Magnetic winding: what is it and what is it good for?

Abstract

Magnetic winding is a fundamental topological quantity that underpins magnetic helicity and measures the entanglement of magnetic field lines. Like magnetic helicity, magnetic winding is also an invariant of ideal magnetohydrodynamics. In this article, we give a detailed description of what magnetic winding describes, how to calculate it and how to interpret it in relation to helicity. We show how magnetic winding provides a clear topological description of magnetic fields (open or closed) and we give examples to show how magnetic winding and helicity can behave differently, thus revealing different and important information about the underlying magnetic field.

Magnetic winding -- a key to unlocking topological complexity in flux emergence

“…For further reading, a detailed description of theoretical aspects of magnetic winding, and its relation to helicity, can be found in [8]. For the behaviour of helicity and winding in flux emergence, we point the reader to [9,10].…”

“…Doing so, we produce a quantity that depends only on the geometry of magnetic field lines. We name this quantity the magnetic winding L [4,8]. The flux of L through P is given by…”

“…Consider the magnetic flux tube shown in Figure 5. The details of how to construct this tube are given in [8].…”

On self and mutual winding helicity

“…Rather than seeking a general gauge invariant measure of helicity, we choose one with a particular gauge that provides a clear topological interpretation (just as the Coulomb gauge, with the Biot-Savart operator, does for classical helicity). We call this measure of helicity winding helicity [10,11,12,13] since it represents the average pairwise winding between all local portions of field lines, weighted by magnetic flux. Definition 1.…”

“…and satisfy H W ij = H W ji . In equation (13), the position vectors x and y refer to different points in the same subdomain S z ∩ Ω i . In equation (14), the position vectors x refer to points in S z ∩ Ω i and the position vectors y refer to those in S z ∩ Ω j .…”