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Loosely speaking, the Navier-Stokes-$\alpha$ model and the Navier-Stokesequations differ by a spatial filtration parametrized by a scale denoted$\alpha$. Starting from a strong two-dimensional solution to theNavier-Stokes-$\alpha$ model driven by a multiplicative noise, we demonstratethat it generates a strong solution to the stochastic Navier-Stokes equationsunder the condition $\alpha$ goes to 0. The initially introduced probabilityspace and the Wiener process are maintained throughout the investigation,thanks to a local monotonicity property that abolishes the use of Skorokhod'stheorem. High spatial regularity a priori estimates for the fluid velocityvector field are carried out within periodic boundary conditions.

Convergence of the stochastic Navier-Stokes-$\alpha$ solutions toward the stochastic Navier-Stokes solutions