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A least Action principle for visco-resistive Hall Magnetohydrodynamic with metriplectic reformulation

Valentin Carlier, Martin Campos-Pinto

TL;DR

This work develops a least-action framework for visco-resistive Hall-MHD ($HMHD$) by first deriving an ideal two-fluid variational principle with electromagnetic coupling, then incorporating viscosity and resistivity via constrained variations and a metriplectic reformulation that combines a Lie-Poisson structure for the ideal part with a metric 4-bracket for dissipation. It yields a canonical Hamiltonian formulation and multiple equivalent Bracket representations of HMHD, including a simplified five-variable HMHD model and a direct link to established HMHD Hamiltonian structures. The viscoresistive extension introduces dissipation through a Drude-like resistive force and viscous stress, recasting the dynamics in a metriplectic form that guarantees entropy production and aligns with Coquinot-Morrison’s metric-bracket approach. These results provide a foundation for structure-preserving numerical schemes and suggest avenues for variational discretizations in HMHD and extended MHD contexts.

Abstract

We present a new variational formulation for Viscous and resistive Hall Magnetohydrodynamic. We first find a variational principle for ideal HMHD by applying the physical assumptions leading to HMHD at the lagrangian level, and then we add the viscous and resistive terms by the means of constrained variations. We also provide a metriplectic reformulation of our formulation, based on two canonical Lie-Poisson brackets for the ideal part and metric 4-brackets for the dissipative part.

A least Action principle for visco-resistive Hall Magnetohydrodynamic with metriplectic reformulation

TL;DR

This work develops a least-action framework for visco-resistive Hall-MHD () by first deriving an ideal two-fluid variational principle with electromagnetic coupling, then incorporating viscosity and resistivity via constrained variations and a metriplectic reformulation that combines a Lie-Poisson structure for the ideal part with a metric 4-bracket for dissipation. It yields a canonical Hamiltonian formulation and multiple equivalent Bracket representations of HMHD, including a simplified five-variable HMHD model and a direct link to established HMHD Hamiltonian structures. The viscoresistive extension introduces dissipation through a Drude-like resistive force and viscous stress, recasting the dynamics in a metriplectic form that guarantees entropy production and aligns with Coquinot-Morrison’s metric-bracket approach. These results provide a foundation for structure-preserving numerical schemes and suggest avenues for variational discretizations in HMHD and extended MHD contexts.

Abstract

We present a new variational formulation for Viscous and resistive Hall Magnetohydrodynamic. We first find a variational principle for ideal HMHD by applying the physical assumptions leading to HMHD at the lagrangian level, and then we add the viscous and resistive terms by the means of constrained variations. We also provide a metriplectic reformulation of our formulation, based on two canonical Lie-Poisson brackets for the ideal part and metric 4-brackets for the dissipative part.

Paper Structure

This paper contains 16 sections, 9 theorems, 138 equations.

Table of Contents

  1. Introduction
  2. Lagrangian variational principle for charged fluids in an electromagnetic field
  3. Euler equation from variational principle
  4. Single charged fluid in an electromagnetic field
  5. Two charged fluids in an electromagnetic field
  6. Hall-MHD
  7. From two charged fluids to Hall MHD
  8. A first variational principle for Hall-MHD
  9. Simplified Hall-MHD
  10. Canonical Hamiltonian and bracket formulation
  11. Comparison with other Hamiltonian formulations
  12. Viscoresistive Hall MHD
  13. Lagrangian Variational approach
  14. Metriplectic bracket, Hamiltonian and Entropy for Viscoresistive Hall MHD
  15. Relation with the metric bracket of Coquinot-Morrison
  16. ...and 1 more sections

Key Result

Theorem 1

Consider the following Lagrangian : where $\boldsymbol{u} \in X(\Omega)$ , $n, s \in F(\Omega)$ and $e$ is the internal energy of the fluid, as a function of $n$ the density and $s$ the entropy. $m_s$ is the mass of a particle constituting the fluid. Then extremizers of the action $\mathfrak{S} = \int_0^T l dt$ under constrained varia (where $\boldsymbol{v} = \boldsymbol{v}(t)$ is a curve in $X(\

Theorems & Definitions (20)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • Remark 1
  • ...and 10 more