Derivation of the Hall and Extended Magnetohydrodynamics Brackets

TL;DR

This paper derives the Hamiltonian structure of Hall and Extended MHD models by formulating their Lagrangian actions, identifying advected fluxes, and obtaining noncanonical Poisson brackets, thus advancing the theoretical understanding of plasma models.

Contribution

It introduces fully Lagrangian actions for Hall and Extended MHD, enabling derivation of their noncanonical Poisson brackets and revealing advected fluxes and conserved helicities.

Findings

Derived Lagrangian actions for Hall and Extended MHD

Identified advected fluxes and Lie-dragged potentials

Established methods to obtain noncanonical brackets from Lagrangian formulations

Abstract

There are several plasma models intermediate in complexity between ideal magnetohydrodynamics (MHD) and two-fluid theory, with Hall and Extended MHD being two important examples. In this paper we investigate several aspects of these theories, with the ultimate goal of deriving the noncanonical Poisson brackets used in their Hamiltonian formulations. We present fully Lagrangian actions for each, as opposed to the fully Eulerian, or mixed Eulerian-Lagrangian, actions that have appeared previously. As an important step in this process we exhibit each theory's two advected fluxes (in analogy to ideal MHD's advected magnetic flux), discovering also that with the correct choice of gauge they have corresponding Lie-dragged potentials resembling the electromagnetic vector potential, and associated conserved helicities. Finally, using the Euler-Lagrange map, we show how to derive the noncanonical Eulerian brackets from canonical Lagrangian ones.