A basis of Casimirs in 3D magnetohydrodynamics

TL;DR

This paper proves that in 3D magnetohydrodynamics, the only regular Casimir invariants are the magnetic and cross-helicity, establishing their fundamental role as the sole independent invariants under the MHD symmetry group.

Contribution

It demonstrates that magnetic and cross-helicity are the only regular Casimir invariants in 3D MHD, clarifying their fundamental importance in the theory.

Findings

Magnetic and cross-helicity are the only regular Casimirs in 3D MHD.

These invariants are the only independent integral invariants under the MHD group.

The result characterizes the structure of invariants in the MHD coadjoint action.

Abstract

We prove that any regular Casimir in 3D magnetohydrodynamics is a function of the magnetic helicity and cross-helicity. In other words, these two helicities are the only independent regular integral invariants of the coadjoint action of the MHD group $\text{SDiff}(M)\ltimes\mathfrak X^*(M)$, which is the semidirect product of the group of volume-preserving diffeomorphisms and the dual space of its Lie algebra.