Spherical winding and helicity

Xiao, D.; Prior, C.B.; Yeates, A.R.

Abstract

In ideal magnetohydrodynamics, magnetic helicity is a conserved dynamical quantity and a topological invariant closely related to Gauss linking numbers. However, for open magnetic fields with non-zero boundary components, the latter geometrical interpretation is complicated by the fact that helicity varies with non-unique choices of a field's vector potential or gauge. Evaluated in a particular gauge called the winding gauge, open-field helicity in Cartesian slab domains has been shown to be the average flux-weighted pairwise winding numbers of field lines, a measure constructed solely from field configurations that manifest its topological origin. In this paper, we derive the spherical analogue of the winding gauge and the corresponding winding interpretation of helicity, in which we formally define the concept of spherical winding of curves. Using a series of examples, we demonstrate novel properties of spherical winding and the validity of spherical winding helicity. We further argue for the canonical status of the winding gauge choice among all vector potentials for magnetic helicity by exhibiting equivalences between local coordinate changes and gauge transformations.

Citation

Xiao, D., Prior, C., & Yeates, A. (2023). Spherical winding and helicity. Journal of Physics A: Mathematical and Theoretical, 56(20), Article 205201. https://doi.org/10.1088/1751-8121/accc17

• Journal Article Type: Article
• Acceptance Date: Apr 11, 2023
• Publication Date: May 19, 2023
• Deposit Date: Apr 18, 2023
• Publicly Available Date: May 31, 2023
• Journal: Journal of Physics A: Mathematical and Theoretical
• Print ISSN: 1751-8113
• Electronic ISSN: 1751-8121
• Publisher: IOP Publishing
• Peer Reviewed: Peer Reviewed
• Volume: 56
• Issue: 20
• Article Number: 205201
• DOI: https://doi.org/10.1088/1751-8121/accc17
• Public URL: https://durham-repository.worktribe.com/output/1176160
• Publisher URL: https://iopscience.iop.org/article/10.1088/1751-8121/accc17