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General equations for conservative yet dissipative (entropy producing)extended magnetohydrodynamics are derived from two-fluid theory. Keeping allterms generates unusual cross-effects, such as thermophoresis and a currentviscosity that mixes with the usual velocity viscosity. While the Poissonbracket of the ideal version of this model have already been discovered, wedetermine its metriplectic counterpart that describes the dissipation. This isdone using a new and general thermodynamic point of view for derivingdissipative brackets, a means of derivation that is natural for understandingand creating dissipative brackets. Finally the formalism is used to studydissipation in the Lagrangian variable picture where, in the context ofextended magnetohydrodynamics, nonlocal dissipative brackets naturally emerge.

A General Metriplectic Framework with Application to Dissipative Extended Magnetohydrodynamics