Topological nature of the Parker magnetostatic theorem

Abstract

The two-plate initial boundary-value problem of Parker is reviewed, treating the relaxation of a 3D magnetic field prescribed with an arbitrary topology to a terminal force-free field in a cold, viscous, electrically perfect fluid conductor. Anchored by their foot-points at the perfectly conducting rigid plates, the relaxing field preserves its topology. The Parker magnetostatic theorem states that for most prescribed field topologies, the terminal field must embed current sheets. The elements of this theorem are reviewed and analyzed to relate this initial boundary-value problem to (i) the variational problem for a force-free field of a given topology and (ii) the direct construction of a force-free field in terms of its pair of Euler flux functions. New insights and understanding are presented on the theorem as the compelling basis of the Parker theory of solar coronal heating.