Coordinate-invariant flux-surface Fourier analysis in tokamaks

This paper establishes that pairing a square-root-area weighted vacuum field perturbation with a full-area-weighted resonant field yields a coupling matrix with coordinate-invariant singular values and consistent real-space patterns, thereby resolving the coordinate dependence issue in tokamak Fourier analysis that previously affected Resonant Magnetic Perturbation coupling and error-field penetration predictions.

The Gist

The Big Picture: Measuring a Magnetic "Handshake"

Imagine a tokamak (a type of nuclear fusion reactor) as a giant, doughnut-shaped room filled with super-hot plasma. To keep this plasma stable and prevent it from crashing into the walls, scientists use external magnets to create a "handshake" between the outside world and the inside plasma.

This handshake happens at specific invisible lines inside the doughnut called rational surfaces. If the external magnets push just right on these lines, they can either stabilize the plasma or, if they push the wrong way, cause it to become unstable.

Scientists use a mathematical tool called a Coupling Matrix to calculate exactly how strong this handshake is. They break the magnetic fields down into waves (Fourier spectra) to see which parts of the external push match up with the internal plasma.

The Problem: The "Map" Changes the Message

The paper identifies a tricky problem: The way we draw the map matters.

To describe the shape of the plasma, scientists use different coordinate systems (like different types of maps: a flat map, a globe, or a Mercator projection). The paper shows that if you use the wrong "map" (coordinate system) to calculate the handshake strength, you get different answers.

• The Analogy: Imagine you are trying to measure how much rain falls in a city.
  • If you use a map that stretches the city out (making it look huge), your rain gauge might say "a lot of rain."
  • If you use a map that squishes the city (making it look tiny), your gauge might say "very little rain."
  • The actual amount of rain hasn't changed, but your measurement depends entirely on how you drew the map.

In the past, scientists sometimes used "maps" that distorted the results. This meant that when they designed magnets to fix the plasma, the design might work on one map but fail on another.

The Solution: The "Square-Root" Rule

The authors discovered a specific mathematical "recipe" to fix this. They found that to get a result that is the same no matter which map you use, you have to weigh your calculations in a very specific way:

1. Inside the Plasma (The Resonant Field): You must weigh the calculation by the full area of the surface. Think of this as counting every drop of rain in every square meter of the city, regardless of how the map stretches it.
2. Outside the Plasma (The Vacuum Field): You must weigh the calculation by the square root of the area.

Why the square root?
Think of it like a dance. If you want two dancers to move in perfect sync (coordinate invariance), and one dancer is moving in a "full area" rhythm, the other dancer must move in a "square-root area" rhythm for them to match up perfectly. If you try to match "full area" with "full area," or "no weight" with "no weight," the dancers stumble, and the results change depending on which map you are looking at.

What They Proved

The team used a powerful computer code called GPEC to test this. They ran simulations using three very different "maps" (PEST, Boozer, and Hamada coordinates):

• The Wrong Way: When they used the standard or "bare" weights (no special math), the results changed wildly. For reactors with weird, squashed shapes (low aspect ratio), the results could differ by a factor of 2 to 3. That means a calculation saying "this magnet will work" could actually be wrong by 200% if the wrong math was used.
• The Right Way: When they applied their new "Square-Root + Full Area" recipe, the results were identical across all three maps. The "handshake" strength was the same, no matter how they drew the map.

Why This Matters

This paper doesn't invent new magnets or new reactors. Instead, it provides the rulebook for the math used to design them.

• For Scientists: It tells them exactly how to write their equations so that their results are real physical truths, not just artifacts of the math they chose.
• For Future Designs: It ensures that when we design magnets for future fusion reactors (like ITER or DEMO), the designs will be robust. We won't accidentally design a magnet that works on a "flat map" but fails on a "curved map."

Summary

The paper says: "If you want to measure the magnetic handshake in a fusion reactor correctly, you must use a specific weighting recipe (Square-Root for outside, Full-Area for inside). If you don't, your measurements will change depending on the coordinate system you use, leading to potentially dangerous errors in magnet design."

Technical Summary

Technical Summary: Coordinate-Invariant Flux-Surface Fourier Analysis in Tokamaks

Problem Statement
The coupling between external magnetic field perturbations and plasma resonant quantities in tokamaks is typically quantified using a coupling matrix, $C$. This matrix relates the Fourier spectrum of a vacuum field perturbation at a control surface (usually the plasma boundary) to resonant metrics (such as resonant flux or normal field) at rational surfaces within the plasma. A critical issue identified in this work is that the Fourier spectra of these resonant quantities depend on the choice of magnetic coordinates (e.g., PEST, Boozer, Hamada). While previous work (Park 2008) established that applying a full-area weighting to the Fourier integrand preserves the resonant coefficients on rational surfaces, this addressed only the "co-domain" (the interior resonant field). The "domain" (the external vacuum field perturbation) remained untreated, leading to coordinate-dependent results for the full coupling matrix. Consequently, the singular values and singular vectors of $C$—which determine the coupling to Resonant Magnetic Perturbation (RMP) coils and error-field penetration—could vary significantly depending on the coordinate system chosen, particularly for strongly shaped, low-aspect-ratio equilibria.

Methodology
The authors employ an analytic proof combined with numerical validation using the Generalized Perturbed Equilibrium Code (GPEC).

1. Analytic Derivation: The paper analyzes the transformation properties of Fourier decompositions under gauge transformations of magnetic coordinates. It defines three potential weightings for the vacuum field perturbation integrand:
   • Bare weighting ($b_x$): No Jacobian factor.
   • Square-root-area weighting ($\tilde{b}_x$): Weighted by $\sqrt{J|\nabla\psi|}$.
   • Full-area weighting ($\bar{b}_x$): Weighted by $J|\nabla\psi|$.
     Using Parseval's theorem, the authors demonstrate that only the square-root-area weighting yields a Fourier coefficient vector that transforms unitarily under coordinate changes, preserving the $L_2$ norm as a physical, coordinate-invariant quantity (the area-averaged squared normal field).
2. Coupling Matrix Construction: The coupling matrix $C$ is constructed by pairing the square-root-area weighted vacuum field ($\tilde{b}_x$) with the full-area weighted resonant field ($\bar{b}_r$), which was previously shown to be invariant.
3. Numerical Validation: Simulations are performed on DIII-D-like and NSTX-U-like equilibria using PEST, Boozer, and Hamada coordinates. The Singular Value Decomposition (SVD) of the coupling matrix $C$ is computed for various weighting combinations to test for coordinate invariance.

Key Contributions and Results

• Coordinate-Invariant SVD: The paper proves that the Singular Value Decomposition (SVD) of the coupling matrix $C$ is invariant under coordinate transformations if and only if the vacuum field perturbation is weighted by the square root of the area element ($\sqrt{J|\nabla\psi|}$) and the resonant field is weighted by the full area element ($J|\nabla\psi|$).
• Physical Interpretation of Dominant Modes: The right singular vectors of this correctly weighted matrix reconstruct to a consistent real-space field pattern across different coordinate systems. This confirms that the "dominant modes" used in plasma control are physical quantities, provided the correct weighting is applied.
• Quantification of Errors: Numerical results show that for high-aspect-ratio, near-circular equilibria, improper weighting leads to minor deviations (1–10%). However, for strongly shaped, low-aspect-ratio equilibria (e.g., NSTX-U-like), improper weighting causes the dominant mode overlap and singular values to differ by a factor of 2–3 between coordinate systems.
• Clarification of Metrics: The paper clarifies the definition of the dominant mode overlap, $\delta$, providing a coordinate-invariant formulation: $\delta = |v_1^\dagger \tilde{b}_x| / B_T$, where $v_1$ is the first right singular vector. It highlights that existing tools like GPEC inherently use the correct weighting, but alternative formulations (such as the Three-Mode Metric/TMEI) and ad-hoc definitions often lack this rigor, leading to coordinate-dependent results.

Significance
The paper completes the coordinate-invariance picture for the plasma-3D-field coupling paradigm. By constraining both the domain (vacuum field) and co-domain (resonant field) of the coupling matrix, the authors provide a rigorous prescription for constructing coupling matrices whose singular values and dominant modes are robust to the choice of magnetic coordinates. This is essential for reliable RMP coil design, error-field correction, and the assessment of error fields in modern, highly shaped tokamaks. The work serves as a corrective guide for future implementations of coupling-matrix formalisms in other codes (e.g., MARS-F) and warns against the use of unweighted or improperly weighted metrics that can yield ambiguous or physically incorrect results, particularly in low-aspect-ratio regimes.