Ocean-forced basal melting has been implicated in the widespread thinning of Antarctic ice shelves, but an understanding of what determines melt rates is hampered by limited knowledge of the buoyancy- and frictionally controlled flows along the ice shelf base that regulate heat transfer from ocean to ice. In an attempt to address this deficiency, a simple model of a buoyant boundary flow, considering only the spatial dimension perpendicular to the boundary, is presented. Results indicate that two possible flow regimes exist: a weakly stratified, geostrophic cross-slope current with upslope flow within a buoyant Ekman layer or a strongly stratified, upslope current with a weak cross-slope flow. The latter regime, which is analogous to the steady solution for a katabatic wind, is most appropriate when the ice–ocean interface is steep. For the gentle slopes typical of Antarctic ice shelves, the buoyant Ekman regime, which has similarities with the case of an unstratified density current on a slope, provides some useful insight. When combined with a background flow, a range of possible near-ice current profiles emerge as a result of arrest or enhancement of the upslope Ekman transport. A simple expression for the upslope transport can be formed that is analogous to that for the wind-forced surface Ekman layer, with curvature of the ice shelf base replacing the wind stress curl in driving exchange between the Ekman layer and the geostrophic current below.