Geoscience Reference
In-Depth Information
Differentiating (15.79) with respect to κ 1 yields
∂F 11
∂F 11
κ 1 E(κ)
κ 3
1
κ 1
E(κ)
κ 3
∂κ 1 =−
dκ,
∂κ 1 =−
dκ,
(15.80)
κ 1
κ 1
the derivative of the lower limit of the integral not contributing since the integrand
vanishes there. Differentiating once more with respect to κ 1 gives
1
κ 1
1
κ 1
.
∂F 11
∂κ 1
∂F 11
∂κ 1
∂κ 1
E(κ 1 )
κ 1
∂κ 1
κ 1
=
,
( 1 )
=
(15.81)
We usually write this result as
1
κ
.
∂F 11
∂κ
∂κ
κ 3
E(κ)
=
(15.82)
15.4.3 The inertial subrange
In the inertial subrange the Kolmogorov ( 1941 ) arguments (Chapter 7) imply that
2 / 3 κ 5 / 3
α 2 / 3 κ 5 / 3 ,
E(κ)
=
(15.83)
where α
1 . 5 is called the Kolmogorov constant. The one-dimensional spectra
also have inertial ranges but with different spectral constants. Equation (15.80) (or
15.82 ) implies that F 11 behaves in the inertial range as
9
F 11 =
55 α 2 / 3 κ 5 / 3 .
(15.84)
Likewise, Eqs. (15.75 )and (15.76) show that in the inertial subrange
4
12
F 11 =
F 22 =
F 33 =
F 11 =
F 33 =
F 22 =
3 F 11 =
55 α 2 / 3 κ 5 / 3 .
(15.85)
The last result - the 4/3 ratio between inertial-subrange levels of “longitudinal”
and “transverse” one-dimensional spectra - is often used as a test of the approach
to isotropy at small scales. In observational work one typically measures frequency
spectra of u and v and/or w , interprets them as the one-dimensional streamwise
wavenumber spectra F 11 and F 22 and/or F 33 , and examines their ratio at inertial-
range κ 1 . Figure 15.4 shows results of this procedure applied to the Kansas data.
 
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