Area and Perimeter Rules of Velocity Circulation in Two-Dimensional Turbulence with Large-scale Absolute Equilibrium

Abstract

We demonstrate that the area rule of velocity circulation -- traditionally associated with the turbulent inertial range but shown not to be exact -- is strictly satisfied in the large-scale absolute equilibrium of two-dimensional (2D) homogeneous isotropic turbulence under enstrophy equipartition. We also derive a novel perimeter rule from the 2D inviscid loop equation, which posits that the probability distribution function (PDF) of velocity circulation depends solely on the loop perimeter rather than its area. This perimeter rule holds strictly in the large-scale absolute equilibrium characterized by energy equipartition. At the intermediate states determined by both enstrophy and energy, these two regimes are separated by a characteristic equilibrium scale $l_\text{eq}$: the area rule governs loop statistics when $l\ll l_\text{eq}$, while the perimeter rule emerges for $l\gg l_\text{eq}$. These statistical laws remain robust even for loops with extreme aspect ratios as low as $0.03$, a value that inertial-range studies never achieved. Our findings provide a new steady-state solution to the 2D loop equation and suggest additional solutions, paving the way for exploring previously undiscovered geometric invariants of turbulence.