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:

Here $\psi =\psi (t,x,y),x\in R/2\pi LZ,y\in R/2\pi Z$. 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_L=ρ→∞lim_β→∞ and lim_κ→0lim_β→∞ we get similar systems. In particular, none of the three limiting systems exhibits the energy cascade to high frequencies. (paper)