Unique Topological Characterization of Braided Magnetic Fields

TL;DR

This paper introduces a topological flux function that uniquely characterizes the magnetic topology of braided magnetic fields, linking magnetic flux, helicity, and field line mappings through a Hamiltonian framework.

Contribution

The paper presents a novel scalar topological flux function that acts as an ideal invariant and uniquely determines the magnetic field line topology in braided magnetic fields.

Findings

The flux function measures average poloidal flux and pairwise crossing numbers.

Its integral yields the relative magnetic helicity.

It uniquely characterizes the magnetic field line mapping.

Abstract

We introduce a topological flux function to quantify the topology of magnetic braids: non-zero, line-tied magnetic fields whose field lines all connect between two boundaries. This scalar function is an ideal invariant defined on a cross-section of the magnetic field, and measures the average poloidal magnetic flux around any given field line, or the average pairwise crossing number between a given field line and all others. Moreover, its integral over the cross-section yields the relative magnetic helicity. Using the fact that the flux function is also an action in the Hamiltonian formulation of the field line equations, we prove that it uniquely characterizes the field line mapping and hence the magnetic topology.