Geoscience Reference
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which indicates that the Kolmogorov constant for the two-dimensional spectrum
of vertical velocity is 0
.
61
α
.
15.6.2.2 Horizontal velocity
From
(15.102)
the spectral densities of the horizontal velocity components are
4
πκ
4
κ
2
+
κ
3
,φ
22
=
4
πκ
4
κ
1
+
κ
3
.
E(κ)
E(κ)
φ
11
=
(15.114)
These are not axisymmetric in the horizontal plane. However, their average
κ
h
+
2
κ
3
φ
11
+
φ
22
2
E(κ)
4
πκ
4
=
(15.115)
2
is axisymmetric. The spectrum of this average is then
φ
(
2
)
(κ
h
)
κ
h
+
dκ
3
.
∞
φ
(
2
)
2
κ
3
+
E(κ)
4
πκ
4
11
22
=
(15.116)
2
2
−∞
The two-dimensional spectrum is, from the axisymmetry,
φ
(
2
)
κ
h
+
dκ
3
.
∞
φ
(
2
)
2
κ
3
+
κ
h
E(κ)
2
κ
4
E
(
2
)
11
22
=
2
πκ
h
=
(15.117)
h
2
2
−∞
Using the geometry of
Figure 15.5
,
this yields in the inertial subrange
π/
2
cos
11
/
3
θdθ
.
π/
2
1
2
E
(
2
)
α
2
/
3
κ
−
5
/
3
cos
5
/
3
θdθ
=
−
(15.118)
h
h
0
0
Evaluating the integrals through
(15.112)
yields
0
.
54
α
2
/
3
κ
−
5
/
3
E
(
2
)
=
.
(15.119)
h
h
This says that the Kolmogorov constant for the two-dimensional spectrum of the
average of the two horizontal velocity spectra is 0
.
54
α
, slightly less than that for
vertical velocity.
