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

TL;DR

This paper analyzes the behavior of the 2D quasigeostrophic equation on the $eta$-plane with random forcing, showing that as $eta$ becomes large, the solution's action-vector law converges to that of an effective integrable system, with no energy cascade observed.

Contribution

It establishes the convergence of the solution's action-vector law to an effective integrable system as $eta o \infty$, including uniformity in dissipation parameter $\kappa$.

Findings

Action-vector law converges to an effective system as $eta o \infty$

Stationary distribution converges to that of the effective system

No energy cascade to high frequencies in the limiting systems

Abstract

We consider the 2d quasigeostrophic equation on the $\beta$-plane for the stream function $\psi$, with dissipation and a random force: $$ (*)\qquad (-\Delta +K)\psi_t - \rho J(\psi, \Delta\psi) -\beta\psi_x= \langle \text{random force}\rangle -\kappa\Delta^2\psi +\Delta\psi, $$ 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 $(*)$ (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 $(*)$ 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 $\le3$; 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.