Geography Reference
In-Depth Information
f
0
∂u
g
∂p
f
0
∂u
g
∂p
D
g
Dt
∂u
g
∂x
+
∂v
g
∂p
∂u
g
∂y
∂v
a
∂p
+
f
0
βy
∂v
g
∂p
f
0
=−
+
(6.43a)
f
0
f
0
∂u
g
∂p
D
g
Dt
∂v
g
∂p
∂v
g
∂x
+
∂v
g
∂p
∂v
g
∂y
∂u
a
∂p
−
f
0
βy
∂u
g
∂p
f
0
=−
−
(6.43b)
However, by the thermal wind relations (6.41a) and (6.41b), the first terms on the
right-hand side in each of these may be expressed, respectively, as
f
0
∂u
g
∂p
∂T
∂y
∂u
g
∂x
+
∂v
g
∂p
∂u
g
∂y
R
p
∂u
g
∂x
−
∂T
∂x
∂u
g
∂y
−
=−
f
0
∂u
g
∂p
∂T
∂y
∂v
g
∂x
+
∂v
g
∂p
∂v
g
∂y
∂v
g
∂x
−
∂v
g
∂y
R
p
∂T
∂x
−
=−
Using the fact that the divergence of the geostrophic wind vanishes,
∂u
g
∂x
∂v
g
∂y
+
=
0
(6.44)
The above terms can be expressed, respectively, as
∂u
g
∂y
R
p
∂T
∂x
+
∂v
g
∂y
∂T
∂y
R
p
∂
V
g
∂y
·∇
Q
2
≡−
=−
T
(6.45a)
∂u
g
∂x
R
p
∂T
∂x
+
∂v
g
∂x
∂T
∂y
R
p
∂
V
g
∂x
·∇
Q
1
≡−
=−
T
(6.45b)
If we now take partial derivatives of (6.40) with respect to x and y, multiply the
results by Rp
−
1
, and again apply the chain rule of differentiation to the advection
terms, we obtain
R
p
∂u
g
∂x
D
g
Dt
∂T
∂x
R
p
∂T
∂x
+
∂v
g
∂x
∂T
∂y
σ
∂ω
κ
p
∂J
∂x
=−
+
∂x
+
(6.46a)
R
p
∂u
g
∂y
D
g
Dt
∂T
∂y
R
p
∂T
∂x
+
∂v
g
∂y
∂T
∂y
σ
∂ω
κ
p
∂J
∂y
=−
+
∂y
+
(6.46b)
Using the definitions of (6.45a,b), we can rewrite (6.43a,b) and (6.46a,b) as
f
0
∂u
g
∂p
D
g
Dt
∂v
a
∂p
+
f
0
βy
∂v
g
∂p
f
0
=−
Q
2
+
(6.47)
R
p
D
g
Dt
∂T
∂y
σ
∂ω
κ
p
∂J
∂y
=
Q
2
+
∂y
+
(6.48)
