Geography Reference
In-Depth Information
where perturbations are assumed in the form given in (8.8). Assuming solu-
tions of the form (8.17), show that the phase speed satisfies a relationship
similar to (8.21) with β replaced everywhere by iµk, and that as a result the
condition for baroclinic instability becomes
U
T
>µ
2λ
2
k
2
−
1/2
−
8.6.
For the case β = 0 determine the phase difference between the 250- and 750-
hPa geopotential fields for the most unstable baroclinic wave (see Problem
8.1). Show that the 500-hPa geopotential and thickness fields (ψ
m
,ψ
T
) are
90
◦
out of phase.
8.7.
For the conditions of Problem 8.6, given that the amplitude of ψ
m
is A
=
10
7
m
2
s
−
1
, solve the system (8.18) and (8.19) to obtain B. Let λ
2
=
2.0
×
10
−
12
m
−
2
15m s
−
1
. Use your results to obtain expressions for
and U
T
=
ψ
1
and ψ
3
.
8.8.
For the situation of Problem 8.7, compute ω
2
using (8.28).
8.9.
Compute the total potential energy per unit cross-sectional area for an atmo-
sphere with an adiabatic lapse rate given that the temperature and pressure
at the ground are p
10
5
Pa and T
300 K, respectively.
8.10.
Consider two air masses at the uniform potential temperatures θ
1
=
=
=
320 K
and θ
2
340 K that are separated by a vertical partition as shown in Fig.
8.7. Each air mass occupies a horizontal area of 10
4
m
2
=
and extends from
10
5
Pa) to the top of the atmosphere. What is the available
potential energy for this system? What fraction of the total potential energy
is available in this case?
8.11.
For the unstable baroclinic wave that satisfies the conditions given in Prob-
lems 8.7 and 8.8, compute the energy conversion terms in (8.37) and (8.38)
and hence obtain the instantaneous rates of change of the perturbation kinetic
and available potential energies.
the surface (p
0
=
8.12.
Starting with (8.62) and (8.63), derive the phase speed c for the Eady wave
given in (8.68).
8.13.
Unstable baroclinic waves play an important role in the global heat budget
by transferring heat poleward. Show that for the Eady wave solution the
poleward heat flux averaged over a wavelength,
L
1
L
v
T
dx
v
T
=
0
