The p.d.f. of the velocity gradients in two-dimensional decaying isotropic turbulence is shown to approach a Cauchy distribution, with algebraic s−2 tails, as the flow becomes dominated by a large number of compact coherent vortices. The statistical argument is independent of the vortex structure, and depends only on general scaling properties. The same argument predicts a Gaussian p.d.f. for the velocity components. The convergence to these limits as a function of the number of vortices is analysed. It is found to be fast in the former case, but slow (logarithmic) in the latter, resulting in residual u−3 tails in all practical cases. The influence of a spread Gaussian vorticity distribution in the cores is estimated, and the relevant dimensionless parameter is identified as the area fraction covered by the cores. A comparison is made with the result of numerical simulations of two-dimensional decaying turbulence. The agreement of the p.d.f.s is excellent in the case of the gradients, and adequate in the case of the velocities. In the latter case the ratio between energy and enstrophy is computed, and agrees with the simulations. All the one-point statistics considered in this paper are consistent with a random arrangement of the vortex cores, with no evidence of energy screening.
Two-dimensional magnetohydrodynamic turbulence with large and small energy-injection length scales
