Energy-Casimir, dynamically accessible, and Lagrangian stability of extended magnetohydrodynamic equilibria

TL;DR

This paper develops stability criteria for extended magnetohydrodynamic (XMHD) equilibria using Hamiltonian methods, providing explicit conditions for various flow configurations and examining Lagrangian stability within a two-fluid model.

Contribution

It introduces explicit stability conditions for XMHD and Hall MHD equilibria, and explores Lagrangian stability analysis within a Hamiltonian framework for two-fluid models.

Findings

Derived explicit stability criteria for axisymmetric XMHD and Hall MHD equilibria.

Obtained the second-order variation of the Hamiltonian for stability analysis.

Formulated the Lagrangian stability problem within a mixed Lagrangian-Eulerian framework.

Abstract

The formal stability analysis of Eulerian extended magnetohydrodynamics (XMHD) equilibria is considered within the noncanonical Hamiltonian framework by means of the energy-Casimir variational principle and the dynamically accessible stability method. Specifically, we find explicit sufficient stability conditions for axisymmetric XMHD and Hall MHD (HMHD) equilibria with toroidal flow and for equilibria with arbitrary flow under constrained perturbations. The dynamically accessible, second-order variation of the Hamiltonian, which can potentially provide explicit stability criteria for generic equilibria, is also obtained. Moreover, we examine the Lagrangian stability of the general quasineutral two-fluid model written in terms of MHD-like variables, by finding the action and the Hamiltonian functionals of the linearized dynamics, working within a mixed Lagrangian-Eulerian framework. Upon neglecting electron mass, we derive a HMHD energy principle, and in addition, the perturbed induction equation arises from Hamilton's equations of motion in view of a consistency condition for the relation between the perturbed magnetic potential and the canonical variables.