Relative magnetic helicity in multiply connected domains

TL;DR

This paper reviews various definitions of relative magnetic helicity in multiply connected domains, demonstrating their common origin, gauge invariance, and conservation in ideal MHD under specific boundary conditions.

Contribution

It introduces a unified, general definition of relative magnetic helicity in multiply connected domains that encompasses existing formulas and proves their gauge invariance and conservation properties.

Findings

Unified definition of relative magnetic helicity

Proved gauge invariance of the general definition

Showed conservation under ideal MHD with specific boundary conditions

Abstract

Magnetic helicity is a conserved quantity of ideal magnetohydrodynamics (MHD) that is related to the topology of the magnetic field, and is widely studied in both laboratory and astrophysical plasmas. When the magnetic field has a non-trivial normal component on the boundary of the domain, the classical definition of helicity must be replaced by relative magnetic helicity. The purpose of this work is to review the various definitions of relative helicity and to show that they have a common origin - a general definition of relative helicity in multiply connected domains. We show that this general definition is both gauge-invariant and is conserved in time under ideal MHD, subject only to closed and line-tied boundary conditions. Other, more specific, formulae for relative helicity, that are used frequently in the literature, are shown to follow from the general expression by imposing extra conditions on the magnetic field or its vector potential.