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The formalisms of Wyld [2] and Martin, Siggia, and Rose (MSR) [3] address theclosure problem of a statistical treatment of homogeneous isotropic turbulence(HIT) based on techniques primarily developed for quantum field theory. In theWyld 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 thiscorrection in a more natural way from the basic equations of the formalism.This leads to what we call the "Improved Wyld-Lee Renormalized PerturbationTheory". MSR had noted that their formalism had more vertex functions thanWyld's formalism and based on this felt Wyld's formalism was incorrect. Howevera careful comparison of both formalisms here shows that the Wyld formalismfollows a different procedure to that of the MSR formalism and so the treatmentof vertex corrections appears in different ways in the two formalisms. Takingthat into account, along with clarifications made to both formalisms, we findthat they are equivalent and we demonstrate this up to fourth order.

Eulerian Field-Theoretic Closure Formalisms for Fluid Turbulence