392
7. Physically Insignificant Fast Waves
encing for the vertical gradients such that
Consider an individual Fourier mode with spatial structure exp i (kx + lz)
and show that the eigenvalues of the amplification matrix for this scheme
are unity and
4 _l2 - if2 ± 2J(k - 2)(k + 2)(kl + l2)
l2 +4
where k = csktl'r and l = cslA-r. What is the stability condition that
ensures that this method will not have any eigenvalues with absolute values
exceeding unity?
5. Compare the errors generated in gravity waves using semi-implicit differencing to those produced in a compressible Boussinesq system in whieh
the true speed of sound C s is artificially reduced to Cs in an effort to increase
efficiency by increasing the maximum stable value for Ar. Hint: Consider
waves with wave numbers on the order of N Ic s but larger than N Ics.
6. The oscillations of the damped-harmonic oscillator (7.88) are "overdamped" when
K
2
> c;.Suppose that the mesh is isotropie with grid
interval A, the Courant number for sound-wave propagation on the small
time step is t,and a = y A 2 I Ar. Estimate the minimum value of a required make the divergence damper over-damp a mode resolved on the numerieal mesh. Do the values of «, and a z used in the test problem shown
in Fig. 7.3d over-damp any of the resolved modes in that test problem?
7. Show that discrete integral of the finite-difference approximation to the vertieal divergence of the vertieal advective flux of kinetic energy,
is zero.
8. Examine the mass-conservation properties of a -coordinate prirnitive-equation
models.
(a) Show that the vertieally integrated mass per unit area in a hydrostatically
balanced atmosphere is Psig.
7. Physically Insignificant Fast Waves
encing for the vertical gradients such that
Consider an individual Fourier mode with spatial structure exp i (kx + lz)
and show that the eigenvalues of the amplification matrix for this scheme
are unity and
4 _l2 - if2 ± 2J(k - 2)(k + 2)(kl + l2)
l2 +4
where k = csktl'r and l = cslA-r. What is the stability condition that
ensures that this method will not have any eigenvalues with absolute values
exceeding unity?
5. Compare the errors generated in gravity waves using semi-implicit differencing to those produced in a compressible Boussinesq system in whieh
the true speed of sound C s is artificially reduced to Cs in an effort to increase
efficiency by increasing the maximum stable value for Ar. Hint: Consider
waves with wave numbers on the order of N Ic s but larger than N Ics.
6. The oscillations of the damped-harmonic oscillator (7.88) are "overdamped" when
K
2
> c;.Suppose that the mesh is isotropie with grid
interval A, the Courant number for sound-wave propagation on the small
time step is t,and a = y A 2 I Ar. Estimate the minimum value of a required make the divergence damper over-damp a mode resolved on the numerieal mesh. Do the values of «, and a z used in the test problem shown
in Fig. 7.3d over-damp any of the resolved modes in that test problem?
7. Show that discrete integral of the finite-difference approximation to the vertieal divergence of the vertieal advective flux of kinetic energy,
is zero.
8. Examine the mass-conservation properties of a -coordinate prirnitive-equation
models.
(a) Show that the vertieally integrated mass per unit area in a hydrostatically
balanced atmosphere is Psig.