278 ANNUAL REPORT SMITHSONIAN INSTITUTION, 195 6 



carries the smoke downstream, and (2) the turbulent velocity fluctua- 

 tions that disperse it in all directions. Figure 6 illustrates some shapes 

 of smoke plumes under the same atmospheric conditions. The nature 

 of atmospheric turbulence makes impossible a correct prediction of 

 any of the individual examples of smoke plumes. One can, however, 

 describe mathematically an average smoke plume and relate the mean 

 concentrations of smoke at each point of this plume to the statistical 

 characteristics of turbulence and to the mean wind velocity. The 

 equations of turbulent diffusion for a continuous point source, which 

 may be used to represent the emission from a smokestack, can be 

 written explicitly under some simplifying assumptions, particularly 

 when the mean wind velocity remains constant and the mass and size 

 of the dispersing particles can be neglected [4]. 



The theoretical equations give the mean concentration distribution 

 of dispersing particles as a function of the statistical characteristics 

 of turbulence and of the mean wind velocity. These equations can 

 be used to determine isoconcentration curves similar to those repre- 

 sented in figure 6 in the framed illustration. Each of these curves 

 is a locus of points at which the mean concentration is the same. 

 These theoretical curves should be compared with the mean concentra- 

 tions measured using a large number of individual smoke plumes 

 (similar to the 10 plumes illustrated in the figure) and observed under 

 the same general meteorological conditions. 



MATHEMATICAL MODEL 



A mathematical model of the atmosphere over an urban area can 

 be used to study the probable pollution patterns. One of the simplest 

 models can be constructed by including in the description of the model 

 the distribution of pollution sources, their emission conditions, and 

 the micrometeorological characteristics that directly affect the dis- 

 persion of pollutants. The mean concentration distribution of pol- 

 lutants due to each source of pollution can be determined, and the 

 effects of the several sources can be added. One can then find the 

 mean concentration pattern of pollution over the urban areas as a 

 function of the time. The relative contributions of each of the sources 

 of pollution to the contamination at various points of the area can 

 then be analyzed. Under some meteorological conditions there exists 

 a thermal inversion above the ground that confines the dispersion of 

 pollutants to the lower levels of the atmosphere. In our mathematical 

 model such an inversion will be represented by proper boundary con- 

 ditions under which the thermal inversion and the ground reflect the 

 dispersing particles. 



In the examples described earlier we have assumed that there exists 

 a well-defined mean wind velocity that is approximately constant in 



