ARTS2D: make 2-D susceptibility consistent with 1-D and correct drift geometry

[joglekara]

Problem

The ARTS 2-D electron susceptibility is inconsistent with the 1-D implementation in two independent ways.

1. Imaginary sign

The 1-D path uses

chiEI = -jnp.pi / (klde**2) * 1j * df

at tsadar/core/physics/form_factor.py:316, while the 2-D helper returns +pi / klde^2 * dfe at line 631 and inserts it as +1j * chiEI at line 820.

For an isotropic Maxwellian, matched projections at normalized phase velocities 0.5, 1, and 2 give matching projected EDF and real principal-value susceptibility, but Im(chi_2D) / Im(chi_1D) is approximately -1 at all three points. The 1-D sign was intentionally corrected in Avi-authored commit 242e4ed0; the 2-D path was missed.

This sign bug is real but is not by itself the small-angle resonance fix: conjugating an electron-only epsilon leaves |epsilon|^2 unchanged.

2. Drift-dependent projection geometry

At form_factor.py:809-818, the 2-D path constructs (omega * k / k^2 - ud) / vTe, replaces the signed resonance coordinate by its magnitude, and derives beta from that drift-shifted vector using atan(y/x).

For longitudinal susceptibility, the projection direction should remain khat; drift enters the signed scalar resonance

xi = (omega / |k| - ue dot khat) / vTe

A drift component perpendicular to k must not rotate the Radon projection. atan(y/x) also has an undefined 0/0 origin and NaN gradients on an axis; use atan2 where an angle is required and define the zero-vector limit.

The current project docstring says beta is the k-vector angle, which conflicts with its caller. The Va and ud frame semantics also differ between the 1-D and 2-D paths.

Acceptance criteria

• Benchmark 1-D and 2-D electron susceptibility against an analytic Maxwellian/Faddeeva reference over positive and negative phase velocities; real and imaginary parts use one documented Fourier/Landau convention and agree within discretization error.
• For an isotropic zero-flow Maxwellian, 1-D and 2-D spectra agree within a documented tolerance.
• Compute the projection direction from khat with atan2 and retain a signed xi.
• Purely perpendicular drift leaves the projected susceptibility unchanged; parallel drift gives only the expected Doppler translation.
• Add an analytic shifted anisotropic-Gaussian Radon test covering mean, variance, coordinate signs, axis-aligned k, xi=0, and finite gradients.
• Document and test the Va/ud frame convention, including multi-ion cases, so 1-D and 2-D use the same Galilean transformation.

Related: #124.

[almilder]

Traced through problem 2 in calc_in_2D / calc_chi_vals — agree it's a real bug, but want to refine the framing a bit before we scope the fix.

Beta rotating with ud is a genuine bug, and it's isolated to one line. The scalar Doppler shift itself is fine — omgdop = omg - vdot(k, Va) (form_factor.py:791) correctly dots the ion flow vector into k. The problem is in how the electron resonance vector is built at form_factor.py:814:

xie = vdiv(vsub(vdot(omgdop / k_mag**2, k), ud), vTe)

vdot(omgdop / k_mag**2, k) = (omgdop/|k|) * k̂, which is correctly a vector along k̂. But the full 2D ud is then subtracted from it, not just ud's projection onto k̂. Any u_perp component rotates the result away from k̂, and since beta is derived from this rotated vector's direction, the projection axis moves with drift when it should stay locked to k̂ regardless of drift.

The "unsigned xie_mag" framing undersells what's actually wrong — no sign is lost in the magnitude/angle split itself. Downstream in calc_chi_vals (form_factor.py:606-633), fe_vphi, dfe, and the ratintn pole location are all evaluated at xie_mag_at along the profile projected at angle beta. Since (xie_mag, beta) is just the polar form of xie, proj_beta(xie_mag) reconstructs the correct value at the point xie — beta carries the orientation that a sign would otherwise encode, so nothing is discarded by taking the magnitude. In other words, "unsigned xie_mag" is a symptom of point 1, not an independent defect: once xie is built correctly (k-only direction, drift entering only as a scalar shift), the magnitude/angle decomposition of it is already fine as-is. Suggest we drop the "signed scalar" framing from the acceptance criteria and state the actual requirement instead: beta must be derived from k̂ alone, and the resonance coordinate must use only the k-projection of drift, ud·k̂.

atan2 is the right fix for the quadrant issue, with one caveat. jnp.arctan(xie[1]/xie[0]) plus the heaviside term at form_factor.py:830 is a hand-rolled 4-quadrant atan2, so replacing it directly with atan2(y, x) removes both the 0/0 division and the NaN-gradient blowup on the x=0 axis. atan2(0, 0) is still mathematically undefined at the true origin though, so we still need to explicitly pick and document a convention for the zero-vector case (e.g. beta = 0, since the projected value there should be direction-independent by symmetry — worth confirming that before locking it in).

Proposed fix:

k_hat = k / k_mag                       # unchanged by drift
beta = atan2(k_hat[1], k_hat[0])        # projection direction: k alone
ud_par = vdot(ud, k_hat)                # only the k-projection of drift
xi = (omgdop / k_mag - ud_par) / vTe    # signed scalar, evaluation point

with the sign of xi folded into which side of the beta-projected profile gets sampled (either via beta + pi when xi < 0, or by having _interp_beta_v accept a signed coordinate directly rather than routing through a separate magnitude/angle split). This directly satisfies the "perpendicular drift leaves the projection unchanged" and "parallel drift gives only the expected Doppler translation" acceptance criteria, since beta no longer depends on drift at all.