The limit of small Rossby numbers for randomly forced quasi-geostrophic equation on β-plane

We consider the 2d quasigeostrophic equation on the β-plane for the stream function ψ, with dissipation and a random force: (+ K)ψ<inf>t</inf> - pJ(ψψ) - βψx = lang;random force〉 ? k²ψ +ψ. (∗) Here ψ = ψ(t, x, y), x εℝ/2πLℤ, y ε ℝ/2πℤ. For typical values of the horizontal period L we prove that the law of the action-vector of a solution for (∗) (formed by the halves of the squared norms of its complex Fourier coefficients) converges, as β → ∞, to the law of an action-vector for solution of an auxiliary effective equation, and the stationary distribution of the action-vector for solutions of (∗) converges to that of the effective equation. Moreover, this convergence is uniform in κ ∈ (0, 1]. The effective equation is an infinite system of stochastic equations which splits into invariant subsystems of complex dimension 3; each of these subsystems is an integrable hamiltonian system, coupled with a Langevin thermostat. Under the iterated limits lim<inf>L=ρ→∞</inf>lim<inf>β→∞</inf> and lim<inf>κ→0</inf>lim<inf>β→∞</inf> we get similar systems. In particular, none of the three limiting systems exhibits the energy cascade to high frequencies.