29.6.10 Checkerboard null mode

As discussed by Killworth et al. (1991), discretization of gravity waves on a B-grid can admit a stationary grid scale checkerboard pattern (see also Messinger 1973 and Jancic 1974 for atmospheric discussions). This pattern is associated with an unsuppressed grid splitting that can be initiated through grid scale forcing, such as topography. That is, the discrete equations admit a checkerboard null mode. Beckers (1999) also discusses subtleties associated with traditional linear stability. In our experience, when the model goes unstable due to CFL violations, this grid mode is the spatial pattern associated with the instability.

Strictly, the grid mode is present only when the surface pressure gradient takes the form $\nabla_{h} \, p_{s} = \overline{\rho} \, g \, \nabla_{h} \, \eta$, where $\overline{\rho}$is an averaged surface density. For example, in the implementation of Killworth et al. (1991), $\nabla_{h} \, p_{s} = \rho_{o} \, g \, \nabla_{h} \, \eta$, and so the null mode is always present. Tests with the alternative expression $p_{s} = \rho_{k=1} \, g \, \eta$ used in MOM, in which surface pressure gradients arise from gradients in both the surface density $\rho_{k=1}$ and surface height, suggest that the null mode is slightly suppressed. A more significant means to suppress the mode, however, is provided through the use of time averaging over the barotropic time steps, as well as with nonlinearity in the shallow water system present when the undulating surface height is fully incorporated to the dynamics.

Hence, for many of our simulations, the grid mode is absent or quite mild. Nonetheless, some experiments did realize the grid mode in the surface height, and so we considered various means of suppressing it. The following relates our experience with this mode and provides suggestions and caveats.