MONTHLY WEATHER REVIEW
VOLUME 97, NUMBER 9
SEPTEMBER 1969
UDC 551.511.2
ON THE’KINETICENERGY SPECTRUM NEAR THE GROUND
ABRAHAM H. OORT
Geophysical Fluid Dynamics Laboratory, ESSA, Princeton, N.J.
ALBION TAYLOR
Atlantic Oceanographic Laboratory (SAIL), ESSA, Silver Spring, Md.
ABSTRACT
For six stations in the northeastern United States, the spectrum of horizontal wind speed was analyzed using
10 yr of 1-min averaged, hourly surface reports. The fast Fourier transform technique was employed to estimate
the spectrum between 1 cycle/2 hr and 1 cycle/2 yr.
The kinetic energy spectra show two major spikes at periods of 24 hr and 1 yr. However, most of the energy is
contained in the traveling cyclones and anticyclones with periods between 2 and 7 days. The apparent discrepancy
between Van der Hoven’s results and our results concerning the existence of an important diurnal cycle in the kinetic
energy can be explained by Blackadar’s theory of the diurnal wind variation with height. Van der Hoven’s spectrum
represents conditions near the top of the surface layer, while our data were taken well within the surface layer. A
line-by-line investigation of the diurnal peak reveals a very sharp line a t 2400 hr with two side lobes 3.9 min away
from the main line. These side lobes are probably caused by an annual modulation of the diurnal cycle.
The spectra tentatively corrected for aliasing give some indication of the existence of a spectral gap between
small-scale turbulence and mesoscale phenomena.
1. INTRODUCTION
Van der Hoven (1957) made a detailed analysis of the
power spectrum of the horizontal wind speed in which he
analyzed measurements taken at Brookhaven National
Laboratory, Long Island, at a height of about 100 m.
By piecing together various sets of observations, he was
able t o present a composite picture of the contribution
to the total variance of the wind speed from different
frequency ranges. The kinetic energy spectrum thus
determined covered periods from 4 sec to about 2 mo.
He convincingly showed that most of the variance of the
wind speed can be explained by the passage of large,
synoptic scale pressure systems with periods of about 4
days. Turbulence of the order of minutes also gave some
contribution, although it was much smaller. However,
between these two regions he found a broad section of the
spectrum centered near the period of 1 hr,. with. very
little energy connected with it; this last: portion of’ the
spectrum was therefore called the “spectral gap” region.
His analysis further showed a small rise in the spectrum
f o r period,s3 of. abwt 12, hr, but surprisingly there was
not. much energy near the 24-hrperiod..
More recent. investigations by Bysova. et al. (1967)
using 40 hr of wind speed data from a 300-m tower at
Obninsk (U.S.S.R.) also show a pronounced minimum
ins the spectrum between the turbulent and mesometeor-
ological wind fluctuations.
Recently, 10 yr of hourly wind records have become
available on magnetic tape for several weather stations
in the United States. This made it feasible to repeat and
extend Van der Hoven’s analysis to include the spectrum
from periods of a few hours up to a few years. Another
contributing factor was the advance in data analysis
made possible by the introduction of the “fast Fourier
transforms” (FFT), recently developed by Cooley and
Tukey (1965) for calculating the Fourier components
directly from a long time series in very little computation
time. This practically eliminates the difficulties connected
with the piecing together of various portions of the
spectrum.
In contrast to Van der Hoven’s work, our analysis
shows a major spike in the ivind spectrum at a period of
24 hr. Therefore, an important part of this paper will
be concerned with an investigation of the diurnal vari-
ability in the kinetic energy.
2. DATAANDDATAANALYSIS
In 1965, records of hourly surface data for several
U.S. stations covering the period Jan. 1, 1949, through
623
624 MONTHLYWEATHERREVIEW
Dec. 31, 1958, were stored on magnetic tape at The
Travelers Research Center, Inc., Conn., under contract
for the U.S. Air Force.
DATA
A group of three stations (Caribou, Old Town, and
Portland) in Maine, representing the northeastern part
of the United States, and another group in the Great
,Lakes Region (Detroit and Sault Sainte Marie, both in
Michigan, and Duluth, Minn.) have been, .investigated..
The climate at all of these stations is influenced to a high
degree by the proximity of water, with the exception
possibly of Caribou. As me shall see later, there appear
to be no major differences in the shape of the spectrum
at the stations considered. Therefore, in this paper we
have arbitrarily selected Caribou for a more detailed
study. In a future study we intend to include other groups
of stations having a more continental climate as well as
stations at a lower latitude.
The wind observations were taken 1 hr apart and
represent 1-min averages. The height of the wind sensor
is not the same for all stations but varies from 30 to 80
ft above the ground. Other factors that make the results
for the different stations not strictly compatible are
1) differences in station elevation above sea level and
2) differences in exposure of the wind sensor due to, e.g.,
neighboring buildings. I n some cases the location of the
wind sensor was appreciably changed during the 10 yr of
record. However, we have included all 10 yr in our analyses.
Although Caribou is located only 150 mi from the
Atlantic coast, its climate can be classified as a typical
continental type. Old Town and Portland have an “east
coast” maritime climate; the winds are generally light.
The climate of the group of stations in the Great
Lakes Region has some maritime characteristics because
of the location of the stations close to the Great Lakes.
Rather frequent changes in the weather pattern occur,
since nearly all atmospheric disturbances that move
eastward across the country pass close enough to affect
the weather.
Further climatological information is given in table 1.
DATAANALYSIS
In the present study we are interested in the contribu-
tion from different frequency bands to the variance of the
horizontal wind speed (i.e.; in the kinetic energy spectrum).
The contribution from eabh frequency range is estimated
by calculating the sum of the squares of the coefficients
in the cosine and sine transforms (the Fourier coefficients)
at the particular frequency. Recently a method for
efficiently computing these coefficients, called the fast
Fourier transform (FFT), has been reported by Cooley
and Tukey (1965). This method produces savings of up
to 99 percent of computer time over conventional methods
of finding the Fourier coefficients. The FFT apparently
not only reduces the computation time but also slightly
reduces round off errors associated with these compu-
Vol. 97, No. 9
TABLE 1.-Climatological information f r m “Local Climatological
Data-With Comparative Data” published by the U.S. Weather
Bureau (1968)
I” I
Caribou, Maine _______ CAR
Old Town, Maine..”. OLD
Portland, Maine .....- PWM
Detroit, Mi,&.. I ___ DET Sault Ste: Marie; Micli. SSM
Duluth, Minn _______
~
DLH
46.9
44.9
43. ’I
42.4
46.5
46.8
68.0 620 33
68.7 124 27
70.3 61 55
83.0 619 81
84.4 721 33
92.2 1409 55
changes in Reported
exposure of wind in- struments
however, more likely that the height was not changed during the 10 yr of record.
elevation above ground was changed from 43 ft to 33 ft on the same date.
Airport.
1 We could find no reports on the height of the wind instruments before 1954. I t is,
2 On June 15. 1949, the wind instruments were relocated on another building. The
3 On July 1,1950, the wind-recording equipment was moved from the city to the Duluth
tations. The computation time is reduced by a factor of
(log2N)/N where N is the number of data points in the
time series (Group on Audio and Electroacoustics (G-AE)
Subcommittee on Measurement Concepts, 1967). For
further details the reader is referred to the papers in a
special issue of the IEEE Transactions on Audio and Elec-
troacoustics (June 1967), entitled “On Fast Fourier
Transform and Its Application to Digital Filtering and
Special Analysis.”
A fast Fourier transform subroutine was coded for the
Univac 1108. This subroutine replaces a time series of
length 2m (m an integer) with the Fourier coefficients for
the time series. Since the maximum value of m allowed by
the program equals 14, the time series may contain a
maximum of 16,384 data points. In the case of hourly
data, one can analyze a series of a t least 1 yr in one pass
(8,760 points). In order to’analyze a series of 10 yr, we
shall replace the values in the original time series by the ;’ -
averages over 6 hr. The number of data points is then’
reduced from 87,600 to 14,600.
Because of the finite length record, it is impossible to
resolve Fourier coefficients corresponding to frequencies
separated by less than a certain amount. This limit of
resolution measured by A F has the value of 1 over the
period of the entire data record, i.e., 1 cycle/l yr’ or
1 cycle/lO yr, according to whether a 1-yr or a 10-yr
period is being analyzed.
In order to clarify these statements, let us follow the
reasoning given by Bingham et al. (1967, pp. 57-58). The
finite Fourier transforms resolve exactly any combination
of sine terms and cosine terms of frequencies Po, F,,
Thus, if a component cj cos(2a Fit++,) were added to
the time series, the transform would be affected in the
coefficients a, and b j (the coefficient a j would be replaced
by a,+c, cos +,, and b , by bi+c, sin +,), but the other
coefficients a, and b,, where k f j , would not be affected.
However, if a component c cos(2rFt++) were added to
,.
FZ, . . . , Fj, . . . (=O, AF, 2AF, . . . , j A F . . .).
September 1969 Abraham H. 0 o .t
the time series, with F not equal to one of the F,, all
Fourier coefficients would be affected by an amount pro-
portional to (]F-Fjl/AF)-l for IF-FjI>2AF. Thus the
influence of a sinusoidal term a t frequency F is largest for
frequencies Fj nearest F, although its influence extends to
coefficients a t frequencies Fj many times AF away. This
“spill-overJJ of the influence may be decreased by de-
creasing AF (which means considering longer records).
Another means of sharpening the resolution lies in a
filtering procedure known as hanning” the Fourier
coefficients (see Blackman and Tukey, 1958, pp. 14-15).
This may be done directly by applying the hanning
weights ->{, x, ->i to the coefficients at frequencies
F- AF, F, F+AF, respectively, or indirectly by multi-
plying the original time series by a data window function
before processing the data. If the time t runs from 0 to
T, a data window can be obtained by multiplying the
data series by a cosine bell:
1I
(1 -cos 2atlT)/2.
This curve has a maximum value of 1 in the middle of
the series and falls off smoothly to 0 at the beginning and
a t the end of the series. After applying the window, the
influence of a sinusoid of frequency F on the other coeffici-
ents tends to (JF-F,I/AF)-3 for JF-FjI>4AF. That is,
the influence of a line a t frequency F is now restricted
to a smaller neighborhood of that frequency.
Prior to applying the data window, the mean of the
series and a least-squares linear trend were computed
and subtracted from the series. The presence of such
long-term variations would tend to bias the spectrum
by introducing extra variance in the lower frequencies.
After the data were conditioned by the mean and
trend removal and by the data window, the time series
was extended by adding sufficient zeros to attain the
number of 214 data points required by the subroutine.
As a result, the number of Fourier components computed
increased from half of 8,760 (or half of 14,600) to 8,193
at frequencies that divide the frequency interval 0 5 F
5 ll(2At) into 8,192 equal parts (in our case At equals 1 hr
or 6 hr). Thus, we have an apparent increase in resolution
over this interval. The increase in resolution is not real;
however, since the Fourier coefficients are no longer
independent, but must be related to each other in such
a manner as to produce the extra zercs. ,This introduces
an interaction between components similar to the spill-
over mentioned above and emphasizes the need of hanning
the data with the cosine bell.
A plot of the individual power estimates versus fre-
quency will in general be very “rough” showing many
individual small peaks. Often, the location and magnitude
of these peaks is climatologically insignificant, being due
to sampling fluctuations rather than any systematic
physical interaction. It is thus desirable to average out
these peaks to obtain a more useful presentation. Of
‘course, we then give up much detail in resolution. How-
and Albion Taylor 625
ever, with a long time series covering nearly three decades
of frequency, the resolution is generally one or two orders
of magnitude greater than required in any case.
There are two possible means of performing the aver-
aging to account for sampling fluctuations. One method
is to break up the entire record into several parts of
equal length, next to compute spectral estimates for each
part, and finally to average these estimates at the cor-
responding frequencies. I n effect, this corresponds to
taking several samples. An alternative method, which we
actually use here, is to calculate the Fourier coefficients
and then to average estimates in several frequency bands
(Hinich and Clay, 1968). Incidentally, the loss of
resolution in the frequency domain produced by this
averaging tends to compensate for the fictitious increased
resolution produced by the subtended zeros.
I n our case,’ we have split up the frequency scale into
about 80 bands. We distributed the band limits according
to a logarithmic scale between the lowest and highest
frequencies attainable. The lowest frequency is evidently
l /(N A t ), and the highest (the Nyquist or folding) fre-
quency 1/(2At), where N is the total number of observa-
tions and At is the time interval between observations.
For example, for a 10-yr record the resolution i n t h e
vicinity of 1 yr is about one-tenth of a year or 36 days;
however, near periods of 1 day the resolution is approxi-
mately 113650 day or 24 sec. I n our logarithmic scale
with 80 bands, the power in the band near 1 day repre-
sents the average over several hundred individual e&-
mates (see table 3 in the Appendix), while in the bands
near 1 yr only one or two estimates are included.
In order to have some assurance that significant in-
formation is not being averaged out along with variations
due to sampling fluctuations, some form of conjidence
statistics would be useful. If the individual es;timates
within each band are distributed very tightly around the
mean for the band, more confidence would be attached
to that mean than if the individual estimates were widely
scattered. Again, more confidence is attached to a mean
if many estimates are used to compose it. ..
A statistic with these properties is given by
..
DF=n M2/(M2+V)
where n is the number of estimates in the band, M is
the mean, and V the variance. The quantity DF is referred’
to as the number of equivalent degrees of freedom and,
according to Blackman and Tukey (1958, pp. 21-25)
the chi-square distribution with DF degrees of freedom
has the same mean-variance relation as the estimates in
that band.
It should be emphasized that a low number of degrees
of freedom does not mean that the data in this band is
unreliable, but merely that the computed mean does not
represent the estimates in the band very well. Thus, if
resonance, for example, produces a very large, narrow’
peak inside the band, say a t the 24-hr period, the de, wees
626 MONTHLY WEATHER Vol. 97, No. 9
HOURLY UINETIC EN ERG^
Iu~st-c-9
CARIBOU. MAINE
PERIOD IHOURS)+
I I l l I I I I l l I l l I l l 1 1
,001 ,002 .w5 a 1 .M .os .I .2 .5 I 2 5 10 20 50 100 200 5w 1000
HORIZONTAL WIND SPEED SPECTRUM
5 - B R O O K H A V E N I O O M CYCLONE-SCPLE -
r 4 -
(AFTER VAN DER HOVEN 1
w SMALL-SCALE TURBULENCE
(HURRICANE CONDITIONS I
18 17 16 15 14 13 12 I1 10 9 8 7 6 5
+ LOG. F
FIGURE 2."Spectrurn wind speed at Brookhaven National Lab-
oratory, Long Island, at about 100-m height (after Van der
Hoven, 1957). Frequency F in cycles/4096 days.
FIGURE 1.-Hourly kinetic energy at Caribou, Maine, for the
first 10 days of January 1949. TABLE 2.--Spectral intensity i n small-scale turbulence maximum as a
junction of roughness length
Cari-
land Town bou Port- Old
of freedom estimate for that band is sharply reduced (see, -
e.g., table 3 in the Appendix of this paper). In such cases ; (ft)
.3 1 .60 indicated. (PXF),., (m2 s e r a ) (zo=M) cm)
a close, line-by-line examination of this band may be (P~F),,,, (m~sec-2) (zo=20cm)
55 27 33
.28
""
(m sec-1) 4.69 3.85 5.75
.42 ,54 1.00 (PXF),., (mt sec-2) (zo=100 c m )
.18 .18 .36
3. THE ALIASING PROBLEM
As mentioned earlier, the reported wind speeds represent
1-min averages and are spaced 1 hr apart. Figure 1 shows
a plot of the reported hourly kinetic energy for Caribou,
Maine, for the first 10 days of January 1949. The graph
gives evidence of high-frequency fluctuations in the data.
This intuitive conclusion is backed up by Van der
Hoven's results a t Brookhaven (fig. 2) which show
that the energy in wind fluctuations below 2 hr cannot be
neglected. The power in the frequency range between 1
cycle/2 hr and 1 cycle/l min causes an aliasing problem.
However, the aliasing is not as bad as it might appear
from figure 2. The observations in the high-frequency
part of t'he spectrum were taken during the passage of a
hurricane near Brookhaven. A more normal situation
would certainly give lower values for the maximum in the
minute range; a maximum value between 0.5 and 1.0
mz sec-2 might be expected (table 2) instead of the
value of 3.0 m2 sec-2 shown in figure 2 .
In the present data sample the power in the non-
resolvable frequencies between 1 cycle/:! hr and 1 cycle/l
min is added to and cannot be distinguished from the real
power in the resolvable range of frequencies between
1 cycle/lO yr and 1 cycle12 hr. The higher frequencies
that are aliased into a resolvable frequency F are:
2FN-F,2FN+F, 4FN-F, 4FN+F, 6FN-F, 6FN+F, etc.,
where F,=Nyquist or folding frequency=l cycle12 hr.
For example, the power in periods of approximately 72,
51, 33, 28, 21, 19, 15.6, 14.4, 12.4, 11.6, 10.3, 9.7, 8.8 min,
etc. will be added to the power in a period of 6 hr.
troit Marie
Sault De-
Ste.
81
.26
.23 .17
4.61 4.95
33
.64 .38
.39
- -
luth DU-
-
55
6.35
.34
.53
.82
-
Z
V
=observation height above ground.
=mean horizoatal wind speed.
(PxF),., =product of power density and frequency in small-scale turbulence
maximum.
70 r=oughnesslength.
Before going into more detail, let us first discuss the
form in which the spectra in this paper will be presented.
For each frequency band, the value of the product of the
mean power density (P) and the mean frequency (F ) will
be plotted as the ordinate and the natural logarithm of
the mean frequency (log,F) as abscissa. This commonly
used scale gives perhaps a better illustration of the con-
tribution of the various ranges of meteorological interest
than a curve of simply the mean power density versus
the frequency. In both cases, the area under the curve
between the two frequencies Fl and F2 gives that portion
of the total variance (kinetic energy) that is explained
by phenomena in this frequency range
In our graphs we chose to let the frequency decrease from
left to right, in order t o have the time scale increase in
that direction.
Let us assume that the aliased part of the spectrum
between 1 cycle/l min and the folding frequency of
1 cycle/:! hr resembles whik noise, ie., the power is not
a function of frequency. The effect of aliasing will then
be t o add to the true power between the folding frequency
September 1969 Abraham H. Oort and Albion Taylor 621
and 1 -cycle/2 yr a constant amount, independent of
frequency. Aliasing as described above is, of course,
independent of the way in which the spectrum is repre-
sented. However, different representations of the spectrum
might give different impressions of the effect of aliasing.
In the graphs in this paper, the product of the power
and the frequency (not the power itself) is plotted along
the y-axis. Each time that one decreases the frequency
by e (in other words, the value of the x-coordinate in the
graphs is decreased by 1) , the contribution to the y-coordi-
nate (power X frequency) due t o aliasing will decrease
by e , simply because the frequency decreases by e . Thus,
the effects of aliasing tend to show up mostly near the
folding frequency of 1 cycle/2 hr. If the true spectrum
were to show a decrease of power with increasing frequency
near the folding frequency instead of being independent
of frequency, the effects of aliasing would be still more
concentrated near this frequency.
For a better understanding of the aliasing effect, we
have performed two experiments for Caribou, Maine. I n
the first experiment we used all hourly surface reports
and analyzed the spectrum in the range from the Nyquist
frequency of 1 cycle12 hr up to 1 cycle/2 yr. In the second
experiment we only used one observation every 6 hr and
could, therefore, determine only the spectrum between the
new Nyquist frequency of 1 cycle/l2 hr and 1 cycle/2 yr.
In this last experiment, the power between 2 hr and 12
hr is aliased further into the rest of the spectrum. The
results shown in figure 3 indicate that most of the distor-
tion is confined to the spectrum in the vicinity of the
Nyquist frequency, i.e., to the range from 12 hr up to
about 2 days. In the next section we shall make use of
this result in order to make a rough correction t o the
spectrum of Caribou for the effects of aliasing from the
range of periods of 1 min to 2 hr.
4. THE KINETIC ENERGY SPECTRUM
I n this section, power spectra of the surface wind speed
will be presented which cover cycles from 1 cycle/2 hr to
1 cycle/2 yr. A split up of the spectral analysis in two
parts was necessary because of the limitation in the total
number of data points in the computer program for the
fast Fourier transform.
METHOD OF ANALYSIS OF THESPECTRUM
The spectrum for periods of 24 hr and up was obtained
by an analysis of the 10 yr of record. The hourly data
were first averaged over nonoverlapping 6-hr intervals.
Next, these 6-hr averages were used as input data for
estimating the spectrum from 12 hr and up. The averaging
process reduces the amplitude of each wave in the spectrum
R(F)=sin(rFT)/(rE'T)
where R(F)=the response function for a wave with
frequency F, F=frequency in cycle/hr, and T=the
filtering interval=6 hr (Holloway, 1958). The original
by
2 nR 12HR 24 HR
FIGURE 3.-Example of the effects of aliasing on the wind speed
spectrum. Dashed line gives the spectrum at Caribou, Maine, if
power between 2 and 12 hr is aliased. Frequency F in cyclee/4096
days.
spectral power was restored by multiplying the computed
power at each frequency by l/R2(F).
The spectrum for periods between 2 hr and 1 day was
obtained by analyzing each year of the 10 yr of record
separately and then averaging the spectral estimates thus
obtained. In general, the effects of aliasing are most
severely felt in this part of the spectrum, i.e., close to the
Nyquist frequency.
THE HIGH FREQUENCYPART OF THESPECTRUM
Our concern at this point is to estimate what the most
probable shape of the spectrum will be between periods
of 2 hr and 1 min. Since Van der Hoven made his study,
further evidence has been accumulated that there exists
a broad gap in the spectrum. For example, Bysova et al.
(1967) analyzed a continuous record of 40 hr of wind
velocity .fluctuations measured at Obninsk (U.S.S.R.).
Their analysis shows a t all levels (25, 75, 150, and 300 m)
a pronounced spectral gap between periods of about 15
min and 7 hr. This gap, which separates the mesoscale
phenomena (periods of the order of a few hours) from the
high-frequency turbulence (periods of the order of
minutes), appears to be centered at a period of about 1 hr.
The value of the high-frequency maximum of P X F
can be estimated using a general relationship found by
Busch and Panofsky (1968) for the maximum
(PXF)/U*2"1
where u*=friction velocity=kV/[log, z/zo) in neutral air,
k=von K&rm&n constant=O.4, z,=roughness length,
z=height observation above ground, and V=mean wind
speed. For simplicity, effects of stability have been ne-
glected in the formula for the friction velocity. Table 2
gives the calculated values of P X F in this maximum,
assuming several different values of the roughness length.
I n the case of Caribou, we rather arbitrarily selected
a value of 0.8m2 sec-2 in the high-frequency maximum
628 MONTHLY WEATHERREVIEW V O l . 97, No. 9
. .. ..
' 5.79
35
1
SPECTRUM CARIBOU , MAINE WIND -
3.0 1 I
2.5 -
FOLDING +; FREQUENCY i \I Y y 2.0
v)
N
1 5 -
1.0 -
f"., $'
,' '. '\, POSSIBLE SHAPE SPECTRUM 0.5 \,,J AT VERY HIGH FREOUENCIES !
,
""""" . '."" j "_..* ; " 4- LOG,F
I I I I
'4 I i
I
13 12 I1 IO 9 8
I t ' t I M 2 M 5 M IO M 20,M IHR 2HR 3 HR 6HR 12HR 24HR 3 0 6 0 I O 0 30 D 112 YR I YR I
FIGURE 4.-Resultsof an attempt to correct the spectrum at Caribou, Maine, for the effects of aliasing from periods between 1 min
(the basic averaging period of the wind reports) and 2 hr (the Nyquist frequency). Frequency F in cycles/4096 days.
corresponding with a roughness length between 50cm
and 100 cm. The location of the maximum does not seem
t o be very well fixed. For higher wind speeds it generally
shifts to higher frequencies. Again arbitrarily, but perhaps
not unreasonably, we selected a location of about 1 cycle/2
min. Having fixed the height and the location of the
high-frequency maximum, we drew freehand a hypothet-
ical line for the "true" spectrum (see dashed line in fig. 4).
This line gives our estimate of how the spectrum would
look if measurements were available for 10 yr with a
1-min instead of a 1-hr separation. In drawing the dashed
curve, we used the following guidelines :
1) The conservation of variance of the wind speed.\n
other words, the area under the dashed curve between
the basic averaging frequency of the data (1 cycle/ 1 min)
and the folding frequency F N (1 cycle/2 hr) should be
approximately equal to the area between the solid (com-
puted) and dashed curves to the right of F N .
2) The assumption of an approximately white power
distribution in the aliased portion of the spectrum to
the left of F N . The contribution t o the spectrum at the
right of F N due to aliasing will then decrease by e , if F
decreases by e (see discussion in section 3). This consider-
ation rather limits the range of acceptable possibilities in
drawing the true spectrum curve to the right of F N . If
one would assume, e.g., significantly more power to the
left of F N than is done in figure 4, the dashed curve would
become negative near the folding frequency-an impossible
situation. After having drawn the dashed curve in the
way described, the total amount of aliased power is
directly determined. However, considerable freedom is
still left in constructing the shape of the spectrum between
the folding frequency and 1 cycle/l min; more information
is needed.
3) The assumption of the existence (which seems to be
well proved, Busch and Panofsky, 1968) and next of
the magnitude and location of the high-frequency maxi-
mum as tentatively derived above in table 2. Finally, one
can infer that very little energy is left to be distributed
between the high-frequency maximum and the folding
frequency.
Although our method of estimating the true spectrum
is of course very approximate, the final curve in figure 4
shows, that our results using a very long time series are
certainly compatible with the existence of a spectral gap
in the vicinity of 1 cycle/:! hr.
DETAILED DISCUSSION OF THE SPECTRUM
Starting on the left side of figure 4 for Caribou, Maine,
one notices first the rise in the spectrum in the vicinity of
a period of 2 min due t o small-scale turbulence with a
maximum value of 0.8 m2 S ~C -~. Then there follows the
spectral gap region with intensities of the order of 0.1
m2 sec-2 or less between roughly 10 min and 2 hr. Next,
in the range from 2 hr up t o 2 days, the level of activity
starts to rise from the low values in the spectral gap up to
a very high level in the "cyclone rise."
Superposed on this part of the spectrum is a minor peak
a t 12 hr and a major peak at '24 hr. A high, broad plateau
of activity of about 3 m2 sec-2 connected with the travel-
ing cyclones and anticyclones is found at periods between
approximately 2 and 8 days. After this, the level ofac-
tivity drops quite rapidly and shows only minor (probably
not significant) bumps near periods of 1 and 2 mo. Finally,
we reach periods of one-half and 1 yr where again im-
portant peaks are found.
A striking feature of the "cyclone rise" is the appearance
of spikes. By making the resolution coarser (or by a little
September 1969 Abraham H. Oort and A l b i o n T a y l o r 629 '
smoothing on the spectrum) i t is relatively easy to get rid
of the spikes. However, we think the picture as it is may
give the reader more insight into the inaccuracies (or
rather the effects of sample fluctuations on the spectrum).
These fluctuations play a role even when one analyzes a
very long sample such as 10 yr of data. It is clear that
one should not interpret the peaks (at 3.1, 3.9, and 5.5
days) as true periodicities comparable to the diurnal and
annual periods or to their higher harmonics. If one looks
at the hundreds of individual power estimates thgt make
up each peak, one soon realizes that each band consists
'of a large number of peaks and valleys. The high average
value in a band means only that the general level of
. activity is high in that portion of the spectrum. If one
studies the spectra for'individual years, spikes seem to
occur each y-ear but not necessarily a t the same frequencies.
Averaging of the 10 individual years would largely smooth
out these spikes.
For:+ore detailed information on the spectrum for
Cai%& e.g., the number of equivalent degrees of freedom
for each band, the reader is referred to table 3 in the
Appendix.
In figure 5, the measured kinetic energy spectra for the
six stations considered in the present study are plotted,
both with (full line) and without (broken line) applying
a correction for aliasing. The area under each solid curve
was normalized; and, because the total variance is not the
same for the different stations, the scale along the y-axis
is also different. The general shape of these spectra is
quite similar, in spite of apparent differences in detail.
For example, in the case of Caribou and Duluth the
intensity in the cyclone rise appears to be much larger
than in any of the other stations. On the other hand,
Detroit shows an exceptionally pronounced annual cycle.
"
The intensity in the cyclone rise for Caribou and Duluth
has a value between 2 and 3 m2 sec-2, while for the other
stations the intensity is only 1 to 1.5 mZ secF2. According
to the description in the Local Climatological Data-With
Comparative Data (US. Weather Bureau, 1958), Caribou
Airport is located on the top of high land which is about
on the same level as most of the surrounding, gently
rolling hills. The exposed location of this airport probably
causes the greater activity in the cyclone and anticyclone
scale, while in the case of Duluth Airport the large value
must be related to its high elevation above sea level (see
table 1). The differences in intensity can thus be explained
by assuming that the observations at Caribou and Duluth
are more representative of the conditions at higher levels
in the atmosphere.
One can expect that, in general, the annual period in
the kinetic energy will show up most prominently at
qontinental stations and a t stations located at high
'latitudes. Since the six stations considered in this study
are located,,in, a relptively narrow latitude belt between
42'. N. and 47' N., only the effect of continentality might
be noticeable. However, there is no clear indication of
. , this effect in the graphs giyen in figure 5, possibly because
' .
the stations are all located close to either the Atlantic
Ocean or to the Great Lakes.
One other important point is t h a t i f one allows for
the effects of aliasing (see dashed curves in fig. 5)"all
spectra tend to show low values near the Nyquist
frequency of 1 cycle/2 hr.
COMPARISONWITH VAN DER HOVEN'S RESULTS - . ' -
If we compare figures 2 and 4, good qualitative agree-
ment is generally found between the spectrum at Brook-
haven, Long Island, determined by Van der Hoven, and
our spectrum for Caribou, Maine. Aswe have pointed
out before, the small-scale turbulence maximum in the
minute range has an abnormally large value in the case
of Brookhaven, due in part to hurricane conditions present
in its determination. The difference in magnitude of the
cyclone peak, i,e., 5 m2 sec.-2 for Brookhaven and less than
3 m2 sec-2 for Caribou, could be related to the difference
in location. However, it appears more likely that it is
mainly due to the difference in elevation above ground
level at which the observations were taken (respectively,
100 m and 10 m). Another contributing factor may be
that Van der Hoven's observations were made. during the
winter half year, while ours are representative of the
entire year.
One important qualitative difference, however, is the
surprising lack of a diurnal peak in the Brookhaven data
even though there^ is a semidiurnal peak of comparable
magnitude. The most probable reason for this discrepancy
lies in the fact that the Brookhaven observations were
taken at a height of 100 m, which is near the top of the
surface layer (see fig. 6, taken from a report by Singer
and Raynor, 1957), while ourdata represent conditions with-
in the surface layer. Both Van der Hoven's results at the
top of the surface layer and our results within the surface
layer are in very close agreement with Blackadar's
description (1959) of the typical diurnal wind variation
with height.
5. THEDIURNAL CYCLE IN ,,THE KINETICENERGY
In table 4 (see Appendix) the hourly kinetic energy
averaged for each of the 10 yr is given as a function of
local time (see also fig. 7). For each station and for each
year there is a large and systematic diurnal variation.
The maximum value is found between 2 and 3 p.m., and
the minimum in the early morning. The phase of this _'
diurnal variation changes rather ,rapidly with height. "
Singer's measurements (fig. 6) show that around 120 m
the phase of the diurnal cycle is shifted by 180' compared
to the phase a t the surface. A t bhis height the maximum'
wind speed is observed at night,, and a minimum around'. .
midday.
The change of wind speed with elevation and time'of '.
the day was clearly explained by Blackadar (1959) as .'
being related to the coupling and decoupling of the surface
and upper layers. Because of the frictional drag exerted
. ..
630 MONTHLYWEATHERREVIEW Vol. 97, No. 9
I I 1 I "
2 0 h
B4a
I 1 :
SPECTRUM WINDSPEED
SAULT ST€. MARIE
ITOTI\L "*RIANCE.'OOM~EEC.~,
I I I
IFIGURE 5.--Spectra wind speed determined from 10 yr of hourly wind reports. The dashed curves represent a rough est,imate of the
spectrum corrected for the effects of aliasing. Frequency F in cycles/4096 days.
on the air by the earth's surface, the wind speed generally
increases with height above the ground. During the day,
convective mixing mill transfer momentum from higher
levels to the surface layer. The wind speed in this layer
mill thus increase until an equilibrium is reached between
the supply of momentum from above and the loss due to
friction with the earth's surface; but this same process
willslow down the upper layers during the daytime.
However, at night convective mixing stops; and the sur-
face and upper layers become effectively decoupled by
the formation of a temperature inversion. The air near
the surface then slows down, while the air in the upper
layers speeds up.
The spectra presented by Bysova et al. (1967) for heights
of 25, 75, 150, and 300 m seem also to show an important
contribution a t frequencies near 1 cycle/24 hr, the highest
contributions being found at 300 m.
The diurnal variation presented in table 4 for the
different stations shows a remarkable variability from
year t o year. We have the impression that all this varia-
bility cannot be explained away by changes in exposure
of the wind instruments and that there may well be a
September 1969 Abraham H. Oort and Albion Taylor 631
70 65 6 5 7 0 75 75
b-
25 10 35 40 40 35 30 25
I I I I I I /l I l l l l l l l l l l l I I I I _
EST 6 7 8 9 IO I1 If 13 14 15 16 17 18 19 20 21 22 23 2: I 2 3 4 5
MldMY MIDNIGHT
FIGURE 6.-Diurnal variation of wind speed (m sec-1) with height
at Brookhaven National Laboratory, Long Island (after Singer
and Raynor, 1957).
PERIWIDAYS) +
O W 015 027 047 OB1 1 4 4 2 9 437 7 6 ~ '13.3 23.2 40.4 707 122. 210: 372
CARIBOU, MAINE
lU'z. 149M2SEC~z)
4 0
3 0 .
KINETIC ENERGY KINETIC ENERGY
4- LOG F
100- SPECTRUM U t SPECTRUM V
CARIBOU. MAINE ,' ,.:. : ,..'"" " ,,, ,, OEi,./' .j' '. .. .\ ..,, ,. 1?+?*300MPSEC4) ... .
I' :.: .. -... '.. . . 10- -._.__,_c, ." ,' i '. ..
I S M ...." - """". , ::
'..... -- -_
8.0 "1.. , Y ". .... . ......,..,... ./
om,. .,' I... ,,
5 . .. . , . . . -, . . , ., . I-
s60-
z _
v)
I
L O C A L LOCAL TlMEI"O"151 I 4 0 A
FIGURE 7.-Ten-year average of the kinetic energy as a function
of local time at Caribou, Old Town, and Portland, Maine; 20-
Detroit and Sault Sninte Marie, Mich.; and Duluth, Minn. C LOG, F
O0 ,A
I I I
9 a 7 6 5 4 3 2
long-term cycle superposed. However, we do not want to
pursue this topic further since this is beyond the aim of
our present investigation. Our purpose in showing the
results for the different years is to make clear that the
diurnal cycle shows up in the results for each year.
THELACK OF A DIURNAL CYCLE IN THE SPECTRUM OF THE
ZONAL ANDMERIDIONALWINDCOMPONENTS
It may be of interest to compare the present results
with the spectrum obtained from separate analyses of
the west-to-east (u) and the south-to-north (v) components
of the wind (fig. 8). In the second type of analysis there is a
significant increase (by a factor of 3) in total variance
because the wind direction adds a new degree of freedom.
However, the most interesting difference (compare figs. 4
358-753 0 - 69 - 2
FIGURE %-Spectra of the west-to-east (u) and the south-to-north
(0 ) components of the wind for Caribou, Maine, determined
from 10 yr of hourly wind reports. Frequency F in cycles/4096
days.
and 8) lies in the fact that, the diurnal peak does not
show more prominently in the u- and v-spectra.
The lack of an important diurnal cycle in the zonal and
meridional wind components is indeed very surprising.
The interpretation is probably that the wind direction
at the stations considered is-at least near the surface-
highly variable and that it does not change systematically
during the course of a day. The 10-yr mean u- and v-
components and the total wind speed as a function of the
time of the day are shown in figure 9. If stations with
strong land- and sea-breeze effects had been selected
632 MONTHLYWEATHERREVIEW Vol. 97, No. 9
1 1 1 1 1 1 1 1 1 1 1 '
CARIBOU. MAINE ..
3 t 4
' t 0""
-I
t i NOON
1 1 1 1 1 1 1 1 1 1 1
0 2 4 6 8 10 12 14 I6 18 20 22 24
E S T (HR 1
FIGURE 9.-Ten-year average of the zonal (u), the meridional (v )
wind component and the total wind speed (Iv() as a function of
local time.
in this study, the diurnal cycle would certainly have been
more prominent in their u- and w-spectra.
We might also mention the strong annual periodicity
in the w-component of the wind, which does not show up
in the u-component. The same is true for the other stations
in Maine . (not shown here). However, in the cases of
Detroit and Sault Sainte Marie the annual period is most
dominant in the u-component, while for Duluth both the
u- and w-components show an important annual cycle. It
is not obvious what causes these differences in the u- and
wspectra.
THE DIURNAL SPIKE IN THE KINETIC ENERGY SPECTRUM
From the kinetic energy spectra as shown in figure 5,
the diurnal band was selected for further study. The
logarithm of the indiwidual power estimates (not multi-
plied by the frequency) is graphed on a linear frequency
scale in figure 10.
A sharp spike a t exactly 24 hr dominates the picture.
However, a little distance away from this spike the, in-
tensity drops off rapidly. I n addition to the main peak,
there are on both sides secondary maxima of about equal
i n t e 4 t y . A plausible explanation for the occurrence of
these s?de lobes, which are located at a distance of about
4 min from the peak, seems to lie in an analogy with the
11
beating" phenomenon in acoustics. Our basic assumption
is that the amplitude A of the diurnal cycle in the kinetic
energy is not constant throughout the year, but has an
annual variation
A=AI+Az COS(~T~/T),
where t is measured in days and T=365.25 days. A
spectral analysis would show a t 24 hr the intensity of that
part (A,) of the diurnal cycle which is constant. However,
the remaining signal would be interpreted as being the
sum effect of two cycles a t (1 + 1/365.25) and (1 - 1/365.25)
cycles per day, corresponding with periods of respectively
24.066 hr and 23.934 hr. These frequencies are indicated
in figure 10 by arrows.
A C O S (~T ~)=(A ~+A ~ C O S (~T ~/T )) C O S (~T ~)
=A1 ~0~(2~t)+(1/2)Az COS 2 ~t (l +I /T )
+(1/2)A2 cos 2Tt(l"1/T).
With some imagination one can also detect a weaker
semiannual modulation of the diurnal cycle a t periods of
24.13 hr and 23.87 hr. The meaning of this effect is that
the modulation during the year is not a pure sine wave
but that also higher harmonics are involved.
I n order to test our hypothesis for this beating phe-
nomenon, we investigated the diurnal cycle in January
and in July (fig. 11). Indeed, the results show that there
is a strong annual modulation of the diurnal cycle. The
largest amplitude is found during July, when there is the
strongest convective exchange between the surface and
higher levels.
Finally, we would like to comment on one other inter-
esting feature shown in figure 10, i.e., the closb cor-
respondence in magnitude and shape of the diurnal spike
and to a lesser degree of the side lobes between the differ-
ent stations. One would expect that the intensity in the
diurnal cycle would be strongly dependent on the latitude
and longitude of the station, in the sense that the intensity
would increase ,with decreasing latitude and also with
increasing continentality. However, the stations used in
this study are not particularly sui'table for investigating
these questions.
6. SUMMARYANDCONCLUSIONS
The spectrum of horizontal wind speed was analyzed
using 10 yr of 1-min-averaged hourly surface reports for
a group of stations in the northeastern United States
and another group near the Great Lakes.
The fast Fourier transform technique was used to
estimate the spectrum between periods of about 2 hr and
2 yr. The results were compared with the earlier work of
Van der Hoven (1957) for Brookhaven, Long Island.
It was found that most of the effects of aliasing are
confined to frequencies between the folding frequency of
1 cycle/2 hr and 1 cycle/l2 hr. After assuming a reasonable
shape for the small-scale turbulence maximum in the
September 1969 Abraham H. Oort and Albion Taylor 63 3
24.07 (HR 1 24.07 24.00 23.93 (HR )
SAULT STE. MARIE
4060 4080 4100 4120 4060 4080 4100 4120 4140
F (CYCLES /4096 DAYS) F (CYCLES /4096 DAYS
FIGURE 10.-Plot of the individual power estimates (m2 sec-2 per elementary frequency interval) versus frequency near the diurnal period
minute range, the spectrum for Caribou, Maine, was was first discussed by Van der Hoven (1957) and has been
corrected for the aliasing from frequencies between 1 most extensively documented by the atmospheric turbu-
cycle/l min and 1 cycle/2 hr. lence group a t T h e Pennsylvania State University under
With regard to the final form of the spectrum as shown Professor Hans A. Panofsky. The present study shows
in figure 4, the following comments can be made: that there is some evidence for a spectral gap even if one
1) Any reasonable method of correcting for the effects uses a very long time series.
of aliasing seems to lead to quite small values for the 2) Most of the variance of the wind speed is explained
spectrum near the folding frequency of 1 cycle12 hr. This by the activity of traveling cyclones and anticyclones a t
finding is compatible with the existence of a wide spectral periods roughly be,tween 2 and 7 days. This agrees quite
gap between the small-scale turbulence maximum near well with Van der Hoven’s results.
(I period of 2 min and the mesoscale phenomena at periods 3) A large peak in the kinetic energy spectrum was
of a few hours. ,The existence of such a gap in the spectrum found at the diurnal period and a minor peak at the semi-
634
30
25
20
5
% ' 5
w cn
10
5
C
MONTHLYWEATHERREVIEW
I KINETIC ENERGY
CARIBOU MAINE
0 JANUARY
X JULY
Noom
2 4 6 8 10 12 14 16 18 20 22 24
E S T (MR )--
FIGURE 11.-Ten-month mean kinetic energy for January and for
July as a function of local time at Caribou, Maine.
TABLE 3.-Data on the power spectrum of the horizontal wind speed
at Caribou, Maine. Hourly data cover the period J a n . I, 1949,
through Dec. S1, 1958.
Band
1
2
3
4
6
5
7
8
10
9
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
N
8630
6900
7710
6180
5520
4950
4420
3540
3960
2840
3170
2530
2270
2040
1810
1630
1450
1310
1160
1040
940
830
442
396
354
317
284
253
227
PT
(days)
"
0.09
.10
. 11
.12
.14
.15
.19
.17
.21
.24
.30
.27
.33
.37
.42
.47
.52
.58
.65
.81
.73
1.03
.91
1.15
1.29
1.44
1.80
1.61
2.01
L o g 2
~-
10.75
10.64
10.53
10.42
10.30
10.19
10.08
9.86
9.97
9.75
9.64
9.53
9.42
9.31
9.19
9.08
8.97
8.86
8.75
8.64
8.53
8.42
8.29
8.18
8.07
7.96
7.85
7.73
7.62
P
0.03
.03
.03
.03
.04
.04
.04
.05
.05
.05
.06
.07
.08
.08
.08
.12
.19
.I7
.14
.28
.19
1.45
.20
.45
.45
.55
.58
.77
.93
1.33
1.22
1.13
1.00
1.02
.99
.98
.98
.90
.94
.86
.91
1.00
.92
1.03
.83
1.47
.96
1.04
1.41
1.07
1.30
5.79
1.61
1.42
1.56
1.49
1.76
1.89
D F
4346
3923
3287
2929
2796
2388
2236
1945
1752
1567
1351
1284
989
996
883
777
243
614
618
508
476
388
6
201
182
148
129
142
104
Band
"_
30
31
32
33
34
35
36
37
38
39
40
42
41
43
44
45
46
47
48
49
50
61
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
68
67
69
70
71
72
73
74
75
76
78
77
79
80
81
N
204
181
163
145
131
116
104
94
83
75
67
59
54
48
42
39
34
31
27
25
22
19
18
16
12
14
12
10
9
8
7
7
5
5
5
4
4
3
3
3
2
2
2
2
2
1
1
1
1
1
1
1
TABLE 3.-Continued
(days)
PT
2.24
2.80
2.51
3.50
3.13
3.91
4.37
4.88
5.46
6.10
6.82
8.52
7.62
9.52
10.6
11.9
13.3
14.9
16.6
18.6
20.8
25.9
23.2
29.0
32.4
36.1
40.4
45.3
50.6
63.1
56.6
70.7
78.8
87.2
97.6
109.
122.
152.
137.
171.
191.
210.
234.
304.
265.
372.
341.
410.
455.
512.
585.
683.
L o g 3
7.51
7.40
7.29
7.18
7.07
6.96
6.84
6.73
6. ti2
6.51
6.40
6.29
6.18
6.06
5.95
5.84
5.62
5.73
5.40
5.51
5.29
5.18
5.06
4.95
4.84
4.74
4.62
4.50
4.39
4.28
4.17
4.06
3.85
3.95
3.62
3.74
3.51
3.40
3.29
3.18
3.07
2.97
2.86
2.74
2.60
2.48
2.40
2.30
2.20
2.08
1.79
1.95
H
1.07
' 1.71
1.38
2.32
2.05
3.08
2.61
2.46
4.10
3.15
2.88
4.03
2.90
4.29
2.57
3.05
3.28
2.78
2.96
2.92
1.49
1.68
2.22
3.14
2.58
4.61
3.53
2.33
4.65
5.42
4.84
3.44
4.93
2.61
4.54
5.17
2.11
3.86
1.83
10.70
2.34
9.80
.65
2.22
9.50
48.94
80.85
7.97
6.01
3.40
2.51
10.86
VOI. 97, No. 9
(m2 sec-2: FXP
1.96
2.26
2.51
3.04
2.41
3.23
2.45
2.06
3.08
2.12
1.73
2.17
1.40
1.85
1.05
.99
1.01
.77
.72
.66
.33
.27
.35
,44
.58
.29
.36
.38
.21
.39
.22
.28
.26
.12
.22
.17
.07
.I2
.05
.06
.23
.19
.ll
.03
.13'
.59
.89
.08
.05
.03
.02 . 01
- pE=mean period in band (days).
N=number of spectral estimates in frequency band.
p=mean power density in band (m2 sec-2 per elementary frequency interval,
meen en frequency in band (cycles per 4,096 days). - -
where elementary frequency interval=l cycle/4,096 days).
FXF=mean spectral intensity in band (m2 sec-1).
Log.B=natural logarithm of mean frequency in band.
DF=number of equivalent degrees of freedom for estimate of P in band and
=-*E 2 (see Blackman and Tukey. 1958, p. 24).
Note that the following relation should hold:
where the constant 0.111 represents the band width in 1og.F units. ".
diurnal period. Van der Woven (1957) did not find the
diurnal peak, probably because his data were taken near
the top of the surface layer (about 100 m). Blackadar's
September 1969
Station
""
CAR
68.0' W.
46.9O N.
(EST)
OLD
44.9O N .
68.7O W.
(EST)
PWM
70.3' W.
43.7' N.
(EST)
DET
83.00 w.
42.4' N.
(EST)
s SM
N.40 w.
46.5' N.
(EST)
DLH
46.8' N.
92.20 w.
(CST)
/i
I
I
I
I
I
I
J
J
I
I
1
Year 1
-
1949 ____...__._.._..__.
10.9 1952 _._...._..__._.._._
11.7 19 53......-"..........
18.7 1950 ._.....___._.......
13.4 1951 -.-. ._._..____ ...-.
15.7
1954. ..-..... . .. .... .__ 11.4
1955 .__.__.....__.._.__ 11.6
1956 .... ....-. ...... ... 10.3
1957 __......_..._....._ 11.8
1958 _.__........_...__. 12.5
1@yr average.. . ... . . 12.8
1949 ..__.____..._....__ 3.9
1950 __...__......_..... 5.7
1951 ................... 4.5
1952 __.._._...____._.._ 4.6
1953 "...."......."" 3.7
1954 _..__._.___.___..._ 3.5
1955 __.._._.___.____... 4.3
1956 _._...-..........._ 3.6
1957. _.__._.___.___..._ 4.2
1958 ..__.._._...___..__ 4.8
lC-yr average-. . .._._ 4.3
1949 _._......""..".. 4.0
1950 _...___...__..__.._ 5.0
1951 ................... 6.3
1952. ....._.___...__... 6.5
1953. ._.____..__..._... 7.9
I954 __...._..._...__... 8.2
1955 _".."""""."_ 8.2
1956. ....._._...._..._. 7.1
.957..". . . -. . . . . . . .". 9.1
.958 _.__.__. ...- .. ..- .. 11.8
10-yr average _._...__ 7.4
1949. ._.__....__..._._. 9.6
1950 _.___..___.._...._. 9.3
1951. ._.__._.._......_. 9.3
,952 __._..__._.._..._ __ 8.9
,953. "."".""..".. 9.4
,954 ___".".""".... 10.1
.955-. . . . -
" . . . . . . . . . . . 9.1
,956 .._.___...__..__.__ 8.8
,958.. _...__...__....__ 12.0
957 ___".... . .._."... 11.3
O-yr average ___..._._. 9.8
949 _._......."....... 8.9
.959 ...."".""...... 7.9
7.7
.952 ___..__...__...____ 7.8
953 _""
~
"........... 7.7
954 ___..........".". 7.7
955" """ ...~ "."" 7.7
957 _.""""......." 9.1
956 ____._...._...__... 7.3
958 __..__..._....__... 8.9
lC-yr average -.-..... 8.1
,950 -.-. .... ..-... .__._ 20.7
949.. . . . . -. . . . . .-. . . - 25.6
,951 __._._.....__....__ 13.5
~952 .._._._.__.._._____ 14.3
1953 -..... .-.. ._ -... ..- 14.9
1954 _.__. . . . ..- - .. .. . . . 16.8
1955 _.__ ...... ..... ...- 17.8
1956. _._.._._.__ __.__. . 16.4
1957 _.__..__..__._.._._ 15.2
1958 __..__.._. .... .. ___ 15.3
1@yr average __....__ 17.0
-
- -
-
- -
-
- -
-
- -
-
- -
-
- -
2
"
15.1
13.5
18.2
10. e
11.2
11.8
11.9
11.2
12.0
11.8
12.7
4.0
5.2
5.9
3.6
4.7
3.7
4.3
3.4
4.3
4.6
4.4
4.1
6.3
5.4
8.2
6.7
7.9
8.2
7.3
11.9
9.0
7.5
9.4
9.1
8.8
8.6
9.9
9.5
8.8
9.0
.o. 7
11.7
9.6
8.5
8.6
7.7
7.3
7.0
7.3
7.6
7.4
8.6
9.0
7.9
16. 7
!O. 6
3.4
4.6
5.1
6.1
7.3
6.1
4.7
.5. 7
.7.0
-
- -
-
- -
-
- -
-
- -
-
- z
"
- -
3
-
14.4
13.0
18.1
10.6
10.7
11.0
11.7
11.6
11.6
11.2
12.4
3.6
4.8
5.8
4.6
3.5
4.5
3.8
3.4
4.1
4.8
4.3
4.3
5.2
6.6
6.6
7.8
8.7
7.5
7.0
8.5
.l. 9
7.4
.o. 0
9.2
8.8
8.3
9.3
9.5
8.4
8.7
0.4
1.7
-
- -
-
- -
-
- -
-
9.4
8.0
8.1
7.9
7.3
7.8
8.1
6.9
6.9
8. 4
7.9
7.7
8.4
3.3
3.7
4.2
5.4
7.0
4.3
6.5
6.5
- -
-
- -
Il. &
-
6.9
- -
4
"
15.1
12.3
19.0
10.4
10.9
11.4
12. 2
10.3
11.5
11.5
12.4
3.7
4.9
5.6
3.4
5.0
4.5
3.4
4.2
3.4
4.9
4.3
4.6
6.8
5.3
6.7
8.1
8.4
7.6
7.1
9.3
.l. 9
7.6
9.5
8.4
9.0
8.9
8.5
8.4
9.0
8.5
0.6
1.5
-
- -
-
- -
-
- -
-
9.2
8.5
8.1
7.5
6.9
7.3
6.7
7.8
8.3
7.2
7.9
7.6
6.4
2.3
2.8
4.3
4.4
5.5
7.0
4.8
6.9
7.1
7.1
- -
-
- -
-
Abraham H. Oort and Albion Taylor
TABLE 4.-Hourly kinetic energy (m2 s e r a ) - -
5
-
15.2
18.7
12.5
10.8
LO. 7
LO. 8
11.8
LO. 7
11. 7
11.6
12.5
3.8
6.0
4.8
3.4
4.8
4.8
3.7
3.8
3.4
4.7
4.3
4.4
7.0
5.6
6.3
8. 1
8.4
7.8
6.7
8.4
1.7
7.4
9.1
9.1
8.6
8.4
9.3
8.9
9.4
0.2
8.7
1.1
-
- -
"
- -
-
- -
-
9.3
8.3
7.8
7.7
7.8
7.4
7.2
7.3
8.2
7.1
7.8
7.7
6.6
3.7
2.7
5.0
4.8
6.5
6.1
6.5
5.4
6.2
- -
-
- -
-
7.4
-
- -
6
-
15.2
19.3
12.7
LO. 7
LO. 5
11.7
[I. 6
LO. 7
11.4
11.1
12.5
4.0
6.3
5.0
5.2
3.6
3.7
5.1
3.6
4.1
4.8
4.5
4.2
5.2
6.8
6.3
7.7
9.0
7.9
9.2
7.1
2.0
7.5
9.5
8.6
8.3
8.9
8.4
8.6
8. 8
0.3
8.8
0.7
9.1
8.4
7.8
8.1
6.9
7.4
7.5
7.3
7.2
8.1
7.7
7.7
6.5
3.3
2.9
4.0
4.7
5.9
6.5
4.9
6.4
5.8
7.1
-
- -
-
- -
-
- -
-
- -
-
- -
-
- -
7
"
16.3
11.0
13.7
11.7
11.5
12.2
11.9
11.6
13.2
11.6
.3.5
5.0
7.3
6.1
4.3
5.8
5.5
4.5
4.3
4.5
5.1
5.2
4.7
7.1
5.7
6.9
8.3
9.2
8.7
9.6
7.3
3.2
-
- -
-
- -
-
8.1
9.3
8.5
9.0
9.3
8.6
9.0
9.7
9.0
0.6
1.0
9.4
8.6
7.8
7.5
7.3
7.4
7.6
7.4
7.5
8.5
7.9
7.8
6.9
3.6
3.9
3.9
4.5
6.0
6.8
7.5
5.6
6.2
7.5
- -
-
- -
-
"
-
-
8 1 9 ~1 0 /1 1 /1 2 ~1 3 )1 4 ~1 5 /1 6 ~1 7
17.9 21.3 23.6 24.2 25.627.0 27.0 27.925.723.7
32.2 26.5 29.1 31.2 33.6 34.7 35.5 32.6 33.1 29.6
14.9 17.4 19.3 19.720.921.622.1 22.2 21.0 19.4
13.5 15.6 17.8 18.9 20.1 21.621.521.920.118.4
13.7 16.2 17.6 19.7 20.020.821.0 21.9 21.0 18.9
13.8 15.5 16.9 18.9 19.5 20.2 21.319.9 18.7 17.3
:3.4 15.5 17.7 19.9 21.222.621.820.9 18.6 18.1
12.8 16.3 17.6 19.9 20.8 21.0 21.8 21.3 20.3 19.1
!4.0 16.4 18.1 20.0 22.423.424.2 24.1 23.220.0
.4.4 15.6 18.620.1 21.7 21.8 22.6 22.5 21.4 18.9
""~-""
.5.1 17.6 19.6 21.2 22.7 23.5 23.923.5 22.3 20.3
-~~~~"__"
5.5 7.1 7.9 8.6 9.4 10.2 10.7 11.0 10.5 9.4
9.5 11.1 14.3 15.4 16.5 18.6 17.4 17.9 17.1 14.2
7.0 8.7 10.6 11.9 13.0 13.4 13.9 13.2 11.8 10.4
5.9 7.3 8.3 10.8 10.7 11.6 11.8 11.8 10.4 9.6
7.5 8.5 10.2 11.8 12.5 13.5 13.9 13.2 12.4 11.3
6.6 8.3 10.7 11.4 13.0 13.5 14.3 13.2 13.0 11.2
5.9 7.1 8.5 9.2 10.1 10.3 10.5 10.7 9.8 8.2
6.0 7.4 8.3 9.3 10.5 11.1 12.2 12.0 11.5 10.4
5.2 6.4 7.6 9.1 10.0 10.3 10.6 10.8 9.9 8.7
6.2 7.5 8.8 10.3 10.9 11.9 11.8 11.3 11.0 10.2
""- ___~"
6.5 7.9 9.5 10.7 11.6 12.4 12.7 12.5 11.7 10.4
_____ ""-" -__~"-__-_-
5.6 6.8 8.1 8.5 9.5 10.9 11.1 10.7 10.4 8.4
8.3 9.5 10.9 11.5 13.2 13.7 14.6 14.2 13.4 11.7
6.7 7.2 8.2 9.6 10.4 11.8 12.5 12.0 12.0 10.4
7.9 9.4 10.9 12.2 12.5 14.3 14.1 14.2 13.5 11.9
9.5 11.5 14.5 15.8 16.7 17.8 18.2 18.3 16.2 14.5
9.6 11.5 13.6 14.8 16.4 16.4 17.7 17.3 15.9 14.2
1.1 12.4 13.5 15.7 17.5 18.5 18.4 18.1 17.3 15.5
8.6 9.3 11.1 12.2 13.413.9 13.6 13.5 12.8 11.8
0.6 12.5 14.6 17.4 19.7 21.3 22.3 22.3 21.7 19.8
4.0 16.8 18.9 20.7 21.0 21.6 23.8 24.1 22.6 21.1
""_" __"
9.2 10.7 12.4 13.8 15.1 16.1 16.616.5 15.6 13.9
""
_-__ ""_ """ ~ ""
0.3 11.7 13.3 14.7 15.4 16.6 18.0 18.1 17.4 16.9
9.8 10.5 11.3 12.3 13.1 13.9 14.6 15.0 14.8 14.3
9.0 9.8 11.4 12.6 12.9 14.0 14.5 14.4 14.2 13.8
0.2 10.7 11.8 12.8 13.9 14.7 15.8 16.1 16.0 14.8
9.1 10.8 11.8 13.4 13.9 15.0 16.0 16.2 15.9 15.4
0.3 11.4 13.6 14.7 15.3 15.6 16.9 17.6 18.0 16.9
9.7 11.2 12.3 13.7 14.5 15.3 16.5 16.8 16.3 15.8
9.7 10.912.213.6 14.5 14.8 15.9 15.8 15.9 15.1
1.2 12.8 13.7 15.1 16.9 17.7 17.9 18.7 18.7 18.3
1.3 12.8 14.3 16.3 18.3 19.4 20.6 21.1 20.9 21.0
" """
_-
0.0 11.2 12.6 13.9 14.9 15.7 16.7 17.0 16.8 16.2
"__-"~"_- "-__"-_____
9.4 9.9 11.0 12.6 14.8 16.416.9 17.1 17.1 17.0
8.7 8.7 10.6 11.6 13.1 13.7 14.9 14.7 14.7 13.9
7.5 8.1 9.6 10.4 11.8 12.9 13.6 15.1 14.4 13.9
7.6 8.6 9.6 10.8 12.5 13.8 14.9 14.9 14.8 14.5
7.5 8.3 9.5 10.5 11.7 13.0 14.0 14.5 14.3 14.0
8.1 8.9 10.2 11.1 12.3 13.8 15.3 15.2 15.0 14.8
8.1 9.0 10.2 11.4 12.8 14.1 14.9 15.5
7.6 8.4 9.1 10.6 11.9 12.8 13.2 13;7 14.6 14.6
8.8 9.6 11.0 12.5 13.8 16.6 17.41910 19.2 18.8
8.4 9.5 10.412.0 13.4 15.3 16.8 18!0 18.6 18.7
15.8 16.0
"_ """_
8.2 8.9 10.1 11.4 12.8 14.3 15.2 15.8 15.9 15.6
-__________"- ""___"" 6.828.830.7 33.3 34.7 34.8 36.7 36.9 36.7 35.5
4.8 25.625.8 26.1 27.427.827.927.727.928.4
4.2 15.1 16.4 19.0.20.6 20.7 21.7 21.2 21.120.4
5.0 15.6 17.9 20.0 21.4 21.9 23.3 22.3 23.1 22.4
6.1 18.3 19.7 22.0 22.9 23.424.4 24.7 24.4 24.1
7.3 19.1 20.6 23.3 23.8 24.3 25.025.624.424.6
7.3 19.1 19.7 21.5 23.5 24.5 25.1 24.1 24.724.4
8.1 19.5 21.6 22.624.2 24.7 26.2 25.1 25.1 25.1
6.3 17.4 18.6 20.822.5 23.6 25.0 24.3 24.8 24.3
7.1 17.9 19.3 21.1 23.1 24.9 25.5 25.025.5 24.7
"" ~-~"-
8.3 19.6 21.123.0 24.4 25.1 26.1 25.725.8 25.4
,... ..
-
._
2
2
1
1
1
1
I
1
I
1
"
1
.. "
1
"
"
"
1
1
1
1
1
1
1
.-
1
" "
1
1
1
1
1
1
1
1
1
2
._
1
.- -
1
1
1
1
1,
1
1
1
1
1
1
3
a
1:
a
2:
2:
21
2:
2:
2:
2:
-
- -
-
-
"
- I -I"--
10.7 17.9 16.7 15.7
!5.7 22.1 22.1 21.3
6.5 14.9 14.4 14.3
6.0 13.4 13.7 13.0
5.5 13.6 13.0 11.8
5.8 14.5 14.2 12.7
5.3 13.9 13.1 12.5
.5.9 12.5 12.4 11.0
7.0 14.7 13.1 12. 7
7.3 15.2 14.7 13.9
7.6 15.3 14.7 13.9
""
-__" "~_
7.5 6.1 5.0 4.6
2.0 9.7 7.7 7.6
8.6 6.5 6.0 5.3
8.8 7.4 6.9 6.1
8.0 6.1 5.9 4.7
7.0 6.0 4.9 4.7
9.1 7.1 6.6 5.7
7.5 6.2 5.5 4.9
8.7 7.1 6.6 5.4
8.9 7.9 6.9 5.8
8 .6 7 .0 6.2 5.5
""
""
6.3 5.0 4.9 4.4
8.7 7.7 6.7 6.3
0.0 8.9 7. 8 7.4
9.7 8.7 8.1 7.2
3.1 10.9 9.7 9.2
1.7 10.3 9.2 8.7
2.4 11.0 10.8 10.3
0.5 9.4 8.5 8.0
6.7 13.7 12.5 10.3
8.7 15.4 13.6 13.3
1.8 10.1 9 .2 8 .5
4.7 14.2 11.7 10.6
3.6 12.4 11.1 10.7
3.4 11.6 10.7 9.7
4.9 12.5 11.7 11.0
3.6 12.2 11.6 10.7
5.7 13.7 12.1 11.0
5.0 13.0 11.7 10.7
4.1 12.7 11.1 10.5
7.1 15.8 14.0 13.3
0.2 18.2 16.7 15.1
"___
""
5.2 13.6 12.2 11.3
__"
5.4 13.1 11.8 10.3
2.3 10.9 9.8 9.5
2.7 11.1 9 .8 8 .7
3.2 11.2 10.0 8.6
3.0 11.1 10.1 9.0
3.1 11.6 10.4 8.7
3.1 11.3 10.2 9.3
2.5 11.3 10.4 9.1
7.0 15.3 13.8 11.9
7.2 16.1 14.7 11.6
""
3.9 12.3 11.1 9.7
___" ""
6.5 23.9 22.4 20.1
1.8 28.2 27.1 24.0
8.8 16.6 15.2 13.5
0.4 18.1 16.2 14.9
2.5 20.5 18.4 16.1
2.8 20.9 19.3 16.9
2.0 20.1 18.7 17.2
3.1 21.919.918.4
2.8 20.5 18.1 16.3
3.3 21.9 19.3 16.4
"-_
5.4 21.3 19.417.4
- -
22
-
15.3
22.5
12.9
11.9
12.2
12.5
12. 1
11.1
12.3
13.0
-
13.6
- -
4.2
6.8
5.1
4.4
5.8
4.2
5.1
4.3
4.8
5.4
5.0
4.7
6.2
7.0
8.5
7.0
8.5
9.2
7.5
LO. 2
12.4
8.1
,o. 8
9.8
LO. 2
9.9
LO. 4
11.0
LO. 6
LO. 0
12.5
13.6
LO. 9
-
- -
-
- -
-
- -
9.9
8.8
8. 8
8.3
7.9
8.7
8.0
0.8
8.5
0.6
9.1
-
- -
x. 8
0.7
3.0
4.8
5.6
6.5
7.0
5.4
7.8
5.3
7.1
-
- -
23
"
15.5
11.0
13.0
11.6
11.8
12.1
11. 1
LO. 5
11.8
12.2
13.1
3.6
6.7
5.2
4.2
5.0
4.6
3.9
3.9
4.7
5.5
4.7
4.2
5.7
6.7
6.8
8.5
9.0
8.7
9.6
7.4
12.8
7.9
-
- -
-
- -
-
- - .o. 7
,o. 2
9.8
9.5
10.2
LO. 3
9.6
9.4
2.2
.3.2
!O. 5
-
- -
9.9
8.3
8.3
8.0
8.1
8.4
7.7
7. 8
0.0
0.4
8.7
16.2
Il. 8
3.8
5.5
4.4
6.5
7.3
7.6
6.1
5.7
7.5
-
- -
-
-
- -
24
"
15. a
19.3
12.9
11: c
11. c
11.3
11.1
LO. 7
11.5
12.2
12. 6
3.9
4.9
6.3
3.7
4.8
3. E
4.2
4.1
4.6
4.9
4. E
-
- -
-
- -
4. c
6.3
5. E
6.8
8.1
8.3
8.7
7.1
2.3
9.0
-
7.7
"
-
0.0
.o. 2
9.6
9.7
9.6
LO. 3
9.3
9.3
1.6
2.3
.o. 2
8.9
7.6
7.9
7.7
7.5
8.3
7.6
9.5
7.6
9.6
8.2
4.8
'1.3
3.4
4.0
5.0
7.7
6.3
6.7
5.3
5. 1
7.0
-
- -
-
- -
-
635
- -
Daily .verage
"
19.5
24.8
16.2
15.0
14.7
14.8
15.1
14.6
15.9
15.7
-
16.6 __ -
6.4
10.5
7.9
6.7
8.1
7.9
6.3
6.3
7.1
7.5
7.5
6.4
7.7
9.2
11.5
9.2
11.1
12.0
13.6
9.5
16.1
-
__ -
"
10.7
" "
12.6
11.3
10.9
11.5
11.7
12.5
11.8
11.5
13.8
15.2
12.3
10.3
11.7
9.9
10.0
9.9
10.1
10.4
12.3
9.8
12.0
10.6
24.2
29.4
16. 2
17.4
18.5
19.5
19.7
20.2
18.6
19.3
20.3
__
"
__
-
__ -
__
636 MONTHLY Vol. 97, No. 9
theory (1959) adequately explains the differences between
Van der Hoven’s and our results.
cyclone rise, most activity is found in the annual and semi-
4) At low frequencies in the spectrum beyond the
annual periods.
The stations used in this study are confined to a latitude
belt between 42” N. and 47” N. One may expect that
there will be a general shift in importance of the diurnal
and annual peaks and of the cyclone rise, if onewould
study stations at a different latitude.
and meridional wind components, it is shown that the
Through a comparison with the spectra of the zonal
diurnal cycle in the wind speed is not accompanied by a
similar cycle in the wind direction. The diurnal variation
in the kinetic energy near the surface with a maximum
between 2 and 3 p.m. is in evidence a t each of the six
stations and for each of the 10 yr.
A closer look at the individual estimates that make up
the diurnal peak in the spectrum shows, in addition to the
mean spike at 24 hr, side lobes which are probably due to
the annual modulation of the diurnal cycle. The diurnal
January. This is what one might expect with a more
cycle is found to be more pronounced in July than in
intense vertical exchange of kinetic energy between the
surface and upper levels during the summer.
REFERENCES
Bingham, C., Godfrey, M. D., and Tukey, J . W., “Modern Tech-
niques of Power Spectrum Estimation,” IEEE Transactions on
5G-66.
Audio and Electroacoustics, Vol. AU-15, No. 2, June 1967. pp.
Blackadar, A. K., “Periodic Wind Variations,” Mineral Indusrries,
Vol. 28, No. 4, College of Mineral Industries, The Pennsylvania
State University, University Park, Jan. 1959, pp. 1-5.
Blackman, R. B., and Tukcy, J. W., The Measurement of Power
Spectra, Dover Publications, Inc., New York, 1958, 190 pp.,
Busch, N. E., and Panofsky, H. A,, “Recent Spectra of Atmospheric
(see pp. 14, 15, and 21-25).
Turbulence,” Quarterly Journal of the Royal Meteorological
Bysova, N. L., Ivanov, V. N., and Morozov, S. A., “Characteristics
of the Wind Velocity and Temperature Fluotuations in the
Colloquium on the Fine-Scale Stwcture, Moscow, June 15-W,
Atmospheric Boundary Layer,” Proceedings of the International
Cooley, J. W., and Tukey, J. W., “An Algorithm for the Machine
1985, Isdatvo Nauke, Moscow, 1967. pp. 76-92.
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ACKNOWLEDGMENTS
to LIS by Mr. Howard M. Frazier of the Travelers Research Center,
Copies of the data tapes used in this study were kindly supplied
Inc. We would like to acknowledge the help of the following people
a t t h e Geophysical Fluid Dynamics Laboratory, ESSA: Mr.
Henry Stambler for writing the unpacking program for the data
tapes, Mr. Richard Stoner for drawing the figures, and Mrs. C. T.
Morgan for typing the manuscript.
Tower Data April ~1950“arch 1952 Brookhaven National
Laboratory,” Brookhaven National Laboratory Report No. BNL
461 (T-IOZ), Contraot No. (AFCKC-TR-57-220), Upton, N.Y.,
June 1957, 93 pp., (see p. 34).
U.S. Weather Bureau, Local Climatological Data-With Cmn-
paratiue Data, for the stations Caribou, Maine, Portland, Maine,
published in Asheville, N.C., 1958.
Detroit, Mich., Sault Sainte Marie, Mich., Duluth, Minn.,
Van der Hoven, I., “Power Spectrum of Horizontal Wind Speed in
the Frequency Range from 0.0007 t o 900 Cycles Per Hour,”
Journal cf Meteorology, Vol. 14, No. 2, Apr. 1957, pp. 16S164.
[Received February 24, 1969; revised April 10, 18691
I I
CORRECTIONNOTICE
Vol. 97, No. 3, March 1969, p. 286, next to the last sentence: “Moscow
perature of record in the State.” is incorrect and should be deleted.
Airport in Idaho reported -50”F, on the 30th, the coldest December tem-