A complete topological invariant for braided magnetic fields

TL;DR

This paper introduces a topological flux function that uniquely characterizes the magnetic topology of braided magnetic fields, providing a new invariant that relates to magnetic helicity and is derived from Hamiltonian field line equations.

Contribution

It presents a novel scalar topological flux function as a complete invariant for magnetic braids, linking magnetic topology with Hamiltonian dynamics.

Findings

The topological flux function is an ideal invariant for magnetic braids.

It uniquely characterizes the magnetic field line mapping.

A simple example demonstrates its application.

Abstract

A topological flux function is introduced 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, whose integral over the cross-section yields the relative magnetic helicity. Recognising that the topological flux function is an action in the Hamiltonian formulation of the field line equations, a simple formula for its differential is obtained. We use this to prove that the topological flux function uniquely characterises the field line mapping and hence the magnetic topology. A simple example is presented.