Variational method for three-dimensional toroidal equilibria

A variational method is developed for three-dimensional toroidal equilibria, characterized by the mapping between two coordinate systems: the cylindrical system (R, φ, Z) and the magnetic system (v, θ, ζ), where v is a radial flux surface label, θ a poloidal angle and ζ a toroidal angle. Two types of mapping, namely, the inverse mapping (v, θ, ζ) → (R, φ, Z) and the mixed mapping (v, θ, φ) → (R, ζ, Z) are considered. The dependent variables are Fourier-analysed in θ and ζ (or φ), and a set of ordinary differential equations are derived for the amplitudes in v from the variational principle. Truncation of the infinite Fourier series leads to a reduced set of equations which we solve numerically by collocation to obtain two- and three-dimensional toroidal equilibria.