Statistical closures from the Martin-Siggia and Rose approach to turbulence

TL;DR

This paper develops statistical closures for turbulence within the Martin-Siggia-Rose (MSR) 2PI effective action framework. It derives a Bethe-Salpeter equation for the three-point velocity correlation $\langle v^l v^n v^m\rangle$ and shows that, at leading order, it closes to a form equivalent to the McComb–Yoffe LET closure under a near-Gaussian equal-time velocity pdf. The authors discuss strategies to go beyond this order via partial resummations and renormalization-group methods, and establish a fluctuation–response relation that underpins the equivalence with LET. Overall, the work validates the MSR functional approach for turbulence closures and links different closure schemes through causality and Gaussianity assumptions, with explicit guidance for systematic improvements.

Abstract

The goal of this paper is to study the statistical closures suggested by the Martin-Siggia and Rose approach to statistical turbulence. We find that the formalism leads to a Bethe-Salpeter equation for the three point correlation of the velocity field. In the leading order approximation this equation becomes an explicit expression. We discuss under which approximations this closure reduces to that proposed in W D McComb and S R Yoffe, A formal derivation of the local energy transfer (LET) theory of homogeneous turbulence, J. Phys. A: Math. Theor. 50, 375501 (2017). This suggests ways to improve upon this closure by dropping these restrictions, resumming the perturbative expansion and/or applying renormalization group techniques.