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Spherical winding and helicity

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

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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.


Xiao, D., Prior, C., & Yeates, A. (2023). Spherical winding and helicity. Journal of Physics A: Mathematical and Theoretical, 56(20), Article 205201.

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
Publisher URL


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As the Version of Record of this article is going to be / has been published on a gold open access basis under a CC BY 4.0 licence, this Accepted Manuscript is available for reuse under a CC BY 4.0 licence immediately.<br /> <br /> Everyone is permitted to use all or part of the original content in this article, provided that they adhere to all the terms of the licence

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