  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.
