Sharp estimates for the number of degrees of freedom for the   damped-driven 2D Navier--Stokes equations
  We derive upper bounds for the number of asymptotic degrees (determining modes and nodes) of freedom for the two-dimensional Navier--Stokes system and Navier-Stokes system with damping. In the first case we obtain the previously known estimates in an explicit form, which are larger than the fractal dimension of the global attractor. However, for the Navier--Stokes system with damping our estimates for the number of the determining modes and nodes are comparable to the sharp estimates for the fractal dimension of the global attractor. Our investigation of the damped-driven 2D Navier--Stokes system is inspired by the Stommel--Charney barotropic model of ocean circulation where the damping represents the Rayleigh friction. We remark that our results equally apply to the Stommel--Charney model. 
Alexei A. Ilyin and Edriss S. Titi