Statistical mechanics of the shallow-water system with a prior potential vorticity distribution

We adapt the statistical mechanics of the shallow-water equations to the case where the flow is forced at small scales. We assume that the statistics of forcing is encoded in a prior potential vorticity distribution which replaces the specification of the Casimir constraints in the case of freely evolving flows. This determines a generalized entropy functional which is maximized by the coarse-grained PV field at statistical equilibrium. Relaxation equations towards equilibrium are derived which conserve the robust constraints (energy, mass and circulation) and increase the generalized entropy.