Statistical Estimates for 2D stochastic Navier-Stokes Equations

Abstract

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 $\mathbb{E}[\ddot{I}‡]$ for 2D flows sustained by a variety of stochastic driving forces. We show that $$\mathbb{E} [ \varepsilon ] \rightarrow 0 \hspace{0.5cm}\mbox{and} \hspace{0.5cm} \mathbb{E} [ χ] \lesssim \mathcal{O}(1)$$ in the inviscid limit, consistent with the dual-cascade in 2D turbulence.

15 pages, accepted in the Journal of Statistical Physics