  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
$\alpha\rightarrow 0^+$. 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 $\alpha\rightarrow 0^+$.

Analysis of PDEs