Hamiltonian magnetohydrodynamics: symmetric formulation, Casimir invariants, and equilibrium variational principles

TL;DR

This paper develops a symmetric Hamiltonian formulation for magnetohydrodynamics (MHD), deriving Casimir invariants and equilibrium principles for symmetric plasma configurations, including helical, axial, and translational symmetries.

Contribution

It introduces a symmetric form of the noncanonical Poisson bracket for MHD and derives equilibrium variational principles using Casimir invariants for symmetric plasma states.

Findings

Derived symmetric Poisson bracket for MHD.

Obtained Casimir invariants directly from the bracket.

Formulated energy-Casimir variational principles for equilibrium configurations.

Abstract

The noncanonical Hamiltonian formulation of magnetohydrodynamics (MHD) is used to construct variational principles for symmetric equilibrium configurations of magnetized plasma including flow. In particular, helical symmetry is considered and results on axial and translational symmetries are retrieved as special cases of the helical configurations. The symmetry condition, which allows the description in terms of a magnetic flux function, is exploited to deduce a symmetric form of the noncanonical Poisson bracket of MHD. Casimir invariants are then obtained directly from the Poisson bracket. Equilibria are obtained from an energy-Casimir principle and reduced forms of this variational principle are obtained by the elimination of algebraic constraints.