Determining modes and nodes of the rotating Navier-Stokes equations

Abstract

We analyse the long-term dynamics of the two-dimensional Navier–Stokes equations on a rotating sphere and the periodic -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 , 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 , where is the rotation rate and depends on up to third derivatives of the forcing. Our bound on the number of determining nodes is slightly weaker, at .