Adaptive guiding-center to full-orbit switching near the wall for exact footprints

[krystophny]

Summary

Hybrid pusher: trace the cheap guiding center in the core, switch to a full
gyro-resolved orbit near the wall, and switch back if the particle returns to the
bulk. The full orbit through the edge gives the exact wall footprint, impact
angle, and energy that a Larmor-circle estimate cannot, because it resolves the
real trajectory through the inhomogeneous edge field.

Not scheduled now. Design record for a later cycle.

Why not the Larmor-circle estimate alone

The tube estimate (separate issue) dresses the guiding center with an analytic
Larmor circle: cheap, but it assumes a uniform field over the gyration and no
scrape-off-layer physics. For exact strike points and impact state, integrate the
actual particle through the edge.

When to switch

Switch guiding-center to full BEFORE the guiding center reaches s = 1, at
s_switch a few Larmor radii inside the last closed flux surface (equivalently
dist(x_gc, wall) < N rho). Switching only after s > 1 is too late: the impact
point and angle are set during the last gyrations across the boundary, and the
guiding center has already lost its FLR information.

How to switch

• Guiding center to full: reconstruct (x_p, v_p) in Cartesian from x_gc,
  vpar, and vperp = sqrt(2 mu B) at a gyrophase. The phase sets the footprint,
  so sample it (random, or carried from a birth-sampled particle, see Particle-birth initialization mode: seed particles, derive guiding centers #414).
  Conserve energy and mu across the switch.
• Full to guiding center: reduce the full state to its guiding center (the
  well-posed particle-to-guiding-center map), recompute mu, resume guiding-center
  tracing.

Re-entry and chattering

A particle can approach the wall, gyrate, and swing back into the bulk without
striking. Switch back to guiding center when dist(x_gc, wall) > M rho with
M > N: a hysteresis band prevents rapid back-and-forth switching at a single
threshold.

Performance

Confined particles that never approach the wall run pure guiding center: zero
overhead. Only the brief edge approaches run full-orbit, a small fraction of the
total trace, so the aggregate slowdown is modest. The worst case is an edge
particle that loiters near the boundary for many bounce times; the hysteresis
band and an optional full-orbit time cap bound it.

Edge and scrape-off-layer field

The full orbit must run from s_switch across s = 1 to the wall, where the
Boozer and VMEC fields are not defined. This needs a field valid in the
scrape-off layer (extended or vacuum field) up to the wall geometry. This is the
main prerequisite, separate from the switching logic.

Full state at impact

As for the tube estimate, the footprint needs the full particle state at the
wall: position, velocity vector, energy.

Relation to other issues

Companion to the FLR loss-tube issue (Level 0/1/2). Both need the
Cartesian particle reconstruction and the full state at impact. Composes with
#414 for the gyrophase distribution.

[krystophny]

Symplecticity of the hybrid

The scheme is piecewise-symplectic, not globally symplectic. Each segment is symplectic in its own phase space (GC on the reduced 4D space, full orbit on 6D), but the switch maps are not symplectic transformations: GC->full lifts by inventing a gyrophase (adds the gyro pair), full->GC projects it away (drops coordinates, cannot be canonical).

Switching back to core does not compound into runaway. Each GC segment stays symplectic, so no secular energy drift within a segment. A switch adds a single bounded O(rho*^2) kick (~2.5e-5 at rho*~1/200), discrete and small, not a growing drift.

Two switch hazards, both fixable:

• mu mismatch: the GC holds mu as an exact invariant; the full orbit's instantaneous mu ripples a few percent at the gyrofrequency. Gyro-average mu over the last gyration before full->GC.
• energy: rescale the reconstructed velocity onto the exact energy surface at every switch. The scheme is then energy-conserving by construction (not symplectic). For loss physics, energy conservation (no spurious heating) is what matters.

Cleaner design: full orbit as a disposable probe

Keep the symplectic GC as the master for all s < 1, never switch into it, so it stays globally symplectic and unperturbed.

• In the edge band s_switch < s < 1, spawn full-orbit probes from the current GC for FLR-loss and footprint diagnostics. Probe strikes wall -> record impact; probe survives the excursion -> discard it, keep advancing the untouched GC.
• Commit to the full orbit only when the GC reaches s = 1 (field ends there; real fate decided in the SOL).

This keeps the master GC symplectic for the entire confined phase with zero back-switch error; probes are pure diagnostics. Back-switching then only happens for a committed orbit re-entering from the SOL.

Caveat this exposes: GC-as-master over-counts losses that FLR would save (GC crosses s=1 but the real particle swings back). The commit-at-s=1 step with a valid SOL field resolves that.