1142 
conditions to warrant treating the coefficient as a con- 
stant with a value equal to the mean for the layer; 
similarly, a mean value for the eddy coefficient for 
horizontal transfer is also taken. Instead of assuming, 
as in previous treatments, a poimt source of infinite 
concentration, an area source of finite concentration is 
postulated. The source is taken to be an elliptical area 
whose major axis lies horizontally crosswind and whose 
minor axis is vertical; the area is located in the vertical 
plane at which the effluent flow first becomes horizontal 
after leaving the top of the stack. With these assump- 
tions and specified boundary conditions the funda- 
mental differential equation is solved. Computed con- 
centrations are substantially smaller near the source 
than those obtained from Roberts’ equation, but the 
difference becomes progressively less with increasing 
distance from the source. As the size of the vertical 
areal source increases, appreciable differences occur for 
greater distances downwind. The theory has not yet 
been checked by measurements of concentrations under 
inversion conditions. The emphasis on areal rather than 
on point sources is valuable, and further investigations 
of diffusion from such sources under various atmos- 
pheric conditions may well lead to significant advances 
of a general nature. Barad’s analysis suggests that 
studies of the effluent from industrial stacks of large 
diameter, considered as vertical areal sources, should 
be made under inversion conditions. 
Relatively few measured values of concentrations 
from individual elevated sources are available to permit 
a comparison with theory; those taken in the course of 
the Trail investigation [19] are of limited applicability 
because they were taken in the Columbia River valley; 
those taken at Brookhaven National Laboratory [53] 
have not been generally available long enough to per- 
mit a full analysis of them; the publication of other 
analyses [16, 92] is in such a form that it is difficult to 
apply the results in an evaluation of the several equa- 
tions which have been proposed, as given above. It is 
thus possible at the present time to make only a few 
tentative remarks concerning the relative merits of 
the various approaches. 
A comparison of the equation of Bosanquet and 
Pearson, (2), with that of Sutton, (4), brings out the 
fact that the expressions are similar in form but differ- 
ent in detail. The numerical parameters p and q are 
closely related to Sutton’s C, and C,, a point empha- 
sized by the fact that under average conditions p = 
0.05 and g = 0.08 whereas C,, = C. = 0.07 for a source 
at a height of 100 m. On the other hand, Sutton’s 
equations involve an additional parameter n which 
gives them a greater flexibility for portraying faithfully 
the diffusion in complex meteorological conditions. Sut- 
ton [84] points out that his treatment supports a num- 
ber of the main conclusions of Bosanquet and Pearson. 
It is stated by Lowry [52] that, in the light of meas- 
ured concentrations at Brookhaven National Labora- 
tory, Sutton’s equation (5) gives concentrations which 
are too high if a time-mean of an hour is used. It ap- 
pears that the measured concentrations may be less 
than one-tenth of the values predicted from (5) using 
ATMOSPHERIC POLLUTION 
Sutton’s values of the parameters, although generally 
the discrepancy is less marked. If we take (5) as apply- 
ing for a time-mean of an hour, then a comparison of 
(5) and (7) indicates that am = C./C,. Sutton assumes 
that, at heights of 25 m and above, turbulence is 
isotropic, in which case it should follow that a,, = 1, but 
Lowry finds that a, < 1. Thus it appears that when 
time-means of an hour are taken, rather than those of 
several minutes, as considered by Sutton, the effective 
turbulence is not isotropic but C, > C.. It has been 
pointed out by Barad [5] that under very stable con- 
ditions at Brookhaven the plume of oil-fog travels 
nearly horizontally for miles with very little increase in 
vertical thickness but with a gradual lateral widening 
and a meandering as of a river. It is not yet clear 
whether the predominance of lateral spreading of the 
smoke is primarily a result of greater horizontal than 
vertical turbulent mixing or a result of the variation 
of the mean wind direction with height, a factor men- 
tioned by Church [16]. The fact that the rate of varia- 
tion of mean wind direction with height in the surface 
layers is a maximum in slowly moving and very stable 
air suggests that this factor should be taken into ac- 
count under such atmospheric conditions. Further in- 
vestigation on this pomt is required. The observed 
meandering of the smoke plume in an approximately 
horizontal plane recalls to mind the work of Parr [67], 
who suggested that the intensity of vertical mixing 
decreases and that of lateral mixing by large eddies 
increases as the vertical stability increases. When such 
meandering is marked, it is clear that the time-mean 
concentration for a period of the order of an hour at a 
fixed point relative to the earth will in general be sub- 
stantially smaller than that for a period of a few 
minutes or the instantaneous maximum value. The 
work of Lowry and Barad at Brookhaven indicates that 
the magnitude of time-mean concentrations is a func- 
tion of the length of the sampling period, and that 
such time-mean concentrations have limited significance 
unless the periods are specified. 
It is not yet possible to compare the analysis of 
Barad [5] with other treatments, because observatiunal 
data are lacking. However, it may be that an adequate 
analysis of the behavior of smoke from an elevated 
source in very stable air will require an assessment of 
the influence of the variation of mean wind direction 
with height, mentioned above. 
To summarize briefly, each of the proposed equa- 
tions must be used with caution, and applied only 
when certain conditions are met. When, under average 
conditions, a general estimate of the mean concentra- 
tion for several minutes is sufficient, either Bosanquet 
and Pearson’s equation (2) or Sutton’s expressions will 
be adequate. On the other hand, if mean concentra- 
tions for a similar period under other meteorologi- 
cal conditions are required, where quantities such as 
the vertical temperature gradient differ markedly from 
the average, then Sutton’s equations are to be preferred. 
When mean concentrations at a point for longer periods 
are required, Lowry’s expression will be more reliable. 
With strong inversions, Barad’s analysis merits atten- 
