The Parker Magnetostatic Theorem

TL;DR

This paper demonstrates the Parker Magnetostatic Theorem by analyzing the topology of force-free magnetic fields in a bounded conducting fluid, showing that certain topologies necessarily involve discontinuities in equilibrium.

Contribution

It provides a rigorous proof of the Parker Magnetostatic Theorem using a neighborhood in solution space and the topology of force-free fields in a bounded domain.

Findings

Force-free fields with certain topologies must contain tangential discontinuities in equilibrium.

Continuous magnetic footpoint displacements produce a restricted subset of field topologies.

The theorem links magnetic topology constraints to the necessity of discontinuities in force-free fields.

Abstract

We demonstrate the Parker Magnetostatic Theorem in terms of a small neighborhood in solution space containing continuous force-free magnetic fields in small deviations from the uniform field. These fields are embedded in a perfectly conducting fluid bounded by a pair of rigid plates where each field is anchored, taking the plates perpendicular to the uniform field. Those force-free fields obtainable from the uniform field by continuous magnetic footpoint displacements at the plates have field topologies that are shown to be a restricted subset of the field topologies similarly created without imposing the force-free equilibirum condition. The theorem then follows from the deduction that a continuous nonequilibrum field with a topology not in that subset must find a force-free state containing tangential discontinuities.