Area Rule for velocity circulation*

The statistical theory of velocity circulation at high Reynolds numbers has witnessed renewed interest following recent studies, both empirical (Iyer et al., arXiv:1902.07326, 2019) and theoretical (Migdal, arXiv:1903.08613, 2019). A central tenet in the scaling theory of circulation is the Area Rule which states that the probability distribution of the circulation around closed contours, whose characteristic dimensions reside in the inertial range, depends solely on the minimal spanning surface of the contour. We examine the Area Rule for both low and high order circulation moments, for different contour shapes and sizes, using the DNS of the three-dimensional Navier-Stokes equations within a cube with periodic boundary conditions, with the integral-scale Reynolds number spanning over two decades. Useful comparisons with other fields such as the Gaussian random fields will be discussed to highlight the simplicity afforded by circulation in the statistical description of small-scale turbulence.