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theorem). If measurements are made in the time domain, the
n
th element of
S
p
in
(3.9)correspondstoanestimateofthevariance(orenergy)inthesignalatfrequency
ω
n
=(
t
isthesampleperiod.Weusethesame“frozenfield”
approximation as earlier to convert frequency to wave number,
k
, in a flow with
mean velocity
U
past the sensor, namely
k
n
−
1
)
/
(
N
∆
t
)
where
∆
U
. Note that this is the angu-
lar wave number, so that the wavelength in a spatially periodic flow with a peak at
k
=
2
πω
/
1m
−
1
wouldbeabout6.3m.
The fundamental frequency interval is
=
1
N
t
so to satisfy the integral con-
straintof(3.7),requiresthatthefrequencyspectrumis
∆ω
=
∆
S
=
S
/
∆ω
=
N
∆
tS
andsimilarlythewavenumberspectrumis
N
∆
tU
S
=
S
/
∆
k
=
S
2
π
kS
are invariant with the frequency
or wave-number interpretation, hence are called the
area-preserving
form of the
spectrum.
The periodogramofa finite time series differsfromthe true spectrumbecauseit
necessarily involves convolution of the spectrum with the transform of a rectangu-
lar window representing the sampling period. Many strategies exist for smoothing
orotherwisemanipulatingtheperiodogramtoaccentuatesalientfeaturesin thefre-
quency(orwavenumber)domain.Theless thetimeseriesis dominatedbyspecific
frequencies,themorevariationisexpectedinparticularspectralestimatesfromone
realization to another (see, e.g., Jenkins and Watts 1968, Chapter 6). Turbulence
is random enough to generally preclude sharp spectral peaks,
4
and we have found
thatsmoothingoftheperiodogramwithsuccessivepassesofamodifiedDaniellfil-
ter, following Bloomfield (1976), improves the estimates at lower wave numbers.
We typicallyfurtheraveragethe estimatesinequallyspacedbinsof
log
10
(
S
=
Consequently the quantities
nS
=
ω
)
which
greatlyreducesvarianceathigherwavenumbers,wheremanyestimatesoccupyone
bin.Byaligningeachindividualspectrum(typicallycalculatedfrom15-minrealiza-
tionsof the flowas describedabove)onto a common
log
10
(
k
grid,it is possibleto
averagespectraforseveralhoursina relativelysteadyflow,thenfitwithhigh-order
polynomials,asshowninFig.3.9,fromMcPheeandMartinson(1994).Confidence
limits forthe actual
w
spectrumaverageareindicatedbytheshadedarea.
In order to explore the utility of spectra like those shown in Fig. 3.9 requires
discussionofthe
inertial subrange
partoftheenergycascade.Westartwithasimple
dimensionalargument.Supposethat somewherebetween the big and tiny “whirls”
of the production and dissipation scales, respectively, there is a range of scales for
which the eddies are not “aware of” their larger and smaller cousins, and that the
k
)
4
Indeed, if a sharp peak appears in the spectrum, it often signals some extraneous source besides
turbulence. Examples are electronic or acoustic noise when the signal-to-noise ratio is small for
acousticcurrentmeters,orstrummingofthemastonwhichtheyaremounted.Usuallyitispossible
tofilterthese effects.
