Instantons in a Lagrangian model of turbulence

TL;DR

This paper explores the application of instanton theory to a Lagrangian turbulence model, using path integrals and numerical methods to analyze velocity gradient statistics and their scaling behavior.

Contribution

It introduces a novel application of instantons to the Recent Fluid Deformation model, combining analytical and numerical techniques to study turbulence statistics.

Findings

Instantons lie on the Vieillefosse line in the RQ-plane.

Analytical pdf evaluation for longitudinal velocity gradients.

Scaling predictions for moments as a function of Reynolds number.

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 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 on the Vieillefosse line concerning the RQ-plane. We illustrate how instantons can be unveiled in the stochastic dynamics performing a conditional statistics.