Steepest-descent moment method for three-dimensional magnetohydrodynamic equilibria

TLDR

The study employs poloidal and toroidal flux coordinates θ and ζ, with pressure p depending on the radial coordinate ρ, to describe nested magnetic flux surfaces. The authors use an energy principle and inverse coordinate representation to derive coupled ordinary differential equations for Fourier moments of the flux surface coordinates, and develop a steepest‑descent iteration—augmented by a renormalization parameter λ and a secondary descent step—to rapidly converge the nonlinear moment equations to an MHD equilibrium. A positive‑definite energy functional guarantees monotonic convergence of the steepest‑descent iteration to an equilibrium solution, provided no magnetic islands form.

Abstract

An energy principle is used to obtain the solution of the magnetohydrodynamic (MHD) equilibrium equation J×B−∇p=0 for nested magnetic flux surfaces that are expressed in the inverse coordinate representation x=x(ρ, θ, ζ). Here, θ are ζ are poloidal and toroidal flux coordinate angles, respectively, and p=p(ρ) labels a magnetic surface. Ordinary differential equations in ρ are obtained for the Fourier amplitudes (moments) in the doubly periodic spectral decomposition of x. A steepest-descent iteration is developed for efficiently solving these nonlinear, coupled moment equations. The existence of a positive-definite energy functional guarantees the monotonic convergence of this iteration toward an equilibrium solution (in the absence of magnetic island formation). A renormalization parameter λ is introduced to ensure the rapid convergence of the Fourier series for x, while simultaneously satisfying the MHD requirement that magnetic field lines are straight in flux coordinates. A descent iteration is also developed for determining the self-consistent value for λ.