  In this paper we consider the Lagrangian Averaged Navier-Stokes Equations,
also known as, LANS-$\alpha$ Navier-Stokes model on the two dimensional torus.
We assume that the noise is a cylindrical Wiener process and its coefficient is
multiplied by $\sqrt{\alpha}$. We then study through the lenses of the large
and moderate deviations principle the behaviour of the trajectories of the
solutions of the stochastic system as $\alpha$ goes to 0. Instead of giving two
separate proofs of the two deviations principles we present a unifying approach
to the proof of the LDP and MDP and express the rate function in term of the
unique solution of the Navier-Stokes equations. Our proof is based on the weak
convergence approach to large deviations principle. As a by-product of our
analysis we also prove that the solutions of the stochastic LANS-$\alpha$ model
converge in probability to the solutions of the deterministic Navier-Stokes
equations.
