A
W=
w 4
where
sinwßt
-u.
k
ßt
This dispersion relation is quadratic in w 2 and has solutions
352
7. Physically Insignificant Fast Waves
fluence of density variations on buoyancy and assumes that buoyancy is conserved
following a fluid parcel. In contrast to the standard Boussinesq system, the compressible Boussinesq system does retain the influence of density fluctuations on
pressure and thereby allows the formation of the prognostic pressure equation
(7.41).
Suppose that the simplified compressible system (7.39)-(7.40) is approximated
using leapfrog time-differencing and that the spatial derivatives are computed using a Fourier pseudospectral method. Waves of the form
(u, W, b, P) = (uo, Wo, bo, Po)ei(kx+iz-wnAt)
are solutions to this system, provided that w, k, and z satisfy the dispersion relation
- e; (k 2 +.e 2 + N 2 je;) w 2 + N 2k2 e; = 0,
The positive root yields the dispersion relation for sound waves; the negative root
yields the dispersion relation for gravity waves. The individual dispersion relations for sound and gravity waves may be greatly simplified whenever the last
term inside the square root in (7.43) is much smaller than the first term. One sufficient condition for this simplification is that
(7.44)
in which case
w 2 = e; (k 2 +.e 2 + N 2j c; ) ,
and the gravity-wave-dispersion relation becomes
(7.45)
In most applications (7.44) is easily satisfied, so the sound-wave-dispersion relation simplifies to
(7.46)