A general metriplectic framework with application to dissipative extended magnetohydrodynamics

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 has already been discovered, we determine its metriplectic counterpart that describes the dissipation. This is done using a new and general thermodynamic point of view to derive dissipative brackets, a means of derivation that is natural for understanding and creating dissipative dynamics without appealing to underlying kinetic theory orderings. Finally, the formalism is used to study dissipation in the Lagrangian variable picture where, in the context of extended magnetohydrodynamics, non-local dissipative brackets naturally emerge.

A class of three-dimensional gyroviscous magnetohydrodynamic models

“…However, given that several variants of extended MHD possess Lagrangian and Hamiltonian formulations (Keramidas Charidakos et al 2014; Abdelhamid, Kawazura & Yoshida 2015; Lingam, Morrison & Miloshevich 2015 a ; Lingam, Morrison & Tassi 2015 b ; D'Avignon, Morrison & Lingam 2016; Lingam, Abdelhamid & Hudson 2016 a ; Lingam, Miloshevich & Morrison 2016 b ; Burby 2017; Miloshevich, Lingam & Morrison 2017), it would seem natural to utilize the gyromap and thus formulate the gyroviscous contributions for this class of models; after doing so, their equilibria and stability can be obtained by using the HAP approach along the lines of Andreussi et al (2010, 2012, 2013, 2016), Morrison et al (2014) and Kaltsas, Throumoulopoulos & Morrison (2017, 2018, 2020) where the stability of a variety of equilibria is analysed using Lagrangian, energy–Casimir and dynamically accessibility methods. Likewise, this approach could also be extended to relativistic MHD and XMHD models with HAP formulations (D'Avignon, Morrison & Pegoraro 2015; Grasso et al 2017; Kawazura, Miloshevich & Morrison 2017; Coquinot & Morrison 2020; Ludwig 2020). We mention in passing that it would be interesting to explore how the time-dependent regauging of Andreussi et al (2013) can be used to produce or remove the -effects, in a manner analogous to how rotation can produce or remove effects of the magnetic field using Larmor's theorem.…”

Geometric Hamilton-Jacobi theory for systems with external forces

“…Remark 2. The dissipative bracket [•, •] defined above resembles the dissipative bracket (•, •) appearing in the metriplectic framework [3,17]. Both brackets are symmetric and bilinear.…”

Singular cotangent models and complexity in fluids with dissipation

“…In [Mor84a], Morrison introduces the metriplectic formalism as an extension of the Hamiltonian formalism in such a way that it includes a dissipation while the essence of a conserved quantity is not lost. It couples Poisson brackets, coming from the Hamiltonian symplectic formalism, with metric brackets, coming from out-of-equilibrium thermodynamics (see also [Mor84b], [Mor98]) and [CM20]). This formalism is able to describe systems with both Hamiltonian and dissipative components, and to model friction, electric resistivity, collisions and more.…”