Geoscience Reference
In-Depth Information
threeterms:
z
T
2
w
+
ν
T
∂
−
w
T
∂
T
T
T
1
2
∂
∂
z
=
(3.6)
∂
∂
∂
x
i
∂
x
i
whereagainthetermontheleftistheproductionofturbulenttemperaturevariance
(nearlyalwayspositive)balancedonthe rightbythegradientofa verticaltransport
termand“dissipation”ofturbulenttemperaturefluctuationsatmolecularscales(
ν
T
isthekinematicheatdiffusivity).Notethatifthetemperaturegradientisknown,and
the thermal dissipation rate may be estimated (say, from microscale measurements
of temperature fluctuations), then (3.6) provides an estimate of vertical heat flux,
providedthe transporttermisnegligible.
3.5 Turbulence Spectra and the Energy Cascade
Spectralanalysisprovidesanimportanttoolforstudyingthecascadeofenergyfrom
largeto small scales in turbulentflows. In many field situations, it is relativelyeas-
ier to measure variance spectra compared with direct covariance between vertical
velocityandthe quantityin question.Spectraltechniquesalsooftenserveasanim-
portant check on the validity of turbulence measurements. Several textbooks treat
thesubjectwell, bothasitpertainstotheturbulencetheory(Batchelor1967;Hinze
1975; Tennekes and Lumley 1972) and for methods of estimating spectra (the fol-
lowingdrawsheavilyontechniquesdescribedbyBloomfield[1976]).
The one-sided spectrum is related to the variance of a quantity (in this case ver-
tical velocity,
w
)
∞
=
σ
w
2
S
w
(
n
)
dn
(3.7)
0
where
n
is frequency(if measurementsaremade in the time domain,as is the most
common case) and
S
w
is the spectral density of the time series
w
. Spectral density
is defined as the Fourier transform of the autocorrelation function. In practice, it
is estimated as follows. Given discrete samples of a deviatory time series
(
=
x
k
;
k
1
...
N
)
performadiscreteFouriertransformtoobtainavector
X
oflength
N
,where
N
k
=
1
x
k
e
−
2π
i
(
k
−
1
)(
n
−
1
)
/
N
;1
≤
n
≤
N
X
(
n
)=
(3.8)
Theone-sidedspectrumisfirstestimatedbythe
periodogram
calculatedby
X
n
·
X
n
/
N
2
n
=[
1
N
/
2
+
1
]
S
p
(
n
)=
(3.9)
X
n
/
N
2
2
X
n
·
n
=[
2
...
N
/
2
]
where the asterisk denotes the complex conjugate. The sum of all the elements
of
S
p
is the variance of the
x
time series (from the discrete form of Parseval's
