Geography Reference
In-Depth Information
f
0
∂v
g
∂p
D
g
Dt
∂u
a
∂p
−
f
0
βy
∂u
g
∂p
f
0
=
Q
1
−
(6.49)
R
p
D
g
Dt
∂T
∂x
σ
∂ω
κ
p
∂J
∂x
=
Q
1
+
∂x
+
(6.50)
Suppose that Q
2
> 0 and that the thermal wind is westerly (∂u
g
/∂p < 0 and
∂T/∂y < 0). Then from (6.47), Q
2
forces an increase in the westerly shear fol-
lowing the geostrophic motion (∂u
g
∂p becomes more negative). However, from
(6.48), Q
2
> 0 forces a positive change in the meridional temperature gradient
following the geostrophic motion (∂T
∂y becomes less negative). Q
2
thus tends
to destroy the thermal wind balance between the vertical shear of the zonal wind
and the meridional temperature gradient. Similarly, Q
1
destroys the thermal wind
balance between vertical shear of the meridional wind and the zonal temperature
gradient. An ageostrophic circulation is thus required to keep the flow in approxi-
mate thermal wind balance.
Subtracting (6.47) from (6.48) and using (6.41a) to eliminate the total derivative
gives
σ
∂ω
∂v
a
∂p
−
f
0
βy
∂v
g
κ
p
∂J
∂y
f
0
∂y
−
∂p
=−
2Q
2
−
(6.51)
Similarly, adding (6.50) to (6.49) and using (6.41b) to eliminate the total deriva-
tive gives
σ
∂ω
∂u
a
∂p
−
f
0
βy
∂u
g
κ
p
∂J
∂x
f
0
∂x
−
∂p
=−
2Q
1
−
(6.52)
∂(6.49)/∂y and use (6.12) to eliminate the
ageostrophic wind, we obtain the
Q
vector form of the omega equation:
If we now take ∂(6.52)/∂x
+
∂
2
ω
∂p
2
f
0
β
∂v
g
κ
p
∇
2
ω
f
0
2
J
σ
∇
+
=−
2
∇·
Q
+
∂p
−
(6.53)
where
T
∂
V
g
∂x
·∇
∂
V
g
∂y
·∇
R
p
R
p
Q
≡
(
Q
1
,Q
2
)
=
−
T,
−
(6.54)
Equation (6.54) shows that vertical motion is forced by the sum of the divergence
of
Q
, the Laplacian of the diabatic heating, and a term related to the β effect that
is generally small for synoptic-scale motions. Unlike the traditional form of the
omega equation, the
Q
vector form does not have forcing terms that partly cancel.
The forcing of ω for adiabatic flow can be represented simply by the pattern of the
Q
vector. By the arguments of the last subsection, the left-hand side in (6.54) is
proportional to the vertical velocity. Hence, a convergent
Q
forces ascent, and a
divergent
Q
forces descent.
