  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 time
and degenerate in Fourier space. The main result is twofold.
  Firstly, we establish a Ruelle-Perron-Frobenius type theorem for the time
inhomogeneous Feynman-Kac evolution operators with regular potentials
associated with the stochastic Navier-Stokes system. The theorem characterizes
asymptotic behaviors of the Feynman-Kac operators in terms of the periodic
family of principal eigenvalues and corresponding unique eigenvectors. The
proof involves a time inhomogeneous version of Ruelle's lower bound technique.
  Secondly, utilizing this Ruelle-Perron-Frobenius type theorem and a Kifer's
criterion, we establish a Donsker-Varadhan type large deviation principle with
a nontrivial good rate function for the occupation measures of the time
inhomogeneous solution processes.
