Asymmetry of MHD equilibria for generic adapted metrics

TL;DR

This paper investigates the symmetry properties of ideal MHD equilibria on 3-manifolds, showing that for generic adapted metrics, such equilibria typically lack continuous symmetries, contrasting with classical conjectures.

Contribution

It proves that on a compact 3-manifold, most MHD equilibria with non-constant pressure do not admit continuous symmetries for generic adapted metrics.

Findings

Most MHD equilibria lack continuous symmetries under generic conditions.

Contrasts with classical conjecture suggesting symmetries in Euclidean domains.

Results apply to both manifolds with and without boundary.

Abstract

Ideal magnetohydrodynamic (MHD) equilibria on a Riemannian 3-manifold satisfy the stationary Euler equations for ideal fluids. A stationary solution $X$ admits a large set of ``adapted" metrics in $M$ for which $X$ solves the corresponding MHD equilibrium equations with the same pressure function. We prove different versions of the following statement: an MHD equilibrium with non-constant pressure on a compact three-manifold with or without boundary admits no continuous Killing symmetries for an open and dense set of adapted metrics. This contrasts with the classical conjecture of Grad which loosely states that an MHD equilibrium on a toroidal Euclidean domain in $\mathbb{R}^3$ with pressure function foliating the domain with nested toroidal surfaces must admit Euclidean symmetries.