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

Abstract

We consider the 2d quasigeostrophic equation on the $\beta$-plane for the stream function $\psi$, with dissipation and a random force: $$(\ast )(-\Delta +K){\psi }_t-\rho J(\psi ,\Delta \psi )-\beta {\psi }_x=⟨\text{random force}⟩-\kappa {\Delta }^2\psi +\Delta \psi ,$$ where $\psi =\psi (t,x,y),x\in \mathbb{R}/2\pi L\mathbb{Z},y\in \mathbb{R}/2\pi \mathbb{Z}$. For typical values of the horizontal period $L$ 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 $\beta \to \infty$, 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 ${\lim }_{L=\rho \to \infty }{\lim }_{\beta \to \infty }$ 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.