Journal of Plasma Physics

Research Article

On the analytic solution of the helical equilibrium equation in the MHD approximation

M. L. Woolleya1

a1 Euratom-UKAEA Association for Fusion Research, Culham Laboratory, Abingdon, Oxfordshire, OX14 3DB, England

Abstract

The second-order elliptic partial differential equation, which describes a class of static ideally conducting magnetohydrodynamic equilibria with helical symmetry, is solved analytically. When the equilibrium is contained within an infinitely long conducting cylinder, the appropriate Dirichiet boundary-value problem may be solved in general in terms of hypergeometric functions. For a countably infinite set of particular cases, these functions are polynomials in the radial co-ordinate; and a solution may be obtained in a closed form. Necessary conditions are given for the existence of the equilibrium, which is described by the simplest of these functions. It is found that the Dirichlet boundary-value problem is not well-posed for these equiilbria; and additional information (equivalent to locating a stationary value of the hydrodynamic pressure) must be provided, in order that the solution be unique.

(Received February 24 1975)

(Revised April 28 1975)