Upper bounds on the dimension of the global attractor of the 2D Navier-Stokes equations on the $\beta -$plane

This article establishes estimates on the dimension of the global attractor of the two-dimensional rotating Navier-Stokes equation for viscous, incompressible fluids on the $\beta$-plane. Previous results in this setting by M.A.H. Al-Jaboori and D. Wirosoetisno (2011) had proved that the global attractor collapses to a single point that depends only the longitudinal coordinate, i.e., zonal flow, when the rotation is sufficiently fast. However, an explicit quantification of the complexity of the global attractor in terms of $\beta$ had remained open. In this paper, such estimates are established which are valid across a wide regime of rotation rates and are consistent with the dynamically degenerate regime previously identified. Additionally, a decomposition of solutions is established detailing the asymptotic behavior of the solutions in the limit of large rotation.