I ~I_
I
__ __
_______
SMrUCTULE E AUJIWEIC TUB3ULCE
MIN T
SURACE IAYL&
by
ROBIR AUMSBRONG DEBICKSOI~ Jr.
Uzited States Military Academy
(1949)
ILS., Neo York University
(1954)
B.8.,
SUBITED = PARTIAL FPUPIIeuT F THE R(QUIRU
FOR TES DEGREa
OF DOCTOR OF PnIUsoarIY
at the
MASSACIUSETTS INSTITUT OF TECBNOLOGY
June, 1964
Signature of Author ...................................................
Department of Meteorology, May 15, 1964
Certified by .. 0,o.....,,..............................,,
....
Thesis Supervisor
Accepted by ............................................
........
Chairman, Deparmeantal Comaittee on
Graduate Students.
S11UC13tRB C AW8MOBE1IC TU1BULENCE IN TIE SRPACE LAYE
by
Robert A, Derrickson
Jr.
SuMbl
ed to tthe Department of Metterology an Mca 15, 1964
in pgatial ftlfillme
t of the ,quirtement for the dgrc
of Doctor of hjilosophy
PoWr spectra and cospectra of the orthoMonae v
Mlocity c ponnts
and temperature for twenty one-hour runs at 16 and 40 mrters height (data
taken by Dr. R.l. Cramsrls group at Round Hill Field Station [M.I.T.,
South Dartmouths, as.) are anaynd, using a var ety of nor~Umit
ations and
coordinate systems. The dependability of the data, particularly in the low
frequency range under convective conditions ps
onsidered.
Algebratc functions for power spectral data in Monil.r-khov
sltLlarity coordinates are obtaind etirically; graphical fmoutions for the
sae data, using
and (A'T) -normanistion, are obtained by egression
analysis. Both these sets of functions have stability as a parameter.
The standard deviation of the a~aunth angle, e , was selected frm a
nuber of possibilities, as the most practical parameter for aqreasita
stratification. Site roughness Is described by the standard deviation of
the asuth angle under neutral condition on,
.
Using three separate criteria, the Isotropic range in the atmosphenic
surface layer is defined.
The observed energy spectral density functions are crmpard to
several theoretical functi'os with the following results: ollgoroff
or Heiacberg spectra ( (n) o" r- 1 3
) agree with thze data within limits
of observational eror, in the isotropic range. Below the lower limit of
3otropy but at higher fraqusences than where there to aEchani al enert
input, the Eraicbnan spectra ( E) o- < ?t 4~
) is the closest to the
observed data. In the low frequency range none of these spectra are valid
since this region Io dominated by stability dependence and lower bboundary
effects,
h1biting very little energ under stable canditions and a large
peak of very low tfrqueacy eneray under unstable conditions.
A model is suges tad viewing the U 2-nomaLsed enersy apectral
density functions as the consequence of mechanical site dependent roughness
energy input and covective stability dependent energy input and the subsequent inertial tvansfer of energy to higher frequencies. Both mechanical
roughness and the convective citation are expressed in terms of the
frequencaes they involve.
Thests Supervisors Edwurd N. Lorena, Sc.D.
Title: Professor of Meteorology
Page
Table of Contenta
1o
II.
I1I.
lY.
V.
Introduction
9
14
Objectives of the research.
A teview of perinent literature
A. B3tef histoUtala revie of research in a ophe
B. Outline of various approaches to the problem.
C. Resumes of selected papers.
turbee.
Data cootlon and processing techniqes.
A. Detailed deascription of the meauremnte
3. kplanation of the data procesi g
C. Data reltability.
Desoription of the research.
A. Method of approach.
B, Defnition of erived ta t enee statistics.
C. Choice of paramaters descIbing effects of stratification
and the lower boundary.
D. Preentation of power Opectra in sJatiirty coordinates.
e. Results
1) Algbraic epreissions for energy epeetral density functions
in smilarity coSodinates.74
2) Varicus athods of estimating stress and heat flu .
3) Regression analysis of paer spectra.
4) Discussion of normalisin tednquee.
5) Lower bound of isotropy In the surfrt e Iayer.
6) Form of spectra at interadiate frequenates,
7) 'Poa of spectra at low frequeniaes.
8) Catparvson of results with those of other ivestigators.
thn rettea predictsies.
roeats
re ts
F. Caf
0. A ozmposto model of energy spectra density fwactions in the
surface layer.
11.
V.
VIII.
IX.
1.
16
16
18
38
38
46
46
55
63
64
74
93
105
116
117
118
122
134
Somnary and coamlusons.
136
Suaggested further research.
140
Aclmkcedg ents.
142
Appndix
A. Resuuas of papers (continuation of part XII e.)
B. List of algebrat expressions for the ortogonal coma~ponesa
ates.
lata y cr
a
in s
and te perat e Po
altw.
Stress -0
C.9 Various methods of es tibt
143
143
ibiography.
145
132
Page
4List of Tables.
Table 1 .
Saple computation sheet for the Mloin-"kumht somilairty
no elisation (Run 67 A, 16 meters)
Table 2.
Sm le computt ion sheet for the mloninC0bukha
noImaU at ona(an 67 A, 40 mtters)
Table 3.
General statiastiCs for the twenmty m , Part A.
Table 4.
General statistics for th teaty Tm e,
Table 5.
Gneal statistics for the twenty ruy , Atsi ia&ry data.
Table 6.
VeaTtial preofle data fo r
horisontal wind.
Table 7.
Total variance of u, v v and T flutatstons before and
after ittar g (hithupai)n
Table 8.
2-nomalssed enery spectral denalty value
tfrequencies (na/I).
Table 9
Derived data for power spectral deseity versus frequency
a.
b.
47
s~m.arity
49
Part: B
ean temperaue and mean
51
52
for selected
88
89
from regrseeton
frao algebrat functions
Isotropi
Table 11.
Values for average slope and values for a/, denn
the stnt of the varws regionV i otropic data and
untifm slope data swmary.
110
A composite of our sall tables (tables 12, 16, 17 and 18).
112
Table 12.
Pe cent probability of observig a coherence as high as
hen the true value is sero.
12a
Table 16
Averag values for 4i/ at the low frequency Iai t of the
unifom elope cregion
112
Average values for the uniform slope, just below the lower
limit of isotropy.
112
Table 17.
rastos as a function of us/
109
Table 10.
Table 18.
Period assooated with oad=
Table 13.
Cohmence "values for-, Wfunctio
flc.u atirrom
period.
for fie nm.
of the7ltonmation.
s-a
of
113
Table 14.
OT coherence as a function of period.
114
Table 156
uT coherence a a
115
function of period.
.1
List of FP1iris.
Figure 1.
Qualitative graph of kinetic enrgy of surface winds as
a f uction of the tifr scale.
Flguea 2.
Scheatic of energy spectral density verwus £sequnoy
riureo
FL~uwe
Theoretica heat fltt as a function of i
iL ta d
maanad1ni
a1 ootrdiates.
Response of stattictal
fiter
verwts frequency, n.
Stndeax deviaton of aalnth angle verus
4 and 32 rmters,
F1iwo 7.
Schnmtic of ?tu)/&I versus log a/
diferent roughness.
itgure 8.
23
32
.
General fuactional fo~ of nIn-d ansional energ spectral
density on nondimenacal frequency and stabilit.
Figure 6.
o10
bet ween
of to sites of
44
54
58
61
Logitudinal enery spectral density at 16 maters in
similarity coordinates, for the individual ms.
65
Figure 9.
Transverse energ spectral denaty at 16 amters in
similarity coordinates for the individual rimn.
66
Efgure 10.
Baong
spectral density of vertcal ca*mpou at 16 mters
in sialar ity coordinates for the individu l uns.
67
FIure 11.
Energy spectral densaty of tmperare fluctuatlos at
16 mters in siilarity coordinates for individual nms.
68
PiguPe 12.
L]a
-tdinal
enegy spectral density at 40 maters in
amilarity cocodinates for individual rns.
69
Fiure 13.
Transverse energy spectral density at 40 meters in stimlarity coordinates for divindual r s.
70
Figure 14.
Enrgy spectral demnsty of vertical comsonent at 40 moters
in sI~alarity coordinates for indivtdual runs.
71
nesrgy spectral density of temperature fluctuations at
40 metrs in sial ity coordinates for individual ine.
72
aEner spectral density ftm algebraic exsresios for
the lonsitdinal and trinsverse cMiponents,
75
Figure 15.
Figure 16.
Figure 17.
Energy spectral density from lgebraie epressioan for
the vertcal component nd temperature flutuations.
1M6-
IYI MIMI
Figure 18.
Figure 19.
Figure 20.
Vigure 21.
Figure
Figure
22.
23.
'0
a III
h
li
i 6
ftm
1u
l
as a function of the MWIwnd
e
lHeat and
spoed and modified stability prmeters.,
78
a) Stress as a function of the mea wind speed and
temperature difference between the surface and 8 meters.
te
b) Stress as a function of wind shear and the
dierence between the surface and 8 mters.
eat flu as a fnction of
, at 16 and 40 metrs.
80
Heat flux and flux divaernce a a function of teS mean
wind and taeperature difernace betmeen the surface
and 8 moaer*.
81
Sample of regreSasan grmaphe for tndividual frequenoo.
82
ontinuation of Figue 22*
83
&aterg spectral density as a fountion of stability for
the u and v-omponenta, obained by restoan.
84
onerVg spectral dens ty as a funation of stab:iLty for
Wand temperature fluctuations, obtained by resression.
as85
Figure 26.
Same data as Figure 24 in saui-log coordinates.
86
Figure 27.
Sam data as Figure 25 n seQt-log coordiate.
87
Figure 28.
of anerg spectral detsty functions at 16
oa
and 40 moters.
91
24.
Figure 25.
Figure 29.
Figure 30.
Figure 31.
Figure 32.
C par~o
of euer
spectral desIty functions of the
u-and v-ciponents.
92
Eneraw spectral dentsity for selected runs as a fwation
stnsag 2 (2 meters)of period for u- and v- ctpo-, t
no~nesIation.
95
Same data as igure 30 uirig coordinates
wbere I Us v
veraw
&,
z 16, 40 meters
(JJ)04
Poter spectra for nine ru in coordinates" %pSu
96
versus
log(period).
Figure 33.
Power spectra for nine runs in coordinates 1
97
-S
it versus
98
Figure 34.
Power spectra for nine runs in coordinates -x
log(period).
4
verasu
,iliImim
Fnure 35.
haliill
,,ii(liiiow
A
1L)ast
a P,
Power spectra for nne rums in coordin
veroanu B
99
Figue 36.
Figutre 37.
Power spectra for nine runs in c oTdin
log(parlod).
e
4o vrsBU
100
roer spectra for nine uns in coordinates N
veranus
101
loG(period).
1igom 38.
Fi&ure 39.
Figpse 40.
Power spectra for nine runs n ooSXrdinates TS
log(period)*.
rower spectra for nine MWo
s in coord4inate
4
vmOw ser
4 1 4o
ra
aotroW ratios as a function o ftrequency.
108
Cprlson of various theoretical spectral funtions with
obsred dataso
Figure 42.
102
123
Non-dimamsional heat flux as a fntion of the tchardson
128
Fi8e 43.
Figure 44.
Period associated with the n =
functionof : a) t4 and b) c/e
value of 7)
*
)
as a
129
of tm~peratre
Vatical profiles of tmperatur3, varianca
and vertical velocity for free convective tuns.
DiagraUmatcal ezplanation of the Uwdified
model*
132
inger
135
At~sph.ric
ulence is
numbers of the order of 10 6 .
an aerodyamcally
enerally fully developed, with Reynild
The usual stuatic
is that of flow over
rough lover boundar with vertical shear of the
horizontal mean windd buoyancy forces present.
Th wind and strai-
fication are both strng functions of the time of day and the synoptic
situation.
In addition, they ae interdependent in a complicated manner.
At a particular site, the type of regim and intensity of turbulence are
highly dependent on the maen wind, its shear and the stratification.
site to site
mubuence
are also dependent on
these charactetce of t
the effect of the love bounday.
From
This effect is vartable depeding oa
the particular turbulence regima that is domant at the time. Anothe
interesting feature of atmospher
turbulenc
is that it involves length
scales ranging from the planetary scale to scales so small that viscous
forces are dandnant over inertal forces.
In t1i
units, the scales span
a range beteen weeks and very sall fractious of a second.
demonstrates this situation scheatically.
laboraotor
Figure 1
~cept under very restricted
conditions, a mathematical solution of the equations of motion
Is out of the question.
Before one can even make Judicious sMaplilying
assumptions, it is necessary to knowC
smething about the spectral distribution of energy sources and sinks, the range of length scales that
be expected to exhibit Isotropic conditi~ouc
enoerg spectral density functions.
ight
and the general shape of the
One aat also knowaE sothing about the
general behavior of the beat and m intum fluxes and under what conditions
the assmption of borsonta homogeity is reasonable.
The first step,
FIG. I :QUALITATI VE 6RAPH OF KIN ETIC ENERG Y OF SURFACE WINOS AS A FUNCTION OF THE
TIME SCALE. INFORMATIO I SOURCES ARE: SALT Z MAN(1I97)' VAN OER HOVEN (1s7) AND UNPUBLISH-WoRK OF H. E. CRAMER.
.-.----P-.-*
-.
~-----*
1~
77-i
SYNO.IC SCALF
t
SURF AcE
TIt
FNER DY
'K 3
KI
I
'N
K'."'
-ONf
"
SUNSTA
O
BLEI
i
•~~~~
|R
~ °
STABLE-
'"
I
T-~-
. r_,.--....
.
'
*
/0
/04
0o
,---rRIOO(HovR* )
I
lo-I
4-i
f/0-
II
ten, becomes the observation of the velocity and temperature fluctuations.
The data used in this research consists offluctuations of the orthogonal
velocity components and temperature taken sLujltaneously by fast response
equipment at the 16-meter and 40-mater levels.
These measuremento were
made with the instruments located on a 40-meter tower surrounded by
reasonably flat terrain at the Round Hill Field Station (M.T.), South
Dartmouth, Mass.
There Are twenty one-hour runs at each level spanning
a frequency range between na
.
.0014 and n a .4 see
From these data,
power spectral and cospectral analyses of the orthogonal velocity component
and temperature fluctuations are obtained.
of
this data sample
(1961),
report.
A more complete description
is given by Cramer, Record, Tillman and Vnughan
and Cramer, Record and Tillman (1962) and in Section iv of this
This represents a larger data sample, taken under carefully
controlled experimental conditions,
Therefore it
then has previously been available.
is believed desirable to supplement work of other investigators
because the results can be stated with greater assurance and the effect of
stratification on the power spectral functions and on the heat and momentum
fluxes can be better defined.
A partial inventory of previous measurements
and some analyses of these, includes the followings
1.
Great Plains data, consisting of approximately fifty runs of
ten minutes length at heights of 1 1/2, 3, 6 and 12 meters.
presented and analysed by Lettau and Davidson (1]957)
These data are
and analysed by
Takeuchi (1961) and others.,
2.
Prairie Grass Data, consisting of approximately fifty runs of
twenty minutes in length at a height of 2 meters.
These datadre presented
___ _
__~~__
Wnd Deland (1959),
by Haugen (1959) and has been analyzed by Panofsky
Cr~er (1960) and others.
3.
Numerous rnie of about one hour each in legth have been taken
at the Brookhaven National Laboratory, Upton, N.Y.
The basic data are
5 sec, averages of velocity components and include temperature fluctuation
data, taken at heights of 23, 46 and 91 meters.
analysed by anofsky and Deland (1959),
Tbese data have been
Panof~ky and McComick (1954),
Panofsky (1962) and others.
4.
There awe a nwmber of analyses of smaller data samples.
of these are: (a) Mloni
(1962) analyzed w-,c mo
unknown record length at the l-mater level.
A se
t data for seven runs of
(b) TaZeuchi (1963) analysed
two runs of 10 minutes duration each at the 16-mter level.
(c) Macready
(1953) analysed five runs, varyinl in length from 10 minutes to one hour.
Blackadar and pnofsite
(1964) gave a review of the status of research
concerning atmospheric turbulence and diffusion, in the Soviet Unon.
The Round Hill data are of sufficiently long tun duration (each
run is approxdrmtely one hour in length) to evaluate the low frequency
convective range with considerable confidence and, at the saw time, the
rapid response of the wind measuring instruments permits one to measure
frequencies high enongh to reach the inertial sub-range for moat runs.
Also, the 16 and 40-mater levels are high enough above ground level so that
vertical velocities of considerable siz
scales,
appear in the larger turbulence
So it was decided as necessary to evaluate these twenty one-hour
rune first; then go on to consider what theoretical works seem to be most
successful in ezplaiuing the behavior of the power spectral density functions.
13
Theoretical work, petinent to atmospheric turbulence, has mainly
boen in three areas:
1.
Isotropic Turbulence.
Work in this area has been extensive and
has led, with the aid of slmilarity assumptions of various forms, to
predctions about the enerWr spectral density functions for those acaas
of atmospheric motion that are sotropic.
Batchelor (1953) and Lin and
Reid (1963) present a conprehenmive description of this area of work.
aitchinan (1958),
Recently
his
direct interaction
(1959) has introduced modelltng In the form of
atppt
aon ftrst, and then operated mathema-
timclly on the resultinS equations with the aid of the isotropic assumption,
Similarity Assumptions In the Anisotropic Case.
2.
meat is the work of Nonin and ObuWhov (1954).
e antum £flEas are constantu
heat and
layer.
Usin
these Pae
tioun
This develop-
They aume that the vertical
a sufficiently shallow turface
and dimensional analysts, predictions are
made about the shape of the vertical wind and temperature profiles;
a meraso£f
3.
also
on-dimensional±sing power spectral density functions is sugpested.
Solution of Equations of N tion under very Restricted Conditions,
assuming that the interaction of one pe rtumbation quantity with another is
small cospated to the interaction of perturbation and man quantities,
works of Deissler (1961); (1962) are examples of this approach.
The
Some very
specialised predictions about the form of the energy spectral density as
a function of frequency are obtained.
For a detailed and rather complete indication of the present state
of theoretical and observational knowledge in the area of atmospheric
turbulence, the following list of books and papers are reconnanded:
Batchelor
" ,,, IIgiIIBMMII
wilA
111oiiYiAi
14
(1953),
Lin and Raid (1963) and Pasquill (1961),
Chapter I.
Tho basis of
the spectral and cospectral analyses of variance and cova~Lance functions
was provided by Taylor (1938).
For a continuation of the discussion,
includinS treatment of the problem with vertical shear and stratification
present, see:
(1962),
Frenkiel and Sheppard (1959), finse (1959) and
?remaiel
With this knowledSe of the nature of atmospheric tarbulence in
the surface layer and of the state of the art regarding both theory and
observationsp the problem can be more specifically stated as follows:
first,a large data sample should be used to learn more about the general
characteristics of atmospheric turbulence.
Then one is in a position to
apply the present theoretical developments judiciously.
For insatnce;
if
it were knomn under what conditions and for what scales that atmospheric
turbulence i. isotropic, then one could make use of the present theoretical
work in this area.
Also one would know what sapitfying assumptions and
modelling are acceptable in treating the problem.
Then the plan of attack
lndicated rs to gather data; analyse it; evaluate the available theoretica
results; and possibly start the whole cycle over again with so n new
predictions.
I.o
BJECTI
COF UI
The objectives of the reearch are to analyse and present the
e2parlmental data from the twenty runs in a manner which is =sat convenient
for reaching theorOtical and observational couclsions.
In order to do this,
it will be necessaay to use several coordinate systems and non-dimansicnalising methods, to find the most appropriate techniques.
An associated
problem to the necessity of selecting pa
rasters to exsp
s the affects of
stratification and lower boundary roughness on the energy spectral density
functions of the orthogonal velocity czl
ents and temperature.
The
problem, specifically, is to express the energy density functions as
functions of:
frequencys n;n
an wind speed at sow height,
expressed as the gradient of potential tepratue,
roughness, so, or so
; buoyancy
; and surface
other prameter descriptive of the lower bonday.
Although prinary emphasis to on the poeor spectral data, sose works designed
to evaluate vertical beat and
-mantm £1umes, is done.
to choose a mini=m nuber of paramters
diftficult to evaluate, mad which
The objective to
which are well behaved, not too
re pertinent to the physics of the problem.
ifth Ua incorporated in the non-dimenasonalising process$ then there are
needed two parameters: one to specify the lover boundary effect and the
other to specify the stratification.
results which can serve two purposes.
When this point is reacheds, one has
TI.
results are of engineering
value, which yield estimates of power spectral densitiesx
momentum ftlum
and heat fl£ume when the necessary paraneters can be evaluated.
The second
and far more Important result, so far as this research is concrned, is a
data sample organized in such a manner that theoretical results can be
critiqued.
The primary thesis objective can now be attained: that is to
choose the theories and models which show the most promise in explaining
observatlonal results.
The last objective to to develop a composite model,
which explains the general shape of the energy spectral density functions
and their dependence on theasal stratification and lower boundary effects.
If these energy spectral donsity functions can be determined with
reasonable accuracy, then the dissipation rate of tuM ulent kinetic
6 , can be determined.
nergy,
Inowlgee of this dissipation rnate of enerzs
its dependence on terrain and meteorological par
nperical weather prediction of the large~
eters is essential for
scales of atmospherc motions
It is also necessary for evaluating the equations of ener
the surface layer, where energy
and
hages ae of prm
balance in
apcSortance.
Estimate
of energy spectral denu ty functions are of basic an rtance in calculatng
diffusion rates of contanants in the atm~ophere.
III.
A REVIEW
A.
1 P&1OETI1
T L.ITDATURE.
rief his orical review- of re8arch in staosp
r c turbulence.
Three papers of G.I. Taylor provide a foundation for the present
day mathamatical analysis of turbulent motions.
In the firat two of these
(Taylor, 1921; 1935), Taylor recognised that the velocity in a fluid is
a continuous function of position and time.
thesis that xa
t, if the turbulent eneroyg
of mean motion (x Is the displacement,
time).
Taylor presented the hyposmallcompare
to the eaergy
i0 She mean velocity and t is the
He made another very tportant contribution to the study of turbu-
lent motion in his 1938 paper when he bsoed that the velocity correlation
tuntion, -X40)-xiXffJ) , and the energy spectral density function,
j ')
symbolav
and
are Fourier transforms of each other.
Expressed in mathemti&cal
this relationship may be stated as follows:
1Y )gaS
(10' e- c L
(3.2)
I_
___I
In 1941,
11111i,1111M
II
iolsgoroff introduced the concept of lunivrsal equi'
libriuaTfor suffEicent by higb
wave
mbers and, with the aid of d
niom~a
arguments, he derived the form of the eneray spectral density function in
this high wave number region,
Batchelor (1953) in his book, "The theory of Homgeneous Turbule ce
presented a comprehensive review of the mathematical theoy of tutbulence,
The treatment is mostly concerned with isotropic turbulnce.
he
The sme year
Batchelor (1953 A)] also showed the Importance of the Richardson numbe-
in evaluatin
the dynamic
tsimilarity of atmospheric motions.
Coswecing with Macready (1953), mtasurents of velocity fluctuations in the atmosphere became available for the first time for empirical
evaluation of the above theoretical predictions.
with the
nreat Plains and Prairie Gras
This trend conrtined
experimental progrs.
For a
more complete inventory, see Section I and Section IIIC.
agnin and Obukhov (1954) proposed a similarity theory desined for
predicting the details of turbulent motions in the boundary layer under
anisotropic conditions.
Many of the more recent papers (see Section IIIC)
analyse the reoently taken data and there is renewed interest in the worke
of KoBmogoroff,
onin and bukhov.
This is about the situation at present
as larger and more satisfactory data samples becom available.
graicnan (1958; 1959) has been doing mportant theoretical w'ork
by first modelling the atmospheric motions, then operating with mathematel
exactness on the siplifed qsyten.
Other recent theoretical work has
Involved the solution o f "liearized" perturbation equatioo
of motion.
Deaisler (1961; 1962) and Dolgiano (1962) are examples of this approach)
0111MI111
114
I
--
,I
, 1 , 111
fil l I ,
This section is a brief chronology of the iportant happenings in
atmospheric turbulence studies for the scales of motion considered hers.
Next, Section III B abows the contrasting approaches to this problem;
Section III C and the Appendix give brief resumes of may pertinent pa~ers~
B.
Outlne of
arious Appoaches to the
roblem.
Due to the chaotic nature of atmospheric turbulence, one wtd
t, given
usually be content to know soae statistics of the flow after tie,
data at time, t =S 0, the valueas of the important parameters and the boundaiy
conditions.
A number of theoretical approaches have been suggested.
The
generally accepted equations of motion are
r
~Non-divergeant flow
(3.3)
Navier-Stokes equation
iaen buoyancy forces are treated, the Boussinesq appromiation is quite
acceptable:
wherea
(0, 0,
of
7'
except when uAutiplying
,
, giving:
)
: coefficient of thesal expansion
of temperature from the value at which
departure
as
e
The symbols used in this report have the same mseaning as ino Batchelorea
-
41
~~~~ IYIIYIYIYIIYIIIIYYIYYlrYIIIIII
Ji
,1 IIIiih
I
lli i
book unleso they are otherwise dcribed,
Thic ivestigattion of amospher c turbulence
st confined to the
surface layer and the regiwos lmIediately adjacent thereto.
necesaary to discuaa
the depth of this layer.
It is threfore
The depth is not only
variable because of dependence on meteorologcal conditions but may also
be d&fined in several ways;
The definition of this layer ha' to be a par
of each tratment and is not defined in general terms.
The follow1ing is a review of the various theoretical approaches to
the probl m
1.
Solving as an znitial value problem, usin
fo
--Z-
a Taylor settea.
1(35)
By taking the divergence of the Xavier-Stokes equation,
Using a Green'e Fmction ,
L
Nest
r
Z4.I ;f!
is evaluated.
As -
di"
a)
beccws larger, it is
necesary to evaluate higher order derivates in thi series and hence to
kno more completely the initial velocity distribution kuncton.
2.
Using the equations of motion and deriving epressions involving
pratuct-me
tbfn
If ir4tial conditions ~nvolve probability ftuntio s,
vaies.
these eauations ar erpressions in joint-probability functions.
MU
de
AdXzdk3
(with suitable restrictions)
anda
In order to evaluate second order product means (say a two point probability function), it to necessary to Iuno
th third order product mean termes
and In order to evaluate the third order product mean tnsw, the fourth
order teram are needed.
So at this point, one can close the set of
equations only by assntag sme evaluation of these hlSer order terms.
Some methods are:
(a) linear solutio,
where the third order terms are set equal
to sero.
(b) pbysical transfer theorems; usins the isotropic assuption, one
ta
the Fourier trandform of the mavier-Stokes equation to get:
S-
where 2(k) is the spectral density of kinetic energy per unit
(3.6)
ass and
T(k) is the defining scalar of the Fourier transform of the complicated
--
--
1116116111
E(k) is also the de fining scalar of
non-linear tem.
ik)
.
See Lin
p. 449, for a more complete description of these terms,
and Reid (1963),
the longitudinal and tranaverse correlation functions and their relation
to each other through the Von bsma-n-Howarth equation.
One makes the
equations of motion determinant by assuming T(k) to be expressible in terms
of 3(k).
(c) quasi-Gaussian asumption: the fourth order teras are ezpressed
as a function of the second-order terms (the value that a Gaussian probability function for the velocity would give), without setting the triple
correlation equal to zero.
the equations of motion first
-odify
3.
deal with the non-linear term
mathematical exactnass.
then operate
This isn
by modelling primarily to
n the resulting system with
reverse order to the procedure in
Section B and is the technique employed by Eratchnan (1958; 1959).
Som
of these methods are presented more fully In Section III C and the Appendix.
Some other authors, who commence by defining a model arez
Thompson (1963),
P iestley (1959) and Lettau (1949).
4.
Hopfos theory of turbulence (see p. 442-444 of the paper by
Lin and Raid).
This method starts with a known joint-probability distri-
buton function,
7 '
Then express (
)
, ffor the velocity, or its Fourier transform, (
)
dwhere the coeEficients in the
series are the complete set of omwnts of t( at a point,
Finally use the
II
ations to ~et ~n
dynmical e
xpression fo
11 ... .
,1
- I l
Nllil
4 lllllinl,MI
ill
This has not yet
(g().
been done for homoigneous turbulence.
5.
S milarity solutimos.
necessary to argmu
state, which is
In order to use similarity, it Is
that the turbulence retme must approach some universal
utepentdnt of the initial conditions and is only affected
by boundary condtions in a very restricted man.ar.
The objective is
to be able to describe the flow reime, using a leser number of physceal
vwas needed to define the solution prior to its approach
paremeters th
The hypothe ss of Rolmgoroff (1941) is the
to the "univereal state".
most important xale of the application of similarity a
atmoopheric tuxbulence.
xeumnts- to
He states that at a suffciently haih Reynolds
nzmber, a region of isotropic turbulence can be spectited in terms of the
dissipation rate of kinetic eners,
and the kixemat
8,
.
viscosity,
If
the Reynolds number is sufficiently high (the usual atmoapheric case), then
the "energy coUalnin
and "dissipatUiv
eaddes
in wave number space to assume that they act
D p
by an "inertial sub-ras
',
wheree
th
are sufficiently separated
undependently and are separated
only
ortant para
er,
Using
this and dimsnsional analysis, he derived an expression for the egergy
spectral density as a function of wave number (or frequency) only.
A
scheatic of the atmospheric case is shown by Figure 2, where the nondirnslonal spectral density of energy is plotted against non-dimoenmonal
rrequency,
og-log coord&inates are used for convanience.
In the non-isotropic case,
onin and Obukhov (1954) as~ed that
the vertical heat flux and m e ntum flux are constant within a shallow
surfane layer,
From these quanties, using d
os~
nal analysia,
a frictlon
i-r-
-- --1-;--------
_ _ ------ ~i-~,- -~e ~
I-lbs
,___
FIG.2: CHEIATIC OF ENERGY SPECTRAL DENSITY VS.
FREQUENCYIN NON-DIMENSIOIVAL COORDINATES,
3HOWING THE VARIOUS SPECTRAL REGIONS.
3
I
321S
3
4
A 56'6-91
4-
..
ILC~bFA(
l
--
. ..
i
..
a
. :1.
..
.
. .. . . I t ....' ~ , ":~ ~
i , " 'i
-,!
t!
TiL
'-- t'!
-+--(" -'-
*- - - -
-
Iit
-
-----
II-
SIt
----- H. . . :I -'-f !: ; ,",
-i
-'C
-
--- -?-
±-t"--
-
-
I • ,. , ;,
"--'
'
~ ~~
,~
-
41t1.07-
.......
~--- -
V-I-
i
,
.1
ci
t
-
-Z--f7
.r~r+iL-;Lf *--
'~
-
~U
' ... -I '
,._
I
- -A -Jt
t
,t------
-T"I
1 - - ----
.It
-
,-
SM'
4-k'
-
•l
il
.
,
,.
0l
;
I
...
-ho
T
Ar
_ _~~_~___~_~___~__ _ ~
~~~~~-~
~~~IIY
YIYYIYIYYI
YIIIIYII
YYIIYVI
iiiur
2:4
lengt,
valocity, V ; a friction tamrature, T ; and a stabilityd
wz'e formed.
L
iotil quantity, a/L, cuclusions
Then using the non-dina
were derived concerning the diabatic wind and temperature profile.
The
data used in this investigation are not well suited to a careful evalua~ti
of "profile theoreme".
The principal srnifcance of this similarity
theorem is that it susgests the functions, V
lise the power
and T
ona-
to non-dinmrs
pectra of velocity and temperature respectively.
A more
detailed explanation of Konin-0bukhov s8rilarity is given in Section III C
Ce Rsumme
of seleed ppe- ras.
In the following review of papers, the selection was determined
by the requirement for an adequate description of research activity in
each of the several separate approaches to the problem.
The papers are
grouped according to type of approach and only the most pertinent ones
to this investigtion are Included in this section.
are included in the Appendix
Additional resmms
Section IX, and a few others are listed only
in the bibliography.
Although the atmospherio
case is one anisotroPff, it to shmwn
later that, under certain conditions, portions of the spectrum aare isotropic,
In order to derive an expression of the fom, 2(k) a £(k), it is necessary
to make certain assumptions about the physics of the problem or about the
higher order productemean teams.
1.
Transfer theories.
A desaription to given by Batchelor (1953)
and Lin and Reid (1963) of several methods for evaluating the uon-linear
transfer term, T(k), In the equation
aeAk}
___7(k p
Lk
__
~__I
__
Probably the best k~own of these Is the Heisenbers theory (Lin and
Reid (1963]), which states that the machanism governing emrner
transfer
between wave nuters is stailar to the one governing viscous dissipation.
In mathematical teras,
The desired equation for 3(k) is:
;.
If the following trans --mationsa
based on maintaining self-similarit:y during initial period decay, are
made, then Figure 13 of Lin and Reid (1963) shows the result* 'Phi
spectrut
appears to be quite similar to that observed under knou
conditions (see Figure 41).
If it
resulting
aisotropic
is assumed that the atmosphere is an
ensemble of these turbulence calls in various states of decay. and that
this transfomation is applied to each cell individually, then the space
average of the result in z - F(x) coordinates would be steady state.
2.
Quasi-Gaustian assumption,
Serous theoretical difficulties
arise in using this assumption, as indicated by Oura (1962),
does not seem worthhile to consider this approach further.
therefore it
240
3.
Linear solutons.
Two papff= by D2180ler (1961; 2962)
tpertAculear interest in pVovldlng an analyOi of thie effects
'e
of OhQe
of
of
the hodrzonta mean wind and buoyancy sepaately on the kinetic onergy
spectr~s of turbulent motion.
The approach of Deaisler (1962) ia to
assume no wind shear and a constant stratification expressed by
quo.
Th folloing equations
isto used to derive two point correlation ftunctions such as IQ
iLj 7",
etc.
correlation tenns
, p
,
Deiseler closed the set of equations by setting all triple
equal to sero and imposed tt
fal conditions at t a 0
of Isotropy and no fluctuations of temperature.
ias conclusions ware:
(a) No steady state was reached
(,) Dissipation occurred mainl
at high frequenries.
(c) Ruqant energW as fed into or rtemoved fro
of frequencies and the effect was
a large rans
omwhat stronger at lower frequencies.
(4) The turbulence is aniootropic at low frequen~a
athis paper represents a demonstration of the fust apr-
es.
tmaion of the
effect of buoyancy on highly developed turbtence Lo the absence of shear
and Wthout non-1near inertial tranasfer of ener
.
The "transfer" that
appears here is merely the net result of buoyany extracting ene y at one
frequency and putting it in at another and bears no relation to the tranfoer
of energy directly between the dearees of freedom that tos
real atmosphere.
oin
on in the
DeLaslers 1961 paper is samaed in the Appendix.
27
Those two papers are isctretiva, even though the sampliftiatlon of the
problem in each case
severe,
The next paper, Rayleigh (1916) treats convection in an unstably
The model consstts of two parallel flat plates of
stratified fluid.
infinite horisontal extent, separated by a distance, f.
The foulation
used is:
(a) Equations of ation without corLolia ter, with and without
viscoasitty.
(b) N=ondiver nt flow.
(c) Bousainesq approximation
(d) A diffusion term in the the
aC equation.
Of particular Interest Is the identification of the modes of greatest
instability.
It
the coes most likely
ins erred that these modes ae
to be observed in the convective case.
In the absence of viscosity,
Sgratest instability occurs when the wave number in the vertical os the
least.
Then the horgwontal wave number,
-. 2 +
'- g
r(P
~
x'
,
is given by:
primarily a function of the depth,
(secondartily
a function of the stabilty
For a fixed depth of fl£uid, the hori ontal wave numbec of mmum convective
energyTi expected to be proportion l to
.
Bu t, k, the thermal diffu
stvityp in this case is also dependent on the turbulent intensityand would
make the wave number of m
m enery less strongly dependent than
~
If R a m, then A , the hortsontal wave length of maxium instability,=,R S
^I
4.
_
Approaches which involve analysis after a model is defined.
(1958; 1959) and ra~chnan and
The cost i~pomtant papers are rraichnan
SpieSel (1962).
The approach here is to model the atmosphere in such a
way that the equations of motion give a closed set of equations.
The modl
is the "direct interaction approximtion" which replaces the non-linear term
in the equations of motion.
This appr0omation states that, for three
Fourier components (p, q,r) giving the aides of a triangle in two dimensLnal wave number space (formung a "tried"), the inertial transfer of
energy can be detemined by the direct transfer between these coponte.s
The "indirect tranafers", those interactions involving an inztemediate
mode, are neglected.
This i
Justified by pointing out that as the region
and therefore the =nmber of permissible Fourier camponents is increased,
the interaction between any three modes (triple correlations), becomes
smaller, but, there are many more of these interactions, and thus the
total effect of the inertial terms remains aportant. Also, it is safer
to neglect the indirect interactions
In the inertial sub-range, this
approaimation gives
In the
raichlan theory, direct interactions are possible between "energy
containing" modes and those of very high wave nuber, whbereas the
oKo
ox~of
theory relies on the "cascade" of energy on a more local scale between
neighboring modes in k-space. Ef1atcMan
auggests that, in the region o,
wave numbers below the inertial subrange, the direct interaction model may
Zq
be coeparatively better than those deperdng on local transfer.
This is
the region where one can expect some interaction between the turbulent
cells and the advecting flow (low of relatively amch lower wave number),
Thompson (1963) used a model of convection between two parallel
infinite planes with themal gradients in a thin boundary layer near each
plate separated by a deep region iwere
statistical stationarity of heat, s
O
0/c
enm
He also imposed
and vorticity transfer.
aan
In
addition, he employed non-divergence and the Bousestq apprasituation,
Thowpsona
results include the following:
(a) The thickness of the
,
boundary layer"s is proportional to4?
bare Ra is the Rayleigh number =n
(a
used in
Thompson (1963).
(b) The thickness of the bounwary layer is independent of the
total depth.
(c) The heat flux is proportional tO(Amp)f( 1 )
) P,)
(d) The variance of temperature reaches a maxztum at the top of
the boundary layer while the vertical velocity reaches a maximan in the
middle of the region.
This model has much in camon with Hterrinj (1963)0a
results.
Lettan (1949) "generaliaeds
mixtlu length theory to include the
diabatic case by defining three parameters of turbulence:
(a)
f (, r zo)
(b) U*f (geoatrophicwind), U
(c)
,U
w
,
With these three parameters (
0a.
,
Who U
a Iw
u , W ), a number of identities
are written,
thich helps explain their meaning.
for isotropic turbulence, ua
stabiity.
Whe
z
These identities are:
the subscrlpt, a Idicates neutral
wa
u ite*- fom A
e-z
* eo)
4a
t,
o
as the parmaeter of
Lettan uses the heat flux, U, rather thnM /
stratification.
ew *
ttpB''
-Z .0
A is aseumed to be the sam for any air property.
For non-isotropic conditions, u
6
.0. Lttau afet the following
assumptions:
-e
t
i
'W
0
LW.4
C41
(Aj'
%04"4
WC!,
nces directly only the vertical
(b) The buoyant accelerationtinfle
SU
turbulence parameters. ohen usin
Ta44
turbulent acceleration gives:
with this basis, he defined,
9S
dOA!L
for=I
h)
au~l
2.)
R;
--(a*,)l
ac~
vertical coaponent of
Othar assocated identities aret
Next he defined,
H
* Zving
Lettan described this system as foll4o:s
ny is defined proportonal to the
heat flux and indapendent of the gradent of potential t~meratare; x is
defined proportional to the gradient of potential temperature and independant
of the hbeat flux."
This leads to the interesting result that y = -
X)
(see p.32 of LIAinu [19491) which is reproduced as Figure 3 to show the
predicted behavior of the heat flux, H. This
figure
ndicates that a
of heat flux occurs at a moderately unstable value and that the
m=rI=m
heat flux decreases as the atmophere becmes more unstable.
There is also
a negative axwdun of heat flux in the slightly stable atmsophere (see
Figure 20).
5.
Similarity
K.olo
orofft (1941) postulated that
in the versal
equilibriuma rane, the satiastics E turbulence are definable as functions
of
and
n-
( /O)
(see Section XIII
From these, he formed a length scale,
,and a velocity scale, 1r'(i.
analysis he showed E(K.)=r'
and
).
$( K
)
Then by dimnsional
, where k is we ntaber
iIs a universal function of a dimensionlese argument, 4
inertial sub-range,
isLa not important, thus
k(
.
K)
dimensional arguments once more, he ccluded that E(9) o(
In the
only.
Usivn
/3
Monin and (bukhov (1954) proposed a s tilarity theoram designed to
e/
~---
I a'L
--IJ~ *~kt-IZYljersRC~TI.~RC~~lrr4Aft~PduXdSiA
_
I
:
I -i--1VTL--l.~~- ..
,~ro
C4--V
1~~V
I--I
.
..
.
-
-
--!-l-i .......
....-- j-~ 1 --1 i j.-..; bi.
-v-.I
."
~-L---.----.11
~ H------
.
' 1
I-1-zI-
-.
1
i11
I
I
4..
I
-1
.i
I--
..
1-
~1~ i.
-!-- -i-
1-.
-.-
-; 1-4-r- --l--- l-- j---
~-i
'"
irzzL
-
I
z-.iz
i..
t.
i..I
-i-i/Il
-i---l-i]--/I-lLlI
::
-
,-
I
II
.-;..I
1--*
_ .i~...l~.~.~l
1
r-----
FLU5
/'
Ii
l'
--
1i
ii i-i-li
I
i
1
I.I
-t
.i
t
I
t
f-
-
zz
I
I
T.i,
-
...
:f I :i ' f i
: ,'' I : i
71F
f
I
k
T
.
--I-HF--
-V-~-''Ti+
apply under conditions of non-lootropy in the surface layer (again see
Section III 3).
They assufed the following paraters ae costant in
this layer:
/E
(a)
(b)
(c)
, buoyancy
, heat flux
' -/ep?'
1
J
, movntzan flux
'dVA
s
rcan these, and using dfmansonal analysis, the following quantities
were formad:
(a) Friction temperature,
,
W/ 77 1
Von
Karmon constant.
La
(b) Stability lengthb,
-
These can be used to derive expressio
(uM
for aon-din
tonal shear and
temperature gradiento
A2
If the ezohange coefficients for wmentum and beat are equal,
universal dimensionless function.
Also, it isduxm
that
L
(
a
) =a
The difficulties arlsing from the necessity of evaluating the above expresson
at
C 0p are avoided by Neuann (1961), who defined the lchardson by the
expression, Ri
S
)
A series expansion of
is chosen appropriate for the stability (or size of a/L) and
upon integration, the following foms are obtained:
V)
_
-
0
where v o 0
* heightbt
I
l
11MIUMIll
34-
helo(193 A) hod h
The analyss o£ Panotall" and
the boa
xeland
(1959)
pe
o bed
a sary
on Prattle
onditio
S; ed the
e1ght dependence
of dat
ehermi
ws
also
milecore
deito Car
ity Prae
inthetb$ent
are
the
papers
cal" ener
(1960)
value spectal
fnsed thedsia
rjselvendatain
and peentaed
thesdrathe
p7oo
L
~~E~uS/b
the noxavlied energy speefral denties.
teos of the standard deviation of th
The Stability ws expressed -i
asi ath angle (see hits Fi~ure 2).
He concluded that the u-spectral energy density (in the lo-1oq coo.rdi=tEac
used above) approxiates a -4/3 alope at high frequency.
Cramr, Record, Till3a
and Vaughan (1961) and Cramer, Record anr
Tillman (1962) gnve a description of the eZprimntal technique used in
collecting the Round H111 data and presented the power spectral and
nts and teperature for fout
cospestral data for the u, v, and v c
one-bour runs at -,the 16-meter level and sCi aonahour runs at the 40-meter
level.
TBey expressed their results in many SosPs
those most closely
associated with the present tavestigation are shown in their Figures
6a, b; 7ab; 8apb, and 9a, b, whAch are based on: log -
i
u, V, and v log
U
support a slope of appro
dependence at km
versus lo
Ise
vermo
log
figurea generally
rtely -5/3 at hibh frequency and a stability
Jrequenty stailar to that shown by Cramer (1960).
In addition to an evaluation of monia-Cbuksho and other profile
thaeorem
, Tsalchi (1961) plotted the Great Plains data in the following
oan-dimmnsional coordinates, using six twenty mute runs:
log
versus logi
; a a tfrequency,
o u, v snd w.
spectral density for
omean wind, 8 1f
(See his Pigues 19&, b, and c).
For two ten minute runs at 16 ~mnters height, Takamchi (1963) gave
the following plots: log PA versus log u, where 7A a variance of the
astuh angle(
al
/
),
and also presented these in the fon:
versus log
In general, both of these works showed spectra approasiating
C7 -
at high frequncies and stability dependence at lower frequencies.
SWotaf
(1963) took data at 26 metera and obtaind (after smoothing)
plots of S2L(nheisa
s t and
?) Lasds 7 . He compared
of local isotropy, which gave on the average uz/ 6
lower limit of isotropy.
Ct
3
at hig
5
=
g, and ras a check
.46
for the
His spectral denafty functions are very close to
frequency.
The analysis involved 6-15 runs.
Priestley (1959 A) obtained n
=
.6 for the lower limit of
lzotropy.
Gurvich (1960) obtained as the loaer lrmit of ootropy:
for unstable case;
ns/i
* .7 for neutral; naz/
na/i =
.4
* 1.9 for the stable case.
Monin (1962) presented data In support of his universal function,
derivd by similarity in 1954.
Ho also gave a plot of the power spectra of
the temperature and w-cionents for six rms as a function of the Richardson
nu*er.
i0l
His coordinates are
Veks
VE/"j
AegS((s
/07
a
These coordinates are also used by Takeuchi and later in this research.
Panofsky (1962) treated the turbulent energy budget and the
vertical flux of turbulent kenetic energy, using the 23, 46 and 91-meter
levels for 9 runs.
(_U
The equation he considered is:
CA
Rf = flux form of the Rchadson
in*
mber.
He sahoed that the vertical div.vge~ce of turbtment kinetic anergy is
i~ortant in unstable casea.
Obukhov (1958) evaluated the validity of the following '"atrcture"
fimction forms (from similarity):
Vp
=)
(Av
a,3
r
)
utNO
'
,
-T
eddy thermal diffusivity.
By using 0I
Evluation gives C,/.I
CV,'9'
1
a plot of
and aWsuming r')l1)
a7
.
VAQS 1
-Z 4C-
He evaluation gives C24
2.4 ;
, he derived
his Figure 7 Sivas
0tFO*
The secoad part of Cbuhov's paper is clncared with the distribution
function
of the oscillations.
density of occurrence;
.r(e .el a
Some results are,
e,At
He graphed To values versus tha probability
also he graphed stability versus
en n i, , ad
.54 (T
a
ter
and 7r = .88 V (indepednd
)
meters)
t of height).
The standard deviation of the elevation angle and particula:ly the
standard deviation of the a;IuEth angle (A ) were dependent on atratification:
- 50 for unstable conditions;
q = 2.50 for stable cond~itio.,
zIV.
AATA COLECC Ht
AiD PROCESSING 2MCHNIQUES
A. Detailed d scrip ion of the
remats.
The maesuremeats were taken at Round Hill Field Station (..T.),
South Dartouth, Mass.
143 ft. in height.
wre mounted to
The tower on thich the instrmsnts
The area imediately surrounding the tower is covered
with beach grass, triaed to 5-10 cm. height.
covered with a mixture of gras,
he adjacent land area is
low shrubs and ams small cedar tree .
Bu$aard"a Bay borders the area to the south and east; 1100 ft. to the west
is a wooded area and about 3/4 mile west of the tower is a north-south
oriented ridge with a maxRa
elevation of 88 ft.
In this data sample there are t wnty one-hour run,
neously at 16
nd 40 meters, of the fluctuations of the orthogonal velocity
components and tempature, spanning a frequency rang
N o .4
ec&l.
talon simulta-
fra
N a .0014 to
The time spacing between measuremants is 1o2 sec.
In additi.on
to the 601~Qa analysa. a 10 lag ~nalysie was done for high frequencies and
a 30 lag analysis (each data point is an averae of 10 points, giving a
spacing of 12 see.) was done for low frequancies.
Power spectral and
cospectral analyses for the three orthogonl velocity eap-tents and tempe
rature and tha possible combinations of these were obtained by digital
computer analysis.
Also total variance and co-variance statistics for
these same elements are available, together with the vertical profile data
'or the zsan wind and temperature associated with each rum.
The fluctuations of wind were measured by heated bead thermistor
anem~oters (speed) and lightweight mechanical bivanes;
the temperature
fluctuations were measured by the change of resistance in platinum wire p-os
II nIIiYUU
'' AW11181111MON
'11IIIII I I1
39
er-filter circuit.
The output of each transducer Is fed into an amplif
An
analos-to-digital encoder taWs the circuit output and converts it to
bnaary nuber representaion.
Finally, a programer samples the encoder
output once per second (represented by a nwrer between 1 and 256) and
prints the data on a punched paper tape. See Section ii
of Cramer. Recovd,
Tillman and Vaughan (1961) for details of this data acquisition system.
In decoding, this information is converted to deoimal data, put on IM.
punched cards or mapetle tape and fed into a computer for spectral and
cospectral analysis.
S. -gplanatio
of tb data procesaag.
Prior to spectral and coapectral analysis, the wind speed and bivane
measurements are cverwd into vtbhogonal velocity components.
procedure is used:
The following
coaJcing with a set of appriot2mtely 3600 data points,
consisting of a velocityi, V; an arimuth angle, AL; and an elevation angle,
Ei, these are converted into fluctuations of each c4onent: aout its mean
value, It is necessary to define a virtual mean a~iuath angle,
A!
VZ CS
(
jl
- E
AS
Al
j~j
and a virtual mean elevation angle, Eas be
4
Then the velocity component fluctuations beome:
a
,
b
i-
46
V
Vi cO(3f - 3*) sin(Ai - A)
where
and u,
) co.(A
m,
ndUa V co(E -
i
N
-
)
and weO0
The data± s amoothed by a 601 point nlS ea and this me
aubstracted from the unaveraged record, in order to rmwve long period
This ditfrenced data forms
fluctuations and drifts in the uman value.
Variance spectra calalatLanas follow procedures
the input to the cmputer.
First an auto-covearance
described by BlaolmEa and Tkey (1958).
mnction,
AL is computed.A
xi
SXcL..
N-M
where
I
I
.L-L1
k)
N
total number of obssrvations
L
the la
.
xi,_
I -"LI
+ t =4/
nu~ber
M = m~xn nutber of lags
Then the follont
equations ae used to obtain moothed power
spectral estimates
rl
z A~,,
rr
d
L~<
-
s
hti
xt [Ao
~L-
'S
41
These moothed values are in munits of vaxience per unit frequeucy interva ,
S
~.
* The quantity, 40, is the time interval between observaticn
and equals about 1.2 sec.
Cospectral estimates are derived in a similar mznuer by replacin
Z by yf in the equation for A,
For a description in complete detail,
rofer to Cramer, Record, Tillman and Vaugban (1961).
&. Data relbility.
in analyzing the lowest frequescies, the initAal data set is reduced
consecutive block averages of each ten points.
tenfold by tak1i
In a new & of 12 see; the mat
nmb e of lags used Is 30.
This rensul
Advauta~e ha
been taken of syasmetSy so that cosie functios up to periodx of 720 sec
can be evaluated.
As habown by Panofsky and Brier (1958), the number of
degrees of freedom for a msing lag vindo
is ~N
the total mnuber of observations, m is the
8 is the spacing of data points.
~r
,here
H is
w~.num
raber of lags, and
Then the degrees of freedom,
F,
for
the 30 lag analysis (8 a 12see.) is 19.5 and for 60 lag analysis (8 a 1.2 sec.)
to 99.5.
ee
is
an 80 per cent chance that the obse
d values, for 30
lag analysls, are between .62 and 1.41 t1en the true value (Tukey and
8al
C(19501).
For sizrlicity and also for desirable motbing the higff-requency
spectral and cospectral estimates ( 60 lag analysis) have been cclbined
into the following list of waye
aer groupeS
4-2
7.5
7 -8
9-11
10
- 20
18
1215
16
13.5
21-27
24
43 - 60
54
28 - 36
37 - 47
32
42
In the diSitally analyzed time series, tho ght has to be given todistortion of spactral data at the high frequency end of the spectrum and
to errors in the spectral density at n
1/360 sec "
and no 1/720 sec "
resuling from those fluctuations in the tiJs serles of periods longor
than 720 se
(m xrtum period that the lag window can see), which has not
been completely fi£tered out.
Aliasin at high Arequencieo is usually relativly easy to idewtif
due to the qffect on the shape of the energy specital density functio.
At high frequency this function (in log-log coordinates) usually has a
unifd~
m slope of about - 5/3, hence aliasing would give a nearly flat
horisoatal graph. Then an abrupt transition from a - 5/3 slope to a much
less slope in going frmm lowr to higher fwrequncies Is an indication of
high frequency aliasing.
Then only the low frequeny eand of the spectrim has to be considared further,
The data-processing techoiques have the effect of a band-
pass filter [see Chapter I of Pasquill (1962)].
This f Uterag action
is represented by the expreaion (variance seen by the system)
(True va iance)
(rnZ
fd
Fi
111111
43
where F(n) a Fourier tranformd normalised covarimce function;
S= sampling interval, t '
maxzim lag, and T a overall record length.
The nature of this f-cetion is see in Figure
, below.
range difficulty atises in deteining where
The low-f~eque
the analysis distribute that variance of periods in excess of 720 sec.,
which the system should not observe.
ttithut including the caMputational
work, an estimate of the "low frequency aliasint
spectral eners
a = + 1/3 as t
density function, E(n) oC
lt
uding the range of E(n) f tntos
was made by assuming a
and taking
- 5/3 and
at low reuncy.
A mmrical integration was carried out and the results are listed below:
1. T o 720 sec.
(a)
(b)
5/3, True value is 03 per Ceam of obsered value.
ca
c a + 1/3, True value is 124 per cent of observed value.
2. T = 360 sec.
(a)
c a
(b)
c a + 1/3
-
5/3, True value tois 132 per cent of observed value.
True value is 102 per cent of observed value.
There is scae error in these, since a more exuct evaluation would
apply to individual data points rather than the assumed smooth function.
In comparing the results above to the spread permitted by the 80 per cent
confidence limits, it to seen that a correction to the data at T w 360 and
720 seea
is not necessary.
For several reasons, it is very d£tficult to do anything about
increasing the reliability of the data without taking simnltaneous tim
series measurements at several points or resorting to measurements over
FI.
:.SYSTEM RESPONSE OFSTATISTICAL
SYSTEM RESPONSE,L)=I-
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SSA MPLIAN INTERVAL a /.., SECS.
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F
:- T'OVERALL RECORD LENGTH P 3600SEC S.
I1+ 1-~* I im'l ittll
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a space network and ralating
;LJ
lt
isceae the number of degreestm of feedom fo
.To O(4',)
a siale time series at a point
one would either bave to shorten the span of the lag window and thus cut
down the maximm
observable perLod, or increase the total length of recordo
If record .lengths of more than an bour are uaed, then changes of the
meteorological parameteras, U and
e6 ~,-may be aIpected and the
observational result would be a cobination of more than one turbulence
regime.
Also, at the longest tiuP scales, Taylores 1ypothesis (x t fit)
is less dependable.
It appears there is no choice left but to moasure
space correlation functions of the fluctuations.
arises beenus
Even then difficulty
although horisontal hocgeneity may be a good assumption
for stable stratification and forced covective regimes,
it is probably
a poor one for free convective regimes (in the very low frequency range).
One can imaltne semi-pemmanent "theamals" over terratin features such as
bare surfaces next to wooded or water areas.
the lack of horiontal
Here horisontal fluxes and
otaogeneity -muld be important.
Despite all these
difficulties, if one had a homgeneous lower boundazy, it seems that
the average valu of the spectral density of energy at a point would be
a reasonable first estito of its spatially avera d value* Then the
low frequency observations here should be reasonable, Including the occurrenca
of a moam
of convective energy at low frequencies, as shown in Figures
8 - 15, 24, 25, 26 and 27.*
ost of these observations were taken under
northwest flow patterns at Round 1ill, where the effect of sea promLity
on horizontal homogeneity is minimied,
The light wind cases with convection
and those having sea trajectories, or trajectories parallel to the shore
0IMiI
are more dt
.
ticult to valte
ghod of appMoon
The investi ation proceeds ai the follofAlg ordert
1.
Defi.nition and euplantion of the necessary derved
statistics,
2.
P sentatio n of the pertinct statistics for the twenty
runs in tabular form.
3,
A discueston of the choice of pirameters to
press the
£ects
of thermal stratification and the lowet boundary.
4.
Presentation of the powr spectral data bt Walonn-bukov
similarity coordinates.
5.
Presentation of the empirical redult,
including the identi-
fication of the isotropic range,
6.
A capareson of these empirical results with those obtained
by other investigators.
7.
A comparison of empirical results with those predicted by
theory and a selection of those theories which are in best agreement with
the avalysed data.
G. A su~8ested model for energy spectral daensty functions.
A,. eitqisn of devgyed turb~uleqge tatishtc
The folloing is a ILst of some of the derived statistics used
in the data analysis.
Additional statistics are eithe
defined in the
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TABLE 7: TOTAL VARIANCE OF ZL, V, W, AND T
BEFORE AND AFTER FILTERING (HIGH-PASS )
TOTA L
RUN
Z-.U'--1 2
M6D 16M.
+
40o.
E /6
G7F 16
0
6 16
+0
678 /6
VI- z
W
T'
t d/
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1..1/
.3.1
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40
16
.E 40
-
G/6
4o
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40
32
I/6
40
s S/6
4o
e- 40
.. ,3S"
4.ra
J. 0.14
3.6ob/
. 093
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2.99 i.
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12.370 .130
4. o30 .74?
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AFTER
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P7.rt SENERAL FUNCTIONAL 'FORA OF THE DEPENDENCE OF NON- DIMENSIONAL
DENSITY ON WNO- DIMENSIONAL FREQUENCY AND STABILITY.
t-
!
t ij
-
4-
-F
D Cll
+--+
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ENER6Y SPECTRAL
I
'
"~"-~i'.;
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t
ti
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7,ii
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t i
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TAWL
47AM
rN&
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ii
V[
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sul
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r
Z
iIi"!-.. :.. .. .... ..... : , : _ __.,
:-:'
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... '.
t; - -' - .. ....
-
-- '
-
- -.
--r ------
'
Iir-
- tt-i
.1
-:---
- --
..... .
,.-.
t:
: i~
-' - -.....- .L - .- -t"1-
gIAN4 ir
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i I i
t
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e
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---
.
I-.
i
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L-t
: _k.: ..f _,
.!
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~--C--
i'
I
K-
-----
U
t
t
:-
S5-
tables and figures *ezOe
they appear or in the resu~sm
of the paprs of
Section III and the Apped2i .
1.
Standard dewation of the asimath angle, a
aQ
j2
and
SV/3
whe
Ta6,(
C
eor aaUiage.
2
Vertical gradi ut of potential temperatureJ
3.
Richrdson number
computed for several layers
in an effort to see if a aingle reprsentative "bulk" value really 0exists
4+ RittonCU velocatty V+c
5. priction emporatune,
k = the von Kanzi
V Id 71 /
constant.
Similarity stability parameter, a/L, where
0.
T1ese and ma
h
L
more general statistics are tabulated by rMa L Tables 3-7.
Tables 1 and 2 are samples of the calculation procedure used in putting
the powe speatrl estmates into the following siatlarity coordinat:
log
S U
versus log
, where i
u v and v
and
log
versus log
w,ere T = t4perature and
SI and ST are power spectral density estimates.
data in this coordinate sytei.
Takeuclh
See Figures 8-15 for the
These are the same coordinates as used by
(1961) and Nonin (1962).
Figure 5 shows the general foa of the powr spectral density
S6
:ncti.ons (in non-dLmnsioal. log-log coordinatea) for both almslarity
Figres 8-15 and 24 and 25 ow the data in these
and U -noxualizaticn.
coordiatoe system.
In quIa tat±ve terms, it can be seen that there are
three apectral recgias.
1.
A hih-equency reSgion of fairly unifom slope, where the
nosmalied spectral density = f(n) only.
This is called the mechanical
and isotropic range.
2.
A lowfrequency regioan,
is an increasilgly stron
wihe
the nomlized spectral deu ity
function of stratification with deasing
frequency.
3.
A transition region betwen these two.
At a particular site (implying constant rougBess or lowr boundary effect),
ou has the eWpirlcal problem of espressins Sis the nery spectral denaity,
as a function of U6,
n, and stratification.
by the interdependence of
Z
This function is complicated
and the stratification.
At this point it
is necessary to choose dosc-Lptive pare8tears representative of the
stability and roumhness.
Batchelor (1953 A) indicated the appropriateness of the PRichar on
uumber in specifying the effect of stratification on turbulence but, as
seen in Tables 3-6 this number is qtite variable with height and is
also difficult to evaluate with sufficient aracacy.
The stability parmaeter, Z/L, has the disadvantage of requiring
that the Bross statistics,
and
T,
be known,
in the preent w~Ok 10 to w5e the mre easily m~a
The approach adopted
aured
quan ties in
determining the power spectral functions and the gross statistics. Althoug
_
it is necssaxy to use the
mentu
and heat flux values later, it is
destrable to keep their use to a min~nam so that th
errors in the power
spectral denity functions, occurring as a result of errors in these
glux values can be Iaol ted.
The gradient of potential temperature can be used, If care is
taken in evaluating it.
Most of the thermal gradient occurs near the
surface below a height of 4 meters, thus, the levels betwo n which
is measured, are critical.
Additionally, in an unstable atmosphere,
small differences in theral gradient m
indicate large differences in
the ecale and intensity of turbulence.
The standard deviation of the amisath angle, (CA, is a sensitive
paramter of stability, as shown by Cramer (1957; 1960), Takeuchi (1963),
Shiot=an
(1963) and Inous (1959).
between (
Inoue found an eapirical relation
end stability for the Prairie Grass data;
degrees ws found for near neutral stratification.
a value of 6.5
The quantity,
O-
is a very voak function of heiSht, a, and is nearly independent of a in
very unstable conditions.
the relation between
Using data from about sixty runs, Figure 6 shows
and
.
For vey unstable conditions,
is a better indicator of the relative difference in turbulence scales
and intensities than o
*
Under very stable conditions, when the
scales are small and the regime is more localized,
d/
mra be a better
measure of the stratif cation (in the absence of filtering), because the
value of
A
is sensitive to the effects of long waves not directly related
to turbulence.
Using igure 6, a modified stabiity parameter, (
cn be defined.
In the mderately -stable and vry stable region,
c <7-
)
dB
in the nea neutral and unstable region (using the "strong
,
: STANDARD
sF.
DEYIATION OF ATIf UTH ANGLE VS.
i
i-
.
--
______
__
t
BETIJEEN f AND
/e
.
--------
iIwr 9
.
*
ITERS.
. . ...
1O M i---TE
.--+
i
-t--%
i
!---I- -~-I
II
Al.
- -----------
I-i
0O
i0
--
10
1
r
i
'
i
I-)
JLe)e
i--t--i
or,&f-
-
t---I-
I
-
p
~+O+
*.4
77
"
J
i
~r
'1
I
I
r
-----
~----- L
L_~__~C___~+_~___C~__
f
-I-*----
0
I--
I
.-
----:
- ..I . -
-i
----
-------
0
. .
-4 ---
-
.-..
-.
I
L--
-.
0r.
I
-
--
--
Li
....
..
-
~
I
_-
+
o~
i'.10
,
LX
I
-1
__
'F--i
_
---1~
-1 I
-
--
-TviJfl
I
-I~t---
I
-
.
(.TEGREES)
04
-*
---_-st
~_--~-i
]]
-_i--- i_ I--i
_
-
-
--.....
*
i~
-
-
I
+. --H--H
i
/
r----
t-H----
I-=-
WI
!,- 1
---
i
e0
hj--8
//
'
/i -- i..- --,~~~
-- -ILI
__
wind
curve),
obtained by placing O on
could be the a
a
the strong wind curve and reading the abscissa°. Thi
parmter in
to 1orporate the best behavior ef each (" and
A modifted
their respective regions of superiority.
easily be obtained, using the ameu
wuse
desigped
fauncton
Cj
can Just as
curves, and s more couvoenint to
s
are also felt in the Richardp
The evaluation difficultiesff 94/
nuaber calculations, aggravated by the taking of a ratio of two wmall
d64
/and
and not accurately detezrmned quantities,
6opecially under neutral and unstable conditions and at a distance from
the lower boundaxy.
It is just under these cond:tons that the Richardson
nu b r is so critical in defining the reg~me
the stability is defined an rIoo
where(
4
of the asiuzth angle under neutral cnditions.
On a purely ampxical basls,
is the standard deviation,
The quantity,
Cjj
used as the description of the roughness or lower boundary effect.
defiiing
V
'x
By
, the lower boundary effect is epressed
in teom of the frequencies it mxcites or,
in terS of length scales.
is then
Y 4/xges)
the rouginess
This assumes that Tslors hypothesis is valid
An the "mechbaical" range (Panofst1y, Rao and Cramer [19581).
The quantity,
elemnts and varies frxn 2-3
Seashould emphasise the larger rou~g4 es
degrees for a very sjooth site or sea surface to 15-20 degrees f-r very
rough ites, such as hilly, wooded areas.
This variability Ls compare=
to the three orders of uagnitude or more variation for the riction length,
r
as s8hown by Sutton (1949), p. 103.
Also V
o to small changes in the lower boundary (se
[1958)., From the follomiag identities,
is no t as sensitive as
Cramer [1960] and Obukhov
~
6,0
2-
2__2-r
coo
it is seen that,
(5.2)
U
which is equal to the area under the curve of the semi-lo plot.
"
The quantity,
,
is appro
mately the ratio of the v- component
gives this ratio
enery to the kinstic eurgW of the mean flow and O(
under neutral conditioa.
Thena
"
gives this ratio from site to aite
and supplies an eatimate, under neutral conditions, of the v-component
variance in terms of U.
If n o 1, where 1 is known to be in the inertial sub-ran8e,
E3
of thi
(see Pasquill and Panofeky (19631 for an evaluation
constant of proportionality), and
=
f /
If the level of turbulent kinetic energy is stedy,
4
+
onerxr per unit ma
at frequency, n
mul,1
_
, where 3 s turbulent kinetic
(E -'L
sumnsd over the componnta and E ms
. and
A if
=0 s then
W&T'
and
w
'
the same quantity
are assmed
except under untable condition when they tend to counterbalance
each other (Taylor [19613, and Panofsky (1962]), then:
This permits one to ostablish the relation between the dissipation rate,
e
_~ __ I
~
_^~
41
/O& VS. Lo 7LE/O AT THv SAME HErIHT FOR
FIC.7? SCHEMATIC OF Rfs
TWO SITES OF OIFFERENT ROUGHNESS, EXPRESSED IN TRRMIS
DEVIATION OF THE
O", .
STANVARD
OP TH
IMUTHH ANGLC
UND4ER
NEUTRAL CONDOItONS,
J
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:, ,,em
,~
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r:1
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iI+t
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!I,
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i--
- :
;
+,
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'.J..L..J
______....
I
iiI
i
. . •. ...
i
;l
j.
i+
:
+ -:
*'--
--
r" ; !
could be #xpressed in terms
siugg sted that
and the strs,tIto
of two fductions, one that is stability dependent and one that is only
A ~anewhat different approach is taken in the
frequency dependent.
following paragraph.
Jusinger (1961) considered the recultain
spectr
if turbulence
were isotropically introduced at two discrete frequenaes (a low frequency
for convective energy and a high frequancy for oschanical ner~r) and then
"cacaded" to higher frequencis as desacribed by the Kologoroff 1rpotheats.
A generaliation of this idea is used below:
Lot 8,74)
be an increment of energy introduced at frequency, ni'
If a "power laF' of aonS form for the enoy spectral density function is
obtained frua asoumptions about the netial energy transers,
then an
increment of energy, introduced at ni, produces an Inrament of energy,
at n,
whiCh
e; d(O'
Is
*
i
for
d
S
!
, or j < L
SO
where c = + 3/2 for the wXrachnan model and c a + 5/3 for the Rolmogoroff
The total energy at j is
hypothesis.
E(
ZO)
)
%
-
C3)
may be thought of as a source strength and (14/d c
h
. an influence Aumction
twhere .is
in
the
inertW
C.~d~c13~,c
ni relative to n:f Let Et
sub-rangeo
be considerd
be a value for H,
Then daeine,
~%
o3
In practice
the integration over the anie my be over a very
restricted range If one paticular roulgneas element is dominant.
Nest
define the total enerv
f-4
.L=f
all i
/.;
Ph.l[n;ln: )Ldn.
4,.
.
The relaton between
convective conditions.
effact is thougft of
f
(5.5)
and 2 ti dfftcult to handle, especially under
These difficultile can be voiLded if the rsrghne
1n3term
introduction
of .nergr
convective "e=citation" as a variable roOugbwis.
this idea sutests
~hese ideas are formally
expressed as:
mechanical rocrugh
(5.6)
,
onvective sortastionl )
;'
(F
*A
$rd~
and
(5.7)
(5.8)
d
This discuosion is expanded in Section V; and more is said about
this model.
ener~,s i,
R(q) and A( ?,A ) are the fractional portion of tha mean
enracted at each Fourier component and is associated with an
actual physica lngth scale of the elements of the loer boundary rouowneo.
D.
Prtesntation of por
seoam&
An taIlarin
coodktes.
,4
Tables I and 2 show the calculations needed to put power spectral
density data into this coordinate system
Figures 8 - 15 give the plotted
result of putting this wenty run data sample into the Monin- bukhov almiThese data are analyzsed in Section V
larity coordinates.
under empirical reaults.
1, and ese~here
Although other investigators have used this
coordinate system, it is believed that the quality and siae of the present;
wch more comprehensive evaluation than has previously
data sample permit a
been possible.
The data introduced here is the basis of the results to
follow.
E.
ResJt.
The results are given in several different foasm and then are
compared.
Each of these forms is a mens of evaluating the enerWY spectral
density functions, if the meteorological parameters of mean wind and stra
and the geometlrcal values of roughness and z are
tification are give
knan.
The first approach is to express the non-dimensional enerY
or
densityS
as a function of s C7
n and a, where the coordinates are those of Figure
or (
spectal
'(
8 - 15 and as schmati-
cally represented in Figure 50
1.
Algebraic expressias for' energy spectral density functions in
similarity coordinates.
The approach here is to express those functions as algebraic
equations;
densities,
then to use these equations to reproduce curves for the spetral
,
n) , for a given velocity component or tempGrature
at a given level for. a given site.
variety of ways to estimate
V
The next step is to use any one of a
and -'* (see the next section), assuarng
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that they are not wailable, to return to dsnsonal coordinates and thus to
give the desired en inerin& estimate.
As shown
n Fisue 5, if the high frequency dependence of SL is
expressed as a function of V,. there is an asymptotic approach to a
unmfom slope at high frequenies and a suggested asymptotic approach to
a difterent uniform slope at low frequencies, with a relatively smooth
transition and a single man
in between these regions.
chown later that this is an over s~plification.
It will be
However this approach
provides resulta, although it does not contribute much enlightenrent.
After experimenting with a number of functions, the following function
is
found to perfom this ' atch!" of low to high frequencies in the
b
desired manner:
eHEnAJS o NJ L
where I is a linear function of
and r'k
a
)
is also used as a stability parameter.
Si at n = 1.0 for the neutral case and serves to position the ordi-
nate, Si
at high fraquency.
n actually stands for
here
b is the slope value at bigh frequency
c is determined by the value of Si at a selected low frequency, say na .1o
b plus d is the slope of Si as n approaches
ero.
a is determined by the low frequency stability dependence of 81
F is determined by the high frequency stability dependence of Si '
NO
- 15.
These conastants were then determined, using the data in Figures
Then using selec ed values of CF)
rh.re CJ
= 4, 5, 8, 10, 12 &ad
18 degrees,
the results are sho
values of
and usuin
the neutral case,
v 10 degrees
the algebraic expreasim n
the corresponding
n nFigures 16 and 17 and by
rt B of the Appendix.
The ~osa serious deficiency of these empirical functions seems
to be in the transition (between-
a .02 and .07) in unstable cases.
Eere, it looks as if a sepmaation occurs between the mecanically and
the convectively Introduced turbulence.
FigurS
s 16, 17, 24 and 25,
Compare Figures 8 - 15 with
If one could tolerate a 50 pr cent error,
then for a large part of the higher frequency scale ( ?L/U >
same 9ucton can be used for either 16 or 40 maters.
J/
)
the
This can be seen
either from Fig ea 8 - 15, 16 and 17 or the algebraic expressions.
Addi-
tiownlly, under near-neutral and unstable conditions, Su and Sv are
ntercsng able with less than a 50 per cent error. in this region,
bo ooe hundred derees of fredom in the data, so that 90
there are about
per cent of the time
the observed value.
re value should be within 15 - 20 per cent of
the tmhe
This difference beteen the 16 and 40-meter .- xvcs
is apparently significant and the use of the more simplifled functions
toild be a matter of accuracy requireients.
In order to return to diEn-
j must he used.
and the heat flux,
;ional quantities, the srrass7t,
2. Various methods of estimating stress and heat flux.
See part C of theb Appendix for a number of methods of deterramining
th 8stre8
and heat flux which are suggested by recent papers.
As an alternative to these methods, the stress and heat flux data
FIG6.1: GRAPHS FOR ENERGY SPECTRAL
COOROINATES.(ALEBRAIC
I~nP.A
COMP ATA4tl
DENSITY lAf SIM'ILAR/TY
L^P
Il
EXPRMSIOAIS)
4. coeA
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The results are
for about sixty rums, have been plotted and analysed.
presente cn F
urea 18, 19, 20 and 21,
Ts
thod would require a now
seat of figmes for each sites If the functional relationships of the
fluxes to the site parmoters are tmll bhaved, these figures may be
quite predictable, after the accumlation of morte data on the site
variability and its effects.
In Figure 21, the .05 labels of divergence
between 16 and 40 mters could be relabelled as cooling
of heat fltu
rates of appro~imately ten degrees c
3.
tigrade per hour.
Regresion analysis of power spectra.
The power spectral data for the twenty runs in sinilarity coordi-
natoe ware initiplied by
qantity
bo
by
a appoPtat
(IT'), is measured betwen the 8s=face and 16 or 40 meters as
reqjuLSd.
Then at each of twelve soloted frequencies (na/4
.015,
.03, .04: .07, .1,
.02,
VAt;
The
o
.2,
a .007,
.01,
.5, and 1.0), a regreson graph of q
feor each velocity ccmponent and temperature at
each level was made.
A eimple curve, usually a straight line, was chosen
by eye to beat represent the data.
See Table 8 and
gures 22
nd 23.
Next, for each velocity component and temperature for each of the twelve
fraquancias, a value of
1
O
J., .J . 4r
was read, from the regreaeson curves for each of si
8, 10, 12, 18) and recorded as shown In Table 9.
-L/--o
stabilities (
4
5
From the derived data o
Table 9, for each velocity comqpoent and temprature at each level, log
or. log S
VS,
log na/U was plotted, giving two figures of the energy
Opctral density of the component fluctuations with stability asWams
a preter
See rigurea 24 - 27.
These same data are also shownbi
appopriate sewi-log
Fg.Its HEAT AND MOMENTUf FLUX AS A FUNCTION OF THE MEAN
oOl FIED STABILITY PARAMETER.
)*, M4~
W1IND SPEED AND (
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6
S
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smErERS -4o
FIG.19l: STRFSS AS A FUNCTION OF WINO ,SHEAR AND THE TFMP.
DIFIERENCS SURFACE TO RM.
SHEAR
.-,-
FIS. .0: HEATPFLU/X AL A FUNCTION
\
A
OF "/
I.
nr -. 01?
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FUNCTION OF 06 AND ATEMsRI.
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F16.22ISAflPLE OP REGRESSION C*APA' FOR AfO1DVIDUAL Frf*49LEC1ES(& oWtS), I-COM,f~~~1.
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FI1.23: CONTINUATION OF /CI.i.
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84
FIG..#: ENERGY SPECTRAL DENSITY AS A FUNCTION OF STABILITY, OBTAINED
. u.-COIP ILONGITOUD!NAL) AT 16M1.
BY REGRESSION.
.
.
.
,,
,
,
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C
t "tt
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c. v-Cof P. (TRANSVERSE)
.I
D. V-COMR AT 40M.
AT 16 M.
-r svn~oL
SAB.
SYIIBOLS
SrAe. sYMO+.s
-
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FIG.JsJ ENERGY SPECTRAL DENSITY AS A FUNCTION
A. W-COMP. (VERTICAL) AT I
.
OF STABILITY, OBTAINED BY REGRESION.
B.
- W-COMP. AT 40bM.
7_1~
1'
t-
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IT
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D. TEMP. AT 40M.
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C. TEPIP. AT 16.
-
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ii
too
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11
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i
;
SHOWN IN
AS
FI6.a26: SAME DATA
FIG.4+, IN
SEMI-LOG. COORDINATES.
SUITABLE
e . o
A. U-COnP. AT I6M.
0 ,*
.
-. 6o
r
AT
M
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rs
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t
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D. V-COMP. AT 40M.
C. V COPIP.AT I/M
SrAA I.. _SyPBOLS
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.
TABLE 9: UO'NORMALIZED ENERGY SPECTRAL DENSITY VALUES
FREQUENCIES(ret/U).(SAMPLE IOF
V- CoMP.
o
fRUN
O
-f
i~rP
.6D
411.
a.8
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440
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.TABLE 1 : DERIVED
DATAa. FROM REGRESSION : Sv /eUVS. SELECTED
.VALUES OF N'*/c, STABILITY EXPRESSED BY o- ; b. FROM ALGEBRAIC
FUNCTIONS: SvO/4v L VS. SELECTED VALUES OF We/G, STABILITY EXPRESSED BY (a/~~T
a.
V- COMP(1/M.)
-
O
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1.39
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I.02.
3Iko
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FROM ALGIEBRAC FUNCTION S
=.1
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30 ,1
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fo m
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L~ W =.0os0
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11
FROM REGRESSION
Sv/U4'-
. ,..007 = .0/
.-
--
-
-
in FIures 26 and 27.
verus /
'1
o
coordinates, '2
From
The data in Figures 22 - 29 show a number of itportant features.
Figures 29 and 30, despite considerable scatter in the graphs of K versus
at a particular frequency, same general relationships are evident.
S
(a) for runs mo
Two reions seem to appear
graph of _Y
C
o lss strongly depedeno
(b) for runs with
- ;
is a rapidly increasing function of
7, the grap of
7, the
stable than (r
t on
At low
.
frequencies, the slope is less strongly dependent on
and becoase
in going to higher frequencies.
With very great
i
independent of
stabilty ('~ <
These
values become insignificantly small.
5), the Y,
graphs auggest two different re~imes of turbulence with the transition
a.06 and 18 also support
Figures
7 or
region near O
In the
som discontinuity in other functions as well, in that region.
case of
, with (r, 4 7, there is no significant energy until 2i)
.03
Otherwise, the behavior of -v
is
.04 at 40 m..
)
at 16 m. or for 2
auch like that of 9
.
shows the same two regions
cw-, 1
a 7 and Ita behavior i~ quite sinmlar
of transition in the vicinity of (4
to?L
.
On C
/46is o
<
(
The stable res7to
1 £Y,
unstable region, the graph of
is
7) shows a very strong dependence of
In tie
=
o.
$I possibly
is less dependent on (A2
A "
. but shows the same decreasing dependence on
frequency, characteristic of the
similar to
with the zone
and V functions.
than
ith increasing
wQ
9w, 4o
is
J-8t1 6 except that no significant energy is seen ii the stable
region until hon
.04.
The
2
i st
-5 behavior is
As shown in Figureas 24, 25, 28 and 29, the
2
like tha1 of
and (AT) fnor al1-
.
FI...
" COMPARISON OF ENERGY SPECTRAL DENSITY FUNCTIONS AT 16 AND 401%.
8. V
Co-PoMENTST
1.
W-CMW~ArNK
16M. -
1.0- .
------..
1.0
0I
F
i
.'d
Owl-
---I.--
1------
- --
.01-
I
4%
.00- 1.0
-
-
1--
Li
.0
* -:
ti.-----.00L
00001
- .0L
Al1.
*
I
U
.ZY: COMPARISON OF ENERGY SPECTRAL DENSITY FUNCTIONS
FI .
OF" ?4 AND V COItPOMENTS
1i---ttj7
1.01
, +j
OY7 -
:'
I t,..
I. . I
7 -.
A-
, c ,-
1
i
4-
.. iii
7...
,.
oK
r t- - I T9'I
- ...
. - ! , +- , , 4. .. -'H4O.I91
Y ...
-.~
,,
,
j~i-{
*f
-
I
/9i~
.4it
04,.
• II,.i
....
~- I-~, %~ ioF' I: -_-,:-fr
I
.0l.'
.
tI
- I- -
lI
-
. . . '3#I
e.i# 0.s
,.
,
/Ll
*i
.001-r
.0l
O ti.
i
I
r-
"
-
:
,
-.
-,
-
-.i
"
+l
-" "- -]
. -i
. . .. ..
LI . +._.
.. .
,
9
J
r.
L--
K-
1t'
l
L'
. .
.,
'
"
i
-4_LL
IsI
i
fT
F- i,
1TTP71
-- -- -
..
i-- "-
4
. .. r.. ....
----T . ... .+
1
' i
+
! .i r
;
c.. L
U"f4-I,
~
8'
-I~44i+
, ,--'*€
-
-. .....-i
I
i
I
4
-~TF1
-'
oo
"
"
........~--.r . - -t
--,-2 -+
I.-
l
I
1-
-- r , +
,' "
......
-- .- . ..
[-
_____
t ',
,
'
"
'. ....
-
-
...
. '
-- ..
... F-- - - ... '- ... ..+
/
i
+"T - - 1 ' -
-
t
.. . .r'"€"
Ii
.
.. .
.
-_
t
. -,- Th7T
I
.oe,
/oO
.1. l---..... i.0
r'
'. ___
1 '_I
";. " ..
'"
:-i •
.I
.01
£1o
1
93
zatlon are successfl at high frequencla for 04
All fuctions show the increamin8
lower frequencies.
>
7, except for
o
),
dependence on stability in goina toward
At the highest frequenoCes, a -5/3 slope will fit well but
toward lower frequencies, the slope takes a less absolute value, which le
nearly uniform over a considerable frequency range for the
- functions.
A cmiparison of
f
,
, and
1igures 28 and 29 show. that, at high
frequenctes, the normalised spectral density fumctton is nearly the same
for
;
/
,
& nd
; 4o
and
Also 9
gj;
4
-o
are in close agree3ent, although the normalisation to not as good at 40m.
Figures 24-27 aho a huge peak of low-fequency "convective" energy,
with uns table stratification, which appears to be separate from the higherfrequency mad~iu,
which is lesa stability-depndent and apparenttly associated
with mechanical energy input.
clearly and it is
Individual runs show this behavor more
asiificent that this feature is retained, after the a
of a~otlAng of the data which is inherent to this analysis method.
seoa-log plots, Figures 26 and 27., give the ener
The
camparison more graphicaplly
Here is said about these two separate areas of suggested enery input late.
The main purpose of these figures grouped under Section V E 3 is to exhibit
the general effect of the ozbinatUio
of ftaquency and stability on noma-
lied power spectral density fuunctonss
4.
Discussion of normaliisng techniques.
Seven runs, selected for their diversty in wind spee
and
stabi-
lity, are roplotted
uithe coordinates indicated below.
Those runs are:
Run 31, O4
p
16 0 36m./sec; Run 34, Oa 18.20,
16
Bru 3S, b
12.70,
3.4,
016 = 4.0m./sect Run 40,, .=
nt
4.7,
= 4.7m/secs
U16 a 3W.a/sec
___
_I
I__
V4
Rm 66E, r.4 a 5.4*,
4 i 6
Run 67E, r C
.4*O U
4.3m/eec; Run 67C0 r
= 9.6m/sec.
6
8.*4
16
=
6.2m/see
The coordnates ae lo(g
versus log T, wh re S = kinetic eaergy spectral dnsity and L = u, v3
j m 16m, 40m; u( 2 )
e n
ind at 2 maters; T a time in seconds (see
Fisure 30).
These sal e rmns ar
log
also presented in the fonm log
versus
where I a u , v, and J a 16m, 40 m. (see Figre 31).
The last set to be presented l se is flS'
T
.
period in seconds, i * ta, v,,
versus lo T.;
Temp.; j = 16,40m., giving eight
pres;
(see Figues 32 - 39)o
Two systems used by C~amer (1960) are of particular Interest because
of thoir success in "normslising" a variety of runs.
of
3 for aormalizatinc
These are (a) use
of spectral density values and (b) log LU
versus log
The Ipux ose of examLing various coordinate system here is to see
which is best suited to describe the functional behavior of the
dezl±ty functions in te
s of physically relevant quantities.
pectral
A nnay
of the six syatems to be compared is:
(a) Siilarity coordinates, lolog
3 - 15).
(i71.uree
(b)
ortmlisation by
qlog
versus log
sz/,
or log ~cT
o (46 )2, log
(Figures 24 and 25.
(c) Sami-log plot of (b),
or
Lversus
26 and 27),
(d) log
L
vru
verOus log
, (Figue 30.)
log
a/U, (Figues
95
RFi. 301 ENERGY SPECTRAL DENSITY FOR A SELECTION OF DIVERSIFILD RUNS AS
A FUNCTION OF PERIOD ; NORMALI ZATION IS WITHr RrsPEC To O' AT-tf.LEVEL.
A. 21-COMP. AT IfM.
S. V-COfP At /In.
1.0;
__ '__LL
4~
:
,.'.L_.I If
:,t..
.0; I?t I,
V
i
-
.
.
t t+-
Ft'
-
+
ir+ '++
*00I(
++_ .....
.. *
... ,
rA
.... #+.., ...+
.. .
..,._
.,,--
$
rt
f
10
,
.00
+-"~~
I
777
4i + 1 : +
i C
:. *
:r ,i *# '
''
C
+
f,
l
g
4'
'
.
1
I++Ti.I
[
.
'
.
't
V-,.--t,
St .
**2
',,L_
I+++- , .V
'
-
0,
4_1
L --
,V,
•
.
_ •
,
T
vf.,+
ijl.1
. 'i..,
,.
Tll-
t+l t , ,,-
i,
L+, +
+
o
I -' + +t i,-l t+,T: r,,
....
I
t,
,'l,
:j3*!
0
',- l-'
C,
i,II
r
r-,e.)(aie.;
.001.
•
._f--.
I
,
~
---
r+
"
+.
T ;+
-
'.
II -..
:'*!
,-
.• :-
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+!.
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: , li::....
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,
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.
--
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i - ' jo
ITT,
+
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. --L- .iiT
m...+
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+'t'
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i
k
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.I- K.-v.Y 1,rric...
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.
eo( e.
D-
, .
,
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,
- -
i
w; !+t~
t,+
!+
¢
-
i,--
Il
.-iiK.F
.-r+a
- 4,
,
•
411
00I
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+
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L
.1 (fc6
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,I
:-
+ oH#.
,
0,
LL
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....,+++ ,.
f
-+E -i -I r-T -'
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.. "l
q +'W'+ '+',1 . , . , .."I
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ttL
++.--
+' I t.r :_.71
t-;
-- +
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i--.....
'-..I i+-,
+ ,
+
l
IIIr:-T Ii,t
;.
.
+ ++-
-- +
,- i
.
i4l;4.+
it!4i. t ...+
J- li, .
T'l-T
-':
i
" +t +I+ .._ ! .
+++e,'- - .
7
'+,
IL ;; I ! :
I' ; :l
:FlTl-I++:
[+Jt
L--+= f+-::
CL<1:,:
--
--
CC-----
!*t
-
I
-+ '
'T
-- i
r
"-
,_ll
,'
~ +,i-. . .It
i
I
'I'--"
i "'
.
I
P
i
:
: Y-IIIIIIIIX.~^-IIYL.II~-~
il^l
I .31:SAME
-COMIP.
A.
s. VS.
DATA ASF16.0, BUT USNG COORDINATES
AT 1.
IQa)
LU: I
10~~~~I
ilii
E-E
,
V-COI
S.
La .U OR
-V
.
. AT 1
~
.. _I
j1'
+> , .
"4 +
rI7 I
"l0"1
1 . "
"++v
+ ),O
?'
i'
I:o
="
wri
I
I
" , ++r . . .
I
[ ',,+
+ ++
,
--
,
,~
.. l , i
,
1. 0
.1
,,
II
-
I.....
'
,
Ir
.....
.
:
f
|
e ;i"
(I
0*
S
:r :. '+'
''
, - - t ' ,' i
S-+
"'+
.
I,+
...
.. .
'; +-- r. .
.
!-
+
,,
..
..sij-+" ' +; -r..............
.
-r
"I' ; +
$7~
.'
WaRt
1E
•
'Aeabl o-."
'
IO'i
3r
i
+'
+
,
m. + +
m
-
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.
t
i •!.
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,
...
4
sw
~
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°
+ ""+"'-" "+)
+-
- .thd--ld-r
.
..
+
i-1
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- - ALv.
• , o
.
____
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I~
~:_i+u +r+
~
-.;
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++ ++
+.--
"
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--
! '; !ti +, I
.
_
;+.
, +
';'
__1IL1
. I , £r
O;L
X-I n t , I
f
.
t + , +
.
. . .
.01
1
1.0
_
.- t
,,
,,
..
C.
L-CoM.
AT 40M.
-_ -
-I
V-COM..AT
D.
~----- 40M.
.P
li-i~
- ,
4.
ci
"
...
-,'
0
--
, " S+
t
--
-
0 ,
+-I..
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at
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.
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l
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.
If :
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Or i-+W
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-
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l.4
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---.
0
1.0II
. . +t-
r------
G-.- ; - . .
tt- 4
t
..
.
- :-
AMAM:11 IT.
~C~
.
I
0~~1
_
+ t
i... . .
,
~
+....
T +_
7+-r
1"
-
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" .
iti~~T7f7
i-i
-t,1 13
.
-i
,
--
i i I fi : +
- -- -
: P+
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,K
a
.
"Ill1-I -----
*,
OTC-- --t--- -,
a i +-
-;o .
,
--I++,.----
" :~~~~r
" + i,+,'
'-
i'
l
U
:
:;,P
-h++ .+. - + +
'~*iI-
i
,-e. i
a,.
+
.
i
+l +.+:
t- .. . ..,+t :+ .
. .
+ '.+
:, _,
-
-.
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. -I
.. , *P+
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;l- -:f
.
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.. -9:JAIY
. . ..
o ' "'
...
, ... .
iv t
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i %.
,
K;b
+ i ,+
iiii
,;I!
--.9
i
-Iaa
v
I ,
"•
.
•
-
Oi.i- -, -;
- -
I
..
.. -' -
01
0
. -+
+.+
- t~i + :.+,+ -
4-,-+
' ji
It V
' +
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.
-
+* -1 -
-
,-pY
. ,
:
.iii,
k,. .
-q@O
It,
'
1
1
-----.-
1-14
...
4
,
+!
. ;.
i|t.
ILIII__^III1_YI~Y__1_1_-~-11111
FIG. 3.21 POWR SPECTRA,. FOR I RUNS /N CORD I NATES
S,16 VS. LOC 7',
7' PER/OD(SEC)
PmfT ITT1TFYAL
J
AI
-
.4
I.0
--
-
-
' '
I1 1
' ' ' '
11
-2
-I----1i' ': ' '
9(
.
W
444
V4
-- 444
H
'
I
-- ---
.
L,
-------i, -
--
i7E
7
-iie
! I.
+l
i
-
"44---
i- ih , :i V4AL
i
kfi
i~.
I.
i~_.i_
-1---
I I
a.m
i..,.
t.1lt~
t
T7777
-I
I
L. J.
-
*-1--
fia,
Wr
___
-$-
I I.~flK:
K
'IF
Thi-'4
-~
j(5i c.)
I
~
e'
I: P n
r
n
9/8
FIG. 33 POWER SPECTRA FOR TRUNS IN COORDINATES
".ey vr, O'7
7LSv,~ V. LOC 7.
I-
-
i"1
~
_
FIG. 34t POWER SPECTRA FOR 9RUNS IN COORDINATES
nSw,I VS LOC 7T.
FIG.35": POF ER SPECTRA FOR IRUNS IN COORDINATES
?n SrE,,is VS. LOG7.
FIG.3:POER SPECTRA FOR ? RUNS IN COORDI NATES
/. 6r
.Su,
qo VS. LOG 'T
viaS
n-~
I
I I
i , ,, ,
i7T-
I.1
~1
/.3
t--1-----t-
3
i.
Hi
It
I
----
-i
.
i.
•
,al
-J
*1
I
i7
,
;
f +A7
~
'
'*
1
-
I
--
~.....
~lH
,- 1
1F1
i
/v
i !~ 1~
i i'
-!*
,,
i)
---. . I
I$,
-
TI
f
i o
-"
i_
'
I-,--
Tj
1-1
/-
A
-
IC
*
6r
-th
~jk
-
"C.)
"
0 - (SE
-.
---
J--
i//
nSvo VS. LOG T.
F/G.37: POWER SPECTRA FOR 9RUNS IN COORDINATES
VWI'
1
".
*
I,
iI
I
IK-II
II
i I
!
---
i
iI
I'ii
i
!
J
ILJK-.
-1 1 1 1 1I
II I i 1 _I I i I I
4+1
1
r-I
I
I
I
I
ii
bLi /
I
I
:11
i-t6
I
7
,
----
A
Z
i-
....
I
IiFiv
tV
I ;
-i
SI,.
nlS-
-
II I 1 i
i
i~
S
i.
,
I
t--
!
Li
.1
v
I
i
___
-I
id'~
t----IS 0--- ---
.I
L~i
All
.t ..
a
L
*
1I
I
t I 1,
I
I cl ,. .
'_JPFT
i
I
2
t
.i r
,
I
I
I1
Pj
- iA
-
i
I
i 3
1
- I
T'(SECc.)
1
t
.
1
1.
. 'I I
--
&
--
I i a.
-
.
rl
i
i
I
A
i I
-I
a
I
I
n
1
''
Ir i ---
FI6.38:POWER SPECTRA FOR YRUNS IN COORDINATES
nSw,,o VS. LOGT.
t
F1G.39: POWER SPECTRA FOR 9RUNS
--
I N COORDINATES
t---,-
-I
;_.
97A
67
670
S67C
,S-mr,, o VS. LOG 7.
XxY
7Srvr
P---
iIi--t
F
!* ,
C
-I- F
i-
-"iL-i--
~1
.-03---I
v!..
-
A
It* --....
F
0 4i
F
iv
I
__ ii~H~
-
- o4-
_• 1_"
- -i i_-
.. 44.
- "
i I
TlsEc4
i
in
M,
-
(0) log -,.
(Fiagure 31).
vermwus log as/,
(g) 77 Sz versus los T, (Figa
es 32 - 39)
Siwilarity coordinates,
(a) although not the most successwl for
"'high frequeucy nomalisation", are probably the beat behaved for a wide
variety of runs, including the very stable ones, as indicated by the
expressions for the spectral
capability of deriving algebraLe and graplhi
density funcIton
in these coordinateso
In the high frequency range, the
B2 and (1)72-. normalisations are the moat sucessful in reding a1 the
j0C
rus to a single function; for 11ig farequenciesp,
except for runs more stable then (
a 7.
onl,
This system is very successful
at separating the frequency raung where
(G
"mrachnical rangeu from the frequency range where
only or the
) prU3
-*
ly
or the "convective raugt", Another consideration of practical importance
io that, in the case of
the U or (47')
w (AT )2 noraisations one needs to now only
to Wotuwn to dimesional quantities
Whereas in the simi-
larity coordinates, the origin lly more reliable estimate of the nondimensional quantity must be multiplied by v
;'wo
or T*2 (which depend on
and O-R); these quantities rumst either be ettated frma
and Sraphs
su my, it
or by means of forulas (see
figures
ection V E 2 and VI11 C).
In
Is difficult to say which of these methods provides better of
estimates the dimensional densities.
Th e e
-log system, (c) has the
advantage of showing the total normalized energy as the area under the
uve.
In (d) and ()
the
2)
normalisation was
sed in an effort to
find a level for the mean wind that would perit the very stable runs to
have tie sa
normalised value at high fequencie
as the more unstable
io4
=-W.
In use: uo
sCseen to be too amall a quantity and the high
Raequency nonmalizatlon was overdone for the stable cases.
However,
use of the mean wind at observation height led to a quantity in the
d
toa for the stabe case that was too large.
Since the vertical
wind profile has more hear in stable than in unstable cases, an intermediate level not far from the surface is suggeated.
Even this techvnque
is not very successful for very stable casess sepecially at 40 meters.
In these cases, the turbulance mzs be very localAed and not fully
developed
and tnd
herefore not definable in this freok.
problem may arise in using
()
An associated
or similar noralization when one tries
to generalise the conclusions to apply to other sites because the level
of best normalization may be
ite dependet.
The a of best normalization
3ay be the effective level of the most important roughness elements, such
as bahec,
tree,
buildings, etc.
Uence, the energy of the mean flow
at that level would be pysically most pertinent in evaluating the
mechaically introduced turbuent energy.
In general, the coordinates
o£ (d) were more successful than (e) in reducang blgh frequency range
varieane among the runs
In coordinates (d) the system was quite successful
(the best of the list) for
and 9v at 16 meters but were less successful
at 40 mater~. The coordinate system (e) was fair at 16 meters but rather
poor at 40 meters.
Coordinate system (f) gives the total energy as the
area under the curve.
y using T o period in seconds, rather than a
reduced" or non/dimsinai frequen y one may study the low
w frquenc7
peaks of convective energy. Tralorts
hypotheas
L less reliable for these
large scales of moion and the period in dimenonal time units shoul be
__
/
(jf
=re velevant to the auer~y proceeses of this region, which are m anly
ynam a1l.
heWdaz~l
absolute untre.
Also, these coordinates give an ener y comparison in
For deriving fnctional relations outide the convective
rogion, these coodinates are probably ineLrior to the othor systems.
5. Lower bound of isotropy in the surface layer.
Knowledje of the eaent of this range is essential in applying
There are three ways this
isotropic theory to atmohenrl motion.
range can be identifled, using Round 1111 data.
They are:
(a) A comparison of the spectral density of energy of the three
velocity componento with that
pmected under iotropic conditiotns,
In
Chapter III of his book, Batchelor (1953) showed that, as a consequence of
defining an isotropic 2-point tensor and Itpsing the conditions of (1)
incoupressibility and (2) '2p
equation is obtained: g
-
%.Z
, the following
£ + 1/2 rO, where f = longi~tudial velocity
correlation coefficient, g
traLnverse velocity correlation coefficient,
r a distance between points used in observing the correlation coeff icients;
f
denotes the derivative of the function, f, with respect to i~ argEnt,
e
r. Then the cosine transform of S = £ + 1/2 rf 0 is
transifErm,
(n) = 4
Thea F(n4,,
The socond ter
~Pc
2
os qarr
4Ra
jss
taken, where the cosine
ZA A.
nk4
& QV ea S
r-L
dA.
on the right equals zero because of the restrictions on
Mr) "!n order to make the e
of the Fourier transform valid.
oO
L ALF(n)
a
F(n
S
+ E
nnA A.
o.
,;.
No
In the isotropic region, the olngoroff bypothesisa
and the
raichnan hypothesis gives F(n)4,0
Then for the
And for the
.40
(n
.2rj
lves F(n)o,
(o
)
olmogoroff bypothaes:
ratchnan model:
., ,. _-
+-.-
.
L ,,
"
So as a consequance of the isotropic condition, L
/7
=tIt 2
the spectral energr density of the longitudinal comp:onent i
3/4 or 4/5
of the spectral enerv density of the transverse component,
In order to
exwa
-
e this property, fourtuen of the twenty nina were selected
SIt
runs wreceliinatoed rom consideration because of suspected hi& frequency
aliasi g at the highest frequecy data points,
The ratio of the energy
density of the v and w components relative to that of the u cmponnt is
cmuted and plotted as a function of as/.
They are
and
oI
ot. levels for six of the fourteen runs are shown in
F~~ure 40.
Values of
and R. of 1.33 or 1.20 would be considered a
verification of the theoratical results. At the very highest frequencaes,
R and R consistently average above 1.0 and certainly quite close to the
desired values of 1.2 to 1.33.
The vcompocent energy to so strongly
167
conatollc
by the heiht 4bove
inreaing eddy siQae
xoud that % drops very quicly with
At the hiU8het frequencies, these data seem to be
~o Wement withal the theoretical reults.
Close Lnupe.tion of Vigrem 40
ahwms that, once the turbulence startB to became antiotropic, it does so
very quickly.
Table 10 (1 of 3 aheets) gives these isotropic ratios to
be .ousedn -locating the lowesttn/
Is nearly isotropic.
the nas/
loes
value where the atmospheric turbulence
The following rather ar trary criterion to used:
value where either transverse
onn
than 80 pew cent of the longtdina valuo
frequencies.
Thee values of ns/
r value
~g
falls to
in golug vro high to low
for each run at each level ware calcu-
lated and are litated in Table 11 under the colum: i(
(colum
At VO
Od
2).
(b)
Location of the polnt where the
function va. frequency (in 1og-lo
lope of the spectral density
coordinates) ceases to be -5/3 as
predicted by the iol~agoroff Lypothees.
This point was located for each
run and the resulto ar entered in Table 11, column 3 for the i-comipnent
energ
function and in colum 4 for the v-capoaent.
This point is not
easy to locate because, in mat runs, there s a gradual transition from
a slope of -5/3 at high frequency to a lesser slope at inteamediate
frequencies.
(c)
Use of theo cospectrL
pairs of fluctutation quantities (u,
and quadrature estimates of the various
sv,, w, vT and wtT).
In the presence of gradients of the mean wind and potential temperature, these fluxes would have finite values even for Lsotropic tutulence,
However, it is reasonable to expect, in examining the coherence (flux or
F/6.40: ISOTROPY RATIOS AS A FUNCTION
R
Rv=SyS/.
fv
Rv,
RwS J/Su. ; Rii
--Rv .,". /,
.
.
/.
/. I
=
.."'"*•
-- ----
...-
AZ -
/.4
S.
.
6 R, --
o
".'"'""
%R **
.,.< ,
OF FREQUENCY
,.
.,N.
- --
Rv, i.
-
--- .**.
...
.u
. .. .
.2.0
Rw
,ewfo
----
Rw,
ow
. .
...--.
" """
.. , .......
.** *..
.
1.:I
*
1.-
1.
/1.0
"
-------
..-
-
..
*.%
0
.
-4
\\
--4
i
5
-
---
...
-'t
*.2 -
;.
..
..
..
,
..
.
.
.
...
.
.
I.
---
___o__
+O
______o___
-
....
...
lk
Rw
~OS
.. o
1.
].-,"a.
. .
.
'..0
'a
-
%.
--------
','
----
!
+
' ,...-0. .
",.-
. . . _...
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.
S.._-
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PR A EWAGESLOPE, AND VAU4S FIOR -vajaDF/iWN6F(l
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.
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lve..
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S_
0
Orr,
I
'CONYEXTIVE" SYJff.ANGE
SLOP[ REGION
UNsE-RM
r
_ IRY
nJATS_7
-
6
7
. 10
1
0
1
.4;A
-----7~iR.NM
?
!_w_1.1
ft
1.67
I
-
I
- --
"42-.01 A.4
am 1c!
.IjI
:.i-.V
J/
I a4 -e
I dJ
.r
-
------
-
_________
14135-I
1
n ROPE
i
1------t------- H-----L--
--Y----~---------
coariance plus the quadrature Lem) as a function of frequency,
valtes of the fl
aus under taotropic conditions are minial and ttt nona
isotropic fluxes would be larger.
naro (Luml e [19643).
The quadrature team actually becomes
So a test for isotropic cond tions, which requires
the coherence to be not significntly different fron sOre,
rily strict, but it
hlat the
is easier to apply.
may be unnecessa-
In this case it gives good agreement
with the other eritera, tbough it has the capability of itsoducing a bias
toward high frequoncy of the lower limit of isotropy.
to locate the lower iir
t of
sotropy as the
Panofsky and Brier [19583) for an explana
significantly frcm zero In Soirg
where the coherence (see
n of this ter) does not differ
on low toward higher frequencies.
-- ~~
Panof~
y and rier (1958) also give the fo=la,
is the probability of obtaining 4 coher
e as high as
coherence. I. sero, and whre df is degrnes of freedopa
Then for 60 lag analysis, f
Then the procedure is
when the true
(see Table 12-beov)wy
.15 was arbtrarily selected because that
value could occur 10 per cent of the t ne when the true coherence i
Then the q# terlon becaomes
where p
aero.
the lower ibdt of isotropy is located at the
point, in going from low to higher frequencies, where the coherence falls
to a value less than .15 and stays below that value for h1gher irequenclies
All esi
of these coherences have been calculated and three of tihu are shown
in Tables 13, 14 and 15, below.
Tables for vTp
w and uv
aoberence
are
not
shmon since these combinations did not yield values significantly. different
from sero.
Then wa V,
and uT coherences were used to determine the naz/
values desired and are listed in columns 5s 6 and 7 of Tabl '11, below.
Fially the throe methods of locating the lower limit of isotropy
A COMPOSITE OF FOUR SMALL TABLES.
/D
o ,
.344
.
/
,
_
STRATI FICATrON -Sv
VERY STABLEa cses) -.15
MOD.STABLEf(cases) .01
NEAR NEUTRAL(icJse) .03
..
_UN3TABLE(
TABLEI 2: PERCENT PROBABILITY OF
OBSERVING A COHERENCE AS HIGH
AS' " WHEN THE TRUE VALUE 15 ZERO.
STRATIFICATION
VERY STABLE
O10DERATEL Y STABLE
--- NEAR NEUTRAL
UNSTABLE
AVERAGE
-
/.o
/.+2.
1.+/
.1-
S,,.mn. Sv,,n. St,+,,.
+Y
.o/
/.i
/..a. /.
/.I
1.31.
1.43
/.13 1.4o 1.36
1.37
1.30
Sui,
1.33
Su,.M
,o, S~ 1
-4.~
-_.
-I-
4.
*
*O-
.
7&4o,4 Z
.3+
67C
.o0
,3 ,
11...
._
TABLEI6 r AVERA6E VALUES FOR nv/U AT THE iLO....
FREQUENCY LIMIT OF THE UNIFOR1 SLOPE
REION.-
'_,
cOssr ea.C. cm.
C.ww. Z-
RU N Tjw.~ PrRioD of _
(_C.S _.
0or.
.ASO
4
16
St_3t.
.
/so
I
V lo pI+ .
.A
. .. -
1t.2 ..
.
I
+.
.
-.17
/.27
----
- 1*
67040/3oo
o. o
-
..
i
*
rs
.0/s"
o
.
*"-'"
1
- ..
di
1LLl,
.
.1/7
i
o
+_."ssu
_
.i
xT
.Leo
.-
't
....
1_C ZR7I
.
.
SI
0B
-
:
_
__
_.. "
__.. _.__L
oo-,,,+Y
/7N--.a
o.~~
o/
- o'o
..1/4
,,
.......i-Tl
..
t*
.1i
*mo umformr Sloepe
-TABLE17: AVERAGE VALUES FOR THE
UNIFORM SLOPE, fUSr BELOW THE
-LOWER LIMIT OF'/SOTROPY.
--
RVUNS
. 30
.*II
.06
.03
cases)
Svj4o.
'-AN
U
TT-
-i --
7TABLE T3COHERENCE VALUES FOR 72,W FLUCTUATIONS AS A FUNCTION
DOF
.tW
RUlN16Re
COHEREN C E
7
Jk,
.
..
.
...
.
PE T-O[Th
.2
36
.2o.
.2+
S......
-.
14.+
10.7
.0
..
2
10
.L.2
.ft
./9*
./7
.01
7_
H-_. --_.
S.3*/0
t
S
-
,
"~"o.3.1
- --- --. i
*/
./ 2
...
,o
/v
.a
0
~
.,,
a/
.17
./s.03
.07
..1
.717r
.04
.ot
.O
.s10
*03.
.. I
./0
.o
./7
. 32
./T
4.5-
.17
./7
.06
.oq
.10
*07
410-
S11,
.01
.3
.01
.08
.
-.r
- 071"
.06
*of
.03
* 03
*06
. =r
*06
.0;
* /0o.JI
.0/
SOY+
.01
.04
.
*o31
.0."
6
.2.7
.07
.04
ii2
.04
3.4
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.1
A
*1s
0I
2./,
./jz_
.10
..
a1
00/3.
.f
./
.
,.o
.21
...
0*
.3
.. 2
.
..
/
.10
6.0
. 0or
.065'
.1*1.3.
.05-
.03
.11
TABLE If: COHERENCE VALUES
OF PERIOD.
FOR h\/T FLUCTUATIONS AS A FUNCTION
WT COH ERENCE
RUN
ERIOD.
SECIS.
14+
72.
ft
36
..
,.
,o.
14.4
.. 23
,7A
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/6
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20
.,21
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.002.
.005
.03
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.19I
.06
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.17
.23
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.43
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0
40
.la.
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./7
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05
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.o47
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130
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76
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.,13
40
33
7E r6 /6
3.4
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4o
40
E7C16
4.5"
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./0
465
CCE 401.
16
6F
6.o
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.14
/7
.17sr
.a
.014
S/sr
.1/
.3
./I'
.of
./o-
. Ifs
./as-
..09
of
.0/
.ar/
.016
.of
.03
.13
.03
.OK
./O
4 16
* OS
.14,
*o
.031
./f
..23
./3
.00/
.o.1
•.0o.
.07
.as-
S.z1/ . Ihf
.43
.1
.02
./0
..0
O03+
.0of
.073
.036
.l
,"*6"
. .
N.
/7
.co3
...
TABLE IS: COHERENCE VALUES
OF PERIOD.
FOR U,T FLUCTUATI ON S AS A FUNCTION
UT
CO H EREN CE
.14+
7;.2
+8
36
. p
2+
678
47C
70
67E
67C
31
32
33
34
36
38
39
40
41
4.2
m20.6
0423
3/
. 34
./oS-
-f,
.o77
.%3
.4/
./swo
.07
* J
•*7
*/8
.
.1+-
14.-
/0.7
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*A o
-17
.46
./s
.0+
./r
.. 7
-17"
.
.17
0./3
.03
.,z
., 6"
.173
.0
.r7
.l
.07
4.5
6.0
0.O
.07-
.003
* 07.
.03
.073
*705
.03
3.4-
.04
. dis.l'es .05
.Iq
o.03
3
.IS0
.o,.
02.
.50
.7:
.46'
.43-
.ol
* 0o7
.
.33
./7
.07
S,33
./3
. &,
S.1.2.
,.o
./4
.31
. o
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* 7
./S "'
.01
.03
.03
.oco3
.307
.43
.075"
.og
.43
.S5-
./o
./2.
.A7
.03
*0/
.04-
.19
6
.ol
.o0
.o07
.11S
.47
.17
.,O.3
.44
.31
S05
*.0
O', -
.o0
.3o
.6
.ol
.037
./77
*4f
.-* +
.34
*1
.41
.OJ3
.Oor
.033
.10
* 006
._07
.197
.42
.07
*.35
,.1
.Ao
.OP
.4.
.01
i,03+
./4
.0o
.94-
. /.24'
10
. 053
./5•3. . _,II .
.0F
S07
.05
.o6
.029
.oZ
.1,
0.16
.175'
.O- S"
.-..
*ol
.03
.o0#bS
.0012..
wie combinrdd to 1ive a "seighted consensus" value.
This consensus value
Is one third depoudent on each of the three methods or: 1/3 dependent on
colunm 2; 1/6 depndendent on coluns 3 and 4 each; and 1/9 dependent on
coluan
5
6 and 7 each.
The esults are listed in coumn 9.
Consideng
the stability as listed in Table 3, these results show that the lower limit
of tsotropy to strongly dependent on the stratftcation.
na/r
1.O for the voey stable cases, .5 or .6 for the modeately stable
cases, and .2 to .3
nas/
At 16 meters,
or neutral and anstable cases.
At 40 meters, the
stability dependence is similar to that at 16 aeters and smewhat
greater
n masptude, varying between 1.5 and .3.
Cramer (1959),
aiesatley (1959)
This
s discuased by
Panofsky and Deland (1959) and othere.
6. Form of the epectra at Intemediate freq
es.
As mentioned eearlera most runs have an extensive region at
intermediate frequencaes, Iamedtately adjacent to the lower limt of
isotropy, where the slope is munifo
(or vory alowly decreassin
toward
lowerequenciesa) and les steep than the -51/3 value of Rolmogoroff theory,
which seems to be valid at higher fequenc4es.
For each runm at each level, starting at the lower limit of teotropy
as indicated in Section V 1 5, the estent and value of this uniform slope
was measured and is recorded in Table 11, colums
,
123, 14 and 15.
The
lower limit of this uniform slope is also stability dependent in about the
sa~
mannner as the lower limit of isotropy.
This is shown in Table 16,
which gives average values for this lower limit for Su and
levels for several classes of stratification.
v
at the two
In genral, this lower lit
decreasb~ with decreasing stability and is larger, for all stabilities
at
I -
kl
/17
40 mater
thai
at 16 mate~e.
The neutral case to uot sgnificantly
diffeurt from the unstable case.
For the most stable mrns, a -5/3
slope may not appear and there assome doubt that an inertial sub-range
occUre.
This s especially true for the v-comp
ent at 40 meters. Whe-
rneme calculations for wT, and uw for Runs 35 and 36 also suggst this,
because even at the hihest frequencies, the coherence values age stil
In the vicinty of .15 rather than following the usual patten of dropping
to very low values.
The value of this slop
(see rai5thnan 19581 1959).
for
u
is also very Important, heonetically,
Table 17 gives average values of this slope
eand s v at both levels and for the same
8strati itio
8groupin
used in Table 16.
The effect of stability on this slope is
s pronounced for
then for Su and also more ponmonced at 40 meters than at 16 maters.
elope of 1.4 is an acceptable value for the slope of 8u 16 (eept
stable cases),
A
very
,40 in neutral and unstable cases and 8,1l6 in unstable
cases.
7.
Porm of spectra at low frequencies.
For unstable runs, there is usually a large peak of energy at low
frequency for S and
.
On the high frequency aide of this peak there
uually occurs a region of fatily usform alope (vey steep).
fQo
The data
the extent and slope of this region to listed in Table 11 unde
columsa
headed: "conventive sub-range".
the
The average values of these slopes
are: 2.31 for u,16j 213 fto SU,40, 2.09 for 9V,16' 179 for v,40 and a
overall average
s 2.09.
The occurrence of these lareo slopea ma
be due to
the fact that the energy density at the largest scales is being augmnted
_ __
//8
more ~pidly by convective enery introduction then it is being decreased
tanafer to ldger frequ na es.
ary
by eu
ncreass with increased lnai
Because the time scale of eddies
scale, it may be that the system does not
e~tablish an energy balance beten input and Uinrtal transfer of energy
for tL es largest scales before the meteorological paanmters Ui
and
have chan9d.
A comparison of vesults with those of other iwnstigators.
8.
(a)
using
o
are no
The results of the r@eression analyis of Section V E 3,
._.
normaslaton and epressed in Iologo and seam-log coordinates,
mpared with the results obtained by Cramr (1960), Panofsky and
Deland (1959) and Cramer, Record, Tillman
There is
these analyses on the general shape of the enegy spectral
ageosnat as
denlties for the various velocity
and, also, on the successful norz~ta
(b)
nd Vaughan (1962).
onts
1mp
s a
on in the
mottaon of frequency
igh g que
Them same data, eperissed in saim arity
cam d to the result
cy ange.
oordinatesP are
shown by Takluchi (1961) in his figwesa
1Ob,
19a, b and by Moin (1962) in his Figures 4, 5 and 7 and by Tuk~oet
4a, b. Takesoh
uihis les
(werds
densitie~
(1963)
(1961) uses the non-dinsional, a/L,
L to the siablity length) to .epress stratlication and his spectral
are calculated fr m data taken at the 2-mater level.
used the Richardsw number and his me
T~dauchi (1963) used 0Q
Caalysis here.
~oin (1962)
ts are at the 1-meter level.
and the 16-moter level, which is the same as the
Again there is good agreement on the general shape of the
t
tion to a -5/3 slope at
asPctral 4dnSity functios and also an approg
ih frOequencat
nes
es even though there are considerable differences in ro99
---
-IYW
.Iwi
m
i
ii
"9
ad ob envation height.
Howvver, as might be epected, the curve shapes
the present result.
at the loest frequenaes differ fro
did not osho
a maa
for the no -dimnaonal energy spectral densities
(in log-log coordinates) at the lowest frequencie,
here.
ran e.
Takeuchi (1961)
which does sho up
The more iartous disagree-eant, hwver, Is in the hIh freaquency
In ths region, Monin (1962) and Takeuchi (1961) shbo
non-dinnsional anergy density,
,
that the
decreases with ireasing stabi-
te result that is *own n this research.
lity, which is the oppositeop
(1963) used no-dimnsional coordinates, log w
versus log fa/U, Where
at the 2 mte level.
V(2) is the man
in his Figure 4a, b.
"A ca
Takec
He also obtained, in the high frequency range, a decrease of this nondi-ensional spectval density with increased stability, in agreement with
the other two papers and again at variance with these results.
the followinS:
nation of this disagroaent
An expla-
in the first two papera
the spectral densities are normalised,
Takeuchi (1961) and Nonin (1962),
using values of the gross atatistics
g'
asesured at the 1 and 2 mter levels.
and T
(derived frm 'TO)
At these levels, very little
contribution to the coverance comes ftam the large eddies compared to what
to observed at 16m. aed 40m.
Those large eddies are very stability sewntive
- sufficiently so to reverse the spectral d4 sity functional dependence on
stabtlity, when the gross statistics are measured at the 16 and 40 meter
level,
as La done with these data.
ICg Us/0.
Since 5 Z U 'P
which is just noimali1ation by
Takeuchd (1963) uses log
thb lis
about the sae
at the 2 moter level.
3F
as log LIp4
vesus
,
When these resulta
are compared with Figure 31, they are found to be more inpeagreement.
lIMilm-1
/
(e) In cmparin
ill
Z)
the total fnctuation ftrgy of each cowmponent
as a z=tion of height (capihna 16 and 40 =ter levels) with the
i£ndi~
ns of Panofelk
and Deland (1959),
asemet wae dubous (ee
there wier
situations where the
the data of Table 7 wd pages 50-58, Advancsa
in Geaopysces, vol.6).
-1. w-comonae
t (vertical)
Lhe data used In this zivestigation show: a) in umstable conditions:
turbulent enory increasoe
with height, b) in modarately stable conditions:
littlo enesgy change with
eight
36 :
c) in very stable conditions, runs 35 and
eanery decrease with he~lght.
anofskW and Deland (1959) obtained a
slow iLcrease with height of energy for the uwrtable case but obtained a
derrease with height in
time caso
Etablecaes.
Apparemttly theikr "stable" or night
correspond to the "very stable" classification used here.
-2. v-coponent (transverse)
1ie itavestigation shov
the lateral tutbutent energy s relatively
constant with height in convtIve cases, An ag reemt ith Panofwaly and
Doland (1959) howeve,
there is an actual
ancrease in the energy with
height for stable and very stable condtions, which ti
dsagream
t ith their findi~gs.
apparently in
Agreement can be achieved only if it
la first a sumud that their "stable" classification corresponds to the
ve-zy stable Runs 35 and 36.
Also, since the energy level Is very low, it
may be more affected by long-wave uon-urbulent fluctuations (fractional
contIbhtion to the total varance) than would be the case in the higher
oanrg
levels associated with lesser stabilities.
by a 601-pouint rnnin8 mean (appr~aut
tely 720 s e.),
If the ensrg
is f£1tered
then the remining
, i
MI0
1ili
Iz!
hig
does
frequency enery reallyo
ecease with height.
-3* ua-compnent (loritudinal)
T~8s investigation showa little variation of the longitudinal total
fluctuation eaeray with height for either stratification, but agrees with
10anot8ky and Deland (1959) in the stable case (decrease with height), If
the sam
filtering is used as for the v-coponent.
This filtering should
he acceptable in the very stable cases, since spectral analyses sugest
very little turbtdeut QU=rgy at the longer tim scales.
(d)
Reco ciliation of a diversity of results for the lo wr liwat
of isotropy in tems of na/6,
A partial lst
of previous findings is: Gurvich (1960) shoma
(1) nra/i a .4 for unstable conditions, (2) ns/r a .7 for neutral conditions,
Shbotant (1963) observed n/U = .46
(3) n/IU : 1.9 for stable conditions.
at 26m
(u and w-cuponet comparison),
nZ~i4.6 maured at 1.5 meters.
Priestley (1959A),
p. 9 7 , gives
Also see p. 98, Adv. in Geophysics,
vol. 6, for several results at contrasting levels and stabilities varyti
>ri/i1.1 for low
between na/U a .4 for unstable conditions at 29 m. to u/
levels and light winds (presumably stable conditions).
the weighted consensus value for nz/i
As shown in Table 11,
increases slightly fnra 16 to 40 moters;
the increase is more marked, the greater the stability.
If one takes Into
c msideration (1) rouSghnos variability in these observational results,
(2) variability In observation he ght, though this effect Is not large,
(3) the defining criteria used, and (4) the variability of stratificationt
then no serious dLsa
t seems to be present.
The results of this
investigation are quite sUaler to those of Gurvich (1960)
but the values
.. -U,0
/22-
zothe neutral and unstable cases are about naIU
S
.3.
(e) Cramer (1960) concluded that for the bi&her freuentes which
include
tba wiform slope and Isotropic regions, for the u-cmqponent:
(1) for levels near the rouaghes
length, say I mter, the slope in loglog
coordinates is - 1.0; (2) at intezmediate levels, say 10-90 meter,
the
slope is near -4/3 and (3) above 90 maters the slope Is nearer -5/3.
Also
Oramer, Record and Tillman (1962), using Round Hill data,, computed an
average value of this
ighr frequency uniform slope, ~cludig the
E8,S
at 16 and 40 meters and obtained a value of -1.5 for this avrae
alope.
Tble 17:, If averaged with the slopes above the isotropy point,
are in agrne t with these values listed above.
olope values do=I nto groupings to asho
on the slope for both the u and v
v-cpoe
Table 17 breaks the
the effect of stability and heiht
tsa.
P. _Comparon of resulta with theorqttcal predictioMs.
1.
Baelow, Is a review of theoretical predictions about the behvio
of E(k) for the high fsequency range to include the inertial sub-range and
the uniform slope region below the lower Ulit
of i otropy.
Figure 41 gives the R(k) function inn log-og coordinate
4: (a)
(
epected
olmooroffWa theory is valid, (b) Kra2einas theory is valid,
HisenbergOs
i)
transfer theory io valid, (d) Bolgianoes'buoyancy sub-range
oaEtes, and (e) 1] [23 (31 are three observed spectra, representing stable,
nQtable and neatral mns.
Lin and Reid (1963) show that, in the inertial
it-range, the lleisenberge
tranaor theory gives the
theory.
Curve c in FiguL
am result as Roos~ oEro
e 41 [taken frm Lin and Reid (1963),
PAgsr
13
-
/23
Fl/.i/tCOMPARISON OF VARIOlS THEORETICAL SPECTRAL
FUNCTION S WITH OBSERVED OATA (t -COiP. ENERGY AT/e, .).
ALL FUNCTIONS ARBITRARILY SUPERIMIPOSED AT THE LOWER LIMIT OF
--I S6dTROPY,
-1-t-t--t"t--I
i * I
I
+
-
-
r
-I
N
iT
--
IIlI.i
I
-
o L _1
ENERGI
SPECTRAL
DENSITY
ao)-it-C--4--3
--- ~'1i--t-
1-F
1 I-
n-
I
If
1......
- -, ~
-7
' fi
,
_i
____
It
lh~p,
li-
r-E
'0ER~MLt-t--
H--r-v-1
l.O
0-
.1
---- -----
.01
4 2L?
0
-t
=uold result from the UeiSaenbergs
theory asssaig initial period eiilnarity
in addition to acme other assutiono
No pretontese is made that these
various curves are exactly placed relative to each other;
of isotropy is used as a gide
ui positioning thmn
unevaluated constants involved in the f=u
to the abscissa may be bad.
the lovet
limit
but because of the many
tots, their scaling relative
Even with these undesireable features, it is
belteved that considerable physcal insight is revealed by this
%CaJ
The CUse
I
are arbitrauily placed coincident at the lower limit of isotopy
merely a an aid in evaluating the relative functional behavior of E(k).
iaCmahnancs theory, curve b, seem to behave better below the lower limit
of isotropy than olwogoroffs theory.
slow this point, the assumption of
Be
independence of the widely separated (in k-space) Fourter CCmnente
bec
suffiatly in emror to cause a sigificant deviation from the
k-5/ 3 forecast.
sWJ
may
lowever the "direct interaction apptzalmacion" of
raieLmnanO
apraitao the actual pWysics at oach lower frequencies, although it
a~o Is based on the isotropic assumption.
On the high wave nuaber side of
the lower liit
womponeunts may well act rather
of isotropy, these Fourier
independently so that the Kolm oroff bypothese8 gives the correct spectrum,
In this research, no frequencies are measured which ate high enough to show
the validity of nleiUobergas theory in the dissipation range where B(k) re
8teamrt, Grant and ooliet (1962) have made msuremernt
i
of turbulence in
water which permit an evaluation of the theoretical functions in this range.
Curve d is for £olgianogs spectra (see Bolgiano (1962) Section IX A).
gives:
a(k)
)
k
11 5
0f/3
for k
for k
<
lower limit of isotropy
le
low
liidt of isotropy
This
--
-----
Oili
One notes the closeness of
Cwos a [I] E2] and [3] are self explanatory.
the various functi±'o to one another over a large range of the spectra
that can be evaluated from the data in this research.
suspicion that the
i
Thee is consderable
mwall differences In slopes uay not be statistixcaly
fgi~tcant and even for a data samplO of this sise1one mw not have hIgh
confidence in ones ability to distingudsh which funtion most closely fits
the data.
iHover, don to the ;tor limit of the nt£*fm slope reon,
there seems to be a best choice, which is very close to the observed spectra
and
acceptably consitent theo= ticaly.
or ReIsenberasg
This choice is: use Kolrrofs
theory in the isotropic region and eraIcnane
theory in the
uniform slope region so long as one remains above the regions of :tchanical
n not too stable
input of enery". Awy £f~a energy source reions an id
stratificatio,
t he spectrum in the uniform slope regi
is very
lose to
ralchnanes forecast of k "3/2 (see Table 17).
Bol~aLno (1962) and Deissler (1962) have both conmsdered the first
orde
effects of buoyancy oa turbulence (neglecting the triple correlation
terM).
The w-componet
issnergy
most directly affected by buoyancy
and also by the distance to the loer boundary.
Batchelor (1953) pointed
out that the effect of "pressure forces" is to tend to distr~t
te energy
equally amon the copoxen- , hence provide a source or sink for the total
energy of the componentss E(k), through the -component, see Deissaler (19621
FiVre 26 (using Prandtl number * *7).
somethat
Although the buoyancy effect tos
ronger at lvw frequenes, a vezy wide range of frequencies are
affected by this energ~
drain (or addition).
In the case of ener
additijo,
this ia not to be confused with the input associated with convective cell
..........
Y
IYIIIii
1i
iU i
n
E
IIIllI
i lu
ilr
evelovment.
Then, for the isotropic and unioArm slope f~Preency regions,
the previousely discussed KRomogroff-taichman combination spectrui seems
to be very close to the observed fwEctioa&l dependence of z(n) on frequency,
u
s and 8 as apptmtions to the total enag in the intermediate
frequency range.
2.
Low ftequency region below the unifoEm slope region (ase Table 17).
In this region, the neryr spectral densty f
dependnt on strat
£cation as the frequency decrease s.
the lowe boundary also becoma Increasingly1
its influeuce on the 8
lbpotheess,
z=
Ufit,
spectra.
potan,
nastios ae inareasingly
the presence of
particularly throu~
This is also the region where Taylors
is questionable.
There is yet another complication:
scale of an Gddy increases as its linear scale increases,
since the tie
there is the possibility that the energy level of these largest convective
eddies is not in equilibrium with the meteorological parameters (i, S'
thls
ould infer varying values of enerr
piarmeerso
);
even for the same m:teorological
Even with these difficulties, it seems profitable to eamine
a list of theoretical predictLons about convection and see if these data
support these predictions.
(a) the primar
effect of instability on cell saze of maxctmu
wculd be through the expansion of the depth of the O
rable to cell sze)
energy
nday layer" (coa&.
n which convection is occurring (see Rayle
bh [19163),
This is not the same boundary layr used by Priestley (1959) or monin and
Obukhov (1954).
ThOei
definition refers to that layer (usually less than
50 meters deep) in which there are no heat flux or mountum flux divergencu.
/z7
This is not so for the layer referred to by Rayleigh.
The 'boundary
1ayo'". has also been thought of as the layer of uanifom ~dings;
this
for steble cases oTr a deep as the d.stance to the bases
ay be very ahall<o
of cutaus clouds in unstable caes,
ThaVoon (1963) and Herring (1963)
are referring to the shallow layer, in convective cases, which contains
On can only inft r that, with greater
most of the temperature gradient.
instability, the
bo'U ndary layer
would be deeper and hence the mwa
m
eeway would shift to longer waves, whose characteritic life times would
be longer.
then, neSlecting wind shear (asawuing ftee convection), one
vould look for a direct relation botmen
and the period of masxam
nergy spectral density (see Figure 43) giving log Tma x ene:
and log T.
Ian
vensus
, whre T
muLpoint of the nx(n) tunction).
is the period asso
versus
ated with the
The data sample ts wall (restricted
to free convective cases) and Figure 43 does not seem to define a relation.
See p. 41-42, Cramer
Record and Tillman (1962), where a relation in the
e-pected sense is indicated.
More research is needed on ths point.
Also
the assumption that a depth of the cow eation can be lnferred from
c a
imy be a bad one.
Certainly
-
would be a safer parameter.
related to the scale of turbulence but 0o
Since it is
could change much more rapidly
~han adjustments in the bcundary layer depth could be mde.
(b)
Priestley (1959) dealt with convective models and made prita~cion
about the heat flux (see also Section III C for a brief suamary ox Pnes4t~s
predictions and see Table 3 for the H?* calculations).
in Figure 42 giving
v8, (rBi but the range of fi
These are plotted
Isto inauficient to
establish any good functioal relation and the results are inconclusive
L1~_~__
~L~_
~l~l~~
__
Z8,
0,
I1
I. I i . l , ' _ ; , , '
I I i I ,.
-- ----- -....
I
-
-
-
Ii "
-" -----+ -i.
I.~.
i,. i':
I---J'''
1i,, I
"""
,.......,.,
:Lt444
, , . I;.l i
-4---
---
iL. 1-_-+-L
---L~-
l ,' l • i
-'~
4
1 1!
,
.
I
I
l
- -.-
i I
I
K4lL
-
I4 2-2
FIG. +.2: NON-DIMENSIONAL HEAT FLUX, H, AT IAND
4o01. AS A FUNCTION OF RICHARDSON NUMBER.
10
-
T 4-4---
--
1.0
.1
* 0,I
-a .
- J'oi
.I
1.0
'+-
.01
,
-
_~^~__
L_
_~~~_~~~~,~~__
_.l-;~u~-----~
----I --)---t --l - 'r^'--l
/iz
FI6.43tPERIOO",TNM, AT I1fM. AND 40M1. ASSOCIATED WITH THE MAXIMU
VALUE OF 7L&9 ) ASA FUNCTION OF & .. V,'A
6. ae/at 4-sat.A
Towt
(SEC.)
P
/00
-
T400
(SEC.)
3oo
. OZ.
~--~-ll-~-^LIIr
l--
(there is alwayr
the difficulty of eveauating
RI
when it to small).
Priestley also suS~j ted that for RI (~ -. 03, the heat flux should be
proportional to z/3.
Table 18 gives resulta for 40 meters in fair
agreesien with this (assumin the 7T16 values a
calculations Wera made as
as
that are at least as good.
Fo
correct).
Also
3 law for the heat Elx with results
his aodel of
astrmetric plumes, Priestley
also made the prediction that the heat flux becomes positive (upward) before
the stratification reac es decreases to neutral conditiois.
Pigure 20 appears to support this prediction.
The data in
However Prof. James Austin
suggested ttht the oe hour length of record, used for obtaining an average
gross statistic for wT
can introduce a bias in the following manner.
If
the averageavstratification for a one hour r=n appears to be neutral but
really contained a co
ination of stable and unstable cwditions, then the
total heat flux would be positive, since the absolute value of the beat
flux under unstable condtioUs is larger than fo the sme amount of stable
stratification (as shaow
n FiUre 20).
All the near neutral runs were
checked with the following results: the two runs (67 C and 111), which were
near neutral for the entire period, did give W'T ^ O. Que run, (15), which
was unstable at: the beginnin
heat flux.
and slightly stable at the end, gave a positive
Several runs having
aO
0, had a positive heat flux, but all
of these had unstable stratification nearer the lower boundary.
From an
inventory of all the runs, no run could be found that was really slightly
stable or neutral throughout
that had a positive heat fl x, so Prof. Austinot
less complicated explanation is preferable.
Despite a lack of good evidence
to support it here, there is still reason to believe that Priestleyes axt-
_I
_
13/
ayit~ric plume model nay be ae;Cul1 one.
The noiar
ni onal heat fl
,
Q , from his similarity aosuptionjs: may not have worked out in thse
calculations because of the troublesme Ri calculations.
(c)
See Sections II
C for a brief discussion of Thd
on (1963)
and also a discussion of Herring (1963) in Section IX A.
These two papaer have in camon similar Seometry, the use of nondivergence, the steady state and the Bouasineaq a-MPationi
heir othe
assumptions are quite different but the results are in good agreement.
Only
the results to which the data is pertinent, are listed belows
l.
wna
o , except negative near the boundary.
A boundary layer
shown by Herring and was assusmd by Thempson.
-2. The temperature variance reaches a 8
at the top of this
layer (Thompson).
The vertical velocity (and supposedly the vertical velocity
-3.
var~t=e," where
2)
reaches a maxc1im
(Thmpson).
deep in the interior of the fluid region
See Figure 44 for the best that this data
sample can do in evaluating the above theoretical re ults.
at a w,
=
With values only
16 and 40 meters for the w- and T-variance, considrable latitude
is available in drawing the vertical profile.
Even so, the siplest analysia
for the w-varance would indicate the ma~uim is far rem.ved from the loer
boundary.
fo ward.
Drawing the profile of temperature variance Is not so straightThe temperature profiles indicate the top of the boundary layer
to be near 8 moters and several of the runs can be drawn to "force" a
ma-imum of
-variance at this level, but it looks like better analysis if
a maxi~um is indicated near the 30 meter level.
I
general
this is
considerable support for t~btheoretical predictions of Thcmpson and HeString
--
--
71[7
::
i
I
-~-
i.
s
-i-
FIG.44: VERTICAL PROFILES OF TIMPERATURE, VARIANCE OF TE1P
::VELOCITY FOR 8RUNS
BELIEVED TOBE "FREE CONVECTIVEN'.
I-
~ TT
r
'-~rr
AND VERTICAL
--'~~'
~~~
II
133
inforation can be obtained from Judicaous modelling
Indlca.ting that weuseu
and also that near the lower boudwar,,lthe tran&fers of vatousw propertAt
e
may be statistically "statonary".
(d)
Lottau (1949) made predictions about the bet flux as a
(see Figore 3 and also the discussio
function of
cton III C)
in
If the adjustment to PVIgr= 20, suggested in Section V I 2b above, was
made (putting -
' a 0 at
theory anddta.
Lttas
<
IBntro
= 0), then there is good
zagreamit beTveen
ction of the third parameter
seem
to give a model quite adaptable to the data.
(e)
A sunary of eh ttieory-data coprte
for the low frequency
esti s the following conclusions:
range G
-1.
There is soae evidenc
to indicate that the period for the
meltm of nF(n) can be roughly estemated from the stability (if
used) - the
-2.
Q( is
g
ratter the instability, the longer the period of the enerj
The use of the model of convection between parallel plates,
with the assumptions indicated in Section (c) above, give good ag~emtn
with th
statisticsa,
though this data sample Ls not well suited for testing
in the low frequency range.
-3.
It is believed that Priestle
type Cnvection is8
a model of saad.ym'tric plum.
good one near the lower boundary, despite the lack
of statistical support here.
-4.
Lettau s derived heat flux function is in good arement wth
data and his use of three parameteras
u , and
, may be adequate
to describe much of the physics of turbulence in a stratifted atmosphere
111
(34
1ear the surface.
G.
Acopsmitenodal
eneroi nectral densit
fctions
in
the Burg-ace laaer.
The next step is to coxbie hethe preceding conclusons in a model
which gives good forecasts for the ener
which is plXs8cally raosonsble.
Tis was begun in Section Va, with the
definitions of the roug~~
ese parameter,
A(a,
spectral density fwnctions and
R(n),and the convective excitation
/,j ) as expesaod in Equation 5-6. Thn R(n) plu A(n, 3/
)
autegrated over all n would give the ratio of turbulent to man flow kinetic
energy, assuming the introduced energy is proportional to
z, which is the affective hbit
In enery 'oduction"
of the most
(crudely approximated by
that the w-fluctuations play a dominant role.
2
at soe level
portant roumhness elaements
'
), it can be seen
Then the problem can be
looked at from the viewpoint of the results of the kae
isnical and convective
excitation of the W-c~mpoaent.
1) = )
)
So the energy introduction at frequency n, is
+
where R(n) becomes a fnction of
Nn
)
only for very stable conditions, and
is very mall under stable conditions.
In the evaluation of
equatiois 5.4 and 5.5, a equals + 5/3 in the isotropic range and equals + 3/2
below the lower litit of isotropy.
Then Figure 45b. shows the resulting
onergy spectral density function "produced" by a simpl£ifed ecitation pattern
Figure 45a.
Comparlson with individual runs, indicates that the general slopl
shown in Figure 45b is a reasonable one.
/3-5
FG7.4 4 DIAGRAM MATI CAL. EXPLANATION or TH F MO0IFIED2
BUS INGER "
--t-tht
ft
I
I
M'ODEL.
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to
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III [
!34
WS
Si5ARY AN CONiCLUSIOQS
A. The non-d
1O
sional ener
spectral density functiona for
u, v, v and T have bean presented In several coordinate systems with
stratification as a paamster.
angle,
The standard deviation of the azlnuth
was chosen as the parameter for stabtiity which had the
p,
mort deairable characterstics.
Associated with this parameter choice,
then, is the choice of (fN(standad deviation of azmstah angle for neutral
conditions) as the desixed way of describiug lowe boundary wougbneso.
),
epreasin, (
An
value) is.
for stratification (or its equivalent
suggested as a way to incorporate the better foatures of the f .Ictional
behavior of %A at nutral and unstable conditions and of
unl
ader stable
conditiLons.
B.
The lower limit of tootropy is deternined, as a function of
stability and height above ground (see Table 11).
This limit was taken
as a consensus average of t~ limits given by : 1.
the lower limit of
the
functmtions for 8 and Sys 2.
the value of rs/O where the anergy
cC the components is no longer within 80 per cent of each other and 3.
the
value of no/iI where the urs w and uT coherence falls below .15 (values lower
than this could occur 10 per cent of the time when the tre
coherence is
zerO).
C. From the plots of non-dimensionalsed energy spectral density,
three adjacent reLons of the spectrum appear:
1.
Isotropic or hih frequency region where 1(n) oC 71-
wh e the
- and (
cQases (((
7).
and
T)2-normalization is tffective e=pt in very stable
The low requency
4limit1
) LY
of this regio2n at the
/37 -
16-meter level varies between 1.0 for very stable runs to .2 for neutral
and unstable runS.
At 40 meters the lower lbats vary between 1,5 and .3
(see Table 11).
QAn
it
2.
2 - and (
ate frequency region of unform slope where
T) 2 - normalisations are still fairly good
stable cases.
eept in very
In this region, spectral density dependence on stability
appears but is not serious. 6n) oC --C
and is saonmhat stability and heiht aabove
where c I between 1.0 and 15
roud dependent (see Tables 16
Toward lower frequenciees the effct of stability is iLcreaing
and 17).
and the slope value is decreasing slowly, unless the run to very unstable.
See Table 16, ~aboing the range of the loer limit of the u ifo=m slope
reSion, varying between
I&
3
for very stable 40-mater cases to
.03 for neutral and unstable cases at 16 meters.
3.
The low requency covective range, whe
the effect of stabi-
lity on the enery spectral density functions is dominant and increasingly
strong in going toward lower frequencies.
region,
No noralisations work n this
A large peak of very low frequency energy, somawbat separated
from the higher frequencies, appears in the convective cases, the lateral
(v) component is most graphically senaitive to stability in thid range,
because the bkgitudinal (u) component containa more mechanically introduced
energy and the w-componen energy is greatly modified by the height aboe
ground.
Thi ra ion includes the region to the low frequey side of the
lower linit of the uniform slope region.
D. Using both 62 or ( T) 2 - normalisation and Nonin-Obukhov
similarity coordinates, the energy
spectral density functions are pResented
(both graph
and algebraic
ncti±cs) as a function of (r , a modified
stability parameter, and are shon in Figures
a
- 15, 24 and 25 and
ti Section IX B. Due to the behavior of the diabatic wind profiloe
"2
U- - noralization as tried with z = 2 mters with the idea that this
level would be near the 'effective
ougnese
heigt" of the most itportant
If the wind speed at this level is a phyaIcally si ificant
element.
quantity, then
normalization might help in the very stable cases.
Actually it wa only partially successful (see figure 31) and eitht
the
two meter level is still too hig or this s mu h too cnude a way to handle
theaeffects of stratification.
E. Caoparisoas of these ezpirical results
ith those of other
This data
investigators, do not show any unreconcileable differences.
9auple, however, due to its size and diverasty of stability conditions,
pomits more to be said about the stability dependence of: 1.
the energy
pectral density funtions, 2. their functional dependence on z, 3. the
location of the lower im t of iootropy and, 4. the unifor
.
slope region.
A cowparisOn of the empirAcal results with those predicted
by theory, gives some very useful information:
-1.
Above the lower limit of isotropy, the Kolmogoroft or Heisenberg
predictions of E() o
2.
is in excellent ageem~et with the data.
In the uniform slope region just below the loer limit of
isotropy, the Eaichnan theory of E(n) o
J135
slope average of between -1.3 and -1.4, but the
was closer to the observed
the
sotropic assuumption
is no longer valid and this region is much closer to the enery
sources and
energy containing eddies.
3.
See Figure 41 for a comparison of various theoretical and
/3q
oebsarved enegy spctral datIty unction.
OvIer much of the obaervable
£equency range hres, these functions are so close together, that it is
Nard to say that the differce between them is statistically significanto
The ccabina
1 tio
the bst
of the
rat
an and Ioangoroff theoies gives
aeemento
4.
In the law requancy couvective reion, very may difficulties
arsde, and despite obseratioal care and sta stical eignlicance of the
data, £ew Inferences can be safely made.
Some of these troubles are:
(a) these meauwements are Eulertin and Taylorqea hpotheas
nyq be only partially valid in this region,
(z = Ut)
Therefore conclusions
involving spatial distributions and length scales are not safe.
(b) horiontal
omoganity cannot be assumed in covetive sia, tions.
(c) the characteristic time scale of a turbulence cell increases
with its linear scale, therefore, these large eddies may not reach equilbri
for -a -particular environment before synoptic and diurnal clhages
(shown by
and
) of the parameters have occurred.
Despite the
difficulties an attempt was made to compare these low frequency results to
theoaetical predictias applicabla to the convective case.
convection there is little low frequency energy.
Wlthout
In most cases, this
camparson yielded nothing statistically dependable.
tWWer, there
cre three results which seem worth stating,
-1.
there was same evidence to suggest that the maxi~ma
of ener
was associated with lower frequenaces, the greater the instability.
-2.
the use of vertical profiles, although they contain only
3 points, of tewperature variance and w-variance and temperature
ive
-- --
---
IYi
/4z
agec%;am t with the epactat3ions of EHarring and Thempson.
They infer
a -maximum far removed from the lower boundary and a tweperatu e variance
nmmau
near the top of the 'boundaqy* layer"
-3.
use of Lotaues three parametrs fhr tuIrblence In the
stratified atmosphere, gi ves good results for heat flux predictions
G.
A model Ls 8ugested uoing the conclusions above and the model
introduced by BusiBng
(1961).
This model contains a description of the
roughness R(n) -uch that the effect of site variability can be accounted
for (see Sections VC. and VG.).
This model relates roughness, R(n) and
), energy introduction
A(7,
total eneray, B;
); : total energy at a frequency, B(n);
"Irand Q in what is believed to be a consatent manner.
;Section VC. and VGO give more details.
VXI.
SL'GGSTBD FWUIRME
A.
ESSARCH
2he model of Section VG. suggests the necessity of investigating
the affect of site variability by comparing energy sp9etral density data
from sites of contrasting rouhness.
This would ebow if the
enerali.d
Businger model is really a good one or not.
B.
Collect and evaluate data especially designed to measure the
csergy and the vertical fluxes of heat and romentum in the very low frequeacy
c~ivective range.
This would have to be done by determl Ing the velocity
correlation functions from a spatial net of masIwng points and siultaneoua measurents.
This experiment would also have to be designed to eva-
late the seriousness of the lack of hori~ontal homogeneity during convection.
Fran this type of data, one might also hope to answer the question of how
--
1100111,
141
1mo
requency It is poasble to measiwe, before
ho "balance
C.
o
aving to worry about
the t3rbuleat syteam with ito enviwramat.
If this very low frequancy convective range behavior were
wall understood near the lower boudcary, thn the next steps would be
to investi~ate how this regaie "matches" into the atmospheric circulations
at the gradient wind level and their relations i
scales.
Equally riportant is the connectio,
any, to convective cloud
it any, of thse large con-
vective eddies with the still larger terrain induced local circulations.
1~
_I
_111 ~
_~
I fa
The autihor wishes to express tcks
merm rss
Prof. Ei..
to the folloun g Faculty
Lores, Prof. C.C. Lira and Prof. J.M. Austinn
and to Dr. H.E. Cramer
Dr. F.A. Record and their Round Hill Field
Station Staff for assistance during tas research.
The author to espe-
cally grateful to Prof. Z.N. Loren and Dr. H.E. Cramer for advice and
guidance of his efforts.
This area of research was chosen and comimnced
at the suggestion of Prof. J.A. Austin.
II_1I__
~_L
Ir
t43
A, RESUMES OF PAPERS (continUatiol of SOction IIe)t
Da tsler (1961),
treated tb1e problem of tle effect of vertical
shear of the horisontal m2an wind in the absence of booyancy.
The approach
was the treating of a single teffects the writing of two-point correlation
fu~ctions and the closing of the set. by -neglecting the triple correlation
terms.
In this ae e the
hmogeneity was
zrposed.
ind shear was asamd uaniform and horizontal
One of is results was:
with increasing shear.
the mvxi u of energy density shf to toward lower frequencies and
8ilarger
in wSaitude.
Solgiano (1959).(1962) also dealt with tiew problem of anisotropic
odel of steady state turbulence in a stably stratified
turb lence, using a
atmosphere, cnmposed of eaisyomtric cellsa.
In the perturbation equations
of motion, only first order tevas are retained and the following energy
equLation is obtained'
p
pressure
static density; (
potential densety.
this equation finally yields his Figur
vRsult,
Operations on
2 (see Figure 41) as the impportant
This steepening of the slope in log E(k) vs log k plot for the
stable atoospher,
just on the low freqency side of the inertial sub--range
was not obsred in this research but then the test certainly is not ideal
b~~caus
of the eS£ffct of the
oi er boundary, particularly on the w-compoent
_II
144
28ao
supporting evdence for hi' theor.tical spectrMi
(1963) did ao
using high elavation data.
Ijk=ley (1964) obtained a theoratical roaflt similar to this, but
with less constraint on the "ne tial transfer" of eneray.
Priestley (1959) dealt with convective regitsm
extensively.
In
Cliapter 4, he differentiated between free and forced convective regimes
and pointsout that although
IZ4P
o
its effect on the turbuluece i nsesa
I<
decreases witb.height,
With observational evidence to
sMpport it, he proposed a model of convection in the surface layer of
aztsyna
tric plumes of penetrative convection.
He used this model to
1e~lain the occuence of an upward heat flux tu the neutral and slightly
stable atmopheree. By using eiilarity,
a assumed asusalan distribution
of velocity profiles cross these cells and dimnsional argmetta, he
obtained a number of interesting predictions,
to cbservations.
1.
A
t
list of these conclustons ae:
for forced convection (RAi
(a) Ipsa rate, d4
R
(b)
me of which are compared
.03)
o((
_
H*
(defition) * p
It C heat Vux k = Von
RI
2.
o
or in the
for free ccvection (RI
(b) I = constant
(C)O OIL;Cr
erman CIantant
(
ux form,
.03)
I_
He o
9
ed
less.
o
it
t haher levelsa
oert"
l
an eMpnattIon of the i = f(R±): "at consoiable wind speeds,
a
achanical y geerated tubaulence
a
_
At asm point, free convection masst tae
convection is the axtymetric plutame
of Bthi
The fo-r
s sufficient to perform the heat traanfer,
Also s~e
weauSger (1955), who dealt with a means of daetezin ng when free or forced
convection will occur,.
Uee 18.5 and 18.6 of Hass (1959), for a presentation of the hRandtl
miing length theowry
the amixLn length
~und profile by letting
which leads to the lgtitLe
I a ka and deftinin
the friction velocity, U
Taylor (1960) ceatned the Mouitbukhov aimilarity ecrltmentally
and suggested a value for "oC" in the evaluation of the universal function:
#(%) I
-
... In the free convect:ve case, he concluded that
y salaty and by friestley (1959) are acceptably
the functions given by
accurate and h
then evaluated the constants.
Hacredy
from 10 min.
aive
(1953) used data talan from
runs.
varying in length
to I hou and under diversifted meteorological conditions, and
showed that at sufficiently high frequenioes, the Kolmogotrff bhpothesis
viva the correct function: R(h) o C
K
or
(t)
I-Cf
A.
for the spatial longitudinal correlation (r = it Is asanead).
In Lettau
and Davidson (1957), Lattau gave an extensave boundary
layer theory, stressing noadimanaional characteriticmrtelatons between wind shear, stratification, and
.
LISTM 0COOM
F
M
RATUU POUM SPECtA IN SIN$
SIONS F
AITY CO
to desrtbe the
tomnt
and heat exchanges
AND TZMI
M
AUTINAS.
_____
s0
ara Figura
Section V3 1. for a general descripdtio
.01, 1 equals 1.0 and at na .02 it
a
.01 fand n m.02, F
and between n
This mens that at
.01 and .02'.
F is introduced betteen n
1ch as :
has the full value shown for F
1iCases8
or decreaCs tnearly along the
log scale farma 1.0 to the high ftrequency value shown for P.
are:
S3zo16
4
14
/.
)
The e preasions
*nS. o
.0
It
.-
Sa/b -
lo
. 3x
3Vo
, -13
1510 -a
lx
S,brodc/cd
'P
W = .0/
X6el4ee)
/. 4C
>.oI 3L
7L& oI
=
F
.3
/
t
*
+
F =14 30o 194,
-/-do
L,4
pII
> *O
~ ~
.3nx
.
00
2
"0/
These
there will be a note
In each ce
form
16 and 17 In algebrai
of those function.
I10 n =
100
/iniodued -. e~-eae
ad,
4npai-r/ eonchbONS
Fop dah/e Oxj ne,2
'd
IL
>9 .0/
2
.
1/
Otke<.
~.
;
nea~r ,qeu
Fotr sta/ke dn
/#~
,
~A-~2 cLdiO I~
-
ft
=
whoda~ce
-
41h
_______=
'Ai
5)Ii /6
cedL Idweell
-
e
)
dwe7O
fleu /-Y' / ecrnd 's :
1,.41focueed
7i~pa
>). Oj
-
/+
4
(/h)2F
AD FAv
)7
N) t
77zf.f
0/0
*15m-o
I
4, So x/40
7we0/c flc(
7
lf/O
ror uns-/-c?,6/4e
&e.
ewid;V43) is :
4o 15,fa6le
exeepll -9.5c
7:;r Siable
Sre 7p
1 6
>0.010
-A
0
03
-4
?)
jliPodc~~e
4 x~
A/ e e)
77". 1O2.
1
frr ewor'.l
no Ye'"i
in
51d
IO
F~/+ 3oo,
T-Or 54,lb/e
Le/sne~7
,,drodh, ed At/
&ref,40 -map
-For ums-hk/e eao1dl'is:
t5 e)Q-)P,
F='S L±OT d I D&4
ICO Z
.90
iz72-4
~ta
Sr~,ne) 'fo
/-S
" =.Oz
He sable- f.p V
dhei £1e1Ze-ii ,
> /0*
)
.o
N
lN
1/4q
C* VARIOUS )WMO1DS OF ESTINATING SRS AND HUAT FLUX
It Lo ntessary to detemie Uw'* and v7T* in order to got
dependable diansatonal enerMy spectral deasity values, from smn larty
coordinatese
Here are some recnt ametods:
Pasquill (1963) gave three methods for evaluatLng the eddy dif usivi ty "Ke,
through which one could compute the stress by using
Z'
K
sa methods are:
1.
The hbtgh frequency
method uses:
Measrm
S~(k) ad eand E.
2.
ttal
8(k) and compaute
.
energy" othod.
dl-t3.
Low frequency mathod.
"
Men square displacement of paratcles,
distribution) T
1ZE
timae, a large value
(asaund Gas an
then solve for
X, kneoang (Jv as before.
PanofB y (1963).
From Monint-dhov slmilaaity, the following
quantities are used:
o-dina
m constant.
LO z L X/q
, uwhere k a Von
onal shear,,)
He has not
o used
(MM
aaan u)
8 (heat) and thus defU.ned
and modified the universal function to be:
0
ek 1~~L
- -~--L~--L"
IL
(5o
A number of epressirons for
can solve for us
Sme
S'.
can be used and then knowing so, one
of thos. functions
1, Monin-Obukhov "log plus linear
2.
are:
profile derived freM:
Kyps function:
-_f-,( )/ )
/
ofuct on was used and the stress
, (this
calculations are showu in Table S)
3.
Webb function:
J 4.-
03/6(- %4'Y'1
?or - 4
Solutions for,V,
/
-fr
0001#3(- V/,'"~ 1
1/ - . o 37
is not tknn, are also discussed.
8o
h
Taiu.hi and Yokyama (1963) presented a more
the s
approach.
used I
kzf (z/L).
o317
/) 4
In place of 1 = hk
eneralized form of
(valid at neural stability), they
The result is
who0
,where
a= constant.
4ear neutral f 'uStable*,
At neutral f
for
1s 1
f
a,
L1Z
/,
s
z
(top
Living the Keyps function,
The more Seneralised
should be better under stable conditions than the Kyps fnction, but
it wa not used because of the presence of two constants, 7
and
,
hich
mnmt be determined from the data.
Taylor (1961) suggested evaluating the "structure fumction" in the
Inertial sub-mange:
)
"a
(XI
( 4 X- ,) 2
e .2/3
.4
_1
__
/5
He then assumed <?
CO
""l-leGle
V
to
=
negelected the vartlow1 divea
.next
sot*
hawaaalty end wvote A COUerVa~to
H~~z"'~Tfix--
n eiaton fog Inte
eI~d
whraH =rheat fl=.
he
nasumed
la
,2
Net, heohowed that the *Oint teu
ce Of
horlsonti?
r& to get:
lrT.
i
and used tbhse valtvueis to solve for V # Also he gmav a second more
Mp1itcal nethod:
a =
b
A7'2)
b
H
,
..
~P
S3IOIGRAMR
arad, MoL., 1958: Project PrafS
e Grass, a field
Iqam
In diffus o
nowm
s~aadaa
no.9, Vle.
V
I W
ANP
-7=Geophy-slcal £ear ch Ditectorate, Air To=e CwibrLdg Resarch
Center, Blford, Nes.
Batchelor, G.L., 1953: The Theory of Hlaoganeos Totblenculen.
University Press, London.
..
C
brid
., 1953a: The conditions for dynmic savilarity of motions
_..
of a frLctaonlss perfect gas atmosphere. ,rart,.,J.Msteor.So_ .
_ -..
B1lackba
R.B. and Takey, V.W*, 1958:
The Meauremnt of Evr Sectre.
Dover Publicatione, Inc., New York.
Blakad,
A.L and P'aofaky, Ii.A.,* 1964:
Turbulence.
Bolgiano, R. Jr., 1959:
,.Geophya.Req.
Recent Soviet Reseazrch in Atmospheric
Rull.AmmKteor.Soc., 45, 2, 80.
3
Tnbulent Spectr. in a stably stratified Atmosphere.
64, 2226-2229.
Dolgiano, RJr, 1962: Structwe of turbatlece in stratified madia.
RA1.2 67, 8, 3015-3023.
BIsi~enr J.A., 1955: On the structure of the atmosphereic sur
S12, 6, 553-561.
c
a
layer.
W
1961: On the relation betwen the apectum of turbulence and
the diabatc wind profile.
GJ
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iorn October 10, 1923 at Chester, Pa. aud was raised near
Taylorea Bridge, Delaark , attending elemntary school there.
Craduated frm John Bassett Moore ridh School, 8yr,
Delaar re in
1941; was a umba of the Art Air Forces £tem May 1943 until June
d tochE~brctan.
an ra Moadvo
1945, corving as a mathe r obsere
Graduated frot Weat Point June 1949 and presently in the U.S. Mr
Fore.
Have eveed orsa to
iIn Spat, Alaska wan also dth
the Fletcher Msland (T - 3) p eoct in the Artic Ocean. Ree.vaed
a. I.S* n Xl teorolofy
ne York UnIverity in 1954; the thesis
waa
Arctic %tetorolosy
(uapublisala4d)e