Navier-Stokes equations on the $β$-plane

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

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