Mean-field approximation and a small parameter in turbulence theory

Abstract

Numerical and physical experiments on two-dimensional (2D) turbulence show that the differences of transverse components of velocity field are well described by Gaussian statistics and Kolmogorov scaling exponents. In this case the dissipation fluctuations are irrelevant in the limit of small viscosity. In general, one can assume the existence of a critical space dimensionality d=d(c), at which the energy flux and all odd-order moments of velocity difference change sign and the dissipation fluctuations become dynamically unimportant. At d<d(c) the flow can be described by the "mean-field theory," leading to the observed Gaussian statistics and Kolmogorov scaling of transverse velocity differences. It is shown that in the vicinity of d=d(c) the ratio of the relaxation and translation characteristic times decreases to zero, thus giving rise to a small parameter of the theory. The expressions for pressure and dissipation contributions to the exact equation for the generating function of transverse velocity differences are derived in the vicinity of d=d(c). The resulting equation describes experimental data on two-dimensional turbulence and demonstrates the onset of intermittency as d-d(c)>0 and r/L-->0 in three-dimensional flows in close agreement with experimental data. In addition, some exact relations between correlation functions of velocity differences are derived. It is also predicted that the single-point probability density function of transverse velocity components in developing as well as in the large-scale stabilized two-dimensional turbulence is a Gaussian.