Statistical Estimates for 2D stochastic Navier-Stokes Equations

The statistical features of homogeneous, isotropic, two-dimensional stochastic turbulence are discussed. We derive some rigorous bounds for the mean value of the bulk energy dissipation rate $\mathbb{E}[\epsilon ]$ and enstrophy dissipation rates $\mathbb{E}[\chi ]$ for 2D flows sustained by a variety of stochastic driving forces. We show that $$\mathbb{E}[\epsilon ]\to 0and\mathbb{E}[\chi ]\lesssim \mathcal{O}(1)$$ in the inviscid limit, consistent with the dual-cascade in 2D turbulence.