Eulerian field-theoretic closure formalisms for fluid turbulence

The formalisms of Wyld [Ann. Phys. 14, 143 (1961)] and Martin, Siggia, and Rose (MSR) [Phys. Rev. A 8, 423 (1973)] 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 ad hoc solution was suggested by Lee [Ann. Phys. 32, 292 (1965)]. 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.