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

We consider the 2d quasigeostrophic equation on the \u3b2-plane for the stream function \u3c8, with dissipation and a random force: (+ K)\u3c8t - pJ(\u3c8\u3c8) - \u3b2\u3c8x = lang;random force\u3009 ? k2\u3c8 +\u3c8. ( 17) Here \u3c8 = \u3c8(t, x, y), x \u3b5\u211d/2\u3c0L\u2124, y \u3b5 \u211d/2\u3c0\u2124. For typical values of the horizontal period L we prove that the law of the action-vector of a solution for ( 17) (formed by the halves of the squared norms of its complex Fourier coefficients) converges, as \u3b2 \u2192 1e, 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 ( 17) converges to that of the effective equation. Moreover, this convergence is uniform in \u3ba 08 (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 limL=\u3c1\u2192 1elim\u3b2\u2192 1e and lim\u3ba\u21920lim\u3b2\u2192 1e we get similar systems. In particular, none of the three limiting systems exhibits the energy cascade to high frequencies