Large deviations principle for the invariant measures of the 2D stochastic Navier-Stokes equations with vanishing noise correlation

We study the two-dimensional incompressible Navier-Stokes equation on the torus, driven by Gaussian noise that is white in time and colored in space. We consider the case where the magnitude of the random forcing $\sqrt{\e}$ and its correlation scale $\delta(\e)$ are both small. We 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.