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

TL;DR

This paper derives bounds on the number of determining modes and nodes for the 2D Navier--Stokes equations on a rotating $eta$-plane, showing fewer degrees of freedom with strong rotation.

Contribution

It provides new bounds on the complexity of the flow on a $eta$-plane, incorporating the effects of variable Coriolis parameter and rotation strength.

Findings

Number of determining modes scales as $cG_0^{1/2} + c'(M/eta)^{1/2}$.

Number of determining nodes scales as $cG_0^{2/3} + c'(M/eta)^{1/2}$.

Fewer degrees of freedom for large $eta$ (strong rotation) compared to classical bounds.

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