Subglacial meltwater draining along the bed of fast-flowing, marine-terminating glaciers emerges at the grounding line, where the ice either goes afloat to form an ice shelf or terminates in a calving face. The input of freshwater to the ocean provides a source of buoyancy and drives convective motion alongside the ice-ocean interface. This process is modeled using the theory of buoyant plumes that has previously been applied to the study of the larger-scale circulation beneath ice shelves. The plume grows through entrainment of ocean waters, and the heat brought into the plume as a result drives melting at the ice-ocean interface. The equations are nondimensionalized by using scales appropriate for the region where the subglacial drainage, rather than the subsequent addition of meltwater, supplies the majority of the buoyancy forcing. It is found that the melt rate within this region can be approximated reasonably well by a function that is linear in ocean temperature, has a cube root dependence on the flux of subglacial meltwater, and has a complex dependency on the slope of the ice-ocean interface. The model is used to investigate variability in melting induced by changes in both ocean temperature and subglacial discharge for a number of realistic examples of ice shelves and tidewater glaciers. The results show how warming ocean waters and increasing subglacial drainage both generate increases in melting near the grounding line.