Topological Interpretations of Open-Field Magnetic Helicity

Abstract

In ideal magnetohydrodynamics, magnetic helicity is a conserved dynamical quantity and a topological invariant closely related to Gauss linking numbers. However, for open magnetic fields with non-zero, normal boundary components, which are common in astro- physical settings, magnetic helicity is not uniquely defined and varies with the non-unique choices of the magnetic vector potentials or gauges.

An explicit interpretation of open-field magnetic helicity based on the entanglement of magnetic field lines has only been known for open Euclidean domains by Prior & Yeates (2014) Astrophys. J. 787 (2). In this thesis, this is proven to be generalisable to open spherical and periodic domains such that open-field magnetic helicity is equivalent to the total, flux-weighted winding of magnetic field lines, an intrinsic measure of magnetic topology. This is achieved by (i) formally constructing novel measures of spherical and periodic winding of open curves and (ii) identifying a particular gauge choice, called the winding gauge from the Hodge decomposition theorem on surfaces in the proof of the generalised poloidal-toroidal decomposition of magnetic fields. The theoretical findings are supplemented and confirmed by a numerical case study on solar observations.

The results obtained will contribute significantly to the field of topological fluid dy- namics, by providing a novel topological interpretation to open-field magnetic helicity using the winding of magnetic field lines and the domain-specific generalised Green’s functions for Laplacian. As open spherical and periodic domains are routinely used in the analytical modelling and numerical simulations for magnetically active regions, the geometry-adapted expressions can improve the modelling accuracy.