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arXiv · 2506.13251

Topological Invariants in Higher-Dimensional Magnetohydrodynamics

Naoki Sato·Ken Abe·Michio Yamada
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Abstract

It is well known that the three-dimensional ideal magnetohydrodynamics (MHD) equations possess three magnetic invariants: (M) magnetic helicity, (C) cross helicity, and (P) the mean-square magnetic potential, in addition to the fundamental invariants of fluid motion. In this paper we construct higher-dimensional generalizations of these invariants for ideal MHD. Specifically, we identify generalized magnetic helicity and generalized cross helicity in all odd spatial dimensions $n=2m+1$, and families of invariants given by integrals of arbitrary functions of the scalar density $B^m/\nu$ of the magnetic field $2$-form $B$, where $B^m$ denotes its $m$-fold wedge product and $\nu$ the fluid-density top form, in all even spatial dimensions $n=2m$. We further establish the existence of invariants for symmetric solutions in arbitrary dimensions, generalizing the mean-square magnetic potential and showing that this invariant arises from symmetry rather than from even dimensionality, in contrast to the enstrophy invariant of the two-dimensional Euler equations.

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Naoki Sato, Ken Abe, Michio Yamada. 2025-06-16. Topological Invariants in Higher-Dimensional Magnetohydrodynamics. https://arxiv.org/abs/2506.13251

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