EPA-600/4-76-030a
July 1976
ATMOSPHERIC DISPERSION PARAMETERS IN GAUSSIAN PLUME MODELING
Part I. Review of Current Systems and Possible Future Developments
by
A. H. Weber
Associate Professor of Meteorology
Department of Geosciences
North Carolina State University
Raleigh, North Carolina 27607
U. S. ENVIRONMENTAL PROTECTION AGENCY
OFFICE OF RESEARCH AND DEVELOPMENT
ENVIRONMENTAL SCIENCES RESEARCH LABORATORY
RESEARCH TRIANGLE PARK, NORTH CAROLINA 27711
Environmental Protection Agency
Region V, Library
230 South Dearborn Street
Chicago, Illinois 6060H
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DISCLAIMER
This report has been reviewed by the Environmental Sciences
Research Laboratory, U. S. Environmental Protection Agency, and
approved for publication. Approval does not signify that the
contents necessarily reflect the views and policies of the
U. S. Environmental Protection Agency, nor does mention of trade
names or commercial products constitute endorsement or recommenda-
tion for use.
ii
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ABSTRACT
A recapitulation of the Gaussian plume model is presented
and Pasquill's technique of assessing the sensitivity of this
model is given. A number of methods for determining dispersion
parameters in the Gaussian plume model are reviewed. Com-
parisons are made with the Fasquill-Gifford curves presently
used in the Turner Workbook. Improved methods resulting from
recent investigations are discussed, in an introductory way
for Part II of this report.
iii
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PREFACE
The increasing concern of the last decade in environmental issues,
and the fuller appreciation that air quality simulation modeling may
provide a unique basis for the objective management of air quality, has
generated an unprecedented interest in the development of techniques for
relating air quality and pollutant emissions through appropriate modeling
of the atmospheric transport and dispersion processes that are involved.
A multitude of recent publications in the U.S.A. and elsewhere points to
the wide interest at many different levels of local, state, regional and
national planning, and testifies to the widespread acceptance of meteoro-
logical type air quality modeling as an important rational basis for air
quality management. This point of view is now internationally recognized
in most industrial countries.
Earlier attitudes towards the quantitative estimation of atmospheric
dispersion of windborne material from industrial and other sources were
strongly influenced by a system introduced in 1958, and published in 1961,
by Dr. F. Pasquill of the Meteorological Office, United Kingdom. This
was followed in 1962 by the publication of his definitive textbook on
"Atmospheric Diffusion," which includes detailed consideration of the
well-known simple "Gaussian-plume" model for the average concentration
distribution in space from an elevated continuous point-source under steady
conditions. The unique feature, however, of the Pasquill system is the
method by which the critical parameters expressing the downwind spread of
the plume might be estimated in terms of the ambient meteorological
conditions. These estimates were later expressed in slightly more convenient,
iv
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although exactly equivalent form, by Dr. F. Gifford, and this so-called
Pasquill-Gifford system for dispersion estimates has been widely used
ever since. It was early given some general endorsement as a valuable
practical scheme by the Public Health Service of the U. S. Department
of Health, Education, and Welfare, by the publication in 1967 of the
"Workbook of Atmospheric Dispersion Estimates" (Public Health Service
Publication No. 999-AP-26) by D. B. Turner, that exclusively utilized
this system of Gaussian-plume dispersion parameters.
In spite of the gradual appearance in recent years of air quality
models based on more sophisticated formulations of the atmospheric pro-
cesses, e.g., through fluid-dynamical equations assumed to govern the
physical processes of transport and turbulent diffusion, great use
continues to be made of the simpler Gaussian-plume models. However, a
direct consequence of the unprecedented interest in the subject has been
the publication of many attempts to confirm or improve the realism of the
dispersion estimates. These and other matters relating to the substantial
progress made in recent years in understanding atmospheric dispersion, are
discussed in a much revised 2nd Edition of the Pasquill book, that was
published late in 1974. Under these circumstances it seemed desirable to
examine critically the possible requirements for change in the Turner
Workbook values for dispersion, that have been so widely used since 1967.
The present two-part report was prepared to meet this need.
It was extremely fortunate that Dr. Pasquill was available for
detailed discussions during its preparation (1975-76), both while he was
a Visiting Professor at the North Carolina State University and also the
Pennsylvania State University (under research grant support of the EPA),
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and also as a Visiting Scientist to the Meteorology and Assessment
Division of the E.P.A., Research Triangle Park, N. G. Even more fortunate
has been Dr. Pasquill's willingness to assume responsibility for prepara-
tion of the second part of the report, which critically examines the
possible requirements for change of the Turner Workbook values. The first
part was prepared by Dr. Allen Weber of the Department of Geosciences at
North Carolina State University, in consultation with Dr. Pasquill and
the undersigned. It provides a reasonably comprehensive review of current
systems and possible future developments, and is a necessary input for the
critical examination of the second part. It is perhaps unnecessary to
emphasize that there are still many problems of dispersion that are unlikely
to be resolved satisfactorily in terms of simple Gaussian-plume models.
However, it is hoped that the present publication will provide a more
up-to-date basis for continuing the successful treatment of the many
important practical problems that can be analyzed by this simple approach.
Research Triangle Park
North Carolina
March 1976
Kenneth L. Calder
Chief Scientist
Meteorology and Assessment Division
Environmental Sciences Research Lab.
vi
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CONTENTS
Preface iv
List of Tables viii
List of Figures ix
Acknowledgements x
1. Introduction 1
2. The Gaussian Plume Model 2
3. Sensitivity of the Gaussian Plume Model 8
4. Review of o-Systems Available for Use 13
5. Improved Methods of Estimating Dispersion Parameters ... 40
6. Effect of Release Altitude on Dispersion Parameters .... 46
Appendix 1 List of Symbols 47
Appendix 2 Relationships Among Turbulence Categorizing Schemes 5C
vii
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LIST OF TABLES
No. Page
1. Applicability of Current Working Theories of Dispersion 14
2. Relationships Among Various Stability Parameters 53
viii
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LIST OF FIGURES
No. Page
1. Idealized Distribution of Concentration at Ground Level
from an Elevated Source 9
2. The Pasquill-Gifford Curves for a Versus Downwind Distance 21
3. The Dispersion Curves of Singer and Smith and the PG Curves 34
4. The TVA Dispersion Curves and the PG Curves 35
5. The St. Louis Dispersion Curves and the PG Curves 36
6. Markee's Dispersion Curves and the PG Curves 37
7. TRC Rural Dispersion Curves and the PG Curves 38
8. Briggs' Rural Dispersion Curves and the PG Curves 39
ix
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ACKNOWLEDGEMENTS
It has been the author's good fortune to have close contact with Dr.
Frank Pasquill for several weeks while writing this review. Dr. Pasquill's
insight on the problem of turbulent dispersion has been extremely valuable.
Mr. Kenneth Calder provided extremely good information and guidance in
pointing out important studies of atmospheric diffusion bearing on this
study.
During the course of the work the author received valuable assistance
from several individuals on the telephone and otherwise. Dr. Frank Gifford
was especially helpful in this way in clarifying important issues. Mr.
Jui-Long Hung and Mr. Richard D'Amato were very helpful in preparing figures
in this report.
The author would like to thank Mr. Larry Niemeyer for providing two
months support under the Intergovernmental Personnel Act to conduct this
s tudy.
x
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SECTION 1
INTRODUCTION
For purposes of simplicity this review will be restricted to a
discussion of the single source dispersion problem (e.g. an isolated
stack) either as a problem in its own right or as an element of an
"area" source. The dispersing material is assumed to be passive i.e.
the material does not react chemically, decay by radioactivity, undergo
washout by precipitation, deposit on the surface of the earth, or
react in any other way to violate the law of conservation of matter.
Furthermore, there will be no discussion of buoyant plume behaviour since
a recent comprehensive review and summary of this problem has been given
by Briggs (1969, 1975).
Discussion of dispersion will be in terms of the so-called Gaussian
plume dispersion model (defined in Section 2). This basic model
follows directly from some of the theoretical models of dispersion. It
is also known that a great deal of the experimental dispersion data now
available for single sources in steady conditions, seems to be reasonably
well described by the Gaussian shape of the spatial distribution pattern,
when properly time-averaged. Gifford (1975, 1976) provided a brief
survey and review of the Gaussian-model methodology as well as clearly
specifying the kinds of situations when the simple Gaussian form is
inappropriate.
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SECTION 2
THE GAUSSIAN PLUME MODEL
*
The Gaussian plume diffusion model for an elevated point source
(assuming reflection at the ground) is represented by the equation
2 ^ s* 2^ f 2\
C(x,y,z;H) = *—- exp\ ^-yifexpy- ~Z' /+ expV- z) fi (1)
2ira a u ( 2a ) ( 2cr i ( 20 1
yz v. y ./ ^ z -' ^ z S
where
C is the time averaged concentration of pollutant;
x, y, and z are the distances downwind, crosswind, and ver-
tically upward, respectively;
H is the effective s< urce height above ground level (H is
equal to the sum of the physical stack height h and the
S
plume rise AH);
Q is the source strength;
0 is the standard deviation of the time-averaged plume
concentration distribution in the crosswind direction;
a is the standard deviation of the time-averaged pluire
Z
concentration distribution in the vertical direction;
u is the time-averaged wind speed at the level H.
The sampling time used in the definitions above is normally
chosen to be about one hour because such a time period admits most of
the turbulent fluctuations and defines time-mean values that are
The historical basis of the Guassian plume model is not of crucial
importance here, although, the interested reader will be able to
intersect the line of development of this model at a significant stage
by referring to Gifford (1960).
**
Mathematical symbols will be defined after their first appearance
in the text and in a special list in the Appendix
2
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quasi-steady. It is also a very convenient time interval to describe
air pollutant concentrations.
It is assumed in the use of the Gaussian plume model that there
is a uniform rate of emission of pollutant during the one hour period.
It is also assumed that diffusion in the x direction is negligible.
This would be literally true if the release were continuous and
effectively so if the time of release is equal to or greater than the
travel time x/u from the source to the location of interest.
The range of application of Eqn. 1 is restricted to within the
downwind distance x where the plume first encounters the top of the
*
mixing layer . The top of the mixing layer is of the order of one
kilometer, but it can extend anywhere from a few hundred meters to
several kilometers depending on several factors including synoptic
scale features of the weather. The top of the mixing layer can be
estimated from the results of Holzworth's (1972) extensive survey or by
direct measurement, e.g. with an acoustic sounder.
The distance downwind where the top of the plume intersects this
"lid" (as it is often called) is quite variable. Similarly the point
downwind where the plume from an elevated source reaches its maximum
ground-level concentration will depend on several factors. Often the
point of maximum ground-level concentration will be closer to the source
than the distance to the plume's reflection from the lid. Since it is
often desired to know what the maximum concentration will be, the com-
plication of the restrictive lid will (under many circumstances) be
*
By considering reflections of the pollutant from the top of the
mixing layer and the ground, the range of the Gaussian plume model can
be extended. Another simpler method of obtaining concentration beyond
the distance where the plume is trapped under the top of the mixing
layer is provided in Turner (1969).
3
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of no consequence. On the other hand, there are circumstances where
the mixing depth will be critically important so this concept must be
borne in mind when doing diffusion calculations.
There are two additional aspects relating to time-averaging that
are of special significance. The first is that the time-average of
a is not usually stationary, i.e. the longer the time average the
larger a tends to become. Therefore, all graphs and tables of this
parameter should make clear the duration of the average.
An attempt has been made (based on experimental studies) to pre-
dict the effect of limited sampling time on the value of a . Several
authors, whose results were summarized by Gifford (1975), suggest
A (2)
a
yB
t
B
where a „ is the standard deviation of concentration averaged over some
yB
reference period t,,, e.g. one hour, and a . is the standard deviation
13 yA
of the concentration over the time period of interest, t . The value
£\.
of q advocated by Gifford is 0.25 to 0.30 which seems to hold for
1 hr. < t. < 100 hrs. By considering this same problem from a (limited)
A.
theoretical viewpoint, it is likely that the effect will be a complex
function of the spectral shape and distance downwind (see Pasquill,
1974).
Similar considerations apply to a from an elevated source,
z
although recognition of the important differences between the spectra
of vertical and horizontal components should make it clear that the
increase of 0 with sampling time may be expected to be terminated
Z
-------
much more decisively at sampling times of 10 minutes or so. Further
considerations along the same lines make it clear that for a ground
level source a sampling time of a few minutes (3 to 5) will yield
steady values of 0 (See also Pasquill's comments on this point in
z
Part II of this report).
Further simpler forms of the concentration from Eqn. 1 are ob-
tainded at special locations. The following formulae and other
related formulae are found in Gifford (1960) and Pasquill (1974).
The concentration at ground level is
2 TT2
C(x,y,0;H)
_ exp - --
ira a u 2a
y z y
(3)
2a
Along the axis of the plume at ground level the concentration is
,,2
C(x,0,0;H) =
exp (-
ira a u
y z
2a
(4)
If one assumes simple power-law forms for the crosswind and
vertical spread, i.e.,
o = a (x.) (—)q (5 a & b)
(where x- is a reference distance downwind)
then, by differentiation the position along the axis where the maximum
occurs is
X =
m
qH2
[Vxi>l
UqJ
2
(p + q)
-|l/2q
(6)
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or alternatively, the distance x equals x when
m
R
(7)
The magnitude of the maximum concentration is
cm = ii/2 «P (- A) _ -^
iru y m H
If p = q the maximum concentration reduces to
c - 2Q ,-!»)
m -2 o J
eiruE y
and a /a = a , a constant independent of x.
With p = q the distance x to the maximum is
l/2q
x =
m
H2
2
raz(Xl)l
x q
. 1 -
2
1/2
or x = x when H/o = 2
m z
The crosswind integrated concentration at ground level is
2
/C(x,y,0;H)dy = I -- exp (-
ir a u
z
2a
(8)
(9)
(10)
(11)
The above formulas now form the basis of a widely used methodology
(Pasquill, 1974 and Slade, 1968) of describing atmospheric dispersion
in many contexts, e.g. diffusion from ground sources, tall stacks,
automobiles, rockets, etc. The method requires additional special
interpretation when the terrain is complicated, e.g., shorelines,
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mountain-valley systems, hills, forests broken by cultivated land,
etc. Gifford (1976) and Egan (1975) have attempted to treat some of
these "special" diffusion problems insofar as this is possible at this
time. For the purpose of this study we will not concern ourselves
with these exceptional flows but rather confine ourselves to the cases
where the simple Gaussian plume model is clearly appropriate.
The critical parameters in the Gaussian model are the effective
source height H and the dispersion parameters a and a . The effective
stack height is composed of the physical stack height h plus the plume
s
rise due to buoyancy and inertia AH. As stated earlier in the review
the formulae by Briggs (1969, 1975) are pertinent to determine AH.
This study will be concerned with the specification of the dispersion
parameters a and a and with related matters affecting dispersion.
y z
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SECTION 3
SENSITIVITY OF THE GAUSSIAN PLUME MODEL
The sensitivity of a model is loosely defined as any measure of
the changes of the dependent variable caused by changes in the
parameters of independent variables of the model.
It is desirable to have some feel for the sensitivity of the
Gaussian plume model and this can be done in a reasonably satisfactory
manner by graphical means. Figure 1 shows the idealized distribution
of concentration at ground level from an elevated source. (This figure
is taken from Pasquill, 1974, Fig. 5-17.) In drawing the isopleths it
has been assumed that a fa = a , a constant irrespective of distance,
and that o « x . The isopleths are labeled according to values of
C(x,y,0)/C . The downwind distance is in terms of x1 a nondimensional
coordinate which uses units of distance that from the source to the
point of maximum concentration. Thus, the value 5 corresponds to a
downwind distance of 5 times the distance from the source to the maxi-
mum concentration. The crosswind distance is in terms of the non-
dimensional coordinate y1 = y/(a H). A typical value of a is two.
If the parameter q = 1 and r /H = 10 the relative crosswind and downwind
stretching of the isopleths shewn in the Figure is correct.
Now by observing Figure 1, one can conclude the following.
(1) Upwind from the point of maximum concentration the concentration
values fall off very rapidly with distance being almost two
orders of magnitude smaller at x' =1/2 than at x1 = 1.
8
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Figure 1. Idealized distribution of concentration at
ground level from an elevated source. (From Figure
5-17 Pasquill, 1974)
-------
(2) Downwind from the point x the fall-off of concentration is
m
much more gradual having decreased cr.e order of magnitude at
x' = 5.
(3) The lateral concentration pattern varies rapidly with y*,
especially so in the vicinity of the point of maximum con-
centration.
Now, in order to evaluate the effect of changes in physical parameters
at the sensitive locations (1) and (3) above, it can be seen that changes
in wind direction will have a dramatic effect on the location of the
areas (1) and (3). Also, given the wind direction, a change in H or a
z
will produce fairly dramatic results.
Pasquill (1974) has analyzed several cases of changes in C caused
by changes in a /E and wind direction at the "most sensitive position"
Z
i.e. where dC/d(c /H) is maximum. The amount of the change in C ranges
Z
between +75% (for a 20% change in a ) to -91% (for a 10° wind direction
change and a 20% change in H). These results indicate the extreme
variation in concentration one can expect for a given sensor location
downwind and relatively close to a stack.
As a further step in demonstrating the sensitivity of the Gaussian
plume model, PaF.quill (1974) estimated errors in C caused by errors in
the dispersion parameters for an elevated source. He considered the
expression for maximum concentration (Eqn. 8). Note that maximum con-
centration depends on source strength Q, wind speed u, the dispersion
parameters c and o , as well as the distance at which the maximum
occurs x . Now for simplicity Pasquill assumed that the wind speed
m
10
-------
and source strength are accurately known. He also assumed that H has 15%
error and that the j_ or a vs x relationships have errors of 15%. Then,
it follows from Eqn. 8
Sn" Ha (x ) *
y m
Pasquill's argument is as follows:
(1) A reasonable estimate of the r.tn.s. (root mean square) error
in H is about 15% (as given above).
(2) a and H are related according to Eqn. (7). The r.m.s. error
z
in a implied by errors in K is therefore 15%.
z
(3) In addition to (2) there is an error in a based on the fact
Z
that we do not know the precise values of p and q to be used in Eqn. (7).
Since most likely 1< p/q< 2 the contribution of this error to a is
about 8%. (This is found by assuming the range between v2 and /3~ is
spanned by four standard deviations and the distribution of departure
over this range is normally distributed.)
(4) By assuming no interrelationships in the errors of (2) and
(3) above the net error in o can be obtained by taking the square root
Z
of the squares of 15 and 8 which is 17%.
(5) There is now a need to know the error in the distance to the
maximum concentration x since 0 is evaluated at this distance (in
m y
2
the expression for C). If x « a (x). then the error in x is found
n m z m / m
by doubling the error in o . From step (4) this results in an error
of 34% in x .
m
11
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9
(6) Since the exact form of the relationship x <* c ~(x ) is not
m z m
known, Pasquill made the reasonable assumption that this error is
about 15%.
(7) The net error in x is found by taking the square root of the
sum of the squares of (5) and (6) yielding 45%.
(8) Now, assuming that a « x and given the results of (7) above
the r.m.s. error in a at x implied by (7) is 45%. Since again the
exact form of the relationship between o and x is unknown, Pasquill
makes the reasonable estimate of 15% error.
(9) The net r.m.s. error in 0 from (8) and (9) above is found
by taking the square root of the sum of squares. This yields 47%.
(10) The total error in the product Ha (x ) is found by taking the
square root of the sum of the squares of 47 and 15. This gives a net
*
r.m.s. error in C of 49%.
m
Thus in the case of a power station plume even knowing the wind
speed and source strength exactly, one can do no better than about
50% error for an individual case even if the errors in a , a , and H
z y
are quite modest.
Moore (1973) has analyzed data collected for the Tilbury and
Northfleet power stations. His results show that the uncertainties
are broadley consistent with those of Pasquill demonstrated above.
12
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SECTION 4
REVIEW OF o-SYSTEMS AVAILABLE FOR USE
PREDICTIONS FROM THEORETICAL BASES
There are three current working theories of atmospheric diffusion;
these are statistical theory, gradient transfer theory, and similarity
theory. Pasquill (1974) has explored the many facets of these theories
and their underlying bases. Table 1 of this report summarizes the
important aspects of each theory with regard to the dispersion parameters
a and a . For homogeneous turbulence the statistical theory is valid
and a can be predicted for an elevated or ground source. The dispersion
y
parameter c can be predicted by one or another of the theories but
z
take special notice that none of the theories listed can be used to
predict o for intermediate range from an elevated source.
A valuable practical result comes from the Hay-Pasquill (1959)
version of the statistical theory. The statistical theory applies as
long as the plume is under the influence of homogeneous turbulence and
would therefore net apply tc calculations of c except for short range
from an elevated plume, while c is small compared with the height of
release. The Hay-Pasquill result is much easier to apply than
Taylor's (1921) original theory because it requires only running averages
rather than power spectra or correlations. It does involve a simpli-
fying assumption; the ramifications of this can be found in Pasquill
(1974) and Gifford (1968). The Hay-Pasquill result states that for
an elevated or ground level source, irrespective of distance and
irrespective of thermal stratification, the value of a can be predicted
13
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Table 1. APPLICABILITY OF CURRENT WORKING THEORIES OF DISPERSION
Theory -*
Dispersion
property
predicted
Limitations
Statistical
0 (elevated or
ground source)
c (elevated source
at short range)
Homogeneous
turbulence
Gradient-transfer
0 (ground source)
z
0 (elevated source
2
at long range)
Dispersive mech-
anism must be small
scale eddies
(no meandering)
Similarity
0 (ground
source at
short range)
Surface stress
layer , near
ground level
sources
14
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from
where the parameter cr is the standard deviation of wind direction
6
measured in radians. The first subscript T refers to the total
sampling period and the second subscript refers to a running average
of interval x/Bu. Pasquill (1974) summarizes several investigations
on the parameter 3; the currently suggested value being
g * 0.44/1
where i is the intensity of turbulence.
The Hay-Pasquill result for 0 (assuming an elevated source) is
Z
-, x/guj
where a is the standard deviation of the elevation angle of the wind
9
measured in radians. Note that this equation appears to be a reasonable
approximation for an elevated source within the distance downwind
to the point where significant interaction of the plume with
the underlying surface has occurred. The main practical problem in
using the above equations is that the required information on wind
fluctuations is seldom readily available.
Although the concept of gradient transfer continues to be challenged
in various ways, there are strong indications that the method is capable
of providing realistic estimates of vertical spread from a surface
release (or more generally even for an elevated release at height H
where a > H) given flow conditions not departing markedly from
neutral stratification. Although the approach does not lead to
any simple general result as in the application of statistical theory,
15
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values of the vertical distribution of concentration (hence a ) may
z
be obtained from solutions (numerical if necessary) of the two-dimensional
diffusion equation with plausible forms of K consistent with experience
on the turbulent structure of the atmospheric boundary layer. These
methods were used by F.E. Smith (1973) and will be given more attention
in Section 5 of this report.
The similarity theory is presently limited in its application
to predictions of c^ in regions of the boundary layer where u^ (friction
velocity) and IL, (the heat flux) are effectively constant. It is also
c
restricted in that the height of release cannot be mere than a few meters
(see Pasquill, 1974). Since o does not scale with Monin-Obukhov length
L (see Appendix 2) there is no way at present to predict the crosswind
dispersion on such a similarity basis (Calder, 1966).
Other significant results relating to the prediction of a based
Z
on the working theories will be found later in this report under
Section 5 entitled Improved Methods for Estimating Dispersion Parameters.
EMPIRICAL SYSTEMS OF SPECIFYING DISPERSION PARAMETERS
None of the working theories mentioned above is universally
applicable over a wide range in x because of various limitations.
either theoretical or practical. Because of this several empirical
systems have come Into widespread use. Gifford C1976) recently
summarized many of these systems.
We will briefly cover the systems mentioned by Gifford and add
some recent ones that have been brought to our attention. The bases of
most of these estimates are measurements of ground level concentration
of some tracer at various distances downwind.
16
-------
Often the basis of the cr determination was the application of
some forn of the Gaussian plume model, e.g., Eqr. 11 with regard to
a , together with a statement of the continuity equation. It is
z
apparent that any particular observational study cannot be adopted
without question as a valid generalization, since there will come into
play local climatological influences and peculiarities of measurement
systems which will not apply at other locations.
Some of the systems can be related in a loose sense to the statis-
tical theory, while others are purely empirical. For convenient practical
application the values of a or a are presented in relation to meteoro-
logical data of a routine nature, and one of the crucial requirements is
that of insuring a realistic representation of the effects of thermal
stratification in terms of such data.
The British Meteorological Office 1958 System
In 1961 Pasquill published a method of determining the dispersion
parameters, on the basis of making simple subjective estimates of the
structure of turbulence of the atmospheric boundary layer from routine
meteorological data, comprised of wind speed, insolation, and cloudiness.
These three meteorological parameters were used to determine a stability
category A through F (A corresponding to very unstable conditions and
F to very stable conditions, D being neutral). Given the stability
category and downwind distance-, one could determine the corresponding
value of the dispersion parameter. Because Pasquill wanted to make
the identification with physical spreading easy for engineers and
other non-meteorologists (principally interested in making quick
17
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estimates of the plume spread) he used an angle 6 to measure "total"
crosswind spread and a height h to measure "total" vertical spread.
The relation of these parameters to o and a is
y z
6 = 4.30 a /x, (14)
and
h = 2.15 0z (15)
A more precise way of measuring atmospheric turbulence is through
the Richardson number Ri (which will be defined later in Appendix 2)
and a qualitative correspondence exists between Pasquill categories and
Ri. The large positive values of Ri correspond to stable conditions
F and large negative values correspond to unstable conditions A, as
pointed out by Islitzer (1965).
In Pasquill's original system no attempt was made to allow for
any special dependence of a growth on elevation of the source, because
Z
at that time the state of knowledge was such that there was no basis on
which to make such a differentiation.
The estimates of vertical dispersion in the Pasquill system are
based on the following: (See also Table 1, Part II of this report
by Pasquill)
(1) Data collected at short range, less than 1 km, during neutral
conditions and consolidated by Calder's (1949) semi-
theoretical treatment. The roughness length implicit in the
Calder treatment was 3 cm.
18
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(2) Data collected at short range during non-neutral conditions,
*
obtained from measurements in Project Prarie Grass (Barad,
1958).
(3) Data collected in the range 1 km to 100 km during unstable
conditions, and calculations based on vertical gustiness.
(z = 30 cm)
o
(4) Data representing stable conditions in the range 1 km to 100 km
were essentially speculative extrapolations from the more
reliable data. (z = 30 cm)
o
Estimates of the crosswind spread were emphasized by Pasquill to
be valid for release times of only a few minutes. If suitable wind
direction fluctuation data were available, Pasquill recommended the
values of the crosswind dispersion parameter to be calculated by Eqn. A.'- •'•
12. When fine-structure data were not available 6 could be calculated
from the difference of the extreme maximum and minimum of the trace and
setting this equal to 6, provided the distance to the point of interest
is less than 0.1 km on the axis of the plume. For distances downwind
of the order of 100 km, Pasquill modified this procedure to be the dif-
ference between the maximum and minimum "15 minute averages" of wind
direction.
For a ground level source the a values can be considered in-
dependent of sampling time up to periods of 1 hr. or so. For an
elevated source (of about 100 m) the estimates should be taken to
Project Prarie Grass was conducted near O'Neill, Nebraska in 1956.
The objective was to determine the rate of diffusion of a tracer gas
as a function of meteorological conditions. The range of experiments
was from 0 to 800 meters and a total of 70 experiments were performed.
The release time for the tracer gas was 10 minutes.
19
-------
apply for 10 min. or more of sampling time.
Giffords (1961) recasting of the Pasquill Systerc
Gifford converted the plume spreading parameters 6 and h of the
Pasquill system to values of a and a by means of Eqn. (14)
and (15). He did this feeling that it was less arbitrary and more
generally understood by people making applications, to use standard
deviations rather than the 10% points of Eqns. 14 and 15. The data
used were exactly those of the Pasquill system mentioned above. The
resulting set of curves are generally known in this country as the
Pasquill-Gifford curves (PG curves) (See Fig. (2)).
Turner's (1961, l':->64) adaptation of Pasquill*s Stability Scheme
Turner adapted the Pasquill stability scheme but added a qualita-
tive specification of insolation in terms of solar altitude rather than
a subjective determination. By specifically allowing for solar elevation
he provided a basis for applying Pasquill's scheme elsewhere than in the
latitude of Britain. This objective system of determining the stability
categories also provided a means of using archived meteorological data
to determine what the stability categories had been at a particular
location. Thus the compute^- could be used with ease to produce a diffusion
climatology for a region for use with the Gaussian model. The
*
Three slightly differing versions of the Pasquill-Gifford curves
exist; Pasquill (1961) (which is the same as the version appearing in
Turner, 1969), Gifford (1961), and Slade (ed.) (1968). This has led
to a small amount of confusion in the published literature, however,
the differences between versions are only in the unstable categories
A, E, and C. The version under consideration here is identical to that
of Turner (1969).
20
-------
10000
IOOO
10
100
PIST^iCE WMliO, r:
Figure 2. The Pasquill-Gifford curves for a versus dowrwind distance
x (From Turner, 1969)
21
-------
a *s and a 's published in the widely used EPA workbook by Turner (1969)
are exactly those of the so-called PG curves.
Klug*s (1969) stability scheme
Klug (1969) developed a highly detailed set of rules to determine
diffusion classes I through V (I corresponding to stable and V to
unstable). The scheme was quite similar to Pasquill's except that it
was more detailed and could handle more complex situations, e.g. the
transition from night to morning when the boundary layer is rapidly
evolving. For dispersion parameters Klug used a and c measurements
from Project Prarie Grass (Barad, 1958).
The Brookhaven National Laboratory (BNL) System by Singer and Smith
Singer and Smith (1966) set forth a stability category system and
data relating to dispersion parameters, based on a variety of data taken
over a 15 year period at the BNL site. These data are noteworthy in
that the plumes were released at 108 m (quite high compared to previous
studies). The stability system recommended was based on wind direction
traces. Singer and Smith gave the stability categories letter designations,
corresponding to a turbulence type indicated on the wind direction trace.
The letters are A, B^, B-, C, and D (A representing very unstable and D
representing very stable). Each of the letter designations corresponds
to a range of wind direction fluctuation in degrees taken over a one
hour period with a Bendix-Friez Aerovane located at source height.
The BNL categories are often referred to as "gustiness classes".
Sources of dispersion data used in the Singer-Smith system were from
tracer experiments using uranine dye, oil fog, and Argon 41 ( a radio-
active isotope), the most important single source of data coming from the
22
-------
oil fog studies. Concentration measurements were obtained in three ways;
photometric densitometers, fluorescence of the oil droplets, and visual
estimates of the dimensions of the plume. The radioactive argon emissions
allowed following the dispersing plume up to distances of more than 50 km.
The uranine dye was used at short range, and in contrast with the rest of
the experiments was released at 2 meter source height. The c values
were not measured directly but were calculated through measurement of
a , C, and u.
V
The sampling times for the dispersion parameters were of the order
of one hour. The dispersion curves in the ASME (American Society of
Mechanical Engineers) Recommended Guide for the Prediction of the
Dispersion of Airborne Effluents; Smith, 1968) are based on the Singer-
Smith formulation and the recommended dispersion parameters are exactly
the same.
One point worth noting with regard to the BNL system is that the
a and a variations with distance downwind are identical.
The TVA (Tennessee Valley Authority) System
Carpenter, et al. (1971) summarized dispersion data representing
helicopter sampling of SO,, emitted from stacks of the TVA system. This
study was the first dealing with dispersion data from large buoyant
plumes. The stacks operational in the TVA system at the time of the
experiments covered a fairly wide range of height between approximately
75 and 150 meters.
The flight paths used by the helicopters were of two basic types:
lateral and vertical cross sections. The lateral cross sections were
made by flying through the plume at about 30 m increments in elevation.
23
-------
A typical number of passes at a given downwind distance would be six
or so. This procedure would be repeated at different downwind distances;
typically these distances were 0.8 km, 6 km, and again at between 16
and 32 km. Vertical traverses were made with the helicopter to determine
the distribution of concentration using the sampler. Then assuming a
Gaussian distribution in the horizontal and vertical, the curves were
integrated by approximating the area under the curve with a rectangle,
and using the relation
Area (where C is the maximum
o or o •- max
v z /
C /2TT measured concentration for (16)
the transect.)
The sampling time involved in the TVA study was basically the time
which it took the helicopter to pass through the plume, i.e. a few
minutes. Shorter times were required to traverse the plume during
stable conditions (the plume width being smaller). Temperature profiles
were determined at the beginning of the sampling period by the heli-
copter temperature probe. The temperature gradient? characterizing
the stability category for the published curves are meant to apply at
plume height. Wind speeds were measured with the pilot balloon technique.
These data were collected over a very long time period (5 to 10
years) so that dispersion values represent a large number of individual
cases in the final averages. The number of individual stacks repre-
sented is of the order of 10 or so.
Some differences relating to the TVA published a-values are
apparent. Since the temperature gradient was maasured at plume height
the data show no super-adiabatic temperature gradients, although there
24
-------
is an extremely high probability that these often existed near the
ground. The reason is that a large portion of the data were
collected during midday hours and during the summer season. It can
be concluded that unstable cases exist in "masked" form amongst the
neutral and slightly stable cases. The tendency for temperature lapse
rate to approach closely the dry adiabatic rate during convective con-
ditions is well known and can be demonstrated from tall tower temper-
ature measurements.
Another difference of the TVA data is the relatively short sampling
times, especially with respect to o .
The McElroy and Pooler (1968) Dispersion Curves for Urban Areas
McElroy and Pooler (1968) studied diffusion of tracer clouds
over St. Louis specifically to obtain dispersion data over urban areas.
This series of experiments consisted of releases of flourescent zinc-
cadmimuin sulfide particles. Twenty six daytime and sixteen evening
experiments were conducted over a two-year period. The samplers
were located on three circular arcs at distances between 0.8 and 16 km.
from the dissemination point. Winds were measured by several methods,
including tracking transponder equipped tetroons by radar and using the
single theodolite technique.
The tracer material was released from near ground level. Sampling
times were usually one hour. Determination of a was by direct means
y
whereas 0 was inferred through the crosswind integrated formula
(Eqn. 11).
A number of different systems were used to describe the turbulence
representative of the various experiments. A "modified gustiness class"
25
-------
similar to the Brookhaven gustiness class was used (different ranges of
wind direction fluctuations were adopted since the wind direction sensor
was located nearer the ground than at Brookhaven). Also used to dif-
ferentiate stability were the Pasquill scheme, CQ, and Ri (Richardson
u
number) values (See Appendix 2). The Richardson number was based on
temperature and wind measurements at the 38 and 138 meter levels of a
television tower.
Markee/s Dispersion Curves
Sigma curves derived by Karkee (Yanskey et.al., 1966) from tracer
experiments at the National Reactor Testing Station (Idaho Falls, Idaho)
are available. Tne diffusion experiments were carried out by Islitzer
and Dumbauld (1963) using uranine dye as a tracer. Releases were made
at ground level with sampling arcs at 100, 200, 400, 800, 1600, and
3200 meters. Sigma y's were determined directly, and sigma z's were
calculated by knowing the maximum of the crosswind distribution for
a ground-level source.
The roughness appropriate to this site is about 3 cm. The release
period and sampling time was 30 minutes. Other data were considered
in Karkee's curves, namely those from Project Green Glow (Fuquay et.al.,
1964).
The stability system used by Markee is the. same as Pasquill's
although different systems could be incorporated since the original
experiments contained measurements of CQ, Ri, u, and AT/Az.
Bultynck and Malet's (1972) Stability System and Dispersion Data
This system was developed for use at a reactor site in Belgium
(the Mol site); it differs from others in that it uses a stability
26
-------
parameter S defined as
36/3z
S =
_ 2 (17)
U69
The symbol u,q stands for the wind speed measured at 69 meters
2 3
height. This parameter (whose dimensions are °C sec /m ) is similar
to a Richardson number. The measurement height of wind is not critical
so long as it is confined to the height interval between 24 and 120
meters. Sixty-nine meters is the height of reactor stacks at the
Mol site.
Turbulence data were used to evaluate the Hay-Pasquill (1959) form
of the Taylor statistical theory (see Eqns. 12 and 13). The parameter
(3 was determined from the expressions of Wandel and Kofoed-Hansen
(1962)
(18)
(19)
The sampling time was one hour in all tests. Two years of data were
used to form the dispersion curves (Jan. 1966 to July 1968) representing
280 hourly observations.
Comparisons of these results with the BNL curves showed favorable
agreement. Dispersion data from experiments at the Mol site also com-
pared very favorably with the predicted values of a . Apparently no
comparisons of o were made. A total of fifteen tracer experiments
z
were done.
27
-------
TRC (The Research Corporation of New England) Curves for Rough and
Inhomogeneous Terrain
Bowne (1974) proposed a new set of dispersion curves based on
previously published data including McElroy and Pooler (1968), Hilst
and Bowne (1971), Bowne, Smith and Entrekin (1969), KcMullen and
Perkins (1963), Smith and Wolf (1963), McCready et.al., (1961),
Hamilton (1963), Haugen and Fuquay (1963), Towrin and Shen (1969),
Church et.al., (1970), Kangos et.al., (1969), and Taylor (1965),
primarily to solve dispersion problems in urban and suburban areas.
Bowne proposed three sets of curves appropriate for rural, suburban,
and urban terrain.
The rural dispersion curves are the same as the PG curves with
the following modifications; first the curves were extrapolated back
towards the source from 100 m to 1 m. There were no measurements of
a or a in this range for rural settings. However, this extrapolation
was thought to be appropriate in order to provide some information,
however crude, in predicting concentrations near highways where con-
centrations in the first 100 m or so are very important. Second, the
PG a curves were modified at distances of 3 to 20 km in unstable
z
conditions to account for the likely occurrence of limited vertical
mixing due to the presence of an elevated inversion layer (finite mixing
depth as mentioned earlier).
The suburban dispersion curves are based on previously published
studies, including experiments at Dugway Proving Ground. The roughness
corresponding to sage covered deserts is comparable to "typical
suburban" areas with trees and bushes. Data were available as close
28
-------
as 50 m from the source, in contrast to the rural curves. No o
y
curves are. published since Bowne felt that there was no significant
difference between rural and suburban areas in this respect. The a
z
values show an extension back towards the source (or x = 0) with all
stability categories approaching the same a value (5m). Bowne
anticipated the effect of the mixing height on the plume by again
showing constant, or flattened-off, dispersion parameters for the un-
stable cases.
The vertical dispersion rates proposed by Bowne for large cities
are based en data obtained from the St. Louis study (McElroy and Pooler,
1968), Ft. Wayne (Csanady, et.al., 1967), and Johnstowne studies (Smith,
1967).
Briggs* Interpolation Formulas
Briggs (1973) proposed a series of interpolation formulas for the
dispersion parameters, using as a basis previously collected dispersion
data. His rationale in forming these curves was tc aid in predicting
maximum ground-level concentrations from an elevated source. He felt
that the PG curves were most accurate at the short ranges and BNL curves
were probably more appropriate at intermediate and longer distances. At
extremely long distances he utilized TVA data.
The transition between the PG curves and the BNL curves was made
when the value of a approached the value of the source height of the BNL
studies, i.e. between 50 and 100 m. When oz approached 300 m, or so, the
TVA curves were given the most weight.
29
-------
Briggs also attempted, as best he could under the circumstances,
to fit the theoretical variation of o with distance predicted by the
1 /o
Taylor statistical theory, i.e. a * x at short distance and a « x at
long rango. Eriggs produced a separate set cf interpolation formulas
for urban conditions based on the St. Louis experiment. Briggs' dispersion
parameters were presented as convenient numerical formulae as well as the
usual dispersion curves.
Dispersion Data for ry in Terms of Travel Time
Fuquay, Simpson, and Hinds (1964) analyzed 46 diffusion experiments
from a ground level source at the Eanford Laboratories near Richland,
Washington. They expressed the. crosswind dispersion parameter a in terms
of travel time (x/u), instead of downwind distance x as is usually the case.
Their dispersion curves are parameterized in terms of aeu" (which is very
nearly equal to cr )r Part II of this report has much additional information
regarding crosswind dispersion.
At the conclusion of this section it is reccprized that r.any other
versions of dispersion parameters exist which rely on experimental work
or models that have not been summarized here. Many of these differ in
a quite insignificant way from the "original" Pasquill-Gifford curves.
Others, while having novel features, are site specific and thus their value
is somewhat limited. (This is recognized to be true of the Mol data
described earlier.) According to Gifford (private communication) at
least 30 additional sets of curves exist that would fall into these
categories. Obviously it would be of little use here to describe all
cf these variations.
30
-------
Comparative curves showing the vertical dispersion coefficient
from a few empirical methods are presented in Figures 3 through 8.
Since the PG curves are the most widely used they are the ones against
which each comparison is made.
Over rural areas during unstable conditions, and for distances down-
wind greater than 1 km, the PG curves are consistently high compared with
all others in the group under consideration. (At short ranges for unstable
conditions only the Markee curves show o less than the PG curves.) For
2
the case of neutral and stable conditions, experimental data for rural sites
tend to lie on both sides of the PG curves, indicating no systematic
disagreement.
In comparing the PG curves with others mentioned in this report,one
needs to be reminded of the fact that there has been no comprehensive
definitive set of data or method amongst those considered, which could be
used as a bench mark. Also,.it is difficult to assess all the factors
which could have contributed to differences in data sets that ought to
be similar.
Because of the earlier discussion on the effect of sampling time on
a , no comparison of these values will be made but some discussion of the
relation between a and o is included in Part II.
The BNL diffusion data (Figure 3, representing an elevated release ,
and indirect measurements of a at generally greater distances thgn were
represented by the basic data of the PG curves) show mostly lower values
of o than the PG curves, except for neutral stability. The slope of the
BNL curves (with the exception of very unstable cases) is not drastically
different from the PG curves over the range 0.1 to about 4 km.
31
-------
The TVA curves show the greatest difference fron the PG curves.
Recalling that there are probably aar.' crises ^i unstable data a^onc the
TVA "neutral" cases, this is not surprising.
The FcElroy-Pooler data for urban areas (Figure 5) show greater
values of a than the PG curves for the range of measurement. No A
category is listed but the B category is higher for McElroy-Pooler than
the PG B category (not shown in Figure 5). There is undoubtedly an effect
of the urban heat flux and greater roughness values, which will be discussed
in Part II of this report. The slope of the curves is again less than the
corresponding PG curves.
The curves proposed by Markee (Figure 6) are somewhat similar to the
PG curves, but are Lower in 0 values (except for neutral conditions) at
almost all downwind distances. The slope of Markee's unstable curve is
less than the PG curves.
Bowne's curves (Figure 7) weight the PG curves heavily at short,
intermediate and long ranges, for rural conditions. The effect of the
"lid" is dominant in the unstable case.
Briggs interpolation curves (Figure 8) for a are designed to follow
the PG curves at short distances and therefore, the agreement is good. It
is obvious from the earlier description of Briggs' system that it will
resemble BNL and TVA curves at longer ranges.
As noted earlier, an interesting aspect of these sets of data is
that all but one show disagreement with the PG curves for very unstable
conditions A (in that the downwind derivative of da /dx is positive).
z
32
-------
This seems to mean that the free convection regime studied by Deardorff
and Willis is somewhat elusive. Pasquill has made a detailed examination of
the model of Eeardorff and Willis (1974a,b) showing comparisons with other
theories in Fart II of this report.
33
-------
100
DISTANC
Figure 3. The dispersion curves of Singer and Finith (Smith, 1968)
(solid lines) and the PC, curves (dashed lines).
34
-------
10000 V~
1000-
100
UJ
21
HETEP£,
S: STAELE (0,64°K/100 KETE'S)
100
Figure 4. The TVA dispersion curves (Carpenter, et. al., 1971)
(solid lines) and the PG curves (dashed lines).
35
-------
10000IUZZEE
1000
iOO
DISTANCE DOWNWIND,
Figure 5. The St. Louis dispersion curves (McElroy and Fooler, 1968)
(solid lines) and the PG curves (dashed lines).
36
-------
10000 V^^
1000
DISTANCE DOWNWIND, K«
Figure 6. Markee's dispersion curves (Yanskey, et. al., 1966)
(solid lines) and the PG curves (dashed lines).
37
-------
IOOOO
1000
100
CO
a:
UJ
DISTANCE DOWNWIND, KM
Figure 7. TRC rural dispersion curves (Bowne, 1974) (solid lines)
and the PG curves (dashed lines).
38
-------
100001=
1000-
100
DISTANCE DOWNWIND, KM
Figure 8. Briggs' rural dispersion curves (Briggs, 1973) (solid
lines) and the PG curves (dashed lines).
39
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SECTION 5
IMPROVED METHODS OF ESTIMATING DISPERSION PARAMETERS
In this section we will examine some new approaches to the method
of estimating dispersion parameters. Five new approachs are considered
in some detail and a sixth is referenced briefly. There may bt: other
methods that we are not aware of because of rapid developments in the
field. Each of these new systems seems to offer something basically
different from what we have considered previously.
Smith's (1973) Systea (Published in Pasquill. 1974)
Smith addressed the problem of predicting concentration patterns
from a ground level source using the gradient transfer theory, incor-
porating numerical solutions 0*. the diffusion equation, and height
dependent diffusivity (K) values computed from the relation
K(z) = r. *m/15 (20)
(for a full discussion of this form of K and complete definitions of the
terms see Pasquill, 1975). (In Eqn. 20, e is the rate of turbulence energy
dissipation and A is a measure of the predominant eddy size.) Smith and
S.A. Mathews obtained the numerical solution of the two dimensional dif-
fusion equation
The height dependent values of e were obtained from a limited amount
of data derived from captive balloon ascents near Cardington, England.
The X profiles vere summarized on the basis cf Busch's and Panof sky's
(1968) studies of spectral scales. Several different atmospheric
stability conditions were represented by the e and A observations.
-------
Briefly, Smith's method involves specifying a geostrophic wind
speed, a roughness value, and (basically) a surface heat flux. These
determine the friction velocity UA in the surface stress layer. Then,
using generally accepted wind profile laws appropriate to the surface
layer, an interpolation is made to match the profile in the surface
layer to the geostrophic wind speed. It should be noted that Smith's
method does not involve any change of wind direction with height.
(The amount of crosswind dispersion which occurs at longer ranges from
the source is influenced by the turning of the wind as has been noted
earlier).
Having solved the two dimensional diffusion equation, Smith used
the concentration profiles to obtain values of dispersion parameters
directly from the relation
a = (cz2dz/ [Cdz .
Z } I )
The practical use of Smith's method is illustrated by Pasquill
(1974). A stability parameter P 3s specified ir terms of wind speed
at 10 m, and vertical heat flux or incoming solar radiation. If
desired, a rough approximation to P can also be obtained from insolation
(slight, moderate, and strong) and cloud amount. If the latter alter-
native is chosen, Pasquill categories A - F can be related to ranges
of the parameter P. Once P is obtained the only remaining parameter
to be specified is the roughness length. Convenient nomograms are
provided to obtain a versus x from z and P.
z o
Smiths a values, having been computed from the gradient transfer
Z
theory, are expected to be valid for vertical dispersion from a ground
41
-------
level source at all distances downwind (over homogeneous flat terrain),
and to be a good approximation for elevated sources at long distances
from the source. When the vertical spread is large compared with the
height of the source, the concentration approaches the value oi/.e would
obtain from a ground level source.
Lagrangian Similarity Theory
Pasquill (1975) has proposed an alternative to the usual formulation
of similarity theory, for which the limitation of restriction to the
surface-stress layer is less important. This proposal is based on
replacing the surface stress and surface heat flux by the standard
deviation of vertical wind speed o and the scale of turbulence £.
w
Both of the latter are to be considered as local, as opposed to surface
parameters, and both are regarded as functions of height in Pasquill*s
formulation.
To make practical use of the equations derived in this approach,
it is necessary to assume that £ is proportional to the maximum of the
vertical velocity spectrum X (a kind of measure of the size of the
predominant eddies). Observations of X are available for use in
m
application of the theory.
Using F.B. Smith's numerical solution of the two dimensional dif-
fusion equation in terms of X and e, these similarity hypotheses have
been tentatively verified for a range of values of z and heat flux Ep.
The theory lacks direct observational testing, however. A useful feature
of the foregoing is that for K profiles differing from those used by
Smith, the vertical spread can be derived from demonstrated similarity
-------
relations, thus avoiding repetition of the numerical solutions of the
diffusion equation.
Moore's Formulation for Power Plant Plumes
Moore (1972, 1974a,b) has given a semi-emperical formulation for
predicting the maximum ground level concentration C and the downwind
distance of this maximum. (Actually a virtual point source is introduced,
which is moved upwind to take account of induced spread, and downwind to
take account of plume trapping in inversions). One of the essential
features of Moore's formulation is the recognition of the different
functional growths of a and o with distance downwind, as follows:
az - L1/2x1/2 (24)
and
cv = Bx (25)
j
where L = 2K/u~ and E is a dimensionless constant relating a to x.
An additional significant feature of Moore's formulation is that he
tried to account for the induced spread of the plume resulting from
heated stack gases. Complete details of this rather complex scheme
including the values of L and B for various conditions are contained in
Moore (1974b).
Deardorff's Free Convection Modeling
Dispersion in the daytime under clear skies and light winds is
frequently dominated by convection. The condition known as "free"
convection is reached when turbulence is independent of surface drag
force. The atmosphere above such a convective mixing layer is capped
by a stable layer (of height z.^). The turbulence and transfer properties
under such conditions have been studied in the laboratory by Deardorff and
43
-------
Willis (1974a,b). Preliminary reports of the laboratory modeling
experiments indicate that the transport properties can be uniquely
represented by a universal relation between a /z. and t where
23. *
t = w /z. and (26)
gHpZ. 1/3
f f j-r -t
* p CpT
v^ being a characteristic vertical velocity.
One of the important qualifications of this approach concerns the
threshold wind speed in the atmosphere, below which one could expect the
free convection model to apply. An example from Pasquill (1975) estimates
that the wind speed below which it would appear reasonable to adopt the
theory nay be as little as 3 m/sec. In relation to the conventional
categories such a wind speed condition would include Category A completely,
but categories E and C only for the lower end of the range of wind speeds
associated with those classes. However, the whole question of applicability
of the Deardorff "convective" model to real atmospheric flow has yet to be
convincingly settled.
Second-Order Closure Modeling - Lewellen (1975)
Recently the problem of atmospheric dispersion, using the full
fluctuation equations coupled with a second-order closure assumption,
has been under investigation and is being actively studied by several
groups. These studies seem to offer considerable promise in improving
the rational basis of specifying c in the Gaussian plume model, or of
Z
predicting the concentration distribution.
Lewellen (1975) has produced several curves of a versus x based on
the second-order closure approach. The curves ere stratified on the
44
-------
basis of Richardson number Ri and Rossby number Ro. Agreement between
Lewellen's values of 0 and F. B. Smith's values is quite good in neutral
conditions. As mentioned above, in Lewellen's solution the dispersion
parameters are dependent on Ri and Ro. These parameters are computed from
observations of wind speed, temperature, and roughness length. There is
a correspondence between Ri and Smith's P parameter which Lewellen presents
in graphical form.
The second-order closure approach offers, what seems to be at present,
the only hope of theoretically determining a for intermediate range for
z
elevated sources, (i.e. a - H/z), at which range the statistical and
gradient transfer theory approaches are both most questionable.
Briggs' Investigations of Very Stable Flows
Recently G. A. Briggs has been working on the problem of dispersion
in very stable conditions. He has adopted the limiting form of K^ from
the Monin-Obukhov similarity theory. Results of his work have not been
published as yet.
45
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SECTION 6
EFFECT OF RELEASE ALTITUDE ON DISPERSION PARAMETERS
HORIZONTAL SPREAD
Pasquill (1975) has summarized our limited understanding of the
properties of the crosswind component of turbulence. In neutral flow
the important eddy size range is independent of height. For crosswind
spread in neutral conditions, the statistical theory ought to be appropriate,
except for large distances downwind where, because of the large depth of
the plume, wind direction change with height induces an enhanced crosswind
spread. Pasquill estimates this distance to be greater than 5 km for an
elevated source. For a gound level source the critical distance is
larger, approaching 12 km.
For non—neutral conditions we do not know enough at present about
the properties of turbulence to be able to predict th° effect of release
altitude.
VERTICAL SPREAD
As noted in Table 2 the statistical theory has proven useful in
describing the vertical spread from an elevated source at short and
intermediate distances, essentially upwind from the point where signi-
ficant portions of the plume touch the ground. Otherwise, we are at
present unable to specify the effect of elevation of the source in any
definitive way.
46
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Appendix 1: Symbols
1 time averaged concentration of passive material
C maximum ground level time averaged concentration of passive
material
C m maximum concentration during a transect of the plume
c specific heat at constant pressure
2 acceleration of gravity
1H plume rise due to bouyancy and inertia
H height of the plume centerline above the ground
':. physical stack height
^^^^^" •*• / ^
i intensity of turbulence; e.g. i = (u1 /u) etc.
r" vertical heat flux
/ scale of turbulence
n frequency
n^ frequency at which nS(n) is a maximum
/ equals uViL.
r' source strength
T total sampling period
'-^ Lagrangian time-scale
T absolute temperature
P Smith's stability parameter
SHFzt 1/3
w Deardorff's scaling velocity = ( —)
t. equals w.t/z.
* * i
t time
z. height of the inversion above the surface
z roughness length
B ratio of Lagrangian and Eulerian time-scales
47
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L equals 2K/u, a scaling length in Moore's formulation for power
plant plumes; also Monin-Obukhov length scale
x,y,z distance downwind, crosswind, and vertically upward, respectively
u average wind speed
t time
x distance downwind to the point of maximum concentration
m
x1 equals a /2/H
y' equals y/a H
.? density
2
a lateral variance of the concentration of the diffused material
2
a vertical variance of the concentration of the diffused material
z
OD variance of wind direction
2
a, variance of elevation angle of the wind
K eddy diffusivity
e rate of dissipation of turbulence energy
Ri Richardson number
Ro Rossby number
6 potential temperature or wind direction or angular crosswind of
a plume
h vertical dimension of a plume of windborne material, conventionally
defined by a concentration of material 1/10 of the ground level or
centerline value
u.,. friction velocity
a a constant independent of x
I -'- / ~ z
S a stability parameter; S = —:—~—
u
48
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2
S(n) spectral density as a function of frequency, / S(n)dn = a
u',v', components of velocity downwind, crosswind, and vertical,
w' respectively, representing fluctuations from the mean, i.e.,
u = u + u1
49
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Appendix 2; Relationships Among Turbulence Categorizing Schemes
For convenience in reading this report we will detail briefly some
of the relationships among turbulence categorizing schemes. Gif ford's
(1976) study contains a table (Table 4) shoving relations among the
various parameters and methods. Table 2 in this report is essentially
the same as Gifford's, except for some deletions and additions; e.g.
since Turner's categories are not used in the Workbook but rather Pasquill
categories, those are deleted.
The first column on the left contains the Pasquill categories. These
are based on solar insolation, surface wind speed, and cloud cover. A
refers to the most unstable category and F to the most stable, D being
neutral.
Category G is sometimes used in the U. S. in cor 1 unction with dis-
persion parameter curves under very stable conditions, (See the discussion
in Gif ford 5 19761' as to the origins of this category.')
The column, labeled BNL represents the Singer-Smith or Brookhaven
method of classification. This method is based on wind-vane direction
traces. The vane is used to measure variations of wind direction in tbe
horizontal plane. The standard deviation of this angle, a , can be related
to Pasquill categories. (See Slade, 1968 for details).
The Richardson number is defined as
.
Ri = T 3z (49)
!§.
3z
(3JK2
W
This is a nondimensional parameter expressing thp. ratio of two turbulence
producing mechanisms in the atmosphere, i.e. buoyancy and mechanical
production. It is often necessary and desirable to approximate the
50
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Richardson number by measuring temperature and windspeed differences
ever a separation of a few meters in height. In that case a simple finite
w — 1 i c 1 d*.«~c. c!p ^ IX XLi-HiciL ILOij j-G
(A8 and Au being the differences in vindspeed and temperature between the
height difference Az. Richardson number, at heights other than near the
ground, can be computed using a similarity law established by using experi-
mental data (Businger, et.al., 1971)
The column label L is the Konin-Obukhov length scale and is defined
pc, — i ' " C
T - _1_ P
" JSB.EF
T
This parameter is a scaling length which is used to describe atmospheric
turbulence, "ote that the value of L corresponds to the distance from
the ground where the mechanical production of turbulence term equals
the buoyancy production term (Pasquill, 1974).
By comparison with the U. S. Nuclear Regulatory r-uide 1.23, one can
see that the temperature change with height column has been omitted.
The use of temperature change with height is incorrect since it contains
provision for only one of the turbulence production mechanisms. However,
as a practical tool it continues to be favored by some groups, because of
a lack of demonstrated inferiority to other systems in dispersion calcu-
lations, and difficulties in measuring many of the other parameters.
For example, Pasquill 's system requires either an observer or instrumentation
recording wind speed, solar insolation, and cloud cover. The BNL and a.
o
methods require a windvane and ^possibly) electronic averaging circuitry.
Often vane response under low wind speed conditions is very unreliable.
51
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The Monin-Obukhov length L requires sophisticated measurements of
turbulence parameters making its direct measurment impractical.
52
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RELATIONS AMONG TURBULENCE-TYPING METHODS
(From Oifford, 1976 Tables 1 and ^
Pasquill (a)
A
B )
c }
D
E
F
BNL
B2
Bl
C
D
ae(c)
25°
20°
15°
10°
5°
2.5°
Ri(at 2m)(d)
-1.0 to -0.7
-0.5 to -0.4
-0.17 to -0.13
0
0.03 to 0.05
0.05 to 0.11
L(e)
-2 to -3
-4 to -5
-12 to -15
00
35 to 75
8 to 35
(a) Pasquill (1961 or 1974)
(b) Philadelphia Electric Co. (1970) (see below)
(c) Slade (1968)
(d) Pasquill and Smith (1971)
(e) Pasquill and Smith (1971)
B~: range of fluctuations of wind azimuth between 40° and 90°
B,: range of fluctuations of wind azimuth between 15° and 40°
C: fluctuations of wind azimuth exceed 15° but trace is "solid"
and unbroken
D: fluctuations of wind azimuth very small (approximating a line)
short-term fluctuations not exceeding 15°
(Fluctuations are recorded over a one hour period)
53
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57
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TECHNICAL REPORT DATA
Please read Inunctions on the reverse before completing)
1 nE^C^T NO.
EPA-600 /4-76-030a
3. RECIPIENT'S ACCESSION NO.
TITLE AND SUBTLE ATMOSPHERIC DISPERSION PARAMETERS IN
GAUSSIAN PLUME MODELING,, Part I. Review of Current
Systems and Possible Future Developments
5 REPORT DATE
July 1976
6. PERFORMING ORGANIZATION CODE
7 AUTHORtS)
A. H. Weber
8 PERFORMING ORGANIZATION REPORT NO.
9. PERFORMING ORGANIZATION NAME AND ADDRESS
Department of Geosciences
North Carolina State University
Raleigh, North Carolina 27607
10. PROGRAM ELEMENT NO.
1AA009
11. CONTRACT/GRANT NO.
12. SPONSORING AGENCV NAME AND ADDRESS
Environmental Sciences Research Laboratory
Office of Research and Development
UoS, Environmental Protection Agency
Research Triangle Park. North Carolina 27711
13. TYPE OF REPORT AND PFRIOD COVERED
In-house Sept. 75 - Mar. 76
14. SPONSORING AGENCY CODE
EPA-ORD
15. SUPPLEMENTARY NOTES
16. ABSTR-ACT
A recapitulation of the Gaussian plume model is presented and Pasquill's
technique of assessing the sensitivity of this model is given. A number of
methods for determining dispersion parameters in the Gaussian plume model
are reviewed. Comparisons are made with the Pasquill-Gifford curves presently
used in the Turner Workbook, Improved methods resulting from recent
investigations are discussed, in an introductory way for Part II of this report.
KEY WORDS AND DOCUMENT ANALYSIS
DESCRIPTORS
Air pollution
*Atmospheric diffusion
*Wind (meteorology)
*Plumes
^Mathematical models
b.IDENTIFIERS/OPEN ENDED TERMS
Gaussian plume
COSATi i icid'Group
13B
04A
04B
2 IB
12A
- '-~= e. ^~ ON STATEMENT
RELEASE TO PUBLIC
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UNCLASSIFIED
21 NO OF PAGES
69
20 SECURITY CLASS (This page)
UNCLASSIFIED
22 PRICE
EPA Form 2220-1 (9-73
59
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