Circulation Statistics in Three-Dimensional Turbulent Flows

TL;DR

This paper analyzes the statistical behavior of circulation in 3D turbulence using field theory and saddle-point methods, revealing asymptotic behaviors and tail asymmetries in the circulation PDF.

Contribution

It introduces a field-theoretic saddle-point approach to turbulence, deriving instanton solutions and analyzing their fluctuations for circulation statistics.

Findings

Asymptotic $Z(\lambda)$ behavior $ o 1/\lambda^2$

Identified subleading $1/\lambda^4$ corrections

Revealed PDF tail asymmetry due to parity breaking

Abstract

We study the large $\lambda$ limit of the loop-dependent characteristic functional $Z(\lambda)=<\exp(i\lambda \oint_c \vec v \cdot d \vec x)>$, related to the probability density function (PDF) of the circulation around a closed contour $c$. The analysis is carried out in the framework of the Martin-Siggia-Rose field theory formulation of the turbulence problem, by means of the saddle-point technique. Axisymmetric instantons, labelled by the component $\sigma_{zz}$ of the strain field -- a partially annealed variable in our formalism -- are obtained for a circular loop in the $xy$ plane, with radius defined in the inertial range. Fluctuations of the velocity field around the saddle-point solutions are relevant, leading to the lorentzian asymptotic behavior $Z(\lambda) \sim 1/{\lambda^2}$. The ${\cal O}(1 / {\lambda^4})$ subleading correction and the asymmetry between right and left PDF tails due to parity breaking mechanisms are also investigated.