Winding and magnetic helicity in periodic domains

Xiao, Daining; Prior, Christopher B.; Yeates, Anthony R.

Abstract

In simply-connected Euclidean domains, it is well-known that the topological complexity of a given magnetic field can be quantified by its magnetic helicity, which is equivalent to the total, flux-weighted winding number of magnetic field lines. Often considered in analytical and numerical studies are domains periodic in two lateral dimensions (periodic domains) which are multiply-connected and homeomorphic to a 2-torus. Whether this equivalence can be generalised to periodic domains remains an open question, first posed by Berger (Berger 1996 J. Geophys. Res. 102, 2637-2644 (doi:10.1029/96JA01896)). In this particle, we answer in the affirmative by defining the novel periodic winding of curves and identifying a vector potential that recovers the topological interpretation of magnetic helicity as winding. Key properties of the topologically defined magnetic helicity in periodic domains are also proved, including its time-conservation in ideal magnetohydrodynamical flows, its connection to Fourier approaches, and its relationship to gauge transformations.

Citation

Xiao, D., Prior, C. B., & Yeates, A. R. (2025). Winding and magnetic helicity in periodic domains. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 481(2307), Article 20240152. https://doi.org/10.1098/rspa.2024.0152

• Journal Article Type: Article
• Acceptance Date: Nov 29, 2024
• Online Publication Date: Feb 21, 2025
• Publication Date: 2025-02
• Deposit Date: Mar 5, 2025
• Publicly Available Date: Mar 5, 2025
• Journal: Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
• Print ISSN: 1364-5021
• Electronic ISSN: 1471-2946
• Publisher: The Royal Society
• Peer Reviewed: Peer Reviewed
• Volume: 481
• Issue: 2307
• Article Number: 20240152
• DOI: https://doi.org/10.1098/rspa.2024.0152
• Public URL: https://durham-repository.worktribe.com/output/3679108