The capability of renormalization group methods (RNG) and Lagrangian renormalized perturbation theories (RPT) to reproduce a renormalized theory of eddy diffusivity for turbulent transport diffusion is discussed in the context of a simplified model with an exact renormalization theory that has recently been developed by the authors. The model problem involves transport diffusion by simple shear flows with turbulent velocity statistics, infrared divergences, and no separation of scales; the exact renormalization theory exhibits a remarkable range of different phenomena as parameters defining the velocity statistics are varied with four distinct regions requiring renormalization so that the model is a stringent test for approximate theories of eddy diffusivity via either RNG or RPT methods. Despite the different philosophy in RNG and RPT methods, all of these different approximations collapse to give the exact theory of eddy diffusivity for one region in the model with infrared divergence that is adjacent to the Kolmogorov value. The RNG methods are very flexible but do not give the exact anomalous scaling exponents for the other three regions with infrared divergence as expected with an ε-expansion procedure. The Lagrangian RPT methods always yield the correct scaling exponents but a much more elaborate analysis of the explicit structure of the model problem is needed to achieve this. In other regions of renormalization, including the Kolmogorov value, the RPT methods predict nonlocal equations for eddy diffusivity while the exact renormalization theory involves local diffusion equations with time-dependent diffusivity; these nonlocal equations are a poor approximation for the actual renormalized dynamics and the Lagrangian direct interaction approximation (DIA) only slightly improves the behavior over the Lagrangian first-order smoothing approximation. On the other hand; RNG methods alway predict a simple local diffusivity in the model and there are regions of renormalization where the rigorous theory for eddy diffusivity is nonlocal.
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