6.4.3 Caveat: inversions with steep topography

In order to solve for the barotropic velocity $\overline{{\bf u}}$, it is necessary to invert the elliptical operator appearing in the relation (6.30) relating the barotropic vorticity to the barotropic streamfunction. The presence of the inverse depth in this operator implies that near regions where H changes rapidly, the elliptic operator also changes rapidly. As emphasized by Smith, Dukowicz, and Malone (1992) and Dukowicz, Smith, and Malone (1993), numerical elliptic inversions will potentially have problems converging in the presence of such regions. This form for the elliptic operator is also the central reason for the Killworth topographic instability (Killworth 1987). This instability can result in significant time step constraints in rigid lid models with non-smoothed topography, and these time step constraints can be more stringent than the typical CFL constraints. More problematic from the perspective of very long-term climate modeling, the Killworth instability can sometimes be slowly growing, and may become problematical only after some few hundreds of years. A slow growing instability is arguably more pernicious than rapidly growing instabilities, since a significant amount of computation time can be used before encountering the problem. Experience at GFDL indicates that the Killworth instability is a nontrivial problem with realistic models, and so it provides motivation to avoid the rigid lid streamfunction method in such models.