UC-53 Meteorology EXPERIMENTAL …BNWL-329 UC-53 Meteorology EXPERIMENTAL INVESTIGATION OF THE...
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PACIFIC m W E S T UBORATORY rtlCHW, WASH1HO'ION
speraw bu I . WTRU MEMORtAL 1 N S M
for the UNlfED,#m N&C E?&RGY UNbER CONTRACT AX&I~IW
BNWL-329
UC-53 Meteorology
EXPERIMENTAL INVESTIGATION OF THE TURBULENCE STRUCTURE
IN THE LOWER ATMOSPHERE
C. E. Elderkin
Atmospheric Sciences Section Environmental and Radiological Sciences Department
December 1966
PACIFIC NORTHWEST LABORATORY RICHLAND, WASHINGTON
P r i n t e d i n USA. P r i c e $5 .00 . A v a i l a b l e f rom t h e C l e a r i n g h o u s e f o r F e d e r a l S c i e n t i f i c and T e c h n i c a l I n f o r m a t i o n
N a t i o n a l Bureau o f S t a n d a r d s U.S. Department of Commerce S p r i n g f i e l d , V i r g i n i a 22151
Abstract
EXPERIMENTAL INVESTIGATION OF THE TURBULENCE STRUCTURE
IN THE LOWER ATMOSPHERE
by CHARLES EDWIN ELDERKIN
Measurements of turbulence in all three components of
the wind were made at several heights, in neutral, unstable,
and stable conditions, and in a variety of wind speeds. The
data were analyzed on an analog computer for power spectral
density distributions, cospectral density distributions and
Reynolds' stresses. Similarity of the power spectral dis-
tributions was observed for each of the components with some
notable deviations. The spectra for the vertical component
measured at 12 meters and above, where bouyancy becomes
effective, demonstrated definite convective energy peaks.
These occurred in addition to the mechanical energy peaks
which corresponded in the normalized spectral distributions
to those for the tests close to the ground where mechanical
energy dominated the structure.
A shift to higher wave numbers was detected for atable
cases in the spectra for all three components. The shift was
mast noticeable at 12 meters where greater stabilities wore more
p o s s i b l e t h a n c l o s e t o t h e q round .
The s p e c t r a l f u n c t i o n f o r t h e i n e r t i a l s u b r a n q e ,
d e s c r i b e d by Ko lmoqoro f f ' s r e l a t i o n , S ( K ) a c 2 l 3 K - 5 / 3 , was
d e t e c t e d c o n s i s t e n t l y i n t h e l o n q i t u d i n a l and v e r t i c a l
component s p e c t r a . Reasonable v a l u e s r e s u l t e d f o r t h e . u n i v e r s a l p o r p o r t i o n a l i t y c o n s t a n t , ' a , e s t i m a t e d b e s t by
a = . 5 1 , and dependence on E 2 / 3 was d e m o n s t r a t e d . The
a d d i t i o n a l f a c t o r o f 4/3 r e q u i r e d between t h e l o n g i t u d i n a l
and v e r t i c a l one-d imens iona l s p e c t r a was m i s s i n q , however.
Only two c a s e s f o r t h e l a t e r a l . wind component showed a minus
f i v e - t h i r d s dependence. The a d d i t i o n a l r e q u i r e m e n t f o r t h e
i n e r t i a l s u b r a n q e , t h a t t h e c o s p e c t r a become z e r o , was
o b s e r v e d t o h o l d o n l y down t o nz /u = 1.0 t o 3.0 w h i l e t h e
minus f i v e - t h i r d s law f o r t h e power s p e c t r a e x t e n d e d some-
what below t h i s v a l u e .
TARLE OF CONTENTS
V I I .
Page
INTRODUCTION . . . . . . . . . . . . . . . . . . . 1
EQUIPMENT . . . . . . . . . . . . . . . . . . . . 5
Turbulence Sensinq Equipment . . . . . . . . . . 5
Recording Equipment . . . . . . . . . . . . . . 11 Analys i s Equipment . . . . . . . . . . . . . . . 1 4
THEORETICAL BACKGROUND FOR ANALYSIS . . . . . . . 33 DATAANALYSISPROGRAMS . . . . . . . . . . . . . . 37
Coordinate Transformat ion Program . . . . . . . 37 ~ e y n o l d s ' S t r e s s Program . . . . . . . . . . . . 4 1
Power S p e c t r a l Program . . . . . . . . . . . . . 4 4
Cospec t r a l Program . . . . . . . . . . . . . . . 50
. . . . . . . A l t e r n a t e S p e c t r a l Densi ty Program 52
S t a t i s t i c a l R e l i a b i l i t y o f S p e c t r a l E s t i m a t e s . 53
FIELD TESTS . . . . . . . . . . . . . . . . . . . 72 THEORETICAL BACKGROUND FOR TURBULENCE RESULTS . . 76
RESULTS OF ANALYSIS . . . . . . . . . . . . . . . 90
. . . . . . . . . Longi tud ina l Component S p e c t r a 90
. . . . . . . . . . . Vertical Component S p e c t r a 92
L a t e r a l Component S p e c t r a . . . . . . . . . . . 95
The Universa l Cons tan t s o f t h e I n e r t i a l
Subrange . . . . . . . . . . . . . . . . . . . 101
Comparison w i t h Prev ious R e s u l t s . . . . . . . . 104
. . . . . . . . . . . . Reynolds 'S t ress R e s u l t s 1 1 4
Paqe
V I I I . SUMMARY AND CONCLUSIONS . . . . . . . . . . . . . 1 6 3
R E F E R E N C E S . . . . . . . . . . . . . . . . . . . . . . . 2.69
. . . . . . . . . . . . . . . . . . . . . . . A P P E N D I X A 1 7 4
Table Paqe
. . . . . . . . 4.1 Power Spectral Proqram Parameters 55
. . . . . . . . . . 4.2 Cospectral Proqram Parameters 56
. . . . . . . . 4.3 Variability of Spectral Estimates 57
5.1 Test Conditions . . . . . . . . . . . . . . . . . 75 7.1 Evaluation of Universal Constant "a" . . . . . . 120
. . . . . . . . . . . . . . . 7.2 Reynolds' Stresses 121
v i i i
LIST OF FIGURES
F i q u r e
2.1
2.2
2 .3
2.4
2.5
2.6
Paqe
Wind Component Meter Turbu lence S e n s i n g P robe . . 20
O r i g i n a l Wind Speed C a l i b r a t i o n Curve . . . . . . 2 1
Wind Speed C a l i b r a t i o n a f t e r I n s t r u m e n t R e v i s i o n 22
C a l i b r a t i o n Wind Tunnel . . . . . . . . . . . . . 23
Ver t ica l Angle C a l i b r a t i o n Curve . . . . . . . . 24
Dependace of V e r t i c a l Anqle C a l i b r a t i o n o n Mind
Speed . . . . . . . . . . . . . . . . . . . . . 25
V e r t i c a l Anqle Response T e s t Arranqement . . . . 26
V e r t i c a l Angle Response Records . . . . . . . . . 27
Speed Response T e s t Arranqement . . . . . . . . . 28
Speed Response T e s t Records . . . . . . . . . . . 29
H o r i z o n t a l Angle Response T e s t Arranqement . . . 30
H o r i z o n t a l Anqle Response T e s t Records . . . . . 31
Basic Analoq Computer Components and Symbols . . 3 2
C o o r d i n a t e T r a n s f o r m a t i o n Proqram . . . . . . . . 58
F u n c t i o n G e n e r a t o r S e t t i n q s f o r Speed C a l i b r a t i o n 6 0
F u n c t i o n G e n e r a t o r S e t t i n g s f o r V e r t i c a l Anqle
C a l i b r a t i o n a t 5 . 0 m p s . . . . . . . . . . . . 61
F u n c t i o n G e n e r a t o r S e t t i n q s f o r V e r t i c a l Angle
C o r r e c t i o n . . . . . . . . . . . . . . . . . . 6 2
C o o r d i n a t e T r a n s f o r m a t i o n Moni tored R e s u l t s . . . h:
Reynolds S t r e s s Proqram . . . . . . . . . . . . . 64
F i q u r e Paqe
. . . . . . . . Reynolds S t r e s s Moni tored R e s u l t s 65
. . . . . . . . . . Power S p e c t r a l F i l t e r Proqram 66
. . . . . . . . Power S p e c t r a l Moni tored R e s u l t s 67
. . . . . . . . . . . . C o s p e c t r a l F i l t e r Proqram 68
. . . . . . . . . . C o s p e c t r a l Moni tored R e s u l t s 69
. . . . . . . . . . . . . . Hete rodyn ing Proqram 70
. . . . . . . . . Heterodyninq Monitored R e s u l t s 7 1
L o n q i t u d i n a l Wind Component S p e c t r a . N e u t r a l
. . . . . . . . . . . . . . . . . . . . . T e s t s 1 2 2
L o n g i t u d i n a l Wind Component S p e c t r a . U n s t a b l e
. . . . . . . . . . . . . . . . . . . . . T e s t s 123
L o n q i t u d i n a l blind Component S p e c t r a . S t a b l e
. . . . . . . . . . . . . . . . . . . . . T e s t s 124
L o n q i t u d i n a l Wind Component S p e c t r a . T e s t s Above
. . . . . . . . . . . . . . . . . . . 12 Meters 125
L o n q i t u d i n a l Wind Component S p e c t r a . N e u t r a l
. . . . . . . . . . . . . . . . . . . . . Tests 126
L o n q i t u d i n a l Wind Component S p e c t r a . U n s t a b l e
. . . . . . . . . . . . . . . . . . . . . T e s t s 127
L o n q i t u d i n a l Wind Component S p e c t r a . S t a b l e
. . . . . . . . . . . . . . . . . . . . . T e s t s 128
L o n q i t u d i n a l Wind Component S p e c t r a . T e s t s Above
. . . . . . . . . . . . . . . . . . . 1 2 M e t e r s 129
Vert ical Wind Component S p e c t r a . N e u t r a l Tests . 130
V e r t i c a l Wind Component S p e c t r a . U n s t a b l e T e s t s 1 3 1
F i g u r e Paqe
V e r t i c a l Wind Component S p e c t r a - S t a b l e T e s t s . 132
V e r t i c a l Wind Component S p e c t r a - T e s t s Above
12 Meters . . . . . . . . . . . . . . . . . . . 133
V e r t i c a l Wind Component S p e c t r a l - N e u t r a l T e s t s 134
V e r t i c a l Wind Component S p e c t r a l - Unstab le T e s t s 135
V e r t i c a l Wind Component S p e c t r a l - S t a b l e T e s t s . 136
V e r t i c a l Wind Component S p e c t r a l - T e s t s Above . v ,
12 Meters . . . . . . . . . . . . . . . . . . . 137
L a t e r a l Wind Component S p e c t r a - N e u t r a l T e s t s . 138
L a t e r a l Wind Component S p e c t r a - Unstab le Tests . 139
L a t e r a l Wind Component S p e c t r a - S t a b l e T e s t s . . 140
L a t e r a l Wind Component S p e c t r a - T e s t s Above 12
Meters . . . . . . . . . . . . . . . . . . . L a t e r a l Wind Component S p e c t r a - N e u t r a l T e s t s
L a t e r a l Wind Component S p e c t r a - Unstab le T e s t s
L a t e r a l Wind Component S p e c t r a - S t a b l e T e s t s . L a t e r a l Wind Component S p e c t r a - T e s t s Above 12
Meters . . . . . . . . . . . . . . . . . . . C o s p e c t r a Retween L o n q i t u d i n a l and V e r t i c a l
V e l o c i t y - N e u t r a l T e s t s . . . . . . . . . . Cospec t ra Between L o n g i t u d i n a l and V e r t i c a l
V e l o c i t y - Unstab le T e s t s . . . . . . . . . . Cospec t ra Between L o n g i t u d i n a l and V e r t i c a l
V e l o c i t y - S t a b l e Tests . . . . . . . . . . . Cospec t ra Between L o n q i t u d i n a l and V e r t i c a l
V e l o c i t y - T e s t s Above 12 Meters . . . . . .
F i g u r e
7.29
Paqe
Cospect ra Between L a t e r a l and V e r t i c a l Ve loc i t y
- N e u t r a l T e s t s . . . . . . . . . . . . . . . . 150
Cospect ra Between L a t e r a l and V e r t i c a l V e l o c i t y
- Unstable Tests . . . . . . . . . . . . . . . 151
Cospec t ra Between L a t e r a l and V e r t i c a l Ve loc i t y
- S t a b l e T e s t s . . . . . . . . . . . . . . . . 152
Cospec t ra Between L a t e r a l and V e r t i c a l Ve loc i t y
- Tests Above 12 Meters . . . . . . . . . . . . 153
Cospec t ra Between Longi tud ina l and L a t e r a l
Ve loc i t y - N e u t r a l Tests . . . . . . . . . . . 154
Cospec t ra Between Lonq i t ud ina l and L a t e r a l
Ve loc i t y - Unstable Tests . . . . . . . . . . . 155
Cospect ra Between Long i t ud ina l and L a t e r a l
V e l o c i t y - S t a b l e T e s t s . . . . . . . . . . . . 156
Cospec t ra Between Long i t du ina l and Lateral
Ve loc i t y - T e s t Above 12 Meters . . . . . . . . 157
Cospec t ra f o r T e s t 13-1 . . . . . . . . . . . . . 158
Average Long i t ud ina l Wind Component Spectrum . . 159
Average V e r t i c a l Wind Component Spectrum . . . . 160
Average L a t e r a l Wind Component Spectrum . . . . . 161
Example o f V a r i a b i l i t y i n C o s p e c t r a l C a l c u l a t i o n 162
EXPERIMENTAL INWSTICATION OF THE TURRIJLFNCE STRUCTtIRE
IN THE LOWER ATMOSPHERF
I. INTRODUCTION
In recent years, since fast response meteoroloqical
instruments have been developed capable of measurinq rapid
fluctuations in the atmospheric variables, data have been
accumulatins on the statistical properties of atmospheric
turbulence. Turbulent intensities of the variables, turbulent
fluxes in the atmosphere and spectral distributions of these
quantities have been measured with increased regularity and
dependability. However, much of the effort remains directed
toward developinq the measurement, data collection and analy-
sis systems, and extensive experimental programs for defininq
atmospheric turbulence structure are still forthcoming.
Individual investiqations have been often limited in several
aspects includinq the number of components measured, the
heiqht of measurement, and the frequency ranqe studied (lim-
ited at low frequencies by the length of record and at high
frequencies by instrument response or samplinq rate). Fur-
thermore, the variations of the statistical characteristics
of the measured quantities with varying average states of
the atmosphere have not been fully explored. Consequently,
much experimental work remains to be done to demonstrate
adequately the structure of turbulence in the atmospheric
boundary layer. Theoretical work1 has been successful in
describinq turbulence primarilv for isotrowic conditions and
direct application to the atmosphere has been li~ited qener-
ally to the high wave number ranqe where local isotrowy and
its consequences in the atmosphere have met with some success
although the extent of these experiments qenerally covered
only a few of the aspects of the locally isotropic turbulence.
The considerable verification of Kolmoqoroff's -5/3 law has . *
generally lacked simultaneous demonstration of the dependence
on c 2 i 3 and the reduction of the cospectra to zero as
required by local isotropy.
More recent theoretical work, the "similarity theory"
of Monin and ~ b o u k h o v ~ ~ and clarifyinq discussions by Panofsky 29
and ~llison'' have contributed insight into the structure of
the anisotropic, energy producinq ranqe of atmospheric turbu-
lence and its relation to the distributions of the mean varia-
bles. An exterision of this dimensional line of reasoninq,
discussed by ~ u r v i c h l ~ has led to some orqanization of mea-
surements of turbulence parameters and spectral distributions
anticipating similarity relationships in turbulence data.
However, for the most part, measurement demonstratinq theo-
retical relationships and orqanization of data defininq the
structure of atmospheric turbulence is far from completed.
The difficulty in obtaining adequate turbulence data
did not end with the development of fast response instrumenta-
tion for measurinq fluctuations of atmospheric variables.
The c o l l e c t i n g , sampl inq , and a n a l y s i s of such d a t a has
i n t r o d u c e d d i f f i c u l t i e s t h a t c o n t i n u e t o i m ~ e d e p r o q r e s s i n
t h e e x p e r i m e n t a l s t u d y of t u r b u l e n c e s t r u c t u r e . D i a i t a l
a n a l y s i s of t u r b u l e n c e d a t a r e q u i r e s samplinq a t a r a t e on
t h e o r d e r o f t e n t i m e s t h e h i q h e s t f r equency o f i n t e r e s t .
D i q i t a l samplinq technoloqy has advanced r a p i d l y , s o t h a t
a u t o m a t i c ana log t o d i q i t a l convers ion can be performed
s i m u l t a n e o u s l y w i t h t h e measurement of v a r i a b l e s a t a r a t e
a d e q u a t e t o i n v e s t i g a t e many t u r b u l e n c e problems of i n t e r e s t .
However, advances have l i k e w i s e been made i n i n s t r u m e n t a t i o n ,
a l lowinq o t h e r t u r b u l e n c e s t u d i e s a t f l u c t u a t i o n f r e q u e n c i e s
p r e s e n t l y beyond p r a c t i c a l d i q i t a l samplinq c a p a b i l i t i e s .
For example, on t h e one hand, t h e t u r b u l e n t f l u x of momentum
a s w e l l a s h e a t and m o i s t u r e i n t h e atmosphere can be s t u d i e d
w i t h d i g i t a l a n a l y s i s t e c h n i q u e s ; l i t t l e d i f f i c u l t v a r i s e s i n
sampling t h e n e c e s s a r y t u r b u l e n t f l u c t u a t i o n measurements a t
an adequa te r a t e . On t h e o t h e r hand, f l u c t u a t i o n measure-
ments i n t h e d i s s i p a t i o n range of eddy s i z e s would be ex t remely
d i f f i c u l t t o sample and a n a l y z e w i t h d i q i t a l e q u i p e n t .
Althouqh d i g i t a l a n a l y s i s methods have been used f o r
most a tmospher ic t u r b u l e n c e s t u d i e s a number o f i n v e s t i g a -
t i o n s have u t i l i z e d s p e c i a l i z e d ana loq d e v i c e s f o r a n a l y z i n g
con t inuous r e a l - t i m e and t ape - recorded measurements. These
d e v i c e s a r e o f t e n l i m i t e d i n t h e i r a c c u r a c y and t h e e x t e n t of
a n a l y s i s t h a t can be c a r r i e d o u t w i t h them. A s i n s t r u m e n t
a c c u r a c y improves and a s more v a r i e d and complex a n a l y s i s
of t u r b u l e n c e d a t a i s d e s i r e d , such d e v i c e s w i l l no l o n q e r he
a d e q u a t e f o r meet inq t h e a n a l y s i s needs .
The q e n e r a l purpose ana loq computer , h i g h l y a c c u r a t e ,
r e s p o n s i v e t o any ranqe o f f r e q u e n c i e s o f i n t e r e s t i n t h e
a tmosphere , and e a s i l y proqrammed t o hand le a wide v a r i e t v
o f problems, i s w e l l s u i t e d t o t h e d e t a i l e d a n a l y s i s o f con-
t i n u o u s t u r b u l e n c e d a t a . The a p p l i c a t i o n of t h i s v e r s a t i l e
t o o l t o a n a l y s i s o f t u r b u l e n c e d a t a h a s n o t been e x t e n s i v e l y
developed.
The a v a i l a b i l i t y o f a g e n e r a l purpose a n a l o q computer
i n R a t t e l l e - N o r t h w e s t L a b o r a t o r i e s a t Hanford, Washinqton (an
AEC i n s t a l l a t i o n and s i t e of t h i s s t u d y ) h a s made it p o s s i b l e
t o i n v e s t i q a t e t h i s means of a n a l y z i n q t u r b u l e n c e d a t a .
The purpose o f t h i s s t u d y , t h e n , i s t o o r q a n i z e t h e
t u r b u l e n c e s t a t i s t i c s such a s t o t a l v a r i a n c e s and c o v a r i a n c e s
of t h e t h r e e wind componcnts i n a d d i t i o n t o s p e c t r a l and
c o s p e c t r a l d i s t r i b u t i o n s , o v e r a wide ranqe of f r e q u e n c i e s ,
a c c o r d i n g t o h e i q h t s , wind speeds , and s t a b i l i t i e s i n o r d e r
t o t e s t t h e Monin-Oboukhov s i m i l a r i t y t h e o r y a s it a p p l i e s
t o them and t o de te rmine t h e forms o f any u n i v e r s a l f u n c t i o n s
invo lved . To accompl ish t h i s a s i z e a b l e q u a n t i t y o f t u r b u -
l e n c e d a t a f o r a l l t h r e e components w i l l be c o l l e c t e d and
t h e ana log computer w i l l be adap ted t o a n a l y z e t h e d a t a f o r
t h e above-mentioned s t a t i s t i c s .
Turbulence Sens inq Equipment - ---- -----
The t u r b u l e n c e d a t a p r e s e n t e d h e r e w e r e c o l l e c t e d a t
Hanford w i t h a f a s t r e sponse s e n s o r , termecl t h e wind component
meter. The s e n s o r s i q n a l s were recorded on a maqnet ic t a p e
r e c o r d e r and l a t e r ana lyzed on an ana loq computer.
The wind component meter, o r i q i n a l l y des iqned by J . J .
Fuquay o f B a t t e l l e Northwest L a b o r a t o r i e s a t Hanford, i s
s i m i l a r i n some r e s p e c t s t o e a r l i e r s e n s o r s d e s c r i b e d by G. C.
~ i 1 1 l ~ and H. E . c ramer7 , u t i l i z i n q h e a t e d thermocouple w i r e s .
M. ~ i ~ a k e ~ ~ h a s developed an improved v e r s i o n o f t h e
wind component meter w i t h h e a t e d thermocouple s e n s o r s s i m i l a r
t o t h e Fuquay model b u t i n c o r p o r a t i n g more complex e l e c t r o n i c
c i r c u i t r y . F u r t h e r r e f e r e n c e w i l l be made t o Miyake 's
i n s t r u m e n t f o l l o w i n q t h e d i s c u s s i o n on t h e response o f t h e
wind component meter.
The i n s t r u m e n t shown i n F i a u r e 2 . 1 s e n s e s t h e t h r e e
components of t h e wind a t a q i v e n s i n q l e p o i n t and produces
c o n t i n u o u s v o l t a g e s i g n a l s r e l a t e d t o t h e t h r e e s p h e r i c a l
wind components. The s i q n a l f o r t h e speed , V , i s a f l u c t u -
a t i n g d c s i g n a l a c r o s s a b r i d g e c i r c u i t , two l e g s o f which
a r e chromel-alumel thermocouple wires h e a t e d by 610 c p s
a l t e r n a t i n g c u r r e n t impressed a c r o s s t h e thermocouple w i r e s
a t a c o n s t a n t , c o n t r o l l e d v o l t a q e . The thermocouples a r e
suspended i n and coo led by t h e a i r s t r e a m w h i l e t h e r e f e r e n c e
junc t io r l s formed a t t h e mountinq s t u d s a r e m a i n t a i n e d a t
ambient t empera tu re . Another thermocouple , unhea ted , removes
t h e f l u c t u a t i o n s i n t h e s i q n a l due t o ambient t e m ~ e r a t u r e
v a r i a t i o n s . The emf produced by t h i s thermocouple a r r a n q e -
ment i s f i l t e r e d t o remove r e s i d u a l h e a t i n q c u r r e n t and
e x t r a n e o u s n o i s e .
The s i q n a l f o r t h e e l e v a t i o n o r v e r t i c a l a n q l e , 4 , i s
a l s o d e r i v e d from h e a t e d thermocouple w i r e s . There a r e two
o f t h e s e i n c l i n e d a t 4 5 ' a n g l e s t o t h e h o r i z o n t a l and 90' t o
e a c h o t h e r , forming a "V" i n t h e v e r t i c a l p l a n e . The thermo-
c o u p l e o u t p u t i s z e r o f o r h o r i z o n t a l winds and v a r i e s p o s i -
t i v e l y and n e q a t i v e l y f o r downward and upward g u s t s , r e s p e c -
t i v e l y . T h i s s i q n a l i s a l s o f i l t e r e d t o remove n o i s e .
The s e n s o r head i s c o n t i n u o u s l y d r i v e n t o remain
o r i e n t e d i n t h e d i r e c t i o n o f t h e h o r i z o n t a l wind, keeping
t h e wind normal t o t h e speed s e n s i n q thermocouple w i r e s a t
a l l t i m e s a ~ d keepinq t h e wind i n t h e p l a n e o f t h e e l e v a t i o n
a n q l e s e n s i n g "V"-shaped thermocouple ar rangement . T h i s i s
accompl ished by a second "V"-shaped ar ranqement of p la t inum
w i r e s i n t h e h o r i z o n t a l p l a n e which p r o v i d e s a n e r r o r s i q n a l
t o a s e r v o a m p l i f i e r and motor . T h i s c o n t i n u a l l y d r i v e s t h e
probe t o t h e n u l l p o s i t i o n , o r i e n t i n g t h e head i n t o t h e
i n s t a n t a n e o u s wind. A p o t e n t i o m e t e r i s a l s o connected t o
t h e s h a f t o f t h e s e r v o motor s o a s i q n a l p r o p o r t i o n a l t o t h e
h o r i z o n t a l a n g l e , 0 , i s produced t o be a m p l i f i e d and recorded
w i t h t h e wind speed and v e r t i c a l a n g l e .
F i q u r e 2 . 1 shows t h e s e n s i n q probc, c o n t a i n i n q t h e
the rmocoup le head w i t h a s s o c i a t e d b r i d q e ne tworks and t h e
s e r v o motor . A 350 f t c a b l e c o n n e c t s t h e s e n s i n q head w i t h
t h e e l e c t r o n i c s , which i n c l u d e s t h e h e a t i n a c u r r e n t o s c i l l a t o r ,
t h e s e r v o a m p l i f i e r , t h e s i q n a l f i l t e r s , and E l e c t r o - I n s t r u -
men t s d i f f e r e n t i a l d c a m p l i f i e r s t o b o o s t t h e s i q n a l s t o
r e c o r d i n g l e v e l . Wiring d i ag rams f o r t h e wind component
meter a r e q i v e n i n an u n p u b l i s h e d r e p o r t by R a t c l i f f e and
4 0 Sheen . C a l i b r a t i o n o f t h e wind component meter w a s accom-
p l i s h e d f i r s t by comparing t h e s ~ e e d s i q n a l w i t h t h e s t a n d a r d
c u p anemometer i n t h e wind t u n n e l o f t h e Department o f
Atmospher ic S c i e n c e s , t h e U n i v e r s i t y o f Washinqton. The
c a l i b r a t i o n c u r v e g a v e t h e n o n l i n e a r r e l a t i o n between t h e
i n s t r u m e n t o u t p u t and t h e t r u e wind speed shown i n F i g u r e
2 . 2 . L a t e r , when t h e i n s t r u m e n t was improved , a chanqe i n
t h e p r o b e h e a t i n g c u r r e n t changed t h e wind speed s i q n a l from
t h e p r e v i o u s s i g n a l l e v e l by a c o n s t a n t f a c t o r f o r a l l wind
s p e e d s . The new c a l i b r a t i o n shown i n F i g u r e 2 . 3 was made
by comparing wind component meter s i g n a l s w i t h c o r r e s p o n d i n q
wind d a t a from new Beckman and W h i t l e y anemometers on t h e
Hanford P o r t a b l e Mast. Speed c a l i b r a t i o n a t v e r y l o w v e l o c i -
t i e s w a s a l s o accompl i shed i n a small wind t u n n e l a t Hanford
made from a 6 - inch d i a m e t e r p l a s t i c t u b e . Turbu lence p u l s e s
w e r e q e n e r a t e d and t h e i r t r a n s p o r t t i m e s o v e r a known t u n n e l
d i s t a n c e w e r e measured. These p o i n t s a r e shown a s circles
i n F iqu re 2 . 3 .
C a l i b r a t i o n of t h e e l e v a t i o n a n q l e s i q n a l was pe r -
formed i n t h e same smal l wind t u n n e l . The thermocouple head,
which can be removed w i t h an e x t e n s i o n c a b l e from t h e s h a f t
of t h e wind component m e t e r , was i n s e r t e d i n t o t h e t u n n e l
and i n c l i n e d a t v a r i o u s a n g l e s t o t h e a i r s t ream. The
c a l i b r a t i o n t u n n e l i s shown w i t h t h e wind component meter
probe i n F igu re 2 . 4 . The r e s u l t i n q c a l i b r a t i o n cu rve pro-
v ided a n o n l i n e a r r e l a t i o n between t h e s i q n a l and t h e e l e v a -
t i o n anq l e . Th i s c a l i b r a t i o n i s shown i n F igu re 2 . 5 . I t
i s no ted t h a t t h e e l e v a t i o n a n s l e c a l i b r a t i o n i s a f u n c t i o n
of wind speed. T h i s dependence, determined from r e p e a t e d
c a l i b r a t i o n s a t v a r i o u s t u n n e l speeds , i s shown i n F i g u r e
2 . 6 .
The h o r i z o n t a l a n q l e s i q n a l is l i n e a r l y r e l a t e d t o
t h e t r u e a n q l e , and t h e c o n s t a n t of p r o p o r t i o n a l i t y i s
e s t a b l i s h e d by t h e known v o l t a q e a c r o s s t h e po t en t i ome te r
p rov id ing t h e h o r i z o n t a l a n g l e s i g n a l .
T e s t s o f t h e response t i m e s of t h e i n s t rumen t were
a l s o made i n t h e 6-inch c a l i b r a t i o n t u n n e l . The v e r t i c a l
a n q l e response was determined by o s c i l l a t i n q t h e thermo-
coup l e head, removed from t h e wind component meter w i t h an
e x t e n s i o n c a b l e and r o t a t e d i n t h e t u n n e l abou t an a x i s con-
c e n t r i c w i t h t h e thermocouple j u n c t i o n s f o r t h e v e r t i c a l
a n g l e measurement. The head was r o t a t e d th rough a 26' a r c
i n t h e v e r t i c a l p l ane a t a number o f speeds w i t h a v a r i a b l e
speed , motor d r i v e n mechanism, s i m u l a t i n q v e r t i c a l a n q l e
f l u c t u a t i o n s o f v a r i o u s f r e q u e n c i e s . F i q u r e 2.7 shows t h e
arranqement f o r t h i s tes t . The v e r t i c a l a n g l e s i q n a l was
r e c o r d e d on a Honeywell V i s i c o r d e r O s c i l l o q r a p h which u t i l i z e s
a m i r r o r qa lvanometer p r o j e c t i n q a l i q h t beam o n t o l i q h t
s e n s i t i v e paper s o t h a t s i g n a l o s c i l l a t i o n f r e q u e n c i e s much
l a r q e r t h a n t h o s e of concern h e r e can he reproduced w i t h o u t
ampl i tude o r phase d i s t o r t i o n . The r e c o r d f o r t h e v e r t i c a l
a n q l e r e sponse i s shown i n F i g u r e 2.8, demons t ra t ing t h a t
t h e i n s t r u m e n t o u t ~ u t i s n o t reduced i n ampl i tude f o r f r e -
q u e n c i e s up t o abou t 4 c p s ; c l o s e t o 5 c p s a s l i q h t reduc-
t i o n i s sugges ted . Also comparinq t h e r e c o t d with r e f e r e n c e s
p u l s e s g e n e r a t e d a t t h e same p o i n t f o r each o s c i l l a t i o n , no
phase s h i f t can be d e t e c t e d f o r t h e v e r t i c a l a n g l e s i g n a l
o v e r t h e ranqe o f f r e q u e n c i e s t e s t e d . The r e c o r d s o f F i g u r e
2.8 w e r e f o r wind speeds of 2.0 mps and 7.0 mps.
The t i m e r e s p o n s e f o r t h e speed s i g n a l was t e s t e d i n
t h e same t u n n e l by sweeping t h e thermocouple head th rough an
8.1° a r c a b o u t an a x i s o u t s i d e t h e t u n n e l a t a r a d i u s of
23-1/4 i n c h s o t h a t t h e s e n s i n g w i r e s w e r e o s c i l l a t e d
e s s e n t i a l l y i n t h e l o q i t u d i n a l d i r e c t i o n . The t es t s e t - u p
i s shown i n F i g u r e 2.9. T h i s mot ion superimposed speed f l u c -
t u a t i o n s on t h e s t e a d y t u n n e l f low, t h e magnitude of which
was de te rmined by t h e o s c i l l a t i o n f requency. The peak v a l u e s
of t h e speed f l u c t u a t i o n s , c a l c u l a t e d f o r v a r i o u s o s c i l l a -
t i o n f r e q u e n c i e s up t o a lmos t 4 c p s a r e compared i n F i g u r e
2.10 w i t h t h e s i q n a l s produced by t h e i n s t r u m e n t .
Fo r 2.0 mps and 7.0 mps t u n n e l s p e e d s , t h e measured
peak v a l u e s o f t h e speed o s c i l l a t i o n s compare w e l l w i t h t h e
c a l c u l a t e d s i q n a l l e v e l s a t a l l f r e q u e n c i e s t e s t e d . The
speed s e n s i n g c a p a b i l i t y a t t h e wind component meter demon-
s t ra tes no s i g n i f i c a n t l o s s i n r e s p o n s e t o a b o u t 4 c p s . The
r e f e r e n c e p u l s e s a g a i n f a i l t o r e v e a l any a p p r e c i a b l e p h a s e
s h i f t .
The h o r i z o n t a l a n q l e r e s p o n s e was t e s t e d by p l a c i n g
t h e wind component meter o n a r o t a t i n g p l a t f o r m w i t h t h e
the rmocoup le h e a d , c o n n e c t e d t o t h e s h a f t , i n s e r t e d i n t o t h e
wind t u n n e l f rom b e n e a t h as shown i n F i q u r e 2.11. The p l a t -
form was o s c i l l a t e d t h r o u g h a 4 3 O arc i n t h e h o r i z o n t a l p l a n e
a t f r e q u e n c i e s up t o 2 c p s . Thus t h e head moved r e l a t i v e t o
t h e body o f t h e wind component m e t e r , b e i n g d r i v e n c o n t i n u -
o u s l y by t h e s e r v o sys tem i n t o t h e a i r f l o w a l o n g t h e a x i s
o f t h e t u n n e l . The h o r i z o n t a l a n g l e s i q n a l , p roduced by t h e
r o t a t i o n o f t h e s h a f t , was r e c o r d e d o n t h e V i s i c o r d e r where
d e v i a t i o n s from a c o n s t a n t peak a m p l i t u d e , a s w e l l a s t h e
o b s e r v e d movement o f t h e the rmocoup le head away from t h e
a x i a l d i r e c t i o n f o r h i g h f r e q u e n c v o s c i l l a t i o n s , would i n d i -
cate a r e d u c e d r e s p o n s e . However, f o r t h e r a n q e o f f r e q u e n c i e s
t e s t e d , l i m i t e d a t a b o u t 2 CDS bv t h e m e c h a n i c a l c a p a b i l i t i e s
o f t h e m o t o r - d r i v e n r o t a t i n q p l a t f o r m , F i q u r e 2.12 shows n o
r e d u c t i o n o f t h e peak a m p l i t u d e . I n f a c t , a t a f r e q u e n c y o f
2 c p s a n o v e r s h o o t w a s d e t e c t e d , i n c r e a s i n g t h e peak a m p l i t u d e
by abou t l o % , a s w e l l a s d i s t o r t i n s t h e wsve form and caus inq
a s m a l l a p p a r e n t phase s h i f t . The p l a t f o r m w a s r o t a t e d
th rough a 1 4 ' a r c by hand t o s i m u l a t e h i s h e r f r equency f l u c -
t u a t i o n s and t h e thermocouple head was o b s e r v e 2 t o remain
o r i e n t e d i n t h e d i r e c t i o n of t h e t u n n e l a x i s an2 no s i q n i f i -
c a n t s i g n a l r e d u c t i o n was n o t i c e d a t l e a s t t o f r e q u e n c i e s
n e a r 4 cps . The s m a l l e r a n q l e of r o t a t i o n used i n t h i s c a s e
i s more c o n s i s t e n t w i t h maqni tudes t o be obse rved i n wind
f l u c t u a t i o n s a t h i g h f r e q u e n c i e s . The r e s u l t s i n d i c a t e t h a t
e r r o r s i n e x c e s s of t h o s e i n t r o d u c e d i n t h e a n a l y s i s o f t h e
d a t a a r e n o t expec ted f o r t h e h o r i z o n t a l a n q l e below 4 cps .
I n t h e t es t s d e s c r i b e d above, comple te r e s p o n s e c u r v e s
of t h e i n s t r u m e n t w e r e n o t de termined b u t it was found t h a t
f l u c t u a t i o n measurements o f a l l t h r e e components cou ld be
used u n c o r r e c t e d up t o 4 c p s and p o s s i b l y h i g h e r w i t h o n l y
minor e r r o r s r e s u l t i n g .
Thouqh t h e f requency response o f t h e wind component
meter a s d e s c r i b e d above i s somewhat f a s t e r t h a n Miyake 's
improved model which b e g i n s t o show l i m i t a t i o n s i n i t s
r e s p o n s e a t 1 c p s , t h e l a t t e r h a s t h e advantage of good
s e n s i t i v i t y a t a l l wind s p e e d s whereas t h e Hanford wind
component meter l o s e s s e n s i t i v i t y a t h igh wind speeds .
Recording Equipment
A seven-channel Ampex FR-1100 magne t i c t a p e r e c o r d e r
w i t h FM e l e c t r o n i c s was used t o r e c o r d t h e t u r b u l e n c e s i g n a l s .
Tape t r a n s p o r t s p e e d s o f 3-3/4, 7-1/2, 1 5 , and 30 i n . p e r set
(ips) a r e s e l e c t a b l e f o r r e c o r d i n q o r p layback. Once r e c o r d e d ,
t h e d a t a from a q i v e n series o f tes ts w e r e s t o r e d u n t i l
a n a l y s i s on t h e ana loq computer c o u l d be schedu led .
A seven-channel Ampex FR-1300 magne t i c t a p e r e c o r d e r /
r e p r o d u c e r w i t h FM e l e c t r o n i c s was used f o r r e p r o d u c i n q t h e
d a t a a t t h e time o f a n a l y s i s . Tape t r a n s p o r t s p e e d s o f
1-7/8, 3-3/4, 7-1/2, 1 5 , 30 and 60 i n . per sec a r e s e l e c t a b l e . I I
Both t a p e r e c o r d e r s r e c o r d and rep roduce o v e r a nominal
i 1 . 0 v o l t rms ranqe . The u s e o f t h e t a p e r e c o r d e r s n o t o n l y
a l lowed t h e t u r b u l e n c e s i g n a l s t o be s t o r e d u n t i l a c o n v e n i e n t
t i m e f o r a n a l y s i s b u t r e c o r d i n q and p l a y i n g back a t d i f f e r e n t
s p e e d s a l lowed t h e d a t a t o be compressed, s h i f t i n g t h e f l u c -
t u a t i o n f r e q u e n c i e s o f i n t e r e s t i n t o a h i q h e r r anqe e a s i l v
hand led by t h e e l e c t r o n i c ana loq computer and q r e a t l y reduc-
i n g t h e a n a l y s i s t i m e .
Both t a p e r e c o r d e r s are h i q h l y a c c u r a t e , c a p a b l e o f
r e p r o d u c i n q a r e c o r d e d s i g n a l w i t h l i t t l e d i s t o r t i o n b u t
t h e i r l i m i t a t i o n s a s o u t l i n e d i n t h e m a n u f a c t u r e r ' s s p e c i -
f i c a t i o n s must be c o n s i d e r e d f o r an a s sessment o f t h e e r r o r s
i n t r o d u c e d . The t a p e t r a n s p o r t i n t r o d u c e s n o i s e due t o
mechan ica l f l u t t e r which does n o t exceed an a m p l i t u d e o f
1 .5% o f t h e f u l l scale a m p l i t u d e f o r f r e q u e n c i e s below 312
c p s a t t h e l o w e s t t r a n s p o r t speed and t h e e r r o r i s c o n s i d e r -
a b l y d e c r e a s e d f o r f a s t e r speeds . Consequent ly , f o r atmos-
p h e r i c f l u c t u a t i o n s o f i n t e r e s t , t h i s e r r o r i s minimal . A
s t a r t t i m e of l e s s t h a n 8 sec b e f o r e t h e t r a n s ~ o r t r e a c h e s
s t a b l e t a p e mot ion i s r e q u i r e d a t t h e f a s t e s t t r a n s p o r t
speed. Care was t a k e n t o a s s u r e t h e motion was s t a b l e i n
o r d e r n o t t o i n t r o d u c e i n i t i a l low f r e q u e n c y o s c i l l a t i o n s i n
t h e d a t a .
D i f f e r e n c e s i n t h e r e l a t i v e p o s i t i o n s , from one
c h a n n e l t o a n o t h e r , o f t h e r e c o r d i n q and rep roduc inq g a p s
on t h e t a p e heads i n t r o d u c e i n t e r c h a n n e l t i m e d i s p l a c e m e n t
e r r o r s . I f l a r q e , such i n c r e m e n t s c o u l d i n t r o d u c e s e r i o u s
e r r o r s when s i g n a l s from d i f f e r e n t c h a n n e l s e n t e r e d i n t o t h e
same computa t ion i n c o o r d i n a t e t r a n s f o r m a t i o n s and c a l c u l a -
t i o n s o f c o v a r i a n c e s and c o s p e c t r a l e s t i m a t e s . However, t h e
t i m e d i s p l a c e m e n t e r r o r does n o t exceed 5 micro-sec a t 60 i n .
p e r sec t a p e speed. Consequent ly , even though t h e a n a l y s i s
t i m e b a s e was reduced by a s much as 1/256 o f t h e o r i q i n a l
measurement t i m e , a t i m e d i s p l a c e m e n t e r r o r between t h e
o r i g i n a l s i g n a l s on t h e o r d e r o f 1 mill i-sec was n o t exceeded
and i s o f no concern i n t h e p r e s e n t s t u d y .
The F M e l e c t r o n i c s f o r t h e FR-1100 have a f l a t f r e -
quency response from 0 t o 625 c p s a t 3-3/4 i p s , e a s i l y cover-
i n q t h e r a n g e o f a tmospher i c f l u c t u a t i o n s o f c o n c e r n d u r i n g
t h e i n i t i a l r e c o r d i n g . The FR-1300 e l e c t r o n i c s have a f l a t
r e s p o n s e from 0 t o 625 c p s a t 1-7/8 i p s , a s s u r i n g no a l tera-
t i o n o f t h e r e l a t i v e ampl i tude of t h e v a r i o u s f r equency
components o v e r t h e r a n g e o f f l u c t u a t i o n s b e i n g s t u d i e d dur-
i n g t h e a n a l y s i s . The s i g n a l t o n o i s e r a t i o f o r t h e
e l e c t r o n i c s of t h e FR-1300 r e c o r d e r does n o t f a l l below 40
db and f o r t h e FR-1100, below 35 db s o t h a t a n o i s e s i q n a l
w i l l n o t exceed 1 t o 2 % of t h e 1 - v o l t r m s f u l l - s c a l e s i q n a l
a t t h e s lowest t a p e speeds and w i l l be improved a t f a s t e r
t a p e speeds , The t o t a l harmonic d i s t o r t i o n does n o t
exceed 2 % f o r e i t h e r r e c o r d e r and t h e l i n e a r i t y i s *1% of
f u l l s c a l e . Thus it can be expec ted t h a t e r r o r s on t h e o r d e r
o f 1 t o 2% w i l l q e n e r a l l y be i n t r o d u c e ? i n t o t h e s i g n a l s by .. t h e t a p e r e c o r d e r s . For t h e h i y h e r f r equency and lower
ampl i tude s i q n a l s , s l i q h t l y l a r q e r e r r o r s can he expec ted .
A n a l y s i s Equipment
The a n a l y s i s of t h e t a p e r e c o r d e r t u r b u l e n c e d a t a was
c a r r i e d o u t on an Ease 1132 q e n e r a l purpose e l e c t r o n i c ana loq
computer. S i n c e t h e proqrams used on t h i s computer f o r t h e
a n a l y s i s w i l l be g i v e n l a t e r , i n v o l v i n g t h e q e n e r a l l y accep ted
symbols f o r t h e v a r i o u s computer components, t h e symbols a r e
i d e n t i f i e d h e r e f o r t h e components used and t h e i r f u n c t i o n s
a r e d e s c r i b e d b r i e f l y . For a more complex d i s c u s s i o n see
~ o h n s o n l ~ and Korn and ~ o r n l ~ .
An e l e c t r o n i c ana loq computer can a n a l y z e a c o n t i n u o u s
v o l t a g e v a r y i n q a s a f u n c t i o n of t i m e . Such a s i q n a l can be
t r e a t e d by c i r c u i t s which sum, m u l t i p l y , d i f f e r e n t i a t e ,
i n t e g r a t e , and f i l t e r t h e v o l t a q e . These c i r c u i t s can be
combined t o q i v e e l e c t r i c a l a n a l o q s of t h e mathemat ica l pro-
c e d u r e s o f i n t e r e s t .
L3asic t o t h e e l e c t r o n i c ana log computer i s t h e h iqh
g a i n d c a m p l i f i e r w i t h r e s i s t i v e and c a p a c i t i v e feedback n e t -
works c a l l e d t h e o p e r a t i o n a l a m p l i f i e r . Alonq w i t h p o t e n t i -
o m e t e r s , d i o d e s and o t h e r e l e c t r o n i c components, t h e
o p e r a t i o n s o f m u l t i p l i c a t i o n , d i v i s i o n , f u n c t i o n v e n e r a t i o n ,
and t r a n s f o r m a t i o n and r o t a t i o n of c o o r d i n a t e s can be
inc luded . B e s i d e s t h e a p p l i c a t i o n o f t h e computer t o d a t a
a n a l y s i s , a s p lanned h e r e , it i s more g e n e r a l l y used f o r
s o l v i n g d i f f e r e n t i a l e q u a t i o n s and s i m u l a t i n ~ complex systems.
F i g u r e 2.13 shows t h e symbols used t o i n d i c a t e each
o f t h e e l e c t r o n i c u n i t s per forming t h e f u n c t i o n s d i s c u s s e d
h e r e . A m p l i f i e r c i r c u i t s o r b lock d iaqrams more comple te ly
d e s c r i b i n g t h e i r o p e r a t i o n a r e a l s o i n c l u d e d .
The summing o f a number o f i n p u t s i q n a l s i s performed
by impress ing each of them a c r o s s one o f t h e i n p u t r e s i s t a n c e s
o f t h e h i g h g a i n d c a m p l i f i e r w i t h r e s i s t i v e feedback shown
i n F i g u r e 2.13a. The n e g a t i v e of t h e o u t p u t v o l t a g e is e q u a l
t o t h e sum o f t h e i n p u t v o l t a q e s each m u l t i p l i e d by a q a i n
f a c t o r which i s s imply t h e r a t i o of t h e feedback r e s i s t a n c e
t o t h e i n p u t r e s i s t a n c e . Gains of 1 and 1 0 , which can be
s e l e c t e d by changing t h e i n p u t r e s i s t a n c e s , a r e v e r y p r e c i s e l y
c o n t r o l l e d w i t h p r e c i s i o n r e s i s t o r s , a c c u r a t e t o w i t h i n 0.01%.
The i n t e g r a t o r shown i n F i g u r e 2.13b u t i l i z e s t h e same
t y p e of h iqh g a i n d c a m p l i f i e r b u t w i t h c a p a c i t i v e r a t h e r
t h a n res i s t ive feedback. The o u t p u t v o l t a g e i s p r o p o r t i o n a l
t o t h e i n t e g r a l of t h e i n p u t v o l t a g e where t h e p r o p o r t i o n a l i t y
-1 c o n s t a n t , - RC ' can v e r y a c c u r a t e l y de te rmine a s e l e c t a b l e
g a i n th rouqh u s e o f p r e c i s i o n i n p u t r e s i s t a n c e s and feedback
c a p a c i t o r s . Bes ides t h e l i m i t s o f p r e c i s i o n de te rmined by
p a s s i v e components, e r r o r s i n i n t e g r a t i o n a r e p o s s i b l e from
a number o f s o u r c e s . However, t h e s e a r e g e n e r a l l y less t h a n
1% f o r computers w i t h h i q h q u a l i t y components. For a thorough
18 d i s c u s s i o n of i n t e g r a t o r e r r o r s see Korn and Korn . The
f requency r e s p o n s e o f h iqh g a i n a m p l i f i e r s , summers and
i n t e g r a t o r s i s f l a t t o 20 k c , f a r more t h a n adequa te f o r
t h e a n a l y s i s o f t h e p r e s e n t problem. The n o i s e l e v e l f o r t h e
a c t i v e components i s abou t 5 mv, peak t o peak, s o s i q n a l
l e v e l s s h o u l d be k e p t a t l e a s t on t h e o r d e r o f a v o l t .
M u l t i p l i c a t i o n by a c o n s t a n t can be performed i n t h e
adding o r i n t e g r a t i n g c i r c u i t s , a s d e s c r i b e d e a r l i e r , by
a d j u s t i n q t h e i n p u t o r feedback r e s i s t a n c e s o r c a p i t a n c e s .
A c o n s t a n t m u l t i p l i e r q i v i n q any d e s i r e d f r a c t i o n o f t h e
s i g n a l i s a l s o p rov ided th rouqh p r e s e t p o t e n t i o m e t e r s i n s e r t e d
i n t h e ana log program a t any p o i n t . The symbol f o r t h i s
o p e r a t i o n is shown i n F i q u r e 2 . 1 3 ~ . Y u l t i p l y i n q two v a r i a -
b l e s t o q e t h e r i s accomplished w i t h t i m e d i v i s i o n m u l t i p l i e r s
i n t h e Ease 1132 computer , shown i n F i g u r e 2.13d, which aver -
a g e s a series o f p u l s e s t h e ampl i tudes o f which a r e de termined
by one of t h e v a r i a b l e s and t h e d u r a t i o n o f which a r e deter-
mined by t h e o t h e r . The p u l s e r a t e i s t h e l i m i t i n g f a c t o r
f o r d e t e r m i n i n q t h e f requency response o f t h e m u l t i p l i e r
which i s f l a t t o 500 c y c l e s . T h i s i s s u f f i c i e n t f o r t h e
p r e s e n t s t u d v a l t h o u s h f o r some ~ r o b l e m s i t can n r c s c n t seri-
o u s l i m i t a t i o n s . The n o i s e l e v e l , ahou t 15 mv (peak t o p e a k ) ,
can a l s o be l a r q e compared t o t h e s i q n a l , exceedinq a l e v e l
of 1% of t h e o u t p u t s i q n a l i f i n p u t s f a l l below 1 0 v o l t s .
The most s e r i o u s of t h e m u l t i ~ l i e r e r r o r s , however, i s t h a t
i n t r o d u c e d by d r i f t of t h e o u t p u t s i q n a l . Low f requency
v a r i a t i o n s of abou t 2 0 mv i n t h e o u t p u t a r e common. The
s e r i o u s n e s s of t h e d r i f t e r r o r r e s u l t s from i t s low f r e -
quency c h a r a c t e r s o t h a t an i n t e q r a t i o n of a m u l t i p l i e r o u t -
p u t can accumula te a s i z e a b l e e r r o r from t h e m u l t i p l i e r d r i f t ,
whereas t h e m u l t i p l i e r n o i s e w i l l c o n t r i b u t e no th inq t o t h e
i n t e q r a l .
A f u n c t i o n q e n e r a t o r can he used t o approximate any
s i n q l e va lued f u n c t i o n of a v a r i a b l e dependent on t i m e w i t h
a series o f s t r a i g h t l i n e segments s o t h a t t h e v a l u e of t h e
f u n c t i o n i s q i v e n c o n t i n u o u s l y from t h e f u n c t i o n q e n e r a t o r a s
t h e t i m e dependent v a r i a b l e i s a p p l i e d t o t h e i n p u t . They
a r e p a r t i c u l a r l y u s e f u l i n r e p r e s e n t i n g n o n - a n a l y t i c f u n c t i o n s
such a s e x p e r i m e n t a l l y o b t a i n e d c u r v e s . Func t ion g e n e r a t o r s
were used i n t h i s s t u d y t o r e p r e s e n t t h e i n s t r u m e n t c a l i b r a -
t i o n curves . A f u n c t i o n g e n e r a t o r i s comprised of a series
o f c i r c u i t s l i k e t h e one shown i n F i g u r e 2.13e t h e o u t p u t s o f
which a r e summed. Each c i r c u i t has a b r e a k p o i n t and a s l o p e
p o t e n t i o m e t e r s e t t i n q which de te rmine s h o r t l i n e a r segments
of t h e f u n c t i o n approximated. The most s e r i o u s e r r o r s
r e s u l t i n g from t h e use of a f u n c t i o n q e n e r a t o r , r a t h e r t h a n
b e i n g i n t r o d u c e d by t h e l i m i t a t i o n s o f t h e e l e c t r o n i c s , a r e
r e l a t e d t o t h e s h a p e o f t h e f u n c t i o n s and how a c c u r a t e l y it
can b e app rox ima ted w i t h s t r a i q h t l i n e seqments . I n t h e
p r e s e n t s t u d y , errors a r i s i n q from q e n e r a t i o n o f c a l i b r a t i o n
c u r v e s i n t h e computer p r o q r a m i n q s h o u l d n o t exceed 5% and
w i l l a v e r a g e less.
C o o r d i n a t e t r a n s f o r m a t i o n s a r e pe r fo rmed hv r e s o l v e r s i
which t a k e a s i n p u t s a s i g n a l r e p r e s e n t i n q a n a n q u l a r measure- , a ;
ment and o n e r e p r e s e n t i n q a r a d i a l measurement and q i v e a s
o u t p u t s t h e c a r t e s i a n components . T h i s i s accompl i shed by
combin ing t h e o p e r a t i o n s o f m u l t i p l i e r s and a f u n c t i o n qen-
e r a t o r . From t h e a n q u l a r measurement i n p u t , s i n e and c o s i n e
f u n c t i o n s a r e q e n e r a t e d by t h e f u n c t i o n q e n e r a t o r . These
and t h e r a d i a l measurements a r e i n p u t s t o t h e m u l t i p l i e r s ,
t h e o u t p u t s o f which a r e t h e n t h e c a r t e s i a n components , a s
shown i n F i g u r e 2 .13 f . The a c c u r a c y o f t h e r e s o l v e r i s
l i m i t e d by t h a t o f t h e m u i t i p l i e r s , p r e v i o u s l v d i s c u s s e d ,
a s w e l l a s by t h e f u n c t i o n q e n e r a t o r . Accord inq t o t h e
computer s p e c i f i c a t i o n s i n t h i s c a s e , t h e f u n c t i o n s e n e r a t o r
i n t h e r e s o l v e r i s a c c u r a t e t o 0 .05%.
Though o t h e r i n t e r n a l l y w i r e d components c a n b e
i n c l u d e d i n a n a l o q compu te r s , t h o s e d i s c u s s e d above a r e most
f u n d a m e n t a l and a r e t h e o n l y components p e r t i n e n t t o t h e
a n a l y s i s o f t h e p r e s e n t p roblem. The manner i n which t h e s e
components a r e i n t e r c o n n e c t e d o r proqrammed f o r t h e a n a l y s i s
w i l l be described l a t e r . A l s o t o be d e s c r i b e d l a t e r a r e
e x t e r n a l l y wired f i l t e r s , necessary components i n t h e pro-
grams t o be d i scussed .
FIGURE 2.1 Wind Component Meter Turbulence Sensing Probe
I - -- - I -- - ---
A -- . - - - saslnd - - -- --- - -- - - - a~ua~ajad - I _i sdu 0.1 = paads a6e~aAy - - -- - -
- -- -
-- .- -
I I I I I I . I J ~ I I 1 1 1 1 1 t t 1 1 r r r .-
-- -- G - .- - -- - - -- I sdw 0.2 = paads a 6 e ~ a ~ v -- - - - -- - - - . - - - - . - - --
-- - - - ---- - -- --
OPERATION CIRCUIT SYMBOL
R3 (a) Summer '3
O+%-- X2 - Ro X1 + X2 + X3
0 x
L - L -
l ntegrator
(c) Multiplication by Constant X1
-$?9 - -
(d) Multiplication of Two Variables
100
X
Y
(e) Function ~ e n e r a t o r :FZH--* F (x) -: =)1 l nverter Summer Slope
-100 v Breakpoint
(f) Resolver rv
oe cos e r cos e
FIGURE 2.13 Basic Analog Computer Components and Symbols
111. THEORETICAL BACKGROUND FOR ANALYSIS
where a l l ( w ) i s t h e t r u e s p e c t r a l d e n s i t y d i s t r i b u t i o n f o r
t h e random t i m e series and Y ( u ) i s t h e t r a n s f e r o r sys tem
f u n c t i o n f o r t h e f i l t e r d e t e r m i n e d from t h e r a t i o o f t h e
I n 1938, G . I . ~ a ~ l o r ~ ~ i n t r o d u c e d t h e c o n c e ~ t o f
a n a l y z i n g t h e e n e r g y i n t u r b u l e n t wind f l u c t u a t i o n s f o r
s p e c t r a l c o n t e n t t h r o u q h F o u r i e r t r a n s f o r m r e l a t i o n s h i p s .
S i n c e t h e n , t h i s m a t h e m a t i c a l t e c h n i q u e h a s been Used f o r
a n a l y z i n g random phenomena o f w i d e l y v a r y i n q p h y s i c a l o r i -
q i n s . A d i s c u s s i o n o f W i e n e r ' s q e n e r a l i z e d harmonic a n a l y -
s is , upon which t h i s t e c h n i q u e i s b a s e d , and i t s a p p l i c a t i o n
t o randomly f l u c t u a t i n q e l e c t r i c a l s i q n a l s such a s t h o s e
a n a l y z e d i n t h e p r e s e n t s t u d y i s q i v e n by Y. K . L e e . l9 The
l i m i t a t i o n s and problems i n t r o d u c e d by t a k i n q f i n i t e random
t i m e series samples o f v a r y i n q l e n g t h h a s been i n v e s t i g a t e d
by Blackman and ~ u k e ~ ~ l e a d i n q t o more mean inqfu l a n a l y s i s
o f e x p e r i m e n t a l d a t a . Mention w i l l b e made h e r e b r i e f l y o f
t h e r e l a t i o n s h i p s i n v o l v e d i n t h e a n a l y s i s o f t u r b u l e n t wind
f l u c t u a t i o n s i q n a l s , and l i m i t a t i o n s i n t r o d u c e d . A more
t h o r o u q h r e v i e w o f t h e m a t h e m a t i c a l background p r e s e n t e d by
Lee and by Blackman and Tukey i s c o v e r e d i n ~ p p e n d i x A.
When a random t i m e series, f ( t ) , i s f i l t e r e d , it can
b e shown eel') t h a t t h e mean s q u a r e d f i l t e r o u t p u t is :
Q l l (u) do (3.1) *o ( t ) = j -OD
Y ( o )
complex i n p u t and o u t p u t a m p l i t u d e s f o r a s t e a d v s t a t e
s i n u s o i d a l s i g n a l . Thus t h e mean s q u a r e o u t p u t o f t h e f i l t e r
i s an a r e a under t h e t r u e s p e c t r a l d e n s i t y f u n c t i o n c u r v e ,
s p e c i f i e d by t h e shape of t h e f i l t e r . I f t h e f i l t e r i s con-
s t r u c t e d t o p r o v i d e low p a s s o r h i q h p a s s f i l t e r i n q , t h e mean
s q u a r e o u t p u t i s s imply t h e v a r i a n c e o f t h e o r i q i n a l s i g n a l
w i t h t h e h i q h end o r low end , r e s p e c t i v e l y , o f t h e spect rum
e l i m i n a t e d . The s h a r p n e s s o f t h e f r equency c u t - o f f and t h e
r a n q e o f f r e q u e n c i e s e l i m i n a t e d can be s e l e c t e d i n t h e con-
s t r u c t i o n o f t h e f i l t e r . S i m i l a r l y , f o r a band p a s s f i l t e r
where t h e mean s u u a r e o u t p u t i n c l u d e s o n l y a p o r t i o n o f t h e
s p e c t r a l d e n s i t y o v e r a narrow ranqe of f r e q u e n c i e s , t h e
c e n t e r f r e q u e n c y and shape o f t h e f i l t e r t r a n s f e r f u n c t i o n
can b e v a r i e d i n t h e c o n s t r u c t i o n o f t h e f i l t e r .
When two d i f f e r e n t random f u n c t i o n s a r e f i l t e r e d
s e p a r a t e l y w i t h matched f i l t e r s , t h e mean p r o d u c t o f t h e
o u t p u t s p r o v i d e s an e s t i m a t e o f t h e c o s p e c t r a l d e n s i t y , i . e . ,
where t h e t r a n s f e r f u n c t i o n f o r t h e f i l t e r i s a q a i n Y ( w ) and
t h e c o s p e c t r a l d e n s i t v , t h e r e a l p a r t o f t h e c r o s s s p e c t r a l
d e n s i t y f u n c t i o n , i s q i v e n by C 1 2 ( u ) . Aqain, h i q h o r low
f r e q u e n c y p o r t i o n s of t h e a r e a under t h e c o s p e c t r a l d e n s i t y
c u r v e can be e l i m i n a t e d o r t h e c o n t r i b u t i o n from o n l y a
nar row band of f r e q u e n c i e s c a n be s e l e c t e d by t h e p r o p e r
c o n s t r u c t i o n of t h e f i l t e r t o p rov ide a s p e c i f i e d f i l t e r
t r a n s f e r f u n c t i o n .
For t h e " d i r e c t o r he te rodyn ing t e c h n i q u e " of s p e c t r a l
a n a l y s i s , t h e F o u r i e r t r a n s f o r m of a t r a n s i e n t f u n c t i o n pro-
duced by t r u n c a t i n g t h e o r i q i n a l t i m e series i s t a k e n .
Such a t r u n c a t i o n can r e s u l t from s imply l i m i t i n q t h e l e n q t h
o f r e c o r d a s d e s c r i b e d by Rlackman and ~ u k e ~ ' . S i m i l a r l y ,
t h e p r o d u c t of t h e t i m e series and t h e u n i t r e sponse func-
t i o n o f a f i l t e r i s a t r a n s i e n t f u n c t i o n and i t s t r a n s f o r m
i s
The averaged p roduc t o f t h i s F o u r i e r t r a n s f o r m w i t h t h e con-
j u g a t e of t h e t r a n s f o r m o f a second t r u n c a t e d random t i m e
series q i v e s
p r o v i d i n g e s t i m a t e s of t h e power s p e c t r a l d e n s i t y when t h e
t w o random f u n c t i o n s are t h e same, and c o s p e c t r a l . a n d quad-
r a t u r e s p e c t r a l d e n s i t y when t h e y a r e d i f f e r e n t . Here, H ( w )
i s t h e power t r a n s f e r f u n c t i o n f o r t h e low pass f i l t e r s and
a 1 2 ( w ) i s t h e t r u e s p e c t r a l d e n s i t y f u n c t i o n . The convolu-
t i o n i n t e g r a l of Equat ion ( 3 . 4 ) i s o f t h e same form a s t h e
s p e c t r a l e s t i m a t e s o b t a i n e d th rough t h e F o u r i e r t r a n s f o r m o f
t h e a u t o c o v a r i a n c e f u n c t i o n f o r a random t i m e series w i t h a
finite lenqth of record, described by Rlackman and ~ u k e ~ ~ .
The relationships described in the foreqoinq discussion
can provide the basis for a number of spectral analysis tech- ,
niques, some of which will he included in the analoq computer - \
I program discussion to follow. I
i
IV. DATA ANALYSIS PROGRAMS
I n proceeding from t h e raw d a t a t o t h e ana lyzed
r e s u l t s t h e sequence i s as f o l l o w s :
( a ) Coord ina te t r a n s f o r m a t i o n chanqes t h e d a t a from
a p o l a r frame t o a c a r t e s i a n frame i n which t h e
mean wind i s a l i g n e d w i t h t h e x a x i s .
(b) Reynolds' stresses a r e computed i n t h e new frame
o f r e f e r e n c e .
( c ) Power s p e c t r a f o r t h e f l u c t u a t i o n s o f a l l t h r e e
wind components a r e computed.
( d ) Cospec t ra f o r p a i r s o f t h e t h r e e wind components
a r e computed.
C o o r d i n a t e Trans fo rmat ion Prosram
The ana log computer h a s been proqrammed, f i r s t o f a l l ,
t o t a k e t h e t a p e recorded i n p u t s (see S e c t i o n V) and t o pro-
v i d e t h e wind f l u c t u a t i o n components u ' , v ' , w ' f o r r e c o r d i n q
on magne t i c t a p e . F i g u r e 4 . 1 shows t h e diaqram for t h i s
proqram i n t o which t h e wind component meter s i q n a l s , o r i q i n -
a l l y t a p e recorded a t 3-3/4 i n c h e s p e r second ( i p s ) , are
p layed back a t 6 0 i p s . The program p r o v i d e s f o r t h e c a l i b r a -
t i o n s o f t h e t u r b u l e n c e i n s t r u m e n t and t r a n s f o r m s c o o r d i n a t e s
from s p h e r i c a l t o c a r t e s i a n . A mathemat ica l r o t a t i o n
o f c o o r d i n a t e s i s made s o t h a t t h e x c o o r d i n a t e i s d i r e c t e d
a long t h e d i r e c t i o n of t h e mean wind. The mean wind i s
removed and t h e t h r e e wind f l u c t u a t i o n components a r e
re - recorded a t 3-3/4 i p s , t h u s c o n t r a c t i n q t h e r e co rd t o
1/16th of t h e o r i q i n a l l enq th .
The f i r s t p a r t of t h e c o o r d i n a t e t r a n s f o r m a t i o n pro--
gram, shown i n F iqure 4 . l a . , i s a p r econd i t i on ing c i r c u i t ,
t o compensate f o r d i f f e r e n c e s i n t h e c a l i b r a t i o n of t h e two
t a p e r e c o r d e r s involved i n hand l ing t h e d a t a . Such d i f f e r -
ences can e a s i l y occu r , p a r t i c u l a r l y s i n c e d i f f e r e n t t a p e
speeds a s w e l l a s d i f f e r e n t t a p e r e c o r d e r s a r e used i n
r eco rd ing and playback. C a l i b r a t i o n s i g n a l s of z e ro and
one v o l t , r ecorded on each channel of t h e o r i g i n a l t a p e
b e f o r e o r a f t e r t h e d a t a a r e p layed back i n t o t h e precondi-
t i o n i n g c i r c u i t . Bias s i g n a l s a r e added t o t h e incominq
ze ro c a l i b r a t i o n s i g n a l s by a d j u s t i n q po t en t i ome te r s h 0,
h 1, and h 2 u n t i l t h e o u t p u t s o f a m p l i f i e r s A 0, A 4 , and
A 8 a r e zero . Then, w i t h t h e one-vo l t c a l i b r a t i o n s i g n a l s
a s i n p u t s a g a i n ad jus tment is made f o r each channel w i th
p o t e n t i o m e t e r s h 3 , h 4 , and h 5 u n t i l t h e o u t p u t s of A 1,
A 5 , and A 9 r each one v o l t . I n t h e speed s i g n a l channel a
b i a s v o l t a g e i s added a t A 5. Th is r e p l a c e s a d c l e v e l which
had been removed from t h e i n s t rumen t speed s i q n a l du r ing
r eco rd ing s o t h a t t h e recorded s i q n a l would be c e n t e r e d i n
t h e i n p u t range of t h e t a p e r e c o r d e r . Now when t h e d a t a a r e
p layed back t h e y a r e p r e sen t ed t o t h e computer beyond t h e
p r econd i t i on inq c i r c u i t , j u s t a s t h e y were g e n e r a t e d by t h e
wind component meter . Also inc luded i n t h e p r econd i t i on inq
c i r c u i t a r e d iode l i m i t e r s a t A 0, A 4 , and A 8. These
3 9
p r e v e n t t h e magnitude of t h e incoming s i q n a l s from r e a c h i n q
a l e v e l t h a t would o v e r l o a d t h e computer a m p l i f i e r s . B y
p r o p e r ad jus tment e x t r a n e o u s n o i s e p u l s e s a r e k e p t a t a
ha rmless l e v e l whi le l e q i t i m a t e s i g n a l s a r e passed u n a l t e r e d .
The n e x t s t a g e i n t h e program p r o v i d e s f o r t h e c a l i b r a -
t i o n of t h e wind component meter. The i n s t r u m e n t speed s i q n a l
i s t h e i n p u t t o f u n c t i o n g e n e r a t o r 12 which i s a d j u s t e d t o
g e n e r a t e t h e speed c a l i b r a t i o n c u r v e a s shown i n F i g u r e 4.2
s o t h a t i t s o u t p u t v o l t a g e i s d i r e c t l y p r o p o r t i o n e d t o t h e
i n s t a n t a n e o u s speed. The i n i t i a l b r e a k p o i n t and s l o p e a r e
de termined from t h e -100 v o l t i n p u t t o A 52 and t h e s e t t i n g
o f p o t e n t i o m e t e r h 6 , r e s p e c t i v e l y .
The c a l i b r a t i o n of t h e v e r t i c a l a n q l e a t a wind speed
of 5 meters p e r second i s se t i n f u n c t i o n g e n e r a t o r 10. The
i n i t i a l b r e a k p o i n t and s l o p e a r e se t w i t h p o t e n t i o m e t e r s
h 9 and h 1 0 , r e s p e c t i v e l y , and t h e o t h e r p o i n t s , set i n t h e
f u n c t i o n g e n e r a t o r , a r e shown i n F i g u r e 4.3 . Then t h e
o u t p u t i n v o l t s from a m p l i f i e r 50 i s e q u a l t o t h e v e r t i c a l
a n g l e i n d e g r e e s f o r any i n s t r u m e n t s i g n a l i n p u t when t h e
wind speed i s 5 mps. A s t h e wind speed v a r i e s from 5 mps
a c o r r e c t i o n i s made t o t h e v e r t i c a l a n q l e r e s u l t i n g from
t h e change i n t h e c a l i b r a t i o n o f t h e v e r t i c a l a n g l e w i t h
change i n wind speed. Func t ion g e n e r a t o r 11, set up a s shown
i n F i g u r e 4 . 4 and w i t h i n i t i a l b r e a k p o i n t and s l o p e g i v e n
A 4 by h 7 and h 8 , produces a s i g n a l p r o p o r t i o n a l t o - f o r any AV
g i v e n i n p u t o f v e r t i c a l a n g l e s i g n a l , E . When t h i s f u n c t i o n Q
q e n e r a t o r o u t p u t is m u l t i p l i e d i n M 30 by t h e d e v i a t i o n i n
wind speed from 5 mps s u p p l i e d from A 32, t h e v e r t i c a l a n q l e
c o r r e c t i o n , A + , i s a v a i l a b l e t o be added i n a m p l i f i e r 1 8 t o
producinq a s i g n a l p r o p o r t i o n a l t o t h e t r u e v e r t i c a l
a n g l e r e q a r d l e s s of wind speed.
The v e r t i c a l a n g l e i s s u p p l i e d th rough a t r u n k l i n e t o
t h e s i n e and c o s i n e f u n c t i o n g e n e r a t o r i n t h e r e s o l v e r o f a
second ana loq computer , t h e Ease 2133, s i n c e o n l y o n e r e s o l v e r . I i s p r o v i d e d i n t h e Ease 1132 and it i s used i n a n o t h e r p a r t
o f t h e proqram. The s i n e and c o s i n e of t h e v e r t i c a l a n q l e .
i s r e t u r n e d th rough t r u n k l i n e s t o t h e Ease 1132 where it
i s m u l t i p l i e d by t h e wind speed i n M 28 and M 29 t o q i v e
t h e v e r t i c a l wind component, w , and t h e wind s p e e d , i n t h e
h o r i z o n t a l p l a n e , VH. The h o r i z o n t a l a n q l e ( d i r e c t l y propor-
t i o n a l t o t h e i n s t r u m e n t s i q n a l s o t h a t no f u n c t i o n q e n e r a t o r
i s r e q u i r e d ) and t h e h o r i z o n t a l wind speed a r e s u p p l i e d t o
t h e r e s o l v e r i n t h e Ease 1132 which p r o v i d e s t h e h o r i z o n t a l
components, u and v , as o u t p u t s . I t i s n e c e s s a r v t o s c a l e
t h e i n p u t s t o b o t h r e s o l v e r s i n o r d e r t h a t t h e o u t p u t s u , v ,
and w g i v e one v o l t f o r each meter p e r second.
The a v e r a g e o f each of t h e components i s o b t a i n e d from
t h e o u t p u t s o f i n t e g r a t o r s A 3 5 , A 38, A 20, and A 21. The
magnitude and d i r e c t i o n of t h e mean wind i s de te rmined from
t h e s e and t h e c o o r d i n a t e system is r e o r i e n t e d w i t h t h e x
a x i s d i r e c t e d a lonq t h e mean wind. To do t h i s , t h e t es t is
p layed back th rough t h e computer a g a i n and t h e mean h o r i z o n t a l
and v e r t i c a l a n g l e s a r e s u b t r a c t e d i n a m p l i f i e r s A 9 and
A 1 8 , r e s p e c t i v e l y , by a d j u s t i n q p o t e n t i o m e t e r s P 9 and
P 18. The mean wind speed i s a l s o s u b t r a c t e d from t h e l o n g i -
t u d i n a l component a t A 30 by s e t t i n g p o t e n t i o m e t e r P 94.
Then t h e f l u c t u a t i o n components, u ' , v ' , w ' w i t h a l l mean
v a l u e s removed, a r e r ecorded on magnetic t a p e a t 3-3/4 i p s
a f t e r a t t e n u a t i o n w i t h p o t e n t i o m e t e r s P 4 1 , P 42, and P 43
t o main ta in t h e i n p u t s t o t h e t a p e r e c o r d e r w i t h i n an
a c c e p t a b l e range . The c o o r d i n a t e t r a n s f o r m a t i o n i s moni tored
on a Brush C h a r t Recorder f o r each test . T y p i c a l r e s u l t s a r e
shown i n F i q u r e 4.5 .
Reynolds S t r e s s Program
The t u r b u l e n c e component s i g n a l s , r ecorded a f t e r
c o o r d i n a t e t r a n s f o r m a t i o n , are p layed back l a t e r i n t o a
number o f a n a l o g computa t iona l programs t o i n v e s t i g a t e t h e
p r o p e r t i e s o f t h e t u r b u l e n c e . The f i r s t program d e t e r m i n e s
t h e v a r i a n c e s o f t h e t u r b u l e n c e components and t h e c o v a r i -
ances between them, i .e . , t h e Reynolds stresses. F i g u r e
4.1 shows t h e diagram f o r t h e program. A f t e r c a l i b r a t i n g
t h e program f o r any t w o o f t h e s i g n a l s th rough t h e precondi-
t i o n i n g c i r c u i t w i t h t h e z e r o and one v o l t c a l i b r a t i o n s i g n a l s
r ecorded on each c h a n n e l o f t h e t a p e w i t h t h e d a t a , each
channe l o f d a t a i s passed th rough an a c t i v e h i g h p a s s f i l t e r
t o remove v e r y long p e r i o d f l u c t u a t i o n s . Each f i l t e r i s
c o n s t r u c t e d from a h i g h g a i n d c a m p l i f i e r w i t h a n i n p u t
resistance and capacitance in series and with resistive feed-
back. The operation of this arrangement is described by a
first order, linear, ordinary differential equation. The
transfer function for such a linear system at any qiven fre-
quency is defined by the ratio of the complex input and out-
put amplitudes for a steady state sinusoidal signal and is
the Fourier transform of its unit impulse response. eel^)
For this arrangement the transfer function is
For the values of the components used in this filter,
The power transfer function is
so that the half-power point is at 0.032 cps. Most of the
data have been analyzed with a playback tape speed of 3-3/4
ips so that a factor of 16 remains between the analysis and
original fluctuation frequencies. Thus in terms of the true
frequency of the atmospheric fluctuations, the half-power
point occurs at 0.002 cps, so that the filter nominally
passes fluctuations with periods smaller than 8-1/3 minutes.
The 80% and 20% power transmission points are at frequencies
a factor of two on either side of the half-power point.
A f t e r f i l t e r i n g , t h e s q u a r e o f each o f t h e s i g n a l s i s
t a k e n and t h e cross p r o d u c t i s c a l c u l a t e d i n m u l t i p l i e r s
M 9 , M 11, and M 10 , r e s p e c t i v e l y . The i n t e g r a l s o f t h e
f i l t e r e d s i g n a l s , t h e i r s q u a r e s and t h e i r c r o s s p roduc t a r e
t a k e n over t h e d u r a t i o n of each t es t . Divid ing t h e f i n a l
o u t p u t s from A 20, A 28 and A 24 by t h e t i m e , t h e Reynolds'
stresses a r e g i v e n . The i n t e g r a t i o n b e g i n s when t h e computer
i s p u t i n t h e "compute" mode. Th i s o c c u r s s i m u l t a n e o u s l y ,
through a s w i t c h i n g ar rangement i n t h e p r e c o n d i t i o n i n g c i r -
c u i t , w i th t h e i n t r o d u c t i o n o f t h e d a t a i n t o t h e program.
S i m i l a r l y , s t o p p i n g t h e i n t e g r a t i o n w i t h t h e "ho ld" mode
o c c u r s s i m u l t a n e o u s l y w i t h removing t h e d a t a . F i g u r e 4.7
shows an example of a moni tored c a l c u l a t i o n .
Two m o d i f i c a t i o n s o f t h i s p rocedure shou ld be mentioned.
The f i r s t t e c h n i q u e a t t e m p t e d d i d n o t u s e t h e f i l t e r d i s -
cussed above, b u t e x t r a n e o u s d c l e v e l s i n t h e s i g n a l s w e r e
removed by supp ly ing a d c b i a s a f t e r t h e average s i g n a l
l e v e l s were de termined a t A 21, and A 25. T h i s invo lved con-
s i d e r a b l e r e c a l c u l a t i o n t o remove e x a c t l y t h e c o n t r i b u t i o n
o f d c l e v e l s i n t h e Reynolds' stress c a l c u l a t i o n s and was
found t o b e i m p r a c t i c a l whereas t h e h i g h p a s s f i l t e r a l lowed
t h e c a l c u l a t i o n s t o be made q u i t e r a p i d l y .
The o t h e r v a r i a t i o n was i n t h e f i l t e r i n g used f o r
c a s e s above 12 meters. The playback speed used f o r t h e
a n a l y s i s was 7-1/2 i p s r a t h e r t h a n 3-3/4 i p s so t h a t t h e
half-power p o i n t of t h e f i l t e r o c c u r r e d a t a r ea l t i m e
f requency o f 0.001 cps . T h i s f i l t e r i n g e s s e n t i a l l y passed
f l u c t u a t i o n s w i th p e r i o d s less than 16-2/3 minu tes s o t h a t
l a r g e r e d d i e s , more impor tan t a t g r e a t e r h e i g h t s , were
inc luded i n t h e stress c a l c u l a t i o n s . T h i s proved t o be
impor t an t f o r d a t a g a t h e r e d a t 12.2 mete r s . See S e c t i o n V I I .
Power S p e c t r a l Program
The power s p e c t r a l e s t i m a t e s a r e o b t a i n e d w i t h a pro-
gram, shown i n F i g u r e 4.8, based on Equat ion (3.1) u t i l i z i n q
e i g h t e e n a c t i v e band pa s s f i l t e r s . The t u r b u l e n c e s i g n a l s
a r e p layed back one a t a t i m e i n t o t h e program, which
i n c l u d e s , f i r s t o f a l l , a p r e c o n d i t i o n i n g c i r c u i t . As b e f o r e ,
t h e program i s c a l i b r a t e d w i t h t h e z e r o and one -vo l t s i g n a l s
accompanying t h e d a t a . I n a d d i t i o n , a sw i t ch and a n o t h e r
g a i n c a l i b r a t i o n po t en t i ome te r a r e p rov ided t o a l l o w a g a i n
s e l e c t i o n du r ing t h e a n a l y s i s . S ince b o t h t h e l e n g t h o f
t h e t es t s and t h e average l e v e l o f t h e s i g n a l s v a r i e s among
tests, t h e g a i n s e l e c t i o n op t imized t h e program o p e r a t i o n
w i thou t t h e n e c e s s i t y of r e - s c a l i n g t h e e n t i r e program f o r
each test . Another sw i t ch ing arrangement i n t h e precondi-
t i o n i n g c i r c u i t i n t r o d u c e s t h e d a t a i n t o t h e program and
s imu l t aneous ly s t a r t s t h e i n t e g r a t i o n o f t h e r e s u l t s .
Another p o s i t i o n o f t h e sw i t ch s imu l t aneous ly removes t h e
d a t a and p u t s t h e computer i n " h o l d , " s t opp ing t h e i n t e g r a -
t i o n and r e t a i n i n g t h e f i n a l v a l u e s a t t h e i n t e g r a t o r o u t p u t s .
Beyond t h e p r econd i t i ong c i r c u i t , t h e program c o n s i s t s
of eighteen filter channels, each channel including a buffer
amplifier, a prefilter gain adjustment potentiometer and
amplifier, the filter with a compensation potentiometer
inserted, a multiplier, a scaling potentiometer and an
integrator. The channels differ only in the values of the
filter components which determine different center frequencies
and filter widths, in the values of the filter compensation
potentiometer settings, and in the other potentiometer
settinqs for a few cases when it is necessary to rescale only
a portion of the channels. The construction of the filter
(~ummins*) is shown only for the first channel but in every
case it is a second order filter, the operation of which is
described by a second order linear ordinary differential
equation which yields a transfer function of the form
where the gain of the filter is K = R2 Cl
R1 (C1 + C2)
the center frequency is = 1
'~1~2ClC2
and a measure of the width is Q = 1
w0R1 (C1 + C2)
The power t r a n s f e r func t ion i s
With t h e approximation t h a t
Where
&i5 2 ( 0 - w 0 ) ,
w e can c l o s e l y approximate t h i s r e l a t i o n s h i p wi th
from which it is apparent t h a t t h e half-power p o i n t occurs
when
demonstrat ing t h a t Q i s a measure of t h e f i l t e r width a s
w e l l a s showing t h a t t h e width is p ropor t iona l t o t h e c e n t e r
frequency. I n t h i s case t h e f i l t e r w a s designed for Q = 10
s o t h a t t h e width would always be 1/10 of t h e c e n t e r f r e -
quency. S ince t h e f i l t e r s are cons t ruc ted with non-preci-
s i o n components t h e a c t u a l widths and g a i n s d e v i a t e s m w h a t
from t h e des ign va lues and were determined exper imenta l ly by
pass ing s i n u s o i d a l s i g n a l s a t a number of f r equenc ie s through
each f i l t e r and then comparing t h e inpu t wi th t h e ou tpu t
ampl i tudes . The c e n t e r f r equency , no, t h e g a i n , K , t h e
wid th a s i n d i c a t e d by t h e v a l u e of Q , and t h e a r e a ,
A = - I l l Y ( u ) l 2 dw under t h e power t r a n s f e r f u n c t i o n ~ 2 , ~ - m
c u r v e normal ized t o t h e q a i n and c e n t e r f r equency , a r e q i v e n
i n Tab le 4 . 1 f o r each f i l t e r i n t h e proqram a s de termined
from t e s t i n c j t h e f i l t e r s . The f i l t e r s o v e r l a p each o t h e r
c l o s e t o t h e o n e - q u a r t e r power p o i n t r a t h e r t h a n t h e more
i d e a l half-power p o i n t . Also , t h e t o t a l r ange of f r e q u e n c i e s
covered i s somewhat l i m i t e d f o r i n v e s t i g a t i n q a tmospher ic
t u r b u l e n c e s p e c t r a . However, p l a y i n g back t h e d a t a a t more
t h a n one t a p e speed a l l o w s t h e program t o sample t h e spect rum
f o r a number of over - l app ing i n t e r v a l s . The o u t p u t of each
f i l t e r , squared i n t h e m u l t i p l i e r , m u l t i p l i e d by t h e f i l t e r
compensat ion p o t e n t i o m e t e r v a l u e , R, as w e l l a s a s c a l i n g
f a c t o r , P , de termined from t h e o t h e r p o t e n t i o m e t e r s e t t i n g s
and a m p l i f i e r g a i n s , and i n t e q r a t e d o v e r t h e l e n g t h o f t h e
t e s t i s a v a i l a b l e a t t h e o u t p u t o f t h e i n t e q r a t o r f o r each
channe l . The f i n a l i n t e g r a l , I , d i v i d e d by t h e i n t e g r a t i o n
t i m e g i v e , a c c o r d i n g t o Equat ion (3.1)
Assuming t h a t t h e s p e c t r a l d e n s i t y i s e s s e n t i a l l y c o n s t a n t
o v e r t h e narrow range of f r e q u e n c i e s passed by t h e f i l t e r , we
can w r i t e
Then, c o n s i d e r i n g t h e g a i n , G , i n t r o d u c e d i n t h e p re -
c o n d i t i o n i n g c i r c u i t and t h e a t t e n u a t i o n , B, of t h e t u r b u -
l e n c e s i g n a l s , i n t r o d u c e d d u r i n g r e c o r d i n g and d e t e r m i n i n g
t h e number o f v o l t s f o r e a c h meter p e r sec o f wind speed ,
t h e p r o d u c t of t h e t r u e wind f l u c t u a t i o n f r e q u e n c y , n , and
t h e s p e c t r a l d e n s i t y of t h e wind f l u c t u a t i o n s , S ( n ) , i s ,
i n u n i t s of ( m p s ) 2
n~ ( n ) = I /T P R A K~ G~ B~ (4.10)
The f i l t e r compensat ion p o t e n t i o m e t e r s e t t i n g , R , i s a d j u s t e d
t o accoun t f o r t h e v a r i a t i o n s i n t h e f i l t e r a r e a normal ized
t o f r equency , A K ~ . Then t h e s p e c t r a l estimates are found
by s imply m u l t i p l y i n g a l l t h e i n t e g r a l s f o r any q i v e n test
by a s i n g l e factor, F i g u r e 4.9 shows a moni tored power
s p e c t r a l c a l c u l a t i o n ,
The c a l c u l a t i o n s f o r some o f t h e first tests a n a l y z e d
o b t a i n e d i n t e g r a l s p r o p o r t i o n a l t o S ( n ) r a t h e r t h a n nS (n ) by
i n c l u d i n g i n t h e f i l t e r compensat ion p o t e n t i o m e t e r s e t t i n g
a n a d d i t i o n a l f a c t o r i n v e r s e l y p r o p o r t i o n a l t o f requency.
T h i s c a l c u l a t i o n proved less e f f e c t i v e t h a n t h e f i n a l t e c h -
n i q u e s i n c e t h e i n t e g r a l s covered a wide range of magni tudes ,
l a r g e enough t o approach t h e o v e r l o a d l i m i t o f t h e computer
f o r t h e low f requency estimates and small enough a t h igh
f r e q u e n c i e s t o approach t h e n o i s e l e v e l of t h e equipment.
To o b t a i n t h e t r u e s p e c t r a l d e n s i t y , i t was n e c e s s a r y t o
m u l t i p l y t h e a n a l y s i s s p e c t r a l d e n s i t y by t h e d a t a t i m e base
change f a c t o r : 256 f o r t h e playback speed of 6 0 i p s i n t o
t h e f i l t e r program and 32 f o r t h e 7-1/2 i p s playback speed.
For t h e l a t e r c a l c u l a t i o n s of nS(n) no such f a c t o r was
r e q u i r e d s i n c e t h e p r o d u c t of a n a l y s i s f r equency and a n a l y s i s
s p e c t r a l d e n s i t y e q u a l s t h e p r o d u c t of t h e f requency and t r u e
s p e c t r a l d e n s i t y r e g a r d l e s s of t h e t a p e speed.
The i n t e g r a l s f o r t h e h i g h e s t f r e q u e n c i e s i n t h e
a n a l y s i s a r e sometimes s m a l l enouqh t o c o n t a i n a c o n s i d e r a b l e
e r r o r r e s u l t i n g p r i n c i p a l l y from t h e m u l t i p l i e r d r i f t .
Numerous checks are made d u r i n g t h e a n a l y s i s p e r i o d , o b t a i n -
i n g i n t e g r a l s w i t h no i n p u t i n t o t h e program. These " n o i s e
i n t e g r a l " checks are compared w i t h t h e t e s t r e s u l t s and
where t h e n o i s e i n t e g r a l i n t r o d u c e s an e r r o r exceeding 10%
o f t h e t es t i n t e g r a l t h e s p e c t r a l e s t i m a t e i s e l i m i n a t e d .
Other e r r o r s i n t r o d u c e d i n t h i s a n a l y s i s s t e m f i r s t
from o n l y be ing a b l e t o c a l i b r a t e t h e program t o w i t h i n p l u s
o r minus one o r two p e r c e n t of t h e o n e - v o l t c a l i b r a t i o n .
Also , s m a l l d i f f e r e n c e s i n t h e l e n g t h o f r e c o r d reproduced
f o r each p a r t of t h e a n a l y s i s of a g i v e n tes t c o n t r i b u t e s t o
small d i f f e r e n c e s i n t h e s p e c t r a l estimates f o r t h e same
f r e q u e n c i e s i n t h e range where t h e a n a l y s i s r e s u l t s o v e r l a p .
F i n a l l y , t h e d e t e r m i n a t i o n of t h e t r u e f i l t e r a r e a s were
made t o w i t h i n abou t p l u s o r minus f i v e p e r c e n t , c o n t r i b u t i n g
e r r o r s o f t h e same magnitude i n t h e s p e c t r a l e s t i m a t e s .
C o s p e c t r a l Program
The c o s p e c t r a l f i l t e r program, based on Equat ion ( 3 . 2 ) ,
i s c o n s t r u c t e d s i m i l a r l y t o t h e power s p e c t r a l program and
i s shown i n F igu re 4.10. A p r e c o n d i t i o n i n q c i r c u i t c a l i -
b r a t e s t h e program, p rov ide s a g a i n s e l e c t i o n and s w i t c h e s
t h e t w o channe l s of d a t a and t h e i n t e q r a t i o n on and o f f
s imu l t aneous ly . Eleven d u a l f i l t e r channe l s f o l l ow , each
w i th a b u f f e r a m p l i f i e r and s c a l i n q po t en t i ome te r f o r each
d a t a l e g o f t h e channe l , a p a i r o f matched f i l t e r s - - o n e f o r
each o f t h e t w o s i g n a l s t - a m u l t i p l i e r , a f i l t e r compensat ion
po t en t i ome te r and an i n t e g r a t o r . The f i l t e r s a r e aga in second
o r d e r w i th t h e same de s ign a s t h e f i l t e r s f o r t h e power
s p e c t r a l program. A l l f i l t e r s were aqa in t e s t e d w i t h
s i n u s o i d a l s i g n a l s o f v a r i o u s f requency. The f i l t e r s a r e
w e l l matched s o t h a t t h e shape o f t h e power t r a n s f e r - f u n c t i o n ,
de termined from t h e o u t p u t o f t h e m u l t i p l i e r f o r t h e v a r i o u s
s i n u s o i d a l i n p u t s , i s q u i t e s i m i l a r t o t h e de s ign f u n c t i o n
shape. However, t h e g a i n and t h e width of each f i l t e r aga in
d i f f e r somewhat from t h e d e s i g n v a l u e s . The f i l t e r test
r e s u l t s were aga in used t o de te rmine t h e t r u e v a l u e s of K~
and Q. These, the c e n t e r f r e q u e n c i e s and t h e power t r a n s f e r
f u n c t i o n a r e a , normal ized t o g a i n and c e n t e r f requency are
t a b u l a t e d i n Tab le 4 .2 f o r each o f t h e e l e v e n f i l t e r s .
The f i n a l i n t e g r a l f o r any channe l d i v i d e d by the t i m e
of i n t e g r a t i o n i s , from Equat ion ( 3 . 2 )
Assuming t h e c o s p e c t r a l d e n s i t y t o be r e l a t i v e l y c o n s t a n t
o v e r t h e narrow f requency ranqe of f i l t e r , w e have
Accounting f o r t h e s e l e c t a b l e q a i n and t h e a t t e n u a t i o n q i v i n q
t h e number of v o l t s p e r mps i n t r o d u c e d i n r e c o r d i n q t h e
ana lyzed wind component f l u c t u a t i o n s , t h e t r u e c o s p e c t r a l
estimates a r e g i v e n i n (mps) by
Again, t h e f i l t e r compensat ion p o t e n t i o m e t e r , P, i s a d j u s t e d
i n each channe l i n v e r s e l y p r o p o r t i o n a l t o A K ~ , t h e f i l t e r
a r e a , normal ized t o c e n t e r f r equency s o t h e f i n a l i n t e g r a l s
f o r a g i v e n test are a l l m u l t i p l i e d by a s i n g l e c o n s t a n t .
For t h e f i r s t tests, ana lyzed f o r c o s p e c t r a l d e n s i t y , C ( n ) ,
r a t h e r t h a n nC ( n ) , t h e f i l t e r compensat ion p o t e n t i o m e t e r
s e t t i n g i n c l u d e d a f a c t o r t o remove t h e dependence of t h e
f i l t e r wid th on c e n t e r f requency. L a t e r , t h i s dependence was
r e t a i n e d , y i e l d i n g estimates o f nC(n) w i t h a na r rower range
o f o u t p u t l e v e l s . The e r r o r s i n t r o d u c e d i n t h i s program are
o f t h e same t y p e as t h o s e found i n t h e power s p e c t r a l a n a l y s i s
proqram and o f t h e same magnitude, g e n e r a l l y on t h e o r d e r o f
f i v e t o t e n p e r c e n t . The example of t h e c o s p e c t r a l f i l t e r
o u t p u t s , F i g u r e 4 . 1 1 , r e s u l t e d from t h e f i r s t arranqernent
c a l . c u l a t i n g C ( n ) .
A l t e r n a t e S p e c t r a l D e n s i t y Proqram ----- - ---- Althouqh t h e f o l l o w i n g program h a s n o t been used i n
computinq t h e r e s u l t s t o be p r e s e n t e d i n t h i s p a p e r , i t h a s
been t e s t e d and shown c a p a b l e of d e t e r m i n i n q power s p e c t r a l
and c r o s s s p e c t r a l e s t i m a t e s . I t u s e s h e t e r o d y n i n g and
f i l t e r i n g , fo l lowed by m u l t i p l i c a t b n and a v e r a g i n q . I t is
an e l e c t r i c a l a n a l o g o f t h e p r o c e d u r e d e s c r i b e d i n E q u a t i o n s
(3 .3) and (3 .4 ) and i n t h e a s s o c i a t e d d i s c u s s i o n .
F i g u r e 4.12 shows two wind component s i q n a l s , a f t e r
p r e c o n d i t i o n i n q , e a c h b e i n g m u l t i p l i e d by t h e s i n e and c o s i n e
o f a g i v e n f r equency , w . The p r o d u c t s are e a c h s u b j e c t e d t o
low p a s s f i l t e r i n g g i v i n g c o n t i n u o u s l y t h e r e a l and imaq ina ry
p a r t s o f t h e F o u r i e r t r a n s f o r m s o f t h e o r i q i n a l s i q n a l s . The
o u t p u t s from A 5 1 and A 12 are t h e imaqinary p a r t s of each
F o u r i e r t r a n s f o r m . The sum of t h e s q u a r e s of t h e r e a l and
imaq ina ry p a r t s f o r e a c h component form t h e power s p e c t r a l
e s t i m a t e s a t t h e o u t p u t o f A 1 8 and A 26. The c r o s s p r o d u c t
o f t h e r e a l p a r t s summed w i t h t h a t o f t h e imag ina ry p a r t s
p r o v i d e s t h e c o s p e c t r a l estimates a t A 19. The c r o s s pro-
d u c t o f t h e r e a l p a r t f o r one component w i t h t h e imag ina ry
p a r t f o r t h e o t h e r component, summed w i t h t h e o t h e r l i k e
p r o d u c t p r o v i d e s t h e q u a d r a t u r e s p e c t r a l d e n s i t y a t t h e o u t -
p u t o f a m p l i f i e r A 22 . S t a t i s t i c a l l y s t a b l e e s t i m a t e s a r e
o b t a i n e d by smoothinq t h e s p e c t r a l d e n s i t i e s and r e c o r d i n q
t h e con t inuous r e s u l t s a t t h e o u t p u t s of A 24, A 29, A 25,
and A 2 8 , a s w e l l a s i n t e q r a t i n q o v e r t h e l e n g t h of r e c o r d
a t A 30, A 35, A 31, and A 34, t o o b t a i n s i n g l e average
v a l u e s f o r t h e e n t i r e tes t . An example of t h e recorded con-
t i n u o u s c a l c u l a t i o n i s g i v e n i n F i g u r e 4.13. With a d d i t i o n a l
equipment i n c l u d i n g a t a p e l o o p mechanism and an o s c i l l a t o r
t h a t can be c o n t i n u o u s l y swept s l o w l y th rough a range of
f r e q u e n c i e s , it w i l l be p r a c t i c a l t o u s e t h i s program i n
p l a c e of t h e m u l t i p l e f i l t e r programs t o p r o v i d e t h e power
s p e c t r a l and c o s p e c t r a l d i s t r i b u t i o n s a s w e l l a s q u a d r a t u r e
s p e c t r a l d i s t r i b u t i o n s n o t p r e s e n t l y o b t a i n e d .
S t a t i s t i c a l R e l i a b i l i t y o f S ~ e c t r a l Estimates
Blackman and Tukey have demonst ra ted t h a t , assuming
a Gauss ian p r o c e s s , t h e s p e c t r a l e s t i m a t e s f o l l o w a c h i -
s q u a r e d i s t r i b u t i o n w i t h t h e number of d e g r e e s o f freedom,
k , e q u a l t o t h e number of e l ementa ry f requency bands i n t h e
e q u i v a l e n t wid th o f t h e s p e c t r a l window, f o r a ' r e l a t i v e l y
smooth s p e c t r a l d i s t r i b u t i o n . Taking t h e wid th of t h e
f i l t e r a t t h e half-power p o i n t a s t h e e q u i v a l e n t w i d t h , We,
and 1/2 T a s t h e width o f t h e e lementa ry f requency bands,
t h e number o f d e g r e e s of freedom i s
Table 4 . 3 g i v e s t h e number of d e a r e c s o f freeclop f o r
each t e s t a t t h e lowes t f r equency o f t h e a n a l y s i s where t h e
f i l t e r i s n a r r o w e s t and consequen t ly t h e s p e c t r a l e s t i m a t e
i s l e a s t s t a b l e , a t t h e h igh f requency end of t h e a n a l y s i s
where t h e e s t i m a t e s a r e most s t a b l e , and a t a f r equency i n
t h e middle o f t h e range o f a n a l y s i s . Also g i v e n i n t h e t a b l e
a r e t h e r a t i o s o f p o s s i b l e s p e c t r a l e s t i m a t e s t o t r u e s p e c t r a l
d e n s i t y between which t h e e s t i m a t e s w i l l f a l l w i t h 80% con- . f i d e n c e . Tab le 4 .3 a l s o a p p l i e s t o t h e c o s p e c t r a l e s t i m a t e s
s i n c e t h e same g e n e r a l range o f f r e q u e n c i e s was covered and
t h e f i l t e r d e s i q n was t h e same a s f o r t h e power s p e c t r a l
e s t i m a t e s . The v a r i a b i l i t y f o r bo th s p e c t r a l and c o s p e c t r a l
e s t i m a t e s was found u s i n g t h e d e s i g n wid th p a r a m e t e r ,
Q = 10. T h i s r e s u l t e d i n s l i g h t l y s m a l l e r v a l u e s f o r t h e
number o f d e g r e e s of freedom th3n t h e somewhat s m a l l e r
w i d t h p a r a m e t e r s , found i n t h e f i l t e r tes ts , would have
g i v e n . Consequent ly , the e s t i m a t e s a r e s l i g h t l y more s t a b l e
t h a n i n d i c a t e d by t h e c a l c u l a t i o n .
TABLE 4.1
POWER SPECTRAL PROGRAM PAPAMETERS
Filter
TABLE 4 . 2
Filter
1
2
3
4
5
6
7
8
9
10
11
COSPECTRAL PROGRAM PARAMETERS
TABLE 4.3
VARIABILITY OF SPECTRAL ESTIMATES
Limiting Ratios of Estimates Degree of Freedom to True Spectral Density
Length (sec.)
1740 2220 960
960
3840 1200 2400
720 960 3540
3840 3540
8040
3180 1980
2700 2700
3420
k~ k~ for 80% Confidence Interval for: n=.012 cps n=.20 cps n=4.9 cps ?i=.012 cps n=.20 cps n=4.9 cps Test
2-1 2-2 2-3
4-1
5-1 5-2 5-3
6-1 6-2 6-3
7-1 7-i
9-1
11-1 11-2
12-1 12-2
13-1
FIGURE 4.la Coordinate Transformation Program
-70 -60 -50 -40 -30 -20 -10 0
10 Ev (volts)
FIGURE 4.2 Function Generator Settings for Speed Calibration
FIGURE 4.3 Function Generator Settings for Vertical Angle Calibration at 5.0 mps
FIGURE 4.4 Function Generator Settings for Vertical Angle Correct ion
FIGURE 4.5 Coordinate Transformation Monitored Results
FILTER CIRCUIT
FILTER COMPENSATION
PRECONDITIONING CIRCUIT
TIMING CIRCUIT
COMPUTER PRINT OUT
FIGURE 4 . 8 Power S p e c t r a l F i l t e r Program
PRECONDITIONING CIRCUIT
zero Ad,
6 8
F I L T E R CIRCUIT
Hold Curnpensatlons Pots are all
Set lo 0.1
0.0100 I 0
GURE 4.10 C o n s p e c t r a l F i l t e r Program
COMPUTER PRINT OUT
V. FIELD TESTS
I n t h e summer o f 1964 a series of t u r b u l e n c e measure-
ments w e r e conducted w i th t h e wind component meter mounted
on a 24.4 meter me teo ro log i ca l tower c a l l e d t h e Hanford
P o r t a b l e Mast a t t h e 3.0 and t h e 6 .1 m e t e r l e v e l s . Measure-
ments w e r e made i n n e u t r a l , u n s t a b l e and s t a b l e c o n d i t i o n s
accompanied by a v a r i e t y o f wind speeds . Mean t empe ra tu r e s ,
wind speeds and d i r e c t i o n s w e r e determined a t l o g a r i t h m i c
i n t e r v a l s from 0.8 meters t o 24.4 meters on t h e tower w i th
a sys tem ( R a t c l i f f e and sheen4') u t i l i z i n g copper-constantan
thermocouples and Beckman and Whit ley cup anemometers and
wind vanes , sampling and logg ing t h e d a t a a u t o m a t i c a l l y
th roughout t h e test series, The wind speeds and tempera-
t u r e s a t 3.0 and 6.1 meters w e r e c o n s i s t e n t l y dependable
and a v a i l a b l e du r ing a l l b u t a few tests f o r Richardson ' s
number c a l c u l a t i o n s . The c a l c u l a t e d g r a d i e n t R icha rdson ' s
number f o r t h e 4.3 meter l e v e l and t h e tower measurements o f
mean wind speed a t t h e h e i g h t o f t h e t u r b u l e n c e measurements,
a r e l i s t e d i n Table 5.1. Also l i s t e d a r e t h e mean longi-
t u d i n a l wind component v a l u e s c a l c u l a t e d i n t h e wind component
meter d a t a a n a l y s i s which ag ree w i th t h e P o r t a b l e Mast mean
wind speed t o w i t h i n 9%. The g e n e r a l l y s l i q h t l y l a r q e r
v a l u e s of mean wind speeds from t h e cup anemometers i s t o be
expected s i n c e t h e s e r e p r e s e n t t h e t o t a l h o r i z o n t a l component
magnitude r a t h e r than t h e h q i t u d i n a l conponent magnitude
only . I n a d d i t i o n , i n t h i s series of tes ts t h e wind compon-
e n t meter o p e r a t i o n was n o t e x a c t l y synchron ized w i t h t h e
mean v a l u e measurements s o t h a t d a t a f o r s l i g h t l y d i f f e r e n t
p e r i o d s o f t i m e were sampled by t h e two equipment svs tems,
a l s o c o n t r i b u t i n g t o t h e s m a l l d i s c r e p a n c i e s i n t h e r e p o r t e d
wind speeds .
S i n c e d a t a c o l l e c t e d a t q r e a t e r h e i g h t s were d e s i r e d ,
l a t e r i n t h e summer o f 1964 t h e wind component meter was
mounted a t 12.2 meters where a s i n g l e t e s t i n s t a b l e condi-
t i o n s was conducted. I n t h e f o l l o w i n g s p r i n q , two add i -
t i o n a l tes ts were conducted a t 12.2 meters and two more a t
6 .1 meters i n n e u t r a l and u n s t a b l e c o n d i t i o n s . Also , t h e
f i n a l t e s t i n u n s t a b l e c o n d i t i o n s was made w i t h t h e wind
component meter mounted a t 87 meters on t h e Hanford 400-
f o o t m e t e o r o l o g i c a l tower. The accompanying m e t e o r o l o g i c a l
d a t a from t h e P o r t a b l e Mast are a g a i n r e p o r t e d i n Tab le 5 . 1
w i t h t h e d a t e s and t i m e s of t h e tests. For t h i s series of
tes ts t h e sampling p e r i o d s f o r t h e win2 component meter and
t h e mean v a l u e system corresponded. However, t h e wind
s p e e d s r e p o r t e d f o r t h e wind component meter i n Tab le 5 .1
d i f f e r s i g n i f i c a n t l y from t h e wind speeds o f t h e cup
anemometer. The e r r o r was f i r s t s u s p e c t e d t o be i n t r o d u c e d
i n t h e a n a l y s i s ; however, r e p r o c e s s i n g o f one o f t h e tests
on t h e ana log computer l e d t o t h e same r e s u l t s . The o n l y
remaining p o s s i b l e e x p l a n a t i o n was t h a t t h e speed c a l i b r a -
t i o n changed somewhat f r o m t h e c a l i b r a t i o n made j u s t p r i o r
t o t h e f i r s t s e r i e s of tu rbu lence measurements. The c a l i b r a -
t i o n appeared t o d r i f t i n one d i r e c t i o n between t h e f i r s t
s e r i e s of t e s t s and t e s t 9-1, conducted a month l a t e r , then
changed i n t h e o t h e r d i r e c t i o n dur inq t h e i n t e rven inq e i q h t
months be fo re t h e f i n a l t e s t s e r i e s . The s p e c t r a l e s t i m a t e s ,
normalized t o t o t a l va r i ance r equ i r ed no c o r r e c t i o n s i n c e
t h e s p e c t r a l e s t i m a t e s and t h e va r i ances were both i n e r r o r
by t h e same f a c t o r . Whenever s p e c t r a l e s t i m a t e s , v a r i a n c e s
o r covar iances a r e r epo r t ed s e p a r a t e l y , t h e d a t a f o r t e s t s
a f t e r t h e f i r s t s e r i e s a r e c o r r e c t e d wi th t h e square of t h e
r a t i o of wind component meter mean wind speed t o tower
measured wind speed. One c o r r e c t i o n f a c t o r was used f o r
test 9-1 and another f a c t o r was used f o r t h e remaining
t e s t s dur inq which t ime t h e c a l i b r a t i o n d i d n o t appear t o
d r i f t f u r t h e r .
The tu rbu lence d a t a were recorded on t h e Ampex FR-1100
t a p e a t 3 - 3 / 4 i p s t a p e t r a n s p o r t speed f o r a l l t e s t s .
C a l i b r a t i o n s i g n a l s of ze ro and one v o l t were recorded wi th
t h e d a t a t o a s s u r e a c c u r a t e reproduc t ion of t h e d a t a dur inq
t h e a n a l y s i s per iod . The d a t a were s t o r e d on t h e t a p e s
u n t i l t h e a n a l y s i s could be c a r r i e d o u t a few months a f t e r
t h e t e s t s , wi th t h e analog computer programs d i scussed
prev ious ly .
TABLE 5.1
TEST CONDITIONS
Measurement Tower - WCY - Height U U
Test Date - Time (meters) (mps) (mps) Ri
?&an wind and temperature data were not available during Pest 4-1 but conditions were estimated to be slightly stable.
ffEvalutbCdwith smoothed data using all mast temperatures.
V I . THEORETICAL BACKGROUND FOR TURBULENCE RESULTS
C e r t a i n f e a t u r e s of t h e measured s p e c t r a can be a n t i -
c i p a t e d and used t o o r g a n i z e s p e c t r a l e s t i m a t e s . From
T a y l o r ' s hypo the s i s43 f o r a f i e l d o f t u rbu l ence advec ted
p a s t a p o i n t o f measurement, i f t h e mean wind speed c a r r y i n q
t h e e d d i e s i s l a r q e compared t o t h e t u r b u l e n t v e l o c i t y , t h e
s p a t i a l d i s t r i b u t i o n of t u r b u l e n c e i s swept by e s s e n t i a l l y
unchanged. Then t h e t i m e dependent au tocova r i ance f u n c t i o n
measured a t a p o i n t can be exp re s sed a s a d i s t a n c e dependent
f u n c t i o n , @(XI = u ( t ) u ( t + x/U) and i t s t r an s fo rm i s
t h e t u r b u l e n t d e n s i t y p e r u n i t wave member.
Thus U S ( n ) p l o t t e d v e r s u s wave number, n /c , should show t h e
c h a r a c t e r o f t u r b u l e n c e d i s t r i b u t e d i n wave number space a lonq
t h e mean wind d i r e c t i o n .
The s c a l e of t h e t u r b u l e n c e is expec ted t o be a func-
t i o n o f h e i g h t , t h e g r e a t e r t h e h e i g h t above t h e qround t h e
l a r g e r a r e t h e edd i e s . I n f a c t , t h e v e r t i c a l s i z e o f t h e
e d d i e s i s expec t ed t o be d i r e c t l y p r o p o r t i o n a l t o h e i g h t a s
proposed by oni in^^ based on s i m i l a r i t y t h e o r y c o n s i d e r a t i o n s .
Then p l o t t i n g t h e s p e c t r a as f u n c t i o n s o f nz/n ( t h e r a t i o of
h e i g h t t o wave l e n g t h ) w i l l t e n d t o normal ize t h e v e r t i c a l
component s p e c t r a l d i s t r i b u t i o n s f o r comparison a t a l l
h e i g h t s . ~ e r m a n * h a s o rgan i zed t h e l o g i t u d i n a l s p e c t r a i n
t h e same manner, b u t found t h a t t h e y behaved d i f f e r e n t l y a t
d i f f e r e n t h e i q h t s opposed t o t h e s i m i l a r i t v h v p o t h e s i s . S i n c e
d i f f e r e n t l e v e l s o f t u r b u l e n c e o c c u r i n d i f f e r e n t tests , t h e
s p e c t r a l e s t i m a t e s a r e normal ized f o r comparison by d i v i d i n q
by t h e t o t a l v a r i a n c e , a 2 . Also, s i n c e t h e t o t a l v a r i - U i
ance i s e x p r e s s e d
= [ n / ~ S ( n ) d ( l n nz/B) ui
w e can p l o t t h e p r o d u c t o f wave number and normal ized s p e c t r a l
d e n s i t y p e r u n i t wave number,
- b U Sui ( n ) n Sui ( n ) - -
2 r
u a u i a u i 2 v e r s u s I n nz/D
and a r e a under t h e c u r v e between two wave numbers r e p r e s e n t s
t h e f r a c t i o n of t h e t u r b u l e n t ene rqy c o n t r i b u t e d by t h a t
r ange o f wave numbers, t h e t o t a l a r e a b e i n q u n i t y .
The shape o f t h e s p e c t r a must depend on t h e manner i n
which enerqy i s i n t r o d u c e d i n t o and removed from t h e t u r b u l e n c e .
The e q u a t i o n f o r t h e t u r b u l e n t ene rgy budget f o r s t e a d y s t a t e
and h o r i z o n t a l homogeneity
d e s c r i b e s t h e c o n t r i b u t i o n and d i s p o s t i o n of t u r b u l e n t
energy by v a r i o u s mechanisms. Here w e have assumed a mean
wind i n t h e x l d i r e c t i o n , and s t a t i o n a r y and homoqeneous
t u r b u l e n c e i n t h e x l - x2 plane .
The f i r s t term on t h e r i q h t of Equat ion ( 6 . 2 ) i s
t h e r a t e a t which energy i s added t o t h e t u r b u l e n c e i n t h e
x l d i r e c t i o n t aken from t h e mean motion th rouqh t h e workinq -
of t h e Reynolds ' stress, - p o u l u 3 , a q a i n s t t h e mean
an v e l o c i t y q r a i d e n t , - . a x 3
The second term d e s c r i b e s t h e r a t e t u r b u l e n t energy i s
i n t r o d u c e d i n t o o r removed from t h e x 3 d i r e c t i o n by t h e
buoyancy f o r c e , r e s u l t i n g from d e n s i t y f l u c t i a t i o n s , workinq
i n t h e g r a v i t a t i o n a l f i e l d . The s i q n of t h e h e a t f l u x , -
cppo u 3 0 , de t e rmines whether t u r b u l e n t energy i s damped or
amp l i f i ed . The t h i r d term i s t h e r a t e t u r b u l e n t energy is
d i s s i p a t e d i n t o h e a t . The f o u r t h term d e s c r i b e s t h e t r a n s -
p o r t of t u r b u l e n t energy ( a s opposed t o p roduc t i on o r removal
a s i n f i r s t t h r e e terms) r e s u l t i n g from t h e d ive rgence of t h e -
f l u x of t u r b u l e n t energy, u 3 e , where e = 1/2 (u i ) (ui).
The l a s t t e r m i s t h e summation o f a l l t h o s e terms which
account f o r t h e t r a n s f e r of t u r b u l e n t energy among components
due t o t h e c o r r e l a t i o n between t h e f l u c t u a t i n q p r e s s u r e f o r c e
and f l u c t u a t i o n s i n v e l o c i t y .
The t u r b u l e n t energy budqet e q u a t i o n i n d i c a t e s t h a t
energy is s u p p l i e d t o t h e t u r b u l e n c e from e x t e r n a l i n f l u e n c e s
only throuqh the mechanical energy term and the convective
energy term. Then, Monin and Oboukhov's similarity theory,
assuming the turbulent reqime is completely determined by
the momentum flux, the heat flux and the buoyancy parameter,
g/To, predicts a dependence for the wind component standard
deviations, aU, a and a on the friction velocity and a v' W
universal function of dimensionless heiqht, z/L o on in*^).
Similarly Panof sky and ~ c ~ o r r n i c k ~ ~ followinq the similarity
theory argument have tested a relationship for the standard
deviation of vertical velocity, proportional to the fric-
tion velocity and a universal function of the dimensionless
height z/L, i.e.
where B is a universal constant, L is Monin's length,
is the friction velocity and H = cp p B " is the turbulent heat flux. Since z/L is uniquely related to the Richardson
number, we can look for the standard deviations of the
velocity component fluctuations to be proportional to
friction velocity and a function of the Richardson number.
In neutral conditions the standard deviations should simply
be proportional to friction velocity. Similarity theory
a i i likewise predicts the dimensionless wind shear, j3 = / 6 , the ratio of wind shear in diabatic conditions to that in
n e u t r a l c o n d i t i o n s , t o be a u n i v e r s a l f u n c t i o n of z/L o r
Richardson ' s number.
The r a n g e s of wave numbers o v e r which t h e t e r m s of
Equa t ion (6.2) a r e e f f e c t i v e w i l l , i n p a r t , de te rmine t h e
s h a p e s o f t h e s p e c t r a a s w e l l a s does t h e manner i n which
t h e energy i s t r a n s f e r r e d ( p r i m a r i l y due t o v o r t e x s t r e t c h -
i n g ) from low wave numbers where it is g e n e r a l l y i n t r o d u c e d ,
t o h i g h wave numbers where it is removed by d i s s i p a t i o n . - D i s s i p a t i o n a c t s p r i n c i p a l l y a t h i q h wave numbers where
- v e l o c i t y g r a d i e n t s a r e v e r y l a r q e thoush f l u c t u a t i o n mag-
n i t u d e s a r e v e r y smal l . I t i s expec ted t h i s w i l l o c c u r
beyond t h e ranqe of wave numbers d e t e c t a b l e w i t h i n s t r u m e n t a -
t i o n used i n t h e p r e s e n t e x p e r i m e n t a l i n v e s t i q a t i o n . Lumley
and panofsky20 c i t e measurements between 1 and 100 meters
which i n d i c a t e t h e Kolmoqoroff m i c r o s c a l e , t h e wave l e n g t h
where v i s c o s i t y becomes dominant , i s of t h e o r d e r o f 1 rnm.
I t i s e x p e c t e d , f u r t h e r , t h a t n e a r t h e qround s u r f a c e
i n t h e range o f wave numbers where energy i s i n t r o d u c e d i n t o
t h e s p e c t r a , t h e r a t e of p r o d u c t i o n of mechanica l t u r b u l e n t
ene rgy outweighs t h e p r o d u c t i o n r a t e o f c o n v e c t i v e t u r b u l e n t
energy. T h i s o c c u r s because t h e q r a d i e n t o f v e l o c i t y is
l a r g e n e a r t h e s u r f a c e and d e c r e a s e s upward w h i l e t h e
Reynolds ' stress remains c o n s t a n t s o t h e mechanica l ene rgy
t e r m d e c r e a s e s upward from a l a r g e v a l u e n e a r t h e s u r f a c e .
Thus, i n t h e t u r b u l e n t l a y e r c l o s e s t t o t h e s u r f a c e t h e
mechanical e n e t g y o r s h e a r p r o d u c t i o n t e r m s h o u l d dominate
t h e s t r u c t u r e o f t h e t u r b u l e n c e i n t h e e n e r g y p r o d u c t i o n
r a n g e o f eddy s i z e s , even i n somewhat d i a b a t i c c o n d i t i o n s ,
w h i l e t h e buoyancy t e r m s h o u l d have a r e l a t i v e l y impercep t -
i b l e e f f e c t . The h e a t f l u x , and c o n s e q u e n t l y t h e buoyancy
t e r m , r ema ins c o n s t a n t w i t h h e i q h t . T h e r e f o r e , a t q r e a t e r
h e i g h t s where v e l o c i t y g r a d i e n t s become s m a l l , t h e buoyancy
t e r m s h o u l d c o n t r i b u t e most s i g n i f i c a n t l y t o t h e t u r b u l e n t
ene rgy . T h i s c o n t r i b u t i o n may o c c u r r e l a t i v e l y i n d e p e n d e n t
o f t h e m e c h a n i c a l e n e r g y c o n t r i b u t i o n and a t lower wave
numbers. A l s o , Lumley and vanofsky20 s u q q e s t t h e s c a l i n g
o f t h e c o n v e c t i v e e d d i e s by t h e h e i g h t and t h e wind s p e e d
s h o u l d n o t h o l d b u t s h o u l d b e d e t e r m i n e d by t h e t h i c k n e s s
o f t h e u n s t a b l e l a y e r and t h e c h a r a c t e r o f t h e s u r f a c e
i n h o m o g e n e i t i e s .
I n u n s t a b l e c o n d i t i o n s , t h e v e r t i c a l t u r b u l e n t t r a n s -
f e r o f h e a t i s o f t e n c o n s i d e r e d o c c u r r i n q a s a r e s u l t e i t h e r
o f f o r c e d c o n v e c t i o n o r o f f r e e c o n v e c t i o n . ~ r i e s t l e ~ ~ ~
describes f o r c e d c o n v e c t i o n a s t h e c o n d i t i o n when t h e
buoyancy does n o t s i g n i f i c a n t l y a f f e c t t h e mot ion o r t h e
h e a t t r a n s f e r c o e f f i c i e n t b u t t h e m e c h a n i c a l l y g e n e r a t e d
t u r b u l e n c e i s r e s p o n s i b l e f o r t h e v e r t i c a l f l u x o f h e a t as
w e l l as momentum. Through t h e s i m i l a r i t y t h e o r y t h e wind
p r o f i l e is shown t o b e a l o g a r i t h m i c p l u s a l i n e a r f u n c t i o n
o f h e i g h t and t h e t e m p e r a t u r e q r a d i e n t i s s i m i l a r l y log-
l i n e a r . Near t h e q round , however, t h e p r o f i l e s o f t e n c a n n o t
b e e x p e r i m e n t a l l y d i s t i n g u i s h e d from t h e s i m p l e l o q a r i t h m i c
n e u t r a l p r o f i l e s . F ree convec t ion occu r s when t h e a i r
motion r e s p o n s i b l e f o r v e r t i c a l f l u x e s r e s u l t f r o m huoyancv,
i . e . , t h e low f requency buoyant e d d i e s , h i g h l y e f f i c i e n t i n
t r a n s f e r r i n g h e a t , are p r i m a r i l y r e s p o n s i b l e f o r t h e h e a t
f l u x and t h e momentum t r a n s f e r a s w e l l . The re fo r e , a r eq ion
o f f o r c e d convec t ion w i l l o f t e n e x i s t i n t h e atmosphere w i t h l a r eg ion of f r e e convec t ion above. The sha rpnes s of t h e
t r a n s i t i o n from fo r ced t o f r e e convec t ion i s open t o q u e s t i o n . b u t c h a r a c t e r i s t i c s of each shou ld be observed i n t h e s p e c t r a .
- Also , when t h e buoyancy term i s s i q n i f i c a n t t h e
a - t r a n s p o r t term - u 3 e may be o f importance. A f e w mea- ax3
30 surements i n u n s t a b l e c o n d i t i o n s , d i s cus sed by Panofsky
have i n d i c a t e d t h a t i n u n s t a b l e c o n d i t i o n s t h e u ~ w a r d f l u x
of t u r b u l e n t energy i n c r e a s e s w i t h h e i g h t s o t h a t t h e f l u x
d ive rgence t e n d s t o compensate f o r t h e a d d i t i o n o f convec-
t i v e energy from t h e working o f t h e d e n s i t y f l u c t u a t i o n s .
T h i s t e r m i s s een t o be e f f e c t i v e o n l y i n t r a n s p o r t i n g enerqy I
from one p l a c e t o ano the r s i n c e it i s a d ive rgence and i t s
i n t e g r a l o v e r a l a r g e enough volume can be r e s t a t e d th rouqh
t h e d ive rgence theorem a s a s u r f a c e i n t e g r a l th rough which
no t r a n s p o r t occu r s . Thus t h e r e is no n e t l o s s o r q a i n o f
ene rgy from t h e flux divergence term.
L i t t l e i s known of t h e ranges of wave numbers o v e r
which t h e t e r m d e s c r i b i n g t h e work done by t h e f l u c t u a t i n g
p r e s s u r e g r a d i e n t is e f f e c t i v e . Express ing t h i s term:
and i n t e q r a t i n q o v e r a volume l a r q e enouqh t o i n c l u d e a l l
eddy s i z e s ,
For t h e f i r s t t e r m on t h e r i q h t w e have t h e i n t e q r a l of a
d i v e r q e n c e r e - e x p r e s s i b l e throuqh t h e d i v e r q e n c e theorem a s
a s u r f a c e i n t e q r a l of a f l u x . Thus t h e f i r s t term i s z e r o
c o n s i d e r i n q h o r i z o n t a l l y homoqeneous c o n d i t i o n s and no -
f l u x of t h e q u a n t i t y , u i p , th rouqh t h e upper and lower s u r -
f a c e s o f t h e boundary l a y e r . The second term i s a l s o z e r o
from t h e e q u a t i o n of c o n t i n u i t y s o t h e r e i s no n e t work done
by t h e f l u c t u a t i n q p r e s s u r e q r a d i e n t b u t ene rqv can be
t r a n s f e r r e d from one r e q i o n w i t h i n t h e volume t o a n o t h e r .
Also , w e can now expand t h i s term, g i v i n q
Thus t h e t o t a l power a s s o c i a t e d w i t h t h e work done by t h e
f l u c t u a t i n g l o n g i t u d i n a l wind component a g a i n s t t h e f l u c t u a t -
i n q p r e s s u r e g r a d i e n t i n t h e same d i r e c t i o n e q u a l s t h e work
done on t h e a i r moving w i t h t h e Other components by t h e i r
r e s p e c t i v e p r e s s u r e q r a d i e n t s . I n t h i s way, t u r b u l e n t enerqy
i s t r a n s f e r r e d o u t of t h e x l component i n t o t h e x2 and x3
components. T h i s t r a n s f e r might appear i n one component ove r
a g iven ranqe of wave numbers wh i l e n o t i n a n o t h e r . I n
view of t h e l i m i t a t i o n s p l aced on bo th t h e s i z e o f t h e
v e r t i c a l e d d i e s and t h e s c a l e of t h e v e r t i c a l p r e s s u r e
g r a d i e n t f l u c t u a t i o n s by t h e p rox imi tv of t h e s u r f a c e , it
might be a n t i c i p a t e d t h a t t h e t u r b u l e n t enerqv g e n e r a t e d i n
t h e x l d i r e c t i o n th rouqh t h e Reynolds' stress t e r m would be
t r a n s f e r r e d t o t h e v e r t i c a l component over a r e l a t i v e l y h iqh
wave number ranqe. The enerqv t r a n s f e r r e t ! i n t o t h e x2
d i r e c t i o n , however, i s n o t s u b j e c t t o such c o n s t r a i n t s and
might occu r o v e r any p a r t of t h e c o s p e c t r a l wave number
range.
The a c t i o n of p r e s s u r e f o r c e s , t r a n s f e r r i n g t u r b u l e n t
energy amonq components, l e a d s t o e q u i p a r t i t i o n of enerqy
a t h i g h e r wave number, and a s t h e c o s p e c t r a f o r each p a i r
of components d e c r e a s e s a t i n c r e a s i n g wave numbers, e v e n t u a l l y
t h e e d d i e s approach i s o t r o p y o v e r t h e upper end of t h e power
spectrum.
The t u r b u l e n t f l u c t u a t i o n s a s s o c i a t e d w i t h t h e s e eddy
s i z e s were d e f i n e d a s l o c a l l y i s o t r o p i c by A. ~ o l m o ~ o r o f f ~ ~ .
H i s d e f i n i t i o n r e q u i r e d s t e a d i n e s s of t h e t u r b u l e n c e i n
t i m e b u t d e a l t o n l y w i t h d i f f e r e n c e s i n v e l o c i t y , from one
p o i n t t o a n o t h e r , t h e p r o b a b i l i t y d i s t r i b u t i o n f o r t h e
v e l o c i t y d i f f e r e n c e s between a l l p o i n t s w i t h i n a g iven space
and time domain beinq invariant with translation, rotation
or reflection of the coordinate axes, allowinq a description
of isotropic eddies existing in a field of turbulence alonq
with anisotropic eddies. Kolmoqoroff then hypothesized that
the probability distribution of the velocity differences or
the average properties of the turbulence for locally
isotropic turbulence was uniquely determined by the rate of
dissipation of energy, E, and the kinematic viscosity, V .
He considered that below the isotropic ranqe energy is fed
from the larqe anistropic eddies, is passed from lower to
hiqher wave numbers by inertial forces and is eventually
dissipated by viscous forces at the same rate it is fed
into the isotropic range, mafntaining an equilibrium state.
Kolmogoroff hypothesized further that the equilibrium ranqe
of eddy sizes throuqh which the energy is passed might have
at its lower wave number end, a ranqe of eddy sizes where
viscous forces are not effective so no dissipation takes
place. Thus, the averaqe properties of the turbulence in
this "inertial subrange" are determined only by E , the
rate at which enerqy is inertially fed through the subranqe
and eventually dissipated beyond it. Then by dimensional
reasoning the variance of the velocity difference or the
structure function, which is a function only of E and the
separation, x, must be proportional to (x) '". Then taking
the Fourier transform,
which i s o f t e n c a l l e d Kolmogoroff ' s minus f i v e - t h i r d s law
f o r t h e t h r e e - d i m e n s i o n a l s p e c t r a l d e n s i t v i n t h e i n e r t i a l
subrange . W e c a n a l s o w r i t e
n S ( n ) = K S ( K ) =
The one-dimensional spec t rum h a s t h e form
where " a " i s a c o n s t a n t which, i n d i c a t e d by t h e m o s t r e c e n t
d e t e r m i n a t i o n s , i s between 0.45 and 0.50 i f K i s q i v e n i n
r a d i a n s p e r u n i t l e n q t h . Lumley and ~ a n o f s k ~ ~ ~ s u q q e s t u s i n q
a v a l u e o f 0.47 f o r t h e l o n g i t u d i n a l v e l o c i t y s p e c t r a l r e l a -
t i o n s h i p . Hencefo r th , wave number w i l l be e x p r e s s e d i n I
c y c l e s p e r meter f o r which t h e s u q q e s t e d v a l u e o f t h e con-
s t a n t , a , i s 0.138.
The p r e c e d i n g e x p r e s s i o n i s f o r t h e d i s t r i b u t i o n o f
t h e v a r i a n c e i n u l among wave numbers, where t h e sub-
s c r i p t , 1 , refers t o t h e a x e s d i r e c t e d a l o n g t h e mean f low.
The d i s t r i b u t i o n o f t h e v a r i a n c e i n ? . o r i n u3among wave
numbers K b e shown ( ~ o l m o ~ o r o f f ' ' ) t o b e i d e n t i c a l i n
form b u t m u l t i p l i e d by 4/3; t h a t i s , t h e c o n s t a n t a p p e a r i n g
i n t h e e x p r e s s i o n i s 4/3a. The l i n e o f r e a s o n i n q on which
t h e proof o f t h i s r e l a t i o n s h i p depends i s t h a t t h e t u r b u l e n c e
be i s o t r o p i c a t t h e wave numbers concerned and t h a t t h e
e q u a t i o n of c o n t i n u i t y f o r an i n c o m p r e s s i b l e f l u i d he v a l i d .
I n t h e remainder of t h i s p r e s e n t a t i o n o n l y t h e s p e c t r a l
d i s t r i b u t i o n s i n t h e one-dimensional wave number s p a c e , K ~ ,
w i l l be c o n s i d e r e d . Hencefor th , K~ w i l l s implv be i n d i c a t e d
by K o r by i t s e s t i m a t e , n/U. Fur thermore , a s has a l r e a d y
been done on a few o c c a s i o n s , t h e n o t a t i o n f o r t h e d i r e c t i o n s
x x and x w i l l be r e p l a c e d w i t h x , y , and z r e s p e c t i v e l v 1 ' 2 ' 3
and t h e cor respond inq wind component f l u c t u a t i o n s u l , u 2 , and
u3 by u ' , v ' and w ' . A l so , u * ~ w i l l o f t e n be used f o r t h e
Reynolds ' t e r m , - u l u 3 .
Then a f u r t h e r e x p e c t a t i o n would be t h a t a r e q i o n of
t h e measured s p e c t r a , above t h e energy addinq ranqe a t low
wave numbers, w i l l f o l l o w t h e r e l a t i o n s h i p f o r t h e i n e r t i a l
subrange. The l e v e l of t u r b u l e n c e i n t h e i n e r t i a l subranqe
i s set by t h e r a t e o f d i s s i p a t i o n which, f o r n e u t r a l condi-
t i o n s , can b e de termined from t h e mechanica l ene rgy produc-
t i o n t e r m . T h i s can b e s e e n from Equa t ion (6.2) where no
o t h e r means o f p r o d u c t i o n o r removal of t u r b u l e n t ene rgy
e x c e p t from t h e f i r s t and t h i r d terms i s s i q n i f i c a n t f o r
t h e s e c o n d i t i o n s . Then it f o l l o w s t h a t Equa t ion (6.2)
becomes
i n view of t h e l o q a r i t h m i c p r o f i l e .
When t h i s r e l a t i o n s h i p h o l d s , w e can see t h a t t h e
i n e r t i a l subranqe is s c a l e d accord ing t o h e i q h t s i n c e Equat ion
(6.3) becomes
and i f t h e spect rum i s normal ized t o t h e s q u a r e of t h e f r i c -
t i o n v e l o c i t y , a u n i v e r s a l f u n c t i o n shou ld be e x p e c t e d f o r
t h e i n e r t i a l subrange i n n e u t r a l c o n d i t i o n s :
where, o f c o u r s e , a ko2l3 i s a u n i v e r s a l c o n s t a n t . For t h e
t r a n s v e r s e component s p e c t r a t h e a d d i t i o n a l f a c t o r o f 4/3 i s
r e q u i r e d . Fur thermore , i f a n o t h e r u n i v e r s a l c o n s t a n t r e l a t e s
t h e t o t a l v a r i a n c e t o t h e f r i c t i o n v e l o c i t y , t h e n w e can
w r i t e
Again f o r t h e t r a n s v e r s e s p e c t r a , t h e 4/3 f a c t o r i s invo lved
and t h e a p p r o p r i a t e c o n s t a n t r e l a t i n q v a r i a n c e t o f r i c t i o n
v e l o c i t y must be used.
~ e r r n a n ~ a l t e r s t h e r e l a t i o n s h i p (6.3a). t o accoun t f o r
d i a b a t i c c o n d i t i o n s g i v i n g
Here, t h e d i s s i p a t i o n i s a q a i n set e q u a l t o t h e mechanica l
e n e r g y p r o d u c t i o n , assuminq t h e buoyancy term i n t h e
t u r b u l e n t ene rgy budqet e q u a t i o n i s ba lanced by t h e f l u x
d i v e r g e n c e t e r m . However, it i s n e c e s s a r v t o account f o r
t h e d e v i a t i o n from t h e l o q a r i t h m i c p r o f i l e f o r d i a h a t i c
c o n d i t i o n s i n t h e mechanical ene rqy p r o d u c t i o n term. T h i s
i s accompl ished by i n c l u d i n q t h e f a c t o r 3 (Ri) , t h e r a t i o of
t h e t r u e wind speed s h e a r t o t h e s h e a r f o r a l o q a r i t h m i c
p r o f i l e , a f u n c t i o n of t h e s t a b i l i t y a s i n d i c a t e d by
Richardson ' s number, R i . C lose t o t h e ground even i n
d i a b a t i c c o n d i t i o n s , s i n c e t h e mechanica l ene rqy p r o d u c t i o n
u s u a l l y predominates , t h e lower p o r t i o n o f t h e wind p r o f i l e
i s q u i t e c l o s e t o l o g a r i t h m i c and Equa t ions (6 .3a ) and (6.3b)
can be used i n t h e f o r c e d convec t ion r e g i o n . Only a s m a l l
e r r o r r e s u l t s , on t h e o r d e r o f t h a t i n t r o d u c e d i n t h e
measurement and a n a l y s i s o f t h e t u r b u l e n c e d a t a .
The o b s e r v a t i o n s made w i t h t h e wind component meter
have been s u b j e c t e d t o a n a l y s i s i n t h e framework of t h e
above d i s c u s s i o n . I n e v a l u a t i n q t h e r e s u l t s it is w e l l t o
b e a r i n mind t h a t some d e t a i l s of t h e t h e o r y a r e n o t
f i r m l y e s t a b l i s h e d . I n p a r t i c u l a r , any t h e o r e t i c a l develop-
ment dependinq on i s o t r o p y i s open t o q u e s t i o n .
VII. RESULTS OF ANALYSIS
L o n a i t u d i n a l Com~onent S ~ e c t r a
The s p e c t r a f o r t h e l o n g i t u d i n a l component were p l o t t e d
i n t h e manner sugges ted by t h e d i s c u s s i o n o f t h e p r ev ious sec-
t i o n , w i t h t h e normal ized s p e c t r a l d e n s i t y , nSu(n) / u u 2 on t h e
o r d i n a t e and nz/n, t h e r a t i o o f h e i g h t t o wave l e n q t h , p l o t t e d
l o g a r i t h m i c a l l y on t h e a b s c i s s a . The tests w e r e gfbuped
acco rd ing t o s t a b i l i t y , F iqu re 7.1 showing t h e s p e c t r a f o r
n e u t r a l c o n d i t i o n s , F iqu re 7.2 f o r u n s t a b l e c o n d i t i o n s and
F igu re 7.3 f o r t h e s t a b l e c a s e s . I n a d d i t i o n , t h e c a s e s
above 12 meters, where buoyancy e f f e c t s a r e expec t ed t o be
most i n ev idence , w e r e p l o t t e d s e p a r a t e l y i n F i g u r e 7 - 4 .
The s i m i l a r i t i e s o f t h e s p e c t r a , p l o t t e d i n t h i s
manner, a r e t o be noted. D i f f e r e n c e s a r e observed between
s t a b i l i t y g roups , however, For a l l s t a b i l i t i e s , peaks i n
t h e s p e c t r a reached maximum normal ized v a l u e s n e a r 0.12 and
were found a t an nz/U v a l u e o f abou t 0.01 to 0 -04 , For
s t a b l e c a s e s an a d d i t i o n a l peak was found n e a r nt/U = 0-1 .
The s t a b l e d a t a were s h i f t e d t o s l i g h t l y h i g h e r wave numbers
t h a n t h e d a t a f o r n t m t r a l and u n s t a b l e c o n d i t i o n s .
One test , 9-1, i n s t a b l e c o n d i t i o n s a t 12.2 meters
conducted above t h e r e g i o n dominated by t h e mechanical
energy p roduc t i on , demons t ra tes a marked d e v i a t i o n &ram t h e
form e x h i b i t e d by the o t h e r d a t a . The s p e c t r a f o r b o t h t h e
h o r i z o n t a l and v e r t i c a l component# f o r t h i s s i n g l e test were
s h i f t e d t o much h i g h e r nz/fl. T h i s i s t o be expec ted s i n c e
a t q r e a t e r h e i g h t s where buoyancy e f f e c t s dominate , t h e
h e i q h t above t h e s u r f a c e shou ld no l o n q e r de te rmine t h e
s c a l e of t h e t u r b u l e n c e f o r t h e s t a b l e c a s e b u t some s t a h i l -
i t y l e n g t h , s m a l l compared w i t h z such a s Monin and Oboukhov's
p a r a m e t e r , L , would c o n t r o l t h e s i z e of t h e e d d i e s .
The enerqy addinq r e q i o n o f t h e l o n g i t u d i n a l wind
component s p e c t r a c o r r e s p o n d s q u i t e w e l l w i t h t h e r e g i o n o f -
maxima f o r t h e u'w' c o s p e c t r a , r e s p o n s i b l e f o r t h e f e e d i n g
o f ene rgy i n t o t h e l o n g i t u d i n a l t u r b u l e n c e from t h e mean
motion. T h i s can be s e e n by comparing t h e s p e c t r a l d i s t r i b u -
t i o n s of F i g u r e s 7 .1 , 7 .2 , 7.3 w i t h t h e c o s p e c t r a l d i s t r i b u -
t i o n s o f F i q u r e s 7.25, 7.26, 7.27, and 7.28.
The s p e c t r a f o r t h e l o n g i t u d i n a l component a r e a l s o
shown w i t h t h e l o q a r i t h m of n ~ ~ ( n ) / o ~ ~ p l o t t e d v e r s u s t h e
l o g a r i t h m of nz/U i n F i g u r e s 7.5 th rouqh 7.8. The u n i v e r s a l
form o f Equa t ion (6.3b) f o r t h e i n e r t i a l subranqe i s a l s o
shown i n F i g u r e s 7.5 and 7.6 and r e a s o n a b l y good v e r i f i c a -
t i o n of t h e i n e r t i a l subrange i s observed f o r n e u t r a l condi-
t i o n s , Equa t ion (6.3b) a p p e a r i n g a b o u t 20% l o w when the v a l u e
a = 0.138 is used. Better agreement is s e e n w i t h t h e
u n s t a b l e d a t a . For b o t h c a s e s o n l y au2/ u * ~ r a t i o s f o r
tests shown i n each f i g u r e were averaged t o o b t a i n t h e
c u r v e s r e p r e s e n t i n g Equat ion (6.3b) . I n F i g u r e 7.8 t h e
obse rved cor respondence o f t h e u n s t a b l e c a s e s above t h e
f o r c e d convec t ion r e g i o n w i t h t h o s e w i t h i n , th rouqhou t t h e
i n e r t i a l subranqe , cou ld n o t have been a n t i c i p a t e d from Equa-
t i o n (6.3b) . I t i s p a r t i c u l a r l y s u r p r i s i n q t o f i n d reason-
a b l e co r respondence o v e r most of t h e spect rum f o r t h e t es t
a t 87 meters where e s s e n t i a l l y no c o r r e l a t i o n e x i s t s between
t h e v e r t i c a l and l o n g i t u d i n a l components and where t h e momentum
f l u x is a p p a r e n t l y accomplished through t h e l a r g e v 'w ' Reynolds
stress.
F o r s t a b l e l o n g i t u d i n a l component s p e c t r a t h e l o w e r
end o f t h e i n e r t i a l subranqe would a p p e a r t o b e a t a b o u t
nz/U = 1.0 w h i l e f o r n e u t r a l and u n s t a b l e c o n d i t i o n s t h e
minus f i v e - t h i r d s law seems t o e x t e n d t o c o n s i d e r a b l y lower
wave numbers, a b o u t 0.2 t o 0.4. However, t h e c o s p e c t r a f o r
a l l p a i r s o f components, shown i n F i q u r e s 7.25 th rouqh 7.37,
show t h e r e q u i r e m e n t s f o r i s o t r o p i c c o n d i t i o n s ( t h a t t h e
c o r r e l a t i o n between a l l p a i r s o f components must be z e r o )
g e n e r a l l y h o l d o n l y a t g r e a t e r wave numbers t h a n nz /u = 1 . 0
t o 3.0.
The o u t s t a n d i n g example o f t h i s i s t h e s table tes t a t
12.2 meters where t h e power spec t rum i s g e n e r a l l y d i s p l a c e d
t o h i g h e r wave numbers b u t t h e minus f i v e - t h i r d s l a w e x t e n d s
t o t h e peak a t nz /n = 0.2, whereas t h e c o s p e c t r a c o n f i r m
i s o t r o p i c c o n d i t i o n s o n l y above 4.0.
V e r t i c a l Component S p e c t r a
The s p e c t r a f o r t h e v e r t i c a l component a r e shown i n
F i g u r e s 7.9 th rouqh 7.16 where t h e normal ized s p e c t r a l
density, nSw(n)/ow2, is plotted versus the logarithm of
nz/U in Figures 7.9 throuqh 7.12 and the loqarithm of each
is plotted in Figures 7.13 throuqh 7.16. The universal
character of the turbulence for this component can be seen
with only a few exceptions, aqain for measurements above the
forced convection reqion shown in Fiqures 7.12 and 7.16.
The maxima for all vertical component spectra occur at a
considerably hiqher value of na/u than for the lonqitudinal
component, a possibility considered earlier, appearing qen-
erally at about na/g = 0.4 with the peak normalized spectral
estimates averaging about 0.2, These values and the same
spectral shape hold within the forced convection reqion for
neutral and unstable conditions with a suqqestion of a slight
shift to hiqher wave numbers for stable cases, below 12
meters. The lower limit for the minus five-thirds law
occurs at about nz/V = 1.2 for the stable cases; for neutral
and unstable conditions it extends to about 0.8 or 1.0.
As mentioned previously, the lack of correlation between
components above nz/u = 1.0 to 3.0 for neutral and unstable
conditions confirms the isotropic condition necessary for
the inertial subrange at slightly hiqher wave numbers than
those over which the minus five-thirds law was observed in
the logarithmic spectral plots.
Again, Equation (6,3b) is shown for neutral and
unstable conditions in Figures 7.13 and 7.14 where only cases
shown in each figure were used to determine the average values
of t h e r a t i o u ~ / u * ~ . Reasonably qood agreement i s a g a i n W
o b t a i n e d u s i n g t h e v a l u e a = 0.138 and i n t h i s c a s e t h e
a d d i t i o n a l f a c t o r of 4/3.
S p e c t r a l peaks i n u n s t a b l e c o n d i t i o n s w i t h i n t h e
f o r c e d c o n v e c t i o n r e g i o n r e s u l t i n q from t h e d i r e c t i n t r o -
d u c t i o n o f c o n v e c t i v e t u r b u l e n t ene rgy i n t o t h e v e r t i c a l
component a r e n o t a b l y a b s e n t . Only f o r test 11-2, a t 6 . 1
meters, w e r e t h e r e a few l a r g e s p e c t r a l e s t i m a t e s a t l o w
wave numbers s u g g e s t i n g c o n v e c t i v e enerqy. T h i s is con-
t r a s t e d i n F i g u r e 7.16 w i t h t h e d e f i n i t e c o n v e c t i v e peak
f o r t e s t 12-2, an u n s t a b l e case a t 12.2 m e t e r s , n o t e d i n t h e
nz/u range below 0.06. The u n s t a b l e t es t a t 87 meters l i k e -
w i s e i n d i c a t e s c o n v e c t i v e enerqy w i t h t h e lowest t w o s p e c t r a l
estimates a t abou t no/n * 0.15. T e s t 12-1, an u n s t a b l e c a s e
f o r t h e 12.2 meter h e i g h t a lso shows a peak, s imi lar t o t h e
o t h e r c o n v e c t i v e peaks , a t a lower wave number t h a n t h a t
i n t r o d u c e d by t h e mechanica l t u r b u l e n c e , i n t h i s case a t
nz/v = 0.13. T h i s test , though conducted under c o v e r o f a
heavy o v e r c a s t , was c a r r i e d o u t on a summer day w i t h a v e r y
l i g h t wind. The r e s u l t i n g t o t a l v e r t i c a l v e l o c i t y v a r i a n c e
w a s so small t h a t even t h e s l i q h t c o n v e c t i v e energy added,
co r respond ing t o a v e r y small t e m p e r a t u r e d i f f e r e n c e , w a s
n o t i c e a b l e when compared w i t h t h e v e r y s m a l l amount o f
mechanica l e n e r g y , as s e e n i n t h e spectrum. A s e x p e c t e d , t h e
c o n v e c t i v e peaks do n o t a p p e a r t o scale a c c o r d i n g t o h e i g h t .
I n terms o f f r equency , t h e 12.2 meter u n s t a b l e case h a s
c o n v e c t i v e enerqy th rough t h e lower end of t h e sampled swec-
trum, below 2.3 cyc les /minu te . The 87 meter u n s t a b l e c a s e
h a s c o n v e c t i v e energy a p p a r e n t below 0 . 9 cyc les /minu te and
t h e l i g h t wind speed c a s e a t 12.2 meters shows c o n v e c t i v e
energy below 1 . 4 cyc les /minu te . Again, t h e correspondence of
t h e s p e c t r a f o r u n s t a b l e c a s e s above w i t h t h o s e w i t h i n t h e
f o r c e d c o n v e c t i o n r e q i o n , th roughou t t h e i n e r t i a l subrange ,
i n s p i t e o f t h e i n a p p l i c a b i l i t y of Equat ion (6.3b) s h o u l d b e
mentioned. Also t o b e noted i s t h e o c c u r r e n c e o f a "mechani-
c a l ene rgy p r o d u c t i o n peak" f o r a l l t h e s e c a s e s a t a b o u t t h e
same nz/u v a l u e a s obse rved w i t h i n t h e f o r c e d convec t ion
r e g i o n ,
The v e r t i c a l component spect rum f o r t h e test i n s t a b l e
c o n d i t i o n s a t 12.2 meters was found, a s w i t h t h e l o n g i t u d i n a l
component, t o be d i s p l a c e d t o h i g h e r wave numbers a s shown
i n F i q u r e 7.16. The i n e r t i a l subrange i s s e e n from s p e c t r a l
and c o s p e c t r a l d i s t r i b u t i o n s t o beg in a t a b o u t nz/u = 4.0,
L a t e r a l Component S p e c t r a
The l a t e r a l component s p e c t r a a r e n o t a s w e l l organ-
i z e d a s t h e v e r t i c a l and l o n g i t u d i n a l s p e c t r a , A l a r g e
f r a c t i o n o f t h e t o t a l v a r i a n c e a p p e a r s a t l o w wave numbers
i n a d i s o r q a n i z e d manner a s a f u n c t i o n of nz /g and w i t h
wide ly v a r y i n q magni tudes , f o r n e u t r a l and u n s t a b l e condi-
t i o n s . However, a d e g r e e o f s i m i l a r i t y i s d e t e c t e d i n t h e
l a t e r a l s p e c t r a shown i n F i g u r e s 7.17 th rough 7.24. A l l
s p e c t r a g e n e r a l l y reduce i n ampl i tude w i t h i n c r e a s i n g wave
number above a b o u t n z / u = 0.1.
F o r n e u t r a l c o n d i t i o n s , a s s e e n i n F i q u r e 7 .17 , mul-
t i p l e p e a k s a t wave numbers below nz /g = 0 . 1 a r e common.
F o r u n s t a b l e c o n d i t i o n s a r a n q e o f even more c o n s i s t e n t l y
h i g h s p e c t r a l e s t i m a t e s below a b o u t n z / a = 0 . 1 i s o b s e r v e d
i n F i g u r e 7.18. The s t a b l e tests d e m o n s t r a t e t h e m o s t con-
s i s t e n t s i m i l a r i t y , s e e n i n F i q u r e 7 .19, e x h i b i t i n q a marke2
a b s e n c e o f t h e e n e r q y a t l o w wave numbers found i n t h e
o t h e r s t a b i l i t y q r o u p s . The s t a b l e g r o u p is a g a i n s h i f t e d
t o somewhat h i q h e r wave numbers t h a n t h e n e u t r a l and u n s t a b l e
c a s e s and t h e 12 .2 meter s t a b l e tes t i n F i q u r e 7.20 i s d i s -
p l a c e d t o c o n s i d e r a b l y h i g h e r nz /e .
The v ' component s p e c t r a are o b s e r v e d from F i q u r e s
7 . 2 1 t h r o u q h 7.24 i n most cases, n o t t o f o l l o w t h e minus
f i v e - t h i r d s law a t t h e h i g h f r e q u e n c y end o f t h e s p e c t r u m
w i t h i n t h e r a n g e o f measurement . Thouqh t h i s i s n o t a u n i -
v e r s a l c h a r a c t e r i s t i c o f t h e t u r b u l e n c e and c o u l d b e p e c u l i a r
t o t h e p a r t i c u l a r t e r r a i n f e a t u r e s o f t h e Hanford s i t e , a
s i m i l a r w ide v a r i a b i l i t y o f t h e i n e r t i a l s u b r a n g e i n c e p t i o n
m i q h t b e a n t i c i p a t e d a t o t h e r sites. I f t u r b u l e n t e n e r q y
i s b e i n g added w i t h i n a wave number r a n g e from a n e x t e r n a l
s o u r c e as w e l l as c a s c a d e d t o h i q h e r wave numbers b y i n e r t i a l
t r a n s f e r , t h e n e g a t i v e s l o p e mus t b e less t h a n t w o - t h i r d s .
However, i f e n e r g y i s b e i n g removed from t h e t u r b u l e n c e i n
a g i v e n component, t h e n e g a t i v e s l o p e o f t h e l o g a r i t h m i c
p l o t w i t h i n t h e r a n q e o f c o n c e r n mus t b e g r e a t e r t h a n
t w o - t h i r d s . The obse rved d e c r e a s e i n l a t e r a l component spec-
t r a l d e n s i t y w i t h i n c r e a s i n q wave number, g r e a t e r t h a n
e x p e c t e d from i n e r t i a l t r a n s f e r , i n d i c a t e s ene rgy i s be ing
removed from t h e v ' component t u r b u l e n c e i n t h e wave number
r e g i o n where t h e o t h e r components demons t ra te no removal o r
a d d i t i o n . C o s p e c t r a l d a t a s u p p o r t t h i s c o n t e n t i o n a s
d e s c r i b e d i n t h e f o l l o w i n g d i s c u s s i o n .
A v e r y t e n t a t i v e p o s s i b i l i t y f o r e x p l a i n i n g t h e
removal o f ene rgy from t h e v ' component t u r b u l e n c e i s t h e -
f e e d i n g o f ene rgy i n t o t h e mean motion. J u s t a s t h e u 'w'
Reynolds' stress withdraws energy from t h e mean motion and -
s u p p l i e s it t o t h e t u r b u l e n c e i n t h e u ' component, t h e v'w'
Reynolds' stress can work w i t h t h e v e r t i c a l q r a d i e n t of t h e
mean l a t e r a l component, t r a n s f e r r i n g energy from t h e tu rbu-
l e n c e i n t h e v ' component t o t h e mean l a t e r a l wind component.
T h i s r e s u l t s i n t h e maintenance o f a wind d i r e c t i o n s h e a r . - Thus t h e mechanica l e n e r g y t e r m , v'w' dti , shou ld n o t
a
n e c e s s a r i l y be ignored i n t h e energy budget Equat ion ( 6 . 2 ) .
The e x i s t e n c e of c o n s i d e r a b l e o r g a n i z e d c o r r e l a t i o n i n t h e
nCw(n) c o s p e c t r a l p l o t s up t o nz/D = 3.0 , beyond t h e p o i n t
where t h e o t h e r c o s p e c t r a have f a l l e n t o z e r o , s u p p o r t s t h e
c o n t e n t i o n t h a t t h i s Reynolds' stress is removing energy
from t h e v ' component t u r b u l e n c e i n t h e s a n e r e g i o n o f wave
numbers. The mean wind d i r e c t i o n s h e a r s , though t h e measure-
ments were n o t adequa te f o r q u a n t i t a t i v e e v a l u a t i o n , were o f
t h e p r o p e r s i g n r e q u i r e d f o r f e e d i n g t h e mean motion. I n a l l
u n s t a b l e and n e u t r a l c a s e s d u r i n g t h e f i r s t series of tes ts
conducted on two c o n s e c u t i v e d a y s , t h e wind backed 10 o r
15 d e g r e e s between 7 and 400 f e e t . The i n c r e a s e o f t h e
l a t e r a l mean wind component w i t h h e i g h t r e q u i r e s a n e t - n e g a t i v e v 'wt c o v a r i a n c e , demonst ra ted by t w o c a l c u l a t i o n s
where t h e h i g h p a s s f i l t e r was l e f t o u t of t h e Reynolds t
stress program a l l o w i n q t h e c o v a r i a n c e a t low wave numbers
t o be i n c l u d e d . The c a l c u l a t i o n s w i t h t h e f i l t e r i n t h e
program gave a p o s i t i v e c o v a r i a n c e f o r t h e s e t w o c a s e s a s
w e l l a s f o r a l l b u t one of t h e o t h e r tests performed on t h e s e
two days because t h e low f requency c o v a r i a n c e was f i l t e r e d -
o u t . The v 'wt c o s p e c t r a l i k e w i s e show t h e n e q a t i v e c o r r e l a -
t i o n a t l o w wave numbers and a p o s i t i v e c o r r e l a t i o n a t h i q h
wave numbers. Such a d i s t r i b u t i o n n o t o n l y removes energy
from t h e t u r b u l e n c e a t t h e upper end of t h e spect rum b u t
s u p p l i e s it a t l o w wave numbers, c o n t r i b u t i n g t o t h e con-
s i d e r a b l e v a r i a b i l i t y d e t e c t e d i n t h e l a t e r a l component
s p e c t r a .
There a r e e x c e p t i o n s t o t h e b e h a v i o r d i s c u s s e d above.
T e s t s 11-1 and 11-2, i n n e u t r a l and u n s t a b l e c o n d i t i o n s
r e s p e c t i v e l y a t 6 . 1 meters, b o t h f o l l o w t h e minus f i v e -
t h i r d s law beyond nz/g = 3.0 and b o t h have e s s e n t i a l l y no
c o r r e l a t i o n between t h e v ' and w ' components o v e r t h e range
o f f r e q u e n c i e s f i l t e r e d f o r c o s p e c t r a l d e n s i t y . Thus t h e
absence o f e n e r g y wi thdrawal from t h e t u r b u l e n c e t o s u p p o r t
a mean l a t e r a l q r a d i e n t a l l o w s t h e t u r b u l e n c e t o become
i s o t r o p i c a t an e a r l i e r s t a q e .
None of t h e s t a b l e tes ts show any s i q n of an i n e r t i a l
subrange even though t h e c o s p e c t r a l d e n s i t i e s a r e q u i t e s m a l l
and no o r g a n i z e d a r e a under t h e v'w' c o s p e c t r a l c u r v e s n e a r
nz/g = 3.0 i s obvious which might t r a n s f e r ene rgy o u t of t h e
t u r b u l e n c e . However, wind d i r e c t i o n s h e a r s , which were n o t
a d e q u a t e l y measured h e r e , a r e o f t e n v e r y l a r g e i n s t a b l e
s i t u a t i o n s and s m a l l c o s p e c t r a l a r e a s miqht y e t be e f f e c t i v e
i n a l t e r i n g t h e low l e v e l of t u r b u l e n c e p r e s e n t i n t h e s t a b l e
c a s e s .
T e s t s above 12 meters show o n l y a l i m i t e d agreement
w i t h t h e minus f i v e - t h i r d s law. The l i q h t wind u n s t a b l e
c a s e , T e s t 12-1 a g r e e s q u i t e w e l l above nz/g = 1 . 0 and t h e
87 meter c a s e , T e s t 13-1 i s n o t i n c o n s i s t e n t w i t h t h e l a w
above nz/n = 10 b u t t h e o t h e r tests d e v i a t e n o t i c e a b l y
th roughou t t h e range of measurement.
Summarizing t h e s i g n i f i c a n t f a c t o r s o f t h e s p e c t r a
b r i e f l y , t h e normal ized s p e c t r a f o r each component have a
r e l a t i v e l y s i m p l e and r e p e a t a b l e dependence on nz/n. There
are, however, d i f f e r e n c e s from one component t o a n o t h e r
which i s t o s a y t h a t a d i f f e r e n t s i m i l a r i t y r e l a t i o n s h i p i s
obse rved f o r each component. T h i s s i m i l a r i t y i s most marked
f o r t h e v e r t i c a l component and p o o r e s t f o r t h e l a t e r a l com-
ponent . A t low wave numbers t h e a p p a r e n t s i m i l a r i t y might
b e q u e s t i o n e d because o f t h e l a r g e v a r i a b i l i t y o f t h e
e s t i m a t e s . A l l , demons t ra te a s l i g h t s h i f t t o h i g h e r wave
numbers o v e r t h e e n t i r e spect rum f o r s t a b l e c o n d i t i o n s n e a r
t h e ground and a l a r q e s h i f t f o r t h e s t a b l e c a s e a t 12.2
meters. Also c o n v e c t i v e peaks i n t h e v e r t i c a l component
s p e c t r a a r e found i n u n s t a b l e c a s e s above 12 meters a t low
wave numbers w h i l e t h e mechanica l ene rqy peak' and t h e upper
end o f t h e normal ized spectrum cor respond t o t h e normal ized
n e u t r a l spectrum. A t r a n s f e r o f t u r b u l e n t e n e r q y t o t h e mean
f l o w i s i n d i c a t e d i n t h e l a t e r a l component s p e c t r a f o r many
tes ts , s h i f t i n g t h e lower end o f t h e i n e r t i a l subrange t o
h i g h e r wave numbers. Otherwise it q e n e r a l l y b e q i n s a t a b o u t
nz/E = 3.0 a l t h o u q h t h e minus f i v e - t h i r d s law f o r t h e v e r t i -
c a l component e x t e n d s down t o nz /u = 0.8 t o 1 .0 and f o r t h e
l o n g i t u d i n a l component, down t o nz /g = 0.2 t o 1.0.
The f e a t u r e s of t h e s p e c t r a f o r t h e tests a t 3.0 and
6 . 1 meters can best be s e e n from pooled r e s u l t s f o r e a c h
component. These a r e shown i n F i q u r e 7.38 f o r t h e l o n g i -
t u d i n a l component, F i g u r e 7.39 f o r t h e v e r t i c a l component,
and F i g u r e 7.40 f o r t h e l a t e r a l component. Averaqe c u r v e s
f o r n e u t r a l , u n s t a b l e and s t a b l e c a s e s a r e shown i n each
f i g u r e w i t h t h e 80% c o n f i d e n c e i n t e r v a l i n d i c a t e d w i t h
shad ing f o r each s t a b i l i t y g roup , r e f l e c t i n g t h e reduced
u n c e r t a i n t y o f t h e d a t a r e s u l t i n g from t h e a v e r a q i n g .
Excluded from t h e l a t e r a l component a v e r a g e s f o r t h e n e u t r a l
and u n s t a b l e c a s e s a r e T e s t s 11-1 and 11-2, r e s p e c t i v e l y ,
which were o b v i o u s l y n o t s i m i l a r t o t h e o t h e r s . I n a d d i t i o n
t o demons t ra t inq more c l e a r l y ( p a r t i c u l a r l y a t low wave
numbers) t h e f e a t u r e s of t h e s p e c t r a p r e v i o u s l v d i s c u s s e d ,
such a s t h e i n s i q n i f i c a n t d i f f e r e n c e between n e u t r a l and
u n s t a b l e c a s e s w i t h i n t h e f o r c e d c o n v e c t i o n r e q i o n and a
n o t i c e a b l e s h i f t t o h i g h e r wave numbers f o r s t a b l e c a s e s ,
it i s a l s o c l e a r t h a t t h e s h i f t o f t h e s t a b l e c u r v e s neces-
s a r y t o m a i n t a i n s i m i l a r i t y i s n o t t h e same f o r a l l compon-
e n t s . For s t a b l e c a s e s t h e normal ized v ' spect rum i s s i m i -
l a r i n shape t o t h a t f o r n e u t r a l c a s e s b u t i s s h i f t e d t o
h i g h e r nz/u by a f a c t o r o f 2.0. For t h e u ' component t h e
f a c t o r i s 1.5 and f o r t h e w' component t h e f a c t o r i s 1.1.
The U n i v e r s a l C o n s t a n t s o f t h e I n e r t i a l Subranqe
E v a l u a t i o n of t h e u n i v e r s a l c o n s t a n t , a , i n t h e
e x p r e s s i o n f o r t h e i n e r t i a l subrange was made f o r i n d i v i d u a l
tests i n a d d i t i o n t o t h e comparison, p r e v i o u s l y d i s c u s s e d ,
between Equa t ion (6.3b) and F i q u r e s 7.5, 7.6, 7.13 and
7.14 u s i n g t h e g e n e r a l l y a c c e p t e d v a l u e o f a = 0.138. The
e v a l u a t i o n s , u s i n g Equat ion ( 6 . 3 b ) , w e r e made i n d e p e n d e n t l y
f o r e a c h tes t from t h e l o n g i t u d i n a l s p e c t r a and w i t h t h e
a d d i t i o n a l f a c t o r of 4/3 i n t h e e q u a t i o n , from t h e v e r t i c a l
s p e c t r a f o r each t e s t conducted a t 3.0 and 6 .1 meters i n
n e u t r a l c o n d i t i o n s . The r e s u l t s are l i s t e d i n T a b l e 7.1.
The e s t i m a t e s o f t h e u n i v e r s a l c o n s t a n t , a , a r e r e a s o n a b l y
c l o s e t o one a n o t h e r f o r a g i v e n component b u t t h e average
v a l u e s from t h e t w o components d i f f e r c o n s i d e r a b l y , t h e
average v a l u e of 0.163 from t h e l o n g i t u d i n a l s p e c t r a be inq
somewhat h i g h and t h e v a l u e of 0 . 1 2 9 from t h e v e r t i c a l
s p e c t r a be inq somewhat low. I t must be s t a t e d t h a t t h e
f a c t o r o f 4 / 3 r e q u i r e d between nsu ( n ) / u * ~ and nSw ( n ) /u*
i n t h e i n e r t i a l subrange f o r n e u t r a l c o n d i t i o n s was n o t
found b u t an t h e average t h e s e q u a n t i t i e s w e r e e s s e n t i a l l y
t h e same f o r t h e two components. The same r e s u l t , n o t y e t
pub l i shed ha s r e c e n t l y been found by R. W. S t ewa r t . T h i s
i n fo rma t ion was r e c e i v e d th rouqh a p e r s o n a l communication
w i t h P r o f . H. A. Panofsky. The dependence o f t h e s p e c t r a l
d e n s i t y i n t h e i n e r t i a l subranqe on ~ ~ 1 3 however, i s i l l u s -
t r a t e d v e r y w e l l s i n c e u* e x t e n d s o v e r a r anqe o f n e a r l y an
o r d e r of magnitude and E , t h e r e f o r e , o v e r a range o f n e a r l y
t h r e e o r d e r s of maqnitude. Indeed, t h e c l o s e o r g a n i z a t i o n
o f t h e s p e c t r a f o r bo th t h e l o n g i t u d i n a l and v e r t i c a l com-
ponen ts i n t h e i n e r t i a l subrange and t h e cons tancy o f t h e
r a t i o s ou/u* and ow/u* f o r n e u t r a l c o n d i t i o n s , t e s t i f i e s
th rough Equat ion (6.3b) t o t h e dependence on E ~ / ?
For t h e u n s t a b l e c a s e s w i t h i n t h e f o r c e d convec t i on
r e g i o n , c o n s i d e r i n g t h a t t h e d i s s i p a t i o n can s t i l l be r e p r e -
s e n t e d by ~ * ~ / k z ( t h e wind speed p r o f i l e s appeared ve ry . c l o s e l y l o g a r i t h m i c ) , t h e v a l u e o f "a" was a g a i n c a l c u l a t e d
from bo th l o n q i t u d i n a l and v e r t i c a l component s p e c t r a and
a r e l i s t e d i n Tab l e 7.1. The v a l u e s o b t a i n e d from v e r t i c a l
s p e c t r a , ave r ag ing 0.123, a r e abou t t h e same a s t h o s e
c a l c u l a t e d f o r t h e n e u t r a l cases, s l i g h t l y lower t h a n
expec ted . The v a l u e s from t h e l o n q i t u d i n a l s p e c t r a ,
a v e r a g i n q 0.131, a r e s e e n t o b e , i n t h i s i n s t a n c e , a u i t e con-
s i s t e n t w i t h t h o s e o b t a i n e d from v e r t i c a l component s p e c t r a .
The r e a s o n a b l e and c o n s i s t e n t r e s u l t s o b t a i n e d i n u n s t a b l e
c o n d i t i o n s i n t h e r e g i o n of f o r c e d c o n v e c t i o n a g a i n demon-
s t ra tes t h e dominance of mechanical ene rqy p r o d u c t i o n o v e r
t h e c o n v e c t i v e enerqy p r o d u c t i o n under t h e s e c o n d i t i o n s .
Only two c a s e s f o r t h e l a t e r a l component cou ld be
used f o r e v a l u a t i n g " a " from t h e s p e c t r a beyond nz/u = 3.0.
These produce a n average v a l u e o f 0.137.
I n s t a b l e c o n d i t i o n s even c l o s e t o t h e qround, a n
e v a l u a t i o n o f t h e c o n s t a n t , a , through Equa t ion (6 .3b) pro-
duces e s t i m a t e s much t o o l a r q e i n d i c a t i n q t h a t Equat ion
( 6 . 3 b ) , w i t h i t s l i m i t a t i o n s , i s i n a p p r o p r i a t e t o t h e s t a b l e
c a s e . I f a l l t h e e f f e c t i v e terms were known f o r t h e e n e r q y
e q u a t i o n i n t h e s t a b l e c a s e s b e s i d e s t h e mechanica l ene rgy
p r o d u c t i o n t e r m , l e a d i n g t o a r e a l i s t i c e x p r e s s i o n f o r t h e
d i s s i p a t i o n , an a d e q u a t e e x p r e s s i o n f o r t h e normal ized
s p e c t r a c o u l d be o b t a i n e d i n t h e i n e r t i a l subrange , compar-
a b l e to Equat ion ( 6 . 3 b ) , f o r cases o t h e r t h a n t h o s e where
mechanica l ene rgy dominates . Then " a " c o u l d be c a l c u l a t e d
s i m i l a r l y f o r s t a b l e cases. F u t u r e tes ts w i t h a d d i t i o n a l
i n s t r u m e n t a t i o n shou ld p r o v i d e t h i s c a p a b i l i t y .
Usinq t h e d i m e n s i o n l e s s wind s h e a r , x(Ri), c a l c u l a t e d
from t h e average Richardson ' s numbers, and t h e average
r a t i o s of t h e v a r i a n c e s t o u * ~ , w i t h t h e a v e r a g e c u r v e s f o r
t h e d i a b a t i c cases i n F i q u r e s 7.38 and 7.39, Equat ion ( 6 . 3 ~ )
was t e s t e d . The aqreement between u n s t a b l e and n e u t r a l
c a s e s was s l i g h t l y improved, y i e l d i n q an average v a l u e f o r
t h e c o n s t a n t , a , of 0.15 and a s e p a r a t e check wi th t h e more
u n s t a b l e tes t 12-2 from 1 2 . 2 meters gave t h e same va lue .
The s t a b l e c a s e s , however, gave v a l u e s of "a" much t o o low,
i n d i c a t i n g t h e assumptions used i n a r r i v i n g a t Equation ( 6 . 3 ~ )
w e r e i n v a l i d f o r t h e s t a b l e ca se .
Comparison wi th Prev ious R e s u l t s
Organiz ing s p e c t r a 1 , e s t i m a t e s a s a f u n c t i o n of wave
number r a t h e r t h a n f requency i s n o t a r e c e n t no t ion .
T a y l o r ' s hypo thes i s , demonstra t ing t h a t f o r a q iven h e i g h t
s p e c t r a l d i s t r i b u t i o n s can be p re sen t ed a s f u n c t i o n s o f
n/n has been v e r i f i e d i n t h e l a b o r a t o r y f o r homogeneous con-
d i t i o n s by Favre , Gavig l io and urna as'^. Panofsky, Cramer
and ~ a o ~ * have shown t h a t , i n t h e f imld , space c o r r e l a t i o n s
and a u t o c o r r e l a t i o n s can be matched through T a y l o r ' s hypo-
t h e s i s c l o s e t o t h e ground f o r l a g d i s t a n c e s less t h a n 90
meters.
F u r t h e r v e r i f i c a t i o n of T a y l o r ' s hypo thes i s is q iven
by a comparison o f a i r c r a f t and tower measurements o f turbu-
l e n c e made by Lappe, Davidson and Notess, d e s c r i b e d by
Panofsky and ~ r . 8 8 ~ ~ . Mea~urements a t 90 and 102 meters
demonstrated t h e v a l i d i t y of t h e hypo thes i s f o r t h e tonqi-
t u d i n a l and v e r t i c a l components excep t f o r d i s c r e p a n c i e s i n
t h e energy a t l o w ' f r e q u e n c i e s f o r t h e v e r t i c a l component t h a t
cou ld be exp l a ined on t h e b a s i s o f s u r f a c e inhomoqenei t ies
o r slow v a r i a t i o n s i n t h e mean wind speed. A s a r e s u l t ,
T a y l o r ' s hypo the s i s would be expected t o app ly th rouqh t h e
range of a n a l y s i s i n t h e p r e s e n t s t udy which ex t ends down
t o 0.012 cps .
The s c a l i n g of t h e t u r b u l e n c e s p e c t r a w i t h h e i q h t
was f i r s t no ted by panofsky31 f o r t h e v e r t i c a l component.
L a t e r Panofsky and ~ c ~ o r m i c k ~ ~ compared v e r t i c a l component
s p e c t r a from v a r i o u s s i tes , d e t e c t i n q s i m i l a r i t i e s when t h e
s p e c t r a were p l o t t e d a s f u n c t i o m o f nz/D f o r h e i q h t s r anq inq
from less t h a n one meter t o s e v e r a l hundred meters.
~ u r v i c h l s l i k e w i s e compumdve r t i c a l component s p e c t r a f o r
h e i g h t s of one and f o u r meters and found matchinq cu rves
when p l o t t e d a s a f u n c t i o n o f nz/g.
An nz/g dependence f o r t h e o t h e r components ha s been
looked f o r by a number of i n v e s t i q a t o r s w i t h mixed r e s u l t s .
Gene ra l l y , t h e a n a l y s e s have i n d i c a t e d agreement w i t h t h e
nz/g dependence r e q u i r e d i n Equat ion (6.3b) f o r t h e i n e r t i a l
subrange b u t a t lower wave numbers some h e i g h t dependence
remains. ~ a v e n ~ o r t ' found l o n g i t u d i n a l s p e c t r a from v a r i -
ous h e i g h t s b e s t o rgan i zed independent o f h e i g h t . ebb^^, 6 Henry, and Cramer , ( a s d e s c r i b e d by Lumley and Panofsky)
a l l f i n d some h e i g h t dependence b u t n o t a s imple l i n e a r
s c a l i n g o f t h e t u r b u l e n t eddy l eng th s . E8erman2 h a s most
e f f e c t i v e l y o rgan i zed e x i s t i n g d a t a f a r t h e l o n g i t u d i n a l
component s p e c t r a , c o l l e c t e d i n p r ev ious f i e l d programs a t
O'Neill, Nebraska, at Brookhaven, Lonq Island, and at Round
Hill, Mass. Also included were a larqe number of spectra
collected by Davenport and by Zubkovsky. Berman plotted
normalized spectral estimates, nSu (n) / u * ~ , as a function of
nz/B and demonstrated an additional heiqht dependence most
pronounced at low wave numbers and disappearinq as the
inertial subranqe was approached where Equation (6.3b) must
hold. The spectral peaks for neutral conditions were found
by interpolation to occur at nz/n values varying proportional
to the 0.75 power of height. Although in the present
investigation no additional heiqht dependence was detected,
the uncertainty of the estimates is too great to contradict
Berman' s conclusions.
The lateral component spectra have been found in pre-
vious studies to be even less dependent on height than the
spectra for the other components. Panofsky and p el and^^
demonstrated that at both Brookhaven and OtNeill, the lateral
spectra were strongly dependent on the stability and rather
insensitive to height changes. However, this effect
occurred at low wave numbers and there is an.indication
from a plot of lateral spectra from B w k h r o s n (Lumley and
~ a n o f s k ~ 2 0 ) in stable conditions that at higher wave numbers,
a dependence on nz/V occurs. The presence of mesoscale
eddies in the lateral and longitudinal turbulence and the
occurrence of the spectral peaks at low frequencies (except
for lateral spectra in stable conditions) causes a large
f r a c t i o n o f t h e t u r b u l e n t ene rqv t o be q u i t e v a r i a b l e i n
maqnitude a t a g i v e n low wave number and d i f f i c u l t t o sample
a d e q u a t e l y . Consequent ly it i s n o t s u r p r i s i n q t o f i n d d i f f e r -
e n c e s i n t h e amount of t u r b u l e n t ene rqy a t low wave numbers,
r e s u l t i n g b o t h from r e a l mesoscale i n f l u e n c e s and from wide
s t a t i s t i c a l v a r i a b i l i t y . I n t h i s s t u d y a l s o , f o r t h e l a t e r a l
component a t wave numbers below nz/u = 0.1 a q r e a t d e a l o f
v a r i a b i l i t y i n t h e s p e c t r a l e s t i m a t e s f o r n e u t r a l and u n s t a b l e
c a s e s i s p r e s e n t , o b s c u r i n q s i m i l a r i t y i n t h i s r e q i o n . The
f i l t e r i n g o u t of t h e v e r y low f requency energv below 0.001
o r 0.002 c p s a l o n g w i t h d c l e v e l s i n t h e s i q n a l s when t h e
t o t a l v a r i a n c e s w e r e c a l c u l a t e d i n t h e p r e s e n t i n v e s t i g a t i o n
may have he lped t o u n i f y t h e d a t a i n each q roup of h o r i z o n t a l
component spectra by removing t h e h i g h l y v a r i a b l e , l a r g e
a m p l i t u d e , low f requency t r e n d s and meander.
The shape of t h e s p e c t r a and t h e l o c a t i o n of t h e
s p e c t r a l peaks f o r t h e v e r t i c a l component i n t h e p r e s e n t
i n v e s t i g a t i o n do n o t d i f f e r g r e a t l y from p r i o r r e p o r t s of
such f e a t u r e s . The comparison by Panofsky and ~ c ~ o r m i c k ~ ~
o f d a t a from v a r i o u s si tes show peaks i n nSw(n) r a n g i n q from
nz/n = 0.2 t o 0.6, a g r e e i n g q u i t e w e l l w i t h t h e o c c u r r e n c e o f
t h e w s p e c t r a peaks , h e r e , a t a b o u t nz/'ti = 0.4. S i m i l a r l y ,
t h e p r e s e n t work a g r e e s w i t h t h e s h a r p drop-off p r e v i o u s l y
found f o r t u r b u l e n t ene rqy i n t h e v e r t i c a l component a t l o w
wave numbers, w e l l b e f o r e nz/n = 0.01. Normalized v e r t i c a l
v e l o c i t y s p e c t r a r e p o r t e d by ~ u r v i c h l ~ a l s o show s p e c t r a l
peaks i n t h e same nz/g r e g i o n , a d i s t r i b u t i o n f o r n e u t r a l
c o n d i t i o n s showing t h e same shape throuqhout b u t s l i q h t l y
h i q h e r i n normal ized s p e c t r a l d e n s i t y t h a n t h e v e r t i c a l
component s p e c t r a r e p o r t e d h e r e . The u n s t a b l e c a s e g iven by
Gurvich a g r e e s o v e r t h e upper end o f t h e spect rum b u t t h e
peak v a l u e of nSw (n ) ow2 = 0.33 i s c o n s i d e r a b l y h i g h e r t han
t h e v a l u e o f 0.22 f o r t h e mechanical enerqy peak i n t h e
p r e s e n t s t udy . The peak found by Gurvich was a l s o a t a
somewhat lower wave number and was q u i t e p o s s i b l y a convec-
t i v e peak o c c u r r i n g n e a r and obscur ing t h e mechanical enerqy
peak. H i s s t a b l e c a s e i s d e f i n i t e l y s h i f t e d t o h i g h e r wave
numbers t h a n f o r n e u t r a l and u n s t a b l e s i t u a t i o n s , more pro-
nounced t han t h e s l i q h t s h i f t no ted i n t h e i n e r t i a l subrange
f o r t h e t h r e e and s i x meter s t a b l e c a s e s i n t h e p r e s e n t s t udy .
The e m p i r i c a l e q u a t i o n sugges ted by Panofsky and
McCormick,
h a s t h e same shape i n t h e energy producing r e g i o n as t h e
v e r t i c a l component s p e c t r a f o r t h e fo r ced convec t ion r e g i o n
i n t h i s r e p o r t bu t i s s h i f t e d t o lower wave numbers by abou t
40% and as panofsky20 p o i n t s o u t , it a l s o does n o t f i t t h e
i n e r t i a l subrange. I t is encourag ing , however, to f i n d a
s i m i l a r shape sugges ted from o t h e r d a t a i n t h e l o w wave
number end o f t h e spec t rum. T h i s i s p a r t j c u l a r l y t r u e s i n c e
c o n v e c t i v e e n e r q y i s added , n o t a s a f u n c t i o n o f nz/U b u t can
a p p e a r , g e n e r a l l y u n d i s t i n q u i s h e d from m e c h a n i c a l t u r b u l e n c e ,
anywhere a t low nz/U v a l u e s and produce v a r y i n q s p e c t r a l
s h a p e s f o r measurements t a k e n above t h e f o r c e d c o n v e c t i o n
r e q i o n . I d e n t i f i c a t i o n o f c o n v e c t i v e e n e r q y peaks h a s been
d i f f i c u l t and s t a t i s t i c a l r e l i a b i l i t y h a s been p o o r , thouqh
t h e i n v e s t i q a t i o n s o f ebb^^ and o f Panofsky and Van d e r ~ o v e n ~ ~
have s u s q e s t e d s e p a r a t e peaks a t low f r e q u e n c i e s s u s p e c t e d
o f r e s u l t i n q from buoyant e n e r q y p r o d u c t i c n . I n t h e p r e s e n t
s t u d y , t h e c o n v e c t i v e peaks n o t e d f o r t h e t i n s t a b l e c a s e a t
12.2 meters and t h a t a t 87 meters miqh t be q u e s t i o n e d f o r
l a c k o f a q u a n t i t y o f s i m i l a r c o n f i r m i n q e v p e r i m e n t a l r e s u l t s
even thouqh t h e y a r e s t a t i s t i c a l l y s i q n i f i c a n t on t h e 8 0 %
l e v e l a ? s e p a r a t e peaks . But t h e d i s p l a c e m e n t o f t h i s
s p e c t r u m a t low wave numbers , w e l l above t h e u n i v e r s a l form
o f t h e s p e c t r a f o r tes ts i n t h e f o r c e d c o n v e c t i o n r e q i o n ,
t e s t i f i e s t o t h e p r e s e n c e o f e n e r q y i n a d d i t i o n t o t h a t
i n t r o d u c e d i n t o t h e t u r b u l e n c e t h r o u q h m e c h a n i c a l f e e d i n q ,
l e a v i n q l i t t l e d o u b t o f i t s c o n v e c t i v e o r i q i n .
F o r t h e h o r i z o n t a l component compar ison of s p e c t r a l
s h a p e s a l l o w some q e n e r a l o b s e r v l t i o n s t o he made. "irst ,
f o r t h e h o r i z o n t a l component s p e - t r a , t h e peaks have been
found a t c o n s i d e r a b l y lower wave numbers t h a n f o r t h e v e r t i -
c a l component. S t . ! b i l i t y h a s been shown t-, e f f e c t t h e s h a p e
o f t h e I l ) w wave n u l b e r end c f tile spec t ru l r , , somewhat more
e n e r g y b e i n g p r e s e n t i n u n s t a b l e c o n d i t i o n s t h a n i n n e u t r a l
and s t a b l e c o n d i t i o n s , b u t f o r a l l s t a b i l i t i e s a preponder-
a n c e o f ene rgy i s found a t low wave numbers. A b i a s o f t h e
l o n g i t u d i n a l s p e c t r a t o low wave numbers, i n r e l a t i o n t o
v e r t i c a l s p e c t r a , w a s a l s o found a t Hanford and t h e i n c r e a s e
i n low wave number energy i n u n s t a b l e c o n d i t i o n s a g r e e d a l s o .
The o c c u r r e n c e of t h e s p e c t r a l peak a t a b o u t nz/g = 0.03
a g r e e s w i t h Berman's a n a l y s i s f o r l o n g i t u d i n a l s p e c t r a a t
6 meters. Also , h i s v a l u e f o r t h e normal ized s p e c t r a l peak
o f 1 . 7 i s n o t g r e a t l y d i f f e r e n t from t h e v a l u e o f 1 .3 found
i n t h e p r e s e n t s tudy . Fur thermore , Berman's a n a l y s i s i n d i -
c a t e d t h e l o n g i t u d i n a l spectra w e r e o n l y s l i q h t l y s h i f t e d
t o lower wave numbers f o r u n s t a b l e c o n d i t i o n s w h i l e a
n o t i c e a b l e s h i f t t o h i g h e r nz /g o c c u r r e d i n s t a b l e condi-
t i o n s , a r e s u l t i n agreement w i t h t h e p r e s e n t s tudy .
For t h e l a t e r a l s p e c t r a , however, a comparison w i t h
Brookhaven and O I N e i l l d a t a r e v e a l s a marked d i f f e r e n c e f o r
t h e s t a b l e c a s e s . Though t h e l a r g e amount o f e n e r g y a t l o w
wave numbers f o r u n s t a b l e c o n d i t i o n s is a comparable f e a t u r e ,
t h e l o w wave number peaks f o r t h e s t a b l e cases a t Hanford
do n o t compare w i t h some o t h e r o b s e r v a t i o n s where peaks i n
s t a b l e l a t e r a l component s p e c t r a o c c u r a t a b o u t t h e same
wave number a s f o r t h e v e r t i c a l component.
A t t h e h i g h wave number end of t h e s p e c t r a , c o n s i d e r -
a b l e e v i d e n c e h a s been accumula ted , s u p p o r t i n g t h e m i n w
f i v e - t h i r d s r e l a t i o n of Kolmogoroff. E a r l y measurement6 by
PTccreadyZ1 u s i n q h o t w i r e t e c h n i q u e s a t s e v e r a l he i . qh t s c l o s e
t o t h e qround a r e c o n s i s t e n t w i t h Kolmoqorof f' s t h e o r y , b o t h
i n t h e r 2 / 3 s h a p e o v e r a r e q i o n o f t h e a u t o c o r r e l a t i o n func -
t i o n and t h r o u q h s p e c t r a l d e n s i t i e s t a k e n a t t h r e e f r e q u e n c i e s .
~ c ~ r e a d ~ ' s ~ ~ r e c e n t work w i t h a s a i l p l a n e l i k e w i s e s u p p o r t s
t h e minus f i v e - t h i r d s s p e c t r a l d i s t r i b u t i o n , B u s i n q e r and
soumi4 i n t h e i r p i o n e e r i n g work w i t h t h e s o n i c anemometer
a l s o show ag reemen t w i t h t h e minus f i v e - t h i r d s law o v e r
e x t e n d e d r e q i o n s of v e r t i c a l component s p e c t r a . Recen t work
w i t h t h e s o n i c anemometer r e p o r t e d by ~ u r v i c h l ~ and oni in"
a l s o s u p p o r t t h e e x i s t e n c e o f a n i n e r t i a l s u b r a n q e i n v e r t i -
c a l component s p e c t r a . F o r t h e h o r i z o n t a l component t h e
r e c e n t measurement o f Pond, e t a1 . 3 8 w i t h a , h o t w i r e ane-
m o m e t e r p r e s e n t v e r y c o n v i n c i n q e v i d e n c e f o r t h e p r e s e n c e
o f a n i n e r t i a l s u b r a n g e a t one h e i q h t o v e r w a t e r . Zubkovsky 's
measurements o f l o n g i t u d i n a l component s p e r t r a a t 4 meters
c o n f i r m t h e minus f i v e - t h i r d s l aw , e x t e n d i n g t o wave l e n g t h s
f i v e t i m e s t h e h e i q h t , a s was s i m i l a r l y found i n t h e p r e s e n t
s t u d y f o r u n s t a b l e c a s e s .
The l o w wave number l i m i t o f t h e i n s r t i a l s u b r a n q e h a s
been i n v e s t i g a t e d p r i m a r i l y by d e t e r m i n i n q t h e p o i n t a t which
t h e minus f i v e - t h i r d law no l o n q e r f i t s o b s e r v e d s p e c t r a l
d i s t r i b u t i o n s . ~ c ~ r e a d ~ * ~ p r e s e n t s summarized r e s u l t s o f a
number o f i n v e s t i g a t i o n s t o show t h a t t h i s p o i n t , f o r t h e
v e r t i c a l and l a t e r a l component s p e c t r a , a p p l i e s t o wave
l e n g t h s on t h e o r d e r o f twice t h e h e i g h t i n t h e r e g i o n below
about 200 meters; nz/g f o r t h i s p o i n t v a r i e s from about
0.8 t o 0 . 4 f o r s t a b l e t o u n s t a b l e c o n d i t i o n , r e s p e c t i v e l y .
H i s summary shows some dependence on h e i g h t ; t h e nz/g
va lue f o r t h e lower l i m i t o f t h e range i n c r e a s e s w i t h h e i g h t .
For t h e l o n g i t u d i n a l component, McCreadyts summary i n d i c a t e s
no s t a b i l i t y dependence f o r t h e lower nz/u l i m i t which o c c u r s
abou t 0.4. ~ u r v i c h l ~ a l s o g i v e s e s t i m a t e s of t h e low wave
number l i m i t f o r t h e minus f i v e - t h i r d s law from h i s one
meter and f o u r m e t e r v e r t i c a l component s p e c t r a . H e found
f o r h i s most s t a b l e c a s e ( R i = 0.28) t h a t t h e l i m i t o ccu r r ed
a t nz/u = 5 .2 , f o r n e u t r a l c o n d i t i o n s , a t nz/B = 0.72, and
f o r h i s m o s t u n s t a b l e c a s e ( R i = -0.76) nz/u = .40.
The r e s u l t s reviewed above a g r e e i n some r e s p e c t s
w i th t h o s e o f t h e p r e s e n t s t udy . The lower l i m i t s o f t h e
minus f i v e - t h i r d s law i s g e n e r a l l y found t o o c c u r a t l o w e r
nz/u f o r l o n g i t u d i n a l s p e c t r a t h a n f o r v e r t i c a l s p e c t r a
a t Hanford a s w e l l a s e l sewhere and some s t a b i l i t y depend-
ence i s a l s o common. However, t h e lower l i m i t o f t h e minus
f i v e - t h i r d s law is observed i n t h e v e r t i c a l component s p e c t r a
a t g e n e r a l l y h i g h e r nz/E for Hanford t h a n e lgewhere . Fu r the r -
more, t h e r e is o n l y a v e r y s l i g h t dependence of t h e occu r r ence
o f t h i s p o i n t on s t a b i l i t y a t 3.0 and 6 .1 meters, w h i l e
t h e r e i s a marked s h i f t of t h e s t a b l e c a s e a t 12.2 meters t o
h i g h e r nz/u a t Hanford. Th i s s u g g e s t s t h a t more ex t remely
s t a b l e and u n s t a b l e s i t u a t i o n s a t t h e 3.0 and 6 .1 meter
l e v e l s might show g r e a t e r s t a b i l i t y dependence. Another
impor tan t f a c t o r i s t h a t t h e s u r f a c e roushness a t Hanford,
c h a r a c t e r i z e d by s a q e b r u s h averaq inq one meter i n h e i q h t , i s
q e n e r a l l y g r e a t e r t h a n a t many o t h e r t u r b u l e n c e t es t s i tes
and c o n s e q u e n t l y a r e g i o n dominated by mechanica l t u r b u l e n c e
i n r e a s o n a b l y s t r o n q winds may be deeper t h a n a t o t h e r
s i tes. Thus d i f f e r e n c e s due t o s t a b i l i t y a t o t h e r s i tes a t
a h e i q h t of f o u r meters miqht n o t b e expec ted a t Hanford
even a t s i x meters.
The s h i f t of t h e lower l i m i t o f t h e minus f i v e - t h i r d s
l o n q i t u d i n a l s p e c t r a w i t h s t a b i l i t y (nz/u = 1.0 f o r s t a b l e
t o nz/U = 0.2 f o r u n s t a b l e ) i s known w i t h abou t 80% c o n f i -
dence. R e s u l t s summarized by McCready show no such s h i f t
b u t it i s b e l i e v e d t h a t t h e r e s u l t s r e p o r t e d h e r e are re l i -
a b l e enouqh t o d e t e c t t h e s m a l l d i f f e r e n c e i n s l o p e of t h e . c u r v e s i n F i g u r e 7.38 which l e a d t o t h e c o n c l u s i o n t h a t
t h e r e i s a s h i f t of t h e lower l i m i t .
The l a c k o f h e i q h t dependence o f t h e minimum nz/B
i n t h e i n e r t i a l subrange f o r 3.0 and 6.1 meters a t Hanford
a l s o d i f f e r s from r e s u l t s r e p o r t e d by McCready.
The o b s e r v a t i o n i n t h e p r e s e n t s t u d y t h a t t h e minus
f i v e - t h i r d s law u s u a l l y e x t e n d s t o lower wave numbers t h a n
t h e t r u e i s o t r o p i c l i m i t (where c o s p e c t r a become z e r o ) a g r e e s
w i t h t h e f i n d i n g s of , pondd2; On t h e o r e t i c a l q rounds ,
if ford'^ h a s a l s o shown t h a t t h e one-dimensional spect rum
i s e x p e c t e d t o f o l l o w t h e minus f i v e - t h i r d s law below t h e
t r u e i s o t r o p i c l i m i t . Confirming measurements o f t h e u t v t
- and v 'w ' c o s p e c t r a a r e l a ck inq i n p r ev ious s t u d i e s and t h e
u'w' c o s p e c t r a l measurements a r e few. However, Monin 27
r e c e n t l y r e p o r t e d r e s u l t s demons t ra t inq t h e nz/g dependence,
a s w e l l a s s t a b i l i t y dependence of t h e u 'w' cospect rum
which i n d i c a t e d t h e i s o t r o p i c l i m i t o ccu r r ed between nz/u
v a l u e s of 1 and 10.
Fo r t h e l o n g i t u d i n a l component from O ' N e i l l d a t a ,
~ 1 ~ 1 1 demonst ra ted t h e dependence of t h e s p e c t r a l e s t i m a t e s
a t h igh wave numbers on t h e 8/3 power o f wind speed. Th i s
dependence i s expec ted from Equat ion (6.3b) w i t h t h e loga-
r i t h m i c p r o f i l e r e l a t i o n s h i p t o u* , i f t h e s p e c t r a l e s t i m a t e s
obey t h e minus f i v e - t h i r d s law a s w e l l a s f o l l o w t h e nz/v
dependence. Thus from
- w i t h u* = Uk/ln(z/z0)
w e have nS(n) = a k 8 1 3 8 1 3
( na ) - 2 / 3 (5) l n z/so) - 2
g i v i n g t h e observed dependence on wind speed.
Reynolds' S t r e s s R e s u l t s
The Reynolds' stresses, c a l c u l a t e d from t h e wind f l u c -
t u a t i o n d a t a w i t h t h e ana log computer program d i s c u s s e d p r e -
v i o u s l y , a r e g i v e n i n Tab le 7.2. With b u t a few e x c e p t i o n s ,
t h e d a t a a r e n o t s i g n i f i c a n t l y d i f f e r e n t from what might be
expec ted . The s t a n d a r d d e v i a t i o n f o r t h e f l u c t u a t i o n s of
e a c h o f t h e wind components i s d i v i d e d bv t h e f r i c t i o n v e l o -
c i t y a s i s t h e mean wind speed a t 3 .0 meters measured inde -
p e n d e n t l y w i t h t h e 80 f o o t tower equ ipmen t , and t h e r a t i o s
a r e l i s t e d i n Tab le 7.2. Thouqh t h e Reynolds ' stresses c o v e r
a c o n s i d e r a b l e r a n g e o f maqn i tudes , t h e r a t i o s d e m o n s t r a t e
t h e y a r e c o n s i s t e n t and r e a s o n a b l e . T h e r a t io o f t h e 3.0
m e t e r wind speed t o f r i c t i o n v e l o c i t y f o r n e u t r a l c o n d i t i o n s
v a r y i n g no more t h a n *5% f rom t h e a v e r a g e r a t i o e x c e p t f o r
one case, d e m o n s t r a t e s agreement w i t h t h e c h a r a c t e r i s t i c
o f t h e l o q a r i t h m i c p r o f i l e where t h e wind speed i s p ropor -
t i o n a l t o t h e f r i c t i o n v e l o c i t y . The a v e r a q e r a t i o also
l e a d s t o a roughness l e n q t h o f z0 - 2.0 c m . , f a i r l y c o n s i s -
t e n t w i t h a roughness l e n g t h , i n d i c a t e d by Hanford wind p ro -
f i l e s , c l o s e r t o 3.0 m. T e s t 6-1 may have u n d e r e s t i m a t e d
t h e f r i c t i o n v e l o c i t y somewhat b e c a u s e o f t h e s h o r t l e n g t h
o f r e c o r d (13 min.) s o t h a t t h e low f r e q u e n c y c o n t r i b u t i o n s
t o t h e u 'w' c o r r e l a t i o n were p o o r l y sampled.
The wind speed a t a small h e i g h t was chosen so t h a t
f o r n o n - n e u t r a l c o n d i t i o n s t h e p r o f i l e would s t i l l b e close
t o l o g a r i t h m i c . The u n s t a b l e c a s e s i n c l u d i n g tes t 12-2 a t
12.2 meters g i v e e s s e n t i a l l y t h e same r a t i o a s t h e n e u t r a l
c a s e s w h i l e t h e s t a b l e c a s e s q e n e r a l l y g i v e v a l u e s somewhat
l a r g e r , i n d i c a t i n g t h e p r o f i l e i s d e v i a t i n q i n t h e e x p e c t e d
d i r e c t i o n from l o g a r i t h m i c a t 3.0 meters f o r t h e s t a b l e
c a s e s . Fo r test 1 2 - 1 a l l r a t i o s l i s t e d are l a r g e , a s migh t
b e e x p e c t e d f o r s u c h a l i g h t wind s p e e d where t h e momentum
f l u x h a s undoubtedly dropped o f f somewhat a t 12.2 meters.
The r a t i o , oW/uf, i s a l s o v e r v c o n s t a n t r e q a r d l e s s
o f h e i g h t o r wind speed f o r t h e n e u t r a l and t h e 3.0 and 6 . 1
meter u n s t a b l e c a s e s w i t h a n a v e r a g e v a l u e o f 1.33. T h i s
v a l u e i s somewhat h i g h e r t h a n t h e v a l u e o f 1.25 de te rmined
by Panofsky and ~ c ~ o r m i c k ~ ~ and c o n s i d e r a b l " h i g h e r t h a n t h e
v a l u e o f 1.05 found i n wind t u n n e l i n v e s t i g a t i o n s c i t e d by
Panofsky and Lumley. However, t h e v a l u e h e r e i s i d e n t i c a l
t o t h a t found by pasquil13 ' . The i n c r e a s e d v a l u e o f aw/u*
i n s t a b l e c o n d i t i o n s n o t e d h e r e h a s n o t been obse rved i n
o t h e r i n v e s t i g a t i o n s . The u n s t a b l e test 12-2 a t 12.2
meters g i v e s r e s u l t s n o t s i g n i f i c a n t l y d i f f e r e n t from t h e
n e u t r a l c a s e s .
The r a t i o s oU/u* and a /u* a r e more v a r i a b l e t h a n v
ow/u* b u t no dependence on h e i g h t , wind speed o r s t a b i l i t y
i s no ted . The l a c k o f dependence o f t h e r a t i o s oU/u* and
a,/u* on wind speed i s i n agreement w i t h p r e v i o u s observa-
t i o n s b u t i n v e s t i g a t o r s have found e l s e w h e r e a s t r o n g
dependence o f ov/u* on s t a b i l i t y a s w e l l a s a n o t i c e a b l e
s t a b i l i t y dependence f o r ou/u*.
Some of t h e d i f f e r e n c e s n o t e d between t h e v a r i a n c e
c a l c u l a t i o n s o f t h i s i n v e s t i g a t i o n and o t h e r r e p o r t e d r e s u l t s
can be a t t r i b u t e d t o t h e h igh-pass f i l t e r i n g performed i n
t h e a n a l o g computer program d e s c r i b e d e a r l i e r . The f i l t e r
removed f l u c t u a t i o n s w i t h p e r i o d s g r e a t e r t h a n a b o u t 8
minu tes f o r t h e tests below 12 meters and removed p e r i o d s
q r e a t e r t h a n abou t 17 minutes f o r tests above 1 2 me te r s .
Thus, c o n t r i b u t i o n s from low f requency h o r i z o n t a l mesoscale
e d d i e s w e r e removed. I f t h e d a t a had n o t been f i l t e r e d i n
t h i s manner, v a r y i n g amounts o f low f requency enerqy would
have appeared i n t h e h o r i z o n t a l component v a r i a n c e s depend-
i n q on t h e l e n q t h o f r e c o r d . A number of f i l t e r s w i t h vary-
i n q c u t - o f f p o i n t s w e r e t r i e d . A f i l t e r was chosen which
would p a s s t h e e n t i r e v e r t i c a l v e l o c i t y v a r i a n c e and t h e
momentum f l u x w i t h o u t l i m i t a t i o n . Consequent ly , it was con-
s i d e r e d t h a t a l l m i c r o s c a l e c o n t r i b u t i o n s t o t h e t u r b u l e n c e
w e r e i n c l u d e d . Another d i f f e r e n c e i s t h e l a r q e c o n t r i b u t i o n
a t low f r e q u e n c i e s i n t h e l a t e r a l wind component v a r i a n c e
found a t Hanford i n s t a b l e c o n d i t i o n s whereas i n v e s t i g a t i o n s
a t o t h e r sites have shown a marked l a c k o f low-frequency
energy f o r s t a b l e s i t u a t i o n s .
The u t w ' c o v a r i a n c e is l a r g e , a s e x p e c t e d , and is
s y s t e m a t i c a l l y r e l a t e d t o t h e wind speed a s d e s c r i b e d above.
Fur thermore , t h e c o s p e c t r a l d i s t r i b u t i o n s f o r t h e 3.0 and
6 .1 meter tests a r e w e l l o r g a n i z e d a s a f u n c t i o n o f nz/n,
demons t ra t inq s i m i l a r i t y a s found i n t h e power spectral
d i s t r i b u t i o n s . The i n t e q r a l s moni tored th rouqhou t t h e
a n a l y s i s (see F i q u r e s 4 .7 and 4 . 1 1 ) i n c r e a s e a t a s t e a d y
r a t e through t h e tes ts f o r b o t h t h e c o s p e c t r a l e s t i m a t e s
and t h e t o t a l c o v a r i a n c e c a l c u l a t i o n s , demons t ra t ing l i t t l e
s t a t i s t i c a l v a r i a b i l i t y . -
The v t w ' c o v a r i a n c e s a r e s m a l l , g e n e r a l l y less t h a n
10% o f t h e cor respond ing u'wT c o v a r i a n c e s , e x c e p t above 12
meters where t h e momentum f l u x h a s d e c r e a s e d o r , f o r t h e 87
meter c a s e , where u'w' i s r e v e r s e d i n s i q n , co r respond inq
t o a d e c r e a s e i n wind speed w i t h h e i q h t . Here t h e v 'w '
stress t r a n s f e r s t h e momentum toward t h e s u r f a c e correspond- -
i n g t o a wind d i r e c t i o n s h e a r a t 400 f e e t . The v 'w'
c o s p e c t r a l d e n s i t i e s a r e l a r g e r t h a n might b e e x p e c t e d from
t h e c o v a r i a n c e v a l u e s s i n c e i n some c a s e s p o s i t i v e c o n t r i b u - . t i o n s a t h igh wave numbers a r e t o a l a r g e e x t e n t ba lanced by
n e g a t i v e c o n t r i b u t i o n s a t l o w wave numbers. Even l a r g e r
n e g a t i v e c o n t r i b u t i o n s a r e e x p e c t e d f o r t h e f i r s t series o f
tests a t wave numbers below t h o s e passed by t h e h igh-pass -
f i l t e r a s ev idenced by tes t 6-1 and 7-2 where t h e v 'w'
c o v a r i a n c e s were a l s o c a l c u l a t e d w i t h o u t t h e f i l t e r . The
r e s u l t i n g n e g a t i v e c o v a r i a n c e s , -0.0864 and -0.0581 respec-
t i v e l y were o p p o s i t e i n s i g n t o t h e f i l t e r e d v a l u e s , corres-
ponding t o t h e r e q u i r e d wind d i r e c t i o n d e c r e a s e w i t h h e i g h t
n o t e d between 7 and 400 f e e t d u r i n g t h e t e s t i n g . The i n t e g r a l s -
moni to red i n t h e a n a l y s i s of b o t h v'w' c o v a r i a n c e and cospec-
t r a l d e n s i t y showed more s t a t i s t i c a l v a r i a b i l i t y t h a n w i t h
t h e u'w' a n a l y s i s . However, t h e i n t e g r a l s g e n e r a l l y t ended
toward t h e f i n a l v a l u e s . F i g u r e 7.41 shows t h e i n c r e a s e d -
v a r i a b i l i t y o v e r t h e u'w' c a l c u l a t i o n shown p r e v i o u s l y i n
F i g u r e 4 . 1 1 f o r t h e same test .
The u ' v ' c o v a r i a n c e s a r e q u i t e l a r q e , b u t a s t h e
c o s p e c t r a l d i s t r i b u t i o n s show, t h e c o n t r i b u t i o n s are made
i n a r a t h e r d i so rgan i zed way, occu r r i nq bo th p o s i t i v e l y and
n e g a t i v e l y a t l a r q e r and l a r g e r magnitudes f o r lower and lower - wave numbers. I n a d d i t i o n , t h e u ' v ' cova r i ance and c o s p e c t r a l
i n t e g r a l s monitored du r inq t h e a n a l y s i s show a d i s t i n c t l a c k
of s t a t i s t i c a l r e g u l a r i t y , a q iven i n t e g r a l o f t e n reach ing
both l a r g e p o s i t i v e and l a r g e n e q a t i v e v a l u e s a t d i f f e r e n t
p o i n t s i n t h e i n t e q r a t i o n pe r i od . The v a r i a b i l i t y i s even
g r e a t e r t h a n see i n F igu re 7.41. Monitoring t h e analoq
computer a n a l y s i s i n t h i s . w a y p rov ided i n s i g h t s i n t o t h e
d e p e n d a b i l i t y of t h e e s t i m a t e s ob t a ined . The conf idence i n - t h e s t a b i l i t y o f t h e u'w' cova r i ance and c o s p e c t r a l r e s u l t s
a s opposed t o t h e h e s i t a n c e t o a t t a c h any s i g n i f i c a n c e t o
t h e u ' v ' r e s u l t s , d e r i v e d from moni to r inq t h e analoq analy-
s is , cou ld n o t have been s u p p l i e d by any o r d i n a r y t e s t of
s t a t i s t i c a l s i g n i f i c a n c e .
TABLE 7 .1
EVALUATION OF UNIVERSAL CONSTANT " a "
( U s i n g E = ~ * ~ / k z o n l y )
From L o n g i t u d i n a l Component
N e u t r a l C a s e s U n s t a b l e C a s e s a a -
T e s t - C y c l e s r a d i a n s T e s t C y c l e s r a d i a n s
11-1 0.190 0 .646 Avg N e u t r a l 0.163 0.555 ~ v g U n s t a b l e
From V e r t i c a l C o m m n e n t
N e u t r a l C a s e s a
T e s t Fyc2es radians
11- 1 0.146 0.497 Avg N e u t r a l 0.439
From L a t e r a l C o m ~ o n e n t
Unstable C a s e s . . a
T e ~ t - C y c l e s r a d i a n s
7- 1 0 . 1 2 1 0 .412 7-2 0 .126 0 .429
11-2 0 .122 0 .415
Avg Unstable 0.123 0.119
N e u t r a l C a s e s U n s t a b f e C a s e s a a
T e s t - tycles radians T e s t - C y c l e s radians
w r l m r l r l m m m m m m N . . . . a .
r l r l r l - 4 m l - I
w o o b m m b m b w r l m r l m m w g r 4 N - J ' 0 0 0 b 0 0 0 . . . . . . . 0 0 0 0 0 0 0
1 1 1 1 1 1
m m N W N b W b CO O m O b 0 0 0 4 o m O N m O O O O O O N O . . . . . . . 0 0 0 0 0 0 0
1
m m w 0 w 0 0 3 N U 0 4 0 4 o o o r l 5
- 3 - 3
. - .
m w o O O C . . . 0 0 0
N m 4 C O w w w m o O N W O 0 0 0 0 0 0 0 0 . . . . 0 0 0 0 I I
In m o w m m w m t n o . . . 0 0 0 I l l
r l o m m a 4 w a d . . * 0 0 0
- 3 - 1 I '
N
3 t,
c, - ' a b CON m k
a 4 4 b C 2 W c . I '4 Q) m m m 4 4 4 ~ .G c
m b w w t n t n w m t n m w 4 0 0 0
* . . * 0 0 0 0
1 1 1 1
0 4 m ~ ~ l z m r n ~ m d 4 0 . . . . 0 0 0 0
m O I N b w o w
0 . 0
N m o
m o w w b w N 0
$ c o r n 4 4 0 * * * . N l 0 0 0 0
N 7 t,
m t n w m w m o N t n N N
* o r . 4 0 0 0
0 . 0 0 5 0 .01 0 .1 1 1 0
n ~ / i ( c y c l e s )
X
N e u t r a l Tes ts
A 5 - 1
0 5 - 2
A 5 - 3 0 0 6 - 1
- X 6 - 2
- 6 - 3
l 1 1 - 1 -
- -
- rn 0
A A
- A A &
l - A -
A a 0
- A rn A
- X
- A 0 l
0 x e.p R
- X x ti X 0 & A 0 A
- 0 0 O r n A A A
0 0 0 -
-
- X x .A0
A e - a O o Q e 0 -
0 6 . . A
- - 'w
x a - - l l l
FIGURE 7.1 Longitudinal Hind Component Spectra - Neutral Tests
I , I l l I I I , I I I I 1 I 1 I 1 I 1 I 1 1 1 1
U n s t a b l e T e s t s
A 7 - 1
FIGURE 7.2 Longitudinal Wind Component Spectra - Unstable Tests
FIGURE 7.3 Longitudinal Wind Component Spectra - Stable Tests
0.25
0.20
0.15
0.10
0.05
- S t a b l e T e s t s
- q 0 2 - 1
-
0 0 A 2 - 2
- 0
q 2-3 A -
A q X 4 - 1
0 O - 0
0 0 O - & x q
O X A A - q
0 X -
q @ o
o 0 x x q - h
q 0 fi A -
q q q
- 0 X ax A
- 0 e0 0 0
- q : A
X 0 - x x
- q 2 00
9( 0 -
q %?: -
x %,A - q x, AO
X A
- xX'b - 0
x yo$ - O C O
- 0 * o P w X
- A
O - ' I " I I 1 I 1 1 I 1 I I I l I l l
0.005 0.01 0 .1 1.0 10
T e s t s A b o v e 1 2 M e t e r s
0 9 - 1
A 1 2 - 1
0 1 2 - 2
X 1 3 - 1
an] s t a b l e T e s t s
U n s t a b l e T e s t s
/ N e u t r a l T e s t s
n z / U ( c y c l e s )
FIGURE 7.4 Longitudinal Wind Component Spectra - Tests Above 12 Meters
FIGURE 7.9 Vertical Wind Component Spectra - Neutral Tests
0.25
0.20
0.15
n SJ n,
ow2
0 .10
0.05
0
- A N e u t r a l T e s t s
- 0 . A x A 5 - 1 A n o
- A
0 5-2 Z A o 6. . -
A 5 - 3 " ". * & O A O -
o 6 - 1 .
e . - A . O O
0 : x 6 - 2
x x AX& A - 6 - 3 x
- d o . %be . 1 1 - 1 A . X. A
- 0. X~ O A W
.X A
- O * + m A A 'b .
OAX - A %'bB A.
- o b Q A *
- :A b ' q x 0 !b - A A 0
A @A o 8 0 AX -
O xA 0 A • - 0
0 0 . Q" A m . -
A a -
A * X m 0
- x ", . . 8 O
. - . . - ~b A .p
A
- 0 A . X O
A '3 px. q
a 0: -
' I " 1 I I , , I I 1 l l l l I I I I I I , _
0.005 0 . 0 1 0 . 1 1.0 10
~ Z I U ( c y c l e s )
X 0
- U n s t a b l e Tests o 0
0 X - A 7-1
0 q 7-2 o A
0 - x 11-2 0 0
X X
- X A X x
- X
A A A X
- 0 0 A A ~ A 0
X
x A q
A
x q A Xa
A A X O
0 A
X q !A X A 0
- A X 0 - q A
- Q X
xA A A X
q X A X
A A X
A A
00 q X - A
d X X
A 0 0
- 0 0
3
1 , , , # I I I I I I I 1 I I , I I I I , , -
0.10 1 .0
~ Z I U ( c y c l e s )
FIGURE 7 . 1 0 V e r t i c a l Wind Companent Spec t r a - Unstable T e s t s
~ Z I U ( c y c l e s )
F I G U R E 7 . 1 1 V e r t i c a l Wind Component S p e c t r a - S t a b l e T e s t s
L a
S t a b l e T e s t s
0 2 - 1 0
A 2 - 2 x 0 A
0 0 2 - 3 x
X 4 - 1 0
o O A
aO- Q Y
0
I
s t a b l e Tests
U n s t a b l e Tests x
0 N e u t r a l Tests X
Tests Above 12 M e t e r s
0 9 - 1
A 1 2 - 1
FIGURE 7 . 1 2 V e r t i c a l Wind Component Spec t r a - T e s t s Above 1 2 Meters
0.5
0.1
n S w ( n '
c w 2
0 .01
FIGURE 7 . 1 4 V e r t i c a l Wind Component S p e c t r a l - Unstable T e s t s
w W in
0 .005 0 .01 0.1 1.0 10
- U n s t a b l e Tests
A 7-1
7 - 2
x 1 1 - 2
X
- -
A x ~ o -
X A - 0 0 0
- A 0 A
A A - 0
0
0
q
11216 ( c y c l e s )
I I l l I I I I , I , I 8 I I I I I I l l I I l l
-
- X N e u t r a l T e s t s
X A - A 5 - 1 . - 0 5 - 2
A 5 - 3 - X .. . .
q 6 - 1
- ' X I x 6 - 2 . X - x 6 - 3
A
X . 1 1 - 1
- x o A x + x 0 a %A A A
0 . I . -
A O x O A " & X O A . 0 .
- x o 8 . A ¤
0 - A - " 4 8 O 0 .
0 - 30 "'Be A x " '
A A 0 .
- . O 43 h . A 0 A 0 0
A r x
A A
A A A
+ . oA - O b . 8
0 A 0 0 A
A 0 A 0
A 0 A 0 A
- 0 x 0 '3 '
0 IC
-X 0 6; . A m . . - .
X O b P
- 0 O X . . A .
PC O A . .
- 0 *o
1 1 , I L I 1 1 1 1 I 1 1 I ,I I I I I I _
0 . 1
n z / U ( c y c l e s )
F I G U R E 7 . 1 7 L a t e r a l Wind Component S p e c t r a - N a t u r a l T e s t s
S t a b l e T e s t s
0.1 1.0
~ Z I U ( c y c l e s )
F I G U R E 7.19 L a t e r a l Wind Component Spec t r a - S t a b l e T e s t s
0
T e s t s A b o v e 12 M e t e r s
0 9 - 1
A 1 2 - 1
0 12-2
X 1 3 - 1
m] s t a b l e T e s t s
a U n s t a b l e T e s t s
N e u t r a l T e s t s
1 .0
nzl6 ( c y c l e s )
FIGURE 7.20 L a t e r a l Wind Component S p e c t r a - T e s t s Above 1 2 Meters
X - -
X -
X -
* - * v
v - a
2 C a
a X
A N N + , I ,
t - t - d X
d a
a
a o x X 0 a
3t a
-
0 - )#
X B - -
X a a
a x B
a a X
a T X C3
a a o
X
44 x @ O
- x
a 0% -
a -
)@ 0
a X B
a -
X 0
X a
a -
0
a
a o
a
- - - -
I I I I
~n ( U ) 3 u -
T e s t s A b o v e 1 2 M e t e r s
0 9 - 1 a 1 2 - 1
1 2 - 2
~ Z I U ( c y c l e s )
FIGURE 7 . 3 6 Cosvectra Between Longi tudinal and L a t e r a l Veloc i ty - Tes t s Above
Average N e u t r a l C u r v e w i t h 80% Conf idence I n t e r v a l
;-A --- Average Unstab le C u r v e - --- with 80% Conf idence I n t e r v a l
: Average Stable C u r v e .. >.:, :'..... ::::.: ...: : :.:.; .. . .. .. ...... :.:. . . .-.:.:..: ................ w i t h 80% Conf idence I n t e r v a l
FIGURE 7 . 3 8 Average L o n g i t u d i n a l Wind Component Spec t rum
A v e r a g e N e u t r a l C u r v e w i t h 80% C o n f i d e n c e I n t e r v a l
- - _ _- - - A v e r a g e U n s t a b l e C u r v e - - ---- w i t h 80% C o n f i d e n c e I n t e r v a l
m - - * A v e r a g e S t a b l e C u r v e w i t h 80% C o n f i d e n c e I n t e r v a l
F I G U R E 7 . 4 0 Average L a t e r a l Wind Component Spectrum
V I I I . SUMMARY AND CONCLUSIONS
Turbulence d a t a ha s been ana lyzed f o r power s p e c t r a l
d i s t r i b u t i o n s , c o s p e c t r a l d i s t r i b u t i o n s and Reynolds' stresses
us ing an analog computer. One of t h e most s i q n i f i c a n t r e s u l t s
found i n t h e s t u d y was t h e degree of s i m i l a r i t y shown between
t h e power s p e c t r a l d i s t r i b u t i o n s for any g iven component a s -
w e l l a s f o r t h e c o s p e c t r a l d i s t r i b u t i o n s of u'w'.. S i m i l a r i t y
of s p e c t r a l d i s t r i b u t i o n s is m o s t obvious and c o n s i s t e n t f o r
t h e v e r t i c a l component w i t h o n l y a few d e v i a t i o n s . Measure-
ments a t 3.0 and 6 .1 metqrs y i e l d t h e same d i s t r i b u t i o n o f
normal ized s p e c t r a l energy , nSw(n)/uy2, as a f u n c t i o n o f
normal ized wave number, nz/n, f o r n e u t r a l and u n s t a b l e con-
d i t i o n s w i t h o n l y one u n s t a b l e c a s e a t 6 .1 meters suqges t i ng
t h e p resence of convec t i ve energy a t low wave numbers. The
u n s t a b l e tests a t 12.2 and 87 meters, however, wh i l e corres-
ponding t o t h e s i m i l a r shape a t h igh wave numbers i n c l u d i n g
t h e mechanical energy peak, demanbtra te a s i g n i f i c a n t departure
due t o t h e convec t i ve energy i n p u t a t low f r e q u e n c i e s , r evea l -
i n g a convec t i ve energy peak n o t c l e a r l y i d e n t i f i e d pre-
v ious ly . The s t a b l e t e e t s show ove r t h e e n t i r e wave number
range o n l y a s l i g h t s h i f t t o h i q h e r nz/D v a l u e s a t 3.0 and
6 .1 meters b u t a t 12.2 meters a v e r y obvious s h i f t t o high
wave numbers demons t ra tes t h a t t h e eddy s i z e s are s c a l e d
n o t s imply accord ing t o h e i g h t b u t a l s o by s t a b i l i t y .
The l o n g i t u d i n a l component s p e c t r a a r e a l s o s i m i l a r
although a t low wave numbers increased v a r i a b i l i t y i s noted.
Again, t h e s t a b l e cases show only a s l i g h t s h i f t t o h iqher
wave numbers f o r t h e low l e v e l tests b u t a l a r q e s h i f t i s
observed f o r t h e 1 2 . 2 meter test. The peaks i n t h e lonqi-
t u d i n a l s p e c t r a occur g e n e r a l l y a t nz/u = 0.03, much lower
than t h e v e r t i c a l component s p e c t r a l peaks nea r nz/g = 0.4,
The l a t e r a l component s p e c t r a a l s o show some s i m i l a r -
i t y , p a r t i c u l a r l y above nz/U = 0.1, Below t h i s p o i n t , how-
e v e r , a g r e a t d e a l of v a r i a b i l i t y is observed. The s t a b l e
tests e x h i b i t t h e b e s t s i m i l a r i t y and, a s wi th t h e o t h e r
components, a r e s h i f t e d t o somewhat h igher nz/g than t h e
n e u t r a l and uns tab le c a s e s ,
The minus f i v e - t h i r d s law i s followed above about
nz/u = 1.0 f o r t h e v e r t i c a l component s p e c t r a and extends t o
a s l o w a s nz/u = 0.2 f o r t h e l o n g i t u d i n a l component s p e c t r a
f o r uns t ab le cases , t o 0.4 f o r t h e n e u t r a l c a s e s , whi le
t h e s t a b l e tests agreed only above nz/u = 1.0. However,
t h e cospec t ra became zero only above nz/u = 1.0 t o 3.0, This
l i m i t a t i o n on t h e i n e r t i a l subrange has n o t g e n e r a l l y been
exper imenta l ly determined. The l a t e r a l compclnent s p e c t r a
f o r t h e 3.0 and 6 . 1 meter l e v e l s f a i l e d t o agree wi th t h e
minus f i v e - t h i r d s l a w wi th in t h e frequency range of a n a l y s i s
except i n t w o caser .where it was observed above nz/U - 3.0.
A t g r e a t e r h e i g h t s t h e v ' r e s u l t s are aga in inconclusive.
The l i g h t wind speed uns tab le t e s t a t 12.2 meters follow8
t h e minus f i v e - t h i r d s law beyond about. 1.0 b u t t h e o t h e r
uns t ab le tes t a t 12.2 meters does n o t f i t t h e minus
f i v e - t h i r d s law w e l l . The 87 meter c a s e f i t s p o s s i b l y on ly
beyond nz/v = 10.
Eva lua t i on of t h e u n i v e r s a l c o n s t a n t , a , f o r t h e
i n e r t i a l subrange r e l a t i o n s h i p l e d t o r e s u l t s , on t h e aver -
aqe , c o n s i s t e n t wi th p r ev ious e v a l u a t i o n s q e n e r a l l y around
a = 0.138. However, t h e r e s u l t s from t h e u ' component
s p e c t r a l e d t o a va lue abou t 20% t o o l a r q e and t h e v e r t i c a l
component s p e c t r a l e d t o v a l u e s abou t 10% low. The t w o
l a t e r a l component s p e c t r a from which e v a l u a t i o n s cou ld be
made qave t h e expec ted r e s u l t .
Dependence of t h e i n e r t i a l subranqe on c2 l3was a l s o
q u i t e w e l l e s t a b l i s h e d through t h e agreement between t h e
measured power s p e c t r a and t h e u n i v e r s a l normal ized expres -
s i o n f o r t h e subranqe, Equat ion (6 .3b) .
Reynolds' stress c a l c u l a t i o n s show t h e measured f r i c -
t i o n v e l o c i t y t o be c o n s i s t e n t w i t h independen t ly measured
wind speeds . R a t i o s of a&* w e r e found f o r n e u t r a l condi-
t i o n s t o be q u i t e c o n s t a n t a t abou t 1.33. Uns tab le t es t
r a t i o s a g r e e w i t h t h i s v a l u e b u t s t a b l e tests o f f e r somewhat
h i g h e r r a t i o s . The r a t i o s o f oU/u* and av/u*, averag ipg
2.9 and 2.0 r e s p e c t i v e l y f o r n e u t r a l c o n d i t i o n s show no
obvious h e i q h t , s t a b i l i t y o r wind speed dependence b u t t h e
v a l u e s va ry somewhat more t h a n t h e a,/u* r a t i o s .
I n t r e a t i n q t h e d a t a some d e f i n i t e advantaqes i n t h e
ana log a n a l y s i s t e chn iques have been noted . The ana loq
computer t r e a t s t h e t u r b u l e n c e s i q n a l s cont inuously , avo id ing
the necessity of discrete samplinq and simplifying the
spectral analysis of turbulence at hiqh frequencies. The
analog computer is easily proqrammed to provide the variety
of analyses necessary and when changes in the programs are
required they can be made easily and tested on the data
immediately with little interruption in the analysis. Any
point in the analysis can be monitored, supplyinq informa-
tion which can suggest program chanqes to improve the
analysis, revealing deficiencies in the data which miqht
have been easily overlooked at the original measurement,
and providing insights into the statistical reliability of
the data. Disadvantages in the analoq analysis techniques
include the considerable time required for analysis and the
limitations on the accuracy of the results. The results of
the present investigation are estimated as being accurate
to within 3 to 5 % in the coordinate transformation and to
within 5 to 10% in the spectral analysis, independent of
the statistical variability. A t the low frequency end of
the spectra the 80% confidence interval only occasionally
falls within *50% of the true average value but at the
center of the range of frequencies analyzed the variations
in the estimates fall within *30% of the true average value
80% of the time for almost all tests. Estimates at the
highest frequency of analysis for all tests are within * 7 %
of the average value 80% of the time, generally being within
t h e l i m i t s of accuracy of t h e a n a l y s i s technique. The
improved d e f i n t i o n of t h e s p e c t r a a t h igh f r equenc i e s due
t o i nc rea sed s t a b i l i t y of t h e e s t i m a t e s was ano ther advan-
t a g e of t h e analoq a n a l y s i s . Because of t h e l a r q e s t a t i s t i -
c a l v a r i a b i l i t y a t low wave numbers t h e s p e c t r a f o r each
component w i th in each s t a b i l i t y qroup w e r e averaged, con-
s i d e r a b l y improving t h e r e l i a b i l i t y of t h e s p e c t r a l d i s -
t r i b u t i o n s .
Fu tu re i n v e s t i g a t i o n s w i l l i nc lude more measurements
of t h e k ind r e p o r t e d here . Dependence of s p e c t r a l charac-
ter is t ics on h e i g h t and on s t a b i l i t y w i l l be f u r t h e r s t u d i e d .
Continued measurement of s p e c t r a l and c o s p e c t r a l d i s t r i b u -
t i o n s f o r t h e t h r e e wind components w i l l be extended t o
s e v e r a l o t h e r h e i g h t s of measurement and t o more extreme
s t a b i l i t i e s . The v a r i a t i o n of Reynolds' stress wi th
s t a b i l i t y should a l s o be c l a r i f i e d i n f u t u r e tests wi th
measurements i n more extreme s t a b i l i t i e s .
Measurements of t empera ture f l u c t u a t i o n s w i l l be
r equ i r ed t o determine p r e c i s e l y s t a b i l i t y a s i n d i c a t e d by
z/L o r t h e f l u x Richardson Number. Furthermore, such
tempera ture measurements, l e ad ing t o t u r b u l e n t h e a t f l u x
de t e rmina t ions , and a d d i t i o n a l t u r b u l e n t wind component
measurements s imul taneous ly t aken a t a number of h e i g h t s ,
l e ad inq t o an a p p r a i s a l of t h e f l u x d ivergence of t u r b u l e n t
energy, can prov ide a more c r i t i c a l e v a l u a t i o n of t h e
t u r b u l e n t energy budget equa t ion and i t s r e l a t i o n t o t h e
form of the spectra.
Instrument comparison studies are also planned for
the future. Comparison of the Hanford wind component meter
with M. Miyake's instrument should be made, identifying the
advantages of each. Comparison with a sonic anemometer is
also desirable.
The Hanford Meteorological tower provides the oppor-
tunity for measuring not only spectra at many heights
through much of the turbulent boundary layer but also the
change in momentum flux and heat flux with height under
various stability conditions. The aircraft operated by
Battelle-Northwest for meteorological studies also offers
opportunities for future comparison of aircraft measured
spectra with tower mounted wind component meter spectra.
The effort directed toward diffusion studies permits
the experimental investigation of turbulence and diffusion
jointly. Experiments with extensive measurements of
turbulence and the resulting turbulent diffusion are planned
for the future at Hanford. Much insight remains to be gained
on the problem of diffusion and its relation,to the turbulent
structure of the atmosphere, particularly in stable condi-
tions (see stewarta2) . Both theoretical and experimental
efforts are required in this direction.
REFERENCES
1. B a t c h e l o r , G . K . ( 1 9 5 6 ) . The t h e o r y o f homoqeneous
t u r b u l e n c e . The U n i v e r s i t y Press, Cambridqe, ~ n q l a n d .
2. Berman, S. (1965) . E s t i m a t i n g t h e l o n g i t u d i n a l wind
spec t rum n e a r t h e ground. Q u a r t . J. Roy. Meteo ro l .
3. Blackman, R. B. and J. W. Tukey ( 1 9 5 8 ) . The measure-
ment o f power s p e c t r a . Dover, New York.
4. B u s i n q e r , J. A. and V. E. Suomi ( 1 9 5 8 ) . V a r i a n c e s p e c t r a
o f t h e v e r t i c a l wind component d e r i v e d from o b s e r v a t i o n s
o f t h e s o n i c anemometer a t O ' N e i l l , Nebraska , i n 1953.
Arch iv f u r Meteor. und K l i m a t . , 1 0 , p. 415.
5. C a l d e r , K . L. ( 1949) . The c r i t e r i o n o f t u r b u l e n c e i n a
f l u i d o f v a r i a b l e d e n s i t y w i t h p a r t i c u l a r r e f e r e n c e t o
c o n d i t i o n s i n t h e a tmosphere . Q u a r t . J. Roy Meteo ro l .
SOC., 75 , pp. 71-78.
6. Cramer, H. E., e t a l . (1962) . S t u d i e s o f t h e s p e c t r a o f
t h e ve r t i ca l f l u x e s o f momentum, h e a t , and m o i s t u r e i n
t h e a t m o s p h e r i c boundary l a y e r . F i n a l Rep., C o n t r a c t
No. DA-36-039-SC-80209. Dept. o f Meteor . , Mass. I n s t .
o f Tech.
7. C r a m e r , H. E. ( 1959) . Measurements o f t u r b u l e n c e s t r u c -
t u r e n e a r t h e ground w i t h i n t h e f r e q u e n c y r a n g e from 0.5
t o 0 .01 c y c l e s sec. " Advances i n Geophys ic s , 6 , p. 75.
8. Cummins, J. C. (1960) . Frequency s p e c t r u m o f alder
H a l l r e a c t o r n o i s e . Rep. AEEW-M-19. Un i t ed Kingdom
Atomic Enerqy A u t h o r i t y , Research Group, W i n f r i t h , D o r s e t ,
England.
Davenpor t , A. G. (1961) . The spect rum o f h o r i z o n t a l
g u s t i n e s s n e a r t h e ground i n h i q h winds. Q u a r t . J . Roy.
Meteoro l . Soc., 87 , p. 194.
E l l i s o n , T. H. ( 1957) . T u r b u l e n t t r a n s p o r t o f h e a t and
momentum from an i n f i n i t e rough p l a n e . J. F l u i d Mech.,
2 , p. 456.
E l y , R. P . , Jr. (1958) . S p e c t r a l a n a l y s i s o f t h e u-com-
ponent o f wind v e l o c i t y a t t h r e e meters. J. Meteorol.,
1 5 , p. 196.
F a v r e , A . , J. G a v i g l i o and R. Dumas (1958) . F u r t h e r
space - t ime c o r r e l a t i o n s o f v e l o c i t y i n a t u r b u l e n t
boundary l a y e r . J. F l u i d Mech., 3 , p. 344.
G i f f o r d , F. , Jr. (1959) . The i n t e r p r e t a t i o n o f meteoro-
l o g i c a l s p e c t r a and c o r r e l a t i o n s . J. Meteoro l . , 1 6 ,
p. 344.
G i l l , G . C. (1954) . A f a s t r e s p o n s e anemometer f o r micro-
m e t e o r o l o g i c a l i n v e s t i q a t i o n , B u l l e t i n A.M.S., 35.
Gurvich , A. S. (1960) . Frequency s p e c t r a and d i s t r i b u -
t i o n f u n c t i o n s o f v e r t i c a l wind components. I zves t ia
ANSSSR, Geophys. S e r . 1960, N o . 7 , p. 1042.
Johnson , C. L. (1956) . Analog computer t e c h n i q u e s .
McGraw-Hill, N e w York.
Kolmogoroff , A. ti. (1941) . The l o c a l s t r u c t u r e o f
t u r b u l e n c e i n i n c o m p r e s s i b l e v i s c o u s f l u i d f o r v e r y
large Reynolds numbers. Doklady ANSSSR, 30, p. 301.
Korn, G. A. and T. M. Korn (1956). ~lectronic analoq
computers. McGraw-Hill,
Lee, Y. W. (1960). Statistical theory of communications.
John Wiley, New York.
Lumley, J. L. and H. A. Panofsky (1964). The structure
of atmospheric turbulence. John Wiley, New York.
MacCready, P. B., Jr. (1953) . Structure of
atmospheric 'turbu~ehee. J- Meteorol. , 10, P- 434. MacCready, P. B., Jr. (1962). The inertial subranqe of
atmospheric turbulence. J. Geophys. Research, 67, p. 1051.
MacCready, P. B., Jr. (1962). Turbulence measurements by
sailplane. J. Geophys. Research, 67, p. 1041.
Miyake, M. (1965) . A constant temperature wind component
meter development and application. Final Rep., Contract
No. AT(45-1)-1545. Dept. of Atmos. Sci., Univ. of
Washington, Seattle.
Monin, A. S. (1959). Smoke propagation in the surface
layer of the atmosphere. Advances in ~eophysics, 6, p.331.
Monin, A. S. (1959) . On the similarity of turbulence in
the presence of a mean vertical temperature gradient. J. - Geo. Res., 64, p. 2196.
Monin, A. S. (1962). Empirical data on turbulence in the
surface layer of the atmosphere. J. Geophys. Research,
67, p. 3103.
Monin, A. S. and A. M. Oboukhov (1954). Basic regularity
i n t u r b u l e n t m i x i n g i n t h e s u r f a c e l a y e r o f t h e atmos-
p h e r e . T rudy Geophys. I n s t . ANSSSR, N o . 2 4 , p. 163 .
29. P a n o f s k y , H. A. ( 1 9 6 1 ) . An a l t e r n a t i v e d e r i v a t i o n o f t h e
d i a b a t i c wind p r o f i l e . Q u a r t . J. Roy. Meteorol. Soc . ,
8 7 , p . 109 .
30. P a n o f s k y , H. A. ( 1 9 6 2 ) . The b u d q e t o f t u r b u l e n t e n e r g y
i n t h e lowest 100 meters. J. Geophys. R e s e a r c h , 6 7 , p.
3161.
31. P a n o f s k y , H. A. ( 1 9 5 3 ) . The v a r i a t i o n o f t h e t u r b u l e n c e
s p e c t r u m w i t h h e i g h t u n d e r s u p e r a d i a b a t i c c o n d i t i o n s .
Q u a r t . J. Roy. Meteorol. S o c . , 79 , p . 150 .
32. P a n o f s k y , H. A . , H. E. Cramer and V. R. K . Rao ( 1 9 5 8 ) .
The r e l a t i o n be tween E u l e r i a n t i m e and s p a c e s p e c t r a .
Q u a r t . J. Roy. Meteorol. S o c . , 8 4 , p. 270.
33. P a n o f s k y , H. A. and R. J. Deland ( 1 9 5 9 ) . One -d imens iona l
s p e c t r a o f a t m o s p h e r i c t u r b u l e n c e i n t h e lowest 100 metres.
Advances i n G e o p h y s i c s , 6 , p. 41.
34. P a n o f s k y , H. A. and R. A. McCormick ( 1 9 6 0 ) . The s p e c t r u m
o f v e r t i c a l v e l o c i t y n e a r t h e s u r f a c e . Q u a r t J. Roy.
Meteorol. Soc . , 86 , p. 495.
35. P a n o f s k y , H. A. and H. P r e s s ( 1 9 6 2 ) . Meteorological a n d
a e r o n a u t i c a l a s p e c t s o f a t m o s p h e r i c t u r b u l e n c e . P rog . i n
Aero. S c i . , 3 , p. 179 . Pergamon P r e s s .
36. P a n o f s k y , H. A. and I. Van d e r Hoven ( 1 9 5 6 ) . S t r u c t u r e o f
s m a l l scale and m i d d l e scale t u r b u l e n c e a t Brookhaven.
S c i . Rep. N o . 1, C o n t r a c t N o . AF 19(604 ) -1027 . P e n n s y l -
v a n i a S t a t e U n i v e r s i t y .
37. P a s q u i l l , F. (1962) . Recent broad-band s p e c t r a l measure-
ments o f t u r b u l e n c e i n t h e lower a tmosphere . J . Geophys.
Research , 67, p. 3025.
38. Pond, S . , R. W. S t e w a r t and R. W. B u r l i n q (1963) . Turbu-
l e n c e s p e c t r a i n wind o v e r waves. J. Atmosph. S c i . , 2 0 ,
39. P r i e s t l e y , C. H. B. (1959) . T u r b u l e n t t r a n s f e r i n t h e
lower a tmosphere . U n i v e r s i t y o f Chicago P r e s s , Chicago.
40. R a t c l i f f e , C. A. and E. M. Sheen (1964) . Wind component
a n a l y z e r improvements. Unpublished r e p o r t . G e n e r a l
Electric C o . , H.A.P.O., R ich land , Washington.
4 1 . R a t c l i f f e , C. A. and E. M. Sheen (1964) . An a u t o m a t i c
d a t a c o l l e c t i o n sys tem f o r m e t e o r o l o g i c a l tower i n s t r u -
m e n t a t i o n . J. Appl. Meteorol., 3 , p. 807.
42 . S t e w a r t , R. W. (1959) . The problem of d i f f u s i o n i n a
s t r a t i f i e d f l u i d . Advances i n Geophysics , 6 , p. 303.
43. T a y l o r , G. I. (1938) . The spec t rum o f t u r b u l e n c e . - Proc.
Roy. Soc., A164, p. 476. . .
4 4 . Webb, E. K. (1955). A u t o c o r r e l a t i o n s and e n e r g y s p e c t r a
o f a t m o s p h e r i c t u r b u l e n c e . C.S.I.R.O. Div. M e t . Phys.
Tech, Pap. N o . 5. Melbourne.
45. Zubkovski, S, L. (1962) . Frequency s p e c t r a o f t h e h o r i -
z o n t a l wind-speed component close t o t h e ground.
I z v e s t i a ANSSSR, Geophys. Se r . 1962, N o . 1 0 , p. 1425.
APPENDIX A
For an a p e r i o d i c o r t r a n s i e n t f u n c t i o n of t i m e , f ( t ) ,
where f ( t ) d t i s f i n i t e , t h e F o u r i e r i n t e q r a l r e p r e s e n t a - -OD
t i o n is g i v e n by
where
(A. 1)
(A. 2 )
Equa t ions ( A . 1 ) and (A.2) form a F o u r i e r t r a n s f o r m p a i r . The
complex con t inuous spect rum, F ( a ) , o f t h e a p e r i o d i c f u n c t i o n ,
f ( t ) , i s a complex q u a n t i t y , i .e.
W e can see t h a t F(w) is i n d e e d t h e c o n t i n u o u s ' spect rum o f
f ( t) from Equa t ion (A. 1) which s t a t e s t h a t f (t) i s composed
of an i n f i n i t e number of s i n u s o i d s , eiwt, o v e r a con t inuous
i n f i n i t e r ange o f f r e q u e n c i e s , each w i t h a n i n f i n i t e s i m a l
ampl i tude F ( o ) do.
C o n s i d e r i n g , now, t w o a p e r i o d i c f u n c t i o n s f l ( t) and
f ( t) w i t h r e s p e c t i v e spect rum f u n c t i o n s F l ( w ) and F2 ( w ) , w e
d e f i n e t h e c o v a r i a n c e between them as
Through t h e Four i e r i n t e g r a l r e p r e s e n t a t i o n s of f l ( t ) and
f ( t) , t h i s can be re-expressed a s
( A . 4 )
where Fl ( o ) r e p r e s e n t s t h e complex conjuga te of F l ( U ) and
where w e have de f ined t h e s p e c t r a l d e n s i t y f u n c t i o n a s
'1 2 (o) 2 s Fl o F2 ( Y ) (A. 5 )
W e s e e from (A.4) t h a t 4 l 2 ( f ) i s t h e F o u r i e r t rans form of
@ 1 2 ( w ) SO t h a t t h e i n v e r s e r e l a t i o n s h i p must ho ld , i .e.
When bo th a p e r i o d i c func t ions a r e t h e same, Equation (A.3)
d e s c r i b e s t h e autocovar iance func t ion f o r an a p e r i o d i c
func t ion .
With t h e energy d e n s i t y spectrum,
(A. 7)
(A. 8 )
t h e au tocova r i ance f u n c t i o n forms a F o u r i e r t r an s fo rm p a i r ,
and
For T = 0, Equat ion (A.9) becomes
(A. LO)
(A. 11)
demons t ra t ing t h a t t h e t o t a l energy, b l l ( 0 ) = f f l ( t )2 d t , -OD
i s d i s t r u b t e d o v e r f requency as d e s c r i b e d by t h e energy
d e n s i t y spectrum. T h i s is P a r s e v a l ' s equa t i on .
When t h e a p e r i o d i c f u n c t i o n s i n (A.3) a r e d i f f e r e n t
f u n c t i o n s o f t i m e , t h e e x p r e s s i o n i s termed t h e c r o s s c o v a r i -
ance and i t s t r ans fo rm i s c a l l e d t h e c r o s s - s p e c t r a l d e n s i t y
f u n c t i o n which, s i n c e t h e c ro s scova r i ance i s n o t n e c e s s a r i l y
an even f u n c t i o n , i s seen t o be a complex q u a n t i t y ,
e 1 2 ( w ) = C 1 2 ( w ) + i Q 1 2 (u) (A. 12)
The r e a l p a r t o f t h e cross s p e c t r a l d e n s i t y f u n c t i o n is
termed t h e c o s p e c t r a l d e n s i t y , and t h e imaginary p a r t t h e
q u a d r a t u r e s p e c t r a l d e n s i t y .
From ( A . 4 ) , t h e t o t a l cova r i ance i s
Before proceedinq t o a d i s c u s s i o n of random c o n t i n u o u s
f u n c t i o n s , it i s a p p r o p r i a t e t.o mention a t t h i s p o i n t t h e
c o n v o l u t i o n i n t e g r a l which w i l l be used l a t e r w i t h a d i s -
c u s s i o n of t r u n c a t e d random f u n c t i o n s . Cons ide r a g a i n , two
a p e r i o d i c f u n c t i o n s of t i m e , f ( t) and f (t) , each w i t h i t s
r e s p e c t i v e t r a n s f o r m Fl (u) and F2 ( w ) , w i t h which t r a n s f o r m
p a i r s such a s (A .1 ) and (A.2) can b e w r i t t e n . The F o u r i e r
t r a n s f o r m o f t h e p r o d u c t of t h e two t i m e f u n c t i o n s i s
Express ing f 2 ( t) w i t h i t s s p e c t r a l r e p r e s e n t a t i o n , r e v e r s i n q
t h e r e s u l t i n g o r d e r of i n t e g r a t i o n and s u b s t i t u t i n g t h e
F o u r i e r t r a n s f o r m of f l ( t) e v a l u a t e d a t t h e f requency (a -c ) ,
w e o b t a i n t h e c o n v o l u t i o n i n t e g r a l ,
( A . 15)
e x p r e s s i n g t h e t r a n s f o r m of a p r o d u c t o f two a p e r i o d i c
f u n c t i o n i n terms o f t h e i n d i v i d u a l t r a n s f o r m s o f t h e func-
t i o n s . T h i s i s o f t e n w r i t t e n s y m b o l i c a l l y as
~ ( w ) = F1 ( w ) * F2 ( a ) (A. 1 6 )
From (A.15) w e can see t h a t t h e c o n v o l u t i o n i n t e g r a l g i v e s
u s a view of one f u n c t i o n of f r equency , a l t e r e d by t h e shape
o f t h e second f u n c t i o n c e n t e r e d a t some g i v e n f requency , m .
Turning now t o random c o n t i n u i n g f u n c t i o n s , w e
cr a c cr Q) 4J ttt 4J w
k Q) t" C C l-i
0 C
C ttt U
$ C U -4 C 3 k 0 w a C rp
V)
2 a 4
4J
(A. 19)
When the two random functions are the same, Equation (A.17)
becomes the autocovariance,
lim - 1 T + 1 1 ( ~ ) =
T+- - fl(t) fl(t+r) dt 2T -T
whose transform is the power spectral density,
(A. 20)
(A. 21)
Again, for T = 0, Equations (A.20) and (A.22) demonstrate
how the average power is distributed over frequency according
to the power spectral density function,
lim 2 /T f 1 2 (t) dt = J ~ ~ ~ ( w ) dw (A. 23) T+m 2T -'I' ,-
When the random functions of time are different, Equation
(A.17) is called the cross-covariance and its transform is
termed the cross-spectral density. Since the cross-covari-
ance is not necessarily an even function (as opposed to the
autocovariance function), the cross spectrum is generally
complex, the real part being the cospectrum and the imaginary
part, the quadrature spectrum, i.e.
(A. 24)
For T = 0 , t h e a v e r a q e c o v a r i a n c e i s seen from Equat ion
(A.17) and (A. 13 ) t o b e t h e i n t e g r a l of c o n t r i b u t i o n s over
t h e e n t i r e r ange o f f r equency
l i m T-+- I 2T $ f , ( t ) f 2 ( t ) d t =
- O D
(A . 2 5 )
When e x p e r i m e n t a l s p e c t r a l d i s t r i b u t i o n s are de te rmined
w i t h d i q i t a l a n a l y s i s methods, E q u a t i o n s (A . 17) th rough (A . 25)
are q e n e r a l l y used , d e t e r m i n i n q c o v a r i a n c e a t v a r i o u s l a q
i n t e r v a l s from t h e d a t a and t a k i n g t r a n s f o r m s o f t h e r e s u l t -
i n q c o v a r i a n c e f u n c t i o n s . However, s i n c e t h e d a t a i s neces-
s a r i l y l i m i t e d i n l e n q t h o f r e c o r d , t h e o o v a r i a n c e f o r a l l
l a g i n t e r v a l s c a n n o t b e de termined and c o n s e q u e n t l y n e i t h e r
can t h e i n t e g r a l o v e r an i n f i n i t e r ange o f l a q s , n e c e s s a r y
f o r t h e t r a n s f o r m a t i o n t o t h e s p e c t r a l d e n s i t y f u n c t i o n .
Blackman and ~ u k e ~ ~ have demons t ra t ed how t h e s p e c t r a l
estimates o b t a i n e d are a c t u a l l y c o n v o l u t i o n s o f t h e t r u e
spec t rum w i t h a " s p e c t r a l window," t h e t r a n s f o r m o f some
" l a g window" which a l t e r s t h e c o v a r i a n c e f u n c t i o n . The l a q
window, which i s m u l t i p l i e d w i t h t h e c o v a r i a n c e f u n c t i o n ,
must a t l e a s t b e a r e c t a n g u l a r f u n c t i o n e q u a l t o one o v e r a
l i m i t e d c o n t i n u o u s r ange o f l a g t i m e and z e r o e l s e w h e r e ,
s imply because o f t h e l i m i t e d l e n g t h o f r e c o r d . However, it
can a r b i t r a r i l y b e s e l e c t e d i n t h e a n a l y s i s i n o r d e r t o
o p t i m i z e t h e t y p e of s p e c t r a l estimate d e s i r e d , which i s
a c t u a l l y an a r e a under t h e s p e c t r a l d e n s i t y f u n c t i o n c u r v e
in the region of some frequency, w, described bv the convolu-
tion inteqral
av[; (u)] =H(u) * @(u) (A. 26)
where av [; (u)] is the ensemble average of the measured
spectral estimates at a given frequency and H(u) is the
spectral window or power transfer function. In addition to
producing spectral estimates containing contributions from
the spectral density distribution over a range of frequency,
the use of a finite length of record, or a sample of the
true time series,. also produces the statistical variability
that accompanies any sampling process.
An equally valid, but less frequently used, technique
for analyzing spectral content of turbulence data is the
filterinq method. This can be performed either electrically
or numerically. However, the following discussion will be
directed toward electrical filterinq, the means by which the
spectral analysis of the present study was carried out.
A filter is a linear system characterized by its
response to a unit impulse function. The unit impulse is
an even function of infinitesimal width and unit area.
With this as an input to a linear system its output is the
unit impulse response. The filtering of a continuous time
series, f(t), can be considered as a convolution of the unit
impulse response, W(t), with the time series so that the
output of the filter can be written as
o r e q u i v a l e n t l y
(A. 2 7 )
For a t i m e series w i t h a z e r o mean v a l u e , t h e a u t o c o v a r i a n c e
of t h e o u t p u t s i g n a l i s
l i m 1
which, a f t e r i n t r o d u c i n g (A.28) and i n t e r c h a n q i n g t h e o r d e r
o f i n t e g r a t i o n , becomes
Express inq t h e a u t o c o v a r i a n c e i n t h e i n t e q r a l above a s t h e
F o u r i e r t r a n s f o r m o f t h e s p e c t r a l d e n s i t y f u n c t i o n , g i v e n
by (A.19), and i n t e r c h a n g i n g t h e o r d e r of i n t e g r a t i o n once
a g a i n , Equat ion (A. 29) becomes
$ o o ( r ) = j Y ( w ) P ( 0 1 Q l 1 ( w ) e i w t d w
-0
(A. 30) 0
= f Iy(w)12 * , , ( w ) e i w t
dw -m
where t h e sys tem o r t r a n s f e r f u n c t i o n , Y ( w ) , i s t h e t r a n s -
form o f W ( t ) ,
(A . 31)
and i s a l s o t h e r a t i o of t h e i n p u t and o u t p u t complex maqni-
t u d e s f o r a s t e a d y s t a t e s i n u s o i d a l i n p u t t o t h e l i n e a r ,
system. I t s con juga t e i s P ( u ) . For T = 0, Equat ion (A.30)
becomes
(A. 3 2
For f i l t e r i n g two t i m e series t o de te rmine t h e c o s p e c t r a l
d e n s i t y d i s t r i b u t i o n , each s i g n a l i s passed through a
s e p a r a t e f i l t e r . Cons ider ing t h a t t h e two f i l t e r s a r e
i d e n t i c a l , t h e o u t p u t s a r e convo lu t i ons of t h e u n i t r e sponse
f u n c t i o n w i th t h e o r i g i n a l t i m e series,
f O 2 ( t) = f W ( E 2 ) f 2 ( t - t 2 ) dC, -0
(A. 33)
(A. 3 4 )
Following t h e same procedure as w i t h t h e au tocova r i ance , t h e
c r o s s cova r i ance f u n c t i o n f o r t h e o u t p u t i s shown t o be
~ W T dw (A. 3 5 ) @ 0 1 2 ( T )
-OD
Aqain , f o r T = 0
and s i n c e t h i s is a r e a l q u a n t i t y on ly , t h e mean o u t p u t
(A. 36)
p roduc t i s a p o r t i o n of t h e a r e a under t h e c o s p e c t r a l
d e n s i t y f u n c t i o n c u r v e , de f ined by t h e shape of t h e f i l t e r
t r a n s f e r f u n c t i o n , o r
f o l ( t ) f o 2 ( t ) = (A . 37) -OD
Up t o t h i s p o i n t , t h e e s t i m a t i o n of s p e c t r a l d e n s i t y
through t h e l i m i t i n q form o f t h e F o u r i e r t r ans form of t h e
random t i m e series h a s n o t been cons idered . The same d i f f i -
c u l t i e s a r i s e w i t h t h i s d i r e c t approach a s t h o s e d i s c u s s e d
by Blackman and Tukey when t h e t rans form of t h e l i m i t i n q
form of t h e au tocovar iance i s used. A s w e t a k e t h e t r a n s -
form of l onge r and l onge r r eco rds of t h e t i m e series, how
c l o s e l y w e a r e approaching t h e l i m i t i n q form of t h e t r a n s -
form i s n o t known; indeed, even though it may be approached
on t h e average , t h e v a r i a n c e of t h e e s t i m a t e s may become
i n f i n i t e . I n s t e a d of cons ide r ing t h e l i m i t i n q form, w e may
cons ide r , u s ing Equat ions (A. 1) through (A , 13) , t h e F o u r i e r
t r ans form of a t r u n c a t e d t i m e series, a t r a n s i e n t f u n c t i o n ,
which i s a c t u a l l y what w e have i n ou r l i m i t e d r ecord .
Blackman and Tukey d e s c r i b e t h i s a s modjfying t h e d a t a w i t h
a " d a t a window," I n f a c t , we may wish t o l i m i t o r a l t e r t h e
r eco rd even beyond what ha s been imposed on it by t h e l i m i t e d
sampling c a p a b i l i t i e s , e .g. t h e o r i g i n a l r e co rd may be passed
through a f i l t e r , I n t h i s case, t h e t i m e series can be
thought of a s be ing a l t e r e d by some weigh t ing f u n c t i o n ,
w (t) , such t h a t
The F o u r i e r t r an s fo rm of t h e t r a n s i e n t f u n c t i o n r e p r e s e n t i n q
t h i s sample of t h e t ime s e r i e s i s
A
F . (o, t) = f W ( r ) f . (t-r) e-iUr d r 3 -OD I ( A . 38)
When t h e o r i g i n a l t i m e series i s m u l t i p l i e d by a s i n e func-
t i o n and by a c o s i n e f u n c t i o n f o r some f i x e d f requency , U ,
and each p roduc t i s passed throuqh a f i l t e r , t h e o u t p u t s a r e
t h e r e a l and imaginary p a r t s o f Equat ion (A.38) a t some
i n s t a n t , t. For a second t i m e series t r e a t e d i n t h e same
manner, t h e f i l t e r o u t p u t s a t some i n s t a n t , t , a r e t h e r e a l
and imaginary components of t h e sample F o u r i e r t r an s fo rm
A Fk(w , t ) = f W ( s ) f k ( t - s ) e - i u s d s (A. 39)
-0
Taking t h e p roduc t of Equat ion (A. 3 8 ) w i t h t h e con juga t e of
Equat ion ( A . 3 9 ) and t hen ave rag ing , i n t h i s c a s e o v e r a l l
t h e sample p r o d u c t s be ing produced con t i nuous ly a s a f u n c t i o n
of t i m e by t h e f i l t e r s , we have , a f t e r changing t h e o r d e r o f
i n t e g r a t i o n
l i m 1 A A - i j ( u , t ) Fk(wrtJ d t = a j k ( w ) T-tw T
-T/2 (A. 40)
A where ( w ) i s t h e measured s p e c t r a l d e n s i t y and D ( r ) , f o r
jk i d e n t i c a l f i l t e r s , i s t h e au tocova r i ance of t h e u n f t r e sponse
f o n c t i o n of t h e f i l t e r , \ r J ( t ) . Then, from t h e convo lu t ion
theorem w e know t h a t t h e measured s p e c t r a l d e n s i t y i s
A
( u ) = H ( u ) * a j k ( u ) = j k
(A. 4 1 ) -OD
where H ( u ) i s t h e power t r a n s f e r f u n c t i o n o r s p e c t r a l window,
t h e t r a n s f o r m of D ( r ) , and a ( u ) i s a g a i n t h e t r u e s p e c t r a l j k
d e n s i t y . Thus, t h e t e c h n i q u e ' d e s c r i b e d above f o r o b t a i n i n g
s p e c t r a l e s t i m a t e s by t h e d i r e c t method, s t a r t i n g w i t h t h e
F o u r i e r t r a n s f o r m of a t r u n c a t i o n o f t h e o r i g i n a l t i m e series
y i e l d s r e s u l t s s i m i l a r t o t h o s e o b t a i n e d w i t h t h e i n d i r e c t
method where t h e t r a n s f o r m of t h e c o v a r i a n c e f u n c t i o n f o r
r e c o r d s of f i n i t e l e n g t h i s used t o o b t a i n s p e c t r a l e s t i m a t e s .
ACKNOWLEDGMENTS
To P r o f e s s o r F r a n k l i n I. Badqley, chairman of my t h e s i s
supe rv i so ry committee, goes my deepes t a p p r e c i a t i o n f o r h i s
p a t i e n c e , f r i e n d l i n e s s , and encouragement a s w e l l a s f o r t h e
t e c h n i c a l i n s i q h t he o f f e r e d i n gu id inq my r e s e a r c h e f f o r t s .
P r o f e s s o r H. A. Panofsky was most h e l p f u l i n reviewing *
t h e s t u d y , o f f e r i n g e n l i g h t e n i n q s u g q e s t i o n s , and r e l a t i n g 6 '
r e c e n t work by o t h e r i n v e s t i g a t o r s n o t y e t pub l i shed .
To M r . James J. Fuquay, who c o l l a b o r a t e d a t Hanford
on t h i s s t u d y , I am most t h a n k f u l . Without h i s i n i t i a l
developmental work on t h e wind component meter, t h i s s t u d y
would n o t have been p o s s i b l e . A s my s u p e r v i s o r a t Hanford,
he gave impetus t o t h e work and a freedom of r e s e a r c h e n a b l i n g
t h e s t u d y t o proceed unimpeded.
My s i n c e r e s t a p p r e c i a t i o n goes t o t h e s t a f f o f t h e
Atmospheric Sc iences S e c t i o n of Ba t te l l e -Nor thwes t who helped
immeasurably i n t h e f i e l d and d a t a r e d u c t i o n work. M r . P au l
Nickola o f f e r e d h i s c h e e r f u l a s s i s t a n c e i n p rov id ing mean
wind and t empe ra tu r e p r o f i l e d a t a . M r . J. W. S l o o t , M r . R. L.
Conley, M r . B. N. Nelson, and M r . D. M. Hughey a ided i n t h e
d a t a r e d u c t i o n most c o n s c i e n t i o u s l y .
Thanks qo t o M r . Pau l Dionne and M r . J. Draper o f
t h e ana log computer f a c i l i t y a t Hanford who were most he lp fu l
and c o n s c i e n t i o u s i n programming and o p e r a t i n g t h e analog
computer d u r i n q t h e t e d i o u s d a t a a n a l y s i s pe r iod .
188
DISTRIBUTION
Number o f C o p i e s
2 A i r F o r c e C a m b r i d ~ e R e s e a r c h L a b o r a t o r i e s
D. A . Haugen J . C . Kaimal
Atomic Ene rgy Commiss ion, Wash ing ton D i v i s i o n o f B i o l o g y and M e d i c i n e
J . Z . H o l l a n d
Bed fo rd I n s t i t u t e o f Oceanography P . 0. Box 1006 Da r tmou th , N . W . Canada
S . D . S m i t h
Booz A l l e n A p p l i e d R e s e a r c h I n c o r p o r a t e d 6151 West C e n t u r y Los A n g e l e s , ~ a l i f o r n i a 90045
R . C a l e
D i v i s i o n o f T e c h n i c a l I n f o r m a t i o n E x t e n s i o n
New York U n i v e r s i t y G e o p h y s i c a l S c i e n c e L a b o r a t o r y 2455 Sedgwick Ave. Bronx , New York
A . D . K i rwan , J r .
P e n n s y l v a n i a S t a t e U n i v e r s i t y D e ~ a r t m e n t o f M e t e o r o l o g y -,
~ n i v e r s i ty P a r k , P e n n ,
H . A . P a n o f s k y
R i c h l a n d O ~ e r a t i o n s O f f i c e
R . K . S h a r p T e c h n i c a l I n f o r m a t i o n L i b r a r y
T r a v e l e r s R e s e a r c h C e n t e r , I n c . H a r t f o r d C o n n e c t i c u t
G . R . H i l s t
189
Number of C o ~ i e s
1 University of British Columbia Institute of Oceanography Vancouver, British Columbia
R. W. Burling
112 Battelle-Northwest
G. M. Dalen P. J. Dionne C. E. Elderkin (100) J. J. Fuquay R. S. Paul C . L. Simpson Technical Information Files (5) Technical Publications (2)
