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 $\omega(x,y,t)=\wb(y,t)+\wt(x,y,t)$, one has $|\wt|_{H^s}^2\le\beta^{-1} M_s(\...)$ as $t\to\infty$. We use this show that, for sufficiently large $\beta$, the global attractor of this system reduces to a point.