Free-boundary high-beta tokamaks. II. Mathematical intermezzo: Hilbert transforms and conformal mapping

Mathematical techniques are described that facilitate the reduction of the stability problem of a toroidal free-boundary high-β tokamak equilibrium with skin currents to one that is basically one-dimensional. This includes the conformal mapping of the simply connected plasma region onto a circular disk and the conformal mapping of the doubly connected vacuum region onto an annulus by means of the Theodorsen and Garrick nonlinear integral equations. Henrici's method of constructing the discretized Hilbert transforms for periodic functions on the boundaries of these domains provides both the basis for constructing the mappings and the tool for the study of the perturbations. The methods are applied to problems of two-dimensional potential flow with a discontinuity of which the stability of sharp-boundary high-β tokamaks is just a special case