356
7. PhysicallyInsignificant Fast Waves
time steps by replacing (7.49) and (7.50) with
(CulIen 1990; Tanguay et al. 1990). Note that the buoyancy forcing in the verticalmomentum equation and the vertical advection of the mean-state buoyancy in the
buoyancy equat ion are now treated by trapezoidal differences. The gravity-wave
dispersion relation for this generalized semi-implicit system is
or
This dispersion relation has the same form as that for the prototype semi-implicit
scheme (7.25), and as discussed in connection with (7.26), stable solutions will
be obtained, provided that IU k Ißt S 1.
7.2.5 Numericallmplementation
The semi-implicit approximation to the compressible Boussinesq system discussed
in the preceding section generates a system of implicit algebraic equations that
must be solved at every time step. The solution procedure will be illustrated in a
relatively simple example using the nonlinear compressible Boussinesq equations
db
2
-+N w=O,
dt
dv
-+VP =bk,
dt
dP
2
-+CsV .v=O.
dt
(7.54)
(7.55)
(7.56)
The definitions of b, P, and N given in (7.42) may be used to show that (7.54)
and (7.55) are identical to the buoyancy and momentum equations in the standard Boussinesq system (7.2) and (7.3). The standard incompressible continuity
equation has been replaced by (7.56) and is recovered in the limit C s ---+ 00.
First consider the situation where only the sound waves are stabilized by semiimplicit differencing and suppose that the spatial derivatives are not discretized.
7. PhysicallyInsignificant Fast Waves
time steps by replacing (7.49) and (7.50) with
(CulIen 1990; Tanguay et al. 1990). Note that the buoyancy forcing in the verticalmomentum equation and the vertical advection of the mean-state buoyancy in the
buoyancy equat ion are now treated by trapezoidal differences. The gravity-wave
dispersion relation for this generalized semi-implicit system is
or
This dispersion relation has the same form as that for the prototype semi-implicit
scheme (7.25), and as discussed in connection with (7.26), stable solutions will
be obtained, provided that IU k Ißt S 1.
7.2.5 Numericallmplementation
The semi-implicit approximation to the compressible Boussinesq system discussed
in the preceding section generates a system of implicit algebraic equations that
must be solved at every time step. The solution procedure will be illustrated in a
relatively simple example using the nonlinear compressible Boussinesq equations
db
2
-+N w=O,
dt
dv
-+VP =bk,
dt
dP
2
-+CsV .v=O.
dt
(7.54)
(7.55)
(7.56)
The definitions of b, P, and N given in (7.42) may be used to show that (7.54)
and (7.55) are identical to the buoyancy and momentum equations in the standard Boussinesq system (7.2) and (7.3). The standard incompressible continuity
equation has been replaced by (7.56) and is recovered in the limit C s ---+ 00.
First consider the situation where only the sound waves are stabilized by semiimplicit differencing and suppose that the spatial derivatives are not discretized.