On three-dimensional toroidal surface current equilibria with rational rotational transform

Abstract

The inverse problem of the existence of magnetohydrostatic equilibria in toroidal geometry with surface currents and vanishing magnetic field inside is considered. The inverse formulation (plasma–vacuum interface given, external wall or conductors to be determined) allows a purely geometric characterization of the problem: Toroidal surfaces with analytic parametrization are admissible plasma–vacuum interfaces if they allow a simple analytic covering by geodesics (field lines) or, equivalently, by equidistant lines (current lines). On the basis of this approach it was shown in a recent publication [Phys. Plasmas 1, 281 (1994)] that equilibria with sufficiently irrational rotational transform exist for configurations with arbitrary but weak deviations from axisymmetry. The present paper completes this work by demonstrating that this result, essentially, cannot be sharpened. More precisely, it is shown that for every rational rotational transform ι there exists a (arbitrarily weak) deformation of the axisymmetric circular torus (with sufficiently large aspect ratio) such that an equilibrium with that toroidal surface as plasma–vacuum interface and that ι does not exist. This does not mean that there are no three-dimensional equilibria with rational ι at all. In fact, in the special case of infinite rotational transform, i.e., field lines are simply closed in the poloidal direction, it is demonstrated that, depending on the type of perturbation, equilibria may survive arbitrarily strong deviations from axisymmetry or may immediately be destroyed. Toroidal surfaces of the former type are the so-called canal surfaces, which are generated by the nonrotating transport of a poloidal section along an arbitrary closed space curve, whereas examples of the latter type are ‘‘bumpy’’ tori, where the poloidal section may vary in magnitude but not in shape along the generating curve. If the poloidal section rotates along the generating curve (‘‘helical’’ torus) so that the rotation velocity vanishes somewhere, the surface is also of the latter type.