Fourier filtering and coefficient tapering at the North Pole in OGCMs

This article considers how some of the measures used to overcome numerical problems near the North Pole affect the ocean solution and computational time step limits. The distortion of the flow and tracer contours produced by a polar island is obviated by implementing a prognostic calculation for a composite polar grid cell, as has been done at NCAR. The severe limitation on time steps caused by small zonal grid spacing near the pole is usually overcome by Fourier filtering, sometimes supplemented by the downward tapering of mixing coefficients as the pole is approached; however, filtering can be expensive, and both measures adversely affect the solution. Fourier filtering produces noise, which manifests itself in such effects as spurious static instabilities and vertical motions; this noise can be due to the separate and different filtering of internal and external momentum modes and tracers, differences in the truncation at different latitudes, and differences in the lengths of filtering rows, horizontally and vertically. Tapering has the effect of concentrating tracer gradients and velocities near the pole, resulting in some deformation of fields. In equilibrium ocean models, these effects are static and localised in the polar region, but with time-varying forcings or coupling to atmosphere and sea ice it is possible that they may seriously affect the global solution. The marginal stability curve in momentum and tracer time-step space should have asymptotes defined by diffusive, viscous, and internal gravity wave stability criteria; at large tracer time steps, tracer advection stability may become limiting. Tests with various time-step combinations and a flat-bottomed Arctic Ocean have confirmed the applicability of these limits and the predicted effects of filtering and tapering on them. They have also shown that the need for tapering is obviated by substituting a truncation which maintains a constant time step limit rather than a constant minimum wave number over the filtering range.