Coordinate-invariant flux-surface Fourier analysis in tokamaks

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Abstract
The Fourier decomposition of a magnetic field perturbation on a tokamak flux surface is not unique; different choices of magnetic coordinates and of the weighting applied to the Fourier integrand yield different spectra, and hence different apparent coupling between an applied 3D field and the plasma. An area weighting of the Fourier integrand preserves the resonant coefficients on rational surfaces [Park et al., Phys. Plasmas 15, 064501 (2008)]. That result constrains only the resonant field components at rational surfaces in the plasma interior; the coordinate dependence of the external Fourier spectrum, or of quantities on arbitrary flux surfaces, which determines the coupling to resonant magnetic perturbation 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. Calculations with the Generalized Perturbed Equilibrium Code 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.
Funding Information
  • U.S. Department of Energy (DE-SC0022270)
  • U.S. Department of Energy (DE-SC0022272)
  • U.S. Department of Energy (DE-SC0024898)
  • National Research Foundation of Korea (RS2023-00281276)
  • Korea Institute of Fusion Energy (CN2502-1)

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