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We consider the incompressible 2D Navier-Stokes equations on the torus,driven by a deterministic time periodic force and a noise that is white in timeand degenerate in Fourier space. The main result is twofold.  Firstly, we establish a Ruelle-Perron-Frobenius type theorem for the timeinhomogeneous Feynman-Kac evolution operators with regular potentialsassociated with the stochastic Navier-Stokes system. The theorem characterizesasymptotic behaviors of the Feynman-Kac operators in terms of the periodicfamily of principal eigenvalues and corresponding unique eigenvectors. Theproof involves a time inhomogeneous version of Ruelle's lower bound technique.  Secondly, utilizing this Ruelle-Perron-Frobenius type theorem and a Kifer'scriterion, we establish a Donsker-Varadhan type large deviation principle witha nontrivial good rate function for the occupation measures of the timeinhomogeneous solution processes.

Large deviations for 2D stochastic Navier-Stokes Equations driven by a periodic force and a degenerate noise