Loop Equation and Area Law in Turbulence

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 $P_C(Γ)$ which allows the scaling solutions in inertial range of spatial scales. The pdf decays as exponential of some power of $ Γ^3/A^2 $ where $A$ is the minimal area inside the loop.