Vortex Gas Model of Turbulent Circulation Statistics and Truncated Scalar Fields*

Statistical properties of circulation — a fluid dynamic observable which shares mathematical properties with the Wilson Loop operator of gauge field theories — encode relevant information about the multi-scale structure of turbulent cascades. Recent massive computational efforts have brought to light the dependence of circulation moments upon Reynolds numbers and length scales, besides the specific shape of heavy-tailed circulation probability distribution functions. We address these focal points in an investigation of circulation statistics for planar cuts of three-dimensional flows. We borrow ideas from the structural approach to turbulence, whereby turbulent flows are depicted as dilute vortex gases, combined with the standard Obukhov-Kolmogorov phenomenological framework of small-scale intermittency.  A refinement of our model takes us to the phenomenon of multifractality breaking, which we propose to describe with the help of truncated two-dimensional scalar fields. It is possible to reproduce, along these lines, key statistical features of circulation, in close agreement with empirical observations compiled from direct numerical simulations.