Navier--Stokes equations on the $\beta$-plane: determining modes and nodes

Abstract

We revisit the 2d Navier--Stokes equations on the periodic $\beta$-plane, with the Coriolis parameter varying as $\beta y$, and obtain bounds on the number of determining modes and nodes of the flow. The number of modes {and nodes} scale as $c{G}_0^{1/2}+c^{\prime }(M/\beta {)}^{1/2}$ and $c{G}_0^{2/3}+c^{\prime }(M/\beta {)}^{1/2}$ respectively, where the Grashof number $G_0=|f_v{|}_{L^2}^{}/({\mu }^2{\kappa }_0^2)$ and $M$ involves higher derivatives of the forcing $f_v$. For large $\beta$ (strong rotation), this results in fewer degrees of freedom than the classical (non-rotating) bound that scales as $c{G}_0$.

15 pages