  We consider the Kolmogorov operator $K$ associated to a stochastic
Navier-Stokes equation driven by space-time white noise on the two-dimensional
torus with periodic boundary conditions and a rotating reference frame,
introducing fictitious forces such as the Coriolis force. This equation then
serves as a simple model for geophysical flows. We prove that the Gaussian
measure induced by the enstrophy is infinitesimally invariant for $K$ on
finitely based cylindrical test functions and moreover $K$ is $L^1$-unique w.
r. t. the enstrophy measure for sufficiently large viscosity.
