Area preserving flows and metriplectic dynamics on an annulus

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.