Geoscience Reference
In-Depth Information
full spectral density over all
κ
3
, as in homogeneous turbulence. Thus, we write
the spectral density tensor and the scalar spectral density in the horizontal plane
(denoted by a superscript (2)) as
∞
∞
φ
(
2
)
(
2
)
(
κ
h
)
(
κ
h
)
=
φ
ij
(
κ
)dκ
3
,
=
φ(
κ
)dκ
3
,
κ
h
=
(κ
1
,κ
2
).
(15.101)
ij
−∞
−∞
In this large wavenumber region we use the isotropic forms,
4
πκ
2
δ
ij
−
κ
2
,
E(κ)
κ
i
κ
j
E
c
(κ)
φ
ij
(
κ
)
=
(
κ
)
=
φ(κ)
=
4
πκ
2
.
(15.102)
We'll consider separately the inertial-range spectra of vertical velocity, horizontal
velocity, and a conserved scalar in the horizontal plane.
Miles
et al.
(
2004
)have
also done the inertial-range spectrum of pressure.
15.6.2.1 Vertical velocity
From
Eq. (15.102)
we have
4
πκ
4
κ
2
κ
3
=
4
πκ
4
κ
1
+
κ
2
=
E(κ)
E(κ)
E(κ)
4
πκ
4
κ
h
.
φ
33
(
κ
)
=
−
(15.103)
From
Eq. (15.101)
the spectrum in the horizontal plane is therefore
∞
E(κ)
φ
(
2
)
φ
(
2
)
4
πκ
4
κ
h
dκ
3
=
=
33
(
κ
h
)
33
(κ
h
).
(15.104)
−∞
Since
φ
(
2
)
is axisymmetric, it is natural to define a two-dimensional spectrum for
w
,
33
∞
E(κ)
2
κ
4
φ
(
2
)
2
πκ
h
φ
(
2
)
E
(
2
)
κ
h
dκ
3
.
=
33
(κ
h
)ds
=
33
(κ
h
)
=
(15.105)
w
κ
h
κ
h
·
κ
h
=
−∞
κ
h
tan
θ
κ
h
dθ/
cos
2
θ
.Wealsohave
κ
we have
dκ
3
=
=
κ
h
/
cos
θ
. Using the inertial
subrange form of the three-dimensional spectrum
E(κ)
=
α
2
/
3
κ
−
5
/
3
(15.106)
in
(15.105)
then yields
π/
2
α
2
/
3
κ
−
5
/
3
E
(
2
)
cos
11
/
3
θdθ.
=
(15.107)
w
h
0
