A geometric cascade for the spectral approximation of the Navier-Stokes equations

We explain some ideas contained in some recent papers, concerning the statistical long time behaviour of the spectral approximation of the Navier-Stokes equations, driven by a highly degenerate white noise forcing. The analysis highlights that the ergodicity of the stochastic system is obtained by a geometric cascade. Such a cascade can be interpreted as the mathematical counterpart of the energy cascade, a well-known phenomenon in turbulence. In the second part of the paper, we analyse the results of some numerical simulations. Such simulations give a hint on the behaviour of the system in the case where the white noise forcing fails the assumptions of the main theorem.