Geography Reference
In-Depth Information
equal zonal and meridional wave numbers (k
=
l), the wavelength of maximum
growth rate turns out to be
2
√
2πL
R
/(H α
m
)
=
L
m
=
5500 km
where α
m
is the value of α for which kc
i
is a maximum.
Substituting this value of α into the solution for the vertical structure of the
streamfunction (8.67) and using the lower boundary condition to express the coef-
ficient B in terms of A, we can determine the vertical structure of the most unstable
mode. As shown in Fig. 8.10, trough and ridge axes slope westward with height, in
agreement with the requirements for extraction of available potential energy from
the mean flow. The axes of the warmest and coldest air, however, tilt eastward with
height, a result that could not be determined from the two-layer model where tem-
perature was given at a single level. Furthermore, Figs. 8.10a and 8.10b show that
east of the upper-level trough axis, where the perturbation meridional velocity is
positive, the vertical velocity is also positive. Thus, parcel motion is poleward and
upward in the region where θ
> 0. Conversely, west of the upper-level trough axis
parcel motion is equatorward and downward where θ
< 0. Both cases are thus
consistent with the energy converting parcel trajectory slopes shown in Fig. 8.8.
8.5
GROWTH AND PROPAGATION OF NEUTRAL MODES
As suggested earlier, baroclinic wave disturbances with certain favorable initial
configurations may amplify rapidly even in the absence of baroclinic instability.
The theory for the optimal initial perturbations that give rise to rapid transient
growth is beyond the scope of this topic. The basic idea of transient growth of
neutral modes can be illustrated quite simply, however, in the two-layer model by
considering disturbances that consist of waves with zonal wave numbers slightly
higher than the short-wave cutoff for baroclinic instability. Similarly, neutral nodes
of the two-layer model can also be used to demonstrate the important phenomenon
of downstream development of disturbances due to the group velocity effect (see
Fig. 7.4).
8.5.1 Transient Growth of Neutral Waves
If we neglect the β effect, let U
m
=
0, and assume that k
2
> 2λ
2
, the two-layer
model of Section 8.2 has two oppositely propagating neutral solutions given by
(8.24) with zonal phase speeds
c
1
=+
U
T
µ
;
c
2
=−
U
T
µ.
(8.71)
where
k
2
1/2
2λ
2
−
µ
=
k
2
2λ
2
+
