Modelling topographic waves in a polar basin

This study is concerned with properties of freely propagating barotropic Rossby waves in a circular polar cap, a prototype model for the Arctic Ocean. The linearised shallow-water equations are used to derive an amplitude equation for the waves in which full spherical geometry is retained. Almost by definition, polar basin dynamics are confined to regions of limited latitudinal extent and this provides a natural small scale which can underpin a rational asymptotic analysis of the amplitude equation. The coefficients of this equation depend on the topography of the basin and, as a simple model of the Arctic basin, we assume that the basin interior is characterised by a constant depth, surrounded by a continental shelf-slope the depth of which has algebraic dependence on co-latitude. Isobaths are therefore a family of concentric circles with centre at the pole. On the shelf and slope regions the leading order amplitude equation is of straightforward Euler type. Asymptotic values of the wave frequencies are derived and these are compared to values computed directly from the full amplitude equation. It is shown that the analytic results are in very good accord with the numerical predictions. Further simulations show that the properties of the waves are not particularly sensitive to the precise details of the underlying topography; this is reassuring as it is difficult to faithfully represent the shelf topography using simple mathematical functions.