For better projections of sea level rise, two things are needed: an improved understanding of the contributing processes and their accurate representation in climate models. A major process is basal melting of ice shelves and glacier tongues by the ocean, which reduces ice sheet stability and increases ice discharge into the ocean. We study marine melting of Greenland’s largest floating ice tongue, the 79° North Glacier, using a high-resolution, 2D-vertical ocean model. While our fjord model is idealized, the results agree with observations of the meltrate and the overturning strength. Our setup is the first application of adaptive vertical coordinates to an ice cavity. Their stratification-zooming allows a vertical resolution finer than 1 m in the entrainment layer of the meltwater plume, which is important for the plume development. In a sensitivity study, we show that the buoyant plume at the ice–ocean interface is responsible for the bulk of basal melting. The melting almost stops when the plume has reached neutral buoyancy. There, the plume detaches from the ice tongue and transports meltwater out of the fjord. The detachment depth depends primarily on the ambient ocean stratification. Our results contribute to the understanding of ice–ocean interactions in glacier cavities. Furthermore, we suggest that our modeling approach with stratification-zooming coordinates will improve the representation of these interactions in global ocean models. Finally, our idealized model topography and forcing are close to a real fjord and completely defined analytically, making the setup an interesting reference case for future model developments.