Statistical Equilibrium of Circulating Fluids

Abstract

We are investigating the inviscid limit of the Navier-Stokes equation, and we find previously unknown anomalous terms in Hamiltonian, Dissipation, and Helicity, which survive this limit and define the turbulent statistics. We find various topologically nontrivial configurations of the confined Clebsch field responsible for vortex sheets and lines. In particular, a stable vortex sheet family is discovered, but its anomalous dissipation vanishes as $\sqrt{\nu }$. Topologically stable stationary singular flows, which we call Kelvinons, are introduced. They have a conserved velocity circulation ${\Gamma }_{\alpha }$ around the loop $C$ and another one ${\Gamma }_{\beta }$ for an infinitesimal closed loop $\tilde{C}$ encircling $C$, leading to a finite helicity. The anomalous dissipation has a finite limit, which we computed analytically. The Kelvinon is responsible for asymptotic PDF tails of velocity circulation, \textbf{perfectly matching numerical simulations}. The loop equation for circulation PDF as functional of the loop shape is derived and studied. This equation is \textbf{exactly} equivalent to the Schrödinger equation in loop space, with viscosity $\nu$ playing the role of Planck's constant. Kelvinons are fixed points of the loop equation at WKB limit $\nu \to 0$. The anomalous Hamiltonian for the Kelvinons contains a large parameter $\log \frac{|{\Gamma }_{\beta }|}{\nu }$. The leading powers of this parameter can be summed up, leading to familiar asymptotic freedom, like in QCD. In particular, the so-called multifractal scaling laws are, as in QCD, modified by the powers of the logarithm.

246 pages, 96 figures, and six appendixes. Submitted to Physics Reports. This version matches the final printed version of Phys. Rep. paper

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