Derivation of the Hall and Extended Magnetohydrodynamics Brackets

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.

36 pages; 2 figures

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Cited by in corpus (18)

• Variational approach to low-frequency kinetic-MHD in the current coupling scheme
• Action Principles for Relativistic Extended Magnetohydrodynamics: A Unified Theory of Magnetofluid Models
• On the structure and statistical theory of turbulence of extended magnetohydrodynamics
• Helical Fluid and (Hall)-MHD Turbulence: a Brief Review
• Structure and computation of two-dimensional incompressible extended MHD
• Hamiltonian Magnetohydrodynamics: Lagrangian, Eulerian, and Dynamically Accessible Stability -- Examples with Translation Symmetry
• Translationally symmetric extended MHD via Hamiltonian reduction: Energy-Casimir equilibria
• Energy-Casimir, dynamically accessible, and Lagrangian stability of extended magnetohydrodynamic equilibria
• Hamiltonian kinetic-Hall Magnetohydrodynamics with fluid and kinetic ions in the current and pressure coupling schemes
• Multi-region relaxed Hall magnetohydrodynamics with flow
• Group properties and solutions for the 1D Hall MHD system in the cold plasma approximation
• A class of three-dimensional gyroviscous magnetohydrodynamic models
• Hall MHD waves: A fundamental departure from their MHD counterparts
• Hamiltonian formulations for perturbed dissipationless plasma equations
• Beatification: Flattening the Poisson Bracket for Two-Dimensional Fluid and Plasma Theories
• Hamiltonian formulation of X-point collapse in an extended magnetohydrodynamics framework
• Nonlinear feedback, double bracket dissipation and port control of Lie-Poisson systems
• A new and alternative look at NonLinear Alfvénic States