Geography Reference
In-Depth Information
7.20.
Derive an expression for the group velocity of a barotropic Rossby wave with
dispersion relation (7.91). Show that for stationary waves the group velocity
always has an eastward zonal component relative to the earth. Hence, Rossby
wave energy propagation must be downstream of topographic sources.
MATLAB EXERCISES
M7.1.
(a) The MATLAB script
phase demo.m
shows that the Fourier series
F(x)
=
A sin(kx)
+
B cos(kx) is equivalent to the form F(x)
=
Re
[
where A and B are real coefficients and C is a complex
coefficient. Modify the MATLAB script to confirm that the expression
F(x)
C exp(ikx)
]
=|
C
|
cos k(x
+
x
0
)
=|
C
|
cos(kx
+
α) represents the same Fourier
sin
−
1
(C
i
/
) and C
i
stands for the imaginary
part of Cand α is the “phase” defined in the MATLAB script. Plot your
results as the third subplot in the script. (b) By running the script for several
input phase angles (such as 0, 30, 60, and 90
◦
), determine the relationship
between α and the location of the maximum of F(x).
M7.2.
In this problem you will examine the formation of a “wave envelope”
for a combination of dispersive waves. The example is that of a deep
water wave in which the group velocity is 1/2 of the phase velocity. The
MATLAB script
grp vel 3.m
has code to show the wave height field at four
different times for a group composed of various numbers of waves with
differing wave numbers and frequencies. Study the code and determine
the period and wavelength of the carrier wave. Then run the script several
times varying the number of wave modes from 4 to 32. Determine the
half-width of the envelope at time t
series where, kx
0
≡
α
=
|
C
|
0 (top line on graph) as a function
of the number of modes in the group. The half-width is here defined as two
times the distance from the point of maximum amplitude (x
=
=
0) to the
point along the envelope where the amplitude is 1/2 the maximum. You can
estimate this from the graph using the
ginput
command to determine the
distance. Use MATLAB to plot a curve of the half-width versus number
of wave modes.
M7.3.
Consider stationary gravity waves forced by flow over a sinusoidal lower
boundary for a case in which the static stability decreases rapidly with
height at about the 6-km level. Thus, the buoyancy frequency is altitude
dependent and the simple analytic solution (7.48) no longer applies. The
MATLAB script named
linear grav wave 1.m
gives a highly accurate
numerical solution for this situation. (a) Describe the qualitative change
in the wave behavior as the zonal wavelength is changed over the range of
10 to 100 km. Be sure to comment on momentum flux as well as on the
vertical velocity. (b) Determine as accurately as you can what the minimum
