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 The IMA Volumes in Mathematics an its Applications, vol. 140