During periods of northward Bz, the electric field applied to the magnetosphere is generally opposite to that occurring during southward Bz, and complicated patterns of convection result, showing some features reversed in comparison with the southward Bz case. We study a simple generalization of early work on idealized convection models, which allows for the coexistence of sunward convection over the central polar cap and antisunward convection elsewhere in the cap. The present model, valid for By ≃0, has a four-cell convection pattern and is based on the combination of ionospheric current conservation with a relation between parallel auroral currents and parallel potential drops. We do not consider global magnetospheric issues involving, e.g., reconnection. The central result of this paper is an expression giving the parallel potential drop for polar cap auroras (with By ≃0) in terms of the polar cap convection field profile.