The Greenland Ice Sheet is an important contributor to global sea level rise. In northern Greenland, floating glacier tongues are primarily melted by ocean currents. These processes are difficult to observe, so we developed a realistic numerical model to study ocean-driven melting at Greenland's largest floating ice tongue, the 79° North Glacier. Our simulation reveals the details of the oceanic currents bringing warm water toward the ice base, melting and shaping the glacier tongue from below.