Relaxed plasma-vacuum systems

Abstract

Taylor’s theory of relaxed toroidal plasmas (states of lowest energy with fixed total magnetic helicity) is extended to include a vacuum between the plasma and the wall. In the extended variational problem, one prescribes, in addition to the helicity and the magnetic fluxes whose conservation follows from the perfect conductivity of the wall, the fluxes whose conservation follows from the assumption that the plasma-vacuum interface is also perfectly conducting (if the wall is a magnetic surface, then one has the toroidal and the poloidal flux in the vacuum). Vanishing of the first energy variation implies a pressureless free-boundary magnetohydrostatic equilibrium with a Beltrami magnetic field in the plasma, and in general with a surface current in the interface. Positivity of the second variation implies that the equilibrium is stable according to ideal magnetohydrodynamics, that it is a relaxed state according to Taylor’s theory if the interface is replaced by a wall, and that the surface current is nonzero (at least if there are no closed magnetic field lines in the interface). The plane slab, with suitable boundary conditions to simulate a genuine torus, is investigated in detail. The relaxed state has the same double symmetry as the vessel if, and only if, the prescribed helicity is in an interval that depends on the prescribed fluxes. This interval is determined in the limit of a thin slab.