Differential-geometrical approach to the dynamics of dissipationless incompressible Hall magnetohydrodynamics I: Lagrangian mechanics on semidirect product of two volume preserving diffeomorphisms and conservation laws

TL;DR

This paper develops a geometric framework for dissipationless incompressible Hall MHD using Lagrangian mechanics on a semidirect product of volume-preserving diffeomorphisms, revealing conserved quantities and simplifying fundamental equations.

Contribution

It introduces a novel geometric formulation of Hall MHD dynamics on a semidirect product space, deriving explicit conservation laws and simplifying the equations using generalized Elsasser variables.

Findings

Four conserved quantities identified, including energy and helicities.

Simplified expressions for Lie algebra structure constants and equations of motion.

Generalized Elsasser variables facilitate the analysis of HMHD dynamics.

Abstract

The dynamics of a dissipationless incompressible Hall magnetohydrodynamic (HMHD) medium is formulated using Lagrangian mechanics on a semidirect product of two volume preserving diffeomorphism groups. In the case of $\mathbb{T}^3$ or $E^3$, the generalized Elsasser variables introduced by Galtier (S. Galtier 2006 J. Plasma Phys. 72 721-769) yield remarkably simple expressions of basic formulas and equations such as the structure constants of Lie algebra, the equation of motion, and the conservation laws. Four constants of motion, where three of the four are independent, are naturally derived from the generalized Elsasser variables representation of the equation of motion for the HMHD system: total plasma energy, magnetic helicity, hybrid helicity, and the modified cross helicity.