external waves were not present. The restriction on the time step can be removed by using an implicit method [11];
Numerical models of ocean circulation admit motions varying on a wide range of time scales. These motions include fast external this approach requires that a large system of algebraic gravity waves, which are approximately independent of depth, and equations be solved at each time step. A commonly used slower internal motions which are fully three-dimensional. Explicit alternative is to split the governing equations into subtime discretizations are impractical for these systems, due to the systems that model the fast and slow motions separately; short timesteps dictated by the fast waves. A commonly used alterthese will be referred to as the barotropic and baroclinic native is to confine the fast waves to a two-dimensional system, via vertical averaging, and then to compute the remaining motions systems, respectively. Such splittings are used, for example, explicitly with a long time step. However, this procedure can lead in [1,5, 9]. Because the fast motions are approximately to numerical instability if the latter system admits sufficiently large independent of z, the barotropic system is two-dimensional residual fast motions due to an inexact splitting. In this paper we and generally resembles the shallow water equations that modify a method developed by R. Bleck and L. T. Smith (J. Geophys.
model the motion of a fluid of constant density. This system Res. C 95, 3273, 1990) in order to obtain a more precise splitting into fast and slow subsystems. In the vertically averaged momentum can be solved explicitly with short time steps or implicitly equation, we use the exact vertical average of the horizontal preswith long time steps; in this case, the costs of such methods sure gradient in place of the approximation used in op cit. We are not great, due to the restriction to two space dimenthen apply natural time discretizations and show that the modified sions. The baroclinic system is fully three-dimensional and splitting produces considerable improvements in stability. แฎ 1997 is solved explicitly with a long time step that is appropriate Academic Press for resolving the slow internal motions. However, if the splitting is inexact, then the baroclinic system can actually