NASA C it - 2 4 38
INVESTIGATION OF THE TURBULENT
WIND FIELD BELOW 500 FEET ALTITUDE
AT THE EASTERN TEST RANGE, FLORIDA
by Alfred K, Blackadar, Ham A. Panofsky,
THE PENNSYLVANIA STATE UNIVERSITY
University Park, Pa. 16801
for George C. Marshall Space Flight Center
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION * WASHINGTON, 0. C
TECHNICAL REPORT STANDARD TITLE PAGE
1. REPORT NO.
2. GOVERNMENT ACCESSION NO.
3. RECIPIENT'S CATALOG NO.
TITLE AND SUBTITLE
Investigation of the Turbulent Wind Field Below 500 Feet
Altitude at the Eastern Test Range, Florida
5. REPORT DATE
6. PERFORMING ORGANIZATION COOE
Alfred K. Blackadar, Hans A. Panofsky, & Franz Fiedler
8 . PERFORMING ORGANIZATION REPORT #
9. PERFORMING ORGANIZATION NAME AND ADDRESS
The Pennsylvania State University
Department of Meteorology
University Park, Pennsylvania 16801
10. WORK UNIT NO.
11. CONTRACT OR GRANT NO.
12. SPONSORING AGENCY NAME ANO ADDRESS
George C. Marshall Space Flight Center
Marshall Space Flight Center, Alabama 35812
13. TYPE OF REPORT & PERIOD COVERED
June 1971 - June 1973
14. SPONSORING AGENCY CODE
15. SUPPLEMENTARY NOTES
This work was conducted under the technical monitorship of Dr. George Fichtl,
Aerospace Environment Division, Marshall Space Flight Center, NASA, in support of the
__ OAST Aeronautical Operating Systems Study Program.
16. ABSTRACT This report contains a detailed analysis of wind profiles and turbulence at th<
150 m Cape Kennedy Meteorological Tower. It appears that the logarithmic law, in neutral
air, and diabatic deviations from this law familiar from the surface layer, apply without
systematic error up to 150 m. This analysis leads to a determination of roughness lengths
which are between 10 and 80 cm at Kennedy, depending jn wind directions. Various methods
are explored for the estimation of wind profiles, wind variances, high-frequency spectra
and coherences between various levels, given roughness length, and either low-level wind
and temperature data, or geostrophic wind and insolation. In the first case, the Monin-
Obukhov length L q is inferred from the winds and temperatures, and surface friction
velocity is determined from the wind profiles. In the second case, theoretical relation-
ships are first explored between geostrophic wind, surface Rossby number and surface
stress. The relation between planetary Richardson number, insolation and geostrophic
wind is explored empirically. The result is a nomogram for the determination of the
planetary Richardson number. Techniques were devised which resulted in surface stesses
reasonably well correlated with the surface stresses obtained from low-level data.
Variances of the velocity components were well correlated with the surface stresses, but
their vertical variations showed random behavior. No clear effect of stability on the
variances was detected. Coherence between winds at different levels followed Davenport
similarity with exponential decay as function of separation or frequency. The "s lopes' 1
for the lateral components were about twice those for the longitudinal components, in
agreement with results elsewhere. Finally, based on all this information, practical
methods are suggested for the estimation of wind profiles and wind statistics.
17. KEY WORDS
Aircraft Design and Operation
19. SECURITY CLASSIF. (of thU r. peril
20. SECURITY CLASSIF. (of thi. p.g.)
21. NO. OF PAGES
MSFC - Form 2292 (Rev December 1 f 7 1 )
For sale by National Technical Information Service* Springfield* Virginia 22151
TABLE OF CONTENTS
I. INTRODUCTION 1
II. DETERMINATION OF ROUGHNESS LENGTH AT CAPE KENNEDY 4
III. DETERMINATION OF L AND » FROM LOW-LEVEL VARIABLES ... 21
IV. DETERMINATION OF PREDICTANDS FROM z , u. , AND L 38
o *o o
4.1 Estimation of Mean Wind 38
4.2 Estimation of Variances 39
4.3 Estimation of Spectra 48
4.4 Estimation of Cross Spectra in the Vertical 52
V. RELATIONS BETWEEN LARGE SCALE PARAMETERS AND u . , L . . . . 56
5.1 Theory 56
5.2 Practical Methods for Determining and L q from
Large-Scale Variables 65
VI. DETERMINATION OF PREDICTANDS FROM LARGE-SCALE VARIABLES . . 72
6.1 Method 72
6.2 Test Results 73
VII. SUMMARY OF PRACTICAL METHODS FOR ESTIMATION OF LOW-LEVEL
WIND STATISTICS 79
7.1 Roughness Length, Friction Velocity and L q When
Large-Scale Information Only Is Available 79
7.2 Friction Velocity and L from Low-Level Data 82
7.3 Estimation of Various statistics from z , L
and u. ? • °. 82
LIST OF FIGURES
Theoretical wind profiles for 20 cm roughness length
and u. = 0.52 m sec - '*'
Neutral wind profiles at Cape Kennedy for wind
directions 225° , 0285°, and 255°
Neutral wind profiles at Cape Kennedy for wind
directions 195°, 165° , and 135°
Neutral wind profiles at Cape Kennedy for wind
directions 315°, 345°
Neutral wind profiles at Cape Kennedy for wind
directions 105° , 75°, 45°
Neutral wind profile at Cape Kennedy for wind
Average wind speed profiles of one-hour runs
at 150 m
Observations of <J> = — — vs. - z/L compared with
Businger's theory *o
Polar diagram of roughness lengths
Nomogram for Richardson number as function of bulk
Richardson number B and z/z (stable)
Nomogram for Richardson number as function of bulk
Richardson number B and z/z q (unstable)
Observed and calculated Ri, O'Neill
Observed and calculated Ri, Cape Kennedy
Pasquill class as function of L and z
H o o
Richardson number as function of wind at 18 m and
Nomogram for wind profile exponent as function of z/z
and bulk Richardson number
Observed values of vs u^ at Cape Kennedy
Observed values of a vs u* at Cape Kennedy
LIST OF FIGURES (Continued)
Observed ratios O /u.vs Ri
Observed ratios a /u. vs Ri
a at 18 m minus a at 150 m as function of wind speed
a¥ 18 m U
a at 18 m minus O at 150 m as function of wind speed
a¥ 18 m V
"Observed" /e vs /e estimated from surface
Decay constant for coherence of u as function of Ri
at 23 m
Decay constant for coherence of v, as fucntion of Ri
at 23 m
"Slope" of u vs Richardson number
"Slope" of v vs Richardson Number
Geos trophic drag coefficient as function of a and
surface Rossby number
Stability parameter 0 as function of insolation and
Stability parameter S as function of a and surface
Comparison of observed and computed insolation,
in 1000 ly min
Comparison of u^ derived from geostrophic wind vs
u* from local data
a as function of u. derived from geostrophic wind
LIST OF TABLES
No . Page
1 Wind profile parameters <j>, ip, and e ^ as
functions of z/L 17
2 Relation of Pasquill stability classes to
weather conditions 30
3 Friction velocities and other relevant statistics
for each analyzed run at Cape Kennedy 32
4 Vertical variation of average standard deviation
of u and v for different stability groups 47
5 Hourly mean values of radiation at Cape Kennedy 69
6 Roughness lengths for various terrain types 79
The research reported herein was supported by NASA Contract NAS8-21140.
Dr. George H. Fichtl of the Aerospace Environment Division, Marshall Space
Flight Center, was the scientific monitor, and support was provided by
Mr. John Enders of the Aeronautical Operating Systems Office, Office of
Advanced Research and Technology, NASA Headquarters.
The research reported in this document is concerned with the results of
studies of wind and turbulence in the first 150 m of the atmospheric
boundary layer. The motivation behind this research is the development of
models of the statistical properties of atmospheric turbulence for the
design and safe operation of aeronautical systems. Atmospheric turbulence
models play a number of crucial roles in the design and operation of
aeronautical systems. First, they provide for the development gust design
criteria; second, they provide for the development of atmospheric turbulence
simulation procedures whereby control systems can be evaluated and pilots
can be trained. Finally, they provide a basis whereby the current require-
ments, criteria, and procedures for reporting winds and turbulence to pilots
prior to take-off or the final approach can be evaluated, updated and
improved, as well as for the development of possibly needed new procedures
It is believed that the models reported herein will contribute significantly
to these areas of aeronautical interest, especially to the development of
atmospheric turbulence simulation procedures.
Certain statistics of airflow near the ground are of special
interest to designers of aeronautical and aerospace vehicles and systems.
These include the mean wind profile; variances of the velocity components;
spectra of the velocity components and of dynamic pressure; and cospectra
between velocity components at different levels. These quantities will be
called "predictands". Two sets of "predictors” will be considered: either,
conditions close to the ground measured locally; or large-scale data available
from weather maps and astronomical tables.
The problem of estimating the predictands at Cape Kennedy from local
variables has been discussed in previous reports, and will be summarized
only briefly here. In this report, emphasis is placed on the problem of
estimating the statistics required from large-scale variables.
Monin-Obukhov theory predicts that the statistics of atmospheric flow
over homogeneous terrain in equilibrium in the "surface layer" are completely
determined by three parameters: the roughness length, z^; the friction
velocity, u^; and the Monin-Obukhov length, L, defined by:
u . c p T
( 1 )
Hence, c p is the specific heat at constant pressure, T temperature, p density,
g gravity, k von Harman' s constant, and H the vertical heat flux (including
the effect of moisture on buoyancy).
The "surface layer" is defined as that region in which the vertical
variation of u^, H and therefore L can be neglected. This is approixmately
true only in the lowest 30 m or so; the possibility remains, however, that
the relationships valid in the surface layer may apply up to the top of the
Kennedy tower (150 m) , if surface values of friction velocity, heat flux and
L could be inserted into these relations. The relations, and tests of their
validity up to 150 m are discussed in Chapter IV.
The roughness length is needed in the estimation of the predictands
both from local and from external variables. Therefore a separate chapter.
Chapter II, is devoted to its determination at Cape Kennedy.
The determination of u^ and L from local variables has been discussed
in earlier reports and will be summarized briefly in Chapter III. The
estimation of the same quantitites from large-scale variables formed a major
part of this project, and is discussed both on the basis of theory and
measurements in Chapter V.
Combination of the results of Chapter II, III, and V leads to techniques
for estimation of the predictands from large-scale variables only. These
methods, and a test of their accuracy, are given in Chapter VI. Chapter VII
gives instructions for the estimation of the various statistics from usually
observed data. Finally, an appendix discusses statistical and mathematical
properties of time series of atmospheric turbulence at Cape Kennedy and else-
The tower and instrumentation at Cape Kennedy will not be described in
detail here, since this has been done by other authors, e.g., by Flchtl and
McVehil (1970). Suffice it to say here that mean and fluctuating wind,
as well as mean temperatures were available at six levels: 18 m, 30 m,
60 m, 90 m, 120 m and 150 m.
Some of the material in this report has been discussed in two previous
reports, NASA CR-1410 and NASA CR-1889; these will be referred to as Reports 1
and 2, respectively.
II. DETERMINATION OF ROUGHNESS LENGTH AT CAPE KENNEDY
Roughness length is usually determined from wind profiles, preferably
under neutral conditions, although alternative methods have been suggested
in earlier reports. Determination of roughness lengths from wind profiles
presupposes a thorough understanding of profile theory, above the surface
layer. At the time of writing of this report, numerous profiles have been
processed so that their properties could be evaluated sufficiently well to
suggest reliable estimates of roughness lengths. To simplify matters, only
neutral conditions are considered first. This is accomplished by considering
only average profiles of individual runs, for which the Richardson number
between 18 and 30 m lies between -.05 and +.05.
Up to 150 m, it is presumably legitimate to obtain equations for the
wind profile under the assumption that the wind direction is invariant with
height. Then, we may derive expressions for the profile over homogeneous
terrain by integration of the differential equation for wind u as a function
of height z:
§ = V* < 2 >
This equation may be considered as the definition of the "mixing length", A.
The quantity u^ is the local friction velocity. Of course in the surface
layer, the friction velocity is constant and A = kz, so that the wind follows
the familiar logarithmic relation.
The behavior of the friction velocity above the friction layer can be
derived rigorously from the equation of motion in the direction of the wind:
Here, v is the component of geos trophic wind at right angles to the surface
wind u, and f is the Coriolis parameter.
From the theory of the geostrophic drag coefficient (see, for example,
Blackadar and Tennekes, 1968), we have
Here B varies with stability. Under neutral conditions the best estimate
for B is about 5 (for further discussion, see Chapter V) . Integration then
shows that, very nearly
u * = u *o ~ ^
where u. is the friction velocity at the surface.
Since the mixing length, A, is kz near the ground, and applying the
scaling appropriate for the neutral planetary boundary layer, we may put:
X = kz F (— ) (6)
Here F is presumably a universal function. For example, Blackadar (1962)
f Z “1
F = (1 + 63 — )
If we now substitute equations (5) and (6) into (2), and integrate, we
may write formally:
( 8 )
where G( ) is another universal function, related to F. In particular,
with Blackadar's hypothesis for F, the expression for the wind profile can
u = : — In — 1 144 fz (9)
Figure 1 shows the logarithmic wind profile, and, for comparision, (9) with
roughness length 20 cm and friction velocity 52 cm/sec. Clearly, the
Blackadar profile in semilogarithmic representation curves significantly
in the lowest 100 m.
Geometrically, the roughness length is given by the intercept of the
profile with the ordinate. In practice, however, the roughness length is
often determined by constructing a tangent to the observed portion of the
profile and determining z q from its intercept with the ordinate, a procedure
based on the assumption of a logarithmic wind profile. Clearly, if the
actual profile is as strongly curved as Blackadar’s profile, the roughness
length would then be strongly overestimated. It is therefore of great
importance to analyze profiles observed under hydrostatically neutral conditions
in order to study their curvature.
Figure 1* Theoretical "wind profiles for 20 cm roughness
length and u* Q * .52 m. • = logarithmic law,
o = log-linear law, A = hyperbolic sine law.
Ten-minute average wind profiles are now available on tape, one for
every hour of 1968. From these, mean "neutral” profiles regardless of wind
speed were computed, for twelve wind direction sectors. Here, "neutral" is
defined by -.05 < Ri < +.05. The resulting profiles are shown in Figure 2.
It is clear that the twelve profiles show considerable irregularities,
but no systematic bending to the right with increasing height. This result
is in agreement with conclusions reached by Thuillier and Lappe (1964) who
analyzed profiles near Dallas, Texas. Many of the irregularities may be
due to instrument effects; in a few cases, as will be seen, the change of
terrain upstream of the tower may be responsible for kinks in the profiles.
In any case, (9) does not provide the best fit to the observations. A
somewhat better fit is provided by the hypothesis
( 10 )
which leads to geostrophic drag coefficient statistics in as good agreement
with observations as (7). In that case, the wind profile can be written,
omitting a term in (fz)
u. sinh 63 fz/u.
*o *o _
k n sinh 63 fz /u.
This wind profile is also shown in Figure 1. It is quite close to the
logarithmic profile and, when fitted to observed profiles, should produce
better roughness lengths than (9); but the results should not necessarily be
superior to roughness lengths determined under the assumption of a logarithmic
profile. Because computations based on the assumption of logarithmic profiles
are by far the simplest, logarithmic relations were assumed.
Actually, two sets of roughness lengths were obtained from the near -neutral
wind profiles: first, lines were fitted to the profiles shown in Figure 2; and
second, average profiles were constructed from fewer individual profiles, the
Richardson number of which was restricted to lie between -.01 and .01. Thus,
usually ten or fewer individual ten-minute profiles were averaged in each wind
direction sector. The random errors in these averages are larger than in the
first sample, but systematic errors should be smaller. Generally, the agreement
between the two sets of roughness lengths was quite good, and they were averaged.
Two other groups of roughness lengths were used, both requiring theoretical
expressions for the correction to the logarithmic wind profiles due to unstable
stratification. The characteristics of these expressions are quite well known
for the surface layer, generally taken as the layer below 30 m or so. Panofsky
and Petersen (1972) have recently tested the hypotheses that the expres-
sions valid below 30 m can be applied up to 100 m, on the basis of observations
at Ris^f, Denmark. The results were favorable to the hypothesis. An independent
test will now be described which was made on Kennedy profiles. This was based
on hour-average mean profiles, in various categories of L q , the Monin-Obukhov
length. The average profiles themselves, along with the mean value of L q for
each, are shown as Figure 3. Again, the "neutral" profile is logarithmic, with
an intercept suggesting a mean roughness length of 0.4 m. The other profiles
have the expected curvatures.
The test was made on the basis of the nondimens ional vertical wind
, kz du
( 12 )
In the surface layer, and, according tu the hypothesis, up to 100 m, in
(1 - 16 z/L )“ 1/4
In practice, normalized wind shears were computed by dividing wind
differences between successive levels Au^ by the corresponding wind dif-
ferences Au under neutral conditions. Since d) = 1 in neutral air,
<L/<L = <L =
u . Au
r u' T n
Subscripts u and n denote unstable and neutral, respectively. The ratio
u, /u. was evaluated from equation (15) (see below) applied at 18 m:
i , 18 v
u . n- 0 L
*no 18 n o v
\I “ 1 O /
This procedure for evaluating <f> has the advantage that it eliminates
systematic errors in wind measurements caused by systematic changes of
roughness with distance from the tower. The determination of the Monin-
Obukhov length L q is described in the next chapter. The same technique had
been used successfully with Ris^ wind profiles.
Figure 4 compares the theoretical dependence on <J) on z /L q with the
observations. Some observations from the Ris$l tower (Denmark) are in-
cluded. The agreement is surprisingly good, and further confirms the hypothesis
that (13) can be used up to 100 m, or even to slightly greater heights. Thus,
the method for determining surface stress and roughness suggested by Panofsky
(1963) can be applied to the tower data.*
First, the diabatic wind profile is written as:
u = — 7 ^ [In (ze ^) - In z ] (15)
Paulson (1970) determined \p for unstable air from (13):
* (f-) = m r ( I=|) - 2 t-" 1 f-> + I < 16 >
o o o
Here x = (1-16 z/L q ) • Equation (15) is a linear equation between
u and In(ze^), with In z q as intercept. In stable air, we take = - 5 z/L q .
Table 1 lists <j>, and e ^ computed from these equations.
*More recent data from Idaho Falls as well as the points plotted suggest
that a better fit is given by the KEYPS equation
which also has the property that, as - z/L , + °°, is independent of z q and
u^ Q , as required by the original theory of free convection. Another excellent
fit, undistinguishable here from the KEYPS equation, is provided by:
<p = a - is f~r 1/3
Wind profile parameters <J>, iJj,
as functions of z/L
z/L 0.1 0.05 0 -.1 -.2 -.3 -.4 -.5 -.6 -.7 -.8 -.9 -1 -2 -3
\p ~0.5 “0.25 0 .28 .45 .59 .70 .79 .87 .94 1.01 1.06 1.12 1.49 1.74
e ^ 1.65 1.28 1.0 .75 .64 .55 .50 .45 .42 .39 .37 .35
.33 .22 .18
4> 1.50 1.25 1.0 .79 .70 .65 .61 .58 .56 .54 .52 .50 .49 .42 .38
Average wind profiles were constructed from the 10-minute average winds
in 1968 for each sector, and (15) was fitted to each by least squares. This
yielded the second set of roughness lengths. The third set was that given
by Fichtl and McVehil (1970) which was based on one-hour average winds, but
only at 18 m and 30 m. The first two sets then effectively represent
average roughness lengths applying to the whole tower, whereas the third set
represents the lowest section of the tower.
All three sets of roughness lengths are shown in polar form in Figure 5.
Also, a line is drawn into the figure, which is taken to be the "best"
distribution of roughness lengths at Cape Kennedy. The three sets agree
quite well with each other, with that by Fichtl and McVehil yielding slightly
lower values than the others. This is because the average logarithmic wind
shear between 18 m and 30 m is very slightly smaller than higher up, sug-
gesting slightly smaller roughness lengths in the immediate neighborhood of
the tower than further away.
Figure 5 • Polar diagram of roughness lengths
A neutral only, ten minute average wind
M neutral and unstable, 1 hour averages
• Fichtl and McVehil
A qualitative test of the roughness lengths involves comparing their
angular distribution with the corresponding distribution of terrain features.
Fichtl and McVehil (1970) distinguish basically two sectors: essentially
smooth terrain (vegetation 0.5 to 1.5 m high) from 300° through 360° to
160°; and a sector with trees about 200 m and more from the tower, from 160°
through south to 300°. The "smooth" area is interrupted by a narrow band
of trees 450 m away to the east and southeast. It is seen that the angular
distribution of roughness lengths shown in Figure 5 agrees fairly well with
West of the tower, and about 200 m from it, there is a strip of forest
extending to a distance about 400 m from it. The wind profiles for wind
directions 255° and 285° in Figure 2 both show a large wind shear between
30 m and 60 m, corresponding to roughness lengths of about 1.5 m. This rough-
ness length is about what would be expected for woods, and the heights of the
large-shear layer are consistent with theory (Elliott, 1958) and observations
elsewhere. Similar irregularities in the wind profiles are found with easterly
winds, which may be due to a small hill in that direction. Nevertheless,
this explanation of profile irregularities must be considered as extremely
In summary, the angular distribution of roughness lengths, as shown
in Figure 5, is in reasonably good agreement with terrain characteristics.
However, it will be seen later that these roughness lengths are generally
larger than what would have been expected from the fluctuation statistics.
The roughness lengths discussed in this chapter are essentially "local"
roughness lengths. Having been obtained from wind profiles up to 150 m at a
tower, they represent approximately a circle of radius 1 km, with the tower
at its center. For applications to the dynamic behavior of rockets on the
pad and. shortly thereafter, such local roughness lengths are needed, in
contrast to requirements for the estimation of the effect of terrain on
airplanes, or on large-scale momentum transfer.
If the relations valid at Cape Kennedy are to be applied elsehwere,
analogous "local" roughness lengths must be estimated. Only approximate
guidelines can be stated for such quantities: 10 cm for unobstructed
water surfaces; 1-10 cm for low grass and low vegetation; 10-80 cm for
fields broken by trees and hours; 80-400 cm for mainly forested regions and
cities. These estimates are still controversial and require considerable
III. DETERMINATION OF L AND u. FROM LOW-LEVEL VARIABLES
According to Monin-Obukhov theory, the meteorological predictands
needed for the design and operation of aeronautical and aerospace systems
can be estimated as function of roughness length, friction velocity and
Monin-Obukhov length in the surface layer over homogeneous terrain. Before
tests can be made to what extent these relations are valid up to the top of
the Kennedy tower, at 150 m, estimates of L , z and u. first have to be
o o *o
made. In the last chapter, the characteristics of roughness length were
described. Here, the question of the estimation of L q and u^ q will be taken up.
According to theory, z/L q in the surface layer is a universal function
of the Richardson number, Ri, defined by
“ - f < y a - - r >/<§> 2 ‘ 17 >
Here, y is the lapse rate of temperature, and y^ the adiabatic lapse rate.
Observations at well instrumented, homogeneous sites (see, e.g. Paulson
1970) confirm the hypothesis, proposed originally independently by Pandolfo,
Dyer and Businger, that Ri and z/L q are essentially equal to each other in
unstable air, so that we may put:
L = z/Ri (18)
In stable air, the Monin-Obukhov hypothesis is more controversial, but a
good relation (for Ri « .20) seems to be
L = z
(see Businger, et al., 1971).
The Richardson number appearing in (18) and (19) has usually been
determined directly from its definition (17) from observations of wind
and temperature at 18 m and 30 m; z in the equations is taken as the
geometric mean of these two heights or 23 m. Although this technique has
actually been used in this project for determining , it probably is not the
best technique because it involves the squares of measured wind shears which
have large observational uncertainties.
A better method for obtaining Ri involves measurement of the bulk
Richardson number. The bulk Richardson number is defined by:
( 20 )
which can be determined with much greater percentage accuracy than Ri,
because u has a much smaller percentage error than du/dz.
Ri is connected with B through
Ri = B [u/ d(ln z) ]2
( 21 )
Given <j> and ip from (12) and (15) respectively, we have:
Ri - B
In z/z -ip
For the surface layer, both iJj and <j) are now quite well known (see
Chapter II), and therefore it is possible to construct a nomogram for Ri
as a function of B and z/z q . This nomogram is shown in Figure 6.
Figure 7 compares values of Ri from (17) with those computed from
Figure 6 for 0 f Neill, where winds are extremely accurate. The agreement
Figure 8 shows the same kind of comparison for Cape Kennedy from winds
and temperatures at 18 m and 30 m. The height z was taken as 23 m, the
geometric mean. The value of was obtained by plotting In z - ty as function
of u from 18 m to 30 m and locating the intercept of the straight line through
the two points.
Figure 8 shows that a line of slope 45° fits the data as well as any
line, suggesting no systematic error. But the scatter is enormous, suggesting
large random errors in the M measured ,, Ri. That Ri- values are uncertain is
confirmed by the wind profiles described in Chapter II.
It is therefore concluded that L can be found from bulk Richardson
number and Figure 6 more accurately than by direct measurement. If lapse
rate is not available, we may use the Pasquill stability classes (Table 2) taken
from Slade (1968). Only rough estimates of wind and radiation conditions
are required to determine these classes. The classes can be combined with
roughness length to arrive at an estimate of L q using Figure 9, which has
been taken from Colder (1972).
Finally, if radiation and wind at a low level are given, L q can be
found from Figure 10, which, however, has been derived from Kennedy data
only and is not necessarily valid elsewhere.
0.04 0.06 0.08
Nomogram for Richardson number as function of
bulk Richardson number B and z/z q (stable).
quill class as function of L and
Figure 10. Richardson number as function of wind
at 18 m and insolation.
Table 2. Relation of Pasquill stability classes to weather conditions
A - Extremely unstable conditions D - Neutral conditions*
B - Moderately unstable conditions E - Slightly stable conditions
C - Slightly unstable conditions F - Moderately stable conditions
Strong Moderate Slight
or ■> 4/8
^Applicable to heavy overcast, day or night.
**The degree of cloudiness is defined as that fraction of the sky above
the local apparent horizon which is covered by clouds.
Fortunately, u^ as determined from wind, roughness and stability is not
very sensitive to the exact value of , so that this quantity has to be known
only approximately, and either of the last simple techniques to estimate it
should be sufficient. The friction velocity is most sensitive to errors in
wind speed, and, to a somewhat smaller extent, to errors in the roughness length.
The surface friction velocity is evaluated from the theory of the
low-level wind profile, equation (15), given the wind near the ground z^ and
L . We write (15) in the form:
= k u
U *o In z/z - ip (z/L )
o T o
As before, we adopt Paulson’s form for ^(z/L q ) in unstable air (16). In
stable air, as before
* <f-> -
Table 3 summarizes the friction velocities computed from (23) and
(16) for all one-hour runs analyzed so far, along with other relevant
statistics for each run. It is almost certain that local friction velocities
estimated from (23) are more reliable than those that can be obtained from
large-scale variables, including geostrophic wind. It is therefore recommended
that, whenever possible, (23) be used for this purpose, given surface wind
reports from hourly sequences.
Table 3. Friction velocities and other relevant statistics for each analyzed run at Cape Kennedy
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• • • •
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vo<tincococor^r^in^DoooN»ncMr^cov > or^vor^oo\o<tc^cM'd'<t;cMinrHvo
H i — I H i — I i — I H H H H i — I H HHHHHHHHHHHHrlHHi IrHrHr-HrH
d) CT3 C\J CO Oh Oh Oh Oh Oh Oh Oh Oh
a c g ^ *3
1 -HrHCMCOCOi — l CM CM CM CM CM CM CM H rH rH rH CM CM CM rHCMCsl rHrH
r^-ON -NtoNcocMcoOcMr^cMinoominvooN on m- no <r cm co on
vO CM M-MCOncMM-M-ONM-HM-inM-COCMM CM CO CM H CO M CM
CO CO rH r-j
o o o o o o o
cm in m on in cm m
st m oo co <r
rH CM CM CM CM
rH rH rH rH CM CO
o o o o o o o
o in o o in O
on co in co <r no oo
CO rH rH rH rH
138 95° 1.8 11 July
Table 3 (continued)
(N CM 00 ON
m n cm in
ncM m m m ^
in CM i£)
cm oo h o vo in
cn o H H vD tn N
h i — oo oo m co oo
CM + + +
I 1 I + I I I
m i rH I i— i
1 I I
srinNrcocoHvr<tH<fO'J-cooinocncoococoNr'T<rsrHNrcM n h
t — i i — | i — | O O O < — t r— H i — I t— I rH rH rH H H rH i — ! * — I i — I t — 1 rH i — I rH rH rH O rH rH O
is. T ^voHco l nvDoovDr s *info\Of r )vo<fvo\Dcn'<r\ONaH<r^‘f f ioo\ocMo
, — IrHi— HOOOrHt-HT-Hi — IHHrlHHHHHrlHrlrlHrHHHO'HrHO
r— T rH* r~? rH* rH* rH* rH^ rH^ rH* rH* r— T tH* rH* rH^ rH* rH* bOOObOOOOOOOOO&OOOOOOOW)
1 — ihhhhhhhhhhcmcmcmcmcmcm h h h cm n cm
<r <r h m o o oo
CM O sf vD
H H CM CM CM CO H CO rH CM CM
H i-H CM CM CM H H rH HrHHCMHHcOrlrlHHNcMHH
cn^<r<t<r<f<tLnininininininin\£)v£)vDv£)v£)v£>rvr > ^r*-.r>.r«*-r^cooo
184 116° 29 Aug
Table 3 (continued)
CM 00 CM
CO 00 CO O
co cm co
r^- cm cc cc co cm <r r- i— !
C\1 CO CM CM CM CO CO CM CO
<J\ LO VO v O Ov CM CO •<!- VO
uouO"<fcrxvor^cr>OT— ir^voovocj>uooor^o<J-o<tvnavc\irHCMT— i^r^r^<tvo
<TH>JH<rcOHH^inMNOHHOCOOM 0044 HOMHinCNCOHO
OOOiHi — I H i — iCMCMCMi — I t — I i — |i — it — It — I i — I O < — I rH i — I t — I f — li — I rH « — I « — I i — I ' 1 < — I' — I* I
vovncoovor^avtHi— tiHvoiHvo<ti— ioovoo<ro<roa\r^T— icMCMavr^omvo
1 — 1
CO T 3
< 1 -
co cm ov av cm cMCMunuoLor— <j- 0 '<i-<rLOOCovnr^rHvocO'<rpHoooo>
COCO COCO CM CM CM CO rH CM CO O CM CM CO CO CM CM CO CM
OOOOOOCOO\OvCT\OVOvCP>OvOOOOi- 1 rHCMOOOOCOoOOOCTvOOOOOOTHfH
HrHi— It— IrHrHr-li— I iHHtHCMCMCMCMCMCMCMCMCMCMCMCMCMCOCOcOCOCOCOCOCO
Wind Average GMT
Direction Speed Date T ime On Time Off
m m cm
on m <r co
oo vo in m
x <* m m
i — i on
cs vo m cn
co cm o
m cm co o
H vO On CO O
CM CO <f O O
NOOO\NCON\OMvOrsOvOO>OMn H CM m VO
CO(OOOOOCMO\r^M , 0 \OONfNOvO X* CO CO Mt 00
O O O X
o o o o o o o
I 1 I I ! I
CO o o
o o o
o o o o
rH rH t-H i — I i — I i — I rH CM CM O O rH CM O CM O
oo\a\CT\OinooN 0 \- 3 -inincMMj*<yi
r^in\ONM , oof s 'inoococoo^(jiM‘vD
1 — I rH rH rH « — I i — I i — I rH i — t t — I rH rH i — I rH rH
mM , r s ‘M , HO\NMtoicooi > »i s *inNcoc\C30voiniDNcovOM , inMM‘\ovo<r
H CM CM CO CO CM CM CM
CM rH t — I rH
o o o O o O o
on oo in o on i — i on
co on in <r m x
CM rH rH i — \ rH
C oo on co vo r".
2 H H CM N N
pcj co co co co CO
O CM in I s - CO H CM
CO CO co CO CO nT
co co co co co co co
m o x m on x <r
m n m in m no no
co co co co co co co
lO M3 O' ON o o
vO M3 M3 M3 O- CO
CO CO CO CO CO CO
H CO <3- ON X
00 00 00 00 ON o
CO CO CO CO CO M"
411 15° 7.6 12 Apr
Wind Average GMT
Direction Speed Date Time On Time Off
n o> <r n
•o- m m cm o\
O in o\ \£)
O H in v£> CM
o o co o co
O O tH O o
m h in m o\ n noN^^o>(Nvon
CM O tH O rH tH rH CMOOOOrHOrHtH
i — 1
i — 1
i — !
>> 4 ->
Cl. 4 ->
n n in n ^ o cm o vo h co o m c-- tH
CM H H n H H H fO H H N CO
C Lnr^O'<fooc^Lnr^. r — imorHodr^oocTN
IV. DETERMINATION OF PREDICTANDS FROM z , u. , AND L
o *o 0
4.1 Estimation of Mean Wind
Wind profiles were discussed in detail in Chapter II. Briefly, the
usual logarithmic expressions appear to be valid in neutral air up to the
top of the tower, 150 m; further, the corrections in unstable air developed
for the surface layer supply good fits at high levels as well. Some systematic
exceptions at Kennedy occurred, however, for example, with westerly winds, the
winds at 30 m and 60 m exceeded those expected from the simple profile theory.
As previously mentioned, the explanation for this apparent anomaly may perhaps
be found in the stretch of forest, about 200 m to 400 m west of the tower.
Once and L q have been determined by methods suggested in Chapters III
or V, then (15) and (16) may be used to estimate the wind profile. However,
simpler procedures can sometimes be used. For example, a frequent practical
problem is that wind, and perhaps temperatures, or radiation, are given near the
surface. The wind at one or more higher levels must be estimated. This
problem can be handled by dividing (12) at a higher level (subscript 2) by (12)
at a low level (subscript 1) . The result is
z 9 e‘*2
In (^- )
U 2 U 1
z e ^1
The roughness length is given, at Kennedy, from Figure 5. At other
locations, the roughness lengths are assumed known, at least approximately.
Equation (16) or Table 1 gives \p in unstable air, and in slightly stable air,
(Ri < .10). For more stable air, (24) is of doubtful value; the more stable
the air, the less winds near the surface are coupled with winds higher up,
so that winds at 100 m or so cannot be estimated from conditions close to the
Engineers often estimate high-level winds from power laws, which fit
profiles reasonably well:
The best fit of such power laws to profiles is obtained if the exponent p
is computed from:
In z/z° - \p
Here, z is the geometric mean height between z ^ and z^. (p is the usual
normalized wind shear and is tabulated above in Table 1. A nomogram for p
as function of z/z and z/L is Figure 11.
Usually, "random” errors in the high-level winds are larger than errors
produced by slight errors in z q or L.
4.2 Estimation of Variances
According to Report 2 and Monin-Obukhov theory, the standard deviations
of the velocity components are well correlated with the friction velocity;
further, the ratio of standard deviations and friction velocity should be
dependent on z/L .
Since friction velocities have been revised in accordance with the
revised roughness lengths, Figures 12 to 15 show new relationships between
the standard deviations and the friction velocities on the one hand, and be-
tween standard deviation-friction velocity ratios and z/L, on the other. The
best estimate for a /u. in neutral air is now 1.6, and that for O /u-» also 1.6.
u w v *
These ratios are smaller than those found on the average (see Lumley and
Panofsky, 1964). However, they are nearly the same as at Brookhaven, which
is located, like Kennedy, in a generally flat countryside. It is likely that
these ratios are not universal, but depend also on a mesoscale roughness which
controls the low frequency portion of the spectrum of the horizontal velocity
components. If the mesoscale variations are pronounced, for example, the
ratio a /u. sometimes exceeds 3. The correlation coefficients between O and O
u * u v
with u* are quite high, being 0.87 for and 0.45 for This suggests that 0^,
and to a smaller extent, O v , can be estimated accurately from u*, provided that
u* can be obtained without systematic error, at least at Kennedy, once u* is
In contrast to earlier results. Figures 13 and 15 show no systematic variation
of a /u. and a /u. with z/L. As to other locations, there are no constant
u * v ”
relations between these ratios and stability.
In spite of the argument that O / u. and O /u. really could be as low as the
observed ratios of 1.6, there is even better evidence that the true ratios should
be about 50% larger. The argument for this will be discussed more fully in
Section 4.3 In that case, the ratios are more nearly equal to those recommended
by Lumley and Panofsky (1964). If this is correct, either the assumed roughness
lengths are too large or the measured standard deviations are too small. Since
Figure 15 • Observed ratios vs
the roughness lengths depend critically on the vertical wind shears, this means
that the statistical fluctuations are too small for the wind shears. There is
no objective way to determine whether the wind shears are too large or the
fluctuations too small. We will make the hypo thesis here that the wind shears
have no systematic errors, but that all measured fluctuation statistics are
about 2/3 of their true value, perhaps due to instrumental imperfections or
problems with the recorder. Hence, for estimations of a and a , we will assume
that - 2.5 u* Q and 0^ - 2.2 u^ as suggested by Lumley and Panofsky. Further,
these factors will be assumed to be relatively independent of stability. As
mentioned before, we recommend that u^ q be estimated from (23) as function of
z , L and wind at a low level,
Report 2 also showed a relation between the difference of high-level and
low-level standard deviations a and the wind at 18 m, in the sense that the
variance decreased most rapidly in strongest winds; for example, there is no
change when the wind is 3 m/sec, and the variance decreases by 7 m sec from
18 m to 150 m with a wind of 8 m/sec. This result has not been explained but
the relation is reproduced as Figures 16, and Figure 17 shows the analogous
relation for a .
Theoretically, one might expect the decrease to be largest in the most
stable air; in unstable air the increase of -z/L with height should diminish
or even reverse this tendency. Table 4 shows that fractional change of average
standard deviation with height (relative to unity at 18 m) .
According to Table 4, the decrease of the longitudinal standard deviations
is indeed largest for the most stable air; further, in unstable air, the
decreases for longitudinal and lateral standard deviations are about the same;
Table 4. Vertical variation of average standard deviation
of u and v for different stability groups
Ri > -
<_ Ri <_ — . 1
Ri < -
however, in the near-neutral class, average lateral standard deivations
actually increase upwards, though not significantly. In fact, upward de-
creases are slightly more common in this category than increases. This
points up the tremendous variability of the change of these statistics with
height, so that the systematic changes shown in the table do not reflect
well the behavior in individual cases. All we can say with certainty is that
the systematic change of the standard deviations is generally small, and that
decreases exceed increases. Wind speed appears better related to the vertical
change than Richardson number (Figures 16 and 17). But, since the relations
of wind speed to changes of the a f s with height have not been explained it is
probably best at present that we assume for estimation purposes that no
significant vertical variation of the c T s exists.
4.3 Estimation of Spectra
Fichtl and McVehil (1970) have discussed spectra of lateral and
longitudional velocity components at Cape Kennedy in some detail. For many
applications to rocket problems, the inertial-subrange portion of the spectra
-! „ ^ C — j ! 1 1 i JJ J 1 -r* l ‘ * — 1 .. ~ J
XO U1 spcuidl IIU^UL LCU1LC, dUU WXXX UC U.XO uuoacu. neit:. rui 1IUJ- XZ.OU LdX WIUU
components, this range extends from wavelengths several times the height to
wavelengths of 1 cm or less. The exact range was discussed in detail in
The equation for the spectra in the inertial subrange is :
S(k) = ae 2/3 k _5/3
Here, "a" represents universal constants, which are about 0.5 for longitudinal
components and 0.67 for lateral components, if the wave number k is measured
in radians per unit length. Hence, the problem of estimation of spectra in
the subrange reduces the problem of estimating the dissipation, e.
In general, the dissipation can be written in terns of the nondimensional
Panofsky, in Report 1, has suggested that, for practical purposes, at
30 m and above, vertical divergence of turbulent energy flux can be neglected,
so that (neglecting the pressure term in the energy budget and assuming
( 29 )
d> = d> - z/L
Fichtl and McVehil (1970) have suggested that, at 18 m and perhaps below,
vertical flux divergence and buoyant energy reproduction cancel (see also
Panofsky, 1962) so that
The cancellation of divergence and buoyant production at low levels
has recently been confirmed by Wyngaard and Cote (1971), but (30) has not.
Wyngaard and Cote f s conclusion is that the pressure term is important to
the turbulent energy budget in unstable air. Nevertheless, (29) and (30)
seem to explain well the ratio of energy dissipation estimates at different
heights quite satisfactorily, as seen in Report 1, where cf> was taken as
(1 - 18 z/L )~ llh .
The problem of estimating E at any level then reduces to estimating
it at a low level, taken here as 30 m (where it is, according to (28) and
*o . ,30, 30,
12 [<j > ( L _) “ L~
Figure 18 compares determined from observed lateral and longitudinal
spectra with the corresponding estimates from (31) , with u^ taken from
Table 3. Apparently the magnitude of the "observed" values of is only
about 2/3 of the "computed on the average. The agreement is best for
weak winds at 30 m and poorest for strong winds. This would be expected if
Figure 18 .
"Observed" 'i'fz- vs estimated from surface
stress, c.g.s. units.
the discrepancy was due to errors in the dissipation estimated from the
spectra; the stronger the wind, the more these estimates depend on high
frequencies, which would be most strongly damped if there was undue friction
in the anemometer. The tentative assumption will then be made that the
"computed" values of £ are better than "observed" values* This assumption
is consistent with the assumption in past sections that "measured" standard
deviations are too low. In other words, (31), with u. determined from (23)
and the roughness lengths of Figure 5 is assumed to result in the best estimates
of e at 30 m.
In Chapter VI, suggestions are made concerning the "best" procedure for
obtaining u* q from geostrophic winds. However, such estimates are likely to
be less accurate than those from (23), based on low-level winds, assumed rough-
ness lengths and L q determined from radiation and wind, as described in
There is at least one flaw in the preceding argument. If e values
estimated from the spectra measured at Kennedy are systematically too low,
the error should decrease with increasing heights , as high frequencies become
less important. Hence, observed ratios of dissipations at 120 m to those at
30 m may be too large. Equation (29) was based on this ratio and should be
re-examined with more reliable estimates of £ from spectra.
Nevertheless, for the time being, it is still suggested that we can use
(29) to estimate e above 30 m:
e = £
a - 16 ty m
4.4 Estimation of Cross Spectra in the Vertical
Cross spectra are usually characterized either by cospectra and
quadrature spectra, or by coherence, Co, and slope, S, defined by
Co ^ = Cos 2 (n) + Q 2 (n)
L 0 ^n' “ c C
b l b 2
where Q is the quadrature spectrum and Cos the cospectrum at frequency n.
Az is the height interval and u the mean speed in Az. In practice, slope
and coherence seem to have simpler properties than cospectrum and quadrature
spectrum. Therefore, the procedure recommended is to estimate coherence and
slope first, and then estimate co- and quadrature spectra, or space- time
correlation function by cosine transform (equation 2.1, Report 2).
As was first suggested by Davenport (1961) , coherence can be well
fitted by exponentials of the form:
Co(n) = e _aAf (34)
Here, Af is the nondimensional frequency, nAz/u and a is a "decay constant 11
which depends on Richardson number. Figures 19 and 20, reproduced from
Report 2, show the relationship between decay constant and Richardson number
at 23 m. The systematic difference between Kennedy and other sites is still
unexplained, but believed to be due to random errors in the Kennedy data.
Figures 21 and 22 reproduce the corresponding figures from Report 2
showing relationships between slopes and Richardson number; here the
agreement between Kennedy and other sites is good. Also, slopes for the
lateral components are about twice those of the longitudinal components.
The slopes are relatively constant up to 100 m, but average about
50 percent less between 120 m and 150 m; for details, see Report 2.
Since the writing of Report 2, considerable work has been done on
decay constants for horizontal separations. The decay constants at right
angles to the wind are about the same as the vertical constants. The
longitudinal constants are much smaller and probably increase with increasing
relative turbulent intensity.
V. RELATIONS BETWEEN LARGE SCALE PARAMETERS AND , L
It has been seen that the mean wind distribution in the surface layers,
together with the distributions of a number of statistics of the turbulent
flow, is determined by the parameters z q , u^ q and L^ . These quantities may
be thought of as predictors of the mean wind and turbulence statistics, since
the relationships between them are now established. However, u^ q and L^ are
not satisfactory for this use, since their determination by ordinary methods
requires a knowledge of the predictands themselves. It is therefore required
to find appropriate bases for estimating the values of u^ q and L q f rom larger
scale variables, such as can be derived from synoptic data or from numerically
calculated properties of the large scale flow.
The basis of such relationships has been laid in theoretical studies
of the planetary boundary layer by many authors. An equation relating the
geostrophic drag coefficient U * Q / V g to the surface Rossby number V^/fz Q was
first derived by Kazanski and Monin (1961) for neutral stratification, and
was later broadened to diabatic boundary layers by Monin and Zilitinkevich
(1967) using the semiempirical theory. Similar relationships have been
derived more recently by Gill (1967), Csanady (1967), Blackadar and Tennekes
(1967) and Blackadar (1969) using singular perturbation methods to match
the flow in the surface (constant stress) layer to that of the outer layer
which is dominated by the earth’s rotation and buoyancy. The theoretical
justification of these latter methods has been discussed in detail by
Blackadar and Tennekes (1968).
In this section we review briefly the theoretical justification and
the probable limitations of these relations as we now know them.
The flow within a neutral barotropic planetary boundary layer (PBL) is
completely determined by the quantities z q , u * c > the Coriolis parameter, f,
and the height z. Because the surface boundary condition requires the
velocity to vanish at the lower surface, the flow close to the surface
is dominated by the surface roughness z^; this condition can be achieved
only by scaling heights close to the surface by z q . On the other hand,
throughout the principal portion of the PBL the appropriate length scale
for the wind distribution must be u^/f. This fact follows directly from
the equations of motion and the necessity for the geostrophic departure
to scale as u^ (Blackadar and Tennekes, 1968). Accordingly, the principal
dimensionless parameter is the ratio of the two length scales, u^ Q /fz Q ,
which may be called the drag Rossby number. In addition to this number
there exists the independent dimensionless ratio zf/u. , and it can be
sh.own from Buckingham’s theorem that all other dimensionless characteristics
of the neutral PBL are functions of these two dimensionless ratios.
Normally, the drag Rossby number is very large, typically 10 to 10 ,
and it is a reasonable hypothesis to treat the case when this parameter
approaches infinity. In this case, the equations of motion demand that
the scale velocity deficits (u - u )/u. and (v - v )/u. be universal
g *o g *o
functions of zf/u^ only as long as z » z q . It can further be shown that
a necessary and sufficient condition for satisfying the boundary conditions
is that the flow near the surface be logarithmic. Also, the logarithmic
solution in the surface layers is compatible with the universal function of
the outer layer only if the Kazanski-Monin relationships are satisfied:
where A and B are constants the values of which must be determined empirically
or from more complete models of the flow. In these equations, the x-direction
is parallel to the surface stress. The equations may be considered as
implicit relations for calculating the values of u* q and the surface wind-
drift angle a when the direction and magnitude of the geostrophic wind are
Model studies of the flow in the PBL make it possible to relate the
two constants A and B to a single disposable constant (Blackadar and Tennekes,
1968). When this constant is chosen so as to achieve the best fit of
empirical data, it is found that A and B are about 0.0 and 4.5, respectively.
The resulting wind distribution is rather similar to the classical Ekman
spiral with a gradient wind level at a height zf/u^ equal to about 0.25.
The argument for the universality of the functions (u - U g)/ U * Q and
(v - v g)/ u * 0 * on neutral equations (35) and (36) depend, rests
on the assumption that there are no relevant parameters other than u *o’ f '
Z Q , and z. Such ideal conditions seldom, if ever, prevail. We must, in
practice, be concerned with such matters as non-steady states, horizontal
temperature gradients (baroclinicity) , and diabatic vertical temperature
The effects of baroclinicity on the surface wind direction appear to
be rather conspicuous, but as far as they affect u* q they are minor. This
conclusion is based on the results of two models studied by Blackadar
(1965a, b). The effect of vertical temperature gradients will be con-
sidered in two parts: (a) the effects of the presence of an inversion at
some level h that effectively prevents the downward flux of momentum and
heat from above, and (b) the effect of a heat source at the surface that
results in the generation of convection by buoyant processes within the PBL.
It frequently happens that a stable layer above the surface is trans-
formed by mechanical mixing into an adiabatic layer surmounted by an inversion,
which must be considered to be impervious to the flux of momentum. If
the inversion is high in comparision, say, to .25 u^^/f, its effect on the flow,
and therefore on u. is negligible, for the momentum flux at these levels
would not be significantly changed by the presence of the inversion. We
expect, therefore that h will be a significant parameter only when it is
small compared to u*/f. We shall study this case in detail. This conclusion
is supported by Deardorff, who found by numerical simulation, values for A
and B of 1.3 and 3.0 when hf/u^. = 0.5. These values are reasonably consistent
with those found with other models where h is infinitely large.
Steady-state flow in the PBL is governed by the equations
£ <v-y ♦ k (f) = o (37)
d T v
- f (u-u ) + JZ (r 1 ) = 0
g dz p
subject to the boundary conditions at the surface
u(z o ) = 0; v(z o ) = 0; t x (z q ) = pu^ 2 ; X y (z Q ) = 0 (38)
and to the condition that at height h the stress vanishes.
The presence of the variable h requires the definition of three
independent dimensionless products, and we choose for these the following
As in the earlier theories, we assume that the flow is independent of z q
except in the immediate vicinity of the surface. Accordingly, the
equations of motion suggest
u - u
(— — *) = a (z, z >
u *0 h
( V !S) - * < 2 ' v
while in the surface layer, the surface boundary condition requires
- t (z/z ) . £ (2R)
and, because of the chosen direction of the x-axis
( 42 )
Since the geos trophic wind is considered to be independent of height, we
We now require that as R approaches infinity, the solutions for the
two layers match each other in a layer the height of which is small compared
to u^/f and h. Accordingly, we have, for such a layer.
f 1 (ZR) = u g (R,Z h ) + u(Z,Z h )
We may now proceed to differentiate this equation successively with
respect to each of the three arguments Z^, Z and R.
9z h 3z h
Rf^CZR) = |f (Z» z h ) = f 2 (Z)
Zf x * (ZR) = gjj 6 - (R,Z h ) = f 3 (R)
where the prime denotes total differentiation with respect to the
argument. That f is a function of Z only follows from the fact that the
left side of (46) is independent of Z h while the right side is independent
of R; in a similar way, f ^ must be a function only of R. By similar reasoning,
one can obtain from (46) and (47) together:
( 48 )
Zf 2 (Z) = Rf 3 (R) = i
where k is an undetermined constant that can be identified with
the von Karman constant. With these substitutions, there result the
- i [in R - A (Z h )]
( 49 )
By entirely analogous reasoning, one obtains
( 50 )
This reasoning shows that the Kazanski-Monin relations are quite
generally valid provided the constants A and B are regarded as functions of
hf/u^. These functions are not known at the present time. It is entirely
possible that some of the scatter in the diagrams for A and B that Clarke has
published is attributable to variations in h, which is generally not
observed. The scatter is most serious in the determined values of B.
The occurrence of surface heating of the PBL introduces still one more
variable, which can be selected to be the Monin- length
where is the surface heat flux. For purposes of nondimensionalizing it,
it is immaterial whether we adopt ku^ o /fL Q , as is generally done, or h/L Q , as
Dear dor ff (1972) has advocated, for either one can be derived from the
other with the use of Z^. To be consistent with general practice, we choose
0 - k “.o /fL o <52)
which may be regarded as a kind of bulk planetary Richardson number.
The geostrophic defects now become functions of three independent
u - u
A = U ( Z , z , a)
V - V
= v (z, z h , a)
= u (R, z h , a)
- 2 - = V (R, Z a)
U *o h
From this point the reasoning proceeds in an entirely analogous way to
the preceding discussion. The result is that the Kazanski-Monin constants
must be regarded as functions of both kh/u. and a to be determined from
The behavior of A and B accompanying variations of a have been studied
by Clarke (1970). The observed values of A do not scatter appreciably, nor
is there any significant variation from the mean value of about 5 for all
values of a in the range of -100 to -1000. Thus, under most typical
unstable conditions, A may be regarded as being well known, even though the
values of its arguments may be uncertain. It must be concluded that whatever
the values of h may have been in Clarke’s data, they had very little effect on
the value of A. Since u^ q is determined primarily by A and is insensitive to
B, it may be inferred that the presence of inversions that limit the height
of the PBL do not normally have to be taken into account in the determination
of u. .
Under stable conditions the scatter in A is much greater. Much of
this scatter can be attributed to larger uncertainties in the determination
of u. in these cases, as well as to the effects of accelerations and other
disturbances. The possibility exists that the scatter might be reduced by
taking h into account. More study of this problem is desirable.
^Calculations based on a two-layer model bounded above by an inversion show
that the inversion height has only a negligible effect on A and B as long
as it is situated above a height of .15 u* Q /f.
The observed scatter in B (a) is enormous, and existing observations give
little useful indication of the true form of this function- It is tempting
to ascribe this scatter to the failure to stratify the data according to h.
Since, however, B is mostly affected by the surface geos trophic drift angle
a, is is most likely that the major portion of the scatter reflects the
difficulty of measuring it and the sensitivity of this angle to local
disturbances. Fortunately, u^ is insensitive to the value of B (a, Z^) and
so, the uncertainty of B is of no great concern for the prediction of u^ .
5 . 2 Practical Methods for Determining 11 ^ and L from Large-Scale Variables
As we have seen, the surface drag coefficient c^ = u*/V^, the surface
Rossby number and the stability parameter 0 are connected by:
k 2 2 1/2
ln(Ro) = A(Cf) - In c d + - B (cr) ]
Here A (a) and B(a) are universal functions recently measured by Clarke.
Any effect of finite inversion height will be neglected. Here, Cf is given by:
Empirically, A(a) and B(c) are well generated by
A (a) = 4.5 for a £ - 50
A(a) = - 14.4*10 -4 a 2 - 14.4*10 -2 a + 0.9 for a > - 50
B(a) = 1.0 for a £ - 75
B(a) = 6.2*10 _4 a 2 + 9. 3*10 -2 a + 4.5 for a > - 75
Since the geostrophic wind speed and the surface Rossby number can be assumed
known, (55) will permit the estimation of the surface stress, provided O is
given. In practice, it is a nuisance to solve (55) numerically for c^.
Therefore, Figure 23 gives isopleths of a as function of c^ and' Ro which allows
a graphical determination of c^, given a, , and Ro. Also, once a is
known, L can be found from the definition of a by (56). Before we can use
Figure 23, we must first design methods for determining O .
As a first approximation, we may assume that a depends only on V and
Figure 24 shows how a is related to these two variables. This figure
is based on measurements at O'Neill (see Lettau and Davisdon, 1957), and Cape
Kennedy. Of course, the graph can be used only during day time and between
latitudes 25° and 45°. On windy nights, a is almost zero and turbulent
processes during lightwind nights play no important role and can be neglected.
Improvements may be possible, when measurements of the long-wave
radiation of the earth’s surface, for example from satellites, are available.
Then, an estimate of the radiation balance during day and night-time hours
can be found.
Where measurements of the incoming radiation are not available, it can
be computed to a certain approximation (dependent on the variability of
cloudiness) from known formulas from the date, the time of the day, the
latitude and the cloudiness.
Some computations have shown that a quasi-empirical formula by
Albrecht (see Moller, 1957) gives better agreement with measurements than
the theoretical formula
( 57 )
1=1 cos 5 a
with I = solar constant
£ = solar zenith angle
a = transmissivity.
Apparently, the difficulties lie in calculating the absorption of
radiation. This absorption changes with the relative air mass and is
dependent on wavelength. It is therefore a complicated function of the
time in the day and location.
The mentioned formula by Albrecht reads
I = (I - I ) cos X. (1 - C M) (1 + 1.19 A c (58)
O W v 1UUU
1^ = part which is absorbed by water vapor
M = absolute air mass (secant of zenith angle)
p = pressure in mb
A = albedo of the earth surface
c = backradiation constant (0.19 at Cape Kennedy)
For Cape Kennedy the simplification
i = 1585 cos c (1 - o.i : 9 M)
gives values which for the present purpose are rather accurate. The very
weak function of pressure (l + 1.19*c
has been contracted with
(i - I ) into one constant.
For March 3, 1968, for example, the hourly mean values in Table 3
have been read off the radiation registrations from Cape Kennedy and have
been compared with computed values from equation (59).
Table 5. Hourly Mean Values of Radiation at Cape Kennedy
m cal /cm min
m cal/cm^ min
So far, this equation by Albrecht gives reliable values for the incoming
radiation only when no clouds are present. If clouds are present, a correction
factor has to be introduced.
Another possibility involving radiation measurements from satellites
exists through use of the temperature 0^ of the earth’s surface. Together
with the temperature 0 at the top of the boundary layer, the temperature
difference A0 = 0 - 0 can be used, instead of insolation, as an external
Equation (56) then can be written in the form
a = JL A
9 fV g c D A0
Since both c^ and T^/A0 (T^ = - — — - — — ; scaling temperature) are functions
P c p u *
of R q and a, a can be expressed as
a = f (S, R q ) (61)
Here S = g- — — is a stability parameter given by external variables. By use
of empirical data for the dependency of c^ and T^/AO upon stability, the
relationship given in equation (61) has been computed. The result is given
in Figure 25. At high Rossby numbers and not too large values of S, O is
nearly independent upon R q and varies only with S. Thus this second method
also gives some justification for the assumption made for the construction
of Figure 24. From the theoretical standpoint, the second method has some
shortcomings. The known relationship for AO/T^ is most likely not correct
for Rossby numbers smaller than 10 . The fact that this ratio changes sign
for smaller Rossby numbers is the reason for intersecting O lines at low S
and R values. Intuitively, a theoretical relationship A0/T. should approach
asymptotically the lines S = 0 at small R q numbers for all O values. The
derivation of such a relationship must remain the task for future work.
Figure 25. Stability parameter S as function of a and surface Rossby number
rol evil —I I —I oJlrol^rlioi
VI. DETERMINATION OF PREDICTANDS FROM LARGE-SCALE VARIABLES
6 . 1 Method
The purpose of this chapter is to test the computational scheme in which
some of the statistics observed at the Kennedy tower are determined from
"external" bulk variables. Equation (55) of Chapter V which gives the
geostrophic drag coefficient c = u./V as a function of Rossby number
u o g
Ro, and the planetary Richardson number 0, was solved by one of the methods
suggested by Fielder (Chapter V) .
First, Figure 24 is used to determine a from I and V ; then, c^ is
determined from Figure 23. Given c^, u. is known; L can then be found
D *0 O
from u^ q and a, given the definition of a.
Values of geopotential height on the 850 mb surface prepared by the
National Meteorological Center were obtained from the Nationl Center for
Atmospheric Research archives in Boulder, Colorado. These 850 mb heights
were available every 12 hours at 00 GMT and 12 GMT, on the regular NMC
Northern Hemisphere grid which has a grid interval of 381 km at 60°N
Smoothed estimates of the geostrophic wind were obtained by
V 2 +
in which Ax is the grid interval at Florida (301 km) . z^ is the averaged
value of the height at the two grid points north of Florida and z^, v
Z E and averaged values of height at the two grid points south, west and
east of Florida. These geos trophic wind values were linearly interpolated
to the tower site.
Insolation measurements, required for the determination of a, were
available at Cape Kennedy. However, inspection of these suggested that
many of these were unrealistically low. Therefore, insolation was
computed by the method discussed in the last chapter. The insolation was
corrected for cloudiness by multiplying it by (1 - .03 C) where C is the
cloudiness in tenths.
Figure 26 compares computed and observed insolations. Clearly, in
many cases the observed values are unrealistic. This is further brought out
by the fact that a-values, estimated from these measurements, are often of the
wrong sign. Therefore, in this test, a was determined from Figure 24 as
functions of computed insolation and measured geostrophic winds. Then,
friction velocities were determined from Figure 23.
6. 2 Test Results
The mean value of the friction velocities observed locally was 0.76
m/sec, about 77% larger than that calculated from V , 0.43 m/sec. The
scattergram depicted in Figure 27 illustrates the relationship between the
two sets of u* q . Some positive correlation is evident. The large amount of
scatter is probably due to the uncertainty in the geostrophic wind calcula-
tions. The systematic difference could easily be due to the fact that an
average geostrophic wind in a (300 km) area is systematically smaller than the
local geos trophic wind over Cape Kennedy, particularly, if the average is
small. Note that the discrepancy disappears for large speeds, where one
might expect smaller systematic differences between the two kinds of wind.
Another difficulty arises from the uncertainty in the roughness lengths
already discussed in earlier chapters. Both sets of u^ q compared in
Figure 26 are based on the same Z Q f s, namely those derived in Chapter II. As
we have seen, these roughness lengths are too large to account for the
statistical fluctuations of the winds at Cape Kennedy.
Now it turns out that the u* *s estimated from V lg are much more sensitive
to the assumed z^'s than those based on geostrophic winds. Therefore the
question was raised what values of z^ would eliminate the systematic
differences between the two sets of u^. These turned out to be two orders
of magnitude less than the observed z^s, and therefore are quite unrealistic.
It is concluded that it is impossible to ascribe the systematic differences
between abscissa and ordinate in Figure 27 to incorrect roughness lengths;
as suggested above, the explanation can be found more probably in the
significantly underestimation of local geostrophic winds.
We shall assume that the u. f s obtained from the winds at 18 m are
correct. Hence, the best estimate of u^ q , given geostrophic winds, would
be based on the line of regression fitting Figure 27.
0.51 + 0.62 u* q (V , a)
These estimates should then form the basis for obtaining wind
profiles eq. (15) and e (eq. (28)) and standard deviations. Test of this
suggestion should be made at other locations since there is no guarantee
that eq. (63) is generally valid.
Figure 28 is a plot of u. (V , cr) calculated from V as abscissa and the
*o g g
standard deviation of the longitudional fluctuation CJ u , as ordinate. The
average value of C^/u^ (V^, a) which agrees quite well with observations made
at other tower sites (see Lumley and Panofsky, 1964). Again, most of the
scatter is probably due to the uncertainty in the geostrophic wind observa-
tions. The large scatter in Figure 28 suggests that the best agreement with
observed standard deviations at Kennedy is obtained by fitting a line of
regression to Figure 27 :
O = 0.80 + 0.77 u. (V , a) (64)
u *o g
Similarly, for the lateral component, at 18 m
O = 0.92 + 0.37 u. (V , a) (65)
v *o g
But, as we have suggested, there are strong indications that the observed
quantities are too small. It is proposed that a better procedure for
estimating O ^ and 0 ^ from geostrophic winds, roughness lengths and
stability information is to use an equation like (63) to estimate u^ q and
then multiply by 2.5 and 2.2, respectively, to obtain cr^ and a This
technique will be tested on observations to be made at Ris^, Denmark, on
the 125 m tower.
Figure 28. as function of u# derived from geostrophic wind.
Still, all estimates based on geostrophic winds are certain to have
large random errors even if the systematic errors can be eliminated, for the
correlation coefficient between observed standard deviations and u* o computed
from geostrophic wind is only about 0.44 for the u-component and 0.24 for the
v-component . As we have seen in Chapter IV, correlation coefficients are
about twice as large between standard deviations and u^'s obtained from
local wind. Hence, standard deviations should be estimated from local
rather than geostrophic wind if at all possible.
\ VII. SUMMARY OF PRACTICAL METHODS FOR ESTIMATION
OF LOW-LEVEL WIND STATISTICS
7*1 Roughness Length , Friction Velocity and When Large-Scale
Information Only Is Available
) . — ^
a. Daytime, When there is only large-scale information, local
statistics can be estimated from the three basic quantitites: geostrophic
wind speed, insolation and roughness length.
The roughness lengths need not be extremely accurate and can be
estimated from the local terrain according to the following table:
Table 6. Roughness lengths for various terrain types, in cm.
Ocean or ice 0.01
Smooth grass 1
Rough grass 5
Farm land, smooth 10
Farm land, rough 50
Forest, cities 200
Insolation with clear sky can be determined from the usual astronomical
formulae, or, if Cape Kennedy is typical, from:
I = 1.585 cos e (1 - 0.19 /seel;)
Here, I is the insolation in ly/min and £ is the zenith distance of the sun,
which can be computed from hour angle t, declination of the sun 6 and
latitude cf> by
cos ^ = cos t cos 6 cos <j) + sin 6 sin (J)
In cloudy skies, this estimate has to be multiplied by (1 - .01 aC)
where C is the cloudiness in tenths and a depends on the type of cloud.
We have used a = 3, though the value should be considerably larger (perhaps
7) with low clouds.
Given I and the geostrophic wind, V , the planetary Richardson
number cf is obtained from Figure 24. Hence, the surface stress is found
from Figure 23, given cr and the geostophic wind. If the Kennedy results
are representative, a better estimate of friction velocity is finally found
u. = 0.51 + 0.62 u* (V , cr)
*o *o g
where u * 0 ( V g> a) is the friction velocity obtained from O and V^.
Next the Pasquill class is determined from Table 2. For this table,
we need a rough estimate of wind speed. This is given sufficiently
accurately by the logarithmic wind law applied at 10 m:
Radiation is taken to be weak if less than 0.5 ly/min, medium if between
0.5 and 1.0 ly/min, and strong if greater than 1.0 ly/min.
Finally, Figure 9 gives an estimate of L q , from Pasquill class and
roughness length. We now have u* , z , and L and can proceed to Section 7.3
«o o o
of this chapter.
b. Night. At night, large-scale variables are not likely to yield
good estimate of L q and friction velocity, unless winds are too strong
and stratification near neutral. The following procedure is recommended,
but has not been tested. First, obtain roughness length as in (a). Then,
determine a first approximation of surface friction velocity from Figure 23,
assuming O - 0. This approximation will be good in strong winds, but too
large in weak-wind cases with strong inversions. In any case, use the
logarithmic law applied at 10 m to obtain a rough wind speed. Use this
to determine Pasquill class from Table 2. Figure 9 gives the first estimate
of L^, as function of roughness length and Pasquill class.
Given L q and u^, we can now estimate the planetary Rossby number from
With this and geostrophic wind, we enter Figure 23 and obtain a second
estimate of surface friction velocity. This procedure can then be iterated
until it converges. It is expected that it will be fairly reliable under
near-neutral conditions, but not in very stable air. But in the latter
case, the winds are likely to be too weak to be of practical consequence.
Again, a better estimate of u. can be found from
u. = 0*51 + 0.62 u . (V , a)
*0 *o g
if the Kennedy results are typical. We then proceed to Section 7.3.
7.2 Friction Velocity and from Low-Level Data
It is expected that better estimates of surface friction velocity and
L q can be obtained when a low-level wind is measured (e.g., given from hourly
sequences) than if geostrophic winds have to be used. First, roughness
length is estimated as before. Next, the Pasquill class is found from
Table 2. Pasquill class and roughness length yield L q according to Figure 9.
The surface friction velocity can now be found from the equation for
the wind profile:
= 0.4 u
u *o In z/z - \p (z/L )
where ip is tabulated in Table 1. It is recommended that the height z be
taken as 10 m, the height recommended for synoptic wind observations. This
method has not been tested.
7.3 Estimation of Various Statistics from z , L and u*_
— o — o *o
a. Wind Profile. The wind profile up to 150 m or so is well
u = 2.5 u* o [In (j~) - i> (f~)]
where is given in Table 1*
b. Variance. At this point, it is recommended that, at all levels,
up to 150 m
= 2.2 u
These equations are not so much based on Kennedy results, but on average
results at various locations. Correction for height or stability appears
premature, since no generally valid behavior has been documented.
c. Spectra at high frequencies. At high frequencies (n > u/z) , the
inertial-subrange formulae appear valid :
S(n) = c e
2/3 2/3 -5/3
Here, e is the dissipation of turbulence into heat and the constant, c,
is .14 for the u-component and .18 for the v-component, n is the frequency.
(Q, quadrature spectrum; C, cospectrum) is about 1 for longitudional and
2 for lateral velocity components below 100 m and half as large between 100 m
and 150 m.
Tests of all these procedures at locations away from Cape Kennedy are
urgently needed to test the generality of these methods.
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