On the Probability Distribution of Velocity Circulation in Three-Dimensional Turbulence

The probability distribution functions of the circulation of velocity in three-dimensional decaying isotropic turbulence are examined by the database of the numerical simulation based on the pseudospectral method. It is shown that the standard deviation increases nearly as $A^{2/3}$ where $A$ is the area of the loop. The PDFs change from exponential to gaussian as the size becomes large. The former agrees well with the prediction by Migdal (1993), although the latter does not match. A modification is proposed to explain the gaussian distribution with the $A^{2/3}$ dependence.