Instantons in a Lagrangian model of turbulence

Abstract

The role of instantons is investigated in the Lagrangian model for the velocity gradient evolution known as the recent fluid deformation approximation. After recasting the model into the path-integral formalism, the probability distribution function (pdf) is computed along with the most probable path in the weak noise limit through the saddle-point approximation. Evaluation of the instanton solution is implemented numerically by means of the iteratively Chernykh–Stepanov method. In the case of the longitudinal velocity gradient statistics, due to symmetry reasons, the number of degrees of freedom can be reduced to one, allowing the pdf to be evaluated analytically as well, thereby enabling a prediction of the scaling of the moments as a function of Reynolds number. It is also shown that the instanton solution lies in the Vieillefosse line concerning the RQ-plane. We illustrate how instantons can be unveiled in the stochastic dynamics performing a conditional statistics.