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

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.

arXiv commentIn this version, some errors, typos, and some dropped references in the previous version are corrected

References in corpus (6)

• Kinetic formulation and global existence for the Hall-Magneto-hydrodynamics system
• Wave turbulence in incompressible Hall MHD
• Energy transfer in Hall-MHD turbulence: cascades, backscatter, and dynamo action
• Action Principles for Extended MHD Models
• Impulse Formulations of Hall Magnetohydrodynamic Equations
• Geodesics and curvature of semidirect product groups

Cited by in corpus (6)

• Chiral Exact Relations for Helicities in Hall Magnetohydrodynamic Turbulence
• Variational formulation of relaxed and multi-region relaxed magnetohydrodynamics
• Time-dependent relaxed magnetohydrodynamics -- inclusion of cross helicity constraint using phase-space action
• Helicity-based, particle-relabeling operator and normal mode expansion of the dissipationless incompressible Hall magnetohydrodynamics
• Differential-geometrical approach to the dynamics of dissipationless incompressible Hall magnetohydrodynamics: II. Geodesic formulation and Riemannian curvature analysis of hydrodynamic and magnetohydrodynamic stabilities
• A new and alternative look at NonLinear Alfvénic States