372
7. Physically InsignificantFast Waves
however, have the advantage of aIlowing a wider choice of methods for the integration of the remaining oscillatory forcing terms, which are approximated using
leapfrog differencing in the conventional semi-irnplicit method .
The elliptic pressure equations that appear when the semi-implicit or projection
methods are used are most efficiently solved by sophisticated algorithms such as
block-cyclic reduction, conjugate gradient, or multigrid methods. One may think
of the srnall-time-step procedure used in the fractional-step methods as a sort of
specialized iterative solver for the Helmholtz equation obtained using the conventional serni-implicit method. The difference in the character of the solution
obtained by the complete- and the partial-splitting methods can be appreciated by
considering the behavior ofthe divergence during the small-time-step integration.
During the small -time-step portion of the completely split method the divergence satisfies
a
2 8 _ c2V28 = a
2 b
ar2
S
araz
The initial conditions for 8 are those at the beginning of each small-time-step cycle, and since divergence is typically generated by the operators evaluated on the
large time step, the initial 8 is nonzero . This divergence is propagated without
loss during the srnall-time-step integration (except for minor modification by the
buoyancy forcing) and tends to accumulate over aseries of large-step-small-step
cycles. The test in which the completely split scheme performs weIl is the case
in which the basic-state horizontal velocity is uniform throughout the fluid. When
U is constant, the linearized advection operator merely produces a Galilean translation of the fluid that does not generate any divergence. (RecaIl that the forcing
from the wave generator was computed on the small time step.) Nonlinear advection can, of course , generate divergence , as can the linearized advection operator
when there is vertical shear in the basic-state wind, and these are the circumstances in which the complete-splitting method produces spurious sound waves.
In contrast, the divergence is almost zero at the start of the first smaIl time step
of the partiaIly split method, and only smaIl changes in the divergence are forced
during each individual smaIl step. Moreover, the divergence forcing on each smaIl
time step closely approximates that which would appear in an explicit srnall-timestep integration of the full compressible equations, provided that the amplitude of
aIl the sound waves is negligible in comparison to slower modes. The divergence
damper ensures that the amplitude of the sound waves remains smaIl and thereby
preserves the stability and accuracy of the solution.
7.5 The Hydrostatic Approximation
Large-scale atmospheric and oceanic motions are very nearly in hydrostatic balance, and as a consequence, they are weIl described by an approximate set of
goveming equations in which the full vertical momentum equation is replaced by
7. Physically InsignificantFast Waves
however, have the advantage of aIlowing a wider choice of methods for the integration of the remaining oscillatory forcing terms, which are approximated using
leapfrog differencing in the conventional semi-irnplicit method .
The elliptic pressure equations that appear when the semi-implicit or projection
methods are used are most efficiently solved by sophisticated algorithms such as
block-cyclic reduction, conjugate gradient, or multigrid methods. One may think
of the srnall-time-step procedure used in the fractional-step methods as a sort of
specialized iterative solver for the Helmholtz equation obtained using the conventional serni-implicit method. The difference in the character of the solution
obtained by the complete- and the partial-splitting methods can be appreciated by
considering the behavior ofthe divergence during the small-time-step integration.
During the small -time-step portion of the completely split method the divergence satisfies
a
2 8 _ c2V28 = a
2 b
ar2
S
araz
The initial conditions for 8 are those at the beginning of each small-time-step cycle, and since divergence is typically generated by the operators evaluated on the
large time step, the initial 8 is nonzero . This divergence is propagated without
loss during the srnall-time-step integration (except for minor modification by the
buoyancy forcing) and tends to accumulate over aseries of large-step-small-step
cycles. The test in which the completely split scheme performs weIl is the case
in which the basic-state horizontal velocity is uniform throughout the fluid. When
U is constant, the linearized advection operator merely produces a Galilean translation of the fluid that does not generate any divergence. (RecaIl that the forcing
from the wave generator was computed on the small time step.) Nonlinear advection can, of course , generate divergence , as can the linearized advection operator
when there is vertical shear in the basic-state wind, and these are the circumstances in which the complete-splitting method produces spurious sound waves.
In contrast, the divergence is almost zero at the start of the first smaIl time step
of the partiaIly split method, and only smaIl changes in the divergence are forced
during each individual smaIl step. Moreover, the divergence forcing on each smaIl
time step closely approximates that which would appear in an explicit srnall-timestep integration of the full compressible equations, provided that the amplitude of
aIl the sound waves is negligible in comparison to slower modes. The divergence
damper ensures that the amplitude of the sound waves remains smaIl and thereby
preserves the stability and accuracy of the solution.
7.5 The Hydrostatic Approximation
Large-scale atmospheric and oceanic motions are very nearly in hydrostatic balance, and as a consequence, they are weIl described by an approximate set of
goveming equations in which the full vertical momentum equation is replaced by