The velocity statistics reveal nonuniversality in both three-dimensional (3D) and two-dimensional (2D) 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 2D and 3D hydrodynamic turbulence and quantum turbulence, where the circulation scale is defined as the square root of the minimum area enclosed by the loop. This definition of scale that is independent of loop shape is based 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 2D instability-driven turbulence without external forcing, 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 3D case [Iyer et al., Proc. Natl. Acad. Sci. USA 118, e2114679118 (2013)], 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.

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• Homogeneous turbulence
• Isotropic turbulence
• Turbulence
• Two-dimensional turbulence