Helically symmetric extended magnetohydrodynamics: Hamiltonian formulation and equilibrium variational principles

TL;DR

This paper develops a Hamiltonian framework for helically symmetric extended magnetohydrodynamics (XMHD), deriving equilibrium equations and invariants, and connects them to known symmetric cases and specific plasma configurations.

Contribution

It extends the Hamiltonian formulation of XMHD to helical symmetry, deriving new equilibrium equations and invariants, and unifies various symmetric cases within a single framework.

Findings

Derived four families of Casimir invariants for helical symmetry.

Formulated generalized equilibrium equations including flow effects.

Presented an example of a double-Beltrami equilibrium with non-planar helical magnetic axis.

Abstract

Hamiltonian extended magnetohydrodynamics (XMHD) is restricted to respect helical symmetry by reducing the Poisson bracket for 3D dynamics to a helically symmetric one, as an extension of the previous study for translationally symmetric XMHD (D.A. Kaltsas et al, Phys. Plasmas 24, 092504 (2017)). Four families of Casimir invariants are obtained directly from the symmetric Poisson bracket and they are used to construct Energy-Casimir variational principles for deriving generalized XMHD equilibrium equations with arbitrary macroscopic flows. The system is then cast into the form of Grad-Shafranov-Bernoulli equilibrium equations. The axisymmetric and the translationally symmetric formulations can be retrieved as geometric reductions of the helical symmetric one. As special cases, the derivation of the corresponding equilibrium equations for incompressible plasmas is discussed and the helically symmetric equilibrium equations for the Hall MHD system are obtained upon neglecting electron inertia. An example of an incompressible double-Beltrami equilibrium is presented in connection with a magnetic configuration having non-planar helical magnetic axis.