This DR is still at a discussion/consultation phase and will receive modifications according to feedbacks.
We want to allow for the parallel component of the magnetic field to fluctuate, i.e. the total magnetic field becomes,
$$B = B_0 + \nabla \times A_\parallel + \delta B_\parallel b,$$
Note: $B$, $B_0$, $b$, and $\nabla$ are vectors while $A_\parallel$ and $\delta B_\parallel$ are scalars.
After the success of the Get EM back project (PR #888), which implements the evolution of $A_\parallel$, we consider the evolution of $\delta B_\parallel$.
We follow Kennedy et al. 2024 and take the perpendicular component of the Ampere law, using the long wavelength limit,
$$\delta B_\parallel = - \frac{\mu_0}{|B|} \delta P_\perp,$$
where $|B|$ is the magnetic field amplitude, $\mu_0$ the free space magnetic permeability, and
$$\delta P_\perp =\sum_s \int d^3 v \frac{1}{2}m_s v_\perp^2 \delta f_s$$
the total perpendicular pressure, obtained by summing the perpendicular kinetic energy, $1/2 m_s v_\perp^2$, moment of the perturbed distribution function $\delta f_s$ for each species $s$ of mass $m_s$.
Open question
While $\delta f$ is accessible in local delta-f codes, it is not clear how we should compute it in a full-f code like Gkeyll. An idea is to build fluctuation against the flux surface average. It could work because the pressure should be a flux function at equilibrium. We would thus define
$$\delta P_\perp = P_\perp - \iint P_\perp dy dz / \iint J dy dz$$
with $P_\perp = \sum_s \int d^3 v \frac{1}{2}m_s v_\perp^2 f$
This DR is still at a discussion/consultation phase and will receive modifications according to feedbacks.
We want to allow for the parallel component of the magnetic field to fluctuate, i.e. the total magnetic field becomes,
Note:$B$ , $B_0$ , $b$ , and $\nabla$ are vectors while $A_\parallel$ and $\delta B_\parallel$ are scalars.
After the success of the Get EM back project (PR #888), which implements the evolution of$A_\parallel$ , we consider the evolution of $\delta B_\parallel$ .
We follow Kennedy et al. 2024 and take the perpendicular component of the Ampere law, using the long wavelength limit,
where$|B|$ is the magnetic field amplitude, $\mu_0$ the free space magnetic permeability, and
the total perpendicular pressure, obtained by summing the perpendicular kinetic energy,$1/2 m_s v_\perp^2$ , moment of the perturbed distribution function $\delta f_s$ for each species $s$ of mass $m_s$ .
Open question
While$\delta f$ is accessible in local delta-f codes, it is not clear how we should compute it in a full-f code like Gkeyll. An idea is to build fluctuation against the flux surface average. It could work because the pressure should be a flux function at equilibrium. We would thus define
with$P_\perp = \sum_s \int d^3 v \frac{1}{2}m_s v_\perp^2 f$