Geoscience Reference
In-Depth Information
We also define the quantities
dZ(
κ
h
)dZ
3
(
κ
h
)
dZ
∗
(
κ
h
)dZ
3
(
κ
h
)
+
C
(
2
)
c,u
3
(
κ
h
)
d
κ
h
=
C
∗
(
2
)
2
Co
(
2
)
=
c,u
3
(
κ
h
)
+
c,u
3
d
κ
h
,
iκ
j
dZ(
κ
h
)dF
j
(
κ
h
)
2
κ
j
Q
(
2
)
iκ
j
dZ
∗
(
κ
h
)dF
j
(
κ
h
)
−
=−
c,cu
j
(
κ
h
)d
κ
h
,
(16.30)
2
Co
(
2
)
dZ
∗
(
κ
h
)dV(
κ
h
)
+
dZ(
κ
h
)dV
∗
(
κ
h
)
=
c,(cu
3
)
,
3
(
κ
h
)d
κ
h
.
The first of the pair of molecular terms in
Eq. (16.28)
produces
2
γκ
h
φ
(
2
)
d
κ
h
,
2
γκ
j
κ
j
dZ dZ
∗
=−
−
(16.31)
which represents destruction through molecular diffusion in the
x
1
-and
x
2
-
directions. The second molecular term produces
γ
dZ
,
33
dZ
∗
+
dZ
,
33
dZ
2
γ dZ
,
3
dZ
,
3
,
(16.32)
which represents destruction through molecular diffusion in the
x
3
-direction. If we
assume local isotropy the molecular destruction rate is 3/2 of that produced by the
horizontal gradients and the spectral evolution equation is
(Problem 16.15)
2
γ dZ
,
3
dZ
,
3
−
γ(dZ dZ
∗
)
,
33
−
=
∂φ
(
2
)
(
κ
h
)
∂t
2
C
,
3
Co
(
2
)
2
κ
j
Q
(
2
)
=−
c,u
3
(
κ
h
)
+
c,cu
j
(
κ
h
)
2
Co
(
2
)
3
γκ
h
φ
(
2
)
(
κ
h
)(j
summed on 1
,
2
).
(16.33)
−
−
c,(cu
3
)
,
3
(
κ
h
)
The terms on the right are ordered as in
Eq. (16.13)
for three-dimensional,
isotropic turbulence. The first term is the rate of production by the interaction
of the turbulent flux and the mean gradient. The second term integrates over the
horizontal plane to the horizontal part of turbulent transport, which is zero; hence
it represents transfer within the horizontal wavenumber plane. The third term inte-
grates to the vertical part of turbulent transport, and the final term is the rate of
molecular destruction.
If we assume isotropy in the plane (also called axisymmetry), as in
Chapter 15
,
φ
(
2
)
,
Co
(
2
)
c,u
3
,
and
Co
(
2
)
c,(cu
3
)
,
3
depend only on wavenumber magnitude
κ
h
, not on its
direction. Furthermore, the isotropic expression
Eq. (16.15)
now becomes
Q
(
2
)
κ
j
F
(
2
)
(κ
h
),
c,cu
j
(
κ
h
)
=
j
=
1
,
2
,
(16.34)
