Metriplectic Framework for Dissipative Magneto-Hydrodynamics

TL;DR

This paper extends the metriplectic framework to dissipative magneto-hydrodynamics, incorporating magnetic fields and resistive effects, and demonstrates its invariance and ability to identify equilibrium states.

Contribution

It develops a metriplectic formulation for visco-resistive MHD, including magnetic and resistive effects, and adapts it to 2D incompressible resistive MHD.

Findings

The framework correctly reproduces the equations of motion.

It identifies thermodynamic entropy as a Casimir.

It demonstrates invariance under the Galileo Group.

Abstract

The metriplectic framework, which permits to formulate an algebraic structure for dissipative systems, is applied to visco-resistive Magneto-Hydrodynamics (MHD), adapting what had already been done for non-ideal Hydrodynamics (HD). The result is obtained by extending the HD symmetric bracket and free energy to include magnetic field dynamics and resistive dissipation. The correct equations of motion are obtained once one of the Casimirs of the Poisson bracket for ideal MHD is identified with the total thermodynamical entropy of the plasma. The metriplectic framework of MHD is shown to be invariant under the Galileo Group. The metriplectic structure also permits to obtain the asymptotic equilibria toward which the dynamics of the system evolves. This scheme is finally adapted to the two-dimensional incompressible resistive MHD, that is of major use in many applications.