Navier--Stokes equations on the $\beta$-plane

We show that, given a sufficiently regular forcing, the solution of the two-dimensional Navier--Stokes equations on the periodic $\beta$-plane (i.e. with the Coriolis force varying as $f_0+\beta y$) will become nearly zonal: with the vorticity $\omega(x,y,t)=\bar\omega(y,t)+\tilde\omega(x,y,t),$ one has $|\tilde\omega|_{H^s}^2 \le \beta^{-1} M_s(\cdots)$ as $t\to\infty$. We use this show that, for sufficiently large $\beta$, the global attractor of this system reduces to a point.