Background
Add a simple geometry-based proxy to VAFT for evaluating whether the initial magnetic configuration in EC-assisted start-up is favorable for mirror confinement of EC-born electrons.
The first implementation should follow the low-energy / low-temperature single-particle model of Ejiri & Takase, Nuclear Fusion 47 (2007) 403–416.
In that model, the low-energy confined-orbit boundary in velocity space is approximated by
[
v_\perp = \alpha |v_\parallel|,
]
with
[
\alpha
\max
\left[
\sqrt{\frac{R_{\rm LIN}}{R_S-R_{\rm LIN}}},
\sqrt{
\frac{2R_CR_S-Z_{\max}^2}
{Z_{\max}^2}
}
\right].
]
Here:
- (R_S): particle starting major radius
- (R_{\rm LIN}): inboard limiter radius
- (R_C): radius of curvature of the poloidal field line on the midplane
- (Z_{\max}): effective vertical extent available for mirror confinement
The corresponding low-temperature geometry factor is
[
F_3(\alpha)
\frac{2+3\alpha^2}
{2(1+\alpha^2)^{3/2}}.
]
The purpose of this issue is not to predict EC breakdown success directly. The first target is a low-cost magnetic mirror / orbit-accessibility proxy for the pre-breakdown or very early start-up configuration.
Goal
Implement only the following first-order flow:
(R_start, Z_start = 0)
↓
local / traced poloidal field-line geometry
↓
R_C, Z_max, R_LIN
↓
Ejiri alpha
↓
F3(alpha)
For later EC analysis, the representative starting point can be chosen as
[
(R_S,Z_S)=(R_{\rm ECR},0),
]
but the EC resonance calculation itself should not be part of the first core implementation. r_start should remain an explicit input.
Formula layer
Add the two pure machine-independent relations, conceptually:
ejiri_loss_cone_alpha(
r_start,
r_inboard_limiter,
curvature_radius,
z_max,
)
ejiri_f3(alpha)The exact final module may follow the existing Formula ownership, but avoid introducing a large new particle-orbit subsystem for this first step.
The Formula functions must:
- accept only physical/numerical inputs;
- not read ODS/IDS;
- not depend on shot number or machine name;
- not trace field lines;
- document the Ejiri definition, units, assumptions, validity, limitations, and reference;
- remain discoverable through the existing Formula catalog.
Do not add a second registry or documentation mechanism.
Geometry process
Add a small reusable processing function for one magnetic snapshot.
Conceptually:
compute_ejiri_mirror_proxy(
r_start,
b_field,
wall_r,
wall_z,
...,
)The exact function name and module ownership may follow the current vaft.process conventions.
Minimum returned physical quantities:
r_start
r_inboard_limiter
curvature_radius
z_max
alpha
f3
Optional diagnostic output may include the traced geometry or termination metadata if useful for validation/debugging.
Reuse the existing field-line tracer
Do not implement another field-line ODE solver.
VAFT already provides vaft.process.equilibrium.trace_field_line(...), which integrates
[
\frac{dR}{d\phi}=R\frac{B_R}{B_\phi},
\qquad
\frac{dZ}{d\phi}=R\frac{B_Z}{B_\phi},
]
with RK4 and already supports:
- forward / backward / both directions;
- wall-polygon termination;
- domain bounds;
- maximum path length.
Reuse that infrastructure rather than duplicating it.
(R_C)
Estimate the local midplane field-line curvature using the Ejiri parabolic approximation
[
R(Z)
\simeq
R_S-\frac{Z^2}{2R_C}.
]
A local quadratic fit around the starting point is sufficient for v1.
(R_{\rm LIN})
Determine the inboard limiter/wall radius at the midplane from the supplied wall geometry.
Do not hard-code a VEST value in the generic Formula or Process API.
(Z_{\max})
Use an explicit geometry-based definition for v1.
The Ejiri model relates (Z_{\max}) to either:
- the top/bottom limiter extent, or
- the position where the field-line geometry no longer permits mirror trapping (described in the paper as the point where the field line becomes vertical).
The initial implementation should define and document a reproducible operational rule based on the traced field line and wall geometry.
Important: in the paper, the effective (Z_{\max}) inferred from numerical confined-orbit geometry can differ somewhat from the literal limiter coordinate or vertical-field-line point because the analytic model uses a parabolic approximation. Therefore v1 should be documented as an Ejiri-inspired geometric mirror proxy, not an exact reproduction of the full numerical particle-orbit boundary.
Relationship to existing VAFT infrastructure
VAFT already owns the required magnetic-field infrastructure:
PF active + passive geometry
↓
Green-function response matrices
↓
active + eddy/passive currents
↓
vacuum psi, Br, Bz
Do not reimplement vacuum-field reconstruction in this issue.
Likewise, do not add a new field-line integrator.
This issue should only add the missing physics relation and the small geometry-to-proxy composition layer.
The intended first contract is therefore:
one magnetic snapshot
↓
one Ejiri mirror-quality result
Time dependence belongs to a later wrapper.
Tests
Add focused tests for:
- Eq. (8) loss-cone parameter (\alpha);
- Eq. (16) (F_3(\alpha));
- scalar and NumPy-compatible behavior where consistent with Formula policy;
- recovery of (R_C) from a synthetic parabolic field line;
- recovery of (R_{\rm LIN}) and (Z_{\max}) from synthetic wall/field-line geometry;
- one end-to-end synthetic geometry test reusing the existing
trace_field_line();
- invalid or degenerate geometry returning an explicit invalid/indeterminate result or error rather than plausible-looking finite values.
Do not duplicate the existing RK4 convergence and generic field-line-tracing tests.
Non-goals
This issue does not implement:
- full guiding-centre particle-orbit integration;
- ((v_\parallel,v_\perp)) velocity-space scans;
- Maxwellian integration;
- toroidal current prediction;
- EC absorption;
- electron-neutral or electron-ion collisions;
- ionization or burn-through modeling;
- EC breakdown success/failure classification;
- EC power dependence;
- (T_e) or (n_e) evolution;
- time-dependent ODS wrappers;
- EC resonance-layer calculation;
- dedicated plotting APIs;
- VEST-specific empirical thresholds.
Future extension
After the one-snapshot proxy is validated, a follow-up can evaluate
[
R_{\rm ECR}(t)
\rightarrow
\alpha(t)
\rightarrow
F_3(t)
]
using time-dependent vacuum fields including passive/eddy currents.
That time trace can then be compared retrospectively with VEST EC discharge data, for example:
PF + passive-current waveform
↓
time-dependent vacuum magnetic geometry
↓
EC resonance starting point
↓
mirror-quality proxy
↓
breakdown / current-initiation observations
The primary validation question would be whether the proxy improves separation of successful and failed EC start-up or current-initiation cases after controlling for operating variables such as EC power and prefill conditions.
Even in that later stage, F3 should remain interpreted as a magnetic confinement/accessibility proxy, not as the breakdown probability itself.
Acceptance criteria
Reference
A. Ejiri and Y. Takase, “Toroidal current initiation in low aspect ratio tokamaks based on single-particle orbit analysis,” Nuclear Fusion 47 (2007) 403–416. DOI: 10.1088/0029-5515/47/5/005.
Background
Add a simple geometry-based proxy to VAFT for evaluating whether the initial magnetic configuration in EC-assisted start-up is favorable for mirror confinement of EC-born electrons.
The first implementation should follow the low-energy / low-temperature single-particle model of Ejiri & Takase, Nuclear Fusion 47 (2007) 403–416.
In that model, the low-energy confined-orbit boundary in velocity space is approximated by
[
v_\perp = \alpha |v_\parallel|,
]
with
[
\alpha
\max
\left[
\sqrt{\frac{R_{\rm LIN}}{R_S-R_{\rm LIN}}},
\sqrt{
\frac{2R_CR_S-Z_{\max}^2}
{Z_{\max}^2}
}
\right].
]
Here:
The corresponding low-temperature geometry factor is
[
F_3(\alpha)
\frac{2+3\alpha^2}
{2(1+\alpha^2)^{3/2}}.
]
The purpose of this issue is not to predict EC breakdown success directly. The first target is a low-cost magnetic mirror / orbit-accessibility proxy for the pre-breakdown or very early start-up configuration.
Goal
Implement only the following first-order flow:
For later EC analysis, the representative starting point can be chosen as
[
(R_S,Z_S)=(R_{\rm ECR},0),
]
but the EC resonance calculation itself should not be part of the first core implementation.
r_startshould remain an explicit input.Formula layer
Add the two pure machine-independent relations, conceptually:
The exact final module may follow the existing Formula ownership, but avoid introducing a large new particle-orbit subsystem for this first step.
The Formula functions must:
Do not add a second registry or documentation mechanism.
Geometry process
Add a small reusable processing function for one magnetic snapshot.
Conceptually:
The exact function name and module ownership may follow the current
vaft.processconventions.Minimum returned physical quantities:
Optional diagnostic output may include the traced geometry or termination metadata if useful for validation/debugging.
Reuse the existing field-line tracer
Do not implement another field-line ODE solver.
VAFT already provides
vaft.process.equilibrium.trace_field_line(...), which integrates[
\frac{dR}{d\phi}=R\frac{B_R}{B_\phi},
\qquad
\frac{dZ}{d\phi}=R\frac{B_Z}{B_\phi},
]
with RK4 and already supports:
Reuse that infrastructure rather than duplicating it.
(R_C)
Estimate the local midplane field-line curvature using the Ejiri parabolic approximation
[
R(Z)
\simeq
R_S-\frac{Z^2}{2R_C}.
]
A local quadratic fit around the starting point is sufficient for v1.
(R_{\rm LIN})
Determine the inboard limiter/wall radius at the midplane from the supplied wall geometry.
Do not hard-code a VEST value in the generic Formula or Process API.
(Z_{\max})
Use an explicit geometry-based definition for v1.
The Ejiri model relates (Z_{\max}) to either:
The initial implementation should define and document a reproducible operational rule based on the traced field line and wall geometry.
Important: in the paper, the effective (Z_{\max}) inferred from numerical confined-orbit geometry can differ somewhat from the literal limiter coordinate or vertical-field-line point because the analytic model uses a parabolic approximation. Therefore v1 should be documented as an Ejiri-inspired geometric mirror proxy, not an exact reproduction of the full numerical particle-orbit boundary.
Relationship to existing VAFT infrastructure
VAFT already owns the required magnetic-field infrastructure:
Do not reimplement vacuum-field reconstruction in this issue.
Likewise, do not add a new field-line integrator.
This issue should only add the missing physics relation and the small geometry-to-proxy composition layer.
The intended first contract is therefore:
Time dependence belongs to a later wrapper.
Tests
Add focused tests for:
trace_field_line();Do not duplicate the existing RK4 convergence and generic field-line-tracing tests.
Non-goals
This issue does not implement:
Future extension
After the one-snapshot proxy is validated, a follow-up can evaluate
[
R_{\rm ECR}(t)
\rightarrow
\alpha(t)
\rightarrow
F_3(t)
]
using time-dependent vacuum fields including passive/eddy currents.
That time trace can then be compared retrospectively with VEST EC discharge data, for example:
The primary validation question would be whether the proxy improves separation of successful and failed EC start-up or current-initiation cases after controlling for operating variables such as EC power and prefill conditions.
Even in that later stage,
F3should remain interpreted as a magnetic confinement/accessibility proxy, not as the breakdown probability itself.Acceptance criteria
trace_field_line()implementation is reused; no second field-line integrator is introduced.r_startexplicitly rather than embedding EC resonance physics.Reference
A. Ejiri and Y. Takase, “Toroidal current initiation in low aspect ratio tokamaks based on single-particle orbit analysis,” Nuclear Fusion 47 (2007) 403–416. DOI: 10.1088/0029-5515/47/5/005.