Statistical mechanics of 2D turbulence with a prior vorticity distribution

We adapt the formalism of the statistical theory of 2D turbulence to 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 for constructing (i) arbitrary statistical equilibrium states in the sense of Ellis, Haven and Turkington, (ii) particular statistical equilibrium states in the sense of Miller, Robert and 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.