Numerical Modeling and Prediction - An Introduction to Dynamic Meteorology - page 488

M13.4. Modify the script advect 1.m by substituting the upstream differenc-
ing scheme described in Problem 13.6 and the Euler backward scheme
described in Problem 13.3. Compare the accuracy in phase and amplitude
of these two schemes with the leapfrog scheme of M13.3.

M13.5. The MATLAB script advect 2.m is similar to the script of M13.3 except
that the initial tracer distribution is a localized positive definite pulse of
width 0.25. Run the script for 100, 200, and 400 grid intervals. Explain
why the pulse changes shape as it is advected. Using the results of Problem
13.10, modify the script to provide a fourth-order accurate approximation
to the advection term and compare the accuracy of the fourth-order ver-
sion to that of the second-order accurate system for the case with 400 grid
intervals.

M13.6. The MATLAB script advect 3.m is a variant of the second-order accurate
script of M13.5 in which the implicit differencing of (13.19) is utilized.
Run this script with σ 1 and 1.25 and 400 grid intervals and then try

running advect 2.m with these values of σ . Give a qualitative explanation
of these results referring to Problem 13.9.
M13.7. The MATLAB script barotropic model.m can be used to solve a finite
difference approximation to the barotropic vorticity equation using the
flux form of the nonlinear terms given in (13.30) and leapfrog time differ-
encing. In this example the initial flow consists of a localized vortex and
a constant zonal mean flow. Run the model for a 10-day simulation using
different values of the time step to determine the maximum time step
permitted to maintain numerical stability. How does this compare with
the CFL criterion (13.18)?
M13.8. The MATLAB script predict.m solves the quadratic difference equa-
tion (13.71) for a specified coefficient a

=

= 3.75 and an initial condition

1.5. (This will be called the control integration.) The user can
specify additional iterations in which small positive and negative pertur-
bations are added to the initial condition. Run this script and note how
the solutions begin to diverge after about 15 iterations. Modify the script
to compute an ensemble average of the perturbed predictions. Show that
the ensemble mean provides a better prediction of the control run than a
random member of the perturbed integrations. [ Hint : Compute the stan-
dard deviation of the ensemble members at each iteration and compare
to the magnitude of the difference between the ensemble mean and the
control.]
M13.9. The MATLAB script Lorenz model.m gives an accurate numerical solu-
tion to the famous three-component Lorenz equations, which are often
used to demonstrate sensitive dependence on initial conditions in chaotic

Y =