The topological properties of magnetic helicity

TLDR

Magnetic helicity quantifies the twist, writhe, and linking of magnetic flux tubes, but no absolute measure exists for subvolumes unless bounded by a magnetic surface. The authors introduce a gauge‑invariant relative helicity that provides a meaningful topological measure for such unbounded volumes. They derive the time derivative of this relative helicity, yielding an expression for the flux of topological structure across boundaries.

Abstract

The relation of magnetic helicity to the topological structure of field lines is discussed. If space is divided into a collection of flux tubes, magnetic helicity arises from internal structure within a flux tube, such as twist and kinking, and external relations between flux tubes, i.e. linking and knotting. The concepts of twist number and writhing number are introduced from the mathematical-biology literature to describe the contributions to helicity from twist about the axis of a flux tube, and from the structure of the axes themselves.There exists no absolute measure of the helicity within a subvolume of space if that subvolume is not bounded by a magnetic surface. However, a topologically meaningful and gauge-invariant relative measure of helicity for such volumes is presented here. The time derivative of this relative measure is calculated, which leads to an expression for the flow of topological structure across boundaries.