Atmospheric Diffusion 1 Eulerian Approach

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

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. 

Note the similarity of (18.32) and (18.28). In fact, if we define a = 2Kxxt, = 2Kyyt, and cj = 2Kzzt, we note that the two expressions are identical. There is, we conclude, evidently a connection between the Eulerian and Lagrangian approaches embodied in a relation between the variances of spread that arise in a Gaussian distribution and the eddy diffusivities in the atmospheric diffusion equation. We will explore this relationship further as we proceed. 

Three-dimensional, time-dependent methods (25, 26) have been recently proposed, but results for reactive atmospheres have not been reported at this writing. Simplified chemistry must be used in each of these approaches because of the emphasis on details of advection and diffusion. The body of data for most aff basins falls short of the input requirements for any transport formulation of this complexity. In some cases it may be difficult to avoid the problem of allowing too many unspecified parameters to obscure the physically based portions of the calculation. One new method uses an Eulerian coordinate frame (La-grangian coordinates refer to a fiuid mass which is followed in time and space in contrast with an Eulerian frame which has fluid moving relative... 

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