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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 


JUNE 1974 








TECHNICAL REPORT STANDARD TITLE PAGE 


1. REPORT NO. 

flASA CR-9A^ 


2. GOVERNMENT ACCESSION NO. 


3. RECIPIENT'S CATALOG NO. 


4 . 


TITLE AND SUBTITLE 

Investigation of the Turbulent Wind Field Below 500 Feet 
Altitude at the Eastern Test Range, Florida 


5. REPORT DATE 

June 197*+ 


6. PERFORMING ORGANIZATION COOE 

M131 


7. AUTHOR(S) 

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. 

NAS8-21140 


12. SPONSORING AGENCY NAME ANO ADDRESS 

George C. Marshall Space Flight Center 
Marshall Space Flight Center, Alabama 35812 


13. TYPE OF REPORT & PERIOD COVERED 

Contractor Final 
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 

Atmospheric Turbulence 
Wind Shear 
Aircraft Response 
Aircraft Design and Operation 

18. DISTRIBUTION 

STATEMENT 


19. SECURITY CLASSIF. (of thU r. peril 

UNCLASSIFIED 

20. SECURITY CLASSIF. (of thi. p.g.) 
UNCLASSIFIED 

21. NO. OF PAGES 

92 

22. PRICE 

$4.00 


MSFC - Form 2292 (Rev December 1 f 7 1 ) 


For sale by National Technical Information Service* Springfield* Virginia 22151 















TABLE OF CONTENTS 


Page 


I. INTRODUCTION 1 

II. DETERMINATION OF ROUGHNESS LENGTH AT CAPE KENNEDY 4 

III. DETERMINATION OF L AND » FROM LOW-LEVEL VARIABLES ... 21 

o *o 

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 

*o o 


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 

*o 


REFERENCES 


85 


LIST OF FIGURES 


No. 


1 


2a 


2b 


2c 


2d 


2e 


3 

4 

5 

6a 


6b 

7 

8 

9 

10 

11 

12 

13 


Theoretical wind profiles for 20 cm roughness length 

and u. = 0.52 m sec - '*' 

*o 

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 
directions 15° 

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) 

o 

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 
insolation 

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 


Page 

7 

9 

9 

10 

10 

11 

13 

16 

18 

24 

25 

26 

27 

28 

29 

40 

42 


43 



LIST OF FIGURES (Continued) 


No. 


Page 

14 

Observed ratios O /u.vs Ri 
u * 

44 

15 

Observed ratios a /u. vs Ri 
v * 

44 

16 

a at 18 m minus a at 150 m as function of wind speed 
a¥ 18 m U 

46 

17 

a at 18 m minus O at 150 m as function of wind speed 
a¥ 18 m V 

46 

18 

"Observed" /e vs /e estimated from surface 
stress 

50 

19 

Decay constant for coherence of u as function of Ri 
at 23 m 

53 

20 

Decay constant for coherence of v, as fucntion of Ri 
at 23 m 

53 

21 

"Slope" of u vs Richardson number 

55 

22 

"Slope" of v vs Richardson Number 

55 

23 

Geos trophic drag coefficient as function of a and 
surface Rossby number 

67 

24 

Stability parameter 0 as function of insolation and 
geostrophic wind 

67 

25 

Stability parameter S as function of a and surface 
Rossby number 

71 

26 

Comparison of observed and computed insolation, 
in 1000 ly min 

74 

27 

Comparison of u^ derived from geostrophic wind vs 
u* from local data 

74 

28 

a as function of u. derived from geostrophic wind 
u * 

77 


v 



LIST OF TABLES 


No . Page 

1 Wind profile parameters <j>, ip, and e ^ as 

functions of z/L 17 

o 

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 


FOREWORD 


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. 



I. INTRODUCTION 


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: 


L 


3 

u . c p T 

-JL E 

kgH 


( 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- 
where. 

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 


2 



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. 


3 



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. 


4 



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: 


2u 



* dz 



(3) 


Here, v is the component of geos trophic wind at right angles to the surface 
S 

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 


v /u. 
g *o 


B 

k 


(4) 


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. 

*o 

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) 

u *o 

Here F is presumably a universal function. For example, Blackadar (1962) 
has suggested: 


f Z “1 

F = (1 + 63 — ) 
u. 

*o 


(7) 


5 



If we now substitute equations (5) and (6) into (2), and integrate, we 


may write formally: 



+ G 


<— )i 

U *o 


( 8 ) 


f z 

where G( ) is another universal function, related to F. In particular, 

U *o 

with Blackadar's hypothesis for F, the expression for the wind profile can 
be written 


* 

u = : — In — 1 144 fz (9) 

k z 

o 

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. 


6 




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. 


7 



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 


X = 


0.0063 



tanh 


kzf 

0.0063 u* 
*o 


( 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, 

2 

omitting a term in (fz) 


u. sinh 63 fz/u. 

*o *o _ 

k n sinh 63 fz /u. 

o *o 


(ID 


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 


8 









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. 


12 



HEIGHT, 




The test was made on the basis of the nondimens ional vertical wind 


shear. 


, kz du 


( 12 ) 


In the surface layer, and, according tu the hypothesis, up to 100 m, in 
unstable air 


* 


(1 - 16 z/L )“ 1/4 
o 


(13) 


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, 
n 


<L/<L = <L = 


u . Au 
*no u 


r u' T n 


u u. 


Au n 


(14) 


Subscripts u and n denote unstable and neutral, respectively. The ratio 

u, /u. was evaluated from equation (15) (see below) applied at 18 m: 
*no *uo 

i , 18 v 

u Kr“) 

u . n- 0 L 

*no 18 n o v 

\I “ 1 O / 


u. u 

*uo u 


18 


i 18 

In — 
z 

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. 


14 



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) 

k o 

Paulson (1970) determined \p for unstable air from (13): 

* (f-) = m r ( I=|) - 2 t-" 1 f-> + I < 16 > 

o o o 

1/4 

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 



o 


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 

o 


15 





Table 1. 


Wind profile parameters <J>, iJj, 


and e 




as functions of z/L 

o 


z/L 0.1 0.05 0 -.1 -.2 -.3 -.4 -.5 -.6 -.7 -.8 -.9 -1 -2 -3 

o 


\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. 


17 



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 
the terrain. 

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 
tentative. 

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 


19 


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 

-2 

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 
refinement . 


20 



III. DETERMINATION OF L AND u. FROM LOW-LEVEL VARIABLES 

o *o 


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) 

o 

In stable air, the Monin-Obukhov hypothesis is more controversial, but a 
good relation (for Ri « .20) seems to be 


21 


(19) 


L = z 
o 



1 ] 


1 


I 


(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: 


B 




( 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 
o 


(.22) 


22 



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 
is excellent. 

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 

o 

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. 


23 



1 

0 

0.02 

0.04 0.06 0.08 

0.10 0.12 



Ri 



Figure 6a. 

Nomogram for Richardson number as function of 
bulk Richardson number B and z/z q (stable). 



24 










i OBSERVED 






0.2 



quill class as function of L and 




U|8m m/sec 


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 


Surface Wind 
speed, m/sec 

Daytime Insolation 
Strong Moderate Slight 

Nighttime 

Thin overcast 
or ■> 4/8 
Cloudiness** 

conditions 
< 3/8 

Cloudiness 

<2 

A 

A-B 

B 



2 

A-B 

B 

C 

E 

F 

4 

B 

B-C 

C 

D 

E 

6 

C 

C-D 

D 

D 

D 

>6 

C 

D 

D 

D 

D 


^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. 


30 


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: 
o 


= k u 

U *o In z/z - ip (z/L ) 
o T o 


(23) 


As before, we adopt Paulson’s form for ^(z/L q ) in unstable air (16). In 
stable air, as before 


* <f-> - 

o 



(24) 


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. 


31 


Table 3. Friction velocities and other relevant statistics for each analyzed run at Cape Kennedy 



rH N(N O CO 00 H JN 00 LO CM O' H CO r-H 

cm o' O' co r^. o> 4 n m m m m 

• • • a a • a • a • a • •• • • 


ctn <t HinooHh.o>roNHcMCMoovooomo>c^r^rNMcoc^HvovDCMfn 

(N-<r oooo<N^cMinrNrsroinr^vovocMcnvDrN><f(NfOfnM'Ct<i-vONc r ) 


co <t vo oo 

O OOH H H 


I I + I I 


cnj ch vo o 
co co m cm o 


i i i I 1 


on cm oo in 
OHiNHO 


I I 1 1 I 


ON O co CO 00 
rH CO CM rH O 


1 ! I I + 


r-» oo rH <r 

in n o si - 



oo <t no 

CO rH CM CO 

• • • • 

III CM 

l 


COCMrH<fCOONOOOrHr^CM»nOONOCMONOinrHOONOinOvOOOCMCM^cM<r<r 

>jONHcoo<rincoocMOOH<fcocooHOcosrcooMsr L nsrsrcnH 

i^.vO'sD<^<j'<foor^inv£)ONONOcooocor^oor^oooovo<J‘OOco<rincoNocMr^ 

HHHHHHHHHHHNHHHHHHHHHHHHHHHHHHH 


<tcocM\ncocooNinoNNOcMtnooocMOO»nrHOvo>novDooNOCMONLn^n- 

OCOinCMCOininrHOOnCOCOCM'd-rHCOrHrHrHOiHOCOMrrHinCOCO-vfO 

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 


COvO<rOrHOOOrHmtnr^OOoOCOOOi-HHr^ONCMcO-<rrHtHONOrHvOOH 
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 


oooooooo 

Noooinomuoo 

^HCNCONOmrO 
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 


oooooooo 

ommincoomcM 

MDCMOnOCNiTl 
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 


HOOONONNOCMinoOOCMiOHiniOOOHsrNOiOCOHstrsJ^ 

OrHcMogooLnvor^r^r^r^oooooooNONONONOOpt-HfHrHcMCMrHcoco 

OOOOOOOOOOOOOOOOOOrHrHrHrHrHrHrHrHrHrHrH 


32 


I 


138 95° 1.8 11 July 


Table 3 (continued) 


(N CM 00 ON 

m n cm in 


vOOiOHM-M 

ncM m m m ^ 


oo in 

in CM i£) 


NjcorNinmcnoNONCMONO\cM(NHMn(M\ocnMrooHco(^cnNHcMOMN 

vtvocMNvo^nnncM vOcor^<tr^unr^cnmrnm<rcor^mro<rrocNiro 


cm oo h o vo in 
cn o H H vD tn N 


HtnHCM\OCMOtn 

OOvOCOOvDOMCMCO 


h i — oo oo m co oo 

(NCMONHiOO 


CM + + + 


I 1 I + I I I 


m i rH I i— i 
1 I I 


HMOcooo<nnoinmOiA^co(Ti^NincooMnoo\oiOsrooNaun 
srinNrcocoHvr<tH<fO'J-cooinocncoococoNr'T<rsrHNrcM n h 

oocMh-tNNtvoNoONh-^ocoNNrvDinr^h-vrinvDN^sj'Nvrc^n'fnH 

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 


OM-oco^sfina\ioinO‘A\oo'OOiDino\ONNrowo\Dino^oun 

COsl'COCMCMOvtOOHOHCOOCOvfcOCOOCOOHHHrlH'd’CMtOH 

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) 
HrMlNMCMCONNOOCO^OvMnin^OOHCMCOCOCOONOOMM'Tin 

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 


ooooooooooooooooooooooooooooo 

MDoooooommoocnoooooooinooooominoooo 
COvO<3-r-ICMOviCMCMO>vOr-0'COcnaMCOOO HmOCMCOtMO^HCOMO 
H i-H CM CM CM H H rH HrHHCMHHcOrlrlHHNcMHH 


^oHcMco<raNOfH<Mcn<finvoaNOcMcn<fini^ocMcn<rvoa>cMcn 

cn^<r<t<r<f<tLnininininininin\£)v£)vDv£)v£)v£>rvr > ^r*-.r>.r«*-r^cooo 

HriHHHHHHHHHrIHrlHHHHHHHHHHHHHHH 


33 


184 116° 29 Aug 



Table 3 (continued) 


vO 

r^ 


vo 

vo 


00 


CM CO 

o\ o 


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 
vOvOCOCOvOP^vOvOCO 


Pd 








rH 

o 

1— 1 

00 

m 

00 






r^ 

o> 

00 

vo 

O'* 

vo 

CM 

m 



CO rH 


vO 

O 

i-H 

o 

m 

i— i 

vO 

00 

<d- 

O 

VO <f 

CO 

O 

• 

• 

• 

• 

• 

• 

• 

• 

• 

• • 

• 

• 

o 

o 

o 

o 

o 



I 1 


1 1 


4 h 

MH 

0 

0 ) 

1 

Eh 


uouO"<fcrxvor^cr>OT— ir^voovocj>uooor^o<J-o<tvnavc\irHCMT— i^r^r^<tvo 
<TH>JH<rcOHH^inMNOHHOCOOM 0044 HOMHinCNCOHO 

iHcocO'd’COMfvorHiHrHooovvooop^oovOrH>^r-*vovomr^vor^ooovavr^invo 
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 


G 

O 

a) 

a 

Eh 


vovncoovor^avtHi— tiHvoiHvo<ti— ioovoo<ro<roa\r^T— icMCMavr^omvo 

HHHCM^HHHH<r(N^fOONOnO'JCOOOvtMOvr^mcMinHO 

iHcMcococM<tLOOiHrHi^.oomoovor^Lnocovomvo^d‘\omvor^oooovo<d-uo 

OOOHHHHMCMMHHHHHHHOHHHHHHHHHHHHHH 


cd 

Q 





n> 

H> 

4-» 

4-» 

H> 

H» 

n> 

U 

H> 

4J 

U 

4-J 

4-> 

















60 

60 

60 

eu 

cx 

a 

cx 

cx 

cx 

ex 

cx 

cx 

CX 

Q* 

PX 

CU 4J 

U 

n> 

4J 

HI 

HI 

HI 

G 

G 

G 

G 

G 

r0 




3 

3 

3 

0) 

a) 

a) 

a> 

(0 

a) 

0) 

<1) 

a) 

a) 

a) 

QJ 

a) 

a 

O 

a 

O 

U 

o 

a 

cd 

cd 

cd 

id 

cd 

<u 

a) 

a) 

0) 

< 

< 

< 

CO 

CO 

CO 

CO 

CO 

CO 

CO 

CO 

CO 

CO 

CO 

CO 

CO 

O 

o 

o 

o 

O 

o 

o 

*s 

*“> 

•“3 

►3 

•-3 

pH 

PH 

pH 

PH 

Ov 

O 

O 

00 

CM 

CM 

CO 

<r 


m 

00 

av 

rH 

i-H 

CM 

00 

CM 

o 

CM 

vo 

00 

ON 

C0 

CO 

<r 

vo 

vO 

O 

vO 


00 


CM 

CO 

CO 


H 

rH 

rH 

i— i 

i— i 

i-H 

t-H 

i— i 

CM 

CM 

CM 

CM 


1 — 1 

i— 1 

rH 

i— 1 

rH 

CM 

CM 

CM 

CM 

CM 

CO 




rH 


0 ) 

CO T 3 
cd <u 
U a> 
<D CX 
> CO 
<2 


< 1 - 

CM 


Ov VO 
CM CO 


CM O 
• • 
CM CO 


vOOOOOO^HOmO'J 


*3 

s 

& 


G 

O 

•H 

U 

O 

<U 

U 

•H 

Q 


OOOOOOOOOOOOOO OOOOOOOOOOOOOOOO 

ivoooooorHrH^oocMOcoooovoinoooiovo^favcomoNcooo 

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 


O 

S 3 

C 

3 


uor^ooovOcMcovor^oocjvcM^fmr^^rvomOi-Hmvor^ovcvimvDr^TOCPOvo 
OOOOOOCOO\OvCT\OVOvCP>OvOOOOi- 1 rHCMOOOOCOoOOOCTvOOOOOOTHfH 
HrHi— It— IrHrHr-li— I iHHtHCMCMCMCMCMCMCMCMCMCMCMCMCMCOCOcOCOCOCOCOCO 


34 


340 


Wind Average GMT 

Direction Speed Date T ime On Time Off 




r>* 


m m cm 
on m <r co 


NTCOcMOOOONCMOOrOOtNOO 

ON'd'ON'^HHNNpN^fOlO 


o 

X 


CO 

X 


co co 

m x 




moo\ocsiHo>o\o«jHH(N<r<tcNr)cMiricMvr 


oo vo in m 
x <* m m 


i — i on 
CO CM 


m 


cs vo m cn 
co cm o 
m cm co o 


H vO On CO O 
COMHmvD 
CM CO <f O O 


NOOO\NCON\OMvOrsOvOO>OMn H CM m VO 

CO(OOOOOCMO\r^M , 0 \OONfNOvO X* CO CO Mt 00 

ooxmxoMrco<rcooooocM oomo 


CM o 
I t 


o o 

I 


O O O X 

till 


o 

I 


o o o o o o o 

I 1 I I ! I 


CM O 

I I 


CO o o 

I 


o o o 


o o o o 


ocriONCM^ooo'OM-inHmnoo 

mMincMMOHin<rHcom^<fN 

oovon'COmoscovoooM-cna'OinN 

HrlrlHHHHHHHHHMrlH 


OMnioHinomoovromoonN^ 

incOincOM-M’M’MHmM-CslHvtCOH 

NOMnM-MnvOCMCOHOOCMM-CMOH 
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 

incMinmomHinoHcooMMin 

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 


ONr^inininCTNino^minooovoo 
momo^o^rooi— i<j-mrHoom 

^(^M'JvOinMtNNHNHCOCMOO 

HHHHHHHCM(NOOH(NOMO 


XI 

X 

X 

X 

X 

X 

U 

u 

u 

u 

u 

u 

Vi 

u 

u u 

U 

U 

U 

U 

U 

U 

a) 

0 ) 

CD 

<D 

<u 

a) 

eg 

cd 

cd 

cd 

cd 

cd 

cd 

cd 

cd cd 

cd 

cd 

cd 

cd 

cd 

cd 

PH 

Ph 

pH 

pH 

PH 

PH 

X 

X 

X 

X 

£ 

X 

X 

X 

X X 

X 

X 

X 

X 

X 

X 

CO 

X 

r- 

oo 

ON 

ON 

rH 

X 

r-. 

oo 

CM 

CM 

co 

X 

m o 

CM 

CM 

r- 

00 

co 

ON 

CM 

CM 

CM 

CM 

CM 

CM 





X 

X 

X 

X 

X CM 

CM 

CM 

CM 

CM 

CM 

CM 


ViMViViViViViViVi 


vooor^cor^OcMincocMCMa\corH<taNOr^rHOrHcocovon-r^oo<rv£>cM>d‘ 

mM , r s ‘M , HO\NMtoicooi > »i s *inNcoc\C30voiniDNcovOM , inMM‘\ovo<r 


o o 

m cm 

O CM 
CM CO 


ooooooooo 
r^XOXCOcOXCMX 
^rrHr^t-HCMOvC-OrH 
H CM CM CO CO CM CM CM 


ooooooo ooo 
OMCONOHOOmON 
aNr'»r--coxxr^aNaNO 
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 


o 

55 


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" 


35 


411 15° 7.6 12 Apr 



Wind Average GMT 

Direction Speed Date Time On Time Off 


*° 

O 


n o> <r n 

•o- m m cm o\ 


mootnH^HrsvfHoon^tnvonNt 
o mvocococMcocMco^j-vomcoinvo^ovo 

N 


O in o\ \£) 

O H in v£> CM 

o o co o co 


O O tH O o 
1 I 


von-tHmooovoovoin<r<M(M<fro>^ 

HninnMOOinHsTNUKNinoro 

m h in m o\ n noN^^o>(Nvon 
CM O tH O rH tH rH CMOOOOrHOrHtH 


r^cMrHvoooavinominincoinoo 

NrHinfn(N4tnoo^Nininin<rH 


rH 

rH 


i — 1 

00 

CM 

m 

CO 

O 

vO 

CO 

CM 

00 

rH 

m 

CO 

CM 

O 

i — 1 

o 

rH 

i— 1 

tH 

CM 

O 

O 

o 

o 

i — ! 

O 

rH 

rH 


rH 

>> 4 -> 
rH CL, 

4 -> 

Cl. 4 -> 


4-1 

4 -> 

4-1 

4-1 

4 -» 

> 

> 

U 

u 

u 

D 

D <U 

CU O 

O 

u 

a 

o 

o 

a 

o 

O 

a) 

Q) 

Q) 

*-5 

•-> to 

CO o 

o 

o 

o 

o 

o 

o 

S3 

S3 

Q 

Q 

Q 

ov 

O O 

n- in 

vO 

<fr 

m 

00 

CTv 

o 

CM 

CT\ 

CO 

CO 

<r 


rH CM 

CM 


rH 

rH 

rH 

tH 

CO 







CO 


rH 

CO 


CM 

CM 

Ov 

o 

rH 

CO 

<r 

rH 

O rH 

o 




CM 

oo 

in 

00 

1^. 

CO 

o 


CM 


3. 

5. 

r^. 


oooooo ooooooooo 

n n in n ^ o cm o vo h co o m c-- tH 

CMvDO'DmNNOn^DHCOin^HO 
CM H H n H H H fO H H N CO 


o 

S3 

C Lnr^O'<fooc^Lnr^. r — imorHodr^oocTN 

a vO\OHHHHCMNnfO^*<r^<f<t^ 

iTiiT|\DvOvDv0^)\OvO\OvO'X)'X>vC>\O'i) 


37 



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 
In ) 


(24) 


o 

The roughness length is given, at Kennedy, from Figure 5. At other 
locations, the roughness lengths are assumed known, at least approximately. 


38 



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 
ground. 

Engineers often estimate high-level winds from power laws, which fit 
profiles reasonably well: 



(25) 


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. 
o o 

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 . 


39 





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 
well estimated. 

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 


41 








o^/u 



Figure 15 • Observed ratios vs 


44 





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 

u v 

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, 
o o 

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 

u 

variance decreased most rapidly in strongest winds; for example, there is no 

2 —2 

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 . 

v 

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; 


45 






Table 4. Vertical variation of average standard deviation 
of u and v for different stability groups 


Height, m 

18 

30 

60 

90 

120 

150 




Ri > - 

-.1 



a 

u 

1.00 

1.01 

.95 

.83 

o 

00 

• 

.77 

a 

V 

1.00 

1.04 

1.00 

1.02 

1.03 

1.05 




o 

Csj 

• 

0 

1 

<_ Ri <_ — . 1 



a 

u 

1.00 

1.01 

.92 

00 

• 

.81 

00 

• 

a 

V 

1.00 

1.01 

.97 

• 

00 

.86 

vO 

00 

• 




Ri < - 

-2.0 



a 

u 

1.00 

.98 

.94 

.90 

00 

.86 

a 

V 

1.00 

1.04 

.94 

.90 

• 

00 

.83 


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 
Report 1. 

The equation for the spectra in the inertial subrange is : 


S(k) = ae 2/3 k _5/3 


(27) 


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 
function, (f)^: 


e 


*° a 


(28) 


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 
equilibrium) 


48 



( 29 ) 


d> = d> - z/L 
e o 

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 


4 > 


- * 


(30) 


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 . 

o 

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 
(29)) 


30 


*o . ,30, 30, 

12 [<j > ( L _) “ L~ 
o o 


(31) 


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 


49 


Ve(FROM SPECTRA) 



Figure 18 . 


"Observed" 'i'fz- vs estimated from surface 

stress, c.g.s. units. 


50 



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) 

*0 

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 
Chapter III. 

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 

O 


z 

L 


3Q m 


30 


o 


-1/4 


30 


51 


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 

and 


S = 


u 

27mAz 


arc tan 


Q(n) 
Cos (n) 


(33) 


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. 


52 








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. 


54 



55 





V. RELATIONS BETWEEN LARGE SCALE PARAMETERS AND , L 

** o 


5.1 Theory 

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). 


56 


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 

*o 

sh.own from Buckingham’s theorem that all other dimensionless characteristics 

of the neutral PBL are functions of these two dimensionless ratios. 

4 6 

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 


57 



the outer layer only if the Kazanski-Monin relationships are satisfied: 


u 

-S_ 


u. 

*o 




f z 

o 


A] 


(35) 


and 



4 

(36) 


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 
known. 

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 


58 



practice, be concerned with such matters as non-steady states, horizontal 
temperature gradients (baroclinicity) , and diabatic vertical temperature 
gradients . 

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 
*o 

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 


59 



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 
set 


Z 




and Z, 
h 



(39) 


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 


(40) 


while in the surface layer, the surface boundary condition requires 


- t (z/z ) . £ (2R) 

*0 

and, because of the chosen direction of the x-axis 



(41) 


( 42 ) 


60 



Since the geos trophic wind is considered to be independent of height, we 
have 


u 




A 


V 

g 


«-y 


(43) 


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. 


A A 

f 1 (ZR) = u g (R,Z h ) + u(Z,Z h ) 


(44) 


We may now proceed to differentiate this equation successively with 
respect to each of the three arguments Z^, Z and R. 


9u 


*u 


9z h 3z h 


= 0 


(45) 


Rf^CZR) = |f (Z» z h ) = f 2 (Z) 


(46) 


3u 


Zf x * (ZR) = gjj 6 - (R,Z h ) = f 3 (R) 


(47) 


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: 


61 


( 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 
solutions 


- i [in R - A (Z h )] 


( 49 ) 


and 



z/z 

o 


By entirely analogous reasoning, one obtains 



B( y 

k 


( 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. 


62 


The occurrence of surface heating of the PBL introduces still one more 


variable, which can be selected to be the Monin- length 


L 

o 


c 

P 


pT u 

kTir 


*o 


3 


(51) 


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 
variables 


u - u 


u 


*o 


A = U ( Z , z , a) 


V - V 


U J 


= v (z, z h , a) 


(53) 


and 


63 



(54) 


*o 


= u (R, z h , a) 


v 

- 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 

*o 

empirical data. 

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. . 

*o 

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 
*o 

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. 


64 



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 

o 

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) ] 


(55) 


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: 


a 



(56) 


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 


65 


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 
o 

Figure 23, we must first design methods for determining O . 

As a first approximation, we may assume that a depends only on V and 

O 

insolation, I. 

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 


66 




( 57 ) 


1=1 cos 5 a 
o 


with I = solar constant 
o 

£ = 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. 
0 w 


68 


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 


Time 

LST 

Measured Radiation 
m cal /cm min 

Calculated Radiation 
m cal/cm^ min 

ratio l_c°!5E 
I meas 

1200 

1080 

1100 

1.02 

1300 

1090 

1068 

0.98 

1400 

1020 

1012 

0.99 

1500 

850 

852 

1.00 

1600 

620 

626 

1.01 

1700 

360 

358 

0.99 

1800 

95 

98 

1.03 

1900 

0 

0 



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. 


69 



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 

8 

difference A0 = 0 - 0 can be used, instead of insolation, as an external 

go’ 

parameter . 

Equation (56) then can be written in the form 


a = JL A 

9 fV g c D A0 


(60) 


H 


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) 

Aq 

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 


70 



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 
o * 


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 

S 

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 
latitude. 

Smoothed estimates of the geostrophic wind were obtained by 


V 

8 


fAx 




V 2 + 




(62) 


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 


72 


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 

S 

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 


73 





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 

*o 

correct. Hence, the best estimate of u^ q , given geostrophic winds, would 
be based on the line of regression fitting Figure 27. 


*o est 


0.51 + 0.62 u* q (V , a) 


063) 


These estimates should then form the basis for obtaining wind 
profiles eq. (15) and e (eq. (28)) and standard deviations. Test of this 


75 


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. 


76 




Figure 28. as function of u# derived from geostrophic wind. 


77 



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. 


8 


I 

f 

f 

\ VII. SUMMARY OF PRACTICAL METHODS FOR ESTIMATION 

OF LOW-LEVEL WIND STATISTICS 

r 

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;) 


79 


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 

S 

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 
from 


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: 



10 m 
z 

o 


80 



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 
its definition: 


0.4 u 


a = 


*o 


f L 


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. 


81 


Again, a better estimate of u. can be found from 

*o 

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 ) 
o o 

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 
described by 


82 



u = 2.5 u* o [In (j~) - i> (f~)] 
o o 

where is given in Table 1* 

b. Variance. At this point, it is recommended that, at all levels, 
up to 150 m 


a 

u 


2.5 u. 
*o 


and 


= 2.2 u 


*o 


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 

u n 


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. 


83 



(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. 


84 


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86 


NASA-Langley, 1974