Geoscience Reference
In-Depth Information
Multiply Equation (17.17) (the divergence equation for turbulent fluctuations in
the Atmospheric Boundary Layer) by
q
, take the time average, then substitute the
resulting equation into the final term in the last equation to give:
′
S
⎡
⎤
∂
q
∂
q
∂
q
∂
q
E
∂
q
′
∂
u
′
∂
q
′
∂
v
′
∂
q
′
∂
w
′
q
+ + +
u
v w q
=∇++−′
n
.
2
u
+′
q
+′
v
+′
q wq
+′
+′
(26.5)
⎢
⎥
q
∂
t
∂
x
∂
y
∂
z
∂
x
∂
x
∂
y
∂
y
∂
z
∂
z
rr
⎢
⎥
⎣
⎦
a
a
Recalling that the chain rule gives:
∂
(
uq
′′
)
∂
q
′
∂
u
′
∂
(
vq
′′
)
∂
q
′
∂
u
′
= ′
u
.
+′
q
.
;
=′
v
.
+′
q
.
;
∂
x
∂
x
∂
x
∂
y
∂
y
∂
y
∂
(
qw
′′
)
∂
q
′
∂
w
′
=′
w
.
+′
q
.
z
z
z
∂
∂
∂
The final term in Equation (26.5) can be re-written to give the required prognostic
equation for mean humidity in the atmosphere, thus:
(
)
(
)
(
)
⎡
⎤
S
∂
uq
′′
∂
vq
′′
∂
wq
′′
∂
q
∂
q
∂
q
∂
q
E
q
2
⎢
⎥
+
u
+
v w q
+
=
n
.
∇+
+
−
+
+
q
∂
t
∂
x
∂
y
∂
z
∂
x
∂
y
∂
z
rr
⎢
⎥
⎣
⎦
a
a
In Equations (26.1) and Equations (26.2) the terms that become negligible with
the assumptions given in the question are as follows.
S
q
(b)
r
a
E
r
E
c
(c)
and
−
ap
r
a
∇
R
c
n
−
(d)
r
ap
2
2
q
uq
(e)
n
.
∇
q
and
∇
q
∂
∂
q
∂
∂
q
(f )
and
w
w
z
z
(g)
∂
(
uq
x
′′
)
,
∂
(
n
′′
q
y
)
,
∂
(
u
′′
q
)
and
∂
(
nq
′′
)
∂
∂
∂
x
∂
y
∂
∂
q
∂
∂
q
y
∂
∂
y
q
∂
∂
q
(h)
u
,
n
,
u
and
n
x
x
(i) After making all of the above simplifying assumptions Equations (26.1)
and (26.2) become:
∂
q
∂
(
w q
′′
)
∂
q
∂
(
w
′′
q
)
=−
and
=−
∂
t
∂
z
∂
t
∂
z
(j) At level (i) the temperature will
be warmer
and the humidity will
change little
.
