Circulation Statistics in Three-Dimensional Turbulent Flows

We study the large λ limit of the loop-dependent characteristic functional Z(λ) =< exp(iλ ∮ c ~v ·d~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 σzz 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 behaviour Z(λ) ∼ 1/λ2. The O(1/λ4) subleading correction and the asymmetry between right and left PDF tails due to parity breaking mechanisms are also investigated. PACS numbers: 47.27.Gs, 11.25.Hf Typeset using REVTEX 1