Circulation Statistics in Three-Dimensional Turbulent Flows

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 $\mathcal{O}(1/{\lambda }^4)$ subleading correction and the asymmetry between right and left PDF tails due to parity breaking mechanisms are also investigated.

Computations are discussed in a more detailed way; accepted for publication in Physical Review E

References in corpus (4)

• Turbulence without pressure
• Instantons in Burgers Equation
• Instantons and Intermittency
• Loop Equation and Area Law in Turbulence

Cited by in corpus (3)

• Eddy-Viscous Modeling and the Topology of Extreme Circulation Events in Three-Dimensional Turbulence
• Circulation in turbulent flow through a contraction
• Circulation-Strain Sum Rule in Stochastic Magnetohydrodynamics