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I will present a series of asymptotic results for the two-dimensional incompressible Navier-Stokes equation, driven by a Gaussian noise that is white in time and colored in space. I will consider the case when the magnitude of the random forcing $\sqrt{\e}$ and its correlation scale $\delta(\e)$ are both small. I will prove a large deviations principle for the solutions, as well as for the family of invariant measures, as $\e$ and $\delta(\e)$ are simultaneously sent to $0$, under a suitable scaling. I will also mention some results on the limiting behavior of the associated quasi-potential and on the exit problem.