Determining modes and nodes of the rotating Navier-Stokes equations

We analyse the long-term dynamics of the two-dimensional Navier-Stokes equations on a rotating sphere and the periodic $\beta$-plane, which can be considered as a planar approximation to the former. It was shown over fifty years ago that the Navier-Stokes equations can be described by a finite number of degrees of freedom, which can be quantified by, for example, the so-called determining modes and determining nodes. After considerable effort, it was shown that, independently of rotation, the number of determining modes and nodes both scale as the Grashof number $\mathcal{G}$, a non-dimensional parameter proportional to the forcing. Using and extending recent results on the behaviour of the rotating Navier-Stokes equations, we prove under reasonable hypotheses that the number of determining modes is bounded by $c\mathcal{G}^{1/2}+ \epsilon^{1/2}M$, where $1/\epsilon$ is the rotation rate and $M$ depends on up to third derivatives of the forcing. Our bound on the number of determining nodes is slightly weaker, at $c \mathcal{G}^{2/3}+ \epsilon^{1/2}M$.