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

TL;DR

This paper estimates the upper bounds of the global attractor dimensions for the three-dimensional Navier-Stokes-$\

Contribution

It provides uniform estimates for attractor dimensions of the Navier-Stokes-$\alpha$ model that remain finite as the regularization parameter approaches zero.

Findings

Upper bounds for attractor dimensions are established.

Estimates are uniform in the regularization parameter.

Attractor dimensions remain finite as $\\alpha \rightarrow 0^+$.

Abstract

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 $\alpha\rightarrow 0^+$.