# A General Metriplectic Framework with Application to Dissipative Extended Magnetohydrodynamics

## TL;DR

This paper develops a comprehensive metriplectic framework for dissipative extended magnetohydrodynamics, deriving new dissipative brackets from thermodynamic principles and exploring their implications in Lagrangian variables.

## Contribution

It introduces a novel thermodynamic approach to derive dissipative brackets in extended magnetohydrodynamics, including nonlocal effects in Lagrangian variables.

## Key findings

- Derived the metriplectic counterpart for dissipative MHD
- Identified new cross-effects such as thermophoresis and current viscosity
- Explored dissipation in Lagrangian variables with nonlocal brackets

## Abstract

General equations for conservative yet dissipative (entropy producing) extended magnetohydrodynamics are derived from two-fluid theory. Keeping all terms generates unusual cross-effects, such as thermophoresis and a current viscosity that mixes with the usual velocity viscosity. While the Poisson bracket of the ideal version of this model have already been discovered, we determine its metriplectic counterpart that describes the dissipation. This is done using a new and general thermodynamic point of view for deriving dissipative brackets, a means of derivation that is natural for understanding and creating dissipative brackets. Finally the formalism is used to study dissipation in the Lagrangian variable picture where, in the context of extended magnetohydrodynamics, nonlocal dissipative brackets naturally emerge.