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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: $$ (*)\qquad (-Δ+K)ψ_t - ρJ(ψ, Δψ) -βψ_x= \langle \text{random force}\rangle -κΔ^2ψ+Δψ, $$ where $ψ=ψ(t,x,y), \ x\in\mathbb{R}/2πL\mathbb{Z}, \ y\in \mathbb{R}/2π\mathbb{Z}$. 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 $β\to\infty$, 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 $κ\in(0,1]$. The effective equation is an infinite system of stochastic equations which splits into invariant subsystems of complex dimension $\le3$; each of these subsystems is an integrable hamiltonian system, coupled with a Langevin thermostat. Under the iterated limits $\lim_{L=ρ\to\infty} \lim_{β\to\infty}$ and $\lim_{κ\to 0} \lim_{β\to\infty}$ we get similar systems. In particular, none of the three limiting systems exhibits the energy cascade to high frequencies.