EPA-600/4-76-042
August 1976
Environmental Monitoring Series
THE "GAUSSIAN-PLUME" MODEL
WITH LIMITED VERTICAL MIXING
Environmental Sciences Research Laboratory
Office of Research and Development
U.S. Environmental Protection Agency
Research Triangle Park, North Carolina 27711
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RESEARCH REPORTING SERIES
Research reports of the Office of Research and Development, U.S. Environmental
Protection Agency, have been grouped into five series. These five broad
categories were established to facilitate further development and application of
environmental technology. Elimination of traditional grouping was consciously
planned to foster technology transfer and a maximum interface in related fields.
The five series are:
1. Environmental Health Effects Research
2. Environmental Protection Technology
3. Ecological Research
4. Environmental Monitoring
5. Socioeconomic Environmental Studies
This report has been assigned to the ENVIRONMENTAL MONITORING series.
This series describes research conducted to develop new or improved methods
and instrumentation 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.
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EPA-600/4-76-042
August 1976
THE "GAUSSIAN-PLUME" MODEL WITH LIMITED VERTICAL MIXING
F. Pasquill, D.Sc
Visiting Scientist
Meteorology and Assessment Division
Environmental Sciences Research Laboratory
Research Triangle Park, North Carolina 27711
U. S. ENVIRONMENTAL PROTECTION AGENCY
OFFICE OF RESEARCH AND DEVELOPMENT
ENVIRONMENTAL SCIENCES RESEARCH LABORATORY
RESEARCH TRIANGLE PARK, NORTH CAROLINA 27711
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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 commerical products constitute endorsement or recommendation for use.
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PREFACE
Restriction of the vertical spread of pollution plumes, as when the
atmospheric surface mixing-layer is of limited depth, is an important effect
that must be quantitatively estimated in air quality modeling. On a recent
visit to the U.S.A., Dr. F. Pasquill kindly agreed to document some
considerations of this question in relation to the practical application of
"Gaussian-plume" modeling, thus extending the discussion in his recent book
Atmospheric Diffusion. The publication of this material as an EPA report
is made in the interests of wide dissemination of the information to air
quality modelers.
Research Triangle Park, Kenneth L. Calder
North Carolina Chief Scientist
July 1976 Meteorology and Assessment Division
Environmental Sciences Research Laboratory
m
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ABSTRACT
Application of the "Gaussian"plume" model for atmospheric dispersion
from an elevated source in a mixing layer of limited depth normally involves
consideration of multiple reflections of the plume between the upper and
lower boundaries. The present analysis considers some simple approximation
formulae that should be useful in practical applications.
iv
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SECTION 1
INTRODUCTION
In the use of the "Gaussian Plume Model" an important requirement is to
make satisfactory allowance for the limiting of vertical spread by an ele-
vated stable layer in the atmosphere. Physically it is to be expected that
in the simple case of a mixing layer of constant depth the vertical distri-
bution downwind of a source will ultimately be uniform, though of course
there will continue to be a variation of concentration acrosswind. On the
other hand, for a source at relatively low level, there will be some distance
downwind within which the upper boundary of the mixed layer has no limiting
effect on the vertical spread and, given the dispersion parameter o , the
progress of vertical spread may then be treated as if the upper boundary did
not exist. Between these two relatively simple regimes the vertical distri-
bution will have a complex form. The present interest is to review briefly
(in the context of the Gaussian Plume Model) the estimation of the limits of
this intermediate regime and the procedure for evaluating ground-level con-
centration within it.
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SECTION 2
SOME ESSENTIAL POINTS IN THE HISTORY OF THE PROBLEM
(a) In the British Meteorological Office system of dispersion estimates (1)
a simple and somewhat intuitive rule was suggested, in which the prescribed
vertical spread was used until the effective "total" spread (as represented
by 2.15 az) became equal to the depth of mixing h', at some distance down-
wind d,. It was then assumed that at distance 2d, the stage of uniform
vertical distribution would be reached, the concentration thereafter being
given by substituting in the Gaussian plume formula a constant vertical spread
in accordance with
2.15cz « 2h' ( or to be exact V.715h' ) (1)
Having derived concentrations for distances d, and 2d, the concentration-
distance graph over the Intermediate distances was completed by interpolation.
(b) The procedure in (a) was offered as a simple way of representing the
essential consequence of multiple reflections of a plume between upper and
lower boundaries. A complete expression for these multiple reflections, in
which the effect of the upper boundary is represented in exactly the same
way as is the ground 1n the familiar treatment of the elevated source without
limitation of upward vertical spread, was given by Bierly and Hewson (2). Their
equation is reproduced in the Turner Workbook (3) [Eq. (5.8)., p. 36 therein].
(c) A compact form of the multiple-reflection equation, for the special
case of the ground-!eve! concentration, has been stated by the writer (4) in
the context of "box models". For the crosswind integrated concentration
(CWID) from a continuous point source (formally equivalent to the concentration
from an infinite line source acrosswind) the ground-level value is given by
the series expression
2
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(2/ir)* (Q/uaz) £ exp -(^/2a^) (2)
where Zp = H ± 2nh', with n = 0,1,2 ... , H the height of the source and Q
the source strength (per unit time for a point source or per unit time and
unit length in the case of a line source). In the further discussion the
infinite line source form will be adopted, the foregoing expression then
being the magnitude of the ground-level concentration C(x,0). For a point
source the concentration C(x,y,0) is given by
C(x,y,0)/C(x,0) = (2Tr)-% ay-% exp -(y*/2o2) (3)
In the absence of an upper boundary to the mixed layer the series at (2)
becomes, with n = 0
C(x,0) = (2/10* (Q/uo2) exp -(H2/2a*) (4)
which with Eq. (3) gives the well-known form for an elevated point source
with no limitation 1n vertical spread. Consecutive reflections from an
upper boundary at h' are represented by n = 1,2,3 etc. Bierly and Hewson's
more extensive expression [for C(x,0,z)] is easily seen to reduce to the
expression at (2) above on setting z to zero, and applying (3) with y = 0.
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SECTION 3
EVALUATION OF THE MULTIPLE-REFLECTION EQUATION
It is convenient to normalize the concentration C(x,0) in the reflection
case, in terms of the ultimate constant value
C(-,0) = Q/uh' (5)
and to write the dimenstonless form
C(x,0)uh'/Q = (2/ir)1* (h'/02)^ exp -(Zjj/20!;) (6)
which may"be evaluated in terms of a /h1 and H/h1. The number of terms
required to give adequate convergence of the series increases with the magni-
tude of oz/h', but experience with the calculation soon shows that the constar
magnitude C(«,0) is very closely approximated, for all H/h1 between 0 and 1,
at the stage a /h1 = 1.0 and then requires terms up to n = 1 or 2 only. In
Table 1 values of the dimensionless concentration are given for intervals of
0.1 in H/h1 and for azh' ranging from 0.01 to 1.4, values very much less than
unity being omitted as of no practical importance.
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SECTION 4
THE UPPER LIMIT OF o_/h' FOR WHICH Eq.(4) IS AN
ADEQUATE APPROXIMATION TO Eq.(6)
The form appropriate to unlimited upward spread, Eq.(4), Is of course
the n = 0 term of Eq.(6). Table 2 shows the fractional contribution of this
term to the total and so represents the fraction of the correct Eq.(6) value
provided by Eq.(4). Note the rapid decrease to 0.5 (actually 0.49 for
a /h1 = 1.0) as H/h1 approaches 1.0, which is associated with the n = 0 and
-1 terms approaching equality and together representing the total effect
(except for c /h1 = 1.0, at which stage the n +1 and -2 terms are just
becoming effective, representing a fractional contribution of 0.02). The
range for adequate approximation from Eq.(4) depends on the magnitude of
H/h1 and for example the practical limit of o /h1 = 0.47, as adopted in the
rule summarized in (a) of Section 2, 1s obviously acceptable for H/h1 up to
say 0.7. The latter magnitude of H/h1 may encompass most of the cases of
practical interest, but there may be interest in plumes rising to near the
top of the mixed layer, for which the full form of Eq.(6) is required below
a limit in o /h1 smaller than 0.47.
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SECTION 5
THE LOWER LIMIT OF oz/h' FOR WHICH Eq.(5) IS AN
ADEQUATE APPROXIMATION TO Eq.(6)
It has already been noted that the "uniform vertical distribution" form
becomes a close approximation when a /h1 1s near unity. It 1s indeed within
1.4% when az/h' is 1.0 and the discrepancy decreases rapidly as oz/h'
Increases. Even for a /h' as low as 0.8 the discrepancy is at most 8.5%,
depending on the magnitude of H/h'. In relation to the working rule recalled
in (a) of Section 2, note that the distance limit (2d,) specified there is
twice the distance at which a /h1 = 0.47 and accordingly corresponds to
.'a /h1 = 0.47 x 2s if az - xs
the latter power law form being expected at least over modest ranges
of x (e.g. see Ref. 4 p. 375). Thus the corresponding limit in a /h1 is
0.8 say when the exponent s is about 0.8, which is an expected value in
near-neutral conditions. For unstable (or stable) conditions s is expected
to be larger (or smaller). It appears therefore that the old rule is
reasonably satisfactory except perhaps in stable conditions. Even then,
with s = 0.58 (as estimated for stability category F, see Ref. 4 p. 375)
the discrepancy is at most 15 - 20%, for H/h' near zero or unity, and decreases
as H/h1 approaches 0.5 and, of course, as distance is increased beyond 2d,.
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SECTION 6
THE INTERMEDIATE RANGE NOT ADEQUATELY
REPRESENTED BY EITHER Eq.(4) OR Eq.(5)
The data in Table 1 may be used to Interpolate, graphically if needs be,
for the normalized concentration at values of o /h' intermediate between the
limits specified in 4. and 5. above. Alternatively a power-law fit to Eq.(6)
between these limits of cz/h' may be adopted, and applied in practice by
taking a linear interpolation in the concentration-distance graph between the
corresponding limits of distance, as in the rule (a) of Section 2. This
equivalence in the a /h1 and "distance" representations is assured if a is
a simple power function of x, as we expect at any rate over modest ranges
of x, and the range involved here is very small in practical terms (i.e. about
•2:1). The question remaining at issue is the choice of optimum values of the
limits in a /h'. To be precise these depend on the magnitude of H/h1 but for
a single rule-encompassing the magnitudes 0 < H/h' <_0.7 the best compromise
would appear to be
Eq.(4) 0 < a,/h1 < 0.5
z
Eq.(5) a,/h1 > 0.85
i """
Linear interpolation of log C v. log x for the
distances corresponding to 0.5 < a /h1 < 0.85
The basis for this procedure is evident in Fig. 1 where for a representative
selection of H/h1 the normalized concentration is shown plotted against a /h'
on a log-log basis. In the linear interpolations shown between the above
limits of a /h1 the greatest discrepancy in relation to the correct curve
is in the case of H/h' =0.7 and is then only about 6%. These limits for the
interpolation procedure are very similar to those implied in the old rule,
namely oz/h' = 0.47 and (in near-neutral conditions) about 0.8 (but greater
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or less in unstable or stable flow).
It is emphasized that the foregoing procedure is advocated only for
H/h' <_Q.7. For H/h' = 1.0 it is evident from the data in Section 4. and
Table 2 that the following applies, exactly for small values of oz/h' and
with a discrepancy of only 2% at 0z/h' = 1.0
C(x,0) = twice the value given by Eq.(4) 0 <.az/h' 1.0
For values of H/h1 between 0.7 and 1.0 there would appear to be no
alternative to the use of values as in Table 1, preferably with smaller
intervals in H/h1 especially for az/h' < 0.5.
Finally it may be recalled that the specially sudden transition between
the regimes in accordance with Eq.(4) and (5) for H/h1 = 0 has already been
noted in a recent review of the procedure of the Turner Workbook. There
(I.e. in Section 3(d)(111) of Ref. 5) a new procedure was recommended which
amounts to
Eq.(4)
Eq.(5)
az/h'
cz/h'
<_ 0.8
> 0.8
The maximum discrepancy from Eq.(6) is at a /h1 =0.8 and there amounts to
8.5%.
8
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REFERENCES
(1) Pasquill, F. (1961). The estimation of the dispersion of wlndborne
material, Met. Mag., 90,33.
(2) Blerly, E.W. and Hewson, E.W., Some restrictive meteorological conditions
to be considered 1n the design of stacks, J. Appl. Met., 1,3,383-390.
(3) Turner, D.B. (1970 revised). Workbook of atmospheric dispersion estimates,
U.S.D.H.E.W., Environmental Health Service.
(4) Pasquill, F. (1974). Atmospheric Diffusion, 2nd. Edn., John Wiley & Sons,
New York.
(5) Pasquill, F., (1976). Atmospheric dispersion parameters in Gaussian plume*
modelling, Pt. II, Possible requirements for change in the Turner Work-
book values. Environmental Protection Agency ESRL-RTP-058 Series No. 4 -
Environmental Monitoring.
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TABLE 1
NORMALIZED CONCENTRATION ACCORDING TO Eq.(6)
The entries are C(x,0)uh'/Q» I.e. ground-level concentration normalized with
reference to the "uniform vertical distribution value" Q/uh1, for an infinite
crosswind line source of strength Q per unit length and time at height H with
in a mixed layer of constant depth h' and with a constant wind speed u .
H/h'+.
oz/h'
0.01
0.02
0.03
0.04
0.05
0.06
0.08
0.10
0.2
0.3
0.4
0.5
0.6
0.8
1.0
1.2
1.4
0
79.79
9.89
26.60
19.95
15.96
13.30
9.97
7.979
3.989
2.660
1.995
1.597
1.340
1.085
1.014
1.002
1.000
. 0.1
1.487
(-4)
1.028
0.877
2. 159
3.317
4.566
4.839
3.520
2.516
1.933
1.566
1.323
1.081
1.014
1.002
1.000
. 0.2
5.35
(-3)
5.14
(-2)
0.438
1.080
2.420
2.130
1.760
1.476
1.274
1.069
1.012
1.001
1.000
. 0.3 .
8.83
(-3)
8.86
(-2)
1.295
1.613
1.506
1.338
1.198
1.050
1.008
1.001
1.000
0.4 .
2.677
(-3)
0.540
1.093
1.211
1.168
1.103
1.026
1.004
1.001
1.000
0.5 .
2.974
(-5)
1.753
(-D
0.663
0.915
0.986
0.998
1.000
1.000
1.000
1.000
0.6 .
1.215
(-7)
4.43
(-2)
0.360
0.652
0.808
0.894
0.974
0.996
0.999
1.000
0.7 .
8.74
(-3)
1.750
0.442
0.653
0.801
0.950
0.991
0.999
1.000
0.8 .
1.336
(-3)
7.69
(-2)
0.292
0.533
0.727
0.931
0.988
0.999
1.000
0.9 .
1.599
(-4)
3.274
(-2)
2.042
(-1)
0.458
0.679
0.919
0.986
0.998
1.000
1.0
2.974
(-5)
2.059
(-2)
1.753
(-1)
0.432
0.663
0.915
0.986
0.998
1.000
Figures in parentheses are exponents (of 10) and refer to the number above.
10
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TABLE 2
THE FRACTIONAL CONTRIBUTION OF THE n = 0 TERM IN Eq.(6)
H/h'-*-. 0 . 0.1 . 0.2 . 0.3 . 0.4 . 0.5 . 0.6 . 0.7 . 0.8 . 0.9 . 1.0
o /h1
o!3
0.4
0.5
0.6
0.8
1.0
1.00
1.00
1.00
0.99
0.92
0.79
1.00
1.00
1.00
0.99
0.92
0.78
1.00
1.00
1.00
0.99
0.90
0.77
1.00
1.00
1.00
0.98
0.89
0.76
1.00
1.00
0.99
0.98
0.86
0.73
1.00
1.00
0.98
0.94
0.82
0.70
1.00
0.99
0.96
0.90
0.77
0.67
1.00
0.98
0.92
0.84
0.72
0.63
0.99
0.92
0.83
0.75
0.65
0.59
0.90
0.78
0.69
0.64
0.58
0.54
0.50
0.50
0.50
0.50
0.50
0.49
11
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Normali zed
Cone.
Normalized
Cone.
(C.4)
1.0
0.5 f
Full lines are from
Eq.(6). Values of H/h1
are in parentheses
Fig. 1. Plots of normalized concentration according to Eq. (6), demon-
strating adequacy of linear interpolation between
o,/h' = 0.5 and 0.85
12
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TECHNICAL REPORT DATA
(Please read Instructions on the reverse before completing)
1. REPORT NO.
EPA-600/4-76-042
3. RECIPIENT'S ACCESSION-NO.
4. TITLE AND SUBTITLE
THE "GAUSSIAN-PLUME"
MIXING
MODEL WITH LIMITED VERTICAL
5. REPORT DATE
August 1976
6. PERFORMING ORGANIZATION CODE
7. AUTHOR(S)
8. PERFORMING ORGANIZATION REPORT NO.
F. Pasquill
9. PERFORMING ORGANIZATION .NAME AND ADDRESS
Meteorology and Assessment Division
Environmental Sciences Research Laboratory
Research Triangle Park, North Carolina 27711
10. PROGRAM ELEMENT NO.
1AA009
11. CONTRACT/GRANT NO.
SPONSORING AGENCY NAME AND ADDRESS
Environmental Sciences Research Laboratory
Office of Research and Development
U.S. Environmental Protection Agency
Research Triangle Park, North Carolina 27711
13. TYPE OF REPORT AND PERIOD COVERED
Final 4/76 - 5/76
14. SPONSORING AGENCY CODE
EPA-ORD
15. SUPPLEMENTARY NOTES
Report prepared by Visiting Scientist
16. ABSTRACT
Application of the "Gaussian-plume" model for atmospheric dispersion from
an elevated source in a mixing layer of limited depth normally involves
consideration of multiple reflections of the plume between the upper and lower
boundaries. The present analysis considers some simple approximation formulae
that should be useful in practical applications.
17.
KEY WORDS AND DOCUMENT ANALYSIS
DESCRIPTORS
b.IDENTIFIERS/OPEN ENDED TERMS
c. COSATI Field/Group
Air pollution
* Atmospheric diffusion
* Wind (meteorology)
* Plumes
* Mathematical models
Gaussian plume
13B
04A
04B
21B
12A
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RELEASE TO PUBLIC
19. SECURITY CLASS (This Report)
UNCLASSIFIED
21. NO. OF PAGES
17
20. SECURITY CLASS (This page)
UNCLASSIFIED
22. PRICE
EPA Form 2220-1 (9-73)
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