5.2.2.2 The non-rigid lid approximation

An ocean model with a free surface strictly has a time dependent domain with a top model grid box possessing a variable upper surface. This added degree of freedom increases the complexity of the numerical scheme for the tracers and the baroclinic mode due to the need to take into account the variable top model grid box. Instead of introducing this complexity, those using a free surface in z-level models have instead considered a model with fixed vertical levels. The upper kinematic, dynamic, and tracer boundary conditions are formulated as open boundary conditions applied at z=0 rather than at $z=\eta$. The application of the upper boundary at z=0 is similar to the rigid lid approach. However, for the non-rigid lid free surface, the upper boundary is open or permeable, whereas the upper boundary in the rigid lid method is closed or impermeable. Most notably, the vertical velocity w(z=0) does not vanish with the free surface. This comparison motivates the name non-rigid lid method for the handling of surface boundary conditions in the free surface method.

An approach such as this was employed by Killworth et al. (1989) and is also used by Dukowicz and Smith (1994). In general, the non-rigid lid approximation is well justified so long as the top model grid box between $z_{1} \le z \le 0$ is much thicker than the maximum free surface height $\eta$. In shallow seas or in models with very fine vertical grid spacings, this assumption is not valid. More crucially, such models do not conserve total tracer or momentum. It is for this reason that MOM has recently (Summer 1999) implemented a full free surface method in which the effects of the undulating surface height has been incorporated into the depth dependent fields. This method is fully documented in Griff ies, Pacanowski, Schmidt, and Balaji (2000).