Hamiltonian Magnetohydrodynamics: Lagrangian, Eulerian, and Dynamically Accessible Stability -- Examples with Translation Symmetry

TL;DR

This paper compares stability conditions for dissipationless fluids and magnetofluids across different frameworks, illustrating the differences with magnetohydrodynamic examples and providing methods to reformulate equilibria with flows.

Contribution

It explicitly analyzes stability frameworks in MHD, introduces a relabeling method for equilibria with flows, and offers a paradigm extendable to various plasma configurations.

Findings

Different stability conditions depend on the chosen framework.

A relabeling technique allows reformulating Eulerian equilibria as Lagrangian.

The procedures are applicable to a broad class of plasma systems.

Abstract

Because different constraints are imposed, stability conditions for dissipationless fluids and magnetofluids may take different forms when derived within the Lagrangian, Eulerian (energy-Casimir), or dynamical accessible frameworks. This is in particular the case when flows are present. These differences are explored explicitly by working out in detail two magnetohydrodynamic examples: convection against gravity in a stratified fluid and translationally invariant perturbations of a rotating magnetized plasma pinch. In this second example we show in explicit form how to perform the time-dependent relabeling introduced in Andreussi {\it et al.}\ [Phys.\ Plasmas {\bf20}, 092104 (2013)] that makes it possible to reformulate Eulerian equilibria with flows as Lagrangian equilibria in the relabeled variables. The procedures detailed in the present article provide a paradigm that can be applied to more general plasma configurations and in addition extended to more general plasma descriptions where dissipation is absent.