Asymptotic theory of diffuse high-beta magnetohydrostatic equilibria in three dimensions

The possibility of constructing asymptotic diffuse high-beta magnetohydrostatic equilibria as solutions of a boundary value problem in arbitrary toroidal domains is demonstrated within the old Scyllac scaling. Given two arbitrary profiles (e.g. the pressure ratio β and the rotation number μ as functions of the volume), and given, within the scaling, an arbitrary boundary at which the magnetic field is required to be tangential, there is a formal power series solution of the magnetohydrostatic equations. If β=O(epsilon), the leading order is obtained from the familiar equilibrium equation in axial symmetry, and the corrugation of the boundary yields higher-order corrections. If β=O(1), this equation is coupled to a linear elliptic equation with boundary data given by the corrugation, so that the latter, no matter how small, has a finite effect upon the equilibrium. If μ=O(1/epsilon), the leading-order problem is elliptic, thus being accessible to standard numerical methods. If μ=O(1) (high-beta stellarators), it is degenerate, and a high-current boundary layer appears unless the corrugation is judiciously chosen. (author)