Action Principles for Extended MHD Models

TL;DR

This paper develops a unified variational framework for deriving extended MHD models from the general two-fluid plasma equations, ensuring Hamiltonian structure and conserved quantities through a novel nonlocal Lagrange-Euler map.

Contribution

It introduces a method to derive reduced plasma models directly within the action principle, preserving Hamiltonian structure and symmetries.

Findings

Recovered various plasma models from a unified action framework

Ensured Hamiltonian structure in reduced models

Derived conserved quantities using Noether's theorem

Abstract

The general, non-dissipative, two-fluid model in plasma physics is Hamiltonian, but this property is sometimes lost or obscured in the process of deriving simplified (or reduced) two-fluid or one-fluid models from the two-fluid equations of motion. To ensure that the reduced models are Hamiltonian, we start with the general two-fluid action functional, and make all the approximations, changes of variables, and expansions directly within the action context. The resulting equations are then mapped to the Eulerian fluid variables using a novel nonlocal Lagrange-Euler map. Using this method, we recover L\"{u}st's general two-fluid model, extended MHD, Hall MHD, and electron MHD from a unified framework. The variational formulation allows us to use Noether's theorem to derive conserved quantities for each symmetry of the action.