Helicity-based, particle-relabeling operator and normal mode expansion of the dissipationless incompressible Hall magnetohydrodynamics

TL;DR

This paper explores the Lagrangian mechanics of dissipationless Hall MHD, revealing helicities from particle relabeling symmetry, and introduces a spectral expansion using double Beltrami flows for analyzing the system.

Contribution

It introduces a novel operator-based spectral representation of Hall MHD dynamics using double Beltrami flows, connecting symmetries to conserved quantities.

Findings

Helicities emerge from particle relabeling symmetry via Noether's theorem.

Eigenfunctions are double Beltrami flows providing an orthogonal basis.

Generalized Elsasser variables effectively handle singularities in the MHD limit.

Abstract

The dynamics of an incompressible, dissipationless Hall magnetohydrodynamic medium are investigated from Lagrangian mechanical viewpoint. The hybrid and magnetic helicities are shown to emerge, respectively, from the application of the particle relabeling symmetry for ion and electron flows to Noether's first theorem, while the constant of motion associated with the theorem is generally given by their arbitrary linear combination. Furthermore, integral path variation associated with the invariant action is expressed by the operation of an integro-differential operator on the reference path. The eigenfunctions of this operator are double Beltrami flows, i.e. force-free stationary solutions to the equation of motion and provide a family of orthogonal function bases that yields the spectral representation of the equation of motion with a remarkably simple form. Among the double Beltrami flows, considering the influence of a uniform background magnetic field and the Hall term effect vanishing limit, the generalized Elsasser variables are found to be the most suitable for avoiding problems with singularities in the standard magnetohydrodynamic limit.