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.
KeywordsGravity Wave Sound Wave Rossby Wave Phase Speed Lamb Wave
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- 1.Explicit time-differencing can still be used for the advection terms because the wind speed normal to the boundary decreases as the fluid approaches the boundary.Google Scholar
- 2.Higher-order implicit schemes are, however, not necessarily more stable than related explicit methods. Backward and trapezoidal differencing are the first- and second-order members of the Adams—Moulton family of implicit time integration schemes. The third- and fourth-order Adams—Moulton schemes generate amplifying solutions to oscillation equation (2.30) for any choice of time step, whereas their explicit cousins, the third- and fourth-order Adams—Bashforth schemes, produce stable nonamplifying solutions whenever the time step is sufficiently small.Google Scholar
- 3.The inclusion of the Coriolis force also requires the inclusion of an additional prognostic equation for the other component of the horizontal velocity.Google Scholar
- 4.The isothermal atmosphere does support a free wave (known as the Lamb wave, see Section 7.5) that disappears in the limit Γ → 0, but it is not necessary to account for the Lamb wave in this discussion of semi-implicit differencing.Google Scholar
- 5.In particular, the most important features may consist of slow-moving Rossby waves, which appear as additional solutions to the Euler equations when latitudinal variations in the Coriolis force are included in the horizontal-momentum equations.Google Scholar
- 6.See Section 3.3 for a discussion of the impact of operator commutativity on the performance of fractional-step schemes.Google Scholar
- 7.One way to appreciate the diffeience in the effectiveness of divergence damping in the completely and partially split schemes is to note the difference in wavelength at which spurious pressure perturbations appear in each solution. The partially split scheme generates errors at much shorter wavelengths than those produced by the completely split method (compare Figs. 7.3a and 7.5b), and the shortwavelength features are removed more rapidly by the divergence damper.Google Scholar
- 8.There has been some concern about the well-posedness of initial-boundary value problems involving the hydrostatic equations (Oliger and SundstrOm 1978). It is not clear how to reconcile these concerns with the successful forecasts obtained twice daily at several operational centers for at least two decades using limited-area weather prediction models based on the hydrostatic governing equations.Google Scholar
- 9.Note that as a consequence of the primitive-equation approximation, the vertical velocity does not appear as part of the kinetic energy.Google Scholar
- 10.The approximations used to obtain (7.142) and (7.143) are motivated by the desire to obtain a clean dispersion relation rather than a straightforward scale analysis.Google Scholar