On the existence of toroidal magnetohydrodynamic equilibria with poloidally closed magnetic field lines

Abstract

The present work explores the existence of nonaxisymmetric toroidal magnetohydrodynamic equilibria when all lines of force of the magnetic field close after a single revolution about a given magnetic axis. It is assumed that the magnetic axis is a closed curve with arbitrary curvature and zero torsion, implying that it is constrained to lie in a plane surface. In addition, it is assumed that the closed magnetic field lines lie in planes that are orthogonal to the magnetic axis. Subject to these conditions, the existence of toroidal magnetic surfaces, F(r)=const, is explored. The governing equilibrium equations of magnetohydrodynamics place a constraint on the function F(r) in the form of two nonlinear partial differential equations that must be simultaneously satisfied. It is demonstrated that this is not always possible without axisymmetry. Two specific cases where toroidal magnetic surfaces do not exist are identified: (I) isodynamic configurations, which implies that the magnetic field strength is constant on each magnetic surface, and (II) configurations with ‘‘bumpy’’ surfaces, which implies that the size but not the geometrical shape of the poloidal section containing the closed field lines depends on s.