Geography Reference
In-Depth Information
systems (such as the atmosphere). The equations for the Lorenz model
can be written as
dX/dt
=−
σX
+
σY
(13.73)
dY /dt
=−
XZ
+
rX
−
Y
(13.74)
dZ/dt
=
XY
−
bZ
(13.75)
Here (X,Y,Z)may be regarded as a vector defining the “climate” and σ ,
r, and b are constants. Run the script letting the initial value of X
=
10
and verify that in the X
Z plane the resulting trajectory of solution
points has the well known “butterfly-wings” shape. Modify the code to
save and plot the time histories of the variables and compare solutions
in which the initial condition on X is increased and decreased by 0.1%.
How long (in nondimensional time units) do the three solutions remain
within 10% of each other?
−
M13.10.
Modify the code in
Lorenz model.m
to include a constant forcing
F
10 on the right side of the equation governing dX/dt . Describe
how the character of the solution changes in this case. [See Palmer
(1993)].
M13.11.
Use the results of Problem 13.7 to write a staggered grid version of the
MATLAB script
forced equatorial mode2.m
(see Problem M11.6) for
the shallow water model on the equatorial β-plane. Set the staggered grid
such that u and are defined at the equator and v is defined at points
δy/2 north and south of the equator. Compare the results of the staggered
grid model with those of
forced equatorial mode2.m
when the latter
has a grid spacing half that of the former.
=
M13.12.
The MATLAB script
nonlinear advect diffuse.m
provides a numeri-
cal approximation to the one-dimensional nonlinear advection-diffusion
equation
K
∂
2
u
∂x
2
∂u
∂t
=−
u
∂u
∂x
+
sin
2πx
Lx
. The script uses leapfrog
differencing for the advective term and forward differencing for the dif-
fusion term. In the absence of diffusion the flow would quickly evolve
to a shock, but diffusion prevents this from occurring. Run the script and
by varying δt determine the maximum time step for a stable solution.
Modify the code by expressing the diffusion term in the Dufort-Frankel
differencing scheme given in Problem 13.11. Determine the maximum
time step allowed for stability in this case. How does the accuracy of the
solution change as the time step is increased?
with initial condition u
(
x, 0
)
=
