Problems
391
2. Suppose that the time derivatives in (7.141)-(7.143) are approximated by
leapfrog differencing and the spatial dependence is represented by a Fourier
spectral or pseudospectral method. Recall that we have assumed U > 0, as
would be the case in the middle latitudes of the Earth 's atmosphere.
(a) Deterrnine the constraints on At required to keep the gravity waves
stable and show that (U + c)kAt I is a necessary condition for stabiIity.
(b) Deterrnine the constraints on At required to keep the Rossby waves stable. Let K be the magnitude of the maximum vector wave number retained
in the truncation, i.e.,
K = max y'k 2 + (2 .
i.e
Show that U K At
waves unless
I is a sufficient condition for the stability of the Rossby
3. Suppose that (7.141)-(7.143) are integrated using the semi-implicit scheme
8t
8z
8P
- + c s
(8U 8W)
- + -
8t
8x
8z
=0,
(a) Detcrrnine the conditions under which the gravity waves are stable.
(b) Detcrmine the conditions under which the Rossby waves are stable.
(c) Discuss the impact of semi-implicit differencing on the accuracy of the
Rossby and gravity wave modes.
4. Two-dimensional sound waves in a neutrally stratified atmosphere satisfy
the Iinearized equations
8u
8P
-+cs-=O.
8t
8x
8w
8P
- + c s - =0.
where P = p' /(Poc s). Let this system be approximated using forwardbackward differencing for the horizontal gradients and trapezoidal differ-