  We adapt the formalism of the statistical theory of 2D turbulence in the case
where the Casimir constraints are replaced by the specification of a prior
vorticity distribution. A phenomenological relaxation equation is obtained for
the evolution of the coarse-grained vorticity. This equation monotonically
increases a generalized entropic functional (determined by the prior) while
conserving circulation and energy. It can be used as a thermodynamical
parametrization of forced 2D turbulence, or as a numerical algorithm to
construct (i) arbitrary statistical equilibrium states in the sense of
Ellis-Haven-Turkington (ii) particular statistical equilibrium states in the
sense of Miller-Robert-Sommeria (iii) arbitrary stationary solutions of the 2D
Euler equation that are formally nonlinearly dynamically stable according to
the Ellis-Haven-Turkington stability criterion refining the Arnold theorems.
