Geoscience Reference
In-Depth Information
2
κ
j
Q
(
2
)
2
κ
h
F
(
2
)
(κ
h
)
. Thus under axisymmetry
Eq. (16.33)
so that
c,cu
j
(
κ
h
)
=
becomes
∂φ
(
2
)
(κ
h
)
∂t
2
Co
(
2
)
2
C
,
3
Co
(
2
)
2
κ
h
F
(
2
)
(κ
h
)
=−
+
−
c,u
3
(κ
h
)
c,(cu
3
)
,
3
(κ
h
)
3
γκ
h
φ
(
2
)
(κ
h
).
−
(16.35)
We now integrate
Eq. (16.35)
over circular rings in the horizontal wavenumber
plane, reintroducing the two-dimensional scalar spectrum defined in
Chapter 15
:
2
π
E
(
2
)
φ
(
2
)
(κ
h
)κ
h
dθ
2
πκ
h
φ
(
2
)
(κ
h
).
(κ
h
)
=
=
(
15
.
97
)
c
0
We define a production spectrum
P
(
2
)
(κ
h
)
,
2
π
P
(
2
)
(κ
h
)
2
C
,
3
Co
(
2
)
=−
c,u
3
(κ
h
)κ
h
dθ,
(16.36)
0
a horizontal transfer spectrum
T
(
2
)
(κ
h
)
,
h
2
π
T
(
2
)
2
κ
h
F
(
2
)
(κ
h
)κ
h
dθ,
(κ
h
)
=
(16.37)
h
0
and a spectrum of vertical turbulent transport,
2
π
2
Co
(
2
)
T
(
2
)
(κ
h
)
=−
c,(cu
3
)
,
3
(κ
h
)κ
h
dθ.
(16.38)
v
0
This gives the spectral scalar variance budget in the horizontal plane,
∂E
(
2
c
(κ
h
)
∂t
T
(
2
)
P
(
2
)
(κ
h
)
T
(
2
)
3
γκ
h
E
(
2
c
(κ
h
).
=
+
(κ
h
)
+
(κ
h
)
−
(16.39)
v
h
This is the counterpart of the isotropic turbulence result,
Eq. (16.19) (
Problem
16.11
)
.
The terms in
Eq. (16.39)
integrate over
κ
h
to the terms in the variance budget
(16.26)
:
∞
∞
T
(
2
)
P
(
2
)
(κ
h
)dκ
h
=−
2
C
,
3
cu
3
=
Pr,
(κ
h
)dκ
h
=
0
;
h
0
0
∞
3
γ
∞
0
T
(
2
)
κ
h
E
(
2
c
(κ
h
)
(c
2
u
3
)
,
3
=
(κ
h
)dκ
h
=−
Tr,
−
=−
χ
c
.
(16.40)
v
0
