  We are dealing with the validity of a large deviation principle for the
two-dimensional Navier-Stokes equation, with periodic boundary conditions,
perturbed by a Gaussian random forcing. We are here interested in the regime
where both the strength of the noise and its correlation are vanishing, on a
length scale $\e$ and $\d(\e)$, respectively, with $0<\e,\ \d(\e)<<1$.
Depending on the relationship between $\e$ and $\d(\e)$ we will prove the
validity of the large deviation principle in different functional spaces.
