Turbulence and Dispersion of Contaminants in the Planetary Boundary Layer 41
Now, changing from frequency ω expressed in radians per second to frequency
= ω π
2
n
in cycles per second, a new spectral density
= πΦ
π
L
L
( ) 2
(2 )
i
i
S n
n can be
introduced, so that
∞
∞
τ = Φ
π
π τ π =
π τ
∫
∫
L
L
L
0
0
( )
(2 ) cos(2
) 2 d
( ) cos(2
) d
i
i
i
R
n
n
n
S n
n n
(3.24)
For τ = 0, Equation 3.24 becomes
∞
∞
σ =
= π Φ
π
=
∫
∫
2
L
L
L
0
0
(0) 2
(2 ) d
( ) d
i
i
i
i
R
n n
S n n
(3.25)
This confi rms that (twice) the kinetic energy per unit of mass is obtained if the
spectrum is integrated over all frequencies.
On the other hand, in terms of the frequency n, Equation 3.21 can be written as
∞
πΦ
π =
τ
π τ τ
∫
L
L
0
2
(2 ) 4
( )cos2
d
i
i
n
R
n
and yields
∞
=
τ
π τ τ
∫
L
L
0
( ) 4
( )cos2
d
i
i
S n
R
n
(3.26)
Equation 3.25 above can be expressed in a slightly changed form:
∞
∞
σ =
=
∫
∫
2
L
L
0
0
( )d
( )d(ln )
i
i
i
S n n
nS n
n
(3.27)
This shows that the area under the Lagrangian one-dimensional (1-D) energy spectrum L ( )
i
S n plotted against n is the same as the area under the curve L ( )
i
nS n plotted
against (ln n).
The use of L ( )
i
nS n has the advantage that this quantity has unit independent of the
unit of frequency selected. The quantity L ( )
i
nS n has the dimension of the variance.
If the variance σ
2
i is fi nite then from Equation 3.27 it follows that
→
L ( ) 0
i
nS n
for
n → ∞. From this and the obvious fact that
→
L ( ) 0
i
nS n
when n → 0, it can be concluded that L ( )
i
nS n vanishes at both the small and large frequencies and must have
its maximum somewhere in between.
Reciprocals of the frequency at maxima obtained from the plot of the function
L ( )
i
nS n are interpreted as the principal timescales of the turbulent fl ow. Then 1/n max
is a useful integral timescale typically six times larger than the Lagrangian integral
timescale (Hanna, 1981).
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