A least-action principle for visco-resistive Hall magnetohydrodynamics with metriplectic reformulation

We present a new variational formulation for viscous and resistive Hall magnetohydrodynamics. We first find a variational principle for ideal Hall magnetohydrodynamics by applying the physical assumptions leading to Hall magnetohydrodynamics at the Lagrangian level, and then we add the viscous and resistive terms by 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.