We consider the vorticity form of the 2D Euler equations which is perturbed by a suitable transport type noise and has white noise initial condition. It is shown that stationary solutions of this equation converge to the unique stationary solution of the 2D Navier–Stokes equation driven by the space-time white noise.
"Convergence of transport noise to Ornstein–Uhlenbeck for 2D Euler equations under the enstrophy measure." Ann. Probab. 48 (1) 264 - 295, January 2020. https://doi.org/10.1214/19-AOP1360