5.2.2.1 The barotropic equation and its two solution methods

Integrating the continuity equation vertically yields a prognostic equation for the sea surface elevation $\eta$, together with the vertically integrated momentum equations. These equations describe the fast barotropic gravity waves. In general, the numerical solution scheme for these equations requires a very small time step if these waves are resolved. There are two methods used in MOM: the option implicit_free_surface of Dukowicz and Smith (1994), and the option explicit_free_surface. The explicit free surface can be time stepped either by the methods of Killworth, Stainforth, Webb and Paterson (1991), or by the methods discussed in Section 29.5 and in Griff ies, Pacanowski, Schmidt, and Balaji (2000). The implicit method as coded in MOM has not been found to be unconditionally stable, although in principle it should be so. In general, if the time step is too large, the barotropic gravity waves are damped out with the implicit scheme. If this damping is not desired, then the explicit method can be used. In this approach, the barotropic mode is integrated with a small time step. While integrating the barotropic equations, the baroclinic flow as well as tracer fields are kept constant. This approximation is justified since the time scale of baroclinic and tracer processes is much larger than the barotropic process. Note that if the time step is smaller than required by the CFL-criterion for barotropic gravity waves, both the implicit and explicit methods have been found to yield similar results.