Two-Dimensional turbulence in the inverse cascade range

Abstract

Numerical and physical experiments on the forced two-dimensional Navier-Stokes equations show that transverse velocity differences are described by ``normal'' Kolmogorov scaling $<(Δv)^{2n}> \propto r^{2n/3}$ and obey a gaussian statistics. Since the non-trivial scaling is a sign of a strong non-linearity of the problem, these two results seem to contradict each other. A theory, explaining this result is presented in this paper. Strong time-dependence of the large-scale features of the flow ($\bar{u^2}\propto t$) results in decoupling of the large-scale dynamics from statistically steady-state small-scale random processes. This time-dependence is also a reason for the localness of the pressure-gradient terms in the equations governing the small-scale velocity difference PDF's. The derived self-consistent expression for the pressure gradient contributions lead to the conclusion that the small-scale transverse velocity differences are governed by a linear Langevin-like equation, strirred by a non-local, universal, solution-depending gaussian random force. This explains the experimentally observed gaussian statistics of transverse velocity differences and their Kolmogorov scaling. The solution for the PDF of longitudinal velocity differences is based on a smallness of the energy flux in two-dimensional turbulence. The theory makes a few quantitative predictions which can be tested experimentally.