7
Physically Insignificant Fast Waves
One reason that explicit time-differencing is widely used in the simulation of
wave-like flows is that accuracy considerations and stability constraints often
yield similar criteria for the maximum time step in numerical integrations of systems that support a single type of wave motion. Many fluid systems, however,
support more than one type of wave motion, and in such circumstances accuracy
considerations and stability constraints can yield very different criteria for the
maximum time step. If explicit time-differencing is used to construct a straightforward numerical approximation to the equations governing a system that supports several types of waves, the maximum stable time step will be limited by
the Courant number associated with the most rapidly propagating wave, yet that
rapidly propagating wave may be of little physical significance.
As an example, consider the Earth 's atmosphere, which supports sound waves,
gravity waves, and Rossby waves. Rossby waves propagate more slowly than
gravity waves which in turn move more slowly than sound waves. The maximum
stable time step with which an explicit numerical method can integrate the full
equations goveming atmospheric motions will be limited by the Courant number associated with sound-wave propagation. If finite differences are used in the
vertical, and the vertical grid spacing is 300 m, the maximum stable time step
will on the order of one second. Since sound waves have no direct meteorological significance, they need not be accurately simulated in order to obtain a good
weather forecast. The quality of the weather forecast depends solelyon the ability
of the model to accurately simulate atmospheric disturbances that evolve on much
slower time scales. Gravity waves can be accurately simulated with time steps on
the order of 10 to 100 seconds; Rossby waves require a time step on the order of
500 to 5000 seconds. In order to obtain a reasonably efficient numerical model for
the simulation of atmospheric circulations, it is necessary to circumvent the staD. R. Durran, Numerical Methods for Wave Equations in Geophysical Fluid Dynamics
© Springer Science+Business Media New York 1999
Physically Insignificant Fast Waves
One reason that explicit time-differencing is widely used in the simulation of
wave-like flows is that accuracy considerations and stability constraints often
yield similar criteria for the maximum time step in numerical integrations of systems that support a single type of wave motion. Many fluid systems, however,
support more than one type of wave motion, and in such circumstances accuracy
considerations and stability constraints can yield very different criteria for the
maximum time step. If explicit time-differencing is used to construct a straightforward numerical approximation to the equations governing a system that supports several types of waves, the maximum stable time step will be limited by
the Courant number associated with the most rapidly propagating wave, yet that
rapidly propagating wave may be of little physical significance.
As an example, consider the Earth 's atmosphere, which supports sound waves,
gravity waves, and Rossby waves. Rossby waves propagate more slowly than
gravity waves which in turn move more slowly than sound waves. The maximum
stable time step with which an explicit numerical method can integrate the full
equations goveming atmospheric motions will be limited by the Courant number associated with sound-wave propagation. If finite differences are used in the
vertical, and the vertical grid spacing is 300 m, the maximum stable time step
will on the order of one second. Since sound waves have no direct meteorological significance, they need not be accurately simulated in order to obtain a good
weather forecast. The quality of the weather forecast depends solelyon the ability
of the model to accurately simulate atmospheric disturbances that evolve on much
slower time scales. Gravity waves can be accurately simulated with time steps on
the order of 10 to 100 seconds; Rossby waves require a time step on the order of
500 to 5000 seconds. In order to obtain a reasonably efficient numerical model for
the simulation of atmospheric circulations, it is necessary to circumvent the staD. R. Durran, Numerical Methods for Wave Equations in Geophysical Fluid Dynamics
© Springer Science+Business Media New York 1999