Flow over Topography and Instability on Beta-Planes: The Effects of Different Expansion Representations

Abstract

We examine the linear instability properties of exact steady solutions for barotropic flow over topography in a beta-plane channel using models with different expansion representations and truncations. We compare our results with those of previous studies and find that in many cases the models used previously contained deficiencies. These range from (i) the stationary basic states in the truncated system not being exact solutions of the original barotropic vorticity equation to (ii) the form drag zonal momentum equation not being satisfied by the truncated scheme to (iii) the boundary conditions not being satisfied by the truncated perturbation to (iv) form drag instability only occurring for basic states and topographies with odd meridional wavenumber. In our study, we use a truncation scheme and expansion of the streamfunction, in terms of a nonorthogonal large-scale flow &ndashUy and meridional sine expansions for both the zonal and wave terms, which ensure that none of the problems occur at each level of truncation. These results are compared with those in a model in which the perturbation has a purely orthogonal function representation including a zonal flow represented by just a cosine expansion. At severe truncation there are significant difference between the two. At high resolution, we find that in some parts of parameter space there are significant qualitative, as well as quantitative, differences between the models while in other parts of parameter space qualitatively similar results may be obtained. The nonlinear stability properties of stationary flow in each model are also examined and differences even at infinite resolution are attributed to Gibbs phenomena with the purely orthogonal function representation.