The limit of small Rossby numbers for randomly forced quasi-geostrophic equation on $\beta$-plane

We consider the 2d quasigeostrophic equation on the $\beta$-plane for the stream function $\psi$, with dissipation and a random force: $$(\ast )(-\mathrm{\Delta }+K){\psi }_t-\rho J(\psi ,\mathrm{\Delta }\psi )-\beta {\psi }_x=⟨\text{random force}⟩-\kappa {\mathrm{\Delta }}^2\psi +\mathrm{\Delta }\psi ,$$ where $\psi=\psi(t,x,y), \ x\in$\mathbb{R}$/2\pi L$\mathbb{Z}$, \ y\in $\mathbb{R}$/2\pi $\mathbb{Z}$$\mathrm{.}Fortypicalvaluesofthehorizontalperiod$L$ we prove that the law of the action-vector of a solution for $(\ast )$ (formed by the halves of the squared norms of its complex Fourier coefficients) converges, as $\beta \to \mathrm{\infty }$, 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 $(\ast )$ converges to that of the effective equation. Moreover, this convergence is uniform in $\kappa \in (0,1]$. The effective equation is an infinite system of stochastic equations which splits into invariant subsystems of complex dimension $\le 3$; each of these subsystems is an integrable hamiltonian system, coupled with a Langevin thermostat. Under the iterated limits ${\lim }_{L=\rho \to \mathrm{\infty }}{\lim }_{\beta \to \mathrm{\infty }}$ and $\lim_{\kappa\to 0} \lim_{\beta\to\infty}$ we get similar systems. In particular, none of the three limiting systems exhibits the energy cascade to high frequencies.