On self and mutual winding helicity

TL;DR

This paper introduces winding helicity as a topological measure of magnetic fields, demonstrating its decomposition into self and mutual components, and applies it to complex magnetic field simulations.

Contribution

It defines winding helicity, shows how to decompose it into self and mutual parts, and demonstrates practical calculation for complex magnetic topologies.

Findings

Winding helicity can be decomposed into self and mutual components.

The decomposition is applicable to complex magnetic geometries.

Application demonstrated on evolving magnetic field simulation.

Abstract

The topological underpinning of magnetic fields connected to a planar boundary is naturally described by field line winding. This observation leads to the definition of winding helicity, which is closely related to the more commonly calculated relative helicity. Winding helicity, however, has several advantages, and we explore some of these in this work. In particular, we show, by splitting the domain into distinct subregions, that winding helicity can be decomposed naturally into "self" and "mutual" components and that these quantities can be calculated, in practice, for magnetic fields with complex geometries and topologies. Further, winding provides a unified topological description from which known expressions for self and mutual helicity can be readily derived and generalized. We illustrate the application of calculating self and mutual winding helicities in a simulation of an evolving magnetic field with non-trivial field line topology.