62 AIR POLLUTION METEOROLOGY
Estimating Concentration of ContaminantsGiven a source of contaminant and meteorological con-
ditions, what is the concentration some distance away?
Originally, this problem was attacked generally by attempt-
ing to solve the diffusion equation:d
d
txK
xyK
yzK
xyzz
.
(2)Here, x is the concentration per unit volume; x, y, and z are
Cartesian coordinates, and the K ’s are diffusion coefficients,
not necessarily equal to each other.
If molecular motions produced the dispersion, the K ’s
would be essentially constant. In the atmosphere, where the
mixing is produced by eddies (molecular mixing is small
enough to be neglected), the K ’s vary in many ways. The diffu-
sion coefficients essentially measure the product of eddy size
and eddy velocity. Eddy size increases with height; so does
K. Eddy velocity varies with lapse rate, roughness length, and
wind speed; so does K. Worst of all, the eddies relevant to dis-
persion probably vary with plume width and depth, and there-
fore with distance from the source. Due to these complications,
solutions of Eq. (2) have not been very successful with atmo-
spheric problems except in some special cases such as continu-
ous line sources at the ground at right angles to the wind.
The more successful methods have been largely empiri-
cal: one assumes that the character of the geometrical distri-
bution of the effluent is known, and postulates that effluent is
conserved during the diffusion process (this can be modified
if there is decay or fall-out), or vertical spread above cities.
The usual assumption is that the distribution of effluent
from a continuous source has a normal (Gaussian) distribu-
tion relative to the center line both in the vertical direction, z
(measured from the ground) and the direction perpendicular
to the wind, y. The rationalization for this assumption is that
the distributions of observed contaminants are also nearly
normal. † Subject to the condition of continuity, the concen-
tration is given by (including reflection at the ground).x
pss sss
Q
Vy
yzH zHyzzz2 22222222expexp exp⎛^2
⎝⎜⎞
⎠⎟⎛ ( ) ( )⎝−
⎜⎜⎞⎠⎟.(3)Here, H is the “effective” height of the source, given by stack
height plus additional rise, σ is the standard deviation of the
distribution of concentration in the y and z -direction, respec-
tively, and V is the wind speed, assumed constant. Q is the
amount of contaminant emitted per unit time.
The various techniques currently in use differ in the ways (^) y and s (^) z are determined. Clearly, these quantities change
with downwind distance x (Figure 3) as well as with rough-
ness and Richardson number.
Quantitative estimation of the Richardson number
requires quite sophisticated instrumentation; approximately,
the Richardson number can be estimated by the wind speed,
the time of the day and year, and the cloudiness. Thus, for
example, on a clear night with little wind, the Richardson
number would be large and positive, and s ’s in Eq. (3) are
small; on the other hand, with strong winds, the Richardson
numbers are near zero, and the dispersion rate as indicated
by the σ would be intermediate.
For many years, standard deviations were obtained by
Sutton’s technique, which is based on a very arbitrary selec-
tion for the mathematical form of Lagrangian correlation func-
tions. More popular at present is the Pasquill–Gifford method
in which s (^) y and s (^) z as function of x are determined by empirical
graphs (Figure 4). Note that the dependence of the standard
deviations on x varies with the “stability category” (from A
to F). These categories are essentially Richardson number cat-
egories, judged more or less subjectively. Thus, A (large dis-
persion) means little wind and strong convection; D is used
in strong winds, hence strong mechanical turbulence and less
dispersion; F applies at night in weak winds.
One drawback of the Pasquill–Gifford method is that it
does not allow for the effect of terrain roughness; the empiri-
cal curves were actually based on experiments over smooth
terrain, and therefore underestimate the dispersion over cities
and other rough regions. Some users of the method suggest
allowing for this by using a different system of categories
over rough terrain than originally recommended.
This difficulty can be avoided if fluctuations of wind
direction and vertical motion are measured. Taylor’s diffu-
sion theorem at right angles to the mean wind can be written
approximately,
ssy F
t
TL
0
⎛
⎝
⎜
⎞
⎠
⎟.
(4)
Here F is a function which is 1 for small diffusion time, t. For
larger t, F decreases slowly; its behavior is fairly well known.
T L is a Lagrangian time scale which is also well known.
X
σz σz
FIGURE 3 Change of vertical effluent
distribution downstream.
† Note added in proof: It now appears that this assumption is not
satisfactory for vertical dispersion, especially if the source is near
the surface.
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