Geography Reference
In-Depth Information
initial trough at time t
m
=
π/ (2kµU
T
). From (8.74) the baroclinic and barotropic
modes at time t
m
are then
µ)
−
1
sin kx
(ψ
m
)
max
=
2A
1
(1
+
(8.75a,b)
µ
)
−
1
sin kx
2A
1
µ
2
(
1
(
ψ
T
)
max
=
+
from which it is shown easily that
2A
1
1
µ
2
(1
µ)
−
1
sin kx
(ψ
1
)
max
=
+
+
;
(ψ
3
)
max
=
2A
1
(1
−
µ) sin kx
so that for small µ the resulting disturbance is nearly barotropic. Thus, the initial
disturbance not only amplifies, but spreads in the vertical. The time for growth
to maximum amplitude is inversely proportional to the basic state thermal wind;
the maximum amplitude, however, depends only on the initial amplitude and the
parameter µ.
Figure 8.11 shows the initial and maximum amplitudes of the barotropic and
baroclinic streamfunctions for f
0
=
10
−
6
m
2
Pa
−
2
s
−
2
,U
T
=
35 m s
−
1
, and a zonal wavelength of 3000 km. In this case the amplitude of the
barotropic disturbance increases by a factor of 8 in about 48 h. Although the most
unstable normal mode for these conditions amplifies by a similar amount in less
than 1 day, if the initial upper level neutral disturbance has a velocity amplitude of
a few meters per second, its growth on the time scale of a few days may dominate
over a normal mode instability that grows exponentially from a much smaller initial
perturbation.
As with normal mode baroclinic instability, the energy source for the transient
amplification of the neutral modes is conversion of the available potential energy
of the mean flow into disturbance potential energy by meridional temperature
advection, followed immediately by conversion to disturbance kinetic energy by
the secondary vertical circulation. This secondary circulation has a vertical velocity
field that lags the upper level streamfunction field by 90
◦
phase. Maximum upward
motion and lower level convergence thus occur to the west of the initial upper level
ridge, and maximum downward motion and lower level divergence occur west of
the initial upper level trough. Vorticity tendencies associated with this convergence
and divergence pattern partly balance the eastward vorticity advection at the upper
level and also act to produce a lower level trough to the east of the initial upper
level trough and lower level ridge to the east of the initial upper level ridge. The
result is that a pattern of nearly barotropic troughs and ridges develops 90
◦
to the
east of the initial trough and ridge pattern. Unlike normal mode instability the
growth in this case does not continue indefinitely. Rather, for a given value of µ
the maximum growth occurs at the time for which t
m
=
10
−
4
s
−
1
,σ
=
2
×
π/(2kU
T
µ). As the zonal
wave number approaches the short-wave instability cutoff, the total amplification
increases, but the amplification time also increases.
