Loop Equation and Area Law in Turbulence

TL;DR

This paper reformulates incompressible fluid dynamics using loop equations, deriving a functional equation for circulation probability distribution that predicts exponential decay related to loop area, advancing understanding of turbulence.

Contribution

It introduces a loop-based reformulation of fluid dynamics and derives a functional equation for circulation distribution, providing new insights into turbulence scaling laws.

Findings

Derived explicit functional equation for circulation pdf

Predicted exponential decay of circulation pdf with area dependence

Connected loop area to turbulence scaling behavior

Abstract

This is the extended version of the preprint \ct{Loop}, based on the lectures given in Cargese Summer School and Chernogolovka Summer School in 93. The incompressible fluid dynamics is reformulated as dynamics of closed loops $C$ in coordinate space. We derive explicit functional equation for the pdf of the circulation $P_C(\Gamma)$ which allows the scaling solutions in inertial range of spatial scales. The pdf decays as exponential of some power of $ \Gamma^3/A^2 $ where $A$ is the minimal area inside the loop.