An Ekman analysis of the surface drainage winds over a sloping ice surface is reported. Ekman pumping by the boundary layer leads to the formation of an upper tropospheric cyclonic vortex above the summit of the ice sheet. The strength and distribution of upper level vorticity is determined by the shape of the underlying ice sheet. The calculation is verified by comparison with the results from a multi-level primitive equation model of flow above an axisymmetric ice sheet. Both models predict that the surface drainage flow will die out on a timescale of a few days, while the upper vortex is predicted to be considerably stronger than observed. Various mechanisms which could lead to the depletion of upper level vorticity, and hence to the retention of a substantial drainage flow, are discussed. It is concluded that disruption of the polar vortex by decaying mid-latitude cyclones, and the consequent export of cyclonic vorticity to lower latitudes, is the most probable mechanism.