Eulerian Field-Theoretic Closure Formalisms for Fluid Turbulence

TL;DR

This paper compares and clarifies two field-theoretic formalisms for modeling homogeneous isotropic turbulence, resolving previous inconsistencies and demonstrating their equivalence up to fourth order.

Contribution

It introduces an improved Wyld-Lee renormalized perturbation theory and clarifies the relationship between Wyld and MSR formalisms for turbulence.

Findings

The Wyld and MSR formalisms are equivalent up to fourth order.

A natural correction to the double-counting problem in Wyld's formalism is proposed.

Clarifications show different procedures but consistent results between the two formalisms.

Abstract

The formalisms of Wyld [2] and Martin, Siggia, and Rose (MSR) [3] address the closure problem of a statistical treatment of homogeneous isotropic turbulence (HIT) based on techniques primarily developed for quantum field theory. In the Wyld formalism, there is a well-known double-counting problem, for which an {\it ad hoc} solution was suggested by Lee [44]. We show how to implement this correction in a more natural way from the basic equations of the formalism. This leads to what we call the "Improved Wyld-Lee Renormalized Perturbation Theory". MSR had noted that their formalism had more vertex functions than Wyld's formalism and based on this felt Wyld's formalism was incorrect. However a careful comparison of both formalisms here shows that the Wyld formalism follows a different procedure to that of the MSR formalism and so the treatment of vertex corrections appears in different ways in the two formalisms. Taking that into account, along with clarifications made to both formalisms, we find that they are equivalent and we demonstrate this up to fourth order.