Variational moment solutions to the Grad–Shafranov equation

A variational method is developed to find approximate solutions to the Grad–Shafranov equation. The surfaces of the constant poloidal magnetic flux ψ(R, Z) are obtained by solving a few ordinary differential equations, which are moments of the Grad–Shafranov equation, for the Fourier amplitudes of the inverse mapping R(ψ, ϑ) and Z(ψ, ϑ). Analytic properties and solutions of the moment equations are considered. Specific calculations using the Impurity Study Experiment (ISX-B) and the Engineering Test Facility (ETF)/International Tokamak Reactor (INTOR) geometries are performed numerically, and the results agree well with those calculated using standard two-dimensional equilibrium codes. The main advantage of the variational moment method is that it significantly reduces the computational time required to determine two-dimensional equilibria without sacrificing accuracy.