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and observe that temperature in the two-level model is represented as
RT
p
∂p
f 0
δp
=−
1
ψ 3 )
Thus, (6.35) becomes
σ
2 ω 2 =−
2
2
∇·
Q
(8.27)
where
ψ 3 )
f 0
δp
V 2
∂x ·∇
V 2
∂y ·∇
Q
=
1
ψ 3 ) ,
1
In order to examine the forcing of vertical motion in baroclinically unstable waves,
we linearize (8.27) by specifying the same basic state and perturbation variables
as in (8.8). For this situation, in which the mean zonal wind and the perturbation
streamfunctions are independent of y, the Q vector has only an x component:
2 ψ 2
∂x 2
U 3 )
f 0
δp
2f 0
δp
U T ζ 2
Q 1 =
(U 1
=
The pattern of the Q vector in this case is similar to that of Fig. 6.13, with eastward
pointing Q centered at the trough and westward pointing Q centered at the ridge.
This is consistent with the fact that Q represents the change of temperature gradient
forced by geostrophic motion alone. In this simple model the temperature gradient
is entirely due to the vertical shear of the mean zonal wind [U T
∂T /∂y] and
the shear of the perturbation meridional velocity tends to advect warm air poleward
east of the 500-hPa trough and cold air equatorward west of the 500-hPa trough so
that there is a tendency to produce a component of temperature gradient directed
eastward at the trough.
The forcing of vertical motion by the Q vector in the linearized model is
from (8.27)
∝−
2
∂x 2
2 ω 2 =−
∂ζ 2
∂x
4f 0
σδp U T
(8.28)
Observing that
2
∂x 2
2 ω 2 ∝−
ω 2
we may interpret (8.28) physically by noting that
∂ζ 2
∂T
∂y
w 2 ∝−
ω 2 ∝−
v 2
U T
∂x ∝−
 
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