Geography Reference
In-Depth Information
where x
0
designates the phase difference. Because ψ
m
is proportional to the
500-hPa geopotential and ψ
T
is proportional to the 500-hPa temperature (or 250-
750-hPa thickness), the phase angle kx
0
gives the phase difference between geopo-
tential and temperature fields at 500 hPa. Furthermore, A
m
and A
T
are measures
of the amplitudes of the 500-hPa disturbance geopotential and thickness fields,
respectively. Using the expressions in (8.39) we obtain
L
∂ψ
m
∂x
=−
k
L
ψ
T
A
T
A
m
cos k (x
+
x
0
−
ct) sin k (x
−
ct) dx
0
L
(8.40)
kA
T
A
m
sin kx
0
L
ct)]
2
dx
=
[sin k (x
−
0
=
(A
T
A
m
k sin kx
0
) /2
Substituting from (8.40) into (8.38) we see that for the usual midlatitude case of
a westerly thermal wind (U
T
> 0) the correlation in (8.40) must be positive if the
perturbation potential energy is to increase. Thus, kx
0
must satisfy the inequality
0 <kx
0
<π. Furthermore, the correlation will be a positive maximum for
kx
0
=
π /2, that is, when the temperature wave lags the geopotential wave by 90˚
in phase at 500 hPa.
This case is shown schematically in Fig. 8.4. Clearly, when the temperature wave
lags the geopotential by one-quarter cycle, the northward advection of warm air by
the geostrophic wind east of the 500-hPa trough and the southward advection of
cold air west of the 500-hPa trough are both maximized. As a result, cold advection
is strong below the 250-hPa trough, and warm advection is strong below the 250-
hPa ridge. In that case, as discussed previously in Section 6.3.1, the upper-level
disturbance will intensify. It should also be noted here that if the temperature
wave lags the geopotential wave, the trough and ridge axes will tilt westward
with height, which, as mentioned in Section 6.1, is observed to be the case for
amplifying midlatitude synoptic systems.
Referring again to Fig. 8.4 and recalling the vertical motion pattern implied by
the omega equation (8.28), we see that the signs of the two terms on the right in
(8.38) cannot be the same. In the westward tilting perturbation of Fig. 8.4, the
vertical motion must be downward in the cold air behind the trough at 500 hPa.
Hence, the correlation between temperature and vertical velocity must be positive
in this situation; that is,
ω
2
ψ
T
< 0
Thus, for quasi-geostrophic perturbations, a westward tilt of the perturbation
with height implies both that the horizontal temperature advection will increase
the available potential energy of the perturbation and that the vertical circulation
will convert perturbation available potential energy to perturbation kinetic energy.
Conversely, an eastward tilt of the system with height would change the signs of
both terms on the right in (8.38).
