specific heat of air, respectively. The simplest way to evaluate the covariance w′ T ′ is computed
from a time series measurement of w′ and T′ by
1
Σ wi′ Ti′ ,
w′ T ′ =
(61)
N
where N is the number of measurements and wi′ and Ti′ are the ith fluctuating vertical velocity and
fluctuating temperatures, respectively. However, this rather elementary method will not shed much
light on the intricate processes of turbulence. We obtain more information by making the same time
series measurements of w′ and T′ and computing the w′ and T′ spectra, which are defined in such a
way that they yield the variance of w′ and T′ and the covariance w′ T ′ , i.e.,
∞
w′2 = σw′ = ∫ φw′ ( f ) df ,
2
0
∞
T ′2 = σT = ∫ φT′ ( f ) df , and
2
(62a,b,c)
′
0
∞
w′ T ′ = ∫
COw′ T′ ( f ) df ,
0
density) of w′ and T′ and the cospectrum of w′T′, respectively. Equation 62 shows not only how
the variances and covariance are distributed with respect to spectra and the cospectra but also
indicate which eddies accounted for most of the variance or covariance. Kaimal et al. (1972) have
established the standard criteria of surface layer spectra. Therefore, based on the collected sonic
data we can compute the spectra and cospectra and see if the calculated spectra are in agreement
with the characteristics reported by Kaimal and his colleagues. By comparing with the given
standard characteristics, we are in a position to learn whether our instruments are operating
properly and whether our measuring techniques are correct. If the computed spectra do not possess
the well-established spectral shapes already reported, it is certain that either the turbulence mea-
surement or the subsequent analysis or both are not operating properly. Evaluations of w′ T ′ from
eq 61 will not provide this built-in verification.
The w′T′ spectrum viewed through eq 62c provides guidelines on how to measure w′ T ′ .
Equation 62c indicates that to measure w′ T ′ we must sample all frequencies between zero and
infinity that contribute to COw′T′. In reality, however, such sampling is neither practical nor
possible, because sampling zero frequency would require an infinitely long time series and, on the
other hand, sampling frequency to infinity would require an infinitely short sampling interval. The
fact is that not all these frequencies contribute to the integral in eq 62c. In dealing with surface-
layer turbulence spectra, it is usual to nondimensionalize frequency f by
z
n= f
,
(63)
u
where z is the height at which the sonic anemometer is located, and u is the mean wind velocity at
height z. Kaimal et al. (1972) indicated that the peak value of COw′T′ is usually near n = 0.1 and falls
off to a value of about 1/10 of its peak value at n > 7. Therefore, we usually need only to sample
frequencies no higher than 7 u / z . For z = 2 m and u = 2 m/s, the Nyquist frequency Fny has to be 7 Hz
and the actual sampling frequency or digitization rate must be twice Fny, or 14 Hz in this example.
To obtain satisfactory and meaningful values of w′ T ′ , Wyngaard (1973) suggested that 30 to 60
min of data collection is necessary. For 30 min of data, the lower integration limit of eq 62 is 0.0056
Hz. Even sampling w′ and T′ for a period of 30 min at a rate of 14 samples per second generates
25,200 data points for each variable. Therefore, to obtain turbulence statistics properly requires
collecting a large amount of data.
15
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