EPA-650/4-75-015
SOME TOPICS
RELATING TO MODELLING OF DISPERSION
IN BOUNDARY LAYER
by
F. Pasquill
North Carolina State University
Raleigh, North Carolina
Research Grant No. 800662
Program Element No. 1AA009
ROAP No. 21ADO
EPA Project Officer: K. L. Calder
National Environmental Research Center
Meteorology Laboratory
Research Triangle Park, North Carolina 27711
Prepared for
U. S. ENVIRONMENTAL PROTECTION AGENCY
Office of Research and Development
Washington, D.C. 20460
April 1975
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EPA REVIEW NOTICE
This grant report has been reviewed by the National Environmental Research
Center - Research Triangle Park, Office of Research and Development, EPA,
and approved for publication. Approval does not signify that the contents
necessarily reflect the views and policies of the Environmental Protection
Agency, nor does mention of trade names or commercial products constitute
endorsement or recommendation for use.
RESEARCH REPORTING SERIES
Research reports of the Office of Research and Development, U. S. Environ-
mental Protection Agency, have been grouped into series. These broad
categories were established to facilitate further development and applica-
tion of environmental technology. Elimination of traditional grouping was
consciously planned to foster technology transfer and maximum interface
in related fields. These 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
9. MISCELLANEOUS
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 report was submitted to the U.S. EPA in fulfillment of Grant No. 800662
by North Carolina State University, Raleigh, North Carolina. The document is
available to the public for sale through the National Technical Information
Service, 5285 Port Royal Road, Springfield, Virginia 22161.
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PREFACE
The several short notes comprising this special report were written
by Dr. F. Pasquill* during the period December 1974-March 1975 while he
was a Visiting Professor of Meteorology in the Department of Geosciences
at North Carolina State University under support of a research grant from
the Meteorology Laboratory of the Environmental Protection Agency. The
six topics are all of major current interest in modeling of dispersion
in the atmospheric boundary layer and are of such importance that they
should be given early and wide publicity in a special report. Many of
the topics continue and extend the discussion of items contained in the
recently published book by Dr. Pasquill (Atmospheric Diffusion, 2nd
Edition, John Wiley & Sons, Inc., 1974) and were the basis for a
series of lectures presented during his recent visit to North Carolina State
University and the Meteorology Laboratory.
K. L. Calder
Chief Scientist, Meteorology Laboratory
U.S. Environmental Protection Agency
* Dr. Pasquill is a Former President of the Royal Meteorological Society
and Retired Head, Boundary Layer Research Branch, Meteorological Office,
Bracknell, England.
iii
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ACKNOWLEDGMENTS
The writer wishes to thank Professor J. L. Lumley for discussion and
correspondence referring to Section 1; Professor H. A. Panofsky and Mr. R.
Draxler for permission to refer to the analysis of dispersion data in Sec-
tion 2; Mr. K. L. Calder, E.P.A., and Br. A. H. Weber, N.C.S.U., for general
discussions concerning this report; and Mrs. Lea Prince, E.P.A., and Mrs.
Althea Peterson, N.C.S.U., for secretarial assistance.
iv
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CONTENTS
LIST OF FIGURES vi
LIST OF TABLES vii
LIST OF SYMBOLS viii
ABSTRACT ix
SUMMARY X
1. ON THE "SECOND-ORDER CLOSURE" APPROACH . 1
2. CROSSWIND DISPERSION AND THE PROPERTIES OF TURBULENCE 7
3. WIND DIRECTION FLUCTUATION STATISTICS OVER LONG SAMPLING TIMES ... 20
4. "LOCAL SIMILARITY" TREATMENT OF VERTICAL SPREAD FROM A GROUND
SOURCE 29
5. REPRESENTATIONS OF DISPERSION IN TERMS OF DISTANCE OR TIME 38
6. MODELLING FOR ELEVATED SOURCES 47
REFERENCES 52
TECHNICAL REPORT DATA SHEET 53
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LIST OF FIGURES
Figure page
1.1 Forenoon build-up of natural evaporation 2
2.1 Lagrangian correlation coefficient R(£) according to (a)
exponential form and (b) Eq. (2.11) 12
2.2 Normalized dispersion S as a function of T/tT for (a) exponential
correlogram (b) Eq. (2.11) (c) Eq. (2.12). The single point
corresponds to Eq. (2.13) 15
2.3 Horizontal dispersion-unstable-ground release (Draxler's analysis). . 17
2.4 Normalized dispersion S against x (as observed in Greenglow and
Hanford 30 Series, vertical bars are extreme range) compared with
calculated curves for T/t^ scales given. References a,b,c as in
Fig. 2.2. The data are for neutral to moderately stable flow - see
Section 5 for further reference • 18
3.1 1200 GMT surface charts, May 1973 21
3.2 3-min average wind directions over 3-hr periods 22
3.3 Standard deviation OQ(T) of the 3-min average wind directions
(after removal of linear trend) as a function of sampling
duration T. Run numbers on left, wind speed m/sec at 44 m (in
parentheses) and incoming solar radiation (mw/cm^) on right 27
3.4 ^a^7) ^-n degrees and a (T) in m/sec for T - 90 min as a function
of u at 44 m (night-time run No. 8 excluded) 28
4.1 Rates of vertical spread dZ /dx or dZ/dx against (K/uz) at Z or Z. . 35
m m
4,2 - and udZm against (X/z) at Z or Z 36
a dx „ , m
w 0 dx
w
5.1 Crosswind spread at x - 200 m as a ratio to ax, and as a function
of u (7 ft) for different values of Ri (data from Fuquay et al. 1964).44
vi
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LIST OF TABLES
Table Page
2.1 Values of R(£) 11
2.2 Calculated values of S = a /a T 14
3.1 Summary of case studies of wind direction fluctuation
(standard deviations of 3-min averages) 25
4.1 Similarity analysis of vertical spread data from numerical
solutions of the two-dimensional diffusion equation 34
vii
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LIST OF SYMBOLS
C concentration (mass of material per unit volume of air)
E rate of evaporation
F(n) normalized one-dimensional spectral density
H height of source
i intensity of turbulence = a /u etc
k von Kfirmfin's constant
K eddy diffusivity
£ integral length-scale of turbulence
L Monin-Obukhov length
n frequency, cycles/sec.
p pressure
R(5) Lagrangian auto-correlation coefficient for time-lag j;
s time of travel of a particle reaching a given distance
S normalized dispersion = a la T
t Lagrangian integral time scale
T temperature or time of travel
T. time of travel for S = 1/2
T1 normalized time of travel = T/t
L
u,v,w velocity components along axes x, y, z
u equivalent advection velocity for diffusing material
u. friction velocity
x,y,z rectangular coordinates, x along the mean wind and z vertical
x downwind distance of ground-level maximum concentration from
an elevated source
z height at which u = u(z)
e e
X mean distance of travel of particles in the x-direction
after a given time
Z mean vertical displacement of particles after a given time
Z vertical dimension of a plume of particles
a exponent in power-law variation of wind with height
a standard deviation - of velocity component or of particle spread
X equivalent wavelength (E u/n ) at which nF(n) is a maximum
primes indicate departures from mean values.
viii
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ABSTRACT
This special report discusses six topics of major current interest in
modeling of dispersion in the atmospheric boundary layer. These are the
second-order closure modeling of turbulence.ap** crosswind dispersion and
the properties of turbulence, wind direction fluctuation statistics over
long sampling times, "local similarity" treatment of vertical spread from
a ground source, representations of dispersion in terms of distance or
time, and modeling for elevated sources.
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SUMMARY
Section 1 contains a restatement of, and an attempt to clarify, an
issue raised previously by the writer and F. B. Smith concerning use of the
2nd-moment equation for a passive material. The point in question is the
feasibility of solving the equation to give the time rate-of-change of the
*\TJI
vertical flux (—) near the surface of an effectively infinite, uniform,
ot
but time-dependent source. A practical example is the diurnal cycle of
natural evaporation, and in this case it is easily demonstrated that the
rvTJI
—— term is of a very small order relative to certain other terms. An
at
analogous case is the distributed pollutant source with strength varying
in the alongwind direction. Generally the issue is a very subtle matter,
depending it seems on ensuring physical stability of the equation by adequate
modelling of the terms. In current practice, however, certain 2nd-order
closure procedures (e.g., that of Aeronautical Research Associates of
3F
Princeton, Inc.) do not depend essentially on evaluating -rr (or the
_ gp
corresponding u — term) when this term is very small, but rather evaluate
oX
F from one of the much larger terms in which F has been introduced explicitly
in the modelling procedure. In such cases, of course, the issue as originally
raised does not apply.
In Section 2 the relation between the crosswind spread from a continuous
point source and the properties of the crosswind component of turbulence in
the atmospheric boundary layer is reviewed, especially in the light of some
recent, unpublished analyses made available by. the Department of Meteorology,
Pennsylvania State University. A major point of interest is the implication
that the Lagrangian correlation function is apparently substantially different
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from a simple exponential form. The difference is in the sense of a much
more rapid fall-off at short lag and a much slower fall at long lag. However,
the precise form of the function remains in doubt, and the point is also re-
emphasized that determination of the Lagrangian integral scale from dispersion
data is subject to considerable uncertainty. Full resolution of both of these
aspects requires continuing basic study.
Another aspect directly relevant to crosswind diffusion, namely the
properties of the standard deviation (a.) of the wind direction fluctuation
o
over long sampling times, is considered in Section 3. Wind direction traces
over three-hour periods near midday in sunny conditions at Cardington, England,
have been analysed. Departures of 3-minute averages from a fitted linear trend
were used to derive aQ as a function of sampling time. The growth curves dif-
fer widely, some approaching maximum value in about 30 min, while in others
significant increase is maintained up to 90 min. The only evident orderly
relation is a systematic decrease of the final a with wind speed. It is
also noteworthy that the 3-hr linear trend was sometimes as much as three
times afl, which further emphasises the point that a simple 'climatology'
of crosswind dispersion can hardly be expected for release times intermediate
between a few minutes and the very long periods to which windrose statistics
refer.
In Section 4 an attempt is made to develop a similarity treatment of
the rate of vertical spread from a ground-level source in a thermally
stratified boundary layer, without restriction to the surface-stress layer,
and directly involving the measurable intensity and scale of the w- component
of turbulence as a function of height. Suitably critical data on vertical
spread and turbulence being at present unavailable, the similarity forms are
xi
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used in an examination of estimates recently obtained from numerical solutions
of the two-dimensional diffusion equation, employing K profiles specified in
terms of the intensity and scale of turbulence. The rates of spread display
orderly relations with the similarity variables over a 100-fold range associ-
ated with a wind range of thermal stratification. One immediately useful con-
sequence is that the similarity relations provide an alternative and more
convenient procedure for further calculations of vertical spread in lieu of
further numerical solutions of the diffusion equation. The results also
encourage the acquisition of critical observational data on which to determine
the similarity relations more satisfactorily and more generally.
The alternatives of using time of travel or distance in describing
dispersion are discussed in Section 5. For completely homogeneous flow, an
argument is given in support of the equivalence which is usually assumed in
terms of the mean wind speed u. For boundary layer flow in which u is a
function of height, identification of the time and distance descriptions
requires an 'equivalent advecting speed' u which increases with distance
as vertical spread increases. With certain assumptions, a rough estimate
is obtained of u and hence of the height z at which u = u(z). The result
is used in further discussion of some of the crosswind dispersion data con-
sidered in Section 2 and of the difficulty of realistically inferring the
magnitude of the Lagrangian time-scale.
Section 6 contains a brief review of the special problems relevant to
the basic treatment of vertical dispersion from an elevated source. The
present lack of an established working treatment for vertical spread that is of
similar quality to those available for crosswind spread or for vertical
spread from a ground-level source is noted. Some difference exists in U.K.
xii
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and U.S. practical systems for dealing with the distribution from elevated
sources. Further work of a basic nature appears to be required to relate
vertical spread (from an elevated source) more satisfactorily to boundary layer
parameters.
xiii
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1. ON THE '2ND-ORDER CLOSURE' APPROACH
This note is an attempt to proceed a little further in clarifying
the issue raised by the writer and F. B. Smith in the note 'Some views
on modelling dispersion and vertical flux' (Meteorological Office
Met. 0.14 TDN 52, also in CCMS Proceedings of the Fifth Meeting of the
Expert Panel on Air Pollution Modelling, 1974).
For the homogeneous time-dependent case of vertical flux of material
(e.g., the non-steady quasi-uniform rate of evaporation from the ground),
which is the time-analogue of the spatially varying but otherwise steady
flux (e.g., the area source of pollution with non-uniform emission), the
2nd moment equation (e.g., see Donaldson, A.M.S. Workshop on Micro-
meteorology) is
9w'C'
9t
(2)
,2 9C
W 9z
(3)
9w|2C'
9z
(5)
C'9p' ,
p9z
(6 + 7)
C'T'
g T
o
(8)
!^ (i.D
(9) (10)
The term numbers are as in T.D.N. 52 and the sum (6) + (7) follows
directly if we assume p = p .
The issue arose from considering the relative magnitudes of
terms (2) and (3) from experience of an extreme case of unsteadiness,
i.e., the forenoon build-up of natural evaporation with strong insolation.
Schematically this is known to be as in Fig. 1.1. For the discussion of
TDN 52, specific data on estimates of E and the quantities in term (3)
1
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SUNRISE
NOON
Figure 1.1. Forenoon build-up of natural evaporation.
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were used, but the argument may be put in more general terms as
2 2
follows, writing a = w' :
w
(K - edd* diffusivity)
2
a
= ~Y~ E for small z (1.2)
2 2
0 (7 11
.W E « -^5 —— (1.3)
ku.z .2 v '
* k u* z
2 2
w
The ratio a /u+ is known to be about 2, u. to be about 1 m/sec, and
** *
for the present practical context we might take z = 10 m. Then term (3)
becomes
* oT4i§Eor°-5E
3E
At the time of maximum — , with sunrise to noon approximately 6 hours,
————— dt
9E E
9t 6 x 3600 sec.
term (3) n , , 0, , n3 in4
80 1 )r\ °-5 x 6 x 3.6 x 10 = 10 .
term (2)
Note that the ratio necessarily decreases with increasing z, but in
setting z at 10 m we can scarcely be significantly overestimating
the ratio for the practical matter of near-ground-level concentration.
Suppose now that the requirement is to solve the hierarchy of
equations for the 1st and 2nd moments in order to derive —-— ,
_ ^~
then w'C' as a function of z and t, thence -r—, thence C as a
ot
function of time. This would correspond to deriving pollutant concen-
tration as a function of distance X over an area source.
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From the foregoing numerical estimates it is clear that in any
solution of the 2nd moment equation for —-— the fractional error
3t
imposed by error in term (3) will be 10 times the latter (fractional
error). At first sight it would appear therefore that the computing
w
of ——— with acceptable accuracy is doomed to failure, for modelling
ot
of terms on the R.H.S. of the 2nd moment equation with an accuracy
4
better than 1 in 10 would seem unthinkable.
It has been pointed out by J. L. Lumley, however (in discussion
and private correspondence), that the foregoing argument does not hold,
and that the equation may be expected to be 'stable' as regards succes-
3w'C'
sive errors in —-— provided the terms on the R.H.S. are adequately
°t ...
8w*C*
modelled. So, although the values of —-— and of w'C'(t) may be
~~^~~~~~~~ ot
wildly in error to start with, they may be expected to settle down to
an 'acceptable' level of accuracy. Apparently this 'stability' is a
common feature of balance equations of the type considered here, and is
dependent on satisfactory modelling of the terms in a physical sense.
The original issue then reduces to two further questions:
(a) accepting that it may not in fact be necessary to model
the R.H.S. terms with the extreme accuracy indicated
above, just how accurately must they be modelled to en-
sure acceptable accuracy in the term (2)?
(b) in the reiterative type of solution which would be
followed, how long is required for the magnitude of
w'C' to settle down to the adequately accurate value?
The answer to (a) presumably requires a progressive experience
in the rational but necessarily 'trial and error' modelling of the
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various terms. For (b) a reasonable criteria would be some large
number, say 10, times the characteristic time-scale of the turbulent
mixing. It is not immediately obvious how, precisely, this time-scale
is to be specified. Presumably it must be a Lagrangian time-scale tT,
Lt
presumably at a height representing the average depth of mixing for the
material which has been released, and this height might typically be
about 100 m. We might then argue roughly as follows from experience
on the scale and spectral properties of the w- component in the atmo-
sphere — with t^ the 'fixed-point' integral-time scale, tT - At^ - 4z/u,
Ci L £<
i.e., approximately 1 minute for z = 100 m and wind speeds of practical
interest.
On the foregoing figures the 'settling-down* time may be some tens
of minutes, which may be acceptably small in relation to the diurnal
change of evaporation. However, for the steady, spatially varying case
over an area of dimension X the relevant time over which the source may
change substantially is presumably X/u, and with X say 10 km this time
is 1 hour or less. In this case it is not so obvious that the solution
for w'C' will have settled down during a typical transit time over an
area source of pollution of typical size, and perhaps this means that
the accuracy required in modelling the terms will be more stringent (in
comparison with that for the case of diurnally varying rate of evaporation).
In practice it also appears that modelling of Eq. (1.1) will usually
entail introduction of the flux w'C' explicitly in one of the larger
terms on the R.H.S. (e.g., the procedure followed by Lewellen (1974)).
2 ?/-!?
When —r— is very small, the solution for w'C' will be essentially
d t
determined by the modelled large term and the procedure will be tantamount
to using a stationary form of the equation (irrespective of whether or
5
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not the small —r—- term is included). It is of course clear that
Ot
in such circumstances doubts about the feasibility of adequately
modelling Eq. (1.1) do not arise to anything like the degree originally
envisaged in this note.
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2. CROSSWIND DISPERSION AND THE PROPERTIES OF TURBULENCE
The relating of crosswind dispersion from a continuous point source to
the flow properties in the atmospheric boundary layer has a fundamental basis
in the Taylor (1921) treatment of diffusion by continuous movements. In the
spectral form as historically developed, this treatment gives the following
result (see Pasquill, 1974, hereafter referred to as Ref. A, p. 125):
2
/
a 2(T) = a V FL(n) dn
7
where a (T) is the crosswind r.m.s. displacement of particles from their mean
position after time of travel T, a is the r.m.s. v-component of the turbulence,
and F (n) is the normalized Lagrangian spectral density function in terms of
r
frequency n, satisfying the integral relation FT (n) dn = 1.
j V"
Eq. (2.1) may be rearranged into a form for a non-dimensional spread S,
a (T)
defined by —^ , as follows:
o T
v
S
2
2
FL(n) Sin 7r(ntL)
d(ntL) (2.2)
in which we have introduced a dimensionless frequency nt (t is the Lagrangian
Lt Li
integral time-scale) and a dimensionless spectrum function F (n)/t . Note that
Li Li
in accordance with the cosine transform relation between F(n) and the correlation
function R the term F (n)/t is equivalent to 4F_(n)/FT(o). For a given
Li i_i L L
T
spectrum function (2.2) is a function of — only, i.e.,
L
S = f(f~) (2.3)
L
-------
which must have the general properties
f = 1 T -»• 0
f = (2tL/T)1/2 T + »
and in which the behaviour at intermediate T is entirely determined by the
shape of the spectrum.
For a prescription of f at intermediate T, two approaches have been followed:
(a) use of conjectured mathematical forms of the spectrum
function F (n) (or in practice the forms of the corresponding
Lagrangian auto-correlation function R(£)> see p. 130 Ref. A)
(b) assumption of a simple form of similarity between Lagrangian
and the (observable) fixed-point frequency spectrum (which is
essentially Eulerian in form).
In the approach (b) suggested by Hay and Pasquill (see p. 135 Ref. A), the
requirement reduces to deriving the variance of the turbulence component,
as observed at a fixed point, after first averaging the turbulent fluctuation
over periods T/3 , where 3 is the ratio of the Lagrangian and "fixed-point"
integral time-scales. In terms of the foregoing quantities
Jv]co,0 (2'4)
where the subscripts °° refer to sampling time (length of record) and T/3
or 0 to the averaging (smoothing) times. In practice finite sampling times (T)
may be used, with an error which may be argued to be small as long as T > T.
Another possible approach, which avoids the commital to particular
correlogram form or the particular assumptions of the Hay-Pasquill approach, is
(c) use of observations of the variation of S with T (or with
equivalent distance x) over a sufficient range to exhibit the
large T limit, to derive tL and so prescribe empirically the
functional form of f.
8
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An approach which Is closely related to (c) has recently been adopted by
R. Draxler at the Department of Meteorology, P.S.U. A practical difficulty
in the application of (c) is that a convincing attainment of the large T form
is very rare, and there are good reasons for not generally expecting the limit
to be attained in practice. In Draxlerfs analysis the method was to
characterize the time-scale by the time T. at which S falls to 0.5 and to fit
the variation of S with T/T. to the form
(2.5)
i
as a simple form consistent with the required limits of f and, of course,
requiring
T±l\ 2 a2 (2.6)
Draxlerfs analysis shows that the data from all the available field
studies on dispersion from a continuous point source may be represented by
(2.5), albeit with very considerable scatter, and a broad specification of
T./t (hence of t ) is thereby provided.
In the process of study of Draxler's analysis, certain features have
emerged which are outlined below.
The form of R(£) implied by Draxler*s form for S
The Taylor relation for dispersion may be stated in differential
form (see p. 124 Ref. A), and in the present notation and context
do, 2(T) , fT
2a R(e)d5 (2.7)
Jo
or 22
d a (T)
2a R(O , 5 - T , (2.8)
v
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so yielding the Lagrangian auto-covariance [a ^ R(S)] from the 2nd
2
differential of o (T). Eq. (2.8) may be written for Draxler's form
(2.5), in terms of T1 = T/tL, as follows:
2 Ji
[1
(2.9)
20
1
2
d2a2
d T
dT
,2
[1
1
2
,,2
d T
111
.2 J
l/2}2
[1 +
•2
(2.10)
On differentiating this becomes:
(1 + T'
-
T'1/2)3
- 1
where
T'
+ T'1/2)4
(2.11)
Values of R(£) according to Eq. (2.11) are shown in Table 2.1 and Fig. 2.1,
in comparison with R(5) of simple exponential form, i.e., R(£) = exp(- — ).
tL
Note that R(£) as in Eq. (2.11) initially falls off much more rapidly than the
exponential form but is then maintained at finite (though very small) value for
a much longer time. It is of interest to consider this difference in relation
to an attitude which has previously been advocated — namely that correlogram
shape is of secondary importance as regards the S(T') function.
The S(T') Function for Different Correlogram or Spectrum Shapes
In earlier analyses of the significance of the form of the correlogram
(p. 130, Ref. A), a certain range of shapes was found not to affect the S(T')
function significantly, and it was accordingly concluded that a and t would
10
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Table 2.1. VALUES OF R(C)
?/tL
0.1
0.2
0.4
1.0
2.0
4.0
10.0
Eq.(2.11)
0.471
0.360
0.254
0.139
0.078
0.040
0.014
exp(- £/t )
0.905
0.819
0.670
0.368
0.135
0.018
5 x 10~5
11
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Figure 2.1. R(£) according to (a) exponential form and (b) Eq. (2.11).
12
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be sufficient to determine a . (Note that in the earlier analysis referred
to dispersion was examined in terms of the dimensionless quantity D = 0 /a t ,
i.e., ST/t ). However, among the forms of R(5) adopted, the simple exponential
Li
form gave the most rapid initial fall. In view of the results in Table 2.1 and
Fig. 2.1 we need to reconsider the point regarding the importance of correlogram
shape.
Table 2.2 and Fig. 2.2 show values of S ( = a /a T) against T1 ( = T/tL)
for the forms of R(£) in Fig. 2.1 and also for a simple form of spectrum
which had previously been found to fit certain 'fixed-point1 data on the
w- component and v- component fluctuations (p. 61, Eq. 2.115 & p. 70, Ref. A).
In Lagrangian terms this spectrum is of the form
4t,
' (1 + 6 tTn)5/3 (
LI
and its consideration in the present context is implicitly on the assumption
of the Hay-Pasquill similarity in spectral shape. Also a single point on
the S,T' curve has been calculated (by graphical integration of Eq. (2.1) )
for another form of spectrum,
4tL
F(n) = * =- (2.13)
(1 + 4t_n)Z
LI
which in a Lagrangian context would be more acceptable than (2.12) in tending
_2
to n at large n (see p. 89 Ref. A).
In comparison with the result for the simple exponential correlogram,
both of the other curves show lower values of S; the maximum discrepancy
being near T1 = 2, as can be seen from the ratios included in Table 2.2.
The single value based on Eq. (2.13) is S = 0.55 at T' = 4, which is virtually
indistinguishable from the curve for Eq. (2.12), suggesting that the difference
in high-frequency behaviour of these two forms of spectrum is unimportant in
the present context.
13
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Table 2.2. CALCULATED VALUES OF S = a /a T
T/tL
0.1
0.4
0.8
1.0
1.2
2.0
4.0
7.0
10.0
12.0
20.0
40.0
Eq.(2.5)
& (2.6)
0.817
0.691
0.613
—
0.564
0.500
0.414
—
—
0.290
—
_—
for
R(£) = exp(- 5/t.)
0.938
(1.36)
__
0.860
__
0.755
(1.51)
0.615
(1.49)
0.495
0.424
0.390
(1.34)
0.308
0.221
for F(n)
in Eq.(2.12
0.848
(1.23)
0.777
(1.27)
—
0.724
(1.28)
0.650
(1.30)
0.543
(1.31)
—
—
0.368
(1.27)
—
__
[figures in parentheses are ratios of S to that for Eq.(2.5-2.6)]
-------
T/tL
Figure 2.2. Normalized dispersion S as a function of T/t[_ for (a) exponential correlogram, (b) Eq.
(2.11), (c) Eq. (2.12). The single point 0 corresponds to Eq. (2.13).
15
-------
The Nature of the Crosswind Dispersion Data
It has already been mentioned that the dispersion data exhibit large scatter,
and it may be seen that this becomes important when considering the fit of the
data to Eq. (2.9). The form of plotting used by Draxler was perforce in terms
of the time T., and as noted previously consistency between his interpolation
2
form Eq. (2.5) and Eq. (2.9) requires T./t = 2a . The magnitude of a is
JL J_i
fairly sensitive to the magnitude adopted for S even at large T since
'— }
Ti
The value of a obtained by Draxler, 0.9, is consistent with S = 0.27 at T/T. = 9,
whereas a value of S = 0.20 would lead to a = 1.3 and T±/tL = 3.4 (instead of 1.6).
From the plot of data given by Draxler for 'ground releases in unstable conditions',
reproduced here as Fig. 2.3, it may be seen that the relative merits of curves
passing through the two values 0.27 and 0.20 are perhaps arguable, especially
as for T/T. >1 the curve passing through S = 0.27 at T/T. = 9 seems
generally to be on the high side of the highest density of points.
Examination of the S,T Data from the /Greenglow' and 'Hanford 30* Field Programmes
Data in neutral to moderately stable conditions obtained in the above-
named programmes (which were included in Draxler's analysis) have been analyzed
to give ensemble average values of S in the form a /a x, where a is the standard
y 8 0
deviation of the wind direction fluctuation and x = uT has been assumed to apply
(u was measured at a height of 7 ft.). The results are shown in Fig. 2.4. Drawn
on Fig. 2.4 are curves corresponding to those of Fig.2.2, with different scales
of T', chosen so as to give approximate fit to the observations. From these
results it is clear that one might fit the data almost equally well in terms
of Eq. 2.5-2.6 or Eq. 2.12, but with very different values of GtL implied —
roughly 5 km or 1.6 km respectively. The fit provided by an exponential form
of correlogram is poor and from the present data it must be concluded that the
16
-------
"CD
X
CO
l_
Q
CD
O)
T3
C
o
0)
CD
C
c
o
'tf>
k_
o.
(/)
45
o
N
'L.
O
X
00
csi
OJ
17
-------
Jt
*x
CO
E-
col
-Q; cc
_ ro
i o>
>
'O
COU-
c ™
CD <->
"O: v
cj
-------
Lagrangian auto-correlation function is not of this simple form.
In the light of the curve shapes in Figure 2.4 there is no very clear-
cut distinction between the interpolation form examined by Draxler and the
form appropriate to the spectrum of Eq. (2.12). Further progress will probably
depend on obtaining better insight into Lagrangian versus 'fixed-point* spectrum
relations, and on making such comparisons as in Figure 2.4 with the added knowl-
edge of the 'fixed-point' spectrum of the v- component (instead of ag alone).
The situation is also complicated in another sense — namely that in the
sheared flow of the atmospheric boundary layer there is expected to be a
contribution to a from the interaction between vertical spread and the mean
turning of wind with height. It has previously been argued (see p. 29
Ref. A), partly on the basis of the very data in Figure 2.3, that this
interaction effect was not evident to an important degree over the range
of distances included in Figure 2.3 (i.e., up to 12.8 km), but that it had
become important at the extreme distance (25.6 km) included in the field
measurements. When one is considering optimum fitting to S- data at large T
it is clear from Figure 2.4 that minor uncertainties in the true value of S
determined directly by the v- component of turbulence, as distinct from the
indirect effects of shear, will sensitively influence the inferred magnitude
of ut .
Zj
The need to use the relation x = uT in interpreting dispersion measure-
ments at a given distance downwind of a source has already been mentioned,
and discussion of this point is continued in Section 5.
19
-------
3. WIND DIRECTION FLUCTUATION STATISTICS OVER LONG SAMPLING TIMES
The application of the relations considered in Section 2 requires a
prescription of the standard deviation 0 of the crosswind component of
turbulence or, as commonly used in practice, the standard deviation OQ of
8
the wind direction fluctuation ( 0 - 0Qu). As afl is not a commonly observed
V o H
quantity there has been much interest in the development of 'climatological'
relations for 0 in terms of other more easily specified parameters. Unfortunately
the properties of the v- component and the customary boundary layer parameters
do not show as much order as those of the w- component (see p. 81-84
of Ref. A). An outstanding feature is the relatively indeterminate nature
of the low-frequency section of the spectrum, around a frequency of about
1 c/hr, a section which is intermediate between well-defined micrometeorological
and macrometeorological sections, and usually termed mesometeorological.
The most obvious appearance of order seems to be in the high-frequency
part of the micrometeorological spectrum. This is evident in relatively orderly
relations for o for short sampling times (1 minute or so, see p. 81 of Ref. A).
H
The properties of 0Q for much longer sampling times (1 hour or more) have
received little critical attention, and so it seemed desirable to carry out
the pilot investigation described below.
A selection was made of ten 3-hr sections of the May 1973 wind direction
records routinely made at a height of 44 m on a tower at Cardington, England.
With one exception the sampling periods were near noon, with moderate to
strong incoming solar radiation and with a good range of wind speed. Small
reproductions of the 1200 Z surface synoptic charts are collected in Fig. 3.1.
Average wind directions over consecutive 3-minute periods were extracted and
these are displayed graphically in Fig. 3.2. For each 3-hr sample a linear
regression of the 3-min average against time was calculated and departures
20
-------
Figure 3.1. 1200 GMT surface charts, May 1973.
21
-------
300
11 MAY 7311-1400
260
280
12 MAY 73 11-1400
240
' 280
o
<
o
u
250
14 MAY 73 0930-1230
010
02Q
080
_ 16 MAY 73 1000-1300
v
I V I
10
20
30
T (3-min STEPS)
40
50
60
Figure 3.2. Three-minute average wind directions over 3-hr periods.
22
-------
120
090
110
17 MAY 73 11-1400
-6-
£ 070
S1
^
CD"
19 MAY 73 1230-1530
5
UJ
oc
g 070
a.
o
0 050
19 MAY 73 20-2300
^_«_ —
-8-
— ^X>^ —
180
26 MAY 73 11-1400
120
150
100
27 MAY 73 1130-1430
10
20
30
T (3-min STEPS)
40
50
60
Figure 3.2.(continued). Three-minute average wind directions over 3-hr periods.
23
-------
from the regression obtained. These 'fluctuations from the linear regression1
were used to calculate standard deviations for sub-sampling-times ranging from
1/2 to 3 hours. The results, with background data including estimates of the
geostrophic wind, are set out in Table 3.1, and graphs of the standard deviations
(a ) as functions of sampling time T are in Fig. 3.3. In Table 3.1 the standard
o
deviations are presented both in the original angular form (a.) and in the
D
form of a , using the approximate conversion already noted.
The main features of the results are as follows:
1. The values of a constitute a significant addition to those
for a sampling time of 3 min (as for example summarized in pp. 81-83
of Ref. A). The addition would, of course, need to be made
in terms of variance.
2. The additional variance represented by the 3-min averages
sometimes appears mainly in the first 30 min of the extended
sampling times but in others it appears progressively with
further increase of sampling time up to 90 min. None of the
samples gives any significant increase with increase of sampling
time beyond 90 min.
3. With reference to the last point of 2. it should be remembered
that a 'linear trend* has been 'extracted' from the basic
variation of 6, and some of this trend would presumably appear
as variance in much longer samples. Referring to Table 3.1
it will be seen that the 'trend' over 3 hrs may be as much as
three times the standard deviation of the 3-min averages. In
terms of short-range crosswind spread from a release over three
-------
^^
o
1 — 1
1—
•=c
1 — 1
UJ
f*~\
Q
o:
car
Q
H-
00
'-^
O
t — 1
1—
ZD
y
_i
U-
z:
0
h-
o
UJ
o;
i— i
Q
Q
2:
1 — 1
3
U_
0
oo
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1 — 1
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s:
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r__
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r—
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1—
00
(••>,
cn
^
2:
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I— i
o:
a
E
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u.
0
t—
CD
UJ
~r"
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<£
1—
O
1—
00
Q
a;
o
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cc
Q
1 — 1
3
^ — .
oo
1 1 1
CD
ef
OC
UJ
^>
.
o
LO o r^
Cn ^*»» ^f r"~ LO
LO ^~ * *
CM 1 - r>. i —
1 ^
o
o
LO *d- i—
r— ^, r— ^J- VO
i — i — LO "Sj- i —
•— r-.
•
O O CM
•z. co E
in o , ,
+J & B £
to re ••- . E -^.
O Q I— p (2 CD
^}-
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r~ cn
i— CM
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r~
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•
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•
LO
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r~~
00
LD
cn
•
o
•
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i.
J=
LO
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•
1^.
cn
o
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•
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00
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cn
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s:
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CO
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c
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i.
h-
-------
hours this means that the arc of spread may contain a
significant (if not dominant) contiibution simply from
the systematic swing of the wind ov^.r the three hours.
4. No variation with intensity of solar radiation is evident
(presumably a consequence of the not very marked range of
this property). However, an inverse relation between a and
wind speed is clearly evident in Fig, 3.4. In terms of a
there appears to be a quite sharp increase with wind speed
up about 7 m/sec, but the two highest wind speeds (near
12 and 14 m/sec) show no further inc'ease in a .
26
-------
10
2
§>
61 (4.1)
•O 74 (5.4)
-• 71 (6.8)
-t 82 (8.3)
-H 47 (7.8)
33 (8.2)
61 (14.2)
30
60 90
r MINUTES
200
Figure 3.3. Standard deviation OQ(T) of the 3-min average wind directions (after removal of
linear trend) as a function of sampling duration T. Run numbers on left, wind speed m/sec
at 44 m (in parentheses) and incoming solar radiation (mw/cm^) on right of diagram.
27
-------
14
12
"s 10
A
A
A
A
i.o
A
.A
1.4
1.2
0.8 ~>
0.6
10 15
u at 44 m (m/sec)
Figure 3.4. a0(r) (•) and av(r) (A) for sampling time T = 90 min (excluding night
observation No. 8).
28
-------
4. 'LOCAL SIMILARITY' TREATMENT OF VERTICAL SPREAD
FROM A GROUND SOURCE
In the 'so-called* Lagrangian similarity theory of diffusion, the
basic principle is to express the rate of dispersion, say dZ/dt (where
Z is the mean displacement of an ensemble of particles after a given
travel time), in terms of the basic parameters of the turbulent boundary
layer. A brief general review of the history and main developments has
been given by the writer (Ref. A). The developments fall into two main
sections — the form of dZ/dt and the form for the resulting decay of
concentration with distance downwind of a source — and here the
principal concern is with the former aspect.
For a stratified atmosphere the general starting point as adopted
by Gifford (1962), following Batchelor's (1959) formulation of the
problem for adiabatic flow, is
f - ta* * (4-i)
The constant b is now widely accepted as near 0.4 (arguments have indeed
been advanced for equality with von Karman's constant), but the form
of $ has not been firmly established, and this is one of the outstanding
limitations in the present stage of development of the theory. Another
limitation is the restriction of the Batchelor-Gifford treatment to the
surface-stress layer and consequently to vertical spreads no more than
some tens of meters.
A Local Similarity Hypothesis for Unlimited Vertical Spread
A possible alternative which avoids some of the difficulties and
limitations of the original approach is to postulate that the rate of
29
-------
increase of Z, or alternatively of some measure Z of the extreme vertical
m
displacement, is always determined by two local properties
(a) ow(z)
(b) the scale of turbulence £(z) prevailing at Z or Z
as the case may be.
e\7
If — is to be expressed in terms of the velocity and length
scales now prescribed, the simplest dimensional hypothesis is
I • Vi
-------
Development of the Local Similarity Hypothesis Into a Form
for Practical Test
We may rearrange Eq. (4.2) and (4.3) into a more practical form as
follows, by taking the further step of assuming & proportional to the
spectral scale X (defined by u/n where n is the frequency for peak
magnitude of n S(n), and S(n) is the spectral energy density at fre-
quency n). Then Eq. (4.2) and (4.3) may be rewritten
dZ c , mN ,. ..
dF = °wf3 (— (4'4)
dZ X
m c f nu /. _N
dF = °wf4 (4'5)
m
r\ V <—
or writing a /u = i and -r— = u(Z)
1 dZ ,. , m. ,. ...
= f, (—) (4.6)
i dX Z
1 dZ X
—T = f4 (-?) (4-7)
i dX Z
m
(a more precise argument would recognize, following Batchelor (1964),
A Y —
that -;— = u (cZ) etc. but this refinement will be omitted in the
dt
present considerations). If now we have specified i and X as functions
of height and are given the form of f» or f, (which would have to be
determined empirically), Eq. (4.6) or (4.7) may be integrated numerically
to give Z or Z as a function of X.
m
An interesting additional relation follows from consideration of
the neutral surface-stress layer, for which Eq. (4.1) may be put in
31
-------
the form (taking b = k)
il = (_K}
dX uz-
where K(= ku^z) is the eddy diffusivity. Equations (4.6) and (4.8) are
formally equivalent if f» is a linear function (for then we have
±\ Iz « K/uz, i.e. K « a \ ,
m w m
a form which may be argued from statistical theory considerations).
This suggests that in general (i.e., irrespective of the extent of
vertical mixing and of thermal stratification) we might expect
— - f- (—) (4.9)
dX uz-
dZ
-^ - fg <:r) (4.io)
dx uz z
m
and it will be seen that these forms do indeed provide a simple
generalization of estimates of vertical spread derived from the two-
dimensional diffusion equation.
Preliminary Tests of the Local Similarity Predictions and Implications
Regarding the Functional Forms
Observational data on vertical spread from a continuous source
at ground-level in relation to the conditions of turbulence are avail-
able from the Prairie Grass project in U.S.A. (see p. 206 Ref. A) and
from N. Thompson's field studies in stable conditions at short range
(1965) and medium range (1966). Unfortunately, in none of these studies
are the profiles of X (in particular) immediately available, nor is it
m
certain that such profiles could now be adequately extracted from the
data records.
32
-------
Pending the possible extraction of such profiles or, as may turn
out to be necessary, the collection of new data, an interim analysis
may be carried out using theoretical estimates of vertical spread derived
by F. B. Smith (see Ref. A p. 363) from numerical solutions of the two-
dimensional diffusion equation. As these solutions necessarily required
specification of the K profiles the resulting data are immediately suit-
able for examination in terms of Eq. (4.9) and (4.10). Also, values of
K used by Smith were derived explicitly from values of X and imply
values of a , so with a little rearrangement of the data the examination
V»
may be conducted in terms of Eq. (4.6) and (4.7). It is noteworthy that
a practical range of thermal stratification is covered by using parameters
appropriate to neutral conditions, to strongly unstable conditions as
represented by a Monin-Obukhov L of -7 m, and to moderately stable con-
ditions represented by L = 4 m. In order to obtain such small magnitudes
of L with realistic turbulent heat fluxes, a relatively light wind was
assumed (geostrophic value 4 m sec ). The numerical details are
collected in Table 4.1 and the final results are plotted in Figures
4.1 and 4.2.
From Figures 4.1 and 4.2 we can conclude that 'diffusion equation'
values of vertical spread follow a near-universal relationship with
'similarity variables'. The present evidence, it will be noted, covers
not only a wide range in stabilities but also a 100-fold range in
vertical spread. The results have two immediate consequences:
(a) should we wish to adopt different K profiles from those
used in the numerical solutions used here we may compute
vertical spread (Z or Z ) much more easily from the
m
33
-------
Table 4.1. SIMILARITY ANALYSES OF VERTICAL SPREAD DATA FROM NUMERICAL
SOLUTION OF THE 2-DIMENSIONAL DIFFUSION EQUATION
(Units in metres and seconds)
L
x
°z
Zm
Z
dZ x 103
m
dlT
dZ
dx
104
103
10
x 103
Data
K
u
K/uZm
Am
ow/u
u dZm
-7
1200
2850
954
45.5
16.8
at Zm
105
4.0
102
1000
263
1.73
00
30x1 O3
320
688
254
11.0
4.07
5.4
4.0
19.6
500
27
4.07
+4
63
135
50
2.25
0.83
0.26
4.0
4.8
104
6.3
3.57
-7
280
602
223
165
61
103
3.95
430
982
264
6.25
3xl03
78
168
62
41.5
15.4
7.5
3.8
118
385
51
8.14
+4
17
36.6
13.5
7.6
2.82
2.24
3.6
18.2
52
13
5.85
-7
27
58
21.5
193
71
13.5
3.6
647
21U
179
10.8
300
12
25.8
9.5
73
27
1.8
3.08
226
75
78
9.36
+4
3.7
7.9
2.92
19.0
7.0
0.102
2.3
56
11
40
4.75
00
300
1.6
3.44
1.27
100
37
0.25
2.18
330
10
113
8.81
aw dx
VZm
103
102
102
Data
K
u
K/uZ
Am
°v/u
u_ d2
w
V2
u,
>m and
0.39
at Z
105
3.98
27.7
1000
26.4
6.4
1.05
0.73
8.15
3.92
8.18
448
4.6
8.8
1.76
K - as used by F.
a - as derived by
Zm"
(height
of cloud)
and Z
0.77
0.26
3.8
1.36
65
1.05
7.9
1.3
B.S. for
F.B.S.
1.63
74
3.88
84.9
755
25.1
24.3
3.39
V-
from
(mean height
2.29
4.2
3.45
19.6
195
6.2
24.7
3.15
1.42
0.142
2.7
3.89
23
2.29
12.3
1.7
03m, Vg=4m/sec,
3.62
4.4
3.28
62.8
80
16.9
42
3.72
taking
2.91
0.67
2.62
26.8
30
8.5
32
3.16
K=eH/1!
1.39
0.049
1.70
9.87
4.1
7.0
10
1.4
3
2.91
0.092
1.71
42.3
3.7
14.5
25.5
2.91
the numerical solutions.
of particles) ,
from az
assuming
Gassian
distribution (i.e. Zm/oz = 2.15, Z/Zm = 0.37).
- as implied by K = yn o A , which is consistent with foregoing
I u w m
expression for K in terms of c.
34
-------
10
:2
10
K/uZmOR K/uZ
10'
Figure 4.1. Rates of vertical spread dZm/dx or dZ/dx against (K/uz) at Zm.
35
-------
0.4
1.0
N
0.1
0.2
10
'in
udZ udZm —
Figure 4.2. — -r- and — — against (7/7.) at Z or Zm.
w
36
-------
similar relations here demonstrated and avoid the labor
of further numerical solutions of the diffusion equation.
(b) The result encourages progress to the next step — namely
the relating of actual values of vertical spread to i and
X/Z or X/Z , so determining the functions more directly and
hopefully more generally.
37
-------
5. REPRESENTATIONS OF DISPERSION IN TERMS OF DISTANCE OR TIME
The Formal Relation Between Time and Distance of Travel
In the description of the concentration field associated with a source
of pollutant, we are normally concerned with source and reception positions
fixed in space. This naturally leads to a consideration of dispersion
parameters (e.g. a or a ) as a function of distance of travel, rather than
y z
time of travel, even though in theoretical treatments we may well be thinking
explicitly in terms of a velocity of dispersion (dZ/dt for example in
Lagrangian similarity theory) and therefore in terms of dispersion after
a given time. However, the latter is not the case in the diffusion equation
approach, in which the representation may be entirely spatial (e.g., the
conventional steady form of the two-dimensional conservation equation
- 3C _ 3 3C ,
u"9^ ~ 8z K 9z >'
For the idealized case of a constant and uniform mean wind it is
customary to relate distance (x) and time of travel (T) by writing
x = u T (5.1)
The physical significance of (5.1) requires careful consideration however.
In a homogeneous field of turbulence the alongwind (x) velocity at any
position is u = u + u', with u identically the time-mean at the position,
or the spatial mean. For an ensemble of passive particles which pass through
the selected position in a long regular sequence (an idealized continuous
point source), the following statements may be made.
(a) Over a travel time T each particle will have a mean speed (u )T>
and for T small (u )_ = u, and for the ensemble of particles the
P T
average of these mean speeds will be
[
-------
If T is very large each particle will experience the whole spectrum
of fluctuations and for each particle (u )„, = u. It is accordingly
P l
a plausible hypothesis that Eq. (5.2) holds for all T and therefore
the mean alongwind displacement X of the particles is
X = [(up)T] T = u T (5.3)
(b) Now consider the ensemble of particles as each achieves a constant
alongwind displacement x after travelling a variable time s. Then
if x (and s) are small
s = x/u = x/(u + u')
x f- u'l ... u1 . ,,
= — 1 - — if — is small
u [ u J u
and the ensemble average of s will be
r i X
[s] = -
u
and for a long distance x each particle will have (u ) = u
P s
by the same argument as in (a). Irrespective of x it is also
a reasonable hypothesis that
[s] = * (5-4)
u
Thus in writing Eq. (5.1) in relation to dispersion there are two alternative
interpretations of T
(c) T(= [s]) is the average time of travel of particles reaching
the position x
(d) T is the particular time of travel after which the particles
have an average displacement X.
The question now to be considered is whether the dispersion characteristics
(say lateral spread) apparent in these two different considerations (with
x = X or T = [s]) are in any way different. One way of looking at this is
39
-------
to note that in the whole ensemble of particles considered as in (c) there
will be a subset (A) which have near-constant time of travel equal to [s](=T).
Likewise in (d) there will be a subset (B) with a near constant displacement
equal to X(=x). These two subsets are by definition the same and will have
the same dispersion characteristics. If each subset is an unbiased sample
of the whole ensemble as regards (say) lateral spread (and there is no
reason to expect otherwise in the conditions defined) then the dispersion
properties of the two whole ensembles will also be identical.
The above argument, though not a formal proof, strongly supports
the equivalence of dispersion properties in the time and distance coordinates
as defined, for the case of homogeneous fj.ow.
In the real case of a boundary layer wind field we may still be able to
assume quasi-homogeneity in the horizontal, but u is now a function of z.
The assumed relation between time and distance of travel may then be as in
the Lagrangian similarity hypotheses, i.e.
~ - u (cZ) (5.5)
where Z is the mean vertical displacement of the ensemble of particles, each
of which has travelled for a given time T. Distance downwind x and mean down-
wind distance of travel X may then be related by making some simple assumptions
as follows. Let
Z(t) = at (5.6)
G(z) - bza (5.7)
Then substituting in Eq. (5.5) and integrating
fT
X(T) = i b(act)a dt
+ a)
40
-------
and on substituting (5.7) for a given reference height z and rearranging,
a
X(T) = - — (u(z ) T) T (5.9)
Thus we may define an 'equivalent homogeneous' u for which X(T) and
x(= u [s]) are identified, such that
the important point being that u increases as T . Alternatively, by
rearranging (5.8) and substituting (5.7) it can be seen that
u = u at z = °Z vx' (5.11)
(l+c01/a
On these arguments we should examine dispersion data (prescribed as
f(x)) in terms of the statistical and similarity theories (which basically
prescribe dispersion as f(T)) by using an equivalent advecting speed as in
(5.10) or (5.11), always remembering however the special assumption made
in (5.6). Suppose, for example, we take a = 0.15 (a typical value for the
atmospheric boundary layer wind profile), then in terms of Eq. (5.11)
z = c^JX)
e ,
- 27J Z(T) (5.12)
and if we take the only estimate that has been made of c, i.e. 0.56 in
neutral flow (see p. 119 of Ref. A) this becomes
ze = 0.22 Z(T) (5.13).
The Physical Involvement of u in the Effects of Thermal Stratification
on Dispersion
Apart from the formal relations considered above, there is complex
-------
involvement of wind speed in the effects of thermal stratification on the
properties of turbulence. Thus, if we accept that the controlling parameter
is z/L or, roughly, the value of the Richardson No. at z, then the grouping
of dispersion data in terms of Ri will necessarily impose some grouping in
wind speeds (since larger values of JRi| tend to be associated with small
wind speeds). Thus the apparent effect of thermal stratification on say a
y
will tend to be different according as the time or distance representation
is adopted. However, provided the time and distance coordinates are related
consistently (as above) there is no obvious reason to expect, for example,
that the two representations will lead to different specifications of the
spatial distribution of pollutant concentration.
An interesting reflection on the foregoing point is provided by the
following considerations of the continuous-point-source dispersion data
reported by Fuquay, Simpson and Hinds. Fuquay et_ al. make much of the
point that their normalized peak exposure (i.e. concentration x wind
speed -5- source strength) shows more variation with 'stability' (as repre-
sented by Ri at a given reference level) when considered in relation to
time of travel (taken to be x/u with u at a given reference level) than
when considered in relation to distance x downwind of the source. This
result is obviously partly a reflection of the inter-relation between Ri
and wind speed already mentioned; and it is difficult to see why on these
grounds one should immediately infer (as do Fuquay et al.) that the disper-
sion v. time representation is basically preferable to dispersion v. distance.
Fuquay et al. make no appeal to the preceding argument concerning the wind
profile, and their conclusion that a is a simple function of aflu(~ cr )
and T, rather than of OQ and T, is to a large degree a statement of the
o
obvious, following the Taylor approach of Section 2. Apart from the
42
-------
subtleties introduced by the wind profile they could equally logically
have asserted that 0 is a simple function of a and x, and not of a and x!
(i.e. in terms of Section 2, we may write a = a Tf(T/t ) or (av/u)
u Tf(uT/utT) i.e. a.x f(x/utT) ).
L t) L
The only important point would appear to be whether, for example,
the testing of either of the foregoing forms in terms of observations at
a distance x is obscured by the wind profile aspect. The point may be
examined in two ways.
The Magnitude of a /a T at Very Short Range
The mangitude of a /a T theoretically tends to unity as T becomes
very small. Failure to confirm values of unity in practice may stem from
two causes:
(a) The time T is not short enough in relation to tT — this
LI
will always result in values lower than unity.
(b) The time T will normally be derived as x/u(z^), i.e. using
a wind speed at some convenient reference level, whereas
the time x/u should be used. The direction of the effect
e
(on the apparent a /a T or a u(z )/a x) will depend on the
relative magnitudes of z, and the required z .
The Fuquay e^t al. data at x = 200 m in groups characterised by
Richardson Number, are plotted against u (7 ft.) in Fig. 5.1. An overall
average of the 'neutral-slightly stable1 group (0 < Ri < 0.08) has already
been derived as 0.98 (Section 2). Data in the more stable group are
generally similar (apart from the isolated point at u near 0.9 m/sec),
but those for the unstable group are relatively lower at the lower wind
43
-------
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speeds in this group. In order to examine the relevance of item (b)
above it is necessary to know the magnitude of vertical spread and in
the absence of observed values these may be estimated roughly (from
p. 375 et seq of Ref. A), for z = 10 cm, x = 200 m and stability categories
as stated, and taking Z/a = 0.78
2
F=5.5 a=4m or Z = 3 m
z
E = 4.5 a = 7 m Z = 5.5 m
z
C = 3.5 o=10m Z = 8 m
z
According to the previous analysis z = 0.22 Z and if this relation is
adopted irrespective of stability the corresponding values of z are
0.7, 1.2 and 1.7 m.
In the Fuquay et al. data zj_ = 7 ft. or approximately 2 m, fortuitously
a satisfactory correspondence to z in unstable condition and being somewhat
high in the stable conditions. This means that u was probably overestimated
in the stable conditions and this could be the explanation of the tendency
to marginally higher values of a /a.x in the stable conditions. Admittedly,
y o
however, the argument cannot be made firm without begging the question as
regards the actual magnitudes of T/t (i.e. in regard to the significance
L
of their departures from zero) for the different conditions of stability.
The Magnitude of 0 /a T at Long Range
1/2
At large x, when a should tend to a variation with x , and if
z > z, (as will usually be the case if z, is a customary low reference
level), the quantity a u(z.)/a x will tend to be an underestimate of the
required a u /a x (for simplicity a is being assumed independent of
height). The time x/u(z,) will correspondingly be an overestimate of the
45
-------
1/2
required time x/u , but in view of the tendency to a « x the net effect
will be that a a u(z-)/a x, x/u(z1) observation will be an underestimate
(of a u la x).
y e v '
In analysing data such as those in Fig. 2.4 the ultimate interest is
in evaluating tT, which on the preceding arguments would be considered
l_i
to be best approximated by
2 _ 2
t_ = a u /2a x
L y e v
at large enough x. If z > z. it follows that use of u(z..) will give an
underestimate of t . As an example the effect may be estimated roughly
LJ
for the case of Fig. 2.4, for which z, was 7 ft. According to the result
in Eq. (5.13) z should be roughly Z/5. In the absence of actual data Z
may be estimated (from p. 375-6 of Ref. A) to be probably no less than
say 40 m at x = 12.8 km (take z = 10 cm and P = 6). Hence the estimated
value of z is 8 m and with zn near 2 m the ratio u /u(z..) could be
e 1 el
considerable — e.g. near 2 if a = 0.5 (a not unreasonably high exponent
for a stable wind profile). Interpretation of dispersion data to yield
estimates of t therefore needs to be carried out with careful attention
Li
to this aspect.
-------
6. MODELLING FOR ELEVATED SOURCES
The basic treatment of the elevated source is still confronted with
certain difficulties, and none of the principal approaches which apparently
provide reasonably satisfactory interpretations of the ground-level source
is obviously acceptable. In considering these difficulties (which are
primarily concerned with vertical spread), it is helpful to consider the
downward spread in three stages, in terms of the scale of turbulence
and the height (H) of the source:
Stage 1. a < H [and therefore <
z
Stage 2. a - H [therefore -
Stage 3. a »H [therefore
In brief the difficulties may be stated as follows:
(a) The Taylor statistical theory, while plausibly applicable to
crosswind spread irrespective of the elevation of the source
and to vertical spread in Stage (IX is in principle not applicable
to vertical spread in Stages (2) and (3), on account of the
systematic change (with height) of the scale of the vertical
component.
(b) The gradient-transfer approach, while plausibly applicable to
vertical spread from a ground-level source (hence to an
elevated source at distances large compared with the distance
of the ground-level maximum concentration, i.e. Stage (3) ), is
in principle not applicable to Stages (1) and (2) of vertical
spread, i.e. those which determine the downwind position x of
the ground-level maximum. This arises from the characteristic
behaviour of a continuous source plume when the scale of
turbulence is not small compared with the magnitude of the
instantaneous plume spread.
47
-------
With regard to (a), as expected, experience does support the
applicability of the statistical theory for Stage (1) (see pp 198-202 Ref. A).
However, it is surprising to find that certain experience with a moderately
elevated source (see p. 203 of Ref. A.) is compatible with the statistical
theory even in Stage (2), when the properties of the w- component at the
height of release are employed as if applicable to the whole height range 0 - H.
The cases in point were for H *= 50 m and were characterised by at least a
moderate degree of vertical mixing (i.e. the maximum value of xffl/H was 20),
and for such cases it is plausible to consider that the effective scale of
turbulence was not very different from that at z = 50 m (bearing in mind
firstly that the variation of I with z was unlikely to be much more rapid
than H oc z, and secondly that the ultimate importance of the smaller scales
close to the ground would tend to be offset by scales even larger than
that at z = H, such larger scales being effective in the downward-moving
'eddies' originating somewhat above z = H).
In respect of the application of these ideas on vertical spread to
the evaluation of the concentration pattern downwind of an elevated
continuous point source, it is worth noting that the pre-eminent requirements
(additional of course to the source strength and wind speed) are
(c) the height H of the source (which really prescribes the vertical
spread associated with the maximum concentration at ground
level) and
(d) the crosswind spread a at the distance of maximum concentra-
tion (x ).
m
The foregoing statement may be formally demonstrated in terms of a
simple plume model (see p. 274 Ref. A). Looked at in this way interest is
48
-------
focussed on the relation between vertical spread and x • Also, as a
* in
further simplification, it may often be advantageous to think initially
in terms of the concentration C(x,z) from an elevated continuous line
source of infinite extent acrosswind (instead of C(x,y,z) from a continuous
point source). In this case C(x,z) is dependent only on vertical dispersion,
but since it is usually reasonable to regard the vertical and crosswind
dispersion processes as acting independently on a continuous point source,
we may write
C(x,y,z) = C(x,z) (6.1)
/27 a
y
when the source strengths, respectively Q per unit time and Q per unit
time and unit length, are identified. Eq. (6.1) assumes (reasonably)
that the time-mean crosswind distribution is Gaussian. With this approach
attention may be concentrated exclusively on the more complex vertical
dispersion and the relatively simple effect of crosswind spread may be
superimposed at a final stage.
Turning to the current state of the practical systems available
for estimating the concentration pattern downwind of an elevated continuous
point source, the following points are especially noteworthy:
(i) The Meteorological Office 1958 system of estimating a (and a )
z y
was basically formulated for a ground-level source, and no attempt
was made to allow for the special properties of vertical spread
in stages 1 and 2 above. This is the system which was reproduced
i
as 'Pasquill-Gifford curves' and adopted in the Turner 'workbook'.
The basic restriction to ground-level source also applies to the
modification and extension of the 1958 system recently formulated
by F. B. Smith (see p. 373 of Ref. A).
49
-------
(ii) The ASME 'Reconnnended Guide for the prediction of the dispersion
of airborne effluents' differs in principle from that in (i)
in that it uses the direct (but necessarily limited) experience
with a passive elevated source at Brookhaven National Laboratory.
One feature of this which requires critical consideration is that,
in line with the general trend in earlier work on diffusion, a
and a are taken to have the same variation with distance x.
z
(iii) It has long been accepted that the growth with distance is
fundamentally different for crosswind and vertical spread from
a ground-level source. For the elevated source a systematic
difference in the two growths was first directly demonstrated
in Hogstrom's study with passive smoke in Sweden (see p. 278
of Ref. A). Moore's (1974) interpretation of the distribution
of sulphur dioxide downwind of power stations in the U.K.
recognizes such a difference and indeed adopts the simple forms
1/2
a .« x, 0 <= x . Although it is probably not difficult to
justify the latter assumption as roughly correct for relatively
stable flow, it is open to question for unstable flow when
the effective scale of turbulence may increase with height for
heights in excess of that reached by hot plumes. (Note also
that the T.V.A. analyses of 'hot plume' data which are quoted
by F. Gifford in an unpublished draft show differences in the
growth curves especially in relatively stable conditions.)
(iv) Moore's study also recognizes an important 'induced1 spread of
hot plumes in the stage of ascent, arising from the relative
vertical motion of plume and ambient air.
50
-------
Moore's interpretation of U.K. data on pollution distribution
downwind of power stations is a commendable attempt to incorporate realistically
the effects of both natural and induced spread. However, the natural spread
is not related explicitly to the field of turbulence. Although the latter
represents a very complex problem it may be that the time is 'ripe' for
a further attempt - possibly by adapting statistical theory for stages 1 and 2
and matching to a similarity treatment of Stage 3 on the lines considered
in Section 4.
51
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REFERENCES
A (reference is as for Pasquill, 1974, below)
Batchelor, G.K., 1959, Note on the diffusion from sources in a turbulent
boundary layer (unpublished).
1964, Diffusion from sources in a turbulent boundary layer,
Archiv Mechaniki Stosowanej, 3, 661.
Donaldson, C duP., 1973, Construction of a dynamic model of the production
of atmospheric turbulence and the dispersal of atmospheric pollutants,
Workshop on Micrometeorology, American Meteorological Society.
Fuquay, J. J., Simpson, C. L. and Hinds, T. H., 1964, Prediction of environ-
mental exposures from sources near the ground based on Hanford experimental
data, J. App. Met., 3, 761-770.
Gifford, F., 1962, Diffusion in the diabatic surface layer, J. Geophys.
Res., 67, 3207.
Lewellen, W. S., et al, 1974, Invariant modelling of turbulence and
diffusion in the planetary boundary layer, Contract Report EPA-650/4-74-
035.
Moore, D. J., 1974, Observed and calculated magnitudes and distances of
maximum ground-level concentration of gaseous effluent material downwind
of a tall stack , Turbulent Diffusion in Environmental Pollution,
Advances in Geophysics, Vol 18B, Academic Press.
Panofsky, H. A., 1965 (with Prasad, B.), Similarity theories and diffusion,
Int. J. Air, Water Pollution, 9, 419-430.
Pasquill, F., 1974, Atmospheric Diffusion, 2nd Edn., Ellis Horwood, Ltd.,
Chichester, and Halsted Press, New York.
Taylor, G. I., 1921, Diffusion by continuous movements, Proc. London Math.
Soc., Ser. 2, 20, 196.
Thompson, N., 1965, Short-range vertical diffusion in stable conditions,
Quart. J. R. Met. Soc., 91, 175.
1966, The estimation of vertical diffusion over medium
distances of travel, Quart. J. R. Met. Soc., 92, 270-276.
52
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TECHNICAL REPORT DATA
(Please read Instructions on the reverse before completing)
1 REPORT NO.
EPA-650/4-75-015
4. TITLE ANDSUSTITLE
Some Topics Relating to Modelling of Dispersion
in Boundary Layer
6. PERFORMING ORGANIZATION CODE
3. RECIPIENT'S ACCESSION NO.
5. REPORT DATE
April 1975
7. AUTHOR(S)
8. PERFORMING ORGANIZATION REPORT NO
F. Pasquill
9 PERFORMING ORGANIZATION NAME AND ADDRESS
North Carolina State University
Raleigh, North Carolina
10. PROGRAM ELEMENT NO.
1AA009
11. CONTRACT/GRANT NO.
800662
12 SPONSORING AGENCY NAME AND ADDRESS
Environmental Protection Agency
National Environmental Research Center
Meteorology Laboratory
Research Triangle Park, North Carolina
13. TYPE OF REPORT AND PERIOD COVERED
Special
14. SPONSORING AGENCY CODE
15 SUPPLEMENTARY NOTES
16. ABSTRACT
This special report discusses six topics all of major current interest in
modelling of dispersion in the atmospheric boundary layer. These are the
second-order closure modelling of turbulence, crosswind dispersion and the
properties of turbulence, wind direction fluctuation statistics over long
sampling times, 'local similarity1 treatment of vertical spread from a ground
source, representations of dispersion in terms of distance or time, and
modelling for elevated sources.
KEY WORDS AND DOCUMENT ANALYSIS
DESCRIPTORS
Atmospheric turbulence
Atmospheric dispersion
b.lDENTIFIERS/OPEN ENDED TERMS
Second-order closure model
Crosswind dispersion, Wind
fluctuation statistics, S
treatment of vertical spread
in terms of distance or t
for elevated sources
ling of turbulen
direction
imilarity
Dispersion
ime, Modelling
3. DISTRIBUTION STATEMEN1
Unlimited
19. SECURITY CLASS (This Report)
Unclassified
21. NO. OF PAGES
66
20. SECURITY CLASS (Thispage)
Unclassified
22. PRICE
EPA Form 2220-1 (9-73)
53
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