United States
Environmental Protection
Agency
Research and Development
f . ironmental Sciences Research fc'PA oOO 4 78021
I 3toratory M,n 1 978
Pt--.earch Triangle Park NC 27711
vvEPA
Atmospheric
Dispersion
Parameters in
Plume Modeling
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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 nine series These nine broad cate-
gories were established to facilitate further development and application of en-
vironmental technology Elimination of traditional grouping was consciously
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The nine series are
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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
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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-600/4-78-021
May 1978
ATMOSPHERIC DISPERSION PARAMETERS IN PLUME MODELING
F. Pasquill
Visiting Scientist
Meteorology and Assessment Division
Environmental Sciences Research Laboratory
Research Triangle Park, North Carolina 27711
ENVIRONMENTAL SCIENCES RESEARCH LABORATORY
OFFICE OF RESEARCH AND DEVELOPMENT
U.S. ENVIRONMENTAL PROTECTION AGENCY
RESEARCH TRIANGLE PARK, NORTH CAROLINA 27711
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DISCLAIMER
This report has been reviewed by the Environmental Sciences Research
Laboratorys U.S. Environmental Protection Agency, and approved for
publication. Mention of trade names or commercial products does not
constitute endorsement or recommendation for use.
n
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ABSTRACT
A brief survey is given of the present position in the specification of
atmospheric dispersion parameters for use in estimating pollutant concentra-
tion from a continuous point release.
The theoretical indications of the distribution to be expected across a
time-mean plume are recalled, with particular reference to the existence of
the Gaussian form. Observational evidence, especially as regards the vertical
distribution from a surface release, is also recalled, and the practical
significance of departure from an assumed Gaussian form is noted.
The use of the Taylor statistical theory in the generalized estimation
of crosswind spread in quasi-ideal boundary layer flow is briefly summarized.
Recent considerations of the behaviour of the crosswind component of turbu-
lence in the surface layer and new developments from laboratory modeling of
horizontal dispersion in convective mixing are noted.
A brief survey is given of the achievements of gradient-transfer theory
and Lagrangian similarity theory in calculating vertical spread from a surface
release. New tests against previous dispersion data underline inadequacies in
the present approaches in very unstable conditions. Promising developments
from the laboratory modeling of a convectively mixed layer and from the
2nd-order-closure modeling of the turbulent fluctuation equations are summarized,
The assimulation of theory and experience into practical systems for the
specification of a and o is briefly reconsidered. For a a practical
procedure based on wind direction fluctuation data is reaffirmed. For a a
new format which may be envisaged for future composite curves is suggested.
Finally, the inherent limitations of practical systems for estimating con-
centration levels are reiterated.
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CONTENTS
Abstract iii
Figures vii
Tables viii
List of Symbols ix-x
1. Introduction 1
2. The Shape of the Distribution of Concentration from a Source . 3
Implications of the classical parabolic equation 3
The special case of steady homogeneous turbulence .... 5
Conformity to Gaussian distribution in reality and the
significance to estimates of concentrations 6
Value of exponent r 6
Non-Gaussian intermediate state 8
Non-Gaussian aspects of instantaneous distribution. ... 8
3. Generalized Estimation of Crosswind Spread in Quasi-Ideal
Boundary Layer Flow 10
The Taylor statistical theory 11
Determination of crosswind spread 11
Fffect of the x-T relationship 11
Practical adaptation of the Taylor theory at various ranges
of T 12
T«tL 12
Immediate range of T 13
Very large T 13
Latest developments relevant to the modeling of crosswind
dispersion 14
Laboratory modeling 14
Second-order closure modeling 15
4. Generalized Examination of Vertical Spread az in Ouasi-Ideal
Boundary Layer Flow 17
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The gradient-transfer and similarity treatments of verti-
cal spread from a surface release 17
Deardorff's mixed-layer similarity approach and laboratory
modeling 21
Second-order closure models 24
Early achievements of the 2nd-order closure technique. . 25
5. The Present Position and Prospects in Practical Systems ... 29
References 35-37
Appendices
A. Similarity-empirical considerations of vertical spread
at short range from a surface release 38
Data from project "prairie grass" 38
Similarity hypotheses on Monin-Obukhov lines. . 40
Similarity in terms of the intensity and scale
of turbulence 46
R. A simple two-layer model for estimating vertical
spread beyond the surface-stress layer in neutral
flow 51
v1
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FIGURES
Number Page
1 Vertical spread from a surface release In a neutral boundary layer 27
2 Envisaged new format of a curves 31
A-l Vertical diffusion similarity functions 45
B-l K-profile in a nuetral boundary layer of depth h 52
B-2 Vertical spread curves for a neutral boundary layer with Surface-
Rossby-Number similarity properties as in Table B-l 56
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TABLES
Number Page
1 Forms of Vertical Distribution and Vertical Spread Implied by
Solutions of the Classical Parabolic Equation of Diffusion. ... 4
2 Observational Evidence for Existence of Gaussian Shape in Time-
Mean Distributions 7
3 Values of 2uC(X,0)a /Q for a Two-Dimensional (Infinite Line
Source) Model with u" Invariant with height and C(X,z)/C(X,0)
= exp-bzr 9
4 Wind Speed Conditions for Applicability of the Convective-Limit
Form for a /u* 16
5 Gradient-Transfer and Similarity Calculations of Vertical Spread
from a Surface Release 19
6 Comparison of Calculated Vertical Spread from a Surface Release
with the Smith-Singer Curve for Height of Release 100 M, in
Neutral Flow 32
A-l Adjustment of Cramer's 1957 Estimates of a from Prairie Grass
Data ? ai-42
A-2 Estimates of (l/ku*)(dl/dt) from Adjusted Vertical Spread
Estimates in Table 1 43-44
A-3 Magnitudes of Boundary Layer Parameters and Dimensionless
Dispersion Velocity $(Z./L). Defined in Eq. (A-9) and (A-10) 47
B-l Parameters and Results in the Estimation of Vertical Spread fro.,.
a Surface Release, Beyond the Surface-Stress Layer, in Neutral
Flow 55
viii
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LIST OF SYMBOLS
The usual subscript notation is used to indicate, for example, axis of
reference (x,y,z), velocity component involved (u,v,w). A zero subscript is
used to denote a ground-level value or an initial value. A subscript g
denotes a geostrophic value.
f Coriolis parameter
c specific heat of air at constant pressure
H vertical heat flux (H surface value)
k von Karman's constant
K eddy diffusivity 3 T
if n
L Mcnin-Obukhov length -- n- -^
source strength, rate of emission (point source) or rate of
emission per unit length (line source)
Lagrangian correlation coefficient for time-lag £
Ri Richardson number
Ro Surface Rossby number V /fz
t, Lagrangian time scale
T absolute temperature or time of travel
v,w velocity components along axes y,z
u* friction velocity =(T /p) '
1/3
w* free convective velocity scale (gH z./pc T) '
x,y,z rectangular coordinates, x along mean wind y across mean wind and
z vertical
z roughness length
X distance of travel downwind of release position
Z vertical displacement of particle
z. mixing depth
zr release height
e rate of dissipation of turbulent kinetic energy per unit mass of
air
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6 wind direction
A spectral scale represented by equivalent wavelength at which
product of frequency and spectral density is a maximum
p air density
a2 variance
4> Monin-Obukhov universal function
T horizontal shearing stress or duration of sampling or release
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SECTION 1
INTRODUCTION
The purpose of this report is to take stock of the present position in
the technique of representing, for air quality modeling, the properties of
crosswind and vertical spread from a single point source, as a function of
distance downwind and of the conditions of flow. Reviewing of this nature
has been continuously in process for many years, and several easily accessible
earlier accounts will be referenced. In the present account the concern will
be partly with some brief reminders of well-established features and partly
with a summary of recent developments.
It should be remembered at the outset that the reliability of the estimates
which are variously made of the familiar properties c and a must be judged
in two respects:
(a) The accuracy, reproducibility, and representativeness of the
available full-scale measurements of dispersion, and
(b) The validity of the theoretical frameworks within which the
results of such measurements may be understood and from which
those inevitably incomplete results may be generalized.
As regards (b) we have long been essentially dependent, for practical
applications, on two classical approaches, the gradient-transfer theory,
requiring appropriate specifications of the eddy diffusivity field, and the
statistical theory as initiated by G. I. Taylor, requiring knowledge of certain
statistical properties of the flow turbulence. For details of the background
of these approaches, references 1 and 2 may be consulted. More recently,
useful and promising theories have been developed in the area of similarity
argument and in the whole sophisticated field of the higher-moment turbulent
fluctuation equations and their solution through 2nd-order-closure assumptions
and hypotheses.
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We shall be concerned (specifically in Sections 3 and 4) with the guidance
provided by all the foregoing approaches in the prescription of a and a .
However, the full description of dispersion also requires a knowledge of the
shapes of the distribution of concentration of material crosswind and ver-
tically, and it seems appropriate to begin (Section 2) with some discussion
of this aspect. Finally, in Section 5, consideration is given to the assimi-
lation of practical experience and theoretical development into the working
formulae and graphs which are currently advocated for estimation of the
dilution of air pollutants.
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SECTION 2
THE SHAPE OF THE DISTRIBUTION OF
CONCENTRATION FROM A SOURCE
One of the most widely used mathematical models of dispersion from a
continuous point source contains the assumption that the crosswind and verti-
cal distributions are of Gaussian form. Various theoretical and empirical
features with bearing on this aspect are now summarized briefly.
IMPLICATIONS OF THE CLASSICAL PARABOLIC EQUATION
For simplicity, and for consistency with the view followed later in this
discussion, attention is first confined to the vertical spread and to a surface
release, though the implications of this particular subsection will apply
equally well to elevated release and to crosswind spread if the gradient-
transfer assumption is considered to apply in those respects also. The appro-
priate equations, in either the one-dimensional time-dependent form (relevant
to an instantaneous plane source of infinite extent) or the two-dimensional
steady form (relevant to a continuous line source of infinite extent crosswind),
and the vertical distribution characteristics which follow from these equations,
with certain assumptions about Hand K, are summarized in Table I.
The first four lines of the table, with steady K, are well-known results,
and the fifth and sixth lines, for arbitrarily time-dependent K, are straight-
forward extensions of the steady-state results. The format of the results and
the selection of cases are such as to bring out certain important points as
follows:
(a) A Gaussian form follows only when r, (or r?) = 2. This includes as
a special case (m = n = o) the so-called Fickian case of constant K
(and constant u"). It is noteworthy, however, that generally the
condition m = n (with n-m * 2 as required for the solution to be
valid) is sufficient. Near the ground the K profile index n is
typically in the region of unity, and falls substantially below
-------
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-------
unity only in stable conditions or with increase in the depth of
layer embraced. As the corresponding wind profile index n is
typically in the range 0.1 to 0.5, the larger values occurring only
with stable conditions and at low heights, it is evident that m = n is
likely to occur only in a combination of the latter conditions.
Otherwise, solution of the diffusion equation for a surface release
in the atmospheric boundary layer implies a decidely non-Gaussian
vertical distribution.
(b) With K and u steady, when a Gaussian distribution does follow from
the diffusion equation, it is necessarily associated with the magni-
tude of the spread growing as T2 or X'2.
(c) For a homogeneous time-dependent K or a steady spatially varying K,
it appears, from lines 5 and 6 of the Table, that a Gaussian distri-
_!' J'
bution will not generally be associated with T2 or X2 forms of
growth. Note especially that if K, (T) °= Ta (or Xa), with positive
and r = 2, it follows that the growth of the spread will behave as
T(l+a)/2 (Qr jjl+oO/2^ We sefi then thflt^ accord1ng to the gradient-
transfer treatment, it is formally possible to have a combination of
non parabolic growth and Gaussian distribution if there are appropriate
time or space variations in the eddy diffusivity K, as might possibly
occur in practice from the diurnal cycle and from systematic changes
in roughness or surface heating.
THE SPECIAL CASE OF STEADY HOMOGENEOUS TURBULENCE
A well-known result of the G. I. Taylor statistical theory, brought out
3
in more detail by Batchelor , is that the spread of particles released
serially from a point grows linearly with time of travel initially, then
progressively less rapidly, tending ultimately to T2. Batchelor concluded
that the particle displacements have a Gaussian probability distribution at
all values of T, though for reasons which must be different for different
ranges of T. With such a universal Gaussian behaviour it then follows that the
diffusion equation does provide a description of the dispersion process for
all T (or X), , provided K varies appropriately with T (or X), indeed in a manner
which satisfies
5
-------
p n
'i/ _ 1 d o u do
K - 2 dT or 2 dT
Obvious similarities exist between this result and that discussed in
the foregoing subsection, but--it cannot be emphasized too stronglythese
are to be regarded only in a formal sense and not in a meaningful physical
sense. In the foregoing subsection the variations in K are to be envisaged
as arising from variations in the fluid properties. On the other hand, the
result for steady homogeneous turbulence is (logically) to be regarded as a
consequence of a dispersive action that is net gradient-transfer in nature.
The crucial point is that except at large enough T or X the displacements of
the particles are caused by turbulent motions of a scale larger than the whole
cross section of the plume of particles.
It may be noted that several workers in dispersion modeling (e.g..
Fortak ) have drawn attention to the result that the conventional Gaussian
plume formula, with arbitrary variation of a with time or distance of travel,
is a solution of the diffusion equation with K defined as above. Although
there is no objection to using this result for homogeneous, steady turbulence
merely as a convenient formality, the important point is that the apparent
K's (associated with a certain nonparabolic growth of a), implicitly constant
with height but varying systematically with T or X, cannot then simply be used
as genuine gradient-transfer K's in some other context, e.g., in the treatment
of the vertical transfer of the particles to an absorbing ground. In the
latter case, logically, the appropriate K must increase with height and
be quasi-constant in time and space.
CONFORMITY TO GAUSSIAN DISTRIBUTION IN REALITY AND THE SIGNIFICANCE TO
ESTIMATES OF CONCENTRATION
A list of sources of observational evidence for the existence or absence
of Gaussian shape in the distribution of dispersed material is given in Table 2,
Outstanding departures from the Gaussian forir are as follows:
Value of Exponent r
For vertical spread at short range from a surface release, the exponent r
in the exponential form is near 1.5 irrespective of thermal stratification.
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TABLE 2. OBSERVATIONAL EVIDENCE FOR EXISTENCE OF
GAUSSIAN SHAPE IN TIME-MEAN DISTRIBUTIONS
1
2
Nature of Dispersion
Crosswind spread from a continuous point
source
Vertical spread from an elevated continuous
Gaussian?
Yes
Yes
Reference
(Dpp 173
& 2?7
(5)
source of passive particles (before ground
becomes effective)
3 Vertical spread in first kilometer downwind
of a surface release
4 Vertical spread from near-surface release
in laboratory convectively-mixed layer
Vertical distribution of power station
plume
No
(r = 1.5)
Only at
very early
stage
Only in
lower half
(6)
(7)
(8)
-------
The significance of the precise value of r to the magnitude of the ground-
level concentration for specified a , wind speed, and source strength is
shown in Table 3; Clearly the departure from Gaussian becomes of practical
importance only when r is less than 1.5 or much larger than 2.
Non-Gaussian Intermediate Stage
A certain intermediate stage in the mixing in a laboratory correctively
mixed layer (even before the distribution is obviously modified by the
presence of the upper boundary to mixing) exhibits non-Gaussian form.
Deardorff and Willis' study shows an early-stace Gaussian vertical distribu-
tion fol.lowed by the appearance of an elevated maximum, which progressively
rises through the mixed layer before the final condition of uniformity with
height is achieved. This means that the concentration at ground level
transiently "undershoots" the value which would be calculated (given a ) on the
existence and degree of this effect in the real atmospheric mixed layer has yet
to be provided.
Non-Gaussian Aspects of Instantaneous Distribution
g
Hamilton's lidar observations of the time-mean vertical distribution
of a power-station plume apparently ir-ay be fitted to a Gaussian shape over
the lower half of the distribution, which, of course, is the significant half
as regards the development of the ground-level concentration to is maximum
value. It is, however, highly questionable whether such an approximation
obtains in the instantaneous distribution, the property that will determind
short-term concentrations in the path of the elevated plume. The fact that
the growth of a rising hot plume is initially dominated by an induced internal
circulation leads one to expect a tendency to a flat rather than a peaked
distribution in the cross section of the plume, but evidence for this is net
immediately available.
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TABLE 3. VALUES OF 2uC(X,0)a /Q FOR_A TWO-DIMENSIONAL
(INFINITE LINE SOURCE) MODEt WITH ii INVARIANT WITH
HEIGHT AND C(X.z)/C(X,0) = exp-bzr
r 1.0 1.5 2.0 2.5
2HC(X,0)az/Q 1.37 0.96 0.8 0.73
N.B. These figures are for unbounded vertical
diffusion; for a source at ground-level they
should be doubled. They may be taken as equiva-
lent to f C(x,y,o)dy from a point source.*
*See p. 350, Ref. 1 for further details
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SECTION 3
GENERALIZED ESTIMATION OF CROSSWIND SPREAD
IN QUASI-IDEAL BOUNDARY LAYER FLOW
In the atmospheric boundary layer both the magnitude and scale of the
crosswird component (v) of turbulence are found to change only slowly with
height (in contrast to the pronounced increase with height of the scale of the
vertical component). Accordingly, when also there are no sharp changes with
time or position, we may treat the flow as quasi-homogeneous as regards the
lateral dispersive action of the turbulence.
For the lateral spread of a continuous point source (and indeed the verti-
cal spread when the plume is elevated), the physical irrelevance of gradient-
transfer action has been discussed on many occasions, and the purely formal
quality of the representation in terms of the parabolic diffusion equation
has been noted in the previous section. The Lagrangian form of Monin-Obukhov
similarity argument, although originally including lateral as well as vertical
spread, has been seriously questioned for the former aspect of dispersion
(p. 119 of Ref. 1). In an Eulerian sense it has become evident that the v-
component is not a simple function of z/L (where L is the Monin-Obukhov
length), and it would be most surprising if the Lagrangian v-properties dif-
fered from the Eulerian in this respect. A possible rationalization cf the
behaviour of the v-component in the surface layer in convective conditions
g
has recently been proposed by Panofsky et al. . Their analysis of several
sets of data on a /u* over uniform surfaces with friction velocity u* demon-
strates a universal dependence, not on z/L, but on Z../L, where z.. is the
effective convective mixing depth. We will be noting the special significance
of their result to lateral spread later in this section. More generally, i.e.
including neutral and stable flows, there are complexities in ^-properties
to be expected from synoptic-scale changes in the flow and from mesoscale
topographical influences, the latter especially in stable conditions and even
in unstable conditions in light winds.
10
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THE TAYLOR STATISTICAL THEORY
Of the three working theories hitherto available to us, the practical
adaptation of the famous Taylor theory of diffusion by continuous movements is
the one most clearly suited to the estimation of a for a continuous point
source. The developments to date have already been extensively reviewed
(p. 123 et seq, p. 185 et seq, Ref. 1, and p. 6 et. seq of Ref. 2), and only
the main points need to be emphasized here.
Determination of Crosswind Spread
In accordance with the assumption of quasi-homogeneous conditicns,
the spread c is related to o and time of travel downwind of the source
in the forms
tL = f(T/V (i)
with f(T/tL) + T/tL as T + 0 (2)
or ay(T) + ovT as T -> 0 (3)
and f(T/tL) + (2T/tL)J* as T + » (4)
or ay(T) + av(2TtL)% as T -> » (5)
tL being the Lagrangian integral time scale. This means that in principle,
neglecting wind direction turning with height (to which reference will be
made later), crosswind spread is determined by a (which is measurable, and
to some extent describable in boundary-layer climatological terms), and by
tL (which is not easily measurable and for which the theoretical and observa-
tional background is only partially helpful).
Effect of the T-T Relationship
Rigorously, the adaptation of the Taylor treatment requires that a
have the limiting value associated with effectively infinite sampling time
(of the turbulence) and corresponding effectively infinite release or
sampling time of the material. Or a qualitative argument, however, (p. 136,
Ref. 1) the result may be expected to be valid for sampling (or release)
11
-------
time T and time of travel T when T < T. As T increases beyond T, the argument
is that the different properties of cluster growth (as distinct from time-mean
plume growth) become increasingly relevant and ultimately dominant.
Inadequacy of the X-T Relationship
In strictly homogeneous conditions, including uniformity of wind speed,
the foregoing a , T relations may be converted to a , X form by substituting
X = lit. However, for real boundary layer flow, even though an assumption of
quasi-homogeneity may be acceptable as regards the properties of the v-component,
the variation of mean wind speed with height makes the simple X, T relation
inadequate. A practical solution is to replace u" by an equivalent advecting
speed, u , which increases with T (as vertical spread increases), and with
certain assumptions a rough estimate of u is derivable in terms of the wind
profile .
PRACTICAL ADAPTATION OF THE TAYLOR THEORY AT VARIOUS RANGES OF T
In principle, and qualifications including those noted in Determination
cf Crosswind Spread and Effect of the -r-TRelationship above, the method
provides for estimates of c in a way that takes into account the properties
cf turbulence for any surface roughness and any thermal stratification. The
practical utility and limitations are most conveniently considered in well-
defined ranges of T/t. .
T«tL.
It has been demonstrated that in this range a approaches the simple
limit in Eq. 3, i.e.
oy(T) - V . (6)
or approximately
ay(X) = - X - a0X (6a)
noting that c?, which is the standard deviation of wind direction fluctuation,
9
should strictly be taken as a function of vertical spread, in accordance with
the definition of u . Thus at short enough time or distance the only require-
ment is an estimate of a_. Abundant evidence supports rough agreement with
o
the very simple form in Eq. (6a), but in more precise terms the crucial point
12
-------
is the behaviour of f(T/t.) in Eq. (1) at small T/t, ; this will be apparent
in the more general considerations which follow.
Intermediate Range of T
For this range the form of f(T/t, ) needs to be specified, basically from
the form of the Lagrangian auto-correlation function R(£)» and earlier
considerations (p. 130 of REf. 1) had led to the impression that f(T/t, ) was
insensitive to such variations of the shape of R(£) as were originally con-
sidered likely. A recent analysis of dispersion data by Draxler has reopened
this question, and it now has to be considered that the initial reduction of
210
R(£) with time-lag £ may be considerably more rapid than exp-5/t. ' , which
was the sharpest fall hitherto considered. It is noteworthy that a form which
has a sharper fall and fits selected dispersion data is provided by
(a) The Hay-Pasquill hypothesis of a simple scale relation between
Lagrangian (moving particle and Eulerian fixed point) turbulent
fluctuations (see p. 135 of Ref. 1), and
(b) An empirical form of the shape of the (Eulerian) v-spectrum (see
p. 70 of Ref. 1).
Full application on the foregoing lines, in terms of a specification
of the actual v-spectrum, is considered to be the most satisfactory approach,
but the requirement for a relatively sophisticated measurement and analysis
of the time-lapse fluctuation of v or 6 is obviously a practical difficulty.
One reasonable practical solution would appear to be the use of an empirical
generalizaticn cf good quality a data accompanied by a data, and a collection
y no y
of such data has been assembled by the writer . According to this data,
ay/XaQ follows a simple function of X, largely irrespective of roughness and
thermal stratification, with departures (for individual samples of data)
which are within a factor of 1.5 at short range (< 1 km) and 2.0 at longer
range (10 km). With reference to the remark at the end of the paragraph or
T t. , note that according to the various U.S. tests a /aQX is on average
L. y o
detectably below unity even at a distance as short as 100 m.
Very Large T
If the flow were ideally homogeneous, the specification of a at very
large T (or corresponding X) would require only an estimate of t.'1- in
13
-------
addition to aQ. Unfortunately, another aspect of departure from homogeneity
then becomes effectivenamely,-the turning of mean wind direction with height--
as a result of which there is, through the process of vertical spreading, a
contribution to the crosswind spread additional to that directly associated
with a . Certain more or less elaborate theoretical treatments of this
hear effect are already available (see pp. 156 and 229 of Ref. 1). With
these and some empirical guidance, first impressions are that the additional
contribution becomes important only for distances of travel exceeding 5-10 km.
LATEST DEVELOPMENTS RELEVANT TO THE MODELING OF CROSSWIND DISPERSION
Laboratory Modeling
Discussion of the laboratory model of Deardorff and Willis for dispersion
in a convectively mixed planetary boundary layer is considered in more detail in
relation to vertical dispersion in the next section. In one of the latest reports
13
of this work , the implication of the model in respect to horizontal dispersion
is also considered. From an ensemble of seven experiments with the water tank
having side-dimension/mixing depth z. ratio 4.0, and with the convective
_i '
scaling velocity w* near 1 m sec , Willis and Deardorff derive statistics of
lateral displacement of nonbuoyant particles released on a line near the bottom
of the tank. As in the work on vertical dispersion, these are expressed in
similarity terms, a /z versus t* = w*t/z.), identification with distance
of travel in a wind being achieved through the relation X = Ut with U constant.
They compare their laboratory results with field data obtained in Idaho for
distances of travel up to 3200 m from a continuous point source, making
plausible estimates of the likely magnitudes of w* and z. during the field
tests. A remarkable degree of agreementwithin about 10% in the ensemble
averagesis found. In noting this very encouraging initial success in
verification of the full-scale applicability of laboratory modeling in
convective conditions, two qualifications deserve special mention.
The a t curves found in the laboratory tank contain a distinct inflection,
tion, with da /dt temporarily reduced and then restored. In full scale
y ] 3
distance terms this appears in the 3- to 4-km range. Willis and Deardorff
ascribe the feature to the delayed appearance of horizontal spread from thermal
outflows at the top of the mixed layer. It is noteworthy that this effect
has not yet been detected in full-scale measurements.
14
-------
The second point concerns the stipulation of the full-scale conditions,
especially of wind speed, for which the laboratory (windless) results may be
reliably adopted. In this connection Willis and Deardorff prescribe an
upper limit cf 12 m sec" . For all lower wind speeds the implication is
that the horizontal dispersive action is essentially controlled by buoyancy
forces and not by the mean shear, and in this respect some further guidance
is now available from the recent generalizations about the behaviour of
the magnitude of the surface layer v-component in convective conditions.
Q
Panofsky et al.'s form for av, in terms of u^.L and z^, m$y be used to
prescribe the combinations of wind speed and heat flux which, for given z
and z., result in a a which is dominantly (say to the extent of 90%) a
consequence of the heat flux. Details are set out in Table 4 for z 20 cm (a
moderate roughness intermediate between smooth plains and urban complexes)
and for z. = 1500 m (a mixing depth typical of afternoon conditions). Note
1 _i
that in such circumstances a surface wind speed of even 4 m sec requires a
vertical heat flux near 500 w m~ to meet the foregoing criterion of dominance
of the buoyancy contribution to a . This is a very strong sensible heat
flux, unlikely to occur except over dry terrain and with the highest sun in
low latitudes. From this standpoint it seems that the 12 m sec" limit pre-
17
scribed by Willis and Deardorff "" may require unrealistically high heat fluxes,
or, alternatively, a very much smoother surface or much larger z..
SECOND-ORDER CLOSURE MODELING
A growing effort is being devoted to the use of 2nd-moment equations,
in which the gradient-transfer assumption is avoided, at least in the 1st-
moment equations such as those considered in Table 1. The progress of the
technique is more comprehensive in relation to vertical transfer as will
be discussed in Section 4. The writer is unaware of any crucial examinations
of the success of the technique in relation to crosswind spread per se.
15
-------
TABLE 4. WIND SPEED CONDITIONS FOR APPLICABILITY OF
THE CONVECTIVE-LIMIT FORM FOR a/u
Taking ay/u* = (12 - z./2L)1/3 Ref. 9
+ (-z./2L)1/3 for large (-z./L)
it follows that
>_ 0.9 when -z^L >_ 65
Associated values of surface heat flux
HQ and if(10m) for -z^L = 65, zi = 1500m, ZQ = 0.2m
iT(lOm) m sec"1 2 2.5 3 3.5 4
HQ wrrf2 67 130 220 350 520
16
-------
SECTION 4
GENERALIZED EXAMINATION OF VERTICAL SPREAD a
IN QUASI-IDEAL BOUNDARY LAYER FLOW Z
THE GRADIENT-TRANSFER AND SIMILARITY TREATMENTS OF VERTICAL SPREAD FROM A
SURFACE RELEASE
The fundamental acceptability of the gradient-transfer relation for
turbulent transfer is often seriously questioned (e.g., see Corrsins's
discussion, ). Empirically, however, the method is undoubtedly successful
in certain applications, a success which Corrsin refers to as "largely
fortuitous and certainly surprising."
It is now a familiar notion that time-mean spread from a continuous
point source is initially dominated by turbulent motions of scale that are
large compared to the cross section of the plume of particles, when the
concern is with lateral spread or even with vertical spread when the plume
is clear of the ground. This type of scale relation, which in an obvious
physical sense is the very opposite of that implied in a gradient-transfer
process, does not exist, however, in the vertical spread action when passive
particles originate at the boundary. The point is simply that at any stage
in the vertical growth of the plume the effective turbulent motions are
constrained in scale by the presence of the underlying boundary, the effective
scale being dependent on height. This presumably is the essential reason for
some success in the K-treatment of the ground-level infinite crosswind line
source.
Also noteworthy at this point is the formal consistency between the K-
treatment, using the familiar momentum-transfer analogy, and the Lagrangian
similarity treatment, for vertical spread as a function of time in the surface-
stress region of the neutral boundary layer (p. 117 of Ref. 1). Associated
with this is a simple relation between the rate of vertical spread and the
eddy diffusivity, in the form
17
-------
dZ/dt = K(Z)/Z (7)
where Z is the mean displacement of particles at a given time after release
at the surface. The result is exact for the neutral surface-stress layer
(p. 118 of Ref. 1) and has been found to be a good approximation, in other
215
thermal stratifications and at greater heights * .
In Table 5 a list is given of applications of the mutually consistent
gradient-transfer and similarity approaches that have led to useful explicit
formulations of the growth of a with time or distance. Unfortunately,
observational data for the critical testing of the theoretical results are still
largely confined to short range, notably the early Porton-Cardington data
at 100 and 229 m (neutral conditions) and the Prairie Grass data at distances
up to 800 m (for a wide range of stratification). Note, however, that in the
latter observations only those at 100 m include measurement of the vertical
distribution of condentration. Those at other distances are confined to
ground level and provide only indirect estimates of a .
A reexamination of the Prairie Grass data in relation to some of the
foregoing methods has recently been attempted by the writer, using the
indirect estimates of a. derived from the concentration measurements at
16
ground level (basically those given by Cramer , but with an adjustment
allowing for the variation of wind speed with height neglected in Cramer's
analysis). Full details are assembled in Appendix A, and a summary of the
principal results follows:
(a) The magnitudes of dZ/dt implied by the Cramer-type analysis of the
Prairie Grass data do not exactly support a similarity relation with
22
Z and the Monin-Obukhov L, as conjectured by Gifford , though
systematic discrepancies from such a relation are not large.
(b) Predicted curves which follow from the assumption that K = ku*z/<|>H
with the different estimates available for <(>M, embrace the range
of the data on dZ/dt in unstable conditions. In stable conditions
the one predicted curve presented tends to be a slight overestimate.
(c) A predicted curve based on a similarity hypothesis consistent
with K = aa X , and evaluated using latest estimate of the
w m
18
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19
-------
properties aw and Xm> provides a rough fit in unstable conditions
though with a curve shape which in detail differs from that based
on K = ku*z/£u. For stable conditions the K = aa X prediction
ri w m
is a gross overestimate. In the latter connection it may be sig-
nificant that the present evaluation uses an empirical generali-
23
zation in which a /u* increases with increasing z/L.
W ' "
(d) At large X/-L the data show a tendency to a growth more rapid than
_ 1/2
the (Z/-L) form predicted on simple similarity grounds, assuming
a regime of free convection with the determining parameters height
and surface heat flux.
' Clearly, several interesting features still need more satisfactory
interpretation, but in the meantime the foregoing results provide at
least a useful empirical step in that, for example, the observed stability
dependence of the function (Z/L) = (l/ku*)(dZ/dt) may now plausibly
be applied to a surface roughness different from that of the Prairie
Grass site. In principle this $ may be converted to the function dZ/dX
by recombining it with the functional form for dX/dt as determined by
the wind profile (see note (c) of Table A-2, Appendix A). Integration may
then be carried out, if necessary numerically, to give I as a function of
distance X" for any specified z and L.
For the consideration of the growth of spread at much longer range from
the source (several kilometres and beyond), no definitive direct measurements
of o are yet available. In any case, the knowledge of the flow characteris-
tics in the upper part of the boundary layer, required for proper interpreta-
tion and generalization of such dispersion measurements, has only recently
begun to accumulate. As noted in Table 5 (Methods 5 and 6) some of these
new data have recently been applied in constructing new estimates of c^.
In nonconvective conditions, for which the gradient-transfer assumption
seems least open to objection, the characteristics of the boundary layer
(including its depth) should be seen within the framework of surface Rossby
Number similarity. This aspect, which has been brought out specifically in
2nd-order closure treatments , was not incorporated in Method 5 of Table 5.
An examination of the implications of simplified assumptions about the neutral
20
-------
K-profile, within the framework of Surface Rossby Number similarity (Method 7
of Table 5) is reported in Appendix B. With the existing uncertainties about
the K-profile in the upper nine-tenths of the boundary layer, this type of
analysis, even if carried out with complete mathematical rigor, can at present
provide no more than an interim solution. However, in one example which will
be noted later in this section, the results are encouraging in being remark-
ably consistent with a completely independent 2nd-order closure calculation.
Furthermore, they do bring out the schematic way in which any extension of the
a curve (beyond the well-established short-range Lagrangian-similarity form)
must reflect the important control exerted on boundary-layer depth by geo-
strophic wind and surface roughness.
In principle the procedure of Appendix B may also be considered for the
stable boundary layer. However, at first sight the shape of the K-profile
25
(as suggested, for example, by the analysis of Businger and Arya, ) seems
unlikely to be suitably approximated by the simple two-layer form adopted in
Appendix B. Instead, the numerically more demanding finite-difference
solution of the diffusion equation would probably be desirable.
DEARDORFF'S MIXED-LAYER SIMILARITY APPROACH AND LABORATORY MODELING
Deardorff's approach is that the structure and transfer properties in a
capped convectively mixed layer are determined completely £y the surface heat
flux H and the depth of mixing z.. The latter parameter is the obvious
characteristic length scale, and on dimensional grounds Deardorff defines a
characteristic velocity scale
1/3
w* = (gH0z./pc T) (8)
From these considerations Deardorff argues that the properties of dispersion
of nc
form
of non buoyant particles, say, the a growth with time, should obey a universal
where t* is a dimensionless time w*t/z.. The form of the function f has been
estimated from both numerical modeling and from the laboratory measurements in
21
-------
a heated water tank referred to briefly in Section 2.
1 ^
Thejnosjt recent publication J of the tank data provides a vertical spread
2
curve, (Z1) against t* where Z1 is the displacement from heights of release
z> for the following conditions:
zr = 0.067 z.
z. = 28.7 cm
tank width/z.
W* -i -1
* - 1 cm sec
and with the following properties
/)
0 Slight acceleration up to (ZT /z. - 0.4, t* - 0.75,
O
0 Bending over to a maximum (Z1) near 0.6, t* - 1.5,
0 Slow descent to the asymptotic value of 0.5 at t^ near 3.
For the range 0.2 <_ t* <_ 0.8 Willis and Deardorff13 give (Z1)2 /z.. * tj'15
In the equivalent Gaussian plume form with idealized reflection of the
distribution at the lower boundary, the familiar a is equal to
h 2
(Z1)2 only at small and large values of crz/zr> and Willis and Deardorff
include a graph of the general relation between these two specifications of
the vertical spread. Usi'ng this graph their vertical spread has been converted
to a and found to fit a linear form
az/z1 = 0.61 t* (10)
with a discrepancy less than 3% over the range 0.2 < t* < 0.8 or 0.1 < az/z^ < 0.5,
Adopting an effective advecting speed ye this may be written in distance
(X) terms as
oz/z. = 0.61(w*/ue)(X/z.) (11)
22
-------
Reference has already been made to a striking feature in Willis and
Deardorff's13 dispersion data, namely, the appearance of a non-gaussian
distribution with a progressively elevated maximum. It is interesting to
note, however, that this departure from Gaussian shape of vertical profile
does not appear at t* values below 0.5 or GZ/Z^ < 0.3); As an essentially
uniform vertical distribution is achieved at t* = 3, it follows that the
'undershoot' in surface concentration (compared with that for a Gaussian
shape and idealized reflection from the upper boundary) is within the range
0.5 < t* < 3. Also, it appears from their Figure 10 that the surface concen-
tration at t* = 1.56 is roughly one-half the magnitude appropriate to a
uniform vertical distribution.
One important feature which remains to be settled is the sensitivity of
13
the vertical spread properties to the height of release. Willis and Deardorff
are careful to emphasize that their result applies to a specific value of z^.
Naturally, one would expect the distribution to be insensitive to zr when
a /z is sufficiently large, and they estimate this condition to be satisfied
for t*<00.5. On this basis we may expect the Gaussian-type vertical spread,
ifor all zr/z. (0.067).
az/z. = = 0.61 t* = 0.61(w*/ue)(X/z.), 0.5 < t* < 0.8 (12)
For t* < 0.5 seme dependence on z z. is to be suspected but has yet to be
specified.
Another important feature to be kept in mind (as in the brief discussion
'relating to crosswind spread) is the condition under which the laboratory
results may be considered directly applicable to an atmospheric mixed layer
1 O
with wind, with full-scale magnitudes of z. and w*. Willis and Deardorff
plausibly take the view that the upper limit of wind speed for such direct
scaling should be associated with a sufficiently large magnitude of z.-L and
for this they adopt 10. In ether words, the stimulation is that when the
negative Mcnin-Obukhov length is less than one-tenth of the mixing depth, the
mixed layer properties are controlled by the similarity laws we have just
13
discussed. On this basis, they estimate an upper-limit wind speed of 12 m
sec" , but this does need qualification in respect to roughness and heat
flux. Thus with z. = 1000 m, hence L < 100 m, the criterion is met at
23
-------
U(10)m = 12 m sec , over a relatively smooth surface (say z = 3 cm) only
-2
when H > 550 mw m , an unusually large heat flux. The wind condition must,
of course, be even more restrictive over the rough surfaces more likely to be
of interest in urban pollution. If for this case we take z = 1 m, and take
-2
a heat flux of 300 mw m as more typical of the range of conditions of
interest, even allowing for artificial heating, the wind speed limit, is
3.7 m sec~ ! In general, of course, the limit depends on the roughness and
intensity of surface heating, but the foregoing example leaves no doubt that
the 12 m sec" value limit constitutes an overstatement of the applicability
of the laboratory results.
SECOND-ORDER CLOSURE MODELS
In the gradient-transfer models which have been considered so far, the
continuity equation relating rate of change of concentration to turbulent
flux divergence is put into the "mean quantity forms appearing in Table 1
by substituting
Flux = K x gradient.
Second moments which represent this flux (e.g., w'C1 where w1 and C1 have the
usual meaningturbulent fluctuations from the mean values) can, however be
expressed in full, without the gradient-transfer assumption, for the momentum,
heat, and material content properties. Solution of these equations requires
"closure," by making certain assumptions for simplifying (modeling) the more
complex terms, including 3rd moments. The idea is that such assumptions,
which may or may not be of the gradient-transfer type, should not introduce
errors as large as might arise from making the gradient-transfer assumption
in the simple equations for dC/dt, etc.
Considerable literature has accumulated on the evolution of this approach,
application of which entails a large effort in numerical solution. The
necessity for such a development, as well as its potential, have been
pc
discussed in a general article by Lumley . The main areas of failure which
must be expected in the simple gradient-transfer model and for which we may
expect the 2nd-order-closure system to provide more correct treatment are
as follows:
24
-------
(a) Violation of scale-relation requirements, in a more subtle fashion
than that already noted as regards relative scales of turbulence
and plume, for example, as a consequence of time-dependent changes
in the turbulence characteristics.
(b) The special action of buoyancy-driven vertical mixing. This point
is taken up in detail in Lumley's article. One of the distortions
which this imposes on the vertical distribution expected from
1 o
gradient-transfer condition has emerged in the Willis and Deardorff
laboratory data discussed in the foregoing subsection, namely, the
appearance of a markedly non-Gaussian distribution (obviously asso-
ciated with a counter-gradient transfer) after the vertical spread-
ing has progressed to a certain state. However, on this last specific
point note that the "certain stage" corresponds to a substantial
magnitude of a /z.., about 0.3!
(c) The vertical spread from an elevated source as distinct from a
surface release. This is the vertical analog of the crosswind
dispersion problem, though with the additional complication of a
variation (in the vertical) of the scale of turbulence. This means
that neither the statistical theory nor the gradient-transfer theory
is applicable for vertical spread from an elevated source, though
for a surface-release the second theory has been provided with some
empirical verification and will be given some further support.
EARLY ACHIEVEMENTS OF THE 2ND-ORDER CLOSURE TECHNIQUE
Some specific achievements in the context of predicting dispersion from
sources have already been reported in the literature. The first and most
24 27 28
extensive application is that by Lewellen and Teske ' ' using the
29
2nd-order closure assumptions advocated by Donaldson . These latter assump-
tions provide expression of unknown terms in relation to characteristic
velocity and length scales and in some respects seem tantamount to a gradient-
transfer assumption, but, of course, then only at the 2nd-order. A brief
summary of the achievements of the Lewellen-Teske work especially relevant to
the present discussion follows:
25
-------
(a) For a ground release the vertical spread growth with distance in a
neutral atmosphere is included in Figure 1 in comparison with
curves which follow from methods 5 and 7 of Table 5. There is
a very reasonable degree of agreement which adds support to the
validity of the simple gradient-transfer method, for distances of
travel (> 1 km) for which no observational test has so far been
developed.
27
(b) In a free-convection mixed layer the vertical distributions
exhibit a remarkable similarity to those observed in the laboratory
13
by Willis and Deardorff . The progressive elevation of the level of
the concentration maximum is well reproduced, but the "undershooting"
of the surface concentration is not so obvious as in the laboratory
data.
(c) Although the expected reduction in a with stability is reproduced,
the particular conditions adopted do not exhibit any significant
change from the spread in neutral flow at distances in the region
of 0.1 km, and the result cannot yet be confirmed in terms of the
Prairie Grass data, which do show a substantial reduction of the
vertical spread at 0.1 km downwind.
(d) The potential of the approach in the treatment of the elevated source
is displayed in a limited way. Interestingly, for a neutral atmos-
phere the 2nd-order closure calculations show that vertical spread
is smaller from a source elevated at some hundreds of metres than
from a surface release, by a factor near two at the range correspond-
ing to a - z /2. The reduction is (not surprisingly) associated
with a fall in a with height.
30
Another recent application is that of Zeman and Lumley (using a closure
scheme differing from that of Donaldson) for the case of the surface input of
pollution into a mixing layer growing as a result of surface heating. From
their calculations they derive effective eddy diffusivities which have a
vertical profile broadly resembling those derived empirically by Crane and
Panofsky from observations of a morning buildup of carbon monoxide pol-
lution in Los Angeles. In these particular calculations, although the
26
-------
loco
100 . _
^H;~-:;r-"'!3v cl' PV-T"? '.1*'"
r.i
1
ice
Figure 1. Vertical spread from a surface release in a
neutral boundary layer-
21
-------
effective values of K became large in the middle of the mixed layer, they do
not become effectively infinite (as implied by Crane and Panofsky's analysis)
32
or negative as found in Deardorff and Willis' laboratory study of dispersion
from an instanteous release.
28
-------
SECTION 5
THE PRESENT POSITION AND PROSPECTS IN PRACTICAL SYSTEMS
For the practical estimation of pollutant concentration arising from a
single continuous point source, two systems in particular have been in popu-
lar use for many yearsthe Pasquill-Gifford (PG) curves incorporated in the
33
Turner Workbook , and the Smith-Singer (SS) curves incorporated in the
34
publication by the American Society of Mechanical Engineers.
The PG curves were based on the understanding and experience accumulated
by the late 1950's on dispersion from surface releases, but they have been
adopted as practical approximations for elevated releases also. Their most
popular feature is the presentation of the a ,a , estimates in terms of
stability categories prescribed in terms of routine meteorological data. The
method has been reviev/ed in the general context of dispersion-estimate
3K
systems by Gifford " and in the specific context of the conventional Gaussian-
plume model by Weber . The need for improvement and updating has been under
consideration for some time, and this was the motivation for Weber's review
12
and for the companion report by the writer on specific requirements for
modification of the parameters and procedure in the Turner Workbook.
The Smith-Singer curves are based on observations of ground-level
concentration from an elevated (z =100 m) release of passive material at
the Brookhaven National Laboratory. Estimates of c and a are given in
groups designated by the width of the wind direction trace and also in
formulae in terms of a. and 0^, the standard deviations of the direction and
inclination of the wind.
It is evident from the progress discussed in the foregoing sections that
the basis for revision of the practical systems is still far from complete.
However, certain obvious evidence points to desirable changes in the PG
curves.
29
-------
12
One interim revision has already been suggested for the estimation of
a . The main points of this are firstly the reminder of the applicability of
the Workbook a parameter to a sampling (or release) time of 3 minutes;
secondly, the reaffirmation of the value of wind direction fluctuation data; and
thirdly, a plausible generalization for variation with distance. Recent pro-
gress in generalizing about the behaviour of a (or the corresponding wind
direction fluctuation OQ) in the atmospheric boundary layer is directly rele-
6
vant to prospects of a climatological basis for o , but certain features of
wind direction variation are not readily included in such a climatology
(notably synoptic and topographically induced variations).
As regards o,, certain details cannot immediately be settled, but the
essentials of a new emerging pattern are already clear enough to suggest
a new format along lines indicated in Figure 2. Explanatory notes are
included with the diagram. At this stage the determining "flow parameters"
are all in their basic form: z , L, w*, z.. For the future, not only is
there the task of confirming and completing the detail of the a ,X curves
branching off the basic neutral form, but also a parallel requirement
for updating the representation of the second and third of those parameters
in more convenient meteorological terms.
Note that the sections of curve given for unstable conditions are in
fact based on elevated source (laboratory) data, using the principle that
the distinction between the concentration distributions from surface and
elevated releases must progressively diminish to a negligible amount as
32
a /z increases. Deardorff and Willis suggest that the approximation
"is adequate at a /z =5, and this means that the curves in rigure 2 are
appropriate -"or -a1,! 7. /z. less than about 0.07. Not only is confirmation o*
this estimate needed, but also the effect of z at shorter distances and in
other stabilities has yet to be convincingly evaluated. One immediate
assurance that the use of surface-release predictions is not severely mis-
leading for moderate elevations of source in neutral conditions is contained
in Table 6. This compares the similarity prediction for ZQ = 1 m with the
Smith-Singer estimates for a release at 100 m, at downwind distances of
0.1-10. Note also that for the unstable conditions Willis and Deardorff
have demonstrated good agreement between their tank values of a and the
30
-------
o
o
o
H
00
-------
TABLE 6. COMPARISON OF CALCULATED VERTICAL SPREAD FROM
A SURFACE RELEASE WITH THE SMITH-SINGER CURVE FOR HEIGHT
OF RELEASE 100 M. IN NEUTRAL FLOW
Distance downwind (km) 0.1
a (Smith -Singer) m 8
a (calculation by method 7 of 18
Table 5, with z 1 m
1 10
50 280
79 302*
* At this distance the calculated az increases slowly with the geostrophic
wind speed V , and the value given is for V =10 m/sec.
32
-------
Smith-Singer estimates, when the data are scaled in terms of z. and w*.
Even when the idealized relations for a are settled, the calculation of
concentration distribution will always be subject to a whole chain of
uncertainties:
(a) Source strength and position,
(b) Departure from the idealized relations between a ,a and the basic
flow parameters, and
(c) Uncertainties in the basic flow parameter values implied by routine
meteorological data.
Features especially relevant to (b) are the controls exerted by topography on
the flow pattern and the effects of site geometry on the behaviour of a plume
near the release point. Discussion of such features will be found in recent
or 07 op
articles by Gifford ' and Egan . An important component of (c) is the
mean wind direction specification, especially as regards individual short-
term averages of concentration when the plume is narrow.
Errors in the ground-level position and magnitude of the maximum
concentration from an elevated source arise especially from uncertainties
in the effective height of source, the estimated growth of vertical spread,
and the wind direction. As a consequence, individual estimates of concentra-
tion at a particular receptor near the ensemble-average position of the
maximum may be grossly in error (see discussion p. 303 of Ref. 1).
Generally, the error statistics for concentration estimates must be
expected to vary widely, according to the particular conditions of source,
site geometry, terrain, and airflow characteristics. From the experience
accumulated in dispersion studies and air pollution surveys, it is possible
to make realistic estimates of the best levels of accuracy which may be
achieved even in the simplest conditions likely to be encountered in practice.
An early exercise OP these lines has been described by the writer (p. 398 of
Ref. 1). It is scarcely necessary to emphasize that laboratory-type accuracy
cannot be expected. Accuracies in the region of +_ 10 percent may be possible
in the most ideal circumstances of source, terrain, and airflow, but for many
circumstances of practical interest there seems at present no basis for
33
-------
expecting uncertainties less than several ten's of percent statistically or
factors of twc individually.
34
-------
REFERENCES
1. Pasquill, F. Atmospheric Diffusion. 2nd Ed. John Wiley & Sons, New York,
N. Y., 1974.
2. Pasqirill, F. The Dispersion of Material in the Atmospheric Boundary
LayerThe Basis for Generalization. In: Lectures on Air Pollution and
Environmental Impact Analyses. Am. Met. Soc., 1975, pp. 1-34.
3. Batchelor, G. K. Diffusion in Field of Homogeneous Turbulence. I.
Eulerian Analysis. Aust. J. Sci. Res., 2:437, 1949.
4, Fortak, H. G. Numerical Simulation of Temporal and Spatial Distributions
of Urban Air Pollution Concentration. In: Proceedings of Symposium on
Multiple-Source Urban Diffusion Models. AP-86, U. S. Air Pollution
Control Office, Research Triangel Park, N. C., 1970.
5. Hay, J. S., and F. Pasquill. Diffusion from a Fixed Source at a Height
of a Few Hundred Feet in the Atmosphere.. J. Fluid Mech., 2:299, 1957.
6. Elliot, W. P. The Vertical Diffusion of Gas from a Continuous Source.
Q. J. Air Water Pollut., 4:33, 1961.
7. Deardorff, J. W., and G. E. Willis. Computer and Laboratory Modeling of
the Vertical Diffusion of Non-Buoyant Particles in the Mixed Layer. Adv.
Geophy., 186:187-200, 1974.
8. Hamilton, P. M. The Application of a Pulsed-Light Rangefinder (Lidar) to
the Study of Chimney Plumes. Philos. Trans. R. Soc. London, (A), 265:
153-172, 1969.
9. Panofsky, H. A., H. Tennekes, D. H. Lenschow, And J. C. Wyngaard. The
Characteristics of Turbulent Velocity Components in the Surface Layer
under Convective Conditions. Boundary Layer Meteorol., 11:355-362, 1977.
10. Pasquill, F. Some Topics Relating to the Modeling of Dispersion in the
Boundary Layer. EPA-650/4-75-015, U. S. Environmental Protection Agency,
Research Triangle Park, N. C., 1975.
11. Draxler, R. R. Determination of Atmospheric Diffusion Parameters. Atmos.
Environ., 10:99-105, 1976.
12. Pasquill, R. Atmospheric Dispersion Parameters in Gaussian Plume Modeling,
Part II. EPA-600/4-76-030b, U. S. Environmental Protection Agency,
35
-------
Research Triangle Park, N. C., 1975.
13. Willis, G. E., and J. W. Deardorff. A Laboratory Model of Diffusion into
the Convective Planetary Boundary Layer. Q. j. R. Meteorol. Soc., 102:
427-446, 1976.
14. Corrsin, S. Limitation of Gradient Transport Models in Random Walks and
in Turbulence. Advances in Geophysics, 18A:25-60, 1974.
15. Chaudrey, F. H., and R. N. Meroney. Similarity Theory of Diffusion and
the Observed Vertical Spread in the Diabatic Surface Layer. Boundary
Layer Meteorol, 3:405-415, 1973.
16. Cramer. H. E. A practical Method for Estimating the Dispersal of
Atmospheric Contaminants. In: Proceeding of the Conference on App. Met.,
Am. Met. Soc., 1957.
17. Calder, K. L. Eddy Diffusion and Evaporation in Flow over Aerodynamical -
ly Smooth and Rough Surfaces. Q. J. R. Meteorol. Soc., 11:153, 1949.
18. Batchelor, G. K. Diffusion from Sources in a Turbulent Boundary Layer.
Archiv. Mechanike Stosowanej, 3:661, 1964.
19. Pasquill, F. Lagrangian Similarity and Vertical Diffusion from a Source
at Ground Level. Q. J. R. Meteorol., 92:185, 1966.
20. Tyldesley, J. B. Contribution to Discussion on "Short Range Vertical
Diffusion in Stable Conditions." Q. J. R. ,1eteorol. Soc., 93-383-385,
1967.
21. Smith, F. B. 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, NATO/CCMS Report No. 14,
1972. (See also Ref. 1, p. 374 and Ref. 2, p. 16.)
22. Gifford, F. A. Diffusion in the Diabatic Surface Layer. .J. Geophys.
Res. , 67:3207, 1962.
23. Merry, M., and H. A. Panofsky. Statistics of Vertical Motion over Land
and Water. Q. J. R. Meteorol. Soc., 102:255-259, 1976.
24. Lewellen, W. S., and M. E. Teske. Second-Order Closure Modeling of
Diffusion in the Atmospheric Boundary Layer. Boundary Layer Meteorol.
10:69-90, 1976.
25. Businger, J. A., and S. P. S. Arya. Height of the Mixed Layer in the
Stably Stratified Planetary Boundary Layer. Adv. Geophy., 18A:73-92,
1974.
26. Lumley, J. L. Simulating Turbulent Transport in Urban Air Pollution
Models. (Unpublished Manuscript 1976.)
36
-------
27. Lewellen, W. S., and M. E. Teske. Atmospheric Pollutant Dispersion
Using Second-Order Closure Modeling of the Turbulence. Environmental
Modeling and Simulation. EPA-600/9-76-016, U. S. Environmental Protection
Agency. Research Triangle Park, N. C., 1975, pp. 714-718.
28. Teske, M.< E., and W. S. Lewellen. Example Calculations of Atmospheric
Dispersion Using Second-Order Closure Modeling. In: Third Symposium on
Atmospheric Turbulence, Diffusion, and Air Quality. Am. Met. Soc., 1973,
pp. 149-154.
29. Donaldson, C. duP. Construction of a Dynamic Model of the Production of
Atmospheric Turbulence and the Dispersal of Atmospheric Pollutants. In:
Workshop on Micrometeorology, Am. Met. Soc., 1973, pp. 313-392.
30. Zeman, 0., and J. L. Lumley. Turbulence and Diffusion Modeling in
Buoyancy Driven Mixed Layers. In: Third Symposium on Atmospheric
Turbulence, Diffusion, and Air Quality. Am. Met. Soc., 1976, pp. 38-45.
31 Crane, G., and H. A. Panofsky. A Dispersion Model for Los Angeles. In:
Third Symposium on Atmospheric Turbulence, Diffusion, and Air Quality.
Am. Met. Soc., 1976, pp. 122-123.
32. Deardorff, J. W., and G. E. Willis. A Parameterization of Diffusion in-
to the Mixed Layer. \ Appl. Meteorol., 14:1451-1458, 1975.
33. Turner, D. Bruce. Workbook of Atmospheric Dispersion Estimates. Offict
of Air Programs Publication No. AP-26. U.S. Environmental Protection
Agency, Research Triangle Park, NC, 84 p, 1970.
34. American Society of Mechanical Engineers. Recommended Guide for the
Prediction of the Dispersion of Airborne Effluents, New York, 1973.
35. Gifford, F. A. Atmospheric Dispersion Models for Environmental
Pollution Applications. In: Lectures on Air Pollution and Environmental
Impact Analyses, Am. Met. Soc., 1975, pp. 35-58.
36. Weber, A. Atmospheric Dispersion Parameters in Gaussian Plume Modeling.
Part I. Review of Current Systems and Possible Future Developments.
EPA-600/4-76-030a, U. S. Environmental Protection Agency, Research
Triangle Park, N. C., 1976.
37. Gifford, F. A. Turbulent Diffusion-Typing Schemes: A Review. Nucl.
Saf., 17:68-86, 1976.
38. Egan, B. A. Turbulent Diffusion in Complex Terrain. In: Lectures on
Air Pollution and Environmental Impact Analyses. Am. Met. Soc., 1975,
pp. 112-135.
37
-------
APPENDIX A
SIMILARITY-EMPIRICAL CONSIDERATIONS OF VERTICAL SPREAD
AT SHORT RANGE FROM A SURFACE RELEASE
DATA FROM PROJECT 1:PRAIRIE GRASS"
The "Prairie Crass" measurements included measurements of vertical distri-
bution at a distance of 100 m from the source. Magnitudes of a derived from
these data have been put together with additional measurements in stable con-
ditions in England by Pasquill (Ref. 1, p 207, Fig. 4.14).
p
Indirect estimates of a have also been derived by Cramer at greater
distances, up to 800 m, from the "Prairie Grass" data by using the measure-
ments of peak concentration and crosswind spread, and substituting these in
the conventional Gaussian plume formula. Also, from this Gaussian formula
(in which implicitly the wind is invariant with height)
C(x,o,o) or l/tfao2 (A-l)
and if C, o are measured as a function of distance the relative variation of
a automatically follows and may be converted into absolute values of a by
adopting the directly measured value at 100 m. The results for a are given
as a function of distance, for various magnitudes of the standard deviation
of wind direction fluctuation a., in Figure 11 of Cramer's paper, and corre-
sponding exponents in the power-law variation of C, a , and a with distance
over distance intervals 50-100, 10C-200, 200-400, and 400-800 m are given in
Tables 4a and 4b of that paper.
The foregoing procedure is questionable on account of the variation of
wind with height, and a more acceptable form of Eq. (A-l) is
C(x.o.o) « I/ ue ay az (A-2)
38
-------
where u is an effective wind speed, increasing with a and therefore with
distance. An expression for u may be obtained by correctly writing the con-
servation equation with u a function of height (« zn for convenience).
With CIC(x,0) denoting the crosswind integrated concentration at the sur-
face, and assuming a Gaussian distribution in the vertical, the conservation
equation for a rate cf release Q is
f_M
Q = CIC(x.O) u(z) exp (-z2/2az2)dz . (A-3)
'o
Taking u~(z) = U"(z()(z/z,)n (A-4)
Q = 2-azz,- ( CIC(x.O) (A-5)
which with z, = az is of the form corresponding' to Eq.(A-2), i.e.,
Q = (ir/2)%az ue CIC(x.O) (A-6)
if we take
up = (27 2- (l+n)/2 u(oj . (A-7)
V » II / |V I II /, ._ «" \>J
The ratio u /u(a ) is of course unity for the case of wind constant with
height (n = 0). As n is increased to unity (which embraces the range found
in practice), the ratio falls to 0.798.
For the present purpose of adjusting the original Cramer estimates of
a , the essential point is that we may now take u directly proportional to
u"(a ), i.e., to a , for a giver wind profile and neglecting the (slow)
variation of n with height in practice. Accordingly, if in Eq. (A-2) we take
C(x,0,0) cr x"c and a « xa, it follows that
x'2 -l/(xaaznaz)
and therefore
oz « xb with b = (c - a)/l+n) (A-8)
39
-------
instead of b = c - a as given by Cramer. Starting with the directly measured
values of az(100 m) and Cramer's values of b, the foregoing result has
been used to derive adjusted values of a in a step-by-step procedure set out
in Table A-l. Note that the original estimates at 800 m are reduced by a
fraction ranging froir only 0.02 in the most unstable conditions to nearly
0.6 in the most stable conditions.
The adjusted empirical values of a have been used to derive a dimension-
less dispersion-velocity function
$(Z/L) = ( l/ku*)( dZ/dt) (A-9)
in which Z is the mean height reached by particles at a given time after leaving
the surface. This form was first postulated by Gifford , with a constant b
in place of the von Karman constant k, and identity of b and k was first
argued by Ellison (see Ref. 1, p 117). Details of the derivation of $(Z~/L)
are summarized in Table A-2, and the results are shown graphically in Figure A-l,
In accordance with the setting of b = k = 0.4 in earlier considerations
of Lagrangian similarity (see Ref. 1, pp. 118 and 206), $(1/1) tends to unity
as I/I tends to zero, and this is reaffirmed in the trend of the empirical
values presented here. However, as the function departs from unity, increas-
ing in unstable flow and decreasing in stable flow, similarity in terms of u*
and L is not entirely supported, in that the set of points for different
magnitudes of L/z do not collapse onto a single curve. Although the depar-
tures from a single curve are not large, compared say with the scatter in
the empirical specification of the Monin-Obukhov velocity profile function
$> , it may be significant that a systematic effect of L/zn is clearly evident.
n 0
SIMILARITY HYPOTHESES ON MONIN-OBUKHOV LINES
Form of $ have already been suggested in terms of the Monin-Obukhov
functions for transfer of momentum and heat (see Table 5 of the text), either
explicitly, or implicitly through gradient transfer considerations. These
forms are reproduced graphically in Figure A-l. Note that the form in terms of
*M seriously underestimates the effect of stable flow, a result which is
partly to be expected from the fact that Tyldesley's setting of the constant
^was determined by fitting to Thompson data, and these data (see Ref 1,
p 207) do show a smaller reduction in a than do the "Prairie Grass" data.
40
-------
TABLE A-1. ADJUSTMENT OF CRAMER'S 1957 E?TIMATES OF az FROM PRAIRIE GMSS DATA
Lengths in
az(100m)
R1(2m)
L
Cramer's
Cramer's
b (100-200)
Est. n
Corrd. b
Adjusted
az(200)
b (200-400)
Est. n
Corrd. b
Adjusted
az(400)
b (400-800)
i
Est. n
Corrd. b
Adjusted
az(800)
Cramer ' s
Original
oz(800)
metres, z
0
10
-0.5
-4
21.5
1.43
0.1
1.30
24.6
1.93
0.1
1.75
82.8
2.46
0.1
2.24
391
400
= 0.008
8
-0.27
-7.4
16
1.24
0.1
1.13
17.5
1.66
0.1
1.51
49.9
2.08
0.1
1.89
185
200
6
-0.09
-22.2
11.5
1.07
0.1
0.97
11.8
1.47
0.1
1.34
29.8
1.75
0.1
1.59
89.7
110
4.5
0
8
1.0
0.16
0.86
8.16
1.20
0.13
1.06
17.0
1.30
0.12
1.16
38.0
50
3
0.06
23.5
5
0.9
0.24
0.73
4.97
0.90
0.29
0.70
8.1
0.80
0.35
0.59
12.2
23
2
0.12
7.1
3
0.8
0.34
0.60
3.03
0.90
0.42
0.63
4.7
0.40
0.45
0.28
5.7
14
(1)
(2)
(3)
(4)
(5)
(6)
(7)
41
-------
TABLE A-I. ADJUSTMENT OF CRAMER-S L(%& ESTIMATES OF az FROM PRAIRIE GRASS DATA
Explanatory Notes
(1) From "eye fit" curve, Fig. 4.14, p 207 of Pasquill1
(2) Taking Ri(z) = z/L in unstable conditions
Ri(z) = (z/L)(l + 4.8Z/L)"1 in stable conditions
(3) Interpolating for a (100) on Fig. 11 of Cramer"
(4) Interpolating in accordance with aft in Cramer's Tables 4a and 4b
(5) In u * zn, for unstable conditions n = C.I is taken as a reasonable
approximation. In stable conditions a power law was fitted to the
similarity form specified in Table A-3, for z = 0.8 cm, over the
height interval corresponding to Cramer's Z values, where 1= 0.776
a,, the latter relation being appropriate to a vertical distribution
form exp-(bzs) with s = 1.5.
(6) Estimated corrected b is b/(l+n) as argued in text
(7) Adjusted az(200) = "directly" measured a (100) x 2b
Adjusted az(400) = Adjusted az(200) x 2
. etc.
42
-------
TABLE A-2. ESTIMATES OF (l/kuJ^Z/cit) FROM ADJUSTED VERTICAL
SPREAD ESTIMATES IN TABLE 1
x Z dl/dx* kU"/uxt (l/kuj(ol/dt)* I/I
(L -4, L/zo -500)
141.4 12.17 0.1119 5.505 3.85 - 3.04
282.2 35.03 0.2168 6.1i8 8.29 - 8.76
565.7 139.7 0.5532 7.026 24.29 -34.9
(L -7.4, L/z -925)
141. 4
282.8
565.7
141.4
282.3
:ec,.7
141.4
2S2.8
565.7
141.4
282.8
565.7
9.17
22.93
74.52
6.52
14.53
"O^O
3.00
4.92
7.71
1.91
2.92
4.01
0.0733
0.1225
0.2490
(L -22.2,
0.0447
0.0688
0.1127
(L 23.5,
0.0155
0.0122
0.00804
(L 7.1,
0.00810
0.00651
0.00199
5.62
6.169
6.9*1
L/Zo
5.85
G . 36
6.S5?
L/2o
5.78
G.46
7.18
L/Zo
5.70
6.52
7.26
2.57
4.72
10.80
-2775)
1.63
2.73
4.90
2937.5)
0.56
0.49
'.2L
387.5)
0.29
0.27
0.090
- 1.24
- 3.10
-10.07
- 0.29
- 0.65
- 1.81
0.128
0.209
0.328
0.269
0.412
0.565
Explanatory Notes:
* In accordance with the analysis of Table A-l, assume over 2:1 intervals
of distance that
Z~(x)/Z| = (x/x-j)
and that Z = 0.776 a . Considering I and d7/dx at the geometric mean of the
distances x1 and 2x1f it fellows that
d7/dx = bZ/2?i x1
T - ob/2^-
L - c. L->
43
-------
TABLE A-2. ESTIMATES OF (lkuJ(dZ/dt) FROM ADJUSTED VERTICAL
SPREAD ESTIMATES IN TABLE L
} For 0.6 Z/L, evaluated from the wind profile equation given in Table 3,
and assuming u « z0-1 for -z/L greater than 1.0.
I Invoking the similarity assumption
dT/dt = "(cZ)
with c = 0.6 (see item 2 of Table 5 of main text for reference).
44
-------
0.2
0.5
5
10
(- VL)
1C
a r-3), O
0 F
\j t <~>
0.2
C.I j
1 ft
12
1
-' '
I/"
,1 14
( -I- Z/L)
Figure A-l. Vertical diffusion similarity functions.
45
-------
The form $ = l/H advocated by Chaudrey and Meroney does provide an encouraging
fit, though there is ambiquity as to the precise numerical values to be adopted
for (f>H in unstable flow, and the one set of values considered in stable flow
overestimates $ somewhat.
SIMILARITY IN TERMS OF THE INTENSITY AND SCALE OF TURBULENCE
As an alternative to the original form of Lagrangian siirilarity treatment,
the writer * has suggested a simple local similarity approach in terms of
the intensity and scale of the vertical component of turbulence, both regarded
as functions of height.
In the practical form to be considered here, the dispersion velocity is
written as follows:
dZ/dt = a
L°wVz
in which X is the equivalent wavelength at which the product cf frequency
and spectral density of the vertical component is a maximum. It turns out
that to a useful approximation the foregoing relation is the same as that
which follows from the assumption of a gradient-transfer relation with
K = ao^ . (A-ll)
On identifying Eq. (A-ll) with the Monin-Obukhov form of KM in the special case
of neutral flofc, and inserting the latest estimates of 0 /u and X /z (see later)
w m
the magnitude of the constant a is 0.154. (Alternatively, note that this mag-
nitude is consistent with the result dT/dt = 0.4 u*.)
The latest estimates of the boundary layer parameters a /u* and X /a* as
w HI
a function of z/L are listed in Table A-3, with the magnitude of the dispersion
function $ which then follows from a combination of Eq. (A-9) and A-10). Also
included are values of the velocity profile function \|>, from which dZ/dt may
be converted to dZ/dX", where X" is the mean distance of travel of the particles.
As can be seen from Figure A-l, the function $ derived in Table A-3 shows
a growth in unstable flow which, over the total range of Z/L covered, is similar
to that found in the "Prairie Grass" measurements of dispersion. However,
there are differences in detail, notably the very sharp increase in the
46
-------
TABLE A-3. MAGNITUDES OF BOUNDARY LAYER PARAMETERS AND DIMENSIONLESS
DISPERSION VELOCITY $(Z/L) DEFINED IN EQ..(A-9) AND (A-10)
z/L
-10
- 5
- 2
- 1
- 0.5
- 0.2
- 0.1
- 0.05
- 0.02
- 0.01
0
0.01
0.02
O.Q5
0.1
0.3
0.5
1
2
^
-1.032
-0.707
-0.354
-0.185
-0.082
-0.024
-0.009
0
0.047
0.094
0.235
0.47
1.41
2.35
4.7
(9.4)
aw/U*
4.084
3.276
2.487
2.064
1.764
1.520
1.419
1.362
1.325
1.313
1.3
1.32
1.34
1.37
1.40
1.57
1.70
1.97
2.36
Vz
6.0
6.0
6.0
6.0
3.5
2.0
2.0
2.0
2.0
2.0
o n
L . \j
(1.97)
(1.94)
(1.90)
1.8
1.25
1.0
0.67
0.45
*U/L)
9.423
7.557
5.737
4.763
2.373
1.170
1.093
1.047
1.017
1.010
1.000
(1.000)
(1.000)
(1.000)
0.970
0.755
0.653
0.507
0.407
ku/u* = ln(z/zn) + iM
9
\p for -ve values of z/L, Dyer and Hicks
t> for +ve values of z/L, 4.7z/L, Merry and Panofsky8
aw/u* = 1>3 L1-0 + 3.0(z/-L)J for -ve values of z/L, Panofsky
et al10 published
r 11/3
=1.3 [1.0+ 2.5(z/L) J for +ve values of z/L, based on
Iierry arid Pc.nofsky°
A /z Based on Kaimal et a! 1972. The "refined" estimates in parentheses
'are consistent with the general shape of the Am, z curve Baking the neutral
value as 2.0, and are used rather than rounded values of 2.0 to avoid values
greater than 1.0 for $.
47
-------
"similarity theory" value fof Z/L in the region of -0.5, which does not appear
in the empirical values. This sharp increase is associated with the sharp
increase of \m with height in that range of z/L. On the other hand, in the
most unstable conditions the empirical values show a much greater rate of
growth than the "similarity theory" values, the latter necessarily tending to
1 / ^
a limiting growth as (Z/-L) ' in accordance with the limits of a /u* and
i /"^
A /z in Table A-3. Note also that this (Z-L) limiting growth also follows
from the similarity hypothesis that in free-convective transfer the dispersion
velocity is determined uniquely by the surface heat flux and the height (7),
so something is clearly calling for explanation in the observed growth of
$ at large values of (Z"-L).
In stable conditions the "similarity theory" function falls away from
unity much more slowly than is found empirically. It may be significant here
that a determining factor in the similarity function is the increase of a/u*
p w
with z/L, now claimed by Merry and Panofsky and included in Table A-3.
48
-------
REFERENCES
1. Pasquill, F. Atmospheric Diffusion, 2nd Ed. John Wiley & Sons, New
York, N. Y., 1974.
2. Cramer, H. E. A Practical Method for Estimating the Dispersal of
Atmospheric Contaminants, Proceedings of the Conference on App. Met.,
Am. Met. Soc., 1957.
3. Gifford, F. A. Diffusion in the Diabatic Surface Layer. J. Geophys.
Res., 67:3207, 1962.
4. Thompson, N. Short-Range Vertical Diffusion in Stable Conditions. Q.
J. R. Meteorol. Soc., 91:175, 1965.
5. Chaudrey, F. H., and R. N. Meroney. Similarity Theory of Diffusion
and the Observed Vertical Spread in the Diabatic Surface Layer. Boundary
Layer Meteorol., 3:405-415, 1973.
6. Pasquill, F. The Dispersion of Material in the Atmospheric Boundary
LayerThe Basis for Generalization. In: Lectures on Air Pollution and
Environmental Impact Analyses, Am. Met. Soc., 1975, pp. 1-34.
7. Pasquill, F. Some Topics Relating to the Modeling of Dispersion in the
Boundary Layer, EPA-650/4-75-015, U. S. Environmental Protection Agency,
Research Triangle Park, N. C., 1975.
8. Merry, M., and H. A. Panofsky. Statistics of Vertical Motion over
Land and Water. Q. J. R. Meteorol. Soc., 102:255-259, 1976.
9. Dyer, A. J., and B. B. Hicks. Flux-Gradient Relationships in the
Constant Flux Layer. Q. J. R. Meteorol. Soc., 96:715-721, 1970.
10. Panofsky, H. A., H. Tennekes, D. H. Lenschow, and J. C. '..'yngaard. The
Characteristics of Turbulent Velocity Components in the Surface Layer
under Convective Conditions. Boundary Layer Meteorol., 11:355-362, 1977.
11. Kaimal. J. C., J. C. Wyngaard, Y. Izumi, and 0. R. Cote. Spectral
Characteristics of Surface Layer Turbulence. Q. J. R. Meteorol. Soc.,
98:563-589, 1972.
12. Dyer. A. J. The Turbulaent Transport of Heat and Water Vapour in an
Unstable Atmosphere. Q. J. R. Meteorol. Soc., 93:501-508, 1967.
49
-------
13. Businger. J. A. Transfer of Momentum and Heat in the Planetary Boundary
Layer. In: Proceedings of the Symposium on Artie Heat Budget and
Atmospheric Circulation (The Rand Corp.), 1966, pp. 305-332.
14. Businger, J. A., J. C. Wyngaard, Y. Izumi, and E. F. Bradley. Flux-
Profile Relationships in the Atmospheric Surface Layer. J. Atmos. Sci.,
28:181-189, 1971.
15. Tyldesley, J. B. Contribution to Discussion on "Short-Range" Vertical
Diffusion in Stable Conditions. Q. J. R. Meteorol. Soc., 93:383-385, 1967.
50
-------
APPENDIX B
A SIMPLE TWO-LAYER MODEL FOR ESTIMATING VERTICAL SPREAD
BEYOND THE SURFACE-STRESS LAYER IN NEUTRAL FLOW
The equivalence of the Lagrangian similarity result, dT/dt = ku*. and
gradient-transfer with K = ku*z has already been noted. These laws, however,
apply only over a shallow surface layer (the surface-stress layer) and for the
higher parts cf the boundary layer a different form of K-profile must apply.
There have been several guides to the form of this profile, and one of the
latest, due to Businger and Arya , is shown in Figure E-l in the dimensionless
form K/u*h versus z/h where h is the depth of the boundary layer. This profile
was derived in a treatment in which von Karman's constant was taken to be 0.35;
bearing this figure in mind the reader can observe that the profile follows the
surface-stress layer form very closely up to z/h = 0.05 and to a reasonable
approximation even up to z/h = 0.1. Thereafter, there is the non familiar
bending of the curve to form a maximum K, in this case with K/u* = 0.033 at
z/h = 0.25, and then a continuous falloff at greater heights. Note also, as
an indication of the present uncertainties, that this maximum K is only about
p
half that previously estimated by Townsend (K/u*h = 0.067).
Notwithstanding the uncertainties it is now suggested that a useful
general guide to the a growth curve throughout a substantial depth of the
atmospheric neutral boundary layer may be obtained from a simple model with
the following properties:
(a) The surface-stress-layer result for d_Z_ and K is taken to apply to
Z1 = h/12. dt
(b) K is taken to be constant at the surface-stress-layer value K(Z") for
h/12
-------
m
"i_L
-trr
-pr
w+i H-tf
£*HI
+-)-r
:~ ,-, \. .
ur
~ '-i
-t~t
-H-4
";i
HH
:,t
:ra
. -. , __.,,
i_i»^-, T.
The continuous curve is from
Businger and Arya's analysis
(Ref. 1) - the y* = 0 curve of
their Fig. 5 combined with
fh/u* = 0.7 as given in their
Fig. 11.
The broken line is the simpli-
fied two-layer form used in
the present analysis.
F
I
i.l U!
rt-f
14+
^
_i..
-r
ir^rn'r
!- '4- , J - I
3 466
f b'.'oi
3 4 5 6 7 a
4 5 6 7 8 9 10
Figure B-l. K-profile in a neutral boundary layer of depth h.
52
-------
A general solution of the two-dimensional equation of diffusion with this
two-layer form of K cculd be obtained numerically, but a short-cut calculation
adequate for bringing out the essential properties of the a growth may be
carried out as follows: For the lower layer (Z <_I.}
dT/dt = ku* (B-l)
dX/dt = u"(cZ~) (B-2)
kF(z)/u* = ln(z/zo). (B-3)
These equations may be integrated to give X~as a function of 7, and the
solution which follows from taking the lower limit of integration at I = z is
kV = Z Inz' - (1 - lnc)(l - iz')
(B-4)
where X = X/z and Z = Z/z (see Ref. 4, p. 117). This result is physically
0 i 0
unacceptable when Z is near 1.0, since with c less than 1.0 (approximately
0.6) dX/dt actually becomes negative. This unacceptable result may be
eliminated simply by making the lower limit of integration z /c, in which the
result is
k2 X"' = Z"'(ln cZ~' - 1) + 1/c. (B-5)
If K were everywhere constant (= K ) we could write
I2 = 2(//az)2 KJ (B-6)
or dl/dt = (Z/az)2 |
-------
From Eq. (B-6), substituting K = ku*Z. and writing (Z/a )2 = r,
C 1 Z
72 - Z2. = 2r2k u*Z. (T - T^ (B-9)
with T. and T the travel times corresponding to 7. and 7, applicable for 7 up
to say 67. in accordance with (b) above. For conversion to a I, X" relation
Eq. (B-9) should be combined with Eq. (B-2) and an appropriate value of c. How-
ever, since a large proportion of the total change of wind speed with height
occurs in the bottom one-tenth of the boundary layer, it will be adequate for
present purposes to take with Eq. (B-9) a constant value of X/T equal to the wind
speed at say z/h =1/6. This constant "effective wind speed" u may be repre-
sented as a fraction p of the geotrophic wind speed, p being drivable as a func-
tion of Ro from F. B. Smith's model. Writing 7 and X" in the dimensionless forms
above, Eq. (B-9) then becomes
- x,)/pV
and at large I/I,
7'2
Given r and k Eqs. (E-10) and (B-ll) are determined by a2> u*/V , p, and V /fz
the first three of these all being functions of the last (the Surface Rossby
Number) and determinable from Smith's model. Table B-l gives, for a practical
range of the Surface Rossby Number, the magnitudes of the foregoing parameters,
of 7,, JC,, and of the coefficients A and B in the 7, X~ relations are specified
in the notes below the table.
The resulting data for 7/z as a function of X/z are shown graphically
in Figure B-2. There is a sharp discontinuity at the junctions of the surface-
stress-layer curve (Eq. (B-5) with the Ro-dependent sections (Eq. (B-10), at
7=7,. This is a consequence of the discontinuity in d7/dt according to
Eq. (B-8) and Eq. (B-7). Note also that the limiting form in Eq. (B-ll) is an
adequate approximation to Eq. (B-10) except near the discontinuity for the
smaller values of Ro. Finally, it is emphasized that the curves are taken
only to 7 = h/2, and even at this stage there will be some error (overestima-
ting 7) if, as expected, there is a reduction of K in the top part of the
boundary layer.
54
-------
TABLE B-l. PARAMETERS AND RESULTS IN THE ESTIMATION OF VERTICAL SPREAD FROM A
SURFACE RELEASE, BEYOND THE SURFACE-STRESS LAYER, IN NEUTRAL FLOW
1
2
3
4
5
6
7
8
Ro
a2
U*/Vg
Ue/Vg
ZJ
X!
A
B
104
0.320
0.080
0.717
21.33
207.6
3.775
xlO2
1.098
3xl04
0.308
0.070
G.790
53.9
835.2
1.200
xlO3
1.555
105
0.290
0.062
0.858
149.8
3.277
xlO3
4.093
xlO3
2.341
3xl05
0.275
0.054
0.870
371.3
1.023
xlO4
1.180
xlO4
3.416
106
0.260
0.0475
0.886
1029
3.490
xlO4
3.790
xlO4
5.285
3xl06
0.247
0.043
0.904
2655
1.058
xlO5
1.102
xlO5
7.996
107
0.236
0.0385
0.910
7572
3.512
xlO5
3.534
xlO5
12.73
Notes
1. Ro = Surface Rossby No. = V /fzQ
2. a« = hf/u*. where h 1s depth of boundary layer
4. u = dX/dt, taken to be u at z = h/6, = pV . evaluated from the interpolation
e 9
form ku(z)/u* = In(z/z0) - z/h
5. ZJ = h/12zo
6. 7| = value of X1 for Z1 = ^ in Eq. (B-5)
7. Coefficient A in X' - X| = A [(Z'2/ Z,'2) - l] (see Eq. (B-10))
8. Coefficient B in Z1 = B X1'2 (see Eq. (B-ll)) (for both A and B, k = 0.4 and
f/o = 0.796, the latter figure being appropriate to a Gaussian distribution)
2,3,4. Data all in accordance with F. B. Smith's3 neutral boundary layer model,
with unpublished minor adjustments.
55
-------
r-
\
0>
o
to
4_
s_
13
O) I
>>ca
Ki
i O
s-
>
GJ S-
> ro
3 !-
O E
O in
10
CU S-
S- CU
O.-Q
oo E
O I
o >,
i j:j
-(-> 1.1
S- 01
CU C
> c;
CVJ
i
OQ
QJ
S-
3
cn
O
O
rH
56
-------
REFERENCES
1. Businger, J. A., and S. P. S. Arya. Height of the Mixed Layer in the
Stably Stratified Planetary Boundary Layer. Adv. Gecphy., 18A:73-92, 1974,
2. Townsend, A. A. The Structure of Shear Flow. Cambridge University Press,
Cambridge, England, 1956.
3. Smith, F. B. Turbulence in the Atmospheric Boundary Layer. Sci. Prog.,
62:127-151, 1975.
4. Pasquill, F. Atmospheric Diffusion, 2nd Ed., John Wiley & Sons, New York,
N. Y., 1974.
57
-------
TECHNICAL REPORT DATA
(Please read Instructions on the reverse before completing)
1. REPORT NO.
EPA-600/4-78-021
3. RECIPIENT'S ACCESSION-NO.
4. TITLE AND SUBTITLE
ATMOSPHERIC DISPERSION PARAMETERS IN PLUME MODELING
5. REPORT DATE
May 1978
6. PERFORMING ORGANIZATION CODE
7. AUTHOR(S)
8. PERFORMING ORGANIZATION REPORT NO.
F. Pasquill
9. PERFORMING ORGANIZATION NAME AND ADDRESS
Environmental Sciences Research Laboratory
Office of Research and Development
U.S. Environmental Protection Agency
Research Triangle Park, North Carolina 27711
10. PROGRAM ELEMENT NO.
1AA603 AB-02 (FY-78)
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 COVFRFD
Inhouse 1/77-8/77
14. SPONSORING AGENCY CODE
EPA/600/09
15. SUPPLEMENTARY NOTES
*Visiting Scientist from United Kingdom
16. ABSTRACT
A survey of the underlying foundations of the present systems for specifi-
cation of the atmospheric dispersion parameters for a continuous point source
leads to the conclusion that the basis for revision of the current systems is
still far from complete, but certain obvious evidence points to desirable
changes. The author's previous recommendations on the crosswind spread are
reiterated, and a suggested revision for vertical spread as a function of
surface roughness, stability, and for the unstable case, the convective velocity
scale and mixing depth, is offered.
17.
KEY WORDS AND DOCUMENT ANALYSIS
DESCRIPTORS
b.lDENTIFIERS/OPEN ENDED TERMS C. COSATI Field/Group
Air pollution
*Atmospheric diffusion
*Plumes
*Meteorology
*Mathematical models
13B
04A
21B
04B
12A
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RELEASE TO PUBLIC
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