Vorticity statistics in the direct cascade of two-dimensional turbulence

Abstract

For the direct cascade of steady two-dimensional (2D) Navier-Stokes turbulence, we derive analytically the probability of strong vorticity fluctuations. When ϖ is the vorticity coarse-grained over a scale R, the probability density function (PDF), P(ϖ), has a universal asymptotic behavior lnP~-ϖ/ϖ(rms) at ϖ≫ϖ(rms)=[Hln(L/R)](1/3), where H is the enstrophy flux and L is the pumping length. Therefore, the PDF has exponential tails and is self-similar, that is, it can be presented as a function of a single argument, ϖ/ϖ(rms), in distinction from other known direct cascades.