Surface current equilibria from a geometric point of view

Abstract

In this paper the inverse problem of the existence of surface current magnetohydrodynamic (MHD) equilibria in toroidal geometry with vanishing magnetic field inside is addressed. Inverse means that the plasma–vacuum interface rather than the external wall or conductors is given, and the latter remain to be determined. This makes a reformulation of the problem possible in geometric terms: what toroidal surfaces with analytic parametrization allow a simple analytic covering by geodesics? If such a covering by geodesics (field lines) exists, their orthogonal trajectories (current lines) also form a simple covering, and are described by a function satisfying a nonlinear partial differential equation of the Hamilton–Jacobi type, whose coefficients are combinations of the metric elements of the surface. All known equilibria—equilibria with zero and infinite rotational transform and the symmetric ones in the case of finite rotational transform—turn out to be solutions of separable cases of that equation, and allow a unified description if the toroidal surface is parametrized in the moving trihedral associated with a closed curve. Analogously to volume current equilibria, the only continuous symmetries compatible with separability are plane, helical, and axial symmetry. In the nonseparable case numerical evidence is presented for cases with chaotic behavior of geodesics, thus restricting possible equilibria for these surfaces. For weak deviation from axisymmetry, Kolmogorov–Arnold–Moser (KAM)-type behavior is observed, i.e., destruction of geodesic coverings with a low rational rotational transform and preservation of those with irrational rotational transform. A previous attempt to establish three-dimensional surface current equilibria on the basis of the KAM theorem is rejected as incomplete, and a complete proof of the existence of equilibria in the weakly nonaxisymmetric case, based on the twist theorem for mappings, is given. Finally, for a certain class of strong deviations from axisymmetry, an analytic criterion is formulated that rules out equilibria for these surfaces.