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The three-dimensional Navier-Stokes-$\alpha$ model for fast rotatinggeophysical fluids is considered. The Navier-Stokes-$\alpha$ model is anonlinear dispersive regularization of the exact Navier-Stokes equationsobtained by Lagrangian averaging and tend to the Navier-Stokes equations as$\alpha\rightarrow 0^+$. We estimate upper bounds for the dimensions of globalattractors and study the dependence of the dimensions on the parameter$\alpha$. All the estimates are uniform in $\alpha$, and our estimate ofattractor dimensions remain finite when $\alpha\rightarrow 0^+$.

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