Geography Reference
In-Depth Information
This complicated relationship between U
T
and k can best be displayed by solving
(8.26) for k
4
/2λ
4
, yielding
k
4
/
2λ
4
1
β
2
/
4λ
4
U
T
1/2
=
1
±
−
In Fig. 8.3 the nondimensional quantity k
2
/2λ
2
, which is proportional to the
square of the zonal wave number, is plotted against the nondimensional parameter
2λ
2
U
T
/β, which is proportional to the thermal wind. As indicated in Fig. 8.3,
the neutral curve separates the unstable region of the U
T
, k plane from the stable
region. It is clear that the inclusion of the β effect serves to stabilize the flow,
for now unstable roots exist only for
>β/(2λ
2
). In addition, the minimum
value of U
T
required for unstable growth depends strongly on k. Thus, the β effect
strongly stabilizes the long wave end of the wave spectrum (k
|
U
T
|
0). Again the
flow is always stable for waves shorter than the critical wavelength L
c
=
→
√
2π/λ.
This long wave stabilization associated with the β effect is caused by the rapid
westward propagation of long waves (i.e., Rossby wave propagation), which occurs
only when the β effect is included in the model. It can be shown that baroclini-
cally unstable waves always propagate at a speed that lies between maximum and
minimum mean zonal wind speeds. Thus, for the two-level model in the usual
midlatitude case where U
1
> U
3
> 0, the real part of the phase speed satisfies the
inequality U
3
< c
r
<U
1
for unstable waves. In a continuous atmosphere, this would
imply that there must be a level where U
=
c
r
. Such a level is called a
critical
Fig. 8.3
Neutral stability curve for the two-level baroclinic model.
