Plasma Relaxation and Topological Aspects in Hall Magnetohydrodynamics

TL;DR

This paper extends Parker’s plasma relaxation theory to Hall MHD, revealing a link between the Beltrami condition and potential vorticity, and showing magnetic field line evolution parallels 2D hydrodynamics.

Contribution

It introduces a novel connection between Hall MHD Beltrami conditions and potential vorticity conservation, expanding understanding of plasma relaxation processes.

Findings

The torsion coefficient alpha is proportional to potential vorticity.

Hall MHD Beltrami condition aligns with potential vorticity conservation.

Magnetic field line evolution mirrors potential vorticity lines in 2D hydrodynamics.

Abstract

Parker's formulation of isotopological plasma relaxation process in magnetohydrodynamics (MHD) is extended to Hall MHD. The torsion coefficient alpha in the Hall MHD Beltrami condition turns out now to be proportional to the "potential vorticity." The Hall MHD Beltrami condition becomes equivalent to the "potential vorticity" conservation equation in two-dimensional (2D) hydrodynamics if the Hall MHD Lagrange multiplier beta is taken to be proportional to the "potential vorticity" as well. The winding pattern of the magnetic field lines in Hall MHD then appears to evolve in the same way as "potential vorticity" lines in 2D hydrodynamics.