For the direct cascade of steady two-dimensional (2D) Navier-Stokes turbulence, we derive analytically the probability of strong vorticity fluctuations. When $\varpi$ is the vorticity coarse-grained over a scale $R$, the probability density function (PDF), $\mathcal{P}(\varpi )$, has a universal asymptotic behavior $\ln \mathcal{P}~-\varpi /{\varpi }_{\mathrm{rms}}$ at $\varpi \gg {\varpi }_{\mathrm{rms}}=[H\ln (L/R)]{}^{1/3}$, 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, $\varpi /{\varpi }_{\mathrm{rms}}$, in distinction from other known direct cascades.

• Received 17 November 2010

DOI:https://doi.org/10.1103/PhysRevE.83.045301