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-