40
Air Pollution and Turbulence: Modeling and Applications
where the inverse transform will be
+∞
− ωτ
−∞
τ =
Φ ω
ω
∫
L
L
1
( )
( )
d
2
i
i
i
R
e
(3.19)
or
+∞
−∞
=
Φ ω ω
∫
L
L
1
(0)
( ) d
2
i
i
R
(3.20)
so that Φ ω
L ( )
i
shows how the turbulent kinetic energy (TKE) is distributed with
respect to frequency.
ω = π = π
2
2
T
n, where T is the period of a sinusoidal oscillation and n is the
frequency in cycles/time or Hertz.
Equations 3.18 and 3.19 defi ne the Wiener–Khinchin theorem (Gardiner, 1983)
and establish a fundamental result that relates the Fourier transform of the autocorrelation function to the spectrum. It means that one may directly measure the autocorrelation function of either a signal or the spectrum, and convert back and forth, which
by means of the fast Fourier transform and computer is relatively straightforward.
We can simplify the expressions (3.18) and (3.19) by noting that because of stationarity of the turbulent velocity fi eld,
τ =
−τ
L
L
( )
( )
i
i
R
R
; that is,
τ
L ( )
i
R
is an even
function. From this property and from Equation 3.18 we obtain
+∞
∞
−∞
Φ ω =
τ
ωτ + ωτ τ =
τ
ωτ τ
π
π
∫
∫
L
L
L
0
1
2
( )
( )(cos
sin ) d
( ) cos d
i
i
i
R
R
(3.21)
since sin ωτ is an odd function.
This shows that Φ ω = Φ −ω
L
L
( )
( )
i
i
, which allows Equations 3.19 and 3.20 to be
written as
∞
τ = Φ ω
ωτ ω
∫
L
L
0
( )
( )cos d
i
i
R
(3.22)
and
∞
= Φ ω ω
∫
L
L
0
(0)
( ) d
i
i
R
(3.23)
The transform of
τ
L ( )
i
R
, Φ ω
L ( )
i
is called the energy spectral density (ESD) function in analogy with the spectra of light studied in physics. The product Φ ω ω
L ( )d
i
is the contribution to variance made by fl uctuations in the interval of width dω
centered at ω.
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