Atmospheric diffusion equation

To arrive at the atmospheric diffusion equation (3.17), a number of constraints must be imposed (Seinfeld, 1975). The deviation between reality and these constraints will thus represent a measure of the inaccuracies associated with the atmospheric diffusion equation. Among these constraints, the following two are most pertinent  

According to measurements made in the atmosphere, the Lagrangian time scale is of the order of 100 sec (Csanady, 1973). Using a characteristic particle velocity of 5 m sec , the above conditions are 100 sec and L 500 m. Since one primary concern is to examine diffusion from point sources such as industrial stacks, which are generally characterized by small T and L, it is apparent that either one (but particularly the second one) or both of the above constraints cannot be satisfied, at least locally, in the vicinity of the point-like source. Therefore, in these situations, it is important to assess the error incurred by the use of the atmospheric diffusion equation. 

This problem has been addressed by Corrsin (1974) for several simple hypothetical cases. Many qualitative and, in some instances, semiquanti- 

for the case of a turbulent boundary layer, although the stationarity condition is fairly well satisfied, the homogeneity condition is probably satisfied only for z/8 0.2. 

This analysis can also be extended to the dispersion of point source emissions in the atmosphere if we assume that the mean concentration gradient is expressed by Cj odz, where a is a constant that is proportional to the emission rate. It then can be shown that 

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D. Solutions of the Steady-State Atmospheric Diffusion Equation. 286... 

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

We will now show that Eq. (4.31) may be obtained by solving the atmospheric diffusion equation in which diffusion in the direction of the mean flow is neglected relative to advection ... 

The object of this section is to derive the Gaussian equations of the previous section as solutions to the atmospheric diffusion equation. Such a relationship has already been demonstrated in Section IV,B for the case of no boundaries. We extend that consideration now to boundaries. We recall that constant eddy diffusivities were assumed in Section IV,B. 

Assuming a uniform wind speed u in the x direction, we begin with the unsteady atmospheric diffusion equation with constant eddy diffusivities ... 

The approach described above is by no means complete or exclusive. For example, Lamb et al. (1975) have proposed an alternative route to assess the adequacy of the atmospheric diffusion equation. Their approach is based on the Lagrangian description of the statistical properties of nonreacting particles released in a turbulent atmosphere. By employing the boundary layer model of Deardorff (1970), the transition probability density p x, y, z, t x, y, z, t ) is determined from the statistics of particles released into the computed flow field. Once p has been obtained, Eq. (3.1) can then be used to derive an estimate of the mean concentration field. Finally, the validity of the atmospheric diffusion equation is assessed by determining the profile of vertical dififiisivity that produced the best fit of the predicted mean concentration field. 

In summary, we conclude that the application of the atmospheric diffusion equation to point sources will introduce an error of the order of 10% into predictions at points reasonably well removed (on the order of 1 km) from the source. 

The Gaussian expressions are not expected to be valid descriptions of turbulent diffusion close to the surface because of spatial inhomogeneities in the mean wind and the turbulence. To deal with diffusion in layers near the surface, recourse is generally had to the atmospheric diffusion equation, in which, as we have noted, the key problem is proper specification of the spatial dependence of the mean velocity and eddy difiusivities. Under steady-state conditions, turbulent diffusion in the direction of the mean wind is usually neglected (the slender-plume approximation), and if the wind direction coincides with the x axis, then = 0. Thus, it is necessary to specify only the lateral (Kyy) and vertical coefficients. It is generally assumed that horizontal homogeneity exists so that u, Kyy, and Ka are independent of y. Hence, Eq. (2.19) becomes... 

Sklarew, R. C., A. J. Fabrick, and J. E. Prager. A Particle-in-Oll Method for Numerical Solution of the Atmospheric Diffusion Equation, and Applications to Air Pollution Problems. Final Report 3SR-844. Vol. 1. (Prepared for the Environmental Protection Agency under Contract 68-02-(X)06) La Jolla, Calif. Systems, Science and Software, 1971. 173 pp. 

McRae J.G., W. R. Goodin, and J. H. Seinfeld (1982) Numerical solution of atmospheric diffusion equation for chemically reactings flows, J. Comput. Phys., 45. 

This equation is termed the semiempirical equation of atmospheric diffusion, or just the atmospheric diffusion equation, and will play an important role in what is to follow. 

SOLUTION OF THE ATMOSPHERIC DIFFUSION EQUATION FOR AN INSTANTANEOUS SOURCE... 

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. 

Before we proceed to solve (18.56), a few comments about the boundary conditions are useful. When the x diffusion term is dropped in the atmospheric diffusion equation, the equation becomes first-order in x, and the natural point for the single boundary condition on x is at x = 0. Since the source is also at x = 0 we have an option of whether to place the source on the RHS of the equation, as in (18.56), or in the x = 0 boundary condition. If we follow the latter course, then the x = 0 boundary condition is obtained by equating material fluxes across the plane at x = 0. The result is... 

Now, if we return to Section 18.5 we see that this Ku is precisely the constant eddy diffusivity used in the atmospheric diffusion equation for stationary, homogeneous turbulence. Since the Lagrangian timescale is defined by... 

Derivation of the Gaussian Plume Equation as a Solution of the Atmospheric Diffusion Equation... 

We saw that by assuming constant eddy diffusivities Kxx. Kyy, and K77, the solution of the atmospheric diffusion equation has a Gaussian form. Thus it should be possible to obtain (18.88) or (18.89) as a solution of an appropriate form of the atmospheric diffusion equation. More importantly, because of the ease in specifying different physical situations in the boundary conditions for the atmospheric diffusion equation, we want to include those situations that we were unable to handle easily in Section 18.9.1, namely, the existence of an inversion layer at height H and partial absorption at the surface. Readers not concerned with the details of this solution may skip directly to Section 18.9.3. 

Let us begin with the atmospheric diffusion equation with eddy diffusivities that are functions of time ... 

While the Gaussian equations have been widely used for atmospheric diffusion calculations, the lack of ability to include changes in windspeed with height and nonlinear chemical reactions limits the situations in which they may be used. The atmospheric diffusion equation provides a more general approach to atmospheric diffusion calculations than do the Gaussian models, since the Gaussian models have been shown to be special cases of that equation when the windspeed is uniform and the eddy diffusivities are constant. The atmospheric diffusion equation in the absence of chemical reaction is... 

SOLUTIONS OF THE STEADY-STATE ATMOSPHERIC DIFFUSION EQUATION 873... 

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