48 R -5 47 R -5 48 R -6 47 R +-6 48 R -5 47 R -6 48 R -6 47 R -5 48 R -6 47 R -6 48 R -6 47 R -6 48 R +-6 47 R -6 48 R -6 47 R -6 48 R -7 47 R -6 48 R -6 47 R -7 48 R -6 48 R +-7 47 R -6 48 R -7 47 R -6 48 R -7 47 R -7 48 R -7 47 R -7 48 R -6 47 R +-7 48 R -7 47 R -7 48 R -8 47 R -7 48 R -7 47 R -7 48 R -8 47 R -7 48 R +-7 47 R D L0 12607 1408 M 8 48 R 8 47 R 9 48 R 8 47 R 8 48 R 8 47 R 9 48 R +8 47 R 8 48 R 9 47 R 8 48 R 8 47 R 9 48 R 8 47 R 8 48 R 9 47 R 8 48 R 8 47 R +9 48 R 8 47 R 8 48 R 9 47 R 8 48 R 8 47 R 8 48 R 9 47 R 8 48 R 8 47 R 9 48 R +8 47 R 8 48 R 9 47 R 8 48 R 8 47 R 9 48 R 8 47 R 8 48 R 9 47 R 8 48 R 8 47 R +9 48 R 8 47 R 8 48 R 8 47 R 9 48 R 8 47 R 8 48 R 9 47 R 8 48 R 8 47 R 9 48 R +8 47 R 8 48 R 9 47 R 8 48 R 8 47 R 9 48 R 8 47 R 8 48 R 9 47 R 8 48 R 8 47 R +8 48 R 9 47 R 8 48 R 8 47 R 9 48 R 8 47 R 8 48 R 9 47 R 8 48 R 8 47 R 9 48 R +8 47 R 8 48 R 9 47 R 8 48 R 8 47 R 9 48 R 8 47 R 8 48 R 8 47 R 33 48 R +33 47 R 33 48 R 33 47 R 33 48 R 33 47 R 6 48 R 6 47 R 5 48 R 6 47 R 5 48 R +5 47 R 6 48 R 5 47 R 5 48 R 6 47 R 5 48 R 5 47 R 5 48 R 5 47 R 5 48 R 5 47 R +5 48 R 5 47 R 4 48 R 5 47 R 5 48 R 4 47 R 5 48 R 4 47 R 5 48 R 4 47 R 5 48 R +4 47 R 4 48 R 4 47 R 4 48 R 4 47 R 5 48 R 4 47 R 3 48 R 4 47 R 4 48 R 4 48 R +4 47 R 3 48 R 4 47 R 4 48 R 3 47 R 4 48 R 3 47 R 3 48 R 4 47 R 3 48 R 3 47 R +3 48 R 4 47 R 3 48 R 3 47 R 3 48 R 2 47 R 3 48 R 3 47 R D 11925 1408 M +0 6888 R D 11925 5303 M 170 0 R D 11646 5384 M -12 -24 R -23 -23 R -93 -70 R +-24 -24 R -12 -23 R D 11634 5360 M -105 0 R -23 -11 R -12 -24 R D +11611 5360 M -47 12 R -35 0 R -11 -12 R D 11611 5360 M -47 24 R -35 0 R +-23 -24 R -12 -35 R D 11494 5243 M 105 0 R 24 12 R 11 23 R D 11518 5243 M +46 -11 R 35 0 R 12 11 R D 11518 5243 M 46 -23 R 35 0 R 24 23 R 11 35 R D +11746 5274 M 0 -14 R 15 0 R 0 14 R -15 0 R D 11753 5274 M 0 -14 R D +11746 5267 M 15 0 R D 11688 5195 M 8 14 R 14 15 R 15 0 R 7 -7 R 7 -15 R +0 -22 R -14 -36 R D 11732 5217 M 0 -29 R -7 -29 R 0 -29 R D 11732 5202 M +-15 -36 R 0 -22 R 8 -14 R 7 -7 R 14 0 R 15 14 R 7 14 R D 11925 5588 M +170 0 R D 11646 5787 M -12 -23 R -23 -23 R -93 -70 R -24 -24 R -12 -23 R D +11634 5764 M -105 0 R -23 -12 R -12 -23 R D 11611 5764 M -47 12 R -35 0 R +-11 -12 R D 11611 5764 M -47 23 R -35 0 R -23 -23 R -12 -35 R D 11494 5647 M +105 0 R 24 12 R 11 23 R D 11518 5647 M 46 -12 R 35 0 R 12 12 R D +11518 5647 M 46 -23 R 35 0 R 24 23 R 11 35 R D 11746 5678 M -21 -79 R +-8 -29 R 0 -22 R 8 -14 R 7 -8 R 14 0 R 15 15 R 7 14 R D 11753 5678 M +-21 -79 R -7 -29 R 0 -36 R D 11746 5678 M 15 0 R -29 -101 R -7 -29 R D +11703 5628 M 72 0 R D L2 10220 1408 M 5114 0 R D 10220 2026 M 5114 0 R D +10220 3071 M 5114 0 R D 10220 4733 M 5114 0 R D 10220 6633 M 5114 0 R D +20.000000 SL L3 12607 1408 M 107 618 R 146 1045 R -488 1662 R -447 1900 R +0 1663 R D 10.000000 SL L0 12718 1519 M -111 -111 R -111 111 R 111 -111 R D +12825 2137 M -222 -222 R 111 111 R -111 111 R 222 -222 R D 12971 3182 M +-222 -222 R 111 111 R -111 111 R 222 -222 R D 12483 4844 M -222 -222 R +111 111 R -111 111 R 222 -222 R D 12036 6744 M -222 -222 R 111 111 R +-111 111 R 222 -222 R D 11925 1519 M 55 -111 R D 11869 1408 M 56 111 R D +11925 2137 M 111 -222 R -222 0 R 111 222 R D 11962 3182 M 111 -222 R +-222 0 R 111 222 R D 13558 4844 M 111 -222 R -222 0 R 111 222 R D +14877 6744 M 111 -222 R -222 0 R 111 222 R D 20.000000 SL L1 11925 1408 M +0 618 R 0 1045 R -816 1662 R -462 1900 R -234 1663 R D 10.000000 SL L0 +11925 1519 M 111 -111 R D 11814 1408 M 111 111 R D 11925 2137 M 111 -111 R +-111 -111 R -111 111 R 111 111 R D 11925 3182 M 111 -111 R -111 -111 R +-111 111 R 111 111 R D 11109 4844 M 111 -111 R -111 -111 R -111 111 R +111 111 R D 10647 6744 M 111 -111 R -111 -111 R -111 111 R 111 111 R D +12607 1519 M 111 0 R 0 -111 R D 12496 1408 M 0 111 R 111 0 R D 12715 2137 M +111 0 R 0 -222 R -222 0 R 0 222 R 111 0 R D 12898 3182 M 111 0 R 0 -222 R +-222 0 R 0 222 R 111 0 R D 13189 4844 M 111 0 R 0 -222 R -222 0 R 0 222 R +111 0 R D 13599 6744 M 111 0 R 0 -222 R -222 0 R 0 222 R 111 0 R D +end restore +showpage +%%Trailer +restore +%%Pages: 1 +%%Trailer +cleartomark +countdictstack +exch sub { end } repeat +restore +%%EOF diff --git a/documentation/source/science_guide/turbulence_schemes/nbldoc_zidiag.eps b/documentation/source/science_guide/turbulence_schemes/nbldoc_zidiag.eps new file mode 100644 index 0000000000..feab8facc1 --- /dev/null +++ b/documentation/source/science_guide/turbulence_schemes/nbldoc_zidiag.eps @@ -0,0 +1,183 @@ +%!PS-Adobe-2.0 EPSF-2.0 +%%BoundingBox: 73 43 287 285 +%%HiResBoundingBox: 74 43.5 286 284.5 +%%Title: Graphics produced by WAVE +%%For: lock@visual +%%Creator: WAVE Version 7.00 (Linux i386) +%%CreationDate: Wed Oct 11 13:31:44 2000 +%%EndComments +% EPSF created by ps2eps 1.68 +%%BeginProlog +save +countdictstack +mark +newpath +/showpage {} def +/setpagedevice {pop} def +%%EndProlog +%%Page 1 1 +%+ wave_prolog.ps -- Prolog for PV-WAVE CL generated PostScript files +%+ $Id: wave_prolog.ps,v 1.5 1998/06/05 17:44:06 thaux Exp $ +%+ Copyright (c) 1989-1992 Precision Visuals, Inc. All Rights Reserved. +%v 1 +save /$WAVE_DICT 60 dict def $WAVE_DICT begin /bdef { bind def } bind def /C +{currentpoint newpath moveto} bdef /D {currentpoint stroke moveto} bdef /F +{closepath fill} bdef /K { setgray } bdef /L {translate} bdef /M {moveto} +bdef /N {rmoveto} bdef /O {show} bdef /P {lineto} bdef /R {rlineto} bdef /S +{gsave show grestore} bdef /Z {gsave currentpoint lineto 20 setlinewidth 1 +setlinecap stroke grestore} bdef /CC {concat} bdef /CD {currentmatrix def} +bdef /CP {currentpoint pop def} bdef /FM {findfont exch makefont} bdef /FS +{findfont exch scalefont} bdef /GS {get setfont} bdef /MD {matrix def} bdef +/MN {mul neg def} bdef /RS {roll put setfont} bdef /SL {setlinewidth} bdef +/SM {setmatrix} bdef /L0 {[] 0 setdash} bdef /L1 {[40 100] 0 setdash} bdef +/L2 {[200 200] 0 setdash} bdef /L3 {[200 100 50 100] 0 setdash} bdef /L4 +{[300 100 50 100 50 100 50 100] 0 setdash} bdef /L5 {[400 200] 0 setdash} +bdef /IO { gsave 0.0 setgray /val exch def moveto /str 20 string def val str +cvs show grestore } bdef /UL { gsave currentpoint /cury exch def /curx exch +def /curmtx matrix currentmatrix def matrix identmatrix setmatrix newpath 0 +0 moveto true charpath flattenpath pathbbox curmtx setmatrix /height exch +def /width exch def /ybase cury height .1 mul sub def pop pop newpath curx +ybase moveto width 0 rlineto closepath stroke grestore } bdef /BG { gsave +currentpoint /cury exch def /curx exch def /curmtx matrix currentmatrix def +matrix identmatrix setmatrix newpath 0 0 moveto true charpath flattenpath +pathbbox curmtx setmatrix /height exch def /width exch def pop pop newpath 0 +0 moveto (M) true charpath flattenpath pathbbox /xdiff exch .1 mul def +/ydiff exch .1 mul def pop pop newpath curx xdiff 2 div sub cury ydiff 2 div +sub moveto width xdiff add 0 rlineto 0 height ydiff add rlineto width neg +xdiff sub 0 rlineto closepath fill grestore } bdef /$T_DICT 20 dict def /T { +$T_DICT begin /align exch def /orien exch def /size exch def /thestring exch +def gsave moveto /chsize 1.0 def /xsize 0.0 def /SUPER 8#330 def /SUBS 8#331 +def /NORM 8#332 def /SCRIPTWID 0.7 def /OFFSET 0.6 def orien rotate size dup +scale /orien false def thestring { /charcode exch def charcode SUPER eq +charcode SUBS eq or { /chsize SCRIPTWID def /orien true def } { charcode +NORM eq { /chsize 1.0 def /orien true def } { ( ) dup 0 charcode put +stringwidth pop chsize mul xsize add /xsize exch def } ifelse } ifelse } +forall xsize align mul neg 0 rmoveto orien { /regularfont currentfont def +/fractionfont currentfont [ SCRIPTWID 0 0 SCRIPTWID 0 0 ] makefont def gsave +newpath 0 0 moveto (1) true charpath flattenpath pathbbox /height exch def +pop pop pop grestore } if /xsize 0 def thestring { /charcode exch def +charcode SUPER eq { 0 OFFSET height mul dup /xsize exch def rmoveto +fractionfont setfont } { charcode SUBS eq { 0 OFFSET height mul neg dup +/xsize exch def rmoveto fractionfont setfont } { charcode NORM eq { 0 xsize +neg rmoveto regularfont setfont } { ( ) dup 0 charcode put show } ifelse } +ifelse } ifelse } forall grestore end } bdef /IsChar { exch /CharStrings get +exch known } bdef /MapCh { 3 -1 roll /Encoding get 3 1 roll put } bdef +/MapDegree { dup 16#b0 exch /degree IsChar { /degree } { /ring } ifelse +MapCh } bdef /MapBB { dup 16#a6 exch /brokenbar IsChar { /brokenbar } { /bar +} ifelse MapCh } bdef /ReEncode { dup findfont begin currentdict dup length +dict begin { 1 index /FID ne {def} {pop pop} ifelse } forall /FontName exch +def dup length 0 ne {/Encoding Encoding 256 array copy def 0 exch { dup type +/nametype eq { Encoding 2 index 2 index put pop 1 add } { exch pop } ifelse +} forall } if pop currentdict dup end end /FontName get exch definefont dup +MapDegree MapBB } bdef /RF { ISOLatin1Encoding exch ReEncode } bdef /Courier +RF /Courier-Bold RF /Courier-Oblique RF /Courier-BoldOblique RF /Helvetica +RF /Helvetica-Bold RF /Helvetica-Oblique RF /Helvetica-BoldOblique RF +/Helvetica-Narrow RF /Helvetica-Narrow-Bold RF /Helvetica-Narrow-Oblique RF +/Helvetica-Narrow-BoldOblique RF /AvantGarde-Book RF /AvantGarde-DemiOblique +RF /AvantGarde-Demi RF /AvantGarde-DemiOblique RF /Bookman-Demi RF +/Bookman-DemiItalic RF /Bookman-Light RF /Bookman-LightItalic RF +/ZapfChancery-MediumItalic RF /NewCenturySchlbk-Roman RF +/NewCenturySchlbk-Bold RF /NewCenturySchlbk-Italic RF +/NewCenturySchlbk-BoldItalic RF /Palatino-Roman RF /Palatino-Bold RF +/Palatino-Italic RF /Palatino-BoldItalic RF /Times-Roman RF /Times-Bold RF +/Times-Italic RF /Times-BoldItalic RF end +%%EndProlog +%%Page: 0 1 +%%BeginPageSetup +save $WAVE_DICT begin 28 28 L 0.028346 dup scale +%%PageBoundingBox: 28 28 311 311 +%%EndPageSetup +/psFontCache 500 array def /psStack 8 array def /psStackInd 0 def +/psTextWidth 0 def /psCurBase 0 def /ASW { stringwidth pop psCurBase add +/psCurBase exch def psCurBase psTextWidth gt +{ /psTextWidth psCurBase def } if } def /PSS { +psStack psStackInd psCurBase put /psStackInd psStackInd 1 add def } def +/PPS { /psStackInd psStackInd 1 sub def +/psCurBase psStack psStackInd get def } def 10.000000 SL L0 0.000 K +3108 1971 M 5959 0 R D 3108 1971 M 0 141 R D 3959 1971 M 0 141 R D +5934 1033 M -50 -16 R -32 -49 R -17 -33 R -16 -49 R -16 -82 R 0 -65 R +16 -33 R 33 -17 R 32 0 R 50 17 R 32 49 R 17 33 R 16 49 R 16 81 R 0 66 R +-16 33 R -33 16 R -32 0 R -33 -16 R -33 -49 R -16 -33 R -17 -49 R -16 -82 R +0 -65 R 16 -33 R 17 -17 R D 5884 689 M 33 17 R 33 49 R 16 33 R 17 49 R +16 81 R 0 66 R -16 33 R -17 16 R D 5835 869 M 148 0 R D 6091 671 M 10 20 R +20 20 R 20 0 R 11 -10 R 10 -20 R 0 -31 R -21 -50 R D 6152 701 M 0 -40 R +-11 -41 R 0 -41 R D 6152 681 M -21 -51 R 0 -30 R 10 -21 R 21 -10 R 20 0 R +20 10 R 20 21 R 21 30 R 10 41 R 0 40 R -10 0 R 0 -10 R 10 -20 R D 6364 782 M +-30 -111 R -10 -41 R 0 -30 R 10 -21 R 10 -10 R 20 0 R 21 21 R 10 20 R D +6374 782 M -30 -111 R -10 -41 R 0 -51 R D 6334 782 M 51 0 R -41 -142 R +-10 -40 R D 6344 782 M 30 -10 R D 6354 782 M 10 -20 R 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0000000000..bbc7b4935c --- /dev/null +++ b/documentation/source/science_guide/turbulence_schemes/new_ktop_shape.eps @@ -0,0 +1,288 @@ +%!PS-Adobe-2.0 EPSF-2.0 +%%BoundingBox: 73 370 332 572 +%%HiResBoundingBox: 74 370.5 331.5 571.5 +%%Title: Graphics produced by WAVE +%%For: frlk@eld206 +%%Creator: WAVE Version 8.00 (Linux i386) +%%CreationDate: Fri Jan 13 11:38:04 2006 +%%EndComments +% EPSF created by ps2eps 1.68 +%%BeginProlog +save +countdictstack +mark +newpath +/showpage {} def +/setpagedevice {pop} def +%%EndProlog +%%Page 1 1 +%+ wave_prolog.ps -- Prolog for PV-WAVE CL generated PostScript files +%+ $Id: wave_prolog.ps,v 1.5 1998/06/05 17:44:06 thaux Exp $ +%+ Copyright (c) 1989-1992 Precision Visuals, Inc. All Rights Reserved. +%v 1 +save /$WAVE_DICT 60 dict def $WAVE_DICT begin /bdef { bind def } bind def /C +{currentpoint newpath moveto} bdef /D {currentpoint stroke moveto} bdef /F +{closepath fill} bdef /K { setgray } bdef /L {translate} bdef /M {moveto} +bdef /N {rmoveto} bdef /O {show} bdef /P {lineto} bdef /R {rlineto} bdef /S +{gsave show grestore} bdef /Z {gsave currentpoint lineto 20 setlinewidth 1 +setlinecap stroke grestore} bdef /CC {concat} bdef /CD {currentmatrix def} +bdef /CP {currentpoint pop def} bdef /FM {findfont exch makefont} bdef /FS +{findfont exch scalefont} bdef /GS {get setfont} bdef /MD {matrix def} bdef +/MN {mul neg def} bdef /RS {roll put setfont} bdef /SL {setlinewidth} bdef +/SM {setmatrix} bdef /L0 {[] 0 setdash} bdef /L1 {[40 100] 0 setdash} bdef +/L2 {[200 200] 0 setdash} bdef /L3 {[200 100 50 100] 0 setdash} bdef /L4 +{[300 100 50 100 50 100 50 100] 0 setdash} bdef /L5 {[400 200] 0 setdash} +bdef /IO { gsave 0.0 setgray /val exch def moveto /str 20 string def val str +cvs show grestore } bdef /UL { gsave currentpoint /cury exch def /curx exch +def /curmtx matrix currentmatrix def matrix identmatrix setmatrix newpath 0 +0 moveto true charpath flattenpath pathbbox curmtx setmatrix /height exch +def /width exch def /ybase cury height .1 mul sub def pop pop newpath curx +ybase moveto width 0 rlineto closepath stroke grestore } bdef /BG { gsave +currentpoint /cury exch def /curx exch def /curmtx matrix currentmatrix def +matrix identmatrix setmatrix newpath 0 0 moveto true charpath flattenpath +pathbbox curmtx setmatrix /height exch def /width exch def pop pop newpath 0 +0 moveto (M) true charpath flattenpath pathbbox /xdiff exch .1 mul def +/ydiff exch .1 mul def pop pop newpath curx xdiff 2 div sub cury ydiff 2 div +sub moveto width xdiff add 0 rlineto 0 height ydiff add rlineto width neg +xdiff sub 0 rlineto closepath fill grestore } bdef /$T_DICT 20 dict def /T { +$T_DICT begin /align exch def /orien exch def /size exch def /thestring exch +def gsave moveto /chsize 1.0 def /xsize 0.0 def /SUPER 8#330 def /SUBS 8#331 +def /NORM 8#332 def /SCRIPTWID 0.7 def /OFFSET 0.6 def orien rotate size dup +scale /orien false def thestring { /charcode exch def charcode SUPER eq +charcode SUBS eq or { /chsize SCRIPTWID def /orien true def } { charcode +NORM eq { /chsize 1.0 def /orien true def } { ( ) dup 0 charcode put +stringwidth pop chsize mul xsize add /xsize exch def } ifelse } ifelse } +forall xsize align mul neg 0 rmoveto orien { /regularfont currentfont def +/fractionfont currentfont [ SCRIPTWID 0 0 SCRIPTWID 0 0 ] makefont def gsave +newpath 0 0 moveto (1) true charpath flattenpath pathbbox /height exch def +pop pop pop grestore } if /xsize 0 def thestring { /charcode exch def +charcode SUPER eq { 0 OFFSET height mul dup /xsize exch def rmoveto +fractionfont setfont } { charcode SUBS eq { 0 OFFSET height mul neg dup +/xsize exch def rmoveto fractionfont setfont } { charcode NORM eq { 0 xsize +neg rmoveto regularfont setfont } { ( ) dup 0 charcode put show } ifelse } +ifelse } ifelse } forall grestore end } bdef /IsChar { exch /CharStrings get +exch known } bdef /MapCh { 3 -1 roll /Encoding get 3 1 roll put } bdef +/MapDegree { dup 16#b0 exch /degree IsChar { /degree } { /ring } ifelse +MapCh } bdef /MapBB { dup 16#a6 exch /brokenbar IsChar { /brokenbar } { /bar +} ifelse MapCh } bdef /ReEncode { dup findfont begin currentdict dup length +dict begin { 1 index /FID ne {def} {pop pop} ifelse } forall /FontName exch +def dup length 0 ne {/Encoding Encoding 256 array copy def 0 exch { dup type +/nametype eq { Encoding 2 index 2 index put pop 1 add } { exch pop } ifelse +} forall } if pop currentdict dup end end /FontName get exch definefont dup +MapDegree MapBB } bdef /RF { ISOLatin1Encoding exch ReEncode } bdef /Courier +RF /Courier-Bold RF /Courier-Oblique RF /Courier-BoldOblique RF /Helvetica +RF /Helvetica-Bold RF /Helvetica-Oblique RF /Helvetica-BoldOblique RF +/Helvetica-Narrow RF /Helvetica-Narrow-Bold RF /Helvetica-Narrow-Oblique RF +/Helvetica-Narrow-BoldOblique RF /AvantGarde-Book RF /AvantGarde-DemiOblique +RF /AvantGarde-Demi RF /AvantGarde-DemiOblique RF /Bookman-Demi RF +/Bookman-DemiItalic RF /Bookman-Light RF /Bookman-LightItalic RF +/ZapfChancery-MediumItalic RF /NewCenturySchlbk-Roman RF +/NewCenturySchlbk-Bold RF /NewCenturySchlbk-Italic RF +/NewCenturySchlbk-BoldItalic RF /Palatino-Roman RF /Palatino-Bold RF +/Palatino-Italic RF /Palatino-BoldItalic RF /Times-Roman RF /Times-Bold RF +/Times-Italic RF /Times-BoldItalic RF end +%%EndProlog +%%Page: 0 1 +%%BeginPageSetup +save $WAVE_DICT begin 54 360 L 0.028346 dup scale +%%PageBoundingBox: 54 360 337 586 +%%EndPageSetup +/psFontCache 500 array def /psStack 8 array def /psStackInd 0 def +/psTextWidth 0 def /psCurBase 0 def /ASW { stringwidth pop psCurBase add +/psCurBase exch def psCurBase psTextWidth gt +{ /psTextWidth psCurBase def } if } def /PSS { +psStack psStackInd psCurBase put /psStackInd 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+-10 9 R -9 19 R 0 28 R 9 18 R 19 19 R 28 9 R 37 10 R 19 9 R 10 19 R 0 19 R +-10 18 R -28 10 R -37 0 R D 1776 7437 M 154 0 R D 1267 7370 M 18 10 R +28 28 R 0 -196 R D 1444 7230 M -9 -9 R 9 -9 R 10 9 R -10 9 R D 1575 7408 M +-28 -9 R -19 -29 R -9 -46 R 0 -28 R 9 -47 R 19 -28 R 28 -9 R 19 0 R 28 9 R +19 28 R 9 47 R 0 28 R -9 46 R -19 29 R -28 9 R -19 0 R D 1776 1442 M 77 0 R +D 1776 1757 M 77 0 R D 1776 2073 M 77 0 R D 1776 2704 M 77 0 R D 1776 3020 M +77 0 R D 1776 3335 M 77 0 R D 1776 3966 M 77 0 R D 1776 4282 M 77 0 R D +1776 4597 M 77 0 R D 1776 5228 M 77 0 R D 1776 5544 M 77 0 R D 1776 5859 M +77 0 R D 1776 6490 M 77 0 R D 1776 6806 M 77 0 R D 1776 7121 M 77 0 R D +823 4086 M 130 -102 R D 823 3984 M 0 102 R D 953 3984 M 0 102 R D 720 4301 M +308 -168 R D 823 4451 M 130 -103 R D 823 4348 M 0 103 R D 953 4348 M 0 103 R +D 906 4506 M 122 0 R D 970 4506 M -17 17 R -6 12 R 0 17 R 6 12 R 17 5 R +58 0 R D 9467 1126 M 0 6311 R D 9467 1126 M -154 0 R D 9467 2389 M -154 0 R +D 9467 3651 M 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R 19 9 R 18 19 R +D 7355 2141 M 0 -131 R D 7355 2085 M 9 28 R 19 18 R 19 10 R 28 0 R D +7486 2206 M 0 -159 R 9 -28 R 19 -9 R 18 0 R D 7458 2141 M 65 0 R D +7579 2085 M 112 0 R 0 18 R -9 19 R -9 9 R -19 10 R -28 0 R -19 -10 R +-18 -18 R -10 -28 R 0 -19 R 10 -28 R 18 -19 R 19 -9 R 28 0 R 19 9 R 18 19 R +D 7860 2206 M 0 -196 R D 7860 2113 M -19 18 R -19 10 R -28 0 R -18 -10 R +-19 -18 R -10 -28 R 0 -19 R 10 -28 R 19 -19 R 18 -9 R 28 0 R 19 9 R 19 19 R +D 8131 2206 M -28 -9 R -19 -28 R -9 -47 R 0 -28 R 9 -47 R 19 -28 R 28 -9 R +18 0 R 28 9 R 19 28 R 10 47 R 0 28 R -10 47 R -19 28 R -28 9 R -18 0 R D +8280 2029 M -9 -10 R 9 -9 R 10 9 R -10 10 R D 8402 2206 M -28 -9 R -10 -19 R +0 -19 R 10 -18 R 18 -10 R 38 -9 R 28 -9 R 19 -19 R 9 -19 R 0 -28 R -9 -18 R +-10 -10 R -28 -9 R -37 0 R -28 9 R -10 10 R -9 18 R 0 28 R 9 19 R 19 19 R +28 9 R 38 9 R 18 10 R 10 18 R 0 19 R -10 19 R -28 9 R -37 0 R D 8701 2206 M +0 -196 R D 8832 2206 M -131 -131 R D 8748 2122 M 84 -112 R D 8962 2256 M +-12 12 R -17 5 R -23 0 R -18 -5 R -11 -12 R 0 -11 R 6 -12 R 5 -6 R 12 -6 R +35 -11 R 11 -6 R 6 -6 R 6 -11 R 0 -18 R -12 -11 R -17 -6 R -23 0 R -18 6 R +-11 11 R D 9002 2233 M 0 -58 R 6 -17 R 12 -6 R 17 0 R 12 6 R 17 17 R D +9066 2233 M 0 -81 R D 9112 2233 M 0 -81 R D 9112 2198 M 6 18 R 12 11 R +11 6 R 17 0 R D 9222 2273 M -11 0 R -12 -5 R -6 -18 R 0 -98 R D 9176 2233 M +40 0 R D +end restore +showpage +%%Trailer +restore +%%Pages: 1 +%%Trailer +cleartomark +countdictstack +exch sub { end } repeat +restore +%%EOF diff --git a/documentation/source/science_guide/turbulence_schemes/newcommand.tex b/documentation/source/science_guide/turbulence_schemes/newcommand.tex new file mode 100644 index 0000000000..cd6ce69576 --- /dev/null +++ b/documentation/source/science_guide/turbulence_schemes/newcommand.tex @@ -0,0 +1,111 @@ +\newcommand{\p}{\partial} +\newcommand{\f}{\frac} +\newcommand{\ti}{\textit} +\newcommand{\fb}{\textbf} +\newcommand{\itz}{\begin{itemize}} +\newcommand{\etz}{\end{itemize}} +\newcommand{\beqn}{\begin{equation}} +\newcommand{\eeqn}{\end{equation}} +\newcommand{\ba}{\begin{align}} +\newcommand{\ea}{\end{align}} +\newcommand{\ol}{\overline} +\newcommand{\rip}{Ri_{p}} +\newcommand{\rfp}{Rf_{p}} +\newcommand{\vk}{\kappa} +\newcommand{\Ical}{{\cal I}} +\newcommand{\Ecal}{{\cal E}} +\hyphenation{para-me-trized} +\newcommand{\etal}{\mbox{\em et al.}} +\newcommand{\tpert}{\theta_{\rm pert}} +\newcommand{\thetavl}{\theta_{v\ell}} +\newcommand{\thetal}{\theta_{\ell}} +\newcommand{\buoyco}{\frac{g}{\theta_0}} +\newcommand{\gperkg}{g$\,$kg$^{-1}$} +\newcommand{\ql}{q_{\ell}} +\newcommand{\qlf}{q_{\ell f}} +\newcommand{\ml}{m_{\ell}} +\newcommand{\rhol}{\rho_{\ell}} +\newcommand{\qlmax}{{\ql}_{\rm ct}} +\newcommand{\qfmax}{{q_f}_{\rm ct}} +\newcommand{\gammaml}{\gamma^{\tiny \rm ML}} +\newcommand{\gammafa}{\gamma^{\scriptsize \rm FA}} +\newcommand{\cms}{cm$\,$s$^{-1}$} +\newcommand{\radf}{F} +\newcommand{\radfnet}{F_{\rm net}} +\newcommand{\abr}{A_{\rm br}} +\newcommand{\vsum}{V_{\rm sum}^3} +\newcommand{\vheat}{V_{\rm heat}^3} +\newcommand{\vsurf}{V_{\rm surf}^3} +\newcommand{\vbr}{V_{\rm br}^3} +\newcommand{\vrad}{V_{\rm rad}^3} +\newcommand{\vsumo}{V_{\rm sum}} +\newcommand{\vheato}{V_{\rm heat}} +\newcommand{\vtopc}{V_{\rm Sc}^3} +\newcommand{\vtopo}{V_{\rm Sc}} +\newcommand{\vtopt}{V_{\rm top}} +\newcommand{\vtoptc}{V_{\rm top}^3} +\newcommand{\vrado}{V_{\rm rad}} +\newcommand{\vbro}{V_{\rm br}} +\newcommand{\vsurfo}{V_{\rm surf}} +\newcommand{\khsurf}{K_h^{\rm surf}} +\newcommand{\khmsurf}{K_{h,m}^{\rm surf}} +\newcommand{\fx}{F_{\chi}} +\newcommand{\fxtot}{\fx^{Tot}} +\newcommand{\fxnt}{\fx^{NT}} +\newcommand{\fxntp}{\fx^{NTP}} +\newcommand{\wxngs}{\wx_{ng}^{\rm surf}} +\newcommand{\wxngt}{\wx_{ng}^{\rm Sc}} +\newcommand{\khsmoke}{K_h^{\rm smoke}} +\newcommand{\kchisurf}{K_{\chi}^{\rm surf}} +\newcommand{\kmsurf}{K_m^{\rm surf}} +\newcommand{\khtop}{K_h^{\rm Sc}} +\newcommand{\kmtop}{K_m^{\rm Sc}} +\newcommand{\khsum}{K_h^{\rm sum}} +\newcommand{\ntml}{\mbox{\tiny \rm NTML}} +\newcommand{\nbdsc}{\mbox{\tiny \rm NBDSC}} +\newcommand{\ntdsc}{\mbox{\tiny \rm NTDSC}} +\newcommand{\ntpar}{\mbox{\tiny \rm NTPAR}} +\newcommand{\nlcl}{\mbox{\tiny \rm NLCL}} +\newcommand{\TKE}{\mbox{TKE}} +\newcommand{\wb}{\ol{w'b}} +\newcommand{\wbs}{\ol{w'b}_S} +\newcommand{\wbsat}{[\ol{w'b'}_S]_{\rm sat}} +\newcommand{\wth}{\ol{w'\theta}} +\newcommand{\wthvs}{\ol{w'\theta_v'}_S} +\newcommand{\wthl}{\ol{w'\thetal'}} +\newcommand{\wthls}{\ol{w'\thetal'}_S} +\newcommand{\wthlzi}{\ol{w'\thetal'}_{z_i}} +\newcommand{\wqi}{\ol{w'q_i'}} +\newcommand{\wqt}{\ol{w'q_t'}} +\newcommand{\wx}{\ol{w'\chi'}} +\newcommand{\wxs}{\ol{w'\chi'}_S} +\newcommand{\wqts}{\ol{w'q_t'}_S} +\newcommand{\wqtzi}{\ol{w'q_t'}_{z_i}} +\newcommand{\wqtngs}{\wqt_{ng}^{\rm surf}} +\newcommand{\zbaseq}{z_{\rm b}} +\newcommand{\zbase}{$ \zbaseq $~} +\newcommand{\zct}{z_{\rm ct}} +\newcommand{\zfl}{z_{\rm fl}} +\newcommand{\zml}{z_{\rm ml}} +\newcommand{\zhe}{z_{\rm h}} +\newcommand{\zh}{$ \zhe $~} +\newcommand{\zhsce}{z_{\rm h}^{\rm Sc}} +\newcommand{\zhsc}{$ \zhsce $~} +\newcommand{\zht}{$z_{\rm top}$~} +\newcommand{\zhpare}{z_{\rm par}} +\newcommand{\zhpar}{$\zhpare$~} +\newcommand{\zloce}{z_{\rm loc}} +\newcommand{\zloc}{$\zloce$~} +\newcommand{\zlcle}{z_{\rm lcl}} +\newcommand{\zlcl}{$\zlcle$~} +\newcommand{\lb}{\left(} +\newcommand{\rb}{\right)} +\newcommand{\heat}{\mathcal{H}} +\newcommand{\zonzi}{\f{z}{z_h}} +\newcommand{\zonzml}{\f{z'}{z_{ml}}} +\newcommand{\kbl}{k_{b}} +\newcommand{\blm}[3]{{\em Bound.~Layer Meteor.},\ {\bf #1},\ #2--#3.} +\newcommand{\qj}[3]{{\em Quart.~J.~Roy.~Meteorol.~Soc.},\ {\bf #1},\ #2--#3.} +\newcommand{\mwr}[3]{{\em Mon.~Weather Rev.},\ {\bf #1},\ #2--#3.} +\newcommand{\jas}[3]{{\em J.~Atmos.~Sci.},\ {\bf #1},\ #2--#3.} +\newcommand{\HRule}{\rule{\linewidth}{1.0mm}} diff --git a/documentation/source/science_guide/turbulence_schemes/stab_dep.eps b/documentation/source/science_guide/turbulence_schemes/stab_dep.eps new file mode 100644 index 0000000000..83eb7f4c26 --- /dev/null +++ b/documentation/source/science_guide/turbulence_schemes/stab_dep.eps @@ -0,0 +1,1398 @@ +%!PS-Adobe-2.0 EPSF-2.0 +%%BoundingBox: 96 66 495 621 +%%HiResBoundingBox: 97 67 494.5 620.5 +%%Title: Graphics produced by WAVE +%%For: frlk@eld206 +%%Creator: WAVE Version 8.00 (Linux i386) +%%CreationDate: Wed Mar 29 18:41:50 2006 +%%EndComments +% EPSF created by ps2eps 1.68 +%%BeginProlog +save +countdictstack +mark +newpath +/showpage {} def +/setpagedevice {pop} def +%%EndProlog +%%Page 1 1 +%+ wave_prolog.ps -- Prolog for PV-WAVE CL generated PostScript files +%+ $Id: wave_prolog.ps,v 1.5 1998/06/05 17:44:06 thaux Exp $ +%+ Copyright (c) 1989-1992 Precision Visuals, Inc. All Rights Reserved. +%v 1 +save /$WAVE_DICT 60 dict def $WAVE_DICT begin /bdef { bind def } bind def /C +{currentpoint newpath moveto} bdef /D {currentpoint stroke moveto} bdef /F +{closepath fill} bdef /K { setgray } bdef /L {translate} bdef /M {moveto} +bdef /N {rmoveto} bdef /O {show} bdef /P {lineto} bdef /R {rlineto} bdef /S +{gsave show grestore} bdef /Z {gsave currentpoint lineto 20 setlinewidth 1 +setlinecap stroke grestore} bdef /CC {concat} bdef /CD {currentmatrix def} +bdef /CP {currentpoint pop def} bdef /FM {findfont exch makefont} bdef /FS +{findfont exch scalefont} bdef /GS {get setfont} bdef /MD {matrix def} bdef +/MN {mul neg def} bdef /RS {roll put setfont} bdef /SL {setlinewidth} bdef +/SM {setmatrix} bdef /L0 {[] 0 setdash} bdef /L1 {[40 100] 0 setdash} bdef +/L2 {[200 200] 0 setdash} bdef /L3 {[200 100 50 100] 0 setdash} bdef /L4 +{[300 100 50 100 50 100 50 100] 0 setdash} bdef /L5 {[400 200] 0 setdash} +bdef /IO { gsave 0.0 setgray /val exch def moveto /str 20 string def val str +cvs show grestore } bdef /UL { gsave currentpoint /cury exch def /curx exch +def /curmtx matrix currentmatrix def matrix identmatrix setmatrix newpath 0 +0 moveto true charpath flattenpath pathbbox curmtx setmatrix /height exch +def /width exch def /ybase cury height .1 mul sub def pop pop newpath curx +ybase moveto width 0 rlineto closepath stroke grestore } bdef /BG { gsave +currentpoint /cury exch def /curx exch def /curmtx matrix currentmatrix def +matrix identmatrix setmatrix newpath 0 0 moveto true charpath flattenpath +pathbbox curmtx setmatrix /height exch def /width exch def pop pop newpath 0 +0 moveto (M) true charpath flattenpath pathbbox /xdiff exch .1 mul def +/ydiff exch .1 mul def pop pop newpath curx xdiff 2 div sub cury ydiff 2 div +sub moveto width xdiff add 0 rlineto 0 height ydiff add rlineto width neg +xdiff sub 0 rlineto closepath fill grestore } bdef /$T_DICT 20 dict def /T { +$T_DICT begin /align exch def /orien exch def /size exch def /thestring exch +def gsave moveto /chsize 1.0 def /xsize 0.0 def /SUPER 8#330 def /SUBS 8#331 +def /NORM 8#332 def /SCRIPTWID 0.7 def /OFFSET 0.6 def orien rotate size dup +scale /orien false def thestring { /charcode exch def charcode SUPER eq +charcode SUBS eq or { /chsize SCRIPTWID def /orien true def } { charcode +NORM eq { /chsize 1.0 def /orien true def } { ( ) dup 0 charcode put +stringwidth pop chsize mul xsize add /xsize exch def } ifelse } ifelse } +forall xsize align mul neg 0 rmoveto orien { /regularfont currentfont def +/fractionfont currentfont [ SCRIPTWID 0 0 SCRIPTWID 0 0 ] makefont def gsave +newpath 0 0 moveto (1) true charpath flattenpath pathbbox /height exch def +pop pop pop grestore } if /xsize 0 def thestring { /charcode exch def +charcode SUPER eq { 0 OFFSET height mul dup /xsize exch def rmoveto +fractionfont setfont } { charcode SUBS eq { 0 OFFSET height mul neg dup +/xsize exch def rmoveto fractionfont setfont } { charcode NORM eq { 0 xsize +neg rmoveto regularfont setfont } { ( ) dup 0 charcode put show } ifelse } +ifelse } ifelse } forall grestore end } bdef /IsChar { exch /CharStrings get +exch known } bdef /MapCh { 3 -1 roll /Encoding get 3 1 roll put } bdef +/MapDegree { dup 16#b0 exch /degree IsChar { /degree } { /ring } ifelse +MapCh } bdef /MapBB { dup 16#a6 exch /brokenbar IsChar { /brokenbar } { /bar +} ifelse MapCh } bdef /ReEncode { dup findfont begin currentdict dup length +dict begin { 1 index /FID ne {def} {pop pop} ifelse } forall /FontName exch +def dup length 0 ne {/Encoding Encoding 256 array copy def 0 exch { dup type +/nametype eq { Encoding 2 index 2 index put pop 1 add } { exch pop } ifelse +} forall } if pop currentdict dup end end /FontName get exch definefont dup +MapDegree MapBB } bdef /RF { ISOLatin1Encoding exch ReEncode } bdef /Courier +RF /Courier-Bold RF /Courier-Oblique RF /Courier-BoldOblique RF /Helvetica +RF /Helvetica-Bold RF /Helvetica-Oblique RF /Helvetica-BoldOblique RF +/Helvetica-Narrow RF /Helvetica-Narrow-Bold RF /Helvetica-Narrow-Oblique RF +/Helvetica-Narrow-BoldOblique RF /AvantGarde-Book RF /AvantGarde-DemiOblique +RF /AvantGarde-Demi RF /AvantGarde-DemiOblique RF /Bookman-Demi RF +/Bookman-DemiItalic RF /Bookman-Light RF /Bookman-LightItalic RF +/ZapfChancery-MediumItalic RF /NewCenturySchlbk-Roman RF +/NewCenturySchlbk-Bold RF /NewCenturySchlbk-Italic RF +/NewCenturySchlbk-BoldItalic RF /Palatino-Roman RF /Palatino-Bold RF +/Palatino-Italic RF /Palatino-BoldItalic RF /Times-Roman RF /Times-Bold RF +/Times-Italic RF /Times-BoldItalic RF end +%%EndProlog +%%Page: 0 1 +%%BeginPageSetup +save $WAVE_DICT begin 54 56 L 0.028346 dup scale +%%PageBoundingBox: 54 56 507 622 +%%EndPageSetup +/psFontCache 500 array def /psStack 8 array def /psStackInd 0 def +/psTextWidth 0 def /psCurBase 0 def /ASW { stringwidth pop psCurBase add +/psCurBase exch def psCurBase psTextWidth gt +{ /psTextWidth psCurBase def } if } def /PSS { +psStack psStackInd psCurBase put /psStackInd psStackInd 1 add def } def +/PPS { /psStackInd psStackInd 1 sub def +/psCurBase psStack psStackInd get def } def 10.000000 SL L0 0.000 K +2983 19589 M 15 29 R 29 29 R 29 0 R 15 -14 R 14 -30 R 0 -43 R -29 -73 R D +3071 19633 M 0 -59 R -15 -58 R 0 -59 R D 3071 19603 M -29 -73 R 0 -43 R +14 -30 R 29 -14 R 30 0 R 29 14 R 29 30 R 15 43 R 29 117 R D 3188 19530 M +0 -43 R 14 -30 R 29 -14 R 30 0 R 29 14 R 29 30 R 29 43 R 15 59 R 0 58 R +-15 0 R 0 -14 R 15 -30 R D 3231 19647 M -29 -117 R 0 -73 R D 3217 19647 M +29 0 R -29 -102 R -15 -58 R D 3479 19511 M -54 -190 R 18 0 R D 3488 19511 M +-54 -190 R D 3452 19511 M 45 0 R -54 -190 R D 3461 19384 M 18 36 R 18 18 R +18 9 R 18 0 R 18 -9 R 9 -18 R 0 -27 R -18 -45 R D 3551 19438 M 0 -36 R +-9 -36 R 0 -36 R D 3551 19420 M -18 -45 R 0 -27 R 9 -18 R 9 -9 R 18 0 R +18 18 R 10 18 R D 3461 19511 M 27 -9 R D 3470 19511 M 9 -18 R D 3898 19808 M +-263 -468 R 14 0 R D 3898 19808 M 14 0 R -263 -468 R D 4175 19808 M +-29 -15 R -44 -29 R -44 -44 R -29 -44 R -29 -58 R -15 -58 R 0 -73 R 15 -59 R +15 -44 R 29 -44 R D 4073 19720 M -29 -44 R -29 -58 R -15 -73 R 0 -117 R D +4175 19808 M -44 -29 R -43 -44 R -30 -44 R -14 -29 R -15 -44 R -14 -58 R +-15 -132 R D 4000 19545 M 15 -132 R 14 -43 R 15 -30 R D 4190 19589 M 14 29 R +30 29 R 29 0 R 14 -14 R 15 -30 R 0 -43 R -29 -73 R D 4277 19633 M 0 -59 R +-14 -58 R 0 -59 R D 4277 19603 M -29 -73 R 0 -43 R 15 -30 R 29 -14 R 29 0 R +29 14 R 30 30 R 29 43 R 29 117 R D 4409 19530 M 0 -43 R 14 -30 R 15 -14 R +29 0 R 29 29 R 15 29 R D 4453 19647 M -30 -117 R 0 -73 R D 4438 19647 M +29 0 R -29 -102 R -15 -58 R D 4598 19511 M -9 -9 R 18 -91 R -9 -9 R 0 109 R +9 -9 R -18 -91 R 9 -9 R D 4553 19484 M 9 0 R 72 -55 R 9 0 R -90 55 R 0 -9 R +90 -37 R 0 -9 R D 4643 19484 M -9 0 R -72 -55 R -9 0 R 90 55 R 0 -9 R +-90 -37 R 0 -9 R D 4734 19804 M 0 9 R 9 0 R 0 -18 R -18 0 R 0 18 R 9 18 R +9 9 R 27 9 R 27 0 R 27 -9 R 9 -18 R 0 -18 R -9 -18 R -9 -9 R -18 -9 R +-27 -9 R D 4815 19840 M 9 -18 R 0 -18 R -9 -18 R -9 -9 R D 4797 19849 M +9 -9 R 9 -18 R 0 -18 R -9 -18 R -18 -18 R -18 -9 R -18 0 R D 4770 19759 M +27 -9 R 9 -9 R 9 -19 R 0 -27 R -9 -18 R -18 -9 R -27 -9 R -27 0 R -27 9 R +-10 9 R -9 18 R 0 18 R 19 0 R 0 -18 R -10 0 R 0 9 R D 4797 19741 M 9 -19 R +0 -27 R -9 -18 R D 4770 19759 M 18 -9 R 9 -19 R 0 -36 R -9 -18 R -9 -9 R +-18 -9 R D 5035 19706 M 0 -249 R 15 0 R D 5035 19706 M 15 0 R 0 -249 R D +4919 19589 M 248 0 R 0 -15 R D 4919 19589 M 0 -15 R 248 0 R D 5240 19589 M +15 29 R 29 29 R 29 0 R 15 -14 R 14 -30 R 0 -43 R -29 -73 R D 5328 19633 M +0 -59 R -15 -58 R 0 -59 R D 5328 19603 M -30 -73 R 0 -43 R 15 -30 R 29 -14 R +29 0 R 30 14 R 29 30 R 14 43 R 30 117 R D 5444 19530 M 0 -43 R 15 -30 R +29 -14 R 29 0 R 30 14 R 29 30 R 29 43 R 15 59 R 0 58 R -15 0 R 0 -14 R +15 -30 R D 5488 19647 M -29 -117 R 0 -73 R D 5474 19647 M 29 0 R -29 -102 R +-15 -58 R D 5736 19511 M -9 -9 R 18 -91 R -9 -9 R 0 109 R 9 -9 R -18 -91 R +9 -9 R D 5691 19484 M 9 0 R 72 -55 R 9 0 R -90 55 R 0 -9 R 90 -37 R 0 -9 R D +5781 19484 M -9 0 R -72 -55 R -9 0 R 90 55 R 0 -9 R -90 -37 R 0 -9 R D +5871 19804 M 0 9 R 9 0 R 0 -18 R -18 0 R 0 18 R 9 18 R 9 9 R 28 9 R 27 0 R +27 -9 R 9 -18 R 0 -18 R -9 -18 R -9 -9 R -18 -9 R -27 -9 R D 5953 19840 M +9 -18 R 0 -18 R -9 -18 R -9 -9 R D 5935 19849 M 9 -9 R 9 -18 R 0 -18 R +-9 -18 R -18 -18 R -18 -9 R -18 0 R D 5908 19759 M 27 -9 R 9 -9 R 9 -19 R +0 -27 R -9 -18 R -18 -9 R -27 -9 R -28 0 R -27 9 R -9 9 R -9 18 R 0 18 R +18 0 R 0 -18 R -9 0 R 0 9 R D 5935 19741 M 9 -19 R 0 -27 R -9 -18 R D +5908 19759 M 18 -9 R 9 -19 R 0 -36 R -9 -18 R -9 -9 R -18 -9 R D +6129 19808 M 30 -44 R 14 -44 R 15 -58 R 0 -73 R -15 -59 R -29 -58 R +-29 -44 R -44 -44 R -44 -29 R -29 -15 R D 6173 19720 M 0 -117 R -14 -73 R +-30 -58 R -29 -44 R D 6129 19808 M 15 -29 R 15 -44 R 14 -132 R D +6173 19720 M -14 -131 R -15 -59 R -15 -43 R -14 -30 R -29 -44 R -44 -43 R +-44 -30 R D 6340 19813 M -45 -154 R 18 0 R D 6367 19849 M -18 -36 R +-45 -154 R D 6367 19849 M -54 -190 R D 6367 19849 M -27 -27 R -27 -18 R +-18 -9 R D 6340 19813 M -18 -9 R -27 -9 R D 6602 19885 M -162 -289 R 9 0 R D +6602 19885 M 9 0 R -162 -289 R D 6693 19804 M 0 9 R 9 0 R 0 -18 R -18 0 R +0 18 R 9 18 R 9 9 R 27 9 R 27 0 R 27 -9 R 9 -18 R 0 -18 R -9 -18 R -9 -9 R +-18 -9 R -27 -9 R D 6774 19840 M 9 -18 R 0 -18 R -9 -18 R -9 -9 R D +6756 19849 M 9 -9 R 9 -18 R 0 -18 R -9 -18 R -18 -18 R -18 -9 R -18 0 R D +6729 19759 M 27 -9 R 9 -9 R 9 -19 R 0 -27 R -9 -18 R -18 -9 R -27 -9 R +-27 0 R -27 9 R -9 9 R -9 18 R 0 18 R 18 0 R 0 -18 R -9 0 R 0 9 R D +6756 19741 M 9 -19 R 0 -27 R -9 -18 R D 6729 19759 M 18 -9 R 9 -19 R 0 -36 R +-9 -18 R -9 -9 R -18 -9 R D 2220 11408 M 5114 0 R D 2220 11408 M 0 158 R D +2197 11196 M -35 -12 R -24 -35 R -11 -58 R 0 -35 R 11 -59 R 24 -35 R +35 -11 R 23 0 R 35 11 R 24 35 R 11 59 R 0 35 R -11 58 R -24 35 R -35 12 R +-23 0 R -24 -12 R -11 -11 R -12 -24 R -12 -58 R 0 -35 R 12 -59 R 12 -23 R +11 -12 R 24 -11 R D 2220 10951 M 23 11 R 12 12 R 12 23 R 12 59 R 0 35 R +-12 58 R -12 24 R -12 11 R -23 12 R D 3499 11408 M 0 158 R D 3428 11196 M +-23 -117 R 23 23 R 36 12 R 35 0 R 35 -12 R 23 -23 R 12 -35 R 0 -23 R +-12 -35 R -23 -24 R -35 -11 R -35 0 R -36 11 R -11 12 R -12 23 R 0 12 R +12 12 R 11 -12 R -11 -12 R D 3499 11114 M 23 -12 R 23 -23 R 12 -35 R 0 -23 R +-12 -35 R -23 -24 R -23 -11 R D 3428 11196 M 117 0 R D 3428 11184 M 59 0 R +58 12 R D 4777 11408 M 0 158 R D 4602 11149 M 23 12 R 35 35 R 0 -245 R D +4649 11184 M 0 -233 R D 4602 10951 M 105 0 R D 4871 11196 M -35 -12 R +-24 -35 R -12 -58 R 0 -35 R 12 -59 R 24 -35 R 35 -11 R 23 0 R 35 11 R +23 35 R 12 59 R 0 35 R -12 58 R -23 35 R -35 12 R -23 0 R -24 -12 R +-11 -11 R -12 -24 R -12 -58 R 0 -35 R 12 -59 R 12 -23 R 11 -12 R 24 -11 R D +4894 10951 M 23 11 R 12 12 R 12 23 R 11 59 R 0 35 R -11 58 R -12 24 R +-12 11 R -23 12 R D 6056 11408 M 0 158 R D 5880 11149 M 24 12 R 35 35 R +0 -245 R D 5927 11184 M 0 -233 R D 5880 10951 M 105 0 R D 6102 11196 M +-23 -117 R 23 23 R 35 12 R 35 0 R 35 -12 R 24 -23 R 12 -35 R 0 -23 R +-12 -35 R -24 -24 R -35 -11 R -35 0 R -35 11 R -11 12 R -12 23 R 0 12 R +12 12 R 11 -12 R -11 -12 R D 6172 11114 M 24 -12 R 23 -23 R 12 -35 R 0 -23 R +-12 -35 R -23 -24 R -24 -11 R D 6102 11196 M 117 0 R D 6102 11184 M 59 0 R +58 12 R D 7334 11408 M 0 158 R D 7135 11149 M 12 -12 R -12 -11 R -11 11 R +0 12 R 11 24 R 12 11 R 35 12 R 47 0 R 35 -12 R 12 -11 R 11 -24 R 0 -23 R +-11 -24 R -35 -23 R -59 -23 R -23 -12 R -24 -23 R -11 -35 R 0 -35 R D +7229 11196 M 23 -12 R 12 -11 R 12 -24 R 0 -23 R -12 -24 R -35 -23 R +-47 -23 R D 7124 10974 M 11 12 R 24 0 R 58 -24 R 35 0 R 24 12 R 11 12 R D +7159 10986 M 58 -35 R 47 0 R 12 11 R 11 24 R 0 23 R D 7428 11196 M -35 -12 R +-24 -35 R -12 -58 R 0 -35 R 12 -59 R 24 -35 R 35 -11 R 23 0 R 35 11 R +23 35 R 12 59 R 0 35 R -12 58 R -23 35 R -35 12 R -23 0 R -24 -12 R +-11 -11 R -12 -24 R -12 -58 R 0 -35 R 12 -59 R 12 -23 R 11 -12 R 24 -11 R D +7451 10951 M 23 11 R 12 12 R 12 23 R 11 59 R 0 35 R -11 58 R -12 24 R +-12 11 R -23 12 R D 2476 11408 M 0 79 R D 2731 11408 M 0 79 R D 2987 11408 M +0 79 R D 3243 11408 M 0 79 R D 3754 11408 M 0 79 R D 4010 11408 M 0 79 R D +4266 11408 M 0 79 R D 4521 11408 M 0 79 R D 5033 11408 M 0 79 R D +5288 11408 M 0 79 R D 5544 11408 M 0 79 R D 5800 11408 M 0 79 R D +6311 11408 M 0 79 R D 6567 11408 M 0 79 R D 6823 11408 M 0 79 R D +7078 11408 M 0 79 R D 4213 10598 M 210 0 R D 4680 10668 M -12 -23 R +-23 -24 R -94 -70 R -23 -23 R -12 -23 R D 4668 10645 M -105 0 R -23 -12 R +-12 -23 R D 4645 10645 M -47 11 R -35 0 R -12 -11 R D 4645 10645 M -47 23 R +-35 0 R -23 -23 R -12 -35 R D 4528 10528 M 105 0 R 24 12 R 11 23 R D +4551 10528 M 47 -12 R 35 0 R 12 12 R D 4551 10528 M 47 -23 R 35 0 R 24 23 R +11 35 R D 4773 10559 M -43 -152 R 14 0 R D 4780 10559 M -43 -152 R D +4751 10559 M 36 0 R -43 -152 R D 4758 10458 M 15 29 R 14 14 R 15 7 R 14 0 R +15 -7 R 7 -14 R 0 -22 R -14 -36 R D 4831 10501 M 0 -29 R -7 -29 R 0 -29 R D +4831 10487 M -15 -36 R 0 -22 R 8 -15 R 7 -7 R 14 0 R 15 15 R 7 14 R D +4758 10559 M 22 -7 R D 4766 10559 M 7 -14 R D 5108 10797 M -210 -374 R +11 0 R D 5108 10797 M 12 0 R -211 -374 R D 5248 10750 M -70 -245 R D +5260 10750 M -70 -245 R D 5271 10750 M -70 -245 R D 5213 10750 M 93 0 R D +5143 10505 M 175 0 R 24 70 R D 5225 10750 M 35 -12 R D 5236 10750 M 12 -23 R +D 5283 10750 M -23 -23 R D 5295 10750 M -35 -12 R D 5190 10516 M -35 -11 R D +5190 10528 M -24 -23 R D 5201 10528 M 12 -23 R D 5190 10516 M 35 -11 R D +5260 10505 M 58 11 R D 5283 10505 M 47 35 R D 5306 10505 M 36 70 R D +2220 19296 M 5114 0 R D 2220 19296 M 0 -158 R D 3499 19296 M 0 -158 R D +4777 19296 M 0 -158 R D 6056 19296 M 0 -158 R D 7334 19296 M 0 -158 R D +2476 19296 M 0 -79 R D 2731 19296 M 0 -79 R D 2987 19296 M 0 -79 R D +3243 19296 M 0 -79 R D 3754 19296 M 0 -79 R D 4010 19296 M 0 -79 R D +4266 19296 M 0 -79 R D 4521 19296 M 0 -79 R D 5033 19296 M 0 -79 R D +5288 19296 M 0 -79 R D 5544 19296 M 0 -79 R D 5800 19296 M 0 -79 R D +6311 19296 M 0 -79 R D 6567 19296 M 0 -79 R D 6823 19296 M 0 -79 R D +7078 19296 M 0 -79 R D 2220 11408 M 0 7888 R D 2220 11408 M 102 0 R D +1618 11653 M -35 -11 R -23 -35 R -12 -59 R 0 -35 R 12 -58 R 23 -35 R +35 -12 R 24 0 R 35 12 R 23 35 R 12 58 R 0 35 R -12 59 R -23 35 R -35 11 R +-24 0 R -23 -11 R -12 -12 R -11 -23 R -12 -59 R 0 -35 R 12 -58 R 11 -24 R +12 -11 R 23 -12 R D 1642 11408 M 23 12 R 12 11 R 11 24 R 12 58 R 0 35 R +-12 59 R -11 23 R -12 12 R -23 11 R D 1805 11431 M -11 -11 R 11 -12 R +12 12 R -12 11 R D 1969 11653 M -35 -11 R -24 -35 R -11 -59 R 0 -35 R +11 -58 R 24 -35 R 35 -12 R 23 0 R 35 12 R 24 35 R 11 58 R 0 35 R -11 59 R +-24 35 R -35 11 R -23 0 R -23 -11 R -12 -12 R -12 -23 R -12 -59 R 0 -35 R +12 -58 R 12 -24 R 12 -11 R 23 -12 R D 1992 11408 M 24 12 R 11 11 R 12 24 R +12 58 R 0 35 R -12 59 R -12 23 R -11 12 R -24 11 R D 2220 13380 M 102 0 R D +1618 13520 M -35 -12 R -23 -35 R -12 -58 R 0 -35 R 12 -59 R 23 -35 R +35 -12 R 24 0 R 35 12 R 23 35 R 12 59 R 0 35 R -12 58 R -23 35 R -35 12 R +-24 0 R -23 -12 R -12 -11 R -11 -24 R -12 -58 R 0 -35 R 12 -59 R 11 -23 R +12 -12 R 23 -12 R D 1642 13274 M 23 12 R 12 12 R 11 23 R 12 59 R 0 35 R +-12 58 R -11 24 R -12 11 R -23 12 R D 1805 13298 M -11 -12 R 11 -12 R +12 12 R -12 12 R D 1922 13520 M -23 -117 R 23 23 R 35 12 R 35 0 R 35 -12 R +24 -23 R 11 -35 R 0 -23 R -11 -35 R -24 -24 R -35 -12 R -35 0 R -35 12 R +-12 12 R -11 23 R 0 12 R 11 12 R 12 -12 R -12 -12 R D 1992 13438 M 24 -12 R +23 -23 R 12 -35 R 0 -23 R -12 -35 R -23 -24 R -24 -12 R D 1922 13520 M +117 0 R D 1922 13508 M 59 0 R 58 12 R D 2220 15352 M 102 0 R D 1583 15445 M +24 12 R 35 35 R 0 -246 R D 1630 15480 M 0 -234 R D 1583 15246 M 105 0 R D +1805 15270 M -11 -12 R 11 -12 R 12 12 R -12 12 R D 1969 15492 M -35 -12 R +-24 -35 R -11 -58 R 0 -35 R 11 -59 R 24 -35 R 35 -12 R 23 0 R 35 12 R +24 35 R 11 59 R 0 35 R -11 58 R -24 35 R -35 12 R -23 0 R -23 -12 R +-12 -11 R -12 -24 R -12 -58 R 0 -35 R 12 -59 R 12 -23 R 12 -12 R 23 -12 R D +1992 15246 M 24 12 R 11 12 R 12 23 R 12 59 R 0 35 R -12 58 R -12 24 R +-11 11 R -24 12 R D 2220 17324 M 102 0 R D 1583 17417 M 24 12 R 35 35 R +0 -246 R D 1630 17452 M 0 -234 R D 1583 17218 M 105 0 R D 1805 17242 M +-11 -12 R 11 -12 R 12 12 R -12 12 R D 1922 17464 M -23 -117 R 23 23 R +35 12 R 35 0 R 35 -12 R 24 -23 R 11 -35 R 0 -23 R -11 -35 R -24 -24 R +-35 -12 R -35 0 R -35 12 R -12 12 R -11 23 R 0 12 R 11 12 R 12 -12 R +-12 -12 R D 1992 17382 M 24 -12 R 23 -23 R 12 -35 R 0 -23 R -12 -35 R +-23 -24 R -24 -12 R D 1922 17464 M 117 0 R D 1922 17452 M 59 0 R 58 12 R D +2220 19296 M 102 0 R D 1560 19213 M 12 -12 R -12 -11 R -12 11 R 0 12 R +12 24 R 12 11 R 35 12 R 46 0 R 35 -12 R 12 -11 R 12 -24 R 0 -23 R -12 -24 R +-35 -23 R -58 -23 R -24 -12 R -23 -23 R -12 -35 R 0 -35 R D 1653 19260 M +24 -12 R 11 -11 R 12 -24 R 0 -23 R -12 -24 R -35 -23 R -46 -23 R D +1548 19038 M 12 12 R 23 0 R 59 -24 R 35 0 R 23 12 R 12 12 R D 1583 19050 M +59 -35 R 46 0 R 12 11 R 12 24 R 0 23 R D 1805 19038 M -11 -12 R 11 -11 R +12 11 R -12 12 R D 1969 19260 M -35 -12 R -24 -35 R -11 -58 R 0 -35 R +11 -59 R 24 -35 R 35 -11 R 23 0 R 35 11 R 24 35 R 11 59 R 0 35 R -11 58 R +-24 35 R -35 12 R -23 0 R -23 -12 R -12 -11 R -12 -24 R -12 -58 R 0 -35 R +12 -59 R 12 -23 R 12 -12 R 23 -11 R D 1992 19015 M 24 11 R 11 12 R 12 23 R +12 59 R 0 35 R -12 58 R -12 24 R -11 11 R -24 12 R D 2220 11802 M 51 0 R D +2220 12197 M 51 0 R D 2220 12591 M 51 0 R D 2220 12986 M 51 0 R D +2220 13774 M 51 0 R D 2220 14169 M 51 0 R D 2220 14563 M 51 0 R D +2220 14958 M 51 0 R D 2220 15746 M 51 0 R D 2220 16141 M 51 0 R D +2220 16535 M 51 0 R D 2220 16930 M 51 0 R D 2220 17719 M 51 0 R D +2220 18113 M 51 0 R D 2220 18507 M 51 0 R D 2220 18902 M 51 0 R D +7334 11408 M 0 7888 R D 7334 11408 M -102 0 R D 7334 13380 M -102 0 R D +7334 15352 M -102 0 R D 7334 17324 M -102 0 R D 7334 19296 M -102 0 R D +7334 11802 M -51 0 R D 7334 12197 M -51 0 R D 7334 12591 M -51 0 R D +7334 12986 M -51 0 R D 7334 13774 M -51 0 R D 7334 14169 M -51 0 R D +7334 14563 M -51 0 R D 7334 14958 M -51 0 R D 7334 15746 M -51 0 R D +7334 16141 M -51 0 R D 7334 16535 M -51 0 R D 7334 16930 M -51 0 R D +7334 17719 M -51 0 R D 7334 18113 M -51 0 R D 7334 18507 M -51 0 R D +7334 18902 M -51 0 R D 2230 14882 M 11 -103 R 10 -65 R 10 -47 R 10 -36 R +10 -28 R 11 -22 R 10 -19 R 10 -15 R 10 -12 R 11 -10 R 10 -9 R 10 -7 R +10 -6 R 10 -5 R 11 -4 R 10 -3 R 10 -2 R 10 -2 R 11 -2 R 10 -1 R 10 -1 R +10 0 R 11 0 R 10 0 R 10 1 R 10 1 R 10 1 R 11 1 R 10 1 R 10 1 R 10 2 R 11 1 R +10 2 R 10 2 R 10 2 R 11 2 R 10 2 R 10 2 R 10 2 R 10 2 R 11 2 R 10 3 R 10 2 R +10 2 R 11 3 R 10 2 R 10 2 R 10 3 R 10 2 R 11 3 R 10 2 R 10 2 R 10 3 R 11 2 R +10 3 R 10 2 R 10 3 R 11 2 R 10 3 R 10 2 R 10 3 R 10 2 R 11 2 R 10 3 R 10 2 R +10 3 R 11 2 R 10 3 R 10 2 R 10 3 R 10 2 R 11 2 R 10 3 R 10 2 R 10 3 R 11 2 R +10 2 R 10 3 R 10 2 R 11 2 R 10 3 R 10 2 R 10 2 R 10 3 R 11 2 R 10 2 R 10 2 R +10 3 R 11 2 R 10 2 R 10 2 R 10 3 R 10 2 R 11 2 R 10 2 R 10 2 R 10 3 R 11 2 R +10 2 R 10 2 R 10 2 R 11 2 R 10 3 R 10 2 R 10 2 R 10 2 R 11 2 R 10 2 R 10 2 R +10 2 R 11 2 R 10 2 R 10 2 R 10 2 R 11 2 R 10 2 R 10 2 R 10 2 R 10 2 R 11 2 R +10 2 R 10 2 R 10 2 R 11 2 R 10 2 R 10 2 R 10 2 R 10 2 R 11 1 R 10 2 R 10 2 R +10 2 R 11 2 R 10 2 R 10 2 R 10 1 R 11 2 R 10 2 R 10 2 R 10 2 R 10 2 R 11 1 R +10 2 R 10 2 R 10 2 R 11 1 R 10 2 R 10 2 R 10 2 R 10 1 R 11 2 R 10 2 R 10 1 R +10 2 R 11 2 R 10 1 R 10 2 R 10 2 R 11 1 R 10 2 R 10 2 R 10 1 R 10 2 R 11 2 R +10 1 R 10 2 R 10 2 R 11 1 R 10 2 R 10 1 R 10 2 R 11 2 R 10 1 R 10 2 R 10 1 R +10 2 R 11 1 R 10 2 R 10 1 R 10 2 R 11 1 R 10 2 R 10 1 R 10 2 R 10 1 R 11 2 R +10 1 R 10 2 R 10 1 R 11 2 R 10 1 R 10 2 R 10 1 R 11 2 R 10 1 R 10 2 R 10 1 R +10 2 R 11 1 R 10 1 R 10 2 R 10 1 R 11 2 R 10 1 R 10 1 R 10 2 R 10 1 R 11 2 R +10 1 R 10 1 R 10 2 R 11 1 R 10 1 R 10 2 R 10 1 R 11 1 R 10 2 R 10 1 R 10 1 R +10 2 R 11 1 R 10 1 R 10 2 R 10 1 R 11 1 R 10 2 R 10 1 R 10 1 R 11 2 R 10 1 R +10 1 R 10 1 R 10 2 R 11 1 R 10 1 R 10 1 R 10 2 R 11 1 R 10 1 R 10 1 R 10 2 R +10 1 R 11 1 R 10 1 R 10 2 R 10 1 R 11 1 R 10 1 R 10 1 R 10 2 R 11 1 R 10 1 R +10 1 R 10 1 R 10 2 R 11 1 R 10 1 R 10 1 R 10 1 R 11 2 R 10 1 R 10 1 R 10 1 R +10 1 R 11 1 R 10 2 R 10 1 R 10 1 R 11 1 R 10 1 R 10 1 R 10 1 R 11 1 R 10 2 R +10 1 R 10 1 R 10 1 R 11 1 R 10 1 R 10 1 R 10 1 R 11 2 R 10 1 R 10 1 R 10 1 R +10 1 R 11 1 R 10 1 R 10 1 R 10 1 R 11 1 R 10 1 R 10 1 R 10 1 R 11 2 R 10 1 R +10 1 R 10 1 R 10 1 R 11 1 R 10 1 R 10 1 R 10 1 R 11 1 R 10 1 R 10 1 R 10 1 R +11 1 R 10 1 R 10 1 R 10 1 R 10 1 R 11 1 R 10 1 R 10 1 R 10 1 R 11 1 R 10 1 R +10 1 R 10 1 R 10 1 R 11 1 R 10 1 R 10 1 R 10 1 R 11 1 R 10 1 R 10 1 R 10 1 R +11 1 R 10 1 R 10 1 R 10 1 R 10 1 R 11 1 R 10 1 R 10 1 R 10 1 R 11 1 R 10 1 R +10 1 R 10 0 R 10 1 R 11 1 R 10 1 R 10 1 R 10 1 R 11 1 R 10 1 R 10 1 R 10 1 R +11 1 R 10 1 R 10 1 R 10 1 R 10 0 R 11 1 R 10 1 R 10 1 R 10 1 R 11 1 R 10 1 R +10 1 R 10 1 R 11 1 R 10 0 R 10 1 R 10 1 R 10 1 R 11 1 R 10 1 R 10 1 R 10 1 R +11 1 R 10 0 R 10 1 R 10 1 R 10 1 R 11 1 R 10 1 R 10 1 R 10 0 R 11 1 R 10 1 R +10 1 R 10 1 R 11 1 R 10 1 R 10 0 R 10 1 R 10 1 R 11 1 R 10 1 R 10 1 R 10 0 R +11 1 R 10 1 R 10 1 R 10 1 R 10 1 R 11 0 R 10 1 R 10 1 R 10 1 R 11 1 R 10 1 R +10 0 R 10 1 R 11 1 R 10 1 R 10 1 R 10 0 R 10 1 R 11 1 R 10 1 R 10 1 R 10 0 R +11 1 R 10 1 R 10 1 R 10 1 R 11 0 R 10 1 R 10 1 R 10 1 R 10 1 R 11 0 R 10 1 R +10 1 R 10 1 R 11 1 R 10 0 R 10 1 R 10 1 R 10 1 R 11 0 R 10 1 R 10 1 R 10 1 R +11 0 R 10 1 R 10 1 R 10 1 R 11 1 R 10 0 R 10 1 R 10 1 R 10 1 R 11 0 R 10 1 R +10 1 R 10 1 R 11 0 R 10 1 R 10 1 R 10 0 R 10 1 R 11 1 R 10 1 R 10 0 R 10 1 R +11 1 R 10 1 R 10 0 R 10 1 R 11 1 R 10 1 R 10 0 R 10 1 R 10 1 R 11 0 R 10 1 R +10 1 R 10 1 R 11 0 R 10 1 R 10 1 R 10 0 R 11 1 R 10 1 R 10 1 R 10 0 R 10 1 R +11 1 R 10 0 R 10 1 R 10 1 R 11 1 R 10 0 R 10 1 R 10 1 R 10 0 R 11 1 R 10 1 R +10 0 R 10 1 R 11 1 R 10 0 R 10 1 R D L1 2230 16565 M 11 -87 R 10 -74 R +10 -65 R 10 -56 R 10 -49 R 11 -43 R 10 -38 R 10 -33 R 10 -30 R 11 -26 R +10 -23 R 10 -21 R 10 -18 R 10 -16 R 11 -14 R 10 -12 R 10 -11 R 10 -9 R +11 -8 R 10 -7 R 10 -5 R 10 -5 R 11 -4 R 10 -3 R 10 -2 R 10 -1 R 10 -1 R +11 0 R 10 1 R 10 1 R 10 1 R 11 2 R 10 2 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10 0 R 10 -1 R 10 0 R 11 0 R +10 0 R 10 0 R 10 0 R 11 0 R 10 -1 R 10 0 R 10 0 R 10 0 R 11 0 R 10 0 R +10 0 R 10 -1 R 11 0 R 10 0 R 10 0 R 10 0 R 11 0 R 10 0 R 10 0 R 10 -1 R +10 0 R 11 0 R 10 0 R 10 0 R 10 0 R 11 0 R 10 0 R 10 0 R 10 -1 R 10 0 R +11 0 R 10 0 R 10 0 R 10 0 R 11 0 R 10 0 R 10 -1 R 10 0 R 11 0 R 10 0 R +10 0 R 10 0 R 10 0 R 11 0 R 10 0 R 10 -1 R 10 0 R 11 0 R 10 0 R 10 0 R +10 0 R 10 0 R 11 0 R 10 0 R 10 0 R 10 -1 R 11 0 R 10 0 R 10 0 R 10 0 R +11 0 R 10 0 R 10 0 R 10 0 R 10 0 R 11 -1 R 10 0 R 10 0 R 10 0 R 11 0 R +10 0 R 10 0 R 10 0 R 11 0 R 10 0 R 10 -1 R 10 0 R 10 0 R 11 0 R 10 0 R +10 0 R 10 0 R 11 0 R 10 0 R 10 0 R 10 0 R 10 -1 R 11 0 R 10 0 R 10 0 R +10 0 R 11 0 R 10 0 R 10 0 R 10 0 R 11 0 R 10 0 R 10 0 R 10 -1 R 10 0 R +11 0 R 10 0 R 10 0 R 10 0 R 11 0 R 10 0 R 10 0 R 10 0 R 10 0 R 11 0 R +10 -1 R 10 0 R 10 0 R 11 0 R 10 0 R 10 0 R 10 0 R 11 0 R 10 0 R 10 0 R +10 0 R 10 0 R 11 -1 R 10 0 R 10 0 R 10 0 R 11 0 R 10 0 R 10 0 R 10 0 R +11 0 R 10 0 R 10 0 R 10 0 R 10 0 R 11 0 R 10 -1 R 10 0 R 10 0 R 11 0 R +10 0 R 10 0 R 10 0 R 10 0 R 11 0 R 10 0 R 10 0 R 10 0 R 11 0 R 10 0 R +10 -1 R D L0 10924 19589 M 15 29 R 29 29 R 29 0 R 15 -14 R 15 -30 R 0 -43 R +-30 -73 R D 11012 19633 M 0 -59 R -15 -58 R 0 -59 R D 11012 19603 M +-29 -73 R 0 -43 R 14 -30 R 30 -14 R 29 0 R 29 14 R 29 30 R 15 43 R 29 117 R +D 11129 19530 M 0 -43 R 14 -30 R 30 -14 R 29 0 R 29 14 R 29 30 R 29 43 R +15 59 R 0 58 R -15 0 R 0 -14 R 15 -30 R D 11173 19647 M -30 -117 R 0 -73 R D +11158 19647 M 29 0 R -29 -102 R -15 -58 R D 11357 19411 M 9 18 R 18 18 R +18 0 R 9 -9 R 9 -18 R 0 -27 R -18 -72 R D 11411 19438 M 0 -45 R -18 -72 R D +11411 19420 M -9 -36 R -18 -63 R 18 0 R D 11420 19393 M 18 27 R 18 18 R +18 9 R 19 0 R 18 -9 R 9 -18 R 0 -27 R -18 -72 R D 11511 19438 M 0 -45 R +-18 -72 R D 11511 19420 M -9 -36 R -18 -63 R 18 0 R D 11520 19393 M 18 27 R +18 18 R 18 9 R 18 0 R 18 -9 R 9 -18 R 0 -27 R -18 -45 R D 11610 19438 M +0 -36 R -9 -36 R 0 -36 R D 11610 19420 M -18 -45 R 0 -27 R 9 -18 R 9 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11 1 R 10 2 R 10 1 R 10 2 R +11 2 R 10 1 R 10 2 R 10 1 R 10 1 R 11 2 R 10 1 R 10 2 R 10 1 R 11 2 R 10 1 R +10 1 R 10 2 R 11 1 R 10 2 R 10 1 R 10 1 R 10 2 R 11 1 R 10 1 R 10 1 R 10 2 R +11 1 R 10 1 R 10 1 R 10 2 R 10 1 R 11 1 R 10 1 R 10 2 R 10 1 R 11 1 R 10 1 R +10 1 R 10 1 R 11 2 R 10 1 R 10 1 R 10 1 R 10 1 R 11 1 R 10 1 R 10 1 R 10 1 R +11 1 R 10 2 R 10 1 R 10 1 R 10 1 R 11 1 R 10 1 R 10 1 R 10 1 R 11 1 R 10 1 R +10 1 R 10 1 R 11 1 R 10 1 R 10 1 R 10 1 R 10 1 R 11 1 R 10 0 R 10 1 R 10 1 R +11 1 R 10 1 R 10 1 R 10 1 R 10 1 R 11 1 R 10 1 R 10 0 R 10 1 R 11 1 R 10 1 R +10 1 R 10 1 R 11 1 R 10 0 R 10 1 R 10 1 R 10 1 R 11 1 R 10 1 R 10 0 R 10 1 R +11 1 R 10 1 R 10 1 R 10 0 R 11 1 R 10 1 R 10 1 R 10 0 R 10 1 R 11 1 R 10 1 R +10 0 R 10 1 R 11 1 R 10 1 R 10 0 R 10 1 R 10 1 R 11 0 R 10 1 R 10 1 R 10 1 R +11 0 R 10 1 R 10 1 R 10 0 R 11 1 R 10 1 R 10 0 R 10 1 R 10 1 R 11 0 R 10 1 R +10 1 R 10 0 R 11 1 R 10 1 R 10 0 R 10 1 R 10 1 R 11 0 R 10 1 R 10 0 R 10 1 R +11 1 R 10 0 R 10 1 R 10 0 R 11 1 R 10 1 R 10 0 R 10 1 R 10 0 R 11 1 R 10 1 R +10 0 R 10 1 R 11 0 R 10 1 R 10 0 R 10 1 R 11 1 R 10 0 R 10 1 R 10 0 R 10 1 R +11 0 R 10 1 R 10 0 R 10 1 R 11 1 R 10 0 R 10 1 R 10 0 R 10 1 R 11 0 R 10 1 R +10 0 R 10 1 R 11 0 R 10 1 R 10 0 R 10 1 R 11 0 R 10 1 R 10 0 R 10 1 R 10 0 R +11 1 R 10 0 R 10 1 R 10 0 R 11 1 R 10 0 R 10 1 R 10 0 R 10 1 R 11 0 R 10 0 R +10 1 R 10 0 R 11 1 R 10 0 R 10 1 R 10 0 R 11 1 R 10 0 R 10 1 R 10 0 R 10 0 R +11 1 R 10 0 R 10 1 R 10 0 R 11 1 R 10 0 R 10 0 R 10 1 R 11 0 R 10 1 R 10 0 R +10 1 R 10 0 R 11 0 R 10 1 R 10 0 R 10 1 R 11 0 R 10 0 R 10 1 R 10 0 R 10 1 R +11 0 R 10 0 R 10 1 R 10 0 R 11 1 R 10 0 R 10 0 R 10 1 R 11 0 R 10 1 R 10 0 R +10 0 R 10 1 R 11 0 R 10 0 R 10 1 R 10 0 R 11 1 R 10 0 R 10 0 R 10 1 R 10 0 R +11 0 R 10 1 R 10 0 R 10 0 R 11 1 R 10 0 R 10 0 R 10 1 R 11 0 R 10 1 R 10 0 R +10 0 R 10 1 R 11 0 R 10 0 R 10 1 R 10 0 R 11 0 R 10 1 R 10 0 R 10 0 R 11 1 R +10 0 R 10 0 R 10 1 R 10 0 R 11 0 R 10 1 R 10 0 R 10 0 R 11 1 R 10 0 R 10 0 R +10 0 R 10 1 R 11 0 R 10 0 R 10 1 R 10 0 R 11 0 R 10 1 R 10 0 R D L1 +10230 3488 M 11 519 R 10 353 R 10 273 R 10 223 R 10 189 R 11 164 R 10 146 R +10 130 R 10 117 R 11 107 R 10 98 R 10 90 R 10 84 R 10 78 R 11 72 R 10 68 R +10 64 R 10 61 R 11 56 R 10 54 R 10 51 R 10 49 R 11 46 R 10 43 R 10 42 R +10 40 R 10 38 R 11 37 R 10 35 R 10 34 R 10 32 R 11 31 R 10 30 R 10 29 R +10 27 R 11 27 R 10 26 R 10 25 R 10 24 R 10 23 R 11 23 R 10 22 R 10 21 R +10 20 R 11 20 R 10 20 R 10 18 R 10 19 R 10 17 R 11 18 R 10 16 R 10 17 R +10 16 R 11 15 R 10 15 R 10 15 R 10 14 R 11 14 R 10 14 R 10 13 R 10 13 R +10 13 R 11 12 R 10 12 R 10 12 R 10 11 R 11 12 R 10 11 R 10 10 R 10 11 R +10 10 R 11 10 R 10 10 R 10 10 R 10 10 R 11 9 R 10 9 R 10 9 R 10 9 R 11 8 R +10 9 R 10 8 R 10 8 R 10 8 R 11 8 R 10 8 R 10 7 R 10 8 R 11 7 R 10 7 R 10 7 R +10 7 R 10 7 R 11 7 R 10 6 R 10 7 R 10 6 R 11 6 R 10 7 R 10 6 R 10 6 R 11 5 R +10 6 R 10 6 R 10 6 R 10 5 R 11 5 R 10 6 R 10 5 R 10 5 R 11 5 R 10 5 R 10 5 R +10 5 R 11 5 R 10 5 R 10 5 R 10 4 R 10 5 R 11 4 R 10 5 R 10 4 R 10 4 R 11 5 R +10 4 R 10 4 R 10 4 R 10 4 R 11 4 R 10 4 R 10 4 R 10 4 R 11 3 R 10 4 R 10 4 R +10 3 R 11 4 R 10 4 R 10 3 R 10 4 R 10 3 R 11 3 R 10 4 R 10 3 R 10 3 R 11 4 R +10 3 R 10 3 R 10 3 R 10 3 R 11 3 R 10 3 R 10 3 R 10 3 R 11 3 R 10 3 R 10 3 R +10 3 R 11 2 R 10 3 R 10 3 R 10 3 R 10 2 R 11 3 R 10 2 R 10 3 R 10 3 R 11 2 R +10 3 R 10 2 R 10 3 R 11 2 R 10 2 R 10 3 R 10 2 R 10 3 R 11 2 R 10 2 R 10 2 R +10 3 R 11 2 R 10 2 R 10 2 R 10 2 R 10 3 R 11 2 R 10 2 R 10 2 R 10 2 R 11 2 R +10 2 R 10 2 R 10 2 R 11 2 R 10 2 R 10 2 R 10 2 R 10 2 R 11 2 R 10 1 R 10 2 R +10 2 R 11 2 R 10 2 R 10 2 R 10 1 R 10 2 R 11 2 R 10 2 R 10 1 R 10 2 R 11 2 R +10 1 R 10 2 R 10 2 R 11 1 R 10 2 R 10 1 R 10 2 R 10 2 R 11 1 R 10 2 R 10 1 R +10 2 R 11 1 R 10 2 R 10 1 R 10 2 R 11 1 R 10 2 R 10 1 R 10 2 R 10 1 R 11 1 R +10 2 R 10 1 R 10 2 R 11 1 R 10 1 R 10 2 R 10 1 R 10 1 R 11 2 R 10 1 R 10 1 R +10 1 R 11 2 R 10 1 R 10 1 R 10 2 R 11 1 R 10 1 R 10 1 R 10 1 R 10 2 R 11 1 R +10 1 R 10 1 R 10 1 R 11 2 R 10 1 R 10 1 R 10 1 R 10 1 R 11 1 R 10 1 R 10 1 R +10 2 R 11 1 R 10 1 R 10 1 R 10 1 R 11 1 R 10 1 R 10 1 R 10 1 R 10 1 R 11 1 R +10 1 R 10 1 R 10 1 R 11 1 R 10 1 R 10 1 R 10 1 R 10 1 R 11 1 R 10 1 R 10 1 R +10 1 R 11 1 R 10 1 R 10 1 R 10 1 R 11 1 R 10 1 R 10 1 R 10 0 R 10 1 R 11 1 R +10 1 R 10 1 R 10 1 R 11 1 R 10 1 R 10 1 R 10 0 R 11 1 R 10 1 R 10 1 R 10 1 R +10 1 R 11 0 R 10 1 R 10 1 R 10 1 R 11 1 R 10 1 R 10 0 R 10 1 R 10 1 R 11 1 R +10 0 R 10 1 R 10 1 R 11 1 R 10 1 R 10 0 R 10 1 R 11 1 R 10 1 R 10 0 R 10 1 R +10 1 R 11 1 R 10 0 R 10 1 R 10 1 R 11 0 R 10 1 R 10 1 R 10 1 R 10 0 R 11 1 R +10 1 R 10 0 R 10 1 R 11 1 R 10 0 R 10 1 R 10 1 R 11 0 R 10 1 R 10 1 R 10 0 R +10 1 R 11 1 R 10 0 R 10 1 R 10 1 R 11 0 R 10 1 R 10 0 R 10 1 R 11 1 R 10 0 R +10 1 R 10 1 R 10 0 R 11 1 R 10 0 R 10 1 R 10 1 R 11 0 R 10 1 R 10 0 R 10 1 R +10 1 R 11 0 R 10 1 R 10 0 R 10 1 R 11 0 R 10 1 R 10 1 R 10 0 R 11 1 R 10 0 R +10 1 R 10 0 R 10 1 R 11 0 R 10 1 R 10 1 R 10 0 R 11 1 R 10 0 R 10 1 R 10 0 R +10 1 R 11 0 R 10 1 R 10 0 R 10 1 R 11 0 R 10 1 R 10 0 R 10 1 R 11 0 R 10 1 R +10 0 R 10 1 R 10 0 R 11 1 R 10 0 R 10 1 R 10 0 R 11 1 R 10 0 R 10 1 R 10 0 R +11 1 R 10 0 R 10 1 R 10 0 R 10 1 R 11 0 R 10 0 R 10 1 R 10 0 R 11 1 R 10 0 R +10 1 R 10 0 R 10 1 R 11 0 R 10 1 R 10 0 R 10 0 R 11 1 R 10 0 R 10 1 R 10 0 R +11 1 R 10 0 R 10 0 R 10 1 R 10 0 R 11 1 R 10 0 R 10 1 R 10 0 R 11 0 R 10 1 R +10 0 R 10 1 R 10 0 R 11 0 R 10 1 R 10 0 R 10 1 R 11 0 R 10 0 R 10 1 R 10 0 R +11 1 R 10 0 R 10 0 R 10 1 R 10 0 R 11 1 R 10 0 R 10 0 R 10 1 R 11 0 R 10 1 R +10 0 R 10 0 R 11 1 R 10 0 R 10 0 R 10 1 R 10 0 R 11 0 R 10 1 R 10 0 R 10 1 R +11 0 R 10 0 R 10 1 R 10 0 R 10 0 R 11 1 R 10 0 R 10 0 R 10 1 R 11 0 R 10 0 R +10 1 R D L2 10230 3870 M 11 600 R 10 399 R 10 302 R 10 243 R 10 203 R +11 173 R 10 151 R 10 133 R 10 119 R 11 107 R 10 96 R 10 88 R 10 81 R 10 74 R +11 69 R 10 64 R 10 59 R 10 55 R 11 52 R 10 48 R 10 46 R 10 43 R 11 41 R +10 38 R 10 36 R 10 35 R 10 33 R 11 31 R 10 30 R 10 28 R 10 27 R 11 26 R +10 25 R 10 24 R 10 23 R 11 22 R 10 21 R 10 20 R 10 19 R 10 19 R 11 18 R +10 18 R 10 16 R 10 17 R 11 15 R 10 16 R 10 14 R 10 15 R 10 13 R 11 14 R +10 13 R 10 12 R 10 12 R 11 12 R 10 12 R 10 11 R 10 11 R 11 10 R 10 11 R +10 10 R 10 9 R 10 10 R 11 9 R 10 9 R 10 9 R 10 8 R 11 9 R 10 8 R 10 8 R +10 8 R 10 7 R 11 8 R 10 7 R 10 7 R 10 7 R 11 7 R 10 6 R 10 7 R 10 6 R 11 7 R +10 6 R 10 6 R 10 5 R 10 6 R 11 6 R 10 5 R 10 6 R 10 5 R 11 5 R 10 5 R 10 5 R +10 5 R 10 5 R 11 5 R 10 4 R 10 5 R 10 4 R 11 5 R 10 4 R 10 4 R 10 5 R 11 4 R +10 4 R 10 4 R 10 4 R 10 3 R 11 4 R 10 4 R 10 4 R 10 3 R 11 4 R 10 3 R 10 4 R +10 3 R 11 3 R 10 4 R 10 3 R 10 3 R 10 3 R 11 3 R 10 3 R 10 3 R 10 3 R 11 3 R +10 3 R 10 3 R 10 3 R 10 2 R 11 3 R 10 3 R 10 2 R 10 3 R 11 3 R 10 2 R 10 3 R +10 2 R 11 3 R 10 2 R 10 2 R 10 3 R 10 2 R 11 2 R 10 3 R 10 2 R 10 2 R 11 2 R +10 2 R 10 2 R 10 2 R 10 3 R 11 2 R 10 2 R 10 2 R 10 2 R 11 2 R 10 1 R 10 2 R +10 2 R 11 2 R 10 2 R 10 2 R 10 2 R 10 1 R 11 2 R 10 2 R 10 2 R 10 1 R 11 2 R +10 2 R 10 1 R 10 2 R 11 2 R 10 1 R 10 2 R 10 1 R 10 2 R 11 1 R 10 2 R 10 1 R +10 2 R 11 1 R 10 2 R 10 1 R 10 2 R 10 1 R 11 1 R 10 2 R 10 1 R 10 2 R 11 1 R +10 1 R 10 1 R 10 2 R 11 1 R 10 1 R 10 2 R 10 1 R 10 1 R 11 1 R 10 2 R 10 1 R +10 1 R 11 1 R 10 1 R 10 2 R 10 1 R 10 1 R 11 1 R 10 1 R 10 1 R 10 1 R 11 1 R +10 1 R 10 2 R 10 1 R 11 1 R 10 1 R 10 1 R 10 1 R 10 1 R 11 1 R 10 1 R 10 1 R +10 1 R 11 1 R 10 1 R 10 1 R 10 1 R 11 1 R 10 1 R 10 1 R 10 0 R 10 1 R 11 1 R +10 1 R 10 1 R 10 1 R 11 1 R 10 1 R 10 1 R 10 0 R 10 1 R 11 1 R 10 1 R 10 1 R +10 1 R 11 1 R 10 0 R 10 1 R 10 1 R 11 1 R 10 1 R 10 0 R 10 1 R 10 1 R 11 1 R +10 0 R 10 1 R 10 1 R 11 1 R 10 0 R 10 1 R 10 1 R 10 1 R 11 0 R 10 1 R 10 1 R +10 0 R 11 1 R 10 1 R 10 1 R 10 0 R 11 1 R 10 1 R 10 0 R 10 1 R 10 1 R 11 0 R +10 1 R 10 1 R 10 0 R 11 1 R 10 0 R 10 1 R 10 1 R 10 0 R 11 1 R 10 1 R 10 0 R +10 1 R 11 0 R 10 1 R 10 1 R 10 0 R 11 1 R 10 0 R 10 1 R 10 1 R 10 0 R 11 1 R +10 0 R 10 1 R 10 0 R 11 1 R 10 0 R 10 1 R 10 1 R 11 0 R 10 1 R 10 0 R 10 1 R +10 0 R 11 1 R 10 0 R 10 1 R 10 0 R 11 1 R 10 0 R 10 1 R 10 0 R 10 1 R 11 0 R +10 1 R 10 0 R 10 1 R 11 0 R 10 1 R 10 0 R 10 1 R 11 0 R 10 1 R 10 0 R 10 1 R +10 0 R 11 1 R 10 0 R 10 1 R 10 0 R 11 0 R 10 1 R 10 0 R 10 1 R 10 0 R 11 1 R +10 0 R 10 1 R 10 0 R 11 0 R 10 1 R 10 0 R 10 1 R 11 0 R 10 0 R 10 1 R 10 0 R +10 1 R 11 0 R 10 1 R 10 0 R 10 0 R 11 1 R 10 0 R 10 1 R 10 0 R 11 0 R 10 1 R +10 0 R 10 1 R 10 0 R 11 0 R 10 1 R 10 0 R 10 0 R 11 1 R 10 0 R 10 1 R 10 0 R +10 0 R 11 1 R 10 0 R 10 0 R 10 1 R 11 0 R 10 0 R 10 1 R 10 0 R 11 0 R 10 1 R +10 0 R 10 1 R 10 0 R 11 0 R 10 1 R 10 0 R 10 0 R 11 1 R 10 0 R 10 0 R 10 1 R +10 0 R 11 0 R 10 1 R 10 0 R 10 0 R 11 1 R 10 0 R 10 0 R 10 0 R 11 1 R 10 0 R +10 0 R 10 1 R 10 0 R 11 0 R 10 1 R 10 0 R 10 0 R 11 1 R 10 0 R 10 0 R 10 0 R +11 1 R 10 0 R 10 0 R 10 1 R 10 0 R 11 0 R 10 0 R 10 1 R 10 0 R 11 0 R 10 1 R +10 0 R 10 0 R 10 0 R 11 1 R 10 0 R 10 0 R 10 1 R 11 0 R 10 0 R 10 0 R 10 1 R +11 0 R 10 0 R 10 0 R 10 1 R 10 0 R 11 0 R 10 1 R 10 0 R 10 0 R 11 0 R 10 1 R +10 0 R 10 0 R 10 0 R 11 1 R 10 0 R 10 0 R 10 0 R 11 1 R 10 0 R 10 0 R 10 0 R +11 1 R 10 0 R 10 0 R 10 0 R 10 1 R 11 0 R 10 0 R 10 0 R 10 1 R 11 0 R 10 0 R +10 0 R 10 0 R 11 1 R 10 0 R 10 0 R 10 0 R 10 1 R 11 0 R 10 0 R 10 0 R 10 1 R +11 0 R 10 0 R 10 0 R 10 0 R 10 1 R 11 0 R 10 0 R 10 0 R 10 0 R 11 1 R 10 0 R +10 0 R D L3 10230 2254 M 11 461 R 10 363 R 10 305 R 10 263 R 10 232 R +11 208 R 10 188 R 10 170 R 10 157 R 11 144 R 10 133 R 10 124 R 10 115 R +10 107 R 11 101 R 10 95 R 10 89 R 10 84 R 11 79 R 10 75 R 10 72 R 10 68 R +11 64 R 10 61 R 10 59 R 10 56 R 10 53 R 11 51 R 10 49 R 10 47 R 10 45 R +11 43 R 10 42 R 10 40 R 10 38 R 11 37 R 10 36 R 10 35 R 10 33 R 10 32 R +11 31 R 10 31 R 10 29 R 10 28 R 11 27 R 10 27 R 10 25 R 10 25 R 10 25 R +11 23 R 10 23 R 10 22 R 10 22 R 11 21 R 10 20 R 10 20 R 10 20 R 11 18 R +10 19 R 10 18 R 10 17 R 10 17 R 11 17 R 10 16 R 10 16 R 10 16 R 11 15 R +10 15 R 10 14 R 10 14 R 10 14 R 11 14 R 10 13 R 10 13 R 10 12 R 11 13 R +10 12 R 10 12 R 10 12 R 11 11 R 10 11 R 10 11 R 10 11 R 10 11 R 11 10 R +10 10 R 10 10 R 10 10 R 11 10 R 10 9 R 10 9 R 10 9 R 10 9 R 11 9 R 10 9 R +10 8 R 10 9 R 11 8 R 10 8 R 10 8 R 10 8 R 11 7 R 10 8 R 10 7 R 10 8 R 10 7 R +11 7 R 10 7 R 10 7 R 10 6 R 11 7 R 10 7 R 10 6 R 10 6 R 11 7 R 10 6 R 10 6 R +10 6 R 10 6 R 11 6 R 10 5 R 10 6 R 10 6 R 11 5 R 10 6 R 10 5 R 10 5 R 10 5 R +11 5 R 10 6 R 10 5 R 10 4 R 11 5 R 10 5 R 10 5 R 10 5 R 11 4 R 10 5 R 10 4 R +10 5 R 10 4 R 11 5 R 10 4 R 10 4 R 10 4 R 11 4 R 10 4 R 10 5 R 10 4 R 10 3 R +11 4 R 10 4 R 10 4 R 10 4 R 11 4 R 10 3 R 10 4 R 10 3 R 11 4 R 10 3 R 10 4 R +10 3 R 10 4 R 11 3 R 10 4 R 10 3 R 10 3 R 11 3 R 10 4 R 10 3 R 10 3 R 11 3 R +10 3 R 10 3 R 10 3 R 10 3 R 11 3 R 10 3 R 10 3 R 10 3 R 11 2 R 10 3 R 10 3 R +10 3 R 10 2 R 11 3 R 10 3 R 10 2 R 10 3 R 11 3 R 10 2 R 10 3 R 10 2 R 11 3 R +10 2 R 10 3 R 10 2 R 10 2 R 11 3 R 10 2 R 10 2 R 10 3 R 11 2 R 10 2 R 10 3 R +10 2 R 10 2 R 11 2 R 10 2 R 10 3 R 10 2 R 11 2 R 10 2 R 10 2 R 10 2 R 11 2 R +10 2 R 10 2 R 10 2 R 10 2 R 11 2 R 10 2 R 10 2 R 10 2 R 11 2 R 10 2 R 10 1 R +10 2 R 11 2 R 10 2 R 10 2 R 10 2 R 10 1 R 11 2 R 10 2 R 10 2 R 10 1 R 11 2 R +10 2 R 10 1 R 10 2 R 10 2 R 11 1 R 10 2 R 10 2 R 10 1 R 11 2 R 10 2 R 10 1 R +10 2 R 11 1 R 10 2 R 10 1 R 10 2 R 10 1 R 11 2 R 10 1 R 10 2 R 10 1 R 11 2 R +10 1 R 10 2 R 10 1 R 10 1 R 11 2 R 10 1 R 10 2 R 10 1 R 11 1 R 10 2 R 10 1 R +10 1 R 11 2 R 10 1 R 10 1 R 10 2 R 10 1 R 11 1 R 10 1 R 10 2 R 10 1 R 11 1 R +10 1 R 10 2 R 10 1 R 10 1 R 11 1 R 10 2 R 10 1 R 10 1 R 11 1 R 10 1 R 10 1 R +10 2 R 11 1 R 10 1 R 10 1 R 10 1 R 10 1 R 11 1 R 10 2 R 10 1 R 10 1 R 11 1 R +10 1 R 10 1 R 10 1 R 11 1 R 10 1 R 10 1 R 10 1 R 10 1 R 11 1 R 10 1 R 10 1 R +10 1 R 11 1 R 10 1 R 10 1 R 10 1 R 10 1 R 11 1 R 10 1 R 10 1 R 10 1 R 11 1 R +10 1 R 10 1 R 10 1 R 11 1 R 10 1 R 10 1 R 10 1 R 10 1 R 11 0 R 10 1 R 10 1 R +10 1 R 11 1 R 10 1 R 10 1 R 10 1 R 10 0 R 11 1 R 10 1 R 10 1 R 10 1 R 11 1 R +10 1 R 10 0 R 10 1 R 11 1 R 10 1 R 10 1 R 10 1 R 10 0 R 11 1 R 10 1 R 10 1 R +10 0 R 11 1 R 10 1 R 10 1 R 10 1 R 11 0 R 10 1 R 10 1 R 10 1 R 10 0 R 11 1 R +10 1 R 10 1 R 10 0 R 11 1 R 10 1 R 10 1 R 10 0 R 10 1 R 11 1 R 10 0 R 10 1 R +10 1 R 11 1 R 10 0 R 10 1 R 10 1 R 11 0 R 10 1 R 10 1 R 10 0 R 10 1 R 11 1 R +10 0 R 10 1 R 10 1 R 11 0 R 10 1 R 10 1 R 10 0 R 10 1 R 11 1 R 10 0 R 10 1 R +10 1 R 11 0 R 10 1 R 10 0 R 10 1 R 11 1 R 10 0 R 10 1 R 10 1 R 10 0 R 11 1 R +10 0 R 10 1 R 10 1 R 11 0 R 10 1 R 10 0 R 10 1 R 11 0 R 10 1 R 10 1 R 10 0 R +10 1 R 11 0 R 10 1 R 10 0 R 10 1 R 11 1 R 10 0 R 10 1 R 10 0 R 10 1 R 11 0 R +10 1 R 10 0 R 10 1 R 11 1 R 10 0 R 10 1 R 10 0 R 11 1 R 10 0 R 10 1 R 10 0 R +10 1 R 11 0 R 10 1 R 10 0 R 10 1 R 11 0 R 10 1 R 10 0 R 10 1 R 10 0 R 11 1 R +10 0 R 10 1 R 10 0 R 11 1 R 10 0 R 10 1 R 10 0 R 11 1 R 10 0 R 10 1 R 10 0 R +10 1 R 11 0 R 10 1 R 10 0 R 10 1 R 11 0 R 10 1 R 10 0 R 10 0 R 11 1 R 10 0 R +10 1 R 10 0 R 10 1 R 11 0 R 10 1 R 10 0 R 10 0 R 11 1 R 10 0 R 10 1 R 10 0 R +10 1 R 11 0 R 10 1 R 10 0 R 10 0 R 11 1 R 10 0 R 10 1 R D +end restore +showpage +%%Trailer +restore +%%Pages: 1 +%%Trailer +cleartomark +countdictstack +exch sub { end } repeat +restore +%%EOF diff --git a/documentation/source/science_guide/turbulence_schemes/subsent_fig7.eps b/documentation/source/science_guide/turbulence_schemes/subsent_fig7.eps new file mode 100644 index 0000000000..4fb826b330 --- /dev/null +++ b/documentation/source/science_guide/turbulence_schemes/subsent_fig7.eps @@ -0,0 +1,323 @@ +%!PS-Adobe-2.0 EPSF-2.0 +%%BoundingBox: 60 38 464 293 +%%HiResBoundingBox: 60.5 39 463.5 292.5 +%%Title: Graphics produced by WAVE +%%For: APRISH::LOCK +%%Creator: WAVE Version 6.21 (vms axp) +%%CreationDate: Tue May 4 15:53:47 1999 + +%%EndComments +% EPSF created by ps2eps 1.68 +%%BeginProlog +save +countdictstack +mark +newpath +/showpage {} def +/setpagedevice {pop} def +%%EndProlog +%%Page 1 1 +%+ wave_prolog.ps -- Prolog for PV-WAVE CL generated PostScript files +%+ $Id: wave_prolog.ps,v 1.5 1998/06/05 17:44:06 thaux Exp $ +%+ Copyright (c) 1989-1992 Precision Visuals, Inc. All Rights Reserved. +%v 1 +save /$WAVE_DICT 60 dict def $WAVE_DICT begin /bdef { bind def } bind def /C +{currentpoint newpath moveto} bdef /D {currentpoint stroke moveto} bdef /F +{closepath fill} bdef /K { setgray } bdef /L {translate} bdef /M {moveto} +bdef /N {rmoveto} bdef /O {show} bdef /P {lineto} bdef /R {rlineto} bdef /S +{gsave show grestore} bdef /Z {gsave currentpoint lineto 20 setlinewidth 1 +setlinecap stroke grestore} bdef /CC {concat} bdef /CD {currentmatrix def} +bdef /CP {currentpoint pop def} bdef /FM {findfont exch makefont} bdef /FS +{findfont exch scalefont} bdef /GS {get setfont} bdef /MD {matrix def} bdef +/MN {mul neg def} bdef /RS {roll put setfont} bdef /SL {setlinewidth} bdef +/SM {setmatrix} bdef /L0 {[] 0 setdash} bdef /L1 {[40 100] 0 setdash} bdef +/L2 {[200 200] 0 setdash} bdef /L3 {[200 100 50 100] 0 setdash} bdef /L4 +{[300 100 50 100 50 100 50 100] 0 setdash} bdef /L5 {[400 200] 0 setdash} +bdef /IO { gsave 0.0 setgray /val exch def moveto /str 20 string def val str +cvs show grestore } bdef /UL { gsave currentpoint /cury exch def /curx exch +def /curmtx matrix currentmatrix def matrix identmatrix setmatrix newpath 0 +0 moveto true charpath flattenpath pathbbox curmtx setmatrix /height exch +def /width exch def /ybase cury height .1 mul sub def pop pop newpath curx +ybase moveto width 0 rlineto closepath stroke grestore } bdef /BG { gsave +currentpoint /cury exch def /curx exch def /curmtx matrix currentmatrix def +matrix identmatrix setmatrix newpath 0 0 moveto true charpath flattenpath +pathbbox curmtx setmatrix /height exch def /width exch def pop pop newpath 0 +0 moveto (M) true charpath flattenpath pathbbox /xdiff exch .1 mul def +/ydiff exch .1 mul def pop pop newpath curx xdiff 2 div sub cury ydiff 2 div +sub moveto width xdiff add 0 rlineto 0 height ydiff add rlineto width neg +xdiff sub 0 rlineto closepath fill grestore } bdef /$T_DICT 20 dict def /T { +$T_DICT begin /align exch def /orien exch def /size exch def /thestring exch +def gsave moveto /chsize 1.0 def /xsize 0.0 def /SUPER 8#330 def /SUBS 8#331 +def /NORM 8#332 def /SCRIPTWID 0.7 def /OFFSET 0.6 def orien rotate size dup +scale /orien false def thestring { /charcode exch def charcode SUPER eq +charcode SUBS eq or { /chsize SCRIPTWID def /orien true def } { charcode +NORM eq { /chsize 1.0 def /orien true def } { ( ) dup 0 charcode put +stringwidth pop chsize mul xsize add /xsize exch def } ifelse } ifelse } +forall xsize align mul neg 0 rmoveto orien { /regularfont currentfont def +/fractionfont currentfont [ SCRIPTWID 0 0 SCRIPTWID 0 0 ] makefont def gsave +newpath 0 0 moveto (1) true charpath flattenpath pathbbox /height exch def +pop pop pop grestore } if /xsize 0 def thestring { /charcode exch def +charcode SUPER eq { 0 OFFSET height mul dup /xsize exch def rmoveto +fractionfont setfont } { charcode SUBS eq { 0 OFFSET height mul neg dup +/xsize exch def rmoveto fractionfont setfont } { charcode NORM eq { 0 xsize +neg rmoveto regularfont setfont } { ( ) dup 0 charcode put show } ifelse } +ifelse } ifelse } forall grestore end } bdef /IsChar { exch /CharStrings get +exch known } bdef /MapCh { 3 -1 roll /Encoding get 3 1 roll put } bdef +/MapDegree { dup 16#b0 exch /degree IsChar { /degree } { /ring } ifelse +MapCh } bdef /MapBB { dup 16#a6 exch /brokenbar IsChar { /brokenbar } { /bar +} ifelse MapCh } bdef /ReEncode { dup findfont begin currentdict dup length +dict begin { 1 index /FID ne {def} {pop pop} ifelse } forall /FontName exch +def dup length 0 ne {/Encoding Encoding 256 array copy def 0 exch { dup type +/nametype eq { Encoding 2 index 2 index put pop 1 add } { exch pop } ifelse +} forall } if pop currentdict dup end end /FontName get exch definefont dup +MapDegree MapBB } bdef /RF { ISOLatin1Encoding exch ReEncode } bdef /Courier +RF /Courier-Bold RF /Courier-Oblique RF /Courier-BoldOblique RF /Helvetica +RF /Helvetica-Bold RF /Helvetica-Oblique RF /Helvetica-BoldOblique RF +/Helvetica-Narrow RF /Helvetica-Narrow-Bold RF /Helvetica-Narrow-Oblique RF +/Helvetica-Narrow-BoldOblique RF /AvantGarde-Book RF /AvantGarde-DemiOblique +RF /AvantGarde-Demi RF /AvantGarde-DemiOblique RF /Bookman-Demi RF +/Bookman-DemiItalic RF /Bookman-Light RF /Bookman-LightItalic RF +/ZapfChancery-MediumItalic RF /NewCenturySchlbk-Roman RF +/NewCenturySchlbk-Bold RF /NewCenturySchlbk-Italic RF +/NewCenturySchlbk-BoldItalic RF /Palatino-Roman RF /Palatino-Bold RF +/Palatino-Italic RF /Palatino-BoldItalic RF /Times-Roman RF /Times-Bold RF +/Times-Italic RF /Times-BoldItalic RF end +%%EndProlog +%%Page: 0 1 +%%BeginPageSetup +save $WAVE_DICT begin 28 28 L 0.028346 dup scale +%%PageBoundingBox: 28 28 481 311 +%%EndPageSetup +/psFontCache 500 array def /psStack 8 array def /psStackInd 0 def +/psTextWidth 0 def /psCurBase 0 def /ASW { stringwidth pop psCurBase add +/psCurBase exch def psCurBase psTextWidth gt +{ /psTextWidth psCurBase def } if } def /PSS { +psStack psStackInd psCurBase put /psStackInd psStackInd 1 add def } def +/PPS { /psStackInd psStackInd 1 sub def +/psCurBase psStack psStackInd get def } def 10.000000 SL L0 0.000 K +2220 1408 M 5114 0 R D 2220 1408 M 0 157 R D 2196 1195 M -35 -11 R -23 -35 R +-12 -59 R 0 -35 R 12 -58 R 23 -35 R 35 -12 R 24 0 R 35 12 R 23 35 R 12 58 R +0 35 R -12 59 R -23 35 R -35 11 R -24 0 R -23 -11 R -12 -12 R -12 -23 R +-11 -59 R 0 -35 R 11 -58 R 12 -24 R 12 -11 R 23 -12 R D 2220 950 M 23 12 R +12 11 R 11 24 R 12 58 R 0 35 R -12 59 R -11 23 R -12 12 R -23 11 R D +5872 1408 M 0 157 R D 5406 1078 M 11 24 R 24 23 R 23 0 R 12 -11 R 11 -24 R +0 -35 R -23 -58 R D 5476 1114 M 0 -47 R -12 -47 R 0 -47 R D 5476 1090 M +-24 -58 R 0 -35 R 12 -24 R 23 -11 R 24 0 R 23 11 R 24 24 R 11 35 R 24 93 R D +5569 1032 M 0 -35 R 12 -24 R 23 -11 R 24 0 R 23 11 R 23 24 R 24 35 R 11 46 R +0 47 R -11 0 R 0 -11 R 11 -24 R D 5604 1125 M -23 -93 R 0 -59 R D +5593 1125 M 23 0 R -23 -82 R -12 -46 R D 5931 1125 M -70 -245 R D +5943 1125 M -70 -245 R D 5931 1125 M 24 0 R -70 -245 R D 5908 1043 M 0 35 R +-12 36 R -23 11 R -23 0 R -35 -11 R -24 -36 R -11 -35 R 0 -23 R 11 -35 R +12 -12 R 23 -11 R 24 0 R 23 11 R 12 12 R 11 23 R 12 35 R D 5815 1102 M +-12 -24 R -12 -35 R 0 -35 R 12 -23 R D 5850 1125 M -24 -23 R -11 -24 R +-12 -35 R 0 -35 R 12 -35 R 11 -11 R D 5826 880 M 94 0 R D 5873 892 M +-35 -12 R D 5873 903 M -23 -23 R D 5885 903 M 11 -23 R D 5873 892 M 35 -12 R +D 6055 1016 M -22 -80 R -7 -28 R 0 -22 R 7 -15 R 7 -7 R 15 0 R 14 15 R +8 14 R D 6062 1016 M -22 -80 R -7 -28 R 0 -37 R D 6055 1016 M 14 0 R +-29 -101 R -7 -29 R D 6012 965 M 72 0 R D 6138 1254 M 0 -374 R D 6322 1002 M +7 0 R 7 14 R -7 -43 R 0 14 R -7 15 R -7 7 R -22 7 R -29 0 R -22 -7 R +-14 -15 R 0 -21 R 7 -15 R 15 -14 R 43 -22 R 7 -14 R 0 -22 R -7 -15 R D +6235 973 M 7 -15 R 51 -29 R 7 -14 R D 6242 1009 M -7 -15 R 0 -14 R 7 -15 R +44 -21 R 14 -15 R 8 -14 R 0 -22 R -8 -14 R -7 -8 R -22 -7 R -29 0 R -21 7 R +-7 8 R -8 14 R 0 15 R -7 -44 R 7 15 R 8 0 R D 4311 668 M -70 -246 R D +4323 668 M -70 -246 R D 4311 668 M 24 0 R -71 -246 R D 4288 586 M 0 35 R +-12 35 R -23 12 R -24 0 R -35 -12 R -23 -35 R -12 -35 R 0 -24 R 12 -35 R +12 -11 R 23 -12 R 23 0 R 24 12 R 11 11 R 12 24 R 12 35 R D 4194 644 M +-11 -23 R -12 -35 R 0 -35 R 12 -24 R D 4229 668 M -23 -24 R -12 -23 R +-11 -35 R 0 -35 R 11 -35 R 12 -12 R D 4206 422 M 93 0 R D 4253 434 M +-35 -12 R D 4253 446 M -24 -24 R D 4264 446 M 12 -24 R D 4253 434 M 35 -12 R +D 4435 558 M -22 -79 R -7 -29 R 0 -22 R 7 -14 R 7 -7 R 15 0 R 14 14 R 7 14 R +D 4442 558 M -22 -79 R -7 -29 R 0 -36 R D 4435 558 M 14 0 R -29 -101 R +-7 -29 R D 4391 508 M 73 0 R D 4763 726 M -12 -12 R 12 -11 R 12 11 R 0 12 R +-12 12 R -23 0 R -24 -12 R -11 -23 R 0 -211 R D 4740 738 M -12 -12 R +-12 -23 R 0 -211 R D 4669 656 M 94 0 R D 4669 492 M 82 0 R D 4856 738 M +0 -246 R D 4868 738 M 0 -246 R D 4821 738 M 47 0 R D 4821 492 M 82 0 R D +4985 656 M 0 -129 R 12 -23 R 35 -12 R 23 0 R 35 12 R 23 23 R D 4997 656 M +0 -129 R 11 -23 R 24 -12 R D 5113 656 M 0 -164 R D 5125 656 M 0 -164 R D +4950 656 M 47 0 R D 5078 656 M 47 0 R D 5113 492 M 47 0 R D 5230 656 M +129 -164 R D 5242 656 M 129 -164 R D 5371 656 M -141 -164 R D 5207 656 M +70 0 R D 5324 656 M 70 0 R D 5207 492 M 70 0 R D 5324 492 M 70 0 R D +2220 1408 M 0 7888 R D 2220 1408 M 102 0 R D 1968 1653 M -35 -12 R -23 -35 R +-12 -58 R 0 -35 R 12 -59 R 23 -35 R 35 -11 R 24 0 R 35 11 R 23 35 R 12 59 R +0 35 R -12 58 R -23 35 R -35 12 R -24 0 R -23 -12 R -12 -11 R -11 -24 R +-12 -58 R 0 -35 R 12 -59 R 11 -23 R 12 -12 R 23 -11 R D 1992 1408 M 23 11 R +12 12 R 11 23 R 12 59 R 0 35 R -12 58 R -11 24 R -12 11 R -23 12 R D +2220 6195 M 102 0 R D 1980 6229 M -12 -23 R -24 -24 R -93 -70 R -23 -23 R +-12 -23 R D 1968 6206 M -105 0 R -24 -12 R -11 -23 R D 1944 6206 M -46 12 R +-35 0 R -12 -12 R D 1944 6206 M -46 23 R -35 0 R -24 -23 R -11 -35 R D +1828 6089 M 105 0 R 23 12 R 12 23 R D 1851 6089 M 47 -12 R 35 0 R 11 12 R D +1851 6089 M 47 -23 R 35 0 R 23 23 R 12 35 R D 2080 6120 M 0 -14 R 14 0 R +0 14 R -14 0 R D 2087 6120 M 0 -14 R D 2080 6113 M 14 0 R D 2022 6041 M +7 14 R 14 14 R 15 0 R 7 -7 R 7 -14 R 0 -22 R -14 -36 R D 2065 6062 M 0 -29 R +-7 -29 R 0 -29 R D 2065 6048 M -14 -36 R 0 -22 R 7 -15 R 7 -7 R 15 0 R +14 15 R 7 14 R D 1180 5340 M 23 -12 R 23 -23 R 71 -94 R 23 -23 R 23 -12 R D +1203 5328 M 0 -105 R 12 -23 R 23 -12 R D 1203 5305 M -12 -47 R 0 -35 R +12 -12 R D 1203 5305 M -23 -47 R 0 -35 R 23 -23 R 35 -12 R D 1320 5188 M +0 105 R -12 23 R -23 12 R D 1320 5211 M 12 47 R 0 35 R -12 12 R D +1320 5211 M 23 47 R 0 35 R -23 23 R -35 12 R D L3 5872 1408 M -8 54 R +-8 54 R -8 55 R -9 54 R -8 55 R -8 54 R -9 54 R -8 55 R -8 54 R -9 55 R +-8 54 R -8 54 R -8 55 R -9 54 R -8 55 R -8 54 R -9 54 R -8 55 R -8 54 R +-9 55 R -8 54 R -8 54 R -9 55 R -8 54 R -8 55 R -8 54 R -9 54 R -8 55 R +-8 54 R -9 55 R -8 54 R -8 54 R -9 55 R -8 54 R -8 55 R -8 54 R -9 54 R +-8 55 R -8 54 R -9 55 R -8 54 R -8 54 R -9 55 R -8 54 R -8 55 R -8 54 R +-9 54 R -8 55 R -8 54 R -9 55 R -8 54 R -8 54 R -9 55 R -8 54 R -8 55 R +-8 54 R -9 54 R -8 55 R -8 54 R -9 55 R -8 54 R -8 54 R -9 55 R -8 54 R +-8 55 R -8 54 R -9 54 R -8 55 R -8 54 R -9 55 R -8 54 R -8 54 R -9 55 R +-8 54 R -8 55 R -9 54 R -8 54 R -8 55 R -8 54 R -9 55 R -8 54 R -8 54 R +-9 55 R -8 54 R -8 55 R -9 54 R -8 54 R -8 55 R -2922 54 R 0 55 R 0 54 R +0 54 R 0 55 R 0 54 R 0 55 R 0 54 R 0 54 R 0 55 R 0 54 R 0 55 R 0 54 R 0 54 R +0 55 R 0 54 R 0 55 R 0 54 R 0 54 R 0 55 R 0 54 R 0 55 R 0 54 R 0 54 R 0 55 R +0 54 R 0 55 R 0 54 R 0 54 R 0 55 R 0 54 R 0 55 R 0 54 R 0 54 R 0 55 R 0 54 R +0 55 R 0 54 R 0 54 R 0 55 R 0 54 R 0 55 R 0 54 R 0 54 R 0 55 R 0 54 R 0 55 R +0 54 R 0 54 R 0 55 R 0 54 R 0 55 R 0 54 R 0 54 R 0 55 R 0 54 R 0 55 R D L2 +2220 1408 M 5114 0 R D 2220 2088 M 5114 0 R D 2220 3312 M 5114 0 R D +2220 5216 M 5114 0 R D 2220 7392 M 5114 0 R D L0 5983 1519 M -111 -111 R +-111 111 R 111 -111 R D 5876 2199 M -222 -222 R 111 111 R -111 111 R +222 -222 R D 5872 1408 M -107 680 R D 5693 3423 M -222 -222 R 111 111 R +-111 111 R 222 -222 R D 5765 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00000000000000000000000000000000000000000000000000000000000000 +% 088888888888888888888888888888888888888888888888888888888888888888888888888888 +% 88888888888888888888888888888888888888888888888888888888000000 +% 000000000000000000000000000000000000000000000000000000000000000000000000000000 +% 00000000000000000000000000000000000000000000000000000000000000 +% 000000000000000000000000000000000000000000000000000000000000000000000000000000 +% 00000000000000000000000000000000000000000000000000000000000000 +% 000000000000000000000000000000000000000000000000000000000000000000000000000000 +% 00000000000000000000000000000000000000000000000000000000000000 +% 000000000000000000000000000000000000000000000000000000000000000000000000000000 +% 00000000000000000000000000000000000000000000000000000000000000 +save /d_sv_obj exch def +userdict /IslandDrawDict 300 dict dup begin put +/bdef {bind def} bind def +/E {exch} bdef +/FF {findfont} bdef +/MF {makefont} bdef +/RO {rotate} bdef +/SC {scale} bdef +/SF {setfont} bdef +/SG {setgray} bdef +/TR {translate} bdef +/bp {lj lw rgb} bdef +/bpbw {lj lw setgray} bdef +/c {curveto} bdef +/cl {closepath} bdef +/fi {eofill} bdef +/g {setgray} bdef +/gr {grestore} bdef +/gs {gsave} bdef +/l {lineto} bdef +/lj {setlinejoin} bdef +/lw {setlinewidth} bdef +/m {moveto} bdef +/n {newpath} bdef +/nx {/x E def} bdef +/r {rmoveto} bdef +/rl {rlineto} bdef +/rgb {setrgbcolor} bdef +/s {show} bdef +/sd {setdash} bdef +/sp {x 0 rmoveto} bdef +/ss {currentpoint pop E m} bdef +/st {stroke} bdef +/BPSIDE 32 def %% pixels per pattern side +/PATFREQ 3.0 def %% pattern pixels per mm +/dp_mat [PATFREQ 0 0 PATFREQ 0 0] def +/dp_pw BPSIDE def %% pattern pixel width +/dp_ph BPSIDE def %% pattern pixel height +/dp_w dp_pw PATFREQ div def %% pattern mm width +/dp_h dp_ph PATFREQ div def %% pattern mm height +/savemat matrix def +/topmat matrix def +/patmat matrix def +/ncpoint errordict /nocurrentpoint get def +errordict begin +/nocurrentpoint { + dup /pathbbox load eq + {pop 0 0 1 1} + {ncpoint} + ifelse +} bdef +end +/ar { %% sa ea sx sy rot tx ty + matrix currentmatrix 8 1 roll TR RO SC + n 0 0 1 5 3 roll arc setmatrix +} bdef +/arn { %% sa ea sx sy rot tx ty + TR RO SC + matrix currentmatrix 8 1 roll + n 0 0 1 5 3 roll arcn setmatrix +} bdef +/el { %% sx sy rot tx ty + matrix currentmatrix 6 1 roll TR RO SC + n 0 0 1 0 360 arc setmatrix cl +} bdef +/image_raster { %% sw sh sd dw dh xs ys + TR SC /sd E def /sh E def /sw E def + /imagebuf sw sd mul 7 add 8 idiv string def + sw sh sd [sw 0 0 sh 0 0] { currentfile imagebuf readhexstring pop} + image +} bdef +/imagemask_raster { + TR SC /sh E def /sw E def + /imagebuf sw 7 add 8 idiv string def + sw sh false [sw 0 0 sh 0 0] + {currentfile imagebuf readhexstring pop} + imagemask +} bdef +/dither_color_raster { % bool sw sh sd dw dh xs ys + TR SC /sd E def /sh E def /sw E def + sd 8 eq and + { + /imagebuf 3 string def + /grayval 1 string def + sw sh sd [sw 0 0 sh 0 0] + { + currentfile imagebuf readhexstring pop pop + imagebuf 0 get 0.299 mul + imagebuf 1 get 0.587 mul add + imagebuf 2 get 0.114 mul add cvi grayval exch 0 exch put grayval + } + image + } + { + /imagebuf sw 3 mul sd mul 7 add 8 idiv string def + sh { currentfile imagebuf readhexstring pop pop } repeat + } ifelse +} bdef +/image_color_raster { % bool sw sh sd dw dh xs ys + /colorimage where not + { dither_color_raster } + { + pop + TR SC /sd E def /sh E def /sw E def pop + /imagebuf sw 3 mul sd mul 7 add 8 idiv string def + sw sh sd [sw 0 0 sh 0 0] { currentfile imagebuf readhexstring pop} + false 3 colorimage + } ifelse +} bdef +/patpath { + /inv E def + topmat setmatrix + pathbbox %% get lo - hi indecies + /hy E dp_h div floor cvi def + /hx E dp_w div floor cvi def + /ly E dp_h div floor cvi def + /lx E dp_w div floor cvi def + lx 1 hx { + dp_w mul + ly 1 hy { + dp_h mul + E dup 3 1 roll E + patmat currentmatrix pop + TR + dp_pw dp_ph inv + dp_mat dp_proc imagemask + patmat setmatrix + } for + pop + } for +} bdef +% setpattern brush of patterns instead of gray +/setpattern { + /blue E def /green E def /red E def + /freq E def /bwidth E def /bpside E def + /bstring E def + /onbits 0 def /offbits 0 def + freq 0 {/y E def /x E def + /xindex x 1 add 2 div bpside mul cvi def + /yindex y 1 add 2 div bpside mul cvi def + bstring yindex bwidth mul xindex 8 idiv add get not + 1 7 xindex 8 mod sub bitshift and 0 ne + {/onbits onbits 1 add def 1} + {/offbits offbits 1 add def 0} + ifelse + } setscreen {} settransfer + systemdict /setcmykcolor known + { /fact 1 onbits offbits onbits add div sub def + 1 red sub fact mul 1 green sub fact mul 1 blue sub fact mul 0 + setcmykcolor + } + { offbits offbits onbits add div setgray} + ifelse +} bdef +/dmatrix matrix def +/dpi 72 0 dmatrix defaultmatrix dtransform + dup mul E dup mul add sqrt +def +/B {gs bp st gr} bdef %% brush: gr lw lj +/Bbw {gs bpbw st gr} bdef %% brush: gr lw lj +/F {gs rgb eofill gr} bdef %% fill: gr +/Fbw {gs setgray eofill gr} bdef %% fill: gr +/PB {gs lj lw setpattern st gr} bdef +/PF {gs eoclip patpath gr} bdef +/BB {gs rgb lj lw strokepath clip patpath gr} bdef +/xdef {exch def} bdef +/clip_region { + /ht xdef + /wd xdef + /bm xdef + /lm xdef + newpath + lm bm moveto + 0 ht rlineto + wd 0 rlineto + 0 ht neg rlineto + closepath clip +} bdef +/reencode_small_dict 12 dict def +/ReencodeSmall { +reencode_small_dict begin +/new_codes_and_names exch def +/new_font_name exch def +/base_font_name exch def +/base_font_dict base_font_name findfont def +/newfont base_font_dict maxlength dict def +base_font_dict { +exch dup /FID ne +{ dup /Encoding eq +{ exch dup length array copy newfont 3 1 roll put } +{ exch newfont 3 1 roll put } +ifelse +} +{ pop pop } +ifelse +} forall +newfont /FontName new_font_name put +new_codes_and_names aload pop +new_codes_and_names length 2 idiv +{ newfont /Encoding get 3 1 roll put } +repeat +new_font_name newfont definefont pop +end %reencode_small_dict +} def +/extended_Zapf [ +8#223 /a89 +8#224 /a90 +8#225 /a93 +8#226 /a94 +8#227 /a91 +8#230 /a92 +8#231 /a205 +8#232 /a85 +8#233 /a206 +8#234 /a86 +8#235 /a87 +8#236 /a88 +8#237 /a95 +8#240 /a96 +] def +/extended_Standard [ +29 /thorn +30 /yacute +31 /divide +128 /Acircumflex +129 /Adieresis +130 /Agrave +131 /Aring +132 /Atilde +133 /Ccedilla +134 /Eacute +135 /Ecircumflex +136 /Edieresis +137 /Egrave +138 /Iacute +139 /Icircumflex +140 /Idieresis +141 /Igrave +142 /Ntilde +143 /Oacute +144 /Ocircumflex +145 /Odieresis +146 /Ograve +147 /Otilde +148 /Scaron +149 /Uacute +150 /Ucircumflex +151 /Udieresis +152 /Ugrave +153 /Ydieresis +154 /Zcaron +155 /aacute +156 /acircumflex +157 /adieresis +158 /agrave +159 /aring +160 /atilde +161 /exclamdown +162 /cent +163 /sterling +164 /fraction +165 /yen +166 /florin +167 /section +168 /currency +169 /quotesingle +170 /quotedblleft +171 /guillemotleft +172 /guilsinglleft +173 /guilsinglright +174 /fi +175 /fl +176 /plusminus +177 /endash +178 /dagger +179 /daggerdbl +180 /periodcentered +181 /twosuperior +182 /paragraph +183 /bullet +184 /quotesinglbase +185 /quotedblbase +186 /quotedblright +187 /guillemotright +188 /ellipsis +189 /perthousand +190 /threesuperior +191 /questiondown +192 /mu +193 /grave +194 /acute +195 /circumflex +196 /tilde +197 /macron +198 /breve +199 /dotaccent +200 /dieresis +201 /onesuperior +202 /ring +203 /cedilla +204 /onequarter +205 /hungarumlaut +206 /ogonek +207 /caron +208 /emdash +209 /ccedilla +210 /copyright +211 /eacute +212 /ecircumflex +213 /edieresis +214 /egrave +215 /iacute +216 /icircumflex +217 /idieresis +218 /igrave +219 /logicalnot +220 /minus +221 /ntilde +222 /oacute +223 /ocircumflex +224 /odieresis +225 /AE +226 /onehalf +227 /ordfeminine +228 /ograve +229 /otilde +230 /registered +231 /scaron +232 /Lslash +233 /Oslash +234 /OE +235 /ordmasculine +236 /trademark +237 /uacute +238 /ucircumflex +239 /udieresis +240 /ugrave +241 /ae +242 /ydieresis +243 /zcaron +244 /Aacute +245 /dotlessi +246 /threequarters +247 /Eth +248 /lslash +249 /oslash +250 /oe +251 /germandbls +252 /multiply +253 /Yacute +254 /Thorn +255 /eth +] def +/extended_Symbol [ +] def +/extend_font { % stack: fontname newfontname +exch dup (ZapfDingbats) eq +{ cvn exch cvn extended_Zapf ReencodeSmall } +{ dup (Symbol) eq +{ cvn exch cvn extended_Symbol ReencodeSmall } +{ cvn exch cvn extended_Standard ReencodeSmall } +ifelse +} +ifelse +} bind def +/extend_font_name { % stack: font_name_string +dup length 1 add string /extended_font_name exch def +extended_font_name 0 (_) putinterval +extended_font_name 1 3 -1 roll putinterval +extended_font_name +} bind def +/gf { +/f exch def f cvn where +{ f exch begin cvn load exec setfont end } +{ f 0 f length 8 sub getinterval dup +/localfont exch extend_font_name def +localfont extend_font +localfont findfont +/xsz f f length 4 sub 4 getinterval cvi def +/ysz f f length 8 sub 4 getinterval cvi def +[ xsz 0 0 ysz neg 0 0 ] makefont dup f cvn exch def +setfont +} +ifelse +} bind def +/gfns { +/f exch def f cvn where +{ f exch begin cvn load exec setfont end } +{ f 0 f length 8 sub getinterval cvn findfont +/xsz f f length 4 sub 4 getinterval cvi def +/ysz f f length 8 sub 4 getinterval cvi def +[ xsz 0 0 ysz neg 0 0 ] makefont dup f cvn exch def +setfont +} +ifelse +} bind def +/ul { % space drop thickness +gs currentpoint currentlinewidth +currentpoint n m 6 -3 roll +lw 0 exch r +0 rl st lw m +gr +} bind def +/nxtab { currentpoint pop 1000.0 mul cvi tab mod + tab exch sub 1000.0 div 0 rmoveto } bind def +/nx { /x exch def } bind def +0. nx +gsave +2.83465 -2.83465 scale 0 -279.4 translate +topmat currentmatrix pop +n 67.155 58.854 m 67.069 59.317 l +66.973 59.786 l +66.865 60.26 l +66.748 60.738 l +66.619 61.22 l +66.48 61.707 l +66.33 62.199 l +66.17 62.693 l +66 63.192 l +65.819 63.694 l +65.628 64.2 l +65.428 64.708 l +65.216 65.22 l +64.994 65.735 l +64.764 66.252 l +64.522 66.771 l +64.271 67.293 l +64.009 67.816 l +63.739 68.341 l 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l +56.356 226.17 l +56.349 226.17 l +56.344 226.18 l +56.339 226.19 l +56.334 226.19 l +56.33 226.2 l +56.327 226.21 l +56.325 226.22 l +56.323 226.23 l +56.323 226.24 l +56.494 226.24 l +cl 0 0 0 F +n 62.891 225.59 m 62.89 225.58 l +62.889 225.57 l +62.886 225.56 l +62.883 225.55 l +62.879 225.54 l +62.874 225.54 l +62.869 225.53 l +62.864 225.52 l +62.858 225.52 l +62.851 225.51 l +62.843 225.51 l +62.836 225.51 l +62.829 225.5 l +62.821 225.5 l +62.813 225.5 l +62.804 225.5 l +62.796 225.5 l +62.789 225.5 l +62.781 225.5 l +62.773 225.51 l +62.765 225.51 l +62.758 225.51 l +62.752 225.52 l +62.746 225.52 l +62.74 225.53 l +62.734 225.54 l +62.726 225.55 l +62.723 225.56 l +62.721 225.57 l +62.72 225.58 l +62.719 225.59 l +62.891 225.59 l +cl 0 0 0 F +n 62.891 225.59 m 62.89 225.58 l +62.889 225.57 l +62.886 225.56 l +62.883 225.55 l +62.879 225.54 l +62.874 225.54 l +62.869 225.53 l +62.864 225.52 l +62.858 225.52 l +62.851 225.51 l +62.843 225.51 l +62.836 225.51 l +62.829 225.5 l 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l +61.56 224.65 l +cl 0 0 0 F +n 61.56 224.65 m 61.559 224.64 l +61.558 224.63 l +61.555 224.62 l +61.552 224.61 l +61.548 224.6 l +61.544 224.59 l +61.539 224.59 l +61.533 224.58 l +61.527 224.58 l +61.52 224.57 l +61.512 224.57 l +61.506 224.57 l +61.498 224.56 l +61.49 224.56 l +61.482 224.56 l +61.474 224.56 l +61.466 224.56 l +61.458 224.56 l +61.45 224.56 l +61.442 224.57 l +61.435 224.57 l +61.428 224.57 l +61.421 224.58 l +61.415 224.58 l +61.409 224.59 l +61.404 224.59 l +61.4 224.6 l +61.396 224.61 l +61.393 224.62 l +61.39 224.63 l +61.389 224.64 l +61.388 224.65 l +61.56 224.65 l +cl 0 0 0 F +n 65.552 225.76 m 65.551 225.75 l +65.55 225.74 l +65.547 225.73 l +65.541 225.72 l +65.536 225.71 l +65.531 225.7 l +65.525 225.7 l +65.519 225.69 l +65.512 225.69 l +65.505 225.68 l +65.498 225.68 l +65.49 225.68 l +65.482 225.68 l +65.474 225.68 l +65.466 225.68 l +65.458 225.68 l +65.45 225.68 l +65.442 225.68 l +65.434 225.68 l +65.427 225.68 l +65.42 225.69 l +65.413 225.69 l 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l +72.259 230.23 l +72.255 230.24 l +72.252 230.25 l +72.249 230.26 l +72.249 230.27 l +72.42 230.27 l +cl 0 0 0 F +n 73.278 230.92 m 73.278 230.91 l +73.277 230.9 l +73.275 230.89 l +73.271 230.88 l +73.267 230.87 l +73.263 230.86 l +73.257 230.86 l +73.251 230.85 l +73.245 230.84 l +73.239 230.84 l +73.232 230.84 l +73.224 230.83 l +73.216 230.83 l +73.208 230.83 l +73.201 230.83 l +73.193 230.83 l +73.185 230.83 l +73.176 230.83 l +73.168 230.83 l +73.161 230.83 l +73.154 230.84 l +73.147 230.84 l +73.14 230.84 l +73.133 230.85 l +73.128 230.86 l +73.123 230.86 l +73.118 230.87 l +73.114 230.88 l +73.111 230.89 l +73.109 230.9 l +73.107 230.91 l +73.107 230.92 l +73.278 230.92 l +cl 0 0 0 F +n 73.278 230.92 m 73.278 230.91 l +73.277 230.9 l +73.275 230.89 l +73.271 230.88 l +73.267 230.87 l +73.263 230.86 l +73.257 230.86 l +73.251 230.85 l +73.245 230.84 l +73.239 230.84 l +73.232 230.84 l +73.224 230.83 l +73.216 230.83 l +73.208 230.83 l +73.201 230.83 l +73.193 230.83 l +73.185 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228.77 m 72.248 228.76 l +72.247 228.75 l +72.244 228.74 l +72.241 228.73 l +72.232 228.72 l +72.227 228.71 l +72.221 228.71 l +72.215 228.7 l +72.209 228.7 l +72.202 228.69 l +72.194 228.69 l +72.186 228.69 l +72.178 228.68 l +72.171 228.68 l +72.163 228.68 l +72.154 228.68 l +72.146 228.68 l +72.139 228.69 l +72.131 228.69 l +72.124 228.69 l +72.116 228.7 l +72.109 228.7 l +72.104 228.71 l +72.098 228.71 l +72.093 228.72 l +72.088 228.73 l +72.081 228.74 l +72.078 228.75 l +72.077 228.76 l +72.076 228.77 l +72.249 228.77 l +cl 0 0 0 F +n 73.751 230.35 m 73.751 230.34 l +73.749 230.33 l +73.747 230.32 l +73.74 230.31 l +73.735 230.3 l +73.73 230.3 l +73.724 230.3 l +73.717 230.28 l +73.711 230.28 l +73.704 230.27 l +73.696 230.27 l +73.688 230.27 l +73.68 230.27 l +73.673 230.27 l +73.665 230.27 l +73.657 230.27 l +73.648 230.27 l +73.641 230.27 l +73.634 230.27 l +73.626 230.27 l +73.619 230.28 l +73.612 230.28 l +73.606 230.3 l +73.6 230.3 l +73.595 230.3 l +73.59 230.31 l +73.586 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+87.638 222.1 l +87.632 222.09 l +87.626 222.09 l +87.619 222.08 l +87.611 222.08 l +87.605 222.08 l +87.597 222.08 l +87.589 222.07 l +87.581 222.07 l +87.572 222.07 l +87.565 222.07 l +87.557 222.07 l +87.549 222.08 l +87.541 222.08 l +87.534 222.08 l +87.527 222.08 l +87.52 222.09 l +87.514 222.09 l +87.508 222.1 l +87.503 222.11 l +87.498 222.11 l +87.495 222.12 l +87.492 222.13 l +87.489 222.14 l +87.488 222.15 l +87.487 222.16 l +87.659 222.16 l +cl 0 0 0 F +n 87.659 222.16 m 87.658 222.15 l +87.657 222.14 l +87.654 222.13 l +87.651 222.12 l +87.647 222.11 l +87.638 222.1 l +87.632 222.09 l +87.626 222.09 l +87.619 222.08 l +87.611 222.08 l +87.605 222.08 l +87.597 222.08 l +87.589 222.07 l +87.581 222.07 l +87.572 222.07 l +87.565 222.07 l +87.557 222.07 l +87.549 222.08 l +87.541 222.08 l +87.534 222.08 l +87.527 222.08 l +87.52 222.09 l +87.514 222.09 l +87.508 222.1 l +87.503 222.11 l +87.498 222.11 l +87.495 222.12 l +87.492 222.13 l +87.489 222.14 l +87.488 222.15 l +87.487 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+82.505 219.85 l +82.498 219.86 l +82.491 219.86 l +82.485 219.87 l +82.48 219.87 l +82.476 219.88 l +82.472 219.89 l +82.469 219.9 l +82.467 219.91 l +82.465 219.92 l +82.465 219.93 l +82.636 219.93 l +cl 0 0 0 F +n 84.439 219.41 m 84.439 219.4 l +84.437 219.39 l +84.435 219.38 l +84.432 219.37 l +84.428 219.36 l +84.423 219.35 l +84.413 219.34 l +84.406 219.34 l +84.399 219.33 l +84.392 219.33 l +84.385 219.33 l +84.378 219.32 l +84.37 219.32 l +84.361 219.32 l +84.353 219.32 l +84.346 219.32 l +84.338 219.32 l +84.33 219.32 l +84.322 219.33 l +84.314 219.33 l +84.308 219.33 l +84.301 219.34 l +84.295 219.34 l +84.289 219.35 l +84.283 219.35 l +84.279 219.36 l +84.275 219.37 l +84.272 219.38 l +84.27 219.39 l +84.268 219.4 l +84.268 219.41 l +84.439 219.41 l +cl 0 0 0 F +n 84.439 219.41 m 84.439 219.4 l +84.437 219.39 l +84.435 219.38 l +84.432 219.37 l +84.428 219.36 l +84.423 219.35 l +84.413 219.34 l +84.406 219.34 l +84.399 219.33 l +84.392 219.33 l +84.385 219.33 l +84.378 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+84.046 220.53 l +84.042 220.52 l +84.037 220.52 l +84.032 220.51 l +84.026 220.5 l +84.019 220.5 l +84.014 220.49 l +84.006 220.49 l +83.999 220.49 l +83.991 220.48 l +83.983 220.48 l +83.976 220.48 l +83.967 220.48 l +83.959 220.48 l +83.951 220.48 l +83.943 220.48 l +83.936 220.49 l +83.928 220.49 l +83.921 220.49 l +83.914 220.5 l +83.908 220.5 l +83.903 220.51 l +83.897 220.52 l +83.889 220.53 l +83.885 220.54 l +83.883 220.55 l +83.882 220.56 l +83.881 220.57 l +84.053 220.57 l +cl 0 0 0 F +n 84.526 221.6 m 84.525 221.59 l +84.524 221.58 l +84.521 221.57 l +84.518 221.56 l +84.514 221.55 l +84.509 221.54 l +84.504 221.54 l +84.498 221.53 l +84.493 221.53 l +84.486 221.52 l +84.478 221.52 l +84.471 221.51 l +84.463 221.51 l +84.456 221.51 l +84.448 221.51 l +84.439 221.51 l +84.431 221.51 l +84.423 221.51 l +84.416 221.51 l +84.408 221.51 l +84.4 221.52 l +84.393 221.52 l +84.386 221.53 l +84.381 221.53 l +84.375 221.54 l +84.369 221.54 l +84.365 221.55 l +84.361 221.56 l +84.358 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224.4 l +81.01 224.4 l +81.002 224.41 l +80.995 224.41 l +80.989 224.42 l +80.983 224.42 l +80.978 224.42 l +80.974 224.43 l +80.967 224.44 l +80.964 224.45 l +80.963 224.46 l +80.962 224.47 l +81.134 224.47 l +cl 0 0 0 F +n 81.134 224.47 m 81.133 224.46 l +81.132 224.45 l +81.129 224.44 l +81.126 224.43 l +81.119 224.42 l +81.113 224.42 l +81.107 224.42 l +81.101 224.41 l +81.094 224.41 l +81.087 224.4 l +81.08 224.4 l +81.072 224.4 l +81.064 224.4 l +81.056 224.39 l +81.048 224.39 l +81.04 224.39 l +81.032 224.4 l +81.024 224.4 l +81.016 224.4 l +81.01 224.4 l +81.002 224.41 l +80.995 224.41 l +80.989 224.42 l +80.983 224.42 l +80.978 224.42 l +80.974 224.43 l +80.967 224.44 l +80.964 224.45 l +80.963 224.46 l +80.962 224.47 l +81.134 224.47 l +cl 0 0 0 F +n 82.293 224.3 m 82.293 224.29 l +82.292 224.28 l +82.289 224.27 l +82.282 224.26 l +82.277 224.25 l +82.272 224.25 l +82.266 224.24 l +82.26 224.23 l +82.254 224.23 l +82.246 224.23 l +82.239 224.22 l +82.231 224.22 l +82.223 224.22 l +82.216 224.22 l +82.207 224.22 l +82.199 224.22 l +82.191 224.22 l +82.183 224.22 l +82.176 224.22 l +82.168 224.23 l +82.161 224.23 l +82.154 224.23 l +82.148 224.24 l +82.143 224.25 l +82.137 224.25 l +82.133 224.26 l +82.129 224.27 l +82.123 224.28 l +82.122 224.29 l +82.121 224.3 l +82.293 224.3 l +cl 0 0 0 F +n 82.293 224.3 m 82.293 224.29 l +82.292 224.28 l +82.289 224.27 l +82.282 224.26 l +82.277 224.25 l +82.272 224.25 l +82.266 224.24 l +82.26 224.23 l +82.254 224.23 l +82.246 224.23 l +82.239 224.22 l +82.231 224.22 l +82.223 224.22 l +82.216 224.22 l +82.207 224.22 l +82.199 224.22 l +82.191 224.22 l +82.183 224.22 l +82.176 224.22 l +82.168 224.23 l +82.161 224.23 l +82.154 224.23 l +82.148 224.24 l +82.143 224.25 l +82.137 224.25 l +82.133 224.26 l +82.129 224.27 l +82.123 224.28 l +82.122 224.29 l +82.121 224.3 l +82.293 224.3 l +cl 0 0 0 F +n 81.864 225.38 m 81.863 225.37 l +81.862 225.36 l +81.859 225.35 l +81.856 225.34 l +81.853 225.33 l +81.848 225.32 l 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+81.692 225.38 l +81.864 225.38 l +cl 0 0 0 F +n 81.22 226.24 m 81.22 226.23 l +81.218 226.22 l +81.216 226.21 l +81.212 226.2 l +81.208 226.19 l +81.198 226.18 l +81.193 226.17 l +81.187 226.17 l +81.18 226.16 l +81.173 226.16 l +81.165 226.16 l +81.157 226.15 l +81.15 226.15 l +81.142 226.15 l +81.134 226.15 l +81.126 226.15 l +81.118 226.15 l +81.11 226.15 l +81.102 226.16 l +81.095 226.16 l +81.088 226.16 l +81.082 226.17 l +81.075 226.17 l +81.069 226.18 l +81.064 226.19 l +81.059 226.19 l +81.055 226.2 l +81.052 226.21 l +81.05 226.22 l +81.048 226.23 l +81.048 226.24 l +81.22 226.24 l +cl 0 0 0 F +n 81.22 226.24 m 81.22 226.23 l +81.218 226.22 l +81.216 226.21 l +81.212 226.2 l +81.208 226.19 l +81.198 226.18 l +81.193 226.17 l +81.187 226.17 l +81.18 226.16 l +81.173 226.16 l +81.165 226.16 l +81.157 226.15 l +81.15 226.15 l +81.142 226.15 l +81.134 226.15 l +81.126 226.15 l +81.118 226.15 l +81.11 226.15 l +81.102 226.16 l +81.095 226.16 l +81.088 226.16 l +81.082 226.17 l +81.075 226.17 l +81.069 226.18 l +81.064 226.19 l +81.059 226.19 l +81.055 226.2 l +81.052 226.21 l +81.05 226.22 l +81.048 226.23 l +81.048 226.24 l +81.22 226.24 l +cl 0 0 0 F +n 87.616 225.59 m 87.615 225.58 l +87.614 225.57 l +87.611 225.56 l +87.608 225.55 l +87.605 225.54 l +87.6 225.54 l +87.595 225.53 l +87.589 225.52 l +87.583 225.52 l +87.576 225.51 l +87.569 225.51 l +87.562 225.51 l +87.554 225.5 l +87.546 225.5 l +87.538 225.5 l +87.53 225.5 l +87.522 225.5 l +87.514 225.5 l +87.506 225.5 l +87.498 225.51 l +87.491 225.51 l +87.484 225.51 l +87.477 225.52 l +87.471 225.52 l +87.465 225.53 l +87.46 225.54 l +87.452 225.55 l +87.449 225.56 l +87.446 225.57 l +87.445 225.58 l +87.444 225.59 l +87.616 225.59 l +cl 0 0 0 F +n 87.616 225.59 m 87.615 225.58 l +87.614 225.57 l +87.611 225.56 l +87.608 225.55 l +87.605 225.54 l +87.6 225.54 l +87.595 225.53 l +87.589 225.52 l +87.583 225.52 l +87.576 225.51 l +87.569 225.51 l +87.562 225.51 l +87.554 225.5 l +87.546 225.5 l 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+86.238 224.57 l +86.231 224.57 l +86.223 224.56 l +86.215 224.56 l +86.208 224.56 l +86.199 224.56 l +86.191 224.56 l +86.183 224.56 l +86.176 224.56 l +86.168 224.57 l +86.16 224.57 l +86.153 224.57 l +86.146 224.58 l +86.141 224.58 l +86.135 224.59 l +86.129 224.59 l +86.125 224.6 l +86.121 224.61 l +86.118 224.62 l +86.115 224.63 l +86.114 224.64 l +86.113 224.65 l +86.286 224.65 l +cl 0 0 0 F +n 86.371 227.66 m 86.37 227.64 l +86.369 227.63 l +86.366 227.62 l +86.36 227.61 l +86.355 227.6 l +86.35 227.6 l +86.344 227.59 l +86.338 227.58 l +86.331 227.58 l +86.325 227.58 l +86.317 227.57 l +86.309 227.57 l +86.301 227.57 l +86.293 227.57 l +86.286 227.57 l +86.277 227.57 l +86.269 227.57 l +86.261 227.57 l +86.253 227.57 l +86.247 227.58 l +86.239 227.58 l +86.232 227.58 l +86.226 227.59 l +86.22 227.6 l +86.215 227.6 l +86.211 227.61 l +86.207 227.62 l +86.201 227.63 l +86.2 227.64 l +86.199 227.66 l +86.371 227.66 l +cl 0 0 0 F +n 86.371 227.66 m 86.37 227.64 l +86.369 227.63 l 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l +86.309 228.69 l +86.301 228.68 l +86.293 228.68 l +86.286 228.68 l +86.277 228.68 l +86.269 228.68 l +86.261 228.69 l +86.253 228.69 l +86.247 228.69 l +86.239 228.7 l +86.232 228.7 l +86.226 228.71 l +86.22 228.71 l +86.215 228.72 l +86.211 228.73 l +86.204 228.74 l +86.201 228.75 l +86.2 228.76 l +86.199 228.77 l +86.371 228.77 l +cl 0 0 0 F +n 86.371 228.77 m 86.37 228.76 l +86.369 228.75 l +86.366 228.74 l +86.363 228.73 l +86.355 228.72 l +86.35 228.71 l +86.344 228.71 l +86.338 228.7 l +86.331 228.7 l +86.325 228.69 l +86.317 228.69 l +86.309 228.69 l +86.301 228.68 l +86.293 228.68 l +86.286 228.68 l +86.277 228.68 l +86.269 228.68 l +86.261 228.69 l +86.253 228.69 l +86.247 228.69 l +86.239 228.7 l +86.232 228.7 l +86.226 228.71 l +86.22 228.71 l +86.215 228.72 l +86.211 228.73 l +86.204 228.74 l +86.201 228.75 l +86.2 228.76 l +86.199 228.77 l +86.371 228.77 l +cl 0 0 0 F +n 88.131 224.74 m 88.13 224.73 l +88.129 224.72 l +88.126 224.71 l +88.123 224.7 l +88.12 224.69 l +88.115 224.68 l +88.11 224.68 l +88.104 224.67 l +88.098 224.67 l +88.091 224.66 l +88.084 224.66 l +88.077 224.65 l +88.069 224.65 l +88.061 224.65 l +88.053 224.65 l +88.046 224.65 l +88.037 224.65 l +88.029 224.65 l +88.021 224.65 l +88.013 224.65 l +88.007 224.66 l +87.999 224.66 l +87.992 224.67 l +87.986 224.67 l +87.98 224.68 l +87.975 224.68 l +87.971 224.69 l +87.967 224.7 l +87.964 224.71 l +87.961 224.72 l +87.96 224.73 l +87.959 224.74 l +88.131 224.74 l +cl 0 0 0 F +n 88.131 224.74 m 88.13 224.73 l +88.129 224.72 l +88.126 224.71 l +88.123 224.7 l +88.12 224.69 l +88.115 224.68 l +88.11 224.68 l +88.104 224.67 l +88.098 224.67 l +88.091 224.66 l +88.084 224.66 l +88.077 224.65 l +88.069 224.65 l +88.061 224.65 l +88.053 224.65 l +88.046 224.65 l +88.037 224.65 l +88.029 224.65 l +88.021 224.65 l +88.013 224.65 l +88.007 224.66 l +87.999 224.66 l +87.992 224.67 l +87.986 224.67 l +87.98 224.68 l +87.975 224.68 l +87.971 224.69 l +87.967 224.7 l +87.964 224.71 l +87.961 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226.58 l +cl 0 0 0 F +n 83.065 226.58 m 83.065 226.57 l +83.063 226.56 l +83.062 226.55 l +83.059 226.54 l +83.05 226.53 l +83.045 226.52 l +83.039 226.52 l +83.032 226.51 l +83.026 226.51 l +83.019 226.5 l +83.011 226.5 l +83.004 226.5 l +82.996 226.5 l +82.988 226.49 l +82.98 226.49 l +82.972 226.49 l +82.964 226.5 l +82.956 226.5 l +82.949 226.5 l +82.941 226.5 l +82.934 226.51 l +82.927 226.51 l +82.92 226.52 l +82.915 226.52 l +82.91 226.53 l +82.905 226.54 l +82.898 226.55 l +82.896 226.56 l +82.894 226.57 l +82.894 226.58 l +83.065 226.58 l +cl 0 0 0 F +n 84.053 228.55 m 84.053 228.54 l +84.051 228.53 l +84.049 228.52 l +84.046 228.51 l +84.042 228.5 l +84.037 228.5 l +84.032 228.49 l +84.026 228.48 l +84.019 228.48 l +84.014 228.48 l +84.006 228.47 l +83.999 228.47 l +83.991 228.47 l +83.983 228.46 l +83.976 228.46 l +83.967 228.46 l +83.959 228.46 l +83.951 228.46 l +83.943 228.47 l +83.936 228.47 l +83.928 228.47 l +83.921 228.48 l +83.914 228.48 l +83.908 228.48 l +83.903 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{setfont} bdef +/SG {setgray} bdef +/TR {translate} bdef +/bp {lj lw rgb} bdef +/bpbw {lj lw setgray} bdef +/c {curveto} bdef +/cl {closepath} bdef +/fi {eofill} bdef +/g {setgray} bdef +/gr {grestore} bdef +/gs {gsave} bdef +/l {lineto} bdef +/lj {setlinejoin} bdef +/lw {setlinewidth} bdef +/m {moveto} bdef +/n {newpath} bdef +/nx {/x E def} bdef +/r {rmoveto} bdef +/rl {rlineto} bdef +/rgb {setrgbcolor} bdef +/s {show} bdef +/sd {setdash} bdef +/sp {x 0 rmoveto} bdef +/ss {currentpoint pop E m} bdef +/st {stroke} bdef +/BPSIDE 32 def %% pixels per pattern side +/PATFREQ 3.0 def %% pattern pixels per mm +/dp_mat [PATFREQ 0 0 PATFREQ 0 0] def +/dp_pw BPSIDE def %% pattern pixel width +/dp_ph BPSIDE def %% pattern pixel height +/dp_w dp_pw PATFREQ div def %% pattern mm width +/dp_h dp_ph PATFREQ div def %% pattern mm height +/savemat matrix def +/topmat matrix def +/patmat matrix def +/ncpoint errordict /nocurrentpoint get def +errordict begin +/nocurrentpoint { + dup /pathbbox load eq + {pop 0 0 1 1} + {ncpoint} + ifelse +} bdef +end +/ar { %% sa ea sx sy rot tx ty + matrix currentmatrix 8 1 roll TR RO SC + n 0 0 1 5 3 roll arc setmatrix +} bdef +/arn { %% sa ea sx sy rot tx ty + TR RO SC + matrix currentmatrix 8 1 roll + n 0 0 1 5 3 roll arcn setmatrix +} bdef +/el { %% sx sy rot tx ty + matrix currentmatrix 6 1 roll TR RO SC + n 0 0 1 0 360 arc setmatrix cl +} bdef +/image_raster { %% sw sh sd dw dh xs ys + TR SC /sd E def /sh E def /sw E def + /imagebuf sw sd mul 7 add 8 idiv string def + sw sh sd [sw 0 0 sh 0 0] { currentfile imagebuf readhexstring pop} + image +} bdef +/imagemask_raster { + TR SC /sh E def /sw E def + /imagebuf sw 7 add 8 idiv string def + sw sh false [sw 0 0 sh 0 0] + {currentfile imagebuf readhexstring pop} + imagemask +} bdef +/dither_color_raster { % bool sw sh sd dw dh xs ys + TR SC /sd E def /sh E def /sw E def + sd 8 eq and + { + /imagebuf 3 string def + /grayval 1 string def + sw sh sd [sw 0 0 sh 0 0] + { + currentfile imagebuf readhexstring pop pop + imagebuf 0 get 0.299 mul + imagebuf 1 get 0.587 mul add + imagebuf 2 get 0.114 mul add cvi grayval exch 0 exch put grayval + } + image + } + { + /imagebuf sw 3 mul sd mul 7 add 8 idiv string def + sh { currentfile imagebuf readhexstring pop pop } repeat + } ifelse +} bdef +/image_color_raster { % bool sw sh sd dw dh xs ys + /colorimage where not + { dither_color_raster } + { + pop + TR SC /sd E def /sh E def /sw E def pop + /imagebuf sw 3 mul sd mul 7 add 8 idiv string def + sw sh sd [sw 0 0 sh 0 0] { currentfile imagebuf readhexstring pop} + false 3 colorimage + } ifelse +} bdef +/patpath { + /inv E def + topmat setmatrix + pathbbox %% get lo - hi indecies + /hy E dp_h div floor cvi def + /hx E dp_w div floor cvi def + /ly E dp_h div floor cvi def + /lx E dp_w div floor cvi def + lx 1 hx { + dp_w mul + ly 1 hy { + dp_h mul + E dup 3 1 roll E + patmat currentmatrix pop + TR + dp_pw dp_ph inv + dp_mat dp_proc imagemask + patmat setmatrix + } for + pop + } for +} bdef +% setpattern brush of patterns instead of gray +/setpattern { + /blue E def /green E def /red E def + /freq E def /bwidth E def /bpside E def + /bstring E def + /onbits 0 def /offbits 0 def + freq 0 {/y E def /x E def + /xindex x 1 add 2 div bpside mul cvi def + /yindex y 1 add 2 div bpside mul cvi def + bstring yindex bwidth mul xindex 8 idiv add get not + 1 7 xindex 8 mod sub bitshift and 0 ne + {/onbits onbits 1 add def 1} + {/offbits offbits 1 add def 0} + ifelse + } setscreen {} settransfer + systemdict /setcmykcolor known + { /fact 1 onbits offbits onbits add div sub def + 1 red sub fact mul 1 green sub fact mul 1 blue sub fact mul 0 + setcmykcolor + } + { offbits offbits onbits add div setgray} + ifelse +} bdef +/dmatrix matrix def +/dpi 72 0 dmatrix defaultmatrix dtransform + dup mul E dup mul add sqrt +def +/B {gs bp st gr} bdef %% brush: gr lw lj +/Bbw {gs bpbw st gr} bdef %% brush: gr lw lj +/F {gs rgb eofill gr} bdef %% fill: gr +/Fbw {gs setgray eofill gr} bdef %% fill: gr +/PB {gs lj lw setpattern st gr} bdef +/PF {gs eoclip patpath gr} bdef +/BB {gs rgb lj lw strokepath clip patpath gr} bdef +/xdef {exch def} bdef +/clip_region { + /ht xdef + /wd xdef + /bm xdef + /lm xdef + newpath + lm bm moveto + 0 ht rlineto + wd 0 rlineto + 0 ht neg rlineto + closepath clip +} bdef +/reencode_small_dict 12 dict def +/ReencodeSmall { +reencode_small_dict begin +/new_codes_and_names exch def +/new_font_name exch def +/base_font_name exch def +/base_font_dict base_font_name findfont def +/newfont base_font_dict maxlength dict def +base_font_dict { +exch dup /FID ne +{ dup /Encoding eq +{ exch dup length array copy newfont 3 1 roll put } +{ exch newfont 3 1 roll put } +ifelse +} +{ pop pop } +ifelse +} forall +newfont /FontName new_font_name put +new_codes_and_names aload pop +new_codes_and_names length 2 idiv +{ newfont /Encoding get 3 1 roll put } +repeat +new_font_name newfont definefont pop +end %reencode_small_dict +} def +/extended_Zapf [ +8#223 /a89 +8#224 /a90 +8#225 /a93 +8#226 /a94 +8#227 /a91 +8#230 /a92 +8#231 /a205 +8#232 /a85 +8#233 /a206 +8#234 /a86 +8#235 /a87 +8#236 /a88 +8#237 /a95 +8#240 /a96 +] def +/extended_Standard [ +29 /thorn +30 /yacute +31 /divide +128 /Acircumflex +129 /Adieresis +130 /Agrave +131 /Aring +132 /Atilde +133 /Ccedilla +134 /Eacute +135 /Ecircumflex +136 /Edieresis +137 /Egrave +138 /Iacute +139 /Icircumflex +140 /Idieresis +141 /Igrave +142 /Ntilde +143 /Oacute +144 /Ocircumflex +145 /Odieresis +146 /Ograve +147 /Otilde +148 /Scaron +149 /Uacute +150 /Ucircumflex +151 /Udieresis +152 /Ugrave +153 /Ydieresis +154 /Zcaron +155 /aacute +156 /acircumflex +157 /adieresis +158 /agrave +159 /aring +160 /atilde +161 /exclamdown +162 /cent +163 /sterling +164 /fraction +165 /yen +166 /florin +167 /section +168 /currency +169 /quotesingle +170 /quotedblleft +171 /guillemotleft +172 /guilsinglleft +173 /guilsinglright +174 /fi +175 /fl +176 /plusminus +177 /endash +178 /dagger +179 /daggerdbl +180 /periodcentered +181 /twosuperior +182 /paragraph +183 /bullet +184 /quotesinglbase +185 /quotedblbase +186 /quotedblright +187 /guillemotright +188 /ellipsis +189 /perthousand +190 /threesuperior +191 /questiondown +192 /mu +193 /grave +194 /acute +195 /circumflex +196 /tilde +197 /macron +198 /breve +199 /dotaccent +200 /dieresis +201 /onesuperior +202 /ring +203 /cedilla +204 /onequarter +205 /hungarumlaut +206 /ogonek +207 /caron +208 /emdash +209 /ccedilla +210 /copyright +211 /eacute +212 /ecircumflex +213 /edieresis +214 /egrave +215 /iacute +216 /icircumflex +217 /idieresis +218 /igrave +219 /logicalnot +220 /minus +221 /ntilde +222 /oacute +223 /ocircumflex +224 /odieresis +225 /AE +226 /onehalf +227 /ordfeminine +228 /ograve +229 /otilde +230 /registered +231 /scaron +232 /Lslash +233 /Oslash +234 /OE +235 /ordmasculine +236 /trademark +237 /uacute +238 /ucircumflex +239 /udieresis +240 /ugrave +241 /ae +242 /ydieresis +243 /zcaron +244 /Aacute +245 /dotlessi +246 /threequarters +247 /Eth +248 /lslash +249 /oslash +250 /oe +251 /germandbls +252 /multiply +253 /Yacute +254 /Thorn +255 /eth +] def +/extended_Symbol [ +] def +/extend_font { % stack: fontname newfontname +exch dup (ZapfDingbats) eq +{ cvn exch cvn extended_Zapf ReencodeSmall } +{ dup (Symbol) eq +{ cvn exch cvn extended_Symbol ReencodeSmall } +{ cvn exch cvn extended_Standard ReencodeSmall } +ifelse +} +ifelse +} bind def +/extend_font_name { % stack: font_name_string +dup length 1 add string /extended_font_name exch def +extended_font_name 0 (_) putinterval +extended_font_name 1 3 -1 roll putinterval +extended_font_name +} bind def +/gf { +/f exch def f cvn where +{ f exch begin cvn load exec setfont end } +{ f 0 f length 8 sub getinterval dup +/localfont exch extend_font_name def +localfont extend_font +localfont findfont +/xsz f f length 4 sub 4 getinterval cvi def +/ysz f f length 8 sub 4 getinterval cvi def +[ xsz 0 0 ysz neg 0 0 ] makefont dup f cvn exch def +setfont +} +ifelse +} bind def +/gfns { +/f exch def f cvn where +{ f exch begin cvn load exec setfont end } +{ f 0 f length 8 sub getinterval cvn findfont +/xsz f f length 4 sub 4 getinterval cvi def +/ysz f f length 8 sub 4 getinterval cvi def +[ xsz 0 0 ysz neg 0 0 ] makefont dup f cvn exch def +setfont +} +ifelse +} bind def +/ul { % space drop thickness +gs currentpoint currentlinewidth +currentpoint n m 6 -3 roll +lw 0 exch r +0 rl st lw m +gr +} bind def +/nxtab { currentpoint pop 1000.0 mul cvi tab mod + tab exch sub 1000.0 div 0 rmoveto } bind def +/nx { /x exch def } bind def +0. nx +gsave +2.83465 -2.83465 scale 0 -279.4 translate +topmat currentmatrix pop +n 77.012 79.367 m 77.482 79.069 l +77.936 78.716 l +78.373 78.309 l +78.791 77.85 l +79.187 77.343 l +79.56 76.788 l +79.907 76.189 l +80.227 75.546 l +80.516 74.863 l +80.774 74.142 l +80.998 73.384 l +81.186 72.593 l +81.336 71.769 l +gsave +0 0 0 0.352 1 B +grestore +n 81.336 71.769 m 81.437 70.994 l +81.5 70.224 l +81.527 69.46 l +81.518 68.707 l +81.474 67.966 l +81.397 67.24 l +81.286 66.531 l +81.144 65.842 l +80.972 65.175 l +80.769 64.534 l +80.537 63.92 l +80.278 63.336 l +79.992 62.785 l +gsave +0 0 0 0.352 1 B +grestore +n 79.341 62.29 m 79.46 62.825 79.499 63.194 79.558 63.644 c +80.955 62.861 l +80.819 62.765 80.307 62.35 79.912 61.97 c +79.49 61.563 79.139 61.175 78.942 60.907 c +79.067 61.216 79.215 61.717 79.341 62.29 c +cl 0 0 0 F +n 151.8 32.807 m 143.22 32.807 l +gsave +0 0 0 0.352 0 B +grestore +n 142.47 33.132 m 143 33.29 143.33 33.436 143.75 33.605 c +143.75 32.004 l +143.6 32.075 143 32.319 142.47 32.477 c +141.91 32.647 141.4 32.763 141.07 32.805 c +141.4 32.846 141.91 32.962 142.47 33.132 c +cl 0 0 0 F +n 151.8 60.29 m 143.22 60.29 l +gsave +0 0 0 0.352 0 B +grestore +n 142.47 60.615 m 143 60.774 143.33 60.92 143.75 61.089 c +143.75 59.487 l +143.6 59.559 143 59.802 142.47 59.961 c +141.91 60.131 141.4 60.247 141.07 60.288 c +141.4 60.33 141.91 60.446 142.47 60.615 c +cl 0 0 0 F +n 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l +86.105 225.92 l +86.103 225.91 l +86.1 225.9 l +86.091 225.89 l +86.086 225.88 l +86.08 225.88 l +86.073 225.87 l +86.066 225.87 l +86.059 225.86 l +86.051 225.86 l +86.043 225.86 l +86.035 225.86 l +86.027 225.86 l +86.019 225.86 l +86.011 225.86 l +86.002 225.86 l +85.994 225.86 l +85.987 225.86 l +85.979 225.86 l +85.972 225.87 l +85.965 225.87 l +85.958 225.88 l +85.952 225.88 l +85.947 225.89 l +85.942 225.9 l +85.935 225.91 l +85.933 225.92 l +85.931 225.93 l +85.931 225.94 l +86.107 225.94 l +cl 0 0 0 F +n 86.107 225.94 m 86.107 225.93 l +86.105 225.92 l +86.103 225.91 l +86.1 225.9 l +86.091 225.89 l +86.086 225.88 l +86.08 225.88 l +86.073 225.87 l +86.066 225.87 l +86.059 225.86 l +86.051 225.86 l +86.043 225.86 l +86.035 225.86 l +86.027 225.86 l +86.019 225.86 l +86.011 225.86 l +86.002 225.86 l +85.994 225.86 l +85.987 225.86 l +85.979 225.86 l +85.972 225.87 l +85.965 225.87 l +85.958 225.88 l +85.952 225.88 l +85.947 225.89 l +85.942 225.9 l +85.935 225.91 l +85.933 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+85.642 240.67 l +85.634 240.68 l +85.626 240.68 l +85.619 240.68 l +85.612 240.69 l +85.606 240.69 l +85.6 240.7 l +85.594 240.71 l +85.586 240.72 l +85.582 240.73 l +85.58 240.74 l +85.578 240.75 l +85.578 240.76 l +85.754 240.76 l +cl 0 0 0 F +n 86.063 240.19 m 86.063 240.18 l +86.061 240.17 l +86.059 240.16 l +86.055 240.15 l +86.051 240.14 l +86.047 240.13 l +86.041 240.13 l +86.035 240.12 l +86.029 240.12 l +86.022 240.11 l +86.015 240.11 l +86.007 240.1 l +85.999 240.1 l +85.991 240.1 l +85.983 240.1 l +85.975 240.1 l +85.967 240.1 l +85.958 240.1 l +85.95 240.1 l +85.942 240.1 l +85.935 240.11 l +85.928 240.11 l +85.921 240.12 l +85.914 240.12 l +85.908 240.13 l +85.903 240.13 l +85.898 240.14 l +85.894 240.15 l +85.891 240.16 l +85.889 240.17 l +85.887 240.18 l +85.887 240.19 l +86.063 240.19 l +cl 0 0 0 F +n 86.063 240.19 m 86.063 240.18 l +86.061 240.17 l +86.059 240.16 l +86.055 240.15 l +86.051 240.14 l +86.047 240.13 l +86.041 240.13 l +86.035 240.12 l +86.029 240.12 l 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l +79.665 215.46 l +79.657 215.45 l +79.649 215.45 l +79.641 215.45 l +79.633 215.45 l +79.625 215.45 l +79.617 215.45 l +79.608 215.45 l +79.6 215.45 l +79.592 215.45 l +79.585 215.46 l +79.578 215.46 l +79.571 215.47 l +79.564 215.47 l +79.558 215.48 l +79.553 215.48 l +79.548 215.49 l +79.544 215.5 l +79.541 215.51 l +79.537 215.53 l +79.537 215.54 l +79.713 215.54 l +cl 0 0 0 F +n 80.022 214.96 m 80.021 214.95 l +80.02 214.94 l +80.017 214.93 l +80.014 214.92 l +80.005 214.91 l +80 214.9 l +79.994 214.9 l +79.988 214.89 l +79.981 214.89 l +79.974 214.88 l +79.966 214.88 l +79.958 214.88 l +79.95 214.88 l +79.942 214.88 l +79.934 214.88 l +79.925 214.88 l +79.917 214.88 l +79.909 214.88 l +79.901 214.88 l +79.894 214.88 l +79.886 214.89 l +79.879 214.89 l +79.873 214.9 l +79.867 214.9 l +79.862 214.91 l +79.857 214.92 l +79.85 214.93 l +79.847 214.94 l +79.846 214.95 l +79.845 214.96 l +80.022 214.96 l +cl 0 0 0 F +n 80.022 214.96 m 80.021 214.95 l +80.02 214.94 l +80.017 214.93 l 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l +83.541 215.84 l +83.533 215.85 l +83.525 215.85 l +83.517 215.85 l +83.509 215.85 l +83.502 215.86 l +83.495 215.86 l +83.489 215.87 l +83.483 215.87 l +83.478 215.88 l +83.473 215.89 l +83.466 215.9 l +83.463 215.91 l +83.462 215.92 l +83.461 215.93 l +83.638 215.93 l +cl 0 0 0 F +n 82.535 218.18 m 82.535 218.17 l +82.533 218.16 l +82.531 218.15 l +82.528 218.14 l +82.519 218.13 l +82.514 218.12 l +82.508 218.12 l +82.501 218.11 l +82.494 218.11 l +82.487 218.1 l +82.479 218.1 l +82.472 218.1 l +82.464 218.09 l +82.455 218.09 l +82.447 218.09 l +82.439 218.09 l +82.431 218.09 l +82.423 218.1 l +82.415 218.1 l +82.407 218.1 l +82.4 218.11 l +82.393 218.11 l +82.386 218.12 l +82.38 218.12 l +82.375 218.13 l +82.37 218.14 l +82.363 218.15 l +82.361 218.16 l +82.359 218.17 l +82.359 218.18 l +82.535 218.18 l +cl 0 0 0 F +n 82.535 218.18 m 82.535 218.17 l +82.533 218.16 l +82.531 218.15 l +82.528 218.14 l +82.519 218.13 l +82.514 218.12 l +82.508 218.12 l +82.501 218.11 l +82.494 218.11 l +82.487 218.1 l +82.479 218.1 l +82.472 218.1 l +82.464 218.09 l +82.455 218.09 l +82.447 218.09 l +82.439 218.09 l +82.431 218.09 l +82.423 218.1 l +82.415 218.1 l +82.407 218.1 l +82.4 218.11 l +82.393 218.11 l +82.386 218.12 l +82.38 218.12 l +82.375 218.13 l +82.37 218.14 l +82.363 218.15 l +82.361 218.16 l +82.359 218.17 l +82.359 218.18 l +82.535 218.18 l +cl 0 0 0 F +n 83.197 217.56 m 83.196 217.55 l +83.195 217.54 l +83.192 217.53 l +83.185 217.52 l +83.18 217.51 l +83.175 217.5 l +83.169 217.5 l +83.163 217.49 l +83.156 217.49 l +83.149 217.48 l +83.141 217.48 l +83.133 217.48 l +83.125 217.48 l +83.117 217.48 l +83.109 217.48 l +83.1 217.48 l +83.092 217.48 l +83.084 217.48 l +83.076 217.48 l +83.069 217.48 l +83.061 217.49 l +83.054 217.49 l +83.048 217.5 l +83.042 217.5 l +83.037 217.51 l +83.032 217.52 l +83.028 217.53 l +83.022 217.54 l +83.021 217.55 l +83.02 217.56 l +83.197 217.56 l +cl 0 0 0 F +n 83.197 217.56 m 83.196 217.55 l +83.195 217.54 l +83.192 217.53 l 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l +83.506 211.6 l +83.505 211.61 l +83.682 211.61 l +cl 0 0 0 F +n 83.682 211.61 m 83.681 211.6 l +83.68 211.59 l +83.677 211.58 l +83.674 211.57 l +83.67 211.56 l +83.665 211.56 l +83.66 211.55 l +83.654 211.55 l +83.648 211.54 l +83.641 211.54 l +83.634 211.53 l +83.626 211.53 l +83.618 211.53 l +83.61 211.53 l +83.602 211.52 l +83.594 211.52 l +83.585 211.52 l +83.577 211.53 l +83.569 211.53 l +83.561 211.53 l +83.554 211.53 l +83.546 211.54 l +83.539 211.54 l +83.533 211.55 l +83.527 211.55 l +83.522 211.56 l +83.517 211.56 l +83.513 211.57 l +83.51 211.58 l +83.507 211.59 l +83.506 211.6 l +83.505 211.61 l +83.682 211.61 l +cl 0 0 0 F +n 81.036 214.21 m 81.035 214.2 l +81.034 214.19 l +81.032 214.18 l +81.028 214.17 l +81.02 214.16 l +81.014 214.15 l +81.008 214.15 l +81.002 214.14 l +80.995 214.14 l +80.988 214.13 l +80.98 214.13 l +80.972 214.13 l +80.964 214.13 l +80.956 214.13 l +80.948 214.13 l +80.939 214.13 l +80.931 214.13 l +80.923 214.13 l +80.915 214.13 l +80.908 214.13 l +80.9 214.14 l +80.894 214.14 l +80.887 214.15 l +80.881 214.15 l +80.876 214.16 l +80.871 214.17 l +80.864 214.18 l +80.862 214.19 l +80.86 214.2 l +80.86 214.21 l +81.036 214.21 l +cl 0 0 0 F +n 81.036 214.21 m 81.035 214.2 l +81.034 214.19 l +81.032 214.18 l +81.028 214.17 l +81.02 214.16 l +81.014 214.15 l +81.008 214.15 l +81.002 214.14 l +80.995 214.14 l +80.988 214.13 l +80.98 214.13 l +80.972 214.13 l +80.964 214.13 l +80.956 214.13 l +80.948 214.13 l +80.939 214.13 l +80.931 214.13 l +80.923 214.13 l +80.915 214.13 l +80.908 214.13 l +80.9 214.14 l +80.894 214.14 l +80.887 214.15 l +80.881 214.15 l +80.876 214.16 l +80.871 214.17 l +80.864 214.18 l +80.862 214.19 l +80.86 214.2 l +80.86 214.21 l +81.036 214.21 l +cl 0 0 0 F +n 79.096 209.72 m 79.095 209.71 l +79.094 209.7 l +79.091 209.69 l +79.088 209.68 l +79.084 209.67 l +79.079 209.66 l +79.074 209.65 l +79.068 209.65 l +79.062 209.64 l +79.055 209.64 l +79.047 209.64 l +79.04 209.63 l +79.032 209.63 l +79.024 209.63 l 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l +78.643 213.02 l +78.638 213.01 l +78.633 213.01 l +78.627 213 l +78.621 212.99 l +78.614 212.99 l +78.607 212.99 l +78.599 212.98 l +78.591 212.98 l +78.583 212.98 l +78.575 212.98 l +78.567 212.98 l +78.558 212.98 l +78.55 212.98 l +78.542 212.98 l +78.534 212.98 l +78.527 212.99 l +78.519 212.99 l +78.512 212.99 l +78.506 213 l +78.5 213.01 l +78.495 213.01 l +78.49 213.02 l +78.486 213.03 l +78.483 213.04 l +78.48 213.05 l +78.479 213.06 l +78.478 213.07 l +78.655 213.07 l +cl 0 0 0 F +n 78.655 213.07 m 78.654 213.06 l +78.653 213.05 l +78.65 213.04 l +78.647 213.03 l +78.643 213.02 l +78.638 213.01 l +78.633 213.01 l +78.627 213 l +78.621 212.99 l +78.614 212.99 l +78.607 212.99 l +78.599 212.98 l +78.591 212.98 l +78.583 212.98 l +78.575 212.98 l +78.567 212.98 l +78.558 212.98 l +78.55 212.98 l +78.542 212.98 l +78.534 212.98 l +78.527 212.99 l +78.519 212.99 l +78.512 212.99 l +78.506 213 l +78.5 213.01 l +78.495 213.01 l +78.49 213.02 l +78.486 213.03 l +78.483 213.04 l 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l +78.512 215.47 l +78.506 215.47 l +78.5 215.48 l +78.495 215.48 l +78.49 215.49 l +78.486 215.5 l +78.483 215.51 l +78.479 215.53 l +78.478 215.54 l +78.655 215.54 l +cl 0 0 0 F +n 78.655 217.74 m 78.654 217.73 l +78.653 217.72 l +78.65 217.71 l +78.647 217.7 l +78.643 217.69 l +78.638 217.69 l +78.633 217.68 l +78.627 217.67 l +78.621 217.67 l +78.614 217.66 l +78.607 217.66 l +78.599 217.66 l +78.591 217.66 l +78.583 217.65 l +78.575 217.65 l +78.567 217.65 l +78.558 217.65 l +78.55 217.65 l +78.542 217.66 l +78.534 217.66 l +78.527 217.66 l +78.519 217.66 l +78.512 217.67 l +78.506 217.67 l +78.5 217.68 l +78.495 217.69 l +78.49 217.69 l +78.486 217.7 l +78.483 217.71 l +78.48 217.72 l +78.479 217.73 l +78.478 217.74 l +78.655 217.74 l +cl 0 0 0 F +n 78.655 217.74 m 78.654 217.73 l +78.653 217.72 l +78.65 217.71 l +78.647 217.7 l +78.643 217.69 l +78.638 217.69 l +78.633 217.68 l +78.627 217.67 l +78.621 217.67 l +78.614 217.66 l +78.607 217.66 l +78.599 217.66 l +78.591 217.66 l 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213.2 l +84.079 213.2 l +cl 0 0 0 F +n 84.079 213.2 m 84.078 213.19 l +84.077 213.18 l +84.074 213.17 l +84.071 213.16 l +84.067 213.15 l +84.062 213.14 l +84.057 213.14 l +84.051 213.13 l +84.045 213.13 l +84.038 213.12 l +84.03 213.12 l +84.023 213.12 l +84.015 213.11 l +84.007 213.11 l +83.999 213.11 l +83.99 213.11 l +83.982 213.11 l +83.974 213.11 l +83.966 213.11 l +83.958 213.12 l +83.95 213.12 l +83.943 213.12 l +83.936 213.13 l +83.93 213.13 l +83.924 213.14 l +83.919 213.14 l +83.914 213.15 l +83.91 213.16 l +83.907 213.17 l +83.904 213.18 l +83.903 213.19 l +83.902 213.2 l +84.079 213.2 l +cl 0 0 0 F +n 79.829 210.27 m 79.823 210.27 l +79.818 210.28 l +79.813 210.29 l +79.806 210.3 l +79.803 210.31 l +79.802 210.32 l +79.801 210.33 l +79.867 210.33 l +79.83 210.27 l +cl 0 0 0 F +n 79.829 210.27 m 79.823 210.27 l +79.818 210.28 l +79.813 210.29 l +79.806 210.3 l +79.803 210.31 l +79.802 210.32 l +79.801 210.33 l +79.867 210.33 l +79.83 210.27 l +cl 0 0 0 F +n 80.815 217.87 m 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l +80.641 217.85 l +80.64 217.86 l +80.639 217.87 l +80.815 217.87 l +cl 0 0 0 F +n 81.433 216.42 m 81.432 216.41 l +81.431 216.4 l +81.428 216.39 l +81.425 216.38 l +81.421 216.37 l +81.417 216.36 l +81.411 216.36 l +81.405 216.35 l +81.399 216.35 l +81.392 216.34 l +81.385 216.34 l +81.377 216.33 l +81.369 216.33 l +81.361 216.33 l +81.353 216.33 l +81.345 216.33 l +81.336 216.33 l +81.328 216.33 l +81.32 216.33 l +81.312 216.33 l +81.305 216.34 l +81.297 216.34 l +81.29 216.35 l +81.284 216.35 l +81.278 216.36 l +81.273 216.36 l +81.268 216.37 l +81.264 216.38 l +81.261 216.39 l +81.258 216.4 l +81.257 216.41 l +81.256 216.42 l +81.433 216.42 l +cl 0 0 0 F +n 81.433 216.42 m 81.432 216.41 l +81.431 216.4 l +81.428 216.39 l +81.425 216.38 l +81.421 216.37 l +81.417 216.36 l +81.411 216.36 l +81.405 216.35 l +81.399 216.35 l +81.392 216.34 l +81.385 216.34 l +81.377 216.33 l +81.369 216.33 l +81.361 216.33 l +81.353 216.33 l +81.345 216.33 l +81.336 216.33 l +81.328 216.33 l +81.32 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216.95 l +cl 0 0 0 F +n 80.066 216.95 m 80.065 216.94 l +80.064 216.93 l +80.061 216.92 l +80.058 216.91 l +80.054 216.9 l +80.05 216.89 l +80.044 216.89 l +80.038 216.88 l +80.032 216.88 l +80.025 216.87 l +80.018 216.87 l +80.01 216.86 l +80.002 216.86 l +79.994 216.86 l +79.986 216.86 l +79.978 216.86 l +79.969 216.86 l +79.961 216.86 l +79.953 216.86 l +79.945 216.86 l +79.938 216.87 l +79.93 216.87 l +79.923 216.88 l +79.917 216.88 l +79.911 216.89 l +79.906 216.89 l +79.901 216.9 l +79.897 216.91 l +79.894 216.92 l +79.891 216.93 l +79.89 216.94 l +79.889 216.95 l +80.066 216.95 l +cl 0 0 0 F +n 82.138 217.39 m 82.138 217.38 l +82.136 217.37 l +82.134 217.36 l +82.131 217.35 l +82.127 217.34 l +82.122 217.33 l +82.117 217.33 l +82.111 217.32 l +82.104 217.32 l +82.097 217.31 l +82.09 217.31 l +82.083 217.31 l +82.075 217.3 l +82.067 217.3 l +82.058 217.3 l +82.05 217.3 l +82.042 217.3 l +82.034 217.3 l +82.026 217.3 l +82.018 217.31 l +82.01 217.31 l +82.003 217.31 l +81.996 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l +84.894 214.24 l +84.889 214.25 l +84.884 214.25 l +84.88 214.26 l +84.877 214.27 l +84.874 214.28 l +84.873 214.29 l +84.872 214.3 l +85.049 214.3 l +cl 0 0 0 F +n 84.696 212.89 m 84.696 212.88 l +84.694 212.87 l +84.692 212.86 l +84.688 212.85 l +84.684 212.84 l +84.674 212.83 l +84.668 212.82 l +84.662 212.82 l +84.655 212.81 l +84.648 212.81 l +84.64 212.81 l +84.632 212.81 l +84.624 212.8 l +84.616 212.8 l +84.608 212.8 l +84.6 212.8 l +84.591 212.8 l +84.583 212.81 l +84.575 212.81 l +84.568 212.81 l +84.561 212.81 l +84.554 212.82 l +84.547 212.82 l +84.541 212.83 l +84.536 212.84 l +84.531 212.84 l +84.527 212.85 l +84.524 212.86 l +84.522 212.87 l +84.52 212.88 l +84.52 212.89 l +84.696 212.89 l +cl 0 0 0 F +n 84.696 212.89 m 84.696 212.88 l +84.694 212.87 l +84.692 212.86 l +84.688 212.85 l +84.684 212.84 l +84.674 212.83 l +84.668 212.82 l +84.662 212.82 l +84.655 212.81 l +84.648 212.81 l +84.64 212.81 l +84.632 212.81 l +84.624 212.8 l +84.616 212.8 l +84.608 212.8 l 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210.31 l +76.692 210.32 l +76.687 210.32 l +76.682 210.33 l +76.678 210.34 l +76.675 210.35 l +76.672 210.36 l +76.671 210.37 l +76.67 210.38 l +76.847 210.38 l +cl 0 0 0 F +n 78.081 209.63 m 78.081 209.62 l +78.079 209.61 l +78.077 209.6 l +78.074 209.59 l +78.07 209.58 l +78.065 209.57 l +78.06 209.57 l +78.054 209.56 l +78.047 209.56 l +78.041 209.55 l +78.033 209.55 l +78.026 209.54 l +78.018 209.54 l +78.01 209.54 l +78.002 209.54 l +77.993 209.54 l +77.985 209.54 l +77.977 209.54 l +77.969 209.54 l +77.961 209.54 l +77.953 209.55 l +77.946 209.55 l +77.939 209.56 l +77.933 209.56 l +77.927 209.57 l +77.921 209.57 l +77.917 209.58 l +77.913 209.59 l +77.909 209.6 l +77.907 209.61 l +77.906 209.62 l +77.905 209.63 l +78.081 209.63 l +cl 0 0 0 F +n 78.081 209.63 m 78.081 209.62 l +78.079 209.61 l +78.077 209.6 l +78.074 209.59 l +78.07 209.58 l +78.065 209.57 l +78.06 209.57 l +78.054 209.56 l +78.047 209.56 l +78.041 209.55 l +78.033 209.55 l +78.026 209.54 l +78.018 209.54 l 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font Helvetica +423.333 /Helvetica STDFONT gsave 11867 1520 translate 0 0 M 1.5 dup scale +-107.279 0 N +(z) show X -101.051 M +%%IncludeResource: font Helvetica +262.467 /Helvetica STDFONT +(h) show grestore +%%IncludeResource: font Helvetica +423.333 /Helvetica STDFONT +%%PageTrailer +end +restore +showpage +%%PageResources: font Helvetica +%%Trailer +restore +%%Pages: 1 +%%DocumentNeededResources: font Helvetica +%%EOF From c9384fb1288e81bd164c0ddfeb52f24ca632e396 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Thu, 23 Apr 2026 13:36:32 +0100 Subject: [PATCH 041/116] Copied-in the bibtex file (and change the .tex file to point to the local copy). --- .../turbulence_schemes/bldoc.tex | 2 +- .../science_guide/turbulence_schemes/refs.bib | 1750 +++++++++++++++++ 2 files changed, 1751 insertions(+), 1 deletion(-) create mode 100644 documentation/source/science_guide/turbulence_schemes/refs.bib diff --git a/documentation/source/science_guide/turbulence_schemes/bldoc.tex b/documentation/source/science_guide/turbulence_schemes/bldoc.tex index 482de14dbb..f233c8da77 100644 --- a/documentation/source/science_guide/turbulence_schemes/bldoc.tex +++ b/documentation/source/science_guide/turbulence_schemes/bldoc.tex @@ -5626,7 +5626,7 @@ \section{Appendix: Notation} \newpage \bibliographystyle{plainnat} -\bibliography{../025/papers_db} +\bibliography{refs} \end{document} % Every document must end with this. diff --git a/documentation/source/science_guide/turbulence_schemes/refs.bib b/documentation/source/science_guide/turbulence_schemes/refs.bib new file mode 100644 index 0000000000..643e41b211 --- /dev/null +++ b/documentation/source/science_guide/turbulence_schemes/refs.bib @@ -0,0 +1,1750 @@ + +@book{Iter_SLS_Saad, + author = {Saad, Y.}, + title = {Iterative Methods for Sparse Linear Systems}, + year = {2003}, + isbn = {0898715342}, + publisher = {Society for Industrial and Applied Mathematics}, + address = {Philadelphia, PA, USA}, + } + +@Article{lock00, + author = {A. 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Br{\"u}mmer", + Year="2003", + Title="On the parameterization of turbulent surface + fluxes over heterogeneous sea ice surfaces", + Journal="J. Geophys. Res.", + Volume="108(C6)", + Pages="3195", + Doi="10.1029/2002JC001385" +} + +@Article{Donelan2004, + Author="M. A. Donelan and B. K. Haus and N. Reul and W. J. Plant and + M. Sriassnie and H. C. Graber and O. B. Brown and + E. S. Saltzman", + Year="2004", + Title="On the limiting aerodynamic roughness of the ocean in + very strong winds", + Journal="Geophys. Res. Lett.", + Volume="31", + Pages="L18306", + Doi="10.1029/2004GL019460" +} + + +@article{Rotta51a, + Author = {Rotta, J.C.}, + Journal = {Zeitschrift f{\"u}r Physik}, + Pages = {547-572}, + Title = {Statistiche Theorie nichthomogener Turbulenz, Part 1}, + Volume = {129}, + Year = {1951}} + +@article{Rotta51b, + Author = {Rotta, J.C.}, + Journal = {Zeitschrift f{\"u}r Physik}, + Pages = {51---77}, + Title = {Statistiche Theorie nichthomogener Turbulenz, Part 2}, + Volume = {131}, + Year = {1951}} + +@article{HanjalicandLaunder72, + Author = {Hanjalic, K. and Launder, B.E.}, + Journal = JFM, + Pages = {609---638}, + Title = {Fully developed asymmetric flow in a plane channel}, + Volume = {52}, + Year = {1972}} + +@article{Hogstrom_1990, +author = {H{\"o}gstr{\"o}m, Ulf}, +title = {Analysis of Turbulence Structure in the Surface Layer with a Modified Similarity Formulation for Near Neutral Conditions}, +journal = {Journal of the Atmospheric Sciences}, +volume = {47}, +number = {16}, +pages = {1949-1972}, +year = {1990}, +doi = {10.1175/1520-0469(1990)047<1949:AOTSIT>2.0.CO;2}, +URL = {https://doi.org/10.1175/1520-0469(1990)047<1949:AOTSIT>2.0.CO;2}, +eprint = {https://doi.org/10.1175/1520-0469(1990)047<1949:AOTSIT>2.0.CO;2}, +abstract = { Abstract Data from a recent detailed surface layer experiment are critically examined in terms of the turbulent kinetic energy budget and the other second order moment budgets formed by the three velocity components and temperature. In moderately unstable and slightly stable conditions nondimensional terms of all the moment budgets studied agree reasonably well with results reported from the Kansas study (after application of a flow distortion correction). In the near-neutral range, where the present experiment contains a large amount of data, results deviate significantly from previous studies in general and, in particular, for ideal, zero-pressure gradient turbulent boundary layers. Several moments, such as u2W, v2w and W2 are not, as expected, constant in the surface layer, but vary logarithmically with height, making instead their non dimensional vertical gradients constant. Some moments scale with the roughness length and others with a length scale containing the large-scale pressure gradient or, with an alternative interpretation the height of the neutral PBL. Evidence is presented that these apparent anomalies are due to so called “inactive” turbulence. From the present analysis, and from some previously published atmospheric studies, it is concluded that the values for the various nondimensional gradients may be universally valid, thus suggesting a modified similarity formulation for the neutral atmospheric surface layer. } +} + +@Article{Donelan2018, + Author = {Donelan, M. A.}, + Year = {2018}, + Title = {On the decrease of the oceanic drag coefficient in high winds}, + Journal = {J. Geophys. Res: Oceans}, + Volume = {123}, + Pages = {1--17}, + Doi = {10.1002/2017JC013394} +} + + +@Article{Hsu2017, + Author = {Hsu, J. and Lien, R. and D'Asaro, E. A. and Sanford, T. B.}, + Year = {2017}, + Title = {Estimates of Surface Wind Stress and Drag Coefficients in + {T}yphoon {M}egi}, + Journal = {J. Phys. Oceanogr.}, + Volume = {47}, + Pages = {545--565}, + Doi = {10.1175/JPO-D-16-0069.1} +} + + + + + From 42b6b0d15e26ec9a83c3bf947f832bc3696dc57d Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Thu, 23 Apr 2026 13:40:54 +0100 Subject: [PATCH 042/116] Converted figures to .svg. --- .../turbulence_schemes/div_r071.svg | 969 ++++++ .../turbulence_schemes/div_r080.svg | 955 ++++++ .../turbulence_schemes/honnert_vs_tanh.svg | 723 +++++ .../turbulence_schemes/ideal_invinteg.svg | 495 +++ .../turbulence_schemes/ideal_revflux.svg | 1224 +++++++ .../turbulence_schemes/nbldoc_zidiag.svg | 383 +++ .../turbulence_schemes/new_ktop_shape.svg | 563 ++++ .../turbulence_schemes/stab_dep.svg | 1320 ++++++++ .../turbulence_schemes/subsent_fig7.svg | 592 ++++ .../turbulence_schemes/wcrp_bltypes1.svg | 2226 +++++++++++++ .../turbulence_schemes/wcrp_bltypes2.svg | 2889 +++++++++++++++++ .../turbulence_schemes/zturb_schem.svg | 386 +++ 12 files changed, 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+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + From 673897332f6a898210be61fe9f372074b06f6bc0 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Thu, 23 Apr 2026 13:43:13 +0100 Subject: [PATCH 043/116] Ran pandoc to convert the .tex file to .rst. --- .../turbulence_schemes/bldoc.rst | 6761 +++++++++++++++++ 1 file changed, 6761 insertions(+) create mode 100644 documentation/source/science_guide/turbulence_schemes/bldoc.rst diff --git a/documentation/source/science_guide/turbulence_schemes/bldoc.rst b/documentation/source/science_guide/turbulence_schemes/bldoc.rst new file mode 100644 index 0000000000..2e403938d7 --- /dev/null +++ b/documentation/source/science_guide/turbulence_schemes/bldoc.rst @@ -0,0 +1,6761 @@ +=============================================== +The Parametrization of Boundary Layer Processes +=============================================== + +:Author: A. Lock, J. Edwards and I. Boutle + +.. role:: raw-latex(raw) + :format: latex +.. + +Introduction and code versions +============================== + +This is the documentation for the “boundary layer” parametrization of +vertical turbulent transports of heat, moisture and horizontal momentum. +It includes surface exchange but *not* the parametrization of the +surface itself. This is covered within the surface (JULES) +documentation. Although commonly referred to as the “boundary layer” +parametrization, it includes a free-tropospheric component. Turbulent +fluxes are calculated up to “BL_LEVELS” which is currently set so that +the entire troposphere is included. + +Generally speaking, only version 9C of the boundary layer +parametrization will be documented here as it is now the only supported +version. However, the documentation still makes occasional references to +previous versions of the scheme (8A, 9B) as it is useful to retain the +history of how we have got to where we are. Version 9 interfaces to the +JULES surface code, which is now the only supported surface code within +the UM. + +Several options for higher order closures are available in the 1A +version of the UM boundary layer scheme and these are documented +separately in . + +.. _`sec:closure`: + +Model variables and turbulence closure +====================================== + +Given source terms, :math:`{\cal S}` say, from processes other than +boundary layer turbulence, Reynolds’ averaging gives the following +equation for conserved scalar variables, :math:`\chi`, and the two +horizontal components of momentum, :math:`{\bf u}` on a sphere gives: + +.. math:: + + \begin{aligned} + \frac{\partial \chi}{\partial t} + &=& - \frac{1}{r^2 \rho} \, \frac{\partial }{\partial z} \left( r^2 \rho \overline{w'\chi'} \right) + + {\cal S} + \label{cons_eqn_scal} \\ + \frac{\partial {\bf u}}{\partial t} + &=& \frac{1}{r^2 \rho} \, \frac{\partial }{\partial z} \left( r^2 {\bf \tau} \right) + + {\cal S} + \label{cons_eqn_uv} + \end{aligned} + +where :math:`\overline{w'\chi'}` and :math:`{\bf \tau}` are the vertical +turbulent fluxes to be parametrized, :math:`r` is the height from the +centre of the planet and :math:`\rho` is density. The scalar variables +treated by (`[cons_eqn_scal] <#cons_eqn_scal>`__), which are +approximately conserved under moist adiabatic ascent, are: + +.. math:: + + \begin{aligned} + \theta_{\ell}&=& T_L + \frac{g}{c_p} z = T - \frac{L}{c_p} q_{\ell} + - \frac{L_s}{c_p} q_f + \frac{g}{c_p} z \label{thetal} \\ + q_t &=& q_v + q_{\ell}+ q_f \label{qt} + \end{aligned} + +where :math:`T` is temperature, :math:`q_v` is specific humidity, +:math:`q_{\ell}` and :math:`q_f` the specific liquid and frozen water +contents respectively, and :math:`L_s=L+L_f` is the latent heat of +sublimation. Note that :math:`\theta_{\ell}` is based on ‘liquid/frozen +water static energy’ (:math:`= c_p T + g z - L q_{\ell}- +L_s q_f`) rather than potential temperature, :math:`\theta`. Note also +that the option to use mixing ratios in the boundary layer code instead +of specific quantities is also available and the details of the +necessary changes are documented in appendix `13 <#app:mixratio>`__. +Ultimately turbulent motions are dissipated as heat and so the source +term :math:`{\cal + S}` in (`[cons_eqn_scal] <#cons_eqn_scal>`__) can include an +approximation for that frictional heating, as described in +appendix `14 <#app:fricheat>`__. Finally, the ice cloud contributions in +(`[thetal] <#thetal>`__) and (`[qt] <#qt>`__) can optionally be ignored +(l_noice_in_turb), which will be more appropriate if the time scales for +ice melting or sublimation are longer than those of the turbulence (and +so may be more appropriate if the ice itself is not being mixed by +parametrized turbulence). In this instance the saturation humidity will +be calculated with respect to liquid water at all temperatures and, for +consistency, only liquid cloud fractions with be considered. + +An additional variable used for diagnostic purposes is + +.. math:: + + \theta_{v\ell}= \theta_{\ell}(1 + c_v q_t) + \label{thetavl} + +where :math:`c_v=(1/\epsilon) -1` and :math:`\epsilon` is the ratio of +the molecular weights of water vapour and dry air (i.e., :math:`\epsilon += M_v/M_a \approx 0.62198`). Thus :math:`\theta_{v\ell}` is a conserved +variable that is equal to virtual potential temperature +(:math:`\theta_v`) in cloud-free air and so is used as a simplified +measure of buoyancy. + +A ‘first-order’ closure is used to parametrize the turbulent fluxes, +although non-local terms are also included. Under the 9C scheme, an +alternative methodology is optionally available, see section +`5.5 <#sec:rev_flux_grad>`__. The standard closures are: + +.. math:: + + \begin{aligned} + \overline{w'\chi'} &=& - K_h \frac{\partial \chi}{\partial z} + K_h^{\rm surf}\gamma_{\chi} + \label{scal_closure} \\ + {\bf \tau} &=& K_m \frac{\partial {\bf u}}{\partial z} + {\bf \tau}^{nl} + \label{uv_closure} + \end{aligned} + +Separate eddy-diffusivities are calculated for momentum, :math:`K_m`, +and for scalar variables, :math:`K_h`. The second term on the right hand +side represents a non-local flux in unstable boundary layers. Currently +it is only applied for transport arising from surface-driven turbulence +(:math:`K_h^{\rm surf}`) and is non-zero only for +:math:`\chi=\theta_{\ell}`, as described in +section `5.3 <#sec:gradadj>`__. + +Thus, the parametrization reduces to determining :math:`K_h`, +:math:`K_m` and :math:`\gamma_{\chi}` and :math:`{\bf \tau}^{nl}`. Two +methods are used to determine :math:`K_h` and :math:`K_m` and how they +are combined for (`[scal_closure] <#scal_closure>`__) and +(`[uv_closure] <#uv_closure>`__) is described in +section `4.2 <#sec:shear>`__. The first method is a local Richardson +number (:math:`Ri`) based scheme. It is calculated for all regimes (but +will be responsible for all mixing in stable conditions), over all +levels up to the specified BL_LEVELS and is described in +section `4 <#sec:local>`__. The second method is a non-locally specified +profile scheme. This is exclusively for unstable boundary layers, is +calculated up to level NL_BL_LEVELS (typically around 6km AMSL) and is +described in more detail in section `5 <#sec:nonlocal>`__. In this +regime, mixing is assumed to occur in (or lead rapidly to the formation +of) well-mixed layers (in which conserved variables are approximately +uniform with height) that are capped by an inversion. Mixing is assumed +to be driven either from the surface in a ‘surface mixed layer’ (SML, by +a positive surface buoyancy flux and by surface stresses) or by +cloud-top buoyancy sources (radiative and evaporative cooling, see +appendix `11 <#app:vscales>`__). As described in section +`5 <#sec:nonlocal>`__, separate :math:`K`-profiles are used for these +two turbulence sources. If the cloud-top sources generate mixing +throughout the SML the layer is said to be ‘coupled’ but if the +:math:`K`-profile representing surface-driven mixing does not extend up +to cloud-top, the layer is referred to as being ‘decoupled’. As +decoupled layers are restricted to being buoyancy driven and typically +below 6km, they are referred to as decoupled stratocumulus (DSC) layers. +The calculation of :math:`\gamma_{\chi}` is described in +section `5.3 <#sec:gradadj>`__ and, finally, fluxes across the top of +both SML and DSC layers (the entrainment fluxes) are specified +explicitly through a separate entrainment parametrization, as described +in section `7 <#sec:entr>`__. + +The strategy used to determine precisely where and when the resulting +eddy-diffusivities should be applied is described in +section `3 <#sec:types>`__. The buoyancy parameters, finite difference +and other notation used here are defined in +appendices `12 <#app:buoyp>`__ and `17 <#app:not>`__. Further papers +describing this scheme and its performance are +:raw-latex:`\cite{lock00}` (noting the corrigendum in +:raw-latex:`\cite{locketal01_corr}`), +:raw-latex:`\cite{martin00:_new_bound_layer_mixin_schem}`, +:raw-latex:`\cite{lock01}`, :raw-latex:`\cite{bushetal1999}` and +:raw-latex:`\cite{brown08:_upgrad_bound_layer_schem_met}`. + +.. _`sec:types`: + +Diagnosis of boundary layer depth and type +========================================== + +The non-locally specified :math:`K`-profiles require the height of the +base and top of the layer to be diagnosed (see +section `5 <#sec:nonlocal>`__). Furthermore, as stated in +section `2 <#sec:closure>`__, the mixing generated by the non-local +:math:`K` profiles is assumed to occur in (or lead rapidly to the +formation of) well-mixed layers capped by an inversion. Thus, the +accurate diagnosis of their vertical extent is crucial. How to make this +diagnosis is dependent on the boundary layer mixing regime which has +been categorised into 7 distinct ‘boundary layer types’: + +- **Type I**: Stable boundary layer (with or without cloud) — turbulent + diffusivities are calculated by the ‘local’ scheme + (section `4 <#sec:local>`__) + +- **Type II**: Boundary layer with stratocumulus over a stable + near-surface layer — as Type I but with a turbulently mixed cloud + layer driven from its top (a DSC layer, diagnosis described in section + `3.2 <#sec:decouple>`__) + +- **Type III**: Well mixed boundary layer — the classic single mixed + layer which may be cloud-topped or clear but is predominantly + buoyancy-driven (c.f. a possible type VII below) — diagnosis described + in section `3.1 <#sec:adiapar>`__) + +- **Type IV**: Unstable boundary layer with a DSC layer not over cumulus + (see section `3.2 <#sec:decouple>`__) — the surface-based and + cloud-top-driven non-local :math:`K` profiles may or may not overlap + and cloud-top entrainment can still include the surface forcing (see + section `7 <#sec:entr>`__) + +- **Type V**: Boundary layer with a DSC layer over cumulus — the cumulus + (treated by the model’s mass-flux convection scheme) provides coupling + with the SML (cumulus diagnosis described in section + `3.1 <#sec:adiapar>`__) + +- **Type VI**: Cumulus-capped boundary layer — no turbulent + diffusivities are allowed [1]_ at or above the LCL as the mass-flux + convection scheme operates here (cumulus diagnosis described in + section `3.1 <#sec:adiapar>`__) + +- **Type VII**: Shear-dominated unstable layer — potentially wind-shear + might allow deeper turbulent mixing in unstable boundary layers than + is apparent purely from the thermodynamic profiles (sufficient even to + inhibit the formation of cumulus); the possibilities are discussed in + section `4.2 <#sec:shear>`__. + +Types I to VI are shown schematically in Fig. `1 <#fig:bltypes>`__. + +.. container:: float + :name: fig:bltypes + +.. _`sec:adiapar`: + +The diagnostic parcel ascent and cumulus diagnosis +-------------------------------------------------- + +**Summary**: the depth of the non-local :math:`K`-profiles for +surface-driven turbulence (with NTML grid-levels in the mixed layer and +top at height :math:`z_{\rm h}` , as required for +(`[kmsurf] <#kmsurf>`__)) is determined from: + +#. a diagnostic moist parcel ascent; top at grid-level NTPAR, height + :math:`z_{\rm par}` :math:`=z_{\mbox{\tiny \rm NTPAR}+\frac{1}{2}}`. + Typically this is an adiabatic parcel but entraining options are + available. + +#. a diagnosis of cumulus-capped layers (if cumulus-capped then NTML and + :math:`z_{\rm h}` are set to the LCL [2]_, if not then to the parcel + top) + +Note that this process is only performed for unstable boundary layers +(defined by a positive surface buoyancy flux, i.e., +:math:`\overline{w'b}_S>0`). + +**Step 1:** the method assumes that the height to which turbulent mixing +driven by surface processes can extend in unstable boundary layers (and +therefore the vertical extent of the :math:`K` profile for +surface-driven turbulence) can be determined solely from the properties +of the thermodynamic profiles. In more detail, the first step in +calculating :math:`z_{\rm h}` is to lift a parcel, with properties from +the first grid-level (:math:`k=k_s`) above the top of the surface layer, +upwards allowing for latent heat release. The top of the surface layer +is taken to be at the lower of :math:`z=0.1`\ :math:`z_{\rm h}` (this is +then consistent with the :math:`K`-profiles, see +section `5.1 <#sec:nlsurf>`__; :math:`z_{\rm h}` is taken from the +previous timestep) and the grid-level above which :math:`\theta_{v\ell}` +starts to increase with height. The ascent is stopped at the grid-level +NTPAR (height +:math:`z_{\rm par}` :math:`=z_{\mbox{\tiny \rm NTPAR}+\frac{1}{2}}`) +above which the parcel becomes more negatively buoyant than a given +threshold, :math:`\theta_v'`. Note that the parcel properties themselves +are not perturbed in order to preserve the height of the mixed-layer’s +lifting condensation level (LCL). The calculation of the parcel’s +buoyancy excess is described in section `3.1.1 <#sec:parxs>`__. +Currently, + +.. math:: + + \theta_v' = \mbox{max} \left[A_{plume}, + \, \mbox{min} \left[ B_{plume} \sigma_{Tv1}, + \, G_{max}z_{\rm h}\right] \right] + \label{parcel_pert} + +where :math:`A_{plume}=0.2`, :math:`B_{plume}=3.26`, +:math:`G_{max}=10^{-3}`\ Km\ :math:`^{-1}`, +:math:`\sigma_{Tv1} = 1.93\, \overline{w'\theta_v'}_S/w_m` and +:math:`w_m^3=u_*^3+0.25\,z_{\rm h}\overline{w'b}_S`. Following +:raw-latex:`\cite{holtslag93:_local_versus_nonloc_bound_layer}`, +:math:`\theta_v'` is related to the magnitude of the gradient +adjustment, :math:`\gamma_{\theta_{\ell}}` (see section +`5.3 <#sec:gradadj>`__). Thus, :math:`B_{plume}=A_{ga}`, although +somewhat arbitrary limits have been placed on the magnitude of +:math:`\theta_v'` for numerical security (the upper limit being +consistent with that applied to :math:`\gamma_{\theta_{\ell}}` in +(`[gradadj] <#gradadj>`__) ). Within limits, then, :math:`\theta_v'` +represents a typical buoyancy excess of boundary layer plumes. + +The pressure at the LCL, :math:`P_{LCL}=P_{k_s} +(T_{LCL}/T_{k_s})^{(1/\kappa)}`, where :math:`\kappa=R/c_p`. The +temperature at the LCL, :math:`T_{LCL}`, is calculated using +approximations in :raw-latex:`\cite{Bolton1980}` as + +.. math:: T_{LCL} = 55 + \frac{2840}{3.5 \log(T_{k_s}) - log(e_{k_s}) - 4.805} + +where the vapour pressure of air in grid-level :math:`k_s`, +:math:`e_{k_s} = +q_{k_s} P_{k_s}/(100 \, \epsilon)`. The full-level below that containing +the LCL is labelled NLCL and +:math:`z_{\rm lcl}` :math:`=z_{\mbox{\tiny \rm NLCL}+\frac{1}{2}}`. If +the parcel rises above the top of the LCL transition zone (defined as +1.1\ :math:`z_{\rm lcl}` , its ascent can also be stopped at the +grid-level at which it has maximum buoyancy excess over the environment. +This is identified as the grid-level above which + +.. math:: + + \frac{d\theta_v}{dz}|_{\rm env} > + \Gamma_{\rm inv}\, \frac{d\theta_v}{dz}|_{\rm par} + +where currently the tolerance for identifying inversions by this method, +:math:`\Gamma_{\rm inv}=1.1`. This use of the height of maximum excess +(if lower than that given by the straight buoyancy threshold, +:math:`\theta_v'`) is typically of little consequence in stratocumulus +regions (which tend to be well-mixed beneath large inversions), but can +be necessary in order to identify the capping inversion in cumulus cases +(e.g. in the trade wind regions). + +**Step 2:** having established :math:`z_{\rm par}` , a crucial +additional test is to determine whether this layer is well-mixed (i.e., +stratocumulus-capped) or cumulus-capped. The parcel ascent can rise to +cloud-top in both cases but cumulus cloud layers are observed not to be +as well-mixed as stratocumulus layers. Recall that application of the +:math:`K` profiles is expected to form or maintain well-mixed layers and +so their current formulation is inappropriate for cumulus cloud layers. +Specifically, a logical flag (CUMULUS) is set to true if + +.. math:: + + \left| \frac{ \Delta_{\rm cld} q_t}{\Delta_{\rm cld} z} \right| > + C_t \, \left| \frac{ \Delta_{\rm sub} q_t}{\Delta_{\rm sub} z} \right| + +where the cloud-layer gradient, :math:`\Delta_{\rm cld}`, is taken +between both NTPAR and NTPAR-1 (to allow for the possibility that a Sc +layer has just deepened by a grid-level) and NLCL and the sub-cloud +layer gradient, :math:`\Delta_{\rm sub}`, between grid-levels NLCL and +:math:`k_s`. Currently the threshold factor, :math:`C_t = 1.1`. If +cumulus is diagnosed, the top of the surface-based mixed layer +(:math:`z_{\rm h}` ) is set to :math:`z_{\rm lcl}` (rather than to +:math:`z_{\rm par}` , as illustrated in Fig. `1 <#fig:bltypes>`__ for +types V and VI). There is then an option to diagnose the thickness of +the LCL transition zone, see section `3.4 <#sec:lclmixing>`__. +Otherwise, the boundary layer surface-driven mixing is capped at +:math:`z_{\rm lcl}` so that mixing into the cumulus cloud layer is only +carried out by the model’s mass-flux convection scheme and not by the +eddy viscosity based boundary layer scheme. Note that basing the CUMULUS +diagnosis on cloud and sub-cloud layer gradients limits the model only +to being able to resolve cumulus with cloud and sub-cloud layers at +least 2 grid-levels (and optionally 400m) thick. Otherwise the layer is +considered well-mixed to :math:`z_{\rm par}` with an option to include a +representation of fluxes into the capping inversion (see +section `3.3 <#sec:dzi>`__). + +If the parcel ascent fails to find an inversion below 3km (or BL_LEVELS) +but the LCL is below BL_LEVELS, then the layer is assumed to be +cumulus-capped. If the LCL is above BL_LEVELS, then again cumulus is +diagnosed with NTML\ :math:`=\mbox{min}[`\ NLCL, BL_LEVELS\ :math:`-1]`, +in the hope that the mass-flux convection scheme (in its moist or dry +mode) will transport the surface fluxes higher! Clearly this restriction +on the boundary layer scheme is not desirable and so a value of +BL_LEVELS above the tropopause is recommended. + +Note that if cumulus is not diagnosed then a further, subgrid estimation +of the height of the capping inversion is attempted for +:math:`z_{\rm h}`  (as described in section `7.1.1 <#sec:sginv>`__). + +.. _`sec:parxs`: + +Calculation of parcel buoyancy excess +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +As described in appendix `12 <#app:buoyp>`__, virtual temperature, +:math:`T_v = +T(1 + c_v q_v - q_{\ell}- q_f)`, is used as the measure of buoyancy. The +condensed water in the parcel at a grid-level :math:`k` +(:math:`q_{\ell f}^p`, the superscript :math:`^p` indicating parcel +properties) is estimated using a Taylor expansion of :math:`q_s` about +the environment at that grid-level +(:math:`q_s^p \approx {q_s}_k + \alpha_L (T^p-T_k)`). Assuming that +:math:`q_{\ell f}^p=q_t^p - q_s^p` gives + +.. math:: + + q_{\ell f}^p = \mbox{max}\left[ 0.0, \, a_L \left( q_t^p - {q_s}_k + - \alpha_L (\theta_{\ell}^p - (g z_k/c_p)-T_k)\right) + \right] + \label{qlpar} + +where the buoyancy parameters :math:`a_L` and :math:`\alpha_L` are +defined in appendix `12 <#app:buoyp>`__. Recall that the parcel has +:math:`q_t` and :math:`\theta_{\ell}` taken from grid-level :math:`k_s` +which are conserved during its ascent. Note that (`[qlpar] <#qlpar>`__) +will not give condensation until the parcel becomes saturated. In the +environment the cloud scheme will allow some condensation (and therefore +warming and stabilisation of the environment profile) to take place +before the grid-level becomes saturated in the mean. To allow for this +in the parcel (without applying the cloud scheme), +(`[qlpar] <#qlpar>`__) is also calculated at each grid-level but using +the environment grid-box mean :math:`q_t` and :math:`\theta_{\ell}` to +give :math:`q_{\ell f}^e`. The difference in the environment’s condensed +water as determined by the UM cloud scheme (i.e., :math:`q_{\ell}+q_f`) +and by (`[qlpar] <#qlpar>`__) (i.e., :math:`q_{\ell f}^e`) is then added +to :math:`q^p_{lf}`. + +Given :math:`q_{\ell f}^p`, (`[thetal] <#thetal>`__) implies +:math:`T^p = \theta_{\ell}^p - (g z_k/c_p) ++ (L q_{\ell f}^p/c_p)` (using :math:`L_s` if :math:`T_k` is below the +melting point) and (`[qt] <#qt>`__) implies +:math:`q_v^p = q_t^p - q_{\ell f}^p` and thus :math:`T_v^p` can be +calculated. Recall that the diagnosis of the parcel’s maximum buoyancy +excess over the environment (described in +section `3.1 <#sec:adiapar>`__) required :math:`\theta_v`. This is +approximated as :math:`\theta_v = T_v + (g z_k/c_p)`. + +.. _`sec:decouple`: + +Diagnosis of the vertical extent of the K-profiles +-------------------------------------------------- + +The diagnosis of mixed layers with turbulence driven from cloud-top has +been separated in to three stages. These are: + +#. diagnose the existence of a decoupled stratocumulus (DSC) layer with + approximately uniform :math:`\theta_{v\ell}` (label the top + grid-level in the mixed-layer NTDSC and diagnose the subgrid height + of its capping inversion, :math:`z_{\rm h}^{\rm Sc}` , see + section `7.1.1 <#sec:sginv>`__) + +#. diagnose an approximate depth of the DSC layer, :math:`z_{\rm ml}`, + in order to be able to calculate the representative turbulent + velocity scales (see appendix `11 <#app:vscales>`__). + +#. calculate the depth of the :math:`K` profiles (see + section `5 <#sec:nonlocal>`__) in both SML and DSC layers using + constraints on the TKE budget of the layer. This includes the + diagnosis of recoupling of DSC layers and decoupling of SMLs + +**Step 1**: the diagnosis of DSC layers depends on whether cumulus +convection has been diagnosed. If a cumulus-capped layer under an +inversion within BL_LEVELS has been diagnosed, grid-levels NTPAR and +NTPAR+1 are tested to see if they contain significant layer cloud +(:math:`C_F>` SC_CFTOL). This threshold for identifying potentially +turbulently-mixed cloud layers is currently SC_CFTOL\ :math:`=0.1`. If +there is significant cloud, NTDSC is set to NTPAR. + +Alternatively, if a well-mixed surface-driven boundary layer was +diagnosed, then from grid-level NTML\ :math:`+2` upwards, a cloud-top +grid-level (:math:`k_{ct}`) is sought such that :math:`{C_F}_{k_{ct}}>` +SC_CFTOL and :math:`{C_F}_{k_{ct}+1}<` SC_CFTOL. If +:math:`\Delta_{k_{ct}} \theta_{v\ell}/ +\Delta_{k_{ct}} z < 10^{-3}`\ Km\ :math:`^{-1}` (i.e., +:math:`\theta_{v\ell}` is approximately well-mixed over at least two +grid-levels), then NTDSC is set to :math:`k_{ct}`. If grid-levels +:math:`k_{ct}` and :math:`k_{ct}-1` are not well-mixed, grid-levels +:math:`k_{ct}-1` and :math:`k_{ct}-2` are tested using the same +criterion. If they are not well-mixed either, the cloud-layer is ignored +for the purposes of turbulent mixing. If grid-levels :math:`k_{ct}-1` +and :math:`k_{ct}-2` are identified as well-mixed a further test is +applied to determine whether the :math:`\theta_v` (rather than +:math:`\theta_{v\ell}`) gradient across grid-levels :math:`k_{ct}` and +:math:`k_{ct}-1` is greater than adiabatic (i.e., whether grid-levels +:math:`k_{ct}` and :math:`k_{ct}-1` actually form part of an inversion — +note that by ignoring the :math:`q_{\ell}` contribution to buoyancy, +:math:`\theta_{v\ell}` is not a good variable to use to measure the +strength of cloud-capping inversions). To do this, the :math:`\theta_v` +gradient between grid-levels :math:`k_{ct}` and :math:`k_{ct}-1` is +compared with that for a parcel lifted adiabatically from grid-level +:math:`k_{ct}-1`, in exactly the same way as for the SML parcel ascent +(see section `3.1.1 <#sec:parxs>`__). If :math:`d\theta_v/dz|_{\rm + env} > \Gamma_{\rm inv} d\theta_v/dz|_{\rm par}` between grid-levels +:math:`k_{ct}` and :math:`k_{ct}-1` then NTDSC is set to +:math:`k_{ct}-1`; if not then NTDSC is set to :math:`k_{ct}` (recall +that grid-levels :math:`k_{ct}-1` and :math:`k_{ct}-2` have already been +identified as well-mixed). + +**Step 2** is to diagnose an approximate depth of the DSC layer, +:math:`z_{\rm ml}`. The bottom grid-level of the mixed-layer (NBDSC) is +diagnosed as the lowest grid-level, descending from NTDSC, where +:math:`{\theta_{v\ell}}_{\mbox{\tiny \rm NTDSC}} + \theta_{v\ell}'` is +less than :math:`\theta_{v\ell}` of the environment. The parcel +perturbation is given by + +.. math:: + + \theta_{v\ell}' = - \, \frac{ \tau_{rc} \Delta_F}{z_{rc}} + \label{dscd_pert} + +where :math:`\Delta_F` (Kms\ :math:`^{-1}`) is the magnitude of the +cloud-top radiative divergence (see appendix `11 <#app:vscales>`__), +:math:`\tau_{rc}` is a timescale for the exposure of boundary layer +eddies to the cloud-top radiative cooling (taken to be 200s) and +:math:`z_{rc}` is a depth-scale for the radiatively cooled layer (taken +to be 50m). These values of :math:`\tau_{rc}` and :math:`z_{rc}` are +only estimates (and will in reality vary from one cloud to another) but +they are consistent with, for example, the observations of +:raw-latex:`\cite{nicholls1986}`. If the parcel failed to fall (i.e., +NBDSC equals NTDSC) in a DSC layer *not* overlying cumulus, then the +layer is assumed not to be well-mixed. At the top of a cumulus layer, +the DSC layer is given a minimum depth of +:math:`\Delta_{\mbox{\tiny \rm NTDSC}+\frac{1}{2}} z`. Otherwise, the +layer depth, :math:`z_{\rm ml}`, is measured from the top of layer NTDSC +to the base of layer NBDSC. + +**Step 3**: the step 2 calculation of :math:`z_{\rm ml}` is used to +calculate the representative velocity scales for the DSC layer but its +calculation is only crude. Here, the vertical extent of the +:math:`K`-profiles is determined more accurately by ensuring that the +magnitude of the integrated buoyancy consumption of TKE within the mixed +layer is less than or equal to a fraction, :math:`D_t`, of the buoyancy +production, following :raw-latex:`\cite{turton1987}`. + +Following appendix `12 <#app:buoyp>`__ the grid-box mean buoyancy flux +can be written as: + +.. math:: + + \overline{w'b}= g \left[ (1-C_F) \left(\beta_T \overline{w'\theta_{\ell}'} + \beta_q \overline{w'q_t'}\right) + + C_F \left( \tilde{\beta_T} \overline{w'\theta_{\ell}'} + \tilde{\beta_q} \overline{w'q_t'}\right) + \right] + \label{eq:wb_cont} + +As standard, the fluxes in (`[eq:wb_cont] <#eq:wb_cont>`__) are then +expanded using the first-order closure in +(`[scal_closure] <#scal_closure>`__) as: + +.. math:: + + \begin{aligned} + \overline{w'\theta_{\ell}'}_k &=& -K_h^{\rm surf}\,\frac{\widetilde{\Delta_k \theta_{\ell}}}{\Delta_k z} + -K_h^{\rm Sc}\,\frac{\Delta_k \theta_{\ell}}{\Delta_k z} \nonumber \\ + \overline{w'q_t'}_k &=& -\left(K_h^{\rm surf}+ K_h^{\rm Sc}\right) \, + \,\frac{\Delta_k q_t}{\Delta_k z} + \label{eq:wx_std} + \end{aligned} + +where +:math:`\widetilde{\Delta_k \theta_{\ell}} = \Delta_k \theta_{\ell}- +\gamma_{\theta_{\ell}} \Delta_k z` in order to include the non-local (or +gradient adjustment) term. If the alternative flux-gradient option is +used, see section `5.5 <#sec:rev_flux_grad>`__, then additional terms +are needed. + +Large-eddy simulations have demonstrated that the crucial region in +determining when decoupling of stratocumulus will occur (i.e., when the +:math:`K` profiles no longer span the entire layer from cloud-top to the +surface) is in a thin layer of unsaturated air just below cloud-base, +where :math:`\overline{w'b}` first becomes negative. Thus, in the above +calculation, it is crucial both to have an accurate measure of +cloud-base height (which will have to be subgrid) and to include +successfully this thin unsaturated layer in the buoyancy consumption +integral. Thus, the :math:`\overline{w'b}` integration is performed over +the cloud and sub-cloud layers separately and the cloud-fraction is +taken to be uniform within the cloud layer (and zero below cloud-base). +The height of cloud-base is given by (`[zc_calc] <#zc_calc>`__). + +An iterative method is then used to find the vertical extent of mixing +(within certain bounds, as described below) such that the magnitude of +buoyancy consumption of TKE within the mixed layer equals a fraction, +:math:`D_t`, of the buoyancy production, i.e., + +.. math:: + + \sum_{z_{k-\frac{1}{2}} > z_i-z_{\rm ml}}^{z_{k-\frac{1}{2}} < z_i} + \left|\left[ \overline{w'b}|_{z_{k-\frac{1}{2}}}<0 \right]\right| \, \Delta_k z \, + \leq \, D_t \, + \sum_{z_{k-\frac{1}{2}} > z_i-z_{\rm ml}}^{z_{k-\frac{1}{2}} < z_i} + \left[ \overline{w'b}|_{z_{k-\frac{1}{2}}}>0 \right] \, \Delta_k z + \label{deccrit} + +Note that, for simplicity, the :math:`{\cal E}_h` factors are not +included in :math:`K_h^{\rm surf}` or :math:`K_h^{\rm Sc}` when +calculating (`[eq:wx_std] <#eq:wx_std>`__) under the assumption that +they will be small. This process is applied to all unstable mixed +layers. For stratocumulus layers, observations and LES suggest a value +of :math:`D_t=0.1`. A separate value of :math:`D_t` can be used for the +sub-cloud layer in cumulus capped boundary layers if this method is used +to determine the LCL transition zone thickness, see section +`3.4 <#sec:lclmixing>`__. For cloud-free mixed layers, :math:`D_t=1` is +used, purely to keep negative buoyancy fluxes down to a reasonably +realistic level (for example, if the parcel top diagnostic returned too +high a boundary layer depth). + +The first step is to test for whether a well-mixed layer is possible +(either decoupling what has so far been diagnosed as a well-mixed layer +or, if one exists, recoupling a decoupled stratocumulus layer), i.e., to +test whether (`[deccrit] <#deccrit>`__) is satisfied with both +:math:`K_h^{\rm surf}` and :math:`K_h^{\rm Sc}` extending from the +surface to the cloud-top. If recoupling is possible then the various +flags identifying the DSC layer are reset (*this includes setting the +cumulus diagnosis to false*), any surface-driven entrainment originally +applied at :math:`z_{\rm h}` is added to the entrainment at +:math:`z_{\rm h}^{\rm Sc}` (after rescaling for the inversion strength +at :math:`z_{\rm h}^{\rm Sc}` ) and :math:`z_{\rm b}` is set to +0.1\ :math:`z_{\rm h}` (for the reason discussed above). If decoupling +is diagnosed, :math:`z_{\rm h}^{\rm Sc}` is set to the original +:math:`z_{\rm h}` (inversion height), although the entrainment across +this inversion is not recalculated (and so keeps any surface-driven +component — the COUPLED flag is therefore set to true, see +section `7 <#sec:entr>`__). + +If a decoupled layer is diagnosed, then an iteration is performed to +find the highest :math:`z_{\rm h}` (so top of the :math:`K_h^{\rm surf}` +profile) that still satisfies (`[deccrit] <#deccrit>`__), but with +:math:`K_h^{\rm Sc}=0` in (`[eq:wx_std] <#eq:wx_std>`__). The iteration +proceeds with :math:`z_{\rm h}` stepping from its lowest permissible +height to its highest (currently 3 steps are used). If at any stage +(`[deccrit] <#deccrit>`__) is violated, then the step below (therefore +containing the height that would give equality in +(`[deccrit] <#deccrit>`__)) is divided by 4 and 3 of those steps are +taken downwards. If (`[deccrit] <#deccrit>`__) is met the step above is +again reduced by a factor of 4 and 3 steps taken upwards. A total of 3 +sweeps are possible, each with a smaller step so that +:math:`z_{\rm h}` approaches the height that gives equality in +(`[deccrit] <#deccrit>`__). The accuracy with which this is achieved +will be the difference in the maximum and minimum permissible heights of +:math:`z_{\rm h}`  divided by :math:`2\times4\times4 = 32`, which will +typically be less than 30m. The top grid-level of the SML, NTML, is +defined as the highest grid-level such that :math:`K_h^{\rm Sc}` is +non-zero at the half-level above. + +The above process is then repeated to find the appropriate +:math:`z_{\rm b}` for :math:`K_h^{\rm Sc}`, i.e., for the base of +top-driven mixing. Some constraints are placed on :math:`z_{\rm b}` , +namely that it should never go below :math:`0.1`\ :math:`z_{\rm h}` (to +avoid affecting the continuity of the :math:`K` profiles at the top of +the surface layer, see (`[ws_defn] <#ws_defn>`__)). If cumulus +convection has been diagnosed then :math:`z_{\rm b}` is not allowed to +go below :math:`z_{\mbox{\tiny \rm NTML}+\frac{1}{2}}` (unless the layer +is diagnosed to recouple completely). Finally, :math:`z_{\rm b}` must +always be at or below :math:`z_{\mbox{\tiny \rm NTDSC}-1}`, so that +mixing in decoupled layers is always resolved, and at least +:math:`\Delta z_{rad}` (the cloud-top radiative cooling depth defined in +section `3.2.2 <#sec:wbint_inv>`__) below the t inversion. The base +grid-level of the DSC layer, NBDSC, is defined (analogously to NTDSC) as +the lowest grid-level such that :math:`K_h^{\rm Sc}` is non-zero at the +half-level below. + +A possible extension to this diagnosis would be to include the shear +contribution to the TKE budget in (`[deccrit] <#deccrit>`__) and so +allow shear-driven mixing to help maintain well-mixed layers. + +Surface layer :math:`\overline{w'b}` integration +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +In the surface layer, below :math:`z_i/10`, the :math:`K` profiles have +a different functional form from the rest of the mixed layer. Rather +than include this additional complexity in the :math:`\overline{w'b}` +integration, the surface layer is treated separately. In place of the +finite-difference form of :math:`\overline{w'b}`, see +(`[eq:wb_cont] <#eq:wb_cont>`__) and (`[eq:wx_std] <#eq:wx_std>`__) +above, :math:`\overline{w'b}` is assumed to be linear between +:math:`\overline{w'b}_S` at the surface and zero at a level which must +be estimated. The surface layer integration is then from the surface up +to :math:`z_{{\rm + K_{SURF}}}`, where :math:`\theta`-level K_SURF is the first above +:math:`z_i/10`. The level where :math:`\overline{w'b}` is zero is found +by linear interpolation across the grid-levels where the diagnosed +cloud-free buoyancy flux would become negative. This is where +:math:`\beta_T +\widetilde{\Delta_k \theta_{\ell}} + \beta_q \Delta_k q_t` becomes +positive and so where the cloud-free part of :math:`\overline{w'b}` +(i.e., that part below cloud-base which is important for decoupling) +becomes negative. + +.. _`sec:wbint_inv`: + +Integration of :math:`\overline{w'b}` close to the inversion +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +Because of the large gradients often seen in fluxes close to the +inversion (in particular, in the LW radiative flux), simple finite +difference flux calculations, (`[eq:wx_std] <#eq:wx_std>`__), can be +significantly inaccurate in this region. An example is shown in +Fig. `2 <#fig:inv_integ>`__. Calculating +:math:`\overline{w'\theta_{\ell}'}_{\mbox{\tiny \rm NTML}+\frac{1}{2}}` +from (`[eq:wx_std] <#eq:wx_std>`__) gives a negative value, largely +because :math:`\Delta_{\mbox{\tiny \rm NTML}+1} \theta_{\ell}` is +positive and so the local flux is large and negative. In reality, +:math:`\overline{w'\theta_{\ell}'}` becomes positive only a short +distance below cloud-top such that the integral here will tend also to +be positive. + +The solution adopted is to integrate :math:`\overline{w'b}` analytically +across the region just below the inversion, labelled +:math:`\Delta z_{rad}` in Fig, `2 <#fig:inv_integ>`__. Since +:math:`\Delta_{\mbox{\tiny \rm NTML}} \theta_{\ell}` can also be +significantly positive (when the grid-level inversion is rising or +falling, for example), the base of this region is taken to be the lower +of the first :math:`\theta`-level below :math:`z_h-100` m (a physically +reasonable depth over which cloud-top radiative cooling might be +expected to occur) and :math:`z_{\mbox{\tiny \rm NTML}-1}`. + +.. container:: float + :name: fig:inv_integ + + .. container:: center + +Then, + +.. math:: + + \begin{aligned} + \int_{z_h-\Delta z_{rad}}^{z_h} \, \overline{w'\theta_{\ell}'}\, dz & = & + \int_{z_h-\Delta z_{rad}}^{z_h} \, F_{\theta_{\ell}}^{Tot} - + F_{\theta_{\ell}}^{NT}\, dz + \nonumber \\ + & = & I^{Tot} - I^{rad} - I^{ppn} + \label{wthl_int} + \end{aligned} + +For the radiative flux, it could be assumed that the subgrid flux +distribution is exponentially dependent on the grid-level LWP, for +example. This would give: + +.. math:: + + I^{rad} = \frac{\Delta z_{rad}} + {\ln(F^{rad}|_{z_h}/F^{rad}|_{z_h-\Delta z_{rad}} ) } + \left(F^{rad}|_{z_h}-F^{rad}|_{z_h-\Delta z_{rad}} \right) + \label{irad} + +However, off-line tests indicated this could give a strong and spurious +sensitivity to :math:`F^{rad}|_{z_h-\Delta z_{rad}}`. Furthermore, for +most realistic scenarios, the logarithmic factor in (`[irad] <#irad>`__) +tends to be close to 3. Consequently, we approximate :math:`I^{rad} = +\Delta z_{rad} ( F^{rad}|_{z_h}-F^{rad}|_{z_h-\Delta z_{rad}} ) /3`. In +addition, :math:`F^{rad}|_{z_h}-F^{rad}|_{z_h-\Delta z_{rad}}` is +approximated as :math:`\Delta F`, the radiative flux change across +cloud-top used in the calculation of :math:`V_{\rm Sc}` +(`[ctraddiv] <#ctraddiv>`__). The precipitation flux is assumed to vary +linearly across this region, as does the total flux, and so its +contribution to :math:`I^{Tot}` cancels with :math:`I^{ppn}` in +(`[wthl_int] <#wthl_int>`__). Finally, for simplicity, the total flux is +taken to be constant and equal to the inversion value, such that +:math:`I^{Tot} = \Delta z_{rad} F^{Tot}|_{z_h}`. With these +approximations, (`[wthl_int] <#wthl_int>`__) becomes + +.. math:: + + \begin{aligned} + \int_{z_h-\Delta z_{rad}}^{z_h} \, \overline{w'\theta_{\ell}'}\, dz & = & + \Delta z_{rad} \left(-w_e \Delta \theta_{\ell}+ \Delta F\right) + - \Delta z_{rad} \Delta F/ 3 \\ + & = & \Delta z_{rad} \left(-w_e \Delta \theta_{\ell}+ \frac{2}{3} \Delta F\right) + \end{aligned} + +For the integral of :math:`\overline{w'q_t'}` across this cloud-top +region, :math:`\overline{w'q_t'}` is also taken to be constant so that: + +.. math:: + + \int_{z_h-\Delta z_{rad}}^{z_h} \, \overline{w'q_t'}\, dz = + - \Delta z_{rad} w_e \Delta q_t + +The integrated buoyancy flux is then found from +(`[eq:wb_cont] <#eq:wb_cont>`__) using the mixed layer cloud fraction +and buoyancy coefficients evaluated at the grid-level above +:math:`z_h -\Delta z_{rad}`. + +.. _`sec:dzi`: + +Diagnosis of inversion thickness +-------------------------------- + +Terminating the diagnostic parcel ascent at its level of neutral +buoyancy ignores any overshooting through the parcel’s own inertia as it +enters the inversion region. This overshooting region effectively +defines the depth of the inversion over which the negative entrainment +heat fluxes are seen. Typically this will be small relative to the model +vertical grid but at higher vertical resolution or when a strongly +surface-heated boundary layer is capped by weak stability inversions +could be resolved. Following :raw-latex:`\cite{beare2008}`, a simple +energetic argument gives a realistic prediction of the top of the +inversion, :math:`z_{top}`, in LES from + +.. math:: + + 6.3 \, w_m^2 = \int_{z_{nb}}^{z_{top}} \, b \, dz + \label{dz_param} + +where :math:`z_{nb}` is the level of neutral buoyancy (found by linear +interpolation between grid-levels), :math:`w_m` is the boundary layer +velocity scale defined in section `5.1 <#sec:nlsurf>`__ and :math:`b` is +the parcel buoyancy. Note that the constant in +(`[dz_param] <#dz_param>`__) is the same as in +:raw-latex:`\cite{beare2008}` because :math:`6.3 = 2.5 * 4^{2/3}` and +:math:`w_m^3` differs by a factor of 4. The buoyancy integration in +(`[dz_param] <#dz_param>`__), that is itself dependent on +:math:`z_{top}`, is performed working upwards from +:math:`z_{\rm par}` assuming piece-wise linear variation of :math:`b` +between grid-levels. Note that the standard definition of the boundary +layer top in the UM is the height of the first flux level below the +level of neutral buoyancy, so +:math:`z_{\rm par}` :math:`=z_{\mbox{\tiny \rm NTPAR}+\frac{1}{2}}`. The +inversion thickness is then defined as + +.. math:: + + \Delta z_i = z_{top} -z_{\rm par} + \label{dz_definition} + +.. _`sec:lclmixing`: + +Diagnosis of the LCL transition zone thickness +---------------------------------------------- + +As described in section `3.1 <#sec:adiapar>`__, when cumulus convection +has been diagnosed surface-driven mixing was originally capped at +:math:`z_{\rm lcl}` so that mixing into the cumulus cloud layer was only +carried out by the model’s mass-flux convection scheme. This was seen to +lead to errors in the mean profiles across the LCL, with superadiabats +being the most extreme manifestation. Using the boundary layer +parametrization to couple cloud and sub-cloud layers would have the +numerical advantage of being implicit. There is also observational and +LES evidence that appropriately-scaled buoyancy fluxes up to the LCL are +indistinguishable from those in cloud-free convective boundary layers +and so the non-local surface-driven mixed layer K-profiles remain +accurate up to this level. To diagnose the depth to which these profiles +should penetrate above the LCL, the algorithm given in +section `3.2 <#sec:decouple>`__ to diagnose the extent of the K-profiles +in decoupled boundary layers can be used (using the switch kprof_cu). +This ensures that the magnitude of the integrated buoyancy consumption +of TKE within the mixed layer is less than or equal to a fraction, +:math:`D_t`, of the buoyancy production. In cumulus layers, cloudy +thermals will generate positive buoyancy fluxes (and are handled by the +convection scheme) but it is assumed that there will also be cloud-free +thermals within the grid box that may penetrate above the grid-box mean +LCL (but are too dry to reach their own LCL). Thus their buoyancy flux +is given by (`[eq:wb_cont] <#eq:wb_cont>`__) with :math:`C_F=0`. +Restricting the negative integral of this buoyancy flux then gives a new +definition for :math:`z_{\rm h}`  that is then used in the calculation +of the surface-driven K-profiles in section `5.1 <#sec:nlsurf>`__ — the +larger the value of :math:`D_t`, the higher :math:`z_{\rm h}` will be. +Typically :math:`D_t=0.1` for decoupled stratocumulus layers while +idealised clear-sky convective boundary layers (where the magnitude of +the entrainment buoyancy flux is a fraction, :math:`A_1`, of the surface +flux) would have :math:`D_t = A_1^2 \sim 0.05`. For GA7 :math:`D_t` has +been set to 0.05 for this cumulus transition zone calculation. Because +the Gregory-Rowntree convection scheme triggers from the LCL, that is +used as a minimum constraint on the boundary layer mixing depth (so that +the massflux and turbulence schemes remain coupled). With other +convection schemes this may not be appropriate and so this minimum +constraint can be relaxed, which is achieved by setting it to half the +height of the LCL (the factor of a half is arbitrary, with no +sensitivity to this choice given that the diagnosis parcel reached the +LCL, but ensures the iteration starts well below the LCL). + +.. _`sec:local`: + +The local scheme +================ + +A first order ‘mixing length’ closure is used: + +.. math:: + + \begin{aligned} + K_m &=& {\cal L}_m^2 \, (S+S_d) \, f_m(Ri) \label{kmlocal}\\ + K_h &=& {\cal L}_h \, {\cal L}_m \, + (S+S_d) \, f_h(Ri) \label{khlocal} + \end{aligned} + +where :math:`{\cal L}_m` and :math:`{\cal L}_h` are the neutral mixing +lengths and :math:`S` is the resolved vertical shear of the horizontal +wind components, :math:`S = \left| \partial {\bf u}/\partial z \right|`. +A representation of the wind shear, :math:`S_d`, generated by drainage +flows in complex terrain can also be included, as described below. Near +the surface simple finite difference calculations for the vertical +gradients can become inaccurate because of the quasi-logarithmic +profiles of variables :raw-latex:`\cite[]{Ayra1991}`. Currently this is +ignored above grid-level 2 and the neutral mixing lengths are given by + +.. math:: + + \begin{aligned} + {\cal L}_m &=& \frac{k(z+z_{0m})}{1+k(z+z_{0m})/\lambda_m} \\ + {\cal L}_h &=& \frac{k(z+z_{0m})}{1+k(z+z_{0m})/\lambda_h} + \end{aligned} + +where :math:`z_{0m}` includes the orographic component. For the lowest +interior grid-level (:math:`k=1`) they are calculated, incorporating +this log profile correction, as + +.. math:: + + \tilde{{\cal L}}_{X,k-1/2} = \frac{k \Delta_{k-1/2} z}{ + ln\left( \frac{z_k + z_{0m}}{z_{k-1} + z_{0m}} \right) + + \frac{k \Delta_{k-1/2} z}{\lambda_X} } + +If near-surface resolution is increased this logarithmic correction +should be considered over more levels. + +The asymptotic mixing lengths are given by + +.. math:: + + \begin{aligned} + \lambda_m &=&\mbox{max}\left[\lambda_0,\, 0.15 z_{\rm loc}, 2 h_B \right] \nonumber\\ + \lambda_h &=&\mbox{max}\left[\lambda_0,\, 0.15 z_{\rm loc}\right] + \label{asymp_ml} + \end{aligned} + +where :math:`\lambda_0` is a minimum mixing length read in from the +namelist and :math:`z_{\rm loc}` is defined below. The orographic +blending height, :math:`h_B` (only used within the boundary layer, as +defined below), is given by + +.. math:: h_B = {\rm max}\left[z_1+(z_{0m})_{\mbox{veg}}, 2^{1/2} \sigma_h \right] + +where :math:`\sigma_h` is the standard deviation of the height of the +subgrid orography and :math:`(z_{0m})_{\mbox{veg}}` is the vegetative +part of the roughness length. The constants in +(`[asymp_ml] <#asymp_ml>`__) can be considered ‘tuned’ (see, in +particular, the operational modifications described in +appendix `15 <#app:opmods>`__). + +The Richardson number, :math:`Ri`, that is used as a local measure of +stability is given by + +.. math:: + + Ri = \frac{\Delta B / \Delta z}{(S+S_d)^2} + \label{ridefn} + +The measure of buoyancy used in :math:`Ri` is + +.. math:: + + \Delta B = g\left( \overline{\beta_T} \Delta \theta_{\ell} + + \overline{\beta_q} \Delta q_t \right) + \label{Bdefn} + +where :math:`\overline{\beta_T}` and :math:`\overline{\beta_q}` are the +grid-box mean (i.e., cloud weighted) buoyancy coefficients, that can be +defined in two different ways, see appendix `12 <#app:buoyp>`__ and +section `4.1 <#sec:fd_ri>`__. Note that (`[Bdefn] <#Bdefn>`__) reduces +to a virtual temperature approximation of buoyancy in cloud-free air and +that neutral buoyancy (in cloudy as well as cloud-free air) is implied +by vertically uniform :math:`\theta_{\ell}` and :math:`q_t`. This is +then entirely consistent with the assumption that :math:`\theta_{\ell}` +and :math:`q_t` are conserved variables within the boundary layer +scheme. + +As described in :raw-latex:`\cite{lock2012}`, the wind shear generated +by drainage flows in complex terrain is thought to lead to additional +vertical mixing. This wind shear can be approximated as + +.. math:: S_d = \frac{\Delta B }{ \Delta z} \, \alpha_d \, t_d \, {\cal Z}_d + +The representative slope of the local terrain, :math:`\alpha_d`, is +given by + +.. math:: \alpha_d^2 = \frac{1.0}{ 25.0 + (l_h/\sigma_h)^2} + +with :math:`l_h` a specified horizontal scale for the terrain, currently +taken to be 1500 m (empirically derived for Scottish orography in the +UKV), and :math:`\sigma_h` the standard deviation of the full subgrid +orographic height. :math:`\sigma_h` should also be taken as the average +over the surrounding area of each grid box (typically 6 to 8 grid +lengths), in order to be representative of the local area over which +such flows will be underresolved. The above formula is used so that +:math:`\alpha_d \sim \sigma_h/l_h` for small :math:`\sigma_h` but only +tends to 0.2 for large values. To limit the vertical extent of +:math:`S_d` to be below approximately :math:`z=\sigma_h`, a +height-dependent factor is included, +:math:`{\cal Z}_d = 0.5( 1 - {\rm tanh}\left[ 4 ((z/\sigma_h)-1) +\right])`. The timescale, :math:`t_d`, takes a fixed value of 30 +minutes, for simplicity. + +Initially, the lowest half-level at which :math:`Ri>Ri_{crit}` is taken +to be a measure of the boundary layer top (:math:`z_{\rm loc}` ) and the +full-level below is designated NTLOC. In general :math:`Ri_{crit}=1` but +a value of 0.25 is recommended for use with the ’SHARPEST’ stability +functions, see below. If the boundary layer was diagnosed as +cumulus-capped by the non-local scheme (see section `3 <#sec:types>`__) +then :math:`z_{\rm loc}` is lowered to :math:`z_{\rm lcl}` (and +:math:`K_h` and :math:`K_m` are set to zero from the base of grid-level +NLCL upwards) so that transports into and within the cumulus cloud layer +can be performed solely by the mass-flux convection scheme. Depending on +the switch local_fa, above NTLOC turbulently-mixed layers (where +:math:`Ri`__) is set to the layer +thickness. Outside of these turbulent layers the mixing lengths are set +to :math:`\lambda_0`. + +For :math:`Ri < 0`, the standard UM stability functions are given by + +.. math:: + + \begin{aligned} + f_m & =& 1 - \frac{g_0 \,Ri} + {1+D_m(\tilde{{\cal L}}_m/\tilde{{\cal L}}_h)|Ri|^{1/2} } \nonumber\\ + f_h & =& \frac{1}{Pr_N}\left(1 - \frac{g_0 \,Ri} + {1+D_h(\tilde{{\cal L}}_m/\tilde{{\cal L}}_h)|Ri|^{1/2} }\right) + \end{aligned} + +with :math:`g_0=10`, :math:`D_m=g_0/4` and :math:`D_h=g_0/25`. If the +stability dependent Prandtl number option is chosen (see below) the +neutral Prandtl number, :math:`Pr_N`, is set to :math:`0.7`; otherwise +:math:`Pr_N=1`. Alternatives are those from the Met Office large-eddy +model (LEM), :raw-latex:`\cite{brown1999}`: + +.. math:: + + \begin{aligned} + f_m & =& (1 - c_{LEM} Ri)^{1/2} \nonumber\\ + f_h & =& \frac{1}{Pr_N}\left(1 - b_{LEM} Ri\right)^{1/2} + \end{aligned} + +where :math:`Pr_N = 0.7`, and the constants :math:`b_{LEM}` and +:math:`c_{LEM}` can take the values 40 and 16 respectively in the +“standard” LEM model or both be 1.43 in the “conventional” model. + +For stable conditions (:math:`Ri > 0`), several forms for the stability +functions are available. The ‘long-tailed’ functions are + +.. math:: f_{\rm stable} = \frac{1}{1+g_0 Ri} + +Alternative functions, which decrease as :math:`1/Ri^2` with increasing +stability are, from :raw-latex:`\cite{louis1979}`: + +.. math:: f_{\rm stable} = \frac{1}{(1+ 5 Ri)^2} + +and the family of “sharp” functions can be written in terms of a +transitional Richardson number, :math:`Ri_{t}`, as: + +.. math:: + + f_{\rm stable} = + \begin{cases} + (1 - 5Ri)^2 & {\rm for}\ 0Ri_{t} + \end{cases} + +where + +.. math:: + + \begin{aligned} + A_{Ri} & = & \left(1-g_0 Ri_{t}\right)/\left(1- g_0 Ri_{t}/2\right)^2 \nonumber\\ + B_{Ri} & = & (g_0/2) /\left(1 - g_0 Ri_{t}/2\right)^2 + \end{aligned} + +For the ‘SHARPEST’ function of :raw-latex:`\cite{derbyshire1997}`, +:math:`Ri_{t}=0.1`, while larger values give even sharper reduction of +turbulence with increasing :math:`Ri`. An additional option, used +operationally in some configurations (originally in the Mesoscale Model, +hence called ’MES tails’), is to blend linearly from Louis functions at +the surface to SHARPEST by 200m. + +A stability dependent Prandtl number (:math:`Pr=f_m/f_h`) is generally +used following :raw-latex:`\cite{MailhotLock2004}` with: + +.. math:: Pr=\min \left( Pr_{\rm max}, \, Pr_N(1+2Ri) \, \right). + +The maximum permitted Prandtl number, :math:`Pr_{\rm max}`, is currently +set to :math:`5` for model stability reasons. The stability functions +for :math:`Ri>0` are then given by: + +.. math:: + + \begin{aligned} + f_m & =& \frac{Pr}{Pr_N} \, f_{\rm stable} \\ + f_h & =& \frac{1}{Pr_N} \, f_{\rm stable} + \end{aligned} + +Note that writing the functions in this way ensures that :math:`f_m=1` +under neutral conditions and the effect of the variation in :math:`Pr` +is for :math:`f_m` to decrease slower with increasing :math:`Ri` than +:math:`f_{\rm stable}`, which can be explained through increasing +gravity-wave activity. + +Finally, the LEM stable functions are also available which cut off all +turbulence beyond a critical Richardson number, :math:`Ri_c=0.25`: + +.. math:: + + \begin{aligned} + f_m & =& \left( 1 - \frac{Ri}{Ri_c} \right)^4 \\ + f_h & =& \frac{1}{Pr_N} \left( 1 - \frac{Ri}{Ri_c} \right)^4 (1 - g_{LEM} Ri) + \end{aligned} + +with :math:`g_{LEM}=1.2`. + +.. _`sec:fd_ri`: + +Finite difference calculations +------------------------------ + +The Charney-Phillips vertical grid staggering used in the UM stores the +horizontal wind components (:math:`u`, :math:`v`) on grid-levels, +:math:`\rho`-levels, that are staggered relative to scalar variables +(such as :math:`\theta_{\ell}` and :math:`q_t`) and vertical velocity, +:math:`w`. While much of the boundary layer scheme is grid-independent, +this has serious implications for the calculation of :math:`Ri`. There +are two obvious possibilities, to calculate :math:`Ri` (and thence +:math:`K(Ri)`) on either :math:`\theta`-levels or :math:`\rho`-levels +and then interpolate either :math:`K_h` or :math:`K_m` to be able to +calculate the required fluxes. To do the former requires averaging the +buoyancy gradient in the numerator (and is referred to by +:raw-latex:`\cite{cullen1994}` as the ‘:math:`\theta`-bar’ method), the +latter the wind shear in the denominator (referred to as the +‘:math:`\rho`-bar’ method). Single-column model and other tests +demonstrated that the ‘:math:`\rho`-bar’ method could readily generate +instabilities just above the top of the boundary layer because averaging +the wind shear into this stable air tended to reduce :math:`Ri` and so +promote mixing. Fortunately, the ‘:math:`\theta`-bar’ method tended to +increase :math:`Ri` above inversions and so damp mixing. Thus, +:math:`Ri` is calculated on :math:`\theta`-levels as + +.. math:: + + Ri_k = \frac{DBDZ_k} + {(\Delta_{k+\frac{1}{2}} {\bf u}/\Delta_{k+\frac{1}{2}} z)^2} + +The buoyancy gradient on :math:`\theta`-level :math:`k` can be +calculated in two different ways, depending on the switch +i_interp_local. The long-standing method is given by + +.. math:: + + DBDZ_k = g\left( \overline{\beta_T}_{k} (D\theta_{\ell}DZ)_k + + \overline{\beta_q}_{k} (Dq_t DZ)_k \right) + +where :math:`\overline{\beta_T}` and :math:`\overline{\beta_q}` are the +grid-box mean (i.e., cloud-fraction weighted) buoyancy coefficients, +defined in appendix `12 <#app:buoyp>`__. Note that because this is +defined on :math:`\theta`-levels, no vertical interpolation of cloud +variables (fractional area and water contents), to which the buoyancy +coefficients are very sensitive, is required. The volume-weighted +gradients of :math:`\theta_{\ell}` and :math:`q_t` are calculated as + +.. math:: + + (D\chi DZ)_k =\left( (z_{k}-z_{k-\frac{1}{2}}) \, \frac{\Delta_{k+1} \chi}{\Delta_{k+1} z} + + (z_{k+\frac{1}{2}}-z_{k}) \, \frac{\Delta_{k} \chi}{\Delta_{k} z} + \right) / \Delta_{k+\frac{1}{2}} z + \label{gradient_interp} + +as long as :math:`\chi_{k-1}` is defined on an atmospheric model level. +To calculate :math:`DBDZ_1`, between the surface and the lowest +:math:`\theta`-level, either the buoyancy gradient from level 1 to 2 can +be extrapolated, i.e., + +.. math:: (D\chi DZ)_1 = \frac{\Delta_{2} \chi}{\Delta_{2} z} + +or surface properties can be used. Over sea, the sea-surface temperature +and :math:`q_{sat}` can be used. Over a heterogeneous (tiled) land +surface the appropriate moisture variable varies between tiles. For the +orographic form drag (`8.8 <#section_2>`__), an average :math:`Ri_{SL}` +of the surface layer is calculated but, as discussed above, subsequent +vertical averaging of :math:`Ri` would potentially be numerically +unstable. In principle, the tile-average of :math:`(Dq_t DZ)_1` could be +calculated but for now, over land, the grid-box average surface +temperature is used to calculate :math:`(D\theta_{\ell}DZ)_1` and +:math:`(Dq_t DZ)_1` is extrapolated from above (i.e., +:math:`(Dq_t DZ)_1 = (Dq_t DZ)_2)`). + +As noted above, a feature of the previous option is that applying the +cloudy buoyancy coefficients at a cloud top level, :math:`k` say, to +strong gradients interpolated between :math:`k-1` and :math:`k+1`, can +yield an unstable :math:`DBDZ_k` despite strong static stability, +especially when the upper level is very dry. This can be related to +cloud-top entrainment instability but this process is intended to be +represented within the non-local scheme. Hence, the alternative method +is to calculate the buoyancy gradient directly on :math:`\rho`-levels +and then interpolate this vertically to give :math:`DBDZ_k`, using +(`[gradient_interp] <#gradient_interp>`__). This then requires a cloud +fraction on :math:`\rho`-levels. The difficulty comes where there is a +change in cloud fraction between levels. For this “edge” fraction, +:math:`f_{edge}` (the fraction of the grid-box that is cloudy in one +level but not in the other), the change in supersaturation +(:math:`s = q_t-q_{sat}`) between levels is used to estimate the +vertical fraction likely to contain cloud, :math:`f_{lev}`. For example, +where :math:`C_F` decreases with height, +:math:`f_{lev}={q_c}_{k-1}/(s_{k-1}-s_{k})`, where :math:`q_c` is the +total condensate, and :math:`f_{lev}` also constrained to be less than +unity. The total cloud volume fraction is then given by +:math:`f_{tot} = {\rm min}[{C_F}_{k-1},{C_F}_k] + f_{edge}f_{lev}` and +this is used to weight the saturated contribution to the buoyancy +parameters on :math:`rho`-levels, e.g., +:math:`\overline{\beta_T}_{k-1/2} = f_{tot} \tilde{\beta_T}_{k-1/2} + (1-f_{tot}){\beta_T}_{k-1/2})`, +where the saturated and unsaturated buoyancy parameters are also +intepolated to :math:`\rho`-levels using +(`[gradient_interp] <#gradient_interp>`__). + +Having calculated :math:`Ri` on :math:`\theta`-levels, :math:`{K_m}_{k}` +and :math:`{K_h}_{k}` are calculated, still on :math:`\theta`-levels, as +in (`[kmlocal] <#kmlocal>`__) and (`[khlocal] <#khlocal>`__). Finally, +:math:`K_h` must be interpolated to :math:`\rho`-levels: + +.. math:: + + {K_h}_{k+\frac{1}{2}} = \left( + (z_{k+\frac{1}{2}}-z_{k}) {K_h}_{k+1} + + (z_{k+1}-z_{k+\frac{1}{2}}) {K_h}_{k} \right) / \Delta_{k} z + +Note that in the code the convention is for fluxes to be held on the +half-level below the variable itself. Consequently, RHOKM(K), and +therefore RI(K), are held on the ‘half-level’ below :math:`\rho`-level +K, which is :math:`\theta`-level K-1. + +In addition to the above, the log profile correction applied to +:math:`{\cal + L}_h` (to give :math:`\tilde{{\cal L}}_h`) must be applied *after* +interpolation of :math:`K_h` to level :math:`k+\frac{1}{2}` in order +that the correct cancellation with the finite difference scalar gradient +in the flux calculation can occur. In the unstable stability functions +(`[unstable_stab] <#unstable_stab>`__), however, +:math:`\tilde{{\cal L}}_h` must be calculated on :math:`\theta`-levels +(i.e., the same as :math:`\tilde{{\cal + L}}_m` and :math:`Ri`) in order to maintain the same stability +dependence. + +.. _`sec:shear`: + +Shear-driven mixing and interaction between the local and non-local schemes +--------------------------------------------------------------------------- + +The general approach is to take :math:`K_{\chi}` in +(`[scal_closure] <#scal_closure>`__) and +(`[uv_closure] <#uv_closure>`__) as + +.. math:: + + K_{\chi} = \mbox{max} \left[ (K_{\chi}^{\rm surf}+K_{\chi}^{\rm Sc}), + K_{\chi}(Ri) \right] + \label{klnl} + +As noted in section `2 <#sec:closure>`__, this implies that mixing in +stable boundary layers is determined exclusively by the local scheme, +:math:`K_{\chi}(Ri)`. Continuing to calculate :math:`K_{\chi}(Ri)` in +unstable boundary layers and using (`[klnl] <#klnl>`__) is seen as the +simplest way of achieving a relatively smooth transition between stable +and unstable boundary layers. + +At the top of unstable mixed layers, great care is taken to ensure the +parametrized entrainment mixing is implemented faithfully, see +section `7 <#sec:entr>`__). Consequently, if a subgrid inversion has +been diagnosed capping a mixed layer (see +section `7.1.1 <#sec:sginv>`__), then :math:`K_{\chi}(Ri)` is set to +zero at the interfaces either side of the inversion grid-level. There +are also options (using the switch Keep_Ri_FA) to set +:math:`K_{\chi}(Ri)` to zero entirely above unstable boundary layers or +across the LCL in cumulus-capped layers. + +However, the mixed-layer depths were only diagnosed from thermodynamic +constraints. In near neutral boundary layers, shear generation of +turbulence might be expected to allow mixing to extend into regions of +weak static stability (and potentially to inhibit the formation of +cumulus). Currently, therefore, if NTLOC\ :math:`>`\ NTML+1 (in layers +that are not cumulus-capped) :math:`K_{\chi}(Ri)` is left unconstrained +by the SML part of the non-local scheme (and so not set to zero from the +SML inversion upwards) and similarly if NTLOC\ :math:`>`\ NTDSC+1. It is +realised that this does not cover the case of shear-driven mixing into +cloud layers that have been diagnosed as cumulus-capped (which would be +poorly represented by the current convection scheme). Several methods +have been introduced that attempt to alleviate this problem, giving rise +to the diagnosis of a “shear-dominated boundary layer” type (type VII), +discussed in section `3 <#sec:types>`__. The first (the “shear-dominated +boundary layer fix”) simply sets the CUMULUS flag to false if NTLOC +:math:`>` NTPAR. This then ensures that the locally-determined :math:`K` +are not set to zero above the LCL. Several more rigorous options are +available that incorporate a “dynamic criteria” in the diagnosis of +boundary layer type. The first of these prohibits the diagnosis of +cumulus boundary layers when the bulk measure of stability, +:math:`-z_i/L`, is small (currently less than 1.6). Here :math:`z_i` is +taken as the top of the diagnosis parcel ascent (or at most 3km) and +:math:`L` is the surface Obukhov length. This test also resets the depth +of the surface-based mixed layer to level 1 since the top of the parcel +ascent may not be suitable (having previously been diagnosed as cumulus +cloud top). The second method effectively increases the importance of +the Richardson number diagnosis and has been developed from analysis of +cold-air outbreaks :raw-latex:`\cite[]{bodas-salcedo2012}`. Because of +the strong surface buoyancy generation of turbulence in these regimes, a +calculation of :math:`Ri` is made that allows for the gradient +adjustment by the non-local scheme, i.e., using +:math:`\widetilde{\Delta_k \theta_{\ell}}` (see +(`[eq:wx_std] <#eq:wx_std>`__)). The height, :math:`z_{\rm loc}` , where +:math:`Ri>Ri_{crit}=0.25` is found. It is then hypothesised that this +level of turbulent instability (that incorporates the effects of shear) +only needs extend some fractional distance into the cloud layer to +disrupt the formation of cumulus elements. Thus, if +:math:`z_{\rm loc}> z_{\rm lcl}+ f_{\rm sh} +\left(z_{\rm par}-z_{\rm lcl}\right)`, where :math:`f_{\rm sh}` is a +tunable parameter (:math:`0 +Ri_{crit}`, :math:`z_{\rm h}^{\rm Sc}` is the top of any stratocumulus +layer and :math:`z_{\rm h}` is the top of surface-based mixed layer, +found by adiabatic parcel ascent but reset to the LCL in cumulus capped +layers. Another diagnostic is available, the “boundary layer depth” +(STASH 25), that is set to :math:`=\mbox{max}[z_{\rm h}, z_{\rm loc}]` +and so represents the depth of the stable boundary layer or “surface” +mixed layer. Also available are three diagnostics that represent the +calculated value of each of the individual terms in STASH 3,304: 3,356 +is set to :math:`z_{\rm h}` ; 3,357 is :math:`z_{\rm h}^{\rm Sc}`  and +3,358 is :math:`z_{\rm loc}` . + +.. _`sec:nonlocal`: + +The non-local scheme +==================== + +This method of calculating :math:`K` values for unstable conditions is +non-local in the sense that, at a given height within the boundary +layer, :math:`K` is determined not by any local properties of the mean +profiles at that height but solely by the magnitude of the turbulence +forcing applied to the layer (as measured by the representative velocity +scales described in appendix `11 <#app:vscales>`__) and the height +within the layer. The non-local scheme is therefore particularly robust +but care must be taken where the profiles are applied. The calculation +of the vertical position and extent of the :math:`K` profiles is +described in section `3 <#sec:types>`__. + +.. _`sec:nlsurf`: + +Surface-driven turbulence +------------------------- + +For turbulence sources at the surface (namely surface drag with velocity +scale :math:`u_*`, and positive surface buoyancy fluxes with velocity +scale :math:`w_*`) in a layer with top at +:math:`z=`\ :math:`z_{\rm h}` , base at :math:`z=0` we set + +.. math:: + + K_m^{\rm surf}= k \ z_{\rm h}\ w_m \ \frac{z}{z_{\rm h}} + \left( 1 - {\cal E}_m^{\rm surf} \frac{z}{z_{\rm h}} \right)^2 + \label{kmsurf} + +where :math:`w_m^3 = u_*^3 + w_s^3`, :math:`u_*` is the friction +velocity (including the orographic roughness component) and :math:`w_s` +is defined below. For the 9C version of the scheme, :math:`z_{\rm h}` is +the diagnosed subgrid inversion height (see +section `7.1.1 <#sec:sginv>`__) for both :math:`K_h^{\rm surf}` and +:math:`K_m^{\rm surf}`. In the 8A version, :math:`K_m^{\rm surf}` uses +:math:`z_{\rm h}` :math:`=z_{\mbox{\tiny \rm NTML}+\frac{1}{2}}`. The +factor :math:`{\cal E}_m^{\rm surf}` is chosen so that +:math:`K_m^{\rm surf}` will tend to +:math:`K_m|_{\mbox{\tiny \rm NTML}+\frac{1}{2}}` as :math:`z` tends to +:math:`z_{\rm h}` , where +:math:`K_m|_{\mbox{\tiny \rm NTML}+\frac{1}{2}}` is the entrainment +eddy-diffusivity (given by (`[khent] <#khent>`__), although, in order to +avoid altering the shape function too much, +:math:`{\cal E}_m^{\rm surf}` is not allowed to fall below :math:`0.7`). +A similar factor, :math:`{\cal E}_h^{\rm + surf}`, is used in the :math:`K_h^{\rm surf}` profile even though the +entrainment fluxes of the thermodynamic variables will usually be +specified explicitly rather than through an eddy-diffusivity (see +section `7 <#sec:entr>`__). + +The form of :math:`w_s` differs between the surface layer +(:math:`z < 0.1`\ :math:`z_{\rm h}` ) and the rest of the mixed-layer: + +.. math:: + + w_s^3 = + \begin{cases} + 2.5 \, \frac{z}{z_{\rm h}} w_*^3 & {\rm surface\ layer} \\ + 0.25 \, w_*^3 & {\rm mixed\ layer} \\ + \end{cases} + \label{ws_defn} + +and :math:`w_*^3=z_{\rm h}\overline{w'b}_S` using :math:`z_{\rm h}` from +the current timestep (note that the use of :math:`w_*` here will be +inconsistent with the use of :math:`V_{\rm heat}` in the entrainment +parametrization in cloudy boundary layers). Note that :math:`w_s` is +continuous across :math:`0.1`\ :math:`z_{\rm h}` and constant with +height in the mixed layer. This form for :math:`w_s` is motivated by a +desire to match the model’s surface transfer formulation within the +surface layer (as described further in section `5.1.1 <#sec:hbcomp>`__) +and to use a cubic sum of velocity scales within the mixed layer +(consistent with dimensional analysis of the TKE equation, see +:raw-latex:`\cite{holtslag93:_local_versus_nonloc_bound_layer}`). + +The formula for :math:`K_h^{\rm surf}` is identical to +(`[kmsurf] <#kmsurf>`__) but with :math:`w_m` replaced by +:math:`w_h=w_m/Pr`, where the turbulent Prandtl number is given by: + +.. math:: + + Pr = 0.75 \frac{u_*^4 + (4/25)w_s^3 w_m}{u_*^4 + (8/25)w_s^3 w_m} + \label{prandtl_nl} + +Thus :math:`Pr` varies from 0.75 in neutral conditions to 0.375 in +convective. The origin of the functional form of +(`[prandtl_nl] <#prandtl_nl>`__) is unknown. + +.. _`sec:hbcomp`: + +Comparison with :raw-latex:`\cite{holtslag93:_local_versus_nonloc_bound_layer}` +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +The surface-driven :math:`K` profiles are the same as those in +:raw-latex:`\cite{holtslag93:_local_versus_nonloc_bound_layer}`, HB93, +except for (`[ws_defn] <#ws_defn>`__) and +(`[prandtl_nl] <#prandtl_nl>`__) and the inclusion of the :math:`{\cal + E}_m^{\rm surf}` terms. For the latter, HB93 effectively set +:math:`{\cal + E}_m^{\rm surf} =1`. To generate entrainment, however, they simply use +:math:`K_m^{\rm surf}|_{\mbox{\tiny \rm NTML}+\frac{1}{2}}`, as +evaluated from (`[kmsurf] <#kmsurf>`__) with a subgrid calculation of +:math:`z_{\rm h}` :math:`>z_{\mbox{\tiny \rm NTML}+\frac{1}{2}}`, rather +than using a separate entrainment parametrization. + +The difference in (`[ws_defn] <#ws_defn>`__) arises from the surface +layer, where HB93 match :math:`w_m` to their surface exchange functions +(i.e., :math:`w_m = u_* / \phi_m`) which results in proportionality +constants of 6 and 0.6 for :math:`w_s` in the surface and mixed layers +respectively. This matching is greatly simplified because their +non-dimensional shear +:math:`\phi_m = ( 1 + 15 k (z/z_i) w_*^3 / u_*^3 )^{-(1/3)}`. To match +:math:`w_m`, through (`[ws_defn] <#ws_defn>`__), to the UM function, +:math:`\phi_m = ( 1 + 16 +k (z/z_i) w_*^3 / u_*^3 )^{-(1/4)}`, would require a complex function of +:math:`u_*` and :math:`w_*` in place of the constant and so this is not +attempted. + +The formula for the Prandtl number used in the interior in HB93 is also +matched to that used in the surface exchange functions (:math:`Pr_{\rm + surf}`, say). For the UM, + +.. math:: + + Pr_{\rm surf} = \frac{\Phi_h}{\Phi_m} + = \left( 1 + 16 \, k \frac{z}{z_{\rm h}} \, \frac{w_*^3}{u_*^3} + \right)^{-1/4} + +giving :math:`Pr_{\rm surf} = 1` in the neutral limit (compared to 0.75 +from (`[prandtl_nl] <#prandtl_nl>`__)). In the convective limit, +:math:`Pr_{\rm surf}|_{0.1\, + z_{\rm h}} \rightarrow 0.9 (w_*/u_*)^{-3/4} = 0.9 \beta^{3/4} = 0.14` +(compared to 0.375 from (`[prandtl_nl] <#prandtl_nl>`__)). Thus, the +Prandtl numbers do not match between the surface layer and interior +formulations in the UM. + +The formulation in HB93 gives :math:`Pr` varying from 1 to 0.6 (for +:math:`-z/L` varying from 0 to 10). In convective conditions +(:math:`-z/L=10`), HB93 have :math:`w_m = 0.85 w_*` and +:math:`w_h=1.4 w_*` while the UM has :math:`w_m = 0.65 +w_*` and :math:`w_h = 1.7 w_*`. The implications of these differences +from HB93 are unknown. The convective LES in :raw-latex:`\cite{lock99}` +suggest :math:`w_h +\approx w_*`; I don’t know where the larger proportionality constants +come from. + +Another difference between the UM and HB93 is that HB93 only apply +gradient adjustment above the surface layer (and this is allowed for in +their mixed layer definition of :math:`Pr`). Simulations in +:raw-latex:`\cite{brown1996}`, however, suggest that this may lead to a +cold bias at the top of the surface layer. It is attempted to alleviate +this in the UM by the application of gradient adjustment down to the +surface (although this will then lead to a dependence on the height of +the lowest grid-level). The implications of this for matching the +Prandtl number between the surface and interior in the UM is not known. + +Cloud-top-driven turbulence +--------------------------- + +For cloud-top-driven turbulence over a layer of depth :math:`z_{\rm ml}` +(with top at :math:`z_{\rm h}` or :math:`z_{\rm h}^{\rm Sc}` and base at +:math:`z_{\rm b}` , determined as in section `3.2 <#sec:decouple>`__), + +.. math:: + + K_m^{\rm Sc}= 0.63 \ k \ z_{\rm ml}\ V_{\rm Sc}\left( \frac{z'}{z_{\rm ml}} \right)^2 + \left( 1 - {\cal E}_m^{\rm Sc} \frac{z'}{z_{\rm ml}} \right)^{0.8} + \label{kmtop} + +where :math:`V_{\rm Sc}^3= V_{\rm rad}^3+V_{\rm br}^3` (see +appendix `11 <#app:vscales>`__) and :math:`z'` is height above +:math:`z_{\rm b}` . Then :math:`K_h = K_m / \mbox{Pr}`, where +:math:`\mbox{Pr}=0.75`. The resulting :math:`K_h` profile was derived +against convective cloudy LES, as described in +:raw-latex:`\cite{lock99_proceedings}`. The appropriate Prandtl number +(and therefore :math:`K_m^{\rm Sc}`) is unknown, 0.75 being chosen +simply as a number in the middle of the range usually quoted for +turbulent mixing in general. As with (`[kmsurf] <#kmsurf>`__), +:math:`z_{\rm h}`  (or :math:`z_{\rm h}^{\rm Sc}` ) are given by the +subgrid diagnosis (see section `7.1.1 <#sec:sginv>`__) except for +:math:`K_m^{\rm Sc}` in the 8A scheme which uses the height of the +half-level below (:math:`z_{\mbox{\tiny \rm NTML}+\frac{1}{2}}` or +:math:`z_{\mbox{\tiny \rm NTDSC}+\frac{1}{2}}`). Again following +(`[kmsurf] <#kmsurf>`__), the factors :math:`{\cal E}_m^{\rm Sc}` and +:math:`{\cal E}_h^{\rm Sc}` are included in (`[kmtop] <#kmtop>`__) so +that :math:`K_m^{\rm Sc}` will tend to +:math:`K_m|_{\mbox{\tiny \rm NTML}+\frac{1}{2}}` (and +:math:`K_h^{\rm Sc}` to +:math:`K_h|_{\mbox{\tiny \rm NTML}+\frac{1}{2}}`), given by +(`[khent] <#khent>`__), as :math:`z` tends to :math:`z_{\rm h}` (and +here no restriction is made on the magnitude of either :math:`{\cal + E}_m^{\rm Sc}` or :math:`{\cal E}_h^{\rm Sc}`). + +.. _`sec:gradadj`: + +Gradient adjustment +------------------- + +Recall that for :math:`\theta_{\ell}` only we use + +.. math:: + + \overline{w'\theta_{\ell}'}= - K_h \frac{\partial \theta_{\ell}}{\partial z} + K_h^{\rm surf}\gamma_{\theta_{\ell}} + \label{wthl} + +where + +.. math:: + + \gamma_{\theta_{\ell}} = + \mbox{min}\left[ A_{ga} \frac{\sigma_{T1}}{z_{\rm h}}, G_{max} \right] + \label{gradadj} + +:math:`A_{ga}=3.26`, :math:`G_{max}=10^{-3}`\ Km\ :math:`^{-1}` and +:math:`\sigma_{T1} = 1.93 \, +\overline{w'\theta_{\ell}'}_S/w_m`, where for this calculation of +:math:`w_m` (given by +:math:`w_m^3=u_*^3+0.25\,z_{\rm h}\overline{w'b}_S`) +:math:`z_{\rm h}` is taken from the previous timestep. The form of +(`[gradadj] <#gradadj>`__) is similar to that used in HB93 and the +magnitude of :math:`\gamma_{\theta_{\ell}}` is the same as in HB93 in +the convective limit — the difference in :math:`A_{ga}` exactly allows +for the different constants in (`[ws_defn] <#ws_defn>`__). + +Consistent with the mixed layer assumptions underlying the non-local +scheme, the flux profile produced by the scheme is assumed to be +essentially determined by the specified surface and entrainment values. +Thus, the effect of including this non-local term (:math:`K_h^{\rm surf} +\gamma_{\theta_{\ell}}`) is to allow the model to maintain more +well-mixed :math:`\theta_{\ell}` profiles (i.e., with +:math:`\partial \theta_{\ell}/ \partial z` less negative or even +positive in a cloud-free surface-heated boundary layer, for example), +subject to an arbitrary upper limit included for numerical safety. Hence +the term ‘gradient adjustment’ rather than non-local flux. When +estimating the buoyancy flux, then (as in +(`[eq:wx_std] <#eq:wx_std>`__)), it is simplest to allow for the +non-local term by adjusting the :math:`\theta_{\ell}` gradient. + +The equivalent term for :math:`q_t` (i.e., :math:`\gamma_{q_t}`) is set +to zero in order to represent crudely the effects on the mixed-layer +:math:`q_t` profile of entrainment drying at the mixed-layer top which +tend to make :math:`q_t` profiles less well mixed than those of +:math:`\theta_{\ell}` :raw-latex:`\cite[]{mahrt1976}`. From UM version +5.5, there is the option to implement the non-gradient stress +parametrization of :raw-latex:`\cite{brown97:_non}`, as described in +section `5.4 <#sec:ngstress>`__. + +.. _`sec:ngstress`: + +Non-gradient stress parametrization +----------------------------------- + +There is an option that is operational in the UM to include an +additional non-gradient (or non-local) stress parametrization, +:math:`{\bf + \tau}^{nl}` in (`[uv_closure] <#uv_closure>`__), as proposed by +:raw-latex:`\cite{brown97:_non}`. They showed that with only a +down-gradient stress parametrization, a one-dimensional model produced +wind profiles in the convective boundary layer that were less well-mixed +than predicted by LES, and underestimated the near surface wind. +Furthermore, :raw-latex:`\cite{brownetal2006}` showed that the +operational verification statistics indicate a slow bias in the 10 m +wind over land by day, especially in spring and summer. + +The non-gradient stress parametrization in the UM is very similar to +that proposed by :raw-latex:`\cite{brown97:_non}`, written + +.. math:: + + (\tau_x^{nl},\tau_y^{nl})= \left[ + \frac{2.7w_*^3}{(u_*^3+0.6w_*^3)}\right] \left[ \left( \frac{z'}{z_{\rm h}'} + \right) \left( 1- \frac{z'}{z_{\rm h}'} \right)^2 \right] + (\tau_x^{s},\tau_y^{s}) + \label{tau_nl} + +Here :math:`w_*` is the convective velocity scale, :math:`u_*` is the +friction velocity, and :math:`(\tau_x^{s},\tau_y^{s})` are the surface +stresses. Note that the surface stresses here have to be diagnosed +explicitly (from time-level n fields) but experience has shown these can +become unrealistically large when the near-surface wind is significantly +out of balance with the surface characteristics. As a safety measure, +these surface stresses can be limited such that the implied stress +gradient across the boundary layer is always less than a parameter, +MAX_STRESS_GRAD, currently set to 0.05 ms\ :math:`^{-2}` (which, for +example, gives a maximum :math:`u_*` of 7 ms\ :math:`^{-1}` in a +boundary layer 1km deep). The term involving :math:`u_*` and :math:`w_*` +is as proposed by :raw-latex:`\cite{brown97:_non}` (although note that +their Table 3 contains a typo), and ensures that the non-gradient stress +is zero in neutral conditions but asymptotes to a stability-independent +fraction of surface stress in convective conditions. The primed +variables in the shape function allow the non-local stress profile to +either be applied across the whole boundary layer (using :math:`z'=z` +and :math:`z_{\rm h}'=z_{\rm h}`), as in +:raw-latex:`\cite{brown97:_non}`, or only above the surface layer (using +:math:`z'=z-0.1z_{\rm h}`, :math:`z_{\rm h}'=z_{\rm h}-0.1z_{\rm h}`). +The motivation for applying the non-local stress above the surface layer +was to ensure that the match to surface layer similarity was maintained +below :math:`0.1z_{\rm h}` (although separate tests suggested that the +impact of this change is small). + +.. _`sec:rev_flux_grad`: + +The revised scalar flux-gradient formulation +-------------------------------------------- + +Following detailed analysis of many large-eddy simulations, including +both surface-heated and cloud-top cooled, a revised flux-gradient +relationship has been developed. In this section, the previous version +will be referred to as the standard one. The formulation is given in +terms of the total flux, + +.. math:: F_{\chi}^{Tot}=\overline{w'\chi'}+ F_{\chi}^{NT} + +where the non-turbulent flux, :math:`F_{\chi}^{NT}`, is the sum of the +radiative (for :math:`\theta_{\ell}`), microphysics and subsidence +fluxes. This is a crucial difference from the old formulation: the mean +profiles in LES are found to respond to the total flux profile (which is +linear in a mixed layer) rather than to the individual components of the +flux. Hence any flux-gradient relationship can never be generic to both +:math:`\overline{w'\theta_{\ell}'}` and :math:`\overline{w'q_t'}` since, +for example, the shape of the :math:`\theta_{\ell}` profile is +determined through interactions with radiation while the :math:`q_t` +profile is not. Physically, this suggests that while processes like +radiation must *locally* generate regions of cold (negatively buoyant) +air at cloud-top, subsequent mixing by turbulent eddies results in a +more-or-less uniformly well-mixed *mean* :math:`\theta_{\ell}` profile +(presumably because these eddies bring locally warm air back up to the +cloud-top region). + +So, the new formulation is written: + +.. math:: + + F_{\chi}^{Tot} = F_{\chi}^{NT}|_{z_{\rm b}} + -\left(K_h^{\rm surf}+ K_h^{\rm Sc}\right)\frac{\partial\overline{\chi}}{\partial z} + + \overline{w'\chi'}_{ng}^{\rm surf}+ \overline{w'\chi'}_{ng}^{\rm Sc} + + f_2 \left(F_{\chi}|_{z_h} - F_{\chi}^{NT}|_{z_{\rm b}} \right) + \label{fg_new} + +where :math:`z_h` and :math:`z_{\rm b}` are the heights of the top and +base of the mixed layer, respectively. It can be seen that +(`[fg_new] <#fg_new>`__) is composed of a local down-gradient component, +two non-gradient flux terms (one generated by surface-driven turbulence +and the other by cloud-top) and a non-local entrainment flux profile. +The turbulent fluxes are then obtained by subtracting off the +non-turbulent component: + +.. math:: \overline{w'\chi'}= F_{\chi}^{Tot} - F_{\chi}^{NT} + +The components of (`[fg_new] <#fg_new>`__) are: + +- :math:`K_{h,m}^{\rm surf}= k z_h w_{h,m} \frac{z}{z_h}\left(1-\frac{z}{z_h}\right)^2` + +- :math:`K_h^{\rm Sc}= 3.6 k V_{\rm Sc}z_{ml} \left(\frac{z'}{z_{ml}}\right)^{3}\left(1-\frac{z'}{z_{ml}} + \right)^{2}` + +- :math:`\overline{w'\chi'}_{ng}^{\rm surf}=K_h^{\rm surf}\gamma_{\chi}` + with :math:`\gamma_{\chi}=A_{ga}\frac{\overline{w'\chi'}_S}{w_h z_h}` + and :math:`A_{ga}=10` + +- :math:`\overline{w'\chi'}_{ng}^{\rm Sc}= f^{Sc} \left(F_{\chi}|_{z_h}- F_{\chi}^{NT}|_{z_{\rm b}} + \right)` with + :math:`f^{Sc}=3.5 \, k \, \frac{V_{\rm Sc}}{V_{\rm sum}} + \left(\frac{z}{z_h}\right)^{3}\left(1-\frac{z}{z_h}\right)` + +- :math:`f_2 = 0.5 \, \frac{z}{z_h}\, 2^{(z/z_h)^4}` + +In the above equations :math:`k` is von Karman’s constant, :math:`z'` +(:math:`=z-z_{\rm b}`) is height above the mixed layer base, +:math:`z_{ml}` (:math:`=z_h-z_{\rm b}`) is the mixed layer depth, +:math:`u_*` is the friction velocity, and :math:`w_*` and +:math:`V_{\rm Sc}` are the velocity scales for surface and cloud-top +buoyancy-driven turbulence. + +Although the structure of the surface-driven non-gradient terms is the +same as for the standard flux-gradient formulation, +(`[scal_closure] <#scal_closure>`__), note that they are now applied to +:math:`q_t` as well as :math:`\theta_{\ell}` and also the empirical +coefficients in the velocity scales have been revised: + +- :math:`w_h = (u_*^3 + C_{ws} w_*^3)^{\frac{1}{3}} / Pr_{\rm neut}` + with :math:`C_{ws}=0.42` for :math:`\frac{z}{z_h}\geq 0.1` and + :math:`C_{ws}=4.2 \frac{z}{z_h}` for :math:`\frac{z}{z_h}<0.1` + +- :math:`w_m = w_h Pr` + +The functional form of the Prandtl number, :math:`Pr`, is unchanged +except that :math:`w_m` is replaced by its neutral value: + +.. math:: + + Pr = Pr_{\rm neut} + \frac{u_*^4 + w_*^3 {w_m}^{\rm neut} / 25} + {u_*^4 + w_*^3 {w_m}^{\rm neut} Pr_{\rm neut}/ (25 Pr_{\rm conv} )} + +and the range is now :math:`Pr_{\rm neut} = 0.75` to +:math:`Pr_{\rm conv} = +0.6`. As with the standard scheme, a constant Prandtl number of 0.75 is +used to calculate :math:`K_m^{\rm Sc}`. + +Discussion of some of the revisions +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +.. container:: center + + .. container:: + :name: tab:vscales + + .. table:: Convective and Neutral limits for velocity scales + + +-------------+-------------+-------------+-------------+-------------+ + | Formulation | Convective | | Neutral | | + | | limit | | limit | | + +-------------+-------------+-------------+-------------+-------------+ + | | :math:`w_m` | :math:`w_h` | :math:`w_m` | :math:`w_h` | + +-------------+-------------+-------------+-------------+-------------+ + | HB | :math:` | :math: | :math:`u_*` | :math:`u_*` | + | | 0.84 \,w_*` | `1.4 \,w_*` | | | + +-------------+-------------+-------------+-------------+-------------+ + | UM standard | :math:` | :math: | :math:`u_*` | :math: | + | | 0.63 \,w_*` | `1.7 \,w_*` | | `1.3 \,u_*` | + +-------------+-------------+-------------+-------------+-------------+ + | UM revised | :math:` | :math:`w_*` | :math:`u_*` | :math: | + | | 0.6 \,w_*` | | | `1.3 \,u_*` | + +-------------+-------------+-------------+-------------+-------------+ + +.. container:: float + :name: fig:stab_dep + +It is useful to compare the velocity scales in the revised scheme with +those in the standard version, as well as those in +:raw-latex:`\cite{holtslag93:_local_versus_nonloc_bound_layer}`, +hereafter HB, on which the parametrization was originally based. Recall +that HB and the standard UM set +:math:`w_m = (u_*^3 + C_{ws} w_*^3)^{\frac{1}{3}}` and :math:`w_h += w_m/Pr`, with :math:`C_{ws} = 0.6` and 0.25, respectively, above the +surface layer. The convective and neutral limits for :math:`w_h` and +:math:`w_m` are given in Table `1 <#tab:vscales>`__ and the stability +dependencies of several parameters are shown in +Fig. `3 <#fig:stab_dep>`__. The parameter :math:`d` in +Fig. `3 <#fig:stab_dep>`__ contains the stability dependence of the +gradient adjustment parameter: + +.. math:: + + \gamma_{\chi}= d \frac{\overline{w'\chi'}_S}{w_* z_h} + \hspace{0.5cm} {\rm with} \hspace{0.5cm} + d^{HB} = 7.2 w_*^2/w_m^2, \hspace{0.2cm} + d^{std} = 6.3 w_*/w_m, \hspace{0.2cm} + d^{rev} = 10 w_*/w_h + \label{grad_adj} + +The inclusion of an extra :math:`w_*/w_m` factor in +:math:`\gamma_{\chi}` was a deliberate change by HB from the original +:raw-latex:`\cite{troen86:_simpl_model_atmos_bound_layer}` formulation +on which the UM was based. This seems an appealing feature (HB’s +:math:`\gamma_{\chi}` will tend to zero as :math:`w_* \rightarrow 0`) +and probably should be considered for the revised scheme (the +dash-dotted line in Fig. `3 <#fig:stab_dep>`__ sets +:math:`d^{std} = 10 w_*^2/w_h^2`). Similarly, :math:`f_2` might benefit +from an additional factor of the form :math:`(V_{\rm surf}^3+ +V_{\rm Sc}^3)/ V_{\rm sum}^3` so that it too tends to zero in the +neutral limit. Further analysis of LES and SCM tests will be required to +verify this. + +Note that the most significant change from the standard UM scheme is the +change to :math:`w_h` in the convective limit. Since +:math:`\gamma_{\chi}` remains unchanged in the convective limit, this +reduction in :math:`w_h` will result in a significantly smaller +:math:`\overline{w'\chi'}_{ng}^{\rm surf}` for the revised scheme which +gives better agreement against LES. + +Compared to the standard scheme, it appears that the revised +:math:`K_h^{\rm Sc}` is very different. However, +Fig.\ `4 <#fig:new_ksc>`__ shows that this actually amounts to a small +adjustment in the shape. In addition, note that the factors +:math:`\varepsilon_h^{surf}` and :math:`\varepsilon_h^{Sc}` have been +removed since the entrainment flux is now carried via the explicit +:math:`f_2` term. + +.. container:: float + :name: fig:new_ksc + + .. container:: center + + |image1| + +.. _`sec:blend`: + +The blended scheme +================== + +For high resolution simulations, the UM has a Smagorinsky-type subgrid +turbulence scheme, described in . However, this scheme is only truly +applicable for horizontal grid-lengths of order :math:`10` m, and any +real-world simulation run at lower resolution than this will inevitably +have unresolved scales somewhere in the domain. Rather than force the +user to make an ad-hoc decision about the scales they are interested in, +and thus grid-length at which to switch from using the boundary-layer +parametrization (1D BL) to the subgrid turbulence scheme (3D Smag), a +method for blending the two parametrizations has been developed. This +blend is regime and scale dependent, allowing a single parametrization +to be used across resolutions, including the completely +unresolved/resolved extremes. This blending process is described in +:raw-latex:`\cite{Boutleetal2014}`, which gives some examples of its use +and comparison to simulations using either the 1D BL or 3D Smag schemes +only. Updated technical details from :raw-latex:`\cite{Boutleetal2014}` +are reproduced below. Several options are available that are selected +using the switch ``blending_option``. These all follow the same +principles but differ in their choice of what should constitute the +boundary layer and how to treat non-turbulent layers of the atmosphere. + +As shown in :raw-latex:`\cite{Honnertetal2011}`, the rate at which +turbulent structures become resolved appears to be different for +different aspects of the flow. For example, moisture fluxes are on a +larger scale than heat or momentum fluxes, and so transition to being +sub-grid at lower resolution. This is just one of many challenges when +creating a truly accurate grey-zone parametrization, and so our aim here +is to start from the simplest possible approach which allows the model +to transition from unresolved to resolved turbulence in a plausible way, +without the user having to decide at which grid-length to switch from a +1D, non-local, to a 3D, local sub-grid scheme. + +Given some function, :math:`W_{1D}`, which tells us how poorly resolved +the turbulence is (:math:`=1` if unresolved, :math:`=0` if well +resolved), we can use this to blend between the 1D BL and 3D Smag +schemes. Both schemes have a local Richardson number formulation: + +.. math:: + + \label{eq-kri} + K_\chi(Ri) = l^2 S f_\chi(Ri), + +where :math:`K_\chi` is the eddy diffusivity, :math:`l` is the mixing +length, :math:`S` is the wind shear, :math:`f_\chi(Ri)` is the stability +function and :math:`\chi` represents conserved heat and moisture +variables, or momentum. Both schemes use the same stability function, +and both schemes can use the full 3D shear for :math:`S`. Therefore the +only difference is in the mixing length, which is calculated as + +.. math:: + + \label{eq-lblend} + l_{\rm blend} = W_{1D}l_{\rm bl}+(1-W_{1D})l_{\rm smag}, + +where :math:`l_{\rm bl}^{-1} = (\kappa z)^{-1} + \lambda_0^{-1}` and +:math:`l_{\rm + smag}^{-2} = (\kappa z)^{-2} + (c_s \Delta x)^{-2}`, :math:`\kappa` is +the von Karman constant and :math:`c_s` is the Smagorinsky constant. +Near the surface :math:`l_{\rm bl}` and :math:`l_{\rm smag}` are +identical, but the asymptotic values are different and this method +weights the asymptotic value according to the weighting of the two +schemes. For example, at :math:`\Delta x=1` km, +:math:`c_s\Delta x=200` m (for :math:`c_s=0.2`), whereas +:math:`\lambda_0=\max(40\ {\rm m}, 0.15z_h)`, which allows for a small +mixing length in shallow unresolved boundary layers (e.g. stable ones). + +The :raw-latex:`\cite{lock00}` scheme also contains a non-local +component to the turbulent flux, and this is simply down-weighted by +:math:`W_{1D}` to ensure that it becomes less significant as the +turbulence becomes better resolved. Therefore the full eddy diffusivity +is given by + +.. math:: K_\chi = \max\left[W_{1D}K_\chi^{\rm NL}, K_\chi(Ri)\right], + +where :math:`K_\chi^{\rm NL}` is the non-local diffusivity and :math:`l` +in Eq. `[eq-kri] <#eq-kri>`__ is given by :math:`l_{\rm blend}` in +Eq. `[eq-lblend] <#eq-lblend>`__. The turbulent flux is then calculated +as + +.. math:: F_\chi=-K_\chi\frac{\partial \chi}{\partial z} + W_{1D}F_\chi^{\rm NL}, + +where :math:`F_\chi^{\rm NL}` is the non-local flux. Therefore when +:math:`W_{1D}=1`, the scheme of :raw-latex:`\cite{lock00}` is recovered, +whilst with :math:`W_{1D}=0` the Smagorinsky-type scheme is recovered. + +Now we need to define the function :math:`W_{1D}` to blend the schemes. +Within the boundary layer this is based on the turbulent kinetic energy +partitioning given by :raw-latex:`\cite{Honnertetal2011}`. We choose the +TKE partitioning because it is most closely linked to the eddy +diffusivity we are trying to parametrize (for example a TKE based scheme +would calculate the eddy diffusivity from the TKE), and simplify the +function slightly, using + +.. math:: + + \label{eq-tanh} + W_{1D} = 1 - \tanh\left(\beta\frac{z_{\rm turb}}{\Delta x}\right)\max\left[0,\min\left[1,r_f\left(l_0-\frac{\Delta x}{z_{\rm turb}}\right)\right] \right], + +where :math:`z_{\rm turb}` is the appropriate lengthscale of the +turbulence, :math:`\beta` is a scaling parameter which controls the +speed of the transition from unresolved to resolved turbulence, +:math:`r_f=\frac{1}{l_0-l_1}`, :math:`l_0=4` and :math:`l_1=0.25` +(N. B. this formula is slightly modified from that given in +:raw-latex:`\cite[]{Boutleetal2014}`). +:raw-latex:`\cite{Malavelleetal2014}` demonstrated that this scaling +method was applicable to any type of unstable boundary layer given an +appropriate choice of :math:`z_{\rm turb}`. In +:raw-latex:`\cite{Boutleetal2014}` this functional form was applied +everywhere, adjusting the values of :math:`z_{\rm turb}` and +:math:`\beta` depending on the regime. The max function is present to +force the lowest resolution simulations to just use the 1D mixing +scheme. An alternative approach that differs above the boundary layer is +described below. + +The simplest case is for a well-mixed boundary layer, where the +appropriate lengthscale is the boundary-layer depth (inversion height). +Therefore we set :math:`z_{\rm turb}=z_h`, which is broadly consistent +with :raw-latex:`\cite{Malavelleetal2014}`, and choose +:math:`\beta=\beta_{\rm + bl}=0.15` to give the best match of our function to that of +:raw-latex:`\cite{Honnertetal2011}`. These functions are shown in +Figure `5 <#fig-blend>`__\ (a) and are only dissimilar for small +:math:`\Delta +x`, where Eq. `[eq-tanh] <#eq-tanh>`__ tends to zero faster. This is by +choice, to force the highest resolution simulations to use the 3D +turbulence scheme. + +.. container:: float + :name: fig-blend + + |image| |image2| + +One of the key benefits of the :raw-latex:`\cite{lock00}` scheme is its +ability to represent decoupled stratocumulus layers, and this is a +feature which needs to be maintained in the blended scheme. Physically +they are similar to well-mixed surface driven boundary layers, and the +:raw-latex:`\cite{lock00}` scheme parametrizes them as such. The +appropriate length scale is now the decoupled cloud mixed layer depth, +:math:`z_{\rm sc}` +:raw-latex:`\cite[i.e.~the depth through which a negatively buoyant parcel +released at cloud top would descend,][]{lock01}`. In this case, below +the decoupled cloud top we set + +.. math:: + + \label{zturb_dsc} + z_{\rm turb}=\min\left[\max\left(z,z_{\rm sml}\right),\max\left(z_{\rm sc},z_h-z\right)\right], + +where :math:`z_{\rm sml}` is the depth of the surface-based mixed layer +:raw-latex:`\cite[i.e.~the depth through which a positively buoyant parcel +released at the surface would ascend,][]{lock00}`. This is shown +schematically in Figure `5 <#fig-blend>`__\ (b), and ensures that +:math:`W_{1D}` has a high value in the poorly resolved surface mixed +layer and cloud layer, and a lower value in between those layers. Again, +this choice of :math:`z_{\rm turb}` is broadly consistent with the +analysis of decoupled stratocumulus LES presented by +:raw-latex:`\cite{Malavelleetal2014}`. Finally, +:raw-latex:`\cite{Honnertetal2011}` also included shallow cumulus +simulations and showed that the relevent length scale there was the +cloud top height. Most of the ``blending_option`` choices apply this to +all regimes diagnosed as cumulus-capped (see section `3 <#sec:types>`__) +but alternatively (``blending_option``\ :math:`=`\ 4) this can be +restricted to strictly shallow cumulus clouds, defined as contiguously +cloudy levels (cloud fraction greater than SC_CFTOL) with cloud top +height below input parameter ``shallow_cu_maxtop``. Note that the +diagnosis of shallow cumulus from the diagnosis parcel ascent (that was +used to identify a cumulus regime) was found frequently to indicate deep +convection even when the resolved clouds were shallow because the +diagnosis parcel, being undilute, would penetrate to the tropopause. +However, having decided the regime is shallow convection, we do still +set :math:`z_{\rm turb}` to the diagnosis parcel top height because, for +current km-scale configurations (without a cumulus convection +parametrization), it was found that the resulting stronger parametrized +vertical mixing was beneficial for the development of the convection, +and that without this a widespread stratiform cloud layer could develop +instead. + +Above the boundary layer top, :raw-latex:`\cite{Boutleetal2014}` aimed +for any free atmospheric mixing to be done by the 3D Smagorinsky scheme. +Therefore, above the boundary layer top they use :math:`z` as the +appropriate length scale, and in general take :math:`z_{\rm turb}` in +Eq. `[eq-tanh] <#eq-tanh>`__ as the greater of that defined by +(`[zturb_dsc] <#zturb_dsc>`__) and :math:`z`. However, this did not give +a particularly fast transition using the value of +:math:`\beta_{\rm bl}`, therefore they used :math:`\beta_{\rm fa}=1` at +a height well above the boundary layer (:math:`z_{\rm fa}=z_h+1` km), +and transitioned between these regimes linearly using + +.. math:: + + \beta = \beta_{\rm bl}\frac{z_{\rm fa}-z}{z_{\rm fa}-z_h} + + \beta_{\rm fa}\frac{z-z_h}{z_{\rm fa}-z_h} + +However, because the above method still uses (`[eq-tanh] <#eq-tanh>`__), +which depends on :math:`z_{\rm turb}/\Delta x`, the rate of transition +to 3D Smagorinsky with height above the boundary layer varies in an +undesirable way with grid size. It might be considered more logical to +think of non-turbulent regions of the free troposphere as unresolved +turbulence and so revert to the 1D mixing scheme there. An alternative +treatment(``blending_option``\ :math:`=`\ 3 or 4), then, is to increase +:math:`W_{1D}` above the boundary layer top smoothly, to reach unity by +some physical height :math:`z_{\rm fa}`, to be independent of both +horizontal and vertical grid sizes. For +:math:`z_{\rm turb} < z < z_{\rm fa}`, then, we set + +.. math:: + + W_{1D} = 1 + \frac{1}{2} \left( W_{1D}|_{z=z_{\rm turb}} - 1 \right) + \left[ 1 + {\rm cos}\left( \pi \, \frac{z-z_{\rm turb}}{z_{\rm fa}-z_{\rm turb}} + \right) \right] + +The cosine term in square brackets transitions smoothly from 2 at +:math:`z=z_{\rm turb}` to zero at :math:`z_{fa}` where, although +somewhat arbitrary, +:math:`z_{\rm fa} = {\rm min}(2 z_{\rm turb}, z_{\rm turb}+1 {\rm km})`. +The former term ensures the transition is well above any shallow +boundary layers while the latter that it does not drift far into the +free atmosphere. In addition, within any layers identified as turbulent, +through having subcritical :math:`Ri`, :math:`z_{\rm turb}` is set to +the layer depth, in the same way as is done for decoupled stratocumulus +in (`[zturb_dsc] <#zturb_dsc>`__). + +For current operational convection-permitting model grid sizes (1.5 km +in the UKV), the representation of cumulus convection remains a +challenge. One option is to include a grey-zone convection +parametrization, described in the documentation of that scheme (see ). +Tests in the UKV, though, showed some detriment to the spin-up of +resolved scale convection (as well as somewhat poor discrimination of +precipitating versus non-precipitating parametrized convection) that led +to the development of an alternative strategy, namely to abandon the +blended turbulence scheme when pure cumulus convection was diagnosed and +leave the representation of cumulus entirely to the resolved scales. +This option (``blending_option``\ :math:`=`\ 2) is also now discouraged. + +.. _`sec:entr`: + +Entrainment fluxes +================== + +**Summary**: parametrized entrainment fluxes (at the top of mixed +layers) are specified for momentum through an eddy-diffusivity, as +described in section `7.4.1 <#sec:ent_K>`__. For scalar variables, if +the inversion is sufficiently sharp so as to be unresolved, the ideal is +to specify the entrainment fluxes explicitly, as described in +section `7.1 <#sec:ent_flux>`__, based on the subgrid inversion +diagnosis described in section `7.1.1 <#sec:sginv>`__. Further details +can be found in :raw-latex:`\cite{lock01}`. If the profiles are such +that the inversion is sharp but a subgrid inversion cannot be diagnosed, +an eddy-diffusivity similar to that for momentum is used (see +section `7.4.1 <#sec:ent_K>`__). If the inversion is thick enough to be +resolved then an eddy diffusivity profile is constructed across the +inversion (see section `7.4.2 <#sec:entr_prof>`__) for both scalars and +momentum fields. For tracer variables (scalars other than +:math:`\theta_{\ell}` and :math:`q_t`), the entrainment fluxes are +specified using an equivalent eddy-diffusivity, as described in +section `7.4.3 <#sec:ent_K_flux>`__. Note that, as indicated below, +several aspects of the implementation of entrainment fluxes were revised +at the 9C scheme and these are documented separately. + +The parametrization of the entrainment rate, :math:`w_e` (given, in the +absence of subsidence, by the rate of rise of the inversion), can be +written (using the notation given in appendix `11 <#app:vscales>`__) + +.. math:: + + w_e = \frac{ A_1 \, V_{\rm sum}^3/ z_{\rm ml}+ g \tilde{\beta_T} \tilde{\alpha_t} + \Delta_F} + {\Delta b + c_T V_{\rm sum}^2/z_{\rm ml}} + \label{we_parm} + +where +:math:`V_{\rm sum}^3= V_{\rm heat}^3+ V_{\rm rad}^3+ V_{\rm br}^3+ A_2 u_*^3`. +The constant :math:`A_1` is given a value 0.23, as in +:raw-latex:`\cite{lock98}`, and :math:`A_1*A_2=5`, as in +:raw-latex:`\cite{driedonks1982}`. To allow for weak inversions, the +:raw-latex:`\cite{zilitinkevich1975}` correction is included in +(`[we_parm] <#we_parm>`__) with the constant, :math:`c_T=1`. A further +parametrization for :math:`\alpha_t`, which is the fraction of the +cloud-top radiative divergence (:math:`\Delta_F`, in Kms\ :math:`^{-1}`) +that occurs across the horizontally-averaged inversion in the LES, can +be written + +.. math:: \alpha_t = 1 - \exp{ \left\{-\Delta z_i / (2 L_{rad})\right\} } + +where the thickness of the inversion is parametrized as +:math:`\Delta z_i = +\mbox{min}[V_{\rm sum}^2/\Delta b, 100]` and :math:`L_{rad}` is a +depth-scale for the radiatively-cooled layer (taken to be 15 +:math:`\times +\,\mbox{max}[200/z_c, 1]`, where :math:`z_c` is the cloud depth). To +allow for a feedback with forcing of entrainment by buoyancy reversal +(see appendix `11 <#app:vscales>`__), +:math:`\tilde{\alpha_t} = \alpha_t+ Br +(1-\alpha_t)`. following :raw-latex:`\cite{lock98}` and +:raw-latex:`\cite{lock09:_factor}`. The calculation of the other +quantities required for (`[we_parm] <#we_parm>`__) is described in +appendix `11 <#app:vscales>`__. At some point during the transition to a +decoupled boundary layer the surface-driven entrainment terms (the terms +in (`[we_parm] <#we_parm>`__) proportional to :math:`V_{\rm heat}^3` and +:math:`u_*`) will no longer contribute to entrainment at cloud top, +because the two layers will have become entirely decoupled. If the +``entr_smooth_dec`` switch is on then the surface contribution to the +parametrized entrainment at :math:`z_{\rm h}^{\rm Sc}` is decreased +linearly as the :math:`\theta_{v\ell}` difference between NTDSC and NTML +increases from 0.5 to 1K. The flag, COUPLED, is set to true and +:math:`z_{\rm h}^{\rm Sc}` is used as the mixed-layer depth in +(`[we_parm] <#we_parm>`__) as long as any surface-driven entrainment +remains. If the ``entr_smooth_dec`` switch is off then this transition +is discontinuous at a :math:`\theta_{v\ell}` difference of 0.5K. + +It should be noted that (`[we_parm] <#we_parm>`__) takes no account of +wind shear anywhere other than at the surface. How to quantify the shear +generation of turbulence in DSC layers is not known. The direct impact +of shear across the inversion is thought to be simply to diffuse the +inversion in the vertical — this wind shear will contribute little to +the mixed layer TKE and so can not contribute to the full process of +mixing across the inversion and down into the mixed layer that is +entrainment. However, important interactions between wind shear across +inversions and cloud-top radiative cooling have been observed that are +not yet accounted for in the UM. + +The least well-determined part of (`[we_parm] <#we_parm>`__) is the +constant :math:`A_2` — the constant in the Zilitinkevich correction, +:math:`c_T`, is also approximate but is included to limit the growth of +layers capped by weak inversions and for numerical safety. A further +limit is applied to the value of :math:`w_e` determined by +(`[we_parm] <#we_parm>`__) such that the inversion cannot rise by more +than one grid-level in a timestep. With current vertical resolutions and +timesteps this is not a serious restriction. The constants :math:`A_1` +and :math:`A_{\rm br}` appeared to be determined within 10-20 % in +:raw-latex:`\cite{lock98}`, although only solid cloud sheets were +simulated (as discussed further in appendix `11 <#app:vscales>`__). +Similarly the parametrizations of :math:`\alpha_t` and +:math:`\Delta z_i` were found to be accurate but the parameter +:math:`L_{rad}` is currently only crudely represented in the UM. + +.. _`sec:ent_flux`: + +Specification of entrainment fluxes in the 9B scheme +---------------------------------------------------- + +Note that the 9C scheme (see next section) differs by generalising the +approach to include all processes operating in the inversion grid-level, +rather than just radiation. + +If it is assumed that the turbulent fluxes reduce from their extremum at +:math:`z=z_i` (the ‘entrainment’ fluxes) to zero at :math:`z=h` a small +distance above, then +:math:`\overline{w'\theta_{\ell}'}_{z_i}= - w_e \Delta \theta_{\ell}+ F|_h - +F|_{z_i}`, so that + +.. math:: + + \begin{aligned} + {\cal H}|_{z_i} & =& - w_e \Delta \theta_{\ell}+ F_{\rm net}|_h \nonumber\\ + \overline{w'q_t'}_{z_i}& =& - w_e \Delta q_t + \end{aligned} + +where the total heat flux +:math:`{\cal H} = \overline{w'\theta_{\ell}'}+ F_{\rm net}` and +:math:`F_{\rm net} += F- F|_{z_{\rm b}}`. The net radiative flux relative to the base of the +mixed layers is simply calculated as + +.. math:: + + F_{\rm net}|_{z_{k+\frac{1}{2}}} = \sum_{k=\mbox{\tiny \rm NBDSC}}^{k} \mbox{max}\left[ + - \Delta_{k+\frac{1}{2}} z \, {\cal S}_F(k), \,0 \right] + +where NBDSC\ :math:`=1` in SMLs, :math:`{\cal S}_F` are the temperature +increments (in Ks\ :math:`^{-1}`) from the radiation scheme and +:math:`F_{\rm net}|_h` is estimated by extrapolating down from +:math:`F|_{z_{\mbox{\tiny \rm NTML}+\frac{3}{2}} +}` using the flux-divergence in grid-level NTML\ :math:`+2` (and +similarly for DSC layers). + +The thermodynamic variables’ entrainment fluxes, then, are imposed +nominally at the subgrid inversion height (:math:`z_i=` +:math:`z_{\rm h}` and/or :math:`z_{\rm h}^{\rm Sc}` ), diagnosed as +described in section `7.1.1 <#sec:sginv>`__. The required grid-level +fluxes (at :math:`z_{\mbox{\tiny \rm NTDSC}+\frac{1}{2}}`, for example) +are then estimated using linear interpolation of :math:`{\cal H}` and +:math:`\overline{w'q_t'}` between :math:`z_{\rm h}^{\rm Sc}` and the +base of the mixed layer: + +.. math:: + + \begin{aligned} + \overline{w'\theta_{\ell}'}|_{ z_{\mbox{\tiny \rm NTDSC}+\frac{1}{2}} } & =& \overline{w'\theta_{\ell}'}|_{z_{\rm b}} + - \frac{ z'_{\mbox{\tiny \rm NTDSC}+\frac{1}{2}} }{z_{\rm ml}} + \left( \tilde{w_e} \Delta \theta_{\ell}+ \overline{w'\theta_{\ell}'}|_{z_{\rm b}} - F_{\rm net}|_{h} \right) + - F_{\rm net}|_{ z_{\mbox{\tiny \rm NTDSC}+\frac{1}{2}} } \nonumber\\ + \overline{w'q_t'}|_{ z_{\mbox{\tiny \rm NTDSC}+\frac{1}{2}} } & =& \overline{w'q_t'}|_{z_{\rm b}} + - \frac{ z'_{\mbox{\tiny \rm NTDSC}+\frac{1}{2}} }{z_{\rm ml}} + \left( \tilde{w_e} \Delta q_t + \overline{w'q_t'}|_{z_{\rm b}} \right) + \end{aligned} + +where :math:`z' = z-z_{\rm b}`, and similarly for the SML entrainment +fluxes (at :math:`z=z_{\mbox{\tiny \rm NTML}+\frac{1}{2}}`). The +turbulent fluxes at the base of the mixed layer are assumed zero except +for the SML where the surface fluxes are used. This interpolation is +illustrated for a SML in Fig. `6 <#fig:fluxinterp>`__. + +.. container:: float + :name: fig:fluxinterp + +Note that, because (`[fluxinterp] <#fluxinterp>`__) includes an explicit +balance between the turbulent and radiative fluxes for +:math:`\overline{w'\theta_{\ell}'}`, it is not possible to parametrize +the entrainment fluxes through a single :math:`K_h` for both +:math:`\overline{w'\theta_{\ell}'}` and :math:`\overline{w'q_t'}`. +Furthermore, the radiative forcing of turbulence in the mixed layer is +fixed through the timestep and so it is consistent to assume the +entrainment fluxes (at :math:`z_i`) are also fixed. Hence +(`[fluxinterp] <#fluxinterp>`__) are implemented explicitly, rather than +via an eddy-diffusivity. This is discussed further, with reference to +tracer fluxes, in section `7.4.3 <#sec:ent_K_flux>`__. + +In order to allow for the long timesteps used in NWP and to facilitate +movement of the subgrid inversion across grid-levels within a timestep, +the parametrization of :math:`w_e` and the model’s subsidence velocity, +:math:`w_S|_{z_i}`, are used to calculate :math:`z_i` at the next +time-level (:math:`z_i^{n+1}`). Currently, the latter is found by linear +interpolation to :math:`z_i` and both are assumed constant in time. If +:math:`z_i^{n+1} < z_{\mbox{\tiny \rm NTDSC}+\frac{1}{2}}`, then the +entrainment fluxes there (given by (`[fluxinterp] <#fluxinterp>`__)) are +multiplied by the fraction of the timestep that :math:`z_i` was above +this grid-level, namely +:math:`(z_i-z_{\mbox{\tiny \rm NTDSC}+\frac{1}{2}})/(z_i - z_i^{n+1})`. +The full entrainment flux at grid-level NTDSC\ :math:`-\frac{1}{2}` must +then also be specified, given by (`[fluxinterp] <#fluxinterp>`__) with +:math:`z_{\mbox{\tiny \rm NTDSC}+ \frac{1}{2}}` replaced by +:math:`z_{\mbox{\tiny \rm NTDSC}- \frac{1}{2}}`. If :math:`z_i` rises +above :math:`z_{\mbox{\tiny \rm NTDSC}+\frac{3}{2}}`, the entrainment +flux is specified only at this higher grid-level (multiplied by the +fraction of the timestep that :math:`z_i` is above this half-level) and +the values of the mixed-layer :math:`K` profiles are used in half-level +NTDSC\ :math:`+\frac{1}{2}` (these will be non-zero because +:math:`z_i>z_{\mbox{\tiny \rm NTDSC}+ \frac{1}{2}}`). Wherever the +entrainment fluxes are specified explicitly, the eddy-diffusivities +(both non-local and local) are set to zero. Also, the mean value of +:math:`z_i` during the timestep is used in +(`[fluxinterp] <#fluxinterp>`__) in order best to approximate the mean +flux gradient across the mixed layer. + +Finally, the entrainment flux is adjusted to allow for numerical +entrainment arising from the model’s resolved vertical advection (as +discussed in :raw-latex:`\cite{lock01}`). This is performed at whichever +grid-level the entrainment fluxes are specified, to allow for any +entrainment implied by a :math:`\theta_{\ell}` subsidence increment, +:math:`\Theta^{\rm + S}` (Ks\ :math:`^{-1}`), at the model grid-level below. The subsidence +increments could be obtained directly in the SCM but in the full 3D UM +advection increments are dominated by the horizontal component. The +subsidence increments are calculated, therefore, from the vertical +velocity field using first order upwind advection (it would clearly be +preferable to use the model’s actual vertical advection algorithm in the +GCM although the errors incurred in this diagnostic calculation should +not be very significant). The interpolated entrainment fluxes given by +(`[fluxinterp] <#fluxinterp>`__) are therefore calculated not using +:math:`w_e` but using an entrainment velocity, :math:`\tilde{w_e}`, that +is reduced to allow for any subsidence increments applied to the +grid-level below the entrainment flux. To take the case of +:math:`z_{\mbox{\tiny \rm NTDSC}+\frac{1}{2}} < +z_i^{n+1} < z_{\mbox{\tiny \rm NTDSC}+\frac{3}{2}}` as an example, this +reduced entrainment velocity is given by + +.. math:: \tilde{w_e} = w_e + \tilde{w_S} + +with :math:`\tilde{w_e}` constrained to lie between 0 and :math:`w_e` +and + +.. math:: + + \tilde{w_S} = - \, \frac{ \Theta^{\rm S}_{\mbox{\tiny \rm NTML}} + ( \Delta_{\mbox{\tiny \rm NTML}+\frac{1}{2}} z ) } + { \Delta \theta_{\ell}} + \label{we_num} + +.. _`sec:sginv`: + +Diagnosis of a sub-grid inversion +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +The profile of :math:`\theta_{v\ell}` is used to diagnose the height of +a discontinuous inversion because it is approximately conserved under +adiabatic vertical motion and is equal to the virtual potential +temperature, :math:`\theta_v`, in the absence of cloud. This should +ensure it is monotonically increasing with height in the statically +stable free-troposphere of a GCM. If :math:`\theta_{v\ell}` does not +increase monotonically between grid-levels NTML and NTML+2 (or NTDSC and +NTDSC+2), then entrainment fluxes are simply specified via +(`[khent] <#khent>`__) and none of the coupling with subsidence or +radiation described above is attempted (the local scheme is also +currently not set to zero above NTML or NTDSC when this occurs to allow +it to diffuse out this static instability). + +.. container:: float + :name: zi_diag + +Having identified the model grid-level at the top of the well-mixed +layer (either level NTML from the parcel ascent, as described in +section `3.1 <#sec:adiapar>`__, or NTDSC for DSC layers, see section +`3.2 <#sec:decouple>`__— the analysis is the same for both), the +grid-level above is designated the inversion level within which the +diagnosis of a subgrid :math:`z_i` will be made. It is assumed that +:math:`\theta_{v\ell}` in grid-level NTML\ :math:`+1` represents a +cell-average value. Thus, :math:`z_i` can be calculated by assuming that +the integral of :math:`\theta_{v\ell}` over grid-level NTML\ :math:`+1` +for the model and for a profile with a discontinuous inversion at +:math:`z_i` are equal, as illustrated by the hatched areas in +Fig. `7 <#zi_diag>`__. To calculate the integral of the discontinuous +profile, the lapse rate of :math:`\theta_{v\ell}` between grid-levels +NTML\ :math:`-1` and :math:`NTML`, :math:`\gamma^{\tiny \rm ML}`, is +extended up to :math:`z_i`, while the stable lapse in the free +atmosphere, between grid-levels NTML\ :math:`+2` and NTML\ :math:`+3`, +:math:`\gamma^{\scriptsize \rm FA}`, is extrapolated down. Equating +these areas gives a quadratic equation in :math:`\Delta z_{disc} = +z_{\mbox{\tiny \rm NTML}+\frac{3}{2}} - z_i` which can be written + +.. math:: + + a (\Delta z_{disc})^2 + b \ \Delta z_{disc} +c =0 + \label{zi_interp} + +The coefficients are given by + +.. math:: + + \begin{aligned} + a & =& 0.5 (\gamma^{\scriptsize \rm FA}- \gamma^{\tiny \rm ML}) \\ + b & =& - \left( {\theta_{v\ell}}_{\mbox{\tiny \rm NTML}+2} + - \gamma^{\scriptsize \rm FA}(z_{\mbox{\tiny \rm NTML}+2}-z_{\mbox{\tiny \rm NTML}+\frac{3}{2}}) \right) + + \left( {\theta_{v\ell}}_{\mbox{\tiny \rm NTML}} + + \gamma^{\tiny \rm ML}(z_{\mbox{\tiny \rm NTML}+\frac{3}{2}}-z_{\mbox{\tiny \rm NTML}}) \right) \\ + c & =& (z_{\mbox{\tiny \rm NTML}+\frac{3}{2}}-z_{\mbox{\tiny \rm NTML}+\frac{1}{2}}) + \left( {\theta_{v\ell}}_{\mbox{\tiny \rm NTML}+1} - + \left( {\theta_{v\ell}}_{\mbox{\tiny \rm NTML}} + + \gamma^{\tiny \rm ML}\left( + \frac{1}{2}(z_{\mbox{\tiny \rm NTML}+\frac{1}{2}}+z_{\mbox{\tiny \rm NTML}+\frac{3}{2}}) + -z_{\mbox{\tiny \rm NTML}} \right) \right) + \right) \\ + \end{aligned} + +Clearly, care must be taken to ensure that :math:`z_i` is not only +well-defined but also sensible (for example, as a rising inversion +encounters more or less stable regions above). If :math:`b>0` this +suggests the estimated lapse rates are inappropriate and these are +therefore set to zero and (`[zi_interp] <#zi_interp>`__) is +recalculated. The case :math:`c<0` suggests the grid-level designated as +the inversion level should have been considered as part of the mixed +layer and so :math:`z_i` is set to be fractionally below +:math:`z_{\mbox{\tiny \rm NTML}+\frac{3}{2}}` (i.e., as high as possible +without attempting to diagnose a subgrid :math:`z_i` in grid-level +NTML\ :math:`+2`). If :math:`b^2-4ac<0` the quadratic equation has no +real roots. In this instance :math:`z_i` is set to fractionally below +:math:`z_{\mbox{\tiny \rm NTML}+\frac{1}{2}}` and NTML (and therefore +the eddy-diffusivity profiles) is lowered by a grid-level. In all other +circumstances, the required root is then +:math:`\Delta z_{disc} = (-b - (b^2-4ac)^{1/2} +)/(2a)`; the other root will either be larger or negative (if +:math:`a<0`). + +In addition, from variations seen in :math:`z_i` during single-column +model simulations, the error in :math:`\Delta z_{disc}` is estimated to +be around 10% of the vertical resolution, +:math:`\Delta_{\mbox{\tiny \rm NTML}+\frac{3}{2}} z`. Accordingly, if +:math:`z_i` is diagnosed as being less than +:math:`z_{\mbox{\tiny \rm NTML}+\frac{1}{2}} + 0.1 \, \Delta_{\mbox{\tiny \rm NTML}+\frac{3}{2}} z`, +NTML is lowered a grid-level and :math:`z_i` is set fractionally below +:math:`z_{\mbox{\tiny \rm NTML}+\frac{1}{2}}`. This small distance below +the grid-level is taken to be :math:`(\Delta t/2) \times 10^{-4}` so +that, were a small rate of rise of :math:`z_i` (of :math:`10^{-4}` +ms\ :math:`^{-1}`, say) to be diagnosed, then :math:`z_i` would spend at +least half the timestep (of length :math:`\Delta t`) in the next +grid-level up. The specified fluxes would then contribute significantly +to that grid-level’s evolution. Conversely, if :math:`z_i` is subsiding, +this technique allows the inversion to drop down a grid-level without +requiring this to be detected by the initial parcel ascent. + +Having calculated :math:`z_i`, the discontinuous jumps in +:math:`\theta_{\ell}` and :math:`q_t` that are used in the entrainment +calculation are calculated from similar integral assumptions: + +.. math:: + + \Delta \chi = \left( {\chi}_{\mbox{\tiny \rm NTML}+1} - {\chi}_{\mbox{\tiny \rm NTML}} \right) \, + \frac{ z_{\mbox{\tiny \rm NTML}+\frac{3}{2}} - z_{\mbox{\tiny \rm NTML}+\frac{1}{2}} } + { z_{\mbox{\tiny \rm NTML}+\frac{3}{2}} - z_i } + \label{dqt_disc} + +with :math:`\chi = \theta_{\ell}` and :math:`q_t`. Note that the lapse +rate above the inversion has been ignored as there is no guarantee of +monotonicity in :math:`q_t` in the atmosphere above the inversion. In +addition, (`[dqt_disc] <#dqt_disc>`__) will become increasingly +inaccurate as :math:`z_i` tends to +:math:`z_{\mbox{\tiny \rm NTML}+\frac{3}{2}}` (and so +:math:`{\chi}_{\mbox{\tiny \rm NTML}+1}` approaches +:math:`{\chi}_{\mbox{\tiny \rm NTML}}`). Consequently, if the fraction +on the right hand side of (`[dqt_disc] <#dqt_disc>`__) is greater than +10, double grid-level jumps are used (i.e., +:math:`\Delta \chi = {\chi}_{\mbox{\tiny \rm NTML}+2} - +{\chi}_{\mbox{\tiny \rm NTML}}`). Finally, note that +(`[dqt_disc] <#dqt_disc>`__) implicitly assumes the structure of the +:math:`\theta_{\ell}` and :math:`q_t` profiles across the inversion +grid-level are consistent with the diagnosed :math:`z_i`. This is very +unlikely to be the case, for example, when running from an analysis so +the 9C scheme uses what has been found to be a more robust algorithm, +see the separate documentation. + +It would clearly be advantageous to pass knowledge of this subgrid +inversion structure to other parametrizations in the UM, particularly +the cloud scheme as currently the cloud fraction in level +NTML\ :math:`+1` is essentially meaningless (being diagnosed from a +mixture of cloudy boundary layer air and typically very dry free +tropospheric air). + +.. _`sec:ent_flux_9c`: + +Specification of entrainment fluxes across sharp inversions in the 9C scheme +---------------------------------------------------------------------------- + +As described in section `7.1 <#sec:ent_flux>`__, when the capping +inversion is thinner than the model vertical grid it is important for +the entrainment flux implementation that the subsidence increments are +realistically and consistently distributed between the inversion +grid-level and the mixed layer. Rather than work with the increments +themselves, though, a more robust solution is to couple the subsidence +and turbulent fluxes across the inversion, exactly analogously to the +coupling of turbulent and radiative fluxes. This allows the total +tendency of the inversion grid-level to be linked to whether the +inversion should be rising or falling (determined from the balance +between the parametrized entrainment rate, :math:`w_e`, and the +large-scale vertical velocity evaluated at the inversion, +:math:`w|_{z_h}`). + +.. container:: float + :name: fig:rev_fluxes + +An idealised subgrid total flux profile is constructed from the +parametrized entrainment flux and the increments from radiation, +precipitation and subsidence, assuming a well-mixed boundary layer +capped by a diagnosed subgrid inversion. The crucial step is to ensure +that the total flux on the model entrainment grid-level equals the +idealised total flux profile interpolated to that level. Consider the +example illustrated in Fig. `8 <#fig:rev_fluxes>`__ of a well-mixed +boundary layer up to :math:`\theta`-level :math:`\mbox{\tiny \rm NTML}`. +For the subgrid :math:`q_t` profiles, the turbulent flux divergence +generates a moistening across the inversion while subsidence generates +drying. For this example it has been assumed the entrainment rate is +slightly larger than the subsidence velocity at the inversion and so +overall there is a weak moistening relative to the mixed layer (the +total flux gradient is more negative across the inversion than in the +mixed layer), consistent with the rising tendency of the inversion. For +the model, the subsidence flux-divergence associated with the inversion +is split across levels :math:`\mbox{\tiny \rm NTML}` and +:math:`\mbox{\tiny \rm NTML}+1`. To keep the *net* moistening of the +model’s boundary layer and inversion consistent with the total subgrid +flux profile, the model’s entrainment flux at +:math:`\mbox{\tiny \rm NTML}+1/2` (shown by the cross in +Fig. `8 <#fig:rev_fluxes>`__) must be found by subtracting the +subsidence flux at :math:`\mbox{\tiny \rm NTML}+1/2` (diamond) from the +total flux interpolated to :math:`\mbox{\tiny \rm NTML}+1/2` (square). +Exactly the same arguments follow for the :math:`\theta_{\ell}` fluxes +except that the situation is complicated by the addition of the +radiative flux. + +The above arguments can be generalised as follows. Writing +:math:`F_{\chi}^{Tot}` as the total flux of a conserved variable +:math:`\chi` (:math:`=q_t` or :math:`\theta_{\ell}`) and +:math:`F_{\chi}^{NTP}` as the flux from physics sources other than +turbulence (i.e., radiation, :math:`F_{\chi}^{rad}`, in the above +examples, but including precipitation fluxes, :math:`F_{\chi}^{ppn}`, in +the full model) and :math:`F_{\chi}^{subs}` as the flux from resolved +scale subsidence, the total flux at the subgrid inversion height is +given by: + +.. math:: + + F_{\chi}^{Tot}|_{z_h} = - w_e \Delta \chi + F_{\chi}^{NTP}|_{z_t} + F_{\chi}^{subs}|_{z_h} + \label{fxtot_zi} + +As in section `7.1 <#sec:ent_flux>`__, (`[fxtot_zi] <#fxtot_zi>`__) is +derived by integrating the conservation equation for :math:`\chi` over +an inversion in which jumps occur over a thin layer with base at a +height :math:`z_h` and top at :math:`z_t` (in the UM, the inversion is +assumed to be infinitesimally thin so that :math:`z_t=z_h`). This +integration gives :math:`- +w_e \Delta \chi = \overline{w'\chi'}|_{z_h} -(F_{\chi}^{NTP}|_{z_t}-F_{\chi}^{NTP}|_{z_h})`. +:raw-latex:`\cite{lock99}` related the non-turbulent flux divergence, +:math:`F_{\chi}^{NTP}|_{z_t}-F_{\chi}^{NTP}|_{z_h}`, to radiative +cooling occurring within undulations of the cloudy boundary layer top. +Similar considerations need to be borne in mind when calculating all the +non-turbulent fluxes in (`[fxtot_zi] <#fxtot_zi>`__). First, the +radiative flux is extrapolated down from +:math:`\mbox{\tiny \rm NTML}+\frac{3}{2}` to :math:`z=z_t` using the +divergence in the grid-level above the inversion as representative of +the free-atmospheric divergence. Second, since the microphysical flux is +generated within the cloud, :math:`F_{\chi}^{ppn}|_{z_t} = +{F_{\chi}}^{ppn}_{\mbox{\tiny \rm NTML}+\frac{3}{2}}`. Finally, the +subsidence flux-divergence across level :math:`\mbox{\tiny \rm NTML}` +and :math:`\mbox{\tiny \rm NTML}+1` is assumed to be associated with the +inversion so :math:`{F_{\chi}}^{Subs}|_{z_h} = +{F_{\chi}}^{Subs}_{\mbox{\tiny \rm NTML}-\frac{1}{2}}`. Thus, the +finite-difference form of (`[fxtot_zi] <#fxtot_zi>`__) becomes + +.. math:: + + F_{\chi}^{Tot}|_{z_h} = - w_e \Delta \chi + + F_{\chi}^{rad}|_{z_t} + {F_{\chi}}^{ppn}_{\mbox{\tiny \rm NTML}+\frac{3}{2}} + + {F_{\chi}}^{subs}_{\mbox{\tiny \rm NTML}-\frac{1}{2}} + \label{fxtot_zi_fd} + +Then, assuming a linear profile of :math:`F_{\chi}^{Tot}` in the mixed +layer, interpolating the total flux to the inversion flux grid-level +gives + +.. math:: + + F_{\chi}^{Tot}|_{ \mbox{\tiny \rm NTML}+\frac{1}{2} } = F_{\chi}^{Tot}|_{z_{\rm b}} + + \frac{ z'_{\mbox{\tiny \rm NTML}+\frac{1}{2}} }{z_{\rm ml}} + \left( F_{\chi}^{Tot}|_{z_h} - F_{\chi}^{Tot}|_{z_{\rm b}} \right) + \label{fxtot_interp} + +where :math:`z'` (:math:`=z-z_{\rm b}`) is height above the base of the +mixed layer at :math:`z=z_{\rm b}`. Finally, the grid-level turbulent +entrainment flux is given by: + +.. math:: + + \overline{w'\chi'}|_{ \mbox{\tiny \rm NTML}+\frac{1}{2} } = F_{\chi}^{Tot}|_{ \mbox{\tiny \rm NTML}+\frac{1}{2} } + - F_{\chi}^{NT}|_{ \mbox{\tiny \rm NTML}+\frac{1}{2} } + \label{rev_entflux} + +This revised algorithm has several advantages over the previous. +Firstly, the fluxes for :math:`q_t` and :math:`\theta_{\ell}` are +coupled independently, whereas in the 9B version the coupling with +subsidence was estimated using only the :math:`\theta_{\ell}` increments +in order to calculate :math:`\tilde{w_e}` in `[we_num] <#we_num>`__. +Secondly, this method makes it much simpler to include all processes, +and precipitation in particular, in a consistent manner. Thirdly, since +the total grid-level flux, +:math:`F_{\chi}^{Tot}|_{\mbox{\tiny \rm NTML}+\frac{1}{2}}` in +(`[fxtot_interp] <#fxtot_interp>`__), is used to calculate the +entrainment fluxes, it is straightforward to ensure that the net budget +of the inversion grid-level, namely :math:`- ( +F_{\chi}^{Tot}|_{\mbox{\tiny \rm NTML}+\frac{3}{2}} - F_{\chi}^{Tot}|_{\mbox{\tiny \rm NTML}+\frac{1}{2}})/\Delta z`, +is consistent with the entrainment/subsidence balance. In other words, +to use :math:`\theta_{\ell}` as an example, if the inversion is rising +(falling) then +:math:`F_{\chi}^{Tot}|_{\mbox{\tiny \rm NTML}+\frac{1}{2}}` is limited +to ensure that the inversion grid-level will cool (warm). Finally, if +the inversion is rising we don’t want the inversion grid-level +:math:`\theta_{\ell}` to cool to less than :math:`\theta_{\ell}` of the +mixed layer by the end of the timestep. In other words, for +:math:`\chi=\theta_{\ell}`, given + +.. math:: + + \begin{aligned} + \chi_{\mbox{\tiny \rm NTML}+1}^{n+1} & =& \chi_{\mbox{\tiny \rm NTML}+1}^{n} + - \frac{\Delta t}{\Delta z} \left( + F_{\chi}^{Tot}|_{ \mbox{\tiny \rm NTML}+\frac{3}{2} } - F_{\chi}^{Tot}|_{ \mbox{\tiny \rm NTML}+\frac{1}{2} } + \right) \\ + \chi_{\mbox{\tiny \rm NTML}}^{n+1} & =& \chi_{\mbox{\tiny \rm NTML}}^{n} + - \frac{\Delta t}{z_{\mbox{\tiny \rm NTML}+\frac{1}{2}}} \left( + F_{\chi}^{Tot}|_{ \mbox{\tiny \rm NTML}+\frac{1}{2} } - F_{\chi}^{Tot}|_{z_{\rm b}} + \right) \\ + \end{aligned} + +where the superscripts :math:`n` and :math:`n+1` refer to the model +timestep, although strictly speaking :math:`n+1` refers to fields after +the boundary layer implicit solver. Requiring that +:math:`\chi_{\mbox{\tiny \rm NTML}+1}^{n+1}\geq\chi_{\mbox{\tiny \rm NTML}}^{n+1}` +implies + +.. math:: + + F_{\chi}^{Tot}|_{ \mbox{\tiny \rm NTML}+\frac{1}{2} } + \left( 1+ \frac{\Delta z}{z_{ml}}\right) + \geq F_{\chi}^{Tot}|_{ \mbox{\tiny \rm NTML}+\frac{3}{2} } + \Delta z \left( + \frac{\chi_{\mbox{\tiny \rm NTML}}^{n}-\chi_{\mbox{\tiny \rm NTML}+1}^{n}}{\Delta t} + + \frac{F_{\chi}^{Tot}|_{z_{\rm b}}}{z_{ml}} \right) + +The same arguments apply for :math:`q_t`, noting that the free +atmosphere can be drier or moister than the mixed layer and so these +cases must be treated separately. If :math:`\theta_{\ell}` of the free +atmosphere is colder than the mixed layer then no subgrid inversion +treatment is attempted and entrainment is modelled using a +straightforward eddy diffusivity. + +Calculation of the inversion jumps in the 9C scheme +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +In the 9B scheme, the discontinuous jumps in :math:`\theta_{\ell}` and +:math:`q_t` that are used in the entrainment calculation were calculated +from integral assumptions similar to those used to diagnose the subgrid +inversion height, :math:`z_h`, and were given by +(`[dqt_disc] <#dqt_disc>`__). Note that +:math:`{\chi}_{\mbox{\tiny \rm NTML}+2}` does not appear in +(`[dqt_disc] <#dqt_disc>`__) and so no direct information from the free +atmosphere is used. Only if the budgets of :math:`\theta_{\ell}` and +:math:`q_t` in level :math:`\mbox{\tiny \rm NTML}+1` are entirely +consistent with the rise and fall of the subgrid inversion will +(`[dqt_disc] <#dqt_disc>`__) give accurate results. This will not be the +case during an assimilation cycle, for example, neither is it likely to +be the case if the convection scheme is detraining into level +:math:`\mbox{\tiny \rm NTML}+1`. + +Instead, a more robust algorithm is used in the 9C scheme and the +subgrid inversion calculation is only attempted where both +:math:`\theta_{\ell}` and :math:`\theta_{v\ell}` are monotonically +increasing and :math:`q_t` is simply monotonic across the inversion. The +formula used is: + +.. math:: + + \Delta \chi = {\chi}_{\mbox{\tiny \rm NTML}+2} - {\chi}_{\mbox{\tiny \rm NTML}} + - \gamma_{\chi} \left( z_{\mbox{\tiny \rm NTML}+2} - z_h \right) + \label{dqt_disc_9c} + +subject to the constraint that the lapse rate adjustment should not +reduce the two grid-length difference by more than half. The +free-atmospheric lapse rates are given by + +.. math:: + + \begin{aligned} + \gamma_{\theta_{\ell}} & =& {\rm max}\left[ \, 0, \, \frac{ {\theta_{\ell}}_{\mbox{\tiny \rm NTML}+3}-{\theta_{\ell}}_{\mbox{\tiny \rm NTML}+2} } + { z_{\mbox{\tiny \rm NTML}+3} - z_{\mbox{\tiny \rm NTML}+2} } + \right] \\ + \gamma_{q_t} & =& {\rm min}\left[ \, 0, \, \frac{ {q_t}_{\mbox{\tiny \rm NTML}+3}-{q_t}_{\mbox{\tiny \rm NTML}+2} } + { z_{\mbox{\tiny \rm NTML}+3} - z_{\mbox{\tiny \rm NTML}+2} } + \right] + \end{aligned} + +.. _`sec:subs_calc`: + +Calculation of the subsidence flux +---------------------------------- + +The vertical advection or subsidence flux, :math:`{F_{\chi}}^{subs}`, is +calculated by integrating estimates of the vertical advection +increments. These estimates are made at 9B from the model’s vertical +velocity field, :math:`w`, using first order upwind advection. As +described above, however, the coupling between different flux profiles +is performed on the model grid and, over land, these coordinate surfaces +follow the underlying terrain. To correct this, the 9C scheme calculates +the subsidence flux in grid-point, rather than physical space, by using +:math:`\dot{\eta}` (where :math:`\eta` is the model’s vertical +coordinate) rather than :math:`w`. + +The following two examples illustrate why this represents an +improvement. First, consider a boundary layer capped by a horizontal +inversion in a horizontal flow over a rising land surface. Here +:math:`w` will be zero and yet the model will be generating a vertical +advection flux across the inversion grid-levels, because +:math:`\dot{\eta}` is negative. Conversely, consider the same boundary +layer but in a flow that follows the coordinate surfaces, going up and +over a hill. Now there will be no vertical advection flux across the +model’s inversion grid-level because :math:`\dot{\eta}` is zero and yet +:math:`w` will be negative on the down-slope thus giving a spurious +subsidence source to the 9B scheme. + +Specification of entrainment eddy diffusivity +--------------------------------------------- + +As discussed above it is considered beneficial to specify the +thermodynamic entrainment fluxes explicitly under the assumption that +both the turbulence forcing and the inversion jumps change slowly +compared to the timestep. Under the circumstance that no subgrid +inversion can be diagnosed, not only is an alternative derivation of the +entrainment fluxes required, but it is also deemed likely that these +assumptions may be violated and so the entrainment fluxes are specified +via an entrainment eddy diffusivity. Currently, this is also the case +for momentum and tracer variables. + +.. _`sec:ent_K`: + +For momentum (and scalars if no subgrid inversion) +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +For momentum, and scalars if a subgrid inversion cannot be diagnosed, +see section `7.1.1 <#sec:sginv>`__, fluxes at the mixed layer top are +specified through an eddy diffusivity which is given by + +.. math:: + + \begin{aligned} + K_h|_{\mbox{\tiny \rm NTML}+\frac{1}{2}} & =& w_e \Delta_{\mbox{\tiny \rm NTML}+1} z \nonumber\\ + K_m|_{\mbox{\tiny \rm NTML}} & =& Pr \, w_e \Delta_{\mbox{\tiny \rm NTML}+\frac{1}{2}} z + \end{aligned} + +noting the Charney-Philips grid implying stresses are staggered from +scalar fluxes. The Prandtl number, :math:`Pr`, takes the same form as +for the non-local :math:`K` profiles, see section `5 <#sec:nonlocal>`__. + +Substituting (`[khent] <#khent>`__) in +(`[scal_closure] <#scal_closure>`__) gives, for example, +:math:`\overline{w'\theta_{\ell}'}|_{\mbox{\tiny \rm NTML}+\frac{1}{2}} = - w_e \Delta_{\mbox{\tiny \rm NTML}+1} \theta_{\ell}`. +Note that this gives entrainment buoyancy fluxes identical to +(`[discinv] <#discinv>`__) as long as there is no buoyancy reversal +generation of turbulence (i.e.,\ :math:`V_{\rm br}=0`) and if variations +in the grid-level jumps across the timestep are ignored. The former is +because the other terms in (`[we_parm] <#we_parm>`__) are inversely +proportional to :math:`\Delta +b`. The latter will never actually be true and can give rise to large +errors if the inversion is rising quickly. Therefore, the thermodynamic +entrainment fluxes are specified explicitly where possible. + +The advantages of diagnosing the subgrid inversion are that it allows +consistency between the turbulent and radiative fluxes and large-scale +vertical advection, it reduces grid-resolution errors arising from the +mixed layer depth calculation and it allows a more accurate calculation +of :math:`V_{\rm br}` and :math:`\alpha_t`. For momentum, because the +jumps across inversions are typically small and variable, it seems +unwise numerically to attempt to specify the inversion stresses +explicitly and so (`[khent] <#khent>`__) is always used. For the 9C +version, the entrainment :math:`K_m` given by (`[khent] <#khent>`__) is +imposed at the height of the temperature inversion +:math:`z_{\rm h}` (either subgrid or at +:math:`z_{\mbox{\tiny \rm NTML}+\frac{1}{2}}`) and +:math:`K_m|_{\mbox{\tiny \rm NTML}+\frac{1}{2}}` is calculated from +(`[kmsurf] <#kmsurf>`__) and (`[kmtop] <#kmtop>`__), noting the use of +the :math:`{\cal + E}` factors. + +.. _`sec:entr_prof`: + +Resolved inversions +~~~~~~~~~~~~~~~~~~~ + +An inversion is defined as being resolved when it extends above the +flux-level above the usual entrainment interface level (see +section `3.3 <#sec:dzi>`__), i.e. when + +.. math:: z_{\mbox{\tiny \rm NTML}+\frac{1}{2}} + \Delta z_i > z_{\mbox{\tiny \rm NTML}+\frac{3}{2}} + +When this happens, there is no subgrid inversion diagnosis and the +entrainment parametrization follows the methodology given in +section `7.4.1 <#sec:ent_K>`__ to give +:math:`K_h|_{\mbox{\tiny \rm NTML}+\frac{1}{2}}`. The diffusion +coefficient profile within the inversion is then calculated assuming the +:math:`\theta_{v\ell}` flux profile within the inversion decreases +following a cosine shape from the standard parametrized entrainment flux +at the inversion base to zero at the inversion top, i.e.: + +.. math:: + + \overline{w'\theta_{v\ell}'} = \overline{w'\theta_{v\ell}'}|_{\mbox{\tiny \rm NTML}+\frac{1}{2}} + cos\left(\pi \frac{z'}{2} \right) + \label{ent_svl} + +where :math:`z'=(z-z_{\rm h})/\Delta z_i` is scaled height within the +inversion. This flux profile is then converted into a diffusion +coefficient profile by inverting the standard flux parametrization: + +.. math:: + + K_h|_{k+\frac{1}{2}}= - \, \frac{\overline{w'\theta_{v\ell}'} } + { ({\theta_{v\ell}}_{k+1}-{\theta_{v\ell}}_{k})/(z_{k+1}-z_k) } + +The diffusion coefficient for momentum entrainment is calculated in the +same way, allowing for the staggered grid, with the same :math:`Pr` as +in (`[khent] <#khent>`__). + +.. _`sec:ent_K_flux`: + +For tracers, when there is a subgrid inversion +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +Here ‘tracers’ refers to scalar variables other than +:math:`\theta_{\ell}` and :math:`q_t`: aerosols, :math:`q_f`, etc. +Ideally, tracer entrainment fluxes would be specified explicitly in the +same way as for :math:`\theta_{\ell}` and :math:`q_t`. However, +specifying the entrainment flux effectively specifies the net change in +mixed-layer tracer concentration across the timestep. Thus, if the +mixed-layer tracer concentration is small at the start of a timestep and +the entrainment flux is larger than the surface flux, the mixed-layer +concentration could go negative (and tests indicated that this did +indeed happen). Specifying the entrainment flux assumes that both the +turbulence forcing and the inversion jump change slowly compared to the +timestep. Whilst this is true for atmospheric :math:`\theta_{\ell}` and +:math:`q_t`, the latter is not true for tracers with a small boundary +layer concentration. Consequently, for a tracer field :math:`\chi`, the +parametrized entrainment fluxes :math:`\overline{w'\chi'}_{ + z_{\mbox{\tiny \rm NTML}+\frac{1}{2}} }` are calculated from +(`[fluxinterp] <#fluxinterp>`__) but are implemented through an +equivalent entrainment eddy-diffusivity given by: + +.. math:: + + K_{\chi}|_{\mbox{\tiny \rm NTML}+\frac{1}{2}} = - \overline{w'\chi'}_{ z_{\mbox{\tiny \rm NTML}+\frac{1}{2}} } + \frac{\Delta_{\mbox{\tiny \rm NTML}+1} z}{\Delta_{\mbox{\tiny \rm NTML}+1} \chi} + \label{K_ent_tracer} + +Note from (`[scal_closure] <#scal_closure>`__) that +(`[K_ent_tracer] <#K_ent_tracer>`__) gives the parametrized flux if +:math:`\Delta_{\mbox{\tiny \rm NTML}+1} \chi` does not change across the +timestep (see section `9 <#sec:implicit>`__ for a description of the +implicit numerical solution of (`[cons_eqn_scal] <#cons_eqn_scal>`__)). +As (`[K_ent_tracer] <#K_ent_tracer>`__) involves the potentially +numerically dangerous calculation of +:math:`\Delta \chi/\Delta_{\mbox{\tiny \rm NTML}+1} \chi` (where +:math:`\Delta +\chi` is the subgrid inversion jump, given by +(`[dqt_disc] <#dqt_disc>`__)), the following constraints are also +ensured: + +.. math:: + + 0 \leq K_{\chi}|_{\mbox{\tiny \rm NTML}+\frac{1}{2}} + \leq 10 \,K_{\chi}|_{\mbox{\tiny \rm NTML}-\frac{1}{2}} + +Surface Exchange +================ + +Note that the surface scheme itself is documented under the JULES +documentation. + +.. _section_1: + +The theoretical basis. +---------------------- + +Making the assumption that **Monin-Obukhov similarity theory** for the +surface layer is valid the gradients of model variables in the surface +layer are related to the surface fluxes by: + +.. math:: + + \begin{aligned} + \frac{\partial T}{\partial z} + \frac{g}{ c_P }&=&-\frac{ H_0 }{ c_P \rho _0 v_\ast } \frac{ \phi _h (z/L)}{kz}\label{1.1.1}\\ + \frac{\partial q}{\partial z}&=&-\frac{ E_0 }{ \rho _0 v_\ast } \frac{ \phi _h (z/L)}{kz}\label{1.1.2}\\ + \frac{\partial {\rm {\bf v}}}{\partial z}&=&\frac{ {\rm {\bf \tau }}_{0} }{ \rho _0 v_\ast } \frac{ \phi _m (z/L)}{kz},\label{1.1.3} + \end{aligned} + +where subscript 0 represents a surface value and subscript \* represents +a surface layer scaling quantity. :math:`\phi _{m}` and :math:`\phi +_{h}` are the Monin-Obukhov stability functions (for the form of these +see section `8.3 <#section_1.3>`__ below). :math:`L` is the +Monin-Obukhov length scale defined by + +.. math:: + + L = \frac{- { v_\ast }^3 }{k F_{B0} / \rho _0 }, + \label{1.1.4} + +where F\ :math:`_{B0}` is the surface buoyancy flux defined by + +.. math:: + + F_{B0} = \frac{ g }{ c_P } \beta _{T1} H_0 + g \beta _{q1} E_0. + \label{1.1.5} + +The buoyancy coefficients in equation (`[1.1.5] <#1.1.5>`__) are given +in appendix `12 <#app:buoyp>`__ with the subscript 1 denoting a value at +the lowest level in the atmosphere model. + +Equations (`[1.1.1] <#1.1.1>`__)–(`[1.1.3] <#1.1.3>`__) can be +integrated from the “surface”, i.e. the roughness height where the +surface variables are defined, to a reference height in the surface +layer, for modelling applications, the height, z\ :math:`_{1}`, of the +bottom model layer above the surface. The resulting expressions for the +surface turbulent fluxes are: + +.. math:: + + \begin{aligned} + \frac{ H_0 }{ c_P \rho _0 }&=&-\frac{ c_H }{ c_D^{1/2} } v_\ast \left( {\Delta T + \frac{g}{c_p }( z_1 + z_{0m} - z_{0h} )} \right)\label{1.1.7}\\ + \frac{ E_0 }{ \rho _0 }&=&-\frac{ c_H }{ c_D^{1/2} } v_\ast \Delta q\label{1.1.8}\\ + \frac{ {\bf \tau }_{0} }{ \rho _{0} }&=& c_D^{1/2} v_\ast \Delta {\rm {\bf v}},\label{1.1.9} + \end{aligned} + +where :math:`\Delta`\ X=X\ :math:`_{1}`-X\ :math:`_{0}`. +From (`[1.1.7] <#1.1.7>`__) and (`[1.1.8] <#1.1.8>`__) the surface +buoyancy flux in definition (`[1.1.4] <#1.1.4>`__) is + +.. math:: + + \frac{ F_{B0} }{ \rho _0 } = -\frac{ c_H }{ c_D^{1/2} } v_\ast \Delta B, + \label{1.1.10} + +.. math:: + + \Delta B = g \beta _{T1} \left( {\Delta T + \frac{g}{ c_P }( z_1 + z_{0m} - z_{0h} )} \right) + + g \beta _{q1} \Delta q + \label{1.1.11} + +The **surface exchange coefficients** in +equations (`[1.1.7] <#1.1.7>`__)–(`[1.1.9] <#1.1.9>`__), c\ :math:`_{D}` +and c\ :math:`_{H}`, are given by + +.. math:: + + \begin{aligned} + c_D^{1/2}&=&\frac{k}{ \Phi _m (L , z_1 + z_{0m} , z_{0m} )} + \label{1.1.12}\\ + \frac{ c_H }{ c_D^{1/2} }&=&\frac{k}{ \Phi _h (L , z_1 + z_{0m} , z_{0h} )}, + \label{1.1.13} + \end{aligned} + +where + +.. math:: + + \begin{aligned} + \Phi _m (L , z_1 + z_{0m} , z_{0m} )&=& \int \limits_{ z_{0m} /L}^{( z_1 + z_{0m} )/L} \frac{ \phi _m (\zeta )}{\zeta } d\zeta\label{1.1.14}\\ + \Phi _h (L , z_1 + z_{0m} , z_{0h} )&=& \int \limits_{ z_{0h} /L}^{( z_1 + z_{0m} )/L} \frac{ \phi _h (\zeta )}{\zeta } d\zeta\label{1.1.15}, + \end{aligned} + +z\ :math:`_{0m}` and z\ :math:`_{0h}` are the **surface roughness +lengths** for momentum and scalars respectively. + +The equations for the **surface turbulent fluxes**, +(`[1.1.7] <#1.1.7>`__)–(`[1.1.9] <#1.1.9>`__), can be written in the +forms + +.. math:: + + \begin{aligned} + \frac{ H_0 }{ c_P \rho _0 }&=&{-c}_H V \left( {\Delta T + \frac{g}{c_p }( z_1 + z_{0m} - z_{0h} )} \right)\nonumber\\ + &=& - C_H \left( {\Delta T + \frac{g}{c_p } ( z_1 + z_{0m} - z_{0h} )} \right)\label{1.1.16}\\ + \frac{ E_0 }{ \rho _0 }&=&- c_H V \Delta q = - C_H \Delta q\label{1.1.17}\\ + \frac{ {\rm {\bf \tau }}_{0} }{ \rho _{0} }&=& c_D V \Delta {\rm {\bf v}}{ }= C_D \Delta {\rm {\bf v}},\label{1.1.18} + \end{aligned} + +where the effective wind speed for surface turbulent exchanges, +:math:`V`, is defined by + +.. math:: V = \frac{ v_\ast }{ c_D^{1/2} } = \frac{ v_\ast ^2 }{ C_D }\label{1.1.19} + +and the **surface conductances** for scalars and momentum are +respectively + +.. math:: + + \begin{aligned} + C_H&=&\frac{k}{ \Phi _h } v_\ast = c_H V\label{1.1.20}\\ + C_D&=&\frac{k}{ \Phi _m } v_\ast = c_D V\label{1.1.21}. + \end{aligned} + +The surface exchange coefficients can then be written in any of the +following forms: + +.. math:: + + \begin{aligned} + c_H&=&\frac{ C_H }{V} = \frac{ C_H C_D }{ v_\ast ^2 } = \frac{ k^2 }{ \Phi _h \Phi _m }\label{1.1.22}\\ + c_D&=&\frac{ C_D }{V} = \frac{ C_D^2 }{ v_\ast ^2 } = \frac{ k^2 }{ \Phi _m^2 }. + \label{1.1.23} + \end{aligned} + +In order to close the system the surface scaling velocity, +v\ :math:`_{\ast +}`, needs to be specified. If + +.. math:: v_\ast = u_\ast \equiv \left| { {\rm {\bf \tau }}_{0} {/} \rho _{0} } \right|^{1/2}\label{1.1.24} + +we have the standard Monin-Obukhov theory and it is easy to deduce that +with this definition :math:`v_{\ast } = c_{D}^{1/2} \Delta`\ **v** and +V=\ :math:`\Delta`\ **v**. To allow for the effect of **turbulent and +cloud-scale gusts** on the surface turbulent fluxes the surface scaling +velocity, v\ :math:`_{\ast }`, can be defined as + +.. math:: + + v_\ast ^2 = u_\ast ^2 + \gamma _t^2 w_\ast ^2 + \gamma _c^2 w_c^2. + \label{1.1.25} + +The second term represents the effects of turbulent eddy-scale +convective gusts and w\ :math:`_{\ast }` is the turbulent convective +scaling velocity defined by + +.. math:: + + w_\ast = {\left( { z_i \frac{ F_{B0} }{ \rho _0 }} \right)}^{1/3} + \label{1.1.26} + +for F\ :math:`_{B0} >` 0 and zero otherwise. z\ :math:`_{i}` is the +height of the top of the surface-based turbulent mixing layer. +:math:`\gamma _{t}` is a dimensionless constant which can be determined +empirically or tuned within empirical limits. The third term represents +the effects of deep convective cloud-scale gusts; the inclusion of this +term is optional. The form implemented is taken from +:raw-latex:`\cite{redelsperger00:_param_mesos_enhan_surfac_fluxes}`, in +which the velocity scale, w\ :math:`_{c}`, is a function of the +convective downdraught mass-flux at cloud base. (Note that the published +expression is given as an adjustment of the 10-m wind and has been +scaled to make it consistent with :math:`v_\ast`.) A further term +:math:`\gamma +_{m}^{2}`\ w\ :math:`_{m}^{2}` could be included in low resolution +models to represent the effects of mesoscale gusts but this is not done +in the Unified Model. The **low wind speed limit**, i.e. as +:math:`\Delta`\ **v** :math:`\to` 0, for unstable conditions (with +w\ :math:`_{c }`\ = 0) can be seen to be + +.. math:: + + v_\ast \sim \gamma _t w_\ast \sim \gamma _t^{3/2} {\left( {\frac{ c_H }{ c_D^{1/2} }} \right)}^{1/2} z_i^{1/2} (-\Delta B )^{1/2} + \label{1.1.27} + +which implies that + +.. math:: + + L \sim -( \gamma _t^3 /k) z_i + \label{1.1.28} + +Thus the low wind speed limits for the sensible and latent heat fluxes +are obtained by substituting (`[1.1.27] <#1.1.27>`__) into +(`[1.1.7] <#1.1.7>`__) and (`[1.1.8] <#1.1.8>`__) with the surface +transfer coefficients evaluated with L given by +(`[1.1.28] <#1.1.28>`__). The finite limit for L implies that the form +of the stability functions, :math:`\phi`, for very large and negative +:math:`\zeta` is unimportant. However, the value of :math:`\Phi_{h}` for +L given by (`[1.1.28] <#1.1.28>`__) is needed if the value of +:math:`\gamma _{t}` is determined from measurements of say the latent +heat flux in very low mean wind conditions. + +.. _section_1.2: + +Comparison with the :raw-latex:`\cite{godfrey1991}` formulation for gustiness +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +We can define the **mean gust speed** at height z\ :math:`_{1}` by + +.. math:: + + v_g = ( V^2 - \left| {\Delta {{\rm {\bf v}}}} \right|^2 {)}^{{1/2}}. + \label{1.2.1} + +Using the definitions of :math:`V` (`[1.1.19] <#1.1.19>`__) and +:math:`v_{\ast }` (`[1.1.25] <#1.1.25>`__) it can be deduced that + +.. math:: + + v_g^2 = W_g^2 ( z_1 ) + \frac{1}{2}\left| {\Delta {{\rm {\bf v}}}} \right|{ }\left[ {\left( {{ } {\left| {\Delta {{\rm {\bf v}}}} \right|}^2 - W_g^2 ( z_1 )} \right)^{1/2} - \left| {\Delta {{\rm {\bf v}}}} \right|} \right] + \label{1.2.2} + +where + +.. math:: + + W_g (z) = \frac{1}{ c_D^{1/2} } {\left( { \gamma _t^2 w_\ast ^2 + \gamma _c^2 w_c^2 } \right)}^{1/2} = \frac{ \Phi _m (L , z + z_{0m} , z_{0m} )}{k} {\left( { \gamma _t^2 w_\ast ^2 + \gamma _c^2 w_c^2 } \right)}^{1/2}. + \label{1.2.3} + +Thus in this formulation the mean gust speed is a function of height +above the surface through the same factor, :math:`\Phi _{m}`\ (z), which +determines the profile of the mean wind **v** in the surface layer (see +Eq. (`[1.1.9] <#1.1.9>`__)). The values of :math:`\Delta`\ **v**, +v\ :math:`_{g}` and :math:`V` thus tend to zero as z :math:`\to` 0. Note +that v\ :math:`_{g} \to` W\ :math:`_{g}` as :math:`\Delta`\ **v** +:math:`\to` 0 and that v\ :math:`_{g} \to` 0 as the convective gustiness +scaling velocities tend to zero. + +Equation (`[1.2.1] <#1.2.1>`__) can be rewritten as + +.. math:: + + V^2 = \left| {\Delta {{\rm {\bf v}}}} \right|^2 + v_g^2, + \label{1.2.4} + +which is exactly the form of :raw-latex:`\cite{godfrey1991}`. However +:raw-latex:`\cite{godfrey1991}` define the mean gust speed as +:math:`\beta`\ w\ :math:`_{\ast +}`. Thus they directly modify the mean surface to air wind difference, +:math:`\Delta`\ **v**, with the gustiness or turbulent convective +scaling velocity, w\ :math:`_{\ast }`, combining a grid dependent +quantity with a constant scaling speed. The two formulations are similar +in that they introduce a mean gust speed but differ in their assumption +about whether this has a non-constant profile in the surface layer. +Although transitory wind gusts may not be as close to equilibrium with +the surface characteristics as the mean wind they should have a profile +in the surface layer which approaches zero at the surface (strictly at +the roughness height z\ :math:`_{0m})`. + +The two formulations can be made equivalent by assuming that +:raw-latex:`\cite{godfrey1991}` :math:`\beta` is not constant but is +given by :math:`\gamma +_{t}(\Phi _{m}`\ (z)/k) which tends to zero as the surface is +approached. However, over sea points where the roughness length is small +(of order 10\ :math:`^{-4}` m) and for which +:raw-latex:`\cite{godfrey1991}` derived their formulation, +:math:`\Phi_{m}`\ (z) varies at most by about 15% between 10 m and 50 m. +Assuming a constant :math:`\beta` does not lead to much inaccuracy in +these circumstances. If gustiness is included over land, as is the case +in the Unified Model, the higher roughness lengths lead to a greater +variation in :math:`\Phi _{m}`\ (z) in the region where models generally +have their lowest level placed. + +If :math:`\gamma _{t}` = 0.08 then in the low wind speed limit of an +unstable tropical maritime surface layer with a virtual temperature +lapse of 1.5 K, a specific humidity lapse of 7x10\ :math:`^{-3}` kg/kg, +SST=303.16 K and a boundary layer depth of 800 m we obtain a latent heat +flux of 37.08 W/m\ :math:`^{2}` assuming the form for the stability +functions given below. + +Making surface exchange consistent with flux differencing +--------------------------------------------------------- + +The boundary layer scheme increments conserved quantities using +differences in fluxes across a layer. This is strictly consistent only +if the conserved quantity is a mass-weighted mean across the layer, +rather than a representative value, such as the value at the middle of +the layer. If the profile of the conserved quantity is linear the mean +of the quantity is the same as the point-value in the middle of the +layer, but this is not so if the profile is not linear. Near the +surface, the profiles will be logarithmic in neutral conditions and so +the values in the middle of the layer will be larger than the mean +values. Surface similarity in the UM is applied treating taking the wind +and temperature in the bottom layer as point values in the calculation +of surface fluxes and is therefore not absolutely consistent with flux +differencing. Whilst the effect of this difference is not large, it is +desirable to have the option of correcting it, which is done by enabling +the option to “make surface exchange consistent with flux differencing.” +The following discussion explains how this is done. + +In effect, the UM takes the displacement height for momentum as +:math:`-z_{0m}`, where :math:`z_{0m}` is the momentum roughness length. +The profile of wind is therefore determined by Monin-Obukhov theory as + +.. math:: \frac{\partial u}{\partial z} = \frac{u_*}{k(z+z_{0m})} \phi_m((z+z_{0m})/L), + +with :math:`u_*` being the friction velocity, :math:`L` the surface +Obukhov length and :math:`\phi_m` the similarity function. It is common +practice to introduce a new function :math:`\psi_m` such that +:math:`\phi_m(\zeta)=1-\zeta \partial \psi_m / \partial \zeta.` +Redefining the vertical coordinate as :math:`\zeta=z/L`, we have + +.. math:: + + \begin{aligned} + u(\zeta) &=& \frac{u_*}{k} \int_{0}^{\zeta} \frac{1}{(\zeta'+\zeta_{0m})} + \phi_m(\zeta'+\zeta_{0m}) \, d\zeta' = + \frac{u_*}{k} \int_{\zeta_{0m}}^{\zeta'+\zeta_{0m}} + \frac{1}{\zeta'} \phi_m(\zeta') \, d\zeta' \\ + &=& \frac{u_*}{k} \int_{\zeta_{0m}}^{\zeta'+\zeta_{0m}} \left ( \frac{1}{\zeta'} - + \frac{d\psi_m}{d\zeta'} \right ) \, d\zeta' \nonumber \\ + &=& \frac{u_*}{k} \left \{ \ln \left ( \frac{\zeta+\zeta_{0m}}{\zeta_{0m}} + \right ) - \psi_m(\zeta+\zeta_{0m}) + \psi_m (\zeta_{0m}) \right \}. + \nonumber + \end{aligned} + +This is also frequently written as + +.. math:: u(\zeta) = \frac{u_*}{k} \Phi_m(\zeta). + +The mean over the lowest layer, of depth :math:`z_1` (or :math:`\zeta_1` +in the rescaled coordinate), is therefore + +.. math:: + + \begin{aligned} + \bar u &=& \frac{u_*}{k\zeta_1} \int_0^{\zeta_1} u(\zeta) \, d \zeta \\ + &=& \frac{u_*}{k\zeta_1} \int_0^{\zeta_1} + \ln \left ( \frac{\zeta+\zeta_{0m}}{\zeta_{0m}} \right ) + - \psi_m(\zeta+\zeta_{0m}) + \psi_m (\zeta_{0m}) \, d \zeta. + \nonumber + \end{aligned} + +We consider the three terms within the integral separately. For the +first, + +.. math:: + + \begin{aligned} + \int_0^{\zeta_1} \ln \left ( \frac{\zeta+\zeta_{0m}}{\zeta_{0m}} \right ) \, d \zeta + &=& \zeta_{0m} \int_1^{1+\zeta_1/\zeta_{0m}} \ln(x) \, dx \\ + &=& \zeta_{0m} \left [ \left ( 1+ \frac{\zeta_1}{\zeta_{0m}} \right ) \ln + \left ( 1+ \frac{\zeta_1}{\zeta_{0m}} \right ) - + \left ( 1+ \frac{\zeta_1}{\zeta_{0m}} \right ) +1 \right ] . \nonumber + \end{aligned} + +For the second, + +.. math:: + + \begin{aligned} + \int_0^{\zeta_1} \psi_m(\zeta+\zeta_{0m}) \, d\zeta &=& + \int_{\zeta_{0m}}^{\zeta_1+\zeta_{0m}} \psi_m(\zeta) \, d\zeta \\ + &=& \left [ \zeta \psi_m + \right ]_{\zeta_{0m}}^{\zeta_1+\zeta_{0m}} + - \int_{\zeta_{0m}}^{\zeta_1+\zeta_{0m}} \zeta \frac{d\psi_m}{d\zeta} + d\zeta \nonumber \\ + &=& (\zeta_1+\zeta_{0m}) \psi_m(\zeta_1+\zeta_{0m}) - \zeta_{0m} + \psi_m(\zeta_{0m}) \nonumber \\ &-& \int_{\zeta_{0m}}^{\zeta_1+\zeta_{0m}} + (1-\phi_m) d\zeta \nonumber \\ + &=& (\zeta_1+\zeta_{0m}) \psi_m(\zeta_1+\zeta_{0m}) - \zeta_{0m} + \psi_m(\zeta_{0m}) \nonumber \\ &+& \int_{\zeta_{0m}}^{\zeta_1+\zeta_{0m}} + (\phi_m -1) \, d\zeta . \nonumber + \end{aligned} + +:math:`\phi_m-1` is retained in the last integral since this will prove +convenient in later algebra. The third integral is trivial. Hence, + +.. math:: + + \begin{aligned} + \bar u &=& \frac{u_*}{k} \left \{ + \left ( 1+ \frac{\zeta_{0m}}{\zeta_1} \right ) \left [ + \ln \left ( 1+ \frac{\zeta_1}{\zeta_{0m}} \right ) \right . \right . \\ + &-& \left . \left . \psi_m(\zeta_1+\zeta_{0m}) + \psi_m(\zeta_{0m}) \right ] -1 + - \frac{1}{\zeta_1} \int_{\zeta_{0m}}^{\zeta_1+\zeta_{0m}} + (\phi_m -1) \, d\zeta \right \} \nonumber \\ + &=& \frac{u_*}{k} \left \{ \left ( 1+ \frac{\zeta_{0m}}{\zeta_1} \right ) + \Phi_m(\zeta_1) - \frac{1}{\zeta_1} \int_{\zeta_{0m}}^{\zeta_1+\zeta_{0m}} + \phi_m \, d\zeta \right \} . \nonumber + \end{aligned} + +Thus, in practical terms, the standard function :math:`\Phi_m` is +evaluated at the top of the layer, scaled by +:math:`1+\zeta_{0m}/\zeta_1` and reduced by the mean value of +:math:`\phi_m`. For standard Monin-Obukhov functions, this last integral +is easy to perform. In the limit :math:`L \rightarrow +\infty` it becomes 1 and to avoid a numerical singularity it is set +equal to 1 in this (nearly neutral) limit. The adjustment of the thermal +Monin-Obukhov function is exactly equivalent. + +Algorithmically, the existing routine ``PHI_M_H`` is replaced by +``PHI_M_H_VOL`` which takes the same inputs except that the heights are +the top of the layers. This is done when the surface fluxes, or +turbulence scales :math:`u_*` and :math:`\theta_*` are calculated. +Monin-Obukhov functions are also used to calculate winds and +temperatures at observed levels. In this case a value at a particular +height is required, so the existing Monin-Obukhov routine should be +used. + +.. _section_1.3: + +The form of the stability functions. +------------------------------------ + +For **stable conditions**, i.e. :math:`\Delta`\ B :math:`\ge` 0, the +stability functions are given by :raw-latex:`\cite{Beljaars1991}`: + +.. math:: + + \begin{aligned} + \Phi _m&=&\ln \left( {\frac{ z_1 + z_{0m} }{ z_{0m} }} \right) - \Psi _m ( \zeta _1 ) + \Psi _m ( \zeta _{0m} ) + \label{1.3.11}\\ + \Phi _h&=&\ln \left( {\frac{ z_1 + z_{0m} }{ z_{0h} }} \right) - \Psi _h ( \zeta _1 ) + \Psi _h ( \zeta _{0h} ) + \label{1.3.12} + \end{aligned} + +where :math:`\zeta _{1}` = (z\ :math:`_{1}` + z\ :math:`_{0m})`/L, +:math:`\zeta _{0m}` = z\ :math:`_{0m}`/L, :math:`\zeta _{0h}` = +z\ :math:`_{0h}`/L and + +.. math:: + + \begin{aligned} + - \Psi _h (\zeta )&=&\left[ { {\left( {1 + \frac{2}{3}a\zeta } \right)}^{3/2} - 1 } \right] + b\left( {\zeta - \frac{c}{d}} \right)\exp (-d\zeta ) + \frac{bc}{d} + \label{1.3.13}\\ + - \Psi _m (\zeta )&=&a\zeta + b\left( {\zeta - \frac{c}{d}} \right)\exp (-d\zeta ) + \frac{bc}{d}, + \label{1.3.14} + \end{aligned} + +with :math:`a = 1`, :math:`b =2/3`, :math:`c = 5`, :math:`d = 0.35`. + +Note that the bulk flux Richardson number for the surface layer is given +by + +.. math:: + + {Ri}_{fB} = \frac{ c_D^{1/2} }{k} \frac{ z_1 }{L} = + \frac{ z_1 /L}{ \Phi _m } + \label{1.3.8} + +so the :raw-latex:`\cite{Beljaars1991}` functions imply +Ri\ :math:`_{f B} \to` 1/a = 1 as z\ :math:`_{1}`/L :math:`\to \infty`. + +For **unstable conditions**, i.e. :math:`\Delta`\ B :math:`<` 0, the +Dyer and Hicks forms :raw-latex:`\cite[]{dyer1974}` are used: + +.. math:: + + \begin{aligned} + \phi _m&=&(1 - 16\zeta )^{-1/4} + \label{1.3.15} \\ + \phi _h&=&(1 - 16\zeta )^{-1/2} + \label{1.3.16} + \end{aligned} + +(Note that :math:`\phi _{h}\prime` is discontinuous at 0.) These are +only empirically verified for :math:`\zeta \ge` -1. Evaluating the +integrals (`[1.1.14] <#1.1.14>`__) and (`[1.1.15] <#1.1.15>`__) we +obtain: + +.. math:: + + \Phi _m = \ln \left( {\frac{ z_1 + z_{0m} }{ z_{0m} }} \right) - 2 \ln \left( {\frac{1 + X_1 }{1 + X_0 }} \right) - \ln \left( {\frac{1 + X_1^2 }{1 + X_0^2 }} \right)+ 2 \left( { {\tan }^{-1} X_1 - {\tan }^{-1} X_0 } \right) + \label{1.3.17} + +where + +.. math:: + + X_1 = (1 - 16 \zeta _1 )^{1/4} , X_0 = (1 - 16 \zeta _{0m} )^{1/4} + \label{1.3.18} + +and + +.. math:: + + \Phi _h = \ln \left( {\frac{ z_1 + z_{0m} }{ z_{0h} }} \right) - 2 \ln \left( {\frac{1 + Y_1 }{1 + Y_0 }} \right) + \label{1.3.19} + +where + +.. math:: + + Y_1 = (1 - 16 \zeta _1 )^{1/2} , Y_0 = (1 - 16 \zeta _{0h} )^{1/2}. + \label{1.3.20} + +.. _section_1.4: + +The iterative algorithm for calculating the surface exchange coefficients +------------------------------------------------------------------------- + +For conditions that are stable, i.e. :math:`\Delta`\ B :math:`\ge` 0, or +near-neutral (taken as :math:`\Delta`\ **v** :math:`\ge` 2 +ms\ :math:`^{-1}`), then start the iteration from the neutral limit, so + +.. math:: + + \begin{aligned} + \Phi _m^{(0)}&=&\ln \left( {\frac{ z_1 + z_{0m} }{ z_{0m} }} \right) + \label{1.4.5} \\ + \Phi _h^{(0)}&=&\ln \left( {\frac{ z_1 + z_{0m} }{ z_{0h} }} \right) + \label{1.4.6} \\ + v_\ast ^{(0)}&=& {\left( {\frac{k}{ \Phi _m^{(0)} }} \right)} \left| {\Delta {{{\rm {\bf v}}}}} \right| + \label{1.4.7} + \end{aligned} + +Otherwise (if :math:`\Delta`\ B :math:`<` 0 and :math:`\Delta`\ **v** +:math:`<` 2 ms\ :math:`^{-1}` ) start from the greater of the neutral +and convective limits for :math:`v_\ast^{(0)}`, so + +.. math:: + + \begin{aligned} + \frac{1}{ L^{(0)} }&=&\frac{-k}{ \gamma _t^3 z_i } + \label{1.4.1}\\ + \Phi _m^{(0)}&=& \Phi _m ( L^{(0)} , z_1 + z_{0m} , z_{0m} ) + \label{1.4.2}\\ + \Phi _h^{(0)}&=& \Phi _h ( L^{(0)} , z_1 + z_{0m} , z_{0h} ) + \label{1.4.3}\\ + v_\ast ^{(0)}&= & MAX{\left[ {\left( {\frac{k}{ \Phi _m^{(0)} }} \right)} \left| {\Delta {{{\rm {\bf v}}}}} \right|, \, + {\left[ { \gamma _t^3 \left( {\frac{k}{ \Phi _h^{(0)} }} \right) z_i \left| {-\Delta B} \right| } \right]}^{ 1/2} \right]} + \label{1.4.4} + \end{aligned} + +Then calculate + +.. math:: + + \begin{aligned} + C_D^{(0)}&=&\frac{k}{ \Phi _m^{(0)} } v_\ast ^{(0)} + \label{1.4.8} \\ + C_H^{(0)}&=&\frac{k}{ \Phi _h^{(0)} } v_\ast ^{(0)} + \label{1.4.9} + \end{aligned} + +Having set up initial values the iteration loop can be entered (this is +the original method used but contains an inconsistency in the treatment +of boundary-layer convective gustiness, as described in +section `8.4.1 <#mo_iter_corrn>`__): + +DO n = 1 to N + +.. math:: + + \begin{aligned} + u_\ast ^{(n)2}&=& C_D^{(n-1)} \left| {\Delta {{\rm {\bf v}}}} \right| + \label{1.4.10} \\ + {\left( {\frac{ F_{B0} }{ \rho _0 }} \right)}^{(n)}&=& { {-C}_H }^{(n-1)} \Delta B + \label{1.4.11}\\ + w_\ast ^{(n)}&=& {\left[ { z_i {\left( {\frac{ F_{B0} }{ \rho _0 }} \right)}^{(n)} } \right]}^{ 1/3} + \label{1.4.12}\\ + v_\ast ^{(n)2}&=& u_\ast ^{(n)2} + \gamma _t^2 w_\ast ^{(n)2} + \gamma _c^2 w_c^2 + \label{1.4.13}\\ + \frac{1}{ L^{(n)} } = \frac{-k( F_{B0} / \rho _0 )^{(n)} }{ v_\ast ^{(n)3} } + \label{1.4.14}\\ + \Phi _m^{(n)}&=& \Phi _m ( L^{(n)} , z_1 + z_{0m} , z_{0m} ) + \label{1.4.15} \\ + \Phi _h^{(n)}&=& \Phi _h ( L^{(n)} , z_1 + z_{0m} , z_{0h} ) + \label{1.4.16} \\ + C_D^{(n)}&=&\frac{k}{ \Phi _m^{(n)} } v_\ast ^{(n)} + \label{1.4.17} \\ + C_H^{(n)}&=&\frac{k}{ \Phi _h^{(n)} } v_\ast ^{(n)} + \label{1.4.18} + \end{aligned} + +END DO. + +For neutral and stable conditions (:math:`\Delta`\ B :math:`\ge` 0) +start the iteration from the neutral values and set +w\ :math:`_{\ast }`\ =0 in the above iteration loop. + +Use the final (N) values of C\ :math:`_{H}` and C\ :math:`_{D}` to +calculate the surface sensible and latent heat fluxes and surface +stress: + +.. math:: + + \begin{aligned} + H_0&=& {-c}_P \rho _0 C_H^{(N)} \left( {\Delta T + \frac{g}{ c_P }( z_1 + z_{0m} - z_{0h} )} \right) + \label{1.4.19} \\ + E_0&=& {-\rho }_0 C_H^{(N)} \Delta q + \label{1.4.20} \\ + {\rm {\bf \tau }}_{0} &=& \rho _0 C_D^{(N)} \Delta {\rm {\bf v}} + \label{1.4.21} + \end{aligned} + +N is the last iteration value. N = 5 is currently used. + +For sea points the momentum roughness length and the wind mixing energy +flux are calculated from v\ :math:`_{\ast }^{(N)}` using the formulae in +subsection `8.6 <#section_1.6>`__ below. + +.. _mo_iter_corrn: + +Correction to the iterative algorithm +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +The above implementation of boundary-layer convective gustiness in the +Monin-Obukhov iteration contains an inconsistency. The overall effect +turns out to be small, but it is desirable to use the corrected form, +which is derived as follows. + +We decompose the wind as +:math:`{\bf u}=\bar{\bf u} + {\bf u}_g + {\bf u}'`, representing, +respectively, the large-scale, gust and small-scale turbulent +contributions to the velocity. Locally, Monin-Obukhov theory then gives + +.. math:: |\bar{\bf u} + {\bf u}_g({\bf x}) | = \frac{u_*({\bf x})}{k} \Phi_m({\bf x}). + +We ignore the spatial variation of :math:`\Phi_m`, expecting that the +principal effect of locally stronger winds is to increase the local +stress – this is exactly true in nearly neutral flow. The local stress +is aligned with the wind so + +.. math:: + + {\bf \tau}({\bf x}) = \rho u_*^2({\bf x}) + \frac{\bar{\bf u} + {\bf u}_g({\bf x})} {|\bar{\bf u} + {\bf + u}_g({\bf x})|}. + +With the assumption that :math:`\Phi_m` does not vary spatially, + +.. math:: + + {\bf \tau}({\bf x}) = \rho \frac{k^2}{\Phi_m^2} + |\bar{\bf u} + {\bf u}_g({\bf x})| (\bar{\bf u} + {\bf u}_g({\bf + x})) \equiv \rho C_D |\bar{\bf u} + {\bf u}_g({\bf x})| (\bar{\bf + u} + {\bf u}_g({\bf x})), + +where :math:`C_D` is the standard drag coefficient, +:math:`\frac{k^2}{\Phi_m^2}`. The grid-box mean effect is + +.. math:: + + \langle{\bf \tau}\rangle = \rho C_D \langle |\bar{\bf u} + {\bf u}_g({\bf x})| + (\bar{\bf u} + {\bf u}_g({\bf x})) \rangle \approx \rho C_D \langle + |\bar{\bf u} + {\bf u}_g({\bf x})| \rangle \bar{\bf u}, + +which is the product of the enhanced wind speed (including gusts) and +the mean velocity. The magnitude of the stress is then +:math:`\langle\tau\rangle = \rho C_D S U`, with the last two symbols +representing the mean wind speed, including gusts, and the mean +background velocity. + +In the model, we want to write this as an effective drag coefficient, +:math:`C_{De}`, so that :math:`\langle\tau\rangle = \rho C_{De}U^2`. Now +define :math:`\tilde u_* = \frac{k}{\Phi_m}U`, the friction velocity due +to large-scale flow, and :math:`\hat u_* = \frac{k}{\Phi_m}S`, the +friction velocity due to the total flow, including gusts (again +implicitly assuming that :math:`\Phi` is unaffected by the gusts). The +representation is :math:`\hat u_*^2 = \tilde u_*^2 +\gamma_t^2 w_*^2`. +Then, + +.. math:: + + C_{De}=\frac{C_DS}{U} = \frac{k^2}{\Phi_m^2} \frac{\hat u_*}{\tilde u_*} + = \frac{k}{\Phi_m}\frac{\hat u_*}{U} + +The variable ``CDV`` in the routine ``FCDCH`` within the Monin-Obukhov +iteration will then be :math:`\frac{k}{\Phi_m}{\hat u_*}` (note that it +is divided by :math:`U` at the end of the routine). This is indeed coded +at the end of the loop; but in the original code ``CDV`` is used to +calculate :math:`\tilde u_*^2` at the beginning of the routine. In fact, +:math:`\tilde u_*^2 = (k/\Phi_m)\tilde u_* U`, which differs from the +coded result by a factor of :math:`\tilde u_*/\hat +u_*`. After this step, the purported :math:`\tilde u_*` is augmented by +the gust contribution to get :math:`\hat u_*`. This has the consequence +of overestimating :math:`C_{De}` and also means that what is described +as ``U_S`` in this loop is not :math:`\tilde u_*`, but +:math:`\sqrt(\tilde u_* \hat +u_*)`. + +If we let :math:`v=\sqrt(\tilde u_* \hat u_*)`, then we get + +.. math:: + + \hat u_*^2 = \frac{1}{2} \left \{ \gamma_t^2 w_*^2 + \sqrt { \gamma_t^4 w_*^4 + + 4 v^2 } \right \}. + +In the corrected version this expression is used to calculate ``V_S``, +namely :math:`\hat u_*` within the iteration. + +.. _section_1.5: + +The interpolation of surface layer variables to standard observation heights +---------------------------------------------------------------------------- + +Integrating (`[1.1.3] <#1.1.3>`__) between the roughness height, +z\ :math:`_{0m}`, and the observation height z\ :math:`_{ob}` we obtain + +.. math:: + + {\rm {\bf v}}_{ob} { = } {\rm {\bf v}}_{0} { + + }\frac{ {\rm {\bf \tau }}_{0} }{ \rho _0 v_\ast k} \Phi _m (L, + z_{ob} + z_{0m} , z_{0m} ) + \label{1.5.1} + +Using the expression for the surface turbulent stress this gives the +interpolation formula + +.. math:: + + {\rm {\bf v}}_{ob} { = } {\rm {\bf v}}_{0} { + + }\frac{ C_D }{k v_\ast } \Phi _m (L, z_{ob} + z_{0m} , z_{0m} ) ( + {\rm {\bf v}}_{1} { - } {\rm {\bf v}}_{0} {)} + \label{1.5.2} + +For wind z\ :math:`_{ob}` is set to 10m and the last iteration (N) +values of C\ :math:`_{D}`, L and :math:`v_{\ast }` are used. +Integrating (`[1.1.1] <#1.1.1>`__) and (`[1.1.2] <#1.1.2>`__) between +the roughness height, z\ :math:`_{0h}`, and the observation height +z\ :math:`_{ob, }` we obtain for the scalar :math:`X` +(:math:`=T+(g/c_{P})z` , :math:`q` or tracer amount) + +.. math:: + + X_{ob} = X_0 + \frac{ F_{X0} }{ \rho _0 v_\ast k} \Phi _h (L, z_{ob} + z_{0h} , z_{0h} ) + \label{1.5.3} + +and using the expression for the surface flux :math:`F_{X0}` of the +scalar quantity :math:`X` this gives the interpolation formula + +.. math:: + + X_{ob} = X_0 + \frac{ C_H }{k v_\ast } \Phi _h (L, z_{ob} + z_{0h} , z_{0h} ) ( X_1 - X_0 ) + \label{1.5.4} + +For temperature and humidity z\ :math:`_{ob}` is set to the screen +height (1.5 m) and the last iteration (N) values of C\ :math:`_{H}`, L +and :math:`v_{\ast }` are used. + +The parametrization of decoupling +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +In the foregoing analysis it is tacitly assumed that the surface layer, +up to the model’s lowest grid level, is in equilibrium with the surface +and lies within the constant flux layer. In light winds, and when the +surface temperature falls quickly, these assumptions are invalid; +equation `[1.5.4] <#1.5.4>`__ then yields temperatures at the height of +observation that are too closely tied to the surface temperature. +Observed temperatures may be significantly warmer: this may be termed +decoupling. Two parametrizations of this effect are available. Both +involve the idea that as the wind becomes very light radiative cooling +comes to determine the temperature profile. + +The first parametrization simply sets the interpolation coefficient +between the surface temperature and that on the model’s lowest level +according to the radiative equilibrium profile when the Richardson +number exceeds 0.25 (a typical criterion for high stability). + +The second more elaborate scheme is directed at the evening transition, +when the surface temperature is falling rapidly: it is under such +conditions that the impact of decoupling on the air temperature is +greatest. For a couple of hours after the surface temperature drops +below the air temperature, radiative cooling directly to the surface +largely determines the atmospheric cooling rate when the wind is light +and this may easily be calculated, provided that a parametrization of +the transmission between the air and the surface is available: this +parametrization depends on the absorbing properties of atmospheric trace +gases. Explicitly, we have + +.. math:: + + \dot T_{ob, \mbox{\tiny rad, surf}} = \frac{4\sigma T_s^3}{c_P} + {\cal K}(z_{ob}) (T_s-T_{ob}), + +where :math:`\dot T_{ob, \mbox{\tiny rad, surf}}` is the cooling rate of +the air at the height of observation due to direct radiative exchanges +with the surface, :math:`T_{ob}` is the air temperature at that height, +:math:`T_s` is the temperature of the surface and +:math:`{\cal K}(z_{ob})` depends on the concentration of trace gases in +the atmosphere, in practice water vapour and carbon dioxide, and their +spectroscopic properties. The contributions of water vapour and carbon +dioxide are, to a good approximation, additive, so we may write + +.. math:: + + {\cal K}(z_{ob}) = \left [ q_w {\cal C}_w(\mu_w, + T_{ob}) + q_c {\cal C}_c(\mu_c, T_{ob}) \right ], + +where :math:`q_w` and :math:`q_c` are the specific concentrations of +water vapour and carbon dioxide and :math:`\mu_w` and :math:`\mu_c` are +the respective pathlengths between the observation height and the +surface. The functions :math:`{\cal C}_w` and :math:`{\cal C}_c` are +parametrized and explicit functional forms are included in the code. + +In stronger winds turbulent cooling will be more important, so the +scheme must approach the standard procedure in that limit. Within the +context of local scaling, it can be shown that the depth of the +atmosphere which feels the impact of surface cooling must scale on +:math:`{\cal L}=(u_*^3/ (g/T_s)\dot T_s)^{1/2}`, where :math:`u_*` is +the surface friction velocity, :math:`\dot T_s` is the surface cooling +rate. This parameter is used as a measure of the strength of turbulence +to define the relaxation back to the strong-wind limit. + +To implement the scheme, the liquid-frozen potential temperature at the +height of observation, :math:`\theta_{ob}=T_{ob}+(g/c_P)z_{ob}-(L/c_P) +q_{cl} - ((L+L_f)/c_P) q_{cf}` is made a prognostic. Whenever the +surface buoyancy flux changes sign and becomes stable, this prognostic +is initialized using standard theory. On subsequent timesteps, it is +updated to allow for radiative cooling to the surface, giving a +provisional value :math:`\theta_{ob}'`, and then relaxed back towards +the result that would be obtained from standard similarity theory, +:math:`\theta_{ob, \mbox{\tiny sim}}`, as in the last section: + +.. math:: + + \begin{aligned} + \theta_{ob}'(t+\delta t) &\leftarrow & \theta_{ob}(t)+ + \delta t \, \dot T_{ob,\mbox{\tiny rad,surf}} \\ + \theta_{ob}(t+\delta t) &\leftarrow & W \theta_{ob}'(t+\delta t) + +(1-W) \theta_{ob, \mbox{\tiny sim}}. + \end{aligned} + +By tuning against an idealized highly vertically resolved model based on +local scaling we set, + +.. math:: + + W = \exp(-(0.4 f)^2 \delta t \, t_{\mbox{\tiny trans}})) / + (1+X({\cal L})\delta t), + +with + +.. math:: + + X({\cal L})= \min \left ( 0.000283 \left \{\frac{{\cal L}}{z_{ob}} + \log \left (1+\frac{z_{ob}}{z_0}\right ) \right \}^2, + \; \frac{0.2 u_* }{z_{ob}} \right ). + +The explicit dependence of :math:`W` on the timestep, :math:`\delta t`, +ensures that the results converge as :math:`\delta t \rightarrow 0`. The +exponential factor involving the Coriolis parameter, :math:`f`, is +intended to represent the recoupling of the surface and atmosphere as +growing directional shear at the top of the incipient stable boundary +layer generates turbulence. This factor is somewhat exaggerated relative +to the results of the model against which it is tuned, as a cautionary +measure to ensure that decoupling is not allowed to persist too long +after the transition. This is the purpose of the inclusion of the factor +:math:`0.4 f t_{\mbox{\tiny trans}}`, where +:math:`t_{\mbox{\tiny trans}}` is the time since the transition. + +It must be stressed that these schemes are heuristic and that the +precise behaviour in weak turbulence is not fully understood. The second +scheme appears to work well during the evening transition, but for +reasons of caution decoupling is suppressed somewhat too rapidly. The +simpler first scheme underestimates decoupling during the transition, +but allows it to persist longer, although tending to overestimate it on +these timescales. Overall, the second scheme is to be preferred. + +.. _section_1.6: + +The surface fluxes for sea and sea-ice gridboxes +------------------------------------------------ + +For gridboxes with sea-ice (i.e. where sea-ice fraction, +f\ :math:`_{I} >` 0) sensible and latent heat fluxes are calculated +separately for the sea and ice parts of the gridbox and combined with +appropriate weighting to obtain the total fluxes into the atmosphere. +This is done because the two surfaces can have very different +temperatures and also differ in their roughness. + +The sea and sea-ice surface fluxes of sensible heat, moisture and +momentum are calculated using gridbox mean surface transfer +coefficients, :math:`<`\ C\ :math:`_{H}>` and +:math:`<`\ C\ :math:`_{D}>`. These are linear combinations of the +corresponding coefficients calculated for the ice-free sea (L), typical +Marginal Ice Zone broken sea-ice (MIZ) and complete ice cover (I). + +The surface sensible heat and moisture fluxes are calculated separately +for the ice-free (leads) and ice-covered parts of the gridbox. Although +the fluxes over the two surfaces are calculated from gridbox mean +surface transfer coefficients, the different surface temperatures give +different fluxes. The ice surface temperature is predicted from a +surface energy balance and ice heat conduction model (see the +documentation for the land and ice surface processes component of the +Unified Model). In current versions of the model the sea surface +temperature (SST) is assumed to be 271.35 K, the freezing point of sea +water, whenever the ice fraction is greater than zero. This is +unrealistic except for genuine leads (i.e. large ice fraction) and a +future version of the model will allow the SST to be larger than 271.35 +K for gridboxes with sea-ice. + +The wind mixing energy flux, F\ :math:`_{WME}` , is the rate of +production of turbulent kinetic energy per unit area in the sea surface +layer by the wind stress at the air-sea interface. In atmosphere-only +configurations of the Unified Model this quantity is a useful +diagnostic. When the atmosphere model is coupled to an ocean model the +wind mixing energy flux is accumulated over an ocean model timestep and +then used in the calculation of the mixing in the upper layers of the +ocean. The gridbox mean **wind mixing energy flux** is given by + +.. math:: + + F_{WME} = ( 1- f_I ) \frac{ \rho _0^{3/2} v_\ast ^3 }{ \rho _{(sea)}^{1/2} } + \label{1.6.9} + +where :math:`v_{\ast }` is calculated using the drag coefficient for the +leads part of the gridbox, c\ :math:`_{D(L)}`, rather than the gridbox +mean value, :math:`<`\ c\ :math:`_{D}>`, when there is partial ice +cover. + +Roughness Lengths over the Sea +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +The roughness lengths for momentum and scalars depend on both the +atmospheric flow and the wave state. The dependence on wave state is not +fully understood and is still a subject of active research. In any case, +it could only fully be represented in a coupled wave-atmosphere model. +Simpler more empirical schemes are therefore currently used in the +Unified Model. + +In all schemes available here the momentum roughness length is given by + +.. math:: + + z_{0m(sea)} = \frac{1.54\times {10}^{-6} }{ v_\ast } + + \frac{\alpha}{g} v_\ast ^2 + \label{eq:z0msea} + +which is a generalisation of Charnock’s formula to include low-wind +conditions :raw-latex:`\cite[]{Smith88}`. :math:`\alpha` is Charnock’s +coefficient, which is determined from field measurements. It is often +taken as a constant, but more elaborate schemes include a dependence on +wind speed. In practice the difference between different +parametrizations of the momentum roughness length therefore comes down +to the specification of Charnock’s coefficient. + +There is greater uncertainty in the roughness lengths for scalars and +the dependencies are described separately for each scheme. Note that +whilst the full versions of some schemes prescribe different roughness +lengths for heat and moisture, in the Unified Model we have only a +single roughness for all scalars. + +Schemes are selected by setting the variable *iseasurfalg*, as now +described. + +#. Option *iseasurfalg=0*. The original and most basic scheme comprises + a fixed value of Charnock’s coefficient and a fixed scalar roughness + length. Typical values of Charnock’s coefficient lie in the range + 0.011–0.018 and a typical value of the thermal roughness length is + :math:`z_{0h(sea)}` = 4x10\ :math:`^{-5}` m. + +#. Option *iseasurfalg=1*. The use of a fixed thermal roughness length, + as above, leads to a rapid increase in the exchange coefficient for + moisture as the wind speed increases that is at variance with + observational evidence. A parametrization of the scalar roughness + length was developed from surface divergence theory + :raw-latex:`\cite[]{csanady2001}`, as described by + :raw-latex:`\cite{edwards2007}`. This involves an inverse dependence + of :math:`z_{0h}` on the friction velocity in the aerodynamically + smooth limit and an inverse dependence of :math:`z_{0h}` on + :math:`z_{0m}` at higher wind speeds that reduces the increase in the + exchange coefficient with wind speed. + + During iteration of the equations of surface transfer to calculate + the Obukhov length, the friction velocity changes, so implicitly + changing the roughness lengths. Historically, in the algorithm + adopted in the Unified Model,roughness lengths have not been modified + within this iteration, with values from the previous timestep being + used. The scheme was therefore originally implemented in a form that + was based on conditions at the previous timestep, but did not require + adding :math:`z_{0h}` to the dump. To cope with conditions of light + winds, this required an iterative calculation of :math:`v_\ast` from + :math:`z_{0m}` from the previous timestep before the calculation of + the Obukhov length and :math:`v_\ast`. Note that although there is + not a 1-1 relationship between :math:`v_\ast` and :math:`z_{0m}`, the + ambiguity is in practice removed by the consideration that the + inversion is only of relevance in conditions of light winds. + + With this scheme a fixed value of Charnock’s coefficient must be + specified as above. + +#. Option *iseasurfalg=2*. An alternative version of the foregoing + scheme has been developed that includes full iteration of the + roughness lengths within the iteration for the Obukhov length. + +#. Option *iseasurfalg=3*. This option provides various forms of the + COARE algorithm. The COARE algorithm exists in various forms and + continues to be developed. Version 3.0 + :raw-latex:`\cite[]{fairall2003}` has been extensively used, while + version 3.5 :raw-latex:`\cite[]{edson2013}` has recently been + released. Whilst the full COARE algorithm provides a complete + description of surface transfer at the sea surface, here we use only + the expressions for the roughness lengths. + + In current versions of the scheme Charnock’s coefficient is specified + using a linear relationship between the 10-m wind speed, valid over a + certain range of wind speeds, with fixed values outside the range: + + .. math:: + + \alpha = a U_{10} +b + \label{eq:charn} + + for :math:`U_{10,min} < U_{10} < U_{10,max}`. The constants + :math:`a`, :math:`b`, :math:`U_{10,min}` and :math:`U_{10,max}` + differ between different versions of the algorithm and are specified + through namelist parameters. Strictly, :math:`U_{10}` here should be + the neutral 10-m wind speed, but over the ocean the difference + between the neutral and stability-adjusted wind speeds is typically + small, so the distinction is often ignored. (Current practice in data + assimilation (2014) is to ignore the distinction). A logical switch + is therefore provided to enable the user to apply the formula using + the true neutral wind or the stability-adjusted wind, as preferred. + + The COARE algorithm does distinguish roughness lengths for heat and + moisture, but this is not currently feasible in the Unified Model, + so, since latent heat fluxes are dominant over the ocean, the scalar + roughness length is set using the expression for the moisture + roughness, + + .. math:: + + z_{0h} = \min(1.15\times 10^{-4}, 5.5\times 10^{-5}/Re_*^{0.6}), + \label{eq:z0h_coare} + + where :math:`Re_*` is the roughness Reynolds number. + + This scheme has been implemented is a form that allows the roughness + lengths to evolve during iteration to obtain the Obukhov length. + +#. Option *iseasurfalg=4*. Equivalent to option *iseasurfalg=1* for a + variable Charnock parameter. A fixed value of Charnock’s coefficient + does not need to be provided. On the other hand, a Charnock field + needs to be provided via wave coupling or initialization. + +#. Option *iseasurfalg=5*. Equivalent to option *iseasurfalg=2* for a + variable Charnock parameter. A fixed value of Charnock’s coefficient + does not need to be provided. On the other hand, a Charnock field + needs to be provided via wave coupling or initialization. + +The observations upon which these schemes are based do not extend to +10-m (neutral) wind speeds much above 20 ms\ :math:`{}^{-1}` and there +is some uncertainty over the behaviour of the drag at the wind speeds +encountered in tropical cyclones: indeed, there is considerable evidence +that it does not continue to increase in the manner predicted by schemes +like those described above and may even decrease. +:raw-latex:`\cite{Donelan2004}` presents some measurements suggesting +that the drag coefficient should not be permitted to increase for 10-m +neutral winds above about 33 ms\ :math:`{}^{-1}`, when the drag +coefficient is about 0.0024. Whilst it is likely that further work will +be required on this topic, the possibility of limiting the drag +coefficient has been allowed for by introducing the option +``i_high_wind_drag`` with the options + +#. Option *i_high_wind_drag=0*. This is the default option of making no + modification to the standard scheme at high winds. + +#. Option *i_high_wind_drag=1*. This option allows the user to specify a + maximum value of the (neutral) drag coefficient, ``cdn_max_sea`` + (called ``cd_limit_sea`` at versions below 11.5). + +#. Option *i_high_wind_drag=2*. Like the previous option, this allows + the user to specify a maximum value of the neutral drag coefficient, + ``cdn_max_sea``, but at higher wind speeds the drag coefficient is + reduced and attains a limiting value, ``cdn_hw_sea``. The reduction + is linear in the wind speed between ``u_cdn_max`` and ``u_cdn_hw``. + This reflects current understanding of the behaviour of the sea + surface at high wind speeds, with the neutral drag coefficient + saturating at around 35 ms\ :math:`{}^{-1}` and declining at higher + wind speeds. Suggested values of these coefficients are based on + :raw-latex:`\cite{donelan2018}` and :raw-latex:`\cite{hsu2017}`. + +It might be thought more logical to subsume the treatment of high winds +under ``iseasurfalg``, but given that standard schemes for surface +exchange at lower wind speeds do not explicitly account for this range +of speeds and that the treatment of high wind speeds is less certain, it +is useful to consider the treatment of high wind speeds as a seperate +option. + +Surface exchange over sea ice +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +As explained above, when sea ice is present, surface exchange involves +exchanges between the atmosphere and the open sea (L), the marginal ice +zone (MIZ) and the zone of pack ice. More mechanistically, one may +consider the interfacial exchanges over the sea and ice surfaces and the +contribution of form drag on the ice freeboard in the marginal ice zone. + +Two approaches are available in uncoupled configurations of the model. + +#. The Original Scheme The exchange coefficients over the sea and sea + ice regions of the gridbox are interpolated between values + representative of pack ice, the marginal ice zone and open sea, using + the ice fraction, f\ :math:`_{I}`. For 0 :math:`\le` + f\ :math:`_{I} <` 0.7 + + .. math:: + + \begin{aligned} + < C_H >&=&( f_I C_{H(MIZ)} + ( 0.7 - f_I ) C_{H(L)} ) / 0.7 + \label{1.6.1} \\ + < C_D >&=&( f_I C_{D(MIZ)} + ( 0.7 - f_I ) C_{D(L)} ) / 0.7 + \label{1.6.2} + \end{aligned} + + and for 0.7 :math:`\le` f\ :math:`_{I} \le` 1 + + .. math:: + + \begin{aligned} + < C_H >&=&( ( 1 - f_I ) C_{H(MIZ)} + ( f_I - 0.7 ) ) C_{H(I)} ) / 0.3 + \label{1.6.3} \\ + < C_D >&=&( ( 1 - f_I ) C_{D(MIZ)} + ( f_I - 0.7 ) ) C_{D(I)} ) / 0.3 + \label{1.6.4} + \end{aligned} + + where + + .. math:: + + \begin{aligned} + C_{H(L)}&= &C_H ( L_{(L)} , z_{0m(sea)} , z_{0h(sea)} ) + \label{1.6.5} \\ + C_{H(MIZ)}&=& C_H ( L_{(I)} , z_{0m(MIZ)} , z_{0h(MIZ)} ) + \label{1.6.6} \\ + C_{H(I)}&=& C_H ( L_{(I)} , z_{0m(sea-ice)} , z_{0h(sea-ice)} ) + \label{1.6.7} + \end{aligned} + + and similarly for the drag coefficient C\ :math:`_{D}`. + + The roughness lengths over open sea are calculated as above, but + those for ice are prescribed and set using the gui or namelists. The + typical roughness length for pack ice, z\ :math:`_{0m(sea-ice)}` = + 5x10\ :math:`^{-4}` m. Historically, z\ :math:`_{0h(sea-ice)}` was + set equal to z\ :math:`_{0m(sea-ice)}`, but more recently it has been + set equal to one fifth of z\ :math:`_{0m(sea-ice)}`, based on + :raw-latex:`\cite{andreas2010}`. The setting for marginal ice is more + problematic. Whilst z\ :math:`_{0m(MIZ)}` should be larger than + z\ :math:`_{0m(sea-ice)}`, good simulations of mean sea-level + pressure are obtained only if z\ :math:`_{0m(MIZ)}` is substantially + greater than z\ :math:`_{0m(sea-ice)}` and a value of 0.1m is + typically used. The ratio of z\ :math:`_{0h(MIZ)}` to + z\ :math:`_{0m(MIZ)}` is standardly set to 0.2 in this case, but + smaller values might be more realistic. However, a better approach in + the longer term is to use an explicit representation of ice form + drag. + +#. Explicit Treatment of Ice Form Drag + + :raw-latex:`\cite{lupkes2012}` have suggested a simple + parametrization of the form drag coefficient of marginal ice that has + been found to perform well in comparison to aircraft measurements + (:raw-latex:`\cite{elvidge2016}`). :raw-latex:`\cite{lupkes2015}` + have extended the parametrization to include the effects of + stability. When coupled to CICE, it is intended that a more elaborate + scheme will be used, but this scheme is useful for application in + atmosphere-only simulations and its implementation is now described. + + The fundamental quantity involved in representing the drag is the + pressure force on the ice free-board in the up-stream flow, + + .. math:: F_p =\int_{z_0}^{h_f} \frac{\rho}{2} [u(z)]^2 \, dz. + + :math:`u(z)` will in general exhibit a mixed character, but it may be + taken as the developed flow over open sea, as in + :raw-latex:`\cite{lupkes2012}`, or may be interpolated between the + developed flows over open sea or pack ice, depending on the ice + fraction, as in :raw-latex:`\cite{lupkes2015}`. In principle, it will + be subject to the effects of stability, but since the free-board does + not much exceed 0.5m, these effects are small + (:raw-latex:`\cite{lupkes2015}`) and the flow may be taken as neutral + up to :math:`h_f`. Hence, + + .. math:: + + F_p \approx \frac{h_f}{2k^2} \rho u_*^2 \left [ (\log(h_f/z_0) -1)^2 +1 + \right ] = + \frac{h_f}{2k^2} \rho C_d U_1^2 \left [(\log(h_f/z_0) -1)^2 +1 \right ]. + \label{eq:int_u2} + + where :math:`C_d` is the upstream drag coefficient and :math:`U_1` is + the wind on the model’s lowest atmospheric level. Because this will + be significantly above :math:`h_f`, the stability dependence of + :math:`C_d` should be considered here (again see + :raw-latex:`\cite{lupkes2015}`). :math:`U_1` may be interpreted as + the wind at a specific height, or, consistenly with the + flux-difference form of the momentum equation, as the layer-averaged + velocity. This distinction affects the numerical value of + :math:`C_d`, but does not otherwise affect the foregoing equation. If + using the original version of the scheme + (:raw-latex:`\cite{lupkes2012}`), :math:`C_d` must be taken as the + neutral drag coefficient. Note also that various approximations may + be made in Equation `[eq:int_u2] <#eq:int_u2>`__. + :raw-latex:`\cite{lupkes2012}` approximate + :math:`(\log(h_f/z_0) -1)^2 +1` as :math:`(\log(h_f/z_0) )^2`; while + :raw-latex:`\cite{lupkes2015}` approximate it as + :math:`(\log(h_f/z_0) -1)^2`. Here we retain the full expression. + + If, in a unit area, there are :math:`N` floes, each of crosswind + dimension :math:`D_i`, the total drag will be + + .. math:: + + F_d = N c_w S_c^2 D_i F_p, + \label{eq:fd_fp} + + where :math:`c_w` is a coefficient and :math:`S_c` is a sheltering + coefficient. The fractional coverage of sea ice within this unit area + is :math:`ND_i^2 c_s`, where :math:`c_s` is related to the shape of + the floe. Overall, the drag per unit area of *ice* is + + .. math:: + + f_d = \frac{h_f}{2k^2} c_e \rho C_d U_1^2 S_c^2 \frac{A}{D_i} + \left [(\log(h_f/z_0) -1)^2 +1 \right ], + + where :math:`c_e=c_w/c_s`. Assuming that :math:`U_1` is blended, it + follows that the form drag coefficient is + + .. math:: + + C_{df} = \frac{h_f}{2k^2} c_e S_c^2 \frac{1}{D_i} + \left [(\log(h_f/z_0) -1)^2 +1 \right ]. + + Defining, :math:`L=(\log(h_f/z_0) -1)^2 +1` and interpolating in the + ice fraction, + + .. math:: + + C_{df} = \frac{c_e}{2}\frac{h_f}{D_i}\frac{S_c^2}{k^2} \left [ + (1-A) C_{ds} L_s + A C_{di} L_i \right ], + + where the sheltering factor is taken to be the same over ice and + water. :raw-latex:`\cite{lupkes2012}` provides parametrizations for + quantities such as :math:`h_f`, while :raw-latex:`\cite{elvidge2016}` + provide suggested values for the constants in the scheme, based on + observations. In using these values in the Unified Model, :math:`c_e` + should be increased by about 30% to represent the effect of differing + approximations of the logarithmic wind profile. + + For scalar transfer :raw-latex:`\cite{lupkes2015}` suggest adding a + contribution to the sensible heat flux to represent the impact of + form drag; however, the mechanistic physical basis of the scheme they + propose is unclear. Moreover, when combined with the interfacial + drag, this suggests scalar transfer much larger than observed by + :raw-latex:`\cite{schroder2003}`. Consequently, no enhancement of the + scalar transfer coefficient by form drag is included. + + The overall drag coefficients are now set by interpolation in the ice + fraction: + + .. math:: + + \begin{aligned} + < C_D >&=& (1 - f_I) C_{D(L)} + f_I (C_{D(I)} + C_{D(FRM)}) + \label{eq:cdice_int} \\ + < C_H >&=& (1 - f_I) C_{H(L)} + f_I C_{H(I)} + \label{eq:chice_int} + \end{aligned} + +.. _`sec:coast`: + +Surface exchange in coastal grid-boxes +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +In coupled ocean-atmosphere modelling the ocean requires appropriate +surface stresses and fluxes over all ocean points. In coastal regions +this means providing sea-surface fluxes from atmospheric grid-boxes that +are partly sea and partly land. This is achieved through coastal tiling, +where the ocean part of the grid box is effectively treated in the same +way as other land surface tiles. However, because of the very different +roughness characteristics of land and sea, this can lead to serious +biases especially in the ocean surface fluxes. One particular problem +that has been identified is that, compared to a neighbouring sea-only +point, coastal points tend to have slower near-surface wind speeds +(because of the rough land surface fraction) but the ocean surface +exchange will still use a typical very small roughness length. Thus the +diagnosed ocean surface stress and sensible and latent heat fluxes are +all significantly smaller than a neighbouring sea point. Although truly +coastal winds are notoriously complex, in a typical climate model grid +box (of 100km or more) the vast majority of the sea area will be +unaffected by the land. Thus a partial solution to this problem is to +take the wind speed over the sea part of coastal points as the average +of that over the neighbouring sea points. The wind speed over the land +component is then slowed (by up to a factor of 5) to maintain the grid +box mean wind speed. + +.. _section_1.7: + +Surface roughness lengths and resistances to evaporation over land +------------------------------------------------------------------ + +For land points the roughness lengths excluding orographic effects are +specified from land use datasets. The vegetative roughness length for +scalars is assumed to be 0.1 of that for momentum. This is a simple +approximation; in reality the factor depends on the land cover type and +the degree of heterogeneity. [Future versions of the Unified Model will +treat surface heterogeneity explicitly by the “tiling” method.] + +The surface moisture flux given by (`[1.1.8] <#1.1.8>`__) or +(`[1.1.17] <#1.1.17>`__) involves a surface humidity value, +q\ :math:`_{0}`. Prior to UM6.3, for evaporation from all of ocean, +sea-ice, lake and snow-covered surfaces as well as from water on +vegetative canopies this surface value is taken to be the saturated +specific humidity at the surface (skin) temperature and pressure, +q\ :math:`_{sat}`\ (T\ :math:`_{0}`,p\ :math:`_{0})`. [Saturation is +respect to liquid water or ice depending on which the surface is.] The +saturation vapour pressure of a liquid, though, is lowered by dissolved +ionic substances. For typical sea salinities the saturated vapour +pressure is only about 90% of the value over pure water. From UM6.3, +therefore, there is the option to include this effect, so that the +parametrization of the surface moisture flux over the sea becomes + +.. math:: E_0 = - \rho_0 c_H V (q_1 - 0.98 q_{sat}(T_0,p_0) ) + +Evapotranspiration through vegetation receives a special treatment +because it is controlled by the physiology of the plants. The +formulation is described in full in the documentation for the land and +ice surface processes component of the Unified Model. The +evapotranspiration for the surface is given by + +.. math:: + + E_t = - \rho _0 \frac{ q_1 - q_{sat} ( T_0 , p_0 )}{( r_a + r_s )} + \label{1.7.1} + +where the aerodynamic resistance, r\ :math:`_{a}` , is given by + +.. math:: + + r_a = \frac{1}{ C_H } = \frac{1}{ c_H V} + \label{1.7.2} + +and r\ :math:`_{s}` is the surface or stomatal resistance to +evaporation. r\ :math:`_{s}` is a function of the available soil +moisture, near surface atmospheric conditions and the radiation +impinging on the plants. [For the formulation see the documentation for +the land and ice surface processes component of the Unified Model.] A +similar formula to (`[1.7.1] <#1.7.1>`__) is used for the evaporation +from the very near surface soil layer. Equation (`[1.7.1] <#1.7.1>`__) +can be written as + +.. math:: + + E_t = - \rho _0 C_E ( q_1 - q_{sat} ( T_0 , p_0 ) ) + \label{1.7.3} + +where + +.. math:: + + C_E = \frac{ C_H }{\left( {1 + \frac{ r_s }{ r_a }} \right)} + \label{1.7.4} + +.. _section_2: + +The modifications needed to incorporate orographic form drag. +------------------------------------------------------------- + +.. _section_2.1: + +Effective roughness lengths +~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +Form drag is included in the surface turbulent flux formulation via +effective roughness lengths for momentum :raw-latex:`\cite[]{wood93}` +and for scalar quantities :raw-latex:`\cite[]{hewer1998}`. The formulae +of section `8.1 <#section_1>`__ are interpreted as relationships between +gridbox mean quantities and fluxes with the roughness lengths replaced +by effective values, z\ :math:`_{0m(eff)}` and z\ :math:`_{0h(eff)}`. + +When form drag is included via effective roughness lengths equations +(`[1.1.7] <#1.1.7>`__)-(`[1.1.9] <#1.1.9>`__) become: + +.. math:: + + \begin{aligned} + \frac{ H_{0(eff)} }{ c_P \rho _0 }&=&\frac{-k}{ \Phi _h (L , z_1 + z_{0m(eff)} , z_{0h(eff)} )} v_{\ast (eff)}\nonumber\\ + && \left( {\Delta T + \frac{g}{c_p }( z_1 + z_{0m(eff)} - z_{0h(eff)} )} \right) + \label{2.1.1} \\ + \frac{ E_{0(eff)} }{ \rho _0 }&=&\frac{-k}{ \Phi _h (L , z_1 + z_{0m(eff)} , z_{0h(eff)} )} v_{\ast (eff)} \Delta q + \label{2.1.2} \\ + \frac{ {\rm {\bf \tau }}_{{0(eff)}} }{ \rho _{0} }&=&\frac{k}{ \Phi _m (L , z_1 + z_{0m(eff)} , z_{0m(eff)} )} v_{\ast (eff)} \Delta {\rm {\bf v}} + \label{2.1.3} + \end{aligned} + +The effective surface scaling velocity, v\ :math:`_{\ast (eff)}` , is +given by (cf. (`[1.1.25] <#1.1.25>`__)) + +.. math:: + + v_{\ast (eff)}^2 = u_{\ast (eff)}^2 + \gamma _t^2 w_\ast ^2 + \gamma _c^2 w_c^2 + \label{2.1.4} + +where + +.. math:: + + u_{\ast (eff)}^2 = \left| { {\rm {\bf \tau }}_{{0(eff)}} {/} \rho _{0} } \right| + \label{2.1.5} + +The effective roughness for momentum is derived by setting the total +effective surface stress, **:math:`\tau`**\ :math:`_{0(eff)}`, to the +sum of the surface stress over a flat surface with the same vegetative +roughness, **:math:`\tau`**\ :math:`_{0(f)}`, and the orographic +pressure drag force at the surface, **:math:`\tau`**\ :math:`_{0(p)}`. +The stresses are evaluated in terms of the velocity at height +z\ :math:`_{c}` above the surface. z\ :math:`_{c}` is currently set to +2\ :math:`^{1/2}\sigma _{h}` where :math:`\sigma _{h}` is the standard +deviation of the unresolved orographic height. Thus + +.. math:: + + \frac{ {\rm {\bf \tau }}_{{0(eff)}} }{ \rho _{0} }{ = }\frac{{k } {v}_{{\ast (eff)}} }{ \Phi _{m} {(L , } {z}_{c} { , } {z}_{{0m(eff)}} {)}}{ }{\rm {\bf v}}{(} {z}_{c} {)} + \label{2.1.6} + +and + +.. math:: + + \frac{ {\rm {\bf \tau }}_{{0(f)}} }{ \rho _{0} }{ = }\frac{{k } {v}_{{\ast (f)}} }{ \Phi _{m} {(L , } {z}_{c} { , } {z}_{{0m}} {)}}{ }{\rm {\bf v}}{(} {z}_{c} {)} + \label{2.1.7} + +where the scaling velocity based on the stress over a flat surface, +v\ :math:`_{\ast +(f)}` , is given by + +.. math:: + + v_{\ast (f)}^2 = u_{\ast (f)}^2 + \gamma _t^2 w_\ast ^2 + \gamma _c^2 w_c^2 + \label{2.1.8} + +with + +.. math:: + + u_{\ast (f)}^2 = \left| { {\rm {\bf \tau }}_{{0(f)}} {/} \rho _{0} } \right| + \label{2.1.9} + +[The scaling velocity which appears in the +expression (`[1.1.4] <#1.1.4>`__) for the Monin-Obukhov length is chosen +to be v\ :math:`_{\ast (eff)}` rather than the flat surface value.] + +The orographic stress is given by + +.. math:: + + \frac{ {\rm {\bf \tau }}_{{0(p)}} }{ \rho _{0} }{ = }\frac{{1}}{{2}}{ } {c}_{{D(orog)}} { } {f}_{D} {(} {{Ri}}_{B} {)}\frac{{A}}{{S}}{ }\left| {{\rm {\bf v}}{(} {z}_{c} {)}} \right|{ }{\rm {\bf v}}{(} {z}_{c} {)} + \label{2.1.10} + +where :math:`A/S` is the total silhouette area of orography in a gridbox +over the flat surface area of the gridbox taken as an average over all +directions. The function f\ :math:`_{D}` is a function of the bulk +Richardson number of the surface layer and is set to 1 for +Ri\ :math:`_{SL} <` 0 and decreases linearly to zero at +Ri\ :math:`_{SL(crit)}` = 0.5. The orographic drag coefficient +c\ :math:`_{D(orog)}` is set to the constant value (typically 0.3, +:raw-latex:`\cite{mason1986}`). + +If the function :math:`\Phi _{m}` and v\ :math:`_{\ast }` are +approximated by their neutral values in (`[2.1.6] <#2.1.6>`__) +and (`[2.1.7] <#2.1.7>`__) then the equation for calculating the +effective momentum roughness is derived + +.. math:: + + \frac{\ln ( z_c / z_{0m(eff)} )}{\ln ( z_c / z_{0m} )} = {\left( {1 + \frac{1}{2} c_{D(orog)} f_D \frac{A}{S} {\left( {\frac{\ln ( z_c / z_{0m} )}{k}} \right)}^2 } \right)}^{-1/2} + \label{2.1.12} + +The stress for the flat surface is related to the total stress by + +.. math:: + + {\rm {\bf \tau }}_{{0(f)}} { = } {\rm {\bf \tau + }}_{{0(eff)}} { } {\left( {{1 + }\frac{{1}}{{2}}{ } + {c}_{{D(orog)}} { } {f}_{D} { }\frac{{A}}{{S}}{ } {\left( + {\frac{\ln {(} {z}_{c} { / } {z}_{{0m}} {)}}{{k}}} + \right)}^{2} } \right)}^{{-1}} + \label{2.1.13} + +which is derived from equations (`[2.1.6] <#2.1.6>`__), +(`[2.1.7] <#2.1.7>`__) and (`[2.1.10] <#2.1.10>`__). +Equation (`[2.1.13] <#2.1.13>`__) implies that + +.. math:: + + C_{D(f)} = C_{D(eff)} {\left( {1 + \frac{1}{2} c_{D(orog)} f_D \frac{A}{S} {\left( {\frac{\ln ( z_c / z_{0m} )}{k}} \right)}^2 } \right)}^{-1} + \label{2.1.14} + +Parametrized orographic drag coefficient +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +:raw-latex:`\cite{wood93}` find that orographic drag coefficient +c\ :math:`_{D(orog)}` depends on A/S via the equation + +.. math:: + + c_{D(orog)} = 2\alpha \beta \pi ^2 \frac{A}{S} \frac{ u_{\ast (f)}^2 }{ v^2 ( z_c )} + \label{2.1.11} + +where :math:`\alpha` and :math:`\beta` are constants +(:math:`\alpha`\ =12 and :math:`\beta`\ =1). + +If (`[2.1.11] <#2.1.11>`__) is used, the formula for the effective +roughness length for momentum becomes + +.. math:: + + \frac{\ln ( z_c / z_{0m(eff)} )}{\ln ( z_c / z_{0m} )} = {\left( {1 + \alpha \beta \pi ^2 f_D {\left( {\frac{A}{S}} \right)}^2 } \right)}^{-1/2} + \label{2.1.15} + +and (`[2.1.13] <#2.1.13>`__) and (`[2.1.14] <#2.1.14>`__) become + +.. math:: + + \begin{aligned} + {\rm {\bf \tau }}_{{0(f)}} &=& {\rm {\bf \tau + }}_{{0(eff)}} {\left( {{1 + }\alpha \beta \pi ^{2} { } {f}_{D} { } + {\left( {\frac{{A}}{{S}}} \right)}^{2} { }} \right)}^{-1} + \label{2.1.16} \\ + C_{D(f)}&= &C_{D(eff)} {\left( {1 + \alpha \beta \pi ^2 f_D {\left( {\frac{A}{S}} \right)}^2 } \right)}^{-1} + \label{2.1.17} + \end{aligned} + +The effective surface flux of scalar X evaluated in terms of values at +z\ :math:`_{c}` is + +.. math:: + + \frac{ F_{X0(eff)} }{ \rho _0 } = \frac{k v_{\ast (eff)} }{ \Phi _h (L , z_c , z_{0h(eff)} )} (X( z_c ) - X_0 ) + \label{2.1.18} + +and the surface flux for the flat surface is given by + +.. math:: + + \frac{ F_{X0(f)} }{ \rho _0 } = \frac{k v_{\ast (f)} }{ \Phi _h (L , z_c , z_{0h} )} (X( z_c ) - X_0 ) + \label{2.1.19} + +:raw-latex:`\cite{hewer1998}` find that the scalar transport is enhanced +when there is orographic form drag such that + +.. math:: + + F_{X0(eff)} = F_{X0(f)} {\left( {1 - 2.2 f_D \frac{A}{S}} \right)}^{-1} + \label{2.1.20} + +Combining (`[2.1.18] <#2.1.18>`__)–(`[2.1.20] <#2.1.20>`__) and using +the neutral values of the stability functions the expression for the +effective scalar roughness length is derived as + +.. math:: + + \frac{\ln ( z_c / z_{0h(eff)} )}{\ln ( z_c / z_{0h} )} = {\left( {1 + \frac{1}{2} c_{D(orog)} f_D \frac{A}{S} {\left( {\frac{\ln ( z_c / z_{0m} )}{k}} \right)}^2 } \right)}^{1/2} \left( {1 - 2.2 f_D \frac{A}{S}} \right) + \label{2.1.21)} + +which becomes + +.. math:: + + \frac{\ln ( z_c / z_{0h(eff)} )}{\ln ( z_c / z_{0h} )} = {\left( {1 + \alpha \beta \pi ^2 f_D {\left( {\frac{A}{S}} \right)}^2 } \right)}^{1/2} \left( {1 - 2.2 f_D \frac{A}{S}} \right) + \label{2.1.22} + +if the :raw-latex:`\cite{wood93}` formulation is used. + +.. _section_2.2: + +The iterative algorithm for calculating the effective surface exchange coefficients +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +For unstable conditions, i.e. :math:`\Delta`\ B :math:`<` 0 : + +IF :math:`\Delta`\ **v** :math:`<` 2 ms\ :math:`^{-1}` then start the +iteration from the convective limit, so + +.. math:: + + \begin{aligned} + \frac{1}{ L^{(0)} }&=&\frac{-k}{ \gamma _t^3 z_i } + \label{2.2.1} \\ + \Phi _m^{(0)}&=& \Phi _m ( L^{(0)} , z_1 + z_{0m(eff)} , z_{0m(eff)} ) + \label{2.2.2} \\ + \Phi _h^{(0)}&=& \Phi _h ( L^{(0)} , z_1 + z_{0m(eff)} , z_{0h} ) + \label{2.2.3} \\ + v_{\ast (eff)}^{(0)}&=& v_{\ast (f)}^{(0)} = {\left[ { \gamma _t^3 \left( {\frac{k}{ \Phi _h^{(0)} }} \right) z_i \left| {-\Delta B} \right| + \gamma _c^2 w_c^2 } \right]}^{ 1/2} + \label{(2.2.4} + \end{aligned} + +ELSE IF (:math:`\Delta`\ **v** :math:`\ge` 2 ms\ :math:`^{-1}` ) start +iteration from the neutral end, so + +.. math:: + + \begin{aligned} + \Phi _m^{(0)}&=&\ln \left( {\frac{ z_1 + z_{0m(eff)} }{ z_{0m(eff)} }} \right) + \label{2.2.5} \\ + \Phi _h^{(0)}&=&\ln \left( {\frac{ z_1 + z_{0m(eff)} }{ z_{0h} }} \right) + \label{2.2.6} \\ + u_{\ast (eff)}^{(0)}&=&\frac{k}{ \Phi _m^{(0)} } \left| {\Delta {{{v}}}} \right| + \label{2.2.7} \\ + v_{\ast (eff)}^{(0)}&=& {\left( { u_{\ast (eff)}^{(0) 2} + \gamma _c^2 w_c^2 } \right)}^{ 1/2} + \label{2.2.8} \\ + u_{\ast (f)}&=& u_{\ast (eff)} \frac{\ln ( z_c / z_{0m(eff)} )}{\ln ( z_c / z_{0m} )} + \label{2.2.9} \\ + v_{\ast (f)}^{(0)}&=& {\left( { u_{\ast (f)}^{(0) 2} + \gamma _c^2 w_c^2 } \right)}^{ 1/2} + \label{2.2.10} + \end{aligned} + +END IF. + +Then calculate: + +.. math:: + + \begin{aligned} + C_{D(eff)}^{(0)}&=&\frac{k}{ \Phi _m^{(0)} } v_{\ast (eff)}^{(0)} + \label{(2.2.11} \\ + C_{H(eff)}^{(0)}&=&\frac{k}{ \Phi _h^{(0)} } v_{\ast (eff)}^{(0)} + \label{(2.2.12} \\ + C_{D(f)}^{(0)}&=& C_{D(eff)}^{(0)} {\left( {1 + \frac{1}{2} c_{D(orog)} f_D \frac{A}{S} {\left( {\frac{\ln ( z_c / z_{0m} )}{k}} \right)}^2 } \right)}^{-1} + \label{(2.2.13} \\ + C_{H(f)}^{(0)}&=& C_{H(eff)}^{(0)} \left( {1 - 2.2 f_D \frac{A}{S}} \right) + \label{(2.2.14} + \end{aligned} + +Having set up initial values the iteration loop can be entered: + +DO n = 1 to N + +.. math:: + + \begin{aligned} + u_{\ast (eff)}^{(n)2}&=& C_{D(eff)}^{(n-1)} \left| {\Delta {{\rm {\bf v}}}} \right| + \label{(2.2.15} \\ + u_{\ast (f)}^{(n)2}&=& C_{D(f)}^{(n-1)} \left| {\Delta {{\rm {\bf v}}}} \right| + \label{(2.2.16} \\ + {\left( {\frac{ F_{B0} }{ \rho _0 }} \right)}^{(n)}&=&- { C_{H(eff)} }^{(n-1)} \Delta B + \label{(2.2.17} \\ + w_\ast ^{(n)}&=& {\left[ { z_i {\left( {\frac{ F_{B0} }{ \rho _0 }} \right)}^{(n)} } \right]}^{ 1/3} + \label{(2.2.18} \\ + v_{\ast (eff)}^{(n)2}&=& u_{\ast (eff)}^{(n)2} + \gamma _t^2 w_\ast ^{(n)2} + \gamma _c^2 w_c^2 + \label{(2.2.19} \\ + v_{\ast (f)}^{(n)2}&= &u_{\ast (f)}^{(n)2} + \gamma _t^2 w_\ast ^{(n)2} + \gamma _c^2 w_c^2 + \label{(2.2.20} \\ + \frac{1}{ L^{(n)} }&=&\frac{-k( F_{B0} / \rho _0 )^{(n)} }{ v_{\ast (eff)}^{(n)3} } + \label{(2.2.21} \\ + \Phi _m^{(n)}&=& \Phi _m ( L^{(n)} , z_1 + z_{0m(eff)} , z_{0m(eff)} ) + \label{(2.2.22} \\ + \Phi _h^{(n)}&=& \Phi _h ( L^{(n)} , z_1 + z_{0m(eff)} , z_{0h} ) + \label{(2.2.23} \\ + C_{D(eff)}^{(n)}&=&\frac{k}{ \Phi _m^{(n)} } v_{\ast (eff)}^{(n)} + \label{(2.2.24} \\ + C_{H(eff)}^{(n)}&=&\frac{k}{ \Phi _h^{(n)} } v_{\ast (eff)}^{(n)} + \label{(2.2.25} \\ + C_{D(f)}^{(n)}&=& C_{D(eff)}^{(n)} {\left( {1 + \frac{1}{2} c_{D(orog)} f_D \frac{A}{S} {\left( {\frac{\ln ( z_c / z_{0m} )}{k}} \right)}^2 } \right)}^{-1} + \label{(2.2.26} \\ + C_{H(f)}^{(n)}&=& C_{H(eff)}^{(n)} \left( {1 - 2.2 f_D \frac{A}{S}} \right) + \label{(2.2.27} + \end{aligned} + +END DO. + +For neutral and stable conditions (:math:`\Delta`\ B :math:`\ge` 0) +start the iteration from the neutral values and set +w\ :math:`_{\ast }`\ =0 in the above iteration loop. Use the final (N) +values of C\ :math:`_{H(eff)}` and C\ :math:`_{D(eff)}` to calculate the +surface sensible and latent heat fluxes and surface stress: + +.. math:: + + \begin{aligned} + H_{0(eff)}&=&- c_P \rho _0 C_{H(eff)}^{(N)} \left( {\Delta T + \frac{g}{c_p }( z_1 + z_{0m(eff)} - z_{0h} )} \right) + \label{2.2.28} \\ + E_{0(eff)}&=& {-\rho }_0 C_{H(eff)}^{(N)} \Delta q + \label{2.2.29} \\ + {\rm {\bf \tau }}_{{0(eff)}} &=& \rho _0 C_{D(eff)}^{(N)} \Delta {\rm {\bf v}} + \label{2.2.30} + \end{aligned} + +The stress for a flat surface, if required for output, is calculated +from + +.. math:: + + {\rm {\bf \tau }}_{{0(f)}} { = } \rho _0 + C_{D(f)}^{(N)} \Delta {\rm {\bf v}} + \label{2.2.31} + +.. _section_2.3: + +Interpolation of surface layer variables to standard observation heights +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +If the observation height wind is assumed to lie on the profile defined +by the effective roughness length and surface scaling velocity then +(c.f. equation (`[1.5.1] <#1.5.1>`__)) + +.. math:: + + {\rm {\bf v}}_{{ob}} { = } {\rm {\bf v}}_{0} { + + }\frac{ {\rm {\bf \tau }}_{{0(eff)}} }{ \rho _0 v_{\ast (eff)} k} + \Phi _m (L, z_{ob} + z_{0m(eff)} , z_{0m(eff)} + \label{2.3.1} + +Using the expression for the surface turbulent stress this becomes + +.. math:: + + {\rm {\bf v}}_{{ob}} { = } {\rm {\bf v}}_{0} { + + }\frac{ C_{D(eff)} }{ {kv}_{\ast (eff)} } \Phi _m (L, z_{ob} + + z_{0m(eff)} , z_{0m(eff)} ) ( {\rm {\bf v}}_1 - {\rm {\bf v}}_{0} + {)} + \label{2.3.2} + +For wind z\ :math:`_{ob}` is set to 10m and the last iteration (N) +values of C\ :math:`_{D(eff)}`, L and v\ :math:`_{\ast (eff)}` are used. +Alternatively if the observation height wind is assumed to lie on a +profile defined by the flat surface roughness length and scaling +velocity then + +.. math:: + + {\rm {\bf v}}_{{ob}} { = } {\rm {\bf v}}_{0} { + + }\frac{ {\rm {\bf \tau }}_{{0(f)}} }{ \rho _0 v_{\ast (f)} k} \Phi + _m (L, z_{ob} + z_{0m} , z_{0m} ) + \label{2.3.3} + +and substituting for the surface stress this becomes + +.. math:: + + {\rm {\bf v}}_{{ob}} { = } {\rm {\bf v}}_{0} { + + }\frac{ C_{D(f)} }{k v_{\ast (f)} } \Phi _m (L, z_{ob} + z_{0m} , + z_{0m} ) ( {\rm {\bf v}}_1 - {\rm {\bf v}}_{0} {)} + \label{2.3.4} + +Most configurations of the Unified Model currently use the latter +assumption with the last iteration value of C\ :math:`_{D(f)}`, L and +:math:`v_{\ast + (f)}` used in the interpolation formula. + +If the observation height scalar quantities are assumed to lie on the +mean profile defined by the effective roughness length and scaling +quantities then (c.f. equation (`[1.5.3] <#1.5.3>`__) we obtain for the +generic scalar :math:`X` (:math:`T+(g/c_{P})z`, :math:`q`, tracer +amount) + +.. math:: + + X_{ob} = X_0 + \frac{ F_{X0(eff)} }{ \rho _0 v_{\ast (eff)} k} \Phi _h (L, z_{ob} + z_{0h(eff)} , z_{0h(eff)} ) + \label{2.3.5} + +and using the expression for the surface flux of the scalar quantity +:math:`X` this becomes + +.. math:: + + X_{ob} = X_0 + \frac{ C_{H(eff)} }{k v_{\ast (eff)} } \Phi _h (L, z_{ob} + z_{0h(eff)} , z_{0h(eff)} ) ( X_1 - X_0 ). + \label{2.3.6} + +Alternatively if the observation height scalar quantities are assumed to +lie on a profile defined by the flat surface roughness length and flux +then + +.. math:: + + X_{ob} = X_0 + \frac{ C_{H(f)} }{k v_{\ast (f)} } \Phi _h (L, z_{ob} + z_{0h} , z_{0h} ) ( X_1 - X_0 ) + \label{2.3.7} + +For temperature and humidity z\ :math:`_{ob}` is set to the screen +height (1.5 m) and the last iteration (N) values of C\ :math:`_{H}`, L +and v\ :math:`_{\ast }` are used. + +.. _section_2.4: + +Distributed form drag – an alternative to the effective roughness length parametrization +---------------------------------------------------------------------------------------- + +An alternative representation of the turbulent form drag due to sub-grid +hills is the explicit orographic stress parametrization proposed by +:raw-latex:`\cite{wood01:_param}`. In this representation the drag is +represented via an orographic stress term, applied directly to the +horizontal momentum equations. The roughness lengths remain at the +vegetative values and no adjustment to the roughness lengths for scalar +quantities is made. + +The turbulent form drag is represented by the term + +.. math:: + + {\bf f}=\frac{1}{\rho}\frac{\partial}{\partial z}{\bf\tau}_{\rm orog} + \label{eq:drag} + +on the right-hand side of the horizontal momentum equation, where +:math:`{\bf\tau}_{\rm orog}` is the horizontal vector containing the +extra stress imparted on the flow by the sub-grid orography This term is +included in the Unified Model as an additional explicit (in terms of +time discretisation) stress. Following :raw-latex:`\cite{wood01:_param}` +we define :math:`{\bf\tau}_{\rm orog}` to be + +.. math:: {\bf\tau}_{\rm orog}(z)=\left({F_p}_x,{F_p}_y\right)e^{-z/\ell}, + +where :math:`{\bf F_p}=({F_p}_x,{F_p}_y)`, :math:`{F_p}_x` and +:math:`{F_p}_y` are the grid-box average :math:`x` and :math:`y` +components of the pressure force on the sub-grid orography, and +:math:`\ell` is a decay scale. We define :math:`\ell` such that + +.. math:: + + \ell={\rm min}\left(\lambda,\frac{z_h}{3}\right), + \label{eq:l} + +where :math:`z_h` is the boundary-layer depth and :math:`\lambda`, a +somewhat ill defined quantity, is related to the horizontal scales of +the sub-grid hills (and set to 300 m). Note that the value of +:math:`\ell` obtained from Eq. (`[eq:l] <#eq:l>`__) is further +constrained to be at least 100 m. + +If the steep-hill expression is to be used, the surface stress applied +is almost identical to that used in the effective roughness +parametrization (Eq. `[2.1.10] <#2.1.10>`__), namely: + +.. math:: + + \frac{\bf F_p}{\rho_0}=\frac{1}{2}c_{D(orog)} f_D (Ri_{B}) \frac{A}{S} + \left\vert{\rm{\bf v}}(\ell) \right\vert{\rm{\bf v}}(\ell), + \label{eq:dragsteep} + +the main difference being the dependence on the height scale +:math:`\ell` rather than :math:`z_c`. Similarly, if the +:raw-latex:`\cite{wood93}` low-hill expression is used, the surface +stress is given by the equivalent of (Eq. `[2.1.16] <#2.1.16>`__), +namely: + +.. math:: + + \frac{\bf F_p}{\rho_0} = {\left( {\frac{\kappa}{\zeta_m}} \right)}^{2} + \alpha \beta \pi ^{2} {f}_{D} (Ri_{B}) + {\left( {\frac{A}{S}} \right)}^{2} + \left\vert{\rm{\bf v}}(\ell) \right\vert{\rm{\bf v}}(\ell), + \label{eq:draglow} + +where :math:`\zeta_m = {\rm log}(\ell/z_{0m})`. There is also an option +to use the low-hill stress (`[eq:draglow] <#eq:draglow>`__) but capped +by that from the steep hill expression +(`[eq:dragsteep] <#eq:dragsteep>`__), to avoid generating huge stresses +at large :math:`A/S`. + +There is also a choice for the Richardson number, :math:`Ri_{B}`, that +appears in the stability dependence, :math:`f_D`, which can either use +:math:`Ri_{B}=Ri_{SL}` (as with the effective roughness length version) +or :math:`Ri_{B \ell}`, a bulk Richardson number between the surface and +the scale height, :math:`\ell`: + +.. math:: + + Ri_{B \ell} = \frac{ \ell \left( g \left( + \overline{\beta_T}_{k\ell} ({\theta_{\ell}}_{k\ell}-{\theta_{\ell}}_{1}) + + \overline{\beta_q}_{k\ell} ({q_t}_{k\ell}-{q_t}_{1}) \right) + + \Delta b_{SL} \right) }{U^2(\ell)} + +where the stability of the atmosphere between the surface and the bottom +model level is included via :math:`\Delta b_{SL}`, which is the +numerator of :math:`Ri_{SL}`, :math:`U` is the wind speed and the +subscript :math:`k \ell` indicates the :math:`\theta`-level containing +:math:`\ell`. + +.. _`sec:implicit`: + +Implicit solution of the diffusion equation +=========================================== + +Unconditionally stable implicit solver +-------------------------------------- + +This is the vertical diffusion scheme of +:raw-latex:`\cite{woodetal2007}` which has the advantages of (i) +unconditional stability and non-oscillatory behaviour for practical NWP +cases and (ii) monotonic damping for suitable choices of a free +parameter :math:`P` which represents the degree of nonlinearity of the +diffusion problem to be solved. If the chosen value of :math:`P` is +equal to the real value of :math:`P` then the scheme is second order +accurate. In practical simulations :math:`P` may vary from timestep to +timestep and from column to column. + +Algorithmic description +~~~~~~~~~~~~~~~~~~~~~~~ + +Consider the non-linear damping equation: + +.. math:: + + \frac{dX}{dt}=-\left(KX^{P}\right)X+S + \label{eq:damp1} + +Here :math:`S` is a constant forcing, or source, term and :math:`KX^{P}` +is the diffusion coefficient, with :math:`K` constant. :math:`P` is +assumed to be positive. The new scheme is written + +.. math:: + + \frac{X^{*}-X^{n}}{\Delta t}=-{\cal I}_{1}\left[K\left(X^{n}\right)^{P}\right] + X^{*}+{\cal E}_{1}\left[K\left(X^{n}\right)^{P}\right] + X^{n}+\left({\cal I}_{1}-{\cal E}_{1}\right)S,\label{eq:sppf1} + +.. math:: + + \frac{X^{n+1}-X^{*}}{\Delta + t}=-{\cal I}_{2}\left[K\left(X^{n}\right)^{P}\right]X^{n+1} + + {\cal E}_{2}\left[K\left(X^{n}\right)^{P}\right]X^{*} + + \left({\cal I}_{2}-{\cal E}_{2}\right)S,\label{eq:sppf2} + +where + +.. math:: + + {\cal E}_{1}=\left(1+\frac{1}{\sqrt{2}}\right) + \left[P+\frac{1}{\sqrt{2}}\pm\sqrt{P + \left(\sqrt{2}-1\right)+\frac{1}{2}}\right] + \label{eq:E1coeff} + +.. math:: + + {\cal E}_{2}=\left(1+\frac{1}{\sqrt{2}}\right) + \left[P+\frac{1}{\sqrt{2}}\mp\sqrt{P\left(\sqrt{2}-1\right)+\frac{1}{2}}\right] + \label{eq:E2coeff} + +.. math:: + + {\cal I}_{1}={\cal I}_{2}=\left(1+\frac{1}{\sqrt{2}}\right)\left(1+P\right) + \label{eq:Icoeff} + +Consider the one-dimensional “forced” boundary layer diffusion equation + +.. math:: + + \frac{\partial X}{\partial t}=\frac{\partial F}{\partial z}+S, + \qquad F=K_{X}\frac{\partial X}{\partial z} + \label{eq:vdiff1} + +where :math:`X` is the scalar variable being diffused, :math:`F` is the +flux of :math:`X`, :math:`t` is the time, :math:`z` is the height from +the earth’s surface, and :math:`K` is the diffusion coefficient which is +often non-constant and depends on :math:`X` (i.e. the PDE is non-linear) +and :math:`S` is a forcing term from other processes preceding the +boundary layer. In the UM these processes are: microphysics, gravity +wave drag, radiation, dynamics and optionally (using the switch +i_impsolve_loc) convection [3]_. :math:`S` represents the total tendency +from these processes. Equations (`[eq:sppf1] <#eq:sppf1>`__), +(`[eq:sppf2] <#eq:sppf2>`__) applied to (`[eq:vdiff1] <#eq:vdiff1>`__) +becomes + +.. math:: + + \begin{aligned} + \frac{X^{*}-X^{n}}{\Delta t} & = & {\cal I}_{1}\frac{\partial F}{\partial z}^{*}-{\cal E}_{1}\frac{\partial F}{\partial z}^{n}+\left({\cal I}_{1}-{\cal E}_{1}\right)S\label{eq:sppf_bl1}\\ + \frac{X^{n+1}-X^{*}}{\Delta t} & = & {\cal I}_{2}\frac{\partial + F}{\partial z}^{n+1}-{\cal E}_{2}\frac{\partial F}{\partial + z}^{*}+\left({\cal I}_{2}-{\cal E}_{2}\right)S\label{eq:sppf_bl2} + \end{aligned} + +where, + +.. math:: + + F^{n}=K_{X}\frac{\partial X}{\partial z}^{n},\; + F^{*}=K_{X}\frac{\partial X}{\partial z}^{*},\; + F^{n+1}=K_{X}\frac{\partial X}{\partial z}^{n+1},\; K_{X}\equiv + K(X^{n}) + +i.e. only one evaluation of the exchange coefficient is required per +timestep. Furthermore, the condition +:math:`I_{1}+I_{2}-({\cal E}_{1}+{\cal E}_{2})=1` ensures that if the +intermediate “starred” quantities are eliminated and the scheme is +reduced into a single equation then the forcing term will be multiplied +by :math:`1`. + +Recall from section `2 <#sec:closure>`__ that the boundary layer solver +computes the increment of :math:`X`, where +:math:`X=u,\; v,\;\theta_{L},\; q_{w}`. Let +:math:`\delta X^{*}=X^{*}-X^{n}`, :math:`\delta +X^{n+1}=X^{n+1}-X^{*}`. Then, + +.. math:: + + F^{*}=F^{n}+K_{X}\frac{\partial\delta X}{\partial z}^{*},\qquad + F^{n+1}=F^{*}+K_{X}\frac{\partial\delta X}{\partial z}^{n+1}. + +Writing equations (`[eq:sppf_bl1] <#eq:sppf_bl1>`__), +(`[eq:sppf_bl2] <#eq:sppf_bl2>`__) in terms of these increments: + +.. math:: + + \begin{aligned} + \frac{\delta X}{\Delta t}^{*} & = & ({\cal I}_{1}-{\cal E}_{1})\left(\frac{\partial F}{\partial z}^{n}+S\right)+{\cal I}_{1}\frac{\partial}{\partial z}\left(K_{X}\frac{\partial\delta X}{\partial z}^{*}\right)\label{eq:sppf_inc1}\\ + \frac{\delta X}{\Delta t}^{n+1} & = & ({\cal I}_{2}-{\cal E}_{2})\left(\frac{\partial F}{\partial z}^{*}+S\right)+{\cal I}_{2}\frac{\partial}{\partial z}\left(K_{X}\frac{\partial\delta X}{\partial z}^{n+1}\right)\label{eq:sppf_inc2}\\ + X^{n+1} & = & X^{n}+\delta X^{*}+\delta X^{n+1}\label{eq:sppf_inc3} + \end{aligned} + +.. _`sec:impsolve`: + +Discrete equations and boundary conditions +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +To derive the boundary conditions for the horizontal wind components we +adapt the technique used in the original scheme. + +**Vertical diffusion solver for momentum variables** + +Consider the following equivalent form of +(`[eq:sppf_bl1] <#eq:sppf_bl1>`__): + +.. math:: \frac{\delta u^{*}}{\Delta t}=\frac{\partial\bar{\tau}_{x}^{*}}{\partial z}+\left({\cal I}_{1}-{\cal E}_{1}\right)S\label{eq:du_star} + +where :math:`\tau_{x}` is the :math:`u` wind component stress (defined +in the same way as the flux in (`[eq:sppf_inc1] <#eq:sppf_inc1>`__) and +:math:`\bar{\tau}_{x}^{*}` its time-average: + +.. math:: + + \bar{\tau}_{x}^{*}={\cal I}_{1}\tau_{x}^{*}-{\cal E}_{1}\tau_{x}^{n},\qquad\tau_{x}^{*}=\tau_{x}^{n}+K_{u}\frac{\partial\delta + u^{*}}{\partial z}.\label{eq:tau_star} + +Substituting (`[eq:tau_star] <#eq:tau_star>`__) into +(`[eq:du_star] <#eq:du_star>`__) the following is obtained: + +.. math:: + + \frac{\delta u^{*}}{\Delta + t}=({\cal I}_{1}-{\cal E}_{1})\left(\frac{\partial\tau_{x}^{n}}{\partial + z}+S\right)+{\cal I}_{1}\frac{\partial}{\partial + z}\left(K_{u}\frac{\partial\delta u^{*}}{\partial z}\right) + +which is identical to (`[eq:sppf_inc1] <#eq:sppf_inc1>`__) for +:math:`X\equiv u,\; +F_{X}\equiv\tau_{x}`. This equivalent derivation is used here as it +presents a more convenient form to express the boundary conditions. +Given that the wind components are defined on :math:`\rho`-levels (half +levels), discretizing the previous equation in :math:`z` on all +:math:`L` half-levels except at the bottom and the top one we obtain + +.. math:: + + \begin{aligned} + \delta u_{k+1/2}^{*} & = & ({\cal I}_{1}-{\cal E}_{1})\Delta t\left(\frac{\tau_{x}^{n}\Big|_{k+1}-\tau_{x}^{n}\Big|_{k}}{z_{k+1}-z_{k}}+S_{k+1/2}\right)\\ + & & +{\cal I}_{1}\frac{\Delta + t}{z_{k+1}-z_{k}}\left[\left(K_{u}\Big|_{k+1}\frac{\delta + u_{k+3/2}^{*}-\delta + u_{k+1/2}^{*}}{z_{k+3/2}-z_{k+1/2}}\right)-\left(K_{u}\Big|_{k}\frac{\delta + u_{k+1/2}^{*}-\delta + u_{k-1/2}^{*}}{z_{k+1/2}-z_{k-1/2}}\right)\right] + \end{aligned} + +or, rearranging + +.. math:: + + A_{k}\delta u_{k+3/2}^{*}+B_{k}\delta + u_{k+1/2}^{*}+C_{k}\delta u_{k-1/2}^{*}=\Delta + t({\cal I}_{1}-{\cal E}_{1})\left(\frac{\tau_{x}^{n}\Big|_{k+1}-\tau_{x}^{n}\Big|_{k}}{z_{k+1}-z_{k}}+S_{k+1/2}\right),\label{eq:tridiag} + +where :math:`k=1,2,\ldots,L-2`, + +.. math:: + + A_{k}=-{\cal I}_{1}\frac{\Delta t\noindent + K_{u}\Big|_{k+1}}{(z_{k+1}-z_{k})(z_{k+3/2}-z_{k+1/2})},\; + C_{k}=-{\cal I}_{1}\frac{\Delta + tK_{u}\Big|_{k}}{(z_{k+1}-z_{k})(z_{k+1/2}-z_{k-1/2})},\; + B_{k}=1-A_{k}-C_{k}. + +(Note that the surface is level :math:`0`). + +For the top :math:`\rho`-level, :math:`k=L-1`, the +:math:`z`-discretization of (`[eq:du_star] <#eq:du_star>`__) is: + +.. math:: B_{L}\delta u_{L-1/2}^{*}+C_{L}\delta u_{L-3/2}^{*}=\Delta t({\cal I}_{1}-{\cal E}_{1})\left(\frac{\tau_{x}^{n}\Big|_{L}-\tau_{x}^{n}\Big|_{L-1}}{z_{L}-z_{L-1}}+S_{L-1/2}\right),\label{eq:tridiag_top} + +where :math:`B_{L}`, :math:`C_{L}` are derived as before setting +:math:`A_{L}=0`. + +For the bottom :math:`\rho`-level, :math:`k=0`, the +:math:`z`-discretization of (`[eq:du_star] <#eq:du_star>`__) is: + +.. math:: + + \begin{aligned} + \delta u_{1/2}^{*} & = & \frac{\Delta t}{z_{1}-0}\left(\bar{\tau}_{x}^{*}\Big|_{1}-\bar{\tau}_{x}^{*}\Big|_{0}\right)+\Delta t\left({\cal I}_{1}-{\cal E}_{1}\right)S_{1/2}\label{eq:u_bc_1} + \end{aligned} + +where, from (`[eq:tau_star] <#eq:tau_star>`__), + +.. math:: \bar{\tau}_{x}^{*}\Big|_{1}=\left({\cal I}_{1}-{\cal E}_{1}\right)\tau_{x}^{n}\Big|_{1}+{\cal I}_{1}\left(K_{u}\frac{\partial\delta u^{*}}{\partial z}\right)\Big|_{1}.\label{eq:u_bc_2} + +Combining (`[eq:u_bc_1] <#eq:u_bc_1>`__), (`[eq:u_bc_2] <#eq:u_bc_2>`__) +the bottom row discretization is obtained: + +.. math:: A_{0}\delta u_{3/2}^{*}+B_{0}\delta u_{1/2}^{*}=\Delta t\left({\cal I}_{1}-{\cal E}_{1}\right)\left(\frac{\tau_{x}^{n}\Big|_{1}}{z_{1}}+S_{1/2}\right)-\frac{\Delta t}{z_{1}}\bar{\tau}_{x}^{*}\Big|_{0}\label{eq:u_bc_3} + +where + +.. math:: A_{0}=-{\cal I}_{1}\frac{\Delta tK_{u}\Big|_{1}}{z_{1}(z_{3/2}-z_{1/2})},\quad B_{0}=1-A_{0}. + +Equations (`[eq:tridiag] <#eq:tridiag>`__), +(`[eq:tridiag_top] <#eq:tridiag_top>`__) and +(`[eq:u_bc_3] <#eq:u_bc_3>`__) form a tridiagonal system of linear +equations. When the elimination procedure takes place +(`[eq:u_bc_3] <#eq:u_bc_3>`__) becomes + +.. math:: \delta u_{1/2}^{*}=\delta u_{1/2}^{'}-\beta\bar{\tau}_{x}^{*}\Big|_{0}\label{eq:du_half} + +where :math:`\delta u_{1/2}^{'}`, :math:`\beta` are available +quantities. Furthermore, + +.. math:: \bar{\tau}_{x}^{*}\Big|_{0}=\left({\cal I}_{1}-{\cal E}_{1}\right)\tau_{x}^{n}\Big|_{0}+{\cal I}_{1}\left(K_{u}\frac{\partial\delta u^{*}}{\partial z}\right)\Big|_{0} + +Approximating :math:`\left(\frac{\partial\delta u^{*}}{\partial + z}\right)\Big|_{0}\approx\frac{\delta u_{1/2}^{*}-\delta + u_{0}^{*}}{z_{1/2}}`, and assuming that :math:`u_{0}=0` the previous +equation becomes + +.. math:: \bar{\tau}_{x}^{*}\Big|_{0}=\left({\cal I}_{1}-{\cal E}_{1}\right)\tau_{x}^{n}\Big|_{0}+{\cal I}_{1}K_{u}\Big|_{0}\frac{\delta u_{1/2}^{*}}{z_{1/2}}.\label{eq:tau_zero} + +From (`[eq:du_half] <#eq:du_half>`__), +(`[eq:tau_zero] <#eq:tau_zero>`__) the following expression for the +implicit surface stress is obtained + +.. math:: + + \bar{\tau}_{x}^{*}\Big|_{0}=\frac{\left({\cal I}_{1}-{\cal E}_{1}\right)\tau_{x}^{n}\Big|_{0}+{\cal I}_{1}(K_{u}\Big|_{0}/z_{1/2})\delta + u_{1/2}^{'}}{1+{\cal I}_{1}(K_{u}\Big|_{0}/z_{1/2})\beta}.\label{eq:imp_tau} + +Then, :math:`\delta u_{1/2}^{*}` can be computed from +(`[eq:imp_tau] <#eq:imp_tau>`__) and (`[eq:du_half] <#eq:du_half>`__). +Similarly the implicit surface stress for :math:`u` which corresponds to +the 2nd stage (`[eq:sppf_inc2] <#eq:sppf_inc2>`__) will be + +.. math:: + + \bar{\tau}_{x}^{n+1}\Big|_{0}=\frac{\left({\cal I}_{2}-{\cal E}_{2}\right)\tau_{x}^{*}\Big|_{0}+{\cal I}_{2}(K_{u}\Big|_{0}/z_{1/2})\delta + u_{1/2}^{'}}{1+{\cal I}_{2}(K_{u}\Big|_{0}/z_{1/2})\beta}.\label{eq:imp_tau2} + +In the same way :math:`\bar{\tau}_{y}^{*}\Big|_{0}`, +:math:`\bar{\tau}_{y}^{n+1}\Big|_{0}` can be derived. + +**Vertical diffusion solver for scalar variables** + +Derivation of boundary conditions for the scalar variables (static +energy and total water content flux) is more difficult: the new scheme +in comparison with the original scheme is more complex and the procedure +for deriving the scalar fluxes described in section 3 of +:raw-latex:`\cite{esseryetal2001}` is also complex. Currently, an +alternative treatment for the boundary conditions has been coded which +works well in practice. The boundary conditions for the scalar +variables, i.e. the surface scalar fluxes for the new scheme are +obtained using the original implicit surface exchange calculation. This +is applied as follows. Consider the equivalent discrete form of +(`[eq:sppf_bl1] <#eq:sppf_bl1>`__) for the thermodynamic variable +:math:`X`: + +.. math:: + + \frac{\delta X^{*}}{\Delta t} + =\frac{\partial\overline{F}^{*}}{\partial z}+\left({\cal I}_{1}-{\cal E}_{1}\right)S\label{eq:dX_star} + +Considering that, + +.. math:: + + \overline{F}^{*}={\cal I}_{1}F^{*}-{\cal E}_{1}F^{n},\qquad + F^{*}=F^{n}+K_{X}\frac{\partial\delta X^{*}}{\partial z}\label{eq:dX_star2} + +(`[eq:dX_star] <#eq:dX_star>`__) would re-produce +(`[eq:sppf_inc1] <#eq:sppf_inc1>`__), which is re-written below, + +.. math:: \frac{\delta X^{*}}{\Delta t}=({\cal I}_{1}-{\cal E}_{1})\left(\frac{\partial F^{n}}{\partial z}+S\right)+{\cal I}_{1}\frac{\partial}{\partial z}\left(K_{X}\frac{\partial\delta X^{*}}{\partial z}\right) + +and thus the following discretization is obtained, on +:math:`\theta`-levels: + +.. math:: + + \begin{aligned} + \frac{\delta X_{k}^{*}}{\Delta t} & = & \left({\cal I}_{1}-{\cal E}_{1}\right)\left(\frac{F_{k+1/2}^{n}-F_{k-1/2}^{n}}{z_{k+1/2}-z_{k-1/2}}+S_{k}\right)\\ + & +&\frac{{\cal I}_{1}}{z_{k+1/2}-z_{k-1/2}}\left[K_{X}\Big|_{k+1/2}\left(\frac{\delta X_{k+1}^{*}-\delta X_{k}^{*}}{z_{k+1}-z_{k}}\right)-K_{X}\Big|_{k-1/2}\left(\frac{\delta X_{k}^{*}-\delta X_{k-1}^{*}}{z_{k}-z_{k-1}}\right)\right],\; k=2,\ldots,L-1. + \end{aligned} + +or, + +.. math:: A_{k}\delta X_{k+1}^{*}+B_{k}\delta X_{k}^{*}+C_{k}\delta X_{k-1}^{*}=\left({\cal I}_{1}-{\cal E}_{1}\right)\left(\frac{F_{k+1/2}^{n}-F_{k-1/2}^{n}}{z_{k+1/2}-z_{k-1/2}}+S_{k}\right),\quad k=2,\ldots,L-1\label{eq:dX_disc} + +where, + +.. math:: A_{k}=-{\cal I}_{1}\frac{\Delta tK_{X}\Big|_{k+1/2}}{(z_{k+1}-z_{k})(z_{k+1/2}-z_{k-1/2})},\; C_{k}=-{\cal I}_{1}\frac{\Delta tK_{X}\Big|_{k-1/2}}{(z_{k+1/2}-z_{k-1/2})(z_{k}-z_{k-1})},\quad B_{k}=1-A_{k}-C_{k}. + +The discrete equation for the top level, :math:`k=L`, will be: + +.. math:: B_{L}\delta X_{L}^{*}+C_{L}\delta X_{L-1}^{*}=\left({\cal I}_{1}-{\cal E}_{1}\right)\left(\frac{-F_{L-1/2}^{n}}{z_{L+1/2}-z_{L-1/2}}+S_{L}\right),\label{eq:dX_disc_top} + +where :math:`B_{L}`, :math:`C_{L}` are derived as before setting +:math:`A_{L}=0`. + +From (`[eq:dX_star] <#eq:dX_star>`__), a bottom interior level +(:math:`k=1`) discretization is + +.. math:: \delta X_{1}^{*}=\frac{\Delta t}{z_{3/2}-0}\left(\overline{F_{3/2}}^{*}-\overline{F_{0}}^{*}\right)+\Delta t\left({\cal I}_{1}-{\cal E}_{1}\right)S_{1}\label{eq:dX1_star} + +:math:`F_{0}` is used instead of :math:`F_{1/2}`. The former is computed +by the implicit surface scheme. This flux gradient is defined in the +same way in the original solver as well. Using +(`[eq:dX_star2] <#eq:dX_star2>`__), (`[eq:dX1_star] <#eq:dX1_star>`__) +becomes + +.. math:: + + \delta + X_{1}^{*}=\frac{\Delta t}{z_{3/2}} + \left[\left({\cal I}_{1}-{\cal E}_{1}\right)F_{3/2}^{n}-\overline{F_{0}}^{*}\right] + +\Delta t\left({\cal I}_{1}-{\cal E}_{1}\right) + S_{1}+\Delta t{\cal I}_{1}\frac{1}{z_{3/2}} + \left(K_{X}\frac{\partial\delta X^{*}}{\partial z}\right)_{3/2} + \label{eq:dX1_star2} + +where :math:`\overline{F}_{0}^{*}` can be approximated as + +.. math:: \overline{F}_{0}^{*}={\cal I}_{1}F_{0}^{*}-{\cal E}_{1}F_{0}^{n}\approx\left({\cal I}_{1}-{\cal E}_{1}\right)F_{JULES}\label{eq:F0_star} + +where, :math:`F_{JULES}` is the implicit flux calculated by the +*implicit surface scheme using the original implicit algorithm*. +Finalising, the discrete equations for the bottom level will be + +.. math:: + + \delta X_{1}^{*}=\Delta t\left({\cal I}_{1}-{\cal E}_{1}\right) + \left(\frac{F_{3/2}^{n}-F_{JULES}}{z_{3/2}}+S_{1}\right) + +\Delta t{\cal I}_{1}\frac{1}{z_{3/2}}K_{X}\Big|_{3/2} + \left(\frac{\delta X_{2}^{*}-\delta X_{1}^{*}}{z_{2}-z_{1}}\right) + +or, + +.. math:: + + A_{1}\delta X_{2}^{*}+B_{1}\delta X_{1}^{*}=\Delta t + \left({\cal I}_{1}-{\cal E}_{1}\right)\left(\frac{F_{3/2}^{n}-F_{JULES}}{z_{3/2}}+S_{1}\right)\label{eq:dX_bottom} + +where, + +.. math:: + + A_{1}=-{\cal I}_{1}\Delta t\frac{K_{X}\Big|_{3/2}}{z_{3/2}(z_{2}-z_{1})}, + \quad B_{1}=1-A_{1}. + +Equations (`[eq:dX_disc] <#eq:dX_disc>`__), +(`[eq:dX_disc_top] <#eq:dX_disc_top>`__) and +(`[eq:dX_bottom] <#eq:dX_bottom>`__) define a tridiagonal system of +equations for :math:`\delta X^{*}`. + +Similarly the corresponding discrete equations for +(`[eq:sppf_inc2] <#eq:sppf_inc2>`__) will be: + +.. math:: B_{L}^{'}\delta X_{L}^{n+1}+C_{L}^{'}\delta X_{L-1}^{n+1}=\left({\cal I}_{2}-{\cal E}_{2}\right)\left(\frac{-F_{L-1/2}^{*}}{z_{L+1/2}-z_{L-1/2}}+S_{L}\right),\label{eq:dXtop_np1} + +.. math:: A_{k}^{'}\delta X_{k+1}^{n+1}+B_{k}^{'}\delta X_{k}^{n+1}+C_{k}^{'}\delta X_{k-1}^{n+1}=\left({\cal I}_{2}-{\cal E}_{2}\right)\left(\frac{F_{k+1/2}^{*}-F_{k-1/2}^{*}}{z_{k+1/2}-z_{k-1/2}}+S_{k}\right),\quad k=L-1,\ldots,2\label{eq:dXk_np1} + +.. math:: A_{1}^{'}\delta X_{2}^{n+1}+B_{1}^{'}\delta X_{1}^{n+1}=\left({\cal I}_{2}-{\cal E}_{2}\right)\left(\frac{F_{3/2}^{*}-F_{JULES}}{z_{3/2}}+S_{1}\right)\label{eq:dX1_np1} + +where, + +.. math:: A_{k}^{'}=-{\cal I}_{2}\frac{\Delta tK_{X}\Big|_{k+1/2}}{(z_{k+1}-z_{k})(z_{k+1/2}-z_{k-1/2})},\; C_{k}^{'}=-{\cal I}_{2}\frac{\Delta tK_{X}\Big|_{k-1/2}}{(z_{k+1/2}-z_{k-1/2})(z_{k}-z_{k-1})},\quad B_{k}^{'}=1-A_{k}^{'}-C_{k}^{'}, + +for :math:`k=L,\ldots,2,\quad A_{L}=0`. + +.. math:: A_{1}^{'}=-{\cal I}_{2}\Delta t\frac{K_{X}\Big|_{3/2}}{z_{3/2}(z_{2}-z_{1})},\quad B_{1}^{'}=1-A_{1}^{'} + +and the approximation + +.. math:: \overline{F}_{0}^{n+1}={\cal I}_{2}F_{0}^{n+1}-{\cal E}_{2}F_{0}^{n}\approx\left({\cal I}_{2}-{\cal E}_{2}\right)F_{JULES}\label{eq:F0_np1} + +has taken place. The same flux :math:`F_{JULES}` will be used for both +(`[eq:F0_star] <#eq:F0_star>`__) and (`[eq:F0_np1] <#eq:F0_np1>`__) and +therefore needs to be computed only once, when the 1st or predictor +stage is computed, i.e. :math:`X^{*}`. Briefly the following +calculations take place for the scalar variables: + +.. container:: center + + +-------------------+-------------------------------------------------+ + | CALL bdy_impl3(): | set up coefficients for | + | | (`[eq:dX_disc_top] <#eq:dX_disc_top>`__), | + | | (`[eq:dX_disc] <#eq:dX_disc>`__) and do a | + | | downward sweep; | + +-------------------+-------------------------------------------------+ + | | do a downward sweep using the original implicit | + | | scheme to | + +-------------------+-------------------------------------------------+ + | | compute information required by the surface | + | | implicit solver; | + +-------------------+-------------------------------------------------+ + | CALL sf_impl2(): | CALL im_sf_pt2(): compute :math:`F_{JULES}` | + | | (scalar implicit fluxes), | + +-------------------+-------------------------------------------------+ + | | using original surface implicit solver; | + +-------------------+-------------------------------------------------+ + | CALL bdy_impl4(): | set up (`[eq:dX_bottom] <#eq:dX_bottom>`__) and | + | | complete downward sweep; | + +-------------------+-------------------------------------------------+ + | | back substitute to compute implicit correction | + | | :math:`\delta X^{*}`; | + +-------------------+-------------------------------------------------+ + | CALL bdy_impl3(): | compute explicit flux | + | | :math:`F^*= | + | | F^n+K_X\frac{\partial \delta X^*}{\partial z}`; | + +-------------------+-------------------------------------------------+ + | | set up coefficients for | + | | (`[eq:dXtop_np1] <#eq:dXtop_np1>`__), | + | | (`[eq:dXk_np1] <#eq:dXk_np1>`__), | + | | (`[eq:dX1_np1] <#eq:dX1_np1>`__) and | + +-------------------+-------------------------------------------------+ + | | do a downward sweep; | + +-------------------+-------------------------------------------------+ + | CALL sf_impl2(): | only momentum variables are affected - no | + | | change in scalars; | + +-------------------+-------------------------------------------------+ + | CALL bdy_impl4(): | back substitute to compute final implicit | + | | correction :math:`\delta X^{n+1}` | + +-------------------+-------------------------------------------------+ + +NB: for CABLE compatibility, sf_impl2 is now called by an intermediate +routine surf_couple_implicit. + +Flux diagnostic formulae +~~~~~~~~~~~~~~~~~~~~~~~~ + +The original boundary layer implicit solver computes the total stress by +time averaging the stresses at :math:`t^{n}` and :math:`t^{n+1}`, where +:math:`[t^{n},t^{n+1}]` denotes the time integration interval for the +vertical diffusion equation being solved. The averaging which takes +place for the zonal wind component stress is: + +.. math:: + + \overline{\tau_{x}}^{n+1}\equiv(1-\gamma)\tau_{x}^{n}+\gamma\tau_{x}^{n+1} + =\tau_{x}^{n}+\gamma + K_{u}\frac{\partial\delta u^{n+1}}{\partial z}\label{eq:taux_tot} + +where :math:`\delta u^{n+1}=u^{n+1}-u^{n}`. Likewise, :math:`\tau_{y}` +and the scalar fluxes are derived. + +For the new scheme the total zonal wind component stress is defined as: + +.. math:: \overline{\tau_{x}}^{n+1}\equiv\overline{\tau_{x}}^{[n,*]}+\overline{\tau_{x}}^{[*,n+1]} + +where, :math:`\overline{\tau_{x}}^{[n,*]}` denotes the total stress for +the 1st stage of the scheme, i.e. the time averaged stress from +:math:`t^{n}` to the pseudo-timelevel :math:`t^{*}` and similarly, +:math:`\overline{\tau_{x}}^{[*,n+1]}` the total stress for the 2nd stage +of the scheme (corrector). The total stress for the predictor and the +corrector are defined as: + +.. math:: + + \begin{aligned} + \overline{\tau_{x}}^{[n,*]}\equiv{\cal I}_{1}\tau_{x}^{*}-{\cal E}_{1}\tau_{x}^{n} & = & \left({\cal I}_{1}-{\cal E}_{1}\right)\tau_{x}^{n}+{\cal I}_{1}K_{u}\frac{\partial\delta u^{*}}{\partial z}\\ + \overline{\tau_{x}}^{[*,n+1]}\equiv{\cal I}_{2}\tau_{x}^{n+1}-{\cal E}_{2}\tau_{x}^{*} & = & \left({\cal I}_{2}-{\cal E}_{2}\right)\tau_{x}^{*}+{\cal I}_{2}K_{u}\frac{\partial\delta u^{n+1}}{\partial z} + \end{aligned} + +where, :math:`\delta u^{*}=u^{*}-u^{n},\;\delta u^{n+1}=u^{n+1}-u^{*}`. +The meridional stress :math:`\tau_{y}` and the scalar fluxes can be +derived in a similar way. These formulae have been validated in SCM +experiments. + +Implicit surface flux and future upgrades +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +This scheme should be incorporated in the calculation of the scalar +implicit fluxes. This would be preferable to the current technique for +calculating the scalar implicit fluxes (use of original implicit scheme +for these). This has been attempted but not yet successfully completed. +In the tested code, 2 calls to the modified im_sf_pt subroutine are +done, one per scheme stage (step), i.e. one for the stage that +:math:`{\delta X}_{1}^{*}` is computed and one for :math:`\delta +X_{1}^{n+1}` where the subscript denotes level number. At each call, a +modified version of the flux formulae (78), (79) of +:raw-latex:`\cite{esseryetal2001}` is used: + +**1st sweep:** + +.. math:: + + \begin{aligned} + \frac{\overline{H^{*}}}{c_{p}} & = & \frac{(1+\beta B_{2})[\gamma_{2}F_{T}^{n}+A_{1}\delta{T'}_{1}+A_{2}\delta{Q'}_{1}]-\beta A_{2}[\gamma_{2}F_{Q}^{n}+B_{1}\delta{T'}_{1}+B_{2}\delta{Q'}_{1}]}{(1+\beta A_{1})(1+\beta B_{2})-\beta^{2}A_{2}B_{1}}\label{eq:FTLstar} + \end{aligned} + +.. math:: + + \begin{aligned} + \overline{E^{*}} & = & \frac{(1+\beta A_{1})[\gamma_{2}F_{Q}^{n}+B_{1}\delta{T'}_{1}+B_{2}\delta{Q'}_{1}]-\beta B_{1}[\gamma_{2}F_{T}^{n}+A_{1}\delta{T'}_{1}+A_{2}\delta{Q'}_{1}]}{(1+\beta A_{1})(1+\beta B_{2})-\beta^{2}A_{2}B_{1}}\label{eq:FQWstar} + \end{aligned} + +where, :math:`F_{T}^{n}`, :math:`F_{Q}^{n}` denote the surface explicit +fluxes, :math:`\gamma_{2}={\cal I}_{1}-{\cal E}_{1}` and the +coefficients :math:`A_{1},\; A_{2},B_{1},\; B_{2}` are given by + +.. math:: + + \begin{equation} + A_1=-\gamma_1\sum_j \nu_j RK_{PMj} + [LD_j\psi_jRK_H(1)_j+A_{*j}], + \end{equation} + \begin{equation} + A_2=\gamma_1\sum_j \nu_j RK_{PMj} + L\psi_jRK_H(1)_j, + \end{equation} + \begin{equation} + B_1=\gamma_1c_p\sum_j \nu_j RK_{PMj} + D_j\psi_jRK_H(1)_j + \end{equation} + \begin{equation} + B_2=-\gamma_1\sum_j \nu_j RK_{PMj} + \psi_j[c_pRK_H(1)_j+A_{*j}]. + \end{equation} + \label{ab_coeffs} + +but with :math:`\gamma_{1}={\cal I}_{1}`. Here +:math:`RK_H(1) =\rho C_H U_1`, +:math:`RK_{PM}={RK_H(1)\over(c_p+LD\psi)RK_H(1)+A_*}`, + +.. math:: + + A_* = (1 - f_r){2\lambda\over\Delta z_s} + {C_c\over\Delta t} + + 4(1 + f_r)\sigma T_s^3, + +.. math:: + + D={q_{\rm sat}(T_*^{(n)},p_*)-q_{\rm sat}(T_1^{(n)},p_*) \over + T_*^{(n)}-T_1^{(n)}}, + +.. math:: \psi=f_a+(1-f_a){g_s\over g_s+C_HU_1}, + +and :math:`\nu_j` represents the fraction of surface tile type +:math:`j`. :math:`f_r` is the radiative canopy fraction, :math:`\lambda` +is the soil conductivity, :math:`\Delta z_s` and :math:`T_s` are the +thickness and temperature of the surface soil layer, :math:`C_c` is the +canopy heat capacity, :math:`f_a` is the saturated fraction of the tile +and :math:`g_s` is the surface conductance. To derive +(`[eq:FTLstar] <#eq:FTLstar>`__), (`[eq:FQWstar] <#eq:FQWstar>`__), the +time-weighted level 1 :math:`T` and :math:`Q` consistent with the +discrete equations of the new scheme is written as follows: + +.. math:: \overline{T_{1}^{*}}=\gamma_{2}T^{n}+\gamma_{1}\delta T_{1}^{*},\quad\overline{Q_{1}^{*}}=\gamma_{2}Q^{n}+\gamma_{1}\delta Q_{1}^{*},\qquad\delta T_{1}^{*}=T^{*}-T^{n},\quad\delta Q_{1}^{*}=Q^{*}-Q^{n} + +and the original derivation is followed. From these expressions the tile +flux for :math:`H` is derived: + +.. math:: + + \begin{aligned} + \frac{H_{j}^{*}}{c_{p}} & = & \gamma_{2}\frac{H_{j}^{(n)}}{c_{p}}-\gamma_{1}RK_{PMj}[LD_{j}\psi_{j}RK_{H}(1)_{j}+A_{*j}][c_{p}\delta{T'}_{1}-\beta\overline{H^{*}}]\\ + & & \qquad\quad+\gamma_{1}RK_{PMj}L\psi_{j}RK_{H}(1)_{j}[\delta{Q'}_{1}-\beta\overline{E^{*}}] + \end{aligned} + +and similarly :math:`E_{j}^{*}`. From these, the tile flux equations +(`[eq:FTLstar] <#eq:FTLstar>`__), (`[eq:FQWstar] <#eq:FQWstar>`__) can +be obtained. + +**2nd sweep:** + +.. math:: + + \begin{aligned} + \frac{\overline{H^{n+1}}}{c_{p}} & = & \frac{(1+\beta B_{2})[\xi_{2}F_{T}^{*}+A_{1}\delta{T'}_{1}+A_{2}\delta{Q'}_{1}]-\beta A_{2}[\xi_{2}F_{Q}^{*}+B_{1}\delta{T'}_{1}+B_{2}\delta{Q'}_{1}]}{(1+\beta A_{1})(1+\beta B_{2})-\beta^{2}A_{2}B_{1}}\label{eq:FTLnp1} + \end{aligned} + +.. math:: + + \begin{aligned} + \overline{E^{n+1}} & = & \frac{(1+\beta A_{1})[\xi_{2}F_{Q}^{*}+B_{1}\delta{T'}_{1}+B_{2}\delta{Q'}_{1}]-\beta B_{1}[\xi_{2}F_{T}^{*}+A_{1}\delta{T'}_{1}+A_{2}\delta{Q'}_{1}]}{(1+\beta A_{1})(1+\beta B_{2})-\beta^{2}A_{2}B_{1}}\label{eq:FQWnp1} + \end{aligned} + +where, the coefficients :math:`A_{1},\; A_{2},B_{1},\; B_{2}` are given +by (`[ab_coeffs] <#ab_coeffs>`__) but with +:math:`\gamma_{1}={\cal I}_{2}`. The above formulae are derived as +explained earlier. The definitions + +.. math:: \overline{T_{1}^{n+1}}=\gamma_{2}T^{*}+\gamma_{1}\delta T_{1}^{n+1},\quad\overline{Q_{1}^{*}}=\gamma_{2}Q^{*}+\gamma_{1}\delta Q_{1}^{n+1},\qquad\delta T_{1}^{n+1}=T^{n+1}-T^{*},\quad\delta Q_{1}^{n+1}=Q^{n+1}-Q^{*} + +are used here. The surface fluxes :math:`F_{T}^{*}\equiv{\displaystyle + \frac{H^{*}}{c_{p}}}`, :math:`F_{Q}^{*}\equiv E^{*}` are computed at +model state :math:`{(T}_{1}^{*},Q_{1}^{*})`. They are the equivalent of +the explicit fluxes +:math:`F_{T}^{n}\equiv{\displaystyle \frac{H^{n}}{c_{p}}}`, +:math:`F_{Q}^{n}\equiv E^{n}`. However, they are not equal to the left +hand-side of (`[eq:FTLstar] <#eq:FTLstar>`__), +(`[eq:FQWstar] <#eq:FQWstar>`__). Both are connected by a linear +relationship, which is simply the definition of the time-weighted +averaging consistent with the new scheme: + +.. math:: \overline{H^{*}}\equiv{\cal I}_{1}H^{*}-{\cal E}_{1}H^{n},\qquad\overline{E^{*}}\equiv{\cal I}_{1}E^{*}-{\cal E}_{1}E^{n}. + +Therefore, once :math:`\overline{H^{*}},\overline{E^{*}}` have been +computed, :math:`H^{*}`, :math:`E^{*}` can be computed as follows: + +.. math:: + + H^{*}=\frac{\overline{H^{*}}+(\gamma_{1}-\gamma_{2})H^{n}}{\gamma_{1}},\qquad + E^{*}=\frac{\overline{E^{*}}+(\gamma_{1}-\gamma_{2})E^{n}}{\gamma_{1}}. + +Note that without the previous transformation the model fails within a +few timesteps. + +The calculation of the surface temperature, evaporation and melting +takes place only in the second sweep. The flux increment obtained from +evaporation and melting is added on :math:`\overline{H^{n+1}}`, +:math:`\overline{E^{n+1}}`. The same relationship is used for the +surface temperature: + +.. math:: + + T_{*}=T_{s}+\frac{1}{A_{*}} + \left[R_{s}-H-LE+\frac{C_{c}}{\Delta t}\left(T_{*}^{n}-T_{s}\right)\right], + +however, the total averaged flux from :math:`t^{n}` to :math:`t^{n+1}` +is used: + +.. math:: H=\overline{H^{*}}+\overline{H^{n+1}},\qquad E=\overline{E^{*}}+\overline{E^{n+1}}. + +The total flux is also kept by the corresponding STASH diagnostic. This +has to be adjusted if evaporation exhausts any of the moisture stores +during the timestep or if the tile has a melting snowcover. + +Limited evaporation +^^^^^^^^^^^^^^^^^^^ + +Downward surface moisture fluxes are added to canopy moisture or, if the +surface temperature is below freezing, snowcover. + +For an upward total moisture flux :math:`E`, the rates of evaporation +from the canopy and soil moisture stores are + +.. math:: E_c = f_a{E\over\psi} + +and + +.. math:: E_s = (1 - f_a)\psi_s {E\over\psi} + +where + +.. math:: \psi_s = {g_s\over g_s+C_HU_1}. + +If the predicted canopy evaporation would exhaust the canopy moisture +store :math:`C` during a timestep, the soil evaporation is recalculated +as + +.. math:: E_s =\psi_s\left(1 - {f_aC\over E_c\Delta t}\right){E\over\psi} + +and :math:`E_c` is reset to :math:`C/\Delta t`. If :math:`E_s` would +then exhaust the available soil moisture :math:`m`, it is limited to +:math:`m/\Delta t`. + +For an adjustment :math:`\Delta(LE)` in the latent heat flux, +repartitioning the surface energy balance gives adjustments + +.. math:: \Delta H = - \left[1 + {A_*\over c_pRK_H(1)}\right]^{-1}\Delta(LE) + +and + +.. math:: \Delta T_* = - {\Delta H+\Delta(LE)\over A_*} + +in the surface sensible heat flux and temperature. + +Evaporation from a lake tile (or the lake fraction of an aggregated +surface) is not limited and does not draw on the conserved moisture +stores. + +Snowmelt +^^^^^^^^ + +Classical surface energy balance neglects snowmelt heat fluxes. If +:math:`T_*>T_m` for a snow-covered tile and sufficient snow is +available, :math:`T_*` is reset to :math:`T_m` by adding an increment + +.. math:: \Delta T_*=T_m-T_*, + +corresponding to a snowmelt heat flux + +.. math:: + + S_m = - [(c_p + L_sD)RK_H(1) + A_*]{\Delta T_*\over L_f}. + \label{eq:Sm} + +The maximum melt rate that can be sustained over a timestep +:math:`\Delta +t`, however, is :math:`S/\Delta t-E`, giving + +.. math:: + + \Delta T_*={L_f(S/\Delta t-E) \over + (c_p+L_cD)RK_H(1) + A_*}. + \label{eq:dTmax} + +:math:`\Delta T_*` is set to the smaller of the values given by +Equations (`[eq:Sm] <#eq:Sm>`__) and (`[eq:dTmax] <#eq:dTmax>`__), and +the surface energy balance is repartitioned by adding increments + +.. math:: \Delta H = c_pRK_H(1)\Delta T_* + +and + +.. math:: \Delta E = DRK_H(1)\Delta T_* + +to the tile heat and moisture fluxes. + +The model with the above changes coded seems to work stably but the +surface fluxes (and therefore the boundary layer increments) are only +qualitatively correct. They seem to be overestimated by the above +scheme. [\ *Could it be that coefficients :math:`D_{j}`, +:math:`\psi_{j}` need to be modified in the second sweep?*] + +Blending height coupling +~~~~~~~~~~~~~~~~~~~~~~~~ + +The same method is used as in section `9.1.2 <#sec:impsolve>`__ to form +two independent tridiagonal systems of linear equations that relate the +increments to momentum, temperature and humidity to the surface fluxes. +The ‘downward sweep’ elimination procedure still takes place to obtain +equation (`[eq:du_half] <#eq:du_half>`__) and a corresponding equation +for the increments to the scalar variables at the bottom model level + +.. math:: \delta X_{1/2}^{*}=\delta X_{1/2}^{'}-\beta_X\frac{\bar{H}_\star}{C_p} + +where :math:`\delta X_{1/2}^{'}` and :math:`\beta_X` are known. An +‘upward sweep’ of this tridiagonal matrix (i.e. back subsitution) then +takes place to obtain equations for the increment to momentum and scalar +variables at a given level :math:`k_{b}` in terms of the surface fluxes + +.. math:: \delta u_{k_{b}}^{*}=\delta u_{k_{b}}^{'}+(-1)^{k_{b}}\beta\bar{\tau}_{x}^{*}\Big|_{0} \prod^{j=2}_{k_{b}} {C_{u}}_{j}^{'} + \sum^{k_{b}-1}_{i=1} \left[(-1)^{k_{b}+i}\delta u_{k_{b}}^{'} \prod^{j=i+1}_{k_{b}} {C_{u}}_{j}^{'}\right] + +.. math:: \delta X_{k_{b}}^{*}=\delta X_{k_{b}}^{'}+(-1)^{k_{b}}\beta_X\frac{\bar{H}_\star}{C_p} \prod^{j=2}_{k_{b}} {C_{X}}_{j}^{'} + \sum^{k_{b}-1}_{i=1} \left[(-1)^{k_{b}+i}\delta X_{k_{b}}^{'} \prod^{j=i+1}_{k_{b}} {C_{X}}_{j}^{'}\right] + +where :math:`\delta u_{k_{b}}^{*}` and :math:`\delta X_{k_{b}}^{*}` are +the only unknowns. The coefficients for these equations are passed to +the surface implicit solver so that the surface fluxes are calculated +using level :math:`k_{b}` (which corresponds to a user-specified +blending height) rather than the bottom model level. The rest of the +implicit solver continues in the same way as for coupling at the bottom +model level. + +By default, this option is switched off as further work is needed (there +is currently a problem with the input data such that this blending +height option crashes on the first time step). + +Derived diagnostics +=================== + +Boundary layer thermal speed: stash 3,355 +----------------------------------------- + +This diagnostic is intended for use in quantifying the strength of +convective thermals for aviation applications. Updraught velocities in +convective boundary layers will scale with the convective velocity +scale, :math:`w_*`, given by :math:`w_*^3 = z_{\rm h}\overline{w'b}_S`. +In addition to the basic convective velocity scale, the strength of +thermals should also depend on the surface stability — it would be +possible to have significant heat flux and boundary layer depth in windy +conditions that should not lead to a strong thermal forecast. This +sensitivity of boundary layer turbulence is already included in the +parametrization of non-local momentum fluxes (see section +`5.4 <#sec:ngstress>`__) through the stability dependence in +(`[tau_nl] <#tau_nl>`__) that can be written as + +.. math:: f_{stab} = - \frac{a_{stab} z_{\rm h}/L }{1 - a_{stab} z_{\rm h}/L} + +for the Obhukov length, (`[1.1.4] <#1.1.4>`__), :math:`<0` (i.e., +unstable boundary layers) and the empirical constant +:math:`a_{stab} = 1.5`. This function tends to unity as :math:`L` +decreases in magnitude (i.e., surface heating increases and wind stress +decreases). The final thermal speed, in units of ms\ :math:`^{-1}`, is +given simply by + +.. math:: {\rm Thermal} \, {\rm Speed} = f_{stab} \, w_* + +Wind gust: stash 3,463 and 3,515 (scale-dependent) +-------------------------------------------------- + +WMO define the wind gust strength as the maximum of the wind averaged +over 3 second intervals. In the boundary layer the strength of gusts is +proportional to the standard deviation of the horizontal wind, +:math:`\sigma_u`, so that + +.. math:: + + U_{gust} = U_{10m} + W_{1D} \, \sigma_u \, \frac{1}{k} \, + {\rm log}\left( \frac{5 \, e^{k \, c_{\rm ugn}} + z_{0m(eff)} } + {5 + z_{0m(eff)}} \right) + \label{windgust} + +The factor :math:`W_{1D}` is included only in the scale-dependent +version of the diagnostic (stash 3,515) to allow for the larger scales +of boundary layer turbulence that are resolved (and so are already +included in :math:`U_{10m}`). The lowest grid-level value of +:math:`W_{1D}`, from (`[eq-tanh] <#eq-tanh>`__), is used, noting that +:math:`W_{1D}` is constant within the boundary layer. The constant +:math:`c_{\rm ugn}` in (`[windgust] <#windgust>`__) is determined from +universal turbulence spectra for a 25% exceeding probability of the +three-second wind gust (:raw-latex:`\cite{beljaars1987}`). It is +included through a function that includes the effective roughness +length, :math:`z_{0m(eff)}`, in order to take into account the very high +effective :math:`u_*` values that occur over mountainous terrain (due to +the orographic form drag parametrization) and so avoid unrealistic high +gust values. Currently the UM takes :math:`c_{\rm ugn}=4` which was +reduced from the value used at ECMWF based on evaluation of the wind +gust performance. The stability dependence of :math:`\sigma_u` is +estimated on the basis of the similarity relation from +:raw-latex:`\cite{panofsky1977}` + +.. math:: + + \sigma_u = + \begin{cases} + A_{gust} u_* (1.0 - z_{\rm h}/ (24 L) )^{1/3} & {\rm for}\ L<0 \\ + A_{gust} u_* & {\rm for}\ L>0 + \end{cases} + +with :math:`A_{gust}=2.29`. For :math:`L` close to zero the wind gust +diagnostic is undefined and so we set :math:`U_{gust} = U_{10m}`. Note +that the friction velocity, :math:`u_*`, must use the implicitly +calculated surface stress components because the explicitly calculated +:math:`u_*` can be erratic, particularly over mountainous regions. Then, +for consistency with the implicit :math:`u_*`, :math:`L` must also be +calculated implicitly and, to avoid potential numerical problems in very +light winds, the unstable (:math:`L<0`) case is rewritten as: + +.. math:: \sigma_u = A_{gust} (u_*^3 + k w_*^3 / 24 )^{1/3} + +TKE: stash 3,473 +---------------- + +A substantial part of the turbulent flux is parametrized in both the +UM’s first order closure and closures involving TKE, :math:`e`, through +a simple down-gradient diffusion term. An estimate of subgrid TKE can +then be made by equating the UM’s diffusion coefficient, +(`[klnl] <#klnl>`__), with that from a typical TKE-closure, i.e. + +.. math:: + + K_m = l \sqrt{e} + \label{tke_closure} + +where :math:`l` is a length scale. Initially it was thought to diagnose +:math:`e` by approximating :math:`l` as the mixing length in +(`[kmlocal] <#kmlocal>`__) but closer inspection reveals that many TKE +closures have diagnostic relationships for :math:`l` that involve the +TKE itself! A common one for stable boundary layers is +:math:`l_{st} \sim \sqrt{e} / N`, where :math:`N` is the Brunt-Vaisala +frequency. :raw-latex:`\cite{Suselj2012}`, for example, also take +:math:`l_{un} = \tau_{un} \sqrt{e}` in unstable boundary layers, where +:math:`\tau_{un}` is a turbulence timescale that they take as a constant +400 seconds. These they combine through :math:`l^{-1}= l_{un}^{-1} + +l_{st}^{-1} = e^{-1/2}( \tau_{un}^{-1} + \tau_{st}^{-1}) \equiv +e^{-1/2}\tau_{turb}^{-1}` and (`[tke_closure] <#tke_closure>`__) +becomes: + +.. math:: K_m = \tau_{turb} e + +To derive a TKE diagnostic then requires a parametrization of the +turbulence timescale, :math:`\tau_{turb}`. + +Basic boundary layer scaling (e.g., Figure 4 of +:raw-latex:`\cite{holtslag91:_eddy_diffus_count_trans_convec}`) shows +that :math:`\overline{w'^2}` from a variety of convective boundary layer +LES and observations nicely follows the relationship + +.. math:: + + \overline{w'^2} = c_{w2} w_*^2 f(z') + \label{w2_scaling} + +where :math:`w_*` is the convective velocity scale and :math:`f` is a +shape function within the boundary layer (:math:`z'=z/z_{\rm h}`). The +shape of this function is very similar to that used in the UM for +:math:`K_m^{\rm surf}` in (`[kmsurf] <#kmsurf>`__). We now assume we can +generalise (`[w2_scaling] <#w2_scaling>`__) by replacing :math:`w_*` +with :math:`w_m` (this really ought to be checked against neutral +boundary layer LES but hasn’t yet been). Setting :math:`f(z')=z' +(1-z')^2` in (`[w2_scaling] <#w2_scaling>`__) and comparing with Fig.4 +of :raw-latex:`\cite{holtslag91:_eddy_diffus_count_trans_convec}` gives +:math:`c_{w2}= 2.66 +/ C_{ws}^{2/3}` (i.e., a constant of 2.66 gives the maximum in +:math:`\overline{w'^2}/w_*^2` at around the observed value of 0.4) so +that we can generalise (`[w2_scaling] <#w2_scaling>`__) to + +.. math:: + + \overline{w'^2} = \frac{2.66 }{C_{ws}^{2/3}} \, w_m^2\, f(z') + \label{gen_w2_scaling} + +where the mixed layer expression for :math:`w_m` is used. + +:raw-latex:`\cite{holtslag91:_eddy_diffus_count_trans_convec}` also show +from analysis of the scalar flux budget that + +.. math:: \overline{w'\theta'} = - \frac{\tau_{turb}}{2} \, \overline{w'^2} \frac{d \theta}{dz} + +where :math:`\tau_{turb}` is a return to isotropy timescale. Ignoring +the non-gradient parametrization in the UM, it follows that + +.. math:: + + K_h^{\rm surf}= \frac{\tau_{turb}}{2} \, \overline{w'^2} + \label{bl_scaling} + +Combining (`[bl_scaling] <#bl_scaling>`__) with +(`[gen_w2_scaling] <#gen_w2_scaling>`__) and (`[kmsurf] <#kmsurf>`__), +and subsuming the Prandtl number into the other constants, for +surface-driven boundary layer mixing we can write: + +.. math:: K_m^{\rm surf}= \frac{\tau_{\rm surf}}{2} \, \overline{w'^2} = \frac{\tau_{turb}}{2} \, \frac{2.66 }{C_{ws}^{2/3}} w_m^2 \, f(z') = k z_{\rm h}w_m f(z') + +which then gives +:math:`\tau_{\rm surf} = C_{ws}^{2/3} k z_{\rm h}/ (1.33 w_m)`. An +analogous timescale can be derived for top-driven mixing in decoupled +stratocumulus layers, :math:`\tau_{\rm Sc} = g_1 k z_{\rm ml}/ (1.33 \, +V_{\rm Sc})`. + +There are two options to derive a TKE diagnosis from the Ri-based scheme +and then combine with the non-local TKE (selected via var_diags_opt). +One is to assume :math:`\tau_{\rm SBL}=0.7/N` as the timescale for +stable boundary layers and combine all these timescales following +:raw-latex:`\cite{Suselj2012}`) to give: + +.. math:: + + e = K_m \tau_{turb}^{-1} + \label{tke_diag} + +where +:math:`\tau_{turb}^{-1} = MAX[ \tau_{\rm surf}^{-1},\tau_{\rm Sc}^{-1}] + \tau_{\rm SBL}^{-1}`. +Note that (`[tke_diag] <#tke_diag>`__) gives :math:`\overline{w'^2}`, +rather than TKE. As a simple fix to improve the near-surface TKE in +convective boundary layers, where the horizontal wind variability often +dominates, the value of :math:`e` given by (`[tke_diag] <#tke_diag>`__) +at the level of the maximum in :math:`K_m^{\rm surf}` is copied to all +levels below that height. + +The second method diagnoses TKE for the local scheme following the Met +Office LEM and MONC, simplifying and parametrizing the terms in the TKE +budget to give + +.. math:: + + e_{loc}^{3/2} = \lambda S^2 K_m (1-Ri/Pr)/C_{e} + \label{tke_diag_loc} + +where :math:`C_{e}=A_{2N}^{3/2}`. Initial tests found that the MONC +value of :math:`A_{2N}=0.23` gave rather large values of :math:`e_{loc}` +and so :math:`C_e=0.41` is used. Note that this is an optional value +used in the higher order closure scheme (see section 2.8.5 of ). It +could also be worth testing the suggested parametrization in (2.300) +there, of :math:`C_e=0.19+0.74 \lambda/\Delta z` but this has not yet +been attempted. The total non-local TKE is computed by adding the TKE +from each non-local component, as is done for the diffusion +coefficients, i.e., + +.. math:: + + e_{nl} = \frac{3}{2} \left( \frac{K_m^{\rm surf}}{\tau_{\rm surf}} + + \frac{K_m^{\rm Sc}}{\tau_{\rm Sc}} \right) + \label{tke_diag_nl} + +The factor of :math:`3/2` in (`[tke_diag_nl] <#tke_diag_nl>`__) arises +because we are really diagnosing :math:`\overline{w'^2}` and so here we +make the assumption of isotropic turbulence to extend this to TKE. As +before, we do also make the simple fix to improve the near-surface TKE +in convective boundary layers, but here we copy only the value of +:math:`e_{nl}` at the level of the maximum in :math:`K_m^{\rm surf}` to +all levels of :math:`e_{nl}` below that height. The final TKE is then +the greater of :math:`e_{nl}` and :math:`e_{loc}` (as is done to combine +the diffusion coefficients). + +The methods above will still underestimate the value of subgrid TKE in +regions of parametrized convection, because the value of :math:`K_m` +will be small (or zero) here as parametrized mixing is assumed to be +done via the convection scheme. Therefore an option *l_conv_tke* is +provided to include an estimate of the TKE due to parametrized +convection within the diagnostic. This is given by: + +.. math:: e_{\rm conv} = \left(\frac{M}{g\rho \times CCA}\right)^2 + +where :math:`M` is the convective updraft mass flux +(Pa s\ :math:`^{-1}`) and CCA is the convective cloud area. The final +diagnostic is then given as the maximum of :math:`e` and +:math:`e_{\rm conv}`. + +If selected, this option also reduces the minimum value used in UKCA by +an order of magnitude. This is possible, because the minimum is no +longer having to provide a realistic estimate of TKE in convective cloud +regions, and is now a genuine numerical minimum. Selecting this option +also corrects a bug in the level indexing of this diagnostic when passed +to UKCA. + +Note that additional diagnostics of the scalar variances are also made +and those are documented in . + +.. _`app:neutwind`: + +Diagnostics of Neutral Winds and Stresses: stash 3,365 to 3,371 +--------------------------------------------------------------- + +Conditions near the ocean’s surface are often described using 10-m +neutral wind and quantities derived from the neutral winds. Such +diagnostics are therefore potentially very useful for evaluation of the +model and have been added to the scheme. + +The equivalent neutral 10-m wind is the wind that would be observed, +given the surface friction velocity and roughness length, if the +stratification were neutral. As such, it is more simply related to the +surface stress than the true stability-dependent wind. Scatterometers +ultimately respond to backscatter from surface capillary waves, which +are driven by the surface stress, so observations from scatterometers +are typically reported as equivalent neutral winds. + +The pseudostress is the product of the wind speed and the vector wind at +a given height (in practice 10m). The kinematic surface stress is +therefore equal to the product of the pseudostress and the drag +coefficient. Pseudostress is sometimes used in observational products, +notably the Cross-Calibrated Multi-Platform (CCMP) surface wind vector +analysis :raw-latex:`\cite[]{atlas2011}`. + +.. _`app:vscales`: + +Appendix: Definitions of the velocity scales +============================================ + +As described in :raw-latex:`\cite{lock00}`, the parametrization of the +entrainment rate in convective boundary layers is based on four velocity +scales, each representative of a turbulence-generating process +(:math:`V_{\rm heat}` for surface heating, :math:`u_*` for surface shear +generation, :math:`V_{\rm rad}` for cloud-top radiative cooling and +:math:`V_{\rm br}` for buoyancy reversal). The velocity scales can be +written + +.. math:: + + \begin{aligned} + V_{\rm heat}^3&=& z_{\rm ml}\! \left( (2-\zeta_s)\zeta_s \overline{w'b}_S+ (1-\zeta_s)^2 [\overline{w'b'}_S]_{\rm sat}\right) + \label{vsurf} \\ + V_{\rm rad}^3&=& z_{\rm ml}\Delta_F\, g \, + \left( \beta_T \zeta_r^2 + \tilde{\beta_T} (1-\zeta_r^2) \right) + \label{vrad} \\ + V_{\rm br}^3&=& A_{\rm br}\chi_s^2 \, \mbox{max}\left[0,-\delta b\right] \, \Delta b ^{1/2} + \, z_c^{3/2} \, C_{fac} \label{vbr} + \end{aligned} + +Here, +:math:`[\overline{w'b'}_S]_{\rm sat}= g ( \tilde{\beta_T} \overline{w'\theta_{\ell}'}_S+ \tilde{\beta_q}\overline{w'q_t'}_S)`, +where the subscript :math:`_S` indicates the surface flux; +:math:`\Delta_F` is the divergence of the net radiative flux, :math:`F` +(in Kms\ :math:`^{-1}`), associated with cloud-top, for which the +calculation is described in section `11.1 <#app:deltaf>`__. + +Various depth parameters are given by :math:`\zeta_s = +(z_{\rm ml}-\tilde{z_c})/z_{\rm ml}`, +:math:`\zeta = (z_{\rm ml}-z_c)/z_{\rm ml}` and :math:`\zeta_r = +\zeta + Br (1-\zeta)`. :math:`z_{\rm ml}` is the mixed-layer depth, +:math:`z_c` is the cloud depth and :math:`\tilde{z_c}` is the +cloud-fraction weighted cloud depth. The former is used in the +calculation of :math:`V_{\rm rad}` as it is assumed the radiative +cooling will occur predominantly in cloudy air. To allow for a feedback +in the presence of buoyancy reversal, the parameter :math:`Br` is +included in :math:`\zeta_r` and :math:`\tilde{\alpha_t}` (in +(`[we_parm] <#we_parm>`__)). It is given in terms of the +:raw-latex:`\cite{siems1990}` parameter, +:math:`D = \chi_s \delta b/\Delta b` and constrained by :math:`0< Br = +10 D < 1`. This gives a linear ramp for this feedback between regimes +where there is no buoyancy reversal (:math:`D \leq 0`) and the feedback +seen in LES of stratocumulus :raw-latex:`\cite[]{lock98}` with +significant buoyancy reversal +(:math:`D \raisebox{-.4ex}{$\ \stackrel{>}{{\scriptstyle \sim}} \ $}0.1`). +Furthermore, the LES of :raw-latex:`\cite{lock09:_factor}` indicated the +presence of cumulus penetrating up into stratocumulus could be +sufficient to enhance the feedback for small :math:`D`. Thus the option +exists to enhance :math:`Br` for :math:`0`__) +within the mixed layer (zero above). Optionally (and currently +implemented as standard) :math:`\gamma_{q_f}` can be set to zero in +which case the third term in (`[zcld_calc] <#zcld_calc>`__) is set to +zero. + +In the 8A scheme, the cloud depth, :math:`z_c`, is calculated as the sum +of :math:`\Delta_{k+\frac{1}{2}} z`, descending from the top mixed-layer +grid-level, as long as the cloud fraction in the layer below is greater +than a tolerance, SC_CFTOL (currently set to 0.1). At the grid-level +:math:`k_b`, say, where :math:`{C_F}_{k_b-1} <` SC_CFTOL, a subgrid +estimate of the height of cloud-base is made using +:math:`\gamma_{q_{\ell}}` and :math:`\gamma_{q_f}` and added to the +grid-level based calculation: + +.. math:: + + \begin{aligned} + z_c &=& z_c + \frac{\Delta_{k_b+\frac{1}{2}} z}{2} + + \mbox{min}\left[ \frac{ \Delta_{k_b+\frac{1}{2}}z + +\Delta_{k_b-\frac{1}{2}}z }{2} C_F^l, + \frac{ q_{\ell}}{ \gamma_{q_{\ell}} } \right]/C_F \nonumber \\ + & & \left. \hspace{2.4cm} + + \mbox{min}\left[ \frac{ \Delta_{k_b+\frac{1}{2}}z + +\Delta_{k_b-\frac{1}{2}}z }{2} C_F^f, + \frac{ q_f }{ \gamma_{q_f} } \right]/C_F \right. + \label{zc_calc} + \end{aligned} + +When :math:`\gamma_{q_f}` is set to zero (currently as standard) the +last term in (`[zc_calc] <#zc_calc>`__) is given by +:math:`(\Delta_{k_b+\frac{1}{2}}z+\Delta_{k_b-\frac{1}{2}}z)C_F^f/(2C_F)`. +Note that, if :math:`k_b=1` in (`[zc_calc] <#zc_calc>`__), then +:math:`\Delta_{k_b-\frac{1}{2}}z` is taken to be zero. The final part of +the 8A calculation of :math:`z_c` is to include the depth to which the +cloud extends into the inversion grid-level. If a subgrid inversion +height, :math:`z_i`, has been diagnosed (see +section `7.1.1 <#sec:sginv>`__) then the height of :math:`z_i` above the +half-level height is added to :math:`z_c` (as long as :math:`C_F>` +SC_CFTOL in grid-levels NTML or NTML\ :math:`+1` or the layer is a +decoupled layer). If no subgrid inversion has been diagnosed and +:math:`C_F(NTML+1)>` SC_CFTOL, then :math:`z_c` is increased by the full +depth of layer :math:`NTML+1` if :math:`C_F(NTML)>` SC_CFTOL and using +(`[zc_calc] <#zc_calc>`__) otherwise (and similarly for DSC layers). The +same ideas are used in the 9C version, extrapolating using the adiabatic +water gradient, except that the grid-level from which this extrapolation +is made is now not the lowest grid-level with :math:`C_F >` SC_CFTOL but +rather the lowest level with :math:`C_F=1` (or the grid-level with the +maximum :math:`C_F`). Most observations of stratocumulus (i.e., mixed +layer clouds) find :math:`q_{\ell}` to be close to adiabatic over the +whole layer and accurately given by the supersaturation of the +well-mixed :math:`q_t`. If :math:`C_F=1` then the Smith cloud scheme +makes :math:`q_{\ell}` equal to the supersaturation and so will be +reasonably accurate. Extrapolating this grid-level :math:`q_{\ell}` to +zero should then give a reasonably accurate measure of cloud-base. + +The formula for :math:`V_{\rm br}` was derived using dimensional +arguments and comparison with LES data: +:math:`\chi_s = -{q_{\ell}}_{\rm ct}(1+(L/c_p)\alpha_L) / ( +\Delta q_t - \alpha_L \Delta \theta_{\ell})`, where +:math:`{q_{\ell}}_{\rm ct}` is the cloud-top liquid water mixing ratio, +:math:`L` is the latent heat of vaporisation of water, :math:`c_p` the +specific heat at constant pressure and T is the temperature; +:math:`\delta b = g(\tilde{\beta_T} \Delta \theta_{\ell} ++ \tilde{\beta_q} \Delta q_t)` and the buoyancy jump across the +inversion is given by + +.. math:: + + \Delta b = g \, \left( \beta_T \Delta \theta_{\ell}+ \beta_q \Delta q_t + + \left( \beta_T \frac{L}{c_p} - + \frac{1+c_v}{c_v}\beta_q \right)\Delta q_{\ell}+ + \left( \beta_T \frac{L_s}{c_p} - + \frac{1+c_v}{c_v}\beta_q \right)\Delta q_f \right) + \label{dbinv} + +The empirical constant :math:`A_{\rm br}= 0.24`. The calculation of +:math:`\Delta +\theta_{\ell}` and :math:`\Delta q_t` is described for a subgrid +inversion in section `7.1.1 <#sec:sginv>`__ or, if one is not diagnosed, +they are taken simply as :math:`\Delta_{\mbox{\tiny \rm NTML}+1}`. For +:math:`\Delta q_{\ell}`, :math:`\Delta q_f` and +:math:`{q_{\ell}}_{\rm ct}`, in-cloud values extrapolated to :math:`z_i` +(either subgrid or :math:`z_{\mbox{\tiny \rm NTML}+\frac{1}{2}}`) from +above and below using the adiabatic lapse rates are calculated as: + +.. math:: + + \begin{aligned} + {q_{\ell}}_{\rm ct}& =& \frac{{q_{\ell}}_{\mbox{\tiny \rm NTML}}}{{C_F}^l_{\mbox{\tiny \rm NTML}}} + + (z_i-z_{\mbox{\tiny \rm NTML}}) \gamma_{q_{\ell}}\\ + q_{\ell}^+ & =& \mbox{max}\left[ 0, \, + \frac{{q_{\ell}}_{\mbox{\tiny \rm NTML}+2}}{{C_F}^l_{\mbox{\tiny \rm NTML}+2}} + - (z_{\mbox{\tiny \rm NTML}+2}-z_i) \gamma_{q_{\ell}} \right] + \end{aligned} + +and similarly for :math:`q_f` (noting that currently +:math:`\gamma_{q_f}=0`) and for DSC layers. Then, + +.. math:: + + \begin{aligned} + \Delta q_{\ell}& =& {C_F^l}_{\mbox{\tiny \rm NTML}+2}\, q_{\ell}^+ - {C_F^l}_{\mbox{\tiny \rm NTML}}\, {q_{\ell}}_{\rm ct}\\ + \Delta q_f & =& {C_F^f}_{\mbox{\tiny \rm NTML}+2}\, q_f^+ - {C_F^f}_{\mbox{\tiny \rm NTML}}\, {q_f}_{\rm ct} + \end{aligned} + +The only other explicit account of variable cloud fraction is in +(`[vbr] <#vbr>`__) for which it is assumed that buoyancy reversal can +only occur for cloudy air underlying cloud-free air (assuming maximum +overlap). Thus, the cloud fraction factor, :math:`C_{fac} = \mbox{max}[ +0.0, -\Delta C_F ]`, where +:math:`\Delta C_F = {C_F}_{\mbox{\tiny \rm NTML}+2} - +{C_F}_{\mbox{\tiny \rm NTML}}` if a subgrid inversion is diagnosed +(because :math:`{C_F}_{\mbox{\tiny \rm NTML}+1}` is currently +meaningless) and :math:`\Delta C_F = +{C_F}_{\mbox{\tiny \rm NTML}+1} - {C_F}_{\mbox{\tiny \rm NTML}}` if not. +A more complete decomposition is not possible given a cloud scheme in +the model :raw-latex:`\cite[]{smith90}` which does not allow discrete +identification of in-cloud and out-of-cloud profiles. The cloud-fraction +dependence of the radiative generation of turbulence is implicitly +treated in (`[vrad] <#vrad>`__) simply by assuming the grid-box mean +radiative flux divergence, :math:`\Delta_F`, occurs solely in the cloudy +air. + +The assumption behind the current cloud fraction dependence is that the +fraction of the boundary layer that is cloud-capped entrains as though +it were an infinite solid cloud sheet (as in the LES used to derive the +parametrization). This essentially assumes the cloud within the grid-box +is continuous. An additional explicit dependence of entrainment on cloud +fraction was implemented in version 4.5 which reduced the cloud-top +source terms of entrainment in partially cloudy boundary layers by a +factor +:math:`\exp{\left\{-(0.9-{C_F}_{\mbox{\tiny \rm NTML}})^3/0.075\right\}}` +for :math:`{C_F}_{\mbox{\tiny \rm NTML}} < 0.9`. It was argued that +partial cloudiness on the scale of the mixed-layer eddies might reduce +the entrainment efficiency of the cloud-top processes. By reducing the +parametrized entrainment warming and drying in partially cloudy boundary +layers it was hoped that the climatological cloudiness of the +sub-tropical marine stratocumulus might be improved. Only marginal +success was observed, though, as more weakly entraining layers became +shallower and their cloud-top therefore warmer. In addition, timeseries +of cloud fraction from New Dynamics climate simulations suggested these +boundary layers either had high total cloud amount or zero. Coupled with +the generally realistic cloud amounts obtained in the New Dynamics this +arbitrary term has currently been dropped from version 5 onwards. The +issue of how the entrainment rate should be parametrized in partially +cloudy boundary layers, however, remains. + +.. _`app:deltaf`: + +Calculation of :math:`\Delta_F` +------------------------------- + +An important term in the entrainment parametrization and +:math:`K_h^{\rm Sc}` is the velocity scale :math:`V_{\rm rad}`, the cube +of which is proportional to the net radiative flux difference associated +with cloud-top, :math:`\Delta_F`. Because radiation tends not to be +called every timestep, the calculation of :math:`\Delta_F` is done +somewhat independently from the cloud-top height. + +In the 9B version, :math:`\Delta_F` is calculated as: + +.. math:: + + \Delta_F= \sum_{k=k_m-1}^{k_m+1} \mbox{max}\left[ + - \Delta_{k+\frac{1}{2}} z \, {\cal S}_F(k), \,0 \right] + \label{ctraddiv} + +where :math:`k_m` is the grid-level with the greatest radiative cooling +increment, :math:`{\cal S}_F`, within 2 grid-levels of cloud-top. In the +8A scheme, :math:`{\cal S}_F` is simply the net (SW+LW) cooling +increment. During the day, though, the net divergence is partly reduced +from the nocturnal (LW) value due to SW warming of the cloud-layer. In +general, the SW warming is more diffuse than the LW cooling (which +occurs mostly within O(50)m of cloud-top). Using grid-level net +radiative increments at the current coarse vertical resolution used in +the UM (:math:`\sim 200`\ m) means that the cancellation between LW and +SW during the day is excessive. + +A slightly more accurate estimate of the net divergence can be obtained +by assuming the SW and LW radiative fluxes at a given height, :math:`z`, +have an exponential shape, dependent on the LWP above :math:`z`, i.e.: + +.. math:: + + F_{LW}(z) = \Delta_F^{LW} \exp^{ - \kappa_{LW} \mbox{LWP}(z) } + \label{eq:explw} + +Then the net divergence can be approximated given the SW flux at the +height where :math:`F_{LW}` becomes some small fraction, :math:`A`, of +:math:`\Delta_F^{LW}` (which implies +:math:`\mbox{LWP} =-ln(A) / \kappa_{LW}`). Then + +.. math:: + + \Delta_F\approx \Delta_F^{LW} + + (1-\exp^{ln(A)\kappa_{SW}/\kappa_{LW}}) \Delta_F^{SW} + +Empirically, see Fig. `9 <#fig:dradts>`__, a reasonable fit to LEM data +is obtained with: + +.. math:: + + \Delta_F\approx \Delta_F^{LW} + 0.35 \Delta_F^{SW} + \label{eq:deltaf_emp} + +Note that in the 9B scheme :math:`\Delta_F^{LW}` and +:math:`\Delta_F^{SW}` are calculated as in (`[ctraddiv] <#ctraddiv>`__) +but with the LW and SW increments separately. + +This change in the calculation of :math:`\Delta_F` is illustrated in +Fig. (`9 <#fig:dradts>`__) from LES of the diurnal cycle of marine +stratocumulus. The top panel is from a simulation which used the code +specified for the EUROCS LES intercomparison, the lower panel used the +Edwards-Slingo radiation scheme in the LES. There are clearly some +differences in the distribution of the SW absorption within the cloud +between these two schemes but the 9B parametrization is clearly an +improvement on the 8A which gives :math:`\Delta_F=0` around midday (and +therefore zero entrainment and turbulent mixing). + +.. container:: float + :name: fig:dradts + + .. container:: centering + + |image3| |image4| + +The 9C version attempted to remove the grid-dependence implied by the +summation over 3 grid-levels in (`[ctraddiv] <#ctraddiv>`__) as follows: + +#. the search for the level with maximum LW radiative cooling, + :math:`k_m`, is restricted to the top half of the mixed layer and no + higher than level NTML\ :math:`+1` + +#. to allow for the case where the radiative cooling is distributed + roughly equally over two grid-levels, if the LW flux divergence in + level :math:`k_m-1` is greater than half that in level :math:`k_m`, + then :math:`k_m` is lowered one grid-level. + +#. then, the cloud-top radiative flux change is initially calculated for + LW and SW fluxes separately as: + + .. math:: + + \Delta_F= F_{k_m+1} - F_{k_{rb}} + \label{ctraddiv_9c} + + where the base grid-level for the calculation, :math:`k_{rb}`, is + taken to be the higher of the base of the LW radiatively cooled layer + and :math:`z_h/2`, since cooling can only generate turbulence if it + occurs in the upper part of the mixed layer. For decoupled + stratocumulus layers, :math:`k_{rb}` is further restricted to be + above the top of the surface mixed layer. + +#. finally, the flux divergence across grid-level :math:`k_m+1` is + separated into cloudy and free-atmospheric contributions by + extrapolating the free-atmospheric flux-gradient downwards. The + cloudy contribution is then included in :math:`\Delta F`: + + .. math:: + + \Delta F= \Delta F+ \Delta_{k_m+\frac{3}{2}} F + - \Delta_{k_m+\frac{5}{2}} \frac{\Delta_{k_m+\frac{3}{2}} z}{k_m+\frac{1}{2}} F + \label{ctraddiv_9c_inv} + +#. As at 9B above, the calculations in + (`[ctraddiv_9c] <#ctraddiv_9c>`__) and + (`[ctraddiv_9c_inv] <#ctraddiv_9c_inv>`__) are performed separately + for LW and SW radiation before the two are combined using the + empirical relationship in (`[eq:deltaf_emp] <#eq:deltaf_emp>`__) + +Further single column model tests with fine vertical resolution have +shown the above calculations can still fail to accurately measure the +radiative flux jump across the top of the cloud. The methodology now +recommended is to identify where the LW radiative cooling profile +transitions from free-tropospheric rates above the cloud to stronger +rates within it. It entails only relatively minor changes to the first +two steps of the algorithm above which become: + +#. the search for the level with maximum LW radiative cooling, + :math:`k_m`, is restricted to the top half of the mixed layer and + below :math:`1.2 \, z_{\rm h}` (rather than level NTML\ :math:`+1` + used above) + +#. if the LW flux divergence in level :math:`k_m+1` is relatively weak + (less than double that in level :math:`k_m+2`), we assume that level + :math:`k_m+1` is actually typical of the free-troposphere and that + :math:`k_m` must therefore be the inversion grid-level (despite + having the strongest LW cooling). Hence we lower :math:`k_m` by one + so that it now marks the top of the mixed layer — note that LW + cooling within the inversion grid-level will be included in step 4 + above (which is unchanged) + +These small changes were found sufficient to give a robust measure of +:math:`\Delta F` for grids varying down to 20m spacing where the +radiative flux profile is well resolved. + +.. _`app:buoyp`: + +Appendix: Derivation and definitions of the buoyancy parameters +=============================================================== + +Buoyancy is measured by the virtual temperature + +.. math:: + + T_v = T(1 + c_v q_v - q_{\ell}- q_f) = T V_{fac} + \label{Tv} + +where :math:`c_v=(1/\epsilon) -1` and :math:`\epsilon` is the ratio of +the molecular weights of water vapour and dry air (i.e., :math:`\epsilon += M_v/M_a \approx 0.62198`). The buoyancy flux is then given by + +.. math:: \overline{w'b}= \frac{g}{T_v}\, \overline{w'T_v'} + +Linearising gives + +.. math:: + + \overline{w'b}= g \left( \beta_T \overline{w'T_L'} + \beta_q \overline{w'q_t'}+ + \left( \beta_T \frac{L}{c_p} - \frac{1+c_v}{c_v} \beta_q \right) + \overline{w'q_{\ell}'} \right) + +where the buoyancy parameters are given by + +.. math:: \beta_T = \frac{1}{T}, \qquad \beta_q = \frac{c_v}{V_{fac}} + +In saturated cloudy air (see, for example, +:raw-latex:`\cite{stage1981}`), the Clausius-Clapeyron equation can be +used to calculate :math:`\overline{w'q_{\ell}'}` (via +:math:`q_{\ell}' = q_t' - q_s' = q_t' - \alpha_L T'`) as + +.. math:: \overline{w'q_{\ell}'} = a_L( \overline{w'q_t'}- \alpha_L \overline{w'T_L'} ) + +where + +.. math:: + + \alpha_L = \frac{\partial q_s}{\partial T} = \frac{\epsilon L q_s(T,p) }{R T^2}, \qquad + a_L = \frac{1}{1+L\alpha_L/c_p} + +where R is the gas constant (:math:`=287.05`). Thus the buoyancy flux +can be written + +.. math:: + + \overline{w'b}= + \begin{cases} + g \left( \beta_T \overline{w'T_L'} + \beta_q \overline{w'q_t'}\right) + & {\rm in\ unsaturated\ air} \\ + g \left( \tilde{\beta_T} \overline{w'T_L'} + \tilde{\beta_q} \overline{w'q_t'} + \right) & {\rm in\ saturated\ air} + \end{cases} + +where + +.. math:: + + \begin{aligned} + \tilde{\beta_T} = \beta_T - \alpha_L \beta_c, & + \tilde{\beta_q} & = \beta_q + \beta_c \\ + {\rm and} & + \beta_c & = a_L \left( \frac{L}{c_p} \beta_T + - \frac{1+c_v}{c_v} \beta_q \right) + \end{aligned} + +Note that here :math:`\tilde{\beta_T}` and :math:`\tilde{\beta_q}` are +strictly *in*-cloud parameters, while their definitions in boundary +layer code prior to 8A were grid-box mean. Thus, here, any necessary +:math:`C_F`-weighting must be included explicitly, as in +(`[eq:wb_cont] <#eq:wb_cont>`__). + +In all the above, if :math:`T` is less than the melting point of ice +then the latent heat of sublimation, :math:`L_s = L + L_f`, is used in +place of :math:`L`. + +.. _`app:mixratio`: + +Appendix: changing between specific humidities and mixing ratios +================================================================ + +Denote wet density by + +.. math:: \rho = \rho_y + \rho_v + \rho_{\ell}+ \rho_{f} + +where :math:`\rho_y` is the density of dry air and the other +:math:`\rho` are vapour, liquid and frozen water respectively. Mixing +ratios and specific humidities are then defined as + +.. math:: m_v = \frac{\rho_v}{\rho_y} q_v = \frac{\rho_v}{\rho} + +When specific quantities are mixed, the turbulent diffusion equations, +(`[cons_eqn_scal] <#cons_eqn_scal>`__) and +(`[cons_eqn_uv] <#cons_eqn_uv>`__), have :math:`\rho` as the wet +density. This is then consistent with the conservation of globally +integrated quantities such as moisture. For example, neglecting +spherical geometry for simplicity: + +.. math:: + + \int (\rho_v + \rho_{\ell}+ \rho_{f})\, d\underline{x} = + \int \rho (q_v + q_{\ell}+ q_{f}) \,d\underline{x} = \int \rho q_t \, d\underline{x} + \label{moisture_cons} + +When mixing ratios are used, the momentum equations, +(`[cons_eqn_uv] <#cons_eqn_uv>`__), remain unchanged and the wet density +still appears. This makes the reasonable assumption that all moisture +components should be included in the momentum budget. For moisture +conservation, (`[moisture_cons] <#moisture_cons>`__) can be rewritten in +terms of mixing ratios as + +.. math:: + + \int (\rho_v + \rho_{\ell}+ \rho_{f})\, d\underline{x} = + \int \rho_y (m_v + m_{\ell}+ m_{f}) \,d\underline{x} = \int \rho_y m_t \, d\underline{x} + +Thus :math:`\rho` in (`[cons_eqn_scal] <#cons_eqn_scal>`__) is replaced +with :math:`\rho_y` for :math:`\chi = m_t` and :math:`\theta_{\ell}` +when mixing ratios are passed into the boundary layer code. + +For surface exchange, JULES initially approximates the surface air +density as :math:`\rho_* = p_S/(R T_S)` (where the subscript :math:`S` +denotes the surface values and :math:`R` the gas constant for dry air, +287 JK\ :math:`^{-1}`\ kg\ :math:`^{-1}`). If a more accurate +calculation of surface air density is requested, following Eqs (1) to +(6) of :raw-latex:`\cite{Webbetal1980}` we then calculate the wet or dry +surface air densities (to be used when the atmospheric humidity is +specific or mixing ratio, respectively) as: + +.. math:: + + \begin{aligned} + \rho_{0} & =& \rho_*/(1+(1/\epsilon-1)q_S) \\ + \rho_{y0} & =& \rho_*/(1+(1/\epsilon)m_{vS}) + \end{aligned} + +where, in each case, the surface humidity is taken as the surface +saturated humidity over open sea but over land and ice surfaces this is +likely to be inappropriate and so the driving level humidity is used +(typically the lowest model level). + +Note that :math:`\theta_{\ell}` itself is defined in terms of mixing +ratios as: + +.. math:: + + \theta_{\ell}= T - \frac{L_c}{c_{pd}} m_{\ell} + - \frac{L_c+L_f}{c_{pd}} m_f + \frac{g}{c_{pd}} z + \label{sl_defn} + +For saturation calculations a version of QSAT is used that is switchable +between input specific and mixing ratio variables. The rate of change of +:math:`q_s` with temperature is also used in the boundary layer code +(see e.g. appendix `12 <#app:buoyp>`__): + +.. math:: \frac{ d q_{sat} }{ dT } = \frac{\epsilon L q_{sat} }{RT^2} + +In fact this expression should really be converted to work for specific +quantities and so simply changing to mixing ratios will improve the +accuracy of this calculation. + +Finally, in appendix `12 <#app:buoyp>`__ virtual temperature is defined +in terms of specific variables as + +.. math:: T_v = T ( 1 + c_v q_v - q_l ) + +In terms of mixing ratios this becomes + +.. math:: T_v = \frac{T ( 1 + m_v/\epsilon )}{ 1 + m_v + m_l } + +Linearising, however, gives + +.. math:: T_v = T ( 1 + c_v m_v - m_l ) + +Thus, the same level of approximation as is currently used is maintained +simply by changing specific variables to mixing ratios. + +Additional points to note are: + +#. the diagnostic of screen humidity, :math:`q`\ 1.5m, will remain as a + specific humidity. Similarly RH1.5m will remain defined in terms of + specific quantities, i.e., :math:`q`\ 1.5m\ :math:`/q_s`\ 1.5m + +#. all other moisture diagnostics (e.g., latent heat fluxes and + increments) will simply switch to being mixing ratios if mixing + ratios are selected - no conversion will be made. + +#. it is important to note that RHOKM will be wet density times + :math:`K_m` while RHOKH will be dry density times :math:`K_h` + +.. _`app:fricheat`: + +Appendix: including the heating from turbulence dissipation +=========================================================== + +An estimate of the true molecular dissipation rate, +:math:`\epsilon_{mol}`, can be obtained by assuming local equilibrium in +the budget of subgrid TKE (SKE). Then the sum of the inputs from +resolved kinetic energy, plus that from subgrid buoyancy effects, must +equal the dissipation. The SKE budget is + +.. math:: + + d SKE/dt = S + T + B + \epsilon_{mol} + \label{ske_budg} + +where the shear production, S, is essentially the resolved KE +dissipation term. Note that the buoyancy term B appears in +(`[ske_budg] <#ske_budg>`__) which indicates that some of the energy +from resolved scale dissipation (i.e. S) should be consumed in doing +work against buoyancy (at least in stable BLs) thus leaving less energy +to be finally dissipated as heat. Note though that CBLs will generate +additional dissipation (and hence heating) through the buoyancy term. +Note that from an atmospheric budget viewpoint the transport term, T, +can be neglected as it will integrate vertically to zero. Locally, +vertical variations could be important but it will be ignored because +finally the heating source is implemented through an integral over the +BL. + +So, this estimate of the molecular dissipation rate should appear as an +additional heating source term, (following +:raw-latex:`\cite{zhang1999}`): + +.. math:: + + \frac{\partial \theta_{\ell}}{\partial t} = + \left[\frac{du}{dz}\tau_x + \frac{dv}{dz}\tau_y + B \right] / (\rho c_p) + +Tests in the SCM showed the heating rate gradients can be very large +near the surface. Hence to avoid stability problems (since this heating +increment must be added after the implicit calculation of the stress +(and heat flux) profiles) the increments are summed over the levels +within the BL (i.e. up to :math:`z_{\rm h}` ) and then that total +heating is applied as a linear decrease from the surface to zero over +:math:`z_{\rm h}` . + +.. _`app:opmods`: + +Appendix: Operational modifications +=================================== + +The operational global forecast model has been found to give improved +performance on NWP Index parameters when the following modifications to +its local :math:`Ri`-based scheme are used. In +(`[asymp_ml] <#asymp_ml>`__), the definition of :math:`\lambda_m` only +is altered to + +.. math:: \lambda_m = \mbox{max}\left[40,\, 0.3 z_{\rm loc}, 2 h_B \right] + +and both :math:`\lambda_m` and :math:`\lambda_h` are not reduced (to +40m) above the boundary layer top. It is possible these modifications +point to problems with the definition of the boundary layer depth and +the use of a Prandtl number of unity with no stability dependence in the +local scheme. Both these issues are under further investigation. + +Appendix: Inputs to UKCA +======================== + +The UKCA chemistry and aerosols sub-model takes a number of boundary +layer diagnostics as input. For a list of these and a brief explanation +of how they are used see the table below. If any changes modify the +results for these variables it will prevent UKCA jobs from regressing. +If the changes are significant it would be prudent to discuss them with +the UKCA code owner before lodging the change. + +.. container:: center + + +-------------------+------+-------------------+-------------------+ + | Boundary layer | | | | + | inputs to UKCA | | | | + +===================+======+===================+===================+ + | Sec | Item | Description | Use in UKCA | + +-------------------+------+-------------------+-------------------+ + | 0 | 24 | SURFACE | dry deposition | + | | | TEMPERATURE AFTER | | + | | | TIMESTEP | | + +-------------------+------+-------------------+-------------------+ + | 0 | 25 | BOUNDARY LAYER | dry deposition, | + | | | DEPTH AFTER | bl nucleation and | + | | | TIMESTEP | call to tr_mix | + +-------------------+------+-------------------+-------------------+ + | 0 | 26 | ROUGHNESS LENGTH | dry deposition | + | | | AFTER TIMESTEP | | + +-------------------+------+-------------------+-------------------+ + | 0 | 233 | SURFACE | dry deposition | + | | | TEMPERATURE ON | | + | | | TILES K | | + +-------------------+------+-------------------+-------------------+ + | 0 | 234 | ROUGHNESS LENGTH | dry deposition | + | | | ON TILES m | | + +-------------------+------+-------------------+-------------------+ + | 3 | 60 | RHOKH_MIX | call to tr_mix | + +-------------------+------+-------------------+-------------------+ + | 3 | 64 | D | call to tr_mix | + | | | TRDZ_CHARNEY_GRID | | + +-------------------+------+-------------------+-------------------+ + | 3 | 65 | GRID-LEVEL OF SML | call to tr_mix | + | | | INVERSION (kent) | | + +-------------------+------+-------------------+-------------------+ + | 3 | 66 | Rho \* | call to tr_mix | + | | | entrainment rate | | + | | | (we_lim) | | + +-------------------+------+-------------------+-------------------+ + | 3 | 67 | Fraction of the | call to tr_mix | + | | | timestep (t_frac) | | + +-------------------+------+-------------------+-------------------+ + | 3 | 68 | zrzi | call to tr_mix | + +-------------------+------+-------------------+-------------------+ + | 3 | 69 | GRID-LEVEL OF DSC | call to tr_mix | + | | | INVERSION (kent) | | + +-------------------+------+-------------------+-------------------+ + | 3 | 70 | Rho \* | call to tr_mix | + | | | entrainment rate | | + | | | dsc | | + +-------------------+------+-------------------+-------------------+ + | 3 | 71 | Fraction of the | call to tr_mix | + | | | timestep dsc | | + +-------------------+------+-------------------+-------------------+ + | 3 | 72 | zrzi dsc | call to tr_mix | + +-------------------+------+-------------------+-------------------+ + | 3 | 73 | ZHSC Top of | call to tr_mix | + | | | decoupled layer | | + +-------------------+------+-------------------+-------------------+ + | 3 | 217 | SURFACE HEAT FLUX | dry deposition | + | | | W/M2 | | + +-------------------+------+-------------------+-------------------+ + | 3 | 230 | 10 METRE WIND | calculate sea | + | | | SPEED ON C-GRID | salt emissions | + +-------------------+------+-------------------+-------------------+ + | 3 | 401 | Dust Emissions | GLOMAP dust | + | | | div 1 | scheme | + +-------------------+------+-------------------+-------------------+ + | 3 | 402 | Dust Emissions | GLOMAP dust | + | | | div 2 | scheme | + +-------------------+------+-------------------+-------------------+ + | 3 | 403 | Dust Emissions | GLOMAP dust | + | | | div 3 | scheme | + +-------------------+------+-------------------+-------------------+ + | 3 | 404 | Dust Emissions | GLOMAP dust | + | | | div 4 | scheme | + +-------------------+------+-------------------+-------------------+ + | 3 | 405 | Dust Emissions | GLOMAP dust | + | | | div 5 | scheme | + +-------------------+------+-------------------+-------------------+ + | 3 | 406 | Dust Emissions | GLOMAP dust | + | | | div 6 | scheme | + +-------------------+------+-------------------+-------------------+ + | 3 | 430 | Dust Friction | dry deposition | + | | | velocity (U\*) on | | + | | | tiles | | + +-------------------+------+-------------------+-------------------+ + | 3 | 462 | STOMATAL | dry deposition | + | | | CONDUCTANCE ON | | + | | | PFTS (M/S) | | + +-------------------+------+-------------------+-------------------+ + | 3 | 465 | FRICTION VELOCITY | dry deposition | + +-------------------+------+-------------------+-------------------+ + | 3 | 473 | TURBULENT KINETIC | ACTIVATE cloud | + | | | ENERGY | scheme | + +-------------------+------+-------------------+-------------------+ + +.. _`app:not`: + +Appendix: Notation +================== + +.. container:: flushleft + + +--------------------------------+------------------------------------+ + | Finite difference notation | | + +================================+====================================+ + | :math:`z_k` | height of the :math:`\theta`-level | + | | :math:`k` | + +--------------------------------+------------------------------------+ + | :math:`z_{k+\frac{1}{2}}` | height of half-level above | + | | :math:`\theta`-level :math:`k` | + +--------------------------------+------------------------------------+ + | :math:`\Delta_k` | indicates a finite difference | + | | between :math:`\theta`-levels | + | | :math:`k` and :math:`k-1` | + +--------------------------------+------------------------------------+ + | :math:`\Delta_{k+\frac{1}{2}}` | indicates a finite difference | + | | between half-levels | + | | :math:`k+\frac{1}{2}` and | + | | :math:`k-\frac{1}{2}` | + +--------------------------------+------------------------------------+ + | :math:`\Delta` | note: real change (i.e., not | + | | necessarily finite-difference) in | + | | a parameter | + +--------------------------------+------------------------------------+ + | | across the capping inversion (see | + | | (`[dbinv] <#dbinv>`__) and | + | | following text) | + +--------------------------------+------------------------------------+ + +.. container:: flushleft + + +----------------------------------+----------------------------------+ + | Model variables | | + +==================================+==================================+ + | :math:`\theta_l`, | thermodynamic variables defined | + | :math:`\theta_{v\ell}` | by (`[thetal] <#thetal>`__) and | + | | (`[thetavl] <#thetavl>`__) | + +----------------------------------+----------------------------------+ + | :math:`T_v`, :math:`\theta_v` | virtual temperature and | + | | potential temperature, | + +----------------------------------+----------------------------------+ + | | defined by (`[Tv] <#Tv>`__) and | + | | in section | + | | (`3.1.1 <#sec:parxs>`__) | + +----------------------------------+----------------------------------+ + | :math:`b` | buoyancy (:math:`=g T_v'/T_v`) | + +----------------------------------+----------------------------------+ + | :math:`q_t`, :math:`q_v`, | specific humidities: | + | :math:`q_s`, :math:`q_{\ell}`, | | + | :math:`q_f` | | + +----------------------------------+----------------------------------+ + | | total, vapour, saturated, liquid | + | | and frozen water, respectively | + +----------------------------------+----------------------------------+ + | :math:`C_F`, :math:`C_F^l`, | cloud fraction and the liquid | + | :math:`C_F^f` | and frozen water parts, | + | | respectively | + +----------------------------------+----------------------------------+ + | :math:`{\cal H}` | total heat flux (net radiative | + | | plus turbulent, | + | | Kms\ :math:`^{-1}`) | + +----------------------------------+----------------------------------+ + +.. container:: flushleft + + +----------------------------------+----------------------------------+ + | Thresholds | | + +==================================+==================================+ + | :math:`C_t` | (:math:`=1.1`) threshold for | + | | ratio of layer | + | | :math:`q_t`-gradients in cumulus | + | | diagnosis | + +----------------------------------+----------------------------------+ + | :math:`\Gamma_{\rm inv}` | (:math:`=1.1`) threshold on | + | | ratio of environment to parcel | + | | :math:`\theta_v` gradients | + +----------------------------------+----------------------------------+ + | | for identifying capping | + | | inversions above the LCL | + +----------------------------------+----------------------------------+ + | SC_CFTOL | (:math:`=0.1`) :math:`C_F` | + | | threshold for recognising the | + | | presence of Sc | + +----------------------------------+----------------------------------+ + | :ma | threshold (in Km\ :math:`^{-1}`) | + | th:`\Delta_{k_{ct}} \theta_{v\el | for initial diagnosis of | + | l}/ \Delta_{k_{ct}} z < 10^{-3}` | *well-mixed* DSC layers | + +----------------------------------+----------------------------------+ + | :math:`D_t` | (:math:`=0.1`) threshold for the | + | | ratio of buoyancy consumption to | + | | production | + +----------------------------------+----------------------------------+ + | | before decoupling occurs | + +----------------------------------+----------------------------------+ + +.. container:: flushleft + + +----------------------------------+----------------------------------+ + | Layer definitions and parameters | | + +==================================+==================================+ + | SML | surface-based mixed layer | + +----------------------------------+----------------------------------+ + | NTML | top :math:`\theta`-level within | + | | SML | + +----------------------------------+----------------------------------+ + | NTPAR | top :math:`\theta`-level reached | + | | by parcel ascent | + +----------------------------------+----------------------------------+ + | DSC | decoupled stratocumulus (mixed | + | | layer) | + +----------------------------------+----------------------------------+ + | NTDSC | top :math:`\theta`-level within | + | | DSC layer | + +----------------------------------+----------------------------------+ + | NBDSC | bottom :math:`\theta`-level | + | | within DSC layer | + +----------------------------------+----------------------------------+ + | NTLOC | top :math:`\theta`-level below | + | | which :math:`Ri<1` | + +----------------------------------+----------------------------------+ + | :math:`z_{\rm h}`  | height of top of SML | + | | (potentially subgrid) | + +----------------------------------+----------------------------------+ + | :math:`z_{\rm h}^{\rm Sc}`  | height of top of DSC layer | + | | (potentially subgrid) | + +----------------------------------+----------------------------------+ + | :math:`z_{\rm b}`  | height of base of DSC layer | + | | (subgrid) | + +----------------------------------+----------------------------------+ + | :math:`z_{\rm par}`  | height of half-level at top of | + | | parcel ascent | + +----------------------------------+----------------------------------+ + | :math:`z_{\rm loc}`  | height of half-level marking | + | | ‘top’ of local :math:`Ri`-based | + | | mixing | + +----------------------------------+----------------------------------+ + | | (where :math:`Ri>1`) | + +----------------------------------+----------------------------------+ + | :math:`z_i` | generic inversion height | + +----------------------------------+----------------------------------+ + | :math:`z_c` | cloud depth | + +----------------------------------+----------------------------------+ + | :math:`z_{\rm ml}` | mixed layer depth | + +----------------------------------+----------------------------------+ + | :math:`K_m^{\rm surf}`, | :math:`K` profiles for | + | :math:`K_h^{\rm surf}` | surface-driven turbulence (in | + | | SML) | + +----------------------------------+----------------------------------+ + | :math:`K_m^{\rm Sc}`, | :math:`K` profiles for | + | :math:`K_h^{\rm Sc}` | cloud-top-driven turbulence | + +----------------------------------+----------------------------------+ + | | (calculated for both DSC and | + | | SML) | + +----------------------------------+----------------------------------+ + | LCL | lifting condensation level | + +----------------------------------+----------------------------------+ + +.. container:: flushleft + + +----------------------------------+----------------------------------+ + | Other parameters | | + +==================================+==================================+ + | :math:`\gamma_{\theta_{\ell}}` | gradient adjustment term, given | + | | by (`[gradadj] <#gradadj>`__) | + +----------------------------------+----------------------------------+ + | :math:`w_m` | scaling velocity for momentum | + | | mixing in the SML | + +----------------------------------+----------------------------------+ + | | (used in :math:`K_m^{\rm surf}`, | + | | :math:`\gamma_{\theta_{\ell}}` | + | | and the SML parcel perturbation, | + | | :math:`\theta_v'`) | + +----------------------------------+----------------------------------+ + | :math:`w_*` | ‘standard’ convective velocity | + | | scale for a cloud-free | + | | convective | + +----------------------------------+----------------------------------+ + | | boundary layer, | + | | :math:`w_ | + | | *^3 = z_{\rm h}\overline{w'b}_S` | + +----------------------------------+----------------------------------+ + | :math:`u_*` | friction velocity (here includes | + | | the orographic component) | + +----------------------------------+----------------------------------+ + | :math:`w_e`, :math:`\tilde{w_e}` | entrainment velocity and | + | | compensated to allow for | + | | subsidence (ms\ :math:`^{-1}`) | + +----------------------------------+----------------------------------+ + | :math:`w_S` | subsidence velocity | + | | (ms\ :math:`^{-1}`) | + +----------------------------------+----------------------------------+ + | :math:`\Delta_F` | cloud-top net radiative | + | | divergence, calculation given in | + | | (`[ctraddiv] <#ctraddiv>`__) | + +----------------------------------+----------------------------------+ + | :math:`\alpha_t` | parameter in entrainment | + | | parametrization, | + | | (`[we_parm] <#we_parm>`__) | + +----------------------------------+----------------------------------+ + | :math:`\tau_{rc}`, | parameters in perturbation | + | :math:`z_{rc}` | calculation, | + | | (`[dscd_pert] <#dscd_pert>`__), | + +----------------------------------+----------------------------------+ + | | for initial identification of | + | | and :math:`z_{\rm ml}` | + | | calculation for DSC layers | + +----------------------------------+----------------------------------+ + | :math:`a_L`, :math:`\alpha_L`, | buoyancy parameters, defined in | + | :math:`\beta_T`, | appendix `12 <#app:buoyp>`__ | + | :math:`\beta_q`, | | + | :math:`\tilde{\beta_T}`, | | + | :math:`\tilde{\beta_q}` | | + +----------------------------------+----------------------------------+ + +.. [1] + unless the option to mix across the LCL is selected, see + section `3.4 <#sec:lclmixing>`__ + +.. [2] + unless the option to mix across the LCL is selected, see + section `3.4 <#sec:lclmixing>`__ + +.. [3] + If i_impsolve_loc = 1, the boundary-layer implicit solver is + performed before the convection call so that :math:`S` excludes the + convection increments. If i_impsolve_loc = 2, it is performed after + the convection call. There are pros and cons to each option. Calling + the implicit solver before convection reduces the accuracy of the + final mixed-layer profile, since convection may alter the mixed-layer + gradients afterwards. On the other hand, allowing convection to act + on a state which includes the heating and moistening by + surface-fluxes over the current timestep may improve the accuracy of + the convective closure. Also the non-turbulent fluxes used to + construct the budgets at entrainment grid-levels (section + `5.5 <#sec:rev_flux_grad>`__) do not include contributions from + convection, so arguably excluding them from :math:`S` is consistent. + In the presence of convective subsidence, the top grid-level of the + sub-cloud mixed layer gets warmed and dried by the subsidence + (consistent with a lowering of the mixed-layer top). However if the + implicit solver is called after convection it does not account for + this lowering, so that all the subsided air is forced to be entrained + into the mixed-layer. + +.. |image1| image:: new_ktop_shape +.. |image| image:: honnert_vs_tanh.eps +.. |image2| image:: zturb_schem.eps +.. |image3| image:: div_r080 + :width: 3.5in +.. |image4| image:: div_r071 + :width: 3.5in From 159e998536f5b7e1a88a1aecdd7c77aa231f6e92 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Thu, 23 Apr 2026 13:52:02 +0100 Subject: [PATCH 044/116] Fixed aligned equation regions. --- .../turbulence_schemes/bldoc.rst | 1232 ++++++++++------- 1 file changed, 751 insertions(+), 481 deletions(-) diff --git a/documentation/source/science_guide/turbulence_schemes/bldoc.rst b/documentation/source/science_guide/turbulence_schemes/bldoc.rst index 2e403938d7..cc45abdd4c 100644 --- a/documentation/source/science_guide/turbulence_schemes/bldoc.rst +++ b/documentation/source/science_guide/turbulence_schemes/bldoc.rst @@ -44,16 +44,18 @@ horizontal components of momentum, :math:`{\bf u}` on a sphere gives: .. math:: - \begin{aligned} - \frac{\partial \chi}{\partial t} - &=& - \frac{1}{r^2 \rho} \, \frac{\partial }{\partial z} \left( r^2 \rho \overline{w'\chi'} \right) - + {\cal S} - \label{cons_eqn_scal} \\ - \frac{\partial {\bf u}}{\partial t} - &=& \frac{1}{r^2 \rho} \, \frac{\partial }{\partial z} \left( r^2 {\bf \tau} \right) - + {\cal S} - \label{cons_eqn_uv} - \end{aligned} + \frac{\partial \chi}{\partial t} + = - \frac{1}{r^2 \rho} \, \frac{\partial }{\partial z} \left( r^2 \rho \overline{w'\chi'} \right) + + {\cal S} + \label{cons_eqn_scal} + +.. math:: + + \frac{\partial {\bf u}}{\partial t} + = \frac{1}{r^2 \rho} \, \frac{\partial }{\partial z} \left( r^2 {\bf \tau} \right) + + {\cal S} + \label{cons_eqn_uv} + where :math:`\overline{w'\chi'}` and :math:`{\bf \tau}` are the vertical turbulent fluxes to be parametrized, :math:`r` is the height from the @@ -63,11 +65,13 @@ approximately conserved under moist adiabatic ascent, are: .. math:: - \begin{aligned} - \theta_{\ell}&=& T_L + \frac{g}{c_p} z = T - \frac{L}{c_p} q_{\ell} - - \frac{L_s}{c_p} q_f + \frac{g}{c_p} z \label{thetal} \\ - q_t &=& q_v + q_{\ell}+ q_f \label{qt} - \end{aligned} + \theta_{\ell}= T_L + \frac{g}{c_p} z = T - \frac{L}{c_p} q_{\ell} + - \frac{L_s}{c_p} q_f + \frac{g}{c_p} z \label{thetal} + +.. math:: + + q_t = q_v + q_{\ell}+ q_f \label{qt} + where :math:`T` is temperature, :math:`q_v` is specific humidity, :math:`q_{\ell}` and :math:`q_f` the specific liquid and frozen water @@ -112,12 +116,14 @@ alternative methodology is optionally available, see section .. math:: - \begin{aligned} - \overline{w'\chi'} &=& - K_h \frac{\partial \chi}{\partial z} + K_h^{\rm surf}\gamma_{\chi} - \label{scal_closure} \\ - {\bf \tau} &=& K_m \frac{\partial {\bf u}}{\partial z} + {\bf \tau}^{nl} - \label{uv_closure} - \end{aligned} + \overline{w'\chi'} = - K_h \frac{\partial \chi}{\partial z} + K_h^{\rm surf}\gamma_{\chi} + \label{scal_closure} + +.. math:: + + {\bf \tau} = K_m \frac{\partial {\bf u}}{\partial z} + {\bf \tau}^{nl} + \label{uv_closure} + Separate eddy-diffusivities are calculated for momentum, :math:`K_m`, and for scalar variables, :math:`K_h`. The second term on the right hand @@ -533,13 +539,15 @@ expanded using the first-order closure in .. math:: - \begin{aligned} - \overline{w'\theta_{\ell}'}_k &=& -K_h^{\rm surf}\,\frac{\widetilde{\Delta_k \theta_{\ell}}}{\Delta_k z} - -K_h^{\rm Sc}\,\frac{\Delta_k \theta_{\ell}}{\Delta_k z} \nonumber \\ - \overline{w'q_t'}_k &=& -\left(K_h^{\rm surf}+ K_h^{\rm Sc}\right) \, - \,\frac{\Delta_k q_t}{\Delta_k z} - \label{eq:wx_std} - \end{aligned} + \overline{w'\theta_{\ell}'}_k = -K_h^{\rm surf}\,\frac{\widetilde{\Delta_k \theta_{\ell}}}{\Delta_k z} + -K_h^{\rm Sc}\,\frac{\Delta_k \theta_{\ell}}{\Delta_k z} + +.. math:: + + \overline{w'q_t'}_k = -\left(K_h^{\rm surf}+ K_h^{\rm Sc}\right) \, + \,\frac{\Delta_k q_t}{\Delta_k z} + \label{eq:wx_std} + where :math:`\widetilde{\Delta_k \theta_{\ell}} = \Delta_k \theta_{\ell}- @@ -707,14 +715,15 @@ Then, .. math:: - \begin{aligned} - \int_{z_h-\Delta z_{rad}}^{z_h} \, \overline{w'\theta_{\ell}'}\, dz & = & - \int_{z_h-\Delta z_{rad}}^{z_h} \, F_{\theta_{\ell}}^{Tot} - - F_{\theta_{\ell}}^{NT}\, dz - \nonumber \\ - & = & I^{Tot} - I^{rad} - I^{ppn} - \label{wthl_int} - \end{aligned} + \int_{z_h-\Delta z_{rad}}^{z_h} \, \overline{w'\theta_{\ell}'}\, dz = + \int_{z_h-\Delta z_{rad}}^{z_h} \, F_{\theta_{\ell}}^{Tot} - + F_{\theta_{\ell}}^{NT}\, dz + +.. math:: + + = I^{Tot} - I^{rad} - I^{ppn} + \label{wthl_int} + For the radiative flux, it could be assumed that the subgrid flux distribution is exponentially dependent on the grid-level LWP, for @@ -745,12 +754,14 @@ approximations, (`[wthl_int] <#wthl_int>`__) becomes .. math:: - \begin{aligned} - \int_{z_h-\Delta z_{rad}}^{z_h} \, \overline{w'\theta_{\ell}'}\, dz & = & - \Delta z_{rad} \left(-w_e \Delta \theta_{\ell}+ \Delta F\right) - - \Delta z_{rad} \Delta F/ 3 \\ - & = & \Delta z_{rad} \left(-w_e \Delta \theta_{\ell}+ \frac{2}{3} \Delta F\right) - \end{aligned} + \int_{z_h-\Delta z_{rad}}^{z_h} \, \overline{w'\theta_{\ell}'}\, dz = + \Delta z_{rad} \left(-w_e \Delta \theta_{\ell}+ \Delta F\right) + - \Delta z_{rad} \Delta F/ 3 + +.. math:: + + = \Delta z_{rad} \left(-w_e \Delta \theta_{\ell}+ \frac{2}{3} \Delta F\right) + For the integral of :math:`\overline{w'q_t'}` across this cloud-top region, :math:`\overline{w'q_t'}` is also taken to be constant so that: @@ -862,11 +873,13 @@ A first order ‘mixing length’ closure is used: .. math:: - \begin{aligned} - K_m &=& {\cal L}_m^2 \, (S+S_d) \, f_m(Ri) \label{kmlocal}\\ - K_h &=& {\cal L}_h \, {\cal L}_m \, - (S+S_d) \, f_h(Ri) \label{khlocal} - \end{aligned} + K_m = {\cal L}_m^2 \, (S+S_d) \, f_m(Ri) \label{kmlocal} + +.. math:: + + K_h = {\cal L}_h \, {\cal L}_m \, + (S+S_d) \, f_h(Ri) \label{khlocal} + where :math:`{\cal L}_m` and :math:`{\cal L}_h` are the neutral mixing lengths and :math:`S` is the resolved vertical shear of the horizontal @@ -880,10 +893,12 @@ ignored above grid-level 2 and the neutral mixing lengths are given by .. math:: - \begin{aligned} - {\cal L}_m &=& \frac{k(z+z_{0m})}{1+k(z+z_{0m})/\lambda_m} \\ - {\cal L}_h &=& \frac{k(z+z_{0m})}{1+k(z+z_{0m})/\lambda_h} - \end{aligned} + {\cal L}_m = \frac{k(z+z_{0m})}{1+k(z+z_{0m})/\lambda_m} + +.. math:: + + {\cal L}_h = \frac{k(z+z_{0m})}{1+k(z+z_{0m})/\lambda_h} + where :math:`z_{0m}` includes the orographic component. For the lowest interior grid-level (:math:`k=1`) they are calculated, incorporating @@ -902,11 +917,13 @@ The asymptotic mixing lengths are given by .. math:: - \begin{aligned} - \lambda_m &=&\mbox{max}\left[\lambda_0,\, 0.15 z_{\rm loc}, 2 h_B \right] \nonumber\\ - \lambda_h &=&\mbox{max}\left[\lambda_0,\, 0.15 z_{\rm loc}\right] - \label{asymp_ml} - \end{aligned} + \lambda_m =\mbox{max}\left[\lambda_0,\, 0.15 z_{\rm loc}, 2 h_B \right] + +.. math:: + + \lambda_h =\mbox{max}\left[\lambda_0,\, 0.15 z_{\rm loc}\right] + \label{asymp_ml} + where :math:`\lambda_0` is a minimum mixing length read in from the namelist and :math:`z_{\rm loc}` is defined below. The orographic @@ -995,12 +1012,14 @@ For :math:`Ri < 0`, the standard UM stability functions are given by .. math:: - \begin{aligned} - f_m & =& 1 - \frac{g_0 \,Ri} - {1+D_m(\tilde{{\cal L}}_m/\tilde{{\cal L}}_h)|Ri|^{1/2} } \nonumber\\ - f_h & =& \frac{1}{Pr_N}\left(1 - \frac{g_0 \,Ri} - {1+D_h(\tilde{{\cal L}}_m/\tilde{{\cal L}}_h)|Ri|^{1/2} }\right) - \end{aligned} + f_m = 1 - \frac{g_0 \,Ri} + {1+D_m(\tilde{{\cal L}}_m/\tilde{{\cal L}}_h)|Ri|^{1/2} } + +.. math:: + + f_h = \frac{1}{Pr_N}\left(1 - \frac{g_0 \,Ri} + {1+D_h(\tilde{{\cal L}}_m/\tilde{{\cal L}}_h)|Ri|^{1/2} }\right) + with :math:`g_0=10`, :math:`D_m=g_0/4` and :math:`D_h=g_0/25`. If the stability dependent Prandtl number option is chosen (see below) the @@ -1010,10 +1029,12 @@ model (LEM), :raw-latex:`\cite{brown1999}`: .. math:: - \begin{aligned} - f_m & =& (1 - c_{LEM} Ri)^{1/2} \nonumber\\ - f_h & =& \frac{1}{Pr_N}\left(1 - b_{LEM} Ri\right)^{1/2} - \end{aligned} + f_m = (1 - c_{LEM} Ri)^{1/2} + +.. math:: + + f_h = \frac{1}{Pr_N}\left(1 - b_{LEM} Ri\right)^{1/2} + where :math:`Pr_N = 0.7`, and the constants :math:`b_{LEM}` and :math:`c_{LEM}` can take the values 40 and 16 respectively in the @@ -1044,10 +1065,12 @@ where .. math:: - \begin{aligned} - A_{Ri} & = & \left(1-g_0 Ri_{t}\right)/\left(1- g_0 Ri_{t}/2\right)^2 \nonumber\\ - B_{Ri} & = & (g_0/2) /\left(1 - g_0 Ri_{t}/2\right)^2 - \end{aligned} + A_{Ri} = \left(1-g_0 Ri_{t}\right)/\left(1- g_0 Ri_{t}/2\right)^2 + +.. math:: + + B_{Ri} = (g_0/2) /\left(1 - g_0 Ri_{t}/2\right)^2 + For the ‘SHARPEST’ function of :raw-latex:`\cite{derbyshire1997}`, :math:`Ri_{t}=0.1`, while larger values give even sharper reduction of @@ -1067,10 +1090,12 @@ for :math:`Ri>0` are then given by: .. math:: - \begin{aligned} - f_m & =& \frac{Pr}{Pr_N} \, f_{\rm stable} \\ - f_h & =& \frac{1}{Pr_N} \, f_{\rm stable} - \end{aligned} + f_m = \frac{Pr}{Pr_N} \, f_{\rm stable} + +.. math:: + + f_h = \frac{1}{Pr_N} \, f_{\rm stable} + Note that writing the functions in this way ensures that :math:`f_m=1` under neutral conditions and the effect of the variation in :math:`Pr` @@ -1083,10 +1108,12 @@ turbulence beyond a critical Richardson number, :math:`Ri_c=0.25`: .. math:: - \begin{aligned} - f_m & =& \left( 1 - \frac{Ri}{Ri_c} \right)^4 \\ - f_h & =& \frac{1}{Pr_N} \left( 1 - \frac{Ri}{Ri_c} \right)^4 (1 - g_{LEM} Ri) - \end{aligned} + f_m = \left( 1 - \frac{Ri}{Ri_c} \right)^4 + +.. math:: + + f_h = \frac{1}{Pr_N} \left( 1 - \frac{Ri}{Ri_c} \right)^4 (1 - g_{LEM} Ri) + with :math:`g_{LEM}=1.2`. @@ -2159,10 +2186,12 @@ F|_{z_i}`, so that .. math:: - \begin{aligned} - {\cal H}|_{z_i} & =& - w_e \Delta \theta_{\ell}+ F_{\rm net}|_h \nonumber\\ - \overline{w'q_t'}_{z_i}& =& - w_e \Delta q_t - \end{aligned} + {\cal H}|_{z_i} = - w_e \Delta \theta_{\ell}+ F_{\rm net}|_h + +.. math:: + + \overline{w'q_t'}_{z_i} = - w_e \Delta q_t + where the total heat flux :math:`{\cal H} = \overline{w'\theta_{\ell}'}+ F_{\rm net}` and @@ -2193,15 +2222,17 @@ base of the mixed layer: .. math:: - \begin{aligned} - \overline{w'\theta_{\ell}'}|_{ z_{\mbox{\tiny \rm NTDSC}+\frac{1}{2}} } & =& \overline{w'\theta_{\ell}'}|_{z_{\rm b}} - - \frac{ z'_{\mbox{\tiny \rm NTDSC}+\frac{1}{2}} }{z_{\rm ml}} - \left( \tilde{w_e} \Delta \theta_{\ell}+ \overline{w'\theta_{\ell}'}|_{z_{\rm b}} - F_{\rm net}|_{h} \right) - - F_{\rm net}|_{ z_{\mbox{\tiny \rm NTDSC}+\frac{1}{2}} } \nonumber\\ - \overline{w'q_t'}|_{ z_{\mbox{\tiny \rm NTDSC}+\frac{1}{2}} } & =& \overline{w'q_t'}|_{z_{\rm b}} - - \frac{ z'_{\mbox{\tiny \rm NTDSC}+\frac{1}{2}} }{z_{\rm ml}} - \left( \tilde{w_e} \Delta q_t + \overline{w'q_t'}|_{z_{\rm b}} \right) - \end{aligned} + \overline{w'\theta_{\ell}'}|_{ z_{\mbox{\tiny \rm NTDSC}+\frac{1}{2}} } = \overline{w'\theta_{\ell}'}|_{z_{\rm b}} + - \frac{ z'_{\mbox{\tiny \rm NTDSC}+\frac{1}{2}} }{z_{\rm ml}} + \left( \tilde{w_e} \Delta \theta_{\ell}+ \overline{w'\theta_{\ell}'}|_{z_{\rm b}} - F_{\rm net}|_{h} \right) + - F_{\rm net}|_{ z_{\mbox{\tiny \rm NTDSC}+\frac{1}{2}} } + +.. math:: + + \overline{w'q_t'}|_{ z_{\mbox{\tiny \rm NTDSC}+\frac{1}{2}} } = \overline{w'q_t'}|_{z_{\rm b}} + - \frac{ z'_{\mbox{\tiny \rm NTDSC}+\frac{1}{2}} }{z_{\rm ml}} + \left( \tilde{w_e} \Delta q_t + \overline{w'q_t'}|_{z_{\rm b}} \right) + where :math:`z' = z-z_{\rm b}`, and similarly for the SML entrainment fluxes (at :math:`z=z_{\mbox{\tiny \rm NTML}+\frac{1}{2}}`). The @@ -2335,20 +2366,25 @@ The coefficients are given by .. math:: - \begin{aligned} - a & =& 0.5 (\gamma^{\scriptsize \rm FA}- \gamma^{\tiny \rm ML}) \\ - b & =& - \left( {\theta_{v\ell}}_{\mbox{\tiny \rm NTML}+2} - - \gamma^{\scriptsize \rm FA}(z_{\mbox{\tiny \rm NTML}+2}-z_{\mbox{\tiny \rm NTML}+\frac{3}{2}}) \right) - + \left( {\theta_{v\ell}}_{\mbox{\tiny \rm NTML}} - + \gamma^{\tiny \rm ML}(z_{\mbox{\tiny \rm NTML}+\frac{3}{2}}-z_{\mbox{\tiny \rm NTML}}) \right) \\ - c & =& (z_{\mbox{\tiny \rm NTML}+\frac{3}{2}}-z_{\mbox{\tiny \rm NTML}+\frac{1}{2}}) - \left( {\theta_{v\ell}}_{\mbox{\tiny \rm NTML}+1} - - \left( {\theta_{v\ell}}_{\mbox{\tiny \rm NTML}} + - \gamma^{\tiny \rm ML}\left( - \frac{1}{2}(z_{\mbox{\tiny \rm NTML}+\frac{1}{2}}+z_{\mbox{\tiny \rm NTML}+\frac{3}{2}}) - -z_{\mbox{\tiny \rm NTML}} \right) \right) - \right) \\ - \end{aligned} + a = 0.5 (\gamma^{\scriptsize \rm FA}- \gamma^{\tiny \rm ML}) + +.. math:: + + b = - \left( {\theta_{v\ell}}_{\mbox{\tiny \rm NTML}+2} + - \gamma^{\scriptsize \rm FA}(z_{\mbox{\tiny \rm NTML}+2}-z_{\mbox{\tiny \rm NTML}+\frac{3}{2}}) \right) + + \left( {\theta_{v\ell}}_{\mbox{\tiny \rm NTML}} + + \gamma^{\tiny \rm ML}(z_{\mbox{\tiny \rm NTML}+\frac{3}{2}}-z_{\mbox{\tiny \rm NTML}}) \right) + +.. math:: + + c = (z_{\mbox{\tiny \rm NTML}+\frac{3}{2}}-z_{\mbox{\tiny \rm NTML}+\frac{1}{2}}) + \left( {\theta_{v\ell}}_{\mbox{\tiny \rm NTML}+1} - + \left( {\theta_{v\ell}}_{\mbox{\tiny \rm NTML}} + + \gamma^{\tiny \rm ML}\left( + \frac{1}{2}(z_{\mbox{\tiny \rm NTML}+\frac{1}{2}}+z_{\mbox{\tiny \rm NTML}+\frac{3}{2}}) + -z_{\mbox{\tiny \rm NTML}} \right) \right) + \right) + Clearly, care must be taken to ensure that :math:`z_i` is not only well-defined but also sensible (for example, as a rising inversion @@ -2565,16 +2601,18 @@ mixed layer by the end of the timestep. In other words, for .. math:: - \begin{aligned} - \chi_{\mbox{\tiny \rm NTML}+1}^{n+1} & =& \chi_{\mbox{\tiny \rm NTML}+1}^{n} - - \frac{\Delta t}{\Delta z} \left( - F_{\chi}^{Tot}|_{ \mbox{\tiny \rm NTML}+\frac{3}{2} } - F_{\chi}^{Tot}|_{ \mbox{\tiny \rm NTML}+\frac{1}{2} } - \right) \\ - \chi_{\mbox{\tiny \rm NTML}}^{n+1} & =& \chi_{\mbox{\tiny \rm NTML}}^{n} - - \frac{\Delta t}{z_{\mbox{\tiny \rm NTML}+\frac{1}{2}}} \left( - F_{\chi}^{Tot}|_{ \mbox{\tiny \rm NTML}+\frac{1}{2} } - F_{\chi}^{Tot}|_{z_{\rm b}} - \right) \\ - \end{aligned} + \chi_{\mbox{\tiny \rm NTML}+1}^{n+1} = \chi_{\mbox{\tiny \rm NTML}+1}^{n} + - \frac{\Delta t}{\Delta z} \left( + F_{\chi}^{Tot}|_{ \mbox{\tiny \rm NTML}+\frac{3}{2} } - F_{\chi}^{Tot}|_{ \mbox{\tiny \rm NTML}+\frac{1}{2} } + \right) + +.. math:: + + \chi_{\mbox{\tiny \rm NTML}}^{n+1} = \chi_{\mbox{\tiny \rm NTML}}^{n} + - \frac{\Delta t}{z_{\mbox{\tiny \rm NTML}+\frac{1}{2}}} \left( + F_{\chi}^{Tot}|_{ \mbox{\tiny \rm NTML}+\frac{1}{2} } - F_{\chi}^{Tot}|_{z_{\rm b}} + \right) + where the superscripts :math:`n` and :math:`n+1` refer to the model timestep, although strictly speaking :math:`n+1` refers to fields after @@ -2633,14 +2671,16 @@ free-atmospheric lapse rates are given by .. math:: - \begin{aligned} - \gamma_{\theta_{\ell}} & =& {\rm max}\left[ \, 0, \, \frac{ {\theta_{\ell}}_{\mbox{\tiny \rm NTML}+3}-{\theta_{\ell}}_{\mbox{\tiny \rm NTML}+2} } - { z_{\mbox{\tiny \rm NTML}+3} - z_{\mbox{\tiny \rm NTML}+2} } - \right] \\ - \gamma_{q_t} & =& {\rm min}\left[ \, 0, \, \frac{ {q_t}_{\mbox{\tiny \rm NTML}+3}-{q_t}_{\mbox{\tiny \rm NTML}+2} } - { z_{\mbox{\tiny \rm NTML}+3} - z_{\mbox{\tiny \rm NTML}+2} } - \right] - \end{aligned} + \gamma_{\theta_{\ell}} = {\rm max}\left[ \, 0, \, \frac{ {\theta_{\ell}}_{\mbox{\tiny \rm NTML}+3}-{\theta_{\ell}}_{\mbox{\tiny \rm NTML}+2} } + { z_{\mbox{\tiny \rm NTML}+3} - z_{\mbox{\tiny \rm NTML}+2} } + \right] + +.. math:: + + \gamma_{q_t} = {\rm min}\left[ \, 0, \, \frac{ {q_t}_{\mbox{\tiny \rm NTML}+3}-{q_t}_{\mbox{\tiny \rm NTML}+2} } + { z_{\mbox{\tiny \rm NTML}+3} - z_{\mbox{\tiny \rm NTML}+2} } + \right] + .. _`sec:subs_calc`: @@ -2694,10 +2734,12 @@ specified through an eddy diffusivity which is given by .. math:: - \begin{aligned} - K_h|_{\mbox{\tiny \rm NTML}+\frac{1}{2}} & =& w_e \Delta_{\mbox{\tiny \rm NTML}+1} z \nonumber\\ - K_m|_{\mbox{\tiny \rm NTML}} & =& Pr \, w_e \Delta_{\mbox{\tiny \rm NTML}+\frac{1}{2}} z - \end{aligned} + K_h|_{\mbox{\tiny \rm NTML}+\frac{1}{2}} = w_e \Delta_{\mbox{\tiny \rm NTML}+1} z + +.. math:: + + K_m|_{\mbox{\tiny \rm NTML}} = Pr \, w_e \Delta_{\mbox{\tiny \rm NTML}+\frac{1}{2}} z + noting the Charney-Philips grid implying stresses are staggered from scalar fluxes. The Prandtl number, :math:`Pr`, takes the same form as @@ -2837,11 +2879,16 @@ layer are related to the surface fluxes by: .. math:: - \begin{aligned} - \frac{\partial T}{\partial z} + \frac{g}{ c_P }&=&-\frac{ H_0 }{ c_P \rho _0 v_\ast } \frac{ \phi _h (z/L)}{kz}\label{1.1.1}\\ - \frac{\partial q}{\partial z}&=&-\frac{ E_0 }{ \rho _0 v_\ast } \frac{ \phi _h (z/L)}{kz}\label{1.1.2}\\ - \frac{\partial {\rm {\bf v}}}{\partial z}&=&\frac{ {\rm {\bf \tau }}_{0} }{ \rho _0 v_\ast } \frac{ \phi _m (z/L)}{kz},\label{1.1.3} - \end{aligned} + \frac{\partial T}{\partial z} + \frac{g}{ c_P }=-\frac{ H_0 }{ c_P \rho _0 v_\ast } \frac{ \phi _h (z/L)}{kz}\label{1.1.1} + +.. math:: + + \frac{\partial q}{\partial z}=-\frac{ E_0 }{ \rho _0 v_\ast } \frac{ \phi _h (z/L)}{kz}\label{1.1.2} + +.. math:: + + \frac{\partial {\rm {\bf v}}}{\partial z}=\frac{ {\rm {\bf \tau }}_{0} }{ \rho _0 v_\ast } \frac{ \phi _m (z/L)}{kz},\label{1.1.3} + where subscript 0 represents a surface value and subscript \* represents a surface layer scaling quantity. :math:`\phi _{m}` and :math:`\phi @@ -2874,11 +2921,16 @@ surface turbulent fluxes are: .. math:: - \begin{aligned} - \frac{ H_0 }{ c_P \rho _0 }&=&-\frac{ c_H }{ c_D^{1/2} } v_\ast \left( {\Delta T + \frac{g}{c_p }( z_1 + z_{0m} - z_{0h} )} \right)\label{1.1.7}\\ - \frac{ E_0 }{ \rho _0 }&=&-\frac{ c_H }{ c_D^{1/2} } v_\ast \Delta q\label{1.1.8}\\ - \frac{ {\bf \tau }_{0} }{ \rho _{0} }&=& c_D^{1/2} v_\ast \Delta {\rm {\bf v}},\label{1.1.9} - \end{aligned} + \frac{ H_0 }{ c_P \rho _0 }=-\frac{ c_H }{ c_D^{1/2} } v_\ast \left( {\Delta T + \frac{g}{c_p }( z_1 + z_{0m} - z_{0h} )} \right)\label{1.1.7} + +.. math:: + + \frac{ E_0 }{ \rho _0 }=-\frac{ c_H }{ c_D^{1/2} } v_\ast \Delta q\label{1.1.8} + +.. math:: + + \frac{ {\bf \tau }_{0} }{ \rho _{0} }= c_D^{1/2} v_\ast \Delta {\rm {\bf v}},\label{1.1.9} + where :math:`\Delta`\ X=X\ :math:`_{1}`-X\ :math:`_{0}`. From (`[1.1.7] <#1.1.7>`__) and (`[1.1.8] <#1.1.8>`__) the surface @@ -2901,21 +2953,25 @@ and c\ :math:`_{H}`, are given by .. math:: - \begin{aligned} - c_D^{1/2}&=&\frac{k}{ \Phi _m (L , z_1 + z_{0m} , z_{0m} )} - \label{1.1.12}\\ - \frac{ c_H }{ c_D^{1/2} }&=&\frac{k}{ \Phi _h (L , z_1 + z_{0m} , z_{0h} )}, - \label{1.1.13} - \end{aligned} + c_D^{1/2}=\frac{k}{ \Phi _m (L , z_1 + z_{0m} , z_{0m} )} + \label{1.1.12} + +.. math:: + + \frac{ c_H }{ c_D^{1/2} }=\frac{k}{ \Phi _h (L , z_1 + z_{0m} , z_{0h} )}, + \label{1.1.13} + where .. math:: - \begin{aligned} - \Phi _m (L , z_1 + z_{0m} , z_{0m} )&=& \int \limits_{ z_{0m} /L}^{( z_1 + z_{0m} )/L} \frac{ \phi _m (\zeta )}{\zeta } d\zeta\label{1.1.14}\\ - \Phi _h (L , z_1 + z_{0m} , z_{0h} )&=& \int \limits_{ z_{0h} /L}^{( z_1 + z_{0m} )/L} \frac{ \phi _h (\zeta )}{\zeta } d\zeta\label{1.1.15}, - \end{aligned} + \Phi _m (L , z_1 + z_{0m} , z_{0m} )= \int \limits_{ z_{0m} /L}^{( z_1 + z_{0m} )/L} \frac{ \phi _m (\zeta )}{\zeta } d\zeta\label{1.1.14} + +.. math:: + + \Phi _h (L , z_1 + z_{0m} , z_{0h} )= \int \limits_{ z_{0h} /L}^{( z_1 + z_{0m} )/L} \frac{ \phi _h (\zeta )}{\zeta } d\zeta\label{1.1.15}, + z\ :math:`_{0m}` and z\ :math:`_{0h}` are the **surface roughness lengths** for momentum and scalars respectively. @@ -2926,12 +2982,20 @@ forms .. math:: - \begin{aligned} - \frac{ H_0 }{ c_P \rho _0 }&=&{-c}_H V \left( {\Delta T + \frac{g}{c_p }( z_1 + z_{0m} - z_{0h} )} \right)\nonumber\\ - &=& - C_H \left( {\Delta T + \frac{g}{c_p } ( z_1 + z_{0m} - z_{0h} )} \right)\label{1.1.16}\\ - \frac{ E_0 }{ \rho _0 }&=&- c_H V \Delta q = - C_H \Delta q\label{1.1.17}\\ - \frac{ {\rm {\bf \tau }}_{0} }{ \rho _{0} }&=& c_D V \Delta {\rm {\bf v}}{ }= C_D \Delta {\rm {\bf v}},\label{1.1.18} - \end{aligned} + \frac{ H_0 }{ c_P \rho _0 }={-c}_H V \left( {\Delta T + \frac{g}{c_p }( z_1 + z_{0m} - z_{0h} )} \right) + +.. math:: + + = - C_H \left( {\Delta T + \frac{g}{c_p } ( z_1 + z_{0m} - z_{0h} )} \right)\label{1.1.16} + +.. math:: + + \frac{ E_0 }{ \rho _0 }=- c_H V \Delta q = - C_H \Delta q\label{1.1.17} + +.. math:: + + \frac{ {\rm {\bf \tau }}_{0} }{ \rho _{0} }= c_D V \Delta {\rm {\bf v}}{ }= C_D \Delta {\rm {\bf v}},\label{1.1.18} + where the effective wind speed for surface turbulent exchanges, :math:`V`, is defined by @@ -2943,21 +3007,25 @@ respectively .. math:: - \begin{aligned} - C_H&=&\frac{k}{ \Phi _h } v_\ast = c_H V\label{1.1.20}\\ - C_D&=&\frac{k}{ \Phi _m } v_\ast = c_D V\label{1.1.21}. - \end{aligned} + C_H=\frac{k}{ \Phi _h } v_\ast = c_H V\label{1.1.20} + +.. math:: + + C_D=\frac{k}{ \Phi _m } v_\ast = c_D V\label{1.1.21}. + The surface exchange coefficients can then be written in any of the following forms: .. math:: - \begin{aligned} - c_H&=&\frac{ C_H }{V} = \frac{ C_H C_D }{ v_\ast ^2 } = \frac{ k^2 }{ \Phi _h \Phi _m }\label{1.1.22}\\ - c_D&=&\frac{ C_D }{V} = \frac{ C_D^2 }{ v_\ast ^2 } = \frac{ k^2 }{ \Phi _m^2 }. + c_H=\frac{ C_H }{V} = \frac{ C_H C_D }{ v_\ast ^2 } = \frac{ k^2 }{ \Phi _h \Phi _m }\label{1.1.22} + +.. math:: + + c_D=\frac{ C_D }{V} = \frac{ C_D^2 }{ v_\ast ^2 } = \frac{ k^2 }{ \Phi _m^2 }. \label{1.1.23} - \end{aligned} + In order to close the system the surface scaling velocity, v\ :math:`_{\ast @@ -3138,17 +3206,21 @@ Redefining the vertical coordinate as :math:`\zeta=z/L`, we have .. math:: - \begin{aligned} - u(\zeta) &=& \frac{u_*}{k} \int_{0}^{\zeta} \frac{1}{(\zeta'+\zeta_{0m})} - \phi_m(\zeta'+\zeta_{0m}) \, d\zeta' = - \frac{u_*}{k} \int_{\zeta_{0m}}^{\zeta'+\zeta_{0m}} - \frac{1}{\zeta'} \phi_m(\zeta') \, d\zeta' \\ - &=& \frac{u_*}{k} \int_{\zeta_{0m}}^{\zeta'+\zeta_{0m}} \left ( \frac{1}{\zeta'} - - \frac{d\psi_m}{d\zeta'} \right ) \, d\zeta' \nonumber \\ - &=& \frac{u_*}{k} \left \{ \ln \left ( \frac{\zeta+\zeta_{0m}}{\zeta_{0m}} - \right ) - \psi_m(\zeta+\zeta_{0m}) + \psi_m (\zeta_{0m}) \right \}. - \nonumber - \end{aligned} + u(\zeta) = \frac{u_*}{k} \int_{0}^{\zeta} \frac{1}{(\zeta'+\zeta_{0m})} + \phi_m(\zeta'+\zeta_{0m}) \, d\zeta' = + \frac{u_*}{k} \int_{\zeta_{0m}}^{\zeta'+\zeta_{0m}} + \frac{1}{\zeta'} \phi_m(\zeta') \, d\zeta' + +.. math:: + + = \frac{u_*}{k} \int_{\zeta_{0m}}^{\zeta'+\zeta_{0m}} \left ( \frac{1}{\zeta'} - + \frac{d\psi_m}{d\zeta'} \right ) \, d\zeta' + +.. math:: + + = \frac{u_*}{k} \left \{ \ln \left ( \frac{\zeta+\zeta_{0m}}{\zeta_{0m}} + \right ) - \psi_m(\zeta+\zeta_{0m}) + \psi_m (\zeta_{0m}) \right \}. + This is also frequently written as @@ -3159,62 +3231,86 @@ in the rescaled coordinate), is therefore .. math:: - \begin{aligned} - \bar u &=& \frac{u_*}{k\zeta_1} \int_0^{\zeta_1} u(\zeta) \, d \zeta \\ - &=& \frac{u_*}{k\zeta_1} \int_0^{\zeta_1} - \ln \left ( \frac{\zeta+\zeta_{0m}}{\zeta_{0m}} \right ) - - \psi_m(\zeta+\zeta_{0m}) + \psi_m (\zeta_{0m}) \, d \zeta. - \nonumber - \end{aligned} + \bar u = \frac{u_*}{k\zeta_1} \int_0^{\zeta_1} u(\zeta) \, d \zeta + +.. math:: + + = \frac{u_*}{k\zeta_1} \int_0^{\zeta_1} + \ln \left ( \frac{\zeta+\zeta_{0m}}{\zeta_{0m}} \right ) + - \psi_m(\zeta+\zeta_{0m}) + \psi_m (\zeta_{0m}) \, d \zeta. + We consider the three terms within the integral separately. For the first, .. math:: - \begin{aligned} - \int_0^{\zeta_1} \ln \left ( \frac{\zeta+\zeta_{0m}}{\zeta_{0m}} \right ) \, d \zeta - &=& \zeta_{0m} \int_1^{1+\zeta_1/\zeta_{0m}} \ln(x) \, dx \\ - &=& \zeta_{0m} \left [ \left ( 1+ \frac{\zeta_1}{\zeta_{0m}} \right ) \ln - \left ( 1+ \frac{\zeta_1}{\zeta_{0m}} \right ) - - \left ( 1+ \frac{\zeta_1}{\zeta_{0m}} \right ) +1 \right ] . \nonumber - \end{aligned} + \int_0^{\zeta_1} \ln \left ( \frac{\zeta+\zeta_{0m}}{\zeta_{0m}} \right ) \, d \zeta + = \zeta_{0m} \int_1^{1+\zeta_1/\zeta_{0m}} \ln(x) \, dx + +.. math:: + + = \zeta_{0m} \left [ \left ( 1+ \frac{\zeta_1}{\zeta_{0m}} \right ) \ln + \left ( 1+ \frac{\zeta_1}{\zeta_{0m}} \right ) - + \left ( 1+ \frac{\zeta_1}{\zeta_{0m}} \right ) +1 \right ] . + For the second, .. math:: - \begin{aligned} - \int_0^{\zeta_1} \psi_m(\zeta+\zeta_{0m}) \, d\zeta &=& - \int_{\zeta_{0m}}^{\zeta_1+\zeta_{0m}} \psi_m(\zeta) \, d\zeta \\ - &=& \left [ \zeta \psi_m - \right ]_{\zeta_{0m}}^{\zeta_1+\zeta_{0m}} - - \int_{\zeta_{0m}}^{\zeta_1+\zeta_{0m}} \zeta \frac{d\psi_m}{d\zeta} - d\zeta \nonumber \\ - &=& (\zeta_1+\zeta_{0m}) \psi_m(\zeta_1+\zeta_{0m}) - \zeta_{0m} - \psi_m(\zeta_{0m}) \nonumber \\ &-& \int_{\zeta_{0m}}^{\zeta_1+\zeta_{0m}} - (1-\phi_m) d\zeta \nonumber \\ - &=& (\zeta_1+\zeta_{0m}) \psi_m(\zeta_1+\zeta_{0m}) - \zeta_{0m} - \psi_m(\zeta_{0m}) \nonumber \\ &+& \int_{\zeta_{0m}}^{\zeta_1+\zeta_{0m}} - (\phi_m -1) \, d\zeta . \nonumber - \end{aligned} + \int_0^{\zeta_1} \psi_m(\zeta+\zeta_{0m}) \, d\zeta = + \int_{\zeta_{0m}}^{\zeta_1+\zeta_{0m}} \psi_m(\zeta) \, d\zeta + +.. math:: + + = \left [ \zeta \psi_m + \right ]_{\zeta_{0m}}^{\zeta_1+\zeta_{0m}} + - \int_{\zeta_{0m}}^{\zeta_1+\zeta_{0m}} \zeta \frac{d\psi_m}{d\zeta} + d\zeta + +.. math:: + + = (\zeta_1+\zeta_{0m}) \psi_m(\zeta_1+\zeta_{0m}) - \zeta_{0m} + \psi_m(\zeta_{0m}) + +.. math:: + + - \int_{\zeta_{0m}}^{\zeta_1+\zeta_{0m}} + (1-\phi_m) d\zeta + +.. math:: + + = (\zeta_1+\zeta_{0m}) \psi_m(\zeta_1+\zeta_{0m}) - \zeta_{0m} + \psi_m(\zeta_{0m}) + +.. math:: + + + \int_{\zeta_{0m}}^{\zeta_1+\zeta_{0m}} + (\phi_m -1) \, d\zeta . + :math:`\phi_m-1` is retained in the last integral since this will prove convenient in later algebra. The third integral is trivial. Hence, .. math:: - \begin{aligned} - \bar u &=& \frac{u_*}{k} \left \{ - \left ( 1+ \frac{\zeta_{0m}}{\zeta_1} \right ) \left [ - \ln \left ( 1+ \frac{\zeta_1}{\zeta_{0m}} \right ) \right . \right . \\ - &-& \left . \left . \psi_m(\zeta_1+\zeta_{0m}) + \psi_m(\zeta_{0m}) \right ] -1 - - \frac{1}{\zeta_1} \int_{\zeta_{0m}}^{\zeta_1+\zeta_{0m}} - (\phi_m -1) \, d\zeta \right \} \nonumber \\ - &=& \frac{u_*}{k} \left \{ \left ( 1+ \frac{\zeta_{0m}}{\zeta_1} \right ) - \Phi_m(\zeta_1) - \frac{1}{\zeta_1} \int_{\zeta_{0m}}^{\zeta_1+\zeta_{0m}} - \phi_m \, d\zeta \right \} . \nonumber - \end{aligned} + \bar u = \frac{u_*}{k} \left \{ + \left ( 1+ \frac{\zeta_{0m}}{\zeta_1} \right ) \left [ + \ln \left ( 1+ \frac{\zeta_1}{\zeta_{0m}} \right ) \right . \right . + +.. math:: + + - \left . \left . \psi_m(\zeta_1+\zeta_{0m}) + \psi_m(\zeta_{0m}) \right ] -1 + - \frac{1}{\zeta_1} \int_{\zeta_{0m}}^{\zeta_1+\zeta_{0m}} + (\phi_m -1) \, d\zeta \right \} + +.. math:: + + = \frac{u_*}{k} \left \{ \left ( 1+ \frac{\zeta_{0m}}{\zeta_1} \right ) + \Phi_m(\zeta_1) - \frac{1}{\zeta_1} \int_{\zeta_{0m}}^{\zeta_1+\zeta_{0m}} + \phi_m \, d\zeta \right \} . + Thus, in practical terms, the standard function :math:`\Phi_m` is evaluated at the top of the layer, scaled by @@ -3244,12 +3340,14 @@ stability functions are given by :raw-latex:`\cite{Beljaars1991}`: .. math:: - \begin{aligned} - \Phi _m&=&\ln \left( {\frac{ z_1 + z_{0m} }{ z_{0m} }} \right) - \Psi _m ( \zeta _1 ) + \Psi _m ( \zeta _{0m} ) - \label{1.3.11}\\ - \Phi _h&=&\ln \left( {\frac{ z_1 + z_{0m} }{ z_{0h} }} \right) - \Psi _h ( \zeta _1 ) + \Psi _h ( \zeta _{0h} ) - \label{1.3.12} - \end{aligned} + \Phi _m=\ln \left( {\frac{ z_1 + z_{0m} }{ z_{0m} }} \right) - \Psi _m ( \zeta _1 ) + \Psi _m ( \zeta _{0m} ) + \label{1.3.11} + +.. math:: + + \Phi _h=\ln \left( {\frac{ z_1 + z_{0m} }{ z_{0h} }} \right) - \Psi _h ( \zeta _1 ) + \Psi _h ( \zeta _{0h} ) + \label{1.3.12} + where :math:`\zeta _{1}` = (z\ :math:`_{1}` + z\ :math:`_{0m})`/L, :math:`\zeta _{0m}` = z\ :math:`_{0m}`/L, :math:`\zeta _{0h}` = @@ -3257,12 +3355,14 @@ z\ :math:`_{0h}`/L and .. math:: - \begin{aligned} - - \Psi _h (\zeta )&=&\left[ { {\left( {1 + \frac{2}{3}a\zeta } \right)}^{3/2} - 1 } \right] + b\left( {\zeta - \frac{c}{d}} \right)\exp (-d\zeta ) + \frac{bc}{d} - \label{1.3.13}\\ - - \Psi _m (\zeta )&=&a\zeta + b\left( {\zeta - \frac{c}{d}} \right)\exp (-d\zeta ) + \frac{bc}{d}, - \label{1.3.14} - \end{aligned} + - \Psi _h (\zeta )=\left[ { {\left( {1 + \frac{2}{3}a\zeta } \right)}^{3/2} - 1 } \right] + b\left( {\zeta - \frac{c}{d}} \right)\exp (-d\zeta ) + \frac{bc}{d} + \label{1.3.13} + +.. math:: + + - \Psi _m (\zeta )=a\zeta + b\left( {\zeta - \frac{c}{d}} \right)\exp (-d\zeta ) + \frac{bc}{d}, + \label{1.3.14} + with :math:`a = 1`, :math:`b =2/3`, :math:`c = 5`, :math:`d = 0.35`. @@ -3283,12 +3383,14 @@ Dyer and Hicks forms :raw-latex:`\cite[]{dyer1974}` are used: .. math:: - \begin{aligned} - \phi _m&=&(1 - 16\zeta )^{-1/4} - \label{1.3.15} \\ - \phi _h&=&(1 - 16\zeta )^{-1/2} - \label{1.3.16} - \end{aligned} + \phi _m=(1 - 16\zeta )^{-1/4} + \label{1.3.15} + +.. math:: + + \phi _h=(1 - 16\zeta )^{-1/2} + \label{1.3.16} + (Note that :math:`\phi _{h}\prime` is discontinuous at 0.) These are only empirically verified for :math:`\zeta \ge` -1. Evaluating the @@ -3332,14 +3434,19 @@ ms\ :math:`^{-1}`), then start the iteration from the neutral limit, so .. math:: - \begin{aligned} - \Phi _m^{(0)}&=&\ln \left( {\frac{ z_1 + z_{0m} }{ z_{0m} }} \right) - \label{1.4.5} \\ - \Phi _h^{(0)}&=&\ln \left( {\frac{ z_1 + z_{0m} }{ z_{0h} }} \right) - \label{1.4.6} \\ - v_\ast ^{(0)}&=& {\left( {\frac{k}{ \Phi _m^{(0)} }} \right)} \left| {\Delta {{{\rm {\bf v}}}}} \right| - \label{1.4.7} - \end{aligned} + \Phi _m^{(0)}=\ln \left( {\frac{ z_1 + z_{0m} }{ z_{0m} }} \right) + \label{1.4.5} + +.. math:: + + \Phi _h^{(0)}=\ln \left( {\frac{ z_1 + z_{0m} }{ z_{0h} }} \right) + \label{1.4.6} + +.. math:: + + v_\ast ^{(0)}= {\left( {\frac{k}{ \Phi _m^{(0)} }} \right)} \left| {\Delta {{{\rm {\bf v}}}}} \right| + \label{1.4.7} + Otherwise (if :math:`\Delta`\ B :math:`<` 0 and :math:`\Delta`\ **v** :math:`<` 2 ms\ :math:`^{-1}` ) start from the greater of the neutral @@ -3347,28 +3454,38 @@ and convective limits for :math:`v_\ast^{(0)}`, so .. math:: - \begin{aligned} - \frac{1}{ L^{(0)} }&=&\frac{-k}{ \gamma _t^3 z_i } - \label{1.4.1}\\ - \Phi _m^{(0)}&=& \Phi _m ( L^{(0)} , z_1 + z_{0m} , z_{0m} ) - \label{1.4.2}\\ - \Phi _h^{(0)}&=& \Phi _h ( L^{(0)} , z_1 + z_{0m} , z_{0h} ) - \label{1.4.3}\\ - v_\ast ^{(0)}&= & MAX{\left[ {\left( {\frac{k}{ \Phi _m^{(0)} }} \right)} \left| {\Delta {{{\rm {\bf v}}}}} \right|, \, - {\left[ { \gamma _t^3 \left( {\frac{k}{ \Phi _h^{(0)} }} \right) z_i \left| {-\Delta B} \right| } \right]}^{ 1/2} \right]} - \label{1.4.4} - \end{aligned} + \frac{1}{ L^{(0)} }=\frac{-k}{ \gamma _t^3 z_i } + \label{1.4.1} -Then calculate +.. math:: + + \Phi _m^{(0)}= \Phi _m ( L^{(0)} , z_1 + z_{0m} , z_{0m} ) + \label{1.4.2} .. math:: - \begin{aligned} - C_D^{(0)}&=&\frac{k}{ \Phi _m^{(0)} } v_\ast ^{(0)} - \label{1.4.8} \\ - C_H^{(0)}&=&\frac{k}{ \Phi _h^{(0)} } v_\ast ^{(0)} - \label{1.4.9} - \end{aligned} + \Phi _h^{(0)}= \Phi _h ( L^{(0)} , z_1 + z_{0m} , z_{0h} ) + \label{1.4.3} + +.. math:: + + v_\ast ^{(0)}= MAX{\left[ {\left( {\frac{k}{ \Phi _m^{(0)} }} \right)} \left| {\Delta {{{\rm {\bf v}}}}} \right|, \, + {\left[ { \gamma _t^3 \left( {\frac{k}{ \Phi _h^{(0)} }} \right) z_i \left| {-\Delta B} \right| } \right]}^{ 1/2} \right]} + \label{1.4.4} + + +Then calculate + +.. math:: + + C_D^{(0)}=\frac{k}{ \Phi _m^{(0)} } v_\ast ^{(0)} + \label{1.4.8} + +.. math:: + + C_H^{(0)}=\frac{k}{ \Phi _h^{(0)} } v_\ast ^{(0)} + \label{1.4.9} + Having set up initial values the iteration loop can be entered (this is the original method used but contains an inconsistency in the treatment @@ -3379,26 +3496,49 @@ DO n = 1 to N .. math:: - \begin{aligned} - u_\ast ^{(n)2}&=& C_D^{(n-1)} \left| {\Delta {{\rm {\bf v}}}} \right| - \label{1.4.10} \\ - {\left( {\frac{ F_{B0} }{ \rho _0 }} \right)}^{(n)}&=& { {-C}_H }^{(n-1)} \Delta B - \label{1.4.11}\\ - w_\ast ^{(n)}&=& {\left[ { z_i {\left( {\frac{ F_{B0} }{ \rho _0 }} \right)}^{(n)} } \right]}^{ 1/3} - \label{1.4.12}\\ - v_\ast ^{(n)2}&=& u_\ast ^{(n)2} + \gamma _t^2 w_\ast ^{(n)2} + \gamma _c^2 w_c^2 - \label{1.4.13}\\ - \frac{1}{ L^{(n)} } = \frac{-k( F_{B0} / \rho _0 )^{(n)} }{ v_\ast ^{(n)3} } - \label{1.4.14}\\ - \Phi _m^{(n)}&=& \Phi _m ( L^{(n)} , z_1 + z_{0m} , z_{0m} ) - \label{1.4.15} \\ - \Phi _h^{(n)}&=& \Phi _h ( L^{(n)} , z_1 + z_{0m} , z_{0h} ) - \label{1.4.16} \\ - C_D^{(n)}&=&\frac{k}{ \Phi _m^{(n)} } v_\ast ^{(n)} - \label{1.4.17} \\ - C_H^{(n)}&=&\frac{k}{ \Phi _h^{(n)} } v_\ast ^{(n)} - \label{1.4.18} - \end{aligned} + u_\ast ^{(n)2}= C_D^{(n-1)} \left| {\Delta {{\rm {\bf v}}}} \right| + \label{1.4.10} + +.. math:: + + {\left( {\frac{ F_{B0} }{ \rho _0 }} \right)}^{(n)}= { {-C}_H }^{(n-1)} \Delta B + \label{1.4.11} + +.. math:: + + w_\ast ^{(n)}= {\left[ { z_i {\left( {\frac{ F_{B0} }{ \rho _0 }} \right)}^{(n)} } \right]}^{ 1/3} + \label{1.4.12} + +.. math:: + + v_\ast ^{(n)2}= u_\ast ^{(n)2} + \gamma _t^2 w_\ast ^{(n)2} + \gamma _c^2 w_c^2 + \label{1.4.13} + +.. math:: + + \frac{1}{ L^{(n)} } = \frac{-k( F_{B0} / \rho _0 )^{(n)} }{ v_\ast ^{(n)3} } + \label{1.4.14} + +.. math:: + + \Phi _m^{(n)}= \Phi _m ( L^{(n)} , z_1 + z_{0m} , z_{0m} ) + \label{1.4.15} + +.. math:: + + \Phi _h^{(n)}= \Phi _h ( L^{(n)} , z_1 + z_{0m} , z_{0h} ) + \label{1.4.16} + +.. math:: + + C_D^{(n)}=\frac{k}{ \Phi _m^{(n)} } v_\ast ^{(n)} + \label{1.4.17} + +.. math:: + + C_H^{(n)}=\frac{k}{ \Phi _h^{(n)} } v_\ast ^{(n)} + \label{1.4.18} + END DO. @@ -3412,14 +3552,19 @@ stress: .. math:: - \begin{aligned} - H_0&=& {-c}_P \rho _0 C_H^{(N)} \left( {\Delta T + \frac{g}{ c_P }( z_1 + z_{0m} - z_{0h} )} \right) - \label{1.4.19} \\ - E_0&=& {-\rho }_0 C_H^{(N)} \Delta q - \label{1.4.20} \\ - {\rm {\bf \tau }}_{0} &=& \rho _0 C_D^{(N)} \Delta {\rm {\bf v}} - \label{1.4.21} - \end{aligned} + H_0= {-c}_P \rho _0 C_H^{(N)} \left( {\Delta T + \frac{g}{ c_P }( z_1 + z_{0m} - z_{0h} )} \right) + \label{1.4.19} + +.. math:: + + E_0= {-\rho }_0 C_H^{(N)} \Delta q + \label{1.4.20} + +.. math:: + + {\rm {\bf \tau }}_{0} = \rho _0 C_D^{(N)} \Delta {\rm {\bf v}} + \label{1.4.21} + N is the last iteration value. N = 5 is currently used. @@ -3642,12 +3787,14 @@ the result that would be obtained from standard similarity theory, .. math:: - \begin{aligned} - \theta_{ob}'(t+\delta t) &\leftarrow & \theta_{ob}(t)+ - \delta t \, \dot T_{ob,\mbox{\tiny rad,surf}} \\ - \theta_{ob}(t+\delta t) &\leftarrow & W \theta_{ob}'(t+\delta t) - +(1-W) \theta_{ob, \mbox{\tiny sim}}. - \end{aligned} + \theta_{ob}'(t+\delta t) \leftarrow \theta_{ob}(t)+ + \delta t \, \dot T_{ob,\mbox{\tiny rad,surf}} + +.. math:: + + \theta_{ob}(t+\delta t) \leftarrow W \theta_{ob}'(t+\delta t) + +(1-W) \theta_{ob, \mbox{\tiny sim}}. + By tuning against an idealized highly vertically resolved model based on local scaling we set, @@ -3925,36 +4072,45 @@ Two approaches are available in uncoupled configurations of the model. .. math:: - \begin{aligned} - < C_H >&=&( f_I C_{H(MIZ)} + ( 0.7 - f_I ) C_{H(L)} ) / 0.7 - \label{1.6.1} \\ - < C_D >&=&( f_I C_{D(MIZ)} + ( 0.7 - f_I ) C_{D(L)} ) / 0.7 - \label{1.6.2} - \end{aligned} + < C_H >=( f_I C_{H(MIZ)} + ( 0.7 - f_I ) C_{H(L)} ) / 0.7 + \label{1.6.1} + + .. math:: + + < C_D >=( f_I C_{D(MIZ)} + ( 0.7 - f_I ) C_{D(L)} ) / 0.7 + \label{1.6.2} + and for 0.7 :math:`\le` f\ :math:`_{I} \le` 1 .. math:: - \begin{aligned} - < C_H >&=&( ( 1 - f_I ) C_{H(MIZ)} + ( f_I - 0.7 ) ) C_{H(I)} ) / 0.3 - \label{1.6.3} \\ - < C_D >&=&( ( 1 - f_I ) C_{D(MIZ)} + ( f_I - 0.7 ) ) C_{D(I)} ) / 0.3 - \label{1.6.4} - \end{aligned} + < C_H >=( ( 1 - f_I ) C_{H(MIZ)} + ( f_I - 0.7 ) ) C_{H(I)} ) / 0.3 + \label{1.6.3} + + .. math:: + + < C_D >=( ( 1 - f_I ) C_{D(MIZ)} + ( f_I - 0.7 ) ) C_{D(I)} ) / 0.3 + \label{1.6.4} + where .. math:: - \begin{aligned} - C_{H(L)}&= &C_H ( L_{(L)} , z_{0m(sea)} , z_{0h(sea)} ) - \label{1.6.5} \\ - C_{H(MIZ)}&=& C_H ( L_{(I)} , z_{0m(MIZ)} , z_{0h(MIZ)} ) - \label{1.6.6} \\ - C_{H(I)}&=& C_H ( L_{(I)} , z_{0m(sea-ice)} , z_{0h(sea-ice)} ) - \label{1.6.7} - \end{aligned} + C_{H(L)}= C_H ( L_{(L)} , z_{0m(sea)} , z_{0h(sea)} ) + \label{1.6.5} + + .. math:: + + C_{H(MIZ)}= C_H ( L_{(I)} , z_{0m(MIZ)} , z_{0h(MIZ)} ) + \label{1.6.6} + + .. math:: + + C_{H(I)}= C_H ( L_{(I)} , z_{0m(sea-ice)} , z_{0h(sea-ice)} ) + \label{1.6.7} + and similarly for the drag coefficient C\ :math:`_{D}`. @@ -4081,12 +4237,14 @@ Two approaches are available in uncoupled configurations of the model. .. math:: - \begin{aligned} - < C_D >&=& (1 - f_I) C_{D(L)} + f_I (C_{D(I)} + C_{D(FRM)}) - \label{eq:cdice_int} \\ - < C_H >&=& (1 - f_I) C_{H(L)} + f_I C_{H(I)} - \label{eq:chice_int} - \end{aligned} + < C_D >= (1 - f_I) C_{D(L)} + f_I (C_{D(I)} + C_{D(FRM)}) + \label{eq:cdice_int} + + .. math:: + + < C_H >= (1 - f_I) C_{H(L)} + f_I C_{H(I)} + \label{eq:chice_int} + .. _`sec:coast`: @@ -4204,15 +4362,23 @@ When form drag is included via effective roughness lengths equations .. math:: - \begin{aligned} - \frac{ H_{0(eff)} }{ c_P \rho _0 }&=&\frac{-k}{ \Phi _h (L , z_1 + z_{0m(eff)} , z_{0h(eff)} )} v_{\ast (eff)}\nonumber\\ - && \left( {\Delta T + \frac{g}{c_p }( z_1 + z_{0m(eff)} - z_{0h(eff)} )} \right) - \label{2.1.1} \\ - \frac{ E_{0(eff)} }{ \rho _0 }&=&\frac{-k}{ \Phi _h (L , z_1 + z_{0m(eff)} , z_{0h(eff)} )} v_{\ast (eff)} \Delta q - \label{2.1.2} \\ - \frac{ {\rm {\bf \tau }}_{{0(eff)}} }{ \rho _{0} }&=&\frac{k}{ \Phi _m (L , z_1 + z_{0m(eff)} , z_{0m(eff)} )} v_{\ast (eff)} \Delta {\rm {\bf v}} - \label{2.1.3} - \end{aligned} + \frac{ H_{0(eff)} }{ c_P \rho _0 }=\frac{-k}{ \Phi _h (L , z_1 + z_{0m(eff)} , z_{0h(eff)} )} v_{\ast (eff)} + +.. math:: + + \left( {\Delta T + \frac{g}{c_p }( z_1 + z_{0m(eff)} - z_{0h(eff)} )} \right) + \label{2.1.1} + +.. math:: + + \frac{ E_{0(eff)} }{ \rho _0 }=\frac{-k}{ \Phi _h (L , z_1 + z_{0m(eff)} , z_{0h(eff)} )} v_{\ast (eff)} \Delta q + \label{2.1.2} + +.. math:: + + \frac{ {\rm {\bf \tau }}_{{0(eff)}} }{ \rho _{0} }=\frac{k}{ \Phi _m (L , z_1 + z_{0m(eff)} , z_{0m(eff)} )} v_{\ast (eff)} \Delta {\rm {\bf v}} + \label{2.1.3} + The effective surface scaling velocity, v\ :math:`_{\ast (eff)}` , is given by (cf. (`[1.1.25] <#1.1.25>`__)) @@ -4343,14 +4509,16 @@ and (`[2.1.13] <#2.1.13>`__) and (`[2.1.14] <#2.1.14>`__) become .. math:: - \begin{aligned} - {\rm {\bf \tau }}_{{0(f)}} &=& {\rm {\bf \tau - }}_{{0(eff)}} {\left( {{1 + }\alpha \beta \pi ^{2} { } {f}_{D} { } - {\left( {\frac{{A}}{{S}}} \right)}^{2} { }} \right)}^{-1} - \label{2.1.16} \\ - C_{D(f)}&= &C_{D(eff)} {\left( {1 + \alpha \beta \pi ^2 f_D {\left( {\frac{A}{S}} \right)}^2 } \right)}^{-1} - \label{2.1.17} - \end{aligned} + {\rm {\bf \tau }}_{{0(f)}} = {\rm {\bf \tau + }}_{{0(eff)}} {\left( {{1 + }\alpha \beta \pi ^{2} { } {f}_{D} { } + {\left( {\frac{{A}}{{S}}} \right)}^{2} { }} \right)}^{-1} + \label{2.1.16} + +.. math:: + + C_{D(f)}= C_{D(eff)} {\left( {1 + \alpha \beta \pi ^2 f_D {\left( {\frac{A}{S}} \right)}^2 } \right)}^{-1} + \label{2.1.17} + The effective surface flux of scalar X evaluated in terms of values at z\ :math:`_{c}` is @@ -4405,36 +4573,58 @@ iteration from the convective limit, so .. math:: - \begin{aligned} - \frac{1}{ L^{(0)} }&=&\frac{-k}{ \gamma _t^3 z_i } - \label{2.2.1} \\ - \Phi _m^{(0)}&=& \Phi _m ( L^{(0)} , z_1 + z_{0m(eff)} , z_{0m(eff)} ) - \label{2.2.2} \\ - \Phi _h^{(0)}&=& \Phi _h ( L^{(0)} , z_1 + z_{0m(eff)} , z_{0h} ) - \label{2.2.3} \\ - v_{\ast (eff)}^{(0)}&=& v_{\ast (f)}^{(0)} = {\left[ { \gamma _t^3 \left( {\frac{k}{ \Phi _h^{(0)} }} \right) z_i \left| {-\Delta B} \right| + \gamma _c^2 w_c^2 } \right]}^{ 1/2} - \label{(2.2.4} - \end{aligned} + \frac{1}{ L^{(0)} }=\frac{-k}{ \gamma _t^3 z_i } + \label{2.2.1} + +.. math:: + + \Phi _m^{(0)}= \Phi _m ( L^{(0)} , z_1 + z_{0m(eff)} , z_{0m(eff)} ) + \label{2.2.2} + +.. math:: + + \Phi _h^{(0)}= \Phi _h ( L^{(0)} , z_1 + z_{0m(eff)} , z_{0h} ) + \label{2.2.3} + +.. math:: + + v_{\ast (eff)}^{(0)}= v_{\ast (f)}^{(0)} = {\left[ { \gamma _t^3 \left( {\frac{k}{ \Phi _h^{(0)} }} \right) z_i \left| {-\Delta B} \right| + \gamma _c^2 w_c^2 } \right]}^{ 1/2} + \label{(2.2.4} + ELSE IF (:math:`\Delta`\ **v** :math:`\ge` 2 ms\ :math:`^{-1}` ) start iteration from the neutral end, so .. math:: - \begin{aligned} - \Phi _m^{(0)}&=&\ln \left( {\frac{ z_1 + z_{0m(eff)} }{ z_{0m(eff)} }} \right) - \label{2.2.5} \\ - \Phi _h^{(0)}&=&\ln \left( {\frac{ z_1 + z_{0m(eff)} }{ z_{0h} }} \right) - \label{2.2.6} \\ - u_{\ast (eff)}^{(0)}&=&\frac{k}{ \Phi _m^{(0)} } \left| {\Delta {{{v}}}} \right| - \label{2.2.7} \\ - v_{\ast (eff)}^{(0)}&=& {\left( { u_{\ast (eff)}^{(0) 2} + \gamma _c^2 w_c^2 } \right)}^{ 1/2} - \label{2.2.8} \\ - u_{\ast (f)}&=& u_{\ast (eff)} \frac{\ln ( z_c / z_{0m(eff)} )}{\ln ( z_c / z_{0m} )} - \label{2.2.9} \\ - v_{\ast (f)}^{(0)}&=& {\left( { u_{\ast (f)}^{(0) 2} + \gamma _c^2 w_c^2 } \right)}^{ 1/2} - \label{2.2.10} - \end{aligned} + \Phi _m^{(0)}=\ln \left( {\frac{ z_1 + z_{0m(eff)} }{ z_{0m(eff)} }} \right) + \label{2.2.5} + +.. math:: + + \Phi _h^{(0)}=\ln \left( {\frac{ z_1 + z_{0m(eff)} }{ z_{0h} }} \right) + \label{2.2.6} + +.. math:: + + u_{\ast (eff)}^{(0)}=\frac{k}{ \Phi _m^{(0)} } \left| {\Delta {{{v}}}} \right| + \label{2.2.7} + +.. math:: + + v_{\ast (eff)}^{(0)}= {\left( { u_{\ast (eff)}^{(0) 2} + \gamma _c^2 w_c^2 } \right)}^{ 1/2} + \label{2.2.8} + +.. math:: + + u_{\ast (f)}= u_{\ast (eff)} \frac{\ln ( z_c / z_{0m(eff)} )}{\ln ( z_c / z_{0m} )} + \label{2.2.9} + +.. math:: + + v_{\ast (f)}^{(0)}= {\left( { u_{\ast (f)}^{(0) 2} + \gamma _c^2 w_c^2 } \right)}^{ 1/2} + \label{2.2.10} + END IF. @@ -4442,16 +4632,24 @@ Then calculate: .. math:: - \begin{aligned} - C_{D(eff)}^{(0)}&=&\frac{k}{ \Phi _m^{(0)} } v_{\ast (eff)}^{(0)} - \label{(2.2.11} \\ - C_{H(eff)}^{(0)}&=&\frac{k}{ \Phi _h^{(0)} } v_{\ast (eff)}^{(0)} - \label{(2.2.12} \\ - C_{D(f)}^{(0)}&=& C_{D(eff)}^{(0)} {\left( {1 + \frac{1}{2} c_{D(orog)} f_D \frac{A}{S} {\left( {\frac{\ln ( z_c / z_{0m} )}{k}} \right)}^2 } \right)}^{-1} - \label{(2.2.13} \\ - C_{H(f)}^{(0)}&=& C_{H(eff)}^{(0)} \left( {1 - 2.2 f_D \frac{A}{S}} \right) - \label{(2.2.14} - \end{aligned} + C_{D(eff)}^{(0)}=\frac{k}{ \Phi _m^{(0)} } v_{\ast (eff)}^{(0)} + \label{(2.2.11} + +.. math:: + + C_{H(eff)}^{(0)}=\frac{k}{ \Phi _h^{(0)} } v_{\ast (eff)}^{(0)} + \label{(2.2.12} + +.. math:: + + C_{D(f)}^{(0)}= C_{D(eff)}^{(0)} {\left( {1 + \frac{1}{2} c_{D(orog)} f_D \frac{A}{S} {\left( {\frac{\ln ( z_c / z_{0m} )}{k}} \right)}^2 } \right)}^{-1} + \label{(2.2.13} + +.. math:: + + C_{H(f)}^{(0)}= C_{H(eff)}^{(0)} \left( {1 - 2.2 f_D \frac{A}{S}} \right) + \label{(2.2.14} + Having set up initial values the iteration loop can be entered: @@ -4459,34 +4657,69 @@ DO n = 1 to N .. math:: - \begin{aligned} - u_{\ast (eff)}^{(n)2}&=& C_{D(eff)}^{(n-1)} \left| {\Delta {{\rm {\bf v}}}} \right| - \label{(2.2.15} \\ - u_{\ast (f)}^{(n)2}&=& C_{D(f)}^{(n-1)} \left| {\Delta {{\rm {\bf v}}}} \right| - \label{(2.2.16} \\ - {\left( {\frac{ F_{B0} }{ \rho _0 }} \right)}^{(n)}&=&- { C_{H(eff)} }^{(n-1)} \Delta B - \label{(2.2.17} \\ - w_\ast ^{(n)}&=& {\left[ { z_i {\left( {\frac{ F_{B0} }{ \rho _0 }} \right)}^{(n)} } \right]}^{ 1/3} - \label{(2.2.18} \\ - v_{\ast (eff)}^{(n)2}&=& u_{\ast (eff)}^{(n)2} + \gamma _t^2 w_\ast ^{(n)2} + \gamma _c^2 w_c^2 - \label{(2.2.19} \\ - v_{\ast (f)}^{(n)2}&= &u_{\ast (f)}^{(n)2} + \gamma _t^2 w_\ast ^{(n)2} + \gamma _c^2 w_c^2 - \label{(2.2.20} \\ - \frac{1}{ L^{(n)} }&=&\frac{-k( F_{B0} / \rho _0 )^{(n)} }{ v_{\ast (eff)}^{(n)3} } - \label{(2.2.21} \\ - \Phi _m^{(n)}&=& \Phi _m ( L^{(n)} , z_1 + z_{0m(eff)} , z_{0m(eff)} ) - \label{(2.2.22} \\ - \Phi _h^{(n)}&=& \Phi _h ( L^{(n)} , z_1 + z_{0m(eff)} , z_{0h} ) - \label{(2.2.23} \\ - C_{D(eff)}^{(n)}&=&\frac{k}{ \Phi _m^{(n)} } v_{\ast (eff)}^{(n)} - \label{(2.2.24} \\ - C_{H(eff)}^{(n)}&=&\frac{k}{ \Phi _h^{(n)} } v_{\ast (eff)}^{(n)} - \label{(2.2.25} \\ - C_{D(f)}^{(n)}&=& C_{D(eff)}^{(n)} {\left( {1 + \frac{1}{2} c_{D(orog)} f_D \frac{A}{S} {\left( {\frac{\ln ( z_c / z_{0m} )}{k}} \right)}^2 } \right)}^{-1} - \label{(2.2.26} \\ - C_{H(f)}^{(n)}&=& C_{H(eff)}^{(n)} \left( {1 - 2.2 f_D \frac{A}{S}} \right) - \label{(2.2.27} - \end{aligned} + u_{\ast (eff)}^{(n)2}= C_{D(eff)}^{(n-1)} \left| {\Delta {{\rm {\bf v}}}} \right| + \label{(2.2.15} + +.. math:: + + u_{\ast (f)}^{(n)2}= C_{D(f)}^{(n-1)} \left| {\Delta {{\rm {\bf v}}}} \right| + \label{(2.2.16} + +.. math:: + + {\left( {\frac{ F_{B0} }{ \rho _0 }} \right)}^{(n)}=- { C_{H(eff)} }^{(n-1)} \Delta B + \label{(2.2.17} + +.. math:: + + w_\ast ^{(n)}= {\left[ { z_i {\left( {\frac{ F_{B0} }{ \rho _0 }} \right)}^{(n)} } \right]}^{ 1/3} + \label{(2.2.18} + +.. math:: + + v_{\ast (eff)}^{(n)2}= u_{\ast (eff)}^{(n)2} + \gamma _t^2 w_\ast ^{(n)2} + \gamma _c^2 w_c^2 + \label{(2.2.19} + +.. math:: + + v_{\ast (f)}^{(n)2}= u_{\ast (f)}^{(n)2} + \gamma _t^2 w_\ast ^{(n)2} + \gamma _c^2 w_c^2 + \label{(2.2.20} + +.. math:: + + \frac{1}{ L^{(n)} }=\frac{-k( F_{B0} / \rho _0 )^{(n)} }{ v_{\ast (eff)}^{(n)3} } + \label{(2.2.21} + +.. math:: + + \Phi _m^{(n)}= \Phi _m ( L^{(n)} , z_1 + z_{0m(eff)} , z_{0m(eff)} ) + \label{(2.2.22} + +.. math:: + + \Phi _h^{(n)}= \Phi _h ( L^{(n)} , z_1 + z_{0m(eff)} , z_{0h} ) + \label{(2.2.23} + +.. math:: + + C_{D(eff)}^{(n)}=\frac{k}{ \Phi _m^{(n)} } v_{\ast (eff)}^{(n)} + \label{(2.2.24} + +.. math:: + + C_{H(eff)}^{(n)}=\frac{k}{ \Phi _h^{(n)} } v_{\ast (eff)}^{(n)} + \label{(2.2.25} + +.. math:: + + C_{D(f)}^{(n)}= C_{D(eff)}^{(n)} {\left( {1 + \frac{1}{2} c_{D(orog)} f_D \frac{A}{S} {\left( {\frac{\ln ( z_c / z_{0m} )}{k}} \right)}^2 } \right)}^{-1} + \label{(2.2.26} + +.. math:: + + C_{H(f)}^{(n)}= C_{H(eff)}^{(n)} \left( {1 - 2.2 f_D \frac{A}{S}} \right) + \label{(2.2.27} + END DO. @@ -4498,14 +4731,19 @@ surface sensible and latent heat fluxes and surface stress: .. math:: - \begin{aligned} - H_{0(eff)}&=&- c_P \rho _0 C_{H(eff)}^{(N)} \left( {\Delta T + \frac{g}{c_p }( z_1 + z_{0m(eff)} - z_{0h} )} \right) - \label{2.2.28} \\ - E_{0(eff)}&=& {-\rho }_0 C_{H(eff)}^{(N)} \Delta q - \label{2.2.29} \\ - {\rm {\bf \tau }}_{{0(eff)}} &=& \rho _0 C_{D(eff)}^{(N)} \Delta {\rm {\bf v}} - \label{2.2.30} - \end{aligned} + H_{0(eff)}=- c_P \rho _0 C_{H(eff)}^{(N)} \left( {\Delta T + \frac{g}{c_p }( z_1 + z_{0m(eff)} - z_{0h} )} \right) + \label{2.2.28} + +.. math:: + + E_{0(eff)}= {-\rho }_0 C_{H(eff)}^{(N)} \Delta q + \label{2.2.29} + +.. math:: + + {\rm {\bf \tau }}_{{0(eff)}} = \rho _0 C_{D(eff)}^{(N)} \Delta {\rm {\bf v}} + \label{2.2.30} + The stress for a flat surface, if required for output, is calculated from @@ -4782,12 +5020,14 @@ becomes .. math:: - \begin{aligned} - \frac{X^{*}-X^{n}}{\Delta t} & = & {\cal I}_{1}\frac{\partial F}{\partial z}^{*}-{\cal E}_{1}\frac{\partial F}{\partial z}^{n}+\left({\cal I}_{1}-{\cal E}_{1}\right)S\label{eq:sppf_bl1}\\ - \frac{X^{n+1}-X^{*}}{\Delta t} & = & {\cal I}_{2}\frac{\partial - F}{\partial z}^{n+1}-{\cal E}_{2}\frac{\partial F}{\partial - z}^{*}+\left({\cal I}_{2}-{\cal E}_{2}\right)S\label{eq:sppf_bl2} - \end{aligned} + \frac{X^{*}-X^{n}}{\Delta t} = {\cal I}_{1}\frac{\partial F}{\partial z}^{*}-{\cal E}_{1}\frac{\partial F}{\partial z}^{n}+\left({\cal I}_{1}-{\cal E}_{1}\right)S\label{eq:sppf_bl1} + +.. math:: + + \frac{X^{n+1}-X^{*}}{\Delta t} = {\cal I}_{2}\frac{\partial + F}{\partial z}^{n+1}-{\cal E}_{2}\frac{\partial F}{\partial + z}^{*}+\left({\cal I}_{2}-{\cal E}_{2}\right)S\label{eq:sppf_bl2} + where, @@ -4821,11 +5061,16 @@ Writing equations (`[eq:sppf_bl1] <#eq:sppf_bl1>`__), .. math:: - \begin{aligned} - \frac{\delta X}{\Delta t}^{*} & = & ({\cal I}_{1}-{\cal E}_{1})\left(\frac{\partial F}{\partial z}^{n}+S\right)+{\cal I}_{1}\frac{\partial}{\partial z}\left(K_{X}\frac{\partial\delta X}{\partial z}^{*}\right)\label{eq:sppf_inc1}\\ - \frac{\delta X}{\Delta t}^{n+1} & = & ({\cal I}_{2}-{\cal E}_{2})\left(\frac{\partial F}{\partial z}^{*}+S\right)+{\cal I}_{2}\frac{\partial}{\partial z}\left(K_{X}\frac{\partial\delta X}{\partial z}^{n+1}\right)\label{eq:sppf_inc2}\\ - X^{n+1} & = & X^{n}+\delta X^{*}+\delta X^{n+1}\label{eq:sppf_inc3} - \end{aligned} + \frac{\delta X}{\Delta t}^{*} = ({\cal I}_{1}-{\cal E}_{1})\left(\frac{\partial F}{\partial z}^{n}+S\right)+{\cal I}_{1}\frac{\partial}{\partial z}\left(K_{X}\frac{\partial\delta X}{\partial z}^{*}\right)\label{eq:sppf_inc1} + +.. math:: + + \frac{\delta X}{\Delta t}^{n+1} = ({\cal I}_{2}-{\cal E}_{2})\left(\frac{\partial F}{\partial z}^{*}+S\right)+{\cal I}_{2}\frac{\partial}{\partial z}\left(K_{X}\frac{\partial\delta X}{\partial z}^{n+1}\right)\label{eq:sppf_inc2} + +.. math:: + + X^{n+1} = X^{n}+\delta X^{*}+\delta X^{n+1}\label{eq:sppf_inc3} + .. _`sec:impsolve`: @@ -4871,15 +5116,17 @@ levels), discretizing the previous equation in :math:`z` on all .. math:: - \begin{aligned} - \delta u_{k+1/2}^{*} & = & ({\cal I}_{1}-{\cal E}_{1})\Delta t\left(\frac{\tau_{x}^{n}\Big|_{k+1}-\tau_{x}^{n}\Big|_{k}}{z_{k+1}-z_{k}}+S_{k+1/2}\right)\\ - & & +{\cal I}_{1}\frac{\Delta - t}{z_{k+1}-z_{k}}\left[\left(K_{u}\Big|_{k+1}\frac{\delta - u_{k+3/2}^{*}-\delta - u_{k+1/2}^{*}}{z_{k+3/2}-z_{k+1/2}}\right)-\left(K_{u}\Big|_{k}\frac{\delta - u_{k+1/2}^{*}-\delta - u_{k-1/2}^{*}}{z_{k+1/2}-z_{k-1/2}}\right)\right] - \end{aligned} + \delta u_{k+1/2}^{*} = ({\cal I}_{1}-{\cal E}_{1})\Delta t\left(\frac{\tau_{x}^{n}\Big|_{k+1}-\tau_{x}^{n}\Big|_{k}}{z_{k+1}-z_{k}}+S_{k+1/2}\right) + +.. math:: + + +{\cal I}_{1}\frac{\Delta + t}{z_{k+1}-z_{k}}\left[\left(K_{u}\Big|_{k+1}\frac{\delta + u_{k+3/2}^{*}-\delta + u_{k+1/2}^{*}}{z_{k+3/2}-z_{k+1/2}}\right)-\left(K_{u}\Big|_{k}\frac{\delta + u_{k+1/2}^{*}-\delta + u_{k-1/2}^{*}}{z_{k+1/2}-z_{k-1/2}}\right)\right] + or, rearranging @@ -5010,10 +5257,12 @@ and thus the following discretization is obtained, on .. math:: - \begin{aligned} - \frac{\delta X_{k}^{*}}{\Delta t} & = & \left({\cal I}_{1}-{\cal E}_{1}\right)\left(\frac{F_{k+1/2}^{n}-F_{k-1/2}^{n}}{z_{k+1/2}-z_{k-1/2}}+S_{k}\right)\\ - & +&\frac{{\cal I}_{1}}{z_{k+1/2}-z_{k-1/2}}\left[K_{X}\Big|_{k+1/2}\left(\frac{\delta X_{k+1}^{*}-\delta X_{k}^{*}}{z_{k+1}-z_{k}}\right)-K_{X}\Big|_{k-1/2}\left(\frac{\delta X_{k}^{*}-\delta X_{k-1}^{*}}{z_{k}-z_{k-1}}\right)\right],\; k=2,\ldots,L-1. - \end{aligned} + \frac{\delta X_{k}^{*}}{\Delta t} = \left({\cal I}_{1}-{\cal E}_{1}\right)\left(\frac{F_{k+1/2}^{n}-F_{k-1/2}^{n}}{z_{k+1/2}-z_{k-1/2}}+S_{k}\right) + +.. math:: + + +\frac{{\cal I}_{1}}{z_{k+1/2}-z_{k-1/2}}\left[K_{X}\Big|_{k+1/2}\left(\frac{\delta X_{k+1}^{*}-\delta X_{k}^{*}}{z_{k+1}-z_{k}}\right)-K_{X}\Big|_{k-1/2}\left(\frac{\delta X_{k}^{*}-\delta X_{k-1}^{*}}{z_{k}-z_{k-1}}\right)\right],\; k=2,\ldots,L-1. + or, @@ -5189,10 +5438,12 @@ corrector are defined as: .. math:: - \begin{aligned} - \overline{\tau_{x}}^{[n,*]}\equiv{\cal I}_{1}\tau_{x}^{*}-{\cal E}_{1}\tau_{x}^{n} & = & \left({\cal I}_{1}-{\cal E}_{1}\right)\tau_{x}^{n}+{\cal I}_{1}K_{u}\frac{\partial\delta u^{*}}{\partial z}\\ - \overline{\tau_{x}}^{[*,n+1]}\equiv{\cal I}_{2}\tau_{x}^{n+1}-{\cal E}_{2}\tau_{x}^{*} & = & \left({\cal I}_{2}-{\cal E}_{2}\right)\tau_{x}^{*}+{\cal I}_{2}K_{u}\frac{\partial\delta u^{n+1}}{\partial z} - \end{aligned} + \overline{\tau_{x}}^{[n,*]}\equiv{\cal I}_{1}\tau_{x}^{*}-{\cal E}_{1}\tau_{x}^{n} = \left({\cal I}_{1}-{\cal E}_{1}\right)\tau_{x}^{n}+{\cal I}_{1}K_{u}\frac{\partial\delta u^{*}}{\partial z} + +.. math:: + + \overline{\tau_{x}}^{[*,n+1]}\equiv{\cal I}_{2}\tau_{x}^{n+1}-{\cal E}_{2}\tau_{x}^{*} = \left({\cal I}_{2}-{\cal E}_{2}\right)\tau_{x}^{*}+{\cal I}_{2}K_{u}\frac{\partial\delta u^{n+1}}{\partial z} + where, :math:`\delta u^{*}=u^{*}-u^{n},\;\delta u^{n+1}=u^{n+1}-u^{*}`. The meridional stress :math:`\tau_{y}` and the scalar fluxes can be @@ -5284,10 +5535,12 @@ flux for :math:`H` is derived: .. math:: - \begin{aligned} - \frac{H_{j}^{*}}{c_{p}} & = & \gamma_{2}\frac{H_{j}^{(n)}}{c_{p}}-\gamma_{1}RK_{PMj}[LD_{j}\psi_{j}RK_{H}(1)_{j}+A_{*j}][c_{p}\delta{T'}_{1}-\beta\overline{H^{*}}]\\ - & & \qquad\quad+\gamma_{1}RK_{PMj}L\psi_{j}RK_{H}(1)_{j}[\delta{Q'}_{1}-\beta\overline{E^{*}}] - \end{aligned} + \frac{H_{j}^{*}}{c_{p}} = \gamma_{2}\frac{H_{j}^{(n)}}{c_{p}}-\gamma_{1}RK_{PMj}[LD_{j}\psi_{j}RK_{H}(1)_{j}+A_{*j}][c_{p}\delta{T'}_{1}-\beta\overline{H^{*}}] + +.. math:: + + \qquad\quad+\gamma_{1}RK_{PMj}L\psi_{j}RK_{H}(1)_{j}[\delta{Q'}_{1}-\beta\overline{E^{*}}] + and similarly :math:`E_{j}^{*}`. From these, the tile flux equations (`[eq:FTLstar] <#eq:FTLstar>`__), (`[eq:FQWstar] <#eq:FQWstar>`__) can @@ -5771,15 +6024,20 @@ written .. math:: - \begin{aligned} - V_{\rm heat}^3&=& z_{\rm ml}\! \left( (2-\zeta_s)\zeta_s \overline{w'b}_S+ (1-\zeta_s)^2 [\overline{w'b'}_S]_{\rm sat}\right) - \label{vsurf} \\ - V_{\rm rad}^3&=& z_{\rm ml}\Delta_F\, g \, - \left( \beta_T \zeta_r^2 + \tilde{\beta_T} (1-\zeta_r^2) \right) - \label{vrad} \\ - V_{\rm br}^3&=& A_{\rm br}\chi_s^2 \, \mbox{max}\left[0,-\delta b\right] \, \Delta b ^{1/2} - \, z_c^{3/2} \, C_{fac} \label{vbr} - \end{aligned} + V_{\rm heat}^3= z_{\rm ml}\! \left( (2-\zeta_s)\zeta_s \overline{w'b}_S+ (1-\zeta_s)^2 [\overline{w'b'}_S]_{\rm sat}\right) + \label{vsurf} + +.. math:: + + V_{\rm rad}^3= z_{\rm ml}\Delta_F\, g \, + \left( \beta_T \zeta_r^2 + \tilde{\beta_T} (1-\zeta_r^2) \right) + \label{vrad} + +.. math:: + + V_{\rm br}^3= A_{\rm br}\chi_s^2 \, \mbox{max}\left[0,-\delta b\right] \, \Delta b ^{1/2} + \, z_c^{3/2} \, C_{fac} \label{vbr} + Here, :math:`[\overline{w'b'}_S]_{\rm sat}= g ( \tilde{\beta_T} \overline{w'\theta_{\ell}'}_S+ \tilde{\beta_q}\overline{w'q_t'}_S)`, @@ -5835,12 +6093,14 @@ approximated as .. math:: - \begin{aligned} - \gamma_{q_{\ell}} &=& -\frac{\gamma_{T_L} \alpha_L + g q_s/(RTV_{fac})} - {1+(L_c/c_p)\alpha_L} \\ - \gamma_{q_f} &=& -\frac{\gamma_{T_L}\alpha_L + g q_s/(RTV_{fac})} - {1+(L_s/c_p)\alpha_L} - \end{aligned} + \gamma_{q_{\ell}} = -\frac{\gamma_{T_L} \alpha_L + g q_s/(RTV_{fac})} + {1+(L_c/c_p)\alpha_L} + +.. math:: + + \gamma_{q_f} = -\frac{\gamma_{T_L}\alpha_L + g q_s/(RTV_{fac})} + {1+(L_s/c_p)\alpha_L} + where :math:`\gamma_{T_L} = -(g/c_p)+ \gamma_{\theta_{\ell}}` and :math:`\gamma_{\theta_{\ell}}` is given by (`[gradadj] <#gradadj>`__) @@ -5860,17 +6120,19 @@ grid-level based calculation: .. math:: - \begin{aligned} - z_c &=& z_c + \frac{\Delta_{k_b+\frac{1}{2}} z}{2} + z_c = z_c + \frac{\Delta_{k_b+\frac{1}{2}} z}{2} + + \mbox{min}\left[ \frac{ \Delta_{k_b+\frac{1}{2}}z + +\Delta_{k_b-\frac{1}{2}}z }{2} C_F^l, + \frac{ q_{\ell}}{ \gamma_{q_{\ell}} } \right]/C_F + +.. math:: + + \left. \hspace{2.4cm} + \mbox{min}\left[ \frac{ \Delta_{k_b+\frac{1}{2}}z - +\Delta_{k_b-\frac{1}{2}}z }{2} C_F^l, - \frac{ q_{\ell}}{ \gamma_{q_{\ell}} } \right]/C_F \nonumber \\ - & & \left. \hspace{2.4cm} - + \mbox{min}\left[ \frac{ \Delta_{k_b+\frac{1}{2}}z - +\Delta_{k_b-\frac{1}{2}}z }{2} C_F^f, - \frac{ q_f }{ \gamma_{q_f} } \right]/C_F \right. - \label{zc_calc} - \end{aligned} + +\Delta_{k_b-\frac{1}{2}}z }{2} C_F^f, + \frac{ q_f }{ \gamma_{q_f} } \right]/C_F \right. + \label{zc_calc} + When :math:`\gamma_{q_f}` is set to zero (currently as standard) the last term in (`[zc_calc] <#zc_calc>`__) is given by @@ -5931,23 +6193,27 @@ above and below using the adiabatic lapse rates are calculated as: .. math:: - \begin{aligned} - {q_{\ell}}_{\rm ct}& =& \frac{{q_{\ell}}_{\mbox{\tiny \rm NTML}}}{{C_F}^l_{\mbox{\tiny \rm NTML}}} - + (z_i-z_{\mbox{\tiny \rm NTML}}) \gamma_{q_{\ell}}\\ - q_{\ell}^+ & =& \mbox{max}\left[ 0, \, - \frac{{q_{\ell}}_{\mbox{\tiny \rm NTML}+2}}{{C_F}^l_{\mbox{\tiny \rm NTML}+2}} - - (z_{\mbox{\tiny \rm NTML}+2}-z_i) \gamma_{q_{\ell}} \right] - \end{aligned} + {q_{\ell}}_{\rm ct} = \frac{{q_{\ell}}_{\mbox{\tiny \rm NTML}}}{{C_F}^l_{\mbox{\tiny \rm NTML}}} + + (z_i-z_{\mbox{\tiny \rm NTML}}) \gamma_{q_{\ell}} + +.. math:: + + q_{\ell}^+ = \mbox{max}\left[ 0, \, + \frac{{q_{\ell}}_{\mbox{\tiny \rm NTML}+2}}{{C_F}^l_{\mbox{\tiny \rm NTML}+2}} + - (z_{\mbox{\tiny \rm NTML}+2}-z_i) \gamma_{q_{\ell}} \right] + and similarly for :math:`q_f` (noting that currently :math:`\gamma_{q_f}=0`) and for DSC layers. Then, .. math:: - \begin{aligned} - \Delta q_{\ell}& =& {C_F^l}_{\mbox{\tiny \rm NTML}+2}\, q_{\ell}^+ - {C_F^l}_{\mbox{\tiny \rm NTML}}\, {q_{\ell}}_{\rm ct}\\ - \Delta q_f & =& {C_F^f}_{\mbox{\tiny \rm NTML}+2}\, q_f^+ - {C_F^f}_{\mbox{\tiny \rm NTML}}\, {q_f}_{\rm ct} - \end{aligned} + \Delta q_{\ell} = {C_F^l}_{\mbox{\tiny \rm NTML}+2}\, q_{\ell}^+ - {C_F^l}_{\mbox{\tiny \rm NTML}}\, {q_{\ell}}_{\rm ct} + +.. math:: + + \Delta q_f = {C_F^f}_{\mbox{\tiny \rm NTML}+2}\, q_f^+ - {C_F^f}_{\mbox{\tiny \rm NTML}}\, {q_f}_{\rm ct} + The only other explicit account of variable cloud fraction is in (`[vbr] <#vbr>`__) for which it is assumed that buoyancy reversal can @@ -6201,13 +6467,15 @@ where .. math:: - \begin{aligned} - \tilde{\beta_T} = \beta_T - \alpha_L \beta_c, & - \tilde{\beta_q} & = \beta_q + \beta_c \\ - {\rm and} & - \beta_c & = a_L \left( \frac{L}{c_p} \beta_T - - \frac{1+c_v}{c_v} \beta_q \right) - \end{aligned} + \tilde{\beta_T} = \beta_T - \alpha_L \beta_c, + \tilde{\beta_q} = \beta_q + \beta_c + +.. math:: + + {\rm and} + \beta_c = a_L \left( \frac{L}{c_p} \beta_T + - \frac{1+c_v}{c_v} \beta_q \right) + Note that here :math:`\tilde{\beta_T}` and :math:`\tilde{\beta_q}` are strictly *in*-cloud parameters, while their definitions in boundary @@ -6274,10 +6542,12 @@ specific or mixing ratio, respectively) as: .. math:: - \begin{aligned} - \rho_{0} & =& \rho_*/(1+(1/\epsilon-1)q_S) \\ - \rho_{y0} & =& \rho_*/(1+(1/\epsilon)m_{vS}) - \end{aligned} + \rho_{0} = \rho_*/(1+(1/\epsilon-1)q_S) + +.. math:: + + \rho_{y0} = \rho_*/(1+(1/\epsilon)m_{vS}) + where, in each case, the surface humidity is taken as the surface saturated humidity over open sea but over land and ice surfaces this is From b7cf81f09318a1ab2452de95fff389fcb3265cb9 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Thu, 23 Apr 2026 13:56:57 +0100 Subject: [PATCH 045/116] Fixed tables. --- .../turbulence_schemes/bldoc.rst | 793 ++++++++++-------- 1 file changed, 421 insertions(+), 372 deletions(-) diff --git a/documentation/source/science_guide/turbulence_schemes/bldoc.rst b/documentation/source/science_guide/turbulence_schemes/bldoc.rst index cc45abdd4c..4d2f610bf3 100644 --- a/documentation/source/science_guide/turbulence_schemes/bldoc.rst +++ b/documentation/source/science_guide/turbulence_schemes/bldoc.rst @@ -1745,28 +1745,39 @@ used to calculate :math:`K_m^{\rm Sc}`. Discussion of some of the revisions ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -.. container:: center - - .. container:: - :name: tab:vscales - - .. table:: Convective and Neutral limits for velocity scales - - +-------------+-------------+-------------+-------------+-------------+ - | Formulation | Convective | | Neutral | | - | | limit | | limit | | - +-------------+-------------+-------------+-------------+-------------+ - | | :math:`w_m` | :math:`w_h` | :math:`w_m` | :math:`w_h` | - +-------------+-------------+-------------+-------------+-------------+ - | HB | :math:` | :math: | :math:`u_*` | :math:`u_*` | - | | 0.84 \,w_*` | `1.4 \,w_*` | | | - +-------------+-------------+-------------+-------------+-------------+ - | UM standard | :math:` | :math: | :math:`u_*` | :math: | - | | 0.63 \,w_*` | `1.7 \,w_*` | | `1.3 \,u_*` | - +-------------+-------------+-------------+-------------+-------------+ - | UM revised | :math:` | :math:`w_*` | :math:`u_*` | :math: | - | | 0.6 \,w_*` | | | `1.3 \,u_*` | - +-------------+-------------+-------------+-------------+-------------+ +.. list-table:: Convective and Neutral limits for velocity scales + :name: tab:vscales + :header-rows: 1 + + * - Formulation + - Convective limit + - + - Neutral limit + - + + * - + - :math:`w_m` + - :math:`w_h` + - :math:`w_m` + - :math:`w_h` + + * - HB + - :math:`0.84 \,w_*` + - :math:`1.4 \,w_*` + - :math:`u_*` + - :math:`u_*` + + * - UM standard + - :math:`0.63 \,w_*` + - :math:`1.7 \,w_*` + - :math:`u_*` + - :math:`1.3 \,u_*` + + * - UM revised + - :math:`0.6 \,w_*` + - :math:`w_*` + - :math:`u_*` + - :math:`1.3 \,u_*` .. container:: float :name: fig:stab_dep @@ -5361,48 +5372,45 @@ therefore needs to be computed only once, when the 1st or predictor stage is computed, i.e. :math:`X^{*}`. Briefly the following calculations take place for the scalar variables: -.. container:: center - - +-------------------+-------------------------------------------------+ - | CALL bdy_impl3(): | set up coefficients for | - | | (`[eq:dX_disc_top] <#eq:dX_disc_top>`__), | - | | (`[eq:dX_disc] <#eq:dX_disc>`__) and do a | - | | downward sweep; | - +-------------------+-------------------------------------------------+ - | | do a downward sweep using the original implicit | - | | scheme to | - +-------------------+-------------------------------------------------+ - | | compute information required by the surface | - | | implicit solver; | - +-------------------+-------------------------------------------------+ - | CALL sf_impl2(): | CALL im_sf_pt2(): compute :math:`F_{JULES}` | - | | (scalar implicit fluxes), | - +-------------------+-------------------------------------------------+ - | | using original surface implicit solver; | - +-------------------+-------------------------------------------------+ - | CALL bdy_impl4(): | set up (`[eq:dX_bottom] <#eq:dX_bottom>`__) and | - | | complete downward sweep; | - +-------------------+-------------------------------------------------+ - | | back substitute to compute implicit correction | - | | :math:`\delta X^{*}`; | - +-------------------+-------------------------------------------------+ - | CALL bdy_impl3(): | compute explicit flux | - | | :math:`F^*= | - | | F^n+K_X\frac{\partial \delta X^*}{\partial z}`; | - +-------------------+-------------------------------------------------+ - | | set up coefficients for | - | | (`[eq:dXtop_np1] <#eq:dXtop_np1>`__), | - | | (`[eq:dXk_np1] <#eq:dXk_np1>`__), | - | | (`[eq:dX1_np1] <#eq:dX1_np1>`__) and | - +-------------------+-------------------------------------------------+ - | | do a downward sweep; | - +-------------------+-------------------------------------------------+ - | CALL sf_impl2(): | only momentum variables are affected - no | - | | change in scalars; | - +-------------------+-------------------------------------------------+ - | CALL bdy_impl4(): | back substitute to compute final implicit | - | | correction :math:`\delta X^{n+1}` | - +-------------------+-------------------------------------------------+ +.. list-table:: Table + :name: table_name + :header-rows: 1 + + * - CALL bdy_impl3(): + - set up coefficients for (`[eq:dX_disc_top] <#eq:dX_disc_top>`__), (`[eq:dX_disc] <#eq:dX_disc>`__) and do a downward sweep; + + * - + - do a downward sweep using the original implicit scheme to + + * - + - compute information required by the surface implicit solver; + + * - CALL sf_impl2(): + - CALL im_sf_pt2(): compute :math:`F_{JULES}` (scalar implicit fluxes), + + * - + - using original surface implicit solver; + + * - CALL bdy_impl4(): + - set up (`[eq:dX_bottom] <#eq:dX_bottom>`__) and complete downward sweep; + + * - + - back substitute to compute implicit correction :math:`\delta X^{*}`; + + * - CALL bdy_impl3(): + - compute explicit flux :math:`F^*=F^n+K_X\frac{\partial \delta X^*}{\partial z}`; + + * - + - set up coefficients for (`[eq:dXtop_np1] <#eq:dXtop_np1>`__), (`[eq:dXk_np1] <#eq:dXk_np1>`__), (`[eq:dX1_np1] <#eq:dX1_np1>`__) and + + * - + - do a downward sweep; + + * - CALL sf_impl2(): + - only momentum variables are affected - no change in scalars; + + * - CALL bdy_impl4(): + - back substitute to compute final implicit correction :math:`\delta X^{n+1}` NB: for CABLE compatibility, sf_impl2 is now called by an intermediate routine surf_couple_implicit. @@ -6678,320 +6686,361 @@ results for these variables it will prevent UKCA jobs from regressing. If the changes are significant it would be prudent to discuss them with the UKCA code owner before lodging the change. -.. container:: center - - +-------------------+------+-------------------+-------------------+ - | Boundary layer | | | | - | inputs to UKCA | | | | - +===================+======+===================+===================+ - | Sec | Item | Description | Use in UKCA | - +-------------------+------+-------------------+-------------------+ - | 0 | 24 | SURFACE | dry deposition | - | | | TEMPERATURE AFTER | | - | | | TIMESTEP | | - +-------------------+------+-------------------+-------------------+ - | 0 | 25 | BOUNDARY LAYER | dry deposition, | - | | | DEPTH AFTER | bl nucleation and | - | | | TIMESTEP | call to tr_mix | - +-------------------+------+-------------------+-------------------+ - | 0 | 26 | ROUGHNESS LENGTH | dry deposition | - | | | AFTER TIMESTEP | | - +-------------------+------+-------------------+-------------------+ - | 0 | 233 | SURFACE | dry deposition | - | | | TEMPERATURE ON | | - | | | TILES K | | - +-------------------+------+-------------------+-------------------+ - | 0 | 234 | ROUGHNESS LENGTH | dry deposition | - | | | ON TILES m | | - +-------------------+------+-------------------+-------------------+ - | 3 | 60 | RHOKH_MIX | call to tr_mix | - +-------------------+------+-------------------+-------------------+ - | 3 | 64 | D | call to tr_mix | - | | | TRDZ_CHARNEY_GRID | | - +-------------------+------+-------------------+-------------------+ - | 3 | 65 | GRID-LEVEL OF SML | call to tr_mix | - | | | INVERSION (kent) | | - +-------------------+------+-------------------+-------------------+ - | 3 | 66 | Rho \* | call to tr_mix | - | | | entrainment rate | | - | | | (we_lim) | | - +-------------------+------+-------------------+-------------------+ - | 3 | 67 | Fraction of the | call to tr_mix | - | | | timestep (t_frac) | | - +-------------------+------+-------------------+-------------------+ - | 3 | 68 | zrzi | call to tr_mix | - +-------------------+------+-------------------+-------------------+ - | 3 | 69 | GRID-LEVEL OF DSC | call to tr_mix | - | | | INVERSION (kent) | | - +-------------------+------+-------------------+-------------------+ - | 3 | 70 | Rho \* | call to tr_mix | - | | | entrainment rate | | - | | | dsc | | - +-------------------+------+-------------------+-------------------+ - | 3 | 71 | Fraction of the | call to tr_mix | - | | | timestep dsc | | - +-------------------+------+-------------------+-------------------+ - | 3 | 72 | zrzi dsc | call to tr_mix | - +-------------------+------+-------------------+-------------------+ - | 3 | 73 | ZHSC Top of | call to tr_mix | - | | | decoupled layer | | - +-------------------+------+-------------------+-------------------+ - | 3 | 217 | SURFACE HEAT FLUX | dry deposition | - | | | W/M2 | | - +-------------------+------+-------------------+-------------------+ - | 3 | 230 | 10 METRE WIND | calculate sea | - | | | SPEED ON C-GRID | salt emissions | - +-------------------+------+-------------------+-------------------+ - | 3 | 401 | Dust Emissions | GLOMAP dust | - | | | div 1 | scheme | - +-------------------+------+-------------------+-------------------+ - | 3 | 402 | Dust Emissions | GLOMAP dust | - | | | div 2 | scheme | - +-------------------+------+-------------------+-------------------+ - | 3 | 403 | Dust Emissions | GLOMAP dust | - | | | div 3 | scheme | - +-------------------+------+-------------------+-------------------+ - | 3 | 404 | Dust Emissions | GLOMAP dust | - | | | div 4 | scheme | - +-------------------+------+-------------------+-------------------+ - | 3 | 405 | Dust Emissions | GLOMAP dust | - | | | div 5 | scheme | - +-------------------+------+-------------------+-------------------+ - | 3 | 406 | Dust Emissions | GLOMAP dust | - | | | div 6 | scheme | - +-------------------+------+-------------------+-------------------+ - | 3 | 430 | Dust Friction | dry deposition | - | | | velocity (U\*) on | | - | | | tiles | | - +-------------------+------+-------------------+-------------------+ - | 3 | 462 | STOMATAL | dry deposition | - | | | CONDUCTANCE ON | | - | | | PFTS (M/S) | | - +-------------------+------+-------------------+-------------------+ - | 3 | 465 | FRICTION VELOCITY | dry deposition | - +-------------------+------+-------------------+-------------------+ - | 3 | 473 | TURBULENT KINETIC | ACTIVATE cloud | - | | | ENERGY | scheme | - +-------------------+------+-------------------+-------------------+ +.. list-table:: Table + :name: table_name + :header-rows: 1 + + * - Boundary layer inputs to UKCA + - + - + - + + * - Sec + - Item + - Description + - Use in UKCA + + * - 0 + - 24 + - SURFACE TEMPERATURE AFTER TIMESTEP + - dry deposition + + * - 0 + - 25 + - BOUNDARY LAYER DEPTH AFTER TIMESTEP + - dry deposition, bl nucleation and call to tr_mix + + * - 0 + - 26 + - ROUGHNESS LENGTH AFTER TIMESTEP + - dry deposition + + * - 0 + - 233 + - SURFACE TEMPERATURE ON TILES K + - dry deposition + + * - 0 + - 234 + - ROUGHNESS LENGTH ON TILES m + - dry deposition + + * - 3 + - 60 + - RHOKH_MIX + - call to tr_mix + + * - 3 + - 64 + - D TRDZ_CHARNEY_GRID + - call to tr_mix + + * - 3 + - 65 + - GRID-LEVEL OF SML INVERSION (kent) + - call to tr_mix + + * - 3 + - 66 + - Rho \* entrainment rate (we_lim) + - call to tr_mix + + * - 3 + - 67 + - Fraction of the timestep (t_frac) + - call to tr_mix + + * - 3 + - 68 + - zrzi + - call to tr_mix + + * - 3 + - 69 + - GRID-LEVEL OF DSC INVERSION (kent) + - call to tr_mix + + * - 3 + - 70 + - Rho \* entrainment rate dsc + - call to tr_mix + + * - 3 + - 71 + - Fraction of the timestep dsc + - call to tr_mix + + * - 3 + - 72 + - zrzi dsc + - call to tr_mix + + * - 3 + - 73 + - ZHSC Top of decoupled layer + - call to tr_mix + + * - 3 + - 217 + - SURFACE HEAT FLUX W/M2 + - dry deposition + + * - 3 + - 230 + - 10 METRE WIND SPEED ON C-GRID + - calculate sea salt emissions + + * - 3 + - 401 + - Dust Emissions div 1 + - GLOMAP dust scheme + + * - 3 + - 402 + - Dust Emissions div 2 + - GLOMAP dust scheme + + * - 3 + - 403 + - Dust Emissions div 3 + - GLOMAP dust scheme + + * - 3 + - 404 + - Dust Emissions div 4 + - GLOMAP dust scheme + + * - 3 + - 405 + - Dust Emissions div 5 + - GLOMAP dust scheme + + * - 3 + - 406 + - Dust Emissions div 6 + - GLOMAP dust scheme + + * - 3 + - 430 + - Dust Friction velocity (U\*) on tiles + - dry deposition + + * - 3 + - 462 + - STOMATAL CONDUCTANCE ON PFTS (M/S) + - dry deposition + + * - 3 + - 465 + - FRICTION VELOCITY + - dry deposition + + * - 3 + - 473 + - TURBULENT KINETIC ENERGY + - ACTIVATE cloud scheme .. _`app:not`: Appendix: Notation ================== -.. container:: flushleft - - +--------------------------------+------------------------------------+ - | Finite difference notation | | - +================================+====================================+ - | :math:`z_k` | height of the :math:`\theta`-level | - | | :math:`k` | - +--------------------------------+------------------------------------+ - | :math:`z_{k+\frac{1}{2}}` | height of half-level above | - | | :math:`\theta`-level :math:`k` | - +--------------------------------+------------------------------------+ - | :math:`\Delta_k` | indicates a finite difference | - | | between :math:`\theta`-levels | - | | :math:`k` and :math:`k-1` | - +--------------------------------+------------------------------------+ - | :math:`\Delta_{k+\frac{1}{2}}` | indicates a finite difference | - | | between half-levels | - | | :math:`k+\frac{1}{2}` and | - | | :math:`k-\frac{1}{2}` | - +--------------------------------+------------------------------------+ - | :math:`\Delta` | note: real change (i.e., not | - | | necessarily finite-difference) in | - | | a parameter | - +--------------------------------+------------------------------------+ - | | across the capping inversion (see | - | | (`[dbinv] <#dbinv>`__) and | - | | following text) | - +--------------------------------+------------------------------------+ - -.. container:: flushleft - - +----------------------------------+----------------------------------+ - | Model variables | | - +==================================+==================================+ - | :math:`\theta_l`, | thermodynamic variables defined | - | :math:`\theta_{v\ell}` | by (`[thetal] <#thetal>`__) and | - | | (`[thetavl] <#thetavl>`__) | - +----------------------------------+----------------------------------+ - | :math:`T_v`, :math:`\theta_v` | virtual temperature and | - | | potential temperature, | - +----------------------------------+----------------------------------+ - | | defined by (`[Tv] <#Tv>`__) and | - | | in section | - | | (`3.1.1 <#sec:parxs>`__) | - +----------------------------------+----------------------------------+ - | :math:`b` | buoyancy (:math:`=g T_v'/T_v`) | - +----------------------------------+----------------------------------+ - | :math:`q_t`, :math:`q_v`, | specific humidities: | - | :math:`q_s`, :math:`q_{\ell}`, | | - | :math:`q_f` | | - +----------------------------------+----------------------------------+ - | | total, vapour, saturated, liquid | - | | and frozen water, respectively | - +----------------------------------+----------------------------------+ - | :math:`C_F`, :math:`C_F^l`, | cloud fraction and the liquid | - | :math:`C_F^f` | and frozen water parts, | - | | respectively | - +----------------------------------+----------------------------------+ - | :math:`{\cal H}` | total heat flux (net radiative | - | | plus turbulent, | - | | Kms\ :math:`^{-1}`) | - +----------------------------------+----------------------------------+ - -.. container:: flushleft - - +----------------------------------+----------------------------------+ - | Thresholds | | - +==================================+==================================+ - | :math:`C_t` | (:math:`=1.1`) threshold for | - | | ratio of layer | - | | :math:`q_t`-gradients in cumulus | - | | diagnosis | - +----------------------------------+----------------------------------+ - | :math:`\Gamma_{\rm inv}` | (:math:`=1.1`) threshold on | - | | ratio of environment to parcel | - | | :math:`\theta_v` gradients | - +----------------------------------+----------------------------------+ - | | for identifying capping | - | | inversions above the LCL | - +----------------------------------+----------------------------------+ - | SC_CFTOL | (:math:`=0.1`) :math:`C_F` | - | | threshold for recognising the | - | | presence of Sc | - +----------------------------------+----------------------------------+ - | :ma | threshold (in Km\ :math:`^{-1}`) | - | th:`\Delta_{k_{ct}} \theta_{v\el | for initial diagnosis of | - | l}/ \Delta_{k_{ct}} z < 10^{-3}` | *well-mixed* DSC layers | - +----------------------------------+----------------------------------+ - | :math:`D_t` | (:math:`=0.1`) threshold for the | - | | ratio of buoyancy consumption to | - | | production | - +----------------------------------+----------------------------------+ - | | before decoupling occurs | - +----------------------------------+----------------------------------+ - -.. container:: flushleft - - +----------------------------------+----------------------------------+ - | Layer definitions and parameters | | - +==================================+==================================+ - | SML | surface-based mixed layer | - +----------------------------------+----------------------------------+ - | NTML | top :math:`\theta`-level within | - | | SML | - +----------------------------------+----------------------------------+ - | NTPAR | top :math:`\theta`-level reached | - | | by parcel ascent | - +----------------------------------+----------------------------------+ - | DSC | decoupled stratocumulus (mixed | - | | layer) | - +----------------------------------+----------------------------------+ - | NTDSC | top :math:`\theta`-level within | - | | DSC layer | - +----------------------------------+----------------------------------+ - | NBDSC | bottom :math:`\theta`-level | - | | within DSC layer | - +----------------------------------+----------------------------------+ - | NTLOC | top :math:`\theta`-level below | - | | which :math:`Ri<1` | - +----------------------------------+----------------------------------+ - | :math:`z_{\rm h}`  | height of top of SML | - | | (potentially subgrid) | - +----------------------------------+----------------------------------+ - | :math:`z_{\rm h}^{\rm Sc}`  | height of top of DSC layer | - | | (potentially subgrid) | - +----------------------------------+----------------------------------+ - | :math:`z_{\rm b}`  | height of base of DSC layer | - | | (subgrid) | - +----------------------------------+----------------------------------+ - | :math:`z_{\rm par}`  | height of half-level at top of | - | | parcel ascent | - +----------------------------------+----------------------------------+ - | :math:`z_{\rm loc}`  | height of half-level marking | - | | ‘top’ of local :math:`Ri`-based | - | | mixing | - +----------------------------------+----------------------------------+ - | | (where :math:`Ri>1`) | - +----------------------------------+----------------------------------+ - | :math:`z_i` | generic inversion height | - +----------------------------------+----------------------------------+ - | :math:`z_c` | cloud depth | - +----------------------------------+----------------------------------+ - | :math:`z_{\rm ml}` | mixed layer depth | - +----------------------------------+----------------------------------+ - | :math:`K_m^{\rm surf}`, | :math:`K` profiles for | - | :math:`K_h^{\rm surf}` | surface-driven turbulence (in | - | | SML) | - +----------------------------------+----------------------------------+ - | :math:`K_m^{\rm Sc}`, | :math:`K` profiles for | - | :math:`K_h^{\rm Sc}` | cloud-top-driven turbulence | - +----------------------------------+----------------------------------+ - | | (calculated for both DSC and | - | | SML) | - +----------------------------------+----------------------------------+ - | LCL | lifting condensation level | - +----------------------------------+----------------------------------+ - -.. container:: flushleft - - +----------------------------------+----------------------------------+ - | Other parameters | | - +==================================+==================================+ - | :math:`\gamma_{\theta_{\ell}}` | gradient adjustment term, given | - | | by (`[gradadj] <#gradadj>`__) | - +----------------------------------+----------------------------------+ - | :math:`w_m` | scaling velocity for momentum | - | | mixing in the SML | - +----------------------------------+----------------------------------+ - | | (used in :math:`K_m^{\rm surf}`, | - | | :math:`\gamma_{\theta_{\ell}}` | - | | and the SML parcel perturbation, | - | | :math:`\theta_v'`) | - +----------------------------------+----------------------------------+ - | :math:`w_*` | ‘standard’ convective velocity | - | | scale for a cloud-free | - | | convective | - +----------------------------------+----------------------------------+ - | | boundary layer, | - | | :math:`w_ | - | | *^3 = z_{\rm h}\overline{w'b}_S` | - +----------------------------------+----------------------------------+ - | :math:`u_*` | friction velocity (here includes | - | | the orographic component) | - +----------------------------------+----------------------------------+ - | :math:`w_e`, :math:`\tilde{w_e}` | entrainment velocity and | - | | compensated to allow for | - | | subsidence (ms\ :math:`^{-1}`) | - +----------------------------------+----------------------------------+ - | :math:`w_S` | subsidence velocity | - | | (ms\ :math:`^{-1}`) | - +----------------------------------+----------------------------------+ - | :math:`\Delta_F` | cloud-top net radiative | - | | divergence, calculation given in | - | | (`[ctraddiv] <#ctraddiv>`__) | - +----------------------------------+----------------------------------+ - | :math:`\alpha_t` | parameter in entrainment | - | | parametrization, | - | | (`[we_parm] <#we_parm>`__) | - +----------------------------------+----------------------------------+ - | :math:`\tau_{rc}`, | parameters in perturbation | - | :math:`z_{rc}` | calculation, | - | | (`[dscd_pert] <#dscd_pert>`__), | - +----------------------------------+----------------------------------+ - | | for initial identification of | - | | and :math:`z_{\rm ml}` | - | | calculation for DSC layers | - +----------------------------------+----------------------------------+ - | :math:`a_L`, :math:`\alpha_L`, | buoyancy parameters, defined in | - | :math:`\beta_T`, | appendix `12 <#app:buoyp>`__ | - | :math:`\beta_q`, | | - | :math:`\tilde{\beta_T}`, | | - | :math:`\tilde{\beta_q}` | | - +----------------------------------+----------------------------------+ +.. list-table:: Table + :name: table_name + :header-rows: 1 + + * - Finite difference notation + - + + * - :math:`z_k` + - height of the :math:`\theta`-level :math:`k` + + * - :math:`z_{k+\frac{1}{2}}` + - height of half-level above :math:`\theta`-level :math:`k` + + * - :math:`\Delta_k` + - indicates a finite difference between :math:`\theta`-levels :math:`k` and :math:`k-1` + + * - :math:`\Delta_{k+\frac{1}{2}}` + - indicates a finite difference between half-levels :math:`k+\frac{1}{2}` and :math:`k-\frac{1}{2}` + + * - :math:`\Delta` + - note: real change (i.e., not necessarily finite-difference) in a parameter + + * - + - across the capping inversion (see (`[dbinv] <#dbinv>`__) and following text) + +.. list-table:: Table + :name: table_name + :header-rows: 1 + + * - Model variables + - + + * - :math:`\theta_l`, :math:`\theta_{v\ell}` + - thermodynamic variables defined by (`[thetal] <#thetal>`__) and (`[thetavl] <#thetavl>`__) + + * - :math:`T_v`, :math:`\theta_v` + - virtual temperature and potential temperature, + + * - + - defined by (`[Tv] <#Tv>`__) and in section (`3.1.1 <#sec:parxs>`__) + + * - :math:`b` + - buoyancy (:math:`=g T_v'/T_v`) + + * - :math:`q_t`, :math:`q_v`, :math:`q_s`, :math:`q_{\ell}`, :math:`q_f` + - specific humidities: + + * - + - total, vapour, saturated, liquid and frozen water, respectively + + * - :math:`C_F`, :math:`C_F^l`, :math:`C_F^f` + - cloud fraction and the liquid and frozen water parts, respectively + + * - :math:`{\cal H}` + - total heat flux (net radiative plus turbulent, Kms\ :math:`^{-1}`) + +.. list-table:: Table + :name: table_name + :header-rows: 1 + + * - Thresholds + - + + * - :math:`C_t` + - (:math:`=1.1`) threshold for ratio of layer :math:`q_t`-gradients in cumulus diagnosis + + * - :math:`\Gamma_{\rm inv}` + - (:math:`=1.1`) threshold on ratio of environment to parcel :math:`\theta_v` gradients + + * - + - for identifying capping inversions above the LCL + + * - SC_CFTOL + - (:math:`=0.1`) :math:`C_F` threshold for recognising the presence of Sc + + * - :math:`\Delta_{k_{ct}} \theta_{v\ell}/ \Delta_{k_{ct}} z < 10^{-3}` + - threshold (in Km\ :math:`^{-1}`) for initial diagnosis of *well-mixed* DSC layers + + * - :math:`D_t` + - (:math:`=0.1`) threshold for the ratio of buoyancy consumption to production + + * - + - before decoupling occurs + +.. list-table:: Table + :name: table_name + :header-rows: 1 + + * - Layer definitions and parameters + - + + * - SML + - surface-based mixed layer + + * - NTML + - top :math:`\theta`-level within SML + + * - NTPAR + - top :math:`\theta`-level reached by parcel ascent + + * - DSC + - decoupled stratocumulus (mixed layer) + + * - NTDSC + - top :math:`\theta`-level within DSC layer + + * - NBDSC + - bottom :math:`\theta`-level within DSC layer + + * - NTLOC + - top :math:`\theta`-level below which :math:`Ri<1` + + * - :math:`z_{\rm h}` + - height of top of SML (potentially subgrid) + + * - :math:`z_{\rm h}^{\rm Sc}` + - height of top of DSC layer (potentially subgrid) + + * - :math:`z_{\rm b}` + - height of base of DSC layer (subgrid) + + * - :math:`z_{\rm par}` + - height of half-level at top of parcel ascent + + * - :math:`z_{\rm loc}` + - height of half-level marking ‘top’ of local :math:`Ri`-based mixing + + * - + - (where :math:`Ri>1`) + + * - :math:`z_i` + - generic inversion height + + * - :math:`z_c` + - cloud depth + + * - :math:`z_{\rm ml}` + - mixed layer depth + + * - :math:`K_m^{\rm surf}`, :math:`K_h^{\rm surf}` + - :math:`K` profiles for surface-driven turbulence (in SML) + + * - :math:`K_m^{\rm Sc}`, :math:`K_h^{\rm Sc}` + - :math:`K` profiles for cloud-top-driven turbulence + + * - + - (calculated for both DSC and SML) + + * - LCL + - lifting condensation level + +.. list-table:: Table + :name: table_name + :header-rows: 1 + + * - Other parameters + - + + * - :math:`\gamma_{\theta_{\ell}}` + - gradient adjustment term, given by (`[gradadj] <#gradadj>`__) + + * - :math:`w_m` + - scaling velocity for momentum mixing in the SML + + * - + - (used in :math:`K_m^{\rm surf}`, :math:`\gamma_{\theta_{\ell}}` and the SML parcel perturbation, :math:`\theta_v'`) + + * - :math:`w_*` + - ‘standard’ convective velocity scale for a cloud-free convective + + * - + - boundary layer, :math:`w_*^3 = z_{\rm h}\overline{w'b}_S` + + * - :math:`u_*` + - friction velocity (here includes the orographic component) + + * - :math:`w_e`, :math:`\tilde{w_e}` + - entrainment velocity and compensated to allow for subsidence (ms\ :math:`^{-1}`) + + * - :math:`w_S` + - subsidence velocity (ms\ :math:`^{-1}`) + + * - :math:`\Delta_F` + - cloud-top net radiative divergence, calculation given in (`[ctraddiv] <#ctraddiv>`__) + + * - :math:`\alpha_t` + - parameter in entrainment parametrization, (`[we_parm] <#we_parm>`__) + + * - :math:`\tau_{rc}`, :math:`z_{rc}` + - parameters in perturbation calculation, (`[dscd_pert] <#dscd_pert>`__), + + * - + - for initial identification of and :math:`z_{\rm ml}` calculation for DSC layers + + * - :math:`a_L`, :math:`\alpha_L`, :math:`\beta_T`, :math:`\beta_q`, :math:`\tilde{\beta_T}`, :math:`\tilde{\beta_q}` + - buoyancy parameters, defined in appendix `12 <#app:buoyp>`__ .. [1] unless the option to mix across the LCL is selected, see From cce11ea4b4f0f276692ca5fc6619730793742fa2 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Thu, 23 Apr 2026 17:49:06 +0100 Subject: [PATCH 046/116] Fixed figures. --- .../turbulence_schemes/blank.svg | 3 + .../turbulence_schemes/bldoc.rst | 98 ++++++++++++++++--- 2 files changed, 86 insertions(+), 15 deletions(-) create mode 100644 documentation/source/science_guide/turbulence_schemes/blank.svg diff --git a/documentation/source/science_guide/turbulence_schemes/blank.svg b/documentation/source/science_guide/turbulence_schemes/blank.svg new file mode 100644 index 0000000000..d5b329f06a --- /dev/null +++ b/documentation/source/science_guide/turbulence_schemes/blank.svg @@ -0,0 +1,3 @@ + + diff --git a/documentation/source/science_guide/turbulence_schemes/bldoc.rst b/documentation/source/science_guide/turbulence_schemes/bldoc.rst index 4d2f610bf3..1a75966f7c 100644 --- a/documentation/source/science_guide/turbulence_schemes/bldoc.rst +++ b/documentation/source/science_guide/turbulence_schemes/bldoc.rst @@ -231,9 +231,17 @@ been categorised into 7 distinct ‘boundary layer types’: Types I to VI are shown schematically in Fig. `1 <#fig:bltypes>`__. -.. container:: float +.. figure:: blank.svg :name: fig:bltypes + Schematic representation of boundary layer types I to VI. The top of the + upward arrows indicate the height \zhpar while the top of their solid line + portions indicate \zh. + + +----------------------------------+----------------------------------+ + | .. image:: wcrp_bltypes1.svg | .. image:: wcrp_bltypes2.svg | + +----------------------------------+----------------------------------+ + .. _`sec:adiapar`: The diagnostic parcel ascent and cumulus diagnosis @@ -706,10 +714,17 @@ of the first :math:`\theta`-level below :math:`z_h-100` m (a physically reasonable depth over which cloud-top radiative cooling might be expected to occur) and :math:`z_{\mbox{\tiny \rm NTML}-1}`. -.. container:: float +.. figure:: blank.svg :name: fig:inv_integ - .. container:: center + Subgrid (lines) and model (symbols) fluxes of $\thetal$: turbulent flux + (dash-dotted, crosses), radiative flux (dashed, triangles) and total flux + (solid). The shaded area illustrates the integrated turbulent flux that + would be obtained were (\protect\mbox{\protect\ref{eq:wx_std}}) used. + + +-----------------------------------+ + | .. image:: ideal_invinteg.svg | + +-----------------------------------+ Then, @@ -1779,9 +1794,19 @@ Discussion of some of the revisions - :math:`u_*` - :math:`1.3 \,u_*` -.. container:: float +.. figure:: blank.svg :name: fig:stab_dep + Stability dependence of the surface velocity scales, Prandtl number + (although I hope something is wrong with my coding of HB here!) and $d$. + Solid lines are from HB, dotted from the standard UM and the dashed from the + revised formulation. The dash-dotted line for $d$ is a potential + modification, as described in the text. + + +------------------------------+ + | .. image:: stab_dep.svg | + +------------------------------+ + It is useful to compare the velocity scales in the revised scheme with those in the standard version, as well as those in :raw-latex:`\cite{holtslag93:_local_versus_nonloc_bound_layer}`, @@ -1833,12 +1858,16 @@ adjustment in the shape. In addition, note that the factors removed since the entrainment flux is now carried via the explicit :math:`f_2` term. -.. container:: float +.. figure:: blank.svg :name: fig:new_ksc - .. container:: center + Standard UM $\khtop$ (solid) and revised (dotted), both scaled by $k z_h + \vtopo$. An upside-down version of $\khsurf$ is also shown (dashed) for + comparison. - |image1| + +-----------------------------------+ + | .. image:: new_ktop_shape.svg | + +-----------------------------------+ .. _`sec:blend`: @@ -1971,10 +2000,20 @@ x`, where Eq. `[eq-tanh] <#eq-tanh>`__ tends to zero faster. This is by choice, to force the highest resolution simulations to use the 3D turbulence scheme. -.. container:: float +.. figure:: blank.svg :name: fig-blend - |image| |image2| + (a) Weighting for the 1D boundary-layer scheme as a function of $\Delta + x/z_{\rm turb}$, showing the function of Equation~\ref{eq-tanh} (blue + solid), the equation in \cite{Boutleetal2014} (black solid) and the TKE + partitioning of \cite{Honnertetal2011} (mean thick dashed, 5th/95th + percentiles thin dashed). (b) Schematic showing the calculation of $z_{\rm + turb}$ used in Eq.~\ref{eq-tanh} for a well-mixed layer (black dotted) and a + decoupled cloud layer (black solid). + + +------------------------------------+------------------------------------+ + | .. image:: honnert_vs_tanh.svg | .. image:: zturb_schem.svg | + +------------------------------------+------------------------------------+ One of the key benefits of the :raw-latex:`\cite{lock00}` scheme is its ability to represent decoupled stratocumulus layers, and this is a @@ -2251,9 +2290,18 @@ turbulent fluxes at the base of the mixed layer are assumed zero except for the SML where the surface fluxes are used. This interpolation is illustrated for a SML in Fig. `6 <#fig:fluxinterp>`__. -.. container:: float +.. figure:: blank.svg :name: fig:fluxinterp + Idealised profiles of (a) $\wqt$ (dash-dotted line) and (b) ${\cal H}$ + (dotted line), $\wthl$ (dash-dotted) and $F$ (dashed). The continuous lines + are the turbulent fluxes on the model grid indicated by the dashed + horizontal lines. + + +---------------------------------+ + | .. image:: subsent_fig7.svg | + +---------------------------------+ + Note that, because (`[fluxinterp] <#fluxinterp>`__) includes an explicit balance between the turbulent and radiative fluxes for :math:`\overline{w'\theta_{\ell}'}`, it is not possible to parametrize @@ -2345,9 +2393,16 @@ radiation described above is attempted (the local scheme is also currently not set to zero above NTML or NTDSC when this occurs to allow it to diffuse out this static instability). -.. container:: float +.. figure:: blank.svg :name: zi_diag + Schematic illustrating the assumptions behind the subgrid diagnosis of + $z_i$. + + +----------------------------------+ + | .. image:: nbldoc_zidiag.svg | + +----------------------------------+ + Having identified the model grid-level at the top of the well-mixed layer (either level NTML from the parcel ascent, as described in section `3.1 <#sec:adiapar>`__, or NTDSC for DSC layers, see section @@ -2489,9 +2544,18 @@ between the parametrized entrainment rate, :math:`w_e`, and the large-scale vertical velocity evaluated at the inversion, :math:`w|_{z_h}`). -.. container:: float +.. figure:: blank.svg :name: fig:rev_fluxes + Subgrid (lines) and model (symbols) profiles and fluxes of, top row, $q_t$ + and, bottom row, $\thetal$: turbulent fluxes (dash-dotted, crosses), + subsidence fluxes (dotted, diamonds), radiative flux (dashed, triangles) and + total flux (solid, squares). + + +----------------------------------+ + | .. image:: ideal_revflux.svg | + +----------------------------------+ + An idealised subgrid total flux profile is constructed from the parametrized entrainment flux and the increments from radiation, precipitation and subsidence, assuming a well-mixed boundary layer @@ -6337,12 +6401,16 @@ between these two schemes but the 9B parametrization is clearly an improvement on the 8A which gives :math:`\Delta_F=0` around midday (and therefore zero entrainment and turbulent mixing). -.. container:: float +.. figure:: blank.svg :name: fig:dradts - .. container:: centering + Time series from LES of $\Delta_\radf$ (solid), $\Delta_\radf^{LW} $ + (dotted), $-\Delta_\radf^{SW} $ (dashed) and the 8A (dash-dot) and 9B (dash- + dot-dot-dot) parametrizations of $\Delta_\radf$. - |image3| |image4| + +------------------------------+------------------------------+ + | .. image:: div_r080.svg | .. image:: div_r071.svg | + +------------------------------+------------------------------+ The 9C version attempted to remove the grid-dependence implied by the summation over 3 grid-levels in (`[ctraddiv] <#ctraddiv>`__) as follows: From 6cb465c196a1b3ac0685ebaa18971f2210e79c8c Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Fri, 24 Apr 2026 15:17:04 +0100 Subject: [PATCH 047/116] Improved automated handling of figures and section cross-referencing. --- .../cloud_schemes/UMDP30_PC2CloudScheme.rst | 528 ++++++++++-------- .../science_guide/cloud_schemes/blank.svg | 3 + 2 files changed, 292 insertions(+), 239 deletions(-) create mode 100644 documentation/source/science_guide/cloud_schemes/blank.svg diff --git a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst index 5d630f081e..3519d2f1ca 100644 --- a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst +++ b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst @@ -64,7 +64,8 @@ straightforward to solve if one is allowed to assume that there is no variability of moisture or temperature on a scale of a model gridbox. In this case the cloud fraction scheme is redundant and only the condensation part remains, which may be solved diagnostically using the -instantaneous condensation assumption in section :ref:`The ‘s’ distribution`. +instantaneous condensation assumption in section :ref:`The ‘s’ distribution +`. However, the ‘no-variability’ assumption is poor until very high resolutions close to, or maybe exceeding, 1 km in the horizontal are reached. Although we may eventually assume that computer power will @@ -76,7 +77,7 @@ parametrization. There are several approaches to take to the solution of the problem, although they are not as independent as often portrayed, since they nearly all require the same instantaneous condensation assumption -(discussed in section :ref:`The ‘s’ distribution`). Hence there are +(discussed in section :ref:`The ‘s’ distribution `). Hence there are mathematical links between all the approaches. *The following are all valid structures to use in this respect.* @@ -118,7 +119,7 @@ and to break the hard diagnostic link between cloud fraction and condensate. These major features of the `Tiedtke (1993)`_ scheme provide the motivation to develop the PC2 cloud scheme. -.. _The ‘s’ distribution: +.. _sec_s_dist: The ‘s’ distribution -------------------- @@ -270,7 +271,8 @@ a purely diagnostic representation such as `Smith (1990)`_ where we explicitly consider distributions of :math:`s`. Strictly, the linear approximation implies that other approximations for :math:`\alpha` are valid: PC2 will do this (see section -:ref:`Numerical application`) since we are concerned in PC2 with the +:ref:`Numerical application `) since we are concerned in PC2 +with the best estimate of the *changes* to :math:`\overline{q_{cl}}`, not the best estimate of :math:`\overline{q_{cl}}` itself. @@ -319,9 +321,8 @@ Concept of PC2 The PC2 scheme develops prognostic expressions for the rates of change of cloud fraction and condensate contents as a result of each process that acts in the model. We consider ice and liquid condensate as two -distinct aspects of clouds, which may or may not overlap. -:numref:`Figure %s ` -provides a schematic summary of the PC2 scheme. +distinct aspects of clouds, which may or may not overlap :numref:`Figure %s +` provides a schematic summary of the PC2 scheme. The equations for the five prognostic cloud variables can be written schematically: @@ -389,7 +390,8 @@ given by :eq:`eq:int_gs_ds` and concept of instantaneous condensation for liquid clouds. Equations :eq:`eq:int_gs_ds` and :eq:`eq:qclbar=int` will form the basis of the -homogeneous forcing methods discussed in section :ref:`Homogeneous forcing`. +homogeneous forcing methods discussed in section :ref:`Homogeneous forcing +`. We note in particular that the convective cloud fraction, previously a quantity that is diagnosed separately from the large-scale cloud fraction calculated by the `Smith (1990)`_ scheme, may, in @@ -407,15 +409,18 @@ specifically developed generic approaches that can be used to calculate expressions for :math:`\frac{\partial \overline{q_{cl}}}{\partial t}` and :math:`\frac{\partial C_l} {\partial t}` . These are referred to as Homogeneous forcing (section -:ref:`Homogeneous forcing`), Injection source (or inhomogeneous forcing, -section :ref:`Injection forcing`) and Width Changing (section -:ref:`Changing the width of the PDF - PC2 erosion`). Two additional modules are -available to assist -with PC2, liquid cloud initiaion (section :ref:`Initiation of cloud`) and the +:ref:`Homogeneous forcing `), Injection source (or inhomogeneous +forcing, +section :ref:`Injection forcing `) and Width Changing (section +:ref:`Changing the width of the PDF - PC2 erosion `). Two additional +modules are available to assist +with PC2, liquid cloud initiaion (section :ref:`Initiation of cloud +`) and the calculation of total cloud fraction changes (section :ref:`Ice cloud and mixed -phase regions`). +phase regions `). At the present time, only the large-scale precipitation (section -:ref:`Large-scale precipitation`) scheme has been rewritten fully to use the PC2 +:ref:`Large-scale precipitation `) scheme has been rewritten fully +to use the PC2 concept of prognostic cloud fractions. The existing mass-flux convection scheme has been modified to enable calculation of the detrained condensate, but direct modification to the cloud fraction is not @@ -444,7 +449,7 @@ of representation of cloud inhomogeneity. Hence the code still exists to enable PC2 to be run with or without a diagnostic convective cloud fraction, although PC2:66 does not include a -diagnostic term. More details are in section :ref:`Convection`. +diagnostic term. More details are in section :ref:`Convection `. Physical basis of the PC2 prognostic cloud scheme ================================================= @@ -453,7 +458,7 @@ In this section we will develop the physical models that PC2 uses in order to calculate its prognostic increment terms. We will also consider the numerical solution of the models. The way in which these are incorporated into the Unifed Model will be discussed in section -:ref:`Implementation in the Unified Model` +:ref:`Implementation in the Unified Model ` Instantaneous condensation -------------------------- @@ -466,7 +471,7 @@ cloud in PC2. We will start by looking at changes to applied to a gridbox, under the assumption of instantaneous condensation. -.. _Homogeneous forcing: +.. _sec_homog: Homogeneous forcing ------------------- @@ -580,7 +585,8 @@ form a complete mathematical set for the solution of :math:`C_l` and initial value of :math:`C_l` is not identically 0 or 1. If :math:`C_l` is 0 or 1 then :math:`G(-Q_c)` remains at zero. The equation set then needs to be initiated in some way. This is discussed further in -`Wilson and Gregory (2003)`_ and in section :ref:`Initiation of cloud`. If +`Wilson and Gregory (2003)`_ and in section :ref:`Initiation of cloud +`. If :math:`G(-Q_c)` is defined, then application of the above equation set may be used to trace out an underlying PDF for any input values of :math:`C_l`, :math:`\overline{q_{cl}}` and :math:`SD`. Although we never @@ -689,7 +695,7 @@ closure removed occasional spurious very large cloud-fraction increments that occur when using :eq:`eqn22`, but did not significantly impact the performance of the model forecast. -.. _Numerical application: +.. _sec_homog_num_app: Numerical application ^^^^^^^^^^^^^^^^^^^^^ @@ -811,7 +817,8 @@ interfere with the possible cancellation of positive and negative increments from different physics schemes. A checking routine is applied, however, in the Unified Model to remove any negative values that are generated, which is discussed in section -:ref:`Bounds checking`. However, the checking routine (Q-Pos) involves a +:ref:`Bounds checking `. However, the checking routine (Q-Pos) +involves a lot of communication between processors and can significantly increase the run-time of the model. The option to “Ensure consistent sinks of qcl and CFL” performs a check at the end of the homogeneous forcing routines @@ -829,7 +836,7 @@ the net change in :math:`\overline{q_T}` minus the net change in calculated simply to account for the latent heat released due to the condensation. -.. _Changing the width of the PDF - PC2 erosion: +.. _sec_width: Changing the width of the PDF - PC2 erosion ------------------------------------------- @@ -940,15 +947,15 @@ of the process that is occuring. Note we don’t need to calculate :math:`b_s` separately, just its *fractional* rate of change. Options for the parameterisation of :math:`\frac{1}{b_s}\frac{\partial b_s}{\partial t}` due to turbulent -“erosion” are described in section :ref:`PC2 erosion`, along with the +“erosion” are described in section :ref:`PC2 erosion `, along with the numerical methods used to integrate the equations. -.. _Initiation of cloud: +.. _sec_init: Initiation of cloud ------------------- -In section :ref:`Homogeneous forcing` we commented that the closure +In section :ref:`Homogeneous forcing ` we commented that the closure :eq:`eqn22` for :math:`G(-Qc)` is only valid if :math:`C_l` is not identically 0 or 1. If :math:`C_l` is 0 or 1 we know that :math:`G(-Q_c)` is equal to 0 but we have lost the information that will @@ -990,7 +997,8 @@ description in :eq:`eqn19` to obtain the expressions We now need to parametrize the PDF width :math:`b_s`. Unlike the `Smith (1990)`_ scheme, this is the only location in the PC2 cloud scheme where the width needs to be defined for the liquid cloud -(although see section :ref:`Deposition and sublimation` for a discussion of an +(although see section :ref:`Deposition and sublimation ` for a +discussion of an equivalent width in the deposition / sublimation relationship for ice cloud). We still choose to define :math:`b_s` in terms of a critical relative humidity parameter, :math:`RH_{crit}`. Like the @@ -1041,7 +1049,7 @@ within this diagnostic calculation of SD it might actually be better to use the representation :eq:`eq:alpha` used by the diagnostic `Smith (1990)`_ scheme. -.. _Numerical Application of the Smith method: +.. _sec_numapp_init: Numerical Application of the Smith method ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ @@ -1069,7 +1077,7 @@ circumstances in which we wish to proceed further: Note: in the UM implementation, the actual conditions for when initiation may occur are more complicated than this, and there are several options depending on a namelist switch. See section -:ref:`Initiation` for details... +:ref:`Initiation ` for details... In the second case, we then make the temporary transformation of variables in order to use the same solution set as in the first case: @@ -1167,7 +1175,7 @@ where the superscript :math:`[i]` labels each iteration. We find that 10 iterations is effective for convergence, with the weighting :math:`f` given by :math:`a_L^{[i]}`. -.. _Initiation using the bimodal scheme: +.. _sec_bimodal_init: Initiation using the bimodal scheme ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ @@ -1198,14 +1206,14 @@ upper and lower values of :math:`Q_N` are then compared to -1 and 1 respectively, to determine whether the saturation boundary lies within the PDF bounds. This is the basic condition for initiation to occur (though there are additional conditions and various options for these in -the soure code; see section :ref:`Initiation`). +the soure code; see section :ref:`Initiation `). If the initiation conditions are met, the diagnostic bimodal cloud scheme code is then called (see UMDP 039), and the diagnosed :math:`C_l` and :math:`q_{cl}` are used to set the prognostic :math:`C_l` and :math:`q_{cl}`. -.. _Injection forcing: +.. _sec_inhomog: Injection forcing ----------------- @@ -1232,8 +1240,8 @@ air. The fractional rate at which existing air is replaced by the injected source air we will write as :math:`\frac{\partial{C_S}}{\partial{t}}`. Provided that only the liquid phase exists (see section -:ref:`Multiple phases in the injection source` for the extention to multiple -phases), we then +:ref:`Multiple phases in the injection source ` for the extention +to multiple phases), we then note that the rate of change of liquid cloud fraction and liquid water content in the gridbox can be written in two parts: firstly the change due to the background, and secondly the change due to the source. @@ -1273,7 +1281,7 @@ completely cloudy air. This equation allows one to calculate the change in :math:`C_l` associated with an injection source change of :math:`\overline{q_{cl}}` for the example of convection. Modifications to the mass-flux convection scheme for PC2 (far from trivial and -discussed in depth in section :ref:`Convection`) allow :math:`Q4_l` +discussed in depth in section :ref:`Convection `) allow :math:`Q4_l` to be calculated (:math:`q_{cl}^S` is already available), and :eq:`eq:dcdt_inhom2` can then be used to calculate the equivalent :math:`C_l` change. We note at this stage that the @@ -1290,7 +1298,7 @@ We could reasonably calculate the change in cloud fraction following the same methods as used to calculate the change in :math:`\overline{q}` or the change in a tracer and we discuss this later. -.. _Multiple phases in the injection source: +.. _sec_multiple: Multiple phases in the injection source ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ @@ -1343,7 +1351,7 @@ equivalently to :eq:`eq:dcdt_inhom` as \frac{\partial{C_l}}{\partial{t}} = - C_l \frac{\partial{C_S}}{\partial{t}} + g_l - \frac{\partial{C_S}}{\partial{t}} . + \frac{\partial{C_S}}{\partial{t}} . Combining :eq:`eq:dcldt_inhom` and :eq:`eq:dctdt_inhom` by eliminating @@ -1429,12 +1437,12 @@ the ice cloud fraction and total cloud fraction. with :math:`\delta_{xi} = h_i = g_i`. These are the expressions that are used within the convection scheme. It still remains to parametrize :math:`\delta_{xl}`, which is given by the convection scheme itself. -This is discussed in section :ref:`Phase of condensate`. +This is discussed in section :ref:`Phase of condensate `. -.. _Numerical application of injection forcing: +.. _sec_multi_numapp: -Numerical application of injection forcing -^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ +Numerical application +^^^^^^^^^^^^^^^^^^^^^ The numerical application using :eq:`eq:dcltdt_almost_final` may be @@ -1502,7 +1510,7 @@ fraction remains within its physical bounds. and similar equations are used for :math:`C_i^{[n+1]}` and :math:`C_t^{[n+1]}`. -.. _A note on the implementation of the cloud fraction change: +.. _sec_conv_imp_note: A note on the implementation of the cloud fraction change ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ @@ -1525,8 +1533,8 @@ detrainment. It is possible to calculate directly the change in :math:`C_l` that should occur due to the detrainment and compensating subsidence treated together, in the same way that :math:`\Delta \overline{q_{cl}}` is calculated (see section -:ref:`Calculation of Grid-Box Averaged Condensate Rate (Q4)`), and this is the -way in which the +:ref:`Calculation of Grid-Box Averaged Condensate Rate (Q4) +`), and this is the way in which the cloud fraction change **should** be done. It is an unfortunate historical emphasis in the early development of PC2 on the derivation of :eq:`eq:dcdt_inhom2` that has led to the treatment @@ -1536,7 +1544,8 @@ convection. The change in :math:`\overline{q_{cl}}` and :math:`C_l` due to the adiabatic warming associated with the compensating subsidence is considered explicitly in the model implementation (see section -:ref:`Background condensation`) for both :math:`\overline{q_{cl}}` and +:ref:`Background condensation `) for both +:math:`\overline{q_{cl}}` and :math:`C_l` after the rest of the convective process has been calculated. It is perhaps arguable that if :eq:`eq:dcdt_inhom2` is going to be applied then the @@ -1547,7 +1556,7 @@ Any major future developments of PC2 for a mass-flux convection scheme would be advised to consider whether it is appropriate to use :eq:`eq:dcdt_inhom2` at all. -.. _Ice cloud and mixed phase regions: +.. _sec_ct: Ice cloud and mixed phase regions --------------------------------- @@ -1556,13 +1565,13 @@ The homogeneous forcing, initiation and PC2 erosion sections described above have only considered the generation and dissipation of liquid clouds. Although the forcing methods will not influence the generation and dissipation of ice cloud (which is primarily performed in the -large-scale precipitation scheme, section :ref:`Large-scale precipitation`) we -are +large-scale precipitation scheme, section :ref:`Large-scale precipitation +`) we are still left with the issue of how created or dissipated liquid cloud overlaps with existing ice cloud in the gridbox. The opposite situation, where changes in ice cloud are specified and changes in the overlap with liquid cloud need to be calculated, is also possible in PC2 (e.g. in the -boundary layer, see section :ref:`Boundary Layer`). +boundary layer, see section :ref:`Boundary Layer `). Here we need a simple assumption to close the problem. The assumption that we now choose is that liquid cloud fraction *changes* are @@ -1756,12 +1765,12 @@ summarised below: and :math:`CCW` in both dry-convective and cumulus-capped boundary-layers. -.. _Turbulence-driven production of subgrid scale liquid cloud: +.. _sec_turb_qcl_scheme: Turbulence-driven production of subgrid scale liquid cloud ---------------------------------------------------------- -.. _Introduction: +.. _sec_sgt_intro: Introduction ^^^^^^^^^^^^ @@ -1782,13 +1791,14 @@ between their theoretically predicted predicted mean cloud properties and the bulk properties of the LES clouds. Subsequently, their model has been used as the basis of subgrid cloud initiation method for use in the Unified Model in conjunction with the PC2 prognostic cloud scheme. In -Section :ref:`Model description` we outline the model of +Section :ref:`Model description ` we outline the model +of `Field et al. (2014)`_. In Section -:ref:`Model implementation and closure relations` we described its -implementation in +:ref:`Model implementation and closure relations ` we +described its implementation in the GCM. -.. _Model description: +.. _sec_sgt_model_describe: Model description ^^^^^^^^^^^^^^^^^ @@ -1889,12 +1899,13 @@ ice supersaturation at water saturation. We use the superscription and water content diagnosed from a parametrization of small-scale turbulent processes. -.. _Model implementation and closure relations: +.. _sec_sgt_model_implement: Model implementation and closure relations ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ -To implement the model of Section :ref:`Model description` in +To implement the model of Section :ref:`Model description +` in the Unified Model, closure relations are needed for the quantities :math:`\sigma_w^2`, :math:`\varepsilon`, :math:`L`, :math:`\tau_{\rm d}` and :math:`S_E`, subject to the constraining relationship given by Eq. @@ -1910,12 +1921,12 @@ prognostic fields, :math:`C_l` and :math:`q_{cl}`. Two methods are available for doing this. In the simplest case, the diagnosed values :math:`C_l^{sgt}` and :math:`q_{cl}^{sgt}` are just treated as increments to model prognostics (option one, in Sec. -:ref:`Options for incrementing model prognostics` below). A more complex option -(see +:ref:`Options for incrementing model prognostics ` below). +A more complex option (see option two, below) is to increment the model fields via the PC2 Erosion functionality. -.. _Closure relations: +.. _sec_sgt_closures: Closure relations ^^^^^^^^^^^^^^^^^ @@ -1932,7 +1943,8 @@ vertical grid spacing in each grid box: :math:`L=\beta_{mix} \Delta z`, where :math:`\Delta z` is calculated as the height different between the :math:`\rho`-levels adjacent to the given :math:`\theta`-point. The parameter, :math:`\beta_{mix}`, is an adjustable constant that the user -can define (see Section :ref:`Other user options` below), however it +can define (see Section :ref:`Other user options ` below), +however it should be of order one. To obtain :math:`\tau_{\rm d}` we impose an eddy size constraint: @@ -1960,12 +1972,13 @@ taken to be the grid box mean values. The first moment of the ice PSD, :math:`{\cal M}_1`, is found from the parametrization, due to `Field et al. (2005)`_, described in Section 4.1 of UMDP26. -.. _Options for incrementing model prognostics: +.. _sec_sgt_increments: Options for incrementing model prognostics ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ -Using the information in Section :ref:`Closure relations` to obtain +Using the information in Section :ref:`Closure relations ` to +obtain closed expressions for the subgrid PDF of :math:`S_i`-fluctuations allows :math:`C_l^{sgt}` and :math:`q_{cl}^{sgt}` to be calculated. These will be non-zero only where there is turbulence as diagnosed by @@ -2044,11 +2057,11 @@ where :math:`q_{cl}` is the liquid cloud amount prior to calling to the turbulent production scheme. The cloud fraction increments are calculated by calling PC2 Erosion with :math:`\left( \Delta q_{cl} \right)_{sgt}` as input. See Section -:ref:`PC2 erosion` for details on how the PC2 Erosion process works. +:ref:`PC2 erosion ` for details on how the PC2 Erosion process works. This method gives cloud fraction increments that are consistent with PC2 cloud scheme. -.. _Other user options: +.. _sec_sgt_options: Other user options ^^^^^^^^^^^^^^^^^^ @@ -2082,7 +2095,7 @@ The following variables and logical switches are optional inputs: #. ``mp_czero`` defines the constant parameter :math:`C_0` (defaults to :math:`C_0=10`). -.. _Application to the Unified Model: +.. _sec_app_um: Application to the Unified Model ================================ @@ -2090,7 +2103,8 @@ Application to the Unified Model This section describes the way in which the physical concepts described in the above section are applied to the sections of the Unified Model, in order to build up the complete prognostic scheme. Description of the -actual subroutines themselves follow in section :ref:`Code Structure`. +actual subroutines themselves follow in section :ref:`Code Structure +`. Note that the large-scale precipitation and convection schemes have considerable documentation below, since these schemes have been heavily modified for PC2. The other schemes use generic forcing scenarios, hence @@ -2099,7 +2113,7 @@ their desciption here is much shorter. Remember, whenever a signficiant able to represent the corresponding condensation and changes in cloud fractions. -.. _Radiation: +.. _sec_rad: Radiation --------- @@ -2108,10 +2122,11 @@ The shortwave and longwave radiation schemes both alter the temperature of the atmosphere, hence we need to calculate the corresponding condensation and cloud fraction changes. For both shortwave and longwave, we use the homogeneous forcing routines (section -:ref:`Homogeneous forcing`) for :math:`\overline{q_{cl}}` and :math:`C_l`, +:ref:`Homogeneous forcing `) for :math:`\overline{q_{cl}}` and +:math:`C_l`, (using eqn. :eq:`eq:deltaqc_exp2` to calculate the :math:`Q_c` forcing) and then the method in section :ref:`Ice cloud and mixed -phase regions` to +phase regions ` to calculate :math:`C_t` changes. There is no :math:`\overline{q_{cf}}` change associated with this process since the deposition / sublimation process is performed within the large-scale precipitation scheme (as it @@ -2122,9 +2137,9 @@ model to use when we know that a large proportion of the heating associated with radiative transfer in the atmosphere comes from the cloudy air and is not evenly spread across the gridbox. Possible developments are discussed in section :ref:`Homogeneous forcing section -improvements`. +improvements `. -.. _Large-scale precipitation: +.. _sec_precip: Large-scale precipitation ------------------------- @@ -2152,7 +2167,7 @@ single ice cloud fraction is stored, the assumption being that the two ice categories are completely overlapped with each other. Graupel is not considered to contribute to the ice cloud fraction. -.. _Fall of ice: +.. _sec_lsp_fall: Fall of ice ^^^^^^^^^^^ @@ -2165,7 +2180,8 @@ spread of ice cloud fraction at a particular level (hence :math:`\overline{q_{cf}}` that leaves a gridbox does not reduce :math:`C_f` in that gridbox). The in-cloud ice content simply reduces due to the fall out of ice - it is the sublimation term (section -:ref:`Deposition and sublimation`) that erodes the fall streaks. However, ice +:ref:`Deposition and sublimation `) that erodes the fall +streaks. However, ice that falls into a clear layer from above may increase the ice cloud fraction. We parametrize this by considering the horizontal overlap of ice clouds between two model layers, and the fall speed of ice between @@ -2205,7 +2221,7 @@ layer below can be filled by ice in the timestep: where :math:`\Delta t` is the timestep. We now choose to assume a minimum overlap between the liquid and the ice phases (as in section -:ref:`Ice cloud and mixed phase regions`). +:ref:`Ice cloud and mixed phase regions `). .. math:: :label: eq:lsp_fall_ct @@ -2220,7 +2236,7 @@ the ice cloud fraction overhang. Consequently, the option not to use the “wind shear value” when calculating the overhang is available in the UMUI (from version 7.6 onwards).** -.. _Homogeneous nucleation: +.. _sec_lsp_homo: Homogeneous nucleation ^^^^^^^^^^^^^^^^^^^^^^ @@ -2263,7 +2279,7 @@ cloud. These give the following changes: \Delta C_t = 0. -.. _Deposition and sublimation: +.. _sec_mp_depsub: Deposition and sublimation ^^^^^^^^^^^^^^^^^^^^^^^^^^ @@ -2272,7 +2288,7 @@ This term exerts one of the most important influences on the ice cloud in the whole model (this applies to the control as well as for PC2). Contained in the formulation is a subgrid-scale assumption that causes equivalent effects to that for a moisture PDF under the ‘:math:`s`’ -framework (section :ref:`The ‘s’ distribution`). However, since +framework (section :ref:`The ‘s’ distribution `). However, since :math:`{q_{cf}}` changes slowly in response to local changes in :math:`q` and :math:`T`, we cannot base the :math:`q_{cf}` response on the same instantaneous condensation framework. It would be useful to @@ -2491,7 +2507,7 @@ a large-scale lifting process will be required in order to condense water from the moistened air. We cannot, therefore, allow any change in cloud fractions to occur as a result, subsequent changes are calculated elsewhere in the model (e.g. by the lifting process, section -:ref:`Response to pressure changes`). +:ref:`Response to pressure changes `). Accretion ^^^^^^^^^ @@ -2551,12 +2567,12 @@ corresponding large reduction in :math:`C_l`. This is an underlying feature of the PC2 scheme (discussed in `Wilson and Gregory (2003)`_), and necessarily implies the skewing of the underlying moisture PDF. Subsequent parts of the model (e.g. the width narrowing, section -:ref:`Changing the width of the PDF - PC2 erosion`) will, of course, act on the -modified fields to +:ref:`Changing the width of the PDF - PC2 erosion `) will, of +course, act on the modified fields to adjust the cloud fractions further, but remember that these are separate processes and modelled elsewhere in the timestep. -.. _PC2 erosion: +.. _sec_turb: PC2 erosion ----------- @@ -2607,10 +2623,10 @@ term “dbsdtbs1” which scales with the rate of homogeneous forcing to zero on input to these routines so is never used. The width-narrowing formulation of section :ref:`Changing the width of the PDF -- PC2 erosion` is used +- PC2 erosion ` is used to calculate increments in :math:`\overline{q_{cl}}` and :math:`C_l`. Using the liquid - ice cloud overlap ideas of section :ref:`Ice cloud and mixed -phase regions` +phase regions ` then gives the associated :math:`C_t` change. This background narrowing term, :math:`\Upsilon`, is originally based upon work by `Stiller and Gregory (2003)`_, although it is a parameter that has been @@ -2621,15 +2637,18 @@ Numerical application of the original width-narrowing method ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Because of the strong link the mathematical expressions for width -narrowing (section :ref:`Changing the width of the PDF - PC2 erosion`) have -with the expressions for -the homogeneous forcing (section :ref:`Homogeneous forcing`), we choose to +narrowing (section :ref:`Changing the width of the PDF - PC2 erosion +`) have with the expressions for +the homogeneous forcing (section :ref:`Homogeneous forcing `), we +choose to represent the timestepping of this process in exactly the same way as for the homogeneous forcing (in fact, in the Unified Model code we use -the same subroutine, see section :ref:`Code Structure`). As before, we use +the same subroutine, see section :ref:`Code Structure `). As before, +we use a simple forward timestepping of :math:`C_l`, with :math:`Q_c` given by :eq:`eq:qc_eq_qt-qs` and :math:`a_L` defined as -discussed in section :ref:`Numerical application` and discretize eq +discussed in section :ref:`Numerical application ` and +discretize eq :eq:`eq:dcdt_width` as: .. math:: :label: eq:dcl_turb_final @@ -2755,7 +2774,7 @@ used to calculate the change in :math:`C_l` using the same moisture PDF assumptions as were used in the original PC2 erosion formulation. To achieve this, we combine equations :eq:`eq:dcdt_width` and :eq:`eq:dqcldt_width` from section -:ref:`Changing the width of the PDF - PC2 erosion` to eliminate +:ref:`Changing the width of the PDF - PC2 erosion ` to eliminate :math:`\frac{1}{b_s} \frac{\partial b_s}{\partial t}` and write :math:`\frac{\partial C_l}{\partial t}` as a function of :math:`\frac{\partial \overline{q_{cl}}}{\partial t}`: @@ -2775,7 +2794,7 @@ for :math:`C_l` and the introduction of some surface area dependence leads to this formulation being referred to as a “hybrid” cloud-surface-area erosion method. -.. _Numerical application of the hybrid erosion method: +.. _sec_erosion_numerics: Numerical application of the hybrid erosion method ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ @@ -2821,7 +2840,7 @@ cumulus regimes. off. This means any cloud detrained by convection during the current timestep cannot be eroded until the following timestep, and so is still present at end-of-timestep. As discussed in section - :ref:`Time-stepping`, this leads to a problematic timestep + :ref:`Time-stepping `, this leads to a problematic timestep sensitivity, since the amount of cloud not subject to erosion is the convection increment, which scales with the timestep length. @@ -2995,7 +3014,8 @@ cumulus regimes. \; \frac{1}{q_{cl}} \frac{\partial q_{cl}}{\partial t} - Under homogeneous forcing (section :ref:`Homogeneous forcing`), we + Under homogeneous forcing (section :ref:`Homogeneous forcing + `), we defined the PDF height at the saturation boundary when near the cloudy end of the PDF as :math:`G(-Q_c) = \frac{n+1}{n+2} \frac{C_l^2}{q_{cl}}` (eq @@ -3056,7 +3076,7 @@ cumulus regimes. where :math:`SD` is the saturation defecit, and :math:`a_L` is the dimensionless factor defined in eq :eq:`eq:a_L`. Following the derivation in section - :ref:`“Smooth” initiation logic` (eq + :ref:`“Smooth” initiation logic ` (eq :eq:`eq:qc_plus_sd`, we can write this in terms of the liquid-water content: :math:`SD = q_{cl} - Q_c` (where :math:`Q_c` was defined in eq @@ -3106,7 +3126,7 @@ cumulus regimes. Note that :math:`q_{cl}` falls to zero after a finite time :math:`\frac{{q_{cl}}_0}{1-b_1} \frac{a_L}{K} - \frac{1}{2 {C_l}_0 (1 - C_l) \, (q_{cl}-Q_c)}`. If the timestep + \frac{1}{2 {C_l}_0 (1 - C_l) \, (q_{cl}-Q_c)}`. If the timestep :math:`\Delta t` is longer than this time, then erosion completely removes the cloud during the current timestep. @@ -3228,7 +3248,7 @@ temperature or moisture content of the model gridboxes, hence PC2 assumes no change in the condensate and cloud fractions as a result of these processes. -.. _Advection: +.. _sec_advec: Advection --------- @@ -3243,7 +3263,8 @@ following each parcel, which will cause an accompanying adiabatic temperature change. These advective pressure and temperature changes imply a homogeneous forcing, which yields a change in :math:`\overline{q_{cl}}` and :math:`C_l` in addition to their transport -by the winds. This is described in section :ref:`Response to pressure changes`. +by the winds. This is described in section :ref:`Response to pressure changes +`. If the UM namelist switch **l_pc2_sl_advection** is turned on, the PC2 homogeneous forcing response to advection is calculated straight after @@ -3252,7 +3273,7 @@ from advection is combined with the Eulerian pressure change from the dynamics Helmholtz solver, and the resulting homogeneous forcing of liquid cloud is computed at the end of the timestep. -.. _Boundary Layer: +.. _sec_bl: Boundary Layer -------------- @@ -3293,7 +3314,8 @@ denominator is small, we will, to avoid numerical problems, set :math:`C_i` to 1 if :math:`q_C^S - \overline{q_{cf}} < 1 \times 10^{-10} kg kg^{-1}`. Note that we do not use the multiple phases injection source expressions -(section :ref:`Multiple phases in the injection source` and equation +(section :ref:`Multiple phases in the injection source ` and +equation :eq:`eq:cff_ts`. This is because the liquid water changes are not associated with the plume model. @@ -3306,7 +3328,7 @@ the ice water content is reduced, the ice cloud fraction is reduced, in such as way as to maintain the same in-cloud ice water content. The :math:`C_t` changes are calculated using the minimum overlap method -of section :ref:`Ice cloud and mixed phase regions`. +of section :ref:`Ice cloud and mixed phase regions `. In *ni-imp-ctl* the control code inhibits the call to the diagnostic cloud scheme if there is deep or shallow convection occurring and the @@ -3322,7 +3344,7 @@ level above the boundary layer mixed layer*. This choice (i.e. ntml) is seen to give improved results in PC2, and is arguably a more physical reasonable choice anyway than using ntml+1. -.. _Convection: +.. _sec_convec: Convection ---------- @@ -3331,11 +3353,12 @@ This section concentrates specifically upon the PC2 interface to the convection scheme. In the current formulation of the UM, only a mass-flux convection scheme exists, and this is what is described below. Work to interface PC2 to the developing turbulence based convection -scheme is commented upon in section :ref:`Turbulence based convection scheme`. +scheme is commented upon in section :ref:`Turbulence based convection scheme +`. An alternative way of calculating cloud fraction increments is currently under development and is described in section -:ref:`Tidier way of coupling convection and PC2`. +:ref:`Tidier way of coupling convection and PC2 `. A traditional view of convective parametrization is a scheme that transports vapour, :math:`q`, heat, :math:`\theta`, and momentum, @@ -3361,7 +3384,8 @@ Introduction to the convective mass flux scheme Within the mass flux scheme the net change in :math:`\overline{q_{cl}}` and :math:`C_l` etc. comes from two distinct sources. Firstly, the condensate and cloud fraction injected from the plume (the :math:`Q4` -terms, section :ref:`Injection forcing`); secondly, the condensation +terms, section :ref:`Injection forcing `); secondly, the +condensation response to the vapour and heat changes associated with the detrainment and compensating subsidence. Strictly, we will see that the :math:`Q4` terms also include the contribution to the condensate transport by the @@ -3382,16 +3406,18 @@ We therefore split the convective contribution in where :math:`Q_{environment}` is the condensation associated with changes in the vapour and temperature from the detrainment and compensating subsidence. Similar splits are made for the cloud -variables, where the injection forcing, section :ref:`Injection forcing`, +variables, where the injection forcing, section :ref:`Injection forcing +`, is used to calculate the first term from :math:`Q4_l`. Section -:ref:`Calculation of Grid-Box Averaged Condensate Rate (Q4)` looks at the issue -of the calculation -of :math:`Q4_l` etc., and section :ref:`Background condensation` looks at +:ref:`Calculation of Grid-Box Averaged Condensate Rate (Q4) +` looks at the issue of the calculation +of :math:`Q4_l` etc., and section :ref:`Background condensation +` looks at the calculation of :math:`Q_{environment}`, and its associated cloud fraction change. We first look at the basic transport equations in a mass flux convection scheme. -.. _Basic Equations for a Convective Mass Flux Scheme: +.. _subsect_basmaseqs: Basic Equations for a Convective Mass Flux Scheme ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ @@ -3406,7 +3432,7 @@ processes (e.g. total water content). In this case, {\frac{\partial \, {\chi}_{\rm{ }}^{\rm{E}}}{\partial \, t}}_{\rm{conv}} = - \frac{1}{\overline{\rho}} \, \frac{\partial \, \overline{\rho w^{'} - {\chi}_{\rm{ }}^{\rm{E'}}}}{\partial \, z} + {\chi}_{\rm{ }}^{\rm{E'}}}}{\partial \, z} To parametrize :eq:`eq:chibasic`, the current UM convection scheme takes a mass flux approximation @@ -3422,7 +3448,7 @@ which can be differentiated to give .. math:: :label: eq:eddyflux - \frac{1}{\overline{\rho}} \, \frac{\partial \, \overline{\rho w^{'} - {\chi}_{\rm{ }}^{\rm{E'}}}}{\partial \, z} = + {\chi}_{\rm{ }}^{\rm{E'}}}}{\partial \, z} = \frac{\partial \, {\chi}_{\rm{ }}^{\rm{P}} \, M^{\rm{P}}}{\partial \, p} - {\chi}_{\rm{ }}^{\rm{E}} \, \frac{\partial \, M^{\rm{P}}}{\partial \, p} - @@ -3439,10 +3465,10 @@ The bulk cloud model plume equations for mass and :math:`{\chi}` are: .. math:: :label: eq:dbydpmfchi - \frac{\partial \, {\chi}_{\rm{ }}^{\rm{P}} \, M^{\rm{P}}}{\partial \, p} - = \left({ + = \left({ \varepsilon \, M^{\rm{P}} \, {\chi}_{\rm{ }}^{\rm{E}} - \mu \, M^{\rm{P}} \, {\chi}_{\rm{ }}^{\rm{R}} - \delta \, M^{\rm{P}} \, - {\chi}_{\rm{ }}^{\rm{P}} + {\chi}_{\rm{ }}^{\rm{P}} } \right) @@ -3519,14 +3545,14 @@ terms for temperature and specific humidity: \equiv \left({ \frac{L}{c_{P}} } \right)\, {\overline{Q}}_{\rm{par}} - \frac{1}{\overline{\rho}} \, \frac{\partial \, \overline{\rho w^{'} - T_{\rm{ }}^{\rm{E'}}}}{\partial \, z} + T_{\rm{ }}^{\rm{E'}}}}{\partial \, z} .. math:: :label: eq:defineq2 {\frac{\partial \, q_{\rm{ }}^{\rm{E}}}{\partial \, t}}_{\rm{conv}} = Q2 \equiv - {\overline{Q}}_{\rm{par}} - \frac{1}{\overline{\rho}} \, \frac{\partial \, \overline{\rho w^{'} - q_{\rm{ }}^{\rm{E'}}}}{\partial \, z} + q_{\rm{ }}^{\rm{E'}}}}{\partial \, z} where :math:`{\overline{Q}}_{\rm{par}}` is the rate of condensation @@ -3590,7 +3616,7 @@ is basic equations = {\overline{Q}}_{\rm{par}} - {\overline{Q}}_{\rm{reset}} - PPN - \frac{1}{\overline{\rho}} \, \frac{\partial \, \overline{\rho w^{'} - l_{\rm{ }}^{\rm{E'}}}}{\partial \, z} + l_{\rm{ }}^{\rm{E'}}}}{\partial \, z} .. math:: :label: eq:basiclold @@ -3607,7 +3633,7 @@ convection scheme :math:`Q4 = 0`. The PC2 scheme requires a reassessment of these assumptions because we wish to allow non-zero environment condensate values and to allow them to change. -.. _Calculation of Grid-Box Averaged Condensate Rate (Q4): +.. _subsect_q4calculation: Calculation of Grid-Box Averaged Condensate Rate (Q4) ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ @@ -3981,7 +4007,7 @@ and } \right] { } -.. _Background condensation: +.. _sec_conv_homog: Background condensation ^^^^^^^^^^^^^^^^^^^^^^^ @@ -4173,8 +4199,9 @@ Homogeneous forcing of the environment by convective-subsidence pressure change To this end, the code includes an option to perform the homogeneous forcing of liquid cloud by convection using the “pressure forcing” from the convective subsidence, consistent with the pressure forcing by -large-scale advection (see sections :ref:`Advection` and -:ref:`Response to pressure changes`). This approach replaces the above method of +large-scale advection (see sections :ref:`Advection ` and +:ref:`Response to pressure changes `). This approach replaces the +above method of homogeneous forcing by convection if the UM namelist switch **l_pc2_homog_conv_pressure** is turned on. By applying the same homogeneous forcing method for advection and convectively-forced @@ -4228,7 +4255,7 @@ by: \Delta T^E = \theta^E \left( \left(\frac{p}{p_{ref}}\right)^\kappa - \left(\frac{p - \Delta - p^E}{p_{ref}}\right)^\kappa + p^E}{p_{ref}}\right)^\kappa \right) where :math:`\theta^E` is the environment potential temperature, @@ -4296,7 +4323,7 @@ temperatures less than around :math:`-42 ^{\circ} C`, with the tuning allowing less precipitation and greater detrainment. This change is necessary in order to produce thick enough anvil clouds. -.. _Phase of condensate: +.. _sec_plume_phase: Phase of condensate ^^^^^^^^^^^^^^^^^^^ @@ -4325,7 +4352,7 @@ is -10 :math:`^{\circ}` C. ^{\circ} C \end{array} \right. -.. _Tidier way of coupling convection and PC2: +.. _sec_conv-simpler: Tidier way of coupling convection and PC2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ @@ -4362,7 +4389,7 @@ temperature thresholds by maintaining a similar linear ramp. For e.g., condensate is assumed to be all-liquid for T :math:`\geq` :math:`tnuc_n` and all-ice for T :math:`\leq` :math:`tnuc_n` - 10.0 -.. _Condensation adjustment in the profiles input to the convection scheme: +.. _sec_conv_input_profs: Condensation adjustment in the profiles input to the convection scheme ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ @@ -4386,17 +4413,17 @@ additional condensation adjustments from PC2 before the convection call: - **l_pc2_sl_advection**: performs homogeneous forcing response to Semi-Lagrangian advection immediately after the advection calculation, instead of at the end of the timestep (see section - :ref:`Response to pressure changes`). + :ref:`Response to pressure changes `). - **l_cloud_call_b4_conv**: performs an additional call to PC2 initiation (and PC2 checks) before the convection scheme (see section - :ref:`Initiation`). This should catch any instances where + :ref:`Initiation `). This should catch any instances where large-scale ascent or other processes have brought the profiles after advection to near or beyond saturation, in grid-points where there was no liquid cloud already present (and so no homogeneous forcing response). -.. _Response to pressure changes: +.. _sec_pres: Response to pressure changes ---------------------------- @@ -4405,7 +4432,7 @@ A pressure change following the parcel during the timestep will result in an adiabatic temperature change which will force condensation, hence we must include this temperature change forcing within PC2. The majority of this pressure change comes from vertical advection (although not -all). Remember that the advection (section :ref:`Advection`), on its +all). Remember that the advection (section :ref:`Advection `), on its own, does not cause condensation, it merely moves the existing cloud field. @@ -4512,32 +4539,36 @@ The splitting of the pressure forcing call under the **l_pc2_sl_advection** switch was originally implemented to make the profiles passed to convection more realistic. -.. _Initiation: +.. _sec_init2: Initiation ---------- -As discussed in section :ref:`Initiation of cloud`, there are occasions when +As discussed in section :ref:`Initiation of cloud `, there are +occasions when :math:`\overline{q_{cl}}` and :math:`C_l` need to be initiated from 0 or 1. The application of the initiation is given in section -:ref:`Initiation of cloud`. The initiation forms a new, separate block of PC2 +:ref:`Initiation of cloud `. The initiation forms a new, separate +block of PC2 code to perform this calculation, and is located immediately following -the pressure change response (section :ref:`Response to pressure changes`). -Also, if the +the pressure change response (section :ref:`Response to pressure changes +`). Also, if the UM namelist switch **l_cloud_call_b4_conv** is set to true, an additional call to PC2 initiation is performed before the convection scheme, to ensure that the condensation response to advection and other forcings earlier in the timestep has been accounted for in the profiles passed to the convection scheme, even if there was no cloud already present for homogeneous forcing to act upon. (see section -:ref:`Condensation adjustment in the profiles input to the convection scheme`). +:ref:`Condensation adjustment in the profiles input to the convection scheme +`). There are currently 3 options for the conditions under-which initiation may occur. For all of these options, if using the bimodal cloud scheme to do initiation within PC2, then the tests on :math:`RH_T` relative to :math:`RH_{crit}` are replaced by equivalent tests for whether the saturation boundary lies within the bounds of the bimodal scheme’s -assumed PDF, as described in section :ref:`Initiation using the bimodal scheme`. +assumed PDF, as described in section :ref:`Initiation using the bimodal scheme +`. “Original” initiation logic ^^^^^^^^^^^^^^^^^^^^^^^^^^^ @@ -4607,7 +4638,7 @@ Or: where :math:`C_{tol}` can be set via the UM namelist; its original standard value is 0.005. Note this threshold is also used to remove small cloud-fractions after initiation; see section -:ref:`Additional checks after PC2 initiation`. +:ref:`Additional checks after PC2 initiation `. This is very similar to the “Original” initiation logic described above, but with the following differences: @@ -4623,7 +4654,7 @@ but with the following differences: - The different threshold when initiating super-cooled cloud is removed. -.. _“Smooth” initiation logic: +.. _sec_smooth_initiation: “Smooth” initiation logic ^^^^^^^^^^^^^^^^^^^^^^^^^ @@ -4668,29 +4699,26 @@ prognostic :math:`C_l` and :math:`q_{cl}` are incremented as follows: - If :math:`{q_{cl}}_{diag} > q_{cl}`: - .. math:: :label: eq:dqcl_init - - \Delta q_{cl} = {q_{cl}}_{diag} - q_{cl} + :math:`\Delta q_{cl} = {q_{cl}}_{diag} - q_{cl} + \quad \refstepcounter{equation}(\theequation)\label{eq:dqcl_init}` - If :math:`Q_C < 0`: - .. math:: :label: eq:dcl_init1 - - \Delta C_{l} = \frac{\Delta q_{cl}}{{q_{cl}}_{diag}} - \left( {C_{l}}_{diag} - C_{l} \right) + :math:`\Delta C_{l} = \frac{\Delta q_{cl}}{{q_{cl}}_{diag}} + \left( {C_{l}}_{diag} - C_{l} \right) + \quad \refstepcounter{equation}(\theequation)\label{eq:dcl_init1}` - If :math:`Q_C > 0`: - .. math:: :label: eq:dcl_init2 - - \Delta C_{l} = \frac{\Delta SD}{{SD}_{diag}} - \left( {C_{l}}_{diag} - C_{l} \right) + :math:`\Delta C_{l} = \frac{\Delta SD}{{SD}_{diag}} + \left( {C_{l}}_{diag} - C_{l} \right) + \quad \refstepcounter{equation}(\theequation)\label{eq:dcl_init2}` - Otherwise: - .. math:: \Delta q_{cl} = 0 + :math:`\Delta q_{cl} = 0` - .. math:: \Delta C_{l} = 0 + :math:`\Delta C_{l} = 0` where the subscript :math:`_{diag}` denotes the liquid cloud water content and fraction predicted by the diagnostic cloud scheme (either @@ -4756,7 +4784,7 @@ implementation of :eq:`eq:dcl_init2` in the code simply uses :math:`q_{cl} - Q_c` in place of :math:`SD`, and :math:`\Delta q_{cl}` in place of :math:`\Delta SD`). -.. _Additional checks after PC2 initiation: +.. _sec_checks2: Additional checks after PC2 initiation ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ @@ -4766,7 +4794,7 @@ For numerical reasons, it is possible to obtain very low, but non zero, values of :math:`C_l` (and equivalently values very close to, but not equal to, 1). The code will reset these clouds to either a fraction of 0 or 1, as appropriate. We choose to apply these terms here and not in the -Bounds Checking part of the code (section :ref:`Bounds checking`) +Bounds Checking part of the code (section :ref:`Bounds checking `) because these are not required to obtain consistency between fields, but are ‘tidying up’ pieces of code, although they may reasonably also be applied in the Bounds Checking. Care needs to be taken when choosing the @@ -4839,10 +4867,10 @@ zero). Note that if these checks are relaxed by lowering the thresholds :math:`C_{tol}` and :math:`C_{tol 2}` to near-zero, similar checks are still performed independently by the bounds checking described in -section :ref:`Bounds checking`, but with a much lower threshold of +section :ref:`Bounds checking `, but with a much lower threshold of :math:`C_{tol 3} = 1 \times 10^{-12}`. -.. _Bounds checking: +.. _sec_checks: Bounds checking --------------- @@ -4859,7 +4887,8 @@ to ensure consistency between these values. The bounds checking is performed three times during the timestep. Firstly, after the parallel part of the physics (*atmos-physics1*) is -complete; secondly, before the initiation (section :ref:`Initiation of cloud`) +complete; secondly, before the initiation (section :ref:`Initiation of cloud +`) is called; thirdly, after the initiation is called. Firstly, if :math:`C_l > 1 - C_{tol 3}` then :math:`C_l` is set to 1. @@ -4880,7 +4909,7 @@ negligible, but non-zero values of :math:`\overline{q_{cl}}` and reset :math:`C_l` to zero. :math:`C_t` gets reset, as it must if there is no liquid cloud, to be equal to :math:`C_i`. -.. _`sec:pc2_checks_sd`: +.. _sec_pc2_checks_sd: The next check complements the first but updates the moisture fields. We firstly calculate :math:`SD` using :eq:`SD2` and @@ -5061,7 +5090,8 @@ overlap situation and then the minimum overlap situation. .. _section-8: Finally, there is a homogeneous nucleation term applied, similar to that -in the large-scale precipitation (section :ref:`Homogeneous nucleation`). +in the large-scale precipitation (section :ref:`Homogeneous nucleation +`). This is a fast microphysics process, and must act to ensure that no liquid cloud created by the initiation is allowed to persist in this phase if the temperature is cold enough. Hence, if @@ -5088,7 +5118,7 @@ phase if the temperature is cold enough. Hence, if C_l \leftarrow 0. -.. _Qpos checks: +.. _sec_qpos: Qpos checks ^^^^^^^^^^^ @@ -5108,7 +5138,7 @@ consistent sinks of qcl and CFL” prevents the QCL increment from trying to remove too much liquid condensate and hence reduces the models reliance on Q-Pos to deal with the inconsistencies. -.. _Data Assimilation: +.. _sec_da: Data Assimilation ----------------- @@ -5162,32 +5192,33 @@ ill-conditioning of this solution near :math:`C_l = 1`. In practice, the ill-conditioning of :eq:`eq:da2` and :eq:`eq:da3` becomes too numerically awkward for us to apply the full solution based on homogeneous forcing, although, for -completeness, we outline it in Appendix :ref:`Appendix; Alternative PC2 - Data -Assimilation formulations`. Hence +completeness, we outline it in Appendix :ref:`Appendix: Alternative PC2 - Data +Assimilation formulations `. Hence we have chosen to apply a much simpler model. Here we use simply the data assimilation increments :math:`\Delta \overline{q}` and :math:`\Delta \overline{T}` within the standard homogeneous forcing -(section :ref:`Homogeneous forcing`), even though we are fully aware that this +(section :ref:`Homogeneous forcing `), even though we are fully +aware that this is inconsistent (because :math:`\Delta \overline{q}` and :math:`\Delta \overline{T}` are not forcings, but are forcings plus the condensation. This allows us an *estimate* of :math:`\Delta \overline{q_{cl}}` and :math:`\Delta{C_l}`, via the homogeneous forcing routine (and :math:`\Delta C_t` via the standard updating described in section -:ref:`Ice cloud and mixed phase regions`). These are the quantities applied as -the equivalent +:ref:`Ice cloud and mixed phase regions `). These are the quantities +applied as the equivalent data assimilation increments for :math:`\Delta \overline{q_{cl}}`, :math:`\Delta{C_l}` and :math:`\Delta C_t`. The increments :math:`\Delta \overline{q}` and :math:`\Delta \overline{T}` remain those that the data assimilation scheme itself calculated. -Appendix :ref:`Appendix; Alternative PC2 - Data Assimilation formulations` -gives, for completeness, the +Appendix :ref:`Appendix: Alternative PC2 - Data Assimilation formulations +` gives, for completeness, the alternative numerical technique for the solution of :eq:`eq:da2` and :eq:`eq:da3`. However, we stress that this technique is not used within the current PC2 formulation. -.. _Implementation in the Unified Model: +.. _sec_um: Implementation in the Unified Model =================================== @@ -5201,7 +5232,7 @@ much as possible in a similar way to the condensate variables. Hence, wherever the condensed water variables :math:`q_{cl}` and :math:`q_{cf}` are updated, the cloud fractions need to be updated consistently. -.. _Area cloud fraction: +.. _sec_acf: Area cloud fraction ------------------- @@ -5225,7 +5256,7 @@ cloud fraction given the volume cloud fraction, taking into account the size of the grid box. The setting of the area cloud fraction is performed at the end of the timestep. -.. _Code Structure: +.. _sec_code: Code Structure -------------- @@ -5257,7 +5288,8 @@ before the microphysics scheme (within *pc2_turbulence_ctl*). Note there is also an optional call to *pc2_turbulence_ctl* after the microphysics scheme, which is used only to estimate the cloud fraction change consistent with the turbulent production of liquid cloud (see -section :ref:`Turbulence-driven production of subgrid scale liquid cloud`). +section :ref:`Turbulence-driven production of subgrid scale liquid cloud +`). Most PC2 code is protected by IF tests on the namelist input *i_cld_vn* = 2 (PC2 in the GUI). However, within the convection scheme, the code is @@ -5435,7 +5467,7 @@ Main Tree from atm_step_4a - | ls_arcld | \* (Smith scheme with area cloud fraction; see - :ref:`Smith scheme with area cloud fraction` for a drill-down + :ref:`Smith scheme with area cloud fraction ` for a drill-down inside this routine) - | bm_ctl @@ -5446,7 +5478,7 @@ Main Tree from atm_step_4a - | pc2_initiation_ctl | \* (interface to PC2 initiation and consistency-checks; see - :ref:`PC2 initiation` for a + :ref:`PC2 initiation ` for a drill-down inside this routine) .. container:: tcolorbox @@ -5520,7 +5552,7 @@ Main Tree from atm_step_4a - | ls_arcld | \* (interface to diagnostic Smith scheme and area cloud fraction; see - :ref:`Smith scheme with area cloud fraction` for a drill-down + :ref:`Smith scheme with area cloud fraction ` for a drill-down inside this routine) - | bm_ctl @@ -5550,7 +5582,7 @@ Main Tree from atm_step_4a - | pc2_assim | \* (PC2 reponse to the analysis increments; see - :ref:`PC2 Data Assimilation` for a drill-down + :ref:`PC2 Data Assimilation ` for a drill-down inside this routine) - ls_acf_brooks (calculate area cloud fraction using @@ -5558,8 +5590,8 @@ Main Tree from atm_step_4a - ls_arcld (call diagnostic Smith scheme with area cloud fraction again to account for the analysis increments; - see :ref:`Smith scheme with area cloud fraction` for a - drill-down + see :ref:`Smith scheme with area cloud fraction + ` for a drill-down inside this routine) .. container:: tcolorbox @@ -5586,7 +5618,7 @@ Main Tree from atm_step_4a - | pc2_initiation_ctl | \* (interface to PC2 initiation and consistency-checks; see - :ref:`PC2 initiation` for a drill-down + :ref:`PC2 initiation ` for a drill-down inside this routine) .. container:: tcolorbox @@ -5597,7 +5629,7 @@ Main Tree from atm_step_4a - | ls_arcld | \* (interface to diagnostic Smith scheme and area cloud - fraction; see :ref:`Smith scheme with area cloud fraction` for a + fraction; see :ref:`Smith scheme with area cloud fraction ` for a drill-down inside this routine) - | bm_ctl @@ -5612,7 +5644,7 @@ Main Tree from atm_step_4a - | pc2_assim | \* (PC2 reponse to the analysis increments; see - :ref:`PC2 Data Assimilation` for a drill-down inside + :ref:`PC2 Data Assimilation ` for a drill-down inside this routine) - | initial_pc2_check @@ -5628,7 +5660,7 @@ Drill-downs within some routines in the call tree are listed separately below, to avoid duplication (since these routines are called in multiple different places in the tree)... -.. _Smith scheme with area cloud fraction: +.. _subsubsec_smith_acf: Smith scheme with area cloud fraction ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ @@ -5667,7 +5699,7 @@ Smith scheme with area cloud fraction | ls_acf_brooks | \* (estimate area cloud fraction) -.. _PC2 initiation: +.. _subsubsec_pc2_initiation: PC2 initiation ^^^^^^^^^^^^^^ @@ -5715,7 +5747,7 @@ PC2 initiation - | pc2_homog_plus_turb | \* (generic homogeneous forcing routine used to interpolate) -.. _PC2 Data Assimilation: +.. _subsubsec_pc2_assim: PC2 Data Assimilation ^^^^^^^^^^^^^^^^^^^^^ @@ -5742,7 +5774,7 @@ PC2 Data Assimilation | \* (self-consistency checks on prognostic cloud fractions and water contents) -.. _Diagnostics: +.. _sec_diags: Diagnostics ----------- @@ -5856,8 +5888,8 @@ rest of the SCM uses the same PC2 code as the full model. Note that the change to PC2 homogeneous forcing from advection under the UM namelist switch **l_pc2_sl_advection** (see section -:ref:`Response to pressure changes`) is also mirrored in the Single-Column -Model. If +:ref:`Response to pressure changes `) is also mirrored in the +Single-Column Model. If this switch is turned on, the PC2 homogeneous forcing call using the SCM forcing increments is moved straight after the call to the forcing routine, so that the condensation adjustment is performed before the @@ -5873,8 +5905,8 @@ The SCM forcings may comprise one or both of the following: For the latter, we can calculate the pressure change experienced by vertically-advected parcels, and so calculate the PC2 homogeneous forcing response in the same way as we do for Semi-Lagrangian advection -in the full model (see section :ref:`Response to pressure changes`). For the -former, we +in the full model (see section :ref:`Response to pressure changes `). +For the former, we don’t know if the prescribed T,q tendencies are due to advection, radiation, or some other process, so we calculate the PC2 homogeneous forcing response as if the tendencies are applied "in-situ". @@ -5939,147 +5971,148 @@ diagnostic output routines. - Number of it erations in in itiation - 10 - p c2-const - - Num: :ref:`Numerical Application of the Smith method` + - Num: :ref:`Numerical Application of the Smith method ` * - :math:`C_{tol}` - cloud -pc2-tol - Bounds checking :math:`C_l` t hreshold - 0.005 - UM namelist - - Num: :ref:`Initiation` + - Num: :ref:`Initiation ` * - :math:`C_{tol 2}` - cloud-p c2-tol-2 - Bounds checking :math:`C_l` t hreshold - 0.001 - UM namelist - - Num: :ref:`Initiation` + - Num: :ref:`Initiation ` * - :math:`RH_{tol}` - rh crit-tol - :math:`RH_{crit}` t olerance in in itiation - 0.01 - p c2-const - - Num: :ref:`Initiation` + - Num: :ref:`Initiation ` * - :math:`q_{cf0\, BL}` - ls-bl0 - Fixed value of BL in-plume :math:`\overline{q_{cf}}` - :math:`1.0\times 10^{-4} \, kg \,kg^{-1}` - imp-ctl - - Clo: :ref:`Boundary Layer` + - Clo: :ref:`Boundary Layer ` * - :math:`q_{cf0}` - one- over-qcf - Fixed in-cloud :math:`\overline{q_{cf}}` if :math:`C_f`\ =0 - :math:`1.0\times 10^{-4} \, kg \,kg^{-1}` - pc2-chck - - Num: :ref:`Bounds checking` + - Num: :ref:`Bounds checking ` * - :math:`m` - pdf-mer ge-power - Merging power for :math:`G(-Q_c)` - 0.5 - p c2-const - - Clo: :ref:`Homogeneous forcing` + - Clo: :ref:`Homogeneous forcing ` * - :math:`n` - p df-power - Shape p arameter for :math:`G(-Q_c)` - 0.0 - p c2-const - - Phy: :ref:`Homogeneous forcing` + - Phy: :ref:`Homogeneous forcing ` * - :math:`w` - w ind-shea r-factor - Wind shear in fallout of ice term - :math:`1.5 \times 10^{-4} \,s^{-1}` - p c2-const - - Phy: :ref:`Fall of ice` + - Phy: :ref:`Fall of ice ` * - :math:`i` - i ce-width - Scaling factor for r eduction in :math:`b_i` - 0.04 - p c2-const - - Phy: :ref:`Deposition and sublimation` + - Phy: :ref:`Deposition and sublimation ` * - :math:`a` - dbsdtb s-turb-0 - Rate of r eduction of PDF width - :math:`-2.25 \times 10^{-5} \,s^{-1}` - UM namelist - - Phy: :ref:`Changing the width of the PDF - PC2 erosion` + - Phy: :ref:`Changing the width of the PDF - PC2 erosion ` * - :math:`b` - dbsdtb s-turb-1 - Rate of r eduction of PDF width - 0 - p c2-const - - Phy: :ref:`Changing the width of the PDF - PC2 erosion` + - Phy: :ref:`Changing the width of the PDF - PC2 erosion ` * - - dbsd tbs-conv - Redn of PDF width in co nvection - 0 - p c2-const - - Phy: :ref:`Changing the width of the PDF - PC2 erosion` + - Phy: :ref:`Changing the width of the PDF - PC2 erosion ` * - - dbs dtbs-exp - V ariation of erosion on RH - 10.05 - p c2-const - - Phy: :ref:`Changing the width of the PDF - PC2 erosion` + - Phy: :ref:`Changing the width of the PDF - PC2 erosion ` * - :math:`RH_{crit}` - RHCRIT - Critical RH for cloud f ormation - - UM namelist - - Phy: :ref:`Initiation of cloud`, :ref:`Deposition and sublimation` + - Phy: :ref:`Initiation of cloud `, :ref:`Deposition and + sublimation ` * - :math:`q_{c0}` - condensa te-limit - Minimum allowed co ndensate - :math:`1 \times 10^{-10} \, kg \,kg^{-1}` - pc2-chck - - Num: :ref:`Bounds checking` + - Num: :ref:`Bounds checking ` * - :math:`q_c^{S0}` - ls0 - Lower limit of plume co ndensate - :math:`5\times 10^{-5} \, kg \,kg^{-1}` - enviro?a - - Num: :ref:`Numerical application` + - Num: :ref:`Numerical application ` * - - *Har d-wired* - Conv cloud fraction for vi sibility - 0.2 - imp-ctl2 - - Diag: :ref:`Diagnostics` + - Diag: :ref:`Diagnostics ` * - - *Har d-wired* - Limit on width of ice dist ribution - 0.001 - lspice3d - - Num: :ref:`Deposition and sublimation` + - Num: :ref:`Deposition and sublimation ` * - - *Har d-wired* - :math:`C_l` limit for init if :math:`T< 0 ^{\circ} C` - 0.05 - pc2-init - - Num: :ref:`Initiation` + - Num: :ref:`Initiation ` * - - *Har d-wired* - T olerance on calc. of :math:`q_C^s` in BL - :math:`1.0 \times 10^{-10} \, kg \,kg^{-1}` - imp-ctl - - Num: :ref:`Boundary Layer` + - Num: :ref:`Boundary Layer ` PC2 also recommends some tunings of the existing convection scheme parameters. These cannot be placed in the library code, since they would @@ -6104,35 +6137,35 @@ PC2:64 and a non-PC2 run. - :math:`-10 ^{\circ} C` - :math:`0^{\circ} C`\ \* - tice.cdk or UMUI - - Phy: :ref:`Convection` + - Phy: :ref:`Convection ` * - QSTICE - App roximate qs at(TICE) - :math:`3.5\times10^{-3}` - :math:`3.5\times10^{-3}` - qs tice.cdk or UMUI - - Phy: :ref:`Convection` + - Phy: :ref:`Convection ` * - *Har d-wired* - Limit on conv. cond. after precip - 0.5 :math:`q_{sat}, 2\times10^{-4}` - :math:`0.5 \,q_{sat}` - cloudw - - Phy: :ref:`Convection` + - Phy: :ref:`Convection ` * - Anvil factor - Shape p arameter for conv. cloud anvil - 0 - 0.3\* - UMUI - - Phy: :ref:`Convection` + - Phy: :ref:`Convection ` * - Tower factor - Shape p arameter for conv. cloud tower - 0 - 0.25\* - UMUI - - Phy: :ref:`Convection` + - Phy: :ref:`Convection ` How to run the PC2 scheme ------------------------- @@ -6169,19 +6202,20 @@ More information Information on results of the scheme and how to run the PC2 code at various model versions is available on the PC2 web site. -.. _Appendix; Alternative PC2 - Data Assimilation formulations: +.. _sec_appendix-da: -Appendix; Alternative PC2 - Data Assimilation formulations +Appendix: Alternative PC2 - Data Assimilation formulations ========================================================== -In this alternative method to section :ref:`Data Assimilation` we will assume +In this alternative method to section :ref:`Data Assimilation ` we will +assume that there exists a homogeneous forcing, :math:`\Delta Q_c`, that gives changes, net of condensation, of :math:`\Delta\overline{q}` and :math:`\Delta\overline{T}`. If we can recover what :math:`\Delta Q_c` is then we can use this to calculate the liquid, :math:`\overline{q_{cl}}`, and liquid cloud fraction, :math:`C_l`, increments. -As in section :ref:`Data Assimilation`, we start by discretising +As in section :ref:`Data Assimilation `, we start by discretising :eq:`dqcldt` to give .. math:: :label: eq:dqcldt_discrete @@ -6252,7 +6286,7 @@ solution. Initially, we calculate :math:`G(-Qc)` and :math:`\Delta Q_c` from the input fields, as in the homogeneous forcing technique (section -:ref:`Homogeneous forcing`) and :eq:`eq:deltaqc_exp2`. +:ref:`Homogeneous forcing `) and :eq:`eq:deltaqc_exp2`. An initial increment, :math:`\Delta C_l^1` is estimated directly using the basic equation @@ -6353,7 +6387,7 @@ the :math:`C_l` terms gives :math:`\Delta \overline{q_{cl \, max}}` as This is a general expression, it is not fixed for a particular PDF. To complete the analysis, we need to estimate :math:`-Q_c+b_s`. To do this, we now make the *assumption* of a power-law type PDF, as in section -:ref:`Initiation of cloud`. If we start from the equivalent of +:ref:`Initiation of cloud `. If we start from the equivalent of :eq:`eqn19` but at the :math:`s=-bs` end of the distribution, equation (B.3) in `Wilson and Gregory (2003)`_ can be equivalently written for :math:`(1-C_l)` as: @@ -6365,7 +6399,7 @@ for :math:`(1-C_l)` as: To derive this from (B.3) note that :math:`C_l` is swapped for :math:`1-C_l` and :math:`(b_s - (-Q_c))` is swapped for :math:`(-Qc - (-b_s))`, as in section :ref:`Numerical Application of the Smith -method`. +method `. Similarly, noting that :math:`\overline{q_{cl}}` can be swapped with :math:`SD`, gives the equivalent to (B.4) in `Wilson and Gregory (2003)`_ as @@ -6476,15 +6510,15 @@ results than simply using the homogeneous forcing method. Further work will be required to enable the implementation of this :math:`\overline{q}` and :math:`\overline{T}` preserving method. -.. _Appendix; Essentials of PC2 for code developers: +.. _sec_code-development: -Appendix; Essentials of PC2 for code developers +Appendix: Essentials of PC2 for code developers =============================================== This section provides some guidance to code developers on the treatment of PC2. Code developers are advised to read the relevant part of section -:ref:`Application to the Unified Model` to understand the way in which the -current PC2 +:ref:`Application to the Unified Model ` to understand the way in +which the current PC2 scheme interacts with their section of code. The essence of a prognostic cloud scheme is that each physical part of @@ -6511,7 +6545,8 @@ inhomogeneous forcing) methods. Homogeneous forcing ------------------- -This is described fully in section :ref:`Homogeneous forcing`. This assumes +This is described fully in section :ref:`Homogeneous forcing `. This +assumes that the distribution of :math:`q_T - q_{sat}(T_L)` about its gridbox mean is unchanged when a process acts. (The mean will change of course, but we assume that the variations in each part of the gridbox from the @@ -6524,7 +6559,8 @@ necessary updates. Injection forcing ----------------- -This is described fully in section :ref:`Injection forcing`. We assume +This is described fully in section :ref:`Injection forcing `. We +assume that we already know a condensate increment :math:`q_{cl}` or :math:`q_{cf}` and that a corresponding cloud fraction increment :math:`C_l` or :math:`C_f` (and :math:`C_t`) remains to be estimated. @@ -6682,12 +6718,12 @@ Convective cloud increments in the mass-flux framework ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ As discussed in section :ref:`A note on the implementation of the cloud -fraction change`, it would be +fraction change `, it would be useful to code up the convective cloud fraction changes to link directly to the mass-flux convection scheme, and not to estimate them from the values of :math:`Q4`, which can introduce errors. -.. _Turbulence based convection scheme: +.. _sec_tbcs: Turbulence based convection scheme ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ @@ -6731,7 +6767,7 @@ particularly poor behaviour, and we do not pick up substantial evidence of problems from this in the full model. This remains something to be investigated. -.. _Homogeneous forcing section improvements: +.. _sec_homog_improve: Homogeneous forcing section improvements ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ @@ -6823,7 +6859,7 @@ developed. However we note two particular issues that relate to PC2. :math:`\overline{q}`, :math:`q_{sat}`, :math:`\overline{q_{cl}}` and :math:`C_l` etc.? -.. _Time-stepping: +.. _sec_timestepping: Time-stepping ^^^^^^^^^^^^^ @@ -6853,7 +6889,7 @@ model anything prognostically when the cycling time is less than the timestep. As discussed in section :ref:`Numerical application of the hybrid erosion -method`, the timestep +method `, the timestep sensitivity of cloud amounts in shallow cumulus regimes can be addressed by using a more accurate numerical method to solve the erosion term. Several options are available under the UM namelist switch @@ -6990,27 +7026,41 @@ against mid-latitude cloud and it is known that tropical clouds have greater vertical coherence. Tuning the parameters in :math:`large_scale_cloud/ls_acf_brooks.F90` may be beneficial. -.. figure:: pc2_process_explanation.svg +.. figure:: blank.svg :name: fig:schematic - :alt: Schematic summary of the PC2 cloud scheme. - :width: 100% Schematic summary of the PC2 cloud scheme. -.. figure:: Timestepping_ctl66.svg + .. list-table:: + :align: center + :widths: 100 + + * - .. image:: pc2_process_explanation.svg + +.. figure:: blank.svg :name: fig:tstep_diag - :alt: Timestepping diagram for the control (non-PC2) scheme - :width: 60% Timestepping diagram for the control (non-PC2) scheme -.. figure:: Timestepping_pc266.svg + .. list-table:: + :align: center + :widths: 60 + + * - .. image:: Timestepping_ctl66.svg + :width: 60% + +.. figure:: blank.svg :name: fig:tstep_prog - :alt: Timestepping diagram for the PC2 scheme - :width: 60% Timestepping diagram for the PC2 scheme + .. list-table:: + :align: center + :widths: 60 + + * - .. image:: Timestepping_pc266.svg + :width: 60% + References ========== diff --git a/documentation/source/science_guide/cloud_schemes/blank.svg b/documentation/source/science_guide/cloud_schemes/blank.svg new file mode 100644 index 0000000000..d5b329f06a --- /dev/null +++ b/documentation/source/science_guide/cloud_schemes/blank.svg @@ -0,0 +1,3 @@ + + From 2d750b7b996127b16b486ecb87def23f60a6955a Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Fri, 24 Apr 2026 15:35:30 +0100 Subject: [PATCH 048/116] Re-applied remaining manual corrections. --- .../cloud_schemes/UMDP30_PC2CloudScheme.rst | 27 ++++++++++--------- 1 file changed, 15 insertions(+), 12 deletions(-) diff --git a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst index 3519d2f1ca..f7512f1d52 100644 --- a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst +++ b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst @@ -3126,7 +3126,7 @@ cumulus regimes. Note that :math:`q_{cl}` falls to zero after a finite time :math:`\frac{{q_{cl}}_0}{1-b_1} \frac{a_L}{K} - \frac{1}{2 {C_l}_0 (1 - C_l) \, (q_{cl}-Q_c)}`. If the timestep + \frac{1}{2 {C_l}_0 (1 - C_l) \, (q_{cl}-Q_c)}`. If the timestep :math:`\Delta t` is longer than this time, then erosion completely removes the cloud during the current timestep. @@ -4699,26 +4699,29 @@ prognostic :math:`C_l` and :math:`q_{cl}` are incremented as follows: - If :math:`{q_{cl}}_{diag} > q_{cl}`: - :math:`\Delta q_{cl} = {q_{cl}}_{diag} - q_{cl} - \quad \refstepcounter{equation}(\theequation)\label{eq:dqcl_init}` + .. math:: :label: eq:dqcl_init + + \Delta q_{cl} = {q_{cl}}_{diag} - q_{cl} - If :math:`Q_C < 0`: - :math:`\Delta C_{l} = \frac{\Delta q_{cl}}{{q_{cl}}_{diag}} - \left( {C_{l}}_{diag} - C_{l} \right) - \quad \refstepcounter{equation}(\theequation)\label{eq:dcl_init1}` + .. math:: :label: eq:dcl_init1 + + \Delta C_{l} = \frac{\Delta q_{cl}}{{q_{cl}}_{diag}} + \left( {C_{l}}_{diag} - C_{l} \right) - If :math:`Q_C > 0`: - :math:`\Delta C_{l} = \frac{\Delta SD}{{SD}_{diag}} - \left( {C_{l}}_{diag} - C_{l} \right) - \quad \refstepcounter{equation}(\theequation)\label{eq:dcl_init2}` + .. math:: :label: eq:dcl_init2 + + \Delta C_{l} = \frac{\Delta SD}{{SD}_{diag}} + \left( {C_{l}}_{diag} - C_{l} \right) - Otherwise: - :math:`\Delta q_{cl} = 0` + .. math:: \Delta q_{cl} = 0 - :math:`\Delta C_{l} = 0` + .. math:: \Delta C_{l} = 0 where the subscript :math:`_{diag}` denotes the liquid cloud water content and fraction predicted by the diagnostic cloud scheme (either @@ -7223,4 +7226,4 @@ References *Modification of the thermodynamic variability closure in the Met Office Unified Model prognostic cloud scheme*. Atmospheric Science Letters. - https://doi.org/10.1002/asl.1021 \ No newline at end of file + https://doi.org/10.1002/asl.1021 From 22d0d553f7f14da948bdb57d2ba196cb03821db5 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Fri, 24 Apr 2026 16:58:58 +0100 Subject: [PATCH 049/116] Fixed script problem that broke table caption, then re-applied. --- .../science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst | 3 ++- 1 file changed, 2 insertions(+), 1 deletion(-) diff --git a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst index f7512f1d52..a87f493e85 100644 --- a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst +++ b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst @@ -6124,7 +6124,8 @@ modification sets. We have included those parameters that have been investigated throughout testing, although only two are different between PC2:64 and a non-PC2 run. -.. list-table:: PC2 parameter values and locations relating to the +.. list-table:: PC2 parameter values and locations relating to the convection. + \*These values are those used in HadGAM :name: tab:pc2_conv_names :header-rows: 1 From dd3429e77d99af0021c202f17f58852feda3e1c4 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Fri, 24 Apr 2026 23:45:38 +0100 Subject: [PATCH 050/116] Updated table fixing script and reran. Now makes list tables instead of grid tables, and cleans up pandoc's leftover |image| blocks. --- .../turbulence_schemes/bldoc.rst | 90 +++++++++++-------- 1 file changed, 55 insertions(+), 35 deletions(-) diff --git a/documentation/source/science_guide/turbulence_schemes/bldoc.rst b/documentation/source/science_guide/turbulence_schemes/bldoc.rst index 1a75966f7c..33a1da2550 100644 --- a/documentation/source/science_guide/turbulence_schemes/bldoc.rst +++ b/documentation/source/science_guide/turbulence_schemes/bldoc.rst @@ -238,9 +238,14 @@ Types I to VI are shown schematically in Fig. `1 <#fig:bltypes>`__. upward arrows indicate the height \zhpar while the top of their solid line portions indicate \zh. - +----------------------------------+----------------------------------+ - | .. image:: wcrp_bltypes1.svg | .. image:: wcrp_bltypes2.svg | - +----------------------------------+----------------------------------+ + .. list-table:: + :align: center + :widths: 46 46 + + * - .. image:: wcrp_bltypes1.svg + :width: 46% + - .. image:: wcrp_bltypes2.svg + :width: 46% .. _`sec:adiapar`: @@ -722,9 +727,11 @@ expected to occur) and :math:`z_{\mbox{\tiny \rm NTML}-1}`. (solid). The shaded area illustrates the integrated turbulent flux that would be obtained were (\protect\mbox{\protect\ref{eq:wx_std}}) used. - +-----------------------------------+ - | .. image:: ideal_invinteg.svg | - +-----------------------------------+ + .. list-table:: + :align: center + :widths: 100 + + * - .. image:: ideal_invinteg.svg Then, @@ -1803,9 +1810,12 @@ Discussion of some of the revisions revised formulation. The dash-dotted line for $d$ is a potential modification, as described in the text. - +------------------------------+ - | .. image:: stab_dep.svg | - +------------------------------+ + .. list-table:: + :align: center + :widths: 80 + + * - .. image:: stab_dep.svg + :width: 80% It is useful to compare the velocity scales in the revised scheme with those in the standard version, as well as those in @@ -1865,9 +1875,11 @@ removed since the entrainment flux is now carried via the explicit \vtopo$. An upside-down version of $\khsurf$ is also shown (dashed) for comparison. - +-----------------------------------+ - | .. image:: new_ktop_shape.svg | - +-----------------------------------+ + .. list-table:: + :align: center + :widths: 100 + + * - .. image:: new_ktop_shape.svg .. _`sec:blend`: @@ -2011,9 +2023,14 @@ turbulence scheme. turb}$ used in Eq.~\ref{eq-tanh} for a well-mixed layer (black dotted) and a decoupled cloud layer (black solid). - +------------------------------------+------------------------------------+ - | .. image:: honnert_vs_tanh.svg | .. image:: zturb_schem.svg | - +------------------------------------+------------------------------------+ + .. list-table:: + :align: center + :widths: 49 49 + + * - .. image:: honnert_vs_tanh.svg + :width: 49% + - .. image:: zturb_schem.svg + :width: 49% One of the key benefits of the :raw-latex:`\cite{lock00}` scheme is its ability to represent decoupled stratocumulus layers, and this is a @@ -2298,9 +2315,11 @@ illustrated for a SML in Fig. `6 <#fig:fluxinterp>`__. are the turbulent fluxes on the model grid indicated by the dashed horizontal lines. - +---------------------------------+ - | .. image:: subsent_fig7.svg | - +---------------------------------+ + .. list-table:: + :align: center + :widths: 100 + + * - .. image:: subsent_fig7.svg Note that, because (`[fluxinterp] <#fluxinterp>`__) includes an explicit balance between the turbulent and radiative fluxes for @@ -2399,9 +2418,11 @@ it to diffuse out this static instability). Schematic illustrating the assumptions behind the subgrid diagnosis of $z_i$. - +----------------------------------+ - | .. image:: nbldoc_zidiag.svg | - +----------------------------------+ + .. list-table:: + :align: center + :widths: 100 + + * - .. image:: nbldoc_zidiag.svg Having identified the model grid-level at the top of the well-mixed layer (either level NTML from the parcel ascent, as described in @@ -2552,9 +2573,11 @@ large-scale vertical velocity evaluated at the inversion, subsidence fluxes (dotted, diamonds), radiative flux (dashed, triangles) and total flux (solid, squares). - +----------------------------------+ - | .. image:: ideal_revflux.svg | - +----------------------------------+ + .. list-table:: + :align: center + :widths: 100 + + * - .. image:: ideal_revflux.svg An idealised subgrid total flux profile is constructed from the parametrized entrainment flux and the increments from radiation, @@ -6408,9 +6431,14 @@ therefore zero entrainment and turbulent mixing). (dotted), $-\Delta_\radf^{SW} $ (dashed) and the 8A (dash-dot) and 9B (dash- dot-dot-dot) parametrizations of $\Delta_\radf$. - +------------------------------+------------------------------+ - | .. image:: div_r080.svg | .. image:: div_r071.svg | - +------------------------------+------------------------------+ + .. list-table:: + :align: center + :widths: 50 50 + + * - .. image:: div_r080.svg + :width: 50% + - .. image:: div_r071.svg + :width: 50% The 9C version attempted to remove the grid-dependence implied by the summation over 3 grid-levels in (`[ctraddiv] <#ctraddiv>`__) as follows: @@ -7138,11 +7166,3 @@ Appendix: Notation implicit solver is called after convection it does not account for this lowering, so that all the subsided air is forced to be entrained into the mixed-layer. - -.. |image1| image:: new_ktop_shape -.. |image| image:: honnert_vs_tanh.eps -.. |image2| image:: zturb_schem.eps -.. |image3| image:: div_r080 - :width: 3.5in -.. |image4| image:: div_r071 - :width: 3.5in From 542e7163d10594943792af1350bd60e7ad7512ca Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Sat, 25 Apr 2026 01:19:30 +0100 Subject: [PATCH 051/116] Fixed bibliography and citations. --- .../turbulence_schemes/bldoc.rst | 606 +++++++++++++++--- 1 file changed, 501 insertions(+), 105 deletions(-) diff --git a/documentation/source/science_guide/turbulence_schemes/bldoc.rst b/documentation/source/science_guide/turbulence_schemes/bldoc.rst index 33a1da2550..192ac604da 100644 --- a/documentation/source/science_guide/turbulence_schemes/bldoc.rst +++ b/documentation/source/science_guide/turbulence_schemes/bldoc.rst @@ -172,11 +172,11 @@ section `3 <#sec:types>`__. The buoyancy parameters, finite difference and other notation used here are defined in appendices `12 <#app:buoyp>`__ and `17 <#app:not>`__. Further papers describing this scheme and its performance are -:raw-latex:`\cite{lock00}` (noting the corrigendum in -:raw-latex:`\cite{locketal01_corr}`), -:raw-latex:`\cite{martin00:_new_bound_layer_mixin_schem}`, -:raw-latex:`\cite{lock01}`, :raw-latex:`\cite{bushetal1999}` and -:raw-latex:`\cite{brown08:_upgrad_bound_layer_schem_met}`. +`Lock et al. (2000)`_ (noting the corrigendum in +`Lock et al. (2001)`_), +`Martin et al. (2000)`_, +`Lock (2001)`_, `Bush et al. (1999)`_ and +`Brown et al. (2008)`_. .. _`sec:types`: @@ -303,7 +303,7 @@ where :math:`A_{plume}=0.2`, :math:`B_{plume}=3.26`, :math:`G_{max}=10^{-3}`\ Km\ :math:`^{-1}`, :math:`\sigma_{Tv1} = 1.93\, \overline{w'\theta_v'}_S/w_m` and :math:`w_m^3=u_*^3+0.25\,z_{\rm h}\overline{w'b}_S`. Following -:raw-latex:`\cite{holtslag93:_local_versus_nonloc_bound_layer}`, +`Holtslag and Boville (1993)`_, :math:`\theta_v'` is related to the magnitude of the gradient adjustment, :math:`\gamma_{\theta_{\ell}}` (see section `5.3 <#sec:gradadj>`__). Thus, :math:`B_{plume}=A_{ga}`, although @@ -316,7 +316,7 @@ represents a typical buoyancy excess of boundary layer plumes. The pressure at the LCL, :math:`P_{LCL}=P_{k_s} (T_{LCL}/T_{k_s})^{(1/\kappa)}`, where :math:`\kappa=R/c_p`. The temperature at the LCL, :math:`T_{LCL}`, is calculated using -approximations in :raw-latex:`\cite{Bolton1980}` as +approximations in `Bolton (1980)`_ as .. math:: T_{LCL} = 55 + \frac{2840}{3.5 \log(T_{k_s}) - log(e_{k_s}) - 4.805} @@ -520,7 +520,7 @@ eddies to the cloud-top radiative cooling (taken to be 200s) and to be 50m). These values of :math:`\tau_{rc}` and :math:`z_{rc}` are only estimates (and will in reality vary from one cloud to another) but they are consistent with, for example, the observations of -:raw-latex:`\cite{nicholls1986}`. If the parcel failed to fall (i.e., +`Nicholls and Turton (1986)`_. If the parcel failed to fall (i.e., NBDSC equals NTDSC) in a DSC layer *not* overlying cumulus, then the layer is assumed not to be well-mixed. At the top of a cumulus layer, the DSC layer is given a minimum depth of @@ -534,7 +534,7 @@ calculation is only crude. Here, the vertical extent of the :math:`K`-profiles is determined more accurately by ensuring that the magnitude of the integrated buoyancy consumption of TKE within the mixed layer is less than or equal to a fraction, :math:`D_t`, of the buoyancy -production, following :raw-latex:`\cite{turton1987}`. +production, following `Turton and Nicholls (1987)`_. Following appendix `12 <#app:buoyp>`__ the grid-box mean buoyancy flux can be written as: @@ -810,7 +810,7 @@ defines the depth of the inversion over which the negative entrainment heat fluxes are seen. Typically this will be small relative to the model vertical grid but at higher vertical resolution or when a strongly surface-heated boundary layer is capped by weak stability inversions -could be resolved. Following :raw-latex:`\cite{beare2008}`, a simple +could be resolved. Following `Beare (2008)`_, a simple energetic argument gives a realistic prediction of the top of the inversion, :math:`z_{top}`, in LES from @@ -824,7 +824,7 @@ interpolation between grid-levels), :math:`w_m` is the boundary layer velocity scale defined in section `5.1 <#sec:nlsurf>`__ and :math:`b` is the parcel buoyancy. Note that the constant in (`[dz_param] <#dz_param>`__) is the same as in -:raw-latex:`\cite{beare2008}` because :math:`6.3 = 2.5 * 4^{2/3}` and +`Beare (2008)`_ because :math:`6.3 = 2.5 * 4^{2/3}` and :math:`w_m^3` differs by a factor of 4. The buoyancy integration in (`[dz_param] <#dz_param>`__), that is itself dependent on :math:`z_{top}`, is performed working upwards from @@ -988,7 +988,7 @@ then entirely consistent with the assumption that :math:`\theta_{\ell}` and :math:`q_t` are conserved variables within the boundary layer scheme. -As described in :raw-latex:`\cite{lock2012}`, the wind shear generated +As described in `Lock (2012)`_, the wind shear generated by drainage flows in complex terrain is thought to lead to additional vertical mixing. This wind shear can be approximated as @@ -1047,7 +1047,7 @@ with :math:`g_0=10`, :math:`D_m=g_0/4` and :math:`D_h=g_0/25`. If the stability dependent Prandtl number option is chosen (see below) the neutral Prandtl number, :math:`Pr_N`, is set to :math:`0.7`; otherwise :math:`Pr_N=1`. Alternatives are those from the Met Office large-eddy -model (LEM), :raw-latex:`\cite{brown1999}`: +model (LEM), `Brown (1999) 2`_: .. math:: @@ -1068,7 +1068,7 @@ functions are available. The ‘long-tailed’ functions are .. math:: f_{\rm stable} = \frac{1}{1+g_0 Ri} Alternative functions, which decrease as :math:`1/Ri^2` with increasing -stability are, from :raw-latex:`\cite{louis1979}`: +stability are, from `Louis (1979)`_: .. math:: f_{\rm stable} = \frac{1}{(1+ 5 Ri)^2} @@ -1094,7 +1094,7 @@ where B_{Ri} = (g_0/2) /\left(1 - g_0 Ri_{t}/2\right)^2 -For the ‘SHARPEST’ function of :raw-latex:`\cite{derbyshire1997}`, +For the ‘SHARPEST’ function of `Derbyshire (1997)`_, :math:`Ri_{t}=0.1`, while larger values give even sharper reduction of turbulence with increasing :math:`Ri`. An additional option, used operationally in some configurations (originally in the Mesoscale Model, @@ -1102,7 +1102,7 @@ hence called ’MES tails’), is to blend linearly from Louis functions at the surface to SHARPEST by 200m. A stability dependent Prandtl number (:math:`Pr=f_m/f_h`) is generally -used following :raw-latex:`\cite{MailhotLock2004}` with: +used following `Mailhot and Lock (2004)`_ with: .. math:: Pr=\min \left( Pr_{\rm max}, \, Pr_N(1+2Ri) \, \right). @@ -1155,7 +1155,7 @@ are two obvious possibilities, to calculate :math:`Ri` (and thence and then interpolate either :math:`K_h` or :math:`K_m` to be able to calculate the required fluxes. To do the former requires averaging the buoyancy gradient in the numerator (and is referred to by -:raw-latex:`\cite{cullen1994}` as the ‘:math:`\theta`-bar’ method), the +`Cullen and James (1994)`_ as the ‘:math:`\theta`-bar’ method), the latter the wind shear in the denominator (referred to as the ‘:math:`\rho`-bar’ method). Single-column model and other tests demonstrated that the ‘:math:`\rho`-bar’ method could readily generate @@ -1437,7 +1437,7 @@ desire to match the model’s surface transfer formulation within the surface layer (as described further in section `5.1.1 <#sec:hbcomp>`__) and to use a cubic sum of velocity scales within the mixed layer (consistent with dimensional analysis of the TKE equation, see -:raw-latex:`\cite{holtslag93:_local_versus_nonloc_bound_layer}`). +`Holtslag and Boville (1993)`_). The formula for :math:`K_h^{\rm surf}` is identical to (`[kmsurf] <#kmsurf>`__) but with :math:`w_m` replaced by @@ -1454,11 +1454,11 @@ convective. The origin of the functional form of .. _`sec:hbcomp`: -Comparison with :raw-latex:`\cite{holtslag93:_local_versus_nonloc_bound_layer}` +Comparison with `Holtslag and Boville (1993)`_ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The surface-driven :math:`K` profiles are the same as those in -:raw-latex:`\cite{holtslag93:_local_versus_nonloc_bound_layer}`, HB93, +`Holtslag and Boville (1993)`_, HB93, except for (`[ws_defn] <#ws_defn>`__) and (`[prandtl_nl] <#prandtl_nl>`__) and the inclusion of the :math:`{\cal E}_m^{\rm surf}` terms. For the latter, HB93 effectively set @@ -1505,7 +1505,7 @@ The formulation in HB93 gives :math:`Pr` varying from 1 to 0.6 (for (:math:`-z/L=10`), HB93 have :math:`w_m = 0.85 w_*` and :math:`w_h=1.4 w_*` while the UM has :math:`w_m = 0.65 w_*` and :math:`w_h = 1.7 w_*`. The implications of these differences -from HB93 are unknown. The convective LES in :raw-latex:`\cite{lock99}` +from HB93 are unknown. The convective LES in `Lock and Macvean (1999)`_ suggest :math:`w_h \approx w_*`; I don’t know where the larger proportionality constants come from. @@ -1513,7 +1513,7 @@ come from. Another difference between the UM and HB93 is that HB93 only apply gradient adjustment above the surface layer (and this is allowed for in their mixed layer definition of :math:`Pr`). Simulations in -:raw-latex:`\cite{brown1996}`, however, suggest that this may lead to a +`Brown (1996)`_, however, suggest that this may lead to a cold bias at the top of the surface layer. It is attempted to alleviate this in the UM by the application of gradient adjustment down to the surface (although this will then lead to a dependence on the height of @@ -1538,7 +1538,7 @@ appendix `11 <#app:vscales>`__) and :math:`z'` is height above :math:`z_{\rm b}` . Then :math:`K_h = K_m / \mbox{Pr}`, where :math:`\mbox{Pr}=0.75`. The resulting :math:`K_h` profile was derived against convective cloudy LES, as described in -:raw-latex:`\cite{lock99_proceedings}`. The appropriate Prandtl number +`Lock (1999)`_. The appropriate Prandtl number (and therefore :math:`K_m^{\rm Sc}`) is unknown, 0.75 being chosen simply as a number in the middle of the range usually quoted for turbulent mixing in general. As with (`[kmsurf] <#kmsurf>`__), @@ -1608,7 +1608,7 @@ to zero in order to represent crudely the effects on the mixed-layer tend to make :math:`q_t` profiles less well mixed than those of :math:`\theta_{\ell}` :raw-latex:`\cite[]{mahrt1976}`. From UM version 5.5, there is the option to implement the non-gradient stress -parametrization of :raw-latex:`\cite{brown97:_non}`, as described in +parametrization of `Brown and Grant (1997)`_, as described in section `5.4 <#sec:ngstress>`__. .. _`sec:ngstress`: @@ -1620,16 +1620,16 @@ There is an option that is operational in the UM to include an additional non-gradient (or non-local) stress parametrization, :math:`{\bf \tau}^{nl}` in (`[uv_closure] <#uv_closure>`__), as proposed by -:raw-latex:`\cite{brown97:_non}`. They showed that with only a +`Brown and Grant (1997)`_. They showed that with only a down-gradient stress parametrization, a one-dimensional model produced wind profiles in the convective boundary layer that were less well-mixed than predicted by LES, and underestimated the near surface wind. -Furthermore, :raw-latex:`\cite{brownetal2006}` showed that the +Furthermore, `Brown et al. (2006)`_ showed that the operational verification statistics indicate a slow bias in the 10 m wind over land by day, especially in spring and summer. The non-gradient stress parametrization in the UM is very similar to -that proposed by :raw-latex:`\cite{brown97:_non}`, written +that proposed by `Brown and Grant (1997)`_, written .. math:: @@ -1650,14 +1650,14 @@ gradient across the boundary layer is always less than a parameter, MAX_STRESS_GRAD, currently set to 0.05 ms\ :math:`^{-2}` (which, for example, gives a maximum :math:`u_*` of 7 ms\ :math:`^{-1}` in a boundary layer 1km deep). The term involving :math:`u_*` and :math:`w_*` -is as proposed by :raw-latex:`\cite{brown97:_non}` (although note that +is as proposed by `Brown and Grant (1997)`_ (although note that their Table 3 contains a typo), and ensures that the non-gradient stress is zero in neutral conditions but asymptotes to a stability-independent fraction of surface stress in convective conditions. The primed variables in the shape function allow the non-local stress profile to either be applied across the whole boundary layer (using :math:`z'=z` and :math:`z_{\rm h}'=z_{\rm h}`), as in -:raw-latex:`\cite{brown97:_non}`, or only above the surface layer (using +`Brown and Grant (1997)`_, or only above the surface layer (using :math:`z'=z-0.1z_{\rm h}`, :math:`z_{\rm h}'=z_{\rm h}-0.1z_{\rm h}`). The motivation for applying the non-local stress above the surface layer was to ensure that the match to surface layer similarity was maintained @@ -1819,7 +1819,7 @@ Discussion of some of the revisions It is useful to compare the velocity scales in the revised scheme with those in the standard version, as well as those in -:raw-latex:`\cite{holtslag93:_local_versus_nonloc_bound_layer}`, +`Holtslag and Boville (1993)`_, hereafter HB, on which the parametrization was originally based. Recall that HB and the standard UM set :math:`w_m = (u_*^3 + C_{ws} w_*^3)^{\frac{1}{3}}` and :math:`w_h @@ -1842,7 +1842,7 @@ gradient adjustment parameter: The inclusion of an extra :math:`w_*/w_m` factor in :math:`\gamma_{\chi}` was a deliberate change by HB from the original -:raw-latex:`\cite{troen86:_simpl_model_atmos_bound_layer}` formulation +`Troen and Mahrt (1986)`_ formulation on which the UM was based. This seems an appealing feature (HB’s :math:`\gamma_{\chi}` will tend to zero as :math:`w_* \rightarrow 0`) and probably should be considered for the revised scheme (the @@ -1898,15 +1898,15 @@ method for blending the two parametrizations has been developed. This blend is regime and scale dependent, allowing a single parametrization to be used across resolutions, including the completely unresolved/resolved extremes. This blending process is described in -:raw-latex:`\cite{Boutleetal2014}`, which gives some examples of its use +`Boutle et al. (2014)`_, which gives some examples of its use and comparison to simulations using either the 1D BL or 3D Smag schemes -only. Updated technical details from :raw-latex:`\cite{Boutleetal2014}` +only. Updated technical details from `Boutle et al. (2014)`_ are reproduced below. Several options are available that are selected using the switch ``blending_option``. These all follow the same principles but differ in their choice of what should constitute the boundary layer and how to treat non-turbulent layers of the atmosphere. -As shown in :raw-latex:`\cite{Honnertetal2011}`, the rate at which +As shown in `Honnert et al. (2011)`_, the rate at which turbulent structures become resolved appears to be different for different aspects of the flow. For example, moisture fluxes are on a larger scale than heat or momentum fluxes, and so transition to being @@ -1951,7 +1951,7 @@ schemes. For example, at :math:`\Delta x=1` km, :math:`\lambda_0=\max(40\ {\rm m}, 0.15z_h)`, which allows for a small mixing length in shallow unresolved boundary layers (e.g. stable ones). -The :raw-latex:`\cite{lock00}` scheme also contains a non-local +The `Lock et al. (2000)`_ scheme also contains a non-local component to the turbulent flux, and this is simply down-weighted by :math:`W_{1D}` to ensure that it becomes less significant as the turbulence becomes better resolved. Therefore the full eddy diffusivity @@ -1967,12 +1967,12 @@ as .. math:: F_\chi=-K_\chi\frac{\partial \chi}{\partial z} + W_{1D}F_\chi^{\rm NL}, where :math:`F_\chi^{\rm NL}` is the non-local flux. Therefore when -:math:`W_{1D}=1`, the scheme of :raw-latex:`\cite{lock00}` is recovered, +:math:`W_{1D}=1`, the scheme of `Lock et al. (2000)`_ is recovered, whilst with :math:`W_{1D}=0` the Smagorinsky-type scheme is recovered. Now we need to define the function :math:`W_{1D}` to blend the schemes. Within the boundary layer this is based on the turbulent kinetic energy -partitioning given by :raw-latex:`\cite{Honnertetal2011}`. We choose the +partitioning given by `Honnert et al. (2011)`_. We choose the TKE partitioning because it is most closely linked to the eddy diffusivity we are trying to parametrize (for example a TKE based scheme would calculate the eddy diffusivity from the TKE), and simplify the @@ -1989,10 +1989,10 @@ speed of the transition from unresolved to resolved turbulence, :math:`r_f=\frac{1}{l_0-l_1}`, :math:`l_0=4` and :math:`l_1=0.25` (N. B. this formula is slightly modified from that given in :raw-latex:`\cite[]{Boutleetal2014}`). -:raw-latex:`\cite{Malavelleetal2014}` demonstrated that this scaling +`Malavelle et al. (2014)`_ demonstrated that this scaling method was applicable to any type of unstable boundary layer given an appropriate choice of :math:`z_{\rm turb}`. In -:raw-latex:`\cite{Boutleetal2014}` this functional form was applied +`Boutle et al. (2014)`_ this functional form was applied everywhere, adjusting the values of :math:`z_{\rm turb}` and :math:`\beta` depending on the regime. The max function is present to force the lowest resolution simulations to just use the 1D mixing @@ -2002,10 +2002,10 @@ described below. The simplest case is for a well-mixed boundary layer, where the appropriate lengthscale is the boundary-layer depth (inversion height). Therefore we set :math:`z_{\rm turb}=z_h`, which is broadly consistent -with :raw-latex:`\cite{Malavelleetal2014}`, and choose +with `Malavelle et al. (2014)`_, and choose :math:`\beta=\beta_{\rm bl}=0.15` to give the best match of our function to that of -:raw-latex:`\cite{Honnertetal2011}`. These functions are shown in +`Honnert et al. (2011)`_. These functions are shown in Figure `5 <#fig-blend>`__\ (a) and are only dissimilar for small :math:`\Delta x`, where Eq. `[eq-tanh] <#eq-tanh>`__ tends to zero faster. This is by @@ -2017,8 +2017,8 @@ turbulence scheme. (a) Weighting for the 1D boundary-layer scheme as a function of $\Delta x/z_{\rm turb}$, showing the function of Equation~\ref{eq-tanh} (blue - solid), the equation in \cite{Boutleetal2014} (black solid) and the TKE - partitioning of \cite{Honnertetal2011} (mean thick dashed, 5th/95th + solid), the equation in `Boutle et al. (2014)`_ (black solid) and the TKE + partitioning of `Honnert et al. (2011)`_ (mean thick dashed, 5th/95th percentiles thin dashed). (b) Schematic showing the calculation of $z_{\rm turb}$ used in Eq.~\ref{eq-tanh} for a well-mixed layer (black dotted) and a decoupled cloud layer (black solid). @@ -2032,11 +2032,11 @@ turbulence scheme. - .. image:: zturb_schem.svg :width: 49% -One of the key benefits of the :raw-latex:`\cite{lock00}` scheme is its +One of the key benefits of the `Lock et al. (2000)`_ scheme is its ability to represent decoupled stratocumulus layers, and this is a feature which needs to be maintained in the blended scheme. Physically they are similar to well-mixed surface driven boundary layers, and the -:raw-latex:`\cite{lock00}` scheme parametrizes them as such. The +`Lock et al. (2000)`_ scheme parametrizes them as such. The appropriate length scale is now the decoupled cloud mixed layer depth, :math:`z_{\rm sc}` :raw-latex:`\cite[i.e.~the depth through which a negatively buoyant parcel @@ -2056,8 +2056,8 @@ schematically in Figure `5 <#fig-blend>`__\ (b), and ensures that layer and cloud layer, and a lower value in between those layers. Again, this choice of :math:`z_{\rm turb}` is broadly consistent with the analysis of decoupled stratocumulus LES presented by -:raw-latex:`\cite{Malavelleetal2014}`. Finally, -:raw-latex:`\cite{Honnertetal2011}` also included shallow cumulus +`Malavelle et al. (2014)`_. Finally, +`Honnert et al. (2011)`_ also included shallow cumulus simulations and showed that the relevent length scale there was the cloud top height. Most of the ``blending_option`` choices apply this to all regimes diagnosed as cumulus-capped (see section `3 <#sec:types>`__) @@ -2077,7 +2077,7 @@ vertical mixing was beneficial for the development of the convection, and that without this a widespread stratiform cloud layer could develop instead. -Above the boundary layer top, :raw-latex:`\cite{Boutleetal2014}` aimed +Above the boundary layer top, `Boutle et al. (2014)`_ aimed for any free atmospheric mixing to be done by the 3D Smagorinsky scheme. Therefore, above the boundary layer top they use :math:`z` as the appropriate length scale, and in general take :math:`z_{\rm turb}` in @@ -2146,7 +2146,7 @@ the inversion is sufficiently sharp so as to be unresolved, the ideal is to specify the entrainment fluxes explicitly, as described in section `7.1 <#sec:ent_flux>`__, based on the subgrid inversion diagnosis described in section `7.1.1 <#sec:sginv>`__. Further details -can be found in :raw-latex:`\cite{lock01}`. If the profiles are such +can be found in `Lock (2001)`_. If the profiles are such that the inversion is sharp but a subgrid inversion cannot be diagnosed, an eddy-diffusivity similar to that for momentum is used (see section `7.4.1 <#sec:ent_K>`__). If the inversion is thick enough to be @@ -2173,9 +2173,9 @@ written (using the notation given in appendix `11 <#app:vscales>`__) where :math:`V_{\rm sum}^3= V_{\rm heat}^3+ V_{\rm rad}^3+ V_{\rm br}^3+ A_2 u_*^3`. The constant :math:`A_1` is given a value 0.23, as in -:raw-latex:`\cite{lock98}`, and :math:`A_1*A_2=5`, as in -:raw-latex:`\cite{driedonks1982}`. To allow for weak inversions, the -:raw-latex:`\cite{zilitinkevich1975}` correction is included in +`Lock (1998)`_, and :math:`A_1*A_2=5`, as in +`Driedonks (1982)`_. To allow for weak inversions, the +`Zilitinkevich (1975)`_ correction is included in (`[we_parm] <#we_parm>`__) with the constant, :math:`c_T=1`. A further parametrization for :math:`\alpha_t`, which is the fraction of the cloud-top radiative divergence (:math:`\Delta_F`, in Kms\ :math:`^{-1}`) @@ -2193,8 +2193,8 @@ depth-scale for the radiatively-cooled layer (taken to be 15 allow for a feedback with forcing of entrainment by buoyancy reversal (see appendix `11 <#app:vscales>`__), :math:`\tilde{\alpha_t} = \alpha_t+ Br -(1-\alpha_t)`. following :raw-latex:`\cite{lock98}` and -:raw-latex:`\cite{lock09:_factor}`. The calculation of the other +(1-\alpha_t)`. following `Lock (1998)`_ and +`Lock (2009)`_. The calculation of the other quantities required for (`[we_parm] <#we_parm>`__) is described in appendix `11 <#app:vscales>`__. At some point during the transition to a decoupled boundary layer the surface-driven entrainment terms (the terms @@ -2230,7 +2230,7 @@ limit is applied to the value of :math:`w_e` determined by than one grid-level in a timestep. With current vertical resolutions and timesteps this is not a serious restriction. The constants :math:`A_1` and :math:`A_{\rm br}` appeared to be determined within 10-20 % in -:raw-latex:`\cite{lock98}`, although only solid cloud sheets were +`Lock (1998)`_, although only solid cloud sheets were simulated (as discussed further in appendix `11 <#app:vscales>`__). Similarly the parametrizations of :math:`\alpha_t` and :math:`\Delta z_i` were found to be accurate but the parameter @@ -2362,7 +2362,7 @@ flux gradient across the mixed layer. Finally, the entrainment flux is adjusted to allow for numerical entrainment arising from the model’s resolved vertical advection (as -discussed in :raw-latex:`\cite{lock01}`). This is performed at whichever +discussed in `Lock (2001)`_). This is performed at whichever grid-level the entrainment fluxes are specified, to allow for any entrainment implied by a :math:`\theta_{\ell}` subsidence increment, :math:`\Theta^{\rm @@ -2629,7 +2629,7 @@ height :math:`z_h` and top at :math:`z_t` (in the UM, the inversion is assumed to be infinitesimally thin so that :math:`z_t=z_h`). This integration gives :math:`- w_e \Delta \chi = \overline{w'\chi'}|_{z_h} -(F_{\chi}^{NTP}|_{z_t}-F_{\chi}^{NTP}|_{z_h})`. -:raw-latex:`\cite{lock99}` related the non-turbulent flux divergence, +`Lock and Macvean (1999)`_ related the non-turbulent flux divergence, :math:`F_{\chi}^{NTP}|_{z_t}-F_{\chi}^{NTP}|_{z_h}`, to radiative cooling occurring within undulations of the cloudy boundary layer top. Similar considerations need to be borne in mind when calculating all the @@ -3157,7 +3157,7 @@ height of the top of the surface-based turbulent mixing layer. empirically or tuned within empirical limits. The third term represents the effects of deep convective cloud-scale gusts; the inclusion of this term is optional. The form implemented is taken from -:raw-latex:`\cite{redelsperger00:_param_mesos_enhan_surfac_fluxes}`, in +`Redelsperger et al. (2000)`_, in which the velocity scale, w\ :math:`_{c}`, is a function of the convective downdraught mass-flux at cloud base. (Note that the published expression is given as an adjustment of the 10-m wind and has been @@ -3194,7 +3194,7 @@ heat flux in very low mean wind conditions. .. _section_1.2: -Comparison with the :raw-latex:`\cite{godfrey1991}` formulation for gustiness +Comparison with the `Godfrey and Beljaars (1991)`_ formulation for gustiness ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ We can define the **mean gust speed** at height z\ :math:`_{1}` by @@ -3235,8 +3235,8 @@ Equation (`[1.2.1] <#1.2.1>`__) can be rewritten as V^2 = \left| {\Delta {{\rm {\bf v}}}} \right|^2 + v_g^2, \label{1.2.4} -which is exactly the form of :raw-latex:`\cite{godfrey1991}`. However -:raw-latex:`\cite{godfrey1991}` define the mean gust speed as +which is exactly the form of `Godfrey and Beljaars (1991)`_. However +`Godfrey and Beljaars (1991)`_ define the mean gust speed as :math:`\beta`\ w\ :math:`_{\ast }`. Thus they directly modify the mean surface to air wind difference, :math:`\Delta`\ **v**, with the gustiness or turbulent convective @@ -3250,12 +3250,12 @@ in the surface layer which approaches zero at the surface (strictly at the roughness height z\ :math:`_{0m})`. The two formulations can be made equivalent by assuming that -:raw-latex:`\cite{godfrey1991}` :math:`\beta` is not constant but is +`Godfrey and Beljaars (1991)`_ :math:`\beta` is not constant but is given by :math:`\gamma _{t}(\Phi _{m}`\ (z)/k) which tends to zero as the surface is approached. However, over sea points where the roughness length is small (of order 10\ :math:`^{-4}` m) and for which -:raw-latex:`\cite{godfrey1991}` derived their formulation, +`Godfrey and Beljaars (1991)`_ derived their formulation, :math:`\Phi_{m}`\ (z) varies at most by about 15% between 10 m and 50 m. Assuming a constant :math:`\beta` does not lead to much inaccuracy in these circumstances. If gustiness is included over land, as is the case @@ -3434,7 +3434,7 @@ The form of the stability functions. ------------------------------------ For **stable conditions**, i.e. :math:`\Delta`\ B :math:`\ge` 0, the -stability functions are given by :raw-latex:`\cite{Beljaars1991}`: +stability functions are given by `Beljaars and Holtslag (1991)`_: .. math:: @@ -3473,7 +3473,7 @@ by \frac{ z_1 /L}{ \Phi _m } \label{1.3.8} -so the :raw-latex:`\cite{Beljaars1991}` functions imply +so the `Beljaars and Holtslag (1991)`_ functions imply Ri\ :math:`_{f B} \to` 1/a = 1 as z\ :math:`_{1}`/L :math:`\to \infty`. For **unstable conditions**, i.e. :math:`\Delta`\ B :math:`<` 0, the @@ -4029,7 +4029,7 @@ described. observational evidence. A parametrization of the scalar roughness length was developed from surface divergence theory :raw-latex:`\cite[]{csanady2001}`, as described by - :raw-latex:`\cite{edwards2007}`. This involves an inverse dependence + `Edwards (2007)`_. This involves an inverse dependence of :math:`z_{0h}` on the friction velocity in the aerodynamically smooth limit and an inverse dependence of :math:`z_{0h}` on :math:`z_{0m}` at higher wind speeds that reduces the increase in the @@ -4118,7 +4118,7 @@ is some uncertainty over the behaviour of the drag at the wind speeds encountered in tropical cyclones: indeed, there is considerable evidence that it does not continue to increase in the manner predicted by schemes like those described above and may even decrease. -:raw-latex:`\cite{Donelan2004}` presents some measurements suggesting +`Donelan et al. (2004)`_ presents some measurements suggesting that the drag coefficient should not be permitted to increase for 10-m neutral winds above about 33 ms\ :math:`{}^{-1}`, when the drag coefficient is about 0.0024. Whilst it is likely that further work will @@ -4142,7 +4142,7 @@ coefficient has been allowed for by introducing the option surface at high wind speeds, with the neutral drag coefficient saturating at around 35 ms\ :math:`{}^{-1}` and declining at higher wind speeds. Suggested values of these coefficients are based on - :raw-latex:`\cite{donelan2018}` and :raw-latex:`\cite{hsu2017}`. + `Donelan (2018)`_ and `Hsu et al. (2017)`_. It might be thought more logical to subsume the treatment of high winds under ``iseasurfalg``, but given that standard schemes for surface @@ -4218,7 +4218,7 @@ Two approaches are available in uncoupled configurations of the model. 5x10\ :math:`^{-4}` m. Historically, z\ :math:`_{0h(sea-ice)}` was set equal to z\ :math:`_{0m(sea-ice)}`, but more recently it has been set equal to one fifth of z\ :math:`_{0m(sea-ice)}`, based on - :raw-latex:`\cite{andreas2010}`. The setting for marginal ice is more + `Andreas et al. (2010)`_. The setting for marginal ice is more problematic. Whilst z\ :math:`_{0m(MIZ)}` should be larger than z\ :math:`_{0m(sea-ice)}`, good simulations of mean sea-level pressure are obtained only if z\ :math:`_{0m(MIZ)}` is substantially @@ -4231,10 +4231,10 @@ Two approaches are available in uncoupled configurations of the model. #. Explicit Treatment of Ice Form Drag - :raw-latex:`\cite{lupkes2012}` have suggested a simple + `L{\ (2012)`_ have suggested a simple parametrization of the form drag coefficient of marginal ice that has been found to perform well in comparison to aircraft measurements - (:raw-latex:`\cite{elvidge2016}`). :raw-latex:`\cite{lupkes2015}` + (`Elvidge et al. (2016)`_). `L{\ (2015)`_ have extended the parametrization to include the effects of stability. When coupled to CICE, it is intended that a more elaborate scheme will be used, but this scheme is useful for application in @@ -4247,12 +4247,12 @@ Two approaches are available in uncoupled configurations of the model. :math:`u(z)` will in general exhibit a mixed character, but it may be taken as the developed flow over open sea, as in - :raw-latex:`\cite{lupkes2012}`, or may be interpolated between the + `L{\ (2012)`_, or may be interpolated between the developed flows over open sea or pack ice, depending on the ice - fraction, as in :raw-latex:`\cite{lupkes2015}`. In principle, it will + fraction, as in `L{\ (2015)`_. In principle, it will be subject to the effects of stability, but since the free-board does not much exceed 0.5m, these effects are small - (:raw-latex:`\cite{lupkes2015}`) and the flow may be taken as neutral + (`L{\ (2015)`_) and the flow may be taken as neutral up to :math:`h_f`. Hence, .. math:: @@ -4266,18 +4266,18 @@ Two approaches are available in uncoupled configurations of the model. the wind on the model’s lowest atmospheric level. Because this will be significantly above :math:`h_f`, the stability dependence of :math:`C_d` should be considered here (again see - :raw-latex:`\cite{lupkes2015}`). :math:`U_1` may be interpreted as + `L{\ (2015)`_). :math:`U_1` may be interpreted as the wind at a specific height, or, consistenly with the flux-difference form of the momentum equation, as the layer-averaged velocity. This distinction affects the numerical value of :math:`C_d`, but does not otherwise affect the foregoing equation. If using the original version of the scheme - (:raw-latex:`\cite{lupkes2012}`), :math:`C_d` must be taken as the + (`L{\ (2012)`_), :math:`C_d` must be taken as the neutral drag coefficient. Note also that various approximations may be made in Equation `[eq:int_u2] <#eq:int_u2>`__. - :raw-latex:`\cite{lupkes2012}` approximate + `L{\ (2012)`_ approximate :math:`(\log(h_f/z_0) -1)^2 +1` as :math:`(\log(h_f/z_0) )^2`; while - :raw-latex:`\cite{lupkes2015}` approximate it as + `L{\ (2015)`_ approximate it as :math:`(\log(h_f/z_0) -1)^2`. Here we retain the full expression. If, in a unit area, there are :math:`N` floes, each of crosswind @@ -4315,19 +4315,19 @@ Two approaches are available in uncoupled configurations of the model. (1-A) C_{ds} L_s + A C_{di} L_i \right ], where the sheltering factor is taken to be the same over ice and - water. :raw-latex:`\cite{lupkes2012}` provides parametrizations for - quantities such as :math:`h_f`, while :raw-latex:`\cite{elvidge2016}` + water. `L{\ (2012)`_ provides parametrizations for + quantities such as :math:`h_f`, while `Elvidge et al. (2016)`_ provide suggested values for the constants in the scheme, based on observations. In using these values in the Unified Model, :math:`c_e` should be increased by about 30% to represent the effect of differing approximations of the logarithmic wind profile. - For scalar transfer :raw-latex:`\cite{lupkes2015}` suggest adding a + For scalar transfer `L{\ (2015)`_ suggest adding a contribution to the sensible heat flux to represent the impact of form drag; however, the mechanistic physical basis of the scheme they propose is unclear. Moreover, when combined with the interfacial drag, this suggests scalar transfer much larger than observed by - :raw-latex:`\cite{schroder2003}`. Consequently, no enhancement of the + `Schr{\ (2003)`_. Consequently, no enhancement of the scalar transfer coefficient by form drag is included. The overall drag coefficients are now set by interpolation in the ice @@ -4549,7 +4549,7 @@ Richardson number of the surface layer and is set to 1 for Ri\ :math:`_{SL} <` 0 and decreases linearly to zero at Ri\ :math:`_{SL(crit)}` = 0.5. The orographic drag coefficient c\ :math:`_{D(orog)}` is set to the constant value (typically 0.3, -:raw-latex:`\cite{mason1986}`). +`Mason (1986)`_). If the function :math:`\Phi _{m}` and v\ :math:`_{\ast }` are approximated by their neutral values in (`[2.1.6] <#2.1.6>`__) @@ -4584,7 +4584,7 @@ Equation (`[2.1.13] <#2.1.13>`__) implies that Parametrized orographic drag coefficient ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -:raw-latex:`\cite{wood93}` find that orographic drag coefficient +`Wood and Mason (1993)`_ find that orographic drag coefficient c\ :math:`_{D(orog)}` depends on A/S via the equation .. math:: @@ -4633,7 +4633,7 @@ and the surface flux for the flat surface is given by \frac{ F_{X0(f)} }{ \rho _0 } = \frac{k v_{\ast (f)} }{ \Phi _h (L , z_c , z_{0h} )} (X( z_c ) - X_0 ) \label{2.1.19} -:raw-latex:`\cite{hewer1998}` find that the scalar transport is enhanced +`Hewer and Wood (1998)`_ find that the scalar transport is enhanced when there is orographic form drag such that .. math:: @@ -4657,7 +4657,7 @@ which becomes \frac{\ln ( z_c / z_{0h(eff)} )}{\ln ( z_c / z_{0h} )} = {\left( {1 + \alpha \beta \pi ^2 f_D {\left( {\frac{A}{S}} \right)}^2 } \right)}^{1/2} \left( {1 - 2.2 f_D \frac{A}{S}} \right) \label{2.1.22} -if the :raw-latex:`\cite{wood93}` formulation is used. +if the `Wood and Mason (1993)`_ formulation is used. .. _section_2.2: @@ -4944,7 +4944,7 @@ Distributed form drag – an alternative to the effective roughness length param An alternative representation of the turbulent form drag due to sub-grid hills is the explicit orographic stress parametrization proposed by -:raw-latex:`\cite{wood01:_param}`. In this representation the drag is +`Wood et al. (2001)`_. In this representation the drag is represented via an orographic stress term, applied directly to the horizontal momentum equations. The roughness lengths remain at the vegetative values and no adjustment to the roughness lengths for scalar @@ -4961,7 +4961,7 @@ on the right-hand side of the horizontal momentum equation, where :math:`{\bf\tau}_{\rm orog}` is the horizontal vector containing the extra stress imparted on the flow by the sub-grid orography This term is included in the Unified Model as an additional explicit (in terms of -time discretisation) stress. Following :raw-latex:`\cite{wood01:_param}` +time discretisation) stress. Following `Wood et al. (2001)`_ we define :math:`{\bf\tau}_{\rm orog}` to be .. math:: {\bf\tau}_{\rm orog}(z)=\left({F_p}_x,{F_p}_y\right)e^{-z/\ell}, @@ -4994,7 +4994,7 @@ parametrization (Eq. `[2.1.10] <#2.1.10>`__), namely: the main difference being the dependence on the height scale :math:`\ell` rather than :math:`z_c`. Similarly, if the -:raw-latex:`\cite{wood93}` low-hill expression is used, the surface +`Wood and Mason (1993)`_ low-hill expression is used, the surface stress is given by the equivalent of (Eq. `[2.1.16] <#2.1.16>`__), namely: @@ -5040,7 +5040,7 @@ Unconditionally stable implicit solver -------------------------------------- This is the vertical diffusion scheme of -:raw-latex:`\cite{woodetal2007}` which has the advantages of (i) +`Wood et al. (2007)`_ which has the advantages of (i) unconditional stability and non-oscillatory behaviour for practical NWP cases and (ii) monotonic damping for suitable choices of a free parameter :math:`P` which represents the degree of nonlinearity of the @@ -5324,7 +5324,7 @@ Derivation of boundary conditions for the scalar variables (static energy and total water content flux) is more difficult: the new scheme in comparison with the original scheme is more complex and the procedure for deriving the scalar fluxes described in section 3 of -:raw-latex:`\cite{esseryetal2001}` is also complex. Currently, an +`Essery et al. (2001)`_ is also complex. Currently, an alternative treatment for the boundary conditions has been coded which works well in practice. The boundary conditions for the scalar variables, i.e. the surface scalar fluxes for the new scheme are @@ -5557,7 +5557,7 @@ done, one per scheme stage (step), i.e. one for the stage that :math:`{\delta X}_{1}^{*}` is computed and one for :math:`\delta X_{1}^{n+1}` where the subscript denotes level number. At each call, a modified version of the flux formulae (78), (79) of -:raw-latex:`\cite{esseryetal2001}` is used: +`Essery et al. (2001)`_ is used: **1st sweep:** @@ -5880,7 +5880,7 @@ included in :math:`U_{10m}`). The lowest grid-level value of :math:`W_{1D}` is constant within the boundary layer. The constant :math:`c_{\rm ugn}` in (`[windgust] <#windgust>`__) is determined from universal turbulence spectra for a 25% exceeding probability of the -three-second wind gust (:raw-latex:`\cite{beljaars1987}`). It is +three-second wind gust (`Beljaars (1987)`_). It is included through a function that includes the effective roughness length, :math:`z_{0m(eff)}`, in order to take into account the very high effective :math:`u_*` values that occur over mountainous terrain (due to @@ -5889,7 +5889,7 @@ gust values. Currently the UM takes :math:`c_{\rm ugn}=4` which was reduced from the value used at ECMWF based on evaluation of the wind gust performance. The stability dependence of :math:`\sigma_u` is estimated on the basis of the similarity relation from -:raw-latex:`\cite{panofsky1977}` +`Panofsky et al. (1977)`_ .. math:: @@ -5930,7 +5930,7 @@ where :math:`l` is a length scale. Initially it was thought to diagnose closures have diagnostic relationships for :math:`l` that involve the TKE itself! A common one for stable boundary layers is :math:`l_{st} \sim \sqrt{e} / N`, where :math:`N` is the Brunt-Vaisala -frequency. :raw-latex:`\cite{Suselj2012}`, for example, also take +frequency. `Suselj et al. (2012)`_, for example, also take :math:`l_{un} = \tau_{un} \sqrt{e}` in unstable boundary layers, where :math:`\tau_{un}` is a turbulence timescale that they take as a constant 400 seconds. These they combine through :math:`l^{-1}= l_{un}^{-1} + @@ -5944,7 +5944,7 @@ To derive a TKE diagnostic then requires a parametrization of the turbulence timescale, :math:`\tau_{turb}`. Basic boundary layer scaling (e.g., Figure 4 of -:raw-latex:`\cite{holtslag91:_eddy_diffus_count_trans_convec}`) shows +`Holtslag and Moeng (1991)`_) shows that :math:`\overline{w'^2}` from a variety of convective boundary layer LES and observations nicely follows the relationship @@ -5961,7 +5961,7 @@ generalise (`[w2_scaling] <#w2_scaling>`__) by replacing :math:`w_*` with :math:`w_m` (this really ought to be checked against neutral boundary layer LES but hasn’t yet been). Setting :math:`f(z')=z' (1-z')^2` in (`[w2_scaling] <#w2_scaling>`__) and comparing with Fig.4 -of :raw-latex:`\cite{holtslag91:_eddy_diffus_count_trans_convec}` gives +of `Holtslag and Moeng (1991)`_ gives :math:`c_{w2}= 2.66 / C_{ws}^{2/3}` (i.e., a constant of 2.66 gives the maximum in :math:`\overline{w'^2}/w_*^2` at around the observed value of 0.4) so @@ -5974,7 +5974,7 @@ that we can generalise (`[w2_scaling] <#w2_scaling>`__) to where the mixed layer expression for :math:`w_m` is used. -:raw-latex:`\cite{holtslag91:_eddy_diffus_count_trans_convec}` also show +`Holtslag and Moeng (1991)`_ also show from analysis of the scalar flux budget that .. math:: \overline{w'\theta'} = - \frac{\tau_{turb}}{2} \, \overline{w'^2} \frac{d \theta}{dz} @@ -6004,7 +6004,7 @@ There are two options to derive a TKE diagnosis from the Ri-based scheme and then combine with the non-local TKE (selected via var_diags_opt). One is to assume :math:`\tau_{\rm SBL}=0.7/N` as the timescale for stable boundary layers and combine all these timescales following -:raw-latex:`\cite{Suselj2012}`) to give: +`Suselj et al. (2012)`_) to give: .. math:: @@ -6109,7 +6109,7 @@ analysis :raw-latex:`\cite[]{atlas2011}`. Appendix: Definitions of the velocity scales ============================================ -As described in :raw-latex:`\cite{lock00}`, the parametrization of the +As described in `Lock et al. (2000)`_, the parametrization of the entrainment rate in convective boundary layers is based on four velocity scales, each representative of a turbulence-generating process (:math:`V_{\rm heat}` for surface heating, :math:`u_*` for surface shear @@ -6152,14 +6152,14 @@ cooling will occur predominantly in cloudy air. To allow for a feedback in the presence of buoyancy reversal, the parameter :math:`Br` is included in :math:`\zeta_r` and :math:`\tilde{\alpha_t}` (in (`[we_parm] <#we_parm>`__)). It is given in terms of the -:raw-latex:`\cite{siems1990}` parameter, +`Siems et al. (1990)`_ parameter, :math:`D = \chi_s \delta b/\Delta b` and constrained by :math:`0< Br = 10 D < 1`. This gives a linear ramp for this feedback between regimes where there is no buoyancy reversal (:math:`D \leq 0`) and the feedback seen in LES of stratocumulus :raw-latex:`\cite[]{lock98}` with significant buoyancy reversal (:math:`D \raisebox{-.4ex}{$\ \stackrel{>}{{\scriptstyle \sim}} \ $}0.1`). -Furthermore, the LES of :raw-latex:`\cite{lock09:_factor}` indicated the +Furthermore, the LES of `Lock (2009)`_ indicated the presence of cumulus penetrating up into stratocumulus could be sufficient to enhance the feedback for small :math:`D`. Thus the option exists to enhance :math:`Br` for :math:`0 Date: Sat, 25 Apr 2026 02:51:49 +0100 Subject: [PATCH 052/116] Fixed equation cross-referencing. --- .../turbulence_schemes/bldoc.rst | 1331 +++++++---------- 1 file changed, 577 insertions(+), 754 deletions(-) diff --git a/documentation/source/science_guide/turbulence_schemes/bldoc.rst b/documentation/source/science_guide/turbulence_schemes/bldoc.rst index 192ac604da..bdf6cb18d6 100644 --- a/documentation/source/science_guide/turbulence_schemes/bldoc.rst +++ b/documentation/source/science_guide/turbulence_schemes/bldoc.rst @@ -42,35 +42,33 @@ boundary layer turbulence, Reynolds’ averaging gives the following equation for conserved scalar variables, :math:`\chi`, and the two horizontal components of momentum, :math:`{\bf u}` on a sphere gives: -.. math:: +.. math:: :label: cons_eqn_scal \frac{\partial \chi}{\partial t} = - \frac{1}{r^2 \rho} \, \frac{\partial }{\partial z} \left( r^2 \rho \overline{w'\chi'} \right) + {\cal S} - \label{cons_eqn_scal} -.. math:: +.. math:: :label: cons_eqn_uv \frac{\partial {\bf u}}{\partial t} = \frac{1}{r^2 \rho} \, \frac{\partial }{\partial z} \left( r^2 {\bf \tau} \right) + {\cal S} - \label{cons_eqn_uv} where :math:`\overline{w'\chi'}` and :math:`{\bf \tau}` are the vertical turbulent fluxes to be parametrized, :math:`r` is the height from the centre of the planet and :math:`\rho` is density. The scalar variables -treated by (`[cons_eqn_scal] <#cons_eqn_scal>`__), which are +treated by :eq:`cons_eqn_scal`, which are approximately conserved under moist adiabatic ascent, are: -.. math:: +.. math:: :label: thetal \theta_{\ell}= T_L + \frac{g}{c_p} z = T - \frac{L}{c_p} q_{\ell} - - \frac{L_s}{c_p} q_f + \frac{g}{c_p} z \label{thetal} + - \frac{L_s}{c_p} q_f + \frac{g}{c_p} z -.. math:: +.. math:: :label: qt - q_t = q_v + q_{\ell}+ q_f \label{qt} + q_t = q_v + q_{\ell}+ q_f where :math:`T` is temperature, :math:`q_v` is specific humidity, @@ -84,10 +82,10 @@ of specific quantities is also available and the details of the necessary changes are documented in appendix `13 <#app:mixratio>`__. Ultimately turbulent motions are dissipated as heat and so the source term :math:`{\cal - S}` in (`[cons_eqn_scal] <#cons_eqn_scal>`__) can include an + S}` in :eq:`cons_eqn_scal` can include an approximation for that frictional heating, as described in appendix `14 <#app:fricheat>`__. Finally, the ice cloud contributions in -(`[thetal] <#thetal>`__) and (`[qt] <#qt>`__) can optionally be ignored +:eq:`thetal` and :eq:`qt` can optionally be ignored (l_noice_in_turb), which will be more appropriate if the time scales for ice melting or sublimation are longer than those of the turbulence (and so may be more appropriate if the ice itself is not being mixed by @@ -97,10 +95,9 @@ consistency, only liquid cloud fractions with be considered. An additional variable used for diagnostic purposes is -.. math:: +.. math:: :label: thetavl \theta_{v\ell}= \theta_{\ell}(1 + c_v q_t) - \label{thetavl} where :math:`c_v=(1/\epsilon) -1` and :math:`\epsilon` is the ratio of the molecular weights of water vapour and dry air (i.e., :math:`\epsilon @@ -114,15 +111,13 @@ although non-local terms are also included. Under the 9C scheme, an alternative methodology is optionally available, see section `5.5 <#sec:rev_flux_grad>`__. The standard closures are: -.. math:: +.. math:: :label: scal_closure \overline{w'\chi'} = - K_h \frac{\partial \chi}{\partial z} + K_h^{\rm surf}\gamma_{\chi} - \label{scal_closure} -.. math:: +.. math:: :label: uv_closure {\bf \tau} = K_m \frac{\partial {\bf u}}{\partial z} + {\bf \tau}^{nl} - \label{uv_closure} Separate eddy-diffusivities are calculated for momentum, :math:`K_m`, @@ -136,8 +131,8 @@ section `5.3 <#sec:gradadj>`__. Thus, the parametrization reduces to determining :math:`K_h`, :math:`K_m` and :math:`\gamma_{\chi}` and :math:`{\bf \tau}^{nl}`. Two methods are used to determine :math:`K_h` and :math:`K_m` and how they -are combined for (`[scal_closure] <#scal_closure>`__) and -(`[uv_closure] <#uv_closure>`__) is described in +are combined for :eq:`scal_closure` and +:eq:`uv_closure` is described in section `4.2 <#sec:shear>`__. The first method is a local Richardson number (:math:`Ri`) based scheme. It is calculated for all regimes (but will be responsible for all mixing in stable conditions), over all @@ -255,7 +250,7 @@ The diagnostic parcel ascent and cumulus diagnosis **Summary**: the depth of the non-local :math:`K`-profiles for surface-driven turbulence (with NTML grid-levels in the mixed layer and top at height :math:`z_{\rm h}` , as required for -(`[kmsurf] <#kmsurf>`__)) is determined from: +:eq:`kmsurf`) is determined from: #. a diagnostic moist parcel ascent; top at grid-level NTPAR, height :math:`z_{\rm par}` :math:`=z_{\mbox{\tiny \rm NTPAR}+\frac{1}{2}}`. @@ -292,12 +287,11 @@ lifting condensation level (LCL). The calculation of the parcel’s buoyancy excess is described in section `3.1.1 <#sec:parxs>`__. Currently, -.. math:: +.. math:: :label: parcel_pert \theta_v' = \mbox{max} \left[A_{plume}, \, \mbox{min} \left[ B_{plume} \sigma_{Tv1}, \, G_{max}z_{\rm h}\right] \right] - \label{parcel_pert} where :math:`A_{plume}=0.2`, :math:`B_{plume}=3.26`, :math:`G_{max}=10^{-3}`\ Km\ :math:`^{-1}`, @@ -310,7 +304,7 @@ adjustment, :math:`\gamma_{\theta_{\ell}}` (see section somewhat arbitrary limits have been placed on the magnitude of :math:`\theta_v'` for numerical security (the upper limit being consistent with that applied to :math:`\gamma_{\theta_{\ell}}` in -(`[gradadj] <#gradadj>`__) ). Within limits, then, :math:`\theta_v'` +:eq:`gradadj` ). Within limits, then, :math:`\theta_v'` represents a typical buoyancy excess of boundary layer plumes. The pressure at the LCL, :math:`P_{LCL}=P_{k_s} @@ -406,33 +400,32 @@ the environment at that grid-level (:math:`q_s^p \approx {q_s}_k + \alpha_L (T^p-T_k)`). Assuming that :math:`q_{\ell f}^p=q_t^p - q_s^p` gives -.. math:: +.. math:: :label: qlpar q_{\ell f}^p = \mbox{max}\left[ 0.0, \, a_L \left( q_t^p - {q_s}_k - \alpha_L (\theta_{\ell}^p - (g z_k/c_p)-T_k)\right) \right] - \label{qlpar} where the buoyancy parameters :math:`a_L` and :math:`\alpha_L` are defined in appendix `12 <#app:buoyp>`__. Recall that the parcel has :math:`q_t` and :math:`\theta_{\ell}` taken from grid-level :math:`k_s` -which are conserved during its ascent. Note that (`[qlpar] <#qlpar>`__) +which are conserved during its ascent. Note that :eq:`qlpar` will not give condensation until the parcel becomes saturated. In the environment the cloud scheme will allow some condensation (and therefore warming and stabilisation of the environment profile) to take place before the grid-level becomes saturated in the mean. To allow for this in the parcel (without applying the cloud scheme), -(`[qlpar] <#qlpar>`__) is also calculated at each grid-level but using +:eq:`qlpar` is also calculated at each grid-level but using the environment grid-box mean :math:`q_t` and :math:`\theta_{\ell}` to give :math:`q_{\ell f}^e`. The difference in the environment’s condensed water as determined by the UM cloud scheme (i.e., :math:`q_{\ell}+q_f`) -and by (`[qlpar] <#qlpar>`__) (i.e., :math:`q_{\ell f}^e`) is then added +and by :eq:`qlpar` (i.e., :math:`q_{\ell f}^e`) is then added to :math:`q^p_{lf}`. -Given :math:`q_{\ell f}^p`, (`[thetal] <#thetal>`__) implies +Given :math:`q_{\ell f}^p`, :eq:`thetal` implies :math:`T^p = \theta_{\ell}^p - (g z_k/c_p) + (L q_{\ell f}^p/c_p)` (using :math:`L_s` if :math:`T_k` is below the -melting point) and (`[qt] <#qt>`__) implies +melting point) and :eq:`qt` implies :math:`q_v^p = q_t^p - q_{\ell f}^p` and thus :math:`T_v^p` can be calculated. Recall that the diagnosis of the parcel’s maximum buoyancy excess over the environment (described in @@ -507,10 +500,9 @@ diagnosed as the lowest grid-level, descending from NTDSC, where less than :math:`\theta_{v\ell}` of the environment. The parcel perturbation is given by -.. math:: +.. math:: :label: dscd_pert \theta_{v\ell}' = - \, \frac{ \tau_{rc} \Delta_F}{z_{rc}} - \label{dscd_pert} where :math:`\Delta_F` (Kms\ :math:`^{-1}`) is the magnitude of the cloud-top radiative divergence (see appendix `11 <#app:vscales>`__), @@ -539,27 +531,25 @@ production, following `Turton and Nicholls (1987)`_. Following appendix `12 <#app:buoyp>`__ the grid-box mean buoyancy flux can be written as: -.. math:: +.. math:: :label: eq:wb_cont \overline{w'b}= g \left[ (1-C_F) \left(\beta_T \overline{w'\theta_{\ell}'} + \beta_q \overline{w'q_t'}\right) + C_F \left( \tilde{\beta_T} \overline{w'\theta_{\ell}'} + \tilde{\beta_q} \overline{w'q_t'}\right) \right] - \label{eq:wb_cont} -As standard, the fluxes in (`[eq:wb_cont] <#eq:wb_cont>`__) are then +As standard, the fluxes in :eq:`eq:wb_cont` are then expanded using the first-order closure in -(`[scal_closure] <#scal_closure>`__) as: +:eq:`scal_closure` as: .. math:: \overline{w'\theta_{\ell}'}_k = -K_h^{\rm surf}\,\frac{\widetilde{\Delta_k \theta_{\ell}}}{\Delta_k z} -K_h^{\rm Sc}\,\frac{\Delta_k \theta_{\ell}}{\Delta_k z} -.. math:: +.. math:: :label: eq:wx_std \overline{w'q_t'}_k = -\left(K_h^{\rm surf}+ K_h^{\rm Sc}\right) \, \,\frac{\Delta_k q_t}{\Delta_k z} - \label{eq:wx_std} where @@ -580,25 +570,24 @@ successfully this thin unsaturated layer in the buoyancy consumption integral. Thus, the :math:`\overline{w'b}` integration is performed over the cloud and sub-cloud layers separately and the cloud-fraction is taken to be uniform within the cloud layer (and zero below cloud-base). -The height of cloud-base is given by (`[zc_calc] <#zc_calc>`__). +The height of cloud-base is given by :eq:`zc_calc`. An iterative method is then used to find the vertical extent of mixing (within certain bounds, as described below) such that the magnitude of buoyancy consumption of TKE within the mixed layer equals a fraction, :math:`D_t`, of the buoyancy production, i.e., -.. math:: +.. math:: :label: deccrit \sum_{z_{k-\frac{1}{2}} > z_i-z_{\rm ml}}^{z_{k-\frac{1}{2}} < z_i} \left|\left[ \overline{w'b}|_{z_{k-\frac{1}{2}}}<0 \right]\right| \, \Delta_k z \, \leq \, D_t \, \sum_{z_{k-\frac{1}{2}} > z_i-z_{\rm ml}}^{z_{k-\frac{1}{2}} < z_i} \left[ \overline{w'b}|_{z_{k-\frac{1}{2}}}>0 \right] \, \Delta_k z - \label{deccrit} Note that, for simplicity, the :math:`{\cal E}_h` factors are not included in :math:`K_h^{\rm surf}` or :math:`K_h^{\rm Sc}` when -calculating (`[eq:wx_std] <#eq:wx_std>`__) under the assumption that +calculating :eq:`eq:wx_std` under the assumption that they will be small. This process is applied to all unstable mixed layers. For stratocumulus layers, observations and LES suggest a value of :math:`D_t=0.1`. A separate value of :math:`D_t` can be used for the @@ -612,7 +601,7 @@ high a boundary layer depth). The first step is to test for whether a well-mixed layer is possible (either decoupling what has so far been diagnosed as a well-mixed layer or, if one exists, recoupling a decoupled stratocumulus layer), i.e., to -test whether (`[deccrit] <#deccrit>`__) is satisfied with both +test whether :eq:`deccrit` is satisfied with both :math:`K_h^{\rm surf}` and :math:`K_h^{\rm Sc}` extending from the surface to the cloud-top. If recoupling is possible then the various flags identifying the DSC layer are reset (*this includes setting the @@ -629,18 +618,18 @@ section `7 <#sec:entr>`__). If a decoupled layer is diagnosed, then an iteration is performed to find the highest :math:`z_{\rm h}` (so top of the :math:`K_h^{\rm surf}` -profile) that still satisfies (`[deccrit] <#deccrit>`__), but with -:math:`K_h^{\rm Sc}=0` in (`[eq:wx_std] <#eq:wx_std>`__). The iteration +profile) that still satisfies :eq:`deccrit`, but with +:math:`K_h^{\rm Sc}=0` in :eq:`eq:wx_std`. The iteration proceeds with :math:`z_{\rm h}` stepping from its lowest permissible height to its highest (currently 3 steps are used). If at any stage -(`[deccrit] <#deccrit>`__) is violated, then the step below (therefore +:eq:`deccrit` is violated, then the step below (therefore containing the height that would give equality in -(`[deccrit] <#deccrit>`__)) is divided by 4 and 3 of those steps are -taken downwards. If (`[deccrit] <#deccrit>`__) is met the step above is +:eq:`deccrit`) is divided by 4 and 3 of those steps are +taken downwards. If :eq:`deccrit` is met the step above is again reduced by a factor of 4 and 3 steps taken upwards. A total of 3 sweeps are possible, each with a smaller step so that :math:`z_{\rm h}` approaches the height that gives equality in -(`[deccrit] <#deccrit>`__). The accuracy with which this is achieved +:eq:`deccrit`. The accuracy with which this is achieved will be the difference in the maximum and minimum permissible heights of :math:`z_{\rm h}`  divided by :math:`2\times4\times4 = 32`, which will typically be less than 30m. The top grid-level of the SML, NTML, is @@ -652,7 +641,7 @@ The above process is then repeated to find the appropriate top-driven mixing. Some constraints are placed on :math:`z_{\rm b}` , namely that it should never go below :math:`0.1`\ :math:`z_{\rm h}` (to avoid affecting the continuity of the :math:`K` profiles at the top of -the surface layer, see (`[ws_defn] <#ws_defn>`__)). If cumulus +the surface layer, see :eq:`ws_defn`). If cumulus convection has been diagnosed then :math:`z_{\rm b}` is not allowed to go below :math:`z_{\mbox{\tiny \rm NTML}+\frac{1}{2}}` (unless the layer is diagnosed to recouple completely). Finally, :math:`z_{\rm b}` must @@ -665,7 +654,7 @@ the lowest grid-level such that :math:`K_h^{\rm Sc}` is non-zero at the half-level below. A possible extension to this diagnosis would be to include the shear -contribution to the TKE budget in (`[deccrit] <#deccrit>`__) and so +contribution to the TKE budget in :eq:`deccrit` and so allow shear-driven mixing to help maintain well-mixed layers. Surface layer :math:`\overline{w'b}` integration @@ -676,7 +665,7 @@ a different functional form from the rest of the mixed layer. Rather than include this additional complexity in the :math:`\overline{w'b}` integration, the surface layer is treated separately. In place of the finite-difference form of :math:`\overline{w'b}`, see -(`[eq:wb_cont] <#eq:wb_cont>`__) and (`[eq:wx_std] <#eq:wx_std>`__) +:eq:`eq:wb_cont` and :eq:`eq:wx_std` above, :math:`\overline{w'b}` is assumed to be linear between :math:`\overline{w'b}_S` at the surface and zero at a level which must be estimated. The surface layer integration is then from the surface up @@ -698,11 +687,11 @@ Integration of :math:`\overline{w'b}` close to the inversion Because of the large gradients often seen in fluxes close to the inversion (in particular, in the LW radiative flux), simple finite -difference flux calculations, (`[eq:wx_std] <#eq:wx_std>`__), can be +difference flux calculations, :eq:`eq:wx_std`, can be significantly inaccurate in this region. An example is shown in Fig. `2 <#fig:inv_integ>`__. Calculating :math:`\overline{w'\theta_{\ell}'}_{\mbox{\tiny \rm NTML}+\frac{1}{2}}` -from (`[eq:wx_std] <#eq:wx_std>`__) gives a negative value, largely +from :eq:`eq:wx_std` gives a negative value, largely because :math:`\Delta_{\mbox{\tiny \rm NTML}+1} \theta_{\ell}` is positive and so the local flux is large and negative. In reality, :math:`\overline{w'\theta_{\ell}'}` becomes positive only a short @@ -741,38 +730,36 @@ Then, \int_{z_h-\Delta z_{rad}}^{z_h} \, F_{\theta_{\ell}}^{Tot} - F_{\theta_{\ell}}^{NT}\, dz -.. math:: +.. math:: :label: wthl_int = I^{Tot} - I^{rad} - I^{ppn} - \label{wthl_int} For the radiative flux, it could be assumed that the subgrid flux distribution is exponentially dependent on the grid-level LWP, for example. This would give: -.. math:: +.. math:: :label: irad I^{rad} = \frac{\Delta z_{rad}} {\ln(F^{rad}|_{z_h}/F^{rad}|_{z_h-\Delta z_{rad}} ) } \left(F^{rad}|_{z_h}-F^{rad}|_{z_h-\Delta z_{rad}} \right) - \label{irad} However, off-line tests indicated this could give a strong and spurious sensitivity to :math:`F^{rad}|_{z_h-\Delta z_{rad}}`. Furthermore, for -most realistic scenarios, the logarithmic factor in (`[irad] <#irad>`__) +most realistic scenarios, the logarithmic factor in :eq:`irad` tends to be close to 3. Consequently, we approximate :math:`I^{rad} = \Delta z_{rad} ( F^{rad}|_{z_h}-F^{rad}|_{z_h-\Delta z_{rad}} ) /3`. In addition, :math:`F^{rad}|_{z_h}-F^{rad}|_{z_h-\Delta z_{rad}}` is approximated as :math:`\Delta F`, the radiative flux change across cloud-top used in the calculation of :math:`V_{\rm Sc}` -(`[ctraddiv] <#ctraddiv>`__). The precipitation flux is assumed to vary +:eq:`ctraddiv`. The precipitation flux is assumed to vary linearly across this region, as does the total flux, and so its contribution to :math:`I^{Tot}` cancels with :math:`I^{ppn}` in -(`[wthl_int] <#wthl_int>`__). Finally, for simplicity, the total flux is +:eq:`wthl_int`. Finally, for simplicity, the total flux is taken to be constant and equal to the inversion value, such that :math:`I^{Tot} = \Delta z_{rad} F^{Tot}|_{z_h}`. With these -approximations, (`[wthl_int] <#wthl_int>`__) becomes +approximations, :eq:`wthl_int` becomes .. math:: @@ -794,7 +781,7 @@ region, :math:`\overline{w'q_t'}` is also taken to be constant so that: - \Delta z_{rad} w_e \Delta q_t The integrated buoyancy flux is then found from -(`[eq:wb_cont] <#eq:wb_cont>`__) using the mixed layer cloud fraction +:eq:`eq:wb_cont` using the mixed layer cloud fraction and buoyancy coefficients evaluated at the grid-level above :math:`z_h -\Delta z_{rad}`. @@ -814,19 +801,18 @@ could be resolved. Following `Beare (2008)`_, a simple energetic argument gives a realistic prediction of the top of the inversion, :math:`z_{top}`, in LES from -.. math:: +.. math:: :label: dz_param 6.3 \, w_m^2 = \int_{z_{nb}}^{z_{top}} \, b \, dz - \label{dz_param} where :math:`z_{nb}` is the level of neutral buoyancy (found by linear interpolation between grid-levels), :math:`w_m` is the boundary layer velocity scale defined in section `5.1 <#sec:nlsurf>`__ and :math:`b` is the parcel buoyancy. Note that the constant in -(`[dz_param] <#dz_param>`__) is the same as in +:eq:`dz_param` is the same as in `Beare (2008)`_ because :math:`6.3 = 2.5 * 4^{2/3}` and :math:`w_m^3` differs by a factor of 4. The buoyancy integration in -(`[dz_param] <#dz_param>`__), that is itself dependent on +:eq:`dz_param`, that is itself dependent on :math:`z_{top}`, is performed working upwards from :math:`z_{\rm par}` assuming piece-wise linear variation of :math:`b` between grid-levels. Note that the standard definition of the boundary @@ -835,10 +821,9 @@ level of neutral buoyancy, so :math:`z_{\rm par}` :math:`=z_{\mbox{\tiny \rm NTPAR}+\frac{1}{2}}`. The inversion thickness is then defined as -.. math:: +.. math:: :label: dz_definition \Delta z_i = z_{top} -z_{\rm par} - \label{dz_definition} .. _`sec:lclmixing`: @@ -867,7 +852,7 @@ thermals will generate positive buoyancy fluxes (and are handled by the convection scheme) but it is assumed that there will also be cloud-free thermals within the grid box that may penetrate above the grid-box mean LCL (but are too dry to reach their own LCL). Thus their buoyancy flux -is given by (`[eq:wb_cont] <#eq:wb_cont>`__) with :math:`C_F=0`. +is given by :eq:`eq:wb_cont` with :math:`C_F=0`. Restricting the negative integral of this buoyancy flux then gives a new definition for :math:`z_{\rm h}`  that is then used in the calculation of the surface-driven K-profiles in section `5.1 <#sec:nlsurf>`__ — the @@ -893,14 +878,14 @@ The local scheme A first order ‘mixing length’ closure is used: -.. math:: +.. math:: :label: kmlocal - K_m = {\cal L}_m^2 \, (S+S_d) \, f_m(Ri) \label{kmlocal} + K_m = {\cal L}_m^2 \, (S+S_d) \, f_m(Ri) -.. math:: +.. math:: :label: khlocal K_h = {\cal L}_h \, {\cal L}_m \, - (S+S_d) \, f_h(Ri) \label{khlocal} + (S+S_d) \, f_h(Ri) where :math:`{\cal L}_m` and :math:`{\cal L}_h` are the neutral mixing @@ -941,10 +926,9 @@ The asymptotic mixing lengths are given by \lambda_m =\mbox{max}\left[\lambda_0,\, 0.15 z_{\rm loc}, 2 h_B \right] -.. math:: +.. math:: :label: asymp_ml \lambda_h =\mbox{max}\left[\lambda_0,\, 0.15 z_{\rm loc}\right] - \label{asymp_ml} where :math:`\lambda_0` is a minimum mixing length read in from the @@ -957,30 +941,28 @@ defined below), is given by where :math:`\sigma_h` is the standard deviation of the height of the subgrid orography and :math:`(z_{0m})_{\mbox{veg}}` is the vegetative part of the roughness length. The constants in -(`[asymp_ml] <#asymp_ml>`__) can be considered ‘tuned’ (see, in +:eq:`asymp_ml` can be considered ‘tuned’ (see, in particular, the operational modifications described in appendix `15 <#app:opmods>`__). The Richardson number, :math:`Ri`, that is used as a local measure of stability is given by -.. math:: +.. math:: :label: ridefn Ri = \frac{\Delta B / \Delta z}{(S+S_d)^2} - \label{ridefn} The measure of buoyancy used in :math:`Ri` is -.. math:: +.. math:: :label: Bdefn \Delta B = g\left( \overline{\beta_T} \Delta \theta_{\ell} + \overline{\beta_q} \Delta q_t \right) - \label{Bdefn} where :math:`\overline{\beta_T}` and :math:`\overline{\beta_q}` are the grid-box mean (i.e., cloud weighted) buoyancy coefficients, that can be defined in two different ways, see appendix `12 <#app:buoyp>`__ and -section `4.1 <#sec:fd_ri>`__. Note that (`[Bdefn] <#Bdefn>`__) reduces +section `4.1 <#sec:fd_ri>`__. Note that :eq:`Bdefn` reduces to a virtual temperature approximation of buoyancy in cloud-free air and that neutral buoyancy (in cloudy as well as cloud-free air) is implied by vertically uniform :math:`\theta_{\ell}` and :math:`q_t`. This is @@ -1026,7 +1008,7 @@ NLCL upwards) so that transports into and within the cumulus cloud layer can be performed solely by the mass-flux convection scheme. Depending on the switch local_fa, above NTLOC turbulently-mixed layers (where :math:`Ri`__) is set to the layer +:math:`z_{\rm loc}` in :eq:`asymp_ml` is set to the layer thickness. Outside of these turbulent layers the mixing lengths are set to :math:`\lambda_0`. @@ -1187,12 +1169,11 @@ variables (fractional area and water contents), to which the buoyancy coefficients are very sensitive, is required. The volume-weighted gradients of :math:`\theta_{\ell}` and :math:`q_t` are calculated as -.. math:: +.. math:: :label: gradient_interp (D\chi DZ)_k =\left( (z_{k}-z_{k-\frac{1}{2}}) \, \frac{\Delta_{k+1} \chi}{\Delta_{k+1} z} + (z_{k+\frac{1}{2}}-z_{k}) \, \frac{\Delta_{k} \chi}{\Delta_{k} z} \right) / \Delta_{k+\frac{1}{2}} z - \label{gradient_interp} as long as :math:`\chi_{k-1}` is defined on an atmospheric model level. To calculate :math:`DBDZ_1`, between the surface and the lowest @@ -1222,7 +1203,7 @@ cloud-top entrainment instability but this process is intended to be represented within the non-local scheme. Hence, the alternative method is to calculate the buoyancy gradient directly on :math:`\rho`-levels and then interpolate this vertically to give :math:`DBDZ_k`, using -(`[gradient_interp] <#gradient_interp>`__). This then requires a cloud +:eq:`gradient_interp`. This then requires a cloud fraction on :math:`\rho`-levels. The difficulty comes where there is a change in cloud fraction between levels. For this “edge” fraction, :math:`f_{edge}` (the fraction of the grid-box that is cloudy in one @@ -1239,11 +1220,11 @@ parameters on :math:`rho`-levels, e.g., :math:`\overline{\beta_T}_{k-1/2} = f_{tot} \tilde{\beta_T}_{k-1/2} + (1-f_{tot}){\beta_T}_{k-1/2})`, where the saturated and unsaturated buoyancy parameters are also intepolated to :math:`\rho`-levels using -(`[gradient_interp] <#gradient_interp>`__). +:eq:`gradient_interp`. Having calculated :math:`Ri` on :math:`\theta`-levels, :math:`{K_m}_{k}` and :math:`{K_h}_{k}` are calculated, still on :math:`\theta`-levels, as -in (`[kmlocal] <#kmlocal>`__) and (`[khlocal] <#khlocal>`__). Finally, +in :eq:`kmlocal` and :eq:`khlocal`. Finally, :math:`K_h` must be interpolated to :math:`\rho`-levels: .. math:: @@ -1263,7 +1244,7 @@ In addition to the above, the log profile correction applied to interpolation of :math:`K_h` to level :math:`k+\frac{1}{2}` in order that the correct cancellation with the finite difference scalar gradient in the flux calculation can occur. In the unstable stability functions -(`[unstable_stab] <#unstable_stab>`__), however, +:eq:`unstable_stab`, however, :math:`\tilde{{\cal L}}_h` must be calculated on :math:`\theta`-levels (i.e., the same as :math:`\tilde{{\cal L}}_m` and :math:`Ri`) in order to maintain the same stability @@ -1275,19 +1256,18 @@ Shear-driven mixing and interaction between the local and non-local schemes --------------------------------------------------------------------------- The general approach is to take :math:`K_{\chi}` in -(`[scal_closure] <#scal_closure>`__) and -(`[uv_closure] <#uv_closure>`__) as +:eq:`scal_closure` and +:eq:`uv_closure` as -.. math:: +.. math:: :label: klnl K_{\chi} = \mbox{max} \left[ (K_{\chi}^{\rm surf}+K_{\chi}^{\rm Sc}), K_{\chi}(Ri) \right] - \label{klnl} As noted in section `2 <#sec:closure>`__, this implies that mixing in stable boundary layers is determined exclusively by the local scheme, :math:`K_{\chi}(Ri)`. Continuing to calculate :math:`K_{\chi}(Ri)` in -unstable boundary layers and using (`[klnl] <#klnl>`__) is seen as the +unstable boundary layers and using :eq:`klnl` is seen as the simplest way of achieving a relatively smooth transition between stable and unstable boundary layers. @@ -1333,7 +1313,7 @@ the strong surface buoyancy generation of turbulence in these regimes, a calculation of :math:`Ri` is made that allows for the gradient adjustment by the non-local scheme, i.e., using :math:`\widetilde{\Delta_k \theta_{\ell}}` (see -(`[eq:wx_std] <#eq:wx_std>`__)). The height, :math:`z_{\rm loc}` , where +:eq:`eq:wx_std`). The height, :math:`z_{\rm loc}` , where :math:`Ri>Ri_{crit}=0.25` is found. It is then hypothesised that this level of turbulent instability (that incorporates the effects of shear) only needs extend some fractional distance into the cloud layer to @@ -1388,11 +1368,10 @@ scale :math:`u_*`, and positive surface buoyancy fluxes with velocity scale :math:`w_*`) in a layer with top at :math:`z=`\ :math:`z_{\rm h}` , base at :math:`z=0` we set -.. math:: +.. math:: :label: kmsurf K_m^{\rm surf}= k \ z_{\rm h}\ w_m \ \frac{z}{z_{\rm h}} \left( 1 - {\cal E}_m^{\rm surf} \frac{z}{z_{\rm h}} \right)^2 - \label{kmsurf} where :math:`w_m^3 = u_*^3 + w_s^3`, :math:`u_*` is the friction velocity (including the orographic roughness component) and :math:`w_s` @@ -1406,7 +1385,7 @@ factor :math:`{\cal E}_m^{\rm surf}` is chosen so that :math:`K_m|_{\mbox{\tiny \rm NTML}+\frac{1}{2}}` as :math:`z` tends to :math:`z_{\rm h}` , where :math:`K_m|_{\mbox{\tiny \rm NTML}+\frac{1}{2}}` is the entrainment -eddy-diffusivity (given by (`[khent] <#khent>`__), although, in order to +eddy-diffusivity (given by :eq:`khent`, although, in order to avoid altering the shape function too much, :math:`{\cal E}_m^{\rm surf}` is not allowed to fall below :math:`0.7`). A similar factor, :math:`{\cal E}_h^{\rm @@ -1418,14 +1397,13 @@ section `7 <#sec:entr>`__). The form of :math:`w_s` differs between the surface layer (:math:`z < 0.1`\ :math:`z_{\rm h}` ) and the rest of the mixed-layer: -.. math:: +.. math:: :label: ws_defn w_s^3 = \begin{cases} 2.5 \, \frac{z}{z_{\rm h}} w_*^3 & {\rm surface\ layer} \\ 0.25 \, w_*^3 & {\rm mixed\ layer} \\ \end{cases} - \label{ws_defn} and :math:`w_*^3=z_{\rm h}\overline{w'b}_S` using :math:`z_{\rm h}` from the current timestep (note that the use of :math:`w_*` here will be @@ -1440,17 +1418,16 @@ and to use a cubic sum of velocity scales within the mixed layer `Holtslag and Boville (1993)`_). The formula for :math:`K_h^{\rm surf}` is identical to -(`[kmsurf] <#kmsurf>`__) but with :math:`w_m` replaced by +:eq:`kmsurf` but with :math:`w_m` replaced by :math:`w_h=w_m/Pr`, where the turbulent Prandtl number is given by: -.. math:: +.. math:: :label: prandtl_nl Pr = 0.75 \frac{u_*^4 + (4/25)w_s^3 w_m}{u_*^4 + (8/25)w_s^3 w_m} - \label{prandtl_nl} Thus :math:`Pr` varies from 0.75 in neutral conditions to 0.375 in convective. The origin of the functional form of -(`[prandtl_nl] <#prandtl_nl>`__) is unknown. +:eq:`prandtl_nl` is unknown. .. _`sec:hbcomp`: @@ -1459,24 +1436,24 @@ Comparison with `Holtslag and Boville (1993)`_ The surface-driven :math:`K` profiles are the same as those in `Holtslag and Boville (1993)`_, HB93, -except for (`[ws_defn] <#ws_defn>`__) and -(`[prandtl_nl] <#prandtl_nl>`__) and the inclusion of the :math:`{\cal +except for :eq:`ws_defn` and +:eq:`prandtl_nl` and the inclusion of the :math:`{\cal E}_m^{\rm surf}` terms. For the latter, HB93 effectively set :math:`{\cal E}_m^{\rm surf} =1`. To generate entrainment, however, they simply use :math:`K_m^{\rm surf}|_{\mbox{\tiny \rm NTML}+\frac{1}{2}}`, as -evaluated from (`[kmsurf] <#kmsurf>`__) with a subgrid calculation of +evaluated from :eq:`kmsurf` with a subgrid calculation of :math:`z_{\rm h}` :math:`>z_{\mbox{\tiny \rm NTML}+\frac{1}{2}}`, rather than using a separate entrainment parametrization. -The difference in (`[ws_defn] <#ws_defn>`__) arises from the surface +The difference in :eq:`ws_defn` arises from the surface layer, where HB93 match :math:`w_m` to their surface exchange functions (i.e., :math:`w_m = u_* / \phi_m`) which results in proportionality constants of 6 and 0.6 for :math:`w_s` in the surface and mixed layers respectively. This matching is greatly simplified because their non-dimensional shear :math:`\phi_m = ( 1 + 15 k (z/z_i) w_*^3 / u_*^3 )^{-(1/3)}`. To match -:math:`w_m`, through (`[ws_defn] <#ws_defn>`__), to the UM function, +:math:`w_m`, through :eq:`ws_defn`, to the UM function, :math:`\phi_m = ( 1 + 16 k (z/z_i) w_*^3 / u_*^3 )^{-(1/4)}`, would require a complex function of :math:`u_*` and :math:`w_*` in place of the constant and so this is not @@ -1493,10 +1470,10 @@ matched to that used in the surface exchange functions (:math:`Pr_{\rm \right)^{-1/4} giving :math:`Pr_{\rm surf} = 1` in the neutral limit (compared to 0.75 -from (`[prandtl_nl] <#prandtl_nl>`__)). In the convective limit, +from :eq:`prandtl_nl`). In the convective limit, :math:`Pr_{\rm surf}|_{0.1\, z_{\rm h}} \rightarrow 0.9 (w_*/u_*)^{-3/4} = 0.9 \beta^{3/4} = 0.14` -(compared to 0.375 from (`[prandtl_nl] <#prandtl_nl>`__)). Thus, the +(compared to 0.375 from :eq:`prandtl_nl`). Thus, the Prandtl numbers do not match between the surface layer and interior formulations in the UM. @@ -1527,11 +1504,10 @@ For cloud-top-driven turbulence over a layer of depth :math:`z_{\rm ml}` (with top at :math:`z_{\rm h}` or :math:`z_{\rm h}^{\rm Sc}` and base at :math:`z_{\rm b}` , determined as in section `3.2 <#sec:decouple>`__), -.. math:: +.. math:: :label: kmtop K_m^{\rm Sc}= 0.63 \ k \ z_{\rm ml}\ V_{\rm Sc}\left( \frac{z'}{z_{\rm ml}} \right)^2 \left( 1 - {\cal E}_m^{\rm Sc} \frac{z'}{z_{\rm ml}} \right)^{0.8} - \label{kmtop} where :math:`V_{\rm Sc}^3= V_{\rm rad}^3+V_{\rm br}^3` (see appendix `11 <#app:vscales>`__) and :math:`z'` is height above @@ -1541,19 +1517,19 @@ against convective cloudy LES, as described in `Lock (1999)`_. The appropriate Prandtl number (and therefore :math:`K_m^{\rm Sc}`) is unknown, 0.75 being chosen simply as a number in the middle of the range usually quoted for -turbulent mixing in general. As with (`[kmsurf] <#kmsurf>`__), +turbulent mixing in general. As with :eq:`kmsurf`, :math:`z_{\rm h}`  (or :math:`z_{\rm h}^{\rm Sc}` ) are given by the subgrid diagnosis (see section `7.1.1 <#sec:sginv>`__) except for :math:`K_m^{\rm Sc}` in the 8A scheme which uses the height of the half-level below (:math:`z_{\mbox{\tiny \rm NTML}+\frac{1}{2}}` or :math:`z_{\mbox{\tiny \rm NTDSC}+\frac{1}{2}}`). Again following -(`[kmsurf] <#kmsurf>`__), the factors :math:`{\cal E}_m^{\rm Sc}` and -:math:`{\cal E}_h^{\rm Sc}` are included in (`[kmtop] <#kmtop>`__) so +:eq:`kmsurf`, the factors :math:`{\cal E}_m^{\rm Sc}` and +:math:`{\cal E}_h^{\rm Sc}` are included in :eq:`kmtop` so that :math:`K_m^{\rm Sc}` will tend to :math:`K_m|_{\mbox{\tiny \rm NTML}+\frac{1}{2}}` (and :math:`K_h^{\rm Sc}` to :math:`K_h|_{\mbox{\tiny \rm NTML}+\frac{1}{2}}`), given by -(`[khent] <#khent>`__), as :math:`z` tends to :math:`z_{\rm h}` (and +:eq:`khent`, as :math:`z` tends to :math:`z_{\rm h}` (and here no restriction is made on the magnitude of either :math:`{\cal E}_m^{\rm Sc}` or :math:`{\cal E}_h^{\rm Sc}`). @@ -1564,18 +1540,16 @@ Gradient adjustment Recall that for :math:`\theta_{\ell}` only we use -.. math:: +.. math:: :label: wthl \overline{w'\theta_{\ell}'}= - K_h \frac{\partial \theta_{\ell}}{\partial z} + K_h^{\rm surf}\gamma_{\theta_{\ell}} - \label{wthl} where -.. math:: +.. math:: :label: gradadj \gamma_{\theta_{\ell}} = \mbox{min}\left[ A_{ga} \frac{\sigma_{T1}}{z_{\rm h}}, G_{max} \right] - \label{gradadj} :math:`A_{ga}=3.26`, :math:`G_{max}=10^{-3}`\ Km\ :math:`^{-1}` and :math:`\sigma_{T1} = 1.93 \, @@ -1583,10 +1557,10 @@ where :math:`w_m` (given by :math:`w_m^3=u_*^3+0.25\,z_{\rm h}\overline{w'b}_S`) :math:`z_{\rm h}` is taken from the previous timestep. The form of -(`[gradadj] <#gradadj>`__) is similar to that used in HB93 and the +:eq:`gradadj` is similar to that used in HB93 and the magnitude of :math:`\gamma_{\theta_{\ell}}` is the same as in HB93 in the convective limit — the difference in :math:`A_{ga}` exactly allows -for the different constants in (`[ws_defn] <#ws_defn>`__). +for the different constants in :eq:`ws_defn`. Consistent with the mixed layer assumptions underlying the non-local scheme, the flux profile produced by the scheme is assumed to be @@ -1599,7 +1573,7 @@ positive in a cloud-free surface-heated boundary layer, for example), subject to an arbitrary upper limit included for numerical safety. Hence the term ‘gradient adjustment’ rather than non-local flux. When estimating the buoyancy flux, then (as in -(`[eq:wx_std] <#eq:wx_std>`__)), it is simplest to allow for the +:eq:`eq:wx_std`), it is simplest to allow for the non-local term by adjusting the :math:`\theta_{\ell}` gradient. The equivalent term for :math:`q_t` (i.e., :math:`\gamma_{q_t}`) is set @@ -1619,7 +1593,7 @@ Non-gradient stress parametrization There is an option that is operational in the UM to include an additional non-gradient (or non-local) stress parametrization, :math:`{\bf - \tau}^{nl}` in (`[uv_closure] <#uv_closure>`__), as proposed by + \tau}^{nl}` in :eq:`uv_closure`, as proposed by `Brown and Grant (1997)`_. They showed that with only a down-gradient stress parametrization, a one-dimensional model produced wind profiles in the convective boundary layer that were less well-mixed @@ -1631,13 +1605,12 @@ wind over land by day, especially in spring and summer. The non-gradient stress parametrization in the UM is very similar to that proposed by `Brown and Grant (1997)`_, written -.. math:: +.. math:: :label: tau_nl (\tau_x^{nl},\tau_y^{nl})= \left[ \frac{2.7w_*^3}{(u_*^3+0.6w_*^3)}\right] \left[ \left( \frac{z'}{z_{\rm h}'} \right) \left( 1- \frac{z'}{z_{\rm h}'} \right)^2 \right] (\tau_x^{s},\tau_y^{s}) - \label{tau_nl} Here :math:`w_*` is the convective velocity scale, :math:`u_*` is the friction velocity, and :math:`(\tau_x^{s},\tau_y^{s})` are the surface @@ -1695,17 +1668,16 @@ cloud-top region). So, the new formulation is written: -.. math:: +.. math:: :label: fg_new F_{\chi}^{Tot} = F_{\chi}^{NT}|_{z_{\rm b}} -\left(K_h^{\rm surf}+ K_h^{\rm Sc}\right)\frac{\partial\overline{\chi}}{\partial z} + \overline{w'\chi'}_{ng}^{\rm surf}+ \overline{w'\chi'}_{ng}^{\rm Sc} + f_2 \left(F_{\chi}|_{z_h} - F_{\chi}^{NT}|_{z_{\rm b}} \right) - \label{fg_new} where :math:`z_h` and :math:`z_{\rm b}` are the heights of the top and base of the mixed layer, respectively. It can be seen that -(`[fg_new] <#fg_new>`__) is composed of a local down-gradient component, +:eq:`fg_new` is composed of a local down-gradient component, two non-gradient flux terms (one generated by surface-driven turbulence and the other by cloud-top) and a non-local entrainment flux profile. The turbulent fluxes are then obtained by subtracting off the @@ -1713,7 +1685,7 @@ non-turbulent component: .. math:: \overline{w'\chi'}= F_{\chi}^{Tot} - F_{\chi}^{NT} -The components of (`[fg_new] <#fg_new>`__) are: +The components of :eq:`fg_new` are: - :math:`K_{h,m}^{\rm surf}= k z_h w_{h,m} \frac{z}{z_h}\left(1-\frac{z}{z_h}\right)^2` @@ -1740,7 +1712,7 @@ buoyancy-driven turbulence. Although the structure of the surface-driven non-gradient terms is the same as for the standard flux-gradient formulation, -(`[scal_closure] <#scal_closure>`__), note that they are now applied to +:eq:`scal_closure`, note that they are now applied to :math:`q_t` as well as :math:`\theta_{\ell}` and also the empirical coefficients in the velocity scales have been revised: @@ -1831,14 +1803,13 @@ Fig. `3 <#fig:stab_dep>`__. The parameter :math:`d` in Fig. `3 <#fig:stab_dep>`__ contains the stability dependence of the gradient adjustment parameter: -.. math:: +.. math:: :label: grad_adj \gamma_{\chi}= d \frac{\overline{w'\chi'}_S}{w_* z_h} \hspace{0.5cm} {\rm with} \hspace{0.5cm} d^{HB} = 7.2 w_*^2/w_m^2, \hspace{0.2cm} d^{std} = 6.3 w_*/w_m, \hspace{0.2cm} d^{rev} = 10 w_*/w_h - \label{grad_adj} The inclusion of an extra :math:`w_*/w_m` factor in :math:`\gamma_{\chi}` was a deliberate change by HB from the original @@ -1922,9 +1893,8 @@ the turbulence is (:math:`=1` if unresolved, :math:`=0` if well resolved), we can use this to blend between the 1D BL and 3D Smag schemes. Both schemes have a local Richardson number formulation: -.. math:: +.. math:: :label: eq-kri - \label{eq-kri} K_\chi(Ri) = l^2 S f_\chi(Ri), where :math:`K_\chi` is the eddy diffusivity, :math:`l` is the mixing @@ -1934,9 +1904,8 @@ variables, or momentum. Both schemes use the same stability function, and both schemes can use the full 3D shear for :math:`S`. Therefore the only difference is in the mixing length, which is calculated as -.. math:: +.. math:: :label: eq-lblend - \label{eq-lblend} l_{\rm blend} = W_{1D}l_{\rm bl}+(1-W_{1D})l_{\rm smag}, where :math:`l_{\rm bl}^{-1} = (\kappa z)^{-1} + \lambda_0^{-1}` and @@ -1960,8 +1929,8 @@ is given by .. math:: K_\chi = \max\left[W_{1D}K_\chi^{\rm NL}, K_\chi(Ri)\right], where :math:`K_\chi^{\rm NL}` is the non-local diffusivity and :math:`l` -in Eq. `[eq-kri] <#eq-kri>`__ is given by :math:`l_{\rm blend}` in -Eq. `[eq-lblend] <#eq-lblend>`__. The turbulent flux is then calculated +in Eq. :eq:`eq-kri` is given by :math:`l_{\rm blend}` in +Eq. :eq:`eq-lblend`. The turbulent flux is then calculated as .. math:: F_\chi=-K_\chi\frac{\partial \chi}{\partial z} + W_{1D}F_\chi^{\rm NL}, @@ -1978,9 +1947,8 @@ diffusivity we are trying to parametrize (for example a TKE based scheme would calculate the eddy diffusivity from the TKE), and simplify the function slightly, using -.. math:: +.. math:: :label: eq-tanh - \label{eq-tanh} W_{1D} = 1 - \tanh\left(\beta\frac{z_{\rm turb}}{\Delta x}\right)\max\left[0,\min\left[1,r_f\left(l_0-\frac{\Delta x}{z_{\rm turb}}\right)\right] \right], where :math:`z_{\rm turb}` is the appropriate lengthscale of the @@ -2008,7 +1976,7 @@ with `Malavelle et al. (2014)`_, and choose `Honnert et al. (2011)`_. These functions are shown in Figure `5 <#fig-blend>`__\ (a) and are only dissimilar for small :math:`\Delta -x`, where Eq. `[eq-tanh] <#eq-tanh>`__ tends to zero faster. This is by +x`, where Eq. :eq:`eq-tanh` tends to zero faster. This is by choice, to force the highest resolution simulations to use the 3D turbulence scheme. @@ -2043,9 +2011,8 @@ appropriate length scale is now the decoupled cloud mixed layer depth, released at cloud top would descend,][]{lock01}`. In this case, below the decoupled cloud top we set -.. math:: +.. math:: :label: zturb_dsc - \label{zturb_dsc} z_{\rm turb}=\min\left[\max\left(z,z_{\rm sml}\right),\max\left(z_{\rm sc},z_h-z\right)\right], where :math:`z_{\rm sml}` is the depth of the surface-based mixed layer @@ -2081,8 +2048,8 @@ Above the boundary layer top, `Boutle et al. (2014)`_ aimed for any free atmospheric mixing to be done by the 3D Smagorinsky scheme. Therefore, above the boundary layer top they use :math:`z` as the appropriate length scale, and in general take :math:`z_{\rm turb}` in -Eq. `[eq-tanh] <#eq-tanh>`__ as the greater of that defined by -(`[zturb_dsc] <#zturb_dsc>`__) and :math:`z`. However, this did not give +Eq. :eq:`eq-tanh` as the greater of that defined by +:eq:`zturb_dsc` and :math:`z`. However, this did not give a particularly fast transition using the value of :math:`\beta_{\rm bl}`, therefore they used :math:`\beta_{\rm fa}=1` at a height well above the boundary layer (:math:`z_{\rm fa}=z_h+1` km), @@ -2093,7 +2060,7 @@ and transitioned between these regimes linearly using \beta = \beta_{\rm bl}\frac{z_{\rm fa}-z}{z_{\rm fa}-z_h} + \beta_{\rm fa}\frac{z-z_h}{z_{\rm fa}-z_h} -However, because the above method still uses (`[eq-tanh] <#eq-tanh>`__), +However, because the above method still uses :eq:`eq-tanh`, which depends on :math:`z_{\rm turb}/\Delta x`, the rate of transition to 3D Smagorinsky with height above the boundary layer varies in an undesirable way with grid size. It might be considered more logical to @@ -2120,7 +2087,7 @@ boundary layers while the latter that it does not drift far into the free atmosphere. In addition, within any layers identified as turbulent, through having subcritical :math:`Ri`, :math:`z_{\rm turb}` is set to the layer depth, in the same way as is done for decoupled stratocumulus -in (`[zturb_dsc] <#zturb_dsc>`__). +in :eq:`zturb_dsc`. For current operational convection-permitting model grid sizes (1.5 km in the UKV), the representation of cumulus convection remains a @@ -2163,12 +2130,11 @@ The parametrization of the entrainment rate, :math:`w_e` (given, in the absence of subsidence, by the rate of rise of the inversion), can be written (using the notation given in appendix `11 <#app:vscales>`__) -.. math:: +.. math:: :label: we_parm w_e = \frac{ A_1 \, V_{\rm sum}^3/ z_{\rm ml}+ g \tilde{\beta_T} \tilde{\alpha_t} \Delta_F} {\Delta b + c_T V_{\rm sum}^2/z_{\rm ml}} - \label{we_parm} where :math:`V_{\rm sum}^3= V_{\rm heat}^3+ V_{\rm rad}^3+ V_{\rm br}^3+ A_2 u_*^3`. @@ -2176,7 +2142,7 @@ The constant :math:`A_1` is given a value 0.23, as in `Lock (1998)`_, and :math:`A_1*A_2=5`, as in `Driedonks (1982)`_. To allow for weak inversions, the `Zilitinkevich (1975)`_ correction is included in -(`[we_parm] <#we_parm>`__) with the constant, :math:`c_T=1`. A further +:eq:`we_parm` with the constant, :math:`c_T=1`. A further parametrization for :math:`\alpha_t`, which is the fraction of the cloud-top radiative divergence (:math:`\Delta_F`, in Kms\ :math:`^{-1}`) that occurs across the horizontally-averaged inversion in the LES, can @@ -2195,10 +2161,10 @@ allow for a feedback with forcing of entrainment by buoyancy reversal :math:`\tilde{\alpha_t} = \alpha_t+ Br (1-\alpha_t)`. following `Lock (1998)`_ and `Lock (2009)`_. The calculation of the other -quantities required for (`[we_parm] <#we_parm>`__) is described in +quantities required for :eq:`we_parm` is described in appendix `11 <#app:vscales>`__. At some point during the transition to a decoupled boundary layer the surface-driven entrainment terms (the terms -in (`[we_parm] <#we_parm>`__) proportional to :math:`V_{\rm heat}^3` and +in :eq:`we_parm` proportional to :math:`V_{\rm heat}^3` and :math:`u_*`) will no longer contribute to entrainment at cloud top, because the two layers will have become entirely decoupled. If the ``entr_smooth_dec`` switch is on then the surface contribution to the @@ -2206,11 +2172,11 @@ parametrized entrainment at :math:`z_{\rm h}^{\rm Sc}` is decreased linearly as the :math:`\theta_{v\ell}` difference between NTDSC and NTML increases from 0.5 to 1K. The flag, COUPLED, is set to true and :math:`z_{\rm h}^{\rm Sc}` is used as the mixed-layer depth in -(`[we_parm] <#we_parm>`__) as long as any surface-driven entrainment +:eq:`we_parm` as long as any surface-driven entrainment remains. If the ``entr_smooth_dec`` switch is off then this transition is discontinuous at a :math:`\theta_{v\ell}` difference of 0.5K. -It should be noted that (`[we_parm] <#we_parm>`__) takes no account of +It should be noted that :eq:`we_parm` takes no account of wind shear anywhere other than at the surface. How to quantify the shear generation of turbulence in DSC layers is not known. The direct impact of shear across the inversion is thought to be simply to diffuse the @@ -2221,12 +2187,12 @@ entrainment. However, important interactions between wind shear across inversions and cloud-top radiative cooling have been observed that are not yet accounted for in the UM. -The least well-determined part of (`[we_parm] <#we_parm>`__) is the +The least well-determined part of :eq:`we_parm` is the constant :math:`A_2` — the constant in the Zilitinkevich correction, :math:`c_T`, is also approximate but is included to limit the growth of layers capped by weak inversions and for numerical safety. A further limit is applied to the value of :math:`w_e` determined by -(`[we_parm] <#we_parm>`__) such that the inversion cannot rise by more +:eq:`we_parm` such that the inversion cannot rise by more than one grid-level in a timestep. With current vertical resolutions and timesteps this is not a serious restriction. The constants :math:`A_1` and :math:`A_{\rm br}` appeared to be determined within 10-20 % in @@ -2321,7 +2287,7 @@ illustrated for a SML in Fig. `6 <#fig:fluxinterp>`__. * - .. image:: subsent_fig7.svg -Note that, because (`[fluxinterp] <#fluxinterp>`__) includes an explicit +Note that, because :eq:`fluxinterp` includes an explicit balance between the turbulent and radiative fluxes for :math:`\overline{w'\theta_{\ell}'}`, it is not possible to parametrize the entrainment fluxes through a single :math:`K_h` for both @@ -2329,7 +2295,7 @@ the entrainment fluxes through a single :math:`K_h` for both Furthermore, the radiative forcing of turbulence in the mixed layer is fixed through the timestep and so it is consistent to assume the entrainment fluxes (at :math:`z_i`) are also fixed. Hence -(`[fluxinterp] <#fluxinterp>`__) are implemented explicitly, rather than +:eq:`fluxinterp` are implemented explicitly, rather than via an eddy-diffusivity. This is discussed further, with reference to tracer fluxes, in section `7.4.3 <#sec:ent_K_flux>`__. @@ -2340,12 +2306,12 @@ the parametrization of :math:`w_e` and the model’s subsidence velocity, time-level (:math:`z_i^{n+1}`). Currently, the latter is found by linear interpolation to :math:`z_i` and both are assumed constant in time. If :math:`z_i^{n+1} < z_{\mbox{\tiny \rm NTDSC}+\frac{1}{2}}`, then the -entrainment fluxes there (given by (`[fluxinterp] <#fluxinterp>`__)) are +entrainment fluxes there (given by :eq:`fluxinterp`) are multiplied by the fraction of the timestep that :math:`z_i` was above this grid-level, namely :math:`(z_i-z_{\mbox{\tiny \rm NTDSC}+\frac{1}{2}})/(z_i - z_i^{n+1})`. The full entrainment flux at grid-level NTDSC\ :math:`-\frac{1}{2}` must -then also be specified, given by (`[fluxinterp] <#fluxinterp>`__) with +then also be specified, given by :eq:`fluxinterp` with :math:`z_{\mbox{\tiny \rm NTDSC}+ \frac{1}{2}}` replaced by :math:`z_{\mbox{\tiny \rm NTDSC}- \frac{1}{2}}`. If :math:`z_i` rises above :math:`z_{\mbox{\tiny \rm NTDSC}+\frac{3}{2}}`, the entrainment @@ -2357,7 +2323,7 @@ NTDSC\ :math:`+\frac{1}{2}` (these will be non-zero because entrainment fluxes are specified explicitly, the eddy-diffusivities (both non-local and local) are set to zero. Also, the mean value of :math:`z_i` during the timestep is used in -(`[fluxinterp] <#fluxinterp>`__) in order best to approximate the mean +:eq:`fluxinterp` in order best to approximate the mean flux gradient across the mixed layer. Finally, the entrainment flux is adjusted to allow for numerical @@ -2374,7 +2340,7 @@ velocity field using first order upwind advection (it would clearly be preferable to use the model’s actual vertical advection algorithm in the GCM although the errors incurred in this diagnostic calculation should not be very significant). The interpolated entrainment fluxes given by -(`[fluxinterp] <#fluxinterp>`__) are therefore calculated not using +:eq:`fluxinterp` are therefore calculated not using :math:`w_e` but using an entrainment velocity, :math:`\tilde{w_e}`, that is reduced to allow for any subsidence increments applied to the grid-level below the entrainment flux. To take the case of @@ -2387,12 +2353,11 @@ reduced entrainment velocity is given by with :math:`\tilde{w_e}` constrained to lie between 0 and :math:`w_e` and -.. math:: +.. math:: :label: we_num \tilde{w_S} = - \, \frac{ \Theta^{\rm S}_{\mbox{\tiny \rm NTML}} ( \Delta_{\mbox{\tiny \rm NTML}+\frac{1}{2}} z ) } { \Delta \theta_{\ell}} - \label{we_num} .. _`sec:sginv`: @@ -2407,7 +2372,7 @@ ensure it is monotonically increasing with height in the statically stable free-troposphere of a GCM. If :math:`\theta_{v\ell}` does not increase monotonically between grid-levels NTML and NTML+2 (or NTDSC and NTDSC+2), then entrainment fluxes are simply specified via -(`[khent] <#khent>`__) and none of the coupling with subsidence or +:eq:`khent` and none of the coupling with subsidence or radiation described above is attempted (the local scheme is also currently not set to zero above NTML or NTDSC when this occurs to allow it to diffuse out this static instability). @@ -2444,10 +2409,9 @@ atmosphere, between grid-levels NTML\ :math:`+2` and NTML\ :math:`+3`, these areas gives a quadratic equation in :math:`\Delta z_{disc} = z_{\mbox{\tiny \rm NTML}+\frac{3}{2}} - z_i` which can be written -.. math:: +.. math:: :label: zi_interp a (\Delta z_{disc})^2 + b \ \Delta z_{disc} +c =0 - \label{zi_interp} The coefficients are given by @@ -2477,7 +2441,7 @@ Clearly, care must be taken to ensure that :math:`z_i` is not only well-defined but also sensible (for example, as a rising inversion encounters more or less stable regions above). If :math:`b>0` this suggests the estimated lapse rates are inappropriate and these are -therefore set to zero and (`[zi_interp] <#zi_interp>`__) is +therefore set to zero and :eq:`zi_interp` is recalculated. The case :math:`c<0` suggests the grid-level designated as the inversion level should have been considered as part of the mixed layer and so :math:`z_i` is set to be fractionally below @@ -2513,26 +2477,25 @@ Having calculated :math:`z_i`, the discontinuous jumps in :math:`\theta_{\ell}` and :math:`q_t` that are used in the entrainment calculation are calculated from similar integral assumptions: -.. math:: +.. math:: :label: dqt_disc \Delta \chi = \left( {\chi}_{\mbox{\tiny \rm NTML}+1} - {\chi}_{\mbox{\tiny \rm NTML}} \right) \, \frac{ z_{\mbox{\tiny \rm NTML}+\frac{3}{2}} - z_{\mbox{\tiny \rm NTML}+\frac{1}{2}} } { z_{\mbox{\tiny \rm NTML}+\frac{3}{2}} - z_i } - \label{dqt_disc} with :math:`\chi = \theta_{\ell}` and :math:`q_t`. Note that the lapse rate above the inversion has been ignored as there is no guarantee of monotonicity in :math:`q_t` in the atmosphere above the inversion. In -addition, (`[dqt_disc] <#dqt_disc>`__) will become increasingly +addition, :eq:`dqt_disc` will become increasingly inaccurate as :math:`z_i` tends to :math:`z_{\mbox{\tiny \rm NTML}+\frac{3}{2}}` (and so :math:`{\chi}_{\mbox{\tiny \rm NTML}+1}` approaches :math:`{\chi}_{\mbox{\tiny \rm NTML}}`). Consequently, if the fraction -on the right hand side of (`[dqt_disc] <#dqt_disc>`__) is greater than +on the right hand side of :eq:`dqt_disc` is greater than 10, double grid-level jumps are used (i.e., :math:`\Delta \chi = {\chi}_{\mbox{\tiny \rm NTML}+2} - {\chi}_{\mbox{\tiny \rm NTML}}`). Finally, note that -(`[dqt_disc] <#dqt_disc>`__) implicitly assumes the structure of the +:eq:`dqt_disc` implicitly assumes the structure of the :math:`\theta_{\ell}` and :math:`q_t` profiles across the inversion grid-level are consistent with the diagnosed :math:`z_i`. This is very unlikely to be the case, for example, when running from an analysis so @@ -2617,12 +2580,11 @@ the full model) and :math:`F_{\chi}^{subs}` as the flux from resolved scale subsidence, the total flux at the subgrid inversion height is given by: -.. math:: +.. math:: :label: fxtot_zi F_{\chi}^{Tot}|_{z_h} = - w_e \Delta \chi + F_{\chi}^{NTP}|_{z_t} + F_{\chi}^{subs}|_{z_h} - \label{fxtot_zi} -As in section `7.1 <#sec:ent_flux>`__, (`[fxtot_zi] <#fxtot_zi>`__) is +As in section `7.1 <#sec:ent_flux>`__, :eq:`fxtot_zi` is derived by integrating the conservation equation for :math:`\chi` over an inversion in which jumps occur over a thin layer with base at a height :math:`z_h` and top at :math:`z_t` (in the UM, the inversion is @@ -2633,7 +2595,7 @@ w_e \Delta \chi = \overline{w'\chi'}|_{z_h} -(F_{\chi}^{NTP}|_{z_t}-F_{\chi}^{NT :math:`F_{\chi}^{NTP}|_{z_t}-F_{\chi}^{NTP}|_{z_h}`, to radiative cooling occurring within undulations of the cloudy boundary layer top. Similar considerations need to be borne in mind when calculating all the -non-turbulent fluxes in (`[fxtot_zi] <#fxtot_zi>`__). First, the +non-turbulent fluxes in :eq:`fxtot_zi`. First, the radiative flux is extrapolated down from :math:`\mbox{\tiny \rm NTML}+\frac{3}{2}` to :math:`z=z_t` using the divergence in the grid-level above the inversion as representative of @@ -2644,46 +2606,43 @@ subsidence flux-divergence across level :math:`\mbox{\tiny \rm NTML}` and :math:`\mbox{\tiny \rm NTML}+1` is assumed to be associated with the inversion so :math:`{F_{\chi}}^{Subs}|_{z_h} = {F_{\chi}}^{Subs}_{\mbox{\tiny \rm NTML}-\frac{1}{2}}`. Thus, the -finite-difference form of (`[fxtot_zi] <#fxtot_zi>`__) becomes +finite-difference form of :eq:`fxtot_zi` becomes -.. math:: +.. math:: :label: fxtot_zi_fd F_{\chi}^{Tot}|_{z_h} = - w_e \Delta \chi + F_{\chi}^{rad}|_{z_t} + {F_{\chi}}^{ppn}_{\mbox{\tiny \rm NTML}+\frac{3}{2}} + {F_{\chi}}^{subs}_{\mbox{\tiny \rm NTML}-\frac{1}{2}} - \label{fxtot_zi_fd} Then, assuming a linear profile of :math:`F_{\chi}^{Tot}` in the mixed layer, interpolating the total flux to the inversion flux grid-level gives -.. math:: +.. math:: :label: fxtot_interp F_{\chi}^{Tot}|_{ \mbox{\tiny \rm NTML}+\frac{1}{2} } = F_{\chi}^{Tot}|_{z_{\rm b}} + \frac{ z'_{\mbox{\tiny \rm NTML}+\frac{1}{2}} }{z_{\rm ml}} \left( F_{\chi}^{Tot}|_{z_h} - F_{\chi}^{Tot}|_{z_{\rm b}} \right) - \label{fxtot_interp} where :math:`z'` (:math:`=z-z_{\rm b}`) is height above the base of the mixed layer at :math:`z=z_{\rm b}`. Finally, the grid-level turbulent entrainment flux is given by: -.. math:: +.. math:: :label: rev_entflux \overline{w'\chi'}|_{ \mbox{\tiny \rm NTML}+\frac{1}{2} } = F_{\chi}^{Tot}|_{ \mbox{\tiny \rm NTML}+\frac{1}{2} } - F_{\chi}^{NT}|_{ \mbox{\tiny \rm NTML}+\frac{1}{2} } - \label{rev_entflux} This revised algorithm has several advantages over the previous. Firstly, the fluxes for :math:`q_t` and :math:`\theta_{\ell}` are coupled independently, whereas in the 9B version the coupling with subsidence was estimated using only the :math:`\theta_{\ell}` increments -in order to calculate :math:`\tilde{w_e}` in `[we_num] <#we_num>`__. +in order to calculate :math:`\tilde{w_e}` in :eq:`we_num`. Secondly, this method makes it much simpler to include all processes, and precipitation in particular, in a consistent manner. Thirdly, since the total grid-level flux, :math:`F_{\chi}^{Tot}|_{\mbox{\tiny \rm NTML}+\frac{1}{2}}` in -(`[fxtot_interp] <#fxtot_interp>`__), is used to calculate the +:eq:`fxtot_interp`, is used to calculate the entrainment fluxes, it is straightforward to ensure that the net budget of the inversion grid-level, namely :math:`- ( F_{\chi}^{Tot}|_{\mbox{\tiny \rm NTML}+\frac{3}{2}} - F_{\chi}^{Tot}|_{\mbox{\tiny \rm NTML}+\frac{1}{2}})/\Delta z`, @@ -2740,13 +2699,13 @@ In the 9B scheme, the discontinuous jumps in :math:`\theta_{\ell}` and :math:`q_t` that are used in the entrainment calculation were calculated from integral assumptions similar to those used to diagnose the subgrid inversion height, :math:`z_h`, and were given by -(`[dqt_disc] <#dqt_disc>`__). Note that +:eq:`dqt_disc`. Note that :math:`{\chi}_{\mbox{\tiny \rm NTML}+2}` does not appear in -(`[dqt_disc] <#dqt_disc>`__) and so no direct information from the free +:eq:`dqt_disc` and so no direct information from the free atmosphere is used. Only if the budgets of :math:`\theta_{\ell}` and :math:`q_t` in level :math:`\mbox{\tiny \rm NTML}+1` are entirely consistent with the rise and fall of the subgrid inversion will -(`[dqt_disc] <#dqt_disc>`__) give accurate results. This will not be the +:eq:`dqt_disc` give accurate results. This will not be the case during an assimilation cycle, for example, neither is it likely to be the case if the convection scheme is detraining into level :math:`\mbox{\tiny \rm NTML}+1`. @@ -2757,11 +2716,10 @@ subgrid inversion calculation is only attempted where both increasing and :math:`q_t` is simply monotonic across the inversion. The formula used is: -.. math:: +.. math:: :label: dqt_disc_9c \Delta \chi = {\chi}_{\mbox{\tiny \rm NTML}+2} - {\chi}_{\mbox{\tiny \rm NTML}} - \gamma_{\chi} \left( z_{\mbox{\tiny \rm NTML}+2} - z_h \right) - \label{dqt_disc_9c} subject to the constraint that the lapse rate adjustment should not reduce the two grid-length difference by more than half. The @@ -2843,14 +2801,14 @@ noting the Charney-Philips grid implying stresses are staggered from scalar fluxes. The Prandtl number, :math:`Pr`, takes the same form as for the non-local :math:`K` profiles, see section `5 <#sec:nonlocal>`__. -Substituting (`[khent] <#khent>`__) in -(`[scal_closure] <#scal_closure>`__) gives, for example, +Substituting :eq:`khent` in +:eq:`scal_closure` gives, for example, :math:`\overline{w'\theta_{\ell}'}|_{\mbox{\tiny \rm NTML}+\frac{1}{2}} = - w_e \Delta_{\mbox{\tiny \rm NTML}+1} \theta_{\ell}`. Note that this gives entrainment buoyancy fluxes identical to -(`[discinv] <#discinv>`__) as long as there is no buoyancy reversal +:eq:`discinv` as long as there is no buoyancy reversal generation of turbulence (i.e.,\ :math:`V_{\rm br}=0`) and if variations in the grid-level jumps across the timestep are ignored. The former is -because the other terms in (`[we_parm] <#we_parm>`__) are inversely +because the other terms in :eq:`we_parm` are inversely proportional to :math:`\Delta b`. The latter will never actually be true and can give rise to large errors if the inversion is rising quickly. Therefore, the thermodynamic @@ -2863,13 +2821,13 @@ mixed layer depth calculation and it allows a more accurate calculation of :math:`V_{\rm br}` and :math:`\alpha_t`. For momentum, because the jumps across inversions are typically small and variable, it seems unwise numerically to attempt to specify the inversion stresses -explicitly and so (`[khent] <#khent>`__) is always used. For the 9C -version, the entrainment :math:`K_m` given by (`[khent] <#khent>`__) is +explicitly and so :eq:`khent` is always used. For the 9C +version, the entrainment :math:`K_m` given by :eq:`khent` is imposed at the height of the temperature inversion :math:`z_{\rm h}` (either subgrid or at :math:`z_{\mbox{\tiny \rm NTML}+\frac{1}{2}}`) and :math:`K_m|_{\mbox{\tiny \rm NTML}+\frac{1}{2}}` is calculated from -(`[kmsurf] <#kmsurf>`__) and (`[kmtop] <#kmtop>`__), noting the use of +:eq:`kmsurf` and :eq:`kmtop`, noting the use of the :math:`{\cal E}` factors. @@ -2893,11 +2851,10 @@ coefficient profile within the inversion is then calculated assuming the following a cosine shape from the standard parametrized entrainment flux at the inversion base to zero at the inversion top, i.e.: -.. math:: +.. math:: :label: ent_svl \overline{w'\theta_{v\ell}'} = \overline{w'\theta_{v\ell}'}|_{\mbox{\tiny \rm NTML}+\frac{1}{2}} cos\left(\pi \frac{z'}{2} \right) - \label{ent_svl} where :math:`z'=(z-z_{\rm h})/\Delta z_i` is scaled height within the inversion. This flux profile is then converted into a diffusion @@ -2910,7 +2867,7 @@ coefficient profile by inverting the standard flux parametrization: The diffusion coefficient for momentum entrainment is calculated in the same way, allowing for the staggered grid, with the same :math:`Pr` as -in (`[khent] <#khent>`__). +in :eq:`khent`. .. _`sec:ent_K_flux`: @@ -2933,26 +2890,25 @@ timestep. Whilst this is true for atmospheric :math:`\theta_{\ell}` and layer concentration. Consequently, for a tracer field :math:`\chi`, the parametrized entrainment fluxes :math:`\overline{w'\chi'}_{ z_{\mbox{\tiny \rm NTML}+\frac{1}{2}} }` are calculated from -(`[fluxinterp] <#fluxinterp>`__) but are implemented through an +:eq:`fluxinterp` but are implemented through an equivalent entrainment eddy-diffusivity given by: -.. math:: +.. math:: :label: K_ent_tracer K_{\chi}|_{\mbox{\tiny \rm NTML}+\frac{1}{2}} = - \overline{w'\chi'}_{ z_{\mbox{\tiny \rm NTML}+\frac{1}{2}} } \frac{\Delta_{\mbox{\tiny \rm NTML}+1} z}{\Delta_{\mbox{\tiny \rm NTML}+1} \chi} - \label{K_ent_tracer} -Note from (`[scal_closure] <#scal_closure>`__) that -(`[K_ent_tracer] <#K_ent_tracer>`__) gives the parametrized flux if +Note from :eq:`scal_closure` that +:eq:`K_ent_tracer` gives the parametrized flux if :math:`\Delta_{\mbox{\tiny \rm NTML}+1} \chi` does not change across the timestep (see section `9 <#sec:implicit>`__ for a description of the -implicit numerical solution of (`[cons_eqn_scal] <#cons_eqn_scal>`__)). -As (`[K_ent_tracer] <#K_ent_tracer>`__) involves the potentially +implicit numerical solution of :eq:`cons_eqn_scal`). +As :eq:`K_ent_tracer` involves the potentially numerically dangerous calculation of :math:`\Delta \chi/\Delta_{\mbox{\tiny \rm NTML}+1} \chi` (where :math:`\Delta \chi` is the subgrid inversion jump, given by -(`[dqt_disc] <#dqt_disc>`__)), the following constraints are also +:eq:`dqt_disc`), the following constraints are also ensured: .. math:: @@ -2975,17 +2931,17 @@ Making the assumption that **Monin-Obukhov similarity theory** for the surface layer is valid the gradients of model variables in the surface layer are related to the surface fluxes by: -.. math:: +.. math:: :label: 1.1.1 - \frac{\partial T}{\partial z} + \frac{g}{ c_P }=-\frac{ H_0 }{ c_P \rho _0 v_\ast } \frac{ \phi _h (z/L)}{kz}\label{1.1.1} + \frac{\partial T}{\partial z} + \frac{g}{ c_P }=-\frac{ H_0 }{ c_P \rho _0 v_\ast } \frac{ \phi _h (z/L)}{kz} -.. math:: +.. math:: :label: 1.1.2 - \frac{\partial q}{\partial z}=-\frac{ E_0 }{ \rho _0 v_\ast } \frac{ \phi _h (z/L)}{kz}\label{1.1.2} + \frac{\partial q}{\partial z}=-\frac{ E_0 }{ \rho _0 v_\ast } \frac{ \phi _h (z/L)}{kz} -.. math:: +.. math:: :label: 1.1.3 - \frac{\partial {\rm {\bf v}}}{\partial z}=\frac{ {\rm {\bf \tau }}_{0} }{ \rho _0 v_\ast } \frac{ \phi _m (z/L)}{kz},\label{1.1.3} + \frac{\partial {\rm {\bf v}}}{\partial z}=\frac{ {\rm {\bf \tau }}_{0} }{ \rho _0 v_\ast } \frac{ \phi _m (z/L)}{kz}, where subscript 0 represents a surface value and subscript \* represents @@ -2994,142 +2950,139 @@ _{h}` are the Monin-Obukhov stability functions (for the form of these see section `8.3 <#section_1.3>`__ below). :math:`L` is the Monin-Obukhov length scale defined by -.. math:: +.. math:: :label: 1.1.4 L = \frac{- { v_\ast }^3 }{k F_{B0} / \rho _0 }, - \label{1.1.4} where F\ :math:`_{B0}` is the surface buoyancy flux defined by -.. math:: +.. math:: :label: 1.1.5 F_{B0} = \frac{ g }{ c_P } \beta _{T1} H_0 + g \beta _{q1} E_0. - \label{1.1.5} -The buoyancy coefficients in equation (`[1.1.5] <#1.1.5>`__) are given +The buoyancy coefficients in equation :eq:`1.1.5` are given in appendix `12 <#app:buoyp>`__ with the subscript 1 denoting a value at the lowest level in the atmosphere model. -Equations (`[1.1.1] <#1.1.1>`__)–(`[1.1.3] <#1.1.3>`__) can be +Equations :eq:`1.1.1`–:eq:`1.1.3` can be integrated from the “surface”, i.e. the roughness height where the surface variables are defined, to a reference height in the surface layer, for modelling applications, the height, z\ :math:`_{1}`, of the bottom model layer above the surface. The resulting expressions for the surface turbulent fluxes are: -.. math:: +.. math:: :label: 1.1.7 - \frac{ H_0 }{ c_P \rho _0 }=-\frac{ c_H }{ c_D^{1/2} } v_\ast \left( {\Delta T + \frac{g}{c_p }( z_1 + z_{0m} - z_{0h} )} \right)\label{1.1.7} + \frac{ H_0 }{ c_P \rho _0 }=-\frac{ c_H }{ c_D^{1/2} } v_\ast \left( {\Delta T + \frac{g}{c_p }( z_1 + z_{0m} - z_{0h} )} \right) -.. math:: +.. math:: :label: 1.1.8 - \frac{ E_0 }{ \rho _0 }=-\frac{ c_H }{ c_D^{1/2} } v_\ast \Delta q\label{1.1.8} + \frac{ E_0 }{ \rho _0 }=-\frac{ c_H }{ c_D^{1/2} } v_\ast \Delta q -.. math:: +.. math:: :label: 1.1.9 - \frac{ {\bf \tau }_{0} }{ \rho _{0} }= c_D^{1/2} v_\ast \Delta {\rm {\bf v}},\label{1.1.9} + \frac{ {\bf \tau }_{0} }{ \rho _{0} }= c_D^{1/2} v_\ast \Delta {\rm {\bf v}}, where :math:`\Delta`\ X=X\ :math:`_{1}`-X\ :math:`_{0}`. -From (`[1.1.7] <#1.1.7>`__) and (`[1.1.8] <#1.1.8>`__) the surface -buoyancy flux in definition (`[1.1.4] <#1.1.4>`__) is +From :eq:`1.1.7` and :eq:`1.1.8` the surface +buoyancy flux in definition :eq:`1.1.4` is -.. math:: +.. math:: :label: 1.1.10 \frac{ F_{B0} }{ \rho _0 } = -\frac{ c_H }{ c_D^{1/2} } v_\ast \Delta B, - \label{1.1.10} -.. math:: +.. math:: :label: 1.1.11 \Delta B = g \beta _{T1} \left( {\Delta T + \frac{g}{ c_P }( z_1 + z_{0m} - z_{0h} )} \right) + g \beta _{q1} \Delta q - \label{1.1.11} The **surface exchange coefficients** in -equations (`[1.1.7] <#1.1.7>`__)–(`[1.1.9] <#1.1.9>`__), c\ :math:`_{D}` +equations :eq:`1.1.7`–:eq:`1.1.9`, c\ :math:`_{D}` and c\ :math:`_{H}`, are given by -.. math:: +.. math:: :label: 1.1.12 c_D^{1/2}=\frac{k}{ \Phi _m (L , z_1 + z_{0m} , z_{0m} )} - \label{1.1.12} -.. math:: +.. math:: :label: 1.1.13 \frac{ c_H }{ c_D^{1/2} }=\frac{k}{ \Phi _h (L , z_1 + z_{0m} , z_{0h} )}, - \label{1.1.13} where -.. math:: +.. math:: :label: 1.1.14 - \Phi _m (L , z_1 + z_{0m} , z_{0m} )= \int \limits_{ z_{0m} /L}^{( z_1 + z_{0m} )/L} \frac{ \phi _m (\zeta )}{\zeta } d\zeta\label{1.1.14} + \Phi _m (L , z_1 + z_{0m} , z_{0m} )= \int \limits_{ z_{0m} /L}^{( z_1 + z_{0m} )/L} \frac{ \phi _m (\zeta )}{\zeta } d\zeta -.. math:: +.. math:: :label: 1.1.15 - \Phi _h (L , z_1 + z_{0m} , z_{0h} )= \int \limits_{ z_{0h} /L}^{( z_1 + z_{0m} )/L} \frac{ \phi _h (\zeta )}{\zeta } d\zeta\label{1.1.15}, + \Phi _h (L , z_1 + z_{0m} , z_{0h} )= \int \limits_{ z_{0h} /L}^{( z_1 + z_{0m} )/L} \frac{ \phi _h (\zeta )}{\zeta } d\zeta, z\ :math:`_{0m}` and z\ :math:`_{0h}` are the **surface roughness lengths** for momentum and scalars respectively. The equations for the **surface turbulent fluxes**, -(`[1.1.7] <#1.1.7>`__)–(`[1.1.9] <#1.1.9>`__), can be written in the +:eq:`1.1.7`–:eq:`1.1.9`, can be written in the forms .. math:: \frac{ H_0 }{ c_P \rho _0 }={-c}_H V \left( {\Delta T + \frac{g}{c_p }( z_1 + z_{0m} - z_{0h} )} \right) -.. math:: +.. math:: :label: 1.1.16 - = - C_H \left( {\Delta T + \frac{g}{c_p } ( z_1 + z_{0m} - z_{0h} )} \right)\label{1.1.16} + = - C_H \left( {\Delta T + \frac{g}{c_p } ( z_1 + z_{0m} - z_{0h} )} \right) -.. math:: +.. math:: :label: 1.1.17 - \frac{ E_0 }{ \rho _0 }=- c_H V \Delta q = - C_H \Delta q\label{1.1.17} + \frac{ E_0 }{ \rho _0 }=- c_H V \Delta q = - C_H \Delta q -.. math:: +.. math:: :label: 1.1.18 - \frac{ {\rm {\bf \tau }}_{0} }{ \rho _{0} }= c_D V \Delta {\rm {\bf v}}{ }= C_D \Delta {\rm {\bf v}},\label{1.1.18} + \frac{ {\rm {\bf \tau }}_{0} }{ \rho _{0} }= c_D V \Delta {\rm {\bf v}}{ }= C_D \Delta {\rm {\bf v}}, where the effective wind speed for surface turbulent exchanges, :math:`V`, is defined by -.. math:: V = \frac{ v_\ast }{ c_D^{1/2} } = \frac{ v_\ast ^2 }{ C_D }\label{1.1.19} +.. math:: :label: 1.1.19 + + V = \frac{ v_\ast }{ c_D^{1/2} } = \frac{ v_\ast ^2 }{ C_D } and the **surface conductances** for scalars and momentum are respectively -.. math:: +.. math:: :label: 1.1.20 - C_H=\frac{k}{ \Phi _h } v_\ast = c_H V\label{1.1.20} + C_H=\frac{k}{ \Phi _h } v_\ast = c_H V -.. math:: +.. math:: :label: 1.1.21 - C_D=\frac{k}{ \Phi _m } v_\ast = c_D V\label{1.1.21}. + C_D=\frac{k}{ \Phi _m } v_\ast = c_D V. The surface exchange coefficients can then be written in any of the following forms: -.. math:: +.. math:: :label: 1.1.22 - c_H=\frac{ C_H }{V} = \frac{ C_H C_D }{ v_\ast ^2 } = \frac{ k^2 }{ \Phi _h \Phi _m }\label{1.1.22} + c_H=\frac{ C_H }{V} = \frac{ C_H C_D }{ v_\ast ^2 } = \frac{ k^2 }{ \Phi _h \Phi _m } -.. math:: +.. math:: :label: 1.1.23 c_D=\frac{ C_D }{V} = \frac{ C_D^2 }{ v_\ast ^2 } = \frac{ k^2 }{ \Phi _m^2 }. - \label{1.1.23} In order to close the system the surface scaling velocity, v\ :math:`_{\ast }`, needs to be specified. If -.. math:: v_\ast = u_\ast \equiv \left| { {\rm {\bf \tau }}_{0} {/} \rho _{0} } \right|^{1/2}\label{1.1.24} +.. math:: :label: 1.1.24 + + v_\ast = u_\ast \equiv \left| { {\rm {\bf \tau }}_{0} {/} \rho _{0} } \right|^{1/2} we have the standard Monin-Obukhov theory and it is easy to deduce that with this definition :math:`v_{\ast } = c_{D}^{1/2} \Delta`\ **v** and @@ -3137,19 +3090,17 @@ V=\ :math:`\Delta`\ **v**. To allow for the effect of **turbulent and cloud-scale gusts** on the surface turbulent fluxes the surface scaling velocity, v\ :math:`_{\ast }`, can be defined as -.. math:: +.. math:: :label: 1.1.25 v_\ast ^2 = u_\ast ^2 + \gamma _t^2 w_\ast ^2 + \gamma _c^2 w_c^2. - \label{1.1.25} The second term represents the effects of turbulent eddy-scale convective gusts and w\ :math:`_{\ast }` is the turbulent convective scaling velocity defined by -.. math:: +.. math:: :label: 1.1.26 w_\ast = {\left( { z_i \frac{ F_{B0} }{ \rho _0 }} \right)}^{1/3} - \label{1.1.26} for F\ :math:`_{B0} >` 0 and zero otherwise. z\ :math:`_{i}` is the height of the top of the surface-based turbulent mixing layer. @@ -3169,26 +3120,24 @@ in the Unified Model. The **low wind speed limit**, i.e. as :math:`\Delta`\ **v** :math:`\to` 0, for unstable conditions (with w\ :math:`_{c }`\ = 0) can be seen to be -.. math:: +.. math:: :label: 1.1.27 v_\ast \sim \gamma _t w_\ast \sim \gamma _t^{3/2} {\left( {\frac{ c_H }{ c_D^{1/2} }} \right)}^{1/2} z_i^{1/2} (-\Delta B )^{1/2} - \label{1.1.27} which implies that -.. math:: +.. math:: :label: 1.1.28 L \sim -( \gamma _t^3 /k) z_i - \label{1.1.28} Thus the low wind speed limits for the sensible and latent heat fluxes -are obtained by substituting (`[1.1.27] <#1.1.27>`__) into -(`[1.1.7] <#1.1.7>`__) and (`[1.1.8] <#1.1.8>`__) with the surface +are obtained by substituting :eq:`1.1.27` into +:eq:`1.1.7` and :eq:`1.1.8` with the surface transfer coefficients evaluated with L given by -(`[1.1.28] <#1.1.28>`__). The finite limit for L implies that the form +:eq:`1.1.28`. The finite limit for L implies that the form of the stability functions, :math:`\phi`, for very large and negative :math:`\zeta` is unimportant. However, the value of :math:`\Phi_{h}` for -L given by (`[1.1.28] <#1.1.28>`__) is needed if the value of +L given by :eq:`1.1.28` is needed if the value of :math:`\gamma _{t}` is determined from measurements of say the latent heat flux in very low mean wind conditions. @@ -3199,41 +3148,37 @@ Comparison with the `Godfrey and Beljaars (1991)`_ formulation for gustiness We can define the **mean gust speed** at height z\ :math:`_{1}` by -.. math:: +.. math:: :label: 1.2.1 v_g = ( V^2 - \left| {\Delta {{\rm {\bf v}}}} \right|^2 {)}^{{1/2}}. - \label{1.2.1} -Using the definitions of :math:`V` (`[1.1.19] <#1.1.19>`__) and -:math:`v_{\ast }` (`[1.1.25] <#1.1.25>`__) it can be deduced that +Using the definitions of :math:`V` :eq:`1.1.19` and +:math:`v_{\ast }` :eq:`1.1.25` it can be deduced that -.. math:: +.. math:: :label: 1.2.2 v_g^2 = W_g^2 ( z_1 ) + \frac{1}{2}\left| {\Delta {{\rm {\bf v}}}} \right|{ }\left[ {\left( {{ } {\left| {\Delta {{\rm {\bf v}}}} \right|}^2 - W_g^2 ( z_1 )} \right)^{1/2} - \left| {\Delta {{\rm {\bf v}}}} \right|} \right] - \label{1.2.2} where -.. math:: +.. math:: :label: 1.2.3 W_g (z) = \frac{1}{ c_D^{1/2} } {\left( { \gamma _t^2 w_\ast ^2 + \gamma _c^2 w_c^2 } \right)}^{1/2} = \frac{ \Phi _m (L , z + z_{0m} , z_{0m} )}{k} {\left( { \gamma _t^2 w_\ast ^2 + \gamma _c^2 w_c^2 } \right)}^{1/2}. - \label{1.2.3} Thus in this formulation the mean gust speed is a function of height above the surface through the same factor, :math:`\Phi _{m}`\ (z), which determines the profile of the mean wind **v** in the surface layer (see -Eq. (`[1.1.9] <#1.1.9>`__)). The values of :math:`\Delta`\ **v**, +Eq. :eq:`1.1.9`). The values of :math:`\Delta`\ **v**, v\ :math:`_{g}` and :math:`V` thus tend to zero as z :math:`\to` 0. Note that v\ :math:`_{g} \to` W\ :math:`_{g}` as :math:`\Delta`\ **v** :math:`\to` 0 and that v\ :math:`_{g} \to` 0 as the convective gustiness scaling velocities tend to zero. -Equation (`[1.2.1] <#1.2.1>`__) can be rewritten as +Equation :eq:`1.2.1` can be rewritten as -.. math:: +.. math:: :label: 1.2.4 V^2 = \left| {\Delta {{\rm {\bf v}}}} \right|^2 + v_g^2, - \label{1.2.4} which is exactly the form of `Godfrey and Beljaars (1991)`_. However `Godfrey and Beljaars (1991)`_ define the mean gust speed as @@ -3436,30 +3381,26 @@ The form of the stability functions. For **stable conditions**, i.e. :math:`\Delta`\ B :math:`\ge` 0, the stability functions are given by `Beljaars and Holtslag (1991)`_: -.. math:: +.. math:: :label: 1.3.11 \Phi _m=\ln \left( {\frac{ z_1 + z_{0m} }{ z_{0m} }} \right) - \Psi _m ( \zeta _1 ) + \Psi _m ( \zeta _{0m} ) - \label{1.3.11} -.. math:: +.. math:: :label: 1.3.12 \Phi _h=\ln \left( {\frac{ z_1 + z_{0m} }{ z_{0h} }} \right) - \Psi _h ( \zeta _1 ) + \Psi _h ( \zeta _{0h} ) - \label{1.3.12} where :math:`\zeta _{1}` = (z\ :math:`_{1}` + z\ :math:`_{0m})`/L, :math:`\zeta _{0m}` = z\ :math:`_{0m}`/L, :math:`\zeta _{0h}` = z\ :math:`_{0h}`/L and -.. math:: +.. math:: :label: 1.3.13 - \Psi _h (\zeta )=\left[ { {\left( {1 + \frac{2}{3}a\zeta } \right)}^{3/2} - 1 } \right] + b\left( {\zeta - \frac{c}{d}} \right)\exp (-d\zeta ) + \frac{bc}{d} - \label{1.3.13} -.. math:: +.. math:: :label: 1.3.14 - \Psi _m (\zeta )=a\zeta + b\left( {\zeta - \frac{c}{d}} \right)\exp (-d\zeta ) + \frac{bc}{d}, - \label{1.3.14} with :math:`a = 1`, :math:`b =2/3`, :math:`c = 5`, :math:`d = 0.35`. @@ -3467,11 +3408,10 @@ with :math:`a = 1`, :math:`b =2/3`, :math:`c = 5`, :math:`d = 0.35`. Note that the bulk flux Richardson number for the surface layer is given by -.. math:: +.. math:: :label: 1.3.8 {Ri}_{fB} = \frac{ c_D^{1/2} }{k} \frac{ z_1 }{L} = \frac{ z_1 /L}{ \Phi _m } - \label{1.3.8} so the `Beljaars and Holtslag (1991)`_ functions imply Ri\ :math:`_{f B} \to` 1/a = 1 as z\ :math:`_{1}`/L :math:`\to \infty`. @@ -3479,47 +3419,41 @@ Ri\ :math:`_{f B} \to` 1/a = 1 as z\ :math:`_{1}`/L :math:`\to \infty`. For **unstable conditions**, i.e. :math:`\Delta`\ B :math:`<` 0, the Dyer and Hicks forms :raw-latex:`\cite[]{dyer1974}` are used: -.. math:: +.. math:: :label: 1.3.15 \phi _m=(1 - 16\zeta )^{-1/4} - \label{1.3.15} -.. math:: +.. math:: :label: 1.3.16 \phi _h=(1 - 16\zeta )^{-1/2} - \label{1.3.16} (Note that :math:`\phi _{h}\prime` is discontinuous at 0.) These are only empirically verified for :math:`\zeta \ge` -1. Evaluating the -integrals (`[1.1.14] <#1.1.14>`__) and (`[1.1.15] <#1.1.15>`__) we +integrals :eq:`1.1.14` and :eq:`1.1.15` we obtain: -.. math:: +.. math:: :label: 1.3.17 \Phi _m = \ln \left( {\frac{ z_1 + z_{0m} }{ z_{0m} }} \right) - 2 \ln \left( {\frac{1 + X_1 }{1 + X_0 }} \right) - \ln \left( {\frac{1 + X_1^2 }{1 + X_0^2 }} \right)+ 2 \left( { {\tan }^{-1} X_1 - {\tan }^{-1} X_0 } \right) - \label{1.3.17} where -.. math:: +.. math:: :label: 1.3.18 X_1 = (1 - 16 \zeta _1 )^{1/4} , X_0 = (1 - 16 \zeta _{0m} )^{1/4} - \label{1.3.18} and -.. math:: +.. math:: :label: 1.3.19 \Phi _h = \ln \left( {\frac{ z_1 + z_{0m} }{ z_{0h} }} \right) - 2 \ln \left( {\frac{1 + Y_1 }{1 + Y_0 }} \right) - \label{1.3.19} where -.. math:: +.. math:: :label: 1.3.20 Y_1 = (1 - 16 \zeta _1 )^{1/2} , Y_0 = (1 - 16 \zeta _{0h} )^{1/2}. - \label{1.3.20} .. _section_1.4: @@ -3530,59 +3464,50 @@ For conditions that are stable, i.e. :math:`\Delta`\ B :math:`\ge` 0, or near-neutral (taken as :math:`\Delta`\ **v** :math:`\ge` 2 ms\ :math:`^{-1}`), then start the iteration from the neutral limit, so -.. math:: +.. math:: :label: 1.4.5 \Phi _m^{(0)}=\ln \left( {\frac{ z_1 + z_{0m} }{ z_{0m} }} \right) - \label{1.4.5} -.. math:: +.. math:: :label: 1.4.6 \Phi _h^{(0)}=\ln \left( {\frac{ z_1 + z_{0m} }{ z_{0h} }} \right) - \label{1.4.6} -.. math:: +.. math:: :label: 1.4.7 v_\ast ^{(0)}= {\left( {\frac{k}{ \Phi _m^{(0)} }} \right)} \left| {\Delta {{{\rm {\bf v}}}}} \right| - \label{1.4.7} Otherwise (if :math:`\Delta`\ B :math:`<` 0 and :math:`\Delta`\ **v** :math:`<` 2 ms\ :math:`^{-1}` ) start from the greater of the neutral and convective limits for :math:`v_\ast^{(0)}`, so -.. math:: +.. math:: :label: 1.4.1 \frac{1}{ L^{(0)} }=\frac{-k}{ \gamma _t^3 z_i } - \label{1.4.1} -.. math:: +.. math:: :label: 1.4.2 \Phi _m^{(0)}= \Phi _m ( L^{(0)} , z_1 + z_{0m} , z_{0m} ) - \label{1.4.2} -.. math:: +.. math:: :label: 1.4.3 \Phi _h^{(0)}= \Phi _h ( L^{(0)} , z_1 + z_{0m} , z_{0h} ) - \label{1.4.3} -.. math:: +.. math:: :label: 1.4.4 v_\ast ^{(0)}= MAX{\left[ {\left( {\frac{k}{ \Phi _m^{(0)} }} \right)} \left| {\Delta {{{\rm {\bf v}}}}} \right|, \, {\left[ { \gamma _t^3 \left( {\frac{k}{ \Phi _h^{(0)} }} \right) z_i \left| {-\Delta B} \right| } \right]}^{ 1/2} \right]} - \label{1.4.4} Then calculate -.. math:: +.. math:: :label: 1.4.8 C_D^{(0)}=\frac{k}{ \Phi _m^{(0)} } v_\ast ^{(0)} - \label{1.4.8} -.. math:: +.. math:: :label: 1.4.9 C_H^{(0)}=\frac{k}{ \Phi _h^{(0)} } v_\ast ^{(0)} - \label{1.4.9} Having set up initial values the iteration loop can be entered (this is @@ -3592,50 +3517,41 @@ section `8.4.1 <#mo_iter_corrn>`__): DO n = 1 to N -.. math:: +.. math:: :label: 1.4.10 u_\ast ^{(n)2}= C_D^{(n-1)} \left| {\Delta {{\rm {\bf v}}}} \right| - \label{1.4.10} -.. math:: +.. math:: :label: 1.4.11 {\left( {\frac{ F_{B0} }{ \rho _0 }} \right)}^{(n)}= { {-C}_H }^{(n-1)} \Delta B - \label{1.4.11} -.. math:: +.. math:: :label: 1.4.12 w_\ast ^{(n)}= {\left[ { z_i {\left( {\frac{ F_{B0} }{ \rho _0 }} \right)}^{(n)} } \right]}^{ 1/3} - \label{1.4.12} -.. math:: +.. math:: :label: 1.4.13 v_\ast ^{(n)2}= u_\ast ^{(n)2} + \gamma _t^2 w_\ast ^{(n)2} + \gamma _c^2 w_c^2 - \label{1.4.13} -.. math:: +.. math:: :label: 1.4.14 \frac{1}{ L^{(n)} } = \frac{-k( F_{B0} / \rho _0 )^{(n)} }{ v_\ast ^{(n)3} } - \label{1.4.14} -.. math:: +.. math:: :label: 1.4.15 \Phi _m^{(n)}= \Phi _m ( L^{(n)} , z_1 + z_{0m} , z_{0m} ) - \label{1.4.15} -.. math:: +.. math:: :label: 1.4.16 \Phi _h^{(n)}= \Phi _h ( L^{(n)} , z_1 + z_{0m} , z_{0h} ) - \label{1.4.16} -.. math:: +.. math:: :label: 1.4.17 C_D^{(n)}=\frac{k}{ \Phi _m^{(n)} } v_\ast ^{(n)} - \label{1.4.17} -.. math:: +.. math:: :label: 1.4.18 C_H^{(n)}=\frac{k}{ \Phi _h^{(n)} } v_\ast ^{(n)} - \label{1.4.18} END DO. @@ -3648,20 +3564,17 @@ Use the final (N) values of C\ :math:`_{H}` and C\ :math:`_{D}` to calculate the surface sensible and latent heat fluxes and surface stress: -.. math:: +.. math:: :label: 1.4.19 H_0= {-c}_P \rho _0 C_H^{(N)} \left( {\Delta T + \frac{g}{ c_P }( z_1 + z_{0m} - z_{0h} )} \right) - \label{1.4.19} -.. math:: +.. math:: :label: 1.4.20 E_0= {-\rho }_0 C_H^{(N)} \Delta q - \label{1.4.20} -.. math:: +.. math:: :label: 1.4.21 {\rm {\bf \tau }}_{0} = \rho _0 C_D^{(N)} \Delta {\rm {\bf v}} - \label{1.4.21} N is the last iteration value. N = 5 is currently used. @@ -3765,45 +3678,41 @@ namely :math:`\hat u_*` within the iteration. The interpolation of surface layer variables to standard observation heights ---------------------------------------------------------------------------- -Integrating (`[1.1.3] <#1.1.3>`__) between the roughness height, +Integrating :eq:`1.1.3` between the roughness height, z\ :math:`_{0m}`, and the observation height z\ :math:`_{ob}` we obtain -.. math:: +.. math:: :label: 1.5.1 {\rm {\bf v}}_{ob} { = } {\rm {\bf v}}_{0} { + }\frac{ {\rm {\bf \tau }}_{0} }{ \rho _0 v_\ast k} \Phi _m (L, z_{ob} + z_{0m} , z_{0m} ) - \label{1.5.1} Using the expression for the surface turbulent stress this gives the interpolation formula -.. math:: +.. math:: :label: 1.5.2 {\rm {\bf v}}_{ob} { = } {\rm {\bf v}}_{0} { + }\frac{ C_D }{k v_\ast } \Phi _m (L, z_{ob} + z_{0m} , z_{0m} ) ( {\rm {\bf v}}_{1} { - } {\rm {\bf v}}_{0} {)} - \label{1.5.2} For wind z\ :math:`_{ob}` is set to 10m and the last iteration (N) values of C\ :math:`_{D}`, L and :math:`v_{\ast }` are used. -Integrating (`[1.1.1] <#1.1.1>`__) and (`[1.1.2] <#1.1.2>`__) between +Integrating :eq:`1.1.1` and :eq:`1.1.2` between the roughness height, z\ :math:`_{0h}`, and the observation height z\ :math:`_{ob, }` we obtain for the scalar :math:`X` (:math:`=T+(g/c_{P})z` , :math:`q` or tracer amount) -.. math:: +.. math:: :label: 1.5.3 X_{ob} = X_0 + \frac{ F_{X0} }{ \rho _0 v_\ast k} \Phi _h (L, z_{ob} + z_{0h} , z_{0h} ) - \label{1.5.3} and using the expression for the surface flux :math:`F_{X0}` of the scalar quantity :math:`X` this gives the interpolation formula -.. math:: +.. math:: :label: 1.5.4 X_{ob} = X_0 + \frac{ C_H }{k v_\ast } \Phi _h (L, z_{ob} + z_{0h} , z_{0h} ) ( X_1 - X_0 ) - \label{1.5.4} For temperature and humidity z\ :math:`_{ob}` is set to the screen height (1.5 m) and the last iteration (N) values of C\ :math:`_{H}`, L @@ -3816,7 +3725,7 @@ In the foregoing analysis it is tacitly assumed that the surface layer, up to the model’s lowest grid level, is in equilibrium with the surface and lies within the constant flux layer. In light winds, and when the surface temperature falls quickly, these assumptions are invalid; -equation `[1.5.4] <#1.5.4>`__ then yields temperatures at the height of +equation :eq:`1.5.4` then yields temperatures at the height of observation that are too closely tied to the surface temperature. Observed temperatures may be significantly warmer: this may be termed decoupling. Two parametrizations of this effect are available. Both @@ -3972,10 +3881,9 @@ wind mixing energy flux is accumulated over an ocean model timestep and then used in the calculation of the mixing in the upper layers of the ocean. The gridbox mean **wind mixing energy flux** is given by -.. math:: +.. math:: :label: 1.6.9 F_{WME} = ( 1- f_I ) \frac{ \rho _0^{3/2} v_\ast ^3 }{ \rho _{(sea)}^{1/2} } - \label{1.6.9} where :math:`v_{\ast }` is calculated using the drag coefficient for the leads part of the gridbox, c\ :math:`_{D(L)}`, rather than the gridbox @@ -3994,11 +3902,10 @@ Unified Model. In all schemes available here the momentum roughness length is given by -.. math:: +.. math:: :label: eq:z0msea z_{0m(sea)} = \frac{1.54\times {10}^{-6} }{ v_\ast } + \frac{\alpha}{g} v_\ast ^2 - \label{eq:z0msea} which is a generalisation of Charnock’s formula to include low-wind conditions :raw-latex:`\cite[]{Smith88}`. :math:`\alpha` is Charnock’s @@ -4070,10 +3977,9 @@ described. using a linear relationship between the 10-m wind speed, valid over a certain range of wind speeds, with fixed values outside the range: - .. math:: + .. math:: :label: eq:charn \alpha = a U_{10} +b - \label{eq:charn} for :math:`U_{10,min} < U_{10} < U_{10,max}`. The constants :math:`a`, :math:`b`, :math:`U_{10,min}` and :math:`U_{10,max}` @@ -4092,10 +3998,9 @@ described. roughness length is set using the expression for the moisture roughness, - .. math:: + .. math:: :label: eq:z0h_coare z_{0h} = \min(1.15\times 10^{-4}, 5.5\times 10^{-5}/Re_*^{0.6}), - \label{eq:z0h_coare} where :math:`Re_*` is the roughness Reynolds number. @@ -4168,46 +4073,39 @@ Two approaches are available in uncoupled configurations of the model. the ice fraction, f\ :math:`_{I}`. For 0 :math:`\le` f\ :math:`_{I} <` 0.7 - .. math:: + .. math:: :label: 1.6.1 < C_H >=( f_I C_{H(MIZ)} + ( 0.7 - f_I ) C_{H(L)} ) / 0.7 - \label{1.6.1} - .. math:: + .. math:: :label: 1.6.2 < C_D >=( f_I C_{D(MIZ)} + ( 0.7 - f_I ) C_{D(L)} ) / 0.7 - \label{1.6.2} and for 0.7 :math:`\le` f\ :math:`_{I} \le` 1 - .. math:: + .. math:: :label: 1.6.3 < C_H >=( ( 1 - f_I ) C_{H(MIZ)} + ( f_I - 0.7 ) ) C_{H(I)} ) / 0.3 - \label{1.6.3} - .. math:: + .. math:: :label: 1.6.4 < C_D >=( ( 1 - f_I ) C_{D(MIZ)} + ( f_I - 0.7 ) ) C_{D(I)} ) / 0.3 - \label{1.6.4} where - .. math:: + .. math:: :label: 1.6.5 C_{H(L)}= C_H ( L_{(L)} , z_{0m(sea)} , z_{0h(sea)} ) - \label{1.6.5} - .. math:: + .. math:: :label: 1.6.6 C_{H(MIZ)}= C_H ( L_{(I)} , z_{0m(MIZ)} , z_{0h(MIZ)} ) - \label{1.6.6} - .. math:: + .. math:: :label: 1.6.7 C_{H(I)}= C_H ( L_{(I)} , z_{0m(sea-ice)} , z_{0h(sea-ice)} ) - \label{1.6.7} and similarly for the drag coefficient C\ :math:`_{D}`. @@ -4255,12 +4153,11 @@ Two approaches are available in uncoupled configurations of the model. (`L{\ (2015)`_) and the flow may be taken as neutral up to :math:`h_f`. Hence, - .. math:: + .. math:: :label: eq:int_u2 F_p \approx \frac{h_f}{2k^2} \rho u_*^2 \left [ (\log(h_f/z_0) -1)^2 +1 \right ] = \frac{h_f}{2k^2} \rho C_d U_1^2 \left [(\log(h_f/z_0) -1)^2 +1 \right ]. - \label{eq:int_u2} where :math:`C_d` is the upstream drag coefficient and :math:`U_1` is the wind on the model’s lowest atmospheric level. Because this will @@ -4274,7 +4171,7 @@ Two approaches are available in uncoupled configurations of the model. using the original version of the scheme (`L{\ (2012)`_), :math:`C_d` must be taken as the neutral drag coefficient. Note also that various approximations may - be made in Equation `[eq:int_u2] <#eq:int_u2>`__. + be made in Equation :eq:`eq:int_u2`. `L{\ (2012)`_ approximate :math:`(\log(h_f/z_0) -1)^2 +1` as :math:`(\log(h_f/z_0) )^2`; while `L{\ (2015)`_ approximate it as @@ -4283,10 +4180,9 @@ Two approaches are available in uncoupled configurations of the model. If, in a unit area, there are :math:`N` floes, each of crosswind dimension :math:`D_i`, the total drag will be - .. math:: + .. math:: :label: eq:fd_fp F_d = N c_w S_c^2 D_i F_p, - \label{eq:fd_fp} where :math:`c_w` is a coefficient and :math:`S_c` is a sheltering coefficient. The fractional coverage of sea ice within this unit area @@ -4333,15 +4229,13 @@ Two approaches are available in uncoupled configurations of the model. The overall drag coefficients are now set by interpolation in the ice fraction: - .. math:: + .. math:: :label: eq:cdice_int < C_D >= (1 - f_I) C_{D(L)} + f_I (C_{D(I)} + C_{D(FRM)}) - \label{eq:cdice_int} - .. math:: + .. math:: :label: eq:chice_int < C_H >= (1 - f_I) C_{H(L)} + f_I C_{H(I)} - \label{eq:chice_int} .. _`sec:coast`: @@ -4383,8 +4277,8 @@ approximation; in reality the factor depends on the land cover type and the degree of heterogeneity. [Future versions of the Unified Model will treat surface heterogeneity explicitly by the “tiling” method.] -The surface moisture flux given by (`[1.1.8] <#1.1.8>`__) or -(`[1.1.17] <#1.1.17>`__) involves a surface humidity value, +The surface moisture flux given by :eq:`1.1.8` or +:eq:`1.1.17` involves a surface humidity value, q\ :math:`_{0}`. Prior to UM6.3, for evaporation from all of ocean, sea-ice, lake and snow-covered surfaces as well as from water on vegetative canopies this surface value is taken to be the saturated @@ -4405,38 +4299,34 @@ formulation is described in full in the documentation for the land and ice surface processes component of the Unified Model. The evapotranspiration for the surface is given by -.. math:: +.. math:: :label: 1.7.1 E_t = - \rho _0 \frac{ q_1 - q_{sat} ( T_0 , p_0 )}{( r_a + r_s )} - \label{1.7.1} where the aerodynamic resistance, r\ :math:`_{a}` , is given by -.. math:: +.. math:: :label: 1.7.2 r_a = \frac{1}{ C_H } = \frac{1}{ c_H V} - \label{1.7.2} and r\ :math:`_{s}` is the surface or stomatal resistance to evaporation. r\ :math:`_{s}` is a function of the available soil moisture, near surface atmospheric conditions and the radiation impinging on the plants. [For the formulation see the documentation for the land and ice surface processes component of the Unified Model.] A -similar formula to (`[1.7.1] <#1.7.1>`__) is used for the evaporation -from the very near surface soil layer. Equation (`[1.7.1] <#1.7.1>`__) +similar formula to :eq:`1.7.1` is used for the evaporation +from the very near surface soil layer. Equation :eq:`1.7.1` can be written as -.. math:: +.. math:: :label: 1.7.3 E_t = - \rho _0 C_E ( q_1 - q_{sat} ( T_0 , p_0 ) ) - \label{1.7.3} where -.. math:: +.. math:: :label: 1.7.4 C_E = \frac{ C_H }{\left( {1 + \frac{ r_s }{ r_a }} \right)} - \label{1.7.4} .. _section_2: @@ -4456,42 +4346,37 @@ gridbox mean quantities and fluxes with the roughness lengths replaced by effective values, z\ :math:`_{0m(eff)}` and z\ :math:`_{0h(eff)}`. When form drag is included via effective roughness lengths equations -(`[1.1.7] <#1.1.7>`__)-(`[1.1.9] <#1.1.9>`__) become: +:eq:`1.1.7`-:eq:`1.1.9` become: .. math:: \frac{ H_{0(eff)} }{ c_P \rho _0 }=\frac{-k}{ \Phi _h (L , z_1 + z_{0m(eff)} , z_{0h(eff)} )} v_{\ast (eff)} -.. math:: +.. math:: :label: 2.1.1 \left( {\Delta T + \frac{g}{c_p }( z_1 + z_{0m(eff)} - z_{0h(eff)} )} \right) - \label{2.1.1} -.. math:: +.. math:: :label: 2.1.2 \frac{ E_{0(eff)} }{ \rho _0 }=\frac{-k}{ \Phi _h (L , z_1 + z_{0m(eff)} , z_{0h(eff)} )} v_{\ast (eff)} \Delta q - \label{2.1.2} -.. math:: +.. math:: :label: 2.1.3 \frac{ {\rm {\bf \tau }}_{{0(eff)}} }{ \rho _{0} }=\frac{k}{ \Phi _m (L , z_1 + z_{0m(eff)} , z_{0m(eff)} )} v_{\ast (eff)} \Delta {\rm {\bf v}} - \label{2.1.3} The effective surface scaling velocity, v\ :math:`_{\ast (eff)}` , is -given by (cf. (`[1.1.25] <#1.1.25>`__)) +given by (cf. :eq:`1.1.25`) -.. math:: +.. math:: :label: 2.1.4 v_{\ast (eff)}^2 = u_{\ast (eff)}^2 + \gamma _t^2 w_\ast ^2 + \gamma _c^2 w_c^2 - \label{2.1.4} where -.. math:: +.. math:: :label: 2.1.5 u_{\ast (eff)}^2 = \left| { {\rm {\bf \tau }}_{{0(eff)}} {/} \rho _{0} } \right| - \label{2.1.5} The effective roughness for momentum is derived by setting the total effective surface stress, **:math:`\tau`**\ :math:`_{0(eff)}`, to the @@ -4503,44 +4388,39 @@ z\ :math:`_{c}` above the surface. z\ :math:`_{c}` is currently set to 2\ :math:`^{1/2}\sigma _{h}` where :math:`\sigma _{h}` is the standard deviation of the unresolved orographic height. Thus -.. math:: +.. math:: :label: 2.1.6 \frac{ {\rm {\bf \tau }}_{{0(eff)}} }{ \rho _{0} }{ = }\frac{{k } {v}_{{\ast (eff)}} }{ \Phi _{m} {(L , } {z}_{c} { , } {z}_{{0m(eff)}} {)}}{ }{\rm {\bf v}}{(} {z}_{c} {)} - \label{2.1.6} and -.. math:: +.. math:: :label: 2.1.7 \frac{ {\rm {\bf \tau }}_{{0(f)}} }{ \rho _{0} }{ = }\frac{{k } {v}_{{\ast (f)}} }{ \Phi _{m} {(L , } {z}_{c} { , } {z}_{{0m}} {)}}{ }{\rm {\bf v}}{(} {z}_{c} {)} - \label{2.1.7} where the scaling velocity based on the stress over a flat surface, v\ :math:`_{\ast (f)}` , is given by -.. math:: +.. math:: :label: 2.1.8 v_{\ast (f)}^2 = u_{\ast (f)}^2 + \gamma _t^2 w_\ast ^2 + \gamma _c^2 w_c^2 - \label{2.1.8} with -.. math:: +.. math:: :label: 2.1.9 u_{\ast (f)}^2 = \left| { {\rm {\bf \tau }}_{{0(f)}} {/} \rho _{0} } \right| - \label{2.1.9} [The scaling velocity which appears in the -expression (`[1.1.4] <#1.1.4>`__) for the Monin-Obukhov length is chosen +expression :eq:`1.1.4` for the Monin-Obukhov length is chosen to be v\ :math:`_{\ast (eff)}` rather than the flat surface value.] The orographic stress is given by -.. math:: +.. math:: :label: 2.1.10 \frac{ {\rm {\bf \tau }}_{{0(p)}} }{ \rho _{0} }{ = }\frac{{1}}{{2}}{ } {c}_{{D(orog)}} { } {f}_{D} {(} {{Ri}}_{B} {)}\frac{{A}}{{S}}{ }\left| {{\rm {\bf v}}{(} {z}_{c} {)}} \right|{ }{\rm {\bf v}}{(} {z}_{c} {)} - \label{2.1.10} where :math:`A/S` is the total silhouette area of orography in a gridbox over the flat surface area of the gridbox taken as an average over all @@ -4552,34 +4432,31 @@ c\ :math:`_{D(orog)}` is set to the constant value (typically 0.3, `Mason (1986)`_). If the function :math:`\Phi _{m}` and v\ :math:`_{\ast }` are -approximated by their neutral values in (`[2.1.6] <#2.1.6>`__) -and (`[2.1.7] <#2.1.7>`__) then the equation for calculating the +approximated by their neutral values in :eq:`2.1.6` +and :eq:`2.1.7` then the equation for calculating the effective momentum roughness is derived -.. math:: +.. math:: :label: 2.1.12 \frac{\ln ( z_c / z_{0m(eff)} )}{\ln ( z_c / z_{0m} )} = {\left( {1 + \frac{1}{2} c_{D(orog)} f_D \frac{A}{S} {\left( {\frac{\ln ( z_c / z_{0m} )}{k}} \right)}^2 } \right)}^{-1/2} - \label{2.1.12} The stress for the flat surface is related to the total stress by -.. math:: +.. math:: :label: 2.1.13 {\rm {\bf \tau }}_{{0(f)}} { = } {\rm {\bf \tau }}_{{0(eff)}} { } {\left( {{1 + }\frac{{1}}{{2}}{ } {c}_{{D(orog)}} { } {f}_{D} { }\frac{{A}}{{S}}{ } {\left( {\frac{\ln {(} {z}_{c} { / } {z}_{{0m}} {)}}{{k}}} \right)}^{2} } \right)}^{{-1}} - \label{2.1.13} -which is derived from equations (`[2.1.6] <#2.1.6>`__), -(`[2.1.7] <#2.1.7>`__) and (`[2.1.10] <#2.1.10>`__). -Equation (`[2.1.13] <#2.1.13>`__) implies that +which is derived from equations :eq:`2.1.6`, +:eq:`2.1.7` and :eq:`2.1.10`. +Equation :eq:`2.1.13` implies that -.. math:: +.. math:: :label: 2.1.14 C_{D(f)} = C_{D(eff)} {\left( {1 + \frac{1}{2} c_{D(orog)} f_D \frac{A}{S} {\left( {\frac{\ln ( z_c / z_{0m} )}{k}} \right)}^2 } \right)}^{-1} - \label{2.1.14} Parametrized orographic drag coefficient ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ @@ -4587,75 +4464,66 @@ Parametrized orographic drag coefficient `Wood and Mason (1993)`_ find that orographic drag coefficient c\ :math:`_{D(orog)}` depends on A/S via the equation -.. math:: +.. math:: :label: 2.1.11 c_{D(orog)} = 2\alpha \beta \pi ^2 \frac{A}{S} \frac{ u_{\ast (f)}^2 }{ v^2 ( z_c )} - \label{2.1.11} where :math:`\alpha` and :math:`\beta` are constants (:math:`\alpha`\ =12 and :math:`\beta`\ =1). -If (`[2.1.11] <#2.1.11>`__) is used, the formula for the effective +If :eq:`2.1.11` is used, the formula for the effective roughness length for momentum becomes -.. math:: +.. math:: :label: 2.1.15 \frac{\ln ( z_c / z_{0m(eff)} )}{\ln ( z_c / z_{0m} )} = {\left( {1 + \alpha \beta \pi ^2 f_D {\left( {\frac{A}{S}} \right)}^2 } \right)}^{-1/2} - \label{2.1.15} -and (`[2.1.13] <#2.1.13>`__) and (`[2.1.14] <#2.1.14>`__) become +and :eq:`2.1.13` and :eq:`2.1.14` become -.. math:: +.. math:: :label: 2.1.16 {\rm {\bf \tau }}_{{0(f)}} = {\rm {\bf \tau }}_{{0(eff)}} {\left( {{1 + }\alpha \beta \pi ^{2} { } {f}_{D} { } {\left( {\frac{{A}}{{S}}} \right)}^{2} { }} \right)}^{-1} - \label{2.1.16} -.. math:: +.. math:: :label: 2.1.17 C_{D(f)}= C_{D(eff)} {\left( {1 + \alpha \beta \pi ^2 f_D {\left( {\frac{A}{S}} \right)}^2 } \right)}^{-1} - \label{2.1.17} The effective surface flux of scalar X evaluated in terms of values at z\ :math:`_{c}` is -.. math:: +.. math:: :label: 2.1.18 \frac{ F_{X0(eff)} }{ \rho _0 } = \frac{k v_{\ast (eff)} }{ \Phi _h (L , z_c , z_{0h(eff)} )} (X( z_c ) - X_0 ) - \label{2.1.18} and the surface flux for the flat surface is given by -.. math:: +.. math:: :label: 2.1.19 \frac{ F_{X0(f)} }{ \rho _0 } = \frac{k v_{\ast (f)} }{ \Phi _h (L , z_c , z_{0h} )} (X( z_c ) - X_0 ) - \label{2.1.19} `Hewer and Wood (1998)`_ find that the scalar transport is enhanced when there is orographic form drag such that -.. math:: +.. math:: :label: 2.1.20 F_{X0(eff)} = F_{X0(f)} {\left( {1 - 2.2 f_D \frac{A}{S}} \right)}^{-1} - \label{2.1.20} -Combining (`[2.1.18] <#2.1.18>`__)–(`[2.1.20] <#2.1.20>`__) and using +Combining :eq:`2.1.18`–:eq:`2.1.20` and using the neutral values of the stability functions the expression for the effective scalar roughness length is derived as -.. math:: +.. math:: :label: 2.1.21) \frac{\ln ( z_c / z_{0h(eff)} )}{\ln ( z_c / z_{0h} )} = {\left( {1 + \frac{1}{2} c_{D(orog)} f_D \frac{A}{S} {\left( {\frac{\ln ( z_c / z_{0m} )}{k}} \right)}^2 } \right)}^{1/2} \left( {1 - 2.2 f_D \frac{A}{S}} \right) - \label{2.1.21)} which becomes -.. math:: +.. math:: :label: 2.1.22 \frac{\ln ( z_c / z_{0h(eff)} )}{\ln ( z_c / z_{0h} )} = {\left( {1 + \alpha \beta \pi ^2 f_D {\left( {\frac{A}{S}} \right)}^2 } \right)}^{1/2} \left( {1 - 2.2 f_D \frac{A}{S}} \right) - \label{2.1.22} if the `Wood and Mason (1993)`_ formulation is used. @@ -4669,154 +4537,127 @@ For unstable conditions, i.e. :math:`\Delta`\ B :math:`<` 0 : IF :math:`\Delta`\ **v** :math:`<` 2 ms\ :math:`^{-1}` then start the iteration from the convective limit, so -.. math:: +.. math:: :label: 2.2.1 \frac{1}{ L^{(0)} }=\frac{-k}{ \gamma _t^3 z_i } - \label{2.2.1} -.. math:: +.. math:: :label: 2.2.2 \Phi _m^{(0)}= \Phi _m ( L^{(0)} , z_1 + z_{0m(eff)} , z_{0m(eff)} ) - \label{2.2.2} -.. math:: +.. math:: :label: 2.2.3 \Phi _h^{(0)}= \Phi _h ( L^{(0)} , z_1 + z_{0m(eff)} , z_{0h} ) - \label{2.2.3} -.. math:: +.. math:: :label: (2.2.4 v_{\ast (eff)}^{(0)}= v_{\ast (f)}^{(0)} = {\left[ { \gamma _t^3 \left( {\frac{k}{ \Phi _h^{(0)} }} \right) z_i \left| {-\Delta B} \right| + \gamma _c^2 w_c^2 } \right]}^{ 1/2} - \label{(2.2.4} ELSE IF (:math:`\Delta`\ **v** :math:`\ge` 2 ms\ :math:`^{-1}` ) start iteration from the neutral end, so -.. math:: +.. math:: :label: 2.2.5 \Phi _m^{(0)}=\ln \left( {\frac{ z_1 + z_{0m(eff)} }{ z_{0m(eff)} }} \right) - \label{2.2.5} -.. math:: +.. math:: :label: 2.2.6 \Phi _h^{(0)}=\ln \left( {\frac{ z_1 + z_{0m(eff)} }{ z_{0h} }} \right) - \label{2.2.6} -.. math:: +.. math:: :label: 2.2.7 u_{\ast (eff)}^{(0)}=\frac{k}{ \Phi _m^{(0)} } \left| {\Delta {{{v}}}} \right| - \label{2.2.7} -.. math:: +.. math:: :label: 2.2.8 v_{\ast (eff)}^{(0)}= {\left( { u_{\ast (eff)}^{(0) 2} + \gamma _c^2 w_c^2 } \right)}^{ 1/2} - \label{2.2.8} -.. math:: +.. math:: :label: 2.2.9 u_{\ast (f)}= u_{\ast (eff)} \frac{\ln ( z_c / z_{0m(eff)} )}{\ln ( z_c / z_{0m} )} - \label{2.2.9} -.. math:: +.. math:: :label: 2.2.10 v_{\ast (f)}^{(0)}= {\left( { u_{\ast (f)}^{(0) 2} + \gamma _c^2 w_c^2 } \right)}^{ 1/2} - \label{2.2.10} END IF. Then calculate: -.. math:: +.. math:: :label: (2.2.11 C_{D(eff)}^{(0)}=\frac{k}{ \Phi _m^{(0)} } v_{\ast (eff)}^{(0)} - \label{(2.2.11} -.. math:: +.. math:: :label: (2.2.12 C_{H(eff)}^{(0)}=\frac{k}{ \Phi _h^{(0)} } v_{\ast (eff)}^{(0)} - \label{(2.2.12} -.. math:: +.. math:: :label: (2.2.13 C_{D(f)}^{(0)}= C_{D(eff)}^{(0)} {\left( {1 + \frac{1}{2} c_{D(orog)} f_D \frac{A}{S} {\left( {\frac{\ln ( z_c / z_{0m} )}{k}} \right)}^2 } \right)}^{-1} - \label{(2.2.13} -.. math:: +.. math:: :label: (2.2.14 C_{H(f)}^{(0)}= C_{H(eff)}^{(0)} \left( {1 - 2.2 f_D \frac{A}{S}} \right) - \label{(2.2.14} Having set up initial values the iteration loop can be entered: DO n = 1 to N -.. math:: +.. math:: :label: (2.2.15 u_{\ast (eff)}^{(n)2}= C_{D(eff)}^{(n-1)} \left| {\Delta {{\rm {\bf v}}}} \right| - \label{(2.2.15} -.. math:: +.. math:: :label: (2.2.16 u_{\ast (f)}^{(n)2}= C_{D(f)}^{(n-1)} \left| {\Delta {{\rm {\bf v}}}} \right| - \label{(2.2.16} -.. math:: +.. math:: :label: (2.2.17 {\left( {\frac{ F_{B0} }{ \rho _0 }} \right)}^{(n)}=- { C_{H(eff)} }^{(n-1)} \Delta B - \label{(2.2.17} -.. math:: +.. math:: :label: (2.2.18 w_\ast ^{(n)}= {\left[ { z_i {\left( {\frac{ F_{B0} }{ \rho _0 }} \right)}^{(n)} } \right]}^{ 1/3} - \label{(2.2.18} -.. math:: +.. math:: :label: (2.2.19 v_{\ast (eff)}^{(n)2}= u_{\ast (eff)}^{(n)2} + \gamma _t^2 w_\ast ^{(n)2} + \gamma _c^2 w_c^2 - \label{(2.2.19} -.. math:: +.. math:: :label: (2.2.20 v_{\ast (f)}^{(n)2}= u_{\ast (f)}^{(n)2} + \gamma _t^2 w_\ast ^{(n)2} + \gamma _c^2 w_c^2 - \label{(2.2.20} -.. math:: +.. math:: :label: (2.2.21 \frac{1}{ L^{(n)} }=\frac{-k( F_{B0} / \rho _0 )^{(n)} }{ v_{\ast (eff)}^{(n)3} } - \label{(2.2.21} -.. math:: +.. math:: :label: (2.2.22 \Phi _m^{(n)}= \Phi _m ( L^{(n)} , z_1 + z_{0m(eff)} , z_{0m(eff)} ) - \label{(2.2.22} -.. math:: +.. math:: :label: (2.2.23 \Phi _h^{(n)}= \Phi _h ( L^{(n)} , z_1 + z_{0m(eff)} , z_{0h} ) - \label{(2.2.23} -.. math:: +.. math:: :label: (2.2.24 C_{D(eff)}^{(n)}=\frac{k}{ \Phi _m^{(n)} } v_{\ast (eff)}^{(n)} - \label{(2.2.24} -.. math:: +.. math:: :label: (2.2.25 C_{H(eff)}^{(n)}=\frac{k}{ \Phi _h^{(n)} } v_{\ast (eff)}^{(n)} - \label{(2.2.25} -.. math:: +.. math:: :label: (2.2.26 C_{D(f)}^{(n)}= C_{D(eff)}^{(n)} {\left( {1 + \frac{1}{2} c_{D(orog)} f_D \frac{A}{S} {\left( {\frac{\ln ( z_c / z_{0m} )}{k}} \right)}^2 } \right)}^{-1} - \label{(2.2.26} -.. math:: +.. math:: :label: (2.2.27 C_{H(f)}^{(n)}= C_{H(eff)}^{(n)} \left( {1 - 2.2 f_D \frac{A}{S}} \right) - \label{(2.2.27} END DO. @@ -4827,30 +4668,26 @@ w\ :math:`_{\ast }`\ =0 in the above iteration loop. Use the final (N) values of C\ :math:`_{H(eff)}` and C\ :math:`_{D(eff)}` to calculate the surface sensible and latent heat fluxes and surface stress: -.. math:: +.. math:: :label: 2.2.28 H_{0(eff)}=- c_P \rho _0 C_{H(eff)}^{(N)} \left( {\Delta T + \frac{g}{c_p }( z_1 + z_{0m(eff)} - z_{0h} )} \right) - \label{2.2.28} -.. math:: +.. math:: :label: 2.2.29 E_{0(eff)}= {-\rho }_0 C_{H(eff)}^{(N)} \Delta q - \label{2.2.29} -.. math:: +.. math:: :label: 2.2.30 {\rm {\bf \tau }}_{{0(eff)}} = \rho _0 C_{D(eff)}^{(N)} \Delta {\rm {\bf v}} - \label{2.2.30} The stress for a flat surface, if required for output, is calculated from -.. math:: +.. math:: :label: 2.2.31 {\rm {\bf \tau }}_{{0(f)}} { = } \rho _0 C_{D(f)}^{(N)} \Delta {\rm {\bf v}} - \label{2.2.31} .. _section_2.3: @@ -4859,24 +4696,22 @@ Interpolation of surface layer variables to standard observation heights If the observation height wind is assumed to lie on the profile defined by the effective roughness length and surface scaling velocity then -(c.f. equation (`[1.5.1] <#1.5.1>`__)) +(c.f. equation :eq:`1.5.1`) -.. math:: +.. math:: :label: 2.3.1 {\rm {\bf v}}_{{ob}} { = } {\rm {\bf v}}_{0} { + }\frac{ {\rm {\bf \tau }}_{{0(eff)}} }{ \rho _0 v_{\ast (eff)} k} \Phi _m (L, z_{ob} + z_{0m(eff)} , z_{0m(eff)} - \label{2.3.1} Using the expression for the surface turbulent stress this becomes -.. math:: +.. math:: :label: 2.3.2 {\rm {\bf v}}_{{ob}} { = } {\rm {\bf v}}_{0} { + }\frac{ C_{D(eff)} }{ {kv}_{\ast (eff)} } \Phi _m (L, z_{ob} + z_{0m(eff)} , z_{0m(eff)} ) ( {\rm {\bf v}}_1 - {\rm {\bf v}}_{0} {)} - \label{2.3.2} For wind z\ :math:`_{ob}` is set to 10m and the last iteration (N) values of C\ :math:`_{D(eff)}`, L and v\ :math:`_{\ast (eff)}` are used. @@ -4884,21 +4719,19 @@ Alternatively if the observation height wind is assumed to lie on a profile defined by the flat surface roughness length and scaling velocity then -.. math:: +.. math:: :label: 2.3.3 {\rm {\bf v}}_{{ob}} { = } {\rm {\bf v}}_{0} { + }\frac{ {\rm {\bf \tau }}_{{0(f)}} }{ \rho _0 v_{\ast (f)} k} \Phi _m (L, z_{ob} + z_{0m} , z_{0m} ) - \label{2.3.3} and substituting for the surface stress this becomes -.. math:: +.. math:: :label: 2.3.4 {\rm {\bf v}}_{{ob}} { = } {\rm {\bf v}}_{0} { + }\frac{ C_{D(f)} }{k v_{\ast (f)} } \Phi _m (L, z_{ob} + z_{0m} , z_{0m} ) ( {\rm {\bf v}}_1 - {\rm {\bf v}}_{0} {)} - \label{2.3.4} Most configurations of the Unified Model currently use the latter assumption with the last iteration value of C\ :math:`_{D(f)}`, L and @@ -4907,31 +4740,28 @@ assumption with the last iteration value of C\ :math:`_{D(f)}`, L and If the observation height scalar quantities are assumed to lie on the mean profile defined by the effective roughness length and scaling -quantities then (c.f. equation (`[1.5.3] <#1.5.3>`__) we obtain for the +quantities then (c.f. equation :eq:`1.5.3` we obtain for the generic scalar :math:`X` (:math:`T+(g/c_{P})z`, :math:`q`, tracer amount) -.. math:: +.. math:: :label: 2.3.5 X_{ob} = X_0 + \frac{ F_{X0(eff)} }{ \rho _0 v_{\ast (eff)} k} \Phi _h (L, z_{ob} + z_{0h(eff)} , z_{0h(eff)} ) - \label{2.3.5} and using the expression for the surface flux of the scalar quantity :math:`X` this becomes -.. math:: +.. math:: :label: 2.3.6 X_{ob} = X_0 + \frac{ C_{H(eff)} }{k v_{\ast (eff)} } \Phi _h (L, z_{ob} + z_{0h(eff)} , z_{0h(eff)} ) ( X_1 - X_0 ). - \label{2.3.6} Alternatively if the observation height scalar quantities are assumed to lie on a profile defined by the flat surface roughness length and flux then -.. math:: +.. math:: :label: 2.3.7 X_{ob} = X_0 + \frac{ C_{H(f)} }{k v_{\ast (f)} } \Phi _h (L, z_{ob} + z_{0h} , z_{0h} ) ( X_1 - X_0 ) - \label{2.3.7} For temperature and humidity z\ :math:`_{ob}` is set to the screen height (1.5 m) and the last iteration (N) values of C\ :math:`_{H}`, L @@ -4952,10 +4782,9 @@ quantities is made. The turbulent form drag is represented by the term -.. math:: +.. math:: :label: eq:drag {\bf f}=\frac{1}{\rho}\frac{\partial}{\partial z}{\bf\tau}_{\rm orog} - \label{eq:drag} on the right-hand side of the horizontal momentum equation, where :math:`{\bf\tau}_{\rm orog}` is the horizontal vector containing the @@ -4971,45 +4800,42 @@ where :math:`{\bf F_p}=({F_p}_x,{F_p}_y)`, :math:`{F_p}_x` and components of the pressure force on the sub-grid orography, and :math:`\ell` is a decay scale. We define :math:`\ell` such that -.. math:: +.. math:: :label: eq:l \ell={\rm min}\left(\lambda,\frac{z_h}{3}\right), - \label{eq:l} where :math:`z_h` is the boundary-layer depth and :math:`\lambda`, a somewhat ill defined quantity, is related to the horizontal scales of the sub-grid hills (and set to 300 m). Note that the value of -:math:`\ell` obtained from Eq. (`[eq:l] <#eq:l>`__) is further +:math:`\ell` obtained from Eq. :eq:`eq:l` is further constrained to be at least 100 m. If the steep-hill expression is to be used, the surface stress applied is almost identical to that used in the effective roughness -parametrization (Eq. `[2.1.10] <#2.1.10>`__), namely: +parametrization (Eq. :eq:`2.1.10`, namely: -.. math:: +.. math:: :label: eq:dragsteep \frac{\bf F_p}{\rho_0}=\frac{1}{2}c_{D(orog)} f_D (Ri_{B}) \frac{A}{S} \left\vert{\rm{\bf v}}(\ell) \right\vert{\rm{\bf v}}(\ell), - \label{eq:dragsteep} the main difference being the dependence on the height scale :math:`\ell` rather than :math:`z_c`. Similarly, if the `Wood and Mason (1993)`_ low-hill expression is used, the surface -stress is given by the equivalent of (Eq. `[2.1.16] <#2.1.16>`__), +stress is given by the equivalent of (Eq. :eq:`2.1.16`, namely: -.. math:: +.. math:: :label: eq:draglow \frac{\bf F_p}{\rho_0} = {\left( {\frac{\kappa}{\zeta_m}} \right)}^{2} \alpha \beta \pi ^{2} {f}_{D} (Ri_{B}) {\left( {\frac{A}{S}} \right)}^{2} \left\vert{\rm{\bf v}}(\ell) \right\vert{\rm{\bf v}}(\ell), - \label{eq:draglow} where :math:`\zeta_m = {\rm log}(\ell/z_{0m})`. There is also an option -to use the low-hill stress (`[eq:draglow] <#eq:draglow>`__) but capped +to use the low-hill stress :eq:`eq:draglow` but capped by that from the steep hill expression -(`[eq:dragsteep] <#eq:dragsteep>`__), to avoid generating huge stresses +:eq:`eq:dragsteep`, to avoid generating huge stresses at large :math:`A/S`. There is also a choice for the Richardson number, :math:`Ri_{B}`, that @@ -5054,55 +4880,50 @@ Algorithmic description Consider the non-linear damping equation: -.. math:: +.. math:: :label: eq:damp1 \frac{dX}{dt}=-\left(KX^{P}\right)X+S - \label{eq:damp1} Here :math:`S` is a constant forcing, or source, term and :math:`KX^{P}` is the diffusion coefficient, with :math:`K` constant. :math:`P` is assumed to be positive. The new scheme is written -.. math:: +.. math:: :label: eq:sppf1 \frac{X^{*}-X^{n}}{\Delta t}=-{\cal I}_{1}\left[K\left(X^{n}\right)^{P}\right] X^{*}+{\cal E}_{1}\left[K\left(X^{n}\right)^{P}\right] - X^{n}+\left({\cal I}_{1}-{\cal E}_{1}\right)S,\label{eq:sppf1} + X^{n}+\left({\cal I}_{1}-{\cal E}_{1}\right)S, -.. math:: +.. math:: :label: eq:sppf2 \frac{X^{n+1}-X^{*}}{\Delta t}=-{\cal I}_{2}\left[K\left(X^{n}\right)^{P}\right]X^{n+1} + {\cal E}_{2}\left[K\left(X^{n}\right)^{P}\right]X^{*} + - \left({\cal I}_{2}-{\cal E}_{2}\right)S,\label{eq:sppf2} + \left({\cal I}_{2}-{\cal E}_{2}\right)S, where -.. math:: +.. math:: :label: eq:E1coeff {\cal E}_{1}=\left(1+\frac{1}{\sqrt{2}}\right) \left[P+\frac{1}{\sqrt{2}}\pm\sqrt{P \left(\sqrt{2}-1\right)+\frac{1}{2}}\right] - \label{eq:E1coeff} -.. math:: +.. math:: :label: eq:E2coeff {\cal E}_{2}=\left(1+\frac{1}{\sqrt{2}}\right) \left[P+\frac{1}{\sqrt{2}}\mp\sqrt{P\left(\sqrt{2}-1\right)+\frac{1}{2}}\right] - \label{eq:E2coeff} -.. math:: +.. math:: :label: eq:Icoeff {\cal I}_{1}={\cal I}_{2}=\left(1+\frac{1}{\sqrt{2}}\right)\left(1+P\right) - \label{eq:Icoeff} Consider the one-dimensional “forced” boundary layer diffusion equation -.. math:: +.. math:: :label: eq:vdiff1 \frac{\partial X}{\partial t}=\frac{\partial F}{\partial z}+S, \qquad F=K_{X}\frac{\partial X}{\partial z} - \label{eq:vdiff1} where :math:`X` is the scalar variable being diffused, :math:`F` is the flux of :math:`X`, :math:`t` is the time, :math:`z` is the height from @@ -5112,19 +4933,19 @@ and :math:`S` is a forcing term from other processes preceding the boundary layer. In the UM these processes are: microphysics, gravity wave drag, radiation, dynamics and optionally (using the switch i_impsolve_loc) convection [3]_. :math:`S` represents the total tendency -from these processes. Equations (`[eq:sppf1] <#eq:sppf1>`__), -(`[eq:sppf2] <#eq:sppf2>`__) applied to (`[eq:vdiff1] <#eq:vdiff1>`__) +from these processes. Equations :eq:`eq:sppf1`, +:eq:`eq:sppf2` applied to :eq:`eq:vdiff1` becomes -.. math:: +.. math:: :label: eq:sppf_bl1 - \frac{X^{*}-X^{n}}{\Delta t} = {\cal I}_{1}\frac{\partial F}{\partial z}^{*}-{\cal E}_{1}\frac{\partial F}{\partial z}^{n}+\left({\cal I}_{1}-{\cal E}_{1}\right)S\label{eq:sppf_bl1} + \frac{X^{*}-X^{n}}{\Delta t} = {\cal I}_{1}\frac{\partial F}{\partial z}^{*}-{\cal E}_{1}\frac{\partial F}{\partial z}^{n}+\left({\cal I}_{1}-{\cal E}_{1}\right)S -.. math:: +.. math:: :label: eq:sppf_bl2 \frac{X^{n+1}-X^{*}}{\Delta t} = {\cal I}_{2}\frac{\partial F}{\partial z}^{n+1}-{\cal E}_{2}\frac{\partial F}{\partial - z}^{*}+\left({\cal I}_{2}-{\cal E}_{2}\right)S\label{eq:sppf_bl2} + z}^{*}+\left({\cal I}_{2}-{\cal E}_{2}\right)S where, @@ -5154,20 +4975,20 @@ X^{n+1}=X^{n+1}-X^{*}`. Then, F^{*}=F^{n}+K_{X}\frac{\partial\delta X}{\partial z}^{*},\qquad F^{n+1}=F^{*}+K_{X}\frac{\partial\delta X}{\partial z}^{n+1}. -Writing equations (`[eq:sppf_bl1] <#eq:sppf_bl1>`__), -(`[eq:sppf_bl2] <#eq:sppf_bl2>`__) in terms of these increments: +Writing equations :eq:`eq:sppf_bl1`, +:eq:`eq:sppf_bl2` in terms of these increments: -.. math:: +.. math:: :label: eq:sppf_inc1 - \frac{\delta X}{\Delta t}^{*} = ({\cal I}_{1}-{\cal E}_{1})\left(\frac{\partial F}{\partial z}^{n}+S\right)+{\cal I}_{1}\frac{\partial}{\partial z}\left(K_{X}\frac{\partial\delta X}{\partial z}^{*}\right)\label{eq:sppf_inc1} + \frac{\delta X}{\Delta t}^{*} = ({\cal I}_{1}-{\cal E}_{1})\left(\frac{\partial F}{\partial z}^{n}+S\right)+{\cal I}_{1}\frac{\partial}{\partial z}\left(K_{X}\frac{\partial\delta X}{\partial z}^{*}\right) -.. math:: +.. math:: :label: eq:sppf_inc2 - \frac{\delta X}{\Delta t}^{n+1} = ({\cal I}_{2}-{\cal E}_{2})\left(\frac{\partial F}{\partial z}^{*}+S\right)+{\cal I}_{2}\frac{\partial}{\partial z}\left(K_{X}\frac{\partial\delta X}{\partial z}^{n+1}\right)\label{eq:sppf_inc2} + \frac{\delta X}{\Delta t}^{n+1} = ({\cal I}_{2}-{\cal E}_{2})\left(\frac{\partial F}{\partial z}^{*}+S\right)+{\cal I}_{2}\frac{\partial}{\partial z}\left(K_{X}\frac{\partial\delta X}{\partial z}^{n+1}\right) -.. math:: +.. math:: :label: eq:sppf_inc3 - X^{n+1} = X^{n}+\delta X^{*}+\delta X^{n+1}\label{eq:sppf_inc3} + X^{n+1} = X^{n}+\delta X^{*}+\delta X^{n+1} .. _`sec:impsolve`: @@ -5181,21 +5002,23 @@ adapt the technique used in the original scheme. **Vertical diffusion solver for momentum variables** Consider the following equivalent form of -(`[eq:sppf_bl1] <#eq:sppf_bl1>`__): +:eq:`eq:sppf_bl1`: + +.. math:: :label: eq:du_star -.. math:: \frac{\delta u^{*}}{\Delta t}=\frac{\partial\bar{\tau}_{x}^{*}}{\partial z}+\left({\cal I}_{1}-{\cal E}_{1}\right)S\label{eq:du_star} + \frac{\delta u^{*}}{\Delta t}=\frac{\partial\bar{\tau}_{x}^{*}}{\partial z}+\left({\cal I}_{1}-{\cal E}_{1}\right)S where :math:`\tau_{x}` is the :math:`u` wind component stress (defined -in the same way as the flux in (`[eq:sppf_inc1] <#eq:sppf_inc1>`__) and +in the same way as the flux in :eq:`eq:sppf_inc1` and :math:`\bar{\tau}_{x}^{*}` its time-average: -.. math:: +.. math:: :label: eq:tau_star \bar{\tau}_{x}^{*}={\cal I}_{1}\tau_{x}^{*}-{\cal E}_{1}\tau_{x}^{n},\qquad\tau_{x}^{*}=\tau_{x}^{n}+K_{u}\frac{\partial\delta - u^{*}}{\partial z}.\label{eq:tau_star} + u^{*}}{\partial z}. -Substituting (`[eq:tau_star] <#eq:tau_star>`__) into -(`[eq:du_star] <#eq:du_star>`__) the following is obtained: +Substituting :eq:`eq:tau_star` into +:eq:`eq:du_star` the following is obtained: .. math:: @@ -5204,7 +5027,7 @@ Substituting (`[eq:tau_star] <#eq:tau_star>`__) into z}+S\right)+{\cal I}_{1}\frac{\partial}{\partial z}\left(K_{u}\frac{\partial\delta u^{*}}{\partial z}\right) -which is identical to (`[eq:sppf_inc1] <#eq:sppf_inc1>`__) for +which is identical to :eq:`eq:sppf_inc1` for :math:`X\equiv u,\; F_{X}\equiv\tau_{x}`. This equivalent derivation is used here as it presents a more convenient form to express the boundary conditions. @@ -5228,11 +5051,11 @@ levels), discretizing the previous equation in :math:`z` on all or, rearranging -.. math:: +.. math:: :label: eq:tridiag A_{k}\delta u_{k+3/2}^{*}+B_{k}\delta u_{k+1/2}^{*}+C_{k}\delta u_{k-1/2}^{*}=\Delta - t({\cal I}_{1}-{\cal E}_{1})\left(\frac{\tau_{x}^{n}\Big|_{k+1}-\tau_{x}^{n}\Big|_{k}}{z_{k+1}-z_{k}}+S_{k+1/2}\right),\label{eq:tridiag} + t({\cal I}_{1}-{\cal E}_{1})\left(\frac{\tau_{x}^{n}\Big|_{k+1}-\tau_{x}^{n}\Big|_{k}}{z_{k+1}-z_{k}}+S_{k+1/2}\right), where :math:`k=1,2,\ldots,L-2`, @@ -5247,42 +5070,50 @@ where :math:`k=1,2,\ldots,L-2`, (Note that the surface is level :math:`0`). For the top :math:`\rho`-level, :math:`k=L-1`, the -:math:`z`-discretization of (`[eq:du_star] <#eq:du_star>`__) is: +:math:`z`-discretization of :eq:`eq:du_star` is: -.. math:: B_{L}\delta u_{L-1/2}^{*}+C_{L}\delta u_{L-3/2}^{*}=\Delta t({\cal I}_{1}-{\cal E}_{1})\left(\frac{\tau_{x}^{n}\Big|_{L}-\tau_{x}^{n}\Big|_{L-1}}{z_{L}-z_{L-1}}+S_{L-1/2}\right),\label{eq:tridiag_top} +.. math:: :label: eq:tridiag_top + + B_{L}\delta u_{L-1/2}^{*}+C_{L}\delta u_{L-3/2}^{*}=\Delta t({\cal I}_{1}-{\cal E}_{1})\left(\frac{\tau_{x}^{n}\Big|_{L}-\tau_{x}^{n}\Big|_{L-1}}{z_{L}-z_{L-1}}+S_{L-1/2}\right), where :math:`B_{L}`, :math:`C_{L}` are derived as before setting :math:`A_{L}=0`. For the bottom :math:`\rho`-level, :math:`k=0`, the -:math:`z`-discretization of (`[eq:du_star] <#eq:du_star>`__) is: +:math:`z`-discretization of :eq:`eq:du_star` is: -.. math:: +.. math:: :label: eq:u_bc_1 \begin{aligned} - \delta u_{1/2}^{*} & = & \frac{\Delta t}{z_{1}-0}\left(\bar{\tau}_{x}^{*}\Big|_{1}-\bar{\tau}_{x}^{*}\Big|_{0}\right)+\Delta t\left({\cal I}_{1}-{\cal E}_{1}\right)S_{1/2}\label{eq:u_bc_1} + \delta u_{1/2}^{*} & = & \frac{\Delta t}{z_{1}-0}\left(\bar{\tau}_{x}^{*}\Big|_{1}-\bar{\tau}_{x}^{*}\Big|_{0}\right)+\Delta t\left({\cal I}_{1}-{\cal E}_{1}\right)S_{1/2} \end{aligned} -where, from (`[eq:tau_star] <#eq:tau_star>`__), +where, from :eq:`eq:tau_star`, -.. math:: \bar{\tau}_{x}^{*}\Big|_{1}=\left({\cal I}_{1}-{\cal E}_{1}\right)\tau_{x}^{n}\Big|_{1}+{\cal I}_{1}\left(K_{u}\frac{\partial\delta u^{*}}{\partial z}\right)\Big|_{1}.\label{eq:u_bc_2} +.. math:: :label: eq:u_bc_2 -Combining (`[eq:u_bc_1] <#eq:u_bc_1>`__), (`[eq:u_bc_2] <#eq:u_bc_2>`__) + \bar{\tau}_{x}^{*}\Big|_{1}=\left({\cal I}_{1}-{\cal E}_{1}\right)\tau_{x}^{n}\Big|_{1}+{\cal I}_{1}\left(K_{u}\frac{\partial\delta u^{*}}{\partial z}\right)\Big|_{1}. + +Combining :eq:`eq:u_bc_1`, :eq:`eq:u_bc_2` the bottom row discretization is obtained: -.. math:: A_{0}\delta u_{3/2}^{*}+B_{0}\delta u_{1/2}^{*}=\Delta t\left({\cal I}_{1}-{\cal E}_{1}\right)\left(\frac{\tau_{x}^{n}\Big|_{1}}{z_{1}}+S_{1/2}\right)-\frac{\Delta t}{z_{1}}\bar{\tau}_{x}^{*}\Big|_{0}\label{eq:u_bc_3} +.. math:: :label: eq:u_bc_3 + + A_{0}\delta u_{3/2}^{*}+B_{0}\delta u_{1/2}^{*}=\Delta t\left({\cal I}_{1}-{\cal E}_{1}\right)\left(\frac{\tau_{x}^{n}\Big|_{1}}{z_{1}}+S_{1/2}\right)-\frac{\Delta t}{z_{1}}\bar{\tau}_{x}^{*}\Big|_{0} where .. math:: A_{0}=-{\cal I}_{1}\frac{\Delta tK_{u}\Big|_{1}}{z_{1}(z_{3/2}-z_{1/2})},\quad B_{0}=1-A_{0}. -Equations (`[eq:tridiag] <#eq:tridiag>`__), -(`[eq:tridiag_top] <#eq:tridiag_top>`__) and -(`[eq:u_bc_3] <#eq:u_bc_3>`__) form a tridiagonal system of linear +Equations :eq:`eq:tridiag`, +:eq:`eq:tridiag_top` and +:eq:`eq:u_bc_3` form a tridiagonal system of linear equations. When the elimination procedure takes place -(`[eq:u_bc_3] <#eq:u_bc_3>`__) becomes +:eq:`eq:u_bc_3` becomes -.. math:: \delta u_{1/2}^{*}=\delta u_{1/2}^{'}-\beta\bar{\tau}_{x}^{*}\Big|_{0}\label{eq:du_half} +.. math:: :label: eq:du_half + + \delta u_{1/2}^{*}=\delta u_{1/2}^{'}-\beta\bar{\tau}_{x}^{*}\Big|_{0} where :math:`\delta u_{1/2}^{'}`, :math:`\beta` are available quantities. Furthermore, @@ -5294,26 +5125,28 @@ Approximating :math:`\left(\frac{\partial\delta u^{*}}{\partial u_{0}^{*}}{z_{1/2}}`, and assuming that :math:`u_{0}=0` the previous equation becomes -.. math:: \bar{\tau}_{x}^{*}\Big|_{0}=\left({\cal I}_{1}-{\cal E}_{1}\right)\tau_{x}^{n}\Big|_{0}+{\cal I}_{1}K_{u}\Big|_{0}\frac{\delta u_{1/2}^{*}}{z_{1/2}}.\label{eq:tau_zero} +.. math:: :label: eq:tau_zero + + \bar{\tau}_{x}^{*}\Big|_{0}=\left({\cal I}_{1}-{\cal E}_{1}\right)\tau_{x}^{n}\Big|_{0}+{\cal I}_{1}K_{u}\Big|_{0}\frac{\delta u_{1/2}^{*}}{z_{1/2}}. -From (`[eq:du_half] <#eq:du_half>`__), -(`[eq:tau_zero] <#eq:tau_zero>`__) the following expression for the +From :eq:`eq:du_half`, +:eq:`eq:tau_zero` the following expression for the implicit surface stress is obtained -.. math:: +.. math:: :label: eq:imp_tau \bar{\tau}_{x}^{*}\Big|_{0}=\frac{\left({\cal I}_{1}-{\cal E}_{1}\right)\tau_{x}^{n}\Big|_{0}+{\cal I}_{1}(K_{u}\Big|_{0}/z_{1/2})\delta - u_{1/2}^{'}}{1+{\cal I}_{1}(K_{u}\Big|_{0}/z_{1/2})\beta}.\label{eq:imp_tau} + u_{1/2}^{'}}{1+{\cal I}_{1}(K_{u}\Big|_{0}/z_{1/2})\beta}. Then, :math:`\delta u_{1/2}^{*}` can be computed from -(`[eq:imp_tau] <#eq:imp_tau>`__) and (`[eq:du_half] <#eq:du_half>`__). +:eq:`eq:imp_tau` and :eq:`eq:du_half`. Similarly the implicit surface stress for :math:`u` which corresponds to -the 2nd stage (`[eq:sppf_inc2] <#eq:sppf_inc2>`__) will be +the 2nd stage :eq:`eq:sppf_inc2` will be -.. math:: +.. math:: :label: eq:imp_tau2 \bar{\tau}_{x}^{n+1}\Big|_{0}=\frac{\left({\cal I}_{2}-{\cal E}_{2}\right)\tau_{x}^{*}\Big|_{0}+{\cal I}_{2}(K_{u}\Big|_{0}/z_{1/2})\delta - u_{1/2}^{'}}{1+{\cal I}_{2}(K_{u}\Big|_{0}/z_{1/2})\beta}.\label{eq:imp_tau2} + u_{1/2}^{'}}{1+{\cal I}_{2}(K_{u}\Big|_{0}/z_{1/2})\beta}. In the same way :math:`\bar{\tau}_{y}^{*}\Big|_{0}`, :math:`\bar{\tau}_{y}^{n+1}\Big|_{0}` can be derived. @@ -5330,23 +5163,23 @@ works well in practice. The boundary conditions for the scalar variables, i.e. the surface scalar fluxes for the new scheme are obtained using the original implicit surface exchange calculation. This is applied as follows. Consider the equivalent discrete form of -(`[eq:sppf_bl1] <#eq:sppf_bl1>`__) for the thermodynamic variable +:eq:`eq:sppf_bl1` for the thermodynamic variable :math:`X`: -.. math:: +.. math:: :label: eq:dX_star \frac{\delta X^{*}}{\Delta t} - =\frac{\partial\overline{F}^{*}}{\partial z}+\left({\cal I}_{1}-{\cal E}_{1}\right)S\label{eq:dX_star} + =\frac{\partial\overline{F}^{*}}{\partial z}+\left({\cal I}_{1}-{\cal E}_{1}\right)S Considering that, -.. math:: +.. math:: :label: eq:dX_star2 \overline{F}^{*}={\cal I}_{1}F^{*}-{\cal E}_{1}F^{n},\qquad - F^{*}=F^{n}+K_{X}\frac{\partial\delta X^{*}}{\partial z}\label{eq:dX_star2} + F^{*}=F^{n}+K_{X}\frac{\partial\delta X^{*}}{\partial z} -(`[eq:dX_star] <#eq:dX_star>`__) would re-produce -(`[eq:sppf_inc1] <#eq:sppf_inc1>`__), which is re-written below, +:eq:`eq:dX_star` would re-produce +:eq:`eq:sppf_inc1`, which is re-written below, .. math:: \frac{\delta X^{*}}{\Delta t}=({\cal I}_{1}-{\cal E}_{1})\left(\frac{\partial F^{n}}{\partial z}+S\right)+{\cal I}_{1}\frac{\partial}{\partial z}\left(K_{X}\frac{\partial\delta X^{*}}{\partial z}\right) @@ -5364,7 +5197,9 @@ and thus the following discretization is obtained, on or, -.. math:: A_{k}\delta X_{k+1}^{*}+B_{k}\delta X_{k}^{*}+C_{k}\delta X_{k-1}^{*}=\left({\cal I}_{1}-{\cal E}_{1}\right)\left(\frac{F_{k+1/2}^{n}-F_{k-1/2}^{n}}{z_{k+1/2}-z_{k-1/2}}+S_{k}\right),\quad k=2,\ldots,L-1\label{eq:dX_disc} +.. math:: :label: eq:dX_disc + + A_{k}\delta X_{k+1}^{*}+B_{k}\delta X_{k}^{*}+C_{k}\delta X_{k-1}^{*}=\left({\cal I}_{1}-{\cal E}_{1}\right)\left(\frac{F_{k+1/2}^{n}-F_{k-1/2}^{n}}{z_{k+1/2}-z_{k-1/2}}+S_{k}\right),\quad k=2,\ldots,L-1 where, @@ -5372,23 +5207,27 @@ where, The discrete equation for the top level, :math:`k=L`, will be: -.. math:: B_{L}\delta X_{L}^{*}+C_{L}\delta X_{L-1}^{*}=\left({\cal I}_{1}-{\cal E}_{1}\right)\left(\frac{-F_{L-1/2}^{n}}{z_{L+1/2}-z_{L-1/2}}+S_{L}\right),\label{eq:dX_disc_top} +.. math:: :label: eq:dX_disc_top + + B_{L}\delta X_{L}^{*}+C_{L}\delta X_{L-1}^{*}=\left({\cal I}_{1}-{\cal E}_{1}\right)\left(\frac{-F_{L-1/2}^{n}}{z_{L+1/2}-z_{L-1/2}}+S_{L}\right), where :math:`B_{L}`, :math:`C_{L}` are derived as before setting :math:`A_{L}=0`. -From (`[eq:dX_star] <#eq:dX_star>`__), a bottom interior level +From :eq:`eq:dX_star`, a bottom interior level (:math:`k=1`) discretization is -.. math:: \delta X_{1}^{*}=\frac{\Delta t}{z_{3/2}-0}\left(\overline{F_{3/2}}^{*}-\overline{F_{0}}^{*}\right)+\Delta t\left({\cal I}_{1}-{\cal E}_{1}\right)S_{1}\label{eq:dX1_star} +.. math:: :label: eq:dX1_star + + \delta X_{1}^{*}=\frac{\Delta t}{z_{3/2}-0}\left(\overline{F_{3/2}}^{*}-\overline{F_{0}}^{*}\right)+\Delta t\left({\cal I}_{1}-{\cal E}_{1}\right)S_{1} :math:`F_{0}` is used instead of :math:`F_{1/2}`. The former is computed by the implicit surface scheme. This flux gradient is defined in the same way in the original solver as well. Using -(`[eq:dX_star2] <#eq:dX_star2>`__), (`[eq:dX1_star] <#eq:dX1_star>`__) +:eq:`eq:dX_star2`, :eq:`eq:dX1_star` becomes -.. math:: +.. math:: :label: eq:dX1_star2 \delta X_{1}^{*}=\frac{\Delta t}{z_{3/2}} @@ -5396,11 +5235,12 @@ becomes +\Delta t\left({\cal I}_{1}-{\cal E}_{1}\right) S_{1}+\Delta t{\cal I}_{1}\frac{1}{z_{3/2}} \left(K_{X}\frac{\partial\delta X^{*}}{\partial z}\right)_{3/2} - \label{eq:dX1_star2} where :math:`\overline{F}_{0}^{*}` can be approximated as -.. math:: \overline{F}_{0}^{*}={\cal I}_{1}F_{0}^{*}-{\cal E}_{1}F_{0}^{n}\approx\left({\cal I}_{1}-{\cal E}_{1}\right)F_{JULES}\label{eq:F0_star} +.. math:: :label: eq:F0_star + + \overline{F}_{0}^{*}={\cal I}_{1}F_{0}^{*}-{\cal E}_{1}F_{0}^{n}\approx\left({\cal I}_{1}-{\cal E}_{1}\right)F_{JULES} where, :math:`F_{JULES}` is the implicit flux calculated by the *implicit surface scheme using the original implicit algorithm*. @@ -5415,10 +5255,10 @@ Finalising, the discrete equations for the bottom level will be or, -.. math:: +.. math:: :label: eq:dX_bottom A_{1}\delta X_{2}^{*}+B_{1}\delta X_{1}^{*}=\Delta t - \left({\cal I}_{1}-{\cal E}_{1}\right)\left(\frac{F_{3/2}^{n}-F_{JULES}}{z_{3/2}}+S_{1}\right)\label{eq:dX_bottom} + \left({\cal I}_{1}-{\cal E}_{1}\right)\left(\frac{F_{3/2}^{n}-F_{JULES}}{z_{3/2}}+S_{1}\right) where, @@ -5427,19 +5267,25 @@ where, A_{1}=-{\cal I}_{1}\Delta t\frac{K_{X}\Big|_{3/2}}{z_{3/2}(z_{2}-z_{1})}, \quad B_{1}=1-A_{1}. -Equations (`[eq:dX_disc] <#eq:dX_disc>`__), -(`[eq:dX_disc_top] <#eq:dX_disc_top>`__) and -(`[eq:dX_bottom] <#eq:dX_bottom>`__) define a tridiagonal system of +Equations :eq:`eq:dX_disc`, +:eq:`eq:dX_disc_top` and +:eq:`eq:dX_bottom` define a tridiagonal system of equations for :math:`\delta X^{*}`. Similarly the corresponding discrete equations for -(`[eq:sppf_inc2] <#eq:sppf_inc2>`__) will be: +:eq:`eq:sppf_inc2` will be: + +.. math:: :label: eq:dXtop_np1 + + B_{L}^{'}\delta X_{L}^{n+1}+C_{L}^{'}\delta X_{L-1}^{n+1}=\left({\cal I}_{2}-{\cal E}_{2}\right)\left(\frac{-F_{L-1/2}^{*}}{z_{L+1/2}-z_{L-1/2}}+S_{L}\right), -.. math:: B_{L}^{'}\delta X_{L}^{n+1}+C_{L}^{'}\delta X_{L-1}^{n+1}=\left({\cal I}_{2}-{\cal E}_{2}\right)\left(\frac{-F_{L-1/2}^{*}}{z_{L+1/2}-z_{L-1/2}}+S_{L}\right),\label{eq:dXtop_np1} +.. math:: :label: eq:dXk_np1 -.. math:: A_{k}^{'}\delta X_{k+1}^{n+1}+B_{k}^{'}\delta X_{k}^{n+1}+C_{k}^{'}\delta X_{k-1}^{n+1}=\left({\cal I}_{2}-{\cal E}_{2}\right)\left(\frac{F_{k+1/2}^{*}-F_{k-1/2}^{*}}{z_{k+1/2}-z_{k-1/2}}+S_{k}\right),\quad k=L-1,\ldots,2\label{eq:dXk_np1} + A_{k}^{'}\delta X_{k+1}^{n+1}+B_{k}^{'}\delta X_{k}^{n+1}+C_{k}^{'}\delta X_{k-1}^{n+1}=\left({\cal I}_{2}-{\cal E}_{2}\right)\left(\frac{F_{k+1/2}^{*}-F_{k-1/2}^{*}}{z_{k+1/2}-z_{k-1/2}}+S_{k}\right),\quad k=L-1,\ldots,2 -.. math:: A_{1}^{'}\delta X_{2}^{n+1}+B_{1}^{'}\delta X_{1}^{n+1}=\left({\cal I}_{2}-{\cal E}_{2}\right)\left(\frac{F_{3/2}^{*}-F_{JULES}}{z_{3/2}}+S_{1}\right)\label{eq:dX1_np1} +.. math:: :label: eq:dX1_np1 + + A_{1}^{'}\delta X_{2}^{n+1}+B_{1}^{'}\delta X_{1}^{n+1}=\left({\cal I}_{2}-{\cal E}_{2}\right)\left(\frac{F_{3/2}^{*}-F_{JULES}}{z_{3/2}}+S_{1}\right) where, @@ -5451,10 +5297,12 @@ for :math:`k=L,\ldots,2,\quad A_{L}=0`. and the approximation -.. math:: \overline{F}_{0}^{n+1}={\cal I}_{2}F_{0}^{n+1}-{\cal E}_{2}F_{0}^{n}\approx\left({\cal I}_{2}-{\cal E}_{2}\right)F_{JULES}\label{eq:F0_np1} +.. math:: :label: eq:F0_np1 + + \overline{F}_{0}^{n+1}={\cal I}_{2}F_{0}^{n+1}-{\cal E}_{2}F_{0}^{n}\approx\left({\cal I}_{2}-{\cal E}_{2}\right)F_{JULES} has taken place. The same flux :math:`F_{JULES}` will be used for both -(`[eq:F0_star] <#eq:F0_star>`__) and (`[eq:F0_np1] <#eq:F0_np1>`__) and +:eq:`eq:F0_star` and :eq:`eq:F0_np1` and therefore needs to be computed only once, when the 1st or predictor stage is computed, i.e. :math:`X^{*}`. Briefly the following calculations take place for the scalar variables: @@ -5464,7 +5312,7 @@ calculations take place for the scalar variables: :header-rows: 1 * - CALL bdy_impl3(): - - set up coefficients for (`[eq:dX_disc_top] <#eq:dX_disc_top>`__), (`[eq:dX_disc] <#eq:dX_disc>`__) and do a downward sweep; + - set up coefficients for :eq:`eq:dX_disc_top`, :eq:`eq:dX_disc` and do a downward sweep; * - - do a downward sweep using the original implicit scheme to @@ -5479,7 +5327,7 @@ calculations take place for the scalar variables: - using original surface implicit solver; * - CALL bdy_impl4(): - - set up (`[eq:dX_bottom] <#eq:dX_bottom>`__) and complete downward sweep; + - set up :eq:`eq:dX_bottom` and complete downward sweep; * - - back substitute to compute implicit correction :math:`\delta X^{*}`; @@ -5488,7 +5336,7 @@ calculations take place for the scalar variables: - compute explicit flux :math:`F^*=F^n+K_X\frac{\partial \delta X^*}{\partial z}`; * - - - set up coefficients for (`[eq:dXtop_np1] <#eq:dXtop_np1>`__), (`[eq:dXk_np1] <#eq:dXk_np1>`__), (`[eq:dX1_np1] <#eq:dX1_np1>`__) and + - set up coefficients for :eq:`eq:dXtop_np1`, :eq:`eq:dXk_np1`, :eq:`eq:dX1_np1` and * - - do a downward sweep; @@ -5511,11 +5359,11 @@ time averaging the stresses at :math:`t^{n}` and :math:`t^{n+1}`, where vertical diffusion equation being solved. The averaging which takes place for the zonal wind component stress is: -.. math:: +.. math:: :label: eq:taux_tot \overline{\tau_{x}}^{n+1}\equiv(1-\gamma)\tau_{x}^{n}+\gamma\tau_{x}^{n+1} =\tau_{x}^{n}+\gamma - K_{u}\frac{\partial\delta u^{n+1}}{\partial z}\label{eq:taux_tot} + K_{u}\frac{\partial\delta u^{n+1}}{\partial z} where :math:`\delta u^{n+1}=u^{n+1}-u^{n}`. Likewise, :math:`\tau_{y}` and the scalar fluxes are derived. @@ -5561,23 +5409,23 @@ modified version of the flux formulae (78), (79) of **1st sweep:** -.. math:: +.. math:: :label: eq:FTLstar \begin{aligned} - \frac{\overline{H^{*}}}{c_{p}} & = & \frac{(1+\beta B_{2})[\gamma_{2}F_{T}^{n}+A_{1}\delta{T'}_{1}+A_{2}\delta{Q'}_{1}]-\beta A_{2}[\gamma_{2}F_{Q}^{n}+B_{1}\delta{T'}_{1}+B_{2}\delta{Q'}_{1}]}{(1+\beta A_{1})(1+\beta B_{2})-\beta^{2}A_{2}B_{1}}\label{eq:FTLstar} + \frac{\overline{H^{*}}}{c_{p}} & = & \frac{(1+\beta B_{2})[\gamma_{2}F_{T}^{n}+A_{1}\delta{T'}_{1}+A_{2}\delta{Q'}_{1}]-\beta A_{2}[\gamma_{2}F_{Q}^{n}+B_{1}\delta{T'}_{1}+B_{2}\delta{Q'}_{1}]}{(1+\beta A_{1})(1+\beta B_{2})-\beta^{2}A_{2}B_{1}} \end{aligned} -.. math:: +.. math:: :label: eq:FQWstar \begin{aligned} - \overline{E^{*}} & = & \frac{(1+\beta A_{1})[\gamma_{2}F_{Q}^{n}+B_{1}\delta{T'}_{1}+B_{2}\delta{Q'}_{1}]-\beta B_{1}[\gamma_{2}F_{T}^{n}+A_{1}\delta{T'}_{1}+A_{2}\delta{Q'}_{1}]}{(1+\beta A_{1})(1+\beta B_{2})-\beta^{2}A_{2}B_{1}}\label{eq:FQWstar} + \overline{E^{*}} & = & \frac{(1+\beta A_{1})[\gamma_{2}F_{Q}^{n}+B_{1}\delta{T'}_{1}+B_{2}\delta{Q'}_{1}]-\beta B_{1}[\gamma_{2}F_{T}^{n}+A_{1}\delta{T'}_{1}+A_{2}\delta{Q'}_{1}]}{(1+\beta A_{1})(1+\beta B_{2})-\beta^{2}A_{2}B_{1}} \end{aligned} where, :math:`F_{T}^{n}`, :math:`F_{Q}^{n}` denote the surface explicit fluxes, :math:`\gamma_{2}={\cal I}_{1}-{\cal E}_{1}` and the coefficients :math:`A_{1},\; A_{2},B_{1},\; B_{2}` are given by -.. math:: +.. math:: :label: ab_coeffs \begin{equation} A_1=-\gamma_1\sum_j \nu_j RK_{PMj} @@ -5595,7 +5443,6 @@ coefficients :math:`A_{1},\; A_{2},B_{1},\; B_{2}` are given by B_2=-\gamma_1\sum_j \nu_j RK_{PMj} \psi_j[c_pRK_H(1)_j+A_{*j}]. \end{equation} - \label{ab_coeffs} but with :math:`\gamma_{1}={\cal I}_{1}`. Here :math:`RK_H(1) =\rho C_H U_1`, @@ -5619,7 +5466,7 @@ is the soil conductivity, :math:`\Delta z_s` and :math:`T_s` are the thickness and temperature of the surface soil layer, :math:`C_c` is the canopy heat capacity, :math:`f_a` is the saturated fraction of the tile and :math:`g_s` is the surface conductance. To derive -(`[eq:FTLstar] <#eq:FTLstar>`__), (`[eq:FQWstar] <#eq:FQWstar>`__), the +:eq:`eq:FTLstar`, :eq:`eq:FQWstar`, the time-weighted level 1 :math:`T` and :math:`Q` consistent with the discrete equations of the new scheme is written as follows: @@ -5638,25 +5485,25 @@ flux for :math:`H` is derived: and similarly :math:`E_{j}^{*}`. From these, the tile flux equations -(`[eq:FTLstar] <#eq:FTLstar>`__), (`[eq:FQWstar] <#eq:FQWstar>`__) can +:eq:`eq:FTLstar`, :eq:`eq:FQWstar` can be obtained. **2nd sweep:** -.. math:: +.. math:: :label: eq:FTLnp1 \begin{aligned} - \frac{\overline{H^{n+1}}}{c_{p}} & = & \frac{(1+\beta B_{2})[\xi_{2}F_{T}^{*}+A_{1}\delta{T'}_{1}+A_{2}\delta{Q'}_{1}]-\beta A_{2}[\xi_{2}F_{Q}^{*}+B_{1}\delta{T'}_{1}+B_{2}\delta{Q'}_{1}]}{(1+\beta A_{1})(1+\beta B_{2})-\beta^{2}A_{2}B_{1}}\label{eq:FTLnp1} + \frac{\overline{H^{n+1}}}{c_{p}} & = & \frac{(1+\beta B_{2})[\xi_{2}F_{T}^{*}+A_{1}\delta{T'}_{1}+A_{2}\delta{Q'}_{1}]-\beta A_{2}[\xi_{2}F_{Q}^{*}+B_{1}\delta{T'}_{1}+B_{2}\delta{Q'}_{1}]}{(1+\beta A_{1})(1+\beta B_{2})-\beta^{2}A_{2}B_{1}} \end{aligned} -.. math:: +.. math:: :label: eq:FQWnp1 \begin{aligned} - \overline{E^{n+1}} & = & \frac{(1+\beta A_{1})[\xi_{2}F_{Q}^{*}+B_{1}\delta{T'}_{1}+B_{2}\delta{Q'}_{1}]-\beta B_{1}[\xi_{2}F_{T}^{*}+A_{1}\delta{T'}_{1}+A_{2}\delta{Q'}_{1}]}{(1+\beta A_{1})(1+\beta B_{2})-\beta^{2}A_{2}B_{1}}\label{eq:FQWnp1} + \overline{E^{n+1}} & = & \frac{(1+\beta A_{1})[\xi_{2}F_{Q}^{*}+B_{1}\delta{T'}_{1}+B_{2}\delta{Q'}_{1}]-\beta B_{1}[\xi_{2}F_{T}^{*}+A_{1}\delta{T'}_{1}+A_{2}\delta{Q'}_{1}]}{(1+\beta A_{1})(1+\beta B_{2})-\beta^{2}A_{2}B_{1}} \end{aligned} where, the coefficients :math:`A_{1},\; A_{2},B_{1},\; B_{2}` are given -by (`[ab_coeffs] <#ab_coeffs>`__) but with +by :eq:`ab_coeffs` but with :math:`\gamma_{1}={\cal I}_{2}`. The above formulae are derived as explained earlier. The definitions @@ -5668,8 +5515,8 @@ model state :math:`{(T}_{1}^{*},Q_{1}^{*})`. They are the equivalent of the explicit fluxes :math:`F_{T}^{n}\equiv{\displaystyle \frac{H^{n}}{c_{p}}}`, :math:`F_{Q}^{n}\equiv E^{n}`. However, they are not equal to the left -hand-side of (`[eq:FTLstar] <#eq:FTLstar>`__), -(`[eq:FQWstar] <#eq:FQWstar>`__). Both are connected by a linear +hand-side of :eq:`eq:FTLstar`, +:eq:`eq:FQWstar`. Both are connected by a linear relationship, which is simply the definition of the time-weighted averaging consistent with the new scheme: @@ -5761,23 +5608,21 @@ available, :math:`T_*` is reset to :math:`T_m` by adding an increment corresponding to a snowmelt heat flux -.. math:: +.. math:: :label: eq:Sm S_m = - [(c_p + L_sD)RK_H(1) + A_*]{\Delta T_*\over L_f}. - \label{eq:Sm} The maximum melt rate that can be sustained over a timestep :math:`\Delta t`, however, is :math:`S/\Delta t-E`, giving -.. math:: +.. math:: :label: eq:dTmax \Delta T_*={L_f(S/\Delta t-E) \over (c_p+L_cD)RK_H(1) + A_*}. - \label{eq:dTmax} :math:`\Delta T_*` is set to the smaller of the values given by -Equations (`[eq:Sm] <#eq:Sm>`__) and (`[eq:dTmax] <#eq:dTmax>`__), and +Equations :eq:`eq:Sm` and :eq:`eq:dTmax`, and the surface energy balance is repartitioned by adding increments .. math:: \Delta H = c_pRK_H(1)\Delta T_* @@ -5801,7 +5646,7 @@ The same method is used as in section `9.1.2 <#sec:impsolve>`__ to form two independent tridiagonal systems of linear equations that relate the increments to momentum, temperature and humidity to the surface fluxes. The ‘downward sweep’ elimination procedure still takes place to obtain -equation (`[eq:du_half] <#eq:du_half>`__) and a corresponding equation +equation :eq:`eq:du_half` and a corresponding equation for the increments to the scalar variables at the bottom model level .. math:: \delta X_{1/2}^{*}=\delta X_{1/2}^{'}-\beta_X\frac{\bar{H}_\star}{C_p} @@ -5844,11 +5689,11 @@ conditions that should not lead to a strong thermal forecast. This sensitivity of boundary layer turbulence is already included in the parametrization of non-local momentum fluxes (see section `5.4 <#sec:ngstress>`__) through the stability dependence in -(`[tau_nl] <#tau_nl>`__) that can be written as +:eq:`tau_nl` that can be written as .. math:: f_{stab} = - \frac{a_{stab} z_{\rm h}/L }{1 - a_{stab} z_{\rm h}/L} -for the Obhukov length, (`[1.1.4] <#1.1.4>`__), :math:`<0` (i.e., +for the Obhukov length, :eq:`1.1.4`, :math:`<0` (i.e., unstable boundary layers) and the empirical constant :math:`a_{stab} = 1.5`. This function tends to unity as :math:`L` decreases in magnitude (i.e., surface heating increases and wind stress @@ -5865,20 +5710,19 @@ over 3 second intervals. In the boundary layer the strength of gusts is proportional to the standard deviation of the horizontal wind, :math:`\sigma_u`, so that -.. math:: +.. math:: :label: windgust U_{gust} = U_{10m} + W_{1D} \, \sigma_u \, \frac{1}{k} \, {\rm log}\left( \frac{5 \, e^{k \, c_{\rm ugn}} + z_{0m(eff)} } {5 + z_{0m(eff)}} \right) - \label{windgust} The factor :math:`W_{1D}` is included only in the scale-dependent version of the diagnostic (stash 3,515) to allow for the larger scales of boundary layer turbulence that are resolved (and so are already included in :math:`U_{10m}`). The lowest grid-level value of -:math:`W_{1D}`, from (`[eq-tanh] <#eq-tanh>`__), is used, noting that +:math:`W_{1D}`, from :eq:`eq-tanh`, is used, noting that :math:`W_{1D}` is constant within the boundary layer. The constant -:math:`c_{\rm ugn}` in (`[windgust] <#windgust>`__) is determined from +:math:`c_{\rm ugn}` in :eq:`windgust` is determined from universal turbulence spectra for a 25% exceeding probability of the three-second wind gust (`Beljaars (1987)`_). It is included through a function that includes the effective roughness @@ -5917,16 +5761,15 @@ A substantial part of the turbulent flux is parametrized in both the UM’s first order closure and closures involving TKE, :math:`e`, through a simple down-gradient diffusion term. An estimate of subgrid TKE can then be made by equating the UM’s diffusion coefficient, -(`[klnl] <#klnl>`__), with that from a typical TKE-closure, i.e. +:eq:`klnl`, with that from a typical TKE-closure, i.e. -.. math:: +.. math:: :label: tke_closure K_m = l \sqrt{e} - \label{tke_closure} where :math:`l` is a length scale. Initially it was thought to diagnose :math:`e` by approximating :math:`l` as the mixing length in -(`[kmlocal] <#kmlocal>`__) but closer inspection reveals that many TKE +:eq:`kmlocal` but closer inspection reveals that many TKE closures have diagnostic relationships for :math:`l` that involve the TKE itself! A common one for stable boundary layers is :math:`l_{st} \sim \sqrt{e} / N`, where :math:`N` is the Brunt-Vaisala @@ -5935,7 +5778,7 @@ frequency. `Suselj et al. (2012)`_, for example, also take :math:`\tau_{un}` is a turbulence timescale that they take as a constant 400 seconds. These they combine through :math:`l^{-1}= l_{un}^{-1} + l_{st}^{-1} = e^{-1/2}( \tau_{un}^{-1} + \tau_{st}^{-1}) \equiv -e^{-1/2}\tau_{turb}^{-1}` and (`[tke_closure] <#tke_closure>`__) +e^{-1/2}\tau_{turb}^{-1}` and :eq:`tke_closure` becomes: .. math:: K_m = \tau_{turb} e @@ -5948,29 +5791,27 @@ Basic boundary layer scaling (e.g., Figure 4 of that :math:`\overline{w'^2}` from a variety of convective boundary layer LES and observations nicely follows the relationship -.. math:: +.. math:: :label: w2_scaling \overline{w'^2} = c_{w2} w_*^2 f(z') - \label{w2_scaling} where :math:`w_*` is the convective velocity scale and :math:`f` is a shape function within the boundary layer (:math:`z'=z/z_{\rm h}`). The shape of this function is very similar to that used in the UM for -:math:`K_m^{\rm surf}` in (`[kmsurf] <#kmsurf>`__). We now assume we can -generalise (`[w2_scaling] <#w2_scaling>`__) by replacing :math:`w_*` +:math:`K_m^{\rm surf}` in :eq:`kmsurf`. We now assume we can +generalise :eq:`w2_scaling` by replacing :math:`w_*` with :math:`w_m` (this really ought to be checked against neutral boundary layer LES but hasn’t yet been). Setting :math:`f(z')=z' -(1-z')^2` in (`[w2_scaling] <#w2_scaling>`__) and comparing with Fig.4 +(1-z')^2` in :eq:`w2_scaling` and comparing with Fig.4 of `Holtslag and Moeng (1991)`_ gives :math:`c_{w2}= 2.66 / C_{ws}^{2/3}` (i.e., a constant of 2.66 gives the maximum in :math:`\overline{w'^2}/w_*^2` at around the observed value of 0.4) so -that we can generalise (`[w2_scaling] <#w2_scaling>`__) to +that we can generalise :eq:`w2_scaling` to -.. math:: +.. math:: :label: gen_w2_scaling \overline{w'^2} = \frac{2.66 }{C_{ws}^{2/3}} \, w_m^2\, f(z') - \label{gen_w2_scaling} where the mixed layer expression for :math:`w_m` is used. @@ -5982,13 +5823,12 @@ from analysis of the scalar flux budget that where :math:`\tau_{turb}` is a return to isotropy timescale. Ignoring the non-gradient parametrization in the UM, it follows that -.. math:: +.. math:: :label: bl_scaling K_h^{\rm surf}= \frac{\tau_{turb}}{2} \, \overline{w'^2} - \label{bl_scaling} -Combining (`[bl_scaling] <#bl_scaling>`__) with -(`[gen_w2_scaling] <#gen_w2_scaling>`__) and (`[kmsurf] <#kmsurf>`__), +Combining :eq:`bl_scaling` with +:eq:`gen_w2_scaling` and :eq:`kmsurf`, and subsuming the Prandtl number into the other constants, for surface-driven boundary layer mixing we can write: @@ -6006,17 +5846,16 @@ One is to assume :math:`\tau_{\rm SBL}=0.7/N` as the timescale for stable boundary layers and combine all these timescales following `Suselj et al. (2012)`_) to give: -.. math:: +.. math:: :label: tke_diag e = K_m \tau_{turb}^{-1} - \label{tke_diag} where :math:`\tau_{turb}^{-1} = MAX[ \tau_{\rm surf}^{-1},\tau_{\rm Sc}^{-1}] + \tau_{\rm SBL}^{-1}`. -Note that (`[tke_diag] <#tke_diag>`__) gives :math:`\overline{w'^2}`, +Note that :eq:`tke_diag` gives :math:`\overline{w'^2}`, rather than TKE. As a simple fix to improve the near-surface TKE in convective boundary layers, where the horizontal wind variability often -dominates, the value of :math:`e` given by (`[tke_diag] <#tke_diag>`__) +dominates, the value of :math:`e` given by :eq:`tke_diag` at the level of the maximum in :math:`K_m^{\rm surf}` is copied to all levels below that height. @@ -6024,10 +5863,9 @@ The second method diagnoses TKE for the local scheme following the Met Office LEM and MONC, simplifying and parametrizing the terms in the TKE budget to give -.. math:: +.. math:: :label: tke_diag_loc e_{loc}^{3/2} = \lambda S^2 K_m (1-Ri/Pr)/C_{e} - \label{tke_diag_loc} where :math:`C_{e}=A_{2N}^{3/2}`. Initial tests found that the MONC value of :math:`A_{2N}=0.23` gave rather large values of :math:`e_{loc}` @@ -6039,13 +5877,12 @@ been attempted. The total non-local TKE is computed by adding the TKE from each non-local component, as is done for the diffusion coefficients, i.e., -.. math:: +.. math:: :label: tke_diag_nl e_{nl} = \frac{3}{2} \left( \frac{K_m^{\rm surf}}{\tau_{\rm surf}} + \frac{K_m^{\rm Sc}}{\tau_{\rm Sc}} \right) - \label{tke_diag_nl} -The factor of :math:`3/2` in (`[tke_diag_nl] <#tke_diag_nl>`__) arises +The factor of :math:`3/2` in :eq:`tke_diag_nl` arises because we are really diagnosing :math:`\overline{w'^2}` and so here we make the assumption of isotropic turbulence to extend this to TKE. As before, we do also make the simple fix to improve the near-surface TKE @@ -6117,21 +5954,19 @@ generation, :math:`V_{\rm rad}` for cloud-top radiative cooling and :math:`V_{\rm br}` for buoyancy reversal). The velocity scales can be written -.. math:: +.. math:: :label: vsurf V_{\rm heat}^3= z_{\rm ml}\! \left( (2-\zeta_s)\zeta_s \overline{w'b}_S+ (1-\zeta_s)^2 [\overline{w'b'}_S]_{\rm sat}\right) - \label{vsurf} -.. math:: +.. math:: :label: vrad V_{\rm rad}^3= z_{\rm ml}\Delta_F\, g \, \left( \beta_T \zeta_r^2 + \tilde{\beta_T} (1-\zeta_r^2) \right) - \label{vrad} -.. math:: +.. math:: :label: vbr V_{\rm br}^3= A_{\rm br}\chi_s^2 \, \mbox{max}\left[0,-\delta b\right] \, \Delta b ^{1/2} - \, z_c^{3/2} \, C_{fac} \label{vbr} + \, z_c^{3/2} \, C_{fac} Here, @@ -6151,7 +5986,7 @@ calculation of :math:`V_{\rm rad}` as it is assumed the radiative cooling will occur predominantly in cloudy air. To allow for a feedback in the presence of buoyancy reversal, the parameter :math:`Br` is included in :math:`\zeta_r` and :math:`\tilde{\alpha_t}` (in -(`[we_parm] <#we_parm>`__)). It is given in terms of the +:eq:`we_parm`). It is given in terms of the `Siems et al. (1990)`_ parameter, :math:`D = \chi_s \delta b/\Delta b` and constrained by :math:`0< Br = 10 D < 1`. This gives a linear ramp for this feedback between regimes @@ -6171,13 +6006,12 @@ stratocumulus over cumulus regime. The cloud-fraction weighted cloud depth, :math:`\tilde{z_c}`, is calculated as -.. math:: +.. math:: :label: zcld_calc \tilde{z_c} = \sum_{k=1}^{NTML+1} \left( {C_F}_k \frac{\Delta_{k+\frac{1}{2}} z}{2} + \mbox{min}\left[C_F^l\frac{\Delta_{k+\frac{1}{2}} z}{2}, \frac{q_{\ell}}{\gamma_{q_{\ell}}} \right] + \mbox{min}\left[C_F^f\frac{\Delta_{k+\frac{1}{2}} z}{2}, \frac{q_f}{\gamma_{q_f}} \right] \right) - \label{zcld_calc} where :math:`C_F` is the cloud fraction, made up of liquid (:math:`C_F^l`) and frozen (:math:`C_F^f`) water parts. The adiabatic @@ -6198,10 +6032,10 @@ approximated as where :math:`\gamma_{T_L} = -(g/c_p)+ \gamma_{\theta_{\ell}}` and -:math:`\gamma_{\theta_{\ell}}` is given by (`[gradadj] <#gradadj>`__) +:math:`\gamma_{\theta_{\ell}}` is given by :eq:`gradadj` within the mixed layer (zero above). Optionally (and currently implemented as standard) :math:`\gamma_{q_f}` can be set to zero in -which case the third term in (`[zcld_calc] <#zcld_calc>`__) is set to +which case the third term in :eq:`zcld_calc` is set to zero. In the 8A scheme, the cloud depth, :math:`z_c`, is calculated as the sum @@ -6220,19 +6054,18 @@ grid-level based calculation: +\Delta_{k_b-\frac{1}{2}}z }{2} C_F^l, \frac{ q_{\ell}}{ \gamma_{q_{\ell}} } \right]/C_F -.. math:: +.. math:: :label: zc_calc \left. \hspace{2.4cm} + \mbox{min}\left[ \frac{ \Delta_{k_b+\frac{1}{2}}z +\Delta_{k_b-\frac{1}{2}}z }{2} C_F^f, \frac{ q_f }{ \gamma_{q_f} } \right]/C_F \right. - \label{zc_calc} When :math:`\gamma_{q_f}` is set to zero (currently as standard) the -last term in (`[zc_calc] <#zc_calc>`__) is given by +last term in :eq:`zc_calc` is given by :math:`(\Delta_{k_b+\frac{1}{2}}z+\Delta_{k_b-\frac{1}{2}}z)C_F^f/(2C_F)`. -Note that, if :math:`k_b=1` in (`[zc_calc] <#zc_calc>`__), then +Note that, if :math:`k_b=1` in :eq:`zc_calc`, then :math:`\Delta_{k_b-\frac{1}{2}}z` is taken to be zero. The final part of the 8A calculation of :math:`z_c` is to include the depth to which the cloud extends into the inversion grid-level. If a subgrid inversion @@ -6243,7 +6076,7 @@ SC_CFTOL in grid-levels NTML or NTML\ :math:`+1` or the layer is a decoupled layer). If no subgrid inversion has been diagnosed and :math:`C_F(NTML+1)>` SC_CFTOL, then :math:`z_c` is increased by the full depth of layer :math:`NTML+1` if :math:`C_F(NTML)>` SC_CFTOL and using -(`[zc_calc] <#zc_calc>`__) otherwise (and similarly for DSC layers). The +:eq:`zc_calc` otherwise (and similarly for DSC layers). The same ideas are used in the 9C version, extrapolating using the adiabatic water gradient, except that the grid-level from which this extrapolation is made is now not the lowest grid-level with :math:`C_F >` SC_CFTOL but @@ -6267,14 +6100,13 @@ specific heat at constant pressure and T is the temperature; + \tilde{\beta_q} \Delta q_t)` and the buoyancy jump across the inversion is given by -.. math:: +.. math:: :label: dbinv \Delta b = g \, \left( \beta_T \Delta \theta_{\ell}+ \beta_q \Delta q_t + \left( \beta_T \frac{L}{c_p} - \frac{1+c_v}{c_v}\beta_q \right)\Delta q_{\ell}+ \left( \beta_T \frac{L_s}{c_p} - \frac{1+c_v}{c_v}\beta_q \right)\Delta q_f \right) - \label{dbinv} The empirical constant :math:`A_{\rm br}= 0.24`. The calculation of :math:`\Delta @@ -6311,7 +6143,7 @@ and similarly for :math:`q_f` (noting that currently The only other explicit account of variable cloud fraction is in -(`[vbr] <#vbr>`__) for which it is assumed that buoyancy reversal can +:eq:`vbr` for which it is assumed that buoyancy reversal can only occur for cloudy air underlying cloud-free air (assuming maximum overlap). Thus, the cloud fraction factor, :math:`C_{fac} = \mbox{max}[ 0.0, -\Delta C_F ]`, where @@ -6324,7 +6156,7 @@ A more complete decomposition is not possible given a cloud scheme in the model :raw-latex:`\cite[]{smith90}` which does not allow discrete identification of in-cloud and out-of-cloud profiles. The cloud-fraction dependence of the radiative generation of turbulence is implicitly -treated in (`[vrad] <#vrad>`__) simply by assuming the grid-box mean +treated in :eq:`vrad` simply by assuming the grid-box mean radiative flux divergence, :math:`\Delta_F`, occurs solely in the cloudy air. @@ -6366,11 +6198,10 @@ somewhat independently from the cloud-top height. In the 9B version, :math:`\Delta_F` is calculated as: -.. math:: +.. math:: :label: ctraddiv \Delta_F= \sum_{k=k_m-1}^{k_m+1} \mbox{max}\left[ - \Delta_{k+\frac{1}{2}} z \, {\cal S}_F(k), \,0 \right] - \label{ctraddiv} where :math:`k_m` is the grid-level with the greatest radiative cooling increment, :math:`{\cal S}_F`, within 2 grid-levels of cloud-top. In the @@ -6387,10 +6218,9 @@ A slightly more accurate estimate of the net divergence can be obtained by assuming the SW and LW radiative fluxes at a given height, :math:`z`, have an exponential shape, dependent on the LWP above :math:`z`, i.e.: -.. math:: +.. math:: :label: eq:explw F_{LW}(z) = \Delta_F^{LW} \exp^{ - \kappa_{LW} \mbox{LWP}(z) } - \label{eq:explw} Then the net divergence can be approximated given the SW flux at the height where :math:`F_{LW}` becomes some small fraction, :math:`A`, of @@ -6405,13 +6235,12 @@ height where :math:`F_{LW}` becomes some small fraction, :math:`A`, of Empirically, see Fig. `9 <#fig:dradts>`__, a reasonable fit to LEM data is obtained with: -.. math:: +.. math:: :label: eq:deltaf_emp \Delta_F\approx \Delta_F^{LW} + 0.35 \Delta_F^{SW} - \label{eq:deltaf_emp} Note that in the 9B scheme :math:`\Delta_F^{LW}` and -:math:`\Delta_F^{SW}` are calculated as in (`[ctraddiv] <#ctraddiv>`__) +:math:`\Delta_F^{SW}` are calculated as in :eq:`ctraddiv` but with the LW and SW increments separately. This change in the calculation of :math:`\Delta_F` is illustrated in @@ -6441,7 +6270,7 @@ therefore zero entrainment and turbulent mixing). :width: 50% The 9C version attempted to remove the grid-dependence implied by the -summation over 3 grid-levels in (`[ctraddiv] <#ctraddiv>`__) as follows: +summation over 3 grid-levels in :eq:`ctraddiv` as follows: #. the search for the level with maximum LW radiative cooling, :math:`k_m`, is restricted to the top half of the mixed layer and no @@ -6455,10 +6284,9 @@ summation over 3 grid-levels in (`[ctraddiv] <#ctraddiv>`__) as follows: #. then, the cloud-top radiative flux change is initially calculated for LW and SW fluxes separately as: - .. math:: + .. math:: :label: ctraddiv_9c \Delta_F= F_{k_m+1} - F_{k_{rb}} - \label{ctraddiv_9c} where the base grid-level for the calculation, :math:`k_{rb}`, is taken to be the higher of the base of the LW radiatively cooled layer @@ -6472,17 +6300,16 @@ summation over 3 grid-levels in (`[ctraddiv] <#ctraddiv>`__) as follows: extrapolating the free-atmospheric flux-gradient downwards. The cloudy contribution is then included in :math:`\Delta F`: - .. math:: + .. math:: :label: ctraddiv_9c_inv \Delta F= \Delta F+ \Delta_{k_m+\frac{3}{2}} F - \Delta_{k_m+\frac{5}{2}} \frac{\Delta_{k_m+\frac{3}{2}} z}{k_m+\frac{1}{2}} F - \label{ctraddiv_9c_inv} #. As at 9B above, the calculations in - (`[ctraddiv_9c] <#ctraddiv_9c>`__) and - (`[ctraddiv_9c_inv] <#ctraddiv_9c_inv>`__) are performed separately + :eq:`ctraddiv_9c` and + :eq:`ctraddiv_9c_inv` are performed separately for LW and SW radiation before the two are combined using the - empirical relationship in (`[eq:deltaf_emp] <#eq:deltaf_emp>`__) + empirical relationship in :eq:`eq:deltaf_emp` Further single column model tests with fine vertical resolution have shown the above calculations can still fail to accurately measure the @@ -6517,10 +6344,9 @@ Appendix: Derivation and definitions of the buoyancy parameters Buoyancy is measured by the virtual temperature -.. math:: +.. math:: :label: Tv T_v = T(1 + c_v q_v - q_{\ell}- q_f) = T V_{fac} - \label{Tv} where :math:`c_v=(1/\epsilon) -1` and :math:`\epsilon` is the ratio of the molecular weights of water vapour and dry air (i.e., :math:`\epsilon @@ -6585,7 +6411,7 @@ Note that here :math:`\tilde{\beta_T}` and :math:`\tilde{\beta_q}` are strictly *in*-cloud parameters, while their definitions in boundary layer code prior to 8A were grid-box mean. Thus, here, any necessary :math:`C_F`-weighting must be included explicitly, as in -(`[eq:wb_cont] <#eq:wb_cont>`__). +:eq:`eq:wb_cont`. In all the above, if :math:`T` is less than the melting point of ice then the latent heat of sublimation, :math:`L_s = L + L_f`, is used in @@ -6607,23 +6433,22 @@ ratios and specific humidities are then defined as .. math:: m_v = \frac{\rho_v}{\rho_y} q_v = \frac{\rho_v}{\rho} When specific quantities are mixed, the turbulent diffusion equations, -(`[cons_eqn_scal] <#cons_eqn_scal>`__) and -(`[cons_eqn_uv] <#cons_eqn_uv>`__), have :math:`\rho` as the wet +:eq:`cons_eqn_scal` and +:eq:`cons_eqn_uv`, have :math:`\rho` as the wet density. This is then consistent with the conservation of globally integrated quantities such as moisture. For example, neglecting spherical geometry for simplicity: -.. math:: +.. math:: :label: moisture_cons \int (\rho_v + \rho_{\ell}+ \rho_{f})\, d\underline{x} = \int \rho (q_v + q_{\ell}+ q_{f}) \,d\underline{x} = \int \rho q_t \, d\underline{x} - \label{moisture_cons} When mixing ratios are used, the momentum equations, -(`[cons_eqn_uv] <#cons_eqn_uv>`__), remain unchanged and the wet density +:eq:`cons_eqn_uv`, remain unchanged and the wet density still appears. This makes the reasonable assumption that all moisture components should be included in the momentum budget. For moisture -conservation, (`[moisture_cons] <#moisture_cons>`__) can be rewritten in +conservation, :eq:`moisture_cons` can be rewritten in terms of mixing ratios as .. math:: @@ -6631,7 +6456,7 @@ terms of mixing ratios as \int (\rho_v + \rho_{\ell}+ \rho_{f})\, d\underline{x} = \int \rho_y (m_v + m_{\ell}+ m_{f}) \,d\underline{x} = \int \rho_y m_t \, d\underline{x} -Thus :math:`\rho` in (`[cons_eqn_scal] <#cons_eqn_scal>`__) is replaced +Thus :math:`\rho` in :eq:`cons_eqn_scal` is replaced with :math:`\rho_y` for :math:`\chi = m_t` and :math:`\theta_{\ell}` when mixing ratios are passed into the boundary layer code. @@ -6661,11 +6486,10 @@ likely to be inappropriate and so the driving level humidity is used Note that :math:`\theta_{\ell}` itself is defined in terms of mixing ratios as: -.. math:: +.. math:: :label: sl_defn \theta_{\ell}= T - \frac{L_c}{c_{pd}} m_{\ell} - \frac{L_c+L_f}{c_{pd}} m_f + \frac{g}{c_{pd}} z - \label{sl_defn} For saturation calculations a version of QSAT is used that is switchable between input specific and mixing ratio variables. The rate of change of @@ -6718,14 +6542,13 @@ the budget of subgrid TKE (SKE). Then the sum of the inputs from resolved kinetic energy, plus that from subgrid buoyancy effects, must equal the dissipation. The SKE budget is -.. math:: +.. math:: :label: ske_budg d SKE/dt = S + T + B + \epsilon_{mol} - \label{ske_budg} where the shear production, S, is essentially the resolved KE dissipation term. Note that the buoyancy term B appears in -(`[ske_budg] <#ske_budg>`__) which indicates that some of the energy +:eq:`ske_budg` which indicates that some of the energy from resolved scale dissipation (i.e. S) should be consumed in doing work against buoyancy (at least in stable BLs) thus leaving less energy to be finally dissipated as heat. Note though that CBLs will generate @@ -6761,7 +6584,7 @@ Appendix: Operational modifications The operational global forecast model has been found to give improved performance on NWP Index parameters when the following modifications to its local :math:`Ri`-based scheme are used. In -(`[asymp_ml] <#asymp_ml>`__), the definition of :math:`\lambda_m` only +:eq:`asymp_ml`, the definition of :math:`\lambda_m` only is altered to .. math:: \lambda_m = \mbox{max}\left[40,\, 0.3 z_{\rm loc}, 2 h_B \right] @@ -6964,7 +6787,7 @@ Appendix: Notation - note: real change (i.e., not necessarily finite-difference) in a parameter * - - - across the capping inversion (see (`[dbinv] <#dbinv>`__) and following text) + - across the capping inversion (see :eq:`dbinv` and following text) .. list-table:: Table :name: table_name @@ -6974,13 +6797,13 @@ Appendix: Notation - * - :math:`\theta_l`, :math:`\theta_{v\ell}` - - thermodynamic variables defined by (`[thetal] <#thetal>`__) and (`[thetavl] <#thetavl>`__) + - thermodynamic variables defined by :eq:`thetal` and :eq:`thetavl` * - :math:`T_v`, :math:`\theta_v` - virtual temperature and potential temperature, * - - - defined by (`[Tv] <#Tv>`__) and in section (`3.1.1 <#sec:parxs>`__) + - defined by :eq:`Tv` and in section (`3.1.1 <#sec:parxs>`__) * - :math:`b` - buoyancy (:math:`=g T_v'/T_v`) @@ -7100,7 +6923,7 @@ Appendix: Notation - * - :math:`\gamma_{\theta_{\ell}}` - - gradient adjustment term, given by (`[gradadj] <#gradadj>`__) + - gradient adjustment term, given by :eq:`gradadj` * - :math:`w_m` - scaling velocity for momentum mixing in the SML @@ -7124,13 +6947,13 @@ Appendix: Notation - subsidence velocity (ms\ :math:`^{-1}`) * - :math:`\Delta_F` - - cloud-top net radiative divergence, calculation given in (`[ctraddiv] <#ctraddiv>`__) + - cloud-top net radiative divergence, calculation given in :eq:`ctraddiv` * - :math:`\alpha_t` - - parameter in entrainment parametrization, (`[we_parm] <#we_parm>`__) + - parameter in entrainment parametrization, :eq:`we_parm` * - :math:`\tau_{rc}`, :math:`z_{rc}` - - parameters in perturbation calculation, (`[dscd_pert] <#dscd_pert>`__), + - parameters in perturbation calculation, :eq:`dscd_pert`, * - - for initial identification of and :math:`z_{\rm ml}` calculation for DSC layers @@ -7561,4 +7384,4 @@ References *Estimates of Surface Wind Stress and Drag Coefficients in {T}yphoon {M}egi*. J. Phys. Oceanogr., 47, 545–565. - https://doi.org/10.1175/JPO-D-16-0069.1 + https://doi.org/10.1175/JPO-D-16-0069.1 \ No newline at end of file From 5d31330bbbfd72db70a6ab4498fa7141a322e924 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Sat, 25 Apr 2026 04:48:19 +0100 Subject: [PATCH 053/116] Restored lost labels on multi-line equation blocks. --- .../science_guide/turbulence_schemes/bldoc.rst | 12 ++++++------ 1 file changed, 6 insertions(+), 6 deletions(-) diff --git a/documentation/source/science_guide/turbulence_schemes/bldoc.rst b/documentation/source/science_guide/turbulence_schemes/bldoc.rst index bdf6cb18d6..ff31151e64 100644 --- a/documentation/source/science_guide/turbulence_schemes/bldoc.rst +++ b/documentation/source/science_guide/turbulence_schemes/bldoc.rst @@ -1019,7 +1019,7 @@ For :math:`Ri < 0`, the standard UM stability functions are given by f_m = 1 - \frac{g_0 \,Ri} {1+D_m(\tilde{{\cal L}}_m/\tilde{{\cal L}}_h)|Ri|^{1/2} } -.. math:: +.. math:: :label: unstable_stab f_h = \frac{1}{Pr_N}\left(1 - \frac{g_0 \,Ri} {1+D_h(\tilde{{\cal L}}_m/\tilde{{\cal L}}_h)|Ri|^{1/2} }\right) @@ -1035,7 +1035,7 @@ model (LEM), `Brown (1999) 2`_: f_m = (1 - c_{LEM} Ri)^{1/2} -.. math:: +.. math:: :label: unstable_stab_lem f_h = \frac{1}{Pr_N}\left(1 - b_{LEM} Ri\right)^{1/2} @@ -1114,7 +1114,7 @@ turbulence beyond a critical Richardson number, :math:`Ri_c=0.25`: f_m = \left( 1 - \frac{Ri}{Ri_c} \right)^4 -.. math:: +.. math:: :label: stable_stab_lem f_h = \frac{1}{Pr_N} \left( 1 - \frac{Ri}{Ri_c} \right)^4 (1 - g_{LEM} Ri) @@ -2221,7 +2221,7 @@ F|_{z_i}`, so that {\cal H}|_{z_i} = - w_e \Delta \theta_{\ell}+ F_{\rm net}|_h -.. math:: +.. math:: :label: discinv \overline{w'q_t'}_{z_i} = - w_e \Delta q_t @@ -2260,7 +2260,7 @@ base of the mixed layer: \left( \tilde{w_e} \Delta \theta_{\ell}+ \overline{w'\theta_{\ell}'}|_{z_{\rm b}} - F_{\rm net}|_{h} \right) - F_{\rm net}|_{ z_{\mbox{\tiny \rm NTDSC}+\frac{1}{2}} } -.. math:: +.. math:: :label: fluxinterp \overline{w'q_t'}|_{ z_{\mbox{\tiny \rm NTDSC}+\frac{1}{2}} } = \overline{w'q_t'}|_{z_{\rm b}} - \frac{ z'_{\mbox{\tiny \rm NTDSC}+\frac{1}{2}} }{z_{\rm ml}} @@ -2792,7 +2792,7 @@ specified through an eddy diffusivity which is given by K_h|_{\mbox{\tiny \rm NTML}+\frac{1}{2}} = w_e \Delta_{\mbox{\tiny \rm NTML}+1} z -.. math:: +.. math:: :label: khent K_m|_{\mbox{\tiny \rm NTML}} = Pr \, w_e \Delta_{\mbox{\tiny \rm NTML}+\frac{1}{2}} z From cec1d90cdd1926764396cb9e0951190c83c6fdc7 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Mon, 27 Apr 2026 11:34:57 +0100 Subject: [PATCH 054/116] Fixed figure cross-referencing. --- .../turbulence_schemes/bldoc.rst | 32 +++++++++---------- 1 file changed, 16 insertions(+), 16 deletions(-) diff --git a/documentation/source/science_guide/turbulence_schemes/bldoc.rst b/documentation/source/science_guide/turbulence_schemes/bldoc.rst index ff31151e64..2793565504 100644 --- a/documentation/source/science_guide/turbulence_schemes/bldoc.rst +++ b/documentation/source/science_guide/turbulence_schemes/bldoc.rst @@ -224,7 +224,7 @@ been categorised into 7 distinct ‘boundary layer types’: inhibit the formation of cumulus); the possibilities are discussed in section `4.2 <#sec:shear>`__. -Types I to VI are shown schematically in Fig. `1 <#fig:bltypes>`__. +Types I to VI are shown schematically in :numref:`Fig. %s `. .. figure:: blank.svg :name: fig:bltypes @@ -358,7 +358,7 @@ layer gradient, :math:`\Delta_{\rm sub}`, between grid-levels NLCL and :math:`k_s`. Currently the threshold factor, :math:`C_t = 1.1`. If cumulus is diagnosed, the top of the surface-based mixed layer (:math:`z_{\rm h}` ) is set to :math:`z_{\rm lcl}` (rather than to -:math:`z_{\rm par}` , as illustrated in Fig. `1 <#fig:bltypes>`__ for +:math:`z_{\rm par}` , as illustrated in :numref:`Fig. %s ` for types V and VI). There is then an option to diagnose the thickness of the LCL transition zone, see section `3.4 <#sec:lclmixing>`__. Otherwise, the boundary layer surface-driven mixing is capped at @@ -689,7 +689,7 @@ Because of the large gradients often seen in fluxes close to the inversion (in particular, in the LW radiative flux), simple finite difference flux calculations, :eq:`eq:wx_std`, can be significantly inaccurate in this region. An example is shown in -Fig. `2 <#fig:inv_integ>`__. Calculating +:numref:`Fig. %s `. Calculating :math:`\overline{w'\theta_{\ell}'}_{\mbox{\tiny \rm NTML}+\frac{1}{2}}` from :eq:`eq:wx_std` gives a negative value, largely because :math:`\Delta_{\mbox{\tiny \rm NTML}+1} \theta_{\ell}` is @@ -700,7 +700,7 @@ be positive. The solution adopted is to integrate :math:`\overline{w'b}` analytically across the region just below the inversion, labelled -:math:`\Delta z_{rad}` in Fig, `2 <#fig:inv_integ>`__. Since +:math:`\Delta z_{rad}` in Fig, :numref:`%s `. Since :math:`\Delta_{\mbox{\tiny \rm NTML}} \theta_{\ell}` can also be significantly positive (when the grid-level inversion is rising or falling, for example), the base of this region is taken to be the lower @@ -1799,8 +1799,8 @@ that HB and the standard UM set surface layer. The convective and neutral limits for :math:`w_h` and :math:`w_m` are given in Table `1 <#tab:vscales>`__ and the stability dependencies of several parameters are shown in -Fig. `3 <#fig:stab_dep>`__. The parameter :math:`d` in -Fig. `3 <#fig:stab_dep>`__ contains the stability dependence of the +:numref:`Fig. %s `. The parameter :math:`d` in +:numref:`Fig. %s ` contains the stability dependence of the gradient adjustment parameter: .. math:: :label: grad_adj @@ -1817,7 +1817,7 @@ The inclusion of an extra :math:`w_*/w_m` factor in on which the UM was based. This seems an appealing feature (HB’s :math:`\gamma_{\chi}` will tend to zero as :math:`w_* \rightarrow 0`) and probably should be considered for the revised scheme (the -dash-dotted line in Fig. `3 <#fig:stab_dep>`__ sets +dash-dotted line in :numref:`Fig. %s ` sets :math:`d^{std} = 10 w_*^2/w_h^2`). Similarly, :math:`f_2` might benefit from an additional factor of the form :math:`(V_{\rm surf}^3+ V_{\rm Sc}^3)/ V_{\rm sum}^3` so that it too tends to zero in the @@ -1833,7 +1833,7 @@ gives better agreement against LES. Compared to the standard scheme, it appears that the revised :math:`K_h^{\rm Sc}` is very different. However, -Fig.\ `4 <#fig:new_ksc>`__ shows that this actually amounts to a small +Fig.\ :numref:`%s ` shows that this actually amounts to a small adjustment in the shape. In addition, note that the factors :math:`\varepsilon_h^{surf}` and :math:`\varepsilon_h^{Sc}` have been removed since the entrainment flux is now carried via the explicit @@ -1974,7 +1974,7 @@ with `Malavelle et al. (2014)`_, and choose :math:`\beta=\beta_{\rm bl}=0.15` to give the best match of our function to that of `Honnert et al. (2011)`_. These functions are shown in -Figure `5 <#fig-blend>`__\ (a) and are only dissimilar for small +:numref:`Figure %s `\ (a) and are only dissimilar for small :math:`\Delta x`, where Eq. :eq:`eq-tanh` tends to zero faster. This is by choice, to force the highest resolution simulations to use the 3D @@ -2018,7 +2018,7 @@ the decoupled cloud top we set where :math:`z_{\rm sml}` is the depth of the surface-based mixed layer :raw-latex:`\cite[i.e.~the depth through which a positively buoyant parcel released at the surface would ascend,][]{lock00}`. This is shown -schematically in Figure `5 <#fig-blend>`__\ (b), and ensures that +schematically in :numref:`Figure %s `\ (b), and ensures that :math:`W_{1D}` has a high value in the poorly resolved surface mixed layer and cloud layer, and a lower value in between those layers. Again, this choice of :math:`z_{\rm turb}` is broadly consistent with the @@ -2271,7 +2271,7 @@ where :math:`z' = z-z_{\rm b}`, and similarly for the SML entrainment fluxes (at :math:`z=z_{\mbox{\tiny \rm NTML}+\frac{1}{2}}`). The turbulent fluxes at the base of the mixed layer are assumed zero except for the SML where the surface fluxes are used. This interpolation is -illustrated for a SML in Fig. `6 <#fig:fluxinterp>`__. +illustrated for a SML in :numref:`Fig. %s `. .. figure:: blank.svg :name: fig:fluxinterp @@ -2400,7 +2400,7 @@ cell-average value. Thus, :math:`z_i` can be calculated by assuming that the integral of :math:`\theta_{v\ell}` over grid-level NTML\ :math:`+1` for the model and for a profile with a discontinuous inversion at :math:`z_i` are equal, as illustrated by the hatched areas in -Fig. `7 <#zi_diag>`__. To calculate the integral of the discontinuous +:numref:`Fig. %s `. To calculate the integral of the discontinuous profile, the lapse rate of :math:`\theta_{v\ell}` between grid-levels NTML\ :math:`-1` and :math:`NTML`, :math:`\gamma^{\tiny \rm ML}`, is extended up to :math:`z_i`, while the stable lapse in the free @@ -2548,7 +2548,7 @@ precipitation and subsidence, assuming a well-mixed boundary layer capped by a diagnosed subgrid inversion. The crucial step is to ensure that the total flux on the model entrainment grid-level equals the idealised total flux profile interpolated to that level. Consider the -example illustrated in Fig. `8 <#fig:rev_fluxes>`__ of a well-mixed +example illustrated in :numref:`Fig. %s ` of a well-mixed boundary layer up to :math:`\theta`-level :math:`\mbox{\tiny \rm NTML}`. For the subgrid :math:`q_t` profiles, the turbulent flux divergence generates a moistening across the inversion while subsidence generates @@ -2563,7 +2563,7 @@ is split across levels :math:`\mbox{\tiny \rm NTML}` and model’s boundary layer and inversion consistent with the total subgrid flux profile, the model’s entrainment flux at :math:`\mbox{\tiny \rm NTML}+1/2` (shown by the cross in -Fig. `8 <#fig:rev_fluxes>`__) must be found by subtracting the +:numref:`Fig. %s `) must be found by subtracting the subsidence flux at :math:`\mbox{\tiny \rm NTML}+1/2` (diamond) from the total flux interpolated to :math:`\mbox{\tiny \rm NTML}+1/2` (square). Exactly the same arguments follow for the :math:`\theta_{\ell}` fluxes @@ -6232,7 +6232,7 @@ height where :math:`F_{LW}` becomes some small fraction, :math:`A`, of \Delta_F\approx \Delta_F^{LW} + (1-\exp^{ln(A)\kappa_{SW}/\kappa_{LW}}) \Delta_F^{SW} -Empirically, see Fig. `9 <#fig:dradts>`__, a reasonable fit to LEM data +Empirically, see :numref:`Fig. %s `, a reasonable fit to LEM data is obtained with: .. math:: :label: eq:deltaf_emp @@ -6244,7 +6244,7 @@ Note that in the 9B scheme :math:`\Delta_F^{LW}` and but with the LW and SW increments separately. This change in the calculation of :math:`\Delta_F` is illustrated in -Fig. (`9 <#fig:dradts>`__) from LES of the diurnal cycle of marine +Fig. (:numref:`%s `) from LES of the diurnal cycle of marine stratocumulus. The top panel is from a simulation which used the code specified for the EUROCS LES intercomparison, the lower panel used the Edwards-Slingo radiation scheme in the LES. There are clearly some From adc8e2f318f4de76b8248c7989d018d614610a33 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Mon, 27 Apr 2026 11:51:46 +0100 Subject: [PATCH 055/116] Corrected multi-panel figure widths (needs to be % of width of current column, so mutliply by number of columns). --- .../source/science_guide/turbulence_schemes/bldoc.rst | 10 ++++------ 1 file changed, 4 insertions(+), 6 deletions(-) diff --git a/documentation/source/science_guide/turbulence_schemes/bldoc.rst b/documentation/source/science_guide/turbulence_schemes/bldoc.rst index 2793565504..02cc5e8388 100644 --- a/documentation/source/science_guide/turbulence_schemes/bldoc.rst +++ b/documentation/source/science_guide/turbulence_schemes/bldoc.rst @@ -238,9 +238,9 @@ Types I to VI are shown schematically in :numref:`Fig. %s `. :widths: 46 46 * - .. image:: wcrp_bltypes1.svg - :width: 46% + :width: 92% - .. image:: wcrp_bltypes2.svg - :width: 46% + :width: 92% .. _`sec:adiapar`: @@ -1996,9 +1996,9 @@ turbulence scheme. :widths: 49 49 * - .. image:: honnert_vs_tanh.svg - :width: 49% + :width: 98% - .. image:: zturb_schem.svg - :width: 49% + :width: 98% One of the key benefits of the `Lock et al. (2000)`_ scheme is its ability to represent decoupled stratocumulus layers, and this is a @@ -6265,9 +6265,7 @@ therefore zero entrainment and turbulent mixing). :widths: 50 50 * - .. image:: div_r080.svg - :width: 50% - .. image:: div_r071.svg - :width: 50% The 9C version attempted to remove the grid-dependence implied by the summation over 3 grid-levels in :eq:`ctraddiv` as follows: From c188b4da940e5004d7dc6bd389a0d1759b83df04 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Mon, 27 Apr 2026 12:15:33 +0100 Subject: [PATCH 056/116] Fixed section cross-referencing. --- .../turbulence_schemes/bldoc.rst | 276 +++++++++--------- 1 file changed, 138 insertions(+), 138 deletions(-) diff --git a/documentation/source/science_guide/turbulence_schemes/bldoc.rst b/documentation/source/science_guide/turbulence_schemes/bldoc.rst index 02cc5e8388..47a4f3521a 100644 --- a/documentation/source/science_guide/turbulence_schemes/bldoc.rst +++ b/documentation/source/science_guide/turbulence_schemes/bldoc.rst @@ -32,7 +32,7 @@ Several options for higher order closures are available in the 1A version of the UM boundary layer scheme and these are documented separately in . -.. _`sec:closure`: +.. _sec_closure: Model variables and turbulence closure ====================================== @@ -79,12 +79,12 @@ water static energy’ (:math:`= c_p T + g z - L q_{\ell}- L_s q_f`) rather than potential temperature, :math:`\theta`. Note also that the option to use mixing ratios in the boundary layer code instead of specific quantities is also available and the details of the -necessary changes are documented in appendix `13 <#app:mixratio>`__. +necessary changes are documented in appendix :ref:`Appendix: changing between specific humidities and mixing ratios `. Ultimately turbulent motions are dissipated as heat and so the source term :math:`{\cal S}` in :eq:`cons_eqn_scal` can include an approximation for that frictional heating, as described in -appendix `14 <#app:fricheat>`__. Finally, the ice cloud contributions in +appendix :ref:`Appendix: including the heating from turbulence dissipation `. Finally, the ice cloud contributions in :eq:`thetal` and :eq:`qt` can optionally be ignored (l_noice_in_turb), which will be more appropriate if the time scales for ice melting or sublimation are longer than those of the turbulence (and @@ -109,7 +109,7 @@ measure of buoyancy. A ‘first-order’ closure is used to parametrize the turbulent fluxes, although non-local terms are also included. Under the 9C scheme, an alternative methodology is optionally available, see section -`5.5 <#sec:rev_flux_grad>`__. The standard closures are: +:ref:`The revised scalar flux-gradient formulation `. The standard closures are: .. math:: :label: scal_closure @@ -126,29 +126,29 @@ side represents a non-local flux in unstable boundary layers. Currently it is only applied for transport arising from surface-driven turbulence (:math:`K_h^{\rm surf}`) and is non-zero only for :math:`\chi=\theta_{\ell}`, as described in -section `5.3 <#sec:gradadj>`__. +section :ref:`Gradient adjustment `. Thus, the parametrization reduces to determining :math:`K_h`, :math:`K_m` and :math:`\gamma_{\chi}` and :math:`{\bf \tau}^{nl}`. Two methods are used to determine :math:`K_h` and :math:`K_m` and how they are combined for :eq:`scal_closure` and :eq:`uv_closure` is described in -section `4.2 <#sec:shear>`__. The first method is a local Richardson +section :ref:`Shear-driven mixing and interaction between the local and non-local schemes `. The first method is a local Richardson number (:math:`Ri`) based scheme. It is calculated for all regimes (but will be responsible for all mixing in stable conditions), over all levels up to the specified BL_LEVELS and is described in -section `4 <#sec:local>`__. The second method is a non-locally specified +section :ref:`The local scheme `. The second method is a non-locally specified profile scheme. This is exclusively for unstable boundary layers, is calculated up to level NL_BL_LEVELS (typically around 6km AMSL) and is -described in more detail in section `5 <#sec:nonlocal>`__. In this +described in more detail in section :ref:`The non-local scheme `. In this regime, mixing is assumed to occur in (or lead rapidly to the formation of) well-mixed layers (in which conserved variables are approximately uniform with height) that are capped by an inversion. Mixing is assumed to be driven either from the surface in a ‘surface mixed layer’ (SML, by a positive surface buoyancy flux and by surface stresses) or by cloud-top buoyancy sources (radiative and evaporative cooling, see -appendix `11 <#app:vscales>`__). As described in section -`5 <#sec:nonlocal>`__, separate :math:`K`-profiles are used for these +appendix :ref:`Appendix: Definitions of the velocity scales `). As described in section +:ref:`The non-local scheme `, separate :math:`K`-profiles are used for these two turbulence sources. If the cloud-top sources generate mixing throughout the SML the layer is said to be ‘coupled’ but if the :math:`K`-profile representing surface-driven mixing does not extend up @@ -156,16 +156,16 @@ to cloud-top, the layer is referred to as being ‘decoupled’. As decoupled layers are restricted to being buoyancy driven and typically below 6km, they are referred to as decoupled stratocumulus (DSC) layers. The calculation of :math:`\gamma_{\chi}` is described in -section `5.3 <#sec:gradadj>`__ and, finally, fluxes across the top of +section :ref:`Gradient adjustment ` and, finally, fluxes across the top of both SML and DSC layers (the entrainment fluxes) are specified explicitly through a separate entrainment parametrization, as described -in section `7 <#sec:entr>`__. +in section :ref:`Entrainment fluxes `. The strategy used to determine precisely where and when the resulting eddy-diffusivities should be applied is described in -section `3 <#sec:types>`__. The buoyancy parameters, finite difference +section :ref:`Diagnosis of boundary layer depth and type `. The buoyancy parameters, finite difference and other notation used here are defined in -appendices `12 <#app:buoyp>`__ and `17 <#app:not>`__. Further papers +appendices :ref:`Appendix: Derivation and definitions of the buoyancy parameters ` and :ref:`Appendix: Notation `. Further papers describing this scheme and its performance are `Lock et al. (2000)`_ (noting the corrigendum in `Lock et al. (2001)`_), @@ -173,15 +173,15 @@ describing this scheme and its performance are `Lock (2001)`_, `Bush et al. (1999)`_ and `Brown et al. (2008)`_. -.. _`sec:types`: +.. _sec_types: Diagnosis of boundary layer depth and type ========================================== The non-locally specified :math:`K`-profiles require the height of the base and top of the layer to be diagnosed (see -section `5 <#sec:nonlocal>`__). Furthermore, as stated in -section `2 <#sec:closure>`__, the mixing generated by the non-local +section :ref:`The non-local scheme `). Furthermore, as stated in +section :ref:`Model variables and turbulence closure `, the mixing generated by the non-local :math:`K` profiles is assumed to occur in (or lead rapidly to the formation of) well-mixed layers capped by an inversion. Thus, the accurate diagnosis of their vertical extent is crucial. How to make this @@ -190,39 +190,39 @@ been categorised into 7 distinct ‘boundary layer types’: - **Type I**: Stable boundary layer (with or without cloud) — turbulent diffusivities are calculated by the ‘local’ scheme - (section `4 <#sec:local>`__) + (section :ref:`The local scheme `) - **Type II**: Boundary layer with stratocumulus over a stable near-surface layer — as Type I but with a turbulently mixed cloud layer driven from its top (a DSC layer, diagnosis described in section - `3.2 <#sec:decouple>`__) + :ref:`Diagnosis of the vertical extent of the K-profiles `) - **Type III**: Well mixed boundary layer — the classic single mixed layer which may be cloud-topped or clear but is predominantly buoyancy-driven (c.f. a possible type VII below) — diagnosis described - in section `3.1 <#sec:adiapar>`__) + in section :ref:`The diagnostic parcel ascent and cumulus diagnosis `) - **Type IV**: Unstable boundary layer with a DSC layer not over cumulus - (see section `3.2 <#sec:decouple>`__) — the surface-based and + (see section :ref:`Diagnosis of the vertical extent of the K-profiles `) — the surface-based and cloud-top-driven non-local :math:`K` profiles may or may not overlap and cloud-top entrainment can still include the surface forcing (see - section `7 <#sec:entr>`__) + section :ref:`Entrainment fluxes `) - **Type V**: Boundary layer with a DSC layer over cumulus — the cumulus (treated by the model’s mass-flux convection scheme) provides coupling with the SML (cumulus diagnosis described in section - `3.1 <#sec:adiapar>`__) + :ref:`The diagnostic parcel ascent and cumulus diagnosis `) - **Type VI**: Cumulus-capped boundary layer — no turbulent diffusivities are allowed [1]_ at or above the LCL as the mass-flux convection scheme operates here (cumulus diagnosis described in - section `3.1 <#sec:adiapar>`__) + section :ref:`The diagnostic parcel ascent and cumulus diagnosis `) - **Type VII**: Shear-dominated unstable layer — potentially wind-shear might allow deeper turbulent mixing in unstable boundary layers than is apparent purely from the thermodynamic profiles (sufficient even to inhibit the formation of cumulus); the possibilities are discussed in - section `4.2 <#sec:shear>`__. + section :ref:`Shear-driven mixing and interaction between the local and non-local schemes `. Types I to VI are shown schematically in :numref:`Fig. %s `. @@ -242,7 +242,7 @@ Types I to VI are shown schematically in :numref:`Fig. %s `. - .. image:: wcrp_bltypes2.svg :width: 92% -.. _`sec:adiapar`: +.. _sec_adiapar: The diagnostic parcel ascent and cumulus diagnosis -------------------------------------------------- @@ -275,7 +275,7 @@ the first grid-level (:math:`k=k_s`) above the top of the surface layer, upwards allowing for latent heat release. The top of the surface layer is taken to be at the lower of :math:`z=0.1`\ :math:`z_{\rm h}` (this is then consistent with the :math:`K`-profiles, see -section `5.1 <#sec:nlsurf>`__; :math:`z_{\rm h}` is taken from the +section :ref:`Surface-driven turbulence `; :math:`z_{\rm h}` is taken from the previous timestep) and the grid-level above which :math:`\theta_{v\ell}` starts to increase with height. The ascent is stopped at the grid-level NTPAR (height @@ -284,7 +284,7 @@ above which the parcel becomes more negatively buoyant than a given threshold, :math:`\theta_v'`. Note that the parcel properties themselves are not perturbed in order to preserve the height of the mixed-layer’s lifting condensation level (LCL). The calculation of the parcel’s -buoyancy excess is described in section `3.1.1 <#sec:parxs>`__. +buoyancy excess is described in section :ref:`Calculation of parcel buoyancy excess `. Currently, .. math:: :label: parcel_pert @@ -300,7 +300,7 @@ where :math:`A_{plume}=0.2`, :math:`B_{plume}=3.26`, `Holtslag and Boville (1993)`_, :math:`\theta_v'` is related to the magnitude of the gradient adjustment, :math:`\gamma_{\theta_{\ell}}` (see section -`5.3 <#sec:gradadj>`__). Thus, :math:`B_{plume}=A_{ga}`, although +:ref:`Gradient adjustment `). Thus, :math:`B_{plume}=A_{ga}`, although somewhat arbitrary limits have been placed on the magnitude of :math:`\theta_v'` for numerical security (the upper limit being consistent with that applied to :math:`\gamma_{\theta_{\ell}}` in @@ -360,7 +360,7 @@ cumulus is diagnosed, the top of the surface-based mixed layer (:math:`z_{\rm h}` ) is set to :math:`z_{\rm lcl}` (rather than to :math:`z_{\rm par}` , as illustrated in :numref:`Fig. %s ` for types V and VI). There is then an option to diagnose the thickness of -the LCL transition zone, see section `3.4 <#sec:lclmixing>`__. +the LCL transition zone, see section :ref:`Diagnosis of the LCL transition zone thickness `. Otherwise, the boundary layer surface-driven mixing is capped at :math:`z_{\rm lcl}` so that mixing into the cumulus cloud layer is only carried out by the model’s mass-flux convection scheme and not by the @@ -370,7 +370,7 @@ to being able to resolve cumulus with cloud and sub-cloud layers at least 2 grid-levels (and optionally 400m) thick. Otherwise the layer is considered well-mixed to :math:`z_{\rm par}` with an option to include a representation of fluxes into the capping inversion (see -section `3.3 <#sec:dzi>`__). +section :ref:`Diagnosis of inversion thickness `). If the parcel ascent fails to find an inversion below 3km (or BL_LEVELS) but the LCL is below BL_LEVELS, then the layer is assumed to be @@ -383,14 +383,14 @@ BL_LEVELS above the tropopause is recommended. Note that if cumulus is not diagnosed then a further, subgrid estimation of the height of the capping inversion is attempted for -:math:`z_{\rm h}`  (as described in section `7.1.1 <#sec:sginv>`__). +:math:`z_{\rm h}`  (as described in section :ref:`Diagnosis of a sub-grid inversion `). -.. _`sec:parxs`: +.. _sec_parxs: Calculation of parcel buoyancy excess ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -As described in appendix `12 <#app:buoyp>`__, virtual temperature, +As described in appendix :ref:`Appendix: Derivation and definitions of the buoyancy parameters `, virtual temperature, :math:`T_v = T(1 + c_v q_v - q_{\ell}- q_f)`, is used as the measure of buoyancy. The condensed water in the parcel at a grid-level :math:`k` @@ -407,7 +407,7 @@ the environment at that grid-level \right] where the buoyancy parameters :math:`a_L` and :math:`\alpha_L` are -defined in appendix `12 <#app:buoyp>`__. Recall that the parcel has +defined in appendix :ref:`Appendix: Derivation and definitions of the buoyancy parameters `. Recall that the parcel has :math:`q_t` and :math:`\theta_{\ell}` taken from grid-level :math:`k_s` which are conserved during its ascent. Note that :eq:`qlpar` will not give condensation until the parcel becomes saturated. In the @@ -429,10 +429,10 @@ melting point) and :eq:`qt` implies :math:`q_v^p = q_t^p - q_{\ell f}^p` and thus :math:`T_v^p` can be calculated. Recall that the diagnosis of the parcel’s maximum buoyancy excess over the environment (described in -section `3.1 <#sec:adiapar>`__) required :math:`\theta_v`. This is +section :ref:`The diagnostic parcel ascent and cumulus diagnosis `) required :math:`\theta_v`. This is approximated as :math:`\theta_v = T_v + (g z_k/c_p)`. -.. _`sec:decouple`: +.. _sec_decouple: Diagnosis of the vertical extent of the K-profiles -------------------------------------------------- @@ -444,14 +444,14 @@ been separated in to three stages. These are: approximately uniform :math:`\theta_{v\ell}` (label the top grid-level in the mixed-layer NTDSC and diagnose the subgrid height of its capping inversion, :math:`z_{\rm h}^{\rm Sc}` , see - section `7.1.1 <#sec:sginv>`__) + section :ref:`Diagnosis of a sub-grid inversion `) #. diagnose an approximate depth of the DSC layer, :math:`z_{\rm ml}`, in order to be able to calculate the representative turbulent - velocity scales (see appendix `11 <#app:vscales>`__). + velocity scales (see appendix :ref:`Appendix: Definitions of the velocity scales `). #. calculate the depth of the :math:`K` profiles (see - section `5 <#sec:nonlocal>`__) in both SML and DSC layers using + section :ref:`The non-local scheme `) in both SML and DSC layers using constraints on the TKE budget of the layer. This includes the diagnosis of recoupling of DSC layers and decoupling of SMLs @@ -486,7 +486,7 @@ strength of cloud-capping inversions). To do this, the :math:`\theta_v` gradient between grid-levels :math:`k_{ct}` and :math:`k_{ct}-1` is compared with that for a parcel lifted adiabatically from grid-level :math:`k_{ct}-1`, in exactly the same way as for the SML parcel ascent -(see section `3.1.1 <#sec:parxs>`__). If :math:`d\theta_v/dz|_{\rm +(see section :ref:`Calculation of parcel buoyancy excess `). If :math:`d\theta_v/dz|_{\rm env} > \Gamma_{\rm inv} d\theta_v/dz|_{\rm par}` between grid-levels :math:`k_{ct}` and :math:`k_{ct}-1` then NTDSC is set to :math:`k_{ct}-1`; if not then NTDSC is set to :math:`k_{ct}` (recall @@ -505,7 +505,7 @@ perturbation is given by \theta_{v\ell}' = - \, \frac{ \tau_{rc} \Delta_F}{z_{rc}} where :math:`\Delta_F` (Kms\ :math:`^{-1}`) is the magnitude of the -cloud-top radiative divergence (see appendix `11 <#app:vscales>`__), +cloud-top radiative divergence (see appendix :ref:`Appendix: Definitions of the velocity scales `), :math:`\tau_{rc}` is a timescale for the exposure of boundary layer eddies to the cloud-top radiative cooling (taken to be 200s) and :math:`z_{rc}` is a depth-scale for the radiatively cooled layer (taken @@ -528,7 +528,7 @@ magnitude of the integrated buoyancy consumption of TKE within the mixed layer is less than or equal to a fraction, :math:`D_t`, of the buoyancy production, following `Turton and Nicholls (1987)`_. -Following appendix `12 <#app:buoyp>`__ the grid-box mean buoyancy flux +Following appendix :ref:`Appendix: Derivation and definitions of the buoyancy parameters ` the grid-box mean buoyancy flux can be written as: .. math:: :label: eq:wb_cont @@ -556,7 +556,7 @@ where :math:`\widetilde{\Delta_k \theta_{\ell}} = \Delta_k \theta_{\ell}- \gamma_{\theta_{\ell}} \Delta_k z` in order to include the non-local (or gradient adjustment) term. If the alternative flux-gradient option is -used, see section `5.5 <#sec:rev_flux_grad>`__, then additional terms +used, see section :ref:`The revised scalar flux-gradient formulation `, then additional terms are needed. Large-eddy simulations have demonstrated that the crucial region in @@ -593,7 +593,7 @@ layers. For stratocumulus layers, observations and LES suggest a value of :math:`D_t=0.1`. A separate value of :math:`D_t` can be used for the sub-cloud layer in cumulus capped boundary layers if this method is used to determine the LCL transition zone thickness, see section -`3.4 <#sec:lclmixing>`__. For cloud-free mixed layers, :math:`D_t=1` is +:ref:`Diagnosis of the LCL transition zone thickness `. For cloud-free mixed layers, :math:`D_t=1` is used, purely to keep negative buoyancy fluxes down to a reasonably realistic level (for example, if the parcel top diagnostic returned too high a boundary layer depth). @@ -614,7 +614,7 @@ is diagnosed, :math:`z_{\rm h}^{\rm Sc}` is set to the original :math:`z_{\rm h}` (inversion height), although the entrainment across this inversion is not recalculated (and so keeps any surface-driven component — the COUPLED flag is therefore set to true, see -section `7 <#sec:entr>`__). +section :ref:`Entrainment fluxes `). If a decoupled layer is diagnosed, then an iteration is performed to find the highest :math:`z_{\rm h}` (so top of the :math:`K_h^{\rm surf}` @@ -648,7 +648,7 @@ is diagnosed to recouple completely). Finally, :math:`z_{\rm b}` must always be at or below :math:`z_{\mbox{\tiny \rm NTDSC}-1}`, so that mixing in decoupled layers is always resolved, and at least :math:`\Delta z_{rad}` (the cloud-top radiative cooling depth defined in -section `3.2.2 <#sec:wbint_inv>`__) below the t inversion. The base +section :ref:`Integration of \overline{w'b} close to the inversion `) below the t inversion. The base grid-level of the DSC layer, NBDSC, is defined (analogously to NTDSC) as the lowest grid-level such that :math:`K_h^{\rm Sc}` is non-zero at the half-level below. @@ -680,7 +680,7 @@ positive and so where the cloud-free part of :math:`\overline{w'b}` (i.e., that part below cloud-base which is important for decoupling) becomes negative. -.. _`sec:wbint_inv`: +.. _sec_wbint_inv: Integration of :math:`\overline{w'b}` close to the inversion ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ @@ -785,7 +785,7 @@ The integrated buoyancy flux is then found from and buoyancy coefficients evaluated at the grid-level above :math:`z_h -\Delta z_{rad}`. -.. _`sec:dzi`: +.. _sec_dzi: Diagnosis of inversion thickness -------------------------------- @@ -807,7 +807,7 @@ inversion, :math:`z_{top}`, in LES from where :math:`z_{nb}` is the level of neutral buoyancy (found by linear interpolation between grid-levels), :math:`w_m` is the boundary layer -velocity scale defined in section `5.1 <#sec:nlsurf>`__ and :math:`b` is +velocity scale defined in section :ref:`Surface-driven turbulence ` and :math:`b` is the parcel buoyancy. Note that the constant in :eq:`dz_param` is the same as in `Beare (2008)`_ because :math:`6.3 = 2.5 * 4^{2/3}` and @@ -825,12 +825,12 @@ inversion thickness is then defined as \Delta z_i = z_{top} -z_{\rm par} -.. _`sec:lclmixing`: +.. _sec_lclmixing: Diagnosis of the LCL transition zone thickness ---------------------------------------------- -As described in section `3.1 <#sec:adiapar>`__, when cumulus convection +As described in section :ref:`The diagnostic parcel ascent and cumulus diagnosis `, when cumulus convection has been diagnosed surface-driven mixing was originally capped at :math:`z_{\rm lcl}` so that mixing into the cumulus cloud layer was only carried out by the model’s mass-flux convection scheme. This was seen to @@ -843,7 +843,7 @@ indistinguishable from those in cloud-free convective boundary layers and so the non-local surface-driven mixed layer K-profiles remain accurate up to this level. To diagnose the depth to which these profiles should penetrate above the LCL, the algorithm given in -section `3.2 <#sec:decouple>`__ to diagnose the extent of the K-profiles +section :ref:`Diagnosis of the vertical extent of the K-profiles ` to diagnose the extent of the K-profiles in decoupled boundary layers can be used (using the switch kprof_cu). This ensures that the magnitude of the integrated buoyancy consumption of TKE within the mixed layer is less than or equal to a fraction, @@ -855,7 +855,7 @@ LCL (but are too dry to reach their own LCL). Thus their buoyancy flux is given by :eq:`eq:wb_cont` with :math:`C_F=0`. Restricting the negative integral of this buoyancy flux then gives a new definition for :math:`z_{\rm h}`  that is then used in the calculation -of the surface-driven K-profiles in section `5.1 <#sec:nlsurf>`__ — the +of the surface-driven K-profiles in section :ref:`Surface-driven turbulence ` — the larger the value of :math:`D_t`, the higher :math:`z_{\rm h}` will be. Typically :math:`D_t=0.1` for decoupled stratocumulus layers while idealised clear-sky convective boundary layers (where the magnitude of @@ -871,7 +871,7 @@ height of the LCL (the factor of a half is arbitrary, with no sensitivity to this choice given that the diagnosis parcel reached the LCL, but ensures the iteration starts well below the LCL). -.. _`sec:local`: +.. _sec_local: The local scheme ================ @@ -943,7 +943,7 @@ subgrid orography and :math:`(z_{0m})_{\mbox{veg}}` is the vegetative part of the roughness length. The constants in :eq:`asymp_ml` can be considered ‘tuned’ (see, in particular, the operational modifications described in -appendix `15 <#app:opmods>`__). +appendix :ref:`Appendix: Operational modifications `). The Richardson number, :math:`Ri`, that is used as a local measure of stability is given by @@ -961,8 +961,8 @@ The measure of buoyancy used in :math:`Ri` is where :math:`\overline{\beta_T}` and :math:`\overline{\beta_q}` are the grid-box mean (i.e., cloud weighted) buoyancy coefficients, that can be -defined in two different ways, see appendix `12 <#app:buoyp>`__ and -section `4.1 <#sec:fd_ri>`__. Note that :eq:`Bdefn` reduces +defined in two different ways, see appendix :ref:`Appendix: Derivation and definitions of the buoyancy parameters ` and +section :ref:`Finite difference calculations `. Note that :eq:`Bdefn` reduces to a virtual temperature approximation of buoyancy in cloud-free air and that neutral buoyancy (in cloudy as well as cloud-free air) is implied by vertically uniform :math:`\theta_{\ell}` and :math:`q_t`. This is @@ -1001,7 +1001,7 @@ to be a measure of the boundary layer top (:math:`z_{\rm loc}` ) and the full-level below is designated NTLOC. In general :math:`Ri_{crit}=1` but a value of 0.25 is recommended for use with the ’SHARPEST’ stability functions, see below. If the boundary layer was diagnosed as -cumulus-capped by the non-local scheme (see section `3 <#sec:types>`__) +cumulus-capped by the non-local scheme (see section :ref:`Diagnosis of boundary layer depth and type `) then :math:`z_{\rm loc}` is lowered to :math:`z_{\rm lcl}` (and :math:`K_h` and :math:`K_m` are set to zero from the base of grid-level NLCL upwards) so that transports into and within the cumulus cloud layer @@ -1121,7 +1121,7 @@ turbulence beyond a critical Richardson number, :math:`Ri_c=0.25`: with :math:`g_{LEM}=1.2`. -.. _`sec:fd_ri`: +.. _sec_fd_ri: Finite difference calculations ------------------------------ @@ -1163,7 +1163,7 @@ i_interp_local. The long-standing method is given by where :math:`\overline{\beta_T}` and :math:`\overline{\beta_q}` are the grid-box mean (i.e., cloud-fraction weighted) buoyancy coefficients, -defined in appendix `12 <#app:buoyp>`__. Note that because this is +defined in appendix :ref:`Appendix: Derivation and definitions of the buoyancy parameters `. Note that because this is defined on :math:`\theta`-levels, no vertical interpolation of cloud variables (fractional area and water contents), to which the buoyancy coefficients are very sensitive, is required. The volume-weighted @@ -1250,7 +1250,7 @@ in the flux calculation can occur. In the unstable stability functions L}}_m` and :math:`Ri`) in order to maintain the same stability dependence. -.. _`sec:shear`: +.. _sec_shear: Shear-driven mixing and interaction between the local and non-local schemes --------------------------------------------------------------------------- @@ -1264,7 +1264,7 @@ The general approach is to take :math:`K_{\chi}` in K_{\chi} = \mbox{max} \left[ (K_{\chi}^{\rm surf}+K_{\chi}^{\rm Sc}), K_{\chi}(Ri) \right] -As noted in section `2 <#sec:closure>`__, this implies that mixing in +As noted in section :ref:`Model variables and turbulence closure `, this implies that mixing in stable boundary layers is determined exclusively by the local scheme, :math:`K_{\chi}(Ri)`. Continuing to calculate :math:`K_{\chi}(Ri)` in unstable boundary layers and using :eq:`klnl` is seen as the @@ -1273,9 +1273,9 @@ and unstable boundary layers. At the top of unstable mixed layers, great care is taken to ensure the parametrized entrainment mixing is implemented faithfully, see -section `7 <#sec:entr>`__). Consequently, if a subgrid inversion has +section :ref:`Entrainment fluxes `). Consequently, if a subgrid inversion has been diagnosed capping a mixed layer (see -section `7.1.1 <#sec:sginv>`__), then :math:`K_{\chi}(Ri)` is set to +section :ref:`Diagnosis of a sub-grid inversion `), then :math:`K_{\chi}(Ri)` is set to zero at the interfaces either side of the inversion grid-level. There are also options (using the switch Keep_Ri_FA) to set :math:`K_{\chi}(Ri)` to zero entirely above unstable boundary layers or @@ -1294,7 +1294,7 @@ cloud layers that have been diagnosed as cumulus-capped (which would be poorly represented by the current convection scheme). Several methods have been introduced that attempt to alleviate this problem, giving rise to the diagnosis of a “shear-dominated boundary layer” type (type VII), -discussed in section `3 <#sec:types>`__. The first (the “shear-dominated +discussed in section :ref:`Diagnosis of boundary layer depth and type `. The first (the “shear-dominated boundary layer fix”) simply sets the CUMULUS flag to false if NTLOC :math:`>` NTPAR. This then ensures that the locally-determined :math:`K` are not set to zero above the LCL. Several more rigorous options are @@ -1342,7 +1342,7 @@ calculated value of each of the individual terms in STASH 3,304: 3,356 is set to :math:`z_{\rm h}` ; 3,357 is :math:`z_{\rm h}^{\rm Sc}`  and 3,358 is :math:`z_{\rm loc}` . -.. _`sec:nonlocal`: +.. _sec_nonlocal: The non-local scheme ==================== @@ -1352,13 +1352,13 @@ non-local in the sense that, at a given height within the boundary layer, :math:`K` is determined not by any local properties of the mean profiles at that height but solely by the magnitude of the turbulence forcing applied to the layer (as measured by the representative velocity -scales described in appendix `11 <#app:vscales>`__) and the height +scales described in appendix :ref:`Appendix: Definitions of the velocity scales `) and the height within the layer. The non-local scheme is therefore particularly robust but care must be taken where the profiles are applied. The calculation of the vertical position and extent of the :math:`K` profiles is -described in section `3 <#sec:types>`__. +described in section :ref:`Diagnosis of boundary layer depth and type `. -.. _`sec:nlsurf`: +.. _sec_nlsurf: Surface-driven turbulence ------------------------- @@ -1377,7 +1377,7 @@ where :math:`w_m^3 = u_*^3 + w_s^3`, :math:`u_*` is the friction velocity (including the orographic roughness component) and :math:`w_s` is defined below. For the 9C version of the scheme, :math:`z_{\rm h}` is the diagnosed subgrid inversion height (see -section `7.1.1 <#sec:sginv>`__) for both :math:`K_h^{\rm surf}` and +section :ref:`Diagnosis of a sub-grid inversion `) for both :math:`K_h^{\rm surf}` and :math:`K_m^{\rm surf}`. In the 8A version, :math:`K_m^{\rm surf}` uses :math:`z_{\rm h}` :math:`=z_{\mbox{\tiny \rm NTML}+\frac{1}{2}}`. The factor :math:`{\cal E}_m^{\rm surf}` is chosen so that @@ -1392,7 +1392,7 @@ A similar factor, :math:`{\cal E}_h^{\rm surf}`, is used in the :math:`K_h^{\rm surf}` profile even though the entrainment fluxes of the thermodynamic variables will usually be specified explicitly rather than through an eddy-diffusivity (see -section `7 <#sec:entr>`__). +section :ref:`Entrainment fluxes `). The form of :math:`w_s` differs between the surface layer (:math:`z < 0.1`\ :math:`z_{\rm h}` ) and the rest of the mixed-layer: @@ -1412,7 +1412,7 @@ parametrization in cloudy boundary layers). Note that :math:`w_s` is continuous across :math:`0.1`\ :math:`z_{\rm h}` and constant with height in the mixed layer. This form for :math:`w_s` is motivated by a desire to match the model’s surface transfer formulation within the -surface layer (as described further in section `5.1.1 <#sec:hbcomp>`__) +surface layer (as described further in section :ref:`Comparison with Holtslag and Boville (1993)_ `) and to use a cubic sum of velocity scales within the mixed layer (consistent with dimensional analysis of the TKE equation, see `Holtslag and Boville (1993)`_). @@ -1429,7 +1429,7 @@ Thus :math:`Pr` varies from 0.75 in neutral conditions to 0.375 in convective. The origin of the functional form of :eq:`prandtl_nl` is unknown. -.. _`sec:hbcomp`: +.. _sec_hbcomp: Comparison with `Holtslag and Boville (1993)`_ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ @@ -1502,7 +1502,7 @@ Cloud-top-driven turbulence For cloud-top-driven turbulence over a layer of depth :math:`z_{\rm ml}` (with top at :math:`z_{\rm h}` or :math:`z_{\rm h}^{\rm Sc}` and base at -:math:`z_{\rm b}` , determined as in section `3.2 <#sec:decouple>`__), +:math:`z_{\rm b}` , determined as in section :ref:`Diagnosis of the vertical extent of the K-profiles `), .. math:: :label: kmtop @@ -1510,7 +1510,7 @@ For cloud-top-driven turbulence over a layer of depth :math:`z_{\rm ml}` \left( 1 - {\cal E}_m^{\rm Sc} \frac{z'}{z_{\rm ml}} \right)^{0.8} where :math:`V_{\rm Sc}^3= V_{\rm rad}^3+V_{\rm br}^3` (see -appendix `11 <#app:vscales>`__) and :math:`z'` is height above +appendix :ref:`Appendix: Definitions of the velocity scales `) and :math:`z'` is height above :math:`z_{\rm b}` . Then :math:`K_h = K_m / \mbox{Pr}`, where :math:`\mbox{Pr}=0.75`. The resulting :math:`K_h` profile was derived against convective cloudy LES, as described in @@ -1519,7 +1519,7 @@ against convective cloudy LES, as described in simply as a number in the middle of the range usually quoted for turbulent mixing in general. As with :eq:`kmsurf`, :math:`z_{\rm h}`  (or :math:`z_{\rm h}^{\rm Sc}` ) are given by the -subgrid diagnosis (see section `7.1.1 <#sec:sginv>`__) except for +subgrid diagnosis (see section :ref:`Diagnosis of a sub-grid inversion `) except for :math:`K_m^{\rm Sc}` in the 8A scheme which uses the height of the half-level below (:math:`z_{\mbox{\tiny \rm NTML}+\frac{1}{2}}` or :math:`z_{\mbox{\tiny \rm NTDSC}+\frac{1}{2}}`). Again following @@ -1533,7 +1533,7 @@ that :math:`K_m^{\rm Sc}` will tend to here no restriction is made on the magnitude of either :math:`{\cal E}_m^{\rm Sc}` or :math:`{\cal E}_h^{\rm Sc}`). -.. _`sec:gradadj`: +.. _sec_gradadj: Gradient adjustment ------------------- @@ -1583,9 +1583,9 @@ tend to make :math:`q_t` profiles less well mixed than those of :math:`\theta_{\ell}` :raw-latex:`\cite[]{mahrt1976}`. From UM version 5.5, there is the option to implement the non-gradient stress parametrization of `Brown and Grant (1997)`_, as described in -section `5.4 <#sec:ngstress>`__. +section :ref:`Non-gradient stress parametrization `. -.. _`sec:ngstress`: +.. _sec_ngstress: Non-gradient stress parametrization ----------------------------------- @@ -1637,7 +1637,7 @@ was to ensure that the match to surface layer similarity was maintained below :math:`0.1z_{\rm h}` (although separate tests suggested that the impact of this change is small). -.. _`sec:rev_flux_grad`: +.. _sec_rev_flux_grad: The revised scalar flux-gradient formulation -------------------------------------------- @@ -1852,7 +1852,7 @@ removed since the entrainment flux is now carried via the explicit * - .. image:: new_ktop_shape.svg -.. _`sec:blend`: +.. _sec_blend: The blended scheme ================== @@ -2027,7 +2027,7 @@ analysis of decoupled stratocumulus LES presented by `Honnert et al. (2011)`_ also included shallow cumulus simulations and showed that the relevent length scale there was the cloud top height. Most of the ``blending_option`` choices apply this to -all regimes diagnosed as cumulus-capped (see section `3 <#sec:types>`__) +all regimes diagnosed as cumulus-capped (see section :ref:`Diagnosis of boundary layer depth and type `) but alternatively (``blending_option``\ :math:`=`\ 4) this can be restricted to strictly shallow cumulus clouds, defined as contiguously cloudy levels (cloud fraction greater than SC_CFTOL) with cloud top @@ -2101,34 +2101,34 @@ blended turbulence scheme when pure cumulus convection was diagnosed and leave the representation of cumulus entirely to the resolved scales. This option (``blending_option``\ :math:`=`\ 2) is also now discouraged. -.. _`sec:entr`: +.. _sec_entr: Entrainment fluxes ================== **Summary**: parametrized entrainment fluxes (at the top of mixed layers) are specified for momentum through an eddy-diffusivity, as -described in section `7.4.1 <#sec:ent_K>`__. For scalar variables, if +described in section :ref:`For momentum (and scalars if no subgrid inversion) `. For scalar variables, if the inversion is sufficiently sharp so as to be unresolved, the ideal is to specify the entrainment fluxes explicitly, as described in -section `7.1 <#sec:ent_flux>`__, based on the subgrid inversion -diagnosis described in section `7.1.1 <#sec:sginv>`__. Further details +section :ref:`Specification of entrainment fluxes in the 9B scheme `, based on the subgrid inversion +diagnosis described in section :ref:`Diagnosis of a sub-grid inversion `. Further details can be found in `Lock (2001)`_. If the profiles are such that the inversion is sharp but a subgrid inversion cannot be diagnosed, an eddy-diffusivity similar to that for momentum is used (see -section `7.4.1 <#sec:ent_K>`__). If the inversion is thick enough to be +section :ref:`For momentum (and scalars if no subgrid inversion) `). If the inversion is thick enough to be resolved then an eddy diffusivity profile is constructed across the -inversion (see section `7.4.2 <#sec:entr_prof>`__) for both scalars and +inversion (see section :ref:`Resolved inversions `) for both scalars and momentum fields. For tracer variables (scalars other than :math:`\theta_{\ell}` and :math:`q_t`), the entrainment fluxes are specified using an equivalent eddy-diffusivity, as described in -section `7.4.3 <#sec:ent_K_flux>`__. Note that, as indicated below, +section :ref:`For tracers, when there is a subgrid inversion `. Note that, as indicated below, several aspects of the implementation of entrainment fluxes were revised at the 9C scheme and these are documented separately. The parametrization of the entrainment rate, :math:`w_e` (given, in the absence of subsidence, by the rate of rise of the inversion), can be -written (using the notation given in appendix `11 <#app:vscales>`__) +written (using the notation given in appendix :ref:`Appendix: Definitions of the velocity scales `) .. math:: :label: we_parm @@ -2157,12 +2157,12 @@ depth-scale for the radiatively-cooled layer (taken to be 15 :math:`\times \,\mbox{max}[200/z_c, 1]`, where :math:`z_c` is the cloud depth). To allow for a feedback with forcing of entrainment by buoyancy reversal -(see appendix `11 <#app:vscales>`__), +(see appendix :ref:`Appendix: Definitions of the velocity scales `), :math:`\tilde{\alpha_t} = \alpha_t+ Br (1-\alpha_t)`. following `Lock (1998)`_ and `Lock (2009)`_. The calculation of the other quantities required for :eq:`we_parm` is described in -appendix `11 <#app:vscales>`__. At some point during the transition to a +appendix :ref:`Appendix: Definitions of the velocity scales `. At some point during the transition to a decoupled boundary layer the surface-driven entrainment terms (the terms in :eq:`we_parm` proportional to :math:`V_{\rm heat}^3` and :math:`u_*`) will no longer contribute to entrainment at cloud top, @@ -2197,12 +2197,12 @@ than one grid-level in a timestep. With current vertical resolutions and timesteps this is not a serious restriction. The constants :math:`A_1` and :math:`A_{\rm br}` appeared to be determined within 10-20 % in `Lock (1998)`_, although only solid cloud sheets were -simulated (as discussed further in appendix `11 <#app:vscales>`__). +simulated (as discussed further in appendix :ref:`Appendix: Definitions of the velocity scales `). Similarly the parametrizations of :math:`\alpha_t` and :math:`\Delta z_i` were found to be accurate but the parameter :math:`L_{rad}` is currently only crudely represented in the UM. -.. _`sec:ent_flux`: +.. _sec_ent_flux: Specification of entrainment fluxes in the 9B scheme ---------------------------------------------------- @@ -2247,7 +2247,7 @@ similarly for DSC layers). The thermodynamic variables’ entrainment fluxes, then, are imposed nominally at the subgrid inversion height (:math:`z_i=` :math:`z_{\rm h}` and/or :math:`z_{\rm h}^{\rm Sc}` ), diagnosed as -described in section `7.1.1 <#sec:sginv>`__. The required grid-level +described in section :ref:`Diagnosis of a sub-grid inversion `. The required grid-level fluxes (at :math:`z_{\mbox{\tiny \rm NTDSC}+\frac{1}{2}}`, for example) are then estimated using linear interpolation of :math:`{\cal H}` and :math:`\overline{w'q_t'}` between :math:`z_{\rm h}^{\rm Sc}` and the @@ -2297,7 +2297,7 @@ fixed through the timestep and so it is consistent to assume the entrainment fluxes (at :math:`z_i`) are also fixed. Hence :eq:`fluxinterp` are implemented explicitly, rather than via an eddy-diffusivity. This is discussed further, with reference to -tracer fluxes, in section `7.4.3 <#sec:ent_K_flux>`__. +tracer fluxes, in section :ref:`For tracers, when there is a subgrid inversion `. In order to allow for the long timesteps used in NWP and to facilitate movement of the subgrid inversion across grid-levels within a timestep, @@ -2359,7 +2359,7 @@ and ( \Delta_{\mbox{\tiny \rm NTML}+\frac{1}{2}} z ) } { \Delta \theta_{\ell}} -.. _`sec:sginv`: +.. _sec_sginv: Diagnosis of a sub-grid inversion ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ @@ -2391,8 +2391,8 @@ it to diffuse out this static instability). Having identified the model grid-level at the top of the well-mixed layer (either level NTML from the parcel ascent, as described in -section `3.1 <#sec:adiapar>`__, or NTDSC for DSC layers, see section -`3.2 <#sec:decouple>`__— the analysis is the same for both), the +section :ref:`The diagnostic parcel ascent and cumulus diagnosis `, or NTDSC for DSC layers, see section +:ref:`Diagnosis of the vertical extent of the K-profiles `— the analysis is the same for both), the grid-level above is designated the inversion level within which the diagnosis of a subgrid :math:`z_i` will be made. It is assumed that :math:`\theta_{v\ell}` in grid-level NTML\ :math:`+1` represents a @@ -2509,12 +2509,12 @@ NTML\ :math:`+1` is essentially meaningless (being diagnosed from a mixture of cloudy boundary layer air and typically very dry free tropospheric air). -.. _`sec:ent_flux_9c`: +.. _sec_ent_flux_9c: Specification of entrainment fluxes across sharp inversions in the 9C scheme ---------------------------------------------------------------------------- -As described in section `7.1 <#sec:ent_flux>`__, when the capping +As described in section :ref:`Specification of entrainment fluxes in the 9B scheme `, when the capping inversion is thinner than the model vertical grid it is important for the entrainment flux implementation that the subsidence increments are realistically and consistently distributed between the inversion @@ -2584,7 +2584,7 @@ given by: F_{\chi}^{Tot}|_{z_h} = - w_e \Delta \chi + F_{\chi}^{NTP}|_{z_t} + F_{\chi}^{subs}|_{z_h} -As in section `7.1 <#sec:ent_flux>`__, :eq:`fxtot_zi` is +As in section :ref:`Specification of entrainment fluxes in the 9B scheme `, :eq:`fxtot_zi` is derived by integrating the conservation equation for :math:`\chi` over an inversion in which jumps occur over a thin layer with base at a height :math:`z_h` and top at :math:`z_t` (in the UM, the inversion is @@ -2738,7 +2738,7 @@ free-atmospheric lapse rates are given by \right] -.. _`sec:subs_calc`: +.. _sec_subs_calc: Calculation of the subsidence flux ---------------------------------- @@ -2779,13 +2779,13 @@ assumptions may be violated and so the entrainment fluxes are specified via an entrainment eddy diffusivity. Currently, this is also the case for momentum and tracer variables. -.. _`sec:ent_K`: +.. _sec_ent_K: For momentum (and scalars if no subgrid inversion) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ For momentum, and scalars if a subgrid inversion cannot be diagnosed, -see section `7.1.1 <#sec:sginv>`__, fluxes at the mixed layer top are +see section :ref:`Diagnosis of a sub-grid inversion `, fluxes at the mixed layer top are specified through an eddy diffusivity which is given by .. math:: @@ -2799,7 +2799,7 @@ specified through an eddy diffusivity which is given by noting the Charney-Philips grid implying stresses are staggered from scalar fluxes. The Prandtl number, :math:`Pr`, takes the same form as -for the non-local :math:`K` profiles, see section `5 <#sec:nonlocal>`__. +for the non-local :math:`K` profiles, see section :ref:`The non-local scheme `. Substituting :eq:`khent` in :eq:`scal_closure` gives, for example, @@ -2831,20 +2831,20 @@ imposed at the height of the temperature inversion the :math:`{\cal E}` factors. -.. _`sec:entr_prof`: +.. _sec_entr_prof: Resolved inversions ~~~~~~~~~~~~~~~~~~~ An inversion is defined as being resolved when it extends above the flux-level above the usual entrainment interface level (see -section `3.3 <#sec:dzi>`__), i.e. when +section :ref:`Diagnosis of inversion thickness `), i.e. when .. math:: z_{\mbox{\tiny \rm NTML}+\frac{1}{2}} + \Delta z_i > z_{\mbox{\tiny \rm NTML}+\frac{3}{2}} When this happens, there is no subgrid inversion diagnosis and the entrainment parametrization follows the methodology given in -section `7.4.1 <#sec:ent_K>`__ to give +section :ref:`For momentum (and scalars if no subgrid inversion) ` to give :math:`K_h|_{\mbox{\tiny \rm NTML}+\frac{1}{2}}`. The diffusion coefficient profile within the inversion is then calculated assuming the :math:`\theta_{v\ell}` flux profile within the inversion decreases @@ -2869,7 +2869,7 @@ The diffusion coefficient for momentum entrainment is calculated in the same way, allowing for the staggered grid, with the same :math:`Pr` as in :eq:`khent`. -.. _`sec:ent_K_flux`: +.. _sec_ent_K_flux: For tracers, when there is a subgrid inversion ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ @@ -2901,7 +2901,7 @@ equivalent entrainment eddy-diffusivity given by: Note from :eq:`scal_closure` that :eq:`K_ent_tracer` gives the parametrized flux if :math:`\Delta_{\mbox{\tiny \rm NTML}+1} \chi` does not change across the -timestep (see section `9 <#sec:implicit>`__ for a description of the +timestep (see section :ref:`Implicit solution of the diffusion equation ` for a description of the implicit numerical solution of :eq:`cons_eqn_scal`). As :eq:`K_ent_tracer` involves the potentially numerically dangerous calculation of @@ -2961,7 +2961,7 @@ where F\ :math:`_{B0}` is the surface buoyancy flux defined by F_{B0} = \frac{ g }{ c_P } \beta _{T1} H_0 + g \beta _{q1} E_0. The buoyancy coefficients in equation :eq:`1.1.5` are given -in appendix `12 <#app:buoyp>`__ with the subscript 1 denoting a value at +in appendix :ref:`Appendix: Derivation and definitions of the buoyancy parameters ` with the subscript 1 denoting a value at the lowest level in the atmosphere model. Equations :eq:`1.1.1`–:eq:`1.1.3` can be @@ -4238,7 +4238,7 @@ Two approaches are available in uncoupled configurations of the model. < C_H >= (1 - f_I) C_{H(L)} + f_I C_{H(I)} -.. _`sec:coast`: +.. _sec_coast: Surface exchange in coastal grid-boxes ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ @@ -4857,7 +4857,7 @@ numerator of :math:`Ri_{SL}`, :math:`U` is the wind speed and the subscript :math:`k \ell` indicates the :math:`\theta`-level containing :math:`\ell`. -.. _`sec:implicit`: +.. _sec_implicit: Implicit solution of the diffusion equation =========================================== @@ -4964,7 +4964,7 @@ intermediate “starred” quantities are eliminated and the scheme is reduced into a single equation then the forcing term will be multiplied by :math:`1`. -Recall from section `2 <#sec:closure>`__ that the boundary layer solver +Recall from section :ref:`Model variables and turbulence closure ` that the boundary layer solver computes the increment of :math:`X`, where :math:`X=u,\; v,\;\theta_{L},\; q_{w}`. Let :math:`\delta X^{*}=X^{*}-X^{n}`, :math:`\delta @@ -4991,7 +4991,7 @@ Writing equations :eq:`eq:sppf_bl1`, X^{n+1} = X^{n}+\delta X^{*}+\delta X^{n+1} -.. _`sec:impsolve`: +.. _sec_impsolve: Discrete equations and boundary conditions ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ @@ -5642,7 +5642,7 @@ scheme. [\ *Could it be that coefficients :math:`D_{j}`, Blending height coupling ~~~~~~~~~~~~~~~~~~~~~~~~ -The same method is used as in section `9.1.2 <#sec:impsolve>`__ to form +The same method is used as in section :ref:`Discrete equations and boundary conditions ` to form two independent tridiagonal systems of linear equations that relate the increments to momentum, temperature and humidity to the surface fluxes. The ‘downward sweep’ elimination procedure still takes place to obtain @@ -5688,7 +5688,7 @@ possible to have significant heat flux and boundary layer depth in windy conditions that should not lead to a strong thermal forecast. This sensitivity of boundary layer turbulence is already included in the parametrization of non-local momentum fluxes (see section -`5.4 <#sec:ngstress>`__) through the stability dependence in +:ref:`Non-gradient stress parametrization `) through the stability dependence in :eq:`tau_nl` that can be written as .. math:: f_{stab} = - \frac{a_{stab} z_{\rm h}/L }{1 - a_{stab} z_{\rm h}/L} @@ -5916,7 +5916,7 @@ to UKCA. Note that additional diagnostics of the scalar variances are also made and those are documented in . -.. _`app:neutwind`: +.. _app_neutwind: Diagnostics of Neutral Winds and Stresses: stash 3,365 to 3,371 --------------------------------------------------------------- @@ -5941,7 +5941,7 @@ coefficient. Pseudostress is sometimes used in observational products, notably the Cross-Calibrated Multi-Platform (CCMP) surface wind vector analysis :raw-latex:`\cite[]{atlas2011}`. -.. _`app:vscales`: +.. _app_vscales: Appendix: Definitions of the velocity scales ============================================ @@ -5974,7 +5974,7 @@ Here, where the subscript :math:`_S` indicates the surface flux; :math:`\Delta_F` is the divergence of the net radiative flux, :math:`F` (in Kms\ :math:`^{-1}`), associated with cloud-top, for which the -calculation is described in section `11.1 <#app:deltaf>`__. +calculation is described in section :ref:`Calculation of \Delta_F `. Various depth parameters are given by :math:`\zeta_s = (z_{\rm ml}-\tilde{z_c})/z_{\rm ml}`, @@ -6070,7 +6070,7 @@ Note that, if :math:`k_b=1` in :eq:`zc_calc`, then the 8A calculation of :math:`z_c` is to include the depth to which the cloud extends into the inversion grid-level. If a subgrid inversion height, :math:`z_i`, has been diagnosed (see -section `7.1.1 <#sec:sginv>`__) then the height of :math:`z_i` above the +section :ref:`Diagnosis of a sub-grid inversion `) then the height of :math:`z_i` above the half-level height is added to :math:`z_c` (as long as :math:`C_F>` SC_CFTOL in grid-levels NTML or NTML\ :math:`+1` or the layer is a decoupled layer). If no subgrid inversion has been diagnosed and @@ -6111,7 +6111,7 @@ inversion is given by The empirical constant :math:`A_{\rm br}= 0.24`. The calculation of :math:`\Delta \theta_{\ell}` and :math:`\Delta q_t` is described for a subgrid -inversion in section `7.1.1 <#sec:sginv>`__ or, if one is not diagnosed, +inversion in section :ref:`Diagnosis of a sub-grid inversion ` or, if one is not diagnosed, they are taken simply as :math:`\Delta_{\mbox{\tiny \rm NTML}+1}`. For :math:`\Delta q_{\ell}`, :math:`\Delta q_f` and :math:`{q_{\ell}}_{\rm ct}`, in-cloud values extrapolated to :math:`z_i` @@ -6184,7 +6184,7 @@ arbitrary term has currently been dropped from version 5 onwards. The issue of how the entrainment rate should be parametrized in partially cloudy boundary layers, however, remains. -.. _`app:deltaf`: +.. _app_deltaf: Calculation of :math:`\Delta_F` ------------------------------- @@ -6335,7 +6335,7 @@ These small changes were found sufficient to give a robust measure of :math:`\Delta F` for grids varying down to 20m spacing where the radiative flux profile is well resolved. -.. _`app:buoyp`: +.. _app_buoyp: Appendix: Derivation and definitions of the buoyancy parameters =============================================================== @@ -6415,7 +6415,7 @@ In all the above, if :math:`T` is less than the melting point of ice then the latent heat of sublimation, :math:`L_s = L + L_f`, is used in place of :math:`L`. -.. _`app:mixratio`: +.. _app_mixratio: Appendix: changing between specific humidities and mixing ratios ================================================================ @@ -6492,7 +6492,7 @@ ratios as: For saturation calculations a version of QSAT is used that is switchable between input specific and mixing ratio variables. The rate of change of :math:`q_s` with temperature is also used in the boundary layer code -(see e.g. appendix `12 <#app:buoyp>`__): +(see e.g. appendix :ref:`Appendix: Derivation and definitions of the buoyancy parameters `): .. math:: \frac{ d q_{sat} }{ dT } = \frac{\epsilon L q_{sat} }{RT^2} @@ -6500,7 +6500,7 @@ In fact this expression should really be converted to work for specific quantities and so simply changing to mixing ratios will improve the accuracy of this calculation. -Finally, in appendix `12 <#app:buoyp>`__ virtual temperature is defined +Finally, in appendix :ref:`Appendix: Derivation and definitions of the buoyancy parameters ` virtual temperature is defined in terms of specific variables as .. math:: T_v = T ( 1 + c_v q_v - q_l ) @@ -6529,7 +6529,7 @@ Additional points to note are: #. it is important to note that RHOKM will be wet density times :math:`K_m` while RHOKH will be dry density times :math:`K_h` -.. _`app:fricheat`: +.. _app_fricheat: Appendix: including the heating from turbulence dissipation =========================================================== @@ -6574,7 +6574,7 @@ within the BL (i.e. up to :math:`z_{\rm h}` ) and then that total heating is applied as a linear decrease from the surface to zero over :math:`z_{\rm h}` . -.. _`app:opmods`: +.. _app_opmods: Appendix: Operational modifications =================================== @@ -6757,7 +6757,7 @@ the UKCA code owner before lodging the change. - TURBULENT KINETIC ENERGY - ACTIVATE cloud scheme -.. _`app:not`: +.. _app_not: Appendix: Notation ================== @@ -6801,7 +6801,7 @@ Appendix: Notation - virtual temperature and potential temperature, * - - - defined by :eq:`Tv` and in section (`3.1.1 <#sec:parxs>`__) + - defined by :eq:`Tv` and in section (:ref:`Calculation of parcel buoyancy excess `) * - :math:`b` - buoyancy (:math:`=g T_v'/T_v`) @@ -6957,15 +6957,15 @@ Appendix: Notation - for initial identification of and :math:`z_{\rm ml}` calculation for DSC layers * - :math:`a_L`, :math:`\alpha_L`, :math:`\beta_T`, :math:`\beta_q`, :math:`\tilde{\beta_T}`, :math:`\tilde{\beta_q}` - - buoyancy parameters, defined in appendix `12 <#app:buoyp>`__ + - buoyancy parameters, defined in appendix :ref:`Appendix: Derivation and definitions of the buoyancy parameters ` .. [1] unless the option to mix across the LCL is selected, see - section `3.4 <#sec:lclmixing>`__ + section :ref:`Diagnosis of the LCL transition zone thickness ` .. [2] unless the option to mix across the LCL is selected, see - section `3.4 <#sec:lclmixing>`__ + section :ref:`Diagnosis of the LCL transition zone thickness ` .. [3] If i_impsolve_loc = 1, the boundary-layer implicit solver is @@ -6979,7 +6979,7 @@ Appendix: Notation surface-fluxes over the current timestep may improve the accuracy of the convective closure. Also the non-turbulent fluxes used to construct the budgets at entrainment grid-levels (section - `5.5 <#sec:rev_flux_grad>`__) do not include contributions from + :ref:`The revised scalar flux-gradient formulation `) do not include contributions from convection, so arguably excluding them from :math:`S` is consistent. In the presence of convective subsidence, the top grid-level of the sub-cloud mixed layer gets warmed and dried by the subsidence @@ -7382,4 +7382,4 @@ References *Estimates of Surface Wind Stress and Drag Coefficients in {T}yphoon {M}egi*. J. Phys. Oceanogr., 47, 545–565. - https://doi.org/10.1175/JPO-D-16-0069.1 \ No newline at end of file + https://doi.org/10.1175/JPO-D-16-0069.1 From 086e26591cfa1bbc6a6d1d4474a80d743f5bc5e5 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Mon, 27 Apr 2026 15:18:06 +0100 Subject: [PATCH 057/116] Fixed multi-line in-line math blocks with incorrect indentation. --- .../turbulence_schemes/bldoc.rst | 70 ++++++------------- 1 file changed, 23 insertions(+), 47 deletions(-) diff --git a/documentation/source/science_guide/turbulence_schemes/bldoc.rst b/documentation/source/science_guide/turbulence_schemes/bldoc.rst index 47a4f3521a..c433957a35 100644 --- a/documentation/source/science_guide/turbulence_schemes/bldoc.rst +++ b/documentation/source/science_guide/turbulence_schemes/bldoc.rst @@ -81,8 +81,7 @@ that the option to use mixing ratios in the boundary layer code instead of specific quantities is also available and the details of the necessary changes are documented in appendix :ref:`Appendix: changing between specific humidities and mixing ratios `. Ultimately turbulent motions are dissipated as heat and so the source -term :math:`{\cal - S}` in :eq:`cons_eqn_scal` can include an +term :math:`{\cal S}` in :eq:`cons_eqn_scal` can include an approximation for that frictional heating, as described in appendix :ref:`Appendix: including the heating from turbulence dissipation `. Finally, the ice cloud contributions in :eq:`thetal` and :eq:`qt` can optionally be ignored @@ -486,8 +485,7 @@ strength of cloud-capping inversions). To do this, the :math:`\theta_v` gradient between grid-levels :math:`k_{ct}` and :math:`k_{ct}-1` is compared with that for a parcel lifted adiabatically from grid-level :math:`k_{ct}-1`, in exactly the same way as for the SML parcel ascent -(see section :ref:`Calculation of parcel buoyancy excess `). If :math:`d\theta_v/dz|_{\rm - env} > \Gamma_{\rm inv} d\theta_v/dz|_{\rm par}` between grid-levels +(see section :ref:`Calculation of parcel buoyancy excess `). If :math:`d\theta_v/dz|_{\rm env} > \Gamma_{\rm inv} d\theta_v/dz|_{\rm par}` between grid-levels :math:`k_{ct}` and :math:`k_{ct}-1` then NTDSC is set to :math:`k_{ct}-1`; if not then NTDSC is set to :math:`k_{ct}` (recall that grid-levels :math:`k_{ct}-1` and :math:`k_{ct}-2` have already been @@ -669,8 +667,7 @@ finite-difference form of :math:`\overline{w'b}`, see above, :math:`\overline{w'b}` is assumed to be linear between :math:`\overline{w'b}_S` at the surface and zero at a level which must be estimated. The surface layer integration is then from the surface up -to :math:`z_{{\rm - K_{SURF}}}`, where :math:`\theta`-level K_SURF is the first above +to :math:`z_{{\rm K_{SURF}}}`, where :math:`\theta`-level K_SURF is the first above :math:`z_i/10`. The level where :math:`\overline{w'b}` is zero is found by linear interpolation across the grid-levels where the diagnosed cloud-free buoyancy flux would become negative. This is where @@ -1239,15 +1236,13 @@ therefore RI(K), are held on the ‘half-level’ below :math:`\rho`-level K, which is :math:`\theta`-level K-1. In addition to the above, the log profile correction applied to -:math:`{\cal - L}_h` (to give :math:`\tilde{{\cal L}}_h`) must be applied *after* +:math:`{\cal L}_h` (to give :math:`\tilde{{\cal L}}_h`) must be applied *after* interpolation of :math:`K_h` to level :math:`k+\frac{1}{2}` in order that the correct cancellation with the finite difference scalar gradient in the flux calculation can occur. In the unstable stability functions :eq:`unstable_stab`, however, :math:`\tilde{{\cal L}}_h` must be calculated on :math:`\theta`-levels -(i.e., the same as :math:`\tilde{{\cal - L}}_m` and :math:`Ri`) in order to maintain the same stability +(i.e., the same as :math:`\tilde{{\cal L}}_m` and :math:`Ri`) in order to maintain the same stability dependence. .. _sec_shear: @@ -1388,8 +1383,7 @@ factor :math:`{\cal E}_m^{\rm surf}` is chosen so that eddy-diffusivity (given by :eq:`khent`, although, in order to avoid altering the shape function too much, :math:`{\cal E}_m^{\rm surf}` is not allowed to fall below :math:`0.7`). -A similar factor, :math:`{\cal E}_h^{\rm - surf}`, is used in the :math:`K_h^{\rm surf}` profile even though the +A similar factor, :math:`{\cal E}_h^{\rm surf}`, is used in the :math:`K_h^{\rm surf}` profile even though the entrainment fluxes of the thermodynamic variables will usually be specified explicitly rather than through an eddy-diffusivity (see section :ref:`Entrainment fluxes `). @@ -1437,10 +1431,8 @@ Comparison with `Holtslag and Boville (1993)`_ The surface-driven :math:`K` profiles are the same as those in `Holtslag and Boville (1993)`_, HB93, except for :eq:`ws_defn` and -:eq:`prandtl_nl` and the inclusion of the :math:`{\cal - E}_m^{\rm surf}` terms. For the latter, HB93 effectively set -:math:`{\cal - E}_m^{\rm surf} =1`. To generate entrainment, however, they simply use +:eq:`prandtl_nl` and the inclusion of the :math:`{\cal E}_m^{\rm surf}` terms. For the latter, HB93 effectively set +:math:`{\cal E}_m^{\rm surf} =1`. To generate entrainment, however, they simply use :math:`K_m^{\rm surf}|_{\mbox{\tiny \rm NTML}+\frac{1}{2}}`, as evaluated from :eq:`kmsurf` with a subgrid calculation of :math:`z_{\rm h}` :math:`>z_{\mbox{\tiny \rm NTML}+\frac{1}{2}}`, rather @@ -1460,8 +1452,7 @@ k (z/z_i) w_*^3 / u_*^3 )^{-(1/4)}`, would require a complex function of attempted. The formula for the Prandtl number used in the interior in HB93 is also -matched to that used in the surface exchange functions (:math:`Pr_{\rm - surf}`, say). For the UM, +matched to that used in the surface exchange functions (:math:`Pr_{\rm surf}`, say). For the UM, .. math:: @@ -1471,8 +1462,7 @@ matched to that used in the surface exchange functions (:math:`Pr_{\rm giving :math:`Pr_{\rm surf} = 1` in the neutral limit (compared to 0.75 from :eq:`prandtl_nl`). In the convective limit, -:math:`Pr_{\rm surf}|_{0.1\, - z_{\rm h}} \rightarrow 0.9 (w_*/u_*)^{-3/4} = 0.9 \beta^{3/4} = 0.14` +:math:`Pr_{\rm surf}|_{0.1\, z_{\rm h}} \rightarrow 0.9 (w_*/u_*)^{-3/4} = 0.9 \beta^{3/4} = 0.14` (compared to 0.375 from :eq:`prandtl_nl`). Thus, the Prandtl numbers do not match between the surface layer and interior formulations in the UM. @@ -1530,8 +1520,7 @@ that :math:`K_m^{\rm Sc}` will tend to :math:`K_h^{\rm Sc}` to :math:`K_h|_{\mbox{\tiny \rm NTML}+\frac{1}{2}}`), given by :eq:`khent`, as :math:`z` tends to :math:`z_{\rm h}` (and -here no restriction is made on the magnitude of either :math:`{\cal - E}_m^{\rm Sc}` or :math:`{\cal E}_h^{\rm Sc}`). +here no restriction is made on the magnitude of either :math:`{\cal E}_m^{\rm Sc}` or :math:`{\cal E}_h^{\rm Sc}`). .. _sec_gradadj: @@ -1592,8 +1581,7 @@ Non-gradient stress parametrization There is an option that is operational in the UM to include an additional non-gradient (or non-local) stress parametrization, -:math:`{\bf - \tau}^{nl}` in :eq:`uv_closure`, as proposed by +:math:`{\bf \tau}^{nl}` in :eq:`uv_closure`, as proposed by `Brown and Grant (1997)`_. They showed that with only a down-gradient stress parametrization, a one-dimensional model produced wind profiles in the convective boundary layer that were less well-mixed @@ -1689,17 +1677,14 @@ The components of :eq:`fg_new` are: - :math:`K_{h,m}^{\rm surf}= k z_h w_{h,m} \frac{z}{z_h}\left(1-\frac{z}{z_h}\right)^2` -- :math:`K_h^{\rm Sc}= 3.6 k V_{\rm Sc}z_{ml} \left(\frac{z'}{z_{ml}}\right)^{3}\left(1-\frac{z'}{z_{ml}} - \right)^{2}` +- :math:`K_h^{\rm Sc}= 3.6 k V_{\rm Sc}z_{ml} \left(\frac{z'}{z_{ml}}\right)^{3}\left(1-\frac{z'}{z_{ml}} \right)^{2}` - :math:`\overline{w'\chi'}_{ng}^{\rm surf}=K_h^{\rm surf}\gamma_{\chi}` with :math:`\gamma_{\chi}=A_{ga}\frac{\overline{w'\chi'}_S}{w_h z_h}` and :math:`A_{ga}=10` -- :math:`\overline{w'\chi'}_{ng}^{\rm Sc}= f^{Sc} \left(F_{\chi}|_{z_h}- F_{\chi}^{NT}|_{z_{\rm b}} - \right)` with - :math:`f^{Sc}=3.5 \, k \, \frac{V_{\rm Sc}}{V_{\rm sum}} - \left(\frac{z}{z_h}\right)^{3}\left(1-\frac{z}{z_h}\right)` +- :math:`\overline{w'\chi'}_{ng}^{\rm Sc}= f^{Sc} \left(F_{\chi}|_{z_h}- F_{\chi}^{NT}|_{z_{\rm b}} \right)` with + :math:`f^{Sc}=3.5 \, k \, \frac{V_{\rm Sc}}{V_{\rm sum}} \left(\frac{z}{z_h}\right)^{3}\left(1-\frac{z}{z_h}\right)` - :math:`f_2 = 0.5 \, \frac{z}{z_h}\, 2^{(z/z_h)^4}` @@ -1909,8 +1894,7 @@ only difference is in the mixing length, which is calculated as l_{\rm blend} = W_{1D}l_{\rm bl}+(1-W_{1D})l_{\rm smag}, where :math:`l_{\rm bl}^{-1} = (\kappa z)^{-1} + \lambda_0^{-1}` and -:math:`l_{\rm - smag}^{-2} = (\kappa z)^{-2} + (c_s \Delta x)^{-2}`, :math:`\kappa` is +:math:`l_{\rm smag}^{-2} = (\kappa z)^{-2} + (c_s \Delta x)^{-2}`, :math:`\kappa` is the von Karman constant and :math:`c_s` is the Smagorinsky constant. Near the surface :math:`l_{\rm bl}` and :math:`l_{\rm smag}` are identical, but the asymptotic values are different and this method @@ -1971,8 +1955,7 @@ The simplest case is for a well-mixed boundary layer, where the appropriate lengthscale is the boundary-layer depth (inversion height). Therefore we set :math:`z_{\rm turb}=z_h`, which is broadly consistent with `Malavelle et al. (2014)`_, and choose -:math:`\beta=\beta_{\rm - bl}=0.15` to give the best match of our function to that of +:math:`\beta=\beta_{\rm bl}=0.15` to give the best match of our function to that of `Honnert et al. (2011)`_. These functions are shown in :numref:`Figure %s `\ (a) and are only dissimilar for small :math:`\Delta @@ -2331,8 +2314,7 @@ entrainment arising from the model’s resolved vertical advection (as discussed in `Lock (2001)`_). This is performed at whichever grid-level the entrainment fluxes are specified, to allow for any entrainment implied by a :math:`\theta_{\ell}` subsidence increment, -:math:`\Theta^{\rm - S}` (Ks\ :math:`^{-1}`), at the model grid-level below. The subsidence +:math:`\Theta^{\rm S}` (Ks\ :math:`^{-1}`), at the model grid-level below. The subsidence increments could be obtained directly in the SCM but in the full 3D UM advection increments are dominated by the horizontal component. The subsidence increments are calculated, therefore, from the vertical @@ -2828,8 +2810,7 @@ imposed at the height of the temperature inversion :math:`z_{\mbox{\tiny \rm NTML}+\frac{1}{2}}`) and :math:`K_m|_{\mbox{\tiny \rm NTML}+\frac{1}{2}}` is calculated from :eq:`kmsurf` and :eq:`kmtop`, noting the use of -the :math:`{\cal - E}` factors. +the :math:`{\cal E}` factors. .. _sec_entr_prof: @@ -2888,8 +2869,7 @@ turbulence forcing and the inversion jump change slowly compared to the timestep. Whilst this is true for atmospheric :math:`\theta_{\ell}` and :math:`q_t`, the latter is not true for tracers with a small boundary layer concentration. Consequently, for a tracer field :math:`\chi`, the -parametrized entrainment fluxes :math:`\overline{w'\chi'}_{ - z_{\mbox{\tiny \rm NTML}+\frac{1}{2}} }` are calculated from +parametrized entrainment fluxes :math:`\overline{w'\chi'}_{ z_{\mbox{\tiny \rm NTML}+\frac{1}{2}} }` are calculated from :eq:`fluxinterp` but are implemented through an equivalent entrainment eddy-diffusivity given by: @@ -4735,8 +4715,7 @@ and substituting for the surface stress this becomes Most configurations of the Unified Model currently use the latter assumption with the last iteration value of C\ :math:`_{D(f)}`, L and -:math:`v_{\ast - (f)}` used in the interpolation formula. +:math:`v_{\ast (f)}` used in the interpolation formula. If the observation height scalar quantities are assumed to lie on the mean profile defined by the effective roughness length and scaling @@ -5120,9 +5099,7 @@ quantities. Furthermore, .. math:: \bar{\tau}_{x}^{*}\Big|_{0}=\left({\cal I}_{1}-{\cal E}_{1}\right)\tau_{x}^{n}\Big|_{0}+{\cal I}_{1}\left(K_{u}\frac{\partial\delta u^{*}}{\partial z}\right)\Big|_{0} -Approximating :math:`\left(\frac{\partial\delta u^{*}}{\partial - z}\right)\Big|_{0}\approx\frac{\delta u_{1/2}^{*}-\delta - u_{0}^{*}}{z_{1/2}}`, and assuming that :math:`u_{0}=0` the previous +Approximating :math:`\left(\frac{\partial\delta u^{*}}{\partial z}\right)\Big|_{0}\approx\frac{\delta u_{1/2}^{*}-\delta u_{0}^{*}}{z_{1/2}}`, and assuming that :math:`u_{0}=0` the previous equation becomes .. math:: :label: eq:tau_zero @@ -5509,8 +5486,7 @@ explained earlier. The definitions .. math:: \overline{T_{1}^{n+1}}=\gamma_{2}T^{*}+\gamma_{1}\delta T_{1}^{n+1},\quad\overline{Q_{1}^{*}}=\gamma_{2}Q^{*}+\gamma_{1}\delta Q_{1}^{n+1},\qquad\delta T_{1}^{n+1}=T^{n+1}-T^{*},\quad\delta Q_{1}^{n+1}=Q^{n+1}-Q^{*} -are used here. The surface fluxes :math:`F_{T}^{*}\equiv{\displaystyle - \frac{H^{*}}{c_{p}}}`, :math:`F_{Q}^{*}\equiv E^{*}` are computed at +are used here. The surface fluxes :math:`F_{T}^{*}\equiv{\displaystyle \frac{H^{*}}{c_{p}}}`, :math:`F_{Q}^{*}\equiv E^{*}` are computed at model state :math:`{(T}_{1}^{*},Q_{1}^{*})`. They are the equivalent of the explicit fluxes :math:`F_{T}^{n}\equiv{\displaystyle \frac{H^{n}}{c_{p}}}`, From 4f1248265547fad34f3e60beb4038e8e143b751b Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Mon, 27 Apr 2026 15:49:48 +0100 Subject: [PATCH 058/116] Fixed tables with no name or title (altered the list-table conversion script so it omits these if not present). --- .../turbulence_schemes/bldoc.rst | 37 ++++++++----------- 1 file changed, 15 insertions(+), 22 deletions(-) diff --git a/documentation/source/science_guide/turbulence_schemes/bldoc.rst b/documentation/source/science_guide/turbulence_schemes/bldoc.rst index c433957a35..494f869ce2 100644 --- a/documentation/source/science_guide/turbulence_schemes/bldoc.rst +++ b/documentation/source/science_guide/turbulence_schemes/bldoc.rst @@ -1726,7 +1726,7 @@ Discussion of some of the revisions .. list-table:: Convective and Neutral limits for velocity scales :name: tab:vscales - :header-rows: 1 + * - Formulation - Convective limit @@ -5284,9 +5284,8 @@ therefore needs to be computed only once, when the 1st or predictor stage is computed, i.e. :math:`X^{*}`. Briefly the following calculations take place for the scalar variables: -.. list-table:: Table - :name: table_name - :header-rows: 1 +.. list-table:: + * - CALL bdy_impl3(): - set up coefficients for :eq:`eq:dX_disc_top`, :eq:`eq:dX_disc` and do a downward sweep; @@ -6579,9 +6578,8 @@ results for these variables it will prevent UKCA jobs from regressing. If the changes are significant it would be prudent to discuss them with the UKCA code owner before lodging the change. -.. list-table:: Table - :name: table_name - :header-rows: 1 +.. list-table:: + * - Boundary layer inputs to UKCA - @@ -6738,9 +6736,8 @@ the UKCA code owner before lodging the change. Appendix: Notation ================== -.. list-table:: Table - :name: table_name - :header-rows: 1 +.. list-table:: + * - Finite difference notation - @@ -6763,9 +6760,8 @@ Appendix: Notation * - - across the capping inversion (see :eq:`dbinv` and following text) -.. list-table:: Table - :name: table_name - :header-rows: 1 +.. list-table:: + * - Model variables - @@ -6794,9 +6790,8 @@ Appendix: Notation * - :math:`{\cal H}` - total heat flux (net radiative plus turbulent, Kms\ :math:`^{-1}`) -.. list-table:: Table - :name: table_name - :header-rows: 1 +.. list-table:: + * - Thresholds - @@ -6822,9 +6817,8 @@ Appendix: Notation * - - before decoupling occurs -.. list-table:: Table - :name: table_name - :header-rows: 1 +.. list-table:: + * - Layer definitions and parameters - @@ -6889,9 +6883,8 @@ Appendix: Notation * - LCL - lifting condensation level -.. list-table:: Table - :name: table_name - :header-rows: 1 +.. list-table:: + * - Other parameters - From 6b86a332c9f4fd89c63c01b08f54da4eb326c276 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Mon, 27 Apr 2026 15:58:23 +0100 Subject: [PATCH 059/116] Added copyright header. --- .../source/science_guide/turbulence_schemes/bldoc.rst | 6 ++++++ 1 file changed, 6 insertions(+) diff --git a/documentation/source/science_guide/turbulence_schemes/bldoc.rst b/documentation/source/science_guide/turbulence_schemes/bldoc.rst index 494f869ce2..9fdf347f45 100644 --- a/documentation/source/science_guide/turbulence_schemes/bldoc.rst +++ b/documentation/source/science_guide/turbulence_schemes/bldoc.rst @@ -1,3 +1,9 @@ +.. ----------------------------------------------------------------------------- + (c) Crown copyright Met Office. All rights reserved. + The file LICENCE, distributed with this code, contains details of the terms + under which the code may be used. + ----------------------------------------------------------------------------- + =============================================== The Parametrization of Boundary Layer Processes =============================================== From 59ba524228670a892760c907f7e5b75915626b96 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Mon, 27 Apr 2026 15:59:57 +0100 Subject: [PATCH 060/116] Changed subsubsection header underline style as-per the style guide. --- .../turbulence_schemes/bldoc.rst | 50 +++++++++---------- 1 file changed, 25 insertions(+), 25 deletions(-) diff --git a/documentation/source/science_guide/turbulence_schemes/bldoc.rst b/documentation/source/science_guide/turbulence_schemes/bldoc.rst index 9fdf347f45..c808e384df 100644 --- a/documentation/source/science_guide/turbulence_schemes/bldoc.rst +++ b/documentation/source/science_guide/turbulence_schemes/bldoc.rst @@ -393,7 +393,7 @@ of the height of the capping inversion is attempted for .. _sec_parxs: Calculation of parcel buoyancy excess -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ As described in appendix :ref:`Appendix: Derivation and definitions of the buoyancy parameters `, virtual temperature, :math:`T_v = @@ -662,7 +662,7 @@ contribution to the TKE budget in :eq:`deccrit` and so allow shear-driven mixing to help maintain well-mixed layers. Surface layer :math:`\overline{w'b}` integration -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ In the surface layer, below :math:`z_i/10`, the :math:`K` profiles have a different functional form from the rest of the mixed layer. Rather @@ -686,7 +686,7 @@ becomes negative. .. _sec_wbint_inv: Integration of :math:`\overline{w'b}` close to the inversion -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Because of the large gradients often seen in fluxes close to the inversion (in particular, in the LW radiative flux), simple finite @@ -1432,7 +1432,7 @@ convective. The origin of the functional form of .. _sec_hbcomp: Comparison with `Holtslag and Boville (1993)`_ -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ The surface-driven :math:`K` profiles are the same as those in `Holtslag and Boville (1993)`_, HB93, @@ -1728,7 +1728,7 @@ and the range is now :math:`Pr_{\rm neut} = 0.75` to used to calculate :math:`K_m^{\rm Sc}`. Discussion of some of the revisions -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ .. list-table:: Convective and Neutral limits for velocity scales :name: tab:vscales @@ -2350,7 +2350,7 @@ and .. _sec_sginv: Diagnosis of a sub-grid inversion -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ The profile of :math:`\theta_{v\ell}` is used to diagnose the height of a discontinuous inversion because it is approximately conserved under @@ -2681,7 +2681,7 @@ treatment is attempted and entrainment is modelled using a straightforward eddy diffusivity. Calculation of the inversion jumps in the 9C scheme -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ In the 9B scheme, the discontinuous jumps in :math:`\theta_{\ell}` and :math:`q_t` that are used in the entrainment calculation were calculated @@ -2770,7 +2770,7 @@ for momentum and tracer variables. .. _sec_ent_K: For momentum (and scalars if no subgrid inversion) -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ For momentum, and scalars if a subgrid inversion cannot be diagnosed, see section :ref:`Diagnosis of a sub-grid inversion `, fluxes at the mixed layer top are @@ -2821,7 +2821,7 @@ the :math:`{\cal E}` factors. .. _sec_entr_prof: Resolved inversions -~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^ An inversion is defined as being resolved when it extends above the flux-level above the usual entrainment interface level (see @@ -2859,7 +2859,7 @@ in :eq:`khent`. .. _sec_ent_K_flux: For tracers, when there is a subgrid inversion -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Here ‘tracers’ refers to scalar variables other than :math:`\theta_{\ell}` and :math:`q_t`: aerosols, :math:`q_f`, etc. @@ -3130,7 +3130,7 @@ heat flux in very low mean wind conditions. .. _section_1.2: Comparison with the `Godfrey and Beljaars (1991)`_ formulation for gustiness -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ We can define the **mean gust speed** at height z\ :math:`_{1}` by @@ -3572,7 +3572,7 @@ subsection `8.6 <#section_1.6>`__ below. .. _mo_iter_corrn: Correction to the iterative algorithm -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ The above implementation of boundary-layer convective gustiness in the Monin-Obukhov iteration contains an inconsistency. The overall effect @@ -3705,7 +3705,7 @@ height (1.5 m) and the last iteration (N) values of C\ :math:`_{H}`, L and :math:`v_{\ast }` are used. The parametrization of decoupling -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ In the foregoing analysis it is tacitly assumed that the surface layer, up to the model’s lowest grid level, is in equilibrium with the surface @@ -3877,7 +3877,7 @@ mean value, :math:`<`\ c\ :math:`_{D}>`, when there is partial ice cover. Roughness Lengths over the Sea -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ The roughness lengths for momentum and scalars depend on both the atmospheric flow and the wave state. The dependence on wave state is not @@ -4043,7 +4043,7 @@ is useful to consider the treatment of high wind speeds as a seperate option. Surface exchange over sea ice -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ As explained above, when sea ice is present, surface exchange involves exchanges between the atmosphere and the open sea (L), the marginal ice @@ -4227,7 +4227,7 @@ Two approaches are available in uncoupled configurations of the model. .. _sec_coast: Surface exchange in coastal grid-boxes -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ In coupled ocean-atmosphere modelling the ocean requires appropriate surface stresses and fluxes over all ocean points. In coastal regions @@ -4322,7 +4322,7 @@ The modifications needed to incorporate orographic form drag. .. _section_2.1: Effective roughness lengths -~~~~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^^^^ Form drag is included in the surface turbulent flux formulation via effective roughness lengths for momentum :raw-latex:`\cite[]{wood93}` @@ -4445,7 +4445,7 @@ Equation :eq:`2.1.13` implies that C_{D(f)} = C_{D(eff)} {\left( {1 + \frac{1}{2} c_{D(orog)} f_D \frac{A}{S} {\left( {\frac{\ln ( z_c / z_{0m} )}{k}} \right)}^2 } \right)}^{-1} Parametrized orographic drag coefficient -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ `Wood and Mason (1993)`_ find that orographic drag coefficient c\ :math:`_{D(orog)}` depends on A/S via the equation @@ -4516,7 +4516,7 @@ if the `Wood and Mason (1993)`_ formulation is used. .. _section_2.2: The iterative algorithm for calculating the effective surface exchange coefficients -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ For unstable conditions, i.e. :math:`\Delta`\ B :math:`<` 0 : @@ -4678,7 +4678,7 @@ from .. _section_2.3: Interpolation of surface layer variables to standard observation heights -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ If the observation height wind is assumed to lie on the profile defined by the effective roughness length and surface scaling velocity then @@ -4861,7 +4861,7 @@ accurate. In practical simulations :math:`P` may vary from timestep to timestep and from column to column. Algorithmic description -~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^ Consider the non-linear damping equation: @@ -4979,7 +4979,7 @@ Writing equations :eq:`eq:sppf_bl1`, .. _sec_impsolve: Discrete equations and boundary conditions -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ To derive the boundary conditions for the horizontal wind components we adapt the technique used in the original scheme. @@ -5333,7 +5333,7 @@ NB: for CABLE compatibility, sf_impl2 is now called by an intermediate routine surf_couple_implicit. Flux diagnostic formulae -~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^ The original boundary layer implicit solver computes the total stress by time averaging the stresses at :math:`t^{n}` and :math:`t^{n+1}`, where @@ -5376,7 +5376,7 @@ derived in a similar way. These formulae have been validated in SCM experiments. Implicit surface flux and future upgrades -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ This scheme should be incorporated in the calculation of the scalar implicit fluxes. This would be preferable to the current technique for @@ -5621,7 +5621,7 @@ scheme. [\ *Could it be that coefficients :math:`D_{j}`, :math:`\psi_{j}` need to be modified in the second sweep?*] Blending height coupling -~~~~~~~~~~~~~~~~~~~~~~~~ +^^^^^^^^^^^^^^^^^^^^^^^^ The same method is used as in section :ref:`Discrete equations and boundary conditions ` to form two independent tridiagonal systems of linear equations that relate the From 8da40a84297406d191f61c5fbcbbdc225e7fbba1 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Mon, 27 Apr 2026 16:38:02 +0100 Subject: [PATCH 061/116] Split lines > 80 characters, as per the style guide. --- .../turbulence_schemes/bldoc.rst | 995 ++++++++++++------ 1 file changed, 684 insertions(+), 311 deletions(-) diff --git a/documentation/source/science_guide/turbulence_schemes/bldoc.rst b/documentation/source/science_guide/turbulence_schemes/bldoc.rst index c808e384df..f4a6dc2368 100644 --- a/documentation/source/science_guide/turbulence_schemes/bldoc.rst +++ b/documentation/source/science_guide/turbulence_schemes/bldoc.rst @@ -51,13 +51,15 @@ horizontal components of momentum, :math:`{\bf u}` on a sphere gives: .. math:: :label: cons_eqn_scal \frac{\partial \chi}{\partial t} - = - \frac{1}{r^2 \rho} \, \frac{\partial }{\partial z} \left( r^2 \rho \overline{w'\chi'} \right) + = - \frac{1}{r^2 \rho} \, \frac{\partial }{\partial z} \left( r^2 \rho + \overline{w'\chi'} \right) + {\cal S} .. math:: :label: cons_eqn_uv \frac{\partial {\bf u}}{\partial t} - = \frac{1}{r^2 \rho} \, \frac{\partial }{\partial z} \left( r^2 {\bf \tau} \right) + = \frac{1}{r^2 \rho} \, \frac{\partial }{\partial z} \left( r^2 {\bf \tau} + \right) + {\cal S} @@ -85,11 +87,13 @@ water static energy’ (:math:`= c_p T + g z - L q_{\ell}- L_s q_f`) rather than potential temperature, :math:`\theta`. Note also that the option to use mixing ratios in the boundary layer code instead of specific quantities is also available and the details of the -necessary changes are documented in appendix :ref:`Appendix: changing between specific humidities and mixing ratios `. +necessary changes are documented in appendix :ref:`Appendix: changing between +specific humidities and mixing ratios `. Ultimately turbulent motions are dissipated as heat and so the source term :math:`{\cal S}` in :eq:`cons_eqn_scal` can include an approximation for that frictional heating, as described in -appendix :ref:`Appendix: including the heating from turbulence dissipation `. Finally, the ice cloud contributions in +appendix :ref:`Appendix: including the heating from turbulence dissipation +`. Finally, the ice cloud contributions in :eq:`thetal` and :eq:`qt` can optionally be ignored (l_noice_in_turb), which will be more appropriate if the time scales for ice melting or sublimation are longer than those of the turbulence (and @@ -114,11 +118,13 @@ measure of buoyancy. A ‘first-order’ closure is used to parametrize the turbulent fluxes, although non-local terms are also included. Under the 9C scheme, an alternative methodology is optionally available, see section -:ref:`The revised scalar flux-gradient formulation `. The standard closures are: +:ref:`The revised scalar flux-gradient formulation `. The +standard closures are: .. math:: :label: scal_closure - \overline{w'\chi'} = - K_h \frac{\partial \chi}{\partial z} + K_h^{\rm surf}\gamma_{\chi} + \overline{w'\chi'} = - K_h \frac{\partial \chi}{\partial z} + K_h^{\rm + surf}\gamma_{\chi} .. math:: :label: uv_closure @@ -138,22 +144,27 @@ Thus, the parametrization reduces to determining :math:`K_h`, methods are used to determine :math:`K_h` and :math:`K_m` and how they are combined for :eq:`scal_closure` and :eq:`uv_closure` is described in -section :ref:`Shear-driven mixing and interaction between the local and non-local schemes `. The first method is a local Richardson +section :ref:`Shear-driven mixing and interaction between the local and +non-local schemes `. The first method is a local Richardson number (:math:`Ri`) based scheme. It is calculated for all regimes (but will be responsible for all mixing in stable conditions), over all levels up to the specified BL_LEVELS and is described in -section :ref:`The local scheme `. The second method is a non-locally specified +section :ref:`The local scheme `. The second method is a non-locally +specified profile scheme. This is exclusively for unstable boundary layers, is calculated up to level NL_BL_LEVELS (typically around 6km AMSL) and is -described in more detail in section :ref:`The non-local scheme `. In this +described in more detail in section :ref:`The non-local scheme `. +In this regime, mixing is assumed to occur in (or lead rapidly to the formation of) well-mixed layers (in which conserved variables are approximately uniform with height) that are capped by an inversion. Mixing is assumed to be driven either from the surface in a ‘surface mixed layer’ (SML, by a positive surface buoyancy flux and by surface stresses) or by cloud-top buoyancy sources (radiative and evaporative cooling, see -appendix :ref:`Appendix: Definitions of the velocity scales `). As described in section -:ref:`The non-local scheme `, separate :math:`K`-profiles are used for these +appendix :ref:`Appendix: Definitions of the velocity scales `). As +described in section +:ref:`The non-local scheme `, separate :math:`K`-profiles are +used for these two turbulence sources. If the cloud-top sources generate mixing throughout the SML the layer is said to be ‘coupled’ but if the :math:`K`-profile representing surface-driven mixing does not extend up @@ -161,16 +172,19 @@ to cloud-top, the layer is referred to as being ‘decoupled’. As decoupled layers are restricted to being buoyancy driven and typically below 6km, they are referred to as decoupled stratocumulus (DSC) layers. The calculation of :math:`\gamma_{\chi}` is described in -section :ref:`Gradient adjustment ` and, finally, fluxes across the top of +section :ref:`Gradient adjustment ` and, finally, fluxes across +the top of both SML and DSC layers (the entrainment fluxes) are specified explicitly through a separate entrainment parametrization, as described in section :ref:`Entrainment fluxes `. The strategy used to determine precisely where and when the resulting eddy-diffusivities should be applied is described in -section :ref:`Diagnosis of boundary layer depth and type `. The buoyancy parameters, finite difference +section :ref:`Diagnosis of boundary layer depth and type `. The +buoyancy parameters, finite difference and other notation used here are defined in -appendices :ref:`Appendix: Derivation and definitions of the buoyancy parameters ` and :ref:`Appendix: Notation `. Further papers +appendices :ref:`Appendix: Derivation and definitions of the buoyancy +parameters ` and :ref:`Appendix: Notation `. Further papers describing this scheme and its performance are `Lock et al. (2000)`_ (noting the corrigendum in `Lock et al. (2001)`_), @@ -186,7 +200,8 @@ Diagnosis of boundary layer depth and type The non-locally specified :math:`K`-profiles require the height of the base and top of the layer to be diagnosed (see section :ref:`The non-local scheme `). Furthermore, as stated in -section :ref:`Model variables and turbulence closure `, the mixing generated by the non-local +section :ref:`Model variables and turbulence closure `, the mixing +generated by the non-local :math:`K` profiles is assumed to occur in (or lead rapidly to the formation of) well-mixed layers capped by an inversion. Thus, the accurate diagnosis of their vertical extent is crucial. How to make this @@ -205,10 +220,12 @@ been categorised into 7 distinct ‘boundary layer types’: - **Type III**: Well mixed boundary layer — the classic single mixed layer which may be cloud-topped or clear but is predominantly buoyancy-driven (c.f. a possible type VII below) — diagnosis described - in section :ref:`The diagnostic parcel ascent and cumulus diagnosis `) + in section :ref:`The diagnostic parcel ascent and cumulus diagnosis + `) - **Type IV**: Unstable boundary layer with a DSC layer not over cumulus - (see section :ref:`Diagnosis of the vertical extent of the K-profiles `) — the surface-based and + (see section :ref:`Diagnosis of the vertical extent of the K-profiles + `) — the surface-based and cloud-top-driven non-local :math:`K` profiles may or may not overlap and cloud-top entrainment can still include the surface forcing (see section :ref:`Entrainment fluxes `) @@ -221,13 +238,15 @@ been categorised into 7 distinct ‘boundary layer types’: - **Type VI**: Cumulus-capped boundary layer — no turbulent diffusivities are allowed [1]_ at or above the LCL as the mass-flux convection scheme operates here (cumulus diagnosis described in - section :ref:`The diagnostic parcel ascent and cumulus diagnosis `) + section :ref:`The diagnostic parcel ascent and cumulus diagnosis + `) - **Type VII**: Shear-dominated unstable layer — potentially wind-shear might allow deeper turbulent mixing in unstable boundary layers than is apparent purely from the thermodynamic profiles (sufficient even to inhibit the formation of cumulus); the possibilities are discussed in - section :ref:`Shear-driven mixing and interaction between the local and non-local schemes `. + section :ref:`Shear-driven mixing and interaction between the local and + non-local schemes `. Types I to VI are shown schematically in :numref:`Fig. %s `. @@ -280,7 +299,8 @@ the first grid-level (:math:`k=k_s`) above the top of the surface layer, upwards allowing for latent heat release. The top of the surface layer is taken to be at the lower of :math:`z=0.1`\ :math:`z_{\rm h}` (this is then consistent with the :math:`K`-profiles, see -section :ref:`Surface-driven turbulence `; :math:`z_{\rm h}` is taken from the +section :ref:`Surface-driven turbulence `; :math:`z_{\rm h}` is +taken from the previous timestep) and the grid-level above which :math:`\theta_{v\ell}` starts to increase with height. The ascent is stopped at the grid-level NTPAR (height @@ -289,7 +309,8 @@ above which the parcel becomes more negatively buoyant than a given threshold, :math:`\theta_v'`. Note that the parcel properties themselves are not perturbed in order to preserve the height of the mixed-layer’s lifting condensation level (LCL). The calculation of the parcel’s -buoyancy excess is described in section :ref:`Calculation of parcel buoyancy excess `. +buoyancy excess is described in section :ref:`Calculation of parcel buoyancy +excess `. Currently, .. math:: :label: parcel_pert @@ -305,7 +326,8 @@ where :math:`A_{plume}=0.2`, :math:`B_{plume}=3.26`, `Holtslag and Boville (1993)`_, :math:`\theta_v'` is related to the magnitude of the gradient adjustment, :math:`\gamma_{\theta_{\ell}}` (see section -:ref:`Gradient adjustment `). Thus, :math:`B_{plume}=A_{ga}`, although +:ref:`Gradient adjustment `). Thus, :math:`B_{plume}=A_{ga}`, +although somewhat arbitrary limits have been placed on the magnitude of :math:`\theta_v'` for numerical security (the upper limit being consistent with that applied to :math:`\gamma_{\theta_{\ell}}` in @@ -365,7 +387,8 @@ cumulus is diagnosed, the top of the surface-based mixed layer (:math:`z_{\rm h}` ) is set to :math:`z_{\rm lcl}` (rather than to :math:`z_{\rm par}` , as illustrated in :numref:`Fig. %s ` for types V and VI). There is then an option to diagnose the thickness of -the LCL transition zone, see section :ref:`Diagnosis of the LCL transition zone thickness `. +the LCL transition zone, see section :ref:`Diagnosis of the LCL transition zone +thickness `. Otherwise, the boundary layer surface-driven mixing is capped at :math:`z_{\rm lcl}` so that mixing into the cumulus cloud layer is only carried out by the model’s mass-flux convection scheme and not by the @@ -388,14 +411,16 @@ BL_LEVELS above the tropopause is recommended. Note that if cumulus is not diagnosed then a further, subgrid estimation of the height of the capping inversion is attempted for -:math:`z_{\rm h}`  (as described in section :ref:`Diagnosis of a sub-grid inversion `). +:math:`z_{\rm h}`  (as described in section :ref:`Diagnosis of a sub-grid +inversion `). .. _sec_parxs: Calculation of parcel buoyancy excess ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ -As described in appendix :ref:`Appendix: Derivation and definitions of the buoyancy parameters `, virtual temperature, +As described in appendix :ref:`Appendix: Derivation and definitions of the +buoyancy parameters `, virtual temperature, :math:`T_v = T(1 + c_v q_v - q_{\ell}- q_f)`, is used as the measure of buoyancy. The condensed water in the parcel at a grid-level :math:`k` @@ -412,7 +437,8 @@ the environment at that grid-level \right] where the buoyancy parameters :math:`a_L` and :math:`\alpha_L` are -defined in appendix :ref:`Appendix: Derivation and definitions of the buoyancy parameters `. Recall that the parcel has +defined in appendix :ref:`Appendix: Derivation and definitions of the buoyancy +parameters `. Recall that the parcel has :math:`q_t` and :math:`\theta_{\ell}` taken from grid-level :math:`k_s` which are conserved during its ascent. Note that :eq:`qlpar` will not give condensation until the parcel becomes saturated. In the @@ -453,10 +479,12 @@ been separated in to three stages. These are: #. diagnose an approximate depth of the DSC layer, :math:`z_{\rm ml}`, in order to be able to calculate the representative turbulent - velocity scales (see appendix :ref:`Appendix: Definitions of the velocity scales `). + velocity scales (see appendix :ref:`Appendix: Definitions of the velocity + scales `). #. calculate the depth of the :math:`K` profiles (see - section :ref:`The non-local scheme `) in both SML and DSC layers using + section :ref:`The non-local scheme `) in both SML and DSC + layers using constraints on the TKE budget of the layer. This includes the diagnosis of recoupling of DSC layers and decoupling of SMLs @@ -491,7 +519,9 @@ strength of cloud-capping inversions). To do this, the :math:`\theta_v` gradient between grid-levels :math:`k_{ct}` and :math:`k_{ct}-1` is compared with that for a parcel lifted adiabatically from grid-level :math:`k_{ct}-1`, in exactly the same way as for the SML parcel ascent -(see section :ref:`Calculation of parcel buoyancy excess `). If :math:`d\theta_v/dz|_{\rm env} > \Gamma_{\rm inv} d\theta_v/dz|_{\rm par}` between grid-levels +(see section :ref:`Calculation of parcel buoyancy excess `). If +:math:`d\theta_v/dz|_{\rm env} > \Gamma_{\rm inv} d\theta_v/dz|_{\rm par}` +between grid-levels :math:`k_{ct}` and :math:`k_{ct}-1` then NTDSC is set to :math:`k_{ct}-1`; if not then NTDSC is set to :math:`k_{ct}` (recall that grid-levels :math:`k_{ct}-1` and :math:`k_{ct}-2` have already been @@ -509,7 +539,8 @@ perturbation is given by \theta_{v\ell}' = - \, \frac{ \tau_{rc} \Delta_F}{z_{rc}} where :math:`\Delta_F` (Kms\ :math:`^{-1}`) is the magnitude of the -cloud-top radiative divergence (see appendix :ref:`Appendix: Definitions of the velocity scales `), +cloud-top radiative divergence (see appendix :ref:`Appendix: Definitions of the +velocity scales `), :math:`\tau_{rc}` is a timescale for the exposure of boundary layer eddies to the cloud-top radiative cooling (taken to be 200s) and :math:`z_{rc}` is a depth-scale for the radiatively cooled layer (taken @@ -532,13 +563,16 @@ magnitude of the integrated buoyancy consumption of TKE within the mixed layer is less than or equal to a fraction, :math:`D_t`, of the buoyancy production, following `Turton and Nicholls (1987)`_. -Following appendix :ref:`Appendix: Derivation and definitions of the buoyancy parameters ` the grid-box mean buoyancy flux +Following appendix :ref:`Appendix: Derivation and definitions of the buoyancy +parameters ` the grid-box mean buoyancy flux can be written as: .. math:: :label: eq:wb_cont - \overline{w'b}= g \left[ (1-C_F) \left(\beta_T \overline{w'\theta_{\ell}'} + \beta_q \overline{w'q_t'}\right) + - C_F \left( \tilde{\beta_T} \overline{w'\theta_{\ell}'} + \tilde{\beta_q} \overline{w'q_t'}\right) + \overline{w'b}= g \left[ (1-C_F) \left(\beta_T \overline{w'\theta_{\ell}'} + + \beta_q \overline{w'q_t'}\right) + + C_F \left( \tilde{\beta_T} \overline{w'\theta_{\ell}'} + \tilde{\beta_q} + \overline{w'q_t'}\right) \right] As standard, the fluxes in :eq:`eq:wb_cont` are then @@ -547,7 +581,8 @@ expanded using the first-order closure in .. math:: - \overline{w'\theta_{\ell}'}_k = -K_h^{\rm surf}\,\frac{\widetilde{\Delta_k \theta_{\ell}}}{\Delta_k z} + \overline{w'\theta_{\ell}'}_k = -K_h^{\rm surf}\,\frac{\widetilde{\Delta_k + \theta_{\ell}}}{\Delta_k z} -K_h^{\rm Sc}\,\frac{\Delta_k \theta_{\ell}}{\Delta_k z} .. math:: :label: eq:wx_std @@ -560,7 +595,8 @@ where :math:`\widetilde{\Delta_k \theta_{\ell}} = \Delta_k \theta_{\ell}- \gamma_{\theta_{\ell}} \Delta_k z` in order to include the non-local (or gradient adjustment) term. If the alternative flux-gradient option is -used, see section :ref:`The revised scalar flux-gradient formulation `, then additional terms +used, see section :ref:`The revised scalar flux-gradient formulation +`, then additional terms are needed. Large-eddy simulations have demonstrated that the crucial region in @@ -584,7 +620,8 @@ buoyancy consumption of TKE within the mixed layer equals a fraction, .. math:: :label: deccrit \sum_{z_{k-\frac{1}{2}} > z_i-z_{\rm ml}}^{z_{k-\frac{1}{2}} < z_i} - \left|\left[ \overline{w'b}|_{z_{k-\frac{1}{2}}}<0 \right]\right| \, \Delta_k z \, + \left|\left[ \overline{w'b}|_{z_{k-\frac{1}{2}}}<0 \right]\right| \, + \Delta_k z \, \leq \, D_t \, \sum_{z_{k-\frac{1}{2}} > z_i-z_{\rm ml}}^{z_{k-\frac{1}{2}} < z_i} \left[ \overline{w'b}|_{z_{k-\frac{1}{2}}}>0 \right] \, \Delta_k z @@ -597,7 +634,8 @@ layers. For stratocumulus layers, observations and LES suggest a value of :math:`D_t=0.1`. A separate value of :math:`D_t` can be used for the sub-cloud layer in cumulus capped boundary layers if this method is used to determine the LCL transition zone thickness, see section -:ref:`Diagnosis of the LCL transition zone thickness `. For cloud-free mixed layers, :math:`D_t=1` is +:ref:`Diagnosis of the LCL transition zone thickness `. For +cloud-free mixed layers, :math:`D_t=1` is used, purely to keep negative buoyancy fluxes down to a reasonably realistic level (for example, if the parcel top diagnostic returned too high a boundary layer depth). @@ -652,7 +690,8 @@ is diagnosed to recouple completely). Finally, :math:`z_{\rm b}` must always be at or below :math:`z_{\mbox{\tiny \rm NTDSC}-1}`, so that mixing in decoupled layers is always resolved, and at least :math:`\Delta z_{rad}` (the cloud-top radiative cooling depth defined in -section :ref:`Integration of \overline{w'b} close to the inversion `) below the t inversion. The base +section :ref:`Integration of \overline{w'b} close to the inversion +`) below the t inversion. The base grid-level of the DSC layer, NBDSC, is defined (analogously to NTDSC) as the lowest grid-level such that :math:`K_h^{\rm Sc}` is non-zero at the half-level below. @@ -673,7 +712,8 @@ finite-difference form of :math:`\overline{w'b}`, see above, :math:`\overline{w'b}` is assumed to be linear between :math:`\overline{w'b}_S` at the surface and zero at a level which must be estimated. The surface layer integration is then from the surface up -to :math:`z_{{\rm K_{SURF}}}`, where :math:`\theta`-level K_SURF is the first above +to :math:`z_{{\rm K_{SURF}}}`, where :math:`\theta`-level K_SURF is the first +above :math:`z_i/10`. The level where :math:`\overline{w'b}` is zero is found by linear interpolation across the grid-levels where the diagnosed cloud-free buoyancy flux would become negative. This is where @@ -772,7 +812,8 @@ approximations, :eq:`wthl_int` becomes .. math:: - = \Delta z_{rad} \left(-w_e \Delta \theta_{\ell}+ \frac{2}{3} \Delta F\right) + = \Delta z_{rad} \left(-w_e \Delta \theta_{\ell}+ \frac{2}{3} \Delta + F\right) For the integral of :math:`\overline{w'q_t'}` across this cloud-top @@ -810,7 +851,8 @@ inversion, :math:`z_{top}`, in LES from where :math:`z_{nb}` is the level of neutral buoyancy (found by linear interpolation between grid-levels), :math:`w_m` is the boundary layer -velocity scale defined in section :ref:`Surface-driven turbulence ` and :math:`b` is +velocity scale defined in section :ref:`Surface-driven turbulence ` +and :math:`b` is the parcel buoyancy. Note that the constant in :eq:`dz_param` is the same as in `Beare (2008)`_ because :math:`6.3 = 2.5 * 4^{2/3}` and @@ -833,7 +875,8 @@ inversion thickness is then defined as Diagnosis of the LCL transition zone thickness ---------------------------------------------- -As described in section :ref:`The diagnostic parcel ascent and cumulus diagnosis `, when cumulus convection +As described in section :ref:`The diagnostic parcel ascent and cumulus +diagnosis `, when cumulus convection has been diagnosed surface-driven mixing was originally capped at :math:`z_{\rm lcl}` so that mixing into the cumulus cloud layer was only carried out by the model’s mass-flux convection scheme. This was seen to @@ -846,7 +889,8 @@ indistinguishable from those in cloud-free convective boundary layers and so the non-local surface-driven mixed layer K-profiles remain accurate up to this level. To diagnose the depth to which these profiles should penetrate above the LCL, the algorithm given in -section :ref:`Diagnosis of the vertical extent of the K-profiles ` to diagnose the extent of the K-profiles +section :ref:`Diagnosis of the vertical extent of the K-profiles +` to diagnose the extent of the K-profiles in decoupled boundary layers can be used (using the switch kprof_cu). This ensures that the magnitude of the integrated buoyancy consumption of TKE within the mixed layer is less than or equal to a fraction, @@ -858,7 +902,8 @@ LCL (but are too dry to reach their own LCL). Thus their buoyancy flux is given by :eq:`eq:wb_cont` with :math:`C_F=0`. Restricting the negative integral of this buoyancy flux then gives a new definition for :math:`z_{\rm h}`  that is then used in the calculation -of the surface-driven K-profiles in section :ref:`Surface-driven turbulence ` — the +of the surface-driven K-profiles in section :ref:`Surface-driven turbulence +` — the larger the value of :math:`D_t`, the higher :math:`z_{\rm h}` will be. Typically :math:`D_t=0.1` for decoupled stratocumulus layers while idealised clear-sky convective boundary layers (where the magnitude of @@ -939,7 +984,8 @@ namelist and :math:`z_{\rm loc}` is defined below. The orographic blending height, :math:`h_B` (only used within the boundary layer, as defined below), is given by -.. math:: h_B = {\rm max}\left[z_1+(z_{0m})_{\mbox{veg}}, 2^{1/2} \sigma_h \right] +.. math:: h_B = {\rm max}\left[z_1+(z_{0m})_{\mbox{veg}}, 2^{1/2} \sigma_h + \right] where :math:`\sigma_h` is the standard deviation of the height of the subgrid orography and :math:`(z_{0m})_{\mbox{veg}}` is the vegetative @@ -964,8 +1010,10 @@ The measure of buoyancy used in :math:`Ri` is where :math:`\overline{\beta_T}` and :math:`\overline{\beta_q}` are the grid-box mean (i.e., cloud weighted) buoyancy coefficients, that can be -defined in two different ways, see appendix :ref:`Appendix: Derivation and definitions of the buoyancy parameters ` and -section :ref:`Finite difference calculations `. Note that :eq:`Bdefn` reduces +defined in two different ways, see appendix :ref:`Appendix: Derivation and +definitions of the buoyancy parameters ` and +section :ref:`Finite difference calculations `. Note that +:eq:`Bdefn` reduces to a virtual temperature approximation of buoyancy in cloud-free air and that neutral buoyancy (in cloudy as well as cloud-free air) is implied by vertically uniform :math:`\theta_{\ell}` and :math:`q_t`. This is @@ -1004,7 +1052,8 @@ to be a measure of the boundary layer top (:math:`z_{\rm loc}` ) and the full-level below is designated NTLOC. In general :math:`Ri_{crit}=1` but a value of 0.25 is recommended for use with the ’SHARPEST’ stability functions, see below. If the boundary layer was diagnosed as -cumulus-capped by the non-local scheme (see section :ref:`Diagnosis of boundary layer depth and type `) +cumulus-capped by the non-local scheme (see section :ref:`Diagnosis of boundary +layer depth and type `) then :math:`z_{\rm loc}` is lowered to :math:`z_{\rm lcl}` (and :math:`K_h` and :math:`K_m` are set to zero from the base of grid-level NLCL upwards) so that transports into and within the cumulus cloud layer @@ -1166,7 +1215,8 @@ i_interp_local. The long-standing method is given by where :math:`\overline{\beta_T}` and :math:`\overline{\beta_q}` are the grid-box mean (i.e., cloud-fraction weighted) buoyancy coefficients, -defined in appendix :ref:`Appendix: Derivation and definitions of the buoyancy parameters `. Note that because this is +defined in appendix :ref:`Appendix: Derivation and definitions of the buoyancy +parameters `. Note that because this is defined on :math:`\theta`-levels, no vertical interpolation of cloud variables (fractional area and water contents), to which the buoyancy coefficients are very sensitive, is required. The volume-weighted @@ -1174,7 +1224,8 @@ gradients of :math:`\theta_{\ell}` and :math:`q_t` are calculated as .. math:: :label: gradient_interp - (D\chi DZ)_k =\left( (z_{k}-z_{k-\frac{1}{2}}) \, \frac{\Delta_{k+1} \chi}{\Delta_{k+1} z} + + (D\chi DZ)_k =\left( (z_{k}-z_{k-\frac{1}{2}}) \, \frac{\Delta_{k+1} + \chi}{\Delta_{k+1} z} + (z_{k+\frac{1}{2}}-z_{k}) \, \frac{\Delta_{k} \chi}{\Delta_{k} z} \right) / \Delta_{k+\frac{1}{2}} z @@ -1220,7 +1271,8 @@ unity. The total cloud volume fraction is then given by :math:`f_{tot} = {\rm min}[{C_F}_{k-1},{C_F}_k] + f_{edge}f_{lev}` and this is used to weight the saturated contribution to the buoyancy parameters on :math:`rho`-levels, e.g., -:math:`\overline{\beta_T}_{k-1/2} = f_{tot} \tilde{\beta_T}_{k-1/2} + (1-f_{tot}){\beta_T}_{k-1/2})`, +:math:`\overline{\beta_T}_{k-1/2} = f_{tot} \tilde{\beta_T}_{k-1/2} + +(1-f_{tot}){\beta_T}_{k-1/2})`, where the saturated and unsaturated buoyancy parameters are also intepolated to :math:`\rho`-levels using :eq:`gradient_interp`. @@ -1248,7 +1300,8 @@ that the correct cancellation with the finite difference scalar gradient in the flux calculation can occur. In the unstable stability functions :eq:`unstable_stab`, however, :math:`\tilde{{\cal L}}_h` must be calculated on :math:`\theta`-levels -(i.e., the same as :math:`\tilde{{\cal L}}_m` and :math:`Ri`) in order to maintain the same stability +(i.e., the same as :math:`\tilde{{\cal L}}_m` and :math:`Ri`) in order to +maintain the same stability dependence. .. _sec_shear: @@ -1265,7 +1318,8 @@ The general approach is to take :math:`K_{\chi}` in K_{\chi} = \mbox{max} \left[ (K_{\chi}^{\rm surf}+K_{\chi}^{\rm Sc}), K_{\chi}(Ri) \right] -As noted in section :ref:`Model variables and turbulence closure `, this implies that mixing in +As noted in section :ref:`Model variables and turbulence closure +`, this implies that mixing in stable boundary layers is determined exclusively by the local scheme, :math:`K_{\chi}(Ri)`. Continuing to calculate :math:`K_{\chi}(Ri)` in unstable boundary layers and using :eq:`klnl` is seen as the @@ -1274,9 +1328,11 @@ and unstable boundary layers. At the top of unstable mixed layers, great care is taken to ensure the parametrized entrainment mixing is implemented faithfully, see -section :ref:`Entrainment fluxes `). Consequently, if a subgrid inversion has +section :ref:`Entrainment fluxes `). Consequently, if a subgrid +inversion has been diagnosed capping a mixed layer (see -section :ref:`Diagnosis of a sub-grid inversion `), then :math:`K_{\chi}(Ri)` is set to +section :ref:`Diagnosis of a sub-grid inversion `), then +:math:`K_{\chi}(Ri)` is set to zero at the interfaces either side of the inversion grid-level. There are also options (using the switch Keep_Ri_FA) to set :math:`K_{\chi}(Ri)` to zero entirely above unstable boundary layers or @@ -1295,7 +1351,8 @@ cloud layers that have been diagnosed as cumulus-capped (which would be poorly represented by the current convection scheme). Several methods have been introduced that attempt to alleviate this problem, giving rise to the diagnosis of a “shear-dominated boundary layer” type (type VII), -discussed in section :ref:`Diagnosis of boundary layer depth and type `. The first (the “shear-dominated +discussed in section :ref:`Diagnosis of boundary layer depth and type +`. The first (the “shear-dominated boundary layer fix”) simply sets the CUMULUS flag to false if NTLOC :math:`>` NTPAR. This then ensures that the locally-determined :math:`K` are not set to zero above the LCL. Several more rigorous options are @@ -1353,11 +1410,13 @@ non-local in the sense that, at a given height within the boundary layer, :math:`K` is determined not by any local properties of the mean profiles at that height but solely by the magnitude of the turbulence forcing applied to the layer (as measured by the representative velocity -scales described in appendix :ref:`Appendix: Definitions of the velocity scales `) and the height +scales described in appendix :ref:`Appendix: Definitions of the velocity scales +`) and the height within the layer. The non-local scheme is therefore particularly robust but care must be taken where the profiles are applied. The calculation of the vertical position and extent of the :math:`K` profiles is -described in section :ref:`Diagnosis of boundary layer depth and type `. +described in section :ref:`Diagnosis of boundary layer depth and type +`. .. _sec_nlsurf: @@ -1378,7 +1437,8 @@ where :math:`w_m^3 = u_*^3 + w_s^3`, :math:`u_*` is the friction velocity (including the orographic roughness component) and :math:`w_s` is defined below. For the 9C version of the scheme, :math:`z_{\rm h}` is the diagnosed subgrid inversion height (see -section :ref:`Diagnosis of a sub-grid inversion `) for both :math:`K_h^{\rm surf}` and +section :ref:`Diagnosis of a sub-grid inversion `) for both +:math:`K_h^{\rm surf}` and :math:`K_m^{\rm surf}`. In the 8A version, :math:`K_m^{\rm surf}` uses :math:`z_{\rm h}` :math:`=z_{\mbox{\tiny \rm NTML}+\frac{1}{2}}`. The factor :math:`{\cal E}_m^{\rm surf}` is chosen so that @@ -1389,7 +1449,8 @@ factor :math:`{\cal E}_m^{\rm surf}` is chosen so that eddy-diffusivity (given by :eq:`khent`, although, in order to avoid altering the shape function too much, :math:`{\cal E}_m^{\rm surf}` is not allowed to fall below :math:`0.7`). -A similar factor, :math:`{\cal E}_h^{\rm surf}`, is used in the :math:`K_h^{\rm surf}` profile even though the +A similar factor, :math:`{\cal E}_h^{\rm surf}`, is used in the :math:`K_h^{\rm +surf}` profile even though the entrainment fluxes of the thermodynamic variables will usually be specified explicitly rather than through an eddy-diffusivity (see section :ref:`Entrainment fluxes `). @@ -1412,7 +1473,8 @@ parametrization in cloudy boundary layers). Note that :math:`w_s` is continuous across :math:`0.1`\ :math:`z_{\rm h}` and constant with height in the mixed layer. This form for :math:`w_s` is motivated by a desire to match the model’s surface transfer formulation within the -surface layer (as described further in section :ref:`Comparison with Holtslag and Boville (1993)_ `) +surface layer (as described further in section :ref:`Comparison with Holtslag +and Boville (1993)_ `) and to use a cubic sum of velocity scales within the mixed layer (consistent with dimensional analysis of the TKE equation, see `Holtslag and Boville (1993)`_). @@ -1437,8 +1499,10 @@ Comparison with `Holtslag and Boville (1993)`_ The surface-driven :math:`K` profiles are the same as those in `Holtslag and Boville (1993)`_, HB93, except for :eq:`ws_defn` and -:eq:`prandtl_nl` and the inclusion of the :math:`{\cal E}_m^{\rm surf}` terms. For the latter, HB93 effectively set -:math:`{\cal E}_m^{\rm surf} =1`. To generate entrainment, however, they simply use +:eq:`prandtl_nl` and the inclusion of the :math:`{\cal E}_m^{\rm surf}` terms. +For the latter, HB93 effectively set +:math:`{\cal E}_m^{\rm surf} =1`. To generate entrainment, however, they simply +use :math:`K_m^{\rm surf}|_{\mbox{\tiny \rm NTML}+\frac{1}{2}}`, as evaluated from :eq:`kmsurf` with a subgrid calculation of :math:`z_{\rm h}` :math:`>z_{\mbox{\tiny \rm NTML}+\frac{1}{2}}`, rather @@ -1458,7 +1522,8 @@ k (z/z_i) w_*^3 / u_*^3 )^{-(1/4)}`, would require a complex function of attempted. The formula for the Prandtl number used in the interior in HB93 is also -matched to that used in the surface exchange functions (:math:`Pr_{\rm surf}`, say). For the UM, +matched to that used in the surface exchange functions (:math:`Pr_{\rm surf}`, +say). For the UM, .. math:: @@ -1468,7 +1533,8 @@ matched to that used in the surface exchange functions (:math:`Pr_{\rm surf}`, s giving :math:`Pr_{\rm surf} = 1` in the neutral limit (compared to 0.75 from :eq:`prandtl_nl`). In the convective limit, -:math:`Pr_{\rm surf}|_{0.1\, z_{\rm h}} \rightarrow 0.9 (w_*/u_*)^{-3/4} = 0.9 \beta^{3/4} = 0.14` +:math:`Pr_{\rm surf}|_{0.1\, z_{\rm h}} \rightarrow 0.9 (w_*/u_*)^{-3/4} = 0.9 +\beta^{3/4} = 0.14` (compared to 0.375 from :eq:`prandtl_nl`). Thus, the Prandtl numbers do not match between the surface layer and interior formulations in the UM. @@ -1498,15 +1564,18 @@ Cloud-top-driven turbulence For cloud-top-driven turbulence over a layer of depth :math:`z_{\rm ml}` (with top at :math:`z_{\rm h}` or :math:`z_{\rm h}^{\rm Sc}` and base at -:math:`z_{\rm b}` , determined as in section :ref:`Diagnosis of the vertical extent of the K-profiles `), +:math:`z_{\rm b}` , determined as in section :ref:`Diagnosis of the vertical +extent of the K-profiles `), .. math:: :label: kmtop - K_m^{\rm Sc}= 0.63 \ k \ z_{\rm ml}\ V_{\rm Sc}\left( \frac{z'}{z_{\rm ml}} \right)^2 + K_m^{\rm Sc}= 0.63 \ k \ z_{\rm ml}\ V_{\rm Sc}\left( \frac{z'}{z_{\rm ml}} + \right)^2 \left( 1 - {\cal E}_m^{\rm Sc} \frac{z'}{z_{\rm ml}} \right)^{0.8} where :math:`V_{\rm Sc}^3= V_{\rm rad}^3+V_{\rm br}^3` (see -appendix :ref:`Appendix: Definitions of the velocity scales `) and :math:`z'` is height above +appendix :ref:`Appendix: Definitions of the velocity scales `) and +:math:`z'` is height above :math:`z_{\rm b}` . Then :math:`K_h = K_m / \mbox{Pr}`, where :math:`\mbox{Pr}=0.75`. The resulting :math:`K_h` profile was derived against convective cloudy LES, as described in @@ -1515,7 +1584,8 @@ against convective cloudy LES, as described in simply as a number in the middle of the range usually quoted for turbulent mixing in general. As with :eq:`kmsurf`, :math:`z_{\rm h}`  (or :math:`z_{\rm h}^{\rm Sc}` ) are given by the -subgrid diagnosis (see section :ref:`Diagnosis of a sub-grid inversion `) except for +subgrid diagnosis (see section :ref:`Diagnosis of a sub-grid inversion +`) except for :math:`K_m^{\rm Sc}` in the 8A scheme which uses the height of the half-level below (:math:`z_{\mbox{\tiny \rm NTML}+\frac{1}{2}}` or :math:`z_{\mbox{\tiny \rm NTDSC}+\frac{1}{2}}`). Again following @@ -1526,7 +1596,8 @@ that :math:`K_m^{\rm Sc}` will tend to :math:`K_h^{\rm Sc}` to :math:`K_h|_{\mbox{\tiny \rm NTML}+\frac{1}{2}}`), given by :eq:`khent`, as :math:`z` tends to :math:`z_{\rm h}` (and -here no restriction is made on the magnitude of either :math:`{\cal E}_m^{\rm Sc}` or :math:`{\cal E}_h^{\rm Sc}`). +here no restriction is made on the magnitude of either :math:`{\cal E}_m^{\rm +Sc}` or :math:`{\cal E}_h^{\rm Sc}`). .. _sec_gradadj: @@ -1537,7 +1608,8 @@ Recall that for :math:`\theta_{\ell}` only we use .. math:: :label: wthl - \overline{w'\theta_{\ell}'}= - K_h \frac{\partial \theta_{\ell}}{\partial z} + K_h^{\rm surf}\gamma_{\theta_{\ell}} + \overline{w'\theta_{\ell}'}= - K_h \frac{\partial \theta_{\ell}}{\partial z} + + K_h^{\rm surf}\gamma_{\theta_{\ell}} where @@ -1602,7 +1674,8 @@ that proposed by `Brown and Grant (1997)`_, written .. math:: :label: tau_nl (\tau_x^{nl},\tau_y^{nl})= \left[ - \frac{2.7w_*^3}{(u_*^3+0.6w_*^3)}\right] \left[ \left( \frac{z'}{z_{\rm h}'} + \frac{2.7w_*^3}{(u_*^3+0.6w_*^3)}\right] \left[ \left( \frac{z'}{z_{\rm + h}'} \right) \left( 1- \frac{z'}{z_{\rm h}'} \right)^2 \right] (\tau_x^{s},\tau_y^{s}) @@ -1665,7 +1738,8 @@ So, the new formulation is written: .. math:: :label: fg_new F_{\chi}^{Tot} = F_{\chi}^{NT}|_{z_{\rm b}} - -\left(K_h^{\rm surf}+ K_h^{\rm Sc}\right)\frac{\partial\overline{\chi}}{\partial z} + -\left(K_h^{\rm surf}+ K_h^{\rm + Sc}\right)\frac{\partial\overline{\chi}}{\partial z} + \overline{w'\chi'}_{ng}^{\rm surf}+ \overline{w'\chi'}_{ng}^{\rm Sc} + f_2 \left(F_{\chi}|_{z_h} - F_{\chi}^{NT}|_{z_{\rm b}} \right) @@ -1681,16 +1755,20 @@ non-turbulent component: The components of :eq:`fg_new` are: -- :math:`K_{h,m}^{\rm surf}= k z_h w_{h,m} \frac{z}{z_h}\left(1-\frac{z}{z_h}\right)^2` +- :math:`K_{h,m}^{\rm surf}= k z_h w_{h,m} + \frac{z}{z_h}\left(1-\frac{z}{z_h}\right)^2` -- :math:`K_h^{\rm Sc}= 3.6 k V_{\rm Sc}z_{ml} \left(\frac{z'}{z_{ml}}\right)^{3}\left(1-\frac{z'}{z_{ml}} \right)^{2}` +- :math:`K_h^{\rm Sc}= 3.6 k V_{\rm Sc}z_{ml} + \left(\frac{z'}{z_{ml}}\right)^{3}\left(1-\frac{z'}{z_{ml}} \right)^{2}` - :math:`\overline{w'\chi'}_{ng}^{\rm surf}=K_h^{\rm surf}\gamma_{\chi}` with :math:`\gamma_{\chi}=A_{ga}\frac{\overline{w'\chi'}_S}{w_h z_h}` and :math:`A_{ga}=10` -- :math:`\overline{w'\chi'}_{ng}^{\rm Sc}= f^{Sc} \left(F_{\chi}|_{z_h}- F_{\chi}^{NT}|_{z_{\rm b}} \right)` with - :math:`f^{Sc}=3.5 \, k \, \frac{V_{\rm Sc}}{V_{\rm sum}} \left(\frac{z}{z_h}\right)^{3}\left(1-\frac{z}{z_h}\right)` +- :math:`\overline{w'\chi'}_{ng}^{\rm Sc}= f^{Sc} \left(F_{\chi}|_{z_h}- + F_{\chi}^{NT}|_{z_{\rm b}} \right)` with + :math:`f^{Sc}=3.5 \, k \, \frac{V_{\rm Sc}}{V_{\rm sum}} + \left(\frac{z}{z_h}\right)^{3}\left(1-\frac{z}{z_h}\right)` - :math:`f_2 = 0.5 \, \frac{z}{z_h}\, 2^{(z/z_h)^4}` @@ -1900,7 +1978,8 @@ only difference is in the mixing length, which is calculated as l_{\rm blend} = W_{1D}l_{\rm bl}+(1-W_{1D})l_{\rm smag}, where :math:`l_{\rm bl}^{-1} = (\kappa z)^{-1} + \lambda_0^{-1}` and -:math:`l_{\rm smag}^{-2} = (\kappa z)^{-2} + (c_s \Delta x)^{-2}`, :math:`\kappa` is +:math:`l_{\rm smag}^{-2} = (\kappa z)^{-2} + (c_s \Delta x)^{-2}`, +:math:`\kappa` is the von Karman constant and :math:`c_s` is the Smagorinsky constant. Near the surface :math:`l_{\rm bl}` and :math:`l_{\rm smag}` are identical, but the asymptotic values are different and this method @@ -1923,7 +2002,8 @@ in Eq. :eq:`eq-kri` is given by :math:`l_{\rm blend}` in Eq. :eq:`eq-lblend`. The turbulent flux is then calculated as -.. math:: F_\chi=-K_\chi\frac{\partial \chi}{\partial z} + W_{1D}F_\chi^{\rm NL}, +.. math:: F_\chi=-K_\chi\frac{\partial \chi}{\partial z} + W_{1D}F_\chi^{\rm + NL}, where :math:`F_\chi^{\rm NL}` is the non-local flux. Therefore when :math:`W_{1D}=1`, the scheme of `Lock et al. (2000)`_ is recovered, @@ -1939,7 +2019,9 @@ function slightly, using .. math:: :label: eq-tanh - W_{1D} = 1 - \tanh\left(\beta\frac{z_{\rm turb}}{\Delta x}\right)\max\left[0,\min\left[1,r_f\left(l_0-\frac{\Delta x}{z_{\rm turb}}\right)\right] \right], + W_{1D} = 1 - \tanh\left(\beta\frac{z_{\rm turb}}{\Delta + x}\right)\max\left[0,\min\left[1,r_f\left(l_0-\frac{\Delta x}{z_{\rm + turb}}\right)\right] \right], where :math:`z_{\rm turb}` is the appropriate lengthscale of the turbulence, :math:`\beta` is a scaling parameter which controls the @@ -1961,7 +2043,8 @@ The simplest case is for a well-mixed boundary layer, where the appropriate lengthscale is the boundary-layer depth (inversion height). Therefore we set :math:`z_{\rm turb}=z_h`, which is broadly consistent with `Malavelle et al. (2014)`_, and choose -:math:`\beta=\beta_{\rm bl}=0.15` to give the best match of our function to that of +:math:`\beta=\beta_{\rm bl}=0.15` to give the best match of our function to +that of `Honnert et al. (2011)`_. These functions are shown in :numref:`Figure %s `\ (a) and are only dissimilar for small :math:`\Delta @@ -2002,7 +2085,8 @@ the decoupled cloud top we set .. math:: :label: zturb_dsc - z_{\rm turb}=\min\left[\max\left(z,z_{\rm sml}\right),\max\left(z_{\rm sc},z_h-z\right)\right], + z_{\rm turb}=\min\left[\max\left(z,z_{\rm sml}\right),\max\left(z_{\rm + sc},z_h-z\right)\right], where :math:`z_{\rm sml}` is the depth of the surface-based mixed layer :raw-latex:`\cite[i.e.~the depth through which a positively buoyant parcel @@ -2016,7 +2100,8 @@ analysis of decoupled stratocumulus LES presented by `Honnert et al. (2011)`_ also included shallow cumulus simulations and showed that the relevent length scale there was the cloud top height. Most of the ``blending_option`` choices apply this to -all regimes diagnosed as cumulus-capped (see section :ref:`Diagnosis of boundary layer depth and type `) +all regimes diagnosed as cumulus-capped (see section :ref:`Diagnosis of +boundary layer depth and type `) but alternatively (``blending_option``\ :math:`=`\ 4) this can be restricted to strictly shallow cumulus clouds, defined as contiguously cloudy levels (cloud fraction greater than SC_CFTOL) with cloud top @@ -2064,7 +2149,8 @@ horizontal and vertical grid sizes. For .. math:: W_{1D} = 1 + \frac{1}{2} \left( W_{1D}|_{z=z_{\rm turb}} - 1 \right) - \left[ 1 + {\rm cos}\left( \pi \, \frac{z-z_{\rm turb}}{z_{\rm fa}-z_{\rm turb}} + \left[ 1 + {\rm cos}\left( \pi \, \frac{z-z_{\rm turb}}{z_{\rm + fa}-z_{\rm turb}} \right) \right] The cosine term in square brackets transitions smoothly from 2 at @@ -2097,31 +2183,39 @@ Entrainment fluxes **Summary**: parametrized entrainment fluxes (at the top of mixed layers) are specified for momentum through an eddy-diffusivity, as -described in section :ref:`For momentum (and scalars if no subgrid inversion) `. For scalar variables, if +described in section :ref:`For momentum (and scalars if no subgrid inversion) +`. For scalar variables, if the inversion is sufficiently sharp so as to be unresolved, the ideal is to specify the entrainment fluxes explicitly, as described in -section :ref:`Specification of entrainment fluxes in the 9B scheme `, based on the subgrid inversion -diagnosis described in section :ref:`Diagnosis of a sub-grid inversion `. Further details +section :ref:`Specification of entrainment fluxes in the 9B scheme +`, based on the subgrid inversion +diagnosis described in section :ref:`Diagnosis of a sub-grid inversion +`. Further details can be found in `Lock (2001)`_. If the profiles are such that the inversion is sharp but a subgrid inversion cannot be diagnosed, an eddy-diffusivity similar to that for momentum is used (see -section :ref:`For momentum (and scalars if no subgrid inversion) `). If the inversion is thick enough to be +section :ref:`For momentum (and scalars if no subgrid inversion) `). +If the inversion is thick enough to be resolved then an eddy diffusivity profile is constructed across the -inversion (see section :ref:`Resolved inversions `) for both scalars and +inversion (see section :ref:`Resolved inversions `) for both +scalars and momentum fields. For tracer variables (scalars other than :math:`\theta_{\ell}` and :math:`q_t`), the entrainment fluxes are specified using an equivalent eddy-diffusivity, as described in -section :ref:`For tracers, when there is a subgrid inversion `. Note that, as indicated below, +section :ref:`For tracers, when there is a subgrid inversion `. +Note that, as indicated below, several aspects of the implementation of entrainment fluxes were revised at the 9C scheme and these are documented separately. The parametrization of the entrainment rate, :math:`w_e` (given, in the absence of subsidence, by the rate of rise of the inversion), can be -written (using the notation given in appendix :ref:`Appendix: Definitions of the velocity scales `) +written (using the notation given in appendix :ref:`Appendix: Definitions of +the velocity scales `) .. math:: :label: we_parm - w_e = \frac{ A_1 \, V_{\rm sum}^3/ z_{\rm ml}+ g \tilde{\beta_T} \tilde{\alpha_t} + w_e = \frac{ A_1 \, V_{\rm sum}^3/ z_{\rm ml}+ g \tilde{\beta_T} + \tilde{\alpha_t} \Delta_F} {\Delta b + c_T V_{\rm sum}^2/z_{\rm ml}} @@ -2146,12 +2240,14 @@ depth-scale for the radiatively-cooled layer (taken to be 15 :math:`\times \,\mbox{max}[200/z_c, 1]`, where :math:`z_c` is the cloud depth). To allow for a feedback with forcing of entrainment by buoyancy reversal -(see appendix :ref:`Appendix: Definitions of the velocity scales `), +(see appendix :ref:`Appendix: Definitions of the velocity scales +`), :math:`\tilde{\alpha_t} = \alpha_t+ Br (1-\alpha_t)`. following `Lock (1998)`_ and `Lock (2009)`_. The calculation of the other quantities required for :eq:`we_parm` is described in -appendix :ref:`Appendix: Definitions of the velocity scales `. At some point during the transition to a +appendix :ref:`Appendix: Definitions of the velocity scales `. At +some point during the transition to a decoupled boundary layer the surface-driven entrainment terms (the terms in :eq:`we_parm` proportional to :math:`V_{\rm heat}^3` and :math:`u_*`) will no longer contribute to entrainment at cloud top, @@ -2186,7 +2282,8 @@ than one grid-level in a timestep. With current vertical resolutions and timesteps this is not a serious restriction. The constants :math:`A_1` and :math:`A_{\rm br}` appeared to be determined within 10-20 % in `Lock (1998)`_, although only solid cloud sheets were -simulated (as discussed further in appendix :ref:`Appendix: Definitions of the velocity scales `). +simulated (as discussed further in appendix :ref:`Appendix: Definitions of the +velocity scales `). Similarly the parametrizations of :math:`\alpha_t` and :math:`\Delta z_i` were found to be accurate but the parameter :math:`L_{rad}` is currently only crudely represented in the UM. @@ -2223,7 +2320,8 @@ mixed layers is simply calculated as .. math:: - F_{\rm net}|_{z_{k+\frac{1}{2}}} = \sum_{k=\mbox{\tiny \rm NBDSC}}^{k} \mbox{max}\left[ + F_{\rm net}|_{z_{k+\frac{1}{2}}} = \sum_{k=\mbox{\tiny \rm NBDSC}}^{k} + \mbox{max}\left[ - \Delta_{k+\frac{1}{2}} z \, {\cal S}_F(k), \,0 \right] where NBDSC\ :math:`=1` in SMLs, :math:`{\cal S}_F` are the temperature @@ -2236,7 +2334,8 @@ similarly for DSC layers). The thermodynamic variables’ entrainment fluxes, then, are imposed nominally at the subgrid inversion height (:math:`z_i=` :math:`z_{\rm h}` and/or :math:`z_{\rm h}^{\rm Sc}` ), diagnosed as -described in section :ref:`Diagnosis of a sub-grid inversion `. The required grid-level +described in section :ref:`Diagnosis of a sub-grid inversion `. The +required grid-level fluxes (at :math:`z_{\mbox{\tiny \rm NTDSC}+\frac{1}{2}}`, for example) are then estimated using linear interpolation of :math:`{\cal H}` and :math:`\overline{w'q_t'}` between :math:`z_{\rm h}^{\rm Sc}` and the @@ -2244,14 +2343,17 @@ base of the mixed layer: .. math:: - \overline{w'\theta_{\ell}'}|_{ z_{\mbox{\tiny \rm NTDSC}+\frac{1}{2}} } = \overline{w'\theta_{\ell}'}|_{z_{\rm b}} + \overline{w'\theta_{\ell}'}|_{ z_{\mbox{\tiny \rm NTDSC}+\frac{1}{2}} } = + \overline{w'\theta_{\ell}'}|_{z_{\rm b}} - \frac{ z'_{\mbox{\tiny \rm NTDSC}+\frac{1}{2}} }{z_{\rm ml}} - \left( \tilde{w_e} \Delta \theta_{\ell}+ \overline{w'\theta_{\ell}'}|_{z_{\rm b}} - F_{\rm net}|_{h} \right) + \left( \tilde{w_e} \Delta \theta_{\ell}+ + \overline{w'\theta_{\ell}'}|_{z_{\rm b}} - F_{\rm net}|_{h} \right) - F_{\rm net}|_{ z_{\mbox{\tiny \rm NTDSC}+\frac{1}{2}} } .. math:: :label: fluxinterp - \overline{w'q_t'}|_{ z_{\mbox{\tiny \rm NTDSC}+\frac{1}{2}} } = \overline{w'q_t'}|_{z_{\rm b}} + \overline{w'q_t'}|_{ z_{\mbox{\tiny \rm NTDSC}+\frac{1}{2}} } = + \overline{w'q_t'}|_{z_{\rm b}} - \frac{ z'_{\mbox{\tiny \rm NTDSC}+\frac{1}{2}} }{z_{\rm ml}} \left( \tilde{w_e} \Delta q_t + \overline{w'q_t'}|_{z_{\rm b}} \right) @@ -2286,7 +2388,8 @@ fixed through the timestep and so it is consistent to assume the entrainment fluxes (at :math:`z_i`) are also fixed. Hence :eq:`fluxinterp` are implemented explicitly, rather than via an eddy-diffusivity. This is discussed further, with reference to -tracer fluxes, in section :ref:`For tracers, when there is a subgrid inversion `. +tracer fluxes, in section :ref:`For tracers, when there is a subgrid inversion +`. In order to allow for the long timesteps used in NWP and to facilitate movement of the subgrid inversion across grid-levels within a timestep, @@ -2320,7 +2423,8 @@ entrainment arising from the model’s resolved vertical advection (as discussed in `Lock (2001)`_). This is performed at whichever grid-level the entrainment fluxes are specified, to allow for any entrainment implied by a :math:`\theta_{\ell}` subsidence increment, -:math:`\Theta^{\rm S}` (Ks\ :math:`^{-1}`), at the model grid-level below. The subsidence +:math:`\Theta^{\rm S}` (Ks\ :math:`^{-1}`), at the model grid-level below. The +subsidence increments could be obtained directly in the SCM but in the full 3D UM advection increments are dominated by the horizontal component. The subsidence increments are calculated, therefore, from the vertical @@ -2379,8 +2483,10 @@ it to diffuse out this static instability). Having identified the model grid-level at the top of the well-mixed layer (either level NTML from the parcel ascent, as described in -section :ref:`The diagnostic parcel ascent and cumulus diagnosis `, or NTDSC for DSC layers, see section -:ref:`Diagnosis of the vertical extent of the K-profiles `— the analysis is the same for both), the +section :ref:`The diagnostic parcel ascent and cumulus diagnosis +`, or NTDSC for DSC layers, see section +:ref:`Diagnosis of the vertical extent of the K-profiles `— the +analysis is the same for both), the grid-level above is designated the inversion level within which the diagnosis of a subgrid :math:`z_i` will be made. It is assumed that :math:`\theta_{v\ell}` in grid-level NTML\ :math:`+1` represents a @@ -2410,17 +2516,20 @@ The coefficients are given by .. math:: b = - \left( {\theta_{v\ell}}_{\mbox{\tiny \rm NTML}+2} - - \gamma^{\scriptsize \rm FA}(z_{\mbox{\tiny \rm NTML}+2}-z_{\mbox{\tiny \rm NTML}+\frac{3}{2}}) \right) + - \gamma^{\scriptsize \rm FA}(z_{\mbox{\tiny \rm NTML}+2}-z_{\mbox{\tiny + \rm NTML}+\frac{3}{2}}) \right) + \left( {\theta_{v\ell}}_{\mbox{\tiny \rm NTML}} + \gamma^{\tiny \rm ML}(z_{\mbox{\tiny \rm NTML}+\frac{3}{2}}-z_{\mbox{\tiny \rm NTML}}) \right) .. math:: - c = (z_{\mbox{\tiny \rm NTML}+\frac{3}{2}}-z_{\mbox{\tiny \rm NTML}+\frac{1}{2}}) + c = (z_{\mbox{\tiny \rm NTML}+\frac{3}{2}}-z_{\mbox{\tiny \rm + NTML}+\frac{1}{2}}) \left( {\theta_{v\ell}}_{\mbox{\tiny \rm NTML}+1} - \left( {\theta_{v\ell}}_{\mbox{\tiny \rm NTML}} + \gamma^{\tiny \rm ML}\left( - \frac{1}{2}(z_{\mbox{\tiny \rm NTML}+\frac{1}{2}}+z_{\mbox{\tiny \rm NTML}+\frac{3}{2}}) + \frac{1}{2}(z_{\mbox{\tiny \rm NTML}+\frac{1}{2}}+z_{\mbox{\tiny \rm + NTML}+\frac{3}{2}}) -z_{\mbox{\tiny \rm NTML}} \right) \right) \right) @@ -2449,7 +2558,8 @@ model simulations, the error in :math:`\Delta z_{disc}` is estimated to be around 10% of the vertical resolution, :math:`\Delta_{\mbox{\tiny \rm NTML}+\frac{3}{2}} z`. Accordingly, if :math:`z_i` is diagnosed as being less than -:math:`z_{\mbox{\tiny \rm NTML}+\frac{1}{2}} + 0.1 \, \Delta_{\mbox{\tiny \rm NTML}+\frac{3}{2}} z`, +:math:`z_{\mbox{\tiny \rm NTML}+\frac{1}{2}} + 0.1 \, \Delta_{\mbox{\tiny \rm +NTML}+\frac{3}{2}} z`, NTML is lowered a grid-level and :math:`z_i` is set fractionally below :math:`z_{\mbox{\tiny \rm NTML}+\frac{1}{2}}`. This small distance below the grid-level is taken to be :math:`(\Delta t/2) \times 10^{-4}` so @@ -2467,8 +2577,10 @@ calculation are calculated from similar integral assumptions: .. math:: :label: dqt_disc - \Delta \chi = \left( {\chi}_{\mbox{\tiny \rm NTML}+1} - {\chi}_{\mbox{\tiny \rm NTML}} \right) \, - \frac{ z_{\mbox{\tiny \rm NTML}+\frac{3}{2}} - z_{\mbox{\tiny \rm NTML}+\frac{1}{2}} } + \Delta \chi = \left( {\chi}_{\mbox{\tiny \rm NTML}+1} - {\chi}_{\mbox{\tiny + \rm NTML}} \right) \, + \frac{ z_{\mbox{\tiny \rm NTML}+\frac{3}{2}} - z_{\mbox{\tiny \rm + NTML}+\frac{1}{2}} } { z_{\mbox{\tiny \rm NTML}+\frac{3}{2}} - z_i } with :math:`\chi = \theta_{\ell}` and :math:`q_t`. Note that the lapse @@ -2502,7 +2614,8 @@ tropospheric air). Specification of entrainment fluxes across sharp inversions in the 9C scheme ---------------------------------------------------------------------------- -As described in section :ref:`Specification of entrainment fluxes in the 9B scheme `, when the capping +As described in section :ref:`Specification of entrainment fluxes in the 9B +scheme `, when the capping inversion is thinner than the model vertical grid it is important for the entrainment flux implementation that the subsidence increments are realistically and consistently distributed between the inversion @@ -2570,15 +2683,18 @@ given by: .. math:: :label: fxtot_zi - F_{\chi}^{Tot}|_{z_h} = - w_e \Delta \chi + F_{\chi}^{NTP}|_{z_t} + F_{\chi}^{subs}|_{z_h} + F_{\chi}^{Tot}|_{z_h} = - w_e \Delta \chi + F_{\chi}^{NTP}|_{z_t} + + F_{\chi}^{subs}|_{z_h} -As in section :ref:`Specification of entrainment fluxes in the 9B scheme `, :eq:`fxtot_zi` is +As in section :ref:`Specification of entrainment fluxes in the 9B scheme +`, :eq:`fxtot_zi` is derived by integrating the conservation equation for :math:`\chi` over an inversion in which jumps occur over a thin layer with base at a height :math:`z_h` and top at :math:`z_t` (in the UM, the inversion is assumed to be infinitesimally thin so that :math:`z_t=z_h`). This integration gives :math:`- -w_e \Delta \chi = \overline{w'\chi'}|_{z_h} -(F_{\chi}^{NTP}|_{z_t}-F_{\chi}^{NTP}|_{z_h})`. +w_e \Delta \chi = \overline{w'\chi'}|_{z_h} +-(F_{\chi}^{NTP}|_{z_t}-F_{\chi}^{NTP}|_{z_h})`. `Lock and Macvean (1999)`_ related the non-turbulent flux divergence, :math:`F_{\chi}^{NTP}|_{z_t}-F_{\chi}^{NTP}|_{z_h}`, to radiative cooling occurring within undulations of the cloudy boundary layer top. @@ -2599,7 +2715,8 @@ finite-difference form of :eq:`fxtot_zi` becomes .. math:: :label: fxtot_zi_fd F_{\chi}^{Tot}|_{z_h} = - w_e \Delta \chi + - F_{\chi}^{rad}|_{z_t} + {F_{\chi}}^{ppn}_{\mbox{\tiny \rm NTML}+\frac{3}{2}} + + F_{\chi}^{rad}|_{z_t} + {F_{\chi}}^{ppn}_{\mbox{\tiny \rm NTML}+\frac{3}{2}} + + {F_{\chi}}^{subs}_{\mbox{\tiny \rm NTML}-\frac{1}{2}} Then, assuming a linear profile of :math:`F_{\chi}^{Tot}` in the mixed @@ -2608,7 +2725,8 @@ gives .. math:: :label: fxtot_interp - F_{\chi}^{Tot}|_{ \mbox{\tiny \rm NTML}+\frac{1}{2} } = F_{\chi}^{Tot}|_{z_{\rm b}} + + F_{\chi}^{Tot}|_{ \mbox{\tiny \rm NTML}+\frac{1}{2} } = + F_{\chi}^{Tot}|_{z_{\rm b}} + \frac{ z'_{\mbox{\tiny \rm NTML}+\frac{1}{2}} }{z_{\rm ml}} \left( F_{\chi}^{Tot}|_{z_h} - F_{\chi}^{Tot}|_{z_{\rm b}} \right) @@ -2618,7 +2736,8 @@ entrainment flux is given by: .. math:: :label: rev_entflux - \overline{w'\chi'}|_{ \mbox{\tiny \rm NTML}+\frac{1}{2} } = F_{\chi}^{Tot}|_{ \mbox{\tiny \rm NTML}+\frac{1}{2} } + \overline{w'\chi'}|_{ \mbox{\tiny \rm NTML}+\frac{1}{2} } = + F_{\chi}^{Tot}|_{ \mbox{\tiny \rm NTML}+\frac{1}{2} } - F_{\chi}^{NT}|_{ \mbox{\tiny \rm NTML}+\frac{1}{2} } This revised algorithm has several advantages over the previous. @@ -2633,7 +2752,8 @@ the total grid-level flux, :eq:`fxtot_interp`, is used to calculate the entrainment fluxes, it is straightforward to ensure that the net budget of the inversion grid-level, namely :math:`- ( -F_{\chi}^{Tot}|_{\mbox{\tiny \rm NTML}+\frac{3}{2}} - F_{\chi}^{Tot}|_{\mbox{\tiny \rm NTML}+\frac{1}{2}})/\Delta z`, +F_{\chi}^{Tot}|_{\mbox{\tiny \rm NTML}+\frac{3}{2}} - +F_{\chi}^{Tot}|_{\mbox{\tiny \rm NTML}+\frac{1}{2}})/\Delta z`, is consistent with the entrainment/subsidence balance. In other words, to use :math:`\theta_{\ell}` as an example, if the inversion is rising (falling) then @@ -2648,29 +2768,34 @@ mixed layer by the end of the timestep. In other words, for \chi_{\mbox{\tiny \rm NTML}+1}^{n+1} = \chi_{\mbox{\tiny \rm NTML}+1}^{n} - \frac{\Delta t}{\Delta z} \left( - F_{\chi}^{Tot}|_{ \mbox{\tiny \rm NTML}+\frac{3}{2} } - F_{\chi}^{Tot}|_{ \mbox{\tiny \rm NTML}+\frac{1}{2} } + F_{\chi}^{Tot}|_{ \mbox{\tiny \rm NTML}+\frac{3}{2} } - F_{\chi}^{Tot}|_{ + \mbox{\tiny \rm NTML}+\frac{1}{2} } \right) .. math:: \chi_{\mbox{\tiny \rm NTML}}^{n+1} = \chi_{\mbox{\tiny \rm NTML}}^{n} - \frac{\Delta t}{z_{\mbox{\tiny \rm NTML}+\frac{1}{2}}} \left( - F_{\chi}^{Tot}|_{ \mbox{\tiny \rm NTML}+\frac{1}{2} } - F_{\chi}^{Tot}|_{z_{\rm b}} + F_{\chi}^{Tot}|_{ \mbox{\tiny \rm NTML}+\frac{1}{2} } - + F_{\chi}^{Tot}|_{z_{\rm b}} \right) where the superscripts :math:`n` and :math:`n+1` refer to the model timestep, although strictly speaking :math:`n+1` refers to fields after the boundary layer implicit solver. Requiring that -:math:`\chi_{\mbox{\tiny \rm NTML}+1}^{n+1}\geq\chi_{\mbox{\tiny \rm NTML}}^{n+1}` +:math:`\chi_{\mbox{\tiny \rm NTML}+1}^{n+1}\geq\chi_{\mbox{\tiny \rm +NTML}}^{n+1}` implies .. math:: F_{\chi}^{Tot}|_{ \mbox{\tiny \rm NTML}+\frac{1}{2} } \left( 1+ \frac{\Delta z}{z_{ml}}\right) - \geq F_{\chi}^{Tot}|_{ \mbox{\tiny \rm NTML}+\frac{3}{2} } + \Delta z \left( - \frac{\chi_{\mbox{\tiny \rm NTML}}^{n}-\chi_{\mbox{\tiny \rm NTML}+1}^{n}}{\Delta t} + \geq F_{\chi}^{Tot}|_{ \mbox{\tiny \rm NTML}+\frac{3}{2} } + \Delta z + \left( + \frac{\chi_{\mbox{\tiny \rm NTML}}^{n}-\chi_{\mbox{\tiny \rm + NTML}+1}^{n}}{\Delta t} + \frac{F_{\chi}^{Tot}|_{z_{\rm b}}}{z_{ml}} \right) The same arguments apply for :math:`q_t`, noting that the free @@ -2706,8 +2831,10 @@ formula used is: .. math:: :label: dqt_disc_9c - \Delta \chi = {\chi}_{\mbox{\tiny \rm NTML}+2} - {\chi}_{\mbox{\tiny \rm NTML}} - - \gamma_{\chi} \left( z_{\mbox{\tiny \rm NTML}+2} - z_h \right) + \Delta \chi = {\chi}_{\mbox{\tiny \rm NTML}+2} - {\chi}_{\mbox{\tiny \rm + NTML}} + - \gamma_{\chi} \left( z_{\mbox{\tiny \rm NTML}+2} - z_h + \right) subject to the constraint that the lapse rate adjustment should not reduce the two grid-length difference by more than half. The @@ -2715,13 +2842,16 @@ free-atmospheric lapse rates are given by .. math:: - \gamma_{\theta_{\ell}} = {\rm max}\left[ \, 0, \, \frac{ {\theta_{\ell}}_{\mbox{\tiny \rm NTML}+3}-{\theta_{\ell}}_{\mbox{\tiny \rm NTML}+2} } + \gamma_{\theta_{\ell}} = {\rm max}\left[ \, 0, \, \frac{ + {\theta_{\ell}}_{\mbox{\tiny \rm NTML}+3}-{\theta_{\ell}}_{\mbox{\tiny \rm + NTML}+2} } { z_{\mbox{\tiny \rm NTML}+3} - z_{\mbox{\tiny \rm NTML}+2} } \right] .. math:: - \gamma_{q_t} = {\rm min}\left[ \, 0, \, \frac{ {q_t}_{\mbox{\tiny \rm NTML}+3}-{q_t}_{\mbox{\tiny \rm NTML}+2} } + \gamma_{q_t} = {\rm min}\left[ \, 0, \, \frac{ {q_t}_{\mbox{\tiny \rm + NTML}+3}-{q_t}_{\mbox{\tiny \rm NTML}+2} } { z_{\mbox{\tiny \rm NTML}+3} - z_{\mbox{\tiny \rm NTML}+2} } \right] @@ -2773,25 +2903,30 @@ For momentum (and scalars if no subgrid inversion) ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ For momentum, and scalars if a subgrid inversion cannot be diagnosed, -see section :ref:`Diagnosis of a sub-grid inversion `, fluxes at the mixed layer top are +see section :ref:`Diagnosis of a sub-grid inversion `, fluxes at the +mixed layer top are specified through an eddy diffusivity which is given by .. math:: - K_h|_{\mbox{\tiny \rm NTML}+\frac{1}{2}} = w_e \Delta_{\mbox{\tiny \rm NTML}+1} z + K_h|_{\mbox{\tiny \rm NTML}+\frac{1}{2}} = w_e \Delta_{\mbox{\tiny \rm + NTML}+1} z .. math:: :label: khent - K_m|_{\mbox{\tiny \rm NTML}} = Pr \, w_e \Delta_{\mbox{\tiny \rm NTML}+\frac{1}{2}} z + K_m|_{\mbox{\tiny \rm NTML}} = Pr \, w_e \Delta_{\mbox{\tiny \rm + NTML}+\frac{1}{2}} z noting the Charney-Philips grid implying stresses are staggered from scalar fluxes. The Prandtl number, :math:`Pr`, takes the same form as -for the non-local :math:`K` profiles, see section :ref:`The non-local scheme `. +for the non-local :math:`K` profiles, see section :ref:`The non-local scheme +`. Substituting :eq:`khent` in :eq:`scal_closure` gives, for example, -:math:`\overline{w'\theta_{\ell}'}|_{\mbox{\tiny \rm NTML}+\frac{1}{2}} = - w_e \Delta_{\mbox{\tiny \rm NTML}+1} \theta_{\ell}`. +:math:`\overline{w'\theta_{\ell}'}|_{\mbox{\tiny \rm NTML}+\frac{1}{2}} = - w_e +\Delta_{\mbox{\tiny \rm NTML}+1} \theta_{\ell}`. Note that this gives entrainment buoyancy fluxes identical to :eq:`discinv` as long as there is no buoyancy reversal generation of turbulence (i.e.,\ :math:`V_{\rm br}=0`) and if variations @@ -2827,11 +2962,13 @@ An inversion is defined as being resolved when it extends above the flux-level above the usual entrainment interface level (see section :ref:`Diagnosis of inversion thickness `), i.e. when -.. math:: z_{\mbox{\tiny \rm NTML}+\frac{1}{2}} + \Delta z_i > z_{\mbox{\tiny \rm NTML}+\frac{3}{2}} +.. math:: z_{\mbox{\tiny \rm NTML}+\frac{1}{2}} + \Delta z_i > z_{\mbox{\tiny + \rm NTML}+\frac{3}{2}} When this happens, there is no subgrid inversion diagnosis and the entrainment parametrization follows the methodology given in -section :ref:`For momentum (and scalars if no subgrid inversion) ` to give +section :ref:`For momentum (and scalars if no subgrid inversion) ` +to give :math:`K_h|_{\mbox{\tiny \rm NTML}+\frac{1}{2}}`. The diffusion coefficient profile within the inversion is then calculated assuming the :math:`\theta_{v\ell}` flux profile within the inversion decreases @@ -2840,7 +2977,8 @@ at the inversion base to zero at the inversion top, i.e.: .. math:: :label: ent_svl - \overline{w'\theta_{v\ell}'} = \overline{w'\theta_{v\ell}'}|_{\mbox{\tiny \rm NTML}+\frac{1}{2}} + \overline{w'\theta_{v\ell}'} = \overline{w'\theta_{v\ell}'}|_{\mbox{\tiny + \rm NTML}+\frac{1}{2}} cos\left(\pi \frac{z'}{2} \right) where :math:`z'=(z-z_{\rm h})/\Delta z_i` is scaled height within the @@ -2850,7 +2988,8 @@ coefficient profile by inverting the standard flux parametrization: .. math:: K_h|_{k+\frac{1}{2}}= - \, \frac{\overline{w'\theta_{v\ell}'} } - { ({\theta_{v\ell}}_{k+1}-{\theta_{v\ell}}_{k})/(z_{k+1}-z_k) } + { + ({\theta_{v\ell}}_{k+1}-{\theta_{v\ell}}_{k})/(z_{k+1}-z_k) } The diffusion coefficient for momentum entrainment is calculated in the same way, allowing for the staggered grid, with the same :math:`Pr` as @@ -2875,19 +3014,23 @@ turbulence forcing and the inversion jump change slowly compared to the timestep. Whilst this is true for atmospheric :math:`\theta_{\ell}` and :math:`q_t`, the latter is not true for tracers with a small boundary layer concentration. Consequently, for a tracer field :math:`\chi`, the -parametrized entrainment fluxes :math:`\overline{w'\chi'}_{ z_{\mbox{\tiny \rm NTML}+\frac{1}{2}} }` are calculated from +parametrized entrainment fluxes :math:`\overline{w'\chi'}_{ z_{\mbox{\tiny \rm +NTML}+\frac{1}{2}} }` are calculated from :eq:`fluxinterp` but are implemented through an equivalent entrainment eddy-diffusivity given by: .. math:: :label: K_ent_tracer - K_{\chi}|_{\mbox{\tiny \rm NTML}+\frac{1}{2}} = - \overline{w'\chi'}_{ z_{\mbox{\tiny \rm NTML}+\frac{1}{2}} } - \frac{\Delta_{\mbox{\tiny \rm NTML}+1} z}{\Delta_{\mbox{\tiny \rm NTML}+1} \chi} + K_{\chi}|_{\mbox{\tiny \rm NTML}+\frac{1}{2}} = - \overline{w'\chi'}_{ + z_{\mbox{\tiny \rm NTML}+\frac{1}{2}} } + \frac{\Delta_{\mbox{\tiny \rm NTML}+1} + z}{\Delta_{\mbox{\tiny \rm NTML}+1} \chi} Note from :eq:`scal_closure` that :eq:`K_ent_tracer` gives the parametrized flux if :math:`\Delta_{\mbox{\tiny \rm NTML}+1} \chi` does not change across the -timestep (see section :ref:`Implicit solution of the diffusion equation ` for a description of the +timestep (see section :ref:`Implicit solution of the diffusion equation +` for a description of the implicit numerical solution of :eq:`cons_eqn_scal`). As :eq:`K_ent_tracer` involves the potentially numerically dangerous calculation of @@ -2919,15 +3062,18 @@ layer are related to the surface fluxes by: .. math:: :label: 1.1.1 - \frac{\partial T}{\partial z} + \frac{g}{ c_P }=-\frac{ H_0 }{ c_P \rho _0 v_\ast } \frac{ \phi _h (z/L)}{kz} + \frac{\partial T}{\partial z} + \frac{g}{ c_P }=-\frac{ H_0 }{ c_P \rho _0 + v_\ast } \frac{ \phi _h (z/L)}{kz} .. math:: :label: 1.1.2 - \frac{\partial q}{\partial z}=-\frac{ E_0 }{ \rho _0 v_\ast } \frac{ \phi _h (z/L)}{kz} + \frac{\partial q}{\partial z}=-\frac{ E_0 }{ \rho _0 v_\ast } \frac{ \phi + _h (z/L)}{kz} .. math:: :label: 1.1.3 - \frac{\partial {\rm {\bf v}}}{\partial z}=\frac{ {\rm {\bf \tau }}_{0} }{ \rho _0 v_\ast } \frac{ \phi _m (z/L)}{kz}, + \frac{\partial {\rm {\bf v}}}{\partial z}=\frac{ {\rm {\bf \tau }}_{0} }{ + \rho _0 v_\ast } \frac{ \phi _m (z/L)}{kz}, where subscript 0 represents a surface value and subscript \* represents @@ -2947,7 +3093,8 @@ where F\ :math:`_{B0}` is the surface buoyancy flux defined by F_{B0} = \frac{ g }{ c_P } \beta _{T1} H_0 + g \beta _{q1} E_0. The buoyancy coefficients in equation :eq:`1.1.5` are given -in appendix :ref:`Appendix: Derivation and definitions of the buoyancy parameters ` with the subscript 1 denoting a value at +in appendix :ref:`Appendix: Derivation and definitions of the buoyancy +parameters ` with the subscript 1 denoting a value at the lowest level in the atmosphere model. Equations :eq:`1.1.1`–:eq:`1.1.3` can be @@ -2959,7 +3106,8 @@ surface turbulent fluxes are: .. math:: :label: 1.1.7 - \frac{ H_0 }{ c_P \rho _0 }=-\frac{ c_H }{ c_D^{1/2} } v_\ast \left( {\Delta T + \frac{g}{c_p }( z_1 + z_{0m} - z_{0h} )} \right) + \frac{ H_0 }{ c_P \rho _0 }=-\frac{ c_H }{ c_D^{1/2} } v_\ast \left( + {\Delta T + \frac{g}{c_p }( z_1 + z_{0m} - z_{0h} )} \right) .. math:: :label: 1.1.8 @@ -2967,7 +3115,8 @@ surface turbulent fluxes are: .. math:: :label: 1.1.9 - \frac{ {\bf \tau }_{0} }{ \rho _{0} }= c_D^{1/2} v_\ast \Delta {\rm {\bf v}}, + \frac{ {\bf \tau }_{0} }{ \rho _{0} }= c_D^{1/2} v_\ast \Delta {\rm {\bf + v}}, where :math:`\Delta`\ X=X\ :math:`_{1}`-X\ :math:`_{0}`. @@ -2980,7 +3129,8 @@ buoyancy flux in definition :eq:`1.1.4` is .. math:: :label: 1.1.11 - \Delta B = g \beta _{T1} \left( {\Delta T + \frac{g}{ c_P }( z_1 + z_{0m} - z_{0h} )} \right) + \Delta B = g \beta _{T1} \left( {\Delta T + \frac{g}{ c_P }( z_1 + z_{0m} - + z_{0h} )} \right) + g \beta _{q1} \Delta q The **surface exchange coefficients** in @@ -3000,11 +3150,13 @@ where .. math:: :label: 1.1.14 - \Phi _m (L , z_1 + z_{0m} , z_{0m} )= \int \limits_{ z_{0m} /L}^{( z_1 + z_{0m} )/L} \frac{ \phi _m (\zeta )}{\zeta } d\zeta + \Phi _m (L , z_1 + z_{0m} , z_{0m} )= \int \limits_{ z_{0m} /L}^{( z_1 + + z_{0m} )/L} \frac{ \phi _m (\zeta )}{\zeta } d\zeta .. math:: :label: 1.1.15 - \Phi _h (L , z_1 + z_{0m} , z_{0h} )= \int \limits_{ z_{0h} /L}^{( z_1 + z_{0m} )/L} \frac{ \phi _h (\zeta )}{\zeta } d\zeta, + \Phi _h (L , z_1 + z_{0m} , z_{0h} )= \int \limits_{ z_{0h} /L}^{( z_1 + + z_{0m} )/L} \frac{ \phi _h (\zeta )}{\zeta } d\zeta, z\ :math:`_{0m}` and z\ :math:`_{0h}` are the **surface roughness @@ -3016,11 +3168,13 @@ forms .. math:: - \frac{ H_0 }{ c_P \rho _0 }={-c}_H V \left( {\Delta T + \frac{g}{c_p }( z_1 + z_{0m} - z_{0h} )} \right) + \frac{ H_0 }{ c_P \rho _0 }={-c}_H V \left( {\Delta T + \frac{g}{c_p }( z_1 + + z_{0m} - z_{0h} )} \right) .. math:: :label: 1.1.16 - = - C_H \left( {\Delta T + \frac{g}{c_p } ( z_1 + z_{0m} - z_{0h} )} \right) + = - C_H \left( {\Delta T + \frac{g}{c_p } ( z_1 + z_{0m} - z_{0h} )} + \right) .. math:: :label: 1.1.17 @@ -3028,7 +3182,8 @@ forms .. math:: :label: 1.1.18 - \frac{ {\rm {\bf \tau }}_{0} }{ \rho _{0} }= c_D V \Delta {\rm {\bf v}}{ }= C_D \Delta {\rm {\bf v}}, + \frac{ {\rm {\bf \tau }}_{0} }{ \rho _{0} }= c_D V \Delta {\rm {\bf v}}{ }= + C_D \Delta {\rm {\bf v}}, where the effective wind speed for surface turbulent exchanges, @@ -3055,11 +3210,13 @@ following forms: .. math:: :label: 1.1.22 - c_H=\frac{ C_H }{V} = \frac{ C_H C_D }{ v_\ast ^2 } = \frac{ k^2 }{ \Phi _h \Phi _m } + c_H=\frac{ C_H }{V} = \frac{ C_H C_D }{ v_\ast ^2 } = \frac{ k^2 }{ \Phi + _h \Phi _m } .. math:: :label: 1.1.23 - c_D=\frac{ C_D }{V} = \frac{ C_D^2 }{ v_\ast ^2 } = \frac{ k^2 }{ \Phi _m^2 }. + c_D=\frac{ C_D }{V} = \frac{ C_D^2 }{ v_\ast ^2 } = \frac{ k^2 }{ \Phi + _m^2 }. In order to close the system the surface scaling velocity, @@ -3068,7 +3225,8 @@ v\ :math:`_{\ast .. math:: :label: 1.1.24 - v_\ast = u_\ast \equiv \left| { {\rm {\bf \tau }}_{0} {/} \rho _{0} } \right|^{1/2} + v_\ast = u_\ast \equiv \left| { {\rm {\bf \tau }}_{0} {/} \rho _{0} } + \right|^{1/2} we have the standard Monin-Obukhov theory and it is easy to deduce that with this definition :math:`v_{\ast } = c_{D}^{1/2} \Delta`\ **v** and @@ -3108,7 +3266,8 @@ w\ :math:`_{c }`\ = 0) can be seen to be .. math:: :label: 1.1.27 - v_\ast \sim \gamma _t w_\ast \sim \gamma _t^{3/2} {\left( {\frac{ c_H }{ c_D^{1/2} }} \right)}^{1/2} z_i^{1/2} (-\Delta B )^{1/2} + v_\ast \sim \gamma _t w_\ast \sim \gamma _t^{3/2} {\left( {\frac{ c_H }{ + c_D^{1/2} }} \right)}^{1/2} z_i^{1/2} (-\Delta B )^{1/2} which implies that @@ -3143,13 +3302,18 @@ Using the definitions of :math:`V` :eq:`1.1.19` and .. math:: :label: 1.2.2 - v_g^2 = W_g^2 ( z_1 ) + \frac{1}{2}\left| {\Delta {{\rm {\bf v}}}} \right|{ }\left[ {\left( {{ } {\left| {\Delta {{\rm {\bf v}}}} \right|}^2 - W_g^2 ( z_1 )} \right)^{1/2} - \left| {\Delta {{\rm {\bf v}}}} \right|} \right] + v_g^2 = W_g^2 ( z_1 ) + \frac{1}{2}\left| {\Delta {{\rm {\bf v}}}} \right|{ + }\left[ {\left( {{ } {\left| {\Delta {{\rm {\bf v}}}} \right|}^2 - W_g^2 ( + z_1 )} \right)^{1/2} - \left| {\Delta {{\rm {\bf v}}}} \right|} \right] where .. math:: :label: 1.2.3 - W_g (z) = \frac{1}{ c_D^{1/2} } {\left( { \gamma _t^2 w_\ast ^2 + \gamma _c^2 w_c^2 } \right)}^{1/2} = \frac{ \Phi _m (L , z + z_{0m} , z_{0m} )}{k} {\left( { \gamma _t^2 w_\ast ^2 + \gamma _c^2 w_c^2 } \right)}^{1/2}. + W_g (z) = \frac{1}{ c_D^{1/2} } {\left( { \gamma _t^2 w_\ast ^2 + \gamma + _c^2 w_c^2 } \right)}^{1/2} = \frac{ \Phi _m (L , z + z_{0m} , z_{0m} + )}{k} {\left( { \gamma _t^2 w_\ast ^2 + \gamma _c^2 w_c^2 } + \right)}^{1/2}. Thus in this formulation the mean gust speed is a function of height above the surface through the same factor, :math:`\Phi _{m}`\ (z), which @@ -3225,7 +3389,8 @@ In effect, the UM takes the displacement height for momentum as :math:`-z_{0m}`, where :math:`z_{0m}` is the momentum roughness length. The profile of wind is therefore determined by Monin-Obukhov theory as -.. math:: \frac{\partial u}{\partial z} = \frac{u_*}{k(z+z_{0m})} \phi_m((z+z_{0m})/L), +.. math:: \frac{\partial u}{\partial z} = \frac{u_*}{k(z+z_{0m})} + \phi_m((z+z_{0m})/L), with :math:`u_*` being the friction velocity, :math:`L` the surface Obukhov length and :math:`\phi_m` the similarity function. It is common @@ -3242,7 +3407,8 @@ Redefining the vertical coordinate as :math:`\zeta=z/L`, we have .. math:: - = \frac{u_*}{k} \int_{\zeta_{0m}}^{\zeta'+\zeta_{0m}} \left ( \frac{1}{\zeta'} - + = \frac{u_*}{k} \int_{\zeta_{0m}}^{\zeta'+\zeta_{0m}} \left ( + \frac{1}{\zeta'} - \frac{d\psi_m}{d\zeta'} \right ) \, d\zeta' .. math:: @@ -3274,7 +3440,8 @@ first, .. math:: - \int_0^{\zeta_1} \ln \left ( \frac{\zeta+\zeta_{0m}}{\zeta_{0m}} \right ) \, d \zeta + \int_0^{\zeta_1} \ln \left ( \frac{\zeta+\zeta_{0m}}{\zeta_{0m}} \right ) \, + d \zeta = \zeta_{0m} \int_1^{1+\zeta_1/\zeta_{0m}} \ln(x) \, dx .. math:: @@ -3369,11 +3536,13 @@ stability functions are given by `Beljaars and Holtslag (1991)`_: .. math:: :label: 1.3.11 - \Phi _m=\ln \left( {\frac{ z_1 + z_{0m} }{ z_{0m} }} \right) - \Psi _m ( \zeta _1 ) + \Psi _m ( \zeta _{0m} ) + \Phi _m=\ln \left( {\frac{ z_1 + z_{0m} }{ z_{0m} }} \right) - \Psi _m ( + \zeta _1 ) + \Psi _m ( \zeta _{0m} ) .. math:: :label: 1.3.12 - \Phi _h=\ln \left( {\frac{ z_1 + z_{0m} }{ z_{0h} }} \right) - \Psi _h ( \zeta _1 ) + \Psi _h ( \zeta _{0h} ) + \Phi _h=\ln \left( {\frac{ z_1 + z_{0m} }{ z_{0h} }} \right) - \Psi _h ( + \zeta _1 ) + \Psi _h ( \zeta _{0h} ) where :math:`\zeta _{1}` = (z\ :math:`_{1}` + z\ :math:`_{0m})`/L, @@ -3382,11 +3551,14 @@ z\ :math:`_{0h}`/L and .. math:: :label: 1.3.13 - - \Psi _h (\zeta )=\left[ { {\left( {1 + \frac{2}{3}a\zeta } \right)}^{3/2} - 1 } \right] + b\left( {\zeta - \frac{c}{d}} \right)\exp (-d\zeta ) + \frac{bc}{d} + - \Psi _h (\zeta )=\left[ { {\left( {1 + \frac{2}{3}a\zeta } \right)}^{3/2} + - 1 } \right] + b\left( {\zeta - \frac{c}{d}} \right)\exp (-d\zeta ) + + \frac{bc}{d} .. math:: :label: 1.3.14 - - \Psi _m (\zeta )=a\zeta + b\left( {\zeta - \frac{c}{d}} \right)\exp (-d\zeta ) + \frac{bc}{d}, + - \Psi _m (\zeta )=a\zeta + b\left( {\zeta - \frac{c}{d}} \right)\exp + (-d\zeta ) + \frac{bc}{d}, with :math:`a = 1`, :math:`b =2/3`, :math:`c = 5`, :math:`d = 0.35`. @@ -3421,7 +3593,10 @@ obtain: .. math:: :label: 1.3.17 - \Phi _m = \ln \left( {\frac{ z_1 + z_{0m} }{ z_{0m} }} \right) - 2 \ln \left( {\frac{1 + X_1 }{1 + X_0 }} \right) - \ln \left( {\frac{1 + X_1^2 }{1 + X_0^2 }} \right)+ 2 \left( { {\tan }^{-1} X_1 - {\tan }^{-1} X_0 } \right) + \Phi _m = \ln \left( {\frac{ z_1 + z_{0m} }{ z_{0m} }} \right) - 2 \ln + \left( {\frac{1 + X_1 }{1 + X_0 }} \right) - \ln \left( {\frac{1 + X_1^2 + }{1 + X_0^2 }} \right)+ 2 \left( { {\tan }^{-1} X_1 - {\tan }^{-1} X_0 } + \right) where @@ -3433,7 +3608,8 @@ and .. math:: :label: 1.3.19 - \Phi _h = \ln \left( {\frac{ z_1 + z_{0m} }{ z_{0h} }} \right) - 2 \ln \left( {\frac{1 + Y_1 }{1 + Y_0 }} \right) + \Phi _h = \ln \left( {\frac{ z_1 + z_{0m} }{ z_{0h} }} \right) - 2 \ln + \left( {\frac{1 + Y_1 }{1 + Y_0 }} \right) where @@ -3460,7 +3636,8 @@ ms\ :math:`^{-1}`), then start the iteration from the neutral limit, so .. math:: :label: 1.4.7 - v_\ast ^{(0)}= {\left( {\frac{k}{ \Phi _m^{(0)} }} \right)} \left| {\Delta {{{\rm {\bf v}}}}} \right| + v_\ast ^{(0)}= {\left( {\frac{k}{ \Phi _m^{(0)} }} \right)} \left| {\Delta + {{{\rm {\bf v}}}}} \right| Otherwise (if :math:`\Delta`\ B :math:`<` 0 and :math:`\Delta`\ **v** @@ -3481,8 +3658,10 @@ and convective limits for :math:`v_\ast^{(0)}`, so .. math:: :label: 1.4.4 - v_\ast ^{(0)}= MAX{\left[ {\left( {\frac{k}{ \Phi _m^{(0)} }} \right)} \left| {\Delta {{{\rm {\bf v}}}}} \right|, \, - {\left[ { \gamma _t^3 \left( {\frac{k}{ \Phi _h^{(0)} }} \right) z_i \left| {-\Delta B} \right| } \right]}^{ 1/2} \right]} + v_\ast ^{(0)}= MAX{\left[ {\left( {\frac{k}{ \Phi _m^{(0)} }} \right)} + \left| {\Delta {{{\rm {\bf v}}}}} \right|, \, + {\left[ { \gamma _t^3 \left( {\frac{k}{ \Phi _h^{(0)} }} \right) z_i + \left| {-\Delta B} \right| } \right]}^{ 1/2} \right]} Then calculate @@ -3509,15 +3688,18 @@ DO n = 1 to N .. math:: :label: 1.4.11 - {\left( {\frac{ F_{B0} }{ \rho _0 }} \right)}^{(n)}= { {-C}_H }^{(n-1)} \Delta B + {\left( {\frac{ F_{B0} }{ \rho _0 }} \right)}^{(n)}= { {-C}_H }^{(n-1)} + \Delta B .. math:: :label: 1.4.12 - w_\ast ^{(n)}= {\left[ { z_i {\left( {\frac{ F_{B0} }{ \rho _0 }} \right)}^{(n)} } \right]}^{ 1/3} + w_\ast ^{(n)}= {\left[ { z_i {\left( {\frac{ F_{B0} }{ \rho _0 }} + \right)}^{(n)} } \right]}^{ 1/3} .. math:: :label: 1.4.13 - v_\ast ^{(n)2}= u_\ast ^{(n)2} + \gamma _t^2 w_\ast ^{(n)2} + \gamma _c^2 w_c^2 + v_\ast ^{(n)2}= u_\ast ^{(n)2} + \gamma _t^2 w_\ast ^{(n)2} + \gamma _c^2 + w_c^2 .. math:: :label: 1.4.14 @@ -3552,7 +3734,8 @@ stress: .. math:: :label: 1.4.19 - H_0= {-c}_P \rho _0 C_H^{(N)} \left( {\Delta T + \frac{g}{ c_P }( z_1 + z_{0m} - z_{0h} )} \right) + H_0= {-c}_P \rho _0 C_H^{(N)} \left( {\Delta T + \frac{g}{ c_P }( z_1 + + z_{0m} - z_{0h} )} \right) .. math:: :label: 1.4.20 @@ -3584,7 +3767,8 @@ We decompose the wind as respectively, the large-scale, gust and small-scale turbulent contributions to the velocity. Locally, Monin-Obukhov theory then gives -.. math:: |\bar{\bf u} + {\bf u}_g({\bf x}) | = \frac{u_*({\bf x})}{k} \Phi_m({\bf x}). +.. math:: |\bar{\bf u} + {\bf u}_g({\bf x}) | = \frac{u_*({\bf x})}{k} + \Phi_m({\bf x}). We ignore the spatial variation of :math:`\Phi_m`, expecting that the principal effect of locally stronger winds is to increase the local @@ -3611,7 +3795,8 @@ where :math:`C_D` is the standard drag coefficient, .. math:: - \langle{\bf \tau}\rangle = \rho C_D \langle |\bar{\bf u} + {\bf u}_g({\bf x})| + \langle{\bf \tau}\rangle = \rho C_D \langle |\bar{\bf u} + {\bf u}_g({\bf + x})| (\bar{\bf u} + {\bf u}_g({\bf x})) \rangle \approx \rho C_D \langle |\bar{\bf u} + {\bf u}_g({\bf x})| \rangle \bar{\bf u}, @@ -3653,7 +3838,8 @@ If we let :math:`v=\sqrt(\tilde u_* \hat u_*)`, then we get .. math:: - \hat u_*^2 = \frac{1}{2} \left \{ \gamma_t^2 w_*^2 + \sqrt { \gamma_t^4 w_*^4 + \hat u_*^2 = \frac{1}{2} \left \{ \gamma_t^2 w_*^2 + \sqrt { \gamma_t^4 + w_*^4 + 4 v^2 } \right \}. In the corrected version this expression is used to calculate ``V_S``, @@ -3691,14 +3877,16 @@ z\ :math:`_{ob, }` we obtain for the scalar :math:`X` .. math:: :label: 1.5.3 - X_{ob} = X_0 + \frac{ F_{X0} }{ \rho _0 v_\ast k} \Phi _h (L, z_{ob} + z_{0h} , z_{0h} ) + X_{ob} = X_0 + \frac{ F_{X0} }{ \rho _0 v_\ast k} \Phi _h (L, z_{ob} + + z_{0h} , z_{0h} ) and using the expression for the surface flux :math:`F_{X0}` of the scalar quantity :math:`X` this gives the interpolation formula .. math:: :label: 1.5.4 - X_{ob} = X_0 + \frac{ C_H }{k v_\ast } \Phi _h (L, z_{ob} + z_{0h} , z_{0h} ) ( X_1 - X_0 ) + X_{ob} = X_0 + \frac{ C_H }{k v_\ast } \Phi _h (L, z_{ob} + z_{0h} , + z_{0h} ) ( X_1 - X_0 ) For temperature and humidity z\ :math:`_{ob}` is set to the screen height (1.5 m) and the last iteration (N) values of C\ :math:`_{H}`, L @@ -3869,7 +4057,8 @@ ocean. The gridbox mean **wind mixing energy flux** is given by .. math:: :label: 1.6.9 - F_{WME} = ( 1- f_I ) \frac{ \rho _0^{3/2} v_\ast ^3 }{ \rho _{(sea)}^{1/2} } + F_{WME} = ( 1- f_I ) \frac{ \rho _0^{3/2} v_\ast ^3 }{ \rho _{(sea)}^{1/2} + } where :math:`v_{\ast }` is calculated using the drag coefficient for the leads part of the gridbox, c\ :math:`_{D(L)}`, rather than the gridbox @@ -4336,19 +4525,23 @@ When form drag is included via effective roughness lengths equations .. math:: - \frac{ H_{0(eff)} }{ c_P \rho _0 }=\frac{-k}{ \Phi _h (L , z_1 + z_{0m(eff)} , z_{0h(eff)} )} v_{\ast (eff)} + \frac{ H_{0(eff)} }{ c_P \rho _0 }=\frac{-k}{ \Phi _h (L , z_1 + + z_{0m(eff)} , z_{0h(eff)} )} v_{\ast (eff)} .. math:: :label: 2.1.1 - \left( {\Delta T + \frac{g}{c_p }( z_1 + z_{0m(eff)} - z_{0h(eff)} )} \right) + \left( {\Delta T + \frac{g}{c_p }( z_1 + z_{0m(eff)} - z_{0h(eff)} )} + \right) .. math:: :label: 2.1.2 - \frac{ E_{0(eff)} }{ \rho _0 }=\frac{-k}{ \Phi _h (L , z_1 + z_{0m(eff)} , z_{0h(eff)} )} v_{\ast (eff)} \Delta q + \frac{ E_{0(eff)} }{ \rho _0 }=\frac{-k}{ \Phi _h (L , z_1 + z_{0m(eff)} , + z_{0h(eff)} )} v_{\ast (eff)} \Delta q .. math:: :label: 2.1.3 - \frac{ {\rm {\bf \tau }}_{{0(eff)}} }{ \rho _{0} }=\frac{k}{ \Phi _m (L , z_1 + z_{0m(eff)} , z_{0m(eff)} )} v_{\ast (eff)} \Delta {\rm {\bf v}} + \frac{ {\rm {\bf \tau }}_{{0(eff)}} }{ \rho _{0} }=\frac{k}{ \Phi _m (L , + z_1 + z_{0m(eff)} , z_{0m(eff)} )} v_{\ast (eff)} \Delta {\rm {\bf v}} The effective surface scaling velocity, v\ :math:`_{\ast (eff)}` , is @@ -4356,13 +4549,15 @@ given by (cf. :eq:`1.1.25`) .. math:: :label: 2.1.4 - v_{\ast (eff)}^2 = u_{\ast (eff)}^2 + \gamma _t^2 w_\ast ^2 + \gamma _c^2 w_c^2 + v_{\ast (eff)}^2 = u_{\ast (eff)}^2 + \gamma _t^2 w_\ast ^2 + \gamma + _c^2 w_c^2 where .. math:: :label: 2.1.5 - u_{\ast (eff)}^2 = \left| { {\rm {\bf \tau }}_{{0(eff)}} {/} \rho _{0} } \right| + u_{\ast (eff)}^2 = \left| { {\rm {\bf \tau }}_{{0(eff)}} {/} \rho _{0} } + \right| The effective roughness for momentum is derived by setting the total effective surface stress, **:math:`\tau`**\ :math:`_{0(eff)}`, to the @@ -4376,13 +4571,17 @@ deviation of the unresolved orographic height. Thus .. math:: :label: 2.1.6 - \frac{ {\rm {\bf \tau }}_{{0(eff)}} }{ \rho _{0} }{ = }\frac{{k } {v}_{{\ast (eff)}} }{ \Phi _{m} {(L , } {z}_{c} { , } {z}_{{0m(eff)}} {)}}{ }{\rm {\bf v}}{(} {z}_{c} {)} + \frac{ {\rm {\bf \tau }}_{{0(eff)}} }{ \rho _{0} }{ = }\frac{{k } {v}_{{\ast + (eff)}} }{ \Phi _{m} {(L , } {z}_{c} { , } {z}_{{0m(eff)}} {)}}{ }{\rm {\bf + v}}{(} {z}_{c} {)} and .. math:: :label: 2.1.7 - \frac{ {\rm {\bf \tau }}_{{0(f)}} }{ \rho _{0} }{ = }\frac{{k } {v}_{{\ast (f)}} }{ \Phi _{m} {(L , } {z}_{c} { , } {z}_{{0m}} {)}}{ }{\rm {\bf v}}{(} {z}_{c} {)} + \frac{ {\rm {\bf \tau }}_{{0(f)}} }{ \rho _{0} }{ = }\frac{{k } {v}_{{\ast + (f)}} }{ \Phi _{m} {(L , } {z}_{c} { , } {z}_{{0m}} {)}}{ }{\rm {\bf v}}{(} + {z}_{c} {)} where the scaling velocity based on the stress over a flat surface, v\ :math:`_{\ast @@ -4390,7 +4589,8 @@ v\ :math:`_{\ast .. math:: :label: 2.1.8 - v_{\ast (f)}^2 = u_{\ast (f)}^2 + \gamma _t^2 w_\ast ^2 + \gamma _c^2 w_c^2 + v_{\ast (f)}^2 = u_{\ast (f)}^2 + \gamma _t^2 w_\ast ^2 + \gamma _c^2 + w_c^2 with @@ -4406,7 +4606,9 @@ The orographic stress is given by .. math:: :label: 2.1.10 - \frac{ {\rm {\bf \tau }}_{{0(p)}} }{ \rho _{0} }{ = }\frac{{1}}{{2}}{ } {c}_{{D(orog)}} { } {f}_{D} {(} {{Ri}}_{B} {)}\frac{{A}}{{S}}{ }\left| {{\rm {\bf v}}{(} {z}_{c} {)}} \right|{ }{\rm {\bf v}}{(} {z}_{c} {)} + \frac{ {\rm {\bf \tau }}_{{0(p)}} }{ \rho _{0} }{ = }\frac{{1}}{{2}}{ } + {c}_{{D(orog)}} { } {f}_{D} {(} {{Ri}}_{B} {)}\frac{{A}}{{S}}{ }\left| {{\rm + {\bf v}}{(} {z}_{c} {)}} \right|{ }{\rm {\bf v}}{(} {z}_{c} {)} where :math:`A/S` is the total silhouette area of orography in a gridbox over the flat surface area of the gridbox taken as an average over all @@ -4424,7 +4626,9 @@ effective momentum roughness is derived .. math:: :label: 2.1.12 - \frac{\ln ( z_c / z_{0m(eff)} )}{\ln ( z_c / z_{0m} )} = {\left( {1 + \frac{1}{2} c_{D(orog)} f_D \frac{A}{S} {\left( {\frac{\ln ( z_c / z_{0m} )}{k}} \right)}^2 } \right)}^{-1/2} + \frac{\ln ( z_c / z_{0m(eff)} )}{\ln ( z_c / z_{0m} )} = {\left( {1 + + \frac{1}{2} c_{D(orog)} f_D \frac{A}{S} {\left( {\frac{\ln ( z_c / + z_{0m} )}{k}} \right)}^2 } \right)}^{-1/2} The stress for the flat surface is related to the total stress by @@ -4442,7 +4646,9 @@ Equation :eq:`2.1.13` implies that .. math:: :label: 2.1.14 - C_{D(f)} = C_{D(eff)} {\left( {1 + \frac{1}{2} c_{D(orog)} f_D \frac{A}{S} {\left( {\frac{\ln ( z_c / z_{0m} )}{k}} \right)}^2 } \right)}^{-1} + C_{D(f)} = C_{D(eff)} {\left( {1 + \frac{1}{2} c_{D(orog)} f_D + \frac{A}{S} {\left( {\frac{\ln ( z_c / z_{0m} )}{k}} \right)}^2 } + \right)}^{-1} Parametrized orographic drag coefficient ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ @@ -4452,7 +4658,8 @@ c\ :math:`_{D(orog)}` depends on A/S via the equation .. math:: :label: 2.1.11 - c_{D(orog)} = 2\alpha \beta \pi ^2 \frac{A}{S} \frac{ u_{\ast (f)}^2 }{ v^2 ( z_c )} + c_{D(orog)} = 2\alpha \beta \pi ^2 \frac{A}{S} \frac{ u_{\ast (f)}^2 }{ v^2 + ( z_c )} where :math:`\alpha` and :math:`\beta` are constants (:math:`\alpha`\ =12 and :math:`\beta`\ =1). @@ -4462,7 +4669,8 @@ roughness length for momentum becomes .. math:: :label: 2.1.15 - \frac{\ln ( z_c / z_{0m(eff)} )}{\ln ( z_c / z_{0m} )} = {\left( {1 + \alpha \beta \pi ^2 f_D {\left( {\frac{A}{S}} \right)}^2 } \right)}^{-1/2} + \frac{\ln ( z_c / z_{0m(eff)} )}{\ln ( z_c / z_{0m} )} = {\left( {1 + + \alpha \beta \pi ^2 f_D {\left( {\frac{A}{S}} \right)}^2 } \right)}^{-1/2} and :eq:`2.1.13` and :eq:`2.1.14` become @@ -4474,7 +4682,8 @@ and :eq:`2.1.13` and :eq:`2.1.14` become .. math:: :label: 2.1.17 - C_{D(f)}= C_{D(eff)} {\left( {1 + \alpha \beta \pi ^2 f_D {\left( {\frac{A}{S}} \right)}^2 } \right)}^{-1} + C_{D(f)}= C_{D(eff)} {\left( {1 + \alpha \beta \pi ^2 f_D {\left( + {\frac{A}{S}} \right)}^2 } \right)}^{-1} The effective surface flux of scalar X evaluated in terms of values at @@ -4482,13 +4691,15 @@ z\ :math:`_{c}` is .. math:: :label: 2.1.18 - \frac{ F_{X0(eff)} }{ \rho _0 } = \frac{k v_{\ast (eff)} }{ \Phi _h (L , z_c , z_{0h(eff)} )} (X( z_c ) - X_0 ) + \frac{ F_{X0(eff)} }{ \rho _0 } = \frac{k v_{\ast (eff)} }{ \Phi _h (L , + z_c , z_{0h(eff)} )} (X( z_c ) - X_0 ) and the surface flux for the flat surface is given by .. math:: :label: 2.1.19 - \frac{ F_{X0(f)} }{ \rho _0 } = \frac{k v_{\ast (f)} }{ \Phi _h (L , z_c , z_{0h} )} (X( z_c ) - X_0 ) + \frac{ F_{X0(f)} }{ \rho _0 } = \frac{k v_{\ast (f)} }{ \Phi _h (L , z_c , + z_{0h} )} (X( z_c ) - X_0 ) `Hewer and Wood (1998)`_ find that the scalar transport is enhanced when there is orographic form drag such that @@ -4503,19 +4714,25 @@ effective scalar roughness length is derived as .. math:: :label: 2.1.21) - \frac{\ln ( z_c / z_{0h(eff)} )}{\ln ( z_c / z_{0h} )} = {\left( {1 + \frac{1}{2} c_{D(orog)} f_D \frac{A}{S} {\left( {\frac{\ln ( z_c / z_{0m} )}{k}} \right)}^2 } \right)}^{1/2} \left( {1 - 2.2 f_D \frac{A}{S}} \right) + \frac{\ln ( z_c / z_{0h(eff)} )}{\ln ( z_c / z_{0h} )} = {\left( {1 + + \frac{1}{2} c_{D(orog)} f_D \frac{A}{S} {\left( {\frac{\ln ( z_c / + z_{0m} )}{k}} \right)}^2 } \right)}^{1/2} \left( {1 - 2.2 f_D \frac{A}{S}} + \right) which becomes .. math:: :label: 2.1.22 - \frac{\ln ( z_c / z_{0h(eff)} )}{\ln ( z_c / z_{0h} )} = {\left( {1 + \alpha \beta \pi ^2 f_D {\left( {\frac{A}{S}} \right)}^2 } \right)}^{1/2} \left( {1 - 2.2 f_D \frac{A}{S}} \right) + \frac{\ln ( z_c / z_{0h(eff)} )}{\ln ( z_c / z_{0h} )} = {\left( {1 + + \alpha \beta \pi ^2 f_D {\left( {\frac{A}{S}} \right)}^2 } \right)}^{1/2} + \left( {1 - 2.2 f_D \frac{A}{S}} \right) if the `Wood and Mason (1993)`_ formulation is used. .. _section_2.2: -The iterative algorithm for calculating the effective surface exchange coefficients +The iterative algorithm for calculating the effective surface exchange +coefficients ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ For unstable conditions, i.e. :math:`\Delta`\ B :math:`<` 0 : @@ -4537,7 +4754,9 @@ iteration from the convective limit, so .. math:: :label: (2.2.4 - v_{\ast (eff)}^{(0)}= v_{\ast (f)}^{(0)} = {\left[ { \gamma _t^3 \left( {\frac{k}{ \Phi _h^{(0)} }} \right) z_i \left| {-\Delta B} \right| + \gamma _c^2 w_c^2 } \right]}^{ 1/2} + v_{\ast (eff)}^{(0)}= v_{\ast (f)}^{(0)} = {\left[ { \gamma _t^3 \left( + {\frac{k}{ \Phi _h^{(0)} }} \right) z_i \left| {-\Delta B} \right| + + \gamma _c^2 w_c^2 } \right]}^{ 1/2} ELSE IF (:math:`\Delta`\ **v** :math:`\ge` 2 ms\ :math:`^{-1}` ) start @@ -4545,7 +4764,8 @@ iteration from the neutral end, so .. math:: :label: 2.2.5 - \Phi _m^{(0)}=\ln \left( {\frac{ z_1 + z_{0m(eff)} }{ z_{0m(eff)} }} \right) + \Phi _m^{(0)}=\ln \left( {\frac{ z_1 + z_{0m(eff)} }{ z_{0m(eff)} }} + \right) .. math:: :label: 2.2.6 @@ -4553,19 +4773,23 @@ iteration from the neutral end, so .. math:: :label: 2.2.7 - u_{\ast (eff)}^{(0)}=\frac{k}{ \Phi _m^{(0)} } \left| {\Delta {{{v}}}} \right| + u_{\ast (eff)}^{(0)}=\frac{k}{ \Phi _m^{(0)} } \left| {\Delta {{{v}}}} + \right| .. math:: :label: 2.2.8 - v_{\ast (eff)}^{(0)}= {\left( { u_{\ast (eff)}^{(0) 2} + \gamma _c^2 w_c^2 } \right)}^{ 1/2} + v_{\ast (eff)}^{(0)}= {\left( { u_{\ast (eff)}^{(0) 2} + \gamma _c^2 w_c^2 + } \right)}^{ 1/2} .. math:: :label: 2.2.9 - u_{\ast (f)}= u_{\ast (eff)} \frac{\ln ( z_c / z_{0m(eff)} )}{\ln ( z_c / z_{0m} )} + u_{\ast (f)}= u_{\ast (eff)} \frac{\ln ( z_c / z_{0m(eff)} )}{\ln ( z_c / + z_{0m} )} .. math:: :label: 2.2.10 - v_{\ast (f)}^{(0)}= {\left( { u_{\ast (f)}^{(0) 2} + \gamma _c^2 w_c^2 } \right)}^{ 1/2} + v_{\ast (f)}^{(0)}= {\left( { u_{\ast (f)}^{(0) 2} + \gamma _c^2 w_c^2 } + \right)}^{ 1/2} END IF. @@ -4582,7 +4806,9 @@ Then calculate: .. math:: :label: (2.2.13 - C_{D(f)}^{(0)}= C_{D(eff)}^{(0)} {\left( {1 + \frac{1}{2} c_{D(orog)} f_D \frac{A}{S} {\left( {\frac{\ln ( z_c / z_{0m} )}{k}} \right)}^2 } \right)}^{-1} + C_{D(f)}^{(0)}= C_{D(eff)}^{(0)} {\left( {1 + \frac{1}{2} c_{D(orog)} f_D + \frac{A}{S} {\left( {\frac{\ln ( z_c / z_{0m} )}{k}} \right)}^2 } + \right)}^{-1} .. math:: :label: (2.2.14 @@ -4595,7 +4821,8 @@ DO n = 1 to N .. math:: :label: (2.2.15 - u_{\ast (eff)}^{(n)2}= C_{D(eff)}^{(n-1)} \left| {\Delta {{\rm {\bf v}}}} \right| + u_{\ast (eff)}^{(n)2}= C_{D(eff)}^{(n-1)} \left| {\Delta {{\rm {\bf v}}}} + \right| .. math:: :label: (2.2.16 @@ -4603,23 +4830,28 @@ DO n = 1 to N .. math:: :label: (2.2.17 - {\left( {\frac{ F_{B0} }{ \rho _0 }} \right)}^{(n)}=- { C_{H(eff)} }^{(n-1)} \Delta B + {\left( {\frac{ F_{B0} }{ \rho _0 }} \right)}^{(n)}=- { C_{H(eff)} }^{(n-1)} + \Delta B .. math:: :label: (2.2.18 - w_\ast ^{(n)}= {\left[ { z_i {\left( {\frac{ F_{B0} }{ \rho _0 }} \right)}^{(n)} } \right]}^{ 1/3} + w_\ast ^{(n)}= {\left[ { z_i {\left( {\frac{ F_{B0} }{ \rho _0 }} + \right)}^{(n)} } \right]}^{ 1/3} .. math:: :label: (2.2.19 - v_{\ast (eff)}^{(n)2}= u_{\ast (eff)}^{(n)2} + \gamma _t^2 w_\ast ^{(n)2} + \gamma _c^2 w_c^2 + v_{\ast (eff)}^{(n)2}= u_{\ast (eff)}^{(n)2} + \gamma _t^2 w_\ast ^{(n)2} + + \gamma _c^2 w_c^2 .. math:: :label: (2.2.20 - v_{\ast (f)}^{(n)2}= u_{\ast (f)}^{(n)2} + \gamma _t^2 w_\ast ^{(n)2} + \gamma _c^2 w_c^2 + v_{\ast (f)}^{(n)2}= u_{\ast (f)}^{(n)2} + \gamma _t^2 w_\ast ^{(n)2} + + \gamma _c^2 w_c^2 .. math:: :label: (2.2.21 - \frac{1}{ L^{(n)} }=\frac{-k( F_{B0} / \rho _0 )^{(n)} }{ v_{\ast (eff)}^{(n)3} } + \frac{1}{ L^{(n)} }=\frac{-k( F_{B0} / \rho _0 )^{(n)} }{ v_{\ast + (eff)}^{(n)3} } .. math:: :label: (2.2.22 @@ -4639,7 +4871,9 @@ DO n = 1 to N .. math:: :label: (2.2.26 - C_{D(f)}^{(n)}= C_{D(eff)}^{(n)} {\left( {1 + \frac{1}{2} c_{D(orog)} f_D \frac{A}{S} {\left( {\frac{\ln ( z_c / z_{0m} )}{k}} \right)}^2 } \right)}^{-1} + C_{D(f)}^{(n)}= C_{D(eff)}^{(n)} {\left( {1 + \frac{1}{2} c_{D(orog)} f_D + \frac{A}{S} {\left( {\frac{\ln ( z_c / z_{0m} )}{k}} \right)}^2 } + \right)}^{-1} .. math:: :label: (2.2.27 @@ -4656,7 +4890,8 @@ surface sensible and latent heat fluxes and surface stress: .. math:: :label: 2.2.28 - H_{0(eff)}=- c_P \rho _0 C_{H(eff)}^{(N)} \left( {\Delta T + \frac{g}{c_p }( z_1 + z_{0m(eff)} - z_{0h} )} \right) + H_{0(eff)}=- c_P \rho _0 C_{H(eff)}^{(N)} \left( {\Delta T + \frac{g}{c_p + }( z_1 + z_{0m(eff)} - z_{0h} )} \right) .. math:: :label: 2.2.29 @@ -4731,14 +4966,16 @@ amount) .. math:: :label: 2.3.5 - X_{ob} = X_0 + \frac{ F_{X0(eff)} }{ \rho _0 v_{\ast (eff)} k} \Phi _h (L, z_{ob} + z_{0h(eff)} , z_{0h(eff)} ) + X_{ob} = X_0 + \frac{ F_{X0(eff)} }{ \rho _0 v_{\ast (eff)} k} \Phi _h + (L, z_{ob} + z_{0h(eff)} , z_{0h(eff)} ) and using the expression for the surface flux of the scalar quantity :math:`X` this becomes .. math:: :label: 2.3.6 - X_{ob} = X_0 + \frac{ C_{H(eff)} }{k v_{\ast (eff)} } \Phi _h (L, z_{ob} + z_{0h(eff)} , z_{0h(eff)} ) ( X_1 - X_0 ). + X_{ob} = X_0 + \frac{ C_{H(eff)} }{k v_{\ast (eff)} } \Phi _h (L, z_{ob} + + z_{0h(eff)} , z_{0h(eff)} ) ( X_1 - X_0 ). Alternatively if the observation height scalar quantities are assumed to lie on a profile defined by the flat surface roughness length and flux @@ -4746,7 +4983,8 @@ then .. math:: :label: 2.3.7 - X_{ob} = X_0 + \frac{ C_{H(f)} }{k v_{\ast (f)} } \Phi _h (L, z_{ob} + z_{0h} , z_{0h} ) ( X_1 - X_0 ) + X_{ob} = X_0 + \frac{ C_{H(f)} }{k v_{\ast (f)} } \Phi _h (L, z_{ob} + + z_{0h} , z_{0h} ) ( X_1 - X_0 ) For temperature and humidity z\ :math:`_{ob}` is set to the screen height (1.5 m) and the last iteration (N) values of C\ :math:`_{H}`, L @@ -4754,7 +4992,8 @@ and v\ :math:`_{\ast }` are used. .. _section_2.4: -Distributed form drag – an alternative to the effective roughness length parametrization +Distributed form drag – an alternative to the effective roughness length +parametrization ---------------------------------------------------------------------------------------- An alternative representation of the turbulent form drag due to sub-grid @@ -4875,7 +5114,8 @@ assumed to be positive. The new scheme is written .. math:: :label: eq:sppf1 - \frac{X^{*}-X^{n}}{\Delta t}=-{\cal I}_{1}\left[K\left(X^{n}\right)^{P}\right] + \frac{X^{*}-X^{n}}{\Delta t}=-{\cal + I}_{1}\left[K\left(X^{n}\right)^{P}\right] X^{*}+{\cal E}_{1}\left[K\left(X^{n}\right)^{P}\right] X^{n}+\left({\cal I}_{1}-{\cal E}_{1}\right)S, @@ -4924,7 +5164,9 @@ becomes .. math:: :label: eq:sppf_bl1 - \frac{X^{*}-X^{n}}{\Delta t} = {\cal I}_{1}\frac{\partial F}{\partial z}^{*}-{\cal E}_{1}\frac{\partial F}{\partial z}^{n}+\left({\cal I}_{1}-{\cal E}_{1}\right)S + \frac{X^{*}-X^{n}}{\Delta t} = {\cal I}_{1}\frac{\partial F}{\partial + z}^{*}-{\cal E}_{1}\frac{\partial F}{\partial z}^{n}+\left({\cal + I}_{1}-{\cal E}_{1}\right)S .. math:: :label: eq:sppf_bl2 @@ -4949,7 +5191,8 @@ intermediate “starred” quantities are eliminated and the scheme is reduced into a single equation then the forcing term will be multiplied by :math:`1`. -Recall from section :ref:`Model variables and turbulence closure ` that the boundary layer solver +Recall from section :ref:`Model variables and turbulence closure ` +that the boundary layer solver computes the increment of :math:`X`, where :math:`X=u,\; v,\;\theta_{L},\; q_{w}`. Let :math:`\delta X^{*}=X^{*}-X^{n}`, :math:`\delta @@ -4965,11 +5208,17 @@ Writing equations :eq:`eq:sppf_bl1`, .. math:: :label: eq:sppf_inc1 - \frac{\delta X}{\Delta t}^{*} = ({\cal I}_{1}-{\cal E}_{1})\left(\frac{\partial F}{\partial z}^{n}+S\right)+{\cal I}_{1}\frac{\partial}{\partial z}\left(K_{X}\frac{\partial\delta X}{\partial z}^{*}\right) + \frac{\delta X}{\Delta t}^{*} = ({\cal I}_{1}-{\cal + E}_{1})\left(\frac{\partial F}{\partial z}^{n}+S\right)+{\cal + I}_{1}\frac{\partial}{\partial z}\left(K_{X}\frac{\partial\delta X}{\partial + z}^{*}\right) .. math:: :label: eq:sppf_inc2 - \frac{\delta X}{\Delta t}^{n+1} = ({\cal I}_{2}-{\cal E}_{2})\left(\frac{\partial F}{\partial z}^{*}+S\right)+{\cal I}_{2}\frac{\partial}{\partial z}\left(K_{X}\frac{\partial\delta X}{\partial z}^{n+1}\right) + \frac{\delta X}{\Delta t}^{n+1} = ({\cal I}_{2}-{\cal + E}_{2})\left(\frac{\partial F}{\partial z}^{*}+S\right)+{\cal + I}_{2}\frac{\partial}{\partial z}\left(K_{X}\frac{\partial\delta X}{\partial + z}^{n+1}\right) .. math:: :label: eq:sppf_inc3 @@ -4991,7 +5240,8 @@ Consider the following equivalent form of .. math:: :label: eq:du_star - \frac{\delta u^{*}}{\Delta t}=\frac{\partial\bar{\tau}_{x}^{*}}{\partial z}+\left({\cal I}_{1}-{\cal E}_{1}\right)S + \frac{\delta u^{*}}{\Delta t}=\frac{\partial\bar{\tau}_{x}^{*}}{\partial + z}+\left({\cal I}_{1}-{\cal E}_{1}\right)S where :math:`\tau_{x}` is the :math:`u` wind component stress (defined in the same way as the flux in :eq:`eq:sppf_inc1` and @@ -4999,7 +5249,8 @@ in the same way as the flux in :eq:`eq:sppf_inc1` and .. math:: :label: eq:tau_star - \bar{\tau}_{x}^{*}={\cal I}_{1}\tau_{x}^{*}-{\cal E}_{1}\tau_{x}^{n},\qquad\tau_{x}^{*}=\tau_{x}^{n}+K_{u}\frac{\partial\delta + \bar{\tau}_{x}^{*}={\cal I}_{1}\tau_{x}^{*}-{\cal + E}_{1}\tau_{x}^{n},\qquad\tau_{x}^{*}=\tau_{x}^{n}+K_{u}\frac{\partial\delta u^{*}}{\partial z}. Substituting :eq:`eq:tau_star` into @@ -5022,7 +5273,8 @@ levels), discretizing the previous equation in :math:`z` on all .. math:: - \delta u_{k+1/2}^{*} = ({\cal I}_{1}-{\cal E}_{1})\Delta t\left(\frac{\tau_{x}^{n}\Big|_{k+1}-\tau_{x}^{n}\Big|_{k}}{z_{k+1}-z_{k}}+S_{k+1/2}\right) + \delta u_{k+1/2}^{*} = ({\cal I}_{1}-{\cal E}_{1})\Delta + t\left(\frac{\tau_{x}^{n}\Big|_{k+1}-\tau_{x}^{n}\Big|_{k}}{z_{k+1}-z_{k}}+S_{k+1/2}\right) .. math:: @@ -5040,7 +5292,8 @@ or, rearranging A_{k}\delta u_{k+3/2}^{*}+B_{k}\delta u_{k+1/2}^{*}+C_{k}\delta u_{k-1/2}^{*}=\Delta - t({\cal I}_{1}-{\cal E}_{1})\left(\frac{\tau_{x}^{n}\Big|_{k+1}-\tau_{x}^{n}\Big|_{k}}{z_{k+1}-z_{k}}+S_{k+1/2}\right), + t({\cal I}_{1}-{\cal + E}_{1})\left(\frac{\tau_{x}^{n}\Big|_{k+1}-\tau_{x}^{n}\Big|_{k}}{z_{k+1}-z_{k}}+S_{k+1/2}\right), where :math:`k=1,2,\ldots,L-2`, @@ -5059,7 +5312,9 @@ For the top :math:`\rho`-level, :math:`k=L-1`, the .. math:: :label: eq:tridiag_top - B_{L}\delta u_{L-1/2}^{*}+C_{L}\delta u_{L-3/2}^{*}=\Delta t({\cal I}_{1}-{\cal E}_{1})\left(\frac{\tau_{x}^{n}\Big|_{L}-\tau_{x}^{n}\Big|_{L-1}}{z_{L}-z_{L-1}}+S_{L-1/2}\right), + B_{L}\delta u_{L-1/2}^{*}+C_{L}\delta u_{L-3/2}^{*}=\Delta t({\cal + I}_{1}-{\cal + E}_{1})\left(\frac{\tau_{x}^{n}\Big|_{L}-\tau_{x}^{n}\Big|_{L-1}}{z_{L}-z_{L-1}}+S_{L-1/2}\right), where :math:`B_{L}`, :math:`C_{L}` are derived as before setting :math:`A_{L}=0`. @@ -5070,25 +5325,31 @@ For the bottom :math:`\rho`-level, :math:`k=0`, the .. math:: :label: eq:u_bc_1 \begin{aligned} - \delta u_{1/2}^{*} & = & \frac{\Delta t}{z_{1}-0}\left(\bar{\tau}_{x}^{*}\Big|_{1}-\bar{\tau}_{x}^{*}\Big|_{0}\right)+\Delta t\left({\cal I}_{1}-{\cal E}_{1}\right)S_{1/2} + \delta u_{1/2}^{*} & = & \frac{\Delta + t}{z_{1}-0}\left(\bar{\tau}_{x}^{*}\Big|_{1}-\bar{\tau}_{x}^{*}\Big|_{0}\right)+\Delta t\left({\cal I}_{1}-{\cal E}_{1}\right)S_{1/2} \end{aligned} where, from :eq:`eq:tau_star`, .. math:: :label: eq:u_bc_2 - \bar{\tau}_{x}^{*}\Big|_{1}=\left({\cal I}_{1}-{\cal E}_{1}\right)\tau_{x}^{n}\Big|_{1}+{\cal I}_{1}\left(K_{u}\frac{\partial\delta u^{*}}{\partial z}\right)\Big|_{1}. + \bar{\tau}_{x}^{*}\Big|_{1}=\left({\cal I}_{1}-{\cal + E}_{1}\right)\tau_{x}^{n}\Big|_{1}+{\cal + I}_{1}\left(K_{u}\frac{\partial\delta u^{*}}{\partial z}\right)\Big|_{1}. Combining :eq:`eq:u_bc_1`, :eq:`eq:u_bc_2` the bottom row discretization is obtained: .. math:: :label: eq:u_bc_3 - A_{0}\delta u_{3/2}^{*}+B_{0}\delta u_{1/2}^{*}=\Delta t\left({\cal I}_{1}-{\cal E}_{1}\right)\left(\frac{\tau_{x}^{n}\Big|_{1}}{z_{1}}+S_{1/2}\right)-\frac{\Delta t}{z_{1}}\bar{\tau}_{x}^{*}\Big|_{0} + A_{0}\delta u_{3/2}^{*}+B_{0}\delta u_{1/2}^{*}=\Delta t\left({\cal + I}_{1}-{\cal + E}_{1}\right)\left(\frac{\tau_{x}^{n}\Big|_{1}}{z_{1}}+S_{1/2}\right)-\frac{\Delta t}{z_{1}}\bar{\tau}_{x}^{*}\Big|_{0} where -.. math:: A_{0}=-{\cal I}_{1}\frac{\Delta tK_{u}\Big|_{1}}{z_{1}(z_{3/2}-z_{1/2})},\quad B_{0}=1-A_{0}. +.. math:: A_{0}=-{\cal I}_{1}\frac{\Delta + tK_{u}\Big|_{1}}{z_{1}(z_{3/2}-z_{1/2})},\quad B_{0}=1-A_{0}. Equations :eq:`eq:tridiag`, :eq:`eq:tridiag_top` and @@ -5103,14 +5364,20 @@ equations. When the elimination procedure takes place where :math:`\delta u_{1/2}^{'}`, :math:`\beta` are available quantities. Furthermore, -.. math:: \bar{\tau}_{x}^{*}\Big|_{0}=\left({\cal I}_{1}-{\cal E}_{1}\right)\tau_{x}^{n}\Big|_{0}+{\cal I}_{1}\left(K_{u}\frac{\partial\delta u^{*}}{\partial z}\right)\Big|_{0} +.. math:: \bar{\tau}_{x}^{*}\Big|_{0}=\left({\cal I}_{1}-{\cal + E}_{1}\right)\tau_{x}^{n}\Big|_{0}+{\cal I}_{1}\left(K_{u}\frac{\partial\delta + u^{*}}{\partial z}\right)\Big|_{0} -Approximating :math:`\left(\frac{\partial\delta u^{*}}{\partial z}\right)\Big|_{0}\approx\frac{\delta u_{1/2}^{*}-\delta u_{0}^{*}}{z_{1/2}}`, and assuming that :math:`u_{0}=0` the previous +Approximating :math:`\left(\frac{\partial\delta u^{*}}{\partial +z}\right)\Big|_{0}\approx\frac{\delta u_{1/2}^{*}-\delta u_{0}^{*}}{z_{1/2}}`, +and assuming that :math:`u_{0}=0` the previous equation becomes .. math:: :label: eq:tau_zero - \bar{\tau}_{x}^{*}\Big|_{0}=\left({\cal I}_{1}-{\cal E}_{1}\right)\tau_{x}^{n}\Big|_{0}+{\cal I}_{1}K_{u}\Big|_{0}\frac{\delta u_{1/2}^{*}}{z_{1/2}}. + \bar{\tau}_{x}^{*}\Big|_{0}=\left({\cal I}_{1}-{\cal + E}_{1}\right)\tau_{x}^{n}\Big|_{0}+{\cal I}_{1}K_{u}\Big|_{0}\frac{\delta + u_{1/2}^{*}}{z_{1/2}}. From :eq:`eq:du_half`, :eq:`eq:tau_zero` the following expression for the @@ -5118,7 +5385,8 @@ implicit surface stress is obtained .. math:: :label: eq:imp_tau - \bar{\tau}_{x}^{*}\Big|_{0}=\frac{\left({\cal I}_{1}-{\cal E}_{1}\right)\tau_{x}^{n}\Big|_{0}+{\cal I}_{1}(K_{u}\Big|_{0}/z_{1/2})\delta + \bar{\tau}_{x}^{*}\Big|_{0}=\frac{\left({\cal I}_{1}-{\cal + E}_{1}\right)\tau_{x}^{n}\Big|_{0}+{\cal I}_{1}(K_{u}\Big|_{0}/z_{1/2})\delta u_{1/2}^{'}}{1+{\cal I}_{1}(K_{u}\Big|_{0}/z_{1/2})\beta}. Then, :math:`\delta u_{1/2}^{*}` can be computed from @@ -5128,7 +5396,8 @@ the 2nd stage :eq:`eq:sppf_inc2` will be .. math:: :label: eq:imp_tau2 - \bar{\tau}_{x}^{n+1}\Big|_{0}=\frac{\left({\cal I}_{2}-{\cal E}_{2}\right)\tau_{x}^{*}\Big|_{0}+{\cal I}_{2}(K_{u}\Big|_{0}/z_{1/2})\delta + \bar{\tau}_{x}^{n+1}\Big|_{0}=\frac{\left({\cal I}_{2}-{\cal + E}_{2}\right)\tau_{x}^{*}\Big|_{0}+{\cal I}_{2}(K_{u}\Big|_{0}/z_{1/2})\delta u_{1/2}^{'}}{1+{\cal I}_{2}(K_{u}\Big|_{0}/z_{1/2})\beta}. In the same way :math:`\bar{\tau}_{y}^{*}\Big|_{0}`, @@ -5152,7 +5421,8 @@ is applied as follows. Consider the equivalent discrete form of .. math:: :label: eq:dX_star \frac{\delta X^{*}}{\Delta t} - =\frac{\partial\overline{F}^{*}}{\partial z}+\left({\cal I}_{1}-{\cal E}_{1}\right)S + =\frac{\partial\overline{F}^{*}}{\partial z}+\left({\cal I}_{1}-{\cal + E}_{1}\right)S Considering that, @@ -5164,14 +5434,18 @@ Considering that, :eq:`eq:dX_star` would re-produce :eq:`eq:sppf_inc1`, which is re-written below, -.. math:: \frac{\delta X^{*}}{\Delta t}=({\cal I}_{1}-{\cal E}_{1})\left(\frac{\partial F^{n}}{\partial z}+S\right)+{\cal I}_{1}\frac{\partial}{\partial z}\left(K_{X}\frac{\partial\delta X^{*}}{\partial z}\right) +.. math:: \frac{\delta X^{*}}{\Delta t}=({\cal I}_{1}-{\cal + E}_{1})\left(\frac{\partial F^{n}}{\partial z}+S\right)+{\cal + I}_{1}\frac{\partial}{\partial z}\left(K_{X}\frac{\partial\delta + X^{*}}{\partial z}\right) and thus the following discretization is obtained, on :math:`\theta`-levels: .. math:: - \frac{\delta X_{k}^{*}}{\Delta t} = \left({\cal I}_{1}-{\cal E}_{1}\right)\left(\frac{F_{k+1/2}^{n}-F_{k-1/2}^{n}}{z_{k+1/2}-z_{k-1/2}}+S_{k}\right) + \frac{\delta X_{k}^{*}}{\Delta t} = \left({\cal I}_{1}-{\cal + E}_{1}\right)\left(\frac{F_{k+1/2}^{n}-F_{k-1/2}^{n}}{z_{k+1/2}-z_{k-1/2}}+S_{k}\right) .. math:: @@ -5182,17 +5456,24 @@ or, .. math:: :label: eq:dX_disc - A_{k}\delta X_{k+1}^{*}+B_{k}\delta X_{k}^{*}+C_{k}\delta X_{k-1}^{*}=\left({\cal I}_{1}-{\cal E}_{1}\right)\left(\frac{F_{k+1/2}^{n}-F_{k-1/2}^{n}}{z_{k+1/2}-z_{k-1/2}}+S_{k}\right),\quad k=2,\ldots,L-1 + A_{k}\delta X_{k+1}^{*}+B_{k}\delta X_{k}^{*}+C_{k}\delta + X_{k-1}^{*}=\left({\cal I}_{1}-{\cal + E}_{1}\right)\left(\frac{F_{k+1/2}^{n}-F_{k-1/2}^{n}}{z_{k+1/2}-z_{k-1/2}}+S_{k}\right),\quad k=2,\ldots,L-1 where, -.. math:: A_{k}=-{\cal I}_{1}\frac{\Delta tK_{X}\Big|_{k+1/2}}{(z_{k+1}-z_{k})(z_{k+1/2}-z_{k-1/2})},\; C_{k}=-{\cal I}_{1}\frac{\Delta tK_{X}\Big|_{k-1/2}}{(z_{k+1/2}-z_{k-1/2})(z_{k}-z_{k-1})},\quad B_{k}=1-A_{k}-C_{k}. +.. math:: A_{k}=-{\cal I}_{1}\frac{\Delta + tK_{X}\Big|_{k+1/2}}{(z_{k+1}-z_{k})(z_{k+1/2}-z_{k-1/2})},\; C_{k}=-{\cal + I}_{1}\frac{\Delta + tK_{X}\Big|_{k-1/2}}{(z_{k+1/2}-z_{k-1/2})(z_{k}-z_{k-1})},\quad + B_{k}=1-A_{k}-C_{k}. The discrete equation for the top level, :math:`k=L`, will be: .. math:: :label: eq:dX_disc_top - B_{L}\delta X_{L}^{*}+C_{L}\delta X_{L-1}^{*}=\left({\cal I}_{1}-{\cal E}_{1}\right)\left(\frac{-F_{L-1/2}^{n}}{z_{L+1/2}-z_{L-1/2}}+S_{L}\right), + B_{L}\delta X_{L}^{*}+C_{L}\delta X_{L-1}^{*}=\left({\cal I}_{1}-{\cal + E}_{1}\right)\left(\frac{-F_{L-1/2}^{n}}{z_{L+1/2}-z_{L-1/2}}+S_{L}\right), where :math:`B_{L}`, :math:`C_{L}` are derived as before setting :math:`A_{L}=0`. @@ -5202,7 +5483,9 @@ From :eq:`eq:dX_star`, a bottom interior level .. math:: :label: eq:dX1_star - \delta X_{1}^{*}=\frac{\Delta t}{z_{3/2}-0}\left(\overline{F_{3/2}}^{*}-\overline{F_{0}}^{*}\right)+\Delta t\left({\cal I}_{1}-{\cal E}_{1}\right)S_{1} + \delta X_{1}^{*}=\frac{\Delta + t}{z_{3/2}-0}\left(\overline{F_{3/2}}^{*}-\overline{F_{0}}^{*}\right)+\Delta + t\left({\cal I}_{1}-{\cal E}_{1}\right)S_{1} :math:`F_{0}` is used instead of :math:`F_{1/2}`. The former is computed by the implicit surface scheme. This flux gradient is defined in the @@ -5214,7 +5497,8 @@ becomes \delta X_{1}^{*}=\frac{\Delta t}{z_{3/2}} - \left[\left({\cal I}_{1}-{\cal E}_{1}\right)F_{3/2}^{n}-\overline{F_{0}}^{*}\right] + \left[\left({\cal I}_{1}-{\cal + E}_{1}\right)F_{3/2}^{n}-\overline{F_{0}}^{*}\right] +\Delta t\left({\cal I}_{1}-{\cal E}_{1}\right) S_{1}+\Delta t{\cal I}_{1}\frac{1}{z_{3/2}} \left(K_{X}\frac{\partial\delta X^{*}}{\partial z}\right)_{3/2} @@ -5223,7 +5507,8 @@ where :math:`\overline{F}_{0}^{*}` can be approximated as .. math:: :label: eq:F0_star - \overline{F}_{0}^{*}={\cal I}_{1}F_{0}^{*}-{\cal E}_{1}F_{0}^{n}\approx\left({\cal I}_{1}-{\cal E}_{1}\right)F_{JULES} + \overline{F}_{0}^{*}={\cal I}_{1}F_{0}^{*}-{\cal + E}_{1}F_{0}^{n}\approx\left({\cal I}_{1}-{\cal E}_{1}\right)F_{JULES} where, :math:`F_{JULES}` is the implicit flux calculated by the *implicit surface scheme using the original implicit algorithm*. @@ -5241,7 +5526,8 @@ or, .. math:: :label: eq:dX_bottom A_{1}\delta X_{2}^{*}+B_{1}\delta X_{1}^{*}=\Delta t - \left({\cal I}_{1}-{\cal E}_{1}\right)\left(\frac{F_{3/2}^{n}-F_{JULES}}{z_{3/2}}+S_{1}\right) + \left({\cal I}_{1}-{\cal + E}_{1}\right)\left(\frac{F_{3/2}^{n}-F_{JULES}}{z_{3/2}}+S_{1}\right) where, @@ -5260,29 +5546,41 @@ Similarly the corresponding discrete equations for .. math:: :label: eq:dXtop_np1 - B_{L}^{'}\delta X_{L}^{n+1}+C_{L}^{'}\delta X_{L-1}^{n+1}=\left({\cal I}_{2}-{\cal E}_{2}\right)\left(\frac{-F_{L-1/2}^{*}}{z_{L+1/2}-z_{L-1/2}}+S_{L}\right), + B_{L}^{'}\delta X_{L}^{n+1}+C_{L}^{'}\delta X_{L-1}^{n+1}=\left({\cal + I}_{2}-{\cal + E}_{2}\right)\left(\frac{-F_{L-1/2}^{*}}{z_{L+1/2}-z_{L-1/2}}+S_{L}\right), .. math:: :label: eq:dXk_np1 - A_{k}^{'}\delta X_{k+1}^{n+1}+B_{k}^{'}\delta X_{k}^{n+1}+C_{k}^{'}\delta X_{k-1}^{n+1}=\left({\cal I}_{2}-{\cal E}_{2}\right)\left(\frac{F_{k+1/2}^{*}-F_{k-1/2}^{*}}{z_{k+1/2}-z_{k-1/2}}+S_{k}\right),\quad k=L-1,\ldots,2 + A_{k}^{'}\delta X_{k+1}^{n+1}+B_{k}^{'}\delta X_{k}^{n+1}+C_{k}^{'}\delta + X_{k-1}^{n+1}=\left({\cal I}_{2}-{\cal + E}_{2}\right)\left(\frac{F_{k+1/2}^{*}-F_{k-1/2}^{*}}{z_{k+1/2}-z_{k-1/2}}+S_{k}\right),\quad k=L-1,\ldots,2 .. math:: :label: eq:dX1_np1 - A_{1}^{'}\delta X_{2}^{n+1}+B_{1}^{'}\delta X_{1}^{n+1}=\left({\cal I}_{2}-{\cal E}_{2}\right)\left(\frac{F_{3/2}^{*}-F_{JULES}}{z_{3/2}}+S_{1}\right) + A_{1}^{'}\delta X_{2}^{n+1}+B_{1}^{'}\delta X_{1}^{n+1}=\left({\cal + I}_{2}-{\cal + E}_{2}\right)\left(\frac{F_{3/2}^{*}-F_{JULES}}{z_{3/2}}+S_{1}\right) where, -.. math:: A_{k}^{'}=-{\cal I}_{2}\frac{\Delta tK_{X}\Big|_{k+1/2}}{(z_{k+1}-z_{k})(z_{k+1/2}-z_{k-1/2})},\; C_{k}^{'}=-{\cal I}_{2}\frac{\Delta tK_{X}\Big|_{k-1/2}}{(z_{k+1/2}-z_{k-1/2})(z_{k}-z_{k-1})},\quad B_{k}^{'}=1-A_{k}^{'}-C_{k}^{'}, +.. math:: A_{k}^{'}=-{\cal I}_{2}\frac{\Delta + tK_{X}\Big|_{k+1/2}}{(z_{k+1}-z_{k})(z_{k+1/2}-z_{k-1/2})},\; C_{k}^{'}=-{\cal + I}_{2}\frac{\Delta + tK_{X}\Big|_{k-1/2}}{(z_{k+1/2}-z_{k-1/2})(z_{k}-z_{k-1})},\quad + B_{k}^{'}=1-A_{k}^{'}-C_{k}^{'}, for :math:`k=L,\ldots,2,\quad A_{L}=0`. -.. math:: A_{1}^{'}=-{\cal I}_{2}\Delta t\frac{K_{X}\Big|_{3/2}}{z_{3/2}(z_{2}-z_{1})},\quad B_{1}^{'}=1-A_{1}^{'} +.. math:: A_{1}^{'}=-{\cal I}_{2}\Delta + t\frac{K_{X}\Big|_{3/2}}{z_{3/2}(z_{2}-z_{1})},\quad B_{1}^{'}=1-A_{1}^{'} and the approximation .. math:: :label: eq:F0_np1 - \overline{F}_{0}^{n+1}={\cal I}_{2}F_{0}^{n+1}-{\cal E}_{2}F_{0}^{n}\approx\left({\cal I}_{2}-{\cal E}_{2}\right)F_{JULES} + \overline{F}_{0}^{n+1}={\cal I}_{2}F_{0}^{n+1}-{\cal + E}_{2}F_{0}^{n}\approx\left({\cal I}_{2}-{\cal E}_{2}\right)F_{JULES} has taken place. The same flux :math:`F_{JULES}` will be used for both :eq:`eq:F0_star` and :eq:`eq:F0_np1` and @@ -5294,7 +5592,8 @@ calculations take place for the scalar variables: * - CALL bdy_impl3(): - - set up coefficients for :eq:`eq:dX_disc_top`, :eq:`eq:dX_disc` and do a downward sweep; + - set up coefficients for :eq:`eq:dX_disc_top`, :eq:`eq:dX_disc` and do a + downward sweep; * - - do a downward sweep using the original implicit scheme to @@ -5315,10 +5614,12 @@ calculations take place for the scalar variables: - back substitute to compute implicit correction :math:`\delta X^{*}`; * - CALL bdy_impl3(): - - compute explicit flux :math:`F^*=F^n+K_X\frac{\partial \delta X^*}{\partial z}`; + - compute explicit flux :math:`F^*=F^n+K_X\frac{\partial \delta + X^*}{\partial z}`; * - - - set up coefficients for :eq:`eq:dXtop_np1`, :eq:`eq:dXk_np1`, :eq:`eq:dX1_np1` and + - set up coefficients for :eq:`eq:dXtop_np1`, :eq:`eq:dXk_np1`, + :eq:`eq:dX1_np1` and * - - do a downward sweep; @@ -5327,7 +5628,8 @@ calculations take place for the scalar variables: - only momentum variables are affected - no change in scalars; * - CALL bdy_impl4(): - - back substitute to compute final implicit correction :math:`\delta X^{n+1}` + - back substitute to compute final implicit correction :math:`\delta + X^{n+1}` NB: for CABLE compatibility, sf_impl2 is now called by an intermediate routine surf_couple_implicit. @@ -5352,7 +5654,8 @@ and the scalar fluxes are derived. For the new scheme the total zonal wind component stress is defined as: -.. math:: \overline{\tau_{x}}^{n+1}\equiv\overline{\tau_{x}}^{[n,*]}+\overline{\tau_{x}}^{[*,n+1]} +.. math:: + \overline{\tau_{x}}^{n+1}\equiv\overline{\tau_{x}}^{[n,*]}+\overline{\tau_{x}}^{[*,n+1]} where, :math:`\overline{\tau_{x}}^{[n,*]}` denotes the total stress for the 1st stage of the scheme, i.e. the time averaged stress from @@ -5363,11 +5666,17 @@ corrector are defined as: .. math:: - \overline{\tau_{x}}^{[n,*]}\equiv{\cal I}_{1}\tau_{x}^{*}-{\cal E}_{1}\tau_{x}^{n} = \left({\cal I}_{1}-{\cal E}_{1}\right)\tau_{x}^{n}+{\cal I}_{1}K_{u}\frac{\partial\delta u^{*}}{\partial z} + \overline{\tau_{x}}^{[n,*]}\equiv{\cal I}_{1}\tau_{x}^{*}-{\cal + E}_{1}\tau_{x}^{n} = \left({\cal I}_{1}-{\cal + E}_{1}\right)\tau_{x}^{n}+{\cal I}_{1}K_{u}\frac{\partial\delta + u^{*}}{\partial z} .. math:: - \overline{\tau_{x}}^{[*,n+1]}\equiv{\cal I}_{2}\tau_{x}^{n+1}-{\cal E}_{2}\tau_{x}^{*} = \left({\cal I}_{2}-{\cal E}_{2}\right)\tau_{x}^{*}+{\cal I}_{2}K_{u}\frac{\partial\delta u^{n+1}}{\partial z} + \overline{\tau_{x}}^{[*,n+1]}\equiv{\cal I}_{2}\tau_{x}^{n+1}-{\cal + E}_{2}\tau_{x}^{*} = \left({\cal I}_{2}-{\cal + E}_{2}\right)\tau_{x}^{*}+{\cal I}_{2}K_{u}\frac{\partial\delta + u^{n+1}}{\partial z} where, :math:`\delta u^{*}=u^{*}-u^{n},\;\delta u^{n+1}=u^{n+1}-u^{*}`. @@ -5394,13 +5703,17 @@ modified version of the flux formulae (78), (79) of .. math:: :label: eq:FTLstar \begin{aligned} - \frac{\overline{H^{*}}}{c_{p}} & = & \frac{(1+\beta B_{2})[\gamma_{2}F_{T}^{n}+A_{1}\delta{T'}_{1}+A_{2}\delta{Q'}_{1}]-\beta A_{2}[\gamma_{2}F_{Q}^{n}+B_{1}\delta{T'}_{1}+B_{2}\delta{Q'}_{1}]}{(1+\beta A_{1})(1+\beta B_{2})-\beta^{2}A_{2}B_{1}} + \frac{\overline{H^{*}}}{c_{p}} & = & \frac{(1+\beta + B_{2})[\gamma_{2}F_{T}^{n}+A_{1}\delta{T'}_{1}+A_{2}\delta{Q'}_{1}]-\beta + A_{2}[\gamma_{2}F_{Q}^{n}+B_{1}\delta{T'}_{1}+B_{2}\delta{Q'}_{1}]}{(1+\beta A_{1})(1+\beta B_{2})-\beta^{2}A_{2}B_{1}} \end{aligned} .. math:: :label: eq:FQWstar \begin{aligned} - \overline{E^{*}} & = & \frac{(1+\beta A_{1})[\gamma_{2}F_{Q}^{n}+B_{1}\delta{T'}_{1}+B_{2}\delta{Q'}_{1}]-\beta B_{1}[\gamma_{2}F_{T}^{n}+A_{1}\delta{T'}_{1}+A_{2}\delta{Q'}_{1}]}{(1+\beta A_{1})(1+\beta B_{2})-\beta^{2}A_{2}B_{1}} + \overline{E^{*}} & = & \frac{(1+\beta + A_{1})[\gamma_{2}F_{Q}^{n}+B_{1}\delta{T'}_{1}+B_{2}\delta{Q'}_{1}]-\beta + B_{1}[\gamma_{2}F_{T}^{n}+A_{1}\delta{T'}_{1}+A_{2}\delta{Q'}_{1}]}{(1+\beta A_{1})(1+\beta B_{2})-\beta^{2}A_{2}B_{1}} \end{aligned} where, :math:`F_{T}^{n}`, :math:`F_{Q}^{n}` denote the surface explicit @@ -5452,14 +5765,17 @@ and :math:`g_s` is the surface conductance. To derive time-weighted level 1 :math:`T` and :math:`Q` consistent with the discrete equations of the new scheme is written as follows: -.. math:: \overline{T_{1}^{*}}=\gamma_{2}T^{n}+\gamma_{1}\delta T_{1}^{*},\quad\overline{Q_{1}^{*}}=\gamma_{2}Q^{n}+\gamma_{1}\delta Q_{1}^{*},\qquad\delta T_{1}^{*}=T^{*}-T^{n},\quad\delta Q_{1}^{*}=Q^{*}-Q^{n} +.. math:: \overline{T_{1}^{*}}=\gamma_{2}T^{n}+\gamma_{1}\delta + T_{1}^{*},\quad\overline{Q_{1}^{*}}=\gamma_{2}Q^{n}+\gamma_{1}\delta + Q_{1}^{*},\qquad\delta T_{1}^{*}=T^{*}-T^{n},\quad\delta Q_{1}^{*}=Q^{*}-Q^{n} and the original derivation is followed. From these expressions the tile flux for :math:`H` is derived: .. math:: - \frac{H_{j}^{*}}{c_{p}} = \gamma_{2}\frac{H_{j}^{(n)}}{c_{p}}-\gamma_{1}RK_{PMj}[LD_{j}\psi_{j}RK_{H}(1)_{j}+A_{*j}][c_{p}\delta{T'}_{1}-\beta\overline{H^{*}}] + \frac{H_{j}^{*}}{c_{p}} = + \gamma_{2}\frac{H_{j}^{(n)}}{c_{p}}-\gamma_{1}RK_{PMj}[LD_{j}\psi_{j}RK_{H}(1)_{j}+A_{*j}][c_{p}\delta{T'}_{1}-\beta\overline{H^{*}}] .. math:: @@ -5475,13 +5791,19 @@ be obtained. .. math:: :label: eq:FTLnp1 \begin{aligned} - \frac{\overline{H^{n+1}}}{c_{p}} & = & \frac{(1+\beta B_{2})[\xi_{2}F_{T}^{*}+A_{1}\delta{T'}_{1}+A_{2}\delta{Q'}_{1}]-\beta A_{2}[\xi_{2}F_{Q}^{*}+B_{1}\delta{T'}_{1}+B_{2}\delta{Q'}_{1}]}{(1+\beta A_{1})(1+\beta B_{2})-\beta^{2}A_{2}B_{1}} + \frac{\overline{H^{n+1}}}{c_{p}} & = & \frac{(1+\beta + B_{2})[\xi_{2}F_{T}^{*}+A_{1}\delta{T'}_{1}+A_{2}\delta{Q'}_{1}]-\beta + A_{2}[\xi_{2}F_{Q}^{*}+B_{1}\delta{T'}_{1}+B_{2}\delta{Q'}_{1}]}{(1+\beta + A_{1})(1+\beta B_{2})-\beta^{2}A_{2}B_{1}} \end{aligned} .. math:: :label: eq:FQWnp1 \begin{aligned} - \overline{E^{n+1}} & = & \frac{(1+\beta A_{1})[\xi_{2}F_{Q}^{*}+B_{1}\delta{T'}_{1}+B_{2}\delta{Q'}_{1}]-\beta B_{1}[\xi_{2}F_{T}^{*}+A_{1}\delta{T'}_{1}+A_{2}\delta{Q'}_{1}]}{(1+\beta A_{1})(1+\beta B_{2})-\beta^{2}A_{2}B_{1}} + \overline{E^{n+1}} & = & \frac{(1+\beta + A_{1})[\xi_{2}F_{Q}^{*}+B_{1}\delta{T'}_{1}+B_{2}\delta{Q'}_{1}]-\beta + B_{1}[\xi_{2}F_{T}^{*}+A_{1}\delta{T'}_{1}+A_{2}\delta{Q'}_{1}]}{(1+\beta + A_{1})(1+\beta B_{2})-\beta^{2}A_{2}B_{1}} \end{aligned} where, the coefficients :math:`A_{1},\; A_{2},B_{1},\; B_{2}` are given @@ -5489,9 +5811,13 @@ by :eq:`ab_coeffs` but with :math:`\gamma_{1}={\cal I}_{2}`. The above formulae are derived as explained earlier. The definitions -.. math:: \overline{T_{1}^{n+1}}=\gamma_{2}T^{*}+\gamma_{1}\delta T_{1}^{n+1},\quad\overline{Q_{1}^{*}}=\gamma_{2}Q^{*}+\gamma_{1}\delta Q_{1}^{n+1},\qquad\delta T_{1}^{n+1}=T^{n+1}-T^{*},\quad\delta Q_{1}^{n+1}=Q^{n+1}-Q^{*} +.. math:: \overline{T_{1}^{n+1}}=\gamma_{2}T^{*}+\gamma_{1}\delta + T_{1}^{n+1},\quad\overline{Q_{1}^{*}}=\gamma_{2}Q^{*}+\gamma_{1}\delta + Q_{1}^{n+1},\qquad\delta T_{1}^{n+1}=T^{n+1}-T^{*},\quad\delta + Q_{1}^{n+1}=Q^{n+1}-Q^{*} -are used here. The surface fluxes :math:`F_{T}^{*}\equiv{\displaystyle \frac{H^{*}}{c_{p}}}`, :math:`F_{Q}^{*}\equiv E^{*}` are computed at +are used here. The surface fluxes :math:`F_{T}^{*}\equiv{\displaystyle +\frac{H^{*}}{c_{p}}}`, :math:`F_{Q}^{*}\equiv E^{*}` are computed at model state :math:`{(T}_{1}^{*},Q_{1}^{*})`. They are the equivalent of the explicit fluxes :math:`F_{T}^{n}\equiv{\displaystyle \frac{H^{n}}{c_{p}}}`, @@ -5501,7 +5827,8 @@ hand-side of :eq:`eq:FTLstar`, relationship, which is simply the definition of the time-weighted averaging consistent with the new scheme: -.. math:: \overline{H^{*}}\equiv{\cal I}_{1}H^{*}-{\cal E}_{1}H^{n},\qquad\overline{E^{*}}\equiv{\cal I}_{1}E^{*}-{\cal E}_{1}E^{n}. +.. math:: \overline{H^{*}}\equiv{\cal I}_{1}H^{*}-{\cal + E}_{1}H^{n},\qquad\overline{E^{*}}\equiv{\cal I}_{1}E^{*}-{\cal E}_{1}E^{n}. Therefore, once :math:`\overline{H^{*}},\overline{E^{*}}` have been computed, :math:`H^{*}`, :math:`E^{*}` can be computed as follows: @@ -5528,7 +5855,8 @@ surface temperature: however, the total averaged flux from :math:`t^{n}` to :math:`t^{n+1}` is used: -.. math:: H=\overline{H^{*}}+\overline{H^{n+1}},\qquad E=\overline{E^{*}}+\overline{E^{n+1}}. +.. math:: H=\overline{H^{*}}+\overline{H^{n+1}},\qquad + E=\overline{E^{*}}+\overline{E^{n+1}}. The total flux is also kept by the corresponding STASH diagnostic. This has to be adjusted if evaporation exhausts any of the moisture stores @@ -5623,7 +5951,8 @@ scheme. [\ *Could it be that coefficients :math:`D_{j}`, Blending height coupling ^^^^^^^^^^^^^^^^^^^^^^^^ -The same method is used as in section :ref:`Discrete equations and boundary conditions ` to form +The same method is used as in section :ref:`Discrete equations and boundary +conditions ` to form two independent tridiagonal systems of linear equations that relate the increments to momentum, temperature and humidity to the surface fluxes. The ‘downward sweep’ elimination procedure still takes place to obtain @@ -5637,9 +5966,15 @@ where :math:`\delta X_{1/2}^{'}` and :math:`\beta_X` are known. An takes place to obtain equations for the increment to momentum and scalar variables at a given level :math:`k_{b}` in terms of the surface fluxes -.. math:: \delta u_{k_{b}}^{*}=\delta u_{k_{b}}^{'}+(-1)^{k_{b}}\beta\bar{\tau}_{x}^{*}\Big|_{0} \prod^{j=2}_{k_{b}} {C_{u}}_{j}^{'} + \sum^{k_{b}-1}_{i=1} \left[(-1)^{k_{b}+i}\delta u_{k_{b}}^{'} \prod^{j=i+1}_{k_{b}} {C_{u}}_{j}^{'}\right] +.. math:: \delta u_{k_{b}}^{*}=\delta + u_{k_{b}}^{'}+(-1)^{k_{b}}\beta\bar{\tau}_{x}^{*}\Big|_{0} \prod^{j=2}_{k_{b}} + {C_{u}}_{j}^{'} + \sum^{k_{b}-1}_{i=1} \left[(-1)^{k_{b}+i}\delta u_{k_{b}}^{'} + \prod^{j=i+1}_{k_{b}} {C_{u}}_{j}^{'}\right] -.. math:: \delta X_{k_{b}}^{*}=\delta X_{k_{b}}^{'}+(-1)^{k_{b}}\beta_X\frac{\bar{H}_\star}{C_p} \prod^{j=2}_{k_{b}} {C_{X}}_{j}^{'} + \sum^{k_{b}-1}_{i=1} \left[(-1)^{k_{b}+i}\delta X_{k_{b}}^{'} \prod^{j=i+1}_{k_{b}} {C_{X}}_{j}^{'}\right] +.. math:: \delta X_{k_{b}}^{*}=\delta + X_{k_{b}}^{'}+(-1)^{k_{b}}\beta_X\frac{\bar{H}_\star}{C_p} \prod^{j=2}_{k_{b}} + {C_{X}}_{j}^{'} + \sum^{k_{b}-1}_{i=1} \left[(-1)^{k_{b}+i}\delta X_{k_{b}}^{'} + \prod^{j=i+1}_{k_{b}} {C_{X}}_{j}^{'}\right] where :math:`\delta u_{k_{b}}^{*}` and :math:`\delta X_{k_{b}}^{*}` are the only unknowns. The coefficients for these equations are passed to @@ -5669,7 +6004,8 @@ possible to have significant heat flux and boundary layer depth in windy conditions that should not lead to a strong thermal forecast. This sensitivity of boundary layer turbulence is already included in the parametrization of non-local momentum fluxes (see section -:ref:`Non-gradient stress parametrization `) through the stability dependence in +:ref:`Non-gradient stress parametrization `) through the +stability dependence in :eq:`tau_nl` that can be written as .. math:: f_{stab} = - \frac{a_{stab} z_{\rm h}/L }{1 - a_{stab} z_{\rm h}/L} @@ -5799,7 +6135,8 @@ where the mixed layer expression for :math:`w_m` is used. `Holtslag and Moeng (1991)`_ also show from analysis of the scalar flux budget that -.. math:: \overline{w'\theta'} = - \frac{\tau_{turb}}{2} \, \overline{w'^2} \frac{d \theta}{dz} +.. math:: \overline{w'\theta'} = - \frac{\tau_{turb}}{2} \, \overline{w'^2} + \frac{d \theta}{dz} where :math:`\tau_{turb}` is a return to isotropy timescale. Ignoring the non-gradient parametrization in the UM, it follows that @@ -5813,7 +6150,9 @@ Combining :eq:`bl_scaling` with and subsuming the Prandtl number into the other constants, for surface-driven boundary layer mixing we can write: -.. math:: K_m^{\rm surf}= \frac{\tau_{\rm surf}}{2} \, \overline{w'^2} = \frac{\tau_{turb}}{2} \, \frac{2.66 }{C_{ws}^{2/3}} w_m^2 \, f(z') = k z_{\rm h}w_m f(z') +.. math:: K_m^{\rm surf}= \frac{\tau_{\rm surf}}{2} \, \overline{w'^2} = + \frac{\tau_{turb}}{2} \, \frac{2.66 }{C_{ws}^{2/3}} w_m^2 \, f(z') = k z_{\rm + h}w_m f(z') which then gives :math:`\tau_{\rm surf} = C_{ws}^{2/3} k z_{\rm h}/ (1.33 w_m)`. An @@ -5832,7 +6171,8 @@ stable boundary layers and combine all these timescales following e = K_m \tau_{turb}^{-1} where -:math:`\tau_{turb}^{-1} = MAX[ \tau_{\rm surf}^{-1},\tau_{\rm Sc}^{-1}] + \tau_{\rm SBL}^{-1}`. +:math:`\tau_{turb}^{-1} = MAX[ \tau_{\rm surf}^{-1},\tau_{\rm Sc}^{-1}] + +\tau_{\rm SBL}^{-1}`. Note that :eq:`tke_diag` gives :math:`\overline{w'^2}`, rather than TKE. As a simple fix to improve the near-surface TKE in convective boundary layers, where the horizontal wind variability often @@ -5937,7 +6277,8 @@ written .. math:: :label: vsurf - V_{\rm heat}^3= z_{\rm ml}\! \left( (2-\zeta_s)\zeta_s \overline{w'b}_S+ (1-\zeta_s)^2 [\overline{w'b'}_S]_{\rm sat}\right) + V_{\rm heat}^3= z_{\rm ml}\! \left( (2-\zeta_s)\zeta_s \overline{w'b}_S+ + (1-\zeta_s)^2 [\overline{w'b'}_S]_{\rm sat}\right) .. math:: :label: vrad @@ -5946,12 +6287,14 @@ written .. math:: :label: vbr - V_{\rm br}^3= A_{\rm br}\chi_s^2 \, \mbox{max}\left[0,-\delta b\right] \, \Delta b ^{1/2} + V_{\rm br}^3= A_{\rm br}\chi_s^2 \, \mbox{max}\left[0,-\delta b\right] \, + \Delta b ^{1/2} \, z_c^{3/2} \, C_{fac} Here, -:math:`[\overline{w'b'}_S]_{\rm sat}= g ( \tilde{\beta_T} \overline{w'\theta_{\ell}'}_S+ \tilde{\beta_q}\overline{w'q_t'}_S)`, +:math:`[\overline{w'b'}_S]_{\rm sat}= g ( \tilde{\beta_T} +\overline{w'\theta_{\ell}'}_S+ \tilde{\beta_q}\overline{w'q_t'}_S)`, where the subscript :math:`_S` indicates the surface flux; :math:`\Delta_F` is the divergence of the net radiative flux, :math:`F` (in Kms\ :math:`^{-1}`), associated with cloud-top, for which the @@ -5989,7 +6332,8 @@ calculated as .. math:: :label: zcld_calc - \tilde{z_c} = \sum_{k=1}^{NTML+1} \left( {C_F}_k \frac{\Delta_{k+\frac{1}{2}} z}{2} + \tilde{z_c} = \sum_{k=1}^{NTML+1} \left( {C_F}_k + \frac{\Delta_{k+\frac{1}{2}} z}{2} + \mbox{min}\left[C_F^l\frac{\Delta_{k+\frac{1}{2}} z}{2}, \frac{q_{\ell}}{\gamma_{q_{\ell}}} \right] + \mbox{min}\left[C_F^f\frac{\Delta_{k+\frac{1}{2}} z}{2}, \frac{q_f}{\gamma_{q_f}} \right] \right) @@ -6051,7 +6395,8 @@ Note that, if :math:`k_b=1` in :eq:`zc_calc`, then the 8A calculation of :math:`z_c` is to include the depth to which the cloud extends into the inversion grid-level. If a subgrid inversion height, :math:`z_i`, has been diagnosed (see -section :ref:`Diagnosis of a sub-grid inversion `) then the height of :math:`z_i` above the +section :ref:`Diagnosis of a sub-grid inversion `) then the height +of :math:`z_i` above the half-level height is added to :math:`z_c` (as long as :math:`C_F>` SC_CFTOL in grid-levels NTML or NTML\ :math:`+1` or the layer is a decoupled layer). If no subgrid inversion has been diagnosed and @@ -6092,7 +6437,8 @@ inversion is given by The empirical constant :math:`A_{\rm br}= 0.24`. The calculation of :math:`\Delta \theta_{\ell}` and :math:`\Delta q_t` is described for a subgrid -inversion in section :ref:`Diagnosis of a sub-grid inversion ` or, if one is not diagnosed, +inversion in section :ref:`Diagnosis of a sub-grid inversion ` or, +if one is not diagnosed, they are taken simply as :math:`\Delta_{\mbox{\tiny \rm NTML}+1}`. For :math:`\Delta q_{\ell}`, :math:`\Delta q_f` and :math:`{q_{\ell}}_{\rm ct}`, in-cloud values extrapolated to :math:`z_i` @@ -6101,13 +6447,15 @@ above and below using the adiabatic lapse rates are calculated as: .. math:: - {q_{\ell}}_{\rm ct} = \frac{{q_{\ell}}_{\mbox{\tiny \rm NTML}}}{{C_F}^l_{\mbox{\tiny \rm NTML}}} + {q_{\ell}}_{\rm ct} = \frac{{q_{\ell}}_{\mbox{\tiny \rm + NTML}}}{{C_F}^l_{\mbox{\tiny \rm NTML}}} + (z_i-z_{\mbox{\tiny \rm NTML}}) \gamma_{q_{\ell}} .. math:: q_{\ell}^+ = \mbox{max}\left[ 0, \, - \frac{{q_{\ell}}_{\mbox{\tiny \rm NTML}+2}}{{C_F}^l_{\mbox{\tiny \rm NTML}+2}} + \frac{{q_{\ell}}_{\mbox{\tiny \rm NTML}+2}}{{C_F}^l_{\mbox{\tiny \rm + NTML}+2}} - (z_{\mbox{\tiny \rm NTML}+2}-z_i) \gamma_{q_{\ell}} \right] @@ -6116,11 +6464,13 @@ and similarly for :math:`q_f` (noting that currently .. math:: - \Delta q_{\ell} = {C_F^l}_{\mbox{\tiny \rm NTML}+2}\, q_{\ell}^+ - {C_F^l}_{\mbox{\tiny \rm NTML}}\, {q_{\ell}}_{\rm ct} + \Delta q_{\ell} = {C_F^l}_{\mbox{\tiny \rm NTML}+2}\, q_{\ell}^+ - + {C_F^l}_{\mbox{\tiny \rm NTML}}\, {q_{\ell}}_{\rm ct} .. math:: - \Delta q_f = {C_F^f}_{\mbox{\tiny \rm NTML}+2}\, q_f^+ - {C_F^f}_{\mbox{\tiny \rm NTML}}\, {q_f}_{\rm ct} + \Delta q_f = {C_F^f}_{\mbox{\tiny \rm NTML}+2}\, q_f^+ - + {C_F^f}_{\mbox{\tiny \rm NTML}}\, {q_f}_{\rm ct} The only other explicit account of variable cloud fraction is in @@ -6282,7 +6632,8 @@ summation over 3 grid-levels in :eq:`ctraddiv` as follows: .. math:: :label: ctraddiv_9c_inv \Delta F= \Delta F+ \Delta_{k_m+\frac{3}{2}} F - - \Delta_{k_m+\frac{5}{2}} \frac{\Delta_{k_m+\frac{3}{2}} z}{k_m+\frac{1}{2}} F + - \Delta_{k_m+\frac{5}{2}} \frac{\Delta_{k_m+\frac{3}{2}} + z}{k_m+\frac{1}{2}} F #. As at 9B above, the calculations in :eq:`ctraddiv_9c` and @@ -6337,7 +6688,8 @@ Linearising gives .. math:: - \overline{w'b}= g \left( \beta_T \overline{w'T_L'} + \beta_q \overline{w'q_t'}+ + \overline{w'b}= g \left( \beta_T \overline{w'T_L'} + \beta_q + \overline{w'q_t'}+ \left( \beta_T \frac{L}{c_p} - \frac{1+c_v}{c_v} \beta_q \right) \overline{w'q_{\ell}'} \right) @@ -6350,13 +6702,15 @@ In saturated cloudy air (see, for example, used to calculate :math:`\overline{w'q_{\ell}'}` (via :math:`q_{\ell}' = q_t' - q_s' = q_t' - \alpha_L T'`) as -.. math:: \overline{w'q_{\ell}'} = a_L( \overline{w'q_t'}- \alpha_L \overline{w'T_L'} ) +.. math:: \overline{w'q_{\ell}'} = a_L( \overline{w'q_t'}- \alpha_L + \overline{w'T_L'} ) where .. math:: - \alpha_L = \frac{\partial q_s}{\partial T} = \frac{\epsilon L q_s(T,p) }{R T^2}, \qquad + \alpha_L = \frac{\partial q_s}{\partial T} = \frac{\epsilon L q_s(T,p) }{R + T^2}, \qquad a_L = \frac{1}{1+L\alpha_L/c_p} where R is the gas constant (:math:`=287.05`). Thus the buoyancy flux @@ -6368,7 +6722,8 @@ can be written \begin{cases} g \left( \beta_T \overline{w'T_L'} + \beta_q \overline{w'q_t'}\right) & {\rm in\ unsaturated\ air} \\ - g \left( \tilde{\beta_T} \overline{w'T_L'} + \tilde{\beta_q} \overline{w'q_t'} + g \left( \tilde{\beta_T} \overline{w'T_L'} + \tilde{\beta_q} + \overline{w'q_t'} \right) & {\rm in\ saturated\ air} \end{cases} @@ -6421,7 +6776,8 @@ spherical geometry for simplicity: .. math:: :label: moisture_cons \int (\rho_v + \rho_{\ell}+ \rho_{f})\, d\underline{x} = - \int \rho (q_v + q_{\ell}+ q_{f}) \,d\underline{x} = \int \rho q_t \, d\underline{x} + \int \rho (q_v + q_{\ell}+ q_{f}) \,d\underline{x} = \int \rho q_t \, + d\underline{x} When mixing ratios are used, the momentum equations, :eq:`cons_eqn_uv`, remain unchanged and the wet density @@ -6433,7 +6789,8 @@ terms of mixing ratios as .. math:: \int (\rho_v + \rho_{\ell}+ \rho_{f})\, d\underline{x} = - \int \rho_y (m_v + m_{\ell}+ m_{f}) \,d\underline{x} = \int \rho_y m_t \, d\underline{x} + \int \rho_y (m_v + m_{\ell}+ m_{f}) \,d\underline{x} = \int \rho_y m_t \, + d\underline{x} Thus :math:`\rho` in :eq:`cons_eqn_scal` is replaced with :math:`\rho_y` for :math:`\chi = m_t` and :math:`\theta_{\ell}` @@ -6473,7 +6830,8 @@ ratios as: For saturation calculations a version of QSAT is used that is switchable between input specific and mixing ratio variables. The rate of change of :math:`q_s` with temperature is also used in the boundary layer code -(see e.g. appendix :ref:`Appendix: Derivation and definitions of the buoyancy parameters `): +(see e.g. appendix :ref:`Appendix: Derivation and definitions of the buoyancy +parameters `): .. math:: \frac{ d q_{sat} }{ dT } = \frac{\epsilon L q_{sat} }{RT^2} @@ -6481,7 +6839,8 @@ In fact this expression should really be converted to work for specific quantities and so simply changing to mixing ratios will improve the accuracy of this calculation. -Finally, in appendix :ref:`Appendix: Derivation and definitions of the buoyancy parameters ` virtual temperature is defined +Finally, in appendix :ref:`Appendix: Derivation and definitions of the buoyancy +parameters ` virtual temperature is defined in terms of specific variables as .. math:: T_v = T ( 1 + c_v q_v - q_l ) @@ -6755,13 +7114,16 @@ Appendix: Notation - height of half-level above :math:`\theta`-level :math:`k` * - :math:`\Delta_k` - - indicates a finite difference between :math:`\theta`-levels :math:`k` and :math:`k-1` + - indicates a finite difference between :math:`\theta`-levels :math:`k` + and :math:`k-1` * - :math:`\Delta_{k+\frac{1}{2}}` - - indicates a finite difference between half-levels :math:`k+\frac{1}{2}` and :math:`k-\frac{1}{2}` + - indicates a finite difference between half-levels :math:`k+\frac{1}{2}` + and :math:`k-\frac{1}{2}` * - :math:`\Delta` - - note: real change (i.e., not necessarily finite-difference) in a parameter + - note: real change (i.e., not necessarily finite-difference) in a + parameter * - - across the capping inversion (see :eq:`dbinv` and following text) @@ -6779,7 +7141,8 @@ Appendix: Notation - virtual temperature and potential temperature, * - - - defined by :eq:`Tv` and in section (:ref:`Calculation of parcel buoyancy excess `) + - defined by :eq:`Tv` and in section (:ref:`Calculation of parcel buoyancy + excess `) * - :math:`b` - buoyancy (:math:`=g T_v'/T_v`) @@ -6803,10 +7166,12 @@ Appendix: Notation - * - :math:`C_t` - - (:math:`=1.1`) threshold for ratio of layer :math:`q_t`-gradients in cumulus diagnosis + - (:math:`=1.1`) threshold for ratio of layer :math:`q_t`-gradients in + cumulus diagnosis * - :math:`\Gamma_{\rm inv}` - - (:math:`=1.1`) threshold on ratio of environment to parcel :math:`\theta_v` gradients + - (:math:`=1.1`) threshold on ratio of environment to parcel + :math:`\theta_v` gradients * - - for identifying capping inversions above the LCL @@ -6815,10 +7180,12 @@ Appendix: Notation - (:math:`=0.1`) :math:`C_F` threshold for recognising the presence of Sc * - :math:`\Delta_{k_{ct}} \theta_{v\ell}/ \Delta_{k_{ct}} z < 10^{-3}` - - threshold (in Km\ :math:`^{-1}`) for initial diagnosis of *well-mixed* DSC layers + - threshold (in Km\ :math:`^{-1}`) for initial diagnosis of *well-mixed* + DSC layers * - :math:`D_t` - - (:math:`=0.1`) threshold for the ratio of buoyancy consumption to production + - (:math:`=0.1`) threshold for the ratio of buoyancy consumption to + production * - - before decoupling occurs @@ -6902,7 +7269,8 @@ Appendix: Notation - scaling velocity for momentum mixing in the SML * - - - (used in :math:`K_m^{\rm surf}`, :math:`\gamma_{\theta_{\ell}}` and the SML parcel perturbation, :math:`\theta_v'`) + - (used in :math:`K_m^{\rm surf}`, :math:`\gamma_{\theta_{\ell}}` and the + SML parcel perturbation, :math:`\theta_v'`) * - :math:`w_*` - ‘standard’ convective velocity scale for a cloud-free convective @@ -6914,7 +7282,8 @@ Appendix: Notation - friction velocity (here includes the orographic component) * - :math:`w_e`, :math:`\tilde{w_e}` - - entrainment velocity and compensated to allow for subsidence (ms\ :math:`^{-1}`) + - entrainment velocity and compensated to allow for subsidence (ms\ + :math:`^{-1}`) * - :math:`w_S` - subsidence velocity (ms\ :math:`^{-1}`) @@ -6929,10 +7298,13 @@ Appendix: Notation - parameters in perturbation calculation, :eq:`dscd_pert`, * - - - for initial identification of and :math:`z_{\rm ml}` calculation for DSC layers + - for initial identification of and :math:`z_{\rm ml}` calculation for DSC + layers - * - :math:`a_L`, :math:`\alpha_L`, :math:`\beta_T`, :math:`\beta_q`, :math:`\tilde{\beta_T}`, :math:`\tilde{\beta_q}` - - buoyancy parameters, defined in appendix :ref:`Appendix: Derivation and definitions of the buoyancy parameters ` + * - :math:`a_L`, :math:`\alpha_L`, :math:`\beta_T`, :math:`\beta_q`, + :math:`\tilde{\beta_T}`, :math:`\tilde{\beta_q}` + - buoyancy parameters, defined in appendix :ref:`Appendix: Derivation and + definitions of the buoyancy parameters ` .. [1] unless the option to mix across the LCL is selected, see @@ -6954,7 +7326,8 @@ Appendix: Notation surface-fluxes over the current timestep may improve the accuracy of the convective closure. Also the non-turbulent fluxes used to construct the budgets at entrainment grid-levels (section - :ref:`The revised scalar flux-gradient formulation `) do not include contributions from + :ref:`The revised scalar flux-gradient formulation `) do + not include contributions from convection, so arguably excluding them from :math:`S` is consistent. In the presence of convective subsidence, the top grid-level of the sub-cloud mixed layer gets warmed and dried by the subsidence From a64ad01495b111d97c72e3496f96f0dc32ada4cc Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Mon, 27 Apr 2026 16:49:43 +0100 Subject: [PATCH 062/116] Removed backticks from the bibliography (copied-in from the bibtex file) as they break rst! --- .../source/science_guide/turbulence_schemes/bldoc.rst | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/documentation/source/science_guide/turbulence_schemes/bldoc.rst b/documentation/source/science_guide/turbulence_schemes/bldoc.rst index f4a6dc2368..5b373b6836 100644 --- a/documentation/source/science_guide/turbulence_schemes/bldoc.rst +++ b/documentation/source/science_guide/turbulence_schemes/bldoc.rst @@ -7547,7 +7547,7 @@ References Lock, A. P. (1999). *A parametrization of turbulent mixing in convective cloud-capped boundary layers derived from large-eddy simulations*. - Proceedings of GCSS-WGNE Workshop on `Cloud processes and cloud feedbacks + Proceedings of GCSS-WGNE Workshop on 'Cloud processes and cloud feedbacks in large-scale models', 9-13 November 1998, ECMWF, Shinfield Park, Reading, Berks., RG2 9AX, U.K.. @@ -7614,7 +7614,7 @@ References .. _Zilitinkevich (1975): Zilitinkevich, S. S. (1975). - *{Comments on ``A model for the dynamics of the inversion above a + *{Comments on ''A model for the dynamics of the inversion above a convective boundary layer''}*. J. Atmos. Sci., 32, 991-992. From de9efdaaab54447bac2ecdf6b9a694021587c324 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Mon, 27 Apr 2026 17:20:28 +0100 Subject: [PATCH 063/116] Manual change: specify 2 header rows in one of the tables. --- documentation/source/science_guide/turbulence_schemes/bldoc.rst | 1 + 1 file changed, 1 insertion(+) diff --git a/documentation/source/science_guide/turbulence_schemes/bldoc.rst b/documentation/source/science_guide/turbulence_schemes/bldoc.rst index 5b373b6836..8ff32c0607 100644 --- a/documentation/source/science_guide/turbulence_schemes/bldoc.rst +++ b/documentation/source/science_guide/turbulence_schemes/bldoc.rst @@ -1810,6 +1810,7 @@ Discussion of some of the revisions .. list-table:: Convective and Neutral limits for velocity scales :name: tab:vscales + :header-rows: 2 * - Formulation From 98be76e38e06e6e1a848e281370da77b10743553 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Wed, 29 Apr 2026 11:09:15 +0100 Subject: [PATCH 064/116] Revert 80-char line-width limit to section headers (it breaks them) --- .../source/science_guide/turbulence_schemes/bldoc.rst | 6 ++---- 1 file changed, 2 insertions(+), 4 deletions(-) diff --git a/documentation/source/science_guide/turbulence_schemes/bldoc.rst b/documentation/source/science_guide/turbulence_schemes/bldoc.rst index 8ff32c0607..44d32c20a7 100644 --- a/documentation/source/science_guide/turbulence_schemes/bldoc.rst +++ b/documentation/source/science_guide/turbulence_schemes/bldoc.rst @@ -4732,8 +4732,7 @@ if the `Wood and Mason (1993)`_ formulation is used. .. _section_2.2: -The iterative algorithm for calculating the effective surface exchange -coefficients +The iterative algorithm for calculating the effective surface exchange coefficients ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ For unstable conditions, i.e. :math:`\Delta`\ B :math:`<` 0 : @@ -4993,8 +4992,7 @@ and v\ :math:`_{\ast }` are used. .. _section_2.4: -Distributed form drag – an alternative to the effective roughness length -parametrization +Distributed form drag – an alternative to the effective roughness length parametrization ---------------------------------------------------------------------------------------- An alternative representation of the turbulent form drag due to sub-grid From d9332300339add6c53050b3a1c5a43ece8819e9d Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Wed, 29 Apr 2026 11:23:59 +0100 Subject: [PATCH 065/116] Fixed table cross-referencing. --- documentation/source/science_guide/turbulence_schemes/bldoc.rst | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/documentation/source/science_guide/turbulence_schemes/bldoc.rst b/documentation/source/science_guide/turbulence_schemes/bldoc.rst index 44d32c20a7..b25e06838d 100644 --- a/documentation/source/science_guide/turbulence_schemes/bldoc.rst +++ b/documentation/source/science_guide/turbulence_schemes/bldoc.rst @@ -1867,7 +1867,7 @@ that HB and the standard UM set :math:`w_m = (u_*^3 + C_{ws} w_*^3)^{\frac{1}{3}}` and :math:`w_h = w_m/Pr`, with :math:`C_{ws} = 0.6` and 0.25, respectively, above the surface layer. The convective and neutral limits for :math:`w_h` and -:math:`w_m` are given in Table `1 <#tab:vscales>`__ and the stability +:math:`w_m` are given in :numref:`Table %s ` and the stability dependencies of several parameters are shown in :numref:`Fig. %s `. The parameter :math:`d` in :numref:`Fig. %s ` contains the stability dependence of the From 0b399d742d48cb6d7b19353ee585b2039de4b64c Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Wed, 29 Apr 2026 11:32:32 +0100 Subject: [PATCH 066/116] Fixed table cross-referencing. --- .../science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst index a87f493e85..5b5d3df6aa 100644 --- a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst +++ b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst @@ -3125,8 +3125,8 @@ cumulus regimes. \right)^\frac{1}{1-b_1} Note that :math:`q_{cl}` falls to zero after a finite time - :math:`\frac{{q_{cl}}_0}{1-b_1} \frac{a_L}{K} - \frac{1}{2 {C_l}_0 (1 - C_l) \, (q_{cl}-Q_c)}`. If the timestep + :math:`\frac{{q_{cl}}_0}{1-b_1} \frac{a_L}{K} \frac{1}{2 {C_l}_0 (1 - + C_l) \, (q_{cl}-Q_c)}`. If the timestep :math:`\Delta t` is longer than this time, then erosion completely removes the cloud during the current timestep. @@ -5946,7 +5946,7 @@ fractions as part of the LBCs file. Parameter values ---------------- -Table `1 <#tab:pc2_names>`__ summarizes the values of parameters used in +:numref:`Table %s ` summarizes the values of parameters used in the PC2 scheme and their location within various comdecks. Those parameters marked as ‘Num’ are those that are not part of the mathematical equation set that is being solved, but are required in From fb4e3f17b85045de380e74245ac7ddcb3722ceb6 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Wed, 29 Apr 2026 11:57:37 +0100 Subject: [PATCH 067/116] Added warning about doc being a work-in-progress / containing UM-isms. --- .../science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst | 6 ++++++ 1 file changed, 6 insertions(+) diff --git a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst index 5b5d3df6aa..4dff5d208b 100644 --- a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst +++ b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst @@ -4,6 +4,12 @@ under which the code may be used. ----------------------------------------------------------------------------- +.. attention:: + + This documentation has been transfered directly from the UM to LFRic; + It is still a work in progress. There are still UM-specific references + and terminology that are yet to be updated. + ==================== The PC2 Cloud Scheme ==================== From cc6ace0b95966b5c8b0a679b821258b4c87aaf59 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Wed, 29 Apr 2026 13:04:12 +0100 Subject: [PATCH 068/116] Added warning at top of file about this being a work-in-progress / containing UM-isms. --- .../source/science_guide/turbulence_schemes/bldoc.rst | 7 ++++++- 1 file changed, 6 insertions(+), 1 deletion(-) diff --git a/documentation/source/science_guide/turbulence_schemes/bldoc.rst b/documentation/source/science_guide/turbulence_schemes/bldoc.rst index b25e06838d..db7a985115 100644 --- a/documentation/source/science_guide/turbulence_schemes/bldoc.rst +++ b/documentation/source/science_guide/turbulence_schemes/bldoc.rst @@ -4,6 +4,12 @@ under which the code may be used. ----------------------------------------------------------------------------- +.. attention:: + + This documentation has been transfered directly from the UM to LFRic; + It is still a work in progress. There are still UM-specific references + and terminology that are yet to be updated. + =============================================== The Parametrization of Boundary Layer Processes =============================================== @@ -1812,7 +1818,6 @@ Discussion of some of the revisions :name: tab:vscales :header-rows: 2 - * - Formulation - Convective limit - From 4bc3b5298542dcaea046242af3393d82ceecd934 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Wed, 29 Apr 2026 13:25:03 +0100 Subject: [PATCH 069/116] Rotated figure so it appears the right way up in the doc. --- .../cloud_schemes/pc2_process_explanation.svg | 18529 ++++++---------- 1 file changed, 6472 insertions(+), 12057 deletions(-) diff --git a/documentation/source/science_guide/cloud_schemes/pc2_process_explanation.svg b/documentation/source/science_guide/cloud_schemes/pc2_process_explanation.svg index 2da9790371..eda5289492 100644 --- 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+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + From 2838a4ee30816b9893c6ad88cda2d6d9f6112550 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Thu, 30 Apr 2026 22:43:32 +0100 Subject: [PATCH 070/116] Ran updated version of the script (reinstates aligned math blocks for long equations split over multiple lines, and tidies-up stray blank lines after math blocks). --- .../cloud_schemes/UMDP30_PC2CloudScheme.rst | 84 ++++++------------- 1 file changed, 24 insertions(+), 60 deletions(-) diff --git a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst index 4dff5d208b..9f1988ebb8 100644 --- a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst +++ b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst @@ -372,7 +372,6 @@ schematically: \frac{\partial C_t}{\partial t} |_{boundary \, layer} + \frac{\partial C_t}{\partial t} |_{precipitation} + ... , - where :math:`\overline{q_{cf}}` is the ice water specfic humidity, :math:`C_l` is the liquid cloud *volume* fraction, :math:`C_i` is the ice cloud volume fraction, and :math:`C_t` is the combined ice or liquid @@ -1837,7 +1836,6 @@ of :math:`p` and :math:`T` given by a_i = \frac{g}{R T}\left( \frac{\epsilon L_s}{c_p T} - 1 \right), - The first term on the right hand side of Eq. :eq:`eqn:squires_eqn` is the sink of vapor due to depositional growth of ice crystals, the second term models entrainment @@ -1881,7 +1879,6 @@ solution PDF is Gaussian with mean and variance given by: \frac{a^2_{\rm i} \sigma^2_w \tau_{\rm d}}{ 2\left(b_i B_0 {\cal M}_1 + \left(\varepsilon/L^2\right)^{1/3}\right)}, - Equation :eq:`eqn:si_avg` and :eq:`eqn:si_var` completely specify the PDF, :math:`F(S_i)`, of steady-state humidity variations for the subgrid @@ -1895,8 +1892,7 @@ given by .. math:: :label: eqn:cloud_liquid q_{cl}^{sgt} = q_{sat\;ice}\int_{S_{i,wat}}^\infty d S_i (S_i -S_{i,wat}) - F(S_i) , - + F(S_i), where :math:`S_{i,wat} = e_{sat\;wat}/e_{sat\;ice}-1` is the value of ice supersaturation at water saturation. We use the superscription @@ -2025,7 +2021,6 @@ increments to the model prognostic fields, :math:`C_l` and \left( \Delta C \right)_{sgt} = C_l^{sgt} - where the left hand sides denote the increments to :math:`C_l`, :math:`q_{cl}`, :math:`T` and the total cloud fraction, :math:`C`, due to the subgrid scheme. Some bounds-checking is then applied to ensure @@ -2058,7 +2053,6 @@ using PC2 Erosion. In this case: \left( \Delta T \right)_{sgt} = \frac{L_v}{c_p} \left( \Delta q_{cl} \right)_{sgt}, - where :math:`q_{cl}` is the liquid cloud amount prior to calling to the turbulent production scheme. The cloud fraction increments are calculated by calling PC2 Erosion with @@ -2263,7 +2257,6 @@ cloud to ice cloud. The cloud fraction changes are: \Delta C_t = 0. - Heterogeneous nucleation ^^^^^^^^^^^^^^^^^^^^^^^^ @@ -2284,7 +2277,6 @@ cloud. These give the following changes: \Delta C_t = 0. - .. _sec_mp_depsub: Deposition and sublimation @@ -2373,7 +2365,6 @@ same assumption). Some algebra retrieves the expressions: q_{ice} = \frac {\overline{q} - C_l q_{sat~liq} - A_{clear} q_{clear} } {A_{ice}}, - where :math:`A_{ice}` is the proportion of the gridbox with ice cloud but not liquid cloud and :math:`A_{clear}` is the proportion of the gridbox without cloud. The numerical application will set @@ -3477,7 +3468,6 @@ The bulk cloud model plume equations for mass and :math:`{\chi}` are: {\chi}_{\rm{ }}^{\rm{P}} } \right) - Equations :eq:`eq:eddyflux`, :eq:`eq:dbydpmassflux` and :eq:`eq:dbydpmfchi` can then be substituted into @@ -3529,7 +3519,6 @@ discretized form of :eq:`eq:chimassflux`, setting - m_{\rm{cb}} \, \left({ {\chi}_{\rm{i,cb}}^{\rm{P}} - {\chi}_{\rm{cb}}^{\rm{E}} } \right) - where the initial parcel value :math:`{\chi}_{\rm{i,cb}}^{\rm{P}}` may be chosen to produce a fixed increment or place a closure condition on the cloud base flux. In fact, the convection equations (see ) differ @@ -3560,7 +3549,6 @@ terms for temperature and specific humidity: - \frac{1}{\overline{\rho}} \, \frac{\partial \, \overline{\rho w^{'} q_{\rm{ }}^{\rm{E'}}}}{\partial \, z} - where :math:`{\overline{Q}}_{\rm{par}}` is the rate of condensation which occurs in the ascending plumes. @@ -3597,7 +3585,6 @@ gradient equations based upon :eq:`eq:gradchipar` }}^{\rm{E}} } \right) - {\overline{Q}}_{\rm{par}} + PPN - The final calculation of rates in the current condensation scheme (, section 10) assumes a further condensation term, :math:`{\overline{Q}}_{\rm{reset}}`, which acts to make the net rate of @@ -3616,21 +3603,20 @@ is basic equations {\frac{\partial \, q_{\rm{ }}^{\rm{E}}}{\partial \, t}}_{\rm{conv}} = Q2 + {\overline{Q}}_{\rm{reset}} -.. math:: +.. math:: :label: eq:basiclold + \begin{aligned} 0 \equiv {\frac{\partial \, l_{\rm{ }}^{\rm{E}}}{\partial \, t}}_{\rm{conv}} - = {\overline{Q}}_{\rm{par}} - + & = & {\overline{Q}}_{\rm{par}} - {\overline{Q}}_{\rm{reset}} - PPN - \frac{1}{\overline{\rho}} \, \frac{\partial \, \overline{\rho w^{'} - l_{\rm{ }}^{\rm{E'}}}}{\partial \, z} - -.. math:: :label: eq:basiclold - - = + l_{\rm{ }}^{\rm{E'}}}}{\partial \, z} + \\ + & = & \mu \, M^{\rm{P}} \, l_{\rm{ }}^{\rm{P}} + \delta \, M^{\rm{P}} \, l_{\rm{ }}^{\rm{P}} - {\overline{Q}}_{\rm{reset}} - + \end{aligned} By analogy with equations :eq:`eq:defineq1` and :eq:`eq:defineq2`, we can define a :math:`Q4` from @@ -3666,7 +3652,6 @@ Define \frac{1}{\overline{\rho}} \, \frac{\partial \, \overline{\rho w^{'} l_{\rm{f}}^{\rm{'}}}}{\partial \, z} - where the PC2 assumption thus far has been that :math:`{\overline{Q}}_{\rm{l, reset}} = 0 = {\overline{Q}}_{\rm{f, reset}}`. @@ -3701,7 +3686,6 @@ condensate is calculated as \frac{{\overline{Q}}_{\rm{f, par}}}{M^{\rm{P}}} - \frac{SNOW}{M^{\rm{P}}} - Following , equations :eq:`eq:dbydpmassflux`, :eq:`eq:vertparl` and :eq:`eq:vertparf` are discretized: @@ -3749,7 +3733,6 @@ are discretized: + 1}} \right) - \left({ SNOW_{\rm{k} + 1} \, / \, M_{\rm{k} + 1} } \right) - where :math:`EPSS_{\rm{k}} = \left({1 + \varepsilon_{\rm{k} + 3 / 4} \, \Delta p_{\rm{k} + 3 / 4}} \right)\, \left({1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4}} \right)`. @@ -3785,7 +3768,6 @@ precipitation terms are suppressed: l_{\rm{f \, k + 1}}^{\rm{E}} } \right)}{EPSS_{\rm{k}}} - At the base of the convective plume (ie. the level immediately above cloud base), :math:`l_{\rm{l \, k}}^{\rm{P}}` is initialized to :math:`l_{\rm{l \, i}}^{\rm{P}}` and :math:`l_{\rm{f \, k}}^{\rm{P}}` to @@ -3807,7 +3789,6 @@ produces zero fluxes at cloud base: M_{\rm{cb}}^{\rm{P}}\, \left({ l_{\rm{f}}^{\rm{P \, i}} - l_{\rm{f}}^{\rm{E}}(\rm{cb}) } \right) - As the convection scheme makes the single phase assumption for parcel condensate, it may be necessary to melt or freeze entrained condensate at this point and adjust the temperature accordingly. @@ -3826,7 +3807,6 @@ at this point and adjust the temperature accordingly. 1}}^{\rm{P}} \; \ldots \; \mbox{ if l_{\rm{l \, k + 1}}^{\rm{P}} is frozen } - Once a final value for the condensation term :math:`{\overline{Q}}_{\rm{x} \, \rm{k} + 1} \, / \, M_{\rm{k} + 1}` has been calculated from the parcel specific humidity equations, it can then @@ -3869,7 +3849,6 @@ This reduces the parcel condensate to : \frac{l_{\rm{f \, k + 1}}^{\rm{P}}}{l_{\rm{k + 1}}^{\rm{P}}} } \right)\, l_{\rm{MIN}}^{\rm{P}} - The final parcel condensate values are then used in the rate calculation based upon eqn :eq:`eq:basiclold`: @@ -3891,7 +3870,6 @@ based upon eqn :eq:`eq:basiclold`: \left({ l_{\rm{f}}^{\rm{P}}(\rm{k}) - l_{\rm{f}}^{\rm{E}}(\rm{k}) } \right)- {\overline{Q}}_{\rm{f, reset}} - Note that, as a side-effect, the environment equations for potential temperature and specific humidity are also altered because the condensate is no longer re-evaporated at the end @@ -3924,7 +3902,6 @@ condensate is no longer re-evaporated at the end \left({ \theta_{\rm{k}}^{\rm{P}} - \theta_{\rm{k}}^{\rm{E}} } \right) } \right] { } - and .. math:: @@ -3953,7 +3930,6 @@ and \left({ q_{\rm{k}}^{\rm{P}} - q_{\rm{k}}^{\rm{E}} } \right) } \right] { } - Similarly, eqns :eq:`eq:q4lmassf` and :eq:`eq:q4fmassf` have a discretized form as follows: @@ -3983,7 +3959,6 @@ Similarly, eqns :eq:`eq:q4lmassf` and \left({ l_{\rm{l \, k}}^{\rm{P}} - l_{\rm{l \, k}}^{\rm{E}} } \right) } \right] { } - and .. math:: @@ -4012,7 +3987,6 @@ and \left({ l_{\rm{f \, k }}^{\rm{P}} - l_{\rm{f \, k}}^{\rm{E}} } \right) } \right] { } - .. _sec_conv_homog: Background condensation @@ -4705,29 +4679,26 @@ prognostic :math:`C_l` and :math:`q_{cl}` are incremented as follows: - If :math:`{q_{cl}}_{diag} > q_{cl}`: - .. math:: :label: eq:dqcl_init - - \Delta q_{cl} = {q_{cl}}_{diag} - q_{cl} + :math:`\Delta q_{cl} = {q_{cl}}_{diag} - q_{cl} \quad + \refstepcounter{equation}(\theequation)\label{eq:dqcl_init}` - If :math:`Q_C < 0`: - .. math:: :label: eq:dcl_init1 - - \Delta C_{l} = \frac{\Delta q_{cl}}{{q_{cl}}_{diag}} - \left( {C_{l}}_{diag} - C_{l} \right) + :math:`\Delta C_{l} = \frac{\Delta q_{cl}}{{q_{cl}}_{diag}} \left( + {C_{l}}_{diag} - C_{l} \right) \quad + \refstepcounter{equation}(\theequation)\label{eq:dcl_init1}` - If :math:`Q_C > 0`: - .. math:: :label: eq:dcl_init2 - - \Delta C_{l} = \frac{\Delta SD}{{SD}_{diag}} - \left( {C_{l}}_{diag} - C_{l} \right) + :math:`\Delta C_{l} = \frac{\Delta SD}{{SD}_{diag}} \left( {C_{l}}_{diag} - + C_{l} \right) \quad + \refstepcounter{equation}(\theequation)\label{eq:dcl_init2}` - Otherwise: - .. math:: \Delta q_{cl} = 0 + :math:`\Delta q_{cl} = 0` - .. math:: \Delta C_{l} = 0 + :math:`\Delta C_{l} = 0` where the subscript :math:`_{diag}` denotes the liquid cloud water content and fraction predicted by the diagnostic cloud scheme (either @@ -4855,7 +4826,6 @@ The thresholds :math:`C_{high}`, :math:`C_{high 2}`, :math:`C_{low}` and C_{low 2} = C_{tol 2}, - where the parameters :math:`C_{tol}` and :math:`C_{tol 2}` can be set via the UM namelist variables **cloud_pc2_tol** and **cloud_pc2_tol_2**. The original standard values of these parameters are @@ -4944,7 +4914,6 @@ condensed to achieve this, so we have: \overline{T} \leftarrow \overline{T} - \frac{L_c}{c_p} SD - The original version of this check on :math:`SD` (which may increase :math:`\overline{q_{cl}}`), made no accompanying changes to liquid cloud fraction. However, increases in :math:`\overline{q_{cl}}` without any @@ -5030,7 +4999,6 @@ behaviour is currently controlled by a temporary logical in the C_t \leftarrow C_i - .. _section-3: The next check is similar to above but for the :math:`C_l = 0` @@ -5052,7 +5020,6 @@ the gridbox: \overline{T} \leftarrow \overline{T} - \frac{L_c}{c_p} \overline{q_{cl}} - .. _section-4: Next, if :math:`C_i > 1` then :math:`C_i` is set to 1. Accordingly, @@ -5095,7 +5062,6 @@ overlap situation and then the minimum overlap situation. C_t \leftarrow \text{Min}( C_t , C_l + C_i, 1) - .. _section-8: Finally, there is a homogeneous nucleation term applied, similar to that @@ -5126,7 +5092,6 @@ phase if the temperature is cold enough. Hence, if C_l \leftarrow 0. - .. _sec_qpos: Qpos checks @@ -5966,7 +5931,7 @@ diagnostic output routines. .. list-table:: PC2 parameter values and locations :name: tab:pc2_names - :header-rows: 1 + * - Symbol - Code variable @@ -6133,7 +6098,7 @@ PC2:64 and a non-PC2 run. .. list-table:: PC2 parameter values and locations relating to the convection. \*These values are those used in HadGAM :name: tab:pc2_conv_names - :header-rows: 1 + * - Code variable - Des cription @@ -6378,13 +6343,12 @@ similar way to above gives .. math:: + \begin{aligned} SD = \int_{-b_s}^{-Q_c} G(s) (-Q_c + b_s) ds + \int_{-b_s}^{-Q_c} - G(s) (-s - b_s) ds - -.. math:: - + G(s) (-s - b_s) ds + \\ = (-Q_c + b_s) (1 - C_l) - I1 , - + \end{aligned} and hence :math:`I1` in terms of :math:`SD`. Using this value of :math:`I1` in :eq:`eqn:deltaqclmax` and cancelling From 8f6563cd8058fce74e35519b26357c8a3944640f Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Thu, 30 Apr 2026 22:49:18 +0100 Subject: [PATCH 071/116] Reinstated manual corrections needed after script. --- .../cloud_schemes/UMDP30_PC2CloudScheme.rst | 27 ++++++++++--------- 1 file changed, 15 insertions(+), 12 deletions(-) diff --git a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst index 9f1988ebb8..c919817634 100644 --- a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst +++ b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst @@ -4679,26 +4679,29 @@ prognostic :math:`C_l` and :math:`q_{cl}` are incremented as follows: - If :math:`{q_{cl}}_{diag} > q_{cl}`: - :math:`\Delta q_{cl} = {q_{cl}}_{diag} - q_{cl} \quad - \refstepcounter{equation}(\theequation)\label{eq:dqcl_init}` + .. math:: :label: eq:dqcl_init + + \Delta q_{cl} = {q_{cl}}_{diag} - q_{cl} - If :math:`Q_C < 0`: - :math:`\Delta C_{l} = \frac{\Delta q_{cl}}{{q_{cl}}_{diag}} \left( - {C_{l}}_{diag} - C_{l} \right) \quad - \refstepcounter{equation}(\theequation)\label{eq:dcl_init1}` + .. math:: :label: eq:dcl_init1 + + \Delta C_{l} = \frac{\Delta q_{cl}}{{q_{cl}}_{diag}} + \left( {C_{l}}_{diag} - C_{l} \right) - If :math:`Q_C > 0`: - :math:`\Delta C_{l} = \frac{\Delta SD}{{SD}_{diag}} \left( {C_{l}}_{diag} - - C_{l} \right) \quad - \refstepcounter{equation}(\theequation)\label{eq:dcl_init2}` + .. math:: :label: eq:dcl_init2 + + \Delta C_{l} = \frac{\Delta SD}{{SD}_{diag}} + \left( {C_{l}}_{diag} - C_{l} \right) - Otherwise: - :math:`\Delta q_{cl} = 0` + .. math:: \Delta q_{cl} = 0 - :math:`\Delta C_{l} = 0` + .. math:: \Delta C_{l} = 0 where the subscript :math:`_{diag}` denotes the liquid cloud water content and fraction predicted by the diagnostic cloud scheme (either @@ -5931,7 +5934,7 @@ diagnostic output routines. .. list-table:: PC2 parameter values and locations :name: tab:pc2_names - + :header-rows: 1 * - Symbol - Code variable @@ -6098,7 +6101,7 @@ PC2:64 and a non-PC2 run. .. list-table:: PC2 parameter values and locations relating to the convection. \*These values are those used in HadGAM :name: tab:pc2_conv_names - + :header-rows: 1 * - Code variable - Des cription From 93fb4a81c1a8f0d19edb93490037c0dd9bb72017 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Thu, 30 Apr 2026 22:56:26 +0100 Subject: [PATCH 072/116] Correction to manual correction: delete stray trailing whitespace. --- .../science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst index c919817634..d0664430ec 100644 --- a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst +++ b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst @@ -4701,7 +4701,7 @@ prognostic :math:`C_l` and :math:`q_{cl}` are incremented as follows: .. math:: \Delta q_{cl} = 0 - .. math:: \Delta C_{l} = 0 + .. math:: \Delta C_{l} = 0 where the subscript :math:`_{diag}` denotes the liquid cloud water content and fraction predicted by the diagnostic cloud scheme (either From ae76786621d0083eff2c50c14c8209429c140b77 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Thu, 30 Apr 2026 23:07:01 +0100 Subject: [PATCH 073/116] Re-applied after further correction to the script to mop-up spurious trailing whitespace. --- .../science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst index d0664430ec..122dc1e07b 100644 --- a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst +++ b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst @@ -3610,7 +3610,7 @@ is basic equations & = & {\overline{Q}}_{\rm{par}} - {\overline{Q}}_{\rm{reset}} - PPN - \frac{1}{\overline{\rho}} \, \frac{\partial \, \overline{\rho w^{'} - l_{\rm{ }}^{\rm{E'}}}}{\partial \, z} + l_{\rm{ }}^{\rm{E'}}}}{\partial \, z} \\ & = & \mu \, M^{\rm{P}} \, l_{\rm{ }}^{\rm{P}} + \delta \, M^{\rm{P}} \, l_{\rm{ @@ -6348,7 +6348,7 @@ similar way to above gives \begin{aligned} SD = \int_{-b_s}^{-Q_c} G(s) (-Q_c + b_s) ds + \int_{-b_s}^{-Q_c} - G(s) (-s - b_s) ds + G(s) (-s - b_s) ds \\ = (-Q_c + b_s) (1 - C_l) - I1 , \end{aligned} From 4619723b393f99c827133df5e26628aea5ef32c6 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Fri, 1 May 2026 00:25:53 +0100 Subject: [PATCH 074/116] Re-applied scripts with corrections to aligned math handling (now preserves the aligned blocks when they contain long equations split over multiple lines, and mops-up spurious blank lines. --- .../turbulence_schemes/bldoc.rst | 264 ++++++------------ 1 file changed, 89 insertions(+), 175 deletions(-) diff --git a/documentation/source/science_guide/turbulence_schemes/bldoc.rst b/documentation/source/science_guide/turbulence_schemes/bldoc.rst index db7a985115..f915e6df0f 100644 --- a/documentation/source/science_guide/turbulence_schemes/bldoc.rst +++ b/documentation/source/science_guide/turbulence_schemes/bldoc.rst @@ -68,7 +68,6 @@ horizontal components of momentum, :math:`{\bf u}` on a sphere gives: \right) + {\cal S} - where :math:`\overline{w'\chi'}` and :math:`{\bf \tau}` are the vertical turbulent fluxes to be parametrized, :math:`r` is the height from the centre of the planet and :math:`\rho` is density. The scalar variables @@ -84,7 +83,6 @@ approximately conserved under moist adiabatic ascent, are: q_t = q_v + q_{\ell}+ q_f - where :math:`T` is temperature, :math:`q_v` is specific humidity, :math:`q_{\ell}` and :math:`q_f` the specific liquid and frozen water contents respectively, and :math:`L_s=L+L_f` is the latent heat of @@ -136,7 +134,6 @@ standard closures are: {\bf \tau} = K_m \frac{\partial {\bf u}}{\partial z} + {\bf \tau}^{nl} - Separate eddy-diffusivities are calculated for momentum, :math:`K_m`, and for scalar variables, :math:`K_h`. The second term on the right hand side represents a non-local flux in unstable boundary layers. Currently @@ -596,7 +593,6 @@ expanded using the first-order closure in \overline{w'q_t'}_k = -\left(K_h^{\rm surf}+ K_h^{\rm Sc}\right) \, \,\frac{\Delta_k q_t}{\Delta_k z} - where :math:`\widetilde{\Delta_k \theta_{\ell}} = \Delta_k \theta_{\ell}- \gamma_{\theta_{\ell}} \Delta_k z` in order to include the non-local (or @@ -773,16 +769,15 @@ expected to occur) and :math:`z_{\mbox{\tiny \rm NTML}-1}`. Then, -.. math:: +.. math:: :label: wthl_int - \int_{z_h-\Delta z_{rad}}^{z_h} \, \overline{w'\theta_{\ell}'}\, dz = + \begin{aligned} + \int_{z_h-\Delta z_{rad}}^{z_h} \, \overline{w'\theta_{\ell}'}\, dz & = & \int_{z_h-\Delta z_{rad}}^{z_h} \, F_{\theta_{\ell}}^{Tot} - F_{\theta_{\ell}}^{NT}\, dz - -.. math:: :label: wthl_int - - = I^{Tot} - I^{rad} - I^{ppn} - + \\ + & = & I^{Tot} - I^{rad} - I^{ppn} + \end{aligned} For the radiative flux, it could be assumed that the subgrid flux distribution is exponentially dependent on the grid-level LWP, for @@ -812,15 +807,14 @@ approximations, :eq:`wthl_int` becomes .. math:: - \int_{z_h-\Delta z_{rad}}^{z_h} \, \overline{w'\theta_{\ell}'}\, dz = + \begin{aligned} + \int_{z_h-\Delta z_{rad}}^{z_h} \, \overline{w'\theta_{\ell}'}\, dz & = & \Delta z_{rad} \left(-w_e \Delta \theta_{\ell}+ \Delta F\right) - \Delta z_{rad} \Delta F/ 3 - -.. math:: - - = \Delta z_{rad} \left(-w_e \Delta \theta_{\ell}+ \frac{2}{3} \Delta - F\right) - + \\ + & = & \Delta z_{rad} \left(-w_e \Delta \theta_{\ell}+ \frac{2}{3} \Delta + F\right) + \end{aligned} For the integral of :math:`\overline{w'q_t'}` across this cloud-top region, :math:`\overline{w'q_t'}` is also taken to be constant so that: @@ -941,7 +935,6 @@ A first order ‘mixing length’ closure is used: K_h = {\cal L}_h \, {\cal L}_m \, (S+S_d) \, f_h(Ri) - where :math:`{\cal L}_m` and :math:`{\cal L}_h` are the neutral mixing lengths and :math:`S` is the resolved vertical shear of the horizontal wind components, :math:`S = \left| \partial {\bf u}/\partial z \right|`. @@ -960,7 +953,6 @@ ignored above grid-level 2 and the neutral mixing lengths are given by {\cal L}_h = \frac{k(z+z_{0m})}{1+k(z+z_{0m})/\lambda_h} - where :math:`z_{0m}` includes the orographic component. For the lowest interior grid-level (:math:`k=1`) they are calculated, incorporating this log profile correction, as @@ -984,7 +976,6 @@ The asymptotic mixing lengths are given by \lambda_h =\mbox{max}\left[\lambda_0,\, 0.15 z_{\rm loc}\right] - where :math:`\lambda_0` is a minimum mixing length read in from the namelist and :math:`z_{\rm loc}` is defined below. The orographic blending height, :math:`h_B` (only used within the boundary layer, as @@ -1082,7 +1073,6 @@ For :math:`Ri < 0`, the standard UM stability functions are given by f_h = \frac{1}{Pr_N}\left(1 - \frac{g_0 \,Ri} {1+D_h(\tilde{{\cal L}}_m/\tilde{{\cal L}}_h)|Ri|^{1/2} }\right) - with :math:`g_0=10`, :math:`D_m=g_0/4` and :math:`D_h=g_0/25`. If the stability dependent Prandtl number option is chosen (see below) the neutral Prandtl number, :math:`Pr_N`, is set to :math:`0.7`; otherwise @@ -1097,7 +1087,6 @@ model (LEM), `Brown (1999) 2`_: f_h = \frac{1}{Pr_N}\left(1 - b_{LEM} Ri\right)^{1/2} - where :math:`Pr_N = 0.7`, and the constants :math:`b_{LEM}` and :math:`c_{LEM}` can take the values 40 and 16 respectively in the “standard” LEM model or both be 1.43 in the “conventional” model. @@ -1133,7 +1122,6 @@ where B_{Ri} = (g_0/2) /\left(1 - g_0 Ri_{t}/2\right)^2 - For the ‘SHARPEST’ function of `Derbyshire (1997)`_, :math:`Ri_{t}=0.1`, while larger values give even sharper reduction of turbulence with increasing :math:`Ri`. An additional option, used @@ -1158,7 +1146,6 @@ for :math:`Ri>0` are then given by: f_h = \frac{1}{Pr_N} \, f_{\rm stable} - Note that writing the functions in this way ensures that :math:`f_m=1` under neutral conditions and the effect of the variation in :math:`Pr` is for :math:`f_m` to decrease slower with increasing :math:`Ri` than @@ -1176,7 +1163,6 @@ turbulence beyond a critical Richardson number, :math:`Ri_c=0.25`: f_h = \frac{1}{Pr_N} \left( 1 - \frac{Ri}{Ri_c} \right)^4 (1 - g_{LEM} Ri) - with :math:`g_{LEM}=1.2`. .. _sec_fd_ri: @@ -2317,7 +2303,6 @@ F|_{z_i}`, so that \overline{w'q_t'}_{z_i} = - w_e \Delta q_t - where the total heat flux :math:`{\cal H} = \overline{w'\theta_{\ell}'}+ F_{\rm net}` and :math:`F_{\rm net} @@ -2363,7 +2348,6 @@ base of the mixed layer: - \frac{ z'_{\mbox{\tiny \rm NTDSC}+\frac{1}{2}} }{z_{\rm ml}} \left( \tilde{w_e} \Delta q_t + \overline{w'q_t'}|_{z_{\rm b}} \right) - where :math:`z' = z-z_{\rm b}`, and similarly for the SML entrainment fluxes (at :math:`z=z_{\mbox{\tiny \rm NTML}+\frac{1}{2}}`). The turbulent fluxes at the base of the mixed layer are assumed zero except @@ -2539,7 +2523,6 @@ The coefficients are given by -z_{\mbox{\tiny \rm NTML}} \right) \right) \right) - Clearly, care must be taken to ensure that :math:`z_i` is not only well-defined but also sensible (for example, as a rising inversion encounters more or less stable regions above). If :math:`b>0` this @@ -2786,7 +2769,6 @@ mixed layer by the end of the timestep. In other words, for F_{\chi}^{Tot}|_{z_{\rm b}} \right) - where the superscripts :math:`n` and :math:`n+1` refer to the model timestep, although strictly speaking :math:`n+1` refers to fields after the boundary layer implicit solver. Requiring that @@ -2861,7 +2843,6 @@ free-atmospheric lapse rates are given by { z_{\mbox{\tiny \rm NTML}+3} - z_{\mbox{\tiny \rm NTML}+2} } \right] - .. _sec_subs_calc: Calculation of the subsidence flux @@ -2923,7 +2904,6 @@ specified through an eddy diffusivity which is given by K_m|_{\mbox{\tiny \rm NTML}} = Pr \, w_e \Delta_{\mbox{\tiny \rm NTML}+\frac{1}{2}} z - noting the Charney-Philips grid implying stresses are staggered from scalar fluxes. The Prandtl number, :math:`Pr`, takes the same form as for the non-local :math:`K` profiles, see section :ref:`The non-local scheme @@ -3081,7 +3061,6 @@ layer are related to the surface fluxes by: \frac{\partial {\rm {\bf v}}}{\partial z}=\frac{ {\rm {\bf \tau }}_{0} }{ \rho _0 v_\ast } \frac{ \phi _m (z/L)}{kz}, - where subscript 0 represents a surface value and subscript \* represents a surface layer scaling quantity. :math:`\phi _{m}` and :math:`\phi _{h}` are the Monin-Obukhov stability functions (for the form of these @@ -3124,7 +3103,6 @@ surface turbulent fluxes are: \frac{ {\bf \tau }_{0} }{ \rho _{0} }= c_D^{1/2} v_\ast \Delta {\rm {\bf v}}, - where :math:`\Delta`\ X=X\ :math:`_{1}`-X\ :math:`_{0}`. From :eq:`1.1.7` and :eq:`1.1.8` the surface buoyancy flux in definition :eq:`1.1.4` is @@ -3151,7 +3129,6 @@ and c\ :math:`_{H}`, are given by \frac{ c_H }{ c_D^{1/2} }=\frac{k}{ \Phi _h (L , z_1 + z_{0m} , z_{0h} )}, - where .. math:: :label: 1.1.14 @@ -3164,7 +3141,6 @@ where \Phi _h (L , z_1 + z_{0m} , z_{0h} )= \int \limits_{ z_{0h} /L}^{( z_1 + z_{0m} )/L} \frac{ \phi _h (\zeta )}{\zeta } d\zeta, - z\ :math:`_{0m}` and z\ :math:`_{0h}` are the **surface roughness lengths** for momentum and scalars respectively. @@ -3172,15 +3148,15 @@ The equations for the **surface turbulent fluxes**, :eq:`1.1.7`–:eq:`1.1.9`, can be written in the forms -.. math:: - - \frac{ H_0 }{ c_P \rho _0 }={-c}_H V \left( {\Delta T + \frac{g}{c_p }( z_1 - + z_{0m} - z_{0h} )} \right) - .. math:: :label: 1.1.16 - = - C_H \left( {\Delta T + \frac{g}{c_p } ( z_1 + z_{0m} - z_{0h} )} + \begin{aligned} + \frac{ H_0 }{ c_P \rho _0 }&=&{-c}_H V \left( {\Delta T + \frac{g}{c_p }( + z_1 + z_{0m} - z_{0h} )} \right) + \\ + &=& - C_H \left( {\Delta T + \frac{g}{c_p } ( z_1 + z_{0m} - z_{0h} )} \right) + \end{aligned} .. math:: :label: 1.1.17 @@ -3191,7 +3167,6 @@ forms \frac{ {\rm {\bf \tau }}_{0} }{ \rho _{0} }= c_D V \Delta {\rm {\bf v}}{ }= C_D \Delta {\rm {\bf v}}, - where the effective wind speed for surface turbulent exchanges, :math:`V`, is defined by @@ -3210,20 +3185,18 @@ respectively C_D=\frac{k}{ \Phi _m } v_\ast = c_D V. - The surface exchange coefficients can then be written in any of the following forms: .. math:: :label: 1.1.22 - c_H=\frac{ C_H }{V} = \frac{ C_H C_D }{ v_\ast ^2 } = \frac{ k^2 }{ \Phi - _h \Phi _m } + c_H=\frac{ C_H }{V} = \frac{ C_H C_D }{ v_\ast ^2 } = \frac{ k^2 }{ \Phi _h + \Phi _m } .. math:: :label: 1.1.23 - c_D=\frac{ C_D }{V} = \frac{ C_D^2 }{ v_\ast ^2 } = \frac{ k^2 }{ \Phi - _m^2 }. - + c_D=\frac{ C_D }{V} = \frac{ C_D^2 }{ v_\ast ^2 } = \frac{ k^2 }{ \Phi _m^2 + }. In order to close the system the surface scaling velocity, v\ :math:`_{\ast @@ -3406,22 +3379,19 @@ Redefining the vertical coordinate as :math:`\zeta=z/L`, we have .. math:: - u(\zeta) = \frac{u_*}{k} \int_{0}^{\zeta} \frac{1}{(\zeta'+\zeta_{0m})} + \begin{aligned} + u(\zeta) &=& \frac{u_*}{k} \int_{0}^{\zeta} \frac{1}{(\zeta'+\zeta_{0m})} \phi_m(\zeta'+\zeta_{0m}) \, d\zeta' = \frac{u_*}{k} \int_{\zeta_{0m}}^{\zeta'+\zeta_{0m}} \frac{1}{\zeta'} \phi_m(\zeta') \, d\zeta' - -.. math:: - - = \frac{u_*}{k} \int_{\zeta_{0m}}^{\zeta'+\zeta_{0m}} \left ( + \\ + &=& \frac{u_*}{k} \int_{\zeta_{0m}}^{\zeta'+\zeta_{0m}} \left ( \frac{1}{\zeta'} - \frac{d\psi_m}{d\zeta'} \right ) \, d\zeta' - -.. math:: - - = \frac{u_*}{k} \left \{ \ln \left ( \frac{\zeta+\zeta_{0m}}{\zeta_{0m}} + \\ + &=& \frac{u_*}{k} \left \{ \ln \left ( \frac{\zeta+\zeta_{0m}}{\zeta_{0m}} \right ) - \psi_m(\zeta+\zeta_{0m}) + \psi_m (\zeta_{0m}) \right \}. - + \end{aligned} This is also frequently written as @@ -3432,87 +3402,72 @@ in the rescaled coordinate), is therefore .. math:: - \bar u = \frac{u_*}{k\zeta_1} \int_0^{\zeta_1} u(\zeta) \, d \zeta - -.. math:: - - = \frac{u_*}{k\zeta_1} \int_0^{\zeta_1} + \begin{aligned} + \bar u &=& \frac{u_*}{k\zeta_1} \int_0^{\zeta_1} u(\zeta) \, d \zeta + \\ + &=& \frac{u_*}{k\zeta_1} \int_0^{\zeta_1} \ln \left ( \frac{\zeta+\zeta_{0m}}{\zeta_{0m}} \right ) - \psi_m(\zeta+\zeta_{0m}) + \psi_m (\zeta_{0m}) \, d \zeta. - + \end{aligned} We consider the three terms within the integral separately. For the first, .. math:: + \begin{aligned} \int_0^{\zeta_1} \ln \left ( \frac{\zeta+\zeta_{0m}}{\zeta_{0m}} \right ) \, d \zeta - = \zeta_{0m} \int_1^{1+\zeta_1/\zeta_{0m}} \ln(x) \, dx - -.. math:: - - = \zeta_{0m} \left [ \left ( 1+ \frac{\zeta_1}{\zeta_{0m}} \right ) \ln + &=& \zeta_{0m} \int_1^{1+\zeta_1/\zeta_{0m}} \ln(x) \, dx + \\ + &=& \zeta_{0m} \left [ \left ( 1+ \frac{\zeta_1}{\zeta_{0m}} \right ) \ln \left ( 1+ \frac{\zeta_1}{\zeta_{0m}} \right ) - \left ( 1+ \frac{\zeta_1}{\zeta_{0m}} \right ) +1 \right ] . - + \end{aligned} For the second, .. math:: - \int_0^{\zeta_1} \psi_m(\zeta+\zeta_{0m}) \, d\zeta = + \begin{aligned} + \int_0^{\zeta_1} \psi_m(\zeta+\zeta_{0m}) \, d\zeta &=& \int_{\zeta_{0m}}^{\zeta_1+\zeta_{0m}} \psi_m(\zeta) \, d\zeta - -.. math:: - - = \left [ \zeta \psi_m + \\ + &=& \left [ \zeta \psi_m \right ]_{\zeta_{0m}}^{\zeta_1+\zeta_{0m}} - \int_{\zeta_{0m}}^{\zeta_1+\zeta_{0m}} \zeta \frac{d\psi_m}{d\zeta} d\zeta - -.. math:: - - = (\zeta_1+\zeta_{0m}) \psi_m(\zeta_1+\zeta_{0m}) - \zeta_{0m} + \\ + &=& (\zeta_1+\zeta_{0m}) \psi_m(\zeta_1+\zeta_{0m}) - \zeta_{0m} \psi_m(\zeta_{0m}) - -.. math:: - - - \int_{\zeta_{0m}}^{\zeta_1+\zeta_{0m}} + \\ &-& \int_{\zeta_{0m}}^{\zeta_1+\zeta_{0m}} (1-\phi_m) d\zeta - -.. math:: - - = (\zeta_1+\zeta_{0m}) \psi_m(\zeta_1+\zeta_{0m}) - \zeta_{0m} + \\ + &=& (\zeta_1+\zeta_{0m}) \psi_m(\zeta_1+\zeta_{0m}) - \zeta_{0m} \psi_m(\zeta_{0m}) - -.. math:: - - + \int_{\zeta_{0m}}^{\zeta_1+\zeta_{0m}} + \\ &+& \int_{\zeta_{0m}}^{\zeta_1+\zeta_{0m}} (\phi_m -1) \, d\zeta . - + \end{aligned} :math:`\phi_m-1` is retained in the last integral since this will prove convenient in later algebra. The third integral is trivial. Hence, .. math:: - \bar u = \frac{u_*}{k} \left \{ + \begin{aligned} + \bar u &=& \frac{u_*}{k} \left \{ \left ( 1+ \frac{\zeta_{0m}}{\zeta_1} \right ) \left [ \ln \left ( 1+ \frac{\zeta_1}{\zeta_{0m}} \right ) \right . \right . - -.. math:: - - - \left . \left . \psi_m(\zeta_1+\zeta_{0m}) + \psi_m(\zeta_{0m}) \right ] -1 + \\ + &-& \left . \left . \psi_m(\zeta_1+\zeta_{0m}) + \psi_m(\zeta_{0m}) \right ] + -1 - \frac{1}{\zeta_1} \int_{\zeta_{0m}}^{\zeta_1+\zeta_{0m}} (\phi_m -1) \, d\zeta \right \} - -.. math:: - - = \frac{u_*}{k} \left \{ \left ( 1+ \frac{\zeta_{0m}}{\zeta_1} \right ) + \\ + &=& \frac{u_*}{k} \left \{ \left ( 1+ \frac{\zeta_{0m}}{\zeta_1} \right ) \Phi_m(\zeta_1) - \frac{1}{\zeta_1} \int_{\zeta_{0m}}^{\zeta_1+\zeta_{0m}} \phi_m \, d\zeta \right \} . - + \end{aligned} Thus, in practical terms, the standard function :math:`\Phi_m` is evaluated at the top of the layer, scaled by @@ -3550,7 +3505,6 @@ stability functions are given by `Beljaars and Holtslag (1991)`_: \Phi _h=\ln \left( {\frac{ z_1 + z_{0m} }{ z_{0h} }} \right) - \Psi _h ( \zeta _1 ) + \Psi _h ( \zeta _{0h} ) - where :math:`\zeta _{1}` = (z\ :math:`_{1}` + z\ :math:`_{0m})`/L, :math:`\zeta _{0m}` = z\ :math:`_{0m}`/L, :math:`\zeta _{0h}` = z\ :math:`_{0h}`/L and @@ -3566,7 +3520,6 @@ z\ :math:`_{0h}`/L and - \Psi _m (\zeta )=a\zeta + b\left( {\zeta - \frac{c}{d}} \right)\exp (-d\zeta ) + \frac{bc}{d}, - with :math:`a = 1`, :math:`b =2/3`, :math:`c = 5`, :math:`d = 0.35`. Note that the bulk flux Richardson number for the surface layer is given @@ -3591,7 +3544,6 @@ Dyer and Hicks forms :raw-latex:`\cite[]{dyer1974}` are used: \phi _h=(1 - 16\zeta )^{-1/2} - (Note that :math:`\phi _{h}\prime` is discontinuous at 0.) These are only empirically verified for :math:`\zeta \ge` -1. Evaluating the integrals :eq:`1.1.14` and :eq:`1.1.15` we @@ -3645,7 +3597,6 @@ ms\ :math:`^{-1}`), then start the iteration from the neutral limit, so v_\ast ^{(0)}= {\left( {\frac{k}{ \Phi _m^{(0)} }} \right)} \left| {\Delta {{{\rm {\bf v}}}}} \right| - Otherwise (if :math:`\Delta`\ B :math:`<` 0 and :math:`\Delta`\ **v** :math:`<` 2 ms\ :math:`^{-1}` ) start from the greater of the neutral and convective limits for :math:`v_\ast^{(0)}`, so @@ -3669,7 +3620,6 @@ and convective limits for :math:`v_\ast^{(0)}`, so {\left[ { \gamma _t^3 \left( {\frac{k}{ \Phi _h^{(0)} }} \right) z_i \left| {-\Delta B} \right| } \right]}^{ 1/2} \right]} - Then calculate .. math:: :label: 1.4.8 @@ -3680,7 +3630,6 @@ Then calculate C_H^{(0)}=\frac{k}{ \Phi _h^{(0)} } v_\ast ^{(0)} - Having set up initial values the iteration loop can be entered (this is the original method used but contains an inconsistency in the treatment of boundary-layer convective gustiness, as described in @@ -3727,7 +3676,6 @@ DO n = 1 to N C_H^{(n)}=\frac{k}{ \Phi _h^{(n)} } v_\ast ^{(n)} - END DO. For neutral and stable conditions (:math:`\Delta`\ B :math:`\ge` 0) @@ -3751,7 +3699,6 @@ stress: {\rm {\bf \tau }}_{0} = \rho _0 C_D^{(N)} \Delta {\rm {\bf v}} - N is the last iteration value. N = 5 is currently used. For sea points the momentum roughness length and the wind mixing energy @@ -3982,7 +3929,6 @@ the result that would be obtained from standard similarity theory, \theta_{ob}(t+\delta t) \leftarrow W \theta_{ob}'(t+\delta t) +(1-W) \theta_{ob, \mbox{\tiny sim}}. - By tuning against an idealized highly vertically resolved model based on local scaling we set, @@ -4262,7 +4208,6 @@ Two approaches are available in uncoupled configurations of the model. < C_D >=( f_I C_{D(MIZ)} + ( 0.7 - f_I ) C_{D(L)} ) / 0.7 - and for 0.7 :math:`\le` f\ :math:`_{I} \le` 1 .. math:: :label: 1.6.3 @@ -4273,7 +4218,6 @@ Two approaches are available in uncoupled configurations of the model. < C_D >=( ( 1 - f_I ) C_{D(MIZ)} + ( f_I - 0.7 ) ) C_{D(I)} ) / 0.3 - where .. math:: :label: 1.6.5 @@ -4288,7 +4232,6 @@ Two approaches are available in uncoupled configurations of the model. C_{H(I)}= C_H ( L_{(I)} , z_{0m(sea-ice)} , z_{0h(sea-ice)} ) - and similarly for the drag coefficient C\ :math:`_{D}`. The roughness lengths over open sea are calculated as above, but @@ -4418,7 +4361,6 @@ Two approaches are available in uncoupled configurations of the model. < C_H >= (1 - f_I) C_{H(L)} + f_I C_{H(I)} - .. _sec_coast: Surface exchange in coastal grid-boxes @@ -4536,8 +4478,8 @@ When form drag is included via effective roughness lengths equations .. math:: :label: 2.1.1 - \left( {\Delta T + \frac{g}{c_p }( z_1 + z_{0m(eff)} - z_{0h(eff)} )} - \right) + \left( {\Delta T + \frac{g}{c_p }( z_1 + z_{0m(eff)} - z_{0h(eff)} )} + \right) .. math:: :label: 2.1.2 @@ -4549,7 +4491,6 @@ When form drag is included via effective roughness lengths equations \frac{ {\rm {\bf \tau }}_{{0(eff)}} }{ \rho _{0} }=\frac{k}{ \Phi _m (L , z_1 + z_{0m(eff)} , z_{0m(eff)} )} v_{\ast (eff)} \Delta {\rm {\bf v}} - The effective surface scaling velocity, v\ :math:`_{\ast (eff)}` , is given by (cf. :eq:`1.1.25`) @@ -4691,7 +4632,6 @@ and :eq:`2.1.13` and :eq:`2.1.14` become C_{D(f)}= C_{D(eff)} {\left( {1 + \alpha \beta \pi ^2 f_D {\left( {\frac{A}{S}} \right)}^2 } \right)}^{-1} - The effective surface flux of scalar X evaluated in terms of values at z\ :math:`_{c}` is @@ -4763,7 +4703,6 @@ iteration from the convective limit, so {\frac{k}{ \Phi _h^{(0)} }} \right) z_i \left| {-\Delta B} \right| + \gamma _c^2 w_c^2 } \right]}^{ 1/2} - ELSE IF (:math:`\Delta`\ **v** :math:`\ge` 2 ms\ :math:`^{-1}` ) start iteration from the neutral end, so @@ -4796,7 +4735,6 @@ iteration from the neutral end, so v_{\ast (f)}^{(0)}= {\left( { u_{\ast (f)}^{(0) 2} + \gamma _c^2 w_c^2 } \right)}^{ 1/2} - END IF. Then calculate: @@ -4819,7 +4757,6 @@ Then calculate: C_{H(f)}^{(0)}= C_{H(eff)}^{(0)} \left( {1 - 2.2 f_D \frac{A}{S}} \right) - Having set up initial values the iteration loop can be entered: DO n = 1 to N @@ -4884,7 +4821,6 @@ DO n = 1 to N C_{H(f)}^{(n)}= C_{H(eff)}^{(n)} \left( {1 - 2.2 f_D \frac{A}{S}} \right) - END DO. For neutral and stable conditions (:math:`\Delta`\ B :math:`\ge` 0) @@ -4906,7 +4842,6 @@ surface sensible and latent heat fluxes and surface stress: {\rm {\bf \tau }}_{{0(eff)}} = \rho _0 C_{D(eff)}^{(N)} \Delta {\rm {\bf v}} - The stress for a flat surface, if required for output, is calculated from @@ -5178,7 +5113,6 @@ becomes F}{\partial z}^{n+1}-{\cal E}_{2}\frac{\partial F}{\partial z}^{*}+\left({\cal I}_{2}-{\cal E}_{2}\right)S - where, .. math:: @@ -5228,7 +5162,6 @@ Writing equations :eq:`eq:sppf_bl1`, X^{n+1} = X^{n}+\delta X^{*}+\delta X^{n+1} - .. _sec_impsolve: Discrete equations and boundary conditions @@ -5277,18 +5210,17 @@ levels), discretizing the previous equation in :math:`z` on all .. math:: - \delta u_{k+1/2}^{*} = ({\cal I}_{1}-{\cal E}_{1})\Delta + \begin{aligned} + \delta u_{k+1/2}^{*} & = & ({\cal I}_{1}-{\cal E}_{1})\Delta t\left(\frac{\tau_{x}^{n}\Big|_{k+1}-\tau_{x}^{n}\Big|_{k}}{z_{k+1}-z_{k}}+S_{k+1/2}\right) - -.. math:: - - +{\cal I}_{1}\frac{\Delta + \\ + & & +{\cal I}_{1}\frac{\Delta t}{z_{k+1}-z_{k}}\left[\left(K_{u}\Big|_{k+1}\frac{\delta u_{k+3/2}^{*}-\delta u_{k+1/2}^{*}}{z_{k+3/2}-z_{k+1/2}}\right)-\left(K_{u}\Big|_{k}\frac{\delta u_{k+1/2}^{*}-\delta u_{k-1/2}^{*}}{z_{k+1/2}-z_{k-1/2}}\right)\right] - + \end{aligned} or, rearranging @@ -5328,10 +5260,8 @@ For the bottom :math:`\rho`-level, :math:`k=0`, the .. math:: :label: eq:u_bc_1 - \begin{aligned} - \delta u_{1/2}^{*} & = & \frac{\Delta - t}{z_{1}-0}\left(\bar{\tau}_{x}^{*}\Big|_{1}-\bar{\tau}_{x}^{*}\Big|_{0}\right)+\Delta t\left({\cal I}_{1}-{\cal E}_{1}\right)S_{1/2} - \end{aligned} + \delta u_{1/2}^{*} = \frac{\Delta + t}{z_{1}-0}\left(\bar{\tau}_{x}^{*}\Big|_{1}-\bar{\tau}_{x}^{*}\Big|_{0}\right)+\Delta t\left({\cal I}_{1}-{\cal E}_{1}\right)S_{1/2} where, from :eq:`eq:tau_star`, @@ -5453,8 +5383,7 @@ and thus the following discretization is obtained, on .. math:: - +\frac{{\cal I}_{1}}{z_{k+1/2}-z_{k-1/2}}\left[K_{X}\Big|_{k+1/2}\left(\frac{\delta X_{k+1}^{*}-\delta X_{k}^{*}}{z_{k+1}-z_{k}}\right)-K_{X}\Big|_{k-1/2}\left(\frac{\delta X_{k}^{*}-\delta X_{k-1}^{*}}{z_{k}-z_{k-1}}\right)\right],\; k=2,\ldots,L-1. - + +\frac{{\cal I}_{1}}{z_{k+1/2}-z_{k-1/2}}\left[K_{X}\Big|_{k+1/2}\left(\frac{\delta X_{k+1}^{*}-\delta X_{k}^{*}}{z_{k+1}-z_{k}}\right)-K_{X}\Big|_{k-1/2}\left(\frac{\delta X_{k}^{*}-\delta X_{k-1}^{*}}{z_{k}-z_{k-1}}\right)\right],\; k=2,\ldots,L-1. or, @@ -5682,7 +5611,6 @@ corrector are defined as: E}_{2}\right)\tau_{x}^{*}+{\cal I}_{2}K_{u}\frac{\partial\delta u^{n+1}}{\partial z} - where, :math:`\delta u^{*}=u^{*}-u^{n},\;\delta u^{n+1}=u^{n+1}-u^{*}`. The meridional stress :math:`\tau_{y}` and the scalar fluxes can be derived in a similar way. These formulae have been validated in SCM @@ -5706,19 +5634,17 @@ modified version of the flux formulae (78), (79) of .. math:: :label: eq:FTLstar - \begin{aligned} - \frac{\overline{H^{*}}}{c_{p}} & = & \frac{(1+\beta - B_{2})[\gamma_{2}F_{T}^{n}+A_{1}\delta{T'}_{1}+A_{2}\delta{Q'}_{1}]-\beta - A_{2}[\gamma_{2}F_{Q}^{n}+B_{1}\delta{T'}_{1}+B_{2}\delta{Q'}_{1}]}{(1+\beta A_{1})(1+\beta B_{2})-\beta^{2}A_{2}B_{1}} - \end{aligned} + \frac{\overline{H^{*}}}{c_{p}} = \frac{(1+\beta + B_{2})[\gamma_{2}F_{T}^{n}+A_{1}\delta{T'}_{1}+A_{2}\delta{Q'}_{1}]-\beta + A_{2}[\gamma_{2}F_{Q}^{n}+B_{1}\delta{T'}_{1}+B_{2}\delta{Q'}_{1}]}{(1+\beta + A_{1})(1+\beta B_{2})-\beta^{2}A_{2}B_{1}} .. math:: :label: eq:FQWstar - \begin{aligned} - \overline{E^{*}} & = & \frac{(1+\beta - A_{1})[\gamma_{2}F_{Q}^{n}+B_{1}\delta{T'}_{1}+B_{2}\delta{Q'}_{1}]-\beta - B_{1}[\gamma_{2}F_{T}^{n}+A_{1}\delta{T'}_{1}+A_{2}\delta{Q'}_{1}]}{(1+\beta A_{1})(1+\beta B_{2})-\beta^{2}A_{2}B_{1}} - \end{aligned} + \overline{E^{*}} = \frac{(1+\beta + A_{1})[\gamma_{2}F_{Q}^{n}+B_{1}\delta{T'}_{1}+B_{2}\delta{Q'}_{1}]-\beta + B_{1}[\gamma_{2}F_{T}^{n}+A_{1}\delta{T'}_{1}+A_{2}\delta{Q'}_{1}]}{(1+\beta + A_{1})(1+\beta B_{2})-\beta^{2}A_{2}B_{1}} where, :math:`F_{T}^{n}`, :math:`F_{Q}^{n}` denote the surface explicit fluxes, :math:`\gamma_{2}={\cal I}_{1}-{\cal E}_{1}` and the @@ -5783,8 +5709,7 @@ flux for :math:`H` is derived: .. math:: - \qquad\quad+\gamma_{1}RK_{PMj}L\psi_{j}RK_{H}(1)_{j}[\delta{Q'}_{1}-\beta\overline{E^{*}}] - + \qquad\quad+\gamma_{1}RK_{PMj}L\psi_{j}RK_{H}(1)_{j}[\delta{Q'}_{1}-\beta\overline{E^{*}}] and similarly :math:`E_{j}^{*}`. From these, the tile flux equations :eq:`eq:FTLstar`, :eq:`eq:FQWstar` can @@ -5794,21 +5719,17 @@ be obtained. .. math:: :label: eq:FTLnp1 - \begin{aligned} - \frac{\overline{H^{n+1}}}{c_{p}} & = & \frac{(1+\beta - B_{2})[\xi_{2}F_{T}^{*}+A_{1}\delta{T'}_{1}+A_{2}\delta{Q'}_{1}]-\beta - A_{2}[\xi_{2}F_{Q}^{*}+B_{1}\delta{T'}_{1}+B_{2}\delta{Q'}_{1}]}{(1+\beta - A_{1})(1+\beta B_{2})-\beta^{2}A_{2}B_{1}} - \end{aligned} + \frac{\overline{H^{n+1}}}{c_{p}} = \frac{(1+\beta + B_{2})[\xi_{2}F_{T}^{*}+A_{1}\delta{T'}_{1}+A_{2}\delta{Q'}_{1}]-\beta + A_{2}[\xi_{2}F_{Q}^{*}+B_{1}\delta{T'}_{1}+B_{2}\delta{Q'}_{1}]}{(1+\beta + A_{1})(1+\beta B_{2})-\beta^{2}A_{2}B_{1}} .. math:: :label: eq:FQWnp1 - \begin{aligned} - \overline{E^{n+1}} & = & \frac{(1+\beta - A_{1})[\xi_{2}F_{Q}^{*}+B_{1}\delta{T'}_{1}+B_{2}\delta{Q'}_{1}]-\beta - B_{1}[\xi_{2}F_{T}^{*}+A_{1}\delta{T'}_{1}+A_{2}\delta{Q'}_{1}]}{(1+\beta - A_{1})(1+\beta B_{2})-\beta^{2}A_{2}B_{1}} - \end{aligned} + \overline{E^{n+1}} = \frac{(1+\beta + A_{1})[\xi_{2}F_{Q}^{*}+B_{1}\delta{T'}_{1}+B_{2}\delta{Q'}_{1}]-\beta + B_{1}[\xi_{2}F_{T}^{*}+A_{1}\delta{T'}_{1}+A_{2}\delta{Q'}_{1}]}{(1+\beta + A_{1})(1+\beta B_{2})-\beta^{2}A_{2}B_{1}} where, the coefficients :math:`A_{1},\; A_{2},B_{1},\; B_{2}` are given by :eq:`ab_coeffs` but with @@ -6295,7 +6216,6 @@ written \Delta b ^{1/2} \, z_c^{3/2} \, C_{fac} - Here, :math:`[\overline{w'b'}_S]_{\rm sat}= g ( \tilde{\beta_T} \overline{w'\theta_{\ell}'}_S+ \tilde{\beta_q}\overline{w'q_t'}_S)`, @@ -6359,7 +6279,6 @@ approximated as \gamma_{q_f} = -\frac{\gamma_{T_L}\alpha_L + g q_s/(RTV_{fac})} {1+(L_s/c_p)\alpha_L} - where :math:`\gamma_{T_L} = -(g/c_p)+ \gamma_{\theta_{\ell}}` and :math:`\gamma_{\theta_{\ell}}` is given by :eq:`gradadj` within the mixed layer (zero above). Optionally (and currently @@ -6385,11 +6304,10 @@ grid-level based calculation: .. math:: :label: zc_calc - \left. \hspace{2.4cm} - + \mbox{min}\left[ \frac{ \Delta_{k_b+\frac{1}{2}}z - +\Delta_{k_b-\frac{1}{2}}z }{2} C_F^f, - \frac{ q_f }{ \gamma_{q_f} } \right]/C_F \right. - + \left. \hspace{2.4cm} + + \mbox{min}\left[ \frac{ \Delta_{k_b+\frac{1}{2}}z + +\Delta_{k_b-\frac{1}{2}}z }{2} C_F^f, + \frac{ q_f }{ \gamma_{q_f} } \right]/C_F \right. When :math:`\gamma_{q_f}` is set to zero (currently as standard) the last term in :eq:`zc_calc` is given by @@ -6462,7 +6380,6 @@ above and below using the adiabatic lapse rates are calculated as: NTML}+2}} - (z_{\mbox{\tiny \rm NTML}+2}-z_i) \gamma_{q_{\ell}} \right] - and similarly for :math:`q_f` (noting that currently :math:`\gamma_{q_f}=0`) and for DSC layers. Then, @@ -6476,7 +6393,6 @@ and similarly for :math:`q_f` (noting that currently \Delta q_f = {C_F^f}_{\mbox{\tiny \rm NTML}+2}\, q_f^+ - {C_F^f}_{\mbox{\tiny \rm NTML}}\, {q_f}_{\rm ct} - The only other explicit account of variable cloud fraction is in :eq:`vbr` for which it is assumed that buoyancy reversal can only occur for cloudy air underlying cloud-free air (assuming maximum @@ -6744,7 +6660,6 @@ where \beta_c = a_L \left( \frac{L}{c_p} \beta_T - \frac{1+c_v}{c_v} \beta_q \right) - Note that here :math:`\tilde{\beta_T}` and :math:`\tilde{\beta_q}` are strictly *in*-cloud parameters, while their definitions in boundary layer code prior to 8A were grid-box mean. Thus, here, any necessary @@ -6817,7 +6732,6 @@ specific or mixing ratio, respectively) as: \rho_{y0} = \rho_*/(1+(1/\epsilon)m_{vS}) - where, in each case, the surface humidity is taken as the surface saturated humidity over open sea but over land and ice surfaces this is likely to be inappropriate and so the driving level humidity is used From 00191b19e709c1b35d12664bc86e66886acc81fc Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Wed, 13 May 2026 14:04:46 +0100 Subject: [PATCH 075/116] Removed latex math \mbox and \tiny commands that aren't supported in mathjax. --- .../turbulence_schemes/bldoc.rst | 335 +++++++++--------- 1 file changed, 164 insertions(+), 171 deletions(-) diff --git a/documentation/source/science_guide/turbulence_schemes/bldoc.rst b/documentation/source/science_guide/turbulence_schemes/bldoc.rst index f915e6df0f..c6f6c97042 100644 --- a/documentation/source/science_guide/turbulence_schemes/bldoc.rst +++ b/documentation/source/science_guide/turbulence_schemes/bldoc.rst @@ -280,7 +280,7 @@ top at height :math:`z_{\rm h}` , as required for :eq:`kmsurf`) is determined from: #. a diagnostic moist parcel ascent; top at grid-level NTPAR, height - :math:`z_{\rm par}` :math:`=z_{\mbox{\tiny \rm NTPAR}+\frac{1}{2}}`. + :math:`z_{\rm par}` :math:`=z_{\mathrm{ \rm NTPAR}+\frac{1}{2}}`. Typically this is an adiabatic parcel but entraining options are available. @@ -307,7 +307,7 @@ taken from the previous timestep) and the grid-level above which :math:`\theta_{v\ell}` starts to increase with height. The ascent is stopped at the grid-level NTPAR (height -:math:`z_{\rm par}` :math:`=z_{\mbox{\tiny \rm NTPAR}+\frac{1}{2}}`) +:math:`z_{\rm par}` :math:`=z_{\mathrm{ \rm NTPAR}+\frac{1}{2}}`) above which the parcel becomes more negatively buoyant than a given threshold, :math:`\theta_v'`. Note that the parcel properties themselves are not perturbed in order to preserve the height of the mixed-layer’s @@ -318,8 +318,8 @@ Currently, .. math:: :label: parcel_pert - \theta_v' = \mbox{max} \left[A_{plume}, - \, \mbox{min} \left[ B_{plume} \sigma_{Tv1}, + \theta_v' = \mathrm{max} \left[A_{plume}, + \, \mathrm{min} \left[ B_{plume} \sigma_{Tv1}, \, G_{max}z_{\rm h}\right] \right] where :math:`A_{plume}=0.2`, :math:`B_{plume}=3.26`, @@ -348,7 +348,7 @@ where the vapour pressure of air in grid-level :math:`k_s`, :math:`e_{k_s} = q_{k_s} P_{k_s}/(100 \, \epsilon)`. The full-level below that containing the LCL is labelled NLCL and -:math:`z_{\rm lcl}` :math:`=z_{\mbox{\tiny \rm NLCL}+\frac{1}{2}}`. If +:math:`z_{\rm lcl}` :math:`=z_{\mathrm{ \rm NLCL}+\frac{1}{2}}`. If the parcel rises above the top of the LCL transition zone (defined as 1.1\ :math:`z_{\rm lcl}` , its ascent can also be stopped at the grid-level at which it has maximum buoyancy excess over the environment. @@ -406,7 +406,7 @@ section :ref:`Diagnosis of inversion thickness `). If the parcel ascent fails to find an inversion below 3km (or BL_LEVELS) but the LCL is below BL_LEVELS, then the layer is assumed to be cumulus-capped. If the LCL is above BL_LEVELS, then again cumulus is -diagnosed with NTML\ :math:`=\mbox{min}[`\ NLCL, BL_LEVELS\ :math:`-1]`, +diagnosed with NTML\ :math:`=\mathrm{min}[`\ NLCL, BL_LEVELS\ :math:`-1]`, in the hope that the mass-flux convection scheme (in its moist or dry mode) will transport the surface fluxes higher! Clearly this restriction on the boundary layer scheme is not desirable and so a value of @@ -435,7 +435,7 @@ the environment at that grid-level .. math:: :label: qlpar - q_{\ell f}^p = \mbox{max}\left[ 0.0, \, a_L \left( q_t^p - {q_s}_k + q_{\ell f}^p = \mathrm{max}\left[ 0.0, \, a_L \left( q_t^p - {q_s}_k - \alpha_L (\theta_{\ell}^p - (g z_k/c_p)-T_k)\right) \right] @@ -533,7 +533,7 @@ identified as well-mixed). **Step 2** is to diagnose an approximate depth of the DSC layer, :math:`z_{\rm ml}`. The bottom grid-level of the mixed-layer (NBDSC) is diagnosed as the lowest grid-level, descending from NTDSC, where -:math:`{\theta_{v\ell}}_{\mbox{\tiny \rm NTDSC}} + \theta_{v\ell}'` is +:math:`{\theta_{v\ell}}_{\mathrm{ \rm NTDSC}} + \theta_{v\ell}'` is less than :math:`\theta_{v\ell}` of the environment. The parcel perturbation is given by @@ -554,7 +554,7 @@ they are consistent with, for example, the observations of NBDSC equals NTDSC) in a DSC layer *not* overlying cumulus, then the layer is assumed not to be well-mixed. At the top of a cumulus layer, the DSC layer is given a minimum depth of -:math:`\Delta_{\mbox{\tiny \rm NTDSC}+\frac{1}{2}} z`. Otherwise, the +:math:`\Delta_{\mathrm{ \rm NTDSC}+\frac{1}{2}} z`. Otherwise, the layer depth, :math:`z_{\rm ml}`, is measured from the top of layer NTDSC to the base of layer NBDSC. @@ -687,9 +687,9 @@ namely that it should never go below :math:`0.1`\ :math:`z_{\rm h}` (to avoid affecting the continuity of the :math:`K` profiles at the top of the surface layer, see :eq:`ws_defn`). If cumulus convection has been diagnosed then :math:`z_{\rm b}` is not allowed to -go below :math:`z_{\mbox{\tiny \rm NTML}+\frac{1}{2}}` (unless the layer +go below :math:`z_{\mathrm{ \rm NTML}+\frac{1}{2}}` (unless the layer is diagnosed to recouple completely). Finally, :math:`z_{\rm b}` must -always be at or below :math:`z_{\mbox{\tiny \rm NTDSC}-1}`, so that +always be at or below :math:`z_{\mathrm{ \rm NTDSC}-1}`, so that mixing in decoupled layers is always resolved, and at least :math:`\Delta z_{rad}` (the cloud-top radiative cooling depth defined in section :ref:`Integration of \overline{w'b} close to the inversion @@ -735,9 +735,9 @@ inversion (in particular, in the LW radiative flux), simple finite difference flux calculations, :eq:`eq:wx_std`, can be significantly inaccurate in this region. An example is shown in :numref:`Fig. %s `. Calculating -:math:`\overline{w'\theta_{\ell}'}_{\mbox{\tiny \rm NTML}+\frac{1}{2}}` +:math:`\overline{w'\theta_{\ell}'}_{\mathrm{ \rm NTML}+\frac{1}{2}}` from :eq:`eq:wx_std` gives a negative value, largely -because :math:`\Delta_{\mbox{\tiny \rm NTML}+1} \theta_{\ell}` is +because :math:`\Delta_{\mathrm{ \rm NTML}+1} \theta_{\ell}` is positive and so the local flux is large and negative. In reality, :math:`\overline{w'\theta_{\ell}'}` becomes positive only a short distance below cloud-top such that the integral here will tend also to @@ -746,12 +746,12 @@ be positive. The solution adopted is to integrate :math:`\overline{w'b}` analytically across the region just below the inversion, labelled :math:`\Delta z_{rad}` in Fig, :numref:`%s `. Since -:math:`\Delta_{\mbox{\tiny \rm NTML}} \theta_{\ell}` can also be +:math:`\Delta_{\mathrm{ \rm NTML}} \theta_{\ell}` can also be significantly positive (when the grid-level inversion is rising or falling, for example), the base of this region is taken to be the lower of the first :math:`\theta`-level below :math:`z_h-100` m (a physically reasonable depth over which cloud-top radiative cooling might be -expected to occur) and :math:`z_{\mbox{\tiny \rm NTML}-1}`. +expected to occur) and :math:`z_{\mathrm{ \rm NTML}-1}`. .. figure:: blank.svg :name: fig:inv_integ @@ -863,7 +863,7 @@ the parcel buoyancy. Note that the constant in between grid-levels. Note that the standard definition of the boundary layer top in the UM is the height of the first flux level below the level of neutral buoyancy, so -:math:`z_{\rm par}` :math:`=z_{\mbox{\tiny \rm NTPAR}+\frac{1}{2}}`. The +:math:`z_{\rm par}` :math:`=z_{\mathrm{ \rm NTPAR}+\frac{1}{2}}`. The inversion thickness is then defined as .. math:: :label: dz_definition @@ -970,11 +970,11 @@ The asymptotic mixing lengths are given by .. math:: - \lambda_m =\mbox{max}\left[\lambda_0,\, 0.15 z_{\rm loc}, 2 h_B \right] + \lambda_m =\mathrm{max}\left[\lambda_0,\, 0.15 z_{\rm loc}, 2 h_B \right] .. math:: :label: asymp_ml - \lambda_h =\mbox{max}\left[\lambda_0,\, 0.15 z_{\rm loc}\right] + \lambda_h =\mathrm{max}\left[\lambda_0,\, 0.15 z_{\rm loc}\right] where :math:`\lambda_0` is a minimum mixing length read in from the namelist and :math:`z_{\rm loc}` is defined below. The orographic @@ -985,7 +985,7 @@ defined below), is given by \right] where :math:`\sigma_h` is the standard deviation of the height of the -subgrid orography and :math:`(z_{0m})_{\mbox{veg}}` is the vegetative +subgrid orography and :math:`(z_{0m})_{\mathrm{veg}}` is the vegetative part of the roughness length. The constants in :eq:`asymp_ml` can be considered ‘tuned’ (see, in particular, the operational modifications described in @@ -1307,7 +1307,7 @@ The general approach is to take :math:`K_{\chi}` in .. math:: :label: klnl - K_{\chi} = \mbox{max} \left[ (K_{\chi}^{\rm surf}+K_{\chi}^{\rm Sc}), + K_{\chi} = \mathrm{max} \left[ (K_{\chi}^{\rm surf}+K_{\chi}^{\rm Sc}), K_{\chi}(Ri) \right] As noted in section :ref:`Model variables and turbulence closure @@ -1377,7 +1377,7 @@ of the non-local surface-based mixing coefficient is reset to diagnosed from the parcel ascent). A diagnostic is calculated, ZHT -:math:`=\mbox{max}[z_{\rm h}, z_{\rm h}^{\rm Sc}, z_{\rm loc}]`, that +:math:`=\mathrm{max}[z_{\rm h}, z_{\rm h}^{\rm Sc}, z_{\rm loc}]`, that gives a measure of the maximum height of turbulent mixing (STASH 3,304). Recall, :math:`z_{\rm loc}` is the boundary layer depth diagnosed by :math:`Ri > @@ -1385,7 +1385,7 @@ Ri_{crit}`, :math:`z_{\rm h}^{\rm Sc}` is the top of any stratocumulus layer and :math:`z_{\rm h}` is the top of surface-based mixed layer, found by adiabatic parcel ascent but reset to the LCL in cumulus capped layers. Another diagnostic is available, the “boundary layer depth” -(STASH 25), that is set to :math:`=\mbox{max}[z_{\rm h}, z_{\rm loc}]` +(STASH 25), that is set to :math:`=\mathrm{max}[z_{\rm h}, z_{\rm loc}]` and so represents the depth of the stable boundary layer or “surface” mixed layer. Also available are three diagnostics that represent the calculated value of each of the individual terms in STASH 3,304: 3,356 @@ -1432,12 +1432,12 @@ the diagnosed subgrid inversion height (see section :ref:`Diagnosis of a sub-grid inversion `) for both :math:`K_h^{\rm surf}` and :math:`K_m^{\rm surf}`. In the 8A version, :math:`K_m^{\rm surf}` uses -:math:`z_{\rm h}` :math:`=z_{\mbox{\tiny \rm NTML}+\frac{1}{2}}`. The +:math:`z_{\rm h}` :math:`=z_{\mathrm{ \rm NTML}+\frac{1}{2}}`. The factor :math:`{\cal E}_m^{\rm surf}` is chosen so that :math:`K_m^{\rm surf}` will tend to -:math:`K_m|_{\mbox{\tiny \rm NTML}+\frac{1}{2}}` as :math:`z` tends to +:math:`K_m|_{\mathrm{ \rm NTML}+\frac{1}{2}}` as :math:`z` tends to :math:`z_{\rm h}` , where -:math:`K_m|_{\mbox{\tiny \rm NTML}+\frac{1}{2}}` is the entrainment +:math:`K_m|_{\mathrm{ \rm NTML}+\frac{1}{2}}` is the entrainment eddy-diffusivity (given by :eq:`khent`, although, in order to avoid altering the shape function too much, :math:`{\cal E}_m^{\rm surf}` is not allowed to fall below :math:`0.7`). @@ -1495,9 +1495,9 @@ except for :eq:`ws_defn` and For the latter, HB93 effectively set :math:`{\cal E}_m^{\rm surf} =1`. To generate entrainment, however, they simply use -:math:`K_m^{\rm surf}|_{\mbox{\tiny \rm NTML}+\frac{1}{2}}`, as +:math:`K_m^{\rm surf}|_{\mathrm{ \rm NTML}+\frac{1}{2}}`, as evaluated from :eq:`kmsurf` with a subgrid calculation of -:math:`z_{\rm h}` :math:`>z_{\mbox{\tiny \rm NTML}+\frac{1}{2}}`, rather +:math:`z_{\rm h}` :math:`>z_{\mathrm{ \rm NTML}+\frac{1}{2}}`, rather than using a separate entrainment parametrization. The difference in :eq:`ws_defn` arises from the surface @@ -1568,8 +1568,8 @@ extent of the K-profiles `), where :math:`V_{\rm Sc}^3= V_{\rm rad}^3+V_{\rm br}^3` (see appendix :ref:`Appendix: Definitions of the velocity scales `) and :math:`z'` is height above -:math:`z_{\rm b}` . Then :math:`K_h = K_m / \mbox{Pr}`, where -:math:`\mbox{Pr}=0.75`. The resulting :math:`K_h` profile was derived +:math:`z_{\rm b}` . Then :math:`K_h = K_m / \mathrm{Pr}`, where +:math:`\mathrm{Pr}=0.75`. The resulting :math:`K_h` profile was derived against convective cloudy LES, as described in `Lock (1999)`_. The appropriate Prandtl number (and therefore :math:`K_m^{\rm Sc}`) is unknown, 0.75 being chosen @@ -1579,14 +1579,14 @@ turbulent mixing in general. As with :eq:`kmsurf`, subgrid diagnosis (see section :ref:`Diagnosis of a sub-grid inversion `) except for :math:`K_m^{\rm Sc}` in the 8A scheme which uses the height of the -half-level below (:math:`z_{\mbox{\tiny \rm NTML}+\frac{1}{2}}` or -:math:`z_{\mbox{\tiny \rm NTDSC}+\frac{1}{2}}`). Again following +half-level below (:math:`z_{\mathrm{ \rm NTML}+\frac{1}{2}}` or +:math:`z_{\mathrm{ \rm NTDSC}+\frac{1}{2}}`). Again following :eq:`kmsurf`, the factors :math:`{\cal E}_m^{\rm Sc}` and :math:`{\cal E}_h^{\rm Sc}` are included in :eq:`kmtop` so that :math:`K_m^{\rm Sc}` will tend to -:math:`K_m|_{\mbox{\tiny \rm NTML}+\frac{1}{2}}` (and +:math:`K_m|_{\mathrm{ \rm NTML}+\frac{1}{2}}` (and :math:`K_h^{\rm Sc}` to -:math:`K_h|_{\mbox{\tiny \rm NTML}+\frac{1}{2}}`), given by +:math:`K_h|_{\mathrm{ \rm NTML}+\frac{1}{2}}`), given by :eq:`khent`, as :math:`z` tends to :math:`z_{\rm h}` (and here no restriction is made on the magnitude of either :math:`{\cal E}_m^{\rm Sc}` or :math:`{\cal E}_h^{\rm Sc}`). @@ -1608,7 +1608,7 @@ where .. math:: :label: gradadj \gamma_{\theta_{\ell}} = - \mbox{min}\left[ A_{ga} \frac{\sigma_{T1}}{z_{\rm h}}, G_{max} \right] + \mathrm{min}\left[ A_{ga} \frac{\sigma_{T1}}{z_{\rm h}}, G_{max} \right] :math:`A_{ga}=3.26`, :math:`G_{max}=10^{-3}`\ Km\ :math:`^{-1}` and :math:`\sigma_{T1} = 1.93 \, @@ -1804,6 +1804,7 @@ Discussion of some of the revisions :name: tab:vscales :header-rows: 2 + * - Formulation - Convective limit - @@ -2311,8 +2312,8 @@ mixed layers is simply calculated as .. math:: - F_{\rm net}|_{z_{k+\frac{1}{2}}} = \sum_{k=\mbox{\tiny \rm NBDSC}}^{k} - \mbox{max}\left[ + F_{\rm net}|_{z_{k+\frac{1}{2}}} = \sum_{k=\mathrm{ \rm NBDSC}}^{k} + \mathrm{max}\left[ - \Delta_{k+\frac{1}{2}} z \, {\cal S}_F(k), \,0 \right] where NBDSC\ :math:`=1` in SMLs, :math:`{\cal S}_F` are the temperature @@ -2327,29 +2328,29 @@ nominally at the subgrid inversion height (:math:`z_i=` :math:`z_{\rm h}` and/or :math:`z_{\rm h}^{\rm Sc}` ), diagnosed as described in section :ref:`Diagnosis of a sub-grid inversion `. The required grid-level -fluxes (at :math:`z_{\mbox{\tiny \rm NTDSC}+\frac{1}{2}}`, for example) +fluxes (at :math:`z_{\mathrm{ \rm NTDSC}+\frac{1}{2}}`, for example) are then estimated using linear interpolation of :math:`{\cal H}` and :math:`\overline{w'q_t'}` between :math:`z_{\rm h}^{\rm Sc}` and the base of the mixed layer: .. math:: - \overline{w'\theta_{\ell}'}|_{ z_{\mbox{\tiny \rm NTDSC}+\frac{1}{2}} } = + \overline{w'\theta_{\ell}'}|_{ z_{\mathrm{ \rm NTDSC}+\frac{1}{2}} } = \overline{w'\theta_{\ell}'}|_{z_{\rm b}} - - \frac{ z'_{\mbox{\tiny \rm NTDSC}+\frac{1}{2}} }{z_{\rm ml}} + - \frac{ z'_{\mathrm{ \rm NTDSC}+\frac{1}{2}} }{z_{\rm ml}} \left( \tilde{w_e} \Delta \theta_{\ell}+ \overline{w'\theta_{\ell}'}|_{z_{\rm b}} - F_{\rm net}|_{h} \right) - - F_{\rm net}|_{ z_{\mbox{\tiny \rm NTDSC}+\frac{1}{2}} } + - F_{\rm net}|_{ z_{\mathrm{ \rm NTDSC}+\frac{1}{2}} } .. math:: :label: fluxinterp - \overline{w'q_t'}|_{ z_{\mbox{\tiny \rm NTDSC}+\frac{1}{2}} } = + \overline{w'q_t'}|_{ z_{\mathrm{ \rm NTDSC}+\frac{1}{2}} } = \overline{w'q_t'}|_{z_{\rm b}} - - \frac{ z'_{\mbox{\tiny \rm NTDSC}+\frac{1}{2}} }{z_{\rm ml}} + - \frac{ z'_{\mathrm{ \rm NTDSC}+\frac{1}{2}} }{z_{\rm ml}} \left( \tilde{w_e} \Delta q_t + \overline{w'q_t'}|_{z_{\rm b}} \right) where :math:`z' = z-z_{\rm b}`, and similarly for the SML entrainment -fluxes (at :math:`z=z_{\mbox{\tiny \rm NTML}+\frac{1}{2}}`). The +fluxes (at :math:`z=z_{\mathrm{ \rm NTML}+\frac{1}{2}}`). The turbulent fluxes at the base of the mixed layer are assumed zero except for the SML where the surface fluxes are used. This interpolation is illustrated for a SML in :numref:`Fig. %s `. @@ -2387,21 +2388,21 @@ the parametrization of :math:`w_e` and the model’s subsidence velocity, :math:`w_S|_{z_i}`, are used to calculate :math:`z_i` at the next time-level (:math:`z_i^{n+1}`). Currently, the latter is found by linear interpolation to :math:`z_i` and both are assumed constant in time. If -:math:`z_i^{n+1} < z_{\mbox{\tiny \rm NTDSC}+\frac{1}{2}}`, then the +:math:`z_i^{n+1} < z_{\mathrm{ \rm NTDSC}+\frac{1}{2}}`, then the entrainment fluxes there (given by :eq:`fluxinterp`) are multiplied by the fraction of the timestep that :math:`z_i` was above this grid-level, namely -:math:`(z_i-z_{\mbox{\tiny \rm NTDSC}+\frac{1}{2}})/(z_i - z_i^{n+1})`. +:math:`(z_i-z_{\mathrm{ \rm NTDSC}+\frac{1}{2}})/(z_i - z_i^{n+1})`. The full entrainment flux at grid-level NTDSC\ :math:`-\frac{1}{2}` must then also be specified, given by :eq:`fluxinterp` with -:math:`z_{\mbox{\tiny \rm NTDSC}+ \frac{1}{2}}` replaced by -:math:`z_{\mbox{\tiny \rm NTDSC}- \frac{1}{2}}`. If :math:`z_i` rises -above :math:`z_{\mbox{\tiny \rm NTDSC}+\frac{3}{2}}`, the entrainment +:math:`z_{\mathrm{ \rm NTDSC}+ \frac{1}{2}}` replaced by +:math:`z_{\mathrm{ \rm NTDSC}- \frac{1}{2}}`. If :math:`z_i` rises +above :math:`z_{\mathrm{ \rm NTDSC}+\frac{3}{2}}`, the entrainment flux is specified only at this higher grid-level (multiplied by the fraction of the timestep that :math:`z_i` is above this half-level) and the values of the mixed-layer :math:`K` profiles are used in half-level NTDSC\ :math:`+\frac{1}{2}` (these will be non-zero because -:math:`z_i>z_{\mbox{\tiny \rm NTDSC}+ \frac{1}{2}}`). Wherever the +:math:`z_i>z_{\mathrm{ \rm NTDSC}+ \frac{1}{2}}`). Wherever the entrainment fluxes are specified explicitly, the eddy-diffusivities (both non-local and local) are set to zero. Also, the mean value of :math:`z_i` during the timestep is used in @@ -2437,8 +2438,8 @@ and .. math:: :label: we_num - \tilde{w_S} = - \, \frac{ \Theta^{\rm S}_{\mbox{\tiny \rm NTML}} - ( \Delta_{\mbox{\tiny \rm NTML}+\frac{1}{2}} z ) } + \tilde{w_S} = - \, \frac{ \Theta^{\rm S}_{\mathrm{ \rm NTML}} + ( \Delta_{\mathrm{ \rm NTML}+\frac{1}{2}} z ) } { \Delta \theta_{\ell}} .. _sec_sginv: @@ -2486,7 +2487,7 @@ for the model and for a profile with a discontinuous inversion at :math:`z_i` are equal, as illustrated by the hatched areas in :numref:`Fig. %s `. To calculate the integral of the discontinuous profile, the lapse rate of :math:`\theta_{v\ell}` between grid-levels -NTML\ :math:`-1` and :math:`NTML`, :math:`\gamma^{\tiny \rm ML}`, is +NTML\ :math:`-1` and :math:`NTML`, :math:`\gamma^{ \rm ML}`, is extended up to :math:`z_i`, while the stable lapse in the free atmosphere, between grid-levels NTML\ :math:`+2` and NTML\ :math:`+3`, :math:`\gamma^{\scriptsize \rm FA}`, is extrapolated down. Equating @@ -2501,26 +2502,25 @@ The coefficients are given by .. math:: - a = 0.5 (\gamma^{\scriptsize \rm FA}- \gamma^{\tiny \rm ML}) + a = 0.5 (\gamma^{\scriptsize \rm FA}- \gamma^{ \rm ML}) .. math:: - b = - \left( {\theta_{v\ell}}_{\mbox{\tiny \rm NTML}+2} - - \gamma^{\scriptsize \rm FA}(z_{\mbox{\tiny \rm NTML}+2}-z_{\mbox{\tiny - \rm NTML}+\frac{3}{2}}) \right) - + \left( {\theta_{v\ell}}_{\mbox{\tiny \rm NTML}} - + \gamma^{\tiny \rm ML}(z_{\mbox{\tiny \rm NTML}+\frac{3}{2}}-z_{\mbox{\tiny \rm NTML}}) \right) + b = - \left( {\theta_{v\ell}}_{\mathrm{ \rm NTML}+2} + - \gamma^{\scriptsize \rm FA}(z_{\mathrm{ \rm NTML}+2}-z_{\mathrm{ \rm + NTML}+\frac{3}{2}}) \right) + + \left( {\theta_{v\ell}}_{\mathrm{ \rm NTML}} + + \gamma^{ \rm ML}(z_{\mathrm{ \rm NTML}+\frac{3}{2}}-z_{\mathrm{ \rm NTML}}) \right) .. math:: - c = (z_{\mbox{\tiny \rm NTML}+\frac{3}{2}}-z_{\mbox{\tiny \rm - NTML}+\frac{1}{2}}) - \left( {\theta_{v\ell}}_{\mbox{\tiny \rm NTML}+1} - - \left( {\theta_{v\ell}}_{\mbox{\tiny \rm NTML}} + - \gamma^{\tiny \rm ML}\left( - \frac{1}{2}(z_{\mbox{\tiny \rm NTML}+\frac{1}{2}}+z_{\mbox{\tiny \rm + c = (z_{\mathrm{ \rm NTML}+\frac{3}{2}}-z_{\mathrm{ \rm NTML}+\frac{1}{2}}) + \left( {\theta_{v\ell}}_{\mathrm{ \rm NTML}+1} - + \left( {\theta_{v\ell}}_{\mathrm{ \rm NTML}} + + \gamma^{ \rm ML}\left( + \frac{1}{2}(z_{\mathrm{ \rm NTML}+\frac{1}{2}}+z_{\mathrm{ \rm NTML}+\frac{3}{2}}) - -z_{\mbox{\tiny \rm NTML}} \right) \right) + -z_{\mathrm{ \rm NTML}} \right) \right) \right) Clearly, care must be taken to ensure that :math:`z_i` is not only @@ -2531,11 +2531,11 @@ therefore set to zero and :eq:`zi_interp` is recalculated. The case :math:`c<0` suggests the grid-level designated as the inversion level should have been considered as part of the mixed layer and so :math:`z_i` is set to be fractionally below -:math:`z_{\mbox{\tiny \rm NTML}+\frac{3}{2}}` (i.e., as high as possible +:math:`z_{\mathrm{ \rm NTML}+\frac{3}{2}}` (i.e., as high as possible without attempting to diagnose a subgrid :math:`z_i` in grid-level NTML\ :math:`+2`). If :math:`b^2-4ac<0` the quadratic equation has no real roots. In this instance :math:`z_i` is set to fractionally below -:math:`z_{\mbox{\tiny \rm NTML}+\frac{1}{2}}` and NTML (and therefore +:math:`z_{\mathrm{ \rm NTML}+\frac{1}{2}}` and NTML (and therefore the eddy-diffusivity profiles) is lowered by a grid-level. In all other circumstances, the required root is then :math:`\Delta z_{disc} = (-b - (b^2-4ac)^{1/2} @@ -2545,12 +2545,12 @@ circumstances, the required root is then In addition, from variations seen in :math:`z_i` during single-column model simulations, the error in :math:`\Delta z_{disc}` is estimated to be around 10% of the vertical resolution, -:math:`\Delta_{\mbox{\tiny \rm NTML}+\frac{3}{2}} z`. Accordingly, if +:math:`\Delta_{\mathrm{ \rm NTML}+\frac{3}{2}} z`. Accordingly, if :math:`z_i` is diagnosed as being less than -:math:`z_{\mbox{\tiny \rm NTML}+\frac{1}{2}} + 0.1 \, \Delta_{\mbox{\tiny \rm +:math:`z_{\mathrm{ \rm NTML}+\frac{1}{2}} + 0.1 \, \Delta_{\mathrm{ \rm NTML}+\frac{3}{2}} z`, NTML is lowered a grid-level and :math:`z_i` is set fractionally below -:math:`z_{\mbox{\tiny \rm NTML}+\frac{1}{2}}`. This small distance below +:math:`z_{\mathrm{ \rm NTML}+\frac{1}{2}}`. This small distance below the grid-level is taken to be :math:`(\Delta t/2) \times 10^{-4}` so that, were a small rate of rise of :math:`z_i` (of :math:`10^{-4}` ms\ :math:`^{-1}`, say) to be diagnosed, then :math:`z_i` would spend at @@ -2566,20 +2566,20 @@ calculation are calculated from similar integral assumptions: .. math:: :label: dqt_disc - \Delta \chi = \left( {\chi}_{\mbox{\tiny \rm NTML}+1} - {\chi}_{\mbox{\tiny - \rm NTML}} \right) \, - \frac{ z_{\mbox{\tiny \rm NTML}+\frac{3}{2}} - z_{\mbox{\tiny \rm + \Delta \chi = \left( {\chi}_{\mathrm{ \rm NTML}+1} - {\chi}_{\mathrm{ \rm + NTML}} \right) \, + \frac{ z_{\mathrm{ \rm NTML}+\frac{3}{2}} - z_{\mathrm{ \rm NTML}+\frac{1}{2}} } - { z_{\mbox{\tiny \rm NTML}+\frac{3}{2}} - z_i } + { z_{\mathrm{ \rm NTML}+\frac{3}{2}} - z_i } with :math:`\chi = \theta_{\ell}` and :math:`q_t`. Note that the lapse rate above the inversion has been ignored as there is no guarantee of monotonicity in :math:`q_t` in the atmosphere above the inversion. In addition, :eq:`dqt_disc` will become increasingly inaccurate as :math:`z_i` tends to -:math:`z_{\mbox{\tiny \rm NTML}+\frac{3}{2}}` (and so -:math:`{\chi}_{\mbox{\tiny \rm NTML}+1}` approaches -:math:`{\chi}_{\mbox{\tiny \rm NTML}}`). Consequently, if the fraction +:math:`z_{\mathrm{ \rm NTML}+\frac{3}{2}}` (and so +:math:`{\chi}_{\mathrm{ \rm NTML}+1}` approaches +:math:`{\chi}_{\mathrm{ \rm NTML}}`). Consequently, if the fraction on the right hand side of :eq:`dqt_disc` is greater than 10, double grid-level jumps are used (i.e., :math:`\Delta \chi = {\chi}_{\mbox{\tiny \rm NTML}+2} - @@ -2639,7 +2639,7 @@ capped by a diagnosed subgrid inversion. The crucial step is to ensure that the total flux on the model entrainment grid-level equals the idealised total flux profile interpolated to that level. Consider the example illustrated in :numref:`Fig. %s ` of a well-mixed -boundary layer up to :math:`\theta`-level :math:`\mbox{\tiny \rm NTML}`. +boundary layer up to :math:`\theta`-level :math:`\mathrm{ \rm NTML}`. For the subgrid :math:`q_t` profiles, the turbulent flux divergence generates a moistening across the inversion while subsidence generates drying. For this example it has been assumed the entrainment rate is @@ -2648,14 +2648,14 @@ overall there is a weak moistening relative to the mixed layer (the total flux gradient is more negative across the inversion than in the mixed layer), consistent with the rising tendency of the inversion. For the model, the subsidence flux-divergence associated with the inversion -is split across levels :math:`\mbox{\tiny \rm NTML}` and -:math:`\mbox{\tiny \rm NTML}+1`. To keep the *net* moistening of the +is split across levels :math:`\mathrm{ \rm NTML}` and +:math:`\mathrm{ \rm NTML}+1`. To keep the *net* moistening of the model’s boundary layer and inversion consistent with the total subgrid flux profile, the model’s entrainment flux at -:math:`\mbox{\tiny \rm NTML}+1/2` (shown by the cross in +:math:`\mathrm{ \rm NTML}+1/2` (shown by the cross in :numref:`Fig. %s `) must be found by subtracting the -subsidence flux at :math:`\mbox{\tiny \rm NTML}+1/2` (diamond) from the -total flux interpolated to :math:`\mbox{\tiny \rm NTML}+1/2` (square). +subsidence flux at :math:`\mathrm{ \rm NTML}+1/2` (diamond) from the +total flux interpolated to :math:`\mathrm{ \rm NTML}+1/2` (square). Exactly the same arguments follow for the :math:`\theta_{\ell}` fluxes except that the situation is complicated by the addition of the radiative flux. @@ -2690,13 +2690,13 @@ cooling occurring within undulations of the cloudy boundary layer top. Similar considerations need to be borne in mind when calculating all the non-turbulent fluxes in :eq:`fxtot_zi`. First, the radiative flux is extrapolated down from -:math:`\mbox{\tiny \rm NTML}+\frac{3}{2}` to :math:`z=z_t` using the +:math:`\mathrm{ \rm NTML}+\frac{3}{2}` to :math:`z=z_t` using the divergence in the grid-level above the inversion as representative of the free-atmospheric divergence. Second, since the microphysical flux is generated within the cloud, :math:`F_{\chi}^{ppn}|_{z_t} = {F_{\chi}}^{ppn}_{\mbox{\tiny \rm NTML}+\frac{3}{2}}`. Finally, the -subsidence flux-divergence across level :math:`\mbox{\tiny \rm NTML}` -and :math:`\mbox{\tiny \rm NTML}+1` is assumed to be associated with the +subsidence flux-divergence across level :math:`\mathrm{ \rm NTML}` +and :math:`\mathrm{ \rm NTML}+1` is assumed to be associated with the inversion so :math:`{F_{\chi}}^{Subs}|_{z_h} = {F_{\chi}}^{Subs}_{\mbox{\tiny \rm NTML}-\frac{1}{2}}`. Thus, the finite-difference form of :eq:`fxtot_zi` becomes @@ -2704,9 +2704,8 @@ finite-difference form of :eq:`fxtot_zi` becomes .. math:: :label: fxtot_zi_fd F_{\chi}^{Tot}|_{z_h} = - w_e \Delta \chi + - F_{\chi}^{rad}|_{z_t} + {F_{\chi}}^{ppn}_{\mbox{\tiny \rm NTML}+\frac{3}{2}} - + - {F_{\chi}}^{subs}_{\mbox{\tiny \rm NTML}-\frac{1}{2}} + F_{\chi}^{rad}|_{z_t} + {F_{\chi}}^{ppn}_{\mathrm{ \rm NTML}+\frac{3}{2}} + + {F_{\chi}}^{subs}_{\mathrm{ \rm NTML}-\frac{1}{2}} Then, assuming a linear profile of :math:`F_{\chi}^{Tot}` in the mixed layer, interpolating the total flux to the inversion flux grid-level @@ -2714,9 +2713,9 @@ gives .. math:: :label: fxtot_interp - F_{\chi}^{Tot}|_{ \mbox{\tiny \rm NTML}+\frac{1}{2} } = - F_{\chi}^{Tot}|_{z_{\rm b}} + - \frac{ z'_{\mbox{\tiny \rm NTML}+\frac{1}{2}} }{z_{\rm ml}} + F_{\chi}^{Tot}|_{ \mathrm{ \rm NTML}+\frac{1}{2} } = F_{\chi}^{Tot}|_{z_{\rm + b}} + + \frac{ z'_{\mathrm{ \rm NTML}+\frac{1}{2}} }{z_{\rm ml}} \left( F_{\chi}^{Tot}|_{z_h} - F_{\chi}^{Tot}|_{z_{\rm b}} \right) where :math:`z'` (:math:`=z-z_{\rm b}`) is height above the base of the @@ -2725,9 +2724,9 @@ entrainment flux is given by: .. math:: :label: rev_entflux - \overline{w'\chi'}|_{ \mbox{\tiny \rm NTML}+\frac{1}{2} } = - F_{\chi}^{Tot}|_{ \mbox{\tiny \rm NTML}+\frac{1}{2} } - - F_{\chi}^{NT}|_{ \mbox{\tiny \rm NTML}+\frac{1}{2} } + \overline{w'\chi'}|_{ \mathrm{ \rm NTML}+\frac{1}{2} } = F_{\chi}^{Tot}|_{ + \mathrm{ \rm NTML}+\frac{1}{2} } + - F_{\chi}^{NT}|_{ \mathrm{ \rm NTML}+\frac{1}{2} } This revised algorithm has several advantages over the previous. Firstly, the fluxes for :math:`q_t` and :math:`\theta_{\ell}` are @@ -2737,7 +2736,7 @@ in order to calculate :math:`\tilde{w_e}` in :eq:`we_num`. Secondly, this method makes it much simpler to include all processes, and precipitation in particular, in a consistent manner. Thirdly, since the total grid-level flux, -:math:`F_{\chi}^{Tot}|_{\mbox{\tiny \rm NTML}+\frac{1}{2}}` in +:math:`F_{\chi}^{Tot}|_{\mathrm{ \rm NTML}+\frac{1}{2}}` in :eq:`fxtot_interp`, is used to calculate the entrainment fluxes, it is straightforward to ensure that the net budget of the inversion grid-level, namely :math:`- ( @@ -2746,7 +2745,7 @@ F_{\chi}^{Tot}|_{\mbox{\tiny \rm NTML}+\frac{1}{2}})/\Delta z`, is consistent with the entrainment/subsidence balance. In other words, to use :math:`\theta_{\ell}` as an example, if the inversion is rising (falling) then -:math:`F_{\chi}^{Tot}|_{\mbox{\tiny \rm NTML}+\frac{1}{2}}` is limited +:math:`F_{\chi}^{Tot}|_{\mathrm{ \rm NTML}+\frac{1}{2}}` is limited to ensure that the inversion grid-level will cool (warm). Finally, if the inversion is rising we don’t want the inversion grid-level :math:`\theta_{\ell}` to cool to less than :math:`\theta_{\ell}` of the @@ -2755,34 +2754,32 @@ mixed layer by the end of the timestep. In other words, for .. math:: - \chi_{\mbox{\tiny \rm NTML}+1}^{n+1} = \chi_{\mbox{\tiny \rm NTML}+1}^{n} + \chi_{\mathrm{ \rm NTML}+1}^{n+1} = \chi_{\mathrm{ \rm NTML}+1}^{n} - \frac{\Delta t}{\Delta z} \left( - F_{\chi}^{Tot}|_{ \mbox{\tiny \rm NTML}+\frac{3}{2} } - F_{\chi}^{Tot}|_{ - \mbox{\tiny \rm NTML}+\frac{1}{2} } + F_{\chi}^{Tot}|_{ \mathrm{ \rm NTML}+\frac{3}{2} } - F_{\chi}^{Tot}|_{ + \mathrm{ \rm NTML}+\frac{1}{2} } \right) .. math:: - \chi_{\mbox{\tiny \rm NTML}}^{n+1} = \chi_{\mbox{\tiny \rm NTML}}^{n} - - \frac{\Delta t}{z_{\mbox{\tiny \rm NTML}+\frac{1}{2}}} \left( - F_{\chi}^{Tot}|_{ \mbox{\tiny \rm NTML}+\frac{1}{2} } - + \chi_{\mathrm{ \rm NTML}}^{n+1} = \chi_{\mathrm{ \rm NTML}}^{n} + - \frac{\Delta t}{z_{\mathrm{ \rm NTML}+\frac{1}{2}}} \left( + F_{\chi}^{Tot}|_{ \mathrm{ \rm NTML}+\frac{1}{2} } - F_{\chi}^{Tot}|_{z_{\rm b}} \right) where the superscripts :math:`n` and :math:`n+1` refer to the model timestep, although strictly speaking :math:`n+1` refers to fields after the boundary layer implicit solver. Requiring that -:math:`\chi_{\mbox{\tiny \rm NTML}+1}^{n+1}\geq\chi_{\mbox{\tiny \rm -NTML}}^{n+1}` +:math:`\chi_{\mathrm{ \rm NTML}+1}^{n+1}\geq\chi_{\mathrm{ \rm NTML}}^{n+1}` implies .. math:: - F_{\chi}^{Tot}|_{ \mbox{\tiny \rm NTML}+\frac{1}{2} } + F_{\chi}^{Tot}|_{ \mathrm{ \rm NTML}+\frac{1}{2} } \left( 1+ \frac{\Delta z}{z_{ml}}\right) - \geq F_{\chi}^{Tot}|_{ \mbox{\tiny \rm NTML}+\frac{3}{2} } + \Delta z - \left( - \frac{\chi_{\mbox{\tiny \rm NTML}}^{n}-\chi_{\mbox{\tiny \rm + \geq F_{\chi}^{Tot}|_{ \mathrm{ \rm NTML}+\frac{3}{2} } + \Delta z \left( + \frac{\chi_{\mathrm{ \rm NTML}}^{n}-\chi_{\mathrm{ \rm NTML}+1}^{n}}{\Delta t} + \frac{F_{\chi}^{Tot}|_{z_{\rm b}}}{z_{ml}} \right) @@ -2801,15 +2798,15 @@ In the 9B scheme, the discontinuous jumps in :math:`\theta_{\ell}` and from integral assumptions similar to those used to diagnose the subgrid inversion height, :math:`z_h`, and were given by :eq:`dqt_disc`. Note that -:math:`{\chi}_{\mbox{\tiny \rm NTML}+2}` does not appear in +:math:`{\chi}_{\mathrm{ \rm NTML}+2}` does not appear in :eq:`dqt_disc` and so no direct information from the free atmosphere is used. Only if the budgets of :math:`\theta_{\ell}` and -:math:`q_t` in level :math:`\mbox{\tiny \rm NTML}+1` are entirely +:math:`q_t` in level :math:`\mathrm{ \rm NTML}+1` are entirely consistent with the rise and fall of the subgrid inversion will :eq:`dqt_disc` give accurate results. This will not be the case during an assimilation cycle, for example, neither is it likely to be the case if the convection scheme is detraining into level -:math:`\mbox{\tiny \rm NTML}+1`. +:math:`\mathrm{ \rm NTML}+1`. Instead, a more robust algorithm is used in the 9C scheme and the subgrid inversion calculation is only attempted where both @@ -2819,10 +2816,8 @@ formula used is: .. math:: :label: dqt_disc_9c - \Delta \chi = {\chi}_{\mbox{\tiny \rm NTML}+2} - {\chi}_{\mbox{\tiny \rm - NTML}} - - \gamma_{\chi} \left( z_{\mbox{\tiny \rm NTML}+2} - z_h - \right) + \Delta \chi = {\chi}_{\mathrm{ \rm NTML}+2} - {\chi}_{\mathrm{ \rm NTML}} + - \gamma_{\chi} \left( z_{\mathrm{ \rm NTML}+2} - z_h \right) subject to the constraint that the lapse rate adjustment should not reduce the two grid-length difference by more than half. The @@ -2831,16 +2826,16 @@ free-atmospheric lapse rates are given by .. math:: \gamma_{\theta_{\ell}} = {\rm max}\left[ \, 0, \, \frac{ - {\theta_{\ell}}_{\mbox{\tiny \rm NTML}+3}-{\theta_{\ell}}_{\mbox{\tiny \rm + {\theta_{\ell}}_{\mathrm{ \rm NTML}+3}-{\theta_{\ell}}_{\mathrm{ \rm NTML}+2} } - { z_{\mbox{\tiny \rm NTML}+3} - z_{\mbox{\tiny \rm NTML}+2} } + { z_{\mathrm{ \rm NTML}+3} - z_{\mathrm{ \rm NTML}+2} } \right] .. math:: - \gamma_{q_t} = {\rm min}\left[ \, 0, \, \frac{ {q_t}_{\mbox{\tiny \rm - NTML}+3}-{q_t}_{\mbox{\tiny \rm NTML}+2} } - { z_{\mbox{\tiny \rm NTML}+3} - z_{\mbox{\tiny \rm NTML}+2} } + \gamma_{q_t} = {\rm min}\left[ \, 0, \, \frac{ {q_t}_{\mathrm{ \rm + NTML}+3}-{q_t}_{\mathrm{ \rm NTML}+2} } + { z_{\mathrm{ \rm NTML}+3} - z_{\mathrm{ \rm NTML}+2} } \right] .. _sec_subs_calc: @@ -2896,12 +2891,11 @@ specified through an eddy diffusivity which is given by .. math:: - K_h|_{\mbox{\tiny \rm NTML}+\frac{1}{2}} = w_e \Delta_{\mbox{\tiny \rm - NTML}+1} z + K_h|_{\mathrm{ \rm NTML}+\frac{1}{2}} = w_e \Delta_{\mathrm{ \rm NTML}+1} z .. math:: :label: khent - K_m|_{\mbox{\tiny \rm NTML}} = Pr \, w_e \Delta_{\mbox{\tiny \rm + K_m|_{\mathrm{ \rm NTML}} = Pr \, w_e \Delta_{\mathrm{ \rm NTML}+\frac{1}{2}} z noting the Charney-Philips grid implying stresses are staggered from @@ -2911,8 +2905,8 @@ for the non-local :math:`K` profiles, see section :ref:`The non-local scheme Substituting :eq:`khent` in :eq:`scal_closure` gives, for example, -:math:`\overline{w'\theta_{\ell}'}|_{\mbox{\tiny \rm NTML}+\frac{1}{2}} = - w_e -\Delta_{\mbox{\tiny \rm NTML}+1} \theta_{\ell}`. +:math:`\overline{w'\theta_{\ell}'}|_{\mathrm{ \rm NTML}+\frac{1}{2}} = - w_e +\Delta_{\mathrm{ \rm NTML}+1} \theta_{\ell}`. Note that this gives entrainment buoyancy fluxes identical to :eq:`discinv` as long as there is no buoyancy reversal generation of turbulence (i.e.,\ :math:`V_{\rm br}=0`) and if variations @@ -2934,8 +2928,8 @@ explicitly and so :eq:`khent` is always used. For the 9C version, the entrainment :math:`K_m` given by :eq:`khent` is imposed at the height of the temperature inversion :math:`z_{\rm h}` (either subgrid or at -:math:`z_{\mbox{\tiny \rm NTML}+\frac{1}{2}}`) and -:math:`K_m|_{\mbox{\tiny \rm NTML}+\frac{1}{2}}` is calculated from +:math:`z_{\mathrm{ \rm NTML}+\frac{1}{2}}`) and +:math:`K_m|_{\mathrm{ \rm NTML}+\frac{1}{2}}` is calculated from :eq:`kmsurf` and :eq:`kmtop`, noting the use of the :math:`{\cal E}` factors. @@ -2955,7 +2949,7 @@ When this happens, there is no subgrid inversion diagnosis and the entrainment parametrization follows the methodology given in section :ref:`For momentum (and scalars if no subgrid inversion) ` to give -:math:`K_h|_{\mbox{\tiny \rm NTML}+\frac{1}{2}}`. The diffusion +:math:`K_h|_{\mathrm{ \rm NTML}+\frac{1}{2}}`. The diffusion coefficient profile within the inversion is then calculated assuming the :math:`\theta_{v\ell}` flux profile within the inversion decreases following a cosine shape from the standard parametrized entrainment flux @@ -2963,8 +2957,8 @@ at the inversion base to zero at the inversion top, i.e.: .. math:: :label: ent_svl - \overline{w'\theta_{v\ell}'} = \overline{w'\theta_{v\ell}'}|_{\mbox{\tiny - \rm NTML}+\frac{1}{2}} + \overline{w'\theta_{v\ell}'} = \overline{w'\theta_{v\ell}'}|_{\mathrm{ \rm + NTML}+\frac{1}{2}} cos\left(\pi \frac{z'}{2} \right) where :math:`z'=(z-z_{\rm h})/\Delta z_i` is scaled height within the @@ -3007,20 +3001,20 @@ equivalent entrainment eddy-diffusivity given by: .. math:: :label: K_ent_tracer - K_{\chi}|_{\mbox{\tiny \rm NTML}+\frac{1}{2}} = - \overline{w'\chi'}_{ - z_{\mbox{\tiny \rm NTML}+\frac{1}{2}} } - \frac{\Delta_{\mbox{\tiny \rm NTML}+1} - z}{\Delta_{\mbox{\tiny \rm NTML}+1} \chi} + K_{\chi}|_{\mathrm{ \rm NTML}+\frac{1}{2}} = - \overline{w'\chi'}_{ + z_{\mathrm{ \rm NTML}+\frac{1}{2}} } + \frac{\Delta_{\mathrm{ \rm NTML}+1} + z}{\Delta_{\mathrm{ \rm NTML}+1} \chi} Note from :eq:`scal_closure` that :eq:`K_ent_tracer` gives the parametrized flux if -:math:`\Delta_{\mbox{\tiny \rm NTML}+1} \chi` does not change across the +:math:`\Delta_{\mathrm{ \rm NTML}+1} \chi` does not change across the timestep (see section :ref:`Implicit solution of the diffusion equation ` for a description of the implicit numerical solution of :eq:`cons_eqn_scal`). As :eq:`K_ent_tracer` involves the potentially numerically dangerous calculation of -:math:`\Delta \chi/\Delta_{\mbox{\tiny \rm NTML}+1} \chi` (where +:math:`\Delta \chi/\Delta_{\mathrm{ \rm NTML}+1} \chi` (where :math:`\Delta \chi` is the subgrid inversion jump, given by :eq:`dqt_disc`), the following constraints are also @@ -3028,8 +3022,8 @@ ensured: .. math:: - 0 \leq K_{\chi}|_{\mbox{\tiny \rm NTML}+\frac{1}{2}} - \leq 10 \,K_{\chi}|_{\mbox{\tiny \rm NTML}-\frac{1}{2}} + 0 \leq K_{\chi}|_{\mathrm{ \rm NTML}+\frac{1}{2}} + \leq 10 \,K_{\chi}|_{\mathrm{ \rm NTML}-\frac{1}{2}} Surface Exchange ================ @@ -3877,10 +3871,10 @@ gases. Explicitly, we have .. math:: - \dot T_{ob, \mbox{\tiny rad, surf}} = \frac{4\sigma T_s^3}{c_P} + \dot T_{ob, \mathrm{ rad, surf}} = \frac{4\sigma T_s^3}{c_P} {\cal K}(z_{ob}) (T_s-T_{ob}), -where :math:`\dot T_{ob, \mbox{\tiny rad, surf}}` is the cooling rate of +where :math:`\dot T_{ob, \mathrm{ rad, surf}}` is the cooling rate of the air at the height of observation due to direct radiative exchanges with the surface, :math:`T_{ob}` is the air temperature at that height, :math:`T_s` is the temperature of the surface and @@ -3917,24 +3911,24 @@ is initialized using standard theory. On subsequent timesteps, it is updated to allow for radiative cooling to the surface, giving a provisional value :math:`\theta_{ob}'`, and then relaxed back towards the result that would be obtained from standard similarity theory, -:math:`\theta_{ob, \mbox{\tiny sim}}`, as in the last section: +:math:`\theta_{ob, \mathrm{ sim}}`, as in the last section: .. math:: \theta_{ob}'(t+\delta t) \leftarrow \theta_{ob}(t)+ - \delta t \, \dot T_{ob,\mbox{\tiny rad,surf}} + \delta t \, \dot T_{ob,\mathrm{ rad,surf}} .. math:: \theta_{ob}(t+\delta t) \leftarrow W \theta_{ob}'(t+\delta t) - +(1-W) \theta_{ob, \mbox{\tiny sim}}. + +(1-W) \theta_{ob, \mathrm{ sim}}. By tuning against an idealized highly vertically resolved model based on local scaling we set, .. math:: - W = \exp(-(0.4 f)^2 \delta t \, t_{\mbox{\tiny trans}})) / + W = \exp(-(0.4 f)^2 \delta t \, t_{\mathrm{ trans}})) / (1+X({\cal L})\delta t), with @@ -3954,8 +3948,8 @@ layer generates turbulence. This factor is somewhat exaggerated relative to the results of the model against which it is tuned, as a cautionary measure to ensure that decoupling is not allowed to persist too long after the transition. This is the purpose of the inclusion of the factor -:math:`0.4 f t_{\mbox{\tiny trans}}`, where -:math:`t_{\mbox{\tiny trans}}` is the time since the transition. +:math:`0.4 f t_{\mathrm{ trans}}`, where +:math:`t_{\mathrm{ trans}}` is the time since the transition. It must be stressed that these schemes are heuristic and that the precise behaviour in weak turbulence is not fully understood. The second @@ -6212,7 +6206,7 @@ written .. math:: :label: vbr - V_{\rm br}^3= A_{\rm br}\chi_s^2 \, \mbox{max}\left[0,-\delta b\right] \, + V_{\rm br}^3= A_{\rm br}\chi_s^2 \, \mathrm{max}\left[0,-\delta b\right] \, \Delta b ^{1/2} \, z_c^{3/2} \, C_{fac} @@ -6258,8 +6252,8 @@ calculated as \tilde{z_c} = \sum_{k=1}^{NTML+1} \left( {C_F}_k \frac{\Delta_{k+\frac{1}{2}} z}{2} - + \mbox{min}\left[C_F^l\frac{\Delta_{k+\frac{1}{2}} z}{2}, \frac{q_{\ell}}{\gamma_{q_{\ell}}} \right] - + \mbox{min}\left[C_F^f\frac{\Delta_{k+\frac{1}{2}} z}{2}, \frac{q_f}{\gamma_{q_f}} \right] + + \mathrm{min}\left[C_F^l\frac{\Delta_{k+\frac{1}{2}} z}{2}, \frac{q_{\ell}}{\gamma_{q_{\ell}}} \right] + + \mathrm{min}\left[C_F^f\frac{\Delta_{k+\frac{1}{2}} z}{2}, \frac{q_f}{\gamma_{q_f}} \right] \right) where :math:`C_F` is the cloud fraction, made up of liquid @@ -6298,14 +6292,14 @@ grid-level based calculation: .. math:: z_c = z_c + \frac{\Delta_{k_b+\frac{1}{2}} z}{2} - + \mbox{min}\left[ \frac{ \Delta_{k_b+\frac{1}{2}}z + + \mathrm{min}\left[ \frac{ \Delta_{k_b+\frac{1}{2}}z +\Delta_{k_b-\frac{1}{2}}z }{2} C_F^l, \frac{ q_{\ell}}{ \gamma_{q_{\ell}} } \right]/C_F .. math:: :label: zc_calc \left. \hspace{2.4cm} - + \mbox{min}\left[ \frac{ \Delta_{k_b+\frac{1}{2}}z + + \mathrm{min}\left[ \frac{ \Delta_{k_b+\frac{1}{2}}z +\Delta_{k_b-\frac{1}{2}}z }{2} C_F^f, \frac{ q_f }{ \gamma_{q_f} } \right]/C_F \right. @@ -6361,37 +6355,36 @@ The empirical constant :math:`A_{\rm br}= 0.24`. The calculation of \theta_{\ell}` and :math:`\Delta q_t` is described for a subgrid inversion in section :ref:`Diagnosis of a sub-grid inversion ` or, if one is not diagnosed, -they are taken simply as :math:`\Delta_{\mbox{\tiny \rm NTML}+1}`. For +they are taken simply as :math:`\Delta_{\mathrm{ \rm NTML}+1}`. For :math:`\Delta q_{\ell}`, :math:`\Delta q_f` and :math:`{q_{\ell}}_{\rm ct}`, in-cloud values extrapolated to :math:`z_i` -(either subgrid or :math:`z_{\mbox{\tiny \rm NTML}+\frac{1}{2}}`) from +(either subgrid or :math:`z_{\mathrm{ \rm NTML}+\frac{1}{2}}`) from above and below using the adiabatic lapse rates are calculated as: .. math:: - {q_{\ell}}_{\rm ct} = \frac{{q_{\ell}}_{\mbox{\tiny \rm - NTML}}}{{C_F}^l_{\mbox{\tiny \rm NTML}}} - + (z_i-z_{\mbox{\tiny \rm NTML}}) \gamma_{q_{\ell}} + {q_{\ell}}_{\rm ct} = \frac{{q_{\ell}}_{\mathrm{ \rm + NTML}}}{{C_F}^l_{\mathrm{ \rm NTML}}} + + (z_i-z_{\mathrm{ \rm NTML}}) \gamma_{q_{\ell}} .. math:: - q_{\ell}^+ = \mbox{max}\left[ 0, \, - \frac{{q_{\ell}}_{\mbox{\tiny \rm NTML}+2}}{{C_F}^l_{\mbox{\tiny \rm - NTML}+2}} - - (z_{\mbox{\tiny \rm NTML}+2}-z_i) \gamma_{q_{\ell}} \right] + q_{\ell}^+ = \mathrm{max}\left[ 0, \, + \frac{{q_{\ell}}_{\mathrm{ \rm NTML}+2}}{{C_F}^l_{\mathrm{ \rm NTML}+2}} + - (z_{\mathrm{ \rm NTML}+2}-z_i) \gamma_{q_{\ell}} \right] and similarly for :math:`q_f` (noting that currently :math:`\gamma_{q_f}=0`) and for DSC layers. Then, .. math:: - \Delta q_{\ell} = {C_F^l}_{\mbox{\tiny \rm NTML}+2}\, q_{\ell}^+ - - {C_F^l}_{\mbox{\tiny \rm NTML}}\, {q_{\ell}}_{\rm ct} + \Delta q_{\ell} = {C_F^l}_{\mathrm{ \rm NTML}+2}\, q_{\ell}^+ - + {C_F^l}_{\mathrm{ \rm NTML}}\, {q_{\ell}}_{\rm ct} .. math:: - \Delta q_f = {C_F^f}_{\mbox{\tiny \rm NTML}+2}\, q_f^+ - - {C_F^f}_{\mbox{\tiny \rm NTML}}\, {q_f}_{\rm ct} + \Delta q_f = {C_F^f}_{\mathrm{ \rm NTML}+2}\, q_f^+ - {C_F^f}_{\mathrm{ \rm + NTML}}\, {q_f}_{\rm ct} The only other explicit account of variable cloud fraction is in :eq:`vbr` for which it is assumed that buoyancy reversal can @@ -6400,7 +6393,7 @@ overlap). Thus, the cloud fraction factor, :math:`C_{fac} = \mbox{max}[ 0.0, -\Delta C_F ]`, where :math:`\Delta C_F = {C_F}_{\mbox{\tiny \rm NTML}+2} - {C_F}_{\mbox{\tiny \rm NTML}}` if a subgrid inversion is diagnosed -(because :math:`{C_F}_{\mbox{\tiny \rm NTML}+1}` is currently +(because :math:`{C_F}_{\mathrm{ \rm NTML}+1}` is currently meaningless) and :math:`\Delta C_F = {C_F}_{\mbox{\tiny \rm NTML}+1} - {C_F}_{\mbox{\tiny \rm NTML}}` if not. A more complete decomposition is not possible given a cloud scheme in @@ -6419,8 +6412,8 @@ is continuous. An additional explicit dependence of entrainment on cloud fraction was implemented in version 4.5 which reduced the cloud-top source terms of entrainment in partially cloudy boundary layers by a factor -:math:`\exp{\left\{-(0.9-{C_F}_{\mbox{\tiny \rm NTML}})^3/0.075\right\}}` -for :math:`{C_F}_{\mbox{\tiny \rm NTML}} < 0.9`. It was argued that +:math:`\exp{\left\{-(0.9-{C_F}_{\mathrm{ \rm NTML}})^3/0.075\right\}}` +for :math:`{C_F}_{\mathrm{ \rm NTML}} < 0.9`. It was argued that partial cloudiness on the scale of the mixed-layer eddies might reduce the entrainment efficiency of the cloud-top processes. By reducing the parametrized entrainment warming and drying in partially cloudy boundary @@ -6451,7 +6444,7 @@ In the 9B version, :math:`\Delta_F` is calculated as: .. math:: :label: ctraddiv - \Delta_F= \sum_{k=k_m-1}^{k_m+1} \mbox{max}\left[ + \Delta_F= \sum_{k=k_m-1}^{k_m+1} \mathrm{max}\left[ - \Delta_{k+\frac{1}{2}} z \, {\cal S}_F(k), \,0 \right] where :math:`k_m` is the grid-level with the greatest radiative cooling @@ -6471,12 +6464,12 @@ have an exponential shape, dependent on the LWP above :math:`z`, i.e.: .. math:: :label: eq:explw - F_{LW}(z) = \Delta_F^{LW} \exp^{ - \kappa_{LW} \mbox{LWP}(z) } + F_{LW}(z) = \Delta_F^{LW} \exp^{ - \kappa_{LW} \mathrm{LWP}(z) } Then the net divergence can be approximated given the SW flux at the height where :math:`F_{LW}` becomes some small fraction, :math:`A`, of :math:`\Delta_F^{LW}` (which implies -:math:`\mbox{LWP} =-ln(A) / \kappa_{LW}`). Then +:math:`\mathrm{LWP} =-ln(A) / \kappa_{LW}`). Then .. math:: From 7ed49f046c8a15a67934df44b868d3041a8f0452 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Wed, 13 May 2026 14:24:32 +0100 Subject: [PATCH 076/116] Replaced \rm with \mathrm{}. --- .../turbulence_schemes/bldoc.rst | 976 +++++++++--------- 1 file changed, 507 insertions(+), 469 deletions(-) diff --git a/documentation/source/science_guide/turbulence_schemes/bldoc.rst b/documentation/source/science_guide/turbulence_schemes/bldoc.rst index c6f6c97042..6973c32f6d 100644 --- a/documentation/source/science_guide/turbulence_schemes/bldoc.rst +++ b/documentation/source/science_guide/turbulence_schemes/bldoc.rst @@ -127,8 +127,8 @@ standard closures are: .. math:: :label: scal_closure - \overline{w'\chi'} = - K_h \frac{\partial \chi}{\partial z} + K_h^{\rm - surf}\gamma_{\chi} + \overline{w'\chi'} = - K_h \frac{\partial \chi}{\partial z} + + K_h^{\mathrm{surf}}\gamma_{\chi} .. math:: :label: uv_closure @@ -138,7 +138,7 @@ Separate eddy-diffusivities are calculated for momentum, :math:`K_m`, and for scalar variables, :math:`K_h`. The second term on the right hand side represents a non-local flux in unstable boundary layers. Currently it is only applied for transport arising from surface-driven turbulence -(:math:`K_h^{\rm surf}`) and is non-zero only for +(:math:`K_h^{\mathrm{surf}}`) and is non-zero only for :math:`\chi=\theta_{\ell}`, as described in section :ref:`Gradient adjustment `. @@ -276,16 +276,16 @@ The diagnostic parcel ascent and cumulus diagnosis **Summary**: the depth of the non-local :math:`K`-profiles for surface-driven turbulence (with NTML grid-levels in the mixed layer and -top at height :math:`z_{\rm h}` , as required for +top at height :math:`z_{\mathrm{h}}` , as required for :eq:`kmsurf`) is determined from: #. a diagnostic moist parcel ascent; top at grid-level NTPAR, height - :math:`z_{\rm par}` :math:`=z_{\mathrm{ \rm NTPAR}+\frac{1}{2}}`. + :math:`z_{\mathrm{par}}` :math:`=z_{\mathrm{ \mathrm{NTPAR}}+\frac{1}{2}}`. Typically this is an adiabatic parcel but entraining options are available. #. a diagnosis of cumulus-capped layers (if cumulus-capped then NTML and - :math:`z_{\rm h}` are set to the LCL [2]_, if not then to the parcel + :math:`z_{\mathrm{h}}` are set to the LCL [2]_, if not then to the parcel top) Note that this process is only performed for unstable boundary layers @@ -297,17 +297,17 @@ driven by surface processes can extend in unstable boundary layers (and therefore the vertical extent of the :math:`K` profile for surface-driven turbulence) can be determined solely from the properties of the thermodynamic profiles. In more detail, the first step in -calculating :math:`z_{\rm h}` is to lift a parcel, with properties from +calculating :math:`z_{\mathrm{h}}` is to lift a parcel, with properties from the first grid-level (:math:`k=k_s`) above the top of the surface layer, upwards allowing for latent heat release. The top of the surface layer -is taken to be at the lower of :math:`z=0.1`\ :math:`z_{\rm h}` (this is +is taken to be at the lower of :math:`z=0.1`\ :math:`z_{\mathrm{h}}` (this is then consistent with the :math:`K`-profiles, see -section :ref:`Surface-driven turbulence `; :math:`z_{\rm h}` is -taken from the +section :ref:`Surface-driven turbulence `; +:math:`z_{\mathrm{h}}` is taken from the previous timestep) and the grid-level above which :math:`\theta_{v\ell}` starts to increase with height. The ascent is stopped at the grid-level NTPAR (height -:math:`z_{\rm par}` :math:`=z_{\mathrm{ \rm NTPAR}+\frac{1}{2}}`) +:math:`z_{\mathrm{par}}` :math:`=z_{\mathrm{ \mathrm{NTPAR}}+\frac{1}{2}}`) above which the parcel becomes more negatively buoyant than a given threshold, :math:`\theta_v'`. Note that the parcel properties themselves are not perturbed in order to preserve the height of the mixed-layer’s @@ -320,12 +320,12 @@ Currently, \theta_v' = \mathrm{max} \left[A_{plume}, \, \mathrm{min} \left[ B_{plume} \sigma_{Tv1}, - \, G_{max}z_{\rm h}\right] \right] + \, G_{max}z_{\mathrm{h}}\right] \right] where :math:`A_{plume}=0.2`, :math:`B_{plume}=3.26`, :math:`G_{max}=10^{-3}`\ Km\ :math:`^{-1}`, :math:`\sigma_{Tv1} = 1.93\, \overline{w'\theta_v'}_S/w_m` and -:math:`w_m^3=u_*^3+0.25\,z_{\rm h}\overline{w'b}_S`. Following +:math:`w_m^3=u_*^3+0.25\,z_{\mathrm{h}}\overline{w'b}_S`. Following `Holtslag and Boville (1993)`_, :math:`\theta_v'` is related to the magnitude of the gradient adjustment, :math:`\gamma_{\theta_{\ell}}` (see section @@ -348,26 +348,26 @@ where the vapour pressure of air in grid-level :math:`k_s`, :math:`e_{k_s} = q_{k_s} P_{k_s}/(100 \, \epsilon)`. The full-level below that containing the LCL is labelled NLCL and -:math:`z_{\rm lcl}` :math:`=z_{\mathrm{ \rm NLCL}+\frac{1}{2}}`. If +:math:`z_{\mathrm{lcl}}` :math:`=z_{\mathrm{ \mathrm{NLCL}}+\frac{1}{2}}`. If the parcel rises above the top of the LCL transition zone (defined as -1.1\ :math:`z_{\rm lcl}` , its ascent can also be stopped at the +1.1\ :math:`z_{\mathrm{lcl}}` , its ascent can also be stopped at the grid-level at which it has maximum buoyancy excess over the environment. This is identified as the grid-level above which .. math:: - \frac{d\theta_v}{dz}|_{\rm env} > - \Gamma_{\rm inv}\, \frac{d\theta_v}{dz}|_{\rm par} + \frac{d\theta_v}{dz}|_{\mathrm{env}} > + \Gamma_{\mathrm{inv}}\, \frac{d\theta_v}{dz}|_{\mathrm{par}} where currently the tolerance for identifying inversions by this method, -:math:`\Gamma_{\rm inv}=1.1`. This use of the height of maximum excess +:math:`\Gamma_{\mathrm{inv}}=1.1`. This use of the height of maximum excess (if lower than that given by the straight buoyancy threshold, :math:`\theta_v'`) is typically of little consequence in stratocumulus regions (which tend to be well-mixed beneath large inversions), but can be necessary in order to identify the capping inversion in cumulus cases (e.g. in the trade wind regions). -**Step 2:** having established :math:`z_{\rm par}` , a crucial +**Step 2:** having established :math:`z_{\mathrm{par}}` , a crucial additional test is to determine whether this layer is well-mixed (i.e., stratocumulus-capped) or cumulus-capped. The parcel ascent can rise to cloud-top in both cases but cumulus cloud layers are observed not to be @@ -378,28 +378,29 @@ Specifically, a logical flag (CUMULUS) is set to true if .. math:: - \left| \frac{ \Delta_{\rm cld} q_t}{\Delta_{\rm cld} z} \right| > - C_t \, \left| \frac{ \Delta_{\rm sub} q_t}{\Delta_{\rm sub} z} \right| + \left| \frac{ \Delta_{\mathrm{cld}} q_t}{\Delta_{\mathrm{cld}} z} \right| > + C_t \, \left| \frac{ \Delta_{\mathrm{sub}} q_t}{\Delta_{\mathrm{sub}} z} + \right| -where the cloud-layer gradient, :math:`\Delta_{\rm cld}`, is taken +where the cloud-layer gradient, :math:`\Delta_{\mathrm{cld}}`, is taken between both NTPAR and NTPAR-1 (to allow for the possibility that a Sc layer has just deepened by a grid-level) and NLCL and the sub-cloud -layer gradient, :math:`\Delta_{\rm sub}`, between grid-levels NLCL and +layer gradient, :math:`\Delta_{\mathrm{sub}}`, between grid-levels NLCL and :math:`k_s`. Currently the threshold factor, :math:`C_t = 1.1`. If cumulus is diagnosed, the top of the surface-based mixed layer -(:math:`z_{\rm h}` ) is set to :math:`z_{\rm lcl}` (rather than to -:math:`z_{\rm par}` , as illustrated in :numref:`Fig. %s ` for +(:math:`z_{\mathrm{h}}` ) is set to :math:`z_{\mathrm{lcl}}` (rather than to +:math:`z_{\mathrm{par}}` , as illustrated in :numref:`Fig. %s ` for types V and VI). There is then an option to diagnose the thickness of the LCL transition zone, see section :ref:`Diagnosis of the LCL transition zone thickness `. Otherwise, the boundary layer surface-driven mixing is capped at -:math:`z_{\rm lcl}` so that mixing into the cumulus cloud layer is only +:math:`z_{\mathrm{lcl}}` so that mixing into the cumulus cloud layer is only carried out by the model’s mass-flux convection scheme and not by the eddy viscosity based boundary layer scheme. Note that basing the CUMULUS diagnosis on cloud and sub-cloud layer gradients limits the model only to being able to resolve cumulus with cloud and sub-cloud layers at least 2 grid-levels (and optionally 400m) thick. Otherwise the layer is -considered well-mixed to :math:`z_{\rm par}` with an option to include a +considered well-mixed to :math:`z_{\mathrm{par}}` with an option to include a representation of fluxes into the capping inversion (see section :ref:`Diagnosis of inversion thickness `). @@ -414,7 +415,7 @@ BL_LEVELS above the tropopause is recommended. Note that if cumulus is not diagnosed then a further, subgrid estimation of the height of the capping inversion is attempted for -:math:`z_{\rm h}`  (as described in section :ref:`Diagnosis of a sub-grid +:math:`z_{\mathrm{h}}`  (as described in section :ref:`Diagnosis of a sub-grid inversion `). .. _sec_parxs: @@ -477,10 +478,10 @@ been separated in to three stages. These are: #. diagnose the existence of a decoupled stratocumulus (DSC) layer with approximately uniform :math:`\theta_{v\ell}` (label the top grid-level in the mixed-layer NTDSC and diagnose the subgrid height - of its capping inversion, :math:`z_{\rm h}^{\rm Sc}` , see + of its capping inversion, :math:`z_{\mathrm{h}}^{\mathrm{Sc}}` , see section :ref:`Diagnosis of a sub-grid inversion `) -#. diagnose an approximate depth of the DSC layer, :math:`z_{\rm ml}`, +#. diagnose an approximate depth of the DSC layer, :math:`z_{\mathrm{ml}}`, in order to be able to calculate the representative turbulent velocity scales (see appendix :ref:`Appendix: Definitions of the velocity scales `). @@ -531,9 +532,9 @@ that grid-levels :math:`k_{ct}-1` and :math:`k_{ct}-2` have already been identified as well-mixed). **Step 2** is to diagnose an approximate depth of the DSC layer, -:math:`z_{\rm ml}`. The bottom grid-level of the mixed-layer (NBDSC) is +:math:`z_{\mathrm{ml}}`. The bottom grid-level of the mixed-layer (NBDSC) is diagnosed as the lowest grid-level, descending from NTDSC, where -:math:`{\theta_{v\ell}}_{\mathrm{ \rm NTDSC}} + \theta_{v\ell}'` is +:math:`{\theta_{v\ell}}_{\mathrm{ \mathrm{NTDSC}}} + \theta_{v\ell}'` is less than :math:`\theta_{v\ell}` of the environment. The parcel perturbation is given by @@ -554,11 +555,11 @@ they are consistent with, for example, the observations of NBDSC equals NTDSC) in a DSC layer *not* overlying cumulus, then the layer is assumed not to be well-mixed. At the top of a cumulus layer, the DSC layer is given a minimum depth of -:math:`\Delta_{\mathrm{ \rm NTDSC}+\frac{1}{2}} z`. Otherwise, the -layer depth, :math:`z_{\rm ml}`, is measured from the top of layer NTDSC +:math:`\Delta_{\mathrm{ \mathrm{NTDSC}}+\frac{1}{2}} z`. Otherwise, the +layer depth, :math:`z_{\mathrm{ml}}`, is measured from the top of layer NTDSC to the base of layer NBDSC. -**Step 3**: the step 2 calculation of :math:`z_{\rm ml}` is used to +**Step 3**: the step 2 calculation of :math:`z_{\mathrm{ml}}` is used to calculate the representative velocity scales for the DSC layer but its calculation is only crude. Here, the vertical extent of the :math:`K`-profiles is determined more accurately by ensuring that the @@ -584,13 +585,14 @@ expanded using the first-order closure in .. math:: - \overline{w'\theta_{\ell}'}_k = -K_h^{\rm surf}\,\frac{\widetilde{\Delta_k - \theta_{\ell}}}{\Delta_k z} - -K_h^{\rm Sc}\,\frac{\Delta_k \theta_{\ell}}{\Delta_k z} + \overline{w'\theta_{\ell}'}_k = + -K_h^{\mathrm{surf}}\,\frac{\widetilde{\Delta_k \theta_{\ell}}}{\Delta_k z} + -K_h^{\mathrm{Sc}}\,\frac{\Delta_k \theta_{\ell}}{\Delta_k z} .. math:: :label: eq:wx_std - \overline{w'q_t'}_k = -\left(K_h^{\rm surf}+ K_h^{\rm Sc}\right) \, + \overline{w'q_t'}_k = -\left(K_h^{\mathrm{surf}}+ K_h^{\mathrm{Sc}}\right) + \, \,\frac{\Delta_k q_t}{\Delta_k z} where @@ -621,15 +623,15 @@ buoyancy consumption of TKE within the mixed layer equals a fraction, .. math:: :label: deccrit - \sum_{z_{k-\frac{1}{2}} > z_i-z_{\rm ml}}^{z_{k-\frac{1}{2}} < z_i} + \sum_{z_{k-\frac{1}{2}} > z_i-z_{\mathrm{ml}}}^{z_{k-\frac{1}{2}} < z_i} \left|\left[ \overline{w'b}|_{z_{k-\frac{1}{2}}}<0 \right]\right| \, \Delta_k z \, \leq \, D_t \, - \sum_{z_{k-\frac{1}{2}} > z_i-z_{\rm ml}}^{z_{k-\frac{1}{2}} < z_i} + \sum_{z_{k-\frac{1}{2}} > z_i-z_{\mathrm{ml}}}^{z_{k-\frac{1}{2}} < z_i} \left[ \overline{w'b}|_{z_{k-\frac{1}{2}}}>0 \right] \, \Delta_k z Note that, for simplicity, the :math:`{\cal E}_h` factors are not -included in :math:`K_h^{\rm surf}` or :math:`K_h^{\rm Sc}` when +included in :math:`K_h^{\mathrm{surf}}` or :math:`K_h^{\mathrm{Sc}}` when calculating :eq:`eq:wx_std` under the assumption that they will be small. This process is applied to all unstable mixed layers. For stratocumulus layers, observations and LES suggest a value @@ -646,25 +648,26 @@ The first step is to test for whether a well-mixed layer is possible (either decoupling what has so far been diagnosed as a well-mixed layer or, if one exists, recoupling a decoupled stratocumulus layer), i.e., to test whether :eq:`deccrit` is satisfied with both -:math:`K_h^{\rm surf}` and :math:`K_h^{\rm Sc}` extending from the +:math:`K_h^{\mathrm{surf}}` and :math:`K_h^{\mathrm{Sc}}` extending from the surface to the cloud-top. If recoupling is possible then the various flags identifying the DSC layer are reset (*this includes setting the cumulus diagnosis to false*), any surface-driven entrainment originally -applied at :math:`z_{\rm h}` is added to the entrainment at -:math:`z_{\rm h}^{\rm Sc}` (after rescaling for the inversion strength -at :math:`z_{\rm h}^{\rm Sc}` ) and :math:`z_{\rm b}` is set to -0.1\ :math:`z_{\rm h}` (for the reason discussed above). If decoupling -is diagnosed, :math:`z_{\rm h}^{\rm Sc}` is set to the original -:math:`z_{\rm h}` (inversion height), although the entrainment across +applied at :math:`z_{\mathrm{h}}` is added to the entrainment at +:math:`z_{\mathrm{h}}^{\mathrm{Sc}}` (after rescaling for the inversion strength +at :math:`z_{\mathrm{h}}^{\mathrm{Sc}}` ) and :math:`z_{\mathrm{b}}` is set to +0.1\ :math:`z_{\mathrm{h}}` (for the reason discussed above). If decoupling +is diagnosed, :math:`z_{\mathrm{h}}^{\mathrm{Sc}}` is set to the original +:math:`z_{\mathrm{h}}` (inversion height), although the entrainment across this inversion is not recalculated (and so keeps any surface-driven component — the COUPLED flag is therefore set to true, see section :ref:`Entrainment fluxes `). If a decoupled layer is diagnosed, then an iteration is performed to -find the highest :math:`z_{\rm h}` (so top of the :math:`K_h^{\rm surf}` +find the highest :math:`z_{\mathrm{h}}` (so top of the +:math:`K_h^{\mathrm{surf}}` profile) that still satisfies :eq:`deccrit`, but with -:math:`K_h^{\rm Sc}=0` in :eq:`eq:wx_std`. The iteration -proceeds with :math:`z_{\rm h}` stepping from its lowest permissible +:math:`K_h^{\mathrm{Sc}}=0` in :eq:`eq:wx_std`. The iteration +proceeds with :math:`z_{\mathrm{h}}` stepping from its lowest permissible height to its highest (currently 3 steps are used). If at any stage :eq:`deccrit` is violated, then the step below (therefore containing the height that would give equality in @@ -672,30 +675,30 @@ containing the height that would give equality in taken downwards. If :eq:`deccrit` is met the step above is again reduced by a factor of 4 and 3 steps taken upwards. A total of 3 sweeps are possible, each with a smaller step so that -:math:`z_{\rm h}` approaches the height that gives equality in +:math:`z_{\mathrm{h}}` approaches the height that gives equality in :eq:`deccrit`. The accuracy with which this is achieved will be the difference in the maximum and minimum permissible heights of -:math:`z_{\rm h}`  divided by :math:`2\times4\times4 = 32`, which will +:math:`z_{\mathrm{h}}`  divided by :math:`2\times4\times4 = 32`, which will typically be less than 30m. The top grid-level of the SML, NTML, is -defined as the highest grid-level such that :math:`K_h^{\rm Sc}` is +defined as the highest grid-level such that :math:`K_h^{\mathrm{Sc}}` is non-zero at the half-level above. The above process is then repeated to find the appropriate -:math:`z_{\rm b}` for :math:`K_h^{\rm Sc}`, i.e., for the base of -top-driven mixing. Some constraints are placed on :math:`z_{\rm b}` , -namely that it should never go below :math:`0.1`\ :math:`z_{\rm h}` (to +:math:`z_{\mathrm{b}}` for :math:`K_h^{\mathrm{Sc}}`, i.e., for the base of +top-driven mixing. Some constraints are placed on :math:`z_{\mathrm{b}}` , +namely that it should never go below :math:`0.1`\ :math:`z_{\mathrm{h}}` (to avoid affecting the continuity of the :math:`K` profiles at the top of the surface layer, see :eq:`ws_defn`). If cumulus -convection has been diagnosed then :math:`z_{\rm b}` is not allowed to -go below :math:`z_{\mathrm{ \rm NTML}+\frac{1}{2}}` (unless the layer -is diagnosed to recouple completely). Finally, :math:`z_{\rm b}` must -always be at or below :math:`z_{\mathrm{ \rm NTDSC}-1}`, so that +convection has been diagnosed then :math:`z_{\mathrm{b}}` is not allowed to +go below :math:`z_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}}` (unless the layer +is diagnosed to recouple completely). Finally, :math:`z_{\mathrm{b}}` must +always be at or below :math:`z_{\mathrm{ \mathrm{NTDSC}}-1}`, so that mixing in decoupled layers is always resolved, and at least :math:`\Delta z_{rad}` (the cloud-top radiative cooling depth defined in section :ref:`Integration of \overline{w'b} close to the inversion `) below the t inversion. The base grid-level of the DSC layer, NBDSC, is defined (analogously to NTDSC) as -the lowest grid-level such that :math:`K_h^{\rm Sc}` is non-zero at the +the lowest grid-level such that :math:`K_h^{\mathrm{Sc}}` is non-zero at the half-level below. A possible extension to this diagnosis would be to include the shear @@ -735,9 +738,9 @@ inversion (in particular, in the LW radiative flux), simple finite difference flux calculations, :eq:`eq:wx_std`, can be significantly inaccurate in this region. An example is shown in :numref:`Fig. %s `. Calculating -:math:`\overline{w'\theta_{\ell}'}_{\mathrm{ \rm NTML}+\frac{1}{2}}` +:math:`\overline{w'\theta_{\ell}'}_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}}` from :eq:`eq:wx_std` gives a negative value, largely -because :math:`\Delta_{\mathrm{ \rm NTML}+1} \theta_{\ell}` is +because :math:`\Delta_{\mathrm{ \mathrm{NTML}}+1} \theta_{\ell}` is positive and so the local flux is large and negative. In reality, :math:`\overline{w'\theta_{\ell}'}` becomes positive only a short distance below cloud-top such that the integral here will tend also to @@ -746,12 +749,12 @@ be positive. The solution adopted is to integrate :math:`\overline{w'b}` analytically across the region just below the inversion, labelled :math:`\Delta z_{rad}` in Fig, :numref:`%s `. Since -:math:`\Delta_{\mathrm{ \rm NTML}} \theta_{\ell}` can also be +:math:`\Delta_{\mathrm{ \mathrm{NTML}}} \theta_{\ell}` can also be significantly positive (when the grid-level inversion is rising or falling, for example), the base of this region is taken to be the lower of the first :math:`\theta`-level below :math:`z_h-100` m (a physically reasonable depth over which cloud-top radiative cooling might be -expected to occur) and :math:`z_{\mathrm{ \rm NTML}-1}`. +expected to occur) and :math:`z_{\mathrm{ \mathrm{NTML}}-1}`. .. figure:: blank.svg :name: fig:inv_integ @@ -796,7 +799,7 @@ tends to be close to 3. Consequently, we approximate :math:`I^{rad} = \Delta z_{rad} ( F^{rad}|_{z_h}-F^{rad}|_{z_h-\Delta z_{rad}} ) /3`. In addition, :math:`F^{rad}|_{z_h}-F^{rad}|_{z_h-\Delta z_{rad}}` is approximated as :math:`\Delta F`, the radiative flux change across -cloud-top used in the calculation of :math:`V_{\rm Sc}` +cloud-top used in the calculation of :math:`V_{\mathrm{Sc}}` :eq:`ctraddiv`. The precipitation flux is assumed to vary linearly across this region, as does the total flux, and so its contribution to :math:`I^{Tot}` cancels with :math:`I^{ppn}` in @@ -859,16 +862,16 @@ the parcel buoyancy. Note that the constant in :math:`w_m^3` differs by a factor of 4. The buoyancy integration in :eq:`dz_param`, that is itself dependent on :math:`z_{top}`, is performed working upwards from -:math:`z_{\rm par}` assuming piece-wise linear variation of :math:`b` +:math:`z_{\mathrm{par}}` assuming piece-wise linear variation of :math:`b` between grid-levels. Note that the standard definition of the boundary layer top in the UM is the height of the first flux level below the level of neutral buoyancy, so -:math:`z_{\rm par}` :math:`=z_{\mathrm{ \rm NTPAR}+\frac{1}{2}}`. The +:math:`z_{\mathrm{par}}` :math:`=z_{\mathrm{ \mathrm{NTPAR}}+\frac{1}{2}}`. The inversion thickness is then defined as .. math:: :label: dz_definition - \Delta z_i = z_{top} -z_{\rm par} + \Delta z_i = z_{top} -z_{\mathrm{par}} .. _sec_lclmixing: @@ -878,7 +881,7 @@ Diagnosis of the LCL transition zone thickness As described in section :ref:`The diagnostic parcel ascent and cumulus diagnosis `, when cumulus convection has been diagnosed surface-driven mixing was originally capped at -:math:`z_{\rm lcl}` so that mixing into the cumulus cloud layer was only +:math:`z_{\mathrm{lcl}}` so that mixing into the cumulus cloud layer was only carried out by the model’s mass-flux convection scheme. This was seen to lead to errors in the mean profiles across the LCL, with superadiabats being the most extreme manifestation. Using the boundary layer @@ -901,10 +904,10 @@ thermals within the grid box that may penetrate above the grid-box mean LCL (but are too dry to reach their own LCL). Thus their buoyancy flux is given by :eq:`eq:wb_cont` with :math:`C_F=0`. Restricting the negative integral of this buoyancy flux then gives a new -definition for :math:`z_{\rm h}`  that is then used in the calculation +definition for :math:`z_{\mathrm{h}}`  that is then used in the calculation of the surface-driven K-profiles in section :ref:`Surface-driven turbulence ` — the -larger the value of :math:`D_t`, the higher :math:`z_{\rm h}` will be. +larger the value of :math:`D_t`, the higher :math:`z_{\mathrm{h}}` will be. Typically :math:`D_t=0.1` for decoupled stratocumulus layers while idealised clear-sky convective boundary layers (where the magnitude of the entrainment buoyancy flux is a fraction, :math:`A_1`, of the surface @@ -970,14 +973,15 @@ The asymptotic mixing lengths are given by .. math:: - \lambda_m =\mathrm{max}\left[\lambda_0,\, 0.15 z_{\rm loc}, 2 h_B \right] + \lambda_m =\mathrm{max}\left[\lambda_0,\, 0.15 z_{\mathrm{loc}}, 2 h_B + \right] .. math:: :label: asymp_ml - \lambda_h =\mathrm{max}\left[\lambda_0,\, 0.15 z_{\rm loc}\right] + \lambda_h =\mathrm{max}\left[\lambda_0,\, 0.15 z_{\mathrm{loc}}\right] where :math:`\lambda_0` is a minimum mixing length read in from the -namelist and :math:`z_{\rm loc}` is defined below. The orographic +namelist and :math:`z_{\mathrm{loc}}` is defined below. The orographic blending height, :math:`h_B` (only used within the boundary layer, as defined below), is given by @@ -1045,19 +1049,19 @@ height-dependent factor is included, minutes, for simplicity. Initially, the lowest half-level at which :math:`Ri>Ri_{crit}` is taken -to be a measure of the boundary layer top (:math:`z_{\rm loc}` ) and the +to be a measure of the boundary layer top (:math:`z_{\mathrm{loc}}` ) and the full-level below is designated NTLOC. In general :math:`Ri_{crit}=1` but a value of 0.25 is recommended for use with the ’SHARPEST’ stability functions, see below. If the boundary layer was diagnosed as cumulus-capped by the non-local scheme (see section :ref:`Diagnosis of boundary layer depth and type `) -then :math:`z_{\rm loc}` is lowered to :math:`z_{\rm lcl}` (and +then :math:`z_{\mathrm{loc}}` is lowered to :math:`z_{\mathrm{lcl}}` (and :math:`K_h` and :math:`K_m` are set to zero from the base of grid-level NLCL upwards) so that transports into and within the cumulus cloud layer can be performed solely by the mass-flux convection scheme. Depending on the switch local_fa, above NTLOC turbulently-mixed layers (where :math:`RiRi_{t} + (1 - 5Ri)^2 & {\mathrm{for}}\ 0Ri_{t} \end{cases} where @@ -1134,22 +1138,22 @@ used following `Mailhot and Lock (2004)`_ with: .. math:: Pr=\min \left( Pr_{\rm max}, \, Pr_N(1+2Ri) \, \right). -The maximum permitted Prandtl number, :math:`Pr_{\rm max}`, is currently +The maximum permitted Prandtl number, :math:`Pr_{\mathrm{max}}`, is currently set to :math:`5` for model stability reasons. The stability functions for :math:`Ri>0` are then given by: .. math:: - f_m = \frac{Pr}{Pr_N} \, f_{\rm stable} + f_m = \frac{Pr}{Pr_N} \, f_{\mathrm{stable}} .. math:: - f_h = \frac{1}{Pr_N} \, f_{\rm stable} + f_h = \frac{1}{Pr_N} \, f_{\mathrm{stable}} Note that writing the functions in this way ensures that :math:`f_m=1` under neutral conditions and the effect of the variation in :math:`Pr` is for :math:`f_m` to decrease slower with increasing :math:`Ri` than -:math:`f_{\rm stable}`, which can be explained through increasing +:math:`f_{\mathrm{stable}}`, which can be explained through increasing gravity-wave activity. Finally, the LEM stable functions are also available which cut off all @@ -1260,7 +1264,7 @@ where :math:`C_F` decreases with height, :math:`f_{lev}={q_c}_{k-1}/(s_{k-1}-s_{k})`, where :math:`q_c` is the total condensate, and :math:`f_{lev}` also constrained to be less than unity. The total cloud volume fraction is then given by -:math:`f_{tot} = {\rm min}[{C_F}_{k-1},{C_F}_k] + f_{edge}f_{lev}` and +:math:`f_{tot} = {\mathrm{min}}[{C_F}_{k-1},{C_F}_k] + f_{edge}f_{lev}` and this is used to weight the saturated contribution to the buoyancy parameters on :math:`rho`-levels, e.g., :math:`\overline{\beta_T}_{k-1/2} = f_{tot} \tilde{\beta_T}_{k-1/2} + @@ -1307,7 +1311,8 @@ The general approach is to take :math:`K_{\chi}` in .. math:: :label: klnl - K_{\chi} = \mathrm{max} \left[ (K_{\chi}^{\rm surf}+K_{\chi}^{\rm Sc}), + K_{\chi} = \mathrm{max} \left[ + (K_{\chi}^{\mathrm{surf}}+K_{\chi}^{\mathrm{Sc}}), K_{\chi}(Ri) \right] As noted in section :ref:`Model variables and turbulence closure @@ -1363,34 +1368,37 @@ the strong surface buoyancy generation of turbulence in these regimes, a calculation of :math:`Ri` is made that allows for the gradient adjustment by the non-local scheme, i.e., using :math:`\widetilde{\Delta_k \theta_{\ell}}` (see -:eq:`eq:wx_std`). The height, :math:`z_{\rm loc}` , where +:eq:`eq:wx_std`). The height, :math:`z_{\mathrm{loc}}` , where :math:`Ri>Ri_{crit}=0.25` is found. It is then hypothesised that this level of turbulent instability (that incorporates the effects of shear) only needs extend some fractional distance into the cloud layer to disrupt the formation of cumulus elements. Thus, if :math:`z_{\rm loc}> z_{\rm lcl}+ f_{\rm sh} -\left(z_{\rm par}-z_{\rm lcl}\right)`, where :math:`f_{\rm sh}` is a -tunable parameter (:math:`0 -Ri_{crit}`, :math:`z_{\rm h}^{\rm Sc}` is the top of any stratocumulus -layer and :math:`z_{\rm h}` is the top of surface-based mixed layer, +Ri_{crit}`, :math:`z_{\mathrm{h}}^{\mathrm{Sc}}` is the top of any stratocumulus +layer and :math:`z_{\mathrm{h}}` is the top of surface-based mixed layer, found by adiabatic parcel ascent but reset to the LCL in cumulus capped layers. Another diagnostic is available, the “boundary layer depth” -(STASH 25), that is set to :math:`=\mathrm{max}[z_{\rm h}, z_{\rm loc}]` +(STASH 25), that is set to :math:`=\mathrm{max}[z_{\mathrm{h}}, +z_{\mathrm{loc}}]` and so represents the depth of the stable boundary layer or “surface” mixed layer. Also available are three diagnostics that represent the calculated value of each of the individual terms in STASH 3,304: 3,356 -is set to :math:`z_{\rm h}` ; 3,357 is :math:`z_{\rm h}^{\rm Sc}`  and -3,358 is :math:`z_{\rm loc}` . +is set to :math:`z_{\mathrm{h}}` ; 3,357 is +:math:`z_{\mathrm{h}}^{\mathrm{Sc}}`  and +3,358 is :math:`z_{\mathrm{loc}}` . .. _sec_nonlocal: @@ -1418,51 +1426,53 @@ Surface-driven turbulence For turbulence sources at the surface (namely surface drag with velocity scale :math:`u_*`, and positive surface buoyancy fluxes with velocity scale :math:`w_*`) in a layer with top at -:math:`z=`\ :math:`z_{\rm h}` , base at :math:`z=0` we set +:math:`z=`\ :math:`z_{\mathrm{h}}` , base at :math:`z=0` we set .. math:: :label: kmsurf - K_m^{\rm surf}= k \ z_{\rm h}\ w_m \ \frac{z}{z_{\rm h}} - \left( 1 - {\cal E}_m^{\rm surf} \frac{z}{z_{\rm h}} \right)^2 + K_m^{\mathrm{surf}}= k \ z_{\mathrm{h}}\ w_m \ \frac{z}{z_{\mathrm{h}}} + \left( 1 - {\cal E}_m^{\mathrm{surf}} \frac{z}{z_{\mathrm{h}}} + \right)^2 where :math:`w_m^3 = u_*^3 + w_s^3`, :math:`u_*` is the friction velocity (including the orographic roughness component) and :math:`w_s` -is defined below. For the 9C version of the scheme, :math:`z_{\rm h}` is +is defined below. For the 9C version of the scheme, :math:`z_{\mathrm{h}}` is the diagnosed subgrid inversion height (see section :ref:`Diagnosis of a sub-grid inversion `) for both -:math:`K_h^{\rm surf}` and -:math:`K_m^{\rm surf}`. In the 8A version, :math:`K_m^{\rm surf}` uses -:math:`z_{\rm h}` :math:`=z_{\mathrm{ \rm NTML}+\frac{1}{2}}`. The -factor :math:`{\cal E}_m^{\rm surf}` is chosen so that -:math:`K_m^{\rm surf}` will tend to -:math:`K_m|_{\mathrm{ \rm NTML}+\frac{1}{2}}` as :math:`z` tends to -:math:`z_{\rm h}` , where -:math:`K_m|_{\mathrm{ \rm NTML}+\frac{1}{2}}` is the entrainment +:math:`K_h^{\mathrm{surf}}` and +:math:`K_m^{\mathrm{surf}}`. In the 8A version, :math:`K_m^{\mathrm{surf}}` uses +:math:`z_{\mathrm{h}}` :math:`=z_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}}`. The +factor :math:`{\cal E}_m^{\mathrm{surf}}` is chosen so that +:math:`K_m^{\mathrm{surf}}` will tend to +:math:`K_m|_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}}` as :math:`z` tends to +:math:`z_{\mathrm{h}}` , where +:math:`K_m|_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}}` is the entrainment eddy-diffusivity (given by :eq:`khent`, although, in order to avoid altering the shape function too much, -:math:`{\cal E}_m^{\rm surf}` is not allowed to fall below :math:`0.7`). -A similar factor, :math:`{\cal E}_h^{\rm surf}`, is used in the :math:`K_h^{\rm -surf}` profile even though the +:math:`{\cal E}_m^{\mathrm{surf}}` is not allowed to fall below :math:`0.7`). +A similar factor, :math:`{\cal E}_h^{\rm surf}`, is used in the +:math:`K_h^{\mathrm{surf}}` profile even though the entrainment fluxes of the thermodynamic variables will usually be specified explicitly rather than through an eddy-diffusivity (see section :ref:`Entrainment fluxes `). The form of :math:`w_s` differs between the surface layer -(:math:`z < 0.1`\ :math:`z_{\rm h}` ) and the rest of the mixed-layer: +(:math:`z < 0.1`\ :math:`z_{\mathrm{h}}` ) and the rest of the mixed-layer: .. math:: :label: ws_defn w_s^3 = \begin{cases} - 2.5 \, \frac{z}{z_{\rm h}} w_*^3 & {\rm surface\ layer} \\ - 0.25 \, w_*^3 & {\rm mixed\ layer} \\ + 2.5 \, \frac{z}{z_{\mathrm{h}}} w_*^3 & {\mathrm{surface}\ layer} \\ + 0.25 \, w_*^3 & {\mathrm{mixed}\ layer} \\ \end{cases} -and :math:`w_*^3=z_{\rm h}\overline{w'b}_S` using :math:`z_{\rm h}` from +and :math:`w_*^3=z_{\mathrm{h}}\overline{w'b}_S` using +:math:`z_{\mathrm{h}}` from the current timestep (note that the use of :math:`w_*` here will be -inconsistent with the use of :math:`V_{\rm heat}` in the entrainment +inconsistent with the use of :math:`V_{\mathrm{heat}}` in the entrainment parametrization in cloudy boundary layers). Note that :math:`w_s` is -continuous across :math:`0.1`\ :math:`z_{\rm h}` and constant with +continuous across :math:`0.1`\ :math:`z_{\mathrm{h}}` and constant with height in the mixed layer. This form for :math:`w_s` is motivated by a desire to match the model’s surface transfer formulation within the surface layer (as described further in section :ref:`Comparison with Holtslag @@ -1471,7 +1481,7 @@ and to use a cubic sum of velocity scales within the mixed layer (consistent with dimensional analysis of the TKE equation, see `Holtslag and Boville (1993)`_). -The formula for :math:`K_h^{\rm surf}` is identical to +The formula for :math:`K_h^{\mathrm{surf}}` is identical to :eq:`kmsurf` but with :math:`w_m` replaced by :math:`w_h=w_m/Pr`, where the turbulent Prandtl number is given by: @@ -1495,9 +1505,9 @@ except for :eq:`ws_defn` and For the latter, HB93 effectively set :math:`{\cal E}_m^{\rm surf} =1`. To generate entrainment, however, they simply use -:math:`K_m^{\rm surf}|_{\mathrm{ \rm NTML}+\frac{1}{2}}`, as +:math:`K_m^{\mathrm{surf}}|_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}}`, as evaluated from :eq:`kmsurf` with a subgrid calculation of -:math:`z_{\rm h}` :math:`>z_{\mathrm{ \rm NTML}+\frac{1}{2}}`, rather +:math:`z_{\mathrm{h}}` :math:`>z_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}}`, rather than using a separate entrainment parametrization. The difference in :eq:`ws_defn` arises from the surface @@ -1519,11 +1529,12 @@ say). For the UM, .. math:: - Pr_{\rm surf} = \frac{\Phi_h}{\Phi_m} - = \left( 1 + 16 \, k \frac{z}{z_{\rm h}} \, \frac{w_*^3}{u_*^3} + Pr_{\mathrm{surf}} = \frac{\Phi_h}{\Phi_m} + = \left( 1 + 16 \, k \frac{z}{z_{\mathrm{h}}} \, + \frac{w_*^3}{u_*^3} \right)^{-1/4} -giving :math:`Pr_{\rm surf} = 1` in the neutral limit (compared to 0.75 +giving :math:`Pr_{\mathrm{surf}} = 1` in the neutral limit (compared to 0.75 from :eq:`prandtl_nl`). In the convective limit, :math:`Pr_{\rm surf}|_{0.1\, z_{\rm h}} \rightarrow 0.9 (w_*/u_*)^{-3/4} = 0.9 \beta^{3/4} = 0.14` @@ -1554,42 +1565,45 @@ Prandtl number between the surface and interior in the UM is not known. Cloud-top-driven turbulence --------------------------- -For cloud-top-driven turbulence over a layer of depth :math:`z_{\rm ml}` -(with top at :math:`z_{\rm h}` or :math:`z_{\rm h}^{\rm Sc}` and base at -:math:`z_{\rm b}` , determined as in section :ref:`Diagnosis of the vertical -extent of the K-profiles `), +For cloud-top-driven turbulence over a layer of depth :math:`z_{\mathrm{ml}}` +(with top at :math:`z_{\mathrm{h}}` or :math:`z_{\mathrm{h}}^{\mathrm{Sc}}` and +base at +:math:`z_{\mathrm{b}}` , determined as in section :ref:`Diagnosis of the +vertical extent of the K-profiles `), .. math:: :label: kmtop - K_m^{\rm Sc}= 0.63 \ k \ z_{\rm ml}\ V_{\rm Sc}\left( \frac{z'}{z_{\rm ml}} - \right)^2 - \left( 1 - {\cal E}_m^{\rm Sc} \frac{z'}{z_{\rm ml}} \right)^{0.8} + K_m^{\mathrm{Sc}}= 0.63 \ k \ z_{\mathrm{ml}}\ V_{\mathrm{Sc}}\left( + \frac{z'}{z_{\mathrm{ml}}} \right)^2 + \left( 1 - {\cal E}_m^{\mathrm{Sc}} \frac{z'}{z_{\mathrm{ml}}} + \right)^{0.8} -where :math:`V_{\rm Sc}^3= V_{\rm rad}^3+V_{\rm br}^3` (see +where :math:`V_{\mathrm{Sc}}^3= V_{\mathrm{rad}}^3+V_{\mathrm{br}}^3` (see appendix :ref:`Appendix: Definitions of the velocity scales `) and :math:`z'` is height above -:math:`z_{\rm b}` . Then :math:`K_h = K_m / \mathrm{Pr}`, where +:math:`z_{\mathrm{b}}` . Then :math:`K_h = K_m / \mathrm{Pr}`, where :math:`\mathrm{Pr}=0.75`. The resulting :math:`K_h` profile was derived against convective cloudy LES, as described in `Lock (1999)`_. The appropriate Prandtl number -(and therefore :math:`K_m^{\rm Sc}`) is unknown, 0.75 being chosen +(and therefore :math:`K_m^{\mathrm{Sc}}`) is unknown, 0.75 being chosen simply as a number in the middle of the range usually quoted for turbulent mixing in general. As with :eq:`kmsurf`, -:math:`z_{\rm h}`  (or :math:`z_{\rm h}^{\rm Sc}` ) are given by the +:math:`z_{\mathrm{h}}`  (or :math:`z_{\mathrm{h}}^{\mathrm{Sc}}` ) are given by +the subgrid diagnosis (see section :ref:`Diagnosis of a sub-grid inversion `) except for -:math:`K_m^{\rm Sc}` in the 8A scheme which uses the height of the -half-level below (:math:`z_{\mathrm{ \rm NTML}+\frac{1}{2}}` or -:math:`z_{\mathrm{ \rm NTDSC}+\frac{1}{2}}`). Again following -:eq:`kmsurf`, the factors :math:`{\cal E}_m^{\rm Sc}` and -:math:`{\cal E}_h^{\rm Sc}` are included in :eq:`kmtop` so -that :math:`K_m^{\rm Sc}` will tend to -:math:`K_m|_{\mathrm{ \rm NTML}+\frac{1}{2}}` (and -:math:`K_h^{\rm Sc}` to -:math:`K_h|_{\mathrm{ \rm NTML}+\frac{1}{2}}`), given by -:eq:`khent`, as :math:`z` tends to :math:`z_{\rm h}` (and +:math:`K_m^{\mathrm{Sc}}` in the 8A scheme which uses the height of the +half-level below (:math:`z_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}}` or +:math:`z_{\mathrm{ \mathrm{NTDSC}}+\frac{1}{2}}`). Again following +:eq:`kmsurf`, the factors :math:`{\cal E}_m^{\mathrm{Sc}}` and +:math:`{\cal E}_h^{\mathrm{Sc}}` are included in :eq:`kmtop` so +that :math:`K_m^{\mathrm{Sc}}` will tend to +:math:`K_m|_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}}` (and +:math:`K_h^{\mathrm{Sc}}` to +:math:`K_h|_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}}`), given by +:eq:`khent`, as :math:`z` tends to :math:`z_{\mathrm{h}}` (and here no restriction is made on the magnitude of either :math:`{\cal E}_m^{\rm -Sc}` or :math:`{\cal E}_h^{\rm Sc}`). +Sc}` or :math:`{\cal E}_h^{\mathrm{Sc}}`). .. _sec_gradadj: @@ -1601,21 +1615,22 @@ Recall that for :math:`\theta_{\ell}` only we use .. math:: :label: wthl \overline{w'\theta_{\ell}'}= - K_h \frac{\partial \theta_{\ell}}{\partial z} - + K_h^{\rm surf}\gamma_{\theta_{\ell}} + + K_h^{\mathrm{surf}}\gamma_{\theta_{\ell}} where .. math:: :label: gradadj \gamma_{\theta_{\ell}} = - \mathrm{min}\left[ A_{ga} \frac{\sigma_{T1}}{z_{\rm h}}, G_{max} \right] + \mathrm{min}\left[ A_{ga} \frac{\sigma_{T1}}{z_{\mathrm{h}}}, G_{max} + \right] :math:`A_{ga}=3.26`, :math:`G_{max}=10^{-3}`\ Km\ :math:`^{-1}` and :math:`\sigma_{T1} = 1.93 \, \overline{w'\theta_{\ell}'}_S/w_m`, where for this calculation of :math:`w_m` (given by -:math:`w_m^3=u_*^3+0.25\,z_{\rm h}\overline{w'b}_S`) -:math:`z_{\rm h}` is taken from the previous timestep. The form of +:math:`w_m^3=u_*^3+0.25\,z_{\mathrm{h}}\overline{w'b}_S`) +:math:`z_{\mathrm{h}}` is taken from the previous timestep. The form of :eq:`gradadj` is similar to that used in HB93 and the magnitude of :math:`\gamma_{\theta_{\ell}}` is the same as in HB93 in the convective limit — the difference in :math:`A_{ga}` exactly allows @@ -1666,9 +1681,9 @@ that proposed by `Brown and Grant (1997)`_, written .. math:: :label: tau_nl (\tau_x^{nl},\tau_y^{nl})= \left[ - \frac{2.7w_*^3}{(u_*^3+0.6w_*^3)}\right] \left[ \left( \frac{z'}{z_{\rm - h}'} - \right) \left( 1- \frac{z'}{z_{\rm h}'} \right)^2 \right] + \frac{2.7w_*^3}{(u_*^3+0.6w_*^3)}\right] \left[ \left( + \frac{z'}{z_{\mathrm{h}}'} + \right) \left( 1- \frac{z'}{z_{\mathrm{h}}'} \right)^2 \right] (\tau_x^{s},\tau_y^{s}) Here :math:`w_*` is the convective velocity scale, :math:`u_*` is the @@ -1688,12 +1703,13 @@ is zero in neutral conditions but asymptotes to a stability-independent fraction of surface stress in convective conditions. The primed variables in the shape function allow the non-local stress profile to either be applied across the whole boundary layer (using :math:`z'=z` -and :math:`z_{\rm h}'=z_{\rm h}`), as in +and :math:`z_{\mathrm{h}}'=z_{\mathrm{h}}`), as in `Brown and Grant (1997)`_, or only above the surface layer (using -:math:`z'=z-0.1z_{\rm h}`, :math:`z_{\rm h}'=z_{\rm h}-0.1z_{\rm h}`). +:math:`z'=z-0.1z_{\mathrm{h}}`, +:math:`z_{\mathrm{h}}'=z_{\mathrm{h}}-0.1z_{\mathrm{h}}`). The motivation for applying the non-local stress above the surface layer was to ensure that the match to surface layer similarity was maintained -below :math:`0.1z_{\rm h}` (although separate tests suggested that the +below :math:`0.1z_{\mathrm{h}}` (although separate tests suggested that the impact of this change is small). .. _sec_rev_flux_grad: @@ -1729,13 +1745,13 @@ So, the new formulation is written: .. math:: :label: fg_new - F_{\chi}^{Tot} = F_{\chi}^{NT}|_{z_{\rm b}} - -\left(K_h^{\rm surf}+ K_h^{\rm - Sc}\right)\frac{\partial\overline{\chi}}{\partial z} - + \overline{w'\chi'}_{ng}^{\rm surf}+ \overline{w'\chi'}_{ng}^{\rm Sc} - + f_2 \left(F_{\chi}|_{z_h} - F_{\chi}^{NT}|_{z_{\rm b}} \right) + F_{\chi}^{Tot} = F_{\chi}^{NT}|_{z_{\mathrm{b}}} + -\left(K_h^{\mathrm{surf}}+ + K_h^{\mathrm{Sc}}\right)\frac{\partial\overline{\chi}}{\partial z} + + \overline{w'\chi'}_{ng}^{\mathrm{surf}}+ \overline{w'\chi'}_{ng}^{\mathrm{Sc}} + + f_2 \left(F_{\chi}|_{z_h} - F_{\chi}^{NT}|_{z_{\mathrm{b}}} \right) -where :math:`z_h` and :math:`z_{\rm b}` are the heights of the top and +where :math:`z_h` and :math:`z_{\mathrm{b}}` are the heights of the top and base of the mixed layer, respectively. It can be seen that :eq:`fg_new` is composed of a local down-gradient component, two non-gradient flux terms (one generated by surface-driven turbulence @@ -1747,13 +1763,14 @@ non-turbulent component: The components of :eq:`fg_new` are: -- :math:`K_{h,m}^{\rm surf}= k z_h w_{h,m} +- :math:`K_{h,m}^{\mathrm{surf}}= k z_h w_{h,m} \frac{z}{z_h}\left(1-\frac{z}{z_h}\right)^2` - :math:`K_h^{\rm Sc}= 3.6 k V_{\rm Sc}z_{ml} \left(\frac{z'}{z_{ml}}\right)^{3}\left(1-\frac{z'}{z_{ml}} \right)^{2}` -- :math:`\overline{w'\chi'}_{ng}^{\rm surf}=K_h^{\rm surf}\gamma_{\chi}` +- + :math:`\overline{w'\chi'}_{ng}^{\mathrm{surf}}=K_h^{\mathrm{surf}}\gamma_{\chi}` with :math:`\gamma_{\chi}=A_{ga}\frac{\overline{w'\chi'}_S}{w_h z_h}` and :math:`A_{ga}=10` @@ -1765,10 +1782,10 @@ The components of :eq:`fg_new` are: - :math:`f_2 = 0.5 \, \frac{z}{z_h}\, 2^{(z/z_h)^4}` In the above equations :math:`k` is von Karman’s constant, :math:`z'` -(:math:`=z-z_{\rm b}`) is height above the mixed layer base, -:math:`z_{ml}` (:math:`=z_h-z_{\rm b}`) is the mixed layer depth, +(:math:`=z-z_{\mathrm{b}}`) is height above the mixed layer base, +:math:`z_{ml}` (:math:`=z_h-z_{\mathrm{b}}`) is the mixed layer depth, :math:`u_*` is the friction velocity, and :math:`w_*` and -:math:`V_{\rm Sc}` are the velocity scales for surface and cloud-top +:math:`V_{\mathrm{Sc}}` are the velocity scales for surface and cloud-top buoyancy-driven turbulence. Although the structure of the surface-driven non-gradient terms is the @@ -1777,7 +1794,7 @@ same as for the standard flux-gradient formulation, :math:`q_t` as well as :math:`\theta_{\ell}` and also the empirical coefficients in the velocity scales have been revised: -- :math:`w_h = (u_*^3 + C_{ws} w_*^3)^{\frac{1}{3}} / Pr_{\rm neut}` +- :math:`w_h = (u_*^3 + C_{ws} w_*^3)^{\frac{1}{3}} / Pr_{\mathrm{neut}}` with :math:`C_{ws}=0.42` for :math:`\frac{z}{z_h}\geq 0.1` and :math:`C_{ws}=4.2 \frac{z}{z_h}` for :math:`\frac{z}{z_h}<0.1` @@ -1788,14 +1805,15 @@ except that :math:`w_m` is replaced by its neutral value: .. math:: - Pr = Pr_{\rm neut} - \frac{u_*^4 + w_*^3 {w_m}^{\rm neut} / 25} - {u_*^4 + w_*^3 {w_m}^{\rm neut} Pr_{\rm neut}/ (25 Pr_{\rm conv} )} + Pr = Pr_{\mathrm{neut}} + \frac{u_*^4 + w_*^3 {w_m}^{\mathrm{neut}} / 25} + {u_*^4 + w_*^3 {w_m}^{\mathrm{neut}} Pr_{\mathrm{neut}}/ (25 + Pr_{\mathrm{conv}} )} -and the range is now :math:`Pr_{\rm neut} = 0.75` to +and the range is now :math:`Pr_{\mathrm{neut}} = 0.75` to :math:`Pr_{\rm conv} = 0.6`. As with the standard scheme, a constant Prandtl number of 0.75 is -used to calculate :math:`K_m^{\rm Sc}`. +used to calculate :math:`K_m^{\mathrm{Sc}}`. Discussion of some of the revisions ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ @@ -1868,7 +1886,7 @@ gradient adjustment parameter: .. math:: :label: grad_adj \gamma_{\chi}= d \frac{\overline{w'\chi'}_S}{w_* z_h} - \hspace{0.5cm} {\rm with} \hspace{0.5cm} + \hspace{0.5cm} {\mathrm{with}} \hspace{0.5cm} d^{HB} = 7.2 w_*^2/w_m^2, \hspace{0.2cm} d^{std} = 6.3 w_*/w_m, \hspace{0.2cm} d^{rev} = 10 w_*/w_h @@ -1890,11 +1908,11 @@ Note that the most significant change from the standard UM scheme is the change to :math:`w_h` in the convective limit. Since :math:`\gamma_{\chi}` remains unchanged in the convective limit, this reduction in :math:`w_h` will result in a significantly smaller -:math:`\overline{w'\chi'}_{ng}^{\rm surf}` for the revised scheme which +:math:`\overline{w'\chi'}_{ng}^{\mathrm{surf}}` for the revised scheme which gives better agreement against LES. Compared to the standard scheme, it appears that the revised -:math:`K_h^{\rm Sc}` is very different. However, +:math:`K_h^{\mathrm{Sc}}` is very different. However, Fig.\ :numref:`%s ` shows that this actually amounts to a small adjustment in the shape. In addition, note that the factors :math:`\varepsilon_h^{surf}` and :math:`\varepsilon_h^{Sc}` have been @@ -1968,18 +1986,18 @@ only difference is in the mixing length, which is calculated as .. math:: :label: eq-lblend - l_{\rm blend} = W_{1D}l_{\rm bl}+(1-W_{1D})l_{\rm smag}, + l_{\mathrm{blend}} = W_{1D}l_{\mathrm{bl}}+(1-W_{1D})l_{\mathrm{smag}}, -where :math:`l_{\rm bl}^{-1} = (\kappa z)^{-1} + \lambda_0^{-1}` and +where :math:`l_{\mathrm{bl}}^{-1} = (\kappa z)^{-1} + \lambda_0^{-1}` and :math:`l_{\rm smag}^{-2} = (\kappa z)^{-2} + (c_s \Delta x)^{-2}`, :math:`\kappa` is the von Karman constant and :math:`c_s` is the Smagorinsky constant. -Near the surface :math:`l_{\rm bl}` and :math:`l_{\rm smag}` are +Near the surface :math:`l_{\mathrm{bl}}` and :math:`l_{\mathrm{smag}}` are identical, but the asymptotic values are different and this method weights the asymptotic value according to the weighting of the two schemes. For example, at :math:`\Delta x=1` km, :math:`c_s\Delta x=200` m (for :math:`c_s=0.2`), whereas -:math:`\lambda_0=\max(40\ {\rm m}, 0.15z_h)`, which allows for a small +:math:`\lambda_0=\max(40\ {\mathrm{m}}, 0.15z_h)`, which allows for a small mixing length in shallow unresolved boundary layers (e.g. stable ones). The `Lock et al. (2000)`_ scheme also contains a non-local @@ -1990,15 +2008,15 @@ is given by .. math:: K_\chi = \max\left[W_{1D}K_\chi^{\rm NL}, K_\chi(Ri)\right], -where :math:`K_\chi^{\rm NL}` is the non-local diffusivity and :math:`l` -in Eq. :eq:`eq-kri` is given by :math:`l_{\rm blend}` in +where :math:`K_\chi^{\mathrm{NL}}` is the non-local diffusivity and :math:`l` +in Eq. :eq:`eq-kri` is given by :math:`l_{\mathrm{blend}}` in Eq. :eq:`eq-lblend`. The turbulent flux is then calculated as .. math:: F_\chi=-K_\chi\frac{\partial \chi}{\partial z} + W_{1D}F_\chi^{\rm NL}, -where :math:`F_\chi^{\rm NL}` is the non-local flux. Therefore when +where :math:`F_\chi^{\mathrm{NL}}` is the non-local flux. Therefore when :math:`W_{1D}=1`, the scheme of `Lock et al. (2000)`_ is recovered, whilst with :math:`W_{1D}=0` the Smagorinsky-type scheme is recovered. @@ -2012,11 +2030,11 @@ function slightly, using .. math:: :label: eq-tanh - W_{1D} = 1 - \tanh\left(\beta\frac{z_{\rm turb}}{\Delta - x}\right)\max\left[0,\min\left[1,r_f\left(l_0-\frac{\Delta x}{z_{\rm - turb}}\right)\right] \right], + W_{1D} = 1 - \tanh\left(\beta\frac{z_{\mathrm{turb}}}{\Delta + x}\right)\max\left[0,\min\left[1,r_f\left(l_0-\frac{\Delta + x}{z_{\mathrm{turb}}}\right)\right] \right], -where :math:`z_{\rm turb}` is the appropriate lengthscale of the +where :math:`z_{\mathrm{turb}}` is the appropriate lengthscale of the turbulence, :math:`\beta` is a scaling parameter which controls the speed of the transition from unresolved to resolved turbulence, :math:`r_f=\frac{1}{l_0-l_1}`, :math:`l_0=4` and :math:`l_1=0.25` @@ -2024,9 +2042,9 @@ speed of the transition from unresolved to resolved turbulence, :raw-latex:`\cite[]{Boutleetal2014}`). `Malavelle et al. (2014)`_ demonstrated that this scaling method was applicable to any type of unstable boundary layer given an -appropriate choice of :math:`z_{\rm turb}`. In +appropriate choice of :math:`z_{\mathrm{turb}}`. In `Boutle et al. (2014)`_ this functional form was applied -everywhere, adjusting the values of :math:`z_{\rm turb}` and +everywhere, adjusting the values of :math:`z_{\mathrm{turb}}` and :math:`\beta` depending on the regime. The max function is present to force the lowest resolution simulations to just use the 1D mixing scheme. An alternative approach that differs above the boundary layer is @@ -2034,7 +2052,7 @@ described below. The simplest case is for a well-mixed boundary layer, where the appropriate lengthscale is the boundary-layer depth (inversion height). -Therefore we set :math:`z_{\rm turb}=z_h`, which is broadly consistent +Therefore we set :math:`z_{\mathrm{turb}}=z_h`, which is broadly consistent with `Malavelle et al. (2014)`_, and choose :math:`\beta=\beta_{\rm bl}=0.15` to give the best match of our function to that of @@ -2071,23 +2089,22 @@ feature which needs to be maintained in the blended scheme. Physically they are similar to well-mixed surface driven boundary layers, and the `Lock et al. (2000)`_ scheme parametrizes them as such. The appropriate length scale is now the decoupled cloud mixed layer depth, -:math:`z_{\rm sc}` +:math:`z_{\mathrm{sc}}` :raw-latex:`\cite[i.e.~the depth through which a negatively buoyant parcel released at cloud top would descend,][]{lock01}`. In this case, below the decoupled cloud top we set .. math:: :label: zturb_dsc - z_{\rm turb}=\min\left[\max\left(z,z_{\rm sml}\right),\max\left(z_{\rm - sc},z_h-z\right)\right], + z_{\mathrm{turb}}=\min\left[\max\left(z,z_{\mathrm{sml}}\right),\max\left(z_{\mathrm{sc}},z_h-z\right)\right], -where :math:`z_{\rm sml}` is the depth of the surface-based mixed layer +where :math:`z_{\mathrm{sml}}` is the depth of the surface-based mixed layer :raw-latex:`\cite[i.e.~the depth through which a positively buoyant parcel released at the surface would ascend,][]{lock00}`. This is shown schematically in :numref:`Figure %s `\ (b), and ensures that :math:`W_{1D}` has a high value in the poorly resolved surface mixed layer and cloud layer, and a lower value in between those layers. Again, -this choice of :math:`z_{\rm turb}` is broadly consistent with the +this choice of :math:`z_{\mathrm{turb}}` is broadly consistent with the analysis of decoupled stratocumulus LES presented by `Malavelle et al. (2014)`_. Finally, `Honnert et al. (2011)`_ also included shallow cumulus @@ -2104,7 +2121,7 @@ used to identify a cumulus regime) was found frequently to indicate deep convection even when the resolved clouds were shallow because the diagnosis parcel, being undilute, would penetrate to the tropopause. However, having decided the regime is shallow convection, we do still -set :math:`z_{\rm turb}` to the diagnosis parcel top height because, for +set :math:`z_{\mathrm{turb}}` to the diagnosis parcel top height because, for current km-scale configurations (without a cumulus convection parametrization), it was found that the resulting stronger parametrized vertical mixing was beneficial for the development of the convection, @@ -2114,46 +2131,48 @@ instead. Above the boundary layer top, `Boutle et al. (2014)`_ aimed for any free atmospheric mixing to be done by the 3D Smagorinsky scheme. Therefore, above the boundary layer top they use :math:`z` as the -appropriate length scale, and in general take :math:`z_{\rm turb}` in +appropriate length scale, and in general take :math:`z_{\mathrm{turb}}` in Eq. :eq:`eq-tanh` as the greater of that defined by :eq:`zturb_dsc` and :math:`z`. However, this did not give a particularly fast transition using the value of -:math:`\beta_{\rm bl}`, therefore they used :math:`\beta_{\rm fa}=1` at -a height well above the boundary layer (:math:`z_{\rm fa}=z_h+1` km), +:math:`\beta_{\mathrm{bl}}`, therefore they used :math:`\beta_{\mathrm{fa}}=1` +at +a height well above the boundary layer (:math:`z_{\mathrm{fa}}=z_h+1` km), and transitioned between these regimes linearly using .. math:: - \beta = \beta_{\rm bl}\frac{z_{\rm fa}-z}{z_{\rm fa}-z_h} + - \beta_{\rm fa}\frac{z-z_h}{z_{\rm fa}-z_h} + \beta = \beta_{\mathrm{bl}}\frac{z_{\mathrm{fa}}-z}{z_{\mathrm{fa}}-z_h} + + \beta_{\mathrm{fa}}\frac{z-z_h}{z_{\mathrm{fa}}-z_h} However, because the above method still uses :eq:`eq-tanh`, -which depends on :math:`z_{\rm turb}/\Delta x`, the rate of transition +which depends on :math:`z_{\mathrm{turb}}/\Delta x`, the rate of transition to 3D Smagorinsky with height above the boundary layer varies in an undesirable way with grid size. It might be considered more logical to think of non-turbulent regions of the free troposphere as unresolved turbulence and so revert to the 1D mixing scheme there. An alternative treatment(``blending_option``\ :math:`=`\ 3 or 4), then, is to increase :math:`W_{1D}` above the boundary layer top smoothly, to reach unity by -some physical height :math:`z_{\rm fa}`, to be independent of both +some physical height :math:`z_{\mathrm{fa}}`, to be independent of both horizontal and vertical grid sizes. For -:math:`z_{\rm turb} < z < z_{\rm fa}`, then, we set +:math:`z_{\mathrm{turb}} < z < z_{\mathrm{fa}}`, then, we set .. math:: - W_{1D} = 1 + \frac{1}{2} \left( W_{1D}|_{z=z_{\rm turb}} - 1 \right) - \left[ 1 + {\rm cos}\left( \pi \, \frac{z-z_{\rm turb}}{z_{\rm - fa}-z_{\rm turb}} + W_{1D} = 1 + \frac{1}{2} \left( W_{1D}|_{z=z_{\mathrm{turb}}} - 1 \right) + \left[ 1 + {\mathrm{cos}}\left( \pi \, + \frac{z-z_{\mathrm{turb}}}{z_{\mathrm{fa}}-z_{\mathrm{turb}}} \right) \right] The cosine term in square brackets transitions smoothly from 2 at -:math:`z=z_{\rm turb}` to zero at :math:`z_{fa}` where, although +:math:`z=z_{\mathrm{turb}}` to zero at :math:`z_{fa}` where, although somewhat arbitrary, -:math:`z_{\rm fa} = {\rm min}(2 z_{\rm turb}, z_{\rm turb}+1 {\rm km})`. +:math:`z_{\mathrm{fa}} = {\mathrm{min}}(2 z_{\mathrm{turb}}, +z_{\mathrm{turb}}+1 {\mathrm{km}})`. The former term ensures the transition is well above any shallow boundary layers while the latter that it does not drift far into the free atmosphere. In addition, within any layers identified as turbulent, -through having subcritical :math:`Ri`, :math:`z_{\rm turb}` is set to +through having subcritical :math:`Ri`, :math:`z_{\mathrm{turb}}` is set to the layer depth, in the same way as is done for decoupled stratocumulus in :eq:`zturb_dsc`. @@ -2207,13 +2226,14 @@ the velocity scales `) .. math:: :label: we_parm - w_e = \frac{ A_1 \, V_{\rm sum}^3/ z_{\rm ml}+ g \tilde{\beta_T} + w_e = \frac{ A_1 \, V_{\mathrm{sum}}^3/ z_{\mathrm{ml}}+ g \tilde{\beta_T} \tilde{\alpha_t} \Delta_F} - {\Delta b + c_T V_{\rm sum}^2/z_{\rm ml}} + {\Delta b + c_T V_{\mathrm{sum}}^2/z_{\mathrm{ml}}} where -:math:`V_{\rm sum}^3= V_{\rm heat}^3+ V_{\rm rad}^3+ V_{\rm br}^3+ A_2 u_*^3`. +:math:`V_{\mathrm{sum}}^3= V_{\mathrm{heat}}^3+ V_{\mathrm{rad}}^3+ +V_{\mathrm{br}}^3+ A_2 u_*^3`. The constant :math:`A_1` is given a value 0.23, as in `Lock (1998)`_, and :math:`A_1*A_2=5`, as in `Driedonks (1982)`_. To allow for weak inversions, the @@ -2242,14 +2262,14 @@ quantities required for :eq:`we_parm` is described in appendix :ref:`Appendix: Definitions of the velocity scales `. At some point during the transition to a decoupled boundary layer the surface-driven entrainment terms (the terms -in :eq:`we_parm` proportional to :math:`V_{\rm heat}^3` and +in :eq:`we_parm` proportional to :math:`V_{\mathrm{heat}}^3` and :math:`u_*`) will no longer contribute to entrainment at cloud top, because the two layers will have become entirely decoupled. If the ``entr_smooth_dec`` switch is on then the surface contribution to the -parametrized entrainment at :math:`z_{\rm h}^{\rm Sc}` is decreased +parametrized entrainment at :math:`z_{\mathrm{h}}^{\mathrm{Sc}}` is decreased linearly as the :math:`\theta_{v\ell}` difference between NTDSC and NTML increases from 0.5 to 1K. The flag, COUPLED, is set to true and -:math:`z_{\rm h}^{\rm Sc}` is used as the mixed-layer depth in +:math:`z_{\mathrm{h}}^{\mathrm{Sc}}` is used as the mixed-layer depth in :eq:`we_parm` as long as any surface-driven entrainment remains. If the ``entr_smooth_dec`` switch is off then this transition is discontinuous at a :math:`\theta_{v\ell}` difference of 0.5K. @@ -2273,7 +2293,7 @@ limit is applied to the value of :math:`w_e` determined by :eq:`we_parm` such that the inversion cannot rise by more than one grid-level in a timestep. With current vertical resolutions and timesteps this is not a serious restriction. The constants :math:`A_1` -and :math:`A_{\rm br}` appeared to be determined within 10-20 % in +and :math:`A_{\mathrm{br}}` appeared to be determined within 10-20 % in `Lock (1998)`_, although only solid cloud sheets were simulated (as discussed further in appendix :ref:`Appendix: Definitions of the velocity scales `). @@ -2298,59 +2318,60 @@ F|_{z_i}`, so that .. math:: - {\cal H}|_{z_i} = - w_e \Delta \theta_{\ell}+ F_{\rm net}|_h + {\cal H}|_{z_i} = - w_e \Delta \theta_{\ell}+ F_{\mathrm{net}}|_h .. math:: :label: discinv \overline{w'q_t'}_{z_i} = - w_e \Delta q_t where the total heat flux -:math:`{\cal H} = \overline{w'\theta_{\ell}'}+ F_{\rm net}` and +:math:`{\cal H} = \overline{w'\theta_{\ell}'}+ F_{\mathrm{net}}` and :math:`F_{\rm net} = F- F|_{z_{\rm b}}`. The net radiative flux relative to the base of the mixed layers is simply calculated as .. math:: - F_{\rm net}|_{z_{k+\frac{1}{2}}} = \sum_{k=\mathrm{ \rm NBDSC}}^{k} - \mathrm{max}\left[ + F_{\mathrm{net}}|_{z_{k+\frac{1}{2}}} = \sum_{k=\mathrm{ + \mathrm{NBDSC}}}^{k} \mathrm{max}\left[ - \Delta_{k+\frac{1}{2}} z \, {\cal S}_F(k), \,0 \right] where NBDSC\ :math:`=1` in SMLs, :math:`{\cal S}_F` are the temperature increments (in Ks\ :math:`^{-1}`) from the radiation scheme and -:math:`F_{\rm net}|_h` is estimated by extrapolating down from +:math:`F_{\mathrm{net}}|_h` is estimated by extrapolating down from :math:`F|_{z_{\mbox{\tiny \rm NTML}+\frac{3}{2}} }` using the flux-divergence in grid-level NTML\ :math:`+2` (and similarly for DSC layers). The thermodynamic variables’ entrainment fluxes, then, are imposed nominally at the subgrid inversion height (:math:`z_i=` -:math:`z_{\rm h}` and/or :math:`z_{\rm h}^{\rm Sc}` ), diagnosed as +:math:`z_{\mathrm{h}}` and/or :math:`z_{\mathrm{h}}^{\mathrm{Sc}}` ), diagnosed +as described in section :ref:`Diagnosis of a sub-grid inversion `. The required grid-level -fluxes (at :math:`z_{\mathrm{ \rm NTDSC}+\frac{1}{2}}`, for example) +fluxes (at :math:`z_{\mathrm{ \mathrm{NTDSC}}+\frac{1}{2}}`, for example) are then estimated using linear interpolation of :math:`{\cal H}` and -:math:`\overline{w'q_t'}` between :math:`z_{\rm h}^{\rm Sc}` and the +:math:`\overline{w'q_t'}` between :math:`z_{\mathrm{h}}^{\mathrm{Sc}}` and the base of the mixed layer: .. math:: - \overline{w'\theta_{\ell}'}|_{ z_{\mathrm{ \rm NTDSC}+\frac{1}{2}} } = - \overline{w'\theta_{\ell}'}|_{z_{\rm b}} - - \frac{ z'_{\mathrm{ \rm NTDSC}+\frac{1}{2}} }{z_{\rm ml}} + \overline{w'\theta_{\ell}'}|_{ z_{\mathrm{ \mathrm{NTDSC}}+\frac{1}{2}} } = + \overline{w'\theta_{\ell}'}|_{z_{\mathrm{b}}} + - \frac{ z'_{\mathrm{ \mathrm{NTDSC}}+\frac{1}{2}} }{z_{\mathrm{ml}}} \left( \tilde{w_e} \Delta \theta_{\ell}+ - \overline{w'\theta_{\ell}'}|_{z_{\rm b}} - F_{\rm net}|_{h} \right) - - F_{\rm net}|_{ z_{\mathrm{ \rm NTDSC}+\frac{1}{2}} } + \overline{w'\theta_{\ell}'}|_{z_{\mathrm{b}}} - F_{\mathrm{net}}|_{h} \right) + - F_{\mathrm{net}}|_{ z_{\mathrm{ \mathrm{NTDSC}}+\frac{1}{2}} } .. math:: :label: fluxinterp - \overline{w'q_t'}|_{ z_{\mathrm{ \rm NTDSC}+\frac{1}{2}} } = - \overline{w'q_t'}|_{z_{\rm b}} - - \frac{ z'_{\mathrm{ \rm NTDSC}+\frac{1}{2}} }{z_{\rm ml}} - \left( \tilde{w_e} \Delta q_t + \overline{w'q_t'}|_{z_{\rm b}} \right) + \overline{w'q_t'}|_{ z_{\mathrm{ \mathrm{NTDSC}}+\frac{1}{2}} } = + \overline{w'q_t'}|_{z_{\mathrm{b}}} + - \frac{ z'_{\mathrm{ \mathrm{NTDSC}}+\frac{1}{2}} }{z_{\mathrm{ml}}} + \left( \tilde{w_e} \Delta q_t + \overline{w'q_t'}|_{z_{\mathrm{b}}} \right) -where :math:`z' = z-z_{\rm b}`, and similarly for the SML entrainment -fluxes (at :math:`z=z_{\mathrm{ \rm NTML}+\frac{1}{2}}`). The +where :math:`z' = z-z_{\mathrm{b}}`, and similarly for the SML entrainment +fluxes (at :math:`z=z_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}}`). The turbulent fluxes at the base of the mixed layer are assumed zero except for the SML where the surface fluxes are used. This interpolation is illustrated for a SML in :numref:`Fig. %s `. @@ -2388,21 +2409,21 @@ the parametrization of :math:`w_e` and the model’s subsidence velocity, :math:`w_S|_{z_i}`, are used to calculate :math:`z_i` at the next time-level (:math:`z_i^{n+1}`). Currently, the latter is found by linear interpolation to :math:`z_i` and both are assumed constant in time. If -:math:`z_i^{n+1} < z_{\mathrm{ \rm NTDSC}+\frac{1}{2}}`, then the +:math:`z_i^{n+1} < z_{\mathrm{ \mathrm{NTDSC}}+\frac{1}{2}}`, then the entrainment fluxes there (given by :eq:`fluxinterp`) are multiplied by the fraction of the timestep that :math:`z_i` was above this grid-level, namely -:math:`(z_i-z_{\mathrm{ \rm NTDSC}+\frac{1}{2}})/(z_i - z_i^{n+1})`. +:math:`(z_i-z_{\mathrm{ \mathrm{NTDSC}}+\frac{1}{2}})/(z_i - z_i^{n+1})`. The full entrainment flux at grid-level NTDSC\ :math:`-\frac{1}{2}` must then also be specified, given by :eq:`fluxinterp` with -:math:`z_{\mathrm{ \rm NTDSC}+ \frac{1}{2}}` replaced by -:math:`z_{\mathrm{ \rm NTDSC}- \frac{1}{2}}`. If :math:`z_i` rises -above :math:`z_{\mathrm{ \rm NTDSC}+\frac{3}{2}}`, the entrainment +:math:`z_{\mathrm{ \mathrm{NTDSC}}+ \frac{1}{2}}` replaced by +:math:`z_{\mathrm{ \mathrm{NTDSC}}- \frac{1}{2}}`. If :math:`z_i` rises +above :math:`z_{\mathrm{ \mathrm{NTDSC}}+\frac{3}{2}}`, the entrainment flux is specified only at this higher grid-level (multiplied by the fraction of the timestep that :math:`z_i` is above this half-level) and the values of the mixed-layer :math:`K` profiles are used in half-level NTDSC\ :math:`+\frac{1}{2}` (these will be non-zero because -:math:`z_i>z_{\mathrm{ \rm NTDSC}+ \frac{1}{2}}`). Wherever the +:math:`z_i>z_{\mathrm{ \mathrm{NTDSC}}+ \frac{1}{2}}`). Wherever the entrainment fluxes are specified explicitly, the eddy-diffusivities (both non-local and local) are set to zero. Also, the mean value of :math:`z_i` during the timestep is used in @@ -2438,8 +2459,8 @@ and .. math:: :label: we_num - \tilde{w_S} = - \, \frac{ \Theta^{\rm S}_{\mathrm{ \rm NTML}} - ( \Delta_{\mathrm{ \rm NTML}+\frac{1}{2}} z ) } + \tilde{w_S} = - \, \frac{ \Theta^{\mathrm{S}}_{\mathrm{ \mathrm{NTML}}} + ( \Delta_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}} z ) } { \Delta \theta_{\ell}} .. _sec_sginv: @@ -2487,10 +2508,10 @@ for the model and for a profile with a discontinuous inversion at :math:`z_i` are equal, as illustrated by the hatched areas in :numref:`Fig. %s `. To calculate the integral of the discontinuous profile, the lapse rate of :math:`\theta_{v\ell}` between grid-levels -NTML\ :math:`-1` and :math:`NTML`, :math:`\gamma^{ \rm ML}`, is +NTML\ :math:`-1` and :math:`NTML`, :math:`\gamma^{ \mathrm{ML}}`, is extended up to :math:`z_i`, while the stable lapse in the free atmosphere, between grid-levels NTML\ :math:`+2` and NTML\ :math:`+3`, -:math:`\gamma^{\scriptsize \rm FA}`, is extrapolated down. Equating +:math:`\gamma^{\scriptsize \mathrm{FA}}`, is extrapolated down. Equating these areas gives a quadratic equation in :math:`\Delta z_{disc} = z_{\mbox{\tiny \rm NTML}+\frac{3}{2}} - z_i` which can be written @@ -2502,25 +2523,26 @@ The coefficients are given by .. math:: - a = 0.5 (\gamma^{\scriptsize \rm FA}- \gamma^{ \rm ML}) + a = 0.5 (\gamma^{\scriptsize \mathrm{FA}}- \gamma^{ \mathrm{ML}}) .. math:: - b = - \left( {\theta_{v\ell}}_{\mathrm{ \rm NTML}+2} - - \gamma^{\scriptsize \rm FA}(z_{\mathrm{ \rm NTML}+2}-z_{\mathrm{ \rm - NTML}+\frac{3}{2}}) \right) - + \left( {\theta_{v\ell}}_{\mathrm{ \rm NTML}} - + \gamma^{ \rm ML}(z_{\mathrm{ \rm NTML}+\frac{3}{2}}-z_{\mathrm{ \rm NTML}}) \right) + b = - \left( {\theta_{v\ell}}_{\mathrm{ \mathrm{NTML}}+2} + - \gamma^{\scriptsize \mathrm{FA}}(z_{\mathrm{ + \mathrm{NTML}}+2}-z_{\mathrm{ \mathrm{NTML}}+\frac{3}{2}}) \right) + + \left( {\theta_{v\ell}}_{\mathrm{ \mathrm{NTML}}} + + \gamma^{ \mathrm{ML}}(z_{\mathrm{ \mathrm{NTML}}+\frac{3}{2}}-z_{\mathrm{ \mathrm{NTML}}}) \right) .. math:: - c = (z_{\mathrm{ \rm NTML}+\frac{3}{2}}-z_{\mathrm{ \rm NTML}+\frac{1}{2}}) - \left( {\theta_{v\ell}}_{\mathrm{ \rm NTML}+1} - - \left( {\theta_{v\ell}}_{\mathrm{ \rm NTML}} + - \gamma^{ \rm ML}\left( - \frac{1}{2}(z_{\mathrm{ \rm NTML}+\frac{1}{2}}+z_{\mathrm{ \rm - NTML}+\frac{3}{2}}) - -z_{\mathrm{ \rm NTML}} \right) \right) + c = (z_{\mathrm{ \mathrm{NTML}}+\frac{3}{2}}-z_{\mathrm{ + \mathrm{NTML}}+\frac{1}{2}}) + \left( {\theta_{v\ell}}_{\mathrm{ \mathrm{NTML}}+1} - + \left( {\theta_{v\ell}}_{\mathrm{ \mathrm{NTML}}} + + \gamma^{ \mathrm{ML}}\left( + \frac{1}{2}(z_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}}+z_{\mathrm{ + \mathrm{NTML}}+\frac{3}{2}}) + -z_{\mathrm{ \mathrm{NTML}}} \right) \right) \right) Clearly, care must be taken to ensure that :math:`z_i` is not only @@ -2531,11 +2553,11 @@ therefore set to zero and :eq:`zi_interp` is recalculated. The case :math:`c<0` suggests the grid-level designated as the inversion level should have been considered as part of the mixed layer and so :math:`z_i` is set to be fractionally below -:math:`z_{\mathrm{ \rm NTML}+\frac{3}{2}}` (i.e., as high as possible +:math:`z_{\mathrm{ \mathrm{NTML}}+\frac{3}{2}}` (i.e., as high as possible without attempting to diagnose a subgrid :math:`z_i` in grid-level NTML\ :math:`+2`). If :math:`b^2-4ac<0` the quadratic equation has no real roots. In this instance :math:`z_i` is set to fractionally below -:math:`z_{\mathrm{ \rm NTML}+\frac{1}{2}}` and NTML (and therefore +:math:`z_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}}` and NTML (and therefore the eddy-diffusivity profiles) is lowered by a grid-level. In all other circumstances, the required root is then :math:`\Delta z_{disc} = (-b - (b^2-4ac)^{1/2} @@ -2545,12 +2567,12 @@ circumstances, the required root is then In addition, from variations seen in :math:`z_i` during single-column model simulations, the error in :math:`\Delta z_{disc}` is estimated to be around 10% of the vertical resolution, -:math:`\Delta_{\mathrm{ \rm NTML}+\frac{3}{2}} z`. Accordingly, if +:math:`\Delta_{\mathrm{ \mathrm{NTML}}+\frac{3}{2}} z`. Accordingly, if :math:`z_i` is diagnosed as being less than -:math:`z_{\mathrm{ \rm NTML}+\frac{1}{2}} + 0.1 \, \Delta_{\mathrm{ \rm -NTML}+\frac{3}{2}} z`, +:math:`z_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}} + 0.1 \, \Delta_{\mathrm{ +\mathrm{NTML}}+\frac{3}{2}} z`, NTML is lowered a grid-level and :math:`z_i` is set fractionally below -:math:`z_{\mathrm{ \rm NTML}+\frac{1}{2}}`. This small distance below +:math:`z_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}}`. This small distance below the grid-level is taken to be :math:`(\Delta t/2) \times 10^{-4}` so that, were a small rate of rise of :math:`z_i` (of :math:`10^{-4}` ms\ :math:`^{-1}`, say) to be diagnosed, then :math:`z_i` would spend at @@ -2566,20 +2588,20 @@ calculation are calculated from similar integral assumptions: .. math:: :label: dqt_disc - \Delta \chi = \left( {\chi}_{\mathrm{ \rm NTML}+1} - {\chi}_{\mathrm{ \rm - NTML}} \right) \, - \frac{ z_{\mathrm{ \rm NTML}+\frac{3}{2}} - z_{\mathrm{ \rm - NTML}+\frac{1}{2}} } - { z_{\mathrm{ \rm NTML}+\frac{3}{2}} - z_i } + \Delta \chi = \left( {\chi}_{\mathrm{ \mathrm{NTML}}+1} - {\chi}_{\mathrm{ + \mathrm{NTML}}} \right) \, + \frac{ z_{\mathrm{ \mathrm{NTML}}+\frac{3}{2}} - z_{\mathrm{ + \mathrm{NTML}}+\frac{1}{2}} } + { z_{\mathrm{ \mathrm{NTML}}+\frac{3}{2}} - z_i } with :math:`\chi = \theta_{\ell}` and :math:`q_t`. Note that the lapse rate above the inversion has been ignored as there is no guarantee of monotonicity in :math:`q_t` in the atmosphere above the inversion. In addition, :eq:`dqt_disc` will become increasingly inaccurate as :math:`z_i` tends to -:math:`z_{\mathrm{ \rm NTML}+\frac{3}{2}}` (and so -:math:`{\chi}_{\mathrm{ \rm NTML}+1}` approaches -:math:`{\chi}_{\mathrm{ \rm NTML}}`). Consequently, if the fraction +:math:`z_{\mathrm{ \mathrm{NTML}}+\frac{3}{2}}` (and so +:math:`{\chi}_{\mathrm{ \mathrm{NTML}}+1}` approaches +:math:`{\chi}_{\mathrm{ \mathrm{NTML}}}`). Consequently, if the fraction on the right hand side of :eq:`dqt_disc` is greater than 10, double grid-level jumps are used (i.e., :math:`\Delta \chi = {\chi}_{\mbox{\tiny \rm NTML}+2} - @@ -2639,7 +2661,7 @@ capped by a diagnosed subgrid inversion. The crucial step is to ensure that the total flux on the model entrainment grid-level equals the idealised total flux profile interpolated to that level. Consider the example illustrated in :numref:`Fig. %s ` of a well-mixed -boundary layer up to :math:`\theta`-level :math:`\mathrm{ \rm NTML}`. +boundary layer up to :math:`\theta`-level :math:`\mathrm{ \mathrm{NTML}}`. For the subgrid :math:`q_t` profiles, the turbulent flux divergence generates a moistening across the inversion while subsidence generates drying. For this example it has been assumed the entrainment rate is @@ -2648,14 +2670,14 @@ overall there is a weak moistening relative to the mixed layer (the total flux gradient is more negative across the inversion than in the mixed layer), consistent with the rising tendency of the inversion. For the model, the subsidence flux-divergence associated with the inversion -is split across levels :math:`\mathrm{ \rm NTML}` and -:math:`\mathrm{ \rm NTML}+1`. To keep the *net* moistening of the +is split across levels :math:`\mathrm{ \mathrm{NTML}}` and +:math:`\mathrm{ \mathrm{NTML}}+1`. To keep the *net* moistening of the model’s boundary layer and inversion consistent with the total subgrid flux profile, the model’s entrainment flux at -:math:`\mathrm{ \rm NTML}+1/2` (shown by the cross in +:math:`\mathrm{ \mathrm{NTML}}+1/2` (shown by the cross in :numref:`Fig. %s `) must be found by subtracting the -subsidence flux at :math:`\mathrm{ \rm NTML}+1/2` (diamond) from the -total flux interpolated to :math:`\mathrm{ \rm NTML}+1/2` (square). +subsidence flux at :math:`\mathrm{ \mathrm{NTML}}+1/2` (diamond) from the +total flux interpolated to :math:`\mathrm{ \mathrm{NTML}}+1/2` (square). Exactly the same arguments follow for the :math:`\theta_{\ell}` fluxes except that the situation is complicated by the addition of the radiative flux. @@ -2690,13 +2712,13 @@ cooling occurring within undulations of the cloudy boundary layer top. Similar considerations need to be borne in mind when calculating all the non-turbulent fluxes in :eq:`fxtot_zi`. First, the radiative flux is extrapolated down from -:math:`\mathrm{ \rm NTML}+\frac{3}{2}` to :math:`z=z_t` using the +:math:`\mathrm{ \mathrm{NTML}}+\frac{3}{2}` to :math:`z=z_t` using the divergence in the grid-level above the inversion as representative of the free-atmospheric divergence. Second, since the microphysical flux is generated within the cloud, :math:`F_{\chi}^{ppn}|_{z_t} = {F_{\chi}}^{ppn}_{\mbox{\tiny \rm NTML}+\frac{3}{2}}`. Finally, the -subsidence flux-divergence across level :math:`\mathrm{ \rm NTML}` -and :math:`\mathrm{ \rm NTML}+1` is assumed to be associated with the +subsidence flux-divergence across level :math:`\mathrm{ \mathrm{NTML}}` +and :math:`\mathrm{ \mathrm{NTML}}+1` is assumed to be associated with the inversion so :math:`{F_{\chi}}^{Subs}|_{z_h} = {F_{\chi}}^{Subs}_{\mbox{\tiny \rm NTML}-\frac{1}{2}}`. Thus, the finite-difference form of :eq:`fxtot_zi` becomes @@ -2704,8 +2726,9 @@ finite-difference form of :eq:`fxtot_zi` becomes .. math:: :label: fxtot_zi_fd F_{\chi}^{Tot}|_{z_h} = - w_e \Delta \chi + - F_{\chi}^{rad}|_{z_t} + {F_{\chi}}^{ppn}_{\mathrm{ \rm NTML}+\frac{3}{2}} + - {F_{\chi}}^{subs}_{\mathrm{ \rm NTML}-\frac{1}{2}} + F_{\chi}^{rad}|_{z_t} + {F_{\chi}}^{ppn}_{\mathrm{ + \mathrm{NTML}}+\frac{3}{2}} + + {F_{\chi}}^{subs}_{\mathrm{ \mathrm{NTML}}-\frac{1}{2}} Then, assuming a linear profile of :math:`F_{\chi}^{Tot}` in the mixed layer, interpolating the total flux to the inversion flux grid-level @@ -2713,20 +2736,21 @@ gives .. math:: :label: fxtot_interp - F_{\chi}^{Tot}|_{ \mathrm{ \rm NTML}+\frac{1}{2} } = F_{\chi}^{Tot}|_{z_{\rm - b}} + - \frac{ z'_{\mathrm{ \rm NTML}+\frac{1}{2}} }{z_{\rm ml}} - \left( F_{\chi}^{Tot}|_{z_h} - F_{\chi}^{Tot}|_{z_{\rm b}} \right) + F_{\chi}^{Tot}|_{ \mathrm{ \mathrm{NTML}}+\frac{1}{2} } = + F_{\chi}^{Tot}|_{z_{\mathrm{b}}} + + \frac{ z'_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}} }{z_{\mathrm{ml}}} + \left( F_{\chi}^{Tot}|_{z_h} - F_{\chi}^{Tot}|_{z_{\mathrm{b}}} \right) -where :math:`z'` (:math:`=z-z_{\rm b}`) is height above the base of the -mixed layer at :math:`z=z_{\rm b}`. Finally, the grid-level turbulent +where :math:`z'` (:math:`=z-z_{\mathrm{b}}`) is height above the base of the +mixed layer at :math:`z=z_{\mathrm{b}}`. Finally, the grid-level turbulent entrainment flux is given by: .. math:: :label: rev_entflux - \overline{w'\chi'}|_{ \mathrm{ \rm NTML}+\frac{1}{2} } = F_{\chi}^{Tot}|_{ - \mathrm{ \rm NTML}+\frac{1}{2} } - - F_{\chi}^{NT}|_{ \mathrm{ \rm NTML}+\frac{1}{2} } + \overline{w'\chi'}|_{ \mathrm{ \mathrm{NTML}}+\frac{1}{2} } = + F_{\chi}^{Tot}|_{ \mathrm{ \mathrm{NTML}}+\frac{1}{2} } + - F_{\chi}^{NT}|_{ \mathrm{ + \mathrm{NTML}}+\frac{1}{2} } This revised algorithm has several advantages over the previous. Firstly, the fluxes for :math:`q_t` and :math:`\theta_{\ell}` are @@ -2736,7 +2760,7 @@ in order to calculate :math:`\tilde{w_e}` in :eq:`we_num`. Secondly, this method makes it much simpler to include all processes, and precipitation in particular, in a consistent manner. Thirdly, since the total grid-level flux, -:math:`F_{\chi}^{Tot}|_{\mathrm{ \rm NTML}+\frac{1}{2}}` in +:math:`F_{\chi}^{Tot}|_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}}` in :eq:`fxtot_interp`, is used to calculate the entrainment fluxes, it is straightforward to ensure that the net budget of the inversion grid-level, namely :math:`- ( @@ -2745,7 +2769,7 @@ F_{\chi}^{Tot}|_{\mbox{\tiny \rm NTML}+\frac{1}{2}})/\Delta z`, is consistent with the entrainment/subsidence balance. In other words, to use :math:`\theta_{\ell}` as an example, if the inversion is rising (falling) then -:math:`F_{\chi}^{Tot}|_{\mathrm{ \rm NTML}+\frac{1}{2}}` is limited +:math:`F_{\chi}^{Tot}|_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}}` is limited to ensure that the inversion grid-level will cool (warm). Finally, if the inversion is rising we don’t want the inversion grid-level :math:`\theta_{\ell}` to cool to less than :math:`\theta_{\ell}` of the @@ -2754,34 +2778,37 @@ mixed layer by the end of the timestep. In other words, for .. math:: - \chi_{\mathrm{ \rm NTML}+1}^{n+1} = \chi_{\mathrm{ \rm NTML}+1}^{n} + \chi_{\mathrm{ \mathrm{NTML}}+1}^{n+1} = \chi_{\mathrm{ + \mathrm{NTML}}+1}^{n} - \frac{\Delta t}{\Delta z} \left( - F_{\chi}^{Tot}|_{ \mathrm{ \rm NTML}+\frac{3}{2} } - F_{\chi}^{Tot}|_{ - \mathrm{ \rm NTML}+\frac{1}{2} } + F_{\chi}^{Tot}|_{ \mathrm{ \mathrm{NTML}}+\frac{3}{2} } - + F_{\chi}^{Tot}|_{ \mathrm{ \mathrm{NTML}}+\frac{1}{2} } \right) .. math:: - \chi_{\mathrm{ \rm NTML}}^{n+1} = \chi_{\mathrm{ \rm NTML}}^{n} - - \frac{\Delta t}{z_{\mathrm{ \rm NTML}+\frac{1}{2}}} \left( - F_{\chi}^{Tot}|_{ \mathrm{ \rm NTML}+\frac{1}{2} } - - F_{\chi}^{Tot}|_{z_{\rm b}} + \chi_{\mathrm{ \mathrm{NTML}}}^{n+1} = \chi_{\mathrm{ \mathrm{NTML}}}^{n} + - \frac{\Delta t}{z_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}}} \left( + F_{\chi}^{Tot}|_{ \mathrm{ \mathrm{NTML}}+\frac{1}{2} } - + F_{\chi}^{Tot}|_{z_{\mathrm{b}}} \right) where the superscripts :math:`n` and :math:`n+1` refer to the model timestep, although strictly speaking :math:`n+1` refers to fields after the boundary layer implicit solver. Requiring that -:math:`\chi_{\mathrm{ \rm NTML}+1}^{n+1}\geq\chi_{\mathrm{ \rm NTML}}^{n+1}` +:math:`\chi_{\mathrm{ \mathrm{NTML}}+1}^{n+1}\geq\chi_{\mathrm{ +\mathrm{NTML}}}^{n+1}` implies .. math:: - F_{\chi}^{Tot}|_{ \mathrm{ \rm NTML}+\frac{1}{2} } + F_{\chi}^{Tot}|_{ \mathrm{ \mathrm{NTML}}+\frac{1}{2} } \left( 1+ \frac{\Delta z}{z_{ml}}\right) - \geq F_{\chi}^{Tot}|_{ \mathrm{ \rm NTML}+\frac{3}{2} } + \Delta z \left( - \frac{\chi_{\mathrm{ \rm NTML}}^{n}-\chi_{\mathrm{ \rm - NTML}+1}^{n}}{\Delta t} - + \frac{F_{\chi}^{Tot}|_{z_{\rm b}}}{z_{ml}} \right) + \geq F_{\chi}^{Tot}|_{ \mathrm{ \mathrm{NTML}}+\frac{3}{2} } + \Delta z + \left( + \frac{\chi_{\mathrm{ \mathrm{NTML}}}^{n}-\chi_{\mathrm{ + \mathrm{NTML}}+1}^{n}}{\Delta t} + + \frac{F_{\chi}^{Tot}|_{z_{\mathrm{b}}}}{z_{ml}} \right) The same arguments apply for :math:`q_t`, noting that the free atmosphere can be drier or moister than the mixed layer and so these @@ -2798,15 +2825,15 @@ In the 9B scheme, the discontinuous jumps in :math:`\theta_{\ell}` and from integral assumptions similar to those used to diagnose the subgrid inversion height, :math:`z_h`, and were given by :eq:`dqt_disc`. Note that -:math:`{\chi}_{\mathrm{ \rm NTML}+2}` does not appear in +:math:`{\chi}_{\mathrm{ \mathrm{NTML}}+2}` does not appear in :eq:`dqt_disc` and so no direct information from the free atmosphere is used. Only if the budgets of :math:`\theta_{\ell}` and -:math:`q_t` in level :math:`\mathrm{ \rm NTML}+1` are entirely +:math:`q_t` in level :math:`\mathrm{ \mathrm{NTML}}+1` are entirely consistent with the rise and fall of the subgrid inversion will :eq:`dqt_disc` give accurate results. This will not be the case during an assimilation cycle, for example, neither is it likely to be the case if the convection scheme is detraining into level -:math:`\mathrm{ \rm NTML}+1`. +:math:`\mathrm{ \mathrm{NTML}}+1`. Instead, a more robust algorithm is used in the 9C scheme and the subgrid inversion calculation is only attempted where both @@ -2816,8 +2843,10 @@ formula used is: .. math:: :label: dqt_disc_9c - \Delta \chi = {\chi}_{\mathrm{ \rm NTML}+2} - {\chi}_{\mathrm{ \rm NTML}} - - \gamma_{\chi} \left( z_{\mathrm{ \rm NTML}+2} - z_h \right) + \Delta \chi = {\chi}_{\mathrm{ \mathrm{NTML}}+2} - {\chi}_{\mathrm{ + \mathrm{NTML}}} + - \gamma_{\chi} \left( z_{\mathrm{ \mathrm{NTML}}+2} - z_h + \right) subject to the constraint that the lapse rate adjustment should not reduce the two grid-length difference by more than half. The @@ -2825,17 +2854,17 @@ free-atmospheric lapse rates are given by .. math:: - \gamma_{\theta_{\ell}} = {\rm max}\left[ \, 0, \, \frac{ - {\theta_{\ell}}_{\mathrm{ \rm NTML}+3}-{\theta_{\ell}}_{\mathrm{ \rm - NTML}+2} } - { z_{\mathrm{ \rm NTML}+3} - z_{\mathrm{ \rm NTML}+2} } + \gamma_{\theta_{\ell}} = {\mathrm{max}}\left[ \, 0, \, \frac{ + {\theta_{\ell}}_{\mathrm{ \mathrm{NTML}}+3}-{\theta_{\ell}}_{\mathrm{ + \mathrm{NTML}}+2} } + { z_{\mathrm{ \mathrm{NTML}}+3} - z_{\mathrm{ \mathrm{NTML}}+2} } \right] .. math:: - \gamma_{q_t} = {\rm min}\left[ \, 0, \, \frac{ {q_t}_{\mathrm{ \rm - NTML}+3}-{q_t}_{\mathrm{ \rm NTML}+2} } - { z_{\mathrm{ \rm NTML}+3} - z_{\mathrm{ \rm NTML}+2} } + \gamma_{q_t} = {\mathrm{min}}\left[ \, 0, \, \frac{ {q_t}_{\mathrm{ + \mathrm{NTML}}+3}-{q_t}_{\mathrm{ \mathrm{NTML}}+2} } + { z_{\mathrm{ \mathrm{NTML}}+3} - z_{\mathrm{ \mathrm{NTML}}+2} } \right] .. _sec_subs_calc: @@ -2891,12 +2920,13 @@ specified through an eddy diffusivity which is given by .. math:: - K_h|_{\mathrm{ \rm NTML}+\frac{1}{2}} = w_e \Delta_{\mathrm{ \rm NTML}+1} z + K_h|_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}} = w_e \Delta_{\mathrm{ + \mathrm{NTML}}+1} z .. math:: :label: khent - K_m|_{\mathrm{ \rm NTML}} = Pr \, w_e \Delta_{\mathrm{ \rm - NTML}+\frac{1}{2}} z + K_m|_{\mathrm{ \mathrm{NTML}}} = Pr \, w_e \Delta_{\mathrm{ + \mathrm{NTML}}+\frac{1}{2}} z noting the Charney-Philips grid implying stresses are staggered from scalar fluxes. The Prandtl number, :math:`Pr`, takes the same form as @@ -2905,11 +2935,11 @@ for the non-local :math:`K` profiles, see section :ref:`The non-local scheme Substituting :eq:`khent` in :eq:`scal_closure` gives, for example, -:math:`\overline{w'\theta_{\ell}'}|_{\mathrm{ \rm NTML}+\frac{1}{2}} = - w_e -\Delta_{\mathrm{ \rm NTML}+1} \theta_{\ell}`. +:math:`\overline{w'\theta_{\ell}'}|_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}} = - +w_e \Delta_{\mathrm{ \mathrm{NTML}}+1} \theta_{\ell}`. Note that this gives entrainment buoyancy fluxes identical to :eq:`discinv` as long as there is no buoyancy reversal -generation of turbulence (i.e.,\ :math:`V_{\rm br}=0`) and if variations +generation of turbulence (i.e.,\ :math:`V_{\mathrm{br}}=0`) and if variations in the grid-level jumps across the timestep are ignored. The former is because the other terms in :eq:`we_parm` are inversely proportional to :math:`\Delta @@ -2921,15 +2951,15 @@ The advantages of diagnosing the subgrid inversion are that it allows consistency between the turbulent and radiative fluxes and large-scale vertical advection, it reduces grid-resolution errors arising from the mixed layer depth calculation and it allows a more accurate calculation -of :math:`V_{\rm br}` and :math:`\alpha_t`. For momentum, because the +of :math:`V_{\mathrm{br}}` and :math:`\alpha_t`. For momentum, because the jumps across inversions are typically small and variable, it seems unwise numerically to attempt to specify the inversion stresses explicitly and so :eq:`khent` is always used. For the 9C version, the entrainment :math:`K_m` given by :eq:`khent` is imposed at the height of the temperature inversion -:math:`z_{\rm h}` (either subgrid or at -:math:`z_{\mathrm{ \rm NTML}+\frac{1}{2}}`) and -:math:`K_m|_{\mathrm{ \rm NTML}+\frac{1}{2}}` is calculated from +:math:`z_{\mathrm{h}}` (either subgrid or at +:math:`z_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}}`) and +:math:`K_m|_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}}` is calculated from :eq:`kmsurf` and :eq:`kmtop`, noting the use of the :math:`{\cal E}` factors. @@ -2949,7 +2979,7 @@ When this happens, there is no subgrid inversion diagnosis and the entrainment parametrization follows the methodology given in section :ref:`For momentum (and scalars if no subgrid inversion) ` to give -:math:`K_h|_{\mathrm{ \rm NTML}+\frac{1}{2}}`. The diffusion +:math:`K_h|_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}}`. The diffusion coefficient profile within the inversion is then calculated assuming the :math:`\theta_{v\ell}` flux profile within the inversion decreases following a cosine shape from the standard parametrized entrainment flux @@ -2957,11 +2987,11 @@ at the inversion base to zero at the inversion top, i.e.: .. math:: :label: ent_svl - \overline{w'\theta_{v\ell}'} = \overline{w'\theta_{v\ell}'}|_{\mathrm{ \rm - NTML}+\frac{1}{2}} + \overline{w'\theta_{v\ell}'} = \overline{w'\theta_{v\ell}'}|_{\mathrm{ + \mathrm{NTML}}+\frac{1}{2}} cos\left(\pi \frac{z'}{2} \right) -where :math:`z'=(z-z_{\rm h})/\Delta z_i` is scaled height within the +where :math:`z'=(z-z_{\mathrm{h}})/\Delta z_i` is scaled height within the inversion. This flux profile is then converted into a diffusion coefficient profile by inverting the standard flux parametrization: @@ -3001,20 +3031,20 @@ equivalent entrainment eddy-diffusivity given by: .. math:: :label: K_ent_tracer - K_{\chi}|_{\mathrm{ \rm NTML}+\frac{1}{2}} = - \overline{w'\chi'}_{ - z_{\mathrm{ \rm NTML}+\frac{1}{2}} } - \frac{\Delta_{\mathrm{ \rm NTML}+1} - z}{\Delta_{\mathrm{ \rm NTML}+1} \chi} + K_{\chi}|_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}} = - \overline{w'\chi'}_{ + z_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}} } + \frac{\Delta_{\mathrm{ \mathrm{NTML}}+1} + z}{\Delta_{\mathrm{ \mathrm{NTML}}+1} \chi} Note from :eq:`scal_closure` that :eq:`K_ent_tracer` gives the parametrized flux if -:math:`\Delta_{\mathrm{ \rm NTML}+1} \chi` does not change across the +:math:`\Delta_{\mathrm{ \mathrm{NTML}}+1} \chi` does not change across the timestep (see section :ref:`Implicit solution of the diffusion equation ` for a description of the implicit numerical solution of :eq:`cons_eqn_scal`). As :eq:`K_ent_tracer` involves the potentially numerically dangerous calculation of -:math:`\Delta \chi/\Delta_{\mathrm{ \rm NTML}+1} \chi` (where +:math:`\Delta \chi/\Delta_{\mathrm{ \mathrm{NTML}}+1} \chi` (where :math:`\Delta \chi` is the subgrid inversion jump, given by :eq:`dqt_disc`), the following constraints are also @@ -3022,8 +3052,8 @@ ensured: .. math:: - 0 \leq K_{\chi}|_{\mathrm{ \rm NTML}+\frac{1}{2}} - \leq 10 \,K_{\chi}|_{\mathrm{ \rm NTML}-\frac{1}{2}} + 0 \leq K_{\chi}|_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}} + \leq 10 \,K_{\chi}|_{\mathrm{ \mathrm{NTML}}-\frac{1}{2}} Surface Exchange ================ @@ -3052,8 +3082,8 @@ layer are related to the surface fluxes by: .. math:: :label: 1.1.3 - \frac{\partial {\rm {\bf v}}}{\partial z}=\frac{ {\rm {\bf \tau }}_{0} }{ - \rho _0 v_\ast } \frac{ \phi _m (z/L)}{kz}, + \frac{\partial {\mathrm{\bf v}}}{\partial z}=\frac{ {\mathrm{\bf \tau }}_{0} + }{ \rho _0 v_\ast } \frac{ \phi _m (z/L)}{kz}, where subscript 0 represents a surface value and subscript \* represents a surface layer scaling quantity. :math:`\phi _{m}` and :math:`\phi @@ -3094,7 +3124,7 @@ surface turbulent fluxes are: .. math:: :label: 1.1.9 - \frac{ {\bf \tau }_{0} }{ \rho _{0} }= c_D^{1/2} v_\ast \Delta {\rm {\bf + \frac{ {\bf \tau }_{0} }{ \rho _{0} }= c_D^{1/2} v_\ast \Delta {\mathrm{\bf v}}, where :math:`\Delta`\ X=X\ :math:`_{1}`-X\ :math:`_{0}`. @@ -3158,8 +3188,8 @@ forms .. math:: :label: 1.1.18 - \frac{ {\rm {\bf \tau }}_{0} }{ \rho _{0} }= c_D V \Delta {\rm {\bf v}}{ }= - C_D \Delta {\rm {\bf v}}, + \frac{ {\mathrm{\bf \tau }}_{0} }{ \rho _{0} }= c_D V \Delta {\mathrm{\bf + v}}{ }= C_D \Delta {\mathrm{\bf v}}, where the effective wind speed for surface turbulent exchanges, :math:`V`, is defined by @@ -3198,7 +3228,7 @@ v\ :math:`_{\ast .. math:: :label: 1.1.24 - v_\ast = u_\ast \equiv \left| { {\rm {\bf \tau }}_{0} {/} \rho _{0} } + v_\ast = u_\ast \equiv \left| { {\mathrm{\bf \tau }}_{0} {/} \rho _{0} } \right|^{1/2} we have the standard Monin-Obukhov theory and it is easy to deduce that @@ -3268,16 +3298,17 @@ We can define the **mean gust speed** at height z\ :math:`_{1}` by .. math:: :label: 1.2.1 - v_g = ( V^2 - \left| {\Delta {{\rm {\bf v}}}} \right|^2 {)}^{{1/2}}. + v_g = ( V^2 - \left| {\Delta {{\mathrm{\bf v}}}} \right|^2 {)}^{{1/2}}. Using the definitions of :math:`V` :eq:`1.1.19` and :math:`v_{\ast }` :eq:`1.1.25` it can be deduced that .. math:: :label: 1.2.2 - v_g^2 = W_g^2 ( z_1 ) + \frac{1}{2}\left| {\Delta {{\rm {\bf v}}}} \right|{ - }\left[ {\left( {{ } {\left| {\Delta {{\rm {\bf v}}}} \right|}^2 - W_g^2 ( - z_1 )} \right)^{1/2} - \left| {\Delta {{\rm {\bf v}}}} \right|} \right] + v_g^2 = W_g^2 ( z_1 ) + \frac{1}{2}\left| {\Delta {{\mathrm{\bf v}}}} + \right|{ }\left[ {\left( {{ } {\left| {\Delta {{\mathrm{\bf v}}}} \right|}^2 + - W_g^2 ( z_1 )} \right)^{1/2} - \left| {\Delta {{\mathrm{\bf v}}}} \right|} + \right] where @@ -3301,7 +3332,7 @@ Equation :eq:`1.2.1` can be rewritten as .. math:: :label: 1.2.4 - V^2 = \left| {\Delta {{\rm {\bf v}}}} \right|^2 + v_g^2, + V^2 = \left| {\Delta {{\mathrm{\bf v}}}} \right|^2 + v_g^2, which is exactly the form of `Godfrey and Beljaars (1991)`_. However `Godfrey and Beljaars (1991)`_ define the mean gust speed as @@ -3589,7 +3620,7 @@ ms\ :math:`^{-1}`), then start the iteration from the neutral limit, so .. math:: :label: 1.4.7 v_\ast ^{(0)}= {\left( {\frac{k}{ \Phi _m^{(0)} }} \right)} \left| {\Delta - {{{\rm {\bf v}}}}} \right| + {{{\mathrm{\bf v}}}}} \right| Otherwise (if :math:`\Delta`\ B :math:`<` 0 and :math:`\Delta`\ **v** :math:`<` 2 ms\ :math:`^{-1}` ) start from the greater of the neutral @@ -3610,7 +3641,7 @@ and convective limits for :math:`v_\ast^{(0)}`, so .. math:: :label: 1.4.4 v_\ast ^{(0)}= MAX{\left[ {\left( {\frac{k}{ \Phi _m^{(0)} }} \right)} - \left| {\Delta {{{\rm {\bf v}}}}} \right|, \, + \left| {\Delta {{{\mathrm{\bf v}}}}} \right|, \, {\left[ { \gamma _t^3 \left( {\frac{k}{ \Phi _h^{(0)} }} \right) z_i \left| {-\Delta B} \right| } \right]}^{ 1/2} \right]} @@ -3633,7 +3664,7 @@ DO n = 1 to N .. math:: :label: 1.4.10 - u_\ast ^{(n)2}= C_D^{(n-1)} \left| {\Delta {{\rm {\bf v}}}} \right| + u_\ast ^{(n)2}= C_D^{(n-1)} \left| {\Delta {{\mathrm{\bf v}}}} \right| .. math:: :label: 1.4.11 @@ -3691,7 +3722,7 @@ stress: .. math:: :label: 1.4.21 - {\rm {\bf \tau }}_{0} = \rho _0 C_D^{(N)} \Delta {\rm {\bf v}} + {\mathrm{\bf \tau }}_{0} = \rho _0 C_D^{(N)} \Delta {\mathrm{\bf v}} N is the last iteration value. N = 5 is currently used. @@ -3802,8 +3833,8 @@ z\ :math:`_{0m}`, and the observation height z\ :math:`_{ob}` we obtain .. math:: :label: 1.5.1 - {\rm {\bf v}}_{ob} { = } {\rm {\bf v}}_{0} { + - }\frac{ {\rm {\bf \tau }}_{0} }{ \rho _0 v_\ast k} \Phi _m (L, + {\mathrm{\bf v}}_{ob} { = } {\mathrm{\bf v}}_{0} { + + }\frac{ {\mathrm{\bf \tau }}_{0} }{ \rho _0 v_\ast k} \Phi _m (L, z_{ob} + z_{0m} , z_{0m} ) Using the expression for the surface turbulent stress this gives the @@ -3811,9 +3842,9 @@ interpolation formula .. math:: :label: 1.5.2 - {\rm {\bf v}}_{ob} { = } {\rm {\bf v}}_{0} { + + {\mathrm{\bf v}}_{ob} { = } {\mathrm{\bf v}}_{0} { + }\frac{ C_D }{k v_\ast } \Phi _m (L, z_{ob} + z_{0m} , z_{0m} ) ( - {\rm {\bf v}}_{1} { - } {\rm {\bf v}}_{0} {)} + {\mathrm{\bf v}}_{1} { - } {\mathrm{\bf v}}_{0} {)} For wind z\ :math:`_{ob}` is set to 10m and the last iteration (N) values of C\ :math:`_{D}`, L and :math:`v_{\ast }` are used. @@ -4482,8 +4513,8 @@ When form drag is included via effective roughness lengths equations .. math:: :label: 2.1.3 - \frac{ {\rm {\bf \tau }}_{{0(eff)}} }{ \rho _{0} }=\frac{k}{ \Phi _m (L , - z_1 + z_{0m(eff)} , z_{0m(eff)} )} v_{\ast (eff)} \Delta {\rm {\bf v}} + \frac{ {\mathrm{\bf \tau }}_{{0(eff)}} }{ \rho _{0} }=\frac{k}{ \Phi _m (L , + z_1 + z_{0m(eff)} , z_{0m(eff)} )} v_{\ast (eff)} \Delta {\mathrm{\bf v}} The effective surface scaling velocity, v\ :math:`_{\ast (eff)}` , is given by (cf. :eq:`1.1.25`) @@ -4497,7 +4528,7 @@ where .. math:: :label: 2.1.5 - u_{\ast (eff)}^2 = \left| { {\rm {\bf \tau }}_{{0(eff)}} {/} \rho _{0} } + u_{\ast (eff)}^2 = \left| { {\mathrm{\bf \tau }}_{{0(eff)}} {/} \rho _{0} } \right| The effective roughness for momentum is derived by setting the total @@ -4512,17 +4543,17 @@ deviation of the unresolved orographic height. Thus .. math:: :label: 2.1.6 - \frac{ {\rm {\bf \tau }}_{{0(eff)}} }{ \rho _{0} }{ = }\frac{{k } {v}_{{\ast - (eff)}} }{ \Phi _{m} {(L , } {z}_{c} { , } {z}_{{0m(eff)}} {)}}{ }{\rm {\bf - v}}{(} {z}_{c} {)} + \frac{ {\mathrm{\bf \tau }}_{{0(eff)}} }{ \rho _{0} }{ = }\frac{{k } + {v}_{{\ast (eff)}} }{ \Phi _{m} {(L , } {z}_{c} { , } {z}_{{0m(eff)}} {)}}{ + }{\mathrm{\bf v}}{(} {z}_{c} {)} and .. math:: :label: 2.1.7 - \frac{ {\rm {\bf \tau }}_{{0(f)}} }{ \rho _{0} }{ = }\frac{{k } {v}_{{\ast - (f)}} }{ \Phi _{m} {(L , } {z}_{c} { , } {z}_{{0m}} {)}}{ }{\rm {\bf v}}{(} - {z}_{c} {)} + \frac{ {\mathrm{\bf \tau }}_{{0(f)}} }{ \rho _{0} }{ = }\frac{{k } + {v}_{{\ast (f)}} }{ \Phi _{m} {(L , } {z}_{c} { , } {z}_{{0m}} {)}}{ + }{\mathrm{\bf v}}{(} {z}_{c} {)} where the scaling velocity based on the stress over a flat surface, v\ :math:`_{\ast @@ -4537,7 +4568,8 @@ with .. math:: :label: 2.1.9 - u_{\ast (f)}^2 = \left| { {\rm {\bf \tau }}_{{0(f)}} {/} \rho _{0} } \right| + u_{\ast (f)}^2 = \left| { {\mathrm{\bf \tau }}_{{0(f)}} {/} \rho _{0} } + \right| [The scaling velocity which appears in the expression :eq:`1.1.4` for the Monin-Obukhov length is chosen @@ -4547,9 +4579,9 @@ The orographic stress is given by .. math:: :label: 2.1.10 - \frac{ {\rm {\bf \tau }}_{{0(p)}} }{ \rho _{0} }{ = }\frac{{1}}{{2}}{ } - {c}_{{D(orog)}} { } {f}_{D} {(} {{Ri}}_{B} {)}\frac{{A}}{{S}}{ }\left| {{\rm - {\bf v}}{(} {z}_{c} {)}} \right|{ }{\rm {\bf v}}{(} {z}_{c} {)} + \frac{ {\mathrm{\bf \tau }}_{{0(p)}} }{ \rho _{0} }{ = }\frac{{1}}{{2}}{ } + {c}_{{D(orog)}} { } {f}_{D} {(} {{Ri}}_{B} {)}\frac{{A}}{{S}}{ }\left| + {{\mathrm{\bf v}}{(} {z}_{c} {)}} \right|{ }{\mathrm{\bf v}}{(} {z}_{c} {)} where :math:`A/S` is the total silhouette area of orography in a gridbox over the flat surface area of the gridbox taken as an average over all @@ -4575,7 +4607,7 @@ The stress for the flat surface is related to the total stress by .. math:: :label: 2.1.13 - {\rm {\bf \tau }}_{{0(f)}} { = } {\rm {\bf \tau + {\mathrm{\bf \tau }}_{{0(f)}} { = } {\rm {\bf \tau }}_{{0(eff)}} { } {\left( {{1 + }\frac{{1}}{{2}}{ } {c}_{{D(orog)}} { } {f}_{D} { }\frac{{A}}{{S}}{ } {\left( {\frac{\ln {(} {z}_{c} { / } {z}_{{0m}} {)}}{{k}}} @@ -4617,7 +4649,7 @@ and :eq:`2.1.13` and :eq:`2.1.14` become .. math:: :label: 2.1.16 - {\rm {\bf \tau }}_{{0(f)}} = {\rm {\bf \tau + {\mathrm{\bf \tau }}_{{0(f)}} = {\rm {\bf \tau }}_{{0(eff)}} {\left( {{1 + }\alpha \beta \pi ^{2} { } {f}_{D} { } {\left( {\frac{{A}}{{S}}} \right)}^{2} { }} \right)}^{-1} @@ -4757,12 +4789,13 @@ DO n = 1 to N .. math:: :label: (2.2.15 - u_{\ast (eff)}^{(n)2}= C_{D(eff)}^{(n-1)} \left| {\Delta {{\rm {\bf v}}}} + u_{\ast (eff)}^{(n)2}= C_{D(eff)}^{(n-1)} \left| {\Delta {{\mathrm{\bf v}}}} \right| .. math:: :label: (2.2.16 - u_{\ast (f)}^{(n)2}= C_{D(f)}^{(n-1)} \left| {\Delta {{\rm {\bf v}}}} \right| + u_{\ast (f)}^{(n)2}= C_{D(f)}^{(n-1)} \left| {\Delta {{\mathrm{\bf v}}}} + \right| .. math:: :label: (2.2.17 @@ -4834,15 +4867,16 @@ surface sensible and latent heat fluxes and surface stress: .. math:: :label: 2.2.30 - {\rm {\bf \tau }}_{{0(eff)}} = \rho _0 C_{D(eff)}^{(N)} \Delta {\rm {\bf v}} + {\mathrm{\bf \tau }}_{{0(eff)}} = \rho _0 C_{D(eff)}^{(N)} \Delta + {\mathrm{\bf v}} The stress for a flat surface, if required for output, is calculated from .. math:: :label: 2.2.31 - {\rm {\bf \tau }}_{{0(f)}} { = } \rho _0 - C_{D(f)}^{(N)} \Delta {\rm {\bf v}} + {\mathrm{\bf \tau }}_{{0(f)}} { = } \rho _0 + C_{D(f)}^{(N)} \Delta {\mathrm{\bf v}} .. _section_2.3: @@ -4855,17 +4889,17 @@ by the effective roughness length and surface scaling velocity then .. math:: :label: 2.3.1 - {\rm {\bf v}}_{{ob}} { = } {\rm {\bf v}}_{0} { + - }\frac{ {\rm {\bf \tau }}_{{0(eff)}} }{ \rho _0 v_{\ast (eff)} k} + {\mathrm{\bf v}}_{{ob}} { = } {\mathrm{\bf v}}_{0} { + + }\frac{ {\mathrm{\bf \tau }}_{{0(eff)}} }{ \rho _0 v_{\ast (eff)} k} \Phi _m (L, z_{ob} + z_{0m(eff)} , z_{0m(eff)} Using the expression for the surface turbulent stress this becomes .. math:: :label: 2.3.2 - {\rm {\bf v}}_{{ob}} { = } {\rm {\bf v}}_{0} { + + {\mathrm{\bf v}}_{{ob}} { = } {\mathrm{\bf v}}_{0} { + }\frac{ C_{D(eff)} }{ {kv}_{\ast (eff)} } \Phi _m (L, z_{ob} + - z_{0m(eff)} , z_{0m(eff)} ) ( {\rm {\bf v}}_1 - {\rm {\bf v}}_{0} + z_{0m(eff)} , z_{0m(eff)} ) ( {\mathrm{\bf v}}_1 - {\mathrm{\bf v}}_{0} {)} For wind z\ :math:`_{ob}` is set to 10m and the last iteration (N) @@ -4876,17 +4910,17 @@ velocity then .. math:: :label: 2.3.3 - {\rm {\bf v}}_{{ob}} { = } {\rm {\bf v}}_{0} { + - }\frac{ {\rm {\bf \tau }}_{{0(f)}} }{ \rho _0 v_{\ast (f)} k} \Phi + {\mathrm{\bf v}}_{{ob}} { = } {\mathrm{\bf v}}_{0} { + + }\frac{ {\mathrm{\bf \tau }}_{{0(f)}} }{ \rho _0 v_{\ast (f)} k} \Phi _m (L, z_{ob} + z_{0m} , z_{0m} ) and substituting for the surface stress this becomes .. math:: :label: 2.3.4 - {\rm {\bf v}}_{{ob}} { = } {\rm {\bf v}}_{0} { + + {\mathrm{\bf v}}_{{ob}} { = } {\mathrm{\bf v}}_{0} { + }\frac{ C_{D(f)} }{k v_{\ast (f)} } \Phi _m (L, z_{ob} + z_{0m} , - z_{0m} ) ( {\rm {\bf v}}_1 - {\rm {\bf v}}_{0} {)} + z_{0m} ) ( {\mathrm{\bf v}}_1 - {\mathrm{\bf v}}_{0} {)} Most configurations of the Unified Model currently use the latter assumption with the last iteration value of C\ :math:`_{D(f)}`, L and @@ -4941,14 +4975,14 @@ The turbulent form drag is represented by the term .. math:: :label: eq:drag - {\bf f}=\frac{1}{\rho}\frac{\partial}{\partial z}{\bf\tau}_{\rm orog} + {\bf f}=\frac{1}{\rho}\frac{\partial}{\partial z}{\bf\tau}_{\mathrm{orog}} on the right-hand side of the horizontal momentum equation, where -:math:`{\bf\tau}_{\rm orog}` is the horizontal vector containing the +:math:`{\bf\tau}_{\mathrm{orog}}` is the horizontal vector containing the extra stress imparted on the flow by the sub-grid orography This term is included in the Unified Model as an additional explicit (in terms of time discretisation) stress. Following `Wood et al. (2001)`_ -we define :math:`{\bf\tau}_{\rm orog}` to be +we define :math:`{\bf\tau}_{\mathrm{orog}}` to be .. math:: {\bf\tau}_{\rm orog}(z)=\left({F_p}_x,{F_p}_y\right)e^{-z/\ell}, @@ -4959,7 +4993,7 @@ components of the pressure force on the sub-grid orography, and .. math:: :label: eq:l - \ell={\rm min}\left(\lambda,\frac{z_h}{3}\right), + \ell={\mathrm{min}}\left(\lambda,\frac{z_h}{3}\right), where :math:`z_h` is the boundary-layer depth and :math:`\lambda`, a somewhat ill defined quantity, is related to the horizontal scales of @@ -4974,7 +5008,7 @@ parametrization (Eq. :eq:`2.1.10`, namely: .. math:: :label: eq:dragsteep \frac{\bf F_p}{\rho_0}=\frac{1}{2}c_{D(orog)} f_D (Ri_{B}) \frac{A}{S} - \left\vert{\rm{\bf v}}(\ell) \right\vert{\rm{\bf v}}(\ell), + \left\vert{\mathrm{\bf v}}(\ell) \right\vert{\mathrm{\bf v}}(\ell), the main difference being the dependence on the height scale :math:`\ell` rather than :math:`z_c`. Similarly, if the @@ -4987,9 +5021,9 @@ namely: \frac{\bf F_p}{\rho_0} = {\left( {\frac{\kappa}{\zeta_m}} \right)}^{2} \alpha \beta \pi ^{2} {f}_{D} (Ri_{B}) {\left( {\frac{A}{S}} \right)}^{2} - \left\vert{\rm{\bf v}}(\ell) \right\vert{\rm{\bf v}}(\ell), + \left\vert{\mathrm{\bf v}}(\ell) \right\vert{\mathrm{\bf v}}(\ell), -where :math:`\zeta_m = {\rm log}(\ell/z_{0m})`. There is also an option +where :math:`\zeta_m = {\mathrm{log}}(\ell/z_{0m})`. There is also an option to use the low-hill stress :eq:`eq:draglow` but capped by that from the steep hill expression :eq:`eq:dragsteep`, to avoid generating huge stresses @@ -5674,7 +5708,7 @@ but with :math:`\gamma_{1}={\cal I}_{1}`. Here .. math:: - D={q_{\rm sat}(T_*^{(n)},p_*)-q_{\rm sat}(T_1^{(n)},p_*) \over + D={q_{\mathrm{sat}}(T_*^{(n)},p_*)-q_{\mathrm{sat}}(T_1^{(n)},p_*) \over T_*^{(n)}-T_1^{(n)}}, .. math:: \psi=f_a+(1-f_a){g_s\over g_s+C_HU_1}, @@ -5916,7 +5950,7 @@ Boundary layer thermal speed: stash 3,355 This diagnostic is intended for use in quantifying the strength of convective thermals for aviation applications. Updraught velocities in convective boundary layers will scale with the convective velocity -scale, :math:`w_*`, given by :math:`w_*^3 = z_{\rm h}\overline{w'b}_S`. +scale, :math:`w_*`, given by :math:`w_*^3 = z_{\mathrm{h}}\overline{w'b}_S`. In addition to the basic convective velocity scale, the strength of thermals should also depend on the surface stability — it would be possible to have significant heat flux and boundary layer depth in windy @@ -5949,7 +5983,8 @@ proportional to the standard deviation of the horizontal wind, .. math:: :label: windgust U_{gust} = U_{10m} + W_{1D} \, \sigma_u \, \frac{1}{k} \, - {\rm log}\left( \frac{5 \, e^{k \, c_{\rm ugn}} + z_{0m(eff)} } + {\mathrm{log}}\left( \frac{5 \, e^{k \, c_{\mathrm{ugn}}} + z_{0m(eff)} } + {5 + z_{0m(eff)}} \right) The factor :math:`W_{1D}` is included only in the scale-dependent @@ -5958,14 +5993,14 @@ of boundary layer turbulence that are resolved (and so are already included in :math:`U_{10m}`). The lowest grid-level value of :math:`W_{1D}`, from :eq:`eq-tanh`, is used, noting that :math:`W_{1D}` is constant within the boundary layer. The constant -:math:`c_{\rm ugn}` in :eq:`windgust` is determined from +:math:`c_{\mathrm{ugn}}` in :eq:`windgust` is determined from universal turbulence spectra for a 25% exceeding probability of the three-second wind gust (`Beljaars (1987)`_). It is included through a function that includes the effective roughness length, :math:`z_{0m(eff)}`, in order to take into account the very high effective :math:`u_*` values that occur over mountainous terrain (due to the orographic form drag parametrization) and so avoid unrealistic high -gust values. Currently the UM takes :math:`c_{\rm ugn}=4` which was +gust values. Currently the UM takes :math:`c_{\mathrm{ugn}}=4` which was reduced from the value used at ECMWF based on evaluation of the wind gust performance. The stability dependence of :math:`\sigma_u` is estimated on the basis of the similarity relation from @@ -5975,8 +6010,9 @@ estimated on the basis of the similarity relation from \sigma_u = \begin{cases} - A_{gust} u_* (1.0 - z_{\rm h}/ (24 L) )^{1/3} & {\rm for}\ L<0 \\ - A_{gust} u_* & {\rm for}\ L>0 + A_{gust} u_* (1.0 - z_{\mathrm{h}}/ (24 L) )^{1/3} & {\mathrm{for}}\ L<0 + \\ + A_{gust} u_* & {\mathrm{for}}\ L>0 \end{cases} with :math:`A_{gust}=2.29`. For :math:`L` close to zero the wind gust @@ -6032,9 +6068,9 @@ LES and observations nicely follows the relationship \overline{w'^2} = c_{w2} w_*^2 f(z') where :math:`w_*` is the convective velocity scale and :math:`f` is a -shape function within the boundary layer (:math:`z'=z/z_{\rm h}`). The +shape function within the boundary layer (:math:`z'=z/z_{\mathrm{h}}`). The shape of this function is very similar to that used in the UM for -:math:`K_m^{\rm surf}` in :eq:`kmsurf`. We now assume we can +:math:`K_m^{\mathrm{surf}}` in :eq:`kmsurf`. We now assume we can generalise :eq:`w2_scaling` by replacing :math:`w_*` with :math:`w_m` (this really ought to be checked against neutral boundary layer LES but hasn’t yet been). Setting :math:`f(z')=z' @@ -6062,7 +6098,7 @@ the non-gradient parametrization in the UM, it follows that .. math:: :label: bl_scaling - K_h^{\rm surf}= \frac{\tau_{turb}}{2} \, \overline{w'^2} + K_h^{\mathrm{surf}}= \frac{\tau_{turb}}{2} \, \overline{w'^2} Combining :eq:`bl_scaling` with :eq:`gen_w2_scaling` and :eq:`kmsurf`, @@ -6074,14 +6110,14 @@ surface-driven boundary layer mixing we can write: h}w_m f(z') which then gives -:math:`\tau_{\rm surf} = C_{ws}^{2/3} k z_{\rm h}/ (1.33 w_m)`. An +:math:`\tau_{\mathrm{surf}} = C_{ws}^{2/3} k z_{\mathrm{h}}/ (1.33 w_m)`. An analogous timescale can be derived for top-driven mixing in decoupled stratocumulus layers, :math:`\tau_{\rm Sc} = g_1 k z_{\rm ml}/ (1.33 \, V_{\rm Sc})`. There are two options to derive a TKE diagnosis from the Ri-based scheme and then combine with the non-local TKE (selected via var_diags_opt). -One is to assume :math:`\tau_{\rm SBL}=0.7/N` as the timescale for +One is to assume :math:`\tau_{\mathrm{SBL}}=0.7/N` as the timescale for stable boundary layers and combine all these timescales following `Suselj et al. (2012)`_) to give: @@ -6090,13 +6126,13 @@ stable boundary layers and combine all these timescales following e = K_m \tau_{turb}^{-1} where -:math:`\tau_{turb}^{-1} = MAX[ \tau_{\rm surf}^{-1},\tau_{\rm Sc}^{-1}] + -\tau_{\rm SBL}^{-1}`. +:math:`\tau_{turb}^{-1} = MAX[ +\tau_{\mathrm{surf}}^{-1},\tau_{\mathrm{Sc}}^{-1}] + \tau_{\mathrm{SBL}}^{-1}`. Note that :eq:`tke_diag` gives :math:`\overline{w'^2}`, rather than TKE. As a simple fix to improve the near-surface TKE in convective boundary layers, where the horizontal wind variability often dominates, the value of :math:`e` given by :eq:`tke_diag` -at the level of the maximum in :math:`K_m^{\rm surf}` is copied to all +at the level of the maximum in :math:`K_m^{\mathrm{surf}}` is copied to all levels below that height. The second method diagnoses TKE for the local scheme following the Met @@ -6119,15 +6155,15 @@ coefficients, i.e., .. math:: :label: tke_diag_nl - e_{nl} = \frac{3}{2} \left( \frac{K_m^{\rm surf}}{\tau_{\rm surf}} - + \frac{K_m^{\rm Sc}}{\tau_{\rm Sc}} \right) + e_{nl} = \frac{3}{2} \left( \frac{K_m^{\mathrm{surf}}}{\tau_{\mathrm{surf}}} + + \frac{K_m^{\mathrm{Sc}}}{\tau_{\mathrm{Sc}}} \right) The factor of :math:`3/2` in :eq:`tke_diag_nl` arises because we are really diagnosing :math:`\overline{w'^2}` and so here we make the assumption of isotropic turbulence to extend this to TKE. As before, we do also make the simple fix to improve the near-surface TKE in convective boundary layers, but here we copy only the value of -:math:`e_{nl}` at the level of the maximum in :math:`K_m^{\rm surf}` to +:math:`e_{nl}` at the level of the maximum in :math:`K_m^{\mathrm{surf}}` to all levels of :math:`e_{nl}` below that height. The final TKE is then the greater of :math:`e_{nl}` and :math:`e_{loc}` (as is done to combine the diffusion coefficients). @@ -6144,7 +6180,7 @@ convection within the diagnostic. This is given by: where :math:`M` is the convective updraft mass flux (Pa s\ :math:`^{-1}`) and CCA is the convective cloud area. The final diagnostic is then given as the maximum of :math:`e` and -:math:`e_{\rm conv}`. +:math:`e_{\mathrm{conv}}`. If selected, this option also reduces the minimum value used in UKCA by an order of magnitude. This is possible, because the minimum is no @@ -6189,29 +6225,29 @@ Appendix: Definitions of the velocity scales As described in `Lock et al. (2000)`_, the parametrization of the entrainment rate in convective boundary layers is based on four velocity scales, each representative of a turbulence-generating process -(:math:`V_{\rm heat}` for surface heating, :math:`u_*` for surface shear -generation, :math:`V_{\rm rad}` for cloud-top radiative cooling and -:math:`V_{\rm br}` for buoyancy reversal). The velocity scales can be +(:math:`V_{\mathrm{heat}}` for surface heating, :math:`u_*` for surface shear +generation, :math:`V_{\mathrm{rad}}` for cloud-top radiative cooling and +:math:`V_{\mathrm{br}}` for buoyancy reversal). The velocity scales can be written .. math:: :label: vsurf - V_{\rm heat}^3= z_{\rm ml}\! \left( (2-\zeta_s)\zeta_s \overline{w'b}_S+ - (1-\zeta_s)^2 [\overline{w'b'}_S]_{\rm sat}\right) + V_{\mathrm{heat}}^3= z_{\mathrm{ml}}\! \left( (2-\zeta_s)\zeta_s + \overline{w'b}_S+ (1-\zeta_s)^2 [\overline{w'b'}_S]_{\mathrm{sat}}\right) .. math:: :label: vrad - V_{\rm rad}^3= z_{\rm ml}\Delta_F\, g \, + V_{\mathrm{rad}}^3= z_{\mathrm{ml}}\Delta_F\, g \, \left( \beta_T \zeta_r^2 + \tilde{\beta_T} (1-\zeta_r^2) \right) .. math:: :label: vbr - V_{\rm br}^3= A_{\rm br}\chi_s^2 \, \mathrm{max}\left[0,-\delta b\right] \, - \Delta b ^{1/2} + V_{\mathrm{br}}^3= A_{\mathrm{br}}\chi_s^2 \, \mathrm{max}\left[0,-\delta + b\right] \, \Delta b ^{1/2} \, z_c^{3/2} \, C_{fac} Here, -:math:`[\overline{w'b'}_S]_{\rm sat}= g ( \tilde{\beta_T} +:math:`[\overline{w'b'}_S]_{\mathrm{sat}}= g ( \tilde{\beta_T} \overline{w'\theta_{\ell}'}_S+ \tilde{\beta_q}\overline{w'q_t'}_S)`, where the subscript :math:`_S` indicates the surface flux; :math:`\Delta_F` is the divergence of the net radiative flux, :math:`F` @@ -6220,11 +6256,11 @@ calculation is described in section :ref:`Calculation of \Delta_F ` Various depth parameters are given by :math:`\zeta_s = (z_{\rm ml}-\tilde{z_c})/z_{\rm ml}`, -:math:`\zeta = (z_{\rm ml}-z_c)/z_{\rm ml}` and :math:`\zeta_r = -\zeta + Br (1-\zeta)`. :math:`z_{\rm ml}` is the mixed-layer depth, +:math:`\zeta = (z_{\mathrm{ml}}-z_c)/z_{\mathrm{ml}}` and :math:`\zeta_r = +\zeta + Br (1-\zeta)`. :math:`z_{\mathrm{ml}}` is the mixed-layer depth, :math:`z_c` is the cloud depth and :math:`\tilde{z_c}` is the cloud-fraction weighted cloud depth. The former is used in the -calculation of :math:`V_{\rm rad}` as it is assumed the radiative +calculation of :math:`V_{\mathrm{rad}}` as it is assumed the radiative cooling will occur predominantly in cloudy air. To allow for a feedback in the presence of buoyancy reversal, the parameter :math:`Br` is included in :math:`\zeta_r` and :math:`\tilde{\alpha_t}` (in @@ -6331,11 +6367,11 @@ makes :math:`q_{\ell}` equal to the supersaturation and so will be reasonably accurate. Extrapolating this grid-level :math:`q_{\ell}` to zero should then give a reasonably accurate measure of cloud-base. -The formula for :math:`V_{\rm br}` was derived using dimensional +The formula for :math:`V_{\mathrm{br}}` was derived using dimensional arguments and comparison with LES data: :math:`\chi_s = -{q_{\ell}}_{\rm ct}(1+(L/c_p)\alpha_L) / ( \Delta q_t - \alpha_L \Delta \theta_{\ell})`, where -:math:`{q_{\ell}}_{\rm ct}` is the cloud-top liquid water mixing ratio, +:math:`{q_{\ell}}_{\mathrm{ct}}` is the cloud-top liquid water mixing ratio, :math:`L` is the latent heat of vaporisation of water, :math:`c_p` the specific heat at constant pressure and T is the temperature; :math:`\delta b = g(\tilde{\beta_T} \Delta \theta_{\ell} @@ -6350,41 +6386,42 @@ inversion is given by \left( \beta_T \frac{L_s}{c_p} - \frac{1+c_v}{c_v}\beta_q \right)\Delta q_f \right) -The empirical constant :math:`A_{\rm br}= 0.24`. The calculation of +The empirical constant :math:`A_{\mathrm{br}}= 0.24`. The calculation of :math:`\Delta \theta_{\ell}` and :math:`\Delta q_t` is described for a subgrid inversion in section :ref:`Diagnosis of a sub-grid inversion ` or, if one is not diagnosed, -they are taken simply as :math:`\Delta_{\mathrm{ \rm NTML}+1}`. For +they are taken simply as :math:`\Delta_{\mathrm{ \mathrm{NTML}}+1}`. For :math:`\Delta q_{\ell}`, :math:`\Delta q_f` and -:math:`{q_{\ell}}_{\rm ct}`, in-cloud values extrapolated to :math:`z_i` -(either subgrid or :math:`z_{\mathrm{ \rm NTML}+\frac{1}{2}}`) from +:math:`{q_{\ell}}_{\mathrm{ct}}`, in-cloud values extrapolated to :math:`z_i` +(either subgrid or :math:`z_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}}`) from above and below using the adiabatic lapse rates are calculated as: .. math:: - {q_{\ell}}_{\rm ct} = \frac{{q_{\ell}}_{\mathrm{ \rm - NTML}}}{{C_F}^l_{\mathrm{ \rm NTML}}} - + (z_i-z_{\mathrm{ \rm NTML}}) \gamma_{q_{\ell}} + {q_{\ell}}_{\mathrm{ct}} = \frac{{q_{\ell}}_{\mathrm{ + \mathrm{NTML}}}}{{C_F}^l_{\mathrm{ \mathrm{NTML}}}} + + (z_i-z_{\mathrm{ \mathrm{NTML}}}) \gamma_{q_{\ell}} .. math:: q_{\ell}^+ = \mathrm{max}\left[ 0, \, - \frac{{q_{\ell}}_{\mathrm{ \rm NTML}+2}}{{C_F}^l_{\mathrm{ \rm NTML}+2}} - - (z_{\mathrm{ \rm NTML}+2}-z_i) \gamma_{q_{\ell}} \right] + \frac{{q_{\ell}}_{\mathrm{ \mathrm{NTML}}+2}}{{C_F}^l_{\mathrm{ + \mathrm{NTML}}+2}} + - (z_{\mathrm{ \mathrm{NTML}}+2}-z_i) \gamma_{q_{\ell}} \right] and similarly for :math:`q_f` (noting that currently :math:`\gamma_{q_f}=0`) and for DSC layers. Then, .. math:: - \Delta q_{\ell} = {C_F^l}_{\mathrm{ \rm NTML}+2}\, q_{\ell}^+ - - {C_F^l}_{\mathrm{ \rm NTML}}\, {q_{\ell}}_{\rm ct} + \Delta q_{\ell} = {C_F^l}_{\mathrm{ \mathrm{NTML}}+2}\, q_{\ell}^+ - + {C_F^l}_{\mathrm{ \mathrm{NTML}}}\, {q_{\ell}}_{\mathrm{ct}} .. math:: - \Delta q_f = {C_F^f}_{\mathrm{ \rm NTML}+2}\, q_f^+ - {C_F^f}_{\mathrm{ \rm - NTML}}\, {q_f}_{\rm ct} + \Delta q_f = {C_F^f}_{\mathrm{ \mathrm{NTML}}+2}\, q_f^+ - + {C_F^f}_{\mathrm{ \mathrm{NTML}}}\, {q_f}_{\mathrm{ct}} The only other explicit account of variable cloud fraction is in :eq:`vbr` for which it is assumed that buoyancy reversal can @@ -6393,7 +6430,7 @@ overlap). Thus, the cloud fraction factor, :math:`C_{fac} = \mbox{max}[ 0.0, -\Delta C_F ]`, where :math:`\Delta C_F = {C_F}_{\mbox{\tiny \rm NTML}+2} - {C_F}_{\mbox{\tiny \rm NTML}}` if a subgrid inversion is diagnosed -(because :math:`{C_F}_{\mathrm{ \rm NTML}+1}` is currently +(because :math:`{C_F}_{\mathrm{ \mathrm{NTML}}+1}` is currently meaningless) and :math:`\Delta C_F = {C_F}_{\mbox{\tiny \rm NTML}+1} - {C_F}_{\mbox{\tiny \rm NTML}}` if not. A more complete decomposition is not possible given a cloud scheme in @@ -6412,8 +6449,8 @@ is continuous. An additional explicit dependence of entrainment on cloud fraction was implemented in version 4.5 which reduced the cloud-top source terms of entrainment in partially cloudy boundary layers by a factor -:math:`\exp{\left\{-(0.9-{C_F}_{\mathrm{ \rm NTML}})^3/0.075\right\}}` -for :math:`{C_F}_{\mathrm{ \rm NTML}} < 0.9`. It was argued that +:math:`\exp{\left\{-(0.9-{C_F}_{\mathrm{ \mathrm{NTML}}})^3/0.075\right\}}` +for :math:`{C_F}_{\mathrm{ \mathrm{NTML}}} < 0.9`. It was argued that partial cloudiness on the scale of the mixed-layer eddies might reduce the entrainment efficiency of the cloud-top processes. By reducing the parametrized entrainment warming and drying in partially cloudy boundary @@ -6434,7 +6471,8 @@ Calculation of :math:`\Delta_F` ------------------------------- An important term in the entrainment parametrization and -:math:`K_h^{\rm Sc}` is the velocity scale :math:`V_{\rm rad}`, the cube +:math:`K_h^{\mathrm{Sc}}` is the velocity scale :math:`V_{\mathrm{rad}}`, the +cube of which is proportional to the net radiative flux difference associated with cloud-top, :math:`\Delta_F`. Because radiation tends not to be called every timestep, the calculation of :math:`\Delta_F` is done @@ -6564,7 +6602,7 @@ two steps of the algorithm above which become: #. the search for the level with maximum LW radiative cooling, :math:`k_m`, is restricted to the top half of the mixed layer and - below :math:`1.2 \, z_{\rm h}` (rather than level NTML\ :math:`+1` + below :math:`1.2 \, z_{\mathrm{h}}` (rather than level NTML\ :math:`+1` used above) #. if the LW flux divergence in level :math:`k_m+1` is relatively weak @@ -6634,10 +6672,10 @@ can be written \overline{w'b}= \begin{cases} g \left( \beta_T \overline{w'T_L'} + \beta_q \overline{w'q_t'}\right) - & {\rm in\ unsaturated\ air} \\ + & {\mathrm{in}\ unsaturated\ air} \\ g \left( \tilde{\beta_T} \overline{w'T_L'} + \tilde{\beta_q} \overline{w'q_t'} - \right) & {\rm in\ saturated\ air} + \right) & {\mathrm{in}\ saturated\ air} \end{cases} where @@ -6649,7 +6687,7 @@ where .. math:: - {\rm and} + {\mathrm{and}} \beta_c = a_L \left( \frac{L}{c_p} \beta_T - \frac{1+c_v}{c_v} \beta_q \right) @@ -6821,9 +6859,9 @@ Tests in the SCM showed the heating rate gradients can be very large near the surface. Hence to avoid stability problems (since this heating increment must be added after the implicit calculation of the stress (and heat flux) profiles) the increments are summed over the levels -within the BL (i.e. up to :math:`z_{\rm h}` ) and then that total +within the BL (i.e. up to :math:`z_{\mathrm{h}}` ) and then that total heating is applied as a linear decrease from the surface to zero over -:math:`z_{\rm h}` . +:math:`z_{\mathrm{h}}` . .. _app_opmods: @@ -7080,7 +7118,7 @@ Appendix: Notation - (:math:`=1.1`) threshold for ratio of layer :math:`q_t`-gradients in cumulus diagnosis - * - :math:`\Gamma_{\rm inv}` + * - :math:`\Gamma_{\mathrm{inv}}` - (:math:`=1.1`) threshold on ratio of environment to parcel :math:`\theta_v` gradients @@ -7128,19 +7166,19 @@ Appendix: Notation * - NTLOC - top :math:`\theta`-level below which :math:`Ri<1` - * - :math:`z_{\rm h}` + * - :math:`z_{\mathrm{h}}` - height of top of SML (potentially subgrid) - * - :math:`z_{\rm h}^{\rm Sc}` + * - :math:`z_{\mathrm{h}}^{\mathrm{Sc}}` - height of top of DSC layer (potentially subgrid) - * - :math:`z_{\rm b}` + * - :math:`z_{\mathrm{b}}` - height of base of DSC layer (subgrid) - * - :math:`z_{\rm par}` + * - :math:`z_{\mathrm{par}}` - height of half-level at top of parcel ascent - * - :math:`z_{\rm loc}` + * - :math:`z_{\mathrm{loc}}` - height of half-level marking ‘top’ of local :math:`Ri`-based mixing * - @@ -7152,13 +7190,13 @@ Appendix: Notation * - :math:`z_c` - cloud depth - * - :math:`z_{\rm ml}` + * - :math:`z_{\mathrm{ml}}` - mixed layer depth - * - :math:`K_m^{\rm surf}`, :math:`K_h^{\rm surf}` + * - :math:`K_m^{\mathrm{surf}}`, :math:`K_h^{\mathrm{surf}}` - :math:`K` profiles for surface-driven turbulence (in SML) - * - :math:`K_m^{\rm Sc}`, :math:`K_h^{\rm Sc}` + * - :math:`K_m^{\mathrm{Sc}}`, :math:`K_h^{\mathrm{Sc}}` - :math:`K` profiles for cloud-top-driven turbulence * - @@ -7180,14 +7218,14 @@ Appendix: Notation - scaling velocity for momentum mixing in the SML * - - - (used in :math:`K_m^{\rm surf}`, :math:`\gamma_{\theta_{\ell}}` and the - SML parcel perturbation, :math:`\theta_v'`) + - (used in :math:`K_m^{\mathrm{surf}}`, :math:`\gamma_{\theta_{\ell}}` and + the SML parcel perturbation, :math:`\theta_v'`) * - :math:`w_*` - ‘standard’ convective velocity scale for a cloud-free convective * - - - boundary layer, :math:`w_*^3 = z_{\rm h}\overline{w'b}_S` + - boundary layer, :math:`w_*^3 = z_{\mathrm{h}}\overline{w'b}_S` * - :math:`u_*` - friction velocity (here includes the orographic component) @@ -7209,8 +7247,8 @@ Appendix: Notation - parameters in perturbation calculation, :eq:`dscd_pert`, * - - - for initial identification of and :math:`z_{\rm ml}` calculation for DSC - layers + - for initial identification of and :math:`z_{\mathrm{ml}}` calculation + for DSC layers * - :math:`a_L`, :math:`\alpha_L`, :math:`\beta_T`, :math:`\beta_q`, :math:`\tilde{\beta_T}`, :math:`\tilde{\beta_q}` From 735e865bf0c945efdeff8bc8706611df4acbd2ce Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Wed, 13 May 2026 14:36:32 +0100 Subject: [PATCH 077/116] Also replace \bf, \cal with \mathrm{}, \mathcal{}. --- .../turbulence_schemes/bldoc.rst | 303 +++++++++--------- 1 file changed, 144 insertions(+), 159 deletions(-) diff --git a/documentation/source/science_guide/turbulence_schemes/bldoc.rst b/documentation/source/science_guide/turbulence_schemes/bldoc.rst index 6973c32f6d..511342e6f0 100644 --- a/documentation/source/science_guide/turbulence_schemes/bldoc.rst +++ b/documentation/source/science_guide/turbulence_schemes/bldoc.rst @@ -49,24 +49,24 @@ separately in . Model variables and turbulence closure ====================================== -Given source terms, :math:`{\cal S}` say, from processes other than +Given source terms, :math:`{\mathcal{S}}` say, from processes other than boundary layer turbulence, Reynolds’ averaging gives the following equation for conserved scalar variables, :math:`\chi`, and the two -horizontal components of momentum, :math:`{\bf u}` on a sphere gives: +horizontal components of momentum, :math:`{\mathbf{u}}` on a sphere gives: .. math:: :label: cons_eqn_scal \frac{\partial \chi}{\partial t} = - \frac{1}{r^2 \rho} \, \frac{\partial }{\partial z} \left( r^2 \rho \overline{w'\chi'} \right) - + {\cal S} + + {\mathcal{S}} .. math:: :label: cons_eqn_uv - \frac{\partial {\bf u}}{\partial t} + \frac{\partial {\mathbf{u}}}{\partial t} = \frac{1}{r^2 \rho} \, \frac{\partial }{\partial z} \left( r^2 {\bf \tau} \right) - + {\cal S} + + {\mathcal{S}} where :math:`\overline{w'\chi'}` and :math:`{\bf \tau}` are the vertical turbulent fluxes to be parametrized, :math:`r` is the height from the @@ -132,7 +132,7 @@ standard closures are: .. math:: :label: uv_closure - {\bf \tau} = K_m \frac{\partial {\bf u}}{\partial z} + {\bf \tau}^{nl} + {\bf \tau} = K_m \frac{\partial {\mathbf{u}}}{\partial z} + {\bf \tau}^{nl} Separate eddy-diffusivities are calculated for momentum, :math:`K_m`, and for scalar variables, :math:`K_h`. The second term on the right hand @@ -630,7 +630,7 @@ buoyancy consumption of TKE within the mixed layer equals a fraction, \sum_{z_{k-\frac{1}{2}} > z_i-z_{\mathrm{ml}}}^{z_{k-\frac{1}{2}} < z_i} \left[ \overline{w'b}|_{z_{k-\frac{1}{2}}}>0 \right] \, \Delta_k z -Note that, for simplicity, the :math:`{\cal E}_h` factors are not +Note that, for simplicity, the :math:`{\mathcal{E}}_h` factors are not included in :math:`K_h^{\mathrm{surf}}` or :math:`K_h^{\mathrm{Sc}}` when calculating :eq:`eq:wx_std` under the assumption that they will be small. This process is applied to all unstable mixed @@ -931,16 +931,16 @@ A first order ‘mixing length’ closure is used: .. math:: :label: kmlocal - K_m = {\cal L}_m^2 \, (S+S_d) \, f_m(Ri) + K_m = {\mathcal{L}}_m^2 \, (S+S_d) \, f_m(Ri) .. math:: :label: khlocal - K_h = {\cal L}_h \, {\cal L}_m \, + K_h = {\mathcal{L}}_h \, {\mathcal{L}}_m \, (S+S_d) \, f_h(Ri) -where :math:`{\cal L}_m` and :math:`{\cal L}_h` are the neutral mixing +where :math:`{\mathcal{L}}_m` and :math:`{\mathcal{L}}_h` are the neutral mixing lengths and :math:`S` is the resolved vertical shear of the horizontal -wind components, :math:`S = \left| \partial {\bf u}/\partial z \right|`. +wind components, :math:`S = \left| \partial {\mathbf{u}}/\partial z \right|`. A representation of the wind shear, :math:`S_d`, generated by drainage flows in complex terrain can also be included, as described below. Near the surface simple finite difference calculations for the vertical @@ -950,11 +950,11 @@ ignored above grid-level 2 and the neutral mixing lengths are given by .. math:: - {\cal L}_m = \frac{k(z+z_{0m})}{1+k(z+z_{0m})/\lambda_m} + {\mathcal{L}}_m = \frac{k(z+z_{0m})}{1+k(z+z_{0m})/\lambda_m} .. math:: - {\cal L}_h = \frac{k(z+z_{0m})}{1+k(z+z_{0m})/\lambda_h} + {\mathcal{L}}_h = \frac{k(z+z_{0m})}{1+k(z+z_{0m})/\lambda_h} where :math:`z_{0m}` includes the orographic component. For the lowest interior grid-level (:math:`k=1`) they are calculated, incorporating @@ -962,7 +962,7 @@ this log profile correction, as .. math:: - \tilde{{\cal L}}_{X,k-1/2} = \frac{k \Delta_{k-1/2} z}{ + \tilde{{\mathcal{L}}}_{X,k-1/2} = \frac{k \Delta_{k-1/2} z}{ ln\left( \frac{z_k + z_{0m}}{z_{k-1} + z_{0m}} \right) + \frac{k \Delta_{k-1/2} z}{\lambda_X} } @@ -1070,12 +1070,13 @@ For :math:`Ri < 0`, the standard UM stability functions are given by .. math:: f_m = 1 - \frac{g_0 \,Ri} - {1+D_m(\tilde{{\cal L}}_m/\tilde{{\cal L}}_h)|Ri|^{1/2} } + {1+D_m(\tilde{{\mathcal{L}}}_m/\tilde{{\mathcal{L}}}_h)|Ri|^{1/2} } .. math:: :label: unstable_stab f_h = \frac{1}{Pr_N}\left(1 - \frac{g_0 \,Ri} - {1+D_h(\tilde{{\cal L}}_m/\tilde{{\cal L}}_h)|Ri|^{1/2} }\right) + {1+D_h(\tilde{{\mathcal{L}}}_m/\tilde{{\mathcal{L}}}_h)|Ri|^{1/2} + }\right) with :math:`g_0=10`, :math:`D_m=g_0/4` and :math:`D_h=g_0/25`. If the stability dependent Prandtl number option is chosen (see below) the @@ -1198,7 +1199,7 @@ increase :math:`Ri` above inversions and so damp mixing. Thus, .. math:: Ri_k = \frac{DBDZ_k} - {(\Delta_{k+\frac{1}{2}} {\bf u}/\Delta_{k+\frac{1}{2}} z)^2} + {(\Delta_{k+\frac{1}{2}} {\mathbf{u}}/\Delta_{k+\frac{1}{2}} z)^2} The buoyancy gradient on :math:`\theta`-level :math:`k` can be calculated in two different ways, depending on the switch @@ -1290,12 +1291,13 @@ therefore RI(K), are held on the ‘half-level’ below :math:`\rho`-level K, which is :math:`\theta`-level K-1. In addition to the above, the log profile correction applied to -:math:`{\cal L}_h` (to give :math:`\tilde{{\cal L}}_h`) must be applied *after* +:math:`{\cal L}_h` (to give :math:`\tilde{{\mathcal{L}}}_h`) must be applied +*after* interpolation of :math:`K_h` to level :math:`k+\frac{1}{2}` in order that the correct cancellation with the finite difference scalar gradient in the flux calculation can occur. In the unstable stability functions :eq:`unstable_stab`, however, -:math:`\tilde{{\cal L}}_h` must be calculated on :math:`\theta`-levels +:math:`\tilde{{\mathcal{L}}}_h` must be calculated on :math:`\theta`-levels (i.e., the same as :math:`\tilde{{\cal L}}_m` and :math:`Ri`) in order to maintain the same stability dependence. @@ -1431,8 +1433,8 @@ scale :math:`w_*`) in a layer with top at .. math:: :label: kmsurf K_m^{\mathrm{surf}}= k \ z_{\mathrm{h}}\ w_m \ \frac{z}{z_{\mathrm{h}}} - \left( 1 - {\cal E}_m^{\mathrm{surf}} \frac{z}{z_{\mathrm{h}}} - \right)^2 + \left( 1 - {\mathcal{E}}_m^{\mathrm{surf}} + \frac{z}{z_{\mathrm{h}}} \right)^2 where :math:`w_m^3 = u_*^3 + w_s^3`, :math:`u_*` is the friction velocity (including the orographic roughness component) and :math:`w_s` @@ -1442,14 +1444,15 @@ section :ref:`Diagnosis of a sub-grid inversion `) for both :math:`K_h^{\mathrm{surf}}` and :math:`K_m^{\mathrm{surf}}`. In the 8A version, :math:`K_m^{\mathrm{surf}}` uses :math:`z_{\mathrm{h}}` :math:`=z_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}}`. The -factor :math:`{\cal E}_m^{\mathrm{surf}}` is chosen so that +factor :math:`{\mathcal{E}}_m^{\mathrm{surf}}` is chosen so that :math:`K_m^{\mathrm{surf}}` will tend to :math:`K_m|_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}}` as :math:`z` tends to :math:`z_{\mathrm{h}}` , where :math:`K_m|_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}}` is the entrainment eddy-diffusivity (given by :eq:`khent`, although, in order to avoid altering the shape function too much, -:math:`{\cal E}_m^{\mathrm{surf}}` is not allowed to fall below :math:`0.7`). +:math:`{\mathcal{E}}_m^{\mathrm{surf}}` is not allowed to fall below +:math:`0.7`). A similar factor, :math:`{\cal E}_h^{\rm surf}`, is used in the :math:`K_h^{\mathrm{surf}}` profile even though the entrainment fluxes of the thermodynamic variables will usually be @@ -1575,8 +1578,8 @@ vertical extent of the K-profiles `), K_m^{\mathrm{Sc}}= 0.63 \ k \ z_{\mathrm{ml}}\ V_{\mathrm{Sc}}\left( \frac{z'}{z_{\mathrm{ml}}} \right)^2 - \left( 1 - {\cal E}_m^{\mathrm{Sc}} \frac{z'}{z_{\mathrm{ml}}} - \right)^{0.8} + \left( 1 - {\mathcal{E}}_m^{\mathrm{Sc}} + \frac{z'}{z_{\mathrm{ml}}} \right)^{0.8} where :math:`V_{\mathrm{Sc}}^3= V_{\mathrm{rad}}^3+V_{\mathrm{br}}^3` (see appendix :ref:`Appendix: Definitions of the velocity scales `) and @@ -1595,15 +1598,15 @@ subgrid diagnosis (see section :ref:`Diagnosis of a sub-grid inversion :math:`K_m^{\mathrm{Sc}}` in the 8A scheme which uses the height of the half-level below (:math:`z_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}}` or :math:`z_{\mathrm{ \mathrm{NTDSC}}+\frac{1}{2}}`). Again following -:eq:`kmsurf`, the factors :math:`{\cal E}_m^{\mathrm{Sc}}` and -:math:`{\cal E}_h^{\mathrm{Sc}}` are included in :eq:`kmtop` so +:eq:`kmsurf`, the factors :math:`{\mathcal{E}}_m^{\mathrm{Sc}}` and +:math:`{\mathcal{E}}_h^{\mathrm{Sc}}` are included in :eq:`kmtop` so that :math:`K_m^{\mathrm{Sc}}` will tend to :math:`K_m|_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}}` (and :math:`K_h^{\mathrm{Sc}}` to :math:`K_h|_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}}`), given by :eq:`khent`, as :math:`z` tends to :math:`z_{\mathrm{h}}` (and here no restriction is made on the magnitude of either :math:`{\cal E}_m^{\rm -Sc}` or :math:`{\cal E}_h^{\mathrm{Sc}}`). +Sc}` or :math:`{\mathcal{E}}_h^{\mathrm{Sc}}`). .. _sec_gradadj: @@ -2318,14 +2321,14 @@ F|_{z_i}`, so that .. math:: - {\cal H}|_{z_i} = - w_e \Delta \theta_{\ell}+ F_{\mathrm{net}}|_h + {\mathcal{H}}|_{z_i} = - w_e \Delta \theta_{\ell}+ F_{\mathrm{net}}|_h .. math:: :label: discinv \overline{w'q_t'}_{z_i} = - w_e \Delta q_t where the total heat flux -:math:`{\cal H} = \overline{w'\theta_{\ell}'}+ F_{\mathrm{net}}` and +:math:`{\mathcal{H}} = \overline{w'\theta_{\ell}'}+ F_{\mathrm{net}}` and :math:`F_{\rm net} = F- F|_{z_{\rm b}}`. The net radiative flux relative to the base of the mixed layers is simply calculated as @@ -2334,9 +2337,9 @@ mixed layers is simply calculated as F_{\mathrm{net}}|_{z_{k+\frac{1}{2}}} = \sum_{k=\mathrm{ \mathrm{NBDSC}}}^{k} \mathrm{max}\left[ - - \Delta_{k+\frac{1}{2}} z \, {\cal S}_F(k), \,0 \right] + - \Delta_{k+\frac{1}{2}} z \, {\mathcal{S}}_F(k), \,0 \right] -where NBDSC\ :math:`=1` in SMLs, :math:`{\cal S}_F` are the temperature +where NBDSC\ :math:`=1` in SMLs, :math:`{\mathcal{S}}_F` are the temperature increments (in Ks\ :math:`^{-1}`) from the radiation scheme and :math:`F_{\mathrm{net}}|_h` is estimated by extrapolating down from :math:`F|_{z_{\mbox{\tiny \rm NTML}+\frac{3}{2}} @@ -2350,7 +2353,7 @@ as described in section :ref:`Diagnosis of a sub-grid inversion `. The required grid-level fluxes (at :math:`z_{\mathrm{ \mathrm{NTDSC}}+\frac{1}{2}}`, for example) -are then estimated using linear interpolation of :math:`{\cal H}` and +are then estimated using linear interpolation of :math:`{\mathcal{H}}` and :math:`\overline{w'q_t'}` between :math:`z_{\mathrm{h}}^{\mathrm{Sc}}` and the base of the mixed layer: @@ -3741,7 +3744,8 @@ turns out to be small, but it is desirable to use the corrected form, which is derived as follows. We decompose the wind as -:math:`{\bf u}=\bar{\bf u} + {\bf u}_g + {\bf u}'`, representing, +:math:`{\mathbf{u}}=\bar{\mathbf{u}} + {\mathbf{u}}_g + {\mathbf{u}}'`, +representing, respectively, the large-scale, gust and small-scale turbulent contributions to the velocity. Locally, Monin-Obukhov theory then gives @@ -3755,28 +3759,30 @@ is aligned with the wind so .. math:: - {\bf \tau}({\bf x}) = \rho u_*^2({\bf x}) - \frac{\bar{\bf u} + {\bf u}_g({\bf x})} {|\bar{\bf u} + {\bf - u}_g({\bf x})|}. + {\bf \tau}({\mathbf{x}}) = \rho u_*^2({\mathbf{x}}) + \frac{\bar{\mathbf{u}} + {\mathbf{u}}_g({\mathbf{x}})} {|\bar{\mathbf{u}} + + {\bf + u}_g({\mathbf{x}})|}. With the assumption that :math:`\Phi_m` does not vary spatially, .. math:: - {\bf \tau}({\bf x}) = \rho \frac{k^2}{\Phi_m^2} - |\bar{\bf u} + {\bf u}_g({\bf x})| (\bar{\bf u} + {\bf u}_g({\bf - x})) \equiv \rho C_D |\bar{\bf u} + {\bf u}_g({\bf x})| (\bar{\bf - u} + {\bf u}_g({\bf x})), + {\bf \tau}({\mathbf{x}}) = \rho \frac{k^2}{\Phi_m^2} + |\bar{\mathbf{u}} + {\mathbf{u}}_g({\mathbf{x}})| (\bar{\mathbf{u}} + {\mathbf{u}}_g({\bf + x})) \equiv \rho C_D |\bar{\mathbf{u}} + {\mathbf{u}}_g({\mathbf{x}})| (\bar{\bf + u} + {\mathbf{u}}_g({\mathbf{x}})), where :math:`C_D` is the standard drag coefficient, :math:`\frac{k^2}{\Phi_m^2}`. The grid-box mean effect is .. math:: - \langle{\bf \tau}\rangle = \rho C_D \langle |\bar{\bf u} + {\bf u}_g({\bf - x})| - (\bar{\bf u} + {\bf u}_g({\bf x})) \rangle \approx \rho C_D \langle - |\bar{\bf u} + {\bf u}_g({\bf x})| \rangle \bar{\bf u}, + \langle{\bf \tau}\rangle = \rho C_D \langle |\bar{\mathbf{u}} + + {\mathbf{u}}_g({\mathbf{x}})| + (\bar{\mathbf{u}} + {\mathbf{u}}_g({\mathbf{x}})) \rangle \approx \rho C_D + \langle + |\bar{\mathbf{u}} + {\mathbf{u}}_g({\mathbf{x}})| \rangle \bar{\mathbf{u}}, which is the product of the enhanced wind speed (including gusts) and the mean velocity. The magnitude of the stress is then @@ -3903,33 +3909,33 @@ gases. Explicitly, we have .. math:: \dot T_{ob, \mathrm{ rad, surf}} = \frac{4\sigma T_s^3}{c_P} - {\cal K}(z_{ob}) (T_s-T_{ob}), + {\mathcal{K}}(z_{ob}) (T_s-T_{ob}), where :math:`\dot T_{ob, \mathrm{ rad, surf}}` is the cooling rate of the air at the height of observation due to direct radiative exchanges with the surface, :math:`T_{ob}` is the air temperature at that height, :math:`T_s` is the temperature of the surface and -:math:`{\cal K}(z_{ob})` depends on the concentration of trace gases in +:math:`{\mathcal{K}}(z_{ob})` depends on the concentration of trace gases in the atmosphere, in practice water vapour and carbon dioxide, and their spectroscopic properties. The contributions of water vapour and carbon dioxide are, to a good approximation, additive, so we may write .. math:: - {\cal K}(z_{ob}) = \left [ q_w {\cal C}_w(\mu_w, - T_{ob}) + q_c {\cal C}_c(\mu_c, T_{ob}) \right ], + {\mathcal{K}}(z_{ob}) = \left [ q_w {\mathcal{C}}_w(\mu_w, + T_{ob}) + q_c {\mathcal{C}}_c(\mu_c, T_{ob}) \right ], where :math:`q_w` and :math:`q_c` are the specific concentrations of water vapour and carbon dioxide and :math:`\mu_w` and :math:`\mu_c` are the respective pathlengths between the observation height and the -surface. The functions :math:`{\cal C}_w` and :math:`{\cal C}_c` are +surface. The functions :math:`{\mathcal{C}}_w` and :math:`{\mathcal{C}}_c` are parametrized and explicit functional forms are included in the code. In stronger winds turbulent cooling will be more important, so the scheme must approach the standard procedure in that limit. Within the context of local scaling, it can be shown that the depth of the atmosphere which feels the impact of surface cooling must scale on -:math:`{\cal L}=(u_*^3/ (g/T_s)\dot T_s)^{1/2}`, where :math:`u_*` is +:math:`{\mathcal{L}}=(u_*^3/ (g/T_s)\dot T_s)^{1/2}`, where :math:`u_*` is the surface friction velocity, :math:`\dot T_s` is the surface cooling rate. This parameter is used as a measure of the strength of turbulence to define the relaxation back to the strong-wind limit. @@ -3960,13 +3966,14 @@ local scaling we set, .. math:: W = \exp(-(0.4 f)^2 \delta t \, t_{\mathrm{ trans}})) / - (1+X({\cal L})\delta t), + (1+X({\mathcal{L}})\delta t), with .. math:: - X({\cal L})= \min \left ( 0.000283 \left \{\frac{{\cal L}}{z_{ob}} + X({\mathcal{L}})= \min \left ( 0.000283 \left + \{\frac{{\mathcal{L}}}{z_{ob}} \log \left (1+\frac{z_{ob}}{z_0}\right ) \right \}^2, \; \frac{0.2 u_* }{z_{ob}} \right ). @@ -4975,7 +4982,8 @@ The turbulent form drag is represented by the term .. math:: :label: eq:drag - {\bf f}=\frac{1}{\rho}\frac{\partial}{\partial z}{\bf\tau}_{\mathrm{orog}} + {\mathbf{f}}=\frac{1}{\rho}\frac{\partial}{\partial + z}{\bf\tau}_{\mathrm{orog}} on the right-hand side of the horizontal momentum equation, where :math:`{\bf\tau}_{\mathrm{orog}}` is the horizontal vector containing the @@ -4986,7 +4994,7 @@ we define :math:`{\bf\tau}_{\mathrm{orog}}` to be .. math:: {\bf\tau}_{\rm orog}(z)=\left({F_p}_x,{F_p}_y\right)e^{-z/\ell}, -where :math:`{\bf F_p}=({F_p}_x,{F_p}_y)`, :math:`{F_p}_x` and +where :math:`{\mathbf{F}_p}=({F_p}_x,{F_p}_y)`, :math:`{F_p}_x` and :math:`{F_p}_y` are the grid-box average :math:`x` and :math:`y` components of the pressure force on the sub-grid orography, and :math:`\ell` is a decay scale. We define :math:`\ell` such that @@ -5007,7 +5015,7 @@ parametrization (Eq. :eq:`2.1.10`, namely: .. math:: :label: eq:dragsteep - \frac{\bf F_p}{\rho_0}=\frac{1}{2}c_{D(orog)} f_D (Ri_{B}) \frac{A}{S} + \frac{\mathbf{F}_p}{\rho_0}=\frac{1}{2}c_{D(orog)} f_D (Ri_{B}) \frac{A}{S} \left\vert{\mathrm{\bf v}}(\ell) \right\vert{\mathrm{\bf v}}(\ell), the main difference being the dependence on the height scale @@ -5018,7 +5026,7 @@ namely: .. math:: :label: eq:draglow - \frac{\bf F_p}{\rho_0} = {\left( {\frac{\kappa}{\zeta_m}} \right)}^{2} + \frac{\mathbf{F}_p}{\rho_0} = {\left( {\frac{\kappa}{\zeta_m}} \right)}^{2} \alpha \beta \pi ^{2} {f}_{D} (Ri_{B}) {\left( {\frac{A}{S}} \right)}^{2} \left\vert{\mathrm{\bf v}}(\ell) \right\vert{\mathrm{\bf v}}(\ell), @@ -5081,34 +5089,34 @@ assumed to be positive. The new scheme is written .. math:: :label: eq:sppf1 - \frac{X^{*}-X^{n}}{\Delta t}=-{\cal - I}_{1}\left[K\left(X^{n}\right)^{P}\right] - X^{*}+{\cal E}_{1}\left[K\left(X^{n}\right)^{P}\right] - X^{n}+\left({\cal I}_{1}-{\cal E}_{1}\right)S, + \frac{X^{*}-X^{n}}{\Delta + t}=-{\mathcal{I}}_{1}\left[K\left(X^{n}\right)^{P}\right] + X^{*}+{\mathcal{E}}_{1}\left[K\left(X^{n}\right)^{P}\right] + X^{n}+\left({\mathcal{I}}_{1}-{\mathcal{E}}_{1}\right)S, .. math:: :label: eq:sppf2 \frac{X^{n+1}-X^{*}}{\Delta - t}=-{\cal I}_{2}\left[K\left(X^{n}\right)^{P}\right]X^{n+1} + - {\cal E}_{2}\left[K\left(X^{n}\right)^{P}\right]X^{*} + - \left({\cal I}_{2}-{\cal E}_{2}\right)S, + t}=-{\mathcal{I}}_{2}\left[K\left(X^{n}\right)^{P}\right]X^{n+1} + + {\mathcal{E}}_{2}\left[K\left(X^{n}\right)^{P}\right]X^{*} + + \left({\mathcal{I}}_{2}-{\mathcal{E}}_{2}\right)S, where .. math:: :label: eq:E1coeff - {\cal E}_{1}=\left(1+\frac{1}{\sqrt{2}}\right) + {\mathcal{E}}_{1}=\left(1+\frac{1}{\sqrt{2}}\right) \left[P+\frac{1}{\sqrt{2}}\pm\sqrt{P \left(\sqrt{2}-1\right)+\frac{1}{2}}\right] .. math:: :label: eq:E2coeff - {\cal E}_{2}=\left(1+\frac{1}{\sqrt{2}}\right) + {\mathcal{E}}_{2}=\left(1+\frac{1}{\sqrt{2}}\right) \left[P+\frac{1}{\sqrt{2}}\mp\sqrt{P\left(\sqrt{2}-1\right)+\frac{1}{2}}\right] .. math:: :label: eq:Icoeff - {\cal I}_{1}={\cal I}_{2}=\left(1+\frac{1}{\sqrt{2}}\right)\left(1+P\right) + {\mathcal{I}}_{1}={\mathcal{I}}_{2}=\left(1+\frac{1}{\sqrt{2}}\right)\left(1+P\right) Consider the one-dimensional “forced” boundary layer diffusion equation @@ -5131,15 +5139,15 @@ becomes .. math:: :label: eq:sppf_bl1 - \frac{X^{*}-X^{n}}{\Delta t} = {\cal I}_{1}\frac{\partial F}{\partial - z}^{*}-{\cal E}_{1}\frac{\partial F}{\partial z}^{n}+\left({\cal - I}_{1}-{\cal E}_{1}\right)S + \frac{X^{*}-X^{n}}{\Delta t} = {\mathcal{I}}_{1}\frac{\partial F}{\partial + z}^{*}-{\mathcal{E}}_{1}\frac{\partial F}{\partial + z}^{n}+\left({\mathcal{I}}_{1}-{\mathcal{E}}_{1}\right)S .. math:: :label: eq:sppf_bl2 - \frac{X^{n+1}-X^{*}}{\Delta t} = {\cal I}_{2}\frac{\partial - F}{\partial z}^{n+1}-{\cal E}_{2}\frac{\partial F}{\partial - z}^{*}+\left({\cal I}_{2}-{\cal E}_{2}\right)S + \frac{X^{n+1}-X^{*}}{\Delta t} = {\mathcal{I}}_{2}\frac{\partial + F}{\partial z}^{n+1}-{\mathcal{E}}_{2}\frac{\partial F}{\partial + z}^{*}+\left({\mathcal{I}}_{2}-{\mathcal{E}}_{2}\right)S where, @@ -5152,7 +5160,7 @@ where, i.e. only one evaluation of the exchange coefficient is required per timestep. Furthermore, the condition -:math:`I_{1}+I_{2}-({\cal E}_{1}+{\cal E}_{2})=1` ensures that if the +:math:`I_{1}+I_{2}-({\mathcal{E}}_{1}+{\mathcal{E}}_{2})=1` ensures that if the intermediate “starred” quantities are eliminated and the scheme is reduced into a single equation then the forcing term will be multiplied by :math:`1`. @@ -5174,17 +5182,17 @@ Writing equations :eq:`eq:sppf_bl1`, .. math:: :label: eq:sppf_inc1 - \frac{\delta X}{\Delta t}^{*} = ({\cal I}_{1}-{\cal - E}_{1})\left(\frac{\partial F}{\partial z}^{n}+S\right)+{\cal - I}_{1}\frac{\partial}{\partial z}\left(K_{X}\frac{\partial\delta X}{\partial - z}^{*}\right) + \frac{\delta X}{\Delta t}^{*} = + ({\mathcal{I}}_{1}-{\mathcal{E}}_{1})\left(\frac{\partial F}{\partial + z}^{n}+S\right)+{\mathcal{I}}_{1}\frac{\partial}{\partial + z}\left(K_{X}\frac{\partial\delta X}{\partial z}^{*}\right) .. math:: :label: eq:sppf_inc2 - \frac{\delta X}{\Delta t}^{n+1} = ({\cal I}_{2}-{\cal - E}_{2})\left(\frac{\partial F}{\partial z}^{*}+S\right)+{\cal - I}_{2}\frac{\partial}{\partial z}\left(K_{X}\frac{\partial\delta X}{\partial - z}^{n+1}\right) + \frac{\delta X}{\Delta t}^{n+1} = + ({\mathcal{I}}_{2}-{\mathcal{E}}_{2})\left(\frac{\partial F}{\partial + z}^{*}+S\right)+{\mathcal{I}}_{2}\frac{\partial}{\partial + z}\left(K_{X}\frac{\partial\delta X}{\partial z}^{n+1}\right) .. math:: :label: eq:sppf_inc3 @@ -5206,7 +5214,7 @@ Consider the following equivalent form of .. math:: :label: eq:du_star \frac{\delta u^{*}}{\Delta t}=\frac{\partial\bar{\tau}_{x}^{*}}{\partial - z}+\left({\cal I}_{1}-{\cal E}_{1}\right)S + z}+\left({\mathcal{I}}_{1}-{\mathcal{E}}_{1}\right)S where :math:`\tau_{x}` is the :math:`u` wind component stress (defined in the same way as the flux in :eq:`eq:sppf_inc1` and @@ -5214,8 +5222,7 @@ in the same way as the flux in :eq:`eq:sppf_inc1` and .. math:: :label: eq:tau_star - \bar{\tau}_{x}^{*}={\cal I}_{1}\tau_{x}^{*}-{\cal - E}_{1}\tau_{x}^{n},\qquad\tau_{x}^{*}=\tau_{x}^{n}+K_{u}\frac{\partial\delta + \bar{\tau}_{x}^{*}={\mathcal{I}}_{1}\tau_{x}^{*}-{\mathcal{E}}_{1}\tau_{x}^{n},\qquad\tau_{x}^{*}=\tau_{x}^{n}+K_{u}\frac{\partial\delta u^{*}}{\partial z}. Substituting :eq:`eq:tau_star` into @@ -5224,8 +5231,8 @@ Substituting :eq:`eq:tau_star` into .. math:: \frac{\delta u^{*}}{\Delta - t}=({\cal I}_{1}-{\cal E}_{1})\left(\frac{\partial\tau_{x}^{n}}{\partial - z}+S\right)+{\cal I}_{1}\frac{\partial}{\partial + t}=({\mathcal{I}}_{1}-{\mathcal{E}}_{1})\left(\frac{\partial\tau_{x}^{n}}{\partial + z}+S\right)+{\mathcal{I}}_{1}\frac{\partial}{\partial z}\left(K_{u}\frac{\partial\delta u^{*}}{\partial z}\right) which is identical to :eq:`eq:sppf_inc1` for @@ -5239,10 +5246,10 @@ levels), discretizing the previous equation in :math:`z` on all .. math:: \begin{aligned} - \delta u_{k+1/2}^{*} & = & ({\cal I}_{1}-{\cal E}_{1})\Delta + \delta u_{k+1/2}^{*} & = & ({\mathcal{I}}_{1}-{\mathcal{E}}_{1})\Delta t\left(\frac{\tau_{x}^{n}\Big|_{k+1}-\tau_{x}^{n}\Big|_{k}}{z_{k+1}-z_{k}}+S_{k+1/2}\right) \\ - & & +{\cal I}_{1}\frac{\Delta + & & +{\mathcal{I}}_{1}\frac{\Delta t}{z_{k+1}-z_{k}}\left[\left(K_{u}\Big|_{k+1}\frac{\delta u_{k+3/2}^{*}-\delta u_{k+1/2}^{*}}{z_{k+3/2}-z_{k+1/2}}\right)-\left(K_{u}\Big|_{k}\frac{\delta @@ -5256,16 +5263,15 @@ or, rearranging A_{k}\delta u_{k+3/2}^{*}+B_{k}\delta u_{k+1/2}^{*}+C_{k}\delta u_{k-1/2}^{*}=\Delta - t({\cal I}_{1}-{\cal - E}_{1})\left(\frac{\tau_{x}^{n}\Big|_{k+1}-\tau_{x}^{n}\Big|_{k}}{z_{k+1}-z_{k}}+S_{k+1/2}\right), + t({\mathcal{I}}_{1}-{\mathcal{E}}_{1})\left(\frac{\tau_{x}^{n}\Big|_{k+1}-\tau_{x}^{n}\Big|_{k}}{z_{k+1}-z_{k}}+S_{k+1/2}\right), where :math:`k=1,2,\ldots,L-2`, .. math:: - A_{k}=-{\cal I}_{1}\frac{\Delta t\noindent + A_{k}=-{\mathcal{I}}_{1}\frac{\Delta t\noindent K_{u}\Big|_{k+1}}{(z_{k+1}-z_{k})(z_{k+3/2}-z_{k+1/2})},\; - C_{k}=-{\cal I}_{1}\frac{\Delta + C_{k}=-{\mathcal{I}}_{1}\frac{\Delta tK_{u}\Big|_{k}}{(z_{k+1}-z_{k})(z_{k+1/2}-z_{k-1/2})},\; B_{k}=1-A_{k}-C_{k}. @@ -5276,9 +5282,8 @@ For the top :math:`\rho`-level, :math:`k=L-1`, the .. math:: :label: eq:tridiag_top - B_{L}\delta u_{L-1/2}^{*}+C_{L}\delta u_{L-3/2}^{*}=\Delta t({\cal - I}_{1}-{\cal - E}_{1})\left(\frac{\tau_{x}^{n}\Big|_{L}-\tau_{x}^{n}\Big|_{L-1}}{z_{L}-z_{L-1}}+S_{L-1/2}\right), + B_{L}\delta u_{L-1/2}^{*}+C_{L}\delta u_{L-3/2}^{*}=\Delta + t({\mathcal{I}}_{1}-{\mathcal{E}}_{1})\left(\frac{\tau_{x}^{n}\Big|_{L}-\tau_{x}^{n}\Big|_{L-1}}{z_{L}-z_{L-1}}+S_{L-1/2}\right), where :math:`B_{L}`, :math:`C_{L}` are derived as before setting :math:`A_{L}=0`. @@ -5289,24 +5294,21 @@ For the bottom :math:`\rho`-level, :math:`k=0`, the .. math:: :label: eq:u_bc_1 \delta u_{1/2}^{*} = \frac{\Delta - t}{z_{1}-0}\left(\bar{\tau}_{x}^{*}\Big|_{1}-\bar{\tau}_{x}^{*}\Big|_{0}\right)+\Delta t\left({\cal I}_{1}-{\cal E}_{1}\right)S_{1/2} + t}{z_{1}-0}\left(\bar{\tau}_{x}^{*}\Big|_{1}-\bar{\tau}_{x}^{*}\Big|_{0}\right)+\Delta t\left({\mathcal{I}}_{1}-{\mathcal{E}}_{1}\right)S_{1/2} where, from :eq:`eq:tau_star`, .. math:: :label: eq:u_bc_2 - \bar{\tau}_{x}^{*}\Big|_{1}=\left({\cal I}_{1}-{\cal - E}_{1}\right)\tau_{x}^{n}\Big|_{1}+{\cal - I}_{1}\left(K_{u}\frac{\partial\delta u^{*}}{\partial z}\right)\Big|_{1}. + \bar{\tau}_{x}^{*}\Big|_{1}=\left({\mathcal{I}}_{1}-{\mathcal{E}}_{1}\right)\tau_{x}^{n}\Big|_{1}+{\mathcal{I}}_{1}\left(K_{u}\frac{\partial\delta u^{*}}{\partial z}\right)\Big|_{1}. Combining :eq:`eq:u_bc_1`, :eq:`eq:u_bc_2` the bottom row discretization is obtained: .. math:: :label: eq:u_bc_3 - A_{0}\delta u_{3/2}^{*}+B_{0}\delta u_{1/2}^{*}=\Delta t\left({\cal - I}_{1}-{\cal - E}_{1}\right)\left(\frac{\tau_{x}^{n}\Big|_{1}}{z_{1}}+S_{1/2}\right)-\frac{\Delta t}{z_{1}}\bar{\tau}_{x}^{*}\Big|_{0} + A_{0}\delta u_{3/2}^{*}+B_{0}\delta u_{1/2}^{*}=\Delta + t\left({\mathcal{I}}_{1}-{\mathcal{E}}_{1}\right)\left(\frac{\tau_{x}^{n}\Big|_{1}}{z_{1}}+S_{1/2}\right)-\frac{\Delta t}{z_{1}}\bar{\tau}_{x}^{*}\Big|_{0} where @@ -5337,9 +5339,7 @@ equation becomes .. math:: :label: eq:tau_zero - \bar{\tau}_{x}^{*}\Big|_{0}=\left({\cal I}_{1}-{\cal - E}_{1}\right)\tau_{x}^{n}\Big|_{0}+{\cal I}_{1}K_{u}\Big|_{0}\frac{\delta - u_{1/2}^{*}}{z_{1/2}}. + \bar{\tau}_{x}^{*}\Big|_{0}=\left({\mathcal{I}}_{1}-{\mathcal{E}}_{1}\right)\tau_{x}^{n}\Big|_{0}+{\mathcal{I}}_{1}K_{u}\Big|_{0}\frac{\delta u_{1/2}^{*}}{z_{1/2}}. From :eq:`eq:du_half`, :eq:`eq:tau_zero` the following expression for the @@ -5347,9 +5347,8 @@ implicit surface stress is obtained .. math:: :label: eq:imp_tau - \bar{\tau}_{x}^{*}\Big|_{0}=\frac{\left({\cal I}_{1}-{\cal - E}_{1}\right)\tau_{x}^{n}\Big|_{0}+{\cal I}_{1}(K_{u}\Big|_{0}/z_{1/2})\delta - u_{1/2}^{'}}{1+{\cal I}_{1}(K_{u}\Big|_{0}/z_{1/2})\beta}. + \bar{\tau}_{x}^{*}\Big|_{0}=\frac{\left({\mathcal{I}}_{1}-{\mathcal{E}}_{1}\right)\tau_{x}^{n}\Big|_{0}+{\mathcal{I}}_{1}(K_{u}\Big|_{0}/z_{1/2})\delta + u_{1/2}^{'}}{1+{\mathcal{I}}_{1}(K_{u}\Big|_{0}/z_{1/2})\beta}. Then, :math:`\delta u_{1/2}^{*}` can be computed from :eq:`eq:imp_tau` and :eq:`eq:du_half`. @@ -5358,9 +5357,8 @@ the 2nd stage :eq:`eq:sppf_inc2` will be .. math:: :label: eq:imp_tau2 - \bar{\tau}_{x}^{n+1}\Big|_{0}=\frac{\left({\cal I}_{2}-{\cal - E}_{2}\right)\tau_{x}^{*}\Big|_{0}+{\cal I}_{2}(K_{u}\Big|_{0}/z_{1/2})\delta - u_{1/2}^{'}}{1+{\cal I}_{2}(K_{u}\Big|_{0}/z_{1/2})\beta}. + \bar{\tau}_{x}^{n+1}\Big|_{0}=\frac{\left({\mathcal{I}}_{2}-{\mathcal{E}}_{2}\right)\tau_{x}^{*}\Big|_{0}+{\mathcal{I}}_{2}(K_{u}\Big|_{0}/z_{1/2})\delta + u_{1/2}^{'}}{1+{\mathcal{I}}_{2}(K_{u}\Big|_{0}/z_{1/2})\beta}. In the same way :math:`\bar{\tau}_{y}^{*}\Big|_{0}`, :math:`\bar{\tau}_{y}^{n+1}\Big|_{0}` can be derived. @@ -5383,14 +5381,14 @@ is applied as follows. Consider the equivalent discrete form of .. math:: :label: eq:dX_star \frac{\delta X^{*}}{\Delta t} - =\frac{\partial\overline{F}^{*}}{\partial z}+\left({\cal I}_{1}-{\cal - E}_{1}\right)S + =\frac{\partial\overline{F}^{*}}{\partial + z}+\left({\mathcal{I}}_{1}-{\mathcal{E}}_{1}\right)S Considering that, .. math:: :label: eq:dX_star2 - \overline{F}^{*}={\cal I}_{1}F^{*}-{\cal E}_{1}F^{n},\qquad + \overline{F}^{*}={\mathcal{I}}_{1}F^{*}-{\mathcal{E}}_{1}F^{n},\qquad F^{*}=F^{n}+K_{X}\frac{\partial\delta X^{*}}{\partial z} :eq:`eq:dX_star` would re-produce @@ -5406,20 +5404,19 @@ and thus the following discretization is obtained, on .. math:: - \frac{\delta X_{k}^{*}}{\Delta t} = \left({\cal I}_{1}-{\cal - E}_{1}\right)\left(\frac{F_{k+1/2}^{n}-F_{k-1/2}^{n}}{z_{k+1/2}-z_{k-1/2}}+S_{k}\right) + \frac{\delta X_{k}^{*}}{\Delta t} = + \left({\mathcal{I}}_{1}-{\mathcal{E}}_{1}\right)\left(\frac{F_{k+1/2}^{n}-F_{k-1/2}^{n}}{z_{k+1/2}-z_{k-1/2}}+S_{k}\right) .. math:: - +\frac{{\cal I}_{1}}{z_{k+1/2}-z_{k-1/2}}\left[K_{X}\Big|_{k+1/2}\left(\frac{\delta X_{k+1}^{*}-\delta X_{k}^{*}}{z_{k+1}-z_{k}}\right)-K_{X}\Big|_{k-1/2}\left(\frac{\delta X_{k}^{*}-\delta X_{k-1}^{*}}{z_{k}-z_{k-1}}\right)\right],\; k=2,\ldots,L-1. + +\frac{{\mathcal{I}}_{1}}{z_{k+1/2}-z_{k-1/2}}\left[K_{X}\Big|_{k+1/2}\left(\frac{\delta X_{k+1}^{*}-\delta X_{k}^{*}}{z_{k+1}-z_{k}}\right)-K_{X}\Big|_{k-1/2}\left(\frac{\delta X_{k}^{*}-\delta X_{k-1}^{*}}{z_{k}-z_{k-1}}\right)\right],\; k=2,\ldots,L-1. or, .. math:: :label: eq:dX_disc A_{k}\delta X_{k+1}^{*}+B_{k}\delta X_{k}^{*}+C_{k}\delta - X_{k-1}^{*}=\left({\cal I}_{1}-{\cal - E}_{1}\right)\left(\frac{F_{k+1/2}^{n}-F_{k-1/2}^{n}}{z_{k+1/2}-z_{k-1/2}}+S_{k}\right),\quad k=2,\ldots,L-1 + X_{k-1}^{*}=\left({\mathcal{I}}_{1}-{\mathcal{E}}_{1}\right)\left(\frac{F_{k+1/2}^{n}-F_{k-1/2}^{n}}{z_{k+1/2}-z_{k-1/2}}+S_{k}\right),\quad k=2,\ldots,L-1 where, @@ -5433,8 +5430,8 @@ The discrete equation for the top level, :math:`k=L`, will be: .. math:: :label: eq:dX_disc_top - B_{L}\delta X_{L}^{*}+C_{L}\delta X_{L-1}^{*}=\left({\cal I}_{1}-{\cal - E}_{1}\right)\left(\frac{-F_{L-1/2}^{n}}{z_{L+1/2}-z_{L-1/2}}+S_{L}\right), + B_{L}\delta X_{L}^{*}+C_{L}\delta + X_{L-1}^{*}=\left({\mathcal{I}}_{1}-{\mathcal{E}}_{1}\right)\left(\frac{-F_{L-1/2}^{n}}{z_{L+1/2}-z_{L-1/2}}+S_{L}\right), where :math:`B_{L}`, :math:`C_{L}` are derived as before setting :math:`A_{L}=0`. @@ -5446,7 +5443,7 @@ From :eq:`eq:dX_star`, a bottom interior level \delta X_{1}^{*}=\frac{\Delta t}{z_{3/2}-0}\left(\overline{F_{3/2}}^{*}-\overline{F_{0}}^{*}\right)+\Delta - t\left({\cal I}_{1}-{\cal E}_{1}\right)S_{1} + t\left({\mathcal{I}}_{1}-{\mathcal{E}}_{1}\right)S_{1} :math:`F_{0}` is used instead of :math:`F_{1/2}`. The former is computed by the implicit surface scheme. This flux gradient is defined in the @@ -5458,18 +5455,16 @@ becomes \delta X_{1}^{*}=\frac{\Delta t}{z_{3/2}} - \left[\left({\cal I}_{1}-{\cal - E}_{1}\right)F_{3/2}^{n}-\overline{F_{0}}^{*}\right] - +\Delta t\left({\cal I}_{1}-{\cal E}_{1}\right) - S_{1}+\Delta t{\cal I}_{1}\frac{1}{z_{3/2}} + \left[\left({\mathcal{I}}_{1}-{\mathcal{E}}_{1}\right)F_{3/2}^{n}-\overline{F_{0}}^{*}\right] + +\Delta t\left({\mathcal{I}}_{1}-{\mathcal{E}}_{1}\right) + S_{1}+\Delta t{\mathcal{I}}_{1}\frac{1}{z_{3/2}} \left(K_{X}\frac{\partial\delta X^{*}}{\partial z}\right)_{3/2} where :math:`\overline{F}_{0}^{*}` can be approximated as .. math:: :label: eq:F0_star - \overline{F}_{0}^{*}={\cal I}_{1}F_{0}^{*}-{\cal - E}_{1}F_{0}^{n}\approx\left({\cal I}_{1}-{\cal E}_{1}\right)F_{JULES} + \overline{F}_{0}^{*}={\mathcal{I}}_{1}F_{0}^{*}-{\mathcal{E}}_{1}F_{0}^{n}\approx\left({\mathcal{I}}_{1}-{\mathcal{E}}_{1}\right)F_{JULES} where, :math:`F_{JULES}` is the implicit flux calculated by the *implicit surface scheme using the original implicit algorithm*. @@ -5477,9 +5472,9 @@ Finalising, the discrete equations for the bottom level will be .. math:: - \delta X_{1}^{*}=\Delta t\left({\cal I}_{1}-{\cal E}_{1}\right) + \delta X_{1}^{*}=\Delta t\left({\mathcal{I}}_{1}-{\mathcal{E}}_{1}\right) \left(\frac{F_{3/2}^{n}-F_{JULES}}{z_{3/2}}+S_{1}\right) - +\Delta t{\cal I}_{1}\frac{1}{z_{3/2}}K_{X}\Big|_{3/2} + +\Delta t{\mathcal{I}}_{1}\frac{1}{z_{3/2}}K_{X}\Big|_{3/2} \left(\frac{\delta X_{2}^{*}-\delta X_{1}^{*}}{z_{2}-z_{1}}\right) or, @@ -5487,14 +5482,14 @@ or, .. math:: :label: eq:dX_bottom A_{1}\delta X_{2}^{*}+B_{1}\delta X_{1}^{*}=\Delta t - \left({\cal I}_{1}-{\cal - E}_{1}\right)\left(\frac{F_{3/2}^{n}-F_{JULES}}{z_{3/2}}+S_{1}\right) + \left({\mathcal{I}}_{1}-{\mathcal{E}}_{1}\right)\left(\frac{F_{3/2}^{n}-F_{JULES}}{z_{3/2}}+S_{1}\right) where, .. math:: - A_{1}=-{\cal I}_{1}\Delta t\frac{K_{X}\Big|_{3/2}}{z_{3/2}(z_{2}-z_{1})}, + A_{1}=-{\mathcal{I}}_{1}\Delta + t\frac{K_{X}\Big|_{3/2}}{z_{3/2}(z_{2}-z_{1})}, \quad B_{1}=1-A_{1}. Equations :eq:`eq:dX_disc`, @@ -5507,21 +5502,18 @@ Similarly the corresponding discrete equations for .. math:: :label: eq:dXtop_np1 - B_{L}^{'}\delta X_{L}^{n+1}+C_{L}^{'}\delta X_{L-1}^{n+1}=\left({\cal - I}_{2}-{\cal - E}_{2}\right)\left(\frac{-F_{L-1/2}^{*}}{z_{L+1/2}-z_{L-1/2}}+S_{L}\right), + B_{L}^{'}\delta X_{L}^{n+1}+C_{L}^{'}\delta + X_{L-1}^{n+1}=\left({\mathcal{I}}_{2}-{\mathcal{E}}_{2}\right)\left(\frac{-F_{L-1/2}^{*}}{z_{L+1/2}-z_{L-1/2}}+S_{L}\right), .. math:: :label: eq:dXk_np1 A_{k}^{'}\delta X_{k+1}^{n+1}+B_{k}^{'}\delta X_{k}^{n+1}+C_{k}^{'}\delta - X_{k-1}^{n+1}=\left({\cal I}_{2}-{\cal - E}_{2}\right)\left(\frac{F_{k+1/2}^{*}-F_{k-1/2}^{*}}{z_{k+1/2}-z_{k-1/2}}+S_{k}\right),\quad k=L-1,\ldots,2 + X_{k-1}^{n+1}=\left({\mathcal{I}}_{2}-{\mathcal{E}}_{2}\right)\left(\frac{F_{k+1/2}^{*}-F_{k-1/2}^{*}}{z_{k+1/2}-z_{k-1/2}}+S_{k}\right),\quad k=L-1,\ldots,2 .. math:: :label: eq:dX1_np1 - A_{1}^{'}\delta X_{2}^{n+1}+B_{1}^{'}\delta X_{1}^{n+1}=\left({\cal - I}_{2}-{\cal - E}_{2}\right)\left(\frac{F_{3/2}^{*}-F_{JULES}}{z_{3/2}}+S_{1}\right) + A_{1}^{'}\delta X_{2}^{n+1}+B_{1}^{'}\delta + X_{1}^{n+1}=\left({\mathcal{I}}_{2}-{\mathcal{E}}_{2}\right)\left(\frac{F_{3/2}^{*}-F_{JULES}}{z_{3/2}}+S_{1}\right) where, @@ -5540,8 +5532,7 @@ and the approximation .. math:: :label: eq:F0_np1 - \overline{F}_{0}^{n+1}={\cal I}_{2}F_{0}^{n+1}-{\cal - E}_{2}F_{0}^{n}\approx\left({\cal I}_{2}-{\cal E}_{2}\right)F_{JULES} + \overline{F}_{0}^{n+1}={\mathcal{I}}_{2}F_{0}^{n+1}-{\mathcal{E}}_{2}F_{0}^{n}\approx\left({\mathcal{I}}_{2}-{\mathcal{E}}_{2}\right)F_{JULES} has taken place. The same flux :math:`F_{JULES}` will be used for both :eq:`eq:F0_star` and :eq:`eq:F0_np1` and @@ -5627,17 +5618,11 @@ corrector are defined as: .. math:: - \overline{\tau_{x}}^{[n,*]}\equiv{\cal I}_{1}\tau_{x}^{*}-{\cal - E}_{1}\tau_{x}^{n} = \left({\cal I}_{1}-{\cal - E}_{1}\right)\tau_{x}^{n}+{\cal I}_{1}K_{u}\frac{\partial\delta - u^{*}}{\partial z} + \overline{\tau_{x}}^{[n,*]}\equiv{\mathcal{I}}_{1}\tau_{x}^{*}-{\mathcal{E}}_{1}\tau_{x}^{n} = \left({\mathcal{I}}_{1}-{\mathcal{E}}_{1}\right)\tau_{x}^{n}+{\mathcal{I}}_{1}K_{u}\frac{\partial\delta u^{*}}{\partial z} .. math:: - \overline{\tau_{x}}^{[*,n+1]}\equiv{\cal I}_{2}\tau_{x}^{n+1}-{\cal - E}_{2}\tau_{x}^{*} = \left({\cal I}_{2}-{\cal - E}_{2}\right)\tau_{x}^{*}+{\cal I}_{2}K_{u}\frac{\partial\delta - u^{n+1}}{\partial z} + \overline{\tau_{x}}^{[*,n+1]}\equiv{\mathcal{I}}_{2}\tau_{x}^{n+1}-{\mathcal{E}}_{2}\tau_{x}^{*} = \left({\mathcal{I}}_{2}-{\mathcal{E}}_{2}\right)\tau_{x}^{*}+{\mathcal{I}}_{2}K_{u}\frac{\partial\delta u^{n+1}}{\partial z} where, :math:`\delta u^{*}=u^{*}-u^{n},\;\delta u^{n+1}=u^{n+1}-u^{*}`. The meridional stress :math:`\tau_{y}` and the scalar fluxes can be @@ -5675,7 +5660,7 @@ modified version of the flux formulae (78), (79) of A_{1})(1+\beta B_{2})-\beta^{2}A_{2}B_{1}} where, :math:`F_{T}^{n}`, :math:`F_{Q}^{n}` denote the surface explicit -fluxes, :math:`\gamma_{2}={\cal I}_{1}-{\cal E}_{1}` and the +fluxes, :math:`\gamma_{2}={\mathcal{I}}_{1}-{\mathcal{E}}_{1}` and the coefficients :math:`A_{1},\; A_{2},B_{1},\; B_{2}` are given by .. math:: :label: ab_coeffs @@ -5697,7 +5682,7 @@ coefficients :math:`A_{1},\; A_{2},B_{1},\; B_{2}` are given by \psi_j[c_pRK_H(1)_j+A_{*j}]. \end{equation} -but with :math:`\gamma_{1}={\cal I}_{1}`. Here +but with :math:`\gamma_{1}={\mathcal{I}}_{1}`. Here :math:`RK_H(1) =\rho C_H U_1`, :math:`RK_{PM}={RK_H(1)\over(c_p+LD\psi)RK_H(1)+A_*}`, @@ -5761,7 +5746,7 @@ be obtained. where, the coefficients :math:`A_{1},\; A_{2},B_{1},\; B_{2}` are given by :eq:`ab_coeffs` but with -:math:`\gamma_{1}={\cal I}_{2}`. The above formulae are derived as +:math:`\gamma_{1}={\mathcal{I}}_{2}`. The above formulae are derived as explained earlier. The definitions .. math:: \overline{T_{1}^{n+1}}=\gamma_{2}T^{*}+\gamma_{1}\delta @@ -6483,11 +6468,11 @@ In the 9B version, :math:`\Delta_F` is calculated as: .. math:: :label: ctraddiv \Delta_F= \sum_{k=k_m-1}^{k_m+1} \mathrm{max}\left[ - - \Delta_{k+\frac{1}{2}} z \, {\cal S}_F(k), \,0 \right] + - \Delta_{k+\frac{1}{2}} z \, {\mathcal{S}}_F(k), \,0 \right] where :math:`k_m` is the grid-level with the greatest radiative cooling -increment, :math:`{\cal S}_F`, within 2 grid-levels of cloud-top. In the -8A scheme, :math:`{\cal S}_F` is simply the net (SW+LW) cooling +increment, :math:`{\mathcal{S}}_F`, within 2 grid-levels of cloud-top. In the +8A scheme, :math:`{\mathcal{S}}_F` is simply the net (SW+LW) cooling increment. During the day, though, the net divergence is partly reduced from the nocturnal (LW) value due to SW warming of the cloud-layer. In general, the SW warming is more diffuse than the LW cooling (which @@ -7105,7 +7090,7 @@ Appendix: Notation * - :math:`C_F`, :math:`C_F^l`, :math:`C_F^f` - cloud fraction and the liquid and frozen water parts, respectively - * - :math:`{\cal H}` + * - :math:`{\mathcal{H}}` - total heat flux (net radiative plus turbulent, Kms\ :math:`^{-1}`) .. list-table:: From ae654880b8384f57e5d7a8710e87776c6bf02838 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Wed, 13 May 2026 14:54:27 +0100 Subject: [PATCH 078/116] Sanitized non-ASCII characters. --- .../turbulence_schemes/bldoc.rst | 612 +++++++++--------- 1 file changed, 306 insertions(+), 306 deletions(-) diff --git a/documentation/source/science_guide/turbulence_schemes/bldoc.rst b/documentation/source/science_guide/turbulence_schemes/bldoc.rst index 511342e6f0..9a55115866 100644 --- a/documentation/source/science_guide/turbulence_schemes/bldoc.rst +++ b/documentation/source/science_guide/turbulence_schemes/bldoc.rst @@ -14,7 +14,7 @@ The Parametrization of Boundary Layer Processes =============================================== -:Author: A. Lock, J. Edwards and I. Boutle +:Author: A. Lock, J. Edwards and I. Boutle .. role:: raw-latex(raw) :format: latex @@ -23,13 +23,13 @@ The Parametrization of Boundary Layer Processes Introduction and code versions ============================== -This is the documentation for the “boundary layer” parametrization of +This is the documentation for the "boundary layer" parametrization of vertical turbulent transports of heat, moisture and horizontal momentum. It includes surface exchange but *not* the parametrization of the surface itself. This is covered within the surface (JULES) -documentation. Although commonly referred to as the “boundary layer” +documentation. Although commonly referred to as the "boundary layer" parametrization, it includes a free-tropospheric component. Turbulent -fluxes are calculated up to “BL_LEVELS” which is currently set so that +fluxes are calculated up to "BL_LEVELS" which is currently set so that the entire troposphere is included. Generally speaking, only version 9C of the boundary layer @@ -50,7 +50,7 @@ Model variables and turbulence closure ====================================== Given source terms, :math:`{\mathcal{S}}` say, from processes other than -boundary layer turbulence, Reynolds’ averaging gives the following +boundary layer turbulence, Reynolds' averaging gives the following equation for conserved scalar variables, :math:`\chi`, and the two horizontal components of momentum, :math:`{\mathbf{u}}` on a sphere gives: @@ -86,17 +86,17 @@ approximately conserved under moist adiabatic ascent, are: where :math:`T` is temperature, :math:`q_v` is specific humidity, :math:`q_{\ell}` and :math:`q_f` the specific liquid and frozen water contents respectively, and :math:`L_s=L+L_f` is the latent heat of -sublimation. Note that :math:`\theta_{\ell}` is based on ‘liquid/frozen -water static energy’ (:math:`= c_p T + g z - L q_{\ell}- +sublimation. Note that :math:`\theta_{\ell}` is based on 'liquid/frozen +water static energy' (:math:`= c_p T + g z - L q_{\ell}- L_s q_f`) rather than potential temperature, :math:`\theta`. Note also that the option to use mixing ratios in the boundary layer code instead of specific quantities is also available and the details of the -necessary changes are documented in appendix :ref:`Appendix: changing between +necessary changes are documented in appendix :ref:`Appendix: changing between specific humidities and mixing ratios `. Ultimately turbulent motions are dissipated as heat and so the source term :math:`{\cal S}` in :eq:`cons_eqn_scal` can include an approximation for that frictional heating, as described in -appendix :ref:`Appendix: including the heating from turbulence dissipation +appendix :ref:`Appendix: including the heating from turbulence dissipation `. Finally, the ice cloud contributions in :eq:`thetal` and :eq:`qt` can optionally be ignored (l_noice_in_turb), which will be more appropriate if the time scales for @@ -119,7 +119,7 @@ variable that is equal to virtual potential temperature (:math:`\theta_v`) in cloud-free air and so is used as a simplified measure of buoyancy. -A ‘first-order’ closure is used to parametrize the turbulent fluxes, +A 'first-order' closure is used to parametrize the turbulent fluxes, although non-local terms are also included. Under the 9C scheme, an alternative methodology is optionally available, see section :ref:`The revised scalar flux-gradient formulation `. The @@ -140,42 +140,42 @@ side represents a non-local flux in unstable boundary layers. Currently it is only applied for transport arising from surface-driven turbulence (:math:`K_h^{\mathrm{surf}}`) and is non-zero only for :math:`\chi=\theta_{\ell}`, as described in -section :ref:`Gradient adjustment `. +section :ref:`Gradient adjustment `. Thus, the parametrization reduces to determining :math:`K_h`, :math:`K_m` and :math:`\gamma_{\chi}` and :math:`{\bf \tau}^{nl}`. Two methods are used to determine :math:`K_h` and :math:`K_m` and how they are combined for :eq:`scal_closure` and :eq:`uv_closure` is described in -section :ref:`Shear-driven mixing and interaction between the local and +section :ref:`Shear-driven mixing and interaction between the local and non-local schemes `. The first method is a local Richardson number (:math:`Ri`) based scheme. It is calculated for all regimes (but will be responsible for all mixing in stable conditions), over all levels up to the specified BL_LEVELS and is described in -section :ref:`The local scheme `. The second method is a non-locally +section :ref:`The local scheme `. The second method is a non-locally specified profile scheme. This is exclusively for unstable boundary layers, is calculated up to level NL_BL_LEVELS (typically around 6km AMSL) and is -described in more detail in section :ref:`The non-local scheme `. +described in more detail in section :ref:`The non-local scheme `. In this regime, mixing is assumed to occur in (or lead rapidly to the formation of) well-mixed layers (in which conserved variables are approximately uniform with height) that are capped by an inversion. Mixing is assumed -to be driven either from the surface in a ‘surface mixed layer’ (SML, by +to be driven either from the surface in a 'surface mixed layer' (SML, by a positive surface buoyancy flux and by surface stresses) or by cloud-top buoyancy sources (radiative and evaporative cooling, see -appendix :ref:`Appendix: Definitions of the velocity scales `). As +appendix :ref:`Appendix: Definitions of the velocity scales `). As described in section :ref:`The non-local scheme `, separate :math:`K`-profiles are used for these two turbulence sources. If the cloud-top sources generate mixing -throughout the SML the layer is said to be ‘coupled’ but if the +throughout the SML the layer is said to be 'coupled' but if the :math:`K`-profile representing surface-driven mixing does not extend up -to cloud-top, the layer is referred to as being ‘decoupled’. As +to cloud-top, the layer is referred to as being 'decoupled'. As decoupled layers are restricted to being buoyancy driven and typically below 6km, they are referred to as decoupled stratocumulus (DSC) layers. The calculation of :math:`\gamma_{\chi}` is described in -section :ref:`Gradient adjustment ` and, finally, fluxes across +section :ref:`Gradient adjustment ` and, finally, fluxes across the top of both SML and DSC layers (the entrainment fluxes) are specified explicitly through a separate entrainment parametrization, as described @@ -183,10 +183,10 @@ in section :ref:`Entrainment fluxes `. The strategy used to determine precisely where and when the resulting eddy-diffusivities should be applied is described in -section :ref:`Diagnosis of boundary layer depth and type `. The +section :ref:`Diagnosis of boundary layer depth and type `. The buoyancy parameters, finite difference and other notation used here are defined in -appendices :ref:`Appendix: Derivation and definitions of the buoyancy +appendices :ref:`Appendix: Derivation and definitions of the buoyancy parameters ` and :ref:`Appendix: Notation `. Further papers describing this scheme and its performance are `Lock et al. (2000)`_ (noting the corrigendum in @@ -202,53 +202,53 @@ Diagnosis of boundary layer depth and type The non-locally specified :math:`K`-profiles require the height of the base and top of the layer to be diagnosed (see -section :ref:`The non-local scheme `). Furthermore, as stated in -section :ref:`Model variables and turbulence closure `, the mixing +section :ref:`The non-local scheme `). Furthermore, as stated in +section :ref:`Model variables and turbulence closure `, the mixing generated by the non-local :math:`K` profiles is assumed to occur in (or lead rapidly to the formation of) well-mixed layers capped by an inversion. Thus, the accurate diagnosis of their vertical extent is crucial. How to make this diagnosis is dependent on the boundary layer mixing regime which has -been categorised into 7 distinct ‘boundary layer types’: +been categorised into 7 distinct 'boundary layer types': -- **Type I**: Stable boundary layer (with or without cloud) — turbulent - diffusivities are calculated by the ‘local’ scheme - (section :ref:`The local scheme `) +- **Type I**: Stable boundary layer (with or without cloud) -- turbulent + diffusivities are calculated by the 'local' scheme + (section :ref:`The local scheme `) - **Type II**: Boundary layer with stratocumulus over a stable - near-surface layer — as Type I but with a turbulently mixed cloud + near-surface layer -- as Type I but with a turbulently mixed cloud layer driven from its top (a DSC layer, diagnosis described in section :ref:`Diagnosis of the vertical extent of the K-profiles `) -- **Type III**: Well mixed boundary layer — the classic single mixed +- **Type III**: Well mixed boundary layer -- the classic single mixed layer which may be cloud-topped or clear but is predominantly - buoyancy-driven (c.f. a possible type VII below) — diagnosis described - in section :ref:`The diagnostic parcel ascent and cumulus diagnosis + buoyancy-driven (c.f. a possible type VII below) -- diagnosis described + in section :ref:`The diagnostic parcel ascent and cumulus diagnosis `) - **Type IV**: Unstable boundary layer with a DSC layer not over cumulus - (see section :ref:`Diagnosis of the vertical extent of the K-profiles - `) — the surface-based and + (see section :ref:`Diagnosis of the vertical extent of the K-profiles + `) -- the surface-based and cloud-top-driven non-local :math:`K` profiles may or may not overlap and cloud-top entrainment can still include the surface forcing (see - section :ref:`Entrainment fluxes `) + section :ref:`Entrainment fluxes `) -- **Type V**: Boundary layer with a DSC layer over cumulus — the cumulus - (treated by the model’s mass-flux convection scheme) provides coupling +- **Type V**: Boundary layer with a DSC layer over cumulus -- the cumulus + (treated by the model's mass-flux convection scheme) provides coupling with the SML (cumulus diagnosis described in section :ref:`The diagnostic parcel ascent and cumulus diagnosis `) -- **Type VI**: Cumulus-capped boundary layer — no turbulent +- **Type VI**: Cumulus-capped boundary layer -- no turbulent diffusivities are allowed [1]_ at or above the LCL as the mass-flux convection scheme operates here (cumulus diagnosis described in - section :ref:`The diagnostic parcel ascent and cumulus diagnosis + section :ref:`The diagnostic parcel ascent and cumulus diagnosis `) -- **Type VII**: Shear-dominated unstable layer — potentially wind-shear +- **Type VII**: Shear-dominated unstable layer -- potentially wind-shear might allow deeper turbulent mixing in unstable boundary layers than is apparent purely from the thermodynamic profiles (sufficient even to inhibit the formation of cumulus); the possibilities are discussed in - section :ref:`Shear-driven mixing and interaction between the local and + section :ref:`Shear-driven mixing and interaction between the local and non-local schemes `. Types I to VI are shown schematically in :numref:`Fig. %s `. @@ -276,16 +276,16 @@ The diagnostic parcel ascent and cumulus diagnosis **Summary**: the depth of the non-local :math:`K`-profiles for surface-driven turbulence (with NTML grid-levels in the mixed layer and -top at height :math:`z_{\mathrm{h}}` , as required for +top at height :math:`z_{\mathrm{h}}` , as required for :eq:`kmsurf`) is determined from: #. a diagnostic moist parcel ascent; top at grid-level NTPAR, height - :math:`z_{\mathrm{par}}` :math:`=z_{\mathrm{ \mathrm{NTPAR}}+\frac{1}{2}}`. + :math:`z_{\mathrm{par}}` :math:`=z_{\mathrm{ \mathrm{NTPAR}}+\frac{1}{2}}`. Typically this is an adiabatic parcel but entraining options are available. #. a diagnosis of cumulus-capped layers (if cumulus-capped then NTML and - :math:`z_{\mathrm{h}}` are set to the LCL [2]_, if not then to the parcel + :math:`z_{\mathrm{h}}` are set to the LCL [2]_, if not then to the parcel top) Note that this process is only performed for unstable boundary layers @@ -297,22 +297,22 @@ driven by surface processes can extend in unstable boundary layers (and therefore the vertical extent of the :math:`K` profile for surface-driven turbulence) can be determined solely from the properties of the thermodynamic profiles. In more detail, the first step in -calculating :math:`z_{\mathrm{h}}` is to lift a parcel, with properties from +calculating :math:`z_{\mathrm{h}}` is to lift a parcel, with properties from the first grid-level (:math:`k=k_s`) above the top of the surface layer, upwards allowing for latent heat release. The top of the surface layer -is taken to be at the lower of :math:`z=0.1`\ :math:`z_{\mathrm{h}}` (this is +is taken to be at the lower of :math:`z=0.1`\ :math:`z_{\mathrm{h}}` (this is then consistent with the :math:`K`-profiles, see -section :ref:`Surface-driven turbulence `; -:math:`z_{\mathrm{h}}` is taken from the +section :ref:`Surface-driven turbulence `; +:math:`z_{\mathrm{h}}` is taken from the previous timestep) and the grid-level above which :math:`\theta_{v\ell}` starts to increase with height. The ascent is stopped at the grid-level NTPAR (height -:math:`z_{\mathrm{par}}` :math:`=z_{\mathrm{ \mathrm{NTPAR}}+\frac{1}{2}}`) +:math:`z_{\mathrm{par}}` :math:`=z_{\mathrm{ \mathrm{NTPAR}}+\frac{1}{2}}`) above which the parcel becomes more negatively buoyant than a given threshold, :math:`\theta_v'`. Note that the parcel properties themselves -are not perturbed in order to preserve the height of the mixed-layer’s -lifting condensation level (LCL). The calculation of the parcel’s -buoyancy excess is described in section :ref:`Calculation of parcel buoyancy +are not perturbed in order to preserve the height of the mixed-layer's +lifting condensation level (LCL). The calculation of the parcel's +buoyancy excess is described in section :ref:`Calculation of parcel buoyancy excess `. Currently, @@ -348,9 +348,9 @@ where the vapour pressure of air in grid-level :math:`k_s`, :math:`e_{k_s} = q_{k_s} P_{k_s}/(100 \, \epsilon)`. The full-level below that containing the LCL is labelled NLCL and -:math:`z_{\mathrm{lcl}}` :math:`=z_{\mathrm{ \mathrm{NLCL}}+\frac{1}{2}}`. If +:math:`z_{\mathrm{lcl}}` :math:`=z_{\mathrm{ \mathrm{NLCL}}+\frac{1}{2}}`. If the parcel rises above the top of the LCL transition zone (defined as -1.1\ :math:`z_{\mathrm{lcl}}` , its ascent can also be stopped at the +1.1\ :math:`z_{\mathrm{lcl}}` , its ascent can also be stopped at the grid-level at which it has maximum buoyancy excess over the environment. This is identified as the grid-level above which @@ -365,9 +365,9 @@ where currently the tolerance for identifying inversions by this method, :math:`\theta_v'`) is typically of little consequence in stratocumulus regions (which tend to be well-mixed beneath large inversions), but can be necessary in order to identify the capping inversion in cumulus cases -(e.g. in the trade wind regions). +(e.g. in the trade wind regions). -**Step 2:** having established :math:`z_{\mathrm{par}}` , a crucial +**Step 2:** having established :math:`z_{\mathrm{par}}` , a crucial additional test is to determine whether this layer is well-mixed (i.e., stratocumulus-capped) or cumulus-capped. The parcel ascent can rise to cloud-top in both cases but cumulus cloud layers are observed not to be @@ -388,21 +388,21 @@ layer has just deepened by a grid-level) and NLCL and the sub-cloud layer gradient, :math:`\Delta_{\mathrm{sub}}`, between grid-levels NLCL and :math:`k_s`. Currently the threshold factor, :math:`C_t = 1.1`. If cumulus is diagnosed, the top of the surface-based mixed layer -(:math:`z_{\mathrm{h}}` ) is set to :math:`z_{\mathrm{lcl}}` (rather than to -:math:`z_{\mathrm{par}}` , as illustrated in :numref:`Fig. %s ` for +(:math:`z_{\mathrm{h}}` ) is set to :math:`z_{\mathrm{lcl}}` (rather than to +:math:`z_{\mathrm{par}}` , as illustrated in :numref:`Fig. %s ` for types V and VI). There is then an option to diagnose the thickness of -the LCL transition zone, see section :ref:`Diagnosis of the LCL transition zone +the LCL transition zone, see section :ref:`Diagnosis of the LCL transition zone thickness `. Otherwise, the boundary layer surface-driven mixing is capped at -:math:`z_{\mathrm{lcl}}` so that mixing into the cumulus cloud layer is only -carried out by the model’s mass-flux convection scheme and not by the +:math:`z_{\mathrm{lcl}}` so that mixing into the cumulus cloud layer is only +carried out by the model's mass-flux convection scheme and not by the eddy viscosity based boundary layer scheme. Note that basing the CUMULUS diagnosis on cloud and sub-cloud layer gradients limits the model only to being able to resolve cumulus with cloud and sub-cloud layers at least 2 grid-levels (and optionally 400m) thick. Otherwise the layer is -considered well-mixed to :math:`z_{\mathrm{par}}` with an option to include a +considered well-mixed to :math:`z_{\mathrm{par}}` with an option to include a representation of fluxes into the capping inversion (see -section :ref:`Diagnosis of inversion thickness `). +section :ref:`Diagnosis of inversion thickness `). If the parcel ascent fails to find an inversion below 3km (or BL_LEVELS) but the LCL is below BL_LEVELS, then the layer is assumed to be @@ -415,7 +415,7 @@ BL_LEVELS above the tropopause is recommended. Note that if cumulus is not diagnosed then a further, subgrid estimation of the height of the capping inversion is attempted for -:math:`z_{\mathrm{h}}`  (as described in section :ref:`Diagnosis of a sub-grid +:math:`z_{\mathrm{h}}` (as described in section :ref:`Diagnosis of a sub-grid inversion `). .. _sec_parxs: @@ -423,7 +423,7 @@ inversion `). Calculation of parcel buoyancy excess ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ -As described in appendix :ref:`Appendix: Derivation and definitions of the +As described in appendix :ref:`Appendix: Derivation and definitions of the buoyancy parameters `, virtual temperature, :math:`T_v = T(1 + c_v q_v - q_{\ell}- q_f)`, is used as the measure of buoyancy. The @@ -452,7 +452,7 @@ before the grid-level becomes saturated in the mean. To allow for this in the parcel (without applying the cloud scheme), :eq:`qlpar` is also calculated at each grid-level but using the environment grid-box mean :math:`q_t` and :math:`\theta_{\ell}` to -give :math:`q_{\ell f}^e`. The difference in the environment’s condensed +give :math:`q_{\ell f}^e`. The difference in the environment's condensed water as determined by the UM cloud scheme (i.e., :math:`q_{\ell}+q_f`) and by :eq:`qlpar` (i.e., :math:`q_{\ell f}^e`) is then added to :math:`q^p_{lf}`. @@ -462,9 +462,9 @@ Given :math:`q_{\ell f}^p`, :eq:`thetal` implies + (L q_{\ell f}^p/c_p)` (using :math:`L_s` if :math:`T_k` is below the melting point) and :eq:`qt` implies :math:`q_v^p = q_t^p - q_{\ell f}^p` and thus :math:`T_v^p` can be -calculated. Recall that the diagnosis of the parcel’s maximum buoyancy +calculated. Recall that the diagnosis of the parcel's maximum buoyancy excess over the environment (described in -section :ref:`The diagnostic parcel ascent and cumulus diagnosis `) required :math:`\theta_v`. This is +section :ref:`The diagnostic parcel ascent and cumulus diagnosis `) required :math:`\theta_v`. This is approximated as :math:`\theta_v = T_v + (g z_k/c_p)`. .. _sec_decouple: @@ -478,8 +478,8 @@ been separated in to three stages. These are: #. diagnose the existence of a decoupled stratocumulus (DSC) layer with approximately uniform :math:`\theta_{v\ell}` (label the top grid-level in the mixed-layer NTDSC and diagnose the subgrid height - of its capping inversion, :math:`z_{\mathrm{h}}^{\mathrm{Sc}}` , see - section :ref:`Diagnosis of a sub-grid inversion `) + of its capping inversion, :math:`z_{\mathrm{h}}^{\mathrm{Sc}}` , see + section :ref:`Diagnosis of a sub-grid inversion `) #. diagnose an approximate depth of the DSC layer, :math:`z_{\mathrm{ml}}`, in order to be able to calculate the representative turbulent @@ -487,7 +487,7 @@ been separated in to three stages. These are: scales `). #. calculate the depth of the :math:`K` profiles (see - section :ref:`The non-local scheme `) in both SML and DSC + section :ref:`The non-local scheme `) in both SML and DSC layers using constraints on the TKE budget of the layer. This includes the diagnosis of recoupling of DSC layers and decoupling of SMLs @@ -516,7 +516,7 @@ and :math:`k_{ct}-2` are identified as well-mixed a further test is applied to determine whether the :math:`\theta_v` (rather than :math:`\theta_{v\ell}`) gradient across grid-levels :math:`k_{ct}` and :math:`k_{ct}-1` is greater than adiabatic (i.e., whether grid-levels -:math:`k_{ct}` and :math:`k_{ct}-1` actually form part of an inversion — +:math:`k_{ct}` and :math:`k_{ct}-1` actually form part of an inversion -- note that by ignoring the :math:`q_{\ell}` contribution to buoyancy, :math:`\theta_{v\ell}` is not a good variable to use to measure the strength of cloud-capping inversions). To do this, the :math:`\theta_v` @@ -543,7 +543,7 @@ perturbation is given by \theta_{v\ell}' = - \, \frac{ \tau_{rc} \Delta_F}{z_{rc}} where :math:`\Delta_F` (Kms\ :math:`^{-1}`) is the magnitude of the -cloud-top radiative divergence (see appendix :ref:`Appendix: Definitions of the +cloud-top radiative divergence (see appendix :ref:`Appendix: Definitions of the velocity scales `), :math:`\tau_{rc}` is a timescale for the exposure of boundary layer eddies to the cloud-top radiative cooling (taken to be 200s) and @@ -567,7 +567,7 @@ magnitude of the integrated buoyancy consumption of TKE within the mixed layer is less than or equal to a fraction, :math:`D_t`, of the buoyancy production, following `Turton and Nicholls (1987)`_. -Following appendix :ref:`Appendix: Derivation and definitions of the buoyancy +Following appendix :ref:`Appendix: Derivation and definitions of the buoyancy parameters ` the grid-box mean buoyancy flux can be written as: @@ -652,22 +652,22 @@ test whether :eq:`deccrit` is satisfied with both surface to the cloud-top. If recoupling is possible then the various flags identifying the DSC layer are reset (*this includes setting the cumulus diagnosis to false*), any surface-driven entrainment originally -applied at :math:`z_{\mathrm{h}}` is added to the entrainment at -:math:`z_{\mathrm{h}}^{\mathrm{Sc}}` (after rescaling for the inversion strength -at :math:`z_{\mathrm{h}}^{\mathrm{Sc}}` ) and :math:`z_{\mathrm{b}}` is set to -0.1\ :math:`z_{\mathrm{h}}` (for the reason discussed above). If decoupling -is diagnosed, :math:`z_{\mathrm{h}}^{\mathrm{Sc}}` is set to the original -:math:`z_{\mathrm{h}}` (inversion height), although the entrainment across +applied at :math:`z_{\mathrm{h}}` is added to the entrainment at +:math:`z_{\mathrm{h}}^{\mathrm{Sc}}` (after rescaling for the inversion strength +at :math:`z_{\mathrm{h}}^{\mathrm{Sc}}` ) and :math:`z_{\mathrm{b}}` is set to +0.1\ :math:`z_{\mathrm{h}}` (for the reason discussed above). If decoupling +is diagnosed, :math:`z_{\mathrm{h}}^{\mathrm{Sc}}` is set to the original +:math:`z_{\mathrm{h}}` (inversion height), although the entrainment across this inversion is not recalculated (and so keeps any surface-driven -component — the COUPLED flag is therefore set to true, see -section :ref:`Entrainment fluxes `). +component -- the COUPLED flag is therefore set to true, see +section :ref:`Entrainment fluxes `). If a decoupled layer is diagnosed, then an iteration is performed to -find the highest :math:`z_{\mathrm{h}}` (so top of the +find the highest :math:`z_{\mathrm{h}}` (so top of the :math:`K_h^{\mathrm{surf}}` profile) that still satisfies :eq:`deccrit`, but with :math:`K_h^{\mathrm{Sc}}=0` in :eq:`eq:wx_std`. The iteration -proceeds with :math:`z_{\mathrm{h}}` stepping from its lowest permissible +proceeds with :math:`z_{\mathrm{h}}` stepping from its lowest permissible height to its highest (currently 3 steps are used). If at any stage :eq:`deccrit` is violated, then the step below (therefore containing the height that would give equality in @@ -675,23 +675,23 @@ containing the height that would give equality in taken downwards. If :eq:`deccrit` is met the step above is again reduced by a factor of 4 and 3 steps taken upwards. A total of 3 sweeps are possible, each with a smaller step so that -:math:`z_{\mathrm{h}}` approaches the height that gives equality in +:math:`z_{\mathrm{h}}` approaches the height that gives equality in :eq:`deccrit`. The accuracy with which this is achieved will be the difference in the maximum and minimum permissible heights of -:math:`z_{\mathrm{h}}`  divided by :math:`2\times4\times4 = 32`, which will +:math:`z_{\mathrm{h}}` divided by :math:`2\times4\times4 = 32`, which will typically be less than 30m. The top grid-level of the SML, NTML, is defined as the highest grid-level such that :math:`K_h^{\mathrm{Sc}}` is non-zero at the half-level above. The above process is then repeated to find the appropriate -:math:`z_{\mathrm{b}}` for :math:`K_h^{\mathrm{Sc}}`, i.e., for the base of -top-driven mixing. Some constraints are placed on :math:`z_{\mathrm{b}}` , -namely that it should never go below :math:`0.1`\ :math:`z_{\mathrm{h}}` (to +:math:`z_{\mathrm{b}}` for :math:`K_h^{\mathrm{Sc}}`, i.e., for the base of +top-driven mixing. Some constraints are placed on :math:`z_{\mathrm{b}}` , +namely that it should never go below :math:`0.1`\ :math:`z_{\mathrm{h}}` (to avoid affecting the continuity of the :math:`K` profiles at the top of the surface layer, see :eq:`ws_defn`). If cumulus -convection has been diagnosed then :math:`z_{\mathrm{b}}` is not allowed to +convection has been diagnosed then :math:`z_{\mathrm{b}}` is not allowed to go below :math:`z_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}}` (unless the layer -is diagnosed to recouple completely). Finally, :math:`z_{\mathrm{b}}` must +is diagnosed to recouple completely). Finally, :math:`z_{\mathrm{b}}` must always be at or below :math:`z_{\mathrm{ \mathrm{NTDSC}}-1}`, so that mixing in decoupled layers is always resolved, and at least :math:`\Delta z_{rad}` (the cloud-top radiative cooling depth defined in @@ -748,7 +748,7 @@ be positive. The solution adopted is to integrate :math:`\overline{w'b}` analytically across the region just below the inversion, labelled -:math:`\Delta z_{rad}` in Fig, :numref:`%s `. Since +:math:`\Delta z_{rad}` in Fig, :numref:`%s `. Since :math:`\Delta_{\mathrm{ \mathrm{NTML}}} \theta_{\ell}` can also be significantly positive (when the grid-level inversion is rising or falling, for example), the base of this region is taken to be the lower @@ -838,7 +838,7 @@ Diagnosis of inversion thickness -------------------------------- Terminating the diagnostic parcel ascent at its level of neutral -buoyancy ignores any overshooting through the parcel’s own inertia as it +buoyancy ignores any overshooting through the parcel's own inertia as it enters the inversion region. This overshooting region effectively defines the depth of the inversion over which the negative entrainment heat fluxes are seen. Typically this will be small relative to the model @@ -862,11 +862,11 @@ the parcel buoyancy. Note that the constant in :math:`w_m^3` differs by a factor of 4. The buoyancy integration in :eq:`dz_param`, that is itself dependent on :math:`z_{top}`, is performed working upwards from -:math:`z_{\mathrm{par}}` assuming piece-wise linear variation of :math:`b` +:math:`z_{\mathrm{par}}` assuming piece-wise linear variation of :math:`b` between grid-levels. Note that the standard definition of the boundary layer top in the UM is the height of the first flux level below the level of neutral buoyancy, so -:math:`z_{\mathrm{par}}` :math:`=z_{\mathrm{ \mathrm{NTPAR}}+\frac{1}{2}}`. The +:math:`z_{\mathrm{par}}` :math:`=z_{\mathrm{ \mathrm{NTPAR}}+\frac{1}{2}}`. The inversion thickness is then defined as .. math:: :label: dz_definition @@ -878,11 +878,11 @@ inversion thickness is then defined as Diagnosis of the LCL transition zone thickness ---------------------------------------------- -As described in section :ref:`The diagnostic parcel ascent and cumulus +As described in section :ref:`The diagnostic parcel ascent and cumulus diagnosis `, when cumulus convection has been diagnosed surface-driven mixing was originally capped at -:math:`z_{\mathrm{lcl}}` so that mixing into the cumulus cloud layer was only -carried out by the model’s mass-flux convection scheme. This was seen to +:math:`z_{\mathrm{lcl}}` so that mixing into the cumulus cloud layer was only +carried out by the model's mass-flux convection scheme. This was seen to lead to errors in the mean profiles across the LCL, with superadiabats being the most extreme manifestation. Using the boundary layer parametrization to couple cloud and sub-cloud layers would have the @@ -892,7 +892,7 @@ indistinguishable from those in cloud-free convective boundary layers and so the non-local surface-driven mixed layer K-profiles remain accurate up to this level. To diagnose the depth to which these profiles should penetrate above the LCL, the algorithm given in -section :ref:`Diagnosis of the vertical extent of the K-profiles +section :ref:`Diagnosis of the vertical extent of the K-profiles ` to diagnose the extent of the K-profiles in decoupled boundary layers can be used (using the switch kprof_cu). This ensures that the magnitude of the integrated buoyancy consumption @@ -904,10 +904,10 @@ thermals within the grid box that may penetrate above the grid-box mean LCL (but are too dry to reach their own LCL). Thus their buoyancy flux is given by :eq:`eq:wb_cont` with :math:`C_F=0`. Restricting the negative integral of this buoyancy flux then gives a new -definition for :math:`z_{\mathrm{h}}`  that is then used in the calculation -of the surface-driven K-profiles in section :ref:`Surface-driven turbulence -` — the -larger the value of :math:`D_t`, the higher :math:`z_{\mathrm{h}}` will be. +definition for :math:`z_{\mathrm{h}}` that is then used in the calculation +of the surface-driven K-profiles in section :ref:`Surface-driven turbulence +` -- the +larger the value of :math:`D_t`, the higher :math:`z_{\mathrm{h}}` will be. Typically :math:`D_t=0.1` for decoupled stratocumulus layers while idealised clear-sky convective boundary layers (where the magnitude of the entrainment buoyancy flux is a fraction, :math:`A_1`, of the surface @@ -927,7 +927,7 @@ LCL, but ensures the iteration starts well below the LCL). The local scheme ================ -A first order ‘mixing length’ closure is used: +A first order 'mixing length' closure is used: .. math:: :label: kmlocal @@ -981,7 +981,7 @@ The asymptotic mixing lengths are given by \lambda_h =\mathrm{max}\left[\lambda_0,\, 0.15 z_{\mathrm{loc}}\right] where :math:`\lambda_0` is a minimum mixing length read in from the -namelist and :math:`z_{\mathrm{loc}}` is defined below. The orographic +namelist and :math:`z_{\mathrm{loc}}` is defined below. The orographic blending height, :math:`h_B` (only used within the boundary layer, as defined below), is given by @@ -991,9 +991,9 @@ defined below), is given by where :math:`\sigma_h` is the standard deviation of the height of the subgrid orography and :math:`(z_{0m})_{\mathrm{veg}}` is the vegetative part of the roughness length. The constants in -:eq:`asymp_ml` can be considered ‘tuned’ (see, in +:eq:`asymp_ml` can be considered 'tuned' (see, in particular, the operational modifications described in -appendix :ref:`Appendix: Operational modifications `). +appendix :ref:`Appendix: Operational modifications `). The Richardson number, :math:`Ri`, that is used as a local measure of stability is given by @@ -1011,9 +1011,9 @@ The measure of buoyancy used in :math:`Ri` is where :math:`\overline{\beta_T}` and :math:`\overline{\beta_q}` are the grid-box mean (i.e., cloud weighted) buoyancy coefficients, that can be -defined in two different ways, see appendix :ref:`Appendix: Derivation and +defined in two different ways, see appendix :ref:`Appendix: Derivation and definitions of the buoyancy parameters ` and -section :ref:`Finite difference calculations `. Note that +section :ref:`Finite difference calculations `. Note that :eq:`Bdefn` reduces to a virtual temperature approximation of buoyancy in cloud-free air and that neutral buoyancy (in cloudy as well as cloud-free air) is implied @@ -1049,19 +1049,19 @@ height-dependent factor is included, minutes, for simplicity. Initially, the lowest half-level at which :math:`Ri>Ri_{crit}` is taken -to be a measure of the boundary layer top (:math:`z_{\mathrm{loc}}` ) and the +to be a measure of the boundary layer top (:math:`z_{\mathrm{loc}}` ) and the full-level below is designated NTLOC. In general :math:`Ri_{crit}=1` but -a value of 0.25 is recommended for use with the ’SHARPEST’ stability +a value of 0.25 is recommended for use with the 'SHARPEST' stability functions, see below. If the boundary layer was diagnosed as -cumulus-capped by the non-local scheme (see section :ref:`Diagnosis of boundary +cumulus-capped by the non-local scheme (see section :ref:`Diagnosis of boundary layer depth and type `) -then :math:`z_{\mathrm{loc}}` is lowered to :math:`z_{\mathrm{lcl}}` (and +then :math:`z_{\mathrm{loc}}` is lowered to :math:`z_{\mathrm{lcl}}` (and :math:`K_h` and :math:`K_m` are set to zero from the base of grid-level NLCL upwards) so that transports into and within the cumulus cloud layer can be performed solely by the mass-flux convection scheme. Depending on the switch local_fa, above NTLOC turbulently-mixed layers (where :math:`Ri 0`), several forms for the stability -functions are available. The ‘long-tailed’ functions are +functions are available. The 'long-tailed' functions are .. math:: f_{\rm stable} = \frac{1}{1+g_0 Ri} @@ -1106,7 +1106,7 @@ stability are, from `Louis (1979)`_: .. math:: f_{\rm stable} = \frac{1}{(1+ 5 Ri)^2} -and the family of “sharp” functions can be written in terms of a +and the family of "sharp" functions can be written in terms of a transitional Richardson number, :math:`Ri_{t}`, as: .. math:: @@ -1127,11 +1127,11 @@ where B_{Ri} = (g_0/2) /\left(1 - g_0 Ri_{t}/2\right)^2 -For the ‘SHARPEST’ function of `Derbyshire (1997)`_, +For the 'SHARPEST' function of `Derbyshire (1997)`_, :math:`Ri_{t}=0.1`, while larger values give even sharper reduction of turbulence with increasing :math:`Ri`. An additional option, used operationally in some configurations (originally in the Mesoscale Model, -hence called ’MES tails’), is to blend linearly from Louis functions at +hence called 'MES tails'), is to blend linearly from Louis functions at the surface to SHARPEST by 200m. A stability dependent Prandtl number (:math:`Pr=f_m/f_h`) is generally @@ -1186,13 +1186,13 @@ are two obvious possibilities, to calculate :math:`Ri` (and thence and then interpolate either :math:`K_h` or :math:`K_m` to be able to calculate the required fluxes. To do the former requires averaging the buoyancy gradient in the numerator (and is referred to by -`Cullen and James (1994)`_ as the ‘:math:`\theta`-bar’ method), the +`Cullen and James (1994)`_ as the ':math:`\theta`-bar' method), the latter the wind shear in the denominator (referred to as the -‘:math:`\rho`-bar’ method). Single-column model and other tests -demonstrated that the ‘:math:`\rho`-bar’ method could readily generate +':math:`\rho`-bar' method). Single-column model and other tests +demonstrated that the ':math:`\rho`-bar' method could readily generate instabilities just above the top of the boundary layer because averaging the wind shear into this stable air tended to reduce :math:`Ri` and so -promote mixing. Fortunately, the ‘:math:`\theta`-bar’ method tended to +promote mixing. Fortunately, the ':math:`\theta`-bar' method tended to increase :math:`Ri` above inversions and so damp mixing. Thus, :math:`Ri` is calculated on :math:`\theta`-levels as @@ -1212,7 +1212,7 @@ i_interp_local. The long-standing method is given by where :math:`\overline{\beta_T}` and :math:`\overline{\beta_q}` are the grid-box mean (i.e., cloud-fraction weighted) buoyancy coefficients, -defined in appendix :ref:`Appendix: Derivation and definitions of the buoyancy +defined in appendix :ref:`Appendix: Derivation and definitions of the buoyancy parameters `. Note that because this is defined on :math:`\theta`-levels, no vertical interpolation of cloud variables (fractional area and water contents), to which the buoyancy @@ -1256,7 +1256,7 @@ is to calculate the buoyancy gradient directly on :math:`\rho`-levels and then interpolate this vertically to give :math:`DBDZ_k`, using :eq:`gradient_interp`. This then requires a cloud fraction on :math:`\rho`-levels. The difficulty comes where there is a -change in cloud fraction between levels. For this “edge” fraction, +change in cloud fraction between levels. For this "edge" fraction, :math:`f_{edge}` (the fraction of the grid-box that is cloudy in one level but not in the other), the change in supersaturation (:math:`s = q_t-q_{sat}`) between levels is used to estimate the @@ -1287,7 +1287,7 @@ in :eq:`kmlocal` and :eq:`khlocal`. Finally, Note that in the code the convention is for fluxes to be held on the half-level below the variable itself. Consequently, RHOKM(K), and -therefore RI(K), are held on the ‘half-level’ below :math:`\rho`-level +therefore RI(K), are held on the 'half-level' below :math:`\rho`-level K, which is :math:`\theta`-level K-1. In addition to the above, the log profile correction applied to @@ -1317,7 +1317,7 @@ The general approach is to take :math:`K_{\chi}` in (K_{\chi}^{\mathrm{surf}}+K_{\chi}^{\mathrm{Sc}}), K_{\chi}(Ri) \right] -As noted in section :ref:`Model variables and turbulence closure +As noted in section :ref:`Model variables and turbulence closure `, this implies that mixing in stable boundary layers is determined exclusively by the local scheme, :math:`K_{\chi}(Ri)`. Continuing to calculate :math:`K_{\chi}(Ri)` in @@ -1327,10 +1327,10 @@ and unstable boundary layers. At the top of unstable mixed layers, great care is taken to ensure the parametrized entrainment mixing is implemented faithfully, see -section :ref:`Entrainment fluxes `). Consequently, if a subgrid +section :ref:`Entrainment fluxes `). Consequently, if a subgrid inversion has been diagnosed capping a mixed layer (see -section :ref:`Diagnosis of a sub-grid inversion `), then +section :ref:`Diagnosis of a sub-grid inversion `), then :math:`K_{\chi}(Ri)` is set to zero at the interfaces either side of the inversion grid-level. There are also options (using the switch Keep_Ri_FA) to set @@ -1349,13 +1349,13 @@ realised that this does not cover the case of shear-driven mixing into cloud layers that have been diagnosed as cumulus-capped (which would be poorly represented by the current convection scheme). Several methods have been introduced that attempt to alleviate this problem, giving rise -to the diagnosis of a “shear-dominated boundary layer” type (type VII), -discussed in section :ref:`Diagnosis of boundary layer depth and type -`. The first (the “shear-dominated -boundary layer fix”) simply sets the CUMULUS flag to false if NTLOC +to the diagnosis of a "shear-dominated boundary layer" type (type VII), +discussed in section :ref:`Diagnosis of boundary layer depth and type +`. The first (the "shear-dominated +boundary layer fix") simply sets the CUMULUS flag to false if NTLOC :math:`>` NTPAR. This then ensures that the locally-determined :math:`K` are not set to zero above the LCL. Several more rigorous options are -available that incorporate a “dynamic criteria” in the diagnosis of +available that incorporate a "dynamic criteria" in the diagnosis of boundary layer type. The first of these prohibits the diagnosis of cumulus boundary layers when the bulk measure of stability, :math:`-z_i/L`, is small (currently less than 1.6). Here :math:`z_i` is @@ -1370,7 +1370,7 @@ the strong surface buoyancy generation of turbulence in these regimes, a calculation of :math:`Ri` is made that allows for the gradient adjustment by the non-local scheme, i.e., using :math:`\widetilde{\Delta_k \theta_{\ell}}` (see -:eq:`eq:wx_std`). The height, :math:`z_{\mathrm{loc}}` , where +:eq:`eq:wx_std`). The height, :math:`z_{\mathrm{loc}}` , where :math:`Ri>Ri_{crit}=0.25` is found. It is then hypothesised that this level of turbulent instability (that incorporates the effects of shear) only needs extend some fractional distance into the cloud layer to @@ -1380,27 +1380,27 @@ disrupt the formation of cumulus elements. Thus, if tunable parameter (:math:`0 -Ri_{crit}`, :math:`z_{\mathrm{h}}^{\mathrm{Sc}}` is the top of any stratocumulus -layer and :math:`z_{\mathrm{h}}` is the top of surface-based mixed layer, +Ri_{crit}`, :math:`z_{\mathrm{h}}^{\mathrm{Sc}}` is the top of any stratocumulus +layer and :math:`z_{\mathrm{h}}` is the top of surface-based mixed layer, found by adiabatic parcel ascent but reset to the LCL in cumulus capped -layers. Another diagnostic is available, the “boundary layer depth” +layers. Another diagnostic is available, the "boundary layer depth" (STASH 25), that is set to :math:`=\mathrm{max}[z_{\mathrm{h}}, z_{\mathrm{loc}}]` -and so represents the depth of the stable boundary layer or “surface” +and so represents the depth of the stable boundary layer or "surface" mixed layer. Also available are three diagnostics that represent the calculated value of each of the individual terms in STASH 3,304: 3,356 -is set to :math:`z_{\mathrm{h}}` ; 3,357 is -:math:`z_{\mathrm{h}}^{\mathrm{Sc}}`  and -3,358 is :math:`z_{\mathrm{loc}}` . +is set to :math:`z_{\mathrm{h}}` ; 3,357 is +:math:`z_{\mathrm{h}}^{\mathrm{Sc}}` and +3,358 is :math:`z_{\mathrm{loc}}` . .. _sec_nonlocal: @@ -1412,12 +1412,12 @@ non-local in the sense that, at a given height within the boundary layer, :math:`K` is determined not by any local properties of the mean profiles at that height but solely by the magnitude of the turbulence forcing applied to the layer (as measured by the representative velocity -scales described in appendix :ref:`Appendix: Definitions of the velocity scales +scales described in appendix :ref:`Appendix: Definitions of the velocity scales `) and the height within the layer. The non-local scheme is therefore particularly robust but care must be taken where the profiles are applied. The calculation of the vertical position and extent of the :math:`K` profiles is -described in section :ref:`Diagnosis of boundary layer depth and type +described in section :ref:`Diagnosis of boundary layer depth and type `. .. _sec_nlsurf: @@ -1428,7 +1428,7 @@ Surface-driven turbulence For turbulence sources at the surface (namely surface drag with velocity scale :math:`u_*`, and positive surface buoyancy fluxes with velocity scale :math:`w_*`) in a layer with top at -:math:`z=`\ :math:`z_{\mathrm{h}}` , base at :math:`z=0` we set +:math:`z=`\ :math:`z_{\mathrm{h}}` , base at :math:`z=0` we set .. math:: :label: kmsurf @@ -1438,16 +1438,16 @@ scale :math:`w_*`) in a layer with top at where :math:`w_m^3 = u_*^3 + w_s^3`, :math:`u_*` is the friction velocity (including the orographic roughness component) and :math:`w_s` -is defined below. For the 9C version of the scheme, :math:`z_{\mathrm{h}}` is +is defined below. For the 9C version of the scheme, :math:`z_{\mathrm{h}}` is the diagnosed subgrid inversion height (see -section :ref:`Diagnosis of a sub-grid inversion `) for both +section :ref:`Diagnosis of a sub-grid inversion `) for both :math:`K_h^{\mathrm{surf}}` and :math:`K_m^{\mathrm{surf}}`. In the 8A version, :math:`K_m^{\mathrm{surf}}` uses -:math:`z_{\mathrm{h}}` :math:`=z_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}}`. The +:math:`z_{\mathrm{h}}` :math:`=z_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}}`. The factor :math:`{\mathcal{E}}_m^{\mathrm{surf}}` is chosen so that :math:`K_m^{\mathrm{surf}}` will tend to :math:`K_m|_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}}` as :math:`z` tends to -:math:`z_{\mathrm{h}}` , where +:math:`z_{\mathrm{h}}` , where :math:`K_m|_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}}` is the entrainment eddy-diffusivity (given by :eq:`khent`, although, in order to avoid altering the shape function too much, @@ -1457,10 +1457,10 @@ A similar factor, :math:`{\cal E}_h^{\rm surf}`, is used in the :math:`K_h^{\mathrm{surf}}` profile even though the entrainment fluxes of the thermodynamic variables will usually be specified explicitly rather than through an eddy-diffusivity (see -section :ref:`Entrainment fluxes `). +section :ref:`Entrainment fluxes `). The form of :math:`w_s` differs between the surface layer -(:math:`z < 0.1`\ :math:`z_{\mathrm{h}}` ) and the rest of the mixed-layer: +(:math:`z < 0.1`\ :math:`z_{\mathrm{h}}` ) and the rest of the mixed-layer: .. math:: :label: ws_defn @@ -1471,14 +1471,14 @@ The form of :math:`w_s` differs between the surface layer \end{cases} and :math:`w_*^3=z_{\mathrm{h}}\overline{w'b}_S` using -:math:`z_{\mathrm{h}}` from +:math:`z_{\mathrm{h}}` from the current timestep (note that the use of :math:`w_*` here will be inconsistent with the use of :math:`V_{\mathrm{heat}}` in the entrainment parametrization in cloudy boundary layers). Note that :math:`w_s` is -continuous across :math:`0.1`\ :math:`z_{\mathrm{h}}` and constant with +continuous across :math:`0.1`\ :math:`z_{\mathrm{h}}` and constant with height in the mixed layer. This form for :math:`w_s` is motivated by a -desire to match the model’s surface transfer formulation within the -surface layer (as described further in section :ref:`Comparison with Holtslag +desire to match the model's surface transfer formulation within the +surface layer (as described further in section :ref:`Comparison with Holtslag and Boville (1993)_ `) and to use a cubic sum of velocity scales within the mixed layer (consistent with dimensional analysis of the TKE equation, see @@ -1510,7 +1510,7 @@ For the latter, HB93 effectively set use :math:`K_m^{\mathrm{surf}}|_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}}`, as evaluated from :eq:`kmsurf` with a subgrid calculation of -:math:`z_{\mathrm{h}}` :math:`>z_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}}`, rather +:math:`z_{\mathrm{h}}` :math:`>z_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}}`, rather than using a separate entrainment parametrization. The difference in :eq:`ws_defn` arises from the surface @@ -1552,7 +1552,7 @@ The formulation in HB93 gives :math:`Pr` varying from 1 to 0.6 (for w_*` and :math:`w_h = 1.7 w_*`. The implications of these differences from HB93 are unknown. The convective LES in `Lock and Macvean (1999)`_ suggest :math:`w_h -\approx w_*`; I don’t know where the larger proportionality constants +\approx w_*`; I don't know where the larger proportionality constants come from. Another difference between the UM and HB93 is that HB93 only apply @@ -1569,9 +1569,9 @@ Cloud-top-driven turbulence --------------------------- For cloud-top-driven turbulence over a layer of depth :math:`z_{\mathrm{ml}}` -(with top at :math:`z_{\mathrm{h}}` or :math:`z_{\mathrm{h}}^{\mathrm{Sc}}` and +(with top at :math:`z_{\mathrm{h}}` or :math:`z_{\mathrm{h}}^{\mathrm{Sc}}` and base at -:math:`z_{\mathrm{b}}` , determined as in section :ref:`Diagnosis of the +:math:`z_{\mathrm{b}}` , determined as in section :ref:`Diagnosis of the vertical extent of the K-profiles `), .. math:: :label: kmtop @@ -1582,16 +1582,16 @@ vertical extent of the K-profiles `), \frac{z'}{z_{\mathrm{ml}}} \right)^{0.8} where :math:`V_{\mathrm{Sc}}^3= V_{\mathrm{rad}}^3+V_{\mathrm{br}}^3` (see -appendix :ref:`Appendix: Definitions of the velocity scales `) and +appendix :ref:`Appendix: Definitions of the velocity scales `) and :math:`z'` is height above -:math:`z_{\mathrm{b}}` . Then :math:`K_h = K_m / \mathrm{Pr}`, where +:math:`z_{\mathrm{b}}` . Then :math:`K_h = K_m / \mathrm{Pr}`, where :math:`\mathrm{Pr}=0.75`. The resulting :math:`K_h` profile was derived against convective cloudy LES, as described in `Lock (1999)`_. The appropriate Prandtl number (and therefore :math:`K_m^{\mathrm{Sc}}`) is unknown, 0.75 being chosen simply as a number in the middle of the range usually quoted for turbulent mixing in general. As with :eq:`kmsurf`, -:math:`z_{\mathrm{h}}`  (or :math:`z_{\mathrm{h}}^{\mathrm{Sc}}` ) are given by +:math:`z_{\mathrm{h}}` (or :math:`z_{\mathrm{h}}^{\mathrm{Sc}}` ) are given by the subgrid diagnosis (see section :ref:`Diagnosis of a sub-grid inversion `) except for @@ -1604,7 +1604,7 @@ that :math:`K_m^{\mathrm{Sc}}` will tend to :math:`K_m|_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}}` (and :math:`K_h^{\mathrm{Sc}}` to :math:`K_h|_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}}`), given by -:eq:`khent`, as :math:`z` tends to :math:`z_{\mathrm{h}}` (and +:eq:`khent`, as :math:`z` tends to :math:`z_{\mathrm{h}}` (and here no restriction is made on the magnitude of either :math:`{\cal E}_m^{\rm Sc}` or :math:`{\mathcal{E}}_h^{\mathrm{Sc}}`). @@ -1633,10 +1633,10 @@ where \overline{w'\theta_{\ell}'}_S/w_m`, where for this calculation of :math:`w_m` (given by :math:`w_m^3=u_*^3+0.25\,z_{\mathrm{h}}\overline{w'b}_S`) -:math:`z_{\mathrm{h}}` is taken from the previous timestep. The form of +:math:`z_{\mathrm{h}}` is taken from the previous timestep. The form of :eq:`gradadj` is similar to that used in HB93 and the magnitude of :math:`\gamma_{\theta_{\ell}}` is the same as in HB93 in -the convective limit — the difference in :math:`A_{ga}` exactly allows +the convective limit -- the difference in :math:`A_{ga}` exactly allows for the different constants in :eq:`ws_defn`. Consistent with the mixed layer assumptions underlying the non-local @@ -1648,7 +1648,7 @@ well-mixed :math:`\theta_{\ell}` profiles (i.e., with :math:`\partial \theta_{\ell}/ \partial z` less negative or even positive in a cloud-free surface-heated boundary layer, for example), subject to an arbitrary upper limit included for numerical safety. Hence -the term ‘gradient adjustment’ rather than non-local flux. When +the term 'gradient adjustment' rather than non-local flux. When estimating the buoyancy flux, then (as in :eq:`eq:wx_std`), it is simplest to allow for the non-local term by adjusting the :math:`\theta_{\ell}` gradient. @@ -1660,7 +1660,7 @@ tend to make :math:`q_t` profiles less well mixed than those of :math:`\theta_{\ell}` :raw-latex:`\cite[]{mahrt1976}`. From UM version 5.5, there is the option to implement the non-gradient stress parametrization of `Brown and Grant (1997)`_, as described in -section :ref:`Non-gradient stress parametrization `. +section :ref:`Non-gradient stress parametrization `. .. _sec_ngstress: @@ -1675,7 +1675,7 @@ down-gradient stress parametrization, a one-dimensional model produced wind profiles in the convective boundary layer that were less well-mixed than predicted by LES, and underestimated the near surface wind. Furthermore, `Brown et al. (2006)`_ showed that the -operational verification statistics indicate a slow bias in the 10 m +operational verification statistics indicate a slow bias in the 10 m wind over land by day, especially in spring and summer. The non-gradient stress parametrization in the UM is very similar to @@ -1784,7 +1784,7 @@ The components of :eq:`fg_new` are: - :math:`f_2 = 0.5 \, \frac{z}{z_h}\, 2^{(z/z_h)^4}` -In the above equations :math:`k` is von Karman’s constant, :math:`z'` +In the above equations :math:`k` is von Karman's constant, :math:`z'` (:math:`=z-z_{\mathrm{b}}`) is height above the mixed layer base, :math:`z_{ml}` (:math:`=z_h-z_{\mathrm{b}}`) is the mixed layer depth, :math:`u_*` is the friction velocity, and :math:`w_*` and @@ -1897,7 +1897,7 @@ gradient adjustment parameter: The inclusion of an extra :math:`w_*/w_m` factor in :math:`\gamma_{\chi}` was a deliberate change by HB from the original `Troen and Mahrt (1986)`_ formulation -on which the UM was based. This seems an appealing feature (HB’s +on which the UM was based. This seems an appealing feature (HB's :math:`\gamma_{\chi}` will tend to zero as :math:`w_* \rightarrow 0`) and probably should be considered for the revised scheme (the dash-dotted line in :numref:`Fig. %s ` sets @@ -1942,7 +1942,7 @@ The blended scheme For high resolution simulations, the UM has a Smagorinsky-type subgrid turbulence scheme, described in . However, this scheme is only truly -applicable for horizontal grid-lengths of order :math:`10` m, and any +applicable for horizontal grid-lengths of order :math:`10` m, and any real-world simulation run at lower resolution than this will inevitably have unresolved scales somewhere in the domain. Rather than force the user to make an ad-hoc decision about the scales they are interested in, @@ -1998,10 +1998,10 @@ the von Karman constant and :math:`c_s` is the Smagorinsky constant. Near the surface :math:`l_{\mathrm{bl}}` and :math:`l_{\mathrm{smag}}` are identical, but the asymptotic values are different and this method weights the asymptotic value according to the weighting of the two -schemes. For example, at :math:`\Delta x=1` km, -:math:`c_s\Delta x=200` m (for :math:`c_s=0.2`), whereas +schemes. For example, at :math:`\Delta x=1` km, +:math:`c_s\Delta x=200` m (for :math:`c_s=0.2`), whereas :math:`\lambda_0=\max(40\ {\mathrm{m}}, 0.15z_h)`, which allows for a small -mixing length in shallow unresolved boundary layers (e.g. stable ones). +mixing length in shallow unresolved boundary layers (e.g. stable ones). The `Lock et al. (2000)`_ scheme also contains a non-local component to the turbulent flux, and this is simply down-weighted by @@ -2012,8 +2012,8 @@ is given by .. math:: K_\chi = \max\left[W_{1D}K_\chi^{\rm NL}, K_\chi(Ri)\right], where :math:`K_\chi^{\mathrm{NL}}` is the non-local diffusivity and :math:`l` -in Eq. :eq:`eq-kri` is given by :math:`l_{\mathrm{blend}}` in -Eq. :eq:`eq-lblend`. The turbulent flux is then calculated +in Eq. :eq:`eq-kri` is given by :math:`l_{\mathrm{blend}}` in +Eq. :eq:`eq-lblend`. The turbulent flux is then calculated as .. math:: F_\chi=-K_\chi\frac{\partial \chi}{\partial z} + W_{1D}F_\chi^{\rm @@ -2041,7 +2041,7 @@ where :math:`z_{\mathrm{turb}}` is the appropriate lengthscale of the turbulence, :math:`\beta` is a scaling parameter which controls the speed of the transition from unresolved to resolved turbulence, :math:`r_f=\frac{1}{l_0-l_1}`, :math:`l_0=4` and :math:`l_1=0.25` -(N. B. this formula is slightly modified from that given in +(N. B. this formula is slightly modified from that given in :raw-latex:`\cite[]{Boutleetal2014}`). `Malavelle et al. (2014)`_ demonstrated that this scaling method was applicable to any type of unstable boundary layer given an @@ -2062,7 +2062,7 @@ that of `Honnert et al. (2011)`_. These functions are shown in :numref:`Figure %s `\ (a) and are only dissimilar for small :math:`\Delta -x`, where Eq. :eq:`eq-tanh` tends to zero faster. This is by +x`, where Eq. :eq:`eq-tanh` tends to zero faster. This is by choice, to force the highest resolution simulations to use the 3D turbulence scheme. @@ -2113,7 +2113,7 @@ analysis of decoupled stratocumulus LES presented by `Honnert et al. (2011)`_ also included shallow cumulus simulations and showed that the relevent length scale there was the cloud top height. Most of the ``blending_option`` choices apply this to -all regimes diagnosed as cumulus-capped (see section :ref:`Diagnosis of +all regimes diagnosed as cumulus-capped (see section :ref:`Diagnosis of boundary layer depth and type `) but alternatively (``blending_option``\ :math:`=`\ 4) this can be restricted to strictly shallow cumulus clouds, defined as contiguously @@ -2135,12 +2135,12 @@ Above the boundary layer top, `Boutle et al. (2014)`_ aimed for any free atmospheric mixing to be done by the 3D Smagorinsky scheme. Therefore, above the boundary layer top they use :math:`z` as the appropriate length scale, and in general take :math:`z_{\mathrm{turb}}` in -Eq. :eq:`eq-tanh` as the greater of that defined by +Eq. :eq:`eq-tanh` as the greater of that defined by :eq:`zturb_dsc` and :math:`z`. However, this did not give a particularly fast transition using the value of :math:`\beta_{\mathrm{bl}}`, therefore they used :math:`\beta_{\mathrm{fa}}=1` at -a height well above the boundary layer (:math:`z_{\mathrm{fa}}=z_h+1` km), +a height well above the boundary layer (:math:`z_{\mathrm{fa}}=z_h+1` km), and transitioned between these regimes linearly using .. math:: @@ -2198,33 +2198,33 @@ Entrainment fluxes **Summary**: parametrized entrainment fluxes (at the top of mixed layers) are specified for momentum through an eddy-diffusivity, as -described in section :ref:`For momentum (and scalars if no subgrid inversion) +described in section :ref:`For momentum (and scalars if no subgrid inversion) `. For scalar variables, if the inversion is sufficiently sharp so as to be unresolved, the ideal is to specify the entrainment fluxes explicitly, as described in -section :ref:`Specification of entrainment fluxes in the 9B scheme +section :ref:`Specification of entrainment fluxes in the 9B scheme `, based on the subgrid inversion -diagnosis described in section :ref:`Diagnosis of a sub-grid inversion +diagnosis described in section :ref:`Diagnosis of a sub-grid inversion `. Further details can be found in `Lock (2001)`_. If the profiles are such that the inversion is sharp but a subgrid inversion cannot be diagnosed, an eddy-diffusivity similar to that for momentum is used (see -section :ref:`For momentum (and scalars if no subgrid inversion) `). +section :ref:`For momentum (and scalars if no subgrid inversion) `). If the inversion is thick enough to be resolved then an eddy diffusivity profile is constructed across the -inversion (see section :ref:`Resolved inversions `) for both +inversion (see section :ref:`Resolved inversions `) for both scalars and momentum fields. For tracer variables (scalars other than :math:`\theta_{\ell}` and :math:`q_t`), the entrainment fluxes are specified using an equivalent eddy-diffusivity, as described in -section :ref:`For tracers, when there is a subgrid inversion `. +section :ref:`For tracers, when there is a subgrid inversion `. Note that, as indicated below, several aspects of the implementation of entrainment fluxes were revised at the 9C scheme and these are documented separately. The parametrization of the entrainment rate, :math:`w_e` (given, in the absence of subsidence, by the rate of rise of the inversion), can be -written (using the notation given in appendix :ref:`Appendix: Definitions of +written (using the notation given in appendix :ref:`Appendix: Definitions of the velocity scales `) .. math:: :label: we_parm @@ -2256,7 +2256,7 @@ depth-scale for the radiatively-cooled layer (taken to be 15 :math:`\times \,\mbox{max}[200/z_c, 1]`, where :math:`z_c` is the cloud depth). To allow for a feedback with forcing of entrainment by buoyancy reversal -(see appendix :ref:`Appendix: Definitions of the velocity scales +(see appendix :ref:`Appendix: Definitions of the velocity scales `), :math:`\tilde{\alpha_t} = \alpha_t+ Br (1-\alpha_t)`. following `Lock (1998)`_ and @@ -2269,10 +2269,10 @@ in :eq:`we_parm` proportional to :math:`V_{\mathrm{heat}}^3` and :math:`u_*`) will no longer contribute to entrainment at cloud top, because the two layers will have become entirely decoupled. If the ``entr_smooth_dec`` switch is on then the surface contribution to the -parametrized entrainment at :math:`z_{\mathrm{h}}^{\mathrm{Sc}}` is decreased +parametrized entrainment at :math:`z_{\mathrm{h}}^{\mathrm{Sc}}` is decreased linearly as the :math:`\theta_{v\ell}` difference between NTDSC and NTML increases from 0.5 to 1K. The flag, COUPLED, is set to true and -:math:`z_{\mathrm{h}}^{\mathrm{Sc}}` is used as the mixed-layer depth in +:math:`z_{\mathrm{h}}^{\mathrm{Sc}}` is used as the mixed-layer depth in :eq:`we_parm` as long as any surface-driven entrainment remains. If the ``entr_smooth_dec`` switch is off then this transition is discontinuous at a :math:`\theta_{v\ell}` difference of 0.5K. @@ -2281,7 +2281,7 @@ It should be noted that :eq:`we_parm` takes no account of wind shear anywhere other than at the surface. How to quantify the shear generation of turbulence in DSC layers is not known. The direct impact of shear across the inversion is thought to be simply to diffuse the -inversion in the vertical — this wind shear will contribute little to +inversion in the vertical -- this wind shear will contribute little to the mixed layer TKE and so can not contribute to the full process of mixing across the inversion and down into the mixed layer that is entrainment. However, important interactions between wind shear across @@ -2289,7 +2289,7 @@ inversions and cloud-top radiative cooling have been observed that are not yet accounted for in the UM. The least well-determined part of :eq:`we_parm` is the -constant :math:`A_2` — the constant in the Zilitinkevich correction, +constant :math:`A_2` -- the constant in the Zilitinkevich correction, :math:`c_T`, is also approximate but is included to limit the growth of layers capped by weak inversions and for numerical safety. A further limit is applied to the value of :math:`w_e` determined by @@ -2298,7 +2298,7 @@ than one grid-level in a timestep. With current vertical resolutions and timesteps this is not a serious restriction. The constants :math:`A_1` and :math:`A_{\mathrm{br}}` appeared to be determined within 10-20 % in `Lock (1998)`_, although only solid cloud sheets were -simulated (as discussed further in appendix :ref:`Appendix: Definitions of the +simulated (as discussed further in appendix :ref:`Appendix: Definitions of the velocity scales `). Similarly the parametrizations of :math:`\alpha_t` and :math:`\Delta z_i` were found to be accurate but the parameter @@ -2314,7 +2314,7 @@ approach to include all processes operating in the inversion grid-level, rather than just radiation. If it is assumed that the turbulent fluxes reduce from their extremum at -:math:`z=z_i` (the ‘entrainment’ fluxes) to zero at :math:`z=h` a small +:math:`z=z_i` (the 'entrainment' fluxes) to zero at :math:`z=h` a small distance above, then :math:`\overline{w'\theta_{\ell}'}_{z_i}= - w_e \Delta \theta_{\ell}+ F|_h - F|_{z_i}`, so that @@ -2346,15 +2346,15 @@ increments (in Ks\ :math:`^{-1}`) from the radiation scheme and }` using the flux-divergence in grid-level NTML\ :math:`+2` (and similarly for DSC layers). -The thermodynamic variables’ entrainment fluxes, then, are imposed +The thermodynamic variables' entrainment fluxes, then, are imposed nominally at the subgrid inversion height (:math:`z_i=` -:math:`z_{\mathrm{h}}` and/or :math:`z_{\mathrm{h}}^{\mathrm{Sc}}` ), diagnosed +:math:`z_{\mathrm{h}}` and/or :math:`z_{\mathrm{h}}^{\mathrm{Sc}}` ), diagnosed as -described in section :ref:`Diagnosis of a sub-grid inversion `. The +described in section :ref:`Diagnosis of a sub-grid inversion `. The required grid-level fluxes (at :math:`z_{\mathrm{ \mathrm{NTDSC}}+\frac{1}{2}}`, for example) are then estimated using linear interpolation of :math:`{\mathcal{H}}` and -:math:`\overline{w'q_t'}` between :math:`z_{\mathrm{h}}^{\mathrm{Sc}}` and the +:math:`\overline{w'q_t'}` between :math:`z_{\mathrm{h}}^{\mathrm{Sc}}` and the base of the mixed layer: .. math:: @@ -2403,12 +2403,12 @@ fixed through the timestep and so it is consistent to assume the entrainment fluxes (at :math:`z_i`) are also fixed. Hence :eq:`fluxinterp` are implemented explicitly, rather than via an eddy-diffusivity. This is discussed further, with reference to -tracer fluxes, in section :ref:`For tracers, when there is a subgrid inversion +tracer fluxes, in section :ref:`For tracers, when there is a subgrid inversion `. In order to allow for the long timesteps used in NWP and to facilitate movement of the subgrid inversion across grid-levels within a timestep, -the parametrization of :math:`w_e` and the model’s subsidence velocity, +the parametrization of :math:`w_e` and the model's subsidence velocity, :math:`w_S|_{z_i}`, are used to calculate :math:`z_i` at the next time-level (:math:`z_i^{n+1}`). Currently, the latter is found by linear interpolation to :math:`z_i` and both are assumed constant in time. If @@ -2434,7 +2434,7 @@ entrainment fluxes are specified explicitly, the eddy-diffusivities flux gradient across the mixed layer. Finally, the entrainment flux is adjusted to allow for numerical -entrainment arising from the model’s resolved vertical advection (as +entrainment arising from the model's resolved vertical advection (as discussed in `Lock (2001)`_). This is performed at whichever grid-level the entrainment fluxes are specified, to allow for any entrainment implied by a :math:`\theta_{\ell}` subsidence increment, @@ -2444,7 +2444,7 @@ increments could be obtained directly in the SCM but in the full 3D UM advection increments are dominated by the horizontal component. The subsidence increments are calculated, therefore, from the vertical velocity field using first order upwind advection (it would clearly be -preferable to use the model’s actual vertical advection algorithm in the +preferable to use the model's actual vertical advection algorithm in the GCM although the errors incurred in this diagnostic calculation should not be very significant). The interpolated entrainment fluxes given by :eq:`fluxinterp` are therefore calculated not using @@ -2498,9 +2498,9 @@ it to diffuse out this static instability). Having identified the model grid-level at the top of the well-mixed layer (either level NTML from the parcel ascent, as described in -section :ref:`The diagnostic parcel ascent and cumulus diagnosis +section :ref:`The diagnostic parcel ascent and cumulus diagnosis `, or NTDSC for DSC layers, see section -:ref:`Diagnosis of the vertical extent of the K-profiles `— the +:ref:`Diagnosis of the vertical extent of the K-profiles `-- the analysis is the same for both), the grid-level above is designated the inversion level within which the diagnosis of a subgrid :math:`z_i` will be made. It is assumed that @@ -2581,7 +2581,7 @@ that, were a small rate of rise of :math:`z_i` (of :math:`10^{-4}` ms\ :math:`^{-1}`, say) to be diagnosed, then :math:`z_i` would spend at least half the timestep (of length :math:`\Delta t`) in the next grid-level up. The specified fluxes would then contribute significantly -to that grid-level’s evolution. Conversely, if :math:`z_i` is subsiding, +to that grid-level's evolution. Conversely, if :math:`z_i` is subsiding, this technique allows the inversion to drop down a grid-level without requiring this to be detected by the initial parcel ascent. @@ -2628,7 +2628,7 @@ tropospheric air). Specification of entrainment fluxes across sharp inversions in the 9C scheme ---------------------------------------------------------------------------- -As described in section :ref:`Specification of entrainment fluxes in the 9B +As described in section :ref:`Specification of entrainment fluxes in the 9B scheme `, when the capping inversion is thinner than the model vertical grid it is important for the entrainment flux implementation that the subsidence increments are @@ -2675,8 +2675,8 @@ mixed layer), consistent with the rising tendency of the inversion. For the model, the subsidence flux-divergence associated with the inversion is split across levels :math:`\mathrm{ \mathrm{NTML}}` and :math:`\mathrm{ \mathrm{NTML}}+1`. To keep the *net* moistening of the -model’s boundary layer and inversion consistent with the total subgrid -flux profile, the model’s entrainment flux at +model's boundary layer and inversion consistent with the total subgrid +flux profile, the model's entrainment flux at :math:`\mathrm{ \mathrm{NTML}}+1/2` (shown by the cross in :numref:`Fig. %s `) must be found by subtracting the subsidence flux at :math:`\mathrm{ \mathrm{NTML}}+1/2` (diamond) from the @@ -2700,7 +2700,7 @@ given by: F_{\chi}^{Tot}|_{z_h} = - w_e \Delta \chi + F_{\chi}^{NTP}|_{z_t} + F_{\chi}^{subs}|_{z_h} -As in section :ref:`Specification of entrainment fluxes in the 9B scheme +As in section :ref:`Specification of entrainment fluxes in the 9B scheme `, :eq:`fxtot_zi` is derived by integrating the conservation equation for :math:`\chi` over an inversion in which jumps occur over a thin layer with base at a @@ -2774,7 +2774,7 @@ to use :math:`\theta_{\ell}` as an example, if the inversion is rising (falling) then :math:`F_{\chi}^{Tot}|_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}}` is limited to ensure that the inversion grid-level will cool (warm). Finally, if -the inversion is rising we don’t want the inversion grid-level +the inversion is rising we don't want the inversion grid-level :math:`\theta_{\ell}` to cool to less than :math:`\theta_{\ell}` of the mixed layer by the end of the timestep. In other words, for :math:`\chi=\theta_{\ell}`, given @@ -2877,13 +2877,13 @@ Calculation of the subsidence flux The vertical advection or subsidence flux, :math:`{F_{\chi}}^{subs}`, is calculated by integrating estimates of the vertical advection -increments. These estimates are made at 9B from the model’s vertical +increments. These estimates are made at 9B from the model's vertical velocity field, :math:`w`, using first order upwind advection. As described above, however, the coupling between different flux profiles is performed on the model grid and, over land, these coordinate surfaces follow the underlying terrain. To correct this, the 9C scheme calculates the subsidence flux in grid-point, rather than physical space, by using -:math:`\dot{\eta}` (where :math:`\eta` is the model’s vertical +:math:`\dot{\eta}` (where :math:`\eta` is the model's vertical coordinate) rather than :math:`w`. The following two examples illustrate why this represents an @@ -2894,7 +2894,7 @@ advection flux across the inversion grid-levels, because :math:`\dot{\eta}` is negative. Conversely, consider the same boundary layer but in a flow that follows the coordinate surfaces, going up and over a hill. Now there will be no vertical advection flux across the -model’s inversion grid-level because :math:`\dot{\eta}` is zero and yet +model's inversion grid-level because :math:`\dot{\eta}` is zero and yet :math:`w` will be negative on the down-slope thus giving a spurious subsidence source to the 9B scheme. @@ -2917,7 +2917,7 @@ For momentum (and scalars if no subgrid inversion) ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ For momentum, and scalars if a subgrid inversion cannot be diagnosed, -see section :ref:`Diagnosis of a sub-grid inversion `, fluxes at the +see section :ref:`Diagnosis of a sub-grid inversion `, fluxes at the mixed layer top are specified through an eddy diffusivity which is given by @@ -2933,7 +2933,7 @@ specified through an eddy diffusivity which is given by noting the Charney-Philips grid implying stresses are staggered from scalar fluxes. The Prandtl number, :math:`Pr`, takes the same form as -for the non-local :math:`K` profiles, see section :ref:`The non-local scheme +for the non-local :math:`K` profiles, see section :ref:`The non-local scheme `. Substituting :eq:`khent` in @@ -2960,7 +2960,7 @@ unwise numerically to attempt to specify the inversion stresses explicitly and so :eq:`khent` is always used. For the 9C version, the entrainment :math:`K_m` given by :eq:`khent` is imposed at the height of the temperature inversion -:math:`z_{\mathrm{h}}` (either subgrid or at +:math:`z_{\mathrm{h}}` (either subgrid or at :math:`z_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}}`) and :math:`K_m|_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}}` is calculated from :eq:`kmsurf` and :eq:`kmtop`, noting the use of @@ -2973,14 +2973,14 @@ Resolved inversions An inversion is defined as being resolved when it extends above the flux-level above the usual entrainment interface level (see -section :ref:`Diagnosis of inversion thickness `), i.e. when +section :ref:`Diagnosis of inversion thickness `), i.e. when .. math:: z_{\mbox{\tiny \rm NTML}+\frac{1}{2}} + \Delta z_i > z_{\mbox{\tiny \rm NTML}+\frac{3}{2}} When this happens, there is no subgrid inversion diagnosis and the entrainment parametrization follows the methodology given in -section :ref:`For momentum (and scalars if no subgrid inversion) ` +section :ref:`For momentum (and scalars if no subgrid inversion) ` to give :math:`K_h|_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}}`. The diffusion coefficient profile within the inversion is then calculated assuming the @@ -3013,7 +3013,7 @@ in :eq:`khent`. For tracers, when there is a subgrid inversion ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ -Here ‘tracers’ refers to scalar variables other than +Here 'tracers' refers to scalar variables other than :math:`\theta_{\ell}` and :math:`q_t`: aerosols, :math:`q_f`, etc. Ideally, tracer entrainment fluxes would be specified explicitly in the same way as for :math:`\theta_{\ell}` and :math:`q_t`. However, @@ -3042,7 +3042,7 @@ equivalent entrainment eddy-diffusivity given by: Note from :eq:`scal_closure` that :eq:`K_ent_tracer` gives the parametrized flux if :math:`\Delta_{\mathrm{ \mathrm{NTML}}+1} \chi` does not change across the -timestep (see section :ref:`Implicit solution of the diffusion equation +timestep (see section :ref:`Implicit solution of the diffusion equation ` for a description of the implicit numerical solution of :eq:`cons_eqn_scal`). As :eq:`K_ent_tracer` involves the potentially @@ -3091,7 +3091,7 @@ layer are related to the surface fluxes by: where subscript 0 represents a surface value and subscript \* represents a surface layer scaling quantity. :math:`\phi _{m}` and :math:`\phi _{h}` are the Monin-Obukhov stability functions (for the form of these -see section `8.3 <#section_1.3>`__ below). :math:`L` is the +see section `8.3 <#section_1.3>`__ below). :math:`L` is the Monin-Obukhov length scale defined by .. math:: :label: 1.1.4 @@ -3104,13 +3104,13 @@ where F\ :math:`_{B0}` is the surface buoyancy flux defined by F_{B0} = \frac{ g }{ c_P } \beta _{T1} H_0 + g \beta _{q1} E_0. -The buoyancy coefficients in equation :eq:`1.1.5` are given -in appendix :ref:`Appendix: Derivation and definitions of the buoyancy +The buoyancy coefficients in equation :eq:`1.1.5` are given +in appendix :ref:`Appendix: Derivation and definitions of the buoyancy parameters ` with the subscript 1 denoting a value at the lowest level in the atmosphere model. -Equations :eq:`1.1.1`–:eq:`1.1.3` can be -integrated from the “surface”, i.e. the roughness height where the +Equations :eq:`1.1.1`-:eq:`1.1.3` can be +integrated from the "surface", i.e. the roughness height where the surface variables are defined, to a reference height in the surface layer, for modelling applications, the height, z\ :math:`_{1}`, of the bottom model layer above the surface. The resulting expressions for the @@ -3131,8 +3131,8 @@ surface turbulent fluxes are: v}}, where :math:`\Delta`\ X=X\ :math:`_{1}`-X\ :math:`_{0}`. -From :eq:`1.1.7` and :eq:`1.1.8` the surface -buoyancy flux in definition :eq:`1.1.4` is +From :eq:`1.1.7` and :eq:`1.1.8` the surface +buoyancy flux in definition :eq:`1.1.4` is .. math:: :label: 1.1.10 @@ -3145,7 +3145,7 @@ buoyancy flux in definition :eq:`1.1.4` is + g \beta _{q1} \Delta q The **surface exchange coefficients** in -equations :eq:`1.1.7`–:eq:`1.1.9`, c\ :math:`_{D}` +equations :eq:`1.1.7`-:eq:`1.1.9`, c\ :math:`_{D}` and c\ :math:`_{H}`, are given by .. math:: :label: 1.1.12 @@ -3172,7 +3172,7 @@ z\ :math:`_{0m}` and z\ :math:`_{0h}` are the **surface roughness lengths** for momentum and scalars respectively. The equations for the **surface turbulent fluxes**, -:eq:`1.1.7`–:eq:`1.1.9`, can be written in the +:eq:`1.1.7`-:eq:`1.1.9`, can be written in the forms .. math:: :label: 1.1.16 @@ -3282,7 +3282,7 @@ which implies that L \sim -( \gamma _t^3 /k) z_i Thus the low wind speed limits for the sensible and latent heat fluxes -are obtained by substituting :eq:`1.1.27` into +are obtained by substituting :eq:`1.1.27` into :eq:`1.1.7` and :eq:`1.1.8` with the surface transfer coefficients evaluated with L given by :eq:`1.1.28`. The finite limit for L implies that the form @@ -3325,13 +3325,13 @@ where Thus in this formulation the mean gust speed is a function of height above the surface through the same factor, :math:`\Phi _{m}`\ (z), which determines the profile of the mean wind **v** in the surface layer (see -Eq. :eq:`1.1.9`). The values of :math:`\Delta`\ **v**, +Eq. :eq:`1.1.9`). The values of :math:`\Delta`\ **v**, v\ :math:`_{g}` and :math:`V` thus tend to zero as z :math:`\to` 0. Note that v\ :math:`_{g} \to` W\ :math:`_{g}` as :math:`\Delta`\ **v** :math:`\to` 0 and that v\ :math:`_{g} \to` 0 as the convective gustiness scaling velocities tend to zero. -Equation :eq:`1.2.1` can be rewritten as +Equation :eq:`1.2.1` can be rewritten as .. math:: :label: 1.2.4 @@ -3389,7 +3389,7 @@ and temperature in the bottom layer as point values in the calculation of surface fluxes and is therefore not absolutely consistent with flux differencing. Whilst the effect of this difference is not large, it is desirable to have the option of correcting it, which is done by enabling -the option to “make surface exchange consistent with flux differencing.” +the option to "make surface exchange consistent with flux differencing." The following discussion explains how this is done. In effect, the UM takes the displacement height for momentum as @@ -3661,7 +3661,7 @@ Then calculate Having set up initial values the iteration loop can be entered (this is the original method used but contains an inconsistency in the treatment of boundary-layer convective gustiness, as described in -section `8.4.1 <#mo_iter_corrn>`__): +section `8.4.1 <#mo_iter_corrn>`__): DO n = 1 to N @@ -3731,7 +3731,7 @@ N is the last iteration value. N = 5 is currently used. For sea points the momentum roughness length and the wind mixing energy flux are calculated from v\ :math:`_{\ast }^{(N)}` using the formulae in -subsection `8.6 <#section_1.6>`__ below. +subsection `8.6 <#section_1.6>`__ below. .. _mo_iter_corrn: @@ -3754,7 +3754,7 @@ contributions to the velocity. Locally, Monin-Obukhov theory then gives We ignore the spatial variation of :math:`\Phi_m`, expecting that the principal effect of locally stronger winds is to increase the local -stress – this is exactly true in nearly neutral flow. The local stress +stress - this is exactly true in nearly neutral flow. The local stress is aligned with the wind so .. math:: @@ -3834,7 +3834,7 @@ namely :math:`\hat u_*` within the iteration. The interpolation of surface layer variables to standard observation heights ---------------------------------------------------------------------------- -Integrating :eq:`1.1.3` between the roughness height, +Integrating :eq:`1.1.3` between the roughness height, z\ :math:`_{0m}`, and the observation height z\ :math:`_{ob}` we obtain .. math:: :label: 1.5.1 @@ -3854,7 +3854,7 @@ interpolation formula For wind z\ :math:`_{ob}` is set to 10m and the last iteration (N) values of C\ :math:`_{D}`, L and :math:`v_{\ast }` are used. -Integrating :eq:`1.1.1` and :eq:`1.1.2` between +Integrating :eq:`1.1.1` and :eq:`1.1.2` between the roughness height, z\ :math:`_{0h}`, and the observation height z\ :math:`_{ob, }` we obtain for the scalar :math:`X` (:math:`=T+(g/c_{P})z` , :math:`q` or tracer amount) @@ -3880,10 +3880,10 @@ The parametrization of decoupling ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ In the foregoing analysis it is tacitly assumed that the surface layer, -up to the model’s lowest grid level, is in equilibrium with the surface +up to the model's lowest grid level, is in equilibrium with the surface and lies within the constant flux layer. In light winds, and when the surface temperature falls quickly, these assumptions are invalid; -equation :eq:`1.5.4` then yields temperatures at the height of +equation :eq:`1.5.4` then yields temperatures at the height of observation that are too closely tied to the surface temperature. Observed temperatures may be significantly warmer: this may be termed decoupling. Two parametrizations of this effect are available. Both @@ -3891,7 +3891,7 @@ involve the idea that as the wind becomes very light radiative cooling comes to determine the temperature profile. The first parametrization simply sets the interpolation coefficient -between the surface temperature and that on the model’s lowest level +between the surface temperature and that on the model's lowest level according to the radiative equilibrium profile when the Richardson number exceeds 0.25 (a typical criterion for high stability). @@ -4066,13 +4066,13 @@ In all schemes available here the momentum roughness length is given by z_{0m(sea)} = \frac{1.54\times {10}^{-6} }{ v_\ast } + \frac{\alpha}{g} v_\ast ^2 -which is a generalisation of Charnock’s formula to include low-wind -conditions :raw-latex:`\cite[]{Smith88}`. :math:`\alpha` is Charnock’s +which is a generalisation of Charnock's formula to include low-wind +conditions :raw-latex:`\cite[]{Smith88}`. :math:`\alpha` is Charnock's coefficient, which is determined from field measurements. It is often taken as a constant, but more elaborate schemes include a dependence on wind speed. In practice the difference between different parametrizations of the momentum roughness length therefore comes down -to the specification of Charnock’s coefficient. +to the specification of Charnock's coefficient. There is greater uncertainty in the roughness lengths for scalars and the dependencies are described separately for each scheme. Note that @@ -4084,9 +4084,9 @@ Schemes are selected by setting the variable *iseasurfalg*, as now described. #. Option *iseasurfalg=0*. The original and most basic scheme comprises - a fixed value of Charnock’s coefficient and a fixed scalar roughness - length. Typical values of Charnock’s coefficient lie in the range - 0.011–0.018 and a typical value of the thermal roughness length is + a fixed value of Charnock's coefficient and a fixed scalar roughness + length. Typical values of Charnock's coefficient lie in the range + 0.011-0.018 and a typical value of the thermal roughness length is :math:`z_{0h(sea)}` = 4x10\ :math:`^{-5}` m. #. Option *iseasurfalg=1*. The use of a fixed thermal roughness length, @@ -4116,7 +4116,7 @@ described. ambiguity is in practice removed by the consideration that the inversion is only of relevance in conditions of light winds. - With this scheme a fixed value of Charnock’s coefficient must be + With this scheme a fixed value of Charnock's coefficient must be specified as above. #. Option *iseasurfalg=2*. An alternative version of the foregoing @@ -4132,7 +4132,7 @@ described. description of surface transfer at the sea surface, here we use only the expressions for the roughness lengths. - In current versions of the scheme Charnock’s coefficient is specified + In current versions of the scheme Charnock's coefficient is specified using a linear relationship between the 10-m wind speed, valid over a certain range of wind speeds, with fixed values outside the range: @@ -4167,24 +4167,24 @@ described. lengths to evolve during iteration to obtain the Obukhov length. #. Option *iseasurfalg=4*. Equivalent to option *iseasurfalg=1* for a - variable Charnock parameter. A fixed value of Charnock’s coefficient + variable Charnock parameter. A fixed value of Charnock's coefficient does not need to be provided. On the other hand, a Charnock field needs to be provided via wave coupling or initialization. #. Option *iseasurfalg=5*. Equivalent to option *iseasurfalg=2* for a - variable Charnock parameter. A fixed value of Charnock’s coefficient + variable Charnock parameter. A fixed value of Charnock's coefficient does not need to be provided. On the other hand, a Charnock field needs to be provided via wave coupling or initialization. The observations upon which these schemes are based do not extend to -10-m (neutral) wind speeds much above 20 ms\ :math:`{}^{-1}` and there +10-m (neutral) wind speeds much above 20 ms\ :math:`{}^{-1}` and there is some uncertainty over the behaviour of the drag at the wind speeds encountered in tropical cyclones: indeed, there is considerable evidence that it does not continue to increase in the manner predicted by schemes like those described above and may even decrease. `Donelan et al. (2004)`_ presents some measurements suggesting that the drag coefficient should not be permitted to increase for 10-m -neutral winds above about 33 ms\ :math:`{}^{-1}`, when the drag +neutral winds above about 33 ms\ :math:`{}^{-1}`, when the drag coefficient is about 0.0024. Whilst it is likely that further work will be required on this topic, the possibility of limiting the drag coefficient has been allowed for by introducing the option @@ -4316,7 +4316,7 @@ Two approaches are available in uncoupled configurations of the model. \frac{h_f}{2k^2} \rho C_d U_1^2 \left [(\log(h_f/z_0) -1)^2 +1 \right ]. where :math:`C_d` is the upstream drag coefficient and :math:`U_1` is - the wind on the model’s lowest atmospheric level. Because this will + the wind on the model's lowest atmospheric level. Because this will be significantly above :math:`h_f`, the stability dependence of :math:`C_d` should be considered here (again see `L{\ (2015)`_). :math:`U_1` may be interpreted as @@ -4327,7 +4327,7 @@ Two approaches are available in uncoupled configurations of the model. using the original version of the scheme (`L{\ (2012)`_), :math:`C_d` must be taken as the neutral drag coefficient. Note also that various approximations may - be made in Equation :eq:`eq:int_u2`. + be made in Equation :eq:`eq:int_u2`. `L{\ (2012)`_ approximate :math:`(\log(h_f/z_0) -1)^2 +1` as :math:`(\log(h_f/z_0) )^2`; while `L{\ (2015)`_ approximate it as @@ -4430,9 +4430,9 @@ specified from land use datasets. The vegetative roughness length for scalars is assumed to be 0.1 of that for momentum. This is a simple approximation; in reality the factor depends on the land cover type and the degree of heterogeneity. [Future versions of the Unified Model will -treat surface heterogeneity explicitly by the “tiling” method.] +treat surface heterogeneity explicitly by the "tiling" method.] -The surface moisture flux given by :eq:`1.1.8` or +The surface moisture flux given by :eq:`1.1.8` or :eq:`1.1.17` involves a surface humidity value, q\ :math:`_{0}`. Prior to UM6.3, for evaporation from all of ocean, sea-ice, lake and snow-covered surfaces as well as from water on @@ -4469,8 +4469,8 @@ evaporation. r\ :math:`_{s}` is a function of the available soil moisture, near surface atmospheric conditions and the radiation impinging on the plants. [For the formulation see the documentation for the land and ice surface processes component of the Unified Model.] A -similar formula to :eq:`1.7.1` is used for the evaporation -from the very near surface soil layer. Equation :eq:`1.7.1` +similar formula to :eq:`1.7.1` is used for the evaporation +from the very near surface soil layer. Equation :eq:`1.7.1` can be written as .. math:: :label: 1.7.3 @@ -4496,7 +4496,7 @@ Effective roughness lengths Form drag is included in the surface turbulent flux formulation via effective roughness lengths for momentum :raw-latex:`\cite[]{wood93}` and for scalar quantities :raw-latex:`\cite[]{hewer1998}`. The formulae -of section `8.1 <#section_1>`__ are interpreted as relationships between +of section `8.1 <#section_1>`__ are interpreted as relationships between gridbox mean quantities and fluxes with the roughness lengths replaced by effective values, z\ :math:`_{0m(eff)}` and z\ :math:`_{0h(eff)}`. @@ -4579,7 +4579,7 @@ with \right| [The scaling velocity which appears in the -expression :eq:`1.1.4` for the Monin-Obukhov length is chosen +expression :eq:`1.1.4` for the Monin-Obukhov length is chosen to be v\ :math:`_{\ast (eff)}` rather than the flat surface value.] The orographic stress is given by @@ -4600,8 +4600,8 @@ c\ :math:`_{D(orog)}` is set to the constant value (typically 0.3, `Mason (1986)`_). If the function :math:`\Phi _{m}` and v\ :math:`_{\ast }` are -approximated by their neutral values in :eq:`2.1.6` -and :eq:`2.1.7` then the equation for calculating the +approximated by their neutral values in :eq:`2.1.6` +and :eq:`2.1.7` then the equation for calculating the effective momentum roughness is derived .. math:: :label: 2.1.12 @@ -4620,9 +4620,9 @@ The stress for the flat surface is related to the total stress by {\frac{\ln {(} {z}_{c} { / } {z}_{{0m}} {)}}{{k}}} \right)}^{2} } \right)}^{{-1}} -which is derived from equations :eq:`2.1.6`, +which is derived from equations :eq:`2.1.6`, :eq:`2.1.7` and :eq:`2.1.10`. -Equation :eq:`2.1.13` implies that +Equation :eq:`2.1.13` implies that .. math:: :label: 2.1.14 @@ -4652,7 +4652,7 @@ roughness length for momentum becomes \frac{\ln ( z_c / z_{0m(eff)} )}{\ln ( z_c / z_{0m} )} = {\left( {1 + \alpha \beta \pi ^2 f_D {\left( {\frac{A}{S}} \right)}^2 } \right)}^{-1/2} -and :eq:`2.1.13` and :eq:`2.1.14` become +and :eq:`2.1.13` and :eq:`2.1.14` become .. math:: :label: 2.1.16 @@ -4687,7 +4687,7 @@ when there is orographic form drag such that F_{X0(eff)} = F_{X0(f)} {\left( {1 - 2.2 f_D \frac{A}{S}} \right)}^{-1} -Combining :eq:`2.1.18`–:eq:`2.1.20` and using +Combining :eq:`2.1.18`-:eq:`2.1.20` and using the neutral values of the stability functions the expression for the effective scalar roughness length is derived as @@ -4892,7 +4892,7 @@ Interpolation of surface layer variables to standard observation heights If the observation height wind is assumed to lie on the profile defined by the effective roughness length and surface scaling velocity then -(c.f. equation :eq:`1.5.1`) +(c.f. equation :eq:`1.5.1`) .. math:: :label: 2.3.1 @@ -4935,7 +4935,7 @@ assumption with the last iteration value of C\ :math:`_{D(f)}`, L and If the observation height scalar quantities are assumed to lie on the mean profile defined by the effective roughness length and scaling -quantities then (c.f. equation :eq:`1.5.3` we obtain for the +quantities then (c.f. equation :eq:`1.5.3` we obtain for the generic scalar :math:`X` (:math:`T+(g/c_{P})z`, :math:`q`, tracer amount) @@ -4967,7 +4967,7 @@ and v\ :math:`_{\ast }` are used. .. _section_2.4: -Distributed form drag – an alternative to the effective roughness length parametrization +Distributed form drag - an alternative to the effective roughness length parametrization ---------------------------------------------------------------------------------------- An alternative representation of the turbulent form drag due to sub-grid @@ -5006,12 +5006,12 @@ components of the pressure force on the sub-grid orography, and where :math:`z_h` is the boundary-layer depth and :math:`\lambda`, a somewhat ill defined quantity, is related to the horizontal scales of the sub-grid hills (and set to 300 m). Note that the value of -:math:`\ell` obtained from Eq. :eq:`eq:l` is further +:math:`\ell` obtained from Eq. :eq:`eq:l` is further constrained to be at least 100 m. If the steep-hill expression is to be used, the surface stress applied is almost identical to that used in the effective roughness -parametrization (Eq. :eq:`2.1.10`, namely: +parametrization (Eq. :eq:`2.1.10`, namely: .. math:: :label: eq:dragsteep @@ -5021,7 +5021,7 @@ parametrization (Eq. :eq:`2.1.10`, namely: the main difference being the dependence on the height scale :math:`\ell` rather than :math:`z_c`. Similarly, if the `Wood and Mason (1993)`_ low-hill expression is used, the surface -stress is given by the equivalent of (Eq. :eq:`2.1.16`, +stress is given by the equivalent of (Eq. :eq:`2.1.16`, namely: .. math:: :label: eq:draglow @@ -5118,7 +5118,7 @@ where {\mathcal{I}}_{1}={\mathcal{I}}_{2}=\left(1+\frac{1}{\sqrt{2}}\right)\left(1+P\right) -Consider the one-dimensional “forced” boundary layer diffusion equation +Consider the one-dimensional "forced" boundary layer diffusion equation .. math:: :label: eq:vdiff1 @@ -5127,7 +5127,7 @@ Consider the one-dimensional “forced” boundary layer diffusion equation where :math:`X` is the scalar variable being diffused, :math:`F` is the flux of :math:`X`, :math:`t` is the time, :math:`z` is the height from -the earth’s surface, and :math:`K` is the diffusion coefficient which is +the earth's surface, and :math:`K` is the diffusion coefficient which is often non-constant and depends on :math:`X` (i.e. the PDE is non-linear) and :math:`S` is a forcing term from other processes preceding the boundary layer. In the UM these processes are: microphysics, gravity @@ -5161,11 +5161,11 @@ where, i.e. only one evaluation of the exchange coefficient is required per timestep. Furthermore, the condition :math:`I_{1}+I_{2}-({\mathcal{E}}_{1}+{\mathcal{E}}_{2})=1` ensures that if the -intermediate “starred” quantities are eliminated and the scheme is +intermediate "starred" quantities are eliminated and the scheme is reduced into a single equation then the forcing term will be multiplied by :math:`1`. -Recall from section :ref:`Model variables and turbulence closure ` +Recall from section :ref:`Model variables and turbulence closure ` that the boundary layer solver computes the increment of :math:`X`, where :math:`X=u,\; v,\;\theta_{L},\; q_{w}`. Let @@ -5889,18 +5889,18 @@ scheme. [\ *Could it be that coefficients :math:`D_{j}`, Blending height coupling ^^^^^^^^^^^^^^^^^^^^^^^^ -The same method is used as in section :ref:`Discrete equations and boundary +The same method is used as in section :ref:`Discrete equations and boundary conditions ` to form two independent tridiagonal systems of linear equations that relate the increments to momentum, temperature and humidity to the surface fluxes. -The ‘downward sweep’ elimination procedure still takes place to obtain +The 'downward sweep' elimination procedure still takes place to obtain equation :eq:`eq:du_half` and a corresponding equation for the increments to the scalar variables at the bottom model level .. math:: \delta X_{1/2}^{*}=\delta X_{1/2}^{'}-\beta_X\frac{\bar{H}_\star}{C_p} where :math:`\delta X_{1/2}^{'}` and :math:`\beta_X` are known. An -‘upward sweep’ of this tridiagonal matrix (i.e. back subsitution) then +'upward sweep' of this tridiagonal matrix (i.e. back subsitution) then takes place to obtain equations for the increment to momentum and scalar variables at a given level :math:`k_{b}` in terms of the surface fluxes @@ -5937,7 +5937,7 @@ convective thermals for aviation applications. Updraught velocities in convective boundary layers will scale with the convective velocity scale, :math:`w_*`, given by :math:`w_*^3 = z_{\mathrm{h}}\overline{w'b}_S`. In addition to the basic convective velocity scale, the strength of -thermals should also depend on the surface stability — it would be +thermals should also depend on the surface stability -- it would be possible to have significant heat flux and boundary layer depth in windy conditions that should not lead to a strong thermal forecast. This sensitivity of boundary layer turbulence is already included in the @@ -6015,9 +6015,9 @@ TKE: stash 3,473 ---------------- A substantial part of the turbulent flux is parametrized in both the -UM’s first order closure and closures involving TKE, :math:`e`, through +UM's first order closure and closures involving TKE, :math:`e`, through a simple down-gradient diffusion term. An estimate of subgrid TKE can -then be made by equating the UM’s diffusion coefficient, +then be made by equating the UM's diffusion coefficient, :eq:`klnl`, with that from a typical TKE-closure, i.e. .. math:: :label: tke_closure @@ -6058,7 +6058,7 @@ shape of this function is very similar to that used in the UM for :math:`K_m^{\mathrm{surf}}` in :eq:`kmsurf`. We now assume we can generalise :eq:`w2_scaling` by replacing :math:`w_*` with :math:`w_m` (this really ought to be checked against neutral -boundary layer LES but hasn’t yet been). Setting :math:`f(z')=z' +boundary layer LES but hasn't yet been). Setting :math:`f(z')=z' (1-z')^2` in :eq:`w2_scaling` and comparing with Fig.4 of `Holtslag and Moeng (1991)`_ gives :math:`c_{w2}= 2.66 @@ -6163,7 +6163,7 @@ convection within the diagnostic. This is given by: .. math:: e_{\rm conv} = \left(\frac{M}{g\rho \times CCA}\right)^2 where :math:`M` is the convective updraft mass flux -(Pa s\ :math:`^{-1}`) and CCA is the convective cloud area. The final +(Pa s\ :math:`^{-1}`) and CCA is the convective cloud area. The final diagnostic is then given as the maximum of :math:`e` and :math:`e_{\mathrm{conv}}`. @@ -6182,7 +6182,7 @@ and those are documented in . Diagnostics of Neutral Winds and Stresses: stash 3,365 to 3,371 --------------------------------------------------------------- -Conditions near the ocean’s surface are often described using 10-m +Conditions near the ocean's surface are often described using 10-m neutral wind and quantities derived from the neutral winds. Such diagnostics are therefore potentially very useful for evaluation of the model and have been added to the scheme. @@ -6237,7 +6237,7 @@ Here, where the subscript :math:`_S` indicates the surface flux; :math:`\Delta_F` is the divergence of the net radiative flux, :math:`F` (in Kms\ :math:`^{-1}`), associated with cloud-top, for which the -calculation is described in section :ref:`Calculation of \Delta_F `. +calculation is described in section :ref:`Calculation of \Delta_F `. Various depth parameters are given by :math:`\zeta_s = (z_{\rm ml}-\tilde{z_c})/z_{\rm ml}`, @@ -6332,7 +6332,7 @@ Note that, if :math:`k_b=1` in :eq:`zc_calc`, then the 8A calculation of :math:`z_c` is to include the depth to which the cloud extends into the inversion grid-level. If a subgrid inversion height, :math:`z_i`, has been diagnosed (see -section :ref:`Diagnosis of a sub-grid inversion `) then the height +section :ref:`Diagnosis of a sub-grid inversion `) then the height of :math:`z_i` above the half-level height is added to :math:`z_c` (as long as :math:`C_F>` SC_CFTOL in grid-levels NTML or NTML\ :math:`+1` or the layer is a @@ -6374,7 +6374,7 @@ inversion is given by The empirical constant :math:`A_{\mathrm{br}}= 0.24`. The calculation of :math:`\Delta \theta_{\ell}` and :math:`\Delta q_t` is described for a subgrid -inversion in section :ref:`Diagnosis of a sub-grid inversion ` or, +inversion in section :ref:`Diagnosis of a sub-grid inversion ` or, if one is not diagnosed, they are taken simply as :math:`\Delta_{\mathrm{ \mathrm{NTML}}+1}`. For :math:`\Delta q_{\ell}`, :math:`\Delta q_f` and @@ -6511,7 +6511,7 @@ Note that in the 9B scheme :math:`\Delta_F^{LW}` and but with the LW and SW increments separately. This change in the calculation of :math:`\Delta_F` is illustrated in -Fig. (:numref:`%s `) from LES of the diurnal cycle of marine +Fig. (:numref:`%s `) from LES of the diurnal cycle of marine stratocumulus. The top panel is from a simulation which used the code specified for the EUROCS LES intercomparison, the lower panel used the Edwards-Slingo radiation scheme in the LES. There are clearly some @@ -6595,7 +6595,7 @@ two steps of the algorithm above which become: :math:`k_m+1` is actually typical of the free-troposphere and that :math:`k_m` must therefore be the inversion grid-level (despite having the strongest LW cooling). Hence we lower :math:`k_m` by one - so that it now marks the top of the mixed layer — note that LW + so that it now marks the top of the mixed layer -- note that LW cooling within the inversion grid-level will be included in step 4 above (which is unchanged) @@ -6764,7 +6764,7 @@ ratios as: For saturation calculations a version of QSAT is used that is switchable between input specific and mixing ratio variables. The rate of change of :math:`q_s` with temperature is also used in the boundary layer code -(see e.g. appendix :ref:`Appendix: Derivation and definitions of the buoyancy +(see e.g. appendix :ref:`Appendix: Derivation and definitions of the buoyancy parameters `): .. math:: \frac{ d q_{sat} }{ dT } = \frac{\epsilon L q_{sat} }{RT^2} @@ -6773,7 +6773,7 @@ In fact this expression should really be converted to work for specific quantities and so simply changing to mixing ratios will improve the accuracy of this calculation. -Finally, in appendix :ref:`Appendix: Derivation and definitions of the buoyancy +Finally, in appendix :ref:`Appendix: Derivation and definitions of the buoyancy parameters ` virtual temperature is defined in terms of specific variables as @@ -6844,9 +6844,9 @@ Tests in the SCM showed the heating rate gradients can be very large near the surface. Hence to avoid stability problems (since this heating increment must be added after the implicit calculation of the stress (and heat flux) profiles) the increments are summed over the levels -within the BL (i.e. up to :math:`z_{\mathrm{h}}` ) and then that total +within the BL (i.e. up to :math:`z_{\mathrm{h}}` ) and then that total heating is applied as a linear decrease from the surface to zero over -:math:`z_{\mathrm{h}}` . +:math:`z_{\mathrm{h}}` . .. _app_opmods: @@ -7164,7 +7164,7 @@ Appendix: Notation - height of half-level at top of parcel ascent * - :math:`z_{\mathrm{loc}}` - - height of half-level marking ‘top’ of local :math:`Ri`-based mixing + - height of half-level marking 'top' of local :math:`Ri`-based mixing * - - (where :math:`Ri>1`) @@ -7207,7 +7207,7 @@ Appendix: Notation the SML parcel perturbation, :math:`\theta_v'`) * - :math:`w_*` - - ‘standard’ convective velocity scale for a cloud-free convective + - 'standard' convective velocity scale for a cloud-free convective * - - boundary layer, :math:`w_*^3 = z_{\mathrm{h}}\overline{w'b}_S` @@ -7237,16 +7237,16 @@ Appendix: Notation * - :math:`a_L`, :math:`\alpha_L`, :math:`\beta_T`, :math:`\beta_q`, :math:`\tilde{\beta_T}`, :math:`\tilde{\beta_q}` - - buoyancy parameters, defined in appendix :ref:`Appendix: Derivation and + - buoyancy parameters, defined in appendix :ref:`Appendix: Derivation and definitions of the buoyancy parameters ` .. [1] unless the option to mix across the LCL is selected, see - section :ref:`Diagnosis of the LCL transition zone thickness ` + section :ref:`Diagnosis of the LCL transition zone thickness ` .. [2] unless the option to mix across the LCL is selected, see - section :ref:`Diagnosis of the LCL transition zone thickness ` + section :ref:`Diagnosis of the LCL transition zone thickness ` .. [3] If i_impsolve_loc = 1, the boundary-layer implicit solver is @@ -7608,7 +7608,7 @@ References W. Fairall and P. S. Guest and R. E. Jordan (2010). *Parametrizing turbulent exchange over summer sea ice and the marginal ice zone*. - Q. J. R. Meteorol. Soc., 136, 927–943. + Q. J. R. Meteorol. Soc., 136, 927-943. .. _L{\ (2012): @@ -7632,7 +7632,7 @@ References Lachlan-Cope and J. C. King (2016). *Observations of surface momentum exchange over the marginal ice zone and recommendations for irs parametrisation*. - Atmos. Chem. Phys., 16, 1545–1563. + Atmos. Chem. Phys., 16, 1545-1563. https://doi.org/10.5194/acp-16-1545-2016 .. _Schr{\ (2003): @@ -7655,7 +7655,7 @@ References Donelan, M. A. (2018). *On the decrease of the oceanic drag coefficient in high winds*. - J. Geophys. Res: Oceans, 123, 1–17. + J. Geophys. Res: Oceans, 123, 1-17. https://doi.org/10.1002/2017JC013394 .. _Hsu et al. (2017): @@ -7663,5 +7663,5 @@ References Hsu, J. and Lien, R. and D'Asaro, E. A. and Sanford, T. B. (2017). *Estimates of Surface Wind Stress and Drag Coefficients in {T}yphoon {M}egi*. - J. Phys. Oceanogr., 47, 545–565. + J. Phys. Oceanogr., 47, 545-565. https://doi.org/10.1175/JPO-D-16-0069.1 From 649a1bd66d9fe312e56f1a9d01594f71abc817c8 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Wed, 13 May 2026 14:57:40 +0100 Subject: [PATCH 079/116] Reran script from scratch (changes line wrapping in a couple of places). --- .../source/science_guide/turbulence_schemes/bldoc.rst | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/documentation/source/science_guide/turbulence_schemes/bldoc.rst b/documentation/source/science_guide/turbulence_schemes/bldoc.rst index 9a55115866..3b72b93559 100644 --- a/documentation/source/science_guide/turbulence_schemes/bldoc.rst +++ b/documentation/source/science_guide/turbulence_schemes/bldoc.rst @@ -302,8 +302,8 @@ the first grid-level (:math:`k=k_s`) above the top of the surface layer, upwards allowing for latent heat release. The top of the surface layer is taken to be at the lower of :math:`z=0.1`\ :math:`z_{\mathrm{h}}` (this is then consistent with the :math:`K`-profiles, see -section :ref:`Surface-driven turbulence `; -:math:`z_{\mathrm{h}}` is taken from the +section :ref:`Surface-driven turbulence `; :math:`z_{\mathrm{h}}` +is taken from the previous timestep) and the grid-level above which :math:`\theta_{v\ell}` starts to increase with height. The ascent is stopped at the grid-level NTPAR (height @@ -1470,8 +1470,8 @@ The form of :math:`w_s` differs between the surface layer 0.25 \, w_*^3 & {\mathrm{mixed}\ layer} \\ \end{cases} -and :math:`w_*^3=z_{\mathrm{h}}\overline{w'b}_S` using -:math:`z_{\mathrm{h}}` from +and :math:`w_*^3=z_{\mathrm{h}}\overline{w'b}_S` using :math:`z_{\mathrm{h}}` +from the current timestep (note that the use of :math:`w_*` here will be inconsistent with the use of :math:`V_{\mathrm{heat}}` in the entrainment parametrization in cloudy boundary layers). Note that :math:`w_s` is From e5af889025985492ef33fa33c3f90f2b906171c0 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Wed, 13 May 2026 15:00:54 +0100 Subject: [PATCH 080/116] Renamed 'bldoc' to 'bl_scheme_doc'. --- .../turbulence_schemes/{bldoc.rst => bl_scheme_doc.rst} | 0 .../turbulence_schemes/{bldoc.tex => bl_scheme_doc.tex} | 0 2 files changed, 0 insertions(+), 0 deletions(-) rename documentation/source/science_guide/turbulence_schemes/{bldoc.rst => bl_scheme_doc.rst} (100%) rename documentation/source/science_guide/turbulence_schemes/{bldoc.tex => bl_scheme_doc.tex} (100%) diff --git a/documentation/source/science_guide/turbulence_schemes/bldoc.rst b/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst similarity index 100% rename from documentation/source/science_guide/turbulence_schemes/bldoc.rst rename to documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst diff --git a/documentation/source/science_guide/turbulence_schemes/bldoc.tex b/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.tex similarity index 100% rename from documentation/source/science_guide/turbulence_schemes/bldoc.tex rename to documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.tex From 509cbfdcd79f91ef338030d9064e136a8b8f2da3 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Wed, 13 May 2026 15:45:07 +0100 Subject: [PATCH 081/116] Re-applied the scripts as they've been updated to fix additional problems. Then re-applied manual corrections (put the diff for these in a local file for next time... --- .../cloud_schemes/UMDP30_PC2CloudScheme.rst | 715 +++++++++--------- .../cloud_schemes/manual_corrections.txt | 62 ++ 2 files changed, 437 insertions(+), 340 deletions(-) create mode 100644 documentation/source/science_guide/cloud_schemes/manual_corrections.txt diff --git a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst index 122dc1e07b..711c7ae3cd 100644 --- a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst +++ b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst @@ -14,8 +14,8 @@ The PC2 Cloud Scheme ==================== -:Author: D. Wilson, A. Bushell, C. Morcrette, V. Varma\ :math:`^{1}`, - M. Whitall +:Author: D. Wilson, A. Bushell, C. Morcrette, V. Varma\ :math:`^{1}`, + M. Whitall .. role:: raw-latex(raw) :format: latex @@ -36,7 +36,7 @@ Introduction This document describes the PC2 *(prognostic cloud, prognostic condensate)* cloud scheme. It should be seen as a complete reference -source for the scheme’s physical assumptions, numerical techniques, +source for the scheme's physical assumptions, numerical techniques, application to the Unified Model and coding within the Unified Model. It does not describe results from the scheme, please refer to the various reports and papers written on this. Except where commented on @@ -70,9 +70,9 @@ straightforward to solve if one is allowed to assume that there is no variability of moisture or temperature on a scale of a model gridbox. In this case the cloud fraction scheme is redundant and only the condensation part remains, which may be solved diagnostically using the -instantaneous condensation assumption in section :ref:`The ‘s’ distribution +instantaneous condensation assumption in section :ref:`The 's' distribution `. -However, the ‘no-variability’ assumption is poor until very high +However, the 'no-variability' assumption is poor until very high resolutions close to, or maybe exceeding, 1 km in the horizontal are reached. Although we may eventually assume that computer power will enable such resolutions to be reached globally, for many years we will @@ -83,7 +83,7 @@ parametrization. There are several approaches to take to the solution of the problem, although they are not as independent as often portrayed, since they nearly all require the same instantaneous condensation assumption -(discussed in section :ref:`The ‘s’ distribution `). Hence there are +(discussed in section :ref:`The 's' distribution `). Hence there are mathematical links between all the approaches. *The following are all valid structures to use in this respect.* @@ -127,7 +127,7 @@ provide the motivation to develop the PC2 cloud scheme. .. _sec_s_dist: -The ‘s’ distribution +The 's' distribution -------------------- Most cloud schemes are based on the concept of a distribution of @@ -469,7 +469,7 @@ Instantaneous condensation -------------------------- Liquid clouds in PC2 use the concept of instantaneous condensation. -Hence the ‘s’ distribution methods are fully applicable to the +Hence the 's' distribution methods are fully applicable to the development of the equations that govern the parametrization of liquid cloud in PC2. We will start by looking at changes to :math:`\overline{q_{cl}}` and :math:`C_l` when a uniform forcing is @@ -520,8 +520,8 @@ rate of change of condensate and cloud fraction based upon choose to develop a parametrization for this quantity based upon the quantities :math:`C_l`, :math:`\overline{q_{cl}}` and the saturation deficit, :math:`SD`, rather than tie :math:`G(-Q_c)` to a process. The -saturation deficit is *defined* here in the ‘s’ framework to be the -first moment of the PDF for ‘s’ values less than :math:`-Q_c`. In this +saturation deficit is *defined* here in the 's' framework to be the +first moment of the PDF for 's' values less than :math:`-Q_c`. In this way it is analogous to the liquid water content, :math:`\overline{q_{cl}}`. Appendix A of `Wilson and Gregory (2003)`_ writes this *definition* as @@ -548,7 +548,7 @@ power law near :math:`s=b_s`. G(s) ~ \propto ~ {(-s + b_s)}^n -provided :math:`s`. However, the checking routine (Q-Pos) involves a lot of communication between processors and can significantly increase -the run-time of the model. The option to “Ensure consistent sinks of qcl -and CFL” performs a check at the end of the homogeneous forcing routines +the run-time of the model. The option to "Ensure consistent sinks of qcl +and CFL" performs a check at the end of the homogeneous forcing routines to ensure that we are not trying to remove more condensate than was there to start with. @@ -852,7 +852,7 @@ is altered. This is, perhaps, the simplest method of representing a process that changes the shape of the PDF, and we will, in PC2, apply it to represent mixing of air within a gridbox, although this is a significant approximation of the process. Its application fulfils the -role of the “cloud erosion” term in the `Tiedtke (1993)`_ scheme. +role of the "cloud erosion" term in the `Tiedtke (1993)`_ scheme. By linking the term to the PDF shape we can place this term on a stronger mathematical footing than the simple reduction term parametrized by `Tiedtke (1993)`_. Equivalent arguments enabled @@ -948,11 +948,11 @@ the final continuous solution To close the solution, we need to parametrize :math:`\frac{1}{b_s} \frac{\partial b_s}{\partial t}` , which could be linked to the physics -of the process that is occuring. Note we don’t need to calculate +of the process that is occuring. Note we don't need to calculate :math:`b_s` separately, just its *fractional* rate of change. Options for the parameterisation of :math:`\frac{1}{b_s}\frac{\partial b_s}{\partial t}` due to turbulent -“erosion” are described in section :ref:`PC2 erosion `, along with the +"erosion" are described in section :ref:`PC2 erosion `, along with the numerical methods used to integrate the equations. .. _sec_init: @@ -980,7 +980,7 @@ a choice of 2 different diagnostic cloud schemes that can be used for this; either a version of the Smith scheme (see UMDP 029), or the bimodal scheme (see UMDP 039). These two options are described below... -Initiation using a “Smith-like” method +Initiation using a "Smith-like" method ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ This option is selected by setting the UM namelist switch @@ -988,7 +988,7 @@ This option is selected by setting the UM namelist switch We will assume the same form of the PDF at its boundaries as is assumed in the derivation of the :math:`G(-Q_c)` closure. For the high -‘:math:`s`’ end of the PDF distribution we integrate the power law +':math:`s`' end of the PDF distribution we integrate the power law description in :eq:`eqn19` to obtain the expressions .. math:: :label: eq:initc @@ -1701,7 +1701,7 @@ Smith scheme with a specified value of :math:`RH_{crit}` while the large :math:`RH` variability associated with these clouds implies much lower values than are typically used. -A profile of “forced cloud fraction”, :math:`C_{forced}`, is +A profile of "forced cloud fraction", :math:`C_{forced}`, is parametrized as linearly varying with height between a cloud-base value, at the lifting condensation level (LCL) from the convection diagnosis parcel ascent, and a cloud-top value of 0.1 at the top of the capping @@ -1742,8 +1742,8 @@ the inversion thickness above. Also, there is an option to treat the calculated forced cumulus cloud fraction and water content as diagnostic quantities passed directly to -the radiation scheme as part of the “convective” cloud, instead of using -them to modify the prognostic “large-scale” cloud variables :math:`C` +the radiation scheme as part of the "convective" cloud, instead of using +them to modify the prognostic "large-scale" cloud variables :math:`C` and :math:`\overline{q_{cl}}`. If this option is used, the convective cloud fraction :math:`CCA` and water content :math:`CCW` output by the convection scheme are updated, by taking the forced cumulus profiles as @@ -1813,13 +1813,13 @@ ice supersaturation :math:`S_i=e_v/e_{sat\;ice}-1`: .. math:: :label: eqn:squires_eqn - \frac{D S_i}{D t} = -b_i B_0 {\cal M}_1 S_i + \frac{D S_i}{D t} = -b_i B_0 {\mathcal{M}}_1 S_i -\left(\frac{\varepsilon}{L^2}\right)^{1/3}(S_i-S_E) + a_i w, -where :math:`{\cal M}_1` is the first moment of ice particle size +where :math:`{\mathcal{M}}_1` is the first moment of ice particle size distribution (PSD), :math:`\varepsilon` is the turbulent dissipation rate, :math:`L` is a prescribed mixing length for the turbulence, -:math:`S_{\rm E}` is the ice supersaturation of the environment +:math:`S_{\mathrm{E}}` is the ice supersaturation of the environment surrounding the cloud and :math:`b_i,B_0` and :math:`a_i` are function of :math:`p` and :math:`T` given by @@ -1837,7 +1837,7 @@ of :math:`p` and :math:`T` given by a_i = \frac{g}{R T}\left( \frac{\epsilon L_s}{c_p T} - 1 \right), The first term on the right hand side of -Eq. :eq:`eqn:squires_eqn` is the sink of vapor due to +Eq. :eq:`eqn:squires_eqn` is the sink of vapor due to depositional growth of ice crystals, the second term models entrainment (mixing) of environmental air into the cloudy volume and the third term is a source term due to vertical air motions. @@ -1851,12 +1851,12 @@ where :math:`\delta` is the Dirac distribution and the intensity of the noise, :math:`\sigma_w^2`, will be called the standard derivation of the vertical velocity fluctuations (due to the white nature of noise, a true expectation value :math:`\overline{w^2}` is not defined) and -:math:`\tau_{\rm d}` a Lagrangian decorrelation time define here by the +:math:`\tau_{\mathrm{d}}` a Lagrangian decorrelation time define here by the relation used by `Rodean (1997)`_: .. math:: :label: eqn:taud - \tau_{\rm d} = \frac{2\sigma_w^2}{\varepsilon C_0}, + \tau_{\mathrm{d}} = \frac{2\sigma_w^2}{\varepsilon C_0}, where :math:`C_0` is a known constant. @@ -1870,14 +1870,14 @@ solution PDF is Gaussian with mean and variance given by: .. math:: :label: eqn:si_avg \overline{S_i} = - S_{\rm E}\frac{ \left(\varepsilon/L^2\right)^{1/3} }{ b_i B_0 {\cal M}_1 - + \left(\varepsilon/L^2\right)^{1/3} }. + S_{\mathrm{E}}\frac{ \left(\varepsilon/L^2\right)^{1/3} }{ b_i B_0 + {\mathcal{M}}_1 + \left(\varepsilon/L^2\right)^{1/3} }. .. math:: :label: eqn:si_var \overline{S_i^2} = - \frac{a^2_{\rm i} \sigma^2_w \tau_{\rm d}}{ 2\left(b_i B_0 {\cal M}_1 + - \left(\varepsilon/L^2\right)^{1/3}\right)}, + \frac{a^2_{\mathrm{i}} \sigma^2_w \tau_{\mathrm{d}}}{ 2\left(b_i B_0 + {\mathcal{M}}_1 + \left(\varepsilon/L^2\right)^{1/3}\right)}, Equation :eq:`eqn:si_avg` and :eq:`eqn:si_var` completely specify the PDF, @@ -1896,7 +1896,7 @@ given by where :math:`S_{i,wat} = e_{sat\;wat}/e_{sat\;ice}-1` is the value of ice supersaturation at water saturation. We use the superscription -‘:math:`sgt`’(=‘*s*\ ub\ *g*\ rid *t*\ urbulence’) to indicate that +':math:`sgt`'(='*s*\ ub\ *g*\ rid *t*\ urbulence') to indicate that :math:`C_l^{sgt}` and :math:`q_{cl}^{sgt}` are values of cloud fraction and water content diagnosed from a parametrization of small-scale turbulent processes. @@ -1909,7 +1909,7 @@ Model implementation and closure relations To implement the model of Section :ref:`Model description ` in the Unified Model, closure relations are needed for the quantities -:math:`\sigma_w^2`, :math:`\varepsilon`, :math:`L`, :math:`\tau_{\rm d}` +:math:`\sigma_w^2`, :math:`\varepsilon`, :math:`L`, :math:`\tau_{\mathrm{d}}` and :math:`S_E`, subject to the constraining relationship given by Eq. :eq:`eqn:taud`. In each model grid box, these parameters specify the subgrid PDF, :math:`F(S_i)`, and from this the liquid cloud @@ -1949,29 +1949,30 @@ can define (see Section :ref:`Other user options ` below), however it should be of order one. -To obtain :math:`\tau_{\rm d}` we impose an eddy size constraint: +To obtain :math:`\tau_{\mathrm{d}}` we impose an eddy size constraint: .. math:: :label: eqn:eddy_size - \tau_{\rm d} = \frac{L}{\sigma_w} = \beta_{mix} \frac{\Delta z}{\sigma_w} + \tau_{\mathrm{d}} = \frac{L}{\sigma_w} = \beta_{mix} \frac{\Delta + z}{\sigma_w} -Eq. :eq:`eqn:taud` then determines the dissipation rate, +Eq. :eq:`eqn:taud` then determines the dissipation rate, :math:`\varepsilon`, that is consistent with the other parameters. The constant :math:`C_0=10` by default, but can be adjusted by the user. The scheme is limited to act only in grid boxes where -:math:`\tau_{\rm d}` is less than a prescribed value, +:math:`\tau_{\mathrm{d}}` is less than a prescribed value, :math:`\tau_{d}^{max}`. The default is -:math:`\tau_d^{max}=1200\;{\rm sec}`, which typically coincides with a +:math:`\tau_d^{max}=1200\;{\mathrm{sec}}`, which typically coincides with a couple of model timesteps. The motivation for this is that a motion that takes longer than a few timestep to decorrelate will be partially -resolved by the dynamics and therefore cannot be considered as ‘subgrid’ +resolved by the dynamics and therefore cannot be considered as 'subgrid' turbulence. Finally, where :math:`T`, :math:`p` and :math:`q` appear in the expressions for :math:`C_l^{sgt}` and :math:`q_{cl}^{sgt}`, these are taken to be the grid box mean values. The first moment of the ice PSD, -:math:`{\cal M}_1`, is found from the parametrization, due to +:math:`{\mathcal{M}}_1`, is found from the parametrization, due to `Field et al. (2005)`_, described in Section 4.1 of UMDP26. .. _sec_sgt_increments: @@ -2074,7 +2075,7 @@ The following variables and logical switches are optional inputs: #. Setting the logical ``l_mixed_phase_t_limit`` to *TRUE* allows the user to use the variable ``mp_t_limit`` to define a temperature limit, :math:`T_{max}`, above which the scheme is not applied. The - default is :math:`T_{max}=0^\circ\;{\rm C}`, so the scheme is only + default is :math:`T_{max}=0^\circ\;{\mathrm{C}}`, so the scheme is only applied to cold clouds. #. The input variable ``mp_tau_d_lim`` defines the upper limit, @@ -2194,8 +2195,8 @@ ice to fall between the levels. O^{[k,k+1]} = \text{Max}( C_{i}^{[k+1]} - C_i^{[k]} , 0) + w \frac{\Delta z^{[k]}}{v_i^{[k]}} -where :math:`O^{[k,k+1]}` is the amount of ice cloud ‘overhanging’ the -current (i.e. :math:`k`\ ’th) layer from the layer above, :math:`w` is a +where :math:`O^{[k,k+1]}` is the amount of ice cloud 'overhanging' the +current (i.e. :math:`k`\ 'th) layer from the layer above, :math:`w` is a parameter that is closely related to the windshear, :math:`\Delta z^{[k]}` is the model layer thickness and :math:`v_i^{[k]}` is the fallspeed of ice in the layer. @@ -2231,9 +2232,9 @@ where :math:`A_{clear}` is the proportion of the gridbox that has neither ice nor liquid cloud present. **An inconsistency has been found in the way that the fall-of-ice term -is linked to the globally constant “wind-shear value” when calculting +is linked to the globally constant "wind-shear value" when calculting the ice cloud fraction overhang. Consequently, the option not to use the -“wind shear value” when calculating the overhang is available in the +"wind shear value" when calculating the overhang is available in the UMUI (from version 7.6 onwards).** .. _sec_lsp_homo: @@ -2262,7 +2263,7 @@ Heterogeneous nucleation This process will freeze a small amount of supercooled liquid water, regardless of the previous presence of ice cloud. This will mean that -previously existing ‘liquid-only’ cloud is converted to mixed phase +previously existing 'liquid-only' cloud is converted to mixed phase cloud. These give the following changes: .. math:: @@ -2285,8 +2286,8 @@ Deposition and sublimation This term exerts one of the most important influences on the ice cloud in the whole model (this applies to the control as well as for PC2). Contained in the formulation is a subgrid-scale assumption that causes -equivalent effects to that for a moisture PDF under the ‘:math:`s`’ -framework (section :ref:`The ‘s’ distribution `). However, since +equivalent effects to that for a moisture PDF under the ':math:`s`' +framework (section :ref:`The 's' distribution `). However, since :math:`{q_{cf}}` changes slowly in response to local changes in :math:`q` and :math:`T`, we cannot base the :math:`q_{cf}` response on the same instantaneous condensation framework. It would be useful to @@ -2313,7 +2314,7 @@ the gridbox due to temperature fluctuations is not significant compared to the fluctuation of :math:`q` described below. We then parametrize a width, :math:`b_i`, to the :math:`q` (not :math:`s`) fluctuations *across the non-liquid cloud part of the gridbox*, based upon -:math:`RH_{crit}`. This is like that for the ‘:math:`s`’ distribution +:math:`RH_{crit}`. This is like that for the ':math:`s`' distribution width, :math:`b_s` but modified: .. math:: :label: eq:b_i @@ -2327,7 +2328,7 @@ limited to a minimum value of zero, but, for numerical reasons, is limited to a minimum value of 0.001. We note that :math:`b_i` has a similar form to :math:`b_s`, except the multiplier :math:`a_L` and the factor in brackets. If we remember from :eq:`eq:s` that the -definition of ‘:math:`s`’ includes a factor :math:`a_L` we see that the +definition of ':math:`s`' includes a factor :math:`a_L` we see that the absence of the :math:`a_L` factor in :eq:`eq:b_i` is consistent. The factor in brackets is a *parametrization* of the effect that, when ice is present, deposition in the moistier parts and @@ -2337,7 +2338,7 @@ function of :math:`\frac{\overline{q_{cf}}}{q_{sat~liq}(\overline{T})}`, and is tunable using the factor :math:`i`, which takes the value of 0.04. -We note that this formulation isn’t totally consistent with the liquid +We note that this formulation isn't totally consistent with the liquid cloud formulation, which considers an underlying PDF across the whole gridbox and does not have, in general, its width prescribed. Remember that we do not calculate on-line the whole of the liquid - vapour PDF, @@ -2548,7 +2549,7 @@ Numerical implementation Note that after each process has been applied, we do *not* recalculate the sizes of the ice-only, liquid-only and mixed phase partitions, but use the values at the start of the microphysics (this includes the -values of :math:`C_i` used in the calculation of ‘in-cloud’ water +values of :math:`C_i` used in the calculation of 'in-cloud' water contents above. However, we do update the cloud fractions themselves sequentially. We also recalculate after each process the overlaps between the rain fraction (see ) and the cloud fractions. @@ -2615,7 +2616,7 @@ is close to :math:`1 \times 10^{-4} s^{-1}`. Note: the source-code for this erosion method (**pc2_hom_conv**, **pc2_homog_plus_turb**, **pc2_delta_hom_turb**) includes an additional -term “dbsdtbs1” which scales with the rate of homogeneous forcing +term "dbsdtbs1" which scales with the rate of homogeneous forcing :math:`\frac{\partial Q_c}{\partial t}`. However this term is always set to zero on input to these routines so is never used. @@ -2669,7 +2670,7 @@ that no more :math:`\overline{q_{cl}}` is removed than the model has available. This was chosen to ensure that the erosion process itself contains this physical limit, not a numerical tidying-up process. -The option “l_fixbug_pc2_qcl_incr” ensures that qcl is set to zero if +The option "l_fixbug_pc2_qcl_incr" ensures that qcl is set to zero if the CFL has reached zero. Cloud-surface-area hybrid erosion method @@ -2788,7 +2789,7 @@ Where the change in liquid water content This combination of a Tiedkte sink term for :math:`q_{cl}`, a PC2 term for :math:`C_l` and the introduction of some surface area dependence -leads to this formulation being referred to as a “hybrid” +leads to this formulation being referred to as a "hybrid" cloud-surface-area erosion method. .. _sec_erosion_numerics: @@ -2818,7 +2819,7 @@ simulations), this discretization can suffer severe numerical overshoot. i.e. the increment based on :math:`C_l^n` is large enough to reduce :math:`q_{cl}` (and hence also :math:`C_l`) to less than zero within a single timestep. If the continuous equation were solved analytically -this wouldn’t happen; as :math:`C_l` declines due to the erosion, so +this wouldn't happen; as :math:`C_l` declines due to the erosion, so will :math:`A(C_l)` and hence the erosion rate, so that :math:`q_{cl}` and :math:`C_l` smoothly decline towards zero. @@ -2829,11 +2830,11 @@ yield much less timestep sensitivity for detrained cloud in shallow cumulus regimes. #. **Retain the explicit discretization, but limit the resulting erosion - increments to ensure :math:`q_{cl}` and :math:`C_l` don’t go + increments to ensure :math:`q_{cl}` and :math:`C_l` don't go negative. (i_pc2_erosion_numerics=1)** Also, to ensure that some cloud remains at end-of-timestep where shallow cumulus is detraining into dry environments, the erosion calculation is fed copies of the - fields with the current timestep’s convection increments subtracted + fields with the current timestep's convection increments subtracted off. This means any cloud detrained by convection during the current timestep cannot be eroded until the following timestep, and so is still present at end-of-timestep. As discussed in section @@ -2937,7 +2938,7 @@ cumulus regimes. :eq:`eq:dcdt_hybrid_discr`, except that the term :math:`\frac{1}{2} \Delta \overline{{q_{cl}}_{ero}}` is omitted (interpolating to the mid-point value of :math:`\overline{q_{cl}}` in - the denominator would be “double-counting” if we are already making + the denominator would be "double-counting" if we are already making an implicit correction to the full increment). In the case where the homogeneous forcing increments have already @@ -2982,7 +2983,7 @@ cumulus regimes. exponentials). When erosion (wrongly) can never entirely remove cloud, this allows tiny values of :math:`q_{cl}` and :math:`C_l` to spuriously spread across the domain via numerical diffusion from - the model’s advection scheme. + the model's advection scheme. Under this option, we attempt to compute an analytic solution to the simultaneous differential equations @@ -3073,7 +3074,7 @@ cumulus regimes. where :math:`SD` is the saturation defecit, and :math:`a_L` is the dimensionless factor defined in eq :eq:`eq:a_L`. Following the derivation in section - :ref:`“Smooth” initiation logic ` (eq + :ref:`"Smooth" initiation logic ` (eq :eq:`eq:qc_plus_sd`, we can write this in terms of the liquid-water content: :math:`SD = q_{cl} - Q_c` (where :math:`Q_c` was defined in eq @@ -3427,45 +3428,49 @@ processes (e.g. total water content). In this case, .. math:: :label: eq:chibasic - {\frac{\partial \, {\chi}_{\rm{ }}^{\rm{E}}}{\partial \, t}}_{\rm{conv}} = + {\frac{\partial \, {\chi}_{\mathrm{ }}^{\mathrm{E}}}{\partial \, + t}}_{\mathrm{conv}} = - \frac{1}{\overline{\rho}} \, \frac{\partial \, \overline{\rho w^{'} - {\chi}_{\rm{ }}^{\rm{E'}}}}{\partial \, z} + {\chi}_{\mathrm{ }}^{\mathrm{E'}}}}{\partial \, z} To parametrize :eq:`eq:chibasic`, the current UM convection scheme takes a mass flux approximation .. math:: :label: eq:massflux - \left({\overline{\rho w^{'} {\chi}_{\rm{ }}^{\rm{E'}}}} \right)_{\rm{conv}} - = M^{\rm{P}} \, - \left({ {\chi}_{\rm{ }}^{\rm{P}} - {\chi}_{\rm{ }}^{\rm{E}} } \right) + \left({\overline{\rho w^{'} {\chi}_{\mathrm{ }}^{\mathrm{E'}}}} + \right)_{\mathrm{conv}} = M^{\mathrm{P}} \, + \left({ {\chi}_{\mathrm{ }}^{\mathrm{P}} - {\chi}_{\mathrm{ }}^{\mathrm{E}} + } \right) which can be differentiated to give .. math:: :label: eq:eddyflux - \frac{1}{\overline{\rho}} \, \frac{\partial \, \overline{\rho w^{'} - {\chi}_{\rm{ }}^{\rm{E'}}}}{\partial \, z} = - \frac{\partial \, {\chi}_{\rm{ }}^{\rm{P}} \, M^{\rm{P}}}{\partial \, p} - - {\chi}_{\rm{ }}^{\rm{E}} \, \frac{\partial \, M^{\rm{P}}}{\partial \, p} - - - M^{\rm{P}} \, \frac{\partial \, {\chi}_{\rm{ }}^{\rm{E}}}{\partial \, p} + {\chi}_{\mathrm{ }}^{\mathrm{E'}}}}{\partial \, z} = + \frac{\partial \, {\chi}_{\mathrm{ }}^{\mathrm{P}} \, + M^{\mathrm{P}}}{\partial \, p} - + {\chi}_{\mathrm{ }}^{\mathrm{E}} \, \frac{\partial \, + M^{\mathrm{P}}}{\partial \, p} - + M^{\mathrm{P}} \, \frac{\partial \, {\chi}_{\mathrm{ + }}^{\mathrm{E}}}{\partial \, p} The bulk cloud model plume equations for mass and :math:`{\chi}` are: .. math:: :label: eq:dbydpmassflux - - \frac{\partial \, M^{\rm{P}}}{\partial \, p} = - \left({ \varepsilon \, M^{\rm{P}} - \mu \, M^{\rm{P}} - \delta \, M^{\rm{P}} - } \right) + - \frac{\partial \, M^{\mathrm{P}}}{\partial \, p} = + \left({ \varepsilon \, M^{\mathrm{P}} - \mu \, M^{\mathrm{P}} - \delta \, + M^{\mathrm{P}} } \right) .. math:: :label: eq:dbydpmfchi - - \frac{\partial \, {\chi}_{\rm{ }}^{\rm{P}} \, M^{\rm{P}}}{\partial \, p} - = \left({ - \varepsilon \, M^{\rm{P}} \, {\chi}_{\rm{ }}^{\rm{E}} - - \mu \, M^{\rm{P}} \, {\chi}_{\rm{ }}^{\rm{R}} - \delta \, M^{\rm{P}} \, - {\chi}_{\rm{ }}^{\rm{P}} + - \frac{\partial \, {\chi}_{\mathrm{ }}^{\mathrm{P}} \, + M^{\mathrm{P}}}{\partial \, p} = \left({ + \varepsilon \, M^{\mathrm{P}} \, {\chi}_{\mathrm{ }}^{\mathrm{E}} + - \mu \, M^{\mathrm{P}} \, {\chi}_{\mathrm{ }}^{\mathrm{R}} - \delta \, + M^{\mathrm{P}} \, {\chi}_{\mathrm{ }}^{\mathrm{P}} } \right) Equations :eq:`eq:eddyflux`, @@ -3475,24 +3480,27 @@ Equations :eq:`eq:eddyflux`, .. math:: :label: eq:chimassflux - {\frac{\partial \, {\chi}_{\rm{ }}^{\rm{E}}}{\partial \, t}}_{\rm{conv}} = - - M^{\rm{P}} \, \frac{\partial \, {\chi}_{\rm{ }}^{\rm{E}}}{\partial \, p} - + \mu \, M^{\rm{P}} \, \left({ {\chi}_{\rm{ }}^{\rm{R}} - {\chi}_{\rm{ }}^{\rm{E}} } \right) - + \delta \, M^{\rm{P}} \, \left({ {\chi}_{\rm{ }}^{\rm{P}} - {\chi}_{\rm{ }}^{\rm{E}} } \right) + {\frac{\partial \, {\chi}_{\mathrm{ }}^{\mathrm{E}}}{\partial \, + t}}_{\mathrm{conv}} = + - M^{\mathrm{P}} \, \frac{\partial \, {\chi}_{\mathrm{ + }}^{\mathrm{E}}}{\partial \, p} + + \mu \, M^{\mathrm{P}} \, \left({ {\chi}_{\mathrm{ }}^{\mathrm{R}} - {\chi}_{\mathrm{ }}^{\mathrm{E}} } \right) + + \delta \, M^{\mathrm{P}} \, \left({ {\chi}_{\mathrm{ }}^{\mathrm{P}} - {\chi}_{\mathrm{ }}^{\mathrm{E}} } \right) -while :math:`{\chi}_{\rm{}}^{\rm{P}}` is obtained from the vertical +while :math:`{\chi}_{\mathrm{}}^{\mathrm{P}}` is obtained from the vertical gradient derived by combining :eq:`eq:dbydpmassflux` and :eq:`eq:dbydpmfchi` : .. math:: :label: eq:gradchipar - M^{\rm{P}} \, \frac{\partial \, {\chi}_{\rm{ }}^{\rm{P}}}{\partial \, p} = - \varepsilon \, M^{\rm{P}} \, \left({ {\chi}_{\rm{ }}^{\rm{P}} - {\chi}_{\rm{ - }}^{\rm{E}} } \right)- - \mu \, M^{\rm{P}} \, \left({ {\chi}_{\rm{ }}^{\rm{P}} - {\chi}_{\rm{ - }}^{\rm{R}} } \right) + M^{\mathrm{P}} \, \frac{\partial \, {\chi}_{\mathrm{ + }}^{\mathrm{P}}}{\partial \, p} = + \varepsilon \, M^{\mathrm{P}} \, \left({ {\chi}_{\mathrm{ }}^{\mathrm{P}} - + {\chi}_{\mathrm{ }}^{\mathrm{E}} } \right)- + \mu \, M^{\mathrm{P}} \, \left({ {\chi}_{\mathrm{ }}^{\mathrm{P}} - + {\chi}_{\mathrm{ }}^{\mathrm{R}} } \right) -Within the model, eqn :eq:`eq:chimassflux` would take +Within the model, eqn :eq:`eq:chimassflux` would take a discretized form which actually depends upon whether the model level, k, is above or at the lowest cloud level (k = cb). Note that the formal cloud base lies at the half-level below, i.e. on the layer boundary @@ -3502,24 +3510,26 @@ discretized form of :eq:`eq:chimassflux`, setting .. math:: :label: eq:chidisck - {\frac{\partial \, {\chi}_{\rm{ }}^{\rm{E}}}{\partial \, t}}_{\rm{conv, \, - k}} = m_{\rm{k+1/2}} \, - \frac{ \left({{\chi}_{\rm{k+1}}^{\rm{E}} - {\chi}_{\rm{k}}^{\rm{E}}} \right)} - {{\Delta z}_{\rm{k \, \rightarrow \, k+1}}} - + {\delta}_{\rm{k}} \, m_{\rm{k}} \, \left({ {\chi}_{\rm{k}}^{\rm{P}} - {\chi}_{\rm{k}}^{\rm{E}} } \right) - \qquad \ldots \; \mbox{for k $>$ cb} + {\frac{\partial \, {\chi}_{\mathrm{ }}^{\mathrm{E}}}{\partial \, + t}}_{\mathrm{conv, \, k}} = m_{\mathrm{k+1/2}} \, + \frac{ \left({{\chi}_{\mathrm{k+1}}^{\mathrm{E}} - + {\chi}_{\mathrm{k}}^{\mathrm{E}}} \right)} + {{\Delta z}_{\mathrm{k \, \rightarrow \, k+1}}} + + {\delta}_{\mathrm{k}} \, m_{\mathrm{k}} \, \left({ {\chi}_{\mathrm{k}}^{\mathrm{P}} - {\chi}_{\mathrm{k}}^{\mathrm{E}} } \right) + \qquad \ldots \; \mathrm{for k $>$ cb} .. math:: :label: eq:chidisccb - {\frac{\partial \, {\chi}_{\rm{ }}^{\rm{E}}}{\partial \, t}}_{\rm{conv, \, - cb}} = m_{\rm{cb+1/2}} \, - \frac{ \left({{\chi}_{\rm{cb+1}}^{\rm{E}} - {\chi}_{\rm{cb}}^{\rm{E}}} - \right)} - {{\Delta z}_{\rm{cb \, \rightarrow \, cb+1}}} - - m_{\rm{cb}} \, - \left({ {\chi}_{\rm{i,cb}}^{\rm{P}} - {\chi}_{\rm{cb}}^{\rm{E}} } \right) + {\frac{\partial \, {\chi}_{\mathrm{ }}^{\mathrm{E}}}{\partial \, + t}}_{\mathrm{conv, \, cb}} = m_{\mathrm{cb+1/2}} \, + \frac{ \left({{\chi}_{\mathrm{cb+1}}^{\mathrm{E}} - + {\chi}_{\mathrm{cb}}^{\mathrm{E}}} \right)} + {{\Delta z}_{\mathrm{cb \, \rightarrow \, cb+1}}} + - m_{\mathrm{cb}} \, + \left({ {\chi}_{\mathrm{i,cb}}^{\mathrm{P}} - + {\chi}_{\mathrm{cb}}^{\mathrm{E}} } \right) -where the initial parcel value :math:`{\chi}_{\rm{i,cb}}^{\rm{P}}` may +where the initial parcel value :math:`{\chi}_{\mathrm{i,cb}}^{\mathrm{P}}` may be chosen to produce a fixed increment or place a closure condition on the cloud base flux. In fact, the convection equations (see ) differ from :eq:`eq:chidisck` and @@ -3530,26 +3540,26 @@ The model convection variables are NOT conserved under moist adiabatic processes because precipitation processes deplete the column moisture and condensation processes affect the temperature, specific humidity and cloud condensate variables. Surprisingly, however, the form of -eqn :eq:`eq:chimassflux` is retained even though the +eqn :eq:`eq:chimassflux` is retained even though the basic equation :eq:`eq:chibasic` acquires additional terms for temperature and specific humidity: .. math:: :label: eq:defineq1 - {\frac{\partial \, T_{\rm{ }}^{\rm{E}}}{\partial \, t}}_{\rm{conv}} = Q1 - \equiv - \left({ \frac{L}{c_{P}} } \right)\, {\overline{Q}}_{\rm{par}} + {\frac{\partial \, T_{\mathrm{ }}^{\mathrm{E}}}{\partial \, + t}}_{\mathrm{conv}} = Q1 \equiv + \left({ \frac{L}{c_{P}} } \right)\, {\overline{Q}}_{\mathrm{par}} - \frac{1}{\overline{\rho}} \, \frac{\partial \, \overline{\rho w^{'} - T_{\rm{ }}^{\rm{E'}}}}{\partial \, z} + T_{\mathrm{ }}^{\mathrm{E'}}}}{\partial \, z} .. math:: :label: eq:defineq2 - {\frac{\partial \, q_{\rm{ }}^{\rm{E}}}{\partial \, t}}_{\rm{conv}} = Q2 - \equiv - {\overline{Q}}_{\rm{par}} + {\frac{\partial \, q_{\mathrm{ }}^{\mathrm{E}}}{\partial \, + t}}_{\mathrm{conv}} = Q2 \equiv - {\overline{Q}}_{\mathrm{par}} - \frac{1}{\overline{\rho}} \, \frac{\partial \, \overline{\rho w^{'} - q_{\rm{ }}^{\rm{E'}}}}{\partial \, z} + q_{\mathrm{ }}^{\mathrm{E'}}}}{\partial \, z} -where :math:`{\overline{Q}}_{\rm{par}}` is the rate of condensation +where :math:`{\overline{Q}}_{\mathrm{par}}` is the rate of condensation which occurs in the ascending plumes. The reason that :eq:`eq:defineq1` and @@ -3562,60 +3572,65 @@ gradient equations based upon :eq:`eq:gradchipar` .. math:: :label: eq:gradtpar - M^{\rm{P}} \, \frac{\partial \, T_{\rm{ }}^{\rm{P}}}{\partial \, p} = - \varepsilon \, M^{\rm{P}} \, \left({ T_{\rm{ }}^{\rm{P}} - T_{\rm{ - }}^{\rm{E}} } \right)- - \mu \, M^{\rm{P}} \, \left({ T_{\rm{ }}^{\rm{P}} - T_{\rm{ - }}^{\rm{R}} } \right)- - \left({ \frac{L}{c_{P}} } \right)\, {\overline{Q}}_{\rm{par}} + M^{\mathrm{P}} \, \frac{\partial \, T_{\mathrm{ }}^{\mathrm{P}}}{\partial \, + p} = + \varepsilon \, M^{\mathrm{P}} \, \left({ T_{\mathrm{ }}^{\mathrm{P}} - + T_{\mathrm{ }}^{\mathrm{E}} } \right)- + \mu \, M^{\mathrm{P}} \, \left({ T_{\mathrm{ }}^{\mathrm{P}} - + T_{\mathrm{ }}^{\mathrm{R}} } \right)- + \left({ \frac{L}{c_{P}} } \right)\, {\overline{Q}}_{\mathrm{par}} .. math:: :label: eq:gradqpar - M^{\rm{P}} \, \frac{\partial \, q_{\rm{ }}^{\rm{P}}}{\partial \, p} = - \varepsilon \, M^{\rm{P}} \, \left({ q_{\rm{ }}^{\rm{P}} - q_{\rm{ - }}^{\rm{E}} } \right)- - \mu \, M^{\rm{P}} \, \left({ q_{\rm{ }}^{\rm{P}} - q_{\rm{ - }}^{\rm{R}} } \right)+ - {\overline{Q}}_{\rm{par}} + M^{\mathrm{P}} \, \frac{\partial \, q_{\mathrm{ }}^{\mathrm{P}}}{\partial \, + p} = + \varepsilon \, M^{\mathrm{P}} \, \left({ q_{\mathrm{ }}^{\mathrm{P}} - + q_{\mathrm{ }}^{\mathrm{E}} } \right)- + \mu \, M^{\mathrm{P}} \, \left({ q_{\mathrm{ }}^{\mathrm{P}} - + q_{\mathrm{ }}^{\mathrm{R}} } \right)+ + {\overline{Q}}_{\mathrm{par}} .. math:: :label: eq:gradlpar - M^{\rm{P}} \, \frac{\partial \, l_{\rm{ }}^{\rm{P}}}{\partial \, p} = - \varepsilon \, M^{\rm{P}} \, \left({ l_{\rm{ }}^{\rm{P}} - l_{\rm{ - }}^{\rm{E}} } \right) - - {\overline{Q}}_{\rm{par}} + PPN + M^{\mathrm{P}} \, \frac{\partial \, l_{\mathrm{ }}^{\mathrm{P}}}{\partial \, + p} = + \varepsilon \, M^{\mathrm{P}} \, \left({ l_{\mathrm{ }}^{\mathrm{P}} - + l_{\mathrm{ }}^{\mathrm{E}} } \right) + - {\overline{Q}}_{\mathrm{par}} + PPN The final calculation of rates in the current condensation scheme (, section 10) assumes a further condensation term, -:math:`{\overline{Q}}_{\rm{reset}}`, which acts to make the net rate of +:math:`{\overline{Q}}_{\mathrm{reset}}`, which acts to make the net rate of change of condensate equal zero, and a final assumption is made that the environment values of condensate remain zero (and also that -:math:`l_{\rm{ }}^{\rm{R}}` = :math:`l_{\rm{ }}^{\rm{P}}`). The result +:math:`l_{\mathrm{ }}^{\mathrm{R}}` = :math:`l_{\mathrm{ }}^{\mathrm{P}}`). The +result is basic equations .. math:: :label: eq:basictold - {\frac{\partial \, T_{\rm{ }}^{\rm{E}}}{\partial \, t}}_{\rm{conv}} = Q1 - - \left({ \frac{L}{c_{P}} } \right)\, {\overline{Q}}_{\rm{reset}} + {\frac{\partial \, T_{\mathrm{ }}^{\mathrm{E}}}{\partial \, + t}}_{\mathrm{conv}} = Q1 - + \left({ \frac{L}{c_{P}} } \right)\, {\overline{Q}}_{\mathrm{reset}} .. math:: :label: eq:basicqold - {\frac{\partial \, q_{\rm{ }}^{\rm{E}}}{\partial \, t}}_{\rm{conv}} = Q2 + - {\overline{Q}}_{\rm{reset}} + {\frac{\partial \, q_{\mathrm{ }}^{\mathrm{E}}}{\partial \, + t}}_{\mathrm{conv}} = Q2 + {\overline{Q}}_{\mathrm{reset}} .. math:: :label: eq:basiclold \begin{aligned} - 0 \equiv {\frac{\partial \, l_{\rm{ }}^{\rm{E}}}{\partial \, t}}_{\rm{conv}} - & = & {\overline{Q}}_{\rm{par}} - - {\overline{Q}}_{\rm{reset}} - PPN + 0 \equiv {\frac{\partial \, l_{\mathrm{ }}^{\mathrm{E}}}{\partial \, + t}}_{\mathrm{conv}} & = & {\overline{Q}}_{\mathrm{par}} - + {\overline{Q}}_{\mathrm{reset}} - PPN - \frac{1}{\overline{\rho}} \, \frac{\partial \, \overline{\rho w^{'} - l_{\rm{ }}^{\rm{E'}}}}{\partial \, z} + l_{\mathrm{ }}^{\mathrm{E'}}}}{\partial \, z} \\ & = & - \mu \, M^{\rm{P}} \, l_{\rm{ }}^{\rm{P}} + \delta \, M^{\rm{P}} \, l_{\rm{ - }}^{\rm{P}} - - {\overline{Q}}_{\rm{reset}} + \mu \, M^{\mathrm{P}} \, l_{\mathrm{ }}^{\mathrm{P}} + \delta \, + M^{\mathrm{P}} \, l_{\mathrm{ }}^{\mathrm{P}} - + {\overline{Q}}_{\mathrm{reset}} \end{aligned} By analogy with equations :eq:`eq:defineq1` and @@ -3638,19 +3653,21 @@ Define .. math:: :label: eq:defineq4l - \left({ \frac{\partial \, l_{\rm{l}}^{\rm{ }}}{\partial \, t} } - \right)_{\rm{conv}} = Q4_{\rm{l}} \equiv - {\overline{Q}}_{\rm{l, par}} - {\overline{Q}}_{\rm{l, reset}} - RAIN - + \left({ \frac{\partial \, l_{\mathrm{l}}^{\mathrm{ }}}{\partial \, t} } + \right)_{\mathrm{conv}} = Q4_{\mathrm{l}} \equiv + {\overline{Q}}_{\mathrm{l, par}} - {\overline{Q}}_{\mathrm{l, reset}} - RAIN + - \frac{1}{\overline{\rho}} \, \frac{\partial \, \overline{\rho w^{'} - l_{\rm{l}}^{\rm{'}}}}{\partial \, z} + l_{\mathrm{l}}^{\mathrm{'}}}}{\partial \, z} .. math:: :label: eq:defineq4f - \left({ \frac{\partial \, l_{\rm{f}}^{\rm{ }}}{\partial \, t} } - \right)_{\rm{conv}} = Q4_{\rm{f}} \equiv - {\overline{Q}}_{\rm{f, par}} - {\overline{Q}}_{\rm{f, reset}} - SNOW - + \left({ \frac{\partial \, l_{\mathrm{f}}^{\mathrm{ }}}{\partial \, t} } + \right)_{\mathrm{conv}} = Q4_{\mathrm{f}} \equiv + {\overline{Q}}_{\mathrm{f, par}} - {\overline{Q}}_{\mathrm{f, reset}} - SNOW + - \frac{1}{\overline{\rho}} \, \frac{\partial \, \overline{\rho w^{'} - l_{\rm{f}}^{\rm{'}}}}{\partial \, z} + l_{\mathrm{f}}^{\mathrm{'}}}}{\partial \, z} where the PC2 assumption thus far has been that :math:`{\overline{Q}}_{\rm{l, reset}} = 0 @@ -3659,8 +3676,8 @@ where the PC2 assumption thus far has been that - The current convection scheme assumes that parcel condensate is single phase (ie. either all liquid or all frozen) and this is seriously hard-wired into the code. Thus we can treat the precipitation and - parcel condensation processes in :math:`Q4_{\rm{l}}` and - :math:`Q4_{\rm{f}}` separately without worrying about cross-transfer + parcel condensation processes in :math:`Q4_{\mathrm{l}}` and + :math:`Q4_{\mathrm{f}}` separately without worrying about cross-transfer between the two because at most only one set will ever be active in a given grid box at one time. However, even for the inactive (zero parcel condensate) phase, convection mixes environmental air into the @@ -3674,17 +3691,19 @@ condensate is calculated as .. math:: :label: eq:vertparl - \frac{\partial \, l_{\rm{l}}^{\rm{P}}}{\partial \, p} = \varepsilon \, - \left({ l_{\rm{l}}^{\rm{P}} - l_{\rm{l}}^{\rm{E}} } \right)- - \frac{{\overline{Q}}_{\rm{l, par}}}{M^{\rm{P}}} - - \frac{RAIN}{M^{\rm{P}}} + \frac{\partial \, l_{\mathrm{l}}^{\mathrm{P}}}{\partial \, p} = + \varepsilon \, + \left({ l_{\mathrm{l}}^{\mathrm{P}} - l_{\mathrm{l}}^{\mathrm{E}} } \right)- + \frac{{\overline{Q}}_{\mathrm{l, par}}}{M^{\mathrm{P}}} - + \frac{RAIN}{M^{\mathrm{P}}} .. math:: :label: eq:vertparf - \frac{\partial \, l_{\rm{f}}^{\rm{P}}}{\partial \, p} = \varepsilon \, - \left({ l_{\rm{f}}^{\rm{P}} - l_{\rm{f}}^{\rm{E}} } \right)- - \frac{{\overline{Q}}_{\rm{f, par}}}{M^{\rm{P}}} - - \frac{SNOW}{M^{\rm{P}}} + \frac{\partial \, l_{\mathrm{f}}^{\mathrm{P}}}{\partial \, p} = + \varepsilon \, + \left({ l_{\mathrm{f}}^{\mathrm{P}} - l_{\mathrm{f}}^{\mathrm{E}} } \right)- + \frac{{\overline{Q}}_{\mathrm{f, par}}}{M^{\mathrm{P}}} - + \frac{SNOW}{M^{\mathrm{P}}} Following , equations :eq:`eq:dbydpmassflux`, :eq:`eq:vertparl` and :eq:`eq:vertparf` @@ -3692,46 +3711,46 @@ are discretized: .. math:: :label: eq:discdmfbydp - M_{\rm{k} + 1} = M_{\rm{k}} \, - \left({ 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right)\, - \left({ 1 - \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right)\, - EPSS_{\rm{k}} + M_{\mathrm{k} + 1} = M_{\mathrm{k}} \, + \left({ 1 - \mu_{\mathrm{k}} \, \Delta p_{\mathrm{k} + 1 / 2} } \right)\, + \left({ 1 - \delta_{\mathrm{k}} \, \Delta p_{\mathrm{k} + 1 / 2} } \right)\, + EPSS_{\mathrm{k}} .. math:: - l_{\rm{l \, k + 1}}^{\rm{P}} = \left({ - l_{\rm{l \, k}}^{\rm{P}} + - \varepsilon_{\rm{k} + 1/4} \, \Delta p_{\rm{k} + 1/4} \, l_{\rm{l \, - k}}^{\rm{E}} + - \varepsilon_{\rm{k} + 3/4} \, \Delta p_{\rm{k} + 3/4} \, - \left[{1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4}} - \right]\, - l_{\rm{l \, k + 1}}^{\rm{E}} - } \right)\, / \, \left({EPSS_{\rm{k}}} \right) + l_{\mathrm{l \, k + 1}}^{\mathrm{P}} = \left({ + l_{\mathrm{l \, k}}^{\mathrm{P}} + + \varepsilon_{\mathrm{k} + 1/4} \, \Delta p_{\mathrm{k} + 1/4} \, + l_{\mathrm{l \, k}}^{\mathrm{E}} + + \varepsilon_{\mathrm{k} + 3/4} \, \Delta p_{\mathrm{k} + 3/4} \, + \left[{1 + \varepsilon_{\mathrm{k} + 1 / 4} \, \Delta p_{\mathrm{k} + 1 / + 4}} \right]\, + l_{\mathrm{l \, k + 1}}^{\mathrm{E}} + } \right)\, / \, \left({EPSS_{\mathrm{k}}} \right) .. math:: :label: eq:discvparl - { } { } + \left({ {\overline{Q}}_{\rm{l} \, \rm{k} + 1} \, / \, M_{\rm{k} - + 1}} \right) - - \left({ RAIN_{\rm{k} + 1} \, / \, M_{\rm{k} + 1} } \right) + { } { } + \left({ {\overline{Q}}_{\mathrm{l} \, \mathrm{k} + 1} \, / \, + M_{\mathrm{k} + 1}} \right) + - \left({ RAIN_{\mathrm{k} + 1} \, / \, M_{\mathrm{k} + 1} } \right) .. math:: - l_{\rm{f \, k + 1}}^{\rm{P}} = \left({ - l_{\rm{f \, k}}^{\rm{P}} + - \varepsilon_{\rm{k} + 1/4} \, \Delta p_{\rm{k} + 1/4} \, l_{\rm{f \, - k}}^{\rm{E}} + - \varepsilon_{\rm{k} + 3/4} \, \Delta p_{\rm{k} + 3/4} \, - \left[{1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4}} - \right]\, - l_{\rm{f \, k + 1}}^{\rm{E}} - } \right)\, / \, \left({EPSS_{\rm{k}}} \right) + l_{\mathrm{f \, k + 1}}^{\mathrm{P}} = \left({ + l_{\mathrm{f \, k}}^{\mathrm{P}} + + \varepsilon_{\mathrm{k} + 1/4} \, \Delta p_{\mathrm{k} + 1/4} \, + l_{\mathrm{f \, k}}^{\mathrm{E}} + + \varepsilon_{\mathrm{k} + 3/4} \, \Delta p_{\mathrm{k} + 3/4} \, + \left[{1 + \varepsilon_{\mathrm{k} + 1 / 4} \, \Delta p_{\mathrm{k} + 1 / + 4}} \right]\, + l_{\mathrm{f \, k + 1}}^{\mathrm{E}} + } \right)\, / \, \left({EPSS_{\mathrm{k}}} \right) .. math:: :label: eq:discvparf - { } { } + \left({ {\overline{Q}}_{\rm{f} \, \rm{k} + 1} \, / \, M_{\rm{k} - + 1}} \right) - - \left({ SNOW_{\rm{k} + 1} \, / \, M_{\rm{k} + 1} } \right) + { } { } + \left({ {\overline{Q}}_{\mathrm{f} \, \mathrm{k} + 1} \, / \, + M_{\mathrm{k} + 1}} \right) + - \left({ SNOW_{\mathrm{k} + 1} \, / \, M_{\mathrm{k} + 1} } \right) where :math:`EPSS_{\rm{k}} = \left({1 + \varepsilon_{\rm{k} + 3 / 4} \, \Delta p_{\rm{k} + 3 / 4}} \right)\, @@ -3746,48 +3765,51 @@ precipitation terms are suppressed: .. math:: :label: eq:discvparldry - l_{\rm{l \, k + 1}}^{\rm{P}} = \frac{\left({ - l_{\rm{l \, k}}^{\rm{P}} + - \varepsilon_{\rm{k} + 1/4} \, \Delta p_{\rm{k} + 1/4} \, l_{\rm{l \, - k}}^{\rm{E}} + - \varepsilon_{\rm{k} + 3/4} \, \Delta p_{\rm{k} + 3/4} \, - \left[{1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4}} - \right]\, - l_{\rm{l \, k + 1}}^{\rm{E}} - } \right)}{EPSS_{\rm{k}}} + l_{\mathrm{l \, k + 1}}^{\mathrm{P}} = \frac{\left({ + l_{\mathrm{l \, k}}^{\mathrm{P}} + + \varepsilon_{\mathrm{k} + 1/4} \, \Delta p_{\mathrm{k} + 1/4} \, + l_{\mathrm{l \, k}}^{\mathrm{E}} + + \varepsilon_{\mathrm{k} + 3/4} \, \Delta p_{\mathrm{k} + 3/4} \, + \left[{1 + \varepsilon_{\mathrm{k} + 1 / 4} \, \Delta p_{\mathrm{k} + 1 / + 4}} \right]\, + l_{\mathrm{l \, k + 1}}^{\mathrm{E}} + } \right)}{EPSS_{\mathrm{k}}} .. math:: :label: eq:discvparfdry - l_{\rm{f \, k + 1}}^{\rm{P}} = \frac{\left({ - l_{\rm{f \, k}}^{\rm{P}} + - \varepsilon_{\rm{k} + 1/4} \, \Delta p_{\rm{k} + 1/4} \, l_{\rm{f \, - k}}^{\rm{E}} + - \varepsilon_{\rm{k} + 3/4} \, \Delta p_{\rm{k} + 3/4} \, - \left[{1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4}} - \right]\, - l_{\rm{f \, k + 1}}^{\rm{E}} - } \right)}{EPSS_{\rm{k}}} + l_{\mathrm{f \, k + 1}}^{\mathrm{P}} = \frac{\left({ + l_{\mathrm{f \, k}}^{\mathrm{P}} + + \varepsilon_{\mathrm{k} + 1/4} \, \Delta p_{\mathrm{k} + 1/4} \, + l_{\mathrm{f \, k}}^{\mathrm{E}} + + \varepsilon_{\mathrm{k} + 3/4} \, \Delta p_{\mathrm{k} + 3/4} \, + \left[{1 + \varepsilon_{\mathrm{k} + 1 / 4} \, \Delta p_{\mathrm{k} + 1 / + 4}} \right]\, + l_{\mathrm{f \, k + 1}}^{\mathrm{E}} + } \right)}{EPSS_{\mathrm{k}}} At the base of the convective plume (ie. the level immediately above -cloud base), :math:`l_{\rm{l \, k}}^{\rm{P}}` is initialized to -:math:`l_{\rm{l \, i}}^{\rm{P}}` and :math:`l_{\rm{f \, k}}^{\rm{P}}` to -:math:`l_{\rm{f \, i}}^{\rm{P}}`, where the initial values are chosen +cloud base), :math:`l_{\mathrm{l \, k}}^{\mathrm{P}}` is initialized to +:math:`l_{\mathrm{l \, i}}^{\mathrm{P}}` and :math:`l_{\mathrm{f \, +k}}^{\mathrm{P}}` to +:math:`l_{\mathrm{f \, i}}^{\mathrm{P}}`, where the initial values are chosen such that the modified form of :eq:`eq:chidisccb` produces zero fluxes at cloud base: .. math:: :label: eq:q4lcbi - Q4_{\rm{l}}(cb) = 0 = M_{\rm{cb+1/2}}^{\rm{P}} \, - \frac{\partial \, l_{\rm{l}}^{\rm{E}}}{\partial \, p} - - M_{\rm{cb}}^{\rm{P}}\, - \left({ l_{\rm{l}}^{\rm{P \, i}} - l_{\rm{l}}^{\rm{E}}(\rm{cb}) } \right) + Q4_{\mathrm{l}}(cb) = 0 = M_{\mathrm{cb+1/2}}^{\mathrm{P}} \, + \frac{\partial \, l_{\mathrm{l}}^{\mathrm{E}}}{\partial \, p} - + M_{\mathrm{cb}}^{\mathrm{P}}\, + \left({ l_{\mathrm{l}}^{\mathrm{P \, i}} - + l_{\mathrm{l}}^{\mathrm{E}}(\mathrm{cb}) } \right) .. math:: :label: eq:q4fcbi - Q4_{\rm{f}}(cb) = 0 = M_{\rm{cb+1/2}}^{\rm{P}} \, - \frac{\partial \, l_{\rm{f}}^{\rm{E}}}{\partial \, p} - - M_{\rm{cb}}^{\rm{P}}\, - \left({ l_{\rm{f}}^{\rm{P \, i}} - l_{\rm{f}}^{\rm{E}}(\rm{cb}) } \right) + Q4_{\mathrm{f}}(cb) = 0 = M_{\mathrm{cb+1/2}}^{\mathrm{P}} \, + \frac{\partial \, l_{\mathrm{f}}^{\mathrm{E}}}{\partial \, p} - + M_{\mathrm{cb}}^{\mathrm{P}}\, + \left({ l_{\mathrm{f}}^{\mathrm{P \, i}} - + l_{\mathrm{f}}^{\mathrm{E}}(\mathrm{cb}) } \right) As the convection scheme makes the single phase assumption for parcel condensate, it may be necessary to melt or freeze entrained condensate @@ -3795,25 +3817,26 @@ at this point and adjust the temperature accordingly. .. math:: :label: eqn:meltlf - \theta_{\rm{k + 1}}^{\rm{P}} = \theta_{\rm{k + 1}}^{\rm{P}} - - \left(\frac{L_{\rm{F}}}{C_{p} \, \Pi_{\rm{k + 1}}} \right)\, l_{\rm{f \, k + - 1}}^{\rm{P}} - \; \ldots \; \mbox{ if l_{\rm{f \, k + 1}}^{\rm{P}} is melted } + \theta_{\mathrm{k + 1}}^{\mathrm{P}} = \theta_{\mathrm{k + 1}}^{\mathrm{P}} - + \left(\frac{L_{\mathrm{F}}}{C_{p} \, \Pi_{\mathrm{k + 1}}} \right)\, + l_{\mathrm{f \, k + 1}}^{\mathrm{P}} + \; \ldots \; \mbox{ if l_{\mathrm{f \, k + 1}}^{\mathrm{P}} is melted } .. math:: :label: eqn:freezell - \theta_{\rm{k + 1}}^{\rm{P}} = \theta_{\rm{k + 1}}^{\rm{P}} + - \left(\frac{L_{\rm{F}}}{C_{p} \, \Pi_{\rm{k + 1}}} \right)\, l_{\rm{l \, k + - 1}}^{\rm{P}} - \; \ldots \; \mbox{ if l_{\rm{l \, k + 1}}^{\rm{P}} is frozen } + \theta_{\mathrm{k + 1}}^{\mathrm{P}} = \theta_{\mathrm{k + 1}}^{\mathrm{P}} + + \left(\frac{L_{\mathrm{F}}}{C_{p} \, \Pi_{\mathrm{k + 1}}} \right)\, + l_{\mathrm{l \, k + 1}}^{\mathrm{P}} + \; \ldots \; \mbox{ if l_{\mathrm{l \, k + 1}}^{\mathrm{P}} is frozen } Once a final value for the condensation term -:math:`{\overline{Q}}_{\rm{x} \, \rm{k} + 1} \, / \, M_{\rm{k} + 1}` has +:math:`{\overline{Q}}_{\mathrm{x} \, \mathrm{k} + 1} \, / \, M_{\mathrm{k} + +1}` has been calculated from the parcel specific humidity equations, it can then be added to the parcel condensate to give a final pre-precipitation value. -- In practice, the rates :math:`{\overline{Q}}_{\rm{x} \, \rm{k} + 1}` +- In practice, the rates :math:`{\overline{Q}}_{\mathrm{x} \, \mathrm{k} + 1}` and :math:`PPN` are not calculated explicitly in the code. Instead, their effect is applied directly as increments to the temperature and moisture fields. @@ -3822,13 +3845,13 @@ The precipitation calculation is unaltered. .. math:: :label: eq:precip - P_{\rm{k} + 1} = \left({ l_{\rm{k + 1}}^{\rm{P}} - l_{\rm{MIN}}^{\rm{P}} } - \right)\, - M_{\rm{k} + 1} \, / \, g + P_{\mathrm{k} + 1} = \left({ l_{\mathrm{k + 1}}^{\mathrm{P}} - + l_{\mathrm{MIN}}^{\mathrm{P}} } \right)\, + M_{\mathrm{k} + 1} \, / \, g -where :math:`l_{\rm{k + 1}}^{\rm{P}}` = -:math:`l_{\rm{l \, k + 1}}^{\rm{P}}` + -:math:`l_{\rm{f \, k + 1}}^{\rm{P}}`. +where :math:`l_{\mathrm{k + 1}}^{\mathrm{P}}` = +:math:`l_{\mathrm{l \, k + 1}}^{\mathrm{P}}` + +:math:`l_{\mathrm{f \, k + 1}}^{\mathrm{P}}`. - Actually, given that the precipitation calculation appears to be based upon the hydrostatic equation, it is debatable whether it is even @@ -3839,36 +3862,38 @@ This reduces the parcel condensate to : .. math:: :label: eq:vparlfinal - l_{\rm{l \, k + 1}}^{\rm{P}} = \left({ - \frac{l_{\rm{l \, k + 1}}^{\rm{P}}}{l_{\rm{k + 1}}^{\rm{P}}} - } \right)\, l_{\rm{MIN}}^{\rm{P}} + l_{\mathrm{l \, k + 1}}^{\mathrm{P}} = \left({ + \frac{l_{\mathrm{l \, k + 1}}^{\mathrm{P}}}{l_{\mathrm{k + 1}}^{\mathrm{P}}} + } \right)\, l_{\mathrm{MIN}}^{\mathrm{P}} .. math:: :label: eq:vparffinal - l_{\rm{f \, k + 1}}^{\rm{P}} = \left({ - \frac{l_{\rm{f \, k + 1}}^{\rm{P}}}{l_{\rm{k + 1}}^{\rm{P}}} - } \right)\, l_{\rm{MIN}}^{\rm{P}} + l_{\mathrm{f \, k + 1}}^{\mathrm{P}} = \left({ + \frac{l_{\mathrm{f \, k + 1}}^{\mathrm{P}}}{l_{\mathrm{k + 1}}^{\mathrm{P}}} + } \right)\, l_{\mathrm{MIN}}^{\mathrm{P}} The final parcel condensate values are then used in the rate calculation -based upon eqn :eq:`eq:basiclold`: +based upon eqn :eq:`eq:basiclold`: .. math:: :label: eq:q4lmassf - Q4_{\rm{l}}(k) = M_{\rm{k+1/2}}^{\rm{P}} \, \frac{\partial \, - l_{\rm{l}}^{\rm{E}}}{\partial \, p} + - \left({ {\mu}_{\rm{k}} \, M_{\rm{k}}^{\rm{P}} + - {\delta}_{\rm{k}} \, M_{\rm{k}}^{\rm{P}} } \right)\, - \left({ l_{\rm{l}}^{\rm{P}}(\rm{k}) - l_{\rm{l}}^{\rm{E}}(\rm{k}) } \right)- - {\overline{Q}}_{\rm{l, reset}} + Q4_{\mathrm{l}}(k) = M_{\mathrm{k+1/2}}^{\mathrm{P}} \, \frac{\partial \, + l_{\mathrm{l}}^{\mathrm{E}}}{\partial \, p} + + \left({ {\mu}_{\mathrm{k}} \, M_{\mathrm{k}}^{\mathrm{P}} + + {\delta}_{\mathrm{k}} \, M_{\mathrm{k}}^{\mathrm{P}} } \right)\, + \left({ l_{\mathrm{l}}^{\mathrm{P}}(\mathrm{k}) - + l_{\mathrm{l}}^{\mathrm{E}}(\mathrm{k}) } \right)- + {\overline{Q}}_{\mathrm{l, reset}} .. math:: :label: eq:q4fmassf - Q4_{\rm{f}}(k) = M_{\rm{k+1/2}}^{\rm{P}} \, \frac{\partial \, - l_{\rm{f}}^{\rm{E}}}{\partial \, p} + - \left({ {\mu}_{\rm{k}} \, M_{\rm{k}}^{\rm{P}} + - {\delta}_{\rm{k}} \, M_{\rm{k}}^{\rm{P}} } \right)\, - \left({ l_{\rm{f}}^{\rm{P}}(\rm{k}) - l_{\rm{f}}^{\rm{E}}(\rm{k}) } \right)- - {\overline{Q}}_{\rm{f, reset}} + Q4_{\mathrm{f}}(k) = M_{\mathrm{k+1/2}}^{\mathrm{P}} \, \frac{\partial \, + l_{\mathrm{f}}^{\mathrm{E}}}{\partial \, p} + + \left({ {\mu}_{\mathrm{k}} \, M_{\mathrm{k}}^{\mathrm{P}} + + {\delta}_{\mathrm{k}} \, M_{\mathrm{k}}^{\mathrm{P}} } \right)\, + \left({ l_{\mathrm{f}}^{\mathrm{P}}(\mathrm{k}) - + l_{\mathrm{f}}^{\mathrm{E}}(\mathrm{k}) } \right)- + {\overline{Q}}_{\mathrm{f, reset}} Note that, as a side-effect, the environment equations for potential temperature and specific humidity are also altered because the @@ -3878,56 +3903,60 @@ condensate is no longer re-evaporated at the end .. math:: - \frac{\Delta \, \theta_{\rm{k}}^{\rm{E}}}{\Delta \, t} = - \left(\frac{ M_{\rm{k}} }{ \Delta \, p_{\rm{k}} } \right) + \frac{\Delta \, \theta_{\mathrm{k}}^{\mathrm{E}}}{\Delta \, t} = + \left(\frac{ M_{\mathrm{k}} }{ \Delta \, p_{\mathrm{k}} } \right) \left[{ - \left({ 1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4} } - \right) - \left({ 1 - \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) - \left({ 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) - \left({ \theta_{\rm{k + 1}}^{\rm{E}} - \theta_{\rm{k}}^{\rm{E}} } \right) + \left({ 1 + \varepsilon_{\mathrm{k} + 1 / 4} \, \Delta p_{\mathrm{k} + 1 / + 4} } \right) + \left({ 1 - \delta_{\mathrm{k}} \, \Delta p_{\mathrm{k} + 1 / 2} } \right) + \left({ 1 - \mu_{\mathrm{k}} \, \Delta p_{\mathrm{k} + 1 / 2} } \right) + \left({ \theta_{\mathrm{k + 1}}^{\mathrm{E}} - + \theta_{\mathrm{k}}^{\mathrm{E}} } \right) } \right . + .. math:: - \left({ \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) - \left({ 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) - \left({ \theta_{\rm{k}}^{\rm{R}} - \theta_{\rm{k}}^{\rm{E}} } \right) + \left({ \delta_{\mathrm{k}} \, \Delta p_{\mathrm{k} + 1 / 2} } \right) + \left({ 1 - \mu_{\mathrm{k}} \, \Delta p_{\mathrm{k} + 1 / 2} } \right) + \left({ \theta_{\mathrm{k}}^{\mathrm{R}} - \theta_{\mathrm{k}}^{\mathrm{E}} + } \right) + .. math:: :label: eq:enviroth \left . { - \left({ \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) - \left({ \theta_{\rm{k}}^{\rm{P}} - \theta_{\rm{k}}^{\rm{E}} } \right) + \left({ \mu_{\mathrm{k}} \, \Delta p_{\mathrm{k} + 1 / 2} } \right) + \left({ \theta_{\mathrm{k}}^{\mathrm{P}} - \theta_{\mathrm{k}}^{\mathrm{E}} + } \right) } \right] { } and .. math:: - \frac{\Delta \, q_{\rm{k}}^{\rm{E}}}{\Delta \, t} = - \left(\frac{ M_{\rm{k}} }{ \Delta \, p_{\rm{k}} } \right) + \frac{\Delta \, q_{\mathrm{k}}^{\mathrm{E}}}{\Delta \, t} = + \left(\frac{ M_{\mathrm{k}} }{ \Delta \, p_{\mathrm{k}} } \right) \left[{ - \left({ 1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4} } + \left({ 1 + \varepsilon_{\mathrm{k} + 1 / 4} \, \Delta p_{\mathrm{k} + 1 / + 4} } \right) + \left({ 1 - \delta_{\mathrm{k}} \, \Delta p_{\mathrm{k} + 1 / 2} } \right) + \left({ 1 - \mu_{\mathrm{k}} \, \Delta p_{\mathrm{k} + 1 / 2} } \right) + \left({ q_{\mathrm{k + 1}}^{\mathrm{E}} - q_{\mathrm{k}}^{\mathrm{E}} } \right) - \left({ 1 - \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) - \left({ 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) - \left({ q_{\rm{k + 1}}^{\rm{E}} - q_{\rm{k}}^{\rm{E}} } \right) } \right . + .. math:: - \left({ \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) - \left({ 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) - \left({ q_{\rm{k}}^{\rm{R}} - q_{\rm{k}}^{\rm{E}} } \right) + \left({ \delta_{\mathrm{k}} \, \Delta p_{\mathrm{k} + 1 / 2} } \right) + \left({ 1 - \mu_{\mathrm{k}} \, \Delta p_{\mathrm{k} + 1 / 2} } \right) + \left({ q_{\mathrm{k}}^{\mathrm{R}} - q_{\mathrm{k}}^{\mathrm{E}} } \right) + .. math:: :label: eq:enviroq \left . { - \left({ \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) - \left({ q_{\rm{k}}^{\rm{P}} - q_{\rm{k}}^{\rm{E}} } \right) + \left({ \mu_{\mathrm{k}} \, \Delta p_{\mathrm{k} + 1 / 2} } \right) + \left({ q_{\mathrm{k}}^{\mathrm{P}} - q_{\mathrm{k}}^{\mathrm{E}} } \right) } \right] { } Similarly, eqns :eq:`eq:q4lmassf` and @@ -3935,56 +3964,62 @@ Similarly, eqns :eq:`eq:q4lmassf` and .. math:: - \frac{\Delta \, l_{\rm{l \, k}}^{\rm{E}}}{\Delta \, t} = - \left(\frac{ M_{\rm{k}} }{ \Delta \, p_{\rm{k}} } \right) + \frac{\Delta \, l_{\mathrm{l \, k}}^{\mathrm{E}}}{\Delta \, t} = + \left(\frac{ M_{\mathrm{k}} }{ \Delta \, p_{\mathrm{k}} } \right) \left[{ - \left({ 1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4} } - \right) - \left({ 1 - \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) - \left({ 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) - \left({ l_{\rm{l \, k + 1}}^{\rm{E}} - l_{\rm{l \, k}}^{\rm{E}} } \right) + \left({ 1 + \varepsilon_{\mathrm{k} + 1 / 4} \, \Delta p_{\mathrm{k} + 1 / + 4} } \right) + \left({ 1 - \delta_{\mathrm{k}} \, \Delta p_{\mathrm{k} + 1 / 2} } \right) + \left({ 1 - \mu_{\mathrm{k}} \, \Delta p_{\mathrm{k} + 1 / 2} } \right) + \left({ l_{\mathrm{l \, k + 1}}^{\mathrm{E}} - l_{\mathrm{l \, + k}}^{\mathrm{E}} } \right) } \right . + .. math:: - \left({ \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) - \left({ 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) - \left({ l_{\rm{l \, k}}^{\rm{P}} - l_{\rm{l \, k}}^{\rm{E}} } \right) + \left({ \delta_{\mathrm{k}} \, \Delta p_{\mathrm{k} + 1 / 2} } \right) + \left({ 1 - \mu_{\mathrm{k}} \, \Delta p_{\mathrm{k} + 1 / 2} } \right) + \left({ l_{\mathrm{l \, k}}^{\mathrm{P}} - l_{\mathrm{l \, k}}^{\mathrm{E}} + } \right) + .. math:: :label: eq:enviroll \left . { - \left({ \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) - \left({ l_{\rm{l \, k}}^{\rm{P}} - l_{\rm{l \, k}}^{\rm{E}} } \right) + \left({ \mu_{\mathrm{k}} \, \Delta p_{\mathrm{k} + 1 / 2} } \right) + \left({ l_{\mathrm{l \, k}}^{\mathrm{P}} - l_{\mathrm{l \, k}}^{\mathrm{E}} + } \right) } \right] { } and .. math:: - \frac{\Delta \, l_{\rm{f \, k}}^{\rm{E}}}{\Delta \, t} = - \left(\frac{ M_{\rm{k}} }{ \Delta \, p_{\rm{k}} } \right) + \frac{\Delta \, l_{\mathrm{f \, k}}^{\mathrm{E}}}{\Delta \, t} = + \left(\frac{ M_{\mathrm{k}} }{ \Delta \, p_{\mathrm{k}} } \right) \left[{ - \left({ 1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4} } - \right) - \left({ 1 - \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) - \left({ 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) - \left({ l_{\rm{f \, k + 1}}^{\rm{E}} - l_{\rm{f \, k}}^{\rm{E}} } \right) + \left({ 1 + \varepsilon_{\mathrm{k} + 1 / 4} \, \Delta p_{\mathrm{k} + 1 / + 4} } \right) + \left({ 1 - \delta_{\mathrm{k}} \, \Delta p_{\mathrm{k} + 1 / 2} } \right) + \left({ 1 - \mu_{\mathrm{k}} \, \Delta p_{\mathrm{k} + 1 / 2} } \right) + \left({ l_{\mathrm{f \, k + 1}}^{\mathrm{E}} - l_{\mathrm{f \, + k}}^{\mathrm{E}} } \right) } \right . + .. math:: - \left({ \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) - \left({ 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) - \left({ l_{\rm{f \, k}}^{\rm{P}} - l_{\rm{f \, k}}^{\rm{E}} } \right) + \left({ \delta_{\mathrm{k}} \, \Delta p_{\mathrm{k} + 1 / 2} } \right) + \left({ 1 - \mu_{\mathrm{k}} \, \Delta p_{\mathrm{k} + 1 / 2} } \right) + \left({ l_{\mathrm{f \, k}}^{\mathrm{P}} - l_{\mathrm{f \, k}}^{\mathrm{E}} + } \right) + .. math:: :label: eq:envirolf \left . { - \left({ \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) - \left({ l_{\rm{f \, k }}^{\rm{P}} - l_{\rm{f \, k}}^{\rm{E}} } \right) + \left({ \mu_{\mathrm{k}} \, \Delta p_{\mathrm{k} + 1 / 2} } \right) + \left({ l_{\mathrm{f \, k }}^{\mathrm{P}} - l_{\mathrm{f \, k}}^{\mathrm{E}} + } \right) } \right] { } .. _sec_conv_homog: @@ -4041,7 +4076,7 @@ Now we recognise that \Delta \overline{q} = Q2~ \Delta t where :math:`Q2` is the rate of moistening of the whole gridbox due to -convection. Remember that, at this stage, we haven’t done any +convection. Remember that, at this stage, we haven't done any condensation outside of the plume. Hence to calculate the condensation we should apply the background change in :math:`\overline{q}` as a uniform forcing for the background air. Hence @@ -4177,7 +4212,7 @@ Homogeneous forcing of the environment by convective-subsidence pressure change ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ To this end, the code includes an option to perform the homogeneous -forcing of liquid cloud by convection using the “pressure forcing” from +forcing of liquid cloud by convection using the "pressure forcing" from the convective subsidence, consistent with the pressure forcing by large-scale advection (see sections :ref:`Advection ` and :ref:`Response to pressure changes `). This approach replaces the @@ -4378,13 +4413,13 @@ The convection scheme itself is highly sensitive to the input environment temperature and moisture profiles *before* the convection increments (or PC2 response) are calculated. In particular, the parcel buoyancy (and hence the CAPE and mass-flux scaling) maybe radically -different depending on whether a “large-scale” condensation / +different depending on whether a "large-scale" condensation / evaporation adjustment is performed before the convection call. Where there is large-scale ascent, the profiles after Semi-Lagrangian advection may have become supersaturated and unrealistically unstable, until the expected condensation adjustment is performed. If the -convection scheme “sees” these unrealistic intermediate profiles, it is +convection scheme "sees" these unrealistic intermediate profiles, it is likely to predict an excessive, unrealistic mass-flux. To address this problem, there are two namelist switches that enable @@ -4546,11 +4581,11 @@ There are currently 3 options for the conditions under-which initiation may occur. For all of these options, if using the bimodal cloud scheme to do initiation within PC2, then the tests on :math:`RH_T` relative to :math:`RH_{crit}` are replaced by equivalent tests for whether the -saturation boundary lies within the bounds of the bimodal scheme’s +saturation boundary lies within the bounds of the bimodal scheme's assumed PDF, as described in section :ref:`Initiation using the bimodal scheme `. -“Original” initiation logic +"Original" initiation logic ^^^^^^^^^^^^^^^^^^^^^^^^^^^ This option is selected by setting the UM namelist switch @@ -4589,7 +4624,7 @@ Equivalently, :math:`C_l` is initiated away from 1 if - :math:`RH_T^{[n+1]} < RH_T^{[n]}` . -“Simplified” initiation logic +"Simplified" initiation logic ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ This option is selected by setting the UM namelist switch @@ -4620,10 +4655,10 @@ standard value is 0.005. Note this threshold is also used to remove small cloud-fractions after initiation; see section :ref:`Additional checks after PC2 initiation `. -This is very similar to the “Original” initiation logic described above, +This is very similar to the "Original" initiation logic described above, but with the following differences: -- The condition that the boundary-layer hasn’t diagnosed cumulus +- The condition that the boundary-layer hasn't diagnosed cumulus convection in the column is removed. Note that this condition spuriously suppressed initiation in the free troposphere *above* any cumulus cloud produced by the convection scheme. @@ -4636,7 +4671,7 @@ but with the following differences: .. _sec_smooth_initiation: -“Smooth” initiation logic +"Smooth" initiation logic ^^^^^^^^^^^^^^^^^^^^^^^^^ This option is selected by setting the UM namelist switch @@ -4673,7 +4708,7 @@ cloud scheme to be called (provided it is expected to predict nonzero cloud water, i.e. :math:`RH_T > RH_{crit}` in the case of the Smith scheme). The :math:`q_{cl}` predicted by the diagnostic cloud scheme is then taken as a minimum limit applied to the prognostic :math:`q_{cl}`. -This amounts to taking the diagnostic cloud scheme’s assumed PDF as a +This amounts to taking the diagnostic cloud scheme's assumed PDF as a minimum allowed width to the actual prognostic moisture PDF. The prognostic :math:`C_l` and :math:`q_{cl}` are incremented as follows: @@ -4779,7 +4814,7 @@ equal to, 1). The code will reset these clouds to either a fraction of 0 or 1, as appropriate. We choose to apply these terms here and not in the Bounds Checking part of the code (section :ref:`Bounds checking `) because these are not required to obtain consistency between fields, but -are ‘tidying up’ pieces of code, although they may reasonably also be +are 'tidying up' pieces of code, although they may reasonably also be applied in the Bounds Checking. Care needs to be taken when choosing the thresholds, since we do not wish to reset small values that are genuinely created by a physics scheme in the model. @@ -5047,7 +5082,7 @@ the microphysics scheme to be maintained. C_i \leftarrow \frac { \overline{q_{cf}} }{q_{cf0}} -where the ‘in-cloud’ ice content +where the 'in-cloud' ice content :math:`q_{cf0} = 1 \times 10^{-4} kg kg^{-1}`. .. _section-7: @@ -5110,8 +5145,8 @@ During trialling prior to operational implementation, it was found that relying on Q-Pos to deal with negative condensate values was very expensive, as the Q-Pos routine does a lot of communications between different processors. It may be preferable to deal with the cause of -negative condensate amounts at their source. The option to “Ensure -consistent sinks of qcl and CFL” prevents the QCL increment from trying +negative condensate amounts at their source. The option to "Ensure +consistent sinks of qcl and CFL" prevents the QCL increment from trying to remove too much liquid condensate and hence reduces the models reliance on Q-Pos to deal with the inconsistencies. @@ -5218,7 +5253,7 @@ Two area cloud fraction parametrizations are available for use with PC2. The area cloud fraction of Cusack (documented in ) has been adapted by `Boutle and Morcrette (2010)`_ so it can be used with PC2 (and -is available from the UMUI as the “Cusack” option from version 7.6 +is available from the UMUI as the "Cusack" option from version 7.6 onwards). This method aims to reproduce some of the detail of the thermodynamic profile lost due to the coarseness of the grid. The interpolation/extrapolation technique is used prior to PC2 initiation @@ -5238,7 +5273,7 @@ performed at the end of the timestep. Code Structure -------------- -A detailed description of the UM’s timestep structure, showing where in +A detailed description of the UM's timestep structure, showing where in the model all the PC2 cloud scheme subroutine calls are made, is given in the subsections below. @@ -5246,7 +5281,7 @@ Note that there are three different subroutines that all do the PC2 homogeneous forcing, with slightly different details: - ***pc2_delta_hom_turb*** outputs increments due to the condensation or - evaporation, but doesn’t update the fields themselves. + evaporation, but doesn't update the fields themselves. - ***pc2_homog_plus_turb*** just updates the fields that are passed in, instead of outputting separate increment arrays. @@ -5318,7 +5353,7 @@ Main Tree from atm_step_4a .. container:: tcolorbox | **atmos_physics1** - | \* (calls explicit “slow” physics routines...) + | \* (calls explicit "slow" physics routines...) .. container:: itemize @@ -5418,7 +5453,7 @@ Main Tree from atm_step_4a .. container:: tcolorbox | **atmos_physics2** - | \* (calls “fast” physics routines...) + | \* (calls "fast" physics routines...) .. container:: itemize @@ -5511,7 +5546,7 @@ Main Tree from atm_step_4a turbulent fluxes) - | pc2_bl_forced_cu - | \* (adds diagnosed “forced cumulus” cloud fraction + | \* (adds diagnosed "forced cumulus" cloud fraction and water content onto the PC2 prognostics) - Calculate area cloud fraction: @@ -5884,7 +5919,7 @@ vertically-advected parcels, and so calculate the PC2 homogeneous forcing response in the same way as we do for Semi-Lagrangian advection in the full model (see section :ref:`Response to pressure changes `). For the former, we -don’t know if the prescribed T,q tendencies are due to advection, +don't know if the prescribed T,q tendencies are due to advection, radiation, or some other process, so we calculate the PC2 homogeneous forcing response as if the tendencies are applied "in-situ". @@ -5922,14 +5957,14 @@ Parameter values :numref:`Table %s ` summarizes the values of parameters used in the PC2 scheme and their location within various comdecks. Those -parameters marked as ‘Num’ are those that are not part of the +parameters marked as 'Num' are those that are not part of the mathematical equation set that is being solved, but are required in order to achieve a stable, realistic, numerical solution. These include, for instance, thresholds for resetting cloud fractions back to 0 or 1. -Those marked ’Phy’ are physical quantities that form an integral part of -the equation set that we wish to solve. Those marked ’Clo’ form part of +Those marked 'Phy' are physical quantities that form an integral part of +the equation set that we wish to solve. Those marked 'Clo' form part of a closure needed to form the equation set, but are less readily related -to physical quantities. Variables marked ’Diag’ form a part of the +to physical quantities. Variables marked 'Diag' form a part of the diagnostic output routines. .. list-table:: PC2 parameter values and locations @@ -6155,10 +6190,10 @@ the options in the UMUI which need to be selected in order to run PC2. No hand-edits are required. - In the LS cloud panel (atmos-science-section-LScloud) push the button - marked ’use the PC2 cloud scheme’. + marked 'use the PC2 cloud scheme'. -- If you wish to use PC2 in the diagnostic only mode, also push ’run the - PC2 scheme in diagnostic only mode’. If you wish to run PC2 fully then +- If you wish to use PC2 in the diagnostic only mode, also push 'run the + PC2 scheme in diagnostic only mode'. If you wish to run PC2 fully then do not push this button - In the large-scale precipitation section @@ -6307,7 +6342,7 @@ Therefore, from :eq:`eq:qclbar=int` .. math:: \overline{q_{cl \, max}} = \int_{s=-b_s}^{\infty} G(s) (b_s + s) ds . -We will use the current value of :math:`Q_c` (which won’t in general to +We will use the current value of :math:`Q_c` (which won't in general to be equal to :math:`b_s`) to split the integral into two ranges of s: .. math:: @@ -6433,7 +6468,7 @@ We have (equivalent to B.6 from `Wilson and Gregory (2003)`_) .. math:: \frac{ (1-C_l)^2 }{SD} = G(-Q_c) \frac{n+2}{n+1}. -If n=0 (i.e. a ‘top-hat’ function) then +If n=0 (i.e. a 'top-hat' function) then :math:`G(-Q_c) = \frac{1}{2 b_s}` and we can write .. math:: C_l = 1 - \sqrt{ \frac{SD}{b_s} } . @@ -6529,7 +6564,7 @@ mean is unchanged when a process acts. (The mean will change of course, but we assume that the variations in each part of the gridbox from the mean do not). Since this is equivalent to every part of the gridbox receiving the same :math:`q_T` and :math:`T_L` increment, we call this -‘Homogeneous Forcing’. We have provided a subroutine +'Homogeneous Forcing'. We have provided a subroutine *pc2-homog-plus-turb*, in deck *pc2-homo* in order to provide the necessary updates. @@ -6544,7 +6579,7 @@ that we already know a condensate increment :math:`q_{cl}` or The injection forcing assumes that new cloud randomly displaces existing cloud in a gridbox, and is designed with detrainment from deep convection in mind, although it is also used elsewhere. It will require -as an input an estimate of the ‘in-cloud’ water content of the new cloud +as an input an estimate of the 'in-cloud' water content of the new cloud that is produced. If you consider that both the homogeneous and injection forcing @@ -6570,7 +6605,7 @@ not alter values of :math:`T` and :math:`q` after the homogeneous forcing section is called* and that the physical interpretation of your :math:`q` and :math:`T` increments does not change. You need to be careful if you are moving code from one subroutine to another that you -don’t inadvertently do this, although the forcing usually sits at the +don't inadvertently do this, although the forcing usually sits at the end of the control subroutine. If your scheme is currently using the injection forcing *subroutine*, @@ -6851,7 +6886,7 @@ We note that in shallow convection at 30 minutes timestep the erosion term is trying to remove most of the cloud that the convective detrainment places into the model. Since the erosion is limited by the amount of cloud fraction and condensate present, what ends up happening -is that the ‘equilibrium’ that is achieved is actually one where the +is that the 'equilibrium' that is achieved is actually one where the cloud fraction and condensate at the end of the timestep are simply the values that were detrained by the convection scheme (and hence depend on the timestep). The CRM suggests a cycling time of around 15 minutes for @@ -6861,7 +6896,7 @@ timestep of 15 minutes upwards. We might just about get away with the 30 minute step of the climate model, but it is not a good situation to try to model. This is demonstrating the difficulty of modelling shallow convective cloud by a prognostic scheme, where the physical lifetime of -the clouds is of order the timestep - ideally we wouldn’t want to try to +the clouds is of order the timestep - ideally we wouldn't want to try to model anything prognostically when the cycling time is less than the timestep. @@ -7123,7 +7158,7 @@ References Boutle, I. A. and C. J. Morcrette (2010). *Parametrization of area cloud fraction*. - Atmos. Sci. Let., 11, 283–289. + Atmos. Sci. Let., 11, 283-289. .. _Field et al. (2005): diff --git a/documentation/source/science_guide/cloud_schemes/manual_corrections.txt b/documentation/source/science_guide/cloud_schemes/manual_corrections.txt new file mode 100644 index 0000000000..0b0ebef109 --- /dev/null +++ b/documentation/source/science_guide/cloud_schemes/manual_corrections.txt @@ -0,0 +1,62 @@ +diff --git a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst +index 9f1988eb..d0664430 100644 +--- a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst ++++ b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst +@@ -4679,26 +4679,29 @@ prognostic :math:`C_l` and :math:`q_{cl}` are incremented as follows: + + - If :math:`{q_{cl}}_{diag} > q_{cl}`: + +- :math:`\Delta q_{cl} = {q_{cl}}_{diag} - q_{cl} \quad +- \refstepcounter{equation}(\theequation)\label{eq:dqcl_init}` ++ .. math:: :label: eq:dqcl_init ++ ++ \Delta q_{cl} = {q_{cl}}_{diag} - q_{cl} + + - If :math:`Q_C < 0`: + +- :math:`\Delta C_{l} = \frac{\Delta q_{cl}}{{q_{cl}}_{diag}} \left( +- {C_{l}}_{diag} - C_{l} \right) \quad +- \refstepcounter{equation}(\theequation)\label{eq:dcl_init1}` ++ .. math:: :label: eq:dcl_init1 ++ ++ \Delta C_{l} = \frac{\Delta q_{cl}}{{q_{cl}}_{diag}} ++ \left( {C_{l}}_{diag} - C_{l} \right) + + - If :math:`Q_C > 0`: + +- :math:`\Delta C_{l} = \frac{\Delta SD}{{SD}_{diag}} \left( {C_{l}}_{diag} - +- C_{l} \right) \quad +- \refstepcounter{equation}(\theequation)\label{eq:dcl_init2}` ++ .. math:: :label: eq:dcl_init2 ++ ++ \Delta C_{l} = \frac{\Delta SD}{{SD}_{diag}} ++ \left( {C_{l}}_{diag} - C_{l} \right) + + - Otherwise: + +- :math:`\Delta q_{cl} = 0` ++ .. math:: \Delta q_{cl} = 0 + +- :math:`\Delta C_{l} = 0` ++ .. math:: \Delta C_{l} = 0 + + where the subscript :math:`_{diag}` denotes the liquid cloud water + content and fraction predicted by the diagnostic cloud scheme (either +@@ -5931,7 +5934,7 @@ diagnostic output routines. + + .. list-table:: PC2 parameter values and locations + :name: tab:pc2_names +- ++ :header-rows: 1 + + * - Symbol + - Code variable +@@ -6098,7 +6101,7 @@ PC2:64 and a non-PC2 run. + .. list-table:: PC2 parameter values and locations relating to the convection. + \*These values are those used in HadGAM + :name: tab:pc2_conv_names +- ++ :header-rows: 1 + + * - Code variable + - Des cription From 768d6c04b8195abb1e6a2d13eb332e90eef68461 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Wed, 13 May 2026 16:40:30 +0100 Subject: [PATCH 082/116] Added file containing manual corrections diff for ease of re-applying. --- .../turbulence_schemes/manual_corrections.txt | 12 ++++++++++++ 1 file changed, 12 insertions(+) create mode 100644 documentation/source/science_guide/turbulence_schemes/manual_corrections.txt diff --git a/documentation/source/science_guide/turbulence_schemes/manual_corrections.txt b/documentation/source/science_guide/turbulence_schemes/manual_corrections.txt new file mode 100644 index 0000000000..b0ade73906 --- /dev/null +++ b/documentation/source/science_guide/turbulence_schemes/manual_corrections.txt @@ -0,0 +1,12 @@ +diff --git a/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst b/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst +index 5b373b68..8ff32c06 100644 +--- a/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst ++++ b/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst +@@ -1810,6 +1810,7 @@ Discussion of some of the revisions + + .. list-table:: Convective and Neutral limits for velocity scales + :name: tab:vscales ++ :header-rows: 2 + + + * - Formulation From a365ab504e4722e1ad30e30e2234842fd05b8332 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Wed, 13 May 2026 17:22:57 +0100 Subject: [PATCH 083/116] Auto-number footnotes. --- .../turbulence_schemes/bl_scheme_doc.rst | 13 +++++++------ 1 file changed, 7 insertions(+), 6 deletions(-) diff --git a/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst b/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst index 3b72b93559..eb204634e7 100644 --- a/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst +++ b/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst @@ -239,7 +239,7 @@ been categorised into 7 distinct 'boundary layer types': :ref:`The diagnostic parcel ascent and cumulus diagnosis `) - **Type VI**: Cumulus-capped boundary layer -- no turbulent - diffusivities are allowed [1]_ at or above the LCL as the mass-flux + diffusivities are allowed [#fnote1]_ at or above the LCL as the mass-flux convection scheme operates here (cumulus diagnosis described in section :ref:`The diagnostic parcel ascent and cumulus diagnosis `) @@ -285,7 +285,8 @@ top at height :math:`z_{\mathrm{h}}` , as required for available. #. a diagnosis of cumulus-capped layers (if cumulus-capped then NTML and - :math:`z_{\mathrm{h}}` are set to the LCL [2]_, if not then to the parcel + :math:`z_{\mathrm{h}}` are set to the LCL [#fnote2]_, if not then to the + parcel top) Note that this process is only performed for unstable boundary layers @@ -5132,7 +5133,7 @@ often non-constant and depends on :math:`X` (i.e. the PDE is non-linear) and :math:`S` is a forcing term from other processes preceding the boundary layer. In the UM these processes are: microphysics, gravity wave drag, radiation, dynamics and optionally (using the switch -i_impsolve_loc) convection [3]_. :math:`S` represents the total tendency +i_impsolve_loc) convection [#fnote3]_. :math:`S` represents the total tendency from these processes. Equations :eq:`eq:sppf1`, :eq:`eq:sppf2` applied to :eq:`eq:vdiff1` becomes @@ -7240,15 +7241,15 @@ Appendix: Notation - buoyancy parameters, defined in appendix :ref:`Appendix: Derivation and definitions of the buoyancy parameters ` -.. [1] +.. [#fnote1] unless the option to mix across the LCL is selected, see section :ref:`Diagnosis of the LCL transition zone thickness ` -.. [2] +.. [#fnote2] unless the option to mix across the LCL is selected, see section :ref:`Diagnosis of the LCL transition zone thickness ` -.. [3] +.. [#fnote3] If i_impsolve_loc = 1, the boundary-layer implicit solver is performed before the convection call so that :math:`S` excludes the convection increments. If i_impsolve_loc = 2, it is performed after From 2f479f5671c5381c89e7bad855c046f1e3973973 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Wed, 13 May 2026 22:48:28 +0100 Subject: [PATCH 084/116] Reinstated affiliation footnote in author list (lost by pandoc). --- .../science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst | 6 +++++- 1 file changed, 5 insertions(+), 1 deletion(-) diff --git a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst index 711c7ae3cd..64469a3296 100644 --- a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst +++ b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst @@ -14,9 +14,13 @@ The PC2 Cloud Scheme ==================== -:Author: D. Wilson, A. Bushell, C. Morcrette, V. Varma\ :math:`^{1}`, +:Author: D. Wilson, A. Bushell, C. Morcrette, V. Varma\ [#affil1]_, M. Whitall + +.. [#affil1] National Institute of Water and Atmospheric Research, Wellington, + New Zealand + .. role:: raw-latex(raw) :format: latex .. From 8bc129be2c196c7d0618c61bfe8d4ed79fbabe1d Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Thu, 14 May 2026 10:04:02 +0100 Subject: [PATCH 085/116] Reinstated placeholders for inter-doc links (using existing custom macro \citeumdp instances in the latex source). --- .../cloud_schemes/UMDP30_PC2CloudScheme.rst | 98 ++++++++++--------- .../cloud_schemes/UMDP30_PC2CloudScheme.tex | 2 + 2 files changed, 54 insertions(+), 46 deletions(-) diff --git a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst index 64469a3296..558d1c349c 100644 --- a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst +++ b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst @@ -93,7 +93,7 @@ valid structures to use in this respect.* - One may diagnose cloud fractions and condensate contents from knowledge of gridbox mean variables. This forms the basis of the - `Smith (1990)`_ scheme, which is described in . + `Smith (1990)`_ scheme, which is described in :umdp:'029'. - A mixed scheme, such as `Sundqvist (1978)`_ uses a prediction of condensate contents, but a diagnostic cloud fraction. @@ -310,8 +310,8 @@ three-dimensional distribution in terms of three separate variables :math:`q_T`, :math:`T_L` and :math:`p`. This is the method used by `Smith (1990)`_, where a symmetric triangular distribution function is used. For further information on the -`Smith (1990)`_ scheme, please refer to . Physics and -dynamics schemes hence only need to provide increments to +`Smith (1990)`_ scheme, please refer to :umdp:'029'. Physics +and dynamics schemes hence only need to provide increments to :math:`\overline{q_T}` and :math:`\overline{T_L}`, provided that a diagnostic scheme (such as `Smith (1990)`_) is called at some point in the timestep to partition :math:`\overline{q_T}` into @@ -768,8 +768,8 @@ choose the dry-bulb temperature :math:`\overline{T}`, and not the liquid-temperature (:math:`\overline{T_L}`), since :math:`q_{sat}` locally is defined by the local dry-bulb temperature (:math:`T`) and we need to consider *changes* in the condensate. This has been confirmed -using simulations using a single gridbox model. contains a longer -discussion of this issue, but we note here that the +using simulations using a single gridbox model. :umdp:'029' contains a +longer discussion of this issue, but we note here that the `Smith (1990)`_ scheme performs best when it does not use :math:`T` to calculate :math:`\alpha` but the gradient of the chord between :math:`(\overline{T_L}, q_{sat}(\overline{T_L}))` and @@ -1011,8 +1011,8 @@ discussion of an equivalent width in the deposition / sublimation relationship for ice cloud). We still choose to define :math:`b_s` in terms of a critical relative humidity parameter, :math:`RH_{crit}`. Like the -`Smith (1990)`_ scheme (see ), we define the value of -:math:`b_s` as +`Smith (1990)`_ scheme (see :umdp:'029'), we define the +value of :math:`b_s` as .. math:: :label: eq:bs @@ -1097,8 +1097,9 @@ value :math:`(2-RH_t)` (which is equivalent to the replacing of respectively. We then solve for the initiated cloud fraction :math:`C_l'`, using the -similar methods as described in , except that we allow the solution to -vary with the PDF shape :math:`n`. We first write :math:`Q_N` as +similar methods as described in :umdp:'029', except that we allow the +solution to vary with the PDF shape :math:`n`. We first write +:math:`Q_N` as .. math:: :label: eq:qn_def @@ -2152,7 +2153,8 @@ Large-scale precipitation Precipitation processes have a large effect on cloud fractions. Here we present the simple physical models that are applied to the transfer terms included in the large-scale precipitation scheme. They are also -presented within the large-scale precipitation documentation (). +presented within the large-scale precipitation documentation +(:umdp:'026'). The basis of the physical model is that microphysical transfer processes can be calculated separately in different partitions of the model cloud @@ -2162,10 +2164,10 @@ consider here separately each process that is modelled in the large-scale precipitation scheme. The changes in :math:`\overline{q_{cl}}`, :math:`\overline{q_{cf}}` and :math:`\overline{q}` remain mathematically the same as in the non-PC2 -version of the code (), we only need to introduce calculations for the -changes in cloud fractions. We will see that many of these changes can -be well modelled by assuming no change to the cloud fractions, and the -others by using simple assumptions. +version of the code (:umdp:'026'), we only need to introduce +calculations for the changes in cloud fractions. We will see that many +of these changes can be well modelled by assuming no change to the cloud +fractions, and the others by using simple assumptions. Although the model may use two ice prognostic ice categories, only a single ice cloud fraction is stored, the assumption being that the two @@ -2297,17 +2299,19 @@ framework (section :ref:`The 's' distribution `). However, since the same instantaneous condensation framework. It would be useful to investigate in the future whether the two descriptions of the moisture variability could be brought together. Because of its importance, we -describe the method below, although we note it is also described in . +describe the method below, although we note it is also described in +:umdp:'026'. We can calculate the local rate of change of :math:`q_{cf}`, given local :math:`T` and :math:`q` etc. using the standard microphysical growth -equations (see ). However, it is critical to know the way in which the -moisture is correlated with the ice in the gridbox. We will assume there -exists a distribution of vapour in the gridbox. We know that the regions -where liquid cloud exists must be saturated with respect to liquid -water, hence we need only consider the part of the gridbox that does not -have liquid water present. The average value, :math:`q_a`, of :math:`q` -within the liquid-free part of the gridbox is thus +equations (see :umdp:'026'). However, it is critical to know the way in +which the moisture is correlated with the ice in the gridbox. We will +assume there exists a distribution of vapour in the gridbox. We know +that the regions where liquid cloud exists must be saturated with +respect to liquid water, hence we need only consider the part of the +gridbox that does not have liquid water present. The average value, +:math:`q_a`, of :math:`q` within the liquid-free part of the gridbox is +thus .. math:: :label: eq:qa @@ -2556,7 +2560,7 @@ use the values at the start of the microphysics (this includes the values of :math:`C_i` used in the calculation of 'in-cloud' water contents above. However, we do update the cloud fractions themselves sequentially. We also recalculate after each process the overlaps -between the rain fraction (see ) and the cloud fractions. +between the rain fraction (see :umdp:'026') and the cloud fractions. There is also a final set of checks that :math:`C_l` and :math:`C_i` lie between 0 and 1 and that :math:`C_t` is bounded between @@ -3535,8 +3539,8 @@ discretized form of :eq:`eq:chimassflux`, setting where the initial parcel value :math:`{\chi}_{\mathrm{i,cb}}^{\mathrm{P}}` may be chosen to produce a fixed increment or place a closure condition on -the cloud base flux. In fact, the convection equations (see ) differ -from :eq:`eq:chidisck` and +the cloud base flux. In fact, the convection equations (see :umdp:'027') +differ from :eq:`eq:chidisck` and :eq:`eq:chidisccb` because a different discretization is used, but the principle is unaltered. @@ -3602,8 +3606,8 @@ gradient equations based upon :eq:`eq:gradchipar` l_{\mathrm{ }}^{\mathrm{E}} } \right) - {\overline{Q}}_{\mathrm{par}} + PPN -The final calculation of rates in the current condensation scheme (, -section 10) assumes a further condensation term, +The final calculation of rates in the current condensation scheme +(:umdp:'027', section 10) assumes a further condensation term, :math:`{\overline{Q}}_{\mathrm{reset}}`, which acts to make the net rate of change of condensate equal zero, and a final assumption is made that the environment values of condensate remain zero (and also that @@ -3709,7 +3713,8 @@ condensate is calculated as \frac{{\overline{Q}}_{\mathrm{f, par}}}{M^{\mathrm{P}}} - \frac{SNOW}{M^{\mathrm{P}}} -Following , equations :eq:`eq:dbydpmassflux`, +Following :umdp:'027', equations +:eq:`eq:dbydpmassflux`, :eq:`eq:vertparl` and :eq:`eq:vertparf` are discretized: @@ -3899,8 +3904,8 @@ based upon eqn :eq:`eq:basiclold`: l_{\mathrm{f}}^{\mathrm{E}}(\mathrm{k}) } \right)- {\overline{Q}}_{\mathrm{f, reset}} -Note that, as a side-effect, the environment equations for potential -temperature and specific humidity are also altered because the +Note that, as a side-effect, the :umdp:'027' environment equations for +potential temperature and specific humidity are also altered because the condensate is no longer re-evaporated at the end (:math:`{\overline{Q}}_{\rm{l, reset}} = 0 = {\overline{Q}}_{\rm{f, reset}}`): @@ -4395,18 +4400,19 @@ A prognostic dust approach is implemented in the micro-physics scheme under large-sale-precipitation where by the heterogeneous nucleation temperature can be defined to vary three dimensionally globally as an arc-tangent function of the mineral dust distribution in the model -(documented in ). By default, both liquid and ice are detrained -simultaneously at the same height, and the fraction of condensate that -is ice linearly ramps as a function of temperature. i.e. condensate is -assumed to be all-liquid when T is greater than one tuneable threshold; -all-ice when T is less than another tuneable threshold, and vary -linearly in-between (the threshold values are given by starticeTkelvin -and alliceTdegC in the UM cloud-scheme namelist. The new heterogeneous -nucleation temperatures calculated in the large-scale-precipitation are -passed to the convection scheme and are used as the above detrainment -temperature thresholds by maintaining a similar linear ramp. For e.g., -condensate is assumed to be all-liquid for T :math:`\geq` :math:`tnuc_n` -and all-ice for T :math:`\leq` :math:`tnuc_n` - 10.0 +(documented in :umdp:'026'). By default, both liquid and ice are +detrained simultaneously at the same height, and the fraction of +condensate that is ice linearly ramps as a function of temperature. i.e. +condensate is assumed to be all-liquid when T is greater than one +tuneable threshold; all-ice when T is less than another tuneable +threshold, and vary linearly in-between (the threshold values are given +by starticeTkelvin and alliceTdegC in the UM cloud-scheme namelist. The +new heterogeneous nucleation temperatures calculated in the +large-scale-precipitation are passed to the convection scheme and are +used as the above detrainment temperature thresholds by maintaining a +similar linear ramp. For e.g., condensate is assumed to be all-liquid +for T :math:`\geq` :math:`tnuc_n` and all-ice for T :math:`\leq` +:math:`tnuc_n` - 10.0 .. _sec_conv_input_profs: @@ -5255,10 +5261,10 @@ Area cloud fraction Two area cloud fraction parametrizations are available for use with PC2. -The area cloud fraction of Cusack (documented in ) has been adapted by -`Boutle and Morcrette (2010)`_ so it can be used with PC2 (and -is available from the UMUI as the "Cusack" option from version 7.6 -onwards). This method aims to reproduce some of the detail of the +The area cloud fraction of Cusack (documented in :umdp:'029') has been +adapted by `Boutle and Morcrette (2010)`_ so it can be used with +PC2 (and is available from the UMUI as the "Cusack" option from version +7.6 onwards). This method aims to reproduce some of the detail of the thermodynamic profile lost due to the coarseness of the grid. The interpolation/extrapolation technique is used prior to PC2 initiation (which is then called with three times as many levels) and it is used, diff --git a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.tex b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.tex index f04d3aa29a..42efc42022 100644 --- a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.tex +++ b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.tex @@ -18,6 +18,8 @@ % Packages needed for the UM subroutine tree diagram in um_call_tree.txt: \input{um_call_tree_preamble} +\newcommand{\citeumdp}[1]{:umdp:`#1`} + \newcommand{\mmax}[1] {\mbox{\footnotesize \sf MAX} \left[#1\right] } %%% These are definitions used in the convection documentation From f8720dc0ade80704d7e965217925e79ad187aeb5 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Thu, 14 May 2026 10:48:06 +0100 Subject: [PATCH 086/116] Added placeholders for inter-doc cross-referencing (using existing instances of \citeumdp custom macro in the latex source. --- .../turbulence_schemes/bl_scheme_doc.rst | 54 ++++++++++--------- .../turbulence_schemes/bl_scheme_doc.tex | 2 + 2 files changed, 30 insertions(+), 26 deletions(-) diff --git a/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst b/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst index eb204634e7..033b5344d1 100644 --- a/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst +++ b/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst @@ -42,7 +42,7 @@ the UM. Several options for higher order closures are available in the 1A version of the UM boundary layer scheme and these are documented -separately in . +separately in :umdp:'025'. .. _sec_closure: @@ -1942,16 +1942,16 @@ The blended scheme ================== For high resolution simulations, the UM has a Smagorinsky-type subgrid -turbulence scheme, described in . However, this scheme is only truly -applicable for horizontal grid-lengths of order :math:`10` m, and any -real-world simulation run at lower resolution than this will inevitably -have unresolved scales somewhere in the domain. Rather than force the -user to make an ad-hoc decision about the scales they are interested in, -and thus grid-length at which to switch from using the boundary-layer -parametrization (1D BL) to the subgrid turbulence scheme (3D Smag), a -method for blending the two parametrizations has been developed. This -blend is regime and scale dependent, allowing a single parametrization -to be used across resolutions, including the completely +turbulence scheme, described in :umdp:'028'. However, this scheme is +only truly applicable for horizontal grid-lengths of order :math:`10` m, +and any real-world simulation run at lower resolution than this will +inevitably have unresolved scales somewhere in the domain. Rather than +force the user to make an ad-hoc decision about the scales they are +interested in, and thus grid-length at which to switch from using the +boundary-layer parametrization (1D BL) to the subgrid turbulence scheme +(3D Smag), a method for blending the two parametrizations has been +developed. This blend is regime and scale dependent, allowing a single +parametrization to be used across resolutions, including the completely unresolved/resolved extremes. This blending process is described in `Boutle et al. (2014)`_, which gives some examples of its use and comparison to simulations using either the 1D BL or 3D Smag schemes @@ -2183,14 +2183,15 @@ in :eq:`zturb_dsc`. For current operational convection-permitting model grid sizes (1.5 km in the UKV), the representation of cumulus convection remains a challenge. One option is to include a grey-zone convection -parametrization, described in the documentation of that scheme (see ). -Tests in the UKV, though, showed some detriment to the spin-up of -resolved scale convection (as well as somewhat poor discrimination of -precipitating versus non-precipitating parametrized convection) that led -to the development of an alternative strategy, namely to abandon the -blended turbulence scheme when pure cumulus convection was diagnosed and -leave the representation of cumulus entirely to the resolved scales. -This option (``blending_option``\ :math:`=`\ 2) is also now discouraged. +parametrization, described in the documentation of that scheme (see +:umdp:'027'). Tests in the UKV, though, showed some detriment to the +spin-up of resolved scale convection (as well as somewhat poor +discrimination of precipitating versus non-precipitating parametrized +convection) that led to the development of an alternative strategy, +namely to abandon the blended turbulence scheme when pure cumulus +convection was diagnosed and leave the representation of cumulus +entirely to the resolved scales. This option +(``blending_option``\ :math:`=`\ 2) is also now discouraged. .. _sec_entr: @@ -6132,12 +6133,13 @@ budget to give where :math:`C_{e}=A_{2N}^{3/2}`. Initial tests found that the MONC value of :math:`A_{2N}=0.23` gave rather large values of :math:`e_{loc}` and so :math:`C_e=0.41` is used. Note that this is an optional value -used in the higher order closure scheme (see section 2.8.5 of ). It -could also be worth testing the suggested parametrization in (2.300) -there, of :math:`C_e=0.19+0.74 \lambda/\Delta z` but this has not yet -been attempted. The total non-local TKE is computed by adding the TKE -from each non-local component, as is done for the diffusion -coefficients, i.e., +used in the higher order closure scheme (see section 2.8.5 of +:umdp:'025'). It could also be worth testing the suggested +parametrization in (2.300) there, of +:math:`C_e=0.19+0.74 \lambda/\Delta z` but this has not yet been +attempted. The total non-local TKE is computed by adding the TKE from +each non-local component, as is done for the diffusion coefficients, +i.e., .. math:: :label: tke_diag_nl @@ -6176,7 +6178,7 @@ also corrects a bug in the level indexing of this diagnostic when passed to UKCA. Note that additional diagnostics of the scalar variances are also made -and those are documented in . +and those are documented in :umdp:'029'. .. _app_neutwind: diff --git a/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.tex b/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.tex index f233c8da77..a2ae72a916 100644 --- a/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.tex +++ b/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.tex @@ -11,6 +11,8 @@ \usepackage{natbib} \usepackage{indentfirst} +\newcommand{\citeumdp}[1]{:umdp:`#1`} + \input{newcommand} \def\gtapp{\raisebox{-.4ex}{$\ \stackrel{>}{{\scriptstyle \sim}} \ $}} From e5fcb284bba2abb7987f812c42d31517d2c0a746 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Thu, 14 May 2026 10:58:07 +0100 Subject: [PATCH 087/116] Deleted stray \noindent in math block (not supported by mathjax). --- .../source/science_guide/turbulence_schemes/bl_scheme_doc.rst | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst b/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst index 033b5344d1..fc3d848ddc 100644 --- a/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst +++ b/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst @@ -5271,7 +5271,7 @@ where :math:`k=1,2,\ldots,L-2`, .. math:: - A_{k}=-{\mathcal{I}}_{1}\frac{\Delta t\noindent + A_{k}=-{\mathcal{I}}_{1}\frac{\Delta t K_{u}\Big|_{k+1}}{(z_{k+1}-z_{k})(z_{k+3/2}-z_{k+1/2})},\; C_{k}=-{\mathcal{I}}_{1}\frac{\Delta tK_{u}\Big|_{k}}{(z_{k+1}-z_{k})(z_{k+1/2}-z_{k-1/2})},\; From ed849bc8fdfc9013f3d6f119750339f337bb65f4 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Thu, 14 May 2026 13:02:10 +0100 Subject: [PATCH 088/116] Fixed spurious nested math. --- .../turbulence_schemes/bl_scheme_doc.rst | 26 +++++++------------ 1 file changed, 10 insertions(+), 16 deletions(-) diff --git a/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst b/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst index fc3d848ddc..1ac5da79d7 100644 --- a/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst +++ b/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst @@ -5667,22 +5667,16 @@ coefficients :math:`A_{1},\; A_{2},B_{1},\; B_{2}` are given by .. math:: :label: ab_coeffs - \begin{equation} - A_1=-\gamma_1\sum_j \nu_j RK_{PMj} - [LD_j\psi_jRK_H(1)_j+A_{*j}], - \end{equation} - \begin{equation} - A_2=\gamma_1\sum_j \nu_j RK_{PMj} - L\psi_jRK_H(1)_j, - \end{equation} - \begin{equation} - B_1=\gamma_1c_p\sum_j \nu_j RK_{PMj} - D_j\psi_jRK_H(1)_j - \end{equation} - \begin{equation} - B_2=-\gamma_1\sum_j \nu_j RK_{PMj} - \psi_j[c_pRK_H(1)_j+A_{*j}]. - \end{equation} + \begin{aligned} + A_1=-\gamma_1\sum_j \nu_j RK_{PMj} + [LD_j\psi_jRK_H(1)_j+A_{*j}], \\ + A_2=\gamma_1\sum_j \nu_j RK_{PMj} + L\psi_jRK_H(1)_j, \\ + B_1=\gamma_1c_p\sum_j \nu_j RK_{PMj} + D_j\psi_jRK_H(1)_j \\ + B_2=-\gamma_1\sum_j \nu_j RK_{PMj} + \psi_j[c_pRK_H(1)_j+A_{*j}]. + \end{aligned} but with :math:`\gamma_{1}={\mathcal{I}}_{1}`. Here :math:`RK_H(1) =\rho C_H U_1`, From 772b6338af2166d738b7c0226d4f6750acebf535 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Thu, 14 May 2026 14:13:40 +0100 Subject: [PATCH 089/116] Manually fixed unsupported \raisebox construct. --- .../turbulence_schemes/bl_scheme_doc.rst | 2 +- .../turbulence_schemes/manual_corrections.txt | 11 +++++++++++ 2 files changed, 12 insertions(+), 1 deletion(-) diff --git a/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst b/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst index 1ac5da79d7..1050244b8e 100644 --- a/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst +++ b/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst @@ -6253,7 +6253,7 @@ included in :math:`\zeta_r` and :math:`\tilde{\alpha_t}` (in where there is no buoyancy reversal (:math:`D \leq 0`) and the feedback seen in LES of stratocumulus :raw-latex:`\cite[]{lock98}` with significant buoyancy reversal -(:math:`D \raisebox{-.4ex}{$\ \stackrel{>}{{\scriptstyle \sim}} \ $}0.1`). +(:math:`D \gtrsim 0.1`). Furthermore, the LES of `Lock (2009)`_ indicated the presence of cumulus penetrating up into stratocumulus could be sufficient to enhance the feedback for small :math:`D`. Thus the option diff --git a/documentation/source/science_guide/turbulence_schemes/manual_corrections.txt b/documentation/source/science_guide/turbulence_schemes/manual_corrections.txt index b0ade73906..27658d4773 100644 --- a/documentation/source/science_guide/turbulence_schemes/manual_corrections.txt +++ b/documentation/source/science_guide/turbulence_schemes/manual_corrections.txt @@ -10,3 +10,14 @@ index 5b373b68..8ff32c06 100644 * - Formulation +--- a/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst ++++ b/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst +@@ -6253,7 +6253,7 @@ included in :math:`\zeta_r` and :math:`\tilde{\alpha_t}` (in + where there is no buoyancy reversal (:math:`D \leq 0`) and the feedback + seen in LES of stratocumulus :raw-latex:`\cite[]{lock98}` with + significant buoyancy reversal +-(:math:`D \raisebox{-.4ex}{$\ \stackrel{>}{{\scriptstyle \sim}} \ $}0.1`). ++(:math:`D \gtrsim 0.1`). + Furthermore, the LES of `Lock (2009)`_ indicated the + presence of cumulus penetrating up into stratocumulus could be + sufficient to enhance the feedback for small :math:`D`. Thus the option From 3b9ff0966dbece578bbe5b313341f9ab6cce442c Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Fri, 15 May 2026 16:56:08 +0100 Subject: [PATCH 090/116] Converted figures using inkscape instead of image magick, to restore colour. --- .../turbulence_schemes/div_r071.svg | 2246 ++-- .../turbulence_schemes/div_r080.svg | 2163 ++-- .../turbulence_schemes/honnert_vs_tanh.svg | 1203 +- .../turbulence_schemes/ideal_invinteg.svg | 1112 +- .../turbulence_schemes/ideal_revflux.svg | 2600 ++-- .../turbulence_schemes/nbldoc_zidiag.svg | 844 +- .../turbulence_schemes/new_ktop_shape.svg | 1195 +- .../turbulence_schemes/stab_dep.svg | 4112 ++++-- .../turbulence_schemes/subsent_fig7.svg | 1368 +- .../turbulence_schemes/wcrp_bltypes1.svg | 8557 +++++++++---- .../turbulence_schemes/wcrp_bltypes2.svg | 10672 +++++++++++----- .../turbulence_schemes/zturb_schem.svg | 605 +- 12 files changed, 23952 insertions(+), 12725 deletions(-) diff --git a/documentation/source/science_guide/turbulence_schemes/div_r071.svg b/documentation/source/science_guide/turbulence_schemes/div_r071.svg index e6072194ee..2a4e33b619 100644 --- a/documentation/source/science_guide/turbulence_schemes/div_r071.svg +++ b/documentation/source/science_guide/turbulence_schemes/div_r071.svg @@ -1,969 +1,1277 @@ - - - - -Created by potrace 1.16, written by Peter Selinger 2001-2019 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + diff --git a/documentation/source/science_guide/turbulence_schemes/div_r080.svg b/documentation/source/science_guide/turbulence_schemes/div_r080.svg index c1c35a09c1..e037496547 100644 --- a/documentation/source/science_guide/turbulence_schemes/div_r080.svg +++ b/documentation/source/science_guide/turbulence_schemes/div_r080.svg @@ -1,955 +1,1208 @@ - - - - -Created by potrace 1.16, written by Peter Selinger 2001-2019 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + diff --git a/documentation/source/science_guide/turbulence_schemes/honnert_vs_tanh.svg b/documentation/source/science_guide/turbulence_schemes/honnert_vs_tanh.svg index 3b0a349a30..3749b90ce5 100644 --- a/documentation/source/science_guide/turbulence_schemes/honnert_vs_tanh.svg +++ b/documentation/source/science_guide/turbulence_schemes/honnert_vs_tanh.svg @@ -1,723 +1,480 @@ - - - - -Created by potrace 1.16, written by Peter Selinger 2001-2019 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + +0.11.010.0∆x/zturb0.00.20.40.60.81.0W1DBoutle et al. (2014)Honnert et al. (2011)vn10.1 diff --git a/documentation/source/science_guide/turbulence_schemes/ideal_invinteg.svg b/documentation/source/science_guide/turbulence_schemes/ideal_invinteg.svg index 05911dc69f..9790d8ac41 100644 --- a/documentation/source/science_guide/turbulence_schemes/ideal_invinteg.svg +++ b/documentation/source/science_guide/turbulence_schemes/ideal_invinteg.svg @@ -1,495 +1,617 @@ - - - - -Created by potrace 1.16, written by Peter Selinger 2001-2019 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + diff --git a/documentation/source/science_guide/turbulence_schemes/ideal_revflux.svg b/documentation/source/science_guide/turbulence_schemes/ideal_revflux.svg index ed46d0ba07..fdec6528c9 100644 --- a/documentation/source/science_guide/turbulence_schemes/ideal_revflux.svg +++ b/documentation/source/science_guide/turbulence_schemes/ideal_revflux.svg @@ -1,1224 +1,1376 @@ - - - - -Created by potrace 1.16, written by Peter Selinger 2001-2019 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + diff --git a/documentation/source/science_guide/turbulence_schemes/nbldoc_zidiag.svg b/documentation/source/science_guide/turbulence_schemes/nbldoc_zidiag.svg index 22253e13f0..eea44a7e14 100644 --- a/documentation/source/science_guide/turbulence_schemes/nbldoc_zidiag.svg +++ b/documentation/source/science_guide/turbulence_schemes/nbldoc_zidiag.svg @@ -1,383 +1,461 @@ - - - - -Created by potrace 1.16, written by Peter Selinger 2001-2019 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + diff --git a/documentation/source/science_guide/turbulence_schemes/new_ktop_shape.svg b/documentation/source/science_guide/turbulence_schemes/new_ktop_shape.svg index f9ffca4895..eeb10629b8 100644 --- a/documentation/source/science_guide/turbulence_schemes/new_ktop_shape.svg +++ b/documentation/source/science_guide/turbulence_schemes/new_ktop_shape.svg @@ -1,563 +1,632 @@ - - - - -Created by potrace 1.16, written by Peter Selinger 2001-2019 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + diff --git a/documentation/source/science_guide/turbulence_schemes/stab_dep.svg b/documentation/source/science_guide/turbulence_schemes/stab_dep.svg index 5e4fa1d41e..d2148054e9 100644 --- a/documentation/source/science_guide/turbulence_schemes/stab_dep.svg +++ b/documentation/source/science_guide/turbulence_schemes/stab_dep.svg @@ -1,1320 +1,2792 @@ - - - - -Created by potrace 1.16, written by Peter Selinger 2001-2019 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + diff --git a/documentation/source/science_guide/turbulence_schemes/subsent_fig7.svg b/documentation/source/science_guide/turbulence_schemes/subsent_fig7.svg index 6d7481680d..f146fb9732 100644 --- a/documentation/source/science_guide/turbulence_schemes/subsent_fig7.svg +++ b/documentation/source/science_guide/turbulence_schemes/subsent_fig7.svg @@ -1,592 +1,776 @@ - - - - -Created by potrace 1.16, written by Peter Selinger 2001-2019 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + diff --git a/documentation/source/science_guide/turbulence_schemes/wcrp_bltypes1.svg b/documentation/source/science_guide/turbulence_schemes/wcrp_bltypes1.svg index cf5118d1ea..e9c9f4d05d 100644 --- a/documentation/source/science_guide/turbulence_schemes/wcrp_bltypes1.svg +++ b/documentation/source/science_guide/turbulence_schemes/wcrp_bltypes1.svg @@ -1,2226 +1,6331 @@ - - - - -Created by potrace 1.16, written by Peter Selinger 2001-2019 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + +θθθ diff --git a/documentation/source/science_guide/turbulence_schemes/wcrp_bltypes2.svg b/documentation/source/science_guide/turbulence_schemes/wcrp_bltypes2.svg index dce1f3b4bf..d94116e8ef 100644 --- a/documentation/source/science_guide/turbulence_schemes/wcrp_bltypes2.svg +++ b/documentation/source/science_guide/turbulence_schemes/wcrp_bltypes2.svg @@ -1,2889 +1,7783 @@ - - - - -Created by potrace 1.16, written by Peter Selinger 2001-2019 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + +θθθ diff --git a/documentation/source/science_guide/turbulence_schemes/zturb_schem.svg b/documentation/source/science_guide/turbulence_schemes/zturb_schem.svg index 894ba49909..3c68567b24 100644 --- a/documentation/source/science_guide/turbulence_schemes/zturb_schem.svg +++ b/documentation/source/science_guide/turbulence_schemes/zturb_schem.svg @@ -1,386 +1,219 @@ - - - - -Created by potrace 1.16, written by Peter Selinger 2001-2019 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + +0zturb0zzsmlzh-zsczhzsmlzsczh From aaec813c1edf034d46d82b00941b0f30a4ddeae5 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Fri, 15 May 2026 17:31:39 +0100 Subject: [PATCH 091/116] Reverted needless splitting up of some math aligned regions (where multiple equations were really meant to be labeled as a group rather than individually), and fixed math within figure captions (still contained latex). --- .../turbulence_schemes/bl_scheme_doc.rst | 185 +++++++++--------- 1 file changed, 93 insertions(+), 92 deletions(-) diff --git a/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst b/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst index 1050244b8e..c6f4022a81 100644 --- a/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst +++ b/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst @@ -584,17 +584,17 @@ As standard, the fluxes in :eq:`eq:wb_cont` are then expanded using the first-order closure in :eq:`scal_closure` as: -.. math:: +.. math:: :label: eq:wx_std - \overline{w'\theta_{\ell}'}_k = + \begin{aligned} + \overline{w'\theta_{\ell}'}_k &=& -K_h^{\mathrm{surf}}\,\frac{\widetilde{\Delta_k \theta_{\ell}}}{\Delta_k z} -K_h^{\mathrm{Sc}}\,\frac{\Delta_k \theta_{\ell}}{\Delta_k z} - -.. math:: :label: eq:wx_std - - \overline{w'q_t'}_k = -\left(K_h^{\mathrm{surf}}+ K_h^{\mathrm{Sc}}\right) - \, + \\ + \overline{w'q_t'}_k &=& -\left(K_h^{\mathrm{surf}}+ + K_h^{\mathrm{Sc}}\right) \, \,\frac{\Delta_k q_t}{\Delta_k z} + \end{aligned} where :math:`\widetilde{\Delta_k \theta_{\ell}} = \Delta_k \theta_{\ell}- @@ -760,10 +760,10 @@ expected to occur) and :math:`z_{\mathrm{ \mathrm{NTML}}-1}`. .. figure:: blank.svg :name: fig:inv_integ - Subgrid (lines) and model (symbols) fluxes of $\thetal$: turbulent flux - (dash-dotted, crosses), radiative flux (dashed, triangles) and total flux - (solid). The shaded area illustrates the integrated turbulent flux that - would be obtained were (\protect\mbox{\protect\ref{eq:wx_std}}) used. + Subgrid (lines) and model (symbols) fluxes of :math:`\thetal`: turbulent + flux (dash-dotted, crosses), radiative flux (dashed, triangles) and total + flux (solid). The shaded area illustrates the integrated turbulent flux that + would be obtained were (\protect\mbox{\protect:eq:`eq:wx_std`}) used. .. list-table:: :align: center @@ -972,14 +972,14 @@ should be considered over more levels. The asymptotic mixing lengths are given by -.. math:: - - \lambda_m =\mathrm{max}\left[\lambda_0,\, 0.15 z_{\mathrm{loc}}, 2 h_B - \right] - .. math:: :label: asymp_ml - \lambda_h =\mathrm{max}\left[\lambda_0,\, 0.15 z_{\mathrm{loc}}\right] + \begin{aligned} + \lambda_m &=&\mathrm{max}\left[\lambda_0,\, 0.15 z_{\mathrm{loc}}, 2 h_B + \right] + \\ + \lambda_h &=&\mathrm{max}\left[\lambda_0,\, 0.15 z_{\mathrm{loc}}\right] + \end{aligned} where :math:`\lambda_0` is a minimum mixing length read in from the namelist and :math:`z_{\mathrm{loc}}` is defined below. The orographic @@ -1068,16 +1068,16 @@ to :math:`\lambda_0`. For :math:`Ri < 0`, the standard UM stability functions are given by -.. math:: - - f_m = 1 - \frac{g_0 \,Ri} - {1+D_m(\tilde{{\mathcal{L}}}_m/\tilde{{\mathcal{L}}}_h)|Ri|^{1/2} } - .. math:: :label: unstable_stab - f_h = \frac{1}{Pr_N}\left(1 - \frac{g_0 \,Ri} + \begin{aligned} + f_m & =& 1 - \frac{g_0 \,Ri} + {1+D_m(\tilde{{\mathcal{L}}}_m/\tilde{{\mathcal{L}}}_h)|Ri|^{1/2} } + \\ + f_h & =& \frac{1}{Pr_N}\left(1 - \frac{g_0 \,Ri} {1+D_h(\tilde{{\mathcal{L}}}_m/\tilde{{\mathcal{L}}}_h)|Ri|^{1/2} }\right) + \end{aligned} with :math:`g_0=10`, :math:`D_m=g_0/4` and :math:`D_h=g_0/25`. If the stability dependent Prandtl number option is chosen (see below) the @@ -1085,13 +1085,13 @@ neutral Prandtl number, :math:`Pr_N`, is set to :math:`0.7`; otherwise :math:`Pr_N=1`. Alternatives are those from the Met Office large-eddy model (LEM), `Brown (1999) 2`_: -.. math:: - - f_m = (1 - c_{LEM} Ri)^{1/2} - .. math:: :label: unstable_stab_lem - f_h = \frac{1}{Pr_N}\left(1 - b_{LEM} Ri\right)^{1/2} + \begin{aligned} + f_m & =& (1 - c_{LEM} Ri)^{1/2} + \\ + f_h & =& \frac{1}{Pr_N}\left(1 - b_{LEM} Ri\right)^{1/2} + \end{aligned} where :math:`Pr_N = 0.7`, and the constants :math:`b_{LEM}` and :math:`c_{LEM}` can take the values 40 and 16 respectively in the @@ -1122,11 +1122,11 @@ where .. math:: - A_{Ri} = \left(1-g_0 Ri_{t}\right)/\left(1- g_0 Ri_{t}/2\right)^2 - -.. math:: - - B_{Ri} = (g_0/2) /\left(1 - g_0 Ri_{t}/2\right)^2 + \begin{aligned} + A_{Ri} & = & \left(1-g_0 Ri_{t}\right)/\left(1- g_0 Ri_{t}/2\right)^2 + \\ + B_{Ri} & = & (g_0/2) /\left(1 - g_0 Ri_{t}/2\right)^2 + \end{aligned} For the 'SHARPEST' function of `Derbyshire (1997)`_, :math:`Ri_{t}=0.1`, while larger values give even sharper reduction of @@ -1861,10 +1861,10 @@ Discussion of some of the revisions :name: fig:stab_dep Stability dependence of the surface velocity scales, Prandtl number - (although I hope something is wrong with my coding of HB here!) and $d$. - Solid lines are from HB, dotted from the standard UM and the dashed from the - revised formulation. The dash-dotted line for $d$ is a potential - modification, as described in the text. + (although I hope something is wrong with my coding of HB here!) and + :math:`d`. Solid lines are from HB, dotted from the standard UM and the + dashed from the revised formulation. The dash-dotted line for :math:`d` is a + potential modification, as described in the text. .. list-table:: :align: center @@ -1926,9 +1926,9 @@ removed since the entrainment flux is now carried via the explicit .. figure:: blank.svg :name: fig:new_ksc - Standard UM $\khtop$ (solid) and revised (dotted), both scaled by $k z_h - \vtopo$. An upside-down version of $\khsurf$ is also shown (dashed) for - comparison. + Standard UM :math:`\khtop` (solid) and revised (dotted), both scaled by + :math:`k z_h \vtopo`. An upside-down version of :math:`\khsurf` is also + shown (dashed) for comparison. .. list-table:: :align: center @@ -2070,13 +2070,13 @@ turbulence scheme. .. figure:: blank.svg :name: fig-blend - (a) Weighting for the 1D boundary-layer scheme as a function of $\Delta - x/z_{\rm turb}$, showing the function of Equation~\ref{eq-tanh} (blue - solid), the equation in `Boutle et al. (2014)`_ (black solid) and the TKE - partitioning of `Honnert et al. (2011)`_ (mean thick dashed, 5th/95th - percentiles thin dashed). (b) Schematic showing the calculation of $z_{\rm - turb}$ used in Eq.~\ref{eq-tanh} for a well-mixed layer (black dotted) and a - decoupled cloud layer (black solid). + (a) Weighting for the 1D boundary-layer scheme as a function of + :math:`\Delta x/z_{\mathrm{turb}}`, showing the function of + Equation~:eq:`eq-tanh` (blue solid), the equation in `Boutle et al. (2014)`_ + (black solid) and the TKE partitioning of `Honnert et al. (2011)`_ (mean + thick dashed, 5th/95th percentiles thin dashed). (b) Schematic showing the + calculation of :math:`z_{\mathrm{turb}}` used in Eq.~:eq:`eq-tanh` for a + well-mixed layer (black dotted) and a decoupled cloud layer (black solid). .. list-table:: :align: center @@ -2321,13 +2321,13 @@ distance above, then :math:`\overline{w'\theta_{\ell}'}_{z_i}= - w_e \Delta \theta_{\ell}+ F|_h - F|_{z_i}`, so that -.. math:: - - {\mathcal{H}}|_{z_i} = - w_e \Delta \theta_{\ell}+ F_{\mathrm{net}}|_h - .. math:: :label: discinv - \overline{w'q_t'}_{z_i} = - w_e \Delta q_t + \begin{aligned} + {\mathcal{H}}|_{z_i} & =& - w_e \Delta \theta_{\ell}+ F_{\mathrm{net}}|_h + \\ + \overline{w'q_t'}_{z_i}& =& - w_e \Delta q_t + \end{aligned} where the total heat flux :math:`{\mathcal{H}} = \overline{w'\theta_{\ell}'}+ F_{\mathrm{net}}` and @@ -2359,21 +2359,21 @@ are then estimated using linear interpolation of :math:`{\mathcal{H}}` and :math:`\overline{w'q_t'}` between :math:`z_{\mathrm{h}}^{\mathrm{Sc}}` and the base of the mixed layer: -.. math:: +.. math:: :label: fluxinterp - \overline{w'\theta_{\ell}'}|_{ z_{\mathrm{ \mathrm{NTDSC}}+\frac{1}{2}} } = - \overline{w'\theta_{\ell}'}|_{z_{\mathrm{b}}} + \begin{aligned} + \overline{w'\theta_{\ell}'}|_{ z_{\mathrm{ \mathrm{NTDSC}}+\frac{1}{2}} } & + =& \overline{w'\theta_{\ell}'}|_{z_{\mathrm{b}}} - \frac{ z'_{\mathrm{ \mathrm{NTDSC}}+\frac{1}{2}} }{z_{\mathrm{ml}}} \left( \tilde{w_e} \Delta \theta_{\ell}+ \overline{w'\theta_{\ell}'}|_{z_{\mathrm{b}}} - F_{\mathrm{net}}|_{h} \right) - F_{\mathrm{net}}|_{ z_{\mathrm{ \mathrm{NTDSC}}+\frac{1}{2}} } - -.. math:: :label: fluxinterp - - \overline{w'q_t'}|_{ z_{\mathrm{ \mathrm{NTDSC}}+\frac{1}{2}} } = + \\ + \overline{w'q_t'}|_{ z_{\mathrm{ \mathrm{NTDSC}}+\frac{1}{2}} } & =& \overline{w'q_t'}|_{z_{\mathrm{b}}} - \frac{ z'_{\mathrm{ \mathrm{NTDSC}}+\frac{1}{2}} }{z_{\mathrm{ml}}} \left( \tilde{w_e} \Delta q_t + \overline{w'q_t'}|_{z_{\mathrm{b}}} \right) + \end{aligned} where :math:`z' = z-z_{\mathrm{b}}`, and similarly for the SML entrainment fluxes (at :math:`z=z_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}}`). The @@ -2384,10 +2384,10 @@ illustrated for a SML in :numref:`Fig. %s `. .. figure:: blank.svg :name: fig:fluxinterp - Idealised profiles of (a) $\wqt$ (dash-dotted line) and (b) ${\cal H}$ - (dotted line), $\wthl$ (dash-dotted) and $F$ (dashed). The continuous lines - are the turbulent fluxes on the model grid indicated by the dashed - horizontal lines. + Idealised profiles of (a) :math:`\wqt` (dash-dotted line) and (b) + :math:`{\mathcal{H}}` (dotted line), :math:`\wthl` (dash-dotted) and + :math:`F` (dashed). The continuous lines are the turbulent fluxes on the + model grid indicated by the dashed horizontal lines. .. list-table:: :align: center @@ -2490,7 +2490,7 @@ it to diffuse out this static instability). :name: zi_diag Schematic illustrating the assumptions behind the subgrid diagnosis of - $z_i$. + :math:`z_i`. .. list-table:: :align: center @@ -2648,10 +2648,10 @@ large-scale vertical velocity evaluated at the inversion, .. figure:: blank.svg :name: fig:rev_fluxes - Subgrid (lines) and model (symbols) profiles and fluxes of, top row, $q_t$ - and, bottom row, $\thetal$: turbulent fluxes (dash-dotted, crosses), - subsidence fluxes (dotted, diamonds), radiative flux (dashed, triangles) and - total flux (solid, squares). + Subgrid (lines) and model (symbols) profiles and fluxes of, top row, + :math:`q_t` and, bottom row, :math:`\thetal`: turbulent fluxes (dash-dotted, + crosses), subsidence fluxes (dotted, diamonds), radiative flux (dashed, + triangles) and total flux (solid, squares). .. list-table:: :align: center @@ -2923,15 +2923,15 @@ see section :ref:`Diagnosis of a sub-grid inversion `, fluxes at the mixed layer top are specified through an eddy diffusivity which is given by -.. math:: - - K_h|_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}} = w_e \Delta_{\mathrm{ - \mathrm{NTML}}+1} z - .. math:: :label: khent - K_m|_{\mathrm{ \mathrm{NTML}}} = Pr \, w_e \Delta_{\mathrm{ + \begin{aligned} + K_h|_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}} & =& w_e \Delta_{\mathrm{ + \mathrm{NTML}}+1} z + \\ + K_m|_{\mathrm{ \mathrm{NTML}}} & =& Pr \, w_e \Delta_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}} z + \end{aligned} noting the Charney-Philips grid implying stresses are staggered from scalar fluxes. The Prandtl number, :math:`Pr`, takes the same form as @@ -4505,15 +4505,15 @@ by effective values, z\ :math:`_{0m(eff)}` and z\ :math:`_{0h(eff)}`. When form drag is included via effective roughness lengths equations :eq:`1.1.7`-:eq:`1.1.9` become: -.. math:: - - \frac{ H_{0(eff)} }{ c_P \rho _0 }=\frac{-k}{ \Phi _h (L , z_1 + - z_{0m(eff)} , z_{0h(eff)} )} v_{\ast (eff)} - .. math:: :label: 2.1.1 - \left( {\Delta T + \frac{g}{c_p }( z_1 + z_{0m(eff)} - z_{0h(eff)} )} + \begin{aligned} + \frac{ H_{0(eff)} }{ c_P \rho _0 }&=&\frac{-k}{ \Phi _h (L , z_1 + + z_{0m(eff)} , z_{0h(eff)} )} v_{\ast (eff)} + \\ + && \left( {\Delta T + \frac{g}{c_p }( z_1 + z_{0m(eff)} - z_{0h(eff)} )} \right) + \end{aligned} .. math:: :label: 2.1.2 @@ -6307,19 +6307,19 @@ estimate of the height of cloud-base is made using :math:`\gamma_{q_{\ell}}` and :math:`\gamma_{q_f}` and added to the grid-level based calculation: -.. math:: +.. math:: :label: zc_calc - z_c = z_c + \frac{\Delta_{k_b+\frac{1}{2}} z}{2} + \begin{aligned} + z_c &=& z_c + \frac{\Delta_{k_b+\frac{1}{2}} z}{2} + \mathrm{min}\left[ \frac{ \Delta_{k_b+\frac{1}{2}}z +\Delta_{k_b-\frac{1}{2}}z }{2} C_F^l, \frac{ q_{\ell}}{ \gamma_{q_{\ell}} } \right]/C_F - -.. math:: :label: zc_calc - - \left. \hspace{2.4cm} - + \mathrm{min}\left[ \frac{ \Delta_{k_b+\frac{1}{2}}z - +\Delta_{k_b-\frac{1}{2}}z }{2} C_F^f, - \frac{ q_f }{ \gamma_{q_f} } \right]/C_F \right. + \\ + & & \left. \hspace{2.4cm} + + \mathrm{min}\left[ \frac{ \Delta_{k_b+\frac{1}{2}}z + +\Delta_{k_b-\frac{1}{2}}z }{2} C_F^f, + \frac{ q_f }{ \gamma_{q_f} } \right]/C_F \right. + \end{aligned} When :math:`\gamma_{q_f}` is set to zero (currently as standard) the last term in :eq:`zc_calc` is given by @@ -6520,9 +6520,10 @@ therefore zero entrainment and turbulent mixing). .. figure:: blank.svg :name: fig:dradts - Time series from LES of $\Delta_\radf$ (solid), $\Delta_\radf^{LW} $ - (dotted), $-\Delta_\radf^{SW} $ (dashed) and the 8A (dash-dot) and 9B (dash- - dot-dot-dot) parametrizations of $\Delta_\radf$. + Time series from LES of :math:`\Delta_\radf` (solid), + :math:`\Delta_\radf^{LW}` (dotted), :math:`-\Delta_\radf^{SW}` (dashed) and + the 8A (dash-dot) and 9B (dash-dot-dot-dot) parametrizations of + :math:`\Delta_\radf`. .. list-table:: :align: center From 566ef5cd47ccbcb805536bbcc5b74c5a76f143e6 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Fri, 15 May 2026 17:37:19 +0100 Subject: [PATCH 092/116] Manual fix: insert F in place of custom macro \radf in caption math. --- .../turbulence_schemes/bl_scheme_doc.rst | 6 +++--- .../turbulence_schemes/manual_corrections.txt | 16 ++++++++++++++++ 2 files changed, 19 insertions(+), 3 deletions(-) diff --git a/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst b/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst index c6f4022a81..f77836e077 100644 --- a/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst +++ b/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst @@ -6520,10 +6520,10 @@ therefore zero entrainment and turbulent mixing). .. figure:: blank.svg :name: fig:dradts - Time series from LES of :math:`\Delta_\radf` (solid), - :math:`\Delta_\radf^{LW}` (dotted), :math:`-\Delta_\radf^{SW}` (dashed) and + Time series from LES of :math:`\Delta_F` (solid), + :math:`\Delta_F^{LW}` (dotted), :math:`-\Delta_F^{SW}` (dashed) and the 8A (dash-dot) and 9B (dash-dot-dot-dot) parametrizations of - :math:`\Delta_\radf`. + :math:`\Delta_F`. .. list-table:: :align: center diff --git a/documentation/source/science_guide/turbulence_schemes/manual_corrections.txt b/documentation/source/science_guide/turbulence_schemes/manual_corrections.txt index 27658d4773..98cbb809d8 100644 --- a/documentation/source/science_guide/turbulence_schemes/manual_corrections.txt +++ b/documentation/source/science_guide/turbulence_schemes/manual_corrections.txt @@ -21,3 +21,19 @@ index 5b373b68..8ff32c06 100644 Furthermore, the LES of `Lock (2009)`_ indicated the presence of cumulus penetrating up into stratocumulus could be sufficient to enhance the feedback for small :math:`D`. Thus the option +--- a/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst ++++ b/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst +@@ -6520,10 +6520,10 @@ therefore zero entrainment and turbulent mixing). + .. figure:: blank.svg + :name: fig:dradts + +- Time series from LES of :math:`\Delta_\radf` (solid), +- :math:`\Delta_\radf^{LW}` (dotted), :math:`-\Delta_\radf^{SW}` (dashed) and ++ Time series from LES of :math:`\Delta_F` (solid), ++ :math:`\Delta_F^{LW}` (dotted), :math:`-\Delta_F^{SW}` (dashed) and + the 8A (dash-dot) and 9B (dash-dot-dot-dot) parametrizations of +- :math:`\Delta_\radf`. ++ :math:`\Delta_F`. + + .. list-table:: + :align: center From 24ca5f08024e7576c0e333a1bd2404a3b1ed8cce Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Fri, 15 May 2026 18:09:22 +0100 Subject: [PATCH 093/116] Further corrections to math and equation references inside re-imported figure captions. --- .../turbulence_schemes/bl_scheme_doc.rst | 32 ++++---- .../turbulence_schemes/manual_corrections.txt | 73 ++++++++++++++++++- 2 files changed, 85 insertions(+), 20 deletions(-) diff --git a/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst b/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst index f77836e077..cd1517eece 100644 --- a/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst +++ b/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst @@ -760,10 +760,10 @@ expected to occur) and :math:`z_{\mathrm{ \mathrm{NTML}}-1}`. .. figure:: blank.svg :name: fig:inv_integ - Subgrid (lines) and model (symbols) fluxes of :math:`\thetal`: turbulent - flux (dash-dotted, crosses), radiative flux (dashed, triangles) and total - flux (solid). The shaded area illustrates the integrated turbulent flux that - would be obtained were (\protect\mbox{\protect:eq:`eq:wx_std`}) used. + Subgrid (lines) and model (symbols) fluxes of :math:`\theta_{\ell}`: + turbulent flux (dash-dotted, crosses), radiative flux (dashed, triangles) + and total flux (solid). The shaded area illustrates the integrated turbulent + flux that would be obtained were :eq:`eq:wx_std` used. .. list-table:: :align: center @@ -1926,9 +1926,9 @@ removed since the entrainment flux is now carried via the explicit .. figure:: blank.svg :name: fig:new_ksc - Standard UM :math:`\khtop` (solid) and revised (dotted), both scaled by - :math:`k z_h \vtopo`. An upside-down version of :math:`\khsurf` is also - shown (dashed) for comparison. + Standard UM :math:`K_h^{\rm Sc}` (solid) and revised (dotted), both scaled + by :math:`k z_h V_{\rm Sc}`. An upside-down version of + :math:`K_h^{\rm surf}` is also shown (dashed) for comparison. .. list-table:: :align: center @@ -2072,10 +2072,10 @@ turbulence scheme. (a) Weighting for the 1D boundary-layer scheme as a function of :math:`\Delta x/z_{\mathrm{turb}}`, showing the function of - Equation~:eq:`eq-tanh` (blue solid), the equation in `Boutle et al. (2014)`_ + Equation :eq:`eq-tanh` (blue solid), the equation in `Boutle et al. (2014)`_ (black solid) and the TKE partitioning of `Honnert et al. (2011)`_ (mean thick dashed, 5th/95th percentiles thin dashed). (b) Schematic showing the - calculation of :math:`z_{\mathrm{turb}}` used in Eq.~:eq:`eq-tanh` for a + calculation of :math:`z_{\mathrm{turb}}` used in Eq. :eq:`eq-tanh` for a well-mixed layer (black dotted) and a decoupled cloud layer (black solid). .. list-table:: @@ -2384,10 +2384,10 @@ illustrated for a SML in :numref:`Fig. %s `. .. figure:: blank.svg :name: fig:fluxinterp - Idealised profiles of (a) :math:`\wqt` (dash-dotted line) and (b) - :math:`{\mathcal{H}}` (dotted line), :math:`\wthl` (dash-dotted) and - :math:`F` (dashed). The continuous lines are the turbulent fluxes on the - model grid indicated by the dashed horizontal lines. + Idealised profiles of (a) :math:`\overline{w'q_t'}` (dash-dotted line) and + (b) :math:`{\mathcal{H}}` (dotted line), :math:`\overline{w'\theta_{\ell}'}` + (dash-dotted) and :math:`F` (dashed). The continuous lines are the turbulent + fluxes on the model grid indicated by the dashed horizontal lines. .. list-table:: :align: center @@ -2649,9 +2649,9 @@ large-scale vertical velocity evaluated at the inversion, :name: fig:rev_fluxes Subgrid (lines) and model (symbols) profiles and fluxes of, top row, - :math:`q_t` and, bottom row, :math:`\thetal`: turbulent fluxes (dash-dotted, - crosses), subsidence fluxes (dotted, diamonds), radiative flux (dashed, - triangles) and total flux (solid, squares). + :math:`q_t` and, bottom row, :math:`\theta_{\ell}`: turbulent fluxes + (dash-dotted, crosses), subsidence fluxes (dotted, diamonds), radiative flux + (dashed, triangles) and total flux (solid, squares). .. list-table:: :align: center diff --git a/documentation/source/science_guide/turbulence_schemes/manual_corrections.txt b/documentation/source/science_guide/turbulence_schemes/manual_corrections.txt index 98cbb809d8..a126a907ba 100644 --- a/documentation/source/science_guide/turbulence_schemes/manual_corrections.txt +++ b/documentation/source/science_guide/turbulence_schemes/manual_corrections.txt @@ -10,8 +10,6 @@ index 5b373b68..8ff32c06 100644 * - Formulation ---- a/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst -+++ b/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst @@ -6253,7 +6253,7 @@ included in :math:`\zeta_r` and :math:`\tilde{\alpha_t}` (in where there is no buoyancy reversal (:math:`D \leq 0`) and the feedback seen in LES of stratocumulus :raw-latex:`\cite[]{lock98}` with @@ -21,8 +19,75 @@ index 5b373b68..8ff32c06 100644 Furthermore, the LES of `Lock (2009)`_ indicated the presence of cumulus penetrating up into stratocumulus could be sufficient to enhance the feedback for small :math:`D`. Thus the option ---- a/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst -+++ b/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst +@@ -760,10 +760,10 @@ expected to occur) and :math:`z_{\mathrm{ \mathrm{NTML}}-1}`. + .. figure:: blank.svg + :name: fig:inv_integ + +- Subgrid (lines) and model (symbols) fluxes of :math:`\thetal`: turbulent +- flux (dash-dotted, crosses), radiative flux (dashed, triangles) and total +- flux (solid). The shaded area illustrates the integrated turbulent flux that +- would be obtained were (\protect\mbox{\protect:eq:`eq:wx_std`}) used. ++ Subgrid (lines) and model (symbols) fluxes of :math:`\theta_{\ell}`: ++ turbulent flux (dash-dotted, crosses), radiative flux (dashed, triangles) ++ and total flux (solid). The shaded area illustrates the integrated turbulent ++ flux that would be obtained were :eq:`eq:wx_std` used. + + .. list-table:: + :align: center +@@ -1926,9 +1926,9 @@ removed since the entrainment flux is now carried via the explicit + .. figure:: blank.svg + :name: fig:new_ksc + +- Standard UM :math:`\khtop` (solid) and revised (dotted), both scaled by +- :math:`k z_h \vtopo`. An upside-down version of :math:`\khsurf` is also +- shown (dashed) for comparison. ++ Standard UM :math:`K_h^{\rm Sc}` (solid) and revised (dotted), both scaled ++ by :math:`k z_h V_{\rm Sc}`. An upside-down version of ++ :math:`K_h^{\rm surf}` is also shown (dashed) for comparison. + + .. list-table:: + :align: center +@@ -2072,10 +2072,10 @@ turbulence scheme. + + (a) Weighting for the 1D boundary-layer scheme as a function of + :math:`\Delta x/z_{\mathrm{turb}}`, showing the function of +- Equation~:eq:`eq-tanh` (blue solid), the equation in `Boutle et al. (2014)`_ ++ Equation :eq:`eq-tanh` (blue solid), the equation in `Boutle et al. (2014)`_ + (black solid) and the TKE partitioning of `Honnert et al. (2011)`_ (mean + thick dashed, 5th/95th percentiles thin dashed). (b) Schematic showing the +- calculation of :math:`z_{\mathrm{turb}}` used in Eq.~:eq:`eq-tanh` for a ++ calculation of :math:`z_{\mathrm{turb}}` used in Eq. :eq:`eq-tanh` for a + well-mixed layer (black dotted) and a decoupled cloud layer (black solid). + + .. list-table:: +@@ -2384,10 +2384,10 @@ illustrated for a SML in :numref:`Fig. %s `. + .. figure:: blank.svg + :name: fig:fluxinterp + +- Idealised profiles of (a) :math:`\wqt` (dash-dotted line) and (b) +- :math:`{\mathcal{H}}` (dotted line), :math:`\wthl` (dash-dotted) and +- :math:`F` (dashed). The continuous lines are the turbulent fluxes on the +- model grid indicated by the dashed horizontal lines. ++ Idealised profiles of (a) :math:`\overline{w'q_t'}` (dash-dotted line) and ++ (b) :math:`{\mathcal{H}}` (dotted line), :math:`\overline{w'\theta_{\ell}'}` ++ (dash-dotted) and :math:`F` (dashed). The continuous lines are the turbulent ++ fluxes on the model grid indicated by the dashed horizontal lines. + + .. list-table:: + :align: center +@@ -2649,9 +2649,9 @@ large-scale vertical velocity evaluated at the inversion, + :name: fig:rev_fluxes + + Subgrid (lines) and model (symbols) profiles and fluxes of, top row, +- :math:`q_t` and, bottom row, :math:`\thetal`: turbulent fluxes (dash-dotted, +- crosses), subsidence fluxes (dotted, diamonds), radiative flux (dashed, +- triangles) and total flux (solid, squares). ++ :math:`q_t` and, bottom row, :math:`\theta_{\ell}`: turbulent fluxes ++ (dash-dotted, crosses), subsidence fluxes (dotted, diamonds), radiative flux ++ (dashed, triangles) and total flux (solid, squares). + + .. list-table:: + :align: center @@ -6520,10 +6520,10 @@ therefore zero entrainment and turbulent mixing). .. figure:: blank.svg :name: fig:dradts From 88d3e8128795d18efa519402ba134ae08fe9a40c Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Wed, 20 May 2026 12:20:48 +0100 Subject: [PATCH 094/116] Removed trailing whitespace. --- .../turbulence_schemes/bl_scheme_doc.rst | 185 +++++++++--------- 1 file changed, 92 insertions(+), 93 deletions(-) diff --git a/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst b/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst index cd1517eece..c47051ee46 100644 --- a/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst +++ b/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst @@ -319,8 +319,8 @@ Currently, .. math:: :label: parcel_pert - \theta_v' = \mathrm{max} \left[A_{plume}, - \, \mathrm{min} \left[ B_{plume} \sigma_{Tv1}, + \theta_v' = \mathrm{max} \left[A_{plume}, + \, \mathrm{min} \left[ B_{plume} \sigma_{Tv1}, \, G_{max}z_{\mathrm{h}}\right] \right] where :math:`A_{plume}=0.2`, :math:`B_{plume}=3.26`, @@ -357,7 +357,7 @@ This is identified as the grid-level above which .. math:: - \frac{d\theta_v}{dz}|_{\mathrm{env}} > + \frac{d\theta_v}{dz}|_{\mathrm{env}} > \Gamma_{\mathrm{inv}}\, \frac{d\theta_v}{dz}|_{\mathrm{par}} where currently the tolerance for identifying inversions by this method, @@ -379,7 +379,7 @@ Specifically, a logical flag (CUMULUS) is set to true if .. math:: - \left| \frac{ \Delta_{\mathrm{cld}} q_t}{\Delta_{\mathrm{cld}} z} \right| > + \left| \frac{ \Delta_{\mathrm{cld}} q_t}{\Delta_{\mathrm{cld}} z} \right| > C_t \, \left| \frac{ \Delta_{\mathrm{sub}} q_t}{\Delta_{\mathrm{sub}} z} \right| @@ -437,8 +437,8 @@ the environment at that grid-level .. math:: :label: qlpar - q_{\ell f}^p = \mathrm{max}\left[ 0.0, \, a_L \left( q_t^p - {q_s}_k - - \alpha_L (\theta_{\ell}^p - (g z_k/c_p)-T_k)\right) + q_{\ell f}^p = \mathrm{max}\left[ 0.0, \, a_L \left( q_t^p - {q_s}_k + - \alpha_L (\theta_{\ell}^p - (g z_k/c_p)-T_k)\right) \right] where the buoyancy parameters :math:`a_L` and :math:`\alpha_L` are @@ -577,7 +577,7 @@ can be written as: \overline{w'b}= g \left[ (1-C_F) \left(\beta_T \overline{w'\theta_{\ell}'} + \beta_q \overline{w'q_t'}\right) + C_F \left( \tilde{\beta_T} \overline{w'\theta_{\ell}'} + \tilde{\beta_q} - \overline{w'q_t'}\right) + \overline{w'q_t'}\right) \right] As standard, the fluxes in :eq:`eq:wb_cont` are then @@ -626,8 +626,8 @@ buoyancy consumption of TKE within the mixed layer equals a fraction, \sum_{z_{k-\frac{1}{2}} > z_i-z_{\mathrm{ml}}}^{z_{k-\frac{1}{2}} < z_i} \left|\left[ \overline{w'b}|_{z_{k-\frac{1}{2}}}<0 \right]\right| \, - \Delta_k z \, - \leq \, D_t \, + \Delta_k z \, + \leq \, D_t \, \sum_{z_{k-\frac{1}{2}} > z_i-z_{\mathrm{ml}}}^{z_{k-\frac{1}{2}} < z_i} \left[ \overline{w'b}|_{z_{k-\frac{1}{2}}}>0 \right] \, \Delta_k z @@ -825,7 +825,7 @@ region, :math:`\overline{w'q_t'}` is also taken to be constant so that: .. math:: - \int_{z_h-\Delta z_{rad}}^{z_h} \, \overline{w'q_t'}\, dz = + \int_{z_h-\Delta z_{rad}}^{z_h} \, \overline{w'q_t'}\, dz = - \Delta z_{rad} w_e \Delta q_t The integrated buoyancy flux is then found from @@ -963,8 +963,8 @@ this log profile correction, as .. math:: - \tilde{{\mathcal{L}}}_{X,k-1/2} = \frac{k \Delta_{k-1/2} z}{ - ln\left( \frac{z_k + z_{0m}}{z_{k-1} + z_{0m}} \right) + \tilde{{\mathcal{L}}}_{X,k-1/2} = \frac{k \Delta_{k-1/2} z}{ + ln\left( \frac{z_k + z_{0m}}{z_{k-1} + z_{0m}} \right) + \frac{k \Delta_{k-1/2} z}{\lambda_X} } If near-surface resolution is increased this logarithmic correction @@ -1112,7 +1112,7 @@ transitional Richardson number, :math:`Ri_{t}`, as: .. math:: - f_{\mathrm{stable}} = + f_{\mathrm{stable}} = \begin{cases} (1 - 5Ri)^2 & {\mathrm{for}}\ 0Ri_{t} @@ -1315,7 +1315,7 @@ The general approach is to take :math:`K_{\chi}` in .. math:: :label: klnl K_{\chi} = \mathrm{max} \left[ - (K_{\chi}^{\mathrm{surf}}+K_{\chi}^{\mathrm{Sc}}), + (K_{\chi}^{\mathrm{surf}}+K_{\chi}^{\mathrm{Sc}}), K_{\chi}(Ri) \right] As noted in section :ref:`Model variables and turbulence closure @@ -1465,7 +1465,7 @@ The form of :math:`w_s` differs between the surface layer .. math:: :label: ws_defn - w_s^3 = + w_s^3 = \begin{cases} 2.5 \, \frac{z}{z_{\mathrm{h}}} w_*^3 & {\mathrm{surface}\ layer} \\ 0.25 \, w_*^3 & {\mathrm{mixed}\ layer} \\ @@ -1533,7 +1533,7 @@ say). For the UM, .. math:: - Pr_{\mathrm{surf}} = \frac{\Phi_h}{\Phi_m} + Pr_{\mathrm{surf}} = \frac{\Phi_h}{\Phi_m} = \left( 1 + 16 \, k \frac{z}{z_{\mathrm{h}}} \, \frac{w_*^3}{u_*^3} \right)^{-1/4} @@ -1619,13 +1619,13 @@ Recall that for :math:`\theta_{\ell}` only we use .. math:: :label: wthl \overline{w'\theta_{\ell}'}= - K_h \frac{\partial \theta_{\ell}}{\partial z} - + K_h^{\mathrm{surf}}\gamma_{\theta_{\ell}} + + K_h^{\mathrm{surf}}\gamma_{\theta_{\ell}} where .. math:: :label: gradadj - \gamma_{\theta_{\ell}} = + \gamma_{\theta_{\ell}} = \mathrm{min}\left[ A_{ga} \frac{\sigma_{T1}}{z_{\mathrm{h}}}, G_{max} \right] @@ -1749,9 +1749,9 @@ So, the new formulation is written: .. math:: :label: fg_new - F_{\chi}^{Tot} = F_{\chi}^{NT}|_{z_{\mathrm{b}}} + F_{\chi}^{Tot} = F_{\chi}^{NT}|_{z_{\mathrm{b}}} -\left(K_h^{\mathrm{surf}}+ - K_h^{\mathrm{Sc}}\right)\frac{\partial\overline{\chi}}{\partial z} + K_h^{\mathrm{Sc}}\right)\frac{\partial\overline{\chi}}{\partial z} + \overline{w'\chi'}_{ng}^{\mathrm{surf}}+ \overline{w'\chi'}_{ng}^{\mathrm{Sc}} + f_2 \left(F_{\chi}|_{z_h} - F_{\chi}^{NT}|_{z_{\mathrm{b}}} \right) @@ -1809,7 +1809,7 @@ except that :math:`w_m` is replaced by its neutral value: .. math:: - Pr = Pr_{\mathrm{neut}} + Pr = Pr_{\mathrm{neut}} \frac{u_*^4 + w_*^3 {w_m}^{\mathrm{neut}} / 25} {u_*^4 + w_*^3 {w_m}^{\mathrm{neut}} Pr_{\mathrm{neut}}/ (25 Pr_{\mathrm{conv}} )} @@ -1829,11 +1829,11 @@ Discussion of some of the revisions * - Formulation - Convective limit - - + - - Neutral limit - - + - - * - + * - - :math:`w_m` - :math:`w_h` - :math:`w_m` @@ -1889,10 +1889,10 @@ gradient adjustment parameter: .. math:: :label: grad_adj - \gamma_{\chi}= d \frac{\overline{w'\chi'}_S}{w_* z_h} - \hspace{0.5cm} {\mathrm{with}} \hspace{0.5cm} - d^{HB} = 7.2 w_*^2/w_m^2, \hspace{0.2cm} - d^{std} = 6.3 w_*/w_m, \hspace{0.2cm} + \gamma_{\chi}= d \frac{\overline{w'\chi'}_S}{w_* z_h} + \hspace{0.5cm} {\mathrm{with}} \hspace{0.5cm} + d^{HB} = 7.2 w_*^2/w_m^2, \hspace{0.2cm} + d^{std} = 6.3 w_*/w_m, \hspace{0.2cm} d^{rev} = 10 w_*/w_h The inclusion of an extra :math:`w_*/w_m` factor in @@ -2146,7 +2146,7 @@ and transitioned between these regimes linearly using .. math:: - \beta = \beta_{\mathrm{bl}}\frac{z_{\mathrm{fa}}-z}{z_{\mathrm{fa}}-z_h} + + \beta = \beta_{\mathrm{bl}}\frac{z_{\mathrm{fa}}-z}{z_{\mathrm{fa}}-z_h} + \beta_{\mathrm{fa}}\frac{z-z_h}{z_{\mathrm{fa}}-z_h} However, because the above method still uses :eq:`eq-tanh`, @@ -2232,7 +2232,7 @@ the velocity scales `) .. math:: :label: we_parm w_e = \frac{ A_1 \, V_{\mathrm{sum}}^3/ z_{\mathrm{ml}}+ g \tilde{\beta_T} - \tilde{\alpha_t} + \tilde{\alpha_t} \Delta_F} {\Delta b + c_T V_{\mathrm{sum}}^2/z_{\mathrm{ml}}} @@ -2338,7 +2338,7 @@ mixed layers is simply calculated as .. math:: F_{\mathrm{net}}|_{z_{k+\frac{1}{2}}} = \sum_{k=\mathrm{ - \mathrm{NBDSC}}}^{k} \mathrm{max}\left[ + \mathrm{NBDSC}}}^{k} \mathrm{max}\left[ - \Delta_{k+\frac{1}{2}} z \, {\mathcal{S}}_F(k), \,0 \right] where NBDSC\ :math:`=1` in SMLs, :math:`{\mathcal{S}}_F` are the temperature @@ -2522,7 +2522,7 @@ z_{\mbox{\tiny \rm NTML}+\frac{3}{2}} - z_i` which can be written .. math:: :label: zi_interp - a (\Delta z_{disc})^2 + b \ \Delta z_{disc} +c =0 + a (\Delta z_{disc})^2 + b \ \Delta z_{disc} +c =0 The coefficients are given by @@ -2742,8 +2742,8 @@ gives .. math:: :label: fxtot_interp F_{\chi}^{Tot}|_{ \mathrm{ \mathrm{NTML}}+\frac{1}{2} } = - F_{\chi}^{Tot}|_{z_{\mathrm{b}}} + - \frac{ z'_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}} }{z_{\mathrm{ml}}} + F_{\chi}^{Tot}|_{z_{\mathrm{b}}} + + \frac{ z'_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}} }{z_{\mathrm{ml}}} \left( F_{\chi}^{Tot}|_{z_h} - F_{\chi}^{Tot}|_{z_{\mathrm{b}}} \right) where :math:`z'` (:math:`=z-z_{\mathrm{b}}`) is height above the base of the @@ -2753,7 +2753,7 @@ entrainment flux is given by: .. math:: :label: rev_entflux \overline{w'\chi'}|_{ \mathrm{ \mathrm{NTML}}+\frac{1}{2} } = - F_{\chi}^{Tot}|_{ \mathrm{ \mathrm{NTML}}+\frac{1}{2} } + F_{\chi}^{Tot}|_{ \mathrm{ \mathrm{NTML}}+\frac{1}{2} } - F_{\chi}^{NT}|_{ \mathrm{ \mathrm{NTML}}+\frac{1}{2} } @@ -2807,7 +2807,7 @@ implies .. math:: - F_{\chi}^{Tot}|_{ \mathrm{ \mathrm{NTML}}+\frac{1}{2} } + F_{\chi}^{Tot}|_{ \mathrm{ \mathrm{NTML}}+\frac{1}{2} } \left( 1+ \frac{\Delta z}{z_{ml}}\right) \geq F_{\chi}^{Tot}|_{ \mathrm{ \mathrm{NTML}}+\frac{3}{2} } + \Delta z \left( @@ -2849,7 +2849,7 @@ formula used is: .. math:: :label: dqt_disc_9c \Delta \chi = {\chi}_{\mathrm{ \mathrm{NTML}}+2} - {\chi}_{\mathrm{ - \mathrm{NTML}}} + \mathrm{NTML}}} - \gamma_{\chi} \left( z_{\mathrm{ \mathrm{NTML}}+2} - z_h \right) @@ -2993,8 +2993,8 @@ at the inversion base to zero at the inversion top, i.e.: .. math:: :label: ent_svl \overline{w'\theta_{v\ell}'} = \overline{w'\theta_{v\ell}'}|_{\mathrm{ - \mathrm{NTML}}+\frac{1}{2}} - cos\left(\pi \frac{z'}{2} \right) + \mathrm{NTML}}+\frac{1}{2}} + cos\left(\pi \frac{z'}{2} \right) where :math:`z'=(z-z_{\mathrm{h}})/\Delta z_i` is scaled height within the inversion. This flux profile is then converted into a diffusion @@ -3039,7 +3039,7 @@ equivalent entrainment eddy-diffusivity given by: K_{\chi}|_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}} = - \overline{w'\chi'}_{ z_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}} } \frac{\Delta_{\mathrm{ \mathrm{NTML}}+1} - z}{\Delta_{\mathrm{ \mathrm{NTML}}+1} \chi} + z}{\Delta_{\mathrm{ \mathrm{NTML}}+1} \chi} Note from :eq:`scal_closure` that :eq:`K_ent_tracer` gives the parametrized flux if @@ -3057,7 +3057,7 @@ ensured: .. math:: - 0 \leq K_{\chi}|_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}} + 0 \leq K_{\chi}|_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}} \leq 10 \,K_{\chi}|_{\mathrm{ \mathrm{NTML}}-\frac{1}{2}} Surface Exchange @@ -3143,7 +3143,7 @@ buoyancy flux in definition :eq:`1.1.4` is .. math:: :label: 1.1.11 \Delta B = g \beta _{T1} \left( {\Delta T + \frac{g}{ c_P }( z_1 + z_{0m} - - z_{0h} )} \right) + z_{0h} )} \right) + g \beta _{q1} \Delta q The **surface exchange coefficients** in @@ -3281,7 +3281,7 @@ which implies that .. math:: :label: 1.1.28 - L \sim -( \gamma _t^3 /k) z_i + L \sim -( \gamma _t^3 /k) z_i Thus the low wind speed limits for the sensible and latent heat fluxes are obtained by substituting :eq:`1.1.27` into @@ -3783,7 +3783,7 @@ where :math:`C_D` is the standard drag coefficient, \langle{\bf \tau}\rangle = \rho C_D \langle |\bar{\mathbf{u}} + {\mathbf{u}}_g({\mathbf{x}})| (\bar{\mathbf{u}} + {\mathbf{u}}_g({\mathbf{x}})) \rangle \approx \rho C_D - \langle + \langle |\bar{\mathbf{u}} + {\mathbf{u}}_g({\mathbf{x}})| \rangle \bar{\mathbf{u}}, which is the product of the enhanced wind speed (including gusts) and @@ -3910,7 +3910,7 @@ gases. Explicitly, we have .. math:: - \dot T_{ob, \mathrm{ rad, surf}} = \frac{4\sigma T_s^3}{c_P} + \dot T_{ob, \mathrm{ rad, surf}} = \frac{4\sigma T_s^3}{c_P} {\mathcal{K}}(z_{ob}) (T_s-T_{ob}), where :math:`\dot T_{ob, \mathrm{ rad, surf}}` is the cooling rate of @@ -4065,8 +4065,8 @@ In all schemes available here the momentum roughness length is given by .. math:: :label: eq:z0msea - z_{0m(sea)} = \frac{1.54\times {10}^{-6} }{ v_\ast } + - \frac{\alpha}{g} v_\ast ^2 + z_{0m(sea)} = \frac{1.54\times {10}^{-6} }{ v_\ast } + + \frac{\alpha}{g} v_\ast ^2 which is a generalisation of Charnock's formula to include low-wind conditions :raw-latex:`\cite[]{Smith88}`. :math:`\alpha` is Charnock's @@ -4565,7 +4565,7 @@ and }{\mathrm{\bf v}}{(} {z}_{c} {)} where the scaling velocity based on the stress over a flat surface, -v\ :math:`_{\ast +v\ :math:`_{\ast (f)}` , is given by .. math:: :label: 2.1.8 @@ -5028,9 +5028,9 @@ namely: .. math:: :label: eq:draglow - \frac{\mathbf{F}_p}{\rho_0} = {\left( {\frac{\kappa}{\zeta_m}} \right)}^{2} - \alpha \beta \pi ^{2} {f}_{D} (Ri_{B}) - {\left( {\frac{A}{S}} \right)}^{2} + \frac{\mathbf{F}_p}{\rho_0} = {\left( {\frac{\kappa}{\zeta_m}} \right)}^{2} + \alpha \beta \pi ^{2} {f}_{D} (Ri_{B}) + {\left( {\frac{A}{S}} \right)}^{2} \left\vert{\mathrm{\bf v}}(\ell) \right\vert{\mathrm{\bf v}}(\ell), where :math:`\zeta_m = {\mathrm{log}}(\ell/z_{0m})`. There is also an option @@ -5047,7 +5047,7 @@ the scale height, :math:`\ell`: .. math:: - Ri_{B \ell} = \frac{ \ell \left( g \left( + Ri_{B \ell} = \frac{ \ell \left( g \left( \overline{\beta_T}_{k\ell} ({\theta_{\ell}}_{k\ell}-{\theta_{\ell}}_{1}) + \overline{\beta_q}_{k\ell} ({q_t}_{k\ell}-{q_t}_{1}) \right) + \Delta b_{SL} \right) }{U^2(\ell)} @@ -5099,8 +5099,8 @@ assumed to be positive. The new scheme is written .. math:: :label: eq:sppf2 \frac{X^{n+1}-X^{*}}{\Delta - t}=-{\mathcal{I}}_{2}\left[K\left(X^{n}\right)^{P}\right]X^{n+1} + - {\mathcal{E}}_{2}\left[K\left(X^{n}\right)^{P}\right]X^{*} + + t}=-{\mathcal{I}}_{2}\left[K\left(X^{n}\right)^{P}\right]X^{n+1} + + {\mathcal{E}}_{2}\left[K\left(X^{n}\right)^{P}\right]X^{*} + \left({\mathcal{I}}_{2}-{\mathcal{E}}_{2}\right)S, where @@ -5176,7 +5176,7 @@ X^{n+1}=X^{n+1}-X^{*}`. Then, .. math:: - F^{*}=F^{n}+K_{X}\frac{\partial\delta X}{\partial z}^{*},\qquad + F^{*}=F^{n}+K_{X}\frac{\partial\delta X}{\partial z}^{*},\qquad F^{n+1}=F^{*}+K_{X}\frac{\partial\delta X}{\partial z}^{n+1}. Writing equations :eq:`eq:sppf_bl1`, @@ -5549,33 +5549,33 @@ calculations take place for the scalar variables: - set up coefficients for :eq:`eq:dX_disc_top`, :eq:`eq:dX_disc` and do a downward sweep; - * - + * - - do a downward sweep using the original implicit scheme to - * - + * - - compute information required by the surface implicit solver; * - CALL sf_impl2(): - CALL im_sf_pt2(): compute :math:`F_{JULES}` (scalar implicit fluxes), - * - + * - - using original surface implicit solver; * - CALL bdy_impl4(): - set up :eq:`eq:dX_bottom` and complete downward sweep; - * - + * - - back substitute to compute implicit correction :math:`\delta X^{*}`; * - CALL bdy_impl3(): - compute explicit flux :math:`F^*=F^n+K_X\frac{\partial \delta X^*}{\partial z}`; - * - + * - - set up coefficients for :eq:`eq:dXtop_np1`, :eq:`eq:dXk_np1`, :eq:`eq:dX1_np1` and - * - + * - - do a downward sweep; * - CALL sf_impl2(): @@ -5963,9 +5963,8 @@ proportional to the standard deviation of the horizontal wind, .. math:: :label: windgust - U_{gust} = U_{10m} + W_{1D} \, \sigma_u \, \frac{1}{k} \, - {\mathrm{log}}\left( \frac{5 \, e^{k \, c_{\mathrm{ugn}}} + z_{0m(eff)} } - + U_{gust} = U_{10m} + W_{1D} \, \sigma_u \, \frac{1}{k} \, + {\mathrm{log}}\left( \frac{5 \, e^{k \, c_{\mathrm{ugn}}} + z_{0m(eff)} } {5 + z_{0m(eff)}} \right) The factor :math:`W_{1D}` is included only in the scale-dependent @@ -6137,7 +6136,7 @@ i.e., .. math:: :label: tke_diag_nl - e_{nl} = \frac{3}{2} \left( \frac{K_m^{\mathrm{surf}}}{\tau_{\mathrm{surf}}} + e_{nl} = \frac{3}{2} \left( \frac{K_m^{\mathrm{surf}}}{\tau_{\mathrm{surf}}} + \frac{K_m^{\mathrm{Sc}}}{\tau_{\mathrm{Sc}}} \right) The factor of :math:`3/2` in :eq:`tke_diag_nl` arises @@ -6269,7 +6268,7 @@ calculated as .. math:: :label: zcld_calc \tilde{z_c} = \sum_{k=1}^{NTML+1} \left( {C_F}_k - \frac{\Delta_{k+\frac{1}{2}} z}{2} + \frac{\Delta_{k+\frac{1}{2}} z}{2} + \mathrm{min}\left[C_F^l\frac{\Delta_{k+\frac{1}{2}} z}{2}, \frac{q_{\ell}}{\gamma_{q_{\ell}}} \right] + \mathrm{min}\left[C_F^f\frac{\Delta_{k+\frac{1}{2}} z}{2}, \frac{q_f}{\gamma_{q_f}} \right] \right) @@ -6363,9 +6362,9 @@ inversion is given by .. math:: :label: dbinv \Delta b = g \, \left( \beta_T \Delta \theta_{\ell}+ \beta_q \Delta q_t + - \left( \beta_T \frac{L}{c_p} - - \frac{1+c_v}{c_v}\beta_q \right)\Delta q_{\ell}+ - \left( \beta_T \frac{L_s}{c_p} - + \left( \beta_T \frac{L}{c_p} - + \frac{1+c_v}{c_v}\beta_q \right)\Delta q_{\ell}+ + \left( \beta_T \frac{L_s}{c_p} - \frac{1+c_v}{c_v}\beta_q \right)\Delta q_f \right) The empirical constant :math:`A_{\mathrm{br}}= 0.24`. The calculation of @@ -6464,7 +6463,7 @@ In the 9B version, :math:`\Delta_F` is calculated as: .. math:: :label: ctraddiv - \Delta_F= \sum_{k=k_m-1}^{k_m+1} \mathrm{max}\left[ + \Delta_F= \sum_{k=k_m-1}^{k_m+1} \mathrm{max}\left[ - \Delta_{k+\frac{1}{2}} z \, {\mathcal{S}}_F(k), \,0 \right] where :math:`k_m` is the grid-level with the greatest radiative cooling @@ -6623,7 +6622,7 @@ Linearising gives .. math:: \overline{w'b}= g \left( \beta_T \overline{w'T_L'} + \beta_q - \overline{w'q_t'}+ + \overline{w'q_t'}+ \left( \beta_T \frac{L}{c_p} - \frac{1+c_v}{c_v} \beta_q \right) \overline{w'q_{\ell}'} \right) @@ -6652,7 +6651,7 @@ can be written .. math:: - \overline{w'b}= + \overline{w'b}= \begin{cases} g \left( \beta_T \overline{w'T_L'} + \beta_q \overline{w'q_t'}\right) & {\mathrm{in}\ unsaturated\ air} \\ @@ -6708,7 +6707,7 @@ spherical geometry for simplicity: .. math:: :label: moisture_cons - \int (\rho_v + \rho_{\ell}+ \rho_{f})\, d\underline{x} = + \int (\rho_v + \rho_{\ell}+ \rho_{f})\, d\underline{x} = \int \rho (q_v + q_{\ell}+ q_{f}) \,d\underline{x} = \int \rho q_t \, d\underline{x} @@ -6721,7 +6720,7 @@ terms of mixing ratios as .. math:: - \int (\rho_v + \rho_{\ell}+ \rho_{f})\, d\underline{x} = + \int (\rho_v + \rho_{\ell}+ \rho_{f})\, d\underline{x} = \int \rho_y (m_v + m_{\ell}+ m_{f}) \,d\underline{x} = \int \rho_y m_t \, d\underline{x} @@ -6757,7 +6756,7 @@ ratios as: .. math:: :label: sl_defn \theta_{\ell}= T - \frac{L_c}{c_{pd}} m_{\ell} - - \frac{L_c+L_f}{c_{pd}} m_f + \frac{g}{c_{pd}} z + - \frac{L_c+L_f}{c_{pd}} m_f + \frac{g}{c_{pd}} z For saturation calculations a version of QSAT is used that is switchable between input specific and mixing ratio variables. The rate of change of @@ -6879,9 +6878,9 @@ the UKCA code owner before lodging the change. * - Boundary layer inputs to UKCA - - - - - - + - + - + - * - Sec - Item @@ -7037,7 +7036,7 @@ Appendix: Notation * - Finite difference notation - - + - * - :math:`z_k` - height of the :math:`\theta`-level :math:`k` @@ -7057,14 +7056,14 @@ Appendix: Notation - note: real change (i.e., not necessarily finite-difference) in a parameter - * - + * - - across the capping inversion (see :eq:`dbinv` and following text) .. list-table:: * - Model variables - - + - * - :math:`\theta_l`, :math:`\theta_{v\ell}` - thermodynamic variables defined by :eq:`thetal` and :eq:`thetavl` @@ -7072,7 +7071,7 @@ Appendix: Notation * - :math:`T_v`, :math:`\theta_v` - virtual temperature and potential temperature, - * - + * - - defined by :eq:`Tv` and in section (:ref:`Calculation of parcel buoyancy excess `) @@ -7082,7 +7081,7 @@ Appendix: Notation * - :math:`q_t`, :math:`q_v`, :math:`q_s`, :math:`q_{\ell}`, :math:`q_f` - specific humidities: - * - + * - - total, vapour, saturated, liquid and frozen water, respectively * - :math:`C_F`, :math:`C_F^l`, :math:`C_F^f` @@ -7095,7 +7094,7 @@ Appendix: Notation * - Thresholds - - + - * - :math:`C_t` - (:math:`=1.1`) threshold for ratio of layer :math:`q_t`-gradients in @@ -7105,7 +7104,7 @@ Appendix: Notation - (:math:`=1.1`) threshold on ratio of environment to parcel :math:`\theta_v` gradients - * - + * - - for identifying capping inversions above the LCL * - SC_CFTOL @@ -7119,14 +7118,14 @@ Appendix: Notation - (:math:`=0.1`) threshold for the ratio of buoyancy consumption to production - * - + * - - before decoupling occurs .. list-table:: * - Layer definitions and parameters - - + - * - SML - surface-based mixed layer @@ -7164,7 +7163,7 @@ Appendix: Notation * - :math:`z_{\mathrm{loc}}` - height of half-level marking 'top' of local :math:`Ri`-based mixing - * - + * - - (where :math:`Ri>1`) * - :math:`z_i` @@ -7182,7 +7181,7 @@ Appendix: Notation * - :math:`K_m^{\mathrm{Sc}}`, :math:`K_h^{\mathrm{Sc}}` - :math:`K` profiles for cloud-top-driven turbulence - * - + * - - (calculated for both DSC and SML) * - LCL @@ -7192,7 +7191,7 @@ Appendix: Notation * - Other parameters - - + - * - :math:`\gamma_{\theta_{\ell}}` - gradient adjustment term, given by :eq:`gradadj` @@ -7200,14 +7199,14 @@ Appendix: Notation * - :math:`w_m` - scaling velocity for momentum mixing in the SML - * - + * - - (used in :math:`K_m^{\mathrm{surf}}`, :math:`\gamma_{\theta_{\ell}}` and the SML parcel perturbation, :math:`\theta_v'`) * - :math:`w_*` - 'standard' convective velocity scale for a cloud-free convective - * - + * - - boundary layer, :math:`w_*^3 = z_{\mathrm{h}}\overline{w'b}_S` * - :math:`u_*` @@ -7229,7 +7228,7 @@ Appendix: Notation * - :math:`\tau_{rc}`, :math:`z_{rc}` - parameters in perturbation calculation, :eq:`dscd_pert`, - * - + * - - for initial identification of and :math:`z_{\mathrm{ml}}` calculation for DSC layers From 7f4dc5618d823ef85b832fa4e29ab6c6aecc0626 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Wed, 20 May 2026 15:06:00 +0100 Subject: [PATCH 095/116] Changed format of UMDP reference placeholders to avoid lint warnings. --- .../science_guide/turbulence_schemes/bl_scheme_doc.rst | 10 +++++----- 1 file changed, 5 insertions(+), 5 deletions(-) diff --git a/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst b/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst index c47051ee46..2e8ac1f56b 100644 --- a/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst +++ b/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst @@ -42,7 +42,7 @@ the UM. Several options for higher order closures are available in the 1A version of the UM boundary layer scheme and these are documented -separately in :umdp:'025'. +separately in `:umdp:`025``. .. _sec_closure: @@ -1942,7 +1942,7 @@ The blended scheme ================== For high resolution simulations, the UM has a Smagorinsky-type subgrid -turbulence scheme, described in :umdp:'028'. However, this scheme is +turbulence scheme, described in `:umdp:`028``. However, this scheme is only truly applicable for horizontal grid-lengths of order :math:`10` m, and any real-world simulation run at lower resolution than this will inevitably have unresolved scales somewhere in the domain. Rather than @@ -2184,7 +2184,7 @@ For current operational convection-permitting model grid sizes (1.5 km in the UKV), the representation of cumulus convection remains a challenge. One option is to include a grey-zone convection parametrization, described in the documentation of that scheme (see -:umdp:'027'). Tests in the UKV, though, showed some detriment to the +`:umdp:`027``). Tests in the UKV, though, showed some detriment to the spin-up of resolved scale convection (as well as somewhat poor discrimination of precipitating versus non-precipitating parametrized convection) that led to the development of an alternative strategy, @@ -6127,7 +6127,7 @@ where :math:`C_{e}=A_{2N}^{3/2}`. Initial tests found that the MONC value of :math:`A_{2N}=0.23` gave rather large values of :math:`e_{loc}` and so :math:`C_e=0.41` is used. Note that this is an optional value used in the higher order closure scheme (see section 2.8.5 of -:umdp:'025'). It could also be worth testing the suggested +`:umdp:`025``). It could also be worth testing the suggested parametrization in (2.300) there, of :math:`C_e=0.19+0.74 \lambda/\Delta z` but this has not yet been attempted. The total non-local TKE is computed by adding the TKE from @@ -6171,7 +6171,7 @@ also corrects a bug in the level indexing of this diagnostic when passed to UKCA. Note that additional diagnostics of the scalar variances are also made -and those are documented in :umdp:'029'. +and those are documented in `:umdp:`029``. .. _app_neutwind: From 8feae1a5121852297763a30a9954a1a47ee7fbe4 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Wed, 20 May 2026 15:36:19 +0100 Subject: [PATCH 096/116] Remove further trailing white-space instances created by splitting long lines at double-spaces. --- .../turbulence_schemes/bl_scheme_doc.rst | 42 +++++++++---------- 1 file changed, 21 insertions(+), 21 deletions(-) diff --git a/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst b/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst index 2e8ac1f56b..40be6bfd3f 100644 --- a/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst +++ b/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst @@ -587,7 +587,7 @@ expanded using the first-order closure in .. math:: :label: eq:wx_std \begin{aligned} - \overline{w'\theta_{\ell}'}_k &=& + \overline{w'\theta_{\ell}'}_k &=& -K_h^{\mathrm{surf}}\,\frac{\widetilde{\Delta_k \theta_{\ell}}}{\Delta_k z} -K_h^{\mathrm{Sc}}\,\frac{\Delta_k \theta_{\ell}}{\Delta_k z} \\ @@ -2786,7 +2786,7 @@ mixed layer by the end of the timestep. In other words, for \chi_{\mathrm{ \mathrm{NTML}}+1}^{n+1} = \chi_{\mathrm{ \mathrm{NTML}}+1}^{n} - \frac{\Delta t}{\Delta z} \left( - F_{\chi}^{Tot}|_{ \mathrm{ \mathrm{NTML}}+\frac{3}{2} } - + F_{\chi}^{Tot}|_{ \mathrm{ \mathrm{NTML}}+\frac{3}{2} } - F_{\chi}^{Tot}|_{ \mathrm{ \mathrm{NTML}}+\frac{1}{2} } \right) @@ -2794,7 +2794,7 @@ mixed layer by the end of the timestep. In other words, for \chi_{\mathrm{ \mathrm{NTML}}}^{n+1} = \chi_{\mathrm{ \mathrm{NTML}}}^{n} - \frac{\Delta t}{z_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}}} \left( - F_{\chi}^{Tot}|_{ \mathrm{ \mathrm{NTML}}+\frac{1}{2} } - + F_{\chi}^{Tot}|_{ \mathrm{ \mathrm{NTML}}+\frac{1}{2} } - F_{\chi}^{Tot}|_{z_{\mathrm{b}}} \right) @@ -3077,7 +3077,7 @@ layer are related to the surface fluxes by: .. math:: :label: 1.1.1 - \frac{\partial T}{\partial z} + \frac{g}{ c_P }=-\frac{ H_0 }{ c_P \rho _0 + \frac{\partial T}{\partial z} + \frac{g}{ c_P }=-\frac{ H_0 }{ c_P \rho _0 v_\ast } \frac{ \phi _h (z/L)}{kz} .. math:: :label: 1.1.2 @@ -3581,7 +3581,7 @@ obtain: .. math:: :label: 1.3.17 - \Phi _m = \ln \left( {\frac{ z_1 + z_{0m} }{ z_{0m} }} \right) - 2 \ln + \Phi _m = \ln \left( {\frac{ z_1 + z_{0m} }{ z_{0m} }} \right) - 2 \ln \left( {\frac{1 + X_1 }{1 + X_0 }} \right) - \ln \left( {\frac{1 + X_1^2 }{1 + X_0^2 }} \right)+ 2 \left( { {\tan }^{-1} X_1 - {\tan }^{-1} X_0 } \right) @@ -3596,7 +3596,7 @@ and .. math:: :label: 1.3.19 - \Phi _h = \ln \left( {\frac{ z_1 + z_{0m} }{ z_{0h} }} \right) - 2 \ln + \Phi _h = \ln \left( {\frac{ z_1 + z_{0m} }{ z_{0h} }} \right) - 2 \ln \left( {\frac{1 + Y_1 }{1 + Y_0 }} \right) where @@ -3718,7 +3718,7 @@ stress: .. math:: :label: 1.4.19 - H_0= {-c}_P \rho _0 C_H^{(N)} \left( {\Delta T + \frac{g}{ c_P }( z_1 + + H_0= {-c}_P \rho _0 C_H^{(N)} \left( {\Delta T + \frac{g}{ c_P }( z_1 + z_{0m} - z_{0h} )} \right) .. math:: :label: 1.4.20 @@ -3863,7 +3863,7 @@ z\ :math:`_{ob, }` we obtain for the scalar :math:`X` .. math:: :label: 1.5.3 - X_{ob} = X_0 + \frac{ F_{X0} }{ \rho _0 v_\ast k} \Phi _h (L, z_{ob} + + X_{ob} = X_0 + \frac{ F_{X0} }{ \rho _0 v_\ast k} \Phi _h (L, z_{ob} + z_{0h} , z_{0h} ) and using the expression for the surface flux :math:`F_{X0}` of the @@ -3871,7 +3871,7 @@ scalar quantity :math:`X` this gives the interpolation formula .. math:: :label: 1.5.4 - X_{ob} = X_0 + \frac{ C_H }{k v_\ast } \Phi _h (L, z_{ob} + z_{0h} , + X_{ob} = X_0 + \frac{ C_H }{k v_\ast } \Phi _h (L, z_{ob} + z_{0h} , z_{0h} ) ( X_1 - X_0 ) For temperature and humidity z\ :math:`_{ob}` is set to the screen @@ -4508,7 +4508,7 @@ When form drag is included via effective roughness lengths equations .. math:: :label: 2.1.1 \begin{aligned} - \frac{ H_{0(eff)} }{ c_P \rho _0 }&=&\frac{-k}{ \Phi _h (L , z_1 + + \frac{ H_{0(eff)} }{ c_P \rho _0 }&=&\frac{-k}{ \Phi _h (L , z_1 + z_{0m(eff)} , z_{0h(eff)} )} v_{\ast (eff)} \\ && \left( {\Delta T + \frac{g}{c_p }( z_1 + z_{0m(eff)} - z_{0h(eff)} )} @@ -4570,7 +4570,7 @@ v\ :math:`_{\ast .. math:: :label: 2.1.8 - v_{\ast (f)}^2 = u_{\ast (f)}^2 + \gamma _t^2 w_\ast ^2 + \gamma _c^2 + v_{\ast (f)}^2 = u_{\ast (f)}^2 + \gamma _t^2 w_\ast ^2 + \gamma _c^2 w_c^2 with @@ -4609,7 +4609,7 @@ effective momentum roughness is derived .. math:: :label: 2.1.12 \frac{\ln ( z_c / z_{0m(eff)} )}{\ln ( z_c / z_{0m} )} = {\left( {1 + - \frac{1}{2} c_{D(orog)} f_D \frac{A}{S} {\left( {\frac{\ln ( z_c / + \frac{1}{2} c_{D(orog)} f_D \frac{A}{S} {\left( {\frac{\ln ( z_c / z_{0m} )}{k}} \right)}^2 } \right)}^{-1/2} The stress for the flat surface is related to the total stress by @@ -4672,7 +4672,7 @@ z\ :math:`_{c}` is .. math:: :label: 2.1.18 - \frac{ F_{X0(eff)} }{ \rho _0 } = \frac{k v_{\ast (eff)} }{ \Phi _h (L , + \frac{ F_{X0(eff)} }{ \rho _0 } = \frac{k v_{\ast (eff)} }{ \Phi _h (L , z_c , z_{0h(eff)} )} (X( z_c ) - X_0 ) and the surface flux for the flat surface is given by @@ -4696,7 +4696,7 @@ effective scalar roughness length is derived as .. math:: :label: 2.1.21) \frac{\ln ( z_c / z_{0h(eff)} )}{\ln ( z_c / z_{0h} )} = {\left( {1 + - \frac{1}{2} c_{D(orog)} f_D \frac{A}{S} {\left( {\frac{\ln ( z_c / + \frac{1}{2} c_{D(orog)} f_D \frac{A}{S} {\left( {\frac{\ln ( z_c / z_{0m} )}{k}} \right)}^2 } \right)}^{1/2} \left( {1 - 2.2 f_D \frac{A}{S}} \right) @@ -4735,7 +4735,7 @@ iteration from the convective limit, so .. math:: :label: (2.2.4 v_{\ast (eff)}^{(0)}= v_{\ast (f)}^{(0)} = {\left[ { \gamma _t^3 \left( - {\frac{k}{ \Phi _h^{(0)} }} \right) z_i \left| {-\Delta B} \right| + + {\frac{k}{ \Phi _h^{(0)} }} \right) z_i \left| {-\Delta B} \right| + \gamma _c^2 w_c^2 } \right]}^{ 1/2} ELSE IF (:math:`\Delta`\ **v** :math:`\ge` 2 ms\ :math:`^{-1}` ) start @@ -4823,7 +4823,7 @@ DO n = 1 to N .. math:: :label: (2.2.20 - v_{\ast (f)}^{(n)2}= u_{\ast (f)}^{(n)2} + \gamma _t^2 w_\ast ^{(n)2} + + v_{\ast (f)}^{(n)2}= u_{\ast (f)}^{(n)2} + \gamma _t^2 w_\ast ^{(n)2} + \gamma _c^2 w_c^2 .. math:: :label: (2.2.21 @@ -4960,7 +4960,7 @@ then .. math:: :label: 2.3.7 - X_{ob} = X_0 + \frac{ C_{H(f)} }{k v_{\ast (f)} } \Phi _h (L, z_{ob} + + X_{ob} = X_0 + \frac{ C_{H(f)} }{k v_{\ast (f)} } \Phi _h (L, z_{ob} + z_{0h} , z_{0h} ) ( X_1 - X_0 ) For temperature and humidity z\ :math:`_{ob}` is set to the screen @@ -5184,14 +5184,14 @@ Writing equations :eq:`eq:sppf_bl1`, .. math:: :label: eq:sppf_inc1 - \frac{\delta X}{\Delta t}^{*} = + \frac{\delta X}{\Delta t}^{*} = ({\mathcal{I}}_{1}-{\mathcal{E}}_{1})\left(\frac{\partial F}{\partial z}^{n}+S\right)+{\mathcal{I}}_{1}\frac{\partial}{\partial z}\left(K_{X}\frac{\partial\delta X}{\partial z}^{*}\right) .. math:: :label: eq:sppf_inc2 - \frac{\delta X}{\Delta t}^{n+1} = + \frac{\delta X}{\Delta t}^{n+1} = ({\mathcal{I}}_{2}-{\mathcal{E}}_{2})\left(\frac{\partial F}{\partial z}^{*}+S\right)+{\mathcal{I}}_{2}\frac{\partial}{\partial z}\left(K_{X}\frac{\partial\delta X}{\partial z}^{n+1}\right) @@ -5406,7 +5406,7 @@ and thus the following discretization is obtained, on .. math:: - \frac{\delta X_{k}^{*}}{\Delta t} = + \frac{\delta X_{k}^{*}}{\Delta t} = \left({\mathcal{I}}_{1}-{\mathcal{E}}_{1}\right)\left(\frac{F_{k+1/2}^{n}-F_{k-1/2}^{n}}{z_{k+1/2}-z_{k-1/2}}+S_{k}\right) .. math:: @@ -5713,7 +5713,7 @@ flux for :math:`H` is derived: .. math:: - \frac{H_{j}^{*}}{c_{p}} = + \frac{H_{j}^{*}}{c_{p}} = \gamma_{2}\frac{H_{j}^{(n)}}{c_{p}}-\gamma_{1}RK_{PMj}[LD_{j}\psi_{j}RK_{H}(1)_{j}+A_{*j}][c_{p}\delta{T'}_{1}-\beta\overline{H^{*}}] .. math:: From 72f006b1df08e72dc88f62ff77641ddc919412cf Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Wed, 20 May 2026 15:56:08 +0100 Subject: [PATCH 097/116] Fixed remaining latex custom macro instances \zh, zhpar imported in figure caption, and remaining instances of latex math \rm (meant to replace with \mathrm). --- .../turbulence_schemes/bl_scheme_doc.rst | 10 +++++----- .../turbulence_schemes/manual_corrections.txt | 17 ++++++++++++++--- 2 files changed, 19 insertions(+), 8 deletions(-) diff --git a/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst b/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst index 40be6bfd3f..63fc5eafe7 100644 --- a/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst +++ b/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst @@ -257,8 +257,8 @@ Types I to VI are shown schematically in :numref:`Fig. %s `. :name: fig:bltypes Schematic representation of boundary layer types I to VI. The top of the - upward arrows indicate the height \zhpar while the top of their solid line - portions indicate \zh. + upward arrows indicate the height :math:`z_{\mathrm{par}}` while the top of + their solid line portions indicate :math:`z_{\mathrm{h}}`. .. list-table:: :align: center @@ -1926,9 +1926,9 @@ removed since the entrainment flux is now carried via the explicit .. figure:: blank.svg :name: fig:new_ksc - Standard UM :math:`K_h^{\rm Sc}` (solid) and revised (dotted), both scaled - by :math:`k z_h V_{\rm Sc}`. An upside-down version of - :math:`K_h^{\rm surf}` is also shown (dashed) for comparison. + Standard UM :math:`K_h^{\mathrm{Sc}}` (solid) and revised (dotted), both + scaled by :math:`k z_h V_{\mathrm{Sc}}`. An upside-down version of + :math:`K_h^{\mathrm{surf}}` is also shown (dashed) for comparison. .. list-table:: :align: center diff --git a/documentation/source/science_guide/turbulence_schemes/manual_corrections.txt b/documentation/source/science_guide/turbulence_schemes/manual_corrections.txt index a126a907ba..00d47bd8da 100644 --- a/documentation/source/science_guide/turbulence_schemes/manual_corrections.txt +++ b/documentation/source/science_guide/turbulence_schemes/manual_corrections.txt @@ -2,6 +2,17 @@ diff --git a/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc index 5b373b68..8ff32c06 100644 --- a/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst +++ b/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst +@@ -257,8 +257,8 @@ Types I to VI are shown schematically in :numref:`Fig. %s `. + :name: fig:bltypes + + Schematic representation of boundary layer types I to VI. The top of the +- upward arrows indicate the height \zhpar while the top of their solid line +- portions indicate \zh. ++ upward arrows indicate the height :math:`z_{\mathrm{par}}` while the top of ++ their solid line portions indicate :math:`z_{\mathrm{h}}`. + + .. list-table:: + :align: center @@ -1810,6 +1810,7 @@ Discussion of some of the revisions .. list-table:: Convective and Neutral limits for velocity scales @@ -41,9 +52,9 @@ index 5b373b68..8ff32c06 100644 - Standard UM :math:`\khtop` (solid) and revised (dotted), both scaled by - :math:`k z_h \vtopo`. An upside-down version of :math:`\khsurf` is also - shown (dashed) for comparison. -+ Standard UM :math:`K_h^{\rm Sc}` (solid) and revised (dotted), both scaled -+ by :math:`k z_h V_{\rm Sc}`. An upside-down version of -+ :math:`K_h^{\rm surf}` is also shown (dashed) for comparison. ++ Standard UM :math:`K_h^{\mathrm{Sc}}` (solid) and revised (dotted), both ++ scaled by :math:`k z_h V_{\mathrm{Sc}}`. An upside-down version of ++ :math:`K_h^{\mathrm{surf}}` is also shown (dashed) for comparison. .. list-table:: :align: center From 65c766e1b37c1cb8c7599eb89a0bfa7c05499572 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Wed, 20 May 2026 21:53:36 +0100 Subject: [PATCH 098/116] Further change to UMDP reference placeholder format. --- .../science_guide/turbulence_schemes/bl_scheme_doc.rst | 10 +++++----- 1 file changed, 5 insertions(+), 5 deletions(-) diff --git a/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst b/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst index 63fc5eafe7..300bbaf98c 100644 --- a/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst +++ b/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst @@ -42,7 +42,7 @@ the UM. Several options for higher order closures are available in the 1A version of the UM boundary layer scheme and these are documented -separately in `:umdp:`025``. +separately in ``:umdp:025``. .. _sec_closure: @@ -1942,7 +1942,7 @@ The blended scheme ================== For high resolution simulations, the UM has a Smagorinsky-type subgrid -turbulence scheme, described in `:umdp:`028``. However, this scheme is +turbulence scheme, described in ``:umdp:028``. However, this scheme is only truly applicable for horizontal grid-lengths of order :math:`10` m, and any real-world simulation run at lower resolution than this will inevitably have unresolved scales somewhere in the domain. Rather than @@ -2184,7 +2184,7 @@ For current operational convection-permitting model grid sizes (1.5 km in the UKV), the representation of cumulus convection remains a challenge. One option is to include a grey-zone convection parametrization, described in the documentation of that scheme (see -`:umdp:`027``). Tests in the UKV, though, showed some detriment to the +``:umdp:027``). Tests in the UKV, though, showed some detriment to the spin-up of resolved scale convection (as well as somewhat poor discrimination of precipitating versus non-precipitating parametrized convection) that led to the development of an alternative strategy, @@ -6127,7 +6127,7 @@ where :math:`C_{e}=A_{2N}^{3/2}`. Initial tests found that the MONC value of :math:`A_{2N}=0.23` gave rather large values of :math:`e_{loc}` and so :math:`C_e=0.41` is used. Note that this is an optional value used in the higher order closure scheme (see section 2.8.5 of -`:umdp:`025``). It could also be worth testing the suggested +``:umdp:025``). It could also be worth testing the suggested parametrization in (2.300) there, of :math:`C_e=0.19+0.74 \lambda/\Delta z` but this has not yet been attempted. The total non-local TKE is computed by adding the TKE from @@ -6171,7 +6171,7 @@ also corrects a bug in the level indexing of this diagnostic when passed to UKCA. Note that additional diagnostics of the scalar variances are also made -and those are documented in `:umdp:`029``. +and those are documented in ``:umdp:029``. .. _app_neutwind: From c6de8149e0f39afe974ad2a1d4844839fd8a60f7 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Wed, 20 May 2026 22:35:11 +0100 Subject: [PATCH 099/116] Applied latest version of post-pandoc corrections script. Changes: (a) Restored aligned math regions where the contained equations are meant to be labeled as a group, (b) Deleted spurious trailing whitespace, (c) Changed format of UMDP cross-reference placeholders to avoid lint warnings. --- .../cloud_schemes/UMDP30_PC2CloudScheme.rst | 342 ++++++++---------- 1 file changed, 152 insertions(+), 190 deletions(-) diff --git a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst index 558d1c349c..a3e5861088 100644 --- a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst +++ b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst @@ -93,7 +93,7 @@ valid structures to use in this respect.* - One may diagnose cloud fractions and condensate contents from knowledge of gridbox mean variables. This forms the basis of the - `Smith (1990)`_ scheme, which is described in :umdp:'029'. + `Smith (1990)`_ scheme, which is described in ``:umdp:029``. - A mixed scheme, such as `Sundqvist (1978)`_ uses a prediction of condensate contents, but a diagnostic cloud fraction. @@ -170,7 +170,7 @@ does not). We now introduce the liquid temperature (:math:`T_L`), where .. math:: :label: eq:tl - T_L = T - \frac{L}{c_p} q_{cl} , + T_L = T - \frac{L}{c_p} q_{cl} , and :math:`L` is the latent heat of vaporization and :math:`c_p` is the heat capacity of air. Note that :math:`T_L` is unaffected by changes of @@ -310,7 +310,7 @@ three-dimensional distribution in terms of three separate variables :math:`q_T`, :math:`T_L` and :math:`p`. This is the method used by `Smith (1990)`_, where a symmetric triangular distribution function is used. For further information on the -`Smith (1990)`_ scheme, please refer to :umdp:'029'. Physics +`Smith (1990)`_ scheme, please refer to ``:umdp:029``. Physics and dynamics schemes hence only need to provide increments to :math:`\overline{q_T}` and :math:`\overline{T_L}`, provided that a diagnostic scheme (such as `Smith (1990)`_) is called at @@ -336,45 +336,39 @@ distinct aspects of clouds, which may or may not overlap :numref:`Figure %s The equations for the five prognostic cloud variables can be written schematically: -.. math:: +.. math:: :label: eq:dqcldt_and_dcdt + \begin{aligned} \frac{\partial \overline{q_{cl}}}{\partial t} = \frac{\partial \overline{q_{cl}}}{\partial t} |_{advection} + \frac{\partial \overline{q_{cl}}}{\partial t} |_{convection} + \frac{\partial \overline{q_{cl}}}{\partial t} |_{boundary \, layer} + \frac{\partial \overline{q_{cl}}}{\partial t} |_{precipitation} + ... - -.. math:: - + \\ \frac{\partial \overline{q_{cf}}}{\partial t} = \frac{\partial \overline{q_{cf}}}{\partial t} |_{advection} + \frac{\partial \overline{q_{cf}}}{\partial t} |_{convection} + \frac{\partial \overline{q_{cf}}}{\partial t} |_{boundary \, layer} + \frac{\partial \overline{q_{cf}}}{\partial t} |_{precipitation} + ... - -.. math:: - + \\ \frac{\partial C_l}{\partial t} = \frac{\partial C_l}{\partial t} |_{advection} + \frac{\partial C_l}{\partial t} |_{convection} + \frac{\partial C_l}{\partial t} |_{boundary \, layer} + \frac{\partial C_l}{\partial t} |_{precipitation} + ... - -.. math:: - + \\ \frac{\partial C_i}{\partial t} = \frac{\partial C_i}{\partial t} |_{advection} + \frac{\partial C_i}{\partial t} |_{convection} + \frac{\partial C_i}{\partial t} |_{boundary \, layer} + \frac{\partial C_i}{\partial t} |_{precipitation} + ... - -.. math:: :label: eq:dqcldt_and_dcdt - + \\ \frac{\partial C_t}{\partial t} = \frac{\partial C_t}{\partial t} |_{advection} + \frac{\partial C_t}{\partial t} |_{convection} + \frac{\partial C_t}{\partial t} |_{boundary \, layer} + \frac{\partial C_t}{\partial t} |_{precipitation} + ... , + \end{aligned} where :math:`\overline{q_{cf}}` is the ice water specfic humidity, :math:`C_l` is the liquid cloud *volume* fraction, :math:`C_i` is the @@ -550,7 +544,7 @@ power law near :math:`s=b_s`. .. math:: :label: eqn19 - G(s) ~ \propto ~ {(-s + b_s)}^n + G(s) ~ \propto ~ {(-s + b_s)}^n provided :math:`s` - * - + * - - dbsd tbs-conv - Redn of PDF width in co nvection - 0 - p c2-const - Phy: :ref:`Changing the width of the PDF - PC2 erosion ` - * - + * - - dbs dtbs-exp - V ariation of erosion on RH - 10.05 @@ -6089,7 +6051,7 @@ diagnostic output routines. * - :math:`RH_{crit}` - RHCRIT - Critical RH for cloud f ormation - - + - - UM namelist - Phy: :ref:`Initiation of cloud `, :ref:`Deposition and sublimation ` @@ -6108,28 +6070,28 @@ diagnostic output routines. - enviro?a - Num: :ref:`Numerical application ` - * - + * - - *Har d-wired* - Conv cloud fraction for vi sibility - 0.2 - imp-ctl2 - Diag: :ref:`Diagnostics ` - * - + * - - *Har d-wired* - Limit on width of ice dist ribution - 0.001 - lspice3d - Num: :ref:`Deposition and sublimation ` - * - + * - - *Har d-wired* - :math:`C_l` limit for init if :math:`T< 0 ^{\circ} C` - 0.05 - pc2-init - Num: :ref:`Initiation ` - * - + * - - *Har d-wired* - T olerance on calc. of :math:`q_C^s` in BL - :math:`1.0 \times 10^{-10} \, kg \,kg^{-1}` @@ -6250,7 +6212,7 @@ and hence, using the discrete form of :math:`\Delta Q_c` from .. math:: - \Delta \overline{q_{cl}} = C_l ( a_L ( \Delta \overline{q} - \alpha \Delta + \Delta \overline{q_{cl}} = C_l ( a_L ( \Delta \overline{q} - \alpha \Delta \overline{T} ) + \Delta \overline{q_{cl}} ) , which rearranges to From fd90e884ff07f8fa972ed156a842a192fae0e22b Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Sat, 23 May 2026 02:14:15 +0100 Subject: [PATCH 100/116] Further format corections: fixed inline math inside bold or italic regions, removed stray curly brackets in bibtex entries. --- .../turbulence_schemes/bl_scheme_doc.rst | 120 +++++++++--------- 1 file changed, 59 insertions(+), 61 deletions(-) diff --git a/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst b/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst index 300bbaf98c..5f7519e555 100644 --- a/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst +++ b/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst @@ -4287,10 +4287,10 @@ Two approaches are available in uncoupled configurations of the model. #. Explicit Treatment of Ice Form Drag - `L{\ (2012)`_ have suggested a simple + `L\ (2012)`_ have suggested a simple parametrization of the form drag coefficient of marginal ice that has been found to perform well in comparison to aircraft measurements - (`Elvidge et al. (2016)`_). `L{\ (2015)`_ + (`Elvidge et al. (2016)`_). `L\ (2015)`_ have extended the parametrization to include the effects of stability. When coupled to CICE, it is intended that a more elaborate scheme will be used, but this scheme is useful for application in @@ -4303,12 +4303,12 @@ Two approaches are available in uncoupled configurations of the model. :math:`u(z)` will in general exhibit a mixed character, but it may be taken as the developed flow over open sea, as in - `L{\ (2012)`_, or may be interpolated between the + `L\ (2012)`_, or may be interpolated between the developed flows over open sea or pack ice, depending on the ice - fraction, as in `L{\ (2015)`_. In principle, it will + fraction, as in `L\ (2015)`_. In principle, it will be subject to the effects of stability, but since the free-board does not much exceed 0.5m, these effects are small - (`L{\ (2015)`_) and the flow may be taken as neutral + (`L\ (2015)`_) and the flow may be taken as neutral up to :math:`h_f`. Hence, .. math:: :label: eq:int_u2 @@ -4321,18 +4321,18 @@ Two approaches are available in uncoupled configurations of the model. the wind on the model's lowest atmospheric level. Because this will be significantly above :math:`h_f`, the stability dependence of :math:`C_d` should be considered here (again see - `L{\ (2015)`_). :math:`U_1` may be interpreted as + `L\ (2015)`_). :math:`U_1` may be interpreted as the wind at a specific height, or, consistenly with the flux-difference form of the momentum equation, as the layer-averaged velocity. This distinction affects the numerical value of :math:`C_d`, but does not otherwise affect the foregoing equation. If using the original version of the scheme - (`L{\ (2012)`_), :math:`C_d` must be taken as the + (`L\ (2012)`_), :math:`C_d` must be taken as the neutral drag coefficient. Note also that various approximations may be made in Equation :eq:`eq:int_u2`. - `L{\ (2012)`_ approximate + `L\ (2012)`_ approximate :math:`(\log(h_f/z_0) -1)^2 +1` as :math:`(\log(h_f/z_0) )^2`; while - `L{\ (2015)`_ approximate it as + `L\ (2015)`_ approximate it as :math:`(\log(h_f/z_0) -1)^2`. Here we retain the full expression. If, in a unit area, there are :math:`N` floes, each of crosswind @@ -4369,19 +4369,19 @@ Two approaches are available in uncoupled configurations of the model. (1-A) C_{ds} L_s + A C_{di} L_i \right ], where the sheltering factor is taken to be the same over ice and - water. `L{\ (2012)`_ provides parametrizations for + water. `L\ (2012)`_ provides parametrizations for quantities such as :math:`h_f`, while `Elvidge et al. (2016)`_ provide suggested values for the constants in the scheme, based on observations. In using these values in the Unified Model, :math:`c_e` should be increased by about 30% to represent the effect of differing approximations of the logarithmic wind profile. - For scalar transfer `L{\ (2015)`_ suggest adding a + For scalar transfer `L\ (2015)`_ suggest adding a contribution to the sensible heat flux to represent the impact of form drag; however, the mechanistic physical basis of the scheme they propose is unclear. Moreover, when combined with the interfacial drag, this suggests scalar transfer much larger than observed by - `Schr{\ (2003)`_. Consequently, no enhancement of the + `Schr\ (2003)`_. Consequently, no enhancement of the scalar transfer coefficient by form drag is included. The overall drag coefficients are now set by interpolation in the ice @@ -4541,10 +4541,10 @@ where \right| The effective roughness for momentum is derived by setting the total -effective surface stress, **:math:`\tau`**\ :math:`_{0(eff)}`, to the +effective surface stress, :math:`\boldsymbol{\tau}`\ :math:`_{0(eff)}`, to the sum of the surface stress over a flat surface with the same vegetative -roughness, **:math:`\tau`**\ :math:`_{0(f)}`, and the orographic -pressure drag force at the surface, **:math:`\tau`**\ :math:`_{0(p)}`. +roughness, :math:`\boldsymbol{\tau}`\ :math:`_{0(f)}`, and the orographic +pressure drag force at the surface, :math:`\boldsymbol{\tau}`\ :math:`_{0(p)}`. The stresses are evaluated in terms of the velocity at height z\ :math:`_{c}` above the surface. z\ :math:`_{c}` is currently set to 2\ :math:`^{1/2}\sigma _{h}` where :math:`\sigma _{h}` is the standard @@ -5013,7 +5013,7 @@ constrained to be at least 100 m. If the steep-hill expression is to be used, the surface stress applied is almost identical to that used in the effective roughness -parametrization (Eq. :eq:`2.1.10`, namely: +parametrization (Eq. :eq:`2.1.10`), namely: .. math:: :label: eq:dragsteep @@ -5023,7 +5023,7 @@ parametrization (Eq. :eq:`2.1.10`, namely: the main difference being the dependence on the height scale :math:`\ell` rather than :math:`z_c`. Similarly, if the `Wood and Mason (1993)`_ low-hill expression is used, the surface -stress is given by the equivalent of (Eq. :eq:`2.1.16`, +stress is given by the equivalent of (Eq. :eq:`2.1.16`), namely: .. math:: :label: eq:draglow @@ -5797,7 +5797,7 @@ has to be adjusted if evaporation exhausts any of the moisture stores during the timestep or if the tile has a melting snowcover. Limited evaporation -^^^^^^^^^^^^^^^^^^^ +""""""""""""""""""" Downward surface moisture fluxes are added to canopy moisture or, if the surface temperature is below freezing, snowcover. @@ -5841,7 +5841,7 @@ surface) is not limited and does not draw on the conserved moisture stores. Snowmelt -^^^^^^^^ +"""""""" Classical surface energy balance neglects snowmelt heat fluxes. If :math:`T_*>T_m` for a snow-covered tile and sufficient snow is @@ -5879,8 +5879,8 @@ to the tile heat and moisture fluxes. The model with the above changes coded seems to work stably but the surface fluxes (and therefore the boundary layer increments) are only qualitatively correct. They seem to be overestimated by the above -scheme. [\ *Could it be that coefficients :math:`D_{j}`, -:math:`\psi_{j}` need to be modified in the second sweep?*] +scheme. [\ *Could it be that coefficients* :math:`D_{j}` *,* +:math:`\psi_{j}` *need to be modified in the second sweep?*] Blending height coupling ^^^^^^^^^^^^^^^^^^^^^^^^ @@ -7275,8 +7275,8 @@ References A. P. Lock and A. R. Brown and M. R. Bush and G. M. Martin and R. N. B. Smith (2000). - *{A New Boundary Layer Mixing Scheme. Part I: Scheme Description and - Single-Column Model Tests}*. + *A New Boundary Layer Mixing Scheme. Part I: Scheme Description and + Single-Column Model Tests*. Mon. Wea. Rev., 128, 3187-3199. .. _Lock et al. (2001): @@ -7290,15 +7290,15 @@ References G. M. Martin and M. R. Bush and A. R. Brown and A. P. Lock and R. N. B. Smith (2000). - *{A New Boundary Layer Mixing Scheme. Part II: Tests in Climate and - Mesoscale Models}*. + *A New Boundary Layer Mixing Scheme. Part II: Tests in Climate and + Mesoscale Models*. Mon. Wea. Rev., 128, 3200-3217. .. _Lock (2001): A. P. Lock (2001). - *{The Numerical Representation of Entrainment in Parametrizations of - Boundary Layer Turbulent Mixing}*. + *The Numerical Representation of Entrainment in Parametrizations of + Boundary Layer Turbulent Mixing*. Mon. Wea. Rev., 129, 1148-1163. .. _Brown and Grant (1997): @@ -7310,16 +7310,16 @@ References .. _Wood et al. (2001): N. Wood and A. R. Brown and F. E. Hewer (2001). - *{Parametrizing the effects of orography on the boundary layer: An - alternative to effective roughness lengths}*. + *Parametrizing the effects of orography on the boundary layer: An + alternative to effective roughness lengths*. Quart. J. Roy. Meteor. Soc., 127, 759-777. .. _Brown et al. (2008): A. R. Brown and R. J. Beare and J. M. Edwards and A. P. Lock and S. J. Keogh and S. F. Milton and D. N. Walters (2008). - *{Upgrades to the Boundary-Layer Scheme in the Met Office Numerical Weather - Prediction Model}*. + *Upgrades to the Boundary-Layer Scheme in the Met Office Numerical Weather + Prediction Model*. Bound.-Layer Meteor., 128, 117-132. .. _Troen and Mahrt (1986): @@ -7332,8 +7332,8 @@ References .. _Holtslag and Moeng (1991): A. A. M. Holtslag and C.-H. Moeng (1991). - *{Eddy Diffusivity and Countergradient Transport in the Convective - Atmospheric Boundary Layer}*. + *Eddy Diffusivity and Countergradient Transport in the Convective + Atmospheric Boundary Layer*. J. Atmos. Sci., 48, 1690-1698. .. _Lock (1998): @@ -7365,7 +7365,7 @@ References .. _Beljaars and Holtslag (1991): A.C.M. Beljaars and A.A.M. Holtslag (1991). - *{Flux Parametrization over Land Surfaces for Atmospheric Models}*. + *Flux Parametrization over Land Surfaces for Atmospheric Models*. J. Appl. Meteor., 30, 327-341. .. _Wood and Mason (1993): @@ -7384,21 +7384,20 @@ References .. _Redelsperger et al. (2000): J.-L. Redelsperger and F. Guichard and S. Mondon (2000). - *{A Parametrization of Mesoscale Enhancement of Surface Fluxes for - Large-Scale Models}*. + *A Parametrization of Mesoscale Enhancement of Surface Fluxes for + Large-Scale Models*. J. Climate, 13, 402-421. .. _Holtslag and Boville (1993): A. A. M. Holtslag and B. A. Boville (1993). - *{Local Versus Nonlocal Boundary-Layer Diffusion in a Global Climate - Model}*. + *Local Versus Nonlocal Boundary-Layer Diffusion in a Global Climate Model*. J. Climate, 6, 1825-1842. .. _Bolton (1980): Bolton, D. (1980). - *{The Computation of Equivalent Potential Temperature}*. + *The Computation of Equivalent Potential Temperature*. Mon. Wea. Rev., 108, 1046-1053. .. _Brown (1996): @@ -7425,7 +7424,7 @@ References .. _Bush et al. (1999): Bush, M. R. and A. P. Lock and R. N. B. Smith (1999). - *{Testing of the new boundary layer scheme in the Mesoscale Model}*. + *Testing of the new boundary layer scheme in the Mesoscale Model*. NWP Tech Report, 260. .. _Cullen and James (1994): @@ -7437,7 +7436,7 @@ References .. _Derbyshire (1997): Derbyshire, S. H. (1997). - *{Recommendations for UM parametrization of stable boundary layers}*. + *Recommendations for UM parametrization of stable boundary layers*. Cardington Tech Note, 38. .. _Driedonks (1982): @@ -7449,8 +7448,8 @@ References .. _Edwards (2007): Edwards, J. M. (2007). - *{Oceanic Latent Heat Fluxes: Consistency with the atmospheric hydrological - and energy cycles and general circulation modeling}*. + *Oceanic Latent Heat Fluxes: Consistency with the atmospheric hydrological + and energy cycles and general circulation modeling*. J. Geophys. Res., 112, D06115. .. _Essery et al. (2001): @@ -7485,7 +7484,7 @@ References .. _Lock (2012): Lock, A. P. (2012). - *{Stable boundary layer modelling at the Met Office}*. + *Stable boundary layer modelling at the Met Office*. ECMWF Workshop on Diurnal Cycles and the Stable Boundary Layer, 137-148. .. _Louis (1979): @@ -7498,21 +7497,21 @@ References Mason, P. J. (1986). *On the parametrization of orographic drag*. - {ECMWF Seminar on Physical Parametrization for Numerical Models of the - Atmosphere}, 2, 139-165. + ECMWF Seminar on Physical Parametrization for Numerical Models of the + Atmosphere, 2, 139-165. .. _Nicholls and Turton (1986): Nicholls, S. and J. D. Turton (1986). - *{An observational study of the structure of stratiform cloud sheets. Part - II: Entrainment}*. + *An observational study of the structure of stratiform cloud sheets. Part + II: Entrainment*. Quart. J. Roy. Meteor. Soc., 112, 461-480. .. _Stage and Businger (1981): Stage, S. A. and J. A. Businger (1981). - *{A model for entrainment into a cloud-topped marine boundary layer. Part - I: Model description and application to a cold-air outbreak episode}*. + *A model for entrainment into a cloud-topped marine boundary layer. Part I: + Model description and application to a cold-air outbreak episode*. J. Atmos. Sci., 38, 2213-2229. .. _Turton and Nicholls (1987): @@ -7545,8 +7544,8 @@ References .. _Zilitinkevich (1975): Zilitinkevich, S. S. (1975). - *{Comments on ''A model for the dynamics of the inversion above a - convective boundary layer''}*. + *Comments on ''A model for the dynamics of the inversion above a convective + boundary layer''*. J. Atmos. Sci., 32, 991-992. .. _Siems et al. (1990): @@ -7560,7 +7559,7 @@ References Mailhot, J. and Lock, A. P. (2004). *An examination of several parametrizations of mixing lengths in a stable - boundary layer: the {GABLS} case*. + boundary layer: the GABLS case*. .. _Honnert et al. (2011): @@ -7607,17 +7606,17 @@ References zone*. Q. J. R. Meteorol. Soc., 136, 927-943. -.. _L{\ (2012): +.. _L\ (2012): - C. L{\ (2012). + C. L\ (2012). *A parametrization, based on sea ice morphology of the neutral atmospheric drag coefficients for weather prediction and climate models*. J. Geophys. Res., 117, D13112. https://doi.org/10.1029/2012JD017630 -.. _L{\ (2015): +.. _L\ (2015): - C. L{\ (2015). + C. L\ (2015). *A stability-dependent parametrization of transfer coefficients for momentum and heat over polar sea to be used in climate models*. J. Geophys. Res. Atmos., 120, 552-581. @@ -7632,9 +7631,9 @@ References Atmos. Chem. Phys., 16, 1545-1563. https://doi.org/10.5194/acp-16-1545-2016 -.. _Schr{\ (2003): +.. _Schr\ (2003): - D. Schr{\ (2003). + D. Schr\ (2003). *On the parameterization of turbulent surface fluxes over heterogeneous sea ice surfaces*. J. Geophys. Res., 108(C6), 3195. @@ -7658,7 +7657,6 @@ References .. _Hsu et al. (2017): Hsu, J. and Lien, R. and D'Asaro, E. A. and Sanford, T. B. (2017). - *Estimates of Surface Wind Stress and Drag Coefficients in {T}yphoon - {M}egi*. + *Estimates of Surface Wind Stress and Drag Coefficients in Typhoon Megi*. J. Phys. Oceanogr., 47, 545-565. https://doi.org/10.1175/JPO-D-16-0069.1 From 649afe81c1c173c85928dd0915626489a8313d8c Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Thu, 28 May 2026 09:19:44 +0100 Subject: [PATCH 101/116] Fixed parsing of bibtex with latex special characters containing quotation marks. --- .../turbulence_schemes/bl_scheme_doc.rst | 36 +++++++++---------- 1 file changed, 18 insertions(+), 18 deletions(-) diff --git a/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst b/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst index 5f7519e555..313000232a 100644 --- a/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst +++ b/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst @@ -4287,10 +4287,10 @@ Two approaches are available in uncoupled configurations of the model. #. Explicit Treatment of Ice Form Drag - `L\ (2012)`_ have suggested a simple + `Lupkes et al. (2012)`_ have suggested a simple parametrization of the form drag coefficient of marginal ice that has been found to perform well in comparison to aircraft measurements - (`Elvidge et al. (2016)`_). `L\ (2015)`_ + (`Elvidge et al. (2016)`_). `Lupkes and Gryanik (2015)`_ have extended the parametrization to include the effects of stability. When coupled to CICE, it is intended that a more elaborate scheme will be used, but this scheme is useful for application in @@ -4303,12 +4303,12 @@ Two approaches are available in uncoupled configurations of the model. :math:`u(z)` will in general exhibit a mixed character, but it may be taken as the developed flow over open sea, as in - `L\ (2012)`_, or may be interpolated between the + `Lupkes et al. (2012)`_, or may be interpolated between the developed flows over open sea or pack ice, depending on the ice - fraction, as in `L\ (2015)`_. In principle, it will + fraction, as in `Lupkes and Gryanik (2015)`_. In principle, it will be subject to the effects of stability, but since the free-board does not much exceed 0.5m, these effects are small - (`L\ (2015)`_) and the flow may be taken as neutral + (`Lupkes and Gryanik (2015)`_) and the flow may be taken as neutral up to :math:`h_f`. Hence, .. math:: :label: eq:int_u2 @@ -4321,18 +4321,18 @@ Two approaches are available in uncoupled configurations of the model. the wind on the model's lowest atmospheric level. Because this will be significantly above :math:`h_f`, the stability dependence of :math:`C_d` should be considered here (again see - `L\ (2015)`_). :math:`U_1` may be interpreted as + `Lupkes and Gryanik (2015)`_). :math:`U_1` may be interpreted as the wind at a specific height, or, consistenly with the flux-difference form of the momentum equation, as the layer-averaged velocity. This distinction affects the numerical value of :math:`C_d`, but does not otherwise affect the foregoing equation. If using the original version of the scheme - (`L\ (2012)`_), :math:`C_d` must be taken as the + (`Lupkes et al. (2012)`_), :math:`C_d` must be taken as the neutral drag coefficient. Note also that various approximations may be made in Equation :eq:`eq:int_u2`. - `L\ (2012)`_ approximate + `Lupkes et al. (2012)`_ approximate :math:`(\log(h_f/z_0) -1)^2 +1` as :math:`(\log(h_f/z_0) )^2`; while - `L\ (2015)`_ approximate it as + `Lupkes and Gryanik (2015)`_ approximate it as :math:`(\log(h_f/z_0) -1)^2`. Here we retain the full expression. If, in a unit area, there are :math:`N` floes, each of crosswind @@ -4369,19 +4369,19 @@ Two approaches are available in uncoupled configurations of the model. (1-A) C_{ds} L_s + A C_{di} L_i \right ], where the sheltering factor is taken to be the same over ice and - water. `L\ (2012)`_ provides parametrizations for + water. `Lupkes et al. (2012)`_ provides parametrizations for quantities such as :math:`h_f`, while `Elvidge et al. (2016)`_ provide suggested values for the constants in the scheme, based on observations. In using these values in the Unified Model, :math:`c_e` should be increased by about 30% to represent the effect of differing approximations of the logarithmic wind profile. - For scalar transfer `L\ (2015)`_ suggest adding a + For scalar transfer `Lupkes and Gryanik (2015)`_ suggest adding a contribution to the sensible heat flux to represent the impact of form drag; however, the mechanistic physical basis of the scheme they propose is unclear. Moreover, when combined with the interfacial drag, this suggests scalar transfer much larger than observed by - `Schr\ (2003)`_. Consequently, no enhancement of the + `Schroder et al. (2003)`_. Consequently, no enhancement of the scalar transfer coefficient by form drag is included. The overall drag coefficients are now set by interpolation in the ice @@ -7606,17 +7606,17 @@ References zone*. Q. J. R. Meteorol. Soc., 136, 927-943. -.. _L\ (2012): +.. _Lupkes et al. (2012): - C. L\ (2012). + C. Lupkes and V. M. Gryanik and J. Hartmann and E. L. Andreas (2012). *A parametrization, based on sea ice morphology of the neutral atmospheric drag coefficients for weather prediction and climate models*. J. Geophys. Res., 117, D13112. https://doi.org/10.1029/2012JD017630 -.. _L\ (2015): +.. _Lupkes and Gryanik (2015): - C. L\ (2015). + C. Lupkes and V. M. Gryanik (2015). *A stability-dependent parametrization of transfer coefficients for momentum and heat over polar sea to be used in climate models*. J. Geophys. Res. Atmos., 120, 552-581. @@ -7631,9 +7631,9 @@ References Atmos. Chem. Phys., 16, 1545-1563. https://doi.org/10.5194/acp-16-1545-2016 -.. _Schr\ (2003): +.. _Schroder et al. (2003): - D. Schr\ (2003). + D. Schroder and T. Vihma and A. Kerber and B. Brummer (2003). *On the parameterization of turbulent surface fluxes over heterogeneous sea ice surfaces*. J. Geophys. Res., 108(C6), 3195. From f17d43e71d7c42f456c2085eb4e84b7998ce07c0 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Thu, 28 May 2026 09:34:04 +0100 Subject: [PATCH 102/116] Re-applied latest version of format corrections scripts (fix in-line math inside emphasis regions, reinstate closing brackets after equation references). --- .../cloud_schemes/UMDP30_PC2CloudScheme.rst | 56 ++++++++++--------- 1 file changed, 29 insertions(+), 27 deletions(-) diff --git a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst index a3e5861088..ae486cfd4f 100644 --- a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst +++ b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst @@ -384,9 +384,9 @@ The idea is to parametrize each of the terms in the above equations. This approach removes the diagnostic method, hence it will be critical that we can write expressions for :math:`\frac{\partial \overline{q_{cl}}}{\partial t}` and -:math:`\frac{\partial C_l}{\partial t}` for *each process that alters -:math:`\overline{T}`, :math:`\overline{p}`, :math:`\overline{q}`, or -:math:`\overline{q_{cl}}` in the model* (and similarly for the ice +:math:`\frac{\partial C_l}{\partial t}` for *each process that alters* +:math:`\overline{T}` *,* :math:`\overline{p}` *,* :math:`\overline{q}` *, or* +:math:`\overline{q_{cl}}` *in the model* (and similarly for the ice terms). In doing so, we will not lose sight of underlying PDF approach given by :eq:`eq:int_gs_ds` and :eq:`eq:qclbar=int` since we will still use the @@ -594,7 +594,7 @@ needs to be initiated in some way. This is discussed further in may be used to trace out an underlying PDF for any input values of :math:`C_l`, :math:`\overline{q_{cl}}` and :math:`SD`. Although we never need to define the complete PDF in PC2 (just the value of -:math:`G(-Q_c)` from equation :eq:`eqn22`, a PDF can be +:math:`G(-Q_c)` from equation :eq:`eqn22`), a PDF can be inferred off-line if required. `Wilson and Gregory (2003)`_ analyse the performance of this parametrization @@ -1202,7 +1202,7 @@ width of the moisture PDF at each point. The positions of the upper and lower truncated bounds of the moisture PDF relative to the saturation threshold are expressed in terms of a normalised :math:`Q_N` = :math:`Q_c` over PDF-width (see equation -:eq:`eq:qn_def`. In entrainment zones, the sum of the two +:eq:`eq:qn_def`). In entrainment zones, the sum of the two Gaussian modes can lead to a highly skewed distribution; hence :math:`Q_N` can have different values for the upper and lower bounds, each normalised by the different widths on either side of the PDF. The @@ -1994,7 +1994,7 @@ Given :math:`C_l^{sgt}` and :math:`q_{cl}^{sgt}`, two options are available for relating these to changes in the model prognostics: Option one: direct increments -^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ +""""""""""""""""""""""""""""" The values of :math:`C_l^{sgt}` and :math:`q_{cl}^{sgt}` can be added as increments to the model prognostic fields, :math:`C_l` and @@ -2029,7 +2029,7 @@ scheme does not condense out more liquid than there is available moisture. Option two: PC2 Erosion method -^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ +"""""""""""""""""""""""""""""" Option one gives a simple method for incrementing the model prognostics, but it gives rise to a potential inconsistency with the PC2 cloud @@ -2341,7 +2341,7 @@ cloud formulation, which considers an underlying PDF across the whole gridbox and does not have, in general, its width prescribed. Remember that we do not calculate on-line the whole of the liquid - vapour PDF, we only parametrize the single point :math:`G(-Qc)`, using equation -:eq:`eqn22`. +:eq:`eqn22`). The width is then limited further to be no greater than :math:`\overline{q}`, to make sure that there are no negative values of @@ -2655,7 +2655,7 @@ discretize eq Similarly to :eq:`eq:c_l^n+1`, we then limit the cloud fraction to 0 and 1 and then apply a mid-point value of :math:`C_l` to calculate the change in :math:`\overline{q_{cl}}` (discretizing eq -:eq:`eq:dqcldt_width`: +:eq:`eq:dqcldt_width`): .. math:: :label: eq:dqcl_turb_final @@ -2678,7 +2678,7 @@ Cloud-surface-area hybrid erosion method `Morcrette and Petch (2010)`_ showed that changes to the erosion parameter (:math:`\Upsilon` in Eqn. -:eq:`eq:dbsbydtbs_turb` did not have as +:eq:`eq:dbsbydtbs_turb`) did not have as significant an impact on the global work done by the erosion process as might be expected. This was due to a feedback process whereby, reducing the erosion parameter leads to more cloud water, more autoconversion of @@ -2828,7 +2828,8 @@ yield much less timestep sensitivity for detrained cloud in shallow cumulus regimes. #. **Retain the explicit discretization, but limit the resulting erosion - increments to ensure :math:`q_{cl}` and :math:`C_l` don't go + increments to ensure** :math:`\boldsymbol{q_{cl}}` **and** + :math:`\boldsymbol{C_l}` **don't go negative. (i_pc2_erosion_numerics=1)** Also, to ensure that some cloud remains at end-of-timestep where shallow cumulus is detraining into dry environments, the erosion calculation is fed copies of the @@ -2859,7 +2860,8 @@ cumulus regimes. erosion, to give some improvement in accuracy. #. **Use an approximate implicit discretisation, which intrinsically - yields a positive solution for :math:`q_{cl}` and :math:`C_l`. + yields a positive solution for** :math:`\boldsymbol{q_{cl}}` **and** + :math:`\boldsymbol{C_l}` **. (i_pc2_erosion_numerics=2)** The copies of the fields passed to the erosion calculation are fully updated with the convection increments. We then write equation :eq:`eq:dqcldt_hybrid` in @@ -2995,7 +2997,7 @@ cumulus regimes. each case. In the code, we first test the value of :math:`Q_c` and compute the erosion increments as follows: - #. **Grid-mean subsaturation (:math:`Q_c < 0`):** + #. **Grid-mean subsaturation (** :math:`\boldsymbol{Q_c < 0}` **):** The relation between the erosion tendencies in liquid-cloud-fraction and liquid water content @@ -3015,9 +3017,9 @@ cumulus regimes. defined the PDF height at the saturation boundary when near the cloudy end of the PDF as :math:`G(-Q_c) = \frac{n+1}{n+2} \frac{C_l^2}{q_{cl}}` (eq - :eq:`eqn20`. In fact, :math:`G(-Q_c)` is set to some + :eq:`eqn20`). In fact, :math:`G(-Q_c)` is set to some blend between this and the value near the clear end of the PDF (eq - :eq:`eqn21`. But we will assume that when eroding cloud + :eq:`eqn21`). But we will assume that when eroding cloud under grid-mean subsaturated conditions (:math:`Q_c < 0`), :math:`G(-Q_c)` follows this scaling with :math:`\frac{C_l^2}{q_{cl}}` even if its value differs somewhat @@ -3073,7 +3075,7 @@ cumulus regimes. dimensionless factor defined in eq :eq:`eq:a_L`. Following the derivation in section :ref:`"Smooth" initiation logic ` (eq - :eq:`eq:qc_plus_sd`, we can write this in terms + :eq:`eq:qc_plus_sd`), we can write this in terms of the liquid-water content: :math:`SD = q_{cl} - Q_c` (where :math:`Q_c` was defined in eq :eq:`eq:qc_eq_qt-qs`, and corresponds to the @@ -3136,11 +3138,11 @@ cumulus regimes. find the solution with the explicitly-treated terms (the exponent :math:`b_1 = \frac{ c_1 }{ 1 - \frac{q_{cl}}{C_l Q_c} }` and the terms :math:`(1 - C_l)` and :math:`(q_{cl}-Q_c)` in eq - :eq:`eq:qcl_int_hybrid` adjusted to values + :eq:`eq:qcl_int_hybrid`) adjusted to values linearly-interpolated to half-way between the start and end of the erosion timestep. - #. **Grid-mean supersaturation (:math:`Q_c > 0`):** + #. **Grid-mean supersaturation (** :math:`\boldsymbol{Q_c > 0}` **):** In this case, erosion does not act to reduce :math:`C_l` and :math:`q_{cl}` towards zero. Instead, the narrow PDF limit it @@ -3209,14 +3211,14 @@ cumulus regimes. iterations are then performed to find the solution with the explicitly-treated terms (the exponent :math:`b_2 = \frac{ c_2 }{ 1 + \frac{SD}{(1-C_l) Q_c} }` and the - term :math:`C_l` in eq :eq:`eq:sd_int_hybrid` + term :math:`C_l` in eq :eq:`eq:sd_int_hybrid`) adjusted to values linearly-interpolated to half-way between the start and end of the erosion timestep. Then the final values of :math:`SD` and :math:`1-C_l` are used to increment :math:`q_{cl} = Q_c + SD` and :math:`C_l`, as prognosed by the rest of the model. - #. **grid-mean saturation (:math:`Q_c` near-zero):** + #. **grid-mean saturation (** :math:`\boldsymbol{Q_c}` **near-zero):** In this case, the PDF is centred on the saturation boundary, so that narrowing it does not change the cloud-fraction. In the limit @@ -3312,7 +3314,7 @@ denominator is small, we will, to avoid numerical problems, set that we do not use the multiple phases injection source expressions (section :ref:`Multiple phases in the injection source ` and equation -:eq:`eq:cff_ts`. This is because the liquid water changes +:eq:`eq:cff_ts`). This is because the liquid water changes are not associated with the plume model. Equation :eq:`eq:qcf_ci` assumes that the change to the ice @@ -4755,7 +4757,7 @@ completely uninitiated state will have zero saturation deficit :math:`SD`, rather than zero :math:`q_{cl}`. Therefore, in this case the increment to :math:`C_l` is calculated based on the fractional increase in :math:`SD` from initiation (equation -:eq:`eq:dcl_init2`, instead of the fractional increase +:eq:`eq:dcl_init2`), instead of the fractional increase in :math:`q_{cl}`. Whether to increment :math:`C_l` based on the increase in :math:`q_{cl}` @@ -6573,7 +6575,7 @@ increment terms. If your scheme is currently using the homogeneous forcing then there is no need to update the cloud part of the scheme, *provided that you do -not alter values of :math:`T` and :math:`q` after the homogeneous +not alter values of* :math:`T` *and* :math:`q` *after the homogeneous forcing section is called* and that the physical interpretation of your :math:`q` and :math:`T` increments does not change. You need to be careful if you are moving code from one subroutine to another that you @@ -6585,7 +6587,7 @@ which necessitates that condensate increments are already calculated by the scheme, then there is also no need to update the cloud part of the scheme. This currently applies to the boundary layer, where :math:`q_{cf}` is altered by tracer mixing. Like for the homogeneous -schemes, this is provided that you *do not alter :math:`T`, :math:`q` or +schemes, this is provided that you *do not alter* :math:`T` *,* :math:`q` *or condensate values after the injection forcing subroutine is called* and that the physical interpretation of your :math:`q` and :math:`T` increments does not change. @@ -7083,7 +7085,7 @@ References .. _Mellor (1977): Mellor, G. (1977). - *The {Gaussian} cloud model relations*. + *The Gaussian cloud model relations*. J. Atmos. Sci., 34, 356-358. .. _Sommeria and Deardorff (1977): @@ -7143,14 +7145,14 @@ References .. _Field et al. (2014): Field, P.R. and Hill, A.A. and Furtado K. and Korolev, A. (2014). - *Mixed-phase clouds in a turbulent environment. {II: A}nalytic treatment*. + *Mixed-phase clouds in a turbulent environment. II: Analytic treatment*. Q. J. Roy. Meteor. Soc., 140, 870-880. .. _Jakob et al. (1999): Jakob, C. and Gregory, D. and Teixeria, J. (1999). *A package of cloud and convection changes for CY21R3*. - Research Department Memorandum, {ECMWF}, Shinfield Park, Reading {RG2 9AX}, + Research Department Memorandum, ECMWF, Shinfield Park, Reading RG2 9AX, United Kingdom. .. _Morcrette and Petch (2010): From 3924d2392b85ff475e129aa5e56dcae138ae6133 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Thu, 28 May 2026 10:05:10 +0100 Subject: [PATCH 103/116] Changed the script to add header-rows: 1 to tables by default, so removed this from manual corrections. --- .../cloud_schemes/manual_corrections.txt | 18 ------------------ 1 file changed, 18 deletions(-) diff --git a/documentation/source/science_guide/cloud_schemes/manual_corrections.txt b/documentation/source/science_guide/cloud_schemes/manual_corrections.txt index 0b0ebef109..33e34bf867 100644 --- a/documentation/source/science_guide/cloud_schemes/manual_corrections.txt +++ b/documentation/source/science_guide/cloud_schemes/manual_corrections.txt @@ -42,21 +42,3 @@ index 9f1988eb..d0664430 100644 where the subscript :math:`_{diag}` denotes the liquid cloud water content and fraction predicted by the diagnostic cloud scheme (either -@@ -5931,7 +5934,7 @@ diagnostic output routines. - - .. list-table:: PC2 parameter values and locations - :name: tab:pc2_names -- -+ :header-rows: 1 - - * - Symbol - - Code variable -@@ -6098,7 +6101,7 @@ PC2:64 and a non-PC2 run. - .. list-table:: PC2 parameter values and locations relating to the convection. - \*These values are those used in HadGAM - :name: tab:pc2_conv_names -- -+ :header-rows: 1 - - * - Code variable - - Des cription From b12187816d1c1b4248804c03a0c90688459d5d7b Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Thu, 28 May 2026 12:25:45 +0100 Subject: [PATCH 104/116] Updated script to add header-rows: 1 in tables by default, then remove this in manual corrections where appropriate. --- .../turbulence_schemes/bl_scheme_doc.rst | 8 --- .../turbulence_schemes/manual_corrections.txt | 61 ++++++++++++++++++- 2 files changed, 59 insertions(+), 10 deletions(-) diff --git a/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst b/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst index 313000232a..4db865d539 100644 --- a/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst +++ b/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst @@ -1826,7 +1826,6 @@ Discussion of some of the revisions :name: tab:vscales :header-rows: 2 - * - Formulation - Convective limit - @@ -5544,7 +5543,6 @@ calculations take place for the scalar variables: .. list-table:: - * - CALL bdy_impl3(): - set up coefficients for :eq:`eq:dX_disc_top`, :eq:`eq:dX_disc` and do a downward sweep; @@ -6876,7 +6874,6 @@ the UKCA code owner before lodging the change. .. list-table:: - * - Boundary layer inputs to UKCA - - @@ -7034,7 +7031,6 @@ Appendix: Notation .. list-table:: - * - Finite difference notation - @@ -7061,7 +7057,6 @@ Appendix: Notation .. list-table:: - * - Model variables - @@ -7092,7 +7087,6 @@ Appendix: Notation .. list-table:: - * - Thresholds - @@ -7123,7 +7117,6 @@ Appendix: Notation .. list-table:: - * - Layer definitions and parameters - @@ -7189,7 +7182,6 @@ Appendix: Notation .. list-table:: - * - Other parameters - diff --git a/documentation/source/science_guide/turbulence_schemes/manual_corrections.txt b/documentation/source/science_guide/turbulence_schemes/manual_corrections.txt index 00d47bd8da..db2f9d01ed 100644 --- a/documentation/source/science_guide/turbulence_schemes/manual_corrections.txt +++ b/documentation/source/science_guide/turbulence_schemes/manual_corrections.txt @@ -13,14 +13,71 @@ index 5b373b68..8ff32c06 100644 .. list-table:: :align: center -@@ -1810,6 +1810,7 @@ Discussion of some of the revisions +@@ -1810,7 +1810,7 @@ Discussion of some of the revisions .. list-table:: Convective and Neutral limits for velocity scales :name: tab:vscales +- :header-rows: 1 + :header-rows: 2 - * - Formulation + - Convective limit +@@ -5542,7 +5542,6 @@ stage is computed, i.e. :math:`X^{*}`. Briefly the following + calculations take place for the scalar variables: + + .. list-table:: +- :header-rows: 1 + + * - CALL bdy_impl3(): + - set up coefficients for :eq:`eq:dX_disc_top`, :eq:`eq:dX_disc` and do a +@@ -6874,7 +6874,6 @@ If the changes are significant it would be prudent to discuss them with + the UKCA code owner before lodging the change. + + .. list-table:: +- :header-rows: 1 + + * - Boundary layer inputs to UKCA + - +@@ -7032,7 +7032,6 @@ Appendix: Notation + ================== + + .. list-table:: +- :header-rows: 1 + + * - Finite difference notation + - +@@ -7059,7 +7059,6 @@ Appendix: Notation + - across the capping inversion (see :eq:`dbinv` and following text) + + .. list-table:: +- :header-rows: 1 + + * - Model variables + - +@@ -7090,7 +7090,6 @@ Appendix: Notation + - total heat flux (net radiative plus turbulent, Kms\ :math:`^{-1}`) + + .. list-table:: +- :header-rows: 1 + + * - Thresholds + - +@@ -7121,7 +7121,6 @@ Appendix: Notation + - before decoupling occurs + + .. list-table:: +- :header-rows: 1 + + * - Layer definitions and parameters + - +@@ -7187,7 +7187,6 @@ Appendix: Notation + - lifting condensation level + + .. list-table:: +- :header-rows: 1 + + * - Other parameters + - @@ -6253,7 +6253,7 @@ included in :math:`\zeta_r` and :math:`\tilde{\alpha_t}` (in where there is no buoyancy reversal (:math:`D \leq 0`) and the feedback seen in LES of stratocumulus :raw-latex:`\cite[]{lock98}` with From 3194c8be32db77164ee93f6e3edf0ce7eff2dd8e Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Thu, 28 May 2026 12:45:53 +0100 Subject: [PATCH 105/116] Deleted original latex source and .eps figures. --- .../cloud_schemes/UMDP30_PC2CloudScheme.tex | 5736 -- .../cloud_schemes/manual_corrections.txt | 44 - .../science_guide/cloud_schemes/refs.bib | 371 - .../cloud_schemes/um_call_tree.tex | 530 - .../cloud_schemes/um_call_tree_preamble.tex | 25 - .../turbulence_schemes/bl_scheme_doc.tex | 5639 -- .../turbulence_schemes/div_r071.eps | 410 - .../turbulence_schemes/div_r080.eps | 402 - .../turbulence_schemes/honnert_vs_tanh.eps | 258 - .../turbulence_schemes/ideal_invinteg.eps | 265 - .../turbulence_schemes/ideal_revflux.eps | 517 - .../turbulence_schemes/manual_corrections.txt | 172 - .../turbulence_schemes/nbldoc_zidiag.eps | 183 - .../turbulence_schemes/new_ktop_shape.eps | 288 - .../turbulence_schemes/newcommand.tex | 111 - .../science_guide/turbulence_schemes/refs.bib | 1750 - .../turbulence_schemes/stab_dep.eps | 1398 - .../turbulence_schemes/subsent_fig7.eps | 323 - .../turbulence_schemes/wcrp_bltypes1.eps | 51115 ------------- .../turbulence_schemes/wcrp_bltypes2.eps | 63242 ---------------- .../turbulence_schemes/zturb_schem.eps | 127 - 21 files changed, 132906 deletions(-) delete mode 100644 documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.tex delete mode 100644 documentation/source/science_guide/cloud_schemes/manual_corrections.txt delete mode 100644 documentation/source/science_guide/cloud_schemes/refs.bib delete mode 100644 documentation/source/science_guide/cloud_schemes/um_call_tree.tex delete mode 100644 documentation/source/science_guide/cloud_schemes/um_call_tree_preamble.tex delete mode 100644 documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.tex delete mode 100644 documentation/source/science_guide/turbulence_schemes/div_r071.eps delete mode 100644 documentation/source/science_guide/turbulence_schemes/div_r080.eps delete mode 100644 documentation/source/science_guide/turbulence_schemes/honnert_vs_tanh.eps delete mode 100644 documentation/source/science_guide/turbulence_schemes/ideal_invinteg.eps delete mode 100644 documentation/source/science_guide/turbulence_schemes/ideal_revflux.eps delete mode 100644 documentation/source/science_guide/turbulence_schemes/manual_corrections.txt delete mode 100644 documentation/source/science_guide/turbulence_schemes/nbldoc_zidiag.eps delete mode 100644 documentation/source/science_guide/turbulence_schemes/new_ktop_shape.eps delete mode 100644 documentation/source/science_guide/turbulence_schemes/newcommand.tex delete mode 100644 documentation/source/science_guide/turbulence_schemes/refs.bib delete mode 100644 documentation/source/science_guide/turbulence_schemes/stab_dep.eps delete mode 100644 documentation/source/science_guide/turbulence_schemes/subsent_fig7.eps delete mode 100644 documentation/source/science_guide/turbulence_schemes/wcrp_bltypes1.eps delete mode 100644 documentation/source/science_guide/turbulence_schemes/wcrp_bltypes2.eps delete mode 100644 documentation/source/science_guide/turbulence_schemes/zturb_schem.eps diff --git a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.tex b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.tex deleted file mode 100644 index 42efc42022..0000000000 --- a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.tex +++ /dev/null @@ -1,5736 +0,0 @@ -\documentclass{UMDP_article} - -\title{The PC2 Cloud Scheme} -\paperno{030} -\umversion{14.0} -\owner{Cyril Morcrette} -\author{D.~Wilson, A.~Bushell, C.~Morcrette, V.~Varma$^{1}$, M.~Whitall} - -\titlecontent{ - \footnotesize - $^1$ National Institute of Water and Atmospheric Research, - Wellington, New Zealand \\ - } - -\usepackage{textcomp} -\usepackage{amstext,natbib} - -% Packages needed for the UM subroutine tree diagram in um_call_tree.txt: -\input{um_call_tree_preamble} - -\newcommand{\citeumdp}[1]{:umdp:`#1`} - -\newcommand{\mmax}[1] {\mbox{\footnotesize \sf MAX} \left[#1\right] } - -%%% These are definitions used in the convection documentation -%%% Definitions running across several chapters are defined in PC2_main. -\newcommand{\gbmll}{\ensuremath{\overline{l}_{\rm{l}}}} -\newcommand{\gbmlf}{\ensuremath{\overline{l}_{\rm{f}}}} -\newcommand{\gbmlm}{\ensuremath{\overline{l}_{\rm{m}}}} -\newcommand{\gbmlx}{\ensuremath{\overline{l}_{\rm{x}}}} -\newcommand{\incml}[1]{\ensuremath{\overline{l}_{\rm c}^{\rm #1}}} -\newcommand{\bigqc}[1]{\ensuremath{Q_{\rm C}^{\rm #1}}} -\newcommand{\GpdfB}{\ensuremath{G^{\rm B}}} -\newcommand{\injectcl}{\ensuremath{C_{\rm{l}}^{\ast}}} -\newcommand{\injectcf}{\ensuremath{C_{\rm{f}}^{\ast}}} -% -\newcommand{\lsubsup}[2]{\ensuremath{l_{\rm{#1}}^{\rm{#2}}}} -\newcommand{\csubsup}[2]{\ensuremath{l_{\rm{#1}}^{\rm{#2}}}} -\newcommand{\qsubsup}[2]{\ensuremath{q_{\rm{#1}}^{\rm{#2}}}} -\newcommand{\tsubsup}[2]{\ensuremath{T_{\rm{#1}}^{\rm{#2}}}} -\newcommand{\thsubsup}[2]{\ensuremath{\theta_{\rm{#1}}^{\rm{#2}}}} -\newcommand{\xsubsup}[2]{\ensuremath{{\chi}_{\rm{#1}}^{\rm{#2}}}} -%%% Private definitions (shorthand commands etc.) -\newcommand{\lc}{\left[} -\newcommand{\rc}{\right]} -\newcommand{\ov}{\overline} -\newcommand{\lp}{\left(} -\newcommand{\rp}{\right)} -\newcommand{\dr}{\partial} - -%%% Define a format for partial derivatives -\newcommand{\pardbyd}[2]{\ensuremath{\frac{\partial \, #1}{\partial \, #2}}} - -%%% Define a few of the commonest variables -\newcommand{\Tliq}{\ensuremath{T_{\rm L}}} -\newcommand{\qtot}{\ensuremath{q_{\rm T}}} -\newcommand{\qsat}{\ensuremath{q_{\rm s}}} -\newcommand{\aliq}{\ensuremath{a_{\rm L}}} -\newcommand{\gbmTliq}{\ensuremath{\overline{T}_{\rm L}}} -\newcommand{\gbmqtot}{\ensuremath{\overline{q}_{\rm T}}} -%Add a new command to do a horizontal line -\newcommand{\HRule}{\rule{\linewidth}{1.0mm}} - -\begin{document} % Every document must start with this. -\maketitle - -\tableofcontents - -\newpage -\section{PC2 developers} -We would like to acknowledge those who developed the PC2 cloud scheme: -Damian Wilson, Andrew Bushell, David Gregory, Amanda Kerr-Munslow, -John Edwards, Jeremy Price, Cyril Morcrette, Martin Sharpe, Thomas Mirfield, Ian Boutle. Many -others offered considerable help, advice and analysis, including Roy Kershaw, -Malcolm Brooks, Richard Forbes and Alejandro Bodas-Salcedo, and we would -like to thank them all for their input. - -\section{Introduction} - -This document describes the PC2 \textit{(prognostic cloud, -prognostic condensate)} cloud scheme. It should be seen as -a complete reference source for the scheme's physical assumptions, -numerical techniques, -application to the Unified Model and coding within the Unified Model. It does -not describe results from the scheme, please refer to the various reports and papers -written on this. Except where commented on explicitly, the description applies -to the PC2:66 version of the PC2 scheme, which is the version that will be -available at UM6.5. The version available at 6.4 is PC2:64. - -This paper will first introduce the concepts that underlie cloud schemes, -before developing a study of the theoretical behaviour of the prognostic PC2 scheme -under certain, well-defined, situations. The next sections shows how -the theory can be applied to the physical and dynamical processes -represented in the Unified Model. Finally, we outline the way in which -the PC2 scheme is implemented within the code of the Unified Model. - -%%\subsection{How to use this documentation} - -\subsection{Cloud schemes} - -The basic requirements of any cloud scheme within a large-scale model are to: -\begin{itemize} -\item{calculate the amount of condensation (from water vapour to liquid water or vice-versa) within each gridbox each timestep} -\item{to calculate or update the cloud fractions for use by the radiation and large-scale precipitation schemes (or any other physics scheme).} -\end{itemize} - -Depending on the model involved, cloud schemes may also treat the -deposition / sublimation process from vapour to ice. The problem is -straightforward to solve if one is allowed to assume that there is no -variability of moisture or temperature on a scale of a model gridbox. -In this case the cloud fraction scheme is redundant and only the condensation -part remains, which may be solved diagnostically using the instantaneous -condensation assumption in section \ref{sec:s_dist}. However, the -`no-variability' assumption is poor until -very high resolutions close to, or maybe exceeding, 1 km in the horizontal -are reached. Although we may eventually assume that computer power -will enable such resolutions to be reached globally, for many years we -will need a subgrid-scale cloud scheme to properly account for the -variability in the atmosphere. This is the principal challenge of -cloud parametrization. - -There are several approaches to take to the solution of the problem, -although they are not as independent as often portrayed, since they -nearly all require the same instantaneous condensation assumption -(discussed in section \ref{sec:s_dist}). Hence there are mathematical -links between -all the approaches. \textit{The following are all valid structures -to use in this respect.} -\begin{itemize} -\item{One may diagnose cloud fractions and condensate contents from knowledge of gridbox mean variables. This forms the basis of the \cite{smith90} scheme, which is described in -\citeumdp{029}.} -\item{A mixed scheme, such as \cite{sundqvist1978} uses a prediction of condensate contents, but a diagnostic cloud fraction.} -\item{Alternatively, one may predict cloud fraction and condensate content changes as a result of each modelled process. This forms the basis of the \cite{t93} scheme and the PC2 scheme.} -\item{Hybrid schemes, such as \cite{t02}, will predict various moments of the subgrid-scale variability, and use this knowledge to diagnose the cloud fraction and condensate contents.} -\end{itemize} - -Many years of experience of the results from the \cite{smith90} -scheme have highlighted deficiences in the diagnosis of cloud from -this scheme, which we feel can only be tackled by adding the -memory of cloud history available by using a prognostic based -scheme. We chose to develop a scheme that directly specified -the impacts on observable prognostics (condensates and -cloud fractions, as in \cite{t93}) rather than on moments of a -probability density function (as in \cite{t02}). This is because -we believe it is easier to physically relate (and hence parametrize) -the effect processes to quantities -such as cloud fraction and condensate rather than to the more abstract -quantities of moments of a probability -density function of moisture. However, although the -PC2 scheme is similar to \cite{t93} in its very basic prognostic variable -structure, the assumptions behind the formulation of the prognostic terms -in PC2 are very different and much improved. The PC2 scheme should not be -considered to be merely an extension of \cite{t93}. - -In particular, we wish to use a prognostic formulation in order to link -the detraiment of moisture from convection directly to cloud fraction, and -to break the hard diagnostic link between cloud fraction and condensate. These -major features of the \cite{t93} scheme provide the motivation to develop -the PC2 cloud scheme. - -\subsection{The `s' distribution} -\label{sec:s_dist} - -Most cloud schemes are based on the concept of a distribution -of fluctuations of moisture and temperature in the gridbox. Here we -mathematically formalize this concept, since it is used both in -the PC2 scheme and the \cite{smith90} scheme. - -This method was first formulated by \cite{m77} and \cite{sommeria_deardorff_1977} -for large-eddy simulations. It can also be applied -to larger scale models. It allows us to calculate vapour and liquid -contents and liquid cloud fraction from knowledge only of the combined -vapour+liquid content, $\overline{q_T}$, and the liquid temperature, -$\overline{T_L}$. These variables are unchanged during condensation -processes, so it is useful to write the cloud scheme in terms of these -variables. - -In this derivation we will consider only liquid condensate. -We assume, as above, that, -locally, the water content in a cloud is such as to remove any -supersaturation. This gives the equation - -\begin{equation} -q_{cl} = q_T - q_{sat}(T,p) -\label{eq:basic_qcl} -\end{equation} - -assuming that $q_T > q_{sat}(T,p)$ ($q_{cl}$ will be zero otherwise). -$q_T$ is the local total water -content, equal to the sum of the condensate $(q_{cl})$ plus the vapour -$(q)$, $T$ is the temperature, $p$ is the pressure and $q_{sat}(T,p)$ -is the saturation specific humidity at temperature T and pressure -p \textit{with respect to liquid water}. (Many earlier diagnostic -cloud schemes use a similar instantaneous condensation assumption -for ice, which would mean that $q_{sat}$ must be taken with respect -to ice when $T < 0 ^{\circ} C$, but the Unified Model does not). -We now introduce the liquid temperature ($T_L$), where $T_L$ is given by - -\begin{equation} -T_L = T - \frac{L}{c_p} q_{cl} , -\label{eq:tl} -\end{equation} - -and $L$ is the latent heat of -vaporization and $c_p$ is the heat capacity of air. -Note that $T_L$ is unaffected by changes of phase between vapour and liquid. -We now write (\ref{eq:basic_qcl}) as an \textit{equality} - -\begin{equation} -q_{cl} = q_T - \left( q_{sat}(T_L,p) + \alpha (T - T_L) \right) -\label{eq:alpha_t_tl} -\end{equation} - -where - -\begin{equation} -\alpha = \frac{ q_{sat}(T,p) - q_{sat}(T_L,p) }{T - T_L } . -\label{eq:alpha} -\end{equation} - -Using (\ref{eq:tl}) in (\ref{eq:alpha_t_tl}) gives the expression - -\begin{equation} -q_{cl} = q_T - q_{sat} (T_L) - \alpha \frac{L}{c_p} q_{cl} -\end{equation} - -or - -\begin{equation} -q_{cl} = a_L \left( q_T - q_{sat}(T_L,p) \right) -\label{eq:l_eq_al} -\end{equation} - -where $a_L$ is given by - -\begin{equation} -a_L = \left( 1 + \alpha \frac{L}{c_p} \right) ^{-1} . -\label{eq:a_L} -\end{equation} - -Thus (\ref{eq:basic_qcl}) has been rewritten \textit{exactly} in terms of the conserved -variables, $q_T$ and $T_L$, although the temperature, $T$, does remain in the -definition of $a_L$. -We will need to consider variations across a gridbox for a -parametrization scheme, -so we expand the expression for condensate (\ref{eq:l_eq_al}) into terms -relating to the gridbox mean and variation from the gridbox mean. - -\begin{equation} -q_{cl} = \overline{ a_L \left( q_T - q_{sat}(T_L,p) \right)} -+ [ a_L \left( q_T - q_{sat}(T_L,p) \right) ]' -\label{eq:bar_plus_pri1} -\end{equation} - -where $\overline{\phi}$ represents the mean of a distibution of $\phi$ and -$\phi = \overline{\phi} + {\phi}'$. The expression (\ref{eq:bar_plus_pri1}) -is \textit{exact} when -using the definition of $\alpha$ given in (\ref{eq:alpha}). - -The idea of a PDF scheme is to calculate the first (mean) term, -$\overline{\phi}$, from the known gridbox mean parameters, $q_T$, $T_L$ -and $p$, and to parametrize the distribution of the second, -variable term, ${\phi}'$. Unfortunately, the mean term is difficult -to write in terms of the gridbox mean variables $\overline{q_T}$ -and $\overline{T_L}$ because -$q_{sat}(T_L,p)$ is not a linear function of $T_L$ (or of $p$). In order to -proceed, we will now make an \textit{approximation} that $q_{sat}(T,p)$ -is a linear function of $T_L$ and $p$. -This equivalently implies that $a_L$ and $\alpha$ are approximated as -being constant across the gridbox. The expression -now becomes more tractable, (\ref{eq:bar_plus_pri1}) becoming: - -\begin{equation} -q_{cl} = a_L \left( \overline{q_T} - q_{sat}(\overline{T_L},\overline{p}) -\right) + a_L \left( {q_T}' - \alpha {T_L}' - \beta {p}' \right) -\label{eq:l_eq_bar_plus_pri} -\end{equation} - -where $\beta = {\frac{\partial q_{sat}}{\partial p}}$ at constant -temperature. The first term is connected with the mean properties of -the gridbox, and is written as $Q_c$, the second term is connected with the -deviation of the local conditions from the mean and is written as -$s$. - -\begin{equation} -Q_c = a_L \left( \overline{q_T} - q_{sat}(\overline{T_L},\overline{p}) \right) -\label{eq:qc_eq_qt-qs} -\end{equation} - -\begin{equation} -s = a_L \left( {q_T}' - \alpha {T_L}' - \beta {p}' \right) -\label{eq:s} -\end{equation} - -This gives the equation - -\begin{equation} -q_{cl} = Q_c + s -\label{eq:l_qc_s} -\end{equation} - -with the assumption that $s \ge -Q_c$ (i.e. $q_{cl} \ge 0$). -If $s < -Q_c$ then $q_{cl} = 0$. The term $a_L$ can be calculated -using (\ref{eq:a_L}) -from (\ref{eq:alpha}) with gridbox mean temperatures, i.e. - -\begin{equation} -\alpha = \frac{ q_{sat}(\overline{T},\overline{p}) -- q_{sat}(\overline{T_L},\overline{p}) }{\overline{T} - -\overline{T_L} } . -\label{eq:alpha_mean} -\end{equation} - -This definition of $\alpha$ and $a_L$ will retrieve an \textit{exact} value for -the gridbox mean $\overline{q_{cl}}$ \textit{if} -the distribution is monodispersed. Hence it is the sensible form to use for a -purely diagnostic representation such as \cite{smith90} where we explicitly -consider distributions of $s$. Strictly, the linear approximation -implies that other -approximations for $\alpha$ are valid: PC2 will do this -(see section \ref{sec:homog_num_app}) since we are concerned in PC2 with -the best estimate of the \textit{changes} to $\overline{q_{cl}}$, not the best -estimate of $\overline{q_{cl}}$ itself. - -We now assume that within -any particular gridbox a distribution $G$ of $s$ occurs (with mean, by -definition, of zero). Considering cloud to be where the water content -is greater than zero (i.e. where $s > -Q_c$) gives an expression for the -liquid cloud \textit{volume} fraction, $C_l$, within the gridbox as - -\begin{equation} -C_l = \int_{s=-Q_c}^{\infty} G(s) ds -\label{eq:int_gs_ds} -\end{equation} - -and the expression for mean condensate, ${\overline{q_{cl}}}$, using -(\ref{eq:l_qc_s}) to expand $q_{cl}$, is - -\begin{equation} -\overline{q_{cl}} = \int_{s=-Q_c}^{\infty} (Q_c + s) G(s) ds . -\label{eq:qclbar=int} -\end{equation} - -If we know (parametrize) the PDF given by $G(s)$ then we can solve -for $C_l$ and $\overline{q_{cl}}$. Note that this distribution is in terms of -$s$, there is no need to know the three-dimensional distribution in terms -of three separate variables $q_T$, $T_L$ and $p$. This is the method used -by \cite{smith90}, where a -symmetric triangular distribution function is used. For further -information on the \cite{smith90} scheme, please refer to \citeumdp{029}. Physics and dynamics schemes hence only need to -provide increments to $\overline{q_T}$ and $\overline{T_L}$, -provided that a diagnostic scheme (such as \cite{smith90}) is called at some point in the -timestep to partition $\overline{q_T}$ into $\overline{q}$ and $\overline{q_{cl}}$, -to calculate the dry bulb temperature $\overline{T}$ (from $\overline{T_L}$ -and $\overline{q_{cl}}$) and to calculate the liquid -cloud fraction, $C_l$. The diagnostic scheme effectively allows a calculation of -condensation associated with any physical process. However, its results remain -tied to the distribution of $G(s)$ that is chosen in (\ref{eq:int_gs_ds}) and -(\ref{eq:qclbar=int}) and it is this tie that we seek to break by the use of -a prognostic scheme. - -\subsection{Concept of PC2} - -The PC2 scheme develops prognostic expressions for the rates of change of -cloud fraction and condensate contents as a result of each process that acts in -the model. We consider ice and liquid condensate as two distinct -aspects of clouds, which may or may not overlap -Figure \ref{fig:schematic} provides a schematic summary of the PC2 scheme. -The equations for the five prognostic cloud variables can be written schematically: - -\begin{eqnarray} -\frac{\partial \overline{q_{cl}}}{\partial t} = -\frac{\partial \overline{q_{cl}}}{\partial t} |_{advection} + -\frac{\partial \overline{q_{cl}}}{\partial t} |_{convection} + -\frac{\partial \overline{q_{cl}}}{\partial t} |_{boundary \, layer} + -\frac{\partial \overline{q_{cl}}}{\partial t} |_{precipitation} + ... \nonumber \\ -\frac{\partial \overline{q_{cf}}}{\partial t} = -\frac{\partial \overline{q_{cf}}}{\partial t} |_{advection} + -\frac{\partial \overline{q_{cf}}}{\partial t} |_{convection} + -\frac{\partial \overline{q_{cf}}}{\partial t} |_{boundary \, layer} + -\frac{\partial \overline{q_{cf}}}{\partial t} |_{precipitation} + ... \nonumber \\ -\frac{\partial C_l}{\partial t} = -\frac{\partial C_l}{\partial t} |_{advection} + -\frac{\partial C_l}{\partial t} |_{convection} + -\frac{\partial C_l}{\partial t} |_{boundary \, layer} + -\frac{\partial C_l}{\partial t} |_{precipitation} + ... \nonumber \\ -\frac{\partial C_i}{\partial t} = -\frac{\partial C_i}{\partial t} |_{advection} + -\frac{\partial C_i}{\partial t} |_{convection} + -\frac{\partial C_i}{\partial t} |_{boundary \, layer} + -\frac{\partial C_i}{\partial t} |_{precipitation} + ... \nonumber \\ -\frac{\partial C_t}{\partial t} = -\frac{\partial C_t}{\partial t} |_{advection} + -\frac{\partial C_t}{\partial t} |_{convection} + -\frac{\partial C_t}{\partial t} |_{boundary \, layer} + -\frac{\partial C_t}{\partial t} |_{precipitation} + ... , -\label{eq:dqcldt_and_dcdt} -\end{eqnarray} - -where $\overline{q_{cf}}$ is the ice water specfic humidity, $C_l$ is the liquid -cloud \textit{volume} fraction, $C_i$ is the ice cloud volume fraction, and $C_t$ -is the combined ice or liquid cloud volume fraction. The amount of mixed -phase cloud, $C_{mp}$, can be calculated by the overlap of the ice -and liquid fractions: - -\begin{equation} -C_{mp} = C_i + C_l - C_t. -\label{eq:mp} -\end{equation} - -The idea is to parametrize each of the terms in the above equations. This -approach removes the diagnostic method, hence it will be critical that -we can write expressions for $\frac{\partial \overline{q_{cl}}}{\partial t}$ -and -$\frac{\partial C_l}{\partial t}$ for \textit{each process that alters -$\overline{T}$, $\overline{p}$, -$\overline{q}$, or $\overline{q_{cl}}$ in the model} (and similarly -for the ice terms). In doing so, -we will not lose sight of underlying PDF approach given by -(\ref{eq:int_gs_ds}) and (\ref{eq:qclbar=int}) since we will still use -the concept of -instantaneous condensation for liquid clouds. Equations -\ref{eq:int_gs_ds} and \ref{eq:qclbar=int} will form the -basis of the homogeneous forcing methods discussed in section -\ref{sec:homog}. -We note in particular that the convective cloud fraction, -previously a quantity that is diagnosed separately from the -large-scale cloud fraction calculated by the \cite{smith90} scheme, -may, in PC2, be included as part of the large-scale -cloud fraction. This aspect is similar to the \cite{t93} approach. - -The final aim of PC2 is -that the parametrization of each term in (\ref{eq:dqcldt_and_dcdt}) -is performed by each part of the model that alters $\overline{T}$, -$\overline{p}$, -$\overline{q}$, $\overline{q_{cl}}$ or $\overline{q_{cf}}$ as an -integral part of that physics or dynamics scheme. -However, in this PC2 scheme we acknowledge that this will not be possible, -at least, not to begin with. -Hence we have specifically developed generic approaches -that can be used to calculate expressions for -$\frac{\partial \overline{q_{cl}}}{\partial t}$ and $\frac{\partial C_l} -{\partial t}$ . -These are referred to as Homogeneous forcing (section \ref{sec:homog}), -Injection source (or inhomogeneous forcing, section \ref{sec:inhomog}) -and Width Changing (section \ref{sec:width}). Two additional modules -are available to assist with PC2, liquid cloud initiaion (section -\ref{sec:init}) and the calculation of total cloud fraction changes -(section \ref{sec:ct}). At -the present time, only the large-scale precipitation (section -\ref{sec:precip}) scheme has been rewritten fully to use the PC2 -concept of prognostic cloud fractions. The existing mass-flux -convection scheme has been modified to enable calculation of the detrained -condensate, but direct modification to the cloud fraction is not -included. All other physics schemes use one of the generic -approaches below. - -\subsubsection{A note on convective cloud fraction} - -It was the original intention that PC2 be able to replace the two -separate diagnostic cloud fractions (large-scale and convective) with -a single cloud fraction, as in \cite{t93}. The hypothesis was that -by detraining cloud -directly from the convection scheme we would no longer need a separate -representation of this cloud type. Our experience with PC2 is that -this is not necessarily the case. We suspect that the basic reason is -that we are unable to truely represent the extreme PDF shapes that -result from convective activity. Additionally, we only create cloud -associated with the detrainment part of the convection scheme, assuming -that cloud associated with the active updraughts in convection is small. -This assumption is not necessarily applicable. Similar arguments, -and model results, come from analysis of the \cite{t93} and -\cite{t02} scheme (Ben Johnson, personal communication). We also note -that with two cloud fraction types and two different optical depths -it is possible to have a basic degree of representation of cloud inhomogeneity. - -Hence the code still exists to enable PC2 to be -run with or without a diagnostic convective cloud fraction, although PC2:66 -does not include a diagnostic term. More details are -in section \ref{sec:convec}. - -\section{Physical basis of the PC2 prognostic cloud scheme} - -In this section we will develop the physical models that PC2 uses in order -to calculate its prognostic increment terms. We will also consider the -numerical solution of the models. The way in which these are incorporated -into the Unifed Model will be discussed in section \ref{sec:um} - -\subsection{Instantaneous condensation} - -Liquid clouds in PC2 use the concept of instantaneous condensation. Hence the -`s' distribution methods are fully applicable to the development of the -equations that govern the parametrization of liquid cloud in PC2. We will -start by looking at changes to $\overline{q_{cl}}$ and $C_l$ when a -uniform forcing -is applied to a gridbox, under the assumption of instantaneous condensation. - -\subsection{Homogeneous forcing} -\label{sec:homog} - -We define the expression -\textit{uniform forcing} (or \textit{homogeneous forcing}) to refer to -changes in local values of $T_L$ -and $q_T$ that occur at a rate independent of the part of the -gridbox in which they are located. This implies that $G(s)$ will not alter -due to such a process. -Uniform forcing simply alters $Q_c$ in (\ref{eq:int_gs_ds}) and -(\ref{eq:qclbar=int}). -In the Unified Model, this concept will be applied to several different sets of -physics increments in order to calculate the condensation and cloud fraction -changes associated with each one, where the physics routine does not allow -the explicit calculation of condensation and cloud fraction changes by -another method. -Large-scale ascent may be considered a meteorological example of such -a process. By differentiating (\ref{eq:int_gs_ds}) and (\ref{eq:qclbar=int}) -with respect to time, assuming uniform forcing -(so ${\frac{\partial G}{\partial t}}$ terms are zero), we obtain - -\begin{equation} -{{\frac{\partial C_l}{\partial t}} = G(-Q_c) {\frac{\partial Q_c} -{\partial t} }.} -\label{dcdt} -\end{equation} - -\begin{equation} -{\frac{\partial \overline{q_{cl}}}{\partial t}} = -C_l {\frac{\partial Q_c}{\partial t}} -\label{dqcldt} -\end{equation} - -The quantity $G(-Q_c)$ is the value of the PDF -of $G$ at $s=-Q_c$, which defines the boundary between -the saturated and unsaturated parts of the distribution. - -If we wish to consider a prognostic cloud scheme with equations for -the rate of change of condensate and cloud fraction based upon (\ref{dqcldt}) -and (\ref{dcdt}) then we need to close (\ref{dcdt}) by specifying the -value of $G(-Q_c)$. We will choose to develop a parametrization for this -quantity based upon the quantities $C_l$, $\overline{q_{cl}}$ and the saturation -deficit, $SD$, rather than tie $G(-Q_c)$ to a process. -The saturation deficit is \textit{defined} here in the `s' -framework to be the first moment of the PDF for `s' values less than $-Q_c$. -In this way it is analogous to the liquid water content, $\overline{q_{cl}}$. -Appendix A of \cite{wg03} writes this \textit{definition} as - -\begin{equation} -{SD = - \int_{-\infty}^{-Q_c} {( s+Q_c ) G(s) ds}} -\label{SD} -\end{equation} - -and shows this is equivalent to - -\begin{equation} -{SD = a_L ( q_{sat}({\overline{T}},{\overline{p}}) - {\overline{q}} ) .} -\label{SD2} -\end{equation} - -The basis behind the parametrization for $G(-Q_c)$ is to consider -an underlying form of the distribution $G(s)$ near the $+b_s$ and -$-b_s$ ends. We borrow the notation of \cite{smith90} and refer -to a quantity $b_s$ that is the value of $s$ when a monomodal -distribution $G(s)$ just equals zero. We suppose that the -distribution G can be described as a power law near $s=b_s$. - -\begin{equation} -G(s) ~ \propto ~ {(-s + b_s)}^n -\label{eqn19} -\end{equation} - -provided $s 1$ then the distribution is narrowed. -For the liquid cloud fraction we therefore have - -\begin{equation} -C_l^{[n+1]} = \int_{s=-Q_c}^{\infty} \xi G(\xi s) ds . -\label{eq:c_l_xi} -\end{equation} - -If we transform variables to $s' = \xi s$ we can rewrite this integral as - -\begin{equation} -C_l^{[n+1]} = \int_{s'=-Q_c \xi}^{\infty} G(s') ds' . -\label{eq:c_l_xi2} -\end{equation} - -Hence the expression for $C_l^{[n+1]}$ is equivalent to using the same -distribution function $G(s)$ as for $C_l^{[n]}$ except that the saturation -boundary has been moved from $-Q_c$ to $-Q_c \xi$. The result is the same as applying -a homogeneous forcing (\ref{eq:deltac}) with a modified forcing, - -\begin{equation} -\Delta Q_c \equiv \xi Q_c - Q_c , -\label{eq:deltac_modified} -\end{equation} - -or the continuous version - -\begin{equation} -\frac{\partial Q_c}{\partial t} \equiv -Q_c \frac{\partial}{\partial t}(\xi - 1) . -\label{eq:xi_equiv} -\end{equation} - -We can write $\xi$ in a slightly more -informative way by linking it to the relative change in width of the PDF -$\frac{1}{b_s} \frac{\partial b_s}{\partial t}$. For a PDF that changes -its width, $\xi$ is defined as - -\begin{equation} -\xi = \frac{b_s}{b_s + \delta b_s} = \frac{1}{1 + \frac{\delta b_s}{b_s}}. -\label{eq:xi_equiv1} -\end{equation} - - -For an infintessimal timestep $\delta t$ we therefore have - -\begin{equation} -\xi = \frac{1}{1 + \frac{1}{b_s} \frac{\partial b_s}{\partial t} \delta t } -\label{eq:xi} -\end{equation} - -and hence, by expanding (\ref{eq:xi}) to give $\xi = 1 - \frac{1}{b_s} -\frac{\partial b_s}{\partial t} \delta t$ and using the homogeneous -forcing expression (\ref{dcdt}) with the modified forcing (\ref{eq:xi_equiv}), -we retrieve the continuous form - -\begin{equation} -\frac{\partial C_l}{\partial t} = - G(-Q_c) Q_c \frac{1}{b_s} -\frac{\partial b_s}{\partial t} . -\label{eq:dcdt_width} -\end{equation} - -A similar analysis can be performed for $\frac{\partial \overline{q_{cl}}} -{\partial t}$ from (\ref{dqcldt}) to give - -\begin{equation} -\overline{q_{cl}}^{[n+1]} = \frac{1}{\xi} \int_{s'=-Q_c \xi}^{\infty} -(- \xi Q_c + s') G(s') ds' . -\label{eq:qcl_xi2} -\end{equation} - -Again, this is equivalent to using the homogeneous forcing with the modified -forcing (\ref{eq:xi_equiv}), but it also includes a scaling -term $\frac{1}{\xi}$. In the infinitessimal limit, this scaling gives -a second term that is proportional to the value of the integral (i.e. -$\overline{q_{cl}}$). Hence we obtain the final continuous solution - -\begin{equation} -\frac{\partial \overline{q_{cl}}}{\partial t} = -(- C_l Q_c+\overline{q_{cl}}) \frac{1}{b_s} \frac{\partial b_s}{\partial t} . -\label{eq:dqcldt_width} -\end{equation} - -To close the solution, we need to parametrize $\frac{1}{b_s} -\frac{\partial b_s}{\partial t}$ , -which could be linked to the physics of the process that is occuring. Note -we don't need to calculate $b_s$ separately, just its \textit{fractional} -rate of change. Options for the parameterisation of -$\frac{1}{b_s}\frac{\partial b_s}{\partial t}$ -due to turbulent ``erosion'' are described in section \ref{sec:turb}, -along with the numerical methods used to integrate the equations. - - -\subsection{Initiation of cloud} -\label{sec:init} -In section \ref{sec:homog} we commented that the closure (\ref{eqn22}) for $G(-Qc)$ -is only valid if $C_l$ is not identically 0 or 1. If $C_l$ -is 0 or 1 we know that $G(-Q_c)$ is equal to 0 but we have lost the -information that will tell us when $G(-Q_c + \Delta Q_c)$ starts -differing from 0. Hence the homogeneous forcing equation set -(\ref{dcdt}), (\ref{dqcldt}) and (\ref{eqn22}) is not complete if -we start from a position where $C_l$ is 0 or 1. To complete this set, -we will need to define a width, $b_s$, to the PDF and provide an initiation -increment to $C_l$ and $\overline{q_{cl}}$ when the value of $-Q_c$ crosses -the limit of the distribution. There is more discussion in -\cite{wg03}. - -To initiate new partial cloud-cover (or new partial clear-sky), we essentially -call a diagnostic cloud scheme to initialise the prognostics -$C_l$ and $\overline{q_{cl}}$. -In the UM there is currently a choice of 2 different diagnostic cloud -schemes that can be used for this; either a version of the Smith scheme -(see UMDP 029), or the bimodal scheme (see UMDP 039). -These two options are described below... - -\subsubsection{Initiation using a ``Smith-like'' method} - -This option is selected by setting the UM namelist switch -{\bf i\_pc2\_init\_method = 1} (Smith). - -We will assume the same form of the PDF at its boundaries as is -assumed in the derivation of the $G(-Q_c)$ closure. For the high `$s$' end -of the PDF distribution we integrate the power law description in -(\ref{eqn19}) to obtain the expressions - -\begin{equation} -C_l = \frac{1}{2 b_s^{n+1}} (b_s + Q_c)^{n+1} , -\label{eq:initc} -\end{equation} - -\begin{equation} -\overline{q_{cl}} = \frac{1}{2 b_s^{n+1}} \frac{(b_s + Q_c)^{n+2}}{n+2} . -\label{eq:initqcl} -\end{equation} - -We now need to parametrize the PDF width $b_s$. Unlike the \cite{smith90} -scheme, this is the only -location in the PC2 cloud scheme where the width needs to be defined -for the liquid cloud (although see section \ref{sec:mp_depsub} for -a discussion of an equivalent width in the deposition / sublimation -relationship for ice cloud). We still choose to define $b_s$ in -terms of a critical relative humidity parameter, $RH_{crit}$. Like -the \cite{smith90} scheme (see \citeumdp{029}), -we define the value of $b_s$ as - -\begin{equation} -b_s = a_L (1 - RH_{crit}) q_{sat} (\overline{T_L}) . -\label{eq:bs} -\end{equation} - -Hence, if the parameter $n$ was the same in PC2 as the equivalent -in \cite{smith90}, the -initial creation of liquid cloud would follow precisely that diagnosed -by the \cite{smith90} scheme (assuming that the numerical implementation of -the calculation is the same). -Its subsequent behaviour in PC2, though, would be different, because -the subsequent physical processes that act are parametrized in different -ways. -Note: for some reason, the implementation in the UM uses a fixed value -of $n = 0$ (corresponding to a top-hat distribution) if a constant $RH_{crit}$ -profile is used, but instead sets $n = 1$ (a triangular distribution) -in the PC2 initiation calculation if a TKE-based variable $RH_{crit}$ is used. -In the latter case, $n = 0$ is still hardwired in the PC2 homogeneous forcing -calculations, so it is not handled consistently. - -An equivalent initiation scheme is required if $C_l$ is 1 and $Q_c$ -is being reduced - at some point we need to introduce clear sky into -the solution. Because we make the choice of symmetry (which could be -relaxed if we used different $RH_{crit}$ values for $C_l$ of 1 and $C_l$ of 0), -the problem is entirely equivalent to that of initiating -from $C_l = 0$, with the exception that $\overline{q_{cl}}$ is replaced by $SD$, -$C_l$ is replaced by $(1-C_l)$, and $Q_c$ is replaced by $-Q_c$. -We hence have the solution - -\begin{equation} -1 - C_l = \frac{1}{2 b_s^{n+1}} (b_s - Q_c)^{n+1} , -\label{eq:init1mc} -\end{equation} - -\begin{equation} -SD = \frac{1}{2 b_s^{n+1}} \frac{(b_s - Q_c)^{n+2}}{n+2} . -\label{eq:initSD} -\end{equation} - -The conversion between $SD$ and $\overline{q_{cl}}$ follows (\ref{SD2}). -We will choose, -as we do throughout PC2, to define $\alpha$ (and hence $a_L$) in terms of -$\frac{\partial q_{sat}(\overline{T})}{\partial t}$, although within -this diagnostic calculation of SD it might actually be better to use -the representation (\ref{eq:alpha}) used by the diagnostic \cite{smith90} scheme. - -\subsubsection{Numerical Application of the Smith method} -\label{sec:numapp_init} - -In order to calculate and compare the state of the model to $b_s$, -we first calculate $T_L$, $q_{sat}(\overline{T_L})$ and calculate -the mean relative total humidity, $RH_T$, where - -\begin{equation} -RH_T = \frac{ \overline{q} + \overline{q_{cl}} } {q_{sat}(\overline{T_L}) } . -\label{eq:rht} -\end{equation} - -We then assess whether initiation is required. There are only two -circumstances in which we wish to proceed further: -\begin{itemize} -\item{If the current cloud fraction $C_l$ is 0 and $-Q_c < b_s$. By dividing the second condition by $a_L q_{sat} (\overline{T_L})$ we see, using the definitions (\ref{eq:qc_eq_qt-qs}) and (\ref{eq:bs}), that this second condition is equivalent to $RH_T > RH_{crit}$.} -\item{If the current cloud fraction $C_l$ is 1 and $-Q_c > -b_s$ (or, equivalently, $RH_T < 2 - RH_{crit})$.} -\end{itemize} -Note: in the UM implementation, the actual conditions for when initiation -may occur are more complicated than this, and there are several options -depending on a namelist switch. See section \ref{sec:init2} for details... - -In the second case, we then make the temporary transformation of variables -in order to use the same solution set as in the first case: $C_l'$ takes the -value $(1-C_l)$ and $RH_t'$ takes the value $(2-RH_t)$ (which is equivalent -to the replacing of $Q_c$ by $-Q_c$). In the first case, $C_l'$ and $RH_t$ take -the same values as $C_l$ and $RH_t$ respectively. - -We then solve for the initiated cloud fraction $C_l'$, using the similar -methods as described in \citeumdp{029}, except -that we allow the solution to vary with the PDF shape $n$. We first -write $Q_N$ as - -\begin{equation} -Q_N = \frac{Q_c}{b_s} = \frac{ a_L (\overline{q_T} - q_{sat}(\overline{T_L})) } -{ a_L (1 - RH_{crit}) q_{sat} (\overline{T_L})} = \frac{RH_T - 1}{1-RH_{crit}} -\label{eq:qn_def} -\end{equation} - -and then use $Q_N$ to solve the initiated cloud fraction. We assume a PDF -described by a power law as in (\ref{eqn19}) (and the equivalent for the -other end of the distribution, the two expressions switching at $Q_c=0$), -which is normalized. The solution to (\ref{eq:int_gs_ds}) is hence - -\begin{equation} -C_l^{init'} = \left\{ \begin{array}{ll} - 0, & Q_N \le -1 \\ - \frac{1}{2} {\left( 1 + Q_N \right)}^{n+1}, & -1 < Q_N \le 0 \\ - 1 - \frac{1}{2} {\left( 1 - Q_N \right)}^{n+1}, & 0 < Q_N < 1 \\ - 1, & 1 \le Q_N . - \end{array} \right. -\label{eq:c_qn} -\end{equation} - -where $C_l^{init'}$ is the initiated value of liquid cloud fraction. -If we had performed the variable transformation we then we need to -transform back, so $C_l^{init} = 1 - C_l^{init'}$, otherwise -$C_l^{init} = C_l^{init'}$. - -In practice, it is likely to be only the second of the -options in (\ref{eq:c_qn}) that the scheme uses, since we will be at -that end of the distribution function, unless previous -parts of the model timestep have resulted in large -forcings to $Q_c$. - -The solution for the initiated liquid water, $\overline{q_{cl}}^{init}$ is -more difficult, since it depends on the width of the distribution $b_s$, -hence on $a_L$ and $\alpha$, and $\alpha$ is a function of the dry-bulb -temperature $\overline{T}$, which is not known until we know the -amount of condensation. -Hence we will need to iterate to a solution. - -We first calculate $q_{sat}(\overline{T})$, $\alpha$, $a_L$ and $b_s$, using -(\ref{eq:alpha_exp}), (\ref{eq:a_L}) and (\ref{eq:bs}). We then solve for the liquid -water content: - -\begin{equation} -\frac{\overline{q_{cl}}^{init'}}{b_s} = \left\{ \begin{array}{ll} - 0, & Q_N \le -1 \\ - \frac{1}{2 (n+2)} {\left( 1 + Q_N \right)}^{n+2}, & -1 < Q_N \le 0 \\ - Q_N + \frac{1}{2 (n+2)} {\left( 1 - Q_N \right)}^{n+2}, & 0 < Q_N < 1 \\ - Q_N, & 1 \le Q_N . - \end{array} \right. -\label{eq:l_bar} -\end{equation} - -If we have been working in transformed variables we now transform -back, so the initiated saturation deficit, $SD^{init}$, takes the -value of $\overline{q_{cl}}^{init'}$. We then use (\ref{SD2}) to -estimate $\overline{q_{cl}}^{init}$ using our initial estimates of -$q_{sat}(\overline{T})$ and $a_L$. If we are not in transformed -variables, we have the first estimate -$\overline{q_{cl}}^{init}=\overline{q_{cl}}^{init'}$. - -We now use this estimate of $\overline{q_{cl}}^{init}$ to -calculate a more accurate estimate of $a_L$ etc. by iteration. In order to -achieve a faster convergence of the iteration, we do not use -(\ref{eq:l_bar}) directly in the estimation of $a_L$ etc., but -use a combination of this value and the one from the previous iteration. - -\begin{equation} -\overline{q_{cl}}^{init~[i+1]} = f \overline{q_{cl}}^{init~[i]} + -(1 - f) \overline{q_{cl}}^{init~[i-1]} -\label{eq:iter} -\end{equation} - -where the superscript $[i]$ labels each iteration. We find that 10 -iterations is effective for convergence, with the weighting -$f$ given by $a_L^{[i]}$. - -\subsubsection{Initiation using the bimodal scheme} -\label{sec:bimodal_init} - -This option is selected by setting the UM namelist switch -{\bf i\_pc2\_init\_method = 2} (Bimodal). - -First, the diagnosis of entrainment zones is performed at all grid-points, -as described in UMDP 039. The parameters of the moisture PDF are then -constructed, assuming either a sum of two Gaussian modes from the top -and bottom of an inversion layer (when within an entrainment zone), -or a single symmetric Gaussian mode (when not in an entrainment zone). -The variance of each Gaussian mode is estimated based on the TKE and other -information output by the boundary-layer scheme -(with a minimum limit applied to the PDF width, consistent with -$RH_{crit}$ = 99\%. -Crucially, each Gaussian mode is truncated to zero at plus and minus -3 standard deviations; this sets the overall width of the moisture -PDF at each point. - -The positions of the upper and lower truncated bounds of the moisture PDF -relative to the saturation threshold are expressed in terms of a normalised -$Q_N$ = $Q_c$ over PDF-width (see equation \ref{eq:qn_def}). -In entrainment zones, the sum of the two Gaussian modes can lead to -a highly skewed distribution; hence $Q_N$ can have different values for the -upper and lower bounds, each normalised by the different widths on -either side of the PDF. The upper and lower values of $Q_N$ are then -compared to -1 and 1 respectively, to determine whether the saturation -boundary lies within the PDF bounds. This is the basic condition for -initiation to occur (though there are additional conditions and various -options for these in the soure code; see section \ref{sec:init2}). - -If the initiation conditions are met, the diagnostic bimodal cloud scheme code -is then called (see UMDP 039), and the diagnosed $C_l$ and $q_{cl}$ are used to -set the prognostic $C_l$ and $q_{cl}$. - -\subsection{Injection forcing} -\label{sec:inhomog} - -Injection forcing (sometimes referred to as inhomogeneous forcing) -uses another concept of how the underlying moisture PDF may change in -order to calculate a change in cloud fraction as a result of a known -injection of condensate into a gridbox. The term was developed in -order to be coupled with a modified mass-flux convection scheme, -but is first presented here in its basic form. - -We will assume a physical model whereby saturated air, containing -condensate, randomly replaces already existing air in the gridbox. -(Such a formulation is designed to represent air detrained from convection -replacing pre-existing air when averaged over a large horizontal domain). -\cite{bwg03} discusses the situation in more detail. Briefly, -we consider two parts to the distribution function $G(s)$. One part -represents the background air. This maintains its PDF shape (in terms -of absolute $q_T$ and $T_L$ values) because we assume it is \textit{randomly} -replaced, but will reduce in amplitude as it is replaced by a -second PDF representing the injected air. - -The fractional rate at which existing air -is replaced by the injected source air we will write as -$\frac{\partial{C_S}}{\partial{t}}$. Provided that only the liquid -phase exists (see section \ref{sec:multiple} for the extention to multiple phases), -we then note that the rate -of change of liquid cloud fraction and liquid water content in -the gridbox can be written in two parts: firstly the change -due to the background, and secondly the change due to the source. - -\begin{equation} -\frac{\partial{C_l}}{\partial{t}} = -- C_l \frac{\partial{C_S}}{\partial{t}} + \frac{\partial{C_S}}{\partial{t}} -\label{eq:dcdt_inhom} -\end{equation} - -\begin{equation} -\frac{\partial{\overline{q_{cl}}}}{\partial{t}} = -- \overline{q_{cl}} \frac{\partial{C_S}}{\partial{t}} -+ q_{cl}^S \frac{\partial{C_S}}{\partial{t}} -\label{eq:dqcldt_inhom} -\end{equation} - -where $q_{cl}^S$ is the liquid water content of the injected -air. Eliminating $\frac{\partial{C_S}}{\partial{t}}$ gives the -relationship - -\begin{equation} -\frac{\partial{C_l}}{\partial{t}} = \frac{1 - C_l}{q_{cl}^S -- \overline{q_{cl}}} Q4_l, -\label{eq:dcdt_inhom2} -\end{equation} - -where $Q4_l$ is the net (\textit{including} the liquid water -in the background distribution -that was randomally replaced) injection source change of $\overline{q_{cl}}$: - -\begin{equation} -Q4_l = \frac{\partial{\overline{q_{cl}}}}{\partial{t}} |_{injection \, source}. -\label{eq:q4} -\end{equation} - -We see that we do not need to know anything about the nature of -the two PDFs involved, except the assumption that the injected -PDF contains completely cloudy air. -This equation allows one to calculate the change in $C_l$ associated -with an injection source change of $\overline{q_{cl}}$ for the example of -convection. Modifications -to the mass-flux convection scheme for PC2 (far from trivial and discussed -in depth in section \ref{sec:convec}) -allow $Q4_l$ to be calculated ($q_{cl}^S$ is already available), -and (\ref{eq:dcdt_inhom2}) can then be used -to calculate the equivalent $C_l$ change. We note at this stage -that the denominator in (\ref{eq:dcdt_inhom2}), being the difference -in two terms that may be close to each other, may cause problems -when we attempt to numerically apply this equation. - -It is reasonable to ask what happens to the air in the distribution that -was replaced. In this mathematical representation of a single gridbox we -need not know anything other than that the air is displaced into a neighbouring -gridbox. In practical use with a mass-flux convection scheme we know more that -this air is displaced downwards in the column. We could reasonably calculate -the change in cloud fraction following the same methods as used to calculate -the change in $\overline{q}$ or the change in a tracer and we discuss this -later. - -\subsubsection{Multiple phases in the injection source} -\label{sec:multiple} - -The injection source formulation can be extended to multiple -phases of condensate. In practice, this will simply be the -two phases ice and liquid, although we need to recognize that -they can overlap with each other. \cite{wilson2001} provides the -background to the derivation and it is briefly presented below. - -We firstly rewrite (\ref{eq:dqcldt_inhom}) but use the net -condensate ($\overline{q_c} = \overline{q_{cl}} + \overline{q_{cf}}$) instead of just the -liquid water expression, and the net cloud amount $C_t$, instead -of the liquid cloud amount $C_l$. The same argument as before leads -to the expressions - -\begin{equation} -\frac{\partial{C_t}}{\partial{t}} = -- C_t \frac{\partial{C_S}}{\partial{t}} + \frac{\partial{C_S}}{\partial{t}} -\label{eq:dctdt_inhom} -\end{equation} - -and - -\begin{equation} -\frac{\partial{\overline{q_{c}}}}{\partial{t}} = -- \overline{q_{c}} \frac{\partial{C_S}}{\partial{t}} -+ q_{c}^S \frac{\partial{C_S}}{\partial{t}} . -\label{eq:dqcdt_inhom} -\end{equation} - -The left hand side of (\ref{eq:dqcdt_inhom}) is written as $Q4_c$. -$q_{c}^S$ is the in-cloud -condensate content (ice plus liquid) of the source. - -Hence eliminating $\frac{\partial{C_S}}{\partial{t}}$ we obtain - -\begin{equation} -\frac{\partial{C_t}}{\partial{t}} = \frac{(1-C_t)}{q_{c}^S - -\overline{q_{c}}} Q4_c . -\label{eq:dctdt_q4} -\end{equation} - -We will assume that the proportion of the injected volume that -contains liquid cloud can be written as $g_l$, and the proportion -that contains ice cloud can be written as $g_i$. Note that it -is not necessary to have $g_l + g_i = 1$ if there is mixed -phase cloud injected. We can write the change in \textit{liquid} cloud -fraction equivalently to (\ref{eq:dcdt_inhom}) as - -\begin{equation} -\frac{\partial{C_l}}{\partial{t}} = -- C_l \frac{\partial{C_S}}{\partial{t}} + g_l \frac{\partial{C_S}}{\partial{t}} . -\label{eq:dcldt_inhom} -\end{equation} - -Combining (\ref{eq:dcldt_inhom}) and (\ref{eq:dctdt_inhom}) by eliminating -$\frac{\partial{C_S}}{\partial{t}}$ gives - -\begin{equation} -\frac{\partial{C_l}}{\partial{t}} = \frac{g_l - C_l}{1 - C_t} -\frac{\partial{C_t}}{\partial{t}} -\label{eq:dctdt_dcdt} -\end{equation} - -and hence from (\ref{eq:dctdt_q4}) we have the result - -\begin{equation} -\frac{\partial{C_l}}{\partial{t}} = -\frac{g_l - C_l}{q_c^S - \overline{q_{c}}} Q4_c . -\label{eq:dcltdt_almost_final} -\end{equation} - -An equivalent expression holds for the ice cloud. Hence the change in the -amount of cloud for each phase may be calculated assuming we know -the volume proportions of the source term that contain each of -the phases and the net increase in the amount of condensate, $Q4_c$ -(regardless of phase). This expression is coded for use in -a generically available inhomogeneous forcing module. However, -we can also write this in a slightly more -accessible form by noting the ratio of (\ref{eq:dqcdt_inhom}) and -(\ref{eq:dqcldt_inhom}) with the $Q4$ definitions following (\ref{eq:q4}). - -\begin{equation} -\frac{Q4_c}{q_c^S - \overline{q_c}} = -\frac{Q4_l}{q_{cl}^S - \overline{q_{cl}}} . -\label{eq:q4_ratios} -\end{equation} - -Using (\ref{eq:q4_ratios}) in -(\ref{eq:dcltdt_almost_final}) gives the final expression - -\begin{equation} -\frac{\partial{C_l}}{\partial{t}} = -\frac{g_l - C_l}{q_{cl}^S - \overline{q_{cl}}} Q4_l -\label{eq:dctdt_final} -\end{equation} - -and similarly for the ice. Note that this expression accounts -for the possibility that liquid cloud is displaced from the -gridbox by added ice cloud. We can further write -$q_{cl}^S$ as a fraction of $q_{c}^S$ - -\begin{equation} -q_{cl}^S = h_l q_{c}^S -\label{eq:qcls_qcs} -\end{equation} - -where $h_l$ is the factor between them (i.e. the \textit{mass} fraction -of the injected condensate that is liquid). It is not necessary -in this theory to have $h_l$ equal to $g_l$: if a mixed -phase plume exists $g_l + g_i$ need not equal 1, but since -$h_l$ and its ice equivalent, $h_i$, refer to mass, $h_l + h_i$ -must equal 1. However, if we do not allow a mixed phase injection -(which is the case in the current mass-flux convection -scheme, where only one phase can be injected), $h_l$ and $g_l$ are equal (and either zero or one in the -current mass-flux convection scheme) and we can write (\ref{eq:dctdt_final}) as - -\begin{equation} -\frac{\partial{C_l}}{\partial{t}} = -\frac{ (\delta_{xl} - C_l) }{ \delta_{xl} q_{c}^S - \overline{q_{cl}} } -Q4_l -\label{eq:dctdt_xl} -\end{equation} - -where $\delta_{xl} = h_l = g_l$. Equivalent expressions exist for the -ice cloud fraction and total cloud fraction. - -\begin{equation} -\frac{\partial{C_i}}{\partial{t}} = -\frac{ (\delta_{xi} - C_l) }{ \delta_{xi} q_{c}^S - \overline{q_{cf}} } -Q4_i -\label{eq:dctdt_xi} -\end{equation} - -\begin{equation} -\frac{\partial{C_t}}{\partial{t}} = -\frac{ (1 - C_t) }{ q_{c}^S - \overline{q_{c}} } Q4_c -\label{eq:dctdt_xc} -\end{equation} - -with $\delta_{xi} = h_i = g_i$. These are the expressions that are used -within the convection scheme. It still remains to parametrize $\delta_{xl}$, -which is given by the convection scheme itself. This is discussed in -section \ref{sec:plume_phase}. - -\subsubsection{Numerical application} -\label{sec:multi_numapp} -The numerical application using (\ref{eq:dcltdt_almost_final}) may be performed -with a basic forward timestep. Each of the three cloud fractions can -be incremented, assuming we know $\Delta{\overline{q_{cl}}}$ and -$\Delta{\overline{q_{cf}}}$, as - -\begin{equation} -\Delta{C_t} = \frac{(1 - C_t)} {q_c^S - \overline{q_{cl}} - \overline{q_{cf}}} -( \Delta{\overline{q_{cl}}} + \Delta{\overline{q_{cf}}} ), -\label{eq:cft_ts} -\end{equation} - -\begin{equation} -\Delta{C_l} = \frac{ (g_l - C_l)} -{q_c^S - \overline{q_{cl}} - \overline{q_{cf}}} -( \Delta{\overline{q_{cl}}} + \Delta{\overline{q_{cf}}} ), -\label{eq:cfl_ts} -\end{equation} - -\begin{equation} -\Delta{C_i} = \frac{ (g_i - C_i)} -{q_c^S - \overline{q_{cl}} - \overline{q_{cf}}} -( \Delta{\overline{q_{cl}}} + \Delta{\overline{q_{cf}}} ). -\label{eq:cff_ts} -\end{equation} - -The application from within the convection scheme is slightly different. -We start with (\ref{eq:dctdt_xl}), but enforce two numerical restrictions -to avoid the equation set becoming ill-conditioned. Firstly, we limit -the denominator $q_c^S - \overline{q_{cl}}$ to a minimum value if we -are considering changes of the same phase as the injected source. - -\begin{equation} -\Delta C_l = \frac {\delta_{xl} - C_l} {\delta_{xl} \text{Max}( q_{c}^S -- \overline{q_{cl}} , q_c^{S0} ) + ( 1 - \delta_{xl} ) (-\overline{q_{cl}}) } -Q4_l -\label{eq:delta_cl} -\end{equation} - -where $q_c^{S0}$ is specified as $5 \times 10^{-5} kg \, kg^{-1}$. -The denominator also has an additional check. If its absolute -value is less than a tolerance value of $1 \times 10^{-10} kg \, kg^{-1}$ -then no change in cloud fraction will be considered. A similar -equation is used for the ice cloud and the change in total cloud fraction - - -\begin{equation} -\Delta C_i = \frac {\delta_{xi} - C_i} {\delta_{xi} \text{Max}( q_{c}^S -- \overline{q_{ci}} , q_c^{S0} ) + ( 1 - \delta_{xi} ) (-\overline{q_{cf}}) } -Q4_i -\label{eq:delta_ci} -\end{equation} - -\begin{equation} -\Delta C_t = \frac {1 - C_t} -{ \text{Max}(q_c^S - \overline{q_c} , q_c^{S0} ) } Q4_c . -\label{eq:delta_ct} -\end{equation} - -We now limit the change in cloud fraction to ensure that the cloud -fraction remains within its physical bounds. - -\begin{equation} -C_l^{[n+1]} = ( 0, C_l^{[n]} + \Delta C_l, 1) -\label{eq:delta_cl_conv_final} -\end{equation} - -and similar equations are used for $C_i^{[n+1]}$ and $C_t^{[n+1]}$. - -\subsubsection{A note on the implementation of the cloud fraction change} -\label{sec:conv_imp_note} - -Equation \ref{eq:dcdt_inhom2} has been derived assuming that the -only change in the cloud properties within the gridbox comes from -the detrainment of air from the convective plume (so that the injection -source is an appropriate model). Attention should be drawn to the fact -that this is not the only source of change from the convection scheme. -Two other terms require consideration, namely advection of the environmental -air downwards by compensating subsidence and the condensation resulting -from the adiabatic warming due to this subsidence. -The former is considered correctly in the calculation of -$\frac{\partial \overline{q_{cl}}}{\partial t}$, which corresponds to $Q4$. -However, the calculation of $\frac{\partial C_l}{\partial t}$ -is then performed using (\ref{eq:dcdt_inhom2}) and \textbf{incorrectly} assuming -that all the $\overline{q_{cl}}$ change comes from the detrainment. It is -possible to calculate directly the change in $C_l$ that should occur due to -the detrainment and compensating subsidence treated together, in the same way -that $\Delta \overline{q_{cl}}$ is calculated (see section -\ref{subsect:q4calculation}), and this is the way in which the cloud fraction -change \textbf{should} be done. -It is an unfortunate historical emphasis in the early development of PC2 -on the derivation of (\ref{eq:dcdt_inhom2}) that has led to the treatment -used within the Unified Model for the change in cloud fractions due to convection. - -The change in $\overline{q_{cl}}$ and $C_l$ due to the adiabatic warming -associated with the compensating subsidence is considered explicitly -in the model implementation (see -section \ref{sec:conv_homog}) for both $\overline{q_{cl}}$ and $C_l$ -after the rest of the convective process has been calculated. It is perhaps -arguable that if (\ref{eq:dcdt_inhom2}) is going to be applied then -the value of $Q4$ used in (\ref{eq:dcdt_inhom2}) should include this term. - -Any major future developments of PC2 for a mass-flux convection scheme would be -advised to consider whether it is appropriate to use (\ref{eq:dcdt_inhom2}) at -all. - -\subsection{Ice cloud and mixed phase regions} -\label{sec:ct} - -The homogeneous forcing, initiation and PC2 erosion sections described -above have only considered the generation and dissipation of liquid -clouds. Although the forcing methods will not influence the generation -and dissipation of ice cloud (which is primarily performed in the -large-scale precipitation scheme, section \ref{sec:precip}) we -are still left with the issue of how created or dissipated liquid -cloud overlaps with existing ice cloud in the gridbox. The -opposite situation, where changes in ice cloud are specified -and changes in the overlap with liquid cloud need to be calculated, -is also possible in PC2 (e.g. in the boundary layer, see -section \ref{sec:bl}). - -Here we -need a simple assumption to close the problem. The assumption -that we now choose is that liquid cloud fraction \textit{changes} are -\textit{minimally} overlapped with ice cloud fraction changes. -This choice is based upon observational evidence that mixed -phase cloud is relatively rare, and also on results from earlier PC2 -development that indicated less supercooled liquid water cloud than -is observed from ground-based lidar. - -With this assumption, the equation set becomes straightforward to -write down. We firstly consider that a change in liquid cloud fraction -$\Delta C_l$ is known and we wish to estimate the resulting change in -the total cloud fraction. There is, of course, no change in the ice -cloud fraction $C_i$, since, from our \textit{definitions} in -(\ref{eq:dqcldt_and_dcdt}) and (\ref{eq:mp}), this includes the mixed phase -contribution. Hence we write - -\begin{equation} -\Delta C_i = 0 . -\label{eq:deltaci_eq_0} -\end{equation} - -The change in the total cloud fraction, $C_t$ will depend upon -the sign of the change of the liquid cloud fraction. If -$\Delta C_l > 0$, then $\Delta C_t$ is going to be the same as $\Delta C_l$ -($C_l$ is being added with minimum overlap to $C_i$), unless the -gridbox becomes completely covered in cloud, when there is no -choice but to generate mixed phase cloud. Hence we have - -\begin{equation} -\Delta C_t = \text{Min} ( \Delta C_l , 1 - C_t ). -\label{eq:deltact_min} -\end{equation} - -If $\Delta C_l < 0$, then we still consider minimum overlap -of the \textit{changes} (this is so that the solution is reversible as -much as possible). Hence $\Delta C_t$ is going to be the same as -$\Delta C_l$ unless $C_l$ is reduced below the existing $C_i$, in -which case no more change to $C_t$ is possible. - -\begin{equation} -\Delta C_t = \text{Max} ( \Delta C_l , C_i - C_t ) , -\label{eq:deltact_min2} -\end{equation} - -remembering that both quantities in the maximum expression in -(\ref{eq:deltact_min2}) have negative values. - -We can write similar expressions if a known amount of ice cloud -is added or removed, and we need to calculate the effect on $C_t$. -Similar to the results above we have: - -\begin{equation} -\Delta C_l = 0 . -\label{eq:deltacl_eq_0} -\end{equation} - -and - -\begin{equation} -\Delta C_t = \left\{ \begin{array}{ll} - \text{Max} ( \Delta C_i , C_l - C_t ), & \Delta C_i < 0 \\ - \text{Min} ( \Delta C_i , 1 - C_t ), & \Delta C_i > 0 . - \end{array} \right. -\label{eq:deltact_min_array} -\end{equation} - -For completeness, we also present here the equation set for -random overlap of changes in liquid cloud with existing ice cloud. -We have, as before, - -\begin{equation} -\Delta C_i = 0 . -\label{eq:deltaci_eq_0_2} -\end{equation} - -For $\Delta C_l > 0$ additional liquid cloud is added -randomly to any location outside that of the current liquid -cloud. A proportion $\frac{1-C_t}{1-C_l}$ of this will be additionally -outside that of existing ice cloud. Hence the net change in -total cloud fraction can be written as - -\begin{equation} -\Delta C_t = \Delta C_l \frac{1 - C_t}{1 - C_l} . -\label{eq:deltact_ran1} -\end{equation} - -Similarly, if $\Delta C_l < 0$, the liquid cloud is removed -randomly from the existing liquid cloud. A proportion -$\frac{C_t - C_i}{C_l}$ of this is from liquid cloud that does not -overlap with existing ice cloud. Hence, - -\begin{equation} -\Delta C_t = \Delta C_l \frac{C_t - C_i}{C_l} . -\label{eq:deltact_ran2} -\end{equation} - -Equivalent equations to (\ref{eq:deltact_ran1}) and -(\ref{eq:deltact_ran2}) but with $C_l$ and $C_i$ swapped apply when -we need to estimate changes in $C_t$ from a known $\Delta C_i$, when -assuming random overlap. - -\subsubsection{Numerical Implementation} - -In general, although the situation does not occur within the current -implementation of PC2 , we might have increments to both $C_l$ and -$C_i$ simultaneously. Hence the implementation is to calculate -$\Delta C_t$ from the sum of that predicted by (\ref{eq:deltact_min}) -or (\ref{eq:deltact_min2}), and (\ref{eq:deltact_min_array}). -For the random overlap situation we also need to apply a check on -the denominator in (\ref{eq:deltact_ran1}) and (\ref{eq:deltact_ran2}) before -calculation, with the result set to the limit $\Delta C_t = 0$ -if the denominator is 0. For the minimum overlap situation a final check -is made that $C_t$ lies between 0 and 1, with the value being reset to -0 or 1 if not. - -\subsection{Forced convective cloud} - -Forced convective clouds are clouds that form at the top of a convective boundary layer -but are too shallow to reach their level of free convection (and become fully fledged -cumulus clouds). These clouds currently require special treatment because initiation -in PC2 uses the Smith scheme with a specified value of $RH_{crit}$ while the large $RH$ -variability associated with these clouds implies much lower values than are typically used. - -A profile of ``forced cloud fraction'', $C_{forced}$, is parametrized as -linearly varying with height between a cloud-base value, at the lifting -condensation level (LCL) from the convection diagnosis parcel ascent, and a cloud-top -value of 0.1 at the top of the capping inversion, $z_i^{top}$. The cloud-base -value of $C_{forced}$ varies linearly between 0.1 and 0.3 for cloud depths -between 100 m and 300 m based loosely on SGP ARM site observations \cite{zk13}. The -inversion top is taken to be the boundary layer depth, $z_h$ plus the inversion -thickness, $\Delta z_i$ parametrized following \cite{rb08} as: -\begin{equation} -\Delta z_i = 6.3 \, w_m^2 / \int_{z_h}^{z_h+\Delta z_i} b \, dz -\label{dz_param} -\end{equation} -where $w_m$ is the boundary layer velocity scale ($w_m^3 = u_*^3 + 0.25 w_*^3$) and $b$ is -the parcel buoyancy that is integrated over the depth of the inversion assuming a -piece-wise linear variation between grid-levels. Note that the constant in (\ref{dz_param}) -is the same as in \cite{rb08} because $6.3 = 2.5 * 4^{2/3}$ and $w_m^3$ differs by a factor of 4. - -The in-cloud water content at the top of the inversion is estimated using the water content -from the diagnostic parcel ascent (used to diagnose boundary layer type and trigger convection), -with linear interpolation used between the lifting condensation level and inversion -top. To allow for sub-adiabatic water content (due to lateral mixing or microphysical -processes) the in-cloud water content can be reduced by a factor, forced\_cu\_fac, that has been -set to 0.5 in GA7. - -These cloud fraction and water content profiles are then used as minimum values and -increments to $C$ and $\overline{q_{cl}}$ calculated if necessary. -This methodology can also optionally be applied to cloud layers diagnosed as cumulus, if the -boundary layer option to mix across the lifting condensation level is selected that generates -a cloud base transition zone thickness which is then treated analgously to the inversion -thickness above. - -Also, there is an option to treat the calculated forced cumulus cloud -fraction and water content as diagnostic quantities passed directly to -the radiation scheme as part of the ``convective'' cloud, instead of -using them to modify the prognostic ``large-scale'' cloud variables -$C$ and $\overline{q_{cl}}$. If this option is used, the convective -cloud fraction $CCA$ and water content $CCW$ output by the convection -scheme are updated, by taking the forced cumulus profiles as their -minimum allowed values. Note that only the convective cloud fields -passed to radiation are updated (i.e. the versions of $CCA$ and $CCW$ -that are stored in the model dump / D1 array). The UM code contains -other copies of the convective cloud fields that are only used for -diagnostics; these are {\em not} updated. - -The different options for how to treat forced cumulus cloud are -controlled by the cloud namelist input $forced\_cu$, and are -summarised below: - -\begin{itemize} -\item $forced\_cu = 0$: No treatment of forced cumulus clouds. -\item $forced\_cu = 1$: Forced cumulus cloud applied to -$C$ and $\overline{q_{cl}}$ only in dry-convective boundary-layers. -\item $forced\_cu = 2$: Forced cumulus cloud applied to -$C$ and $\overline{q_{cl}}$ in both dry-convective and -cumulus-capped boundary-layers. -\item $forced\_cu = 3$: Forced cumulus cloud applied to -$CCA$ and $CCW$ in both dry-convective and -cumulus-capped boundary-layers. -\end{itemize} - - -\subsection{Turbulence-driven production of subgrid scale liquid cloud} -\label{sec:turb_qcl_scheme} - -\subsubsection{Introduction}\label{sec:sgt_intro} - -\cite{fhfk14} developed a model for -subgrid liquid water production by turbulent motions. -Their method uses an exactly soluble -stochastic process to describe -subgrid relative humidity (RH) fluctuations. -The probability density function (PDF) of the fluctuations -can be diagnosed in terms of the local turbulent local state -and any pre-existing ice cloud. The -liquid cloud properties (cloud fraction and liquid water content) -can be then be calculated as truncated moments of the PDF. - -\cite{fhfk14} initially used their model to -understand and parametrize the results of Large Eddy Simulations (LES) -of shear-induced, Altostratus clouds. They obtained excellent -agreement between their theoretically predicted predicted mean -cloud properties and the bulk properties of the LES clouds. -Subsequently, their model has been used as the basis of -subgrid cloud initiation method for use in the Unified Model -in conjunction with the PC2 prognostic cloud scheme. In Section \ref{sec:sgt_model_describe} -we outline the model of \cite{fhfk14}. In Section \ref{sec:sgt_model_implement} -we described its implementation in the GCM. - - -\subsubsection{Model description} -\label{sec:sgt_model_describe} - -\cite{fhfk14} started from the equation for the dynamics of -ice supersaturation $S_i=e_v/e_{sat\;ice}-1$: -\begin{equation}\label{eqn:squires_eqn} - \frac{D S_i}{D t} = -b_i B_0 {\cal M}_1 S_i - -\left(\frac{\varepsilon}{L^2}\right)^{1/3}(S_i-S_E) + a_i w, -\end{equation} -where ${\cal M}_1$ is the first moment of ice particle size distribution (PSD), -$\varepsilon$ is the turbulent dissipation rate, $L$ is a prescribed mixing length -for the turbulence, $S_{\rm E}$ is the ice supersaturation of the -environment surrounding the cloud and $b_i,B_0$ and $a_i$ are function of $p$ and $T$ given by -\begin{eqnarray} - b_i &=& \frac{1}{q} + \frac{\epsilon L_s^2}{c_p R T^2}, \\ - B_0 &=& 4\pi C \left( \frac{\epsilon L_s^2}{K_a R T^2} + \frac{R T}{\epsilon e_{si} \psi} \right)^{-1}, \\ - a_i &=& \frac{g}{R T}\left( \frac{\epsilon L_s}{c_p T} - 1 \right), \\ -\end{eqnarray} -The first term on the right hand side of Eq.~\ref{eqn:squires_eqn} is -the sink of vapor due to depositional growth of ice crystals, the second -term models entrainment (mixing) of environmental air into the cloudy -volume and the third term is a source term due to vertical air motions. - -\cite{fhfk14} modeled vertical velocity as a white-noise process -with autocorrelation function: -\begin{equation} - \overline{w(t)w(s)} = \sigma_w^2 \tau_{\rm d} \delta(t-s), -\end{equation} -where $\delta$ is the Dirac distribution and the intensity of the -noise, $\sigma_w^2$, will be called the -standard derivation of the vertical velocity fluctuations (due to the white nature of -noise, a true expectation value $\overline{w^2}$ is not defined) and $\tau_{\rm d}$ -a Lagrangian decorrelation time define here by the relation used by \cite{rodean1997}: -\begin{equation} - \tau_{\rm d} = \frac{2\sigma_w^2}{\varepsilon C_0}, -\label{eqn:taud} -\end{equation} -where $C_0$ is a known constant. - -Because it is linear in $S_i$, Equation \ref{eqn:squires_eqn} can be solved exactly, -for any given realisation of the noise term. By averaging the solutions over the the noise -and taking a steady-state limit (see \cite{fhfk14} for details) it can be shown -that the solution PDF is Gaussian with mean and variance given by: -\begin{eqnarray} - \overline{S_i} &=& - S_{\rm E}\frac{ \left(\varepsilon/L^2\right)^{1/3} }{ b_i B_0 {\cal M}_1 + \left(\varepsilon/L^2\right)^{1/3} }. - \label{eqn:si_avg} \\ - \overline{S_i^2} &=& - \frac{a^2_{\rm i} \sigma^2_w \tau_{\rm d}}{ 2\left(b_i B_0 {\cal M}_1 + \left(\varepsilon/L^2\right)^{1/3}\right)}, - \label{eqn:si_var} -\end{eqnarray} - -Equation \ref{eqn:si_avg} and \ref{eqn:si_var} completely specify the PDF, $F(S_i)$, of -steady-state humidity variations for the subgrid model. The liquid cloud fraction and -liquid water mass mixing ratio are given by -\begin{eqnarray} - C_l^{sgt} &=& \int_{S_{i,wat}}^\infty d S_i F(S_i), \label{eqn:cloud_fraction} \\ - q_{cl}^{sgt} &=& q_{sat\;ice}\int_{S_{i,wat}}^\infty d S_i (S_i -S_{i,wat}) F(S_i) \label{eqn:cloud_liquid}, -\end{eqnarray} -where $S_{i,wat} = e_{sat\;wat}/e_{sat\;ice}-1$ is the value of ice -supersaturation at water saturation. -We use the superscription `$sgt$'(=`{\it s}ub{\it g}rid {\it t}urbulence') to indicate -that $C_l^{sgt}$ and $q_{cl}^{sgt}$ are values of cloud fraction and water content -diagnosed from a parametrization of small-scale turbulent processes. - - -\subsubsection{Model implementation and closure relations} -\label{sec:sgt_model_implement} - -To implement the model of Section \ref{sec:sgt_model_describe} in the -Unified Model, closure relations are needed for the quantities $\sigma_w^2$, -$\varepsilon$, $L$, $\tau_{\rm d}$ and $S_E$, subject to the constraining relationship -given by Eq. \ref{eqn:taud}. -In each model grid box, these parameters specify the subgrid PDF, $F(S_i)$, and -from this the liquid cloud fraction and water content produced by turbulence -can be found using Eqs \ref{eqn:cloud_fraction} and \ref{eqn:cloud_liquid}. - -In addition we need to make some assumptions about how the diagnosed values -$C_l^{sgt}$ and $q_{cl}^{sgt}$ relate to the model prognostic fields, $C_l$ and $q_{cl}$. -Two methods are available for doing this. In the simplest case, the diagnosed -values $C_l^{sgt}$ and $q_{cl}^{sgt}$ are just treated as increments to model prognostics -(option one, in Sec. \ref{sec:sgt_increments} below). -A more complex option (see option two, below) is to increment the -model fields via the PC2 Erosion functionality. - -\subsubsection{Closure relations}\label{sec:sgt_closures} - -The vertical velocity variance, $\sigma_w^2$, is available as a diagnostic from the -Boundary Layer scheme. Because the Boundary Layer scheme is called after the -Microphysics on each model timestep, the diagnostic value is stored in a (non-advected) -model prognostic field. The scheme will operate only where there is diagnosed turbulence, -i.e., non-zero $\sigma_w^2$. - -We take the mixing length scale, $L$, to be proportional to the vertical -grid spacing in each grid box: $L=\beta_{mix} \Delta z$, where $\Delta z$ is -calculated as the height different between the $\rho$-levels adjacent -to the given $\theta$-point. The parameter, $\beta_{mix}$, -is an adjustable constant that the user can define (see Section \ref{sec:sgt_options} below), -however it should be of order one. - -To obtain $\tau_{\rm d}$ we impose an eddy size constraint: -\begin{equation} - \tau_{\rm d} = \frac{L}{\sigma_w} = \beta_{mix} \frac{\Delta z}{\sigma_w} -\label{eqn:eddy_size} -\end{equation} -Eq.~\ref{eqn:taud} then determines the dissipation rate, $\varepsilon$, that is consistent -with the other parameters. The constant $C_0=10$ by default, but can be adjusted by the user. - -The scheme is limited to act only in grid boxes where $\tau_{\rm d}$ is less than a -prescribed value, $\tau_{d}^{max}$. The default is $\tau_d^{max}=1200\;{\rm sec}$, which typically -coincides with a couple of model timesteps. The motivation for this is that a -motion that takes longer than a few timestep to decorrelate will be partially resolved by -the dynamics and therefore cannot be considered as `subgrid' turbulence. - -Finally, where $T$, $p$ and $q$ appear in the expressions for $C_l^{sgt}$ and $q_{cl}^{sgt}$, -these are taken to be the grid box mean values. The first moment of the ice PSD, ${\cal M}_1$, -is found from the parametrization, due to \cite{fhbicc05}, described -in Section 4.1 of UMDP26. - - -\subsubsection{Options for incrementing model prognostics}\label{sec:sgt_increments} - -Using the information in Section \ref{sec:sgt_closures} to obtain closed expressions -for the subgrid PDF of $S_i$-fluctuations allows $C_l^{sgt}$ and $q_{cl}^{sgt}$ to be -calculated. These will be non-zero only where there is turbulence as diagnosed by the -Boundary Layer scheme (and hence non-zero $\sigma_w^2$). To calculate $C_l^{sgt}$ and $q_{cl}^{sgt}$ -the integrals in Eqs \ref{eqn:cloud_fraction} and \ref{eqn:cloud_liquid} are evaluated -numerically using discretisation based on user-specified number of bins. - -Given $C_l^{sgt}$ and $q_{cl}^{sgt}$, two options are available for relating these -to changes in the model prognostics: - -\paragraph{Option one: direct increments} - -The values of $C_l^{sgt}$ and $q_{cl}^{sgt}$ can be added as increments to the -model prognostic fields, $C_l$ and $q_{cl}$. In this case -\begin{eqnarray} - \left( \Delta C_l \right)_{sgt} &=& C_l^{sgt} \\ - \left( \Delta q_{cl} \right)_{sgt} &=& q_{cl}^{sgt}, \\ - \left( \Delta q \right)_{sgt} &=& -\left( \Delta q_{cl} \right)_{sgt}, \\ - \left( \Delta T \right)_{sgt} &=& \frac{L_v}{c_p} \left( \Delta q_{cl} \right)_{sgt}, \\ - \left( \Delta C \right)_{sgt} &=& C_l^{sgt} \\ -\end{eqnarray} -where the left hand sides denote the increments to $C_l$, $q_{cl}$, $T$ and the -total cloud fraction, $C$, due to -the subgrid scheme. Some bounds-checking is then applied to ensure that: -(a) the resultant cloud fractions to not exceed one; (b) the scheme does not -condense out more liquid than there is available moisture. - -\paragraph{Option two: PC2 Erosion method} - -Option one gives a simple method for incrementing the model prognostics, but -it gives rise to a potential inconsistency with the PC2 cloud scheme. This arises because -the subgrid production scheme can elevate cloud fraction to unity in grid boxes -that are subsequently diagnosed by PC2 Initiation to meet the criteria for clear-sky initiation. -PC2 then counteracts the scheme by removing some of the liquid cloud. To try to mitigate -against this issue, cloud fraction increments can be applied using PC2 Erosion. In this case: -\begin{eqnarray} - \left( \Delta q_{cl} \right)_{sgt} &=& q_{cl}^{sgt} - q_{cl}, \\ - \left( \Delta q \right)_{sgt} &=& -\left( \Delta q_{cl} \right)_{sgt}, \\ - \left( \Delta T \right)_{sgt} &=& \frac{L_v}{c_p} \left( \Delta q_{cl} \right)_{sgt}, \\ -\end{eqnarray} -where $q_{cl}$ is the liquid cloud amount prior to calling to the -turbulent production scheme. The cloud fraction increments are calculated -by calling PC2 Erosion with $\left( \Delta q_{cl} \right)_{sgt}$ as input. -See Section \ref{sec:turb} for details on how the PC2 Erosion process works. -This method gives cloud fraction increments that are consistent with -PC2 cloud scheme. - - -\subsubsection{Other user options}\label{sec:sgt_options} - -The following variables and logical switches are optional inputs: -\begin{enumerate} - \item The logical \verb!l_dcfl_by_erosion! provides a switch to - apply cloud fraction increments using PC2 Erosion. Defaults to {\it FALSE}. - \item Setting the logical \verb!l_mixed_phase_t_limit! to {\it TRUE} - allows the user to use the variable \verb!mp_t_limit! to define a temperature limit, $T_{max}$, - above which the scheme is not applied. The default is $T_{max}=0^\circ\;{\rm C}$, so the - scheme is only applied to cold clouds. - \item The input variable \verb!mp_tau_d_lim! defines the - upper limit, $\tau_d^{max}$, on the value of $\tau_d$ above which the scheme is not applied. - The default value is $\tau_d^{max}=1200.0$, so the scheme is not applied in grid boxes - where the decorrelation time scale exceeds $1200$ seconds. - \item \verb!nbins_mp! is the number of bins used in the discretisation of the integrals in - Eqs \ref{eqn:cloud_fraction} and \ref{eqn:cloud_liquid} for $C_l^{sgt}$ and $q_{cl}^{sgt}$. - The default value is $100$ bins. - \item \verb!mp_dz_scal! is the scale parameter, $\beta_{mix}$, in the definition of the mixing length, $L=\beta_{mix}\Delta z$. - \item \verb!mp_czero! defines the constant parameter $C_0$ (defaults to $C_0=10$). -\end{enumerate} - - -\section{Application to the Unified Model} -\label{sec:app_um} - -This section describes the way in which the physical concepts described in the above -section are applied to the sections of the Unified Model, in order to build -up the complete prognostic scheme. Description of the actual subroutines -themselves follow in section \ref{sec:code}. -Note that the large-scale precipitation -and convection schemes have considerable documentation below, since these -schemes have been heavily modified for PC2. The other schemes use generic -forcing scenarios, hence their desciption here is much shorter. Remember, -whenever a signficiant $\overline{T}$ or $\overline{q}$ change occurs, -PC2 must be able to -represent the corresponding condensation and changes in cloud fractions. - -\subsection{Radiation} -\label{sec:rad} - -The shortwave and longwave radiation schemes both alter the temperature -of the atmosphere, hence we need to calculate the corresponding condensation -and cloud fraction changes. For both shortwave and longwave, we use -the homogeneous forcing routines (section \ref{sec:homog}) -for $\overline{q_{cl}}$ and $C_l$, (using eqn. -\ref{eq:deltaqc_exp2} to calculate the $Q_c$ forcing) and then the method in -section \ref{sec:ct} to calculate $C_t$ changes. There is no -$\overline{q_{cf}}$ change associated with this process since the -deposition / sublimation process is performed within the large-scale -precipitation scheme (as it also is in the absence of PC2). - -It is reasonable to question whether homogeneous forcing is a reasonable -model to use when we know that a large proportion of the heating -associated with radiative transfer in the atmosphere comes from the -cloudy air and is not evenly spread across the gridbox. Possible developments -are discussed in section \ref{sec:homog_improve}. - -\subsection{Large-scale precipitation} -\label{sec:precip} - -Precipitation processes have a large effect on cloud fractions. Here we -present the simple physical models that are applied to the transfer -terms included in the large-scale precipitation scheme. They are also -presented within the large-scale precipitation documentation (\citeumdp{026}). - -The basis of the physical model is that microphysical transfer processes can -be calculated separately in different partitions of the model cloud -(i.e. mixed phase cloud, liquid phase cloud, ice phase cloud or clear sky). -However, processes may change the size of these partitions. We consider -here separately each process that is modelled in the large-scale -precipitation scheme. The changes in $\overline{q_{cl}}$, $\overline{q_{cf}}$ -and $\overline{q}$ remain mathematically the same as in the non-PC2 version -of the code (\citeumdp{026}), we only need -to introduce calculations for the changes in cloud fractions. We will see -that many of these -changes can be well modelled by assuming no change to the cloud fractions, -and the others by using simple assumptions. - -Although the model may use two ice prognostic ice categories, only -a single ice cloud fraction is stored, the assumption being that the -two ice categories are completely overlapped with each other. Graupel -is not considered to contribute to the ice cloud fraction. - -\subsubsection{Fall of ice} -\label{sec:lsp_fall} - -The fall of ice is the process that contributes most to the growth of -ice cloud fraction in the model. The model results are therefore sensitive -to the formulation of this process. We will make the basic assumption -that a trail of falling ice does not reduce the horizontal spread of -ice cloud fraction at a particular level (hence $\overline{q_{cf}}$ -that leaves a gridbox does not reduce $C_f$ in that gridbox). The in-cloud -ice content simply reduces due to the fall out of ice - it is the -sublimation term (section \ref{sec:mp_depsub}) that erodes the fall streaks. -However, ice that falls into a clear layer from above may increase -the ice cloud fraction. We parametrize this by considering the -horizontal overlap of ice clouds between two model layers, and the -fall speed of ice between them. We will assume an overlap that -is nearly, but not quite, maximum, the difference being dependent -upon the windshear and the time taken for ice to fall between the -levels. - -\begin{equation} -O^{[k,k+1]} = \text{Max}( C_{i}^{[k+1]} - C_i^{[k]} , 0) -+ w \frac{\Delta z^{[k]}}{v_i^{[k]}} -\label{eq:overhang} -\end{equation} - -where $O^{[k,k+1]}$ is the amount of ice cloud `overhanging' the current -(i.e. $k$'th) layer -from the layer above, $w$ is a parameter that is closely related to the -windshear, $\Delta z^{[k]}$ is the model layer thickness and $v_i^{[k]}$ is -the fallspeed of ice in the layer. $v_i^{[k]}$ is calculated in the microphysics -scheme and, if two ice prognostics are used, is the mass-weighted average fall -speed of the two categories. -The factor $\frac{\Delta z}{v_i}$ is simply the time -taken for the ice to fall through one model layer. Multiplying this -by the windshear would give an estimate to the amount -of overlap between a cloud source and its fall streak in the layer -below (it is an \textit{estimate} since we assume that the cloud -source is continuous and unbroken). Although it is quite possible within -PC2 to do this, to date we have not programmed this link, and we -use a constant, but tunable, value of $1.5 \times 10^{-4} s^{-1}$ for $w$. - -The change in $C_i$ over the timestep is then given by the overlap proportion -multiplied by the how much (in the vertical dimension) of the layer below -can be filled by ice in the timestep: - -\begin{equation} -\Delta C_i = \text{Max}(O^{[k,k+1]} , 1) \text{Min} (v_i \frac{\Delta t}{\Delta z^{[k]}} , 1) -\label{eq:lsp_fall} -\end{equation} - -where $\Delta t$ is the timestep. We now choose to assume a minimum overlap -between the liquid and the ice phases (as in section \ref{sec:ct}). - -\begin{equation} -\Delta C_t = \text{Min} ( \Delta C_i , A_{clear} ) -\label{eq:lsp_fall_ct} -\end{equation} - -where $A_{clear}$ is the proportion of the gridbox that has neither -ice nor liquid cloud present. - -\textbf{An inconsistency has been found in the way that the fall-of-ice term is linked to the globally constant ``wind-shear value'' -when calculting the ice cloud fraction overhang. Consequently, -the option not to use the ``wind shear value'' when calculating the overhang is available in -the UMUI (from version 7.6 onwards).} - -\subsubsection{Homogeneous nucleation} -\label{sec:lsp_homo} -This will freeze all supercooled liquid water when a temperature threshold -is exceeded. Hence we turn all existing liquid and mixed phase cloud to -ice cloud. The cloud fraction changes are: - -\begin{eqnarray} -C_l \leftarrow 0 \nonumber \\ -C_i \leftarrow C_t \nonumber \\ -\Delta C_t = 0. -\label{eq:lsp_homo} -\end{eqnarray} - -\subsubsection{Heterogeneous nucleation} -This process will freeze a small amount of supercooled liquid water, -regardless of the previous presence of ice cloud. This will mean that -previously existing `liquid-only' cloud is converted to mixed phase -cloud. These give the following changes: - -\begin{eqnarray} -\Delta C_l = 0 \nonumber \\ -C_i \leftarrow C_t \nonumber \\ -\Delta C_t = 0. -\label{eq:lsp_het} -\end{eqnarray} - -\subsubsection{Deposition and sublimation} -\label{sec:mp_depsub} -This term exerts one of the most important influences on the ice cloud in -the whole model (this applies to the control as well as for PC2). Contained -in the formulation is a subgrid-scale assumption that causes equivalent -effects to that for a moisture PDF under the `$s$' framework -(section \ref{sec:s_dist}). However, since ${q_{cf}}$ changes -slowly in response to local changes in $q$ and $T$, we cannot base the -$q_{cf}$ response on the same instantaneous condensation framework. It would -be useful to investigate in the future whether the two descriptions of the -moisture variability could be brought together. -Because of its importance, -we describe the method below, although we note it is also described in -\citeumdp{026}. - -We can calculate the local rate of change of $q_{cf}$, given local $T$ and $q$ -etc. using the standard microphysical growth equations (see \citeumdp{026}). -However, it is critical to know the way in which -the moisture is correlated with the ice in the gridbox. We will assume -there exists a distribution of vapour in the gridbox. We know that -the regions where liquid cloud exists must be saturated with respect -to liquid water, hence we need only consider the part of the gridbox -that does not have liquid water present. The average value, $q_a$, -of $q$ within the liquid-free part of the gridbox is thus - -\begin{equation} -q_a = \frac{ \overline{q} - C_l q_{sat \, liq}(\overline{T}) } {1 - C_l} -\label{eq:qa} -\end{equation} - -where we have assumed that the fluctuation of $q_{sat~liq}$ across -the gridbox due to temperature fluctuations is not significant compared -to the fluctuation of $q$ described below. We then parametrize a width, $b_i$, -to the $q$ (not $s$) fluctuations \textit{across the non-liquid cloud part -of the gridbox}, based upon $RH_{crit}$. This is like that for the `$s$' -distribution width, $b_s$ but modified: - -\begin{equation} -b_i = (1 - RH_{crit} ) q_{sat \, liq} ( 1 - \frac{1}{2} -~ \frac{\overline{q_{cf}}} {i q_{sat \, liq}(\overline{T})} ) . -\label{eq:b_i} -\end{equation} - -where the factor $( 1 - \frac{1}{2} -\frac{\overline{q_{cf}}} {i ~ q_{sat \, liq}(\overline{T})})$ should be limited -to a minimum value of zero, but, for numerical reasons, is limited to -a minimum value of 0.001. We note that $b_i$ has a similar form to $b_s$, -except the multiplier $a_L$ and the factor in brackets. If we remember -from (\ref{eq:s}) that the definition of `$s$' includes a factor $a_L$ -we see that the absence of the $a_L$ factor in (\ref{eq:b_i}) is -consistent. The factor in brackets is a \textit{parametrization} of the -effect that, when ice -is present, deposition in the moistier parts and sublimation in the -drier parts of the gridbox must reduce the width of the distribution -of $q$ across the gridbox. It is a simple linear function of -$\frac{\overline{q_{cf}}}{q_{sat~liq}(\overline{T})}$, and is tunable -using the factor $i$, which takes the value of 0.04. - -We note that this formulation isn't totally consistent with the liquid -cloud formulation, which considers an underlying PDF across the whole -gridbox and does not have, in general, its width prescribed. -Remember that we do not calculate on-line the whole of the liquid -- vapour PDF, we only parametrize the single point $G(-Qc)$, -using equation \ref{eqn22}). - -The width is then limited further to be no greater than $\overline{q}$, -to make sure that there are no negative values of $q$ predicted within the -gridbox (possible at low -temperatures where $q_{sat~liq}(\overline{T})$ diverges from -$q_{sat~ice}(\overline{T})$). - -We then calculate the average value of $q$ in the ice-only and clear-sky -partitions of the gridbox. To do this, we make the further assumption -that the ice is correlated with the moistest part of the distribution -(an instantaneous condensation formulation would make the -same assumption). Some algebra retrieves the expressions: - -\begin{eqnarray} -q_{clear} = q_a - b_i A_{ice} ; \\ -q_{ice} = \frac {\overline{q} - C_l q_{sat~liq} - A_{clear} q_{clear} } -{A_{ice}}, -\label{eq:q_clear_and_q_ice} -\end{eqnarray} - -where $A_{ice}$ is the proportion of the gridbox with ice cloud but not -liquid cloud and $A_{clear}$ is the proportion of the gridbox without cloud. -The numerical application will set $q_{clear}$ to $q_a$ if $A_{ice}$ -is zero. We now have a representation of the $q$ values in each of the -gridbox cloud partitions, and can solve the microphysical transfer equation -in each partition. - -The cloud fraction changes now need to be parametrized. We use the -model that deposition will \textit{not} adjust the ice cloud -\textit{fraction} (increases will be done within the fall-of-ice microphysics -section). However, deposition can -decrease the liquid cloud fraction (locally, $q$ can be reduced by -deposition to below $q_{sat~liq}$, hence this is not inconsistent with -the assumptions for the riming term below. This is the principal sink -of supercooled liquid cloud fraction in the model. Sublimation will -be allowed to decrease the ice cloud fraction (since sublimation cannot -act in liquid cloud, there is no impact on the liquid cloud). To solve -for these models, we will need to further split the ice-only partition -to give the proportion of that partition that is above and below ice -saturation. This gives, in general, an area of the gridbox $A_{ice1}$ -that contains ice and is above saturation where - -\begin{equation} - A_{ice1} = \frac{1}{2} A_{ice} + \frac{1}{2} - \frac{ (q_{ice}-q_{sat~ice}(\overline{T})) } {b_i}, -\label{eq:q_ice_above_sat} -\end{equation} - -having assumed that $A_{ice1}$ is between 0 and $A_{ice}$. -If not, it is trivial to partition the gridbox, since the moisture in the ice-only -partition is either completely above or completely below $q_{sat~ice}(\overline{T})$. -The corresponding -area that contains ice and is below saturation is given by -$A_{ice2} = A_{ice} - A_{ice1}$. -We can now parametrize the change in cloud fractions. For deposition, -we shall assume a uniform distribution of local values of $q_{cl}$ about -the local mean. If we assume a uniform removal of local $q_{cl}$ then, with -a little algebra, we can obtain an expression for the change in $C_l$: - -\begin{equation} -\Delta C_l = C_l ( 1 - \frac {\Delta \overline{q_{cl}}} {\overline{q_{cl}}} ) -^{\frac{1}{2}} - C_l -\label{eq:deltacfl_dep} -\end{equation}. - -Since this occurs only in the mixed phase part of the gridbox, we can say -that $\Delta C_t = 0$. We will also note that the change in $\overline{q_{cl}}$ -due to deposition is limited by the amount of $\overline{q_{cl}}$ that is in -the mixed phase partition in the gridbox, hence (\ref{eq:deltacfl_dep}), -although it formally allows removal of $C_l$ from an ice-free partition, will -be unlikely to do so. - -The sublimation forms the main method by which ice cloud is destroyed in PC2, -hence PC2 results are relatively sensitive to its formulation. Here we -make a similar assumption to that used for liquid in the deposition term, -except that we limit the changes only to the region of the gridbox where -ice is subliming. - -\begin{equation} -\Delta C_i = A_{ice2} ( 1 + \frac{\Delta \overline{q_{cf}} } -{ \overline{q_{cf}} ( \frac{A_{ice2}}{C_i} ) } - )^{\frac{1}{2}} - A_{ice2} -\label{eq:deltacfi_sub} -\end{equation}. - -The term $\overline{q_{cf}} ( \frac{A_{ice2}}{C_i} )$ is the amount of -$\overline{q_{cf}}$ that is present in the subliming ice region, hence its -ratio with $\Delta \overline{q_{cf}}$ is the fractional change in that region. The -change in the total cloud fraction must also be equal to the change above, -since sublimation cannot occur in the presence of liquid cloud: - -\begin{equation} -\Delta C_t = \Delta C_i . -\label{eq:deltacft_sub} -\end{equation} - -\subsubsection{Riming} -This process acts only where mixed phase cloud occurs - although, in theory, -it could remove any supercooled liquid totally, the air would remain -saturated with respect to liquid water. Hence any subsequent cooling would -regenerate the same amount of liquid cloud. Hence we choose to model this -process as having \textit{no effect} on the cloud fractions. - -\subsubsection{Capture} -This is the freezing of raindrops onto ice crystals by collision. This does -not alter the ice cloud \textit{fraction} in the gridbox (although it does -alter $\overline{q_{cf}}$, and it has no interaction with the liquid cloud. -Again, we therefore choose to model this process as having \textit{no effect} -on the cloud fractions. - -\subsubsection{Evaporation of melting ice} -Here we simply assume that ice cloud fraction is removed in proportion to the -ice content that is removed. - -\begin{equation} -\Delta C_i = C_i \frac{ \Delta \overline{q_{cf}}}{\overline{q_{cf}}} . -\label{eq:lsp_evapmeltsnow} -\end{equation} - -Because the evaporation cannot occur in the liquid part of the gridbox, -there is no change to $C_t$ (or to $C_l$). - -\subsubsection{Melting} -Again, the change in $C_i$ is calculated using the method -in (\ref{eq:lsp_evapmeltsnow}). - -\begin{equation} -\Delta C_i = C_i \frac{ \Delta \overline{q_{cf}}}{\overline{q_{cf}}} . -\label{eq:lsp_melt} -\end{equation} - -The change in $C_t$ is calculated assuming that there is no correlation -in the gridbox between where the ice melts and the liquid cloud. Hence -we must multiply (\ref{eq:lsp_melt}) by the proportion of ice cloud -fraction that exists without liquid cloud (i.e. $\frac{A_{ice}}{C_i}$). - -\begin{equation} -\Delta C_t = C_i \frac{ \Delta \overline{q_{cf}}}{\overline{q_{cf}}} -\frac{A_{ice}}{C_i} . -\label{eq:lsp_melt2} -\end{equation} - -\subsubsection{Evaporation of rain} -Evaporation of rain will not, \textit{on its own}, -generate liquid cloud, since a -large-scale lifting process will be required in order to condense water -from the moistened air. We cannot, therefore, allow any change in cloud -fractions to occur as a result, subsequent changes are calculated elsewhere -in the model (e.g. by the lifting process, section \ref{sec:pres}). - -\subsubsection{Accretion} -Accretion is the sweep-out of liquid water droplets by rain. We argue -in a similar way to the riming term, that this will not remove any liquid -cloud fraction, since a small amount of lifting will regenerate the same -amount of liquid cloud. Hence we choose to model this process as having -\textit{no effect} on the cloud fractions. The arguments underlying the -formulation of the evaporation of rain and the accretion cloud fraction -changes may appear to be inconsistent in their limiting cases and the -subsequent response to lifting. However, when the limiting case is -not reached the formulations are both correct. For the moment, it is -not considered necessary to increase the complexity of the current, -simple representations. - -\subsubsection{Autoconversion} -As for accretion, the generation of rain directly from collision -and coalescence of liquid water droplets will not alter the cloud fractions. - -\subsubsection{Other microphysics terms} -There are already (i.e. also in the control) -two numerical tidy-up terms at the end of the microphysics -section that remove small rain amounts and provide an additional -melting term for the snow. These do not change the cloud fractions. - -If there are small amounts of ice present at the end of the -microphysics then these are removed at the end of the microphysics -timestep (also in the control). PC2 responds by resetting the -cloud fractions appropriately, so $C_t$ is reset to $C_l$ etc. - -\subsubsection{Numerical implementation} - -Note that after each process has been applied, we do \textit{not} -recalculate the sizes of the ice-only, liquid-only and mixed phase -partitions, but use the values at the start of the microphysics (this -includes the values of $C_i$ used in the calculation of `in-cloud' -water contents above. However, we do update the cloud fractions -themselves sequentially. We also recalculate after each process -the overlaps between the rain fraction (see \citeumdp{026}) and the cloud fractions. - -There is also a final set of checks that $C_l$ and $C_i$ lie -between 0 and 1 and that $C_t$ is bounded between $\text{Max}(C_l, C_i)$ -(maximum overlap of liquid and ice) and $\text{Min}(C_l+C_i,1)$ -(minimum overlap of liquid and ice). - -We should note in particular, that these parametrizations allow a -considerable reduction in $\overline{q_{cl}}$ without a corresponding -large reduction in $C_l$. This is an underlying feature of the PC2 scheme -(discussed in \cite{wg03}), and necessarily implies the -skewing of the underlying moisture PDF. Subsequent parts of -the model (e.g. the width narrowing, section \ref{sec:width}) will, of -course, act on the modified fields to adjust the cloud fractions further, -but remember that these are separate processes and modelled elsewhere in the -timestep. - -\subsection{PC2 erosion} -\label{sec:turb} - -\subsubsection{Original width-narrowing method} -(selected by setting {\bf i\_pc2\_erosion\_method = 1} in the UM namelist). - -In parallel with the homogeneous forcing part of the PC2 response to -convection, we introduce -a new block of code that allows a background change of the PDF width. -At earlier versions of PC2 (PC2:65 and earlier) this block was included as -a separate section of code that was called in as part of the atmphya parallel -timestepping. This was later moved to better numerically balance -increments from -the convection scheme with the cloud fraction erosion term. From VN8.1 onwards, a further option was introduced to implement the erosion prior to the microphysics parametrization. This was primarily to allow PC2 to be run at convection-resolving scales, at which the convection scheme is not called and therefore the erosion is not called. -We have empirically selected a rate of change of width that depends upon the -relative total humidity of the grid box, such that there is more -erosion in drier gridboxes. This promotes more rapid erosion of -shallow convective cloud, which is the main effect that we seek to -include, although the physical implication that dry air is -more turbulent than moist air does not match the way the real atmosphere -works, especially in the -stratosphere. No doubt the link can be improved upon with more -research. The formulation used is: - -\begin{equation} -\frac{1}{b_s} \frac{\partial b_s}{\partial t} = \Upsilon exp ( - \frac{2.01 Q_c}{0.2 a_L q_{sat liq}(T_L)} ) -\label{eq:dbsbydtbs_turb} -\end{equation} - -where the 0.2 factor is chosen to be closely equivalent to $1 - RH_{crit}$ -and the value of 2.01 has been selected through tuning. The code merges the -two numerical values into a single quantity (dbsdtbs-exp), equal to 10.05. -We note that in PC2:64 -(the library 6.4 code, a value of 0.62 is used rather than 2.01). As a guide -to the $RH_T$ dependence, note -that when the value of $RH_T$ is 0.85, the value of -$\frac{1}{b_s} \frac{\partial b_s}{\partial t}$ is close to $1 \times 10^{-4} s^{-1}$. - -Note: the source-code for this erosion method ({\bf pc2\_hom\_conv}, -{\bf pc2\_homog\_plus\_turb}, {\bf pc2\_delta\_hom\_turb}) -includes an additional term ``dbsdtbs1'' -which scales with the rate of homogeneous forcing -$\frac{\partial Q_c}{\partial t}$. However this term is always -set to zero on input to these routines so is never used. - -The width-narrowing formulation of section \ref{sec:width} is used to -calculate increments in $\overline{q_{cl}}$ and $C_l$. Using the liquid - -ice cloud overlap ideas of section \ref{sec:ct} then gives the associated -$C_t$ change. This background narrowing term, $\Upsilon$, is originally based upon work -by \cite{sg03}, although it is a parameter that has been -extensively tuned during PC2 development, a typical value would be $\Upsilon=-2.25 \times 10^{-5} s^{-1}$. - -\subsubsection{Numerical application of the original width-narrowing method} - -Because of the strong link the mathematical expressions for width narrowing -(section \ref{sec:width}) have with the expressions for the -homogeneous forcing (section \ref{sec:homog}), we choose to represent -the timestepping of this process in exactly the same way as for -the homogeneous forcing (in fact, in the Unified Model code we use -the same subroutine, see section \ref{sec:code}). As -before, we use a simple forward timestepping of $C_l$, with -$Q_c$ given by (\ref{eq:qc_eq_qt-qs}) and $a_L$ defined as discussed -in section \ref{sec:homog_num_app} and discretize eq \ref{eq:dcdt_width} as: - -\begin{equation} -\Delta C_l^{[n+1]} = - G(-Q_c) Q_c \frac{1}{b_s} -\frac{\partial b_s}{\partial t} \Delta t. -\label{eq:dcl_turb_final} -\end{equation} - -Similarly to (\ref{eq:c_l^n+1}), we then limit the cloud fraction to 0 and -1 and then apply a mid-point value of $C_l$ to calculate the change in -$\overline{q_{cl}}$ (discretizing eq \ref{eq:dqcldt_width}): - -\begin{equation} -\Delta q_{cl}^{[n+1]} = (q_{cl}^{[n]} - Q_c \frac{1}{2}(C_l^{[n]}+C_l^{[n+1]})) -\frac{1}{b_s} \frac{\partial b_s}{\partial t} \Delta t. -\label{eq:dqcl_turb_final} -\end{equation} - -In this case the value of $\Delta q_{cl}$ \textit{is} limited to ensure that -no more $\overline{q_{cl}}$ is removed than the model has available. This was -chosen to ensure that the erosion process itself contains this physical limit, -not a numerical tidying-up process. - -The option ``l\_fixbug\_pc2\_qcl\_incr'' ensures that qcl is set to zero -if the CFL has reached zero. - - -\subsubsection{Cloud-surface-area hybrid erosion method} -(selected by setting {\bf i\_pc2\_erosion\_method = 3} in the UM namelist). - -\cite{morcrette_petch} showed that changes to the erosion parameter ($\Upsilon$ in Eqn. \ref{eq:dbsbydtbs_turb}) did not have as significant an impact -on the global work done by the erosion process as might be expected. This was due to a feedback process -whereby, reducing the erosion parameter leads to more cloud water, more autoconversion of cloud water to rain, more -fall-out of rain and more drying of the layer, hence increasing the $exp ( - \frac{2.01 Q_c}{0.2 a_L q_{sat liq}(T_L)} )$ part of -Eqn. \ref{eq:dbsbydtbs_turb}. Although the feedback is physically plausible it crucially depends on the formulation of -Eqn. \ref{eq:dbsbydtbs_turb} and the dependence of the rate of narrowing of the PDF on the moisture, a dependence that was developed -from a pragmatic rather than theoretical stand-point. -The option for an alternative way of calculating the erosion was introduced at vn8.0 - -We use equation 30 from \cite{t93} to specify the sink of $q_{cl}$ due to erosion, i.e. -\begin{equation} -\frac{\partial q_{cl}}{\partial t}=-A K(q_{sat}-q_v) -\label{eq:dqcldt_hybrid} -\end{equation} -(note we have changed the sign as we have replaced the evaporation rate $E_2$ -in \cite{t93} with $-\frac{\partial q_{cl}}{\partial t}$ on the left-hand-side). -In the \cite{t93} scheme, $A$ is set to the cloud fraction (i.e. $A=C_l$). -Here we recall that "cloud erosion" is meant to represent the evaporation of cloud water due to the -mixing of clear and cloudy air and that this can only happen on the edges of cloud, where saturated air is exposed to sub-saturated air. -If the cloud fraction is small (e.g. 5$\%$), then there are not many clouds, so there is only a small surface area from which evaporation can occur. -Similarly if the cloud cover is very high (e.g. 95$\%$) then there is again not much surface area exposed to clear sky. -A maximum in exposed surface area is expected when the cloud cover is 50$\%$. - -By imagining that the grid-box is broken up into cubes whose horizontal dimension equal the layer depth it is possible -to work out what the maximum lateral surface area would be, as a function of cloud fraction, for different arrangements of cloudy cubes. -The maximum lateral surface area, is when the clear and cloudy cubes are arranged in a chess-board pattern, and the minimum is when then are all grouped -together into a circular clump. Numerical tests using randomly distributed cloudy cube shows that the variation -in lateral surface area, $S$, as a function of cloud fraction can be expressed as: -\begin{equation} -S= - 2 C_l ^{2} + 2 C_l -\label{eq:S_Cl} \end{equation} -The maximum normalised surface area of 0.5 occurs at a cloud fraction of 0.5. -Using a cloud mask derived from satellite imagery shows that real cloud fields do follow this kind of dependence, -but that the peak surface area is nearer to 0.35, meaning that real clouds are not as randomly distributed as random -ones and that there is some kind of clumping together, which is what we might have expected. -When it comes to implementing such a scheme in the model, there will need to be a tunable parameter to govern the -rate of evaporation. This will not affect the shape of the lateral surface area function. -As a result the details of whether the peak lateral surface area is 0.5 or 0.35 are simply absorbed into the tunable parameter $K$, -which is supplied from the UMUI (using the same text box as was used for supplying $\Upsilon$). - -The exposed surface area associated with the tops and bottom of the clouds is calculated assuming maximum overlap in adjacent layers and is added to the lateral -surface area to give a total surface area, -\begin{equation} -A=max(C_l(k)-C_l(k+1),0.0)+max(C_l(k)-C_l(k-1),0.0)+S -\label{eq:A_top_and_bottom} \end{equation} -it is this value of $A$ which we use in Eqn. \ref{eq:dqcldt_hybrid}. - -Note that the contributions from the top and bottom interfaces of the current -model-level $max(C_l(k)-C_l(k+1),0.0)$ and $max(C_l(k)-C_l(k-1),0.0)$ -may optionally either be included or excluded, depending on the -UM namelist switch \textbf{i\_pc2\_erosion\_method}. -Further note: at present these contributions are hardwired to be excluded, -as they prevented the erosion calculations from being parallelised -in the vertical direction using OpenMP, and no operational model configurations -were using them. - -Having calculated a reduction in $q_{cl}$ using the Tiedtke-surface-area method, we -then work out the relative rate of narrowing that would have given the same sink of $q_{cl}$. This value of $\frac{1}{b_s} \frac{\partial b_s}{\partial t}$ -is then used to calculate the change in $C_l$ using the same moisture PDF assumptions as were used in the original PC2 erosion formulation. -To achieve this, we combine equations \ref{eq:dcdt_width} and -\ref{eq:dqcldt_width} from section \ref{sec:width} to eliminate -$\frac{1}{b_s} \frac{\partial b_s}{\partial t}$ and write -$\frac{\partial C_l}{\partial t}$ as a function of -$\frac{\partial \overline{q_{cl}}}{\partial t}$: - -\begin{equation} -\frac{\partial C_l}{\partial t} - = - \frac{ G(-Q_c) Q_c \frac{\partial \overline{q_{cl}}}{\partial t} } - { (- C_l Q_c+\overline{q_{cl}}) } -\label{eq:dcdt_hybrid} -\end{equation} - -Where the change in liquid water content -$\frac{\partial \overline{q_{cl}}}{\partial t}$ -is given by eq \ref{eq:dqcldt_hybrid} above. - -This combination of a Tiedkte sink term for $q_{cl}$, a PC2 term for $C_l$ and the introduction of some surface area dependence leads to this formulation -being referred to as a ``hybrid'' cloud-surface-area erosion method. - -\subsubsection{Numerical application of the hybrid erosion method} -\label{sec:erosion_numerics} - -Next, we consider how to numerically discretise equations -\ref{eq:dqcldt_hybrid} and \ref{eq:dcdt_hybrid} -to compute cloud increments due to erosion. -The simplest approach is an explicit forwards-in-time discretisation: - -\begin{equation} -\frac{ \Delta {q_{cl}}_{ero}}{\Delta t} = A(C_l^n) K(q_{sat}-q_v) -\label{eq:hybrid_erosion_expl} \end{equation} - -i.e. the increment is calculated by evaluating the term $A$ from equations -\ref{eq:S_Cl} and \ref{eq:A_top_and_bottom} using the value of cloud-fraction -$C_l$ \textit{before} erosion has been applied. - -However, when the environment is significantly subsaturated -(so that the term $(q_{sat}-q_v)$ is large and negative), -and long timesteps $\Delta t$ are used -(e.g. order 1000 s used in global climate simulations), -this discretization can suffer severe numerical overshoot. -i.e. the increment based on $C_l^n$ is large enough to reduce $q_{cl}$ -(and hence also $C_l$) to less than zero within a single timestep. -If the continuous equation were solved analytically this wouldn't happen; -as $C_l$ declines due to the erosion, so will $A(C_l)$ -and hence the erosion rate, so that $q_{cl}$ and $C_l$ smoothly decline -towards zero. - -Three options are available in the code to address this problem, -selected by the UM namelist switch \textbf{i\_pc2\_erosion\_numerics}, -detailed below. -Single-Column Model tests indicate that the 2nd and 3rd options yield much less -timestep sensitivity for detrained cloud in shallow cumulus regimes. - -\begin{enumerate} - -\item \textbf{Retain the explicit discretization, but limit the resulting -erosion increments to ensure $q_{cl}$ and $C_l$ don't go negative. -(i\_pc2\_erosion\_numerics=1)} -Also, to ensure that some cloud remains at end-of-timestep where -shallow cumulus is detraining into dry environments, the erosion -calculation is fed copies of the fields with the current timestep's -convection increments subtracted off. This means any cloud detrained -by convection during the current timestep cannot be eroded until the -following timestep, and so is still present at end-of-timestep. -As discussed in section \ref{sec:timestepping}, -this leads to a problematic timestep sensitivity, -since the amount of cloud not subject to erosion is the convection increment, -which scales with the timestep length. - -Having computed the erosion $q_{cl}$ increment using -\ref{eq:hybrid_erosion_expl}, the consistent $C_l$ increment is computed -by discretising \ref{eq:dcdt_hybrid} as: - -\begin{equation} -\frac{\Delta {C_l}_{ero}}{\Delta t} - = - \frac{ G(-Q_c)^n Q_c^n \frac{\Delta \overline{{q_{cl}}_{ero}}}{\Delta t} } - { (- C_l^n Q_c^n + ( \overline{q_{cl}^n} - + \frac{1}{2} \Delta \overline{{q_{cl}}_{ero}} ) ) } -\label{eq:dcdt_hybrid_discr} -\end{equation} - -i.e. all terms are treated explicitly (using the values before erosion), -except for $\overline{q_{cl}}$ which takes the mid-point interpolated -half-way between its values before and after erosion, -to give some improvement in accuracy. - -\item \textbf{Use an approximate implicit discretisation, -which intrinsically yields a positive solution for $q_{cl}$ and $C_l$. -(i\_pc2\_erosion\_numerics=2)} -The copies of the fields passed to the erosion calculation are -fully updated with the convection increments. -We then write equation \ref{eq:dqcldt_hybrid} in the form: - -\[ -\frac{\partial q_{cl}}{\partial t} = q_{cl} f(q_{cl},C_l,(q_{sat}-q_v)) -\] - -(where the term $f(q_{cl},C_l,(q_{sat}-q_v)) = \frac{A K(q_{sat}-q_v)}{q_{cl}}$ -will be treated explicitly, under the assumption that this ratio will -evolve more slowly while erosion rapidly reduces both the numerator -and the denominator). - -We then take a backwards-in-time implicit discretisation in terms of the -leading factor $q_{cl}$: - -\[ -\frac{ q_{cl}^{n+1} - q_{cl}^{n}}{\Delta t} = q_{cl}^{n+1} f^n -\] - -Now, the problem is somewhat complicated by the fact that in the code, -erosion is calculated in parallel with the homogeneous forcing by convection, -and we need to account for the homogeneous forcing increment -$\Delta q_{cl}^{hom}$ in our implicit solution. We therefore write the above as: - -\[ -\Delta q_{cl}^{ero} = \Delta t - ( q_{cl}^n + \Delta q_{cl}^{hom} + \Delta q_{cl}^{ero} ) f^n -\] - -Rearranging: - -\[ -\Delta q_{cl}^{ero} = \Delta t f^n q_{cl}^n \frac{ q_{cl}^n + \Delta q_{cl}^{hom} } - { q_{cl}^n - \Delta t f^n q_{cl}^n } -\] - -Note that the term $\Delta t f^n q_{cl}^n$ is the erosion increment -we would obtain from the purely explicit discretisation, -$\Delta q_{cl}^{ero\,expl}$. The implicit discretisation is implemented by -first calculating $\Delta q_{cl}^{ero\,expl}$ using equation -\ref{eq:hybrid_erosion_expl} -(as we do for \textbf{i\_pc2\_erosion\_numerics=1}) -but then rescaling it using the above expression, which becomes: - -\begin{equation} -\Delta q_{cl}^{ero} = \Delta q_{cl}^{ero\,expl} \frac{ q_{cl}^n + \Delta q_{cl}^{hom} } - { q_{cl}^n - \Delta q_{cl}^{ero\,expl} } -\label{eq:hybrid_erosion_impl_qcl} \end{equation} - -Provided erosion is acting to reduce cloud-water ($\Delta q_{cl}^{ero\,expl} < 0$), -and homogeneous forcing by convection has not already completely removed -the cloud ($q_{cl}^n + \Delta q_{cl}^{hom} > 0$), \ref{eq:hybrid_erosion_impl_qcl} -is guaranteed to yield a stable, positive solution for $q_{cl}$. - -We also apply exactly the same argument to the equation for the cloud-fraction -increment $C_l$, and obtain: - -\begin{equation} -\Delta C_l^{ero} = \Delta C_l^{ero\,expl} \frac{ C_l^n + \Delta C_l^{hom} } - { C_l^n - \Delta C_l^{ero\,expl} } -\label{eq:hybrid_erosion_impl_Cl} \end{equation} - -Where $\Delta C_l^{ero\,expl}$ is computed using eq \ref{eq:dcdt_hybrid_discr}, -except that the term $\frac{1}{2} \Delta \overline{{q_{cl}}_{ero}}$ -is omitted (interpolating to the mid-point value of $\overline{q_{cl}}$ -in the denominator would be ``double-counting'' if we are already making -an implicit correction to the full increment). - -In the case where the homogeneous forcing increments have already removed -all of the cloud water content or fraction, erosion is not performed, -and $q_{cl}$ and $C_l$ are both set to zero. In the case where erosion is -actually acting to increase cloud-fraction, the code defaults to retaining -the explicit discretisation solution $\Delta q_{cl}^{ero\,expl}$ and -$\Delta C_l^{ero\,expl}$. Otherwise, equations \ref{eq:hybrid_erosion_impl_qcl} -and \ref{eq:hybrid_erosion_impl_Cl} are applied to yield the implicit solution. - -\item \textbf{Use an analytic solution to the integration of the -time-derivatives in (\ref{eq:dqcldt_hybrid}) and (\ref{eq:dcdt_hybrid}) -for greater accuracy. -(i\_pc2\_erosion\_numerics=3)} - -Two problems have been identified with the above implicit numerical method: -\begin{itemize} - -\item The implicit correction is applied completely independently to the -increments for $q_{cl}$ and $C_l$. So as with the explicit method, -differing numerical error in the increments for the two variables -can lead to them becoming inconsistent with eachother. -It was found by experimentation that even with the implicit correction, -it is possible for erosion to reduce $C_l$ by a bigger fraction than $q_{cl}$, -so that the in-cloud water content $\frac{q_{cl}}{C_l}$ is {\em increased}. -Narrowing the PDF should only {\em decrease} the in-cloud water-content; -occasional large increases due to numerical error can lead to spurious -precipitation being produced by the microphysics scheme. - -\item The implicit correction makes it impossible for erosion to reduce -$q_{cl}$, $C_l$ to zero. As we will show below, the analytic solution -to the equations posed does in fact go to zero after a finite time -under grid-mean subsaturation -(although the erosion rate declines with $C_l$ as it approaches zero, -$C_l$ approaches zero more slowly than $q_{cl}$, so that both variables decrease -following power-law curves not exponentials). -When erosion (wrongly) can never entirely remove cloud, this allows -tiny values of $q_{cl}$ and $C_l$ to spuriously spread across the domain -via numerical diffusion from the model's advection scheme. - -\end{itemize} - -Under this option, we attempt to compute an analytic solution to the -simultaneous differential equations \ref{eq:dqcldt_hybrid} and -\ref{eq:dcdt_hybrid} so that $q_{cl}$ and $C_l$ both decrease smoothly and -consistently. -The equations lead to somewhat different behaviour depending on whether -the grid-mean state is subsaturated ($Q_c < 0$), supersaturated ($Q_c > 0$), -or close to saturation ($Q_c$ near-zero). We can employ different -approximations to integrate the equations in each case. -In the code, we first test the value of $Q_c$ and compute the erosion -increments as follows: - -\begin{enumerate} - -\item {\bf Grid-mean subsaturation ($Q_c < 0$):} - -The relation between the erosion tendencies in liquid-cloud-fraction and -liquid water content (\ref{eq:dcdt_hybrid}) can be expressed in terms of -{\em fractional} rates of change -(dividing the top and bottom by $-C_l Q_c$, and dividing both sides by $C_l$): - -\begin{equation} -\frac{1}{C_l} \frac{\partial C_l}{\partial t} - = \frac{ G(-Q_c) \frac{q_{cl}}{C_l^2} }{ 1 - \frac{q_{cl}}{C_l Q_c} } \; - \frac{1}{q_{cl}} \frac{\partial q_{cl}}{\partial t} -\label{eq:dcdt_hybrid_1} -\end{equation} - -Under homogeneous forcing (section \ref{sec:homog}), we defined the PDF height -at the saturation boundary when near the cloudy end of the PDF as -$G(-Q_c) = \frac{n+1}{n+2} \frac{C_l^2}{q_{cl}}$ (eq \ref{eqn20}). -In fact, $G(-Q_c)$ is set to some blend between this and the value near -the clear end of the PDF (eq \ref{eqn21}). But we will assume that -when eroding cloud under grid-mean subsaturated conditions ($Q_c < 0$), -$G(-Q_c)$ follows this scaling with $\frac{C_l^2}{q_{cl}}$ even if its -value differs somewhat from eq \ref{eqn20}. -Therefore the quantity $c_1 = G(-Q_c) \frac{q_{cl}}{C_l^2}$ remains constant -during the erosion process, and eq \ref{eq:dcdt_hybrid_1} becomes: - -\begin{equation} -\frac{1}{C_l} \frac{\partial C_l}{\partial t} - = \frac{ c_1 }{ 1 - \frac{q_{cl}}{C_l Q_c} } \; - \frac{1}{q_{cl}} \frac{\partial q_{cl}}{\partial t} -\label{eq:dcdt_hybrid_2} -\end{equation} - -The term $1 - \frac{q_{cl}}{C_l Q_c}$ (which is $> 1$ since we are considering -grid-mean subsaturation $Q_c < 0$) usually remains close to 1 in practice, -so we can assume its fractional variation over the timestep is small -compared to the other terms, and treat it explicitly. -We can therefore straightforwardly integrate eq \ref{eq:dcdt_hybrid_2} -to obtain the scaling of $C_l$ with $q_{cl}$ as both are reduced by erosion: - -\begin{equation} -\frac{C_l}{{C_l}_0} = \left( \frac{q_{cl}}{{q_{cl}}_0} \right)^{b_1} -\label{eq:cl_qcl_scaling} -\end{equation} - -where ${C_l}_0$, ${q_{cl}}_0$ are the values before erosion is applied, -and the exponent is $b_1 = \frac{ c_1 }{ 1 - \frac{q_{cl}}{C_l Q_c} }$. -When $G(-Q_c)$ takes its value from the cloudy end of the PDF, we have -$c_1 = \frac{n+1}{n+2}$. Since the PDF power $n > 0$ and $Q_c < 0$ -under the considered grid-mean subsaturation, we always have $b_1 < 1$. -This ensures that erosion reduces $C_l$ at a slower fractional rate than -$q_{cl}$, so that in-cloud water content $\frac{q_{cl}}{C_l}$ always decreases. - -Next we derive an integral solution for the decline of $q_{cl}$ with time. -Ignoring the cloud surface-area contributions from the levels above and below -(they are disabled in the code anyway), the erosion liquid water content -tendency is obtained by combining \ref{eq:dqcldt_hybrid} and \ref{eq:S_Cl}: - -\begin{equation} -\frac{\partial q_{cl}}{\partial t} = -K \, 2 C_l (1 - C_l) \, (q_{sat}(T)-q_v) -\label{eq:dqcldt_hybrid_1} -\end{equation} - -From eq \ref{SD2}, $q_{sat}(T)-q_v = \frac{SD}{a_L}$, where $SD$ is the -saturation defecit, and $a_L$ is the dimensionless factor defined in -eq \ref{eq:a_L}. Following the derivation in section -\ref{sec:smooth_initiation} (eq \ref{eq:qc_plus_sd}), -we can write this in terms of the liquid-water content: $SD = q_{cl} - Q_c$ -(where $Q_c$ was defined in eq \ref{eq:qc_eq_qt-qs}, and corresponds to the -grid-mean supersaturation converted to an equivalent liquid water content). -Substituting this into (\ref{eq:dqcldt_hybrid_1}) above, we obtain: - -\begin{equation} -\frac{\partial q_{cl}}{\partial t} = -\frac{K}{a_L} \, 2 C_l (1 - C_l) \, - (q_{cl}-Q_c) -\label{eq:dqcldt_hybrid_2} -\end{equation} - -Substituting eq \ref{eq:cl_qcl_scaling} for the leading factor of $C_l$ -on the right-hand-side and rearranging: - -\[ -\left( \frac{q_{cl}}{{q_{cl}}_0} \right)^{-b_1} \frac{\partial q_{cl}}{\partial t} - = -\frac{K}{a_L} \, 2 {C_l}_0 (1 - C_l) \, (q_{cl}-Q_c) -\] - -In significantly subsaturated conditions the r.h.s. has only weak dependence -on $C_l$ and $q_{cl}$ ($C_l << 1$, $q_{cl} << -Q_c$), so we can treat the -whole r.h.s. explicitly (i.e. neglect its variation during each timestep), -so that the above integrates to: - -\[ -\left[ \frac{{q_{cl}}_0}{1-b_1} \left( \frac{q_{cl}}{{q_{cl}}_0} \right)^{1-b_1} -\right]_{{q_{cl}}_0}^{{q_{cl}}_{\Delta t}} - = -\frac{K}{a_L} \, 2 {C_l}_0 (1 - C_l) \, (q_{cl}-Q_c) \Delta t -\] - -Inserting the limits of the integral on the l.h.s. and rearranging, -we obtain our analytical solution for $q_{cl}$ after time $\Delta t$: - -\begin{equation} -{q_{cl}}_{\Delta t} = {q_{cl}}_0 \left( 1 - \frac{1-b_1}{{q_{cl}}_0} - \frac{K}{a_L} \, 2 {C_l}_0 (1 - C_l) \, (q_{cl}-Q_c) \Delta t - \right)^\frac{1}{1-b_1} -\label{eq:qcl_int_hybrid} -\end{equation} - -Note that $q_{cl}$ falls to zero after a finite time -$\frac{{q_{cl}}_0}{1-b_1} \frac{a_L}{K} - \frac{1}{2 {C_l}_0 (1 - C_l) \, (q_{cl}-Q_c)}$. -If the timestep $\Delta t$ is longer than this time, then erosion -completely removes the cloud during the current timestep. - -We first set ${q_{cl}}_0$ and ${C_l}_0$ to the values already updated -by homogeneous forcing, and then sequentially compute the updated -$q_{cl}$ after erosion using (\ref{eq:qcl_int_hybrid}). -Then we substitute this value into (\ref{eq:cl_qcl_scaling}) to compute -the consistent updated value of $C_l$. -Finally, to improve accuracy, a small number of iterations are performed -to find the solution with the explicitly-treated terms -(the exponent $b_1 = \frac{ c_1 }{ 1 - \frac{q_{cl}}{C_l Q_c} }$ -and the terms $(1 - C_l)$ and $(q_{cl}-Q_c)$ in eq \ref{eq:qcl_int_hybrid}) -adjusted to values linearly-interpolated to half-way between the -start and end of the erosion timestep. - -\item {\bf Grid-mean supersaturation ($Q_c > 0$):} - -In this case, erosion does not act to reduce $C_l$ and $q_{cl}$ towards zero. -Instead, the narrow PDF limit it adjusts towards has -no remaining subsaturated air, so that $C_l = 1$ and $q_{cl} = Q_c$. -In this case, we can repeat the above derivation, but considering the -equations for the clear-fraction $1-C_l$ in place of $C_l$, -and the saturation defecit $SD = q_{cl}-Q_c$ in place of $q_{cl}$. -Assuming that $G(-Qc)$ follows the scaling for the clear end of the PDF -(\ref{eqn21}), this leads to a similar equation to (\ref{eq:cl_qcl_scaling}) -but for the scaling as erosion reduces $1-C_l$ and $SD$ towards zero .: - -\begin{equation} -\frac{1-C_l}{1-{C_l}_0} = \left( \frac{SD}{SD_0} \right)^{b_2} -\label{eq:ca_sd_scaling} -\end{equation} - -with $b_2 = \frac{ c_2 }{ 1 + \frac{SD}{(1-C_l) Q_c} }$ -and $c_2 = G(-Q_c) \frac{SD}{(1-C_l)^2}$ -(note we must have $0 < b_2 < 1$). - -And then the tendency equation for $SD$ is: - -\begin{equation} -\frac{\partial SD}{\partial t} = -\frac{K}{a_L} \, 2 (1 - C_l) C_l \, SD -\label{eq:dsddt_hybrid_2} -\end{equation} - -The one asymmetry between this and the $q_{cl}$ tendency equation -(\ref{eq:dqcldt_hybrid_2}) is that for $SD$ the r.h.s. is directly proportional -to the quantity in the time-derivative, whereas for $q_{cl}$ there is an -additional $Q_c$ term which is constant during erosion. -Substituting (\ref{eq:ca_sd_scaling}) for the leading factor of -$(1 - C_l)$ in (\ref{eq:dsddt_hybrid_2}), integrating over time $\Delta t$ -(neglecting the fractional variation of $C_l$ over the timestep) -and rearranging, we obtain: - -\begin{equation} -{SD}_{\Delta t} = {SD}_0 \left( 1 + b_2 - \frac{K}{a_L} \, 2 (1-{C_l}_0) C_l \, \Delta t - \right)^{-\frac{1}{b_2}} -\label{eq:sd_int_hybrid} -\end{equation} - -Note that the additional power of $SD$ on the r.h.s. of -(\ref{eq:dsddt_hybrid_2}) leads to the integral solution having -a negative exponent. This means that under grid-mean supersaturation, -erosion makes $SD$ and $1-C_l$ approach but never quite reach zero, -which is quite different behaviour to grid-mean subsaturation where -$q_{cl}$ and $C_l$ go to zero over a finite time. -This asymmetry is because the erosion rate is parameterised to be -proportional to $SD$, and this tends to zero as the PDF is narrowed -under supersaturation, but remains finite positive under subsaturation. - -We first set ${SD}_0 = {q_{cl}}_0 - Q_c$ (where as above ${q_{cl}}_0$ is -the value already updated by homogeneous forcing), -then compute the value of $SD$ updated by erosion using -(\ref{eq:sd_int_hybrid}). -Then we substitute this value into (\ref{eq:ca_sd_scaling}) to compute -the consistent updated value of $1-C_l$. -A small number of iterations are then performed -to find the solution with the explicitly-treated terms -(the exponent $b_2 = \frac{ c_2 }{ 1 + \frac{SD}{(1-C_l) Q_c} }$ -and the term $C_l$ in eq \ref{eq:sd_int_hybrid}) -adjusted to values linearly-interpolated to half-way between the -start and end of the erosion timestep. -Then the final values of $SD$ and $1-C_l$ are used to increment -$q_{cl} = Q_c + SD$ and $C_l$, as prognosed by the rest of the model. - -\item {\bf grid-mean saturation ($Q_c$ near-zero):} - -In this case, the PDF is centred on the -saturation boundary, so that narrowing it does not change the cloud-fraction. -In the limit $Q_c = 0$, we have $q_{cl} = SD$, and (\ref{eq:dqcldt_hybrid_2}) -or (\ref{eq:dsddt_hybrid_2}) becomes: - -\begin{equation} -\frac{1}{q_{cl}} \frac{\partial q_{cl}}{\partial t} - = -\frac{K}{a_L} \, 2 C_l (1 - C_l) -\end{equation} - -where everything on the r.h.s. is constant under erosion. -This simply integrates to give exponential decline of $q_{cl}$ -(and $SD$) towards zero: - -\begin{equation} -{q_{cl}}_{\Delta t} = {q_{cl}}_0 e^{ -\frac{K}{a_L} \, 2 C_l (1 - C_l) \Delta t } -\end{equation} - -\end{enumerate} - -\end{enumerate} - - -\subsection{Orographic and Gravity Wave Drag} -The Orographic and Gravity Wave Drag sections do not alter the temperature -or moisture content of the model gridboxes, hence PC2 assumes no change in the -condensate and cloud fractions as a result of these processes. - -\subsection{Advection} -\label{sec:advec} - -The advection of $\overline{q_{cl}}$ and $\overline{q_{cf}}$ are already -performed separately by the semi-Lagrangian advection scheme. -Advection of the three cloud fractions $C_l$, $C_i$ and $C_t$ are all -performed by PC2 in the same way. - -Note that ascent or subsidence by advection entails a pressure change following -each parcel, which will cause an accompanying adiabatic temperature change. -These advective pressure and temperature changes imply a homogeneous forcing, -which yields a change in $\overline{q_{cl}}$ and $C_l$ in addition to their -transport by the winds. This is described in section \ref{sec:pres}. - -If the UM namelist switch \textbf{l\_pc2\_sl\_advection} is turned on, -the PC2 homogeneous forcing response to advection is calculated -straight after the call to Semi-Lagrangian advection. -Otherwise, the pressure change from advection is combined with the -Eulerian pressure change from the dynamics Helmholtz solver, and the resulting -homogeneous forcing of liquid cloud is computed at the end of the timestep. - -\subsection{Boundary Layer} -\label{sec:bl} -At a basic level, the boundary layer scheme works by -mixing $\overline{q_T}$ and $\overline{T_L}$, and tracer mixing -$\overline{q_{cf}}$. The condensation and $C_l$ changes are represented -using the homogeneous forcing representation. The forcing of $Q_c$ can be -written in $\Delta \overline{q_T}$ and $\Delta \overline{T_L}$ terms -using (\ref{eq:deltaqc_exp}). - -$\overline{q_{cf}}$ is already mixed using the tracer mixing scheme. PC2 -will calculate the corresponding $C_i$ change assuming the inhomogeneous -forcing scenario. Although this is not necessarily an appropriate physical -model to use, it is the only generic physical model we have currently -developed in order to convert increments in a condensate to increments in -a cloud fraction. We use a value of the in-cloud water content $q_c^S$ -based upon a linear combination of the current in-cloud ice water -content, $\frac{\overline{q_{cf}}}{C_i}$, and a fixed value. - -\begin{equation} -q_C^S = C_i \frac{\overline{q_{cf}}}{C_i} + ( 1 - C_i) q_{cf0 \, BL} -\label{eq:qcf_ci} -\end{equation} - -where $q_{cf0 \, BL}$ is a specified value of $1 \times 10^{-4} \, kg \, kg^{-1}$. -We then use the inhomogeneous forcing equation based upon -(\ref{eq:dcdt_inhom2}) but for ice water content to write - -\begin{equation} -\Delta C_i = \frac{(1 - C_i)}{q_C^S - \overline{q_{cf}}} Q4_i . -\label{eq:deltaci_bl} -\end{equation} - -Since the physical model will have $C_i$ tend to 1 if the -denominator is small, we will, to avoid numerical problems, set -$C_i$ to 1 if $q_C^S - \overline{q_{cf}} < 1 \times 10^{-10} kg kg^{-1}$. -Note that we do not use the multiple phases injection source -expressions (section \ref{sec:multiple} and equation \ref{eq:cff_ts}). -This is because the liquid water changes are not associated with the plume model. - -Equation \ref{eq:qcf_ci} assumes that the change to the ice water content has led to an increase in ice water content. -However, if the ince water content has reduced, the change to the ice cloud fraction is not consistent. -The option to "Use consistent formulation of ice cloud fraction changes due to boundary-layer processes" ensure that -if the ice water content is reduced, the ice cloud fraction is reduced, in such as way as to maintain -the same in-cloud ice water content. - -The $C_t$ changes are calculated using the minimum overlap method of -section \ref{sec:ct}. - -In \textit{ni-imp-ctl} the control code inhibits the -call to the diagnostic cloud scheme -if there is deep or shallow convection occurring and the model level -is less than \textit{or equal to} the layer immediately above the top of the -boundary layer mixed layer (i.e. level ntml+1). This is in order to -ensure that there is -no large-scale cloud present below the base of the convective cloud, but -additionally performs this calculation at the level above, probably -in order that latent heating from large-scale condensation does not -inhibit the convection. A similar thing is performed for PC2, with -any large-scale cloud being evaporated if the same criteria are met, -\textit{except that it is not performed on the level above the boundary -layer mixed layer}. This choice (i.e. ntml) is seen to give improved -results in PC2, and is arguably a more physical reasonable choice -anyway than using ntml+1. - -\subsection{Convection} -\label{sec:convec} - -This section concentrates specifically upon the PC2 interface to the -convection scheme. In the current formulation of the UM, only a -mass-flux convection scheme exists, and this is what is described -below. Work to interface PC2 to the developing turbulence based -convection scheme is commented upon in section \ref{sec:tbcs}. - -An alternative way of calculating cloud fraction increments is currently under development and -is described in section \ref{sec:conv-simpler}. - -A traditional view of convective parametrization is a scheme that -transports vapour, $q$, -heat, $\theta$, and momentum, $u$ and $v$ winds within a single column. It -does not consider transport sideways to adjoining columns, and (at least -in the Gregory-Rowntree scheme used in the UM) is considered independent of -any resolved scale vertical air motions. This necessitates the view of -compensating subsidence within the column, whereas some conceptual models -of tropical convection would have the bulk of the ascent in the -convective cores and the -associated descent thousands of miles away in the downward branch of the -Hadley circulation. The parametrization schemes traditionally overlook the -existence of condensate in the model column. The non-PC2 version -of the mass-flux convection scheme used in the UM would have the same -large-scale liquid and ice prognostics before and after convection occurs -(apart from a bolt-on evaporation below convective cloud base), with no -regard at all to what happens to it or its effect on the rest of the -convection. Within PC2 we have had to work to more fully incorporate -the condensate into the convection scheme. - -\subsubsection{Introduction to the convective mass flux scheme} - -Within the mass flux scheme the net change in -$\overline{q_{cl}}$ and $C_l$ etc. -comes from two distinct sources. Firstly, the condensate and cloud -fraction injected from the plume (the $Q4$ terms, section \ref{sec:inhomog}); -secondly, the condensation response to the vapour and heat changes associated -with the detrainment and compensating subsidence. Strictly, we will see that the -$Q4$ terms also include the contribution to the condensate transport -by the compensating subsidence - this casts doubt on the validity of -the application of the injection forcing scenario to calculate the -equivalent cloud fraction change, since ideally the cloud fractions -ought to be transported by the compensating subsidence in a similar -way to the condensate transport (which is documented below). - -We therefore split the convective contribution in (\ref{eq:dqcldt_and_dcdt}) -into two parts: - -\begin{equation} -\frac{\partial \overline{q_{cl}}}{\partial t} |_{convection} = -Q4_l + Q_{environment} -\label{eq:inhomg_plus_homog} -\end{equation} - -where $Q_{environment}$ is the condensation associated with changes -in the vapour and temperature from the detrainment and compensating -subsidence. Similar splits are made for the cloud variables, where -the injection forcing, section \ref{sec:inhomog}, is used to calculate -the first term from $Q4_l$. Section \ref{subsect:q4calculation} looks -at the issue of the -calculation of $Q4_l$ etc., and section \ref{sec:conv_homog} looks at -the calculation of $Q_{environment}$, and its associated cloud -fraction change. We first look at the basic transport equations in a -mass flux convection scheme. - -\subsubsection{Basic Equations for a Convective Mass Flux Scheme} -\label{subsect:basmaseqs} - -We first consider a generic mass-flux scheme before its application to PC2. -As discussed by Grant and Stirling (personal communication), -the equations for convective -tendencies are most simply applied to a variable, ${\chi}$, that is conserved -under moist adiabatic processes (e.g. total water content). In this case, -% -\begin{equation} -{\pardbyd{\xsubsup{ }{E}}{t}}_{\rm{conv}} = -- \frac{1}{\ov{\rho}} \, \pardbyd{\ov{\rho w^{'} \xsubsup{ }{E'}}}{z} -\label{eq:chibasic} \end{equation} - -To parametrize \ref{eq:chibasic}, the current UM convection scheme takes a -mass flux approximation -% -\begin{equation} -\lp {\ov{\rho w^{'} \xsubsup{ }{E'}}} \rp_{\rm{conv}} = M^{\rm{P}} \, -\lp { \xsubsup{ }{P} - \xsubsup{ }{E} } \rp -\label{eq:massflux} \end{equation} -% -which can be differentiated to give -% -\begin{equation} -- \frac{1}{\ov{\rho}} \, \pardbyd{\ov{\rho w^{'} \xsubsup{ }{E'}}}{z} = -\pardbyd{\xsubsup{ }{P} \, M^{\rm{P}}}{p} - -\xsubsup{ }{E} \, \pardbyd{M^{\rm{P}}}{p} - -M^{\rm{P}} \, \pardbyd{\xsubsup{ }{E}}{p} -\label{eq:eddyflux} \end{equation} - -The bulk cloud model plume equations for mass and ${\chi}$ are: -% -\begin{eqnarray} -- \pardbyd{M^{\rm{P}}}{p} & = & -\lp { \varepsilon \, M^{\rm{P}} - \mu \, M^{\rm{P}} - \delta \, M^{\rm{P}} } \rp -\label{eq:dbydpmassflux} \\ -- \pardbyd{\xsubsup{ }{P} \, M^{\rm{P}}}{p} & = & \lp { -\varepsilon \, M^{\rm{P}} \, \xsubsup{ }{E} -- \mu \, M^{\rm{P}} \, \xsubsup{ }{R} - \delta \, M^{\rm{P}} \, \xsubsup{ }{P} -} \rp \label{eq:dbydpmfchi} -\end{eqnarray} - -Equations \ref{eq:eddyflux}, \ref{eq:dbydpmassflux} and -\ref{eq:dbydpmfchi} can then be substituted into \ref{eq:chibasic} to give: -% -\begin{equation} -{\pardbyd{\xsubsup{ }{E}}{t}}_{\rm{conv}} = -- M^{\rm{P}} \, \pardbyd{\xsubsup{ }{E}}{p} -+ \mu \, M^{\rm{P}} \, \lp { \xsubsup{ }{R} - \xsubsup{ }{E} } \rp -+ \delta \, M^{\rm{P}} \, \lp { \xsubsup{ }{P} - \xsubsup{ }{E} } \rp -\label{eq:chimassflux} \end{equation} -% -while \xsubsup{}{P} is obtained from the vertical gradient derived by combining -\ref{eq:dbydpmassflux} and \ref{eq:dbydpmfchi} : -% -\begin{equation} -M^{\rm{P}} \, \pardbyd{\xsubsup{ }{P}}{p} = -\varepsilon \, M^{\rm{P}} \, \lp { \xsubsup{ }{P} - \xsubsup{ }{E} } \rp - -\mu \, M^{\rm{P}} \, \lp { \xsubsup{ }{P} - \xsubsup{ }{R} } \rp -\label{eq:gradchipar} \end{equation} - -Within the model, eqn~\ref{eq:chimassflux} would take a discretized form -which actually depends upon whether the model level, k, is above or at the -lowest cloud level (k = cb). Note that the formal cloud base lies at the -half-level below, i.e. on the layer boundary which is also the top of the -turbulent mixed boundary layer. A simple discretized form of -\ref{eq:chimassflux}, setting ${ \mu = 0 }$, is: -% -\begin{eqnarray} -{\pardbyd{\xsubsup{ }{E}}{t}}_{\rm{conv, \, k}} & = & m_{\rm{k+1/2}} \, -\frac{ \lp {\xsubsup{k+1}{E} - \xsubsup{k}{E}} \rp } -{{\Delta z}_{\rm{k \, \rightarrow \, k+1}}} -+ {\delta}_{\rm{k}} \, m_{\rm{k}} \, \lp { \xsubsup{k}{P} - \xsubsup{k}{E} } \rp -\qquad \ldots \; \mbox{for k $>$ cb} \label{eq:chidisck} \\ -{\pardbyd{\xsubsup{ }{E}}{t}}_{\rm{conv, \, cb}} & = & m_{\rm{cb+1/2}} \, -\frac{ \lp {\xsubsup{cb+1}{E} - \xsubsup{cb}{E}} \rp } -{{\Delta z}_{\rm{cb \, \rightarrow \, cb+1}}} -- m_{\rm{cb}} \, -\lp { \xsubsup{i,cb}{P} - \xsubsup{cb}{E} } \rp \label{eq:chidisccb} -\end{eqnarray} -% -where the initial parcel value \xsubsup{i,cb}{P} may be chosen to produce a -fixed increment or place a closure condition on the cloud base flux. In fact, -the convection equations (see \citeumdp{027}) differ from \ref{eq:chidisck} and -\ref{eq:chidisccb} because a different discretization is used, but the -principle is unaltered. - -The model convection variables are NOT conserved under moist adiabatic processes -because precipitation processes deplete the column moisture and condensation -processes affect the temperature, specific humidity and cloud condensate -variables. Surprisingly, however, the form of eqn~\ref{eq:chimassflux} is -retained even though the basic equation \ref{eq:chibasic} acquires additional -terms for temperature and specific humidity: -% -\begin{eqnarray} -{\pardbyd{\tsubsup{ }{E}}{t}}_{\rm{conv}} = Q1 & \equiv & -\lp { \frac{L}{c_{P}} } \rp \, {\ov{Q}}_{\rm{par}} -- \frac{1}{\ov{\rho}} \, \pardbyd{\ov{\rho w^{'} \tsubsup{ }{E'}}}{z} -\label{eq:defineq1} \\ -{\pardbyd{\qsubsup{ }{E}}{t}}_{\rm{conv}} = Q2 & \equiv & - {\ov{Q}}_{\rm{par}} -- \frac{1}{\ov{\rho}} \, \pardbyd{\ov{\rho w^{'} \qsubsup{ }{E'}}}{z} -\label{eq:defineq2} -\end{eqnarray} -% -where ${\ov{Q}}_{\rm{par}}$ is the rate of condensation which occurs in the -ascending plumes. - -The reason that \ref{eq:defineq1} and \ref{eq:defineq2} retain this form -is due to cancellation from the bulk cloud terms equivalent to -\ref{eq:dbydpmfchi} which are modified in the same way as -\ref{eq:defineq1} and \ref{eq:defineq2}. The change is seen in the -vertical gradient equations based upon \ref{eq:gradchipar} -% -\begin{eqnarray} -M^{\rm{P}} \, \pardbyd{\tsubsup{ }{P}}{p} & = & -\varepsilon \, M^{\rm{P}} \, \lp { \tsubsup{ }{P} - \tsubsup{ }{E} } \rp - -\mu \, M^{\rm{P}} \, \lp { \tsubsup{ }{P} - \tsubsup{ }{R} } \rp - -\lp { \frac{L}{c_{P}} } \rp \, {\ov{Q}}_{\rm{par}} \label{eq:gradtpar} \\ -M^{\rm{P}} \, \pardbyd{\qsubsup{ }{P}}{p} & = & -\varepsilon \, M^{\rm{P}} \, \lp { \qsubsup{ }{P} - \qsubsup{ }{E} } \rp - -\mu \, M^{\rm{P}} \, \lp { \qsubsup{ }{P} - \qsubsup{ }{R} } \rp + -{\ov{Q}}_{\rm{par}} \label{eq:gradqpar} \\ -M^{\rm{P}} \, \pardbyd{\lsubsup{ }{P}}{p} & = & -\varepsilon \, M^{\rm{P}} \, \lp { \lsubsup{ }{P} - \lsubsup{ }{E} } \rp -- {\ov{Q}}_{\rm{par}} + PPN \label{eq:gradlpar} -\end{eqnarray} - -The final calculation of rates in the current condensation scheme (\citeumdp{027}, -section 10) assumes a further condensation term, ${\ov{Q}}_{\rm{reset}}$, which -acts to make the net rate of change of condensate equal zero, and a final -assumption is made that the environment values of condensate remain zero (and -also that \lsubsup{ }{R} = \lsubsup{ }{P}). The -result is basic equations -% -\begin{eqnarray} -{\pardbyd{\tsubsup{ }{E}}{t}}_{\rm{conv}} & = & Q1 - -\lp { \frac{L}{c_{P}} } \rp \, {\ov{Q}}_{\rm{reset}} -\label{eq:basictold} \\ -{\pardbyd{\qsubsup{ }{E}}{t}}_{\rm{conv}} & = & Q2 + {\ov{Q}}_{\rm{reset}} -\label{eq:basicqold} \\ -0 \equiv {\pardbyd{\lsubsup{ }{E}}{t}}_{\rm{conv}} & = & {\ov{Q}}_{\rm{par}} - -{\ov{Q}}_{\rm{reset}} - PPN -- \frac{1}{\ov{\rho}} \, \pardbyd{\ov{\rho w^{'} \lsubsup{ }{E'}}}{z} \nonumber \\ -& = & -\mu \, M^{\rm{P}} \, \lsubsup{ }{P} + \delta \, M^{\rm{P}} \, \lsubsup{ }{P} - -{\ov{Q}}_{\rm{reset}} -\label{eq:basiclold} -\end{eqnarray} - -By analogy with equations \ref{eq:defineq1} and \ref{eq:defineq2}, we can -define a $Q4$ from \ref{eq:basiclold} and state that for the control -convection scheme $Q4 = 0$. The PC2 scheme requires a reassessment of these -assumptions because we wish to allow non-zero environment condensate values and -to allow them to change. - -\subsubsection{Calculation of Grid-Box Averaged Condensate Rate (Q4)} -\label{subsect:q4calculation} - -The PC2 condensation scheme allows convection to feed cloud condensate (ice or -liquid) directly into the large scale and to update the cloud amount accordingly. - -Define -% -\begin{eqnarray} -\lp { \pardbyd{\lsubsup{l}{ }}{t} } \rp_{\rm{conv}} = Q4_{\rm{l}} & \equiv & -{\ov{Q}}_{\rm{l, par}} - {\ov{Q}}_{\rm{l, reset}} - RAIN - -\frac{1}{\ov{\rho}} \, \pardbyd{\ov{\rho w^{'} \lsubsup{l}{'}}}{z} -\label{eq:defineq4l} \\ -\lp { \pardbyd{\lsubsup{f}{ }}{t} } \rp_{\rm{conv}} = Q4_{\rm{f}} & \equiv & -{\ov{Q}}_{\rm{f, par}} - {\ov{Q}}_{\rm{f, reset}} - SNOW - -\frac{1}{\ov{\rho}} \, \pardbyd{\ov{\rho w^{'} \lsubsup{f}{'}}}{z} -\label{eq:defineq4f} -\end{eqnarray} -% -where the PC2 assumption thus far has been that ${\ov{Q}}_{\rm{l, reset}} = 0 -= {\ov{Q}}_{\rm{f, reset}}$. - -\begin{itemize} -\item{The current convection scheme assumes that parcel -condensate is single phase -(ie. either all liquid or all frozen) and this is seriously hard-wired into the -code. Thus we can treat the precipitation and parcel condensation -processes in $ Q4_{\rm{l}} $ and $ Q4_{\rm{f}} $ separately without worrying -about cross-transfer between the two because at most only one set will ever be -active in a given grid box at one time. However, even for the inactive (zero -parcel condensate) phase, convection mixes environmental air into the -parcel and -can therefore maintain a non-zero $Q4$. Enablement of multiple phase condensate -in the current scheme is a task requiring great caution as the formulations -are extremely sensitive to errors in assignment of condensate phase.} -\end{itemize} - -Based on \ref{eq:gradlpar}, the vertical dependence of condensate is -calculated as -% -\begin{eqnarray} -\pardbyd{\lsubsup{l}{P}}{p} & = & \varepsilon \, -\lp { \lsubsup{l}{P} - \lsubsup{l}{E} } \rp - -\frac{{\ov{Q}}_{\rm{l, par}}}{M^{\rm{P}}} - -\frac{RAIN}{M^{\rm{P}}} \label{eq:vertparl} \\ -\pardbyd{\lsubsup{f}{P}}{p} & = & \varepsilon \, -\lp { \lsubsup{f}{P} - \lsubsup{f}{E} } \rp - -\frac{{\ov{Q}}_{\rm{f, par}}}{M^{\rm{P}}} - -\frac{SNOW}{M^{\rm{P}}}\label{eq:vertparf} -\end{eqnarray} - -Following \citeumdp{027}, equations \ref{eq:dbydpmassflux}, \ref{eq:vertparl} and -\ref{eq:vertparf} are discretized: -% -\begin{eqnarray} -M_{\rm{k} + 1} & = & M_{\rm{k}} \, -\lp { 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp \, -\lp { 1 - \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp \, -EPSS_{\rm{k}} \label{eq:discdmfbydp} \\ -\lsubsup{l \, k + 1}{P} & = & \lp { -\lsubsup{l \, k}{P} + -\varepsilon_{\rm{k} + 1/4} \, \Delta p_{\rm{k} + 1/4} \, \lsubsup{l \, k}{E} + -\varepsilon_{\rm{k} + 3/4} \, \Delta p_{\rm{k} + 3/4} \, -\lc {1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4}} \rc \, -\lsubsup{l \, k + 1}{E} -} \rp \, / \, \lp {EPSS_{\rm{k}}} \rp \nonumber \\ -{ } & { } & + \lp { {\ov{Q}}_{\rm{l} \, \rm{k} + 1} \, / \, M_{\rm{k} + 1}} \rp -- \lp { RAIN_{\rm{k} + 1} \, / \, M_{\rm{k} + 1} } \rp -\label{eq:discvparl} \\ -\lsubsup{f \, k + 1}{P} & = & \lp { -\lsubsup{f \, k}{P} + -\varepsilon_{\rm{k} + 1/4} \, \Delta p_{\rm{k} + 1/4} \, \lsubsup{f \, k}{E} + -\varepsilon_{\rm{k} + 3/4} \, \Delta p_{\rm{k} + 3/4} \, -\lc {1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4}} \rc \, -\lsubsup{f \, k + 1}{E} -} \rp \, / \, \lp {EPSS_{\rm{k}}} \rp \nonumber \\ -{ } & { } & + \lp { {\ov{Q}}_{\rm{f} \, \rm{k} + 1} \, / \, M_{\rm{k} + 1}} \rp -- \lp { SNOW_{\rm{k} + 1} \, / \, M_{\rm{k} + 1} } \rp -\label{eq:discvparf} -\end{eqnarray} -% -where $EPSS_{\rm{k}} = -\lp {1 + \varepsilon_{\rm{k} + 3 / 4} \, \Delta p_{\rm{k} + 3 / 4}} \rp \, -\lp {1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4}} \rp $. - -The condensation and precipitation terms in equations \ref{eq:discdmfbydp}, -\ref{eq:discvparl} and \ref{eq:discvparf} make the equations implicit. -They are therefore solved by starting with an ascent in which condensation and -precipitation terms are suppressed: -% -\begin{eqnarray} -\lsubsup{l \, k + 1}{P} & = & \frac{\lp { -\lsubsup{l \, k}{P} + -\varepsilon_{\rm{k} + 1/4} \, \Delta p_{\rm{k} + 1/4} \, \lsubsup{l \, k}{E} + -\varepsilon_{\rm{k} + 3/4} \, \Delta p_{\rm{k} + 3/4} \, -\lc {1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4}} \rc \, -\lsubsup{l \, k + 1}{E} -} \rp}{EPSS_{\rm{k}}} \label{eq:discvparldry} \\ -\lsubsup{f \, k + 1}{P} & = & \frac{\lp { -\lsubsup{f \, k}{P} + -\varepsilon_{\rm{k} + 1/4} \, \Delta p_{\rm{k} + 1/4} \, \lsubsup{f \, k}{E} + -\varepsilon_{\rm{k} + 3/4} \, \Delta p_{\rm{k} + 3/4} \, -\lc {1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4}} \rc \, -\lsubsup{f \, k + 1}{E} -} \rp}{EPSS_{\rm{k}}} \label{eq:discvparfdry} -\end{eqnarray} -% -At the base of the convective plume (ie. the level immediately above cloud -base), \lsubsup{l \, k}{P} is initialized to \lsubsup{l \, i}{P} and -\lsubsup{f \, k}{P} to \lsubsup{f \, i}{P}, where the initial values are chosen -such that the modified form of \ref{eq:chidisccb} produces zero fluxes at -cloud base: -% -\begin{eqnarray} -Q4_{\rm{l}}(cb) = 0 & = & M_{\rm{cb+1/2}}^{\rm{P}} \, -\pardbyd{\lsubsup{l}{E}}{p} - M_{\rm{cb}}^{\rm{P}}\, -\lp { \lsubsup{l}{P \, i} - \lsubsup{l}{E}(\rm{cb}) } \rp \label{eq:q4lcbi} \\ -Q4_{\rm{f}}(cb) = 0 & = & M_{\rm{cb+1/2}}^{\rm{P}} \, -\pardbyd{\lsubsup{f}{E}}{p} - M_{\rm{cb}}^{\rm{P}}\, -\lp { \lsubsup{f}{P \, i} - \lsubsup{f}{E}(\rm{cb}) } \rp \label{eq:q4fcbi} -\end{eqnarray} - - -As the convection scheme makes the single phase assumption for parcel -condensate, it may be necessary to melt or freeze entrained condensate at this -point and adjust the temperature accordingly. -% -\begin{eqnarray} -\theta_{\rm{k + 1}}^{\rm{P}} = \theta_{\rm{k + 1}}^{\rm{P}} - -\lp \frac{L_{\rm{F}}}{C_{p} \, \Pi_{\rm{k + 1}}} \rp \, \lsubsup{f \, k + 1}{P} -& \; \ldots \; & \mbox{ if \lsubsup{f \, k + 1}{P} is melted } -\label{eqn:meltlf} \\ -\theta_{\rm{k + 1}}^{\rm{P}} = \theta_{\rm{k + 1}}^{\rm{P}} + -\lp \frac{L_{\rm{F}}}{C_{p} \, \Pi_{\rm{k + 1}}} \rp \, \lsubsup{l \, k + 1}{P} -& \; \ldots \; & \mbox{ if \lsubsup{l \, k + 1}{P} is frozen } -\label{eqn:freezell} -\end{eqnarray} - -Once a final value for the condensation term -$ {\ov{Q}}_{\rm{x} \, \rm{k} + 1} \, / \, M_{\rm{k} + 1} $ has been calculated -from the parcel specific humidity equations, it can then be added to the parcel -condensate to give a final pre-precipitation value. - -\begin{itemize} -\item{In practice, the rates $ {\ov{Q}}_{\rm{x} \, \rm{k} + 1}$ and -$ PPN $ are not calculated explicitly in the code. -Instead, their effect is applied directly as increments to the temperature and -moisture fields.} -\end{itemize} - -The precipitation calculation is unaltered. -% -\begin{equation} -P_{\rm{k} + 1} = \lp { \lsubsup{k + 1}{P} - \lsubsup{MIN}{P} } \rp \, -M_{\rm{k} + 1} \, / \, g -\label{eq:precip} \end{equation} -% -where \lsubsup{k + 1}{P} = \lsubsup{l \, k + 1}{P} + \lsubsup{f \, k + 1}{P}. - -\begin{itemize} -\item{Actually, given that the precipitation calculation appears to be -based upon the hydrostatic equation, it is debatable whether it is even suitable -for use with the New Dynamics model and I guess therefore that this needs -revisiting at some point.} -\end{itemize} - -This reduces the parcel condensate to : -% -\begin{eqnarray} -\lsubsup{l \, k + 1}{P} & = & \lp { -\frac{\lsubsup{l \, k + 1}{P}}{\lsubsup{k + 1}{P}} -} \rp \, \lsubsup{MIN}{P} \label{eq:vparlfinal} \\ -\lsubsup{f \, k + 1}{P} & = & \lp { -\frac{\lsubsup{f \, k + 1}{P}}{\lsubsup{k + 1}{P}} -} \rp \, \lsubsup{MIN}{P} \label{eq:vparffinal} -\end{eqnarray} - -The final parcel condensate values are then used in the rate calculation based -upon eqn~\ref{eq:basiclold}: -% -\begin{eqnarray} -Q4_{\rm{l}}(k) & = & M_{\rm{k+1/2}}^{\rm{P}} \, \pardbyd{\lsubsup{l}{E}}{p} + -\lp { {\mu}_{\rm{k}} \, M_{\rm{k}}^{\rm{P}} + -{\delta}_{\rm{k}} \, M_{\rm{k}}^{\rm{P}} } \rp \, -\lp { \lsubsup{l}{P}(\rm{k}) - \lsubsup{l}{E}(\rm{k}) } \rp - -{\ov{Q}}_{\rm{l, reset}} \label{eq:q4lmassf} \\ -Q4_{\rm{f}}(k) & = & M_{\rm{k+1/2}}^{\rm{P}} \, \pardbyd{\lsubsup{f}{E}}{p} + -\lp { {\mu}_{\rm{k}} \, M_{\rm{k}}^{\rm{P}} + -{\delta}_{\rm{k}} \, M_{\rm{k}}^{\rm{P}} } \rp \, -\lp { \lsubsup{f}{P}(\rm{k}) - \lsubsup{f}{E}(\rm{k}) } \rp - -{\ov{Q}}_{\rm{f, reset}} \label{eq:q4fmassf} -\end{eqnarray} - - -Note that, as a side-effect, the \citeumdp{027} environment equations for potential -temperature and specific humidity are also altered because the condensate is no -longer re-evaporated at the end (${\ov{Q}}_{\rm{l, reset}} = 0 -= {\ov{Q}}_{\rm{f, reset}}$): -% -\begin{eqnarray} -\frac{\Delta \, \theta_{\rm{k}}^{\rm{E}}}{\Delta \, t} = -\lp \frac{ M_{\rm{k}} }{ \Delta \, p_{\rm{k}} } \rp -\lc { -\lp { 1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4} } \rp -\lp { 1 - \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp -\lp { 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp -\lp { \theta_{\rm{k + 1}}^{\rm{E}} - \theta_{\rm{k}}^{\rm{E}} } \rp -} \right . & + & \nonumber \\ -\lp { \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp -\lp { 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp -\lp { \theta_{\rm{k}}^{\rm{R}} - \theta_{\rm{k}}^{\rm{E}} } \rp -& + & \nonumber \\ -\left . { -\lp { \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp -\lp { \theta_{\rm{k}}^{\rm{P}} - \theta_{\rm{k}}^{\rm{E}} } \rp -} \rc & { } & \label{eq:enviroth} -\end{eqnarray} -% -and -% -\begin{eqnarray} -\frac{\Delta \, q_{\rm{k}}^{\rm{E}}}{\Delta \, t} = -\lp \frac{ M_{\rm{k}} }{ \Delta \, p_{\rm{k}} } \rp -\lc { -\lp { 1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4} } \rp -\lp { 1 - \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp -\lp { 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp -\lp { q_{\rm{k + 1}}^{\rm{E}} - q_{\rm{k}}^{\rm{E}} } \rp -} \right . & + & \nonumber \\ -\lp { \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp -\lp { 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp -\lp { q_{\rm{k}}^{\rm{R}} - q_{\rm{k}}^{\rm{E}} } \rp -& + & \nonumber \\ -\left . { -\lp { \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp -\lp { q_{\rm{k}}^{\rm{P}} - q_{\rm{k}}^{\rm{E}} } \rp -} \rc & { } & \label{eq:enviroq} -\end{eqnarray} - -Similarly, eqns \ref{eq:q4lmassf} and \ref{eq:q4fmassf} have -a discretized form as follows: -% -\begin{eqnarray} -\frac{\Delta \, \lsubsup{l \, k}{E}}{\Delta \, t} = -\lp \frac{ M_{\rm{k}} }{ \Delta \, p_{\rm{k}} } \rp -\lc { -\lp { 1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4} } \rp -\lp { 1 - \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp -\lp { 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp -\lp { \lsubsup{l \, k + 1}{E} - \lsubsup{l \, k}{E} } \rp -} \right . & + & \nonumber \\ -\lp { \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp -\lp { 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp -\lp { \lsubsup{l \, k}{P} - \lsubsup{l \, k}{E} } \rp -& + & \nonumber \\ -\left . { -\lp { \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp -\lp { \lsubsup{l \, k}{P} - \lsubsup{l \, k}{E} } \rp -} \rc & { } & \label{eq:enviroll} -\end{eqnarray} -% -and -% -\begin{eqnarray} -\frac{\Delta \, \lsubsup{f \, k}{E}}{\Delta \, t} = -\lp \frac{ M_{\rm{k}} }{ \Delta \, p_{\rm{k}} } \rp -\lc { -\lp { 1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4} } \rp -\lp { 1 - \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp -\lp { 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp -\lp { \lsubsup{f \, k + 1}{E} - \lsubsup{f \, k}{E} } \rp -} \right . & + & \nonumber \\ -\lp { \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp -\lp { 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp -\lp { \lsubsup{f \, k}{P} - \lsubsup{f \, k}{E} } \rp -& + & \nonumber \\ -\left . { -\lp { \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \rp -\lp { \lsubsup{f \, k }{P} - \lsubsup{f \, k}{E} } \rp -} \rc & { } & \label{eq:envirolf} -\end{eqnarray} - -\subsubsection{Background condensation} -\label{sec:conv_homog} -The modification to the convective plume will result in the transport, -detrainment and entrainment of condensate, in addition to the -transport of vapour and heat. Although condensation processes within -the plume are treated, it does not treat condensation in the -environment, which is forced by the compensating subsidence. We wish -to relate the environmental increments of vapour and temperature -to a forcing that can be applied in the environment. Because we -know that any detrained air associated with detrained liquid water -from the plume must be saturated with respect to liquid water, we -are able to translate the environmental changes into forcings. - -Here we will consider that the vapour change in the gridbox is as a result of -\textit{saturated with respect to liquid water} air being injected -from the plume and background air being displaced. -We do not consider whether the background air is at saturation yet, for we wish -to derive the expression for the required condensation if this is not the case. -We consider only liquid water clouds, ice clouds have no background condensation -applied as we do not make the instantaneous condensation assumption. - -Hence we can write - -\begin{equation} -\Delta \overline{q} = \Delta C_S ( q_{sat liq}(\overline{T_{s}}) - \overline{q} ) -+ (1 - \Delta C_S) \Delta \overline{q_{background}} -\end{equation} - -where $\Delta C_S$ is the volume of plume air that is detrained into the gridbox, -as discussed by \cite{bwg03}. $T_s$ is the temperature of the air injected into -the gridbox by the plume. The first term is simply the difference -between the value of $q$ in the plume and what was previously in the gridbox, and -the second term is the effect of a background change of $q$ that will be applied -across the part of the gridbox that is not associated with the injected air. We write this as: - -\begin{equation} -(1 - \Delta C_S) \Delta \overline{q_{background}} = \Delta \overline{q} - -\Delta C_S ( q_{sat liq}(\overline{T_{s}}) - \overline{q} ) . -\label{eqn:1mcs} -\end{equation} - -Now we recognise that - -\begin{equation} -\Delta \overline{q} = Q2~ \Delta t -\label{eqn:Q2} -\end{equation} - -where $Q2$ is the rate of moistening of the whole gridbox due to convection. -Remember that, at this stage, we haven't done any condensation outside of the plume. -Hence to calculate the condensation we should apply the background change in $\overline{q}$ -as a uniform forcing for the background air. Hence (\ref{eqn:1mcs}) becomes, using -(\ref{eqn:Q2}), - -\begin{equation} -(1 - \Delta C_S) A_q |_{background} \Delta t = Q2 ~ \Delta t - \Delta C_S -( q_{sat liq}(T_{s}) - \overline{q} ) . -\label{eqn:Aq} -\end{equation} - -where $A_q |_{background}$ is the currently unknown background forcing of -$q$ (see \cite{gwb02}) and $\Delta t$ is the timestep. -We can do the same analysis for the temperature change, and obtain - -\begin{equation} -(1 - \Delta C_S) A_T |_{background} \Delta t = Q1~ \Delta t - -\Delta C_S (T_s - \overline{T} ) -\label{eqn:AT} -\end{equation} - -where Q1 is the rate of warming in the gridbox due to convection and $A_T |_{background}$ is -the currently unknown background forcing of temperature. - -The full change of liquid water content in the gridbox is that injected, -$Q4~\Delta t$, plus the amount of condensation in the background from -the uniform forcings (see -\cite{wg03}). Note that the uniform forcings are only applied across -a proportion $1 - \Delta C_S$ of the gridbox. Hence these two terms give, using the -homogeneous forcing equations (\ref{dqcldt}) and (\ref{eq:deltaqc_exp2}), - -\begin{equation} -\Delta \overline{q_{cl}} |_{convection} = Q4 \Delta t -+ (1 - \Delta C_S) a_L C_l (A_q |_{background} \Delta t -- \alpha A_T |_{background} \Delta t ). -\label{eqn:qclconv} -\end{equation} - -Using (\ref{eqn:Aq}) and (\ref{eqn:AT}) to expand the forcing terms in (\ref{eqn:qclconv}) gives - -\begin{equation} -\Delta \overline{q_{cl}} |_{convection} = Q4 \Delta t -+ \Delta t a_L C_l ( Q2 - \alpha Q1) - \Delta C_S a_L C_l -(q_{sat liq}(T_s)-\overline{q} - \alpha (T_s - \overline{T})) . -\end{equation} - -We now note that - -\begin{equation} -q_{sat} (T_s) - q_{sat liq} (\overline{T}) = \alpha (T_s - \overline{T} ) -\end{equation} - -and hence the final result - -\begin{equation} -\Delta \overline{q_{cl}} |_{convection} = Q4 \Delta t -+ \Delta t ~ a_L C_l ( ( Q2 - \alpha Q1) - \Delta C_S -(q_{sat liq}(\overline{T}) - \overline{q} ) ) . -\label{eqn:dqcl} -\end{equation} - -There is thus an extra term, $-\Delta C_S (q_{sat}(\overline{T})-\overline{q} )$, -which needs to be included in addition to the standard application of the homogeneous -forcing of $Q1$ and $Q2$ (this is represented -by the second term of the expression). This has arisen from the requirement that -the vapour injected by the plume is saturated. We need simply -to retrieve the value of $\Delta C_S$ to complete the parametrization. -This can be straightforwardly obtained -from (\ref{eq:dcldt_inhom}), which links the net change of liquid cloudy volume -due to the injection, $\Delta C_{injection}$, with $\Delta C_S$. - -\begin{equation} -\Delta C_{injection} = (g_l - C_l) \Delta C_S -\end{equation} - -where $g_l$ is 1 if the injected cloud is of liquid phase and 0 if it -is of ice phase. We already know $\Delta C_{injection}$ from the -injection forcing arguments (\ref{eq:dctdt_xl}) above that link it to $Q4$. -We therefore complete the parametrization by calculating $\Delta C_S$ based -on whether $\Delta C$ is positive or negative. If $\Delta C$ is positive, -we assume that the plume must be of liquid phase and hence - -\begin{equation} -\Delta C_S = \frac{\Delta C_{injection}} {1 -C_l} . -\label{eqn:cs1} -\end{equation} - -If $\Delta C_l$ is negative, we assume that the plume must be of ice phase and hence - -\begin{equation} -\Delta C_S = - \frac{\Delta C_{injection}} {C_l} . -\label{eqn:cs2} -\end{equation} - -Here we have still assumed that the vapour content in the detrained plume is equal to -$q_{sat liq}$. A better assumption may be to replace the $q_{sat liq}$ term in -(\ref{eqn:dqcl}) with a $q_{sat}$ expression that depends on the volume fraction -of detrained condensate that is liquid phase, $g_l$. - -If $\Delta C_{injection}$ is zero, we assume that $\Delta C_S$ is 0 also. Equations -(\ref{eqn:dqcl}),(\ref{eqn:cs1}), and (\ref{eqn:cs2}) form the parametrization -for $\Delta \overline{q_{cl}}|_{convection}$. The -representation of $\Delta C_{convection}$ is similar in form to -$\Delta \overline{q_{cl}}|_{convection}$: - -\begin{equation} -\Delta C |_{convection} = \Delta C_{injection} -+ \Delta t ~ a_L G(-Q_c) ( ( Q2 - \alpha Q1) - \Delta C_S (q_{sat}(\overline{T}) -- \overline{q} ) ) . -\end{equation} - -where the specification of $G(-Q_c)$ follows (\ref{eqn22}). Note -that the code includes the numerical limit restriction that -$\Delta C_S$ is between 0 and 1. - -Thus we are able to parametrize the net condensation and cloud changes -associated with the $Q1$ and $Q2$ terms in a physically more consistent way than -using simple homogeneous application of these terms. - -As an aside, we note that in the \cite{t93} scheme the condensation and cloud -fraction change associated with the compensating subsidence is taken out of -the convection term by adding the vertical motion associated with the compensating -subsidence to the large-scale vertical velocity before the -\cite{t93} equivalent of the homogeneous forcing term is applied. By doing -so it ensures that any balance between these two terms (as the tropical circulation -is commonly analysed to show) is removed before the net effect is calculated, -leading to more accurate numerical behaviour. - -\subsubsection{Homogeneous forcing of the environment by -convective-subsidence pressure change} - -To this end, the code includes an option to perform the homogeneous forcing -of liquid cloud by convection using the ``pressure forcing'' from the -convective subsidence, consistent with the pressure forcing by large-scale -advection (see sections \ref{sec:advec} and \ref{sec:pres}). -This approach replaces the above method of homogeneous forcing by convection -if the UM namelist switch \textbf{l\_pc2\_homog\_conv\_pressure} is turned on. -By applying the same homogeneous forcing method for advection and -convectively-forced subsidence, we should get the correct zero net -change in liquid cloud in the common situation where the large-scale ascent -and convective subsidence are in balance (implying no net vertical displacement -of environment parcels). - -Under this option, the increments to $\overline{q_{cl}}$ and $C_l$ produced -by the convection scheme are assumed to already include the effects -of entrainment, detrainment (i.e. injection) and compensating subsidence -(i.e. vertical advection) as expressed by equation \ref{eq:chimassflux}, -but exclude the effects of homogeneous forcing of clouds in the enviroment. -Note that taking equation \ref{eq:chimassflux} with $\chi$ set to -water vapour $q$, detrainment of saturated air into a subsaturated -environment will imply a positive tendency of $\overline{q}$, -but this is \textit{not} a homogeneous forcing, since the increase -in $\overline{q}$ is entirely due to injecting new parcels of saturated -air without altering the existing environment parcels. -Setting $\chi$ to be $\overline{q_{cl}}$ or $C_l$ in equation -\ref{eq:chimassflux}, there is a simply-calculated source of cloud -water and fraction wherever the detrained air is cloudy -($C_l=1$ in the detrained parcel), and we assume -these terms have been calculated this way inside the convection scheme. - -Since entrainment and detrainment do not constitute a homgeneous forcing -and are already accounted for in the convection scheme, - the only component of the convective forcing of liquid cloud -that needs to be done by the PC2 call after convection is the homogeneous -forcing by the subsidence term. This is in essence a vertical advection -(environmental forced descent by updrafts, or forced ascent by downdrafts). -The homogeneous forcing can be calculated from the expected pressure change -(and accompaying adiabatic temperature change) following the environment -as it is vertically displaced. -Conveniently, the UM already holds the convective mass-flux in units -of Pa s$^{-1}$, so it already expresses the pressure vertical velocity forced -by subsidence in the environment: - -\begin{equation} -\Delta p^E = \Delta t \left( M_{up} - M_{dwn} \right) -\label{eq:delta_p_conv} \end{equation} - -where $M_{up}$ is the updraft mass-flux, $M_{dwn}$ is the downdraft mass-flux, -and $\Delta t$ is the model timestep length. -The adiabatic temperature change following an environment parcel -subsided from pressure $p - \Delta p^E$ to $p$ is then given by: - -\begin{equation} -\Delta T^E = \theta^E \left( \left(\frac{p}{p_{ref}}\right)^\kappa - - \left(\frac{p - \Delta p^E}{p_{ref}}\right)^\kappa - \right) -\label{eq:delta_t_conv} \end{equation} - -where $\theta^E$ is the environment potential temperature, -$p_{ref}$ is the reference pressure used to define potential temperature, -and $\kappa = \frac{R_d}{c_p}$ is the ratio of the gas constant for dry air -over its heat capacity at constant pressure. -\ref{eq:delta_p_conv} and \ref{eq:delta_t_conv} are passed into the -PC2 homogeneous forcing routine after convection as the forcings -to be applied (with the forcings to all other variables set to zero). - - -\subsubsection{Convective cloud amount} -It is a debatable point whether the convective cloud fraction should -be set to zero. Although this was one of the original key concepts of -PC2, the cloud that is detrained from the convection scheme is into -the \textit{environment}, and does not represent the tower cloud. However, -it should be able to represent recently detrained cloudy air in a more -accurate way than by simply appealing to a diagnostic large-scale cloud scheme. -There are similar issues associated with the cloud fraction predicted -from the Tiedtke scheme. Probably the most consistent interpretation -is the inclusion of a tower cloud fraction within PC2, but not an -anvil cloud. However, we need to consider carefully any double -counting (or non-counting) implications. In the PC2:64 formulation, -we can represent the large optical depths associated -with new anvils, although we also tend to overestimate the optical -depth of shallow convective clouds. -Hence we choose to apply neither a diagnostic anvil or tower cloud, -so similar to Tiedtke, and let the large-scale cloud fraction represent -the convection completely. - -Strictly speaking these choices are independent of the PC2 scheme, -being simply choices that are available as part of the existing convection -scheme, but they are clearly directly related to the rest of the -cloud scheme formulation. - -\subsubsection{CAPE scaling} -The CAPE scaling option in the mass-flux convection scheme scales its -increments by the calculated values of $\frac{1}{CAPE} \frac{dCAPE}{dt}$. -This applies -also to all the PC2 calculated condensate and cloud fraction increments. -Additionally, in order to achieve reasonable mass flux profiles, -it has proved necessary to adjust the calculation of -$\frac{dCAPE}{dt}$ to use increments of -$\Delta \theta$ (potential -temperature) and $\Delta q$ calculated using a non-PC2 calculation -of these terms. Hence we consider any detrained condensate to have been -evaporated when we calculate $\frac{dCAPE}{dt}$. - -\subsubsection{Convective precipitation} -The amount of condensate detrained from convective plumes, and hence -the amount of moisture in the upper levels of the atmosphere, is very -dependent upon the amount of convective precipitation that is allowed -to fall from the column. The standard parametrization of this is that -any condensate greater than a specified value (dependent on $T$) -is precipitated, leaving the rest to be detrained. - -PC2 incorporates a tuning to this function of temperature by applying -the additional restriction that the limit may not fall to less than -$2 \times 10^{-4}~kg~kg^{-1}$. This implies a difference at temperatures -less than around $-42 ^{\circ} C$, with the tuning allowing less -precipitation and greater detrainment. This change is necessary -in order to produce thick enough anvil clouds. - -\subsubsection{Phase of condensate} -\label{sec:plume_phase} -The phase of the convective condensate \textit{carried in the plume} -is determined by a single phase change temperature TICE, with -condensate entirely in the -ice phase at colder temperatures and condensate entirely in the liquid -phase at warmer temperatures. For PC2:66, this temperature is -10 $^{\circ}$ C. - -\begin{equation} -\delta_{xl} = \left\{ \begin{array}{ll} - 1, & T_{plume} \ge -10 ^{\circ} C \\ - 0, & T_{plume} < -10 ^{\circ} C - \end{array} \right. -\end{equation} - -\begin{equation} -\delta_{xi} = \left\{ \begin{array}{ll} - 0, & T_{plume} \ge -10 ^{\circ} C \\ - 1, & T_{plume} < -10 ^{\circ} C - \end{array} \right. -\end{equation} - -\subsubsection{Tidier way of coupling convection and PC2} -\label{sec:conv-simpler} - -This area is still under development. But in brief, work is udner way to ensure that the -convective plume smoothly transitions from detraining liquid to detraining ice, rather -than using the abrupt change implied by the current formulation of the convection scheme. -Additionally, rather than using inhomogeneous increments to condensate (combining detrainment -and subsidence advection) to calculate cloud fraction increments, an alternative is to use -the detrainment of condensate to simply grow cloud fraction to ensure a specified in-cloud -liquid water content. The cloud fraction are then advected downwards byt he subsidence advection. - The increments to cloud fraction from detrainment and subsidence are then combined. - -\subsubsection{Prognostic dust approach} -A prognostic dust approach is implemented in the micro-physics scheme under -large-sale-precipitation where by the heterogeneous nucleation temperature -can be defined to vary three dimensionally globally as an arc-tangent -function of the mineral dust distribution in the model (documented -in \citeumdp{026}). By default, both liquid and ice are detrained simultaneously at the same -height, and the fraction of condensate that is ice linearly ramps as a function of temperature. -i.e. condensate is assumed to be all-liquid when T is greater than one tuneable threshold; all-ice -when T is less than another tuneable threshold, and vary linearly in-between (the threshold values are -given by starticeTkelvin and alliceTdegC in the UM cloud-scheme namelist. The new heterogeneous -nucleation temperatures calculated in the large-scale-precipitation are passed to the convection -scheme and are used as the above detrainment temperature thresholds by -maintaining a similar linear ramp. For e.g., condensate is -assumed to be all-liquid for T $\geq$ $tnuc_n$ and all-ice for T -$\leq$ $tnuc_n$ - 10.0 - -\subsubsection{Condensation adjustment in the profiles input to the -convection scheme} -\label{sec:conv_input_profs} - -The convection scheme itself is highly sensitive to the input environment -temperature and moisture profiles {\it before} the convection increments -(or PC2 response) are calculated. In particular, the parcel buoyancy -(and hence the CAPE and mass-flux scaling) maybe radically different -depending on whether a ``large-scale'' condensation / evaporation adjustment -is performed before the convection call. - -Where there is large-scale ascent, the profiles after Semi-Lagrangian advection -may have become supersaturated and unrealistically unstable, until the -expected condensation adjustment is performed. If the convection scheme -``sees'' these unrealistic intermediate profiles, it is likely to -predict an excessive, unrealistic mass-flux. - -To address this problem, there are two namelist switches that enable -additional condensation adjustments from PC2 before the convection call: - -\begin{itemize} -\item {\bf l\_pc2\_sl\_advection}: performs homogeneous forcing response -to Semi-Lagrangian advection immediately after the advection calculation, -instead of at the end of the timestep (see section \ref{sec:pres}). -\item {\bf l\_cloud\_call\_b4\_conv}: performs an additional call to -PC2 initiation (and PC2 checks) before the convection scheme -(see section \ref{sec:init2}). -This should catch any instances where large-scale ascent or other processes -have brought the profiles after advection to near or beyond saturation, -in grid-points where there was no liquid cloud already present -(and so no homogeneous forcing response). -\end{itemize} - -\subsection{Response to pressure changes} - \label{sec:pres} - -A pressure change following the parcel during the timestep will result -in an adiabatic temperature change which will force condensation, -hence we must include this temperature change forcing within PC2. -The majority of this pressure change comes from vertical advection -(although not all). -Remember that the advection (section \ref{sec:advec}), on its own, -does not cause condensation, it merely moves the existing cloud field. - - Using the semi-Lagrangian advection in the same way as is performed -for $\overline{q_{cl}}$ etc., the PC2 scheme will obtain the value of the -model prognostic \textit{Exner}, ($\prod$) on the departure points -($\prod_{dep}$). \textit{Exner} is defined as - -\begin{equation} -\label{eq:exner} -\prod = \frac{T}{\theta} = \left( \frac{p}{p_{ref}} \right)^{\kappa} -\end{equation} - -where $\theta$ is the potential temperature, $p_{ref}$ is a reference -pressure set to 1000 hPa, and $\kappa = -\frac{c_p - c_v}{c_p}$ , where $c_v$ is the heat capacity of dry -air at constant volume. The \textit{Exner} quantity is kept as a prognostic -variable in the model (this is unchanged from the control model), and the -value of $\prod$ on the departure points represents the initial value -in the timestep, since there is no update to $\prod$ until the end of -the timestep. After the second physics updates have been performed -(\textit{atmos-physics2}), the -model (including the control) recalculates the value of \textit{Exner} -($\prod^{[n+1]}$). -From $\prod_{dep}$ and $\prod^{[n+1]}$ we can calculate, using the definition -(\ref{eq:exner}), the values of departure pressure and temperature: - -\[ -\overline{p}_{dep} = p_{ref} {\prod_{dep}}^{\frac{1}{\kappa}} -\] - -\[ -\overline{T}_{dep} = \theta \prod_{dep} -\] - -Hence we obtain the net forcing values - -\begin{equation} -\Delta \overline{T} = \overline{T}^{[n+1]} - \overline{T}_{dep} -\label{eq:deltatsl} -\end{equation} - -and - -\begin{equation} -\Delta \overline{p} = \overline{p}^{[n+1]} - \overline{p}_{dep} . -\label{eq:deltapsl} -\end{equation} - -where $\overline{T}^{[n+1]}$ and $\overline{p}^{[n+1]}$ are the temperature -and pressure at the arrival point, after the dynamics call. -(\ref{eq:deltatsl}) and (\ref{eq:deltapsl}) are passed to the homogeneous -forcing routine in order to calculate -the condensation and cloud fraction changes associated with the pressure -change. - -We include this forcing towards the end of the timestep. There are two -reasons for this: -firstly, values of $\prod^{[n+1]}$ are not calculated by the control model -until after the physics is complete; secondly, it makes sense to locate this -process in the timestep in a similar location -to where the large-scale cloud scheme is included in the control (i.e. -after the implicit part of the boundary layer has finished). - -However, there -is a counter argument that says we should include this process immediately -after the dynamics, since we can then apply a forcing on an initial state -that has not already been modified by the dynamics, boundary layer and -convection schemes. This improves the numerics of the problem, since the -homogeneous forcing is designed to take time level n values as inputs. - -These issue are optionally addressed by turning on the UM namelist switch -\textbf{l\_pc2\_sl\_advection}. Under this switch, the PC2 homogeneous -forcing response to pressure change is split: -\begin{enumerate} -\item Forcing by the \textit{Lagrangian} component of pressure change, -performed immediately after the Semil-Lagrangian advection scheme -(before the call to atmos\_physics2). -This calculates the pressure change from the departure point value of -\textit{Exner} described above, to the start-of-timestep value of -\textit{Exner} at the arrival point. -\item Forcing by the \textit{Eulerian} component of pressure change, -performed at the end of the timestep (after the dynamics Helmholtz solver). -This calculates the pressure change from the start-of-timestep \textit{Exner} -at the arrival point, to the end-of-timestep \textit{Exner}. -\end{enumerate} - -Having to calculate the pressure forcing twice obviously adds some -computational cost, but has several advantages: -\begin{itemize} -\item As noted above, the PC2 homogeneous forcing calls can now take -as input the temperature and water-vapour content \textit{before} -the pressure change has been applied, as intended. This should improve -the numerical accuracy. -\item Most of the condensation or evaporation from the dynamics comes from -the \textit{Lagrangian} component of the pressure change, which has now moved -from the end of the timestep to before the dynamics Helmholtz solver. -This means that any latent heating from condensation forced by ascent is now -accounted for by the solver within the same timestep. -This improves the numerical accuracy of the dynamics-physics coupling. -\item If the condensation forced by resolved ascent is only added on at the -end of the timestep, the profiles passed into atmos\_physics2 can contain -out-of-balance thermodynamic states (e.g. if the profile has been lifted -by advection, it maybe supersaturated / unrealistically unstable before -the resulting condensation is added on). This may adversely affect the -convection scheme, which must act upon the profiles passed into -atmos\_physics2. -\end{itemize} - -The splitting of the pressure forcing call under the -\textbf{l\_pc2\_sl\_advection} switch was originally implemented to make the -profiles passed to convection more realistic. - -\subsection{Initiation} -\label{sec:init2} -As discussed in section \ref{sec:init}, there are occasions when -$\overline{q_{cl}}$ and $C_l$ need to be initiated from 0 or 1. -The application of the initiation is given in section \ref{sec:init}. -The initiation forms a new, separate block of PC2 code to perform this -calculation, and is located immediately following the pressure change -response (section \ref{sec:pres}). -Also, if the UM namelist switch {\bf l\_cloud\_call\_b4\_conv} is set to -true, an additional call to PC2 initiation is performed before the -convection scheme, to ensure that the condensation response to -advection and other forcings earlier in the timestep has been accounted for -in the profiles passed to the convection scheme, even if there was no -cloud already present for homogeneous forcing to act upon. -(see section \ref{sec:conv_input_profs}). - -There are currently 3 options for the conditions under-which initiation -may occur. For all of these options, -if using the bimodal cloud scheme to do initiation within PC2, -then the tests on $RH_T$ relative to $RH_{crit}$ are replaced by equivalent -tests for whether the saturation boundary lies within the bounds -of the bimodal scheme's assumed PDF, as described in section -\ref{sec:bimodal_init}. - -\subsubsection{``Original'' initiation logic} - -This option is selected by setting the UM namelist switch -{\bf i\_pc2\_init\_logic = 1} (Original) - -The initiation will be called if the liquid cloud fraction is either -0 or 1 and appropriate $RH$ criteria hold, along with other restrictions. -$C_l$ is initiated away from 0 if - -\begin{itemize} -\item{ $RH_T > RH_{crit} + RH_{crit \, tol}$ \textbf{and} } -\item{ Cumulus convection has {\em not} been diagnosed from the - boundary-layer in the current column \textbf{and} } -\item{ The current level is not below the surface mixed-layer LCL \textbf{and} } -\item{ $C_l = 0$ \textbf{and} } -\item{ $RH_T^{[n+1]} > RH_T^{[n]}$ ,} -\end{itemize} - -where $RH_{crit \, tol}$ is a specified tolerance parameter, of value 0.01, -and $RH_T$ is defined in (\ref{eq:rht}). $RH_T^{[n]}$ is the start of -timestep value of $RH_T$ (i.e. at time level n) and $RH_T^{[n+1]}$ is the -value when initiation is called. -Additionally, there is another possibility for the last of the relations. -This second option also allows initiation when the water -is supercooled: - -\begin{itemize} -\item{ $C_l < 0.05$ \textit{and} $\overline{T} < 0 ^{\circ} C$ .} -\end{itemize} - -Equivalently, $C_l$ is initiated away from 1 if - -\begin{itemize} -\item{ $RH_T < 2 - RH_{crit} - RH_{crit \, tol}$ \textbf{and} } -\item{ $C_l = 1$ \textbf{and}} -\item{ $RH_T^{[n+1]} < RH_T^{[n]}$ .} -\end{itemize} - -\subsubsection{``Simplified'' initiation logic} - -This option is selected by setting the UM namelist switch -{\bf i\_pc2\_init\_logic = 2} (Simplified) - -Under this option, the conditions for initiation are: - -Either: -\begin{itemize} -\item $RH_T > RH_{crit} + RH_{crit \, tol}$ \textbf{and} -\item $C_l < C_{tol}$ \textbf{and} -\item The current level is not below the surface mixed-layer LCL \textbf{and} -\item $RH_T^{[n+1]} > RH_T^{[n]}$ -\end{itemize} -Or: -\begin{itemize} -\item $RH_T < 2 - RH_{crit} - RH_{crit \, tol}$ \textbf{and} -\item $C_l > 1 - C_{tol}$ \textbf{and} -\item $RH_T^{[n+1]} < RH_T^{[n]}$ -\end{itemize} - -where $C_{tol}$ can be set via the UM namelist; its original standard value -is 0.005. Note this threshold is also used to remove small cloud-fractions -after initiation; see section \ref{sec:checks2}. - -This is very similar to the ``Original'' initiation logic described above, -but with the following differences: -\begin{itemize} -\item The condition that the boundary-layer hasn't diagnosed cumulus - convection in the column is removed. - Note that this condition spuriously suppressed initiation in the free - troposphere {\em above} any cumulus cloud produced by the convection - scheme. -\item $C_l$ only needs to be within a numerical tolerance $C_{tol}$ from - 0 or 1, rather than having to be {\it exactly} 0 or 1. -\item The different threshold when initiating super-cooled cloud is removed. -\end{itemize} - -\subsubsection{``Smooth'' initiation logic} -\label{sec:smooth_initiation} - -This option is selected by setting the UM namelist switch -{\bf i\_pc2\_init\_logic = 3} (Smooth) - -There is a fundamental numerical problem with the above options, in that -the initiation process is not permitted to have any effect at all unless -$C_l$ goes to (near) 0 or 1, but can predict values of $C_l$ very different -to 0 or 1 when it does activate. This leads to unphysical sudden noisy jumps -in $C_l$ and $q_{cl}$ when initiation occurs. -For example, if erosion causes $C_l$ to steadily decline, it will continue -to decline (even when the grid-mean $RH_T$ exceeds $RH_{crit}$) until -it reaches the threshold (0 or $C_{tol}$). At this point, initiation suddenly -increases $C_l$ and $q_{cl}$ to the values predicted by the diagnostic cloud -scheme. Erosion may then gradually remove them again, and the cycle repeats. -There is no physical reason for this internal mode of variability in the -scheme. - -Another problem arises if we consider the sensitivity to model resolution. -Suppose we have many adjacent small grid-boxes with similar $RH_T$, -a few containing cloud, the rest containing no cloud. If the whole -region cools to the point where $RH_T > RH_{crit}$, then new cloud -will initiate in the cloud-free grid-boxes, but not in the cloudy grid-boxes. -Now suppose we run a coarse-grained version of the same simulation; -the many small grid-boxes are replaced by a single grid-box containing the -average $C_l$ over the small grid-boxes. Since we now have just one -grid-box already containing partial cloud-cover, initiation of new cloud -can no longer occur anywhere. - -To address these problems, there is an option to use a much simpler / -numerically better-posed initiation method; -always allow the diagnostic cloud scheme to be called -(provided it is expected to predict nonzero cloud water, -i.e. $RH_T > RH_{crit}$ in the case of the Smith scheme). -The $q_{cl}$ predicted by the diagnostic cloud scheme is then taken -as a minimum limit applied to the prognostic $q_{cl}$. -This amounts to taking the diagnostic cloud scheme's assumed PDF as a minimum -allowed width to the actual prognostic moisture PDF. -The prognostic $C_l$ and $q_{cl}$ are incremented as follows: - -\begin{itemize} - -\item If ${q_{cl}}_{diag} > q_{cl}$: - -$\Delta q_{cl} = {q_{cl}}_{diag} - q_{cl} -\quad \refstepcounter{equation}(\theequation)\label{eq:dqcl_init}$ - - - \begin{itemize} - - \item If $Q_C < 0$: - - $\Delta C_{l} = \frac{\Delta q_{cl}}{{q_{cl}}_{diag}} - \left( {C_{l}}_{diag} - C_{l} \right) - \quad \refstepcounter{equation}(\theequation)\label{eq:dcl_init1}$ - - \item If $Q_C > 0$: - - $\Delta C_{l} = \frac{\Delta SD}{{SD}_{diag}} - \left( {C_{l}}_{diag} - C_{l} \right) - \quad \refstepcounter{equation}(\theequation)\label{eq:dcl_init2}$ - - \end{itemize} - -\item Otherwise: - -$\Delta q_{cl} = 0$ - -$\Delta C_{l} = 0$ - -\end{itemize} - -where the subscript $_{diag}$ denotes the liquid cloud water content and -fraction predicted by the diagnostic cloud scheme (either Smith or Bimodal). - -Equation \ref{eq:dcl_init1} simply sets the cloud-fraction to a weighted -mean of the pre-existing and diagnostic-scheme cloud-fractions, in proportion -to the fraction of the water content that was created by initiation -versus that which was already there. -If the pre-existing $q_{cl}$ is zero, \ref{eq:dqcl_init} and \ref{eq:dcl_init1} -simply set $q_{cl}$ and $C_l$ to their new diagnosed values, -as in the previous options. -Crucially, in the limit that the pre-existing $q_{cl}$ approaches -${q_{cl}}_{diag}$, the increments to $q_{cl}$ and $C_l$ smoothly go to zero. -This is important to make the initiation process numerically well-posed, -so that it yields a smooth, continuous solution. - -Note that when we are initiating from $C_l = 1$ instead of $C_l = 0$, -we expect the pre-existing $q_{cl}$ to be nonzero even when there is -no pre-existing sub-grid PDF width. In this case, the completely -uninitiated state will have zero saturation deficit $SD$, rather than -zero $q_{cl}$. Therefore, in this case the increment to $C_l$ is calculated -based on the fractional increase in $SD$ from initiation -(equation \ref{eq:dcl_init2}), instead of the fractional increase in $q_{cl}$. - -Whether to increment $C_l$ based on the increase in $q_{cl}$ or $SD$ is -determined based on the sign of $Q_C$, which is defined as in equation -\ref{eq:qc_eq_qt-qs} (reproduced here for clarity): - -\[ -Q_c = a_L \left( \overline{q_T} - q_{sat}(\overline{T_L}) \right) -\] - -The saturation deficit $SD$ is defined by equation \ref{SD2}: - -\[ -SD = a_L \left( q_{sat}(\overline{T}) - \overline{q} \right) -\] - -Under the reasonable approximation that $q_{sat}$ varies linearly between -$\overline{T}$ and $\overline{T_L}$, so that the values of -$\alpha$ and $a_L$ are the same in -both of these equations, and: - -\[ -q_{sat}(\overline{T_L}) = q_{sat}(\overline{T}) - \alpha \frac{L}{c_p} q_{cl} -\] - -we obtain: - -\begin{equation} -q_{cl} = Q_c + SD -\label{eq:qc_plus_sd} -\end{equation} - -It can be seen that when $Q_C > 0$ (total-water super-saturation), -it represents the value $q_{cl}$ would have if the whole grid-box -were saturated ($SD = 0$, $C_l = 1$). Note that $q_{cl}$ cannot fall below -$Q_C$, since $SD$ cannot be negative. -Since $Q_c$ is invariant under condensation / evaporation, we must have -$\Delta SD = \Delta q_{cl}$ -(hence the implementation of \ref{eq:dcl_init2} in the code simply uses -$q_{cl} - Q_c$ in place of $SD$, and $\Delta q_{cl}$ in place of $\Delta SD$). - -\subsubsection{Additional checks after PC2 initiation} - \label{sec:checks2} - -The initiation is followed immediately by a section of resetting code. -For numerical reasons, it is possible to obtain very low, but non zero, -values of $C_l$ (and equivalently values very close to, but not equal to, -1). The code will reset these clouds to either a fraction of 0 or 1, as -appropriate. We choose to apply these terms here and not in the -Bounds Checking part of the code (section \ref{sec:checks}) because -these are not required to obtain consistency between fields, but are -`tidying up' pieces of code, although they may reasonably also be -applied in the Bounds Checking. Care needs to be taken when choosing -the thresholds, since -we do not wish to reset small values that are genuinely created -by a physics scheme in the model. - -We first calculate $RH_T$ using (\ref{eq:rht}) and compare -this to the critical relative humidity, $RH_{crit}$. The liquid -cloud fraction will be reset to 1 if: -\begin{itemize} -\item{ $RH_T > 2 - RH_{crit}$ and $C_l \ge C_{high}$} -\item{ or $C_l \ge C_{high 2}$ } -\end{itemize} -where $C_{high}$ and $C_{high 2}$ are defined in \ref{eq:chigh-chigh2}. -The evaporation is done by calculating $SD$ using (\ref{SD2}) with (\ref{eq:a_L}) and -(\ref{eq:alpha_exp}) and evaporating the equivalent amount of liquid -into the gridbox to take it to saturation, according to -(\ref{eq:qsdcheck1}) below. - -Similarly, the equivalent check for low values of $RH_T$ is performed. -The liquid -cloud fraction will be reset to 0 if: -\begin{itemize} -\item{ $RH_T < RH_{crit}$ and $C_l \le C_{low}$} -\item{ or $C_l \le C_{low 2}$ .} -\end{itemize} -The remaining $\overline{q_{cl}}$ is evaporated into the gridbox -using (\ref{eq:qclcheck}) below. - -The thresholds $C_{high}$, $C_{high 2}$, $C_{low}$ and $C_{low 2}$ are -set using the parameters $C_{tol}$ and $C_{tol 2}$, according to: - -\begin{eqnarray} -C_{high} = 1 - C_{tol}, \nonumber \\ -C_{high 2} = 1 - C_{tol 2}, \nonumber \\ -C_{low} = C_{tol}, \nonumber \\ -C_{low 2} = C_{tol 2}, -\label{eq:chigh-chigh2} -\end{eqnarray} - -where the parameters $C_{tol}$ and $C_{tol 2}$ can be set via the UM namelist -variables {\bf cloud\_pc2\_tol} and {\bf cloud\_pc2\_tol\_2}. -The original standard values of these parameters are -$C_{tol} = 0.005$ and a lower value $C_{tol 2} = 0.001$. - -Investigations in SCM runs using the comorph convection scheme -(which behaves more smoothly and so typically gives smaller increments -to $C_l$ over a single timestep than other schemes which exhibit intermittent -behaviour) suggested these thresholds are too high to avoid spuriously -resetting physical values of $C_l$ to zero. Detrainment from sparse -shallow cumulus, or advection of cloud into a neighbouring grid-box -under light winds, commonly give increments which increase $C_l$ from zero -to a value less than $0.005$ in one timestep (but would eventually increase -$C_l$ to a significant value over subsequent timesteps if the checks did -not keep resetting $C_l$ to zero). - -Note that if these checks are relaxed by lowering the thresholds -$C_{tol}$ and $C_{tol 2}$ to near-zero, -similar checks are still performed independently by the bounds checking -described in section \ref{sec:checks}, but with a much lower -threshold of $C_{tol 3} = 1 \times 10^{-12}$. - -\subsection{Bounds checking} - \label{sec:checks} -Ideally, model prognostics would never become inconsistent with one another. -However, even although the mathematical solution of the governing equations -may be well behaved, due to numerical inaccuracies values may become -inconsistent. For the cloud and condensate quantities, there are a number -of consistencies that must apply. The bounds checking forms a subroutine -that will, if necessary, adjust $\overline{q}$, $\overline{q_{cl}}$, -$\overline{q_{cf}}$, $C_l$, $C_i$, $C_t$ and, for latent heating, -$\overline{T}$, to ensure consistency between these values. - -The bounds checking is performed three times during the timestep. Firstly, -after the parallel part of the physics (\textit{atmos-physics1}) is complete; -secondly, before the initiation (section \ref{sec:init}) is called; thirdly, -after the initiation is called. - -\subsubsection{} -Firstly, if $C_l > 1 - C_{tol 3}$ then $C_l$ is set to 1. -Accordingly, $C_t$ is set to 1 as well. -$C_{tol 3}$ is a tiny numerical tolerance set to $1 \times 10^{-12}$, -a value intended to be in the realm of floating point rounding error rather -than anything that represents a physical solution. - -\subsubsection{} -The second check is to reset $\overline{C_l}$ to zero. This may be performed -for two reasons. Firstly, if the amount of $\overline{q_{cl}}$ is very small -($\overline{q_{cl}} < q_{c0}$, where $q_{c0} = 1 \times 10^{-10} kg kg^{-1}$), -so we avoid carrying negligible, but non-zero values of $\overline{q_{cl}}$ -and $C_l$. Secondly, if $C_l < C_{tol 3}$ then we reasonably reset $C_l$ to zero. -$C_t$ gets reset, as it must if there is no liquid cloud, to be equal to $C_i$. - -\subsubsection{} -\label{sec:pc2_checks_sd} -The next check complements the first but updates the moisture fields. -We firstly calculate $SD$ using (\ref{SD2}) and -(\ref{eq:alpha_exp}). We then check whether $SD < 0$. -This check catches instances where we have grid-mean supersaturation, -which ought to be impossible (under the instantaneous condensation -assumption made by PC2, condensation should occur to instantly adjust -any supersaturated regions of the gridbox to saturation, so we -{\it must always} have $SD \ge 0$. -When this condition is violated, we condense water vapour to adjust to -grid-mean saturation. $-SD$ corresponds to the amount of vapour that must be -condensed to achieve this, so we have: - -\begin{eqnarray} -\overline{q} \leftarrow \overline{q} + SD \nonumber \\ -\overline{q_{cl}} \leftarrow \overline{q_{cl}} - SD \nonumber \\ -\overline{T} \leftarrow \overline{T} - \frac{L_c}{c_p} SD -\label{eq:qsdcheck1} -\end{eqnarray} - -The original version of this check on $SD$ -(which may increase $\overline{q_{cl}}$), made no accompanying changes to -liquid cloud fraction. However, increases in $\overline{q_{cl}}$ -without any increase in $C_l$ can lead to spurious high in-cloud condensate -which is then converted to rain by the microphysics at the next time-step. -There are currently 4 options for how to treat $C_l$ when increasing -$\overline{q_{cl}}$ under this saturation adjustment, selected by the -UM large-scale cloud namelist switch {\bf i\_pc2\_checks\_cld\_frac\_method}: -\begin{itemize} -\item {\bf i\_pc2\_checks\_cld\_frac\_method = 0} - -Original method; $C_l$ is left unaltered. -\item {\bf i\_pc2\_checks\_cld\_frac\_method = 1} - -Set $C_l$ and $C_t$ to 1. -\item {\bf i\_pc2\_checks\_cld\_frac\_method = 2} - -If $\overline{q_{cl}}$ and $C_l$ were already nonzero before the adjustment, -increase $C_l$ at the same fractional rate as $\overline{q_{cl}}$, so that -the in-cloud water content $\frac{\overline{q_{cl}}}{C_l}$ is conserved. -Otherwise, increase $C_l$ so-as to yield a prescribed in-cloud water -content set to 0.5 g kg$^{-1}$. $C_t$ is then increased by the same -amount as $C_l$, to maintain consistency. -\item {\bf i\_pc2\_checks\_cld\_frac\_method = 3} - -This is the same as option 2 above, except in the case where -$\overline{q_{cl}}$ or $C_l$ was zero before the adjustment. In this case, -$C_l$ is set based on an empirical power-law function of $\overline{q_{cl}}$. -\end{itemize} - -\subsubsection{} - -Next we check whether $SD > 0$, {\it and} $C_l = 1$ -(the first of our checks has ensured that $C_l$ is no greater than 1). -This check catches instances where we have total cloud-cover in a subsaturated -grid-box, which ought to be impossible (if the whole grid-box is full of liquid -cloud, then it must be at grid-mean saturation, i.e. $SD = 0$). -When this happens, we adjust $\overline{q}$ and $\overline{q_{cl}}$ -to take $SD$ to zero, -\textit{provided} that $\overline{q_{cl}} > SD$. Remember that $SD$ -corresponds to the amount of vapour that must be -\textit{evaporated} into the gridbox to give saturation, so we simply -make exactly the same adjustments as we do for removing supersaturated -states above (\ref{eq:qsdcheck1}), except that here $SD$ is positive rather -than negative. - -Our proviso that $\overline{q_{cl}} > SD$ ensures that we do not make -$\overline{q_{cl}}$ negative by this adjustment. -If $\overline{q_{cl}} < SD$, then we cannot bring the gridbox to saturation, -but it is still wrong to allow $C_l = 1$ in a subsaturated gridbox! -This was identified as a bug in the bounds-checking code, which sometimes -caused instances of $C_l = 1$ to spuriously persist in dry environments. -This behaviour is currently controlled by a temporary logical in the -{\bf temp\_fixes} namelist: -\begin{itemize} -\item If {\bf l\_pc2\_checks\_sdfix} is set to false, the code simply does -nothing when it finds instances of $C_l = 1$, $SD > 0$ and -$SD > \overline{q_{cl}}$, allowing such artefacts to persist. -\item If {\bf l\_pc2\_checks\_sdfix} is set to true, in these instances -we simply evaporate all the remaining liquid water, and reset $C_l$ to zero: -\begin{eqnarray} -\overline{q} \leftarrow \overline{q} + \overline{q_{cl}} \nonumber \\ -\overline{T} \leftarrow \overline{T} - \frac{L_c}{c_p} \overline{q_{cl}} -\nonumber \\ -\overline{q_{cl}} \leftarrow 0 \nonumber \\ -C_l \leftarrow 0\nonumber \\ -C_t \leftarrow C_i -\label{eq:qsdcheck2} -\end{eqnarray} -\end{itemize} - -\subsubsection{} -The next check is similar to above but for the $C_l = 0$ situation. - -If $\overline{q_{cl}} < q_{c0}$ or $C_l = 0$ then we evaporate the -small amount of $\overline{q_{cl}}$ that remains in the gridbox: - -\begin{eqnarray} -\overline{q} \leftarrow \overline{q} + \overline{q_{cl}} \nonumber \\ -\overline{q_{cl}} \leftarrow 0 \nonumber \\ -\overline{T} \leftarrow \overline{T} - \frac{L_c}{c_p} \overline{q_{cl}} -\label{eq:qclcheck} -\end{eqnarray} - -\subsubsection{} -Next, if $C_i > 1$ then $C_i$ is set to 1. Accordingly, $C_t$ is set to -1 as well. - -\subsubsection{} -The following check is on the ice water content, $\overline{q_{cf}}$, and -the ice fraction $C_i$. If $\overline{q_{cf}} < q_{c0}$ we simply condense some -vapour to remove the negative quantity. - -\subsubsection{} -However, instead of removing small amounts of -$\overline{q_{cf}}$ when $C_i = 0$ but $\overline{q_{cf}} > 0$, we choose instead to create -some $C_i$ to keep consistency. This is to allow small, but significant, -amounts of $\overline{q_{cf}}$ created by the microphysics scheme to -be maintained. - -\begin{equation} -C_i \leftarrow \frac { \overline{q_{cf}} }{q_{cf0}} -\label{eq:cf_reset} -\end{equation} - -where the `in-cloud' ice content $q_{cf0} = 1 \times 10^{-4} kg kg^{-1}$. - -\subsubsection{} -The next two checks are on the total cloud fraction, $C_t$, to ensure -that it takes on a value that is physically possible, given the values -of $C_l$ and $C_i$. We have, firstly, the maximum overlap situation and -then the minimum overlap situation. - -\begin{eqnarray} -C_t \leftarrow \text{Max}( C_t, C_i, C_l ) \nonumber \\ -C_t \leftarrow \text{Min}( C_t , C_l + C_i, 1) -\label{eq:ctchecks} -\end{eqnarray} - -\subsubsection{} -Finally, there is a homogeneous nucleation term applied, similar -to that in the large-scale precipitation (section \ref{sec:lsp_homo}). This is -a fast microphysics process, and must act to ensure that no liquid cloud -created by the initiation is allowed to persist in this phase if the -temperature is cold enough. Hence, if $\overline{T} < T_{homo}$ then - -\begin{eqnarray} -\overline{q_{cf}} \leftarrow \overline{q_{cf}} + \overline{q_{cl}} \nonumber \\ -\overline{q_{cl}} \leftarrow 0 \nonumber \\ -\overline{T} \leftarrow \overline{T} + \frac{L_f}{c_p} \overline{q_{cl}} \nonumber \\ -C_i \leftarrow C_t \nonumber \\ -C_l \leftarrow 0. -\label{eq:homochecks} -\end{eqnarray} - -\subsubsection{Qpos checks} -\label{sec:qpos} - -The implementation of the PC2 code includes an additional bounds check after -the \textit{atmos-physics-2} part of the model timestep has been completed. This -check is necessary to trap a rare failure, and uses the \textit{Qpos} subroutines -to check that $\overline{q_{cl}}$ is greater or equal to 0. - -During trialling prior to operational implementation, it was found that relying on Q-Pos -to deal with negative condensate values was very expensive, as the Q-Pos routine does a lot of communications between -different processors. It may be preferable to deal with the cause of negative condensate amounts at their source. -The option to ``Ensure consistent sinks of qcl and CFL'' -prevents the QCL increment from -trying to remove too much liquid condensate and hence reduces the models reliance on Q-Pos to -deal with the inconsistencies. - -\subsection{Data Assimilation} -\label{sec:da} - -The data assimilation section in the model will output assimilation increments -that represent changes to $\overline{q}$ and $\overline{T}$ which -\textit{include} the condensation contributions. We hence need to calculate -equivalent increments to $\overline{q_{cl}}$, $C_l$ and $C_t$. We assume -that the assimilation has not calculated these using a different method. -We consider the homogeneous framework and assume that there is a forcing -value of $Q_c$ that exists that will produce the known increment to -$\overline{q}$ and $\overline{T}$. - -Discritising (\ref{dqcldt}) we have, using (\ref{eq:deltaqc_exp}) and -expanding $\Delta T_L$ in terms of $\Delta T$ and $\Delta q_{cl}$, - -\begin{equation} -\Delta \overline{q_{cl}} = C_l ( a_L ( \Delta \overline{q} - -\alpha \Delta \overline{T} - \beta \Delta \overline{p}) + \Delta \overline{q_{cl}} ). -\label{eq:da1} -\end{equation} - -Remember that $Q_c$ (and hence $\Delta Q_c$) is independent of condensation. -Rearranging, we obtain - -\begin{equation} -\Delta \overline{q_{cl}} = \frac{1}{1 - C_l} C_l -a_L ( \Delta \overline{q} - \alpha \Delta \overline{T} - \beta \Delta \overline{p}) -\label{eq:da2} -\end{equation} - -and hence an expression for the condensate increment, -$\Delta \overline{q_{cl}}$, that accompanies the known increments -to $\overline{q}$ and $\overline{T}$. The similar analysis, from -(\ref{dcdt}) and (\ref{eq:da1}) gives - -\begin{equation} -\Delta C_l = \frac{1}{1 - C_l} G(-Q_c) - a_L ( \Delta \overline{q} - \alpha \Delta \overline{T} -- \beta \Delta \overline{p} ) . -\label{eq:da3} -\end{equation} - -Hence the equation set is equivalent to the use of the homogeneous -forcing set, except for the multiplier $\frac{1}{1 - C_l}$. Although this -is a clean solution, we -need to be very careful with the ill-conditioning of this solution -near $C_l = 1$. - -In practice, the ill-conditioning of (\ref{eq:da2}) and (\ref{eq:da3}) becomes too -numerically awkward for us to apply the full solution based on homogeneous -forcing, although, for completeness, we outline it in Appendix -\ref{sec:appendix-da}. Hence we have -chosen to apply a much simpler model. Here we use simply the data assimilation -increments $\Delta \overline{q}$ and $\Delta \overline{T}$ within the standard -homogeneous forcing (section \ref{sec:homog}), even though we are fully -aware that this is inconsistent (because $\Delta \overline{q}$ and $\Delta -\overline{T}$ are not forcings, but are forcings plus the condensation. -This allows us an \textit{estimate} of $\Delta \overline{q_{cl}}$ and $\Delta{C_l}$, -via the homogeneous forcing routine (and $\Delta C_t$ via the standard -updating described in section \ref{sec:ct}). These are the quantities applied -as the equivalent data assimilation increments for $\Delta \overline{q_{cl}}$, -$\Delta{C_l}$ and $\Delta C_t$. The increments $\Delta \overline{q}$ and -$\Delta \overline{T}$ remain those that the data assimilation scheme itself -calculated. - -Appendix \ref{sec:appendix-da} gives, for completeness, the alternative -numerical technique for the solution of (\ref{eq:da2}) and (\ref{eq:da3}). -However, we stress that this technique is not used within the current -PC2 formulation. - -\section{Implementation in the Unified Model} -\label{sec:um} - -This section considers the implementation of PC2 within the Unified Model -code and provides a brief guide to its use. - -In general, we have written PC2 so that the timestepping of the -cloud fraction variables within the \textit{atm\_step\_4a} -subroutine is treated as much as possible in a similar way to -the condensate variables. -Hence, wherever the condensed water variables $q_{cl}$ and $q_{cf}$ -are updated, the cloud fractions need to be updated consistently. - -\subsection{Area cloud fraction} -\label{sec:acf} - -Two area cloud fraction parametrizations are available for use with PC2. - -The area cloud fraction of Cusack (documented in \citeumdp{029}) has been -adapted by \cite{boutle_morcrette10} so it can be used with PC2 (and is available from the UMUI as the -``Cusack'' option from version 7.6 onwards). This method aims to -reproduce some of the detail of the thermodynamic -profile lost due to the coarseness of the grid. The interpolation/extrapolation technique is -used prior to PC2 initiation (which is then called with three times as many levels) -and it is used, along with the homogeneous forcing idea at the start of the timestep to -allow more cloud to be seen by radiation. - -The diagnostic area cloud fraction of \cite{bhi05} -has also been implemented in the model (available from the UMUI at version 6.4 onwards), -and this is used in PC2:64. This method diagnoses the area cloud fraction -given the volume cloud fraction, taking into account the size of the grid -box. The setting of the area cloud fraction is performed at the end of the timestep. - -\subsection{Code Structure} -\label{sec:code} - -A detailed description of the UM's timestep structure, -showing where in the model all the PC2 cloud scheme subroutine calls are made, -is given in the subsections below. - -Note that there are three different subroutines that all do -the PC2 homogeneous forcing, with slightly different details: - -\begin{itemize} -\item {\bf{\it pc2\_delta\_hom\_turb}} outputs increments due to the -condensation or evaporation, but doesn't update the fields themselves. -\item {\bf{\it pc2\_homog\_plus\_turb}} just updates the fields that -are passed in, instead of outputting separate increment arrays. -\item {\bf{\it pc2\_hom\_conv}} outputs increments but includes additional -calculations for various cloud erosion formulations. -\end{itemize} - -Note that code exists in the first two of these routines to do erosion, -but they can only do it via an input fixed rate of narrowing of the -moisture PDF (which is currently set to zero in all instances). -PC2 development has settled on a more complicated treatment of erosion, -which has only been implemented in {\it pc2\_hom\_conv}. -This can either be called after the convection scheme -(within {\it pc2\_from\_conv\_ctl}), -or before the microphysics scheme (within {\it pc2\_turbulence\_ctl}). - -Note there is also an optional call to {\it pc2\_turbulence\_ctl} -after the microphysics scheme, which is used only to estimate the -cloud fraction change consistent with the turbulent production of -liquid cloud (see section \ref{sec:turb_qcl_scheme}). - -Most PC2 code is protected by IF tests on the namelist input -{\it i\_cld\_vn} = 2 (PC2 in the GUI). -However, within the convection scheme, the code is controlled by logicals -{\it l\_calc\_dxek} (which is just set to true if using PC2, and set false -otherwise), and {\it l\_q\_interact}, which controls -whether to allow the interactive detrainment and entrainment of condensate. - -There is also a switch (currently hardwired to .false. in the code) called -{\it l\_pc2\_reset}. Turning this on (not recommended!) does 2 things: - -\begin{itemize} -\item Convective entrainment and detrainment of condensate is disabled, -by setting {\it l\_q\_interact} to false. -\item The prognostic cloud variables are overwritten by a call to -the diagnostic cloud scheme at the end of the timestep, -in subroutine {\it qt\_bal\_cld}. -NOTE: this functionality will no longer work, because inside {\it qt\_bal\_cld} -the call to the diagnostic cloud scheme is now protected by IF tests on -using either the Smith or bimodal cloud schemes. If using PC2, no cloud scheme -is called here, and required output variables are just left unset! -\end{itemize} - -The location of the various cloud scheme routine calls within the UM -is summarised in the list below. - -% The latex source input here contains a colour-coded itemize list -% of the UM subroutine tree, showing the locations of all the cloud-scheme -% routines. To edit this, open the source file source/029/um_call_tree.tex -\input{um_call_tree} - -\subsection{Diagnostics} -\label{sec:diags} - -Nearly all diagnostics retain their meaning when PC2 is run. However, there -are a few that are subtly modified. - -The convective diagnostics that use the convective cloud base and top -calculations remain the same if PC2 is used with a zeroed convective -cloud fraction. These values are not reset by the convection scheme, since -the model is still predicting convection between the diagnosed levels. - -The visibility diagnostics need modifying if the convective cloud -fraction is switched off, since they use the convective cloud fraction -within their calculation. Here we use a value of 0.2 for the convective -cloud amount if there is convective precipitation but the two-dimensional -convective cloud amount is zero. This will be the case if the PC2 -scheme has zeroed the convective cloud amount. - -There are a number of increment diagnostics that are required to -fully diagnose the moisture cycle within PC2. Since most physics -sections can cause condensation, condensate and cloud fraction increment -diagnostics have been written for each of these sections. - -\begin{itemize} -\item{ $\overline{T}$, $\overline{q}$, $\overline{q_{cl}}$, $\overline{C_t}$ and $\overline{C_l}$ increments from SW radiation, $\overline{T}$ increment from SW Radiation without including the condensation: \textbf{Section 1} .} -\item{ $\overline{T}$, $\overline{q}$, $\overline{q_{cl}}$, $\overline{C_t}$ and $\overline{C_l}$ increments from LW radiation, $\overline{T}$ increment from LW Radiation without including the condensation: \textbf{Section 2} .} -\item{ $\overline{T}$, $\overline{q}$, $\overline{q_{cl}}$, $\overline{q_{cf}}$, $\overline{C_t}$, $\overline{C_l}$ and $\overline{C_f}$ increments from Boundary Layer: \textbf{Section 3} .} -\item{ $\overline{T}$, $\overline{q}$, $\overline{q_{cl}}$, $\overline{q_{cf}}$, $\overline{C_t}$, $\overline{C_l}$ and $\overline{C_f}$ increments from Large-scale precipitation: \textbf{Section 4} .} -\item{ $\overline{T}$, $\overline{q}$, $\overline{q_{cl}}$, $\overline{q_{cf}}$, $\overline{C_t}$, $\overline{C_l}$ and $\overline{C_f}$ increments from Convection, $\overline{q}$, $\overline{q_{cl}}$, $\overline{q_{cf}}$, $\overline{C_t}$, $\overline{C_l}$ and $\overline{C_f}$ increments from the inhomogeneous part of the Convection scheme only: \textbf{Section 5} .} -\item{ $\overline{T}$, $\overline{q}$, $\overline{q_{cl}}$, $\overline{q_{cf}}$, $\overline{C_t}$, $\overline{C_l}$ and $\overline{C_f}$ increments from the Advection: \textbf{Section 12} .} -\end{itemize} - -However, there -are a number of parts of PC2 that do not fit into a pre-existing section of -code, and hence the associated increment diagnostics are not easily placed -within the UM framework. These increments were available using a -modification set or branch and a user-STASHmaster file up to version 7.5. From version 7.6 these diagnostics are available as standard. - -\begin{itemize} -\item{ $\overline{T}$, $\overline{q}$, $\overline{q_{cl}}$, $\overline{C_t}$, and $\overline{C_l}$ increments from the PC2 erosion section: \textbf{Section 4} or {\bf Section 5} depending on where the erosion is called.} -\item{ $\overline{T}$, $\overline{q}$, $\overline{q_{cl}}$, $\overline{q_{cf}}$, $\overline{C_t}$, $\overline{C_l}$ and $\overline{C_f}$ increments from the Bounds Checking after atmphya: \textbf{Section 4} .} -\item{ $\overline{T}$, $\overline{q}$, $\overline{q_{cl}}$, $\overline{q_{cf}}$, $\overline{C_t}$, $\overline{C_l}$ and $\overline{C_f}$ increments from the Initiation and Bounds checking at the end of the timestep: \textbf{Section 16} .} -\item{ $\overline{T}$, $\overline{q}$, $\overline{q_{cl}}$, $\overline{q_{cf}}$, $\overline{C_t}$, $\overline{C_l}$ and $\overline{C_f}$ increments from the Pressure Forcing section: \textbf{Section 16} .} -\end{itemize} - -\subsection{Single Column Model} - -The updating in the single column model follows the same timestepping as -that in the full model, but the changes to atm-step are mirrored -within scm\_main. The method used is to store the driving SCM forcing -increments of vapour, liquid and temperature across the forcing subroutine. -The forcing of pressure is -set to zero. After atmos\_physics2 has been called, -a PC2 section of code calls the -homogeneous forcing subroutine with these increments. This therefore -treats the response of PC2 to the prescribed dynamical forcing in the -SCM as homogeneous. Following -this calculation, the initiation scheme is called, as usual. Finally, the area -cloud fraction is set to the bulk cloud fraction and $\overline{\Theta}$ -(potential temperature) -is made consistent with $\overline{T}$ (dry-bulb temperature) which was -changed by the condensation in the PC2 response to the homogeneous forcing. -The rest of the SCM uses the same PC2 code as the full model. - -Note that the change to PC2 homogeneous forcing from advection under the UM -namelist switch \textbf{l\_pc2\_sl\_advection} (see section \ref{sec:pres}) -is also mirrored in the Single-Column Model. If this switch is turned on, -the PC2 homogeneous forcing call using the SCM forcing increments is moved -straight after the call to the forcing routine, so that the condensation -adjustment is performed before the call to atmos\_physics2. -If \textbf{l\_pc2\_sl\_advection} is turned on, the PC2 response to SCM -forcings is also improved as follows... - -The SCM forcings may comprise one or both of the following: -\begin{itemize} -\item (a) Prescribed tendencies or relaxation applied to T,q. -\item (b) Interactive vertical advection applied to T,q. -\end{itemize} -For the latter, we can calculate the pressure change experienced by -vertically-advected parcels, and so calculate the PC2 homogeneous forcing -response in the same way as we do for Semi-Lagrangian advection in the -full model (see section \ref{sec:pres}). -For the former, we don't know if the prescribed T,q tendencies are due to -advection, radiation, or some other process, so we calculate the PC2 -homogeneous forcing response as if the tendencies are applied "in-situ". - -To split the PC2 homogeneous response into these 2 components, the SCM -forcing routine outputs: -\begin{itemize} -\item (a) The forcing increments to T,q excluding the contribution from -interactive vertical advection. -\item (b) The value of exner pressure at departure points, consistent with -the vertical advection. -\end{itemize} -The PC2 homogeneous forcing responses to these 2 forcing components -are then calculated by 2 separate PC2 calls in scm\_main. - -\subsection{Limited Area Boundary Conditions} - -Cloud fractions on the limited area boundaries are fully updateable. -Writing of cloud fraction Limited Area Boundary Conditions (LBCs) -will be automatic if PC2 is selected. -A PC2 LAM may be run from an LBC file with or without cloud fraction LBCs -(this is specified by the logical l-pc2-lbc, which is set in the UMUI). -If there are no cloud fraction lbcs then around the edge of the domain the -checking and initiation routines will be applying significant increments -to the cloud and condensate fields near the boundaries, but this does -not have an adverse effect well away from the boundaries. If there are no -cloud fraction LBCs the cloud fraction fields themselves are not forced to -zero around the edge of the domain but are allowed to freely find their -own value. A PC2 run that outputs lbcs will, by default, always -output cloud fractions as part of the LBCs file. - -\subsection{Parameter values} - -Table \ref{tab:pc2_names} summarizes the values of parameters used in the PC2 -scheme and their location within various comdecks. Those parameters marked -as `Num' are those that are not part of the mathematical equation -set that is being solved, but are required in order to achieve a stable, -realistic, numerical solution. These include, for instance, thresholds for -resetting cloud fractions back to 0 or 1. Those marked 'Phy' are physical -quantities that form an integral part of the equation set that we wish to solve. -Those marked 'Clo' form part of a closure needed to form the equation -set, but are less readily related to physical quantities. -Variables marked 'Diag' form a part of the diagnostic output routines. -\begin{table}[ht] -\begin{center} -\footnotesize -\begin{tabular}{llllll} -\hline -Symbol & Code variable & Description & Value & Location & Notes and ref. \\ \hline -- & init-iterations & Number of iterations in initiation & 10 & pc2-const & Num: \ref{sec:numapp_init} \\ -$C_{tol}$ & cloud-pc2-tol & Bounds checking $C_l$ threshold & 0.005 & UM namelist & Num: \ref{sec:init2} \\ -$C_{tol 2}$ & cloud-pc2-tol-2 & Bounds checking $C_l$ threshold & 0.001 & UM namelist & Num: \ref{sec:init2} \\ -$RH_{tol}$ & rhcrit-tol & $RH_{crit}$ tolerance in initiation & 0.01 & pc2-const & Num: \ref{sec:init2} \\ -$q_{cf0 \, BL}$ & ls-bl0 & Fixed value of BL in-plume $\overline{q_{cf}}$ & $1.0 \times 10^{-4} \, kg \, kg^{-1}$ & imp-ctl & Clo: \ref{sec:bl} \\ -$q_{cf0}$ & one-over-qcf & Fixed in-cloud $\overline{q_{cf}}$ if $C_f$=0 & $1.0 \times 10^{-4} \, kg \, kg^{-1}$ & pc2-chck & Num: \ref{sec:checks} \\ -$m$ & pdf-merge-power & Merging power for $G(-Q_c)$ & 0.5 & pc2-const & Clo: \ref{sec:homog} \\ -$n$ & pdf-power & Shape parameter for $G(-Q_c)$ & 0.0 & pc2-const & Phy: \ref{sec:homog} \\ -$w$ & wind-shear-factor & Wind shear in fallout of ice term & $1.5 \times 10^{-4} \, s^{-1}$ & pc2-const & Phy: \ref{sec:lsp_fall} \\ -$i$ & ice-width & Scaling factor for reduction in $b_i$ & 0.04 & pc2-const & Phy: \ref{sec:mp_depsub} \\ -$a$ & dbsdtbs-turb-0 & Rate of reduction of PDF width & $-2.25 \times 10^{-5} \, s^{-1}$ & UM namelist & Phy: \ref{sec:width} \\ -$b$ & dbsdtbs-turb-1 & Rate of reduction of PDF width & 0 & pc2-const & Phy: \ref{sec:width} \\ - & dbsdtbs-conv & Redn of PDF width in convection & 0 & pc2-const & Phy: \ref{sec:width} \\ - & dbsdtbs-exp & Variation of erosion on RH & 10.05 & pc2-const & Phy: \ref{sec:width} \\ -$RH_{crit}$ & RHCRIT & Critical RH for cloud formation & & UM namelist & Phy: \ref{sec:init}, \ref{sec:mp_depsub} \\ -$q_{c0}$ & condensate-limit& Minimum allowed condensate & $1 \times 10^{-10} \, kg \, kg^{-1}$ & pc2-chck & Num: \ref{sec:checks} \\ -$q_c^{S0}$ & ls0 & Lower limit of plume condensate & $5 \times 10^{-5} \, kg \, kg^{-1} $ & enviro?a & Num: \ref{sec:multi_numapp} \\ - & \textit{Hard-wired} & Conv cloud fraction for visibility& 0.2 & imp-ctl2 & Diag: \ref{sec:diags} \\ - & \textit{Hard-wired} & Limit on width of ice distribution& 0.001 & lspice3d & Num: \ref{sec:mp_depsub} \\ - & \textit{Hard-wired} & $C_l$ limit for init if $T < 0 ^{\circ} C$ & 0.05 & pc2-init & Num: \ref{sec:init2} \\ - & \textit{Hard-wired} & Tolerance on calc. of $q_C^s$ in BL & $1.0 \times 10^{-10} \, kg \, kg^{-1}$ & imp-ctl & Num: \ref{sec:bl} \\ -\hline -\end{tabular} -\end{center} -\caption{PC2 parameter values and locations } -\label{tab:pc2_names} -\end{table} - -PC2 also recommends some tunings of the existing convection -scheme parameters. These cannot be placed in the library code, since they -would interact with non-PC2 simulations, hence would need to be specified -with modification sets. We have included those parameters that have been -investigated throughout testing, although only two are different between -PC2:64 and a non-PC2 run. - -\begin{table}[ht] -\begin{center} -\tiny -\begin{tabular}{llllll} -\hline -Code variable & Description & Value in PC2 & Value in Control & Location & Notes and reference \\ \hline -TICE & Temperature at which plume freezes & $-10 ^{\circ} C$ & $0 ^{\circ} C$* & tice.cdk or UMUI & Phy: \ref{sec:convec} \\ -QSTICE & Approximate qsat(TICE) & $3.5 \times 10^{-3}$ & $3.5 \times 10^{-3}$ & qstice.cdk or UMUI & Phy: \ref{sec:convec} \\ -\textit{Hard-wired} & Limit on conv. cond. after precip & 0.5 $q_{sat}, 2 \times 10^{-4}$ & $0.5 \, q_{sat}$ & cloudw & Phy: \ref{sec:convec} \\ -Anvil factor & Shape parameter for conv. cloud anvil & 0 & 0.3* & UMUI & Phy: \ref{sec:convec} \\ -Tower factor & Shape parameter for conv. cloud tower & 0 & 0.25* & UMUI & Phy: \ref{sec:convec} \\ -\hline -\end{tabular} -\end{center} -\caption{PC2 parameter values and locations relating to the convection. *These values are those used in HadGAM} -\label{tab:pc2_conv_names} -\end{table} - -\subsection{How to run the PC2 scheme} -Running PC2 is straightforward, but you should seek advice as to -modification sets that you need to include to ensure you are -running the most up-to-date version of PC2. -The following is a brief checklist of the options in the UMUI which need -to be selected in order to run PC2. No hand-edits are required. -\begin{itemize} -\item{In the LS cloud panel (atmos-science-section-LScloud) push the button marked 'use the PC2 cloud scheme'.} -\item{If you wish to use PC2 in the diagnostic only mode, also push 'run the PC2 scheme in diagnostic only mode'. If you wish to run PC2 fully then do not push this button} -\item{In the large-scale precipitation section (atmos-science-section-LSprecip) select the 3D large-scale precipitation scheme.} -\item{The specification of the LA boundary conditions can be set in the atmos-InFiles-OtherAncil-LBC panel.} -\item{You will need to select modsets to include update the library code to the PC2 version described here. Seek advice on these.} -\item{You may wish to adjust the convective anvil parameters in atmos-science-section-convec. Again, seek advice.} -\end{itemize} - -\subsection{More information} - -Information on results of the scheme and how to run the PC2 code at -various model versions is available on the PC2 web site. - -\section{Appendix: Alternative PC2 - Data Assimilation formulations} -\label{sec:appendix-da} - -In this alternative method to section \ref{sec:da} we will assume that there -exists a homogeneous forcing, $\Delta Q_c$, -that gives changes, net of condensation, of $\Delta\overline{q}$ and -$\Delta\overline{T}$. If we can recover -what $\Delta Q_c$ is then we can use this to calculate the liquid, -$\overline{q_{cl}}$, and liquid cloud fraction, $C_l$, increments. - -As in section \ref{sec:da}, we start by discretising (\ref{dqcldt}) to give - -\begin{equation} -\Delta \overline{q_{cl}} = C_l \Delta Q_c -\label{eq:dqcldt_discrete} -\end{equation} - -and hence, using the discrete form of $\Delta Q_c$ from -(\ref{eq:deltaqc_exp2}) gives - -\begin{equation} -\Delta \overline{q_{cl}} = C_l ( a_L ( \Delta \overline{q} - \alpha \Delta -\overline{T} ) + \Delta \overline{q_{cl}} ) , -\end{equation} - -which rearranges to - -\begin{equation} -\Delta \overline{q_{cl}} = \frac{1}{1-C_l} C_l a_L ( \Delta \overline{q} -- \alpha \Delta \overline{T} - \beta \Delta \overline{p}) . -\label{eqn:delataqcl} -\end{equation} - -Comparing to (\ref{eq:deltaqc_exp2}) and (\ref{eq:dqcldt_discrete}) we see that -$\Delta \overline{q_{cl}} $ is the same as if we had applied the -homogeneous forcing technique -using $\Delta \overline{q}$, $\Delta \overline{T}$ and $\Delta \overline{p}$ as -forcings, except multiplied by a factor of $\frac{1}{1-C_l}$. - -We can calculate $\Delta C$ in a similar way. From (\ref{eq:deltac}) - -\begin{equation} -\Delta C_l = G(-Q_c) \Delta Q_c -\end{equation} - -and hence, using our value of $\Delta Q_c$ from (\ref{eq:deltaqc_exp2}) -and $\Delta \overline{q_{cl}}$ from (\ref{eqn:delataqcl}) - -\begin{equation} -\Delta C_l = G(-Q_c) (a_L ( \Delta \overline{q} - \alpha \Delta \overline{T} ) -+ \frac{1}{1-C_l} C_l a_L ( \Delta \overline{q} - \alpha \Delta \overline{T} ) ) -\end{equation} - -which rearranges to - -\begin{equation} -\Delta C_l = \frac{1}{1-C_l} G(-Q_c) a_L ( \Delta \overline{q} -- \alpha \Delta \overline{T} -\beta \Delta \overline{p}) . -\label{eqn:c1mc} -\end{equation} - -This is also a factor of $\frac{1}{1-C_l}$ different from using -$\Delta \overline{q}$, -$\Delta \overline{T}$ and $\Delta \overline{p}$ -directly as forcings (the factor must be the same, as we are still -using the homogeneous forcing hypothesis). This equation forms the basis -for the more advanced technique discussed in this section. However, -it is undefined at $C_l=1$ and becomes ill-conditioned near $C_l=1$, -hence there must be care taken when this expression is solved numerically. - -\subsection{Numerical solution} - -The timestepping applied is picked as a result of numerical tests -forcing a single gridbox with uniform increments. Many numerical techniques -were tested, this gives a fast but reasonably well behaved solution. - -Initially, we calculate $G(-Qc)$ and $\Delta Q_c$ from the input fields, -as in the homogeneous forcing -technique (section \ref{sec:homog}) and (\ref{eq:deltaqc_exp2}). - -An initial increment, $\Delta C_l^1$ is estimated directly using the -basic equation - -\begin{equation} -\Delta C_l^1 = \frac{1}{1-C_l^(n)} G(-Q_c) \Delta Q_c . -\end{equation} - -We then recalculate this expression, using a mid-timestep estimate -for $\Delta C_l$; - -\begin{equation} -\Delta C_l = \frac{1}{1-(C_l^{[n]} + \frac{1}{2} \Delta C_l^1)} -G(-Q_c) \Delta Q_c -\end{equation} - -where the term $C_l^{[n]} + \frac{1}{2} \Delta C_l^1$ is limited to be no more -than 0.9999 to avoid divide by zero problems. The final, updated value of -cloud fraction, $C_l^{[n+1]}$, is then - -\begin{equation} -C_l^{[n+1]} = C_l^{[n]} + \Delta C_l -\end{equation} - -and this value is limited to 0 or 1. - -The liquid water term simply uses the final version of $C_l$ in its -calculation. - -\begin{equation} -\Delta \overline{q_{cl}} = \frac{1}{1-C_l^{[n+1]}} C_l^{[n+1]} \Delta Q_c -\end{equation} - -and will be set to 0 if $C^{[n+1]}$ is 0. There is an additional limit, -see below, applied to the liquid -water term, which will prevent the value of $\Delta \overline{q_{cl}}$ -increasing to a large number if $C_l^{[n+1]}$ is very close to 1. - -\subsection{Limit on the liquid water content} - -We will choose a limit on $\overline{q_{cl}}$ to be equal to its value when -the underlying PDF just corresponds to total cloud cover. Therefore, from -(\ref{eq:qclbar=int}) - -\begin{equation} -\overline{q_{cl \, max}} = \int_{s=-b_s}^{\infty} G(s) (b_s + s) ds . -\end{equation} - -We will use the current value of $Q_c$ (which won't in general to be equal to -$b_s$) to split the integral into two ranges of s: - -\begin{equation} -\overline{q_{cl \, max}} = \int_{s=-b_s}^{-Q_c} G(s) (b_s + s) ds -+ \int_{s=-Q_c}^{\infty} G(s) (b_s + s) ds . -\end{equation} - -For the moment we write the first of these integrals as $I1$, and split the -second integral whilst introducing a $(+ Q_c - Q_c)$ term to the integrand: - -\begin{equation} -\overline{q_{cl \, max}} = I1 + \int_{s=-Q_c}^{\infty} G(s) (b_s - Q_c) ds -+ \int_{s=-Q_c}^{\infty} G(s) (s + Q_c) ds . -\end{equation} - -The last of the integrals is now the current liquid water content, -$\overline{q_{cl}}$. -The second integral is proportional to the liquid cloud fraction $C_l$. - -\begin{equation} -\overline{q_{cl \, max}} = I1 + C_l (b_s - Q_c) + \overline{q_{cl}} -\end{equation} - -or - -\begin{equation} -\overline{\Delta q_{cl \, max}} = I1 + C_l (b_s - Q_c) . -\label{eqn:deltaqclmax} -\end{equation} - -Now consider the expression for the saturation deficit, which we have -defined, from (\ref{SD}) as - -\begin{equation} -SD = \int_{-b_s}^{-Q_c} G(s) (-Q_c - s) ds . -\end{equation} - -Splitting and adding the term $(+b_s - b_s)$ to the integrand in a -similar way to above gives - -\begin{eqnarray} -SD = \int_{-b_s}^{-Q_c} G(s) (-Q_c + b_s) ds + \int_{-b_s}^{-Q_c} -G(s) (-s - b_s) ds \nonumber \\ -= (-Q_c + b_s) (1 - C_l) - I1 , -\end{eqnarray} - -and hence $I1$ in terms of $SD$. Using this value of $I1$ in -(\ref{eqn:deltaqclmax}) and cancelling the $C_l$ terms gives -$\Delta \overline{q_{cl \, max}}$ as - -\begin{equation} -\Delta \overline{q_{cl \, max}} = (-Q_c + b_s) - SD . -\label{eqn:delta2} -\end{equation} - -This is a general expression, it is not fixed for a particular PDF. To -complete the analysis, we need to estimate $-Q_c+b_s$. To do this, we now -make the \textit{assumption} of a power-law type PDF, as in section -\ref{sec:init}. If we start from the equivalent of -(\ref{eqn19}) but at the $s=-bs$ end of the distribution, equation (B.3) -in \cite{wg03} can be equivalently written for $(1-C_l)$ as: - -\begin{equation} -(1-C_l) = \frac{ A (-Q_c + b_s)^{n+1} }{n+1} . -\label{eqn:1mc} -\end{equation} - -To derive this from (B.3) note that $C_l$ is swapped for $1-C_l$ and -$(b_s - (-Q_c))$ is swapped for $(-Qc - (-b_s))$, as in section -\ref{sec:numapp_init}. Similarly, noting that $\overline{q_{cl}}$ can be -swapped with $SD$, gives the equivalent to (B.4) in \cite{wg03} -as - -\begin{equation} -SD = \frac{ A (-Q_c + b_s)^{n+2} }{(n+1)(n+2)}. -\label{eqn:sd} -\end{equation} - -Using the value $(1-C_l)$ from (\ref{eqn:1mc}) in (\ref{eqn:sd}) gives - -\begin{equation} -\frac{SD}{1-C_l} = \frac {-Q_c + b_s}{n+2} . -\end{equation} - -Finally, we use this expression for $(-Q_c + b_s)$ in (\ref{eqn:delta2}) to -parametrize $\Delta \overline{q_{cl \, max}}$ in terms -of the saturation deficit - -\begin{equation} -\Delta \overline{q_{cl \, max}} = SD ( \frac{n+2}{1-C_l} - 1 ) . -\label{eqn:sdr1mc} -\end{equation} - -This is the expression that is used for the limit on $\overline{q_{cl}}$. -We subsequently apply a second limit, since numerically this expression is -still not well behaved when $C_l$ is close to 1. Here we note that just at -complete cloud cover for a symmetric PDF we have -$\overline{q_{cl}} = b_s$. Hence we estimate $b_s$ as in \cite{smith90}, - -\begin{equation} -b_s = a_L ( 1 - RH_{crit} ) q_{sat}(\overline{T_L}) , -\label{eqn:bs} -\end{equation} - -and take the smaller value for -of (\ref{eqn:sdr1mc}) and (\ref{eqn:bs}) for $\Delta \overline{q_{cl \, max}}$. - - -\subsubsection{Initiation from $C_l=1$} -The equations are not defined when $C_l=1$. (Note that when $C_l=0$ we -will calculate $G(-Q_c)=0$ so there is no change in cloud fraction or liquid water -content in this case). The assimilation is capable of lowering $\overline{q}$ -below $q_{sat}(\overline{T})$ and hence there should be a corresponding -change in cloud fraction and liquid water content. In theory, we can use -the expression for $\Delta \overline{q_{cl \, max}}$ and assume that the -initial liquid water is equal to $b_s$. However, this produces -a tricky set of simulataneous equations, which are not easily solvable -except in the case where $n=0$. We proceed by making this assumption for $n$, -acknowledging that this is not necessarily entirely consistent with the -rest of the model (although it is in the PC2:64 formulation). - -We have (equivalent to B.6 from \cite{wg03}) - -\begin{equation} -\frac{ (1-C_l)^2 }{SD} = G(-Q_c) \frac{n+2}{n+1}. -\end{equation} - -If n=0 (i.e. a `top-hat' function) then $G(-Q_c) = \frac{1}{2 b_s}$ -and we can write - -\begin{equation} -C_l = 1 - \sqrt{ \frac{SD}{b_s} } . -\end{equation} - -We now assume $b_s$ is equal to our current value of $\overline{q_{cl}}$ -and hence - -\begin{equation} -C_l^{[n+1]} = 1 - \sqrt{ \frac{SD^{[n+1]}}{\overline{q_{cl}^{[n]}}} } -\label{eqn:1msqrt} -\end{equation} - -where $C_l^{[n+1]}$ and $SD^{[n+1]}$ are the values of $C_l$ and $SD$ after -this initiation has been applied. -Using our previous expression (\ref{eqn:sdr1mc}) for -$\Delta \overline{q_{cl max}}$ gives (remembering that we are considering -the reverse process, so the sign is opposite), - -\begin{equation} -\Delta \overline{q_{cl}} = - SD^{[n+1]} ( \frac{2}{1-C_l^{[n+1]}} - 1 ) -\end{equation} - -(remembering that $n=0$ is assumed). Hence, replacing $C_l^{[n+1]}$ by -(\ref{eqn:1msqrt}) we have - -\begin{equation} -\Delta \overline{q_{cl}} = SD^{[n+1]} - 2 \sqrt{ SD^{[n+1]} -\overline{q_{cl}}^{[n]} } . -\end{equation} - -This is the expression we use, $SD^{[n+1]}$ is calculated after the -ssimilation increments have been applied, using (\ref{SD2}): - -\begin{equation} -SD^{(n+1)} = a_L^{[n+1]} ( q_{sat}(\overline{T}^{[n+1]}, -\overline{p}^{[n+1]}) - \overline{q}^{[n+1]} ). -\end{equation} - -\subsection{Results} - -Results demonstrate a problem in that there is a distinct asymmetry -between changes when $\Delta Q_c$ is large and positive and when -$\Delta Q_c$ is large and negative, when -cloud fractions start near 1. In the former case, the limit to the amount of -liquid and cloud fraction that can be created means that changes must be -kept relatively small, whereas in the latter case, all the cloud and -liquid water can be removed easily. (The $1/(1-C_l)$ term allows this -to be done relatively quickly). Hence this assimilation -method has a net tendency to remove cloud from the simulation, which, -at the moment, gives poorer results than simply using the homogeneous -forcing method. - -Further work will be required to enable the implementation of -this $\overline{q}$ -and $\overline{T}$ preserving method. - -\section{Appendix: Essentials of PC2 for code developers} -\label{sec:code-development} -This section provides some guidance to code developers on the treatment -of PC2. Code developers are advised to read the relevant part of section -\ref{sec:app_um} to understand the way in which the current PC2 scheme -interacts with their section of code. - -The essence of a prognostic cloud scheme is that each physical part of the -model is able to calculate increments to the cloud fractions and condensate -contents. These form an integral part of each physics scheme and should be -considered by code owners as such, hence any alteration to a scheme -\textit{must} consider also the impact on $q_{cl}$, $q_{cf}$, $C_t$, -$C_l$ and $C_f$, as -well as on the more traditional $T$, $q$ and wind prognostics. Often -there should be no impact, but this cannot be assumed without consideration. -There is no diagnostic cloud fraction and condensation scheme which can be -run in PC2, since this would reset any effect of the cloud prognostics used -elsewhere in the model. (The diagnostic scheme can still be used for model -\textit{diagnostics}, such as visibility and fog fraction, and will be -kept in later versions of the UM). - -Since this places a significant burden on code developers, the PC2 -developers have produced two generic representations which can take increments -to $q$ and $T$ etc. and produce an estimate of the condensation and -cloud fraction changes associated with the increments. These are the -homogeneous forcing and injection forcing (or inhomogeneous forcing) -methods. - -\subsection{Homogeneous forcing} -This is described fully in section \ref{sec:homog}. This assumes that the -distribution of $q_T - q_{sat}(T_L)$ about its gridbox mean is unchanged when -a process acts. (The mean will change of course, but we assume that the -variations in each part of the gridbox from the mean do not). Since this -is equivalent to every part of the gridbox receiving the same $q_T$ and $T_L$ -increment, we call this `Homogeneous Forcing'. We have provided a subroutine -\textit{pc2-homog-plus-turb}, in deck \textit{pc2-homo} in order to -provide the necessary updates. - -\subsection{Injection forcing} -This is described fully in section \ref{sec:inhomog}. We assume that -we already know a condensate increment $q_{cl}$ or $q_{cf}$ and that a -corresponding cloud fraction increment $C_l$ or $C_f$ (and $C_t$) remains -to be estimated. The injection forcing assumes that new cloud randomly -displaces existing cloud in a gridbox, and is designed with detrainment -from deep convection in mind, although it is also used elsewhere. It will -require as an input an estimate of the `in-cloud' water content of -the new cloud that is produced. - -If you consider that both the homogeneous and injection forcing representations -are both poor assumptions for your scheme, you will need to provide -another method for calculating the condensation and cloud fraction changes. -The PC2 team can advise, but you should not expect them to do the work. -You can, of course, replace existing homogeneous and inhomogeneous forcing -calls with new representations of changes to the prognostics if you think -you have improved representations available. This is part of the -development of any prognostic variable representation. - -\subsection{Do I need to modify anything when I change a parametrization scheme?} - -Here we assume that you wish to do the minimum work possible to get -PC2 to work, rather than a full reconsideration of the physics of the PC2 -increment terms. - -If your scheme is currently using the homogeneous forcing -then there is no need to update the cloud part of the scheme, -\textit{provided that -you do not alter values of $T$ and $q$ after the homogeneous forcing -section is called} and that the physical interpretation of your $q$ and $T$ -increments does not change. You need to be careful if you are moving code from -one subroutine to another that you don't inadvertently do this, although -the forcing usually sits at the end of the control subroutine. - -If your scheme is currently using the injection forcing \textit{subroutine}, -which necessitates that condensate -increments are already calculated by the scheme, then there is also no need -to update the cloud part of the scheme. This currently applies to the boundary -layer, where $q_{cf}$ is altered by tracer mixing. Like for the -homogeneous schemes, this -is provided that you \textit{do not alter $T$, $q$ or condensate values after -the injection forcing subroutine is called} and that the physical -interpretation of your $q$ and $T$ increments does not change. - -Changes to winds do \textit{not} need to have a condensation or -cloud fraction increment -associated with them. There may be future scope for developing an -orographic cloud representation (probably diagnostic), but this is not -an essential part of the scheme as it stands. - -If your scheme uses hardwired assumptions about what is happening e.g. -convection or microphysics, then you \textit{do} need to be careful that -$T$, $q$ and condensates -are still calculated correctly after you have performed your changes. -Currently there are many PC2 assumptions hard-wired into the mass-flux -convection scheme: -\begin{itemize} -\item{Any change to the scientific basis by which changes to $T$, $q$, $q_{cl}$ and $q_{cf}$ are calculated requires careful consideration} -\item{Simple changes to convective parameters, such as detrainment rates, should not require a change to the PC2 code} -\item{Be particularly careful if you move code around, \textit{especially the calculation of convective cloud fractions}, since PC2 incorporates a set-to-zero in the code. This will need to be replicated or there is a risk that the diagnostic cloud fraction is no longer set to zero correctly by PC2.} -\end{itemize} -Each microphysics transfer term has been considered individually for PC2 and this -should remain the case. - -Be especially careful when you do anything in the atmphy and atmstep levels of -the code that includes additional changes $T$, $q$, $q_{cl}$ or $q_{cf}$, since -they may need cloud fraction or condensation changes to go along with them. - -In summary, changes to existing increments of $T$, $q$ etc. within the current -UM structure are unlikely to -necessitate a modification for PC2 if their physical interpretation has not -changed. However, new methods of generating $T$ and -$q$ increments will require new code to be added for PC2. - -\subsection{Further PC2 development work} -There are a number of areas in which the PC2:66 formulation can be -developed further, and many of these have been mentioned in the documentation -above. Some -of these are simple sensitivity studies which have not been fully explored in -development, others are more complex alterations. It is fair to say -that the link to the convection has proved the most problematic issue -so far with PC2 development. - -\subsubsection{PC2 cloud erosion} -The cloud erosion is a critical term for the simulation of shallow convective -cloud. A large amount of erosion is required to keep the cloud fractions relatively -low in shallow convection, which is why we have linked the erosion to the relative -humidity. We recognise, however, that this is more an empirical choice than a -physically informed choice. In particular, a low relative humidity (e.g. in the -stratosphere) would imply a very high erosion rate - although the net effect -is to remove any cloud, which is a reasonable thing to do, there is an implication of -the parametrization that mixing within the stratosphere is high, which is -clearly incorrect. We have also seen relatively low cloud fractions in the -mid-levels of deep convection in PC2, and presume that this is influenced -by the erosion formulation. A link to mass flux has also been proposed, but tests -with CRMs do not support a clear link. Perhaps it is more natural to compare the -erosion with the turbulent kinetic energy. This should be available within the -boundary layer and convection schemes, but not outside of these in the current -UM. - -The erosion formulation in PC2:66 is one where the width of the PDF is -always narrowed (developed following \cite{sg03}). -It may be advantageous to think whether there are unmodelled -processes in the atmosphere that result in an increase in width. Clearly -convection is likely to be one, but this is already represented in PC2. -There may be other models entirely for the way in which the PDF changes as a result -of mixing of air within a gridbox or within the column, these may prove -fruitful to explore. - -Another issue is whether width-narrowing (or widening) is really an effective -way of representing the erosion process. CRM evidence suggests that the required -erosion rates to balance convective cloud generation are larger for -liquid cloud fraction than liquid water (by up to a factor of 2), suggesting -that the real atmospheric erosion favours removal of cloud fraction -over liquid water more strongly than the model. - -The in-cloud condensate that is detrained from convective plumes is high. -We might think that the mixing in of environmental air in reality is -likely to lead to more cloud around the plumes and lower condensate within -the plumes. However, the width narrowing scheme is not a good model of mixing -in this situation, always reducing the amount of cloud because it is incorrectly -assumed that much of the detrained plume has condensate contents only just above zero -and that the shape of the moisture PDF remains unchanged. This may have a -bearing on the problem of the lack of mid-level cloud in the model (although I -think there are many reasons for this). A different -mixing method may give significantly different results for the areas around -convective plumes. - -\subsubsection{Narrowing of the moisture PDF} -Most of the parametrized terms in PC2 act to reduce the width of the -moisture PDF. The only terms that can increase the width are the convection, -and the initiation (which can reset the width). This may not be the -best way to describe the way in which the PDF evolves, in particular it -is sensible to ask whether the erosion term should actually increase -the width in the presence of large vertical gradients of moisture. - -\subsubsection{Convective cloud increments in the mass-flux framework} -As discussed in section \ref{sec:conv_imp_note}, it would be useful -to code up the convective cloud fraction changes to link directly to -the mass-flux convection scheme, and not to estimate them from the values -of $Q4$, which can introduce errors. - -\subsubsection{Turbulence based convection scheme} -\label{sec:tbcs} -We will need to properly consider the links between PC2 and the -turbulence based convection scheme. In essence, we can use the diagnosed -cloud fraction and condensate values from the convection scheme to -start off the cloud again when convection has ceased. This has been -tested to some degree but will need proper analysis. The difficult -decision comes in choosing what to do with the condensate and cloud fraction -that is present \textit{before} the convection starts, since we must -ensure conservation of moisture. This is not helped by the traditional -view of convective parametrization that ignores the existence of the condensate -phase in the atmosphere (i.e. it is only concerned with transport of $q$ and -$\theta$, not of $q_{cl}$ and $q_{cf}$) despite the phase changes forming -an integral part of the convection scheme. - -\subsubsection{Detailed convective comparisons with CRM/LEM data} -This work is already underway at the Met Office, in order to properly -evaluate the performance of the convective cloud parametrization -in PC2 against high resolution research models. - -\subsubsection{Choice of PDF parameters} -Work by Dan Tang at Leeds University has highlighted an interesting -and undesirable property of the choice of $m$ and $n$ parameters in the -homogeneous forcing formulation. If a distribution is homogeneously -forced to $C_l = 0$, then we do not necessarily get $\overline{q_{cl}}$ -tending to zero. This is because there is enough influence from the -$\frac{{(1-C_l)}^2}{SD}$ term in the combination (\ref{eqn22}) to -stop the natural convergence of the $\frac{{C_l}^2}{\overline{q_{cl}}}$ term -to $C_l =0$ and $\overline{q_{cl}}=0$. Increasing the power of $m$ should -help. However, we note that the tests that have been done on the chosen -$n$ and $m$ values (0 and 0.5 respectively) do not show particularly -poor behaviour, and we do not pick up substantial evidence of problems -from this in the full model. This remains something to be investigated. - -\subsubsection{Homogeneous forcing section improvements} -\label{sec:homog_improve} -Although the homogeneous forcing provides a convenient method to -calculate increments to $C_l$ and $\overline{q_{cl}}$, it is clearly -not the best representation possible of the processes that use it. -For example, although the clear-sky radiative heating may perhaps -best be considered as a homogeneous process, the part of the -radiative heating influenced by clouds should, ideally, be applied -to the cloudy part of the gridbox and not the clear part. Vertical -advection is likely to be correlated with where there is already -cloud, rather than being uniform throughout the gridbox. There is -no reason that a process that uses homogeneous forcing as its -condensation model should not be looked at with a view to using -something better. This is one of the strengths of the PC2 framework and -is an intention of the project. - -\subsubsection{Overlap of ice and liquid cloud changes} -We have assumed within PC2 that ice and liquid cloud changes are -minimally overlapped with each other (within the same gridbox) in -order to maintain as much supercooled liquid water as possible. Although -there is good observational evidence to say that the two condensate -phases tend not to coexist together in a cloud, it may be possible to -characterise and apply this overlap in a more quantiative way. - -\subsubsection{Parameter tuning} -The sensitivity of some of the parameters in PC2 have not been properly -tested, mainly due to a lack of resources rather than a physical reason. -We have seen that the most effective method of tuning cloud is with the -erosion term, which has been increased to high values in order to remove -enough cloud and is probably as high as we reasonably wish to take it given -the length of the timestep. -\begin{itemize} -\item{The phase change temperature (between liquid and ice) in the -convective plume, TICE, is known to influence the strength of the convection -through the latent heat differences. It also impacts on the amount of -supercooled liquid water in the model. The quantitative impact of altering -this could be explored. We note that CRM simulations of deep convection -suggest that some supercooled liquid water exists within the plumes to -$-40 ^{\circ} C$ and that a representation with partial liquid and partial -ice phase would be more appropriate, based possibly on the current diagnosed -convective cloud phase in the non-PC2 model. Although the theoretical work -has been done to allow partial phases, we repeat the caution that care -must be taken when doing the work and appropriate testing done to ensure -that heat and moisture are properly conserved within the convection scheme.} - -\item{The growth of $C_f$ due to the fall-out of ice term in the microphysics -is parametrized with a dependence on windshear. We have never linked this -directly to the windshear, instead we have used estimated the windshear -as a fixed value. There is no reason why the actual model windshear cannot -be passed into the scheme in order to properly calculate this term.} -\item{$RH_{crit}$ remains a tunable parameter. Although its impact is less -than in a non-PC2 simulation, it is still significant in initiating cloud -and in determining the evolution of the ice cloud. There is also an implicit -overlap assumption regarding the ice cloud fractions, again this might be -improved upon.} -\item{$n$ and $m$ values in the homogeneous forcing have not been -thoroughly investigated for a long time now, and may yield some sensitivities}. -\end{itemize} - -\subsubsection{Cloud inhomogeneities} -A cloud generator approach to cloud inhomogeneities is currently -being developed. However we note two particular issues that relate -to PC2. -\begin{itemize} -\item{The first is that in the diagnostic scheme, the two cloud -fractions (convective and large-scale) allows, to some degree, a -representation of cloud inhomogeneity. This is absent from PC2, -although we note that the convective cloud fraction variable has -not been removed from the radiative transfer code for PC2, it is merely set -to zero, so it is easy to put back.} -\item{The generation of inhomogeneities using a cloud generator -requires some estimate of the variance (and possibly skewness) -of the condensate in the -gridbox. It is possible to back out the full moisture PDF at -each grid point by homogeneous forcing (providing $C_l$ is not equal -to 0 or 1), but this is very expensive and cannot be done on-line. Is -there a quick \textit{estimate} of the variance or skewness that it is -possible to obtain from knowledge only of $\overline{q}$, -$q_{sat}$, $\overline{q_{cl}}$ and $C_l$ etc.?} -\end{itemize} - -\subsubsection{Time-stepping} -\label{sec:timestepping} -A proper analysis of timestep sensitivities of PC2 (as opposed to -microphysics, convection etc) in the full UM -or SCM has not been done for a long time. -In the early development stages much effort was -placed in developing good numerical techniques for each of the terms -in PC2, and to explore the way in which they coupled together. An example -is the homogeneous forcing timestep investigated by \cite{wg03}. -We note that in shallow convection at 30 minutes timestep the erosion -term is trying to remove -most of the cloud that the convective detrainment places into the model. -Since the erosion is limited by the amount of cloud fraction and -condensate present, what ends -up happening is that the `equilibrium' that is achieved is actually one where -the cloud fraction and condensate at the end of the timestep are simply -the values that were detrained by the convection scheme (and hence depend -on the timestep). The CRM suggests a cycling time of around 15 minutes for -liquid water content and just less than half and hour for the cloud fraction, -so we would expect timestep dependency to occur from around a timestep of -15 minutes upwards. We might just about get away with the 30 minute step -of the climate model, but it is not a good situation to try to model. -This is demonstrating the difficulty of modelling shallow convective cloud -by a prognostic scheme, where the physical lifetime of the clouds is -of order the timestep - ideally we wouldn't want to try to model anything -prognostically when the cycling time is less than the timestep. - -As discussed in section \ref{sec:erosion_numerics}, the timestep sensitivity -of cloud amounts in shallow cumulus regimes can be addressed by using -a more accurate numerical method to solve the erosion term. -Several options are available under the UM namelist switch -\textbf{i\_pc2\_erosion\_numerics}. - -In the early development of PC2 we chose to incorporate the PC2 cloud -and condensation increments in the same location where the increments -were calculated (e.g. the microphysics cloud fraction increments -get added along with the microphysics $\overline{T}$ and $\overline{q}$ -increments). This choice was made in order not to confuse the timestepping -method in the UM, which has been carefully developed over a number of -years to achieve numerical accuracy. However, we note that the rapidly -varying nature (in space and time) of variables such as $\overline{q_{cl}}$ -and $C_l$ is very different from the smooth fields of $\overline{q_T}$ and -$\overline{T}$, for which the timestepping was developed, and it may -not be appropriate to implement these in the same locations. In -particular, we might wish to store the increments through the timestep -and update values of $\overline{q_{cl}}$ and $C_l$ etc. at the end -of the timestep, where many of the balances can be cancelled. - -One issue is that we are calculating the increments due to condensation -associated with the adiabatic response to pressure changes after the -Helmholtz solver. Pragmatically, we need to do it here since we do not know -the arrival value of pressure until after the Helmholtz solver has been -used. However, in order to achieve balanced dynamical fields, it is useful -the Helmholtz solver to be called after all the latent heating terms have -been calculated (which not only includes the adiabatic response to -lifting but the cloud initiation term). We have shown that PC2 can run with -the two terms switched over, but this implies that we are missing part of -the pressure change following the parcel (the time changing part -rather than the spatially changing adiabatic part). Although the adiabatic -change is usually likely to dominate, it may be a significant loss. -Under the UM namelist switch \textbf{l\_pc2\_sl\_advection}, -we can call the PC2 response twice, once before the Helmholtz solver and -once afterwards in order to pick up most of the latent heat change before -the solver, but not to have PC2 miss some of the pressure change. -The call for the advective part (before the Helmholtz solver) is -actually done before the call to atmos\_physics2 as well, and so results in -more realistic, saturation-adjusted, profiles being passed to the convection -scheme. - -We have placed the initiation at the end of the timestep, but it is sensible to -ask whether this could ideally be located elsewhere. - -\subsubsection{Initiation formulation} -Ideally this should be a relatively infrequent part of the model -but remains an essential part of the code. It is reasonable to ask whether -the initiation is optimal, particular in the diagnosis of when it is -applied. For example, we note that the initiation is currently -symmetrical, with initiation from $C_l=1$ occuring with the same -$RH_{crit}$ value as from $C_l=0$. However, the \cite{wf00} -observations hint that a higher $RH_{crit}$ might be more appropriate -for initiation from $C_l=1$. - -\subsubsection{70-levels performance} -The performance of PC2:66 in the 70-levels model is not good as -far as shallow convective cloud is concerned (there is far too -much of it in the trade regions). It may be that PC2 is latching onto -a convection sensitivity that is present on going from L38 to L70 but -had little effect in a non-PC2 simulation. It may also be related -to a reduction in timestep from 30 minutes to 20 minutes. -Investigations have not -made much progress in identifying the reasons for the differences, -or producing effective tunings to counter the problem. - -\subsubsection{High horizontal resolution performace} -PC2 has only been tested once at 4 km horizontal resolution. This -produced excessive shallow convective cloud (this may or may not be related -to the 70-levels problem above). Since this simulation the erosion -term has been increased dramatically, which may help. We note that -one of the main advantages of PC2, that of a prognostic link of -cloud to convection, is reduced at high resolution, as convection -becomes more explicit rather than diagnosed. We hence see a -resolution limit beyond which it is no longer appropriate to use -PC2. Results look acceptable at 12 km resolution, but we have not -quantitatively explored this limit. - -\subsubsection{Diagnostic evaluation} -One of the principal areas for future cloud scheme development -work planned in the future is in the area of detailed evaluation against -a number of data sources, such as CloudSat, ground based radar, -or case study campaigns. The quantitative evaluation has been -lacking to a significant degree in the development of the scheme, -as the focus has been on tackling qualitatively poor results. -Hence new sources of evaluation work on PC2 would be very welcome. - -\subsubsection{Moisture distribution within the deposition/sublimation term} -The liquid cloud changes in PC2 (or in a non-PC2 run) are based upon -a moisture PDF, as are the deposition/sublimation changes. However, it is -not the same PDF. It has always been the case with the prognostic ice -microphysics term that its PDF, whether explicit or implicit, has not -been rigorously consistent with the PDF used in the calculation of liquid -water, because it was most easily developed that way and produced reasonable -results. It may be useful to investigate whether the two PDF -representations can be brought together in a rigourous way, both for the PC2 -scheme and the \cite{smith90} scheme. - -We have similarly noted potential inconsistencies in the parametrization -of cloud fraction changes between the evaporation of rain term and the -riming (or accretion) term. Again, it might be possible to bring together -these formulations into a single consistent framework. - -\subsubsection{Area cloud fraction representation} -The current area cloud fraction representation is not used when -convection is taking place (signified by the \textit{cumulus} logical). -This inevitably leads to a potential switching between two different values -of the cloud fields if the convective boundary layer (not whether the -convection is shallow or deep) switches on and off, which is undesirable, -although not as bad as switching cloud on and off completely (as for the -current convective cloud formulation). Additionally, it is reasonable to argue that -having an area cloud fraction for cirrus cloud depend upon whether the boundary -layer is well mixed or has shallow convection occuring is not a reasonable link. - -Work in Australia on a TWP-ICE single column model case study using PC2 -suggests the area cloud fraction scheme over estimates the area cloud coverage -for tropical anvil clouds (which exist long after the convection itself has -ceased). This is perhaps not surprising since the \cite{bhi05} area -cloud fraction scheme was evaluated against mid-latitude cloud and it is -known that tropical clouds have greater vertical coherence. Tuning the -parameters in $large_scale_cloud/ls_acf_brooks.F90$ may be beneficial. - -%%\subsection{Acknowledgements} - -\begin{figure} -\begin{center} -\includegraphics[scale=1.0]{pc2_process_explanation} -\caption{Schematic summary of the PC2 cloud scheme.} -\label{fig:schematic} -\end{center} -\end{figure} - -\begin{figure} -\begin{center} -\includegraphics[scale=0.6]{Timestepping_ctl66.epsi} -\caption{Timestepping diagram for the control (non-PC2) scheme} -\label{fig:tstep_diag} -\end{center} -\end{figure} - -\begin{figure} -\begin{center} -\includegraphics[scale=0.6]{Timestepping_pc266.epsi} -\caption{Timestepping diagram for the PC2 scheme} -\label{fig:tstep_prog} -\end{center} -\end{figure} - -\bibliography{refs} -\bibliographystyle{plainnat} - -\end{document} diff --git a/documentation/source/science_guide/cloud_schemes/manual_corrections.txt b/documentation/source/science_guide/cloud_schemes/manual_corrections.txt deleted file mode 100644 index 33e34bf867..0000000000 --- a/documentation/source/science_guide/cloud_schemes/manual_corrections.txt +++ /dev/null @@ -1,44 +0,0 @@ -diff --git a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst -index 9f1988eb..d0664430 100644 ---- a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst -+++ b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst -@@ -4679,26 +4679,29 @@ prognostic :math:`C_l` and :math:`q_{cl}` are incremented as follows: - - - If :math:`{q_{cl}}_{diag} > q_{cl}`: - -- :math:`\Delta q_{cl} = {q_{cl}}_{diag} - q_{cl} \quad -- \refstepcounter{equation}(\theequation)\label{eq:dqcl_init}` -+ .. math:: :label: eq:dqcl_init -+ -+ \Delta q_{cl} = {q_{cl}}_{diag} - q_{cl} - - - If :math:`Q_C < 0`: - -- :math:`\Delta C_{l} = \frac{\Delta q_{cl}}{{q_{cl}}_{diag}} \left( -- {C_{l}}_{diag} - C_{l} \right) \quad -- \refstepcounter{equation}(\theequation)\label{eq:dcl_init1}` -+ .. math:: :label: eq:dcl_init1 -+ -+ \Delta C_{l} = \frac{\Delta q_{cl}}{{q_{cl}}_{diag}} -+ \left( {C_{l}}_{diag} - C_{l} \right) - - - If :math:`Q_C > 0`: - -- :math:`\Delta C_{l} = \frac{\Delta SD}{{SD}_{diag}} \left( {C_{l}}_{diag} - -- C_{l} \right) \quad -- \refstepcounter{equation}(\theequation)\label{eq:dcl_init2}` -+ .. math:: :label: eq:dcl_init2 -+ -+ \Delta C_{l} = \frac{\Delta SD}{{SD}_{diag}} -+ \left( {C_{l}}_{diag} - C_{l} \right) - - - Otherwise: - -- :math:`\Delta q_{cl} = 0` -+ .. math:: \Delta q_{cl} = 0 - -- :math:`\Delta C_{l} = 0` -+ .. math:: \Delta C_{l} = 0 - - where the subscript :math:`_{diag}` denotes the liquid cloud water - content and fraction predicted by the diagnostic cloud scheme (either diff --git a/documentation/source/science_guide/cloud_schemes/refs.bib b/documentation/source/science_guide/cloud_schemes/refs.bib deleted file mode 100644 index 136ad7991e..0000000000 --- a/documentation/source/science_guide/cloud_schemes/refs.bib +++ /dev/null @@ -1,371 +0,0 @@ -%@string{qj="Q. 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Morcrette}, - title = {Modification of the thermodynamic variability closure in the Met Office Unified Model prognostic cloud scheme}, - journal = {Atmospheric Science Letters}, - year = {2020}, - volume = {}, - pages = {}, - doi = {10.1002/asl.1021}, -} - -@Article{vanweverberg2020, - author = {Van Weverberg, K. and C.J. Morcrette and I.A. Boutle}, - title = {Bi-modal diagnostic cloud fraction parameterization. Part I: Motivating analysis and scheme description}, - journal = mwr, - year = {2020}, - volume = {}, - pages = {-}, -} - diff --git a/documentation/source/science_guide/cloud_schemes/um_call_tree.tex b/documentation/source/science_guide/cloud_schemes/um_call_tree.tex deleted file mode 100644 index 97d6a73f15..0000000000 --- a/documentation/source/science_guide/cloud_schemes/um_call_tree.tex +++ /dev/null @@ -1,530 +0,0 @@ - -% Latex source to make a diagram of the UM subroutine call tree, showing the -% locations of all cloud scheme calls. This diagram is included in both -% UMDP 029 (large-scale cloud scheme) and UMDP 030 (PC2). - -% NOTE: any preamble text required for this should be put in the file -% um_call_tree_preamble.tex, which is also inlcuded in both the UMDPs. - -Subroutines only called for the \textcolor{blue}{Smith} scheme are highlighted -in \textcolor{blue}{blue}, those only called for \textcolor{mygreen}{PC2} are -in \textcolor{mygreen}{green}, and those only called for -the \textcolor{purple}{bimodal} scheme are in \textcolor{purple}{purple}. - -\subsubsection{Main Tree from atm\_step\_4a} - -\begin{itemize} - -\item {\bf atm\_step\_4a} \\* -(performs one timestep of the Unified Model...) - \begin{itemize} - - \begin{tcolorbox}[enhanced jigsaw, breakable] - \item {\bf atm\_step\_alloc\_4a} \\* - (does miscellaneous initialisations in atm\_step) - - \begin{itemize} - - \item \textcolor{mygreen}{\bf pc2\_rhtl} \\* - (calculate start-of-timestep Relative Humidity, used by PC2 initiation) - - \end{itemize} - \end{tcolorbox} - - \begin{tcolorbox}[enhanced jigsaw, breakable] - \item {\bf atmos\_physics1} \\* - (calls explicit ``slow'' physics routines...) - \begin{itemize} - - \begin{tcolorbox} - \item {\bf microphys\_ctl} \\* - (interface to microphysics scheme) - \begin{itemize} - - \item \textcolor{mygreen}{\bf pc2\_turbulence\_ctl} \\* - (Perform optional erosion of liquid-cloud; - done here if NOT doing erosion after convection, - e.g. if no convection scheme is used). - \begin{itemize} - - \item \textcolor{mygreen}{\bf pc2\_hom\_conv} \\* - (called here just to do erosion) - - \end{itemize} - - \item \textcolor{blue}{\bf ls\_cld} \\* - (Smith scheme without area cloud fraction calculation, - to set initial cloud fields passed into microphysics) - - \item {\bf ls\_ppn} \\* - (microphysics scheme) - - \item {\bf mphys\_turb\_gen\_mixed\_phase} \\* - (turbulent production of liquid cloud) - - \item \textcolor{mygreen}{\bf pc2\_turbulence\_ctl} \\* - (optionally use the PC2 pdf-width-change code to calculate - the cloud-fraction change from the above turbulent production - of liquid cloud) - - \begin{itemize} - - \item \textcolor{mygreen}{\bf pc2\_hom\_conv} \\* - (called here just to calculate the cloud fraction increment - consistent with the turbulent qcl increment) - - \end{itemize} - - \end{itemize} - \end{tcolorbox} - - \begin{tcolorbox} - \item {\bf rad\_ctl} \\* - (interface to radiation scheme) - \begin{itemize} - - \item {\bf sw\_rad} \\* - (short-wave radiation scheme) - - \item \textcolor{mygreen}{\bf pc2\_homog\_plus\_turb} \\* - (PC2 homogeneous forcing of liquid-cloud by SW radiation heating) - - \item {\bf lw\_rad} \\* - (long-wave radiation scheme) - - \item \textcolor{mygreen}{\bf pc2\_homog\_plus\_turb} \\* - (PC2 homogeneous forcing of liquid-cloud by LW radiation tendency) - - \end{itemize} - \end{tcolorbox} - - \begin{tcolorbox} - \item {\bf atmos\_physics1\_alloc\_pc2} - (wrapper for PC2 self-consistency checks at end of atmos\_physics1) - - \begin{itemize} - - \item Add increments from microphysics + radiation onto - start-of-timestep fields to form updated fields. - - \item \textcolor{mygreen}{\bf pc2\_checks} \\* - (self-consistency checks on cloud fractions and water contents) - - \item Convert corrected updated fields back to increments. - - \end{itemize} - \end{tcolorbox} - - \end{itemize} - \end{tcolorbox} - - Begin loop over solver outer cycles - - \begin{itemize} - - \begin{tcolorbox}[enhanced jigsaw, breakable] - \item {\bf atm\_step\_phys\_reset} \\* - (for PC2, on subsequent solver outer cycles, - reset cloud-fractions to saved values after atmos\_physics1) - \end{tcolorbox} - - \begin{tcolorbox}[enhanced jigsaw, breakable] - \item {\bf eg\_sl\_moisture} \\* - (large-scale advection of cloud water contents and fractions) - \end{tcolorbox} - - \begin{tcolorbox}[enhanced jigsaw, breakable] - \item \textcolor{mygreen}{\bf pc2\_pressure\_forcing\_only} \\* - (Optionally calculate homogeneous forcing of liquid cloud by the - pressure change along the trajectory from departure point to - arrival point). - \begin{itemize} - - \item \textcolor{mygreen}{\bf pc2\_homog\_plus\_turb} \\* - (generic homogeneous forcing routine used here). - - \end{itemize} - \end{tcolorbox} - - \begin{tcolorbox}[enhanced jigsaw, breakable] - \item {\bf atmos\_physics2} \\* - (calls ``fast'' physics routines...) - - \begin{itemize} - - \begin{tcolorbox} - \item {\bf ni\_bl\_ctl} \\* - (interface to explicit boundary-layer and surface scheme calls, - including calculation of TKE and TKE-based $RH_{crit}$) - \end{tcolorbox} - - \begin{tcolorbox} - \item \textcolor{purple}{\bf bm\_calc\_tau} \\* - (calculates turbulence properties used in the bimodal cloud scheme, - based on the boundary-layer scheme TKE and mixing-length) - \end{tcolorbox} - - \begin{tcolorbox} - \item {\bf cloud\_call\_b4\_conv} \\* - (routine for optional cloud-scheme calls before convection) - \begin{itemize} - - \item \textcolor{blue}{\bf ls\_arcld} \\* - (Smith scheme with area cloud fraction; - see \ref{subsubsec:smith_acf} for a drill-down inside this routine) - - \item \textcolor{purple}{\bf bm\_ctl} \\* - (bimodal scheme) - - \item \textcolor{purple}{Set area cloud fraction equal to bulk - cloud fraction} - - \item \textcolor{mygreen}{\bf pc2\_initiation\_ctl} \\* - (interface to PC2 initiation and consistency-checks; - see \ref{subsubsec:pc2_initiation} for a drill-down inside this - routine) - - \end{itemize} - \end{tcolorbox} - - \begin{tcolorbox} - \item {\bf ni\_conv\_ctl} or {\bf other\_conv\_ctl} \\* - (interface routines to various convection schemes...) - \begin{itemize} - - \item {\bf glue\_conv\_5a/6a} \\* - (calls deep, shallow and mid-level convection schemes) - \begin{itemize} - - \item{\bf deep/shallow/mid\_conv} \\* - (convection scheme main routines) - \begin{itemize} - - \item {\bf convec2} - (completes lifting of the convective parcel by one model-level) - \begin{itemize} - - \item {\bf parcel} - (calculates new parcel properties at next level) - - \item {\bf environ} - (calculates grid-mean increments to primary fields; - includes PC2 partitioning of detrained condensate mass - between liquid and ice phases) - - \item \textcolor{mygreen}{\bf pc2\_environ} - (calculates increments to PC2 cloud fractions due to - convective detrainment and subsidence) - - \end{itemize} - - \end{itemize} - - \end{itemize} - - \item \textcolor{mygreen}{\bf pc2\_from\_conv\_ctl} \\* - (PC2 calculations after convection) - \begin{itemize} - - \item \textcolor{mygreen}{\bf pc2\_hom\_conv} \\* - (homogeneous forcing by convection, and erosion of liquid-cloud) - - \end{itemize} - - \end{itemize} - \end{tcolorbox} - - \begin{tcolorbox} - \item {\bf ni\_imp\_ctl} \\* - (interface to boundary-layer implicit solver) - \begin{itemize} - - \item {\bf imp\_solver} \\* - (implicitly solves vertical diffusion to find $T_l$ and $q_T$ - updated by turbulent fluxes). - - \item \textcolor{mygreen}{\bf pc2\_bl\_inhom\_ice} \\* - (inhomogeneous forcing of ice-cloud) - - \item \textcolor{mygreen}{\bf pc2\_delta\_hom\_turb} \\* - (homogeneous forcing of liquid cloud by the turbulent fluxes) - - \item \textcolor{mygreen}{\bf pc2\_bl\_forced\_cu} \\* - (adds diagnosed ``forced cumulus'' cloud fraction and water content - onto the PC2 prognostics) - - \item Calculate area cloud fraction: - - \textcolor{mygreen}{\bf ls\_acf\_brooks} \\* - (for the Brooks epirical method) - - \textcolor{mygreen}{\bf pc2\_hom\_arcld} \\* - (for the Cusack vertical interpolation method) - \begin{itemize} - - \item \textcolor{mygreen}{\bf pc2\_homog\_plus\_turb} \\* - (generic homogeneous forcing routine used to interpolate) - - \end{itemize} - - \item \textcolor{blue}{\bf ls\_arcld} \\* - (interface to diagnostic Smith scheme and area cloud fraction; - see \ref{subsubsec:smith_acf} for a drill-down inside this routine) - - \item \textcolor{purple}{\bf bm\_ctl} \\* - (bimodal cloud scheme) - - \item \textcolor{purple}{Set area cloud fraction equal to bulk - cloud fraction} - - \item {\bf diagnostics\_bl} \\* - (outputs boundary-layer diagnostics to STASH) - \begin{itemize} - - \item {\bf ls\_cld} \\* - (Smith scheme used here to calculate various diagnostics of - near-surface temperature and humidity, by extrapolating pressure, - $T_l$ and $q_t$ down to the desired height and then - re-diagnosing $q_{cl}$. - - \end{itemize} - - \end{itemize} - \end{tcolorbox} - - \end{itemize} - \end{tcolorbox} - - \begin{tcolorbox}[enhanced jigsaw, breakable] - \item {\bf atm\_step\_ac\_assim} \\* - (interface to Data Assimilation analysis increments...) - - \begin{itemize} - - \item {\bf ac\_ctl} - (control routine for Data Assimilation analysis increments...) - \begin{itemize} - - \item{\bf ac} - (main analysis increment routine) - - \item \textcolor{mygreen}{\bf pc2\_assim} \\* - (PC2 reponse to the analysis increments; - see \ref{subsubsec:pc2_assim} for a drill-down inside this routine) - - \item \textcolor{mygreen}{\bf ls\_acf\_brooks} - (calculate area cloud fraction using Brooks empirical method if active) - - \item \textcolor{blue}{\bf ls\_arcld} - (call diagnostic Smith scheme with area cloud fraction again to - account for the analysis increments; - see \ref{subsubsec:smith_acf} for a drill-down inside this routine) - - \end{itemize} - - \end{itemize} - \end{tcolorbox} - - \begin{tcolorbox}[enhanced jigsaw, breakable] - \item {\bf eg\_sl\_helmholtz} \\* - (dynamics pressure solver; updates pressure, and the winds used - to perform advection on the next solver outer cycle) - \end{tcolorbox} - - \end{itemize} - - End loop over solver outer cycles - - \begin{tcolorbox}[enhanced jigsaw, breakable] - \item \textcolor{mygreen}{\bf pc2\_pressure\_forcing} \\* - (interface to miscellaneous PC2 calculations at end-of-timestep) - \begin{itemize} - - \item \textcolor{mygreen}{\bf pc2\_homog\_plus\_turb} \\* - (homogeneous forcing of liquid-cloud by the dynamics pressure change; - optionally either uses total pressure change including the - Lagrangian component following the winds, or only the Eulerian - component from the dynamics solver) - - \item \textcolor{mygreen}{\bf pc2\_initiation\_ctl} \\* - (interface to PC2 initiation and consistency-checks; - see \ref{subsubsec:pc2_initiation} for a drill-down inside this routine) - - \end{itemize} - \end{tcolorbox} - - \begin{tcolorbox}[enhanced jigsaw, breakable] - \item {\bf qt\_bal\_cld} \\* - (calculates end-of-timestep cloud state consistent with final pressure...) - \begin{itemize} - - \item \textcolor{blue}{\bf ls\_arcld} \\* - (interface to diagnostic Smith scheme and area cloud fraction; - see \ref{subsubsec:smith_acf} for a drill-down inside this routine) - - \item \textcolor{purple}{\bf bm\_ctl} \\* - (bimodal cloud scheme) - - \item \textcolor{purple}{Set area cloud fraction equal to bulk - cloud fraction} - - \end{itemize} - \end{tcolorbox} - - \begin{tcolorbox}[enhanced jigsaw, breakable] - \item {\bf iau} \\* - (incremental analysis update; part of data assimilation) - \begin{itemize} - - \item \textcolor{mygreen}{\bf pc2\_assim} \\* - (PC2 reponse to the analysis increments; - see \ref{subsubsec:pc2_assim} for a drill-down inside this routine) - - \item \textcolor{mygreen}{\bf initial\_pc2\_check} \\* - (wrapper for optional self-consistency checks on prognostic cloud variables - if not doing PC2 response to analysis increments) - - \begin{itemize} - - \item \textcolor{mygreen}{\bf pc2\_checks} \\* - (self-consistency checks on cloud fractions and water contents) - - \end{itemize} - - \end{itemize} - \end{tcolorbox} - - \end{itemize} - -\end{itemize} - - -Drill-downs within some routines in the call tree are listed separately below, -to avoid duplication -(since these routines are called in multiple different places in the tree)... - -\subsubsection{Smith scheme with area cloud fraction} -\label{subsubsec:smith_acf} - -\begin{itemize} - -\begin{tcolorbox}[enhanced jigsaw, breakable] -\item \textcolor{blue}{\bf ls\_arcld} \\* -(interface to diagnostic Smith scheme and area cloud fraction) - \begin{itemize} - - \item If no area cloud fraction scheme: - - \textcolor{blue}{\bf ls\_cld} \\* - (just directly call Smith scheme) - - Set area cloud fraction equal to bulk cloud fraction. - - \item If using Cusack vertical interpolation method: - - Interpolate fields onto finer vertical grid - - \textcolor{blue}{\bf ls\_cld} \\* - (call Smith scheme using higher vertical resolution fields) - - Coarse-grain cloud fields back to model grid, but set area cloud - fraction to max of bulk cloud fraction over corresponding fine-grid levels. - - \item If using Brooks empirical area cloud fraction method: - - \textcolor{blue}{\bf ls\_cld} \\* - (just directly call Smith scheme) - - \textcolor{blue}{\bf ls\_acf\_brooks} \\* - (estimate area cloud fraction) - - \end{itemize} -\end{tcolorbox} - -\end{itemize} - - -\subsubsection{PC2 initiation} -\label{subsubsec:pc2_initiation} - -\begin{itemize} - -\begin{tcolorbox}[enhanced jigsaw, breakable] -\item \textcolor{mygreen}{\bf pc2\_initiation\_ctl} \\* -(interface to PC2 initiation and consistency-checks) - \begin{itemize} - - \item \textcolor{mygreen}{\bf pc2\_checks} \\* - (self-consistency checks on cloud fractions and water contents) - - \item PC2 initiation of liquid-cloud: - - \textcolor{mygreen}{\bf pc2\_bm\_initiate} \\* - (using the bimodal cloud scheme) - - \textcolor{mygreen}{\bf pc2\_arcld} \\* - (using the Smith scheme with the Cusack vertical interpolation method) - - \begin{itemize} - - \item \textcolor{mygreen}{\bf pc2\_initiate} \\* - (initiation using the Smith scheme, - called here on a finer vertical grid as per the Cusack method) - - \end{itemize} - - \textcolor{mygreen}{\bf pc2\_initiate} \\* - (using the Smith scheme with no area cloud representation) - - \item \textcolor{mygreen}{\bf pc2\_checks2} \\* - (further self-consistency checks on cloud-fractions) - - \item \textcolor{mygreen}{\bf pc2\_checks} \\* - (repeat the first lot of self-consistency checks again, - just in case we broke something in the mean-time!) - - \item \textcolor{mygreen}{\bf pc2\_hom\_arcld} \\* - (finds area cloud fraction using a version of the Cusack method, - where the cloud fraction on the finer vertical grid is estimated by - applying homogeneous forcing relative to the original grid fields) - - \begin{itemize} - - \item \textcolor{mygreen}{\bf pc2\_homog\_plus\_turb} \\* - (generic homogeneous forcing routine used to interpolate) - - \end{itemize} - - \end{itemize} -\end{tcolorbox} - -\end{itemize} - - -\subsubsection{PC2 Data Assimilation} -\label{subsubsec:pc2_assim} - -\begin{itemize} - -\begin{tcolorbox}[enhanced jigsaw, breakable] -\item \textcolor{mygreen}{\bf pc2\_assim} \\* -(PC2 reponse to the analysis increments) - \begin{itemize} - - \item \textcolor{mygreen}{\bf pc2\_homog\_plus\_turb} \\* - (generic PC2 homogeneous forcing routine used here for liquid-cloud) - - \item Estimate change in ice-cloud fraction from the assimilation - increment to ice-cloud mass. - - \item \textcolor{mygreen}{\bf pc2\_total\_cf} \\* - (update bulk cloud fraction due to change in ice cloud fraction) - - \item \textcolor{mygreen}{\bf pc2\_checks} \\* - (self-consistency checks on prognostic cloud fractions and - water contents) - - \end{itemize} -\end{tcolorbox} - -\end{itemize} diff --git a/documentation/source/science_guide/cloud_schemes/um_call_tree_preamble.tex b/documentation/source/science_guide/cloud_schemes/um_call_tree_preamble.tex deleted file mode 100644 index 009537991a..0000000000 --- a/documentation/source/science_guide/cloud_schemes/um_call_tree_preamble.tex +++ /dev/null @@ -1,25 +0,0 @@ - -% Packages needed for the UM subroutine tree diagram in um_call_tree.txt - -% Used to colour-code things in the subroutine call tree diagram: -\usepackage{xcolor} -% Define a darker green, as in some pdf viewers the standard green -% is too bright to be readable on the grey background. -\definecolor{mygreen}{rgb}{0.0, 0.667, 0.0} - -% Allow more deeply nested lists, for writing the subroutine call tree: -\usepackage{enumitem} -\setlistdepth{20} -\renewlist{itemize}{itemize}{20} -\setlist[itemize]{label=$\cdot$} -\setlist[itemize,1]{label=\textcolor{black}{$\bullet$}} -\setlist[itemize,2]{label=\textcolor{blue}{$\bullet$}} -\setlist[itemize,3]{label=\textcolor{purple}{$\bullet$}} -\setlist[itemize,4]{label=\textcolor{red}{$\bullet$}} -\setlist[itemize,5]{label=\textcolor{orange}{$\bullet$}} -\setlist[itemize,6]{label=\textcolor{yellow}{$\bullet$}} -\setlist[itemize,7]{label=\textcolor{green}{$\bullet$}} -\setlist[itemize,8]{label=\textcolor{cyan}{$\bullet$}} - -% Used to draw boxes around subroutines in the call tree diagram: -\usepackage[most]{tcolorbox} \ No newline at end of file diff --git a/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.tex b/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.tex deleted file mode 100644 index a2ae72a916..0000000000 --- a/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.tex +++ /dev/null @@ -1,5639 +0,0 @@ -\documentclass{UMDP_article} - -\title{The Parametrization of Boundary Layer Processes} -\paperno{024} -\umversion{13.8} -\owner{Adrian Lock} -\author{A.~Lock, J.~Edwards and I.~Boutle} - -\usepackage{amsmath,amssymb} -\usepackage{algorithmic} -\usepackage{natbib} -\usepackage{indentfirst} - -\newcommand{\citeumdp}[1]{:umdp:`#1`} - -\input{newcommand} -\def\gtapp{\raisebox{-.4ex}{$\ \stackrel{>}{{\scriptstyle \sim}} \ $}} - -%\def\captionarb#1{\caption{#1}} -\def\captionarb#1{ - \protect\footnotesize - \caption{\protect\footnotesize#1} - \protect\normalsize\protect\vspace{0.7cm}} -\def\comment#1{({\em #1})} -%\def\comment#1{} % suppress comments -\def\nocomment#1{} - -\begin{document} -\maketitle - -\tableofcontents - -\newpage - -\section{Introduction and code versions} - -This is the documentation for the ``boundary layer'' parametrization -of vertical turbulent transports of heat, moisture and horizontal -momentum. It includes surface exchange but \emph{not} the -parametrization of the surface itself. This is covered within the -surface (JULES) documentation. Although commonly referred to as the -``boundary layer'' parametrization, it includes a free-tropospheric -component. Turbulent fluxes are calculated up to ``BL\_LEVELS'' which -is currently set so that the entire troposphere is included. - -Generally speaking, only version 9C of the boundary layer -parametrization will be documented here as it is now the only -supported version. However, the documentation still makes occasional -references to previous versions of the scheme (8A, 9B) as it is useful -to retain the history of how we have got to where we are. Version 9 -interfaces to the JULES surface code, which is now the only supported -surface code within the UM. - -Several options for higher order closures are available in the 1A -version of the UM boundary layer scheme and these are documented -separately in \citeumdp{025}. - -\section{Model variables and turbulence closure} -\label{sec:closure} - -Given source terms, ${\cal S}$ say, from processes other than boundary -layer turbulence, Reynolds' averaging gives the following equation for -conserved scalar variables, $\chi$, and the two horizontal components -of momentum, ${\bf u}$ on a sphere gives: -\begin{eqnarray} - \frac{\partial \chi}{\partial t} - &=& - \frac{1}{r^2 \rho} \, \frac{\partial }{\partial z} \left( r^2 \rho \overline{w'\chi'} \right) - + {\cal S} - \label{cons_eqn_scal} \\ - \frac{\partial {\bf u}}{\partial t} - &=& \frac{1}{r^2 \rho} \, \frac{\partial }{\partial z} \left( r^2 {\bf \tau} \right) - + {\cal S} - \label{cons_eqn_uv} -\end{eqnarray} -where $\overline{w'\chi'}$ and ${\bf \tau}$ are the vertical turbulent -fluxes to be parametrized, $r$ is the height from the centre of the -planet and $\rho$ is density. The scalar variables treated by -(\ref{cons_eqn_scal}), which are approximately conserved under moist -adiabatic ascent, are: -\begin{eqnarray} - \thetal &=& T_L + \frac{g}{c_p} z = T - \frac{L}{c_p} \ql - - \frac{L_s}{c_p} q_f + \frac{g}{c_p} z \label{thetal} \\ - q_t &=& q_v + \ql + q_f \label{qt} -\end{eqnarray} -where $T$ is temperature, $q_v$ is specific humidity, $\ql$ and $q_f$ -the specific liquid and frozen water contents respectively, and -$L_s=L+L_f$ is the latent heat of sublimation. Note that $\thetal$ is -based on `liquid/frozen water static energy' ($= c_p T + g z - L \ql - -L_s q_f$) rather than potential temperature, $\theta$. Note also that -the option to use mixing ratios in the boundary layer code instead of -specific quantities is also available and the details of the necessary -changes are documented in appendix~\ref{app:mixratio}. Ultimately -turbulent motions are dissipated as heat and so the source term ${\cal - S}$ in (\ref{cons_eqn_scal}) can include an approximation for that -frictional heating, as described in appendix~\ref{app:fricheat}. -Finally, the ice cloud contributions in (\ref{thetal}) and (\ref{qt}) can -optionally be ignored (l\_noice\_in\_turb), which will be more appropriate if -the time scales for ice melting or sublimation are longer than those of the -turbulence (and so may be more appropriate if the ice itself is not being -mixed by parametrized turbulence). In this instance the saturation humidity -will be calculated with respect to liquid water at all temperatures and, for -consistency, only liquid cloud fractions with be considered. - -An additional variable used for diagnostic purposes is -\begin{equation} - \thetavl = \thetal (1 + c_v q_t) - \label{thetavl} -\end{equation} -where $c_v=(1/\epsilon) -1$ and $\epsilon$ is the ratio of the -molecular weights of water vapour and dry air (\mbox{i.e.}, $\epsilon -= M_v/M_a \approx 0.62198$). Thus $\thetavl$ is a conserved variable -that is equal to virtual potential temperature ($\theta_v$) in -cloud-free air and so is used as a simplified measure of buoyancy. - -A `first-order' closure is used to parametrize the turbulent fluxes, -although non-local terms are also included. Under the 9C scheme, an -alternative methodology is optionally available, see section -\ref{sec:rev_flux_grad}. The standard closures are: -\begin{eqnarray} - \overline{w'\chi'} &=& - K_h \frac{\partial \chi}{\partial z} + \khsurf \gamma_{\chi} - \label{scal_closure} \\ - {\bf \tau} &=& K_m \frac{\partial {\bf u}}{\partial z} + {\bf \tau}^{nl} - \label{uv_closure} -\end{eqnarray} - -Separate eddy-diffusivities are calculated for momentum, $K_m$, and -for scalar variables, $K_h$. The second term on the right hand side -represents a non-local flux in unstable boundary layers. Currently it -is only applied for transport arising from surface-driven turbulence -($\khsurf$) and is non-zero only for $\chi=\thetal$, as described in -section~\ref{sec:gradadj}. - -Thus, the parametrization reduces to determining $K_h$, $K_m$ and -$\gamma_{\chi}$ and ${\bf \tau}^{nl}$. Two methods are used to -determine $K_h$ and $K_m$ and how they are combined for -(\ref{scal_closure}) and (\ref{uv_closure}) is described in -section~\ref{sec:shear}. The first method is a local Richardson -number ($Ri$) based scheme. It is calculated for all regimes (but -will be responsible for all mixing in stable conditions), over all -levels up to the specified BL\_LEVELS and is described in -section~\ref{sec:local}. The second method is a non-locally specified -profile scheme. This is exclusively for unstable boundary layers, is -calculated up to level NL\_BL\_LEVELS (typically around 6km AMSL) and -is described in more detail in section~\ref{sec:nonlocal}. In this -regime, mixing is assumed to occur in (or lead rapidly to the -formation of) well-mixed layers (in which conserved variables are -approximately uniform with height) that are capped by an inversion. -Mixing is assumed to be driven either from the surface in a `surface -mixed layer' (SML, by a positive surface buoyancy flux and by surface -stresses) or by cloud-top buoyancy sources (radiative and evaporative -cooling, see appendix~\ref{app:vscales}). As described in section -\ref{sec:nonlocal}, separate $K$-profiles are used for these two -turbulence sources. If the cloud-top sources generate mixing -throughout the SML the layer is said to be `coupled' but if the -$K$-profile representing surface-driven mixing does not extend up to -cloud-top, the layer is referred to as being `decoupled'. As -decoupled layers are restricted to being buoyancy driven and typically -below 6km, they are referred to as decoupled stratocumulus (DSC) -layers. The calculation of $\gamma_{\chi}$ is described in -section~\ref{sec:gradadj} and, finally, fluxes across the top of both -SML and DSC layers (the entrainment fluxes) are specified explicitly -through a separate entrainment parametrization, as described in -section \ref{sec:entr}. - -The strategy used to determine precisely where and when the resulting -eddy-diffusivities should be applied is described in -section~\ref{sec:types}. The buoyancy parameters, finite difference -and other notation used here are defined in appendices~\ref{app:buoyp} -and \ref{app:not}. Further papers describing this scheme and its -performance are \cite{lock00} (noting the corrigendum in -\cite{locketal01_corr}), \cite{martin00:_new_bound_layer_mixin_schem}, -\cite{lock01}, \cite{bushetal1999} and -\cite{brown08:_upgrad_bound_layer_schem_met}. - -%------------------------------------------------------------------------ -% DIAGNOSIS OF TYPE AND DEPTHS -%------------------------------------------------------------------------ -\section{Diagnosis of boundary layer depth and type} -\label{sec:types} - -The non-locally specified $K$-profiles require the height of the base -and top of the layer to be diagnosed (see section~\ref{sec:nonlocal}). -Furthermore, as stated in section~\ref{sec:closure}, the mixing -generated by the non-local $K$ profiles is assumed to occur in (or -lead rapidly to the formation of) well-mixed layers capped by an -inversion. Thus, the accurate diagnosis of their vertical extent is -crucial. How to make this diagnosis is dependent on the boundary -layer mixing regime which has been categorised into 7 distinct -`boundary layer types': -\begin{itemize} -\item{\bf Type I}: Stable boundary layer (with or without cloud) --- - turbulent diffusivities are calculated by the `local' scheme - (section~\ref{sec:local}) -\item{\bf Type II}: Boundary layer with stratocumulus over a stable - near-surface layer --- as Type I but with a turbulently mixed cloud - layer driven from its top (a DSC layer, diagnosis described in - section \ref{sec:decouple}) -\item{\bf Type III}: Well mixed boundary layer --- the classic single - mixed layer which may be cloud-topped or clear but is predominantly - buoyancy-driven (\mbox{c.f.} a possible type VII below) --- - diagnosis described in section~\ref{sec:adiapar}) -\item{\bf Type IV}: Unstable boundary layer with a DSC layer not over - cumulus (see section~\ref{sec:decouple}) --- the surface-based and - cloud-top-driven non-local $K$ profiles may or may not overlap and - cloud-top entrainment can still include the surface forcing (see - section~\ref{sec:entr}) -\item{\bf Type V}: Boundary layer with a DSC layer over cumulus --- - the cumulus (treated by the model's mass-flux convection scheme) - provides coupling with the SML (cumulus diagnosis described in - section \ref{sec:adiapar}) -\item{\bf Type VI}: Cumulus-capped boundary layer --- no turbulent - diffusivities are allowed\footnote{unless the option to mix across - the LCL is selected, see section~\ref{sec:lclmixing}} at or above - the LCL as the mass-flux convection scheme operates here (cumulus - diagnosis described in section~\ref{sec:adiapar}) -\item{\bf Type VII}: Shear-dominated unstable layer --- potentially - wind-shear might allow deeper turbulent mixing in unstable boundary - layers than is apparent purely from the thermodynamic profiles - (sufficient even to inhibit the formation of cumulus); the - possibilities are discussed in section~\ref{sec:shear}. -\end{itemize} -Types I to VI are shown schematically in Fig.~\ref{fig:bltypes}. - -\begin{figure}[tbh] - \centering - \scalebox{0.46}{\includegraphics{wcrp_bltypes1}} - \scalebox{0.46}{\includegraphics{wcrp_bltypes2}} - \captionarb{Schematic representation of boundary layer types I to - VI. The top of the upward arrows indicate the height \zhpar while - the top of their solid line portions indicate \zh.} - \label{fig:bltypes} -\end{figure} - -\subsection{The diagnostic parcel ascent and cumulus diagnosis} -\label{sec:adiapar} - -{\bf Summary}: the depth of the non-local $K$-profiles for -surface-driven turbulence (with NTML grid-levels in the mixed layer -and top at height \zh, as required for (\ref{kmsurf})) is determined -from: -\begin{enumerate} -\item a diagnostic moist parcel ascent; top at grid-level NTPAR, - height \zhpar$=z_{\ntpar+\frac{1}{2}}$. Typically this is an - adiabatic parcel but entraining options are available. -\item a diagnosis of cumulus-capped layers (if cumulus-capped then - NTML and \zh are set to the LCL\footnote{unless the option to mix - across the LCL is selected, see section~\ref{sec:lclmixing}}, if - not then to the parcel top) -\end{enumerate} -Note that this process is only performed for unstable boundary layers -(defined by a positive surface buoyancy flux, \mbox{i.e.}, $\wbs>0$). - -{\bf Step 1:} the method assumes that the height to which turbulent -mixing driven by surface processes can extend in unstable boundary -layers (and therefore the vertical extent of the $K$ profile for -surface-driven turbulence) can be determined solely from the -properties of the thermodynamic profiles. In more detail, the first -step in calculating \zh is to lift a parcel, with properties from the -first grid-level ($k=k_s$) above the top of the surface layer, upwards -allowing for latent heat release. The top of the surface layer is -taken to be at the lower of $z=0.1$\zh (this is then consistent with -the $K$-profiles, see section~\ref{sec:nlsurf}; \zh is taken from the -previous timestep) and the grid-level above which $\thetavl$ starts to -increase with height. The ascent is stopped at the grid-level NTPAR -(height \zhpar$=z_{\ntpar+\frac{1}{2}}$) above which the parcel -becomes more negatively buoyant than a given threshold, $\theta_v'$. -Note that the parcel properties themselves are not perturbed in order -to preserve the height of the mixed-layer's lifting condensation level -(LCL). The calculation of the parcel's buoyancy excess is described -in section~\ref{sec:parxs}. Currently, -\begin{equation} - \theta_v' = \mbox{max} \left[A_{plume}, - \, \mbox{min} \left[ B_{plume} \sigma_{Tv1}, - \, G_{max}\zhe \right] \right] - \label{parcel_pert} -\end{equation} -where $A_{plume}=0.2$, $B_{plume}=3.26$, $G_{max}=10^{-3}$Km$^{-1}$, -$\sigma_{Tv1} = 1.93\, \wthvs/w_m$ and $w_m^3=u_*^3+0.25\,\zhe \wbs$. -Following \cite{holtslag93:_local_versus_nonloc_bound_layer}, -$\theta_v'$ is related to the magnitude of the gradient adjustment, -$\gamma_{\thetal}$ (see section \ref{sec:gradadj}). Thus, -$B_{plume}=A_{ga}$, although somewhat arbitrary limits have been -placed on the magnitude of $\theta_v'$ for numerical security (the -upper limit being consistent with that applied to $\gamma_{\thetal}$ -in (\ref{gradadj}) ). Within limits, then, $\theta_v'$ represents a -typical buoyancy excess of boundary layer plumes. - -The pressure at the LCL, $P_{LCL}=P_{k_s} -(T_{LCL}/T_{k_s})^{(1/\kappa)}$, where $\kappa=R/c_p$. The -temperature at the LCL, $T_{LCL}$, is calculated using approximations -in \cite{Bolton1980} as -\begin{equation*} - T_{LCL} = 55 + \frac{2840}{3.5 \log(T_{k_s}) - log(e_{k_s}) - 4.805} -\end{equation*} -where the vapour pressure of air in grid-level $k_s$, $e_{k_s} = -q_{k_s} P_{k_s}/(100 \, \epsilon)$. The full-level below that -containing the LCL is labelled NLCL and \zlcl$=z_{\nlcl+\frac{1}{2}}$. -If the parcel rises above the top of the LCL transition zone (defined -as 1.1\zlcl, its ascent can also be stopped at the grid-level at which -it has maximum buoyancy excess over the environment. This is -identified as the grid-level above which -\begin{equation*} - \frac{d\theta_v}{dz}|_{\rm env} > - \Gamma_{\rm inv}\, \frac{d\theta_v}{dz}|_{\rm par} -\end{equation*} -where currently the tolerance for identifying inversions by this -method, $\Gamma_{\rm inv}=1.1$. This use of the height of maximum -excess (if lower than that given by the straight buoyancy threshold, -$\theta_v'$) is typically of little consequence in stratocumulus -regions (which tend to be well-mixed beneath large inversions), but -can be necessary in order to identify the capping inversion in cumulus -cases (e.g.~in the trade wind regions). - -{\bf Step 2:} having established \zhpar, a crucial additional test is -to determine whether this layer is well-mixed (\mbox{i.e.}, -stratocumulus-capped) or cumulus-capped. The parcel ascent can rise -to cloud-top in both cases but cumulus cloud layers are observed not -to be as well-mixed as stratocumulus layers. Recall that application -of the $K$ profiles is expected to form or maintain well-mixed layers -and so their current formulation is inappropriate for cumulus cloud -layers. Specifically, a logical flag (CUMULUS) is set to true if -\begin{equation*} - \left| \frac{ \Delta_{\rm cld} q_t}{\Delta_{\rm cld} z} \right| > - C_t \, \left| \frac{ \Delta_{\rm sub} q_t}{\Delta_{\rm sub} z} \right| -\end{equation*} -where the cloud-layer gradient, $ \Delta_{\rm cld} $, is taken between -both NTPAR and NTPAR-1 (to allow for the possibility that a Sc layer -has just deepened by a grid-level) and NLCL and the sub-cloud layer -gradient, $\Delta_{\rm sub}$, between grid-levels NLCL and $k_s$. -Currently the threshold factor, $C_t = 1.1$. If cumulus is diagnosed, -the top of the surface-based mixed layer (\zh) is set to \zlcl (rather -than to \zhpar, as illustrated in Fig.~\ref{fig:bltypes} for types V -and VI). There is then an option to diagnose the thickness of the LCL -transition zone, see section~\ref{sec:lclmixing}. Otherwise, the -boundary layer surface-driven mixing is capped at \zlcl so that mixing -into the cumulus cloud layer is only carried out by the model's -mass-flux convection scheme and not by the eddy viscosity based -boundary layer scheme. Note that basing the CUMULUS diagnosis on -cloud and sub-cloud layer gradients limits the model only to being -able to resolve cumulus with cloud and sub-cloud layers at least 2 -grid-levels (and optionally 400m) thick. Otherwise the layer is -considered well-mixed to \zhpar with an option to include a -representation of fluxes into the capping inversion (see -section~\ref{sec:dzi}). - -If the parcel ascent fails to find an inversion below 3km (or -BL\_LEVELS) but the LCL is below BL\_LEVELS, then the layer is assumed -to be cumulus-capped. If the LCL is above BL\_LEVELS, then again -cumulus is diagnosed with NTML$=\mbox{min}[$NLCL, BL\_LEVELS$-1]$, in -the hope that the mass-flux convection scheme (in its moist or dry -mode) will transport the surface fluxes higher! Clearly this -restriction on the boundary layer scheme is not desirable and so a -value of BL\_LEVELS above the tropopause is recommended. - -Note that if cumulus is not diagnosed then a further, subgrid -estimation of the height of the capping inversion is attempted for \zh -(as described in section~\ref{sec:sginv}). - -\subsubsection{Calculation of parcel buoyancy excess} -\label{sec:parxs} - -As described in appendix~\ref{app:buoyp}, virtual temperature, $T_v = -T(1 + c_v q_v - \ql - q_f)$, is used as the measure of buoyancy. The -condensed water in the parcel at a grid-level $k$ ($\qlf^p$, the -superscript $^p$ indicating parcel properties) is estimated using a -Taylor expansion of $q_s$ about the environment at that grid-level -($q_s^p \approx {q_s}_k + \alpha_L (T^p-T_k) $). Assuming that -$\qlf^p=q_t^p - q_s^p$ gives -\begin{equation} - \qlf^p = \mbox{max}\left[ 0.0, \, a_L \left( q_t^p - {q_s}_k - - \alpha_L (\thetal^p - (g z_k/c_p)-T_k)\right) - \right] - \label{qlpar} -\end{equation} -where the buoyancy parameters $a_L$ and $\alpha_L$ are defined in -appendix \ref{app:buoyp}. Recall that the parcel has $q_t$ and -$\thetal$ taken from grid-level $k_s$ which are conserved during its -ascent. Note that (\ref{qlpar}) will not give condensation until the -parcel becomes saturated. In the environment the cloud scheme will -allow some condensation (and therefore warming and stabilisation of -the environment profile) to take place before the grid-level becomes -saturated in the mean. To allow for this in the parcel (without -applying the cloud scheme), (\ref{qlpar}) is also calculated at each -grid-level but using the environment grid-box mean $q_t$ and $\thetal$ -to give $\qlf^e$. The difference in the environment's condensed water -as determined by the UM cloud scheme (\mbox{i.e.}, $\ql+q_f$) and by -(\ref{qlpar}) (\mbox{i.e.}, $\qlf^e$) is then added to $q^p_{lf}$. - -Given $\qlf^p$, (\ref{thetal}) implies $T^p = \thetal^p - (g z_k/c_p) -+ (L \qlf^p/c_p)$ (using $L_s$ if $T_k$ is below the melting point) -and (\ref{qt}) implies $q_v^p = q_t^p - \qlf^p $ and thus $T_v^p$ can -be calculated. Recall that the diagnosis of the parcel's maximum -buoyancy excess over the environment (described in -section~\ref{sec:adiapar}) required $\theta_v$. This is approximated -as $\theta_v = T_v + (g z_k/c_p)$. - -\subsection{Diagnosis of the vertical extent of the K-profiles} -\label{sec:decouple} - -The diagnosis of mixed layers with turbulence driven from cloud-top -has been separated in to three stages. These are: -\begin{enumerate} -\item diagnose the existence of a decoupled stratocumulus (DSC) layer - with approximately uniform $\thetavl$ (label the top grid-level in - the mixed-layer NTDSC and diagnose the subgrid height of its capping - inversion, \zhsc, see section~\ref{sec:sginv}) -\item diagnose an approximate depth of the DSC layer, $\zml$, in order - to be able to calculate the representative turbulent velocity scales - (see appendix \ref{app:vscales}). -\item calculate the depth of the $K$ profiles (see - section~\ref{sec:nonlocal}) in both SML and DSC layers using - constraints on the TKE budget of the layer. This includes the - diagnosis of recoupling of DSC layers and decoupling of SMLs -\end{enumerate} - -{\bf Step 1}: the diagnosis of DSC layers depends on whether cumulus -convection has been diagnosed. If a cumulus-capped layer under an -inversion within BL\_LEVELS has been diagnosed, grid-levels NTPAR and -NTPAR+1 are tested to see if they contain significant layer cloud -($C_F>$ SC\_CFTOL). This threshold for identifying potentially -turbulently-mixed cloud layers is currently SC\_CFTOL$=0.1$. If there -is significant cloud, NTDSC is set to NTPAR. - -Alternatively, if a well-mixed surface-driven boundary layer was -diagnosed, then from grid-level NTML$+2$ upwards, a cloud-top -grid-level ($k_{ct}$) is sought such that ${C_F}_{k_{ct}}>$ SC\_CFTOL -and ${C_F}_{k_{ct}+1}<$ SC\_CFTOL. If $\Delta_{k_{ct}} \thetavl / -\Delta_{k_{ct}} z < 10^{-3}$Km$^{-1}$ (\mbox{i.e.}, $\thetavl$ is -approximately well-mixed over at least two grid-levels), then NTDSC is -set to $k_{ct}$. If grid-levels $k_{ct}$ and $k_{ct}-1$ are not -well-mixed, grid-levels $k_{ct}-1$ and $k_{ct}-2$ are tested using the -same criterion. If they are not well-mixed either, the cloud-layer is -ignored for the purposes of turbulent mixing. If grid-levels -$k_{ct}-1$ and $k_{ct}-2$ are identified as well-mixed a further test -is applied to determine whether the $\theta_v$ (rather than -$\thetavl$) gradient across grid-levels $k_{ct}$ and $k_{ct}-1$ is -greater than adiabatic (\mbox{i.e.}, whether grid-levels $k_{ct}$ and -$k_{ct}-1$ actually form part of an inversion --- note that by -ignoring the $\ql$ contribution to buoyancy, $\thetavl$ is not a good -variable to use to measure the strength of cloud-capping inversions). -To do this, the $\theta_v$ gradient between grid-levels $k_{ct}$ and -$k_{ct}-1$ is compared with that for a parcel lifted adiabatically -from grid-level $k_{ct}-1$, in exactly the same way as for the SML -parcel ascent (see section \ref{sec:parxs}). If $d\theta_v/dz|_{\rm - env} > \Gamma_{\rm inv} d\theta_v/dz|_{\rm par} $ between -grid-levels $k_{ct}$ and $k_{ct}-1$ then NTDSC is set to $k_{ct}-1$; -if not then NTDSC is set to $k_{ct}$ (recall that grid-levels -$k_{ct}-1$ and $k_{ct}-2$ have already been identified as well-mixed). - -{\bf Step 2} is to diagnose an approximate depth of the DSC layer, -$\zml$. The bottom grid-level of the mixed-layer (NBDSC) is diagnosed -as the lowest grid-level, descending from NTDSC, where -${\thetavl}_{\ntdsc} + \thetavl'$ is less than $\thetavl$ of the -environment. The parcel perturbation is given by -\begin{equation} - \thetavl' = - \, \frac{ \tau_{rc} \Delta_\radf }{z_{rc}} - \label{dscd_pert} -\end{equation} -where $\Delta_\radf$ (Kms$^{-1}$) is the magnitude of the cloud-top -radiative divergence (see appendix~\ref{app:vscales}), $\tau_{rc}$ is -a timescale for the exposure of boundary layer eddies to the cloud-top -radiative cooling (taken to be 200s) and $z_{rc}$ is a depth-scale for -the radiatively cooled layer (taken to be 50m). These values of -$\tau_{rc}$ and $z_{rc}$ are only estimates (and will in reality vary -from one cloud to another) but they are consistent with, for example, -the observations of \cite{nicholls1986}. If the parcel failed to fall -(\mbox{i.e.}, NBDSC equals NTDSC) in a DSC layer {\em not} overlying -cumulus, then the layer is assumed not to be well-mixed. At the top -of a cumulus layer, the DSC layer is given a minimum depth of -$\Delta_{\ntdsc+\frac{1}{2}} z$. Otherwise, the layer depth, $\zml$, -is measured from the top of layer NTDSC to the base of layer NBDSC. - -{\bf Step 3}: the step 2 calculation of $\zml$ is used to calculate -the representative velocity scales for the DSC layer but its -calculation is only crude. Here, the vertical extent of the -$K$-profiles is determined more accurately by ensuring that the -magnitude of the integrated buoyancy consumption of TKE within the -mixed layer is less than or equal to a fraction, $D_t$, of the -buoyancy production, following \cite{turton1987}. - -Following appendix~\ref{app:buoyp} the grid-box mean buoyancy flux can -be written as: -\begin{equation} - \wb = g \left[ (1-C_F) \left(\beta_T \overline{w'\thetal'} + \beta_q \wqt \right) + - C_F \left( \tilde{\beta_T} \overline{w'\thetal'} + \tilde{\beta_q} \wqt \right) - \right] - \label{eq:wb_cont} -\end{equation} - -As standard, the fluxes in (\ref{eq:wb_cont}) are then expanded using -the first-order closure in (\ref{scal_closure}) as: -\begin{eqnarray} - \wthl_k &=& -\khsurf \,\frac{\widetilde{\Delta_k \thetal}}{\Delta_k z} - -\khtop \,\frac{\Delta_k \thetal}{\Delta_k z} \nonumber \\ - \wqt_k &=& -\left(\khsurf + \khtop \right) \, - \,\frac{\Delta_k q_t}{\Delta_k z} - \label{eq:wx_std} -\end{eqnarray} -where $\widetilde{\Delta_k \thetal} = \Delta_k \thetal - -\gamma_{\thetal} \Delta_k z$ in order to include the non-local (or -gradient adjustment) term. If the alternative flux-gradient option is -used, see section \ref{sec:rev_flux_grad}, then additional terms are -needed. - -Large-eddy simulations have demonstrated that the crucial region in -determining when decoupling of stratocumulus will occur (\mbox{i.e.}, -when the $K$ profiles no longer span the entire layer from cloud-top -to the surface) is in a thin layer of unsaturated air just below -cloud-base, where $\wb$ first becomes negative. Thus, in the above -calculation, it is crucial both to have an accurate measure of -cloud-base height (which will have to be subgrid) and to include -successfully this thin unsaturated layer in the buoyancy consumption -integral. Thus, the $\wb$ integration is performed over the cloud and -sub-cloud layers separately and the cloud-fraction is taken to be -uniform within the cloud layer (and zero below cloud-base). The -height of cloud-base is given by (\ref{zc_calc}). - -An iterative method is then used to find the vertical extent of mixing -(within certain bounds, as described below) such that the magnitude of -buoyancy consumption of TKE within the mixed layer equals a fraction, -$D_t$, of the buoyancy production, \mbox{i.e.}, -\begin{equation} - \sum_{z_{k-\frac{1}{2}} > z_i-\zml}^{z_{k-\frac{1}{2}} < z_i} - \left|\left[ \wb|_{z_{k-\frac{1}{2}}}<0 \right]\right| \, \Delta_k z \, - \leq \, D_t \, - \sum_{z_{k-\frac{1}{2}} > z_i-\zml}^{z_{k-\frac{1}{2}} < z_i} - \left[ \wb|_{z_{k-\frac{1}{2}}}>0 \right] \, \Delta_k z - \label{deccrit} -\end{equation} -Note that, for simplicity, the ${\cal E}_h$ factors are not included -in $\khsurf$ or $\khtop$ when calculating (\ref{eq:wx_std}) under the -assumption that they will be small. This process is applied to all -unstable mixed layers. For stratocumulus layers, observations and LES -suggest a value of $D_t=0.1$. A separate value of $D_t$ can be used for -the sub-cloud layer in cumulus capped boundary layers if this method is -used to determine the LCL transition zone thickness, see -section \ref{sec:lclmixing}. For cloud-free mixed layers, $D_t=1$ is -used, purely to keep negative buoyancy fluxes down to a reasonably -realistic level (for example, if the parcel top diagnostic returned -too high a boundary layer depth). - -The first step is to test for whether a well-mixed layer is possible -(either decoupling what has so far been diagnosed as a well-mixed -layer or, if one exists, recoupling a decoupled stratocumulus layer), -\mbox{i.e.}, to test whether (\ref{deccrit}) is satisfied with both -$\khsurf$ and $\khtop$ extending from the surface to the cloud-top. -If recoupling is possible then the various flags identifying the DSC -layer are reset ({\em this includes setting the cumulus diagnosis to - false}), any surface-driven entrainment originally applied at \zh is -added to the entrainment at \zhsc (after rescaling for the inversion -strength at \zhsc) and \zbase is set to 0.1\zh (for the reason -discussed above). If decoupling is diagnosed, \zhsc is set to the -original \zh (inversion height), although the entrainment across this -inversion is not recalculated (and so keeps any surface-driven -component --- the COUPLED flag is therefore set to true, see -section~\ref{sec:entr}). - -If a decoupled layer is diagnosed, then an iteration is performed to -find the highest \zh (so top of the $\khsurf$ profile) that still -satisfies (\ref{deccrit}), but with $\khtop=0$ in (\ref{eq:wx_std}). -The iteration proceeds with \zh stepping from its lowest permissible -height to its highest (currently 3 steps are used). If at any stage -(\ref{deccrit}) is violated, then the step below (therefore containing -the height that would give equality in (\ref{deccrit})) is divided by -4 and 3 of those steps are taken downwards. If (\ref{deccrit}) is met -the step above is again reduced by a factor of 4 and 3 steps taken -upwards. A total of 3 sweeps are possible, each with a smaller step -so that \zh approaches the height that gives equality in -(\ref{deccrit}). The accuracy with which this is achieved will be the -difference in the maximum and minimum permissible heights of \zh -divided by $2\times4\times4 = 32$, which will typically be less than -30m. The top grid-level of the SML, NTML, is defined as the highest -grid-level such that $\khtop$ is non-zero at the half-level above. - -The above process is then repeated to find the appropriate \zbase for -$\khtop$, \mbox{i.e.}, for the base of top-driven mixing. Some -constraints are placed on \zbase, namely that it should never go below -$0.1$\zh (to avoid affecting the continuity of the $K$ profiles at the -top of the surface layer, see (\ref{ws_defn})). If cumulus convection -has been diagnosed then \zbase is not allowed to go below -$z_{\ntml+\frac{1}{2}}$ (unless the layer is diagnosed to recouple -completely). Finally, \zbase must always be at or below -$z_{\ntdsc-1}$, so that mixing in decoupled layers is always resolved, -and at least $\Delta z_{rad}$ (the cloud-top radiative cooling depth -defined in section \ref{sec:wbint_inv}) below the t inversion. The -base grid-level of the DSC layer, NBDSC, is defined (analogously to -NTDSC) as the lowest grid-level such that $\khtop$ is non-zero at the -half-level below. - -A possible extension to this diagnosis would be to include the shear -contribution to the TKE budget in (\ref{deccrit}) and so allow -shear-driven mixing to help maintain well-mixed layers. - -\subsubsection{Surface layer $\wb$ integration} - -In the surface layer, below $z_i/10$, the $K$ profiles have a -different functional form from the rest of the mixed layer. Rather -than include this additional complexity in the $\wb$ integration, the -surface layer is treated separately. In place of the -finite-difference form of $\wb$, see (\ref{eq:wb_cont}) and -(\ref{eq:wx_std}) above, $\wb$ is assumed to be linear between $\wbs$ -at the surface and zero at a level which must be estimated. The -surface layer integration is then from the surface up to $z_{{\rm - K_{SURF}}}$, where $\theta$-level K\_SURF is the first above -$z_i/10$. The level where $\wb$ is zero is found by linear -interpolation across the grid-levels where the diagnosed cloud-free -buoyancy flux would become negative. This is where $\beta_T -\widetilde{\Delta_k \thetal} + \beta_q \Delta_k q_t$ becomes positive -and so where the cloud-free part of $\wb$ (\mbox{i.e.}, that part -below cloud-base which is important for decoupling) becomes negative. - - -\subsubsection{Integration of $\wb$ close to the inversion} -\label{sec:wbint_inv} - -Because of the large gradients often seen in fluxes close to the -inversion (in particular, in the LW radiative flux), simple finite -difference flux calculations, (\ref{eq:wx_std}), can be significantly -inaccurate in this region. An example is shown in -Fig.~\ref{fig:inv_integ}. Calculating $\wthl_{\ntml+\frac{1}{2}}$ -from (\ref{eq:wx_std}) gives a negative value, largely because -$\Delta_{\ntml+1} \thetal$ is positive and so the local flux is large -and negative. In reality, $\wthl$ becomes positive only a short -distance below cloud-top such that the integral here will tend also to -be positive. - -The solution adopted is to integrate $\wb$ analytically across the -region just below the inversion, labelled $\Delta z_{rad}$ in -Fig,~\ref{fig:inv_integ}. Since $\Delta_{\ntml} \thetal$ can also be -significantly positive (when the grid-level inversion is rising or -falling, for example), the base of this region is taken to be the -lower of the first $\theta$-level below $z_h-100$ m (a physically -reasonable depth over which cloud-top radiative cooling might be -expected to occur) and $z_{\ntml-1}$. -\begin{figure}[tb] - \begin{center} - \scalebox{1}{\includegraphics{ideal_invinteg}} - \caption{Subgrid (lines) and model (symbols) fluxes of $\thetal$: - turbulent flux (dash-dotted, crosses), radiative flux (dashed, - triangles) and total flux (solid). The shaded area illustrates - the integrated turbulent flux that would be obtained were - (\protect\mbox{\protect\ref{eq:wx_std}}) used.} - \label{fig:inv_integ} - \end{center} -\end{figure} -Then, -\begin{eqnarray} - \int_{z_h-\Delta z_{rad}}^{z_h} \, \wthl \, dz & = & - \int_{z_h-\Delta z_{rad}}^{z_h} \, F_{\thetal}^{Tot} - - F_{\thetal}^{NT}\, dz - \nonumber \\ - & = & I^{Tot} - I^{rad} - I^{ppn} - \label{wthl_int} -\end{eqnarray} -For the radiative flux, it could be assumed that the subgrid flux -distribution is exponentially dependent on the grid-level LWP, for -example. This would give: -\begin{equation} - I^{rad} = \frac{\Delta z_{rad}} - {\ln(F^{rad}|_{z_h}/F^{rad}|_{z_h-\Delta z_{rad}} ) } - \lb F^{rad}|_{z_h}-F^{rad}|_{z_h-\Delta z_{rad}} \rb - \label{irad} -\end{equation} -However, off-line tests indicated this could give a strong and -spurious sensitivity to $F^{rad}|_{z_h-\Delta z_{rad}}$. Furthermore, -for most realistic scenarios, the logarithmic factor in (\ref{irad}) -tends to be close to 3. Consequently, we approximate $I^{rad} = -\Delta z_{rad} ( F^{rad}|_{z_h}-F^{rad}|_{z_h-\Delta z_{rad}} ) /3 $. -In addition, $F^{rad}|_{z_h}-F^{rad}|_{z_h-\Delta z_{rad}} $ is -approximated as $\Delta \radf$, the radiative flux change across -cloud-top used in the calculation of $\vtopo$ (\ref{ctraddiv}). The -precipitation flux is assumed to vary linearly across this region, as -does the total flux, and so its contribution to $I^{Tot}$ cancels with -$I^{ppn}$ in (\ref{wthl_int}). Finally, for simplicity, the total -flux is taken to be constant and equal to the inversion value, such -that $I^{Tot} = \Delta z_{rad} F^{Tot}|_{z_h}$. With these -approximations, (\ref{wthl_int}) becomes -\begin{eqnarray*} - \int_{z_h-\Delta z_{rad}}^{z_h} \, \wthl \, dz & = & - \Delta z_{rad} \lb -w_e \Delta \thetal + \Delta \radf \rb - - \Delta z_{rad} \Delta \radf / 3 \\ - & = & \Delta z_{rad} \lb -w_e \Delta \thetal + \frac{2}{3} \Delta \radf \rb -\end{eqnarray*} -For the integral of $\wqt$ across this cloud-top region, $\wqt$ is -also taken to be constant so that: -\begin{equation} - \int_{z_h-\Delta z_{rad}}^{z_h} \, \wqt \, dz = - - \Delta z_{rad} w_e \Delta q_t -\end{equation} -The integrated buoyancy flux is then found from (\ref{eq:wb_cont}) -using the mixed layer cloud fraction and buoyancy coefficients -evaluated at the grid-level above $ z_h -\Delta z_{rad} $. - -\subsection{Diagnosis of inversion thickness} -\label{sec:dzi} - -Terminating the diagnostic parcel ascent at its level of neutral -buoyancy ignores any overshooting through the parcel's own inertia as -it enters the inversion region. This overshooting region effectively -defines the depth of the inversion over which the negative entrainment -heat fluxes are seen. Typically this will be small relative to the -model vertical grid but at higher vertical resolution or when a -strongly surface-heated boundary layer is capped by weak stability -inversions could be resolved. Following \cite{beare2008}, a simple -energetic argument gives a realistic prediction of the top of the -inversion, $z_{top}$, in LES from -\begin{equation} - 6.3 \, w_m^2 = \int_{z_{nb}}^{z_{top}} \, b \, dz - \label{dz_param} -\end{equation} -where $z_{nb}$ is the level of neutral buoyancy (found by linear -interpolation between grid-levels), $w_m$ is the boundary layer -velocity scale defined in section \ref{sec:nlsurf} and $b$ is the -parcel buoyancy. Note that the constant in (\ref{dz_param}) is the -same as in \cite{beare2008} because $6.3 = 2.5 * 4^{2/3}$ and $w_m^3$ -differs by a factor of 4. The buoyancy integration in -(\ref{dz_param}), that is itself dependent on $z_{top}$, is performed -working upwards from \zhpar assuming piece-wise linear variation of -$b$ between grid-levels. Note that the standard definition of the -boundary layer top in the UM is the height of the first flux level -below the level of neutral buoyancy, so -\zhpar$=z_{\ntpar+\frac{1}{2}}$. The inversion thickness is then -defined as -\begin{equation} - \Delta z_i = z_{top} -\zhpare - \label{dz_definition} -\end{equation} - -\subsection{Diagnosis of the LCL transition zone thickness} -\label{sec:lclmixing} - -As described in section~\ref{sec:adiapar}, when cumulus convection has -been diagnosed surface-driven mixing was originally capped at \zlcl so -that mixing into the cumulus cloud layer was only carried out by the -model's mass-flux convection scheme. This was seen to lead to errors -in the mean profiles across the LCL, with superadiabats being the most -extreme manifestation. Using the boundary layer parametrization to -couple cloud and sub-cloud layers would have the numerical advantage -of being implicit. There is also observational and LES evidence that -appropriately-scaled buoyancy fluxes up to the LCL are -indistinguishable from those in cloud-free convective boundary layers -and so the non-local surface-driven mixed layer K-profiles remain -accurate up to this level. To diagnose the depth to which these -profiles should penetrate above the LCL, the algorithm given in -section~\ref{sec:decouple} to diagnose the extent of the K-profiles in -decoupled boundary layers can be used (using the switch kprof\_cu). -This ensures that the magnitude of the integrated buoyancy consumption -of TKE within the mixed layer is less than or equal to a fraction, $D_t$, -of the buoyancy production. In cumulus layers, cloudy thermals will -generate positive buoyancy fluxes (and are handled by the convection -scheme) but it is assumed that there will also be cloud-free thermals -within the grid box that may penetrate above the grid-box mean LCL -(but are too dry to reach their own LCL). Thus their buoyancy flux is -given by (\ref{eq:wb_cont}) with $C_F=0$. Restricting the negative -integral of this buoyancy flux then gives a new definition for \zh -that is then used in the calculation of the surface-driven K-profiles -in section~\ref{sec:nlsurf} --- the larger the value of $D_t$, the -higher \zh will be. Typically $D_t=0.1$ for decoupled stratocumulus -layers while idealised clear-sky convective boundary layers (where -the magnitude of the entrainment buoyancy flux is a fraction, $A_1$, -of the surface flux) would have $D_t = A_1^2 \sim 0.05$. For GA7 $D_t$ -has been set to 0.05 for this cumulus transition zone calculation. -Because the Gregory-Rowntree convection scheme triggers from the LCL, that -is used as a minimum constraint on the boundary layer mixing depth (so that -the massflux and turbulence schemes remain coupled). With other convection -schemes this may not be appropriate and so this minimum constraint can be -relaxed, which is achieved by setting it to half the height of the LCL (the -factor of a half is arbitrary, with no sensitivity to this choice given that -the diagnosis parcel reached the LCL, but ensures the iteration starts well -below the LCL). - -%------------------------------------------------------------------------ -% LOCAL SCHEME -%------------------------------------------------------------------------ -\section{The local scheme} -\label{sec:local} - -A first order `mixing length' closure is used: -\begin{eqnarray} - K_m &=& {\cal L}_m^2 \, (S+S_d) \, f_m(Ri) \label{kmlocal}\\ - K_h &=& {\cal L}_h \, {\cal L}_m \, - (S+S_d) \, f_h(Ri) \label{khlocal} -\end{eqnarray} -where ${\cal L}_m$ and ${\cal L}_h$ are the neutral mixing lengths and -$S$ is the resolved vertical shear of the horizontal wind components, -$S = \left| \partial {\bf u}/\partial z \right|$. A representation of -the wind shear, $S_d$, generated by drainage flows in complex terrain -can also be included, as described below. Near the surface simple -finite difference calculations for the vertical gradients can become -inaccurate because of the quasi-logarithmic profiles of variables -\cite[]{Ayra1991}. Currently this is ignored above grid-level 2 and -the neutral mixing lengths are given by -\begin{eqnarray*} - {\cal L}_m &=& \frac{k(z+z_{0m})}{1+k(z+z_{0m})/\lambda_m} \\ - {\cal L}_h &=& \frac{k(z+z_{0m})}{1+k(z+z_{0m})/\lambda_h} -\end{eqnarray*} -where $z_{0m}$ includes the orographic component. For the lowest -interior grid-level ($k=1$) they are calculated, incorporating this -log profile correction, as -\begin{equation*} - \tilde{{\cal L}}_{X,k-1/2} = \frac{k \Delta_{k-1/2} z}{ - ln\left( \frac{z_k + z_{0m}}{z_{k-1} + z_{0m}} \right) - + \frac{k \Delta_{k-1/2} z}{\lambda_X} } -\end{equation*} -If near-surface resolution is increased this logarithmic correction -should be considered over more levels. - -The asymptotic mixing lengths are given by -\begin{eqnarray} - \lambda_m &=&\mbox{max}\left[\lambda_0,\, 0.15 \zloce, 2 h_B \right] \nonumber\\ - \lambda_h &=&\mbox{max}\left[\lambda_0,\, 0.15 \zloce \right] - \label{asymp_ml} -\end{eqnarray} -where $\lambda_0$ is a minimum mixing length read in from the namelist -and \zloc is defined below. The orographic blending height, $h_B$ (only -used within the boundary layer, as defined below), is given by -\begin{equation*} - h_B = {\rm max}\left[z_1+(z_{0m})_{\mbox{veg}}, 2^{1/2} \sigma_h \right] -\end{equation*} -where $\sigma_h$ is the standard deviation of the height of the -subgrid orography and $(z_{0m})_{\mbox{veg}}$ is the vegetative part -of the roughness length. The constants in (\ref{asymp_ml}) can be -considered `tuned' (see, in particular, the operational modifications -described in appendix~\ref{app:opmods}). - -The Richardson number, $Ri$, that is used as a local measure of -stability is given by -\begin{equation} - Ri = \frac{\Delta B / \Delta z}{(S+S_d)^2} - \label{ridefn} -\end{equation} - -The measure of buoyancy used in $Ri$ is -\begin{equation} - \Delta B = g\left( \overline{\beta_T} \Delta \thetal - + \overline{\beta_q} \Delta q_t \right) - \label{Bdefn} -\end{equation} -where $\overline{\beta_T}$ and $\overline{\beta_q}$ are the grid-box -mean (\mbox{i.e.}, cloud weighted) buoyancy coefficients, that can be -defined in two different ways, see appendix~\ref{app:buoyp} and -section~\ref{sec:fd_ri}. Note that (\ref{Bdefn}) reduces -to a virtual temperature approximation of buoyancy in cloud-free air -and that neutral buoyancy (in cloudy as well as cloud-free air) is -implied by vertically uniform $\thetal$ and $q_t$. This is then -entirely consistent with the assumption that $\thetal$ and $q_t$ are -conserved variables within the boundary layer scheme. - -As described in \cite{lock2012}, the wind shear generated by drainage -flows in complex terrain is thought to lead to additional vertical -mixing. This wind shear can be approximated as -\begin{equation*} - S_d = \frac{\Delta B }{ \Delta z} \, \alpha_d \, t_d \, {\cal Z}_d -\end{equation*} -The representative slope of the local terrain, $\alpha_d$, is given by -\begin{equation*} - \alpha_d^2 = \frac{1.0}{ 25.0 + (l_h/\sigma_h)^2} -\end{equation*} -with $l_h$ a specified horizontal scale for the terrain, currently -taken to be 1500 m (empirically derived for Scottish orography in the -UKV), and $\sigma_h$ the standard deviation of the full subgrid -orographic height. $\sigma_h$ should also be taken as the average -over the surrounding area of each grid box (typically 6 to 8 grid -lengths), in order to be representative of the local area over which -such flows will be underresolved. The above formula is used so that -$\alpha_d \sim \sigma_h/l_h $ for small $\sigma_h$ but only tends to -0.2 for large values. To limit the vertical extent of $S_d$ to be -below approximately $z=\sigma_h$, a height-dependent factor is -included, ${\cal Z}_d = 0.5( 1 - {\rm tanh}\left[ 4 ((z/\sigma_h)-1) -\right])$. The timescale, $t_d$, takes a fixed value of 30 minutes, -for simplicity. - -Initially, the lowest half-level at which $Ri>Ri_{crit}$ is taken to -be a measure of the boundary layer top (\zloc) and the full-level -below is designated NTLOC. In general $Ri_{crit}=1$ but a value of -0.25 is recommended for use with the 'SHARPEST' stability functions, -see below. If the boundary layer was diagnosed as cumulus-capped by -the non-local scheme (see section~\ref{sec:types}) then \zloc is -lowered to \zlcl (and $K_h$ and $K_m$ are set to zero from the base of -grid-level NLCL upwards) so that transports into and within the -cumulus cloud layer can be performed solely by the mass-flux -convection scheme. Depending on the switch local\_fa, above NTLOC -turbulently-mixed layers (where $Ri 0$), several forms for the stability -functions are available. The `long-tailed' functions are -\begin{equation*} - f_{\rm stable} = \frac{1}{1+g_0 Ri} -\end{equation*} -Alternative functions, which decrease as $1/Ri^2$ with increasing -stability are, from \cite{louis1979}: -\begin{equation*} - f_{\rm stable} = \frac{1}{(1+ 5 Ri)^2} -\end{equation*} -and the family of ``sharp'' functions can be written in terms of a transitional -Richardson number, $Ri_{t}$, as: -\begin{equation*} - f_{\rm stable} = -\begin{cases} - (1 - 5Ri)^2 & {\rm for}\ 0Ri_{t} -\end{cases} -\end{equation*} -where -\begin{eqnarray} -A_{Ri} & = & \left(1-g_0 Ri_{t}\right)/\left(1- g_0 Ri_{t}/2\right)^2 \nonumber\\ -B_{Ri} & = & (g_0/2) /\left(1 - g_0 Ri_{t}/2\right)^2 -\end{eqnarray} -For the `SHARPEST' function of \cite{derbyshire1997}, $Ri_{t}=0.1$, while -larger values give even sharper reduction of turbulence with increasing $Ri$. -An additional option, used operationally in some configurations (originally -in the Mesoscale Model, hence called 'MES tails'), is to blend linearly -from Louis functions at the surface to SHARPEST by 200m. - -A stability dependent Prandtl number ($Pr=f_m/f_h$) is generally used -following \cite{MailhotLock2004} with: -\begin{equation*} - Pr=\min \left( Pr_{\rm max}, \, Pr_N(1+2Ri) \, \right). -\end{equation*} -The maximum permitted Prandtl number, $Pr_{\rm max}$, is currently set -to $5$ for model stability reasons. The stability functions for $Ri>0$ are -then given by: -\begin{eqnarray*} - f_m & =& \frac{Pr}{Pr_N} \, f_{\rm stable} \\ - f_h & =& \frac{1}{Pr_N} \, f_{\rm stable} -\end{eqnarray*} -Note that writing the functions in this way ensures that $f_m=1$ under -neutral conditions and the effect of the variation in $Pr$ is for $f_m$ -to decrease slower with increasing $Ri$ than $f_{\rm stable}$, which can -be explained through increasing gravity-wave activity. - -Finally, the LEM stable functions are also available which cut off all -turbulence beyond a critical Richardson number, $Ri_c=0.25$: -\begin{eqnarray*} - f_m & =& \left( 1 - \frac{Ri}{Ri_c} \right)^4 \\ - f_h & =& \frac{1}{Pr_N} \left( 1 - \frac{Ri}{Ri_c} \right)^4 (1 - g_{LEM} Ri) -\end{eqnarray*} -\label{stable_stab_lem} -with $g_{LEM}=1.2$. - -\subsection{Finite difference calculations} -\label{sec:fd_ri} - -The Charney-Phillips vertical grid staggering used in the UM stores -the horizontal wind components ($u$, $v$) on grid-levels, -$\rho$-levels, that are staggered relative to scalar variables (such -as $\thetal$ and $q_t$) and vertical velocity, $w$. While much of the -boundary layer scheme is grid-independent, this has serious -implications for the calculation of $Ri$. There are two obvious -possibilities, to calculate $Ri$ (and thence $K(Ri)$) on either -$\theta$-levels or $\rho$-levels and then interpolate either $K_h$ or -$K_m$ to be able to calculate the required fluxes. To do the former -requires averaging the buoyancy gradient in the numerator (and is -referred to by \cite{cullen1994} as the `$\theta$-bar' method), the -latter the wind shear in the denominator (referred to as the -`$\rho$-bar' method). Single-column model and other tests -demonstrated that the `$\rho$-bar' method could readily generate -instabilities just above the top of the boundary layer because -averaging the wind shear into this stable air tended to reduce $Ri$ -and so promote mixing. Fortunately, the `$\theta$-bar' method tended -to increase $Ri$ above inversions and so damp mixing. Thus, $Ri$ is -calculated on $\theta$-levels as -\begin{equation*} - Ri_k = \frac{DBDZ_k} - {(\Delta_{k+\frac{1}{2}} {\bf u}/\Delta_{k+\frac{1}{2}} z)^2} -\end{equation*} -The buoyancy gradient on $\theta$-level $k$ can be calculated in two different ways, -depending on the switch {\rm i\_interp\_local}. The long-standing method is given by -\begin{equation*} - DBDZ_k = g\left( \overline{\beta_T}_{k} (D\thetal DZ)_k - + \overline{\beta_q}_{k} (Dq_t DZ)_k \right) -\end{equation*} -where $\overline{\beta_T}$ and $\overline{\beta_q}$ are the grid-box -mean (\mbox{i.e.}, cloud-fraction weighted) buoyancy coefficients, -defined in appendix~\ref{app:buoyp}. Note that because this is -defined on $\theta$-levels, no vertical interpolation of cloud -variables (fractional area and water contents), to which the buoyancy -coefficients are very sensitive, is required. The volume-weighted -gradients of $\thetal$ and $q_t$ are calculated as -\begin{equation} - (D\chi DZ)_k =\left( (z_{k}-z_{k-\frac{1}{2}}) \, \frac{\Delta_{k+1} \chi}{\Delta_{k+1} z} + - (z_{k+\frac{1}{2}}-z_{k}) \, \frac{\Delta_{k} \chi}{\Delta_{k} z} - \right) / \Delta_{k+\frac{1}{2}} z -\label{gradient_interp} -\end{equation} -as long as $\chi_{k-1}$ is defined on an atmospheric model level. To calculate -$DBDZ_1$, between the surface and the lowest $\theta$-level, either the buoyancy gradient -from level 1 to 2 can be extrapolated, \mbox{i.e.}, -\begin{equation*} - (D\chi DZ)_1 = \frac{\Delta_{2} \chi}{\Delta_{2} z} -\end{equation*} -or surface properties can be used. Over sea, the sea-surface temperature -and $q_{sat}$ can be used. Over a heterogeneous (tiled) land surface the -appropriate moisture variable varies between tiles. For the orographic form drag -(\ref{section_2}), an average $Ri_{SL}$ of the surface layer is calculated but, as -discussed above, subsequent vertical averaging of $Ri$ would potentially be -numerically unstable. In principle, the tile-average of $(Dq_t DZ)_1$ could -be calculated but for now, over land, the grid-box average surface temperature is -used to calculate $(D\thetal DZ)_1$ and $(Dq_t DZ)_1$ is extrapolated from -above (\mbox{i.e.}, $(Dq_t DZ)_1 = (Dq_t DZ)_2)$). - -As noted above, a feature of the previous option is that applying the cloudy -buoyancy coefficients at a cloud top level, $k$ say, to strong gradients -interpolated between $k-1$ and $k+1$, can yield an unstable $DBDZ_k$ despite -strong static stability, especially when the upper level is very dry. This can be -related to cloud-top entrainment instability but this process is intended to be -represented within the non-local scheme. Hence, the alternative method is to -calculate the buoyancy gradient directly on $\rho$-levels and then interpolate this -vertically to give $DBDZ_k$, using (\ref{gradient_interp}). This then -requires a cloud fraction on $\rho$-levels. The difficulty comes where there is -a change in cloud fraction between levels. For this ``edge'' fraction, $f_{edge}$ -(the fraction of the grid-box that is cloudy in one level but not in the other), the -change in supersaturation ($ s = q_t-q_{sat}$) between -levels is used to estimate the vertical fraction likely to contain cloud, $f_{lev}$. -For example, where $C_F$ decreases with height, $f_{lev}={q_c}_{k-1}/(s_{k-1}-s_{k})$, -where $q_c$ is the total condensate, and $f_{lev}$ also constrained to be less -than unity. The total cloud volume fraction -is then given by $f_{tot} = {\rm min}[{C_F}_{k-1},{C_F}_k] + f_{edge}f_{lev}$ and this -is used to weight the saturated contribution to the buoyancy parameters on -$rho$-levels, \mbox{e.g.}, -$\overline{\beta_T}_{k-1/2} = f_{tot} \tilde{\beta_T}_{k-1/2} + (1-f_{tot}){\beta_T}_{k-1/2})$, -where the saturated and unsaturated buoyancy parameters are also intepolated to -$\rho$-levels using (\ref{gradient_interp}). - -Having calculated $Ri$ on $\theta$-levels, ${K_m}_{k}$ and ${K_h}_{k}$ -are calculated, still on $\theta$-levels, as in (\ref{kmlocal}) and -(\ref{khlocal}). Finally, $K_h$ must be interpolated to -$\rho$-levels: -\begin{equation*} - {K_h}_{k+\frac{1}{2}} = \left( - (z_{k+\frac{1}{2}}-z_{k}) {K_h}_{k+1} + - (z_{k+1}-z_{k+\frac{1}{2}}) {K_h}_{k} \right) / \Delta_{k} z -\end{equation*} -Note that in the code the convention is for fluxes to be held on the -half-level below the variable itself. Consequently, RHOKM(K), and -therefore RI(K), are held on the `half-level' below $\rho$-level K, -which is $\theta$-level K-1. - -In addition to the above, the log profile correction applied to ${\cal - L}_h$ (to give $\tilde{{\cal L}}_h$) must be applied {\em after} -interpolation of $K_h$ to level $k+\frac{1}{2}$ in order that the -correct cancellation with the finite difference scalar gradient in the -flux calculation can occur. In the unstable stability functions -(\ref{unstable_stab}), however, $\tilde{{\cal L}}_h$ must be -calculated on $\theta$-levels (\mbox{i.e.}, the same as $\tilde{{\cal - L}}_m$ and $Ri$) in order to maintain the same stability -dependence. - -\subsection{Shear-driven mixing and interaction between the local and - non-local schemes} -\label{sec:shear} -The general approach is to take $K_{\chi}$ in (\ref{scal_closure}) and -(\ref{uv_closure}) as -\begin{equation} - K_{\chi} = \mbox{max} \left[ (K_{\chi}^{\rm surf}+K_{\chi}^{\rm Sc}), - K_{\chi}(Ri) \right] - \label{klnl} -\end{equation} -As noted in section~\ref{sec:closure}, this implies that mixing in -stable boundary layers is determined exclusively by the local scheme, -$K_{\chi}(Ri)$. Continuing to calculate $K_{\chi}(Ri)$ in unstable -boundary layers and using (\ref{klnl}) is seen as the simplest way of -achieving a relatively smooth transition between stable and unstable -boundary layers. - -At the top of unstable mixed layers, great care is taken to ensure the -parametrized entrainment mixing is implemented faithfully, see -section~\ref{sec:entr}). Consequently, if a subgrid inversion has -been diagnosed capping a mixed layer (see section~\ref{sec:sginv}), -then $K_{\chi}(Ri)$ is set to zero at the interfaces either side of -the inversion grid-level. There are also options (using the switch -Keep\_Ri\_FA) to set $K_{\chi}(Ri)$ to zero entirely above unstable -boundary layers or across the LCL in cumulus-capped layers. - -However, the mixed-layer depths were only diagnosed from thermodynamic -constraints. In near neutral boundary layers, shear generation of -turbulence might be expected to allow mixing to extend into regions of -weak static stability (and potentially to inhibit the formation of -cumulus). Currently, therefore, if NTLOC$>$NTML+1 (in layers that are -not cumulus-capped) $K_{\chi}(Ri)$ is left unconstrained by the SML -part of the non-local scheme (and so not set to zero from the SML -inversion upwards) and similarly if NTLOC$>$NTDSC+1. It is realised -that this does not cover the case of shear-driven mixing into cloud -layers that have been diagnosed as cumulus-capped (which would be -poorly represented by the current convection scheme). Several methods -have been introduced that attempt to alleviate this problem, giving -rise to the diagnosis of a ``shear-dominated boundary layer'' type -(type VII), discussed in section~\ref{sec:types}. The first (the -``shear-dominated boundary layer fix'') simply sets the CUMULUS flag -to false if NTLOC $>$ NTPAR. This then ensures that the -locally-determined $K$ are not set to zero above the LCL. Several -more rigorous options are available that incorporate a ``dynamic -criteria'' in the diagnosis of boundary layer type. The first of -these prohibits the diagnosis of cumulus boundary layers when the bulk -measure of stability, $-z_i/L$, is small (currently less than 1.6). -Here $z_i$ is taken as the top of the diagnosis parcel ascent (or at -most 3km) and $L$ is the surface Obukhov length. This test also resets -the depth of the surface-based mixed layer to level 1 since the top of -the parcel ascent may not be suitable (having previously -been diagnosed as cumulus cloud top). The second method -effectively increases the importance of the Richardson number -diagnosis and has been developed from analysis of cold-air outbreaks -\cite[]{bodas-salcedo2012}. Because of the strong surface buoyancy -generation of turbulence in these regimes, a calculation of $Ri$ is -made that allows for the gradient adjustment by the non-local scheme, -\mbox{i.e.}, using $\widetilde{\Delta_k \thetal}$ (see -(\ref{eq:wx_std})). The height, \zloc, where $Ri>Ri_{crit}=0.25$ is -found. It is then hypothesised that this level of turbulent -instability (that incorporates the effects of shear) only needs extend -some fractional distance into the cloud layer to disrupt the formation -of cumulus elements. Thus, if $\zloce > \zlcle + f_{\rm sh} -\left(\zhpare-\zlcle\right) $, where $f_{\rm sh}$ is a tunable -parameter ($0 -Ri_{crit}$, \zhsc is the top of any stratocumulus layer and \zh is the -top of surface-based mixed layer, found by adiabatic parcel ascent but -reset to the LCL in cumulus capped layers. Another diagnostic is -available, the ``boundary layer depth'' (STASH 25), that is set to -$=\mbox{max}[\zhe, \zloce]$ and so represents the depth of the stable -boundary layer or ``surface'' mixed layer. Also available are three -diagnostics that represent the calculated value of each of the -individual terms in STASH 3,304: 3,356 is set to \zh; 3,357 is \zhsc -and 3,358 is \zloc. - -%------------------------------------------------------------------------ -% NON-LOCAL SCHEME -%------------------------------------------------------------------------ -\section{The non-local scheme} -\label{sec:nonlocal} - -This method of calculating $K$ values for unstable conditions is -non-local in the sense that, at a given height within the boundary -layer, $K$ is determined not by any local properties of the mean -profiles at that height but solely by the magnitude of the turbulence -forcing applied to the layer (as measured by the representative -velocity scales described in appendix~\ref{app:vscales}) and the -height within the layer. The non-local scheme is therefore -particularly robust but care must be taken where the profiles are -applied. The calculation of the vertical position and extent of the -$K$ profiles is described in section~\ref{sec:types}. - -\subsection{Surface-driven turbulence} -\label{sec:nlsurf} - -For turbulence sources at the surface (namely surface drag with -velocity scale $u_*$, and positive surface buoyancy fluxes with -velocity scale $w_*$) in a layer with top at $z=$\zh, base at $z=0$ we -set -\begin{equation} - \kmsurf = k \ \zhe \ w_m \ \frac{z}{\zhe} - \left( 1 - {\cal E}_m^{\rm surf} \frac{z}{\zhe} \right)^2 - \label{kmsurf} -\end{equation} -where $w_m^3 = u_*^3 + w_s^3$, $u_*$ is the friction velocity -(including the orographic roughness component) and $w_s$ is defined -below. For the 9C version of the scheme, \zh is the -diagnosed subgrid inversion height (see section~\ref{sec:sginv}) for -both $\khsurf$ and $\kmsurf$. In the 8A version, $\kmsurf$ uses -\zh$=z_{\ntml+\frac{1}{2}}$. The factor ${\cal E}_m^{\rm surf}$ is -chosen so that $\kmsurf$ will tend to $ K_m|_{\ntml+\frac{1}{2}}$ as -$z$ tends to \zh, where $K_m|_{\ntml+\frac{1}{2}}$ is the entrainment -eddy-diffusivity (given by (\ref{khent}), although, in order to avoid -altering the shape function too much, ${\cal E}_m^{\rm surf}$ is not -allowed to fall below $0.7$). A similar factor, ${\cal E}_h^{\rm - surf}$, is used in the $\khsurf$ profile even though the entrainment -fluxes of the thermodynamic variables will usually be specified -explicitly rather than through an eddy-diffusivity (see -section~\ref{sec:entr}). - -The form of $w_s$ differs between the surface layer ($ z < 0.1 $\zh) -and the rest of the mixed-layer: -\begin{equation} - w_s^3 = - \begin{cases} - 2.5 \, \frac{z}{\zhe} w_*^3 & {\rm surface\ layer} \\ - 0.25 \, w_*^3 & {\rm mixed\ layer} \\ - \end{cases} - \label{ws_defn} -\end{equation} -and $w_*^3=\zhe \wbs$ using \zh from the current timestep (note that -the use of $w_*$ here will be inconsistent with the use of $\vheato$ -in the entrainment parametrization in cloudy boundary layers). Note -that $w_s$ is continuous across $0.1$\zh and constant with height in -the mixed layer. This form for $w_s$ is motivated by a desire to -match the model's surface transfer formulation within the surface -layer (as described further in section~\ref{sec:hbcomp}) and to use a -cubic sum of velocity scales within the mixed layer (consistent with -dimensional analysis of the TKE equation, see -\cite{holtslag93:_local_versus_nonloc_bound_layer}). - -The formula for $\khsurf$ is identical to (\ref{kmsurf}) but with -$w_m$ replaced by $w_h=w_m/Pr$, where the turbulent Prandtl number is -given by: -\begin{equation} - Pr = 0.75 \frac{u_*^4 + (4/25)w_s^3 w_m}{u_*^4 + (8/25)w_s^3 w_m} - \label{prandtl_nl} -\end{equation} -Thus $Pr$ varies from 0.75 in neutral conditions to 0.375 in -convective. The origin of the functional form of (\ref{prandtl_nl}) -is unknown. - -\subsubsection[Comparison with Holtslag and Boville (1993)]{Comparison with - \cite{holtslag93:_local_versus_nonloc_bound_layer}} -\label{sec:hbcomp} -The surface-driven $K$ profiles are the same as those in -\cite{holtslag93:_local_versus_nonloc_bound_layer}, HB93, except for -(\ref{ws_defn}) and (\ref{prandtl_nl}) and the inclusion of the ${\cal - E}_m^{\rm surf}$ terms. For the latter, HB93 effectively set ${\cal - E}_m^{\rm surf} =1$. To generate entrainment, however, they simply -use $\kmsurf|_{\ntml+\frac{1}{2}}$, as evaluated from (\ref{kmsurf}) -with a subgrid calculation of \zh$>z_{\ntml+\frac{1}{2}}$, rather than -using a separate entrainment parametrization. - -The difference in (\ref{ws_defn}) arises from the surface layer, where -HB93 match $w_m$ to their surface exchange functions (\mbox{i.e.}, -$w_m = u_* / \phi_m $) which results in proportionality constants of 6 -and 0.6 for $w_s$ in the surface and mixed layers respectively. This -matching is greatly simplified because their non-dimensional shear -$\phi_m = ( 1 + 15 k (z/z_i) w_*^3 / u_*^3 )^{-(1/3)}$. To match -$w_m$, through (\ref{ws_defn}), to the UM function, $\phi_m = ( 1 + 16 -k (z/z_i) w_*^3 / u_*^3 )^{-(1/4)}$, would require a complex function -of $u_*$ and $w_*$ in place of the constant and so this is not -attempted. - -The formula for the Prandtl number used in the interior in HB93 is -also matched to that used in the surface exchange functions ($Pr_{\rm - surf}$, say). For the UM, -\begin{equation*} - Pr_{\rm surf} = \frac{\Phi_h}{\Phi_m} - = \left( 1 + 16 \, k \frac{z}{\zhe} \, \frac{w_*^3}{u_*^3} - \right)^{-1/4} -\end{equation*} -giving $Pr_{\rm surf} = 1$ in the neutral limit (compared to 0.75 from -(\ref{prandtl_nl})). In the convective limit, $Pr_{\rm surf}|_{0.1\, - \zhe} \rightarrow 0.9 (w_*/u_*)^{-3/4} = 0.9 \beta^{3/4} = 0.14 $ -(compared to 0.375 from (\ref{prandtl_nl})). Thus, the Prandtl -numbers do not match between the surface layer and interior -formulations in the UM. - -The formulation in HB93 gives $Pr$ varying from 1 to 0.6 (for $-z/L$ -varying from 0 to 10). In convective conditions ($-z/L=10$), HB93 -have $w_m = 0.85 w_*$ and $w_h=1.4 w_*$ while the UM has $w_m = 0.65 -w_*$ and $w_h = 1.7 w_*$. The implications of these differences from -HB93 are unknown. The convective LES in \cite{lock99} suggest $ w_h -\approx w_*$; I don't know where the larger proportionality constants -come from. - -Another difference between the UM and HB93 is that HB93 only apply -gradient adjustment above the surface layer (and this is allowed for -in their mixed layer definition of $Pr$). Simulations in -\cite{brown1996}, however, suggest that this may lead to a cold bias -at the top of the surface layer. It is attempted to alleviate this in -the UM by the application of gradient adjustment down to the surface -(although this will then lead to a dependence on the height of the -lowest grid-level). The implications of this for matching the Prandtl -number between the surface and interior in the UM is not known. - -\subsection{Cloud-top-driven turbulence} - -For cloud-top-driven turbulence over a layer of depth $\zml$ (with top -at \zh or \zhsc and base at \zbase, determined as in -section~\ref{sec:decouple}), -\begin{equation} - \kmtop = 0.63 \ k \ \zml \ \vtopo \left( \frac{z'}{\zml} \right)^2 - \left( 1 - {\cal E}_m^{\rm Sc} \frac{z'}{\zml} \right)^{0.8} - \label{kmtop} -\end{equation} -where $\vtopc = \vrad+\vbr$ (see appendix~\ref{app:vscales}) and $z'$ -is height above \zbase. Then $K_h = K_m / \mbox{Pr}$, where -$\mbox{Pr}=0.75$. The resulting $K_h$ profile was derived against -convective cloudy LES, as described in \cite{lock99_proceedings}. The -appropriate Prandtl number (and therefore $\kmtop$) is unknown, 0.75 -being chosen simply as a number in the middle of the range usually -quoted for turbulent mixing in general. As with (\ref{kmsurf}), \zh -(or \zhsc) are given by the subgrid diagnosis (see section -\ref{sec:sginv}) except for $\kmtop$ in the 8A scheme which uses the -height of the half-level below ($z_{\ntml+\frac{1}{2}}$ or -$z_{\ntdsc+\frac{1}{2}}$). Again following (\ref{kmsurf}), the -factors ${\cal E}_m^{\rm Sc}$ and ${\cal E}_h^{\rm Sc}$ are included -in (\ref{kmtop}) so that $\kmtop$ will tend to -$K_m|_{\ntml+\frac{1}{2}}$ (and $\khtop$ to -$K_h|_{\ntml+\frac{1}{2}}$), given by (\ref{khent}), as $z$ tends to -\zh (and here no restriction is made on the magnitude of either ${\cal - E}_m^{\rm Sc}$ or ${\cal E}_h^{\rm Sc}$). - -\subsection{Gradient adjustment} -\label{sec:gradadj} -Recall that for $\thetal$ only we use -\begin{equation} - \wthl = - K_h \frac{\partial \thetal}{\partial z} + \khsurf \gamma_{\thetal} -\label{wthl} -\end{equation} -where -\begin{equation} - \gamma_{\thetal} = - \mbox{min}\left[ A_{ga} \frac{\sigma_{T1}}{\zhe}, G_{max} \right] - \label{gradadj} -\end{equation} -$A_{ga}=3.26$, $G_{max}=10^{-3}$Km$^{-1}$ and $\sigma_{T1} = 1.93 \, -\wthls/w_m$, where for this calculation of $w_m$ (given by -$w_m^3=u_*^3+0.25\,\zhe \wbs$) \zh is taken from the previous -timestep. The form of (\ref{gradadj}) is similar to that used in HB93 -and the magnitude of $\gamma_{\thetal}$ is the same as in HB93 in the -convective limit --- the difference in $A_{ga}$ exactly allows for the -different constants in (\ref{ws_defn}). - -Consistent with the mixed layer assumptions underlying the non-local -scheme, the flux profile produced by the scheme is assumed to be -essentially determined by the specified surface and entrainment -values. Thus, the effect of including this non-local term ($ \khsurf -\gamma_{\thetal} $) is to allow the model to maintain more well-mixed -$\thetal$ profiles (\mbox{i.e.}, with $\partial \thetal / \partial z$ -less negative or even positive in a cloud-free surface-heated boundary -layer, for example), subject to an arbitrary upper limit included for -numerical safety. Hence the term `gradient adjustment' rather than -non-local flux. When estimating the buoyancy flux, then (as in -(\ref{eq:wx_std})), it is simplest to allow for the non-local term by -adjusting the $\thetal$ gradient. - -The equivalent term for $q_t$ (\mbox{i.e.}, $\gamma_{q_t}$) is set to -zero in order to represent crudely the effects on the mixed-layer -$q_t$ profile of entrainment drying at the mixed-layer top which tend -to make $q_t$ profiles less well mixed than those of $\thetal$ -\cite[]{mahrt1976}. From UM version 5.5, there is the option to -implement the non-gradient stress parametrization of -\cite{brown97:_non}, as described in section~\ref{sec:ngstress}. - -\subsection{Non-gradient stress parametrization} -\label{sec:ngstress} - -There is an option that is operational in the UM to include an -additional non-gradient (or non-local) stress parametrization, ${\bf - \tau}^{nl}$ in (\ref{uv_closure}), as proposed by -\cite{brown97:_non}. They showed that with only a down-gradient -stress parametrization, a one-dimensional model produced wind profiles -in the convective boundary layer that were less well-mixed than -predicted by LES, and underestimated the near surface wind. -Furthermore, \cite{brownetal2006} showed that the operational -verification statistics indicate a slow bias in the 10~m wind over -land by day, especially in spring and summer. - -The non-gradient stress parametrization in the UM is very similar to -that proposed by \cite{brown97:_non}, written -\begin{equation} - (\tau_x^{nl},\tau_y^{nl})= \left[ - \frac{2.7w_*^3}{(u_*^3+0.6w_*^3)}\right] \left[ \left( \frac{z'}{\zhe'} - \right) \left( 1- \frac{z'}{\zhe'} \right)^2 \right] - (\tau_x^{s},\tau_y^{s}) - \label{tau_nl} -\end{equation} -Here $w_*$ is the convective velocity scale, $u_*$ is the friction -velocity, and $(\tau_x^{s},\tau_y^{s})$ are the surface stresses. -Note that the surface stresses here have to be diagnosed explicitly -(from time-level n fields) but experience has shown these can become -unrealistically large when the near-surface wind is significantly out -of balance with the surface characteristics. As a safety measure, -these surface stresses can be limited such that the implied stress -gradient across the boundary layer is always less than a parameter, -MAX\_STRESS\_GRAD, currently set to 0.05 ms$^{-2}$ (which, for example, -gives a maximum $u_*$ of 7 ms$^{-1}$ in a boundary layer 1km deep). The -term involving $u_*$ and $w_*$ is as proposed by \cite{brown97:_non} -(although note that their Table 3 contains a typo), and ensures that -the non-gradient stress is zero in neutral conditions but asymptotes -to a stability-independent fraction of surface stress in convective -conditions. The primed variables in the shape function allow the -non-local stress profile to either be applied across the whole boundary -layer (using $z'=z$ and $\zhe'=\zhe$), as in \cite{brown97:_non}, -or only above the surface layer (using $z'=z-0.1\zhe$, $\zhe'=\zhe-0.1\zhe$). -The motivation for applying the non-local stress above the surface layer -was to ensure that the match to surface layer similarity was maintained below -$0.1\zhe$ (although separate tests suggested that the impact of this -change is small). - -\subsection{The revised scalar flux-gradient formulation} -\label{sec:rev_flux_grad} - -Following detailed analysis of many large-eddy simulations, including -both surface-heated and cloud-top cooled, a revised flux-gradient -relationship has been developed. In this section, the previous -version will be referred to as the standard one. The formulation is -given in terms of the total flux, -\begin{equation*} - F_{\chi}^{Tot}=\wx + F_{\chi}^{NT} -\end{equation*} -where the non-turbulent flux, $F_{\chi}^{NT}$, is the sum of the -radiative (for $\thetal$), microphysics and subsidence fluxes. This -is a crucial difference from the old formulation: the mean profiles in -LES are found to respond to the total flux profile (which is linear in -a mixed layer) rather than to the individual components of the flux. -Hence any flux-gradient relationship can never be generic to both -$\wthl$ and $\wqt$ since, for example, the shape of the $\thetal$ -profile is determined through interactions with radiation while the -$q_t$ profile is not. Physically, this suggests that while processes -like radiation must {\em locally} generate regions of cold (negatively -buoyant) air at cloud-top, subsequent mixing by turbulent eddies -results in a more-or-less uniformly well-mixed {\em mean} $\thetal$ -profile (presumably because these eddies bring locally warm air back -up to the cloud-top region). - -So, the new formulation is written: -\begin{equation} - F_{\chi}^{Tot} = F_{\chi}^{NT}|_{\zbaseq} - -\lb \khsurf + \khtop \rb \f{\p \ol{\chi}}{\p z} - + \wxngs + \wxngt - + f_2 \lb F_{\chi}|_{z_h} - F_{\chi}^{NT}|_{\zbaseq} \rb -\label{fg_new} -\end{equation} -where $z_h$ and $\zbaseq$ are the heights of the top and base of the -mixed layer, respectively. It can be seen that (\ref{fg_new}) is -composed of a local down-gradient component, two non-gradient flux -terms (one generated by surface-driven turbulence and the other by -cloud-top) and a non-local entrainment flux profile. The turbulent -fluxes are then obtained by subtracting off the non-turbulent -component: -\begin{equation*} - \wx = F_{\chi}^{Tot} - F_{\chi}^{NT} -\end{equation*} - -The components of (\ref{fg_new}) are: -\begin{itemize} -\item{$\khmsurf = k z_h w_{h,m} \zonzi \lb 1-\zonzi \rb^2$} -\item{$\khtop = 3.6 k \vtopo z_{ml} \lb \zonzml \rb^{3}\lb 1-\zonzml - \rb^{2}$} -\item{$\wxngs=\khsurf \gamma_{\chi}$ with - $\gamma_{\chi}=A_{ga}\f{\wxs}{w_h z_h}$ and $A_{ga}=10$} -\item{$\wxngt = f^{Sc} \lb F_{\chi}|_{z_h}- F_{\chi}^{NT}|_{\zbaseq} - \rb $ with $f^{Sc}=3.5 \, k \, \f{\vtopo}{\vsumo} - \lb\zonzi\rb^{3}\lb1-\zonzi \rb$} -\item{$f_2 = 0.5 \, \zonzi \, 2^{(z/z_h)^4}$} -\end{itemize} -In the above equations $k$ is von Karman's constant, $z'$ -($=z-\zbaseq$) is height above the mixed layer base, $z_{ml}$ -($=z_h-\zbaseq$) is the mixed layer depth, $u_*$ is the friction -velocity, and $w_*$ and $\vtopo$ are the velocity scales for surface -and cloud-top buoyancy-driven turbulence. - -Although the structure of the surface-driven non-gradient terms is the -same as for the standard flux-gradient formulation, -(\ref{scal_closure}), note that they are now applied to $q_t$ as well -as $\thetal$ and also the empirical coefficients in the velocity -scales have been revised: -\begin{itemize} -\item{$w_h = (u_*^3 + C_{ws} w_*^3)^{\f{1}{3}} / Pr_{\rm neut} $ with - $C_{ws}=0.42$ for $\zonzi \geq 0.1$ and $C_{ws}=4.2 \zonzi$ for - $\zonzi<0.1$ } -\item{$w_m = w_h Pr $} -\end{itemize} -The functional form of the Prandtl number, $Pr$, is unchanged except -that $w_m$ is replaced by its neutral value: -\begin{equation*} - Pr = Pr_{\rm neut} - \frac{u_*^4 + w_*^3 {w_m}^{\rm neut} / 25} - {u_*^4 + w_*^3 {w_m}^{\rm neut} Pr_{\rm neut}/ (25 Pr_{\rm conv} )} -\end{equation*} -and the range is now $ Pr_{\rm neut} = 0.75$ to $ Pr_{\rm conv} = -0.6$. As with the standard scheme, a constant Prandtl number of 0.75 -is used to calculate $\kmtop$. - -\subsubsection{Discussion of some of the revisions} - -\begin{table}[h] -\begin{center} -{\begin{tabular}{c|cc|cc} -Formulation & \multicolumn{2}{c}{Convective limit} & \multicolumn{2}{c}{Neutral limit} \\ - & $w_m$ & $w_h$ & $w_m$ & $w_h$ \\ -\hline -HB & $0.84 \,w_*$ & $1.4 \,w_*$ & $u_*$ & $u_*$ \\ -UM standard & $0.63 \,w_*$ & $1.7 \,w_*$ & $u_*$ & $1.3 \,u_*$ \\ -UM revised & $0.6 \,w_*$ & $ w_*$ & $u_*$ & $1.3 \,u_*$ \\ -\end{tabular}} -\end{center} -\caption{Convective and Neutral limits for velocity scales} -\label{tab:vscales} -\end{table} - -\begin{figure}[p] - \centering - \scalebox{0.8}{\includegraphics{stab_dep}} - \caption{Stability dependence of the surface velocity scales, - Prandtl number (although I hope something is wrong with my coding - of HB here!) and $d$. Solid lines are from HB, dotted from the - standard UM and the dashed from the revised formulation. The - dash-dotted line for $d$ is a potential modification, as described - in the text. } - \label{fig:stab_dep} -\end{figure} - -It is useful to compare the velocity scales in the revised scheme with -those in the standard version, as well as those in -\cite{holtslag93:_local_versus_nonloc_bound_layer}, hereafter HB, on -which the parametrization was originally based. Recall that HB and -the standard UM set $w_m = (u_*^3 + C_{ws} w_*^3)^{\f{1}{3}}$ and $w_h -= w_m/Pr$, with $C_{ws} = 0.6$ and 0.25, respectively, above the -surface layer. The convective and neutral limits for $w_h$ and $w_m$ -are given in Table~\ref{tab:vscales} and the stability dependencies of -several parameters are shown in Fig.~\ref{fig:stab_dep}. The -parameter $d$ in Fig.~\ref{fig:stab_dep} contains the stability -dependence of the gradient adjustment parameter: -\begin{equation} - \gamma_{\chi}= d \f{\wxs}{w_* z_h} - \hspace{0.5cm} {\rm with} \hspace{0.5cm} - d^{HB} = 7.2 w_*^2/w_m^2, \hspace{0.2cm} - d^{std} = 6.3 w_*/w_m, \hspace{0.2cm} - d^{rev} = 10 w_*/w_h - \label{grad_adj} -\end{equation} -The inclusion of an extra $w_*/w_m$ factor in $\gamma_{\chi}$ was a -deliberate change by HB from the original -\cite{troen86:_simpl_model_atmos_bound_layer} formulation on which the -UM was based. This seems an appealing feature (HB's $\gamma_{\chi}$ -will tend to zero as $w_* \rightarrow 0$) and probably should be -considered for the revised scheme (the dash-dotted line in -Fig.~\ref{fig:stab_dep} sets $d^{std} = 10 w_*^2/w_h^2$). Similarly, -$f_2$ might benefit from an additional factor of the form $(\vsurf + -\vtopc )/ \vsum $ so that it too tends to zero in the neutral limit. -Further analysis of LES and SCM tests will be required to verify this. - -Note that the most significant change from the standard UM scheme is -the change to $w_h$ in the convective limit. Since $\gamma_{\chi}$ -remains unchanged in the convective limit, this reduction in $w_h$ -will result in a significantly smaller $\wxngs$ for the revised scheme -which gives better agreement against LES. - -Compared to the standard scheme, it appears that the revised $\khtop$ -is very different. However, Fig.\ref{fig:new_ksc} shows that this -actually amounts to a small adjustment in the shape. In addition, -note that the factors $\varepsilon_h^{surf}$ and $\varepsilon_h^{Sc}$ -have been removed since the entrainment flux is now carried via the -explicit $f_2$ term. - -\begin{figure}[tbh] - \begin{center} - \includegraphics{new_ktop_shape} - \caption{Standard UM $\khtop$ (solid) and revised (dotted), both - scaled by $k z_h \vtopo$. An upside-down version of $\khsurf$ - is also shown (dashed) for comparison.} - \label{fig:new_ksc} - \end{center} -\end{figure} - -%------------------------------------------------------------------------ -% The blended scheme -%------------------------------------------------------------------------ -\section{The blended scheme} -\label{sec:blend} - -For high resolution simulations, the UM has a Smagorinsky-type -subgrid turbulence scheme, described in \citeumdp{028}. However, this scheme -is only truly applicable for horizontal grid-lengths of order $10$~m, -and any real-world simulation run at lower resolution than this will -inevitably have unresolved scales somewhere in the domain. Rather than -force the user to make an ad-hoc decision about the scales they are -interested in, and thus grid-length at which to switch from using the -boundary-layer parametrization (1D BL) to the subgrid turbulence -scheme (3D Smag), a method for blending the two parametrizations has -been developed. This blend is regime and scale dependent, allowing a -single parametrization to be used across resolutions, including the -completely unresolved/resolved extremes. This blending process is -described in \cite{Boutleetal2014}, which gives some examples of its -use and comparison to simulations using either the 1D BL or 3D Smag -schemes only. Updated technical details from \cite{Boutleetal2014} are -reproduced below. Several options are available that are selected using -the switch {\tt blending\_option}. These all follow the same principles -but differ in their choice of what should constitute the boundary layer and -how to treat non-turbulent layers of the atmosphere. - -As shown in \cite{Honnertetal2011}, the rate at which turbulent -structures become resolved appears to be different for different -aspects of the flow. For example, moisture fluxes are on a larger -scale than heat or momentum fluxes, and so transition to being -sub-grid at lower resolution. This is just one of many challenges when -creating a truly accurate grey-zone parametrization, and so our aim -here is to start from the simplest possible approach which allows the -model to transition from unresolved to resolved turbulence in a -plausible way, without the user having to decide at which grid-length -to switch from a 1D, non-local, to a 3D, local sub-grid scheme. - -Given some function, $W_{1D}$, which tells us how poorly resolved the -turbulence is ($=1$ if unresolved, $=0$ if well resolved), we can use -this to blend between the 1D BL and 3D Smag schemes. Both schemes have -a local Richardson number formulation: -\begin{equation}\label{eq-kri} - K_\chi(Ri) = l^2 S f_\chi(Ri), -\end{equation} -where $K_\chi$ is the eddy diffusivity, $l$ is the mixing length, $S$ -is the wind shear, $f_\chi(Ri)$ is the stability function and $\chi$ -represents conserved heat and moisture variables, or momentum. Both -schemes use the same stability function, and both schemes can use the -full 3D shear for $S$. Therefore the only difference is in the mixing -length, which is calculated as -\begin{equation}\label{eq-lblend} - l_{\rm blend} = W_{1D}l_{\rm bl}+(1-W_{1D})l_{\rm smag}, -\end{equation} -where $l_{\rm bl}^{-1} = (\kappa z)^{-1} + \lambda_0^{-1}$ and $l_{\rm - smag}^{-2} = (\kappa z)^{-2} + (c_s \Delta x)^{-2}$, $\kappa$ is the -von Karman constant and $c_s$ is the Smagorinsky constant. Near the -surface $l_{\rm bl}$ and $l_{\rm smag}$ are identical, but the -asymptotic values are different and this method weights the asymptotic -value according to the weighting of the two schemes. For example, at -$\Delta x=1$~km, $c_s\Delta x=200$~m (for $c_s=0.2$), whereas -$\lambda_0=\max(40\ {\rm m}, 0.15z_h)$, which allows for a small -mixing length in shallow unresolved boundary layers (e.g.~stable -ones). - -The \cite{lock00} scheme also contains a non-local component to the -turbulent flux, and this is simply down-weighted by $W_{1D}$ to ensure -that it becomes less significant as the turbulence becomes better -resolved. Therefore the full eddy diffusivity is given by -\begin{equation} - K_\chi = \max\left[W_{1D}K_\chi^{\rm NL}, K_\chi(Ri)\right], -\end{equation} -where $K_\chi^{\rm NL}$ is the non-local diffusivity and $l$ in -Eq.~\ref{eq-kri} is given by $l_{\rm blend}$ in -Eq.~\ref{eq-lblend}. The turbulent flux is then calculated as -\begin{equation} - F_\chi=-K_\chi\frac{\partial \chi}{\partial z} + W_{1D}F_\chi^{\rm NL}, -\end{equation} -where $F_\chi^{\rm NL}$ is the non-local flux. Therefore when -$W_{1D}=1$, the scheme of \cite{lock00} is recovered, whilst with -$W_{1D}=0$ the Smagorinsky-type scheme is recovered. - -Now we need to define the function $W_{1D}$ to blend the schemes. Within the -boundary layer this is based on the turbulent kinetic energy partitioning -given by \cite{Honnertetal2011}. We choose the TKE -partitioning because it is most closely linked to the eddy diffusivity -we are trying to parametrize (for example a TKE based scheme would -calculate the eddy diffusivity from the TKE), and simplify the -function slightly, using -\begin{equation}\label{eq-tanh} - W_{1D} = 1 - \tanh\left(\beta\frac{z_{\rm turb}}{\Delta x}\right)\max\left[0,\min\left[1,r_f\left(l_0-\frac{\Delta x}{z_{\rm turb}}\right)\right] \right], -\end{equation} -where $z_{\rm turb}$ is the appropriate lengthscale of the turbulence, -$\beta$ is a scaling parameter which controls the speed of the -transition from unresolved to resolved -turbulence, $r_f=\frac{1}{l_0-l_1}$, $l_0=4$ and $l_1=0.25$ (N.~B.~this -formula is slightly modified from that given in \cite[]{Boutleetal2014}). -\cite{Malavelleetal2014} demonstrated that this scaling -method was applicable to any type of unstable boundary layer given an -appropriate choice of $z_{\rm turb}$. In \cite{Boutleetal2014} this -functional form was applied everywhere, adjusting the values of -$z_{\rm turb}$ and $\beta$ depending on the regime. -The max function is present to force the lowest resolution simulations -to just use the 1D mixing scheme. An alternative approach that differs -above the boundary layer is described below. - -The simplest case is for a well-mixed boundary layer, where the -appropriate lengthscale is the boundary-layer depth (inversion -height). Therefore we set $z_{\rm turb}=z_h$, which is broadly -consistent with \cite{Malavelleetal2014}, and choose $\beta=\beta_{\rm - bl}=0.15$ to give the best match of our function to that of -\cite{Honnertetal2011}. These functions are shown in -Figure~\ref{fig-blend}(a) and are only dissimilar for small $\Delta -x$, where Eq.~\ref{eq-tanh} tends to zero faster. This is by choice, -to force the highest resolution simulations to use the 3D turbulence -scheme. -\begin{figure}[tbh] - \centering - \noindent\includegraphics[width=0.49\columnwidth]{honnert_vs_tanh.eps} - \noindent\includegraphics[width=0.49\columnwidth]{zturb_schem.eps} - \caption{(a) Weighting for the 1D boundary-layer scheme as a - function of $\Delta x/z_{\rm turb}$, showing the function of - Equation~\ref{eq-tanh} (blue solid), the equation in - \cite{Boutleetal2014} (black solid) and the TKE partitioning of - \cite{Honnertetal2011} (mean thick dashed, 5th/95th percentiles - thin dashed). (b) Schematic showing the calculation of $z_{\rm - turb}$ used in Eq.~\ref{eq-tanh} for a well-mixed layer (black - dotted) and a decoupled cloud layer (black solid).} - \label{fig-blend} -\end{figure} - -One of the key benefits of the \cite{lock00} scheme is its ability to -represent decoupled stratocumulus layers, and this is a feature which -needs to be maintained in the blended scheme. Physically they are -similar to well-mixed surface driven boundary layers, and the -\cite{lock00} scheme parametrizes them as such. The appropriate length -scale is now the decoupled cloud mixed layer depth, $z_{\rm sc}$ -\cite[i.e.~the depth through which a negatively buoyant parcel -released at cloud top would descend,][]{lock01}. In this case, below -the decoupled cloud top we set -\begin{equation}\label{zturb_dsc} - z_{\rm turb}=\min\left[\max\left(z,z_{\rm sml}\right),\max\left(z_{\rm sc},z_h-z\right)\right], -\end{equation} -where $z_{\rm sml}$ is the depth of the surface-based mixed layer -\cite[i.e.~the depth through which a positively buoyant parcel -released at the surface would ascend,][]{lock00}. This is shown -schematically in Figure~\ref{fig-blend}(b), and ensures that $W_{1D}$ -has a high value in the poorly resolved surface mixed layer and cloud -layer, and a lower value in between those layers. Again, this choice -of $z_{\rm turb}$ is broadly consistent with the analysis of decoupled -stratocumulus LES presented by \cite{Malavelleetal2014}. Finally, -\cite{Honnertetal2011} also included shallow cumulus simulations and -showed that the relevent length scale there was the cloud top height. Most -of the {\tt blending\_option} choices apply this to all regimes diagnosed -as cumulus-capped (see section~\ref{sec:types}) but alternatively -({\tt blending\_option}$=$4) this can be restricted to strictly shallow -cumulus clouds, defined as contiguously cloudy levels (cloud fraction -greater than SC\_CFTOL) with cloud top height below input parameter -{\tt shallow\_cu\_maxtop}. Note that the diagnosis of shallow cumulus -from the diagnosis parcel ascent (that was used to identify a cumulus regime) -was found frequently to indicate deep convection even when the resolved -clouds were shallow because the diagnosis parcel, being undilute, would -penetrate to the tropopause. However, having decided the regime is shallow -convection, we do still set $z_{\rm turb}$ to the diagnosis parcel top height -because, for current km-scale configurations (without a cumulus convection -parametrization), it was found that the resulting stronger parametrized -vertical mixing was beneficial for the development of the convection, and -that without this a widespread stratiform cloud layer could develop instead. - -Above the boundary layer top, \cite{Boutleetal2014} aimed for any free -atmospheric mixing to be done by the 3D Smagorinsky scheme. Therefore, above -the boundary layer top they use $z$ as the appropriate length scale, and in -general take $z_{\rm turb}$ in Eq.~\ref{eq-tanh} as the greater of that defined -by (\ref{zturb_dsc}) and $z$. However, this did not give a particularly -fast transition using the value of $\beta_{\rm bl}$, therefore they used -$\beta_{\rm fa}=1$ at a height well above the boundary layer -($z_{\rm fa}=z_h+1$~km), and transitioned between these regimes linearly using -\begin{equation} - \beta = \beta_{\rm bl}\frac{z_{\rm fa}-z}{z_{\rm fa}-z_h} + - \beta_{\rm fa}\frac{z-z_h}{z_{\rm fa}-z_h} -\end{equation} -However, because the above method still uses (\ref{eq-tanh}), which depends on -$z_{\rm turb}/\Delta x$, the rate of transition to 3D Smagorinsky with height -above the boundary layer varies in an undesirable way with grid size. It -might be considered more logical to think of non-turbulent regions of the free -troposphere as unresolved turbulence and so revert to the 1D mixing scheme -there. An alternative treatment({\tt blending\_option}$=$3 or 4), then, is to -increase $W_{1D}$ above the boundary layer top smoothly, to reach unity by -some physical height $z_{\rm fa}$, to be independent of both horizontal and -vertical grid sizes. For $z_{\rm turb} < z < z_{\rm fa}$, then, we set -\begin{equation} - W_{1D} = 1 + \frac{1}{2} \left( W_{1D}|_{z=z_{\rm turb}} - 1 \right) - \left[ 1 + {\rm cos}\left( \pi \, \frac{z-z_{\rm turb}}{z_{\rm fa}-z_{\rm turb}} - \right) \right] -\end{equation} -The cosine term in square brackets transitions smoothly from 2 at -$z=z_{\rm turb}$ to zero at $z_{fa}$ where, although somewhat -arbitrary, $z_{\rm fa} = {\rm min}(2 z_{\rm turb}, z_{\rm turb}+1 {\rm km})$. -The former term ensures the transition is well above any shallow boundary -layers while the latter that it does not drift far into the free atmosphere. -In addition, within any layers identified as turbulent, through having -subcritical $Ri$, $z_{\rm turb}$ is set to the layer depth, in the same way -as is done for decoupled stratocumulus in (\ref{zturb_dsc}). - -For current operational convection-permitting model grid sizes (1.5 km -in the UKV), the representation of cumulus convection remains a -challenge. One option is to include a grey-zone convection -parametrization, described in the documentation of that scheme -(see \citeumdp{027}). Tests in the UKV, though, showed some -detriment to the spin-up of resolved scale convection (as well as -somewhat poor discrimination of precipitating versus non-precipitating -parametrized convection) that led to the development of an alternative -strategy, namely to abandon the blended turbulence scheme when pure -cumulus convection was diagnosed and leave the representation of -cumulus entirely to the resolved scales. This option -({\tt blending\_option}$=$2) is also now discouraged. - -%------------------------------------------------------------------------ -% ENTRAINMENT -%------------------------------------------------------------------------ -\section{Entrainment fluxes} -\label{sec:entr} - -{\bf Summary}: parametrized entrainment fluxes (at the top of mixed -layers) are specified for momentum through an eddy-diffusivity, as -described in section~\ref{sec:ent_K}. For scalar variables, if the -inversion is sufficiently sharp so as to be unresolved, the ideal is -to specify the entrainment fluxes explicitly, as described in -section~\ref{sec:ent_flux}, based on the subgrid inversion diagnosis -described in section~\ref{sec:sginv}. Further details can be found in -\cite{lock01}. If the profiles are such that the inversion is sharp -but a subgrid inversion cannot be diagnosed, an eddy-diffusivity -similar to that for momentum is used (see section~\ref{sec:ent_K}). -If the inversion is thick enough to be resolved then an eddy -diffusivity profile is constructed across the inversion (see -section~\ref{sec:entr_prof}) for both scalars and momentum fields. -For tracer variables (scalars other than $\thetal$ and $q_t$), the -entrainment fluxes are specified using an equivalent eddy-diffusivity, -as described in section~\ref{sec:ent_K_flux}. Note that, as indicated -below, several aspects of the implementation of entrainment fluxes -were revised at the 9C scheme and these are documented separately. - -The parametrization of the entrainment rate, $w_e$ (given, in the -absence of subsidence, by the rate of rise of the inversion), can be -written (using the notation given in appendix~\ref{app:vscales}) -\begin{equation} - w_e = \frac{ A_1 \, \vsum / \zml + g \tilde{\beta_T} \tilde{\alpha_t} - \Delta_\radf } - {\Delta b + c_T \vsumo^2/\zml } - \label{we_parm} -\end{equation} -where $ \vsum = \vheat + \vrad + \vbr + A_2 u_*^3 $. The constant -$A_1$ is given a value 0.23, as in \cite{lock98}, and $A_1*A_2=5$, as -in \cite{driedonks1982}. To allow for weak inversions, the -\cite{zilitinkevich1975} correction is included in (\ref{we_parm}) -with the constant, $c_T=1$. A further parametrization for $\alpha_t$, -which is the fraction of the cloud-top radiative divergence -($\Delta_\radf $, in Kms$^{-1}$) that occurs across the -horizontally-averaged inversion in the LES, can be written -\begin{equation*} - \alpha_t = 1 - \exp{ \left\{-\Delta z_i / (2 L_{rad})\right\} } -\end{equation*} -where the thickness of the inversion is parametrized as $\Delta z_i = -\mbox{min}[\vsumo^2/\Delta b, 100]$ and $L_{rad}$ is a depth-scale for -the radiatively-cooled layer (taken to be 15 $\times -\,\mbox{max}[200/z_c, 1]$, where $z_c$ is the cloud depth). To allow -for a feedback with forcing of entrainment by buoyancy reversal (see -appendix~\ref{app:vscales}), $\tilde{\alpha_t} = \alpha_t+ Br -(1-\alpha_t) $. following \cite{lock98} and \cite{lock09:_factor}. -The calculation of the other quantities required for (\ref{we_parm}) -is described in appendix \ref{app:vscales}. At some point during the -transition to a decoupled boundary layer the surface-driven -entrainment terms (the terms in (\ref{we_parm}) proportional to -$\vheat$ and $u_*$) will no longer contribute to entrainment at cloud -top, because the two layers will have become entirely decoupled. If -the {\tt entr\_smooth\_dec} switch is on then the surface contribution -to the parametrized entrainment at \zhsc is decreased linearly as the -$\thetavl$ difference between NTDSC and NTML increases from 0.5 to 1K. -The flag, COUPLED, is set to true and \zhsc is used as the mixed-layer -depth in (\ref{we_parm}) as long as any surface-driven entrainment -remains. If the {\tt entr\_smooth\_dec} switch is off then this -transition is discontinuous at a $\thetavl$ difference of 0.5K. - -It should be noted that (\ref{we_parm}) takes no account of wind shear -anywhere other than at the surface. How to quantify the shear -generation of turbulence in DSC layers is not known. The direct -impact of shear across the inversion is thought to be simply to -diffuse the inversion in the vertical --- this wind shear will -contribute little to the mixed layer TKE and so can not contribute to -the full process of mixing across the inversion and down into the -mixed layer that is entrainment. However, important interactions -between wind shear across inversions and cloud-top radiative cooling -have been observed that are not yet accounted for in the UM. - -The least well-determined part of (\ref{we_parm}) is the constant -$A_2$ --- the constant in the Zilitinkevich correction, $c_T$, is also -approximate but is included to limit the growth of layers capped by -weak inversions and for numerical safety. A further limit is applied -to the value of $w_e$ determined by (\ref{we_parm}) such that the -inversion cannot rise by more than one grid-level in a timestep. With -current vertical resolutions and timesteps this is not a serious -restriction. The constants $A_1$ and $A_{\rm br}$ appeared to be -determined within 10-20 \% in \cite{lock98}, although only solid cloud -sheets were simulated (as discussed further in -appendix~\ref{app:vscales}). Similarly the parametrizations of -$\alpha_t$ and $\Delta z_i$ were found to be accurate but the -parameter $L_{rad}$ is currently only crudely represented in the UM. - -\subsection{Specification of entrainment fluxes in the 9B scheme} -\label{sec:ent_flux} -Note that the 9C scheme (see next section) differs by generalising the -approach to include all processes operating in the inversion -grid-level, rather than just radiation. - -If it is assumed that the turbulent fluxes reduce from their extremum -at $z=z_i$ (the `entrainment' fluxes) to zero at $z=h$ a small -distance above, then $ \wthlzi = - w_e \Delta \thetal + \radf|_h - -\radf|_{z_i}$, so that -\begin{eqnarray} - {\cal H}|_{z_i} & =& - w_e \Delta \thetal + \radfnet|_h \nonumber\\ - \wqtzi & =& - w_e \Delta q_t -\end{eqnarray} -\label{discinv} -where the total heat flux ${\cal H} = \wthl + \radfnet$ and $\radfnet -= \radf - \radf|_{\zbaseq}$. The net radiative flux relative to the -base of the mixed layers is simply calculated as -\begin{equation*} - \radfnet|_{z_{k+\frac{1}{2}}} = \sum_{k=\nbdsc}^{k} \mbox{max}\left[ - - \Delta_{k+\frac{1}{2}} z \, {\cal S}_\radf(k), \,0 \right] -\end{equation*} -where NBDSC$=1$ in SMLs, ${\cal S}_\radf$ are the temperature -increments (in Ks$^{-1}$) from the radiation scheme and $\radfnet|_h$ -is estimated by extrapolating down from $\radf|_{z_{\ntml+\frac{3}{2}} -}$ using the flux-divergence in grid-level NTML$+2$ (and similarly for -DSC layers). - -The thermodynamic variables' entrainment fluxes, then, are imposed -nominally at the subgrid inversion height ($z_i=$ \zh and/or \zhsc), -diagnosed as described in section~\ref{sec:sginv}. The required -grid-level fluxes (at $z_{\ntdsc+\frac{1}{2}}$, for example) are then -estimated using linear interpolation of ${\cal H}$ and $\wqt$ between -\zhsc and the base of the mixed layer: -\begin{eqnarray} - \wthl|_{ z_{\ntdsc+\frac{1}{2}} } & =& \wthl|_{\zbaseq} - - \frac{ z'_{\ntdsc+\frac{1}{2}} }{\zml} - \left( \tilde{w_e} \Delta \thetal + \wthl|_{\zbaseq} - \radfnet|_{h} \right) - - \radfnet|_{ z_{\ntdsc+\frac{1}{2}} } \nonumber\\ - \wqt|_{ z_{\ntdsc+\frac{1}{2}} } & =& \wqt|_{\zbaseq} - - \frac{ z'_{\ntdsc+\frac{1}{2}} }{\zml} - \left( \tilde{w_e} \Delta q_t + \wqt|_{\zbaseq} \right) -\end{eqnarray} -\label{fluxinterp} -where $z' = z-\zbaseq$, and similarly for the SML entrainment fluxes -(at $z=z_{\ntml+\frac{1}{2}}$). The turbulent fluxes at the base of -the mixed layer are assumed zero except for the SML where the surface -fluxes are used. This interpolation is illustrated for a SML in -Fig.~\ref{fig:fluxinterp}. -\begin{figure}[tbh] - \centering - \scalebox{1.0}{\includegraphics{subsent_fig7}} - \captionarb{Idealised profiles of (a) $\wqt$ (dash-dotted line) and - (b) ${\cal H}$ (dotted line), $\wthl$ (dash-dotted) and $F$ - (dashed). The continuous lines are the turbulent fluxes on the - model grid indicated by the dashed horizontal lines.} - \label{fig:fluxinterp} -\end{figure} -Note that, because (\ref{fluxinterp}) includes an explicit balance -between the turbulent and radiative fluxes for $\wthl$, it is not -possible to parametrize the entrainment fluxes through a single $K_h$ -for both $\wthl$ and $\wqt$. Furthermore, the radiative forcing of -turbulence in the mixed layer is fixed through the timestep and so it -is consistent to assume the entrainment fluxes (at $z_i$) are also -fixed. Hence (\ref{fluxinterp}) are implemented explicitly, rather -than via an eddy-diffusivity. This is discussed further, with -reference to tracer fluxes, in section~\ref{sec:ent_K_flux}. - -In order to allow for the long timesteps used in NWP and to facilitate -movement of the subgrid inversion across grid-levels within a -timestep, the parametrization of $w_e$ and the model's subsidence -velocity, $w_S|_{z_i}$, are used to calculate $z_i$ at the next -time-level ($z_i^{n+1}$). Currently, the latter is found by linear -interpolation to $z_i$ and both are assumed constant in time. If -$z_i^{n+1} < z_{\ntdsc +\frac{1}{2}}$, then the entrainment fluxes -there (given by (\ref{fluxinterp})) are multiplied by the fraction of -the timestep that $z_i$ was above this grid-level, namely -$(z_i-z_{\ntdsc +\frac{1}{2}})/(z_i - z_i^{n+1}) $. The full -entrainment flux at grid-level NTDSC$-\frac{1}{2}$ must then also be -specified, given by (\ref{fluxinterp}) with $z_{\ntdsc + \frac{1}{2}}$ -replaced by $z_{\ntdsc - \frac{1}{2}}$. If $z_i$ rises above -$z_{\ntdsc+\frac{3}{2}}$, the entrainment flux is specified only at -this higher grid-level (multiplied by the fraction of the timestep -that $z_i$ is above this half-level) and the values of the mixed-layer -$K$ profiles are used in half-level NTDSC$+\frac{1}{2}$ (these will be -non-zero because $z_i>z_{\ntdsc + \frac{1}{2}}$). Wherever the -entrainment fluxes are specified explicitly, the eddy-diffusivities -(both non-local and local) are set to zero. Also, the mean value of -$z_i$ during the timestep is used in (\ref{fluxinterp}) in order best -to approximate the mean flux gradient across the mixed layer. - -Finally, the entrainment flux is adjusted to allow for numerical -entrainment arising from the model's resolved vertical advection (as -discussed in \cite{lock01}). This is performed at whichever -grid-level the entrainment fluxes are specified, to allow for any -entrainment implied by a $\thetal$ subsidence increment, $\Theta^{\rm - S}$ (Ks$^{-1}$), at the model grid-level below. The subsidence -increments could be obtained directly in the SCM but in the full 3D UM -advection increments are dominated by the horizontal component. The -subsidence increments are calculated, therefore, from the vertical -velocity field using first order upwind advection (it would clearly be -preferable to use the model's actual vertical advection algorithm in -the GCM although the errors incurred in this diagnostic calculation -should not be very significant). The interpolated entrainment fluxes -given by (\ref{fluxinterp}) are therefore calculated not using $w_e$ -but using an entrainment velocity, $\tilde{w_e}$, that is reduced to -allow for any subsidence increments applied to the grid-level below -the entrainment flux. To take the case of $z_{\ntdsc+\frac{1}{2}} < -z_i^{n+1} < z_{\ntdsc+\frac{3}{2}}$ as an example, this reduced -entrainment velocity is given by -\begin{equation*} - \tilde{w_e} = w_e + \tilde{w_S} -\end{equation*} -with $\tilde{w_e}$ constrained to lie between 0 and $w_e$ and -\begin{equation} - \tilde{w_S} = - \, \frac{ \Theta^{\rm S}_{\ntml} - ( \Delta_{\ntml+\frac{1}{2}} z ) } - { \Delta \thetal } - \label{we_num} -\end{equation} - -\subsubsection{Diagnosis of a sub-grid inversion} -\label{sec:sginv} - -The profile of $\thetavl$ is used to diagnose the height of a -discontinuous inversion because it is approximately conserved under -adiabatic vertical motion and is equal to the virtual potential -temperature, $\theta_v$, in the absence of cloud. This should ensure -it is monotonically increasing with height in the statically stable -free-troposphere of a GCM. If $\thetavl$ does not increase -monotonically between grid-levels NTML and NTML+2 (or NTDSC and -NTDSC+2), then entrainment fluxes are simply specified via -(\ref{khent}) and none of the coupling with subsidence or radiation -described above is attempted (the local scheme is also currently not -set to zero above NTML or NTDSC when this occurs to allow it to -diffuse out this static instability). - -\begin{figure}[tbh] - \centering - \scalebox{1.0}{\includegraphics{nbldoc_zidiag}} - \captionarb{Schematic illustrating the assumptions behind the - subgrid diagnosis of $z_i$.} - \label{zi_diag} -\end{figure} -Having identified the model grid-level at the top of the well-mixed -layer (either level NTML from the parcel ascent, as described in -section~\ref{sec:adiapar}, or NTDSC for DSC layers, see section -\ref{sec:decouple}--- the analysis is the same for both), the -grid-level above is designated the inversion level within which the -diagnosis of a subgrid $z_i$ will be made. It is assumed that -$\thetavl$ in grid-level NTML$+1$ represents a cell-average value. -Thus, $z_i$ can be calculated by assuming that the integral of -$\thetavl$ over grid-level NTML$+1$ for the model and for a profile -with a discontinuous inversion at $z_i$ are equal, as illustrated by -the hatched areas in Fig.~\ref{zi_diag}. To calculate the integral of -the discontinuous profile, the lapse rate of $\thetavl$ between -grid-levels NTML$-1$ and $NTML$, $\gammaml$, is extended up to $z_i$, -while the stable lapse in the free atmosphere, between grid-levels -NTML$+2$ and NTML$+3$, $\gammafa$, is extrapolated down. Equating -these areas gives a quadratic equation in $\Delta z_{disc} = -z_{\ntml+\frac{3}{2}} - z_i$ which can be written -\begin{equation} - a (\Delta z_{disc})^2 + b \ \Delta z_{disc} +c =0 -\label{zi_interp} -\end{equation} -The coefficients are given by -\begin{eqnarray*} - a & =& 0.5 (\gammafa - \gammaml) \\ - b & =& - \left( {\thetavl}_{\ntml+2} - - \gammafa (z_{\ntml+2}-z_{\ntml+\frac{3}{2}}) \right) - + \left( {\thetavl}_{\ntml} - + \gammaml (z_{\ntml+\frac{3}{2}}-z_{\ntml}) \right) \\ - c & =& (z_{\ntml+\frac{3}{2}}-z_{\ntml+\frac{1}{2}}) - \left( {\thetavl}_{\ntml+1} - - \left( {\thetavl}_{\ntml} + - \gammaml \left( - \frac{1}{2}(z_{\ntml+\frac{1}{2}}+z_{\ntml+\frac{3}{2}}) - -z_{\ntml} \right) \right) - \right) \\ -\end{eqnarray*} - -Clearly, care must be taken to ensure that $z_i$ is not only -well-defined but also sensible (for example, as a rising inversion -encounters more or less stable regions above). If $b>0$ this suggests -the estimated lapse rates are inappropriate and these are therefore -set to zero and (\ref{zi_interp}) is recalculated. The case $c<0$ -suggests the grid-level designated as the inversion level should have -been considered as part of the mixed layer and so $z_i$ is set to be -fractionally below $z_{\ntml+\frac{3}{2}}$ (\mbox{i.e.}, as high as -possible without attempting to diagnose a subgrid $z_i$ in grid-level -NTML$+2$). If $b^2-4ac<0$ the quadratic equation has no real roots. -In this instance $z_i$ is set to fractionally below -$z_{\ntml+\frac{1}{2}}$ and NTML (and therefore the eddy-diffusivity -profiles) is lowered by a grid-level. In all other circumstances, the -required root is then $\Delta z_{disc} = (-b - (b^2-4ac)^{1/2} -)/(2a)$; the other root will either be larger or negative (if $a<0$). - -In addition, from variations seen in $z_i$ during single-column model -simulations, the error in $\Delta z_{disc}$ is estimated to be around -10\% of the vertical resolution, $\Delta_{\ntml+\frac{3}{2}} z$. -Accordingly, if $z_i$ is diagnosed as being less than -$z_{\ntml+\frac{1}{2}} + 0.1 \, \Delta_{\ntml+\frac{3}{2}} z$, NTML is -lowered a grid-level and $z_i$ is set fractionally below -$z_{\ntml+\frac{1}{2}}$. This small distance below the grid-level is -taken to be $(\Delta t/2) \times 10^{-4}$ so that, were a small rate -of rise of $z_i$ (of $10^{-4}$ ms$^{-1}$, say) to be diagnosed, then -$z_i$ would spend at least half the timestep (of length $\Delta t$) in -the next grid-level up. The specified fluxes would then contribute -significantly to that grid-level's evolution. Conversely, if $z_i$ is -subsiding, this technique allows the inversion to drop down a -grid-level without requiring this to be detected by the initial parcel -ascent. - -Having calculated $z_i$, the discontinuous jumps in $\thetal$ and -$q_t$ that are used in the entrainment calculation are calculated from -similar integral assumptions: -\begin{equation} - \Delta \chi = \left( {\chi}_{\ntml+1} - {\chi}_{\ntml} \right) \, - \frac{ z_{\ntml+\frac{3}{2}} - z_{\ntml+\frac{1}{2}} } - { z_{\ntml+\frac{3}{2}} - z_i } -\label{dqt_disc} -\end{equation} -with $\chi = \thetal$ and $q_t$. Note that the lapse rate above the -inversion has been ignored as there is no guarantee of monotonicity in -$q_t$ in the atmosphere above the inversion. In addition, -(\ref{dqt_disc}) will become increasingly inaccurate as $z_i$ tends to -$z_{\ntml+\frac{3}{2}}$ (and so ${\chi}_{\ntml+1}$ approaches -${\chi}_{\ntml}$). Consequently, if the fraction on the right hand -side of (\ref{dqt_disc}) is greater than 10, double grid-level jumps -are used (\mbox{i.e.}, $\Delta \chi = {\chi}_{\ntml+2} - -{\chi}_{\ntml} $). Finally, note that (\ref{dqt_disc}) implicitly -assumes the structure of the $\thetal$ and $q_t$ profiles across the -inversion grid-level are consistent with the diagnosed $z_i$. This is -very unlikely to be the case, for example, when running from an -analysis so the 9C scheme uses what has been found to be a more robust -algorithm, see the separate documentation. - -It would clearly be advantageous to pass knowledge of this subgrid -inversion structure to other parametrizations in the UM, particularly -the cloud scheme as currently the cloud fraction in level NTML$+1$ is -essentially meaningless (being diagnosed from a mixture of cloudy -boundary layer air and typically very dry free tropospheric air). - -\subsection{Specification of entrainment fluxes across sharp inversions in the 9C scheme} -\label{sec:ent_flux_9c} - -As described in section~\ref{sec:ent_flux}, when the capping inversion -is thinner than the model vertical grid it is important for the -entrainment flux implementation that the subsidence increments are -realistically and consistently distributed between the inversion -grid-level and the mixed layer. Rather than work with the increments -themselves, though, a more robust solution is to couple the subsidence -and turbulent fluxes across the inversion, exactly analogously to the -coupling of turbulent and radiative fluxes. This allows the total -tendency of the inversion grid-level to be linked to whether the -inversion should be rising or falling (determined from the balance -between the parametrized entrainment rate, $w_e$, and the large-scale -vertical velocity evaluated at the inversion, $w|_{z_h}$). - -\begin{figure}[p] - \centering - \scalebox{1}{\includegraphics{ideal_revflux}} - \caption{Subgrid (lines) and model (symbols) profiles and fluxes of, - top row, $q_t$ and, bottom row, $\thetal$: turbulent fluxes - (dash-dotted, crosses), subsidence fluxes (dotted, diamonds), - radiative flux (dashed, triangles) and total flux (solid, - squares).} - \label{fig:rev_fluxes} -\end{figure} -An idealised subgrid total flux profile is constructed from the -parametrized entrainment flux and the increments from radiation, -precipitation and subsidence, assuming a well-mixed boundary layer -capped by a diagnosed subgrid inversion. The crucial step is to -ensure that the total flux on the model entrainment grid-level equals -the idealised total flux profile interpolated to that level. Consider -the example illustrated in Fig.~\ref{fig:rev_fluxes} of a well-mixed -boundary layer up to $\theta$-level $\ntml$. For the subgrid $q_t$ -profiles, the turbulent flux divergence generates a moistening across -the inversion while subsidence generates drying. For this example it -has been assumed the entrainment rate is slightly larger than the -subsidence velocity at the inversion and so overall there is a weak -moistening relative to the mixed layer (the total flux gradient is -more negative across the inversion than in the mixed layer), -consistent with the rising tendency of the inversion. For the model, -the subsidence flux-divergence associated with the inversion is split -across levels $\ntml$ and $\ntml+1$. To keep the {\em net} moistening -of the model's boundary layer and inversion consistent with the total -subgrid flux profile, the model's entrainment flux at $\ntml+1/2$ -(shown by the cross in Fig.~\ref{fig:rev_fluxes}) must be found by -subtracting the subsidence flux at $\ntml+1/2$ (diamond) from the -total flux interpolated to $\ntml+1/2$ (square). Exactly the same -arguments follow for the $\thetal$ fluxes except that the situation is -complicated by the addition of the radiative flux. - -The above arguments can be generalised as follows. Writing $\fxtot$ -as the total flux of a conserved variable $\chi$ ($=q_t$ or $\thetal$) -and $\fxntp$ as the flux from physics sources other than turbulence -(\mbox{i.e.}, radiation, $\fx^{rad}$, in the above examples, but -including precipitation fluxes, $\fx^{ppn}$, in the full model) and -$\fx^{subs}$ as the flux from resolved scale subsidence, the total -flux at the subgrid inversion height is given by: -\begin{equation} - \fxtot|_{z_h} = - w_e \Delta \chi + \fxntp|_{z_t} + \fx^{subs}|_{z_h} - \label{fxtot_zi} -\end{equation} -As in section~\ref{sec:ent_flux}, (\ref{fxtot_zi}) is derived by -integrating the conservation equation for $\chi$ over an inversion in -which jumps occur over a thin layer with base at a height $z_h$ and -top at $z_t$ (in the UM, the inversion is assumed to be -infinitesimally thin so that $z_t=z_h$). This integration gives $ - -w_e \Delta \chi = \wx|_{z_h} -(\fxntp|_{z_t}-\fxntp|_{z_h})$. -\cite{lock99} related the non-turbulent flux divergence, -$\fxntp|_{z_t}-\fxntp|_{z_h}$, to radiative cooling occurring within -undulations of the cloudy boundary layer top. Similar considerations -need to be borne in mind when calculating all the non-turbulent fluxes -in (\ref{fxtot_zi}). First, the radiative flux is extrapolated down -from $\ntml+\frac{3}{2}$ to $z=z_t$ using the divergence in the -grid-level above the inversion as representative of the -free-atmospheric divergence. Second, since the microphysical flux is -generated within the cloud, $\fx^{ppn}|_{z_t} = -{\fx}^{ppn}_{\ntml+\frac{3}{2}}$. Finally, the subsidence -flux-divergence across level $\ntml$ and $\ntml+1$ is assumed to be -associated with the inversion so ${\fx}^{Subs}|_{z_h} = -{\fx}^{Subs}_{\ntml-\frac{1}{2}}$. Thus, the finite-difference form -of (\ref{fxtot_zi}) becomes \beqn \fxtot|_{z_h} = - w_e \Delta \chi + -\fx^{rad}|_{z_t} + {\fx}^{ppn}_{\ntml+\frac{3}{2}} + -{\fx}^{subs}_{\ntml-\frac{1}{2}} -\label{fxtot_zi_fd} -\eeqn - -Then, assuming a linear profile of $\fxtot$ in the mixed layer, -interpolating the total flux to the inversion flux grid-level gives -\begin{equation} - \fxtot|_{ \ntml+\frac{1}{2} } = \fxtot|_{\zbaseq} + - \frac{ z'_{\ntml+\frac{1}{2}} }{\zml} - \left( \fxtot|_{z_h} - \fxtot|_{\zbaseq} \right) - \label{fxtot_interp} -\end{equation} -where $z'$ ($=z-\zbaseq$) is height above the base of the mixed layer -at $z=\zbaseq$. Finally, the grid-level turbulent entrainment flux is -given by: -\begin{equation} - \wx|_{ \ntml+\frac{1}{2} } = \fxtot|_{ \ntml+\frac{1}{2} } - - \fxnt|_{ \ntml+\frac{1}{2} } - \label{rev_entflux} -\end{equation} -This revised algorithm has several advantages over the previous. -Firstly, the fluxes for $q_t$ and $\thetal$ are coupled independently, -whereas in the 9B version the coupling with subsidence was estimated -using only the $\thetal$ increments in order to calculate -$\tilde{w_e}$ in \ref{we_num}. Secondly, this method makes it much -simpler to include all processes, and precipitation in particular, in -a consistent manner. Thirdly, since the total grid-level flux, -$\fxtot|_{\ntml+\frac{1}{2}}$ in (\ref{fxtot_interp}), is used to -calculate the entrainment fluxes, it is straightforward to ensure that -the net budget of the inversion grid-level, namely $- ( -\fxtot|_{\ntml+\frac{3}{2}} - \fxtot|_{\ntml+\frac{1}{2}})/\Delta z $, -is consistent with the entrainment/subsidence balance. In other -words, to use $\thetal$ as an example, if the inversion is rising -(falling) then $\fxtot|_{\ntml+\frac{1}{2}}$ is limited to ensure that -the inversion grid-level will cool (warm). Finally, if the inversion -is rising we don't want the inversion grid-level $\thetal$ to cool to -less than $\thetal$ of the mixed layer by the end of the timestep. In -other words, for $\chi=\thetal$, given -\begin{eqnarray*} - \chi_{\ntml+1}^{n+1} & =& \chi_{\ntml+1}^{n} - - \frac{\Delta t}{\Delta z} \left( - \fxtot|_{ \ntml+\frac{3}{2} } - \fxtot|_{ \ntml+\frac{1}{2} } - \right) \\ - \chi_{\ntml}^{n+1} & =& \chi_{\ntml}^{n} - - \frac{\Delta t}{z_{\ntml+\frac{1}{2}}} \left( - \fxtot|_{ \ntml+\frac{1}{2} } - \fxtot|_{\zbaseq} - \right) \\ -\end{eqnarray*} -where the superscripts $n$ and $n+1$ refer to the model timestep, -although strictly speaking $n+1$ refers to fields after the boundary -layer implicit solver. Requiring that -$\chi_{\ntml+1}^{n+1}\geq\chi_{\ntml}^{n+1}$ implies -\begin{equation} - \fxtot|_{ \ntml+\frac{1}{2} } - \left( 1+ \frac{\Delta z}{z_{ml}}\right) - \geq \fxtot|_{ \ntml+\frac{3}{2} } + \Delta z \left( - \frac{\chi_{\ntml}^{n}-\chi_{\ntml+1}^{n}}{\Delta t} - + \frac{\fxtot|_{\zbaseq}}{z_{ml}} \right) -\end{equation} -The same arguments apply for $q_t$, noting that the free atmosphere -can be drier or moister than the mixed layer and so these cases must -be treated separately. If $\thetal$ of the free atmosphere is colder -than the mixed layer then no subgrid inversion treatment is attempted -and entrainment is modelled using a straightforward eddy diffusivity. - -\subsubsection{Calculation of the inversion jumps in the 9C scheme} - -In the 9B scheme, the discontinuous jumps in $\thetal$ and $q_t$ that -are used in the entrainment calculation were calculated from integral -assumptions similar to those used to diagnose the subgrid inversion -height, $z_h$, and were given by (\ref{dqt_disc}). Note that -${\chi}_{\ntml+2}$ does not appear in (\ref{dqt_disc}) and so no -direct information from the free atmosphere is used. Only if the -budgets of $\thetal$ and $q_t$ in level $\ntml+1$ are entirely -consistent with the rise and fall of the subgrid inversion will -(\ref{dqt_disc}) give accurate results. This will not be the case -during an assimilation cycle, for example, neither is it likely to be -the case if the convection scheme is detraining into level $\ntml+1$. - -Instead, a more robust algorithm is used in the 9C scheme and the -subgrid inversion calculation is only attempted where both $\thetal$ -and $\thetavl$ are monotonically increasing and $q_t$ is simply -monotonic across the inversion. The formula used is: -\begin{equation} - \Delta \chi = {\chi}_{\ntml+2} - {\chi}_{\ntml} - - \gamma_{\chi} \left( z_{\ntml+2} - z_h \right) -\label{dqt_disc_9c} -\end{equation} -subject to the constraint that the lapse rate adjustment should not -reduce the two grid-length difference by more than half. The -free-atmospheric lapse rates are given by - -\begin{eqnarray*} - \gamma_{\thetal} & =& {\rm max}\left[ \, 0, \, \f{ {\thetal}_{\ntml+3}-{\thetal}_{\ntml+2} } - { z_{\ntml+3} - z_{\ntml+2} } - \right] \\ - \gamma_{q_t} & =& {\rm min}\left[ \, 0, \, \f{ {q_t}_{\ntml+3}-{q_t}_{\ntml+2} } - { z_{\ntml+3} - z_{\ntml+2} } - \right] -\end{eqnarray*} - -\subsection{Calculation of the subsidence flux} -\label{sec:subs_calc} - -The vertical advection or subsidence flux, ${\fx}^{subs}$, is -calculated by integrating estimates of the vertical advection -increments. These estimates are made at 9B from the model's vertical -velocity field, $w$, using first order upwind advection. As described -above, however, the coupling between different flux profiles is -performed on the model grid and, over land, these coordinate surfaces -follow the underlying terrain. To correct this, the 9C scheme -calculates the subsidence flux in grid-point, rather than physical -space, by using $\dot{\eta}$ (where $\eta$ is the model's vertical -coordinate) rather than $w$. - -The following two examples illustrate why this represents an -improvement. First, consider a boundary layer capped by a horizontal -inversion in a horizontal flow over a rising land surface. Here $w$ -will be zero and yet the model will be generating a vertical advection -flux across the inversion grid-levels, because $\dot{\eta}$ is -negative. Conversely, consider the same boundary layer but in a flow -that follows the coordinate surfaces, going up and over a hill. Now -there will be no vertical advection flux across the model's inversion -grid-level because $\dot{\eta}$ is zero and yet $w$ will be negative -on the down-slope thus giving a spurious subsidence source to the 9B -scheme. - - -\subsection{Specification of entrainment eddy diffusivity} - -As discussed above it is considered beneficial to specify the -thermodynamic entrainment fluxes explicitly under the assumption that -both the turbulence forcing and the inversion jumps change slowly -compared to the timestep. Under the circumstance that no subgrid -inversion can be diagnosed, not only is an alternative derivation of -the entrainment fluxes required, but it is also deemed likely that -these assumptions may be violated and so the entrainment fluxes are -specified via an entrainment eddy diffusivity. Currently, this is -also the case for momentum and tracer variables. - -\subsubsection{For momentum (and scalars if no subgrid inversion)} -\label{sec:ent_K} - -For momentum, and scalars if a subgrid inversion cannot be diagnosed, -see section~\ref{sec:sginv}, fluxes at the mixed layer top are -specified through an eddy diffusivity which is given by -\begin{eqnarray} - K_h|_{\ntml+\frac{1}{2}} & =& w_e \Delta_{\ntml+1} z \nonumber\\ - K_m|_{\ntml} & =& Pr \, w_e \Delta_{\ntml+\frac{1}{2}} z -\end{eqnarray} -\label{khent} -noting the Charney-Philips grid implying stresses are staggered from -scalar fluxes. The Prandtl number, $Pr$, takes the same form as for -the non-local $K$ profiles, see section~\ref{sec:nonlocal}. - -Substituting (\ref{khent}) in (\ref{scal_closure}) gives, for example, -$\wthl|_{\ntml+\frac{1}{2}} = - w_e \Delta_{\ntml+1} \thetal$. Note -that this gives entrainment buoyancy fluxes identical to -(\ref{discinv}) as long as there is no buoyancy reversal generation of -turbulence (\mbox{i.e.},$\vbro=0$) and if variations in the grid-level -jumps across the timestep are ignored. The former is because the -other terms in (\ref{we_parm}) are inversely proportional to $\Delta -b$. The latter will never actually be true and can give rise to large -errors if the inversion is rising quickly. Therefore, the -thermodynamic entrainment fluxes are specified explicitly where -possible. - -The advantages of diagnosing the subgrid inversion are that it allows -consistency between the turbulent and radiative fluxes and large-scale -vertical advection, it reduces grid-resolution errors arising from the -mixed layer depth calculation and it allows a more accurate -calculation of $\vbro$ and $\alpha_t$. For momentum, because the -jumps across inversions are typically small and variable, it seems -unwise numerically to attempt to specify the inversion stresses -explicitly and so (\ref{khent}) is always used. For the 9C -version, the entrainment $K_m$ given by (\ref{khent}) is imposed at -the height of the temperature inversion \zh (either subgrid or at -$z_{\ntml+\frac{1}{2}}$) and $K_m|_{\ntml+\frac{1}{2}}$ is calculated -from (\ref{kmsurf}) and (\ref{kmtop}), noting the use of the ${\cal - E}$ factors. - -\subsubsection{Resolved inversions} -\label{sec:entr_prof} -An inversion is defined as being resolved when it extends above the -flux-level above the usual entrainment interface level (see -section~\ref{sec:dzi}), \mbox{i.e.} when -\begin{equation*} - z_{\ntml+\frac{1}{2}} + \Delta z_i > z_{\ntml+\frac{3}{2}} -\end{equation*} -When this happens, there is no subgrid inversion diagnosis and the -entrainment parametrization follows the methodology given in -section~\ref{sec:ent_K} to give $K_h|_{\ntml+\frac{1}{2}}$. The -diffusion coefficient profile within the inversion is then calculated -assuming the $\thetavl$ flux profile within the inversion decreases -following a cosine shape from the standard parametrized entrainment -flux at the inversion base to zero at the inversion top, \mbox{i.e.}: -\begin{equation} - \overline{w'\thetavl'} = \overline{w'\thetavl'}|_{\ntml+\frac{1}{2}} - cos\left(\pi \frac{z'}{2} \right) - \label{ent_svl} -\end{equation} -where $z'=(z-\zhe)/\Delta z_i$ is scaled height within the inversion. -This flux profile is then converted into a diffusion coefficient -profile by inverting the standard flux parametrization: -\begin{equation*} - K_h|_{k+\frac{1}{2}}= - \, \frac{\overline{w'\thetavl'} } - { ({\thetavl}_{k+1}-{\thetavl}_{k})/(z_{k+1}-z_k) } -\end{equation*} -The diffusion coefficient for momentum entrainment is calculated in -the same way, allowing for the staggered grid, with the same $Pr$ as -in (\ref{khent}). - -\subsubsection{For tracers, when there is a subgrid inversion} -\label{sec:ent_K_flux} -Here `tracers' refers to scalar variables other than $\thetal$ and -$q_t$: aerosols, $q_f$, etc. Ideally, tracer entrainment fluxes would -be specified explicitly in the same way as for $\thetal$ and $q_t$. -However, specifying the entrainment flux effectively specifies the net -change in mixed-layer tracer concentration across the timestep. Thus, -if the mixed-layer tracer concentration is small at the start of a -timestep and the entrainment flux is larger than the surface flux, the -mixed-layer concentration could go negative (and tests indicated that -this did indeed happen). Specifying the entrainment flux assumes that -both the turbulence forcing and the inversion jump change slowly -compared to the timestep. Whilst this is true for atmospheric -$\thetal$ and $q_t$, the latter is not true for tracers with a small -boundary layer concentration. Consequently, for a tracer field -$\chi$, the parametrized entrainment fluxes $\overline{w'\chi'}_{ - z_{\ntml+\frac{1}{2}} }$ are calculated from (\ref{fluxinterp}) but -are implemented through an equivalent entrainment eddy-diffusivity -given by: -\begin{equation} - K_{\chi}|_{\ntml+\frac{1}{2}} = - \overline{w'\chi'}_{ z_{\ntml+\frac{1}{2}} } - \frac{\Delta_{\ntml+1} z}{\Delta_{\ntml+1} \chi} - \label{K_ent_tracer} -\end{equation} -Note from (\ref{scal_closure}) that (\ref{K_ent_tracer}) gives the -parametrized flux if $\Delta_{\ntml+1} \chi$ does not change across -the timestep (see section~\ref{sec:implicit} for a description of the -implicit numerical solution of (\ref{cons_eqn_scal})). As -(\ref{K_ent_tracer}) involves the potentially numerically dangerous -calculation of $\Delta \chi/\Delta_{\ntml+1} \chi$ (where $\Delta -\chi$ is the subgrid inversion jump, given by (\ref{dqt_disc})), the -following constraints are also ensured: -\begin{equation*} - 0 \leq K_{\chi}|_{\ntml+\frac{1}{2}} - \leq 10 \,K_{\chi}|_{\ntml-\frac{1}{2}} -\end{equation*} - -%------------------------------------------------------------------------ -\section{Surface Exchange} - -Note that the surface scheme itself is documented under the JULES -documentation. - -\subsection{The theoretical basis.}\label{section_1} -Making the assumption that \textbf{Monin-Obukhov similarity theory} -for the surface layer is valid the gradients of model variables in the -surface layer are related to the surface fluxes by: -\begin{eqnarray} - \frac{\partial T}{\partial z} + \frac{g}{ c_P }&=&-\frac{ H_0 }{ c_P \rho _0 v_\ast } \frac{ \phi _h (z/L)}{kz}\label{1.1.1}\\ - \frac{\partial q}{\partial z}&=&-\frac{ E_0 }{ \rho _0 v_\ast } \frac{ \phi _h (z/L)}{kz}\label{1.1.2}\\ - \frac{\partial {\rm {\bf v}}}{\partial z}&=&\frac{ {\rm {\bf \tau }}_{0} }{ \rho _0 v_\ast } \frac{ \phi _m (z/L)}{kz},\label{1.1.3} -\end{eqnarray} -where subscript 0 represents a surface value and subscript * -represents a surface layer scaling quantity. $\phi _{m}$ and $\phi -_{h}$ are the Monin-Obukhov stability functions (for the form of these -see section~\ref{section_1.3} below). $L$ is the Monin-Obukhov length -scale defined by -\begin{equation} - L = \frac{- { v_\ast }^3 }{k F_{B0} / \rho _0 }, - \label{1.1.4} -\end{equation} -where F$_{B0}$ is the surface buoyancy flux defined by -\begin{equation} - F_{B0} = \frac{ g }{ c_P } \beta _{T1} H_0 + g \beta _{q1} E_0. - \label{1.1.5} -\end{equation} -The buoyancy coefficients in equation~(\ref{1.1.5}) are given in -appendix~\ref{app:buoyp} with the subscript 1 denoting a value at the -lowest level in the atmosphere model. - -Equations~(\ref{1.1.1})--(\ref{1.1.3}) can be integrated from the -``surface'', i.e. the roughness height where the surface variables are -defined, to a reference height in the surface layer, for modelling -applications, the height, z$_{1}$, of the bottom model layer above the -surface. The resulting expressions for the surface turbulent fluxes -are: -\begin{eqnarray} - \frac{ H_0 }{ c_P \rho _0 }&=&-\frac{ c_H }{ c_D^{1/2} } v_\ast \left( {\Delta T + \frac{g}{c_p }( z_1 + z_{0m} - z_{0h} )} \right)\label{1.1.7}\\ - \frac{ E_0 }{ \rho _0 }&=&-\frac{ c_H }{ c_D^{1/2} } v_\ast \Delta q\label{1.1.8}\\ - \frac{ {\bf \tau }_{0} }{ \rho _{0} }&=& c_D^{1/2} v_\ast \Delta {\rm {\bf v}},\label{1.1.9} -\end{eqnarray} -where $\Delta $X=X$_{1}$-X$_{0}$. From~(\ref{1.1.7}) and~(\ref{1.1.8}) -the surface buoyancy flux in definition~(\ref{1.1.4}) is -\begin{equation} - \frac{ F_{B0} }{ \rho _0 } = -\frac{ c_H }{ c_D^{1/2} } v_\ast \Delta B, - \label{1.1.10} -\end{equation} -\begin{equation} - \Delta B = g \beta _{T1} \left( {\Delta T + \frac{g}{ c_P }( z_1 + z_{0m} - z_{0h} )} \right) - + g \beta _{q1} \Delta q - \label{1.1.11} -\end{equation} -The \textbf{surface exchange coefficients }in -equations~(\ref{1.1.7})--(\ref{1.1.9}), c$_{D}$ and c$_{H}$, are given -by -\begin{eqnarray} - c_D^{1/2}&=&\frac{k}{ \Phi _m (L , z_1 + z_{0m} , z_{0m} )} - \label{1.1.12}\\ - \frac{ c_H }{ c_D^{1/2} }&=&\frac{k}{ \Phi _h (L , z_1 + z_{0m} , z_{0h} )}, - \label{1.1.13} -\end{eqnarray} -where -\begin{eqnarray} - \Phi _m (L , z_1 + z_{0m} , z_{0m} )&=& \int \limits_{ z_{0m} /L}^{( z_1 + z_{0m} )/L} \frac{ \phi _m (\zeta )}{\zeta } d\zeta\label{1.1.14}\\ - \Phi _h (L , z_1 + z_{0m} , z_{0h} )&=& \int \limits_{ z_{0h} /L}^{( z_1 + z_{0m} )/L} \frac{ \phi _h (\zeta )}{\zeta } d\zeta\label{1.1.15}, -\end{eqnarray} -z$_{0m}$ and z$_{0h}$ are the \textbf{surface roughness lengths }for -momentum and scalars respectively. - -The equations for the \textbf{surface turbulent fluxes}, -(\ref{1.1.7})--(\ref{1.1.9}), can be written in the forms -\begin{eqnarray} - \frac{ H_0 }{ c_P \rho _0 }&=&{-c}_H V \left( {\Delta T + \frac{g}{c_p }( z_1 + z_{0m} - z_{0h} )} \right)\nonumber\\ - &=& - C_H \left( {\Delta T + \frac{g}{c_p } ( z_1 + z_{0m} - z_{0h} )} \right)\label{1.1.16}\\ - \frac{ E_0 }{ \rho _0 }&=&- c_H V \Delta q = - C_H \Delta q\label{1.1.17}\\ - \frac{ {\rm {\bf \tau }}_{0} }{ \rho _{0} }&=& c_D V \Delta {\rm {\bf v}}{ }= C_D \Delta {\rm {\bf v}},\label{1.1.18} -\end{eqnarray} -where the effective wind speed for surface turbulent exchanges, $V$, -is defined by -\begin{equation} - V = \frac{ v_\ast }{ c_D^{1/2} } = \frac{ v_\ast ^2 }{ C_D }\label{1.1.19} -\end{equation} -and the \textbf{surface conductances} for scalars and momentum are -respectively -\begin{eqnarray} - C_H&=&\frac{k}{ \Phi _h } v_\ast = c_H V\label{1.1.20}\\ - C_D&=&\frac{k}{ \Phi _m } v_\ast = c_D V\label{1.1.21}. -\end{eqnarray} - -The surface exchange coefficients can then be written in any of the -following forms: -\begin{eqnarray} - c_H&=&\frac{ C_H }{V} = \frac{ C_H C_D }{ v_\ast ^2 } = \frac{ k^2 }{ \Phi _h \Phi _m }\label{1.1.22}\\ - c_D&=&\frac{ C_D }{V} = \frac{ C_D^2 }{ v_\ast ^2 } = \frac{ k^2 }{ \Phi _m^2 }. -\label{1.1.23} -\end{eqnarray} -In order to close the system the surface scaling velocity, v$_{\ast -}$, needs to be specified. If -\begin{equation} - v_\ast = u_\ast \equiv \left| { {\rm {\bf \tau }}_{0} {/} \rho _{0} } \right|^{1/2}\label{1.1.24} -\end{equation} -we have the standard Monin-Obukhov theory and it is easy to deduce -that with this definition $v_{\ast } = c_{D}^{1/2} \Delta $\textbf{v} -and V=$\Delta $\textbf{v}. To allow for the effect of -\textbf{turbulent and cloud-scale gusts }on the surface turbulent -fluxes the surface scaling velocity, v$_{\ast }$, can be defined as -\begin{equation} - v_\ast ^2 = u_\ast ^2 + \gamma _t^2 w_\ast ^2 + \gamma _c^2 w_c^2. - \label{1.1.25} -\end{equation} -The second term represents the effects of turbulent eddy-scale -convective gusts and w$_{\ast }$ is the turbulent convective scaling -velocity defined by -\begin{equation} - w_\ast = {\left( { z_i \frac{ F_{B0} }{ \rho _0 }} \right)}^{1/3} - \label{1.1.26} -\end{equation} -for F$_{B0} >$ 0 and zero otherwise. z$_{i}$ is the height of the top -of the surface-based turbulent mixing layer. $\gamma _{t}$ is a -dimensionless constant which can be determined empirically or tuned -within empirical limits. The third term represents the effects of -deep convective cloud-scale gusts; the inclusion of this term is -optional. The form implemented is taken from -\cite{redelsperger00:_param_mesos_enhan_surfac_fluxes}, in which the -velocity scale, w$_{c}$, is a function of the convective downdraught -mass-flux at cloud base. (Note that the published expression is given -as an adjustment of the 10-m wind and has been scaled to make it -consistent with $v_\ast$.) A further term $\gamma -_{m}^{2}$w$_{m}^{2}$ could be included in low resolution models to -represent the effects of mesoscale gusts but this is not done in the -Unified Model. The\textbf{ low wind speed limit}, i.e. as $\Delta -$\textbf{v} $\to $ 0, for unstable conditions (with w$_{c }$= 0) can -be seen to be -\begin{equation} - v_\ast \sim \gamma _t w_\ast \sim \gamma _t^{3/2} {\left( {\frac{ c_H }{ c_D^{1/2} }} \right)}^{1/2} z_i^{1/2} (-\Delta B )^{1/2} - \label{1.1.27} -\end{equation} -which implies that -\begin{equation} - L \sim -( \gamma _t^3 /k) z_i - \label{1.1.28} -\end{equation} - -Thus the low wind speed limits for the sensible and latent heat fluxes -are obtained by substituting~(\ref{1.1.27}) into (\ref{1.1.7}) and -(\ref{1.1.8}) with the surface transfer coefficients evaluated with L -given by (\ref{1.1.28}). The finite limit for L implies that the form -of the stability functions, $\phi $, for very large and negative -$\zeta $ is unimportant. However, the value of $\Phi_{h}$ for L given -by (\ref{1.1.28}) is needed if the value of $\gamma _{t}$ is -determined from measurements of say the latent heat flux in very low -mean wind conditions. - -\subsubsection[Comparison with Godfrey and Beljaars (1991) formulation for gustiness] -{Comparison with the \cite{godfrey1991} formulation for gustiness} -\label{section_1.2} -We can define the \textbf{mean gust speed} at height z$_{1}$ by -\begin{equation} - v_g = ( V^2 - \left| {\Delta {{\rm {\bf v}}}} \right|^2 {)}^{{1/2}}. - \label{1.2.1} -\end{equation} -Using the definitions of $V$ (\ref{1.1.19}) and $v_{\ast }$ -(\ref{1.1.25}) it can be deduced that -\begin{equation} - v_g^2 = W_g^2 ( z_1 ) + \frac{1}{2}\left| {\Delta {{\rm {\bf v}}}} \right|{ }\left[ {\left( {{ } {\left| {\Delta {{\rm {\bf v}}}} \right|}^2 - W_g^2 ( z_1 )} \right)^{1/2} - \left| {\Delta {{\rm {\bf v}}}} \right|} \right] - \label{1.2.2} -\end{equation} -where -\begin{equation} - W_g (z) = \frac{1}{ c_D^{1/2} } {\left( { \gamma _t^2 w_\ast ^2 + \gamma _c^2 w_c^2 } \right)}^{1/2} = \frac{ \Phi _m (L , z + z_{0m} , z_{0m} )}{k} {\left( { \gamma _t^2 w_\ast ^2 + \gamma _c^2 w_c^2 } \right)}^{1/2}. - \label{1.2.3} -\end{equation} -Thus in this formulation the mean gust speed is a function of height -above the surface through the same factor, $\Phi _{m}$(z), which -determines the profile of the mean wind \textbf{v} in the surface -layer (see Eq.~(\ref{1.1.9})). The values of $\Delta $\textbf{v}, -v$_{g}$ and $V$ thus tend to zero as z $\to $ 0. Note that v$_{g} \to -$ W$_{g}$ as $\Delta $\textbf{v} $\to $ 0 and that v$_{g} \to $ 0 as -the convective gustiness scaling velocities tend to zero. - -Equation~(\ref{1.2.1}) can be rewritten as -\begin{equation} - V^2 = \left| {\Delta {{\rm {\bf v}}}} \right|^2 + v_g^2, - \label{1.2.4} -\end{equation} -which is exactly the form of \cite{godfrey1991}. However -\cite{godfrey1991} define the mean gust speed as $\beta $w$_{\ast -}$. Thus they directly modify the mean surface to air wind difference, -$\Delta $\textbf{v}, with the gustiness or turbulent convective -scaling velocity, w$_{\ast }$, combining a grid dependent quantity -with a constant scaling speed. The two formulations are similar in -that they introduce a mean gust speed but differ in their assumption -about whether this has a non-constant profile in the surface -layer. Although transitory wind gusts may not be as close to -equilibrium with the surface characteristics as the mean wind they -should have a profile in the surface layer which approaches zero at -the surface (strictly at the roughness height z$_{0m})$. - -The two formulations can be made equivalent by assuming that -\cite{godfrey1991} $\beta $ is not constant but is given by $\gamma -_{t}(\Phi _{m}$(z)/k) which tends to zero as the surface is -approached. However, over sea points where the roughness length is -small (of order 10$^{-4}$ m) and for which \cite{godfrey1991} derived -their formulation, $\Phi_{m}$(z) varies at most by about 15{\%} -between 10 m and 50 m. Assuming a constant $\beta $ does not lead to -much inaccuracy in these circumstances. If gustiness is included over -land, as is the case in the Unified Model, the higher roughness -lengths lead to a greater variation in $\Phi _{m}$(z) in the region -where models generally have their lowest level placed. - -If $\gamma _{t}$ = 0.08 then in the low wind speed limit of an -unstable tropical maritime surface layer with a virtual temperature -lapse of 1.5 K, a specific humidity lapse of 7x10$^{-3}$ kg/kg, -SST=303.16 K and a boundary layer depth of 800 m we obtain a latent -heat flux of 37.08 W/m$^{2}$ assuming the form for the stability -functions given below. - - -\subsection{Making surface exchange consistent with flux differencing} - -The boundary layer scheme increments conserved quantities using -differences in fluxes across a layer. This is strictly consistent only -if the conserved quantity is a mass-weighted mean across the layer, -rather than a representative value, such as the value at the middle of -the layer. If the profile of the conserved quantity is linear the mean -of the quantity is the same as the point-value in the middle of the -layer, but this is not so if the profile is not linear. Near the -surface, the profiles will be logarithmic in neutral conditions and so -the values in the middle of the layer will be larger than the mean -values. Surface similarity in the UM is applied treating taking the -wind and temperature in the bottom layer as point values in the -calculation of surface fluxes and is therefore not absolutely -consistent with flux differencing. Whilst the effect of this -difference is not large, it is desirable to have the option of -correcting it, which is done by enabling the option to ``make surface -exchange consistent with flux differencing.'' The following -discussion explains how this is done. - -In effect, the UM takes the displacement height for momentum as -$-z_{0m}$, where $z_{0m}$ is the momentum roughness length. The -profile of wind is therefore determined by Monin-Obukhov theory as -\begin{equation} - \frac{\partial u}{\partial z} = \frac{u_*}{k(z+z_{0m})} \phi_m((z+z_{0m})/L), -\end{equation} -with $u_*$ being the friction velocity, $L$ the surface Obukhov length -and $\phi_m$ the similarity function. It is common practice to -introduce a new function $\psi_m$ such that -$\phi_m(\zeta)=1-\zeta \partial \psi_m / \partial \zeta.$ Redefining -the vertical coordinate as $\zeta=z/L$, we have -\begin{eqnarray} - u(\zeta) &=& \frac{u_*}{k} \int_{0}^{\zeta} \frac{1}{(\zeta'+\zeta_{0m})} - \phi_m(\zeta'+\zeta_{0m}) \, d\zeta' = - \frac{u_*}{k} \int_{\zeta_{0m}}^{\zeta'+\zeta_{0m}} - \frac{1}{\zeta'} \phi_m(\zeta') \, d\zeta' \\ - &=& \frac{u_*}{k} \int_{\zeta_{0m}}^{\zeta'+\zeta_{0m}} \left ( \frac{1}{\zeta'} - - \frac{d\psi_m}{d\zeta'} \right ) \, d\zeta' \nonumber \\ - &=& \frac{u_*}{k} \left \{ \ln \left ( \frac{\zeta+\zeta_{0m}}{\zeta_{0m}} - \right ) - \psi_m(\zeta+\zeta_{0m}) + \psi_m (\zeta_{0m}) \right \}. - \nonumber -\end{eqnarray} -This is also frequently written as -\begin{equation} - u(\zeta) = \frac{u_*}{k} \Phi_m(\zeta). -\end{equation} -The mean over the lowest layer, of depth $z_1$ (or $\zeta_1$ in the -rescaled coordinate), is therefore -\begin{eqnarray} - \bar u &=& \frac{u_*}{k\zeta_1} \int_0^{\zeta_1} u(\zeta) \, d \zeta \\ - &=& \frac{u_*}{k\zeta_1} \int_0^{\zeta_1} - \ln \left ( \frac{\zeta+\zeta_{0m}}{\zeta_{0m}} \right ) - - \psi_m(\zeta+\zeta_{0m}) + \psi_m (\zeta_{0m}) \, d \zeta. - \nonumber -\end{eqnarray} -We consider the three terms within the integral separately. For the -first, -\begin{eqnarray} - \int_0^{\zeta_1} \ln \left ( \frac{\zeta+\zeta_{0m}}{\zeta_{0m}} \right ) \, d \zeta - &=& \zeta_{0m} \int_1^{1+\zeta_1/\zeta_{0m}} \ln(x) \, dx \\ - &=& \zeta_{0m} \left [ \left ( 1+ \frac{\zeta_1}{\zeta_{0m}} \right ) \ln - \left ( 1+ \frac{\zeta_1}{\zeta_{0m}} \right ) - - \left ( 1+ \frac{\zeta_1}{\zeta_{0m}} \right ) +1 \right ] . \nonumber -\end{eqnarray} -For the second, -\begin{eqnarray} - \int_0^{\zeta_1} \psi_m(\zeta+\zeta_{0m}) \, d\zeta &=& - \int_{\zeta_{0m}}^{\zeta_1+\zeta_{0m}} \psi_m(\zeta) \, d\zeta \\ - &=& \left [ \zeta \psi_m - \right ]_{\zeta_{0m}}^{\zeta_1+\zeta_{0m}} - - \int_{\zeta_{0m}}^{\zeta_1+\zeta_{0m}} \zeta \frac{d\psi_m}{d\zeta} - d\zeta \nonumber \\ - &=& (\zeta_1+\zeta_{0m}) \psi_m(\zeta_1+\zeta_{0m}) - \zeta_{0m} - \psi_m(\zeta_{0m}) \nonumber \\ &-& \int_{\zeta_{0m}}^{\zeta_1+\zeta_{0m}} - (1-\phi_m) d\zeta \nonumber \\ - &=& (\zeta_1+\zeta_{0m}) \psi_m(\zeta_1+\zeta_{0m}) - \zeta_{0m} - \psi_m(\zeta_{0m}) \nonumber \\ &+& \int_{\zeta_{0m}}^{\zeta_1+\zeta_{0m}} - (\phi_m -1) \, d\zeta . \nonumber -\end{eqnarray} -$\phi_m-1$ is retained in the last integral since this will prove -convenient in later algebra. The third integral is trivial. Hence, -\begin{eqnarray} - \bar u &=& \frac{u_*}{k} \left \{ - \left ( 1+ \frac{\zeta_{0m}}{\zeta_1} \right ) \left [ - \ln \left ( 1+ \frac{\zeta_1}{\zeta_{0m}} \right ) \right . \right . \\ - &-& \left . \left . \psi_m(\zeta_1+\zeta_{0m}) + \psi_m(\zeta_{0m}) \right ] -1 - - \frac{1}{\zeta_1} \int_{\zeta_{0m}}^{\zeta_1+\zeta_{0m}} - (\phi_m -1) \, d\zeta \right \} \nonumber \\ - &=& \frac{u_*}{k} \left \{ \left ( 1+ \frac{\zeta_{0m}}{\zeta_1} \right ) - \Phi_m(\zeta_1) - \frac{1}{\zeta_1} \int_{\zeta_{0m}}^{\zeta_1+\zeta_{0m}} - \phi_m \, d\zeta \right \} . \nonumber -\end{eqnarray} -Thus, in practical terms, the standard function $\Phi_m$ is evaluated -at the top of the layer, scaled by $1+\zeta_{0m}/\zeta_1$ and reduced -by the mean value of $\phi_m$. For standard Monin-Obukhov functions, -this last integral is easy to perform. In the limit $L \rightarrow -\infty$ it becomes 1 and to avoid a numerical singularity it is set -equal to 1 in this (nearly neutral) limit. The adjustment of the -thermal Monin-Obukhov function is exactly equivalent. - -Algorithmically, the existing routine {\tt PHI\_M\_H} is replaced by -{\tt PHI\_M\_H\_VOL} which takes the same inputs except that the -heights are the top of the layers. This is done when the surface -fluxes, or turbulence scales $u_*$ and $\theta_*$ are -calculated. Monin-Obukhov functions are also used to calculate winds -and temperatures at observed levels. In this case a value at a -particular height is required, so the existing Monin-Obukhov routine -should be used. - -\subsection{The form of the stability functions.} -\label{section_1.3} -For \textbf{stable conditions}, i.e. $\Delta $B $\ge $ 0, the -stability functions are given by \cite{Beljaars1991}: -\begin{eqnarray} - \Phi _m&=&\ln \left( {\frac{ z_1 + z_{0m} }{ z_{0m} }} \right) - \Psi _m ( \zeta _1 ) + \Psi _m ( \zeta _{0m} ) - \label{1.3.11}\\ - \Phi _h&=&\ln \left( {\frac{ z_1 + z_{0m} }{ z_{0h} }} \right) - \Psi _h ( \zeta _1 ) + \Psi _h ( \zeta _{0h} ) - \label{1.3.12} -\end{eqnarray} -where $\zeta _{1}$ = (z$_{1}$ + z$_{0m})$/L, $\zeta _{0m}$ = -z$_{0m}$/L, $\zeta _{0h}$ = z$_{0h}$/L and -\begin{eqnarray} - - \Psi _h (\zeta )&=&\left[ { {\left( {1 + \frac{2}{3}a\zeta } \right)}^{3/2} - 1 } \right] + b\left( {\zeta - \frac{c}{d}} \right)\exp (-d\zeta ) + \frac{bc}{d} - \label{1.3.13}\\ - - \Psi _m (\zeta )&=&a\zeta + b\left( {\zeta - \frac{c}{d}} \right)\exp (-d\zeta ) + \frac{bc}{d}, - \label{1.3.14} -\end{eqnarray} -with $a = 1$, $b =2/3$, $c = 5$, $d = 0.35$. - -Note that the bulk flux Richardson number for the surface layer is -given by -\begin{equation} {Ri}_{fB} = \frac{ c_D^{1/2} }{k} \frac{ z_1 }{L} = - \frac{ z_1 /L}{ \Phi _m } - \label{1.3.8} -\end{equation} -so the \cite{Beljaars1991} functions imply Ri$_{f B} \to $ 1/a = 1 as -z$_{1}$/L $\to \infty $. - -For \textbf{unstable conditions}, i.e. $\Delta $B $<$ 0, the Dyer and -Hicks forms \cite[]{dyer1974} are used: -\begin{eqnarray} - \phi _m&=&(1 - 16\zeta )^{-1/4} - \label{1.3.15} \\ - \phi _h&=&(1 - 16\zeta )^{-1/2} - \label{1.3.16} -\end{eqnarray} -(Note that $\phi _{h}\prime $ is discontinuous at 0.) These are only -empirically verified for $\zeta \ge $ -1. Evaluating the integrals -(\ref{1.1.14}) and (\ref{1.1.15}) we obtain: -\begin{equation} - \Phi _m = \ln \left( {\frac{ z_1 + z_{0m} }{ z_{0m} }} \right) - 2 \ln \left( {\frac{1 + X_1 }{1 + X_0 }} \right) - \ln \left( {\frac{1 + X_1^2 }{1 + X_0^2 }} \right)+ 2 \left( { {\tan }^{-1} X_1 - {\tan }^{-1} X_0 } \right) - \label{1.3.17} -\end{equation} -where -\begin{equation} - X_1 = (1 - 16 \zeta _1 )^{1/4} , X_0 = (1 - 16 \zeta _{0m} )^{1/4} - \label{1.3.18} -\end{equation} -and -\begin{equation} - \Phi _h = \ln \left( {\frac{ z_1 + z_{0m} }{ z_{0h} }} \right) - 2 \ln \left( {\frac{1 + Y_1 }{1 + Y_0 }} \right) - \label{1.3.19} -\end{equation} -where -\begin{equation} - Y_1 = (1 - 16 \zeta _1 )^{1/2} , Y_0 = (1 - 16 \zeta _{0h} )^{1/2}. - \label{1.3.20} -\end{equation} - -\subsection{The iterative algorithm for calculating the surface - exchange coefficients}\label{section_1.4} - -For conditions that are stable, i.e. $\Delta $B $\ge$ 0, or near-neutral (taken as -$\Delta $\textbf{v} $\ge$ 2 ms$^{-1}$), then start the iteration from the neutral limit, so -\begin{eqnarray} - \Phi _m^{(0)}&=&\ln \left( {\frac{ z_1 + z_{0m} }{ z_{0m} }} \right) - \label{1.4.5} \\ - \Phi _h^{(0)}&=&\ln \left( {\frac{ z_1 + z_{0m} }{ z_{0h} }} \right) - \label{1.4.6} \\ - v_\ast ^{(0)}&=& {\left( {\frac{k}{ \Phi _m^{(0)} }} \right)} \left| {\Delta {{{\rm {\bf v}}}}} \right| - \label{1.4.7} -\end{eqnarray} -Otherwise (if $\Delta $B $<$ 0 and $\Delta $\textbf{v} $<$ 2 ms$^{-1}$ ) start from the greater of the neutral and convective limits for $v_\ast^{(0)}$, so -\begin{eqnarray} - \frac{1}{ L^{(0)} }&=&\frac{-k}{ \gamma _t^3 z_i } - \label{1.4.1}\\ - \Phi _m^{(0)}&=& \Phi _m ( L^{(0)} , z_1 + z_{0m} , z_{0m} ) - \label{1.4.2}\\ - \Phi _h^{(0)}&=& \Phi _h ( L^{(0)} , z_1 + z_{0m} , z_{0h} ) - \label{1.4.3}\\ - v_\ast ^{(0)}&= & MAX{\left[ {\left( {\frac{k}{ \Phi _m^{(0)} }} \right)} \left| {\Delta {{{\rm {\bf v}}}}} \right|, \, - {\left[ { \gamma _t^3 \left( {\frac{k}{ \Phi _h^{(0)} }} \right) z_i \left| {-\Delta B} \right| } \right]}^{ 1/2} \right]} - \label{1.4.4} -\end{eqnarray} - -Then calculate -\begin{eqnarray} - C_D^{(0)}&=&\frac{k}{ \Phi _m^{(0)} } v_\ast ^{(0)} - \label{1.4.8} \\ - C_H^{(0)}&=&\frac{k}{ \Phi _h^{(0)} } v_\ast ^{(0)} - \label{1.4.9} -\end{eqnarray} - -Having set up initial values the iteration loop can be entered (this -is the original method used but contains an inconsistency in the -treatment of boundary-layer convective gustiness, as described in -section~\ref{mo_iter_corrn}): - -DO n = 1 to N -\begin{eqnarray} - u_\ast ^{(n)2}&=& C_D^{(n-1)} \left| {\Delta {{\rm {\bf v}}}} \right| - \label{1.4.10} \\ - {\left( {\frac{ F_{B0} }{ \rho _0 }} \right)}^{(n)}&=& { {-C}_H }^{(n-1)} \Delta B - \label{1.4.11}\\ - w_\ast ^{(n)}&=& {\left[ { z_i {\left( {\frac{ F_{B0} }{ \rho _0 }} \right)}^{(n)} } \right]}^{ 1/3} - \label{1.4.12}\\ - v_\ast ^{(n)2}&=& u_\ast ^{(n)2} + \gamma _t^2 w_\ast ^{(n)2} + \gamma _c^2 w_c^2 - \label{1.4.13}\\ - \frac{1}{ L^{(n)} } = \frac{-k( F_{B0} / \rho _0 )^{(n)} }{ v_\ast ^{(n)3} } - \label{1.4.14}\\ - \Phi _m^{(n)}&=& \Phi _m ( L^{(n)} , z_1 + z_{0m} , z_{0m} ) - \label{1.4.15} \\ - \Phi _h^{(n)}&=& \Phi _h ( L^{(n)} , z_1 + z_{0m} , z_{0h} ) - \label{1.4.16} \\ - C_D^{(n)}&=&\frac{k}{ \Phi _m^{(n)} } v_\ast ^{(n)} - \label{1.4.17} \\ - C_H^{(n)}&=&\frac{k}{ \Phi _h^{(n)} } v_\ast ^{(n)} - \label{1.4.18} -\end{eqnarray} - -END DO. - -For neutral and stable conditions ($\Delta $B $\ge $ 0) start the -iteration from the neutral values and set w$_{\ast }$=0 in the above -iteration loop. - -Use the final (N) values of C$_{H}$ and C$_{D}$ to calculate the -surface sensible and latent heat fluxes and surface stress: - -\begin{eqnarray} - H_0&=& {-c}_P \rho _0 C_H^{(N)} \left( {\Delta T + \frac{g}{ c_P }( z_1 + z_{0m} - z_{0h} )} \right) - \label{1.4.19} \\ - E_0&=& {-\rho }_0 C_H^{(N)} \Delta q - \label{1.4.20} \\ - {\rm {\bf \tau }}_{0} &=& \rho _0 C_D^{(N)} \Delta {\rm {\bf v}} - \label{1.4.21} -\end{eqnarray} -N is the last iteration value. N = 5 is currently used. - -For sea points the momentum roughness length and the wind mixing -energy flux are calculated from v$_{\ast }^{(N)}$ using the formulae -in subsection~\ref{section_1.6} below. - -\subsubsection{Correction to the iterative algorithm}\label{mo_iter_corrn} -The above implementation of boundary-layer convective gustiness in the -Monin-Obukhov iteration contains an inconsistency. The overall effect -turns out to be small, but it is desirable to use the corrected form, -which is derived as follows. - -We decompose the wind as ${\bf u}=\bar{\bf u} + {\bf u}_g + {\bf u}'$, -representing, respectively, the large-scale, gust and small-scale -turbulent contributions to the velocity. Locally, Monin-Obukhov theory -then gives -\begin{equation} - |\bar{\bf u} + {\bf u}_g({\bf x}) | = \frac{u_*({\bf x})}{k} \Phi_m({\bf x}). -\end{equation} -We ignore the spatial variation of $\Phi_m$, expecting that the -principal effect of locally stronger winds is to increase the local -stress -- this is exactly true in nearly neutral flow. The local -stress is aligned with the wind so -\begin{equation} {\bf \tau}({\bf x}) = \rho u_*^2({\bf x}) - \frac{\bar{\bf u} + {\bf u}_g({\bf x})} {|\bar{\bf u} + {\bf - u}_g({\bf x})|}. -\end{equation} -With the assumption that $\Phi_m$ does not vary spatially, -\begin{equation} {\bf \tau}({\bf x}) = \rho \frac{k^2}{\Phi_m^2} - |\bar{\bf u} + {\bf u}_g({\bf x})| (\bar{\bf u} + {\bf u}_g({\bf - x})) \equiv \rho C_D |\bar{\bf u} + {\bf u}_g({\bf x})| (\bar{\bf - u} + {\bf u}_g({\bf x})), -\end{equation} -where $C_D$ is the standard drag coefficient, $\frac{k^2}{\Phi_m^2}$. -The grid-box mean effect is -\begin{equation} - \langle{\bf \tau}\rangle = \rho C_D \langle |\bar{\bf u} + {\bf u}_g({\bf x})| - (\bar{\bf u} + {\bf u}_g({\bf x})) \rangle \approx \rho C_D \langle - |\bar{\bf u} + {\bf u}_g({\bf x})| \rangle \bar{\bf u}, -\end{equation} -which is the product of the enhanced wind speed (including gusts) and -the mean velocity. The magnitude of the stress is then -$\langle\tau\rangle = \rho C_D S U$, with the last two symbols -representing the mean wind speed, including gusts, and the mean -background velocity. - -In the model, we want to write this as an effective drag coefficient, -$C_{De}$, so that $\langle\tau\rangle = \rho C_{De}U^2$. Now define -$\tilde u_* = \frac{k}{\Phi_m}U$, the friction velocity due to -large-scale flow, and $\hat u_* = \frac{k}{\Phi_m}S$, the friction -velocity due to the total flow, including gusts (again implicitly -assuming that $\Phi$ is unaffected by the gusts). The representation -is $\hat u_*^2 = \tilde u_*^2 +\gamma_t^2 w_*^2$. Then, -\begin{equation} - C_{De}=\frac{C_DS}{U} = \frac{k^2}{\Phi_m^2} \frac{\hat u_*}{\tilde u_*} - = \frac{k}{\Phi_m}\frac{\hat u_*}{U} -\end{equation} -The variable {\tt CDV} in the routine {\tt FCDCH} within the -Monin-Obukhov iteration will then be $\frac{k}{\Phi_m}{\hat u_*}$ -(note that it is divided by $U$ at the end of the routine). This is -indeed coded at the end of the loop; but in the original code {\tt - CDV} is used to calculate $\tilde u_*^2$ at the beginning of the -routine. In fact, $\tilde u_*^2 = (k/\Phi_m)\tilde u_* U$, which -differs from the coded result by a factor of $\tilde u_*/\hat -u_*$. After this step, the purported $\tilde u_*$ is augmented by the -gust contribution to get $\hat u_*$. This has the consequence of -overestimating $C_{De}$ and also means that what is described as {\tt - U\_S} in this loop is not $\tilde u_*$, but $\sqrt(\tilde u_* \hat -u_*)$. - -If we let $v=\sqrt(\tilde u_* \hat u_*)$, then we get -\begin{equation} - \hat u_*^2 = \frac{1}{2} \left \{ \gamma_t^2 w_*^2 + \sqrt { \gamma_t^4 w_*^4 - + 4 v^2 } \right \}. -\end{equation} -In the corrected version this expression is used to calculate {\tt - V\_S}, namely $\hat u_*$ within the iteration. - - -\subsection{The interpolation of surface layer variables to standard - observation heights}\label{section_1.5} -Integrating~(\ref{1.1.3}) between the roughness height, z$_{0m}$, and -the observation height z$_{ob}$ we obtain -\begin{equation} - {\rm {\bf v}}_{ob} { = } {\rm {\bf v}}_{0} { + - }\frac{ {\rm {\bf \tau }}_{0} }{ \rho _0 v_\ast k} \Phi _m (L, - z_{ob} + z_{0m} , z_{0m} ) - \label{1.5.1} -\end{equation} -Using the expression for the surface turbulent stress this gives the -interpolation formula -\begin{equation} - {\rm {\bf v}}_{ob} { = } {\rm {\bf v}}_{0} { + - }\frac{ C_D }{k v_\ast } \Phi _m (L, z_{ob} + z_{0m} , z_{0m} ) ( - {\rm {\bf v}}_{1} { - } {\rm {\bf v}}_{0} {)} - \label{1.5.2} -\end{equation} -For wind z$_{ob}$ is set to 10m and the last iteration (N) values of -C$_{D}$, L and $v_{\ast }$ are used. Integrating~(\ref{1.1.1}) and -(\ref{1.1.2}) between the roughness height, z$_{0h}$, and the -observation height z$_{ob, }$ we obtain for the scalar $X$ -($=T+(g/c_{P})z$ , $q$ or tracer amount) -\begin{equation} - X_{ob} = X_0 + \frac{ F_{X0} }{ \rho _0 v_\ast k} \Phi _h (L, z_{ob} + z_{0h} , z_{0h} ) - \label{1.5.3} -\end{equation} -and using the expression for the surface flux $F_{X0}$ of the scalar -quantity $X$ this gives the interpolation formula -\begin{equation} - X_{ob} = X_0 + \frac{ C_H }{k v_\ast } \Phi _h (L, z_{ob} + z_{0h} , z_{0h} ) ( X_1 - X_0 ) - \label{1.5.4} -\end{equation} -For temperature and humidity z$_{ob}$ is set to the screen height (1.5 -m) and the last iteration (N) values of C$_{H}$, L and $v_{\ast }$ are -used. - -\subsubsection{The parametrization of decoupling} - -In the foregoing analysis it is tacitly assumed that the surface -layer, up to the model's lowest grid level, is in equilibrium with the -surface and lies within the constant flux layer. In light winds, and -when the surface temperature falls quickly, these assumptions are -invalid; equation~\ref{1.5.4} then yields temperatures at the height -of observation that are too closely tied to the surface -temperature. Observed temperatures may be significantly warmer: this -may be termed decoupling. Two parametrizations of this effect are -available. Both involve the idea that as the wind becomes very light -radiative cooling comes to determine the temperature profile. - -The first parametrization simply sets the interpolation coefficient -between the surface temperature and that on the model's lowest level -according to the radiative equilibrium profile when the Richardson -number exceeds 0.25 (a typical criterion for high stability). - -The second more elaborate scheme is directed at the evening -transition, when the surface temperature is falling rapidly: it is -under such conditions that the impact of decoupling on the air -temperature is greatest. For a couple of hours after the surface -temperature drops below the air temperature, radiative cooling -directly to the surface largely determines the atmospheric cooling -rate when the wind is light and this may easily be calculated, -provided that a parametrization of the transmission between the air -and the surface is available: this parametrization depends on the -absorbing properties of atmospheric trace gases. Explicitly, we have -\begin{equation} - \dot T_{ob, \mbox{\tiny rad, surf}} = \frac{4\sigma T_s^3}{c_P} - {\cal K}(z_{ob}) (T_s-T_{ob}), -\end{equation} -where $\dot T_{ob, \mbox{\tiny rad, surf}}$ is the cooling rate of the -air at the height of observation due to direct radiative exchanges -with the surface, $T_{ob}$ is the air temperature at that height, -$T_s$ is the temperature of the surface and ${\cal K}(z_{ob})$ depends -on the concentration of trace gases in the atmosphere, in practice -water vapour and carbon dioxide, and their spectroscopic -properties. The contributions of water vapour and carbon dioxide are, -to a good approximation, additive, so we may write -\begin{equation} - {\cal K}(z_{ob}) = \left [ q_w {\cal C}_w(\mu_w, - T_{ob}) + q_c {\cal C}_c(\mu_c, T_{ob}) \right ], -\end{equation} -where $q_w$ and $q_c$ are the specific concentrations of water vapour -and carbon dioxide and $\mu_w$ and $\mu_c$ are the respective -pathlengths between the observation height and the surface. The -functions ${\cal C}_w$ and ${\cal C}_c$ are parametrized and explicit -functional forms are included in the code. - -In stronger winds turbulent cooling will be more important, so the -scheme must approach the standard procedure in that limit. Within the -context of local scaling, it can be shown that the depth of the -atmosphere which feels the impact of surface cooling must scale on -${\cal L}=(u_*^3/ (g/T_s)\dot T_s)^{1/2}$, where $u_*$ is the surface -friction velocity, $\dot T_s$ is the surface cooling rate. This -parameter is used as a measure of the strength of turbulence to define -the relaxation back to the strong-wind limit. - -To implement the scheme, the liquid-frozen potential temperature at -the height of observation, $\theta_{ob}=T_{ob}+(g/c_P)z_{ob}-(L/c_P) -q_{cl} - ((L+L_f)/c_P) q_{cf}$ is made a prognostic. Whenever the -surface buoyancy flux changes sign and becomes stable, this prognostic -is initialized using standard theory. On subsequent timesteps, it is -updated to allow for radiative cooling to the surface, giving a -provisional value $\theta_{ob}'$, and then relaxed back towards the -result that would be obtained from standard similarity theory, -$\theta_{ob, \mbox{\tiny sim}}$, as in the last section: -\begin{eqnarray} - \theta_{ob}'(t+\delta t) &\leftarrow & \theta_{ob}(t)+ - \delta t \, \dot T_{ob,\mbox{\tiny rad,surf}} \\ - \theta_{ob}(t+\delta t) &\leftarrow & W \theta_{ob}'(t+\delta t) - +(1-W) \theta_{ob, \mbox{\tiny sim}}. -\end{eqnarray} -By tuning against an idealized highly vertically resolved model based -on local scaling we set, -\begin{equation} - W = \exp(-(0.4 f)^2 \delta t \, t_{\mbox{\tiny trans}})) / - (1+X({\cal L})\delta t), -\end{equation} -with -\begin{equation} - X({\cal L})= \min \left ( 0.000283 \left \{\frac{{\cal L}}{z_{ob}} - \log \left (1+\frac{z_{ob}}{z_0}\right ) \right \}^2, - \; \frac{0.2 u_* }{z_{ob}} \right ). -\end{equation} -The explicit dependence of $W$ on the timestep, $\delta t$, ensures -that the results converge as $\delta t \rightarrow 0$. The exponential -factor involving the Coriolis parameter, $f$, is intended to represent -the recoupling of the surface and atmosphere as growing directional -shear at the top of the incipient stable boundary layer generates -turbulence. This factor is somewhat exaggerated relative to the -results of the model against which it is tuned, as a cautionary -measure to ensure that decoupling is not allowed to persist too long -after the transition. This is the purpose of the inclusion of the -factor $0.4 f t_{\mbox{\tiny trans}}$, where $t_{\mbox{\tiny trans}}$ -is the time since the transition. - -It must be stressed that these schemes are heuristic and that the -precise behaviour in weak turbulence is not fully understood. The -second scheme appears to work well during the evening transition, but -for reasons of caution decoupling is suppressed somewhat too -rapidly. The simpler first scheme underestimates decoupling during the -transition, but allows it to persist longer, although tending to -overestimate it on these timescales. Overall, the second scheme is to -be preferred. - -\subsection{The surface fluxes for sea and sea-ice - gridboxes}\label{section_1.6} -For gridboxes with sea-ice (i.e. where sea-ice fraction, f$_{I} >$ 0) -sensible and latent heat fluxes are calculated separately for the sea -and ice parts of the gridbox and combined with appropriate weighting -to obtain the total fluxes into the atmosphere. This is done because -the two surfaces can have very different temperatures and also differ -in their roughness. - -The sea and sea-ice surface fluxes of sensible heat, moisture and -momentum are calculated using gridbox mean surface transfer -coefficients, $<$C$_{H}>$ and $<$C$_{D}>$. These are linear -combinations of the corresponding coefficients calculated for the -ice-free sea (L), typical Marginal Ice Zone broken sea-ice (MIZ) and -complete ice cover (I). - -The surface sensible heat and moisture fluxes are calculated -separately for the ice-free (leads) and ice-covered parts of the -gridbox. Although the fluxes over the two surfaces are calculated from -gridbox mean surface transfer coefficients, the different surface -temperatures give different fluxes. The ice surface temperature is -predicted from a surface energy balance and ice heat conduction model -(see the documentation for the land and ice surface processes -component of the Unified Model). In current versions of the model the -sea surface temperature (SST) is assumed to be 271.35 K, the freezing -point of sea water, whenever the ice fraction is greater than -zero. This is unrealistic except for genuine leads (i.e. large ice -fraction) and a future version of the model will allow the SST to be -larger than 271.35 K for gridboxes with sea-ice. - -The wind mixing energy flux, F$_{WME}$ , is the rate of production of -turbulent kinetic energy per unit area in the sea surface layer by the -wind stress at the air-sea interface. In atmosphere-only -configurations of the Unified Model this quantity is a useful -diagnostic. When the atmosphere model is coupled to an ocean model the -wind mixing energy flux is accumulated over an ocean model timestep -and then used in the calculation of the mixing in the upper layers of -the ocean. The gridbox mean \textbf{wind mixing energy flux} is given -by - -\begin{equation} - F_{WME} = ( 1- f_I ) \frac{ \rho _0^{3/2} v_\ast ^3 }{ \rho _{(sea)}^{1/2} } - \label{1.6.9} -\end{equation} -where $v_{\ast }$ is calculated using the drag coefficient for the -leads part of the gridbox, c$_{D(L)}$, rather than the gridbox mean -value, $<$c$_{D}>$, when there is partial ice cover. - -\subsubsection{Roughness Lengths over the Sea} - -The roughness lengths for momentum and scalars depend on both the -atmospheric flow and the wave state. The dependence on wave state is -not fully understood and is still a subject of active research. In any -case, it could only fully be represented in a coupled wave-atmosphere -model. Simpler more empirical schemes are therefore currently used in -the Unified Model. - -In all schemes available here the momentum roughness length is given by -\begin{equation} - z_{0m(sea)} = \frac{1.54\times {10}^{-6} }{ v_\ast } + - \frac{\alpha}{g} v_\ast ^2 - \label{eq:z0msea} -\end{equation} -which is a generalisation of Charnock's formula to include low-wind -conditions \cite[]{Smith88}. $\alpha$ is Charnock's coefficient, which -is determined from field measurements. It is often taken as a -constant, but more elaborate schemes include a dependence on wind -speed. In practice the difference between different parametrizations -of the momentum roughness length therefore comes down to the -specification of Charnock's coefficient. - -There is greater uncertainty in the roughness lengths for scalars and -the dependencies are described separately for each scheme. Note that -whilst the full versions of some schemes prescribe different roughness -lengths for heat and moisture, in the Unified Model we have only a -single roughness for all scalars. - -Schemes are selected by setting the variable {\sl iseasurfalg}, as now -described. -\begin{enumerate} -\item Option {\sl iseasurfalg=0}. The original and most basic scheme - comprises a fixed value of Charnock's coefficient and a fixed scalar - roughness length. Typical values of Charnock's coefficient lie in - the range 0.011--0.018 and a typical value of the thermal roughness - length is $z_{0h(sea)}$ = 4x10$^{-5}$ m. - -\item Option {\sl iseasurfalg=1}. The use of a fixed thermal - roughness length, as above, leads to a rapid increase in the - exchange coefficient for moisture as the wind speed increases that - is at variance with observational evidence. A parametrization of the - scalar roughness length was developed from surface divergence theory - \cite[]{csanady2001}, as described by \cite{edwards2007}. This - involves an inverse dependence of $z_{0h}$ on the friction velocity - in the aerodynamically smooth limit and an inverse dependence of - $z_{0h}$ on $z_{0m}$ at higher wind speeds that reduces the increase - in the exchange coefficient with wind speed. - - During iteration of the equations of surface transfer to calculate - the Obukhov length, the friction velocity changes, so implicitly - changing the roughness lengths. Historically, in the algorithm - adopted in the Unified Model,roughness lengths have not been - modified within this iteration, with values from the previous - timestep being used. The scheme was therefore originally implemented - in a form that was based on conditions at the previous timestep, but - did not require adding $z_{0h}$ to the dump. To cope with conditions - of light winds, this required an iterative calculation of $v_\ast$ - from $z_{0m}$ from the previous timestep before the calculation of - the Obukhov length and $v_\ast$. Note that although there is not a - 1-1 relationship between $v_\ast$ and $z_{0m}$, the ambiguity is in - practice removed by the consideration that the inversion is only of - relevance in conditions of light winds. - - With this scheme a fixed value of Charnock's coefficient must be - specified as above. - -\item Option {\sl iseasurfalg=2}. An alternative version of the - foregoing scheme has been developed that includes full iteration of - the roughness lengths within the iteration for the Obukhov length. - -\item Option {\sl iseasurfalg=3}. This option provides various forms - of the COARE algorithm. The COARE algorithm exists in various forms - and continues to be developed. Version 3.0 \cite[]{fairall2003} has - been extensively used, while version 3.5 \cite[]{edson2013} has - recently been released. Whilst the full COARE algorithm provides a - complete description of surface transfer at the sea surface, here we - use only the expressions for the roughness lengths. - - In current versions of the scheme Charnock's coefficient is - specified using a linear relationship between the 10-m wind speed, - valid over a certain range of wind speeds, with fixed values outside - the range: - \begin{equation} - \alpha = a U_{10} +b - \label{eq:charn} - \end{equation} - for $U_{10,min} < U_{10} < U_{10,max}$. The constants $a$, $b$, - $U_{10,min}$ and $U_{10,max}$ differ between different versions of - the algorithm and are specified through namelist - parameters. Strictly, $U_{10}$ here should be the neutral 10-m wind - speed, but over the ocean the difference between the neutral and - stability-adjusted wind speeds is typically small, so the - distinction is often ignored. (Current practice in data assimilation - (2014) is to ignore the distinction). A logical switch is therefore - provided to enable the user to apply the formula using the true - neutral wind or the stability-adjusted wind, as preferred. - - The COARE algorithm does distinguish roughness lengths for heat and - moisture, but this is not currently feasible in the Unified Model, - so, since latent heat fluxes are dominant over the ocean, the scalar - roughness length is set using the expression for the moisture - roughness, - \begin{equation} - z_{0h} = \min(1.15\times 10^{-4}, 5.5\times 10^{-5}/Re_*^{0.6}), - \label{eq:z0h_coare} - \end{equation} - where $Re_*$ is the roughness Reynolds number. - - This scheme has been implemented is a form that allows the roughness - lengths to evolve during iteration to obtain the Obukhov length. - -\item Option {\sl iseasurfalg=4}. Equivalent to option {\sl iseasurfalg=1} - for a variable Charnock parameter. A fixed value of Charnock's coefficient - does not need to be provided. On the other hand, a Charnock field needs - to be provided via wave coupling or initialization. - -\item Option {\sl iseasurfalg=5}. Equivalent to option {\sl iseasurfalg=2} - for a variable Charnock parameter. A fixed value of Charnock's coefficient - does not need to be provided. On the other hand, a Charnock field needs - to be provided via wave coupling or initialization. - -\end{enumerate} - -The observations upon which these schemes are based do not extend to -10-m (neutral) wind speeds much above 20~ms${}^{-1}$ and there is -some uncertainty -over the behaviour of the drag at the wind speeds encountered in -tropical cyclones: indeed, there is considerable evidence that it -does not continue to increase in the manner predicted by schemes -like those described above and may even decrease. \cite{Donelan2004} -presents some measurements suggesting that the drag coefficient should -not be permitted to increase for 10-m neutral winds above about -33~ms${}^{-1}$, when the drag coefficient is about 0.0024. Whilst it is -likely that further work will be required on this topic, the possibility -of limiting the drag coefficient has been allowed for by introducing -the option {\tt i\_high\_wind\_drag} with the options -\begin{enumerate} -\item Option {\sl i\_high\_wind\_drag=0}. This is the default option -of making no modification to the standard scheme at high winds. -\item Option {\sl i\_high\_wind\_drag=1}. This option allows the -user to specify a maximum value of the (neutral) drag coefficient, -{\tt cdn\_max\_sea} (called {\tt cd\_limit\_sea} at versions below 11.5). -\item Option {\sl i\_high\_wind\_drag=2}. Like the previous option, -this allows the user to specify a maximum value of the neutral drag -coefficient, {\tt cdn\_max\_sea}, but at higher wind speeds the drag -coefficient is reduced and attains a limiting value, {\tt cdn\_hw\_sea}. -The reduction is linear in the wind speed between {\tt u\_cdn\_max} and -{\tt u\_cdn\_hw}. This reflects current understanding of the behaviour -of the sea surface at high wind speeds, with the neutral drag coefficient -saturating at around 35 ms${}^{-1}$ and declining at higher wind speeds. -Suggested values of these coefficients are based on \cite{donelan2018} -and \cite{hsu2017}. -\end{enumerate} -It might be thought more logical to subsume the treatment of high winds -under {\tt iseasurfalg}, but given that standard schemes for surface -exchange at lower wind speeds do not explicitly account for this range -of speeds and that the treatment of high wind speeds is less certain, -it is useful to consider the treatment of high wind speeds as a seperate -option. - -\subsubsection{Surface exchange over sea ice} - -As explained above, when sea ice is present, surface exchange involves -exchanges between the atmosphere and the open sea (L), the marginal -ice zone (MIZ) and the zone of pack ice. More mechanistically, one may -consider the interfacial exchanges over the sea and ice surfaces and the -contribution of form drag on the ice freeboard in the marginal ice zone. - -Two approaches are available in uncoupled configurations of the model. - -\begin{enumerate} - -\item -{The Original Scheme} -The exchange coefficients over the sea and sea ice regions of the -gridbox are interpolated between values representative of pack ice, -the marginal ice zone and open sea, using the ice -fraction, f$_{I}$. For 0 $\le $ f$_{I} <$ 0.7 -\begin{eqnarray} - < C_H >&=&( f_I C_{H(MIZ)} + ( 0.7 - f_I ) C_{H(L)} ) / 0.7 - \label{1.6.1} \\ - < C_D >&=&( f_I C_{D(MIZ)} + ( 0.7 - f_I ) C_{D(L)} ) / 0.7 - \label{1.6.2} -\end{eqnarray} -and for 0.7 $\le $ f$_{I} \le $ 1 -\begin{eqnarray} - < C_H >&=&( ( 1 - f_I ) C_{H(MIZ)} + ( f_I - 0.7 ) ) C_{H(I)} ) / 0.3 - \label{1.6.3} \\ - < C_D >&=&( ( 1 - f_I ) C_{D(MIZ)} + ( f_I - 0.7 ) ) C_{D(I)} ) / 0.3 - \label{1.6.4} -\end{eqnarray} -where -\begin{eqnarray} - C_{H(L)}&= &C_H ( L_{(L)} , z_{0m(sea)} , z_{0h(sea)} ) - \label{1.6.5} \\ - C_{H(MIZ)}&=& C_H ( L_{(I)} , z_{0m(MIZ)} , z_{0h(MIZ)} ) - \label{1.6.6} \\ - C_{H(I)}&=& C_H ( L_{(I)} , z_{0m(sea-ice)} , z_{0h(sea-ice)} ) - \label{1.6.7} -\end{eqnarray} -and similarly for the drag coefficient C$_{D}$. - -The roughness lengths over open sea are calculated as above, but those for -ice are prescribed and set using the gui or namelists. The typical -roughness length for pack ice, z$_{0m(sea-ice)}$ = 5x10$^{-4}$ m. -Historically, z$_{0h(sea-ice)}$ was set equal to z$_{0m(sea-ice)}$, -but more recently it has been set equal to one fifth of z$_{0m(sea-ice)}$, -based on \cite{andreas2010}. The setting for marginal ice is more -problematic. Whilst z$_{0m(MIZ)}$ should be larger than z$_{0m(sea-ice)}$, -good simulations of mean sea-level pressure are obtained only if -z$_{0m(MIZ)}$ is substantially greater than z$_{0m(sea-ice)}$ and a -value of 0.1m is typically used. The ratio of z$_{0h(MIZ)}$ to -z$_{0m(MIZ)}$ is standardly set to 0.2 in this case, but smaller values -might be more realistic. However, a better approach in the longer term -is to use an explicit representation of ice form drag. - -\item -{Explicit Treatment of Ice Form Drag} - -\cite{lupkes2012} have suggested a simple parametrization of the -form drag coefficient of marginal ice that has been found to -perform well in comparison to aircraft measurements (\cite{elvidge2016}). -\cite{lupkes2015} have extended the parametrization to include the effects -of stability. When coupled to CICE, it is intended that a more elaborate -scheme will be used, but this scheme is useful for application in -atmosphere-only simulations and its implementation is now described. - -The fundamental quantity involved in representing the drag is the -pressure force on the ice free-board in the up-stream flow, -\begin{equation} -F_p =\int_{z_0}^{h_f} \frac{\rho}{2} [u(z)]^2 \, dz. -\end{equation} -$u(z)$ will in general exhibit a mixed character, but it may be taken as -the developed flow over open sea, as in \cite{lupkes2012}, or may be -interpolated between the developed flows over open sea or pack ice, depending -on the ice fraction, as in \cite{lupkes2015}. -In principle, it will be subject to the effects of stability, but -since the free-board does not much exceed 0.5m, these effects are small -(\cite{lupkes2015}) and the flow may be taken as neutral up to $h_f$. -Hence, -\begin{equation} -F_p \approx \frac{h_f}{2k^2} \rho u_*^2 \left [ (\log(h_f/z_0) -1)^2 +1 -\right ] = -\frac{h_f}{2k^2} \rho C_d U_1^2 \left [(\log(h_f/z_0) -1)^2 +1 \right ]. -\label{eq:int_u2} -\end{equation} -where $C_d$ is the upstream drag coefficient and $U_1$ is the wind -on the model's lowest atmospheric level. Because this will be -significantly above $h_f$, the stability dependence of $C_d$ should -be considered here (again see \cite{lupkes2015}). $U_1$ may be interpreted -as the wind at a specific height, or, consistenly with the flux-difference -form of the momentum equation, as the layer-averaged velocity. This -distinction affects the numerical value of $C_d$, but does not otherwise -affect the foregoing equation. If using the original version of the -scheme (\cite{lupkes2012}), $C_d$ must be taken as the neutral drag -coefficient. Note also that various approximations may be made in -Equation~\ref{eq:int_u2}. \cite{lupkes2012} approximate -$(\log(h_f/z_0) -1)^2 +1$ as $(\log(h_f/z_0) )^2 $; while \cite{lupkes2015} -approximate it as $(\log(h_f/z_0) -1)^2 $. Here we retain the full expression. - -If, in a unit area, there are $N$ floes, each of crosswind dimension $D_i$, -the total drag will be -\begin{equation} -F_d = N c_w S_c^2 D_i F_p, -\label{eq:fd_fp} -\end{equation} -where $c_w$ is a coefficient and $S_c$ is a sheltering coefficient. The -fractional coverage of sea ice within this unit area is $ND_i^2 c_s$, -where $c_s$ is related to the shape of the floe. Overall, the drag per -unit area of {\em ice} is -\begin{equation} -f_d = \frac{h_f}{2k^2} c_e \rho C_d U_1^2 S_c^2 \frac{A}{D_i} -\left [(\log(h_f/z_0) -1)^2 +1 \right ], -\end{equation} -where $c_e=c_w/c_s$. Assuming that $U_1$ is blended, it follows that the -form drag coefficient is -\begin{equation} -C_{df} = \frac{h_f}{2k^2} c_e S_c^2 \frac{1}{D_i} -\left [(\log(h_f/z_0) -1)^2 +1 \right ]. -\end{equation} -Defining, $L=(\log(h_f/z_0) -1)^2 +1$ and interpolating in the ice fraction, -\begin{equation} -C_{df} = \frac{c_e}{2}\frac{h_f}{D_i}\frac{S_c^2}{k^2} \left [ -(1-A) C_{ds} L_s + A C_{di} L_i \right ], -\end{equation} -where the sheltering factor is taken to be the same over ice and water. -\cite{lupkes2012} provides parametrizations for quantities such as -$h_f$, while \cite{elvidge2016} provide suggested values for the -constants in the scheme, based on observations. In using these values -in the Unified Model, $c_e$ should be increased by about 30\% to represent -the effect of differing approximations of the logarithmic wind profile. - -For scalar transfer \cite{lupkes2015} suggest adding a contribution to -the sensible heat flux to represent the impact of form drag; however, -the mechanistic physical basis of the scheme they propose is unclear. -Moreover, when combined with the interfacial drag, this suggests scalar -transfer much larger than observed by \cite{schroder2003}. Consequently, -no enhancement of the scalar transfer coefficient by form drag is -included. - -The overall drag coefficients are now set by interpolation in the ice fraction: -\begin{eqnarray} - < C_D >&=& (1 - f_I) C_{D(L)} + f_I (C_{D(I)} + C_{D(FRM)}) - \label{eq:cdice_int} \\ - < C_H >&=& (1 - f_I) C_{H(L)} + f_I C_{H(I)} - \label{eq:chice_int} -\end{eqnarray} - -\end{enumerate} - - -\subsubsection{Surface exchange in coastal grid-boxes} -\label{sec:coast} - -In coupled ocean-atmosphere modelling the ocean requires appropriate -surface stresses and fluxes over all ocean points. In coastal regions -this means providing sea-surface fluxes from atmospheric grid-boxes -that are partly sea and partly land. This is achieved through coastal -tiling, where the ocean part of the grid box is effectively treated in -the same way as other land surface tiles. However, because of the -very different roughness characteristics of land and sea, this can -lead to serious biases especially in the ocean surface fluxes. One -particular problem that has been identified is that, compared to a -neighbouring sea-only point, coastal points tend to have slower -near-surface wind speeds (because of the rough land surface fraction) -but the ocean surface exchange will still use a typical very small -roughness length. Thus the diagnosed ocean surface stress and -sensible and latent heat fluxes are all significantly smaller than a -neighbouring sea point. Although truly coastal winds are notoriously -complex, in a typical climate model grid box (of 100km or more) the -vast majority of the sea area will be unaffected by the land. Thus a -partial solution to this problem is to take the wind speed over the -sea part of coastal points as the average of that over the -neighbouring sea points. The wind speed over the land component is -then slowed (by up to a factor of 5) to maintain the grid box mean -wind speed. - -\subsection{Surface roughness lengths and resistances to evaporation - over land}\label{section_1.7} -For land points the roughness lengths excluding orographic effects are -specified from land use datasets. The vegetative roughness length for -scalars is assumed to be 0.1 of that for momentum. This is a simple -approximation; in reality the factor depends on the land cover type -and the degree of heterogeneity. [Future versions of the Unified Model -will treat surface heterogeneity explicitly by the ``tiling'' method.] - -The surface moisture flux given by~(\ref{1.1.8}) or (\ref{1.1.17}) -involves a surface humidity value, q$_{0}$. Prior to UM6.3, for -evaporation from all of ocean, sea-ice, lake and snow-covered surfaces -as well as from water on vegetative canopies this surface value is -taken to be the saturated specific humidity at the surface (skin) -temperature and pressure, q$_{sat}$(T$_{0}$,p$_{0})$. [Saturation is -respect to liquid water or ice depending on which the surface is.] -The saturation vapour pressure of a liquid, though, is lowered by -dissolved ionic substances. For typical sea salinities the saturated -vapour pressure is only about 90\% of the value over pure water. From -UM6.3, therefore, there is the option to include this effect, so that -the parametrization of the surface moisture flux over the sea becomes -\begin{equation*} - E_0 = - \rho_0 c_H V (q_1 - 0.98 q_{sat}(T_0,p_0) ) -\end{equation*} -Evapotranspiration through vegetation receives a special treatment -because it is controlled by the physiology of the plants. The -formulation is described in full in the documentation for the land and -ice surface processes component of the Unified Model. The -evapotranspiration for the surface is given by -\begin{equation} - E_t = - \rho _0 \frac{ q_1 - q_{sat} ( T_0 , p_0 )}{( r_a + r_s )} - \label{1.7.1} -\end{equation} -where the aerodynamic resistance, r$_{a}$ , is given by -\begin{equation} - r_a = \frac{1}{ C_H } = \frac{1}{ c_H V} - \label{1.7.2} -\end{equation} -and r$_{s}$ is the surface or stomatal resistance to -evaporation. r$_{s}$ is a function of the available soil moisture, -near surface atmospheric conditions and the radiation impinging on the -plants. [For the formulation see the documentation for the land and -ice surface processes component of the Unified Model.] A similar -formula to~(\ref{1.7.1}) is used for the evaporation from the very -near surface soil layer. Equation~(\ref{1.7.1}) can be written as -\begin{equation} - E_t = - \rho _0 C_E ( q_1 - q_{sat} ( T_0 , p_0 ) ) - \label{1.7.3} -\end{equation} -where -\begin{equation} - C_E = \frac{ C_H }{\left( {1 + \frac{ r_s }{ r_a }} \right)} - \label{1.7.4} -\end{equation} - -\subsection{The modifications needed to incorporate orographic form - drag.} -\label{section_2} -\subsubsection{Effective roughness lengths}\label{section_2.1} -Form drag is included in the surface turbulent flux formulation via -effective roughness lengths for momentum \cite[]{wood93} and for -scalar quantities \cite[]{hewer1998}. The formulae of -section~\ref{section_1} are interpreted as relationships between -gridbox mean quantities and fluxes with the roughness lengths replaced -by effective values, z$_{0m(eff)}$ and z$_{0h(eff)}$. - -When form drag is included via effective roughness lengths equations -(\ref{1.1.7})-(\ref{1.1.9}) become: -\begin{eqnarray} - \frac{ H_{0(eff)} }{ c_P \rho _0 }&=&\frac{-k}{ \Phi _h (L , z_1 + z_{0m(eff)} , z_{0h(eff)} )} v_{\ast (eff)}\nonumber\\ - && \left( {\Delta T + \frac{g}{c_p }( z_1 + z_{0m(eff)} - z_{0h(eff)} )} \right) - \label{2.1.1} \\ - \frac{ E_{0(eff)} }{ \rho _0 }&=&\frac{-k}{ \Phi _h (L , z_1 + z_{0m(eff)} , z_{0h(eff)} )} v_{\ast (eff)} \Delta q - \label{2.1.2} \\ - \frac{ {\rm {\bf \tau }}_{{0(eff)}} }{ \rho _{0} }&=&\frac{k}{ \Phi _m (L , z_1 + z_{0m(eff)} , z_{0m(eff)} )} v_{\ast (eff)} \Delta {\rm {\bf v}} - \label{2.1.3} -\end{eqnarray} -The effective surface scaling velocity, v$_{\ast (eff)}$ , is given by -(cf. (\ref{1.1.25})) -\begin{equation} - v_{\ast (eff)}^2 = u_{\ast (eff)}^2 + \gamma _t^2 w_\ast ^2 + \gamma _c^2 w_c^2 - \label{2.1.4} -\end{equation} -where -\begin{equation} - u_{\ast (eff)}^2 = \left| { {\rm {\bf \tau }}_{{0(eff)}} {/} \rho _{0} } \right| - \label{2.1.5} -\end{equation} -The effective roughness for momentum is derived by setting the total -effective surface stress, \textbf{$\tau $}$_{0(eff)}$, to the sum of -the surface stress over a flat surface with the same vegetative -roughness, \textbf{$\tau $}$_{0(f)}$, and the orographic pressure drag -force at the surface, \textbf{$\tau $}$_{0(p)}$. The stresses are -evaluated in terms of the velocity at height z$_{c}$ above the -surface. z$_{c}$ is currently set to 2$^{1/2}\sigma _{h}$ where -$\sigma _{h}$ is the standard deviation of the unresolved orographic -height. Thus -\begin{equation} - \frac{ {\rm {\bf \tau }}_{{0(eff)}} }{ \rho _{0} }{ = }\frac{{k } {v}_{{\ast (eff)}} }{ \Phi _{m} {(L , } {z}_{c} { , } {z}_{{0m(eff)}} {)}}{ }{\rm {\bf v}}{(} {z}_{c} {)} - \label{2.1.6} -\end{equation} -and -\begin{equation} - \frac{ {\rm {\bf \tau }}_{{0(f)}} }{ \rho _{0} }{ = }\frac{{k } {v}_{{\ast (f)}} }{ \Phi _{m} {(L , } {z}_{c} { , } {z}_{{0m}} {)}}{ }{\rm {\bf v}}{(} {z}_{c} {)} - \label{2.1.7} -\end{equation} -where the scaling velocity based on the stress over a flat surface, v$_{\ast -(f)}$ , is given by -\begin{equation} - v_{\ast (f)}^2 = u_{\ast (f)}^2 + \gamma _t^2 w_\ast ^2 + \gamma _c^2 w_c^2 - \label{2.1.8} -\end{equation} -with -\begin{equation} - u_{\ast (f)}^2 = \left| { {\rm {\bf \tau }}_{{0(f)}} {/} \rho _{0} } \right| - \label{2.1.9} -\end{equation} -[The scaling velocity which appears in the expression~(\ref{1.1.4}) for the -Monin-Obukhov length is chosen to be v$_{\ast (eff)}$ rather than the flat -surface value.] - -The orographic stress is given by -\begin{equation} - \frac{ {\rm {\bf \tau }}_{{0(p)}} }{ \rho _{0} }{ = }\frac{{1}}{{2}}{ } {c}_{{D(orog)}} { } {f}_{D} {(} {{Ri}}_{B} {)}\frac{{A}}{{S}}{ }\left| {{\rm {\bf v}}{(} {z}_{c} {)}} \right|{ }{\rm {\bf v}}{(} {z}_{c} {)} - \label{2.1.10} -\end{equation} -where $A/S$ is the total silhouette area of orography in a gridbox -over the flat surface area of the gridbox taken as an average over all -directions. The function f$_{D}$ is a function of the bulk Richardson -number of the surface layer and is set to 1 for Ri$_{SL} <$ 0 and -decreases linearly to zero at Ri$_{SL(crit)}$ = 0.5. The orographic -drag coefficient c$_{D(orog)}$ is set to the constant value (typically -0.3, \cite{mason1986}). - -If the function $\Phi _{m}$ and v$_{\ast }$ are approximated by their -neutral values in~(\ref{2.1.6}) and~(\ref{2.1.7}) then the equation -for calculating the effective momentum roughness is derived -\begin{equation} - \frac{\ln ( z_c / z_{0m(eff)} )}{\ln ( z_c / z_{0m} )} = {\left( {1 + \frac{1}{2} c_{D(orog)} f_D \frac{A}{S} {\left( {\frac{\ln ( z_c / z_{0m} )}{k}} \right)}^2 } \right)}^{-1/2} - \label{2.1.12} -\end{equation} -The stress for the flat surface is related to the total stress by -\begin{equation} - {\rm {\bf \tau }}_{{0(f)}} { = } {\rm {\bf \tau - }}_{{0(eff)}} { } {\left( {{1 + }\frac{{1}}{{2}}{ } - {c}_{{D(orog)}} { } {f}_{D} { }\frac{{A}}{{S}}{ } {\left( - {\frac{\ln {(} {z}_{c} { / } {z}_{{0m}} {)}}{{k}}} - \right)}^{2} } \right)}^{{-1}} - \label{2.1.13} -\end{equation} -which is derived from equations~(\ref{2.1.6}), (\ref{2.1.7}) and -(\ref{2.1.10}). Equation~(\ref{2.1.13}) implies that -\begin{equation} - C_{D(f)} = C_{D(eff)} {\left( {1 + \frac{1}{2} c_{D(orog)} f_D \frac{A}{S} {\left( {\frac{\ln ( z_c / z_{0m} )}{k}} \right)}^2 } \right)}^{-1} - \label{2.1.14} -\end{equation} - - -\subsubsection{Parametrized orographic drag coefficient} - -\cite{wood93} find that orographic drag coefficient c$_{D(orog)}$ -depends on A/S via the equation -\begin{equation} - c_{D(orog)} = 2\alpha \beta \pi ^2 \frac{A}{S} \frac{ u_{\ast (f)}^2 }{ v^2 ( z_c )} - \label{2.1.11} -\end{equation} -where $\alpha $ and $\beta $ are constants ($\alpha $=12 and $\beta $=1). - -If (\ref{2.1.11}) is used, the formula for the effective roughness length -for momentum becomes -\begin{equation} - \frac{\ln ( z_c / z_{0m(eff)} )}{\ln ( z_c / z_{0m} )} = {\left( {1 + \alpha \beta \pi ^2 f_D {\left( {\frac{A}{S}} \right)}^2 } \right)}^{-1/2} - \label{2.1.15} -\end{equation} -and~(\ref{2.1.13}) and (\ref{2.1.14}) become -\begin{eqnarray} - {\rm {\bf \tau }}_{{0(f)}} &=& {\rm {\bf \tau - }}_{{0(eff)}} {\left( {{1 + }\alpha \beta \pi ^{2} { } {f}_{D} { } - {\left( {\frac{{A}}{{S}}} \right)}^{2} { }} \right)}^{-1} - \label{2.1.16} \\ - C_{D(f)}&= &C_{D(eff)} {\left( {1 + \alpha \beta \pi ^2 f_D {\left( {\frac{A}{S}} \right)}^2 } \right)}^{-1} - \label{2.1.17} -\end{eqnarray} -The effective surface flux of scalar X evaluated in terms of values at -z$_{c}$ is -\begin{equation} - \frac{ F_{X0(eff)} }{ \rho _0 } = \frac{k v_{\ast (eff)} }{ \Phi _h (L , z_c , z_{0h(eff)} )} (X( z_c ) - X_0 ) - \label{2.1.18} -\end{equation} -and the surface flux for the flat surface is given by -\begin{equation} - \frac{ F_{X0(f)} }{ \rho _0 } = \frac{k v_{\ast (f)} }{ \Phi _h (L , z_c , z_{0h} )} (X( z_c ) - X_0 ) - \label{2.1.19} -\end{equation} - -\cite{hewer1998} find that the scalar transport is enhanced when there -is orographic form drag such that -\begin{equation} - F_{X0(eff)} = F_{X0(f)} {\left( {1 - 2.2 f_D \frac{A}{S}} \right)}^{-1} - \label{2.1.20} -\end{equation} -Combining~(\ref{2.1.18})--(\ref{2.1.20}) and using the neutral values -of the stability functions the expression for the effective scalar -roughness length is derived as -\begin{equation} - \frac{\ln ( z_c / z_{0h(eff)} )}{\ln ( z_c / z_{0h} )} = {\left( {1 + \frac{1}{2} c_{D(orog)} f_D \frac{A}{S} {\left( {\frac{\ln ( z_c / z_{0m} )}{k}} \right)}^2 } \right)}^{1/2} \left( {1 - 2.2 f_D \frac{A}{S}} \right) - \label{2.1.21)} -\end{equation} -which becomes -\begin{equation} - \frac{\ln ( z_c / z_{0h(eff)} )}{\ln ( z_c / z_{0h} )} = {\left( {1 + \alpha \beta \pi ^2 f_D {\left( {\frac{A}{S}} \right)}^2 } \right)}^{1/2} \left( {1 - 2.2 f_D \frac{A}{S}} \right) - \label{2.1.22} -\end{equation} -if the \cite{wood93} formulation is used. - -\subsubsection{The iterative algorithm for calculating the effective - surface exchange coefficients}\label{section_2.2} - -For unstable conditions, i.e. $\Delta $B $<$ 0 : - -IF $\Delta $\textbf{v} $<$ 2 ms$^{-1}$ then start the iteration from -the convective limit, so -\begin{eqnarray} - \frac{1}{ L^{(0)} }&=&\frac{-k}{ \gamma _t^3 z_i } - \label{2.2.1} \\ - \Phi _m^{(0)}&=& \Phi _m ( L^{(0)} , z_1 + z_{0m(eff)} , z_{0m(eff)} ) - \label{2.2.2} \\ - \Phi _h^{(0)}&=& \Phi _h ( L^{(0)} , z_1 + z_{0m(eff)} , z_{0h} ) - \label{2.2.3} \\ - v_{\ast (eff)}^{(0)}&=& v_{\ast (f)}^{(0)} = {\left[ { \gamma _t^3 \left( {\frac{k}{ \Phi _h^{(0)} }} \right) z_i \left| {-\Delta B} \right| + \gamma _c^2 w_c^2 } \right]}^{ 1/2} - \label{(2.2.4} -\end{eqnarray} - -ELSE IF ($\Delta $\textbf{v} $\ge $ 2 ms$^{-1}$ ) start iteration from -the neutral end, so -\begin{eqnarray} - \Phi _m^{(0)}&=&\ln \left( {\frac{ z_1 + z_{0m(eff)} }{ z_{0m(eff)} }} \right) - \label{2.2.5} \\ - \Phi _h^{(0)}&=&\ln \left( {\frac{ z_1 + z_{0m(eff)} }{ z_{0h} }} \right) - \label{2.2.6} \\ - u_{\ast (eff)}^{(0)}&=&\frac{k}{ \Phi _m^{(0)} } \left| {\Delta {{{v}}}} \right| - \label{2.2.7} \\ - v_{\ast (eff)}^{(0)}&=& {\left( { u_{\ast (eff)}^{(0) 2} + \gamma _c^2 w_c^2 } \right)}^{ 1/2} - \label{2.2.8} \\ - u_{\ast (f)}&=& u_{\ast (eff)} \frac{\ln ( z_c / z_{0m(eff)} )}{\ln ( z_c / z_{0m} )} - \label{2.2.9} \\ - v_{\ast (f)}^{(0)}&=& {\left( { u_{\ast (f)}^{(0) 2} + \gamma _c^2 w_c^2 } \right)}^{ 1/2} - \label{2.2.10} -\end{eqnarray} - -END IF. - -Then calculate: -\begin{eqnarray} - C_{D(eff)}^{(0)}&=&\frac{k}{ \Phi _m^{(0)} } v_{\ast (eff)}^{(0)} - \label{(2.2.11} \\ - C_{H(eff)}^{(0)}&=&\frac{k}{ \Phi _h^{(0)} } v_{\ast (eff)}^{(0)} - \label{(2.2.12} \\ - C_{D(f)}^{(0)}&=& C_{D(eff)}^{(0)} {\left( {1 + \frac{1}{2} c_{D(orog)} f_D \frac{A}{S} {\left( {\frac{\ln ( z_c / z_{0m} )}{k}} \right)}^2 } \right)}^{-1} - \label{(2.2.13} \\ - C_{H(f)}^{(0)}&=& C_{H(eff)}^{(0)} \left( {1 - 2.2 f_D \frac{A}{S}} \right) - \label{(2.2.14} -\end{eqnarray} - -Having set up initial values the iteration loop can be entered: - -DO n = 1 to N -\begin{eqnarray} - u_{\ast (eff)}^{(n)2}&=& C_{D(eff)}^{(n-1)} \left| {\Delta {{\rm {\bf v}}}} \right| - \label{(2.2.15} \\ - u_{\ast (f)}^{(n)2}&=& C_{D(f)}^{(n-1)} \left| {\Delta {{\rm {\bf v}}}} \right| - \label{(2.2.16} \\ - {\left( {\frac{ F_{B0} }{ \rho _0 }} \right)}^{(n)}&=&- { C_{H(eff)} }^{(n-1)} \Delta B - \label{(2.2.17} \\ - w_\ast ^{(n)}&=& {\left[ { z_i {\left( {\frac{ F_{B0} }{ \rho _0 }} \right)}^{(n)} } \right]}^{ 1/3} - \label{(2.2.18} \\ - v_{\ast (eff)}^{(n)2}&=& u_{\ast (eff)}^{(n)2} + \gamma _t^2 w_\ast ^{(n)2} + \gamma _c^2 w_c^2 - \label{(2.2.19} \\ - v_{\ast (f)}^{(n)2}&= &u_{\ast (f)}^{(n)2} + \gamma _t^2 w_\ast ^{(n)2} + \gamma _c^2 w_c^2 - \label{(2.2.20} \\ - \frac{1}{ L^{(n)} }&=&\frac{-k( F_{B0} / \rho _0 )^{(n)} }{ v_{\ast (eff)}^{(n)3} } - \label{(2.2.21} \\ - \Phi _m^{(n)}&=& \Phi _m ( L^{(n)} , z_1 + z_{0m(eff)} , z_{0m(eff)} ) - \label{(2.2.22} \\ - \Phi _h^{(n)}&=& \Phi _h ( L^{(n)} , z_1 + z_{0m(eff)} , z_{0h} ) - \label{(2.2.23} \\ - C_{D(eff)}^{(n)}&=&\frac{k}{ \Phi _m^{(n)} } v_{\ast (eff)}^{(n)} - \label{(2.2.24} \\ - C_{H(eff)}^{(n)}&=&\frac{k}{ \Phi _h^{(n)} } v_{\ast (eff)}^{(n)} - \label{(2.2.25} \\ - C_{D(f)}^{(n)}&=& C_{D(eff)}^{(n)} {\left( {1 + \frac{1}{2} c_{D(orog)} f_D \frac{A}{S} {\left( {\frac{\ln ( z_c / z_{0m} )}{k}} \right)}^2 } \right)}^{-1} - \label{(2.2.26} \\ - C_{H(f)}^{(n)}&=& C_{H(eff)}^{(n)} \left( {1 - 2.2 f_D \frac{A}{S}} \right) - \label{(2.2.27} -\end{eqnarray} - -END DO. - -For neutral and stable conditions ($\Delta $B $\ge $ 0) start the -iteration from the neutral values and set w$_{\ast }$=0 in the above -iteration loop. Use the final (N) values of C$_{H(eff)}$ and -C$_{D(eff)}$ to calculate the surface sensible and latent heat fluxes -and surface stress: -\begin{eqnarray} - H_{0(eff)}&=&- c_P \rho _0 C_{H(eff)}^{(N)} \left( {\Delta T + \frac{g}{c_p }( z_1 + z_{0m(eff)} - z_{0h} )} \right) - \label{2.2.28} \\ - E_{0(eff)}&=& {-\rho }_0 C_{H(eff)}^{(N)} \Delta q - \label{2.2.29} \\ - {\rm {\bf \tau }}_{{0(eff)}} &=& \rho _0 C_{D(eff)}^{(N)} \Delta {\rm {\bf v}} - \label{2.2.30} -\end{eqnarray} -The stress for a flat surface, if required for output, is calculated from -\begin{equation} - {\rm {\bf \tau }}_{{0(f)}} { = } \rho _0 - C_{D(f)}^{(N)} \Delta {\rm {\bf v}} - \label{2.2.31} -\end{equation} - -\subsubsection{Interpolation of surface layer variables to standard - observation heights}\label{section_2.3} -If the observation height wind is assumed to lie on the profile -defined by the effective roughness length and surface scaling velocity -then (c.f. equation~(\ref{1.5.1})) -\begin{equation} - {\rm {\bf v}}_{{ob}} { = } {\rm {\bf v}}_{0} { + - }\frac{ {\rm {\bf \tau }}_{{0(eff)}} }{ \rho _0 v_{\ast (eff)} k} - \Phi _m (L, z_{ob} + z_{0m(eff)} , z_{0m(eff)} - \label{2.3.1} -\end{equation} -Using the expression for the surface turbulent stress this becomes -\begin{equation} - {\rm {\bf v}}_{{ob}} { = } {\rm {\bf v}}_{0} { + - }\frac{ C_{D(eff)} }{ {kv}_{\ast (eff)} } \Phi _m (L, z_{ob} + - z_{0m(eff)} , z_{0m(eff)} ) ( {\rm {\bf v}}_1 - {\rm {\bf v}}_{0} - {)} - \label{2.3.2} -\end{equation} -For wind z$_{ob}$ is set to 10m and the last iteration (N) values of -C$_{D(eff)}$, L and v$_{\ast (eff)}$ are used. Alternatively if the -observation height wind is assumed to lie on a profile defined by the -flat surface roughness length and scaling velocity then -\begin{equation} - {\rm {\bf v}}_{{ob}} { = } {\rm {\bf v}}_{0} { + - }\frac{ {\rm {\bf \tau }}_{{0(f)}} }{ \rho _0 v_{\ast (f)} k} \Phi - _m (L, z_{ob} + z_{0m} , z_{0m} ) - \label{2.3.3} -\end{equation} -and substituting for the surface stress this becomes -\begin{equation} - {\rm {\bf v}}_{{ob}} { = } {\rm {\bf v}}_{0} { + - }\frac{ C_{D(f)} }{k v_{\ast (f)} } \Phi _m (L, z_{ob} + z_{0m} , - z_{0m} ) ( {\rm {\bf v}}_1 - {\rm {\bf v}}_{0} {)} - \label{2.3.4} -\end{equation} -Most configurations of the Unified Model currently use the latter -assumption with the last iteration value of C$_{D(f)}$, L and $v_{\ast - (f)}$ used in the interpolation formula. - -If the observation height scalar quantities are assumed to lie on the -mean profile defined by the effective roughness length and scaling -quantities then (c.f. equation~(\ref{1.5.3}) we obtain for the generic -scalar $X$ ($T+(g/c_{P})z$, $q$, tracer amount) -\begin{equation} - X_{ob} = X_0 + \frac{ F_{X0(eff)} }{ \rho _0 v_{\ast (eff)} k} \Phi _h (L, z_{ob} + z_{0h(eff)} , z_{0h(eff)} ) - \label{2.3.5} -\end{equation} -and using the expression for the surface flux of the scalar quantity -$X$ this becomes -\begin{equation} - X_{ob} = X_0 + \frac{ C_{H(eff)} }{k v_{\ast (eff)} } \Phi _h (L, z_{ob} + z_{0h(eff)} , z_{0h(eff)} ) ( X_1 - X_0 ). - \label{2.3.6} -\end{equation} -Alternatively if the observation height scalar quantities are assumed -to lie on a profile defined by the flat surface roughness length and -flux then -\begin{equation} - X_{ob} = X_0 + \frac{ C_{H(f)} }{k v_{\ast (f)} } \Phi _h (L, z_{ob} + z_{0h} , z_{0h} ) ( X_1 - X_0 ) - \label{2.3.7} -\end{equation} -For temperature and humidity z$_{ob}$ is set to the screen height (1.5 -m) and the last iteration (N) values of C$_{H}$, L and v$_{\ast }$ are -used. - -\subsection{Distributed form drag -- an alternative to the effective - roughness length parametrization}\label{section_2.4} -An alternative representation of the turbulent form drag due to -sub-grid hills is the explicit orographic stress parametrization -proposed by \cite{wood01:_param}. In this representation the drag is -represented via an orographic stress term, applied directly to the -horizontal momentum equations. The roughness lengths remain at the -vegetative values and no adjustment to the roughness lengths for -scalar quantities is made. - -The turbulent form drag is represented by the term -\begin{equation} - {\bf f}=\frac{1}{\rho}\frac{\partial}{\partial z}{\bf\tau}_{\rm orog} - \label{eq:drag} -\end{equation} -on the right-hand side of the horizontal momentum equation, where -${\bf\tau}_{\rm orog}$ is the horizontal vector containing the extra -stress imparted on the flow by the sub-grid orography This term is -included in the Unified Model as an additional explicit (in terms of -time discretisation) stress. Following \cite{wood01:_param} we define -${\bf\tau}_{\rm orog}$ to be -\begin{equation} - {\bf\tau}_{\rm orog}(z)=\left({F_p}_x,{F_p}_y\right)e^{-z/\ell}, -\end{equation} -where ${\bf F_p}=({F_p}_x,{F_p}_y)$, ${F_p}_x$ and ${F_p}_y$ are the -grid-box average $x$ and $y$ components of the pressure force on the -sub-grid orography, and $\ell$ is a decay scale. We define $\ell$ such -that -\begin{equation} - \ell={\rm min}\left(\lambda,\frac{z_h}{3}\right), - \label{eq:l} -\end{equation} -where $z_h$ is the boundary-layer depth and $\lambda$, a somewhat ill -defined quantity, is related to the horizontal scales of the sub-grid -hills (and set to 300 m). Note that the value of $\ell$ obtained from -Eq.~(\ref{eq:l}) is further constrained to be at least 100 m. - -If the steep-hill expression is to be used, the surface stress applied is -almost identical to that used in the effective roughness parametrization -(Eq.~\ref{2.1.10}), namely: -\begin{equation} - \frac{\bf F_p}{\rho_0}=\frac{1}{2}c_{D(orog)} f_D (Ri_{B}) \frac{A}{S} - \left\vert{\rm{\bf v}}(\ell) \right\vert{\rm{\bf v}}(\ell), - \label{eq:dragsteep} -\end{equation} -the main difference being the dependence on the height scale $\ell$ rather -than $z_c$. Similarly, if the \cite{wood93} low-hill expression is used, the -surface stress is given by the equivalent of (Eq.~\ref{2.1.16}), namely: -\begin{equation} - \frac{\bf F_p}{\rho_0} = {\left( {\frac{\kappa}{\zeta_m}} \right)}^{2} - \alpha \beta \pi ^{2} {f}_{D} (Ri_{B}) - {\left( {\frac{A}{S}} \right)}^{2} - \left\vert{\rm{\bf v}}(\ell) \right\vert{\rm{\bf v}}(\ell), - \label{eq:draglow} -\end{equation} -where $\zeta_m = {\rm log}(\ell/z_{0m})$. There is also an option to use the -low-hill stress (\ref{eq:draglow}) but capped by that from the steep hill -expression (\ref{eq:dragsteep}), to avoid generating huge stresses at large $A/S$. - -There is also a choice for the Richardson number, $Ri_{B}$, that appears -in the stability dependence, $f_D$, which can either use $Ri_{B}=Ri_{SL}$ -(as with the effective roughness length version) or $Ri_{B \ell}$, a bulk -Richardson number between the surface and the scale height, $\ell$: -\begin{equation*} - Ri_{B \ell} = \frac{ \ell \left( g \left( - \overline{\beta_T}_{k\ell} ({\thetal}_{k\ell}-{\thetal}_{1}) - + \overline{\beta_q}_{k\ell} ({q_t}_{k\ell}-{q_t}_{1}) \right) - + \Delta b_{SL} \right) }{U^2(\ell)} -\end{equation*} -where the stability of the atmosphere between the surface and the bottom -model level is included via $\Delta b_{SL}$, which is the numerator of -$Ri_{SL}$, $U$ is the wind speed and the subscript $k \ell$ indicates -the $\theta$-level containing $\ell$. - -%------------------------------------------------------------------------ -% Section: implicit solver -%------------------------------------------------------------------------ -\section{Implicit solution of the diffusion equation} -\label{sec:implicit} - -\subsection{Unconditionally stable implicit solver} - -This is the vertical diffusion scheme of \cite{woodetal2007} which has -the advantages of (i) unconditional stability and non-oscillatory -behaviour for practical NWP cases and (ii) monotonic damping for -suitable choices of a free parameter $P$ which represents the degree -of nonlinearity of the diffusion problem to be solved. If the chosen -value of $P$ is equal to the real value of $P$ then the scheme is -second order accurate. In practical simulations $P$ may vary from -timestep to timestep and from column to column. - -\subsubsection{Algorithmic description} - -Consider the non-linear damping equation: -\begin{equation} - \frac{dX}{dt}=-\left(KX^{P}\right)X+S - \label{eq:damp1} -\end{equation} -Here $S$ is a constant forcing, or source, term and $KX^{P}$ is the -diffusion coefficient, with $K$ constant. $P$ is assumed to be -positive. The new scheme is written -\begin{equation} - \frac{X^{*}-X^{n}}{\Delta t}=-\Ical_{1}\left[K\left(X^{n}\right)^{P}\right] - X^{*}+\Ecal_{1}\left[K\left(X^{n}\right)^{P}\right] - X^{n}+\left(\Ical_{1}-\Ecal_{1}\right)S,\label{eq:sppf1} -\end{equation} -\begin{equation} - \frac{X^{n+1}-X^{*}}{\Delta - t}=-\Ical_{2}\left[K\left(X^{n}\right)^{P}\right]X^{n+1} + - \Ecal_{2}\left[K\left(X^{n}\right)^{P}\right]X^{*} + - \left(\Ical_{2}-\Ecal_{2}\right)S,\label{eq:sppf2} -\end{equation} -where -\begin{equation} - \Ecal_{1}=\left(1+\frac{1}{\sqrt{2}}\right) - \left[P+\frac{1}{\sqrt{2}}\pm\sqrt{P - \left(\sqrt{2}-1\right)+\frac{1}{2}}\right] - \label{eq:E1coeff} -\end{equation} -\begin{equation} - \Ecal_{2}=\left(1+\frac{1}{\sqrt{2}}\right) - \left[P+\frac{1}{\sqrt{2}}\mp\sqrt{P\left(\sqrt{2}-1\right)+\frac{1}{2}}\right] - \label{eq:E2coeff} -\end{equation} -\begin{equation} - \Ical_{1}=\Ical_{2}=\left(1+\frac{1}{\sqrt{2}}\right)\left(1+P\right) - \label{eq:Icoeff} -\end{equation} -Consider the one-dimensional ``forced'' boundary layer diffusion -equation -\begin{equation} - \frac{\partial X}{\partial t}=\frac{\partial F}{\partial z}+S, - \qquad F=K_{X}\frac{\partial X}{\partial z} - \label{eq:vdiff1} -\end{equation} -where $X$ is the scalar variable being diffused, $F$ is the flux of -$X$, $t$ is the time, $z$ is the height from the earth's surface, and -$K$ is the diffusion coefficient which is often non-constant and -depends on $X$ (i.e. the PDE is non-linear) and $S$ is a forcing term -from other processes preceding the boundary layer. In the UM these -processes are: microphysics, gravity wave drag, radiation, dynamics -and optionally (using the switch i\_impsolve\_loc) convection\footnote{If -i\_impsolve\_loc = 1, the boundary-layer implicit solver is -performed before the convection call so that $S$ excludes the convection -increments. If i\_impsolve\_loc = 2, it is performed after the convection -call. There are pros and cons to each option. Calling the implicit solver -before convection reduces the accuracy of the final mixed-layer profile, -since convection may alter the mixed-layer gradients afterwards. -On the other hand, allowing convection to act on a state which includes -the heating and moistening by surface-fluxes over the current timestep -may improve the accuracy of the convective closure. -Also the non-turbulent fluxes used to construct the budgets at entrainment -grid-levels (section \ref{sec:rev_flux_grad}) do not include contributions -from convection, so arguably excluding them from $S$ is consistent. -In the presence of convective subsidence, the top grid-level of the -sub-cloud mixed layer gets warmed and dried by the subsidence -(consistent with a lowering of the mixed-layer top). However if the -implicit solver is called after convection it does not account for this -lowering, so that all the subsided air is forced to be entrained into the -mixed-layer.}. $S$ represents the total tendency from these processes. -Equations (\ref{eq:sppf1}), (\ref{eq:sppf2}) applied to -(\ref{eq:vdiff1}) becomes -\begin{eqnarray} - \frac{X^{*}-X^{n}}{\Delta t} & = & \Ical_{1}\frac{\partial F}{\partial z}^{*}-\Ecal_{1}\frac{\partial F}{\partial z}^{n}+\left(\Ical_{1}-\Ecal_{1}\right)S\label{eq:sppf_bl1}\\ - \frac{X^{n+1}-X^{*}}{\Delta t} & = & \Ical_{2}\frac{\partial - F}{\partial z}^{n+1}-\Ecal_{2}\frac{\partial F}{\partial - z}^{*}+\left(\Ical_{2}-\Ecal_{2}\right)S\label{eq:sppf_bl2} -\end{eqnarray} -where, -\begin{equation*} - F^{n}=K_{X}\frac{\partial X}{\partial z}^{n},\; - F^{*}=K_{X}\frac{\partial X}{\partial z}^{*},\; - F^{n+1}=K_{X}\frac{\partial X}{\partial z}^{n+1},\; K_{X}\equiv - K(X^{n}) -\end{equation*} -i.e. only one evaluation of the exchange coefficient is -required per timestep. Furthermore, the condition -$I_{1}+I_{2}-(\Ecal_{1}+\Ecal_{2})=1$ ensures that if the intermediate -``starred'' quantities are eliminated and the scheme is reduced into a -single equation then the forcing term will be multiplied by $1$. - -Recall from section~\ref{sec:closure} that the boundary layer solver -computes the increment of $X$, where $X=u,\; v,\;\theta_{L},\; q_{w}$. -Let $\delta X^{*}=X^{*}-X^{n}$, $\delta -X^{n+1}=X^{n+1}-X^{*}$. Then, -\begin{equation*} - F^{*}=F^{n}+K_{X}\frac{\partial\delta X}{\partial z}^{*},\qquad - F^{n+1}=F^{*}+K_{X}\frac{\partial\delta X}{\partial z}^{n+1}. -\end{equation*} -Writing equations (\ref{eq:sppf_bl1}), -(\ref{eq:sppf_bl2}) in terms of these increments: -\begin{eqnarray} - \frac{\delta X}{\Delta t}^{*} & = & (\Ical_{1}-\Ecal_{1})\left(\frac{\partial F}{\partial z}^{n}+S\right)+\Ical_{1}\frac{\partial}{\partial z}\left(K_{X}\frac{\partial\delta X}{\partial z}^{*}\right)\label{eq:sppf_inc1}\\ - \frac{\delta X}{\Delta t}^{n+1} & = & (\Ical_{2}-\Ecal_{2})\left(\frac{\partial F}{\partial z}^{*}+S\right)+\Ical_{2}\frac{\partial}{\partial z}\left(K_{X}\frac{\partial\delta X}{\partial z}^{n+1}\right)\label{eq:sppf_inc2}\\ - X^{n+1} & = & X^{n}+\delta X^{*}+\delta X^{n+1}\label{eq:sppf_inc3} -\end{eqnarray} - -\subsubsection{Discrete equations and boundary conditions} -\label{sec:impsolve} - -To derive the boundary conditions for the horizontal wind components -we adapt the technique used in the original scheme. - -\noindent -{\bf Vertical diffusion solver for momentum variables } - -\noindent -Consider the following equivalent form of (\ref{eq:sppf_bl1}): -\begin{equation} - \frac{\delta u^{*}}{\Delta t}=\frac{\partial\bar{\tau}_{x}^{*}}{\partial z}+\left(\Ical_{1}-\Ecal_{1}\right)S\label{eq:du_star} -\end{equation} -where $\tau_{x}$ is the $u$ wind component stress (defined in the same -way as the flux in (\ref{eq:sppf_inc1}) and $\bar{\tau}_{x}^{*}$ its -time-average: -\begin{equation} - \bar{\tau}_{x}^{*}=\Ical_{1}\tau_{x}^{*}-\Ecal_{1}\tau_{x}^{n},\qquad\tau_{x}^{*}=\tau_{x}^{n}+K_{u}\frac{\partial\delta - u^{*}}{\partial z}.\label{eq:tau_star} -\end{equation} -Substituting (\ref{eq:tau_star}) into (\ref{eq:du_star}) the following -is obtained: -\begin{equation*} - \frac{\delta u^{*}}{\Delta - t}=(\Ical_{1}-\Ecal_{1})\left(\frac{\partial\tau_{x}^{n}}{\partial - z}+S\right)+\Ical_{1}\frac{\partial}{\partial - z}\left(K_{u}\frac{\partial\delta u^{*}}{\partial z}\right) -\end{equation*} -which is identical to (\ref{eq:sppf_inc1}) for $X\equiv u,\; -F_{X}\equiv\tau_{x}$. This equivalent derivation is used here as it -presents a more convenient form to express the boundary -conditions. Given that the wind components are defined on -$\rho$-levels (half levels), discretizing the previous equation in $z$ -on all $L$ half-levels except at the bottom and the top one we obtain -\begin{eqnarray*} - \delta u_{k+1/2}^{*} & = & (\Ical_{1}-\Ecal_{1})\Delta t\left(\frac{\tau_{x}^{n}\Big|_{k+1}-\tau_{x}^{n}\Big|_{k}}{z_{k+1}-z_{k}}+S_{k+1/2}\right)\\ - & & +\Ical_{1}\frac{\Delta - t}{z_{k+1}-z_{k}}\left[\left(K_{u}\Big|_{k+1}\frac{\delta - u_{k+3/2}^{*}-\delta - u_{k+1/2}^{*}}{z_{k+3/2}-z_{k+1/2}}\right)-\left(K_{u}\Big|_{k}\frac{\delta - u_{k+1/2}^{*}-\delta - u_{k-1/2}^{*}}{z_{k+1/2}-z_{k-1/2}}\right)\right] -\end{eqnarray*} -or, rearranging -\begin{equation} - A_{k}\delta u_{k+3/2}^{*}+B_{k}\delta - u_{k+1/2}^{*}+C_{k}\delta u_{k-1/2}^{*}=\Delta - t(\Ical_{1}-\Ecal_{1})\left(\frac{\tau_{x}^{n}\Big|_{k+1}-\tau_{x}^{n}\Big|_{k}}{z_{k+1}-z_{k}}+S_{k+1/2}\right),\label{eq:tridiag} -\end{equation} -where $k=1,2,\ldots,L-2$, -\begin{equation*} - A_{k}=-\Ical_{1}\frac{\Delta t\noindent - K_{u}\Big|_{k+1}}{(z_{k+1}-z_{k})(z_{k+3/2}-z_{k+1/2})},\; - C_{k}=-\Ical_{1}\frac{\Delta - tK_{u}\Big|_{k}}{(z_{k+1}-z_{k})(z_{k+1/2}-z_{k-1/2})},\; - B_{k}=1-A_{k}-C_{k}. -\end{equation*} -(Note that the surface is level $0$). - -For the top $\rho$-level, $k=L-1$, the $z$-discretization of -(\ref{eq:du_star}) is: -\begin{equation} - B_{L}\delta u_{L-1/2}^{*}+C_{L}\delta u_{L-3/2}^{*}=\Delta t(\Ical_{1}-\Ecal_{1})\left(\frac{\tau_{x}^{n}\Big|_{L}-\tau_{x}^{n}\Big|_{L-1}}{z_{L}-z_{L-1}}+S_{L-1/2}\right),\label{eq:tridiag_top} -\end{equation} -where $B_{L}$, $C_{L}$ are derived as before setting $A_{L}=0$. - -For the bottom $\rho$-level, $k=0$, the $z$-discretization of -(\ref{eq:du_star}) is: -\begin{eqnarray} - \delta u_{1/2}^{*} & = & \frac{\Delta t}{z_{1}-0}\left(\bar{\tau}_{x}^{*}\Big|_{1}-\bar{\tau}_{x}^{*}\Big|_{0}\right)+\Delta t\left(\Ical_{1}-\Ecal_{1}\right)S_{1/2}\label{eq:u_bc_1} -\end{eqnarray} -where, from (\ref{eq:tau_star}), -\begin{equation} - \bar{\tau}_{x}^{*}\Big|_{1}=\left(\Ical_{1}-\Ecal_{1}\right)\tau_{x}^{n}\Big|_{1}+\Ical_{1}\left(K_{u}\frac{\partial\delta u^{*}}{\partial z}\right)\Big|_{1}.\label{eq:u_bc_2} -\end{equation} -Combining (\ref{eq:u_bc_1}), (\ref{eq:u_bc_2}) the bottom row -discretization is obtained: -\begin{equation} - A_{0}\delta u_{3/2}^{*}+B_{0}\delta u_{1/2}^{*}=\Delta t\left(\Ical_{1}-\Ecal_{1}\right)\left(\frac{\tau_{x}^{n}\Big|_{1}}{z_{1}}+S_{1/2}\right)-\frac{\Delta t}{z_{1}}\bar{\tau}_{x}^{*}\Big|_{0}\label{eq:u_bc_3} -\end{equation} -where -\begin{equation*} - A_{0}=-\Ical_{1}\frac{\Delta tK_{u}\Big|_{1}}{z_{1}(z_{3/2}-z_{1/2})},\quad B_{0}=1-A_{0}. -\end{equation*} - -Equations (\ref{eq:tridiag}), (\ref{eq:tridiag_top}) and -(\ref{eq:u_bc_3}) form a tridiagonal system of linear equations. When -the elimination procedure takes place (\ref{eq:u_bc_3}) becomes -\begin{equation} - \delta u_{1/2}^{*}=\delta u_{1/2}^{'}-\beta\bar{\tau}_{x}^{*}\Big|_{0}\label{eq:du_half} -\end{equation} -where $\delta u_{1/2}^{'}$, $\beta$ are available -quantities. Furthermore, -\begin{equation*} - \bar{\tau}_{x}^{*}\Big|_{0}=\left(\Ical_{1}-\Ecal_{1}\right)\tau_{x}^{n}\Big|_{0}+\Ical_{1}\left(K_{u}\frac{\partial\delta u^{*}}{\partial z}\right)\Big|_{0} -\end{equation*} -Approximating $\left(\frac{\partial\delta u^{*}}{\partial - z}\right)\Big|_{0}\approx\frac{\delta u_{1/2}^{*}-\delta - u_{0}^{*}}{z_{1/2}}$, and assuming that $u_{0}=0$ the previous -equation becomes -\begin{equation} - \bar{\tau}_{x}^{*}\Big|_{0}=\left(\Ical_{1}-\Ecal_{1}\right)\tau_{x}^{n}\Big|_{0}+\Ical_{1}K_{u}\Big|_{0}\frac{\delta u_{1/2}^{*}}{z_{1/2}}.\label{eq:tau_zero} -\end{equation} -From (\ref{eq:du_half}), (\ref{eq:tau_zero}) the following expression -for the implicit surface stress is obtained -\begin{equation} - \bar{\tau}_{x}^{*}\Big|_{0}=\frac{\left(\Ical_{1}-\Ecal_{1}\right)\tau_{x}^{n}\Big|_{0}+\Ical_{1}(K_{u}\Big|_{0}/z_{1/2})\delta - u_{1/2}^{'}}{1+\Ical_{1}(K_{u}\Big|_{0}/z_{1/2})\beta}.\label{eq:imp_tau} -\end{equation} -Then, $\delta u_{1/2}^{*}$ can be computed from (\ref{eq:imp_tau}) and -(\ref{eq:du_half}). Similarly the implicit surface stress for $u$ -which corresponds to the 2nd stage (\ref{eq:sppf_inc2}) will be -\begin{equation} - \bar{\tau}_{x}^{n+1}\Big|_{0}=\frac{\left(\Ical_{2}-\Ecal_{2}\right)\tau_{x}^{*}\Big|_{0}+\Ical_{2}(K_{u}\Big|_{0}/z_{1/2})\delta - u_{1/2}^{'}}{1+\Ical_{2}(K_{u}\Big|_{0}/z_{1/2})\beta}.\label{eq:imp_tau2} -\end{equation} -In the same way $\bar{\tau}_{y}^{*}\Big|_{0}$, -$\bar{\tau}_{y}^{n+1}\Big|_{0}$ can be derived. - -\noindent -{\bf Vertical diffusion solver for scalar variables} - -\noindent -Derivation of boundary conditions for the scalar variables (static -energy and total water content flux) is more difficult: the new scheme -in comparison with the original scheme is more complex and the -procedure for deriving the scalar fluxes described in section 3 of -\cite{esseryetal2001} is also complex. Currently, an alternative -treatment for the boundary conditions has been coded which works well -in practice. The boundary conditions for the scalar variables, -i.e. the surface scalar fluxes for the new scheme are obtained using -the original implicit surface exchange calculation. This is applied as -follows. Consider the equivalent discrete form of (\ref{eq:sppf_bl1}) -for the thermodynamic variable $X$: -\begin{equation} - \frac{\delta X^{*}}{\Delta t} - =\frac{\partial\overline{F}^{*}}{\partial z}+\left(\Ical_{1}-\Ecal_{1}\right)S\label{eq:dX_star} -\end{equation} -Considering that, -\begin{equation} - \overline{F}^{*}=\Ical_{1}F^{*}-\Ecal_{1}F^{n},\qquad - F^{*}=F^{n}+K_{X}\frac{\partial\delta X^{*}}{\partial z}\label{eq:dX_star2} -\end{equation} (\ref{eq:dX_star}) -would re-produce (\ref{eq:sppf_inc1}), which is re-written below, -\begin{equation*} - \frac{\delta X^{*}}{\Delta t}=(\Ical_{1}-\Ecal_{1})\left(\frac{\partial F^{n}}{\partial z}+S\right)+\Ical_{1}\frac{\partial}{\partial z}\left(K_{X}\frac{\partial\delta X^{*}}{\partial z}\right) -\end{equation*} -and thus the following discretization is obtained, on $\theta$-levels: -\begin{eqnarray*} - \frac{\delta X_{k}^{*}}{\Delta t} & = & \left(\Ical_{1}-\Ecal_{1}\right)\left(\frac{F_{k+1/2}^{n}-F_{k-1/2}^{n}}{z_{k+1/2}-z_{k-1/2}}+S_{k}\right)\\ - & +&\frac{\Ical_{1}}{z_{k+1/2}-z_{k-1/2}}\left[K_{X}\Big|_{k+1/2}\left(\frac{\delta X_{k+1}^{*}-\delta X_{k}^{*}}{z_{k+1}-z_{k}}\right)-K_{X}\Big|_{k-1/2}\left(\frac{\delta X_{k}^{*}-\delta X_{k-1}^{*}}{z_{k}-z_{k-1}}\right)\right],\; k=2,\ldots,L-1.\end{eqnarray*} -or, -\begin{equation} - A_{k}\delta X_{k+1}^{*}+B_{k}\delta X_{k}^{*}+C_{k}\delta X_{k-1}^{*}=\left(\Ical_{1}-\Ecal_{1}\right)\left(\frac{F_{k+1/2}^{n}-F_{k-1/2}^{n}}{z_{k+1/2}-z_{k-1/2}}+S_{k}\right),\quad k=2,\ldots,L-1\label{eq:dX_disc} -\end{equation} -where, -\begin{equation*} - A_{k}=-\Ical_{1}\frac{\Delta tK_{X}\Big|_{k+1/2}}{(z_{k+1}-z_{k})(z_{k+1/2}-z_{k-1/2})},\; C_{k}=-\Ical_{1}\frac{\Delta tK_{X}\Big|_{k-1/2}}{(z_{k+1/2}-z_{k-1/2})(z_{k}-z_{k-1})},\quad B_{k}=1-A_{k}-C_{k}. -\end{equation*} -The discrete equation for the top level, $k=L$, will be: -\begin{equation} - B_{L}\delta X_{L}^{*}+C_{L}\delta X_{L-1}^{*}=\left(\Ical_{1}-\Ecal_{1}\right)\left(\frac{-F_{L-1/2}^{n}}{z_{L+1/2}-z_{L-1/2}}+S_{L}\right),\label{eq:dX_disc_top} -\end{equation} -where $B_{L}$, $C_{L}$ are derived as before setting $A_{L}=0$. - -From (\ref{eq:dX_star}), a bottom interior level ($k=1$) -discretization is -\begin{equation} - \delta X_{1}^{*}=\frac{\Delta t}{z_{3/2}-0}\left(\overline{F_{3/2}}^{*}-\overline{F_{0}}^{*}\right)+\Delta t\left(\Ical_{1}-\Ecal_{1}\right)S_{1}\label{eq:dX1_star} -\end{equation} -$F_{0}$ is used instead of $F_{1/2}$. The former is computed by the -implicit surface scheme. This flux gradient is defined in the same way -in the original solver as well. Using (\ref{eq:dX_star2}), -(\ref{eq:dX1_star}) becomes -\begin{equation} - \delta - X_{1}^{*}=\frac{\Delta t}{z_{3/2}} - \left[\left(\Ical_{1}-\Ecal_{1}\right)F_{3/2}^{n}-\overline{F_{0}}^{*}\right] - +\Delta t\left(\Ical_{1}-\Ecal_{1}\right) - S_{1}+\Delta t\Ical_{1}\frac{1}{z_{3/2}} - \left(K_{X}\frac{\partial\delta X^{*}}{\partial z}\right)_{3/2} - \label{eq:dX1_star2} -\end{equation} -where $\overline{F}_{0}^{*}$ can be approximated as -\begin{equation} - \overline{F}_{0}^{*}=\Ical_{1}F_{0}^{*}-\Ecal_{1}F_{0}^{n}\approx\left(\Ical_{1}-\Ecal_{1}\right)F_{JULES}\label{eq:F0_star} -\end{equation} -where, $F_{JULES}$ is the implicit flux calculated by the -\textit{implicit surface scheme using the original implicit - algorithm}. Finalising, the discrete equations for the bottom level -will be -\begin{equation*} - \delta X_{1}^{*}=\Delta t\left(\Ical_{1}-\Ecal_{1}\right) - \left(\frac{F_{3/2}^{n}-F_{JULES}}{z_{3/2}}+S_{1}\right) - +\Delta t\Ical_{1}\frac{1}{z_{3/2}}K_{X}\Big|_{3/2} - \left(\frac{\delta X_{2}^{*}-\delta X_{1}^{*}}{z_{2}-z_{1}}\right) -\end{equation*} -or, -\begin{equation} - A_{1}\delta X_{2}^{*}+B_{1}\delta X_{1}^{*}=\Delta t - \left(\Ical_{1}-\Ecal_{1}\right)\left(\frac{F_{3/2}^{n}-F_{JULES}}{z_{3/2}}+S_{1}\right)\label{eq:dX_bottom} -\end{equation} -where, -\begin{equation*} - A_{1}=-\Ical_{1}\Delta t\frac{K_{X}\Big|_{3/2}}{z_{3/2}(z_{2}-z_{1})}, - \quad B_{1}=1-A_{1}. -\end{equation*} -Equations (\ref{eq:dX_disc}), (\ref{eq:dX_disc_top}) and -(\ref{eq:dX_bottom}) define a tridiagonal system of equations for -$\delta X^{*}$. - -Similarly the corresponding discrete equations for -(\ref{eq:sppf_inc2}) will be: -\begin{equation} - B_{L}^{'}\delta X_{L}^{n+1}+C_{L}^{'}\delta X_{L-1}^{n+1}=\left(\Ical_{2}-\Ecal_{2}\right)\left(\frac{-F_{L-1/2}^{*}}{z_{L+1/2}-z_{L-1/2}}+S_{L}\right),\label{eq:dXtop_np1} -\end{equation} -\begin{equation} - A_{k}^{'}\delta X_{k+1}^{n+1}+B_{k}^{'}\delta X_{k}^{n+1}+C_{k}^{'}\delta X_{k-1}^{n+1}=\left(\Ical_{2}-\Ecal_{2}\right)\left(\frac{F_{k+1/2}^{*}-F_{k-1/2}^{*}}{z_{k+1/2}-z_{k-1/2}}+S_{k}\right),\quad k=L-1,\ldots,2\label{eq:dXk_np1} -\end{equation} -\begin{equation} - A_{1}^{'}\delta X_{2}^{n+1}+B_{1}^{'}\delta X_{1}^{n+1}=\left(\Ical_{2}-\Ecal_{2}\right)\left(\frac{F_{3/2}^{*}-F_{JULES}}{z_{3/2}}+S_{1}\right)\label{eq:dX1_np1} -\end{equation} -where, -\begin{equation*} - A_{k}^{'}=-\Ical_{2}\frac{\Delta tK_{X}\Big|_{k+1/2}}{(z_{k+1}-z_{k})(z_{k+1/2}-z_{k-1/2})},\; C_{k}^{'}=-\Ical_{2}\frac{\Delta tK_{X}\Big|_{k-1/2}}{(z_{k+1/2}-z_{k-1/2})(z_{k}-z_{k-1})},\quad B_{k}^{'}=1-A_{k}^{'}-C_{k}^{'}, -\end{equation*} -for $k=L,\ldots,2,\quad A_{L}=0$. -\begin{equation*} - A_{1}^{'}=-\Ical_{2}\Delta t\frac{K_{X}\Big|_{3/2}}{z_{3/2}(z_{2}-z_{1})},\quad B_{1}^{'}=1-A_{1}^{'} -\end{equation*} -and the approximation -\begin{equation} - \overline{F}_{0}^{n+1}=\Ical_{2}F_{0}^{n+1}-\Ecal_{2}F_{0}^{n}\approx\left(\Ical_{2}-\Ecal_{2}\right)F_{JULES}\label{eq:F0_np1} -\end{equation} -has taken place. The same flux $F_{JULES}$ will be used for both -(\ref{eq:F0_star}) and (\ref{eq:F0_np1}) and therefore needs to be -computed only once, when the 1st or predictor stage is computed, -i.e. $X^{*}$. Briefly the following calculations take place for the -scalar variables: -\begin{center} -{\begin{tabular}{ll} -CALL bdy\_impl3(): & set up coefficients for (\ref{eq:dX_disc_top}), (\ref{eq:dX_disc}) and do a downward sweep; \\ - & do a downward sweep using the original implicit scheme to \\ - & compute information required by the surface implicit solver; \\ -CALL sf\_impl2(): & CALL im\_sf\_pt2(): compute $F_{JULES}$ (scalar implicit fluxes), \\ - & using original surface implicit solver; \\ -CALL bdy\_impl4(): & set up (\ref{eq:dX_bottom}) and complete downward sweep; \\ - & back substitute to compute implicit correction $\delta X^{*}$; \\ -CALL bdy\_impl3(): & compute explicit flux -$F^*=F^n+K_X\frac{\partial \delta X^*}{\partial z}$; \\ - & set up coefficients for (\ref{eq:dXtop_np1}), (\ref{eq:dXk_np1}), (\ref{eq:dX1_np1}) and \\ - & do a downward sweep; \\ -CALL sf\_impl2(): & only momentum variables are affected - no change in scalars; \\ -CALL bdy\_impl4(): & back substitute to compute final implicit correction $\delta X^{n+1}$ \\ -\end{tabular}} -\end{center} -NB: for CABLE compatibility, sf\_impl2 is now called by an -intermediate routine surf\_couple\_implicit. - -\subsubsection{Flux diagnostic formulae} - -The original boundary layer implicit solver computes the total stress -by time averaging the stresses at $t^{n}$ and $t^{n+1}$, where -$[t^{n},t^{n+1}]$ denotes the time integration interval for the -vertical diffusion equation being solved. The averaging which takes -place for the zonal wind component stress is: -\begin{equation} - \overline{\tau_{x}}^{n+1}\equiv(1-\gamma)\tau_{x}^{n}+\gamma\tau_{x}^{n+1} - =\tau_{x}^{n}+\gamma - K_{u}\frac{\partial\delta u^{n+1}}{\partial z}\label{eq:taux_tot} -\end{equation} -where $\delta u^{n+1}=u^{n+1}-u^{n}$. Likewise, $\tau_{y}$ and the -scalar fluxes are derived. - -For the new scheme the total zonal wind component stress is defined -as: -\begin{equation*} - \overline{\tau_{x}}^{n+1}\equiv\overline{\tau_{x}}^{[n,*]}+\overline{\tau_{x}}^{[*,n+1]} -\end{equation*} -where, $\overline{\tau_{x}}^{[n,*]}$ denotes the total stress for the -1st stage of the scheme, i.e. the time averaged stress from $t^{n}$ to -the pseudo-timelevel $t^{*}$ and similarly, -$\overline{\tau_{x}}^{[*,n+1]}$ the total stress for the 2nd stage of -the scheme (corrector). The total stress for the predictor and the -corrector are defined as: -\begin{eqnarray*} - \overline{\tau_{x}}^{[n,*]}\equiv\Ical_{1}\tau_{x}^{*}-\Ecal_{1}\tau_{x}^{n} & = & \left(\Ical_{1}-\Ecal_{1}\right)\tau_{x}^{n}+\Ical_{1}K_{u}\frac{\partial\delta u^{*}}{\partial z}\\ - \overline{\tau_{x}}^{[*,n+1]}\equiv\Ical_{2}\tau_{x}^{n+1}-\Ecal_{2}\tau_{x}^{*} & = & \left(\Ical_{2}-\Ecal_{2}\right)\tau_{x}^{*}+\Ical_{2}K_{u}\frac{\partial\delta u^{n+1}}{\partial z} -\end{eqnarray*} -where, $\delta u^{*}=u^{*}-u^{n},\;\delta u^{n+1}=u^{n+1}-u^{*}$. The -meridional stress $\tau_{y}$ and the scalar fluxes can be derived in a -similar way. These formulae have been validated in SCM experiments. - -\subsubsection{Implicit surface flux and future upgrades} - -This scheme should be incorporated in the calculation of the scalar -implicit fluxes. This would be preferable to the current technique for -calculating the scalar implicit fluxes (use of original implicit -scheme for these). This has been attempted but not yet successfully -completed. In the tested code, 2 calls to the modified im\_sf\_pt -subroutine are done, one per scheme stage (step), i.e. one for the -stage that ${\delta X}_{1}^{*}$ is computed and one for $\delta -X_{1}^{n+1}$ where the subscript denotes level number. At each call, a -modified version of the flux formulae (78), (79) of -\cite{esseryetal2001} is used: - -\noindent \textbf{1st sweep:} -\begin{eqnarray} - \frac{\overline{H^{*}}}{c_{p}} & = & \frac{(1+\beta B_{2})[\gamma_{2}F_{T}^{n}+A_{1}\delta{T'}_{1}+A_{2}\delta{Q'}_{1}]-\beta A_{2}[\gamma_{2}F_{Q}^{n}+B_{1}\delta{T'}_{1}+B_{2}\delta{Q'}_{1}]}{(1+\beta A_{1})(1+\beta B_{2})-\beta^{2}A_{2}B_{1}}\label{eq:FTLstar} -\end{eqnarray} -\begin{eqnarray} - \overline{E^{*}} & = & \frac{(1+\beta A_{1})[\gamma_{2}F_{Q}^{n}+B_{1}\delta{T'}_{1}+B_{2}\delta{Q'}_{1}]-\beta B_{1}[\gamma_{2}F_{T}^{n}+A_{1}\delta{T'}_{1}+A_{2}\delta{Q'}_{1}]}{(1+\beta A_{1})(1+\beta B_{2})-\beta^{2}A_{2}B_{1}}\label{eq:FQWstar} -\end{eqnarray} -where, $F_{T}^{n}$, $F_{Q}^{n}$ denote the surface explicit fluxes, -$\gamma_{2}=\Ical_{1}-\Ecal_{1}$ and the coefficients -$A_{1},\; A_{2},B_{1},\; B_{2}$ are given by -\begin{subequations} - \begin{equation} - A_1=-\gamma_1\sum_j \nu_j RK_{PMj} - [LD_j\psi_jRK_H(1)_j+A_{*j}], - \end{equation} - \begin{equation} - A_2=\gamma_1\sum_j \nu_j RK_{PMj} - L\psi_jRK_H(1)_j, - \end{equation} - \begin{equation} - B_1=\gamma_1c_p\sum_j \nu_j RK_{PMj} - D_j\psi_jRK_H(1)_j - \end{equation} - \begin{equation} - B_2=-\gamma_1\sum_j \nu_j RK_{PMj} - \psi_j[c_pRK_H(1)_j+A_{*j}]. - \end{equation} - \label{ab_coeffs} -\end{subequations} -but with $\gamma_{1}=\Ical_{1}$. Here $RK_H(1) =\rho C_H U_1$, -$RK_{PM}={RK_H(1)\over(c_p+LD\psi)RK_H(1)+A_*}$, -\begin{equation*} - A_* = (1 - f_r){2\lambda\over\Delta z_s} + {C_c\over\Delta t} + - 4(1 + f_r)\sigma T_s^3, -\end{equation*} -\begin{equation*} - D={q_{\rm sat}(T_*^{(n)},p_*)-q_{\rm sat}(T_1^{(n)},p_*) \over - T_*^{(n)}-T_1^{(n)}}, -\end{equation*} -\begin{equation*} - \psi=f_a+(1-f_a){g_s\over g_s+C_HU_1}, -\end{equation*} -and $\nu_j$ represents the fraction of surface tile type $j$. $f_r$ is -the radiative canopy fraction, $\lambda$ is the soil conductivity, -$\Delta z_s$ and $T_s$ are the thickness and temperature of the -surface soil layer, $C_c$ is the canopy heat capacity, $f_a$ is the -saturated fraction of the tile and $g_s$ is the surface -conductance. To derive (\ref{eq:FTLstar}), (\ref{eq:FQWstar}), the -time-weighted level 1 $T$ and $Q$ consistent with the discrete -equations of the new scheme is written as follows: -\begin{equation*} - \overline{T_{1}^{*}}=\gamma_{2}T^{n}+\gamma_{1}\delta T_{1}^{*},\quad\overline{Q_{1}^{*}}=\gamma_{2}Q^{n}+\gamma_{1}\delta Q_{1}^{*},\qquad\delta T_{1}^{*}=T^{*}-T^{n},\quad\delta Q_{1}^{*}=Q^{*}-Q^{n} -\end{equation*} -and the original derivation is followed. From these expressions the -tile flux for $H$ is derived: -\begin{eqnarray*} - \frac{H_{j}^{*}}{c_{p}} & = & \gamma_{2}\frac{H_{j}^{(n)}}{c_{p}}-\gamma_{1}RK_{PMj}[LD_{j}\psi_{j}RK_{H}(1)_{j}+A_{*j}][c_{p}\delta{T'}_{1}-\beta\overline{H^{*}}]\\ - & & \qquad\quad+\gamma_{1}RK_{PMj}L\psi_{j}RK_{H}(1)_{j}[\delta{Q'}_{1}-\beta\overline{E^{*}}] -\end{eqnarray*} -and similarly $E_{j}^{*}$. From these, the tile flux equations -(\ref{eq:FTLstar}), (\ref{eq:FQWstar}) can be obtained. - -\noindent \textbf{2nd sweep:} -\begin{eqnarray} - \frac{\overline{H^{n+1}}}{c_{p}} & = & \frac{(1+\beta B_{2})[\xi_{2}F_{T}^{*}+A_{1}\delta{T'}_{1}+A_{2}\delta{Q'}_{1}]-\beta A_{2}[\xi_{2}F_{Q}^{*}+B_{1}\delta{T'}_{1}+B_{2}\delta{Q'}_{1}]}{(1+\beta A_{1})(1+\beta B_{2})-\beta^{2}A_{2}B_{1}}\label{eq:FTLnp1} -\end{eqnarray} -\begin{eqnarray} - \overline{E^{n+1}} & = & \frac{(1+\beta A_{1})[\xi_{2}F_{Q}^{*}+B_{1}\delta{T'}_{1}+B_{2}\delta{Q'}_{1}]-\beta B_{1}[\xi_{2}F_{T}^{*}+A_{1}\delta{T'}_{1}+A_{2}\delta{Q'}_{1}]}{(1+\beta A_{1})(1+\beta B_{2})-\beta^{2}A_{2}B_{1}}\label{eq:FQWnp1} -\end{eqnarray} -where, the coefficients $A_{1},\; A_{2},B_{1},\; B_{2}$ are given by -(\ref{ab_coeffs}) but with $\gamma_{1}=\Ical_{2}$. The above -formulae are derived as explained earlier. The definitions -\begin{equation*} - \overline{T_{1}^{n+1}}=\gamma_{2}T^{*}+\gamma_{1}\delta T_{1}^{n+1},\quad\overline{Q_{1}^{*}}=\gamma_{2}Q^{*}+\gamma_{1}\delta Q_{1}^{n+1},\qquad\delta T_{1}^{n+1}=T^{n+1}-T^{*},\quad\delta Q_{1}^{n+1}=Q^{n+1}-Q^{*} -\end{equation*} -are used here. The surface fluxes $F_{T}^{*}\equiv{\displaystyle - \frac{H^{*}}{c_{p}}}$, $F_{Q}^{*}\equiv E^{*}$ are computed at model -state ${(T}_{1}^{*},Q_{1}^{*})$. They are the equivalent of the -explicit fluxes $F_{T}^{n}\equiv{\displaystyle \frac{H^{n}}{c_{p}}}$, -$F_{Q}^{n}\equiv E^{n}$. However, they are not equal to the left -hand-side of (\ref{eq:FTLstar}), (\ref{eq:FQWstar}). Both are -connected by a linear relationship, which is simply the definition of -the time-weighted averaging consistent with the new scheme: -\begin{equation*} - \overline{H^{*}}\equiv\Ical_{1}H^{*}-\Ecal_{1}H^{n},\qquad\overline{E^{*}}\equiv\Ical_{1}E^{*}-\Ecal_{1}E^{n}. -\end{equation*} -Therefore, once $\overline{H^{*}},\overline{E^{*}}$ have been -computed, $H^{*}$, $E^{*}$ can be computed as follows: -\begin{equation*} - H^{*}=\frac{\overline{H^{*}}+(\gamma_{1}-\gamma_{2})H^{n}}{\gamma_{1}},\qquad - E^{*}=\frac{\overline{E^{*}}+(\gamma_{1}-\gamma_{2})E^{n}}{\gamma_{1}}. -\end{equation*} -Note that without the previous transformation the model fails within a -few timesteps. - -The calculation of the surface temperature, evaporation and melting -takes place only in the second sweep. The flux increment obtained from -evaporation and melting is added on $\overline{H^{n+1}}$, -$\overline{E^{n+1}}$. The same relationship is used for the surface -temperature: -\begin{equation*} - T_{*}=T_{s}+\frac{1}{A_{*}} - \left[R_{s}-H-LE+\frac{C_{c}}{\Delta t}\left(T_{*}^{n}-T_{s}\right)\right], -\end{equation*} -however, the total averaged flux from $t^{n}$ to $t^{n+1}$ is used: -\begin{equation*} - H=\overline{H^{*}}+\overline{H^{n+1}},\qquad E=\overline{E^{*}}+\overline{E^{n+1}}. -\end{equation*} -The total flux is also kept by the corresponding STASH -diagnostic. This has to be adjusted if evaporation exhausts any of the -moisture stores during the timestep or if the tile has a melting -snowcover. - -\paragraph{Limited evaporation} - -Downward surface moisture fluxes are added to canopy moisture or, if -the surface temperature is below freezing, snowcover. - -For an upward total moisture flux $E$, the rates of evaporation from -the canopy and soil moisture stores are -\begin{equation} - E_c = f_a{E\over\psi} -\end{equation} -and -\begin{equation} - E_s = (1 - f_a)\psi_s {E\over\psi} -\end{equation} -where -\begin{equation} - \psi_s = {g_s\over g_s+C_HU_1}. -\end{equation} -If the predicted canopy evaporation would exhaust the canopy moisture -store $C$ during a timestep, the soil evaporation is recalculated as -\begin{equation} - E_s =\psi_s\left(1 - {f_aC\over E_c\Delta t}\right){E\over\psi} -\end{equation} -and $E_c$ is reset to $C/\Delta t$. If $E_s$ would then exhaust the -available soil moisture $m$, it is limited to $m/\Delta t$. - -For an adjustment $\Delta(LE)$ in the latent heat flux, repartitioning -the surface energy balance gives adjustments -\begin{equation} - \Delta H = - \left[1 + {A_*\over c_pRK_H(1)}\right]^{-1}\Delta(LE) -\end{equation} -and -\begin{equation} - \Delta T_* = - {\Delta H+\Delta(LE)\over A_*} -\end{equation} -in the surface sensible heat flux and temperature. - -Evaporation from a lake tile (or the lake fraction of an aggregated -surface) is not limited and does not draw on the conserved moisture -stores. - -\paragraph{Snowmelt} - -Classical surface energy balance neglects snowmelt heat fluxes. If -$T_*>T_m$ for a snow-covered tile and sufficient snow is available, -$T_*$ is reset to $T_m$ by adding an increment -\begin{equation} - \Delta T_*=T_m-T_*, -\end{equation} -corresponding to a snowmelt heat flux -\begin{equation} - S_m = - [(c_p + L_sD)RK_H(1) + A_*]{\Delta T_*\over L_f}. - \label{eq:Sm} -\end{equation} -The maximum melt rate that can be sustained over a timestep $\Delta -t$, however, is $S/\Delta t-E$, giving -\begin{equation} - \Delta T_*={L_f(S/\Delta t-E) \over - (c_p+L_cD)RK_H(1) + A_*}. - \label{eq:dTmax} -\end{equation} -$\Delta T_*$ is set to the smaller of the values given by Equations -(\ref{eq:Sm}) and (\ref{eq:dTmax}), and the surface energy balance is -repartitioned by adding increments -\begin{equation} - \Delta H = c_pRK_H(1)\Delta T_* -\end{equation} -and -\begin{equation} - \Delta E = DRK_H(1)\Delta T_* -\end{equation} -to the tile heat and moisture fluxes. - -The model with the above changes coded seems to work stably but the -surface fluxes (and therefore the boundary layer increments) are only -qualitatively correct. They seem to be overestimated by the above -scheme. {[}\textit{Could it be that coefficients $D_{j}$, $\psi_{j}$ - need to be modified in the second sweep?}] - -\subsubsection{Blending height coupling} - -The same method is used as in section~\ref{sec:impsolve} to form two -independent tridiagonal systems of linear equations that relate the -increments to momentum, temperature and humidity to the surface fluxes. -The `downward sweep' elimination procedure still takes place to obtain -equation (\ref{eq:du_half}) and a corresponding equation for the -increments to the scalar variables at the bottom model level - -\begin{equation} - \delta X_{1/2}^{*}=\delta X_{1/2}^{'}-\beta_X\frac{\bar{H}_\star}{C_p} -\end{equation} - -where $\delta X_{1/2}^{'}$ and $\beta_X$ are known. An `upward sweep' of -this tridiagonal matrix (i.e. back subsitution) then takes place to -obtain equations for the increment to momentum and scalar variables at a -given level $\kbl$ in terms of the surface fluxes - -\begin{equation} - \delta u_{\kbl}^{*}=\delta u_{\kbl}^{'}+(-1)^{\kbl}\beta\bar{\tau}_{x}^{*}\Big|_{0} \prod^{j=2}_{\kbl} {C_{u}}_{j}^{'} + \sum^{\kbl-1}_{i=1} \left[(-1)^{\kbl+i}\delta u_{\kbl}^{'} \prod^{j=i+1}_{\kbl} {C_{u}}_{j}^{'}\right] -\end{equation} - -\begin{equation} - \delta X_{\kbl}^{*}=\delta X_{\kbl}^{'}+(-1)^{\kbl}\beta_X\frac{\bar{H}_\star}{C_p} \prod^{j=2}_{\kbl} {C_{X}}_{j}^{'} + \sum^{\kbl-1}_{i=1} \left[(-1)^{\kbl+i}\delta X_{\kbl}^{'} \prod^{j=i+1}_{\kbl} {C_{X}}_{j}^{'}\right] -\end{equation} - -where $\delta u_{\kbl}^{*}$ and $\delta X_{\kbl}^{*}$ are the only unknowns. -The coefficients for these equations are passed to the surface implicit -solver so that the surface fluxes are calculated using level $\kbl$ -(which corresponds to a user-specified blending height) rather than -the bottom model level. The rest of the implicit solver continues in the -same way as for coupling at the bottom model level. - -By default, this option is switched off as further work is needed -(there is currently a problem with the input data such that this -blending height option crashes on the first time step). - - -\section{Derived diagnostics} - -\subsection{Boundary layer thermal speed: stash 3,355} - -This diagnostic is intended for use in quantifying the strength of -convective thermals for aviation applications. Updraught velocities -in convective boundary layers will scale with the convective velocity -scale, $w_*$, given by $ w_*^3 = \zhe \wbs $. In addition to the -basic convective velocity scale, the strength of thermals should also -depend on the surface stability --- it would be possible to have -significant heat flux and boundary layer depth in windy conditions -that should not lead to a strong thermal forecast. This sensitivity of -boundary layer turbulence is already included in the parametrization -of non-local momentum fluxes (see section \ref{sec:ngstress}) through -the stability dependence in (\ref{tau_nl}) that can be written as -\begin{equation} - f_{stab} = - \frac{a_{stab} \zhe/L }{1 - a_{stab} \zhe/L} -\end{equation} -for the Obhukov length, (\ref{1.1.4}), $<0$ (\mbox{i.e.}, unstable -boundary layers) and the empirical constant $a_{stab} = 1.5$. This -function tends to unity as $L$ decreases in magnitude (\mbox{i.e.}, -surface heating increases and wind stress decreases). The final -thermal speed, in units of ms$^{-1}$, is given simply by -\begin{equation} - {\rm Thermal} \, {\rm Speed} = f_{stab} \, w_* -\end{equation} - -\subsection{Wind gust: stash 3,463 and 3,515 (scale-dependent)} - -WMO define the wind gust strength as the maximum of the wind averaged -over 3 second intervals. In the boundary layer the strength of gusts -is proportional to the standard deviation of the horizontal wind, -$\sigma_u$, so that -\begin{equation} - U_{gust} = U_{10m} + W_{1D} \, \sigma_u \, \frac{1}{k} \, - {\rm log}\left( \frac{5 \, e^{k \, c_{\rm ugn}} + z_{0m(eff)} } - {5 + z_{0m(eff)}} \right) -\label{windgust} -\end{equation} -The factor $W_{1D}$ is included only in the scale-dependent version of -the diagnostic (stash 3,515) to allow for the larger scales of boundary -layer turbulence that are resolved (and so are already included in -$U_{10m}$). The lowest grid-level value of $W_{1D}$, from (\ref{eq-tanh}), -is used, noting that $W_{1D}$ is constant within the boundary layer. The -constant $c_{\rm ugn}$ in (\ref{windgust}) is determined from universal -turbulence spectra for a 25\% exceeding probability of the three-second -wind gust (\cite{beljaars1987}). It is included through a function that -includes the effective roughness length, $z_{0m(eff)}$, in order to -take into account the very high effective $u_*$ values that occur over -mountainous terrain (due to the orographic form drag parametrization) -and so avoid unrealistic high gust values. Currently the UM takes -$c_{\rm ugn}=4$ which was reduced from the value used at ECMWF based -on evaluation of the wind gust performance. The stability dependence -of $\sigma_u$ is estimated on the basis of the similarity relation -from \cite{panofsky1977} -\begin{equation*} - \sigma_u = -\begin{cases} - A_{gust} u_* (1.0 - \zhe / (24 L) )^{1/3} & {\rm for}\ L<0 \\ - A_{gust} u_* & {\rm for}\ L>0 -\end{cases} -\end{equation*} -with $A_{gust}=2.29$. For $L$ close to zero the wind gust diagnostic -is undefined and so we set $U_{gust} = U_{10m}$. Note that the -friction velocity, $u_*$, must use the implicitly calculated surface -stress components because the explicitly calculated $u_*$ can be -erratic, particularly over mountainous regions. Then, for consistency -with the implicit $u_*$, $L$ must also be calculated implicitly and, to -avoid potential numerical problems in very light winds, the unstable -($L<0$) case is rewritten as: -\begin{equation*} - \sigma_u = A_{gust} (u_*^3 + k w_*^3 / 24 )^{1/3} -\end{equation*} - -\subsection{TKE: stash 3,473} - -A substantial part of the turbulent flux is parametrized in both the -UM's first order closure and closures involving TKE, $e$, through a -simple down-gradient diffusion term. An estimate of subgrid TKE can -then be made by equating the UM's diffusion coefficient, -(\ref{klnl}), with that from a typical TKE-closure, \mbox{i.e.} -\begin{equation} - K_m = l \sqrt{e} - \label{tke_closure} -\end{equation} -where $l$ is a length scale. Initially it was thought to diagnose $e$ -by approximating $l$ as the mixing length in (\ref{kmlocal}) but -closer inspection reveals that many TKE closures have diagnostic -relationships for $l$ that involve the TKE itself! A common one for -stable boundary layers is $l_{st} \sim \sqrt{e} / N$, where $N$ is the -Brunt-Vaisala frequency. \cite{Suselj2012}, for example, also take -$l_{un} = \tau_{un} \sqrt{e} $ in unstable boundary layers, where -$\tau_{un}$ is a turbulence timescale that they take as a constant 400 -seconds. These they combine through $l^{-1}= l_{un}^{-1} + -l_{st}^{-1} = e^{-1/2}( \tau_{un}^{-1} + \tau_{st}^{-1}) \equiv -e^{-1/2}\tau_{turb}^{-1}$ and (\ref{tke_closure}) becomes: -\begin{equation*} - K_m = \tau_{turb} e -\end{equation*} -To derive a TKE diagnostic then requires a parametrization of the -turbulence timescale, $\tau_{turb}$. - -Basic boundary layer scaling (\mbox{e.g.}, Figure 4 of -\cite{holtslag91:_eddy_diffus_count_trans_convec}) shows that -$\ol{w'^2}$ from a variety of convective boundary layer LES and -observations nicely follows the relationship -\begin{equation} - \ol{w'^2} = c_{w2} w_*^2 f(z') - \label{w2_scaling} -\end{equation} -where $w_*$ is the convective velocity scale and $f$ is a shape -function within the boundary layer ($z'=z/\zhe$). The shape of this -function is very similar to that used in the UM for $\kmsurf$ in -(\ref{kmsurf}). We now assume we can generalise (\ref{w2_scaling}) by -replacing $w_*$ with $w_m$ (this really ought to be checked against -neutral boundary layer LES but hasn't yet been). Setting $ f(z')=z' -(1-z')^2$ in (\ref{w2_scaling}) and comparing with Fig.4 of -\cite{holtslag91:_eddy_diffus_count_trans_convec} gives $c_{w2}= 2.66 -/ C_{ws}^{2/3}$ (\mbox{i.e.}, a constant of 2.66 gives the maximum in -$\ol{w'^2}/w_*^2$ at around the observed value of 0.4) so that we can -generalise (\ref{w2_scaling}) to -\begin{equation} - \ol{w'^2} = \frac{2.66 }{C_{ws}^{2/3}} \, w_m^2\, f(z') - \label{gen_w2_scaling} -\end{equation} -where the mixed layer expression for $w_m$ is used. - -\cite{holtslag91:_eddy_diffus_count_trans_convec} also show from -analysis of the scalar flux budget that -\begin{equation*} - \ol{w'\theta'} = - \frac{\tau_{turb}}{2} \, \ol{w'^2} \frac{d \theta}{dz} -\end{equation*} -where $\tau_{turb}$ is a return to isotropy timescale. Ignoring the -non-gradient parametrization in the UM, it follows that -\begin{equation} - \khsurf = \frac{\tau_{turb}}{2} \, \ol{w'^2} - \label{bl_scaling} -\end{equation} -Combining (\ref{bl_scaling}) with (\ref{gen_w2_scaling}) and -(\ref{kmsurf}), and subsuming the Prandtl number into the other -constants, for surface-driven boundary layer mixing we can write: -\begin{equation*} - \kmsurf = \frac{\tau_{\rm surf}}{2} \, \ol{w'^2} = \frac{\tau_{turb}}{2} \, \frac{2.66 }{C_{ws}^{2/3}} w_m^2 \, f(z') = k \zhe w_m f(z') -\end{equation*} -which then gives $\tau_{\rm surf} = C_{ws}^{2/3} k \zhe / (1.33 w_m) -$. An analogous timescale can be derived for top-driven mixing in -decoupled stratocumulus layers, $\tau_{\rm Sc} = g_1 k \zml / (1.33 \, -\vtopo) $. - -There are two options to derive a TKE diagnosis from the Ri-based scheme -and then combine with the non-local TKE (selected via var\_diags\_opt). One -is to assume $\tau_{\rm SBL}=0.7/N$ as the timescale for -stable boundary layers and combine all these timescales following -\cite{Suselj2012}) to give: -\begin{equation} - e = K_m \tau_{turb}^{-1} - \label{tke_diag} -\end{equation} -where $\tau_{turb}^{-1} = MAX[ \tau_{\rm surf}^{-1},\tau_{\rm Sc}^{-1}] + \tau_{\rm SBL}^{-1}$. -Note that (\ref{tke_diag}) gives $\ol{w'^2}$, rather than TKE. As a simple fix -to improve the near-surface TKE in convective boundary layers, where the horizontal -wind variability often dominates, the value of $e$ given by (\ref{tke_diag}) at -the level of the maximum in $\kmsurf$ is copied to all levels below that height. - -The second method diagnoses TKE for the local scheme following the Met Office -LEM and MONC, simplifying and parametrizing the terms in the TKE budget to give -\begin{equation} - e_{loc}^{3/2} = \lambda S^2 K_m (1-Ri/Pr)/C_{e} - \label{tke_diag_loc} -\end{equation} -where $C_{e}=A_{2N}^{3/2}$. Initial tests found that the MONC value of -$A_{2N}=0.23$ gave rather large values of $e_{loc}$ and so $C_e=0.41$ is used. Note -that this is an optional value used in the higher order closure scheme (see section -2.8.5 of \citeumdp{025}). It could also be worth testing the suggested parametrization in -(2.300) there, of $C_e=0.19+0.74 \lambda/\Delta z$ but this has not yet been attempted. -The total non-local TKE is computed by adding the TKE from each non-local component, -as is done for the diffusion coefficients, \mbox{i.e.}, -\begin{equation} - e_{nl} = \frac{3}{2} \left( \frac{\kmsurf}{\tau_{\rm surf}} - + \frac{\kmtop}{\tau_{\rm Sc}} \right) - \label{tke_diag_nl} -\end{equation} -The factor of $3/2$ in (\ref{tke_diag_nl}) arises because we are really diagnosing -$\ol{w'^2}$ and so here we make the assumption of isotropic turbulence to extend -this to TKE. As before, we do also make the simple fix to improve the near-surface -TKE in convective boundary layers, but here we copy only the value of $e_{nl}$ at the -level of the maximum in $\kmsurf$ to all levels of $e_{nl}$ below that height. The -final TKE is then the greater of $e_{nl}$ and $e_{loc}$ (as is done to combine the -diffusion coefficients). - -The methods above will still underestimate the value of subgrid TKE in -regions of parametrized convection, because the value of $K_m$ will be -small (or zero) here as parametrized mixing is assumed to be done via -the convection scheme. Therefore an option {\sl l\_conv\_tke} is -provided to include an estimate of the TKE due to parametrized -convection within the diagnostic. This is given by: -\begin{equation} -e_{\rm conv} = \left(\frac{M}{g\rho \times CCA}\right)^2 -\end{equation} -where $M$ is the convective updraft mass flux (Pa~s$^{-1}$) and CCA is -the convective cloud area. The final diagnostic is then given as the -maximum of $e$ and $e_{\rm conv}$. - -If selected, this option also reduces the minimum value used in UKCA -by an order of magnitude. This is possible, because the minimum is no -longer having to provide a realistic estimate of TKE in convective -cloud regions, and is now a genuine numerical minimum. Selecting this -option also corrects a bug in the level indexing of this diagnostic -when passed to UKCA. - -Note that additional diagnostics of the scalar variances are also made and those -are documented in \citeumdp{029}. - -\subsection{Diagnostics of Neutral Winds and Stresses: stash 3,365 to 3,371} -\label{app:neutwind} - -Conditions near the ocean's surface are often described using 10-m -neutral wind and quantities derived from the neutral winds. Such -diagnostics are therefore potentially very useful for evaluation of -the model and have been added to the scheme. - -The equivalent neutral 10-m wind is the wind that would be observed, -given the surface friction velocity and roughness length, if the -stratification were neutral. As such, it is more simply related to the -surface stress than the true stability-dependent wind. Scatterometers -ultimately respond to backscatter from surface capillary waves, which -are driven by the surface stress, so observations from scatterometers -are typically reported as equivalent neutral winds. - -The pseudostress is the product of the wind speed and the vector wind -at a given height (in practice 10m). The kinematic surface stress is -therefore equal to the product of the pseudostress and the drag -coefficient. Pseudostress is sometimes used in observational products, -notably the Cross-Calibrated Multi-Platform (CCMP) surface wind vector -analysis \cite[]{atlas2011}. - -%------------------------------------------------------------------------ -% APPENDIX: VELOCITY SCALES -%------------------------------------------------------------------------ -%\setcounter{section}{0} -%\renewcommand{\thesection}{\Alph{section}} -%\renewcommand{\theequation}{\Alph{section}.\arabic{equation}} -%\setcounter{equation}{0} - -\section{Appendix: Definitions of the velocity scales} -\label{app:vscales} - -As described in \cite{lock00}, the parametrization of the entrainment -rate in convective boundary layers is based on four velocity scales, -each representative of a turbulence-generating process ($\vheato$ for -surface heating, $u_*$ for surface shear generation, $\vrado$ for -cloud-top radiative cooling and $\vbro$ for buoyancy reversal). The -velocity scales can be written -\begin{eqnarray} - \vheat &=& \zml \! \left( (2-\zeta_s)\zeta_s \wbs + (1-\zeta_s)^2 \wbsat \right) - \label{vsurf} \\ - \vrad &=& \zml \Delta_\radf \, g \, - \left( \beta_T \zeta_r^2 + \tilde{\beta_T} (1-\zeta_r^2) \right) - \label{vrad} \\ - \vbr &=& \abr \chi_s^2 \, \mbox{max}\left[0,-\delta b\right] \, \Delta b ^{1/2} - \, z_c^{3/2} \, C_{fac} \label{vbr} -\end{eqnarray} -Here, $\wbsat = g ( \tilde{\beta_T} \wthls + \tilde{\beta_q}\wqts )$, -where the subscript $_S$ indicates the surface flux; $\Delta_\radf$ is -the divergence of the net radiative flux, $\radf$ (in Kms$^{-1}$), -associated with cloud-top, for which the calculation is described in -section~\ref{app:deltaf}. - -Various depth parameters are given by $\zeta_s = -(\zml-\tilde{z_c})/\zml $, $\zeta = (\zml-z_c)/\zml $ and $ \zeta_r = -\zeta + Br (1-\zeta) $. $\zml$ is the mixed-layer depth, $z_c$ is the -cloud depth and $\tilde{z_c}$ is the cloud-fraction weighted cloud -depth. The former is used in the calculation of $\vrado$ as it is -assumed the radiative cooling will occur predominantly in cloudy air. -To allow for a feedback in the presence of buoyancy reversal, the -parameter $Br$ is included in $ \zeta_r $ and $\tilde{\alpha_t} $ (in -(\ref{we_parm})). It is given in terms of the \cite{siems1990} -parameter, $D = \chi_s \delta b/\Delta b $ and constrained by $0< Br = -10 D < 1$. This gives a linear ramp for this feedback between regimes -where there is no buoyancy reversal ($D \leq 0$) and the feedback seen -in LES of stratocumulus \cite[]{lock98} with significant buoyancy -reversal ($D \gtapp 0.1$). Furthermore, the LES of -\cite{lock09:_factor} indicated the presence of cumulus penetrating up -into stratocumulus could be sufficient to enhance the feedback for -small $D$. Thus the option exists to enhance $Br$ for $0$ SC\_CFTOL in grid-levels NTML or NTML$+1$ or the layer is a -decoupled layer). If no subgrid inversion has been diagnosed and -$C_F(NTML+1)>$ SC\_CFTOL, then $z_c$ is increased by the full depth of -layer $NTML+1$ if $C_F(NTML)>$ SC\_CFTOL and using (\ref{zc_calc}) -otherwise (and similarly for DSC layers). The same ideas are used in -the 9C version, extrapolating using the adiabatic -water gradient, except that the grid-level from which this -extrapolation is made is now not the lowest grid-level with $C_F > $ -SC\_CFTOL but rather the lowest level with $C_F=1$ (or the grid-level -with the maximum $C_F$). Most observations of stratocumulus (i.e., -mixed layer clouds) find $\ql$ to be close to adiabatic over the whole -layer and accurately given by the supersaturation of the well-mixed -$q_t$. If $C_F=1$ then the Smith cloud scheme makes $\ql$ equal to the -supersaturation and so will be reasonably accurate. Extrapolating this -grid-level $\ql$ to zero should then give a reasonably accurate -measure of cloud-base. - -The formula for $\vbro$ was derived using dimensional arguments and -comparison with LES data: $\chi_s = -\qlmax (1+(L/c_p)\alpha_L) / ( -\Delta q_t - \alpha_L \Delta \thetal)$, where $\qlmax$ is the -cloud-top liquid water mixing ratio, $L$ is the latent heat of -vaporisation of water, $c_p$ the specific heat at constant pressure -and T is the temperature; $\delta b = g(\tilde{\beta_T} \Delta \thetal -+ \tilde{\beta_q} \Delta q_t)$ and the buoyancy jump across the -inversion is given by -\begin{equation} - \Delta b = g \, \left( \beta_T \Delta \thetal + \beta_q \Delta q_t + - \left( \beta_T \frac{L}{c_p} - - \frac{1+c_v}{c_v}\beta_q \right)\Delta \ql + - \left( \beta_T \frac{L_s}{c_p} - - \frac{1+c_v}{c_v}\beta_q \right)\Delta q_f \right) - \label{dbinv} -\end{equation} -The empirical constant $\abr = 0.24$. The calculation of $\Delta -\thetal$ and $ \Delta q_t$ is described for a subgrid inversion in -section~\ref{sec:sginv} or, if one is not diagnosed, they are taken -simply as $\Delta_{\ntml+1}$. For $\Delta \ql$, $\Delta q_f$ and -$\qlmax$, in-cloud values extrapolated to $z_i$ (either subgrid or -$z_{\ntml+\frac{1}{2}}$) from above and below using the adiabatic -lapse rates are calculated as: -\begin{eqnarray*} - \qlmax & =& \frac{{\ql}_{\ntml}}{{C_F}^l_{\ntml}} - + (z_i-z_{\ntml}) \gamma_{\ql}\\ - \ql^+ & =& \mbox{max}\left[ 0, \, - \frac{{\ql}_{\ntml+2}}{{C_F}^l_{\ntml+2}} - - (z_{\ntml+2}-z_i) \gamma_{\ql} \right] -\end{eqnarray*} -and similarly for $q_f$ (noting that currently $\gamma_{q_f}=0$) and -for DSC layers. Then, -\begin{eqnarray*} - \Delta \ql & =& {C_F^l}_{\ntml+2}\, \ql^+ - {C_F^l}_{\ntml}\, \qlmax \\ - \Delta q_f & =& {C_F^f}_{\ntml+2}\, q_f^+ - {C_F^f}_{\ntml}\, \qfmax -\end{eqnarray*} - -The only other explicit account of variable cloud fraction is in -(\ref{vbr}) for which it is assumed that buoyancy reversal can only -occur for cloudy air underlying cloud-free air (assuming maximum -overlap). Thus, the cloud fraction factor, $C_{fac} = \mbox{max}[ -0.0, -\Delta C_F ] $, where $\Delta C_F = {C_F}_{\ntml+2} - -{C_F}_{\ntml} $ if a subgrid inversion is diagnosed (because -${C_F}_{\ntml+1}$ is currently meaningless) and $\Delta C_F = -{C_F}_{\ntml+1} - {C_F}_{\ntml} $ if not. A more complete -decomposition is not possible given a cloud scheme in the model -\cite[]{smith90} which does not allow discrete identification of -in-cloud and out-of-cloud profiles. The cloud-fraction dependence of -the radiative generation of turbulence is implicitly treated in -(\ref{vrad}) simply by assuming the grid-box mean radiative flux -divergence, $\Delta_\radf$, occurs solely in the cloudy air. - -The assumption behind the current cloud fraction dependence is that -the fraction of the boundary layer that is cloud-capped entrains as -though it were an infinite solid cloud sheet (as in the LES used to -derive the parametrization). This essentially assumes the cloud -within the grid-box is continuous. An additional explicit dependence -of entrainment on cloud fraction was implemented in version 4.5 which -reduced the cloud-top source terms of entrainment in partially cloudy -boundary layers by a factor -$\exp{\left\{-(0.9-{C_F}_{\ntml})^3/0.075\right\}}$ for $ -{C_F}_{\ntml} < 0.9$. It was argued that partial cloudiness on the -scale of the mixed-layer eddies might reduce the entrainment -efficiency of the cloud-top processes. By reducing the parametrized -entrainment warming and drying in partially cloudy boundary layers it -was hoped that the climatological cloudiness of the sub-tropical -marine stratocumulus might be improved. Only marginal success was -observed, though, as more weakly entraining layers became shallower -and their cloud-top therefore warmer. In addition, timeseries of -cloud fraction from New Dynamics climate simulations suggested these -boundary layers either had high total cloud amount or zero. Coupled -with the generally realistic cloud amounts obtained in the New -Dynamics this arbitrary term has currently been dropped from version 5 -onwards. The issue of how the entrainment rate should be parametrized -in partially cloudy boundary layers, however, remains. - -\subsection{Calculation of $\Delta_\radf$} -\label{app:deltaf} - -An important term in the entrainment parametrization and $\khtop$ is -the velocity scale $\vrado$, the cube of which is proportional to the -net radiative flux difference associated with cloud-top, $\Delta_F$. -Because radiation tends not to be called every timestep, the -calculation of $\Delta_\radf$ is done somewhat independently from the -cloud-top height. - -In the 9B version, $\Delta_\radf$ is calculated as: -\begin{equation} - \Delta_\radf = \sum_{k=k_m-1}^{k_m+1} \mbox{max}\left[ - - \Delta_{k+\frac{1}{2}} z \, {\cal S}_\radf(k), \,0 \right] - \label{ctraddiv} -\end{equation} -where $k_m$ is the grid-level with the greatest radiative cooling -increment, ${\cal S}_\radf$, within 2 grid-levels of cloud-top. In -the 8A scheme, ${\cal S}_\radf$ is simply the net (SW+LW) cooling -increment. During the day, though, the net divergence is partly -reduced from the nocturnal (LW) value due to SW warming of the -cloud-layer. In general, the SW warming is more diffuse than the LW -cooling (which occurs mostly within O(50)m of cloud-top). Using -grid-level net radiative increments at the current coarse vertical -resolution used in the UM ($\sim 200$m) means that the cancellation -between LW and SW during the day is excessive. - -A slightly more accurate estimate of the net divergence can be -obtained by assuming the SW and LW radiative fluxes at a given height, -$z$, have an exponential shape, dependent on the LWP above $z$, -\mbox{i.e.}: -\begin{equation} - F_{LW}(z) = \Delta_\radf^{LW} \exp^{ - \kappa_{LW} \mbox{LWP}(z) } - \label{eq:explw} -\end{equation} -Then the net divergence can be approximated given the SW flux at the -height where $F_{LW}$ becomes some small fraction, $A$, of $ -\Delta_\radf^{LW} $ (which implies $ \mbox{LWP} =-ln(A) / \kappa_{LW} -$). Then -\begin{equation*} - \Delta_\radf \approx \Delta_\radf^{LW} + - (1-\exp^{ln(A)\kappa_{SW}/\kappa_{LW}}) \Delta_\radf^{SW} -\end{equation*} -Empirically, see Fig.~\ref{fig:dradts}, a reasonable fit to LEM data -is obtained with: -\begin{equation} - \Delta_\radf \approx \Delta_\radf^{LW} + 0.35 \Delta_\radf^{SW} - \label{eq:deltaf_emp} -\end{equation} -Note that in the 9B scheme $\Delta_\radf^{LW} $ and $\Delta_\radf^{SW} -$ are calculated as in (\ref{ctraddiv}) but with the LW and SW -increments separately. - -This change in the calculation of $\Delta_\radf$ is illustrated in -Fig.~(\ref{fig:dradts}) from LES of the diurnal cycle of marine -stratocumulus. The top panel is from a simulation which used the code -specified for the EUROCS LES intercomparison, the lower panel used the -Edwards-Slingo radiation scheme in the LES. There are clearly some -differences in the distribution of the SW absorption within the cloud -between these two schemes but the 9B parametrization is clearly an -improvement on the 8A which gives $\Delta_\radf=0$ around midday (and -therefore zero entrainment and turbulent mixing). - -\begin{figure}[tbh] - \begin{centering} - \includegraphics[width=3.5in]{div_r080} - \includegraphics[width=3.5in]{div_r071} - \end{centering} - \captionarb{Time series from LES of $\Delta_\radf$ (solid), - $\Delta_\radf^{LW} $ (dotted), $-\Delta_\radf^{SW} $ (dashed) and - the 8A (dash-dot) and 9B (dash-dot-dot-dot) parametrizations of - $\Delta_\radf$.} - \label{fig:dradts} -\end{figure} - -The 9C version attempted to remove the grid-dependence implied by the -summation over 3 grid-levels in (\ref{ctraddiv}) as follows: -\begin{enumerate} -\item the search for the level with maximum LW radiative - cooling, $k_m$, is restricted to the top half of the mixed layer and - no higher than level NTML$+1$ -\item to allow for the case where the radiative cooling is distributed - roughly equally over two grid-levels, if the LW flux divergence in - level $k_m-1$ is greater than half that in level $k_m$, then $k_m$ - is lowered one grid-level. -\item then, the cloud-top radiative flux change is initially - calculated for LW and SW fluxes separately as: - \begin{equation} - \Delta_\radf = \radf_{k_m+1} - \radf_{k_{rb}} - \label{ctraddiv_9c} - \end{equation} - where the base grid-level for the calculation, $k_{rb}$, is taken to - be the higher of the base of the LW radiatively cooled layer and - $z_h/2$, since cooling can only generate turbulence if it occurs in - the upper part of the mixed layer. For decoupled stratocumulus - layers, $k_{rb}$ is further restricted to be above the top of the - surface mixed layer. -\item finally, the flux divergence across grid-level $k_m+1$ is - separated into cloudy and free-atmospheric contributions by - extrapolating the free-atmospheric flux-gradient downwards. The - cloudy contribution is then included in $\Delta \radf$: - \begin{equation} - \Delta \radf = \Delta \radf + \Delta_{k_m+\f{3}{2}} \radf - - \Delta_{k_m+\f{5}{2}} \f{\Delta_{k_m+\f{3}{2}} z}{k_m+\f{1}{2}} \radf - \label{ctraddiv_9c_inv} - \end{equation} -\item As at 9B above, the calculations in (\ref{ctraddiv_9c}) and - (\ref{ctraddiv_9c_inv}) are performed separately for LW and SW - radiation before the two are combined using the empirical - relationship in (\ref{eq:deltaf_emp}) -\end{enumerate} - -Further single column model tests with fine vertical resolution have shown -the above calculations can still fail to accurately measure the radiative -flux jump across the top of the cloud. The methodology now recommended is -to identify where the LW radiative cooling profile transitions from -free-tropospheric rates above the cloud to stronger rates within it. It -entails only relatively minor changes to the first two steps of the -algorithm above which become: -\begin{enumerate} -\item the search for the level with maximum LW radiative cooling, $k_m$, is -restricted to the top half of the mixed layer and below $1.2 \, \zhe$ (rather -than level NTML$+1$ used above) -\item if the LW flux divergence in level $k_m+1$ is relatively weak (less than -double that in level $k_m+2$), we assume that level $k_m+1$ is actually typical -of the free-troposphere and that $k_m$ must therefore be the inversion -grid-level (despite having the strongest LW cooling). Hence we lower -$k_m$ by one so that it now marks the top of the mixed layer --- note that -LW cooling within the inversion grid-level will be included in step 4 above -(which is unchanged) -\end{enumerate} -These small changes were found sufficient to give a robust measure of -$\Delta \radf$ for grids varying down to 20m spacing where the radiative -flux profile is well resolved. - -%------------------------------------------------------------------------ -% APPENDIX: BUOYANCY PARAMETERS -%------------------------------------------------------------------------ -\newpage -\section{Appendix: Derivation and definitions of the buoyancy parameters} -\label{app:buoyp} - -Buoyancy is measured by the virtual temperature -\begin{equation} - T_v = T(1 + c_v q_v - \ql - q_f) = T V_{fac} - \label{Tv} -\end{equation} -where $c_v=(1/\epsilon) -1$ and $\epsilon$ is the ratio of the -molecular weights of water vapour and dry air (\mbox{i.e.}, $\epsilon -= M_v/M_a \approx 0.62198$). The buoyancy flux is then given by -\begin{equation*} - \wb = \frac{g}{T_v}\, \overline{w'T_v'} -\end{equation*} -Linearising gives -\begin{equation*} - \wb = g \left( \beta_T \overline{w'T_L'} + \beta_q \wqt + - \left( \beta_T \frac{L}{c_p} - \frac{1+c_v}{c_v} \beta_q \right) - \overline{w'\ql'} \right) -\end{equation*} -where the buoyancy parameters are given by - -\begin{equation*} - \beta_T = \frac{1}{T}, \qquad \beta_q = \frac{c_v}{V_{fac}} -\end{equation*} - -In saturated cloudy air (see, for example, \cite{stage1981}), the -Clausius-Clapeyron equation can be used to calculate $ -\overline{w'\ql'} $ (via $\ql' = q_t' - q_s' = q_t' - \alpha_L T'$) as -\begin{equation*} - \overline{w'\ql'} = a_L( \wqt - \alpha_L \overline{w'T_L'} ) -\end{equation*} -where -\begin{equation*} - \alpha_L = \frac{\partial q_s}{\partial T} = \frac{\epsilon L q_s(T,p) }{R T^2}, \qquad - a_L = \frac{1}{1+L\alpha_L/c_p} -\end{equation*} -where R is the gas constant ($=287.05$). Thus the buoyancy flux can -be written -\begin{equation*} - \wb = -\begin{cases} - g \left( \beta_T \overline{w'T_L'} + \beta_q \wqt \right) - & {\rm in\ unsaturated\ air} \\ - g \left( \tilde{\beta_T} \overline{w'T_L'} + \tilde{\beta_q} \wqt - \right) & {\rm in\ saturated\ air} -\end{cases} -\end{equation*} -where -\begin{eqnarray*} - \tilde{\beta_T} = \beta_T - \alpha_L \beta_c, & - \tilde{\beta_q} & = \beta_q + \beta_c \\ - {\rm and} & - \beta_c & = a_L \left( \frac{L}{c_p} \beta_T - - \frac{1+c_v}{c_v} \beta_q \right) -\end{eqnarray*} - -Note that here $\tilde{\beta_T}$ and $\tilde{\beta_q}$ are strictly -{\em in}-cloud parameters, while their definitions in boundary layer -code prior to 8A were grid-box mean. Thus, here, any necessary -$C_F$-weighting must be included explicitly, as in (\ref{eq:wb_cont}). - -In all the above, if $T$ is less than the melting point of ice then -the latent heat of sublimation, $L_s = L + L_f$, is used in place of -$L$. - -%------------------------------------------------------------------------ -% APPENDIX: MIXING RATIOS -%------------------------------------------------------------------------ -\newpage -\section{Appendix: changing between specific humidities and mixing ratios} -\label{app:mixratio} - -Denote wet density by -\begin{equation*} - \rho = \rho_y + \rho_v + \rhol + \rho_{f} -\end{equation*} -where $\rho_y$ is the density of dry air and the other $\rho$ are -vapour, liquid and frozen water respectively. Mixing ratios and -specific humidities are then defined as -\begin{equation*} - m_v = \frac{\rho_v}{\rho_y} q_v = \frac{\rho_v}{\rho} -\end{equation*} - -When specific quantities are mixed, the turbulent diffusion equations, -(\ref{cons_eqn_scal}) and (\ref{cons_eqn_uv}), have $\rho$ as the wet -density. This is then consistent with the conservation of globally -integrated quantities such as moisture. For example, neglecting -spherical geometry for simplicity: -\begin{equation} - \int (\rho_v + \rhol + \rho_{f})\, d\underline{x} = - \int \rho (q_v + \ql + q_{f}) \,d\underline{x} = \int \rho q_t \, d\underline{x} -\label{moisture_cons} -\end{equation} - -When mixing ratios are used, the momentum equations, -(\ref{cons_eqn_uv}), remain unchanged and the wet density still -appears. This makes the reasonable assumption that all moisture -components should be included in the momentum budget. For moisture -conservation, (\ref{moisture_cons}) can be rewritten in terms of -mixing ratios as -\begin{equation*} - \int (\rho_v + \rhol + \rho_{f})\, d\underline{x} = - \int \rho_y (m_v + \ml + m_{f}) \,d\underline{x} = \int \rho_y m_t \, d\underline{x} -\end{equation*} -Thus $\rho$ in (\ref{cons_eqn_scal}) is replaced with $\rho_y$ for -$\chi = m_t$ and $\thetal$ when mixing ratios are passed into the -boundary layer code. - -For surface exchange, JULES initially approximates the surface air density as -$\rho_* = p_S/(R T_S)$ (where the subscript $S$ denotes the surface values and $R$ -the gas constant for dry air, 287 JK$^{-1}$kg$^{-1}$). If a more accurate calculation -of surface air density is requested, following Eqs (1) to (6) of -\cite{Webbetal1980} we then calculate the wet or dry surface air densities (to be -used when the atmospheric humidity is specific or mixing ratio, respectively) as: -\begin{eqnarray*} - \rho_{0} & =& \rho_*/(1+(1/\epsilon-1)q_S) \\ - \rho_{y0} & =& \rho_*/(1+(1/\epsilon)m_{vS}) -\end{eqnarray*} -where, in each case, the surface humidity is taken as the surface saturated humidity -over open sea but over land and ice surfaces this is likely to be inappropriate -and so the driving level humidity is used (typically the lowest model level). - -Note that $\thetal$ itself is defined in terms -of mixing ratios as: -\begin{equation} - \thetal = T - \frac{L_c}{c_{pd}} \ml - - \frac{L_c+L_f}{c_{pd}} m_f + \frac{g}{c_{pd}} z - \label{sl_defn} -\end{equation} - -For saturation calculations a version of QSAT is used that is -switchable between input specific and mixing ratio variables. The -rate of change of $q_s$ with temperature is also used in the boundary -layer code (see e.g. appendix~\ref{app:buoyp}): -\begin{equation*} - \frac{ d q_{sat} }{ dT } = \frac{\epsilon L q_{sat} }{RT^2} -\end{equation*} -In fact this expression should really be converted to work for -specific quantities and so simply changing to mixing ratios will -improve the accuracy of this calculation. - -Finally, in appendix~\ref{app:buoyp} virtual temperature is defined in -terms of specific variables as -\begin{equation*} - T_v = T ( 1 + c_v q_v - q_l ) -\end{equation*} -In terms of mixing ratios this becomes -\begin{equation*} - T_v = \frac{T ( 1 + m_v/\epsilon )}{ 1 + m_v + m_l } -\end{equation*} -Linearising, however, gives -\begin{equation*} - T_v = T ( 1 + c_v m_v - m_l ) -\end{equation*} -Thus, the same level of approximation as is currently used is -maintained simply by changing specific variables to mixing ratios. - -Additional points to note are: -\begin{enumerate} -\item the diagnostic of screen humidity, $q$1.5m, will remain as a - specific humidity. Similarly RH1.5m will remain defined in terms of - specific quantities, \mbox{i.e.}, $q$1.5m$/q_s$1.5m -\item all other moisture diagnostics (\mbox{e.g.}, latent heat fluxes - and increments) will simply switch to being mixing ratios if mixing - ratios are selected - no conversion will be made. -\item it is important to note that RHOKM will be wet density times - $K_m$ while RHOKH will be dry density times $K_h$ -\end{enumerate} - -%------------------------------------------------------------------------ -% APPENDIX: frictional heating -%------------------------------------------------------------------------ -\section{Appendix: including the heating from turbulence dissipation} -\label{app:fricheat} - -An estimate of the true molecular dissipation rate, $\epsilon_{mol}$, -can be obtained by assuming local equilibrium in the budget of subgrid -TKE (SKE). Then the sum of the inputs from resolved kinetic energy, -plus that from subgrid buoyancy effects, must equal the dissipation. -The SKE budget is -\begin{equation} - d SKE/dt = S + T + B + \epsilon_{mol} -\label{ske_budg} -\end{equation} -where the shear production, S, is essentially the resolved KE -dissipation term. Note that the buoyancy term B appears in -(\ref{ske_budg}) which indicates that some of the energy from resolved -scale dissipation (i.e. S) should be consumed in doing work against -buoyancy (at least in stable BLs) thus leaving less energy to be -finally dissipated as heat. Note though that CBLs will generate -additional dissipation (and hence heating) through the buoyancy term. -Note that from an atmospheric budget viewpoint the transport term, T, -can be neglected as it will integrate vertically to zero. Locally, -vertical variations could be important but it will be ignored because -finally the heating source is implemented through an integral over the -BL. - -So, this estimate of the molecular dissipation rate should appear as -an additional heating source term, (following \cite{zhang1999}): -\begin{equation*} - \frac{\partial \thetal}{\partial t} = - \left[\frac{du}{dz}\tau_x + \frac{dv}{dz}\tau_y + B \right] / (\rho c_p) -\end{equation*} -Tests in the SCM showed the heating rate gradients can be very large -near the surface. Hence to avoid stability problems (since this -heating increment must be added after the implicit calculation of the -stress (and heat flux) profiles) the increments are summed over the -levels within the BL (i.e. up to \zh) and then that total heating is -applied as a linear decrease from the surface to zero over \zh. - -%------------------------------------------------------------------------ -% APPENDIX: OPERATIONAL FUDGES -%------------------------------------------------------------------------ -\section{Appendix: Operational modifications} -\label{app:opmods} - -The operational global forecast model has been found to give improved -performance on NWP Index parameters when the following modifications -to its local $Ri$-based scheme are used. In (\ref{asymp_ml}), the -definition of $\lambda_m$ only is altered to -\begin{equation*} - \lambda_m = \mbox{max}\left[40,\, 0.3 \zloce, 2 h_B \right] -\end{equation*} -and both $\lambda_m$ and $\lambda_h$ are not reduced (to 40m) above -the boundary layer top. It is possible these modifications point to -problems with the definition of the boundary layer depth and the use -of a Prandtl number of unity with no stability dependence in the local -scheme. Both these issues are under further investigation. - -\newpage -%------------------------------------------------------------------------ -% APPENDIX: UKCA INPUTS -%------------------------------------------------------------------------ -\section{Appendix: Inputs to UKCA} - -The UKCA chemistry and aerosols sub-model takes a number of boundary -layer diagnostics as input. For a list of these and a brief -explanation of how they are used see the table below. If any changes -modify the results for these variables it will prevent UKCA jobs from -regressing. If the changes are significant it would be prudent to -discuss them with the UKCA code owner before lodging the change. - -\begin{table}[htp] - \begin{center} - \begin{tabular}{|p{0.5cm}|p{0.7cm}|p{6cm}|p{6cm}|} - \hline - \multicolumn{4}{|c|}{Boundary layer inputs to UKCA} \\ - \hline - Sec & Item & Description & Use in UKCA \\ - \hline - 0 & 24 & SURFACE TEMPERATURE AFTER TIMESTEP & dry deposition \\ - 0 & 25 & BOUNDARY LAYER DEPTH AFTER TIMESTEP & dry deposition, bl nucleation and call to tr\_mix \\ - 0 & 26 & ROUGHNESS LENGTH AFTER TIMESTEP & dry deposition \\ - 0 & 233 & SURFACE TEMPERATURE ON TILES K & dry deposition \\ - 0 & 234 & ROUGHNESS LENGTH ON TILES m & dry deposition \\ - 3 & 60 & RHOKH\_MIX & call to tr\_mix \\ - 3 & 64 & DTRDZ\_CHARNEY\_GRID & call to tr\_mix \\ - 3 & 65 & GRID-LEVEL OF SML INVERSION (kent) & call to tr\_mix \\ - 3 & 66 & Rho * entrainment rate (we\_lim) & call to tr\_mix \\ - 3 & 67 & Fraction of the timestep (t\_frac) & call to tr\_mix \\ - 3 & 68 & zrzi & call to tr\_mix \\ - 3 & 69 & GRID-LEVEL OF DSC INVERSION (kent) & call to tr\_mix \\ - 3 & 70 & Rho * entrainment rate dsc & call to tr\_mix \\ - 3 & 71 & Fraction of the timestep dsc & call to tr\_mix \\ - 3 & 72 & zrzi dsc & call to tr\_mix \\ - 3 & 73 & ZHSC Top of decoupled layer & call to tr\_mix \\ - 3 & 217 & SURFACE HEAT FLUX W/M2 & dry deposition \\ - 3 & 230 & 10 METRE WIND SPEED ON C-GRID & calculate sea salt emissions \\ - 3 & 401 & Dust Emissions div 1 & GLOMAP dust scheme \\ - 3 & 402 & Dust Emissions div 2 & GLOMAP dust scheme \\ - 3 & 403 & Dust Emissions div 3 & GLOMAP dust scheme \\ - 3 & 404 & Dust Emissions div 4 & GLOMAP dust scheme \\ - 3 & 405 & Dust Emissions div 5 & GLOMAP dust scheme \\ - 3 & 406 & Dust Emissions div 6 & GLOMAP dust scheme \\ - 3 & 430 & Dust Friction velocity (U*) on tiles & dry deposition \\ - 3 & 462 & STOMATAL CONDUCTANCE ON PFTS (M/S) & dry deposition \\ - 3 & 465 & FRICTION VELOCITY & dry deposition \\ - 3 & 473 & TURBULENT KINETIC ENERGY & ACTIVATE cloud scheme \\ - \hline - \end{tabular} - \end{center} -\end{table} - -%------------------------------------------------------------------------ -% APPENDIX: NOTATION -%------------------------------------------------------------------------ -\section{Appendix: Notation} -\label{app:not} - -\begin{flushleft} - \begin{tabular}{|l|l|} - \hline - \multicolumn{2}{|c|}{Finite difference notation} \\ - \hline - $z_k$ & height of the $\theta$-level $k$ \\ - $z_{k+\frac{1}{2}}$ & height of half-level above $\theta$-level $k$ \\ - $\Delta_k$ & indicates a finite difference between $\theta$-levels $k$ and - $k-1$ \\ - $\Delta_{k+\frac{1}{2}}$ & indicates a finite difference between - half-levels $k+\frac{1}{2}$ and $k-\frac{1}{2}$ \\ - $\Delta$ & note: real change (\mbox{i.e.}, not necessarily - finite-difference) in a parameter \\ - & across the capping inversion (see (\ref{dbinv}) and following text) \\ - \hline - \end{tabular} -\end{flushleft} - -\begin{flushleft} - \begin{tabular}{|l|l|} - \hline - \multicolumn{2}{|c|}{Model variables} \\ - \hline - $\theta_l$, $\thetavl$ & thermodynamic variables defined by (\ref{thetal}) - and (\ref{thetavl}) \\ - $T_v$, $\theta_v$ & virtual temperature and potential temperature, \\ - & defined by (\ref{Tv}) and in section (\ref{sec:parxs}) \\ - $b$ & buoyancy ($=g T_v'/T_v$) \\ - $q_t$, $q_v$, $q_s$, $\ql$, $q_f$ & specific humidities: \\ - & total, vapour, saturated, liquid and frozen water, respectively \\ - $C_F$, $C_F^l$, $C_F^f$ & cloud fraction and the liquid and frozen water - parts, respectively \\ - ${\cal H}$ & total heat flux (net radiative plus turbulent, Kms$^{-1}$) \\ - \hline - \end{tabular} -\end{flushleft} - -\begin{flushleft} - \begin{tabular}{|l|l|} - \hline - \multicolumn{2}{|c|}{Thresholds} \\ - \hline - $C_t$ & ($=1.1$) threshold for ratio of layer $q_t$-gradients in cumulus - diagnosis \\ - $\Gamma_{\rm inv}$ & ($=1.1$) threshold on ratio of environment to parcel - $\theta_v$ gradients \\ - & for identifying capping inversions above the LCL \\ - SC\_CFTOL & ($=0.1$) $C_F$ threshold for recognising the presence of Sc \\ - $\Delta_{k_{ct}} \thetavl / \Delta_{k_{ct}} z < 10^{-3}$ & threshold - (in Km$^{-1}$) for initial diagnosis of {\em well-mixed} DSC layers \\ - $D_t$ & ($=0.1$) threshold for the ratio of buoyancy consumption to - production \\ - & before decoupling occurs \\ - \hline - \end{tabular} -\end{flushleft} - -\begin{flushleft} - \begin{tabular}{|l|l|} - \hline - \multicolumn{2}{|c|}{Layer definitions and parameters} \\ - \hline - SML & surface-based mixed layer \\ - NTML & top $\theta$-level within SML \\ - NTPAR & top $\theta$-level reached by parcel ascent \\ - DSC & decoupled stratocumulus (mixed layer) \\ - NTDSC & top $\theta$-level within DSC layer \\ - NBDSC & bottom $\theta$-level within DSC layer \\ - NTLOC & top $\theta$-level below which $Ri<1$ \\ - \zh & height of top of SML (potentially subgrid) \\ - \zhsc & height of top of DSC layer (potentially subgrid) \\ - \zbase & height of base of DSC layer (subgrid) \\ - \zhpar & height of half-level at top of parcel ascent \\ - \zloc & height of half-level marking `top' of local $Ri$-based mixing \\ - & (where $Ri>1$) \\ - $z_i$ & generic inversion height \\ - $z_c$ & cloud depth \\ - $\zml$ & mixed layer depth \\ - $\kmsurf$, $\khsurf$ & $K$ profiles for surface-driven turbulence (in SML) \\ - $\kmtop$, $\khtop$ & $K$ profiles for cloud-top-driven turbulence \\ - & (calculated for both DSC and SML) \\ - LCL & lifting condensation level \\ - \hline - \end{tabular} -\end{flushleft} - -\begin{flushleft} - \begin{tabular}{|l|l|} - \hline - \multicolumn{2}{|c|}{Other parameters} \\ - \hline - $\gamma_{\thetal}$ & gradient adjustment term, given by (\ref{gradadj}) \\ - $w_m$ & scaling velocity for momentum mixing in the SML \\ - & (used in $\kmsurf$, $\gamma_{\thetal}$ - and the SML parcel perturbation, $\theta_v'$) \\ - $w_*$ & `standard' convective velocity scale for a cloud-free convective \\ - & boundary layer, $ w_*^3 = \zhe \wbs $ \\ - $u_*$ & friction velocity (here includes the orographic component) \\ - $w_e$, $\tilde{w_e}$ & entrainment velocity and compensated to allow for - subsidence (ms$^{-1}$) \\ - $w_S$ & subsidence velocity (ms$^{-1}$) \\ - $\Delta_\radf $ & cloud-top net radiative divergence, calculation given in - (\ref{ctraddiv}) \\ - $\alpha_t$ & parameter in entrainment parametrization, (\ref{we_parm}) \\ - $\tau_{rc}$, $z_{rc}$ & parameters in perturbation calculation, - (\ref{dscd_pert}), \\ - & for initial identification of and $\zml$ calculation for DSC layers \\ - $a_L$, $\alpha_L$, $\beta_T$, $\beta_q$, $\tilde{\beta_T}$, $\tilde{\beta_q}$ - & buoyancy parameters, defined in appendix~\ref{app:buoyp} \\ - \hline - \end{tabular} -\end{flushleft} - -\newpage - -\bibliographystyle{plainnat} -\bibliography{refs} - -\end{document} % Every document must end with this. - - - - - - diff --git a/documentation/source/science_guide/turbulence_schemes/div_r071.eps b/documentation/source/science_guide/turbulence_schemes/div_r071.eps deleted file mode 100644 index 8e6f414f4c..0000000000 --- a/documentation/source/science_guide/turbulence_schemes/div_r071.eps +++ /dev/null @@ -1,410 +0,0 @@ -%!PS-Adobe-2.0 EPSF-2.0 -%%BoundingBox: 48 68 493 335 -%%HiResBoundingBox: 48.5 69 492 334 -%%Title: Graphics produced by WAVE -%%For: lock@visual -%%Creator: WAVE Version 7.00 (Linux i386) -%%CreationDate: Tue Aug 20 09:23:13 2002 -%%EndComments -% EPSF created by ps2eps 1.68 -%%BeginProlog -save -countdictstack -mark -newpath -/showpage {} def -/setpagedevice {pop} def -%%EndProlog -%%Page 1 1 -%+ wave_prolog.ps -- Prolog for PV-WAVE CL generated PostScript files -%+ $Id: wave_prolog.ps,v 1.5 1998/06/05 17:44:06 thaux Exp $ -%+ Copyright (c) 1989-1992 Precision Visuals, Inc. 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/Helvetica-Narrow-Bold RF /Helvetica-Narrow-Oblique RF -/Helvetica-Narrow-BoldOblique RF /AvantGarde-Book RF /AvantGarde-DemiOblique -RF /AvantGarde-Demi RF /AvantGarde-DemiOblique RF /Bookman-Demi RF -/Bookman-DemiItalic RF /Bookman-Light RF /Bookman-LightItalic RF -/ZapfChancery-MediumItalic RF /NewCenturySchlbk-Roman RF -/NewCenturySchlbk-Bold RF /NewCenturySchlbk-Italic RF -/NewCenturySchlbk-BoldItalic RF /Palatino-Roman RF /Palatino-Bold RF -/Palatino-Italic RF /Palatino-BoldItalic RF /Times-Roman RF /Times-Bold RF -/Times-Italic RF /Times-BoldItalic RF end -%%EndProlog -%%Page: 0 1 -%%BeginPageSetup -save $WAVE_DICT begin 28 56 L 0.028346 dup scale -%%PageBoundingBox: 28 56 509 339 -%%EndPageSetup -/psFontCache 500 array def /psStack 8 array def /psStackInd 0 def -/psTextWidth 0 def /psCurBase 0 def /ASW { stringwidth pop psCurBase add -/psCurBase exch def psCurBase psTextWidth gt -{ /psTextWidth psCurBase def } if } def /PSS { -psStack psStackInd psCurBase put /psStackInd psStackInd 1 add def } def -/PPS { /psStackInd psStackInd 1 sub def -/psCurBase psStack psStackInd get def } def 10.000000 SL L0 0.000 K -6903 9734 M 0 -306 R D 6918 9734 M 0 -306 R D 6859 9734 M 176 0 R 43 -14 R -15 -15 R 15 -29 R 0 -29 R -15 -30 R -15 -14 R -43 -15 R -117 0 R D -7035 9734 M 29 -14 R 14 -15 R 15 -29 R 0 -29 R -15 -30 R -14 -14 R -29 -15 R -D 6859 9428 M 103 0 R D 6991 9588 M 29 -14 R 15 -15 R 43 -102 R 15 -15 R -15 0 R 14 15 R D 7020 9574 M 15 -30 R 29 -102 R 14 -14 R 30 0 R 14 29 R -0 14 R D 7283 9734 M -44 -14 R -29 -44 R -15 -73 R 0 -44 R 15 -73 R 29 -44 R -44 -14 R 29 0 R 44 14 R 29 44 R 15 73 R 0 44 R -15 73 R -29 44 R -44 14 R --29 0 R -29 -14 R -15 -15 R -14 -29 R -15 -73 R 0 -44 R 15 -73 R 14 -29 R -15 -15 R 29 -14 R D 7312 9428 M 29 14 R 15 15 R 15 29 R 14 73 R 0 44 R --14 73 R -15 29 R -15 15 R -29 14 R D 7487 9734 M 0 -87 R D 7487 9676 M -15 29 R 29 29 R 29 0 R 73 -44 R 30 0 R 14 15 R 15 29 R D 7502 9705 M 29 15 R -29 0 R 73 -30 R D 7692 9734 M 0 -44 R -15 -43 R -58 -73 R -15 -30 R --14 -43 R 0 -73 R D 7677 9647 M -73 -73 R -14 -30 R -15 -43 R 0 -73 R D -7823 9676 M 30 14 R 43 44 R 0 -306 R D 7882 9720 M 0 -292 R D 7823 9428 M -132 0 R D 8320 9559 M 263 0 R D 8948 9734 M 0 -306 R D 8963 9734 M 0 -306 R -D 9050 9647 M 0 -117 R D 8904 9734 M 234 0 R 0 -87 R -15 87 R D 8963 9588 M -87 0 R D 8904 9428 M 234 0 R 0 87 R -15 -87 R D 9474 9793 M -263 -482 R D -9737 9690 M 14 44 R 0 -87 R -14 43 R -30 30 R -43 14 R -44 0 R -44 -14 R --29 -30 R 0 -29 R 14 -29 R 15 -15 R 29 -14 R 88 -29 R 29 -15 R 29 -29 R D -9547 9661 M 29 -29 R 29 -15 R 88 -29 R 29 -14 R 15 -15 R 14 -29 R 0 -59 R --29 -29 R -44 -14 R -44 0 R -43 14 R -30 29 R -14 44 R 0 -87 R 14 43 R D -10102 9734 M 0 -306 R D 10116 9734 M 0 -306 R D 10204 9647 M 0 -117 R D -10058 9734 M 234 0 R 0 -87 R -15 87 R D 10116 9588 M 88 0 R D 10058 9428 M -234 0 R 0 87 R -15 -87 R D 10408 9734 M 0 -219 R 15 -44 R 29 -29 R 44 -14 R -29 0 R 44 14 R 29 29 R 15 44 R 0 219 R D 10423 9734 M 0 -219 R 15 -44 R -29 -29 R 29 -14 R D 10365 9734 M 102 0 R D 10569 9734 M 88 0 R D -10759 9734 M 0 -306 R D 10774 9734 M 0 -306 R D 10715 9734 M 175 0 R -44 -14 R 15 -15 R 14 -29 R 0 -29 R -14 -30 R -15 -14 R -44 -15 R -116 0 R D -10890 9734 M 30 -14 R 14 -15 R 15 -29 R 0 -29 R -15 -30 R -14 -14 R --30 -15 R D 10715 9428 M 102 0 R D 10847 9588 M 29 -14 R 14 -15 R 44 -102 R -15 -15 R 14 0 R 15 15 R D 10876 9574 M 14 -30 R 30 -102 R 14 -14 R 29 0 R -15 29 R 0 14 R D 11153 9734 M -43 -14 R -30 -30 R -14 -29 R -15 -58 R -0 -44 R 15 -58 R 14 -30 R 30 -29 R 43 -14 R 30 0 R 43 14 R 30 29 R 14 30 R -15 58 R 0 44 R -15 58 R -14 29 R -30 30 R -43 14 R -30 0 R -29 -14 R --29 -30 R -15 -29 R -14 -58 R 0 -44 R 14 -58 R 15 -30 R 29 -29 R 29 -14 R D -11183 9428 M 29 14 R 29 29 R 15 30 R 14 58 R 0 44 R -14 58 R -15 29 R --29 30 R -29 14 R D 11577 9690 M 15 -43 R 0 87 R -15 -44 R -29 30 R -44 14 R --29 0 R -44 -14 R -29 -30 R -15 -29 R -15 -44 R 0 -73 R 15 -43 R 15 -30 R -29 -29 R 44 -14 R 29 0 R 44 14 R 29 29 R 15 30 R D 11475 9734 M -30 -14 R --29 -30 R -14 -29 R -15 -44 R 0 -73 R 15 -43 R 14 -30 R 29 -29 R 30 -14 R D -11869 9690 M 15 44 R 0 -87 R -15 43 R -29 30 R -44 14 R -44 0 R -44 -14 R --29 -30 R 0 -29 R 15 -29 R 14 -15 R 30 -14 R 87 -29 R 29 -15 R 30 -29 R D -11679 9661 M 29 -29 R 30 -15 R 87 -29 R 29 -14 R 15 -15 R 15 -29 R 0 -59 R --30 -29 R -43 -14 R -44 0 R -44 14 R -29 29 R -15 44 R 0 -87 R 15 43 R D -2220 1408 M 14114 0 R D 2220 1408 M 0 157 R D 2196 1195 M -35 -11 R --23 -35 R -12 -59 R 0 -35 R 12 -58 R 23 -35 R 35 -12 R 24 0 R 35 12 R -23 35 R 12 58 R 0 35 R -12 59 R -23 35 R -35 11 R -24 0 R -23 -11 R --12 -12 R -12 -23 R -11 -59 R 0 -35 R 11 -58 R 12 -24 R 12 -11 R 23 -12 R D -2220 950 M 23 12 R 12 11 R 11 24 R 12 58 R 0 35 R -12 59 R -11 23 R -12 12 R --23 11 R D 6371 1408 M 0 157 R D 6301 1195 M -24 -117 R 24 24 R 35 12 R -35 0 R 35 -12 R 23 -24 R 12 -35 R 0 -23 R -12 -35 R -23 -23 R -35 -12 R --35 0 R -35 12 R -12 11 R -12 24 R 0 11 R 12 12 R 12 -12 R -12 -11 R D -6371 1114 M 23 -12 R 24 -24 R 11 -35 R 0 -23 R -11 -35 R -24 -23 R -23 -12 R -D 6301 1195 M 117 0 R D 6301 1184 M 58 0 R 59 11 R D 10522 1408 M 0 157 R D -10347 1149 M 23 11 R 35 35 R 0 -245 R D 10393 1184 M 0 -234 R D 10347 950 M -105 0 R D 10615 1195 M -35 -11 R -23 -35 R -12 -59 R 0 -35 R 12 -58 R -23 -35 R 35 -12 R 24 0 R 35 12 R 23 35 R 12 58 R 0 35 R -12 59 R -23 35 R --35 11 R -24 0 R -23 -11 R -12 -12 R -11 -23 R -12 -59 R 0 -35 R 12 -58 R -11 -24 R 12 -11 R 23 -12 R D 10639 950 M 23 12 R 12 11 R 12 24 R 11 58 R -0 35 R -11 59 R -12 23 R -12 12 R -23 11 R D 14673 1408 M 0 157 R D -14498 1149 M 23 11 R 35 35 R 0 -245 R D 14545 1184 M 0 -234 R D 14498 950 M -105 0 R D 14720 1195 M -23 -117 R 23 24 R 35 12 R 35 0 R 35 -12 R 23 -24 R -12 -35 R 0 -23 R -12 -35 R -23 -23 R -35 -12 R -35 0 R -35 12 R -12 11 R --11 24 R 0 11 R 11 12 R 12 -12 R -12 -11 R D 14790 1114 M 23 -12 R 24 -24 R -11 -35 R 0 -23 R -11 -35 R -24 -23 R -23 -12 R D 14720 1195 M 117 0 R D -14720 1184 M 58 0 R 59 11 R D 3050 1408 M 0 78 R D 3880 1408 M 0 78 R D -4710 1408 M 0 78 R D 5541 1408 M 0 78 R D 7201 1408 M 0 78 R D 8031 1408 M -0 78 R D 8861 1408 M 0 78 R D 9692 1408 M 0 78 R D 11352 1408 M 0 78 R D -12182 1408 M 0 78 R D 13013 1408 M 0 78 R D 13843 1408 M 0 78 R D -15503 1408 M 0 78 R D 8032 738 M 0 -246 R D 8044 738 M 0 -246 R D 7962 738 M --12 -70 R 0 70 R 176 0 R 0 -70 R -12 70 R D 7997 492 M 82 0 R D 8207 738 M --11 -12 R 11 -12 R 12 12 R -12 12 R D 8207 656 M 0 -164 R D 8219 656 M -0 -164 R D 8172 656 M 47 0 R D 8172 492 M 82 0 R D 8336 656 M 0 -164 R D -8348 656 M 0 -164 R D 8348 621 M 23 23 R 35 12 R 23 0 R 36 -12 R 11 -23 R -0 -129 R D 8429 656 M 24 -12 R 12 -23 R 0 -129 R D 8476 621 M 24 23 R -35 12 R 23 0 R 35 -12 R 12 -23 R 0 -129 R D 8558 656 M 23 -12 R 12 -23 R -0 -129 R D 8301 656 M 47 0 R D 8301 492 M 82 0 R D 8429 492 M 82 0 R D -8558 492 M 82 0 R D 8710 586 M 140 0 R 0 23 R -12 24 R -11 11 R -24 12 R --35 0 R -35 -12 R -23 -23 R -12 -35 R 0 -24 R 12 -35 R 23 -23 R 35 -12 R -24 0 R 35 12 R 23 23 R D 8838 586 M 0 35 R -11 23 R D 8768 656 M -23 -12 R --23 -23 R -12 -35 R 0 -24 R 12 -35 R 23 -23 R 23 -12 R D 9131 738 M --12 -12 R 12 -12 R 11 12 R -11 12 R D 9131 656 M 0 -164 R D 9142 656 M -0 -164 R D 9095 656 M 47 0 R D 9095 492 M 82 0 R D 9259 656 M 0 -164 R D -9271 656 M 0 -164 R D 9271 621 M 23 23 R 35 12 R 24 0 R 35 -12 R 11 -23 R -0 -129 R D 9353 656 M 23 -12 R 12 -23 R 0 -129 R D 9224 656 M 47 0 R D -9224 492 M 82 0 R D 9353 492 M 81 0 R D 9703 738 M 0 -246 R D 9715 738 M -0 -246 R D 9715 621 M 23 23 R 35 12 R 24 0 R 35 -12 R 11 -23 R 0 -129 R D -9797 656 M 23 -12 R 12 -23 R 0 -129 R D 9668 738 M 47 0 R D 9668 492 M -82 0 R D 9797 492 M 81 0 R D 10007 656 M -35 -12 R -24 -23 R -11 -35 R -0 -24 R 11 -35 R 24 -23 R 35 -12 R 23 0 R 35 12 R 24 23 R 11 35 R 0 24 R --11 35 R -24 23 R -35 12 R -23 0 R -24 -12 R -23 -23 R -12 -35 R 0 -24 R -12 -35 R 23 -23 R 24 -12 R D 10030 492 M 24 12 R 23 23 R 12 35 R 0 24 R --12 35 R -23 23 R -24 12 R D 10194 656 M 0 -129 R 11 -23 R 36 -12 R 23 0 R -35 12 R 23 23 R D 10205 656 M 0 -129 R 12 -23 R 24 -12 R D 10322 656 M -0 -164 R D 10334 656 M 0 -164 R D 10159 656 M 46 0 R D 10287 656 M 47 0 R D -10322 492 M 47 0 R D 10451 656 M 0 -164 R D 10463 656 M 0 -164 R D -10463 586 M 11 35 R 24 23 R 23 12 R 35 0 R 12 -12 R 0 -11 R -12 -12 R --12 12 R 12 11 R D 10416 656 M 47 0 R D 10416 492 M 82 0 R D 10743 633 M -12 23 R 0 -47 R -12 24 R -12 11 R -23 12 R -47 0 R -23 -12 R -12 -11 R -0 -24 R 12 -11 R 23 -12 R 59 -24 R 23 -11 R 12 -12 R D 10626 621 M 12 -12 R -23 -11 R 59 -24 R 23 -12 R 12 -11 R 0 -35 R -12 -12 R -23 -12 R -47 0 R --24 12 R -11 12 R -12 23 R 0 -47 R 12 24 R D 2220 9296 M 14114 0 R D -2220 9296 M 0 -158 R D 6371 9296 M 0 -158 R D 10522 9296 M 0 -158 R D -14673 9296 M 0 -158 R D 3050 9296 M 0 -79 R D 3880 9296 M 0 -79 R D -4710 9296 M 0 -79 R D 5541 9296 M 0 -79 R D 7201 9296 M 0 -79 R D -8031 9296 M 0 -79 R D 8861 9296 M 0 -79 R D 9692 9296 M 0 -79 R D -11352 9296 M 0 -79 R D 12182 9296 M 0 -79 R D 13013 9296 M 0 -79 R D -13843 9296 M 0 -79 R D 15503 9296 M 0 -79 R D 2220 1408 M 0 7888 R D -2220 1408 M 282 0 R D 1968 1653 M -35 -12 R -23 -35 R -12 -58 R 0 -35 R -12 -59 R 23 -35 R 35 -11 R 24 0 R 35 11 R 23 35 R 12 59 R 0 35 R -12 58 R --23 35 R -35 12 R -24 0 R -23 -12 R -12 -11 R -11 -24 R -12 -58 R 0 -35 R -12 -59 R 11 -23 R 12 -12 R 23 -11 R D 1992 1408 M 23 11 R 12 12 R 11 23 R -12 59 R 0 35 R -12 58 R -11 24 R -12 11 R -23 12 R D 2220 2985 M 282 0 R D -1676 3043 M 12 -12 R -12 -11 R -11 11 R 0 12 R 11 23 R 12 12 R 35 12 R -47 0 R 35 -12 R 11 -12 R 12 -23 R 0 -23 R -12 -24 R -35 -23 R -58 -23 R --23 -12 R -24 -24 R -11 -35 R 0 -35 R D 1770 3090 M 23 -12 R 12 -12 R -11 -23 R 0 -23 R -11 -24 R -35 -23 R -47 -23 R D 1665 2868 M 11 11 R 24 0 R -58 -23 R 35 0 R 23 12 R 12 11 R D 1700 2879 M 58 -35 R 47 0 R 11 12 R -12 23 R 0 24 R D 1968 3090 M -35 -12 R -23 -35 R -12 -58 R 0 -35 R 12 -59 R -23 -35 R 35 -12 R 24 0 R 35 12 R 23 35 R 12 59 R 0 35 R -12 58 R -23 35 R --35 12 R -24 0 R -23 -12 R -12 -12 R -11 -23 R -12 -58 R 0 -35 R 12 -59 R -11 -23 R 12 -12 R 23 -12 R D 1992 2844 M 23 12 R 12 12 R 11 23 R 12 59 R -0 35 R -12 58 R -11 23 R -12 12 R -23 12 R D 2220 4563 M 282 0 R D -1770 4644 M 0 -222 R D 1781 4667 M 0 -245 R D 1781 4667 M -128 -175 R -187 0 R D 1735 4422 M 81 0 R D 1968 4667 M -35 -11 R -23 -35 R -12 -59 R -0 -35 R 12 -58 R 23 -35 R 35 -12 R 24 0 R 35 12 R 23 35 R 12 58 R 0 35 R --12 59 R -23 35 R -35 11 R -24 0 R -23 -11 R -12 -12 R -11 -23 R -12 -59 R -0 -35 R 12 -58 R 11 -24 R 12 -11 R 23 -12 R D 1992 4422 M 23 12 R 12 11 R -11 24 R 12 58 R 0 35 R -12 59 R -11 23 R -12 12 R -23 11 R D 2220 6140 M -282 0 R D 1805 6210 M -12 -12 R 12 -11 R 11 11 R 0 12 R -11 23 R -24 12 R --35 0 R -35 -12 R -23 -23 R -12 -23 R -11 -47 R 0 -70 R 11 -35 R 24 -24 R -35 -11 R 23 0 R 35 11 R 23 24 R 12 35 R 0 11 R -12 35 R -23 24 R -35 11 R --12 0 R -35 -11 R -23 -24 R -12 -35 R D 1746 6245 M -23 -12 R -23 -23 R --12 -23 R -12 -47 R 0 -70 R 12 -35 R 23 -24 R 24 -11 R D 1758 6000 M 23 11 R -24 24 R 11 35 R 0 11 R -11 35 R -24 24 R -23 11 R D 1968 6245 M -35 -12 R --23 -35 R -12 -58 R 0 -35 R 12 -59 R 23 -35 R 35 -11 R 24 0 R 35 11 R -23 35 R 12 59 R 0 35 R -12 58 R -23 35 R -35 12 R -24 0 R -23 -12 R --12 -11 R -11 -24 R -12 -58 R 0 -35 R 12 -59 R 11 -23 R 12 -12 R 23 -11 R D -1992 6000 M 23 11 R 12 12 R 11 23 R 12 59 R 0 35 R -12 58 R -11 24 R --12 11 R -23 12 R D 2220 7718 M 282 0 R D 1723 7823 M -35 -12 R -12 -24 R -0 -35 R 12 -23 R 35 -12 R 47 0 R 35 12 R 11 23 R 0 35 R -11 24 R -35 12 R --47 0 R -23 -12 R -12 -24 R 0 -35 R 12 -23 R 23 -12 R D 1770 7717 M 23 12 R -12 23 R 0 35 R -12 24 R -23 12 R D 1723 7717 M -35 -11 R -12 -12 R -11 -23 R -0 -47 R 11 -23 R 12 -12 R 35 -12 R 47 0 R 35 12 R 11 12 R 12 23 R 0 47 R --12 23 R -11 12 R -35 11 R D 1723 7717 M -23 -11 R -12 -12 R -12 -23 R -0 -47 R 12 -23 R 12 -12 R 23 -12 R D 1770 7577 M 23 12 R 12 12 R 11 23 R -0 47 R -11 23 R -12 12 R -23 11 R D 1968 7823 M -35 -12 R -23 -35 R --12 -59 R 0 -35 R 12 -58 R 23 -35 R 35 -12 R 24 0 R 35 12 R 23 35 R 12 58 R -0 35 R -12 59 R -23 35 R -35 12 R -24 0 R -23 -12 R -12 -12 R -11 -23 R --12 -59 R 0 -35 R 12 -58 R 11 -23 R 12 -12 R 23 -12 R D 1992 7577 M 23 12 R -12 12 R 11 23 R 12 58 R 0 35 R -12 59 R -11 23 R -12 12 R -23 12 R D -2220 9296 M 282 0 R D 1466 9213 M 23 11 R 35 35 R 0 -245 R D 1513 9248 M -0 -234 R D 1466 9014 M 105 0 R D 1735 9259 M -35 -11 R -24 -35 R -11 -59 R -0 -35 R 11 -58 R 24 -35 R 35 -12 R 23 0 R 35 12 R 23 35 R 12 58 R 0 35 R --12 59 R -23 35 R -35 11 R -23 0 R -24 -11 R -11 -12 R -12 -23 R -12 -59 R -0 -35 R 12 -58 R 12 -24 R 11 -11 R 24 -12 R D 1758 9014 M 23 12 R 12 11 R -12 24 R 11 58 R 0 35 R -11 59 R -12 23 R -12 12 R -23 11 R D 1968 9259 M --35 -11 R -23 -35 R -12 -59 R 0 -35 R 12 -58 R 23 -35 R 35 -12 R 24 0 R -35 12 R 23 35 R 12 58 R 0 35 R -12 59 R -23 35 R -35 11 R -24 0 R -23 -11 R --12 -12 R -11 -23 R -12 -59 R 0 -35 R 12 -58 R 11 -24 R 12 -11 R 23 -12 R D -1992 9014 M 23 12 R 12 11 R 11 24 R 12 58 R 0 35 R -12 59 R -11 23 R --12 12 R -23 11 R D 2220 1802 M 141 0 R D 2220 2196 M 141 0 R D 2220 2591 M -141 0 R D 2220 3380 M 141 0 R D 2220 3774 M 141 0 R D 2220 4168 M 141 0 R D -2220 4957 M 141 0 R D 2220 5352 M 141 0 R D 2220 5746 M 141 0 R D -2220 6535 M 141 0 R D 2220 6929 M 141 0 R D 2220 7324 M 141 0 R D -2220 8112 M 141 0 R D 2220 8507 M 141 0 R D 2220 8901 M 141 0 R D 829 3655 M -246 0 R D 829 3666 M 246 0 R D 829 3620 M 0 116 R 12 35 R 23 24 R 24 11 R -35 12 R 58 0 R 35 -12 R 23 -11 R 24 -24 R 12 -35 R 0 -116 R D 829 3736 M -12 24 R 23 23 R 24 12 R 35 11 R 58 0 R 35 -11 R 23 -12 R 24 -23 R 12 -24 R D -829 3912 M 12 -12 R 12 12 R -12 11 R -12 -11 R D 911 3912 M 164 0 R D -911 3923 M 164 0 R D 911 3877 M 0 46 R D 1075 3877 M 0 81 R D 911 4017 M -164 70 R D 911 4028 M 140 59 R D 911 4157 M 164 -70 R D 911 3993 M 0 71 R D -911 4110 M 0 70 R D 981 4239 M 0 140 R -23 0 R -24 -12 R -11 -11 R -12 -24 R -0 -35 R 12 -35 R 23 -23 R 35 -12 R 23 0 R 35 12 R 24 23 R 12 35 R 0 24 R --12 35 R -24 23 R D 981 4367 M -35 0 R -23 -11 R D 911 4297 M 12 -23 R -23 -24 R 35 -11 R 23 0 R 35 11 R 24 24 R 12 23 R D 911 4472 M 164 0 R D -911 4484 M 164 0 R D 981 4484 M -35 12 R -23 23 R -12 24 R 0 35 R 12 11 R -11 0 R 12 -11 R -12 -12 R -11 12 R D 911 4437 M 0 47 R D 1075 4437 M 0 82 R -D 911 4706 M 12 -23 R 11 -12 R 24 -12 R 23 0 R 23 12 R 12 12 R 12 23 R -0 24 R -12 23 R -12 12 R -23 11 R -23 0 R -24 -11 R -11 -12 R -12 -23 R -0 -24 R D 923 4683 M 23 -12 R 47 0 R 23 12 R D 1016 4753 M -23 12 R -47 0 R --23 -12 R D 934 4765 M -11 11 R -12 24 R 12 0 R 0 -24 R D 1004 4671 M -12 -12 R 23 -11 R 12 0 R 24 11 R 23 35 R 0 59 R 12 35 R 11 12 R D -1051 4648 M 12 11 R 12 35 R 0 59 R 23 35 R 23 12 R 12 0 R 23 -12 R 12 -35 R -0 -70 R -12 -35 R -23 -12 R -12 0 R -23 12 R -23 35 R D 981 4881 M 0 141 R --23 0 R -24 -12 R -11 -12 R -12 -23 R 0 -35 R 12 -35 R 23 -24 R 35 -11 R -23 0 R 35 11 R 24 24 R 12 35 R 0 23 R -12 35 R -24 24 R D 981 5010 M -35 0 R --23 -12 R D 911 4940 M 12 -24 R 23 -23 R 35 -12 R 23 0 R 35 12 R 24 23 R -12 24 R D 911 5115 M 164 0 R D 911 5127 M 164 0 R D 946 5127 M -23 23 R --12 35 R 0 24 R 12 35 R 23 11 R 129 0 R D 911 5209 M 12 23 R 23 12 R 129 0 R -D 911 5080 M 0 47 R D 1075 5080 M 0 82 R D 1075 5209 M 0 81 R D 946 5489 M -12 -12 R 11 12 R -11 12 R -12 0 R -23 -24 R -12 -23 R 0 -35 R 12 -35 R -23 -24 R 35 -11 R 23 0 R 35 11 R 24 24 R 12 35 R 0 23 R -12 35 R -24 24 R D -911 5419 M 12 -23 R 23 -24 R 35 -12 R 23 0 R 35 12 R 24 24 R 12 23 R D -981 5582 M 0 141 R -23 0 R -24 -12 R -11 -12 R -12 -23 R 0 -35 R 12 -35 R -23 -24 R 35 -11 R 23 0 R 35 11 R 24 24 R 12 35 R 0 23 R -12 35 R -24 24 R D -981 5711 M -35 0 R -23 -12 R D 911 5641 M 12 -23 R 23 -24 R 35 -12 R 23 0 R -35 12 R 24 24 R 12 23 R D 782 6073 M 24 -23 R 35 -24 R 47 -23 R 58 -12 R -47 0 R 58 12 R 59 23 R 35 24 R 23 23 R D 806 6050 M 47 -24 R 35 -11 R -58 -12 R 47 0 R 58 12 R 47 11 R 47 24 R D 829 6155 M 246 47 R D 829 6167 M -187 35 R D 829 6248 M 246 -46 R D 829 6248 M 246 47 R D 829 6260 M 187 35 R -D 829 6342 M 246 -47 R D 829 6120 M 0 82 R D 829 6307 M 0 70 R D 911 6447 M -164 0 R D 911 6459 M 164 0 R D 946 6459 M -23 23 R -12 35 R 0 24 R 12 35 R -23 11 R 129 0 R D 911 6541 M 12 23 R 23 12 R 129 0 R D 946 6587 M -23 24 R --12 35 R 0 23 R 12 35 R 23 12 R 129 0 R D 911 6669 M 12 23 R 23 12 R 129 0 R -D 911 6412 M 0 47 R D 1075 6412 M 0 82 R D 1075 6541 M 0 81 R D 1075 6669 M -0 82 R D 832 6808 M 0 130 R D 774 6996 M 7 7 R 7 -7 R -7 -8 R -7 0 R -14 8 R --8 7 R -7 22 R 0 29 R 7 21 R 8 7 R 14 8 R 14 0 R 15 -8 R 14 -21 R 15 -36 R -7 -15 R 15 -14 R 21 -8 R 22 0 R D 745 7054 M 7 14 R 8 7 R 14 7 R 14 0 R -15 -7 R 14 -21 R 15 -29 R D 882 6988 M -7 8 R 0 14 R 15 36 R 0 22 R -8 14 R --7 8 R D 875 7010 M 22 36 R 0 29 R -7 7 R -15 8 R -14 0 R D 782 7142 M -24 23 R 35 24 R 47 23 R 58 12 R 47 0 R 58 -12 R 59 -23 R 35 -24 R 23 -23 R D -806 7165 M 47 24 R 35 11 R 58 12 R 47 0 R 58 -12 R 47 -11 R 47 -24 R D -16334 1408 M 0 7888 R D 16334 1408 M -283 0 R D 16334 2985 M -283 0 R D -16334 4563 M -283 0 R D 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24 24 R 35 11 R 35 0 R 35 -11 R 23 -24 R 12 -35 R -0 -23 R -12 -35 R -23 -24 R -35 -11 R -35 0 R -35 11 R -12 12 R -12 23 R -0 12 R 12 12 R 12 -12 R -12 -12 R D 5551 7408 M 23 -11 R 24 -24 R 11 -35 R -0 -23 R -11 -35 R -24 -24 R -23 -11 R D 5481 7490 M 117 0 R D 5481 7478 M -58 0 R 59 12 R D 5691 7537 M 23 -24 R 24 -35 R 23 -46 R 12 -59 R 0 -47 R --12 -58 R -23 -58 R -24 -35 R -23 -24 R D 5714 7513 M 24 -46 R 12 -35 R -11 -59 R 0 -47 R -11 -58 R -12 -47 R -24 -46 R D -end restore -showpage -%%Trailer -restore -%%Pages: 1 -%%Trailer -cleartomark -countdictstack -exch sub { end } repeat -restore -%%EOF diff --git a/documentation/source/science_guide/turbulence_schemes/div_r080.eps b/documentation/source/science_guide/turbulence_schemes/div_r080.eps deleted file mode 100644 index e9f21baab6..0000000000 --- a/documentation/source/science_guide/turbulence_schemes/div_r080.eps +++ /dev/null @@ -1,402 +0,0 @@ -%!PS-Adobe-2.0 EPSF-2.0 -%%BoundingBox: 48 68 493 333 -%%HiResBoundingBox: 48.5 69 492 332.5 -%%Title: Graphics produced by WAVE -%%For: lock@visual -%%Creator: WAVE Version 7.00 (Linux i386) -%%CreationDate: Tue Aug 20 09:24:05 2002 -%%EndComments -% EPSF created by ps2eps 1.68 -%%BeginProlog -save -countdictstack -mark -newpath -/showpage {} def -/setpagedevice {pop} def -%%EndProlog -%%Page 1 1 -%+ wave_prolog.ps -- Prolog for PV-WAVE CL generated PostScript files -%+ $Id: wave_prolog.ps,v 1.5 1998/06/05 17:44:06 thaux Exp $ -%+ Copyright (c) 1989-1992 Precision Visuals, Inc. All Rights Reserved. -%v 1 -save /$WAVE_DICT 60 dict def $WAVE_DICT begin /bdef { bind def } bind def /C -{currentpoint newpath moveto} bdef /D {currentpoint stroke moveto} bdef /F -{closepath fill} bdef /K { setgray } bdef /L {translate} bdef /M {moveto} -bdef /N {rmoveto} bdef /O {show} bdef /P {lineto} bdef /R {rlineto} bdef /S -{gsave show grestore} bdef /Z {gsave currentpoint lineto 20 setlinewidth 1 -setlinecap stroke grestore} bdef /CC {concat} bdef /CD {currentmatrix def} -bdef /CP {currentpoint pop def} bdef /FM {findfont exch makefont} bdef /FS -{findfont exch scalefont} bdef /GS {get setfont} bdef /MD {matrix def} bdef -/MN {mul neg def} bdef /RS {roll put setfont} bdef /SL {setlinewidth} bdef -/SM {setmatrix} bdef /L0 {[] 0 setdash} bdef /L1 {[40 100] 0 setdash} bdef -/L2 {[200 200] 0 setdash} bdef /L3 {[200 100 50 100] 0 setdash} bdef /L4 -{[300 100 50 100 50 100 50 100] 0 setdash} bdef /L5 {[400 200] 0 setdash} -bdef /IO { gsave 0.0 setgray /val exch def moveto /str 20 string def val str -cvs show grestore } bdef /UL { gsave currentpoint /cury exch def /curx exch -def /curmtx matrix currentmatrix def matrix identmatrix setmatrix newpath 0 -0 moveto true charpath flattenpath pathbbox curmtx setmatrix /height exch -def /width exch def /ybase cury height .1 mul sub def pop pop newpath curx -ybase moveto width 0 rlineto closepath stroke grestore } bdef /BG { gsave -currentpoint /cury exch def /curx exch def /curmtx matrix currentmatrix def -matrix identmatrix setmatrix newpath 0 0 moveto true charpath flattenpath -pathbbox curmtx setmatrix /height exch def /width exch def pop pop newpath 0 -0 moveto (M) true charpath flattenpath pathbbox /xdiff exch .1 mul def -/ydiff exch .1 mul def pop pop newpath curx xdiff 2 div sub cury ydiff 2 div -sub moveto width xdiff add 0 rlineto 0 height ydiff add rlineto width neg -xdiff sub 0 rlineto closepath fill grestore } bdef /$T_DICT 20 dict def /T { -$T_DICT begin /align exch def /orien exch def /size exch def /thestring exch -def gsave moveto /chsize 1.0 def /xsize 0.0 def /SUPER 8#330 def /SUBS 8#331 -def /NORM 8#332 def /SCRIPTWID 0.7 def /OFFSET 0.6 def orien rotate size dup -scale /orien false def thestring { /charcode exch def charcode SUPER eq -charcode SUBS eq or { /chsize SCRIPTWID def /orien true def } { charcode -NORM eq { /chsize 1.0 def /orien true def } { ( ) dup 0 charcode put -stringwidth pop chsize mul xsize add /xsize exch def } ifelse } ifelse } -forall xsize align mul neg 0 rmoveto orien { /regularfont currentfont def -/fractionfont currentfont [ SCRIPTWID 0 0 SCRIPTWID 0 0 ] makefont def gsave -newpath 0 0 moveto (1) true charpath flattenpath pathbbox /height exch def -pop pop pop grestore } if /xsize 0 def thestring { /charcode exch def -charcode SUPER eq { 0 OFFSET height mul dup /xsize exch def rmoveto -fractionfont setfont } { charcode SUBS eq { 0 OFFSET height mul neg dup -/xsize exch def rmoveto fractionfont setfont } { charcode NORM eq { 0 xsize -neg rmoveto regularfont setfont } { ( ) dup 0 charcode put show } ifelse } -ifelse } ifelse } forall grestore end } bdef /IsChar { exch /CharStrings get -exch known } bdef /MapCh { 3 -1 roll /Encoding get 3 1 roll put } bdef -/MapDegree { dup 16#b0 exch /degree IsChar { /degree } { /ring } ifelse -MapCh } bdef /MapBB { dup 16#a6 exch /brokenbar IsChar { /brokenbar } { /bar -} ifelse MapCh } bdef /ReEncode { dup findfont begin currentdict dup length -dict begin { 1 index /FID ne {def} {pop pop} ifelse } forall /FontName exch -def dup length 0 ne {/Encoding Encoding 256 array copy def 0 exch { dup type -/nametype eq { Encoding 2 index 2 index put pop 1 add } { exch pop } ifelse -} forall } if pop currentdict dup end end /FontName get exch definefont dup -MapDegree MapBB } bdef /RF { ISOLatin1Encoding exch ReEncode } bdef /Courier -RF /Courier-Bold RF /Courier-Oblique RF /Courier-BoldOblique RF /Helvetica -RF /Helvetica-Bold RF /Helvetica-Oblique RF /Helvetica-BoldOblique RF -/Helvetica-Narrow RF /Helvetica-Narrow-Bold RF /Helvetica-Narrow-Oblique RF -/Helvetica-Narrow-BoldOblique RF /AvantGarde-Book RF /AvantGarde-DemiOblique -RF /AvantGarde-Demi RF /AvantGarde-DemiOblique RF /Bookman-Demi RF -/Bookman-DemiItalic RF /Bookman-Light RF /Bookman-LightItalic RF -/ZapfChancery-MediumItalic RF /NewCenturySchlbk-Roman RF -/NewCenturySchlbk-Bold RF /NewCenturySchlbk-Italic RF -/NewCenturySchlbk-BoldItalic RF /Palatino-Roman RF /Palatino-Bold RF -/Palatino-Italic RF /Palatino-BoldItalic RF /Times-Roman RF /Times-Bold RF -/Times-Italic RF /Times-BoldItalic RF end -%%EndProlog -%%Page: 0 1 -%%BeginPageSetup -save $WAVE_DICT begin 28 56 L 0.028346 dup scale -%%PageBoundingBox: 28 56 509 339 -%%EndPageSetup -/psFontCache 500 array def /psStack 8 array def /psStackInd 0 def -/psTextWidth 0 def /psCurBase 0 def /ASW { stringwidth pop psCurBase add -/psCurBase exch def psCurBase psTextWidth gt -{ /psTextWidth psCurBase def } if } def /PSS { -psStack psStackInd psCurBase put /psStackInd psStackInd 1 add def } def -/PPS { /psStackInd psStackInd 1 sub def -/psCurBase psStack psStackInd get def } def 10.000000 SL L0 0.000 K -7480 9734 M 0 -306 R D 7495 9734 M 0 -306 R D 7436 9734 M 176 0 R 43 -14 R -15 -15 R 15 -29 R 0 -29 R -15 -30 R -15 -14 R -43 -15 R -117 0 R D -7612 9734 M 29 -14 R 14 -15 R 15 -29 R 0 -29 R -15 -30 R -14 -14 R -29 -15 R -D 7436 9428 M 103 0 R D 7568 9588 M 29 -14 R 15 -15 R 43 -102 R 15 -15 R -15 0 R 14 15 R D 7597 9574 M 15 -30 R 29 -102 R 14 -14 R 30 0 R 14 29 R -0 14 R D 7860 9734 M -44 -14 R -29 -44 R -15 -73 R 0 -44 R 15 -73 R 29 -44 R -44 -14 R 29 0 R 44 14 R 29 44 R 15 73 R 0 44 R -15 73 R -29 44 R -44 14 R --29 0 R -29 -14 R -15 -15 R -15 -29 R -14 -73 R 0 -44 R 14 -73 R 15 -29 R -15 -15 R 29 -14 R D 7889 9428 M 29 14 R 15 15 R 15 29 R 14 73 R 0 44 R --14 73 R -15 29 R -15 15 R -29 14 R D 8137 9734 M -43 -14 R -15 -30 R -0 -43 R 15 -30 R 43 -14 R 59 0 R 44 14 R 14 30 R 0 43 R -14 30 R -44 14 R --59 0 R -29 -14 R -14 -30 R 0 -43 R 14 -30 R 29 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656 M 0 -164 R D 8172 656 M 47 0 R D -8172 492 M 82 0 R D 8336 656 M 0 -164 R D 8348 656 M 0 -164 R D 8348 621 M -23 23 R 35 12 R 23 0 R 36 -12 R 11 -23 R 0 -129 R D 8429 656 M 24 -12 R -12 -23 R 0 -129 R D 8476 621 M 24 23 R 35 12 R 23 0 R 35 -12 R 12 -23 R -0 -129 R D 8558 656 M 23 -12 R 12 -23 R 0 -129 R D 8301 656 M 47 0 R D -8301 492 M 82 0 R D 8429 492 M 82 0 R D 8558 492 M 82 0 R D 8710 586 M -140 0 R 0 23 R -12 24 R -11 11 R -24 12 R -35 0 R -35 -12 R -23 -23 R --12 -35 R 0 -24 R 12 -35 R 23 -23 R 35 -12 R 24 0 R 35 12 R 23 23 R D -8838 586 M 0 35 R -11 23 R D 8768 656 M -23 -12 R -23 -23 R -12 -35 R -0 -24 R 12 -35 R 23 -23 R 23 -12 R D 9131 738 M -12 -12 R 12 -12 R 11 12 R --11 12 R D 9131 656 M 0 -164 R D 9142 656 M 0 -164 R D 9095 656 M 47 0 R D -9095 492 M 82 0 R D 9259 656 M 0 -164 R D 9271 656 M 0 -164 R D 9271 621 M -23 23 R 35 12 R 24 0 R 35 -12 R 11 -23 R 0 -129 R D 9353 656 M 23 -12 R -12 -23 R 0 -129 R D 9224 656 M 47 0 R D 9224 492 M 82 0 R D 9353 492 M -81 0 R D 9703 738 M 0 -246 R D 9715 738 M 0 -246 R D 9715 621 M 23 23 R -35 12 R 24 0 R 35 -12 R 11 -23 R 0 -129 R D 9797 656 M 23 -12 R 12 -23 R -0 -129 R D 9668 738 M 47 0 R D 9668 492 M 82 0 R D 9797 492 M 81 0 R D -10007 656 M -35 -12 R -24 -23 R -11 -35 R 0 -24 R 11 -35 R 24 -23 R 35 -12 R -23 0 R 35 12 R 24 23 R 11 35 R 0 24 R -11 35 R -24 23 R -35 12 R -23 0 R --24 -12 R -23 -23 R -12 -35 R 0 -24 R 12 -35 R 23 -23 R 24 -12 R D -10030 492 M 24 12 R 23 23 R 12 35 R 0 24 R -12 35 R -23 23 R -24 12 R D -10194 656 M 0 -129 R 11 -23 R 36 -12 R 23 0 R 35 12 R 23 23 R D 10205 656 M -0 -129 R 12 -23 R 24 -12 R D 10322 656 M 0 -164 R D 10334 656 M 0 -164 R D -10159 656 M 46 0 R D 10287 656 M 47 0 R D 10322 492 M 47 0 R D 10451 656 M -0 -164 R D 10463 656 M 0 -164 R D 10463 586 M 11 35 R 24 23 R 23 12 R 35 0 R -12 -12 R 0 -11 R -12 -12 R -12 12 R 12 11 R D 10416 656 M 47 0 R D -10416 492 M 82 0 R D 10743 633 M 12 23 R 0 -47 R -12 24 R -12 11 R -23 12 R --47 0 R -23 -12 R -12 -11 R 0 -24 R 12 -11 R 23 -12 R 59 -24 R 23 -11 R -12 -12 R D 10626 621 M 12 -12 R 23 -11 R 59 -24 R 23 -12 R 12 -11 R 0 -35 R --12 -12 R -23 -12 R -47 0 R -24 12 R -11 12 R -12 23 R 0 -47 R 12 24 R D -2220 9296 M 14114 0 R D 2220 9296 M 0 -158 R D 7260 9296 M 0 -158 R D -12301 9296 M 0 -158 R D 3228 9296 M 0 -79 R D 4236 9296 M 0 -79 R D -5244 9296 M 0 -79 R D 6252 9296 M 0 -79 R D 8268 9296 M 0 -79 R D -9277 9296 M 0 -79 R D 10285 9296 M 0 -79 R D 11293 9296 M 0 -79 R D -13309 9296 M 0 -79 R D 14317 9296 M 0 -79 R D 15325 9296 M 0 -79 R D -2220 1408 M 0 7888 R D 2220 1408 M 282 0 R D 1968 1653 M -35 -12 R -23 -35 R --12 -58 R 0 -35 R 12 -59 R 23 -35 R 35 -11 R 24 0 R 35 11 R 23 35 R 12 59 R -0 35 R -12 58 R -23 35 R -35 12 R -24 0 R -23 -12 R -12 -11 R -11 -24 R --12 -58 R 0 -35 R 12 -59 R 11 -23 R 12 -12 R 23 -11 R D 1992 1408 M 23 11 R -12 12 R 11 23 R 12 59 R 0 35 R -12 58 R -11 24 R -12 11 R -23 12 R D -2220 2985 M 282 0 R D 1676 3043 M 12 -12 R -12 -11 R -11 11 R 0 12 R 11 23 R -12 12 R 35 12 R 47 0 R 35 -12 R 11 -12 R 12 -23 R 0 -23 R -12 -24 R --35 -23 R -58 -23 R -23 -12 R -24 -24 R -11 -35 R 0 -35 R D 1770 3090 M -23 -12 R 12 -12 R 11 -23 R 0 -23 R -11 -24 R -35 -23 R -47 -23 R D -1665 2868 M 11 11 R 24 0 R 58 -23 R 35 0 R 23 12 R 12 11 R D 1700 2879 M -58 -35 R 47 0 R 11 12 R 12 23 R 0 24 R D 1968 3090 M -35 -12 R -23 -35 R --12 -58 R 0 -35 R 12 -59 R 23 -35 R 35 -12 R 24 0 R 35 12 R 23 35 R 12 59 R -0 35 R -12 58 R -23 35 R -35 12 R -24 0 R -23 -12 R -12 -12 R -11 -23 R --12 -58 R 0 -35 R 12 -59 R 11 -23 R 12 -12 R 23 -12 R D 1992 2844 M 23 12 R -12 12 R 11 23 R 12 59 R 0 35 R -12 58 R -11 23 R -12 12 R -23 12 R D -2220 4563 M 282 0 R D 1770 4644 M 0 -222 R D 1781 4667 M 0 -245 R D -1781 4667 M -128 -175 R 187 0 R D 1735 4422 M 81 0 R D 1968 4667 M -35 -11 R --23 -35 R -12 -59 R 0 -35 R 12 -58 R 23 -35 R 35 -12 R 24 0 R 35 12 R -23 35 R 12 58 R 0 35 R -12 59 R -23 35 R -35 11 R -24 0 R -23 -11 R --12 -12 R -11 -23 R -12 -59 R 0 -35 R 12 -58 R 11 -24 R 12 -11 R 23 -12 R D -1992 4422 M 23 12 R 12 11 R 11 24 R 12 58 R 0 35 R -12 59 R -11 23 R --12 12 R -23 11 R D 2220 6140 M 282 0 R D 1805 6210 M -12 -12 R 12 -11 R -11 11 R 0 12 R -11 23 R -24 12 R -35 0 R -35 -12 R -23 -23 R -12 -23 R --11 -47 R 0 -70 R 11 -35 R 24 -24 R 35 -11 R 23 0 R 35 11 R 23 24 R 12 35 R -0 11 R -12 35 R -23 24 R -35 11 R -12 0 R -35 -11 R -23 -24 R -12 -35 R D -1746 6245 M -23 -12 R -23 -23 R -12 -23 R -12 -47 R 0 -70 R 12 -35 R -23 -24 R 24 -11 R D 1758 6000 M 23 11 R 24 24 R 11 35 R 0 11 R -11 35 R --24 24 R -23 11 R D 1968 6245 M -35 -12 R -23 -35 R -12 -58 R 0 -35 R -12 -59 R 23 -35 R 35 -11 R 24 0 R 35 11 R 23 35 R 12 59 R 0 35 R -12 58 R --23 35 R -35 12 R -24 0 R -23 -12 R -12 -11 R -11 -24 R -12 -58 R 0 -35 R -12 -59 R 11 -23 R 12 -12 R 23 -11 R D 1992 6000 M 23 11 R 12 12 R 11 23 R -12 59 R 0 35 R -12 58 R -11 24 R -12 11 R -23 12 R D 2220 7718 M 282 0 R D -1723 7823 M -35 -12 R -12 -24 R 0 -35 R 12 -23 R 35 -12 R 47 0 R 35 12 R -11 23 R 0 35 R -11 24 R -35 12 R -47 0 R -23 -12 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12 0 R 0 -24 R D 1004 4671 M 12 -12 R 23 -11 R -12 0 R 24 11 R 23 35 R 0 59 R 12 35 R 11 12 R D 1051 4648 M 12 11 R 12 35 R -0 59 R 23 35 R 23 12 R 12 0 R 23 -12 R 12 -35 R 0 -70 R -12 -35 R -23 -12 R --12 0 R -23 12 R -23 35 R D 981 4881 M 0 141 R -23 0 R -24 -12 R -11 -12 R --12 -23 R 0 -35 R 12 -35 R 23 -24 R 35 -11 R 23 0 R 35 11 R 24 24 R 12 35 R -0 23 R -12 35 R -24 24 R D 981 5010 M -35 0 R -23 -12 R D 911 4940 M -12 -24 R 23 -23 R 35 -12 R 23 0 R 35 12 R 24 23 R 12 24 R D 911 5115 M -164 0 R D 911 5127 M 164 0 R D 946 5127 M -23 23 R -12 35 R 0 24 R 12 35 R -23 11 R 129 0 R D 911 5209 M 12 23 R 23 12 R 129 0 R D 911 5080 M 0 47 R D -1075 5080 M 0 82 R D 1075 5209 M 0 81 R D 946 5489 M 12 -12 R 11 12 R --11 12 R -12 0 R -23 -24 R -12 -23 R 0 -35 R 12 -35 R 23 -24 R 35 -11 R -23 0 R 35 11 R 24 24 R 12 35 R 0 23 R -12 35 R -24 24 R D 911 5419 M -12 -23 R 23 -24 R 35 -12 R 23 0 R 35 12 R 24 24 R 12 23 R D 981 5582 M -0 141 R -23 0 R -24 -12 R -11 -12 R -12 -23 R 0 -35 R 12 -35 R 23 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1008 -15 R 1008 22 R -1008 -8 R 1008 13 R 1009 0 R D L2 2220 1408 M 1008 0 R 1008 391 R 1008 975 R -1008 1236 R 1008 1295 R 1008 1120 R 1009 711 R 1008 155 R 1008 -379 R -1008 -702 R 1008 -754 R 1008 -852 R 1008 -1017 R 1009 -1045 R D L3 -2220 6894 M 1008 18 R 1008 -392 R 1008 -975 R 1008 -1239 R 1008 -1286 R -1008 -1133 R 674 -479 R D 11328 1408 M 973 664 R 1008 776 R 1008 843 R -1008 1031 R 1009 1044 R D L0 2220 6894 M 1008 17 R 1008 -191 R 1008 -370 R -1008 -383 R 1008 -301 R 1008 -182 R 1009 -308 R 1008 -207 R 1008 -77 R -1008 21 R 1008 153 R 1008 273 R 1008 402 R 1009 495 R D L4 2220 6894 M -1008 18 R 1008 -137 R 1008 -341 R 1008 -436 R 1008 -444 R 1008 -406 R -1009 -254 R 1008 -49 R 1008 117 R 1008 231 R 1008 286 R 1008 290 R -1008 370 R 1009 365 R D L0 3161 8901 M 940 0 R D 4242 9068 M 0 -246 R D -4254 9068 M 140 -222 R D 4254 9044 M 140 -222 R 0 246 R D 4207 9068 M 47 0 R -D 4359 9068 M 70 0 R D 4207 8822 M 70 0 R D 4511 9068 M 0 -246 R D -4523 9068 M 0 -246 R D 4593 8998 M 0 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82 0 R D 5364 7884 M 70 0 R -D 5598 7849 M 0 -210 R D 5492 7744 M 211 0 R D 5936 7849 M 12 35 R 0 -70 R --12 35 R -23 24 R -35 11 R -35 0 R -35 -11 R -23 -24 R 0 -23 R 11 -23 R -12 -12 R 23 -12 R 70 -23 R 24 -12 R 23 -23 R D 5785 7826 M 23 -23 R 23 -12 R -70 -23 R 24 -12 R 11 -12 R 12 -23 R 0 -47 R -23 -23 R -35 -12 R -35 0 R --35 12 R -24 23 R -11 35 R 0 -70 R 11 35 R D 6030 7884 M 47 -245 R D -6042 7884 M 35 -187 R D 6123 7884 M -46 -245 R D 6123 7884 M 47 -245 R D -6135 7884 M 35 -187 R D 6217 7884 M -47 -245 R D 5995 7884 M 82 0 R D -6182 7884 M 70 0 R D L4 3161 7324 M 940 0 R D L0 4359 7455 M -12 -12 R -12 -11 R 12 11 R 0 12 R -12 23 R -23 12 R -35 0 R -35 -12 R -24 -23 R --11 -23 R -12 -47 R 0 -70 R 12 -35 R 23 -24 R 35 -11 R 23 0 R 35 11 R -24 24 R 11 35 R 0 11 R -11 36 R -24 23 R -35 12 R -11 0 R -35 -12 R --24 -23 R -11 -36 R D 4301 7490 M -24 -12 R -23 -23 R -12 -23 R -11 -47 R -0 -70 R 11 -35 R 24 -24 R 23 -11 R D 4312 7245 M 24 11 R 23 24 R 12 35 R -0 11 R -12 36 R -23 23 R 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12 R 0 11 R 11 24 R 12 11 R 35 12 R 47 0 R 35 -12 R 12 -23 R -0 -35 R -12 -23 R -35 -12 R -35 0 R D 5329 7490 M 23 -12 R 12 -23 R 0 -35 R --12 -23 R -23 -12 R 23 -12 R 24 -23 R 11 -24 R 0 -35 R -11 -23 R -12 -12 R --35 -11 R -47 0 R -35 11 R -12 12 R -11 23 R 0 12 R 11 12 R 12 -12 R --12 -12 R D 5364 7362 M 12 -36 R 0 -35 R -12 -23 R -12 -12 R -23 -11 R D -5481 7490 M -24 -117 R 24 24 R 35 11 R 35 0 R 35 -11 R 23 -24 R 12 -35 R -0 -23 R -12 -35 R -23 -24 R -35 -11 R -35 0 R -35 11 R -12 12 R -12 23 R -0 12 R 12 12 R 12 -12 R -12 -12 R D 5551 7408 M 23 -11 R 24 -24 R 11 -35 R -0 -23 R -11 -35 R -24 -24 R -23 -11 R D 5481 7490 M 117 0 R D 5481 7478 M -58 0 R 59 12 R D 5691 7537 M 23 -24 R 24 -35 R 23 -46 R 12 -59 R 0 -47 R --12 -58 R -23 -58 R -24 -35 R -23 -24 R D 5714 7513 M 24 -46 R 12 -35 R -11 -59 R 0 -47 R -11 -58 R -12 -47 R -24 -46 R D -end restore -showpage -%%Trailer -restore -%%Pages: 1 -%%Trailer -cleartomark -countdictstack -exch sub { end } repeat -restore -%%EOF diff --git a/documentation/source/science_guide/turbulence_schemes/honnert_vs_tanh.eps b/documentation/source/science_guide/turbulence_schemes/honnert_vs_tanh.eps deleted file mode 100644 index 1066d61d43..0000000000 --- a/documentation/source/science_guide/turbulence_schemes/honnert_vs_tanh.eps +++ /dev/null @@ -1,258 +0,0 @@ -%!PS-Adobe-3.0 -%%BoundingBox: 45 68 611 634 -%%Title: Graphics produced by IDL -%%For: frib@eld638, /net/home/h04/frib/idl -%%Creator: IDL Version 8.2 (linux x86_64 m64) -%%CreationDate: Thu Feb 5 14:07:43 2015 -%%DocumentData: Clean7bit -%%Requirements: color -%%LanguageLevel: 1 -%%PageOrder: Ascend -%%Pages: (atend) -%%DocumentNeededResources: (atend) -%%EndComments -%%BeginProlog -save -%+ prolog.ps -- Prolog for IDL generated PostScript files -%+ Copyright (c) 1988-2012 Exelis Visual Information Solutions, Inc. All Rights Reserved. -%v 5 -/$IDL_DICT 40 dict def $IDL_DICT begin /bdef { bind def } bind def /C -{currentpoint newpath moveto} bdef /CP {currentpoint} bdef /D {currentpoint -stroke moveto} bdef /F {closepath fill} bdef /K { setgray } bdef /M {moveto} -bdef /N {rmoveto} bdef /P {lineto} bdef /R {rlineto} bdef /S {gsave show -grestore} bdef /X {currentpoint pop} bdef /Z {gsave currentpoint lineto 20 -setlinewidth 1 setlinecap stroke grestore} bdef /L0 {[] 0 setdash} bdef /L1 -{[40 100] 0 setdash} bdef /L2 {[200 200] 0 setdash} bdef /L3 {[200 100 50 -100] 0 setdash} bdef /L4 {[300 100 50 100 50 100 50 100] 0 setdash} bdef /L5 -{[400 200] 0 setdash} bdef /STDFONT { findfont exch scalefont setfont } bdef -/ISOFONT { findfont dup length dict begin { 1 index /FID ne {def} {pop pop} -ifelse } forall /Encoding ISOLatin1Encoding def currentdict end /idltmpfont -exch definefont exch scalefont setfont } bdef /ISOBULLET { gsave /Helvetica -findfont exch scalefont setfont (\267) show currentpoint 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M 0 168 R D 11531 2112 M 0 168 R D -11879 2112 M 0 168 R D 14241 2112 M 0 168 R D 15440 2112 M 0 168 R D -16291 2112 M 0 168 R D 16951 2112 M 0 168 R D 17490 2112 M 0 168 R D -17946 2112 M 0 168 R D 18341 2112 M 0 168 R D 18689 2112 M 0 168 R D -gsave 11166 634 translate 0 0 M 1.5 dup scale -626.165 0 N -%%IncludeResource: font Symbol -423.333 /Symbol STDFONT -(D) show -%%IncludeResource: font Helvetica -423.333 /Helvetica STDFONT -(x/z) show X -101.051 M -%%IncludeResource: font Helvetica -262.467 /Helvetica STDFONT -(turb) show grestore -%%IncludeResource: font Helvetica -423.333 /Helvetica STDFONT 3330 18944 M 15671 0 R D 5380 18944 M 0 -337 R D -12191 18944 M 0 -337 R D 19001 18944 M 0 -337 R D 3330 18944 M 0 -168 R D -3869 18944 M 0 -168 R D 4325 18944 M 0 -168 R D 4720 18944 M 0 -168 R D -5069 18944 M 0 -168 R D 7430 18944 M 0 -168 R D 8630 18944 M 0 -168 R D -9481 18944 M 0 -168 R D 10141 18944 M 0 -168 R D 10680 18944 M 0 -168 R D -11136 18944 M 0 -168 R D 11531 18944 M 0 -168 R D 11879 18944 M 0 -168 R D -14241 18944 M 0 -168 R D 15440 18944 M 0 -168 R D 16291 18944 M 0 -168 R D -16951 18944 M 0 -168 R D 17490 18944 M 0 -168 R D 17946 18944 M 0 -168 R D -18341 18944 M 0 -168 R D 18689 18944 M 0 -168 R D 3330 2112 M 0 16832 R D -3330 2112 M 314 0 R D gsave 3164 2112 translate 0 0 M 1.5 dup scale --588.433 0 N -(0.0) show grestore 3330 5479 M 314 0 R D gsave 3164 5267 translate 0 0 M -1.5 dup scale -588.433 0 N -(0.2) show grestore 3330 8845 M 314 0 R D gsave 3164 8634 translate 0 0 M -1.5 dup scale -588.433 0 N -(0.4) show grestore 3330 12211 M 314 0 R D gsave 3164 12000 translate 0 0 M -1.5 dup scale -588.433 0 N -(0.6) show grestore 3330 15578 M 314 0 R D gsave 3164 15367 translate 0 0 M -1.5 dup scale -588.433 0 N -(0.8) show grestore 3330 18944 M 314 0 R D gsave 3164 18575 translate 0 0 M -1.5 dup scale -588.433 0 N -(1.0) show grestore 3330 2954 M 157 0 R D 3330 3795 M 157 0 R D 3330 4637 M -157 0 R D 3330 6320 M 157 0 R D 3330 7162 M 157 0 R D 3330 8003 M 157 0 R D -3330 9687 M 157 0 R D 3330 10528 M 157 0 R D 3330 11370 M 157 0 R D -3330 13053 M 157 0 R D 3330 13895 M 157 0 R D 3330 14736 M 157 0 R D -3330 16419 M 157 0 R D 3330 17261 M 157 0 R D 3330 18103 M 157 0 R D -gsave 1782 10528 translate 0 0 M 90 rotate 1.5 dup scale -367.53 0 N -(W) show X -101.051 M -%%IncludeResource: font Helvetica -262.467 /Helvetica STDFONT -(1D) show grestore -%%IncludeResource: font Helvetica -423.333 /Helvetica STDFONT 19001 2112 M 0 16832 R D 19001 2112 M -313 0 R D -19001 5479 M -313 0 R D 19001 8845 M -313 0 R D 19001 12211 M -313 0 R D -19001 15578 M -313 0 R D 19001 18944 M -313 0 R D 19001 2954 M -157 0 R D -19001 3795 M -157 0 R D 19001 4637 M -157 0 R D 19001 6320 M -157 0 R D -19001 7162 M -157 0 R D 19001 8003 M -157 0 R D 19001 9687 M -157 0 R D -19001 10528 M -157 0 R D 19001 11370 M -157 0 R D 19001 13053 M -157 0 R D -19001 13895 M -157 0 R D 19001 14736 M -157 0 R D 19001 16419 M -157 0 R D -19001 17261 M -157 0 R D 19001 18103 M 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33 R 51 32 R 51 32 R -49 32 R 49 33 R 48 32 R 48 32 R 46 32 R 46 32 R 45 32 R 45 32 R 44 32 R -43 32 R 42 32 R 42 32 R 42 32 R 40 32 R 41 32 R 39 32 R 40 32 R 38 32 R -38 32 R 38 32 R 37 32 R 37 32 R 36 31 R 36 32 R 36 32 R 35 32 R 34 32 R -34 31 R 34 32 R 34 32 R 33 32 R 32 31 R 33 32 R 32 32 R 31 31 R 32 32 R -31 31 R 30 32 R 30 31 R 31 32 R 29 31 R 282 314 R 258 311 R 236 308 R -219 304 R 205 299 R 190 295 R 180 289 R 169 284 R 160 277 R 151 272 R -145 264 R 137 259 R 132 251 R 126 245 R 120 238 R 116 231 R 112 225 R -108 217 R 103 212 R 101 205 R 97 198 R 94 192 R 91 187 R 88 180 R 86 175 R -83 169 R 81 163 R 79 159 R 77 153 R 75 148 R 73 144 R 71 139 R 69 134 R -68 130 R 67 126 R 65 122 R 64 119 R 62 114 R 61 111 R 60 107 R 58 104 R -58 101 R 56 97 R 55 95 R 54 92 R 54 89 R 52 86 R 52 84 R 50 81 R 50 79 R -49 77 R 48 74 R 47 72 R 47 70 R 46 68 R 45 66 R 44 65 R 44 62 R 43 61 R -43 59 R 42 58 R 41 56 R 41 54 R 40 53 R 40 52 R 39 50 R 39 49 R 38 47 R -37 47 R 38 45 R 36 44 R 37 43 R 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32 24 R 32 24 R 31 24 R 31 23 R -31 22 R 30 22 R 30 21 R 30 20 R 282 180 R 257 141 R 237 112 R 219 91 R -204 73 R 191 60 R 179 50 R 169 41 R 160 35 R 152 30 R 144 25 R 138 21 R -131 18 R 126 16 R 121 14 R 116 12 R 111 10 R 108 9 R 104 8 R 100 7 R 97 6 R -94 5 R 91 5 R 88 4 R 86 4 R 83 4 R 81 3 R 79 2 R 77 3 R 75 2 R 73 2 R 71 1 R -70 2 R 68 1 R 66 2 R 65 1 R 64 1 R 62 1 R 61 1 R 60 0 R 59 1 R 57 1 R 56 0 R -56 1 R 54 0 R 53 0 R 53 1 R 51 0 R 51 0 R 49 1 R 49 0 R 48 0 R 48 0 R 46 0 R -46 1 R 45 0 R 45 0 R 43 0 R 44 0 R 42 0 R 42 0 R 41 0 R 41 0 R 41 0 R 39 1 R -39 0 R 39 0 R 38 0 R 38 0 R 37 0 R 37 0 R 36 0 R 36 0 R 35 0 R 35 0 R 35 0 R -34 0 R 34 0 R 33 0 R 33 0 R 33 0 R 32 0 R 32 0 R 32 0 R 31 0 R 31 0 R 31 0 R -30 0 R 30 0 R 30 0 R D 3330 2688 M 59 19 R 57 19 R 56 20 R 56 19 R 54 21 R -53 20 R 53 21 R 51 21 R 51 21 R 49 22 R 49 22 R 48 22 R 48 23 R 46 22 R -46 23 R 45 23 R 45 24 R 44 24 R 43 23 R 42 25 R 42 24 R 42 24 R 40 25 R -41 25 R 39 25 R 40 25 R 38 26 R 38 25 R 38 26 R 37 26 R 37 26 R 36 26 R -36 27 R 36 26 R 35 27 R 34 27 R 34 27 R 34 27 R 34 27 R 33 27 R 32 28 R -33 27 R 32 28 R 31 28 R 32 27 R 31 28 R 30 28 R 30 29 R 31 28 R 29 28 R -282 287 R 258 293 R 236 296 R 219 298 R 205 298 R 190 296 R 180 294 R -169 291 R 160 286 R 151 283 R 145 277 R 137 271 R 132 266 R 126 259 R -120 254 R 116 247 R 112 241 R 108 234 R 103 228 R 101 222 R 97 216 R -94 209 R 91 204 R 88 197 R 86 192 R 83 186 R 81 180 R 79 175 R 77 170 R -75 165 R 73 160 R 71 155 R 69 150 R 68 146 R 67 142 R 65 137 R 64 134 R -62 129 R 61 126 R 60 122 R 58 118 R 58 115 R 56 112 R 55 109 R 54 105 R -54 103 R 52 99 R 52 97 R 50 94 R 50 92 R 49 89 R 48 87 R 47 84 R 47 83 R -46 80 R 45 78 R 44 75 R 44 74 R 43 72 R 43 71 R 42 68 R 41 67 R 41 65 R -40 64 R 40 62 R 39 60 R 39 59 R 38 58 R 37 56 R 38 55 R 36 54 R 37 52 R -36 51 R 35 51 R 35 49 R 35 47 R 34 47 R 34 46 R 33 45 R 33 44 R 33 43 R -32 42 R 32 41 R 32 40 R 31 39 R 31 39 R 31 38 R 30 37 R 30 36 R 30 36 R -282 320 R 257 266 R 237 224 R 219 190 R 204 164 R 191 142 R 179 124 R -169 109 R 160 97 R 152 86 R 144 78 R 138 69 R 131 63 R 126 58 R 121 52 R -116 48 R 111 44 R 108 40 R 104 38 R 100 34 R 97 32 R 94 30 R 91 28 R 88 26 R -86 24 R 83 23 R 81 21 R 79 20 R 77 19 R 75 18 R 73 17 R 71 16 R 70 15 R -68 14 R 66 14 R 65 13 R 64 12 R 62 12 R 61 11 R 60 10 R 59 10 R 57 10 R -56 9 R 56 9 R 54 9 R 53 8 R 53 8 R 51 7 R 51 7 R 49 7 R 49 7 R 48 6 R 48 6 R -46 6 R 46 6 R 45 5 R 45 6 R 43 5 R 44 5 R 42 5 R 42 4 R 41 5 R 41 4 R 41 4 R -39 4 R 39 4 R 39 4 R 38 4 R 38 3 R 37 4 R 37 3 R 36 3 R 36 3 R 35 3 R 35 3 R -35 3 R 34 3 R 34 3 R 33 2 R 33 3 R 33 3 R 32 2 R 32 2 R 32 3 R 31 2 R 31 2 R -31 2 R 30 3 R 30 2 R 30 2 R D 40 setlinewidth L0 -0.000 0.000 1.000 setrgbcolor 3330 2195 M 59 11 R 57 11 R 56 12 R 56 13 R -54 13 R 53 15 R 53 16 R 51 16 R 51 17 R 49 18 R 49 20 R 48 20 R 48 20 R -46 22 R 46 23 R 45 24 R 45 24 R 44 26 R 43 26 R 42 27 R 42 28 R 42 29 R -40 30 R 41 30 R 39 32 R 40 32 R 38 33 R 38 33 R 38 35 R 37 35 R 37 35 R -36 37 R 36 37 R 36 38 R 35 38 R 34 39 R 34 40 R 34 40 R 34 40 R 33 42 R -32 41 R 33 43 R 32 42 R 31 43 R 32 44 R 31 44 R 30 44 R 30 45 R 31 45 R -29 46 R 282 470 R 258 487 R 236 492 R 219 489 R 205 478 R 190 463 R -180 446 R 169 426 R 160 407 R 151 386 R 145 367 R 137 348 R 132 329 R -126 312 R 120 295 R 116 312 R 112 295 R 108 280 R 103 265 R 101 251 R -97 239 R 94 227 R 91 215 R 88 206 R 86 195 R 83 187 R 81 178 R 79 169 R -77 163 R 75 155 R 73 149 R 71 142 R 69 137 R 68 131 R 67 126 R 65 121 R -64 116 R 62 112 R 61 108 R 60 103 R 58 101 R 58 96 R 56 93 R 55 90 R 54 87 R -54 84 R 52 81 R 52 79 R 50 76 R 50 73 R 49 72 R 48 69 R 47 67 R 47 65 R -46 63 R 45 62 R 44 60 R 44 58 R 43 56 R 43 55 R 42 53 R 41 52 R 41 51 R -40 49 R 40 48 R 39 47 R 39 45 R 38 45 R 37 43 R 38 42 R 36 42 R 37 40 R -36 40 R 35 38 R 35 38 R 35 36 R 34 36 R 34 35 R 33 35 R 33 33 R 33 33 R -32 33 R 32 31 R 32 31 R 31 30 R 31 30 R 31 29 R 30 28 R 30 28 R 30 28 R -282 246 R 257 206 R 237 175 R 219 150 R 204 130 R 191 114 R 179 101 R -169 89 R 160 81 R 152 72 R 144 65 R 138 60 R 131 54 R 126 50 R 121 46 R -116 42 R 111 39 R 108 37 R 104 33 R 100 32 R 97 0 R 94 0 R 91 0 R 88 0 R -86 0 R 83 0 R 81 0 R 79 0 R 77 0 R 75 0 R 73 0 R 71 0 R 70 0 R 68 0 R 66 0 R -65 0 R 64 0 R 62 0 R 61 0 R 60 0 R 59 0 R 57 0 R 56 0 R 56 0 R 54 0 R 53 0 R -53 0 R 51 0 R 51 0 R 49 0 R 49 0 R 48 0 R 48 0 R 46 0 R 46 0 R 45 0 R 45 0 R -43 0 R 44 0 R 42 0 R 42 0 R 41 0 R 41 0 R 41 0 R 39 0 R 39 0 R 39 0 R 38 0 R -38 0 R 37 0 R 37 0 R 36 0 R 36 0 R 35 0 R 35 0 R 35 0 R 34 0 R 34 0 R 33 0 R -33 0 R 33 0 R 32 0 R 32 0 R 32 0 R 31 0 R 31 0 R 31 0 R 30 0 R 30 0 R 30 0 R -D 60 setlinewidth 0.000 0.000 0.000 setrgbcolor 9481 7162 M 1199 0 R D -gsave 10917 6993 translate 0 0 M 1.5 dup scale -(Boutle et al. \(2014\)) show grestore L2 9481 6320 M 1199 0 R D -gsave 10917 6152 translate 0 0 M 1.5 dup scale -(Honnert et al. \(2011\)) show grestore L0 0.000 0.000 1.000 setrgbcolor -9481 5479 M 1199 0 R D gsave 10917 5310 translate 0 0 M 1.5 dup scale -(vn10.1) show grestore -%%PageTrailer -end -restore -showpage -%%PageResources: font Helvetica -%%+ font Symbol -%%Trailer -restore -%%Pages: 1 -%%DocumentNeededResources: font Helvetica -%%+ font Symbol -%%EOF diff --git a/documentation/source/science_guide/turbulence_schemes/ideal_invinteg.eps b/documentation/source/science_guide/turbulence_schemes/ideal_invinteg.eps deleted file mode 100644 index 443becd7aa..0000000000 --- a/documentation/source/science_guide/turbulence_schemes/ideal_invinteg.eps +++ /dev/null @@ -1,265 +0,0 @@ -%!PS-Adobe-2.0 EPSF-2.0 -%%BoundingBox: 56 40 254 236 -%%HiResBoundingBox: 57 41 253.5 235.5 -%%Title: Graphics produced by WAVE -%%For: frlk@eld206 -%%Creator: WAVE Version 8.00 (Linux i386) -%%CreationDate: Thu Mar 30 16:29:43 2006 -%%EndComments -% EPSF created by ps2eps 1.68 -%%BeginProlog -save -countdictstack -mark -newpath -/showpage {} def -/setpagedevice {pop} def -%%EndProlog -%%Page 1 1 -%+ wave_prolog.ps -- Prolog for PV-WAVE CL generated PostScript files -%+ $Id: wave_prolog.ps,v 1.5 1998/06/05 17:44:06 thaux Exp $ -%+ Copyright (c) 1989-1992 Precision Visuals, Inc. All Rights Reserved. -%v 1 -save /$WAVE_DICT 60 dict def $WAVE_DICT begin /bdef { bind def } bind def /C -{currentpoint newpath moveto} bdef /D {currentpoint stroke moveto} bdef /F -{closepath fill} bdef /K { setgray } bdef /L {translate} bdef /M {moveto} -bdef /N {rmoveto} bdef /O {show} bdef /P {lineto} bdef /R {rlineto} bdef /S -{gsave show grestore} bdef /Z {gsave currentpoint lineto 20 setlinewidth 1 -setlinecap stroke grestore} bdef /CC {concat} bdef /CD {currentmatrix def} -bdef /CP {currentpoint pop def} bdef /FM {findfont exch makefont} bdef /FS -{findfont exch scalefont} bdef /GS {get setfont} bdef /MD {matrix def} bdef -/MN {mul neg def} bdef /RS {roll put setfont} bdef /SL {setlinewidth} bdef -/SM {setmatrix} bdef /L0 {[] 0 setdash} bdef /L1 {[40 100] 0 setdash} bdef -/L2 {[200 200] 0 setdash} bdef /L3 {[200 100 50 100] 0 setdash} bdef /L4 -{[300 100 50 100 50 100 50 100] 0 setdash} bdef /L5 {[400 200] 0 setdash} -bdef /IO { gsave 0.0 setgray /val exch def moveto /str 20 string def val str -cvs show grestore } bdef /UL { gsave currentpoint /cury exch def /curx exch -def /curmtx matrix currentmatrix def matrix identmatrix setmatrix newpath 0 -0 moveto true charpath flattenpath pathbbox curmtx setmatrix /height exch -def /width exch def /ybase cury height .1 mul sub def pop pop newpath curx -ybase moveto width 0 rlineto closepath stroke grestore } bdef /BG { gsave -currentpoint /cury exch def /curx exch def /curmtx matrix currentmatrix def -matrix identmatrix setmatrix newpath 0 0 moveto true charpath flattenpath -pathbbox curmtx setmatrix /height exch def /width exch def pop pop newpath 0 -0 moveto (M) true charpath flattenpath pathbbox /xdiff exch .1 mul def -/ydiff exch .1 mul def pop pop newpath curx xdiff 2 div sub cury ydiff 2 div -sub moveto width xdiff add 0 rlineto 0 height ydiff add rlineto width neg -xdiff sub 0 rlineto closepath fill grestore } bdef /$T_DICT 20 dict def /T { -$T_DICT begin /align exch def /orien exch def /size exch def /thestring exch -def gsave moveto /chsize 1.0 def /xsize 0.0 def /SUPER 8#330 def /SUBS 8#331 -def /NORM 8#332 def /SCRIPTWID 0.7 def /OFFSET 0.6 def orien rotate size dup -scale /orien false def thestring { /charcode exch def charcode SUPER eq -charcode SUBS eq or { /chsize SCRIPTWID def /orien true def } { charcode -NORM eq { /chsize 1.0 def /orien true def } { ( ) dup 0 charcode put -stringwidth pop chsize mul xsize add /xsize exch def } ifelse } ifelse } -forall xsize align mul neg 0 rmoveto orien { /regularfont currentfont def -/fractionfont currentfont [ SCRIPTWID 0 0 SCRIPTWID 0 0 ] makefont def gsave -newpath 0 0 moveto (1) true charpath flattenpath pathbbox /height exch def -pop pop pop grestore } if /xsize 0 def thestring { /charcode exch def -charcode SUPER eq { 0 OFFSET height mul dup /xsize exch def rmoveto -fractionfont setfont } { charcode SUBS eq { 0 OFFSET height mul neg dup -/xsize exch def rmoveto fractionfont setfont } { charcode NORM eq { 0 xsize -neg rmoveto regularfont setfont } { ( ) dup 0 charcode put show } ifelse } -ifelse } ifelse } forall grestore end } bdef /IsChar { exch /CharStrings get -exch known } bdef /MapCh { 3 -1 roll /Encoding get 3 1 roll put } bdef -/MapDegree { dup 16#b0 exch /degree IsChar { /degree } { /ring } ifelse -MapCh } bdef /MapBB { dup 16#a6 exch /brokenbar IsChar { /brokenbar } { /bar -} ifelse MapCh } bdef /ReEncode { dup findfont begin currentdict dup length -dict begin { 1 index /FID ne {def} {pop pop} ifelse } forall /FontName exch -def dup length 0 ne {/Encoding Encoding 256 array copy def 0 exch { dup type -/nametype eq { Encoding 2 index 2 index put pop 1 add } { exch pop } ifelse -} forall } if pop currentdict dup end end /FontName get exch definefont dup -MapDegree MapBB } bdef /RF { ISOLatin1Encoding exch ReEncode } bdef /Courier -RF /Courier-Bold RF /Courier-Oblique RF /Courier-BoldOblique RF /Helvetica -RF /Helvetica-Bold RF /Helvetica-Oblique RF /Helvetica-BoldOblique RF -/Helvetica-Narrow RF /Helvetica-Narrow-Bold RF /Helvetica-Narrow-Oblique RF -/Helvetica-Narrow-BoldOblique RF /AvantGarde-Book RF /AvantGarde-DemiOblique -RF /AvantGarde-Demi RF /AvantGarde-DemiOblique RF /Bookman-Demi RF -/Bookman-DemiItalic RF /Bookman-Light RF /Bookman-LightItalic RF -/ZapfChancery-MediumItalic RF /NewCenturySchlbk-Roman RF -/NewCenturySchlbk-Bold RF /NewCenturySchlbk-Italic RF -/NewCenturySchlbk-BoldItalic RF /Palatino-Roman RF /Palatino-Bold RF -/Palatino-Italic RF /Palatino-BoldItalic RF /Times-Roman RF /Times-Bold RF -/Times-Italic RF /Times-BoldItalic RF end -%%EndProlog -%%Page: 0 1 -%%BeginPageSetup -save $WAVE_DICT begin 28 28 L 0.028346 dup scale -%%PageBoundingBox: 28 28 254 254 -%%EndPageSetup -/psFontCache 500 array def /psStack 8 array def /psStackInd 0 def -/psTextWidth 0 def /psCurBase 0 def /ASW { stringwidth pop psCurBase add -/psCurBase exch def psCurBase psTextWidth gt -{ /psTextWidth psCurBase def } if } def /PSS { -psStack psStackInd psCurBase put /psStackInd psStackInd 1 add def } def -/PPS { /psStackInd psStackInd 1 sub def -/psCurBase psStack psStackInd get def } def 10.000000 SL L0 0.000 K -2220 1408 M 5114 0 R D 3925 1408 M 0 118 R D 3901 1196 M -35 -12 R -23 -35 R --12 -58 R 0 -35 R 12 -59 R 23 -35 R 35 -12 R 24 0 R 35 12 R 23 35 R 12 59 R -0 35 R -12 58 R -23 35 R -35 12 R -24 0 R -23 -12 R -12 -12 R -11 -23 R --12 -58 R 0 -35 R 12 -59 R 11 -23 R 12 -12 R 23 -12 R D 3925 950 M 23 12 R -12 12 R 11 23 R 12 59 R 0 35 R -12 58 R -11 23 R -12 12 R -23 12 R D -4607 1408 M 0 118 R D 3772 738 M 0 -245 R D 3784 738 M 0 -245 R D 3924 738 M -0 -245 R D 3936 738 M 0 -245 R D 3737 738 M 82 0 R D 3889 738 M 82 0 R D -3784 621 M 140 0 R D 3737 493 M 82 0 R D 3889 493 M 82 0 R D 4041 586 M -140 0 R 0 24 R -12 23 R -11 12 R -24 11 R -35 0 R -35 -11 R -23 -24 R --12 -35 R 0 -23 R 12 -35 R 23 -23 R 35 -12 R 24 0 R 35 12 R 23 23 R D -4169 586 M 0 35 R -11 24 R D 4099 656 M -23 -11 R -23 -24 R -12 -35 R -0 -23 R 12 -35 R 23 -23 R 23 -12 R D 4275 633 M 0 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R D 1240 3654 M 84 -117 R 0 117 R -7 0 R 0 -137 R D 1409 3647 M 0 -130 R D 1415 3647 M 0 -130 R -6 0 R D -1370 3654 M 84 0 R 0 -7 R D 1370 3654 M 0 -7 R 84 0 R D 1493 3654 M 0 -137 R -D 1500 3621 M 0 -104 R -7 0 R D 1500 3621 M 46 -104 R D 1493 3654 M -53 -117 R 52 117 R D 1591 3621 M -45 -104 R D 1591 3621 M 0 -104 R 7 0 R -0 137 R D 1650 3654 M 0 -137 R D 1650 3654 M 6 0 R 0 -130 R 72 0 R 0 -7 R --78 0 R D 1760 3582 M 111 0 R 0 -6 R D 1760 3582 M 0 -6 R 111 0 R D -1936 3628 M 13 6 R 20 20 R 0 -137 R D 1936 3628 M 0 -7 R 13 7 R 13 13 R -0 -124 R 7 0 R D 1984 5105 M -93 -245 R D 1984 5105 M 94 -245 R D -1984 5070 M 82 -210 R D 1902 4871 M 164 0 R D 1891 4860 M 187 0 R D -2311 5035 M -11 -23 R -24 -24 R -93 -70 R -23 -23 R -12 -24 R D 2300 5012 M --105 0 R -24 -12 R -11 -24 R D 2276 5012 M -46 11 R -35 0 R -12 -11 R D -2276 5012 M -46 23 R -35 0 R -24 -23 R -11 -36 R D 2160 4895 M 105 0 R -23 11 R 12 24 R D 2183 4895 M 47 -12 R 35 0 R 11 12 R D 2183 4895 M 47 -24 R -35 0 R 23 24 R 12 35 R D 2375 4868 M 0 -101 R D 2383 4868 M 0 -101 R D -2383 4824 M 7 22 R 14 15 R 15 7 R 21 0 R 8 -7 R 0 -8 R -8 -7 R -7 7 R 7 8 R -D 2354 4868 M 29 0 R D 2354 4767 M 50 0 R D 2498 4853 M 0 -7 R -7 0 R 0 7 R -7 8 R 15 7 R 29 0 R 14 -7 R 7 -8 R 8 -14 R 0 -51 R 7 -14 R 7 -7 R D -2563 4853 M 0 -65 R 8 -14 R 14 -7 R 7 0 R D 2563 4839 M -7 -7 R -43 -8 R --22 -7 R -7 -14 R 0 -15 R 7 -14 R 22 -7 R 22 0 R 14 7 R 14 14 R D -2513 4824 M -15 -7 R -7 -14 R 0 -15 R 7 -14 R 15 -7 R D 2715 4918 M 0 -151 R -D 2723 4918 M 0 -151 R D 2715 4846 M -14 15 R -15 7 R -14 0 R -22 -7 R --14 -15 R -7 -22 R 0 -14 R 7 -22 R 14 -14 R 22 -7 R 14 0 R 15 7 R 14 14 R D -2672 4868 M -15 -7 R -14 -15 R -7 -22 R 0 -14 R 7 -22 R 14 -14 R 15 -7 R D -2694 4918 M 29 0 R D 2715 4767 M 29 0 R D 20.000000 SL 2288 3560 M 0 1178 R -D 2288 5266 M 0 609 R -68 -122 R D 2288 5875 M 68 -122 R D 2220 3682 M -68 -122 R 68 122 R D 10.000000 SL 4718 1519 M -111 -111 R -111 111 R -111 -111 R D 4823 2189 M -222 -222 R 111 111 R -111 111 R 222 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41 R 0 40 R 0 41 R 0 41 R 0 40 R 0 41 R 0 40 R 0 41 R -0 41 R 0 40 R 0 41 R 0 40 R 0 41 R 0 41 R 0 40 R 0 41 R 0 40 R 1 41 R 0 41 R -1 40 R 2 41 R 2 40 R 5 41 R 6 41 R 10 40 R 14 41 R 20 40 R 28 41 R 39 41 R -51 40 R 68 41 R 87 41 R 108 40 R 131 41 R 155 40 R 180 41 R 201 41 R -218 40 R 231 41 R 234 40 R 228 41 R 213 41 R 186 40 R 149 41 R 105 40 R -54 41 R 10 41 R 11 40 R 10 41 R 10 40 R 10 41 R 11 41 R 10 40 R 10 41 R -10 40 R 10 41 R 11 41 R 10 40 R 10 41 R 10 41 R 11 40 R 10 41 R 10 40 R -10 41 R 10 41 R 11 40 R 10 41 R 10 40 R 10 41 R 11 41 R 10 40 R 10 41 R -10 40 R 11 41 R 10 41 R 10 40 R 10 41 R 10 40 R 11 41 R D L3 4607 1408 M -6 41 R 6 40 R 6 41 R 6 40 R 7 41 R 6 41 R 6 40 R 6 41 R 6 40 R 7 41 R 6 41 R -6 40 R 6 41 R 6 41 R 7 40 R 6 41 R 6 40 R 6 41 R 6 41 R 7 40 R 6 41 R 6 40 R -6 41 R 6 41 R 7 40 R 6 41 R 6 40 R 6 41 R 6 41 R 7 40 R 6 41 R 6 40 R 6 41 R -6 41 R 7 40 R 6 41 R 6 40 R 6 41 R 6 41 R 7 40 R 6 41 R 6 41 R 6 40 R 6 41 R -7 40 R 6 41 R 6 41 R 6 40 R 6 41 R 7 40 R 6 41 R 6 41 R 6 40 R 6 41 R 6 40 R -7 41 R 6 41 R 6 40 R 6 41 R 6 40 R 7 41 R 6 41 R 6 40 R 6 41 R 6 40 R 7 41 R -6 41 R 6 40 R 6 41 R 6 41 R 7 40 R 6 41 R 6 40 R 6 41 R 6 41 R 7 40 R 6 41 R -6 40 R 6 41 R 6 41 R 7 40 R 6 41 R 6 40 R 5 41 R 6 41 R 5 40 R 5 41 R 3 40 R -2 41 R 0 41 R -3 40 R -8 41 R -14 40 R -22 41 R -32 41 R -46 40 R -62 41 R --80 41 R -102 40 R -125 41 R -149 40 R -173 41 R -195 41 R -212 40 R --225 41 R -227 40 R -223 41 R -206 41 R -180 40 R -143 41 R 577 40 R -628 41 R 0 41 R 0 40 R 0 41 R 0 40 R 0 41 R 0 41 R 0 40 R 0 41 R 0 40 R -0 41 R 0 41 R 0 40 R 0 41 R 0 41 R 0 40 R 0 41 R 0 40 R 0 41 R 0 41 R 0 40 R -0 41 R 0 40 R 0 41 R 0 41 R 0 40 R 0 41 R 0 40 R 0 41 R 0 41 R 0 40 R 0 41 R -0 40 R 0 41 R D L0 4607 1408 M 6 41 R 6 40 R 6 41 R 6 40 R 7 41 R 6 41 R -6 40 R 6 41 R 6 40 R 7 41 R 6 41 R 6 40 R 6 41 R 6 41 R 7 40 R 6 41 R 6 40 R -6 41 R 6 41 R 7 40 R 6 41 R 6 40 R 6 41 R 6 41 R 7 40 R 6 41 R 6 40 R 6 41 R -6 41 R 7 40 R 6 41 R 6 40 R 6 41 R 6 41 R 7 40 R 6 41 R 6 40 R 6 41 R 6 41 R -7 40 R 6 41 R 6 41 R 6 40 R 6 41 R 7 40 R 6 41 R 6 41 R 6 40 R 6 41 R 7 40 R -6 41 R 6 41 R 6 40 R 6 41 R 6 40 R 7 41 R 6 41 R 6 40 R 6 41 R 6 40 R 7 41 R -6 41 R 6 40 R 6 41 R 6 40 R 7 41 R 6 41 R 6 40 R 6 41 R 6 41 R 7 40 R 6 41 R -6 40 R 6 41 R 6 41 R 7 40 R 6 41 R 6 40 R 6 41 R 6 41 R 7 40 R 6 41 R 6 40 R -6 41 R 6 41 R 7 40 R 6 41 R 6 40 R 6 41 R 6 41 R 7 40 R 6 41 R 6 40 R 6 41 R -6 41 R 7 40 R 6 41 R 6 41 R 6 40 R 6 41 R 7 40 R 6 41 R 6 41 R 6 40 R 6 41 R -7 40 R 6 41 R 6 41 R 6 40 R 6 41 R 682 40 R 682 41 R 10 41 R 11 40 R 10 41 R -10 40 R 10 41 R 11 41 R 10 40 R 10 41 R 10 40 R 10 41 R 11 41 R 10 40 R -10 41 R 10 41 R 11 40 R 10 41 R 10 40 R 10 41 R 10 41 R 11 40 R 10 41 R -10 40 R 10 41 R 11 41 R 10 40 R 10 41 R 10 40 R 11 41 R 10 41 R 10 40 R -10 41 R 10 40 R 11 41 R D -end restore -showpage -%%Trailer -restore -%%Pages: 1 -%%Trailer -cleartomark -countdictstack -exch sub { end } repeat -restore -%%EOF diff --git a/documentation/source/science_guide/turbulence_schemes/ideal_revflux.eps b/documentation/source/science_guide/turbulence_schemes/ideal_revflux.eps deleted file mode 100644 index b3e8cec02e..0000000000 --- a/documentation/source/science_guide/turbulence_schemes/ideal_revflux.eps +++ /dev/null @@ -1,517 +0,0 @@ -%!PS-Adobe-2.0 EPSF-2.0 -%%BoundingBox: 60 38 464 520 -%%HiResBoundingBox: 60.5 39 463.5 519 -%%Title: Graphics produced by WAVE -%%For: frlk@eld206 -%%Creator: WAVE Version 8.00 (Linux i386) -%%CreationDate: Thu Mar 30 11:10:47 2006 -%%EndComments -% EPSF created by ps2eps 1.68 -%%BeginProlog -save -countdictstack -mark -newpath -/showpage {} def -/setpagedevice {pop} def -%%EndProlog -%%Page 1 1 -%+ wave_prolog.ps -- Prolog for PV-WAVE CL generated PostScript files -%+ $Id: wave_prolog.ps,v 1.5 1998/06/05 17:44:06 thaux Exp $ -%+ Copyright (c) 1989-1992 Precision Visuals, Inc. All Rights Reserved. -%v 1 -save /$WAVE_DICT 60 dict def $WAVE_DICT begin /bdef { bind def } bind def /C -{currentpoint newpath moveto} bdef /D {currentpoint stroke moveto} bdef /F -{closepath fill} bdef /K { setgray } bdef /L {translate} bdef /M {moveto} -bdef /N {rmoveto} bdef /O {show} bdef /P {lineto} bdef /R {rlineto} bdef /S -{gsave show grestore} bdef /Z {gsave currentpoint lineto 20 setlinewidth 1 -setlinecap stroke grestore} bdef /CC {concat} bdef /CD {currentmatrix def} -bdef /CP {currentpoint pop def} bdef /FM {findfont exch makefont} bdef /FS -{findfont exch scalefont} bdef /GS {get setfont} bdef /MD {matrix def} bdef -/MN {mul neg def} bdef /RS {roll put setfont} bdef /SL {setlinewidth} bdef -/SM {setmatrix} bdef /L0 {[] 0 setdash} bdef /L1 {[40 100] 0 setdash} bdef -/L2 {[200 200] 0 setdash} bdef /L3 {[200 100 50 100] 0 setdash} bdef /L4 -{[300 100 50 100 50 100 50 100] 0 setdash} bdef /L5 {[400 200] 0 setdash} -bdef /IO { gsave 0.0 setgray /val exch def moveto /str 20 string def val str -cvs show grestore } bdef /UL { gsave currentpoint /cury exch def /curx exch -def /curmtx matrix currentmatrix def matrix identmatrix setmatrix newpath 0 -0 moveto true charpath flattenpath pathbbox curmtx setmatrix /height exch -def /width exch def /ybase cury height .1 mul sub def pop pop newpath curx -ybase moveto width 0 rlineto closepath stroke grestore } bdef /BG { gsave -currentpoint /cury exch def /curx exch def /curmtx matrix currentmatrix def -matrix identmatrix setmatrix newpath 0 0 moveto true charpath flattenpath -pathbbox curmtx setmatrix /height exch def /width exch def pop pop newpath 0 -0 moveto (M) true charpath flattenpath pathbbox /xdiff exch .1 mul def -/ydiff exch .1 mul def pop pop newpath curx xdiff 2 div sub cury ydiff 2 div -sub moveto width xdiff add 0 rlineto 0 height ydiff add rlineto width neg -xdiff sub 0 rlineto closepath fill grestore } bdef /$T_DICT 20 dict def /T { -$T_DICT begin /align exch def /orien exch def /size exch def /thestring exch -def gsave moveto /chsize 1.0 def /xsize 0.0 def /SUPER 8#330 def /SUBS 8#331 -def /NORM 8#332 def /SCRIPTWID 0.7 def /OFFSET 0.6 def orien rotate size dup -scale /orien false def thestring { /charcode exch def charcode SUPER eq -charcode SUBS eq or { /chsize SCRIPTWID def /orien true def } { charcode -NORM eq { /chsize 1.0 def /orien true def } { ( ) dup 0 charcode put -stringwidth pop chsize mul xsize add /xsize exch def } ifelse } ifelse } -forall xsize align mul neg 0 rmoveto orien { /regularfont currentfont def -/fractionfont currentfont [ SCRIPTWID 0 0 SCRIPTWID 0 0 ] makefont def gsave -newpath 0 0 moveto (1) true charpath flattenpath pathbbox /height exch def -pop pop pop grestore } if /xsize 0 def thestring { /charcode exch def -charcode SUPER eq { 0 OFFSET height mul dup /xsize exch def rmoveto -fractionfont setfont } { charcode SUBS eq { 0 OFFSET height mul neg dup -/xsize exch def rmoveto fractionfont setfont } { charcode NORM eq { 0 xsize -neg rmoveto regularfont setfont } { ( ) dup 0 charcode put show } ifelse } -ifelse } ifelse } forall grestore end } bdef /IsChar { exch /CharStrings get -exch known } bdef /MapCh { 3 -1 roll /Encoding get 3 1 roll put } bdef -/MapDegree { dup 16#b0 exch /degree IsChar { /degree } { /ring } ifelse -MapCh } bdef /MapBB { dup 16#a6 exch /brokenbar IsChar { /brokenbar } { /bar -} ifelse MapCh } bdef /ReEncode { dup findfont begin currentdict dup length -dict begin { 1 index /FID ne {def} {pop pop} ifelse } forall /FontName exch -def dup length 0 ne {/Encoding Encoding 256 array copy def 0 exch { dup type -/nametype eq { Encoding 2 index 2 index put pop 1 add } { exch pop } ifelse -} forall } if pop currentdict dup end end /FontName get exch definefont dup -MapDegree MapBB } bdef /RF { ISOLatin1Encoding exch ReEncode } bdef /Courier -RF /Courier-Bold RF /Courier-Oblique RF /Courier-BoldOblique RF /Helvetica -RF /Helvetica-Bold RF /Helvetica-Oblique RF /Helvetica-BoldOblique RF -/Helvetica-Narrow RF /Helvetica-Narrow-Bold RF /Helvetica-Narrow-Oblique RF -/Helvetica-Narrow-BoldOblique RF /AvantGarde-Book RF /AvantGarde-DemiOblique -RF /AvantGarde-Demi RF /AvantGarde-DemiOblique RF /Bookman-Demi RF -/Bookman-DemiItalic RF /Bookman-Light RF /Bookman-LightItalic RF -/ZapfChancery-MediumItalic RF /NewCenturySchlbk-Roman RF -/NewCenturySchlbk-Bold RF /NewCenturySchlbk-Italic RF -/NewCenturySchlbk-BoldItalic RF /Palatino-Roman RF /Palatino-Bold RF -/Palatino-Italic RF /Palatino-BoldItalic RF /Times-Roman RF /Times-Bold RF -/Times-Italic RF /Times-BoldItalic RF end -%%EndProlog -%%Page: 0 1 -%%BeginPageSetup -save $WAVE_DICT begin 28 28 L 0.028346 dup scale -%%PageBoundingBox: 28 28 481 538 -%%EndPageSetup -/psFontCache 500 array def /psStack 8 array def /psStackInd 0 def -/psTextWidth 0 def /psCurBase 0 def /ASW { stringwidth pop psCurBase add -/psCurBase exch def psCurBase psTextWidth gt -{ /psTextWidth psCurBase def } if } def /PSS { -psStack psStackInd psCurBase put /psStackInd psStackInd 1 add def } def -/PPS { /psStackInd psStackInd 1 sub def -/psCurBase psStack psStackInd get def } def 10.000000 SL L0 0.000 K -2220 10408 M 5114 0 R D 2220 10408 M 0 138 R D 2197 10196 M -35 -12 R --24 -35 R -11 -58 R 0 -35 R 11 -59 R 24 -35 R 35 -12 R 23 0 R 35 12 R -24 35 R 11 59 R 0 35 R -11 58 R -24 35 R -35 12 R -23 0 R -24 -12 R --11 -12 R -12 -23 R -12 -58 R 0 -35 R 12 -59 R 12 -23 R 11 -12 R 24 -12 R D -2220 9950 M 23 12 R 12 12 R 12 23 R 12 59 R 0 35 R -12 58 R -12 23 R --12 12 R -23 12 R D 15005 10408 M 0 138 R D 4697 9668 M -70 -245 R D -4709 9668 M -70 -245 R D 4697 9668 M 24 0 R -70 -245 R D 4674 9586 M 0 35 R --12 35 R -23 12 R -24 0 R -35 -12 R -23 -35 R -12 -35 R 0 -23 R 12 -35 R -12 -12 R 23 -11 R 23 0 R 24 11 R 12 12 R 11 23 R 12 35 R D 4580 9645 M --11 -24 R -12 -35 R 0 -35 R 12 -23 R D 4615 9668 M -23 -23 R -12 -24 R --11 -35 R 0 -35 R 11 -35 R 12 -11 R D 4592 9423 M 94 0 R D 4639 9434 M --35 -11 R D 4639 9446 M -24 -23 R D 4651 9446 M 11 -23 R D 4639 9434 M -35 -11 R D 4821 9559 M -22 -80 R -7 -29 R 0 -21 R 7 -15 R 7 -7 R 15 0 R -14 15 R 7 14 R D 4828 9559 M -22 -80 R -7 -29 R 0 -36 R D 4821 9559 M 14 0 R --29 -101 R -7 -29 R D 4777 9508 M 73 0 R D 2220 10408 M 0 6888 R D -2220 10408 M 102 0 R D 1969 10653 M -35 -11 R -24 -35 R -11 -59 R 0 -35 R -11 -58 R 24 -35 R 35 -12 R 23 0 R 35 12 R 24 35 R 11 58 R 0 35 R -11 59 R --24 35 R -35 11 R -23 0 R -23 -11 R -12 -12 R -12 -23 R -12 -59 R 0 -35 R -12 -58 R 12 -24 R 12 -11 R 23 -12 R D 1992 10408 M 24 12 R 11 11 R 12 24 R -12 58 R 0 35 R -12 59 R -12 23 R -11 12 R -24 11 R D 2220 14303 M 102 0 R D -1915 14373 M -12 -23 R -23 -24 R -94 -70 R -23 -23 R -12 -24 R D -1903 14350 M -105 0 R -23 -12 R -12 -23 R D 1880 14350 M -47 11 R -35 0 R --12 -11 R D 1880 14350 M -47 23 R -35 0 R -23 -23 R -12 -35 R D 1763 14233 M -105 0 R 24 12 R 11 23 R D 1786 14233 M 47 -12 R 35 0 R 12 12 R D -1786 14233 M 47 -24 R 35 0 R 24 24 R 11 35 R D 2008 14264 M -44 -152 R -15 0 R D 2015 14264 M -43 -152 R D 1986 14264 M 36 0 R -43 -152 R D -1993 14163 M 15 29 R 14 14 R 15 7 R 14 0 R 15 -7 R 7 -14 R 0 -22 R -15 -36 R -D 2066 14206 M 0 -29 R -8 -29 R 0 -29 R D 2066 14192 M -15 -37 R 0 -21 R -7 -15 R 8 -7 R 14 0 R 15 14 R 7 15 R D 1993 14264 M 22 -7 R D 2001 14264 M -7 -15 R D 2220 14588 M 102 0 R D 1973 14658 M -12 -23 R -23 -24 R -94 -70 R --23 -23 R -12 -24 R D 1961 14635 M -105 0 R -23 -12 R -12 -23 R D -1938 14635 M -47 11 R -35 0 R -12 -11 R D 1938 14635 M -47 23 R -35 0 R --23 -23 R -12 -35 R D 1821 14518 M 105 0 R 23 12 R 12 23 R D 1844 14518 M -47 -12 R 35 0 R 12 12 R D 1844 14518 M 47 -24 R 35 0 R 23 24 R 12 35 R D -2073 14549 M -22 -80 R -7 -29 R 0 -21 R 7 -15 R 7 -7 R 15 0 R 14 14 R 8 15 R -D 2080 14549 M -22 -80 R -7 -29 R 0 -36 R D 2073 14549 M 14 0 R -29 -101 R --7 -29 R D 2030 14498 M 72 0 R D 1180 13840 M 24 -11 R 23 -24 R 70 -93 R -23 -23 R 24 -12 R D 1204 13829 M 0 -105 R 11 -24 R 24 -11 R D 1204 13805 M --12 -46 R 0 -35 R 12 -12 R D 1204 13805 M -24 -46 R 0 -35 R 24 -24 R -35 -11 R D 1320 13689 M 0 105 R -11 23 R -24 12 R D 1320 13712 M 12 47 R -0 35 R -12 11 R D 1320 13712 M 24 47 R 0 35 R -24 23 R -35 12 R D L1 -6056 10408 M 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R -0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R -0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R -0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R -0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R -0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R -0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R -0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R -341 48 R -341 47 R --341 48 R -341 47 R -341 48 R -341 47 R -3 48 R -2 47 R -3 48 R -2 47 R --3 48 R -2 47 R -3 48 R -2 47 R -3 48 R -3 47 R -2 48 R -3 47 R -2 48 R --3 47 R -2 48 R -3 47 R -3 48 R -2 47 R -3 48 R -2 47 R -3 48 R -2 47 R --3 48 R -2 47 R -3 48 R -3 47 R -2 48 R -3 47 R -2 48 R -3 47 R -2 48 R --3 47 R -2 48 R -3 47 R -3 48 R -2 47 R -3 48 R -2 48 R -3 47 R -2 48 R --3 47 R -2 48 R -3 47 R -3 48 R -2 47 R -3 48 R -2 47 R -3 48 R -2 47 R --3 48 R -2 47 R -3 48 R -3 47 R -2 48 R -3 47 R -2 48 R -3 47 R D L0 -6056 10757 M 111 0 R 0 -222 R -222 0 R 0 222 R 111 0 R D 6056 11469 M -111 0 R 0 -222 R -222 0 R 0 222 R 111 0 R D 6056 12894 M 111 0 R 0 -222 R --222 0 R 0 222 R 111 0 R D 4735 14794 M 111 0 R 0 -222 R -222 0 R 0 222 R -111 0 R D 3901 16695 M 111 0 R 0 -222 R -222 0 R 0 222 R 111 0 R D -6056 11026 M 0 -618 R D 6056 11026 M Z L2 2220 10408 M 5114 0 R D L0 -6056 12071 M 0 -1045 R D 6056 12071 M Z L2 2220 11026 M 5114 0 R D L0 -6056 13733 M 0 -1662 R D 6056 13733 M -1321 0 R D L2 2220 12071 M 5114 0 R D -L0 4735 15633 M 0 -1900 R D 4735 15633 M -834 0 R D L2 2220 13733 M 5114 0 R -D L0 3901 17296 M 0 -1663 R D L2 2220 15633 M 5114 0 R D L0 3692 12934 M -0 -246 R D 3704 12934 M 0 -246 R D 3657 12934 M 82 0 R D 3657 12688 M -176 0 R 0 70 R -12 -70 R D 3891 12782 M 140 0 R 0 23 R -12 23 R -11 12 R --24 12 R -35 0 R -35 -12 R -23 -23 R -12 -35 R 0 -24 R 12 -35 R 23 -23 R -35 -12 R 24 0 R 35 12 R 23 23 R D 4019 12782 M 0 35 R -11 23 R D -3949 12852 M -23 -12 R -23 -23 R -12 -35 R 0 -24 R 12 -35 R 23 -23 R -23 -12 R D 4101 12852 M 70 -164 R D 4113 12852 M 58 -140 R D 4241 12852 M --70 -164 R D 4078 12852 M 70 0 R D 4195 12852 M 70 0 R D 4323 12782 M -140 0 R 0 23 R -11 23 R -12 12 R -23 12 R -35 0 R -35 -12 R -24 -23 R --11 -35 R 0 -24 R 11 -35 R 24 -23 R 35 -12 R 23 0 R 35 12 R 23 23 R D -4452 12782 M 0 35 R -12 23 R D 4382 12852 M -24 -12 R -23 -23 R -12 -35 R -0 -24 R 12 -35 R 23 -23 R 24 -12 R D 4557 12934 M 0 -246 R D 4569 12934 M -0 -246 R D 4522 12934 M 47 0 R D 4522 12688 M 82 0 R D 4872 12934 M 0 -246 R -D 4884 12934 M 140 -222 R D 4884 12910 M 140 -222 R 0 246 R D 4837 12934 M -47 0 R D 4989 12934 M 70 0 R D 4837 12688 M 70 0 R D 5188 12934 M 0 -246 R D -5200 12934 M 0 -246 R D 5118 12934 M -12 -70 R 0 70 R 175 0 R 0 -70 R --11 70 R D 5153 12688 M 82 0 R D 5363 12934 M 0 -246 R D 5375 12934 M -70 -211 R D 5363 12934 M 82 -246 R 82 246 R 0 -246 R D 5538 12934 M 0 -246 R -D 5328 12934 M 47 0 R D 5527 12934 M 46 0 R D 5328 12688 M 70 0 R D -5492 12688 M 81 0 R D 5655 12934 M 0 -246 R D 5667 12934 M 0 -246 R D -5620 12934 M 82 0 R D 5620 12688 M 175 0 R 0 70 R -11 -70 R D 10220 10408 M -5114 0 R D 10220 10408 M 0 138 R D 10197 10196 M -35 -12 R -24 -35 R --11 -58 R 0 -35 R 11 -59 R 24 -35 R 35 -12 R 23 0 R 35 12 R 24 35 R 11 59 R -0 35 R -11 58 R -24 35 R -35 12 R -23 0 R -24 -12 R -11 -12 R -12 -23 R --12 -58 R 0 -35 R 12 -59 R 12 -23 R 11 -12 R 24 -12 R D 10220 9950 M 23 12 R -12 12 R 12 23 R 12 59 R 0 35 R -12 58 R -12 23 R -12 12 R -23 12 R D -13873 10408 M 0 138 R D 12101 9668 M -70 -245 R D 12113 9668 M -70 -245 R D -12101 9668 M 24 0 R -70 -245 R D 12078 9586 M 0 35 R -12 35 R -23 12 R --23 0 R -35 -12 R -24 -35 R -12 -35 R 0 -23 R 12 -35 R 12 -12 R 23 -11 R -24 0 R 23 11 R 12 12 R 11 23 R 12 35 R D 11985 9645 M -12 -24 R -12 -35 R -0 -35 R 12 -23 R D 12020 9668 M -24 -23 R -11 -24 R -12 -35 R 0 -35 R -12 -35 R 11 -11 R D 11996 9423 M 94 0 R D 12043 9434 M -35 -11 R D -12043 9446 M -23 -23 R D 12055 9446 M 11 -23 R D 12043 9434 M 35 -11 R D -12225 9559 M -22 -80 R -7 -29 R 0 -21 R 7 -15 R 7 -7 R 15 0 R 14 15 R 8 14 R -D 12232 9559 M -22 -80 R -7 -29 R 0 -36 R D 12225 9559 M 14 0 R -29 -101 R --7 -29 R D 12181 9508 M 73 0 R D 12553 9727 M -12 -12 R 12 -12 R 12 12 R -0 12 R -12 11 R -23 0 R -24 -11 R -11 -24 R 0 -210 R D 12530 9738 M --12 -11 R -12 -24 R 0 -210 R D 12460 9656 M 93 0 R D 12460 9493 M 81 0 R D -12647 9738 M 0 -245 R D 12658 9738 M 0 -245 R D 12612 9738 M 46 0 R D -12612 9493 M 81 0 R D 12775 9656 M 0 -128 R 12 -23 R 35 -12 R 23 0 R 35 12 R -24 23 R D 12787 9656 M 0 -128 R 11 -23 R 24 -12 R D 12904 9656 M 0 -163 R D -12915 9656 M 0 -163 R D 12740 9656 M 47 0 R D 12869 9656 M 46 0 R D -12904 9493 M 46 0 R D 13020 9656 M 129 -163 R D 13032 9656 M 129 -163 R D -13161 9656 M -141 -163 R D 12997 9656 M 70 0 R D 13114 9656 M 70 0 R D -12997 9493 M 70 0 R D 13114 9493 M 70 0 R D 13254 9586 M 140 0 R 0 24 R --11 23 R -12 12 R -23 11 R -35 0 R -35 -11 R -24 -24 R -12 -35 R 0 -23 R -12 -35 R 24 -23 R 35 -12 R 23 0 R 35 12 R 23 23 R D 13383 9586 M 0 35 R --12 24 R D 13313 9656 M -24 -11 R -23 -24 R -12 -35 R 0 -23 R 12 -35 R -23 -23 R 24 -12 R D 13581 9633 M 12 23 R 0 -46 R -12 23 R -11 12 R -24 11 R --46 0 R -24 -11 R -12 -12 R 0 -23 R 12 -12 R 24 -12 R 58 -23 R 23 -12 R -12 -11 R D 13464 9621 M 12 -11 R 24 -12 R 58 -23 R 23 -12 R 12 -12 R 0 -35 R --12 -11 R -23 -12 R -47 0 R -23 12 R -12 11 R -12 24 R 0 -47 R 12 23 R D -10220 10408 M 0 6888 R D 10220 10408 M 102 0 R D 9969 10653 M -35 -11 R --24 -35 R -11 -59 R 0 -35 R 11 -58 R 24 -35 R 35 -12 R 23 0 R 35 12 R -24 35 R 11 58 R 0 35 R -11 59 R -24 35 R -35 11 R -23 0 R -23 -11 R --12 -12 R -12 -23 R -12 -59 R 0 -35 R 12 -58 R 12 -24 R 12 -11 R 23 -12 R D -9992 10408 M 24 12 R 11 11 R 12 24 R 12 58 R 0 35 R -12 59 R -12 23 R --11 12 R -24 11 R D 10220 14303 M 102 0 R D 9915 14373 M -12 -23 R -23 -24 R --94 -70 R -23 -23 R -12 -24 R D 9903 14350 M -105 0 R -23 -12 R -12 -23 R D -9880 14350 M -47 11 R -35 0 R -12 -11 R D 9880 14350 M -47 23 R -35 0 R --23 -23 R -12 -35 R D 9763 14233 M 105 0 R 24 12 R 11 23 R D 9786 14233 M -47 -12 R 35 0 R 12 12 R D 9786 14233 M 47 -24 R 35 0 R 24 24 R 11 35 R D -10008 14264 M -44 -152 R 15 0 R D 10015 14264 M -43 -152 R D 9986 14264 M -36 0 R -43 -152 R D 9993 14163 M 15 29 R 14 14 R 15 7 R 14 0 R 15 -7 R -7 -14 R 0 -22 R -15 -36 R D 10066 14206 M 0 -29 R -8 -29 R 0 -29 R D -10066 14192 M -15 -37 R 0 -21 R 7 -15 R 8 -7 R 14 0 R 15 14 R 7 15 R D -9993 14264 M 22 -7 R D 10001 14264 M 7 -15 R D 10220 14588 M 102 0 R D -9973 14658 M -12 -23 R -23 -24 R -94 -70 R -23 -23 R -12 -24 R D -9961 14635 M -105 0 R -23 -12 R -12 -23 R D 9938 14635 M -47 11 R -35 0 R --12 -11 R D 9938 14635 M -47 23 R -35 0 R -23 -23 R -12 -35 R D 9821 14518 M -105 0 R 23 12 R 12 23 R D 9844 14518 M 47 -12 R 35 0 R 12 12 R D -9844 14518 M 47 -24 R 35 0 R 23 24 R 12 35 R D 10073 14549 M -22 -80 R --7 -29 R 0 -21 R 7 -15 R 7 -7 R 15 0 R 14 14 R 8 15 R D 10080 14549 M --22 -80 R -7 -29 R 0 -36 R D 10073 14549 M 14 0 R -29 -101 R -7 -29 R D -10030 14498 M 72 0 R D L3 13873 10408 M -9 48 R -9 47 R -9 48 R -9 47 R --9 48 R -9 47 R -8 48 R -9 47 R -9 48 R -9 47 R -9 48 R -9 47 R -9 48 R --9 47 R -9 48 R -9 47 R -9 48 R -8 47 R -9 48 R -9 47 R -9 48 R -9 47 R --9 48 R -9 47 R -9 48 R -9 47 R -9 48 R -9 47 R -8 48 R -9 47 R -9 48 R --9 47 R -9 48 R -9 47 R -9 48 R -9 47 R -9 48 R -9 47 R -9 48 R -8 47 R --9 48 R -9 47 R -9 48 R -9 47 R -9 48 R -9 47 R -9 48 R -9 47 R -9 48 R --9 47 R -8 48 R -9 47 R -9 48 R -9 47 R -9 48 R -9 47 R -9 48 R -9 47 R --9 48 R -9 47 R -9 48 R -8 47 R -9 48 R -9 47 R -9 48 R -9 47 R -9 48 R --9 47 R -9 48 R -9 47 R -9 48 R -9 47 R -8 48 R -9 47 R -9 48 R -9 47 R --9 48 R -9 47 R -9 48 R -9 47 R -9 48 R -9 47 R -487 48 R -487 47 R --487 48 R -487 47 R -487 48 R -487 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R -0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R -0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R -0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 48 R -0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R -0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R D L1 10220 10408 M -0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R -0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R -0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R -0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R -0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R -0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R -0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 0 48 R -0 47 R 0 48 R 0 47 R 0 48 R 0 47 R 404 48 R 409 47 R 414 48 R 419 47 R -424 48 R 429 47 R 3 48 R 3 47 R 3 48 R 4 47 R 3 48 R 4 47 R 3 48 R 4 47 R -3 48 R 4 47 R 3 48 R 4 47 R 4 48 R 3 47 R 4 48 R 4 47 R 4 48 R 4 47 R 4 48 R -4 47 R 4 48 R 4 47 R 4 48 R 4 47 R 4 48 R 4 47 R 4 48 R 4 47 R 5 48 R 4 47 R -4 48 R 5 47 R 4 48 R 5 47 R 4 48 R 5 47 R 4 48 R 5 48 R 4 47 R 5 48 R 5 47 R -5 48 R 4 47 R 5 48 R 5 47 R 5 48 R 5 47 R 5 48 R 5 47 R 5 48 R 5 47 R 5 48 R -5 47 R 5 48 R 6 47 R 5 48 R 5 47 R D L0 13873 10408 M -9 48 R -9 47 R --9 48 R -9 47 R -9 48 R -9 47 R -8 48 R -9 47 R -9 48 R -9 47 R -9 48 R --9 47 R -9 48 R -9 47 R -9 48 R -9 47 R -9 48 R -8 47 R -9 48 R -9 47 R --9 48 R -9 47 R -9 48 R -9 47 R -9 48 R -9 47 R -9 48 R -9 47 R -8 48 R --9 47 R -9 48 R -9 47 R -9 48 R -9 47 R -9 48 R -9 47 R -9 48 R -9 47 R --9 48 R -8 47 R -9 48 R -9 47 R -9 48 R -9 47 R -9 48 R -9 47 R -9 48 R --9 47 R -9 48 R -9 47 R -8 48 R -9 47 R -9 48 R -9 47 R -9 48 R -9 47 R --9 48 R -9 47 R -9 48 R -9 47 R -9 48 R -8 47 R -9 48 R -9 47 R -9 48 R --9 47 R -9 48 R -9 47 R -9 48 R -9 47 R -9 48 R -9 47 R -8 48 R -9 47 R --9 48 R -9 47 R -9 48 R -9 47 R -9 48 R -9 47 R -9 48 R -9 47 R -82 48 R --78 47 R -73 48 R -69 47 R -63 48 R -58 47 R 3 48 R 3 47 R 3 48 R 4 47 R -3 48 R 4 47 R 3 48 R 4 47 R 3 48 R 4 47 R 3 48 R 4 47 R 4 48 R 3 47 R 4 48 R -4 47 R 4 48 R 4 47 R 4 48 R 4 47 R 4 48 R 4 47 R 4 48 R 4 47 R 4 48 R 4 47 R -4 48 R 4 47 R 5 48 R 4 47 R 4 48 R 5 47 R 4 48 R 5 47 R 4 48 R 5 47 R 4 48 R -5 48 R 4 47 R 5 48 R 5 47 R 5 48 R 4 47 R 5 48 R 5 47 R 5 48 R 5 47 R 5 48 R -5 47 R 5 48 R 5 47 R 5 48 R 5 47 R 5 48 R 6 47 R 5 48 R 5 47 R D L2 -10220 10408 M 5114 0 R D 10220 11026 M 5114 0 R D 10220 12071 M 5114 0 R D -10220 13733 M 5114 0 R D 10220 15633 M 5114 0 R D 20.000000 SL L3 -13873 10408 M -116 618 R -196 1045 R -2061 1662 R -1280 1900 R 0 1663 R D -10.000000 SL L0 13984 10519 M -111 -111 R -111 111 R 111 -111 R D -13868 11137 M -222 -222 R 111 111 R -111 111 R 222 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a/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst b/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst -index 5b373b68..8ff32c06 100644 ---- a/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst -+++ b/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst -@@ -257,8 +257,8 @@ Types I to VI are shown schematically in :numref:`Fig. %s `. - :name: fig:bltypes - - Schematic representation of boundary layer types I to VI. The top of the -- upward arrows indicate the height \zhpar while the top of their solid line -- portions indicate \zh. -+ upward arrows indicate the height :math:`z_{\mathrm{par}}` while the top of -+ their solid line portions indicate :math:`z_{\mathrm{h}}`. - - .. list-table:: - :align: center -@@ -1810,7 +1810,7 @@ Discussion of some of the revisions - - .. list-table:: Convective and Neutral limits for velocity scales - :name: tab:vscales -- :header-rows: 1 -+ :header-rows: 2 - - * - Formulation - - Convective limit -@@ -5542,7 +5542,6 @@ stage is computed, i.e. :math:`X^{*}`. Briefly the following - calculations take place for the scalar variables: - - .. list-table:: -- :header-rows: 1 - - * - CALL bdy_impl3(): - - set up coefficients for :eq:`eq:dX_disc_top`, :eq:`eq:dX_disc` and do a -@@ -6874,7 +6874,6 @@ If the changes are significant it would be prudent to discuss them with - the UKCA code owner before lodging the change. - - .. list-table:: -- :header-rows: 1 - - * - Boundary layer inputs to UKCA - - -@@ -7032,7 +7032,6 @@ Appendix: Notation - ================== - - .. list-table:: -- :header-rows: 1 - - * - Finite difference notation - - -@@ -7059,7 +7059,6 @@ Appendix: Notation - - across the capping inversion (see :eq:`dbinv` and following text) - - .. list-table:: -- :header-rows: 1 - - * - Model variables - - -@@ -7090,7 +7090,6 @@ Appendix: Notation - - total heat flux (net radiative plus turbulent, Kms\ :math:`^{-1}`) - - .. list-table:: -- :header-rows: 1 - - * - Thresholds - - -@@ -7121,7 +7121,6 @@ Appendix: Notation - - before decoupling occurs - - .. list-table:: -- :header-rows: 1 - - * - Layer definitions and parameters - - -@@ -7187,7 +7187,6 @@ Appendix: Notation - - lifting condensation level - - .. list-table:: -- :header-rows: 1 - - * - Other parameters - - -@@ -6253,7 +6253,7 @@ included in :math:`\zeta_r` and :math:`\tilde{\alpha_t}` (in - where there is no buoyancy reversal (:math:`D \leq 0`) and the feedback - seen in LES of stratocumulus :raw-latex:`\cite[]{lock98}` with - significant buoyancy reversal --(:math:`D \raisebox{-.4ex}{$\ \stackrel{>}{{\scriptstyle \sim}} \ $}0.1`). -+(:math:`D \gtrsim 0.1`). - Furthermore, the LES of `Lock (2009)`_ indicated the - presence of cumulus penetrating up into stratocumulus could be - sufficient to enhance the feedback for small :math:`D`. Thus the option -@@ -760,10 +760,10 @@ expected to occur) and :math:`z_{\mathrm{ \mathrm{NTML}}-1}`. - .. figure:: blank.svg - :name: fig:inv_integ - -- Subgrid (lines) and model (symbols) fluxes of :math:`\thetal`: turbulent -- flux (dash-dotted, crosses), radiative flux (dashed, triangles) and total -- flux (solid). The shaded area illustrates the integrated turbulent flux that -- would be obtained were (\protect\mbox{\protect:eq:`eq:wx_std`}) used. -+ Subgrid (lines) and model (symbols) fluxes of :math:`\theta_{\ell}`: -+ turbulent flux (dash-dotted, crosses), radiative flux (dashed, triangles) -+ and total flux (solid). The shaded area illustrates the integrated turbulent -+ flux that would be obtained were :eq:`eq:wx_std` used. - - .. list-table:: - :align: center -@@ -1926,9 +1926,9 @@ removed since the entrainment flux is now carried via the explicit - .. figure:: blank.svg - :name: fig:new_ksc - -- Standard UM :math:`\khtop` (solid) and revised (dotted), both scaled by -- :math:`k z_h \vtopo`. An upside-down version of :math:`\khsurf` is also -- shown (dashed) for comparison. -+ Standard UM :math:`K_h^{\mathrm{Sc}}` (solid) and revised (dotted), both -+ scaled by :math:`k z_h V_{\mathrm{Sc}}`. An upside-down version of -+ :math:`K_h^{\mathrm{surf}}` is also shown (dashed) for comparison. - - .. list-table:: - :align: center -@@ -2072,10 +2072,10 @@ turbulence scheme. - - (a) Weighting for the 1D boundary-layer scheme as a function of - :math:`\Delta x/z_{\mathrm{turb}}`, showing the function of -- Equation~:eq:`eq-tanh` (blue solid), the equation in `Boutle et al. (2014)`_ -+ Equation :eq:`eq-tanh` (blue solid), the equation in `Boutle et al. (2014)`_ - (black solid) and the TKE partitioning of `Honnert et al. (2011)`_ (mean - thick dashed, 5th/95th percentiles thin dashed). (b) Schematic showing the -- calculation of :math:`z_{\mathrm{turb}}` used in Eq.~:eq:`eq-tanh` for a -+ calculation of :math:`z_{\mathrm{turb}}` used in Eq. :eq:`eq-tanh` for a - well-mixed layer (black dotted) and a decoupled cloud layer (black solid). - - .. list-table:: -@@ -2384,10 +2384,10 @@ illustrated for a SML in :numref:`Fig. %s `. - .. figure:: blank.svg - :name: fig:fluxinterp - -- Idealised profiles of (a) :math:`\wqt` (dash-dotted line) and (b) -- :math:`{\mathcal{H}}` (dotted line), :math:`\wthl` (dash-dotted) and -- :math:`F` (dashed). The continuous lines are the turbulent fluxes on the -- model grid indicated by the dashed horizontal lines. -+ Idealised profiles of (a) :math:`\overline{w'q_t'}` (dash-dotted line) and -+ (b) :math:`{\mathcal{H}}` (dotted line), :math:`\overline{w'\theta_{\ell}'}` -+ (dash-dotted) and :math:`F` (dashed). The continuous lines are the turbulent -+ fluxes on the model grid indicated by the dashed horizontal lines. - - .. list-table:: - :align: center -@@ -2649,9 +2649,9 @@ large-scale vertical velocity evaluated at the inversion, - :name: fig:rev_fluxes - - Subgrid (lines) and model (symbols) profiles and fluxes of, top row, -- :math:`q_t` and, bottom row, :math:`\thetal`: turbulent fluxes (dash-dotted, -- crosses), subsidence fluxes (dotted, diamonds), radiative flux (dashed, -- triangles) and total flux (solid, squares). -+ :math:`q_t` and, bottom row, :math:`\theta_{\ell}`: turbulent fluxes -+ (dash-dotted, crosses), subsidence fluxes (dotted, diamonds), radiative flux -+ (dashed, triangles) and total flux (solid, squares). - - .. list-table:: - :align: center -@@ -6520,10 +6520,10 @@ therefore zero entrainment and turbulent mixing). - .. figure:: blank.svg - :name: fig:dradts - -- Time series from LES of :math:`\Delta_\radf` (solid), -- :math:`\Delta_\radf^{LW}` (dotted), :math:`-\Delta_\radf^{SW}` (dashed) and -+ Time series from LES of :math:`\Delta_F` (solid), -+ :math:`\Delta_F^{LW}` (dotted), :math:`-\Delta_F^{SW}` (dashed) and - the 8A (dash-dot) and 9B (dash-dot-dot-dot) parametrizations of -- :math:`\Delta_\radf`. -+ :math:`\Delta_F`. - - .. list-table:: - :align: center diff --git a/documentation/source/science_guide/turbulence_schemes/nbldoc_zidiag.eps b/documentation/source/science_guide/turbulence_schemes/nbldoc_zidiag.eps deleted file mode 100644 index feab8facc1..0000000000 --- a/documentation/source/science_guide/turbulence_schemes/nbldoc_zidiag.eps +++ /dev/null @@ -1,183 +0,0 @@ -%!PS-Adobe-2.0 EPSF-2.0 -%%BoundingBox: 73 43 287 285 -%%HiResBoundingBox: 74 43.5 286 284.5 -%%Title: Graphics produced by WAVE -%%For: lock@visual -%%Creator: WAVE Version 7.00 (Linux i386) -%%CreationDate: Wed Oct 11 13:31:44 2000 -%%EndComments -% EPSF created by ps2eps 1.68 -%%BeginProlog -save -countdictstack -mark -newpath -/showpage {} def -/setpagedevice {pop} def -%%EndProlog -%%Page 1 1 -%+ wave_prolog.ps -- Prolog for PV-WAVE CL generated PostScript files -%+ $Id: wave_prolog.ps,v 1.5 1998/06/05 17:44:06 thaux Exp $ -%+ Copyright (c) 1989-1992 Precision Visuals, Inc. 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file mode 100644 index 643e41b211..0000000000 --- a/documentation/source/science_guide/turbulence_schemes/refs.bib +++ /dev/null @@ -1,1750 +0,0 @@ - -@book{Iter_SLS_Saad, - author = {Saad, Y.}, - title = {Iterative Methods for Sparse Linear Systems}, - year = {2003}, - isbn = {0898715342}, - publisher = {Society for Industrial and Applied Mathematics}, - address = {Philadelphia, PA, USA}, - } - -@Article{lock00, - author = {A. 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In moderately unstable and slightly stable conditions nondimensional terms of all the moment budgets studied agree reasonably well with results reported from the Kansas study (after application of a flow distortion correction). In the near-neutral range, where the present experiment contains a large amount of data, results deviate significantly from previous studies in general and, in particular, for ideal, zero-pressure gradient turbulent boundary layers. Several moments, such as u2W, v2w and W2 are not, as expected, constant in the surface layer, but vary logarithmically with height, making instead their non dimensional vertical gradients constant. Some moments scale with the roughness length and others with a length scale containing the large-scale pressure gradient or, with an alternative interpretation the height of the neutral PBL. Evidence is presented that these apparent anomalies are due to so called “inactive” turbulence. From the present analysis, and from some previously published atmospheric studies, it is concluded that the values for the various nondimensional gradients may be universally valid, thus suggesting a modified similarity formulation for the neutral atmospheric surface layer. } -} - -@Article{Donelan2018, - Author = {Donelan, M. A.}, - Year = {2018}, - Title = {On the decrease of the oceanic drag coefficient in high winds}, - Journal = {J. Geophys. Res: Oceans}, - Volume = {123}, - Pages = {1--17}, - Doi = {10.1002/2017JC013394} -} - - -@Article{Hsu2017, - Author = {Hsu, J. and Lien, R. and D'Asaro, E. A. and Sanford, T. B.}, - Year = {2017}, - Title = {Estimates of Surface Wind Stress and Drag Coefficients in - {T}yphoon {M}egi}, - Journal = {J. Phys. Oceanogr.}, - Volume = {47}, - Pages = {545--565}, - Doi = {10.1175/JPO-D-16-0069.1} -} - - - - - diff --git a/documentation/source/science_guide/turbulence_schemes/stab_dep.eps b/documentation/source/science_guide/turbulence_schemes/stab_dep.eps deleted file mode 100644 index 83eb7f4c26..0000000000 --- a/documentation/source/science_guide/turbulence_schemes/stab_dep.eps +++ /dev/null @@ -1,1398 +0,0 @@ -%!PS-Adobe-2.0 EPSF-2.0 -%%BoundingBox: 96 66 495 621 -%%HiResBoundingBox: 97 67 494.5 620.5 -%%Title: Graphics produced by WAVE -%%For: frlk@eld206 -%%Creator: WAVE Version 8.00 (Linux i386) -%%CreationDate: Wed Mar 29 18:41:50 2006 -%%EndComments -% EPSF created by ps2eps 1.68 -%%BeginProlog -save -countdictstack -mark -newpath -/showpage {} def -/setpagedevice {pop} def -%%EndProlog -%%Page 1 1 -%+ wave_prolog.ps -- Prolog for PV-WAVE CL generated PostScript files -%+ $Id: wave_prolog.ps,v 1.5 1998/06/05 17:44:06 thaux Exp $ -%+ Copyright (c) 1989-1992 Precision Visuals, Inc. 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17417 M 24 12 R 35 35 R -0 -246 R D 1630 17452 M 0 -234 R D 1583 17218 M 105 0 R D 1805 17242 M --11 -12 R 11 -12 R 12 12 R -12 12 R D 1922 17464 M -23 -117 R 23 23 R -35 12 R 35 0 R 35 -12 R 24 -23 R 11 -35 R 0 -23 R -11 -35 R -24 -24 R --35 -12 R -35 0 R -35 12 R -12 12 R -11 23 R 0 12 R 11 12 R 12 -12 R --12 -12 R D 1992 17382 M 24 -12 R 23 -23 R 12 -35 R 0 -23 R -12 -35 R --23 -24 R -24 -12 R D 1922 17464 M 117 0 R D 1922 17452 M 59 0 R 58 12 R D -2220 19296 M 102 0 R D 1560 19213 M 12 -12 R -12 -11 R -12 11 R 0 12 R -12 24 R 12 11 R 35 12 R 46 0 R 35 -12 R 12 -11 R 12 -24 R 0 -23 R -12 -24 R --35 -23 R -58 -23 R -24 -12 R -23 -23 R -12 -35 R 0 -35 R D 1653 19260 M -24 -12 R 11 -11 R 12 -24 R 0 -23 R -12 -24 R -35 -23 R -46 -23 R D -1548 19038 M 12 12 R 23 0 R 59 -24 R 35 0 R 23 12 R 12 12 R D 1583 19050 M -59 -35 R 46 0 R 12 11 R 12 24 R 0 23 R D 1805 19038 M -11 -12 R 11 -11 R -12 11 R -12 12 R D 1969 19260 M -35 -12 R -24 -35 R -11 -58 R 0 -35 R -11 -59 R 24 -35 R 35 -11 R 23 0 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0 R D 7334 16930 M -51 0 R D -7334 17719 M -51 0 R D 7334 18113 M -51 0 R D 7334 18507 M -51 0 R D -7334 18902 M -51 0 R D 2230 14882 M 11 -103 R 10 -65 R 10 -47 R 10 -36 R -10 -28 R 11 -22 R 10 -19 R 10 -15 R 10 -12 R 11 -10 R 10 -9 R 10 -7 R -10 -6 R 10 -5 R 11 -4 R 10 -3 R 10 -2 R 10 -2 R 11 -2 R 10 -1 R 10 -1 R -10 0 R 11 0 R 10 0 R 10 1 R 10 1 R 10 1 R 11 1 R 10 1 R 10 1 R 10 2 R 11 1 R -10 2 R 10 2 R 10 2 R 11 2 R 10 2 R 10 2 R 10 2 R 10 2 R 11 2 R 10 3 R 10 2 R -10 2 R 11 3 R 10 2 R 10 2 R 10 3 R 10 2 R 11 3 R 10 2 R 10 2 R 10 3 R 11 2 R -10 3 R 10 2 R 10 3 R 11 2 R 10 3 R 10 2 R 10 3 R 10 2 R 11 2 R 10 3 R 10 2 R -10 3 R 11 2 R 10 3 R 10 2 R 10 3 R 10 2 R 11 2 R 10 3 R 10 2 R 10 3 R 11 2 R -10 2 R 10 3 R 10 2 R 11 2 R 10 3 R 10 2 R 10 2 R 10 3 R 11 2 R 10 2 R 10 2 R -10 3 R 11 2 R 10 2 R 10 2 R 10 3 R 10 2 R 11 2 R 10 2 R 10 2 R 10 3 R 11 2 R -10 2 R 10 2 R 10 2 R 11 2 R 10 3 R 10 2 R 10 2 R 10 2 R 11 2 R 10 2 R 10 2 R -10 2 R 11 2 R 10 2 R 10 2 R 10 2 R 11 2 R 10 2 R 10 2 R 10 2 R 10 2 R 11 2 R -10 2 R 10 2 R 10 2 R 11 2 R 10 2 R 10 2 R 10 2 R 10 2 R 11 1 R 10 2 R 10 2 R -10 2 R 11 2 R 10 2 R 10 2 R 10 1 R 11 2 R 10 2 R 10 2 R 10 2 R 10 2 R 11 1 R -10 2 R 10 2 R 10 2 R 11 1 R 10 2 R 10 2 R 10 2 R 10 1 R 11 2 R 10 2 R 10 1 R -10 2 R 11 2 R 10 1 R 10 2 R 10 2 R 11 1 R 10 2 R 10 2 R 10 1 R 10 2 R 11 2 R -10 1 R 10 2 R 10 2 R 11 1 R 10 2 R 10 1 R 10 2 R 11 2 R 10 1 R 10 2 R 10 1 R -10 2 R 11 1 R 10 2 R 10 1 R 10 2 R 11 1 R 10 2 R 10 1 R 10 2 R 10 1 R 11 2 R -10 1 R 10 2 R 10 1 R 11 2 R 10 1 R 10 2 R 10 1 R 11 2 R 10 1 R 10 2 R 10 1 R -10 2 R 11 1 R 10 1 R 10 2 R 10 1 R 11 2 R 10 1 R 10 1 R 10 2 R 10 1 R 11 2 R -10 1 R 10 1 R 10 2 R 11 1 R 10 1 R 10 2 R 10 1 R 11 1 R 10 2 R 10 1 R 10 1 R -10 2 R 11 1 R 10 1 R 10 2 R 10 1 R 11 1 R 10 2 R 10 1 R 10 1 R 11 2 R 10 1 R -10 1 R 10 1 R 10 2 R 11 1 R 10 1 R 10 1 R 10 2 R 11 1 R 10 1 R 10 1 R 10 2 R -10 1 R 11 1 R 10 1 R 10 2 R 10 1 R 11 1 R 10 1 R 10 1 R 10 2 R 11 1 R 10 1 R -10 1 R 10 1 R 10 2 R 11 1 R 10 1 R 10 1 R 10 1 R 11 2 R 10 1 R 10 1 R 10 1 R -10 1 R 11 1 R 10 2 R 10 1 R 10 1 R 11 1 R 10 1 R 10 1 R 10 1 R 11 1 R 10 2 R -10 1 R 10 1 R 10 1 R 11 1 R 10 1 R 10 1 R 10 1 R 11 2 R 10 1 R 10 1 R 10 1 R -10 1 R 11 1 R 10 1 R 10 1 R 10 1 R 11 1 R 10 1 R 10 1 R 10 1 R 11 2 R 10 1 R -10 1 R 10 1 R 10 1 R 11 1 R 10 1 R 10 1 R 10 1 R 11 1 R 10 1 R 10 1 R 10 1 R -11 1 R 10 1 R 10 1 R 10 1 R 10 1 R 11 1 R 10 1 R 10 1 R 10 1 R 11 1 R 10 1 R -10 1 R 10 1 R 10 1 R 11 1 R 10 1 R 10 1 R 10 1 R 11 1 R 10 1 R 10 1 R 10 1 R -11 1 R 10 1 R 10 1 R 10 1 R 10 1 R 11 1 R 10 1 R 10 1 R 10 1 R 11 1 R 10 1 R -10 1 R 10 0 R 10 1 R 11 1 R 10 1 R 10 1 R 10 1 R 11 1 R 10 1 R 10 1 R 10 1 R -11 1 R 10 1 R 10 1 R 10 1 R 10 0 R 11 1 R 10 1 R 10 1 R 10 1 R 11 1 R 10 1 R -10 1 R 10 1 R 11 1 R 10 0 R 10 1 R 10 1 R 10 1 R 11 1 R 10 1 R 10 1 R 10 1 R -11 1 R 10 0 R 10 1 R 10 1 R 10 1 R 11 1 R 10 1 R 10 1 R 10 0 R 11 1 R 10 1 R -10 1 R 10 1 R 11 1 R 10 1 R 10 0 R 10 1 R 10 1 R 11 1 R 10 1 R 10 1 R 10 0 R -11 1 R 10 1 R 10 1 R 10 1 R 10 1 R 11 0 R 10 1 R 10 1 R 10 1 R 11 1 R 10 1 R -10 0 R 10 1 R 11 1 R 10 1 R 10 1 R 10 0 R 10 1 R 11 1 R 10 1 R 10 1 R 10 0 R -11 1 R 10 1 R 10 1 R 10 1 R 11 0 R 10 1 R 10 1 R 10 1 R 10 1 R 11 0 R 10 1 R -10 1 R 10 1 R 11 1 R 10 0 R 10 1 R 10 1 R 10 1 R 11 0 R 10 1 R 10 1 R 10 1 R -11 0 R 10 1 R 10 1 R 10 1 R 11 1 R 10 0 R 10 1 R 10 1 R 10 1 R 11 0 R 10 1 R -10 1 R 10 1 R 11 0 R 10 1 R 10 1 R 10 0 R 10 1 R 11 1 R 10 1 R 10 0 R 10 1 R -11 1 R 10 1 R 10 0 R 10 1 R 11 1 R 10 1 R 10 0 R 10 1 R 10 1 R 11 0 R 10 1 R -10 1 R 10 1 R 11 0 R 10 1 R 10 1 R 10 0 R 11 1 R 10 1 R 10 1 R 10 0 R 10 1 R -11 1 R 10 0 R 10 1 R 10 1 R 11 1 R 10 0 R 10 1 R 10 1 R 10 0 R 11 1 R 10 1 R -10 0 R 10 1 R 11 1 R 10 0 R 10 1 R D L1 2230 16565 M 11 -87 R 10 -74 R -10 -65 R 10 -56 R 10 -49 R 11 -43 R 10 -38 R 10 -33 R 10 -30 R 11 -26 R -10 -23 R 10 -21 R 10 -18 R 10 -16 R 11 -14 R 10 -12 R 10 -11 R 10 -9 R -11 -8 R 10 -7 R 10 -5 R 10 -5 R 11 -4 R 10 -3 R 10 -2 R 10 -1 R 10 -1 R -11 0 R 10 1 R 10 1 R 10 1 R 11 2 R 10 2 R 10 3 R 10 3 R 11 4 R 10 3 R 10 4 R -10 5 R 10 4 R 11 5 R 10 5 R 10 5 R 10 6 R 11 6 R 10 5 R 10 6 R 10 6 R 10 7 R -11 6 R 10 7 R 10 6 R 10 7 R 11 7 R 10 7 R 10 6 R 10 7 R 11 8 R 10 7 R 10 7 R -10 7 R 10 8 R 11 7 R 10 7 R 10 8 R 10 7 R 11 8 R 10 7 R 10 8 R 10 7 R 10 8 R -11 7 R 10 8 R 10 8 R 10 7 R 11 8 R 10 7 R 10 8 R 10 8 R 11 7 R 10 8 R 10 7 R -10 8 R 10 7 R 11 8 R 10 8 R 10 7 R 10 8 R 11 7 R 10 7 R 10 8 R 10 7 R 10 8 R -11 7 R 10 7 R 10 8 R 10 7 R 11 7 R 10 8 R 10 7 R 10 7 R 11 7 R 10 7 R 10 7 R -10 8 R 10 7 R 11 7 R 10 7 R 10 7 R 10 7 R 11 6 R 10 7 R 10 7 R 10 7 R 11 7 R -10 6 R 10 7 R 10 7 R 10 6 R 11 7 R 10 7 R 10 6 R 10 7 R 11 6 R 10 7 R 10 6 R -10 7 R 10 6 R 11 6 R 10 6 R 10 7 R 10 6 R 11 6 R 10 6 R 10 6 R 10 6 R 11 7 R -10 6 R 10 6 R 10 6 R 10 5 R 11 6 R 10 6 R 10 6 R 10 6 R 11 6 R 10 5 R 10 6 R -10 6 R 10 5 R 11 6 R 10 5 R 10 6 R 10 6 R 11 5 R 10 5 R 10 6 R 10 5 R 11 6 R -10 5 R 10 5 R 10 6 R 10 5 R 11 5 R 10 5 R 10 5 R 10 5 R 11 5 R 10 6 R 10 5 R -10 5 R 11 5 R 10 4 R 10 5 R 10 5 R 10 5 R 11 5 R 10 5 R 10 5 R 10 4 R 11 5 R -10 5 R 10 4 R 10 5 R 10 5 R 11 4 R 10 5 R 10 4 R 10 5 R 11 4 R 10 5 R 10 4 R -10 5 R 11 4 R 10 4 R 10 5 R 10 4 R 10 4 R 11 5 R 10 4 R 10 4 R 10 4 R 11 4 R -10 5 R 10 4 R 10 4 R 10 4 R 11 4 R 10 4 R 10 4 R 10 4 R 11 4 R 10 4 R 10 4 R -10 4 R 11 3 R 10 4 R 10 4 R 10 4 R 10 4 R 11 3 R 10 4 R 10 4 R 10 4 R 11 3 R -10 4 R 10 4 R 10 3 R 11 4 R 10 3 R 10 4 R 10 4 R 10 3 R 11 4 R 10 3 R 10 3 R -10 4 R 11 3 R 10 4 R 10 3 R 10 4 R 10 3 R 11 3 R 10 4 R 10 3 R 10 3 R 11 3 R -10 4 R 10 3 R 10 3 R 11 3 R 10 3 R 10 4 R 10 3 R 10 3 R 11 3 R 10 3 R 10 3 R -10 3 R 11 3 R 10 3 R 10 3 R 10 3 R 10 3 R 11 3 R 10 3 R 10 3 R 10 3 R 11 3 R -10 3 R 10 3 R 10 2 R 11 3 R 10 3 R 10 3 R 10 3 R 10 3 R 11 2 R 10 3 R 10 3 R -10 2 R 11 3 R 10 3 R 10 3 R 10 2 R 10 3 R 11 3 R 10 2 R 10 3 R 10 2 R 11 3 R -10 3 R 10 2 R 10 3 R 11 2 R 10 3 R 10 2 R 10 3 R 10 2 R 11 3 R 10 2 R 10 3 R -10 2 R 11 2 R 10 3 R 10 2 R 10 3 R 11 2 R 10 2 R 10 3 R 10 2 R 10 2 R 11 3 R -10 2 R 10 2 R 10 3 R 11 2 R 10 2 R 10 2 R 10 3 R 10 2 R 11 2 R 10 2 R 10 3 R -10 2 R 11 2 R 10 2 R 10 2 R 10 2 R 11 3 R 10 2 R 10 2 R 10 2 R 10 2 R 11 2 R -10 2 R 10 2 R 10 2 R 11 2 R 10 2 R 10 2 R 10 2 R 10 2 R 11 2 R 10 2 R 10 2 R -10 2 R 11 2 R 10 2 R 10 2 R 10 2 R 11 2 R 10 2 R 10 2 R 10 2 R 10 2 R 11 2 R -10 1 R 10 2 R 10 2 R 11 2 R 10 2 R 10 2 R 10 1 R 11 2 R 10 2 R 10 2 R 10 2 R -10 1 R 11 2 R 10 2 R 10 2 R 10 1 R 11 2 R 10 2 R 10 2 R 10 1 R 10 2 R 11 2 R -10 1 R 10 2 R 10 2 R 11 2 R 10 1 R 10 2 R 10 2 R 11 1 R 10 2 R 10 1 R 10 2 R -10 2 R 11 1 R 10 2 R 10 1 R 10 2 R 11 2 R 10 1 R 10 2 R 10 1 R 10 2 R 11 1 R -10 2 R 10 2 R 10 1 R 11 2 R 10 1 R 10 2 R 10 1 R 11 2 R 10 1 R 10 2 R 10 1 R -10 2 R 11 1 R 10 1 R 10 2 R 10 1 R 11 2 R 10 1 R 10 2 R 10 1 R 11 1 R 10 2 R -10 1 R 10 2 R 10 1 R 11 1 R 10 2 R 10 1 R 10 2 R 11 1 R 10 1 R 10 2 R 10 1 R -10 1 R 11 2 R 10 1 R 10 1 R 10 2 R 11 1 R 10 1 R 10 2 R 10 1 R 11 1 R 10 1 R -10 2 R 10 1 R 10 1 R 11 2 R 10 1 R 10 1 R 10 1 R 11 2 R 10 1 R 10 1 R 10 1 R -10 2 R 11 1 R 10 1 R 10 1 R 10 1 R 11 2 R 10 1 R 10 1 R 10 1 R 11 1 R 10 2 R -10 1 R 10 1 R 10 1 R 11 1 R 10 1 R 10 2 R 10 1 R 11 1 R 10 1 R 10 1 R 10 1 R -11 1 R 10 2 R 10 1 R 10 1 R 10 1 R 11 1 R 10 1 R 10 1 R 10 1 R 11 1 R 10 1 R -10 2 R 10 1 R 10 1 R 11 1 R 10 1 R 10 1 R 10 1 R 11 1 R 10 1 R 10 1 R D L2 -2230 16573 M 11 -81 R 10 -71 R 10 -62 R 10 -56 R 10 -49 R 11 -44 R 10 -41 R -10 -36 R 10 -34 R 11 -30 R 10 -28 R 10 -26 R 10 -25 R 10 -22 R 11 -21 R -10 -19 R 10 -19 R 10 -17 R 11 -16 R 10 -15 R 10 -15 R 10 -13 R 11 -13 R -10 -12 R 10 -12 R 10 -11 R 10 -10 R 11 -10 R 10 -10 R 10 -9 R 10 -8 R -11 -9 R 10 -8 R 10 -7 R 10 -8 R 11 -7 R 10 -6 R 10 -7 R 10 -6 R 10 -6 R -11 -6 R 10 -6 R 10 -5 R 10 -5 R 11 -6 R 10 -4 R 10 -5 R 10 -5 R 10 -4 R -11 -5 R 10 -4 R 10 -4 R 10 -4 R 11 -4 R 10 -3 R 10 -4 R 10 -3 R 11 -4 R -10 -3 R 10 -3 R 10 -4 R 10 -3 R 11 -3 R 10 -3 R 10 -2 R 10 -3 R 11 -3 R -10 -3 R 10 -2 R 10 -3 R 10 -2 R 11 -2 R 10 -3 R 10 -2 R 10 -2 R 11 -3 R -10 -2 R 10 -2 R 10 -2 R 11 -2 R 10 -2 R 10 -2 R 10 -2 R 10 -2 R 11 -1 R -10 -2 R 10 -2 R 10 -2 R 11 -1 R 10 -2 R 10 -2 R 10 -1 R 10 -2 R 11 -1 R -10 -2 R 10 -1 R 10 -2 R 11 -1 R 10 -1 R 10 -2 R 10 -1 R 11 -1 R 10 -2 R -10 -1 R 10 -1 R 10 -2 R 11 -1 R 10 -1 R 10 -1 R 10 -1 R 11 -1 R 10 -2 R -10 -1 R 10 -1 R 11 -1 R 10 -1 R 10 -1 R 10 -1 R 10 -1 R 11 -1 R 10 -1 R -10 -1 R 10 -1 R 11 -1 R 10 -1 R 10 -1 R 10 -1 R 10 0 R 11 -1 R 10 -1 R -10 -1 R 10 -1 R 11 -1 R 10 -1 R 10 0 R 10 -1 R 11 -1 R 10 -1 R 10 0 R -10 -1 R 10 -1 R 11 -1 R 10 0 R 10 -1 R 10 -1 R 11 -1 R 10 0 R 10 -1 R -10 -1 R 10 0 R 11 -1 R 10 -1 R 10 0 R 10 -1 R 11 0 R 10 -1 R 10 -1 R 10 0 R -11 -1 R 10 0 R 10 -1 R 10 -1 R 10 0 R 11 -1 R 10 0 R 10 -1 R 10 0 R 11 -1 R -10 -1 R 10 0 R 10 -1 R 11 0 R 10 -1 R 10 0 R 10 -1 R 10 0 R 11 -1 R 10 0 R -10 -1 R 10 0 R 11 -1 R 10 0 R 10 0 R 10 -1 R 10 0 R 11 -1 R 10 0 R 10 -1 R -10 0 R 11 -1 R 10 0 R 10 0 R 10 -1 R 11 0 R 10 -1 R 10 0 R 10 0 R 10 -1 R -11 0 R 10 -1 R 10 0 R 10 0 R 11 -1 R 10 0 R 10 -1 R 10 0 R 10 0 R 11 -1 R -10 0 R 10 0 R 10 -1 R 11 0 R 10 0 R 10 -1 R 10 0 R 11 0 R 10 -1 R 10 0 R -10 0 R 10 -1 R 11 0 R 10 0 R 10 -1 R 10 0 R 11 0 R 10 -1 R 10 0 R 10 0 R -11 -1 R 10 0 R 10 0 R 10 -1 R 10 0 R 11 0 R 10 0 R 10 -1 R 10 0 R 11 0 R -10 -1 R 10 0 R 10 0 R 10 0 R 11 -1 R 10 0 R 10 0 R 10 0 R 11 -1 R 10 0 R -10 0 R 10 -1 R 11 0 R 10 0 R 10 0 R 10 -1 R 10 0 R 11 0 R 10 0 R 10 -1 R -10 0 R 11 0 R 10 0 R 10 0 R 10 -1 R 10 0 R 11 0 R 10 0 R 10 -1 R 10 0 R -11 0 R 10 0 R 10 -1 R 10 0 R 11 0 R 10 0 R 10 0 R 10 -1 R 10 0 R 11 0 R -10 0 R 10 0 R 10 -1 R 11 0 R 10 0 R 10 0 R 10 0 R 10 -1 R 11 0 R 10 0 R -10 0 R 10 0 R 11 -1 R 10 0 R 10 0 R 10 0 R 11 0 R 10 -1 R 10 0 R 10 0 R -10 0 R 11 0 R 10 -1 R 10 0 R 10 0 R 11 0 R 10 0 R 10 0 R 10 -1 R 11 0 R -10 0 R 10 0 R 10 0 R 10 0 R 11 -1 R 10 0 R 10 0 R 10 0 R 11 0 R 10 0 R -10 -1 R 10 0 R 10 0 R 11 0 R 10 0 R 10 0 R 10 -1 R 11 0 R 10 0 R 10 0 R -10 0 R 11 0 R 10 0 R 10 -1 R 10 0 R 10 0 R 11 0 R 10 0 R 10 0 R 10 0 R -11 -1 R 10 0 R 10 0 R 10 0 R 10 0 R 11 0 R 10 0 R 10 -1 R 10 0 R 11 0 R -10 0 R 10 0 R 10 0 R 11 0 R 10 -1 R 10 0 R 10 0 R 10 0 R 11 0 R 10 0 R -10 0 R 10 -1 R 11 0 R 10 0 R 10 0 R 10 0 R 11 0 R 10 0 R 10 0 R 10 -1 R -10 0 R 11 0 R 10 0 R 10 0 R 10 0 R 11 0 R 10 0 R 10 0 R 10 -1 R 10 0 R -11 0 R 10 0 R 10 0 R 10 0 R 11 0 R 10 0 R 10 -1 R 10 0 R 11 0 R 10 0 R -10 0 R 10 0 R 10 0 R 11 0 R 10 0 R 10 -1 R 10 0 R 11 0 R 10 0 R 10 0 R -10 0 R 10 0 R 11 0 R 10 0 R 10 0 R 10 -1 R 11 0 R 10 0 R 10 0 R 10 0 R -11 0 R 10 0 R 10 0 R 10 0 R 10 0 R 11 -1 R 10 0 R 10 0 R 10 0 R 11 0 R -10 0 R 10 0 R 10 0 R 11 0 R 10 0 R 10 -1 R 10 0 R 10 0 R 11 0 R 10 0 R -10 0 R 10 0 R 11 0 R 10 0 R 10 0 R 10 0 R 10 -1 R 11 0 R 10 0 R 10 0 R -10 0 R 11 0 R 10 0 R 10 0 R 10 0 R 11 0 R 10 0 R 10 0 R 10 -1 R 10 0 R -11 0 R 10 0 R 10 0 R 10 0 R 11 0 R 10 0 R 10 0 R 10 0 R 10 0 R 11 0 R -10 -1 R 10 0 R 10 0 R 11 0 R 10 0 R 10 0 R 10 0 R 11 0 R 10 0 R 10 0 R -10 0 R 10 0 R 11 -1 R 10 0 R 10 0 R 10 0 R 11 0 R 10 0 R 10 0 R 10 0 R -11 0 R 10 0 R 10 0 R 10 0 R 10 0 R 11 0 R 10 -1 R 10 0 R 10 0 R 11 0 R -10 0 R 10 0 R 10 0 R 10 0 R 11 0 R 10 0 R 10 0 R 10 0 R 11 0 R 10 0 R -10 -1 R D L0 10924 19589 M 15 29 R 29 29 R 29 0 R 15 -14 R 15 -30 R 0 -43 R --30 -73 R D 11012 19633 M 0 -59 R -15 -58 R 0 -59 R D 11012 19603 M --29 -73 R 0 -43 R 14 -30 R 30 -14 R 29 0 R 29 14 R 29 30 R 15 43 R 29 117 R -D 11129 19530 M 0 -43 R 14 -30 R 30 -14 R 29 0 R 29 14 R 29 30 R 29 43 R -15 59 R 0 58 R -15 0 R 0 -14 R 15 -30 R D 11173 19647 M -30 -117 R 0 -73 R D -11158 19647 M 29 0 R -29 -102 R -15 -58 R D 11357 19411 M 9 18 R 18 18 R -18 0 R 9 -9 R 9 -18 R 0 -27 R -18 -72 R D 11411 19438 M 0 -45 R -18 -72 R D -11411 19420 M -9 -36 R -18 -63 R 18 0 R D 11420 19393 M 18 27 R 18 18 R -18 9 R 19 0 R 18 -9 R 9 -18 R 0 -27 R -18 -72 R D 11511 19438 M 0 -45 R --18 -72 R D 11511 19420 M -9 -36 R -18 -63 R 18 0 R D 11520 19393 M 18 27 R -18 18 R 18 9 R 18 0 R 18 -9 R 9 -18 R 0 -27 R -18 -45 R D 11610 19438 M -0 -36 R -9 -36 R 0 -36 R D 11610 19420 M -18 -45 R 0 -27 R 9 -18 R 9 -9 R -18 0 R 18 18 R 9 18 R D 11956 19808 M -262 -468 R 14 0 R D 11956 19808 M -15 0 R -263 -468 R D 12234 19808 M -29 -15 R -44 -29 R -44 -44 R -29 -44 R --29 -58 R -15 -58 R 0 -73 R 15 -59 R 14 -44 R 29 -44 R D 12132 19720 M --30 -44 R -29 -58 R -14 -73 R 0 -117 R D 12234 19808 M -44 -29 R -44 -44 R --29 -44 R -15 -29 R -14 -44 R -15 -58 R -14 -132 R D 12059 19545 M 14 -132 R -15 -43 R 14 -30 R D 12249 19589 M 14 29 R 29 29 R 30 0 R 14 -14 R 15 -30 R -0 -43 R -29 -73 R D 12336 19633 M 0 -59 R -14 -58 R 0 -59 R D 12336 19603 M --29 -73 R 0 -43 R 15 -30 R 29 -14 R 29 0 R 29 14 R 29 30 R 30 43 R 29 117 R -D 12468 19530 M 0 -43 R 14 -30 R 15 -14 R 29 0 R 29 29 R 15 29 R D -12511 19647 M -29 -117 R 0 -73 R D 12497 19647 M 29 0 R -29 -102 R -15 -58 R -D 12657 19511 M -9 -9 R 18 -91 R -9 -9 R 0 109 R 9 -9 R -18 -91 R 9 -9 R D -12612 19484 M 9 0 R 72 -55 R 9 0 R -90 55 R 0 -9 R 90 -37 R 0 -9 R D -12702 19484 M -9 0 R -72 -55 R -9 0 R 90 55 R 0 -9 R -90 -37 R 0 -9 R D -12792 19804 M 0 9 R 9 0 R 0 -18 R -18 0 R 0 18 R 9 18 R 9 9 R 28 9 R 27 0 R -27 -9 R 9 -18 R 0 -18 R -9 -18 R -9 -9 R -18 -9 R -27 -9 R D 12874 19840 M -9 -18 R 0 -18 R -9 -18 R -9 -9 R D 12856 19849 M 9 -9 R 9 -18 R 0 -18 R --9 -18 R -18 -18 R -18 -9 R -19 0 R D 12829 19759 M 27 -9 R 9 -9 R 9 -19 R -0 -27 R -9 -18 R -18 -9 R -27 -9 R -28 0 R -27 9 R -9 9 R -9 18 R 0 18 R -18 0 R 0 -18 R -9 0 R 0 9 R D 12856 19741 M 9 -19 R 0 -27 R -9 -18 R D -12829 19759 M 18 -9 R 9 -19 R 0 -36 R -9 -18 R -9 -9 R -18 -9 R D -13094 19706 M 0 -249 R 15 0 R D 13094 19706 M 15 0 R 0 -249 R D -12977 19589 M 249 0 R 0 -15 R D 12977 19589 M 0 -15 R 249 0 R D -13299 19589 M 14 29 R 30 29 R 29 0 R 14 -14 R 15 -30 R 0 -43 R -29 -73 R D -13386 19633 M 0 -59 R -14 -58 R 0 -59 R D 13386 19603 M -29 -73 R 0 -43 R -15 -30 R 29 -14 R 29 0 R 29 14 R 30 30 R 14 43 R 29 117 R D 13503 19530 M -0 -43 R 15 -30 R 29 -14 R 29 0 R 29 14 R 30 30 R 29 43 R 14 59 R 0 58 R --14 0 R 0 -14 R 14 -30 R D 13547 19647 M -29 -117 R 0 -73 R D 13532 19647 M -30 0 R -30 -102 R -14 -58 R D 13795 19511 M -9 -9 R 18 -91 R -9 -9 R 0 109 R -9 -9 R -18 -91 R 9 -9 R D 13749 19484 M 9 0 R 73 -55 R 9 0 R -91 55 R 0 -9 R -91 -37 R 0 -9 R D 13840 19484 M -9 0 R -73 -55 R -9 0 R 91 55 R 0 -9 R --91 -37 R 0 -9 R D 13930 19804 M 0 9 R 9 0 R 0 -18 R -18 0 R 0 18 R 9 18 R -9 9 R 27 9 R 27 0 R 28 -9 R 9 -18 R 0 -18 R -9 -18 R -9 -9 R -19 -9 R --27 -9 R D 14012 19840 M 9 -18 R 0 -18 R -9 -18 R -9 -9 R D 13993 19849 M -10 -9 R 9 -18 R 0 -18 R -9 -18 R -19 -18 R -18 -9 R -18 0 R D 13966 19759 M -27 -9 R 10 -9 R 9 -19 R 0 -27 R -9 -18 R -19 -9 R -27 -9 R -27 0 R -27 9 R --9 9 R -9 18 R 0 18 R 18 0 R 0 -18 R -9 0 R 0 9 R D 13993 19741 M 10 -19 R -0 -27 R -10 -18 R D 13966 19759 M 18 -9 R 9 -19 R 0 -36 R -9 -18 R -9 -9 R --18 -9 R D 14188 19808 M 29 -44 R 15 -44 R 15 -58 R 0 -73 R -15 -59 R --29 -58 R -29 -44 R -44 -44 R -44 -29 R -29 -15 R D 14232 19720 M 0 -117 R --15 -73 R -29 -58 R -29 -44 R D 14188 19808 M 15 -29 R 14 -44 R 15 -132 R D -14232 19720 M -15 -131 R -14 -59 R -15 -43 R -14 -30 R -30 -44 R -43 -43 R --44 -30 R D 14399 19813 M -45 -154 R 18 0 R D 14426 19849 M -18 -36 R --45 -154 R D 14426 19849 M -54 -190 R D 14426 19849 M -27 -27 R -27 -18 R --18 -9 R D 14399 19813 M -18 -9 R -27 -9 R D 14661 19885 M -163 -289 R 9 0 R -D 14661 19885 M 9 0 R -163 -289 R D 14752 19804 M 0 9 R 9 0 R 0 -18 R --19 0 R 0 18 R 10 18 R 9 9 R 27 9 R 27 0 R 27 -9 R 9 -18 R 0 -18 R -9 -18 R --9 -9 R -18 -9 R -27 -9 R D 14833 19840 M 9 -18 R 0 -18 R -9 -18 R -9 -9 R D -14815 19849 M 9 -9 R 9 -18 R 0 -18 R -9 -18 R -18 -18 R -18 -9 R -18 0 R D -14788 19759 M 27 -9 R 9 -9 R 9 -19 R 0 -27 R -9 -18 R -18 -9 R -27 -9 R --27 0 R -28 9 R -9 9 R -9 18 R 0 18 R 18 0 R 0 -18 R -9 0 R 0 9 R D -14815 19741 M 9 -19 R 0 -27 R -9 -18 R D 14788 19759 M 18 -9 R 9 -19 R -0 -36 R -9 -18 R -9 -9 R -18 -9 R D 10220 11408 M 5114 0 R D 10220 11408 M -0 158 R D 10197 11196 M -35 -12 R -24 -35 R -11 -58 R 0 -35 R 11 -59 R -24 -35 R 35 -11 R 23 0 R 35 11 R 24 35 R 11 59 R 0 35 R -11 58 R -24 35 R --35 12 R -23 0 R -24 -12 R -11 -11 R -12 -24 R -12 -58 R 0 -35 R 12 -59 R -12 -23 R 11 -12 R 24 -11 R D 10220 10951 M 23 11 R 12 12 R 12 23 R 12 59 R -0 35 R -12 58 R -12 24 R -12 11 R -23 12 R D 11499 11408 M 0 158 R D -11428 11196 M -23 -117 R 23 23 R 36 12 R 35 0 R 35 -12 R 23 -23 R 12 -35 R -0 -23 R -12 -35 R -23 -24 R -35 -11 R -35 0 R -36 11 R -11 12 R -12 23 R -0 12 R 12 12 R 11 -12 R -11 -12 R D 11499 11114 M 23 -12 R 23 -23 R 12 -35 R -0 -23 R -12 -35 R -23 -24 R -23 -11 R D 11428 11196 M 117 0 R D -11428 11184 M 59 0 R 58 12 R D 12777 11408 M 0 158 R D 12602 11149 M 23 12 R -35 35 R 0 -245 R D 12649 11184 M 0 -233 R D 12602 10951 M 105 0 R D -12871 11196 M -35 -12 R -24 -35 R -12 -58 R 0 -35 R 12 -59 R 24 -35 R -35 -11 R 23 0 R 35 11 R 23 35 R 12 59 R 0 35 R -12 58 R -23 35 R -35 12 R --23 0 R -24 -12 R -11 -11 R -12 -24 R -12 -58 R 0 -35 R 12 -59 R 12 -23 R -11 -12 R 24 -11 R D 12894 10951 M 23 11 R 12 12 R 12 23 R 11 59 R 0 35 R --11 58 R -12 24 R -12 11 R -23 12 R D 14056 11408 M 0 158 R D 13880 11149 M -24 12 R 35 35 R 0 -245 R D 13927 11184 M 0 -233 R D 13880 10951 M 105 0 R D -14102 11196 M -23 -117 R 23 23 R 35 12 R 35 0 R 35 -12 R 24 -23 R 12 -35 R -0 -23 R -12 -35 R -24 -24 R -35 -11 R -35 0 R -35 11 R -11 12 R -12 23 R -0 12 R 12 12 R 11 -12 R -11 -12 R D 14172 11114 M 24 -12 R 23 -23 R 12 -35 R -0 -23 R -12 -35 R -23 -24 R -24 -11 R D 14102 11196 M 117 0 R D -14102 11184 M 59 0 R 58 12 R D 15334 11408 M 0 158 R D 15135 11149 M -12 -12 R -12 -11 R -11 11 R 0 12 R 11 24 R 12 11 R 35 12 R 47 0 R 35 -12 R -12 -11 R 11 -24 R 0 -23 R -11 -24 R -35 -23 R -59 -23 R -23 -12 R -24 -23 R --11 -35 R 0 -35 R D 15229 11196 M 23 -12 R 12 -11 R 12 -24 R 0 -23 R --12 -24 R -35 -23 R -47 -23 R D 15124 10974 M 11 12 R 24 0 R 58 -24 R 35 0 R -24 12 R 11 12 R D 15159 10986 M 58 -35 R 47 0 R 12 11 R 11 24 R 0 23 R D -15428 11196 M -35 -12 R -24 -35 R -12 -58 R 0 -35 R 12 -59 R 24 -35 R -35 -11 R 23 0 R 35 11 R 23 35 R 12 59 R 0 35 R -12 58 R -23 35 R -35 12 R --23 0 R -24 -12 R -11 -11 R -12 -24 R -12 -58 R 0 -35 R 12 -59 R 12 -23 R -11 -12 R 24 -11 R D 15451 10951 M 23 11 R 12 12 R 12 23 R 11 59 R 0 35 R --11 58 R -12 24 R -12 11 R -23 12 R D 10476 11408 M 0 79 R D 10731 11408 M -0 79 R D 10987 11408 M 0 79 R D 11243 11408 M 0 79 R D 11754 11408 M 0 79 R -D 12010 11408 M 0 79 R D 12266 11408 M 0 79 R D 12521 11408 M 0 79 R D -13033 11408 M 0 79 R D 13288 11408 M 0 79 R D 13544 11408 M 0 79 R D -13800 11408 M 0 79 R D 14311 11408 M 0 79 R D 14567 11408 M 0 79 R D -14823 11408 M 0 79 R D 15078 11408 M 0 79 R D 12213 10598 M 210 0 R D -12680 10668 M -12 -23 R -23 -24 R -94 -70 R -23 -23 R -12 -23 R D -12668 10645 M -105 0 R -23 -12 R -12 -23 R D 12645 10645 M -47 11 R -35 0 R --12 -11 R D 12645 10645 M -47 23 R -35 0 R -23 -23 R -12 -35 R D -12528 10528 M 105 0 R 24 12 R 11 23 R D 12551 10528 M 47 -12 R 35 0 R -12 12 R D 12551 10528 M 47 -23 R 35 0 R 24 23 R 11 35 R D 12773 10559 M --43 -152 R 14 0 R D 12780 10559 M -43 -152 R D 12751 10559 M 36 0 R --43 -152 R D 12758 10458 M 15 29 R 14 14 R 15 7 R 14 0 R 15 -7 R 7 -14 R -0 -22 R -14 -36 R D 12831 10501 M 0 -29 R -7 -29 R 0 -29 R D 12831 10487 M --15 -36 R 0 -22 R 8 -15 R 7 -7 R 14 0 R 15 15 R 7 14 R D 12758 10559 M -22 -7 R D 12766 10559 M 7 -14 R D 13108 10797 M -210 -374 R 11 0 R D -13108 10797 M 12 0 R -211 -374 R D 13248 10750 M -70 -245 R D 13260 10750 M --70 -245 R D 13271 10750 M -70 -245 R D 13213 10750 M 93 0 R D 13143 10505 M -175 0 R 24 70 R D 13225 10750 M 35 -12 R D 13236 10750 M 12 -23 R D -13283 10750 M -23 -23 R D 13295 10750 M -35 -12 R D 13190 10516 M -35 -11 R -D 13190 10528 M -24 -23 R D 13201 10528 M 12 -23 R D 13190 10516 M 35 -11 R -D 13260 10505 M 58 11 R D 13283 10505 M 47 35 R D 13306 10505 M 36 70 R D -10220 19296 M 5114 0 R D 10220 19296 M 0 -158 R D 11499 19296 M 0 -158 R D -12777 19296 M 0 -158 R D 14056 19296 M 0 -158 R D 15334 19296 M 0 -158 R D -10476 19296 M 0 -79 R D 10731 19296 M 0 -79 R D 10987 19296 M 0 -79 R D -11243 19296 M 0 -79 R D 11754 19296 M 0 -79 R D 12010 19296 M 0 -79 R D -12266 19296 M 0 -79 R D 12521 19296 M 0 -79 R D 13033 19296 M 0 -79 R D -13288 19296 M 0 -79 R D 13544 19296 M 0 -79 R D 13800 19296 M 0 -79 R D -14311 19296 M 0 -79 R D 14567 19296 M 0 -79 R D 14823 19296 M 0 -79 R D -15078 19296 M 0 -79 R D 10220 11408 M 0 7888 R D 10220 11408 M 102 0 R D -9618 11653 M -35 -11 R -23 -35 R -12 -59 R 0 -35 R 12 -58 R 23 -35 R -35 -12 R 24 0 R 35 12 R 23 35 R 12 58 R 0 35 R -12 59 R -23 35 R -35 11 R --24 0 R -23 -11 R -12 -12 R -11 -23 R -12 -59 R 0 -35 R 12 -58 R 11 -24 R -12 -11 R 23 -12 R D 9642 11408 M 23 12 R 12 11 R 11 24 R 12 58 R 0 35 R --12 59 R -11 23 R -12 12 R -23 11 R D 9805 11431 M -11 -11 R 11 -12 R -12 12 R -12 11 R D 9969 11653 M -35 -11 R -24 -35 R -11 -59 R 0 -35 R -11 -58 R 24 -35 R 35 -12 R 23 0 R 35 12 R 24 35 R 11 58 R 0 35 R -11 59 R --24 35 R -35 11 R -23 0 R -23 -11 R -12 -12 R -12 -23 R -12 -59 R 0 -35 R -12 -58 R 12 -24 R 12 -11 R 23 -12 R D 9992 11408 M 24 12 R 11 11 R 12 24 R -12 58 R 0 35 R -12 59 R -12 23 R -11 12 R -24 11 R D 10220 12986 M 102 0 R D -9618 13125 M -35 -11 R -23 -35 R -12 -59 R 0 -35 R 12 -58 R 23 -35 R -35 -12 R 24 0 R 35 12 R 23 35 R 12 58 R 0 35 R -12 59 R -23 35 R -35 11 R --24 0 R -23 -11 R -12 -12 R -11 -23 R -12 -59 R 0 -35 R 12 -58 R 11 -24 R -12 -11 R 23 -12 R D 9642 12880 M 23 12 R 12 11 R 11 24 R 12 58 R 0 35 R --12 59 R -11 23 R -12 12 R -23 11 R D 9805 12903 M -11 -11 R 11 -12 R -12 12 R -12 11 R D 9910 13079 M 12 -12 R -12 -12 R -11 12 R 0 12 R 11 23 R -12 12 R 35 11 R 47 0 R 35 -11 R 12 -12 R 11 -23 R 0 -24 R -11 -23 R --35 -23 R -59 -24 R -23 -11 R -24 -24 R -11 -35 R 0 -35 R D 10004 13125 M -23 -11 R 12 -12 R 12 -23 R 0 -24 R -12 -23 R -35 -23 R -47 -24 R D -9899 12903 M 11 12 R 24 0 R 58 -23 R 35 0 R 24 11 R 11 12 R D 9934 12915 M -58 -35 R 47 0 R 12 12 R 11 23 R 0 24 R D 10220 14563 M 102 0 R D -9618 14703 M -35 -12 R -23 -35 R -12 -58 R 0 -35 R 12 -59 R 23 -35 R -35 -11 R 24 0 R 35 11 R 23 35 R 12 59 R 0 35 R -12 58 R -23 35 R -35 12 R --24 0 R -23 -12 R -12 -11 R -11 -24 R -12 -58 R 0 -35 R 12 -59 R 11 -23 R -12 -12 R 23 -11 R D 9642 14458 M 23 11 R 12 12 R 11 23 R 12 59 R 0 35 R --12 58 R -11 24 R -12 11 R -23 12 R D 9805 14481 M -11 -12 R 11 -11 R -12 11 R -12 12 R D 10004 14680 M 0 -222 R D 10016 14703 M 0 -245 R D -10016 14703 M -129 -175 R 187 0 R D 9969 14458 M 82 0 R D 10220 16141 M -102 0 R D 9618 16281 M -35 -12 R -23 -35 R -12 -58 R 0 -36 R 12 -58 R -23 -35 R 35 -12 R 24 0 R 35 12 R 23 35 R 12 58 R 0 36 R -12 58 R -23 35 R --35 12 R -24 0 R -23 -12 R -12 -12 R -11 -23 R -12 -58 R 0 -36 R 12 -58 R -11 -23 R 12 -12 R 23 -12 R D 9642 16035 M 23 12 R 12 12 R 11 23 R 12 58 R -0 36 R -12 58 R -11 23 R -12 12 R -23 12 R D 9805 16059 M -11 -12 R 11 -12 R -12 12 R -12 12 R D 10039 16246 M -12 -12 R 12 -12 R 12 12 R 0 12 R -12 23 R --23 12 R -35 0 R -35 -12 R -24 -23 R -12 -24 R -11 -46 R 0 -71 R 11 -35 R -24 -23 R 35 -12 R 23 0 R 35 12 R 24 23 R 11 35 R 0 12 R -11 35 R -24 24 R --35 11 R -11 0 R -35 -11 R -24 -24 R -12 -35 R D 9981 16281 M -24 -12 R --23 -23 R -12 -24 R -12 -46 R 0 -71 R 12 -35 R 24 -23 R 23 -12 R D -9992 16035 M 24 12 R 23 23 R 12 35 R 0 12 R -12 35 R -23 24 R -24 11 R D -10220 17719 M 102 0 R D 9618 17858 M -35 -11 R -23 -35 R -12 -59 R 0 -35 R -12 -58 R 23 -35 R 35 -12 R 24 0 R 35 12 R 23 35 R 12 58 R 0 35 R -12 59 R --23 35 R -35 11 R -24 0 R -23 -11 R -12 -12 R -11 -23 R -12 -59 R 0 -35 R -12 -58 R 11 -24 R 12 -11 R 23 -12 R D 9642 17613 M 23 12 R 12 11 R 11 24 R -12 58 R 0 35 R -12 59 R -11 23 R -12 12 R -23 11 R D 9805 17636 M -11 -11 R -11 -12 R 12 12 R -12 11 R D 9957 17858 M -35 -11 R -12 -24 R 0 -35 R -12 -23 R 35 -12 R 47 0 R 35 12 R 12 23 R 0 35 R -12 24 R -35 11 R -47 0 R --23 -11 R -12 -24 R 0 -35 R 12 -23 R 23 -12 R D 10004 17753 M 23 12 R -12 23 R 0 35 R -12 24 R -23 11 R D 9957 17753 M -35 -12 R -12 -11 R --11 -24 R 0 -46 R 11 -24 R 12 -11 R 35 -12 R 47 0 R 35 12 R 12 11 R 11 24 R -0 46 R -11 24 R -12 11 R -35 12 R D 9957 17753 M -23 -12 R -12 -11 R --12 -24 R 0 -46 R 12 -24 R 12 -11 R 23 -12 R D 10004 17613 M 23 12 R 12 11 R -12 24 R 0 46 R -12 24 R -12 11 R -23 12 R D 10220 19296 M 102 0 R D -9583 19213 M 24 12 R 35 35 R 0 -245 R D 9630 19248 M 0 -233 R D 9583 19015 M -105 0 R D 9805 19038 M -11 -12 R 11 -11 R 12 11 R -12 12 R D 9969 19260 M --35 -12 R -24 -35 R -11 -58 R 0 -35 R 11 -59 R 24 -35 R 35 -11 R 23 0 R -35 11 R 24 35 R 11 59 R 0 35 R -11 58 R -24 35 R -35 12 R -23 0 R -23 -12 R --12 -11 R -12 -24 R -12 -58 R 0 -35 R 12 -59 R 12 -23 R 12 -12 R 23 -11 R D -9992 19015 M 24 11 R 11 12 R 12 23 R 12 59 R 0 35 R -12 58 R -12 24 R --11 11 R -24 12 R D 10220 11802 M 51 0 R D 10220 12197 M 51 0 R D -10220 12591 M 51 0 R D 10220 13380 M 51 0 R D 10220 13774 M 51 0 R D -10220 14169 M 51 0 R D 10220 14958 M 51 0 R D 10220 15352 M 51 0 R D -10220 15746 M 51 0 R D 10220 16535 M 51 0 R D 10220 16930 M 51 0 R D -10220 17324 M 51 0 R D 10220 18113 M 51 0 R D 10220 18507 M 51 0 R D -10220 18902 M 51 0 R D 15334 11408 M 0 7888 R D 15334 11408 M -102 0 R D -15334 12986 M -102 0 R D 15334 14563 M -102 0 R D 15334 16141 M -102 0 R D -15334 17719 M -102 0 R D 15334 19296 M -102 0 R D 15334 11802 M -51 0 R D -15334 12197 M -51 0 R D 15334 12591 M -51 0 R D 15334 13380 M -51 0 R D -15334 13774 M -51 0 R D 15334 14169 M -51 0 R D 15334 14958 M -51 0 R D -15334 15352 M -51 0 R D 15334 15746 M -51 0 R D 15334 16535 M -51 0 R D -15334 16930 M -51 0 R D 15334 17324 M -51 0 R D 15334 18113 M -51 0 R D -15334 18507 M -51 0 R D 15334 18902 M -51 0 R D 10230 19199 M 11 -82 R -10 -72 R 10 -62 R 10 -54 R 10 -49 R 11 -43 R 10 -39 R 10 -35 R 10 -32 R -11 -30 R 10 -26 R 10 -25 R 10 -22 R 10 -21 R 11 -20 R 10 -18 R 10 -17 R -10 -16 R 11 -15 R 10 -14 R 10 -13 R 10 -12 R 11 -12 R 10 -11 R 10 -10 R -10 -10 R 10 -10 R 11 -9 R 10 -8 R 10 -8 R 10 -8 R 11 -7 R 10 -8 R 10 -6 R -10 -7 R 11 -6 R 10 -6 R 10 -6 R 10 -5 R 10 -6 R 11 -5 R 10 -5 R 10 -5 R -10 -4 R 11 -5 R 10 -4 R 10 -4 R 10 -4 R 10 -4 R 11 -4 R 10 -4 R 10 -3 R -10 -4 R 11 -3 R 10 -3 R 10 -3 R 10 -3 R 11 -3 R 10 -3 R 10 -3 R 10 -3 R -10 -2 R 11 -3 R 10 -2 R 10 -3 R 10 -2 R 11 -3 R 10 -2 R 10 -2 R 10 -2 R -10 -2 R 11 -2 R 10 -2 R 10 -2 R 10 -2 R 11 -2 R 10 -2 R 10 -2 R 10 -2 R -11 -1 R 10 -2 R 10 -2 R 10 -1 R 10 -2 R 11 -1 R 10 -2 R 10 -1 R 10 -2 R -11 -1 R 10 -2 R 10 -1 R 10 -1 R 10 -2 R 11 -1 R 10 -1 R 10 -2 R 10 -1 R -11 -1 R 10 -1 R 10 -1 R 10 -1 R 11 -2 R 10 -1 R 10 -1 R 10 -1 R 10 -1 R -11 -1 R 10 -1 R 10 -1 R 10 -1 R 11 -1 R 10 -1 R 10 -1 R 10 -1 R 11 -1 R -10 0 R 10 -1 R 10 -1 R 10 -1 R 11 -1 R 10 -1 R 10 -1 R 10 0 R 11 -1 R -10 -1 R 10 -1 R 10 0 R 10 -1 R 11 -1 R 10 -1 R 10 0 R 10 -1 R 11 -1 R 10 0 R -10 -1 R 10 -1 R 11 0 R 10 -1 R 10 -1 R 10 0 R 10 -1 R 11 -1 R 10 0 R 10 -1 R -10 0 R 11 -1 R 10 -1 R 10 0 R 10 -1 R 10 0 R 11 -1 R 10 0 R 10 -1 R 10 -1 R -11 0 R 10 -1 R 10 0 R 10 -1 R 11 0 R 10 -1 R 10 0 R 10 -1 R 10 0 R 11 0 R -10 -1 R 10 0 R 10 -1 R 11 0 R 10 -1 R 10 0 R 10 -1 R 11 0 R 10 -1 R 10 0 R -10 0 R 10 -1 R 11 0 R 10 -1 R 10 0 R 10 0 R 11 -1 R 10 0 R 10 -1 R 10 0 R -10 0 R 11 -1 R 10 0 R 10 0 R 10 -1 R 11 0 R 10 -1 R 10 0 R 10 0 R 11 -1 R -10 0 R 10 0 R 10 -1 R 10 0 R 11 0 R 10 -1 R 10 0 R 10 0 R 11 -1 R 10 0 R -10 0 R 10 0 R 10 -1 R 11 0 R 10 0 R 10 -1 R 10 0 R 11 0 R 10 -1 R 10 0 R -10 0 R 11 0 R 10 -1 R 10 0 R 10 0 R 10 -1 R 11 0 R 10 0 R 10 0 R 10 -1 R -11 0 R 10 0 R 10 0 R 10 -1 R 11 0 R 10 0 R 10 0 R 10 -1 R 10 0 R 11 0 R -10 0 R 10 -1 R 10 0 R 11 0 R 10 0 R 10 -1 R 10 0 R 10 0 R 11 0 R 10 0 R -10 -1 R 10 0 R 11 0 R 10 0 R 10 -1 R 10 0 R 11 0 R 10 0 R 10 0 R 10 -1 R -10 0 R 11 0 R 10 0 R 10 0 R 10 -1 R 11 0 R 10 0 R 10 0 R 10 0 R 10 -1 R -11 0 R 10 0 R 10 0 R 10 0 R 11 -1 R 10 0 R 10 0 R 10 0 R 11 0 R 10 0 R -10 -1 R 10 0 R 10 0 R 11 0 R 10 0 R 10 -1 R 10 0 R 11 0 R 10 0 R 10 0 R -10 0 R 10 -1 R 11 0 R 10 0 R 10 0 R 10 0 R 11 0 R 10 0 R 10 -1 R 10 0 R -11 0 R 10 0 R 10 0 R 10 0 R 10 -1 R 11 0 R 10 0 R 10 0 R 10 0 R 11 0 R -10 0 R 10 -1 R 10 0 R 11 0 R 10 0 R 10 0 R 10 0 R 10 0 R 11 -1 R 10 0 R -10 0 R 10 0 R 11 0 R 10 0 R 10 0 R 10 -1 R 10 0 R 11 0 R 10 0 R 10 0 R -10 0 R 11 0 R 10 0 R 10 -1 R 10 0 R 11 0 R 10 0 R 10 0 R 10 0 R 10 0 R -11 0 R 10 -1 R 10 0 R 10 0 R 11 0 R 10 0 R 10 0 R 10 0 R 10 0 R 11 -1 R -10 0 R 10 0 R 10 0 R 11 0 R 10 0 R 10 0 R 10 0 R 11 0 R 10 -1 R 10 0 R -10 0 R 10 0 R 11 0 R 10 0 R 10 0 R 10 0 R 11 0 R 10 -1 R 10 0 R 10 0 R -11 0 R 10 0 R 10 0 R 10 0 R 10 0 R 11 0 R 10 0 R 10 -1 R 10 0 R 11 0 R -10 0 R 10 0 R 10 0 R 10 0 R 11 0 R 10 0 R 10 0 R 10 -1 R 11 0 R 10 0 R -10 0 R 10 0 R 11 0 R 10 0 R 10 0 R 10 0 R 10 0 R 11 0 R 10 -1 R 10 0 R -10 0 R 11 0 R 10 0 R 10 0 R 10 0 R 10 0 R 11 0 R 10 0 R 10 0 R 10 0 R -11 -1 R 10 0 R 10 0 R 10 0 R 11 0 R 10 0 R 10 0 R 10 0 R 10 0 R 11 0 R -10 0 R 10 0 R 10 -1 R 11 0 R 10 0 R 10 0 R 10 0 R 11 0 R 10 0 R 10 0 R -10 0 R 10 0 R 11 0 R 10 0 R 10 0 R 10 -1 R 11 0 R 10 0 R 10 0 R 10 0 R -10 0 R 11 0 R 10 0 R 10 0 R 10 0 R 11 0 R 10 0 R 10 0 R 10 -1 R 11 0 R -10 0 R 10 0 R 10 0 R 10 0 R 11 0 R 10 0 R 10 0 R 10 0 R 11 0 R 10 0 R 10 0 R -10 0 R 10 0 R 11 -1 R 10 0 R 10 0 R 10 0 R 11 0 R 10 0 R 10 0 R 10 0 R -11 0 R 10 0 R 10 0 R 10 0 R 10 0 R 11 0 R 10 0 R 10 -1 R 10 0 R 11 0 R -10 0 R 10 0 R 10 0 R 11 0 R 10 0 R 10 0 R 10 0 R 10 0 R 11 0 R 10 0 R 10 0 R -10 0 R 11 0 R 10 0 R 10 -1 R 10 0 R 10 0 R 11 0 R 10 0 R 10 0 R 10 0 R -11 0 R 10 0 R 10 0 R D L1 10230 19113 M 11 -160 R 10 -141 R 10 -125 R -10 -112 R 10 -101 R 11 -92 R 10 -83 R 10 -77 R 10 -70 R 11 -65 R 10 -60 R -10 -56 R 10 -52 R 10 -48 R 11 -46 R 10 -42 R 10 -41 R 10 -37 R 11 -36 R -10 -34 R 10 -32 R 10 -31 R 11 -28 R 10 -28 R 10 -26 R 10 -25 R 10 -24 R -11 -23 R 10 -22 R 10 -21 R 10 -20 R 11 -19 R 10 -19 R 10 -17 R 10 -17 R -11 -17 R 10 -16 R 10 -15 R 10 -15 R 10 -14 R 11 -14 R 10 -13 R 10 -13 R -10 -13 R 11 -12 R 10 -12 R 10 -11 R 10 -11 R 10 -11 R 11 -10 R 10 -10 R -10 -10 R 10 -9 R 11 -10 R 10 -9 R 10 -9 R 10 -8 R 11 -8 R 10 -8 R 10 -8 R -10 -8 R 10 -8 R 11 -7 R 10 -7 R 10 -7 R 10 -7 R 11 -6 R 10 -7 R 10 -6 R -10 -7 R 10 -6 R 11 -6 R 10 -5 R 10 -6 R 10 -6 R 11 -5 R 10 -5 R 10 -6 R -10 -5 R 11 -5 R 10 -5 R 10 -4 R 10 -5 R 10 -5 R 11 -4 R 10 -5 R 10 -4 R -10 -4 R 11 -5 R 10 -4 R 10 -4 R 10 -4 R 10 -4 R 11 -4 R 10 -3 R 10 -4 R -10 -4 R 11 -3 R 10 -4 R 10 -3 R 10 -4 R 11 -3 R 10 -3 R 10 -4 R 10 -3 R -10 -3 R 11 -3 R 10 -3 R 10 -3 R 10 -3 R 11 -3 R 10 -3 R 10 -3 R 10 -2 R -11 -3 R 10 -3 R 10 -2 R 10 -3 R 10 -3 R 11 -2 R 10 -3 R 10 -2 R 10 -3 R -11 -2 R 10 -2 R 10 -3 R 10 -2 R 10 -2 R 11 -3 R 10 -2 R 10 -2 R 10 -2 R -11 -2 R 10 -2 R 10 -2 R 10 -2 R 11 -2 R 10 -2 R 10 -2 R 10 -2 R 10 -2 R -11 -2 R 10 -2 R 10 -2 R 10 -2 R 11 -2 R 10 -1 R 10 -2 R 10 -2 R 10 -2 R -11 -1 R 10 -2 R 10 -2 R 10 -1 R 11 -2 R 10 -1 R 10 -2 R 10 -2 R 11 -1 R -10 -2 R 10 -1 R 10 -2 R 10 -1 R 11 -2 R 10 -1 R 10 -2 R 10 -1 R 11 -1 R -10 -2 R 10 -1 R 10 -2 R 11 -1 R 10 -1 R 10 -2 R 10 -1 R 10 -1 R 11 -1 R -10 -2 R 10 -1 R 10 -1 R 11 -1 R 10 -2 R 10 -1 R 10 -1 R 10 -1 R 11 -1 R -10 -2 R 10 -1 R 10 -1 R 11 -1 R 10 -1 R 10 -1 R 10 -1 R 11 -1 R 10 -1 R -10 -1 R 10 -2 R 10 -1 R 11 -1 R 10 -1 R 10 -1 R 10 -1 R 11 -1 R 10 -1 R -10 -1 R 10 -1 R 10 -1 R 11 -1 R 10 0 R 10 -1 R 10 -1 R 11 -1 R 10 -1 R -10 -1 R 10 -1 R 11 -1 R 10 -1 R 10 -1 R 10 -1 R 10 0 R 11 -1 R 10 -1 R -10 -1 R 10 -1 R 11 -1 R 10 0 R 10 -1 R 10 -1 R 11 -1 R 10 -1 R 10 0 R -10 -1 R 10 -1 R 11 -1 R 10 0 R 10 -1 R 10 -1 R 11 -1 R 10 0 R 10 -1 R -10 -1 R 10 -1 R 11 0 R 10 -1 R 10 -1 R 10 -1 R 11 0 R 10 -1 R 10 -1 R 10 0 R -11 -1 R 10 -1 R 10 0 R 10 -1 R 10 -1 R 11 0 R 10 -1 R 10 -1 R 10 0 R 11 -1 R -10 0 R 10 -1 R 10 -1 R 10 0 R 11 -1 R 10 -1 R 10 0 R 10 -1 R 11 0 R 10 -1 R -10 0 R 10 -1 R 11 -1 R 10 0 R 10 -1 R 10 0 R 10 -1 R 11 0 R 10 -1 R 10 -1 R -10 0 R 11 -1 R 10 0 R 10 -1 R 10 0 R 10 -1 R 11 0 R 10 -1 R 10 0 R 10 -1 R -11 0 R 10 -1 R 10 0 R 10 -1 R 11 0 R 10 -1 R 10 0 R 10 -1 R 10 0 R 11 -1 R -10 0 R 10 -1 R 10 0 R 11 -1 R 10 0 R 10 -1 R 10 0 R 11 -1 R 10 0 R 10 -1 R -10 0 R 10 0 R 11 -1 R 10 0 R 10 -1 R 10 0 R 11 -1 R 10 0 R 10 -1 R 10 0 R -10 0 R 11 -1 R 10 0 R 10 -1 R 10 0 R 11 -1 R 10 0 R 10 0 R 10 -1 R 11 0 R -10 -1 R 10 0 R 10 0 R 10 -1 R 11 0 R 10 -1 R 10 0 R 10 0 R 11 -1 R 10 0 R -10 0 R 10 -1 R 10 0 R 11 -1 R 10 0 R 10 0 R 10 -1 R 11 0 R 10 0 R 10 -1 R -10 0 R 11 -1 R 10 0 R 10 0 R 10 -1 R 10 0 R 11 0 R 10 -1 R 10 0 R 10 0 R -11 -1 R 10 0 R 10 0 R 10 -1 R 11 0 R 10 0 R 10 -1 R 10 0 R 10 0 R 11 -1 R -10 0 R 10 0 R 10 -1 R 11 0 R 10 0 R 10 -1 R 10 0 R 10 0 R 11 -1 R 10 0 R -10 0 R 10 0 R 11 -1 R 10 0 R 10 0 R 10 -1 R 11 0 R 10 0 R 10 -1 R 10 0 R -10 0 R 11 0 R 10 -1 R 10 0 R 10 0 R 11 -1 R 10 0 R 10 0 R 10 0 R 10 -1 R -11 0 R 10 0 R 10 -1 R 10 0 R 11 0 R 10 0 R 10 -1 R 10 0 R 11 0 R 10 0 R -10 -1 R 10 0 R 10 0 R 11 -1 R 10 0 R 10 0 R 10 0 R 11 -1 R 10 0 R 10 0 R -10 0 R 11 -1 R 10 0 R 10 0 R 10 0 R 10 -1 R 11 0 R 10 0 R 10 0 R 10 -1 R -11 0 R 10 0 R 10 0 R 10 -1 R 10 0 R 11 0 R 10 0 R 10 0 R 10 -1 R 11 0 R -10 0 R 10 0 R 10 -1 R 11 0 R 10 0 R 10 0 R 10 -1 R 10 0 R 11 0 R 10 0 R -10 0 R 10 -1 R 11 0 R 10 0 R 10 0 R 10 -1 R 10 0 R 11 0 R 10 0 R 10 0 R -10 -1 R 11 0 R 10 0 R 10 0 R 10 -1 R 11 0 R 10 0 R 10 0 R 10 0 R 10 -1 R -11 0 R 10 0 R 10 0 R 10 0 R 11 -1 R 10 0 R 10 0 R 10 0 R 11 0 R 10 -1 R -10 0 R 10 0 R 10 0 R 11 0 R 10 -1 R 10 0 R 10 0 R 11 0 R 10 0 R 10 -1 R -10 0 R 10 0 R 11 0 R 10 0 R 10 0 R 10 -1 R 11 0 R 10 0 R 10 0 R D L2 -10230 19147 M 11 -129 R 10 -114 R 10 -101 R 10 -90 R 10 -82 R 11 -74 R -10 -67 R 10 -62 R 10 -58 R 11 -53 R 10 -50 R 10 -46 R 10 -43 R 10 -41 R -11 -38 R 10 -37 R 10 -34 R 10 -33 R 11 -31 R 10 -30 R 10 -29 R 10 -27 R -11 -26 R 10 -25 R 10 -24 R 10 -23 R 10 -23 R 11 -21 R 10 -21 R 10 -21 R -10 -19 R 11 -19 R 10 -19 R 10 -18 R 10 -17 R 11 -17 R 10 -16 R 10 -16 R -10 -16 R 10 -15 R 11 -15 R 10 -15 R 10 -14 R 10 -14 R 11 -13 R 10 -13 R -10 -13 R 10 -13 R 10 -13 R 11 -12 R 10 -12 R 10 -11 R 10 -12 R 11 -11 R -10 -11 R 10 -11 R 10 -11 R 11 -10 R 10 -10 R 10 -11 R 10 -9 R 10 -10 R -11 -10 R 10 -9 R 10 -10 R 10 -9 R 11 -9 R 10 -8 R 10 -9 R 10 -9 R 10 -8 R -11 -9 R 10 -8 R 10 -8 R 10 -8 R 11 -8 R 10 -7 R 10 -8 R 10 -7 R 11 -8 R -10 -7 R 10 -7 R 10 -8 R 10 -7 R 11 -6 R 10 -7 R 10 -7 R 10 -7 R 11 -6 R -10 -7 R 10 -6 R 10 -6 R 10 -7 R 11 -6 R 10 -6 R 10 -6 R 10 -6 R 11 -6 R -10 -6 R 10 -5 R 10 -6 R 11 -6 R 10 -5 R 10 -6 R 10 -5 R 10 -5 R 11 -6 R -10 -5 R 10 -5 R 10 -5 R 11 -5 R 10 -5 R 10 -5 R 10 -5 R 11 -5 R 10 -5 R -10 -4 R 10 -5 R 10 -5 R 11 -4 R 10 -5 R 10 -4 R 10 -5 R 11 -4 R 10 -4 R -10 -5 R 10 -4 R 10 -4 R 11 -4 R 10 -4 R 10 -4 R 10 -4 R 11 -4 R 10 -4 R -10 -4 R 10 -4 R 11 -4 R 10 -4 R 10 -3 R 10 -4 R 10 -4 R 11 -4 R 10 -3 R -10 -4 R 10 -3 R 11 -4 R 10 -3 R 10 -4 R 10 -3 R 10 -4 R 11 -3 R 10 -3 R -10 -4 R 10 -3 R 11 -3 R 10 -3 R 10 -3 R 10 -4 R 11 -3 R 10 -3 R 10 -3 R -10 -3 R 10 -3 R 11 -3 R 10 -3 R 10 -3 R 10 -3 R 11 -3 R 10 -2 R 10 -3 R -10 -3 R 11 -3 R 10 -3 R 10 -2 R 10 -3 R 10 -3 R 11 -2 R 10 -3 R 10 -3 R -10 -2 R 11 -3 R 10 -2 R 10 -3 R 10 -2 R 10 -3 R 11 -2 R 10 -3 R 10 -2 R -10 -3 R 11 -2 R 10 -2 R 10 -3 R 10 -2 R 11 -2 R 10 -3 R 10 -2 R 10 -2 R -10 -3 R 11 -2 R 10 -2 R 10 -2 R 10 -2 R 11 -3 R 10 -2 R 10 -2 R 10 -2 R -10 -2 R 11 -2 R 10 -2 R 10 -2 R 10 -2 R 11 -2 R 10 -2 R 10 -2 R 10 -2 R -11 -2 R 10 -2 R 10 -2 R 10 -2 R 10 -2 R 11 -2 R 10 -2 R 10 -1 R 10 -2 R -11 -2 R 10 -2 R 10 -2 R 10 -2 R 11 -1 R 10 -2 R 10 -2 R 10 -2 R 10 -1 R -11 -2 R 10 -2 R 10 -1 R 10 -2 R 11 -2 R 10 -1 R 10 -2 R 10 -2 R 10 -1 R -11 -2 R 10 -2 R 10 -1 R 10 -2 R 11 -1 R 10 -2 R 10 -1 R 10 -2 R 11 -1 R -10 -2 R 10 -1 R 10 -2 R 10 -1 R 11 -2 R 10 -1 R 10 -2 R 10 -1 R 11 -2 R -10 -1 R 10 -2 R 10 -1 R 10 -1 R 11 -2 R 10 -1 R 10 -2 R 10 -1 R 11 -1 R -10 -2 R 10 -1 R 10 -1 R 11 -2 R 10 -1 R 10 -1 R 10 -2 R 10 -1 R 11 -1 R -10 -1 R 10 -2 R 10 -1 R 11 -1 R 10 -1 R 10 -2 R 10 -1 R 10 -1 R 11 -1 R -10 -1 R 10 -2 R 10 -1 R 11 -1 R 10 -1 R 10 -1 R 10 -2 R 11 -1 R 10 -1 R -10 -1 R 10 -1 R 10 -1 R 11 -1 R 10 -2 R 10 -1 R 10 -1 R 11 -1 R 10 -1 R -10 -1 R 10 -1 R 11 -1 R 10 -1 R 10 -1 R 10 -1 R 10 -1 R 11 -1 R 10 -1 R -10 -1 R 10 -1 R 11 -1 R 10 -1 R 10 -2 R 10 -1 R 10 0 R 11 -1 R 10 -1 R -10 -1 R 10 -1 R 11 -1 R 10 -1 R 10 -1 R 10 -1 R 11 -1 R 10 -1 R 10 -1 R -10 -1 R 10 -1 R 11 -1 R 10 -1 R 10 -1 R 10 -1 R 11 0 R 10 -1 R 10 -1 R -10 -1 R 10 -1 R 11 -1 R 10 -1 R 10 -1 R 10 0 R 11 -1 R 10 -1 R 10 -1 R -10 -1 R 11 -1 R 10 -1 R 10 0 R 10 -1 R 10 -1 R 11 -1 R 10 -1 R 10 0 R -10 -1 R 11 -1 R 10 -1 R 10 -1 R 10 0 R 11 -1 R 10 -1 R 10 -1 R 10 -1 R -10 0 R 11 -1 R 10 -1 R 10 -1 R 10 0 R 11 -1 R 10 -1 R 10 -1 R 10 0 R 10 -1 R -11 -1 R 10 -1 R 10 0 R 10 -1 R 11 -1 R 10 0 R 10 -1 R 10 -1 R 11 -1 R 10 0 R -10 -1 R 10 -1 R 10 0 R 11 -1 R 10 -1 R 10 0 R 10 -1 R 11 -1 R 10 0 R 10 -1 R -10 -1 R 10 0 R 11 -1 R 10 -1 R 10 0 R 10 -1 R 11 -1 R 10 0 R 10 -1 R 10 -1 R -11 0 R 10 -1 R 10 0 R 10 -1 R 10 -1 R 11 0 R 10 -1 R 10 0 R 10 -1 R 11 -1 R -10 0 R 10 -1 R 10 0 R 11 -1 R 10 -1 R 10 0 R 10 -1 R 10 0 R 11 -1 R 10 -1 R -10 0 R 10 -1 R 11 0 R 10 -1 R 10 0 R 10 -1 R 10 -1 R 11 0 R 10 -1 R 10 0 R -10 -1 R 11 0 R 10 -1 R 10 0 R 10 -1 R 11 -1 R 10 0 R 10 -1 R 10 0 R 10 -1 R -11 0 R 10 -1 R 10 0 R 10 -1 R 11 0 R 10 -1 R 10 0 R 10 -1 R 10 0 R 11 -1 R -10 0 R 10 -1 R 10 0 R 11 -1 R 10 0 R 10 -1 R 10 0 R 11 -1 R 10 0 R 10 -1 R -10 0 R 10 -1 R 11 0 R 10 -1 R 10 0 R 10 -1 R 11 0 R 10 -1 R 10 0 R 10 -1 R -11 0 R 10 -1 R 10 0 R 10 0 R 10 -1 R 11 0 R 10 -1 R 10 0 R 10 -1 R 11 0 R -10 -1 R 10 0 R 10 0 R 10 -1 R 11 0 R 10 -1 R 10 0 R 10 -1 R 11 0 R 10 -1 R -10 0 R D L0 4726 9749 M -88 -306 R D 4741 9749 M -88 -306 R D 4755 9749 M --87 -306 R D 4682 9749 M 175 0 R 44 -14 R 15 -29 R 0 -30 R -15 -43 R --29 -30 R -58 -14 R -117 0 R D 4887 9735 M 14 -29 R 0 -30 R -14 -43 R --30 -30 R D 4857 9749 M 15 -14 R 15 -29 R 0 -30 R -15 -43 R -29 -30 R --29 -14 R D 4595 9443 M 116 0 R D 4697 9749 M 44 -14 R D 4711 9749 M -15 -29 R D 4770 9749 M -29 -29 R D 4784 9749 M -43 -14 R D 4653 9457 M --44 -14 R D 4653 9472 M -29 -29 R D 4668 9472 M 14 -29 R D 4653 9457 M -44 -14 R D 4945 9589 M 15 29 R 29 29 R 29 0 R 15 -14 R 14 -30 R 0 -58 R --29 -102 R D 5033 9633 M 0 -88 R -30 -102 R D 5033 9603 M -15 -58 R --29 -102 R 29 0 R D 5164 9618 M 0 15 R -14 0 R 0 -30 R 29 0 R 0 30 R --15 14 R -29 0 R -29 -14 R -30 -30 R -29 -58 R D 2220 1408 M 5114 0 R D -2220 1408 M 0 158 R D 2197 1196 M -35 -12 R -24 -35 R -11 -58 R 0 -35 R -11 -59 R 24 -35 R 35 -11 R 23 0 R 35 11 R 24 35 R 11 59 R 0 35 R -11 58 R --24 35 R -35 12 R -23 0 R -24 -12 R -11 -11 R -12 -24 R -12 -58 R 0 -35 R -12 -59 R 12 -23 R 11 -12 R 24 -11 R D 2220 951 M 23 11 R 12 12 R 12 23 R -12 59 R 0 35 R -12 58 R -12 24 R -12 11 R -23 12 R D 3499 1408 M 0 158 R D -3428 1196 M -23 -117 R 23 23 R 36 12 R 35 0 R 35 -12 R 23 -23 R 12 -35 R -0 -23 R -12 -35 R -23 -24 R -35 -11 R -35 0 R -36 11 R -11 12 R -12 23 R -0 12 R 12 12 R 11 -12 R -11 -12 R D 3499 1114 M 23 -12 R 23 -23 R 12 -35 R -0 -23 R -12 -35 R -23 -24 R -23 -11 R D 3428 1196 M 117 0 R D 3428 1184 M -59 0 R 58 12 R D 4777 1408 M 0 158 R D 4602 1149 M 23 12 R 35 35 R 0 -245 R -D 4649 1184 M 0 -233 R D 4602 951 M 105 0 R D 4871 1196 M -35 -12 R --24 -35 R -12 -58 R 0 -35 R 12 -59 R 24 -35 R 35 -11 R 23 0 R 35 11 R -23 35 R 12 59 R 0 35 R -12 58 R -23 35 R -35 12 R -23 0 R -24 -12 R --11 -11 R -12 -24 R -12 -58 R 0 -35 R 12 -59 R 12 -23 R 11 -12 R 24 -11 R D -4894 951 M 23 11 R 12 12 R 12 23 R 11 59 R 0 35 R -11 58 R -12 24 R -12 11 R --23 12 R D 6056 1408 M 0 158 R D 5880 1149 M 24 12 R 35 35 R 0 -245 R D -5927 1184 M 0 -233 R D 5880 951 M 105 0 R D 6102 1196 M -23 -117 R 23 23 R -35 12 R 35 0 R 35 -12 R 24 -23 R 12 -35 R 0 -23 R -12 -35 R -24 -24 R --35 -11 R -35 0 R -35 11 R -11 12 R -12 23 R 0 12 R 12 12 R 11 -12 R --11 -12 R D 6172 1114 M 24 -12 R 23 -23 R 12 -35 R 0 -23 R -12 -35 R --23 -24 R -24 -11 R D 6102 1196 M 117 0 R D 6102 1184 M 59 0 R 58 12 R D -7334 1408 M 0 158 R D 7135 1149 M 12 -12 R -12 -11 R -11 11 R 0 12 R 11 24 R -12 11 R 35 12 R 47 0 R 35 -12 R 12 -11 R 11 -24 R 0 -23 R -11 -24 R --35 -23 R -59 -23 R -23 -12 R -24 -23 R -11 -35 R 0 -35 R D 7229 1196 M -23 -12 R 12 -11 R 12 -24 R 0 -23 R -12 -24 R -35 -23 R -47 -23 R D -7124 974 M 11 12 R 24 0 R 58 -24 R 35 0 R 24 12 R 11 12 R D 7159 986 M -58 -35 R 47 0 R 12 11 R 11 24 R 0 23 R D 7428 1196 M -35 -12 R -24 -35 R --12 -58 R 0 -35 R 12 -59 R 24 -35 R 35 -11 R 23 0 R 35 11 R 23 35 R 12 59 R -0 35 R -12 58 R -23 35 R -35 12 R -23 0 R -24 -12 R -11 -11 R -12 -24 R --12 -58 R 0 -35 R 12 -59 R 12 -23 R 11 -12 R 24 -11 R D 7451 951 M 23 11 R -12 12 R 12 23 R 11 59 R 0 35 R -11 58 R -12 24 R -12 11 R -23 12 R D -2476 1408 M 0 79 R D 2731 1408 M 0 79 R D 2987 1408 M 0 79 R D 3243 1408 M -0 79 R D 3754 1408 M 0 79 R D 4010 1408 M 0 79 R D 4266 1408 M 0 79 R D -4521 1408 M 0 79 R D 5033 1408 M 0 79 R D 5288 1408 M 0 79 R D 5544 1408 M -0 79 R D 5800 1408 M 0 79 R D 6311 1408 M 0 79 R D 6567 1408 M 0 79 R D -6823 1408 M 0 79 R D 7078 1408 M 0 79 R D 4213 598 M 210 0 R D 4680 668 M --12 -23 R -23 -24 R -94 -70 R -23 -23 R -12 -23 R D 4668 645 M -105 0 R --23 -12 R -12 -23 R D 4645 645 M -47 11 R -35 0 R -12 -11 R D 4645 645 M --47 23 R -35 0 R -23 -23 R -12 -35 R D 4528 528 M 105 0 R 24 12 R 11 23 R D -4551 528 M 47 -12 R 35 0 R 12 12 R D 4551 528 M 47 -23 R 35 0 R 24 23 R -11 35 R D 4773 559 M -43 -152 R 14 0 R D 4780 559 M -43 -152 R D 4751 559 M -36 0 R -43 -152 R D 4758 458 M 15 29 R 14 14 R 15 7 R 14 0 R 15 -7 R 7 -14 R -0 -22 R -14 -36 R D 4831 501 M 0 -29 R -7 -29 R 0 -29 R D 4831 487 M --15 -36 R 0 -22 R 8 -15 R 7 -7 R 14 0 R 15 15 R 7 14 R D 4758 559 M 22 -7 R -D 4766 559 M 7 -14 R D 5108 797 M -210 -374 R 11 0 R D 5108 797 M 12 0 R --211 -374 R D 5248 750 M -70 -245 R D 5260 750 M -70 -245 R D 5271 750 M --70 -245 R D 5213 750 M 93 0 R D 5143 505 M 175 0 R 24 70 R D 5225 750 M -35 -12 R D 5236 750 M 12 -23 R D 5283 750 M -23 -23 R D 5295 750 M -35 -12 R -D 5190 516 M -35 -11 R D 5190 528 M -24 -23 R D 5201 528 M 12 -23 R D -5190 516 M 35 -11 R D 5260 505 M 58 11 R D 5283 505 M 47 35 R D 5306 505 M -36 70 R D 2220 9296 M 5114 0 R D 2220 9296 M 0 -158 R D 3499 9296 M 0 -158 R -D 4777 9296 M 0 -158 R D 6056 9296 M 0 -158 R D 7334 9296 M 0 -158 R D -2476 9296 M 0 -79 R D 2731 9296 M 0 -79 R D 2987 9296 M 0 -79 R D -3243 9296 M 0 -79 R D 3754 9296 M 0 -79 R D 4010 9296 M 0 -79 R D -4266 9296 M 0 -79 R D 4521 9296 M 0 -79 R D 5033 9296 M 0 -79 R D -5288 9296 M 0 -79 R D 5544 9296 M 0 -79 R D 5800 9296 M 0 -79 R D -6311 9296 M 0 -79 R D 6567 9296 M 0 -79 R D 6823 9296 M 0 -79 R D -7078 9296 M 0 -79 R D 2220 1408 M 0 7888 R D 2220 1408 M 102 0 R D -1618 1653 M -35 -11 R -23 -35 R -12 -59 R 0 -35 R 12 -58 R 23 -35 R 35 -12 R -24 0 R 35 12 R 23 35 R 12 58 R 0 35 R -12 59 R -23 35 R -35 11 R -24 0 R --23 -11 R -12 -12 R -11 -23 R -12 -59 R 0 -35 R 12 -58 R 11 -24 R 12 -11 R -23 -12 R D 1642 1408 M 23 12 R 12 11 R 11 24 R 12 58 R 0 35 R -12 59 R --11 23 R -12 12 R -23 11 R D 1805 1431 M -11 -11 R 11 -12 R 12 12 R -12 11 R -D 1969 1653 M -35 -11 R -24 -35 R -11 -59 R 0 -35 R 11 -58 R 24 -35 R -35 -12 R 23 0 R 35 12 R 24 35 R 11 58 R 0 35 R -11 59 R -24 35 R -35 11 R --23 0 R -23 -11 R -12 -12 R -12 -23 R -12 -59 R 0 -35 R 12 -58 R 12 -24 R -12 -11 R 23 -12 R D 1992 1408 M 24 12 R 11 11 R 12 24 R 12 58 R 0 35 R --12 59 R -12 23 R -11 12 R -24 11 R D 2220 2723 M 102 0 R D 1618 2863 M --35 -12 R -23 -35 R -12 -59 R 0 -35 R 12 -58 R 23 -35 R 35 -12 R 24 0 R -35 12 R 23 35 R 12 58 R 0 35 R -12 59 R -23 35 R -35 12 R -24 0 R -23 -12 R --12 -12 R -11 -23 R -12 -59 R 0 -35 R 12 -58 R 11 -23 R 12 -12 R 23 -12 R D -1642 2617 M 23 12 R 12 12 R 11 23 R 12 58 R 0 35 R -12 59 R -11 23 R --12 12 R -23 12 R D 1805 2641 M -11 -12 R 11 -12 R 12 12 R -12 12 R D -1910 2816 M 12 -12 R -12 -12 R -11 12 R 0 12 R 11 23 R 12 12 R 35 12 R -47 0 R 35 -12 R 12 -12 R 11 -23 R 0 -24 R -11 -23 R -35 -23 R -59 -24 R --23 -11 R -24 -24 R -11 -35 R 0 -35 R D 2004 2863 M 23 -12 R 12 -12 R -12 -23 R 0 -24 R -12 -23 R -35 -23 R -47 -24 R D 1899 2641 M 11 11 R 24 0 R -58 -23 R 35 0 R 24 12 R 11 11 R D 1934 2652 M 58 -35 R 47 0 R 12 12 R -11 23 R 0 24 R D 2220 4037 M 102 0 R D 1618 4177 M -35 -11 R -23 -36 R --12 -58 R 0 -35 R 12 -58 R 23 -35 R 35 -12 R 24 0 R 35 12 R 23 35 R 12 58 R -0 35 R -12 58 R -23 36 R -35 11 R -24 0 R -23 -11 R -12 -12 R -11 -24 R --12 -58 R 0 -35 R 12 -58 R 11 -24 R 12 -11 R 23 -12 R D 1642 3932 M 23 12 R -12 11 R 11 24 R 12 58 R 0 35 R -12 58 R -11 24 R -12 12 R -23 11 R D -1805 3955 M -11 -11 R 11 -12 R 12 12 R -12 11 R D 2004 4154 M 0 -222 R D -2016 4177 M 0 -245 R D 2016 4177 M -129 -175 R 187 0 R D 1969 3932 M 82 0 R -D 2220 5352 M 102 0 R D 1618 5492 M -35 -12 R -23 -35 R -12 -58 R 0 -35 R -12 -59 R 23 -35 R 35 -12 R 24 0 R 35 12 R 23 35 R 12 59 R 0 35 R -12 58 R --23 35 R -35 12 R -24 0 R -23 -12 R -12 -11 R -11 -24 R -12 -58 R 0 -35 R -12 -59 R 11 -23 R 12 -12 R 23 -12 R D 1642 5246 M 23 12 R 12 12 R 11 23 R -12 59 R 0 35 R -12 58 R -11 24 R -12 11 R -23 12 R D 1805 5270 M -11 -12 R -11 -12 R 12 12 R -12 12 R D 2039 5457 M -12 -12 R 12 -12 R 12 12 R 0 12 R --12 23 R -23 12 R -35 0 R -35 -12 R -24 -23 R -12 -24 R -11 -46 R 0 -70 R -11 -35 R 24 -24 R 35 -12 R 23 0 R 35 12 R 24 24 R 11 35 R 0 11 R -11 35 R --24 24 R -35 11 R -11 0 R -35 -11 R -24 -24 R -12 -35 R D 1981 5492 M --24 -12 R -23 -23 R -12 -24 R -12 -46 R 0 -70 R 12 -35 R 24 -24 R 23 -12 R D -1992 5246 M 24 12 R 23 24 R 12 35 R 0 11 R -12 35 R -23 24 R -24 11 R D -2220 6667 M 102 0 R D 1618 6807 M -35 -12 R -23 -35 R -12 -59 R 0 -35 R -12 -58 R 23 -35 R 35 -12 R 24 0 R 35 12 R 23 35 R 12 58 R 0 35 R -12 59 R --23 35 R -35 12 R -24 0 R -23 -12 R -12 -12 R -11 -23 R -12 -59 R 0 -35 R -12 -58 R 11 -23 R 12 -12 R 23 -12 R D 1642 6561 M 23 12 R 12 12 R 11 23 R -12 58 R 0 35 R -12 59 R -11 23 R -12 12 R -23 12 R D 1805 6585 M -11 -12 R -11 -12 R 12 12 R -12 12 R D 1957 6807 M -35 -12 R -12 -24 R 0 -35 R 12 -23 R -35 -12 R 47 0 R 35 12 R 12 23 R 0 35 R -12 24 R -35 12 R -47 0 R -23 -12 R --12 -24 R 0 -35 R 12 -23 R 23 -12 R D 2004 6701 M 23 12 R 12 23 R 0 35 R --12 24 R -23 12 R D 1957 6701 M -35 -11 R -12 -12 R -11 -23 R 0 -47 R -11 -23 R 12 -12 R 35 -12 R 47 0 R 35 12 R 12 12 R 11 23 R 0 47 R -11 23 R --12 12 R -35 11 R D 1957 6701 M -23 -11 R -12 -12 R -12 -23 R 0 -47 R -12 -23 R 12 -12 R 23 -12 R D 2004 6561 M 23 12 R 12 12 R 12 23 R 0 47 R --12 23 R -12 12 R -23 11 R D 2220 7981 M 102 0 R D 1583 8074 M 24 12 R -35 35 R 0 -245 R D 1630 8110 M 0 -234 R D 1583 7876 M 105 0 R D 1805 7899 M --11 -11 R 11 -12 R 12 12 R -12 11 R D 1969 8121 M -35 -11 R -24 -36 R --11 -58 R 0 -35 R 11 -58 R 24 -35 R 35 -12 R 23 0 R 35 12 R 24 35 R 11 58 R -0 35 R -11 58 R -24 36 R -35 11 R -23 0 R -23 -11 R -12 -12 R -12 -24 R --12 -58 R 0 -35 R 12 -58 R 12 -24 R 12 -11 R 23 -12 R D 1992 7876 M 24 12 R -11 11 R 12 24 R 12 58 R 0 35 R -12 58 R -12 24 R -11 12 R -24 11 R D -2220 9296 M 102 0 R D 1583 9213 M 24 12 R 35 35 R 0 -245 R D 1630 9248 M -0 -233 R D 1583 9015 M 105 0 R D 1805 9038 M -11 -12 R 11 -11 R 12 11 R --12 12 R D 1910 9213 M 12 -12 R -12 -11 R -11 11 R 0 12 R 11 24 R 12 11 R -35 12 R 47 0 R 35 -12 R 12 -11 R 11 -24 R 0 -23 R -11 -24 R -35 -23 R --59 -23 R -23 -12 R -24 -23 R -11 -35 R 0 -35 R D 2004 9260 M 23 -12 R -12 -11 R 12 -24 R 0 -23 R -12 -24 R -35 -23 R -47 -23 R D 1899 9038 M -11 12 R 24 0 R 58 -24 R 35 0 R 24 12 R 11 12 R D 1934 9050 M 58 -35 R 47 0 R -12 11 R 11 24 R 0 23 R D 2220 1737 M 51 0 R D 2220 2065 M 51 0 R D -2220 2394 M 51 0 R D 2220 3051 M 51 0 R D 2220 3380 M 51 0 R D 2220 3709 M -51 0 R D 2220 4366 M 51 0 R D 2220 4695 M 51 0 R D 2220 5023 M 51 0 R D -2220 5681 M 51 0 R D 2220 6009 M 51 0 R D 2220 6338 M 51 0 R D 2220 6995 M -51 0 R D 2220 7324 M 51 0 R D 2220 7653 M 51 0 R D 2220 8310 M 51 0 R D -2220 8639 M 51 0 R D 2220 8967 M 51 0 R D 7334 1408 M 0 7888 R D 7334 1408 M --102 0 R D 7334 2723 M -102 0 R D 7334 4037 M -102 0 R D 7334 5352 M --102 0 R D 7334 6667 M -102 0 R D 7334 7981 M -102 0 R D 7334 9296 M --102 0 R D 7334 1737 M -51 0 R D 7334 2065 M -51 0 R D 7334 2394 M -51 0 R D -7334 3051 M -51 0 R D 7334 3380 M -51 0 R D 7334 3709 M -51 0 R D -7334 4366 M -51 0 R D 7334 4695 M -51 0 R D 7334 5023 M -51 0 R D -7334 5681 M -51 0 R D 7334 6009 M -51 0 R D 7334 6338 M -51 0 R D -7334 6995 M -51 0 R D 7334 7324 M -51 0 R D 7334 7653 M -51 0 R D -7334 8310 M -51 0 R D 7334 8639 M -51 0 R D 7334 8967 M -51 0 R D -2230 8780 M 11 144 R 10 78 R 10 46 R 10 29 R 10 17 R 11 10 R 10 4 R 10 0 R -10 -3 R 11 -6 R 10 -7 R 10 -8 R 10 -10 R 10 -10 R 11 -11 R 10 -11 R 10 -12 R -10 -12 R 11 -12 R 10 -12 R 10 -13 R 10 -12 R 11 -13 R 10 -12 R 10 -13 R -10 -12 R 10 -13 R 11 -12 R 10 -12 R 10 -12 R 10 -12 R 11 -12 R 10 -11 R -10 -12 R 10 -11 R 11 -12 R 10 -11 R 10 -11 R 10 -11 R 10 -11 R 11 -10 R -10 -11 R 10 -10 R 10 -11 R 11 -10 R 10 -10 R 10 -10 R 10 -10 R 10 -9 R -11 -10 R 10 -9 R 10 -10 R 10 -9 R 11 -9 R 10 -9 R 10 -9 R 10 -9 R 11 -9 R -10 -8 R 10 -9 R 10 -8 R 10 -9 R 11 -8 R 10 -8 R 10 -8 R 10 -8 R 11 -8 R -10 -8 R 10 -8 R 10 -7 R 10 -8 R 11 -7 R 10 -8 R 10 -7 R 10 -7 R 11 -7 R -10 -8 R 10 -7 R 10 -7 R 11 -7 R 10 -6 R 10 -7 R 10 -7 R 10 -6 R 11 -7 R -10 -7 R 10 -6 R 10 -6 R 11 -7 R 10 -6 R 10 -6 R 10 -6 R 10 -7 R 11 -6 R -10 -6 R 10 -6 R 10 -5 R 11 -6 R 10 -6 R 10 -6 R 10 -6 R 11 -5 R 10 -6 R -10 -5 R 10 -6 R 10 -5 R 11 -6 R 10 -5 R 10 -6 R 10 -5 R 11 -5 R 10 -5 R -10 -5 R 10 -6 R 11 -5 R 10 -5 R 10 -5 R 10 -5 R 10 -5 R 11 -5 R 10 -4 R -10 -5 R 10 -5 R 11 -5 R 10 -4 R 10 -5 R 10 -5 R 10 -4 R 11 -5 R 10 -5 R -10 -4 R 10 -5 R 11 -4 R 10 -4 R 10 -5 R 10 -4 R 11 -5 R 10 -4 R 10 -4 R -10 -4 R 10 -5 R 11 -4 R 10 -4 R 10 -4 R 10 -4 R 11 -4 R 10 -4 R 10 -4 R -10 -4 R 10 -4 R 11 -4 R 10 -4 R 10 -4 R 10 -4 R 11 -4 R 10 -4 R 10 -4 R -10 -3 R 11 -4 R 10 -4 R 10 -4 R 10 -3 R 10 -4 R 11 -4 R 10 -3 R 10 -4 R -10 -4 R 11 -3 R 10 -4 R 10 -3 R 10 -4 R 11 -3 R 10 -4 R 10 -3 R 10 -4 R -10 -3 R 11 -3 R 10 -4 R 10 -3 R 10 -3 R 11 -4 R 10 -3 R 10 -3 R 10 -4 R -10 -3 R 11 -3 R 10 -3 R 10 -4 R 10 -3 R 11 -3 R 10 -3 R 10 -3 R 10 -3 R -11 -3 R 10 -4 R 10 -3 R 10 -3 R 10 -3 R 11 -3 R 10 -3 R 10 -3 R 10 -3 R -11 -3 R 10 -3 R 10 -3 R 10 -2 R 10 -3 R 11 -3 R 10 -3 R 10 -3 R 10 -3 R -11 -3 R 10 -3 R 10 -2 R 10 -3 R 11 -3 R 10 -3 R 10 -2 R 10 -3 R 10 -3 R -11 -3 R 10 -2 R 10 -3 R 10 -3 R 11 -2 R 10 -3 R 10 -3 R 10 -2 R 11 -3 R -10 -3 R 10 -2 R 10 -3 R 10 -2 R 11 -3 R 10 -3 R 10 -2 R 10 -3 R 11 -2 R -10 -3 R 10 -2 R 10 -3 R 10 -2 R 11 -3 R 10 -2 R 10 -3 R 10 -2 R 11 -2 R -10 -3 R 10 -2 R 10 -3 R 11 -2 R 10 -2 R 10 -3 R 10 -2 R 10 -2 R 11 -3 R -10 -2 R 10 -2 R 10 -3 R 11 -2 R 10 -2 R 10 -3 R 10 -2 R 10 -2 R 11 -2 R -10 -3 R 10 -2 R 10 -2 R 11 -2 R 10 -3 R 10 -2 R 10 -2 R 11 -2 R 10 -2 R -10 -3 R 10 -2 R 10 -2 R 11 -2 R 10 -2 R 10 -2 R 10 -2 R 11 -3 R 10 -2 R -10 -2 R 10 -2 R 10 -2 R 11 -2 R 10 -2 R 10 -2 R 10 -2 R 11 -2 R 10 -2 R -10 -2 R 10 -2 R 11 -2 R 10 -2 R 10 -2 R 10 -2 R 10 -2 R 11 -2 R 10 -2 R -10 -2 R 10 -2 R 11 -2 R 10 -2 R 10 -2 R 10 -2 R 11 -2 R 10 -2 R 10 -2 R -10 -2 R 10 -2 R 11 -2 R 10 -2 R 10 -1 R 10 -2 R 11 -2 R 10 -2 R 10 -2 R -10 -2 R 10 -2 R 11 -1 R 10 -2 R 10 -2 R 10 -2 R 11 -2 R 10 -2 R 10 -1 R -10 -2 R 11 -2 R 10 -2 R 10 -2 R 10 -1 R 10 -2 R 11 -2 R 10 -2 R 10 -1 R -10 -2 R 11 -2 R 10 -2 R 10 -1 R 10 -2 R 10 -2 R 11 -2 R 10 -1 R 10 -2 R -10 -2 R 11 -1 R 10 -2 R 10 -2 R 10 -1 R 11 -2 R 10 -2 R 10 -1 R 10 -2 R -10 -2 R 11 -1 R 10 -2 R 10 -2 R 10 -1 R 11 -2 R 10 -2 R 10 -1 R 10 -2 R -11 -1 R 10 -2 R 10 -2 R 10 -1 R 10 -2 R 11 -1 R 10 -2 R 10 -2 R 10 -1 R -11 -2 R 10 -1 R 10 -2 R 10 -1 R 10 -2 R 11 -2 R 10 -1 R 10 -2 R 10 -1 R -11 -2 R 10 -1 R 10 -2 R 10 -1 R 11 -2 R 10 -1 R 10 -2 R 10 -1 R 10 -2 R -11 -1 R 10 -2 R 10 -1 R 10 -2 R 11 -1 R 10 -2 R 10 -1 R 10 -2 R 10 -1 R -11 -2 R 10 -1 R 10 -1 R 10 -2 R 11 -1 R 10 -2 R 10 -1 R 10 -2 R 11 -1 R -10 -1 R 10 -2 R 10 -1 R 10 -2 R 11 -1 R 10 -2 R 10 -1 R 10 -1 R 11 -2 R -10 -1 R 10 -2 R 10 -1 R 11 -1 R 10 -2 R 10 -1 R 10 -1 R 10 -2 R 11 -1 R -10 -1 R 10 -2 R 10 -1 R 11 -2 R 10 -1 R 10 -1 R 10 -2 R 10 -1 R 11 -1 R -10 -2 R 10 -1 R 10 -1 R 11 -2 R 10 -1 R 10 -1 R 10 -1 R 11 -2 R 10 -1 R -10 -1 R 10 -2 R 10 -1 R 11 -1 R 10 -2 R 10 -1 R 10 -1 R 11 -1 R 10 -2 R -10 -1 R 10 -1 R 10 -1 R 11 -2 R 10 -1 R 10 -1 R 10 -1 R 11 -2 R 10 -1 R -10 -1 R 10 -1 R 11 -2 R 10 -1 R 10 -1 R 10 -1 R 10 -2 R 11 -1 R 10 -1 R -10 -1 R 10 -2 R 11 -1 R 10 -1 R 10 -1 R 10 -1 R 11 -2 R 10 -1 R 10 -1 R -10 -1 R 10 -1 R 11 -2 R 10 -1 R 10 -1 R 10 -1 R 11 -1 R 10 -1 R 10 -2 R -10 -1 R 10 -1 R 11 -1 R 10 -1 R 10 -1 R 10 -2 R 11 -1 R 10 -1 R 10 -1 R D L1 -2230 6318 M 11 -19 R 10 -20 R 10 -20 R 10 -19 R 10 -20 R 11 -19 R 10 -20 R -10 -19 R 10 -19 R 11 -20 R 10 -19 R 10 -19 R 10 -18 R 10 -19 R 11 -19 R -10 -18 R 10 -18 R 10 -19 R 11 -18 R 10 -17 R 10 -18 R 10 -18 R 11 -17 R -10 -17 R 10 -17 R 10 -17 R 10 -17 R 11 -16 R 10 -17 R 10 -16 R 10 -16 R -11 -16 R 10 -16 R 10 -16 R 10 -15 R 11 -15 R 10 -15 R 10 -15 R 10 -15 R -10 -15 R 11 -14 R 10 -15 R 10 -14 R 10 -14 R 11 -14 R 10 -13 R 10 -14 R -10 -13 R 10 -14 R 11 -13 R 10 -13 R 10 -13 R 10 -12 R 11 -13 R 10 -12 R -10 -12 R 10 -12 R 11 -12 R 10 -12 R 10 -12 R 10 -12 R 10 -11 R 11 -11 R -10 -12 R 10 -11 R 10 -11 R 11 -10 R 10 -11 R 10 -11 R 10 -10 R 10 -10 R -11 -11 R 10 -10 R 10 -10 R 10 -10 R 11 -9 R 10 -10 R 10 -10 R 10 -9 R -11 -9 R 10 -10 R 10 -9 R 10 -9 R 10 -9 R 11 -9 R 10 -8 R 10 -9 R 10 -9 R -11 -8 R 10 -8 R 10 -9 R 10 -8 R 10 -8 R 11 -8 R 10 -8 R 10 -8 R 10 -8 R -11 -7 R 10 -8 R 10 -7 R 10 -8 R 11 -7 R 10 -7 R 10 -8 R 10 -7 R 10 -7 R -11 -7 R 10 -7 R 10 -7 R 10 -6 R 11 -7 R 10 -7 R 10 -6 R 10 -7 R 11 -6 R -10 -7 R 10 -6 R 10 -6 R 10 -6 R 11 -6 R 10 -6 R 10 -6 R 10 -6 R 11 -6 R -10 -6 R 10 -6 R 10 -5 R 10 -6 R 11 -6 R 10 -5 R 10 -6 R 10 -5 R 11 -6 R -10 -5 R 10 -5 R 10 -5 R 11 -6 R 10 -5 R 10 -5 R 10 -5 R 10 -5 R 11 -5 R -10 -5 R 10 -4 R 10 -5 R 11 -5 R 10 -5 R 10 -4 R 10 -5 R 10 -5 R 11 -4 R -10 -5 R 10 -4 R 10 -4 R 11 -5 R 10 -4 R 10 -4 R 10 -5 R 11 -4 R 10 -4 R -10 -4 R 10 -4 R 10 -4 R 11 -4 R 10 -4 R 10 -4 R 10 -4 R 11 -4 R 10 -4 R -10 -4 R 10 -4 R 11 -3 R 10 -4 R 10 -4 R 10 -4 R 10 -3 R 11 -4 R 10 -3 R -10 -4 R 10 -3 R 11 -4 R 10 -3 R 10 -4 R 10 -3 R 10 -4 R 11 -3 R 10 -3 R -10 -4 R 10 -3 R 11 -3 R 10 -3 R 10 -3 R 10 -4 R 11 -3 R 10 -3 R 10 -3 R -10 -3 R 10 -3 R 11 -3 R 10 -3 R 10 -3 R 10 -3 R 11 -3 R 10 -3 R 10 -3 R -10 -2 R 10 -3 R 11 -3 R 10 -3 R 10 -3 R 10 -2 R 11 -3 R 10 -3 R 10 -3 R -10 -2 R 11 -3 R 10 -2 R 10 -3 R 10 -3 R 10 -2 R 11 -3 R 10 -2 R 10 -3 R -10 -2 R 11 -3 R 10 -2 R 10 -3 R 10 -2 R 11 -2 R 10 -3 R 10 -2 R 10 -2 R -10 -3 R 11 -2 R 10 -2 R 10 -3 R 10 -2 R 11 -2 R 10 -2 R 10 -3 R 10 -2 R -10 -2 R 11 -2 R 10 -2 R 10 -2 R 10 -3 R 11 -2 R 10 -2 R 10 -2 R 10 -2 R -11 -2 R 10 -2 R 10 -2 R 10 -2 R 10 -2 R 11 -2 R 10 -2 R 10 -2 R 10 -2 R -11 -2 R 10 -2 R 10 -2 R 10 -2 R 10 -1 R 11 -2 R 10 -2 R 10 -2 R 10 -2 R -11 -2 R 10 -1 R 10 -2 R 10 -2 R 11 -2 R 10 -1 R 10 -2 R 10 -2 R 10 -2 R -11 -1 R 10 -2 R 10 -2 R 10 -1 R 11 -2 R 10 -2 R 10 -1 R 10 -2 R 10 -2 R -11 -1 R 10 -2 R 10 -2 R 10 -1 R 11 -2 R 10 -1 R 10 -2 R 10 -1 R 11 -2 R -10 -1 R 10 -2 R 10 -1 R 10 -2 R 11 -1 R 10 -2 R 10 -1 R 10 -2 R 11 -1 R -10 -2 R 10 -1 R 10 -2 R 11 -1 R 10 -2 R 10 -1 R 10 -1 R 10 -2 R 11 -1 R -10 -1 R 10 -2 R 10 -1 R 11 -2 R 10 -1 R 10 -1 R 10 -2 R 10 -1 R 11 -1 R -10 -1 R 10 -2 R 10 -1 R 11 -1 R 10 -2 R 10 -1 R 10 -1 R 11 -1 R 10 -2 R -10 -1 R 10 -1 R 10 -1 R 11 -2 R 10 -1 R 10 -1 R 10 -1 R 11 -1 R 10 -1 R -10 -2 R 10 -1 R 10 -1 R 11 -1 R 10 -1 R 10 -1 R 10 -2 R 11 -1 R 10 -1 R -10 -1 R 10 -1 R 11 -1 R 10 -1 R 10 -1 R 10 -1 R 10 -2 R 11 -1 R 10 -1 R -10 -1 R 10 -1 R 11 -1 R 10 -1 R 10 -1 R 10 -1 R 11 -1 R 10 -1 R 10 -1 R -10 -1 R 10 -1 R 11 -1 R 10 -1 R 10 -1 R 10 -1 R 11 -1 R 10 -1 R 10 -1 R -10 -1 R 10 -1 R 11 -1 R 10 -1 R 10 -1 R 10 -1 R 11 -1 R 10 -1 R 10 -1 R -10 -1 R 11 -1 R 10 0 R 10 -1 R 10 -1 R 10 -1 R 11 -1 R 10 -1 R 10 -1 R -10 -1 R 11 -1 R 10 0 R 10 -1 R 10 -1 R 10 -1 R 11 -1 R 10 -1 R 10 -1 R -10 -1 R 11 0 R 10 -1 R 10 -1 R 10 -1 R 11 -1 R 10 -1 R 10 0 R 10 -1 R -10 -1 R 11 -1 R 10 -1 R 10 0 R 10 -1 R 11 -1 R 10 -1 R 10 -1 R 10 0 R -11 -1 R 10 -1 R 10 -1 R 10 0 R 10 -1 R 11 -1 R 10 -1 R 10 0 R 10 -1 R -11 -1 R 10 -1 R 10 0 R 10 -1 R 10 -1 R 11 -1 R 10 0 R 10 -1 R 10 -1 R -11 -1 R 10 0 R 10 -1 R 10 -1 R 11 0 R 10 -1 R 10 -1 R 10 0 R 10 -1 R 11 -1 R -10 -1 R 10 0 R 10 -1 R 11 -1 R 10 0 R 10 -1 R 10 -1 R 10 0 R 11 -1 R 10 -1 R -10 0 R 10 -1 R 11 -1 R 10 0 R 10 -1 R 10 0 R 11 -1 R 10 -1 R 10 0 R 10 -1 R -10 -1 R 11 0 R 10 -1 R 10 0 R 10 -1 R 11 -1 R 10 0 R 10 -1 R 10 -1 R 11 0 R -10 -1 R 10 0 R 10 -1 R 10 -1 R 11 0 R 10 -1 R 10 0 R 10 -1 R 11 0 R 10 -1 R -10 -1 R 10 0 R 10 -1 R 11 0 R 10 -1 R 10 0 R 10 -1 R 11 -1 R 10 0 R 10 -1 R -D L2 2230 6333 M 11 -5 R 10 -5 R 10 -5 R 10 -6 R 10 -5 R 11 -5 R 10 -6 R -10 -5 R 10 -5 R 11 -6 R 10 -5 R 10 -6 R 10 -5 R 10 -6 R 11 -5 R 10 -6 R -10 -5 R 10 -6 R 11 -6 R 10 -5 R 10 -6 R 10 -5 R 11 -6 R 10 -5 R 10 -5 R -10 -6 R 10 -5 R 11 -6 R 10 -5 R 10 -5 R 10 -6 R 11 -5 R 10 -5 R 10 -6 R -10 -5 R 11 -5 R 10 -5 R 10 -5 R 10 -5 R 10 -5 R 11 -5 R 10 -5 R 10 -5 R -10 -5 R 11 -5 R 10 -5 R 10 -5 R 10 -5 R 10 -4 R 11 -5 R 10 -5 R 10 -4 R -10 -5 R 11 -5 R 10 -4 R 10 -5 R 10 -4 R 11 -4 R 10 -5 R 10 -4 R 10 -5 R -10 -4 R 11 -4 R 10 -4 R 10 -5 R 10 -4 R 11 -4 R 10 -4 R 10 -4 R 10 -4 R -10 -4 R 11 -4 R 10 -4 R 10 -4 R 10 -4 R 11 -3 R 10 -4 R 10 -4 R 10 -4 R -11 -3 R 10 -4 R 10 -4 R 10 -3 R 10 -4 R 11 -3 R 10 -4 R 10 -3 R 10 -4 R -11 -3 R 10 -3 R 10 -4 R 10 -3 R 10 -3 R 11 -4 R 10 -3 R 10 -3 R 10 -3 R -11 -3 R 10 -4 R 10 -3 R 10 -3 R 11 -3 R 10 -3 R 10 -3 R 10 -3 R 10 -3 R -11 -3 R 10 -3 R 10 -2 R 10 -3 R 11 -3 R 10 -3 R 10 -3 R 10 -2 R 11 -3 R -10 -3 R 10 -2 R 10 -3 R 10 -3 R 11 -2 R 10 -3 R 10 -3 R 10 -2 R 11 -3 R -10 -2 R 10 -3 R 10 -2 R 10 -2 R 11 -3 R 10 -2 R 10 -3 R 10 -2 R 11 -2 R -10 -3 R 10 -2 R 10 -2 R 11 -3 R 10 -2 R 10 -2 R 10 -2 R 10 -2 R 11 -3 R -10 -2 R 10 -2 R 10 -2 R 11 -2 R 10 -2 R 10 -2 R 10 -2 R 10 -2 R 11 -2 R -10 -2 R 10 -2 R 10 -2 R 11 -2 R 10 -2 R 10 -2 R 10 -2 R 11 -2 R 10 -2 R -10 -2 R 10 -2 R 10 -1 R 11 -2 R 10 -2 R 10 -2 R 10 -2 R 11 -1 R 10 -2 R -10 -2 R 10 -2 R 11 -1 R 10 -2 R 10 -2 R 10 -1 R 10 -2 R 11 -2 R 10 -1 R -10 -2 R 10 -1 R 11 -2 R 10 -2 R 10 -1 R 10 -2 R 10 -1 R 11 -2 R 10 -1 R -10 -2 R 10 -1 R 11 -2 R 10 -1 R 10 -2 R 10 -1 R 11 -2 R 10 -1 R 10 -2 R -10 -1 R 10 -2 R 11 -1 R 10 -1 R 10 -2 R 10 -1 R 11 -1 R 10 -2 R 10 -1 R -10 -1 R 10 -2 R 11 -1 R 10 -1 R 10 -2 R 10 -1 R 11 -1 R 10 -2 R 10 -1 R -10 -1 R 11 -1 R 10 -2 R 10 -1 R 10 -1 R 10 -1 R 11 -1 R 10 -2 R 10 -1 R -10 -1 R 11 -1 R 10 -1 R 10 -1 R 10 -2 R 11 -1 R 10 -1 R 10 -1 R 10 -1 R -10 -1 R 11 -1 R 10 -1 R 10 -1 R 10 -2 R 11 -1 R 10 -1 R 10 -1 R 10 -1 R -10 -1 R 11 -1 R 10 -1 R 10 -1 R 10 -1 R 11 -1 R 10 -1 R 10 -1 R 10 -1 R -11 -1 R 10 -1 R 10 -1 R 10 -1 R 10 -1 R 11 -1 R 10 -1 R 10 -1 R 10 -1 R -11 0 R 10 -1 R 10 -1 R 10 -1 R 10 -1 R 11 -1 R 10 -1 R 10 -1 R 10 -1 R -11 -1 R 10 0 R 10 -1 R 10 -1 R 11 -1 R 10 -1 R 10 -1 R 10 -1 R 10 0 R -11 -1 R 10 -1 R 10 -1 R 10 -1 R 11 -1 R 10 0 R 10 -1 R 10 -1 R 10 -1 R -11 -1 R 10 0 R 10 -1 R 10 -1 R 11 -1 R 10 0 R 10 -1 R 10 -1 R 11 -1 R 10 0 R -10 -1 R 10 -1 R 10 -1 R 11 0 R 10 -1 R 10 -1 R 10 -1 R 11 0 R 10 -1 R -10 -1 R 10 0 R 11 -1 R 10 -1 R 10 0 R 10 -1 R 10 -1 R 11 -1 R 10 0 R 10 -1 R -10 -1 R 11 0 R 10 -1 R 10 -1 R 10 0 R 10 -1 R 11 0 R 10 -1 R 10 -1 R 10 0 R -11 -1 R 10 -1 R 10 0 R 10 -1 R 11 -1 R 10 0 R 10 -1 R 10 0 R 10 -1 R 11 -1 R -10 0 R 10 -1 R 10 0 R 11 -1 R 10 -1 R 10 0 R 10 -1 R 10 0 R 11 -1 R 10 0 R -10 -1 R 10 -1 R 11 0 R 10 -1 R 10 0 R 10 -1 R 11 0 R 10 -1 R 10 0 R 10 -1 R -10 -1 R 11 0 R 10 -1 R 10 0 R 10 -1 R 11 0 R 10 -1 R 10 0 R 10 -1 R 11 0 R -10 -1 R 10 0 R 10 -1 R 10 0 R 11 -1 R 10 0 R 10 -1 R 10 0 R 11 -1 R 10 0 R -10 -1 R 10 0 R 10 -1 R 11 0 R 10 -1 R 10 0 R 10 -1 R 11 0 R 10 -1 R 10 0 R -10 -1 R 11 0 R 10 -1 R 10 0 R 10 0 R 10 -1 R 11 0 R 10 -1 R 10 0 R 10 -1 R -11 0 R 10 -1 R 10 0 R 10 0 R 10 -1 R 11 0 R 10 -1 R 10 0 R 10 -1 R 11 0 R -10 0 R 10 -1 R 10 0 R 11 -1 R 10 0 R 10 -1 R 10 0 R 10 0 R 11 -1 R 10 0 R -10 -1 R 10 0 R 11 0 R 10 -1 R 10 0 R 10 -1 R 11 0 R 10 0 R 10 -1 R 10 0 R -10 -1 R 11 0 R 10 0 R 10 -1 R 10 0 R 11 0 R 10 -1 R 10 0 R 10 -1 R 10 0 R -11 0 R 10 -1 R 10 0 R 10 0 R 11 -1 R 10 0 R 10 -1 R 10 0 R 11 0 R 10 -1 R -10 0 R 10 0 R 10 -1 R 11 0 R 10 0 R 10 -1 R 10 0 R 11 0 R 10 -1 R 10 0 R -10 0 R 10 -1 R 11 0 R 10 0 R 10 -1 R 10 0 R 11 0 R 10 -1 R 10 0 R 10 0 R -11 -1 R 10 0 R 10 0 R 10 -1 R 10 0 R 11 0 R 10 -1 R 10 0 R 10 0 R 11 -1 R -10 0 R 10 0 R 10 -1 R 11 0 R 10 0 R 10 0 R 10 -1 R 10 0 R 11 0 R 10 -1 R -10 0 R 10 0 R 11 -1 R 10 0 R 10 0 R 10 0 R 10 -1 R 11 0 R 10 0 R 10 -1 R -10 0 R 11 0 R 10 0 R 10 -1 R D L0 12996 9749 M -44 -160 R -14 -59 R 0 -43 R -14 -30 R 15 -14 R 29 0 R 29 29 R 15 29 R D 13011 9749 M -44 -160 R -15 -59 R -0 -73 R D 12952 9749 M 73 0 R -58 -204 R -15 -58 R D 12938 9545 M 0 44 R --15 44 R -29 14 R -29 0 R -44 -14 R -29 -44 R -15 -44 R 0 -29 R 15 -44 R -14 -15 R 30 -14 R 29 0 R 29 14 R 15 15 R 14 29 R 15 44 R D 12821 9618 M --15 -29 R -14 -44 R 0 -44 R 14 -29 R D 12865 9647 M -29 -29 R -15 -29 R --15 -44 R 0 -44 R 15 -44 R 15 -14 R D 12967 9749 M 44 -14 R D 12982 9749 M -14 -29 R D 10220 1408 M 5114 0 R D 10220 1408 M 0 158 R D 10197 1196 M --35 -12 R -24 -35 R -11 -58 R 0 -35 R 11 -59 R 24 -35 R 35 -11 R 23 0 R -35 11 R 24 35 R 11 59 R 0 35 R -11 58 R -24 35 R -35 12 R -23 0 R -24 -12 R --11 -11 R -12 -24 R -12 -58 R 0 -35 R 12 -59 R 12 -23 R 11 -12 R 24 -11 R D -10220 951 M 23 11 R 12 12 R 12 23 R 12 59 R 0 35 R -12 58 R -12 24 R --12 11 R -23 12 R D 11499 1408 M 0 158 R D 11428 1196 M -23 -117 R 23 23 R -36 12 R 35 0 R 35 -12 R 23 -23 R 12 -35 R 0 -23 R -12 -35 R -23 -24 R --35 -11 R -35 0 R -36 11 R -11 12 R -12 23 R 0 12 R 12 12 R 11 -12 R --11 -12 R D 11499 1114 M 23 -12 R 23 -23 R 12 -35 R 0 -23 R -12 -35 R --23 -24 R -23 -11 R D 11428 1196 M 117 0 R D 11428 1184 M 59 0 R 58 12 R D -12777 1408 M 0 158 R D 12602 1149 M 23 12 R 35 35 R 0 -245 R D 12649 1184 M -0 -233 R D 12602 951 M 105 0 R D 12871 1196 M -35 -12 R -24 -35 R -12 -58 R -0 -35 R 12 -59 R 24 -35 R 35 -11 R 23 0 R 35 11 R 23 35 R 12 59 R 0 35 R --12 58 R -23 35 R -35 12 R -23 0 R -24 -12 R -11 -11 R -12 -24 R -12 -58 R -0 -35 R 12 -59 R 12 -23 R 11 -12 R 24 -11 R D 12894 951 M 23 11 R 12 12 R -12 23 R 11 59 R 0 35 R -11 58 R -12 24 R -12 11 R -23 12 R D 14056 1408 M -0 158 R D 13880 1149 M 24 12 R 35 35 R 0 -245 R D 13927 1184 M 0 -233 R D -13880 951 M 105 0 R D 14102 1196 M -23 -117 R 23 23 R 35 12 R 35 0 R -35 -12 R 24 -23 R 12 -35 R 0 -23 R -12 -35 R -24 -24 R -35 -11 R -35 0 R --35 11 R -11 12 R -12 23 R 0 12 R 12 12 R 11 -12 R -11 -12 R D 14172 1114 M -24 -12 R 23 -23 R 12 -35 R 0 -23 R -12 -35 R -23 -24 R -24 -11 R D -14102 1196 M 117 0 R D 14102 1184 M 59 0 R 58 12 R D 15334 1408 M 0 158 R D -15135 1149 M 12 -12 R -12 -11 R -11 11 R 0 12 R 11 24 R 12 11 R 35 12 R -47 0 R 35 -12 R 12 -11 R 11 -24 R 0 -23 R -11 -24 R -35 -23 R -59 -23 R --23 -12 R -24 -23 R -11 -35 R 0 -35 R D 15229 1196 M 23 -12 R 12 -11 R -12 -24 R 0 -23 R -12 -24 R -35 -23 R -47 -23 R D 15124 974 M 11 12 R 24 0 R -58 -24 R 35 0 R 24 12 R 11 12 R D 15159 986 M 58 -35 R 47 0 R 12 11 R -11 24 R 0 23 R D 15428 1196 M -35 -12 R -24 -35 R -12 -58 R 0 -35 R 12 -59 R -24 -35 R 35 -11 R 23 0 R 35 11 R 23 35 R 12 59 R 0 35 R -12 58 R -23 35 R --35 12 R -23 0 R -24 -12 R -11 -11 R -12 -24 R -12 -58 R 0 -35 R 12 -59 R -12 -23 R 11 -12 R 24 -11 R D 15451 951 M 23 11 R 12 12 R 12 23 R 11 59 R -0 35 R -11 58 R -12 24 R -12 11 R -23 12 R D 10476 1408 M 0 79 R D -10731 1408 M 0 79 R D 10987 1408 M 0 79 R D 11243 1408 M 0 79 R D -11754 1408 M 0 79 R D 12010 1408 M 0 79 R D 12266 1408 M 0 79 R D -12521 1408 M 0 79 R D 13033 1408 M 0 79 R D 13288 1408 M 0 79 R D -13544 1408 M 0 79 R D 13800 1408 M 0 79 R D 14311 1408 M 0 79 R D -14567 1408 M 0 79 R D 14823 1408 M 0 79 R D 15078 1408 M 0 79 R D -12213 598 M 210 0 R D 12680 668 M -12 -23 R -23 -24 R -94 -70 R -23 -23 R --12 -23 R D 12668 645 M -105 0 R -23 -12 R -12 -23 R D 12645 645 M -47 11 R --35 0 R -12 -11 R D 12645 645 M -47 23 R -35 0 R -23 -23 R -12 -35 R D -12528 528 M 105 0 R 24 12 R 11 23 R D 12551 528 M 47 -12 R 35 0 R 12 12 R D -12551 528 M 47 -23 R 35 0 R 24 23 R 11 35 R D 12773 559 M -43 -152 R 14 0 R -D 12780 559 M -43 -152 R D 12751 559 M 36 0 R -43 -152 R D 12758 458 M -15 29 R 14 14 R 15 7 R 14 0 R 15 -7 R 7 -14 R 0 -22 R -14 -36 R D -12831 501 M 0 -29 R -7 -29 R 0 -29 R D 12831 487 M -15 -36 R 0 -22 R 8 -15 R -7 -7 R 14 0 R 15 15 R 7 14 R D 12758 559 M 22 -7 R D 12766 559 M 7 -14 R D -13108 797 M -210 -374 R 11 0 R D 13108 797 M 12 0 R -211 -374 R D -13248 750 M -70 -245 R D 13260 750 M -70 -245 R D 13271 750 M -70 -245 R D -13213 750 M 93 0 R D 13143 505 M 175 0 R 24 70 R D 13225 750 M 35 -12 R D -13236 750 M 12 -23 R D 13283 750 M -23 -23 R D 13295 750 M -35 -12 R D -13190 516 M -35 -11 R D 13190 528 M -24 -23 R D 13201 528 M 12 -23 R D -13190 516 M 35 -11 R D 13260 505 M 58 11 R D 13283 505 M 47 35 R D -13306 505 M 36 70 R D 10220 9296 M 5114 0 R D 10220 9296 M 0 -158 R D -11499 9296 M 0 -158 R D 12777 9296 M 0 -158 R D 14056 9296 M 0 -158 R D -15334 9296 M 0 -158 R D 10476 9296 M 0 -79 R D 10731 9296 M 0 -79 R D -10987 9296 M 0 -79 R D 11243 9296 M 0 -79 R D 11754 9296 M 0 -79 R D -12010 9296 M 0 -79 R D 12266 9296 M 0 -79 R D 12521 9296 M 0 -79 R D -13033 9296 M 0 -79 R D 13288 9296 M 0 -79 R D 13544 9296 M 0 -79 R D -13800 9296 M 0 -79 R D 14311 9296 M 0 -79 R D 14567 9296 M 0 -79 R D -14823 9296 M 0 -79 R D 15078 9296 M 0 -79 R D 10220 1408 M 0 7888 R D -10220 1408 M 102 0 R D 9969 1653 M -35 -11 R -24 -35 R -11 -59 R 0 -35 R -11 -58 R 24 -35 R 35 -12 R 23 0 R 35 12 R 24 35 R 11 58 R 0 35 R -11 59 R --24 35 R -35 11 R -23 0 R -23 -11 R -12 -12 R -12 -23 R -12 -59 R 0 -35 R -12 -58 R 12 -24 R 12 -11 R 23 -12 R D 9992 1408 M 24 12 R 11 11 R 12 24 R -12 58 R 0 35 R -12 59 R -12 23 R -11 12 R -24 11 R D 10220 2842 M 102 0 R D -9910 2935 M 12 -11 R -12 -12 R -11 12 R 0 11 R 11 24 R 12 11 R 35 12 R -47 0 R 35 -12 R 12 -11 R 11 -24 R 0 -23 R -11 -23 R -35 -24 R -59 -23 R --23 -12 R -24 -23 R -11 -35 R 0 -35 R D 10004 2982 M 23 -12 R 12 -11 R -12 -24 R 0 -23 R -12 -23 R -35 -24 R -47 -23 R D 9899 2760 M 11 12 R 24 0 R -58 -24 R 35 0 R 24 12 R 11 12 R D 9934 2772 M 58 -35 R 47 0 R 12 11 R -11 24 R 0 23 R D 10220 4276 M 102 0 R D 10004 4393 M 0 -222 R D 10016 4416 M -0 -245 R D 10016 4416 M -129 -175 R 187 0 R D 9969 4171 M 82 0 R D -10220 5711 M 102 0 R D 10039 5815 M -12 -11 R 12 -12 R 12 12 R 0 11 R --12 24 R -23 11 R -35 0 R -35 -11 R -24 -24 R -12 -23 R -11 -47 R 0 -70 R -11 -35 R 24 -23 R 35 -12 R 23 0 R 35 12 R 24 23 R 11 35 R 0 12 R -11 35 R --24 23 R -35 12 R -11 0 R -35 -12 R -24 -23 R -12 -35 R D 9981 5850 M --24 -11 R -23 -24 R -12 -23 R -12 -47 R 0 -70 R 12 -35 R 24 -23 R 23 -12 R D -9992 5605 M 24 12 R 23 23 R 12 35 R 0 12 R -12 35 R -23 23 R -24 12 R D -10220 7145 M 102 0 R D 9957 7285 M -35 -12 R -12 -23 R 0 -36 R 12 -23 R -35 -12 R 47 0 R 35 12 R 12 23 R 0 36 R -12 23 R -35 12 R -47 0 R -23 -12 R --12 -23 R 0 -36 R 12 -23 R 23 -12 R D 10004 7179 M 23 12 R 12 23 R 0 36 R --12 23 R -23 12 R D 9957 7179 M -35 -11 R -12 -12 R -11 -23 R 0 -47 R -11 -23 R 12 -12 R 35 -12 R 47 0 R 35 12 R 12 12 R 11 23 R 0 47 R -11 23 R --12 12 R -35 11 R D 9957 7179 M -23 -11 R -12 -12 R -12 -23 R 0 -47 R -12 -23 R 12 -12 R 23 -12 R D 10004 7039 M 23 12 R 12 12 R 12 23 R 0 47 R --12 23 R -12 12 R -23 11 R D 10220 8579 M 102 0 R D 9700 8672 M 24 12 R -35 35 R 0 -246 R D 9747 8707 M 0 -234 R D 9700 8473 M 105 0 R D 9969 8719 M --35 -12 R -24 -35 R -11 -58 R 0 -35 R 11 -59 R 24 -35 R 35 -12 R 23 0 R -35 12 R 24 35 R 11 59 R 0 35 R -11 58 R -24 35 R -35 12 R -23 0 R -23 -12 R --12 -12 R -12 -23 R -12 -58 R 0 -35 R 12 -59 R 12 -23 R 12 -12 R 23 -12 R D -9992 8473 M 24 12 R 11 12 R 12 23 R 12 59 R 0 35 R -12 58 R -12 23 R --11 12 R -24 12 R D 10220 1767 M 51 0 R D 10220 2125 M 51 0 R D 10220 2484 M -51 0 R D 10220 3201 M 51 0 R D 10220 3559 M 51 0 R D 10220 3918 M 51 0 R D -10220 4635 M 51 0 R D 10220 4994 M 51 0 R D 10220 5352 M 51 0 R D -10220 6069 M 51 0 R D 10220 6428 M 51 0 R D 10220 6786 M 51 0 R D -10220 7503 M 51 0 R D 10220 7862 M 51 0 R D 10220 8220 M 51 0 R D -10220 8938 M 51 0 R D 10220 9296 M 51 0 R D 15334 1408 M 0 7888 R D -15334 1408 M -102 0 R D 15334 2842 M -102 0 R D 15334 4276 M -102 0 R D -15334 5711 M -102 0 R D 15334 7145 M -102 0 R D 15334 8579 M -102 0 R D -15334 1767 M -51 0 R D 15334 2125 M -51 0 R D 15334 2484 M -51 0 R D -15334 3201 M -51 0 R D 15334 3559 M -51 0 R D 15334 3918 M -51 0 R D -15334 4635 M -51 0 R D 15334 4994 M -51 0 R D 15334 5352 M -51 0 R D -15334 6069 M -51 0 R D 15334 6428 M -51 0 R D 15334 6786 M -51 0 R D -15334 7503 M -51 0 R D 15334 7862 M -51 0 R D 15334 8220 M -51 0 R D -15334 8938 M -51 0 R D 15334 9296 M -51 0 R D 10230 2478 M 11 567 R 10 435 R -10 357 R 10 302 R 10 261 R 11 230 R 10 204 R 10 183 R 10 165 R 11 150 R -10 137 R 10 126 R 10 116 R 10 107 R 11 100 R 10 92 R 10 87 R 10 81 R 11 75 R -10 71 R 10 68 R 10 63 R 11 60 R 10 56 R 10 54 R 10 51 R 10 48 R 11 46 R -10 44 R 10 42 R 10 40 R 11 38 R 10 37 R 10 35 R 10 34 R 11 32 R 10 31 R -10 30 R 10 29 R 10 28 R 11 26 R 10 26 R 10 25 R 10 24 R 11 23 R 10 22 R -10 22 R 10 21 R 10 20 R 11 20 R 10 19 R 10 18 R 10 18 R 11 18 R 10 17 R -10 16 R 10 16 R 11 16 R 10 15 R 10 14 R 10 15 R 10 14 R 11 13 R 10 13 R -10 13 R 10 13 R 11 12 R 10 12 R 10 12 R 10 11 R 10 11 R 11 11 R 10 11 R -10 10 R 10 10 R 11 10 R 10 10 R 10 9 R 10 10 R 11 9 R 10 9 R 10 8 R 10 9 R -10 8 R 11 8 R 10 9 R 10 7 R 10 8 R 11 8 R 10 7 R 10 7 R 10 8 R 10 7 R 11 6 R -10 7 R 10 7 R 10 6 R 11 7 R 10 6 R 10 6 R 10 6 R 11 6 R 10 6 R 10 6 R 10 6 R -10 5 R 11 6 R 10 5 R 10 5 R 10 6 R 11 5 R 10 5 R 10 5 R 10 5 R 11 5 R 10 4 R -10 5 R 10 5 R 10 4 R 11 5 R 10 4 R 10 4 R 10 5 R 11 4 R 10 4 R 10 4 R 10 4 R -10 4 R 11 4 R 10 4 R 10 4 R 10 4 R 11 3 R 10 4 R 10 4 R 10 3 R 11 4 R 10 3 R -10 4 R 10 3 R 10 3 R 11 4 R 10 3 R 10 3 R 10 3 R 11 3 R 10 4 R 10 3 R 10 3 R -10 3 R 11 3 R 10 2 R 10 3 R 10 3 R 11 3 R 10 3 R 10 3 R 10 2 R 11 3 R 10 3 R -10 2 R 10 3 R 10 2 R 11 3 R 10 2 R 10 3 R 10 2 R 11 3 R 10 2 R 10 3 R 10 2 R -11 2 R 10 3 R 10 2 R 10 2 R 10 2 R 11 2 R 10 3 R 10 2 R 10 2 R 11 2 R 10 2 R -10 2 R 10 2 R 10 2 R 11 2 R 10 2 R 10 2 R 10 2 R 11 2 R 10 2 R 10 2 R 10 2 R -11 2 R 10 2 R 10 1 R 10 2 R 10 2 R 11 2 R 10 2 R 10 1 R 10 2 R 11 2 R 10 1 R -10 2 R 10 2 R 10 1 R 11 2 R 10 2 R 10 1 R 10 2 R 11 1 R 10 2 R 10 1 R 10 2 R -11 2 R 10 1 R 10 2 R 10 1 R 10 1 R 11 2 R 10 1 R 10 2 R 10 1 R 11 2 R 10 1 R -10 1 R 10 2 R 11 1 R 10 2 R 10 1 R 10 1 R 10 2 R 11 1 R 10 1 R 10 1 R 10 2 R -11 1 R 10 1 R 10 1 R 10 2 R 10 1 R 11 1 R 10 1 R 10 2 R 10 1 R 11 1 R 10 1 R -10 1 R 10 1 R 11 2 R 10 1 R 10 1 R 10 1 R 10 1 R 11 1 R 10 1 R 10 1 R 10 1 R -11 1 R 10 2 R 10 1 R 10 1 R 10 1 R 11 1 R 10 1 R 10 1 R 10 1 R 11 1 R 10 1 R -10 1 R 10 1 R 11 1 R 10 1 R 10 1 R 10 1 R 10 1 R 11 1 R 10 0 R 10 1 R 10 1 R -11 1 R 10 1 R 10 1 R 10 1 R 10 1 R 11 1 R 10 1 R 10 0 R 10 1 R 11 1 R 10 1 R -10 1 R 10 1 R 11 1 R 10 0 R 10 1 R 10 1 R 10 1 R 11 1 R 10 1 R 10 0 R 10 1 R -11 1 R 10 1 R 10 1 R 10 0 R 11 1 R 10 1 R 10 1 R 10 0 R 10 1 R 11 1 R 10 1 R -10 0 R 10 1 R 11 1 R 10 1 R 10 0 R 10 1 R 10 1 R 11 0 R 10 1 R 10 1 R 10 1 R -11 0 R 10 1 R 10 1 R 10 0 R 11 1 R 10 1 R 10 0 R 10 1 R 10 1 R 11 0 R 10 1 R -10 1 R 10 0 R 11 1 R 10 1 R 10 0 R 10 1 R 10 1 R 11 0 R 10 1 R 10 0 R 10 1 R -11 1 R 10 0 R 10 1 R 10 0 R 11 1 R 10 1 R 10 0 R 10 1 R 10 0 R 11 1 R 10 1 R -10 0 R 10 1 R 11 0 R 10 1 R 10 0 R 10 1 R 11 1 R 10 0 R 10 1 R 10 0 R 10 1 R -11 0 R 10 1 R 10 0 R 10 1 R 11 1 R 10 0 R 10 1 R 10 0 R 10 1 R 11 0 R 10 1 R -10 0 R 10 1 R 11 0 R 10 1 R 10 0 R 10 1 R 11 0 R 10 1 R 10 0 R 10 1 R 10 0 R -11 1 R 10 0 R 10 1 R 10 0 R 11 1 R 10 0 R 10 1 R 10 0 R 10 1 R 11 0 R 10 0 R -10 1 R 10 0 R 11 1 R 10 0 R 10 1 R 10 0 R 11 1 R 10 0 R 10 1 R 10 0 R 10 0 R -11 1 R 10 0 R 10 1 R 10 0 R 11 1 R 10 0 R 10 0 R 10 1 R 11 0 R 10 1 R 10 0 R -10 1 R 10 0 R 11 0 R 10 1 R 10 0 R 10 1 R 11 0 R 10 0 R 10 1 R 10 0 R 10 1 R -11 0 R 10 0 R 10 1 R 10 0 R 11 1 R 10 0 R 10 0 R 10 1 R 11 0 R 10 1 R 10 0 R -10 0 R 10 1 R 11 0 R 10 0 R 10 1 R 10 0 R 11 1 R 10 0 R 10 0 R 10 1 R 10 0 R -11 0 R 10 1 R 10 0 R 10 0 R 11 1 R 10 0 R 10 0 R 10 1 R 11 0 R 10 1 R 10 0 R -10 0 R 10 1 R 11 0 R 10 0 R 10 1 R 10 0 R 11 0 R 10 1 R 10 0 R 10 0 R 11 1 R -10 0 R 10 0 R 10 1 R 10 0 R 11 0 R 10 1 R 10 0 R 10 0 R 11 1 R 10 0 R 10 0 R -10 0 R 10 1 R 11 0 R 10 0 R 10 1 R 10 0 R 11 0 R 10 1 R 10 0 R D L1 -10230 3488 M 11 519 R 10 353 R 10 273 R 10 223 R 10 189 R 11 164 R 10 146 R -10 130 R 10 117 R 11 107 R 10 98 R 10 90 R 10 84 R 10 78 R 11 72 R 10 68 R -10 64 R 10 61 R 11 56 R 10 54 R 10 51 R 10 49 R 11 46 R 10 43 R 10 42 R -10 40 R 10 38 R 11 37 R 10 35 R 10 34 R 10 32 R 11 31 R 10 30 R 10 29 R -10 27 R 11 27 R 10 26 R 10 25 R 10 24 R 10 23 R 11 23 R 10 22 R 10 21 R -10 20 R 11 20 R 10 20 R 10 18 R 10 19 R 10 17 R 11 18 R 10 16 R 10 17 R -10 16 R 11 15 R 10 15 R 10 15 R 10 14 R 11 14 R 10 14 R 10 13 R 10 13 R -10 13 R 11 12 R 10 12 R 10 12 R 10 11 R 11 12 R 10 11 R 10 10 R 10 11 R -10 10 R 11 10 R 10 10 R 10 10 R 10 10 R 11 9 R 10 9 R 10 9 R 10 9 R 11 8 R -10 9 R 10 8 R 10 8 R 10 8 R 11 8 R 10 8 R 10 7 R 10 8 R 11 7 R 10 7 R 10 7 R -10 7 R 10 7 R 11 7 R 10 6 R 10 7 R 10 6 R 11 6 R 10 7 R 10 6 R 10 6 R 11 5 R -10 6 R 10 6 R 10 6 R 10 5 R 11 5 R 10 6 R 10 5 R 10 5 R 11 5 R 10 5 R 10 5 R -10 5 R 11 5 R 10 5 R 10 5 R 10 4 R 10 5 R 11 4 R 10 5 R 10 4 R 10 4 R 11 5 R -10 4 R 10 4 R 10 4 R 10 4 R 11 4 R 10 4 R 10 4 R 10 4 R 11 3 R 10 4 R 10 4 R -10 3 R 11 4 R 10 4 R 10 3 R 10 4 R 10 3 R 11 3 R 10 4 R 10 3 R 10 3 R 11 4 R -10 3 R 10 3 R 10 3 R 10 3 R 11 3 R 10 3 R 10 3 R 10 3 R 11 3 R 10 3 R 10 3 R -10 3 R 11 2 R 10 3 R 10 3 R 10 3 R 10 2 R 11 3 R 10 2 R 10 3 R 10 3 R 11 2 R -10 3 R 10 2 R 10 3 R 11 2 R 10 2 R 10 3 R 10 2 R 10 3 R 11 2 R 10 2 R 10 2 R -10 3 R 11 2 R 10 2 R 10 2 R 10 2 R 10 3 R 11 2 R 10 2 R 10 2 R 10 2 R 11 2 R -10 2 R 10 2 R 10 2 R 11 2 R 10 2 R 10 2 R 10 2 R 10 2 R 11 2 R 10 1 R 10 2 R -10 2 R 11 2 R 10 2 R 10 2 R 10 1 R 10 2 R 11 2 R 10 2 R 10 1 R 10 2 R 11 2 R -10 1 R 10 2 R 10 2 R 11 1 R 10 2 R 10 1 R 10 2 R 10 2 R 11 1 R 10 2 R 10 1 R -10 2 R 11 1 R 10 2 R 10 1 R 10 2 R 11 1 R 10 2 R 10 1 R 10 2 R 10 1 R 11 1 R -10 2 R 10 1 R 10 2 R 11 1 R 10 1 R 10 2 R 10 1 R 10 1 R 11 2 R 10 1 R 10 1 R -10 1 R 11 2 R 10 1 R 10 1 R 10 2 R 11 1 R 10 1 R 10 1 R 10 1 R 10 2 R 11 1 R -10 1 R 10 1 R 10 1 R 11 2 R 10 1 R 10 1 R 10 1 R 10 1 R 11 1 R 10 1 R 10 1 R -10 2 R 11 1 R 10 1 R 10 1 R 10 1 R 11 1 R 10 1 R 10 1 R 10 1 R 10 1 R 11 1 R -10 1 R 10 1 R 10 1 R 11 1 R 10 1 R 10 1 R 10 1 R 10 1 R 11 1 R 10 1 R 10 1 R -10 1 R 11 1 R 10 1 R 10 1 R 10 1 R 11 1 R 10 1 R 10 1 R 10 0 R 10 1 R 11 1 R -10 1 R 10 1 R 10 1 R 11 1 R 10 1 R 10 1 R 10 0 R 11 1 R 10 1 R 10 1 R 10 1 R -10 1 R 11 0 R 10 1 R 10 1 R 10 1 R 11 1 R 10 1 R 10 0 R 10 1 R 10 1 R 11 1 R -10 0 R 10 1 R 10 1 R 11 1 R 10 1 R 10 0 R 10 1 R 11 1 R 10 1 R 10 0 R 10 1 R -10 1 R 11 1 R 10 0 R 10 1 R 10 1 R 11 0 R 10 1 R 10 1 R 10 1 R 10 0 R 11 1 R -10 1 R 10 0 R 10 1 R 11 1 R 10 0 R 10 1 R 10 1 R 11 0 R 10 1 R 10 1 R 10 0 R -10 1 R 11 1 R 10 0 R 10 1 R 10 1 R 11 0 R 10 1 R 10 0 R 10 1 R 11 1 R 10 0 R -10 1 R 10 1 R 10 0 R 11 1 R 10 0 R 10 1 R 10 1 R 11 0 R 10 1 R 10 0 R 10 1 R -10 1 R 11 0 R 10 1 R 10 0 R 10 1 R 11 0 R 10 1 R 10 1 R 10 0 R 11 1 R 10 0 R -10 1 R 10 0 R 10 1 R 11 0 R 10 1 R 10 1 R 10 0 R 11 1 R 10 0 R 10 1 R 10 0 R -10 1 R 11 0 R 10 1 R 10 0 R 10 1 R 11 0 R 10 1 R 10 0 R 10 1 R 11 0 R 10 1 R -10 0 R 10 1 R 10 0 R 11 1 R 10 0 R 10 1 R 10 0 R 11 1 R 10 0 R 10 1 R 10 0 R -11 1 R 10 0 R 10 1 R 10 0 R 10 1 R 11 0 R 10 0 R 10 1 R 10 0 R 11 1 R 10 0 R -10 1 R 10 0 R 10 1 R 11 0 R 10 1 R 10 0 R 10 0 R 11 1 R 10 0 R 10 1 R 10 0 R -11 1 R 10 0 R 10 0 R 10 1 R 10 0 R 11 1 R 10 0 R 10 1 R 10 0 R 11 0 R 10 1 R -10 0 R 10 1 R 10 0 R 11 0 R 10 1 R 10 0 R 10 1 R 11 0 R 10 0 R 10 1 R 10 0 R -11 1 R 10 0 R 10 0 R 10 1 R 10 0 R 11 1 R 10 0 R 10 0 R 10 1 R 11 0 R 10 1 R -10 0 R 10 0 R 11 1 R 10 0 R 10 0 R 10 1 R 10 0 R 11 0 R 10 1 R 10 0 R 10 1 R -11 0 R 10 0 R 10 1 R 10 0 R 10 0 R 11 1 R 10 0 R 10 0 R 10 1 R 11 0 R 10 0 R -10 1 R D L2 10230 3870 M 11 600 R 10 399 R 10 302 R 10 243 R 10 203 R -11 173 R 10 151 R 10 133 R 10 119 R 11 107 R 10 96 R 10 88 R 10 81 R 10 74 R -11 69 R 10 64 R 10 59 R 10 55 R 11 52 R 10 48 R 10 46 R 10 43 R 11 41 R -10 38 R 10 36 R 10 35 R 10 33 R 11 31 R 10 30 R 10 28 R 10 27 R 11 26 R -10 25 R 10 24 R 10 23 R 11 22 R 10 21 R 10 20 R 10 19 R 10 19 R 11 18 R -10 18 R 10 16 R 10 17 R 11 15 R 10 16 R 10 14 R 10 15 R 10 13 R 11 14 R -10 13 R 10 12 R 10 12 R 11 12 R 10 12 R 10 11 R 10 11 R 11 10 R 10 11 R -10 10 R 10 9 R 10 10 R 11 9 R 10 9 R 10 9 R 10 8 R 11 9 R 10 8 R 10 8 R -10 8 R 10 7 R 11 8 R 10 7 R 10 7 R 10 7 R 11 7 R 10 6 R 10 7 R 10 6 R 11 7 R -10 6 R 10 6 R 10 5 R 10 6 R 11 6 R 10 5 R 10 6 R 10 5 R 11 5 R 10 5 R 10 5 R -10 5 R 10 5 R 11 5 R 10 4 R 10 5 R 10 4 R 11 5 R 10 4 R 10 4 R 10 5 R 11 4 R -10 4 R 10 4 R 10 4 R 10 3 R 11 4 R 10 4 R 10 4 R 10 3 R 11 4 R 10 3 R 10 4 R -10 3 R 11 3 R 10 4 R 10 3 R 10 3 R 10 3 R 11 3 R 10 3 R 10 3 R 10 3 R 11 3 R -10 3 R 10 3 R 10 3 R 10 2 R 11 3 R 10 3 R 10 2 R 10 3 R 11 3 R 10 2 R 10 3 R -10 2 R 11 3 R 10 2 R 10 2 R 10 3 R 10 2 R 11 2 R 10 3 R 10 2 R 10 2 R 11 2 R -10 2 R 10 2 R 10 2 R 10 3 R 11 2 R 10 2 R 10 2 R 10 2 R 11 2 R 10 1 R 10 2 R -10 2 R 11 2 R 10 2 R 10 2 R 10 2 R 10 1 R 11 2 R 10 2 R 10 2 R 10 1 R 11 2 R -10 2 R 10 1 R 10 2 R 11 2 R 10 1 R 10 2 R 10 1 R 10 2 R 11 1 R 10 2 R 10 1 R -10 2 R 11 1 R 10 2 R 10 1 R 10 2 R 10 1 R 11 1 R 10 2 R 10 1 R 10 2 R 11 1 R -10 1 R 10 1 R 10 2 R 11 1 R 10 1 R 10 2 R 10 1 R 10 1 R 11 1 R 10 2 R 10 1 R -10 1 R 11 1 R 10 1 R 10 2 R 10 1 R 10 1 R 11 1 R 10 1 R 10 1 R 10 1 R 11 1 R -10 1 R 10 2 R 10 1 R 11 1 R 10 1 R 10 1 R 10 1 R 10 1 R 11 1 R 10 1 R 10 1 R -10 1 R 11 1 R 10 1 R 10 1 R 10 1 R 11 1 R 10 1 R 10 1 R 10 0 R 10 1 R 11 1 R -10 1 R 10 1 R 10 1 R 11 1 R 10 1 R 10 1 R 10 0 R 10 1 R 11 1 R 10 1 R 10 1 R -10 1 R 11 1 R 10 0 R 10 1 R 10 1 R 11 1 R 10 1 R 10 0 R 10 1 R 10 1 R 11 1 R -10 0 R 10 1 R 10 1 R 11 1 R 10 0 R 10 1 R 10 1 R 10 1 R 11 0 R 10 1 R 10 1 R -10 0 R 11 1 R 10 1 R 10 1 R 10 0 R 11 1 R 10 1 R 10 0 R 10 1 R 10 1 R 11 0 R -10 1 R 10 1 R 10 0 R 11 1 R 10 0 R 10 1 R 10 1 R 10 0 R 11 1 R 10 1 R 10 0 R -10 1 R 11 0 R 10 1 R 10 1 R 10 0 R 11 1 R 10 0 R 10 1 R 10 1 R 10 0 R 11 1 R -10 0 R 10 1 R 10 0 R 11 1 R 10 0 R 10 1 R 10 1 R 11 0 R 10 1 R 10 0 R 10 1 R -10 0 R 11 1 R 10 0 R 10 1 R 10 0 R 11 1 R 10 0 R 10 1 R 10 0 R 10 1 R 11 0 R -10 1 R 10 0 R 10 1 R 11 0 R 10 1 R 10 0 R 10 1 R 11 0 R 10 1 R 10 0 R 10 1 R -10 0 R 11 1 R 10 0 R 10 1 R 10 0 R 11 0 R 10 1 R 10 0 R 10 1 R 10 0 R 11 1 R -10 0 R 10 1 R 10 0 R 11 0 R 10 1 R 10 0 R 10 1 R 11 0 R 10 0 R 10 1 R 10 0 R -10 1 R 11 0 R 10 1 R 10 0 R 10 0 R 11 1 R 10 0 R 10 1 R 10 0 R 11 0 R 10 1 R -10 0 R 10 1 R 10 0 R 11 0 R 10 1 R 10 0 R 10 0 R 11 1 R 10 0 R 10 1 R 10 0 R -10 0 R 11 1 R 10 0 R 10 0 R 10 1 R 11 0 R 10 0 R 10 1 R 10 0 R 11 0 R 10 1 R -10 0 R 10 1 R 10 0 R 11 0 R 10 1 R 10 0 R 10 0 R 11 1 R 10 0 R 10 0 R 10 1 R -10 0 R 11 0 R 10 1 R 10 0 R 10 0 R 11 1 R 10 0 R 10 0 R 10 0 R 11 1 R 10 0 R -10 0 R 10 1 R 10 0 R 11 0 R 10 1 R 10 0 R 10 0 R 11 1 R 10 0 R 10 0 R 10 0 R -11 1 R 10 0 R 10 0 R 10 1 R 10 0 R 11 0 R 10 0 R 10 1 R 10 0 R 11 0 R 10 1 R -10 0 R 10 0 R 10 0 R 11 1 R 10 0 R 10 0 R 10 1 R 11 0 R 10 0 R 10 0 R 10 1 R -11 0 R 10 0 R 10 0 R 10 1 R 10 0 R 11 0 R 10 1 R 10 0 R 10 0 R 11 0 R 10 1 R -10 0 R 10 0 R 10 0 R 11 1 R 10 0 R 10 0 R 10 0 R 11 1 R 10 0 R 10 0 R 10 0 R -11 1 R 10 0 R 10 0 R 10 0 R 10 1 R 11 0 R 10 0 R 10 0 R 10 1 R 11 0 R 10 0 R -10 0 R 10 0 R 11 1 R 10 0 R 10 0 R 10 0 R 10 1 R 11 0 R 10 0 R 10 0 R 10 1 R -11 0 R 10 0 R 10 0 R 10 0 R 10 1 R 11 0 R 10 0 R 10 0 R 10 0 R 11 1 R 10 0 R -10 0 R D L3 10230 2254 M 11 461 R 10 363 R 10 305 R 10 263 R 10 232 R -11 208 R 10 188 R 10 170 R 10 157 R 11 144 R 10 133 R 10 124 R 10 115 R -10 107 R 11 101 R 10 95 R 10 89 R 10 84 R 11 79 R 10 75 R 10 72 R 10 68 R -11 64 R 10 61 R 10 59 R 10 56 R 10 53 R 11 51 R 10 49 R 10 47 R 10 45 R -11 43 R 10 42 R 10 40 R 10 38 R 11 37 R 10 36 R 10 35 R 10 33 R 10 32 R -11 31 R 10 31 R 10 29 R 10 28 R 11 27 R 10 27 R 10 25 R 10 25 R 10 25 R -11 23 R 10 23 R 10 22 R 10 22 R 11 21 R 10 20 R 10 20 R 10 20 R 11 18 R -10 19 R 10 18 R 10 17 R 10 17 R 11 17 R 10 16 R 10 16 R 10 16 R 11 15 R -10 15 R 10 14 R 10 14 R 10 14 R 11 14 R 10 13 R 10 13 R 10 12 R 11 13 R -10 12 R 10 12 R 10 12 R 11 11 R 10 11 R 10 11 R 10 11 R 10 11 R 11 10 R -10 10 R 10 10 R 10 10 R 11 10 R 10 9 R 10 9 R 10 9 R 10 9 R 11 9 R 10 9 R -10 8 R 10 9 R 11 8 R 10 8 R 10 8 R 10 8 R 11 7 R 10 8 R 10 7 R 10 8 R 10 7 R -11 7 R 10 7 R 10 7 R 10 6 R 11 7 R 10 7 R 10 6 R 10 6 R 11 7 R 10 6 R 10 6 R -10 6 R 10 6 R 11 6 R 10 5 R 10 6 R 10 6 R 11 5 R 10 6 R 10 5 R 10 5 R 10 5 R -11 5 R 10 6 R 10 5 R 10 4 R 11 5 R 10 5 R 10 5 R 10 5 R 11 4 R 10 5 R 10 4 R -10 5 R 10 4 R 11 5 R 10 4 R 10 4 R 10 4 R 11 4 R 10 4 R 10 5 R 10 4 R 10 3 R -11 4 R 10 4 R 10 4 R 10 4 R 11 4 R 10 3 R 10 4 R 10 3 R 11 4 R 10 3 R 10 4 R -10 3 R 10 4 R 11 3 R 10 4 R 10 3 R 10 3 R 11 3 R 10 4 R 10 3 R 10 3 R 11 3 R -10 3 R 10 3 R 10 3 R 10 3 R 11 3 R 10 3 R 10 3 R 10 3 R 11 2 R 10 3 R 10 3 R -10 3 R 10 2 R 11 3 R 10 3 R 10 2 R 10 3 R 11 3 R 10 2 R 10 3 R 10 2 R 11 3 R -10 2 R 10 3 R 10 2 R 10 2 R 11 3 R 10 2 R 10 2 R 10 3 R 11 2 R 10 2 R 10 3 R -10 2 R 10 2 R 11 2 R 10 2 R 10 3 R 10 2 R 11 2 R 10 2 R 10 2 R 10 2 R 11 2 R -10 2 R 10 2 R 10 2 R 10 2 R 11 2 R 10 2 R 10 2 R 10 2 R 11 2 R 10 2 R 10 1 R -10 2 R 11 2 R 10 2 R 10 2 R 10 2 R 10 1 R 11 2 R 10 2 R 10 2 R 10 1 R 11 2 R -10 2 R 10 1 R 10 2 R 10 2 R 11 1 R 10 2 R 10 2 R 10 1 R 11 2 R 10 2 R 10 1 R -10 2 R 11 1 R 10 2 R 10 1 R 10 2 R 10 1 R 11 2 R 10 1 R 10 2 R 10 1 R 11 2 R -10 1 R 10 2 R 10 1 R 10 1 R 11 2 R 10 1 R 10 2 R 10 1 R 11 1 R 10 2 R 10 1 R -10 1 R 11 2 R 10 1 R 10 1 R 10 2 R 10 1 R 11 1 R 10 1 R 10 2 R 10 1 R 11 1 R -10 1 R 10 2 R 10 1 R 10 1 R 11 1 R 10 2 R 10 1 R 10 1 R 11 1 R 10 1 R 10 1 R -10 2 R 11 1 R 10 1 R 10 1 R 10 1 R 10 1 R 11 1 R 10 2 R 10 1 R 10 1 R 11 1 R -10 1 R 10 1 R 10 1 R 11 1 R 10 1 R 10 1 R 10 1 R 10 1 R 11 1 R 10 1 R 10 1 R -10 1 R 11 1 R 10 1 R 10 1 R 10 1 R 10 1 R 11 1 R 10 1 R 10 1 R 10 1 R 11 1 R -10 1 R 10 1 R 10 1 R 11 1 R 10 1 R 10 1 R 10 1 R 10 1 R 11 0 R 10 1 R 10 1 R -10 1 R 11 1 R 10 1 R 10 1 R 10 1 R 10 0 R 11 1 R 10 1 R 10 1 R 10 1 R 11 1 R -10 1 R 10 0 R 10 1 R 11 1 R 10 1 R 10 1 R 10 1 R 10 0 R 11 1 R 10 1 R 10 1 R -10 0 R 11 1 R 10 1 R 10 1 R 10 1 R 11 0 R 10 1 R 10 1 R 10 1 R 10 0 R 11 1 R -10 1 R 10 1 R 10 0 R 11 1 R 10 1 R 10 1 R 10 0 R 10 1 R 11 1 R 10 0 R 10 1 R -10 1 R 11 1 R 10 0 R 10 1 R 10 1 R 11 0 R 10 1 R 10 1 R 10 0 R 10 1 R 11 1 R -10 0 R 10 1 R 10 1 R 11 0 R 10 1 R 10 1 R 10 0 R 10 1 R 11 1 R 10 0 R 10 1 R -10 1 R 11 0 R 10 1 R 10 0 R 10 1 R 11 1 R 10 0 R 10 1 R 10 1 R 10 0 R 11 1 R -10 0 R 10 1 R 10 1 R 11 0 R 10 1 R 10 0 R 10 1 R 11 0 R 10 1 R 10 1 R 10 0 R -10 1 R 11 0 R 10 1 R 10 0 R 10 1 R 11 1 R 10 0 R 10 1 R 10 0 R 10 1 R 11 0 R -10 1 R 10 0 R 10 1 R 11 1 R 10 0 R 10 1 R 10 0 R 11 1 R 10 0 R 10 1 R 10 0 R -10 1 R 11 0 R 10 1 R 10 0 R 10 1 R 11 0 R 10 1 R 10 0 R 10 1 R 10 0 R 11 1 R -10 0 R 10 1 R 10 0 R 11 1 R 10 0 R 10 1 R 10 0 R 11 1 R 10 0 R 10 1 R 10 0 R -10 1 R 11 0 R 10 1 R 10 0 R 10 1 R 11 0 R 10 1 R 10 0 R 10 0 R 11 1 R 10 0 R -10 1 R 10 0 R 10 1 R 11 0 R 10 1 R 10 0 R 10 0 R 11 1 R 10 0 R 10 1 R 10 0 R -10 1 R 11 0 R 10 1 R 10 0 R 10 0 R 11 1 R 10 0 R 10 1 R D -end restore -showpage -%%Trailer -restore -%%Pages: 1 -%%Trailer -cleartomark -countdictstack -exch sub { end } repeat -restore -%%EOF diff --git a/documentation/source/science_guide/turbulence_schemes/subsent_fig7.eps b/documentation/source/science_guide/turbulence_schemes/subsent_fig7.eps deleted file mode 100644 index 4fb826b330..0000000000 --- a/documentation/source/science_guide/turbulence_schemes/subsent_fig7.eps +++ /dev/null @@ -1,323 +0,0 @@ -%!PS-Adobe-2.0 EPSF-2.0 -%%BoundingBox: 60 38 464 293 -%%HiResBoundingBox: 60.5 39 463.5 292.5 -%%Title: Graphics produced by WAVE -%%For: APRISH::LOCK -%%Creator: WAVE Version 6.21 (vms axp) -%%CreationDate: Tue May 4 15:53:47 1999 - -%%EndComments -% EPSF created by ps2eps 1.68 -%%BeginProlog -save -countdictstack -mark -newpath -/showpage {} def -/setpagedevice {pop} def -%%EndProlog -%%Page 1 1 -%+ wave_prolog.ps -- Prolog for PV-WAVE CL generated PostScript files -%+ $Id: wave_prolog.ps,v 1.5 1998/06/05 17:44:06 thaux Exp $ -%+ Copyright (c) 1989-1992 Precision Visuals, Inc. All Rights Reserved. -%v 1 -save /$WAVE_DICT 60 dict def $WAVE_DICT begin /bdef { bind def } bind def /C -{currentpoint newpath moveto} bdef /D {currentpoint stroke moveto} bdef /F -{closepath fill} bdef /K { setgray } bdef /L {translate} bdef /M {moveto} -bdef /N {rmoveto} bdef /O {show} bdef /P {lineto} bdef /R {rlineto} bdef /S -{gsave show grestore} bdef /Z {gsave currentpoint lineto 20 setlinewidth 1 -setlinecap stroke grestore} bdef /CC {concat} bdef /CD {currentmatrix def} -bdef /CP {currentpoint pop def} bdef /FM {findfont exch makefont} bdef /FS -{findfont exch scalefont} bdef /GS {get setfont} bdef /MD {matrix def} bdef -/MN {mul neg def} bdef /RS {roll put setfont} bdef /SL {setlinewidth} bdef -/SM {setmatrix} bdef /L0 {[] 0 setdash} bdef /L1 {[40 100] 0 setdash} bdef -/L2 {[200 200] 0 setdash} bdef /L3 {[200 100 50 100] 0 setdash} bdef /L4 -{[300 100 50 100 50 100 50 100] 0 setdash} bdef /L5 {[400 200] 0 setdash} -bdef /IO { gsave 0.0 setgray /val exch def moveto /str 20 string def val str -cvs show grestore } bdef /UL { gsave currentpoint /cury exch def /curx exch -def /curmtx matrix currentmatrix def matrix identmatrix setmatrix newpath 0 -0 moveto true charpath flattenpath pathbbox curmtx setmatrix /height exch -def /width exch def /ybase cury height .1 mul sub def pop pop newpath curx -ybase moveto width 0 rlineto closepath stroke grestore } bdef /BG { gsave -currentpoint /cury exch def /curx exch def /curmtx matrix currentmatrix def -matrix identmatrix setmatrix newpath 0 0 moveto true charpath flattenpath -pathbbox curmtx setmatrix /height exch def /width exch def pop pop newpath 0 -0 moveto (M) true charpath flattenpath pathbbox /xdiff exch .1 mul def -/ydiff exch .1 mul def pop pop newpath curx xdiff 2 div sub cury ydiff 2 div -sub moveto width xdiff add 0 rlineto 0 height ydiff add rlineto width neg -xdiff sub 0 rlineto closepath fill grestore } bdef /$T_DICT 20 dict def /T { -$T_DICT begin /align exch def /orien exch def /size exch def /thestring exch -def gsave moveto /chsize 1.0 def /xsize 0.0 def /SUPER 8#330 def /SUBS 8#331 -def /NORM 8#332 def /SCRIPTWID 0.7 def /OFFSET 0.6 def orien rotate size dup -scale /orien false def thestring { /charcode exch def charcode SUPER eq -charcode SUBS eq or { /chsize SCRIPTWID def /orien true def } { charcode -NORM eq { /chsize 1.0 def /orien true def } { ( ) dup 0 charcode put -stringwidth pop chsize mul xsize add /xsize exch def } ifelse } ifelse } -forall xsize align mul neg 0 rmoveto orien { /regularfont currentfont def -/fractionfont currentfont [ SCRIPTWID 0 0 SCRIPTWID 0 0 ] makefont def gsave -newpath 0 0 moveto (1) true charpath flattenpath pathbbox /height exch def -pop pop pop grestore } if /xsize 0 def thestring { /charcode exch def -charcode SUPER eq { 0 OFFSET height mul dup /xsize exch def rmoveto -fractionfont setfont } { charcode SUBS eq { 0 OFFSET height mul neg dup -/xsize exch def rmoveto fractionfont setfont } { charcode NORM eq { 0 xsize -neg rmoveto regularfont setfont } { ( ) dup 0 charcode put show } ifelse } -ifelse } ifelse } forall grestore end } bdef /IsChar { exch /CharStrings get -exch known } bdef /MapCh { 3 -1 roll /Encoding get 3 1 roll put } bdef -/MapDegree { dup 16#b0 exch /degree IsChar { /degree } { /ring } ifelse -MapCh } bdef /MapBB { dup 16#a6 exch /brokenbar IsChar { /brokenbar } { /bar -} ifelse MapCh } bdef /ReEncode { dup findfont begin currentdict dup length -dict begin { 1 index /FID ne {def} {pop pop} ifelse } forall /FontName exch -def dup length 0 ne {/Encoding Encoding 256 array copy def 0 exch { dup type -/nametype eq { Encoding 2 index 2 index put pop 1 add } { exch pop } ifelse -} forall } if pop currentdict dup end end /FontName get exch definefont dup -MapDegree MapBB } bdef /RF { ISOLatin1Encoding exch ReEncode } bdef /Courier -RF /Courier-Bold RF /Courier-Oblique RF /Courier-BoldOblique RF /Helvetica -RF /Helvetica-Bold RF /Helvetica-Oblique RF /Helvetica-BoldOblique RF -/Helvetica-Narrow RF /Helvetica-Narrow-Bold RF /Helvetica-Narrow-Oblique RF -/Helvetica-Narrow-BoldOblique RF /AvantGarde-Book RF /AvantGarde-DemiOblique -RF /AvantGarde-Demi RF /AvantGarde-DemiOblique RF /Bookman-Demi RF -/Bookman-DemiItalic RF /Bookman-Light RF /Bookman-LightItalic RF -/ZapfChancery-MediumItalic RF /NewCenturySchlbk-Roman RF -/NewCenturySchlbk-Bold RF /NewCenturySchlbk-Italic RF -/NewCenturySchlbk-BoldItalic RF /Palatino-Roman RF /Palatino-Bold RF -/Palatino-Italic RF /Palatino-BoldItalic RF /Times-Roman RF /Times-Bold RF -/Times-Italic RF /Times-BoldItalic RF end -%%EndProlog -%%Page: 0 1 -%%BeginPageSetup -save $WAVE_DICT begin 28 28 L 0.028346 dup scale -%%PageBoundingBox: 28 28 481 311 -%%EndPageSetup -/psFontCache 500 array def /psStack 8 array def /psStackInd 0 def -/psTextWidth 0 def /psCurBase 0 def /ASW { stringwidth pop psCurBase add -/psCurBase exch def psCurBase psTextWidth gt -{ /psTextWidth psCurBase def } if } def /PSS { -psStack psStackInd psCurBase put /psStackInd psStackInd 1 add def } def -/PPS { /psStackInd psStackInd 1 sub def -/psCurBase psStack psStackInd get def } def 10.000000 SL L0 0.000 K -2220 1408 M 5114 0 R D 2220 1408 M 0 157 R D 2196 1195 M -35 -11 R -23 -35 R --12 -59 R 0 -35 R 12 -58 R 23 -35 R 35 -12 R 24 0 R 35 12 R 23 35 R 12 58 R -0 35 R -12 59 R -23 35 R -35 11 R -24 0 R -23 -11 R -12 -12 R -12 -23 R --11 -59 R 0 -35 R 11 -58 R 12 -24 R 12 -11 R 23 -12 R D 2220 950 M 23 12 R -12 11 R 11 24 R 12 58 R 0 35 R -12 59 R -11 23 R -12 12 R -23 11 R D -5872 1408 M 0 157 R D 5406 1078 M 11 24 R 24 23 R 23 0 R 12 -11 R 11 -24 R -0 -35 R -23 -58 R D 5476 1114 M 0 -47 R -12 -47 R 0 -47 R D 5476 1090 M --24 -58 R 0 -35 R 12 -24 R 23 -11 R 24 0 R 23 11 R 24 24 R 11 35 R 24 93 R D -5569 1032 M 0 -35 R 12 -24 R 23 -11 R 24 0 R 23 11 R 23 24 R 24 35 R 11 46 R -0 47 R -11 0 R 0 -11 R 11 -24 R D 5604 1125 M -23 -93 R 0 -59 R D -5593 1125 M 23 0 R -23 -82 R -12 -46 R D 5931 1125 M -70 -245 R D -5943 1125 M -70 -245 R D 5931 1125 M 24 0 R -70 -245 R D 5908 1043 M 0 35 R --12 36 R -23 11 R -23 0 R -35 -11 R -24 -36 R -11 -35 R 0 -23 R 11 -35 R -12 -12 R 23 -11 R 24 0 R 23 11 R 12 12 R 11 23 R 12 35 R D 5815 1102 M --12 -24 R -12 -35 R 0 -35 R 12 -23 R D 5850 1125 M -24 -23 R -11 -24 R --12 -35 R 0 -35 R 12 -35 R 11 -11 R D 5826 880 M 94 0 R D 5873 892 M --35 -12 R D 5873 903 M -23 -23 R D 5885 903 M 11 -23 R D 5873 892 M 35 -12 R -D 6055 1016 M -22 -80 R -7 -28 R 0 -22 R 7 -15 R 7 -7 R 15 0 R 14 15 R -8 14 R D 6062 1016 M -22 -80 R -7 -28 R 0 -37 R D 6055 1016 M 14 0 R --29 -101 R -7 -29 R D 6012 965 M 72 0 R D 6138 1254 M 0 -374 R D 6322 1002 M -7 0 R 7 14 R -7 -43 R 0 14 R -7 15 R -7 7 R -22 7 R -29 0 R -22 -7 R --14 -15 R 0 -21 R 7 -15 R 15 -14 R 43 -22 R 7 -14 R 0 -22 R -7 -15 R D -6235 973 M 7 -15 R 51 -29 R 7 -14 R D 6242 1009 M -7 -15 R 0 -14 R 7 -15 R -44 -21 R 14 -15 R 8 -14 R 0 -22 R -8 -14 R -7 -8 R -22 -7 R -29 0 R -21 7 R --7 8 R -8 14 R 0 15 R -7 -44 R 7 15 R 8 0 R D 4311 668 M -70 -246 R D -4323 668 M -70 -246 R D 4311 668 M 24 0 R -71 -246 R D 4288 586 M 0 35 R --12 35 R -23 12 R -24 0 R -35 -12 R -23 -35 R -12 -35 R 0 -24 R 12 -35 R -12 -11 R 23 -12 R 23 0 R 24 12 R 11 11 R 12 24 R 12 35 R D 4194 644 M --11 -23 R -12 -35 R 0 -35 R 12 -24 R D 4229 668 M -23 -24 R -12 -23 R --11 -35 R 0 -35 R 11 -35 R 12 -12 R D 4206 422 M 93 0 R D 4253 434 M --35 -12 R D 4253 446 M -24 -24 R D 4264 446 M 12 -24 R D 4253 434 M 35 -12 R -D 4435 558 M -22 -79 R -7 -29 R 0 -22 R 7 -14 R 7 -7 R 15 0 R 14 14 R 7 14 R -D 4442 558 M -22 -79 R -7 -29 R 0 -36 R D 4435 558 M 14 0 R -29 -101 R --7 -29 R D 4391 508 M 73 0 R D 4763 726 M -12 -12 R 12 -11 R 12 11 R 0 12 R --12 12 R -23 0 R -24 -12 R -11 -23 R 0 -211 R D 4740 738 M -12 -12 R --12 -23 R 0 -211 R D 4669 656 M 94 0 R D 4669 492 M 82 0 R D 4856 738 M -0 -246 R D 4868 738 M 0 -246 R D 4821 738 M 47 0 R D 4821 492 M 82 0 R D -4985 656 M 0 -129 R 12 -23 R 35 -12 R 23 0 R 35 12 R 23 23 R D 4997 656 M -0 -129 R 11 -23 R 24 -12 R D 5113 656 M 0 -164 R D 5125 656 M 0 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bdef -/FF {findfont} bdef -/MF {makefont} bdef -/RO {rotate} bdef -/SC {scale} bdef -/SF {setfont} bdef -/SG {setgray} bdef -/TR {translate} bdef -/bp {lj lw rgb} bdef -/bpbw {lj lw setgray} bdef -/c {curveto} bdef -/cl {closepath} bdef -/fi {eofill} bdef -/g {setgray} bdef -/gr {grestore} bdef -/gs {gsave} bdef -/l {lineto} bdef -/lj {setlinejoin} bdef -/lw {setlinewidth} bdef -/m {moveto} bdef -/n {newpath} bdef -/nx {/x E def} bdef -/r {rmoveto} bdef -/rl {rlineto} bdef -/rgb {setrgbcolor} bdef -/s {show} bdef -/sd {setdash} bdef -/sp {x 0 rmoveto} bdef -/ss {currentpoint pop E m} bdef -/st {stroke} bdef -/BPSIDE 32 def %% pixels per pattern side -/PATFREQ 3.0 def %% pattern pixels per mm -/dp_mat [PATFREQ 0 0 PATFREQ 0 0] def -/dp_pw BPSIDE def %% pattern pixel width -/dp_ph BPSIDE def %% pattern pixel height -/dp_w dp_pw PATFREQ div def %% pattern mm width -/dp_h dp_ph PATFREQ div def %% pattern mm height -/savemat matrix def -/topmat matrix def -/patmat matrix def -/ncpoint errordict /nocurrentpoint get def -errordict begin -/nocurrentpoint { - dup /pathbbox load eq - {pop 0 0 1 1} - {ncpoint} - ifelse -} bdef -end -/ar { %% sa ea sx sy rot tx ty - matrix currentmatrix 8 1 roll TR RO SC - n 0 0 1 5 3 roll arc setmatrix -} bdef -/arn { %% sa ea sx sy rot tx ty - TR RO SC - matrix currentmatrix 8 1 roll - n 0 0 1 5 3 roll arcn setmatrix -} bdef -/el { %% sx sy rot tx ty - matrix currentmatrix 6 1 roll TR RO SC - n 0 0 1 0 360 arc setmatrix cl -} bdef -/image_raster { %% sw sh sd dw dh xs ys - TR SC /sd E def /sh E def /sw E def - /imagebuf sw sd mul 7 add 8 idiv string def - sw sh sd [sw 0 0 sh 0 0] { currentfile imagebuf readhexstring pop} - image -} bdef -/imagemask_raster { - TR SC /sh E def /sw E def - /imagebuf sw 7 add 8 idiv string def - sw sh false [sw 0 0 sh 0 0] - {currentfile imagebuf readhexstring pop} - imagemask -} bdef -/dither_color_raster { % bool sw sh sd dw dh xs ys - TR SC /sd E def /sh E def /sw E def - sd 8 eq and - { - /imagebuf 3 string def - /grayval 1 string def - sw sh sd [sw 0 0 sh 0 0] - { - currentfile imagebuf readhexstring pop pop - imagebuf 0 get 0.299 mul - imagebuf 1 get 0.587 mul add - imagebuf 2 get 0.114 mul add cvi grayval exch 0 exch put grayval - } - image - } - { - /imagebuf sw 3 mul sd mul 7 add 8 idiv string def - sh { currentfile imagebuf readhexstring pop pop } repeat - } ifelse -} bdef -/image_color_raster { % bool sw sh sd dw dh xs ys - /colorimage where not - { dither_color_raster } - { - pop - TR SC /sd E def /sh E def /sw E def pop - /imagebuf sw 3 mul sd mul 7 add 8 idiv string def - sw sh sd [sw 0 0 sh 0 0] { currentfile imagebuf readhexstring pop} - false 3 colorimage - } ifelse -} bdef -/patpath { - /inv E def - topmat setmatrix - pathbbox %% get lo - hi indecies - /hy E dp_h div floor cvi def - /hx E dp_w div floor cvi def - /ly E dp_h div floor cvi def - /lx E dp_w div floor cvi def - lx 1 hx { - dp_w mul - ly 1 hy { - dp_h mul - E dup 3 1 roll E - patmat currentmatrix pop - TR - dp_pw dp_ph inv - dp_mat dp_proc imagemask - patmat setmatrix - } for - pop - } for -} bdef -% setpattern brush of patterns instead of gray -/setpattern { - /blue E def /green E def /red E def - /freq E def /bwidth E def /bpside E def - /bstring E def - /onbits 0 def /offbits 0 def - freq 0 {/y E def /x E def - /xindex x 1 add 2 div bpside mul cvi def - /yindex y 1 add 2 div bpside mul cvi def - bstring yindex bwidth mul xindex 8 idiv add get not - 1 7 xindex 8 mod sub bitshift and 0 ne - {/onbits onbits 1 add def 1} - {/offbits offbits 1 add def 0} - ifelse - } setscreen {} settransfer - systemdict /setcmykcolor known - { /fact 1 onbits offbits onbits add div sub def - 1 red sub fact mul 1 green sub fact mul 1 blue sub fact mul 0 - setcmykcolor - } - { offbits offbits onbits add div setgray} - ifelse -} bdef -/dmatrix matrix def -/dpi 72 0 dmatrix defaultmatrix dtransform - dup mul E dup mul add sqrt -def -/B {gs bp st gr} bdef %% brush: gr lw lj -/Bbw {gs bpbw st gr} bdef %% brush: gr lw lj -/F {gs rgb eofill gr} bdef %% fill: gr -/Fbw {gs setgray eofill gr} bdef %% fill: gr -/PB {gs lj lw setpattern st gr} bdef -/PF {gs eoclip patpath gr} bdef -/BB {gs rgb lj lw strokepath clip patpath gr} bdef -/xdef {exch def} bdef -/clip_region { - /ht xdef - /wd xdef - /bm xdef - /lm xdef - newpath - lm bm moveto - 0 ht rlineto - wd 0 rlineto - 0 ht neg rlineto - closepath clip -} bdef -/reencode_small_dict 12 dict def -/ReencodeSmall { -reencode_small_dict begin -/new_codes_and_names exch def -/new_font_name exch def -/base_font_name exch def -/base_font_dict base_font_name findfont def -/newfont base_font_dict maxlength dict def -base_font_dict { -exch dup /FID ne -{ dup /Encoding eq -{ exch dup length array copy newfont 3 1 roll put } -{ exch newfont 3 1 roll put } -ifelse -} -{ pop pop } -ifelse -} forall -newfont /FontName new_font_name put -new_codes_and_names aload pop -new_codes_and_names length 2 idiv -{ newfont /Encoding get 3 1 roll put } -repeat -new_font_name newfont definefont pop -end %reencode_small_dict -} def -/extended_Zapf [ -8#223 /a89 -8#224 /a90 -8#225 /a93 -8#226 /a94 -8#227 /a91 -8#230 /a92 -8#231 /a205 -8#232 /a85 -8#233 /a206 -8#234 /a86 -8#235 /a87 -8#236 /a88 -8#237 /a95 -8#240 /a96 -] def -/extended_Standard [ -29 /thorn -30 /yacute -31 /divide -128 /Acircumflex -129 /Adieresis -130 /Agrave -131 /Aring -132 /Atilde -133 /Ccedilla -134 /Eacute -135 /Ecircumflex -136 /Edieresis -137 /Egrave -138 /Iacute -139 /Icircumflex -140 /Idieresis -141 /Igrave -142 /Ntilde -143 /Oacute -144 /Ocircumflex -145 /Odieresis -146 /Ograve -147 /Otilde -148 /Scaron -149 /Uacute -150 /Ucircumflex -151 /Udieresis -152 /Ugrave -153 /Ydieresis -154 /Zcaron -155 /aacute -156 /acircumflex -157 /adieresis -158 /agrave -159 /aring -160 /atilde -161 /exclamdown -162 /cent -163 /sterling -164 /fraction -165 /yen -166 /florin -167 /section -168 /currency -169 /quotesingle -170 /quotedblleft -171 /guillemotleft -172 /guilsinglleft -173 /guilsinglright -174 /fi -175 /fl -176 /plusminus -177 /endash -178 /dagger -179 /daggerdbl -180 /periodcentered -181 /twosuperior -182 /paragraph -183 /bullet -184 /quotesinglbase -185 /quotedblbase -186 /quotedblright -187 /guillemotright -188 /ellipsis -189 /perthousand -190 /threesuperior -191 /questiondown -192 /mu -193 /grave -194 /acute -195 /circumflex -196 /tilde -197 /macron -198 /breve -199 /dotaccent -200 /dieresis -201 /onesuperior -202 /ring -203 /cedilla -204 /onequarter -205 /hungarumlaut -206 /ogonek -207 /caron -208 /emdash -209 /ccedilla -210 /copyright -211 /eacute -212 /ecircumflex -213 /edieresis -214 /egrave -215 /iacute -216 /icircumflex -217 /idieresis -218 /igrave -219 /logicalnot -220 /minus -221 /ntilde -222 /oacute -223 /ocircumflex -224 /odieresis -225 /AE -226 /onehalf -227 /ordfeminine -228 /ograve -229 /otilde -230 /registered -231 /scaron -232 /Lslash -233 /Oslash -234 /OE -235 /ordmasculine -236 /trademark -237 /uacute -238 /ucircumflex -239 /udieresis -240 /ugrave -241 /ae -242 /ydieresis -243 /zcaron -244 /Aacute -245 /dotlessi -246 /threequarters -247 /Eth -248 /lslash -249 /oslash -250 /oe -251 /germandbls -252 /multiply -253 /Yacute -254 /Thorn -255 /eth -] def -/extended_Symbol [ -] def -/extend_font { % stack: fontname newfontname -exch dup (ZapfDingbats) eq -{ cvn exch cvn extended_Zapf ReencodeSmall } -{ dup (Symbol) eq -{ cvn exch cvn extended_Symbol ReencodeSmall } -{ cvn exch cvn extended_Standard ReencodeSmall } -ifelse -} -ifelse -} bind def -/extend_font_name { % stack: font_name_string -dup length 1 add string /extended_font_name exch def -extended_font_name 0 (_) putinterval -extended_font_name 1 3 -1 roll putinterval -extended_font_name -} bind def -/gf { -/f exch def f cvn where -{ f exch begin cvn load exec setfont end } -{ f 0 f length 8 sub getinterval dup -/localfont exch extend_font_name def -localfont extend_font -localfont findfont -/xsz f f length 4 sub 4 getinterval cvi def -/ysz f f length 8 sub 4 getinterval cvi def -[ 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000000000000000000000000000000000000000000000000000000000000000000000000000000 -% 00000000 -% 002000200020002000200020002000200020002000200020002000200020002000200020002000 -% 200020002000200020002000200020002000200020002000200020002000000000000000000000 -% 00000000 -% 000000000000000000000000000000000000000000000000000000000000000000000000000000 -% 000000000000000000000000000000000000000000000000000000000000000000000000000000 -% 00000000 -% 000800080008000800080008000800080008000800080008000800080008000800080008000800 -% 080008000800080008000800080008000800080008000800080008000800000000000000000000 -% 00000000 -% 000000000000000000000000000000000000000000000000000000000000000000000000000000 -% 000000000000000000000000000000000000000000000000000000000000000000000000000000 -% 00000000 -save /d_sv_obj exch def -userdict /IslandDrawDict 300 dict dup begin put -/bdef {bind def} bind def -/E {exch} bdef -/FF {findfont} bdef -/MF {makefont} bdef -/RO {rotate} bdef -/SC {scale} bdef -/SF {setfont} bdef -/SG {setgray} bdef -/TR {translate} bdef -/bp {lj lw rgb} bdef -/bpbw {lj lw setgray} bdef -/c {curveto} bdef -/cl {closepath} bdef -/fi {eofill} bdef -/g {setgray} bdef -/gr {grestore} bdef -/gs {gsave} bdef -/l {lineto} bdef -/lj {setlinejoin} bdef -/lw {setlinewidth} bdef -/m {moveto} bdef -/n {newpath} bdef -/nx {/x E def} bdef -/r {rmoveto} bdef -/rl {rlineto} bdef -/rgb {setrgbcolor} bdef -/s {show} bdef -/sd {setdash} bdef -/sp {x 0 rmoveto} bdef -/ss {currentpoint pop E m} bdef -/st {stroke} bdef -/BPSIDE 32 def %% pixels per pattern side -/PATFREQ 3.0 def %% pattern pixels per mm -/dp_mat [PATFREQ 0 0 PATFREQ 0 0] def -/dp_pw BPSIDE def %% pattern pixel width -/dp_ph BPSIDE def %% pattern pixel height -/dp_w dp_pw PATFREQ div def %% pattern mm width -/dp_h dp_ph PATFREQ div def %% pattern mm height -/savemat matrix def -/topmat matrix def -/patmat matrix def -/ncpoint errordict /nocurrentpoint get def -errordict begin -/nocurrentpoint { - dup /pathbbox load eq - {pop 0 0 1 1} - {ncpoint} - ifelse -} bdef -end -/ar { %% sa ea sx sy rot tx ty - matrix currentmatrix 8 1 roll TR RO SC - n 0 0 1 5 3 roll arc setmatrix -} bdef -/arn { %% sa ea sx sy rot tx ty - TR RO SC - matrix currentmatrix 8 1 roll - n 0 0 1 5 3 roll arcn setmatrix -} bdef -/el { %% sx sy rot tx ty - matrix currentmatrix 6 1 roll TR RO SC - n 0 0 1 0 360 arc setmatrix cl -} bdef -/image_raster { %% sw sh sd dw dh xs ys - TR SC /sd E def /sh E def /sw E def - /imagebuf sw sd mul 7 add 8 idiv string def - sw sh sd [sw 0 0 sh 0 0] { currentfile imagebuf readhexstring pop} - image -} bdef -/imagemask_raster { - TR SC /sh E def /sw E def - /imagebuf sw 7 add 8 idiv string def - sw sh false [sw 0 0 sh 0 0] - {currentfile imagebuf readhexstring pop} - imagemask -} bdef -/dither_color_raster { % bool sw sh sd dw dh xs ys - TR SC /sd E def /sh E def /sw E def - sd 8 eq and - { - /imagebuf 3 string def - /grayval 1 string def - sw sh sd [sw 0 0 sh 0 0] - { - currentfile imagebuf readhexstring pop pop - imagebuf 0 get 0.299 mul - imagebuf 1 get 0.587 mul add - imagebuf 2 get 0.114 mul add cvi grayval exch 0 exch put grayval - } - image - } - { - /imagebuf sw 3 mul sd mul 7 add 8 idiv string def - sh { currentfile imagebuf readhexstring pop pop } repeat - } ifelse -} bdef -/image_color_raster { % bool sw sh sd dw dh xs ys - /colorimage where not - { dither_color_raster } - { - pop - TR SC /sd E def /sh E def /sw E def pop - /imagebuf sw 3 mul sd mul 7 add 8 idiv string def - sw sh sd [sw 0 0 sh 0 0] { currentfile imagebuf readhexstring pop} - false 3 colorimage - } ifelse -} bdef -/patpath { - /inv E def - topmat setmatrix - pathbbox %% get lo - hi indecies - /hy E dp_h div floor cvi def - /hx E dp_w div floor cvi def - /ly E dp_h div floor cvi def - /lx E dp_w div floor cvi def - lx 1 hx { - dp_w mul - ly 1 hy { - dp_h mul - E dup 3 1 roll E - patmat currentmatrix pop - TR - dp_pw dp_ph inv - dp_mat dp_proc imagemask - patmat setmatrix - } for - pop - } for -} bdef -% setpattern brush of patterns instead of gray -/setpattern { - /blue E def /green E def /red E def - /freq E def /bwidth E def /bpside E def - /bstring E def - /onbits 0 def /offbits 0 def - freq 0 {/y E def /x E def - /xindex x 1 add 2 div bpside mul cvi def - /yindex y 1 add 2 div bpside mul cvi def - bstring yindex bwidth mul xindex 8 idiv add get not - 1 7 xindex 8 mod sub bitshift and 0 ne - {/onbits onbits 1 add def 1} - {/offbits offbits 1 add def 0} - ifelse - } setscreen {} settransfer - systemdict /setcmykcolor known - { /fact 1 onbits offbits onbits add div sub def - 1 red sub fact mul 1 green sub fact mul 1 blue sub fact mul 0 - setcmykcolor - } - { offbits offbits onbits add div setgray} - ifelse -} bdef -/dmatrix matrix def -/dpi 72 0 dmatrix defaultmatrix dtransform - dup mul E dup mul add sqrt -def -/B {gs bp st gr} bdef %% brush: gr lw lj -/Bbw {gs bpbw st gr} bdef %% brush: gr lw lj -/F {gs rgb eofill gr} bdef %% fill: gr -/Fbw {gs setgray eofill gr} bdef %% fill: gr -/PB {gs lj lw setpattern st gr} bdef -/PF {gs eoclip patpath gr} bdef -/BB {gs rgb lj lw strokepath clip patpath gr} bdef -/xdef {exch def} bdef -/clip_region { - /ht xdef - /wd xdef - /bm xdef - /lm xdef - newpath - lm bm moveto - 0 ht rlineto - wd 0 rlineto - 0 ht neg rlineto - closepath clip -} bdef -/reencode_small_dict 12 dict def -/ReencodeSmall { -reencode_small_dict begin -/new_codes_and_names exch def -/new_font_name exch def -/base_font_name exch def -/base_font_dict base_font_name findfont def -/newfont base_font_dict maxlength dict def -base_font_dict { -exch dup /FID ne -{ dup /Encoding eq -{ exch dup length array copy newfont 3 1 roll put } -{ exch newfont 3 1 roll put } -ifelse -} -{ pop pop } -ifelse -} forall -newfont /FontName new_font_name put -new_codes_and_names aload pop -new_codes_and_names length 2 idiv -{ newfont /Encoding get 3 1 roll put } -repeat -new_font_name newfont definefont pop -end %reencode_small_dict -} def -/extended_Zapf [ -8#223 /a89 -8#224 /a90 -8#225 /a93 -8#226 /a94 -8#227 /a91 -8#230 /a92 -8#231 /a205 -8#232 /a85 -8#233 /a206 -8#234 /a86 -8#235 /a87 -8#236 /a88 -8#237 /a95 -8#240 /a96 -] def -/extended_Standard [ -29 /thorn -30 /yacute -31 /divide -128 /Acircumflex -129 /Adieresis -130 /Agrave -131 /Aring -132 /Atilde -133 /Ccedilla -134 /Eacute -135 /Ecircumflex -136 /Edieresis -137 /Egrave -138 /Iacute -139 /Icircumflex -140 /Idieresis -141 /Igrave -142 /Ntilde -143 /Oacute -144 /Ocircumflex -145 /Odieresis -146 /Ograve -147 /Otilde -148 /Scaron -149 /Uacute -150 /Ucircumflex -151 /Udieresis -152 /Ugrave -153 /Ydieresis -154 /Zcaron -155 /aacute -156 /acircumflex -157 /adieresis -158 /agrave -159 /aring -160 /atilde -161 /exclamdown -162 /cent -163 /sterling -164 /fraction -165 /yen -166 /florin -167 /section -168 /currency -169 /quotesingle -170 /quotedblleft -171 /guillemotleft -172 /guilsinglleft -173 /guilsinglright -174 /fi -175 /fl -176 /plusminus -177 /endash -178 /dagger -179 /daggerdbl -180 /periodcentered -181 /twosuperior -182 /paragraph -183 /bullet -184 /quotesinglbase -185 /quotedblbase -186 /quotedblright -187 /guillemotright -188 /ellipsis -189 /perthousand -190 /threesuperior -191 /questiondown -192 /mu -193 /grave -194 /acute -195 /circumflex -196 /tilde -197 /macron -198 /breve -199 /dotaccent -200 /dieresis -201 /onesuperior -202 /ring -203 /cedilla -204 /onequarter -205 /hungarumlaut -206 /ogonek -207 /caron -208 /emdash -209 /ccedilla -210 /copyright -211 /eacute -212 /ecircumflex -213 /edieresis -214 /egrave -215 /iacute -216 /icircumflex -217 /idieresis -218 /igrave -219 /logicalnot -220 /minus -221 /ntilde -222 /oacute -223 /ocircumflex -224 /odieresis -225 /AE -226 /onehalf -227 /ordfeminine -228 /ograve -229 /otilde -230 /registered -231 /scaron -232 /Lslash -233 /Oslash -234 /OE -235 /ordmasculine -236 /trademark -237 /uacute -238 /ucircumflex -239 /udieresis -240 /ugrave -241 /ae -242 /ydieresis -243 /zcaron -244 /Aacute -245 /dotlessi -246 /threequarters -247 /Eth -248 /lslash -249 /oslash -250 /oe -251 /germandbls -252 /multiply -253 /Yacute -254 /Thorn -255 /eth -] def -/extended_Symbol [ -] def -/extend_font { % stack: fontname newfontname -exch dup (ZapfDingbats) eq -{ cvn exch cvn extended_Zapf ReencodeSmall } -{ dup (Symbol) eq -{ cvn exch cvn extended_Symbol ReencodeSmall } -{ cvn exch cvn extended_Standard ReencodeSmall } -ifelse -} -ifelse -} bind def -/extend_font_name { % stack: font_name_string -dup length 1 add string /extended_font_name exch def -extended_font_name 0 (_) putinterval -extended_font_name 1 3 -1 roll putinterval -extended_font_name -} bind def -/gf { -/f exch def f cvn where -{ f exch begin cvn load exec setfont end } -{ f 0 f length 8 sub getinterval dup -/localfont exch extend_font_name def -localfont extend_font -localfont findfont -/xsz f f length 4 sub 4 getinterval cvi def -/ysz f f length 8 sub 4 getinterval cvi def -[ xsz 0 0 ysz neg 0 0 ] makefont dup f cvn exch def -setfont -} 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font Helvetica -423.333 /Helvetica STDFONT gsave 11867 1520 translate 0 0 M 1.5 dup scale --107.279 0 N -(z) show X -101.051 M -%%IncludeResource: font Helvetica -262.467 /Helvetica STDFONT -(h) show grestore -%%IncludeResource: font Helvetica -423.333 /Helvetica STDFONT -%%PageTrailer -end -restore -showpage -%%PageResources: font Helvetica -%%Trailer -restore -%%Pages: 1 -%%DocumentNeededResources: font Helvetica -%%EOF From af31f34baa01921d480ab04cd61f5385c9f74426 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Thu, 28 May 2026 13:18:30 +0100 Subject: [PATCH 106/116] Changed label for dz_param equation in the PC2 doc to avoid clash with the same equation label used in the BL doc. --- .../science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst index ae486cfd4f..d211258767 100644 --- a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst +++ b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst @@ -1711,7 +1711,7 @@ varies linearly between 0.1 and 0.3 for cloud depths between 100 m and layer depth, :math:`z_h` plus the inversion thickness, :math:`\Delta z_i` parametrized following `Beare (2008)`_ as: -.. math:: :label: dz_param +.. math:: :label: dz_param_inv \Delta z_i = 6.3 \, w_m^2 / \int_{z_h}^{z_h+\Delta z_i} b \, dz @@ -1719,7 +1719,7 @@ where :math:`w_m` is the boundary layer velocity scale (:math:`w_m^3 = u_*^3 + 0.25 w_*^3`) and :math:`b` is the parcel buoyancy that is integrated over the depth of the inversion assuming a piece-wise linear variation between grid-levels. Note that the constant -in :eq:`dz_param` is the same as in +in :eq:`dz_param_inv` is the same as in `Beare (2008)`_ because :math:`6.3 = 2.5 * 4^{2/3}` and :math:`w_m^3` differs by a factor of 4. From bf8992b182616dcaf078571336accda63a918e2b Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Fri, 29 May 2026 23:53:12 +0100 Subject: [PATCH 107/116] Tightened-up script to replace missed instances of \rm with \mathrm etc. --- .../cloud_schemes/UMDP30_PC2CloudScheme.rst | 18 ++++++++++-------- 1 file changed, 10 insertions(+), 8 deletions(-) diff --git a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst index ae486cfd4f..09c10d54c4 100644 --- a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst +++ b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst @@ -1844,7 +1844,7 @@ is a source term due to vertical air motions. `Field et al. (2014)`_ modeled vertical velocity as a white-noise process with autocorrelation function: -.. math:: \overline{w(t)w(s)} = \sigma_w^2 \tau_{\rm d} \delta(t-s), +.. math:: \overline{w(t)w(s)} = \sigma_w^2 \tau_{\mathrm{d}} \delta(t-s), where :math:`\delta` is the Dirac distribution and the intensity of the noise, :math:`\sigma_w^2`, will be called the standard derivation of the @@ -3670,8 +3670,8 @@ Define l_{\mathrm{f}}^{\mathrm{'}}}}{\partial \, z} where the PC2 assumption thus far has been that -:math:`{\overline{Q}}_{\rm{l, reset}} = 0 -= {\overline{Q}}_{\rm{f, reset}}`. +:math:`{\overline{Q}}_{\mathrm{l, reset}} = 0 += {\overline{Q}}_{\mathrm{f, reset}}`. - The current convection scheme assumes that parcel condensate is single phase (ie. either all liquid or all frozen) and this is seriously @@ -3753,9 +3753,11 @@ are discretized: - \left({ SNOW_{\mathrm{k} + 1} \, / \, M_{\mathrm{k} + 1} } \right) \end{aligned} -where :math:`EPSS_{\rm{k}} = -\left({1 + \varepsilon_{\rm{k} + 3 / 4} \, \Delta p_{\rm{k} + 3 / 4}} \right)\, -\left({1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4}} \right)`. +where :math:`EPSS_{\mathrm{k}} = +\left({1 + \varepsilon_{\mathrm{k} + 3 / 4} \, \Delta p_{\mathrm{k} + 3 / 4}} +\right)\, +\left({1 + \varepsilon_{\mathrm{k} + 1 / 4} \, \Delta p_{\mathrm{k} + 1 / 4}} +\right)`. The condensation and precipitation terms in equations :eq:`eq:discdmfbydp`, @@ -3899,8 +3901,8 @@ based upon eqn :eq:`eq:basiclold`: Note that, as a side-effect, the ``:umdp:027`` environment equations for potential temperature and specific humidity are also altered because the condensate is no longer re-evaporated at the end -(:math:`{\overline{Q}}_{\rm{l, reset}} = 0 -= {\overline{Q}}_{\rm{f, reset}}`): +(:math:`{\overline{Q}}_{\mathrm{l, reset}} = 0 += {\overline{Q}}_{\mathrm{f, reset}}`): .. math:: :label: eq:enviroth From f78e5c3cd91618f15c3749e2da5beffa84446529 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Sat, 30 May 2026 00:11:57 +0100 Subject: [PATCH 108/116] Tightened-up script to replace missed instances of \rm with \mathrm etc. --- .../turbulence_schemes/bl_scheme_doc.rst | 201 +++++++++--------- 1 file changed, 103 insertions(+), 98 deletions(-) diff --git a/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst b/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst index 4db865d539..ef494c000d 100644 --- a/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst +++ b/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst @@ -94,7 +94,7 @@ of specific quantities is also available and the details of the necessary changes are documented in appendix :ref:`Appendix: changing between specific humidities and mixing ratios `. Ultimately turbulent motions are dissipated as heat and so the source -term :math:`{\cal S}` in :eq:`cons_eqn_scal` can include an +term :math:`{\mathcal{S}}` in :eq:`cons_eqn_scal` can include an approximation for that frictional heating, as described in appendix :ref:`Appendix: including the heating from turbulence dissipation `. Finally, the ice cloud contributions in @@ -525,8 +525,8 @@ gradient between grid-levels :math:`k_{ct}` and :math:`k_{ct}-1` is compared with that for a parcel lifted adiabatically from grid-level :math:`k_{ct}-1`, in exactly the same way as for the SML parcel ascent (see section :ref:`Calculation of parcel buoyancy excess `). If -:math:`d\theta_v/dz|_{\rm env} > \Gamma_{\rm inv} d\theta_v/dz|_{\rm par}` -between grid-levels +:math:`d\theta_v/dz|_{\mathrm{env}} > \Gamma_{\mathrm{inv}} +d\theta_v/dz|_{\mathrm{par}}` between grid-levels :math:`k_{ct}` and :math:`k_{ct}-1` then NTDSC is set to :math:`k_{ct}-1`; if not then NTDSC is set to :math:`k_{ct}` (recall that grid-levels :math:`k_{ct}-1` and :math:`k_{ct}-2` have already been @@ -718,8 +718,8 @@ finite-difference form of :math:`\overline{w'b}`, see above, :math:`\overline{w'b}` is assumed to be linear between :math:`\overline{w'b}_S` at the surface and zero at a level which must be estimated. The surface layer integration is then from the surface up -to :math:`z_{{\rm K_{SURF}}}`, where :math:`\theta`-level K_SURF is the first -above +to :math:`z_{{\mathrm{K}_{SURF}}}`, where :math:`\theta`-level K_SURF is the +first above :math:`z_i/10`. The level where :math:`\overline{w'b}` is zero is found by linear interpolation across the grid-levels where the diagnosed cloud-free buoyancy flux would become negative. This is where @@ -986,8 +986,8 @@ namelist and :math:`z_{\mathrm{loc}}` is defined below. The orographic blending height, :math:`h_B` (only used within the boundary layer, as defined below), is given by -.. math:: h_B = {\rm max}\left[z_1+(z_{0m})_{\mbox{veg}}, 2^{1/2} \sigma_h - \right] +.. math:: h_B = {\mathrm{max}}\left[z_1+(z_{0m})_{\mathrm{veg}}, 2^{1/2} + \sigma_h \right] where :math:`\sigma_h` is the standard deviation of the height of the subgrid orography and :math:`(z_{0m})_{\mathrm{veg}}` is the vegetative @@ -1027,7 +1027,8 @@ As described in `Lock (2012)`_, the wind shear generated by drainage flows in complex terrain is thought to lead to additional vertical mixing. This wind shear can be approximated as -.. math:: S_d = \frac{\Delta B }{ \Delta z} \, \alpha_d \, t_d \, {\cal Z}_d +.. math:: S_d = \frac{\Delta B }{ \Delta z} \, \alpha_d \, t_d \, + {\mathcal{Z}}_d The representative slope of the local terrain, :math:`\alpha_d`, is given by @@ -1045,7 +1046,7 @@ such flows will be underresolved. The above formula is used so that tends to 0.2 for large values. To limit the vertical extent of :math:`S_d` to be below approximately :math:`z=\sigma_h`, a height-dependent factor is included, -:math:`{\cal Z}_d = 0.5( 1 - {\rm tanh}\left[ 4 ((z/\sigma_h)-1) +:math:`{\mathcal{Z}}_d = 0.5( 1 - {\mathrm{tanh}}\left[ 4 ((z/\sigma_h)-1) \right])`. The timescale, :math:`t_d`, takes a fixed value of 30 minutes, for simplicity. @@ -1100,12 +1101,12 @@ where :math:`Pr_N = 0.7`, and the constants :math:`b_{LEM}` and For stable conditions (:math:`Ri > 0`), several forms for the stability functions are available. The 'long-tailed' functions are -.. math:: f_{\rm stable} = \frac{1}{1+g_0 Ri} +.. math:: f_{\mathrm{stable}} = \frac{1}{1+g_0 Ri} Alternative functions, which decrease as :math:`1/Ri^2` with increasing stability are, from `Louis (1979)`_: -.. math:: f_{\rm stable} = \frac{1}{(1+ 5 Ri)^2} +.. math:: f_{\mathrm{stable}} = \frac{1}{(1+ 5 Ri)^2} and the family of "sharp" functions can be written in terms of a transitional Richardson number, :math:`Ri_{t}`, as: @@ -1138,7 +1139,7 @@ the surface to SHARPEST by 200m. A stability dependent Prandtl number (:math:`Pr=f_m/f_h`) is generally used following `Mailhot and Lock (2004)`_ with: -.. math:: Pr=\min \left( Pr_{\rm max}, \, Pr_N(1+2Ri) \, \right). +.. math:: Pr=\min \left( Pr_{\mathrm{max}}, \, Pr_N(1+2Ri) \, \right). The maximum permitted Prandtl number, :math:`Pr_{\mathrm{max}}`, is currently set to :math:`5` for model stability reasons. The stability functions @@ -1292,14 +1293,14 @@ therefore RI(K), are held on the 'half-level' below :math:`\rho`-level K, which is :math:`\theta`-level K-1. In addition to the above, the log profile correction applied to -:math:`{\cal L}_h` (to give :math:`\tilde{{\mathcal{L}}}_h`) must be applied -*after* +:math:`{\mathcal{L}}_h` (to give :math:`\tilde{{\mathcal{L}}}_h`) must be +applied *after* interpolation of :math:`K_h` to level :math:`k+\frac{1}{2}` in order that the correct cancellation with the finite difference scalar gradient in the flux calculation can occur. In the unstable stability functions :eq:`unstable_stab`, however, :math:`\tilde{{\mathcal{L}}}_h` must be calculated on :math:`\theta`-levels -(i.e., the same as :math:`\tilde{{\cal L}}_m` and :math:`Ri`) in order to +(i.e., the same as :math:`\tilde{{\mathcal{L}}}_m` and :math:`Ri`) in order to maintain the same stability dependence. @@ -1376,8 +1377,9 @@ adjustment by the non-local scheme, i.e., using level of turbulent instability (that incorporates the effects of shear) only needs extend some fractional distance into the cloud layer to disrupt the formation of cumulus elements. Thus, if -:math:`z_{\rm loc}> z_{\rm lcl}+ f_{\rm sh} -\left(z_{\rm par}-z_{\rm lcl}\right)`, where :math:`f_{\mathrm{sh}}` is a +:math:`z_{\mathrm{loc}}> z_{\mathrm{lcl}}+ f_{\mathrm{sh}} +\left(z_{\mathrm{par}}-z_{\mathrm{lcl}}\right)`, where :math:`f_{\mathrm{sh}}` +is a tunable parameter (:math:`0z_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}}`, rather @@ -1528,8 +1531,8 @@ k (z/z_i) w_*^3 / u_*^3 )^{-(1/4)}`, would require a complex function of attempted. The formula for the Prandtl number used in the interior in HB93 is also -matched to that used in the surface exchange functions (:math:`Pr_{\rm surf}`, -say). For the UM, +matched to that used in the surface exchange functions +(:math:`Pr_{\mathrm{surf}}`, say). For the UM, .. math:: @@ -1540,8 +1543,8 @@ say). For the UM, giving :math:`Pr_{\mathrm{surf}} = 1` in the neutral limit (compared to 0.75 from :eq:`prandtl_nl`). In the convective limit, -:math:`Pr_{\rm surf}|_{0.1\, z_{\rm h}} \rightarrow 0.9 (w_*/u_*)^{-3/4} = 0.9 -\beta^{3/4} = 0.14` +:math:`Pr_{\mathrm{surf}}|_{0.1\, z_{\mathrm{h}}} \rightarrow 0.9 +(w_*/u_*)^{-3/4} = 0.9 \beta^{3/4} = 0.14` (compared to 0.375 from :eq:`prandtl_nl`). Thus, the Prandtl numbers do not match between the surface layer and interior formulations in the UM. @@ -1606,8 +1609,8 @@ that :math:`K_m^{\mathrm{Sc}}` will tend to :math:`K_h^{\mathrm{Sc}}` to :math:`K_h|_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}}`), given by :eq:`khent`, as :math:`z` tends to :math:`z_{\mathrm{h}}` (and -here no restriction is made on the magnitude of either :math:`{\cal E}_m^{\rm -Sc}` or :math:`{\mathcal{E}}_h^{\mathrm{Sc}}`). +here no restriction is made on the magnitude of either +:math:`{\mathcal{E}}_m^{\mathrm{Sc}}` or :math:`{\mathcal{E}}_h^{\mathrm{Sc}}`). .. _sec_gradadj: @@ -1643,7 +1646,7 @@ for the different constants in :eq:`ws_defn`. Consistent with the mixed layer assumptions underlying the non-local scheme, the flux profile produced by the scheme is assumed to be essentially determined by the specified surface and entrainment values. -Thus, the effect of including this non-local term (:math:`K_h^{\rm surf} +Thus, the effect of including this non-local term (:math:`K_h^{\mathrm{surf}} \gamma_{\theta_{\ell}}`) is to allow the model to maintain more well-mixed :math:`\theta_{\ell}` profiles (i.e., with :math:`\partial \theta_{\ell}/ \partial z` less negative or even @@ -1770,7 +1773,7 @@ The components of :eq:`fg_new` are: - :math:`K_{h,m}^{\mathrm{surf}}= k z_h w_{h,m} \frac{z}{z_h}\left(1-\frac{z}{z_h}\right)^2` -- :math:`K_h^{\rm Sc}= 3.6 k V_{\rm Sc}z_{ml} +- :math:`K_h^{\mathrm{Sc}}= 3.6 k V_{\mathrm{Sc}}z_{ml} \left(\frac{z'}{z_{ml}}\right)^{3}\left(1-\frac{z'}{z_{ml}} \right)^{2}` - @@ -1778,9 +1781,9 @@ The components of :eq:`fg_new` are: with :math:`\gamma_{\chi}=A_{ga}\frac{\overline{w'\chi'}_S}{w_h z_h}` and :math:`A_{ga}=10` -- :math:`\overline{w'\chi'}_{ng}^{\rm Sc}= f^{Sc} \left(F_{\chi}|_{z_h}- - F_{\chi}^{NT}|_{z_{\rm b}} \right)` with - :math:`f^{Sc}=3.5 \, k \, \frac{V_{\rm Sc}}{V_{\rm sum}} +- :math:`\overline{w'\chi'}_{ng}^{\mathrm{Sc}}= f^{Sc} \left(F_{\chi}|_{z_h}- + F_{\chi}^{NT}|_{z_{\mathrm{b}}} \right)` with + :math:`f^{Sc}=3.5 \, k \, \frac{V_{\mathrm{Sc}}}{V_{\mathrm{sum}}} \left(\frac{z}{z_h}\right)^{3}\left(1-\frac{z}{z_h}\right)` - :math:`f_2 = 0.5 \, \frac{z}{z_h}\, 2^{(z/z_h)^4}` @@ -1815,7 +1818,7 @@ except that :math:`w_m` is replaced by its neutral value: Pr_{\mathrm{conv}} )} and the range is now :math:`Pr_{\mathrm{neut}} = 0.75` to -:math:`Pr_{\rm conv} = +:math:`Pr_{\mathrm{conv}} = 0.6`. As with the standard scheme, a constant Prandtl number of 0.75 is used to calculate :math:`K_m^{\mathrm{Sc}}`. @@ -1902,8 +1905,8 @@ on which the UM was based. This seems an appealing feature (HB's and probably should be considered for the revised scheme (the dash-dotted line in :numref:`Fig. %s ` sets :math:`d^{std} = 10 w_*^2/w_h^2`). Similarly, :math:`f_2` might benefit -from an additional factor of the form :math:`(V_{\rm surf}^3+ -V_{\rm Sc}^3)/ V_{\rm sum}^3` so that it too tends to zero in the +from an additional factor of the form :math:`(V_{\mathrm{surf}}^3+ +V_{\mathrm{Sc}}^3)/ V_{\mathrm{sum}}^3` so that it too tends to zero in the neutral limit. Further analysis of LES and SCM tests will be required to verify this. @@ -1992,7 +1995,7 @@ only difference is in the mixing length, which is calculated as l_{\mathrm{blend}} = W_{1D}l_{\mathrm{bl}}+(1-W_{1D})l_{\mathrm{smag}}, where :math:`l_{\mathrm{bl}}^{-1} = (\kappa z)^{-1} + \lambda_0^{-1}` and -:math:`l_{\rm smag}^{-2} = (\kappa z)^{-2} + (c_s \Delta x)^{-2}`, +:math:`l_{\mathrm{smag}}^{-2} = (\kappa z)^{-2} + (c_s \Delta x)^{-2}`, :math:`\kappa` is the von Karman constant and :math:`c_s` is the Smagorinsky constant. Near the surface :math:`l_{\mathrm{bl}}` and :math:`l_{\mathrm{smag}}` are @@ -2009,15 +2012,15 @@ component to the turbulent flux, and this is simply down-weighted by turbulence becomes better resolved. Therefore the full eddy diffusivity is given by -.. math:: K_\chi = \max\left[W_{1D}K_\chi^{\rm NL}, K_\chi(Ri)\right], +.. math:: K_\chi = \max\left[W_{1D}K_\chi^{\mathrm{NL}}, K_\chi(Ri)\right], where :math:`K_\chi^{\mathrm{NL}}` is the non-local diffusivity and :math:`l` in Eq. :eq:`eq-kri` is given by :math:`l_{\mathrm{blend}}` in Eq. :eq:`eq-lblend`. The turbulent flux is then calculated as -.. math:: F_\chi=-K_\chi\frac{\partial \chi}{\partial z} + W_{1D}F_\chi^{\rm - NL}, +.. math:: F_\chi=-K_\chi\frac{\partial \chi}{\partial z} + + W_{1D}F_\chi^{\mathrm{NL}}, where :math:`F_\chi^{\mathrm{NL}}` is the non-local flux. Therefore when :math:`W_{1D}=1`, the scheme of `Lock et al. (2000)`_ is recovered, @@ -2057,8 +2060,8 @@ The simplest case is for a well-mixed boundary layer, where the appropriate lengthscale is the boundary-layer depth (inversion height). Therefore we set :math:`z_{\mathrm{turb}}=z_h`, which is broadly consistent with `Malavelle et al. (2014)`_, and choose -:math:`\beta=\beta_{\rm bl}=0.15` to give the best match of our function to -that of +:math:`\beta=\beta_{\mathrm{bl}}=0.15` to give the best match of our function +to that of `Honnert et al. (2011)`_. These functions are shown in :numref:`Figure %s `\ (a) and are only dissimilar for small :math:`\Delta @@ -2252,10 +2255,10 @@ be written where the thickness of the inversion is parametrized as :math:`\Delta z_i = -\mbox{min}[V_{\rm sum}^2/\Delta b, 100]` and :math:`L_{rad}` is a +\mathrm{min}[V_{\mathrm{sum}}^2/\Delta b, 100]` and :math:`L_{rad}` is a depth-scale for the radiatively-cooled layer (taken to be 15 :math:`\times -\,\mbox{max}[200/z_c, 1]`, where :math:`z_c` is the cloud depth). To +\,\mathrm{max}[200/z_c, 1]`, where :math:`z_c` is the cloud depth). To allow for a feedback with forcing of entrainment by buoyancy reversal (see appendix :ref:`Appendix: Definitions of the velocity scales `), @@ -2330,8 +2333,8 @@ F|_{z_i}`, so that where the total heat flux :math:`{\mathcal{H}} = \overline{w'\theta_{\ell}'}+ F_{\mathrm{net}}` and -:math:`F_{\rm net} -= F- F|_{z_{\rm b}}`. The net radiative flux relative to the base of the +:math:`F_{\mathrm{net}} += F- F|_{z_{\mathrm{b}}}`. The net radiative flux relative to the base of the mixed layers is simply calculated as .. math:: @@ -2343,7 +2346,7 @@ mixed layers is simply calculated as where NBDSC\ :math:`=1` in SMLs, :math:`{\mathcal{S}}_F` are the temperature increments (in Ks\ :math:`^{-1}`) from the radiation scheme and :math:`F_{\mathrm{net}}|_h` is estimated by extrapolating down from -:math:`F|_{z_{\mbox{\tiny \rm NTML}+\frac{3}{2}} +:math:`F|_{z_{\mathrm{ \mathrm{NTML}}+\frac{3}{2}} }` using the flux-divergence in grid-level NTML\ :math:`+2` (and similarly for DSC layers). @@ -2439,8 +2442,8 @@ entrainment arising from the model's resolved vertical advection (as discussed in `Lock (2001)`_). This is performed at whichever grid-level the entrainment fluxes are specified, to allow for any entrainment implied by a :math:`\theta_{\ell}` subsidence increment, -:math:`\Theta^{\rm S}` (Ks\ :math:`^{-1}`), at the model grid-level below. The -subsidence +:math:`\Theta^{\mathrm{S}}` (Ks\ :math:`^{-1}`), at the model grid-level below. +The subsidence increments could be obtained directly in the SCM but in the full 3D UM advection increments are dominated by the horizontal component. The subsidence increments are calculated, therefore, from the vertical @@ -2452,8 +2455,8 @@ not be very significant). The interpolated entrainment fluxes given by :math:`w_e` but using an entrainment velocity, :math:`\tilde{w_e}`, that is reduced to allow for any subsidence increments applied to the grid-level below the entrainment flux. To take the case of -:math:`z_{\mbox{\tiny \rm NTDSC}+\frac{1}{2}} < -z_i^{n+1} < z_{\mbox{\tiny \rm NTDSC}+\frac{3}{2}}` as an example, this +:math:`z_{\mathrm{ \mathrm{NTDSC}}+\frac{1}{2}} < +z_i^{n+1} < z_{\mathrm{ \mathrm{NTDSC}}+\frac{3}{2}}` as an example, this reduced entrainment velocity is given by .. math:: \tilde{w_e} = w_e + \tilde{w_S} @@ -2517,7 +2520,7 @@ extended up to :math:`z_i`, while the stable lapse in the free atmosphere, between grid-levels NTML\ :math:`+2` and NTML\ :math:`+3`, :math:`\gamma^{\scriptsize \mathrm{FA}}`, is extrapolated down. Equating these areas gives a quadratic equation in :math:`\Delta z_{disc} = -z_{\mbox{\tiny \rm NTML}+\frac{3}{2}} - z_i` which can be written +z_{\mathrm{ \mathrm{NTML}}+\frac{3}{2}} - z_i` which can be written .. math:: :label: zi_interp @@ -2608,8 +2611,8 @@ inaccurate as :math:`z_i` tends to :math:`{\chi}_{\mathrm{ \mathrm{NTML}}}`). Consequently, if the fraction on the right hand side of :eq:`dqt_disc` is greater than 10, double grid-level jumps are used (i.e., -:math:`\Delta \chi = {\chi}_{\mbox{\tiny \rm NTML}+2} - -{\chi}_{\mbox{\tiny \rm NTML}}`). Finally, note that +:math:`\Delta \chi = {\chi}_{\mathrm{ \mathrm{NTML}}+2} - +{\chi}_{\mathrm{ \mathrm{NTML}}}`). Finally, note that :eq:`dqt_disc` implicitly assumes the structure of the :math:`\theta_{\ell}` and :math:`q_t` profiles across the inversion grid-level are consistent with the diagnosed :math:`z_i`. This is very @@ -2720,11 +2723,11 @@ radiative flux is extrapolated down from divergence in the grid-level above the inversion as representative of the free-atmospheric divergence. Second, since the microphysical flux is generated within the cloud, :math:`F_{\chi}^{ppn}|_{z_t} = -{F_{\chi}}^{ppn}_{\mbox{\tiny \rm NTML}+\frac{3}{2}}`. Finally, the +{F_{\chi}}^{ppn}_{\mathrm{ \mathrm{NTML}}+\frac{3}{2}}`. Finally, the subsidence flux-divergence across level :math:`\mathrm{ \mathrm{NTML}}` and :math:`\mathrm{ \mathrm{NTML}}+1` is assumed to be associated with the inversion so :math:`{F_{\chi}}^{Subs}|_{z_h} = -{F_{\chi}}^{Subs}_{\mbox{\tiny \rm NTML}-\frac{1}{2}}`. Thus, the +{F_{\chi}}^{Subs}_{\mathrm{ \mathrm{NTML}}-\frac{1}{2}}`. Thus, the finite-difference form of :eq:`fxtot_zi` becomes .. math:: :label: fxtot_zi_fd @@ -2768,8 +2771,8 @@ the total grid-level flux, :eq:`fxtot_interp`, is used to calculate the entrainment fluxes, it is straightforward to ensure that the net budget of the inversion grid-level, namely :math:`- ( -F_{\chi}^{Tot}|_{\mbox{\tiny \rm NTML}+\frac{3}{2}} - -F_{\chi}^{Tot}|_{\mbox{\tiny \rm NTML}+\frac{1}{2}})/\Delta z`, +F_{\chi}^{Tot}|_{\mathrm{ \mathrm{NTML}}+\frac{3}{2}} - +F_{\chi}^{Tot}|_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}})/\Delta z`, is consistent with the entrainment/subsidence balance. In other words, to use :math:`\theta_{\ell}` as an example, if the inversion is rising (falling) then @@ -2965,7 +2968,7 @@ imposed at the height of the temperature inversion :math:`z_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}}`) and :math:`K_m|_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}}` is calculated from :eq:`kmsurf` and :eq:`kmtop`, noting the use of -the :math:`{\cal E}` factors. +the :math:`{\mathcal{E}}` factors. .. _sec_entr_prof: @@ -2976,8 +2979,8 @@ An inversion is defined as being resolved when it extends above the flux-level above the usual entrainment interface level (see section :ref:`Diagnosis of inversion thickness `), i.e. when -.. math:: z_{\mbox{\tiny \rm NTML}+\frac{1}{2}} + \Delta z_i > z_{\mbox{\tiny - \rm NTML}+\frac{3}{2}} +.. math:: z_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}} + \Delta z_i > z_{\mathrm{ + \mathrm{NTML}}+\frac{3}{2}} When this happens, there is no subgrid inversion diagnosis and the entrainment parametrization follows the methodology given in @@ -3028,8 +3031,8 @@ turbulence forcing and the inversion jump change slowly compared to the timestep. Whilst this is true for atmospheric :math:`\theta_{\ell}` and :math:`q_t`, the latter is not true for tracers with a small boundary layer concentration. Consequently, for a tracer field :math:`\chi`, the -parametrized entrainment fluxes :math:`\overline{w'\chi'}_{ z_{\mbox{\tiny \rm -NTML}+\frac{1}{2}} }` are calculated from +parametrized entrainment fluxes :math:`\overline{w'\chi'}_{ z_{\mathrm{ +\mathrm{NTML}}+\frac{1}{2}} }` are calculated from :eq:`fluxinterp` but are implemented through an equivalent entrainment eddy-diffusivity given by: @@ -3750,8 +3753,8 @@ representing, respectively, the large-scale, gust and small-scale turbulent contributions to the velocity. Locally, Monin-Obukhov theory then gives -.. math:: |\bar{\bf u} + {\bf u}_g({\bf x}) | = \frac{u_*({\bf x})}{k} - \Phi_m({\bf x}). +.. math:: |\bar{\mathbf{u}} + {\mathbf{u}}_g({\mathbf{x}}) | = + \frac{u_*({\mathbf{x}})}{k} \Phi_m({\mathbf{x}}). We ignore the spatial variation of :math:`\Phi_m`, expecting that the principal effect of locally stronger winds is to increase the local @@ -4993,7 +4996,7 @@ included in the Unified Model as an additional explicit (in terms of time discretisation) stress. Following `Wood et al. (2001)`_ we define :math:`{\bf\tau}_{\mathrm{orog}}` to be -.. math:: {\bf\tau}_{\rm orog}(z)=\left({F_p}_x,{F_p}_y\right)e^{-z/\ell}, +.. math:: {\bf\tau}_{\mathrm{orog}}(z)=\left({F_p}_x,{F_p}_y\right)e^{-z/\ell}, where :math:`{\mathbf{F}_p}=({F_p}_x,{F_p}_y)`, :math:`{F_p}_x` and :math:`{F_p}_y` are the grid-box average :math:`x` and :math:`y` @@ -5313,7 +5316,7 @@ the bottom row discretization is obtained: where -.. math:: A_{0}=-{\cal I}_{1}\frac{\Delta +.. math:: A_{0}=-{\mathcal{I}}_{1}\frac{\Delta tK_{u}\Big|_{1}}{z_{1}(z_{3/2}-z_{1/2})},\quad B_{0}=1-A_{0}. Equations :eq:`eq:tridiag`, @@ -5329,9 +5332,8 @@ equations. When the elimination procedure takes place where :math:`\delta u_{1/2}^{'}`, :math:`\beta` are available quantities. Furthermore, -.. math:: \bar{\tau}_{x}^{*}\Big|_{0}=\left({\cal I}_{1}-{\cal - E}_{1}\right)\tau_{x}^{n}\Big|_{0}+{\cal I}_{1}\left(K_{u}\frac{\partial\delta - u^{*}}{\partial z}\right)\Big|_{0} +.. math:: + \bar{\tau}_{x}^{*}\Big|_{0}=\left({\mathcal{I}}_{1}-{\mathcal{E}}_{1}\right)\tau_{x}^{n}\Big|_{0}+{\mathcal{I}}_{1}\left(K_{u}\frac{\partial\delta u^{*}}{\partial z}\right)\Big|_{0} Approximating :math:`\left(\frac{\partial\delta u^{*}}{\partial z}\right)\Big|_{0}\approx\frac{\delta u_{1/2}^{*}-\delta u_{0}^{*}}{z_{1/2}}`, @@ -5395,10 +5397,10 @@ Considering that, :eq:`eq:dX_star` would re-produce :eq:`eq:sppf_inc1`, which is re-written below, -.. math:: \frac{\delta X^{*}}{\Delta t}=({\cal I}_{1}-{\cal - E}_{1})\left(\frac{\partial F^{n}}{\partial z}+S\right)+{\cal - I}_{1}\frac{\partial}{\partial z}\left(K_{X}\frac{\partial\delta - X^{*}}{\partial z}\right) +.. math:: \frac{\delta X^{*}}{\Delta + t}=({\mathcal{I}}_{1}-{\mathcal{E}}_{1})\left(\frac{\partial F^{n}}{\partial + z}+S\right)+{\mathcal{I}}_{1}\frac{\partial}{\partial + z}\left(K_{X}\frac{\partial\delta X^{*}}{\partial z}\right) and thus the following discretization is obtained, on :math:`\theta`-levels: @@ -5421,9 +5423,9 @@ or, where, -.. math:: A_{k}=-{\cal I}_{1}\frac{\Delta - tK_{X}\Big|_{k+1/2}}{(z_{k+1}-z_{k})(z_{k+1/2}-z_{k-1/2})},\; C_{k}=-{\cal - I}_{1}\frac{\Delta +.. math:: A_{k}=-{\mathcal{I}}_{1}\frac{\Delta + tK_{X}\Big|_{k+1/2}}{(z_{k+1}-z_{k})(z_{k+1/2}-z_{k-1/2})},\; + C_{k}=-{\mathcal{I}}_{1}\frac{\Delta tK_{X}\Big|_{k-1/2}}{(z_{k+1/2}-z_{k-1/2})(z_{k}-z_{k-1})},\quad B_{k}=1-A_{k}-C_{k}. @@ -5518,15 +5520,15 @@ Similarly the corresponding discrete equations for where, -.. math:: A_{k}^{'}=-{\cal I}_{2}\frac{\Delta - tK_{X}\Big|_{k+1/2}}{(z_{k+1}-z_{k})(z_{k+1/2}-z_{k-1/2})},\; C_{k}^{'}=-{\cal - I}_{2}\frac{\Delta +.. math:: A_{k}^{'}=-{\mathcal{I}}_{2}\frac{\Delta + tK_{X}\Big|_{k+1/2}}{(z_{k+1}-z_{k})(z_{k+1/2}-z_{k-1/2})},\; + C_{k}^{'}=-{\mathcal{I}}_{2}\frac{\Delta tK_{X}\Big|_{k-1/2}}{(z_{k+1/2}-z_{k-1/2})(z_{k}-z_{k-1})},\quad B_{k}^{'}=1-A_{k}^{'}-C_{k}^{'}, for :math:`k=L,\ldots,2,\quad A_{L}=0`. -.. math:: A_{1}^{'}=-{\cal I}_{2}\Delta +.. math:: A_{1}^{'}=-{\mathcal{I}}_{2}\Delta t\frac{K_{X}\Big|_{3/2}}{z_{3/2}(z_{2}-z_{1})},\quad B_{1}^{'}=1-A_{1}^{'} and the approximation @@ -5759,8 +5761,8 @@ hand-side of :eq:`eq:FTLstar`, relationship, which is simply the definition of the time-weighted averaging consistent with the new scheme: -.. math:: \overline{H^{*}}\equiv{\cal I}_{1}H^{*}-{\cal - E}_{1}H^{n},\qquad\overline{E^{*}}\equiv{\cal I}_{1}E^{*}-{\cal E}_{1}E^{n}. +.. math:: + \overline{H^{*}}\equiv{\mathcal{I}}_{1}H^{*}-{\mathcal{E}}_{1}H^{n},\qquad\overline{E^{*}}\equiv{\mathcal{I}}_{1}E^{*}-{\mathcal{E}}_{1}E^{n}. Therefore, once :math:`\overline{H^{*}},\overline{E^{*}}` have been computed, :math:`H^{*}`, :math:`E^{*}` can be computed as follows: @@ -5940,7 +5942,8 @@ parametrization of non-local momentum fluxes (see section stability dependence in :eq:`tau_nl` that can be written as -.. math:: f_{stab} = - \frac{a_{stab} z_{\rm h}/L }{1 - a_{stab} z_{\rm h}/L} +.. math:: f_{stab} = - \frac{a_{stab} z_{\mathrm{h}}/L }{1 - a_{stab} + z_{\mathrm{h}}/L} for the Obhukov length, :eq:`1.1.4`, :math:`<0` (i.e., unstable boundary layers) and the empirical constant @@ -5949,7 +5952,7 @@ decreases in magnitude (i.e., surface heating increases and wind stress decreases). The final thermal speed, in units of ms\ :math:`^{-1}`, is given simply by -.. math:: {\rm Thermal} \, {\rm Speed} = f_{stab} \, w_* +.. math:: {\mathrm{Thermal}} \, {\mathrm{Speed}} = f_{stab} \, w_* Wind gust: stash 3,463 and 3,515 (scale-dependent) -------------------------------------------------- @@ -6083,15 +6086,16 @@ Combining :eq:`bl_scaling` with and subsuming the Prandtl number into the other constants, for surface-driven boundary layer mixing we can write: -.. math:: K_m^{\rm surf}= \frac{\tau_{\rm surf}}{2} \, \overline{w'^2} = - \frac{\tau_{turb}}{2} \, \frac{2.66 }{C_{ws}^{2/3}} w_m^2 \, f(z') = k z_{\rm - h}w_m f(z') +.. math:: K_m^{\mathrm{surf}}= \frac{\tau_{\mathrm{surf}}}{2} \, + \overline{w'^2} = \frac{\tau_{turb}}{2} \, \frac{2.66 }{C_{ws}^{2/3}} w_m^2 \, + f(z') = k z_{\mathrm{h}}w_m f(z') which then gives :math:`\tau_{\mathrm{surf}} = C_{ws}^{2/3} k z_{\mathrm{h}}/ (1.33 w_m)`. An analogous timescale can be derived for top-driven mixing in decoupled -stratocumulus layers, :math:`\tau_{\rm Sc} = g_1 k z_{\rm ml}/ (1.33 \, -V_{\rm Sc})`. +stratocumulus layers, :math:`\tau_{\mathrm{Sc}} = g_1 k z_{\mathrm{ml}}/ (1.33 +\, +V_{\mathrm{Sc}})`. There are two options to derive a TKE diagnosis from the Ri-based scheme and then combine with the non-local TKE (selected via var_diags_opt). @@ -6154,7 +6158,7 @@ done via the convection scheme. Therefore an option *l_conv_tke* is provided to include an estimate of the TKE due to parametrized convection within the diagnostic. This is given by: -.. math:: e_{\rm conv} = \left(\frac{M}{g\rho \times CCA}\right)^2 +.. math:: e_{\mathrm{conv}} = \left(\frac{M}{g\rho \times CCA}\right)^2 where :math:`M` is the convective updraft mass flux (Pa s\ :math:`^{-1}`) and CCA is the convective cloud area. The final @@ -6234,7 +6238,7 @@ where the subscript :math:`_S` indicates the surface flux; calculation is described in section :ref:`Calculation of \Delta_F `. Various depth parameters are given by :math:`\zeta_s = -(z_{\rm ml}-\tilde{z_c})/z_{\rm ml}`, +(z_{\mathrm{ml}}-\tilde{z_c})/z_{\mathrm{ml}}`, :math:`\zeta = (z_{\mathrm{ml}}-z_c)/z_{\mathrm{ml}}` and :math:`\zeta_r = \zeta + Br (1-\zeta)`. :math:`z_{\mathrm{ml}}` is the mixed-layer depth, :math:`z_c` is the cloud depth and :math:`\tilde{z_c}` is the @@ -6348,7 +6352,7 @@ zero should then give a reasonably accurate measure of cloud-base. The formula for :math:`V_{\mathrm{br}}` was derived using dimensional arguments and comparison with LES data: -:math:`\chi_s = -{q_{\ell}}_{\rm ct}(1+(L/c_p)\alpha_L) / ( +:math:`\chi_s = -{q_{\ell}}_{\mathrm{ct}}(1+(L/c_p)\alpha_L) / ( \Delta q_t - \alpha_L \Delta \theta_{\ell})`, where :math:`{q_{\ell}}_{\mathrm{ct}}` is the cloud-top liquid water mixing ratio, :math:`L` is the latent heat of vaporisation of water, :math:`c_p` the @@ -6405,13 +6409,13 @@ and similarly for :math:`q_f` (noting that currently The only other explicit account of variable cloud fraction is in :eq:`vbr` for which it is assumed that buoyancy reversal can only occur for cloudy air underlying cloud-free air (assuming maximum -overlap). Thus, the cloud fraction factor, :math:`C_{fac} = \mbox{max}[ +overlap). Thus, the cloud fraction factor, :math:`C_{fac} = \mathrm{max}[ 0.0, -\Delta C_F ]`, where -:math:`\Delta C_F = {C_F}_{\mbox{\tiny \rm NTML}+2} - -{C_F}_{\mbox{\tiny \rm NTML}}` if a subgrid inversion is diagnosed +:math:`\Delta C_F = {C_F}_{\mathrm{ \mathrm{NTML}}+2} - +{C_F}_{\mathrm{ \mathrm{NTML}}}` if a subgrid inversion is diagnosed (because :math:`{C_F}_{\mathrm{ \mathrm{NTML}}+1}` is currently meaningless) and :math:`\Delta C_F = -{C_F}_{\mbox{\tiny \rm NTML}+1} - {C_F}_{\mbox{\tiny \rm NTML}}` if not. +{C_F}_{\mathrm{ \mathrm{NTML}}+1} - {C_F}_{\mathrm{ \mathrm{NTML}}}` if not. A more complete decomposition is not possible given a cloud scheme in the model :raw-latex:`\cite[]{smith90}` which does not allow discrete identification of in-cloud and out-of-cloud profiles. The cloud-fraction @@ -6854,7 +6858,8 @@ its local :math:`Ri`-based scheme are used. In :eq:`asymp_ml`, the definition of :math:`\lambda_m` only is altered to -.. math:: \lambda_m = \mbox{max}\left[40,\, 0.3 z_{\rm loc}, 2 h_B \right] +.. math:: \lambda_m = \mathrm{max}\left[40,\, 0.3 z_{\mathrm{loc}}, 2 h_B + \right] and both :math:`\lambda_m` and :math:`\lambda_h` are not reduced (to 40m) above the boundary layer top. It is possible these modifications From 7e1672045b8f71d14aa4c5f9622f8d1f71e136ee Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Fri, 21 Aug 2026 11:10:26 +0100 Subject: [PATCH 109/116] Repaired the broken UM code-structure diagram and made the coloured text work. --- documentation/source/_static/custom.css | 8 + .../cloud_schemes/UMDP30_PC2CloudScheme.rst | 625 +++++++------- .../cloud_schemes/manual_corrections.txt | 807 ++++++++++++++++++ 3 files changed, 1146 insertions(+), 294 deletions(-) diff --git a/documentation/source/_static/custom.css b/documentation/source/_static/custom.css index 2a908a932c..bd819c80df 100644 --- a/documentation/source/_static/custom.css +++ b/documentation/source/_static/custom.css @@ -24,3 +24,11 @@ html[data-theme="dark"] { color: var(--pst-color-link-hover); } } + +/* Set colours in text colour roles */ +.blue { color: #0000ff; } +.green { color: #00b000; } +.purple { color: #ff00ff; } +.blue-lbl { color: #0000ff; font-weight: bold; } +.green-lbl { color: #00b000; font-weight: bold; } +.purple-lbl { color: #ff00ff; font-weight: bold; } diff --git a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst index 09c10d54c4..5f83815ad7 100644 --- a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst +++ b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst @@ -5249,6 +5249,13 @@ performed at the end of the timestep. Code Structure -------------- +.. role:: blue +.. role:: purple +.. role:: green +.. role:: blue-lbl +.. role:: green-lbl +.. role:: purple-lbl + A detailed description of the UM's timestep structure, showing where in the model all the PC2 cloud scheme subroutine calls are made, is given in the subsections below. @@ -5303,465 +5310,495 @@ called *l_pc2_reset*. Turning this on (not recommended!) does 2 things: The location of the various cloud scheme routine calls within the UM is summarised in the list below. -Subroutines only called for the Smith scheme are highlighted in blue, -those only called for PC2 are in green, and those only called for the -bimodal scheme are in purple. +Subroutines only called for the :blue:`Smith` scheme are highlighted in +:blue:`blue`, +those only called for :green:`PC2` are in :green:`green`, +and those only called for the :purple:`bimodal` scheme are in :purple:`purple`. Main Tree from atm_step_4a ^^^^^^^^^^^^^^^^^^^^^^^^^^ -.. container:: itemize | **atm_step_4a** - | \* (performs one timestep of the Unified Model...) + | (performs one timestep of the Unified Model...) + + | **atm_step_alloc_4a** + | (does miscellaneous initialisations in atm_step) + + | :green-lbl:`pc2_rhtl` + | (calculate start-of-timestep Relative Humidity, used by + PC2 initiation) + + | + + | **atmos_physics1** + | (calls explicit "slow" physics routines...) + + | **microphys_ctl** + | (interface to microphysics scheme) + + | :green-lbl:`pc2_turbulence_ctl` + | (Perform optional erosion of liquid-cloud; done + here if NOT doing erosion after convection, e.g. if + no convection scheme is used). + + | :green-lbl:`pc2_hom_conv` + | (called here just to do erosion) + + | + + | :blue-lbl:`ls_cld` + | (Smith scheme without area cloud fraction + calculation, to set initial cloud fields passed into + microphysics) + + | + + | **ls_ppn** + | (microphysics scheme) + + | + + | **mphys_turb_gen_mixed_phase** + | (turbulent production of liquid cloud) + + | - .. container:: itemize + | :green-lbl:`pc2_turbulence_ctl` + | (optionally use the PC2 pdf-width-change code to + calculate the cloud-fraction change from the above + turbulent production of liquid cloud) - .. container:: tcolorbox + | :green-lbl:`pc2_hom_conv` + | (called here just to calculate the cloud + fraction increment consistent with the turbulent + qcl increment) - | **atm_step_alloc_4a** - | \* (does miscellaneous initialisations in atm_step) + | - - | pc2_rhtl - | \* (calculate start-of-timestep Relative Humidity, used by - PC2 initiation) + | **rad_ctl** + | (interface to radiation scheme) - .. container:: tcolorbox + | **sw_rad** + | (short-wave radiation scheme) - | **atmos_physics1** - | \* (calls explicit "slow" physics routines...) + | - .. container:: itemize + | :green-lbl:`pc2_homog_plus_turb` + | (PC2 homogeneous forcing of liquid-cloud by SW + radiation heating) - .. container:: tcolorbox + | - | **microphys_ctl** - | \* (interface to microphysics scheme) + | **lw_rad** + | (long-wave radiation scheme) - - | pc2_turbulence_ctl - | \* (Perform optional erosion of liquid-cloud; done - here if NOT doing erosion after convection, e.g. if - no convection scheme is used). + | - - | pc2_hom_conv - | \* (called here just to do erosion) + | :green-lbl:`pc2_homog_plus_turb` + | (PC2 homogeneous forcing of liquid-cloud by LW + radiation tendency) - - | ls_cld - | \* (Smith scheme without area cloud fraction - calculation, to set initial cloud fields passed into - microphysics) + | - - | **ls_ppn** - | \* (microphysics scheme) + | **atmos_physics1_alloc_pc2** + | (wrapper for PC2 + self-consistency checks at end of atmos_physics1) - - | **mphys_turb_gen_mixed_phase** - | \* (turbulent production of liquid cloud) + | Add increments from microphysics + radiation onto + start-of-timestep fields to form updated fields. - - | pc2_turbulence_ctl - | \* (optionally use the PC2 pdf-width-change code to - calculate the cloud-fraction change from the above - turbulent production of liquid cloud) + | :green-lbl:`pc2_checks` + | (self-consistency checks on cloud fractions and + water contents) - - | pc2_hom_conv - | \* (called here just to calculate the cloud - fraction increment consistent with the turbulent - qcl increment) + | Convert corrected updated fields back to increments. - .. container:: tcolorbox + | Begin loop over solver outer cycles - | **rad_ctl** - | \* (interface to radiation scheme) - - | **sw_rad** - | \* (short-wave radiation scheme) + | **atm_step_phys_reset** + | (for PC2, on subsequent solver outer cycles, reset + cloud-fractions to saved values after atmos_physics1) - - | pc2_homog_plus_turb - | \* (PC2 homogeneous forcing of liquid-cloud by SW - radiation heating) + | - - | **lw_rad** - | \* (long-wave radiation scheme) + | **eg_sl_moisture** + | (large-scale advection of cloud water contents and fractions) - - | pc2_homog_plus_turb - | \* (PC2 homogeneous forcing of liquid-cloud by LW - radiation tendency) + | - .. container:: tcolorbox + | :green-lbl:`pc2_pressure_forcing_only` + | (Optionally calculate homogeneous forcing of liquid + cloud by the pressure change along the trajectory from + departure point to arrival point). - **atmos_physics1_alloc_pc2** (wrapper for PC2 - self-consistency checks at end of atmos_physics1) + | :green-lbl:`pc2_homog_plus_turb` + | (generic homogeneous forcing routine used here). - - Add increments from microphysics + radiation onto - start-of-timestep fields to form updated fields. + | - - | pc2_checks - | \* (self-consistency checks on cloud fractions and - water contents) + | **atmos_physics2** + | (calls "fast" physics routines...) - - Convert corrected updated fields back to increments. + | **ni_bl_ctl** + | (interface to explicit boundary-layer and surface + scheme calls, including calculation of TKE and + TKE-based :math:`RH_{crit}`) - Begin loop over solver outer cycles + | - .. container:: itemize + | :purple-lbl:`bm_calc_tau` + | (calculates turbulence properties used in the + bimodal cloud scheme, based on the boundary-layer + scheme TKE and mixing-length) - .. container:: tcolorbox + | - | **atm_step_phys_reset** - | \* (for PC2, on subsequent solver outer cycles, reset - cloud-fractions to saved values after atmos_physics1) + | **cloud_call_b4_conv** + | (routine for optional cloud-scheme calls before convection) - .. container:: tcolorbox + | :blue-lbl:`ls_arcld` + | (Smith scheme with area cloud fraction; see + :ref:`Smith scheme with area cloud fraction + ` for a drill-down inside this routine) - | **eg_sl_moisture** - | \* (large-scale advection of cloud water contents and - fractions) + | - .. container:: tcolorbox + | :purple-lbl:`bm_ctl` + | (bimodal scheme) - | pc2_pressure_forcing_only - | \* (Optionally calculate homogeneous forcing of liquid - cloud by the pressure change along the trajectory from - departure point to arrival point). + | Set area cloud fraction equal to bulk cloud fraction - - | pc2_homog_plus_turb - | \* (generic homogeneous forcing routine used here). + | :green-lbl:`pc2_initiation_ctl` + | (interface to PC2 initiation and consistency-checks; see + :ref:`PC2 initiation ` for a + drill-down inside this routine) - .. container:: tcolorbox + | - | **atmos_physics2** - | \* (calls "fast" physics routines...) + | **ni_conv_ctl** or **other_conv_ctl** + | (interface routines to various convection schemes...) - .. container:: itemize + | **glue_conv_5a/6a** + | (calls deep, shallow and mid-level convection schemes) - .. container:: tcolorbox + | **deep/shallow/mid_conv** + | (convection scheme main routines) - | **ni_bl_ctl** - | \* (interface to explicit boundary-layer and surface - scheme calls, including calculation of TKE and - TKE-based :math:`RH_{crit}`) + | **convec2** + | (completes lifting of the convective + parcel by one model-level) - .. container:: tcolorbox + | **parcel** + | (calculates new parcel properties at next level) - | bm_calc_tau - | \* (calculates turbulence properties used in the - bimodal cloud scheme, based on the boundary-layer - scheme TKE and mixing-length) + | - .. container:: tcolorbox + | **environ** + | (calculates grid-mean increments + to primary fields; includes PC2 partitioning + of detrained condensate mass between liquid + and ice phases) - | **cloud_call_b4_conv** - | \* (routine for optional cloud-scheme calls before - convection) + | - - | ls_arcld - | \* (Smith scheme with area cloud fraction; see - :ref:`Smith scheme with area cloud fraction ` for a drill-down - inside this routine) + | :green-lbl:`pc2_environ` + | (calculates increments to PC2 + cloud fractions due to convective detrainment + and subsidence) - - | bm_ctl - | \* (bimodal scheme) + | - - Set area cloud fraction equal to bulk cloud fraction + | :green-lbl:`pc2_from_conv_ctl` + | (PC2 calculations after convection) - - | pc2_initiation_ctl - | \* (interface to PC2 initiation and - consistency-checks; see - :ref:`PC2 initiation ` for a - drill-down inside this routine) + | :green-lbl:`pc2_hom_conv` + | (homogeneous forcing by convection, and + erosion of liquid-cloud) - .. container:: tcolorbox + | - | **ni_conv_ctl** or **other_conv_ctl** - | \* (interface routines to various convection - schemes...) + | **ni_imp_ctl** + | (interface to boundary-layer implicit solver) - - | **glue_conv_5a/6a** - | \* (calls deep, shallow and mid-level convection - schemes) + | **imp_solver** + | (implicitly solves vertical diffusion to find + :math:`T_l` and :math:`q_T` updated by turbulent fluxes). - - | **deep/shallow/mid_conv** - | \* (convection scheme main routines) + | - - **convec2** (completes lifting of the convective - parcel by one model-level) + | :green-lbl:`pc2_bl_inhom_ice` + | (inhomogeneous forcing of ice-cloud) - - **parcel** (calculates new parcel properties - at next level) + | - - **environ** (calculates grid-mean increments - to primary fields; includes PC2 partitioning - of detrained condensate mass between liquid - and ice phases) + | :green-lbl:`pc2_delta_hom_turb` + | (homogeneous forcing of liquid cloud by the turbulent fluxes) - - pc2_environ (calculates increments to PC2 - cloud fractions due to convective detrainment - and subsidence) + | - - | pc2_from_conv_ctl - | \* (PC2 calculations after convection) + | :green-lbl:`pc2_bl_forced_cu` + | (adds diagnosed "forced cumulus" cloud fraction + and water content onto the PC2 prognostics) - - | pc2_hom_conv - | \* (homogeneous forcing by convection, and - erosion of liquid-cloud) + | Calculate area cloud fraction: - .. container:: tcolorbox + | :green-lbl:`ls_acf_brooks` + | (for the Brooks epirical method) - | **ni_imp_ctl** - | \* (interface to boundary-layer implicit solver) + | - - | **imp_solver** - | \* (implicitly solves vertical diffusion to find - :math:`T_l` and :math:`q_T` updated by turbulent - fluxes). + | :green-lbl:`pc2_hom_arcld` + | (for the Cusack vertical interpolation method) - - | pc2_bl_inhom_ice - | \* (inhomogeneous forcing of ice-cloud) + | :green-lbl:`pc2_homog_plus_turb` + | (generic homogeneous forcing routine used to interpolate) - - | pc2_delta_hom_turb - | \* (homogeneous forcing of liquid cloud by the - turbulent fluxes) + | - - | pc2_bl_forced_cu - | \* (adds diagnosed "forced cumulus" cloud fraction - and water content onto the PC2 prognostics) + | :blue-lbl:`ls_arcld` + | (interface to diagnostic Smith scheme and area + cloud fraction; see + :ref:`Smith scheme with area cloud fraction + ` for a drill-down inside this routine) - - Calculate area cloud fraction: + | - | ls_acf_brooks - | \* (for the Brooks epirical method) + | :purple-lbl:`bm_ctl` + | (bimodal cloud scheme) - | pc2_hom_arcld - | \* (for the Cusack vertical interpolation method) + | Set area cloud fraction equal to bulk cloud fraction - - | pc2_homog_plus_turb - | \* (generic homogeneous forcing routine used to - interpolate) + | **diagnostics_bl** + | (outputs boundary-layer diagnostics to STASH) - - | ls_arcld - | \* (interface to diagnostic Smith scheme and area - cloud fraction; see - :ref:`Smith scheme with area cloud fraction ` for a drill-down - inside this routine) + | **ls_cld** + | (Smith scheme used here to calculate various + diagnostics of near-surface temperature and + humidity, by extrapolating pressure, :math:`T_l` + and :math:`q_t` down to the desired height and + then re-diagnosing :math:`q_{cl}`. - - | bm_ctl - | \* (bimodal cloud scheme) + | - - Set area cloud fraction equal to bulk cloud fraction + | **atm_step_ac_assim** + | (interface to Data Assimilation analysis increments...) - - | **diagnostics_bl** - | \* (outputs boundary-layer diagnostics to STASH) + | **ac_ctl** + | (control routine for Data Assimilation analysis increments...) - - | **ls_cld** - | \* (Smith scheme used here to calculate various - diagnostics of near-surface temperature and - humidity, by extrapolating pressure, :math:`T_l` - and :math:`q_t` down to the desired height and - then re-diagnosing :math:`q_{cl}`. + | **ac** + | (main analysis increment routine) - .. container:: tcolorbox + | - | **atm_step_ac_assim** - | \* (interface to Data Assimilation analysis increments...) + | :green-lbl:`pc2_assim` + | (PC2 reponse to the analysis increments; see + :ref:`PC2 Data Assimilation ` + for a drill-down inside this routine) - - **ac_ctl** (control routine for Data Assimilation analysis - increments...) + | - - **ac** (main analysis increment routine) + | :green-lbl:`ls_acf_brooks` + | (calculate area cloud fraction using + Brooks empirical method if active) - - | pc2_assim - | \* (PC2 reponse to the analysis increments; see - :ref:`PC2 Data Assimilation ` for a drill-down - inside this routine) + | - - ls_acf_brooks (calculate area cloud fraction using - Brooks empirical method if active) + | :blue-lbl:`ls_arcld` + | (call diagnostic Smith scheme with area cloud + fraction again to account for the analysis increments; + see :ref:`Smith scheme with area cloud fraction + ` for a drill-down inside this routine) - - ls_arcld (call diagnostic Smith scheme with area cloud - fraction again to account for the analysis increments; - see :ref:`Smith scheme with area cloud fraction - ` for a drill-down - inside this routine) + | - .. container:: tcolorbox + | **eg_sl_helmholtz** + | (dynamics pressure solver; updates pressure, and the + winds used to perform advection on the next solver outer cycle) - | **eg_sl_helmholtz** - | \* (dynamics pressure solver; updates pressure, and the - winds used to perform advection on the next solver outer - cycle) + | End loop over solver outer cycles - End loop over solver outer cycles + | :green-lbl:`pc2_pressure_forcing` + | (interface to miscellaneous PC2 calculations at end-of-timestep) - .. container:: tcolorbox + | :green-lbl:`pc2_homog_plus_turb` + | (homogeneous forcing of liquid-cloud by the dynamics + pressure change; optionally either uses total pressure + change including the Lagrangian component following the + winds, or only the Eulerian component from the dynamics + solver) - | pc2_pressure_forcing - | \* (interface to miscellaneous PC2 calculations at - end-of-timestep) + | - - | pc2_homog_plus_turb - | \* (homogeneous forcing of liquid-cloud by the dynamics - pressure change; optionally either uses total pressure - change including the Lagrangian component following the - winds, or only the Eulerian component from the dynamics - solver) + | :green-lbl:`pc2_initiation_ctl` + | (interface to PC2 initiation and consistency-checks; see + :ref:`PC2 initiation ` for a drill-down + inside this routine) - - | pc2_initiation_ctl - | \* (interface to PC2 initiation and consistency-checks; see - :ref:`PC2 initiation ` for a drill-down - inside this routine) + | - .. container:: tcolorbox + | **qt_bal_cld** + | (calculates end-of-timestep cloud state consistent with + final pressure...) - | **qt_bal_cld** - | \* (calculates end-of-timestep cloud state consistent with - final pressure...) + | :blue-lbl:`ls_arcld` + | (interface to diagnostic Smith scheme and area cloud + fraction; see :ref:`Smith scheme with area cloud fraction + ` for a drill-down inside this routine) - - | ls_arcld - | \* (interface to diagnostic Smith scheme and area cloud - fraction; see :ref:`Smith scheme with area cloud fraction ` for a - drill-down inside this routine) + | - - | bm_ctl - | \* (bimodal cloud scheme) + | :purple-lbl:`bm_ctl` + | (bimodal cloud scheme) - - Set area cloud fraction equal to bulk cloud fraction + | Set area cloud fraction equal to bulk cloud fraction - .. container:: tcolorbox + | - | **iau** - | \* (incremental analysis update; part of data assimilation) + | **iau** + | (incremental analysis update; part of data assimilation) - - | pc2_assim - | \* (PC2 reponse to the analysis increments; see - :ref:`PC2 Data Assimilation ` for a drill-down inside - this routine) + | :green-lbl:`pc2_assim` + | (PC2 reponse to the analysis increments; see + :ref:`PC2 Data Assimilation ` + for a drill-down inside this routine) - - | initial_pc2_check - | \* (wrapper for optional self-consistency checks on + | + + | :green-lbl:`initial_pc2_check` + | (wrapper for optional self-consistency checks on prognostic cloud variables if not doing PC2 response to analysis increments) - - | pc2_checks - | \* (self-consistency checks on cloud fractions and water - contents) + | :green-lbl:`pc2_checks` + | (self-consistency checks on cloud fractions and water + contents) Drill-downs within some routines in the call tree are listed separately below, to avoid duplication (since these routines are called in multiple different places in the tree)... + .. _subsubsec_smith_acf: Smith scheme with area cloud fraction ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ -.. container:: itemize + | :blue-lbl:`ls_arcld` + | (interface to diagnostic Smith scheme and area cloud fraction) - .. container:: tcolorbox + | If no area cloud fraction scheme: - | ls_arcld - | \* (interface to diagnostic Smith scheme and area cloud - fraction) + | :blue-lbl:`ls_cld` + | (just directly call Smith scheme) - - If no area cloud fraction scheme: + | Set area cloud fraction equal to bulk cloud fraction. - | ls_cld - | \* (just directly call Smith scheme) + | If using Cusack vertical interpolation method: - Set area cloud fraction equal to bulk cloud fraction. + | Interpolate fields onto finer vertical grid - - If using Cusack vertical interpolation method: + | :blue-lbl:`ls_cld` + | (call Smith scheme using higher vertical resolution fields) - Interpolate fields onto finer vertical grid + | Coarse-grain cloud fields back to model grid, but set area cloud + fraction to max of bulk cloud fraction over corresponding + fine-grid levels. - | ls_cld - | \* (call Smith scheme using higher vertical resolution fields) + | If using Brooks empirical area cloud fraction method: - Coarse-grain cloud fields back to model grid, but set area cloud - fraction to max of bulk cloud fraction over corresponding - fine-grid levels. + | :blue-lbl:`ls_cld` + | (just directly call Smith scheme) - - If using Brooks empirical area cloud fraction method: + | - | ls_cld - | \* (just directly call Smith scheme) + | :blue-lbl:`ls_acf_brooks` + | (estimate area cloud fraction) - | ls_acf_brooks - | \* (estimate area cloud fraction) .. _subsubsec_pc2_initiation: PC2 initiation ^^^^^^^^^^^^^^ -.. container:: itemize - - .. container:: tcolorbox - - | pc2_initiation_ctl - | \* (interface to PC2 initiation and consistency-checks) + | :green-lbl:`pc2_initiation_ctl` + | (interface to PC2 initiation and consistency-checks) - - | pc2_checks - | \* (self-consistency checks on cloud fractions and water + | :green-lbl:`pc2_checks` + | (self-consistency checks on cloud fractions and water contents) - - PC2 initiation of liquid-cloud: + | PC2 initiation of liquid-cloud: - | pc2_bm_initiate - | \* (using the bimodal cloud scheme) + | :green-lbl:`pc2_bm_initiate` + | (using the bimodal cloud scheme) - | pc2_arcld - | \* (using the Smith scheme with the Cusack vertical + | + + | :green-lbl:`pc2_arcld` + | (using the Smith scheme with the Cusack vertical interpolation method) - - | pc2_initiate - | \* (initiation using the Smith scheme, called here on a - finer vertical grid as per the Cusack method) + | :green-lbl:`pc2_initiate` + | (initiation using the Smith scheme, called here on a + finer vertical grid as per the Cusack method) + + | + + | :green-lbl:`pc2_initiate` + | (using the Smith scheme with no area cloud representation) - | pc2_initiate - | \* (using the Smith scheme with no area cloud representation) + | - - | pc2_checks2 - | \* (further self-consistency checks on cloud-fractions) + | :green-lbl:`pc2_checks2` + | (further self-consistency checks on cloud-fractions) - - | pc2_checks - | \* (repeat the first lot of self-consistency checks again, + | + + | :green-lbl:`pc2_checks` + | (repeat the first lot of self-consistency checks again, just in case we broke something in the mean-time!) - - | pc2_hom_arcld - | \* (finds area cloud fraction using a version of the Cusack + | + + | :green-lbl:`pc2_hom_arcld` + | (finds area cloud fraction using a version of the Cusack method, where the cloud fraction on the finer vertical grid is estimated by applying homogeneous forcing relative to the original grid fields) - - | pc2_homog_plus_turb - | \* (generic homogeneous forcing routine used to interpolate) + | :green-lbl:`pc2_homog_plus_turb` + | (generic homogeneous forcing routine used to interpolate) + .. _subsubsec_pc2_assim: PC2 Data Assimilation ^^^^^^^^^^^^^^^^^^^^^ -.. container:: itemize - - .. container:: tcolorbox + | :green-lbl:`pc2_assim` + | (PC2 reponse to the analysis increments) - | pc2_assim - | \* (PC2 reponse to the analysis increments) - - - | pc2_homog_plus_turb - | \* (generic PC2 homogeneous forcing routine used here for + | :green-lbl:`pc2_homog_plus_turb` + | (generic PC2 homogeneous forcing routine used here for liquid-cloud) - - Estimate change in ice-cloud fraction from the assimilation + | Estimate change in ice-cloud fraction from the assimilation increment to ice-cloud mass. - - | pc2_total_cf - | \* (update bulk cloud fraction due to change in ice cloud + | :green-lbl:`pc2_total_cf` + | (update bulk cloud fraction due to change in ice cloud fraction) - - | pc2_checks - | \* (self-consistency checks on prognostic cloud fractions and + | + + | :green-lbl:`pc2_checks` + | (self-consistency checks on prognostic cloud fractions and water contents) + .. _sec_diags: Diagnostics diff --git a/documentation/source/science_guide/cloud_schemes/manual_corrections.txt b/documentation/source/science_guide/cloud_schemes/manual_corrections.txt index 33e34bf867..4ceb16f92c 100644 --- a/documentation/source/science_guide/cloud_schemes/manual_corrections.txt +++ b/documentation/source/science_guide/cloud_schemes/manual_corrections.txt @@ -42,3 +42,810 @@ index 9f1988eb..d0664430 100644 where the subscript :math:`_{diag}` denotes the liquid cloud water content and fraction predicted by the diagnostic cloud scheme (either +index 09c10d54..5f83815a 100644 +--- a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst ++++ b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst +@@ -5249,6 +5249,13 @@ performed at the end of the timestep. + Code Structure + -------------- + ++.. role:: blue ++.. role:: purple ++.. role:: green ++.. role:: blue-lbl ++.. role:: green-lbl ++.. role:: purple-lbl ++ + A detailed description of the UM's timestep structure, showing where in + the model all the PC2 cloud scheme subroutine calls are made, is given + in the subsections below. +@@ -5303,465 +5310,495 @@ called *l_pc2_reset*. Turning this on (not recommended!) does 2 things: + The location of the various cloud scheme routine calls within the UM is + summarised in the list below. + +-Subroutines only called for the Smith scheme are highlighted in blue, +-those only called for PC2 are in green, and those only called for the +-bimodal scheme are in purple. ++Subroutines only called for the :blue:`Smith` scheme are highlighted in ++:blue:`blue`, ++those only called for :green:`PC2` are in :green:`green`, ++and those only called for the :purple:`bimodal` scheme are in :purple:`purple`. + + Main Tree from atm_step_4a + ^^^^^^^^^^^^^^^^^^^^^^^^^^ + +-.. container:: itemize + + | **atm_step_4a** +- | \* (performs one timestep of the Unified Model...) ++ | (performs one timestep of the Unified Model...) ++ ++ | **atm_step_alloc_4a** ++ | (does miscellaneous initialisations in atm_step) ++ ++ | :green-lbl:`pc2_rhtl` ++ | (calculate start-of-timestep Relative Humidity, used by ++ PC2 initiation) ++ ++ | ++ ++ | **atmos_physics1** ++ | (calls explicit "slow" physics routines...) ++ ++ | **microphys_ctl** ++ | (interface to microphysics scheme) ++ ++ | :green-lbl:`pc2_turbulence_ctl` ++ | (Perform optional erosion of liquid-cloud; done ++ here if NOT doing erosion after convection, e.g. if ++ no convection scheme is used). ++ ++ | :green-lbl:`pc2_hom_conv` ++ | (called here just to do erosion) ++ ++ | ++ ++ | :blue-lbl:`ls_cld` ++ | (Smith scheme without area cloud fraction ++ calculation, to set initial cloud fields passed into ++ microphysics) ++ ++ | ++ ++ | **ls_ppn** ++ | (microphysics scheme) ++ ++ | ++ ++ | **mphys_turb_gen_mixed_phase** ++ | (turbulent production of liquid cloud) ++ ++ | + +- .. container:: itemize ++ | :green-lbl:`pc2_turbulence_ctl` ++ | (optionally use the PC2 pdf-width-change code to ++ calculate the cloud-fraction change from the above ++ turbulent production of liquid cloud) + +- .. container:: tcolorbox ++ | :green-lbl:`pc2_hom_conv` ++ | (called here just to calculate the cloud ++ fraction increment consistent with the turbulent ++ qcl increment) + +- | **atm_step_alloc_4a** +- | \* (does miscellaneous initialisations in atm_step) ++ | + +- - | pc2_rhtl +- | \* (calculate start-of-timestep Relative Humidity, used by +- PC2 initiation) ++ | **rad_ctl** ++ | (interface to radiation scheme) + +- .. container:: tcolorbox ++ | **sw_rad** ++ | (short-wave radiation scheme) + +- | **atmos_physics1** +- | \* (calls explicit "slow" physics routines...) ++ | + +- .. container:: itemize ++ | :green-lbl:`pc2_homog_plus_turb` ++ | (PC2 homogeneous forcing of liquid-cloud by SW ++ radiation heating) + +- .. container:: tcolorbox ++ | + +- | **microphys_ctl** +- | \* (interface to microphysics scheme) ++ | **lw_rad** ++ | (long-wave radiation scheme) + +- - | pc2_turbulence_ctl +- | \* (Perform optional erosion of liquid-cloud; done +- here if NOT doing erosion after convection, e.g. if +- no convection scheme is used). ++ | + +- - | pc2_hom_conv +- | \* (called here just to do erosion) ++ | :green-lbl:`pc2_homog_plus_turb` ++ | (PC2 homogeneous forcing of liquid-cloud by LW ++ radiation tendency) + +- - | ls_cld +- | \* (Smith scheme without area cloud fraction +- calculation, to set initial cloud fields passed into +- microphysics) ++ | + +- - | **ls_ppn** +- | \* (microphysics scheme) ++ | **atmos_physics1_alloc_pc2** ++ | (wrapper for PC2 ++ self-consistency checks at end of atmos_physics1) + +- - | **mphys_turb_gen_mixed_phase** +- | \* (turbulent production of liquid cloud) ++ | Add increments from microphysics + radiation onto ++ start-of-timestep fields to form updated fields. + +- - | pc2_turbulence_ctl +- | \* (optionally use the PC2 pdf-width-change code to +- calculate the cloud-fraction change from the above +- turbulent production of liquid cloud) ++ | :green-lbl:`pc2_checks` ++ | (self-consistency checks on cloud fractions and ++ water contents) + +- - | pc2_hom_conv +- | \* (called here just to calculate the cloud +- fraction increment consistent with the turbulent +- qcl increment) ++ | Convert corrected updated fields back to increments. + +- .. container:: tcolorbox ++ | Begin loop over solver outer cycles + +- | **rad_ctl** +- | \* (interface to radiation scheme) + +- - | **sw_rad** +- | \* (short-wave radiation scheme) ++ | **atm_step_phys_reset** ++ | (for PC2, on subsequent solver outer cycles, reset ++ cloud-fractions to saved values after atmos_physics1) + +- - | pc2_homog_plus_turb +- | \* (PC2 homogeneous forcing of liquid-cloud by SW +- radiation heating) ++ | + +- - | **lw_rad** +- | \* (long-wave radiation scheme) ++ | **eg_sl_moisture** ++ | (large-scale advection of cloud water contents and fractions) + +- - | pc2_homog_plus_turb +- | \* (PC2 homogeneous forcing of liquid-cloud by LW +- radiation tendency) ++ | + +- .. container:: tcolorbox ++ | :green-lbl:`pc2_pressure_forcing_only` ++ | (Optionally calculate homogeneous forcing of liquid ++ cloud by the pressure change along the trajectory from ++ departure point to arrival point). + +- **atmos_physics1_alloc_pc2** (wrapper for PC2 +- self-consistency checks at end of atmos_physics1) ++ | :green-lbl:`pc2_homog_plus_turb` ++ | (generic homogeneous forcing routine used here). + +- - Add increments from microphysics + radiation onto +- start-of-timestep fields to form updated fields. ++ | + +- - | pc2_checks +- | \* (self-consistency checks on cloud fractions and +- water contents) ++ | **atmos_physics2** ++ | (calls "fast" physics routines...) + +- - Convert corrected updated fields back to increments. ++ | **ni_bl_ctl** ++ | (interface to explicit boundary-layer and surface ++ scheme calls, including calculation of TKE and ++ TKE-based :math:`RH_{crit}`) + +- Begin loop over solver outer cycles ++ | + +- .. container:: itemize ++ | :purple-lbl:`bm_calc_tau` ++ | (calculates turbulence properties used in the ++ bimodal cloud scheme, based on the boundary-layer ++ scheme TKE and mixing-length) + +- .. container:: tcolorbox ++ | + +- | **atm_step_phys_reset** +- | \* (for PC2, on subsequent solver outer cycles, reset +- cloud-fractions to saved values after atmos_physics1) ++ | **cloud_call_b4_conv** ++ | (routine for optional cloud-scheme calls before convection) + +- .. container:: tcolorbox ++ | :blue-lbl:`ls_arcld` ++ | (Smith scheme with area cloud fraction; see ++ :ref:`Smith scheme with area cloud fraction ++ ` for a drill-down inside this routine) + +- | **eg_sl_moisture** +- | \* (large-scale advection of cloud water contents and +- fractions) ++ | + +- .. container:: tcolorbox ++ | :purple-lbl:`bm_ctl` ++ | (bimodal scheme) + +- | pc2_pressure_forcing_only +- | \* (Optionally calculate homogeneous forcing of liquid +- cloud by the pressure change along the trajectory from +- departure point to arrival point). ++ | Set area cloud fraction equal to bulk cloud fraction + +- - | pc2_homog_plus_turb +- | \* (generic homogeneous forcing routine used here). ++ | :green-lbl:`pc2_initiation_ctl` ++ | (interface to PC2 initiation and consistency-checks; see ++ :ref:`PC2 initiation ` for a ++ drill-down inside this routine) + +- .. container:: tcolorbox ++ | + +- | **atmos_physics2** +- | \* (calls "fast" physics routines...) ++ | **ni_conv_ctl** or **other_conv_ctl** ++ | (interface routines to various convection schemes...) + +- .. container:: itemize ++ | **glue_conv_5a/6a** ++ | (calls deep, shallow and mid-level convection schemes) + +- .. container:: tcolorbox ++ | **deep/shallow/mid_conv** ++ | (convection scheme main routines) + +- | **ni_bl_ctl** +- | \* (interface to explicit boundary-layer and surface +- scheme calls, including calculation of TKE and +- TKE-based :math:`RH_{crit}`) ++ | **convec2** ++ | (completes lifting of the convective ++ parcel by one model-level) + +- .. container:: tcolorbox ++ | **parcel** ++ | (calculates new parcel properties at next level) + +- | bm_calc_tau +- | \* (calculates turbulence properties used in the +- bimodal cloud scheme, based on the boundary-layer +- scheme TKE and mixing-length) ++ | + +- .. container:: tcolorbox ++ | **environ** ++ | (calculates grid-mean increments ++ to primary fields; includes PC2 partitioning ++ of detrained condensate mass between liquid ++ and ice phases) + +- | **cloud_call_b4_conv** +- | \* (routine for optional cloud-scheme calls before +- convection) ++ | + +- - | ls_arcld +- | \* (Smith scheme with area cloud fraction; see +- :ref:`Smith scheme with area cloud fraction ` for a drill-down +- inside this routine) ++ | :green-lbl:`pc2_environ` ++ | (calculates increments to PC2 ++ cloud fractions due to convective detrainment ++ and subsidence) + +- - | bm_ctl +- | \* (bimodal scheme) ++ | + +- - Set area cloud fraction equal to bulk cloud fraction ++ | :green-lbl:`pc2_from_conv_ctl` ++ | (PC2 calculations after convection) + +- - | pc2_initiation_ctl +- | \* (interface to PC2 initiation and +- consistency-checks; see +- :ref:`PC2 initiation ` for a +- drill-down inside this routine) ++ | :green-lbl:`pc2_hom_conv` ++ | (homogeneous forcing by convection, and ++ erosion of liquid-cloud) + +- .. container:: tcolorbox ++ | + +- | **ni_conv_ctl** or **other_conv_ctl** +- | \* (interface routines to various convection +- schemes...) ++ | **ni_imp_ctl** ++ | (interface to boundary-layer implicit solver) + +- - | **glue_conv_5a/6a** +- | \* (calls deep, shallow and mid-level convection +- schemes) ++ | **imp_solver** ++ | (implicitly solves vertical diffusion to find ++ :math:`T_l` and :math:`q_T` updated by turbulent fluxes). + +- - | **deep/shallow/mid_conv** +- | \* (convection scheme main routines) ++ | + +- - **convec2** (completes lifting of the convective +- parcel by one model-level) ++ | :green-lbl:`pc2_bl_inhom_ice` ++ | (inhomogeneous forcing of ice-cloud) + +- - **parcel** (calculates new parcel properties +- at next level) ++ | + +- - **environ** (calculates grid-mean increments +- to primary fields; includes PC2 partitioning +- of detrained condensate mass between liquid +- and ice phases) ++ | :green-lbl:`pc2_delta_hom_turb` ++ | (homogeneous forcing of liquid cloud by the turbulent fluxes) + +- - pc2_environ (calculates increments to PC2 +- cloud fractions due to convective detrainment +- and subsidence) ++ | + +- - | pc2_from_conv_ctl +- | \* (PC2 calculations after convection) ++ | :green-lbl:`pc2_bl_forced_cu` ++ | (adds diagnosed "forced cumulus" cloud fraction ++ and water content onto the PC2 prognostics) + +- - | pc2_hom_conv +- | \* (homogeneous forcing by convection, and +- erosion of liquid-cloud) ++ | Calculate area cloud fraction: + +- .. container:: tcolorbox ++ | :green-lbl:`ls_acf_brooks` ++ | (for the Brooks epirical method) + +- | **ni_imp_ctl** +- | \* (interface to boundary-layer implicit solver) ++ | + +- - | **imp_solver** +- | \* (implicitly solves vertical diffusion to find +- :math:`T_l` and :math:`q_T` updated by turbulent +- fluxes). ++ | :green-lbl:`pc2_hom_arcld` ++ | (for the Cusack vertical interpolation method) + +- - | pc2_bl_inhom_ice +- | \* (inhomogeneous forcing of ice-cloud) ++ | :green-lbl:`pc2_homog_plus_turb` ++ | (generic homogeneous forcing routine used to interpolate) + +- - | pc2_delta_hom_turb +- | \* (homogeneous forcing of liquid cloud by the +- turbulent fluxes) ++ | + +- - | pc2_bl_forced_cu +- | \* (adds diagnosed "forced cumulus" cloud fraction +- and water content onto the PC2 prognostics) ++ | :blue-lbl:`ls_arcld` ++ | (interface to diagnostic Smith scheme and area ++ cloud fraction; see ++ :ref:`Smith scheme with area cloud fraction ++ ` for a drill-down inside this routine) + +- - Calculate area cloud fraction: ++ | + +- | ls_acf_brooks +- | \* (for the Brooks epirical method) ++ | :purple-lbl:`bm_ctl` ++ | (bimodal cloud scheme) + +- | pc2_hom_arcld +- | \* (for the Cusack vertical interpolation method) ++ | Set area cloud fraction equal to bulk cloud fraction + +- - | pc2_homog_plus_turb +- | \* (generic homogeneous forcing routine used to +- interpolate) ++ | **diagnostics_bl** ++ | (outputs boundary-layer diagnostics to STASH) + +- - | ls_arcld +- | \* (interface to diagnostic Smith scheme and area +- cloud fraction; see +- :ref:`Smith scheme with area cloud fraction ` for a drill-down +- inside this routine) ++ | **ls_cld** ++ | (Smith scheme used here to calculate various ++ diagnostics of near-surface temperature and ++ humidity, by extrapolating pressure, :math:`T_l` ++ and :math:`q_t` down to the desired height and ++ then re-diagnosing :math:`q_{cl}`. + +- - | bm_ctl +- | \* (bimodal cloud scheme) ++ | + +- - Set area cloud fraction equal to bulk cloud fraction ++ | **atm_step_ac_assim** ++ | (interface to Data Assimilation analysis increments...) + +- - | **diagnostics_bl** +- | \* (outputs boundary-layer diagnostics to STASH) ++ | **ac_ctl** ++ | (control routine for Data Assimilation analysis increments...) + +- - | **ls_cld** +- | \* (Smith scheme used here to calculate various +- diagnostics of near-surface temperature and +- humidity, by extrapolating pressure, :math:`T_l` +- and :math:`q_t` down to the desired height and +- then re-diagnosing :math:`q_{cl}`. ++ | **ac** ++ | (main analysis increment routine) + +- .. container:: tcolorbox ++ | + +- | **atm_step_ac_assim** +- | \* (interface to Data Assimilation analysis increments...) ++ | :green-lbl:`pc2_assim` ++ | (PC2 reponse to the analysis increments; see ++ :ref:`PC2 Data Assimilation ` ++ for a drill-down inside this routine) + +- - **ac_ctl** (control routine for Data Assimilation analysis +- increments...) ++ | + +- - **ac** (main analysis increment routine) ++ | :green-lbl:`ls_acf_brooks` ++ | (calculate area cloud fraction using ++ Brooks empirical method if active) + +- - | pc2_assim +- | \* (PC2 reponse to the analysis increments; see +- :ref:`PC2 Data Assimilation ` for a drill-down +- inside this routine) ++ | + +- - ls_acf_brooks (calculate area cloud fraction using +- Brooks empirical method if active) ++ | :blue-lbl:`ls_arcld` ++ | (call diagnostic Smith scheme with area cloud ++ fraction again to account for the analysis increments; ++ see :ref:`Smith scheme with area cloud fraction ++ ` for a drill-down inside this routine) + +- - ls_arcld (call diagnostic Smith scheme with area cloud +- fraction again to account for the analysis increments; +- see :ref:`Smith scheme with area cloud fraction +- ` for a drill-down +- inside this routine) ++ | + +- .. container:: tcolorbox ++ | **eg_sl_helmholtz** ++ | (dynamics pressure solver; updates pressure, and the ++ winds used to perform advection on the next solver outer cycle) + +- | **eg_sl_helmholtz** +- | \* (dynamics pressure solver; updates pressure, and the +- winds used to perform advection on the next solver outer +- cycle) ++ | End loop over solver outer cycles + +- End loop over solver outer cycles ++ | :green-lbl:`pc2_pressure_forcing` ++ | (interface to miscellaneous PC2 calculations at end-of-timestep) + +- .. container:: tcolorbox ++ | :green-lbl:`pc2_homog_plus_turb` ++ | (homogeneous forcing of liquid-cloud by the dynamics ++ pressure change; optionally either uses total pressure ++ change including the Lagrangian component following the ++ winds, or only the Eulerian component from the dynamics ++ solver) + +- | pc2_pressure_forcing +- | \* (interface to miscellaneous PC2 calculations at +- end-of-timestep) ++ | + +- - | pc2_homog_plus_turb +- | \* (homogeneous forcing of liquid-cloud by the dynamics +- pressure change; optionally either uses total pressure +- change including the Lagrangian component following the +- winds, or only the Eulerian component from the dynamics +- solver) ++ | :green-lbl:`pc2_initiation_ctl` ++ | (interface to PC2 initiation and consistency-checks; see ++ :ref:`PC2 initiation ` for a drill-down ++ inside this routine) + +- - | pc2_initiation_ctl +- | \* (interface to PC2 initiation and consistency-checks; see +- :ref:`PC2 initiation ` for a drill-down +- inside this routine) ++ | + +- .. container:: tcolorbox ++ | **qt_bal_cld** ++ | (calculates end-of-timestep cloud state consistent with ++ final pressure...) + +- | **qt_bal_cld** +- | \* (calculates end-of-timestep cloud state consistent with +- final pressure...) ++ | :blue-lbl:`ls_arcld` ++ | (interface to diagnostic Smith scheme and area cloud ++ fraction; see :ref:`Smith scheme with area cloud fraction ++ ` for a drill-down inside this routine) + +- - | ls_arcld +- | \* (interface to diagnostic Smith scheme and area cloud +- fraction; see :ref:`Smith scheme with area cloud fraction ` for a +- drill-down inside this routine) ++ | + +- - | bm_ctl +- | \* (bimodal cloud scheme) ++ | :purple-lbl:`bm_ctl` ++ | (bimodal cloud scheme) + +- - Set area cloud fraction equal to bulk cloud fraction ++ | Set area cloud fraction equal to bulk cloud fraction + +- .. container:: tcolorbox ++ | + +- | **iau** +- | \* (incremental analysis update; part of data assimilation) ++ | **iau** ++ | (incremental analysis update; part of data assimilation) + +- - | pc2_assim +- | \* (PC2 reponse to the analysis increments; see +- :ref:`PC2 Data Assimilation ` for a drill-down inside +- this routine) ++ | :green-lbl:`pc2_assim` ++ | (PC2 reponse to the analysis increments; see ++ :ref:`PC2 Data Assimilation ` ++ for a drill-down inside this routine) + +- - | initial_pc2_check +- | \* (wrapper for optional self-consistency checks on ++ | ++ ++ | :green-lbl:`initial_pc2_check` ++ | (wrapper for optional self-consistency checks on + prognostic cloud variables if not doing PC2 response to + analysis increments) + +- - | pc2_checks +- | \* (self-consistency checks on cloud fractions and water +- contents) ++ | :green-lbl:`pc2_checks` ++ | (self-consistency checks on cloud fractions and water ++ contents) + + Drill-downs within some routines in the call tree are listed separately + below, to avoid duplication (since these routines are called in multiple + different places in the tree)... + ++ + .. _subsubsec_smith_acf: + + Smith scheme with area cloud fraction + ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +-.. container:: itemize ++ | :blue-lbl:`ls_arcld` ++ | (interface to diagnostic Smith scheme and area cloud fraction) + +- .. container:: tcolorbox ++ | If no area cloud fraction scheme: + +- | ls_arcld +- | \* (interface to diagnostic Smith scheme and area cloud +- fraction) ++ | :blue-lbl:`ls_cld` ++ | (just directly call Smith scheme) + +- - If no area cloud fraction scheme: ++ | Set area cloud fraction equal to bulk cloud fraction. + +- | ls_cld +- | \* (just directly call Smith scheme) ++ | If using Cusack vertical interpolation method: + +- Set area cloud fraction equal to bulk cloud fraction. ++ | Interpolate fields onto finer vertical grid + +- - If using Cusack vertical interpolation method: ++ | :blue-lbl:`ls_cld` ++ | (call Smith scheme using higher vertical resolution fields) + +- Interpolate fields onto finer vertical grid ++ | Coarse-grain cloud fields back to model grid, but set area cloud ++ fraction to max of bulk cloud fraction over corresponding ++ fine-grid levels. + +- | ls_cld +- | \* (call Smith scheme using higher vertical resolution fields) ++ | If using Brooks empirical area cloud fraction method: + +- Coarse-grain cloud fields back to model grid, but set area cloud +- fraction to max of bulk cloud fraction over corresponding +- fine-grid levels. ++ | :blue-lbl:`ls_cld` ++ | (just directly call Smith scheme) + +- - If using Brooks empirical area cloud fraction method: ++ | + +- | ls_cld +- | \* (just directly call Smith scheme) ++ | :blue-lbl:`ls_acf_brooks` ++ | (estimate area cloud fraction) + +- | ls_acf_brooks +- | \* (estimate area cloud fraction) + + .. _subsubsec_pc2_initiation: + + PC2 initiation + ^^^^^^^^^^^^^^ + +-.. container:: itemize +- +- .. container:: tcolorbox +- +- | pc2_initiation_ctl +- | \* (interface to PC2 initiation and consistency-checks) ++ | :green-lbl:`pc2_initiation_ctl` ++ | (interface to PC2 initiation and consistency-checks) + +- - | pc2_checks +- | \* (self-consistency checks on cloud fractions and water ++ | :green-lbl:`pc2_checks` ++ | (self-consistency checks on cloud fractions and water + contents) + +- - PC2 initiation of liquid-cloud: ++ | PC2 initiation of liquid-cloud: + +- | pc2_bm_initiate +- | \* (using the bimodal cloud scheme) ++ | :green-lbl:`pc2_bm_initiate` ++ | (using the bimodal cloud scheme) + +- | pc2_arcld +- | \* (using the Smith scheme with the Cusack vertical ++ | ++ ++ | :green-lbl:`pc2_arcld` ++ | (using the Smith scheme with the Cusack vertical + interpolation method) + +- - | pc2_initiate +- | \* (initiation using the Smith scheme, called here on a +- finer vertical grid as per the Cusack method) ++ | :green-lbl:`pc2_initiate` ++ | (initiation using the Smith scheme, called here on a ++ finer vertical grid as per the Cusack method) ++ ++ | ++ ++ | :green-lbl:`pc2_initiate` ++ | (using the Smith scheme with no area cloud representation) + +- | pc2_initiate +- | \* (using the Smith scheme with no area cloud representation) ++ | + +- - | pc2_checks2 +- | \* (further self-consistency checks on cloud-fractions) ++ | :green-lbl:`pc2_checks2` ++ | (further self-consistency checks on cloud-fractions) + +- - | pc2_checks +- | \* (repeat the first lot of self-consistency checks again, ++ | ++ ++ | :green-lbl:`pc2_checks` ++ | (repeat the first lot of self-consistency checks again, + just in case we broke something in the mean-time!) + +- - | pc2_hom_arcld +- | \* (finds area cloud fraction using a version of the Cusack ++ | ++ ++ | :green-lbl:`pc2_hom_arcld` ++ | (finds area cloud fraction using a version of the Cusack + method, where the cloud fraction on the finer vertical grid is + estimated by applying homogeneous forcing relative to the + original grid fields) + +- - | pc2_homog_plus_turb +- | \* (generic homogeneous forcing routine used to interpolate) ++ | :green-lbl:`pc2_homog_plus_turb` ++ | (generic homogeneous forcing routine used to interpolate) ++ + + .. _subsubsec_pc2_assim: + + PC2 Data Assimilation + ^^^^^^^^^^^^^^^^^^^^^ + +-.. container:: itemize +- +- .. container:: tcolorbox ++ | :green-lbl:`pc2_assim` ++ | (PC2 reponse to the analysis increments) + +- | pc2_assim +- | \* (PC2 reponse to the analysis increments) +- +- - | pc2_homog_plus_turb +- | \* (generic PC2 homogeneous forcing routine used here for ++ | :green-lbl:`pc2_homog_plus_turb` ++ | (generic PC2 homogeneous forcing routine used here for + liquid-cloud) + +- - Estimate change in ice-cloud fraction from the assimilation ++ | Estimate change in ice-cloud fraction from the assimilation + increment to ice-cloud mass. + +- - | pc2_total_cf +- | \* (update bulk cloud fraction due to change in ice cloud ++ | :green-lbl:`pc2_total_cf` ++ | (update bulk cloud fraction due to change in ice cloud + fraction) + +- - | pc2_checks +- | \* (self-consistency checks on prognostic cloud fractions and ++ | ++ ++ | :green-lbl:`pc2_checks` ++ | (self-consistency checks on prognostic cloud fractions and + water contents) + ++ + .. _sec_diags: + + Diagnostics From efd895ffa4555cf76d80aaa3458f6eefc58b45d9 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Fri, 21 Aug 2026 22:02:33 +0100 Subject: [PATCH 110/116] Moved the specification of text colours into the .rst file where they're used. --- documentation/source/_static/custom.css | 8 ----- .../cloud_schemes/UMDP30_PC2CloudScheme.rst | 26 ++++++++++---- .../cloud_schemes/manual_corrections.txt | 34 +++++++++++++------ 3 files changed, 42 insertions(+), 26 deletions(-) diff --git a/documentation/source/_static/custom.css b/documentation/source/_static/custom.css index bd819c80df..2a908a932c 100644 --- a/documentation/source/_static/custom.css +++ b/documentation/source/_static/custom.css @@ -24,11 +24,3 @@ html[data-theme="dark"] { color: var(--pst-color-link-hover); } } - -/* Set colours in text colour roles */ -.blue { color: #0000ff; } -.green { color: #00b000; } -.purple { color: #ff00ff; } -.blue-lbl { color: #0000ff; font-weight: bold; } -.green-lbl { color: #00b000; font-weight: bold; } -.purple-lbl { color: #ff00ff; font-weight: bold; } diff --git a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst index 5f83815ad7..858160875d 100644 --- a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst +++ b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst @@ -4,6 +4,25 @@ under which the code may be used. ----------------------------------------------------------------------------- +.. raw:: html + + + +.. role:: blue +.. role:: green +.. role:: purple + +.. role:: blue-lbl(strong) + :class: blue +.. role:: green-lbl(strong) + :class: green +.. role:: purple-lbl(strong) + :class: purple + .. attention:: This documentation has been transfered directly from the UM to LFRic; @@ -5249,13 +5268,6 @@ performed at the end of the timestep. Code Structure -------------- -.. role:: blue -.. role:: purple -.. role:: green -.. role:: blue-lbl -.. role:: green-lbl -.. role:: purple-lbl - A detailed description of the UM's timestep structure, showing where in the model all the PC2 cloud scheme subroutine calls are made, is given in the subsections below. diff --git a/documentation/source/science_guide/cloud_schemes/manual_corrections.txt b/documentation/source/science_guide/cloud_schemes/manual_corrections.txt index 4ceb16f92c..9833a5057c 100644 --- a/documentation/source/science_guide/cloud_schemes/manual_corrections.txt +++ b/documentation/source/science_guide/cloud_schemes/manual_corrections.txt @@ -42,23 +42,35 @@ index 9f1988eb..d0664430 100644 where the subscript :math:`_{diag}` denotes the liquid cloud water content and fraction predicted by the diagnostic cloud scheme (either -index 09c10d54..5f83815a 100644 +index 5f83815a..85816087 100644 --- a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst +++ b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst -@@ -5249,6 +5249,13 @@ performed at the end of the timestep. - Code Structure - -------------- +@@ -4,6 +4,25 @@ + under which the code may be used. + ----------------------------------------------------------------------------- ++.. raw:: html ++ ++ ++ +.. role:: blue -+.. role:: purple +.. role:: green -+.. role:: blue-lbl -+.. role:: green-lbl -+.. role:: purple-lbl ++.. role:: purple ++ ++.. role:: blue-lbl(strong) ++ :class: blue ++.. role:: green-lbl(strong) ++ :class: green ++.. role:: purple-lbl(strong) ++ :class: purple + - A detailed description of the UM's timestep structure, showing where in - the model all the PC2 cloud scheme subroutine calls are made, is given - in the subsections below. + .. attention:: + + This documentation has been transfered directly from the UM to LFRic; @@ -5303,465 +5310,495 @@ called *l_pc2_reset*. Turning this on (not recommended!) does 2 things: The location of the various cloud scheme routine calls within the UM is summarised in the list below. From cd540316fb912259a39b66d4ed5c030e29221736 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Thu, 3 Sep 2026 11:44:42 +0100 Subject: [PATCH 111/116] Fixed error in the python scripts when parsing latex cite commands with square brackets (regex failed to match these so they were still broken). --- .../turbulence_schemes/bl_scheme_doc.rst | 97 ++++++++++++++++--- .../turbulence_schemes/manual_corrections.txt | 2 +- 2 files changed, 84 insertions(+), 15 deletions(-) diff --git a/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst b/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst index ef494c000d..24c7b4f5a2 100644 --- a/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst +++ b/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst @@ -946,7 +946,7 @@ A representation of the wind shear, :math:`S_d`, generated by drainage flows in complex terrain can also be included, as described below. Near the surface simple finite difference calculations for the vertical gradients can become inaccurate because of the quasi-logarithmic -profiles of variables :raw-latex:`\cite[]{Ayra1991}`. Currently this is +profiles of variables `Ayra (1991)`_. Currently this is ignored above grid-level 2 and the neutral mixing lengths are given by .. math:: @@ -1367,7 +1367,7 @@ of the surface-based mixed layer to level 1 since the top of the parcel ascent may not be suitable (having previously been diagnosed as cumulus cloud top). The second method effectively increases the importance of the Richardson number diagnosis and has been developed from analysis of -cold-air outbreaks :raw-latex:`\cite[]{bodas-salcedo2012}`. Because of +cold-air outbreaks `Bodas-Salcedo et al. (2012)`_. Because of the strong surface buoyancy generation of turbulence in these regimes, a calculation of :math:`Ri` is made that allows for the gradient adjustment by the non-local scheme, i.e., using @@ -1661,7 +1661,7 @@ The equivalent term for :math:`q_t` (i.e., :math:`\gamma_{q_t}`) is set to zero in order to represent crudely the effects on the mixed-layer :math:`q_t` profile of entrainment drying at the mixed-layer top which tend to make :math:`q_t` profiles less well mixed than those of -:math:`\theta_{\ell}` :raw-latex:`\cite[]{mahrt1976}`. From UM version +:math:`\theta_{\ell}` `Mahrt (1976)`_. From UM version 5.5, there is the option to implement the non-gradient stress parametrization of `Brown and Grant (1997)`_, as described in section :ref:`Non-gradient stress parametrization `. @@ -2045,7 +2045,7 @@ turbulence, :math:`\beta` is a scaling parameter which controls the speed of the transition from unresolved to resolved turbulence, :math:`r_f=\frac{1}{l_0-l_1}`, :math:`l_0=4` and :math:`l_1=0.25` (N. B. this formula is slightly modified from that given in -:raw-latex:`\cite[]{Boutleetal2014}`). +`Boutle et al. (2014)`_). `Malavelle et al. (2014)`_ demonstrated that this scaling method was applicable to any type of unstable boundary layer given an appropriate choice of :math:`z_{\mathrm{turb}}`. In @@ -3566,7 +3566,7 @@ so the `Beljaars and Holtslag (1991)`_ functions imply Ri\ :math:`_{f B} \to` 1/a = 1 as z\ :math:`_{1}`/L :math:`\to \infty`. For **unstable conditions**, i.e. :math:`\Delta`\ B :math:`<` 0, the -Dyer and Hicks forms :raw-latex:`\cite[]{dyer1974}` are used: +Dyer and Hicks forms `Dyer (1974)`_ are used: .. math:: :label: 1.3.15 @@ -4071,7 +4071,7 @@ In all schemes available here the momentum roughness length is given by \frac{\alpha}{g} v_\ast ^2 which is a generalisation of Charnock's formula to include low-wind -conditions :raw-latex:`\cite[]{Smith88}`. :math:`\alpha` is Charnock's +conditions `Smith (1988)`_. :math:`\alpha` is Charnock's coefficient, which is determined from field measurements. It is often taken as a constant, but more elaborate schemes include a dependence on wind speed. In practice the difference between different @@ -4098,7 +4098,7 @@ described. moisture as the wind speed increases that is at variance with observational evidence. A parametrization of the scalar roughness length was developed from surface divergence theory - :raw-latex:`\cite[]{csanady2001}`, as described by + `Csanady (2001)`_, as described by `Edwards (2007)`_. This involves an inverse dependence of :math:`z_{0h}` on the friction velocity in the aerodynamically smooth limit and an inverse dependence of :math:`z_{0h}` on @@ -4130,8 +4130,8 @@ described. #. Option *iseasurfalg=3*. This option provides various forms of the COARE algorithm. The COARE algorithm exists in various forms and continues to be developed. Version 3.0 - :raw-latex:`\cite[]{fairall2003}` has been extensively used, while - version 3.5 :raw-latex:`\cite[]{edson2013}` has recently been + `Fairall et al. (2003)`_ has been extensively used, while + version 3.5 `Edson et al. (2013)`_ has recently been released. Whilst the full COARE algorithm provides a complete description of surface transfer at the sea surface, here we use only the expressions for the roughness lengths. @@ -4498,8 +4498,8 @@ Effective roughness lengths ^^^^^^^^^^^^^^^^^^^^^^^^^^^ Form drag is included in the surface turbulent flux formulation via -effective roughness lengths for momentum :raw-latex:`\cite[]{wood93}` -and for scalar quantities :raw-latex:`\cite[]{hewer1998}`. The formulae +effective roughness lengths for momentum `Wood and Mason (1993)`_ +and for scalar quantities `Hewer and Wood (1998)`_. The formulae of section `8.1 <#section_1>`__ are interpreted as relationships between gridbox mean quantities and fluxes with the roughness lengths replaced by effective values, z\ :math:`_{0m(eff)}` and z\ :math:`_{0h(eff)}`. @@ -6198,7 +6198,7 @@ a given height (in practice 10m). The kinematic surface stress is therefore equal to the product of the pseudostress and the drag coefficient. Pseudostress is sometimes used in observational products, notably the Cross-Calibrated Multi-Platform (CCMP) surface wind vector -analysis :raw-latex:`\cite[]{atlas2011}`. +analysis `Atlas et al. (2011)`_. .. _app_vscales: @@ -6252,7 +6252,7 @@ included in :math:`\zeta_r` and :math:`\tilde{\alpha_t}` (in :math:`D = \chi_s \delta b/\Delta b` and constrained by :math:`0< Br = 10 D < 1`. This gives a linear ramp for this feedback between regimes where there is no buoyancy reversal (:math:`D \leq 0`) and the feedback -seen in LES of stratocumulus :raw-latex:`\cite[]{lock98}` with +seen in LES of stratocumulus `Lock (1998)`_ with significant buoyancy reversal (:math:`D \gtrsim 0.1`). Furthermore, the LES of `Lock (2009)`_ indicated the @@ -6417,7 +6417,7 @@ overlap). Thus, the cloud fraction factor, :math:`C_{fac} = \mathrm{max}[ meaningless) and :math:`\Delta C_F = {C_F}_{\mathrm{ \mathrm{NTML}}+1} - {C_F}_{\mathrm{ \mathrm{NTML}}}` if not. A more complete decomposition is not possible given a cloud scheme in -the model :raw-latex:`\cite[]{smith90}` which does not allow discrete +the model `Smith (1990)`_ which does not allow discrete identification of in-cloud and out-of-cloud profiles. The cloud-fraction dependence of the radiative generation of turbulence is implicitly treated in :eq:`vrad` simply by assuming the grid-box mean @@ -7326,6 +7326,13 @@ References evaporation*. Bound.-Layer Meteor., 37, 129-148. +.. _Smith (1990): + + R. N. B. Smith (1990). + *A scheme for predicting layer clouds and their water content in a general + circulation model*. + Quart. J. Roy. Meteor. Soc., 116, 435-460. + .. _Holtslag and Moeng (1991): A. A. M. Holtslag and C.-H. Moeng (1991). @@ -7391,6 +7398,20 @@ References *Local Versus Nonlocal Boundary-Layer Diffusion in a Global Climate Model*. J. Climate, 6, 1825-1842. +.. _Ayra (1991): + + Ayra, S. P. (1991). + *Finite-Difference Errors in Estimating of Gradients in the Atmospheric + Surface Layer*. + J. Appl. Meteor., 30, 251-253. + +.. _Bodas-Salcedo et al. (2012): + + Bodas-Salcedo, A. and K. D. Williams and P. R. Field and A. P. Lock (2012). + *Contribution of midlatitude cyclone clouds to the short-wave deficit in + the Southern Ocean*. + J. Climate, 25, 7467-7486. + .. _Bolton (1980): Bolton, D. (1980). @@ -7442,6 +7463,12 @@ References *Models and observations of the growth of the atmospheric boundary layer*. Bound.-Layer Meteor., 23, 283-306. +.. _Dyer (1974): + + Dyer, A. J. (1974). + *A review of flux-profile relationships*. + Bound.-Layer Meteor., 7, 363-372. + .. _Edwards (2007): Edwards, J. M. (2007). @@ -7490,6 +7517,12 @@ References *A parametric model of vertical eddy fluxes in the atmosphere*. Bound.-Layer Meteor., 17, 187-202. +.. _Mahrt (1976): + + Mahrt, L. (1976). + *Mixed layer moisture structure*. + Mon. Wea. Rev., 104, 1403-1407. + .. _Mason (1986): Mason, P. J. (1986). @@ -7504,6 +7537,13 @@ References II: Entrainment*. Quart. J. Roy. Meteor. Soc., 112, 461-480. +.. _Smith (1988): + + Smith, S. D. (1988). + *Coefficients for sea surface wind stress, heat flux and wind profiles as a + function of wind speed and temperature*. + J. Geophys. Res., 93, 15467-15472. + .. _Stage and Businger (1981): Stage, S. A. and J. A. Businger (1981). @@ -7595,6 +7635,35 @@ References PDF Multiple Mass Flux Scheme*. J. Atmos. Sci., 69, 1513-1533. +.. _Csanady (2001): + + G. T. Csanady (2001). + *Air-sea interaction: Laws and Mechanisms*. + +.. _Fairall et al. (2003): + + C. W. Fairall and E. F. Bradley and J. E. Hare and A. A. Grachev and J. B. + Edson (2003). + *Bulk parametrization of air-sea fluxes: Updates and Verification for the + COARE Algorithm*. + J. Climate, 16, 571-591. + +.. _Edson et al. (2013): + + J. B. Edson and V. Jampana and R. A. Weller and S. P. Biggore and A. J. + Plueddemann and C. W. Fairall and S. D. Miller and L. Mahrt and D. VIckers + and H. Hersbach (2013). + *On the exchange of momentum over the open ocean*. + J. Phys. Oceanogr., 43, 1589-1610. + +.. _Atlas et al. (2011): + + R. Atlas and R. N. Hoffman and J. Ardizzone and S. M. Leidner and J. C. + Jusem and D. K. Smith and D. Gombos (2011). + *A cross-calibrated, multiplatform ocean surface wind velocity product for + meteorological and oceanographic applications*. + Bull. Amer. Meteorol. Soc., 92, 157-174, doi:10.1175/2010BAMS2946.1. + .. _Andreas et al. (2010): E. L. Andreas and T. W. Horst and A. A. Grachev and P. O. G. Persson and C. diff --git a/documentation/source/science_guide/turbulence_schemes/manual_corrections.txt b/documentation/source/science_guide/turbulence_schemes/manual_corrections.txt index db2f9d01ed..3719354a0b 100644 --- a/documentation/source/science_guide/turbulence_schemes/manual_corrections.txt +++ b/documentation/source/science_guide/turbulence_schemes/manual_corrections.txt @@ -80,7 +80,7 @@ index 5b373b68..8ff32c06 100644 - @@ -6253,7 +6253,7 @@ included in :math:`\zeta_r` and :math:`\tilde{\alpha_t}` (in where there is no buoyancy reversal (:math:`D \leq 0`) and the feedback - seen in LES of stratocumulus :raw-latex:`\cite[]{lock98}` with + seen in LES of stratocumulus `Lock (1998)`_ with significant buoyancy reversal -(:math:`D \raisebox{-.4ex}{$\ \stackrel{>}{{\scriptstyle \sim}} \ $}0.1`). +(:math:`D \gtrsim 0.1`). From 40107c6bc1273c7cf99ca9cc1cf57e5392b06536 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Thu, 3 Sep 2026 12:15:38 +0100 Subject: [PATCH 112/116] Re-applied script after further fixing it to correctly handle instances of cite commands with square brackets / optional arguments containing note text. --- .../turbulence_schemes/bl_scheme_doc.rst | 36 +++++++++---------- .../turbulence_schemes/manual_corrections.txt | 2 +- 2 files changed, 19 insertions(+), 19 deletions(-) diff --git a/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst b/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst index 24c7b4f5a2..dcd480fabe 100644 --- a/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst +++ b/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst @@ -946,7 +946,7 @@ A representation of the wind shear, :math:`S_d`, generated by drainage flows in complex terrain can also be included, as described below. Near the surface simple finite difference calculations for the vertical gradients can become inaccurate because of the quasi-logarithmic -profiles of variables `Ayra (1991)`_. Currently this is +profiles of variables [`Ayra (1991)`_]. Currently this is ignored above grid-level 2 and the neutral mixing lengths are given by .. math:: @@ -1367,7 +1367,7 @@ of the surface-based mixed layer to level 1 since the top of the parcel ascent may not be suitable (having previously been diagnosed as cumulus cloud top). The second method effectively increases the importance of the Richardson number diagnosis and has been developed from analysis of -cold-air outbreaks `Bodas-Salcedo et al. (2012)`_. Because of +cold-air outbreaks [`Bodas-Salcedo et al. (2012)`_]. Because of the strong surface buoyancy generation of turbulence in these regimes, a calculation of :math:`Ri` is made that allows for the gradient adjustment by the non-local scheme, i.e., using @@ -1661,7 +1661,7 @@ The equivalent term for :math:`q_t` (i.e., :math:`\gamma_{q_t}`) is set to zero in order to represent crudely the effects on the mixed-layer :math:`q_t` profile of entrainment drying at the mixed-layer top which tend to make :math:`q_t` profiles less well mixed than those of -:math:`\theta_{\ell}` `Mahrt (1976)`_. From UM version +:math:`\theta_{\ell}` [`Mahrt (1976)`_]. From UM version 5.5, there is the option to implement the non-gradient stress parametrization of `Brown and Grant (1997)`_, as described in section :ref:`Non-gradient stress parametrization `. @@ -2045,7 +2045,7 @@ turbulence, :math:`\beta` is a scaling parameter which controls the speed of the transition from unresolved to resolved turbulence, :math:`r_f=\frac{1}{l_0-l_1}`, :math:`l_0=4` and :math:`l_1=0.25` (N. B. this formula is slightly modified from that given in -`Boutle et al. (2014)`_). +[`Boutle et al. (2014)`_]). `Malavelle et al. (2014)`_ demonstrated that this scaling method was applicable to any type of unstable boundary layer given an appropriate choice of :math:`z_{\mathrm{turb}}`. In @@ -2096,8 +2096,8 @@ they are similar to well-mixed surface driven boundary layers, and the `Lock et al. (2000)`_ scheme parametrizes them as such. The appropriate length scale is now the decoupled cloud mixed layer depth, :math:`z_{\mathrm{sc}}` -:raw-latex:`\cite[i.e.~the depth through which a negatively buoyant parcel -released at cloud top would descend,][]{lock01}`. In this case, below +[i.e.~the depth through which a negatively buoyant parcel +released at cloud top would descend, `Lock (2001)`_]. In this case, below the decoupled cloud top we set .. math:: :label: zturb_dsc @@ -2105,8 +2105,8 @@ the decoupled cloud top we set z_{\mathrm{turb}}=\min\left[\max\left(z,z_{\mathrm{sml}}\right),\max\left(z_{\mathrm{sc}},z_h-z\right)\right], where :math:`z_{\mathrm{sml}}` is the depth of the surface-based mixed layer -:raw-latex:`\cite[i.e.~the depth through which a positively buoyant parcel -released at the surface would ascend,][]{lock00}`. This is shown +[i.e.~the depth through which a positively buoyant parcel +released at the surface would ascend, `Lock et al. (2000)`_]. This is shown schematically in :numref:`Figure %s `\ (b), and ensures that :math:`W_{1D}` has a high value in the poorly resolved surface mixed layer and cloud layer, and a lower value in between those layers. Again, @@ -3566,7 +3566,7 @@ so the `Beljaars and Holtslag (1991)`_ functions imply Ri\ :math:`_{f B} \to` 1/a = 1 as z\ :math:`_{1}`/L :math:`\to \infty`. For **unstable conditions**, i.e. :math:`\Delta`\ B :math:`<` 0, the -Dyer and Hicks forms `Dyer (1974)`_ are used: +Dyer and Hicks forms [`Dyer (1974)`_] are used: .. math:: :label: 1.3.15 @@ -4071,7 +4071,7 @@ In all schemes available here the momentum roughness length is given by \frac{\alpha}{g} v_\ast ^2 which is a generalisation of Charnock's formula to include low-wind -conditions `Smith (1988)`_. :math:`\alpha` is Charnock's +conditions [`Smith (1988)`_]. :math:`\alpha` is Charnock's coefficient, which is determined from field measurements. It is often taken as a constant, but more elaborate schemes include a dependence on wind speed. In practice the difference between different @@ -4098,7 +4098,7 @@ described. moisture as the wind speed increases that is at variance with observational evidence. A parametrization of the scalar roughness length was developed from surface divergence theory - `Csanady (2001)`_, as described by + [`Csanady (2001)`_], as described by `Edwards (2007)`_. This involves an inverse dependence of :math:`z_{0h}` on the friction velocity in the aerodynamically smooth limit and an inverse dependence of :math:`z_{0h}` on @@ -4130,8 +4130,8 @@ described. #. Option *iseasurfalg=3*. This option provides various forms of the COARE algorithm. The COARE algorithm exists in various forms and continues to be developed. Version 3.0 - `Fairall et al. (2003)`_ has been extensively used, while - version 3.5 `Edson et al. (2013)`_ has recently been + [`Fairall et al. (2003)`_] has been extensively used, while + version 3.5 [`Edson et al. (2013)`_] has recently been released. Whilst the full COARE algorithm provides a complete description of surface transfer at the sea surface, here we use only the expressions for the roughness lengths. @@ -4498,8 +4498,8 @@ Effective roughness lengths ^^^^^^^^^^^^^^^^^^^^^^^^^^^ Form drag is included in the surface turbulent flux formulation via -effective roughness lengths for momentum `Wood and Mason (1993)`_ -and for scalar quantities `Hewer and Wood (1998)`_. The formulae +effective roughness lengths for momentum [`Wood and Mason (1993)`_] +and for scalar quantities [`Hewer and Wood (1998)`_]. The formulae of section `8.1 <#section_1>`__ are interpreted as relationships between gridbox mean quantities and fluxes with the roughness lengths replaced by effective values, z\ :math:`_{0m(eff)}` and z\ :math:`_{0h(eff)}`. @@ -6198,7 +6198,7 @@ a given height (in practice 10m). The kinematic surface stress is therefore equal to the product of the pseudostress and the drag coefficient. Pseudostress is sometimes used in observational products, notably the Cross-Calibrated Multi-Platform (CCMP) surface wind vector -analysis `Atlas et al. (2011)`_. +analysis [`Atlas et al. (2011)`_]. .. _app_vscales: @@ -6252,7 +6252,7 @@ included in :math:`\zeta_r` and :math:`\tilde{\alpha_t}` (in :math:`D = \chi_s \delta b/\Delta b` and constrained by :math:`0< Br = 10 D < 1`. This gives a linear ramp for this feedback between regimes where there is no buoyancy reversal (:math:`D \leq 0`) and the feedback -seen in LES of stratocumulus `Lock (1998)`_ with +seen in LES of stratocumulus [`Lock (1998)`_] with significant buoyancy reversal (:math:`D \gtrsim 0.1`). Furthermore, the LES of `Lock (2009)`_ indicated the @@ -6417,7 +6417,7 @@ overlap). Thus, the cloud fraction factor, :math:`C_{fac} = \mathrm{max}[ meaningless) and :math:`\Delta C_F = {C_F}_{\mathrm{ \mathrm{NTML}}+1} - {C_F}_{\mathrm{ \mathrm{NTML}}}` if not. A more complete decomposition is not possible given a cloud scheme in -the model `Smith (1990)`_ which does not allow discrete +the model [`Smith (1990)`_] which does not allow discrete identification of in-cloud and out-of-cloud profiles. The cloud-fraction dependence of the radiative generation of turbulence is implicitly treated in :eq:`vrad` simply by assuming the grid-box mean diff --git a/documentation/source/science_guide/turbulence_schemes/manual_corrections.txt b/documentation/source/science_guide/turbulence_schemes/manual_corrections.txt index 3719354a0b..d0ed9efeec 100644 --- a/documentation/source/science_guide/turbulence_schemes/manual_corrections.txt +++ b/documentation/source/science_guide/turbulence_schemes/manual_corrections.txt @@ -80,7 +80,7 @@ index 5b373b68..8ff32c06 100644 - @@ -6253,7 +6253,7 @@ included in :math:`\zeta_r` and :math:`\tilde{\alpha_t}` (in where there is no buoyancy reversal (:math:`D \leq 0`) and the feedback - seen in LES of stratocumulus `Lock (1998)`_ with + seen in LES of stratocumulus [`Lock (1998)`_] with significant buoyancy reversal -(:math:`D \raisebox{-.4ex}{$\ \stackrel{>}{{\scriptstyle \sim}} \ $}0.1`). +(:math:`D \gtrsim 0.1`). From 826dee612de8a2db1da9174c80ac7d41033dcae3 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Thu, 10 Sep 2026 14:52:35 +0100 Subject: [PATCH 113/116] Restored grey-scale in the PC2 process figure instead of black-and-white (sci/tech review request). --- .../cloud_schemes/pc2_process_explanation.svg | 6487 +---------------- 1 file changed, 31 insertions(+), 6456 deletions(-) diff --git a/documentation/source/science_guide/cloud_schemes/pc2_process_explanation.svg b/documentation/source/science_guide/cloud_schemes/pc2_process_explanation.svg index eda5289492..d5e9ca7388 100644 --- a/documentation/source/science_guide/cloud_schemes/pc2_process_explanation.svg +++ b/documentation/source/science_guide/cloud_schemes/pc2_process_explanation.svg @@ -1,6475 +1,50 @@ + + - - - -Created by potrace 1.16, written by Peter Selinger 2001-2019 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 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- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + inkscape:current-layer="g8" /> From c8de1d69cbad70046229ddd4b0ff53c28945a7e5 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Thu, 10 Sep 2026 15:21:19 +0100 Subject: [PATCH 114/116] Fixed equation that was broken due to nested {} inside mbox (script now updated to handle this). --- .../science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst index 858160875d..2e5c3d7d33 100644 --- a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst +++ b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst @@ -3842,14 +3842,14 @@ at this point and adjust the temperature accordingly. \theta_{\mathrm{k + 1}}^{\mathrm{P}} = \theta_{\mathrm{k + 1}}^{\mathrm{P}} - \left(\frac{L_{\mathrm{F}}}{C_{p} \, \Pi_{\mathrm{k + 1}}} \right)\, l_{\mathrm{f \, k + 1}}^{\mathrm{P}} - \; \ldots \; \mbox{ if l_{\mathrm{f \, k + 1}}^{\mathrm{P}} is melted } + \; \ldots \; \mathrm{ if l_{\mathrm{f \, k + 1}}^{\mathrm{P}} is melted } .. math:: :label: eqn:freezell \theta_{\mathrm{k + 1}}^{\mathrm{P}} = \theta_{\mathrm{k + 1}}^{\mathrm{P}} + \left(\frac{L_{\mathrm{F}}}{C_{p} \, \Pi_{\mathrm{k + 1}}} \right)\, l_{\mathrm{l \, k + 1}}^{\mathrm{P}} - \; \ldots \; \mbox{ if l_{\mathrm{l \, k + 1}}^{\mathrm{P}} is frozen } + \; \ldots \; \mathrm{ if l_{\mathrm{l \, k + 1}}^{\mathrm{P}} is frozen } Once a final value for the condensation term :math:`{\overline{Q}}_{\mathrm{x} \, \mathrm{k} + 1} \, / \, M_{\mathrm{k} + From 8c31e25868d89ee99db4f74f3d90c446d578cd33 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Thu, 10 Sep 2026 15:29:05 +0100 Subject: [PATCH 115/116] Manual fix to the equation that had brackets nested inside mbox (needed extra spacing and removal of mathrm from the l). --- .../cloud_schemes/UMDP30_PC2CloudScheme.rst | 6 ++++-- .../cloud_schemes/manual_corrections.txt | 19 +++++++++++++++++++ 2 files changed, 23 insertions(+), 2 deletions(-) diff --git a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst index 2e5c3d7d33..2723468ee0 100644 --- a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst +++ b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst @@ -3842,14 +3842,16 @@ at this point and adjust the temperature accordingly. \theta_{\mathrm{k + 1}}^{\mathrm{P}} = \theta_{\mathrm{k + 1}}^{\mathrm{P}} - \left(\frac{L_{\mathrm{F}}}{C_{p} \, \Pi_{\mathrm{k + 1}}} \right)\, l_{\mathrm{f \, k + 1}}^{\mathrm{P}} - \; \ldots \; \mathrm{ if l_{\mathrm{f \, k + 1}}^{\mathrm{P}} is melted } + \; \ldots \; \mathrm{if} \; l_{\mathrm{f \, k + 1}}^{\mathrm{P}} + \; \mathrm{is \; melted} .. math:: :label: eqn:freezell \theta_{\mathrm{k + 1}}^{\mathrm{P}} = \theta_{\mathrm{k + 1}}^{\mathrm{P}} + \left(\frac{L_{\mathrm{F}}}{C_{p} \, \Pi_{\mathrm{k + 1}}} \right)\, l_{\mathrm{l \, k + 1}}^{\mathrm{P}} - \; \ldots \; \mathrm{ if l_{\mathrm{l \, k + 1}}^{\mathrm{P}} is frozen } + \; \ldots \; \mathrm{if} \; l_{\mathrm{l \, k + 1}}^{\mathrm{P}} + \; \mathrm{is \; frozen} Once a final value for the condensation term :math:`{\overline{Q}}_{\mathrm{x} \, \mathrm{k} + 1} \, / \, M_{\mathrm{k} + diff --git a/documentation/source/science_guide/cloud_schemes/manual_corrections.txt b/documentation/source/science_guide/cloud_schemes/manual_corrections.txt index 9833a5057c..bea4cb6312 100644 --- a/documentation/source/science_guide/cloud_schemes/manual_corrections.txt +++ b/documentation/source/science_guide/cloud_schemes/manual_corrections.txt @@ -861,3 +861,22 @@ index 5f83815a..85816087 100644 .. _sec_diags: Diagnostics +@@ -3842,14 +3842,16 @@ at this point and adjust the temperature accordingly. + \theta_{\mathrm{k + 1}}^{\mathrm{P}} = \theta_{\mathrm{k + 1}}^{\mathrm{P}} - + \left(\frac{L_{\mathrm{F}}}{C_{p} \, \Pi_{\mathrm{k + 1}}} \right)\, + l_{\mathrm{f \, k + 1}}^{\mathrm{P}} +- \; \ldots \; \mathrm{ if l_{\mathrm{f \, k + 1}}^{\mathrm{P}} is melted } ++ \; \ldots \; \mathrm{if} \; l_{\mathrm{f \, k + 1}}^{\mathrm{P}} ++ \; \mathrm{is \; melted} + + .. math:: :label: eqn:freezell + + \theta_{\mathrm{k + 1}}^{\mathrm{P}} = \theta_{\mathrm{k + 1}}^{\mathrm{P}} + + \left(\frac{L_{\mathrm{F}}}{C_{p} \, \Pi_{\mathrm{k + 1}}} \right)\, + l_{\mathrm{l \, k + 1}}^{\mathrm{P}} +- \; \ldots \; \mathrm{ if l_{\mathrm{l \, k + 1}}^{\mathrm{P}} is frozen } ++ \; \ldots \; \mathrm{if} \; l_{\mathrm{l \, k + 1}}^{\mathrm{P}} ++ \; \mathrm{is \; frozen} + + Once a final value for the condensation term + :math:`{\overline{Q}}_{\mathrm{x} \, \mathrm{k} + 1} \, / \, M_{\mathrm{k} + From 48e8bf32c0e766b1845c67dd49142ebacf275163 Mon Sep 17 00:00:00 2001 From: MichaelWhitall <43407209+MichaelWhitall@users.noreply.github.com> Date: Thu, 10 Sep 2026 15:38:25 +0100 Subject: [PATCH 116/116] Fixed subroutine-name that was spuriously in math mode (that turned out to be a mistake inherited as-is from the original latex source so fixing manually). --- .../cloud_schemes/UMDP30_PC2CloudScheme.rst | 2 +- .../science_guide/cloud_schemes/manual_corrections.txt | 9 +++++++++ 2 files changed, 10 insertions(+), 1 deletion(-) diff --git a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst index 2723468ee0..ce68d8ca81 100644 --- a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst +++ b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst @@ -7063,7 +7063,7 @@ convection itself has ceased). This is perhaps not surprising since the `Brooks et al. (2005)`_ area cloud fraction scheme was evaluated against mid-latitude cloud and it is known that tropical clouds have greater vertical coherence. Tuning the parameters in -:math:`large_scale_cloud/ls_acf_brooks.F90` may be beneficial. +**large_scale_cloud/ls_acf_brooks.F90** may be beneficial. .. figure:: blank.svg :name: fig:schematic diff --git a/documentation/source/science_guide/cloud_schemes/manual_corrections.txt b/documentation/source/science_guide/cloud_schemes/manual_corrections.txt index bea4cb6312..f890ca4cee 100644 --- a/documentation/source/science_guide/cloud_schemes/manual_corrections.txt +++ b/documentation/source/science_guide/cloud_schemes/manual_corrections.txt @@ -880,3 +880,12 @@ index 5f83815a..85816087 100644 Once a final value for the condensation term :math:`{\overline{Q}}_{\mathrm{x} \, \mathrm{k} + 1} \, / \, M_{\mathrm{k} + +@@ -7063,7 +7063,7 @@ convection itself has ceased). This is perhaps not surprising since the + `Brooks et al. (2005)`_ area cloud fraction scheme was evaluated + against mid-latitude cloud and it is known that tropical clouds have + greater vertical coherence. Tuning the parameters in +-:math:`large_scale_cloud/ls_acf_brooks.F90` may be beneficial. ++**large_scale_cloud/ls_acf_brooks.F90** may be beneficial. + + .. figure:: blank.svg + :name: fig:schematic