EPA-600/4-75-004
July 1975
Environmental Monitoring
ATMOSPHERIC
TURBULENCE PROPERTIES
IN THE LOWEST 300 METERS
U.S. Environmental Protection Agency
Office of Research and Development
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EPA-600/4-7 5-004
ATMOSPHERIC
TURBULENCE PROPERTIES
IN THE LOWEST 300 METERS
by
A.M. Weber, J.S. Irwin,
J.P. Kahler, and W.B. Petersen
North Carolina State University
Raleigh, N. C.
Grant No. 800662
ROAP No. 21ADO-33
Program Element No. 1AA009
EPA Project Officer: George W. Griffing
Environmental Sciences Research Laboratory
Office of Air, Land, and Water Use
Research Triangle Park, N. C. 27711
Prepared for
U.S. ENVIRONMENTAL PROTECTION AGENCY
Office of Research and Development
Washington, D. C. 20460
July 1975
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EPA REVIEW NOTICE
This report has been reviewed by the National Environmental Research
Center - Research Triangle Park, Office of Research and Development,
EPA, and approved for publication. Approval does not signify thnt the
contents necessarily reflect the views and policies of the Environmental
Protection Agency, nor does mention of trade names or commercial
products constitute endorsement or recommendation for use.
RESEARCH REPORTING SERIES
Research reports of the Office of Research and Development, U.S . Environ-
mental Protection Agency, have been grouped into series. These broad
categories were established to facilitate further development and applica-
tion of environmental technology. Elimination of traditional grouping was
consciously planned to foster technology transfer and maximum interface
in related fields. These series are:
1. ENVIRONMENTAL HEALTH EFFECTS RESEARCH
2. ENVIRONMENTAL PROTECTION TECHNOLOGY
3. ECOLOGICAL RESEARCH
4. ENVIRONMENTAL MONITORING
5. SOCIOECONOMIC ENVIRONMENTAL STUDIES
6. SCIENTIFIC AND TECHNICAL ASSESSMENT REPORTS
9. MISCELLANEOUS
This report has been assigned to the ENVIRONMENTAL MONITORING
series. This series describes research conducted to develop new or
improved methods and instrumentation for the identification and quanti-'
fication of environmental pollutants at the lowest conceivably significant
concentrations. It also includes studies to determine the ambient concen-
I rations of pollutants in the environment and/or the variance of pollutants
as a function of time or meteorological factors.
This document is available to the public for sale through the National
Technical Information Service, Springfield, Virginia 22161.
Publication No. EPA-600/4-75-OD4
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ABSTRACT
Results of analyses of atmospheric turbulence data collected from
the SRL-WJBF Meteorological Facility at Beach Island, South Carolina,
are presented. The results include the variation of turbulence parameters
both within and above the surface layer. The nondimensional wind shear,
, is plotted with respect to the ratio of height, z, and Monin-Obukhov
length, L. Roughness lengths for two fetches are obtained. Standard
deviations of wind elevation, a , and azimuth, a., and a /a are plotted
£j A ZL A.
versus the stability parameter z/L. The approximate relationships
involving standard deviations of the vertical and lateral wind components,
CT and o , respectively, and mean vector wind speed U, e.g., a - Ua
and a = Ua , are investigated and found to be very accurate in almost
V A
all circumstances. The ratio a /u., where u.. is surface friction velocity,
w * *
is found to agree with previous measurements. The variable 0 is found
E
to be a function of the scaling parameter fz/u^, where f is the Coriolis
parameter, in near neutral stability conditions. Averages of eddy
viscosity are plotted with height and stability. The ratio of momentum
and heat diffusivities, K /K, , is presented as a function of z/L. The
spectral scale, A , and the mixing length are studied and found to be
consistent with the results of Pasquill (1972). Properties of spectra
measured at 18.3 meters are investigated and found to agree well with
previously published results. The turbulence energy budget and dimen-
sionless temperature gradient are compared with other investigators'
results.
111
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CONTENTS
List of Figures v
List of Tables viii
Acknowledgments i*
Sections
I
II
III
IV
V
VI
VII
VIII
IX
X
XI
XII
XIII
XIV
XV
XVI
XVII
XVIII
XIX
XX
XXI
XXII
XXIII
XXIV
XXV
XXVI
XXVII
Glossary
Conclusions
Recommendations
Introduction
Application to Diffusion
Tower and Terrain
Instruments
Experiment
Data Processing
Surface Friction Velocity
Nondimensional Wind Shear, cj>
m
versus z/L For Near Neutral Stabilities
m
Roughness Lengths
a,., a., and a /a. versus z/L
111 A LJ A
Transformation Ratios
a /u.
w *
The Scaling Parameter fz/u^
Eddy Viscosity
Ratio of Eddy Dif fusivities
Spectral Scale, A , of the Vertical Component and Mixing
Length
Spectral Ratios
Normalized Turbulence Velocity Spectra
The Kolmogorov Constant
Energy Budget Theory and Dissipation of Turbulence Energy
Dimensionless Temperature Gradient, 4>
References
Appendices
1
4
7
8
9
11
14
16
17
21
24
39
42
45
54
66
74
88
91
94
98
106
112
115
126
140
144
IV
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LIST OF FIGURES
No. Page
1. Terrain at SRL-WJBF Tower 13
2. Plot of ufc - Bivane and Cup Anemometer, Versus 22
UA - Gill Anemometer
3. Nondimensional Wind Shear, , Versus z/L, z = 45.4 m - 26
Method 1 m
4. m Versus z/L, z = 113.0 m - Method 1 27
5. Versus z/L, z = 158.9 m - Method 1 28
6. Versus z/L, z = 211.8 m - Method 1 29
7. <|> Versus z/L, z = 273.2 m - Method 1 30
8. Versus z/L, 54.9 m - Method 2 33
9. Versus z/L, 114.3 m - Method 2 34
m
10. Versus z/L, 160.1 m - Method 2 35
m
11. Versus z/L, 213.0 m - Method 2 36
12. Versus z/L, Near Neutral - Method 1 40
14. <|> Versus z/L, Near Neutral - Method 2 41
m
15. Ln(z) - iJ)(z/L) Versus Mean Wind Speed 44
16. a. Versus z/L for South Winds 46
A
17. CTA Versus z/L for Southwest Winds 47
18. o_ Versus z/L for South Winds 49
£i
19. a,. Versus z/L for Southwest Winds 50
r.
20. °E/aA Versus z/L for South Winds 52
21. aE/°A Versus z/L for Southwest Winds 53
22. a /ap Versus Magnitude of Mean Vector Wind, U 55
23. a /a. Versus Magnitude of Mean Vector Wind, U 56
V /\
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No. Page
24. a la Versus Magnitude of Mean Vector Wind, U 57
25. a la Versus U - Data Removed by Editing 61
26. a la Versus U - Data Removed by Editing 62
27. o la. Versus U - Data Removed by Editing 63
U A
28. 0 /u. (Local) Versus z/L at 18.3 m 68
w 3C
29. 0 /u. (Local) Versus z/L at 91.4 m 69
w 7C
30. o /uA(Local) Versus z/L at 304.8 m 70
31. o /u, (Surface) Versus z/L at 18.3 m 71
w *
32. o /u.(Surface) Versus z/L at 91.4 m 72
w "
33. o /u. (Surface) Versus z/L at 304.8 m 73
w "
34. o,. Versus fz/u, For All Levels - Near Neutral 76
E *
35. 0. Versus fz/u^ For All Levels - Near Neutral 77
36. a_ Versus fz/u. For All Levels - Stable 83
E *
37. 0 Versus fz/u^ For All Levels - Stable 84
38. a^ Versus fz/u. For All Levels - Unstable 85
E *
39. o. Versus fz/u^ For All Levels - Unstable 86
40. Eddy Viscosity Profile 90
41. Ratio of Momentum and Heat Diffusivity Versus z/L 93
42. A (Max) Profile 96
43. Mixing Length Profile 97
44. S (n)/S (n) Versus Nondimensional Frequency - Logarithmic Plot 102
w u
45. S (n)/S (n) Versus Nondimensional Frequency - Linear Plot 103
w u
46. S (n)/S (n) Versus Nondimensional Frequency - Logarithmic Plot 104
47. S (n)/S (n) Versus Nondimensional Frequency - Linear Plot 105
48. Normalized w Spectra Versus Nondimensional Frequency 108
49. Normalized v Spectra Versus Nondimensional Frequency 109
VI
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No. Page
50. Normalized u Spectra Versus Nondimensional Frequency 111
51. Observed e Versus Calculated e 121
52. Nondimensional Dissipation Versus z/L at 18.3 m - Sonic 123
53. Nondimensional Dissipation Versus z/L - Cup & Bivane 124
54. . Versus z/L at 18.3 m - Method 1 128
h
55. <(>h Versus z/L at 91.4 m - Method 1 129
56. <}>h Versus z/L at 137.2 m - Method 1 130
57. h Versus z/L at 182.9 m - Method 1 131
58. h Versus z/L at 243.8 m - Method 1 132
59. ij>h Versus z/L at 304.8 m - Method 1 133
60. h Versus z/L at 13.5 m - Method 2 134
61. . Versus z/L at 59.9 m - Method 2 135
62. . Versus z/L at 112.8 m - Method 2 136
63. <}>, Versus z/L at 185.4 m - Method 2 137
64. 4>, Versus z/L at 289.1 m - Method 2 138
65. <(>m Versus z/L, using <|>m = (1 - 15 z/L)~1/4, 2/L < 0 and 152
m
(1 - 16 z/L)"1/4, Z/L < 0
vn
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LIST OF TABLES
No. Page
1. Points Not Plotted on Figures 3 through 7 31
2. Points Not Plotted on Figures 8 through 12 38
3. Analysis of Variance for a /a Versus Mean Wind Velocity 58
U A
4. Analysis of Variance for a /a Versus Mean Wind Velocity 59
V A
5. Analysis of Variance for o /a_ Versus Mean Wind Velocity 60
w E
6. Analysis of Variance for o_ Versus a /U 64
r, W
1. Analysis of Variance for a Versus a /U 64
8. Average a /UA for Local u^ ' s and Surface u.v's 67
9. Linear Regression and Variance Analysis of o on In fz/uA 73
10. Linear Regression and Variance Analysis of o. on In fz/u^ 79
11. Quadratic Regression and Variance Analysis of
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ACKNOWLEDGMENTS
Sincere appreciation is expressed to Dr. Todd Crawford, who
graciously provided the use of the SRL-WJBF meteorological facility
during the experiment. Several Savannah River Laboratory scientists,
especially Dr. M. M. Pendergast, assisted in many ways. Thanks is also
due to Battelle Northwest Laboratory personnel, including Tom Horst,
who provided support with several specialized instruments. Larry
Rainey, Ray Hollowell and Michael Shipman did much of the original
computer programming. A special thanks is due to David Delong who
provided assistance and advice to the project from its inception.
Appreciation is expressed to Dr. Hans A. Panofsky who provided many
suggestions in the analysis of the data.
This research was supported by the Environmental Protection
Agency, Meteorology Laboratory, under EPA Grant No. R-800662.
IX
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SECTION I
GLOSSARY OF TERMS, ABBREVIATIONS, AND SYMBOLS
The following is a list of symbols used in this report:
g acceleration due to gravity,
k von Karman's constant,
u1, v', w' longitudinal, tranverse, and vertical components of the
fluctuating part of the wind,
U magnitude of the mean vector wind,
S mean horizontal wind speed,
z vertical coordinate,
6 mean potential temperature,
91 fluctuating part of potential temperature,
1/2
u^ friction velocity [t/p] ,
L Monin-Obukhov length scale,
p mean density of the air,
T surface turbulent shearing stress,
S (n), S (n), S (n) vertical, transverse, and longitudinal component
of the power spectral density function,
n frequency measured in Hertz,
K wave number measured in radians per meter,
a,, a, a- Kolmogorov constants for u, v, and w components of the
inertial subrange power spectral density function,
e. dissipation rate for turbulent energy,
a , a standard deviation of the wind elevation and azimuth angles
respectively,
a , a , a standard deviation of the u, v, and w component of
the wind,
2 222
q /2 = (u1 + v1 + w1 )/2 total turbulent kinetic energy,
q specific humidity,
0
£, mixing length.
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Some of the results were analyzed in dimensionless form using the
dimensionless quantities below, where the overbar symbol designates a
time average:
~k2U* 36
(>, = - 7 ..... a dimensionless potential temperature
dZ
" i i , .
w 6 gradient,
Icz c
<}> = r ..... a dimensionless dissipation of
u^ turbulent energy,
= - ..... a dimensionless wind shear,
m u 3z
z/L = : ^ ,. a dimensionless height,
V*
r z/L
<|> = [!-<(> (z/L)]/(z/L)d(z/L) . . . correction to be subtracted
' o from logarithmic wind pro-
file to account for non-
adiabatic conditions,
fz/u (where f is the Coriolis parameter) ... a dimensionless height,
K, /K = /d>, = - r - - . . ratio of eddy transfer coefficients,
hmrmh 2 . / ,
u 30/c)z and
a dimensionless frequency.
Statistical Terms
Degrees of Freedom (D.F.) - The total degrees of freedom equals the number
of independent observations. Every coefficient estimated in the regres-
sion model accounts for one degree of freedom. The number of degrees of
freedom for the deviation from the regression equals the total minus the
number of coefficients estimated in the regression. In the analysis of
variance tables, degrees of freedom will be abbreviated D.F.
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F-test - F is the ratio of two mean squares such that the expected
value of the numerator under the null hypothesis is equal to the
expected value of the demoninator. For all tests in this report the
null hypothesis is that the particular regression coefficient(s) under
the source of variation is zero. F values that are not statistically
significant at the 5% level are noted by (NS). A value of F that was
significant at the 5% level indicates that if the null hypothesis is
true less than 1 out of 20 samples will give an F value of this
magnitude.
Mean Square - The sum of squares divided by the degrees of freedom.
R^ - The multiple regression coefficient. It is the proportion of the
total variation that can be attributed or explained by the variables
in the regression model. An R^ equal to 1 implies that all of the
sample observations lie on the regression equation.
Ninety-five Percent Confidence Interval - The 95% confidence interval
of a regression coefficient b is determined by
b±t.05
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SECTION II
CONCLUSIONS
Friction velocities measured by the cup anemometer and bivane systems
at the SRL-WJBF tower agree reasonably well with independent measurements
by a Gill - u, v, w anemometer and a sonic anemometer. The nondimensional
wind shear calculated from the data varies with the stability parameter
z/L in a manner that is well modeled by the Businger, Wyngaard, Izumi,
Bradley (1971) expressions based on the 1968 Kansas experiment. The
departure from BWIB curves seems to start above 160 meters and become
progressively worse. The implication is that the simple equations pre-
dicted by similarity theory for the surface layer cannot be extended much
above 160 m except as a first approximation.
Roughness lengths were found to be about 8 cm and 36 cm for the
two predominate wind directions at the tower site.
Both o_ and a decreased with increasing stability. Height depen-
dencies were characterized by a decrease in the rate of change of ap and
0 with z/L as the height of measurement increased. The value of 0 and
A £
0. at the lowest level seemed to be a predictor for values at higher levels.
Flow over terrain characterized by larger roughness lengths, z , clearly in-
creased the magnitudes of o and 0. in comparison to flow corresponding
£1 A
to smaller z . The variation of 0p and a. with z, z , and L agree
qualitatively with the nomograms prepared by Panofsky and Prasad (1965).
The approximations,
0w/0£ = U and (2.1)
ov/aA = U , (2.2)
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were demonstrated to be very good under any of the conditions encountered
in this experiment including "quasi-laminar" flow which occurs at night.
The mean over all samples of a /u. (using friction velocities mea-
w
sured near the surface) for neutral conditions (|z/L,| < 0.05) is 1.27.
This value agrees well with previously published values.
Under near neutral stability conditions, a and 0 were found to
Lj A
scale with fz/u^. A quadratic regression of a on In fz/u^ accounted
for 70 percent of the variation of a . A linear regression of a on
Ct A
In fz/u. accounted for 67 percent of the variation of o, with In fz/u..
* A *
Average measured eddy viscosities in neutral conditions (|Z/L| < 0.05)
agree well with the values predicted by the equation:
\i = u*kz (2'3)
where u^ is taken from the average of the 18.3 m measured values.
The data supported the contention that the ratios of momentum and
heat diffusivities, K /K, approach zero for increasing stability when
z/L< 0. The near neutral value for the ratios was less than 1.
Although there is considerable scatter in the results, the measured
spectral scales of the vertical component of velocity agree with the
results of Pasquill (1972).
The spectral ratios, S (n)/S (n) and S (n)/S (n) calculated from
18.3 m sonic anemometer data corresponding to a stability range of
-0.3 <_ z/L <^ 0.1 were found to exhibit a distinct trend toward the 4/3
value expected in the inertial subrange.
The normalized spectra measured at 18.3 meters exhibited a syste-
matic progression with z/L during stable conditions but were randomly
clustered during unstable conditions. A region separating the stable
5
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from unstable spectra was detected on the v spectra plot indicating an
abrupt shift in the scale of turbulence. The spectral peaks and cor-
responding frequencies were found to be in agreement with the results
of the 1968 Kansas experiment. Under stable conditions z/L was found
to be an adequate scaling parameter for the spectra.
A Kolmogorov constant calculated from six sonic u spectra correspond-
ing to near neutral stability was found to be 0.5 if a value of 0.4 was
assumed for the von Karman constant.
The dissipation of turbulence energy obtained from the 18.3 m sonic
spectra was found to be reasonably estimated by
e = =^- MX (1 - 16 z/L) ' , (z/L£0) or (2.4)
Z
e= u.3(l + 4.7 z/L), (z/L^O). (2.5)
Similarity theory has proven particularly effective in describing
turbulence statistics in the atmospheric surface layer and above up to
100 meters. This was found to be the case in this experiment even though
the terrain was nonhomogeneous. In fact a tendency for the atmospheric
turbulence statistics to be described by the relatively simple concepts
of similarity theory was found at levels well above 100 meters.
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SECTION III
RECOMMENDATIONS
The objective of this research program was to measure and study
atmospheric turbulence statistics which are relevant to dispersion of
air pollutants in the atmosphere. In order to achieve this end it was
essential to demonstrate that the quality of data was good. Agreement
between different instruments at the same level and concurrence between
these measurements and others in the surface layer give resonable as-
surance that the quality of these data is good. Having extablished data
quality makes it possible to have confidence in the findings, i.e. simi-
larity theory can be extended upward beyond the usually defined surface
layer over nonhomogeneous terrain. It is also possible to recommend that
new investigations using these data be attempted.
New investigations should be directed toward more thoroughly estab-
lishing the effect of nonhomogeneous terrain on atmospheric turbulence
statistics. Also, the data removed during the editing procedure should
be studied, especially those cases demonstrating laminar flow at night.
Spectra should be studied for changes with height and stability.
It is gratifying that as an approximation the relatively simple
concepts of similarity theory give a reasonable description of atmospheric
turbulence properties several meters above the surface layer and for non-
homogeneous fetches.
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SECTION IV
INTRODUCTION
The diffusion of gaseous pollutants from tall stacks is directly
related to atmospheric turbulence in the planetary boundary layer (PEL).
Tall stacks are frequently used in modern fossil-fuel power generating
plants and individual stacks can be 1000 ft. (304.8 m) or more in height.
Several such stacks are being planned for construction in the future.
Since our knowledge of atmospheric turbulence in the PEL is relatively
inadequate, it is of great importance to carry out studies of turbulence in
the field and compare findings with theoretical predictions when possible.
Many studies of atmospheric turbulence have been accomplished in the
surface layer (i.e., the region below 100 meters) but few have examined
data above 100 meters. Also, most theories specify homogeneous terrain
upwind from the sensor locations. While most towers are sited in non-
homogeneous terrain the ideal site for testing theory is perfectly flat
and without changes in roughness. There have been few attempts to test
turbulence theories for nonhomogeneous fetches and at heights above
the surface layer.
Information obtained in this investigation is expected to be
ultimately applicable to the design of tall stacks, in particular, to
help answer questions regarding the advantage or disadvantage of a
particular height of stack for various meteorological conditions.
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SECTION V
APPLICATION TO DIFFUSION
Application of the results of this study is found in the Gaussian
Plume Diffusion Model. In this model the concentration at a point
(x, y) at ground level downwind from a source at effective stack
height H is:
X (x,y,; H) =
Q
TTO- a u exp
y z
2
"^
( y J
exp
1 H2
2 2
a
z
(5.1)
where
U = mean wind speed,
X = the concentration of the gas or aerosol,
x,y = distance along-wind and cross-wind measured from the source,
H = effective stack height,
Q = source strength, and
a ,a = horizontal and vertical standard deviations of concentration.
y z
Cramer (1964) has shown that relationships of the form
cp
,9
°y " CTA x and
* °
(5.2)
(5.3)
where a. and a are the standard deviations of wind azimuth and eleva-
A E
tion angle, respectively, and p and q are functions of stability, are
very useful in predicting the downwind spread of a plume in various
stability conditions. A knowledge of the statistical behavior of a.
and a,., with height and stability can possibly aid in the use of this
iij
model for tall stacks or, perhaps, lead to better models of diffusion
in the future.
9
-------
If one assumes that a = aa , i.e., where the vertical and
y z
horizontal cloud growths are simply proportional, then at ground
level the maximum concentration is given by:
2Qa
irH eUo
.y
where e is the base of the natural system of logarithms. Thus from
Eqs. (5.2), (5.3), and (5.4) one can see that the ratio, a /a , is
fcj A
important in determining the maximum concentration.
Because of the close relationship between atmospheric turbulence
and plume dispersion it is of interest to see if theories of atmospheric
turbulence such as the Monin-Obukhov similarity theory hold at levels
higher than the surface layer.
10
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SECTION VI
TOWER AND TERRAIN
The WJBF-TV Savannah River Laboratory Meteorological Facility,
(hereafter SRL-WJBF tower) is 11 kilometers southeast of Augusta, Georgia,
and 23 kilometers southwest of Aiken, South Carolina. The 366 meter tower,
whose base stands 121 meters above mean sea level (MSL), has been in use
as a meteorological tower since October, 1965. It was then that Savannah
River Laboratory (SRL) began its data collection program for reactor
safety studies. The instrumentation and several features of the SRL
program were summarized in Cooper and Rusche (1968). The instruments
are mounted on heavy booms extending about 3 meters outward (direction
225°) from the tower.
In 1973, changes to the system were made to improve the quality
of the data. The details of the system's renovations are contained in
Crawford (1974). The most significant improvement was the addition
of a digital data acquisition system (DAS). The system monitors and
records information from 32 channels of data directly on magnetic tape
in a format compatible with SRL's IBM 360/195 computer facility.
The WJBF tower is one of three TV towers near the rural community
of Beach Island, South Carolina. There is no one dominant form of land
use in the immediate vicinity of the tower. Along any given azimuth
from the tower, the rolling terrain varies with pine tree forests
(average height approximately 12 meters), pastures and fields, and
clearings of waist-high scrub and young saplings.
Two major topographical features are the Savannah River and an
intermittent stream called Long Branch. The broad Savannah River flows
within 5.6 kilometers to the west and within 9.7 kilometers to the
11
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south of the instrumented tower. Long Branch slopes from northwest
to southeast and its closest approach to the tower is about 610 meters
to the southwest (see Figure 1).
12
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RADIO
TOWER
WRDW
RADIO TOWER
WJBF
' 3
Figure 1. Terrain at SRL-WJBF tower. Dashed equilateral triangle shows
location and orientation of pibal observation sites with 304.8
meter baselines. Scale is 2 5/8 inches equals one mile. Con-
tours are for every 50 feet.
13
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SECTION VII
INSTRUMENTS
The temperature sensors used to calculate mean temperature at the
six tower levels were (100 ohm) platinum resistance wire thermometers.
These thermometers were wind aspirated except at the lowest level which
was mechanically aspirated. The response time of the thermometers is
on the order of several seconds and they are of no value in measuring
turbulent heat flux.
A thermistor attached to a Gill u-v-w anemometer was supplied by
Battelle Northwest to measure the heat flux. The fast response thermistor
was found (by Battelle) to have a time constant of less than or equal to
0.12 sec (depending on wind speed). This instrument configuration has
been used to compute turbulent heat flux on previous occasions by Battelle
Northwest with results that compared favorably with simultaneous measure-
ments by the sonic anemometer.
Comparisons between the Gill u-v-w anemometer and the sonic anemometer
for measuring stresses are documented in Horst (1973). The instruments
seem to agree well for measured turbulence spectra for wavelengths longer
than 25 meters (0.2 Hertz at wind speed 5 mps). A correction was applied
to the wind measurements of the Gill anemometer to account for cosine
response as suggested in Horst (1973).
The wind measuring instruments are Climet cup and bivane systems.
The distance constant of the cup anemometers is on the order of 1 m.
Averaging circuits for the photo-chopper reduce the frequency response
(-3 db point for sine wave input) to about 0.1 Hertz. The characteristics
of the bivanes were measured by Brookhaven National Laboratory and found
to have a damping ratio of 0.55 and a delay distance of 0.9 m.
14
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Both the sonic anemometer and the Gill anemometer give best results
when the mean wind direction is into the open end of the sensor arrays.
The sonic anemometer is somewhat more critical in this respect than is
the Gill instrument. Light and variable winds are thus a problem in that
gusts reach the sensors from unfavorable directions. Constant monitoring
of wind direction was necessary during the experiment so that when large
excursions of azimuth angle did occur data collection could be interrupted.
15
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SECTION VIII
EXPERIMENT
Climet cup and bivane systems were mounted and leveled with respect
to gravity at six levels on the tower (18.3, 91.4, 137.2, 182.9, 243.8,
and 304.8 meters). (Refer to Appendix A for further details). Slow
response aspirated temperature sensors (platinum resistance wire ther-
mometers) were located at 3.0, 36.6, 91.4, 137.2, 182.8, 335.3 meters.
(For further details on instrumentation refer to Cooper and Rusche, 1968,
and Crawford, 1974.) Battelle-Northwest maintained and operated three
instruments which were attached to two aluminum booms mounted at about
18.3 meters. Included were a Gill u-v-w propeller anemometer with a fast
response thermistor (refer to Appendix B for further details) mounted on
its vertical arm, and a sonic anemometer.
During the sixteen day period between 13 May and 29 May, 1973, the
instruments on the SRL-WJBF tower recorded data unless conditions indicated
rain, fog, or other disturbing influences. Also, to further examine the
planetary boundary layer, pibal measurements of the wind were taken using
double theodolite techniques.
16
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SECTION IX
DATA PROCESSING
To monitor simultaneously all instruments on the tower required 31
of the available 32 channels of the DAS. The DAS had the capability of
scanning all 31 channels at several pre-set rates; however, during this
experiment either a five or a ten scans-per-second mode was selected.
Preceding each scan, information as to the year, day, hour, minute, and
second was recorded on the tape. The data on tape were compacted and
calibrated at SRL's computer facility and then sent to North Carolina
State University for processing at the Triangle Universities Computation
Center (TUCC).
Wind velocity measurements were referred to a natural coordinate
system, i.e., the x axis was oriented along the mean wind vector, the
y axis was oriented perpendicular to the x axis and in the horizontal
plane, the z axis was oriented perpendicular to the x and y axes so
as to form a right-hand coordinate system. It is important to note
that in the natural coordinate system used the mean wind vector did
not necessarily lie in the horizontal plane and that at each level of
the tower the wind components u, v, and w refer to the natural coordi-
nate system appropriate to that level and 40-minute time period.
The data were time averaged over 40-minute periods. Averages
and variances of the turbulence parameters of interest were edited and
then analyzed. Out of a total of 239 blocks of data,119 were retained
after editing for further analysis.
The data were edited as follows: (1) data blocks were removed if
a particular block did not contain a full 40 minutes of data; (2) 40-
minute blocks were removed if an upward transport of momentum was
17
-------
evidenced, i.e., -u'w' < 0, because similarity theory does not model
such situations; (3) blocks were rejected during conditions where the
2 2
variance of the vertical velocity fell below 0.01 m /sec (during very
stable conditions the vertical velocity fluctuations are heavily suppressed
leading to a kind of laminar flow); and (5) blocks were rejected during the
transition period at sunrise when the rapid surface heating and resultant
momentum and heat flux progressively affects higher levels in the PEL.
The Monin-Obukhov length scale, L, for such transition conditions computed
\
at the 18.3 meter level, indicated dynamic instability when in fact higher
levels of the tower still reflected nighttime conditions, i.e., low vertical
velocity variances, positive potential temperature lapse rates and large
vertical wind shears; thus the 18.3 meter L value did not characterize
the entire region between the ground and the top of the tower. Details
of the data block editing procedure are given in Appendix C.
Both the mean wind speed, S, and the magnitude of the mean wind
vector, U, were calculated for each 40-minute block. The value of S
is always greater than U and this difference can be significant.
The turbulent shear stress was calculated using the expression
o o 1/2 2
T = p (v'w1 + u'w1 ) = puA . (9.1)
The value of (T/P) at 18.3 meters was assumed to be the surface friction
velocity. The term |v'w' | was included in the calculation, since it was
greater in magnitude than 10 percent of |u'w' | values in 59 to 75 percent
of the cases (depending upon the level of measurement).
The Monin-Obukhov length scale,
L = g (9.2)
k £ w'e'
o
18
-------
was calculated using a friction velocity from the Gill u-v-w anemometer
mounted at 18.3 meters. Von Karman's constant, k, was assumed to have
the value 0.35. The parameter 9 was obtained from the 3 meter level.
The temperature flux, w'61, was calculated from the Gill u-v-w anemometer
and the fast response thermistor at 18.3 meters.
Atmospheric turbulence spectra were calculated using the Fast Fourier
Transform (FFT) technique designed by Singleton (1969). Wind speed and
direction were converted to components in a Cartesian coordinate system
where the x-axis was the direction of the mean wind vector, the y-axis
was perpendicular to the mean wind vector and in the horizontal plane,
and the z-axis was oriented to form a right handed system. It should be
noted that the z-axis was not always oriented along a plumb line. The use
of this coordinate system was recommended by Kaimal and Touart (1967).
The time series composed of fluctuating velocity components were then
multiplied by a cosine taper data window. This resulted in a cosine taper
over the beginning and ending 10 percent of the 40-minute time series.
The FFT was then applied to the tapered data resulting in a raw spectrum.
Squaring and adding the Fourier transformed coefficients resulted in a
raw power spectrum estimate. Autocorrelation values were then obtained
by applying an inverse FFT. A Parzen lag window was applied to the
autocorrelation values followed by another application of the FFT. This
was done in order to obtain smoothed power spectral density estimates.
These values were then multiplied by 1/0.875 to correct for the effect of
the cosine taper data window (Bendat and Piersol, 1971). The standardized
frequency bandwidth corresponding to the Parzen lag window (1.86 Hertz)
was converted to nondimensional frequency bandwidths with the expression
B = i'862 (9.3)
At(lags)U *
19
-------
When the sonic anemometer was in operation the sampling rate was
10 times per second. The resulting 24,000 data points for each 40-minute
sampling period were then reduced to 8,000 data points by 3-point block
averaging. In effect, this changed the sampling interval from 0.1 sec.
to 0.3 sec. for all data taken while the sonic anemometer was in
operation.
20
-------
SECTION X
SURFACE FRICTION VELOCITY
The instruments nearest to the surface were at a height of 18.3 meters.
The data derived from those instruments were used in computing surface
friction velocities. Computed friction velocity from the cup and bivane
system and the Gill anemometer differed slightly. One can ask if the
difference is due only to random error or if there is a real statistical
difference in the two measurements. The sonic anemometer also was capable
of measuring friction velocity. However, because of difficulties with
the sonic anemometer mentioned in Section VII and the fact that the sonic
was only present for about one half of the experimental period, this sec-
tion is limited to a discussion of the differences in friction velocity
between the cup and bivane system and the Gill anemometer. A plot of u^
calculated from measurements by the Gill anemometer versus u^ calculated
from measurements by the cup and bivane system is shown in Figure 2.
Assuming that the true relationship is:
u*bivane = al + 61 U*Gill + e
-------
CM
CD
O U^
UJ
CO CD
UJ
en
OQ
vJ
*
CD
CM
CD
CD..
I
i r
D
D
j j j j____ j
. H C . 6 0.8
I I
1.0
U.CGILL) CM/SEC)
Figure 2. Comparison of friction velocity, u*, measured by the bivane
and cup anemometer at 18.3 meters versus UA measured by the
Gill propeller anemometer at the same height.
22
-------
From the confidence intervals one can see that a., is small but
not equal to zero and $.. is close to but not equal to one as it would
be if the true model were of the form:
U*bivane = U*Gill + £»
where the friction velocities calculated from both systems differed by
only random error. Thus, the sample results indicate a real statistical
difference between the two calculated friction velocities. Eq. (10.1)
assumes UA_..... as the independent variable (or correct variable) and
indicates the predicted friction velocity for the cup and bivane system.
However, it is possible to reverse the dependent and independent variables
and fit a straight line of the form
u.0>11 = a + 6 u. + e. (10.6)
*Gill 2 2 *bivane
Doing this the regression equation fit is
U*,-TI = --00891 + 1.1376 ii... (10.7)
*Gill *bivane ,
with 95% confidence intervals about a and 3~ of
- .0251 < (*2 < .00724, and (10.8)
1.1051 < S2 < 1.1701 , (10.9)
2 -4
and an estimate of a of 8.04 x 10
Because cups have more inertia than the Gill propellers, the Gill
instrument was expected to be more responsive and this seems to be
reflected in the fact that friction velocities calculated by the Gill
anemometer were higher than the bivane values.
23
-------
SECTION XI
NONDIMENSIONAL WIND SHEAR,
m
A cornerstone of Monin-Obukhov similarity theory is that given
steady conditions and homogeneous terrain, nondimensional wind shear, <$> ,
is a function of z/L for the surface layer. Results by Businger et al.
(1971), show that
V = Ir fi = (1 " 16 z/L)~1/4 ' z/L < ° and
((> = (1 + 4.7 z/L), z/L > 0
m
(see Appendix E) in the surface layer.
Measurements in this experiment were mostly outside the surface
layer and representative of rather nonhomogeneous fetches . Nevertheless
plots of vs z/L were made with varying degrees of "success", the lower
3 levels being better than the upper 3 levels of the tower. Two methods
of computing m were used as described below. In all cases u^ is computed
from the 18.3 meter Gill u-v-w anemometer according to the formula:
*- / 4
[f i i~\ i / t i~\ i
(u'w1) + (v'w1) ]
A series of graphs is presented for which z/L is the abscissa, <(> is the
ordinate and each separate plot refers to a different reference level in
the calculation of d> Each plot contains solid lines which represent the
m
empirical expressions obtained by Businger et al. (1971) , hereafter referred
to as the BWIB lines (see Appendix E) ,
m = (1 - 16 z/L)~lM , z/L < 0 (11-4)
m
(1 + 4.7 z/L) , z/L > 0,
(11.5)
and the KEYPS equation (lower line on the unstable side),
<(>m4 - (18 z/L) <(>m3 = 1 (Lumley and Panofsky, 1964). (11.6)
24
-------
Method 1. The expression for $ was approximated by
*m = ^ ll ' Where ~Z = Az/(ln Z2/Z1}-
The variable, z, represents the appropriate reference level between
z2 and z. for . The value of z lies between the arithmetic and
geometric means. Stearns (1970) used a similar expression.
The difference in mean wind, All, was obtained from values of U
corresponding to the 6 levels on the tower equipped with cup and bivane
systems and referenced to 5 intermediate levels represented by the z
given above. No correction was made for overspeeding of the cup anemometers.
The resulting plots (Figures 3 through 7 ) indicate that is reason-
m
ably modeled by similarity theory throughout the height of the tower.
Scatter is small at the lower levels and increases with height.
At the 273.2 m level (Figure 7 ) there is a tendency for the points
to fall below or to the right of the line on the stable side and above
both lines on the unstable side. This indicates some departure from
similarity theory at higher levels of the tower, particularly above 212 m;
however the fit was fairly good at the lower levels (Figures 3 through 5 ).
On the 45.4 m plot (Figure 3 ) the <() 's on the unstable side can be seen
to decrease more rapidly near neutral than either the KEYPS or the BWIB
expressions predict. A number of points were not plotted because they
fell outside the boundaries of the plots (see Table 1 ).
Method 2. The technique used by Businger et al. (1971) was followed
in producing the next set of plots. In order to more precisely duplicate
the method used by Businger, mean wind speeds, S, were used in place of
magnitudes of the mean vector wind. A quadratic least squares second
25
-------
Figure 3. Nondimensional wind shear versus z/L using z = Az/(ln z./z ),
Method 1. A total of 119 points are plotted for z = 45.4 m.
Solid lines represent the BWIB equations and the KEYPS equation
(lower solid line for z/L < 0).
26
-------
Figure 4. Nondimensional wind shear versus z/L using z = Az/(ln z_/z1),
Method 1. A total of 114 points are plotted for z = 113.0 m.
Solid lines represent the BWIB equations and the KEYPS equation
(lower solid line for z/L < 0).
27
-------
Figure 5. Nondimensional wind shear versus z/L using z = Az/(In z?/z..),
Method 1. A total of 112 points are plotted for z = 158.9 m.
Solid lines represent the BWIB equations and the KEYPS equation
(lower solid line for z/L < 0).
28
-------
-f.O
Figure 6. Nondimensional wind shear versus z/L using z = Az/(ln z9/z1),
Method 1. A total of 112 points are plotted for z = 211.8 m.
Solid lines represent the BWIB equations and the KEYPS equation
(lower solid line for z/L < 0).
29
-------
9.O..
-f.O
-2.0
0
z/L
2.0
Figure 7. Nondimensional wind shear versus z/L using z = Az/(ln z./z..),
Method 1. A total of 102 points are plotted for z = 273.2 m.
Solid lines represent the BWIB equations and the KEYPS equation
(lower solid line for z/L < 0).
30
-------
Table 1. CALCULATED NONDIMENSIONAL WIND SHEARS WHICH WERE NOT PLOTTED
ON FIGURES 3 THROUGH 7. COMMENTS REFER TO THE LAYER IN WHICH
WAS CALCULATED.
Level Comments
113.0 Fit the trend in the data.
113.0 i Fit the trend in the data.
113.0 Large positive wind shear.
113.0 Large positive wind shear.
113.0 Large positive wind shear.
158.9 ,' Negative wind shear.
158.9 Fit the trend in the data.
158.9 Negative wind shear.
158.9 Negative wind shear.
158.9 Negative wind shear.
158.9 Fit the trend in the data.
158.9 Fit the trend in the data.
211.8 Small positive wind shear.
211.8 Fit the trend in the data.
211.8 Negative wind shear.
211.8 Fit the trend in the data.
211.8 Negative wind shear.
211.8 Fit the trend in the data.
211.8 Fit the trend in the data.
273.2 Negative wind shear.
273.2 Negative wind shear.
273.2 Negative wind shear.
273.2 Fit the trend in the data
273.2 Negative wind shear.
273.2 Negative wind shear.
273.2 Fit the trend in the data.
273.2 Fit the trend in the data.
273.2 Fit the trend in the data.
273.2 Negative wind shear.
273.2 Negative wind shear.
273.2 Fit the trend in the data.
273.2 Fit the trend in the data.
273.2 Large change in wind direction.
273.2 Large change in wind direction.
273.2 Negative wind shear.
273.2 Large change in wind direction.
31
-------
2
order polynomial regression of the form S = A(ln z) + B In z + C was
fitted to the mean cup speeds using the 6 tower measurement heights of
18.3, 91.4, 137.2, 182.3, 243.8 and 304.8 meters. This was done for each
40-minute time block yielding 119 regression equations for the wind pro-
files corresponding to each time block. Each regression equation was
then differentiated with respect to In z yielding 119 expressions for
dS/d (In z). Numerical values for dS/d (In z) were then obtained for
each time block and tower level. These were reduced 10 percent to
compensate for overspeeding of the cups. The dimensionless shear was
calculated for each level and time using the expression
, _ k dS
u d(ln z)
= - (2A In z + B). (11.8)
Plots of 4> versus z/L were produced for five intermediate levels
using this technique (Figures 8 through 12) . These plots are very
similar in appearance to those produced by Method 1 but with a small
reduction in the amount of scatter. As with the previous method the
scatter increases with height. A number of points were not plotted
because they fell outside the boundaries of the plot (see Table 2) .
32
-------
Figure 8. Nondimensional wind shear versus z/L using Businger's method
(Method 2). A total of 119 points are plotted for the 54.9 m
level. Solid lines represent the BWIB equations and the KEYPS
equation (lower solid line for z/L < 0).
33
-------
9.0.
-f.O
-2.0
0
2.0
4.0
Figure 9. Nondimensional wind shear versus z/L using Businger's method
(Method 2). A total of 118 points are plotted for the 114.3 m
level. Solid lines represent the BWIB equations and the KEYPS
equation (lower solid line for z/L< 0).
34
-------
Figure 10. Nondlmensional wind shear versus z/L using Buslnger's method
(Method 2). A total of 116 points are plotted for the 160.1 m
level. Solid lines represent the BWIB equations and the KEYPS
equation (lower solid line for z/L < 0).
35
-------
9.0
B
6.0
3.0
D
a
D a a
m
Figure 11. Nondimensional wind shear versus z/L using Businger's method
(Method 2). A total of 115 points are plotted for the 213.4 m
level. Solid lines represent the BWIB equations and the KEYPS
equation (lower solid line for z/L < 0).
36
-------
Figure 12. Nondimensional wind shear versus z/L using Businger's method
(Method 2). A total of 107 points are plotted for the 274.3 m
level. Solid lines represent the BWIB equations and the KEYPS
equation (lower solid line for z/L < 0).
37
-------
Table 2. CALCULATED NONDIMENSIONAL WIND SHEARS WHICH WERE NOT PLOTTED
ON FIGURES 8 THROUGH 12. COMMENTS REFER TO THE LAYER IN WHICH
d> WAS CALCULATED.
Level Comments
114.3 Fit the trend in the data.
160.1 Fit the trend in the data.
160.1 Fit the trend in the data.
160.1 Fit the trend in the data.
213.4 Did not fit trend in the data.
213.4 Fit the trend in the data.
213.4 Fit the trend in the data.
213.4 Fit the trend in the data.
274.3 Did not fit trend in the data.
274.3 Fit the trend in the data.
274.3 Did not fit the trend in the data.
274.3 Fit the trend in the data.
274.3 Fit the trend in the data.
274.3 Fit the trend in the data.
274.3 Fit the trend in the data.
274.3 Fit the trend in the data.
274.3 Fit the trend in the data.
274.3 Did not fit the trend in the data.
274.3 Did not fit the trend in the data.
274.3 Fit trend - large positive wind shears.
38
-------
m
SECTION XII
vs z/L FOR NEAR NEUTRAL STABILITIES (-0.1 < z/L < 0..1)
It is generally assumed that if the appropriate conditions for
similarity theory are met, the intercept of at z/L = 0 should equal one.
Based upon this assumption, values for von Karman's constant have been
determined by choosing a value that would force <{> to one at z/L = 0.
Ah expanded view of those points corresponding to near neutral z/L values
is presented in the next series of plots so that a visual assessment can
be made of what the intercept may be. The plots have two scales labeled
according to the value of von Karman's constant that is assumed (i.e.,
0.35 or 0.4). The solid line on all plots represents the BWIB line.
Plotting the <(> 's calculated using Method 1 against z/L resulted
in relatively small scatter at low levels for near neutral stability
(Figure 13). At the lowest level (45.5 m) the points fall below the
BWIB line indicating a k near 0.43. The points are very nearly parallel
in trend to the BWIB line. At the higher levels, few points fall in the
near neutral range but those points plotted have a higher center of
gravity than those referenced to lower levels .
The near neutral plot of points (Figure 14 ) calculated using
Method 2 referenced to the lowest level are below but nearly parallel
to the BWIB line. The position of the points seems to support a k ~ 0.4.
The center of gravity of the points moves progressively upward as the
reference level increases.
39
-------
k=0.35
-0.10
i r
-0.10 -0.05
0
0.05 0.10
z/L,k=0.35
Figure 13. Nondimensional wind shear versus z/L using Method 1.
Reference levels are 45.4 m (squares), 113 m (triangles),
and 158.9 m (crosses). The solid line is the BWIB line.
Upper and right side scales are for k = 0.40 whereas
lower and left side scales are for k = 0.35.
40
-------
k=0.35
-0.10 -0.05
I i
-0.10 -0.05
z/L,k=0.40
0.1.0
I
0.05 0.10
z/L,k=0.35
Figure 14. Nondlmensional wind shear versus z/L using Method 2.
Reference levels are 54.9 m (squares), 114.3 m (triangles),
and 160.1 m (crosses). The solid line represents the
BWIB line.
41
-------
SECTION XIII
ROUGHNESS LENGTHS
The terrain to the south was relatively flat, rising less than
8 meters in 900 meters, and more homogeneous than the terrain to the
southwest. The roughness elements to the south varied from 60 centimeter
grass and shrubs to a plowed field about 300 meters from the tower. The
plowed field extended out to 900 meters .where tall trees dominated the
landscape.
The terrain to the southwest sloped down from the tower about 2.5
degrees in the first 700 meters from the tower. The foliage within the
first 300 meters of the tower is similar to that to the south. Beyond
300 meters the terrain consists of trees and patches of pasture land.
Roughness lengths z can be determined from the following
equation:
Inz -
-------
is a plot of the average of all profiles of wind speed versus
ln(z) - i|;(z/L) for each height. The intercept value of south wind
profiles using the 18.3 meter and 91.4 meter levels was -2.25, which
corresponds to a roughness length near 8 centimeters. The z from the
southwest was determined similarly to be 36 centimeters. These results
seem reasonable considering the terrain features. Panofsky and
Townsend (1964) showed that the turbulence parameters measured at a
given height reflect the roughness elements of terrain at a distance
upwind approximately 10 times the height of the instrument.
Surprisingly, the average profile for the southwest winds yielded
a linear relation for all levels of the tower with no abrupt or even
noticeable changes in slope. This seems to imply that the z for
southwest winds experiences no changes in value at distances beyond
300 meters from the tower (this contradicts visual observations of the
roughness elements). It is possible that profile averaging has obscured
properties of the individual profiles so that this contradiction appears.
The same effect may have produced the kink in the mean profile
representing the south mode.
43
-------
N
i
s\
N
c
MERN WIND SPEED CM/SEC)
Figure 15. ln(z) - iJ*(z/L) versus mean wind speed. Triangles and
squares are wind from the southwest and south, respectively.
44
-------
SECTION XIV
o , o , AND a /a VERSUS z/L
ij A jj A
According to similarity theory (see Section XXIV)
and are universal functions of z/L. Using (13.1), (15.3),
w v
and (15.4), we obtain
a = a /U and (14.3)
Hi W
OA £ av/U . (14.4)
k(|>w(z/L) kv(z/L) (14.5 &
ThuS ° =
E In Z/ZQ - *(z/L) A In z/zo - *(z/L) ' 14.6)
u*
if U = r [In z/z - 41 (z/L)]. It is seen that av and a are universal
K O LI &
functions of z/z and z/L. Panofsky and Prasad (1965) produced nomograms
relating a and a to In z/z and z/L.
ill A O o
In this analysis aE> a., and OE/
-------
LUJ
0
c
b
CD
CD
CD
LO
CO
CD
CD
CO
CD
LO
CM
CD
CD
CM
CD
LO
* i
CD
CD
« l
CD
LO
CD
I i I i I 1
D 1
- A 2
+ 3
X r
r 5
~m 6
a
A
+
o
~ X *
X+*JIA+
x \ *** S^f*
"v"i^-«Y"
~" " " iW
"^r*
i i i i i i
12.0 -9.0 -6.0 -3.0 0. 3.0 6.0 9
.0
z/L
Figure 16. a. versus z/L for mode 1 (south winds). Symbols correspond
A
to measurement heights of 18 m (1), 91 m (2), 137 m (3),
183 m (4), 244 m (5), and 305 m (6).
46
-------
CD
LU
CD
O
10
CO
0
CO
CD
IT)
CM
O
OJ
CD
LO
i i
CD
CD
l
CD
LO
a
A
+
h- x
Y
X
1
2
3
4
5
6
D
ft
-12.0 -9.0 -6.0 -3.0 0.
z/L
3.0 6.0 9.0
Figure 17. a versus z/L for mode 2 (southwest winds). Symbols
A
correspond to measurement heights of 18 m (1), 91 m (2),
137 m (3), 183 m (4), 244 m (5), and 305 m (6).
47
-------
than for flow over the terrain associated with the smaller z , (mode 1).
The a values In Figure 16 for z/L < 0 seem to indicate a systematic
A
progression of a. to lower values as the reference height increases.
A
This might reflect the effect of sampling times on a., i.e. longer
A.
samples of data are needed at the higher levels to obtain a stationary
value for the average. If sample lengths are not critical at these
heights then a. at the lowest levels may be a predictor for the a.
** A
values higher up on the tower. For z/L > 0, o. decreases as z/L
A
increases.
The large amount of scatter in a. values near z/L = 0 is puzzling
A
especially since extremely light winds are absent from the data.
W. S. Lewellen has pointed out (personal communication) that according
to invariant modeling the atmosphere seems to possess hysteresis, i.e.
it remembers its previous stability condition when passing through
neutral. This might account for the larger scatter. Other possible
explanations are that slight shifts in azimuth reflect strikingly
different terrain and that some levels of the tower are affected more
than others. This problem is still undergoing investigation.
Values of o\, were plotted against z/L using data corresponding to
Ci
modes 1 and 2 (Figures 18 and 19 respectively). The different rough-
nesses corresponding to modes 1 and 2 apparently have a very significant
effect on cr_ (i.e. the highest values for c_ are approximately twice as
Ci Jl
large on the mode 2 plot as compared with those on the plot corresponding
to mode 1). a., tends to take on higher values at the lower reference
£i
levels and lower values at higher levels. The mode 1 plot exhibits a
tendency for a_ to be nearly constant with height when o_, is near its
48
-------
o
ro
O
/">
CD
UJ
a
00
D 1
h * 2
+ 3
y 5
-12.0 -9.0 -6.0 -3.0 0.
z/L
3.0 6.0 9.0
Figure 18. a_, versus z/L for mode 1 (south winds). Symbols correspond
£i
to measurement heights of 18 m (1), 91 m (2), 137 m (3),
183 m (4), 244 m (5), and 305 m (6).
49
-------
CD
CM
CD
CD
£ CM
UJ ^
Q
V /
U.
b
CD
00
CD
CD
I 1 1 I 1 1
Y
* x
Y
*" «
» »D
X +
» Y Y »
°X
* m
x y-fe
Y *» *» +
*** °
"«*«*3
* V «vji$
!!^0
D 1 *«^X^
X 4 * WX X Y
* 6 /.** *
1 1 1 1 1 1
12.0 -9.0 -6.0 -3.0 0. 3.0 6.0 9
z/L
Figure 19. a., versus z/L for mode 2 (southwest winds). Symbols
E
correspond to measurement heights of 18 m (1), 91 m (2),
137 m (3), 183 m (4), 244 m (5), and 305 m (6).
50
-------
maximum value and for a to decrease with height when a is slightly
£i E
less than the maximum. Figure 19 (mode 2) displays considerably more
scatter than Figure 18 (mode 1) which, apparently, is an effect of the
rougher fetch traversed by the air.
A qualitative comparison of the plotted o and a. trends with the
1£ A
nomograms prepared by Panofsky and Prasad (1965) resulted in fair
agreement. If a and a were functions of z/L only, then for any z/L
value these parameters should each equal a constant (i.e. no additional
height dependency whatsoever). Although there seems to be a dependency
on z/L, there are also systematic changes due to other effects that
could be attributed to a need for additional length scales.
A result of similarity theory is the relationship,
OE/OA = w(z/L)/v(z/L), where (14.5)
cf> and are universal functions of z/L. This result implies that if
the conditions for similarity theory are met (i.e. steady-state, hori-
zontally homogeneous, atmospheric surface layer), then cr_/aA should be
t A
a function of z/L only. Horizontal homogeneity is not a valid assumption
considering the terrain surrounding the tower. Also, most of the levels
at which the measurements were made exceed those generally included in
the surface layer. Because of these restrictions, a_/a. may be dependent
Cl A
upon additional scaling parameters.
Plots of a.,/0. vs z/L were produced corresponding to data from
lj A
modes 1 and 2 (Figures 20 and 21). Although both plots display roughly
similar trends, the ratios corresponding to the rougher terrain (mode 2)
have a greater range of values than the mode 1 ratios. It is likely that
the difference is terrain induced. In general, the ratios tend to be
higher under unstable conditions and lower during stable conditions
regardless of mode.
51
-------
, 1
"-*
o
1 1
^-,
s °°
Q o
V /
b -
\ CD
UJ
b
st-
0
CV
CD
.
1 1 1 1 I 1
X
K V *
a
* x I *. ~
Y K '^^fei
~ V 4 * T 4 y^fg^. ( ~
Y v *^HBSr«
x * Y'-^wP*
~"~ jf y Jj^f&C' " ~~
.oS'1--*
* Y «+« *«*
* x * SP^. Y
»
1 v"
D 1 X
~A 2
3x T
. .
n *
x
Y 5
~* 6
1 1 1 1 1 1
°-12.0 -9.0 -6.0 -3.0 0. 3.0 6.0 9.0
z/L
Figure 20. a^/a versus z/L for mode 1 (south winds). Symbols correspond
E A
to measurement heights of 18 m (1), 91 m (2), 137 m (3),
183 m (4), 244 m (5), and 305 m (6).
52
-------
CM
00
en to
UJ
b
CM
CD
o
D 1
A 2
+ 3
n
x T
Y 5
m 6
1
»
Y X
Y
*
1
« *
*
1
-12.0 -9.0 -6.0 -3.0 0.
z/L
3.0 6.0 9.0
Figure 21. a /a. versus z/L for mode 2 (southwest winds). Symbols
£ A
correspond to measurement heights of 18 m (1), 91 m (2),
137 m (3), 183 m (4), 244 m (5), and 305 m (6).
53
-------
SECTION XV
TRANSFORMATION RATIOS
In this section the validity of the following approximations is
analyzed:
°v a °AU» (15.1)
ow - oEU . (15.2)
The rationale justifying these approximations discussed by Irwin (1974)
is based upon the assumption that double correlation terms such as v'v'
are much larger than the triple and higher correlation terms such as
v'v'u1 and others. If the rather questionable assumption that a = a
is made, then a = CTAU. These relationships have a practical signifi-
U A
cance in that the terms on the right can be measured directly from
standard equipment making the transformation to a streamline coordinate
system unnecessary.
Regression and variance analyses and plots of o /a , a /a and
V A U A
-------
O
OJ
LU
Q OJ
UJ
b
I
8 12
U CM/SEC)
16
20
Figure 22. The ratio, a /
-------
CD
OJ
CD
cr
b
00
i
8 12
U CM/SEC)
I I
16
20
Figure 23. The ratio, 0 /a , versus the magnitude of the mean vector
wind, U, using edited data.
56
-------
o
CM
O CM
cc
b
1
i .°T *~rr
o a
.
1 1 1
8 12
U CM/SEC)
16
20
Figure 24. The ratio, o^, versus the magnitude of the mean vector
wind, U, using edited data.
57
-------
Table 3. ANALYSIS OF VARIANCE FOR 0 /a VERSUS THE MEAN WIND VELOCITY U
U A
Source of
variation
Average
Linear regression
Deviations from
D.F.
1
1
712
Sum of Squares
57506.9
11068.5
3156.5
Mean Square
57506.9
11068.5
4.4
regression
Total 714 71731.9
2
R = Multiple regression coefficient = 0.778
Assuming the true population model is given by
aw/CTA = al + e!U + £ >
2*
where e is normally distributed with mean 0 and variance a
The regression model is a /a. = a.. + b,U where a., is the estimate of
and b, is the estimate of 3-1
The regression coefficients are
an = -1.083 with a standard deviation 0 = 0.216,
1 al
b, = 1.286 with a standard deviation 0, = 0.026.
1 bx
The 95 percent confidence intervals for a., and fL are given by
-1.506 < o^ < -0.660 ,
1.235 < < 1-337 .
* This assumption will be the same for all population models included
in this report and will be omitted from all following tables.
58
-------
Table 4. ANALYSIS OF VARIANCE FOR 0 /a. VERSUS THE MEAN WIND VELOCITY U
V A
Source of
variation D.F.
Average 1
Linear regression 1
Deviations from
regression 712
Total 714
R2 = 0.995
Sum of Squares
41955.0
6533.7
33.0
48521.7
Mean Square
41955.0
6533.7
0.046
Assuming the true population model is a /a. = a_ + $2U + e,
then the regression model is a /a = a + b,U,
V A £* <-
where a» = -0.063 and the standard deviation a = 0.022,
and b0 = 0.988 and the standard deviation a. = 0.003.
2 b2
The 95 percent confidence intervals for a_ and 62 are
-0.106 < o.n < -0.020,
0.982 < 62 < 0.994.
59
-------
Table 5. ANALYSIS OF VARIANCE FOR a /a,, VERSUS THE MEAN WIND VELOCITY U
W E
Source of
variation
Average
Linear regression
Deviations from
regression
Total
D.F.
1
1
712
714
Sum of Squares
41169.4
6357.4
46.1
47572.9
Mean Square
41169.4
6357.4
0.065
R2 = 0.993
True population model: a /OE = a_ + 3_u + e
Regression model: a /a = a~ + b-U
v7 Hi -J J
a = -0.028 and a = 0.026 b0 = 0.975 and o, = 0.003
3 a_ 3 b_
The 95 percent confidence intervals for a» and 6, are
-0.079 < ct3 < -.023 and 0.969 < $3 <_ 0.981.
been removed, (Figures 25, 26 and 27). Amazingly little scatter v;as
detected on the a la and a /a plots indicating that this relationship
is very good under any of the conditions encountered in this experiment
including "quasi-laminar" flow which occurs at night. The a /a plot
showed a fair increase in scatter for the plot using the points removed
during editing. Thus editing seemed to improve this relationship more
than it did the other two.
Linear regressions of a on a /U and 0A on a /U were done in order
E w A v
to obtain regression equations for a_ and cr. in terms of a /U and a /U
fc. A w v
2
directly. The regressions yielded R 's of 0.991 and 0.982, respectively.
The regression and variance analyses are contained in Tables 6
and 7.
60
-------
CD
OJ
CD
LU
Q OJ
LU
b
b co
D
1 T
I I
a
8 12
U CM/SEC)
16
20
Figure 25. The ratio, a /
-------
CD
OJ
OJ
o:
b
00
0
I i i r
j i
8 12
U CM/SEC}
I I
16
20
Figure 26. The ratio, a^/a , versus the magnitude of the mean vector
wind, U, using data removed during editing.
62
-------
CD
OJ
lO
^ ,
a
o
a o B o
B o - a
S "f "
'.' : .1
B OB O
-0° ?: ***' .-
00 a B
0 0°
_ " ° *" * OB ° °a
"."o'0" " S""", °SS ',' " " »o "
B '"'""a, " B° f
° ^"l"*^, ^C B° ° o ~
> *^*?^ - "
h^blf "*" 0 ° * ° B °
. ^^^i**"a " OD
°" ^^^Ef1*'1*" Si0" " * *
qJ-atf&A't'i If0 B °B B
:a^r^ *" :
>* r i i i i i a. i i i
^f 8 12 16 2
U (M/SEC)
Figure 27. The ratio, a /a., versus the magnitude of the mean vector
U A
wind, U, using data removed during editing.
63
-------
Table 6. ANALYSIS OF VARIANCE FOR a,, VERSUS a /U.
JL W
Source of
variation
Average
Linear Regression
Deviations from
regression
Total
True Population Model :
Regression Model: a =
D.F.
1
1
712
714
a = a1
lli -L
al + bl
Sum of Squares
8.9865
2.4386
0.0229
11.448
+ B, a /U + e
1 w
o /U
w
Mean Squares
8.9865
2.4386
0.3219 x 10~4
R2 = 0.991
0.00059 and a = 0.00046.
b. = 1.028 and a, = 0.0037.
The 95 percent confidence intervals of a. and (3-, are
.00031_< a <_ .00149
and
1.273
1.287
Table 7. ANALYSIS
OF VARIANCE
FOR OA VERSUS a /U.
A v
Source of
variation
Average
Linear Regression
Deviations from
regression
Total
D.F.
1
1
712
714
Sum of Squares
17.2573
4.2286
0.0783
21.5642
Mean Squares
17.2573
4.2286
0.00011
R2 = 0.982
True Population Model: a. = « + 3_ a /U + e
Regression Model: a. = a« + b« a /U
-0.00336 and
0.00090.
b, = 1.049 and
i b.
0.0054.
The 95 percent confidence intervals of « and £ are
-0.00186 <.a2£ 0.00512 and
1.038< B0< 1.059
64
-------
It can be concluded from these results that the approximations,
av = aAu and (15.3)
°w = CTEU , (15.4)
are extremely good In the range of velocities used. The approximation,
*u " *AU . (15.5)
Is not nearly as good as the first two and should probably be written as
o « o (a + b D) , (15.6)
U A
where for this study
a = -1.08 and (15.7)
b = 1.29. (15.8)
It is also implied from these results that triple and higher
correlation terms are indeed very small and in most cases can be
ignored.
65
-------
SECTION XVI
°w/U*
Similarity theory predicts that in neutral conditions (|z/L,| < 0.05)
cr /u. is a constant, about 1.2 to 1.3 (see Panofsky, 1972). Irwin (1974)
W **
using the SRL-WJBF tower data determined the ratio to be 1.45. Figures
28 through 30 are scatter plots of a /UA using local u^ values. It is
important to note the distinction in the manner in which UA was calculated.
Irwin did not include the v'w* component in his calculation of u^. Figures
31 through 33 are scatter plots of a /UA, where UA is the surface u^.
The u^'s calculated from the Gill anemometer at the 18.3 meter level were
used as surface u^'s. In Table 8 are the mean values of a /u*> local
and surface, for neutral conditions. The weighted mean value for a /u^
(local) for all levels is 1.49. The weighted mean for a /UA (surface)
is 1.27, in good agreement with Panofsky (1972).
66
-------
Table 8. THE AVERAGE RATIOS OF a /u. FOR LOCAL u 's AND SURFACE u '
yf n
-------
CD
CD
CD
CO
« CD
D o
\
3 CM
CD
CD
a
a
I
I
-2.00
-1.00
0
z/L
1.00
2.CO
Figure 28. Ratio of standard deviation of vertical velocity to
local friction velocity as a function of z/L at 18.3
meters.
68
-------
CD
O
CD
CD
CO
« CD
D CD
\
3 c\j
CD
CD
CD
-10.0
I
I
-5.00
0
z/L
5.00
10.0
Figure 29. Ratio of standard deviation of vertical velocity to
local friction velocity as a function of z/L at 91.4
meters.
69
-------
o
CO
cv
a
a
D a
D
a D
n
ODO
1
-!0.0 -5.00
0
z/L
5.00
10.0
Figure 30. Ratio of standard deviation of vertical velocity to
local friction velocity as a function of z/L at 304.8
meters.
70
-------
CD
O
O
O
00
\
2
b
CD
CD
CM
CD
CD
a
a
D a
I
I
-2.00
-1.00
0
z/L
1.00
2.00
Figure 31. Ratio of standard deviation of vertical velocity to
surface friction velocity as a function of z/L at 18.3
meters.
71
-------
CD
CD
CD
CD
00
« CD
D CD
\
3 C\J
CD
CD
CD
D
0
I
I
-10.0
-5.00
0
z/L
5.00
10.0
Figure 32. Ratio of standard deviation of vertical velocity to
surface friction velocity as a function of z/L at 91.4
meters.
72
-------
CD
sJ-
CD
CO
o
D
° ° °Dqi °o_
D DO »
D AlO £1
flpD glBOO,
t»fe) °*0
a ° % HD[ji
0 ,,00*0 0
0 o OB o
o
a
I
-10.0
-5.00
0
z/L
5.00
10.0
Figure 33. Ratio of standard deviation of vertical velocity to
surface friction velocity as a function of z/L at 304.8
meters.
73
-------
SECTION XVII
THE SCALING PARAMETER,
It is generally accepted that u^/(Bf) is the thickness of the neutral
Ekman layer, where g is a constant between 0.3 and 0.5 and UA and f
denote the surface friction velocity and Coriolis parameter, respectively,
If a and a were functions of z/L only, then for any z/L value these
parameters should each equal a constant (i.e., no additional height
dependency whatsoever). This, however, has not been found to be the
case. Although there see,r.is to be a dependency on z/L, there are also
systen/atic changes due. to other effects that could be attributed to a
need for additional length scales.
Above the surface layer (i.e., the Ekman layer) the Coriolis
parameter becomes important. This suggests a scaling height related
to the thickness of the Ektnan layer under neutral conditions, fz/u.v.
Other choices of additional scaling parameters include z/z where z
is the surface roughness length and z/z ' where z ' is the upstream or
bulk roughness length associated with a sufficiently long fetch (possibly
15 km) traversed by the air. Blackadar and Tennekes (1968) derived a
velocity defect law valid .in the Ekman layer under conditions of steady,
horizontally homogeneous flow and for neutral atmospheric stability that
was a function of In fz/ufc. The relationship between a and In fz/uA is
d in Appendix D.
In this analysis data corresponding to neutral stabilities, defined
as z/L <_ | 0.05 | were selected resulting in 54 time blocks of data, two
of whifh were discarded because both the wind azimuth and the elevation
angle standard deviations were anomalously large at all levels of the
74
-------
tower. For convenience, values for the same time blocks were used for
all levels of the tower. This implies a linear increase in ranges of
z/L values with height. For example, at the 304.8 m level, instead of
a range of z/L values between ±0.05, the range is ±0.83. It was expected
that this biasing effect would produce some scatter in the plots for the
higher levels. Values of fz/u^ were computed for each set of a and a
using a Coriolis parameter for 33° latitude and the surface friction
velocity, UA.
Figure 34 is a scatter plot of ap versus fz/u^ for six levels
(18.3 to 304.8 meters) and neutral stabilities (-0.05<^ z/L <_ +0.05)
defined at 18.3 m. The solid regression line accounting for 70 percent
of the variation of a_ with f z/u, is described by
b *
OE = -5.9 - 3.8 In (fz/u*) - 0.2 [In (fz/u*)]2. (17.1)
Tables 9 and 11 provide regression and variance analysis data from
linear and quadratic regression of a_ on In fz/u...
E "
Figure 35 is an analogous plot for a. versus f z/u^ again for all
levels (18.3 meters to 304.8 meters) and neutral stability (-0.05 <_ z/L <_
+0.05) defined at 18.3 meters. The fitted line accounting for more than
67 percent of the variation of a with f z/u^ is described by
OA = -1.75 - 2.35 In fz/u^.
The plots show a distinct dependency on fz/u^. Some striking
features are: (1) a similarity in trend and in scatter between the
plot of aF and the plot of a.. (2) An extremely rapid drop in the
ordinate value with a relatively minute increase in the abscissa at the
low end of the scale. (3) a and a seem to approach fixed values
JL A
asymptotically at large values of fz/u^.
75
-------
CD,
0
0.020 0.040 0.060
fz/u,
0.080 0.10 0.12
Figure 34. o versus the scaling parameter, fz/u., for all levels
h w
(18.3 to 304.8 m) under near neutral stability conditions
(-0.05 < z/L < 0.05) as defined at 18.3 m. The solid line is
a regression fit that accounts for 70 percent of the variation
of o with In fz/uA. The regression equation is:
or = -5.9 - 3.8 In fz/u. - 0.2(ln fz/u.)2.
76
-------
CD
CM
CD
UJ
Q
cr
b
LO
18
D
CD,
0
I
I
0.020 O.OfO 0.060 0.080 0.10
fz/u*
0.12
Figure 35. o versus the scaling parameter, fz/u^, for all levels
A
(18.3 to 304.8 m) under near neutral stability conditions
(-0.05 1 z/L <_ 0.05) as defined at 18.3 m. The solid line is
a regression fit that accounts for 67 percent of the variation
of a with In fz/u.. The regression equation is:
E *
o = -1.75 - 2.35 In fz/u..
A **
77
-------
Table 9. LINEAR REGRESSION AND VARIANCE ANALYSIS OF a^ ON ln(fz/u.J FOR
E *
NEAR NEUTRAL STABILITY.
Source of
variation
Average
Linear Term
Total due to
regression
Deviations from
regression
Total
Sum of Squares
9604.750
906.099
10510.848
420.846
10931.695
D.F.
1
1
2
310
312
Mean Square
9604.750
906.099
5255.422
1.358
35.037
F
2251.09
667.44
3871.21
R2 = 0.684
True Population Model: o = a + 3,ln(fz/u^) + e
Regression Model: a = a + bnln(fz/u.)
r* J- _L *
a. = -1.465 and a = 0.028. b. = -1.717 and a, = 0.066
1 al * bl
The 95 percent confidence intervals for «1 and 3. are
-1.520 < a, < -1.410 and -1.846 < (3. < -1.588
78
-------
Table 10. LINEAR REGRESSION AND VARIANCE ANALYSIS OF OA ON ln(fz/u^) FOR
NEAR NEUTRAL STABILITY.
Source of
variation
Average
Linear Term
Total due to
regression
Deviations from
regression
Total
Sum of Squares
19271.445
1701.895
20973.305
831.020
21804.324
D.F.
1
1
2
310
312
Mean Square
19271.455
1701.895
10486.652
2.681
69.886
F
2366.25
634.85
3911.90
R2 = 0.672
True Population Model: a = a + 3 ln(fz/u^) +
Regression Model: a
hi
a + b,ln(fz/u.)
1 X x
-1.753 and a = 0.393.
-2.353 and a, = 0.093
The 95 percent confidence intervals for a. and $1 are
-2.523 <_ "a £ -0.982
and
-2.535 <_ 3.,^ 1 -2.170
79
-------
Table 11. QUADRATIC REGRESSION AND VARIANCE ANALYSIS OF a ON ln(fz/uA)
FOR NEAR NEUTRAL STABILITY.
Source of
variation
Average
Linear Term
Quadratic Term
Total due to
regression
Deviations from
regression
Total
True Population
Sum of Squares
9604.750
906.099
19.960
10530.809
400.886
10931.695
Model: a = a + (
Hi J-
D.F.
1
1
1
3
309
312
^InCfz/u,
Mean Square
9604.750
906.099
19.960
3510.270
1.297
35.037
c) + f32[ln(fz/u,
F
2251.09
667.44
15.38
2705.69
R2 = 0.70
.>i2 + «
o
Regression Model: a_ = a. + b.,ln(f z/u.) + b~[ln(fz/u.)]
Ell * / *
an = -5.941 and a = 1.173. b. = -3.811 and a, = 0.059.
1 31 l bl
bn = 0.231 and a = 0.059.
2 b2
The 95 percent confidence intervals for a , S, and 3? are
-8.240 < a.. < -3.642, -4.865 < 3, < -2.756,
X *"~ i ~~*
and - 0.347 < 3 < -0.172
80
-------
Table 12. QUADRATIC REGRESSION AND VARIANCE ANALYSIS OF OA ON ln(fz/u^)
FOR NEAR NEUTRAL STABILITY.
Source of
variation
Average
Linear Term
Quadratic Term
Total due to
regression
Deviations from
regression
Total
Sum of Squares
19271.445
1701.859
1.690
20974.992
829.329
21804.324
D.F.
1
1
1
3
309
312
Mean Square
19271.445
1701.859
1.690
6991.664
2.684
69.886
F
2366.25
634.85
0.63(NS)
2605.03
R2 = 0.672
(NS) indicates the term is not statistically significant at the .05 level
as determined by the F-Test. This indicates the true model should be a
linear relation.
2
True Population Model: a. = an + 3,ln(fz/u.,) + B0[ln(fz/u.)] + e
All x Z *
Regression Model: a = a1 + 1
cV _L
a, = -0.451 and a = 1.687.
+ b,[ln(fz/u.)r
z "
bn = -1.744 and cr
-L D,
0.773.
b0 = 0.067 and a,
2. b.
0.085
The 95 percent confidence intervals for a.., 3, and 3? are
-3.758 <_ a^ <_ ' 2.856,
and -0.100 < 3^ < 0.234.
-3.260 < 3, < - 0.229,
^ J.
81
-------
Tables 10 and 12 provide regression and variance analysis data from
linear and quadratic regressions of a. on In fz/u^.
Figure 36 is a plot of a versus fz/uft for stable conditions
(0.45 < z/L < 0.55) for all levels. The regression line accounting
for nearly 45 percent of the variation of 0_ with fz/u.. is
K *
aE = 1.75 - 1.02 In fz/uA. (17.3)
Figure 37 is a plot of a. versus fZ/UA for stable conditions
(0.45 < z/L < 0.55). The regression line in this case accounts for
only 17.2 percent of the variation of o. with fz/u^. The plot seems to
indicate that there is very little trend of the standard deviation of
the azimuth angle with fZ/UA and appears to be nearly a constant value
approximately equal to 5 degrees.
Figure 38 is a plot of o versus fZ/UA for unstable conditions in
the range (-0.55 < z/L < -0.45). The regression line accounts for only
18.4 percent of variation of 0_ with In fz/u.. In this case the range
t *
of values for fz/uA is fairly small and the scatter on the plot appears
to be quite large.
Figure 39 is a plot of a. versus In fz/u. for unstable conditions
A *
(-0.55 < z/L < -0.45). The regression line in this case accounts for
24.5 percent of variation of a. with In fz/u^. Scatter on this plot
is somewhat smaller than that of the standard deviation of the elevation
angle.
In conclusion, it has been found that under neutral atmospheric
conditions the standard deviation of the elevation and azimuth angle
of the wind is scaled quite well with the parameter fz/u^. Under
stable atmospheric conditions the relationship is not as clearly
82
-------
en
UJ
o
V 1
oo
I
I
I
I
0.020 0.010 0.060
fz/u.
0.080 0.10 0.12
Figure 36. o versus the scaling parameter, fz/u., for all levels
Ci *
(18.3 to 304.8 m) under stable conditions (0.45 1 z/L 1 0.55)
defined at all levels. The solid line is a regression fit
that accounts for nearly 45 percent of the variation of c^
with In fz/u^.
83
-------
CD
OJ
O
LU
Q
cc
b
LO
I
I
I
U.02G O.OfO O.U6G
fz/u.
0.08C 0.10 0.12
Figure 37. a versus the scaling parameter, fz/u^, for all levels
A
(18.3 to 304.8 m) under stable conditions (0.45 1 z/L 1 0.55)
defined at all levels. The solid line is a regression fit that
accounts for 17 percent of the variation of <*A with In fz/u^.
84
-------
OJ
en
CD
UJ
CD
°a
uJ
b
00
0.0.20 0..050 0.060 0.080 0..10
fz/u*
0.12
Figure 38. a versus the scaling parameter, fz/u^, for all levels
(18.3 to 304.8 m) under unstable conditions (-0.55 1 z/L 1 -0.45)
defined at all levels. The solid line is a regression fit that
accounts for 18 percent of the variation of o with In fz/uA.
85
-------
OJ
LU
Q
CE
b
fff
a a
I
I
1
I
0.020 O.OfO 0.060
fz/u*
0.080 0.1.0 0.12
Figure 39. o versus the scaling parameter, fz/u^, for all levels
(18.3 to 304.8 m) under unstable conditions (-0.55 1 z/L 1 -0.45)
defined at all levels. The solid line is a regression fit that
accounts for 24.5 percent of the variation of o". with In fz/u
*
86
-------
defined although a trend can still be seen in the a,, relationship. The
K
least responsiveness to this scaling parameter was generally found under
unstable atmospheric conditions where there seemed to be little variation
with fz/u^. This implies that the parameter is irrelevant for these con-
ditions as suggested by Deardorff (1970).
87
-------
SECTION XVIII
EDDY VISCOSITY
The eddy viscosity, K , is defined by:
m
aua
Under neutral conditions with homogeneous terrain the eddy viscosity can
be expressed as:
K = u,(z)kz (18.2)
in ~
Using the results of Blackadar and Tennekes (1968) on the change of fric-
tion velocity with height, it can be shown that the vertical gradient of
friction velocity in neutral, barotropic conditions can be expressed as:
3uA/9z = -3.83 x 10~4 sec"1. (18.3)
If the above equation is integrated with respect to height and substituted
back into the expression for K for neutral conditions one obtains:
m
= uA(0)kz - 3.83 x 10 Hkz . (18.4)
Eddy viscosities for neutral (|Z/L| < 0.05) conditions were
calculated from Eq. (18.1) using
3U/9z = AU/Az, (18.5)
where AU was the difference in mean speeds between levels measured by the
cup and bivane systems. The value Az was the difference between levels
and the derivative was assumed to apply at the mean height between levels.
The friction velocity was obtained by taking the measured values at
two consecutive levels, finding their mean, and squaring the result.
Figure 40 is the ensemble averaged eddy viscosity profile for
neutral conditions. The solid line is the theoretical change of K
m
88
-------
with height for neutral conditions using the change predicted by
Blackadar and Tennekes. The average value of the friction velocity
at the lower level for neutral conditions was 0.599 m/sec.
89
-------
CD
CD
CO
CD
CD
CM
IXI
CD
CD
I I I I I
I
30 60
K CMVSEC)
90
120
Figure 40. Eddy viscosity profile for neutral cases.
90
-------
SECTION XIX
RATIO OF EDDY DIFFUSIVITIES,
In the constant stress layer
VK» - ^/^^
It is frequently assumed that the ratio of heat and momentum diffusivities,
VK , is approximately unity in near neutral conditions. Businger et al.
m --
(1971) found that this ratio was roughly 1.35 and attributed the differ-
ence to k = 0.35 rather than the customarily accepted value of 0.4.
Kitaigorodskii (1973) presented a plot containing K, /K ratios from
several different sources. The value near z/L = 0 appeared to be less
than unity. Nickerson (1973) summarized results from various authors
who found K, /K ranging between 0.8 and 1.35 for near neutral stabilities.
For convenience in plotting the ratio, Panofsky has pointed out
(personal communication) that it is preferable to use K /K, instead of
VK , the reason being that the limiting value of the ratio as
m
z/L -> -<*> is zero. In particular, if the expressions derived by Businger
et al. (1971) for $ and , are the correct ones, the limiting values
m h
of K /K, at extreme ranges of stabilities are
m h
lim ,, _ lim .74 + 4.7 z/L _ , n
z/L - » V*m - z/L - - 1 + 4.7 z/L ~ 1<0 '
and
-1/2
!*» */* = "» 0.74(1 - 9 z/L) ' ^
z/L + -» V*m z/L H. - (1 _ 15 z/L)-l/4
Panofsky also points out that , becomes indeterminate at z/L = 0.
This causes considerable scatter of the data for near neutral conditions.
91
-------
Figures 41 and Ala are plots of K /K, vs z/L for the combined
levels of 91.4 and 137.2 (Figure 41) and 182.8, 247.4, and 304.8
meters (Figure 4 la) . The ratios were calculated using the relationship,
where (j> and were obtained from the regression technique. Although
the scatter is large, a qualitative examination reveals that the ratios
approach zero with increasing |z/LJ on the unstable side and seem to
approach a discontinuity at z/L = 0. On the stable side the ratios are
larger than on the unstable side and approach a value between 1 and 2
for large z/L. The large amount of scatter makes it difficult to see any
height dependencies in the data.
92
-------
oo
UD
CM
C\J
1
tD
CO
A 1
+ 2
*
$
1
-3 -2
-1 0 1
z/L
Figure 41. Ratio of momentum and heat diffusivity versus z/L (modes 1
and 2). Symbols correspond to measurement heights of
91 m (1) and 137 m (2).
93
-------
oo
CM
CM
1
vO
1
OO
1
-4 -3
-2
-1
0
z/L
Figure 41a. Ratio of momentum and heat diffusivity versus z/L (modes 1
and 2). Symbols correspond to measurement heights of
183 m (1), 244 m (2), and 305 m (3).
93a
-------
SECTION XX
SPECTRAL SCALE, A , OF THE VERTICAL COMPONENT AND MIXING LENGTH
m
Figure 42 is a vertical profile of X , where A is the wavelength
mm
corresponding to the maximum in the nS (n) vs ln(n) spectrum. The peak
nondimensional frequencies obtained from subjective analysis are related
to A , through the equations,
f = I;5- and A = , (20.1)
U m n
m
where
f is the nondimensional frequency,
n is the frequency in (hertz),
U is the magnitude of the mean vector wind (meters/sec)
n is the frequency at the spectral maximum (hertz),
A is the wavelength (meters) corresponding to the peak.
Due to flatness of some of the spectra there was considerable
scatter in the peak frequencies. The f values were determined by eye
estimates of the peak in the vertical velocity spectra. In some cases
aliasing made it impossible to determine the peak frequency. This
involved only 27 cases out of 618 vertical velocity spectra analyzed.
The data were separated into different stability classes using
z/L at 18.3 meters. The A 's were then averaged for each height and
stability. Figure 42 shows that the results are consistent with
Pasquill, 1972, for neutral, stable, and unstable conditions.
94
-------
The wind profile over homogeneous terrain and neutral conditions
can be obtained from:
£ - u*/* (20-2)
where £ is the mixing length. According to Pasquill (1972) , the relation
between £ and X in neutral conditions is £ = X /5. Thus, by relabeling
m m
the scale on the abscissa in Figure 42, i.e., by dividing abscissa values
by five one can estimate mixing lengths for neutral conditions.
Blackadar (1974) used concepts from Blackadar and Tennekes (1968)
to show that the mixing length for the neutral planetary boundary layer
could be functionally represented as:
£ = kzF(fz/uA) , (20.3)
where F is a universal function given by:
F = (1+63 fz/u^r1 . (20.4)
However, Blackadar was able to obtain a better fit to the observed wind
profile by using the expression:
0.0063 tanh . (20.5)
WJBF-SRL tower data was used to compute £ from Eqn. (20.5). The
averaged profile is shown in Fig. 43. Non-neutral stability classes
are shown for interest only. The estimates of mixing length by
Eqn. (20.5) are smaller than those by Pasquill 's A /5 estimates.
95
-------
103
D
D
D
D
O
UJ
HI
D
10*
101
, . I
102
XCmax) CM)
Figure A2. Vertical profile of the mean wavelength of the maximum of
the vertical velocity spectra. Squares, triangles, and
crosses are stable, neutral, and unstable cases, respectively.
96
-------
1C3
102
10C
10°
O /
o
o
oa
101
102
CM)
10a
Figure 43. Mixing length profile for Blackadar's interpolation
formula. Triangles, squares and octagons are unstable,
neutral and stable cases, respectively.
97
-------
SECTION XXI
SPECTRAL RATIOS
According to Kolmogorov's hypothesis all turbulent motions with
sufficiently high Reynolds number possess local isotropy in the high
frequency end of the spectrum. In this locally isotropic region, the
turbulent properties of the fluid are determined by viscosity, v, and
dissipation, e. At the low frequency end of the local isotropy range,
viscosity has little effect on the power spectral density function,
F(K). This region is called the inertial subrange. Therefore, in the
inertial subrange, the power spectral density function is related to
dissipation and wave number in the following manner:
2/3 -5/1
F(K) = Ce ' K ' . (21.1)
A theoretical result of Kolmogorov's hypothesis is that the con-
stant for the v and w components of the one dimensional power spectral
density function is equal to 4/3 of that for the u component.
As a consequence of Taylor's hypothesis, the inertial subrange
power spectral density function (PSDF) can also be written in terms
of frequency (Hertz) . If can be shown that
nS(n) - KS(K) = C^ K~2/3 = C2e (J)~, (21.2)
so that, plotting spectra versus frequency on a log-log plot, causes
the inertial subrange portion of the spectra to be indicated by a
-2/3 slope. In order to confirm that local isotropy exists in this
expected inertial subrange, the ratios of the v spectra to the u spectra,
and of the w spectra to the u spectra, should yield a value of 4/3. In
fact, confirmation of isotropy at high frequencies in the first 20 meters
has been somewhat scarce. The 4/3 ratio between inertial subrange
98
-------
spectral levels of the v and u components was not found in earlier
experiments (Kaimal et al., 1972). A trend toward the 4/3 ratio in
the Round Hill data was noted by Busch and Panofsky (1968). More
recently, Kaimal et al. (1972) found good agreement with the 4/3 pre-
diction for all but the most stable cases for dimensionless frequencies
greater than one. Busch (1973) found that the frequency range in which
the 4/3 ratio is obtained is shorter than the range where -5/3 slope
is obtained. The conclusion is that isotropy exists only over the
shorter of these two ranges. This has generally been found by those
who have investigated. In fact, many times the -5/3 slope has been
found while at the same time the required 4/3 spectral ratios have not.
This is significant, since much of the theory concerning atmospheric
turbulence is based upon the assumption of the existence of an inertial
subrange and, therefore, of local isotropy at the high frequency end of
the spectrum.
To determine whether the sonic spectra confirmed the existence
of isotropy at the high frequency end of the spectrum and, therefore,
the existence of an inertial subrange (this also requires a -2/3 slope
for the logarithmic spectra), the ratios, S (n)/S (n) and S (n)/S (n),
were calculated and plotted against nondimensional frequency, f, with
the expectation that any trends in the ratios toward 4/3 could be de-
tected. The plots contain symbols which identify each set of ratios
with a particular z/L range. The nondimensional frequency, f, repre-
sents the abscissa and the spectral ratio is the ordinate. The magni-
tude of the mean vector wind, U, used in determining f ranged between
2.5 and 6.5 meters per second for this set of spectra. The ratios
99
-------
were plotted using a semilog plot and again using a logarithmic plot.
The linear scale was used to present a more graphic display of the
amount of scatter.
Figure 44 is a plot of Sw(n)/Su(n) versus f. No systematic changes
due to stability differences can be detected. This is probably due to
the small range of stabilities used. Although there is considerable
scatter in this plot, the ratios can be seen to approach 4/3 near f = 1.
A realistic display of the scatter in the plot can be seen in Figure 45
An average of S (n)/S (n) for f greater than 1, yielded a value of 1.23.
The ratio, S (n)/S (n), is plotted in Figure 46. Kaimal et al.
v u
(1972) found that S (n)/S (n) approached the 4/3 ratio at lower values of
frequency than S (n)/S (n) and this tendency can be seen on Figure 46 as
well. The ratios on this plot appear to approach and nearly reach the
4/3 line around a nondimensional frequency of 0.5. The mean value of
the ratios for frequencies greater than 1.0 is 1.27 with a variance of
0.07. As with the previous plot, there does not appear to be any syste-
matic changes with z/L between the various spectra. This is probably
because the range of z/L values is small. Figure 47 is a plot of
S (n)/S (n) versus f with a linearly scaled ordlnate which emphasizes
the amount of scatter.
In summary, it has been shown that the spectral ratios (S (n)/S (n)
and S (n)/S (n) ) obtained from the one dimensional atmospheric turbu-
lence velocity spectra computed from velocity fluctuations, which were
measured on a sonic anemometer at a height of 18.3 meters under condi-
tions of near neutral stability, exhibit a distinct trend towards the
4/3 ratio that should be expected at the high frequency end of the
100
-------
spectrum if local Isotropy and therefore the inertial subrange exists.
The results displayed here are in general agreement with those of
Kaimal e£ .al/ (1972) for a comparative stability range.
101
-------
o
ro
101
10"
c
v_->
3
CO
CO
10
-I
io-
+ 1
o
- x 3
A 4
10
-3
I I I
xxxxxxx***
10
-2
i o°
f=nz/U
101
10a
Figure 44. Ratio of w spectra to u spectra versus nondimensional frequency using logarithmic
ordinate. Dashed line indicates an ordinate value of 4/3. Symbols designate
ratios corresponding to the following z/L values: 1 - (z/L = -0.305),
2 - (-0.196 < z/L < -0.190), 3 - (-0.096 < z/L <- 0.053), 4 - (z/L = 0.103).
-------
3
CO
CM
S co
U)
1 I I
+ 1
n 2
x 3
-A -
f=nz/U
Figure 45. Ratio of w spectra to u spectra versus nondimensional frequency using linear
ordinate. Dashed line indicates an ordinate value of 4/3. Symbols designate
ratios corresponding to the following z/L values: 1 - (z/L = -0.305),
2 - (-0.196 < z/L < -0.190), 3 - (-0.096 < z/L < 0.053), 4 - (z/L = 0.103).
-------
10C
3
00
c
*>
>
10
-1
10
-2
D 2
-x 3
DDDDDDDDDDDOn
cxxxxxxxsgx"^
J I
I
I
. I
10-
10
-2
10'1 10°
f=nz/U
Figure 46. Ratio of v spectra to u spectra versus nondimensional frequency using logarithmic
ordinate. Dashed line indicates an ordinate value of 4/3. Symbols designate
ratios corresponding to the following z/L values: 1 - (z/L = -0.305),
2 - (-0.196 < z/L < -0.190), 3 - (-0.096 < z/L < 0.053), 4 - (z/L = 0.103).
10a
-------
LO
C
v_^
CO
\
C
CO
CO
o
Ol
~~r
+' 1
D 2
x 3
*
-------
SECTION XXII
NORMALIZED TURBULENCE VELOCITY SPECTRA
Atmospheric turbulence velocity spectra were analyzed and plotted
using a procedure which was used by Kaimal et al. (1972) in which the
spectra were collapsed into universal curves in the inertial subrange.
The spectral behavior at the lower frequencies were then observed as
a function of z/L.
The inertial subrange one-dimensional velocity spectra normalized
2
with u^ (surface stress) can be written in the form,
nS (n) _ q -kze, ,na\ (22.1)
2 - _ ,.2/3 < 3' Hi }
u (2uk) u
where
H£ = f (22.2)
U
kze _
-3 = *e . and (22.3)
u*
a = a1, q_, or a, depending upon the component of velocity, u, v,
or w, respectively, being used.
The normalized velocity spectra rewritten in this notation become
nS(n) q . 2/3 .-2/3
2 * 2~7T *c f ' (22.4)
u* (2irkr/J £
The only parameter on the right that is dependent upon z/L is .
2/3
Dividing both sides of the equation by results in an equation
which is a function of the nondimensional frequency only. Normalizing
the velocity spectra in this manner effectively removes the z/L depen-
dence from the equations for the inertial subrange portion of the
spectrum. This brings all spectra, regardless of stability, into
106
-------
coincidence in Che inertial subrange* The differences between these
spectra at the low frequency end of the spectrum outside of the inertial
subrange can then be attributed to z/L differences. It should be noted
2
that, although UA appears in the normalization term, applying the defi-
2/3 2 2/3
nition of <|> results in cancelling the u^ term leaving (kze) as
the normalizing factor.
Figure 48 is a plot of the w spectra versus nondimensional frequency
where the spectra have been normalized using the method described above.
Each individual spectrum is shown with a different symbol corresponding
to a z/L value. This method of normalization seems to give reasonable
results. The spectra have collapsed in the inertial subrange and ex-
hibit a slope very nearly equal to -2/3. As indicated in Section XXI
on spectral ratios, the inertial subrange can be expected to begin near
a nondimensional frequency of one. There are 13 sets of spectra on
this plot with stabilities in the range (-0.3 < z/L < 0.1). The two
most stable spectra are arranged according to z/L at the low frequency
end of the spectrum but the third stable spectrum, which is very near
neutral (z/L = .0006) lies among the main group of unstable spectra.
The unstable spectra do not seem to exhibit any particular trend with
decreasing stability. A spectral maximum representative of the full
range of stabilities plotted appears to be near 0.4 for f and 0.4 to
0.5 for the magnitude of the normalized spectra. These results are
generally consistent with those from the 1968 Kansas experiment (Kaimal
et al., 1972) for an equivalent stability range.
f.
Figure 49 is a plot of the normalized spectra for the v component
of the velocity. Again, as in Figure 48, the spectra collapse in the
107
-------
to1
e
D
10
-
10
-2
* 1
x 2
x 3
J3 5
D 6
x 7
+ 8
A 9
o 10
Y 11
* 12
* 13
1 1 1
10
-3
10
-2
i o°
f=nz/U
101
Figure 48. Normalized w spectra versus nondimensional frequency. Symbols designate
spectra corresponding to the following z/L values: 1 - (-0.305), 2 - (-0.251),
3 - (-0.196), 4 - (-0.136), 5 - (-0.109), 6 - (-0.096), 7 - (-0.065), 8 - (-0.063),
9 - (-0.042), 10 - (-0.020), 11 - (0.0006), 12 - (0.053), and 13 - (0.103).
102
-------
5! 10C
CM
-------
inertial subrange and a slope of -2/3 is observed. The stable spectra
are arranged according to z/L. Of particular interest is the separation
between the areas occupied by the stable and unstable spectra as though
the spectra were excluded from this region. Kaimal et^ ad. (1972) noted
a similar property in the v component of the normalized spectra. This
seems to indicate a sudden shift in the predominant scale of motion as
z/L changes sign. The unstable spectra do not display any systematic
changes with z/L and tend to be randomly clustered. A spectral peak
representative of the stable spectra exists at approximately 0.7 and
corresponds to a nondimensional frequency near 0.2. This value is to
be compared with a peak value of 0.5 and a nondimensional frequency of
0.25 obtained in the 1968 Kansas experiment. The spectral peak for the
unstable side which represents a z/L range of 0 to -0.3 falls between
1 and 5 and corresponds to a range of frequencies between 0.05 and 0.09.
Figure 50is a plot of the normalized logarithmic u spectra. As
with the w and v spectra, these exhibit the collapsing in the inertial
subrange and also some progression with stability at the low frequency
end of the spectrum for positive z/L values. The unstable spectra do
not reveal a progression with z/L but appear to be clustered in a ran-
dom fashion. A stable spectral peak of 1 to 1.5 is observed near a
nondimensicnal frequency of approximately 0.1. The spectral peaks cor-
responding to the unstable range of z/L lie between 2 and 4 in a non-
dimensional frequency range of .05 to .08. These results compare
favorably with those presented by Kaimal et_ a!L. (1972).
110
-------
to1
CO
CM
e
cvj
10C
-------
SECTION XXIII
THE KOLMOGOROV CONSTANT
The logarithmic u spectra for the inertial subrange in terms of
frequency in Hertz can be expressed as
o t \ 2 2/3 ,2Trn.-5/3 ,~- ..
nSu(n) = c^ -^- e (-^-) . (23.1)
Under conditions of steady state, horizontal uniformity, and neutral
atmospheric stability, the dissipation is generally assumed to be
e = ~ . (23.2)
Thus the logarithmic u spectra for the inertial subrange can be re-
written as 2
al u* -2/T
nSu(n) -- 2/3 -- 2/3 f ' (23'3)
U (2Trr/J It /J
From the above equation it is apparent that determination of the value
of the Kolrcogorov constant depends upon the value assumed for the von
Kantian constant. In order to examine the relationship between these
two constants, six sonic u spectra corresponding to near neutral stabil-
ity were used. Selected inertial subrange u spectra values corresponding
to nondimensional frequencies greater than 1 were required to correspond
to a point where S (n)/S (n) and S (n)/S (n) were both near 4/3 in value.
Table 13 lists the set of u spectra values that were used along with the
corresponding nondimensional frequency and time of observation. Also
listed is the u^ corresponding to each spectrum and a term labeled
"factor" (defined in the table) . The factor for each time period was
calculated and the average value was determined. Three different
values of k were then used to estimate three values for a.. . The results
are shown in Table 14.
112
-------
Table 13. PARAMETERS USED TO CALCULATE c^ AS A FUNCTION OF k.
Time
139 1235
139 1355
139 1555
140 1740
140 1820
144 1334
al
z/L
-0.110
-0.048
-0.072
0.001
0.060
-0.022
-, r
I
f
3.14116
4.20622
4.08560
4.07796
3.06275
1.20109
/3 nS (n)
u l
2 ]
nS (n)
0.02405
0.02976
0.02860
0.02789
0.03129
0.19433
2/3
«*
0.66405
0.42556
0.57572
0.58587
0.38838
1.01300
the bracketed
»*<,«. «,.»
Factor
0.48450
1.63002
1.06410
0.86167
1.84713
1.00473
term repre-
Mean Factor = 0.92195, Stnd. Error, Factor = 0.44655
Table 14. VALUES OF o CORRESPONDING TO SELECTED VALUES OF k.
k
0.35
0.40
0.44
al
0.458
0.501
0.533
al Stnd. Error
0.222
0.242
0.238
113 .
-------
Frenzen (1973) in his investigations of Kblmogorov-von Karman
products found that for k = 0.393, a. = 0.50. Kaimal e£ ad. (1972)
showed that a was 0.50 ±0.05. Boston (1970) obtained a value of 0.51
for a-. The result presented in this investigation of a^ - 0.50 which
corresponds to a von Karman constant of 0.40 is in general agreement with
those cited above. Others have found values of a. ranging from 0.53
(Wyngaard and Pao, 1972) to 0.69 (Gibson et_ al., 1970).
114
-------
SECTION XXIV
ENERGY BUDGET THEORY AND DISSIPATION OF TURBULENCE ENERGY
In high Reynolds number atmospheric turbulence, the budget of
turbulent kinetic energy per unit mass is expressed (Lumley and Panof
sky, 1964):
. £.
2 at 2 3X i j 3X e
3tV3u/ i 3P'u '
2 3x. i i j 3x« 3x. p 3x.
j J J *
In the above expression, the primes designate fluctuating parts and
the unprimed terms represent the mean parts of the velocity in the x.
direction. Repeated indices are summed. The overbar designates a
time average.
Assuming horizontal homogeneity and steady-state conditions
and taking the coordinate system so that the x-axis lies parallel
to the mean vector wind, results in the much simplified energy
budget equation:
_ _ SIT & - i 3 ..._.-- -r -i...
u*wf ** wf6' 4- f^u'u1 + vfvf + w'wMw1!
U 3z 6 W 2 3z IV -rvv -rww;wj
0. (24.2,
and after further simplification:
2
, , 3U g ,Q, , 1 3w'q 1 3p'w* _
u w "^T ~ o w 9 + T ?T" + e + r a., = °
3z 6 2 9z p 9z
The terrain surrounding the tower where the turbulence statistics were
obtained for this experiment cast serious doubt upon the assumption of
horizontal homogeneity used to simplify the energy budget equation.
115
-------
nw
The error due to this assumption will be combined with the *-r
p dz
term and be designated, 1, (wyngaard and Cote, 1971) for whatever
imbalance may exist as a result of these assumptions and unmeasured
terms.
3
Multiplying the equation by kz/u^ results in the nondimensional
equation:
. 2.
kz 3U g TST kz kz 3 (w q /2) kze , kzl _ n
~ ^r 37 ~ e w e ~T + ~T 3Z + ~T + T ~ °»
* ° u* u* u* u*
(24.4)
where
-u'w1 * u^ = T/p. (24.5)
This approximation is generally made based upon the assumption that
the term, -v'w1, is small.
The first term is called nondimensional wind shear and is
designated
kz 3U
The second term represents the nondimensional form of the buoyant
production of turbulent energy. It is designated
2
z/L = ~ .
(24'6>
(24.7)
kg/e w'e1
The third term represents the divergence of the vertical flux of
kinetic energy or turbulent transport. It is designated
kz 3(w'q2/2)
*D = ~T - £ - (24.8)
116
-------
The fourth term is the nondimenslonalized form of dissipation desig-
nated :
+e - ^ (24.9)
u*
The last term is the nondimensionalized imbalance term which includes
the unmeasured pressure transport term and errors.
Monin and Obukhov (1954), using methods of the theory of similitude,
related the averaged characteristics of the surface layer of the atmo-
sphere to momentum, heat, and humidity fluxes (u^, w'01., w'q ' respec-
tively) . The Monin-Obukhov similarity theory predicts that the dimen-
sionless quantities (<|) and ) are functions of z/L in the surface
layer up to a height of near 50 m. Other similarity hypotheses that
have been subsequently proposed are that the quantities ( , a /UA,
a /UA, and o /u^) are functions of z/L only in the surface layer. Many
experimental results have shown that similarity theory may be followed
as high as 150 m, e.g. Panofsky (1972).
Dissipation of turbulence energy, e, was calculated from 13 sonic
spectra corresponding to z/L values ranging from -0.3 to +0.1 as
follows. The inertial subrange had been determined to exist at non-
dimensional frequencies greater than 1. This determination was based
upon a -2/3 slope for the logarithmic spectra, as well as the trend for
the spectral ratios to approach 4/3 with increasing frequency in accord-
ance with Kolmogorov's hypothesis. Points within the inertial subrange
were selected using criteria of spectral ratios near 4/3 while requiring
that the w and v spectra be approximately equal. The logarithmic u.
spectra can be written in the following form:
nSu(n) = [a.L/(2u)2/3] £2/3 (nU)'273 . (24.10)
117
-------
If Kolmogorov's constant for the u spectra is taken as 0.5, (Kaimal et al.,
1972) (also see Kolmogorov Constant section) then the logarithmic u
spectra can be written as
nSu(n) = 0.147(ez)2/3 f~2/3 . (24.11)
As a result of Kolmogorov's hypothesis the universal constants for the
v and w spectra are 4/3 times the constant for the u spectra, so that
nS (n) = nS (n) = 0.196 (ez)2/3 f~2/3 . (24.12)
v w
These equations were solved for e, the dissipation, after substituting
the appropriate values for the logarithmic spectra, the nondimensional
frequency, and the height, which was 18.3 meters. Since e should equal
the same value regardless of which component of the spectra it was de-
rived from, the three values of dissipation which were obtained were
then averaged and the resulting value was taken to represent dissipation
for the particular data time block under consideration.
Blackadar et_ aL. (1974) found that at 30 meters and below, e is
estimated by
e = u^3 (1 - 16 z/L)~1/4 (z/L < 0) , (24.13)
where a value of 0.4 is being used for the von Karman constant. This
expression was used to calculate e values which were then compared with
the corresponding values obtained from the sonic spectra. For z/L > 0
the expression,
e . lil u 3 + 4j7 z/ (24.14)
z x
was used to calculate the dissipation values for comparing with the
sonic derived values of e. Table 15 is a listing of the z/L values
along with the corresponding calculated and observed dissipations.
118
-------
Table 15. z/L, CALCULATED e, AND e FROM SONIC SPECTRA. THESE VALUES
CORRESPONDING TO THE INDICATED Z/L VALUES WERE PLOTTED IN
FIGURE 51.
z/L
-0.110
-0.048
-0.072
-0.074
0.001
0.060
-0.022
-0.287
-0.224
-0.155
-0.124
0.118
-0.348
Calculated e
0.031
0.009
0.022
0.033
0.028
0.010
0.132
0.013
0.015
0.019
0.031
0.008
0.012
Sonic Observed e
0.011
0.020
0.019
0.027
0.018
0.017
0.100
0.014
0.017
0.018
0.024
0.009
0.016
The "Calculated e" were obtained from the relation
-> c 3
* 1 IL
(1 - 16 z/L) ' , z/L < 0
z
2.5 u 3
- (1 + 4.7 z/L) , z/L > 0.
119
-------
A visual comparison of dissipation obtained from the two sources is
shown in Figure 51. In this figure the ordinate represents observed
dissipation obtained from the sonic spectra and the abscissa indicates
dissipation calculated using the above two equations. The dashed line
is thG line along which dissipation obtained from both methods is equal.
The scatter of points is centered about the dashed line indicating that
the equations used to estimate e represent a reasonable approximation
for the observed values.
By making appropriate assumptions and simplifications, many authors
have attempted to reduce the energy budget to fewer and more easily ob-
tained terns. In particular, two kinds of simplification have been
suggested. Lumley and Panofsky (1964) suggest that under unstable
conditions as an approximate result, dissipation equals production of
mechanical energy, so that
<|>m = 4>e, z/L < 0 . (24.15)
It has also been suggested that the sum of buoyant and mechanical energy
production is equal to dissipation,
- z/L = $ , z/L < 0 . (24.16)
Busch and Panofsky (1968) found in checking the validity of these
equations for the range, -0.4 < z/L < 0.5, that did not vary with
z/T. on the unstable side. On the stable side they discovered that the
line represented by
<|) = - z/L = 1+9 z/L , (24.17)
em '
fits a collection of data from several different sites and heights up
to 91 in. Blackadar eit _al. (1974) implied that for unstable conditions,
dissipation was balanced by buoyant and mechanical energy production
120
-------
1U
OI
LU
Q_
CO
O
F 1
O
CO
O
QZ 1 n-2
u_ lu
10-;
1 ' ' ' 1 ' . i | . . y
- /° -
/
/
0 ^
/ D
r &
/ | 1 1 1 1 1 1 1 1 1 1 1 1 _L.
o-3 icr2 icr1 i
CflLCULRTED e
Figure 51. Dissipation, e, obtained from sonic anemometer spectra versus e
calculated using empirical relationships. Reference height is
18.3 meters. The dashed line is the line along which the
dissipation from either source is equal.
121
-------
at 30 meters and above and that dissipation was equal to turbulent
energy production below 30 meters. Fichtl and McVehil (1970) determined
that the data collected at Cape Kennedy tended to favor a balance be-
tween dissipation and production of energy at 18 meters.
Because-the sonic spectra apparently support a k F 0.4 (see Sec-
tion XXIII on Kolmogorov Constant) if a Kolmogorov constant of about 0.5 is
assumed, the z/L values used in this particular analysis were calculated
using this value for k. Nondimensional dissipation, $ , was then plotted
against, z/L. The resulting plot is shown in Figure 52. The solid line
represents the results of Busch and Panofsky (1968), where
$ = 1, z/L < 0 and (24.18)
=1 + 9 z/L, z/L > 0. (24.19)
The dashed line represents the relationship, A = where
e m
m
(1 - 16 z/L)~1/4 , z/L < 0 and (24.20)
= (1 + 4.7 z/L) , z/L > 0 (Businger et al., 1971). (24.21)
m ~ ~~~
The points are generally low on the unstable side and seem to be nearly
constant with z/L. The two points on the stable side are nearly centered
about the solid line in apparent agreement with the results of Busch
and Panofsky (1968).
Another plot of <|> versus z/L was produced using a UA from 18
meter level cup and bivane measurements (Figure 53). The solid line
is the same as on the previous plot. The dashed line represents the
relation,
*e = *m ~ Z/L' (24.22)
where is represented by the previously stated empirical equations.
m
On the unstable side, the data tend to follow either line equally well.
122
-------
CD
CD
LO
CD
CM
n
0 -0- 0- £? D D
0.3 -0.2 -0.1
z/L
0.
0.1
0.2
Figure 52. Nondimensional dissipation, , versus z/L at 18.3 meters (sonic
data). The dashed line represents the relation
,-1/4
where d>
y
(1 - 16 z/L)~i'*t for z/L < 0 and = (1 + 4.7 z/L) for z/L > 0.
The solid line represents the results of Bush and Panofsky (1968):
e
<|> = 1 for z/L < 0 and
- z/L = 1+9 z/L for z/L > 0.
m ~
123
-------
CD
LO
CD
© co
CD
CM
CD
D
I
-O.f -0.3 -0.2
-0.1
z/L
0
0.1
0.2
Figure 53. Nondimensional dissipation, e, versus z/L at 18.3 meters
using cup and bivane measured friction velocities. The
dashed line represents the relation, = m - z/L where
4> = (1 - 16 z/L)"1'4 for z/L < 0 and <|> = 1 + 4.7 z/L
m m
for z/L i 0. The solid line represents the results of
Bush and Panofsky (1968): =1 for z/L < 0 and
m
- z/L =1+9 z/L for z/L > 0.
124
-------
The stable side suffers from a shortage of points. This plot seems to
support the contention that dissipation is balanced by the sum of
buoyant and mechanical production at 18.3 m in agreement with the
results of Busch and Panofsky (1968).
125
-------
SECTION XV
DIMENSIONLESS TEMPERATURE GRADIENT,- <(>,_
h
Dimensional reasoning dictates that if the momentum and heat flux
are used to nondimensionalize the potential temperature gradient, the
dimensionless temperature gradient, , , must be in the following form:
-kZU^ g Q
h w^?1" 9z
Similarity theory predicts that . is a function of z/L in the surface
layer of the atmosphere. To test the validity of this hypothesis and
to determine whether it could be extended to levels above the surface
layer, <(>, was calculated using two different techniques and then plotted
against z/L. The two techniques were used in order to determine whether
one produced superior results.
Method 1. The potential temperatures for the 6 levels of the
tower (3, 36.6, 91.4, 137.2, 182.8 and 335.3 m) were fitted on In z
with a least squares polynomial regression of degree 2 for each of the
119 edited time periods. The resulting regression equations were dif-
ferentiated with respect to In z obtaining equations for 36/91n z
representing each time block. The dimensionless temperature gradients
were then calculated for 91.4, 137.2, 182.9, 243.8, and 304.8 m using
the relation below:
~ku* an
99 - , (25.2)
where calculations of UA and w'61 from the 18.3 meter level were used
as representative of the surface or constant flux layer.
Method 2. A finite difference approximation for 90/9z was used
in terms of the same potential temperatures as in Method 1. The
126
-------
dimensionless temperature gradient was calculated using the
expression:
-kz A6
*h = -& (25.3)
w 9
Z2 " Zl
where z = - - is the height to which <{>. is referenced.
i Z2
In
Zl
Values of , were calculated for 5 levels intermediate (as determined
by z) to the 6 levels from which the potential temperatures were measured.
Plots of , versus z/L were produced corresponding to each level
for which <}>, was calculated using Method 1 (Figures 54 through 59) .
For positive z/L values the plots display considerable scatter which
increases with height. For z/L values less than zero the 4>,'s are
suppressed and tend to fall slightly below the line. The solid lines
on all of the plots are from the empirical expressions (Businger _et
al., 1971):
* = - i Z/L < o and (25.4)
h (1-9 z/L)1'2
<(>. = 0.74 + 4.7 z/L z/L _> 0- (25.5)
Plots of , calculated using Method 2 appear in Figures 60 through 64.
These are similar in most respects to those using Method 1. Of parti-
cular interest is the fact that only one point appears at the highest
level plotted (289.1 m) for z/L < 0. In closely examining all plots
using either method, it can be seen that the number of points plotted
decreases with height. The reason for this in most cases is because
the stability as indicated by the vertical heat flux (-w'61) does not
agree in sign with the stability as indicated by the potential
127
-------
9.0.
-4.0
-2.0
0
z/L
2.0
Figure 54. Nondimensional temperature gradient, h (Method 1) versus z/L
at 18.3 meters. The solid line represents the results of
Businger et^ a^. (1971): h =0.74 (1-9 z/L)
and =0.74 + 4.7 z/L for z/L > 0.
128
-1/2
for z/L < 0
-------
Figure 55. Nondimensional temperature gradient, (ji^ (Method 1) versus z/L
at 91.4 meters. The solid line represents the results of
-1/2
Businger et al. (1971): . =0.74 (1-9 z/L) ' for z/L < 0
il ~~
and <|>h = 0.74 + 4.7 z/L for z/L _> 0.
129
-------
Figure 56. Nondimensional temperature gradient, h, (Method 1) versus z/L
at 137.2 meters. The solid line represents the results of
Businger et al. (1971): $ =0.74 (1-9 z/L)~1/2 for z/L < 0
and h = 0.74 + 4.7 z/L for z/L > 0.
130
-------
9.O..
6.O..
3.O..
-2.0
0
z/L
2.0
Figure 57. Nondimensional temperature gradient, (Ji^, (Method 1) versus z/L
at 182.9 meters. The solid line represents the results of
-l /?
Businger et al. (1971): <{., =0.74 (1-9 z/L) ' for z/L < 0
~i n ~~
and h = 0.74 + 4.7 z/L for z/L > 0.
131
-------
-.0."
9.O..
-f.O
-2.0
0
z/L
2.0
f.O
Figure 58. Nondimensional temperature gradient, ^* (Method 1) versus z/L
at 243.8 meters. The solid line represents the results of
-1/2
Businger et al. (1971): , = 0.74 (1-9 z/L) for z/L < 0
1 - n *~
and , = 0.74 + 4.7 z/L for z/L > 0.
n ~
132
-------
Figure 59. Nondlmensional temperature gradient, h» (Method 1) versus z/L
at 304.8 meters. The solid line represents the results of
-1 /?
Businger et al. (1971): , = 0.74 (1 - 9 z/L) ' for z/L < 0
n ~
and (j), =0.74+4.7 z/L for z/L > 0.
n -
133
-------
Figure 60. Nondimensional temperature gradient, h» (Method 2) versus z/L
at 13.5 meters. The solid line represents the results of
Businger et al. (1971): , =0.74 (1-9 z/L)~1/2 for z/L < 0
~~~ n
and . = 0.74 + 4.7 z/L for z/L >0).
n
134
-------
f.O
Figure 61. Nondimensional temperature gradient, ^, (Method 2) versus z/L
at 59.9 meters. The solid line represents the results of
Businger et al. (1971): <|>, = 0.74 (1 - 9 z/L)-1/2 for z/L < 0
1
and
. = 0.74 + 4.7 z/L for z/L > 0.
135
-------
Figure 62. Nondimensional temperature gradient, 4^, (Method 2) versus z/L
at 112.8 meters. The solid line represents the results of
Businger et al . (1971): ^ = 0.74 (1-9 z/L)
and (f = 0.74 + 4.7 z/L for z/L > 0.
-1 /?
'
for z/L < 0
136
-------
Figure 63. Nondimensional temperature gradient, (ji^, (Method 2) versus z/L
at 185.4 meters. The solid line represents the results of
-11">
Businger et al. (1971): <|>h = 0.74 (1-9 z/L) ' for z/L < 0
and <(). = 0.74 + 4.7 z/L for z/L > 0.
137
-------
-4.0
Figure 64. Nondimensional temperature gradient, ^, (Method 2) versus z/L
at 289.1 meters. The solid line represents the results of
-1 /?
Businger et al. (1971): , = 0.74 (1 - 9 z/L) ' for z/L < 0
"" ' ll
and , = 0.74 + 4.7 z/L for z/L > 0.
138
-------
temperature gradient (36/9z). The heat flux indicates a dynamic or
turbulent stability while the potential temperature indicates a static
stability. It can be seen from the definition of the dimensionless
temperature gradient that, ideally, <|>, approaches an indeterminate value
as z/L approaches 0 (i.e. , _ - -» 77 ). This explains the large
Z/lf "*" U | ~ | U
amount of scatter near zero.
Neither method of calculating <|>, produced clearly superior results.
In general, the plots from both methods contain considerable scatter and
roughly 1/3 of the total number of points were not plotted because of
negative , values (in most cases) and also because *. fell outside of
n h
the bounds of the plot in the case of those non-negative values not
plotted.
139
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SECTION XXVI
REFERENCES
1. Air Ministry, 1961: Handbook of Meteorological Instruments, Part II,
Instruments for Upper Air Observations. Her Majesty's Stationery
Office, London.
2. Bendat, J. S., and A. G. Piersol, 1971: Random Data; Analysis and
Measurement Procedures. Wiley-Intersciences, New York.
3. Blackadar, A. K., and H. Tennekes, 1968: Asymptotic similarity in
neutral barotropic planetary boundary layers. Journal of the
Atmospheric Sciences, vol. 25, pp. 1015-1020.
4. Blackadar, A. K., H. A. Panofsky, and F. Fiedler, 1974: Investigation
of the turbulent wind field below 500 feet altitude at the Eastern
Test Range, Florida. NASA Contractor Report, NASA CR-2438.
5. Boston, N., 1970: An investigation of high wave number temperature
and velocity spectra in air. Ph.D. Thesis, University of British
Columbia, Vancouver, British Columbia.
6. Busch, N. E., 1973: The surface boundary layer. Boundary-Layer
Meteorology, vol. 4, pp. 213-240.
7. Busch, N. E., and H. A. Panofsky, 1968: Recent spectra of atmospheric
turbulence. Quarterly Journal of the Royal Meteorological Society,
vol. 94, pp. 132-148.
8. Businger, J. A., 1972: Turbulent transfer in the atmospheric surface
layer. Workshop on Micrometeorology, American Meteorological Society,
Boston, Massachusetts, pp. 1-69.
9. Businger, J. A., J. C. Wyngaard, Y. Izumi, and E. F. Bradley, 1971:
Flux-profile relationships in the atmospheric surface layer.
Journal of the Atmospheric Sciences, vol. 28, pp. 181-189.
10. Clark, R. H., 1970: Observational studies in the atmospheric boundary
layer. Quarterly Journal of the Royal Meteorological Society, vol.
96, pp. 91-119.
11. Cooper, R. E., and B. C. Rusche, 1968: The SRL Meteorological Program
and Off-Site Dose Calculations. E. I. DuPont de Nemours & Company,
Aiken, South Carolina.
12. Cramer, H. E., 1964: Meteorological Prediction on Techniques and
Data Systems. Report GCA-64-3-6, Geophysical Corporation of America,
Bedford, Massachusetts.
13. Crawford, T. V., 1974: Progress Report Dose-To-Man FY 1973. E. I.
DuPont de Nemours & Company, Aiken, South Carolina.
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14. Deardorff, J. W., 1970: Preliminary results from numerical integra-
tions of the unstable planetary boundary layer. Journal of the
Atmospheric Sciences, vol. 27, pp. 1209-1211.
15. Fichtl, G. H., and G. E. McVehil, 1970: Longitudinal and lateral
spectra of turbulence in the atmospheric boundary layer at the
Kennedy Space Center. Journal of Applied Meteorology, vol. 9,
pp. 51-63.
16. Frenzen, P., 1973: The observed relation between the Kolmogorov
and von Karman constants in the surface boundary layer. Boundary-
Layer Meteorology, vol. 3, pp. 348-358.
17. Gibson, C. H., G. R. Stegen, and R. B. Williams, 1970: Statistics
of the fine structure of turbulent velocity and temperature fields
measured at high Reynolds number. Journal of Fluid Mechanics,
vol. 41, pp. 153-167.
18. Gill, G. C. jet_ ^1_., 1967: Accuracy of wind measurements on towers
and stacks. Bulletin, American Meteorological Society, vol. 48,
pp. 665-674.
19. Gurvic, A. S., 1960: An experimental investigation of the frequency
spectra of the vertical component of the wind velocity in the
bottom layer of the atmosphere. Academy Science, Union of Soviet
Socialist Republic, vol. 130, pp. 806-809.
20. Horst, T. W., 1973: Corrections for Response Errors in a Three
Component Propeller Anemometer. BNWL-SA-4262, Battelle Memorial
Institute, Pacific Northwest Laboratories, Richland, Washington.
21. Irwin, J. S., 1974: Analysis of turbulence parameters in the lowest
300 meters of the atmosphere. Master's Thesis (unpublished),
Department of Geosciences, North Carolina State University, Raleigh,
North Carolina.
22. Kaimal, J. C., and D. A. Haugen, 1967: Characteristics of vertical
velocity fluctuations observed on a 430 m. tower. Quarterly Journal
of the Royal Meteorological Society, vol. 93, pp. 305-317.
23. Kaimal, J. C., and C. N. Touart, 1967: Critical examination of a
transformation to vector-mean coordinates. Journal of Applied
Meteorology, vol. 6, pp. 583-587.
24. Kaimal, J. C., J. C. Wyngaard, Y. Izumi, and 0. R. Cote", 1972:
Spectral characteristics of surface-layer turbulence. Quarterly
Journal of the Royal Meteorological Society, vol. 98, pp. 563-589.
25. Kitaigorodskii, S. A., 1973: The Physics of Air-Sea Interaction.
Israel Program for Scientific Translations Ltd., Jerusalem,
Israel, p. 19.
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26. Lumley, J. L., and H. A. Panofsky, 1964: The Structure of Atmospheric
Turbulence. Interscience Publishers, New York.
27. Mitsuta, Y., M. Miyake, and Y. Kobori, 1967: Three dimensional sonic
anemometer-thermometer for atmospheric turbulence measurement.
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to the Study of Atmospheric Surface Layer. Kyoto University, Kyoto,
Japan, pp. 37-56.
28. Monin, A. S., and A. M. Obukhov, 1954: Basic regularity in turbulent
mixing in the surface layer of the atmosphere. Academy Science,
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Knoxville, Tennessee.
30. Panofsky, H. A., 1972: Tower micrometeorlogy. Workshop on Micro-
meteorology, American Meteorological Society, Boston, Massachusetts,
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31. Panofsky, H. A., and G. W. Brier, 1958: Some Applications of Statistics
to Meteorology. The Pennsylvania State University, University Park,
Pennsylvania.
32. Panofsky, H. A., and R. A. McCormick, 1960: The spectrum of vertical
velocity near the surface. Quarterly Journal of the Royal Meteorological
Society, vol. 86, pp. 495-503.
33. Panofsky, H. A., and B. Prasad, 1965: Similarity theories and diffusion.
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34. Panofsky, H. A., and A. A. Townsend, 1964: Change of terrain rough-
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cal Society, vol. 90, pp. 147-155.
35. Pasquill, F., 1962: Recent broad-band spectral measurements of
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36. Pasquill, F., 1972: Some aspects of boundary layer description.
Quarterly Journal of the Royal Meteorological Society, vol. 98,
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37. Paulson, C. A., 1970: The mathematical representation of wind speed
and temperature profiles in the unstable atmospheric surface layer.
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38. Petersen, W. B., 1974: Analysis of turbulence parameters from the
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39. Singleton, R. C., 1969: An algorithm for computing the Mixed Radix
Fast Fourier Transform. IEEE Transactions on Audio Electro-
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40. Slade, D. H., 1969: Wind measurements on a tall tower in rough
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41. Smith, M., 1968: ASME Guide for Prediction of the Dispersion of
Airborn Effluents. American Society of Mechanical Engineers,
New York, New York.
42. Stearns, C. R., 1970: Conversion of profile differences to true
gradients at the geometric mean height in the surface layer.
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kinetic energy and temperature variance in the atmospheric surface
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143
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SECTION XXVII
APPENDICES
Page
A. Leveling Procedure 141
B. Thermistor Discussion 142
C. Details of Data Editing 143
D. o_, 0, Functions of fz/u. 146
E A *
144
-------
APPENDIX A
Leveling Procedure
The cup and bivane system was mounted on the end of an extremely
rigid, heavy steel boom whose length was 6 meters. The actual mounting
of the booms and installation of instruments was performed by subcon-
tractors for SRL. Prior to collecting data for this experiment, it was
found that the instruments had no guarantee of being level with respect
to gravity. To eliminate this source of error a means was devised for
leveling the instruments. First the booms were retracted and the cup
and bivane masts were leveled. (Each boom had two degrees of freedom
of movement; up and down and rotational). Then a portable reference
mast was strapped at approximately 2.5 meters from the back end of the
boom. The mast was attached to the boom by means of a saddle which
securely held its position fixed. The reference mast was leveled and
the boom was then pushed out in its extended position. By using the
portable mast as a reference the boom was leveled and bolted in position.
The sensitivity of the level was within 1/500 of a degree.
The Gill and Sonic anemometers had electrostatic levels installed
on the base of each instrument. Guy-wires were attached to hold the
flexible aluminum booms steady. Using a remote sensor and by tightening
and loosening turnbuckles mounted on the guy-wires the Gill and Sonic
instruments were leveled to 1/10 of a degree.
145
-------
APPENDIX B
Thermistor Discussion
The thermistor and bridge circuit were provided by Battelle North-
west. The thermistor, a Ceco microbead thermistor (//31A401C), was
purchased from Victory Engineering Corporation. The voltage signals
from the thermistor bridge circuit were amplified and connected to the
DAS. The DAS amplified by a factor of 10 with a maximum range of ±10
volts. Thus the output from the bridge circuit was limited to be less
than one volt by setting the initial amplifier gain. The bridge circuit
was designed by Battelle Northwest to yield a nearly linear relationship
between voltage output and temperature fluctuations.
The characteristic time (T) and dissipation constant of the ther-
mistor were 0.03 sec. (in water) and 0.045 watts/sec., respectively.
The low value of the dissipation constant means that self heat effects
were negligible and the low value of T means that the response of the
thermistor was fast enough to obtain meaningful temperature fluctuations
while sampling at 10 times per second.
146
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Details of Data Editing
APPENDIX C
Time
Code
Time
Code
Time
Code
139/10:40:22
139/11:55:04
139/12:35:04
139/13:15:04
139/13:55:16
139/14:35:40
139/15:15:40
139/15:55:52
139/16:35:52
140/12:18:12
140/12:58:12
140/13:38:12
140/14:18:12
140/14:58:12
140/15:38:12
140/16:18:12
140/17:40:11
140/18:20:11
140/19:00:11
140/19:40:11
140/20:20:11
140/21:00:11
140/21:40:11
140/22:30:25
140/23:10:25
140/23:50:25
141/00:30:25
141/01:10:25
141/01:50:25
141/02:30:25
141/03:10:25
141/03:50:25
141/04:30:25
141/05:10:25
141/05:50:25
141/06:30:25
141/18:20:24
141/19:00:24
141/19:40:24
141/20:20:24
141/21:00:24
141/21:40:24
141/22:20:24
141/23:00:24
141/23:40:24
142/00:20:24
3
U
U
1
N
1
U
U
4
U
1
U
1
U
U
4
N
S
S
2
2
2
4
1
2
2
2
2
2
1
S
1
2
1
1
6
3
3
1
1
1
1
1
1
1
3
142/01:00:24
142/01:40:24
142/02:20:24
142/03:00:24
142/03:40:24
142/04:20:24
142/12:54:49
142/13:34:49
142/14:14:49
142/14:54:49
142/15:34:49
142/16:14:47
142/16:54:47
142/17:34:47
142/18:14:47
142/18:54:47
142/19:34:47
142/20:43:11
142/21:23:11
142/22:03:11
142/22:29:24
142/23:09:24
142/23:49:24
143/00:29:24
143/01:09:24
143/01:49:24
143/02:29:24
143/03:09:24
143/03:49:24
143/04:29:24
143/05:09:24
143/05:49:24
143/06:29:24
143/07:09:23
143/07:49:23
143/08:29:23
143/09:09:23
143/09:49:23
143/10:29:23
143/11:09:23
143/11:49:23
143/12:29:23
143/13:09:23
143/13:49:23
143/14:29:23
143/15:09:23
1
1
3
3
2
4
U
U
1
U
1
U
N
1
1
1
1
1
1
4
1
2
1
2
1
1
1
1
1
1
S
1
4
1
5
U
N
U
U
N
N
U
N
N
U
U
143/15:49:23
144/07:59:23
144/08:39:23
144/09:19:23
144/09:59:23
144/10:39:47
144/12:08:57
144/12:48:57
144/13:34:47
144/14:14:59
144/14:54:59
144/15:34:59
144/16:40:11
144/17:20:11
144/18:41:25
144/19:21:25
144/20:01:25
144/21:03:01
144/21:43:01
144/22:23:01
144/23:03:01
144/23:43:01
145/00:23:01
145/01:03:01
145/01:43:01
145/08:20:42
145/10:11:28
145/11:31:05
145/12:57:10
145/14:09:41
145/14:49:41
145/15:42:24
145/16:22:48
145/17:02:48
145/17:43:12
145/18:23:12
145/19:03:12
145/19:43:12
145/20:23:12
145/21:03:12
145/21:43:36
145/22:23:36
145/23:03:36
145/23:43:36
146/00:23:36
146/01:03:36
4
5
N
U
U
N
U
N
N
N
N
4
N
N
N
N
S
S
S
S
S
S
S
1
4
5
U
U
1
U
4
1
1
U
U
3
2
1
1
4
1
2
1
3
3
1
147
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Time
Code
Time
Code
Time
Code
146/01:43:36
146/02:23:36
146/12:21:24
146/13:01:24
146/13:41:24
146/14:21:24
146/15:01:24
146/15:41:24
146/16:21:24
146/17:01:24
146/17:41:24
146/18:21:24
146/19:01:24
146/19:41:24
146/20:21:24
146/21:37:02
146/22:17:02
146/22:57:02
146/23:37:02
147/10:44:24
147/11:29:05
147/12:09:05
147/12:49:29
147/13:29:29
147/14:09:29
147/15:30:05
147/16:10:05
147/16:50:05
147/17:30:05
147/18:10:05
147/18:50:05
147/20:26:08
147/21:06:08
147/21:46:08
1
1
U
u
1
u
u
u
1
1
N
1
1
1
1
1
1
1
1
3
3
U
U
U
u
u
u
N
N
s
4
N
S
S
147/23:08:13
147/23:48:13
148/00:28:13
148/01:08:13
148/01:48:13
148/02:40:17
148/03:20:17
148/04:00:17
148/04:40:41
148/05:20:41
148/06:00:41
148/06:40:41
148/07:20:41
148/08:00:24
148/08:40:24
148/09:20:24
148/10:00:24
148/11:22:17
148/12:02:17
148/12:42:17
148/13:22:17
148/14:02:17
148/14:42:17
148/15:22:17
148/16:02:17
148/16:42:17
148/17:22:17
148/18:02:17
148/18:42:17
148/20:20:46
148/21:00:46
148/21:40:46
148/22:20:46
148/23:00:46
S
S
S
S
S
N
N
N
N
N
N
N
4
N
N
N
N
U
N
N
U
N
N
N
N
N
N
N
4
N
N
N
S
S
148/23:40:46
149/00:52:48
149/01:32:48
149/02:12:48
149/02:52:48
149/03:32:48
149/04:12:48
149/04:52:48
149/06:37:35
149/07:17:35
149/07:57:35
149/09:03:24
149/16:15:08
149/16:55:08
149/17:35:08
149/18:15:08
149/18:55:08
149/19:35:08
149/20:25:10
149/21:05:10
149/21:42:24
149/22:22:24
149/23:02:24
149/23:42:24
150/00:22:24
150/01:02:24
150/01:42:24
150/02:22:24
150/03:39:10
150/04:19:10
150/04:59:10
150/05:39:10
150/06:19:10
S
N
N
N
N
N
N
N
1
S
4
1
N
N
U
1
1
1
1
1
1
1
1
S
1
S
1
2
1
1
1
1
2
148
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Code
Time: day/hour:minutes:seconds
1 : u^ less than zero
2 : variance of the vertical velocity fluctuations less
than 0.01 m~2s~2
3 : bad thermistor values
4 .: period does not contain full 40 minutes of data
5 : morning time situation where z/L was not representative
of the stability through the tower
6 : fog
N : neutral
S : stable
U : unstable
149
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APPENDIX D
0_, a. Functions of fz/u,
£i A 2_
That a and a. are functions of In fz/uft can be justified on the
basis of theory after making some appropriate assumptions and simplifi-
cations. It has been shown that
a
a~, - (see Section XV) .
c. U
In the Ekman layer,
U ° 1 ,
= a + T- In fz/u^ + const
under conditions of a neutral, barotropic, horizontally homogeneous
planetary boundary layer (see Blackadar and Tennekes, 1968). Since
the tower heights represent the lower part of the Ekman layer, V = 0
is a valid assumption. This was assumed to be a boundary condition
when the velocity defect law was derived.
A result of similarity theory is that the variance of the vertical
velocity is a universal function of z/L. For near neutral conditions
o
= C (f> (z/L) = 1.25
u* w
(see Panofsky 1972; Busch and Panofsky, 1968). Thus
1.25 u. , nr 1.25 cj-/2
E U U . 1 , fz , k ,,1/2 ,, '
_£ + _1 + const x + ^ fj + c
k
where CT is in radians and CL = (u^/U )~ is the geostrophic drag coef-
ficient. Thus 'a_ can be expressed as a function of In fz/u... and C_ after
*ii ** D
making a few simplifying assumptions and approximations. Knowing a ,
C_, and fz/u^ simultaneously will allow the unknown constant in the
above equation to be calculated.
An analogous expression can be derived for a..
150
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APPENDIX E
The BWIB Lines
The expressions used for <|> in this report are often referred to
as the BWIB lines. They are:
-11U
m = (1 - y z/L) , z/L < 0 (E.I)
and
<|> = (1 + 4.7 z/L) , z/L > 0. (E.2)
m
The value of the coefficient, y> °f z/L in Eqn. (E.I) is sometimes used
as 16 and sometimes 15. The value, 15, is the most recent one by
Businger (1971). However, for the purposes of this report, there is no
significant difference between the plots using either value. Figure 65
is a plot of the Eqn. (E.I) and (E.2) using values of 15 and 16. The
difference is too small to be seen on the computer produced plot.
151
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9.0
6.0
3.O..
-f .0
-2.0
0
z/L
2.0
Figure 65. Nondimensional wind shear, $ versus z/L computed from the
m -1/4
equations = (1 + 4.7 z/L), z/L > 0; <|> = (1 - 15 z/L) ,
m -1/4
z/L < 0; and <(> = (1 - 16 z/L) , z/L < 0. The two curves
on the unstable side are almost indistinguishable.
152
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TECHNICAL REPORT DATA
(Please read Instructions on the reverse before completing)
1. REPORT NO.
EPA-600/4-75-004
2.
3. RECIPIENT'S ACCESSION- NO.
4. TITLE AND SUBTITLE 5. REPORT DATE
ATMOSPHERIC TURBULENCE PROPERTIES IN THE LOWEST 300 July 1975
METERS
7. AUTHOR(S)
A.M. Weber, J.P. Kahler, J.S. Irwin, and W.
9. PERFORMING ORGANIZATION NAME AND ADDRESS
North Carolina State University
Raleigh, N. C. 27607
12. SPONSORING AGENCY NAME AND ADDRESS
Meteorology & Assessment Division, ESRL (M
Environmental Protection Agency
Research Triangle Park, N. C. 27711
6. PERFORMING ORGANIZATION CODE
65050
8. PERFORMING ORGANIZATION REPORT NO.
B. Petersen
10. PROGRAM ELEMENT NO.
1AA009
11. CONTRACT/GRANT NO.
800662
13. TYPE OF REPORT AND PERIOD COVERED
D-80) F1nal
" ""' 14. SPONSORING AGENCY CODE
15. SUPPLEMENTARY NOTES
16. ABSTRACT
Analyses of atmospheric turbulence data for a nonhomogeneous terrain
the SRL-WJBJ Meteorological Facility at Beach Island, South Carolina
Variations of the turbulence parameters are studied within and above
layer.
17.
collected at
, are presented.
the surface
KEY WORDS AND DOCUMENT ANALYSIS
a. DESCRIPTORS
Turbulence 2004
Atmospheric Diffusion
13. DISTRIBUTION STATEMENT
Release Unlimited
b. IDENTIFIERS/OPEN ENDED TERMS
Atmospheric Turbulence
Properties-Lowest
300 Meters
j
19. SECURITY CLASS (This Report)
None
2O. SECURITY CLASS (This page)
None
c. COSATI Field/Group
Atmospheric
Diffusion
Atmospheric
Physics
21. NO. OF PAGES
161
22. PRICE
EPA Form 2220-1 (9-73)
153
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