Geoscience Reference
In-Depth Information
The last term disappears on taking the time average of this last equation and
applying Reynolds averaging rules, and the equation then becomes:
⎛
⎞
⎛
⎞
∂′
()
w
2
∂′
()
w
2
∂′
()
w
2
∂′
()
w
2
∂
w
∂
w
∂
w
u
+
+
v
+
w
+
2(
wu
′ ′
)
+
2(
wv
′ ′
)
+
2(
ww
′
′
)
⎜
⎟
⎜
⎟
∂
t
∂
x
∂
y
∂
z
∂
x
∂
y
∂
z
⎝
⎠
⎝
⎠
⎛
⎞
2
2
2
(
w
′′
)
∂′
()
w
∂′
()
w
∂′
()
w
q
2
⎛
∂′
P
⎞
+′
u
+′
v
+′
w
=
2
g
v
−
w
′
⎜
⎟
⎜
⎟
r
⎝
⎠
∂
x
∂
y
∂
z
∂
z
⎝
⎠
q
(18.8)
a
v
⎛
⎞
2
2
w
′
2
∂′
w
∂
∂′
w
+
2
u
w
′
+
w
′
+
w
′
⎜
⎟
x
2
y
2
z
2
∂
∂
∂
⎝
⎠
Recall that the divergence of turbulent fluctuations is zero in the ABL. Consequently
Equation (18.8) still holds if the time average of the product of (
w
)
2
with the diver-
gence of turbulent fluctuations is added into the left hand side of Equation (18.8).
When this is done, the equation becomes:
′
⎛
⎞
⎛
⎞
∂′
()
w
2
∂′
()
w
2
∂′
()
w
2
∂′
()
w
2
∂
w
∂
w
∂
w
+
+
v
+
w
+
2(
w u
′ ′
)
+
2(
w v
′ ′
)
+
2(
w w
′
′
)
u
⎜
⎟
⎜
⎟
∂
t
∂
x
∂
y
∂
z
∂
x
∂
y
∂
z
⎝
⎠
⎝
⎠
⎛
⎞
∂
u
′
∂
v
′
∂
u
2
2
2
′
∂′
()
w
∂′
()
w
∂′
()
w
(18.9)
2
2
2
+′
()
w
+′
()
w
+′
()
w
+′
u
+′
v
+′
w
⎜
⎟
∂
x
∂
x
∂
x
∂
x
∂
y
∂
z
⎝
⎠
(
w
′′
)
P
q
2
⎛
∂′
⎞
⎛
⎞
w
′
2
w
2
2
w
∂′
∂
∂′
=
2
g
v
−
w
′
⎜
⎟ +
2
w
′
+′
w
+′
w
u
⎜
⎟
r
⎝
⎠
∂
z
q
2
2
2
∂
x
∂
y
∂
z
⎝
⎠
a
v
The product rule of calculus gives the four identities:
2
2
∂′ ′
uw
()
∂ ′
()
w
∂′
u
=′
u
+ ′
()
w
2
∂
x
∂
x
∂
x
2
2
∂′ ′
vw
()
∂ ′
()
w
∂′
v
=′
v
+ ′
()
w
2
∂
y
∂
y
∂
y
(18.10)
2
2
∂′ ′
ww
()
∂ ′
()
w
∂′
w
=′
w
+ ′
()
w
2
∂
z
∂
z
∂
z
wP
P
w
∂′
(
′
)
∂′
∂′
=′
w
+′
P
z
z
z
∂
∂
∂
