Attractor Dimensions of Three-Dimensional Navier-Stokes-$\alpha$ Model for Fast Rotating Fluids on Generic-Period Domains: Comparison with Navier-Stokes Equations

The three-dimensional Navier-Stokes-$\alpha$ model for fast rotating geophysical fluids is considered. The Navier-Stokes-$\alpha$ model is a nonlinear dispersive regularization of the exact Navier-Stokes equations obtained by Lagrangian averaging and tend to the Navier-Stokes equations as $\alpha \to 0^{+}$. We estimate upper bounds for the dimensions of global attractors and study the dependence of the dimensions on the parameter $\alpha$. All the estimates are uniform in $\alpha$, and our estimate of attractor dimensions remain finite when $\alpha \to 0^{+}$.