Ergodicity of the Finite Dimensional Approximation of the 3D Navier–Stokes Equations Forced by a Degenerate Noise

We prove ergodicity of the finite dimensional approximations of the three dimensional Navier–Stokes equations, driven by a random force. The forcing noise acts only on a few modes and some algebraic conditions on the forced modes are found that imply the ergodicity. The convergence rate to the unique invariant measure is shown to be exponential.