600382062
AN EXPERIMENTAL STUDY OF TURBULENCE IN AN URBAN ENVIRONMENT
John F. Clarke, Jason K.S. Ching, and James M. Godowitch
Meteorology and Assessment Division
Environmental Sciences Research Laboratory
U.S. Environmental Protection Agency
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
Laboratory, U.S. Environmental Protection Agency, and approved for publication.
Mention of trade names or commercial products does not constitute endorsement
or recommendation for use.
AFFILIATION
Dr. Clarke, Dr. Ching, and Mr. Godowitch are meteorologists in the
Meteorology and Assessment Division, Environmental Sciences Research Labora
tory, U.S. Environmental Protection Agency, Research Triangle Park, North
Carolina. They are on assignment from the National Oceanic and Atmospheric
Administration, U.S. Department of Commerce.
^ ••
n
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ABSTRACT
The structure of turbulence in the urban surface boundary layer is
discussed. Wind and temperature fluctuations were measured with fast-
response sensors at a height of 31 m in four areas of varied land use in
the St. Louis environs (a rural and three urban sites). The second
moments of the fluctuations were computed for one-hour time series and
analyzed within the framework of Monin-Obukhov similarity theory (i.e.,
normalized by appropriate velocity and temperature scales). The results
are discussed relative to observed land-use features and calculated
surface roughness lengths for each of the sites.
Average surface roughness lengths ranged from 0.7 to 1.7 m for the
urban sites, varying by several meters as a function of wind direction
at individual sites. The normalized velocity and temperature variances
for the rural site were consistent with similarity theory. For the
urban sites, plots of the normalized velocity variances showed an
orderly departure from similarity theory for both neutral and unstable
stratifications; they were smaller than the corresponding normalized
variance for the rural site.
The urban anomalies are discussed relative to the terms in the
turbulent kinetic energy budget equation. For neutral stratification,
the normalized velocity variances are up to 15% lower at the urban sites
compared to the rural site. They appear to be inversely proportional to
surface roughness length. The nondimensionallzed dissipation rate of
turbulent energy was less than unity at an urban site with tall rough-
f * "
111
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ness elements, and spectral peak wavelengths were generally larger at
the urban sites. These anomalies to similarity theory are suggested to
be due to the wake region of the roughness elements extending to near
the height of the measurements. For unstable stratification, the
normalized velocity variances and the nondimensionalized energy dissipa-
tion rate for the urban sites are about 50% lower than for the rural
site. This anomaly is suggested to be due to increased importance of
vertical transport processes within the urban area.
Ancillary analyses suggest that the spectral peak wavelength,
not the mixed layer height, is the proper scaling length for free
convection similarity. During the afternoon transition period the
two scales may differ significantly.
fv
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TABLE OF CONTENTS
ABSTRACT 11 ii
LIST OF TABLES vi 1
LIST OF FIGURES .v111
LIST OF SYMBOLS xi 1
ACKNOWLEDGEMENTS . ".. xv
1.0 INTRODUCTION
1
1.1 Probl em Area 1
1.2 Research Objectives 3
1.3 Simi 1 arity Theory 4
1.4 Urban Turbulence Measurements 8
1.5 Scope and Conduct of study 10
2.0 EXPERIMENTAL PROGRAM 12
2.1 Regional Air Pollution Study 12
2.2 St. Louis Environs 12
2.3 Turbulence Program 14
2.3.1 Description of Sites And Measurements 14
2.3.2 Instrumentation 25
a. Gill Anemometer 25
b. Temperature System 26
c. Moisture and Net Radiation 26
2.3.3 Data Processing 27
3.0 RESULTS
28
3.1 Surface Roughness Length .....28
3.1.1 Land Use Evaluation of z0 30
3.1.2 z0 from Similarity Profile Assumption 35
3.1.3 Application 42
3.2 Reynold Stresses and Temperature Variances 43
3.2.1 Velocity Variances ...43
a. Neutral Stratification 43
b. Dependence on z'/L .....47
c. Dependence on z-j/L 64
d. Ratio of Variances 73
e. Intensity of Turbulence 75
3.2.2 Temperature Variance 84
3.2.3 Covariances 86
a. Momentum Flux 87
b. Heat Flux 87
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3.3 Spectral Characteristics 93
3.3.1 Introduction 93
3.3.2 Background 95
3.3.3 Dimensionless Dissipation Rate 96
3.3.4 Vertical Velocity Spectra 101
3.3.5 Horizontal Velocity Components 107
3.3.6 Temperature Spectra 115
4.0 SYNTHESIS OF RESULTS 119
4.1 Introduction 119
4.2 Boundary Layer Phenomena 119
4.3 Similarity Theory 122
4.4 Conclusions 128
REFERENCES 130
APPENDICES 137
A. Gill UVW Anemometer 137
A-l Level ing 137
A-2 Cal ibrat ion 139
A-3 Response Characteristics 141
B. Data Processing 144
B-l In it i al Process ing 144
B-2 Spectral Computations 146
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LIST OF TABLES
Table
Pa^e
1.1. Similarity Prediction for the Relevant Second Moments
and Form of the Asymptotic Limits 7
2.1. Starting and Ending Dates of Instrument Operation and
Total Hours of Data Obtained for Each Site and Experi-
mental Peri od... . 16
3.1. Site Land-Use Characteristics and Estimated Displacement
Lengths and Roughness Lengths Based on the Empirical
Results of Kutzbach (1961) and Counihan (1971), and
Average Calculated Roughness Lengths 34
3.2. Averages and Standard Deviations of the Normalized
Velocity Standard Deviations for Neutral Stratifica-
tion, Number of Observations, and Average Roughness
Lengths for the Summer and Fall Data Sets 45
3.3 The Ratio aw/u* as a Function of z'/L and Correla-
tion Coefficients R*, for Summer and Fall Data Sets 58
3.4 Same as for Table 3.3 but for av/u* 64
3.5 The Ratio au/u* as a Function of Lm(u)/L and
Correlation Coefficients R*, for the Summer Data Set 73
3.6. Averages and Standard Deviations of Intensity of
Turbulence Components for Neutral and Slightly
Unstable Conditions, Number of Observations, and
Average Roughness Lengths for Summer and Fall Data Sets 81
3.7. Average Hourly Values of -u'w1 and -v'w"* X100
for the Summer Data Set ,
A-l. Effect of Instrument Response on Second Moments 143
B-l. Number of Spectra for Each Site, Spectral
Component, and Stabality Class Used in Analyses 149
vii
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LIST OF FIGURES
Figure
2.1. Map of St. Louis Metropolitan area showing the
25-station RAMS network and major population centers 13
2.2. Topographic map of the city of St. Louis 15
2.3. Aerial photograph of site 105 looking southwest 17
2.4. Photograph of tower at site 105 19
2.5. Photograph from tower at site 107 looking north 20
2.6. Aerial photograph of site 109 looking northwest 22
2.7. Photograph from tower at site 111 looking east 23
2.8. Photograph from tower at site 111 looking southwest 24
3.1. The ratios z0/h and d/h as a function of Ar/A
from the empirical data of Kutzbach (1961) and
Counihan (1971) 33
3.2. Schematic of land-use features and average heights
for site 107 33
3.3. Surface roughness length vs. wind direction for
neutral, unstable, and stable conditions for site 107 37
3.4, Surface roughness length vs. wind direction 38
3.5. Surface roughness length averaged over 20 degree wind
direction sectors for the summer data 39
3.6. Same as for Figure 3.5 for fall data set 41
3.7. Site averaged values of cr/u* vs. the corresponding
average Z0 and linear regression fits 46
3.8. Plots for aw/u* vs. Z0 for neutral stratification.... 48
3.9. Individual data plots for aw/u* vs. U for neutral
stratification for the four sites 49
3.10. aw/u* vs. z'/L for summer data set 50
3.11 0W vs. u* for z'/L < 1.0 52
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Figure
3.12. aw vs. u* for z'/L < -1
Page
L—
..54
3.13. aw vs. C(u*3+0.4uf3)l/3 for z'/L
between 0 and -1 ............................................... 56
3.14. Same as for Figure 3.13 but for z'/L < -1 ...................... 57
3.15. aw vs. u* for stable stratification ............................ 59
3.16. Plots of a/u* vs. z'/L for site 105 ........................... 61
3.17. av/u* averaged over intervals of z'/L of 0.25 vs.
z'/L for sites 105 and 109 ......................... . ........... 62
3.18. av vs. Uf for z'/L < -1
63
3.19. Average mixing heights near the urban center and
average surface heat flux for sites 105 and 107 for
26 July through 13 August ...................... . ............... 66
3.20. The ratio ay/u* vs. Zj/L for sites 105, 107,
and 109 for the period 0900 to 1200 h and 1300 to
1600 h ...... . .................................. . ............... 68
3-21 au vs. (gwTTLm(u)/T)1/3 for z'/L < -1 ......................... 70
3.22. au vs. right-hand side of Eq. 3.12 for z'/L
between 0 and -1 ............................................... 71
3.23. Same as for Figure 3.22 but for z'/L < -1 ...................... 72
3.24. Plots of av/au vs. z'/L for the summer data set.. ...... . ....... 74
3.25. Plots of aw/au vs. z'/L for the summer data set ................ 76
3.26. Diurnal variation of vertical and lateral intensities
of turbulence for the summer data set .......................... 77
3.27. Vertical and lateral intensities of turbulence vs.
z ' /L for the summer data set ................................... 79
3.28. Plots of site averaged values of aw/U (a) and av/U
(b) vs. Z'/ZQ for neutral and slightly unstable
stratifications ................................................ 82
3.29. Individual plots of aw/U vs. Z0 for neutral
stratification ................................................. 83
3.30. The ratio aT/T* for the summer data set ........................ 85
x
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Figure
Page
M*u^»
3.31. Plots of u* vs. U for site 107 and estimated linear
fit to plots for all four sites 89
3.32. Plots of U/u* vs. z'/L
90
3.33. Diurnal variation of the average heat flux for sites
105, 107, and 109 and net radiation for site 105 91
3.34. Ratio -tT/y1 vs. z'/L 94
3.35. Turbulent energy dissipation rate e vs. z'/L 97
3.36. The ratio e/gwTVT vs. z(/L ................................... 99
3.37. Estimated fit to plots of the nondimensionalized
energy dissipation rate vs. z'/L for a) site 105 and
the Kansas data and b) for sites 105, 107, and 109 ............ 100
3.38. Vertical velocity spectra for sites and stability
classes indicated ............................................. 102
3.39. Plots of z'/Lm(w) vs. z'/L for summer data set ................ 105
3.40. Estimated fit to plots of the diurnal variation of
Lm(w) for sites 105, 107, and 109 ............................. 106
3.41. Lm(w) vs. wind direction for site 107 for neutral
stratification ................................................ 108
3.42. Lateral velocity spectra for sites and stability
classes indicated ............................................. 109
3.43. Same as for Figure 3.42 for the longitudinal velocity
component [[[ 110
3.44. Estimated fit to plots of diurnal variation of
for sites 105, 107, and 109 ............................. 113
3.45. Same as for Figure 3.44 for the L(u) ......................... 114
3.46. Comparison of z^, Lm, w'T1, and av for the
urban sites and av for the rural site for the
heating period of the day 116
3.47. Temperature spectra for sites and stability classes
indicated 118
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Figure
Page
A-l. Photograph of Gill anemometer showing attached plumb
bob leveling device .................. ... - ..................... 138
A-2. Time constants for temperature system and Gill
anemometer for four wind angles to the propeller .............. 142
B-l. Simplified flow diagram of data processing .................... 145
xl
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ABBREVIATIONS AND SYMBOLS
Abbreviation
A —constant
A —area
—constant
Ar —area of roughness elements
b —constant
B —constant
BR — Bowen ratio
C —regression slope
—constants
CD —specific heat at constant pressure
P
—displacement length
D,D' --regression constants
E»E' —regression constants
f --normalized frequency
fm —normalized frequency associated with peak
in the nS(n) spectrum
Fn —net radiation
F(n) —power spectrum function
g —acceleration due to gravity
f
g/T —buoyancy parameter
h —height of roughness elements
k —von Karman constant
—mixing length
L —Monin-Qbukhov length
""ill
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LE —latent heat flux
Lm --spectral length scale (peak wave length)
l_i —integral length scale
L£ —dissipation length scale
n —cyclic frequency
nm —frequency associated with peak in nS(n) spectrum
N* —dissipation rate for temperature variance
N --number of sampling points
p ' --fluctuating pressure
—turbulent kinetic energy
R —sample rate
R1 —residual term in energy equation
R* —correlation coefficient
R(t) —auto-correlation coefficient
s —silhouette area of roughness elements
S —specific area of roughness elements
S(n) —absolute spectrum function
t —time
T —temperature
T1 --fluctuating temperature
T* —scaling temperature
u',v',w' —fluctuating velocity components
u* --friction velocity
--convection velocity scale
U —mean wind speed
w* —free convection velocity scale
xm
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w*1 --modified free convection velocity scale
z —height above the surface
z1 --height minus displacement length (z-d)
z-j —planetary boundary layer height
z0 --roughness length
Z0 —effective roughness length
Symbol
—spectral constant
B —spectral constant
e --dissipation rate of turbulent energy
tc —wave number
v —count
P —density of air
a --standard deviation
T —time constant
—surface shear stress
*e —dimension dissipation rate of turbulence
—dimensionless dissipation rate for temperature
variance
—dimensionless wind shear
—diabatic correction to logarithmic wind profile
xiv
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ACKNOWLEDGEMENTS
The authors would like to express their appreciation to Dr. F. S.
Binkowski for his frequent and very helpful advice and Dr. S. P. S. Arya
for his critical review of the manuscript.
Our graditude also extends to J. W. Ashley, A. Busse and D. H.
Coventry for their assistance in the computer aspects of the research,
and to C. Rodriques and B. Poole for help in preparation of the manuscript
xv
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1.0 INTRODUCTION
1.1 PROBLEM AREA
The study of turbulence or eddy motion as a transport process for
momentum, heat, and water vapor is basic to understanding and predict-
ing of the structure of the atmospheric boundary layer. Under station-
ary, homogeneous flow conditions the spectra and cross-spectra of the
turbulent components of the wind, temperature, and moisture provide
a complete description of the turbulence structure at the height of
measurement, and are usually analyzed in terms of three features:
a) the shape of the spectrum or cospectrum as a measure of the
distribution of the variance or covariance as a function of
wavelength or frequency;
b) the integral scale, which is usually assumed proportional to
the location of the peak in the spectrum; and
c) the total variance or covariance.
These features, particularly the latter two, are directly relevant to
diffusion of pollutants in the atmosphere.
Obtaining turbulent fluctuation data and subsequent computing of
spectra and cospectra are difficult and costly. The study of turbu-
lence, therefore, is often a process of defining physical concepts that
are based largely on the interpretation of empirical data, and organiz-
ing these results into a theoretical or empirical framework such that
the turbulent structure can be deduced from the relevant driving parame-
ters. This process is usually accomplished within the framework of sim-
ilarity theory, where turbulence is assumed dynamically similar over any
surface when properly scaled.
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For flow in the surface boundary layer above low and homogeneous
roughness features, Monin and Obukhov (1954) similarity theory pro-
vides a reasonable description and parameterization of mean flow and
turbulent processes (see Ariel1 and Nadezhina, 1976; Monin and Yaglom,
1965), By contrast, the turbulent structure of boundary layer flows
over large and irregular roughness features is poorly understood.
Monin-Obukhov similarity theory does not contain a length scale charac-
teristic of the roughness elements, and is thus valid only away from the
direct influence of the roughness features (Tennekes, 1973). Wind
tunnel studies have provided valuable insight on the nature of ideal
flows over rough surfaces (see Counihan, 1971; Raupach et al., 1980),
and the effect of change of roughness on mean profiles and turbulent
characteristics of the flow field. Numerical models of change-of-rough-
ness flows (e.g., see Peterson, 1969) are helpful to understanding such
flows. However, like wind tunnel studies, the models have been applied
mostly for neutral conditions, and the necessary closure assumptions are
inadequately tested with atmospheric data. Meteorological towers
located in nonhomogeneous roughness fields provide a source of atmos-
pheric data; with few exceptions these have been limited to wind and
temperature profiles and lacked measurement of turbulence parameters.
Relatively little turbulence data have been obtained in the surface
boundary layer over the very complex roughness features of an urban
area. Such data are important for modeling the mean flow and dispersion
processes in these environs.
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1.2 RESEARCH OBJECTIVES
The research reported here is concerned with the structure of tur-
bulence in the surface boundary layer over a city. It is based on ex-
*
tensive observations of the turbulent wind and temperature above four
varied land-use areas in the St. Louis, Missouri, environs. The purpose
of the study is to suggest a framework for parameterizing urban turbu-
lence statistics.
The research approach was to seek relations between turbulence pa-
rameters based on the interpretation of empirical data. The form of se-
lected nondimensionalized urban turbulence statistics as a function of
atmospheric stratification is tested against the form predicted by Monin
and Obukhov (1954) similarity theory. In this respect, the empirical
specification of similarity relationships resulting from the Kansas
(e.g., see Wyngaard et a!., 1971) and Minnesota (Izumi and Caughey,
1976) boundary layer experiments are used as a standard for comparing
the urban results.
The Monin-Obukhov similarity relationships cannot be expected to
hold for urban areas a priori due to the large and nonhomogeneous sur-
face features. The similarity relationships may also be invalid in the
turbulent wake region of the roughness elements; at the urban sites in
the present study the wake region may extend to the height of measure-
ment (31 m). Thus the specific objectives of this study are:
1) to determine how extensively the similarity relationships, as
verified empirically for ideal rural sites, apply to urban
data; and
2) to discuss significant and orderly differences between the
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urban results and the similarity predictions in terms of site
land-use, i.e., surface scaling, features.
1.3 SIMILARITY THEORY
Within the constant stress layer, Monin and Obukhov (1954) similar-
ity theory is a useful tool for making predictions about certain statis-
tics of atmospheric turbulence. According to similarity theory the mean
velocity gradients and turbulence characteristics are completely deter-
mined by the height z, the surface momentum flux TO/P, the kinematic
heat flux H/pCp, and the buoyancy parameter g/T. From these parame-
ters velocity, temperature, and length scales can be defined as:
u* = -
T* = -w'T'/u*
L = -Tu*3/gb7TT
(1.1)
It follows that any other parameter describing the structure of ideal
flow in the surface boundary layer, nondimensionalized by the above
scaling parameters, should be a universal function of the only other
dimensionless quantity that can be formed, i.e., z/L. Such parameters
include the velocity and temperature gradients, the second moments of
the fluctuations of the velocity components and temperature, spectra and
cospectra, and other higher-order quantities. The meaning of L is that
when z is small compared to the magnitude of L, mechanical turbulence
predominates. For z > |L|, buoyancy effects become important. Thus z/L
indicates the relative importance of mechanical and buoyancy effects
similar to the Richardson number (Ri) and can indeed be specified in
terms of Ri (Monin and Yaglom, 1965; Binkowski, 1974; Businger et a!.,
1971).
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The velocity gradient when nondimensionalized by kz/u* is a univer-
sal function of z/L:
(1-2)
W
ith the right-hand side of the above expression equal to unity, Eq.
1.2 is the familiar differential equation of the logarithmic wind pro-
file that has been established and verified for neutral stratification
by laboratory and field experiments. Thus as z/L tends to zero, i.e.,
neutral stratification, m(z/L) + 1. In the asymptotic limits as z/L
-»• -*« and z/L * +», similarity theory predicts 4>m(z/L) to be propor-
tional to (-z/L)"l/3 ancj a linear function, respectively (Monin and
Obukhov 1954). Businger et al. (1971) presented semi-empirical rela-
•
tionships for m(z/L) that in the unstable limit, indicate m(z/L) *
While this form is in conflict with the free convective
similarity prediction, it is the current basis for estimating the wind
profile for homogeneous, stationary conditions. Paulson (1970),
Nickerson and Smiley (1975), and Benoit (1977) have integrated Eq. 1.2
(using the Businger et al, form) to obtain mathematical expressions for
wind speed in the diabatic surface layer. The Nickerson-Smiley integra-
tion was used in this study.
Similarity theory also predicts that the second moments of the
fluctuations of velocity and temperature, when appropriately normalized
by u* and T*, are a function of z/L. Those of interest here are -u'w' =
u*2 and w'T' = H/pCn, which have constant values of unity when
normalized; the horizontal heat flux u'T1; and the variance of the
fluctuations, which are usually considered in terms of the standard
deviations au, av, aw, and cry.
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The form of the similarity predictions in the asymptotic limits of
z/l + -», 0, and +°° can be obtained through simple dimensional analysis
of the relevant variables. Consider for example:
, w'T1, z).
(1-3)
In the limit as w'T' + 0 (neutral stratification) the buoyancy param-
eter must disappear and it can be easily shown that aw/u* should be
constant. Similarly, in the limit of free convection the dependence on
u* must vanish and from dimensional considerations alone it can be shown
that aw/u* should be proportional to (-z/L)l/3. For extremely stable
stratification the turbulent eddies are very small such that little
exchange occurs between different levels. Thus aw, away from the
immediate influence of the surface, is independent of z and from
dimensional considerations is proportional to u*. The similarity
predictions for the relevant second moments are discussed in detail by
Monin and Yaglom (1965) and are summarized in Table 1.1.
The empirical verification of the similarity predictions are as yet
inconclusive, as well as determination of the functional form of the
relations at intervening values of z/l. Excellent summaries of recent
progress in empirical determination of the functional forms are provided
by Ariel' and Nadezhina (1976) and Monin and Yaglom (1965). The form of
the relationship is agreed upon in general: however, scatter of the data
points among the many investigators is large. This scatter is partially
due to nonstandard techniques for measuring the stress and heat flux,
vertical variation of these parameters in the boundary layer, variations
of sampling times, and possibly to the influence of mesoscale circula-
tion features.
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Table 1.1. Similarity Predictions for the Relevant Second Moments and Form of
the Asymptotic Limits.
GENERAL MOIMIN - OBUKHOV
SIMILARITY PREDICTION
FORM AS z/L GOES TO
-oc
00
w
uT
u»2f-| (z/L)
u»2f2(z/D
u»2f3(z/L)
T*2f4(z/L)
-u*2 f5 (z/L)
-u* T* f6 (z/L)
u« T* f7 (z/L)
u*2 (-z/L)2/3
A2 u»2 (-z/L)2/3
AS u*2 (-z/L)2/3
-U*
•u* T*
A-i'u
AS' u*2
A4'T*2
•u* T*
Ay' u* T*
A2"u*2
A3" u*2
A4"u*2
-u*2
-u* T*
A/' u* T*
7
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Similarity theory can be extended to specify relationship between
parameters (e.g., av/au, aw/au), correlation coefficients, turbulence
intensities, third moments, spectral 'forms, etc. Thus, to determine any
of the normalized parameters in the surface boundary layer, it is suf-
ficient to specify u* and w'T', and to know the functional form of the
similarity relationship. These relationships are often the basis for
evaluating the importance of the various terms in the set of equations
of the turbulent covariances (see Wyngaard and Cot&, 1971 and Wyngaard
et al., 1971). These equations, when coupled with the equations for
mean quantities and for the conservation of species, form a consistent
set for the prediction of turbulence and the dispersal of pollutants in
the atmospheric surface layer (see Donaldson, 1973).
1.4 URBAN TURBULENCE MEASUREMENTS
Relatively few studies of turbulence within urban environments have
been conducted, and these studies for the most part lack temporal and
spatial resolution as well as a complete set of parameters for the eval-
uation of the similarity functions. Graham (1968) reported on the spa-
tial variation of the u and v components of turbulent intensity in the
urban and rural environs of Fort Wayne, Indiana. The data represented
twenty sampling periods and were obtained during nocturnal hours and
generally with northwest winds. The standard deviations of the velocity
fluctuations were roughly the same for both urban and rural sites. How-
ever, wind speeds at the city stations were only about half those at
the rural site, such that turbulence intensities were about twice as
great for the city sites.
8
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Bowne and Ball (1970) made a more extensive analysis of the same
data set, concentrating on data from an urban and a rural tower equipped
to measure the stress,' mean wind, turbulent components of the wind, and
Richardson number. Ratios °u/u*> av/u*> ancl aw/u* dt the rural site
were comparable to those measured in other homogeneous terrain studies
and were slightly lower than those for the urban site. Spectral analy-
sis of the wind fluctuations indicated a shift of energy to higher fre-
quencies at the urban site. Urban studies by Brook (1972) and Peschier
(1972) showed turbulence intensity to be primarily dependent on rough-
ness features, which varied as a function of wind direction. Jones et
al. (1971), reporting on tethered balloon measurements to about 300 m
above two urban sites in Liverpool, England, found the neutral wind
velocity power law exponent to be about 0.21.
Yokoyama (1971) made an extensive study of turbulence parameters
at 45, 180, and 313 m on a tower in a suburban area north of Tokyo; how-
ever, the tower was surrounded by open areas to a minimum distance of
1 km. His results indicate the similarity relationship for aw/u*
(where u* was determined locally) for neutral stratification holds to
313 m, with the proportionality constant being about 1.16. Wamser and
Miiller (1977), reporting on vertical spectral scale lengths at a sub-
urban tower near Hamburg, Germany, found that roughness lengths in the
vicinity of the tower were a function of wind direction. Their results
indicate a decrease of the vertical spectral length scale with increas-
ing roughness, and a dependence of the spectral length scale on z/L
similar to that for sites with small homogeneous roughness features.
Jackson (1978) discussed turbulence parameters and wind and temperature
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profiles from a 70 m tower at an urban site in Wellington, New Zealand;
he found the power law velocity profile exponent to be about 0.5 for
neutral stratification. Jackson's analyses of turbulence intensities,
local friction velocity, and spectral scale lengths suggest a universal
structure of turbulence over the exceptionally rough terrain.
Ramsdell (1975) reported on power spectral shapes, variances, and
length scales from three-dimensional wind velocity measurements at an
urban site, an open suburban area, and an airport in Seattle, Washing-
ton. Ramsdell concluded that the urban turbulence structure does not
differ significantly from that observed over flat terrain. His analy-
ses, however, were based on only 25 hours of data, and he lacked a
direct measurement of stratification.
The papers noted above represent the bulk of turbulence measure-
ments over urban areas in the past 10 years. Although hardly defini-
tive, many of the results are qualitatively consistent with similarity
theory to the extent that some urban turbulence statistics can be de-
scribed as a function of z/L and the intensity of turbulence is a func-
tion of surface roughness. Similarity theory, however, does not predict
a dependence of the vertical scale length on surface roughness.
1.5 SCOPE AND CONDUCT OF STUDY
The analyses in this study are based on high resolution measure-
ments of the three components of the wind and temperature, at 31 m
above four land-use areas in the St, Louis environs. Profile data were
not obtained and thus the study is limited to turbulent quantities. In
other respects the data are extensive, covering a total of nine weeks
during two seasonal periods; about 3800 hours of data were obtained and
10
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processed. With few exceptions (to be discussed later) all the data
were used in the analyses, i.e., the data were not screened to eliminate
nonstationary periods or nonhomogeneous flow situations.
A primary objective of the study is to relate the turbulent struc-
ture to land-use features. Site land-use characteristics are described
and surface roughness lengths are determined for each of the sites at
the outset. The relevant wind and temperature variances and covariances
are presented for the four sites and analyzed within the framework of
Monin-Obukhov similarity theory. The results are discussed with respect
to the surface roughness lengths determined for the individual sites,
and are compared to similar observations and analyses presented by
other investigators as representative of turbulent flow above small and
homogeneous roughness elements. The spectral characteristics of the
turbulent fluctuations (spectral shapes, peak wavelengths, and energy
dissipation rates) are determined and compared with similar analyses
from the Kansas and Minnesota boundary layer studies. Emphasis is
placed on the vertical velocity component. Finally, the results of
the analyses are summarized and discussed with respect to the turbulent
energy budget.
11
-------
2.0 EXPERIMENTAL PROGRAM
2.1 REGIONAL AIR POLLUTION STUDY
The data for this study were obtained during intensive field pro-
grams within the framework of the Regional Air Pollution Study (RAPS).
The RAPS program, conducted in St. Louis, Missouri, from 1973 through
early 1977, had as its prime purpose the collection of a complete and
accurate data base for use in development, modification, and evaluation
of urban air quality simulation models (Schiermeier, 1978). The basic
component of RAPS, the Regional Air Monitoring System (RAMS), consisted
of a 25-station aerometric network (Figure 2.1). In addition to exten-
sive air quality measurements, all stations were equipped to measure
wind direction and wind speed. Temperature difference between 5 and 30
meters, atmospheric pressure, direct and diffuse solar radiation, and
infrared sky radiation were measured at selected stations. Special
intensive field studies lasting about six weeks were conducted within
RAPS about twice yearly. During these periods, research support was
provided to principal investigators on special projects related to the
overall purpose of RAPS. One such project was the study of the turbu-
lent structure of the urban surface boundary layer reported herein.
2.2 ST. LOUIS ENVIRONS
The city of St. Louis, Missouri, lies on the west bank of the
Mississippi River about 16 km south of the conjunction of the Missis-
sippi and Missouri Rivers (see Figure 2.1). The city proper has a popu-
lation of about 622,000 in an area of 158 km^, and extends along the
Mississippi River for 31 km and westward approximately 11 km. The
12
-------
• REGIONAL AIR MONITORING STATIONS
05 10 15 20
SCALE IN KILOMETERS
ALTON
JV\^
GRANITE CITY
ST LOUIS
4 - • i i
"'"' • *ii2
109
EAST
.LOUIS
117
BALLWIN
123
BELLEVILLE
124
Figure 2.1. Map of St Louis metropolitan area showing the 25-station RAMS network
and major population centers (shaded).
13
-------
metropolitan area, which includes communities on both sides of the Mis-
sissippi, has a population of 2.4 million in an area of 7,203 Km^. The
major population areas are shaded in Figure 2.1.
The elevation of the terrain in the St. Louis area varies from 122
m MSL (elevation of the river) to about 184 m. The west bank of the
Mississippi rises abruptly to the city. A flat area, known as the
American Bottoms, with an average elevation of about 128 m, extends to
about 15 km east of the river. Topographic features for the city of St
Louis are shown in Figure 2.2.
2.3 TURBULENCE PROGRAM
During the RAPS 1976 summer intensive field program, special in-
strumentation was mounted atop 30-m RAMS towers at sites 105, 107, 109,
and 111 (see Figure 2.1) to measure the three-component turbulent wind
and temperature fluctuations. In addition, humidity fluctuations and
net radiation were measured at sites 105 and 109. Starting and ending
dates of instrument operation and total hours of data obtained are given
in Table 2.1.
2.3.1 Description of Sites and Measurements
Site 105 was located in a high density urban commercial area 3 km
south of the urban center and 1 km west of the Mississippi River. Land
in the vicinity of the station was used for trucking, warehousing, and
commercial operations. Buildings, predominately two-story and of large
aerial extent, contributed about 25% of the land-use features. About
60% of the area was paved; the remainder was primarily lawn with a few
small trees along the streets. Figure 2.3 is an aerial view from the
14
-------
DOWNTOWN^
ST. LOUIS
ALTITUDE (MSL), meters
>178
Km
Figure 2.2. Topographic map of the city of St. Louis
15
-------
Table 2.1.
Starting
Hours of
Period.
and Ending Dates of Instrument Operation and Total
Data Obtained for Each Site and Experimental
SUMMER
SITE
105
107
109
111
INSTRUMENT
Gill UVW
Temperature
Hum i d i ty
Net Radiation
Gill UVW
Temperature
Gill UVW
Temperature
Humidity
Net Radiation
Gill UVW
Temperature
DATES
July 21 to Aug 31
July 28 to Aug 31
July 28 to Aug 7
July 18 to Aug 31
July 28 to Aug 31
July 26 to Aug 31
July 20 to Aug 31
July 20 to Aug 30
Aug 9 to Aug 13
July 15 to Aug 28
July 21 to Aug 13
Aug 1 to Aug 13
HOURS
855
673
63
1030
603
593
695
691
26
1038
321
208
FALL
105
107
109
Gill UVW
Temperature
Gill UVW
Temperature
Gill UVW
Temperature
Oct 26 to Nov 20
Oct 26 to Nov 20
Oct 26 to Nov 20
Oct 26 to nov 20
Oct 26 to Nov 20
Oct 26 to Nov 20
449
447
419
386
459
453
16
-------
Figure 2.3. Aerial photograph of site 105 looking southwest.
17
-------
northeast showing the surface features in the vicinity of the tower.
The elevation of the site was about 131 m. The terrain is relatively
flat to the northeast through south, but rises abruptly to the southwest
through north. Topographic features for sites 105, 107, and 111 are
shown in Figure 2.2. Special instrumentation at site 105 included a
Gill UVW anemometer, a fast response temperature system of in-house
design, and a Lyman-alpha humidiometer. All instruments were located at
the top of the tower. Figure 2.4 is a photograph of the tower at site
105 with the Gill anemometer mounted at the top. The temperature sensor
was located on the w-propeller arm, which was approximately 31 m above
the surface. Values from these systems were recorded every 1/2 second
in the RAMS data acquisition system. A Swissteco net radiometer was
extended 2 m from the tower about 29 m above the surface. Output from
this system was recorded on strip charts.
Site 107 was located in the northwest section of St. Louis about 6
km from the center of the city. Land use for several kilometers sur-
rounding the site consisted mostly of older single family and duplex
two-story dwellings. Population density is high and the area is consid-
ered urban in nature. However, in contrast to site 105, about 60% of
the land area is covered by trees or grass. Twenty-five percent of the
land is used for buildings; streets and other paved surfaces make up the
remaining 15%. Figure 2.5 is a photograph taken from the tower looking
north. The site was located on a small north-south ridge at an eleva-
tion of 158 m (Figure 2.2). Special instrumentation at the station
consisted of a Gill anemometer and a fast-response temperature system
exposed as described for site 105.
18
-------
Figure 2.4. Photograph of tower at site 105.
19
-------
**"- *
O
O)
C
o
O
0)
CO
0)
o
4-*
o
Q.
CO
O
4-»
O
J=
a.
0)
.£?
iZ
20
-------
Site 109 was located in a rural agricultural area about 10 km east
of the city. Farm land generally surrounded the station; however, a
group of farm buildings was located in the immediate northeast quadrant,
and small trees and underbrush in the immediate southeast quadrant.
Small fields separated by hedgerows and scattered homes characterized
the land use at greater distances in the easterly quadrants. Figure 2.6
is an aerial view of the station looking northwest, showing these
features and the relatively flat land of the American Bottoms (elevation
128 m MSL). A bluff rising 67 m lies approximately 5 km to the east.
Special instrumentation and exposure were the same as for station 105.
Only one Lyman-alpha humidiometer was available to the program. This
instrument was moved from site 105 to site 109 on 9 August and removed
13 August.
Site 111 was located in an older residential community approximate-
ly 9 km southwest of the urban center. The area immediately surrounding
the site was utilized primarily for high-density single family resi-
dences. A lumber yard was located about 65 m east of the station and a
small sand and gravel plant 0.5 km to the north. Figure 2.7 is a view
from the tower looking east over the lumber yard and some of the larger
roughness features. Figure 2.8 is a view looking southwest over the
residential community. Buildings at an average height of 7.5 m covered
about 15% of the area and trees averaging about 13.5 m made up about 25%
of the land use. The site was located in a small valley at an elevation
of 134 m (Figure 2.2) with terrain rising 30 m within 3 km to the north-
west through the northeast. Special instrumentation at the site con-
sisted of a Gill UVW anemometer and a fast response temperature sensor.
21
-------
VI
1
o
O)
Ci
o
v»
w
O
a
2
O)
o
a
CO
csi
.s*
m
22
-------
CO
0)
CD
c
o
o
03
CO
O
o
a
CO
O)
o
o
a.
0)
.S*
iZ
23
-------
0)
O
V)
O)
o
o
0)
(A
-»
CO
o
o
a
CO
O)
o
o
OL
CO
CM"
0)
3
O>
LZ
24
-------
An abbreviated turbulence measurement program was conducted during
the RAPS 1976 fall intensive field study. The three-component turbulent
wind and temperature fluctuations were obtained at sites 105, 107, and
109, as described for the summer program, from 26 October to 20 Novem-
ber. Operational statistics for this data set are included in Table
2.1.
2.3.2 Instrumentation
a. Gill Anemometer--
The Gill UVW anemometer is a three-component wind instrument de-
signed for direct measurements of three orthogonal components of the
wind. It employs foamed polystyrene propellers molded in the form of a
true generated helicoid. The propellers drive miniature d.c. tachometer
generators which provide an analog voltage output directly proportional
to the rotation of the propellers. Details of the instrument are given
by Gill (1975). The instruments, using maximum-response, 23-cm propel-
lers (catalog numbers 21281 and 21180/27105), had an output of about
2400 mv at 1800 rpm (approximately 9.15 m/sec). The signal, when pro-
cessed through RAMS data acquisition system, had an output of about
115 mv/m/sec for the horizontal components and 145 mv/m/sec for the
vertical component. The RAMS system could detect a change of 2.4 mv
providing a resolution of 0.021 m/sec and 0.017 m/sec for the vertical
and horizontal components, respectively. Each component of each instru-
ment was calibrated in the EPA wind tunnel and in the field, and indi-
vidual calibrations varied slightly from the above representative
values. The Gill UVW anemometer has been used extensively in micro-
meteorological studies by other investigators and thus considerable
25
-------
information on its limitations and operation for optimum response is
available (e.g., see Hicks, 1972; Horst, 1973; Fichtl and Kumar, 1974;
Wesely and Hicks, 1975). The response characteristics, calibration, and
operation of the instruments used in this study are discussed briefly in
Appendix A.
m
b. Temperature System—
The fast response temperature systems used in the field study con-
sisted of a bead thermistor in a Wheatstone bridge circuit. The ther-
istor, a Fenwall type 6112, had resistances of approximately 23,000,
8,000, and 3,000 ohms at 0, 25, and 50°C, respectively. The voltage
output of the bridge circuit was linear from 0 to 5 volts over two pre-
selected ranges of 0 to 25°C and 15 to 40°C. (The latter range was
used in the summer field study and the former in the fall study.) Thus
the system had an output of 200 mv/°C and, when applied through the
RAMS data acquistion system, a resolution of 0.012°C.
The time constant of the temperature system, determined from wind
tunnel tests, was a function of wind speed as shown in Figure A-2. For
speeds between 2 and 5 m/sec, the time constant averaged 0.63 sec, and
thus was reasonably matched to the w sensor (see Appendix A). The bead
was extremely small, and atmospheric tests indicated that it could be
freely exposed without a radiation shield for measurement of temperature
fluctuations. The bead thermistor was exposed on the w-arm of the Gill
anemometer about 20 cm from the propeller.
c. Moisture and Net Radiation--
The Lyman-alpha humidiometer (see Buck, 1976) and Swissteco net
26
-------
radiometer measurements were used in an ancillary capacity in this
study. The manufacturers1 recommendations for calibration and exposure
were followed and both instruments were cleaned about every other day of
operation. The calibration of the Lyman-alpha changed significantly
during the course of the summer field study. The change in calibration,
however, was determined to be nearly linear and appropriate adjustments
were made to the data.
2.3.3 Data Processing
The data acquired during the field studies were computer processed
and analysed. Data processing was carried out in three phases. The
first phase involved processing the data to obtain the basic turbulence
parameters (these procedures are outlined in Appendix B). The second
phase was essentially one of printing, plotting, and performing sta-
tistical analyses to test theoretical forms and determine empirical
relationships between the basic turbulence parameters. The programming
and procedures were standard and will not be discussed further. In
phase 3, the data were subjected to time series analyses; spectra and
turbulent length scales were determined for each of the hourly data
series. Computational techniques are outlined in Section 3.3 and
Appendix B.
27
-------
3.0 RESULTS
3.1 SURFACE ROUGHNESS LENGTH
Integration of Eq. 1.2 for neutral stratification gives the familar
logarithmic wind profile equation. For flow above tall roughness ele-
ments, relative to the height of wind measurement, a zero plane dis-
placement length d, is customarily introduced, giving the wind profile
for neutral stratification by the empirical modification:
7 ' (I 1 \
U = " in ±__ v°*-1 /
K z0
where z' = z-d. Thus the wind profile and the stress depend on two
physical parameters, the roughness length z0, and the displacement
length d, which need be specified for each of the sites at the outset.
The roughness length z0, a physical parameter dependent only on
the characteristics of the surface, enters (along with d) as a boundary
condition into the logarithmic velocity profile equation. The roughness
length is usually determined from measurements of the wind profile
through the neutral surface boundary layer as the height at which
extrapolation of the mean profile towards the surface produces a zero
velocity. The profile method, although sensitive to errors in the wind
measurements, has been used extensively for obtaining z0. Counihan's
(1975) summary of the results indicate z0 to vary from 0.0001 cm for
calm seas to as great as 7.5 m for urban areas. Oke's (1974) summary of
z0 values for suburban-urban areas ranged from 0.4 m to 4.5 m. Several
empirical techniques have been suggested for estimating z0 from the
density and/or height of roughness elements, e.g., see Counihan (1971),
Brutsaert (1975), Lettau (1969).
28
-------
Displacement length is poorly defined and parameterized by compari-
son with z0. This problem is undoubtedly due to the insignificance of
d in Eq. 3.1 above surfaces of relatively small roughness, as was the
case in most field programs designed to define the nature of the atmos-
pheric boundary layer. Physically, d is the displacement of the height
of the surface seen by the air flow and resulting turbulent exchange
processes, which is due to the presence of the roughness elements.
Empirically, d is the vertical distance that the profile must be shifted
such that a straight line can be drawn through a logarithmic plot of the
neutral velocity profile. Hanna (1969) suggested a displacement length
between 6 and 9 m for the urban environment; it should therefore not be
ignored in describing the structure of the surface boundary layer over a
city. Attempts to assign a d value to urban areas using profile data
have generally presented considerable problems (see Pasquill, 1970),
although efforts to obtain d above dense vegetation have proven more
successful. Brutsaert (1975), in summarizing this work, suggested an
empirical form of d = 2h/3, where h is the average height of the vegeta-
tion.
The above concepts and empirical procedures for determining z0 and
d may be appropriate for flows of extended fetch over low and uniform
roughness elements (i.e., to a surface boundary layer in equilibrium
flow). Their validity is questionable at low heights above tall rough-
ness elements, to abrupt changes of roughness, and to roughness elements
of varying height and density. All of these conditions exist in an
urban area and the flow is likely to "see" different surface character-
istics at a fixed location as a function of wind direction and speed and
29
-------
the height of observation. Thus the adequacy of z0 values derived for
an urban area must be judged in several respects; the consistency of the
results with previous studies, the extent to which equilibrium flow con-
ditions exist, and the extent to which the results can be described
within a theoretical or empirical framework of the total flow character-
istics.
Profile data were not obtained in the present study. Values of
z0 and d, however, can be estimated based on a visual site survey as
suggested by Lettau (1969), using the empirical formulas of both Lettau
and Counihan (1971). Derivation of z0 can also be achieved through
the logarithmic profile law (Eq. 3.1), using measured values of U and u*
and an estimate of d. Both techniques were applied in this study and
both gave results within the range of z0 values reported in the liter-
ature for urban areas by Oke (1974).
3.1.1 Land Use Evaluation of z_Q
In concept, both z0 and d are determined by surface structural
features. The behavior of these parameters is obvious in two asymp-
totic limits of roughness density. As the ratio of the area covered by
larger roughness elements Ar to the total area A approaches zero, z0
approaches a finite but small value appropriate to the underlying sur-
face, and d goes to zero for all practical purposes. As Ar/A approaches
1.0 (larger roughness elements completely cover the surface) z0 ap-
proaches a small value appropriate to the top surface of the larger ele-
ments, and d tends to the height of the larger roughness elements.
Functional relationships for z0 and d at intermediate values of sur-
face structural parameters have long been sought by micrometeorologists
and fluid dynamicists. 30
-------
Kutzbach (1961), using bushel baskets on an ice-covered lake,
found the ratio of z0/h (h is the height of the roughness elements)
proportional to the basket density, to the 1.1 power. Displacement
length was determined to be proportional to the same quantity, to the
0.29 power. Lettau (1969) described Kutzbach 's data by suggesting the
relationship, z0/h = 0.5s/S. In this expression, s is the silhouette
area of the average obstacle in the vertical-crosswind plane, and S is
the specific area of roughness elements (total site area divided by the
number of roughness elements) measured in the horizontal plane.
Businger (1974), also using Kutzbach's data, suggested the empirical
form z0/h = 0.5Cih2(l-C2h2/S)3/S. The constants GI and G£ relate to
the geometry of the roughness elements, i.e., their width, height, and
separation distance. When the roughness elements are uniformly spaced
cubes, h2/S = s/S = Ar/A.
Kutzbach's results are based on a maximum ratio Ar/A of 0,25.
Counihan (1971) created roughness densities ranging from 0 to 1QO% in a
wind tunnel. He found z0/h increased with increasing Ar/A to a maximum
value of 0.3 at Ar/A = 0.25 and then decreased asymptotically to the
roughness length for the underlying baseboard at Ar/A = 1.0. Counihan
has shown that equilibrium flow conditions were not established in the
limited fetch of his experiments with the result that z0 was probably
overestimated. He determined the displacement length to be on the order
of z0 up to Ar/A = 0.25. However, d/h continues to increase with
increasing density, approaching unity at Ar/A = 1.0. Curves repre-
senting the results of both Kutzbach and Counihan for z0/h and d/h as
31
-------
function of Ar/A are presented in Figure 3.1. The values of d from the
two experiments are obviously quite different at low roughness densi-
ties. It is not clear if this difference is entirely due to the dif-
ferent shapes of roughness elements used in the two cases or to differ-
ences in the techniques.
Actual roughness features in urban areas vary significantly from
the simple forms for which the above techniques are valid. Figure 3.2
is a diagram of percent land-use type and height estimated for site 107.
The miscellaneous category is a gross estimate of vehicles, small
structures, and small vegetation. The distribution of roughness ele-
ments is very complex, being nonuniform in space, and even in time
(roughness effects of trees and other vegetation will vary with season).
However, Z0 and d values were estimated from the curves in Figure 3.1
based on the average height of the higher roughness features and their
spatial coverage.
The percentage of each of four land-use categories (buildings,
trees, paved area, and low grass area) in the immediate vicinity of
each site was estimated from site visits and studies of photographs
taken from a helicopter and from the top of the towers. The average
heights of buildings and trees were also estimated. The results for
the four sites are given in Table 3.1. They are mainly representative
of average features within a 200 to 300 m radius of the sites. Surface
features farther upwind may affect the wind structure; however, dif-
ferentiation of surface features as a function of distance and wind
direction was not practical, except for site 109.
Displacement and roughness lengths, estimated for each of the
32
-------
Af/A
Figure 3.1. The ratios of zo/h and d/h as a function of Ar/A from the empirical data of
Kutzbach (1961) (dashed line) and Counihan (1971) (solid lines).
18
16
14
1 12
I
s 10
ui 8
X
6
4
2
n
• !••! "
• ii ni
•
BUILDINGS
•^-^^^
TREES
PAVED
GRASS
I MISC.
10
20
30
40
50
60
70
80
90
100
LAND USE, percent
Figure 3.2. Schematic of land-use features and average heights for site 107.
33
-------
Table 3.1. Site Land-Use Characteristics and Estimated Displacement Lengths (d)
and Roughness Lengths(zj Based on the Empirical Results of Kutzbachj1961)
(K) and Counihan (1971) (C), and Average Calculated Values.
SITE
LAND USE
BUILDINGS
TREES
PAVED
GRASS
d(K)
d(C)
ZQ(K)
Z0
105
Ar/A
.25
.01
.59
.15
h
(m)
5.5
5.5
0
0
4.0
1.65
1.2
1.65
107
Ar/A
.25
.25
.16
.34
h
(m)
7.5
12
0
0
8.4
6.3
*
1.17
111
Ar/A
.16
.25
.14
.45
h
(m)
7.5
13.5
0
0
9.2
5.8
*
1.89
109
Ar/A
.05
.05
.01
.89
h
(m)
4.5
3.0
0
0
.84
.19
.04
.06
CALCULATED VALUES
d(1)
Z0(2)
Z0(3)
2
0.67
0.67
6
1.39
1.20
6
1.71
1.37
0
0.33
0.46
*METHOD OF CALCULATION NOT VALID FOR THIS CATEGORY OF Ar/A
(1) ESTIMATED FOR USE IN WIND PROFILE EQUATION (EQ. 3.1)
(2) CALCULATED FROM PROFILE EQUATION.
(3)
= h/8.15.
34
-------
sites from the empirical relations of both Kutzbach and Counihan, are
given in Table 3.1. The values for z0 for the urban sites are consis-
tent with previously reported values of z0 for urban-suburban areas
(see Oke, 1974). The values for rural site 109 represent an average of
individual estimates for four wind direction quadrants. Higher values
of z0 and d were obtained for the northeast quadrant, which contained
4.5-m buildings, and for the southeast quadrant, which contained 3-m
trees.
3.1.2 zQ From Similarity Profile Assumption
Because of the height and spatial variation of roughness features,
the total stress may result from a combination of form drag on the
higher roughness elements (trees and buildings), and skin friction drag
on the intervening surfaces. Arya (1975) suggested a region of merging
of the two drag effects above the higher roughness elements. Within and
above the region of merging, the flow characteristics depend on an
"overall roughness parameter11 (to be denoted by upper case Z0). This
parameter, with sufficient height and fetch for equilibrium flow to be
established and under neutral stability, satisfies the modified loga-
rithmic profile law, Eq. 3.1. This equation in the form:
(3.2)
° exp(kU/u*)
was used to calculate Z0 for the four sites, using hourly average
values of U and u* derived from the tower UVW anemometers and an esti-
mated value of d. Values of d used in the calculations are given in
Table 3.1. They generally reflect Counihan 's results, and like Z0, may
vary at a given site with wind direction. However, at the height of the
35
-------
instrumentation of this study, Z0 is relatively insensitive to varia-
tions of d; a change in d of three meters about any of the table values
resulted in less than 15% variation in the calculated value of Z0.
Surface roughness lengths from Eq. 3.2 using summer data for urban
site 107 are shown in Figure 3.3 (circles) as a function of wind direc-
tion. The data points represent hourly values for u* > 0.15 m/sec and U
> 2 m/sec (to eliminate possible high noise to signal ratio in the
measurement system), and neutral conditions (defined as 0.05 > z'/L >
-0.05). The Nickerson-Smiley (1975) integration of Eq. 1.1, using the
form of m(z/L) suggested by Businger et al. (1971), was used to extend
the Z0 calculations over the diabatic range of the data. Results of
these computations for z'/L < -0.1 (squares) and z'/L > 0.1 (triangles)
are also given for site 107 in Figure 3.3. No obvious dependence of
Z0 on stability is found. Surface roughness lengths over the total
range of z'/L for the four sites are shown in Figure 3.4. An obvious
wind direction dependence occurs at all sites, but Z0 also varies
considerably for any given wind direction. This latter variation may
represent noise in the technique.
The variation of derived roughness lengths averaged over 20 degree
wind direction sectors is given in Figure 3.5 for the summer data set.
The variation is large and suggests that a single roughness length can-
not characterize an urban site (or the rural site). The derived rough-
ness lengths, however, do reflect the general site features, (i.e.,
average Z0 values for each of the sites - given on the inset - appear
to be a function of the height of buildings and trees). Brutsaert
(1975) suggested that h/z0 = 8.15 to a first approximation. Average
36
-------
en
O
UJ
cc
Q
Q
1
CO
**
g
0>
1
3
3
0)
O
0)
•a
v 09
"I
£
II
13
"
s
« 8
2 •
M +-•
37
-------
WIND DIRECTION,degrees
Figure 3.4. Surface roughness length vs. wind direction (summer data)
38
-------
3.0
2.0
E
*
e
Ni
1.0
3.22
I
\ 3.73
•
\
\
\
v\
\
\
•\
' 9
V-.
\
SITE
105
107
109
111
ZD
0.67
1.39
0.33
1.71
NO.
471
366
353
131
/\
V ' \ 111
\/ \
\' \
r
107
~. S \
Ox.' \
••...-' 109
i
c®
40
80
120
160
200
240
280
320
360
WIND DIRECTION, degrees
Figure 3.5. Surface roughness length averaged over 20 degree wind direction sectors for
the summer data. The inset gives the site average ZQ and number of data points for each
site.
39
-------
calculated Z0 values for the summer data set and those obtained by
Brutsaert's relationship, both shown in Table 3.1, have a definite
correspondence.
Roughness lengths similarly calculated for the fall data set
(sites 105, 107, and 109) and averaged for 20 degree wind direction
sectors are plotted in Figure 3.6. Average values for each site are
given on the inset. A seasonal variation of Z0 is not unexpected at
site 107, where deciduous trees comprise 25 percent of the roughness
features, and at the rural site (109). Differences in the site average
values between summer and fall also reflect changes in the prevailing
seasonal wind direction.
The calculated values of Z0 are extremely sensitive to those
parameters contained in the exponential term of Eq. 3.2; u*, k, and U.
The value of u* appropriate to the logarithmic wind profile equation
(and also for normalizing the velocity standard deviations in the next
section) is the spatially averaged surface value. The values used in
this study are determined by the eddy correlation technique from meas-
urements at 31 m above the surface, and thus are "local" values, u*[_.
They may differ from the average surface values, especially at low
heights above tall roughness elements.
The similarity wind profile does not contain parameters representa-
tive of the surface roughness features and consequently is valid only at
heights away from the direct wake influence of surface features.
Tennekes (1973) suggested that the minimum height at which the logarith-
mic law becomes valid is on the order of 2 = 100Z0, Plate and Quraishi
(1965) indicated this height to be z > 2h (h is the height of the
40
-------
oo o
CM o a
en CM
a;
to
CB
-a
10
CO
0)
3
O)
iE
CO
CO
CO
CO
CO
41
-------
roughness elements), and Garratt (1978a) suggested 4.5h, The nature of
the boundary layer in the wake region is uncertain. However, a depar-
ture of u* or U by as little as 5% from the profile assumption can
result in 25% variation in the value of Z0. The calculations are equal-
ly sensitive to the value of k, which has been the subject of recent
controversy. There is considerable experimental basis to retain the
value at 0.40 (See Hicks, 1976; Garratt, 1977) as was done here.
In spite of these potentially large sources of error, the calcu-
lated roughness lengths are in good agreement with those expected from
the land-use features and those reported in the literature for urban
areas.
3.1.3 Application
Urban roughness lengths estimated from land-use features are in
general agreement with those calculated through the logarithmic wind
profile law. Roughness and displacement lengths may vary by a factor of
at least three over an urban area and Z0 apparently may vary by that
magnitude as a function of wind direction at a site selected to be
representative of the surrounding land use. The results suggest that an
average site roughness length may be obtained from knowledge of land-use
features. Although the condition of equilibrium flow has not been
demonstrated, Z0 values calculated from the tower turbulence measure-
ments appear to be reasonable and are used in subsequent attempts to
show order to the observed structure of turbulence over St. Louis. In
some cases, the average site Z0 values are used in the analyses.
Generally, however, Z0 values are averaged for the data set corres-
ponding to the turbulence parameters being analyzed.
42
-------
3.2 REYNOLD STRESSES AND TEMPERATURE VARIANCES
The second moments of the velocity and temperature fluctuations are
presented in this section and are analyzed and discussed within the
framework of Monin-Obukhov similarity theory. The variances u'2,
w
, and T'2 and the longitudinal heat flux u'T1, normalized by u* and
and T*, are assumed to be functions of z'/L. The stress u'w1 and heat
flux w'T1 are unity when normalized, and the lateral component of stress
v'w1 is theoretically zero in the horizontally homogeneous surface
layer. Also discussed are turbulent characteristics specified as a con-
sequence of the similarity relationships, such as the ratios av/au
and ow/au, and the turbulence intensities ou/U, av/U, and aw/U. These
computed quantities are compared for the various sites/seasons and to
corresponding relationships reported by other investigators for flow
above small and homogeneous roughness elements.
3.2.1 Velocity Variances
a. Neutral Stratif ication--
The similarity predictions for the normalized velocity standard
deviations are given in Table 1.1. For neutral stratification (the
heat flux approaches zero or at a height very near the surface) the
buoyancy parameter has no effect and the velocity standard deviations
are given by:
au = AIU*, av = A2U*, aw = A3u* (3.3)
where Aj_, A2, and A3 are constants independent of height, wind speed,
and surface roughness. The data summary of Ariel1 and Nadezhina (1976)
show this to be the case within limits that can be reasonably attributed
43
-------
to experimental error. Their results for neutral stratification as
given by Binkowski (1979) are:
/u* = 2.3(0.4)
av/u* = 1.9(0.3)
" = 1.3(0.2).
(3.4)
The numbers in parentheses are the standard deviation of the observa-
tions making up the mean value.
Averages and standard deviations (s.d.) of au/u*» av/u*> and Ow/u*
are given in Table 3.2 for neutral stratification (defined as 0.05 >
z'/L > -0.05) for each of the sites for both summer (S) and fall (F)
data sets. Also given are the number of observations, average surface
roughness length for the specific data sets, and the overall weighted
averages of the normalized velocity standard deviations. The weighted
averages are in good agreement with the values referenced above. The
individual means for each of the components were tested against the
corresponding component for site 109S for independence of the samples
using the Student t distribution. The null hypothesis was that the mean
for site 109S was equal to the mean for the other sites at a level of
significance of 0.05. The superscripts on Table 3.2 denote those data
for which the null hypothesis was rejected. The urban sites generally
have significantly smaller values of the ratios than site 109.
The average values of the normalized velocity components (from
Table 3.2) for each of the sites/seasons are plotted against the corres-
ponding average Z0 in Figure 3.7. An apparent inverse relationship
exists; as Z0 increases from 0.2 to 1.6 m, au/u* and av/u* decrease
about 12% and aw/u* by about 16%. The slope of the regression lines for
44
-------
Table 3.2. Averages and Standard Deviations (s.d.) of the Normalized Velocity
Standard Deviations for Neutral Stratification, Number of Observations (No.),
and Average Roughness Lengths for Summer (S) and Fall (Fj Data Sets. See
Text for Explanation of Superscript.
105S
107S
109S
111S
105F
107F
109F
NO.
97
107
29
36
125
111
28
Zo
.7
1.3
.2
1.6
.8
1.1
.4
WEIGHTED AVERAGE
au/u*
2.36*
2.39*
2.57
2.19*
2.41*
2.39*
2.54
2.39
s.d.
.43
.26
.47
.16
.22
.18
.37
av/u*
1.81*
1.78*
2.0
1.72*
1.78*
1.74*
*
1.85
1.79
s.d.
.39
.24
.41
.21
.22
.21
.44
aw/u*
1.32
1.17*
1.33
1.28
1.29
1.20*
1.45
1.26
s.d.
.18
.08
.20
.09
.15
.09
.35
-------
2.8
2.4
2.2
2.0
1.8
1.6
1.4
1.2
1.0
D
D
D
a
a
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Zn, meters
1.6
1.8
Figure 3.7. Site averaged values of a/u* vs. the corresponding average Zo and linear
regression fits. The solid symbols represent Z0 = h/8.15 for each of the sites.
46
-------
u/u* and
-------
Z0,m
Figure 3.8 Plots for ow/u* vs. Zo for neutral stratification (summer data set)
48
-------
0
2
+
+
+ +
r+
, + ,+
SITE 109
f + ^
WIND SPEED, m/sec
Figure 3.9 Individual data plots for aw/u* vs. U for neutral stratification for the
four sites (summer data set).
49
-------
*
3
-5 -4 -3 -2
z'/L
-1
0
1
Figure 3.10 aw/u* vs. z'/L for summer data set. The solid line represents the
Panofsky et al. (1977) empirical fit to tower and aircraft data (aw/u* =
*fl /^J*
4 *^ f 4 *~J*^ /I 1 I / ^J1
50
-------
data set). The solid line, given by aw/u* = 1.3(l-3z'/L)l/3s represents
the Panofsky et al. (1977) empirical fit to tower and aircraft data.
There is a difference in the plots for the rural site (109) and the
urban sites. The data for site 109 are in reasonable agreement with
the Panofsky et al. curve, whereas the data for the urban sites are
lower in magnitude, especially in the region of forced convection
(i.e., small negative values of z'/L). The data plots for site 105 ex-
tend to large negative values of z'/L, and although the values are smal-
ler than expected from the Panofsky et al. form, the slope approaches a
1/3 power law as suggested for free convection similarity (Deardorff and
Willis, 1975).
The trend in
-------
1.00
0.75
0.50
0.25
0.00
1.00
0.75
o
0)
t/t
0.50
0.25
0.00
r
/
V
/*
& +
f
i
1
*
»
SITE
1 1 1
1 1
1
109
i
T
0.0
0.2
0.4
0.6
0.8
1.0
w'T'z'
Figure 3.11 aw vs. Uf for z'/L< -1.0.
52
-------
the fall season. Plots of aw versus u* for the summer data set are
shown in Figure 3.12. These plots, along with the site and seasonal
shift of the slopes on Figure 3.11, suggest that the friction velocity
(as well as the heat flux) may be important for parameterizing the
vertical velocity variance, even under conditions approaching free
convection.
A functional dependence of aw on both u* and uf can be derived
out of the turbulent kinetic energy budget equation which for the
simplified conditions of homogeneity and stationarity is given by:
3
'1
u*J gw'T1 1 dw'q^ 1 dp'w
(3.6)
The first term on the right-hand side represents the shear production,
with i being the mixing length; the second term represents the buoyancy
production; the last two terms represent turbulent and pressure trans-
ports of energy due to flux divergence, e is the dissipation rate of
turbulent energy per unit volume and can be expressed as (Tennekes and
Lumley, 1972):
e = aw
3/1 (w)
(3.7)
where Le is the dissipation length scale (see Appendix B). Neglecting
the last two terms of Eq. 3.6 (Wyngaard and Cotg, 1971, indicated the
net effect of these two terms to be small) and assuming a s kz1, Eqs.
3.6 and 3.7 yield:
aw = C(u*3 + 0.4uf3)l/3 (3.8)
where C is on the order (L-/£)l/3. Eq. 3.8 can be expressed as
aw
/u* = C(l-z1/L)l/35 a form similar to that suggested by Panofsky
53
-------
2.0
1.5
CJ
0)
v>
E 1.0
*
b
0.5
0.0
&> +
SITE 105
2.0 -
1.5
u
o>
1 1-0
0.5
0.0
0.0
0.2
0.4
0.6
SITE 109
0.8
1.0
u», m/sec
Figure 3.12. a... vs. u* for z'/L<-1
w
54
-------
et al. (1977), but with less dependence on buoyancy-generated turbu-
lence. Figure 3.13 shows plots of aw versus the right-hand side of
Eq. 3.8 for z'/L between 0 and -1 for the summer data set. These data
•
represent hours between 0700 and 2000 h with u* > 0.15 m/sec and U >
2.0 m/sec. The solid lines represent the estimated linear fit to the
data forced to go through the origin. A linear relationship is sug-
gested for all three sites, with the constant C being about 1.28, 1.19,
and 1.18 for sites 109, 105, and 107, respectively. The dashed lines
represent the estimated fit to the fall data, and indicate the general
consistency of the relationship for an independent data set. Plots of
the same quantities are given in Figure 3.14 for z'/L < -1. The appar-
ent differences in the slopes for the two figures may be due to neglect
of the flux divergence terms in Eq. 3.6, or to the failure of Eq. 3.8 to
adequately describe the relative importances of the shear and buoyancy
production processes.
The relative importance of the two processes was sought through a
least-squares fit to the data in the form aw3 = QU*3 +Euf3, over
the range 0 > z'/L > -5. These results are given in Table 3.3 in a form
consistent with that suggested by Panofsky et al. (1977), i.e., °
= Dt(l-E'zt/L)1/3. in this equation, D1 = D*/3 and E1 = 2.5E/D.
(The values of D1 and E1 given by Panofsky et al. are 1.3 and 3.0,
respectively,) Note that the values of E' in Table 3.3 are consistently
smaller for the urban sites. Correlation coefficients are large for all
sites and for both seasons, suggesting that the neglected terms of Eq.
3.6 are unimportant or that a high correlation exists between the neg-
lected terms and the buoyancy production term.
55
-------
1.00
0.75 -
o
V
t/t
e 0.50 -
0.25 -
0.00
1.00
SITE 105
0.75
o
»
tf>
3
to
0.50 *-
0.25 -
0.00
1.00
0.75 -
QJ
0.50 -
0.25 -
O1.18
SITE 107
0.00
SITE 109
0.0
0.2
0.4
0.6
0.8
1.0
(u*3
1/3
, m/sec
Figure 3.13.
-------
/'GO -j
5-^00
500
I
f ,-v
\
«
\
\
s
-------
z'/L < -1
1.00
u
o
CO
0.75
S 0.50
*
§
b
0.25
0.00
[•
r
SITE 105
u
0>
c/i
3
b
1.00
0.75
0.50 h
-i
4
i
-i
i
0.25 "r-
0.00
0.0
O1.43
0.2
0.4
0.6
(u*3 + 0.4uf3)1/3 ,m/sec
]
I
1
SITE 109
0.8
1.0
Figure 3.14. Same as for Figure 3,13 but for z'/L <• 1.
57
-------
Table 3.3. The Ratio ow/u* as a Function of z'/L and Correlation
Coefficients R*, for Summer (S) and Fall (F) Data Sets. (The Expression
•f \ i \ t •*"} T
for aw/u* was Derived out of a Least-Squares Fit to awj = Du*J +
for 0 > z'/L > -5.)
SITE aw/u* R*
w
105S 1.16(1-1.Zz'/L)1/3 0.96
107S 1.10(1-1.Qz'/L)1/3 0.96
109S 1.07(1-3.8z'/L)l/3 0.96
HIS 1.20(1-2.5z'/L)l/3 o.95
105F 1.10(1-1.Iz'/L)1/3 0.95
107F 1.11(1-1.lz'/L)l/3 0.98
109F 1.25(1-2.lz'/L)l/3 0.95
In the limit as z'/L ->• +» (stable stratification) similarity
theory predicts each of the normalized velocity standard deviations to
approach a constant value. The plots in Figure 3.10 suggest aw/u*
increases with increasingly positive z'/L. Results of Haugen et al.
(1971) and the data summaries by Merry and Panofsky (1976) and Ariel1
and Nadezhina (1976) also suggest that aw/u* may increase with z'/L
in the range of small positive zl/L. Merry and Panofsky (1976) sug-
gested the increase may be due in part to errors in aw and u*, when
these quantities become very small under the buoyancy suppression of
turbulence associated with stable stratification. The data plots in
the present study were generally limited to u* > 0.15 m/sec and U > 2
m/sec, values well above the noise level of the system, to minimize
instrument response errors. An alternative presentation to Figure
3.10 is given in Figure 3.15, where aw is plotted against u* for
58
-------
1.00
0.75
o
0)
S 0.50
0.25
0.00
O1.33
SITE 105 -
o
03
(A
1,00
0.75
E 0.50
0.25
0.00
SITE 107 -
o
1.00
0.75
E 0.50
m
0.25
0.00
O1.33
0.0
0.2
0.4
0.6
u*, m/sec
SITE 109 -i
0.8
1.0
Figure 3.15. aw vs. u* for stable stratification.
w
59
-------
stable stratification (w'f' < 0)- These plots suggest aw/u* is constant
for stable stratification as predicted by similarity theory. The
estimated slopes through the data points are about 1.3 for sites 105 and
109 and 1.2 for site 107. These values are consistent with those
obtained for neutral stratification. Estimated slopes for the fall data
set (dashed lines on Figure 3.15) are consistent with those for the
summer data set.
Experimental data for the horizontal components
-------
N
CO
X
-a
E
crt
*
3
O
(A
_o
Q.
CO
CO
0)
.5*
iZ
61
-------
1
I I I
SITE 109
» = 3.5(-z/L)
I I
1/3
I I I f
O
SITE 105
= 2.5{-z/L)
1/3
1 _ I 1 I I I 1 I
O —
-5
-2
z/L
-1
+1
Figure 3.17. 0v/u* averaged over intervals of z'/L of 0,25 vs. z'/L for sites 105 and
109 (summer data).
62
-------
2.0
\ 1.5
i 0
as
1.5
- I
1.0,
I 0.5
j o.6ii
'~~~7
0.0
•H-
+
SITE 105
*
*
•f
SITE 109
i 0.6
) 0.8
j 1.0
, m/sec
Figure 3.18. a vs. ti for z'/L <-1.
63
-------
a least-squares fit of the form av = Du*3 + Euf for 0 > z'/L > -5
are given for sites 105 and 109 in Table 3.4. As for ow/u* the coeffi-
cients of the buoyancy term are smaller at the urban site for both sea-
sons. Correlations coefficients are smaller than for ow/u* (Table 3.3)
Table 3.4. Same as for Table 3.3 but for ow/u*.
v
SITE o,y/u* R*
v
105S 1.63(1-2.6z'/L)l/3 0.86
109S 1.81(1-4.Oz'/L)l/3 0.82
105F 1.66(1-3.4z'/L)l/3 Q.91
109F 1.84(1-5.2z'/L)l/3 0.86
To the extent that the normalized velocity variances can be
described as a function of z'/L, Monin-Obukhov similarity theory
apparently holds for the data obtained in both urban and rural environs.
However, the empirical form of the relationship differs: for any z'/L
the normalized velocity variances are generally larger for the rural
site compared to the urban site. The data scatter for the horizontal
components was generally larger than for the vertical component. An
alternative approach, i.e., free convection similarity theory is applied
to the horizontal components below.
c. Dependence on z-j/L--
The large scatter typically associated with plots of av/u* and
ou/u* as functions of z/L has generally led to the conclusion that
Monin-Obukhov similarity does not hold for these parameters under
convective conditions (Lumley and Panofsky, 1964). Kaimal et al. (1972)
64
-------
show the peak wavelength of the w spectra to scale with z/L, but not the
peak in the horizontal velocity components. From later experiments in
northwestern Minnesota, Kaimal et al . (1976) concluded that the scaling
height appropriate to the spectral peak of the horizontal velocity com-
ponents in the convective boundary layer is the height of the mixed
layer z-j. This scaling length also apparently applies in the surface
layer. However, Kaimal (1978) noted that the interaction between ter-
rain length scales and spectral length scales at low heights above tall
roughness elements is an unknown factor.
For the limit approaching free convection, Deardorff (1970) sug-
gested the horizontal velocity components to scale with the free convec-
tive velocity scale:
w
* =
(3.9)
Since ^u ^ w* and av ^ w*, even in the surface layer, the normalized
velocity standard deviations are given by:
Vu* * au/u*
(3.10)
Empirical support for this form is given by Panofsky et al . (1977).
Mixed layer height estimates were available from ancillary lidar
measurements (Endlich et al., 1978) on 15 days during the summer experi-
mental period for testing the convective boundary layer similarity rela-
tionship. The lidar data were obtained near the urban center and were
generally available from sunrise through early evening. Hourly average
values of mixing height estimated from the lidar returns are shown in
Figure 3.19 for the 15 day period, along with smoothed average heat flux
for sites 105 and 107. The spatial variation of the mixed layer over
65
-------
1600
1400
«j 1200
O»
X
2
UJ
X
1000
800
Z 600
X
400
200
100
I I
I I
O
r^Toi
o
O
o
o
o
o
1
I
400
300
m
200
r-
C
X
N>
100
0700 0800 0900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300
TIME (CST), hour
Figure 3,19. Average mixing heights near the urban center (open circles) and
average surface heat flux for sites 105 and 107 (solid line) for 26 July through
13 August.
66
-------
the St. Louis area is complex, especially during the morning transition
period (Ching et al., 1978); however, the lidar estimated values were
applied to both urban and rural sites.
Figure 3.19 shows two distinctly different periods in the evolution
of the convective boundary layer that were reasonably consistent on all
of the days. The first is the morning transition period when both heat
flux and the height of the mixed layer increase rapidly. The second
period begins about 1300 h and extends through the late afternoon and is
characterized by a nearly constant mixed layer height and decreasing
heat flux. The time scale of the changing boundary layer parameters for
both periods is long compared to the convective time scale (zi/w*), and
thus the requirement of stationarity inherent in the similarity formula-
tion should be met. However, it is of interest to test the consistency
of the similarity relationship for the two periods; the data were strat-
ified according to time periods: 0900 to 1200 h and 1300 to 1600 h.
Plots of av/u* versus Zj/L for sites 105, 107, and 109 are
shown in Figure 3.20. The solid line represents the form postulated by
Wyngaard and Cote (1974) (o/u* = (3.06 + 0.6(zi/-L)2/3)l/2) and the
dashed line for site 107 is the form suggested by Panofsky al. (1977)
(a/u* = (12+Q.5z-j/-L)l/3) m The free convection similarity form is
generally supported by the present data set. However, for small values
of -Zi/L, the plots for site 105 are inconsistent with those for site
109 (and the forms postulated by Wyngaard and Cote, and Panofsky et
al.). There is no apparent stratification as a function of time period.
An alternative test of mixed layer scaling is derived out of Eq.
3.9, using the peak in the longitudinal velocity spectrum, Lm(u), as a
67
-------
*
3
0900-1200
01300-1600
-4»y\
f*» f V
• o o
SITE 109
50
100
150 200
Zj/L
250
300
350
Figure 3.20. The ratio ay/u* vs. Zj/L for sites 105,107, and 109 (summer data)
for the period 0900 to 1200 hrs. (solid dots) and 1300 to 1600 hrs. (open circles)
The solid line represents the form postulated by Wyngaard and Cote (1974) and
the dotted line that of Panofsky et al, (1976).
68
-------
surrogate for Zj (Kaimal et a!., 1976, has indicated the peak
wavelength of the horizontal components to be proportional to z-j):
PI
w*1 = (gwTTrLm(u)/T)l/3. (3.11)
ots of au versus the right-hand side of the above expression for
z'/L < -1 (given in Figure 3.21) support the use of Lm(u) as a convec-
tive scaling length. The longitudinal velocity component is discussed
here in preference to the lateral component because Lm(u) can be
determined with greater confidence than Lm(v) (see Section 3.3).
The inclusion of a mechanical production term through Eq. 3.6,
with the assumption that e * au^/t-m(u) (Hanna,1968), yields:
au = C(u*3l_m(u)A + w*'3)l/3. (3.12)
Plots of au versus the right-hand side of Eq. 3.12 (?, was assumed pro-
portional to kz1) are given in Figure 3.22 for z'/L between 0 and -1
and in Figure 3,23 for z'/L < -1, Generally, Eq. 3.12 appears to be an
appropriate predictor of ou; however, total consistency does not exist
for all sites and stratifications as indicated by the different values
of C. A least-squares analysis of the form ou3 = ou*3 + Ew*'3 is given
in Table 3.5 for the summer data set. The result for site 109 is
similar to the empirical form suggested by Panofsky et al. (1977)
(°u/u* = 2.3(l-0.04zi/L)1/3) when z1 is assumed given by Lm(u)/1.5.
Correlations coefficients are slightly higher than obtained for av/u*
using the convective velocity scale (Table 3.4) suggesting that w*1,
rather than Uf, may be the preferred scaling velocity. Consistent with
previous results for °w/u* and °v/u*, the coefficient of the buoy-
ancy term is smaller for the urban site (105).
69
-------
tn
•**.
2.0
I._ i i __1 i
] i 1 I I
1.5
1.0
0.5
0.0
SITE 105
= 0.5
i i
I I I I If
If I I 1 I
2.0
1 _L
1.5
1.0
0.5
0.0
i
SITE 109
C = 0.63
+
l
i i i r i i i i
o.o 0.5 1.0
f I I
\ \ I 1
1.5
I I I I 1
2.0 2.5
3.0
'lwT'Lm(u)J
1 /3 m/sec
Figure 3.21 au vs.
for z'/L < -1.
70
-------
2.0
I i i \
0}
l/t
3
b
o
OJ
M
o
V
(ft
1.5 -
1.0
0.5 -
0.0
2.0
I \
I I I
1.5 \-
C"*0.53
SITE 107
1 T
2.0
1.5 -
1.0
0.5 -
0.0
1
<>\1/3
+ w*3\ m/sec
\ kz'
Figure 3.22 au vs. right hand of Eq. 3.12 for z'/L between 0 and -
71
-------
u
0)
3
to
w
8
2.0
b
1.5
1.0
0.5
0.0
2.0
L J I
c * 0.43
i i i
I I t 1(11
1.5
c = 0.56
£ 1.0 h
0.5 -
0.0
I I I i
T \ I
0
1
1/3
Till
1 .L 1 _. I .. I
SITE 109
I l * I
kz
, m/sec
Figure 3.23 Same as for Figure 3.22 but for z'/L < -1.
72
-------
Table 3.5. The ratio °u/u* as a Function of Lm(u)/L and Correlation
Coefficients R*, for the Summer Data Set. (The Expression for <7u/u*
was Derived out of a Least-Squares Fit to au3 = ou*3 + w*-3 for
0 > z'/L >-5.)
SITE au/u* R*
105 2.0(l-0.02Lm(u)/L)l/3 0.89
109 2.2(l-0.04Lm(u)/L)1/3 0.91
d. Ratio of the Variances--
In the regime of buoyancy dominated turbulence, ou ^ av and thus
av should behave as discussed above for au. The ratio av/au versus
z'/L is shown in Figure 3.24 for the summer data set. For z'/L < -1,
the ratio behaves as expected from free convective similarity; i.e.,
both normalized standard deviations scale with mixed layer height such
that the average ratio av/au i£ sbout unity.
The ratio crw/ou for the summer data set is given in Figure 3.25.
The theoretical form of this ratio is derived out of both Monin-Obukhov
and free convective similarity:
°w/°u = f(z'/L)/f(zi/L) = C(z'/Zi) (3.13)
Thus aw/au> which characterizes the anisotropy of the turbulence,
depends on both the height of measurement and the depth of the convec-
tive mixed layer (or alternately Lm(u)). For neutral and stable
stratifications, the dependence on mixed layer height and height van-
ishes, and the ratio approaches a constant value which for neutral
stratification is about 0.53 (from Table 3.2). Busch (1973), for
measurements at 5.66 m, found the ratio essentially constant for stable
conditions at a value of 0.6, a sharp transition near z/L = 0, and
73
-------
0
Figure 3.24 Plots of av/au vs. z'/L for summer data set
74
-------
--+++
Figure 3.25 Plots of ffw/au vs. z'/L for the summer data set
75
-------
constant again at a value of about 0.5 for z/L < -1. The present data
(Figure 3.25) indicate ow/au to increase from the neutral value of
0.53 to about 0.67 at z'/L = -0.75, and remain essentially constant out
to large negative values of z'/L. The influence of boundary layer depth
was not apparent, although it may be represented by the scatter of the
data points.
e. Intensity of Turbulence ~
The vertical and horizontal components of turbulence intensity can
be expressed through the similarity wind profile equation in the form:
aw/U = kaw/u*/[1n(z'/Z0)-*(z'/L)] (3.14)
and
= kaU5V/u*/[ln(z'/Z0)-*(z'/L)] (3.15)
where Y(zVL) is the correction to the logarithmic wind profile for
diabatic conditions. The turbulence intensity components depend on the
height of observation and surface roughness. They are nevertheless
useful and frequently measured turbulence parameters and often recom-
mended for parameterizations of dispersion in the surface boundary
layer.
The diurnal variations of av/U and aw/U for sites 105, 107,
and 109, given in Figure 3.26 for the summer experimental period, indi-
cate these quantities are a function of atmospheric stability, whereas
the differences in the relative magnitude at the three sites reflect
a land-use dependence. Results for the fall experimental period were
similar; however, the turbulence components were noticeably smaller
during the midday period, reflecting the decrease in convective activity
76
-------
0.30
0.20
0.10
i I I 1 i
m
SUMMER
» • * •
109
0.50
0.40
0.30
0.20
.10
av/0
109 '.
1 I I I I I I I I I I J I I I I I
i I I
0000
0300
0600
0900
1200
1500
1800
2100
TIME,hrs
Figure 3.26. Diurnal variation of vertical (upper) and lateral (lower) intensities of
turbulence for the summer data set.
77
-------
during the fall season. Urban-rural turbulence intensity differences
(sites 105 and 107 contrasted with site 109) are small during the
morning transition period (0700-1000 h) and relatively large during the
afternoon transition period (1300-1700 h). The similarity of turbulence
intensities at all of the sites during the morning transition period is
possibly the result of more rapid conversion of solar radiation to
turbulent energy in the rural environment (relative to the urban
environment), owing to the lower heat capacity and conductivity of the
rural substrates. The larger values of turbulence intensity at the
urban site during the afternoon period represent a true urban anomaly
arising from the larger heat capacity and conductivity of the urban
substrate.
Effects of atmospheric stratification on ow/U and ov/U are
shown on Figure 3.27, where these quantities are plotted against z'/L.
Each data point represents the average of at least five values for the
given site over a discrete interval of z'/L (the scatter of the individ-
ual data points was large due to the influence of surface roughness
which varied markedly as a function of wind direction as described in
Section 3.1). The plots for sites 105, 107, and 109 each represent a
total of about 400 data values with 80, 94, and 60 percent, respec-
tively, occurring for 0.5 > z'/L > - 0.5. The plots for site 111,
which was in operation less than two weeks, represent only 131 hourly
data values - most of which were in the above range of z'/L. Both aw/U
and av/U appear to be functions of z'/L for small absolute values of
z'/L. For unstable stratification, the scatter of the class-interval-
averaged data points is too large to have confidence in the shape of the
78
-------
0.40
0.30
1=5
0.20
0.10
o
w
V^M
u
oo
O
o
A - Q
00°°0
A AA O
A
A
O
O
'A
1 I I
a
°A
o
O SITE 105
D SITE 107
A SITE 109
SITE 111
a
U
a
O O O
A O
1 ! I
I I
-2.0 -1.6 -1.2 -0.8 -0.4 0 +0.4 +0.8 -2.0 -1.6 -1.2 -0.8 -0.4 0 +0.4 +0.8
Z'/L
Z/L
Figure 3.27. Vertical and lateral intensities of turbulence vs. Z'/L for summer data set.
79
-------
curves; however, they do not continue at the near neutral (z'/L = 0)
value of the slopes. Near neutral stratification, the slope for a
is greater than for aw/U; av/U responds more directly to increasing
buoyancy-generated turbulence, whereas aw/U is somewhat constrained
by the proximity of the surface. A striking feature of Figure 3.27 is
the apparent dependence of intensities on the specific site and thus
Z0; turbulence intensities for the four sites are ordered roughly as
expected from the computed surface roughness lengths.
Turbulence intensity statistics for neutral and slightly unstable
stratifications are given in Table 3.6. Also given are average Z0
values and the number of observations making up the statistics. Note
that some of the average turbulent intensities in the unstable data
groups for the fall are less than for the corresponding neutral group;
the former group is made up of relatively few observations and average
Z0 values are smaller. Plots of the average site/season values of
aw/U against the corresponding average z'/Z0 are shown in Figure 3.28a.
The solid line represents the evaluation of Eq. 3.14 for neutral strat-
ification (aw/u*(0) = 1.3, i()(z/L) - 0.0) and agrees well with the
observed data points (plain symbols). This correspondence extends to
the individual values of aw/U and Z0, shown for the summer data set
in Figure 3.29. The dashed line represents the evaluation of Eq. 3.14
for neutral stratification. The departure of the plots for site 107
from the expected form is due to aw/u* being significantly less (1.20)
than the value of 1.3 assumed in Eq. 3.14. Overall the plots demon-
strate a dependence of av/U on z'/Z0 inherent in Eq, 3.14.
The dashed line on Figure 3.28a represents the evaluation of Eq.
3.14 for slightly unstable stratification (z'/L = -0.5, for which
according to Panofsky et al . (1977), ow/u*(-0.5) = 1.76). The site
80
-------
Table 3.6. Averages and Standard Deviations (s.d.) of Intensity of Turbulence
Components for Neutral and Slightly Unstable Conditions, Number of Observations,
and Average Roughness Lengths for Summer (S) and Fall (F) Data Sets.
SITE
NO.
V
1
1
1
1
Zo
<7U/U
J
1
1
1
t
s.d.
i
i
(Ty/U ' S.d.
1
trw/U
1
|
1
1
1
s.d.
NEUTRAL (.05>z/L>-.05)
105S
107S
109S
111S
105F
107F
109F,
97
107
29
36
125
111
28
0.7
1.3
0.2
1.6
0.8
1 1
1. 1
0.4
.239
.316
.197
.323
.260
.298
.203
.06
.05
.04
.04
.05
.04
.05
.187
.237
.154
.250
.193
.217
.149
.06
.04
.04
.04
.04
.03
.03
I
.136
.156
.102.
.190
.139
.150
.116
i
.03
.02
.02
.02
.03
.02
.03
SLIGHTLY UNSTABLE (-.6
-------
o
^
0.34
0.32
0.30
0.28
0.26
0.24
0.22
0.20
0.18
0.16
0.14
0.12
0.10
0.08
a.
O
A
105
107
109
111
I I I I I I
I I I! I I
10
20 30 50
100 200 300 500 1000
10
20 30 50
100 200 300 500 1000
z'/z.
Figure 3.28. Plots of the site averaged values of aw/U (a) and av/U (b) vs. z'/zo for neutral
(plain symbols) and slightly unstable (circled symbols) stratifications. The curves represent
expected form from Eqs, 3,14 and 3.15.
82
-------
Z0, m
Figure 3.29. Individual plots of aw/U vs. Zo for neutral stratification (summer data set)
The dashed lines represent expected form from Eq. 3.14.
83
-------
averaged data for -0.6 < Z'/L < -0.4, indicated by the circled symbols,
are well below their expected values, but still suggest a strong depen-
dence on Z0.
Site averaged values of av/U are plotted against corresponding
values of z'/Z0 in Figure 3.28b. The solid line is the evaluation
of Eq. 3.15 for neutral stratification (av/u*(0) = 1.8) and again
is a good fit to the data. Since Monin-Obukhov similarity theory may
not hold for av/u* in unstable stratification, no attempt was made
to evaluate Eq. 3.12 for z'/L = -0.5. The data points for -0.6 < z'/L
< -0.4 are, however, given on Figure 3.28b. As for neutral stratifica-
tion, av/U appears to have a strong dependence on surface features
for z'/Z0 < 100.
3.2.2 Temperature Variance
The similarity prediction for T'2 (see Table 1.1) is discussed
here in terms of the standard deviation ay normalized by a scaling
temperature T* = -w'T'/u*. Dimensional considerations show that in the
limit of free convection, the ratio aj/T* should be proportional to
. For neutral stratification, z/L + 0, both ay and w'T' + 0,
and the ratio is constant. In the limit as z/L •* +», similarity theory
predicts the ratio to be a constant.
The ratio aj/T* for the summer data set is shown in Figure 3.30.
For z'/L < 0, the data plots are similar at all four sites and are in
good agreement with the ratio for the Kansas data. The solid line on
the plots for site 105 represents the Wyngaard et al. (1971) fit to the
Kansas data and is given by aj/T* = 0.95(-z/L)~l/3. In the limit
z'/L + 0, aj/T* for the St. Louis data is significantly larger than
84
-------
8
8
-5
SITE 105
-
SITE 111
I
-tl
+
hT
-4-
-4
-3
-2
1
1
-3
-2
-1
zVL
1
t
Figure 3.30. The ratio aj/T* for the summer data set. The solid line on site 105 represents
the Wyngaard et al. (1971) fit to the Kansas data (aT/T* = 0.95(Z'/-L)-1/3)-
85
-------
for the Kansas or Minnesota data (these latter data sets give maximum
ratios of about 3.0). We believe this is due to the 1-hour averaging
time of the data often being over an unstationary period. For example,
near the time of transition from stable to unstable conditions, the
hourly average value of heat flux may be near zero, where in reality
neutral conditions existed for a relatively short period of time. The
temperature standard deviation on the other hand is based on periods of
both the positive and negative heat flux and thus aj/T* may obtain a
large value.
For small positive values of z'/L, the Kansas and Minnesota data
show ay/"1"* to be between 1.5 and 2.0, Although insufficient to
establish a definite trend, oj/T* for the present data set appears to
approach a value greater than 2.0 for stable stratification.
3.2.3 Covariance
The important covariances describing the nature of the momentum
and heat transport are u'w* and w'r, respectively. These when appropri-
ately normalized are unity in the surface boundary layer. However,
along with the buoyancy parameter g/T, they completely describe (through
Monin-Obukhov similarity theory) the structure of the surface boundary
layer. It is thus of interest to briefly describe their behavior for
the four sites. Two other covariances are also discussed; the cross
wind component of the stress v'w', which will be shown to be approximate-
ly zero, and the horizontal heat flux u'T1. The final two covariances
u1v1 and v'T' were not calculated, but by definition are identically
zero for horizontally homogeneous flow.
86
-------
a. Momentum Flux —
The along and cross wind components of hourly-averaged vertical
momentum flux are given in Table 3.7 for even hours of the summer data
set. The values are given as -u'w'xlOO. A value of 23, for example,
indicates a downward momentum flux of magnitude 0.23 m/sec. The
values of u'w1 appear to reflect the expected diurnal variation of
wind speed and stability, and surface roughness (as expected from the
similarity wind profile).
The v^w1 component of momentum flux is significant only during the
afternoon hours (in a deep convective boundary layer). The maximum con-
tribution to the average friction velocity, u* = (u'w^2 + 7TwT2)l/4,
from v'w' is less than 1%; this lateral component was ignored in deter-
mination of u*. Plots of u* versus U for neutral stratification are
given in Figure 3.31 for site 107. The slope of the estimated linear
fit (solid line) is the square root of the drag coefficient and repre-
sents the effect of surface roughness. Estimated linear fits to the
data for the other sites are also given in Figure 3.31. The slopes are
about 0.11, 0.13, 0.07, and 0.15 for sites 105, 1075 109, and 111,
respectively. Plots of U/u* versus z'/L for sites 105, 107, and 109 are
given in Figure 3.32 for the summer data set. The value of U/u* de-
creases abruptly from stable stratification to a different constant
value for each site at z'/L = -0.5. The scatter of the data points is
in part due to the variation of Z0.
b. Heat Flux —
The diurnal march of average heat fluxes (H = pCnW'T') for sites
105, 107, and 109 (summer data set) are depicted in Figure 3.33 along
87
-------
Table 3.7. Average Hourly Values of -u'w' and Vw' x100
Data Set.
for Summer
SITE
*
3
I
m
3
>
I
105
107
109
111
105
107
109
111
0000
9
10
2
11
0
1
0
0
0200
8
8
1
9
0
0
0
0
0400
5
5
1
9
0
0
0
0
0600
9
9
4
13
1
-1
0
-1
HOUR, CST
0800 1000 1200
13 18 17
17 24 23
6 9 10
27 33 36
0 0 1
-1 1 2
•1 -1 0
-1 1 0
1400
18
23
12
44
2
3
2
3
1600
23
26
11
49
3
2
0
1
1800
19
22
5
32
2
0
0
0
2000
12
13
3
19
1
0
0
0
2200
9
12
3
17
0
0
0
1
-------
1.00
o
4)
trt
*
3
0.75
£ 0.50
0.25
0.00
SITE 105
SITE 107
SITE 109
SITE 111
4
U , m/sec
8
Figure 3,31. Plots of u* vs. U for site 107 (summer data set, neutral stratification) and
estimated linear fit to plots for all four sites.
89
-------
30
SITE 105
20
*
3
10
0
-f-
_t-
3
D
30
SITE 107
20
10
0
30
3
3
SITE 109
-- r
20
10
-6
-2
z'/L
Figure 3.32. Plots of U/u* vs. z'/L
90
-------
480
432
384
306
288
CVJ
240
192
LL
144
96
48
•48
•96
SITE 105
SITE 107
SITE 109
SITE 111
NET RADIATION
(SITE 105)
0 2 4 6 8 10 12 14 16 18 20 22
»
HOUR, CST
Figure 3.33. Diurnal variation of average heat flux for sites 105,107, 109, and 111,and
net radiation for site 105 (summer data set).
91
-------
with the net radiation Fn for site 105 (Fn for site 109 was very
similar). The peak afternoon heat flux for site 105, about 300 w/m2,
is significantly larger than for sites 107 and 109. The 105 heat flux
remains positive throughout the night whereas at 107 and 109 it goes0
slightly negative. The contrast between the heat flux at site 105 and
the heat flux at sites 107 and 109 is believed due to the characteris-
tics of the surface and the resulting energy budgets. The few moisture
flux measurements at sites 105 and 109, using the Lyman-alpha humidio-
meter, are helpful in this respect. Afternoon Bowen ratios, BR = H/LE,
(LE is the latent flux) were on the order of 2.0 at site 105 and 0.5 at
site 109. Ching et al. (1978) constructed a surface energy budget from
this information in which the ground heat flux and anthropogenic heat
sources constituted the residual term. These calculations suggest that
the high afternoon heat flux at site 105 is not unreasonable considering
the relatively small vegetated area available for evapotranspiration
(See Table 3.1). Similarly, the differences between sites 105 and 107
are not unreasonable.
The horizontal heat flux, which is specified as a result of Monin-
Obukhov similarity theory (see Table 1.1), appears as a horizontal gra-
dient term in the thermal energy equation, and also appears in the
buoyancy production term of the shear stress budget. While these terms
are generally assumed small, they have not been measured in an urban
area where advective processes may dominate boundary layer heating,
especially during transition periods in the diurnal cycle (Ching et al.,
1978).
The horizontal heat flux, normalized by u*T*, can be expressed as
92
-------
-u 'T'/WT1. This ratio is plotted as function of z'/L in Figure 3.34,
wherein the solid line (site 105) represents the theoretical form postu-
lated by Wyngaard et al. (1971) for z/L < 0, and is an excellent fit to
the Kansas data. The ratio for the present data set is significantly
larger than for the Kansas data. For large negative z'/L, -u'T' is on
the order of w'T' in the present data set; for Kansas data, it is about
0.25w'T', The larger values may result from horizontal temperature
gradients associated with the urban area- The ratio in Figure 3.34 is
similar to that found by Zubkovskii and Tsvang (1966) for a site near an
urban area. The rural site (Figure 3.34) has similarly large values of
the ratio, suggesting possible nonhomogeneous conditions may also exist
there.
3.3 SPECTRAL REPRESENTATION
3.3.1 Introduction
While the total variance is a useful turbulence parameter, the
distribution of variance as a function of frequency, i.e., the energy
spectrum, provides a better description of the turbulent structure.
The energy spectrum is obtained through the Fourier transform of the
auto-correlation coefficient R(t) by:
oo
F(n) = 4J R(t)cos(2Trntdt). (3.16)
o
F(n) represents the fractional contribution of the variance at frequency
n to the total variance such that the integral of F(n) over all frequen-
cies is equal to unity. It is customary in dealing with geophysical
time series to work with the absolute spectrum, S(n) = a^F(n), multi-
plied by frequency n and plotted against log n. Such a representation
93
-------
^
»
-5
1
Figure 3.34. Ratio -u'T'/w'T vs. z'/L (summer data set). The solid line (site 105)
represents the theoretical form of Wyngaard et al.(1971).
94
-------
usually results in a distinct peak in the spectrum that is apparently a
function of stratification (see Kaimal et al., 1972), and usually con-
sidered to be the length scale of the predominate eddies contributing
to the total variance. This length scale and other spectral features
determined for the St. Louis summer data set are discussed here relative
to the underlying surface roughness features and to similar determina-
tions for the Kansas and Minnesota data sets.
3.3.2 Background
Monin-Obukhov similarity theory predicts:
F(n) =! = G(z/L,f) (3.17)
where f = nz/U is reduced frequency. Thus, spectral representations of
the velocity components have generally been normalized by a (or alter-
nately by u*2 since the two are related as a function of z/L). The
procedure used here follows that of Kaimal et al . (1972) where the
spectra are normalized by u*2 2/3. T^ dimensionless dissipation rate
for turbulent energy £ is defined as:
= kze/u*3.
(3.18)
The dissipation rate of turbulent energy per unit volume e is obtained
through the Kolmogorov hypothesis for the inertia! subrange of the one-
dimensional wave number spectrum:
F(ic) =
(3.19)
Here < is the one-dimensional wave number and a is an universal constant
equal to approximately 0.5 (Kaimal et al., 1972; Wyngaard and Cot£,
1971; Kaimal, 1973). Wave number spectra are converted to frequency
95
-------
spectra through the Taylor (1938) frozen turbulence hypothesis, which
implies that the statistics of a homogeneous stationary turbulent field
at a fixed point are essentially the same as if the spatial pattern is
frozen and moved past the fixed point with the mean wind speed U. With
K = 2-irn/U and <¥(<) = nS(n), Eq. 3.19 yields:
nS(n) = ae2/3(2irn/U)-2/3 (3.20)
where e can be obtained by:
e = (2irn/U)(nS(n)/a)3/2 (3.21)
when nS(n) is evaluated at a frequency n within the inertia! subrange.
Combining Eqs. 3.18 and 3.21, we obtain a normalization which
brings all spectra into coincidence in the inertial subrange; spectra
for the longitudinal velocity component can then be expressed as:
= 0.27f-2/3. (3.22)
The spectral forms for the lateral and vertical velocity components,
which differ by a factor of 4/3 (as a consequence of local isotropy),
are given by:
u*2* 2/3 u*2*p2/3
. 0.36f-2/3. (3. 23)
3.3.3 Dimensionless Dissipation Rate
The dissipation rate of turbulent kinetic energy, calculated
from the vertical velocity spectra through Eq. 3.21, is given in
Figure 3.35 for sites 105, 107, and 109. . Each data point represents a
one-hour data sample with u* > 0.15 m/sec, U > 2.0 m/sec, and an apparent
96
-------
0.04
CO
o
CM
0.02
0.00
CO
o
03
01
0.04
0.02
0.00
SITE 107
CO
u
a
C/l
E
U/
0.02
0.00
SITE 109
-6
-I-
-4
-1-
-2
0
zVL
Figure 3.35. Turbulent energy dissipation rate € vs. z'/L.
97
-------
-2/3 spectral slope in the inertia! subrange (subjectively determined).
The dissipation rate for unstable stratification appears independent of
z'/L and lower in magnitude at the urban sites. The ratio e/gw T /T,
given in Figure 3.36, decreases with -zl/L. For site 105, the ratio is
less than unity for z'/L < -0.5 and decreases to about 0.5 at z'/L = -6
Since the shear production term is positive but small as compared to
gw'TVT, the transport terms apparently remove energy from the surface
layer at a rate greater than the dissipation rate at large z */-!_.
Wyngaard and Cote (1971) suggested an upward transport of turbulent
energy during convective conditions at about the same rate as the
buoyancy production.
It is convenient to nondimensionalize all terms in energy equation
by kz/u*3 (k is the Kantian constant), to obtain:
= 4m - z'/L + R1 (3.24)
where %i is the nondimensionalized mechanical production rate, and R1
represents the residual due to the vertical transport terms and also
includes any inhomogeneity effects.
Plots of $ versus z'/L for site 105 are shown on Figure 3.37a.
The solid curve is the estimated fit to the data points. The dashed
line represents the dimensionless dissipation rate for the Kansas data
(Wyngaard and Cot£, 1971) expressed as e2/3 = l+0.5(z/-L)2/3.
Except for neutral stratification (z'/L = 0), the two curves differ
markedly. The estimated fit to the plots for sites 105, 107, and 109
98
-------
7.5
SITE 105
5.0
2.5
0.0
7.5
5.0
VU
2.5
0.0
7.5
SITE 107
-f"
5.0
2.5
0.0
SITE 109
-*•
-4
4-'I
i
-2
0
z'/L
Figure 3.36. The ratio e/gw'T'/T vs. z'/L.
99
-------
(A
V) -*.
.2 JQ
•°-o
5: c
O) CO
"S
-------
are given on Figure 3.37b. For neutral stratification, 4>e is about
1.0, 0.7, and 1.0 for sites 105, 107, and 109, respectively. For un-
stable stratification, the curve for site 109 has the same basic magni-
tude and trend as the Kansas data. Note that for small negative values
of z'/L, £ is less than unity at all three sites.
3.3.4 Vertical Velocity Spectra
The nS(n) spectra for the velocity components and temperature were
computed for hourly data samples from which a linear trend was removed.
The procedure for obtaining the spectrum from a time series is outlined
in Appendix B. The result was eleven spectral estimates each averaged
over 2X (x=0,10) raw spectral points over equally spaced log-frequency
intervals. Thus, the first spectral estimate consists of the first raw
spectral point, whereas the eleventh estimate represents an average over
1024 points.
Vertical velocity spectra, normalized by u*2£2/3 and plotted
against reduced frequency f = nz/u, are given in Figure 3,38 for three
stratification classes for sites 105, 107, and 109. The data screening
procedure for the spectral plots was the same as given above for the
dissipation plots except the criteria for stable spectra were relaxed to
u* > 0.10 m/sec and U > 1.0 m/sec to obtain a larger sample. All spec-
tra are in coincidence in the inertia subrange (because of the normali-
zation procedure) and the scatter increases toward the low frequency
end. The scatter, however, is not larger than expected from the uncer-
tainty of the individual spectral estimates. The dashed lines on the
plots represent a composite Kansas spectrum for neutral stratification
(Kaimal et al., 1972). The solid lines above or below represent the
101
-------
10
04
c
*-
+ ¥
io-
10
-2
SITE 105
UNSTABLE
NEUTRAL
i i • i j 11 \ »
ID"3
10 r~
1
k
~
10-1 r
e
L
^
10-2
m-3
-
^
-
"• '"i ' ' ' " "" ' ' ' '
* +
* V +. * ^
- * ^**-*^*WV
* +** ^^^
* X^ /^ ^
T^ * /
* *• *^* /
-f ^ -*-+ /
lS /
* SITE 105
STABLE
| F , , . iJ . . . , J L I . I 1 J tl . Jit
:
-.
1
-
*tH
^
•
1
-
1 d . J
(=
-
SITE 107
UNSTABLE
L . . I J I
1 ' I t I J . L I J
SITE 107 l
NEUTRAL :
i i i i iinJ i j i i 1111\
10
-3
10
•2
10
-1
10 1C*3 ID'2
-1
10
f = nz'/U
SITE 109
UNSTABLE
SITE 109 1
NEUTRAL
1 • • a
t-
SITE 109
STABLE
10 10*3
10'
10
-1
10
Figure 3.38. Vertical velocity spectra for sites and stability classes indicated.
Dashed lines are Kaimal et al. (1974) composite spectrum for z7L=O. Solid
lines above and below are the Kaimal spectra for z'/L=—2 and 1 respectively.
102
-------
Kaimal spectra for z'/L = -2 or +1, respectively. The peak of the St.
Louis spectra are shifted towards lower frequencies compared to the
Kansas spectra, especially for neutral and stable stratifications. For
a given stratification class, no obvious differences occur among the
spectra for the St. Louis sites. In general, these spectra are in
agreement with the Monin-Obukhov similarity prediction (i.e., the spec-
tral form appears to be a function of z'/L and f).
The spectral representations shown in Figure 3.38 are inadequate
for comparison of the spectra for different sites. Turbulence length
scales derived from the spectral form provide a quantitative means of
comparison of the spectra for different sites and atmospheric influ-
ences. Three commonly referenced length scales are the wavelength cor-
responding to the peak in the logarithmic power spectra Lm = U/n
m
(m refers to the value associated with the peak in the spectrum), the
CO
Eulerian integral length scale L-j = Uf R(t)dt, and the dissipation
o
length scale Le = aw/e- For homogeneous, fully-developed turbulence
these length scales are not independent of each other. For the w
spectra and neutral-to-stable horizontal velocity spectra, which are
expressible in the form:
S(n) = A
(3.25)
Bn5/3 '
Kaimal (1973) has shown that:
Lm = 2*11 = TT
(3.26)
Kaimal (1973), Hanna (1968), and Wamser and Miiller (1977) have shown
empirically the relationship between Lm and Le to be valid for w
spectra. The procedure for this study was to fit Eq. 3.25 to the raw
103
-------
spectra estimates to determine A and B and subsequently the length scale
and dissipation rate (see Appendix B). However, Eq. 3.25 was fitted
over different ranges of the spectrum such that independent estimates of
Lm, L-J, and Le could be obtained (e.g., L-j was determined by giving
greater weight to the low frequency end of the spectrum and Le by
giving more weight to the high frequency end). The relationships be-
tween these several length scales were approximately as given by Eq.
3.26 and thus only Lm is discussed here.
Information on the maximum wavelength of the spectrum is conven-
iently and historically presented in terms of the associated normalized
frequency fm = nmz'/U = z'/Lm. This inverse normalized length
on
scale is plotted against z'/L in Figure 3.39 for site 105, 107, and
109. The ratio is essentially constant for very unstable stratificati
at a value of about 0.18 for site 109 and 0.15 for site 105. For neu-
tral stratification the ratio is 0.31, 0.29, and 0.36 for sites 105,
107, and 109, respectively. The data scatter for stable stratification
is large; however, it is obvious that the ratio increases with increas-
ing stability and the slope is greater for site 109 than 107. The solid
line shown on the plots for site 109 is the empirical fit to the Kansas
data (Kaimal et al., 1972), which has a value of 0.5 at z'/L = 0 and
0.17 for unstable stratification. Eversole (1979) found z'/Lm(w) to
be about 0.13 for unstable stratification at the Boulder tower.
The diurnal variation of Lm(w), visually estimated from plots, is
given in Figure 3.40. A land-use influence is apparent; Lm is general-
ly larger at the urban sites during the nocturnal hours, reflecting the
less stable conditions associated with the urban heat island and larger
104
-------
1
0
SITE 105
1
N
0
SITE 107
N
_ SITE 109
-6
-4
-2
0
z'/L
Figure3.39. Plots of z'/Lm(w) vs. z'/L for summer data set. The solid
lines represent results for Kansas data set (Kaimal et al, 1972),
105
-------
200
E
*
^•fc
5
100
I I I I I I I I I I I I I I I I I I I I I I I I
SITE 105
SITE 107
SITE 109
•-. \
t * •
I I I 1 1 I II J I L L_L I _1 I I I I 1 I I I I
0
8
10
12 14 16 18
20 22
TIME (CST), hour
Figure 3.40 Estimated fit to plots of diurnal variation of Lm(w) for sites
105, 107, and 109.
106
-------
surface roughness. During the convective period, Lm for site 107 is
significantly lower than for sites 105 and 109. This, we believe,
reflects the difference in stratification at the sites which are
strongly influenced by surface features. In other words, the midday
values of Lm at site 107 are driven by z'/L values in the range of
0 to -1, while those for sites 105 and 109 are driven by z'/L values
extending to -6. Surface features strongly influence boundary layer
structure at the height of measurement; however, the differences in
Lm(w) between sites 107 and 105 would be expected to decrease at
greater distances above the surface. The more rapid decrease of Lm(w)
at site 109 during the late afternoon hours suggests a boundary layer
gradually decreasing in height in the urban area, but collapsing rather
quickly in the rural environs.
Wamser and Mtiller (1977), using data at 50 m from a tower in sub-
urban Hamburg, Germany, found Lm(w) to be inversely proportional to
the surface roughness length during neutral and convective conditions.
This is in contrast to the St. Louis data set, where Lm(w) is larger
for the urban area. The data, however, do not appear to be strongly
influenced by surface roughness as suggested in Figure 3.41, where
Lm(w) is plotted as a function of wind direction for site 107. In
contrast to Figure 3.3 (z0 versus wind direction) essentially no
f
variation of Lm(w) with wind direction exists.
3.3.5 Horizontal Velocity Components
The spectral plots for the lateral and longitudinal velocity com-
ponents are given in Figures 3.42 and 3.43, respectively. The dashed
lines again represent the generalized spectra of Kaimal et al. (1972)
107
-------
4-
4-
4-4-
o
CO
f>
CO
O
O
o
CO
CO
CO
|
0)
O
CN
O
UJ
o:
5
o
z
S
o
S
£
= -a
•o
CO
CD
•5
a>
iT
UJ '
108
-------
c
*•*
>
c
SITE 105
UNSTABLE
H
1
10
10
,-3L
TO-3
10
SITE 105 1
NEUTRAL
10
-1
|r
E
t
SITE 105
STABLE
10*3
SITE 107 1
UNSTABLE
SITE 107 1
NEUTRAL !
I . 1 I I I t
Ttr
TTtfl
SITE 107
STABLE
,.,,,! , - , ,
SITE 109 1
UNSTABLE :
I ' r • i I \
SITE 109
NEUTRAL
r-
^
C
-
1
c~
—
+
^
*
* •*•
*
* * _<.
+ ^T^ * * ^ i ,
* J- 1 • If* J
'"' ' ' ' '"3
-i
"
_
, :
: ^-^.^i-^^
r^^f^
: * - „ _*•>
SITE 109
STABLE
10
,-3
10
-2
10
-1
1
10 1(T3 10'2
-1
10
f - nz'/U
1
10 10'3 10-2
10
,-1
10
Figure 3.42. Lateral velocity spectra for sites and stability classes indicated. Dashed and
solid lines are as indicated for Figure 3.38. The upper dashed line for site 107 is an exclu-
sive zone (see text).
109
-------
SPECTRA FOR LONGITUDINAL VELOCITY COMPONENT
10
icr
i ^
NT
L 4-
SITE 105
UNSTABLE
10
10
-3
L. _
SITE 107
UNSTABLE
SITE 109
UNSTABLE j
a • *
JJJ
(h
•0-
*
a
10-2 .-
10
SITE 105 i
NEUTRAL :
SITE 107 1
NEUTRAL :
-3
SITE 109 1
—
NEUTRAL i
L . _ * 1 L
10 g
r - *:
SITE 107
STABLE
f - nz'/U
Figure 3.43. Same as for Figure 3.42 for longitudinal velocity component
110
-------
for neutral stratification, and the solid lines for z'/L = -2 or +1.
The upper dashed line for site 107 (unstable stratification) on both
figures is what Kaimal et al. describe as the upper limit of an exclu-
sion zone given by z'/L = -0.0. They suggest a step shift in energy as
the atmosphere passes from neutral to unstable stratification; conse-
quently no spectra should fall within this region.
The St. Louis spectra, while qualitatively compatible with the
Kaimal form, exhibit some obvious anomalies. For example, the peak in
the stable spectra appears to be shifted to lower frequencies, and a
low frequency mesoscale component occurs for both stable and neutral
stratifications. The mesoscale component, which is more pronounced for
the lateral velocity, corresponds to wavelengths on the order 3 to 10
km. These longer wavelengths may be due to both land use and urban
scale induced perturbations on the flow. However, the mesoscale com-
ponent is also present in the data for the rural site.
The existence of the mesoscale spectral components makes identify-
ing a representative peak in the spectral plots difficult for unstable
stratification. Kaimal (1978) discussed the unstable horizontal veloc-
ity spectra as consisting of three regions: an inertia! subrange
region that falls off as f-2/3 (this region follows Monin-Obukhov
similarity, i.e., nS(n)=f(z'/L,f)); a low frequency energy generation
region governed by the height of the mixed layer z-j; and a transition
or matching region between the above two. The inertia! subrange
and the matching region described by Kaimal are generally apparent in
i
the St. Louis unstable spectra. The low frequency region is masked by
the mesoscale energy, and the lack of a mixed layer scaling length in
•
the analysis.
Ill
-------
The diurnal variation of the peak wavelength in the lateral and
longitudinal velocity components are shown on Figures 3.44 and 3.45.
These visually smoothed and averaged plots are based on Eq. 3.25, which
may not be completely appropriate for the horizontal velocity components
(especially the lateral component), and may also be contaminated by
energy at mesoscale frequencies. Thus they should be interpreted with
caution and only in a relative sense, and may be considered an underes-
timate.
Urban influences appear in the data. Spectral peaks for the noc-
turnal hours are highest at site 105 and during the convective hours
lowest at site 107. The former feature is likely associated with the
urban heat island and the latter with the relatively less unstable
conditions existing at site 107 during the afternoon period.
The afternoon transition of Lm(u,v) at all sites is in contrast
to that indicated by Caughey and Kaimal (1977). They suggest a rapid
collapse of convection near sunset. Our data (which are supported by
aircraft measurements of turbulence over St. Louis during the same
period) suggest the layer of vigorous mixing decreases more gradually
over a 3 to 4 hour period. Urban lidar mixing heights from Figure 3.19
are superimposed on Figures 3.44 and 3.45. The lidar mixing heights are
approximately equal to Lm during the morning transition period and
considerably larger than Lm during the evening transition. The peak
wavelength /is believed to more accurately reflect the actual height of
mixing in the atmosphere than the lidar mixing heights. The decrease in
the lidar mixing heights later than the spectral peak is believed due
112
-------
1600
1400
1200
1000
;» soo
600
400
200
i i i i i i i i i \ i \ i i
SITE 105
SITE 107
SfTE 109
' '0' ' ' ' ' '
MIXING HEIGHT O
i i i i . i i i i i i i i i i i i I i i i. i i
8
10
12
14
16
18 20
22
TIME (CST), hour
Figure 3.44. Estimated fit to plots of diurnal variation of Lm(v) for sites 105, 107, and
109. Open circles-represent urban mixing heights.
113
-------
o
o
o
to
o
o
I
I
I
I
o
o o
ui LLI ui
t t t
lo
.--X'
o
LU
I
a
o
o
o
o
CN
o
o
o
o
o
CD
o
o
CM
CN
O
CN
00
3
C
CN
CO
CN
O
o
CM
3
-------
to residual aerosol, which may take several hours to be advected out of
the city.
Urban mixing heights and heat flux from Figure 3.19, Lm(v) for
site 105, and av for both sites 105 and 109 are given in Figure 3.46
for the heating period of the day. The peak wavelength appears to be a
better indicator of av than z\. Also note the similarities between
the diurnal anomalies of av for sites 105 and 109 on Figure 3.46 and
the diurnal anomalies of Lm(v) on Figure 3.44. These features suggest
that mixed layer turbulence responds more directly to Lm than to Zj.
Kaimal et al. (1976) and Kaimal (1978) determined Lm(u,v) to be
about 1.3 to l.Szi. While Figures 3.45 and 3.46 do not indicate these
values, the form assumed for the spectral shape from which the peak was
determined (Eq. 3.25) would generally underestimate the peak. Visual
inspection of the unstable spectral plots on Figures 3.42 and 3.43 sug-
gest that spectral peaks on the order of 1500 to 3000 m would not be
unreasonable. However, the observations were made at a height of about
0.022^ and could be responding to a mixture of surface layer and mixed
layer seal ing.
3.3.6 Temperature Spectra
Within the inertia! subrange, Corrsin (1951) proposed the tempera-
ture spectrum as given by:
(3.27)
where B is a constant with a value of approximately 0.8 (Wyngaard and
CotS, 1971) and N* is the dissipation rate of T^/2. Kaimal et al .
(1972) developed the following normalization based on this form, which
115
-------
1600
1500
1400
1300
1200
1100
1000
E
K- 900
X
o
I 800
o
I 700
s
600
500
400
300
200
100
I I
MIXING HEIGHT
SITE 105
SITE 109
HEAT FLUX O
Lm(v) A
I
I
I.I I 01 I
1.3
1.2
1.1
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
o
CA
E
0.24
0.20
0.18
0.16
0.14
0.12
0.10
0.08
0.06
0.04
0.02
O
0)
X
D
<
8 10 12 14 16
TIME(CST) hr.
18 20 22 24
Figure 3.46. Comparison of z;, Lm/ w'T, and av for urban sites and av for rural
site for heating period of the day.
116
-------
collapses all spectra to a common form within the inertial subrange:
17, = 0.43f2/3. (3.28)
l3
Here $ = kzN*/u*T*2 is the normalized dissipation rate of temperature
and all other symbols have been defined previously.
The normalized temperature spectra for three stratification
classes and three sites are shown in Figure 3.47. Again the Kaimal et
al . (1972) composite spectra are given for comparison. Similar to the
velocity spectra, the peaks in St. Louis stable temperature spectra
appear shifted to lower frequency, but to a larger extent, for site 107
as compared to site 109. The neutral spectra are generally ill -defined
and, along with the stable spectra, have a significant mesoscale com-
ponent. The spectra for the unstable stratification consist of a broad
peak and generally lack a mesoscale component. Again, differences
between the urban sites and site 109 are not obvious if they exist at
all.
117
-------
h-
*
10
1 r
10-1 -r
10
-2
SITE 105 1
UNSTABLE 1
10
10
-3
10*1 r
10-2 r
10'3
10
SITE 105
NEUTRAL
10
10
-2 ^
SITE 105
STABLE
10
-2
10
-1
SITE 107 i
UNSTABLE I
SITE 107
NEUTRAL
SITE 107 1
-•
STABLE :
10
-3
10
-2
SITE 109
UNSTABLE
SITE 109 1
NEUTRAL \
r- -
SITE 109 I
STABLE :
10-
f - nz'/U
10 10-3 10*2
10
-1
10
Figure 3.47. Temperature spectra for sites and stability classes indicated. Dashed and solid
lines are as indicated for figure 3.38.
118
-------
4.0 SYNTHESIS OF RESULTS
4.1 INTRODUCTION
The basic purpose of this study was the evaluation of Monin-
Obukhov similarity theory for description of turbulence above the com-
plex and rough urban surface. Discussions relative to this topic are
contained in Section 4.3. However, the analyses presented in Section 3
also encourage discussion with respect to general boundary layer phenom-
ena and urban anomalies not directly addressed by similarity theory.
These latter topics are briefly addressed below.
4.2 BOUNDARY LAYER PHENOMENA
Land-use features varied significantly among the four sites. Thus
at the outset each site was characterized numerically by an estimated
displacement length and a site-averaged roughness length calculated
through the similarity wind profile formulation. Estimated displacement
lengths ranged from 2 to 6 m at the urban sites and site-averaged rough-
ness lengths from 0.7 to 1.7 m (Table 3.1). Surface roughness length
varied significantly with wind direction at both urban and rural sites,
suggesting the surface features were not homogeneous in space. The sur-
face wind stress was proportional to Z0 (as expected from the method
of calculation of Z0). Relatively large values of stress occurred at
the urban sites throughout the diurnal cycle. The value of u* for the
convective period of the day was 0.2 or larger at all sites.
The surface energy budget also varied with the composition of land-
use features. Afternoon values of heat flux at the urban commercial
site (105), which had a high percentage of paved areas and few trees,
119
-------
were about twice those at the rural site (109). During the nocturnal
hours, the heat flux was generally negative at site 109, but was seldom
negative at site 105. Latent heat flux was significantly greater at
site 109 than at site 105; afternoon Bowen ratios of 0.5 and 2.0 were
characteristic of sites 109 and 105, respectively. The heat flux at
urban site 107, which had numerous tall trees, was similar to that at
site 109 during daylight hours. At night, site 107 had a zero or very
small negative heat flux characteristic of an urban site.
The boundary layer stratification reflected the land-use features
responding to the ambient air flow and solar radiation. Based on compu-
tations of z'/L, which includes the effects of both heat flux and sur-
face stress, site 109 was strongly stable at night and strongly unstable
during the afternoon. Site 105 was neutral and strongly unstable for
the two periods, respectively. Site 107 was essentially neutral at
night but only slightly to moderately unstable during the convective
period of the day (due to the large surface stress and relatively small
heat flux).
Partly in response to the temporal and spatial variation of strati-
fication, the diurnal variation of most turbulence parameters differed
significantly between the urban and rural environs. The turbulent wind
standard deviations, turbulence intensities, and the spectral peak wave-
lengths were without exception higher at the urban sites during noctur-
nal hours due to the urban heat island and associated deeper momentum
boundary layer. The turbulence parameters tended to converge during the
morning transition period (i.e.,the normalized turbulence structure was
similar in both the urban and rural environs between 0800 and 1000 h)
120
-------
and diverged during the afternoon transition of the boundary layer to
stable stratification.
The afternoon transition of the boundary layer from unstable to
stable stratification in both urban and rural environs occurred over a
relatively long period of time. Both horizontal and vertical turbulent
intensity components peaked out about noon and declined steadily to near
their nocturnal equilibrium value by 1800 h. The velocity variances
while peaking about noon declined only slightly to 1400 and then steadi-
ly to 1900 h. The peak wavelength in all of the velocity components ex-
hibited behavior similar to the velocity variances, i.e., they decline
steadily after 1400 h and well in advance of decline in lidar-determined
mixing heights. These observations suggest that free convection turbu-
lence should be scaled with the peak wavelength of the horizontal veloc-
ity components rather than the height of the mixed layer. During the
late afternoon period these two scale lengths may differ significantly.
Turbulent mixing to the top of the "mixed layer", as specified by lidar
or temperature-dewpoint profiles, probably does not cease abruptly after
the heat flux peaks. We suggest, however, that the probability of any
thermal reaching the top of the "mixed layer" decreases significantly
past 1300 to 1400 h and continues to decrease to a near zero value prior
to sunset, such that the peak in the energy spectrum is continually
shifting to higher frequencies. The probability of a thermal reaching
z-j or any height within the mixed layer after 1400 h likely depends on
the height and strength of the mixed layer capping inversion and on the
surface energy budget, which may have significant spatial variability in
urban environs.
121
-------
4.3 SIMILARITY THEORY
Results of the validation tests of current similarity parameteri-
zations using this data set were mixed. The nondimensionalized turbu-
lence parameters (i.e., the velocity and temperature variances, turbu-
lence intensities, and spectra) for site 109 generally behaved as ex-
pected from similarity theory; the average magnitude of the data plots
as a function of z'/L was consistent with corresponding plots for ideal
sites. However, the scatter of the data points was large; probably due
to the nonhomogeneous distribution of land-use features and the abrupt
change in roughness features near the tower in the easterly quadrants.
A fully developed turbulent boundary layer may not have existed with
easterly winds. The observational scatter for site 109 is representa-
tive of other nonideal sites (e.g., see Weber et al. 1975) and is indic-
ative of the uncertainty inherent in the application of the similarity
approach to practical diffusion problems.
The nondimensionalized turbulence parameters for the urban sites
were generally an orderly function of z'/L; the data plots exhibited
less scatter than the corresponding ratio for site 109. The plots of
some urban parameterizations (e.g., see aj/T* in Figure 3.30) were in
very good agreement with the empirical expressions derived by others for
flat homogeneous sites (e.g., Kansas). Other nondimensionalized ratios
for the urban sites, for example aw/u*, depart noticeably from Monin-
Obukhov similarity theory as empirically verified for homogeneous sites
of small roughness (Figure 3.10). The departure is most apparent in the
region of forced convection where the slope of aw/u* with -z'/L is
smaller than expected. For large -z'/L (approaching free convection),
122
-------
the ratio was lower than expected; however, the slope is approximately
proportional to (-z'/L)l/3 as predicted by similarity theory. Even
under neutral stratification the data suggest site specific differences;
the normalized vertical velocity variance decreases with increasing
Z0. Similar anomalies occurred with ay/u* and QU/U*; a decrease in
the ratios with increasing roughness under neutral conditions Figure 3.7,
and lower than predicte d values for slightly unstable stratification
(Figure 3.10). The lateral and vertic al turbulence intensities were
essentially as expected from the similarity wind profile equation for
neutral stratification, but much lower in magnitude than expected for
z'/L = -0.5 (Figure 3.28). The nondimensionalized dissipation rate of
turbulent kinetic energy behaved much like the ratio aw/u*- At urban
site 107, * was significantly less than the expected value of unity
* Cr
for neutral stratification, and at site 105 it was lower than expected
throughout the range of unstable stratification (Figure 3.37). The data
plots for site 109 were in general agreement with similarity theory;
however, the scatter of the plots was large.
The differences between the derived empirical similarity forms for
the urban sites and those for the rural site are about 10 to 15% for
neutral and stable stratifications and about a factor of two for unsta-
ble stratification. For many applications (e.g., atmospheric diffusion
estimates) these differences are within the reliability of the application
form such that Monin-Obukhov similarity theory, or a simple modification
thereof, can be applied to urban areas (such as the forms given in Ta-
bles 3.3 to 3.5 for the urban sites).
The general consistency of the departure of the urban data from
123
-------
similarity theory suggests that it may be possible to describe a physi-
cal basis for the departure. The normalized eddy energy budget (Eq.
3.24), along with the expression for e given by Eq. 3.7 and Le given
by Eq. B-8 relates the velocity and dissipation parameterizations to
other relevant parameters through:
V"* = <(*m - z'/L +R')/1.2fm)1/3. (4.1)
Thus the anomalies in aw/u* and e may reflect corresponding anomalies
1n *m> fm» Z'/L, and in the residual term R1 representing primarily
vertical transport and advection of turbulent energy. The possibility
that errors in calculating these terms contribute to the anomalies in
the similarity parameterizations is examined below.
The reduced frequencies fm = z'/Lm(w) averaged for neutral strati-
fication were 0.31, 0.29, and 0.36 for sites 105, 107, and 109, respec-
tively. Average peak wavelengths for the three sites were 93, 89, and
86 m, respectively. For unstable stratification, fm was again smaller
and Lm larger for the urban sites. This trend is opposite to that re-
quired to give the observed variation of average aw/u* for the three
sites (Figure 3.7). The existence of a larger spectral length scale
above the rough urban surface, however, raises the possibility that m
may be less than unity for neutral stratification and may account in
part for the lower values of aw/u* and e at the urban sites. Assuming
R1 = 0 for neutral stratification (z'/L = 0), values of $e = 0.65 and
1.03 are required for sites 107 and 109, respectively, to satisfy Eq.
4.1 for the observed values of aw/u* and fm. These values of *m are
consistent with the values of for neutral stratification shown in
Figure 3.37.
124
-------
Mixing length may be defined as (Sutton, 1960):
£ = u*/dU/dz
(4.2)
where £ is conventionally equated to kz1 to obtain the differential
equation of the logarithmic wind profile for neutral stratification (Eq.
1.2). This formulation of the wind profile does not contain a scaling
parameter characteristic of the roughness elements and thus is valid
only at heights significantly above the roughness elements. Tennekes
(1973) suggested this minimum height to be on the order of 100z0. The
urban data reported herein were obtained at heights of 20 to 50Z0-
A schematic comparison of the normalized neutral velocity profile
and that suggested by Garratt (1978a) for flow in the immediate vicinity
of the roughness elements is shown in Figure 4.1. The velocity does not
increase as rapidly with height above the rough surface as predicted by
the logarithmic law. Consequently m determined using kz1 as the mixing
length may be less than unity. Garratt (1978a, 1978b) has shown this to
be the case above a rough surface of trees and shrubs (z0 = 0.4 m). He
found £ for
neutral stratification). However, Mulhearn (1979), in a review of wind
tunnel studies, found m to be generally greater than unity above rough
surfaces.
For unstable stratification the departure of the urban data from
empirically accepted forms is more obvious and also more difficult to
125
-------
C
UJ
Z
ku
Figure 4.1. Normalized velocity profile in vicinity of roughness elements (neutral
stratification).
126
-------
explain. Comparison of the plots of aw and av versus Uf, for sites 105
and 109, given in Figures 3.11 and 3.18, suggests either the heat flux
is less effective in the production of turbulent energy above a rough
surface, or the heat flux at site 105, which is larger than at site 109
by a factor of two, is in error. This is further confirmed in Figures
3.14 and 3.23. If the heat flux at site 105 were halved a universal
similarity form could be specified without regards to surface features.
We have examined and rejected the possibility of significant error in
the heat flux at site 105 (and consequently in z'/L, Eq. 4.1). This
conclusion is supported by the fall data set, where the average midday
heat flux at site 105 was smaller than the heat flux at sites 107 and
109. Yet the derived empirical similarity relationships are consistent
with the summer data set (Figures 3.13 and 3,14). However, the buoyancy
parameter for sites 109 and 107 could be underestimated by as much as
20% during the afternoon hours due to neglecting the effects of water
vapor fluctuations (Brook, 1978).
Eq. 4.1 was evaluated for R1 for unstable stratification using ob-
served values of aw/u* and fm, and the Businger et al. (1971) expression
form. For site 109 the residual term was essentially constant at a
value of -0.15, for z'/L between -1 and -5. For site 105, the residual
term was -0.8, -1.4, and -3.2 for z'/L = -1, -2, and -5, respectively.
Thus there appears to be a significant net removal of turbulent energy
from the surface layer at site 105 of about 60 to 80% of the buoyancy
production.
The residual term (Eq. 4.1) includes both vertical flux of turbu-
lent energy and advective effects. The data plots for sites 105 have
127
-------
relatively little scatter, suggesting that advection and nonhomogeneity
of the turbulent field are not significant features contributing to the
anomaly in the urban turbulence structure. It is unlikely that the
advective component would have the same magnitude and sign regardless
of wind direction. Thus the anomalies in the similarity parameters for
site 105 may be associated with vertical transport of eddy energy away
from the surface layer (possibly associated with urban-scale circula-
tion features) or other undetermined effects.
4.4 CONCLUSIONS
The major thrust of this study was to describe the structure of
turbulence in the surface boundary layer of an urban area. The approach
was through evaluation of the present data base in light of recent em-
pirical verifications of Monin-Obukhov and free convection similarity
theories (i.e., the results of the Kansas and Minnesota boundary layer
t
experiments). From the extensive analyses of the turbulence data ob-
tained in the St. Louis environs, it is concluded that many nondimen-
sionalized turbulence parameterizations (e.g., aw/u*, au/u*, ov/u*) for
the urban sites differ significantly from those for the rural site: the
parameterizations for the rural site were in general agreement with
similarity theory. The following more specific findings amplify this
general conclusion:
* The standard deviation of the vertical velocity at both urban and
rural sites can be described as a function of u* and Uf. The hor-
izontal velocity components scale with u* and w*1. In this respect
the urban data can be described within the framework of similarity
theory.
128
-------
* The empirical similarity constants derived for the urban sites were
smaller than those for the rural sites.
* For neutral stratification the normalized velocity standard devia-
tions were inversely proportional to surface roughness. The nondi-
mensionalized dissipation rate had a similar tendency; it was con-
siderably less than unity at site 107. These anomalies from simi-
larity theory are believed due to the roughness wake region extend-
ing to the height of the instrumentation at site 107.
* For unstable stratification the urban velocity standard deviations.
turbulence intensities, and <>e were smaller than expected from sim-
ilarity theory. Flux divergence of turbulent energy due to organized
and possibly stationary vertical motions over portions of the city
is the likely cause of the anomalies.
* For stable stratification the velocity variances were a linear
function of u* (i.e., °w/u* = constant) at each of the sites. The
individual slopes (for each site) appear to be a function of Z0.
* Temperature spectra at all sites compared well with the Kansas
empirical form of Monin-Obukhov similarity theory.
* Turbulence length scales were larger for the urban site suggesting
that 4>m may be correspondingly smaller above the rough urban surface.
* The peak wavelength of the longitudinal velocity spectrum appears
more appropriate for free convection scaling than z-j. During the
afternoon transition of the boundary layer to stable stratification,
the two length scales may differ significantly.
129
-------
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136
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APPENDIX A
Gill UVW ANEMOMETER
Computed values of stress, heat flux, and vertical velocity vari-
ance are extremely sensitive to the condition, exposure, and leveling
precision of the Gill anemometer. Thus a special leveling device was
attached to the Gill instruments used in this study and extensive pre-
and post-field wind tunnel calibration tests conducted. The leveling
method, calibration, and response characteristics of the Gill instrument
are discussed below.
A-l Leveling
A number of investigators have discussed errors in the stress and
heat flux resulting from tilt of the w sensor (Kaimal and Haugen, 1971;
Dyer, et al., 1970; Dyer and Hicks, 1972; Wesely and Hicks, 1975). Er-
rors on the order of 14% per degree of tilt for the stress and 4% for the
heat flux are typical under ideal conditions. Kaimal and Haugen (1969)
concluded that tilt should be less than 0.1° for sensors used in the
measurement of stress. The instruments used in this study were aligned
on the towers using a high-precision plumb bob level attached to the
shaft of the anemometer, as shown in Figure A-l. The leveling system,
aligned with the w-arm, had a precision of 0.1°. The alignment of the
w-arm on the towers was further checked with a transit from the ground.
In all cases, the w-arm was vertical within the accuracy of the transit.
The plumb bob leveling device interfered with the flow over the
back of the u and v arms. Consideration was given for this interference
137
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Figure A-1. Photograph of Gill anemometer showing attached plumb bob leveling device.
138
-------
in the calibration of the instrument and in the orientation of the in-
struments on the towers with respect to the prevailing wind. For the
summer experimental period the u and v arms were oriented to 135° and
225°, respectively; the orientation for the fall period was 190° and
280°, respectively.
A-2 Calibration
The Gill UVW anemometer was designed to respond only to the compo-
nent of the wind parallel to the propeller shaft, i.e., a true cosine
response. However, the manufacturer's calibration (R. M. Young Company,
1973) indicates a deviation from the cosine response which increases as
the wind becomes more normal to the shaft. A computer algorithm for
correcting for noncosine response in the Gill anemometer based on the
manufacturer's calibration is given by Horst (1972, 1973). Hicks (1972)
showed that the cosine correction for the horizontal components was a
function of wind speed. The vertical correction was essentially linear
over a range of elevation angles of ±30° and equal to approximately
1.25 times the indicated velocity. Hicks also found that the response
of the Gill instrument deteriorates significantly when employed in the
field for an extended period. Because of such characteristics of the
instrument and the addition of the leveling device to the shaft, the
Gill instruments used in this study were extensively calibrated in the
EPA wind tunnel. Conclusions and operational procedures resulting from
the calibration program are:
1. The correction function for noncosine response was relatively con-
stant for wind speeds greater than 2.0 m/sec. Those hours with
139
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wind speeds less than 2 m/sec were normally not used in the analyses,
and a single cosine correction function was applied to all the data.
2, The anemometer had a much better response at small angles (±10°) to
the normal when the flow was from the shaft side of the propeller.
This was especially evident at higher tunnel speeds and is due to
turbulence created by the flow over the shaft. To eliminate this
source of error, shaft extenders were used on all anemometers as sug-
gested by Hicks (1972), A shaft extender is a small cylindrical
»
piece of plastic about 8 cm long and the same diameter as the shaft.
It is affixed to the propeller such that the physical configuration
is symmetrical on both sides of the propeller. Dyer et al. (1967)
found that use of the shaft extender provided a more symmetrical
response and reduced the stall angle from 4° to 2°.
3. The leveling system attached to anemometer affected the response
when the wind was from the back side. This was accounted for in the
cosine correction for the instrument.
4. Pre- and post-field speed calibrations were not significantly differ-
ent. However, the bearing friction of the instrument increased ap-
preciably. The average starting speeds were about 41 cm/sec and 72
cm/sec for the pre- and post-field tests, respectively. Average
stalling speeds were considerably lower, about 28 and 41 cm/sec,
respectively.
5. The distant constant of the Gill instrument is about 1 m for the wind
parallel to the shaft, and 2.5 m for the wind normal to the shaft
(Gill, 1975; Hicks, 1972). The EPA wind tunnel tests gave similar
results. These are expressed as time constants as a function of
tunnel speed and angle of attack to the propeller in Figure
140
-------
A-3 Response Characteristics
The Gill anemometer is usually treated as a simple linear system
having a time constant T responding to a sine function input. For two
in-phase systems, the fractional response as a function of frequency cu
is given by:
(A-l)
Eq. A-l is evaluated in Table A-l for selected variances and covar-
iances for three cyclic frequencies (n = 0)72*), and for the time con-
stants appropriate to 2 and 5 m/sec (Figure A-2). The Gill anemometer
has an obvious high-frequency loss in response, which is larger for low
wind speeds and for the w component. This problem is not considered
serious in the present study since, at the height of instrument exposure
(31 m), little turbulent energy is contained at reduced frequencies (f =
nz/U) greater than 3 (n * 0.3. Horst (1973) has shown that data
corrected only for cosine response give estimates of the second moments
comparable to that of a sonic anemometer, up to frequencies of 0.3.
141
-------
u
a)
o
CJ
WIND SPEED, m/sec
Figure A-2. Time constant as a function of wind speed for the temperature system (dashed
line) and Gill anemometer for four wind angles to the propeller.
142
-------
Table A-1. Effect of Instrument Response on Second Moments
f
.15
1.5
15
f
.06
.6
6
n
.01
.1
1.0
n
.01
.1
1.0
u'2
w'2
u'w'
w'T'
u = 2 m sec"1
1.0
.85
.05
.99
.64
.02
*
1.0
.71
.03
1.0
.70
.03
u = 5 m sec"^
1.0
.98
.23
.99
.93
.12
1.0
.95
.18
1.0
.91
.10
143
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APPENDIX B
DATA PROCESSING
B-l Initial Processing
A simplified flow diagram of data processing is shown in Figure
B-l. The RAMS tapes contained both 1-min average RAMS data and 1/2-sec
turbulence data in POP 8 computer language. The turbulence data were
subsequently extracted through TRANSLATOR, converted to UNIVAC language
and packed on the TURB tapes. Both RAMS and TURB tapes have been
archived.
TURBCALC is the basic data processing program. One-hour data
blocks (7200, 1/2-sec values) from each of the five sensors (tempera-
ture, humidity, and three components of the wind) were read by TURBCALC
in integer form, along with initialization and calibration factors. The
following operations and computations were then carried out within
TURBCALC or by subroutines called by TURBCALC in the listed order:
1. The integer format of the data was converted back to voltage values
as originally input to the data acquisition system (with a loss of
resolution of 2.4 mv). All subsequent processing was in voltages and
the calibration factors were applied just prior to the output stage
to convert the results to engineering units.
2. The anemometer data were corrected for noncosine response through
subroutine COSCO.
3, Hourly average values of the voltages were computed for each of the
sensors along with quantities representing the mean wind speed and
standard deviation of elevation angle. Mean wind direction and
144
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RAMS
TAPES
TRANSLATOR
SPECTRA
I
TURB
TAPES
TURBNEW
COSCO I
DQiR
JFC SITE 105
JFC SITE 107
JFC SITE 109
JFC SITE 111
MEANS AND VARIANCES
TURBCALC
JFC STATS
COSCO
SIGMA
LTSQ
H
DEV
Figure B-1. Simplified flow diagram of data processing.
145
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standard deviation of azimuth angle were obtained through subroutine
SIGMA.
4. The basic data were then subjected to trend removal through subrou-
tine LTSQ, which applies a 2nc*-order least-squares fit to the hour-
ly time series.
5. Subroutine DEV takes the results of LTSQ and calculates the departure
of the points in the time series from the trend vector. New time
series of the turbulent components of u1, v1, w', T', q1 were
obtained in which u1 is in the direction of the mean wind and v1 is
in the cross wind direction.
6. Hourly variances and covariances were computed from the turbulence
time series and surface boundary layer parameters calculated (e.g.,
heat flux, stress, Monin-Obukhov length, etc.).
7. Calibration factors were applied and the computed parameters printed
out and written to a disk file, JFCSTATS.
B-2 Spectral Computations
Processing of the data for the spectral computations was accom-
plished through the left-hand branch of the flow diagram in Figure B-l.
TURBNEW is the driving program which calculates the mean wind speed and
covariances for normalizing the spectra; it then calls DQIR. DQIR in
turn calls several subroutines which detrend the data, compute the
spectrum by a FFT routine, fit a preassumed spectral shape to spectral
estimates, and write the output for each site to an individual file.
The time series were linearly, rather than quadratically, detrended as
was done for TURBCALC. The rationale for this approach was that a more
146
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severe filter could be applied to the spectral estimates if necessary.
The detrended time series were applied to a mixed-radix fast Fourier
transform (FFT) algorithm (Singleton, 1969). The 7200, 1/2 second hour-
ly data samples applied to the FFT resulted in 3600 spectral estimates
for each component (u, v, w, and T). A subsequent processing routine
provided 14 values of S(v) and vS(v) averaged over 256 raw spec-
tral values, and 11 values of vS(v) averaged on a logarithmic scale
(i.e., over consecutive 2X points for x = 0 to 10). This averaging
procedure was the only smoothing applied to the spectral estimates. The
latter output was used for visual representation of the data (see
Figures 3.38, 3.42, 3.43, and 3.47).
The large volume of data mandated some form of objective analysis.
It was assumed at the outset that the form of the w spectra and the
neutral-to-stable horizontal velocity spectra could be expressed as:
Av
vS(v) «
(B-l)
(Kaimal, 1972, 1978; Kaimal et al., 1972). VS(V) represents the spec-
tral estimate at count v. Eq. B-l was fitted to the raw spectral esti-
mates by a least-squares technique over 1 to 1800 points and 4 to 1800
points. The latter range, which represents an extreme filtering of the
low frequency component, was used for the w spectra; the former range
was used for horizontal component spectra. Thus two sets of values for
A and B were generated for each hourly time series.
The procedure for fitting Eq. B-l to the spectrum assumed a -2/3
slope in the inertia! subrange. Spectra which by a subjective analysis
147
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obviously did not have a -2/3 slope in the inertia! subrange were not
included in the analyses of section 3.3. The number of spectra for each
site and component used in the analyses are given in Table B-l. Site
105 had the highest percentage of spectra with an apparent -2/3 slope.
Site 111 had a very low percentage of spectra with a -2/3 stope and thus
was not included in the analyses of Section 3.3.
Given the spectral form of Eq. B-l, all statistics and length
scales associated with the spectrum can be specified from the values of
A and B. At the low frequency end of the spectrum, i.e., as v * 0, S(v)
= A; at the high frequency end (in the inertia! subrange),VS(V) =
. The peak in the spectrum, obtained through differentia-
tion of Eq. B-l, is at vmax s 1.275b, where b =
Count v is converted to frequency through v = Nn/R; N is the number
of data points in the time series (7200) and R is the sampling rate
(2 per sec). The wavelength of the spectral peak Lm is given by:
I - 1
m * nm 1.275bR
where nm is the frequency corresponding to the peak in the nS(n)
spectrum.
.^•-^^
The variance of the time series is given by a2 =
Applying this to Eq. B-l gives:
a2 = 1.98A5 - A/2.
(B-2)
S(n)dn
(B-3)
For large b, the last term may be dropped with very minor effect on the
total variance.
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Table B-l. Number of spectra for each site, spectral component, and
stability class used in the analyses.
SPECTRAL TOTAL AND PERCENT SPECTRA USED IN ANALYSES
COMPONENT z'/L SITE 105 SITE 107 SITE 109
TOTAL % TOTAL % TOTAL %
-.5 to -2 70 90 16 38 74 55
u1 +.05 to -.05 83 92 115 28 33 41
+ .5 to +1 7 86 24 33 51 25
-.5 to -2 70 55 16 63 74 24
v1 +.05 to -.05 83 77 115 62 33 48
+.5 to +1 7 86 24 50 51 51
-.5 to -2 70 80 16 38 74 55
w1 +.05 to -.05 83 86 115 77 33 64
+.5 to +1 7 100 24 96 51 75
-.5 to -2 70 63 16 81 74 99
T1 +.05 to -.05 83 93 115 94 33 91
+.5 to +1 7 71 24 92 51 94
149
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The integral length scale can be obtained through:
i . - S(n)U
LI - «—
4a^
(B-4)
(Builtjes, 1975). From Eqs. B-2, B-3, and B-4 it can be shown that:
Li " /.92bR F^T*
(B-5)
This result is very close to the theoretical form derived by Webb
(1955).
Tennekes and Lumley (1972) derived a length scale from consider-
ation of the rate of dissipation of kinetic energy as:
(B-6)
Using the expression for given by Eq. 3.19, which can be expressed as:
e = (A/Ba)3/22TrR/UN (B-7)
and using Eqs. B-2 and B-3, we obtain for the w and v components (a =
0.667):
L£=0-2414UN-Lm/3.25. (B-8)
bR
Because of the redundancy of the three length scales by the above
calculation techniques, only Lm and e are discussed in Section 3.3.
150
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TECHNICAL REPORT DATA
(Please read faxwctions on the reverse before completing)
i REPORT NO.
2,
3. RECIPIENT'S ACCESSION»NO.
4. TITLE AND SUBTITLE
5. REPORT DATE
AN EXPERIMENTAL STUDY OF TURBULENCE IN AN
URBAN ENVIRONMENT
6. PERFORMING ORGANIZATION CODE
7. AUTHOR(S)
8, PERFORMING ORGANIZATION REPORT NO
J.F. Clarke, J.K.S. Ching and J.M. Godowitch
9. PERFORMING ORGANIZATION NAME AND ADDRESS
10. PROGRAM ELEMENT NO.
CDWA1A/02-1324 (FY-82)
same as 12
11. CONTRACT/GRANT NO.
12. SPONSORING AGENCY NAME AND ADDRESS
13. TYPE OF REPORT AND PERIOD COVERED
Environmental Sciences Research Laboratory - RTF, NC
Office of Research and Development
U.S. Environmental Protection Agency
Research Triangle Park, NC 27711
In-house
14. SPONSORING AGENCY CODE
EPA/600/09
15. SUPPLEMENTARY NOTES
16. ABSTRACT
The structure of turbulence in the urban surface boundary layer is discussed.
Wind and temperature fluctuations were measured with fast-response sensors at a
height of 31 m at a rural and three urban sites in the St. Louis environs. The
second moments of the fluctuations were computed for one-hour time series and
analyzed within the framework of Monin-Obukhov similarity theory. The results
are discussed relative to observed land-use features and calculated surface rough-
ness lengths for each of the sites.
Average surface roughness lengths ranged from 0.7 to 1.7 m for the urban sites.
The normalized velocity and temperature variances for the rural site were consistent
with similarity theory. For the urban sites, the normalized velocity variances
showed an orderly departure from similarity theory for both neutral and unstable
stratifications.
The urban anomalies are discussed relative to the terms in the turbulent
kinetic energy budget equation. For neutral stratification, the normalized velocity
variances are up to 15% lower at the urban sites compared to the rural site. They
appear to be inversely proportional to surface roughness length. For unstable
stratification, the normalized velocity variances for the urban sites are about 50%
lower than for the rural site.
7.
KEY WORDS AND DOCUMENT ANALYSIS
DESCRIPTORS
b.lDENTIFIERS/OPEN ENDED TERMS
c. COSATi Field/Group
13. DISTRIBUTION STATEMENT
RELEASE TO PUBLIC
19. SECURITY CLASS (This Report)
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21. NO. OF PAGES
167
20 SECURITY CLASS (This page)
UNCLASSIFIED
22. PRICE
EPA Form 2220-1 (9-73)
151
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