Circulation statistics in three-dimensional turbulent flows

We study the large $\ensuremath{\lambda}$ limit of the loop-dependent characteristic functional $Z(\ensuremath{\lambda})=〈\mathrm{exp}(i\ensuremath{\lambda}{\ensuremath{\oint}}_{c}\stackrel{\ensuremath{\rightarrow}}{v}\ensuremath{\cdot}d\stackrel{\ensuremath{\rightarrow}}{x})〉,$ 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, labeled by the component ${\ensuremath{\sigma}}_{\mathrm{zz}}$ of the strain field\char22{}a partially annealed variable in our formalism\char22{}are obtained for a circular loop in the $x\ensuremath{-}y$ 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 $Z(\ensuremath{\lambda})\ensuremath{\sim}1/{\ensuremath{\lambda}}^{2}.$ The $O(1/{\ensuremath{\lambda}}^{4})$ subleading correction and the asymmetry between right and left PDF tails due to parity breaking mechanisms are also investigated.