  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.
