Hamiltonian Magnetohydrodynamics: Lagrangian, Eulerian, and Dynamically Accessible Stability - Theory

TL;DR

This paper develops and compares three Hamiltonian-based energy principles for analyzing the stability of magnetized plasma flows, providing new criteria for stability in different variable formulations.

Contribution

It introduces and compares Lagrangian, Eulerian, and dynamically accessible energy principles for MHD stability analysis, expanding the theoretical framework.

Findings

Derived sufficient stability conditions for symmetric equilibria.

Extended the energy-Casimir principle to second order.

Provided general stability criteria and comparison of approaches.

Abstract

Stability conditions of magnetized plasma flows are obtained by exploiting the Hamiltonian structure of the magnetohydrodynamics (MHD) equations and, in particular, by using three kinds of energy principles. First, the Lagrangian variable energy principle is described and sufficient stability conditions are presented. Next, plasma flows are described in terms of Eulerian variables and the noncanonical Hamiltonian formulation of MHD is exploited. For symmetric equilibria, the energy-Casimir principle is expanded to second order and sufficient conditions for stability to symmetric perturbation are obtained. Then, dynamically accessible variations, i.e. variations that explicitly preserve invariants of the system, are introduced and the respective energy principle is considered. General criteria for stability are obtained, along with comparisons between the three different approaches.