Strong Turbulence in d-Dimensions

Abstract

In the limit $d\to\infty$ the role of pressure gradients and that of the incompressibility constraint decreases, thus blurring the difference between transverse and longitudinal velocity correlation functions. Using Polyakov's expression for the dissipation anomaly the closed equation for the probability density function is obtained. This model for the dissipation terms is the only one satisfying both equations of motion and a set of dynamical constraints. The resulting equations show that when $d\to\infty$, the predictions of Kolmogorov theory are exact. It is also shown that the $O(1/d)$ pressure effects, producing the regularization of the equations of motion for the PDF are responsible for the distinction between the Navier-Stokes and Burgers dynamics.