Geoscience Reference
In-Depth Information
Expanding each atmospheric variable in mean and fluctuating parts this becomes:
(
)
(
)
(
)
(
)
∂
ww
+′
∂
ww
+′
∂
ww
+′
∂
ww
+′
(
)
(
)
(
)
++′
uu
++′
vv
+ +′
ww
(17.19)
t
x
y
z
∂
∂
∂
∂
(
)
(
)
(
)
(
)
PP
⎛
2
2
2
⎞
∂
+′
∂
ww ww
ww
+′
∂
+′
∂
+′
⎛
q
′
⎞
1
=−
g
1
−
−
+
u
+
+
⎜
⎟
⎜
⎟
⎝
⎠
r
2
2
2
q
∂
z
∂
x
∂
y
∂
z
⎝
⎠
a
Multiplying out and averaging the last equation, gives:
ww w w w
∂∂
′
∂
w
∂
′
∂
∂
′
++ +
u
u
+′
u
+′
u
t
t
x
x
x
x
∂
∂
∂
∂
∂
∂
∂
w
∂
w
′
∂
w
∂
w
′
∂
w
∂
w
′
∂
w
∂
w
′
v
+
v
+
+′
v
+′
v
+
w
+
w
+ ′
w
+ ′
w
(17.20)
∂
y
∂
y
∂
y
∂
y
∂
z
∂
z
∂
z
∂
z
⎛
⎞
2
w
2
2
2
2
2
q
′
1
∂
P
1
∂
P
′
∂
∂
w
′
∂
w
∂
w
′
∂
w
∂
w
′
=− +
gg
−
−
+
u
+
+
+
+
+
⎜
⎟
r
r
2
2
2
2
2
2
∂
z
∂
z
∂
x
∂
x
∂
y
∂
y
∂
z
∂
z
q
⎝
⎠
a
a
Applying Reynolds averaging rules removes terms 2, 4, 5, 8, 9, 12, 13, 16, 18, 20, 22,
and 24 from this equation. Rearranging the remaining terms then gives:
⎛
⎞
⎛
2
2
2
⎞
⎛
⎞
∂
w www P www w w w
u
∂
∂
∂
1
∂
∂
∂
∂
∂
′
∂
′
∂
′
(17.21)
+
+ +
v w g
−−
+
u
+ +
−′
u
+′
v w
+′
⎜
⎟
⎜
⎟
⎜
⎟
r
2
2
2
∂
t
⎝
∂
x
∂
y
∂
z
⎠
∂
z
⎝
∂
x
∂
y
∂
z
⎠
∂
x
∂
y
∂
z
⎝
⎠
a
It is now appropriate to re-write the final term in this last equation so that its
relationship to turbulent fluxes becomes more obvious, as follows.
Multiplying Equation (17.17) by
w
′
and taking the time average gives:
∂
u
′
∂
v
′
∂
w
′
(17.22)
w
′
+′
w
+′
w
=
0
x
y
z
∂
∂
∂
Subtracting this
zero identity
from Equation (17.21) and re-expressing the resulting
equation in more concise form gives:
⎛
⎞
∂
w
′
∂
w
′
∂
w
′
∂
u
′
∂
v
′
∂
w
′
dw
P
1
∂
∂
2
−−
g
+∇−′
u
w
u
+′
v
+′
w
+′
w
+′
w
+′
w
⎜
⎟
(17.23)
dt
r
z
x
y
z
x
y
z
∂
∂
∂
∂
∂
∂
⎝
⎠
a
