Slender plume approximation

Physically, this condition states that the spread of the individual puffs is small compared to the downwind distance. An alternative way of expressing this condition is that the rate of spreading of a puff in the direction of the mean flow is small compared to the rate of advection of the puff by the mean flow. This assumption is termed the slender-plume approximation. 

Summarizing, we have evaluated the general expression (4.3) in the special case in which the standard deviations of the puff distribution are the same in the three coordinate directions and in which the standard deviation is much smaller than the distance from the source at any x, the so-called slender-plume approximation. 

Equation (4.17) can be simplified somewhat for most situations of practical interest by the slender-plume approximation. In dealing with Eq. (4.17), this approximation implies that advection dominates plume dispersion so that only the concentrations close to the plume centerline are of importance. In the case of Eq. (4.17) we M e interested in values of x, y, and z that satisfy... 

Equation (4.22) is the expression for the mean concentration from a continuous point source of strength q at the origin in an infinite fluid when the standard deviations of plume spread are different in the different coordinate directions and when the slender-plume approximation is invoked. 

If we invoke the slender-plume approximation, we are interested only in the solution close to the plume centerline. Thus, as in Eq. (4.20), Eq. 

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... 

To illustrate the application of the Monte Carlo method, we consider the problem of simulating the dispersion of material emitted from a continuous line source located between the ground and an inversion layer. A similar case has been considered by Runca et al. (1981). We assume that the mean wind u is constant and that the slender-plume approximation holds. The line source is located at a height h between the ground (z = 0) and an inversion layer (z = Zi). If the ground is perfectly reflecting, the analytical expression for the mean concentration is found by integrating the last entry of Table II over y from -< to -Hoo. The result can be expressed as... 

Physically, the mean concentration emanating from a point source is a plume that can be visualized to be composed of many puffs, each of whose concentration distributions is sharply peaked about its centroid at all travel distances. Thus the spread of each puff is small compared to the downwind distance it has traveled. This assumption is called the slender plume approximation. 

An Alternate Derivation of (18.55) Equation (18.55) is based on the slender plume approximation as expressed by (18.54). We will now show that the slender plume approximation is equivalent to neglecting diffusion in the direction of the mean flow in the atmospheric diffusion equation. Thus (c) is governed by... 

In this problem we wish to examine two aspects of atmospheric diffusion theory (1) the slender plume approximation and (2) surface deposition. To do so, consider an infinitely long, continuously emitting, ground-level crosswind line source of strength qi. We will assume that the mean concentration is described by the atmospheric diffusion equation,... 

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