Steepest‐descent moment method for three‐dimensional magnetohydrodynamic equilibria

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.