Publication
In this paper, we present metriplectic extensions of the Lie–Poisson dynamics of fluid flows over the unit annulus for both the Lebesgue [Formula: see text]-pairing and the Sobolev [Formula: see text]-pairing. Several examples are provided including 2D Navier–Stokes equations, potential vorticity flows with super-diffusivity, averaged 2D Navier–Stokes equations, and a second grade fluid model. We extend the theory to semi-direct products. Accordingly, a metriplectic extension of the 2D electron MHD equation is proposed. In particular, Casimir dissipations are added to the Lie–Poisson systems for both the [Formula: see text]-pairing and the [Formula: see text]-pairing as well as their semidirect product extensions.