Modeling error of $ \alpha $-models of turbulence on a two-dimensional torus

This paper is devoted to study the rate of convergence weak solutions \begin{document}$ {\bf u}_\alpha $\end{document} id="M3">\begin{document}$ \alpha $\end{document}-regularization models solution id="M4">\begin{document}$ u} Navier-Stokes equations in two-dimensional periodic case, as regularization parameter id="M5">\begin{document}$ goes zero. More specifically, we will consider Leray-\begin{document}$ $\end{document}, simplified Bardina, and modified id="M7">\begin{document}$ models. Our aim improve known results terms rates also show estimates valid over long-time intervals. The hold case bounded domain with homogeneous Dirichlet boundary conditions.