EPA-600/4-76-030a
                                                July 1976
   ATMOSPHERIC DISPERSION PARAMETERS IN GAUSSIAN PLUME MODELING
Part I.   Review of Current Systems and Possible Future Developments
                               by
                           A. H.  Weber
                Associate Professor of Meteorology
                    Department of Geosciences
                 North Carolina State University
                  Raleigh, North Carolina  27607
              U. S. ENVIRONMENTAL PROTECTION AGENCY
               OFFICE OF RESEARCH AND DEVELOPMENT
           ENVIRONMENTAL SCIENCES RESEARCH LABORATORY
          RESEARCH TRIANGLE PARK, NORTH CAROLINA  27711
             Environmental  Protection Agency
             Region V,  Library
             230 South  Dearborn Street
             Chicago, Illinois  6060H

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                               DISCLAIMER






     This report has been reviewed by the Environmental Sciences




Research Laboratory, U. S. Environmental Protection Agency, and




approved for publication.  Approval does not signify that the




contents necessarily reflect the views and policies of the




U. S. Environmental Protection Agency, nor does mention of trade




names or commercial products constitute endorsement or recommenda-




tion for use.
                                   ii

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                                ABSTRACT






     A recapitulation of the Gaussian plume model is presented




and Pasquill's technique of assessing the sensitivity of this




model is given.  A number of methods for determining dispersion




parameters in the Gaussian plume model are reviewed.  Com-




parisons are made with the Fasquill-Gifford curves presently




used in the Turner Workbook.  Improved methods resulting from




recent investigations are discussed, in an introductory way




for Part II of this report.
                                   iii

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                               PREFACE





     The increasing concern of the last decade in environmental issues,




and the fuller appreciation that air quality simulation modeling may




provide a unique basis for the objective management of air quality,  has




generated an unprecedented interest in the development of techniques for




relating air quality and pollutant emissions through appropriate modeling




of the atmospheric transport and dispersion processes that are involved.




A multitude of recent publications in the U.S.A.  and elsewhere points to




the wide interest at many different levels of local, state, regional and




national planning, and testifies to the widespread acceptance of meteoro-




logical type air quality modeling as an important rational basis for air




quality management.  This point of view is now internationally recognized




in most industrial countries.




     Earlier attitudes towards the quantitative estimation of atmospheric




dispersion of windborne material from industrial and other sources were




strongly influenced by a system introduced in 1958, and published in 1961,




by Dr. F. Pasquill of the Meteorological Office,  United Kingdom.  This




was followed in 1962 by the publication of his definitive textbook on




"Atmospheric Diffusion," which includes detailed consideration of the




well-known simple "Gaussian-plume" model for the average concentration




distribution in space from an elevated continuous point-source under steady




conditions.  The unique feature, however, of the Pasquill system is the




method by which the critical parameters expressing the downwind spread of




the plume might be estimated in terms of the ambient meteorological




conditions.  These estimates were later expressed in slightly more convenient,






                                      iv

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although exactly equivalent form, by Dr. F. Gifford, and this so-called




Pasquill-Gifford system for dispersion estimates has been widely used




ever since.  It was early given some general endorsement as a valuable




practical scheme by the Public Health Service of the U. S. Department




of Health, Education, and Welfare, by the publication in 1967 of the




"Workbook of Atmospheric Dispersion Estimates" (Public Health Service




Publication No. 999-AP-26) by D. B. Turner, that exclusively utilized




this system of Gaussian-plume dispersion parameters.




     In spite of the gradual appearance in recent years of air quality




models based on more sophisticated formulations of the atmospheric pro-




cesses, e.g., through fluid-dynamical equations assumed to govern the




physical processes of transport and turbulent diffusion, great use




continues to be made of the simpler Gaussian-plume models.  However, a




direct consequence of the unprecedented interest in the subject has been




the publication of many attempts to confirm or improve the realism of the




dispersion estimates.  These and other matters relating to the substantial




progress made in recent years in understanding atmospheric dispersion, are




discussed in a much revised 2nd Edition of the Pasquill book, that was




published late in 1974.  Under these circumstances it seemed desirable to




examine critically the possible requirements for change in the Turner




Workbook values for dispersion, that have been so widely used since 1967.




The present two-part report was prepared to meet this need.




     It was extremely fortunate that Dr. Pasquill was available for




detailed discussions during its preparation (1975-76), both while he was




a Visiting Professor at the North Carolina State University and also the




Pennsylvania State University (under research grant support of the EPA),

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and also as a Visiting Scientist to the Meteorology and Assessment

Division of the E.P.A., Research Triangle Park,  N.  G.   Even more fortunate

has been Dr. Pasquill's willingness to assume responsibility for prepara-

tion of the second part of the report, which critically examines the

possible requirements for change of the Turner Workbook values.   The first

part was prepared by Dr. Allen Weber of the Department of Geosciences at

North Carolina State University, in consultation with Dr. Pasquill and

the undersigned.  It provides a reasonably comprehensive review of current

systems and possible future developments, and is a  necessary input for the

critical examination of the second part.  It is perhaps unnecessary to

emphasize that there are still many problems of dispersion that are unlikely

to be resolved satisfactorily in terms of simple Gaussian-plume models.

However, it is hoped that the present publication will provide a more

up-to-date basis for continuing the successful treatment of the many

important practical problems that can be analyzed by this simple approach.
Research Triangle Park
North Carolina
March 1976
Kenneth L. Calder
Chief Scientist
Meteorology and Assessment Division
Environmental Sciences Research Lab.
                                      vi

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                               CONTENTS









Preface	      iv




List of Tables	    viii




List of Figures	      ix




Acknowledgements	       x






1.  Introduction	       1




2.  The Gaussian Plume Model  	       2




3.  Sensitivity of the Gaussian Plume Model 	       8




4.  Review of o-Systems Available for Use	      13




5.  Improved Methods of Estimating Dispersion Parameters  ...      40




6.  Effect of Release Altitude on Dispersion Parameters ....      46






Appendix 1  List of Symbols	      47




Appendix 2  Relationships Among Turbulence Categorizing Schemes      5C
                                  vii

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                           LIST OF TABLES




No.                                                                 Page




 1.  Applicability of Current Working Theories of Dispersion         14




 2.  Relationships Among Various Stability Parameters                53
                                    viii

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                          LIST OF FIGURES

No.                                                                  Page

 1.  Idealized Distribution of Concentration at  Ground Level
     from an Elevated Source                                           9

 2.  The Pasquill-Gifford Curves for a  Versus Downwind Distance       21

 3.  The Dispersion Curves of  Singer and  Smith and  the PG Curves       34

 4.  The TVA Dispersion Curves and the  PG  Curves                      35

 5.  The St. Louis Dispersion  Curves and the PG  Curves                 36

 6.  Markee's Dispersion Curves and the PG Curves                      37

 7.  TRC Rural Dispersion Curves and the PG Curves                     38

 8.  Briggs' Rural Dispersion  Curves and  the PG  Curves                 39
                                   ix

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                             ACKNOWLEDGEMENTS









     It has been the author's good fortune to have close contact with Dr.




Frank Pasquill for several weeks while writing this review.   Dr. Pasquill's




insight on the problem of turbulent dispersion has been extremely valuable.




Mr. Kenneth Calder provided extremely good information and guidance in




pointing out important studies of atmospheric diffusion bearing on this




study.









     During the course of the work the author received valuable assistance




from several individuals on the telephone and otherwise.  Dr. Frank Gifford




was especially helpful in this way in clarifying important issues.  Mr.




Jui-Long Hung and Mr. Richard D'Amato were very helpful in preparing figures




in this report.









     The author would like to thank Mr. Larry Niemeyer for providing two




months support under the Intergovernmental Personnel Act to conduct this




s tudy.
                                      x

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                                SECTION 1




                              INTRODUCTION






    For purposes of simplicity this review will be restricted to a




discussion of the single source dispersion problem (e.g. an isolated




stack) either as a problem in its own right or as an element of an




"area" source.  The dispersing material is assumed to be passive i.e.




the material does not react chemically, decay by radioactivity, undergo




washout by precipitation, deposit on the surface of the earth, or




react in any other way to violate the law of conservation of matter.




Furthermore, there will be no discussion of buoyant plume behaviour since




a recent comprehensive review and summary of this problem has been given




by Briggs (1969, 1975).




    Discussion of dispersion will be in terms of the so-called Gaussian




plume dispersion model (defined in Section 2).  This basic model




follows directly from some of the theoretical models of dispersion.  It




is also known that a great deal of the experimental dispersion data now




available for single sources in steady conditions, seems to be reasonably




well described by the Gaussian shape of the spatial distribution pattern,




when properly time-averaged.  Gifford (1975, 1976) provided a brief




survey and review of the Gaussian-model methodology as well as clearly




specifying the kinds of situations when the simple Gaussian form is




inappropriate.

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                              SECTION 2

                      THE GAUSSIAN PLUME MODEL

                                       *
     The Gaussian plume diffusion model  for an elevated point source

 (assuming reflection at the ground) is represented by the equation

                                  2 ^    s*      2^     f      2\
     C(x,y,z;H) = 	*—- exp\	^-yifexpy-   ~Z' /+ expV-    z) fi   (1)
                  2ira a u    (  2a   )    (    2cr    i     (   20   1
                     yz     v.    y ./    ^      z  -'     ^     z S


where

          C is the time averaged concentration of pollutant;

          x, y, and z are the distances downwind, crosswind, and ver-

          tically upward, respectively;

          H is the effective s< urce height above ground level (H  is

          equal to the sum of the physical stack height h  and the
                                                         S

          plume rise AH);

          Q is the source strength;

          0  is the standard deviation of the time-averaged plume

          concentration distribution in the crosswind direction;

          a  is the standard deviation of the time-averaged pluire
           Z

          concentration distribution in the vertical direction;

          u is the time-averaged wind speed at the level H.

     The sampling time used in the definitions above is normally

chosen to be  about  one hour because such a time period admits most  of

the turbulent fluctuations and defines time-mean values that are
   The historical basis  of  the Guassian plume model  is not of  crucial
 importance here, although,  the interested reader will be able  to
 intersect the  line  of  development of  this model at a significant  stage
 by referring to Gifford  (1960).
 **
   Mathematical symbols  will be defined after their  first appearance
 in the text and in  a special list in  the Appendix
                                       2

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quasi-steady.  It is also a very convenient time interval to describe

air pollutant concentrations.

     It is assumed in the use of the Gaussian plume model that there

is a uniform rate of emission of pollutant during the one hour period.

It is also assumed that diffusion in the x direction is negligible.

This would be literally true if the release were continuous and

effectively so if the time of release is equal to or greater than the

travel time x/u from the source to the location of interest.

     The range of application  of  Eqn. 1 is restricted to within the

downwind distance x where the plume first encounters the top of the
            *
mixing layer .  The top of the mixing layer is of the order of one

kilometer, but it can extend anywhere from a few hundred meters to

several kilometers depending on several factors including synoptic

scale features of the weather.  The top of the mixing layer can be

estimated from the results of Holzworth's (1972) extensive survey or by

direct measurement,  e.g. with an acoustic sounder.

     The distance downwind where the top of the plume intersects this

"lid" (as it is often called) is quite variable.  Similarly the point

downwind where the plume from an elevated source reaches its maximum

ground-level concentration will depend on several factors.  Often the

point of maximum ground-level concentration will be closer to the source

than the distance to the plume's reflection from the lid.  Since it is

often desired to know what the maximum concentration will be, the com-

plication of the restrictive lid will (under many circumstances) be
*
   By considering reflections of the pollutant from the top of the
mixing layer and the ground, the range of the Gaussian plume model can
be extended.  Another simpler method of obtaining concentration beyond
the distance where the plume is trapped under the top of the mixing
layer is provided in Turner (1969).
                                      3

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of no consequence.  On the other hand, there are circumstances where



the mixing depth will be critically important so this concept must be



borne in mind when doing diffusion calculations.



     There are two additional aspects relating to time-averaging that



are of special significance.  The first is that the time-average of



a  is not usually stationary, i.e. the longer the time average the



larger a  tends to become.  Therefore, all graphs and tables of this



parameter should make clear the duration of the average.



     An attempt has been made (based on experimental studies) to pre-



dict the effect of limited sampling time on the value of a .  Several



authors, whose results were summarized by Gifford (1975), suggest





                 A                                               (2)
         a
          yB
                t
B
where a „ is the standard deviation of concentration averaged over some
       yB


reference period t,,, e.g. one hour, and a .  is the standard deviation
                  13                      yA


of the concentration over the time period of interest, t .   The value
                                                        £\.


of q advocated by Gifford is 0.25 to 0.30 which seems to hold for



1 hr. <  t.  <  100 hrs.  By considering this same problem from a (limited)
         A.


theoretical viewpoint, it is likely that the effect will be a complex



function of the spectral shape and distance downwind  (see Pasquill,



1974).



     Similar considerations apply to a  from an elevated source,
                                      z


although recognition of the important differences between the spectra



of vertical and horizontal components should make it clear that the



increase of 0  with sampling time may be expected to be terminated
             Z

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much more decisively at sampling times of 10 minutes  or  so.   Further



considerations along the same lines make it clear  that for a  ground



level source a sampling time of a few minutes  (3 to 5) will yield



steady values of 0  (See also Pasquill's comments  on  this point  in
                  z


Part II of this report).



     Further simpler forms of the concentration from  Eqn. 1 are  ob-



tainded at special locations.  The following formulae and other



related formulae are found in Gifford (1960) and Pasquill (1974).



The concentration at ground level is



                                    2     TT2
       C(x,y,0;H)
                  _ exp  - --

             ira a u        2a
               y z           y
(3)
                                         2a
Along the axis of the plume at ground level the concentration  is



                                   ,,2
C(x,0,0;H) =
                           exp  (-
                    ira a u
                      y z
                                  2a
(4)
     If one assumes simple power-law forms for the crosswind and



vertical spread, i.e.,





                                 o  = a  (x.)  (—)q         (5 a & b)
(where x- is a reference distance downwind)



then, by differentiation the position along the axis where the maximum



occurs is
X =
m
qH2

[Vxi>l
UqJ
2
(p + q)
                            -|l/2q
                                                                  (6)

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or alternatively, the distance x equals x  when
                                         m
         R
                                                                 (7)
The magnitude of the maximum concentration is
cm = ii/2 «P (- A) _ -^

                    iru   y  m     H
If p = q the maximum concentration reduces to
c  -   2Q   ,-!»)
 m     -2   o J
     eiruE     y
and a /a  = a , a constant independent of x.



With p = q the distance x to the maximum is


                         l/2q
       x  =
        m
H2
2
raz(Xl)l
x q
. 1 -
2
                       1/2
or x = x  when H/o  = 2
        m         z



The crosswind integrated concentration at ground level is



                                                2
       /C(x,y,0;H)dy =  I  -- exp (-
                                ir  a u
                                    z
                                              2a
                                                                 (8)
                                                                 (9)
                                                         (10)
                                                                (11)
     The above formulas now form the basis of a widely used methodology



(Pasquill, 1974 and Slade, 1968) of describing atmospheric dispersion



in many contexts, e.g. diffusion from ground sources, tall stacks,



automobiles, rockets, etc.  The method requires additional special



interpretation when the terrain is complicated, e.g., shorelines,

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mountain-valley systems, hills, forests broken by cultivated land,



etc.  Gifford (1976) and Egan (1975) have attempted to treat some of



these "special" diffusion problems insofar as this is possible at this



time.  For the purpose of this study we will not concern ourselves



with these exceptional flows but rather confine ourselves to the cases



where the simple Gaussian plume model is clearly appropriate.



     The critical parameters in the Gaussian model are the effective



source height H and the dispersion parameters a  and a .  The effective



stack height is composed of the physical stack height h  plus the plume
                                                       s


rise due to buoyancy and inertia AH.  As stated earlier in the review



the formulae by Briggs (1969, 1975) are pertinent to determine AH.



This study will be concerned with the specification of the dispersion



parameters a  and a  and with related matters affecting dispersion.
            y      z

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                              SECTION 3




               SENSITIVITY OF THE GAUSSIAN PLUME MODEL






     The sensitivity of a model is loosely defined as any measure of




the changes of the dependent variable caused by changes in the




parameters of independent variables of the model.




     It is desirable to have some feel for the sensitivity of the




Gaussian plume model and this can be done in a reasonably satisfactory




manner by graphical means.  Figure 1 shows the idealized distribution




of concentration at ground level from an elevated source.  (This figure




is taken from Pasquill, 1974, Fig. 5-17.)  In drawing the isopleths it




has been assumed that a fa  = a , a constant irrespective of distance,




and that o « x .  The isopleths are labeled according to values of




C(x,y,0)/C .  The downwind distance is in terms of x1 a nondimensional




coordinate which uses units of distance that from the source to the




point of maximum concentration.  Thus, the value 5 corresponds to a




downwind distance of 5 times the distance from the source to the maxi-




mum concentration.  The crosswind distance is in terms of the non-




dimensional coordinate y1 = y/(a H).  A typical value of a  is two.




If the parameter q = 1 and r /H = 10 the relative crosswind and downwind




stretching of the isopleths shewn in the Figure is correct.




     Now by observing Figure 1, one can conclude the following.




     (1) Upwind from the point of maximum concentration the concentration




         values fall off very rapidly with distance being almost two




         orders of magnitude smaller at x' =1/2 than at x1 = 1.
                                      8

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Figure 1.  Idealized distribution of concentration at
ground level from an elevated source.  (From Figure
5-17 Pasquill, 1974)

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     (2) Downwind from the point x  the fall-off of concentration is
                                  m


         much more gradual having decreased cr.e order of magnitude at



         x' = 5.



     (3) The lateral concentration pattern varies rapidly with y*,



         especially so in the vicinity of the point of maximum con-



         centration.



     Now, in order to evaluate the effect of changes in physical parameters



at the sensitive locations (1) and (3) above, it can be seen that changes



in wind direction will have a dramatic effect on the location of the



areas (1) and (3).  Also, given the wind direction, a change in H or a
                                                                      z


will produce fairly dramatic results.



     Pasquill (1974) has analyzed several cases of changes in C caused



by changes in a /E and wind direction at the "most sensitive position"
               Z


i.e. where dC/d(c /H) is maximum.  The amount of the change in C ranges
                 Z


between +75% (for a 20% change in a ) to -91% (for a 10° wind direction



change and a 20% change in H). These results indicate the extreme



variation in concentration one can expect for a given sensor location



downwind and relatively close to a stack.



     As a further step in demonstrating the sensitivity of the Gaussian



plume model, PaF.quill (1974) estimated errors in C  caused by errors in



the dispersion parameters for an elevated source.  He considered the



expression for maximum concentration  (Eqn. 8).  Note that maximum con-



centration depends on source strength Q, wind speed u, the dispersion



parameters c  and o , as well as the distance at which the maximum



occurs x .  Now for simplicity Pasquill assumed that the wind speed
        m




                                     10

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 and  source  strength are  accurately known.   He also assumed that H has 15%



error and that the j_ or a  vs x relationships have errors of 15%.  Then,



 it follows  from  Eqn.  8





          Sn" Ha (x )   *
                  y  m




 Pasquill's  argument is as  follows:



      (1)  A reasonable estimate of the r.tn.s. (root mean square) error



 in H is  about  15% (as given above).



      (2)  a  and H are related  according to Eqn.  (7).   The r.m.s. error
            z


 in a  implied  by errors  in K  is therefore 15%.
     z


      (3)  In addition to (2)  there is  an error in a  based on the fact
                                                    Z


 that we  do  not know the  precise values of  p and q to be used in Eqn.  (7).



 Since most  likely 1< p/q< 2 the  contribution of this error to a  is



 about 8%.   (This is found  by  assuming  the range between v2 and  /3~ is



 spanned  by  four  standard deviations and the distribution of departure



 over this range  is normally distributed.)



      (4)  By assuming no interrelationships in the errors of (2) and



 (3)  above the  net error  in o  can  be obtained by taking the square root
                            Z


 of the squares of 15  and 8 which is 17%.



      (5)  There  is now a need to know  the  error in the distance to the



 maximum  concentration x  since  0  is evaluated at this distance (in
                       m        y

                                    2
 the  expression for C).  If x « a  (x). then the error in x  is found
                    n        m     z   m /                    m


 by doubling the  error in o .  From step (4) this  results in an  error
of  34%  in x  .
           m
                                     11

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                                                          9
     (6)  Since the exact form of the relationship x  <* c ~(x ) is not
                                                    m    z   m
known, Pasquill made the reasonable assumption that this error is
about 15%.
     (7)  The net error in x  is found by taking the square root of  the
sum of the squares of (5) and (6) yielding 45%.
     (8)  Now, assuming that a  « x and given the results of  (7) above
the r.m.s. error in a  at x  implied by (7) is 45%.  Since again the
exact form of the relationship between o  and x is unknown, Pasquill
makes the reasonable estimate of 15% error.
     (9)  The net r.m.s. error in 0  from (8) and (9) above is found
by taking the square root of the sum of squares.  This yields 47%.
    (10)  The total error in the product Ha (x ) is found by taking  the
square root of the sum of the squares of 47 and 15.  This gives a net
                          *
r.m.s. error in C  of 49%.
                 m
     Thus in the case of a power station plume even knowing the wind
speed and source strength exactly, one can do no better than about
50% error for an individual case even if the errors in a , a  , and H
                                                        z   y
are quite modest.
   Moore  (1973) has analyzed data collected for the Tilbury and
Northfleet power stations.  His results show that the uncertainties
are broadley consistent with those of Pasquill demonstrated above.
                                     12

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                             SECTION 4



               REVIEW OF o-SYSTEMS AVAILABLE FOR USE



PREDICTIONS FROM THEORETICAL BASES






     There are three current working theories of atmospheric diffusion;



these are statistical theory, gradient transfer theory, and similarity



theory.  Pasquill (1974) has explored the many facets of these theories



and their underlying bases.  Table 1 of this report summarizes the



important aspects of each theory with regard to the dispersion parameters



a  and a .  For homogeneous turbulence the statistical theory is valid



and a  can be predicted for an elevated or ground source.  The dispersion
     y


parameter c  can be predicted by one or another of the theories but
           z


take special notice that none of the theories listed can be used to



predict o  for intermediate range from an elevated source.



     A valuable practical result comes from the Hay-Pasquill (1959)




version of the statistical theory.  The statistical theory applies as



long as the plume is under the influence of homogeneous turbulence and



would therefore net apply tc calculations of c  except for short range



from an elevated plume,  while c  is small compared with the height of




release.  The Hay-Pasquill result is much easier to apply than



Taylor's (1921) original theory because it requires only running averages



rather than power spectra or correlations.  It does involve a simpli-



fying assumption; the ramifications of this can be found in Pasquill



(1974) and Gifford (1968).  The Hay-Pasquill result states that for



an elevated or ground level source, irrespective of distance and



irrespective of thermal stratification, the value of a  can be predicted
                                     13

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Table 1.  APPLICABILITY OF CURRENT WORKING THEORIES OF DISPERSION
Theory -*
Dispersion
property
predicted
Limitations
Statistical
0 (elevated or
ground source)
c (elevated source
at short range)
Homogeneous
turbulence
Gradient-transfer
0 (ground source)
z
0 (elevated source
2
at long range)
Dispersive mech-
anism must be small
scale eddies
(no meandering)
Similarity
0 (ground
source at
short range)
Surface stress
layer , near
ground level
sources
                                14

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from
where the parameter cr  is the standard deviation of wind direction
                     6


measured in radians.  The first subscript  T  refers to the total



sampling period and the second subscript refers to a running average



of interval x/Bu.  Pasquill (1974) summarizes several investigations



on the parameter 3; the currently suggested value being



          g * 0.44/1



where i is the intensity of turbulence.



The Hay-Pasquill result for 0  (assuming an elevated source) is
                             Z
                    -, x/guj
where a  is the standard deviation of the elevation angle of the wind
       9


measured in radians.  Note that this equation appears to be a reasonable



approximation for an elevated source within the distance downwind



to the point where significant interaction of the plume with



the underlying surface has occurred.  The main practical problem in



using the above equations is that the required information on wind



fluctuations is seldom readily available.



     Although the concept of gradient transfer continues to be challenged



in various ways, there are strong indications that the method is capable



of providing realistic estimates of vertical spread from a surface



release (or more generally even for an elevated release at height H



where a  > H) given flow conditions not departing markedly from



neutral stratification.  Although the approach does not lead to



any simple general result as in the application of statistical theory,




                                     15

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values of the vertical distribution of concentration (hence a ) may
                                                             z


be obtained from solutions (numerical if necessary) of the two-dimensional



diffusion equation with plausible forms of K consistent with experience



on the turbulent structure of the atmospheric boundary layer.  These



methods were used by F.E. Smith (1973) and will be given more attention



in Section 5 of this report.



     The similarity theory is presently limited in its application



to predictions of c^ in regions of the boundary layer where u^ (friction



velocity) and IL, (the heat flux) are effectively constant.  It is also
               c


restricted in that the height of release cannot be mere than a few meters



(see Pasquill, 1974).  Since o  does not scale with Monin-Obukhov length



L (see Appendix 2) there is no way at present to predict the crosswind



dispersion on such a similarity basis (Calder, 1966).



     Other significant results relating to the prediction of a  based
                                                              Z


on the working theories will be found later in this report under



Section 5 entitled Improved Methods for Estimating Dispersion Parameters.



EMPIRICAL SYSTEMS OF SPECIFYING DISPERSION PARAMETERS



     None of the working theories mentioned above is universally



applicable over a wide range in x because of various limitations.



either theoretical or practical.  Because of this several empirical



systems have come Into widespread use.  Gifford C1976)  recently



summarized many of these systems.



     We will briefly cover the systems mentioned by Gifford and add



some recent ones that have been brought to our attention.  The bases of



most of these estimates are measurements of ground level concentration



of some tracer at various distances downwind.




                                      16

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     Often the basis of the cr determination was the application of



some forn of the Gaussian plume model, e.g., Eqr. 11 with regard to



a , together with a statement of the continuity equation.  It is
 z


apparent that any particular observational study cannot be adopted



without question as a valid generalization, since there will come into



play local climatological influences and peculiarities of measurement



systems which will not apply at other locations.



     Some of the systems can be related in a loose sense to the statis-



tical theory, while others are purely empirical.  For convenient practical



application the values of a  or a  are presented in relation to meteoro-



logical data of a routine nature, and one of the crucial requirements is



that of insuring a realistic representation of the effects of thermal



stratification in terms of such data.



The British Meteorological Office 1958 System



     In 1961 Pasquill published a method of determining the dispersion



parameters, on the basis of making simple subjective estimates of the



structure of turbulence of the atmospheric boundary layer from routine



meteorological data, comprised of wind speed, insolation, and cloudiness.



These three meteorological parameters were used to determine a stability



category A through F (A corresponding to very unstable conditions and



F to very stable conditions, D being neutral).  Given the stability



category and downwind distance-, one could determine the corresponding



value of the dispersion parameter.  Because Pasquill wanted to make



the identification with physical spreading easy for engineers and



other non-meteorologists (principally interested in making quick





                                     17

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estimates of the plume spread) he used an angle 6 to measure "total"




crosswind spread and a height h to measure "total" vertical spread.




The relation of these parameters to o  and a  is
                                     y      z




          6 = 4.30 a /x,                                      (14)




and




          h = 2.15 0z                                         (15)





     A more precise way of measuring atmospheric turbulence is through




the Richardson number Ri (which will be defined later in Appendix 2)




and a qualitative correspondence exists between Pasquill categories and




Ri.  The large positive values of Ri correspond to stable conditions




F and large negative values correspond to unstable conditions A, as




pointed out by Islitzer (1965).




     In Pasquill's original system no attempt was made to allow for




any special dependence of a  growth on elevation of the source,  because
                           Z



at that time the state of knowledge was such that there was no basis on




which to make such a differentiation.




     The estimates of vertical dispersion in the Pasquill system are




based on the following:  (See also Table 1, Part II of this report




by Pasquill)




     (1)  Data collected at short range, less than 1 km, during neutral




          conditions and consolidated by Calder's (1949) semi-




          theoretical treatment.  The roughness length implicit in the




          Calder treatment was 3 cm.
                                     18

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     (2)  Data collected at short range during non-neutral conditions,
                                                            *
          obtained from measurements in Project Prarie Grass   (Barad,

          1958).

     (3)  Data collected in the range 1 km to 100 km during unstable

          conditions, and calculations based on vertical gustiness.

          (z   = 30 cm)
            o
     (4)  Data representing stable conditions in the range 1 km to 100 km

          were essentially speculative extrapolations from the more

          reliable data.  (z  = 30 cm)
                            o
     Estimates of the crosswind spread were emphasized by Pasquill to

be valid for release times of only a few minutes.  If suitable wind

direction fluctuation data were available, Pasquill recommended the

values of the crosswind dispersion parameter to be calculated by Eqn.  A.'- •'•

12.  When fine-structure data were not available 6 could be calculated

from the difference of the extreme maximum and minimum of the trace and

setting this equal to 6, provided the distance to the point of interest

is less than 0.1 km on the axis of the plume.  For distances downwind

of the order of 100 km, Pasquill modified this procedure to be the dif-

ference between the maximum and minimum "15 minute averages" of wind

direction.

     For a ground level source the a  values can be considered in-

dependent of sampling time up to periods of 1 hr. or so.  For an

elevated source (of about 100 m) the estimates should be taken to
   Project Prarie Grass was conducted near O'Neill, Nebraska in 1956.
The objective was to determine the rate of diffusion of a tracer gas
as a function of meteorological conditions.  The range of experiments
was from 0 to 800 meters and a total of 70 experiments were performed.
The release time for the tracer gas was 10 minutes.

                                    19

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apply for 10 min. or more of sampling time.

Giffords (1961) recasting of the Pasquill Systerc

     Gifford converted the plume spreading parameters 6 and h of the

Pasquill system to values of a  and a  by means of Eqn. (14)

and (15).  He did this feeling that it was less arbitrary and more

generally understood by people making applications, to use standard

deviations rather than the 10% points of Eqns. 14 and 15.   The data

used were exactly those of the Pasquill system mentioned above.  The

resulting set of curves are generally known in this country as the

Pasquill-Gifford curves (PG curves) (See Fig. (2)).

Turner's (1961, l':->64) adaptation of Pasquill*s Stability Scheme

     Turner adapted the Pasquill stability scheme but added a qualita-

tive specification of insolation in terms of solar altitude rather than

a subjective determination.  By specifically allowing for solar elevation

he provided a basis for applying Pasquill's scheme elsewhere than in the

latitude of Britain.  This objective system of determining the stability

categories also provided a means of using archived meteorological data

to determine what the stability categories had been at a particular

location.  Thus the compute^- could be used with ease to produce a diffusion

climatology for a region for use with the Gaussian model.  The
*
   Three slightly differing versions of the Pasquill-Gifford curves
exist; Pasquill (1961)  (which is the same as the version appearing in
Turner, 1969), Gifford  (1961), and Slade (ed.)  (1968).  This has led
to a small amount of confusion in the published literature, however,
the differences between versions are only in the unstable categories
A, E, and C.  The version under consideration here is identical to that
of Turner (1969).

                                     20

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10000
 IOOO
                                                 10
100
                              PIST^iCE WMliO, r:
Figure 2.   The Pasquill-Gifford curves for a  versus dowrwind distance
           x (From Turner, 1969)
                                    21

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a *s and a 's published in the widely used EPA workbook by Turner (1969)




are exactly those of the so-called PG curves.




Klug*s  (1969) stability scheme



     Klug (1969) developed a highly detailed set of rules to determine




diffusion classes I through V (I corresponding to stable and V to




unstable).  The scheme was quite similar to Pasquill's except that it




was more detailed and could handle more complex situations, e.g. the




transition from night to morning when the boundary layer is rapidly




evolving.  For dispersion parameters Klug used a  and c  measurements




from Project Prarie Grass (Barad, 1958).




The Brookhaven National Laboratory (BNL) System by Singer and Smith




     Singer and Smith (1966) set forth a stability category system and




data relating to dispersion parameters, based on a variety of data taken




over a 15 year period at the BNL site.  These data are noteworthy in




that the plumes were released at 108 m  (quite high compared to previous




studies).  The stability system recommended was based on wind direction




traces.  Singer and Smith gave the stability categories letter designations,




corresponding to a turbulence type indicated on the wind direction trace.




The letters are A, B^, B-, C, and D  (A  representing very unstable and D




representing very stable).  Each of the letter designations corresponds




to a range of wind direction fluctuation in degrees taken over a one




hour period with a Bendix-Friez Aerovane located at source height.




The BNL categories are often referred  to as "gustiness classes".




     Sources of dispersion data used  in the Singer-Smith system were from




tracer  experiments using uranine dye,  oil fog, and Argon 41  ( a radio-




active  isotope),  the most important  single source of data coming from the





                                      22

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oil fog studies.  Concentration measurements were obtained in three ways;




photometric densitometers, fluorescence of the oil droplets, and visual




estimates of the dimensions of the plume.  The radioactive argon emissions




allowed following the dispersing plume up to distances of more than 50 km.




The uranine dye was used at short range, and in contrast with the rest of




the experiments was released at 2 meter source height.  The c  values




were not measured directly but were calculated through measurement of




a , C, and u.
 V



     The sampling times for the dispersion parameters were of the order




of one hour.  The dispersion curves in the ASME (American Society of




Mechanical Engineers) Recommended Guide for the Prediction of the




Dispersion of Airborne Effluents; Smith, 1968) are based on the Singer-




Smith formulation and the recommended dispersion parameters are exactly




the same.




     One point worth noting with regard to the BNL system is that the




a  and a  variations with distance downwind are identical.




The TVA (Tennessee Valley Authority) System




     Carpenter, et al. (1971) summarized dispersion data representing




helicopter sampling of SO,, emitted from stacks of the TVA system.  This




study was the first dealing with dispersion data from large buoyant




plumes.  The stacks operational in the TVA system at the time of the




experiments covered a fairly wide range of height between approximately




75 and 150 meters.




     The flight paths used by the helicopters were of two basic types:




lateral and vertical cross sections.  The lateral cross sections were




made by flying through the plume at about 30 m increments in elevation.




                                     23

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A typical number of passes at a given downwind distance would be six




or so.  This procedure would be repeated at different downwind distances;




typically these distances were 0.8 km, 6 km, and again at between 16




and 32 km.  Vertical traverses were made with the helicopter to determine



the distribution of concentration using the sampler.  Then assuming a




Gaussian distribution in the horizontal and vertical, the curves were




integrated by approximating the area under the curve with a rectangle,




and using the relation





                       Area          (where C    is the maximum
          o  or o   •- 	                 max
           v     z       /
                     C   /2TT          measured concentration for     (16)

                                      the transect.)




     The sampling time involved in the TVA study was basically the time




which it took the helicopter to pass through the plume, i.e. a few




minutes.  Shorter times were required to traverse the plume during




stable conditions (the plume width being smaller).  Temperature profiles




were determined at the beginning of the sampling period by the heli-




copter temperature probe.  The temperature gradient? characterizing



the stability category for the published curves are meant to apply at



plume height.  Wind speeds were measured with the pilot balloon technique.



     These data were collected over a very long time period (5 to 10



years) so that dispersion values represent a large number of individual




cases in the final averages.  The number of individual stacks repre-




sented is of the order of 10 or so.




     Some differences relating to the TVA published a-values are




apparent.  Since the temperature gradient was maasured at plume height




the data show no super-adiabatic temperature gradients, although there






                                     24

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is an extremely high probability that these often existed near the


ground.  The reason is that a large portion of the data were


collected during midday hours and during the summer season.  It can


be concluded that unstable cases exist in "masked" form amongst the


neutral and slightly stable cases.  The tendency for temperature lapse


rate to approach closely the dry adiabatic rate during convective con-


ditions is well known and can be demonstrated from tall tower temper-


ature measurements.


     Another difference of the TVA data is the relatively short sampling


times, especially with respect to o .


The McElroy and Pooler (1968) Dispersion Curves for Urban Areas


     McElroy and Pooler (1968) studied diffusion of tracer clouds


over St. Louis specifically to obtain dispersion data over urban areas.


This series of experiments consisted of releases of flourescent zinc-


cadmimuin sulfide particles.  Twenty six daytime and sixteen evening


experiments were conducted over a two-year period.  The samplers


were located on three circular arcs at distances between 0.8 and 16 km.


from the dissemination point.  Winds were measured by several methods,


including tracking transponder equipped tetroons by radar and using the


single theodolite technique.


     The tracer material was released from near ground level.  Sampling


times were usually one hour.  Determination of a  was by direct means
                                                y

whereas 0  was inferred through the crosswind integrated formula


(Eqn. 11).


     A number of different systems were used to describe the turbulence


representative of the various experiments.   A "modified gustiness class"




                                     25

-------
similar to the Brookhaven gustiness class was used (different ranges of



wind direction fluctuations were adopted since the wind direction sensor



was located nearer the ground than at Brookhaven).  Also used to dif-



ferentiate stability were the Pasquill scheme, CQ, and Ri  (Richardson
                                                u


number) values (See Appendix 2).  The Richardson number was based on



temperature and wind measurements at the 38 and 138 meter levels of a



television tower.



Markee/s Dispersion Curves



     Sigma curves derived by Karkee (Yanskey et.al., 1966) from tracer



experiments at the National Reactor Testing Station (Idaho Falls, Idaho)



are available.  Tne diffusion experiments were carried out by Islitzer



and Dumbauld (1963) using uranine dye as a tracer.  Releases were made



at ground level with sampling arcs at 100, 200, 400, 800, 1600, and



3200 meters.  Sigma y's were determined directly, and sigma z's were



calculated by knowing the maximum of the crosswind distribution for



a ground-level source.



     The roughness appropriate to this site is about 3 cm.  The release



period and sampling time was 30 minutes.  Other data were considered



in Karkee's curves, namely those from Project Green Glow (Fuquay et.al.,



1964).



     The stability system used by Markee is the. same as Pasquill's



although different systems could be incorporated since the original



experiments contained measurements of CQ, Ri, u, and AT/Az.



Bultynck and Malet's  (1972) Stability System and Dispersion Data



     This system was developed for use at a reactor site in Belgium



(the Mol site); it differs from others in that it uses a stability




                                     26

-------
parameter S defined as



               36/3z
           S =
               _  2                                            (17)

               U69
     The symbol u,q stands for the wind speed measured at 69 meters


                                                    2  3
height.  This parameter  (whose dimensions are °C sec /m ) is similar



to a Richardson number.  The measurement height of wind is not critical



so long as it is confined to the height interval between 24 and 120



meters.  Sixty-nine meters is the height of reactor stacks at the



Mol site.




     Turbulence data were used to evaluate the Hay-Pasquill (1959) form


of the Taylor statistical theory (see Eqns. 12 and 13).  The parameter



(3 was determined from the expressions of Wandel and Kofoed-Hansen



(1962)




                                                              (18)





                                                              (19)





The sampling time was one hour in all tests.  Two years of data were



used to form the dispersion curves (Jan. 1966 to July 1968) representing



280 hourly observations.



     Comparisons of these results with the BNL curves showed favorable



agreement.  Dispersion data from experiments at the Mol site also com-



pared very favorably with the predicted values of a .  Apparently no



comparisons of o  were made.  A total of fifteen tracer experiments
                z


were done.
                                     27

-------
TRC (The Research Corporation of New England) Curves for Rough and



Inhomogeneous Terrain



     Bowne (1974) proposed a new set of dispersion curves based on



previously published data including McElroy and Pooler (1968), Hilst



and Bowne (1971), Bowne, Smith and Entrekin (1969), KcMullen and



Perkins (1963), Smith and Wolf (1963), McCready et.al., (1961),



Hamilton (1963), Haugen and Fuquay (1963), Towrin and Shen (1969),



Church et.al.,  (1970), Kangos et.al., (1969), and Taylor (1965),



primarily to solve dispersion problems in urban and suburban areas.



Bowne proposed three sets of curves appropriate for rural, suburban,



and urban terrain.



     The rural dispersion curves are the same as the PG curves with



the following modifications; first the curves were extrapolated back



towards the source from 100 m to 1 m.  There were no measurements of



a  or a  in this range for rural settings.  However, this extrapolation



was thought to be appropriate in order to provide some information,



however crude, in predicting concentrations near highways where con-



centrations in the first 100 m or so are very important.  Second, the



PG a  curves were modified at distances of 3 to 20 km in unstable
    z


conditions to account for the likely occurrence of limited vertical



mixing due to the presence of an elevated inversion layer (finite mixing



depth as mentioned earlier).



     The suburban dispersion curves are based on previously published



studies, including experiments at Dugway Proving Ground.   The roughness



corresponding to sage covered deserts is comparable to "typical



suburban" areas with trees and bushes.  Data were available as close




                                     28

-------
as 50 m  from the source, in contrast to the rural curves.  No o
                                                                y


curves are. published since Bowne felt that there was no significant



difference between rural and suburban areas in this respect.  The a
                                                                   z


values show an extension back towards the source (or x = 0) with all



stability categories approaching the same a  value (5m).  Bowne



anticipated the effect of the mixing height on the plume by again



showing constant, or flattened-off,  dispersion parameters for the un-



stable cases.



     The vertical dispersion rates proposed by Bowne for large cities



are based en data obtained from the St. Louis study  (McElroy and Pooler,



1968), Ft. Wayne (Csanady, et.al., 1967), and Johnstowne studies (Smith,



1967).



Briggs* Interpolation Formulas



     Briggs  (1973) proposed a series of interpolation formulas for the



dispersion parameters, using as a basis previously collected dispersion



data.  His rationale in forming these curves was tc aid in predicting



maximum ground-level concentrations from an elevated source.  He felt



that the PG curves were most accurate at the short ranges and BNL curves



were probably more appropriate at intermediate and longer distances.  At



extremely long distances he utilized TVA data.



     The transition between the PG curves and the BNL curves was made



when the value of a  approached the value of the source height of the BNL



studies, i.e. between 50 and 100 m.  When oz approached 300 m, or so, the



TVA curves were given the most weight.
                                     29

-------
      Briggs also attempted,  as  best  he could  under  the  circumstances,


 to fit the theoretical variation of  o  with distance predicted by the

                                                                    1 /o
 Taylor statistical theory,  i.e. a *  x at short distance and a «  x    at


 long rango.  Eriggs produced a  separate set cf interpolation formulas


 for urban conditions based  on the St. Louis experiment.  Briggs' dispersion


 parameters were presented as convenient numerical formulae as well as  the


 usual dispersion curves.


 Dispersion Data for ry in Terms of Travel Time


      Fuquay, Simpson,  and Hinds (1964) analyzed 46  diffusion experiments


 from a ground level source  at the Eanford Laboratories  near Richland,


 Washington.  They expressed the. crosswind dispersion parameter a  in  terms


 of travel time (x/u),  instead of downwind distance  x as is usually the case.


 Their dispersion curves are parameterized in  terms  of aeu" (which is very


 nearly equal to cr )r  Part II of this report has much additional information


 regarding crosswind dispersion.
     At the conclusion of this section it is reccprized that r.any other


versions of dispersion parameters exist which rely on experimental work


or models that have not been summarized here.  Many of these differ in


a quite insignificant way from the "original" Pasquill-Gifford curves.


Others, while having novel features, are site specific and thus their value


is somewhat limited.  (This is recognized to be true of the Mol data


described earlier.)  According to Gifford (private communication) at


least 30 additional sets of curves exist that would fall into these


categories.  Obviously it would be of little use here to describe all


cf these variations.



                                      30

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     Comparative curves showing the vertical dispersion coefficient




from  a few  empirical methods are presented in Figures 3 through 8.




Since  the PG curves are the most widely used they are the ones against




which  each comparison is made.




     Over rural areas during unstable conditions,  and for distances down-




wind greater than 1 km,  the PG curves are consistently high compared with




all others in the group under consideration.  (At short ranges for unstable




conditions only the Markee curves show o  less than the PG curves.)  For
                                        2



the case of neutral and stable conditions, experimental data for rural sites




tend to lie on both sides of the PG curves,  indicating no systematic




disagreement.




     In comparing the PG curves with others mentioned in this report,one




needs  to be reminded of the fact that there has been no comprehensive




definitive set of data or method amongst those considered, which could be




used as a bench mark.  Also,.it is difficult to assess all the factors




which  could have contributed to differences in data sets that ought to




be similar.




     Because of the earlier discussion on the effect of sampling time on




a  , no comparison of these values will be made but some discussion of the




relation between a  and o  is included in Part II.




     The BNL diffusion data  (Figure 3, representing an elevated release ,




and indirect measurements of a  at generally greater distances thgn were




represented by the basic data of the PG curves) show mostly lower values




of o   than the PG curves, except for neutral stability.  The slope of the




BNL curves (with the exception of very unstable cases) is not drastically




different from the PG curves over the range 0.1 to about 4 km.





                                      31

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     The TVA curves show the greatest difference fron the PG curves.



Recalling that there are probably aar.'  crises ^i  unstable data a^onc the



TVA "neutral" cases, this is not surprising.



     The FcElroy-Pooler data for urban areas (Figure 5) show  greater



values of a  than the PG curves for the range of measurement.  No A



category is listed but the B category is higher for McElroy-Pooler than



the PG B category (not shown in Figure 5).  There is undoubtedly an effect



of the urban heat flux and greater roughness values, which will be discussed



in Part II of this report.  The slope of the curves is again less than the



corresponding PG curves.



     The curves proposed by Markee (Figure 6) are somewhat similar to the



PG curves, but are Lower in 0  values (except for neutral conditions)  at



almost all downwind distances.   The slope  of Markee's  unstable curve is



less than the PG curves.



     Bowne's curves (Figure 7)  weight the PG curves heavily  at short,



intermediate and long ranges, for rural conditions.  The effect of the



"lid" is dominant in the unstable case.



     Briggs interpolation curves (Figure 8) for a  are designed to follow



the PG curves at short distances and therefore, the agreement is good.  It



is obvious from the earlier description of Briggs' system that it will



resemble BNL and TVA curves at longer ranges.



     As noted earlier, an interesting aspect of these sets of data is



that all but one show disagreement with the PG curves for very unstable



conditions A (in that the downwind derivative of da /dx is positive).
                                                   z
                                      32

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This seems to mean that the free convection regime studied by Deardorff




and Willis is somewhat elusive.  Pasquill has made a detailed examination of




the model of Eeardorff and Willis (1974a,b) showing comparisons with other




theories in Fart II of this report.
                                     33

-------
                                                                       100
                               DISTANC
Figure 3.  The dispersion curves  of  Singer and Finith (Smith, 1968)
           (solid lines) and  the  PC,  curves (dashed lines).
                                       34

-------
10000 V~
 1000-
   100
UJ
21
                                                     HETEP£,

                                  S: STAELE (0,64°K/100  KETE'S)
                                                                     100
Figure 4.  The TVA dispersion curves  (Carpenter, et. al., 1971)
           (solid lines) and the PG curves  (dashed lines).
                                      35

-------
 10000IUZZEE
  1000
   iOO
                               DISTANCE DOWNWIND,
Figure 5.  The St.  Louis dispersion curves (McElroy and Fooler, 1968)
           (solid lines) and the PG curves (dashed lines).
                                      36

-------
10000 V^^
 1000
                              DISTANCE DOWNWIND, K«
Figure 6.  Markee's dispersion curves (Yanskey,  et.  al.,  1966)
           (solid lines) and the PG curves (dashed lines).
                                      37

-------
IOOOO
 1000
   100
CO
a:
UJ
                               DISTANCE DOWNWIND, KM
 Figure  7.  TRC rural dispersion curves  (Bowne, 1974) (solid lines)
           and the PG curves  (dashed lines).
                                      38

-------
 100001=
  1000-
   100
                               DISTANCE DOWNWIND,  KM


Figure 8.   Briggs'  rural dispersion curves (Briggs, 1973)  (solid
           lines) and  the PG curves (dashed lines).
                                     39

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                              SECTION  5




        IMPROVED METHODS  OF  ESTIMATING DISPERSION PARAMETERS




      In this section we will examine some new approaches to the method




of estimating dispersion parameters.  Five new approachs are considered




in some detail and a sixth is referenced briefly.   There may bt: other




methods that we are not aware of because of rapid developments in the




field.   Each of these new systems seems to offer something basically




different from what we have considered previously.




Smith's  (1973) Systea  (Published in Pasquill. 1974)




      Smith addressed the problem of predicting concentration patterns




from  a ground level source using the  gradient transfer  theory, incor-




porating numerical solutions  0*. the diffusion equation, and height




dependent diffusivity  (K) values computed from the relation
           K(z) =  r.     *m/15                                (20)





(for a full discussion of this form of K and complete definitions of the




terms see Pasquill, 1975).  (In Eqn. 20, e is the rate of turbulence energy




dissipation and A  is a measure of the predominant eddy size.)  Smith and




S.A. Mathews obtained the numerical solution of the two dimensional dif-




fusion equation
 The height dependent values of e were obtained  from a limited amount




 of  data derived from captive balloon ascents near Cardington, England.




 The X  profiles vere summarized on the basis cf Busch's and Panof sky's




 (1968)  studies of spectral scales.  Several different atmospheric




 stability conditions were represented by the e  and A  observations.

-------
     Briefly, Smith's method involves specifying a geostrophic wind
speed, a roughness value, and (basically) a surface heat flux.  These
determine the friction velocity UA in the surface stress layer.  Then,
using generally accepted wind profile laws appropriate to the surface
layer, an interpolation is made to match the profile in the surface
layer to the geostrophic wind speed.  It should be noted that Smith's
method does not involve any change of wind direction with height.
(The amount of crosswind dispersion which occurs at longer ranges from
the source is influenced by the turning of the wind as has been noted
earlier).
     Having solved the two dimensional diffusion equation, Smith used
the concentration profiles to obtain values of dispersion parameters
directly from the relation

          a  =  (cz2dz/  [Cdz  .
           Z    }     I )
     The practical use of Smith's method is illustrated by Pasquill
(1974).  A stability parameter P 3s specified ir terms of wind speed
at 10 m, and vertical heat flux or incoming solar radiation.  If
desired, a rough approximation to P can also be obtained from insolation
(slight, moderate, and strong) and cloud amount.  If the latter alter-
native is chosen,  Pasquill categories A - F can be related to ranges
of the parameter P.  Once P is obtained the only remaining parameter
to be specified is the roughness length.  Convenient nomograms are
provided to obtain a  versus x from z  and P.
                    z                o
     Smiths a  values, having been computed from the gradient transfer
             Z
theory, are expected to be valid for vertical dispersion from a ground

                                     41

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level source at all distances downwind (over homogeneous flat terrain),



and to be a good approximation for elevated sources at long distances



from the source.  When the vertical spread is large compared with the



height of the source, the concentration approaches the value oi/.e would



obtain from a ground level source.







Lagrangian Similarity Theory



     Pasquill (1975) has proposed an alternative to the usual formulation



of similarity theory, for which the limitation of restriction to the



surface-stress layer is less important.  This proposal is based on



replacing the surface stress and surface heat flux by the standard



deviation of vertical wind speed o  and the scale of turbulence £.
                                  w


Both of the latter are to be considered as local, as opposed to surface



parameters, and both are regarded as functions of height in Pasquill*s



formulation.



     To make practical use of the equations derived in this approach,



it is necessary to assume that £ is proportional to the maximum of the



vertical velocity spectrum X  (a kind of measure of the size of the



predominant eddies).  Observations of X  are available for use in
                                       m


application of the theory.



     Using F.B. Smith's numerical solution of the two dimensional dif-



fusion equation in terms of X  and e, these similarity hypotheses have



been tentatively verified for a range of values of z  and heat flux Ep.



The theory lacks direct observational testing, however.  A useful feature



of the foregoing is that for K profiles differing from those used by



Smith, the vertical spread can be derived from demonstrated similarity

-------
relations, thus avoiding repetition of the numerical solutions of the


diffusion equation.


Moore's Formulation for Power Plant Plumes


     Moore (1972, 1974a,b) has given a semi-emperical formulation for


predicting the maximum ground level concentration C  and the downwind


distance of this maximum.  (Actually a virtual point source is introduced,


which is moved upwind to take account of induced spread, and downwind to


take account of plume trapping in inversions).  One of the essential


features of Moore's formulation is the recognition of the different


functional growths of a  and o  with distance downwind, as follows:



                     az - L1/2x1/2                                  (24)



and


                     cv = Bx                                        (25)
                      j



where L = 2K/u~ and E is a dimensionless constant relating a  to x.


     An additional significant feature of Moore's formulation is that he


tried to account for the induced spread of the plume resulting from


heated stack gases.  Complete details of this rather complex scheme


including the values of L and B for various conditions are contained in


Moore (1974b).


Deardorff's Free Convection Modeling


     Dispersion in the daytime under clear skies and light winds is


frequently dominated by convection.  The condition known as "free"


convection is reached when turbulence is independent of surface drag


force.  The atmosphere above such a convective mixing layer is capped


by a stable layer (of height z.^).  The turbulence and transfer properties


under such conditions have been studied in the laboratory by Deardorff and


                                      43

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Willis (1974a,b).  Preliminary reports of the laboratory modeling



experiments indicate that the transport properties can be uniquely



represented by a universal relation between a /z.  and t  where
                                             23.       *



              t  = w /z.     and                                    (26)






                    gHpZ. 1/3
                   f  f j-r -t

               *    p CpT



 v^  being a characteristic vertical velocity.



     One of the important qualifications of this approach concerns the



threshold wind speed in the atmosphere, below which one could expect the



free convection model to apply.  An example from Pasquill (1975)  estimates



that the wind speed below which it would appear reasonable to adopt the



theory nay be as little as 3 m/sec.  In relation to the conventional



categories such a wind speed condition would include Category A completely,



but categories E and C only for the lower end of the range of wind speeds



associated with those classes.  However, the whole question of applicability



of the Deardorff "convective" model to real atmospheric flow has yet to be



convincingly settled.



Second-Order Closure Modeling - Lewellen (1975)




     Recently the problem of atmospheric dispersion, using the full



fluctuation equations  coupled with a second-order closure assumption,



has been under investigation and is being actively studied by several



groups.  These studies seem to offer considerable promise in improving



the rational basis of  specifying c  in the Gaussian plume model, or of
                                  Z


predicting the concentration distribution.



     Lewellen (1975) has produced several curves of a  versus x based on



the second-order closure approach.  The curves ere stratified on the
                                      44

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basis of Richardson number Ri and Rossby number Ro.  Agreement between



Lewellen's values of 0  and F. B. Smith's values is quite good in neutral




conditions.  As mentioned above, in Lewellen's solution the dispersion




parameters are dependent on Ri and Ro.  These parameters are computed from




observations of wind speed, temperature, and roughness length.  There is




a correspondence between Ri and Smith's P parameter which Lewellen presents




in graphical form.



     The second-order closure approach offers, what seems to be at present,




the only hope of theoretically determining a  for intermediate range for
                                            z



elevated sources, (i.e. a  - H/z), at which range the statistical and




gradient transfer theory approaches are both most questionable.




Briggs' Investigations of Very Stable Flows




     Recently G. A.  Briggs has been working on the problem of dispersion




in very stable conditions.  He has adopted the limiting form of K^ from




the Monin-Obukhov  similarity theory.  Results of his work have not been



published as yet.
                                      45

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                              SECTION 6




         EFFECT OF RELEASE ALTITUDE ON DISPERSION PARAMETERS




HORIZONTAL SPREAD




     Pasquill (1975) has summarized our limited understanding of the




properties of the crosswind component of turbulence.   In neutral flow




the important eddy size range is independent of height.   For  crosswind




spread in neutral conditions, the statistical theory  ought to be appropriate,




except for large distances downwind where, because of the large depth of




the plume, wind direction change with height induces  an enhanced crosswind




spread.  Pasquill estimates this distance to be greater than  5 km for an




elevated source.  For a gound level source the critical distance is




larger, approaching 12 km.




     For non—neutral conditions we do not know enough at present about




the properties of turbulence to be able to predict th° effect of release




altitude.




VERTICAL  SPREAD




     As noted in Table 2 the statistical theory has proven useful in




describing the vertical spread from an elevated source at short and




intermediate distances, essentially upwind from the point where signi-




ficant portions of the plume touch the ground.  Otherwise, we are at




present unable to specify the effect of elevation of  the source in any




definitive way.
                                      46

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Appendix 1:  Symbols

1       time averaged concentration of passive material

C       maximum ground level time averaged concentration of passive
        material

C m     maximum concentration during a transect of the plume

c       specific heat at constant pressure

2       acceleration of gravity

1H      plume rise due to bouyancy and inertia

H       height of the plume centerline above the ground

':.       physical stack height
                                           ^^^^^" •*• / ^
i       intensity of turbulence; e.g. i = (u1     /u) etc.

r"       vertical heat flux

/       scale of turbulence

n       frequency

n^      frequency at which nS(n) is a maximum

/       equals uViL.

r'       source strength

T       total sampling period

'-^      Lagrangian time-scale

T       absolute temperature

P       Smith's stability parameter
                                        SHFzt 1/3
w       Deardorff's scaling velocity = (	—)
t.       equals w.t/z.
 *              *   i
t       time

z.      height of the inversion above the surface

z       roughness length

B       ratio of Lagrangian and Eulerian time-scales

                                   47

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L       equals 2K/u,  a scaling length in Moore's formulation for power

        plant plumes; also Monin-Obukhov length scale


x,y,z   distance downwind, crosswind, and vertically upward, respectively


u       average wind  speed


t       time


x       distance downwind to the point of maximum concentration
 m


x1      equals a  /2/H


y'      equals y/a H


.?       density

 2
a       lateral variance of the concentration of the diffused material



 2
a       vertical variance of the concentration of the diffused material
 z



OD      variance of wind direction



 2
a,      variance of elevation angle of the wind



K       eddy diffusivity


e       rate of dissipation of turbulence energy


Ri      Richardson number


Ro      Rossby number


6       potential temperature or wind direction or angular crosswind of

        a plume


h       vertical dimension of a plume of windborne material, conventionally

        defined by a concentration of material 1/10 of the ground level or

        centerline value


u.,.      friction velocity


a       a constant independent of x

                                   I -'- / ~ z
S       a stability parameter; S = —:—~—

                                     u
                                  48

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                                                                  2
S(n)    spectral density as a function of frequency,  / S(n)dn = a

u',v',  components of velocity downwind, crosswind, and vertical,
w'      respectively, representing fluctuations from the mean, i.e.,
        u = u + u1
                                      49

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Appendix 2;  Relationships Among Turbulence Categorizing Schemes

     For convenience in reading this report we will detail briefly some

of the relationships among turbulence categorizing schemes.  Gif ford's

(1976) study contains a table (Table 4) shoving relations among the

various parameters and methods.  Table 2 in this report is essentially

the same as Gifford's, except for some deletions and additions; e.g.

since Turner's categories are not used in the Workbook but rather Pasquill

categories, those are deleted.

     The first column on the left contains the Pasquill categories.  These

are based on solar insolation, surface wind speed, and cloud cover.  A

refers to the most unstable category and F to the most stable, D being

neutral.

     Category G is sometimes used in the U. S. in cor 1 unction with dis-

persion parameter curves under very stable conditions,  (See the discussion

in Gif ford 5 19761' as to the origins of this category.')

     The column, labeled BNL represents the Singer-Smith or Brookhaven

method of classification.  This method is based on wind-vane direction

traces.  The vane is used to measure variations of wind direction in tbe

horizontal plane.  The standard deviation of this angle, a , can be related

to Pasquill categories.   (See Slade, 1968 for details).

     The Richardson number is defined as
           .
Ri =  T  3z                                             (49)
                      !§.
                       3z
                      (3JK2
                      W
This is a nondimensional parameter expressing thp. ratio of two turbulence

producing mechanisms in  the atmosphere, i.e. buoyancy and mechanical

production.  It is often necessary and desirable to approximate the

                                   50

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Richardson number by measuring temperature and windspeed differences


ever a separation of a few meters in height.  In that case a  simple finite


w — 1 i c 1 d*.«~c. c!p ^ IX XLi-HiciL ILOij j-G
(A8 and Au being the differences in vindspeed and temperature between  the

height difference Az.  Richardson number, at heights other than near the

ground, can be computed using a similarity law established by using experi-

mental data (Businger, et.al., 1971)

     The column label L is the Konin-Obukhov length scale and is defined

pc,                — i '  " C
              T - _1_  P
                " JSB.EF
                   T


This parameter is a scaling length which is used to describe atmospheric

turbulence,  "ote that the value of L corresponds to the distance from

the ground where the mechanical production of turbulence term equals

the buoyancy production term  (Pasquill, 1974).

     By comparison with the U. S. Nuclear Regulatory r-uide 1.23, one can

see that the temperature change with height column has been omitted.

The use of temperature change with height is incorrect since it contains

provision for only one of the turbulence production mechanisms.  However,

as a practical tool it continues to be favored by some groups, because of

a lack of demonstrated inferiority to other systems in dispersion calcu-

lations, and difficulties in measuring many of the other parameters.

For example, Pasquill 's system requires either an observer or instrumentation

recording wind speed, solar insolation, and cloud cover.  The BNL and  a.
                                                                       o

methods require a windvane and ^possibly) electronic averaging circuitry.

Often vane response under low wind speed conditions is very unreliable.

                                  51

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The Monin-Obukhov length L requires sophisticated measurements of




turbulence parameters making its direct measurment impractical.
                                  52

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          RELATIONS AMONG TURBULENCE-TYPING METHODS
              (From Oifford, 1976 Tables 1 and ^
Pasquill (a)
A
B )
c }
D
E
F
BNL
B2
Bl

C

D
ae(c)
25°
20°
15°
10°
5°
2.5°
Ri(at 2m)(d)
-1.0 to -0.7
-0.5 to -0.4
-0.17 to -0.13
0
0.03 to 0.05
0.05 to 0.11
L(e)
-2 to -3
-4 to -5
-12 to -15
00
35 to 75
8 to 35
(a) Pasquill (1961 or 1974)

(b) Philadelphia Electric Co. (1970) (see below)

(c) Slade (1968)

(d) Pasquill and Smith (1971)

(e) Pasquill and Smith (1971)

B~:  range of fluctuations of wind azimuth between 40° and 90°

B,:  range of fluctuations of wind azimuth between 15° and 40°

 C:  fluctuations of wind azimuth exceed 15° but trace is "solid"
     and unbroken

 D:  fluctuations of wind azimuth very small (approximating a line)
     short-term fluctuations not exceeding 15°

(Fluctuations are recorded over a one hour period)
                               53

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                               55

-------
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                                56

-------
Smith, F.B., 1973:  A scheme for estimating the vertical dispersion of a
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Smith, M.E» (ed.)> 1968:  Recommended Guide for the prediction of the
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                                58

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                                57

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                                   TECHNICAL REPORT DATA
                            Please read Inunctions on the reverse before completing)
 1 nE^C^T NO.
 EPA-600 /4-76-030a
                                                           3. RECIPIENT'S ACCESSION NO.
  TITLE AND SUBTLE   ATMOSPHERIC DISPERSION PARAMETERS  IN
  GAUSSIAN PLUME MODELING,,   Part I.  Review of Current
  Systems and Possible Future Developments
             5 REPORT DATE
                   July 1976
             6. PERFORMING ORGANIZATION CODE
7 AUTHORtS)

     A.  H.  Weber
                                                           8 PERFORMING ORGANIZATION REPORT NO.
9. PERFORMING ORGANIZATION NAME AND ADDRESS
     Department of Geosciences
     North Carolina State  University
     Raleigh, North Carolina  27607
             10. PROGRAM ELEMENT NO.
                1AA009
             11. CONTRACT/GRANT NO.
 12. SPONSORING AGENCV NAME AND ADDRESS
     Environmental Sciences  Research Laboratory
     Office of Research  and  Development
     UoS, Environmental  Protection Agency
     Research Triangle Park.  North Carolina  27711
                                                           13. TYPE OF REPORT AND PFRIOD COVERED
                                                           In-house     Sept.  75 - Mar.  76
             14. SPONSORING AGENCY CODE
                EPA-ORD
 15. SUPPLEMENTARY NOTES
 16. ABSTR-ACT
          A recapitulation  of  the Gaussian plume model is presented and Pasquill's
     technique of assessing the sensitivity of this model is given.  A number  of
     methods for determining dispersion parameters in the Gaussian plume model
     are reviewed.  Comparisons are made with the Pasquill-Gifford curves presently
     used in the Turner Workbook,  Improved methods resulting from recent
     investigations are discussed, in an introductory way for Part II of this  report.
                                KEY WORDS AND DOCUMENT ANALYSIS
                  DESCRIPTORS
     Air pollution
    *Atmospheric diffusion
    *Wind (meteorology)
    *Plumes
    ^Mathematical models
                                              b.IDENTIFIERS/OPEN ENDED TERMS
   Gaussian plume
                                                                           COSATi i icid'Group
   13B
   04A
   04B
   2 IB
   12A
   - '-~= e. ^~ ON STATEMENT

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