Synoptic-Scale Motions II: Baroclinic Instability - An Introduction to Dynamic Meteorology - page 266

is independent of height and is positive for a growing wave. How does the
magnitude of the heat flux at a given instant change if the mean wind shear
is doubled?

8.14. Assuming that the coefficient A in (8.67) is real, obtain an expression for
the geostrophic streamfunction ψ (x, y, z ∗ ,t) for the most unstable mode
in the Eady stability problem for the case k

l. Use this result to derive an
expression for the corresponding vertical velocity w* in terms of A.

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8.15. For the neutral baroclinic wave disturbance in the two-layer model given by
(8.75a,b), derive the corresponding ω 2 field. Describe how the convergence
and divergence fields associated with this secondary circulation influence
the evolution of the disturbance.

8.16. For the situation of Problem 8.15, derive expressions for the conversion of
zonal available potential energy to eddy available potential energy and the
conversion of eddy available potential energy to eddy kinetic energy.

MATLAB EXERCISES

M8.1. By varying the input zonal wavelength in the MATLAB script
twolayer model 1A.m , find the shortest zonal wavelength for which the
two-layer model is baroclinically unstable and the wavelength of maxi-
mum instability (i.e., most rapid growth rate) in the case where the basic
state thermal wind is 15 ms - 1 . The case given corresponds to the situation
given in Section 8.2.1 in the textbook. The code of twolayer model 1B.m
differs only in that a finite meridional width is assumed with meridional
wave number m π/(3000 km). Compare the zonal wavelengths for

short-wave cutoff and for maximum growth rate for these two cases.

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M8.2. Use the MATLAB script twolayer model 2A.m in order to examine the
transient growth associated with a disturbance that is initially entirely
confined to the 250-hPa level. Let the basic state thermal wind be 25 m
s - 1 . By examining zonal wavelengths shorter than the instability cutoff
determined in Problem M8.1, find the zonal wavelength that gives the
largest amplification of the disturbance. Repeat this exercise using the
script twolayer model 2B.m , which

includes a meridional dependence

with wave number m π/(3000 km). [Note that the program will termi-

nate if you choose a wavelength that is long enough to be baroclinically
unstable.] Discuss the vertical structure of the transiently growing stable
modes. What sort of situation in the real atmosphere does this solution
crudely model?

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