Hamiltonian formulations for perturbed dissipationless plasma equations

TL;DR

This paper develops Hamiltonian formulations for perturbed plasma equations, emphasizing the roles of polarization and magnetization, and introduces a framework for functional perturbation methods to analyze plasma stability.

Contribution

It presents a novel Hamiltonian perturbation framework for Vlasov-Maxwell and MHD equations, incorporating polarization and magnetization effects, and extends to higher-order stability analysis.

Findings

Framework for functional perturbation methods in plasma physics

Hamiltonian formulations for perturbed plasma equations

Generalization of plasma stability analysis to higher-order perturbations

Abstract

The Hamiltonian formulations for the perturbed Vlasov-Maxwell equations and the perturbed ideal magnetohydrodynamics (MHD) equations are expressed in terms of the perturbation derivative $\partial{\cal F}/\partial\epsilon \equiv [{\cal F}, {\cal S}]$ of an arbitrary functional ${\cal F}[\vb{\psi}]$ of the Vlasov-Maxwell fields $\vb{\psi} = ({\sf f},{\bf E},{\bf B})$ or the ideal MHD fields $\vb{\psi} = (\rho,{\bf u},s,{\bf B})$, which are assumed to depend continuously on the (dimensionless) perturbation parameter $\epsilon$. Here, $[\;,\;]$ denotes the functional Poisson bracket for each set of plasma equations and the perturbation {\it action} functional ${\cal S}$ is said to generate dynamically accessible perturbations of the plasma fields. The new Hamiltonian perturbation formulation introduces a framework for functional perturbation methods in plasma physics and highlights the crucial roles played by polarization and magnetization in Vlasov-Maxwell and ideal MHD perturbation theories. One application considered in this paper is a formulation of plasma stability that guarantees dynamical accessibility and leads to a natural generalization to higher-order perturbation theory.