64 AIR POLLUTION METEOROLOGY
An equation similar to (4) also exists for vertical spread-
ing; however, it is theoretically less valid, since turbulence is
not homogeneous in the vertical.
As the plume expands vertically, the vertical distribution
cannot remain normal indefinitely. At the bottom, the plume
is limited by the ground. At the top, the plume will be lim-
ited by an elevated inversion layer. Eventually, the vertical
distribution becomes uniform. In that case, the concentration
is given by the equation:xQ
VDyy y
2 22
2
ps sexp (5)where D is the height of the inversion layer, which is also
the thickness of the “mixed layer.” Note that the concentra-
tion is inversely proportional to VD, the “ventilation factor,”
which is the product of D, and V, the average wind in the
mixed layer.
The lateral spread is often limited by topography. In a
valley of width W, the factor (2)()exp y^22 spsyy 2 in
Eqs. (3) and (5) is replaced by 1/ W, after the contaminant
concentration fills the valley uniformly in the y -direction
(the direction perpendicular to the valley). The effect of this
change is that relatively large concentrations are maintained at
large distances from the sources.
Although the Pasquill–Gifford graphs are still popular
in practical applications, evaluation in diffusion experiments
have suggested serious deficiencies. Thus, the research com-
munity is groping for alternate methods. In particular, ver-
tical distributions are far from Gaussian, particularly for
ground sources. Significant progress has been made only for
the important case of light-wind, sunny conditions. Then, the
basic predictors are the thickness of the planetary boundary
layer (PBL), z i ; another important predictor is a vertical-
velocity parameter, w * which is proportional to ( z i H ) 1/3 where
H is the vertical heat flux at the surface. H is not usually mea-
sured, but must be estimated independently; fortunately, it is
raised to the 1/3 power. Lateral dispersion is still Gaussian,but with s (^) y given by
s (^) y / z i f ( tw / z i ) f ( X ) where X tw / z i
f is presumably universal and fairly well known.
The vertical distribution is definitely not Gaussian;
for example, the center line of the plume rises for ground
sources. More important, the center line comes down
toward the surface for elevated sources, unless the sources
are buoyant.
If vertical diffusion is normalized by the new variables,
it depends on z / z i , X and h / z i where h is stack height. The
distributions have been measured for different h / z i , and com-
plicated formulas exist to fit the observations. The results
are believed to be quite reliable, because numerical models,
laboratory experiments and full-scale observations are all in
satisfactory agreement.
The results of this research should be used in practi-
cal applications, but have not been. For more detail, see
Panofsky and Dutton, 1984.
City Models
These different methods give the pollutant concentrations
downwind from a single source. In order to obtain the total
picture of air pollution from a city, the concentrations result-
ing from all sources must be added together, separately for
all different wind directions, different meteorological condi-
tions, and for each contaminant. Such a procedure is expen-
sive, even if carried out with an electronic computer, and
even if, as is usually done, all small sources in square-mile
areas are combined. Therefore, complete city models of air
pollutant concentrations have only been constructed for very
few locations. It is necessary, however, to have city models
in order to understand the distribution of contaminants; only
then it is possible to determine the most economical strategy
to reduce the pollution, and to evaluate the effects of expan-
sion of housing and industry.
Because the construction of a complete city model is so
expensive, city models are often simplified. For example, if
the city is represented by a series of parallel line sources,
the computations are greatly reduced. Many other simplifi-
cations have been introduced; for a summary of many city
models now in existence, see Stern (1968).
Diurnal Variation of Air Pollution
Equation (5) which shows that concentrations at consider-
able distances from individual sources are inversely propor-
tional to the ventilation factor ( VD ), can be used to explain
some of the variations in air pollution caused by meteoro-
logical factors. First, we shall consider the diurnal variation
of air pollution. Of course, the actual variation of pollution
may be different if the source strength varies systematically
with time of day. The diurnal variation is different in cities
and in the country. Consider typical vertical temperature
distributions as seen in Figure 5. During the day, both over
cities and country, the ground temperature is high, giving a
deep mixed layer. After sunset, the air temperature near the
surface in the country falls, producing an inversion reaching
down to the ground. After air moves from the country out
over the relatively warmer and rougher city, a thin mixed
layer is formed near the ground. The thickness of this mixed
10,000
5,000
COUNTRY
CITY
CITY
COUNTRY
NIGHT DAY
TEMPERATURE
HEIGHT, ft
FIGURE 5 Vertical temperature distribution
(schematic) over city and country, day and night.
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