Eulerian approach, turbulent diffusion

The Eulerian approach to turbulent diffusion was shown to lead to the atmospheric diffusion equation (2.19) ... 

The Eulerian (bottom-up) approach is to start with the convective-diffusion equation and through Reynolds averaging, obtain time-smoothed transport equations that describe micromixing effectively. Several schemes have been proposed to close the two terms in the time-smoothed equations, namely, scalar turbulent flux in reactive mixing, and the mean reaction rate (Bourne and Toor, 1977 Brodkey and Lewalle, 1985 Dutta and Tarbell, 1989 Fox, 1992 Li and Toor, 1986). However, numerical solution of the three-dimensional transport equations for reacting flows using CFD codes are prohibitive in terms of the numerical effort required, especially for the case of multiple reactions with... 

Having demonstrated that exact solution for the mean concentrations (c, (x, t j) even of inert species in a turbulent fluid is not possible in general by either the Eulerian or Lagrangian approaches, we now consider what assumptions and approximations can be invoked to obtain practical descriptions of atmospheric diffusion. In Section 18.4 we shall proceed from the two basic equations for (c,), (18.4) and (18.8), to obtain the equations commonly used for atmospheric diffusion. A particularly important aspect is the delineation of the assumptions and limitations inherent in each description. 

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