Application of an Approximate R-N-G Theory, to a Model for Turbulent Transport, with Exact Renormalization

  • Marco Avellaneda
  • Andrew J. Majda
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 55)


The important practical problem of “eddy diffusivity” and renormalization for turbulent transport is discussed. Then a simple model for turbulent transport with rigorous renormalization developed recently by the authors is described. The simple form of the model problem is deceptive; the renormalization theory for this problem exhibits a remarkable range of different phenomena as parameters in the velocity statistics are varied. Thus, the model problem is an interesting test problem for renormalized perturbation theories. In this paper the approximate R-N-G method of Yakhot and Orszag is applied to this exactly solvable model problem and developed in a detailed fashion. The predictions of the approximate R-N-G theory are compared with the exact renormalization theory. In one of the four different regions of nontrivial renormalization the R-N-G theory is exact both for predicting the anomalous scaling exponents and the Green’s function for the equation for eddy diffusivity. The reasons explaining this spectacular success are given in Section 4. For the other three regions, the anomalous scaling exponents and Green’s function predicted by R-N-G are a rather poor approximation to the actual renormalization theory. However, in the one region adjacent to the mean field boundary, the R-N-G theory predicts the first order Taylor expansion of the scaling exponent in agreement with the expected behavior of an ε-expansion procedure from critical phenomena.


Model Problem Eddy Diffusivity Velocity Statistic Turbulent Transport Infrared Divergence 
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Copyright information

© Springer-Verlag New York, Inc. 1993

Authors and Affiliations

  • Marco Avellaneda
    • 1
  • Andrew J. Majda
    • 2
  1. 1.Department of Mathematics, Courant InstituteNew York, UniversityNew YorkUSA
  2. 2.Department of Mathematics and Program in Applied and Computational MathematicsPrinceton UniversityPrincetonUSA

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