Statistical mechanics of 2D turbulence with a prior vorticity distribution

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 new relaxation equation is obtained for the evolution of the coarse-grained vorticity. It can be used as a thermodynamical parametrization of forced 2D turbulence (determined by the prior), or as a numerical algorithm to construct arbitrary nonlinearly dynamically stable stationary solutions of the 2D Euler equation. Two-dimensional incompressible flows with high Reynolds numbers are described by the 2D Euler equations ∂ω ∂t + u · ∇ω = 0, ω = −∆ψ, u = −z×∇ψ, (1) where ω is the vorticity and ψ the streamfunction. The 2D Euler equations are known to develop a complicated mixing process which ultimately leads to the emergence of a large-scale coherent structure, typically a jet or a vortex [1]. Jovian atmosphere shows a wide diversity of structures: Jupiter’s great red spot, white ovals, brown barges,... One question of fundamental interest is to understand and predict the structure and the stability of these equilibrium states. To that purpose, Miller [2] and Robert & Sommeria [3] have proposed a statistical mechanics of the 2D Euler equation. The idea is to replace the deterministic description of the flow ω(r, t) by a probabilistic description where ρ(r, σ, t) gives the density probability of finding the vorticity level ω = σ in r at time t. The observed (coarse-grained) vorticity field is then expressed as ω(r, t) = ∫ ρσdσ. To apply the statistical theory, one must first specify the constraints attached to the 2D Euler equation. The circulation Γ = ∫ ωdr and the energy E = 12 ∫ ωψdr will be called robust constraints because they can be expressed in terms of the coarse-grained field ω (the energy of the fluctuations can be neglected). These integrals can be calculated at any time from the coarse-grained field ω(r, t) and they are conserved by the dynamics. By contrast, the Casimir invariants If = ∫ f(ω)dr, or equivalently the fine-grained moments of the vorticity Γ n>1 = ∫ ωndr = ∫ ρσndσdr, will be called fragile constraints because they must be expressed in terms of the fine-grained vorticity. Indeed, the moments of the coarse-grained vorticity Γ n>1 = ∫ ωndr are not conserved since ωn 6= ωn (part of the coarse-grained moments goes into fine-grained fluctuations). Therefore, the moments Γ n>1 must be calculated from the fine-grained field ω(r, t) or from the initial conditions, i.e. before the vorticity has mixed. Since we often do not know the initial conditions nor the fine-grained field, the Casimir invariants often appear as “hidden constraints” [4].