Geoscience Reference
In-Depth Information
Answer 8
(a). Start from Equation (16. 46), the basic equation for conservation of water
vapor in the atmosphere, i.e.
S
q
q
q
q
⎡
2
q
2
q
2
q
⎤
∂
∂
∂
∂
∂
∂
∂
E
q
+
u
+
v w
+
=
ν
+
+
+
+
⎢
⎥
q
∂
t
∂
x
∂
y
∂
z
∂
x
2
∂
y
2
∂
z
2
rr
⎣
⎦
a
a
and expand the variables
u, q
and
r
a
as mean and fluctuating part, thus:
∂
(
qq
+′
)
∂
(
qq
+′
)
∂
(
qq
+′
)
∂
(
qq
+′
)
++′
(
uu
)
++′
(
vv
)
+ +′
(
www
)
∂
t
∂
x
∂
y
∂
z
S
⎡
∂
2
(
qq qq qq
+′
)
∂
2
(
+′
)
∂
2
(
+′
)
⎤
E
q
=
n
+
+
+
+
⎢
⎥
(
) (
)
q
2
2
2
∂
x
∂
y
∂
z
⎣
⎦
rr
+′
rr
+′
a
a
a
a
Multiply out and separate the factors, thus:
∂
q
∂
q
∂
q
∂
q
′
∂
q
∂
q
′
⎡
∂
q
′
⎤
⎡
∂
q
′
∂
q
∂
q
′
⎤
⎡
⎤
+
+
u
+
u
+
u
'
+′
u
+
n
+
v
+′
v
+′
v
⎢
⎥
⎢
⎥
⎢
⎥
t
t
x
x
x
x
y
y
y
y
∂∂
∂
∂
∂
∂
∂
∂
∂
∂
⎣
⎦
⎣
⎦
⎣
⎦
S
⎡
∂
q
∂
q
′
∂
q
∂
q
′
⎤
⎡
2
2
2
2
2
2
⎤
∂
q
∂
q
′
∂
q
∂
q
′
∂
q
∂
q
′
E
q
wwww
z
+
+
+′
+′
=
n
+
+
+
+
+
+
+
⎢
⎥
⎢
⎥
(
) (
)
q
∂
∂
z
∂
z
∂
z
∂
x
2
∂
x
2
∂
y
2
∂
y
2
∂
z
2
∂
z
2
rr
+′
rr
+′
⎣
⎦
⎣
⎦
a
a
a
a
Average this equation and apply the Boussinesq approximation (in this case this just
means using average values for density because there is no buoyancy term), thus:
⎡
⎤
⎡
⎤
⎡
⎤
∂∂
qq q q q q qq q q
u
′
∂
∂
′
∂
∂
′
∂ ∂
′
∂
∂
′
+
+
+
u
+
u
′
+
u
′
+
⎢
n
+
v
+
v
′
+
v
′
⎥
⎢
⎥
⎢
⎥
∂∂
t
t
∂
x
∂
x
∂
x
∂
x
∂ ∂
y
y
∂
y
∂
y
⎢
⎥
⎢
⎥
⎢
⎥
⎣
⎦
⎣
⎦
⎣
⎦
⎡
⎤
⎡
⎤
S
∂
q
∂
q
′
∂
q
∂
q
′
∂
2
q
∂
2
q
′
∂
2
q
∂
2
q
′
∂
2
q
∂
2
q
′
E
q
⎢
⎥
⎢
⎥
+
wwww
z
+
+
′
+
′
=
n
+
+
+
+
+
+
+
q
2
2
2
2
2
2
∂
∂
z
∂
z
∂
z
∂
x
∂
x
∂
y
∂
y
∂
z
∂
z
⎢
⎥
⎢
⎥
rr
⎣
⎦
⎣
⎦
a
a
Apply Reynolds averaging to eliminate terms 2, 4, 5, 8, 9, 12, 13, 16, 18, and 20 and
to eliminate overbars on already averaged terms, thus:
S
⎡
⎤
⎡
⎤
⎡
⎤
⎡
2
2
2
⎤
∂
q
∂
q
∂
q
′
∂
q
∂
q
′
∂
q
∂
q
′
∂
q
∂
q
∂
q
E
⎡⎤
+
q
u
+
u
′
+
n
+
v ww
′
+
+
′
=
n
+
+
+
+
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢⎥
⎣⎦⎢
q
t
x
x
y
y
z
z
2
2
2
∂
∂
∂
∂
∂
∂
∂
∂
x
∂
y
∂
z
⎥
⎢
⎥
⎢
⎥
rr
⎣
⎦
⎣
⎦
⎣
⎦
⎣
⎦
a
a
Re-order the terms and rewrite the viscosity term in vector format, thus:
S
⎡
⎤
∂
q
∂
q
∂
q
∂
q
E
∂
q
′
∂
q
′
∂
q
′
q
2
+ + +
u
v w q
=∇++−′
n
.
u
+′
v w
+′
⎢
⎥
q
∂
t
∂
x
∂
y
∂
z
∂
x
∂
y
∂
z
rr
⎢
⎥
⎣
⎦
a
a
