\>EPA
United States
Environmental Protection
Agency
Environmental Sciences Research
Laboratory
Research.Triapgle Park NC 27711
EPA-600 4-79-062
October 1979
Research and Development
Scheme for
Estimating Dispersion
Parameters as a
Function of Release
Height ^
-------
RESEARCH REPORTING SERIES
Research reports of the Office of Research and Development, U.S. Environmental
Protection Agency, have been grouped into nine series These nine broad cate-
gories were established to facilitate further development and application of en-
vironmen al technology Elimination of traditional grouping was consciously
planned to foster technology transfer and a maximum interface in related fields
The nine series are.
1 Environmental Health Effects Research
2 Environmental Protection Technology
3. Ecological Research
4 Environmental Monitoring
5. Socioeconomic Environmental Studies
6 Scientific and Technical Assessment Reports (STAR)
7 Interagency Energy-Environment Research and Development
8. "Special" Reports
9. Miscellaneous Reports
This repoit has been assigned to the ENVIRONMENTAL MONITORING series.
This serie:; describes research conducted to develop new or improved methods
and insta mentation for the identification and quantification of environmental
pollutants at the lowest conceivably significant concentrations. It also includes
studies to determine the ambient concentrations of pollutants in the environment
and/or the variance of pollutants as a function of time or meteorological factors.
This document is available to the public through the National Technical Informa-
tion Service, Springfield, Virginia 22161.
-------
EPA-600/4-79-062
October 1979
SCHEME FOR ESTIMATING DISPERSION
PARAMETERS AS A FUNCTION OF
RELEASE HEIGHT
by
John S. Irwin
Meteorology and Assessment Division
Environmental Sciences Research Laboratory
Research Triangle Park, North Carolina 27711
ENVRIONMENTAL SCIENCES RESEARCH LABORATORY
OFFICE OF RESEARCH AND DEVELOPMENT
U.S. ENVIRONMENTAL PROTECTION AGENCY
RESEARCH TRINAGLE PARK, NORTH CAROLINA 27711
-------
DISCLAIMER
This report has been reviewed by the Environmental Sciences Research
Laboratory, U. S. Environmental Protection Agency, and approved for publication.
Mention of trade names or commercial products does not constitute endorsement
or recommendation for use.
AFFILIATION
Mr. Irwin is a meteorologist in the Meteorology and Assessment Division,
Environmental Sciences Research Laboratory, U. S. Environmental Protection
Agency, Research Triangle Park, North Carolina. He is on assignment from the
National Oceanic and Atmospheric Administration, U. S. Department of Commerce.
-------
FOREWORD
As discussed by Mr. Ken Calder the increasing concern of the last decade
in environmental issues, and the fuller appreciation that air quality simulation
modeling may provide a basis for the objective management of air quality, has
generated an unprecedented inter est 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 multi-
tude of recent publications testify to the widespread acceptance of meteor-
ological-type air quality modeling as an important rational basis for air
quality management.
Earlier attitudes towards the quantitative estimation of atmospheric dis-
persion of windborne material from industrial and other sources were strongly
influenced by a Gaussian-plume dispersion parameter scheme introduced in 1958,
and published in 1961, by Dr. F. Pasquill of the Meteorological Office, United
Kingdom. In spite of the gradual appearance in recent years of air quality
models based on more sophisticated formulations of the atmospheric processes,
great use continues to be made of the simpler Gaussian-plume models. As a
direct consequence of the unprecedented interest in the subject, a Workshop on
Stability Classification Schemes and Sigma Curves was held at the American
Meteorological Society Headquarters in Boston on 27-29 June 1977.
About 25 scientists representing a cross-section of regulatory agencies,
public utilities, universities, research laboratories, and consulting corporations
were invited by the AMS to the workshop. One of the conclusions reached at the
workshop was that the present characterizations of steady-state dispersion over
flat, homogeneous sites represent an unfinished state of the art. They suggested
that examination of dispersion data in terms of parameters now recognized as
important might resolve many of the differences so evident between presently
available Sigma schemes. However, they recognized that more observations are
required in order to confirm existing hypotheses of the more appropriate para-
iii
-------
meterization of dispersion from elevated releases.
This report presents one such investigation in which an attempt was made
to synthesize existing dispersion data into one generalized scheme using
currently hypothesized parameters of interest. The resulting generalized
scheme was developed especially with tall sources in mind but needs further
refinement and confirmation before it can be used in routine assessments of air
quality. In this regard, the scheme (or some simplification thereof) does
provide a framework for the analysis of future field data. It is perhaps
unnecessary to emphasize that the author resoundly corroborates the workshop
participants' recommendation for more field data of dispersion from elevated
releases accompanied by comprehensive meteorological measurements. Such data
is deemed necessary far the development of generalized scheme on a more sure
basis than current hypotheses concerning the parameterization of dispersion.
>/
Meteorology and Assessment Division
Environmental Sciences Research Laboratory
Research Triangle Park, North Carolina
March 1979
iv
-------
ABSTRACT
Recent concern about properly characterizing the Gaussian plume dispersion
parameters as a function of release height and surface roughness prompted the
American Meteorological Society (AMS) to sponsor a workshop on stability
classification schemes and sigma curves. Since sufficient field data was
lacking, a generalized scheme was not recommended at the workshop.
Based on an investigation where the dispersion parameters are assumed to
have the form a = a t F , a generalized scheme is presented for
z»y w, v z,y
estimating the dispersion parameters as a function of release height. Further
development is needed to refine the scheme for more generalized applicability,
since, as documented in this discussion, the scheme requires as input meteor-
ological data not routinely available. The scheme incorporates results from
various studies, and once it is more practically structured it will prove
useful for characterizing dispersion from tall soruces in a variety of situa-
tions. The generalized scheme was developed particularly for Gaussian plume
modeling; therefore, it is restricted to modeling applications having flat
terrain and having steady-state meteorological conditions.
This report covers a period from July 15, 1977 to March 15, 1979 and work
was completed as of June 15, 1979.
-------
CONTENTS
Foreword iii
Abstract v
Figures viii
Tables ix
Symbols x
Acknowledgment xi
1. Introduction 1
2. Theory and Development - Vertical Dispersion 3
Estimation of a 3
Estimation of a1 12
3. Theory and Development - Lateral Dispersion 13
Estimation of a 13
Estimation of a1 15
Estimation of a1' 16
4. Overview 17
Vertical dispersion 17
Lateral dispersion 22
5. Conclusion 25
References 27
Appendices
A. List of data 31
B. Subroutine SZSY 47
Vll
-------
FIGURES
Number
*
1 Values of F versus t for elevated releases
during convectively unstable conditions 7
*
2 Values of F versus t for near-surface releases
during convectively unstable conditions 7
3 Characterization of the T parameter for
use in Draxler's equations for estimating
the dispersion during netural and stable
conditions 10
4 Values of F versus t/T for surface and elevated
releases during neutral and stable conditions 11
5 Summary diagram of the characterization of the
vertical dispersion function, F , as a function
of effective release height, H , and stability 19
6 Summary diagram of the characterization of the
lateral dispersion function, F , as a function
of effective release height, H , and stability 23
A-l Vertical profile of a /w* for fully convective
W
conditions 33
vi 11
-------
TABLES
Number Page
A-l Porton Data Set 34
A-2 Tank Data Set 36
A-3 Numerical Data Set 38
A-4 Hanford Data Set 42
A-5 Studsvik and Agesta Data Set 44
A-6 Second Order Closure Data Set 46
IX
-------
LIST OF SYMBOLS
h — the depth of the convectively mixed layer
h — the release height (stack height) of the emissions
Ah -- the plume rise of emissions
is -- the von-Karman constant, 0.4
O _
L --0 u*/(k g w'e1), the Monin-Obukhov scaling length where
g is the acceleration due to gravity and w'e1 is the mean
surface kinematic heat flux
t — the travel time downwind of the release position
u -- the horizontal wind speed in the direction of pollutant
transport
u* -- the surface friction velocity
w* -- the convective velocity scale
x,y,z -- rectangular coordinates; x along the mean wind, y across mean
wind, z vertical. Origin located at ground level at release
position
H -- h + Ah, the effective release height
t. O
A0 -- the horizontal wind direction shear through the vertical
extent of the plume
~Q — the mean potential temperature
a — the standard deviations of the vertical and lateral wind
w,v
component fluctuations
-- the standard deviation of the vertical and lateral pollutant
z'y distributions
-------
ACKNOWLEDGMENT
The author is especially indebted to Dr. Francis Binkowski and Dr.
Robert Lamb, whose knowledge of dispersion processes and of atmospheric
turbulence structure were essential to the completion of this study. Both
provided perceptive counsel, and both generously donated research results and
data. The author is also grateful for Mr. D. Bruce Turner's advice throughout
the study. Finally, my thanks go to Pam Hinton and Joan £mory for their
secretarial assistance in the preparation of this report.
XI
-------
SECTION 1
INTRODUCTION
The following discussion documents the basis of a practical scheme to
estimate dispersion parameters that would be used in a Gaussian plume model
for estimating the surface concentrations of a pollutant emitted at an
elevated height. In a summary review of the workshop sponsored by the
American Meteorological Society on Stability Classification Schemes and
Sigma Curves, Hanna et al. (1977) suggest that a generalized scheme, explicitly
handling most of the processes affecting dispersion, might promote more
consistent modeling than presently exists and might provide more insight
than presently exists for directing further research. Workshop participants
noted that several schemes are currently used to estimate the dispersion of
material as a function of downwind transport. Each scheme has its inherent
limitations because of the data base used to construct the estimation scheme.
Reviewed at the AMS workshop were: (1) the data bases of current estimation
schemes and (2) several current rules of thumb that account for dispersion
processes not explicitly ascribable to the current schemes, such as surface
roughness and buoyant plume rise. Even when properly applied, the schemes
and rules of thumb represent an unfinished state-of-the-art.
Using the AMS workshop recommendations as a basis, a generalized scheme
was constructed. In the development and construction of the generalized
scheme it became apparent that more information would be needed in order to
complete the characterization of dispersion. Hence, the generalized scheme
was designed to facilitate future refinement and improvement through the
incorporation of new field data and new research results. The scheme views
dispersive processes such as buoyant plume rise, as independent of the other
ongoing dispersive processes. With this viewpoint, the total dispersion is
then the summation of the individual effects of each process (p. 207, Hbgstrbm
[1964] and p. 23, Pasquill [1976]).
1
-------
Hence, the total vertical dispersion a becomes:
2 2 '2
a - a + a
z zo z
where a = vertical dispersion due to atmospheric turbulence
az = induced vertical dispersion due to buoyancy effects
Currently, OZQ is viewed as a function of the surface roughness, of the
effective release height, of a surface stability parameter (such as the
Monin-Obukhov scaling length), and of the depth of the convectively mixed
layer (which represents the upper limit of vertical dispersion during unstable
conditions). Also, aZQ is affected by urban "heat island" effects; however,
the degree of urban effects likely depend on the size and layout of the
city, and further studies are needed to predict/estimate urban effects in a
generalized manner.
The total lateral dispersion, a , becomes:
p n ip lip
(7 — (J "4" 0 T CJ
y yo y y
where a = lateral dispersion due to atmospheric turbulence
a = induced lateral dispersion due to buoyancy effects
CT = induced lateral dispersion due to the turning with
height of the horizontal wind direction
Currently, a is viewed as a function of a surface stability parameter
(such as the Monin-Obukhov scaling length), of the depth of the convectively
mixed layer, and of the lateral velocity fluctuations (which are a function
of averaging time, of stability, and of terrain characteristics).
Sections 2 and 3 review the various techniques mentioned at the AMS
workshop for modeling the vertical and lateral dispersion as expressed by
Equations 1 and 2. The terms within Equations 1 and 2 are discussed individ-
ually in order to explain the rationale for the particular modeling techniques
selected for use in the construction of the generalized scheme. Section 4
summarizes the generalized scheme and presents an overview of the scheme in
its basic form. Section 5 concludes the discussion with an outline of the
analysis required to cast the scheme in a form practical for routine use.
-------
SECTION 2
THEORY AND DEVELOPMENT-
VERTICAL DISPERSION
ESTIMATION OF a^Q
Pasquill (1971) derived from Taylor's (1921) equation an expression for
estimating the vertical dispersion (neglecting buoyancy) a which Draxler
(1976) investigated in the following form:
azo = CTw t Fz(t/V (3)
where a = total standard deviation of the vertical wind
w
component fluctuations calculated or estimated over
a duration of about 1 hour
t = the travel time downwind of the release point
t. = the Lagrangian time scale
For F to be a universal function, independent of stability and effective
release height, and to satisfy Taylor's limits for small and large travel
times, conditions must be stationary and homogeneous both vertically and
horizontally. With such conditions, a would be proportional to /F at
large travel times. In reality the vertical turbulence structure is not
homogeneous. In discussing this point, Pasquill (1971) notes that for small
travel times the effects of nonhomogeneity are minimal and a should still
be proportional to t. But for large travel times the effects of nonhomo-
geneity can not be neglected, and F becomes a function of release height
and stability as well as travel time. Furthermore, it is reasonable to
expect a to be proportional to something other than /F for large travel
-------
times. For these reasons, as recommended at the AMS workshop, the following
parameters are needed to characterize a :
I = surface roughness length
H = h + Ah, the effective release height, where h is the
Co 5
stack height and Ah is the buoyant plume rise
L = Monin-Obukhov length scale (a stability parameter)
h = depth of the convectively mixed layer
The results of similarity theory suggest that cr can be envisioned as a
W
function of H , L, and Z (see Binkowski, 1979), where H is restricted to
heights within the surface layer where u* (the surface friction velocity) is
constant. Studies by Kaimal et al. (1976) and by Willis and Deardorff
(1974) suggest that a is a function of h, L, H , and u*, where H is now
W 66
within the convectively mixed layer. Combining these results with the
characterizations of FZ by Draxler (1976) and by Nieuwstadt and Van Duuren
(1979) presents much evidence that OZQ can be considered functionally as:
azo = 'w^e'1*^'"*) ^("e'1""'11*'1!'^
H , L, h, and u* seem necessary to characterize F , because the basic assump-
tions of homogeneity used to develop Equation 3 are in practice violated.
In simplistic terms, F describes the growth of a as a function of downwind
transport and responds to the overall structure of the turbulence; whereas
a describes the local features of the problem, assuming the structure of
W
turbulence remains fixed over the transport downwind.
Within the context of this discussion, the various schemes for estimating
dispersion can be examined as special cases of Equation 4. In the case of
the Pasquill-Gifford (P-G) scheme (Turner, 1970), values for Hg and Z have
been specified as surface release height and 3 cm, respectively; however,
values for h, L, and u* are not available. Similarly, the Brookhaven (BNL)
scheme (Smith, 1968) values for H and Z are 100 m and 1 m, respectively;
however, values for h, L, and u* are not available. Conceivably, rules of
thumb could be employed to adjust these schemes for other modeling situations,
but in practice many questions arise. For instance, a description of the
-------
effects of surface roughness on vertical dispersion was developed by F. B.
Smith (1972) using K-theory. These results are for surface releases and address
only vertical dispersion. Hence, these rerults are questionable if applied to
the BNL scheme. Furthermore, to handle release heights other than at surface
and 100 m using these two schemes: (1) the schemes would individually have to
be adjusted to the particular situation with respect to L, Z , h, u*, and t^,
and (2) adjustment would have to be made for H . The required adjustment would
be very difficult since most of the original values of the parameters are
unknown. In addition, the appropriate manner to extrapolate to other release
heights is unclear. Because of the many uncertainties involved, most current
schemes for estimating dispersion are used unadjusted.
Considering these uncertainties, it was decided to characterize a in
some other manner than by tailoring existing schemes (such as the BNL or P-G
schemes). Pasquill's (1978) review of the various theoretical models charac-
terizing dispersion suggests that each of these models is valid for only a
limited range of conditions. However, these models do provide valuable results,
and even their piecemeal inclusion is important in formulating a generalized
model. Hence, investigation began into further development of existing charac-
terization of F , using results gained from laboratory studies and theoretical
characterizations of dispersion. Using Equation 4 to characterize a allows
most of the factors affecting a to be handled explicitly; see Draxler (1976)
and Nieuwstadt and Van Duuren (1979). And, as shown by Nieuwstadt and Van
Duuren, this basic model lends itself to ready adaptation and improvement as
data become available from new field studies. Furthermore, Cramer (1976)
reported the utility of characterizing a for elevated releases in this manner.
Cramer used a very simple characterization of the terms in Equation 4. A power-
law relationship was used to characterize a as a function of height where the
w
power-law exponent was varied as a function of stability and F was set to a
constant value of one. Hence, Draxler1s results and Nieuwstadt and Van Duuren1s
results represent refinements to an already proven modeling technique. The
fact that Cramer's model performed as well as it did, suggest that for elevated
releases F may not have to be specified in detail. This mitigates to some
extent the concern expressed at the AMS workshop regarding the routine use of
models using characterizations of F before a consensus as to the detailed form
of F has been reached.
-------
The following discussion describes the characterization of F and is
divided by stability into two parts - unstable conditions and neutral-to-stable
conditions. The division was made because the set of parameters found useful
for characterizing FZ during unstable conditions was found to be different than
the set of parameters found useful for neutral and stable conditions.
Unstable Conditions
Although Draxler (1976) did specify a form for F for both elevated and
near-surface releases during unstable conditions, he did so on a limited data
base. The Porton data, Hay and Pasquill (1957), were the only data for elevated
releases providing direct measurements of both the vertical dispersion and the
standard deviation of the elevation angle a . Unfortunately, other extensive
sets of field data for possibly improving Draxler's results do not exist.
However, laboratory data do exist that contain direct measurements of the
vertical dispersion during convectively unstable conditions, resulting from the
tank experiments performed by Willis and Deardorff (1976, 1978). Their data
were for scaled release heights, H /h, of 0.067 and 0.24, where H was the
actual release height since Ah was zero. To these data can be added the numerical
results reported by Lamb (1978) for convectively unstable conditions. The data
presented by Lamb were for a scaled release height of 0.26 where Ah was zero.
Lamb (1979) also h^s results for scaled release heights of 0.025, 0.50, and
0.71 where Ah was zero. The Porton data, the tank data, and the numerical data
are listed in Appendix A.
For the Porton data, h was assumed to be 1000 m. Values of F were computed
* z
as a /(a t) and depicted graphically as a function of t = w*t/h, where w* is
£* W
the convective velocity scale and is equal to:
w* = u*( -h/(k L) )1/3 (5)
In Equation 5, k is the von Karman constant. The results are depicted separately
in Figures 1 and 2, since the data seem to suggest a different variation for F
•it *-
as a function of t for releases where H /h was less than or equal to 0.67
(Figure 2) from the variation suggested for releases where He/h was greater
than or equal to 0.15 (Figure 1). The curves shown on Figures 1 and 2 are
hand-drawn best fits to the trend reflected in the data values. For t greater
than 1, the distribution of the material by dispersion is effectively becoming
-------
10
Iff
- O He/h = 0.24
" D He/h = 0.26
" A He/h = 0.50
" • He/h = 0.71
— • He/h = 0.15
0.01
0.10
Figure 1. Values of Fz versus t* for elevated releases during convectively
unstable conditions.
10
1.0
Mill
_OHe/h = 0.025
— DHP/h = 0.067
I I I Mitt
0.01
0.10
1.0
10
Figure 2. Fz versus t* for near-surface releases during convectively
unstable conditions.
7
-------
limited by the presence of a finite vertical domain, h, in which dispersion
can take place. The best method to characterize a for t greater than 1
depends on the type of Gaussian plume dispersion model to be used. If the
dispersion model assumes reflections occur at the surface and at h, then no
limit need be placed on the growth of a . For models which assume such
multiple reflections, a continued rate of growth proportional to /t* could
be used, as shown in Figures 1 and 2. Such an assumption is admittedly
arbitrary, since we would not expect the growth rate of the vertical dis-
persion to obey the limits suggested by Taylor's theory for long travel
times, a point discussed in the introduction of this section.
The standard deviations of the pollutant distribution used in developing
Figures 1 and 2 were Computed using vertical profiles of the pollutant
distribution. For these particular data sets, Ah was zero; hence, a was
assumed to be equal to a . The height used in the computation of a from
the profiles was the release height. As long as the distributions are
Gaussian, we can use Figures 1 and 2 to estimate a and use these estimates
directly in a Gaussian plume model to estimate concentrations.
During convectively unstable conditions, the vertical dispersion of the
pollutant material is controlled by the vigorous updrafts of the hot thermals
and by the broad areas of slowly decending air. The effect of this convective
structure on the turbulence structure has been reported by Kaimal et al.
(1976), Willis and Deardorff (1976, 1978), and Lamb (1978, 1979). In brief,
the distribution of vertical velocity fluctuations at most any height within
the convectively mixed mixed layer is not Gaussian; as a result, for nonbuoyant
releases of material, Lamb (1979) reports a systematic, non-Gaussian distri-
bution of material in the vertical. For buoyant releases, the preliminary
findings by Lamb (personal communication) indicate that the pollutant distri-
bution tends to be more Gaussian in nature.
More research is needed to determine if buoyant releases when modeled
using a Gaussian plume model require special handling similar to that required
(Lamb, 1979) for nonbuoyant releases. In any case, these findings do raise
questions about how to best use 0 values inferred from the surface concentra-
tion pattern by employing a Gaussian plume assumption. Some field studies have
8
-------
characterized the vertical dispersion based on surface measurements by
assuming the distribution in the vertical to be Gaussian. If, as Lamb's
results suggest, the departure from a Gauss-'an distribution in the vertical
is a function of the source characteristics, then characterizations of
vertical dispersion from surface observations can be used directly only for
sources having like characteristics as those used in the field study. To
generalize such results to handle other source types would require rarely
available data concerning the true vertical distribution taking place during
the field study. For this reason, the caracterization of F has been based
solely on a values measured directly. If future studies indicate adjustments
are needed in order to correct for non-Gaussian pollutant distribution,
these adjustments can be made within the Gaussian plume model as Lamb suggests
without altering the characterization of F . Another benefit of characterizing
FZ from az values measured directly is that the characterization of F can be
improved using field experiments without having to invoke any assumptions
regarding the vertical distribution of material.
Neutral and Stable Conditions
For stable and neutral conditions, the Porton data were the only field
data used by Draxler that contained direct measurements of both the vertical
dispersion and the standard deviation of the elevation angle a . Of the
other field data used by Draxler, several cases of near-neutral data presented
by Hogstrom (1964) and by Hilst and Simpson (1958) were adapted for use in
this study by assuming that conditions were neutral and estimating a . In
addition to these field data, a portion of the second-order closure results
of the vertical dispersion versus downwind travel time reported by Lewellen
and Teske (1975) were used. Lewellen and Teske depicted their profiles of
turbulence during neutral conditions; therefore, their estimates of the
vertical dispersion were usable for a surface release and an elevated release
in this study. The second-order closure results and the mentioned field
data are listed in Appendix A.
Values of F = a /(a x), where x is the downwind distance, were
plotted as a function of travel time from which the time T (when F was
approximately equal to 1/2) was subjectively determined. This type of analysis
-------
seemed appropriate, considering Draxler's analysis. The small travel times
(<40 s) of the Porton data did not prove useful in defining the variation of
FZ as a function of travel time. Hence, the Porton data were not used in
succeeding analyses.
Figure 3 depicts the results of the above analysis, where T has been
plotted versus the release height of the data set from which T was determined.
200
in
-a
0
6
100
I I
/
7 He(m)
/ 0570
/
/ D 87
\?—
-------
T (sec) =
50
(3He - 50)/2
200
He £ 50 m
50m< Hg > 150 m
He £ 150 m
(6)
Using this characterization of T , values of F were plotted as a function of
t/T along with the Draxler (1976) characterizations of F (Figure 4). The
10
1.0
0.1
0.945(T/T0)°-806]
1/[1+0.90(T/T0)1/2]
O He = 570m
O He = 87m
A Hri = 56m
He= 50m
Om —
0.01
0.1
1.0
10.0
100
1000
Figure 4. Values of Fz versus t/To for surface and elevated releases during
neutral and stable conditions.
dashed line in Figure 4 is Draxler1s recommendation for r for surface releases;
Fz =
0.9(t/T)1/2
(7)
and the solid line is Draxler1s recommendation for F for elevated releases:
0.945(t/To)°'806}
(8)
Draxler1s (1976) characterization of FZ for elevated releases, Equation 8,
seems to perform well for effective release heights above 50 m. And his
characterization for surface releases, Equation 7, seems to follow Lewellen and
Teske's (1975) results for a surface release. All the data do not fit these
characterizations, as some deviations can be seen. However, whether the
11
-------
deviations occur because of data problems or because of characterization
problems is difficult to determine. More data are needed before a complete
characterization of a during neutral and stable conditions can be developed.
ESTIMATION OF a*
Pasquill (1976, 1979) gives an expression for estimating the induced
i
dispersion due to buoyancy, a , as:
a'z = Ah/3.5 (9)
In Equation 9, Ah equals the buoyant plume rise. In discussing buoyancy-
induced dispersion, Pasquill (1974) notes that during buoyant plume rise
strong evidence indicates a linear relationship exists between the instant-
aneous plume depth a.id the plume rise; however, no data base exists allowing
direct comparisons between the induced dispersion of a rising buoyant plume
and the natural dispersion and time-averaged dispersion of a passive plume.
The linear relationship is evident in Figure 4.1 of Briggs (1969), which
depicts plume depth versus plume rise as inferred from photographs taken at
three TVA power plants. Pasquill (1974) notes that further evidence regarding
the magnitude of the induced dispersion is available from Hogstrom's
(1964) measurements. Hbgstrbm1s data were developed from successively
released puffs of smoke photographed from the point of release, making it
possible to compare the instantaneous size and scatter of the puff centers.
The TVA studies of plume dispersion and Hogstrom's results suggest that the
total induced plume depth due to buoyancy can be approximated as equal to
the plume rise. Pasquill assumes the distribution of concentration within a
buoyancy-dominated plume can be approximated as uniform. On this basis, the
equivalent standard deviation becomes Ah/(2 /3~ ), which leads to the expression
given in Equation 9.
12
-------
SECTION 3
THEORY AND DEVELOPMENT-
LATERAL DISPERSION
ESTIMATION OF a
As a starting point the following formula was suggested (Hanna et al.,
1977) for estimating the lateral dispersion (neglecting buoyancy and direc-
tional shear effects), a :
ayo = av * Fy(t/tL} (10)
In Equation 10, a is the standard deviation of the horizontal crosswind
component of the wind. The estimate of a should have an averaging time
comparable to the averaging time to be associated with the concentration
estimate. The above expression for a was derived by Pasquill (1971) from
Taylor's (1921) equation and assumes that conditions are stationary and
homogeneous, vertically and horizontally.
Various estimates of F are available (e.g., Pasquill, 1976; Draxler,
1976; Deardorff and Willis, 1975). Even though different data bases have
been used, the various estimates of F are quite similar.
Recently Nieuwstadt and Van Duuren (1979) proposed a generalized form
for F , valid for unstable conditions, which, in the limit of convectively
unstable conditions (when -h/L -> °°), approximates Deardorff and Willis's
1975 result:
a /h = 0.51 t*/{l+ 0.91 t*}^ (11)
t* = w*t/h
13
-------
In the limit of neutral conditions (when -h/L -> 0), the form approximates
Draxler's (1976) result:
ayo = av
TQ = 1000 s
The appeal of Nieuwstadt and Van Duuren's form for F is that it explicitly
incorporates the scaling parameters appropriate for both convectively unstable
conditions and neutral conditions.
Their proposed characterization of F was:
Fy = !/{!+ t/T.r (13)
In Equation 13, T.. = (c u*/h){l+ d(-h/L)} ' . They completed their charac-
terization of <
et al. (1977):
terization of a by using a form for a similar to that proposed by Pasquill
av = a u*{l+ b(-h/L)}1/3 (14)
The constants a, b, c, and d were solved for by constraining a to conform
to Deardorff and Willis's result when -h/L -> °° and constraining a to
approximate Draxler's result when -h/L -> 0. They used Tennekes's (1969)
relationship to estimate the boundary layer height in the neutral limit as:
h = 0.3 u*/f (15)
-4 -1
In Equation 15, the Coriolis parameter f was estimated to be 10 s .
Nieuwstadt and Van Duuren compared estimates of a using their generali-
zed characterization of a and F with lateral dispersion measurements from
six tracer experiments performed at Cabauw in the Netherlands. These data
were for conditions between convectively unstable conditions and neutral
conditions, when 0 < -h/L < 15 and the horizontal wind speed at release
height (generally 200 m) varied between 4 and 10 m/s. The few cases of
field data available compared quite well with their estimates of a .
More for cosmetic than technical reasons, a minor embellishment to the
Nieuwstadt and Van Duuren characterization of F is suggested. Since the
forms of Equations 11 and 12 differ, it is suggested that Deardorff and
Willis's result be approximated as:
14
-------
a Q/h = 0.51 t*/n+ 0.45 t* } (16)
and allow F to have the form:
Fy = !/{!+ (t/T.f2) (17)
Now Equations 12, 16, and 17 have the same form. This allows the constants
a, b, c, and d to be specified such that in the convective limit of -h/L
-»- ~, a will equal Equation 16 and in the neutral limit of -h/L -> 0, a
will equal Equation 12.
In solving for the constants, the neutral value for a /u* of 1.78 suggested
by Binkowski (1979) can be employed, in which case the von Karman constant, k,
should be 0.4. According to Nieuwstadt and Van Duuren, (1979) Equation 15 is
used to estimate h in the limit as -h/L -> 0. The procedure to solve for the
constants is the same as outlined by Nieuwstadt and Van Duuren and results in:
a = 1.78
b = 0.059
c = 2.5
d = 0.00133
The preceding discussion outlined a characterization of a for unstable
conditions that, as conditions become neutral, matches Draxler's characteriza-
tion for a valid for neutral and stable conditions. As with the characteri-
yo
zation in Section 2 for a , the characterization presented for a explicitly
handles many of the parameters considered important for lateral dispersion.
By using data from studies in which these parameters are known or measured,
much as Nieuwstadt and Van Duuren did, it is expected that the characteri-
zation can be further developed and verified.
ESTIMATION OF a'
In buoyancy-dominated plumes, the plume cross section concentration
pattern, although complex, is usually considered symmetrical with the lateral
and vertical dimensions being equal; see, for example, the discussion on page
63 of Briggs (1969). Although allowances were recommended (Hanna et al.,1977)
for buoyancyinduced dispersion in the lateral direction, the problem was not
discussed explicitly. However, Briggs (personal conversations) suggests, for
15
-------
for simple Gaussian plume modeling, it is appropriate to assume that the induced
lateral dispersion due to buoyancy a' is equal to the induced vertical dispersion
due to buoyancy a1. Hence, a' can be estimated as:
c'y = o'z = Ah/3.5 (18)
ESTIMATION OF a"
•j
Pasquill (1976, 1974) discusses the effects of wind direction shear in the
vertical on the time-averaged lateral dispersion. Interaction between
the vertical spread and--the turning of the wind direction with height enhances
the lateral dispersion. Inspection of instantaneous cross sections of plumes,
such as developed by aircraft sampling, reveals systematic displacement of the
plumes in accordance with the veering of the wind direction with height; see,
for example, Schiermeier and Niemeyer (1970) or Brown et al. (1972). However,
time-averaged crosswind distributions at various levels do not reveal similar
structure in accordance with the veering action of the wind direction; see
discussion on page 229 of Pasquill (1974). Pasquill notes that a complete
understanding of the importance of the induced dispersion due to wind direction
shear in the vertical a" has not been reached. However, he suggests as a rough
J
rule, that the total instantaneous crosswind spread (expressed in an angular
sense) at large downwind distances, on the order of 20 to 100 km, is equal to
0.75 A0, where A0 is the change of wind direction over the entire vertical
extent of the plume. Hence, a" can be estimated as:
a" = (0.75/4.3) x A0 = 0.174 x AQ (19)
Here, x is the distance of transport downwind.
For estimates of the total time-averaged lateral dispersion a , a" would
have to be combined with the contribution to the lateral dispersion due to wind
n
direction fluctuations a . Existing data suggest a contributes little to the
total lateral dispersion for downwind distances of less than 10 km. For downwind
distances more than 20 km, the effect is more likely to be important in character-
izing the total lateral dispersion.
16
-------
SECTION 4
OVERVIEW
In Section 2 and Section 3, some of the techniques available for modeling
various processes affecting dispersion from elevated releases were discussed.
Based on the currently available data and results, a technique was selected
for estimating each of the processes. Further development and refinement is
needed in all of the techniques selected. Indeed, some of the techniques re-
present approximations developed more through a combination of theory and in-
ferences than from analysis of field data designed specifically to investigate
the particular dispersive process. The discussion presented in this section
is devoted to an overview of the generalized scheme which can be formed through
the synthesis of the selected techniques. Some of the difficulties of the
various techniques selected have been noted in Sections 2 and 3. Other diffi-
culties that arise from synthesizing the techniques into one scheme will be
discussed in this section. Appendix B presents a FORTRAN listing that performs
the computations outlined in the following discussion.
VERTICAL DISPERSION
Based on the discussions within Sections 1 and 2, the total vertical
dispersion, a , can be approximated as:
CTz = (ffw * Fz)2 + (Ah/3-5)2 (2°)
where a = total standard deviation of the vertical wind component at the
w
effective release height H
t = travel time downwind, usually approximated as x/u. where x is the
downwind distance and u. is the wind speed at thesrelease height
»s
Ah = buoyant plume rise
H = h + Ah, the effective release height
c 5
17
-------
The dispersion function, F , is a function of stability, effective release
height, and travel time downwind.
As discussed in Section 2, the parameters useful for characterizing F
are different for convectively unstable conditions than for neutral and stable
conditions. During convective conditions, the convective velocity scale, w*,
is an appropriate scaling parameter; however, as conditions become neutral
(-L -*-«>), w* -> 0 (Equation 5). Furthermore, w* is appropriate for use only
within the convectively mixed layer. This convective layer extends from a
height |L| from the surface to a height h from the surface. The layer extending
from the surface to height |L| is considered the surface layer, or constant
stress layer, where u* (the surface friction velocity) is nearly invariant
with height. The characterizations of F presented in Figures 1 and 2 are for
dispersion within the convective layer when conditions are convectively unstable
(roughly, when -h/L > 10). Inspection of Figure 1 suggests that FZ is independ-
ent of effective release height for scaled heights H /h greater than 0.25.
For releases where H /h is less than 0.25, F seems to be a function of height.
A comparison of the results depicted in Figure 1 with those depicted in Figure
2, suggests that when H /h = 0.067, F could be estimated using a linear inter-
polation between FZ for H /h of 0.25 and F for H /h of 0.025.
For releases where Hg/h is less than 0.025, FZ could be estimated by
extrapolation. Care must be taken in such extrapolations that H is not
within the surface layer; in other words, -H /L should be greater than 1.
However, this restriction may not be sufficient. For instance, convectively
unstable parameterizations of the vertical velocity fluctuations seem to fail
for heights less than 0.005 h; see Figure 4 of Irwin (1979). Hence, the
results presented in Figures 1 and 2 are considered useful for characterizing
F for effective release heights greater than 0.005 h and less than h when
-h/L is greater than 10.
The preceding paragraph discussed some of the limitations of the
characterization presented for F for unstable conditions. The following
discussion addresses some of the limitations of the FZ characterizations
presented for neutral and stable conditions. During stable conditions, it is
convenient to envision the atmosphere as being divided into a number of horizontal
18
-------
layers. In the surface layer, extending to a height of order |L| the
surface friction velocity is nearly invariant with height. Above this
surface layer is the transitional layer (Planetary Boundary Layer) between
the disturbed flow within the surface layer and the smooth frictionless flow
of the free atmosphere. The characterization of F presented for neutral and
stable conditions is considered useful within the Planetary Boundary Layer
and the surface layer. Of the dispersion data available, the majority is for
near-neutral conditions. Hence, the characterization of F is tentative,
especially for stable conditions. Fortunately, the vertical dispersion due
to atmospheric turbulence, a , is at a minimum during stable conditions.
Hence, for elevated releases, the tentative nature of F will be of little
concern so long as the terrain is flat and conditions are steady-state. If
conditions are not steady-state during the transport downwind or the terrain
is not flat, none of the characterizations presented for F are likely to be
relevant.
TOP OF PLANETARY BOUNDARY LAYER
v>
ft
-------
Figure 5 summarizes the preceding discussion on characterizing F . Based
e preceding discussion, FZ for unstable conditions when -h/L is greater
than 10:
a fc + 3 fQ when 0.005 < H /h < 0.25
r — s e e
(21)
when
where a = (10/9)0 - 4He/h)
= 1 - a 3
{(1+1.88 t*)/(l+ t* )}** - 4.5 t*(l- t*)4
V
0.44/t*)*2
* *<
1 - 0.7 t + 0.2 t
(0.25/t*)*8
He/h 1 0.25
for t < 1
for t >_ 1
for t* < 1
for t > 1
t = w*t/h
For unstable conditions when -h/L is less than 10, w* is no longer appro-
priate for use in characterizing the dispersion. No simple device has been
found to transition from Equation 21 and asymptotically approach the results
valid for neutral conditions. At present, it is suggested that the character-
ization of F for neutral conditions be used for the slightly unstable conditions,
Hence, the characterization of FZ for slightly unstable conditions (-h/L < 10)
and for neutral and stable conditions is:
a
6-
for
He < 50 m
(22)
for 50m <_ H < hpB,
where a1 = 1 - He/50m
3' = 1 - a1
= the depth of the Planetary Boundary Layer
= !/{!+ 0.9(t/To)i'2}
fe =
0.945(t/T0)0'806}
20
-------
50 s
(3He - 50m)/2m
200 s
50m<
50 m
150 m
150 m
The gray hatched area in Figure 5 for H /h less than 0.005 denotes the situa-
tions where the characterization of F (Equations 21 and 22) has not been
specified.
For nonbuoyant releases when H /h is greater than 0.25, Lamb (1979) suggests
that H in the Gaussian plume model be adjusted in order to properly estimate
surface concentrations. For instance, for an elevated nonbuoyant release,
the Gaussian plume equation (for estimating maximum surface concentrations
beneath the centerline of the plume, for downwind distances where the dispersion
has not been affected by the presence of a limit to dispersion h) could be
written as:
exp
- VzV
" 1 /RH
- o 1 f
\°z
2'
X =
where x = the surface concentration, g/m
Q = the emission rate, g/s
H = the effective release height; in this case, simply hc
C -•
Ah is zero
(23)
if
R is the correction factor to H needed to force Equation 23 to yield proper
estimates of the surface concentrations resulting from dispersion from a non-
buoyant elevated release. R varies as a function of downwind distance as
depicted in Figure 6 of Lamb (1979). Simple approximations to Lamb's results
are, for H /h >^ 0.25
R =
1
h/4 + (He - h/4)(2 - x1)
h/4
for x1
for 1 < x1
for x1
£ 1
< 2
> 2
(24)
where x1 = (x/H
For He/h < 0.25,
R = 1
(25)
21
-------
Equation 24 and 25 simplify Lamb's results. This is especially true for
Equation 25, since the locus of maximum concentration tends to rise off the
surface for surface releases in convectively unstable conditions. However, the
maximum surface concentration for surface releases occurs so close to the
release point that R is essentially equal to 1. It is important to stress that
Equations 24 and 25 have been shown necessary only for nonbuoyant releases when
conditions are convectively unstable. Later studies may suggest that R needs
further characterization to handle other situations which have a non-Gaussian
distribution of pollutant material in the vertical or for certain types of
buoyant releases. At present, R is set to 1 for all cases except those cases
meeting the criteria set forth when Equation 24 is appropriate.
LATERAL DISPERSION
Based on the discussions within Sections 1 and 3, the total lateral dis-
persion a can be approximated as;
CTy = (av * Fy)2 + (Ah/3-5)2 + (°-174 x A0)2 (26)
where a = the total standard deviation of the lateral wind component
at the effective release height (should have an averaging
time the same as the averaging time to be associated with the
concentration estimate)
A0 = the average change in the horizontal wind direction over the
vertical extent of the plume (in radians)
The dispersion function F is mainly a function of stability and travel time
J
downwind.
As discussed in Section 3, the parameters useful for characterizing F
change in going from convectively unstable conditions to neutral and stable
conditions. Whereas w* is the appropriate scaling velocity during convectively
unstable conditions, u* is the appropriate scaling velocity within the surface
layer (during stable or unstable conditions) and within the Planetary Boundary
Layer during neutral and stable conditions. Hence, the characterization of F
presented in Section 3 is appropriate except for effective release heights less
than 0.005 h when conditions are convectively unstable. Also, T was chosen to
22
-------
be 1000 seconds. (Draxler (1976) suggests T is 300 s for surface releases
during stable conditions). Hence, the characterization of F presented is not
suggested for releases less than 50 m.
TOP OF PLANETARY BOUNDARY LAYER
UJ
CO
UJ
HI
a:
O
\-
X
g
UJ
X
^L =
avT
I.
1/3
Hp/h = 0.005
qy
avT
0.9(T/T0)1/2
T0 =1000 sec
He = 50 meters
UNSTABLE
NEUTRAL
STABLE
Figure 6. Summary diagram of the characterization of the lateral dispersion function, Fy,
as a function of effective release height, He, and stability.
Figure 6 summarizes the preceding discussions regarding the characteri-
zation of F . Specfically, for unstable conditions when H /h is greater than
0.005 (or Hg is greater than 50 m - whichever is the larger height), F can be
approximated as:
(27)
where T1 = (2.5 u*/h){l + 0.00133(-h/L)}1/3
23
-------
The value of the von Karman constant assumed in the above expressions is
0.4. For neutral and stable conditions:
F =
*j
0.90(t/1000s)'
(28)
The gray hatched areas in Figure 6 denote where the above characterization
of F is not appropriate.
The third term in Equation 26 requires A0 to be evaluated over the
vertical extent of the plume. The total vertical extent of the plume can be
approximated by assuming a Gaussian distribution in the vertical, centered
about the effective release height. The top of the plume Tp and the bottom
of the plume Tg are assumed to occur at the points where the concentration
is one-tenth of the center!ine value. On this basis, A9 can be estimated,
provided information is available regarding the average change in the wind
direction within the Planetary Boundary Layer, 80/3z, as:
A6 = (96/3z) (Tp - TB)
where
the lesser of h or Hg + 2.15 az for L <_ 0
He + 2.15 az for L > 0
TB = the greater of zero or H - 2.15 a
In most applications, this third term in Equation 26 can be neglected for
downwind distances less than 10 km.
24
-------
SECTION 5
CONCLUSION
The dispersion parameter estimation scheme presented was designed parti-
cularly for Gaussian plume modeling of elevated releases from tall sources.
The scheme incorporates results from various studies of dispersion. Once the
scheme is cast in a more practical form, it is anticipated to prove useful for
characterizing dispersion in a variety of situations, even though the scheme
is restricted to modeling applications having flat, homogeneous terrain and
having steady-state meteorological conditions. Through the incorporation of
dispersion results from elevated releases in unstable to slightly unstable
conditons, the characterization of F might be simplified, much as the character-
ization of F was by Nieuwstadt and Van Duuren (1979) results. The characteri-
zations of F and F might also be expanded to include surface releases; the
scheme could then be used with no release height restrictions. This report
documents the scheme in its most basic form. Further analysis is required
before the input parameters can be fully discussed. The scheme in its basic
form requires meteorological data not routinely available, such as the Monin-
Obukhov length scale, the surface friction velocity, and the standard deviations
of the vertical and lateral velocity components at the effective height of
release.
In order to further develop the scheme for routine use, information is
needed concerning the sensitivity of the scheme to the various input parameters.
The sensitivity analysis will help to reveal
• the necessary precision and accuracy for specifying input
parameters,
• input parameters to be simplified or eliminated.
25
-------
Once the sensitivity analysis has been performed, then a field data set can
be sought to evaluate the performance of the scheme for characterizing the
dispersion from an elevated release. The results of the sensitivity analysis
will be used to evaluate existing data bases and decide which data bases have
the input specified sufficiently in order to properly evaluate the performance
of the scheme.
For widespread use of the scheme (and perhaps even for the performance
evaluation), some of the input., such as u*, L, a , o , and h, most likely will
have to be estimated from routinely available meteorological data. Here again,
the results of the sensitivity analysis will prove useful. The characteri-
zations selected to estimate the input parameters from routinely available
data can be developed to satisfy accuracy and precision constraints specified
by the sensitivity analysis.
26
-------
REFERENCES
Benoit, R. 1977. On the Integral of the Surface Layer Profile-Gradient
Functions. J. Appl. Meteorol. 16(8):859-860.
Binkowski, F. S. 1979. A Simple Semi-Empirical Theory for Turbulence in the
the Atmospheric Surface Layer. Atmos. Environ. 13(2):247-253.
Brown, R. M., L. A. Cohen, and M. E. Smith. 1972. Diffusion Measurements in
the 10 - 100 km Range. J. Appl. Meteorol. ll(2):323-334.
Briggs, G. A. 1969. Plume Rise. NTIS Pub. No. TID-25075, NTIS, U.S. Dept. of
Commerce, Springfield, Virginia. 81 pp.
Cramer, H. E. 1976. Improved Techniques for Modeling the Dispersion of Tall
Stack Plumes. In: Proceedings of the Seventh International Technical
Meeting on Air Pollution Modeling and its Application. N. 51, NATO/CCMS,
Airlie House, Virginia, pp. 731-780.
Deardorff, J. W. and G. E. Willis. 1975. A Parameterization of Diffusion
into the Mixed Layer. J. Appl. Meteorol. 14:1451-1458.
Draxler, R. R. 1976. Determination of Atmospheric Diffusion Parameters.
Atmos. Environ. (10):363-372.
Hanna, S. R., G. A. Briggs, J. Deardorff, B. A. Egan, F. A. Gifford, and F.
Pasquill. 1977. AMS-Workshop on Stability Classification Schemes and
Sigma Curves - Summary of Recommendations. Bull. Am. Meteorol. Soc.
58(12):1305-1309.
Hay, J. S., and F. Pasquill. 1957. Diffusion from a Fixed Source at a Height
of a Few Hundred Feet in the Atmosphere. J. Fluid Mech. (2)-.299-310.
Hilst, G. R., and C. L. Simpson, 1958. Observations of Vertical Diffusion
Rates in Stable Atmospheres. J. Meteorol. (15):125-126.
27
-------
Hb'gstrb'm, U. 1964. An Experimental Study on Atmospheric Diffusion. Tellus
(16): 205-251.
Irwin, J. S. 1979. Estimating Plume Dispersion - A Recommended Generalized
Scheme. In: Proceedings of the Fourth Symposium on Turbulence, Diffusion,
and Air Pollution. Am. Meteorol. Soc., Boston, Massachusetts pp. 62-69.
Izumi, Y., and J. S. Caughey. 1976. Minnesota 1973 Atmospheric Boundary
Layer Experiment Data Report. AFCRL-TR-76-0038, Hanscom AFB, Massachusetts,
28 pp.
*.
Kaimal, J. C., J. C. Wyngaard, D. A. Haugen, 0. R. Cote, and Y. Izumi. 1976.
Turbulence Structure in the Convective Boundary Layer. J. Atmos. Sci.,
(33):2152-2169.
Lamb, R. G. 1978. A Numerical Simulation of Dispersion from an Elevated
Point Source in the Convective Planetary Boundary Layer. Atmos. Environ.v
12:1297-1304.
Lamb, R. G. 1979. The Effects of Release Height on Material Dispersion in
the Convective Planetary Boundary Layer. In: Proceedings of the Fourth
Symposium on Turbulence, Diffusion, and Air Pollution. Am. Meteorol.
Soc., Boston, Massachusetts, pp. 27-33.
Lewellen, W. S., and M. Teske. 1975. Turbulence Modeling and Its Application
to Atmospheric Diffusion: Part I. Recent Program Development, Verifica-
tion and Application. EPA-600/4-75-016a, U.S. Environmental Protection
Agency, Research Triangle Park, North Carolina. 79 pp.
Nieuwstadt, F. T. M., and H. Van Duuren. 1979. Dispersion Experiments with
SFg from the 213 m High Meteorological Mast at Cabauw in the Netherlands.
In: Proceedings of the Fourth Symposium on Turbulence, Diffusion, and
Air Pollution. Am. Meteorol. Soc., Boston, Massachusetts, pp. 34-40.
Panofsky, H. A., H. Tennekes, D. H. Lenschow, and J. C. Wyngaard. 1977. The
Characteristics of Turbulent Velocity Components in the Surface Layer
under Convective Conditions. Boundary-Layer Meteorol. (11):355-361.
Pasquill, F. 1971. Atmospheric Dispersion of Pollutant. Q. J. R. Meteorol.
Soc. 97:369-395.
28
-------
Pasquill, F. 1974. Atmospheric Diffusion. 2nd ed. Halsted Press, New York.
429 pp.
Pasquill, F. 1976. Atmospheric Dispersion Parameters in Gaussian Plume
Modeling: Part II. Possible Requirements for Change in the Turner
Workbook Values. EPA-600/4-76-030b, U.S. Environmental Protection Agency,
Research Triangle Park, North Carolina. 53 pp.
Pasquill, F. 1978. Atmospheric Dispersion Parameters in Plume Modeling. EPA-
600/4-78-021, U.S. Environmental Protection Agency, Research Triangle
Park, North Carolina. 68 pp.
Pasquill, F. 1979. Atmospheric Dispersion Modeling. J. Air Poll. Cont. Assn.
29(2):117-119.
Schiermeier, F. A., and L. E. Niemeyer. 1970. Large Power Plant Effluent
Study (LAPPES) Volume 1 - Instrumentation, Procedures, and Data Tabula-
tions (1968). APTD 70-2, U.S. Department of Health, Education, and
Welfare, National Air Pollution Control Administration, Raleigh, North
Carolina. 410 pp.
Smith, F. B. 1972. A Scheme for Estimating the Vertical Dispersion of a Plume
from a Source Near Ground Level. In: Proceedings of the Third Meeting of
the Expert Panel on Air Pollution Modeling. N.14, NATO/CCMS, Paris,
France. Chap. 17, pp. 1-14.
Smith, M. E. 1968. Recommended Guide for the Prediction of the Dispersion of
Airborne Effluents. The American Society of Mechanical Engineers, New
York, New York. 85 pp.
Taylor, G. I. 1921. Diffusion by Continuous Movements. Proc. Math. SL20:
196-212.
Tennekes, H. 1969. Similarity Laws and Scale Relations in the Planetary
Boundary Layer. Workshop on Micrometeorology. Am. Meteorol. Soc.,
Boston, Massachusetts, pp. 177-214.
Turner, D. B. 1970. Workbook of Atmospheric Dispersion Estimates. Office of
Air Programs Pub. No. AP-26, Environmental Protection Agency, Research
Triangle Park, North Carolina. 84 pp.
29
-------
Willis, G. E., and J. W. Deardorff. 1974. A Laboratory Model of the Unstable
Planetary Boundary Layer. J. Attnos. Sci. 31(5): 1297-1307.
Willis, G. E., and J. W. Deardorff. 1976. A Laboratory Model of Diffusion
into the Convective Planetary Boundary Layer. Q. J. R. Meteorol. Soc.
102(432):427-445.
Willis, G. E., and J. W. Deardorff. 1978. A Laboratory Study of Dispersion
from an Elevated Source within a Modeled Convective Planetary Boundary
Layer. Atmos. Environ. 12:1305-1311.
30
-------
APPENDIX A
LIST OF DATA
The following discussion presents the dispersion data from various
studies that were used to estimate the change in the vertical dispersion
versus travel time. For each data set some assumptions were necessary,
since at present no comprehensive data set exists that contains all the
required information. The assumptions made are felt to be consistent
with the intent of developing a generalized characterization of dispersion
for use in the interim until a more complete scheme can be developed
through the acquisition of new data.
Of the parameters required to characterize the vertical dispersion
function FZ = az/(a t), typically a (the standard deviation of the
vertical wind component at the effective release height) is the one
parameter that most often is not explicitly given. During convectively
unstable conditions, the variation of a within the convective layer has
been characterized by several authors using field data, numerical data,
and tank data. These results were summarized in Figure 4 of Irwin
(1979) and reproduced here as Figure A-l. This figure can be used to
estimate a /w* during convectively unstable conditions as a function of
H /h, where H is the effective release height and h is the depth of the
convectively mixed layer. The semi-empirical relationships employing
similarity scaling theory can be used to estimate u* (the surface friction
velocity) and a (within the surface layer), given an estimate of Z
(the surface roughness length), L (the Monin-Obukhov scaling length),
and a measurement of the horizontal wind speed at a given height within
the surface layer. For neutral conditions where 1/L is zero, the nondi-
mensional wind shear can be estimated as (Benoit, 1977):
u/u* = (1/k) ln(Z/ZQ) (A-l)
and a within the surface layer can be estimated as (Binkowski, 1979):
aw/u* = 1.277 (A-2)
31
-------
If the vertical dispersion a is given as a function of downwind distance, F
values can be estimated as:
Fz = az/(awt) " az/(aex) (A"3)
where the standard deviation of the vertical wind direction a could be estimated
as:
ae * aw/u = 1.277 k/ln(Z/ZQ) (A-4)
using Equations A-l and A-2.
32
-------
10
10-1
10
I I I I I I I I •
O 1970 LENSHOW AIRCRAFT DATA, FIGURE 3 OF WILLIS AND DEARDORFF (1974) ;
D 1964 TELFORD AND WARNER AIRCRAFT DATA, FIGURE 3 OF WILLIS AND
A 1973 MINNESOTA DATA, IZUMI AND CAUGHEY (1976) DEARDORFF (1974)
NUMERICAL SOLUTION, FIGURE 3 OF WILLIS AND DEARDORFF (1974)
TANK DATA, FIGURE 3 OF WILLIS AND DEARDORFF (1974)
a
Z/h = 0.03 "
-aw/w*=1.342(Z/h)0-333
I
0.1
0.2
0.3
0.4
0.5
0.6 0.7 0.8 0.9 1
Figure A-1. Vertical profile of QW/W for fully convective conditions. aw is the stand-
ard deviation of the vertical velocity fluctuations, w* is the convective velocity scale,
and h is the depth of the convectively mixed layer. Solution shown as solid line was de-
rived, for Z/h greater than 0.03, from hand-drawn fits to the data values. Solid line,
for Z/h less than 0.03, is prediction for free convection conditions (Equation 17 of
Kaimal et al., 1976).
33
-------
PORTON DATA SET (Hay and Pasquill, 1957)
In order to estimate parameters needed to characterize F during unstable
conditions, the depth of the convectively mixed layer h was estimated for the
Porton data as 1000 m. For unstable conditions H /h is then approximately 0.15.
This suggests, based on Figure A-l, that a /w* is approximately 0.55. Since
w
values for a and u were given at the effective release height, w* was estimated
as:
W* - U
-------
TANK DATA (Willis and Deardorff, 1976, 1978)
The results reported by Willis and Deardorff (1976, 1978) were for
tank studies that modeled convectively unstable conditions. Using
Figure 3 of Willis and Deardorff (1976), values of nondimensional vertical
*
dispersion, a = a /h, were extracted as a function of nondimensional
*
time, t = w*t/h. Similarly, using Figure 2 of Willis and Deardorff (1978),
* *
values were extracted for a as a function of t . The results of Willis
and Deardorff (1976) were for scaled release height, H /h, of 0.067,
and, using the results of Figure A-l, a /w* was estimated to be 0.50.
w
The results of Willis and Deardorff (1978) were for a scaled release
height of 0.24, and, using Figure A-l, a /w* was estimated to be 0.63
(Table A-2).
It is important to realize that the values of a. used to compute
* z
a , were the actual standard deviation of the observed particles. The
distribution of the particles may have been other than Gaussian. If
*
particle distribution is not Gaussian, then use of these values of a
may not accurately estimate surface concentrations if used unadjusted in
a Gaussian dispersion model.
35
-------
TABLE A-2. TANK DATA SET*
*
t
0.13
0.18
0.25
0.37
0.44
0.50
0.55
0.64
0.70
0.78
0.12
0.21
0.27
0.33
0.39
0.45
0.50
0.57
0.63
0.71
He/h
*
a
z
0.05
0.07
0.11
0.19
0.23
0.26
0.29
0.33
0.37
0.41
He/h
0.085
0.120
0.150
0.170
0.186
0.204
0.210
0.225
0.236
0.250
= 0.067
F
z
0.77
0'.78
0.88
1.03
1.05
1.04
1.06
1.03
1.06
1.06
= 0.24
1.12
0.91
0.88
0.82
0.76
0.72
0.67
0.63
0.59
0.56
aw/W*
*
t
0.82
0.88
0.95
1.02
1.08
1.24
1.89
2.23
2.61
2.89
VW*
0.82
0.93
1.05
1.17
1.47
1.82
2.23
2.80
3.24
= 0.50
*
a
Z
0.44
0.46
0.48
0.50
0.52
0.56
0.57
0.55
0.53
0.53
= 0.63
0.260
0.268
0.282
0.295
0.312
0.350
0.388
0.390
0.374
F
z
1.07
1.05
1.01
0.98
0.97
0.90
0.60
0.49
0.40
0.37
0.50
0.46
0.43
0.40
0.34
0.31
0.28
0.22
0.18
* Data taken from Willis and Deardorff (1976, 1978).
36
-------
NUMERICAL DATA SET (Lamb, 1978)
The results of Lamb (1978) were for a numerical simulation of convectively
unstable conditions where the scaled release height, H /h,
was 0.26. In conversations with Lamb, I found that studies for other
release heights had been accomplished but were as yet unpublished.
These unpublished results, in part, were used in this analysis and are
listed (Table A-3). For each release height the scaled release height
was used to estimate the scaled vertical velocity fluctuations from
Figure A-l.
*
The values of a , used to compute a , were the actual standard
z z *
deviations of the modeled particles. Use of these values of a , without
adjustment in a Gaussian dispersion model may not yield valid estimates
of surface concentrations. My conversations with Lamb revealed that the
distributions of the particle were not Gaussian for the elevated releases.
The distributions were skewed with more particles towards the surface;
*
hence, use of these az values in a Gaussian dispersion model (without
adjustment) tends to underestimate surface concentrations. The skewed
distributions were noticed by Lamb for his simulations with nonbuoyant
releases. However, the distributions for buoyant releases appear to be
more Gaussian. Lamb has only begun to investigate dispersion from
buoyant releases. Hence, the latter result reported is of a very pre-
liminary nature.
37
-------
TABLE A-3. NUMERICAL DATA SET3
*
t
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
013
026
040
053
066
079
106
132
198
251
304
410
506
611
106
132
202
268
347
387
439
506
558
625
677
*
a
z
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
He
0047
0090
0130
0167
0203
0240
0321
0409
0666
0905
1160
1704
2219
2822
He
0614
0762
1131
1449
1792
1948
2130
2353
2444
2575
2663
/h
F
z
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
1.
1.
1.
/h
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
= 0.025
922
883
829
804
785
775
773
790
858
920
973
060
119
178
= 0.26
92
92
89
86
82
80
77
74
70
65
62
aw
*
t
0.
0.
0.
1.
1.
1.
1.
1.
1.
1.
2.
2.
°w
0.
0.
0.
1.
1.
1.
1.
2.
2.
2.
/w* =
717
810
902
008
100
281
323
442
766
951
083
232
/w* =
743
796
927
059
178
462
806
201
746
905
0.
*
a
z
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
392
3439
3953
4391
4806
5118
5583
5670
5875
6198
6239
6245
6235
63
2758
2824
2974
3128
3269
3638
4006
4183
4277
4310
F
z
1.
1.
1.
1.
1.
1.
1.
1.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
224
245
242
216
187
112
093
039
895
816
765
713
59
56
51
47
44
39
35
30
25
24
38
-------
TABLE A-3. (continued)
He/h = 0.50
t
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
*
.053
.066
.079
.093
.106
.119
.132
.162
.202
.254
.307
.347
.400
.453
.506
.545
0
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
*
0310
0387
0463
0537
0610
0682
0753
0908
1106
1349
1571
1731
1939
2134
2311
2428
Fz
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
93
93
93
92
91
91
90
89
87
84
81
79
77
74
72
70
aw
t
0.
0.
0.
0.
0.
0.
0.
1.
1.
1.
1.
1.
1.
2.
2.
/w* =
*
598
651
704
757
796
849
902
006
125
234
327
449
555
069
373
= 0.
*
az
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
63
2568
2694
2807
2905
2970
3042
3096
3146
3180
3164
3148
3123
3093
3086
3107
F
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
z
.68
.65
.63
.61
.59
.57
.54
.49
.45
.41
.38
.34
.31
.24
.21
39
-------
TABLE A-3. (continued)
*
t
0.013
0.026
0.040
0.053
0.066
0.079
0.093
0.106
0.119
0.132
0.198
0.251
0.304
He
*
a
Z
0.0075
0.0146
0.0214
0.0282
0.0349
0.0412
0.0475
0.0533
0.0591
0.0646
0.0894
0.1064
0.1213
/h = 0.50
F
z
1.05
1.02
0.97
0.97
0.96
0.95
0.93
0.91
0.90
0.89
0.82
0.77
0.73
av
*
t
0.357
0.423
0.506
0.545
0.625
0.704
0.757
0.849
0.902
1.008
1.508
1.819
2.004
/w = 0 5 5
./ W j. \J . *J *J
*
cr
z
0.1354
0.1531
0.1762
0.1873
0.2097
0.2329
0.2491
0.2772
0.2937
0.3246
0.4342
0.4540
0.4502
F
z
0.69
0.66
0.63
0.62
0.61
0.60
0.60
0.59
0.59
0.59
0.52
0.45
0.41
Data are from Lamb (1978).
40
-------
HANFORD DATA SKf (Hilst and Simp-.on, 1958)
Using tr* field data reported by Hilst and Simpson (1958), tests C
and D were determined to be closest to neutral conditions. The data did
not report the values of the standard deviation of the vertical velocity
fluctuations, but data was given for the horizontal wind speed, u, at
60.96 m. UsJ r: n'ogstrb'm's 0-964) estimate of the Hanford surface rough-
ness length, f , of 0,05 m, /al'ies of the surface friction velocity u*
were astimd* -J u;!ng the reportai ,-:ind speeds and the assumption that
the stability w.s neutral. For tn;t C, u* was estimated as 0.295 m/s;
by means of equation A-l .vnd the wind speed at the tracer release height
of 56.4 was 3Si:3mated to le 5.17 c./s. Using the estimated u* and Equation
A-2, CT was eU!mated to he 0.37 m/s. A similar analysis was performed
w
on test D, yis! ling estimates with u and a at 56.4 m of 4.57 m/s and
W
0.33 m/s, rrr.pectively (Tablo A-4).
41
-------
TABLE A-4. HANFORD DATA SET3
Travel
distance
(m)
Travel
time
(sec)
az
(m)
Fz
He = 56.40 m
u* = 0.29 m/s
152.4
304.8
762.0
1524.0
Hg = 56.40 m
u* = 0.26 m/s
152.4
304.8
762.0
1542.0
Field test C
ZQ = 0.05 m
a ,(56.4 m) = 0.37 m/s
w
29.9
59.8
149.4
298.8
Field test D
ZQ = 0.05 m
aw(56.4 m) = 0.33 m/s
33.4
66.7
166.7
333.5
u(60.96 m) =
u(56.40 m) =
5.5
8.9
13.6
15.7
u(60.96 m) =
u(56.40 m)
5.5
8.1
12.3
14.8
5.23 m/s
5.17 m/s
0.50
0.40
0.25
0.14
4.60 m/s
= 4.57 m/s
0.50
0.37
0.22
0.13
Data are from Hi 1st and Simpson (1958).
42
-------
0
STUDSVIK AND AGESTA DATA SET (Hogstrb'm 1964)
Table 2 of Hogstrom (1964) reports the dispersion results for
Studsvik. The release height was 87 m for these data. The results for
A equal to 0.3 in the table are the closest to near-neutral conditions.
Hb'lstrb'm (p. 221) suggests that the roughness length at Studsvik was
0.6 m and that a /u was 0.079 at 87 m. Using the reported value for
w
a /u, Equation A-3 would suggest that Z equals about 0.10 m at Studsvik.
W 0
The discrepancy between Hogstrom's reported value for the surface rough-
ness and the estimated value is evidence that the flow at Studsvik was
not ideal. Hogstrom noticed that terrain-induced anomolies existed in
the turbulence profiles and wind speed profiles at Studsvik. For this
study, Z is assumed to be 0.1 m. If the flow was not steady-state,
assuming Z equals 0.1 m allows the Studsvik data to be used for the
shorter travel times. Hence, the resulting estimates of F, for the
o Z
longer travel times are considered suspect. The Agesta site had more
ideal flow; hence, the Agesta wind information has been used to estimate
the wind speed at Studsvik. Hogstrom (p 226) suggests that u(122m)/uf =
0.71 where uf = 10 m/s. Using Equation A-l and this estimate of wind
speed at 122 m, u* is estimated to be 0.4 m/s, and u(87m) is estimated
to be 6.8 m/s. With this estimate of u at 87 m and the reported value
for a /u, a is estimated to be 0.54 m/s (Table A-5).
W W
Table 3 of Hogstrom (1964) reports the results of dispersion studies
O
at Agesta. The release height for these data was 50 m. Unlike the
o
Studsvik data, the Agestasite apparently presented no major terrain
problems. For the neutral data (x = -°°), Hogstrom reports a /u at 50 m
W
as 0.12. Using Equation A-3, this suggests Z is 0.59 m, which agrees
with Hogstrom's estimate of Z . Assuming u(122m) is 7.1 m/s and Z is
0.59 m, Equation A-l estimates u* to be 0.53 m/s and u(50m) to be 5.9 m/s.
With the given value for a /u and this e;
W
is estimated to be 0.71 m/s (Table A-5).
With the given value for a /u and this estimate of u(50m), a
W W
43
-------
TABLE A-5. STUDSVIK AND AGESTA DATA SET*
Studsvik - 87 m (X = 0.3) Agesta - 50 m (x = -»)
Travel
distance
(m)
100
250
500
1000
2000
3000
4000
5000
Gz
(m)
7.0
15.0
24.7
38.3
57.6
69.5
80.5
88.0
Travel
time
(s)
14.7
36.8
73.5
147.1
294.1
441.2
588.2
735.3
Fz
0.89
0.76
0.63
0.49
0.37
0.29
0.26
0.22
Travel
distance
(m)
50
100
150
250
500
750
1000
CTz
(m)
5.0
8.3
12.2
17.8
31.6
38.5
46.2
Travel
time
(s)
8.5
17.0
25.4
42.4
84.8
127.1
169.5
Fz
0.83
0.69
0.68
0.59
0.53
0.43
0.39
Data are from Hb'gstrom (1964).
44
-------
SECOND ORDER CLOSURE DATA SET (Lewellen and Teske, 1975)
Figure 22 of Lewellen and Teske (1975) depicts the variation of
vertical dispersion as a function of downwind distance for a surface
release and an elevated release during neutral conditions. The plot has
been normalized using scaling parameters. Figure 11 of their report
suggests that results for neutral conditions can be interpreted dimen-
sionally using:
u = the geostrophic wind speed, 10 m/s
Z = the surface roughness length, 10 cm
-4 -1
f = the Coriolis parameter, 10 s
RQ = the Rossby number, uq/(Z f), 106
A typical wind speed at 10 m during neutral conditioons is 5 m/s. The
*,
velocity scale u as defined by Lewellen and Teske would then be 0.5 m/s
and RQ = u/(ZQf) would be 50,000.
Figure 10 of their report suggests that for a surface release
during neutral conditions a /u = 1.1; hence, a equals approximately
w w
0.55 m/s. For the surface release, the transport wind speed was assumed
to be 5 m/s.
For the elevated release (H f/u = 0.144), H is 570 m if u equals
2
0.5 m/s. Using Figure 7 of their report (H f/u = 0.00057), (a/uj
eg w g
is estimated to be 0.0011 and u(570m)/u is estimated to be 0.96. Hence,
a is estimated to be 0.332 m/s, and the transport wind speed is estimated
W
to be 9.6 m/s.
With the above information, data values were extracted from Figure
22 of Lewellen and Teske's (1975) report and expressed in dimensional
form (Table A-6).
45
-------
TABLE A-6. SECOND ORDER CLOSURE DATA SET*
He =
Travel
distance
(km)
0.10
0.25
0.50
1.00
2.50
5.00
10.00
25.00
50.00
0 u = 5 m/s
az
(m)
5.60
11.75
21.25
36.25
75.00
132.50
212.50
375.00
550.00
Z = 0.1 m
0
Travel
time
(sec)
20
50
100
200
500
1000
2000
5000
10000
a =0.55 m/s
w
Fz
0.51
0.43
0.39
0.33
0.27
0.24
0.19
0.14
0.10
He =
0.14
0.25
0.50
1.00
2.50
5.00
10.00
25.00
50.00
570 m u = 9.6 m/s
5.00
7.60
12.75
21.50
41.50
71.00
110.00
200.00
280.00
Z = 0.1 m
0
14.6
26.0
52.1
104.2
260.4
520.8
1041.7
2604.2
5208.3
a = 0.33 m/s
W
1.02
0.87
0.73
0.61
0.47
0.41
0.31
0.23
0.16
Data are from Lewellen and Teske (1975).
46
-------
APPENDIX B
SUBROUTINE SZSY
The following FORTRAN subroutine performs the computations, as outlined
in Section 4, for estimating the total vertical and lateral dispersion from
an elevated point source. The subroutine was constructed in preparation for
the further development and evaluations needed.
47
-------
SUBROUTINE SZSY(XKM,OLL,HS,DH,U,HL,DTHETA,USTAR,SE,SA,FSZ,FSY,
CSZ,SY,RZR,IKEY,XTI,XXO)
C
C THIS SUBROUTINE CALCULATES THE VERTICAL AND HORIZONTAL DISPERSION
C (SZ AND SY, RESPECTIVELY) IN METERS. ALL UNITS OF INPUT PARA-
C -METERS ARE IN METERS, SECONDS, AND RADIANS EXCEPT FOR XKM WHICH
C IS IN KILOMETERS.
C
C THIS SUBROUTINE INCLUDES UPDATES AS OF 14 MARCH 1979.
C PROGRAMMER: JOHN S. IRWIN
C EPA (MD 80)
C RTF, NC 27711
C PHONE NUMBERS: COM. (919) 541-4564
C FTS 629-4564
C
C
C GLOSSARY OF TERMS
C
C INPUT PARAMETERS:
C
C XKM = DOWNWIND DISTANCE IN KILOMETERS;
C OLL = 1/L, WHERE L IS THE MONIN-OBUKHOV SCALING LENGTH
C IN METERS. THIS ROUTINE ASSUMES THAT THE VON
C KARMAN CONSTANT IS EQUAL TO 0.4.
C HS = STACK HEIGHT IN METERS.
C DH = THE PLUME RISE IN METERS, (NOTE DH MUST BE GREATER THAN OR
C EQUAL TO ZERO).
C U = TRANSPORT WIND SPEED IN METERS PER SECOND, (NOTE MUST BE
C GREATER THAN ZERO).
C HL = IF L IS LE. ZERO, HL IS THE DEPTH OF THE CONVECTIVELY
C MIXED LAYER IN METERS. IF L IS GT. ZERO, HL IS THE
C DEPTH OF THE STABLE SURFACE LAYER IN METERS.
C DTHETA = THE RATE OF TURNING OF THE HORIZONTAL WIND DIRECTION IN
C RADIANS PER METER.
C USTAR = THE SURFACE FRICTION VELOCITY SCALE IN METERS PER SECOND.
C FOR UNSTABLE CONDITIONS WE USE WSTAR INSTEAD OF USTAR.
C WSTAR = USTAR*(-HL/(0.4*L) )**0.33333
C SE = THE STANDARD DEVIATION OF THE VERTICAL WIND DIRECTION IN
C RADIANS.
C SA r THE STANDARD DEVIATION OF THE HORIZONTAL WIND DIRECTION IN
C RADIANS.
C
C OUTPUT PARAMETERS
C
C FSZ = SZ / ( SE*X) IN RADIANS** -1.
C FSY = SY / ( SA*X) IN RADIANS** -1.
C SZ r THE VERTICAL DISPERSION PARAMETER IN METERS.
C SY = THE HORIZONTAL DISPERSION PARAMETER IN METERS.
C RZR = THE FRACTIONAL ADJUSTMENT TO BE APPLIED TO THE EFFECTIVE
C STACK HEIGHT IN THE GAUSSIAN PLUME DISPERSION MODEL.
48
-------
C RZR IS ALWAYS 1.0 UNLESS PLUME IS NOMBUOYANT AND
C MEETS SOME OTHER CONDITIONS, SEE PARA. 3 IN SECTION OF
C OF COMMENTS ENTITLED 'INTERPOLATION SCHEMES'.
C IKEY = IF NO ERRORS ARE DETECTED IN THE INPUT DATA, IKEY IS +1.
C IF ERRORS ARE DETECTED IN THE INPUT DATA, EXECUTION IS
C STOPPED AND IKEY IS SET TO -1.
C XTI = U*TI, SEE PARA. 7 IN SECTION OF COMMENTS ENTITLED
C 'EQUATIONS OF INTEREST' FOR DEFINITION OF TI.
C XTO = U*TO, SEE PARA. 7 IN SECTION OF COMMENTS ENTITLED
C 'EQUATIONS OF INTEREST' FOR DEFINITION OF TO.
C
C MATHEMATICAL MODEL
C
C SZ = ( SZO**2 + SZ1**2 )**0.5
C WHERE SZO = THE DISPERSION DUE TO TURBULENCE AND TRANSPORT.
C SZ1 = THE INDUCED DISPERSION DUE TO BUOYANT PLUME RISE,
C DH/3.5 .
C
C SY = ( SYO**2 + SY1**2 + SY2**2 )**0.5
C WHERE SYO - THE DISPERSION DUE TO TURBULENCE AND TRANSPORT,
C SY1 = THE INDUCED DISPERSION DUE TO BUOYANT PLUME RISE,
C DH/3.5 .
C SY2 = THE INDUCED DISPERSION DUE TO HORIZONTAL WIND
C DIRECTION SHEAR IN THE VERTICAL,
C 0.173*DTHETA*(H1 - H2)*X
C AND H1 IS THE UPPER EXTENT OF THE PLUME AND H2
C IS THE LOWER EXTENT OF THE PLUME. THE UPPER AND
C LOWER EXTENT OF THE PLUME ARE DEFINED AS THE
C EFFECTIVE PLUME HEIGHT, ZR=HS+DH, PLUS AND MINUS
C 2.15SZ, UNLESS SUCH WOULD CAUSE H1 TO BE ABOVE
C THE CONVECTIVE MIXED LAYER LID HEIGHT OR H2 TO BE
C LESS THAN THE GROUND HEIGHT.
C
C INTERNAL PARAMETERS
C
C X = XKM*1000, DOWNWIND DISTANCE IN METERS,
C ZR = HS+DH, EFFECTIVE HEIGHT OF THE PLUME IN METERS,
C NOTE, ZR IS SET TO 10.0 METERS IF IT IS LESS THAN 10 METERS.
C SEE = SE, UNLESS SE IS LESS THAN 0.01. 1N SUCH CASES, SEE IS SET
C TO 0.01 RADIANS.
C
C SPECIFIC TO UNSTABLE ANALYSES
C
C TSTAR r (X/U)*(WSTAR/HL), SCALED TRANSPORT TIME,
C ZROHL = ZR/HL, SCALED EFFECTIVE RELEASE HEIGHT,
C RZR = A FRACTION WHICH IS 1.0 UNLESS ZROHL IS GREATER THAN 0.25-
C RZR IS THE FRACTIONAL ADJUSTMENT TO THE EFFECTIVE RELEASE
C HEIGHT SUGGESTED BY R. LAMB'S MUMMERICAL MODEL RESULTS.
C
49
-------
C SPECIFIC TO NEUTRAL/STABLE ANALYSES
C
C XO = U*OTO, ANALOGOUS TO DRAXLER'S TO EXCEPT HERE WE ARE
C IN METERS. TO AND XO ARE EMPIRICAL PARAMETERS RELATED
C TO THE LAGRANGIAN TIME SCALE.
C OTO = AN EMPIRICAL PARAMETER RELATED TO THE LAGRANGIAN TIME SCALE.
C IT IS ASSIGNED A VALUE BASED ON THE EFFECTIVE RELEASE
C HEIGHT. WHERE THE EFFECTIVE RELEASE HEIGHT (ZR)
C IS EXPRESSED IN METERS AND OTO IS EVALUATED AS:
C 50 SEC IF ZR IS LESS THAN 50 METERS,
C 200 SEC IF ZR IS GREATER THAN 150 METERS, AND
C (3*ZR - 50)/2 SEC FOR ZR VALUES INBETWEEN.
C
C EQUATIONS OF INTEREST
C
C 1. INTERPOLATION CONSTANTS USED FOR UNSTABLE CONDITIONS WHEN
C ZROHL IS LESS THAN 0.25.
C AU = (10/9)*(1 - 4ZR/HL)
C BU = 1 - AU
C
C 2. INTERPOLATION CONSTANTS USED FOR NEUTRAL/STABLE ANALYSES WHEN
C ZR IS LESS THAN 50 METERS.
C AS = 1 - BS
C BS = ZR/50.0
C
C 3- EQUATION " FSP "
C IF TSTAR LE. 1.0, THEN
C FSP = SQRTC (1.0+1.88*TSTAR)/(1.0+TSTAR**3) )
C - 4.5*TSTAR*(1 - TSTAR)**4.0
C IF TSTAR GT. 1.0, THEN
C FSP = SQRTC 1.WTSTAR )
C
C 4. EQUATION " FEP "
C IF TSTAR LE. 1.0, THEN
C FEP = 1 - 0.7*TSTAR + 0.2*TSTAR**2
C IF TSTAR GT. 1.0, THEN
C FEP = SQRTC 1/(4*TSTAR) )
C
C 5. EQUATION " FS "
C FS = 1/( 1 + 0.9*SQRT(X/XO) )
C
C 6. EQUATION " FE "
C FE = 1/( 1 + 0.945*(X/XO)**0.806 )
C
C 7. EQUATION FOR FSY (APPROXIMATION TO NIEUSTADT AND DUUREN,1979 RESULTS)
C IF UNSTABLE:
C OTI = (C*USTAR/HL)*(1+D(-HL/L))**1/3
C WHERE C = 2.5
C D = 0.00133
50
-------
C FSY = 1/(1+SQRT(T/TD)
C NOTE TI = 1/OTI
C IF NEUTRAL OR STABLE (DRAXLER'S RESULTS FOR ELEVATED RELEASES):
C TTO = 1000 SEC
C XXO = U*TTO, WHERE U IS IN M/S.
C FSY = 1/(U0.9*SQRT(X/XXO))
C
C INTERPOLATION SCHEMES
C
C 1. IF CONDITIONS ARE UNSTABLE AND ZROHL IS LT. 0.25, THEN
C
C FSZ = AU*FSP + BU*FEP
C
C OTHERWISE,
C
C FSZ = FEP
C
C 2. IF CONDITIONS ARE NEUTRAL/STABLE AND ZR IS LT. 50 METERS, THEN
C
C FSZ = AS*FS + BS*FE
C
C OTHERWISE,
C
C FSZ = FE
C
C 3. APPROXIMATION TO LAMB'S ADJUSTMENT TO EFFECTIVE RELEASE HEIGHT
C WHEN CONDITIONS ARE UNSTABLE AND ZROHL IS GT. 0.25 .
C
C IF TSTAR/ZROHL IS LT. 1.0, RZR =1.0.
C IF TSTAR/ZROHL IS GT 2.0, RZR = 0.25/ZROHL.
C ALL OTHER CASES, RZR = 0.25/ZROHL +
C ( (ZROHL-0.25)/ZROHL )*( 2 - TSTAR/ZROHL ).
C
IKEY = 1
IF(DH.LT.O.O) GO TO 900
IF(U.LE.O.O) GO TO 900
X = XKM*1000.0
ZR = HS + DH
IFUR.LE.10.0) ZR = 10.0
OL = OLL
TEST = HL*OL
C
C WE USE SEE FOR SE (INPUT) AND OL FOR OLL (INPUT)
C BECAUSE WE MODIFY SEE AND OL IN THE
C FOLLOWING ANALYSIS AND DO NOT WISH TO
C CHANGE THE USER INPUT TO THE SUBROUTINE.
C
51
-------
c *
c *
C START VERTICAL DISPERSION *
C CALCULATIONS *
C *
C *
£**********#*********************#*#********##***************##***
C
C THE UNSTABLE ANALYSIS FOR DISPERSION IN THIS
C ROUTINE IS KNOWN TO BE VALID FOR UNSTABLE
C CONDITIONS WHERE ABS(HL/L) IS GREATER THAN OR EQUAL
C TO 10.0. WHEN ABS(HLXL) IS LESS THAN 10.0
C AND L IS NEGATIVE (IE. UNSTABLE CONDITIONS)
C IT MAY BE BETTER TO TREAT THE DISPERSION
C AS IF IT WHERE NEUTRAL CONDITIONS BUT USE
C THE TURBULENCE INTENSITIES APPROPRIATE
C FOR THE STABILITY CONDITIONS (IE. THE ACTUAL
C L IS USED TO DETERMINE SE AND SA).
C
C
SEE = SE
IFCSEE.LT.0.01) SEE = 0.01
IF(ZR.LT.HL) GO TO 10
C
C WE HAVE A CASE WHEN THE EFFECTIVE RELEASE
C IS ABOVE THE SURFACE MIXED LAYER. WE WILL
C MODEL THIS AS IF CONDITIONS WERE NEUTRAL.
C
OL = 0.0
SEE = 0.01
TEST =0.0
10 RZR = 1.0
C
C TEST FOR STABILITY REGIME
C
IFCTEST.LT.-10.0) GO TO 100
C
C*** MUST BE STABLE
C
XO = 0.0
OTO = 50.0
IF(ZR.GT.50.0) OTO = (3.0*ZR-50.0)/2.0
IFCZR.GT.150.0) OTO = 200.0
XO = U*OTO
FE = 1.0/(1.0+0.9M5*(X/XO)**0.806)
IF(ZR.LT.50.0) GO TO 50
FSZ = FE
GO TO 200
52
-------
50 CONTINUE
FS = 1.07(1.0+0.90*SQRT(X/XO))
BS = ZR/50,0
AS = 1 • BS
FSZ = AS*FS + BS*FE
75 GO TO 200
100 CONTINUE
C
C*** MUST F^l UNSTABLE CONDITIONS
C
IF(HL.LE.O.O) GO TO 900
IF(USTAR.LE.O.O) GO TO 900
WSTAR = USTAR*(-HL*OL/0.4)**0.33333
TSTAR=(X/U)»(WSTAR/HL)
ZROHL = ZR/HL
IFCZROKL.GE.1.0) GO TO 900
IFCTSTAR.LT.1.0) GO TO 120
FEP= SQRT(1.0/(4.0*TSTAR))
GO TO 12C
120 FEP= 1.0-0.70*TSTAR+0.20*TSTAR*TSTAR
125 IFCZROHL.LT.0.25) GO TO 145
FSZ = FEP
IF(DH.GT.O.O) GO TO 200
C
C NOTICE WE APPLY THE CORRECTIONS TO THE
C SCALING FACTOR FOR THE EFFECTIVE RELEASE
C HEIGHT (RZR) AS LAMB SUGGEST ONLY WHEN
C THE PLUME IS NONBUOYANT.
C
XTSTAR = TSTAR/ZROHL
IF(XTSTAR.LE.1.0) GO TO 200
IF(XTSTAR.LT.2.0) GO TO 135
RZR = 0.25/ZROHL
GO TO 200
135 RZR = (0.25/ZROHL) + ((ZROHL-0.25)/ZROHL)*(2.0-XTSTAR)
GO TO 200
145 IF(TSTAR.LT.I.O) GO TO 150
FSP=SQRT(1.44/TSTAR)
GO TO 160
150 FSP = SQRT((1.0+1.88*TSTAR)/(1.0+TSTAR**3.0))
C - 4.5*TSTAR«(1.0-TSTAR)**4.0
160 AU = (10.0/9.0)»(1.0 - 4.0*ZROHL)
BU = 1.0 - AU
FSZ = AU*FSP+BU*FEP
200 SZO = SEE*X*FSZ
C PLUME RISE EFFECTS
SZ1 = DH/3.5
SZ = SQRT(SZO*SZO+SZ1*SZ1)
SY1 = SZ1
53
-------
c
£XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
c *
c *
C FINISH VERTICAL DISPERSION *
C CALCULATIONS AND START LATERAL DISPERSION *
C CALCULATIONS *
C *
C *
QXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
c
C THE FORMULA FOR UNSTABLE CONDITIONS GIVEN BELOW
C FOR FSY BECOMES IDENTICAL OR NEARLY SO TO THE
C FORMULA GIVEN BELOW FOR FSY FOR NEUTRAL CONDITIONS
C AS CONDITIONS BECOME LESS UNSTABLE.
C AS CONDITIONS BECOME CONVECTIVELY UNSTABLE, THE
C FORMULA FOR FSY APPROXIMATES THE RESULTS FOUND
C BY WILLIS AND DEARDORFF.
C
XXO =0.0
C
C 'WE ZERO XXO EVERY TIME THROUGH BECAUSE IT IS USED
C ONLY FOR STABLE CALCULATIONS. IT LOOKS A BIT STRANGE
C IF XXO DOES MOT VARY IN VALUE WHICH CAN HAPPEN IF
C THE FIRST TIME THROUGH THE LOOP WE HAVE STABLE
C CONDITIONS AMD THE NEXT SEVERAL TIMES THROUGH WE
C HAVE UNSTABLE CONDITIONS. HENCE, ZERO XXO EVERY
C TIME, LESS QUESTIONS THAT WAY.
C
XTI = 0.0
C
C AS WITH XXO SO ALSO WITH XTI.
C
IF(TEST.LT.O.O) GO TO 300
C
C WE TEST ON L HERE. THIS IS OK SINCE
C THE ROUTINE CAN HANDLE UNSTABLE DISPERSION
C IN THE LATERAL DIMENSIONS RIGHT UP TO
C NEUTRAL CONDITIONS. THIS WAS NOT SO
C FOR THE VERTICAL DISPERSION CALC. SINCE WE
C 'CALLED' UNSTABLE CONDITONS WITH -HL/L'S
C OF 10 OR LESS 'NEUTRAL' IN SO FAR AS THE
C ESTIMATION OF THE DISPERSION FUNCTION WAS
C CONCERNED.
C
C*** MUST BE STABLE (DRAXLER,1976)
C
XXO = U*1000.0
FSY = 1.0/C 1.0+0.9*SQRT(X/XXO) )
GO TO 400
54
-------
c
C*** MUST BE UNSTABLE (NIEUWSTADT AND DUUREN.1979)
C
300 CONTINUE
C = 2.5
D = 0.00133
OTI = (C*USTAR/HL)*(1.0-D*HL*OL)**0.33333
XT I = I!/OTI
FSY = 1.0/( 1.0+SQRT(X/XTI) )
400 SYO = SA*X*FSY
C
C HORIZONTAL WIND SHEAR EFFECTS (PASQUILL, 1976;1974)
C
H1 = ZR + 2.15*SZ
IF(TEST.GT.O.O) GO TO 420
C
C WE HAVE AN UPPER LIMIT TO THE VERTICAL EXTENT OF
C THE PLUME (H1) DURING UNSTABLE AND NEUTRAL CONDITIONS.
C HOWEVER, DURING STABLE CONDITIONS THIS IS NOT SO.
C
IF(HLGT.HL) H1 = HL
420 H2 = ZR - 2.15*SZ
IFCH2.LT.O.O) H2 = 0.0
C
C THE ABOVE CHECK KEEPS THE LOWER EXTENT OF THE
C PLUME ABOVE THE GROUND.
C
SY2 = 0.173*DTHETA*(H1 - H2)*X
SY = SQRT(SYO*SYO+SY1*SY1+SY2*SY2)
C
QXXX»*XXXXXXXXXXXXXXXXXX*XXXXXXXXXXXXXXXXXXXXXXXXXXXXX
C *
C FINISH *
C LATERAL DISPERSION CALCULATIONS *
C *
QXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
GO TO 1000
900 CONTINUE
C
C ERROR DETECTED IN SPECIFICATION OF THE INPUT
C
FSZ = 0.0
FSY = 0.0
SZ = 0.0
SY = 0.0
RZR = 0.0
IKEY r -1
1000 RETURN
END
55
-------
TECHNICAL REPORT DATA
(Please read Instructions on the reverse before completing)
2.
3. RECIPIENT'S ACCESSION-NO.
4. TITLE AND SUBTITLE
SCHEME FOR ESTIMATING DISPERSION
PARAMETERS AS A FUNCTION OF RELEASE HEIGHT
5. REPORT DATE
October 1979
6. PERFORMING ORGANIZATION CODE
7. AUTHOR(S)
John S. Irwin
8. PERFORMING ORGANIZATION REPORT NO.
9. PERFORMING ORGANIZATION NAME AND ADDRESS
Same as block 12
10. PROGRAM ELEMENT NO.
1AA603A AB-025 (FY-79)
11. CONTRACT/GRANT NO.
12. SPONSORING AGENCY NAME AND ADDRESS
Environmental Sciences Research Laboratory - RTP, NC
Office of Research and Development
U.S. Environmental Protection Agency
Research Triangle Park, North Carolina 27711
13. TYPE OF REPORT AND PERIOD COVERED
In housp 7/77 - fi/7Q
14. SPONSORING AGENCY CODE
EPA/600/09
15. SUPPLEMENTARY NOTES
16. ABSTRACT
Based on an investigation where the dispersion parameters are assumed to
have the form o^ = QW v t FZ , a generalized scheme is presented for estimating
the dispersion parameters as a function of release height. Further development is
needed to refine the scheme for more generalized applicability, since, as documented
in this discussion, the scheme requires as input meteorological data not routinely
available. The scheme incorporates results from various studies, and once it is more
practically structured it will prove useful for characterizing dispersion from tall
sources in a variety of situations. The generalized scheme was developed particularly
for Gaussian plume modeling; therefore, it is restricted to modeling applications
having flat terrain and having steady-state meteorological conditions.
7.
KEY WORDS AND DOCUMENT ANALYSIS
DESCRIPTORS
b.lDENTIFIERS/OPEN ENDED TERMS C. COS AT I Field/Group
* Air pollution
* Meteorology
* Atmospheric diffusion
* Estimating
Mathematical Models
* Height
13B
04B
04A
12A
8. DISTRIBUTION STATEMENT
RELEASE TO PUBLIC
19. SECURITY CLASS (ThisReport)
UNCLASSIFIED
21. NO. OF PAGES
20. SECURITY CLASS (Thispage)
UNCLASSIFIED
22. PRICE
EPA Form 2220-1 (9-73)
56
------- |