PE toroidal torque nonzero for ideal-MHD runs — track imag/real ratio as a Hermiticity check

[logan-nc]

Summary

perturbed_equilibrium/energies/toroidal_torque is nonzero (~0.7% of the plasma energy) for ideal-MHD runs, where it should be identically zero. It should be tracked as a Hermiticity diagnostic rather than as a physical torque.

Physics

In src/PerturbedEquilibrium/Response.jl:107:

state.toroidal_torque = -2 * nn * imag(py)      # py = ⟨Φ_resp, Λ⁻¹·Φ_resp⟩ / 4

The torque is the imaginary part of the plasma-energy quadratic form, which is nonzero only if the plasma inductance Λ has a nonzero anti-Hermitian part (see the comment at Response.jl:54). In pure ideal MHD the energy principle is self-adjoint → δW real → Λ Hermitian → imag(py) = 0 → torque ≡ 0. A real toroidal torque arises only from the drift-kinetic dissipation added to the Euler–Lagrange operator when kinetic_factor > 0.

Observation

The diiid_n1 case is a pure ideal run (kinetic_source = "fixed", kinetic_factor = 0.0), yet reports toroidal_torque ≈ ±0.05. With plasma_energy = real(py) ≈ 3.4 J, that is |imag(py)/real(py)| ≈ 0.7% — residual numerical anti-Hermiticity in the computed ideal Λ (vacuum grri/grre coupling, the BVP solve, spline error, possibly reg_spot), not physics.

Evidence it is numerical noise: during #313 a grid change (removing the Δ′ knot ladder) flipped its sign at unchanged magnitude (−0.051 → +0.051) — a near-zero quantity whose sign is numerically ill-defined.

Suggested fix (harness)

regression-harness/cases/diiid_n1.toml ([quantities.pe_toroidal_torque], ~L342) currently has noise_threshold = 1e-12, so it tracks ~0.7% numerical noise and reports grid perturbations as large percentage swings. For ideal runs, track the ratio |imag(py)/real(py)| as a Hermiticity check with a physically meaningful floor (e.g. 1e-2), rather than the raw torque. This turns a noisy tracked value into a genuine correctness invariant (ideal Λ should be Hermitian to within tolerance).

Optional (code)

Consider zeroing toroidal_torque (or Hermitian-projecting Λ) when kinetic_factor == 0, and/or documenting the field as a Hermiticity diagnostic in the ideal regime.

References

• src/PerturbedEquilibrium/Response.jl:54,99-107
• regression-harness/cases/diiid_n1.toml:342-348

Surfaced during review of #313.