plume buoyancy (vbuoy), and 3) the mean updraft (w ) or downdraft ( ) velocity. The mean 1 w2

height of each trajectory z or , can be found by averaging eq. (53). These parcel (or c1 zc2

particle) height distributions are thus related to concentration and are characterized by Fz1 (=

Fw1x/u) and Fz2 (= Fw2x/u), the standard deviations of the two concentration distributions

comprising the bi-Gaussian form as derived in Weil et al. (1997).

Figure 15: AERMOD’s pdf approach for plume dispersion in the CBL.

AERMOD approximates the skewed distribution by superimposing two

Gaussian distributions, the updraft and downdraft distributions.

Figure 16 compares the bi-Gaussian pdf with the Gaussian form, which is symmetric about

w = 0. As can be seen, for the negative and positive tails of the distributions, the bi-Gaussian pdf

is biased towards smaller and larger pw values, respectively, than the Gaussian. In addition, for

the bi-Gaussian forms, approximately 60% of the area under the pw curve is on the negative side

of the w axis and approximately 40% on the positive side. This is consistent with the results of

numerical simulations and field observations (Lamb 1982; Weil 1988a).

Figure 16: Probability density functon of the vertical velocity. While the Gaussian

curve is unskewed the bi-Gaussian curve has a skewness of S = 1.

In the pdf approach used here (Weil et al. 1997), there are, as mentioned in the previous

section, three primary sources that contribute to the modeled concentration field: 1) the “direct”

or real source at the stack, 2) an “indirect” source that the model locates above the CBL top to

account for the slow downward dispersion of buoyant plumes that “loft” or remain near, but

below, zi , and 3) a “penetrated source” that contains the portion of plume material that has

penetrated into the stable layer above zi. The direct source describes the dispersion of plume

material that reaches the ground directly from the source via downdrafts. The indirect source is

included to treat the first interaction of the “updraft” plume with the elevated inversion - that is,

for plume sections that initially rise to the CBL top in updrafts and return to the ground via

downdrafts. Image sources are added to treat the subsequent plume interactions with the ground

and inversion and to satisfy the zero-flux conditions at z = 0 and at z = zi. This source plays the

same role as the first image source above zi in the standard Gaussian model, but differs in the

treatment of plume buoyancy. For the indirect source, a modified reflection approach is adopted

in which the vertical velocity is reflected at z = zi, but an “indirect” source plume rise )hi is added

to delay the downward dispersion of plume material from the CBL top. This is intended to mimic

the lofting behavior. The penetrated source is included to account for material that initially

penetrates the elevated inversion but subsequently can reenter the CBL via turbulent mixing of



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