Geography Reference
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where
0
is a constant geopotential and L a constant length scale.
Hint:
Assume that ψ(x, y) has the same functional dependence on (x, y)as
does .
11.5.
Starting from the perturbation equations (11.29)-(11.31), show that the
sum of kinetic plus available potential energy is conserved for equatorial
waves. Hence, show that for the Kelvin wave there is an equipartition of
energy between kinetic and available potential energy.
11.6.
Solve for the meridional dependence of the zonal wind and geopotential
perturbations for a Rossby-gravity mode in terms of the meridional veloc-
ity distribution (11.40).
11.7.
Use the linearized model (11.50) and (11.51) to compute the meridional
distribution of divergence in the mixed layer for a situation in which the
geostrophic wind is given by u
g
u
o
exp
−
βy
2
/2c
, v
g
=
=
0 where u
0
and c are constants.
11.8.
Show that the frequency of the n
=
1
equa
torial Rossby mode is given
kβ
k
2
3β/
√
gh
e
−
1
and use this result to solve
approximately by ν
=−
+
ˆ
for the
u (y) and
ˆ
(y) fields in terms of
v(y).
Hin
t:
Use the fact that the
ˆ
Rossby wave phase speed is much less than
√
gh
e
.
MATLAB EXERCISES
M11.1.
The MATLAB script
profile 2.m
and the function
Tmoist.m
use
December-March seasonal mean pressure and temperature data for
Samoa (14
◦
S, 171
◦
W) in the file
tropical temp.dat
, together with an
assumed constant relative humidity of 80%, to calculate and graph verti-
cal profiles of temperature, dewpoint, and the temperature corresponding
to pseudoadiabatic ascent from the lifting condensation level for a par-
cel lifted from the lowest level of the sounding. Modify this script to
plot profiles of the potential temperature, the equivalent potential tem-
perature, and the saturation equivalent potential temperature. (
Hint:
See
Problem M9.2.)
M11.2.
For the thermodynamic sounding of Exercise M11.1, compute the CAPE
and the profile of vertical velocity for a parcel lifted from the lowest layer
in the absence of entrainment. What is the maximum vertical velocity
and what is the distance that the parcel overshoots its level of neutral
buoyancy? What can you conclude about convection associated with this
mean tropical sounding?
M11.3.
Consider a simple nondivergent flow given by the streamfunction ψ
=
A sin kx sin ly. The MATLAB script
nonlinear balance.m
solves the
