plume buoyancy (vbuoy), and 3) the mean updraft (w ) or downdraft ( ) velocity. The mean 1 w2
height of each trajectoryz or , can be found by averaging eq. (53). These parcel (or c1 zc2
particle) height distributions are thus related to concentration and are characterized byFz1 (=
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 largerpw values, respectively, than the Gaussian. In addition, for
the bi-Gaussian forms, approximately 60% of the area under thepw curve is on the negative side
of thew 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 ofS = 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 abovezi. 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 atz = 0 and at z = zi. This source plays the
same role as the first image source abovezi 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 atz = 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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