On the rate of convergence of the 2-D stochastic Leray-$\alpha$ model to the 2-D stochastic Navier-Stokes equations with multiplicative noise

In the present paper we study the convergence of the solution of the two dimensional (2-D) stochastic Leray-$\alpha$ model to the solution of the 2-D stochastic Navier-Stokes equations. We are mainly interested in the rate of convergence, as $\alpha$ tends to 0, of the error function which is the difference between the two solutions in an appropriate topology. We show that when properly localized the error function converges in mean square as $\alpha \to 0$ and the convergence is of order $O(\alpha )$. We also prove that the error converges in probability to zero with order at most $O(\alpha )$.