Geoscience Reference
In-Depth Information
Using the first three identities in the fourth term of Equation (18.8) and
rearranging gives:
⎛
⎞
2
2
2
2
∂′
()
w
∂′
()
w
∂′
()
w
∂′
()
w
(
w
′′
q
)
2(
∂′
w P
′
) 2
∂′
w
u
v
+
+
v
+
w
=
2
g
−
+
P
′
⎜
⎟
∂
t
∂
x
∂
y
∂
z
r
∂
z
r
∂
z
q
⎝
⎠
a
a
v
⎛
⎞
⎛
⎞
(18.11)
w
w
w
u
w
2
v
w
2
w
w
2
∂
∂
∂
∂′
()
′
∂′
()
′
∂′
()
′
−
2(
wu
′ ′
)
+
2(
wv
′ ′
)
+
2(
ww
′
′
)
−
+
+
⎜
⎟
⎜
⎟
∂
x
∂
y
∂
z
∂
x
∂
y
∂
z
⎝
⎠
⎝
⎠
⎛
⎞
2
2
2
∂′
w
∂′
w
∂′
w
+
2
u
w
′
+′
w
+′
w
⎜
⎟
∂
x
2
∂
y
2
∂
z
2
⎝
⎠
The product rule of calculus gives the results:
2
∂′
2
()
w
2
∂′
2
w
∂′
w
⎛
⎞
=′
w
+
⎜
⎟
∂
x
w
2
∂
x
⎝
∂
x
⎠
2
∂′
2
()
2
∂′
2
w
⎛
∂′
w
⎞
(18.12)
=′
w
+
⎜
⎟
∂
y
w
2
∂
y
∂
y
⎝
⎠
2
∂′
2
()
2
∂′
2
w
∂′
w
⎛
⎞
=′
w
+
⎜
⎟
z
2
∂
z
⎝
∂
z
⎠
∂
but observations in the ABL show:
2
2
2
2
w
2
w
2
w
2
⎛
w
⎞
2
w
2
w
∂′
()
⎛
∂′
⎞
∂′
()
∂′
∂′
()
⎛
∂′
⎞
<<
;
<<
;
<<
⎜
⎟
⎜
⎟
(18.13)
⎜
⎟
2
⎝
x
⎠
2
y
2
⎝
z
⎠
∂
x
∂
∂
y
⎝
∂
⎠
∂
z
∂
Together Equations (18.12) and (18.13) imply that the last term in Equation
(18.11) can be re-written such that the equation becomes:
⎛
⎞
2
2
2
2
(
w
′′
)
∂′
()
w
∂′
()
w
∂′
()
w
∂′
()
w
q
2(
∂′
w P
′
) 2
∂′
w
+
u
+
v
+
w
=
2
g
v
−
+
P
′
⎜
⎟
r
r
∂
t
∂
x
∂
y
∂
z
∂
z
∂
z
q
⎝
⎠
a
a
v
(18.14)
⎛
⎛
⎞
∂
w
∂
w
∂
w
∂ ′
u
()
w
′
2
∂ ′
v
()
w
′
2
∂
w
′
(
w
−
2
(
wu
′ ′
)
+
(
wv
′ ′
)
+
(
ww
′
′
)
−
+
+
⎜
⎜
⎟
x
y
z
x
y
z
∂
∂
∂
∂
∂
∂
⎝
⎠
⎝
⎛
⎞
2
2
2
′
′
′
∂
w
⎛
∂
w
⎞
∂
w
⎛
⎞
⎛
⎞
u
⎜
⎟
−
2
+
+
⎜
⎟
⎜
⎟
⎜
⎟
⎝
⎠
⎝
⎠
∂
x
⎝
∂
y
⎠
∂
z
⎜
⎟
⎝
⎠
