In line with the above concepts there are three main mathematical sources that contribute to

the modeled concentration field: 1) the direct source (at the stack), 2) the indirect source, and 3)

the penetrated source. The strength of the direct source is fpQ, where Q is the source emission

rate and fp is the calculated fraction of the plume mass trapped in the CBL (0 # fp # 1). Likewise,

the indirect source strength is fpQ since this (modified image) source is included to satisfy the noflux

boundary condition at z = zi for the trapped material. The strength of the penetrated source is

(1- fp)Q, which is the fraction of the source emission that initially penetrates into the elevated

stable layer. In addition to the three main sources, other image sources are included to satisfy the

no-flux conditions at z = 0 and zi.

For material dispersing within a convective layer, the conceptual picture (see Figure 13) is a

plume embedded within a field of updrafts and downdrafts that are sufficiently large to displace

the plume section within it. The relationship between the particle (or air parcel) height, zc and w

is found by superposing the plume rise ()h) and the vertical displacement due to w (i.e., wx/u), as


where hs is the stack height (corrected for stack tip downwash), u is the mean wind speed (a

vertical average over the convective boundary layer) and x is the downwind distance. The )h

above includes source momentum and buoyancy effects as given by eq. (91) below (see Briggs

(1984)). The Fz or pdf of zc is found from the vertical velocity pdf pw as described in Weil et al.

(1997). In the CBL a good approximation to pw is the superposition of two Gaussian distributions

(Baerentsen and Berkowicz 1984; Weil 1988a) such that

where 8

1 and 8

2 are weighting coefficients for the two distributions with 8

1 + 8

2 = 1(the subscripts

1 and 2 refer to the updraft and downdraft distributions, respectively). The parameters of the pdf

(w1, w2, Fw1, Fw2, 8

1, 8

2) are functions of Fw (the “total” or overall root mean square vertical

turbulent velocity), the vertical velocity skewness S w (where is the third moment of w = 3 σ 3 w3

w), and a parameter R w w . An expanded discussion of the pdf w w = σ = −σ = 1 1 2 2 2

parameters is given in Weil et al. (1997).

The instantaneous plume is assumed to have a Gaussian concentration distribution about its

randomly varying centerline. The mean or average concentration is found by summing the

concentrations due to all of the random centerline displacements. This averaging process results

in a skewed distribution which AERMOD represents as a bi-Gaussian pdf (i.e., one for updrafts

and the other for downdrafts). Figure 15 illustrates the bi-Gaussian approach to approximate the

skewed vertical concentration distribution in the CBL. The figure shows two mean trajectories,

each representing the average of many individual trajectories of parcels (or particles) released into

downdrafts (the downdraft plume) or updrafts (the updraft plume). The velocities determining

these mean trajectories are: 1) the mean horizontal wind speed (u), 2) the vertical velocity due to



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