Topological Invariants in Higher-Dimensional Magnetohydrodynamics

TL;DR

This paper extends known three-dimensional magnetic invariants in ideal magnetohydrodynamics to higher odd and even dimensions, revealing new conserved quantities and their origins related to symmetry and dimensionality.

Contribution

It constructs higher-dimensional generalizations of magnetic invariants and identifies their dependence on symmetry, expanding the theoretical framework of ideal MHD.

Findings

Generalized magnetic helicity and cross helicity in all odd dimensions.

Families of invariants involving functions of magnetic field density in even dimensions.

Existence of invariants for symmetric solutions across all dimensions.

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.