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

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 $cG_0^{1/2} + c'(M/\beta)^{1/2}$ and $cG_0^{2/3} + c'(M/\beta)^{1/2}$ 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 $cG_0$.