# Translationally symmetric extended MHD via Hamiltonian reduction: Energy-Casimir equilibria

## TL;DR

This paper develops a Hamiltonian reduction approach to ideal translationally symmetric extended MHD, deriving equilibrium equations and invariants, and analyzing special cases like static plasmas and Hall MHD, including a numerical equilibrium example.

## Contribution

It introduces a Hamiltonian reduction method for extended MHD with symmetry, deriving generalized equilibrium equations and invariants, and explores special cases with numerical results.

## Key findings

- Derived energy-Casimir variational principles for XMHD equilibria.
- Identified Casimir invariants for symmetric XMHD.
- Presented a numerical equilibrium configuration showing ion-flow separation.

## Abstract

The Hamiltonian structure of ideal translationally symmetric extended MHD (XMHD) is obtained by employing a method of Hamiltonian reduction on the three-dimensional noncanonical Poisson bracket of XMHD. The existence of the continuous spatial translation symmetry allows the introduction of Clebsch-like forms for the magnetic and velocity fields. Upon employing the chain rule for functional derivatives, the 3D Poisson bracket is reduced to its symmetric counterpart. The sets of symmetric Hall, Inertial, and extended MHD Casimir invariants are identified, and used to obtain energy-Casimir variational principles for generalized XMHD equilibrium equations with arbitrary macroscopic flows. The obtained set of generalized equations is cast into Grad-Shafranov-Bernoulli (GSB) type, and special cases are investigated: static plasmas, equilibria with longitudinal flows only, and Hall MHD equilibria, where the electron inertia is neglected. The barotropic Hall MHD equilibrium equations are derived as a limiting case of the XMHD GSB system, and a numerically computed equilibrium configuration is presented that shows the separation of ion-flow from electromagnetic surfaces.