336
7. Physically Insignificant Fast Waves
bility constraint associated with sound-wave propagation and bring the maximum
stable time step into closer agreement with the time step limitations arising from
accuracy considerations.
There are two basic approaches for circumventing the time step constraint imposed by a rapidly moving, physically insignificant wave. The first approach is
to approximate the full goveming equations with a set of "filtered" equations that
does not support the rapidly moving wave. As an example, the full equations for
stratified compressible flow might be approximated by the Boussinesq equations
for incompressible flow. In this approach fundamental approximations are introduced to the original continuous equations prior to any numerical approximations
that may be subsequently employed to generate finite-difference or spectral solutions to the filtered goveming equations. The use of the filtered equation set may
be motivated entirely by numerical considerations, or it may arise naturally from
the standard approximations used in the study of a given physical phenomenon.
Gravity waves, for example, are often studied in the context of Boussinesq incompressible flow in order to simplify and streamline the mathematical description of
the problem.
The second approach for circumventing the time step constraint imposed by a
rapidly moving, physically insignificant wave leaves the continuous goveming
equations unmodified and relies on numerical techniques to stabilize the fastmoving wave. These numerical techniques achieve efficiency by sacrificing the
accuracy with which the fast-rnoving wave is represented. Note that neither one
ofthese approaches is appropriate in situations where the fast-moving wave needs
to be accurately simulated, since small time steps are required to adequately resolve the fast-moving wave.
This chapter begins by examining techniques for the numerical solution of the
Boussinesq equations, which is one of the most fundamental systems of filtered
equations . Methods for the solution of a second system of filtered equations, the
primitive equations, are presented in Sections 7.5-7 .6. Numerical methods for
stabilizing the solution to problems that simultaneously support fast- and slowmoving waves are considered in Sections 7.2-7.4. One of these techn iques, the
semi-implicit method, is frequently used to integrate the primitive equations in
applications where the phenomena of primary interest are slow-rnoving Rossby
waves. In such applications the numerical integration is stabilized with respect
to two different types of physically insignificant, rapidly moving waves. Sound
waves are filtered by the primitive-equation approximation, and the most rapidly
moving gravity waves (and the Lamb wave) are stabilized by the semi-implicit
time integration.
7.1 The Projection Method
The unapproximated mass conservation equation (1.32) is a prognostic equation
for the density that can be combined with the equation of state to form a prognostic equation for pressure . The equat ion of state can also be used to eliminate
7. Physically Insignificant Fast Waves
bility constraint associated with sound-wave propagation and bring the maximum
stable time step into closer agreement with the time step limitations arising from
accuracy considerations.
There are two basic approaches for circumventing the time step constraint imposed by a rapidly moving, physically insignificant wave. The first approach is
to approximate the full goveming equations with a set of "filtered" equations that
does not support the rapidly moving wave. As an example, the full equations for
stratified compressible flow might be approximated by the Boussinesq equations
for incompressible flow. In this approach fundamental approximations are introduced to the original continuous equations prior to any numerical approximations
that may be subsequently employed to generate finite-difference or spectral solutions to the filtered goveming equations. The use of the filtered equation set may
be motivated entirely by numerical considerations, or it may arise naturally from
the standard approximations used in the study of a given physical phenomenon.
Gravity waves, for example, are often studied in the context of Boussinesq incompressible flow in order to simplify and streamline the mathematical description of
the problem.
The second approach for circumventing the time step constraint imposed by a
rapidly moving, physically insignificant wave leaves the continuous goveming
equations unmodified and relies on numerical techniques to stabilize the fastmoving wave. These numerical techniques achieve efficiency by sacrificing the
accuracy with which the fast-rnoving wave is represented. Note that neither one
ofthese approaches is appropriate in situations where the fast-moving wave needs
to be accurately simulated, since small time steps are required to adequately resolve the fast-moving wave.
This chapter begins by examining techniques for the numerical solution of the
Boussinesq equations, which is one of the most fundamental systems of filtered
equations . Methods for the solution of a second system of filtered equations, the
primitive equations, are presented in Sections 7.5-7 .6. Numerical methods for
stabilizing the solution to problems that simultaneously support fast- and slowmoving waves are considered in Sections 7.2-7.4. One of these techn iques, the
semi-implicit method, is frequently used to integrate the primitive equations in
applications where the phenomena of primary interest are slow-rnoving Rossby
waves. In such applications the numerical integration is stabilized with respect
to two different types of physically insignificant, rapidly moving waves. Sound
waves are filtered by the primitive-equation approximation, and the most rapidly
moving gravity waves (and the Lamb wave) are stabilized by the semi-implicit
time integration.
7.1 The Projection Method
The unapproximated mass conservation equation (1.32) is a prognostic equation
for the density that can be combined with the equation of state to form a prognostic equation for pressure . The equat ion of state can also be used to eliminate