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arxiv: 2606.02901 · v1 · pith:XV7AHV56new · submitted 2026-06-01 · ⚛️ physics.plasm-ph

Coordinate-invariant flux-surface Fourier analysis in tokamaks

Matthew Pharr , Evan Bursch , Nikolas Logan , Priyansh Lunia , Jong-Kyu Park , Carlos Paz-Soldan This is my paper

Pith reviewed 2026-06-28 11:45 UTC · model grok-4.3

classification ⚛️ physics.plasm-ph
keywords tokamaksFourier analysisflux surfacesresonant magnetic perturbationscoordinate invarianceplasma responsevacuum field perturbations
0
0 comments X

The pith

Pairing square-root-area weighted vacuum perturbations with full-area-weighted resonant fields yields a coordinate-invariant coupling matrix for tokamak RMP analysis.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that the external Fourier spectrum of vacuum field perturbations, which controls coupling to resonant magnetic perturbation coils, depends on the choice of magnetic coordinates unless a specific weighting is applied. By combining square-root-area weighting on the vacuum side with full-area weighting on the resonant side, the resulting coupling matrix has singular values that stay the same under coordinate changes and right singular vectors that map back to the same real-space pattern. This completes an invariance result that previously covered only the resonant interior. The approach matters for reliable prediction of error-field penetration and RMP-driven plasma responses in shaped, low-aspect-ratio plasmas where coordinate choice otherwise shifts dominant-mode overlap by factors of two to three. The result is demonstrated analytically and confirmed with GPEC calculations, and it applies to any code that computes flux-surface Fourier spectra of resonant or external quantities.

Core claim

Pairing a square-root-area weighted vacuum field perturbation with a full-area-weighted resonant field produces a coupling matrix C whose singular values are invariant under coordinate transformations and whose right singular vectors reconstruct to a consistent real-space field pattern across coordinate systems, completing the coordinate-invariance picture for the plasma-3D-field coupling paradigm.

What carries the argument

The coupling matrix C formed by the square-root-area weighted vacuum perturbation paired with the full-area-weighted resonant field.

If this is right

  • Singular values of C can be compared or ranked reliably without regard to the coordinate system selected for the calculation.
  • Right singular vectors of C map to the same physical field pattern no matter which flux-surface coordinates are chosen.
  • Improper weighting produces dominant modes whose vacuum-field overlap differs by factors of 2-3 between coordinates in strongly shaped, low-aspect-ratio equilibria.
  • The same coordinate dependence affects alternative formulations such as the three-mode metric or direct zeroing of a resonant component without proper weighting.
  • The weighted construction applies to any computational tool that extracts Fourier spectra of resonant or external quantities on flux surfaces.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The invariant singular values could serve as a standardized metric for comparing RMP coil effectiveness across different equilibrium codes.
  • The same weighting recipe may remove coordinate artifacts when analyzing error-field penetration or neoclassical toroidal viscosity in non-axisymmetric fields.
  • One could test the method by repeating a single RMP penetration calculation in both straight-field-line and Boozer coordinates and verifying that the dominant coupled mode is identical only after weighting.

Load-bearing premise

The vacuum field perturbation can be treated as an external quantity whose Fourier spectrum is computed independently of the plasma response on the flux surface.

What would settle it

Recompute the singular values of the coupling matrix in two different magnetic coordinate systems both with and without the square-root weighting on the vacuum side; invariance should appear only when the weighting is used.

Figures

Figures reproduced from arXiv: 2606.02901 by Carlos Paz-Soldan, Evan Bursch, Jong-Kyu Park, Matthew Pharr, Nikolas Logan, Priyansh Lunia.

Figure 1
Figure 1. Figure 1: Constant θ lines shown at zero machine angle in PEST, Boozer, and Hamada coordinates for a DIII-D-like equilibrium used in previous works20,39. Note that PEST and Boozer coordinates are similar, while Hamada coordinates show significant deviation, espe￾cially a tendency to group near the x-point. work. II. COORDINATE INVARIANCE OF RESONANT COUPLING In the presence of closed flux surfaces, an ideal MHD equi… view at source ↗
Figure 2
Figure 2. Figure 2: Coordinate-invariance of the full-area-weighted resonant metric, [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: First right-singular vector, i.e. the dominant mode, of the [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Singular values of the coupling matrix C defined for the three different weightings of the vacuum field perturbation and an area￾weighted resonant field calculated for n = 1 in a DIII-D-like equilibrium shown in [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Aspect ratio and shaping scan plotted versus [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Last closed flux surface of equilibria used in the aspect ratio and shaping scan, shown in Fig. [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
read the original abstract

The Fourier spectra of resonant quantities in tokamaks depend on the choice of magnetic coordinates, and an area weighting of the Fourier integrand preserves the resonant coefficients on rational surfaces. That result constrains only the resonant interior; the coordinate dependence of the external Fourier spectrum, which determines the coupling to Resonant Magnetic Perturbation (RMP) coils and error-field penetration, was left untreated. This paper shows that pairing a square-root-area weighted vacuum field perturbation with a full-area-weighted resonant field yields a coupling matrix C whose singular values are invariant under coordinate transformations and whose right singular vectors reconstruct to a consistent real-space field pattern across coordinate systems, completing the coordinate-invariance picture for the plasma-3D-field coupling paradigm. GPEC calculations confirm the analytic result and show that improperly weighted coupling matrices can produce dominant modes whose overlap with the vacuum field perturbation differs by a factor of $2--3$ between coordinate systems for strongly shaped, low aspect ratio equilibria, with the discrepancy growing with inverse aspect ratio. The same coordinate dependence afflicts alternative formulations such as the three-mode metric or zeroing the $q=2$ resonant field without proper weighting. The result applies to any tool computing Fourier spectra of resonant or external quantities on flux surfaces.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper claims that pairing a square-root-area weighted vacuum field perturbation with a full-area-weighted resonant field on tokamak flux surfaces produces a coupling matrix C whose singular values are invariant under coordinate reparametrization and whose right singular vectors reconstruct to a consistent real-space field pattern. An analytic derivation of this invariance property is presented, together with numerical confirmation via GPEC calculations on shaped, low-aspect-ratio equilibria that demonstrate factor-of-2--3 discrepancies in dominant-mode overlap when improper weightings are used.

Significance. If the result holds, it supplies the missing coordinate-invariant treatment of the external Fourier spectrum in the plasma-3D-field coupling paradigm, with immediate applicability to RMP and error-field calculations. The manuscript's strengths include an explicit analytic derivation of the invariance from the chosen weightings and direct numerical verification across coordinate systems in GPEC for realistic equilibria.

minor comments (3)
  1. Abstract: the factor-of-2--3 discrepancy is stated without a pointer to the specific GPEC run or figure that quantifies it for the shaped equilibria.
  2. The equilibria employed for the GPEC confirmation (aspect ratio, shaping parameters) are not tabulated or referenced by name, which would aid reproducibility.
  3. Notation for the weighting operators (square-root-area versus full-area) should be introduced with an explicit equation early in the text rather than only in the abstract.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment, significance statement, and recommendation of minor revision. No major comments appear in the provided report, so there are no specific points requiring point-by-point response or manuscript changes at this stage.

Circularity Check

0 steps flagged

No significant circularity; derivation is a direct mathematical property of the weighting definition

full rationale

The central result is an analytic demonstration that a coupling matrix C constructed from square-root-area weighting on the vacuum perturbation and full-area weighting on the resonant field has singular values invariant under coordinate reparametrization, with right singular vectors reconstructing consistently in real space. This follows directly from the definitions and the properties of the Fourier integrals and singular value decomposition; the paper supplies both the proof and independent numerical verification in GPEC for shaped equilibria. No steps reduce by construction to fitted parameters, self-citations, or prior ansatzes from the same authors. The separation of vacuum and resonant fields is a standard modeling choice external to the invariance claim itself. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The result rests on standard Fourier analysis on flux surfaces and the separation of vacuum versus resonant contributions; no free parameters, ad-hoc axioms, or new entities are introduced in the abstract.

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