Summary: We consider the damped and driven two-dimensional Navier-Stokes equations at the limit of small viscosity coefficient \(\nu\to 0^+\). In particular, we obtain upper bounds of the order \(\nu^{-1}\) on the fractal and Hausdorff dimensions of the global attractor for the system on the torus \(T^2\), on the sphere \(S^2\) and in a bounded domain. Furthermore, in the case of the torus, we establish a lower bound of the order \(\nu^{-1}\).
This sharp estimate is remarkably smaller than the well established sharp bound for the dimension of the global attractor of the Navier-Stokes equations on the torus \(T^2\), which is of the order \(\nu^{-4/3}\). This means that the damping/friction term plays a significant role in reducing the number of degrees of freedom in this two-dimensional model. This, we believe, is done by dissipating the energy at the large spatial scales which is transferred to these scales via the inverse cascade mechanism. Finally, we remark that the system of equations studied here is related to the Stommel-Charney barotropic ocean circulation model of the gulf stream.