Numerical models of ocean circulation typically admit motions varying on a wide range of time scales. These include external gravity waves, which can travel with speeds of up to hundreds of meters per second. On the other hand, motions such as currents and internal gravity waves exhibit speeds on the order of a meter per second or less. In the case of external waves, the disturbances are nearly independent of depth, and for the sake of efficiency it is common practice to represent these motions with a two-dimensional subsystem which is solved by techniques that are different from those used to simulate the remaining three-dimensional motions. However, if this splitting of fast and slow time scales is insufficiently accurate, then the resulting algorithm can be unstable. The instability can be removed by using a more precise vertical averaging when deriving the two-dimensional system that models the fast waves. One goal of this talk is to outline these results.
A second goal is to describe some other issues related to time stepping schemes. A time discretization that is commonly used in geophysical fluid dynamics is the leap-frog method, which is a three-level scheme based on centered differencing. One drawback of this method is a computational mode consisting of nonphysical, grid-scale oscillations with respect to time. These oscillations contaminate the solution and can be strongly stimulated by impulsive forcing. This is the case, for example, with a prominent ocean model which allows sudden transfers of mass between the uppermost ``mixed'' layer and the stratified interior. These problems can be avoided by using a time stepping method that uses only two time levels. I will sketch some preliminary results for a two-level scheme which incorporates the improved splitting mentioned above.