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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 melt rate and 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. We find that the plume development is dominated by entrainment only initially. In the stratified upper part of the cavity, the subglacial plume shows continuous detrainment. It reaches neutral buoyancy near 100 m depth, detaches from the ice, and transports meltwater out of the fjord. Melting almost stops there. In a sensitivity study, we show that the detachment depth depends primarily on 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. Plain Language Summary: The global increase of sea levels is a consequence of human‐induced climate change. It presents a threat to coastal regions and demands action to protect human life and infrastructure near the coast. Planning protective measures requires projections of sea level rise, computed with climate models. We present an approach to improve the simulation of an important contributor to sea level rise: melting of floating ice shelves by ocean circulation. Our modeling approach uses a vertical model grid that evolves over time. The temporal evolution depends on the density structure of the ocean. Large density differences appear just below an ice shelf, where fresh meltwater mixes with salty seawater. The adaptive grid of our model resolves this mixing process in great detail. This is important for an accurate computation of the melt rate and enables us to study in depth the ice shelf–ocean interactions. We study them at the glacier tongue of the 79° North Glacier, which is Greenland's largest ice shelf. The physical understanding gained from our simulations is also applicable to other floating glacier tongues and ice shelves. We suggest that using the presented model technique in global ocean models can improve projections of melting and sea level rise. Key Points: Melting of the 79° North Glacier ice tongue by turbulent ocean currents is studied with an idealized 2D‐vertical fjord modelThe subglacial plume behaves like an entraining plume close to the grounding line and like a detraining gravity current further downstreamA vertical resolution finer than 1 m is achieved in the subglacial plume by using adaptive vertical coordinates that zoom to stratification [ABSTRACT FROM AUTHOR]