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

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