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such parcels the correlations between disturbance meridional velocity and tem-
perature and between disturbance vertical velocity and temperature will both be
positive as required for baroclinically unstable disturbances. For parcels that have
trajectory slopes greater than the mean potential temperature slope, however, both
of these correlations will be negative. Such parcels must then convert disturbance
kinetic energy to disturbance available potential energy, which is in turn converted
to zonal mean available potential energy. Therefore, in order that perturbations
are able to extract potential energy from the mean flow, the perturbation parcel
trajectories in the meridional plane must have slopes less than the slopes of the
potential temperature surfaces, and a permanent rearrangement of air must take
place for there to be a net heat transfer.
Since we have previously seen that poleward-moving air must rise and
equatorward-moving air must sink, it is clear that the rate of energy generation
can be greater for an atmosphere in which the meridional slope of the poten-
tial temperature surfaces is large. We can also see more clearly why baroclinic
instability has a short-wave cutoff. As mentioned previously, the intensity of the
vertical circulation must increase as the wavelength decreases. Thus, the slopes
of the parcel trajectories must increase with decreasing wavelength, and for some
critical wavelength the trajectory slopes will become greater than the slopes of the
potential temperature surfaces. Unlike convective instability, where the most rapid
amplification occurs for the smallest possible scales, baroclinic instability is most
effective at an intermediate range of scales.
The energy flow for quasi-geostrophic perturbations is summarized in Fig. 8.9
by means of a block diagram. In this type of energy diagram each block represents a
reservoir of a particular type of energy, and arrows designate the direction of energy
flow. The complete energy cycle cannot be derived in terms of linear perturbation
theory but will be discussed qualitatively in Chapter 10.
8.4
BAROCLINIC INSTABILITY OF A CONTINUOUSLY STRATIFIED
ATMOSPHERE
In the two previous sections some basic aspects of baroclinic instability were elu-
cidated in the context of a simple two-layer model. The dependence of growth
rate on vertical shear and the existence of a short-wave cutoff were clearly demon-
strated. The two-layer model, however, does have one severe constraint: it assumes
that the altitude dependence of large-scale systems can be adequately represented
with only two degrees of freedom in the vertical (i.e., the streamfunctions at the
250- and 750-hPa levels). Although most synoptic-scale systems in midlatitudes
are observed to have vertical scales comparable to the depth of the troposphere,
observed vertical structures do differ. Disturbances that are concentrated near the
ground or near the tropopause can hardly be represented accurately in the two-layer
model.
