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 Vector x B Vector - del p = 0 for nested magnetic flux surfaces that are expressed in the inverse coordinate representation x Vector = x Vector(rho, theta, zeta). Here, theta and zeta are poloidal and toroidal flux coordinate angles, respectively, and p = p(rho) labels a magnetic surface. Ordinary differential equations in rho are obtained for the Fourier amplitudes (moments) in the doubly periodic spectral decomposition of x Vector. 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 lambda is introduced to ensure the rapid convergence of the Fourier series for x Vector, 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 lambda