Area rule of velocity circulation in two-dimensional instability-driven turbulence beyond the inertial range

Abstract

The velocity statistics reveal non-universality in both three-dimensional (3-D) and two-dimensional (2-D) turbulence, despite both prototype systems containing an energy inertial range with constant energy flux. Recently, statistics of scale-dependent velocity circulation exhibit universal bifractal behavior in 2-D and 3-D hydrodynamic turbulence and quantum turbulence, where the circulation scale is defined as the square root of the minimum area enclosed by the loop. This loop-shape independent definition of scale bases on the area rule of circulation first proposed by Migdal: the probability density function (PDF) of circulation is only a function of the minimal surface area enclosed by the loop but not the shape of the loop. This paper demonstrates that the derivation of the circulation area rule can be generalized to all scales in 2-D instability-driven turbulence, not limited to the inertial range. However, the area rule is not the only solution to the loop equation, so it may not be observed. Another necessary condition for the validity of the area rule is that the second-order momentum of circulation is loop-shape independent. By deriving the relationship between the second-order moment of circulation on a rectangular loop and the energy spectrum, we prove that the area rule cannot be satisfied in the classic inertial-range turbulence with -5/3 or -3 spectral scalings. As in the 3-D case, the second-order moment of circulation is size-dependent. Compared with the circulation PDFs, the PDFs normalized by the second-order moment of circulation exhibit significantly weaker dependence on loop shape.