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

TL;DR

This paper demonstrates that the two-dimensional Navier-Stokes equations on a $eta$-plane tend to become nearly zonal over time, with the solution's non-zonal component diminishing as $eta$ increases, leading to a simplified long-term behavior.

Contribution

It proves that solutions of the 2D Navier-Stokes equations on the $eta$-plane become nearly zonal and that the global attractor reduces to a point for large $eta$, revealing long-term dynamics.

Findings

Solutions become nearly zonal as time progresses.

The non-zonal component diminishes with increasing $eta$.

The global attractor reduces to a point for sufficiently large $eta$.

Abstract

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.