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Grounding lines exist where land-based glacial ice flows on to a body of water. Accurately modelling grounding-line migration at the ice–ocean interface is essential for estimating future ice-sheet mass change. On the interior of ice sheets, the shores of subglacial lakes are also grounding lines. Grounding-line positions are sensitive to water volume changes such as sea-level rise or subglacial-lake drainage. Here, we introduce numerical methods for simulating grounding-line dynamics in the marine ice sheet and subglacial-lake settings. Variational inequalities arise from contact conditions that relate normal stress, water pressure and velocity at the base. Existence and uniqueness of solutions to these problems are established using a minimisation argument. A penalty method is used to replace the variational inequalities with variational equations that are solved using a finite-element method. We illustrate the grounding-line response to tidal cycles in the marine ice-sheet problem and filling–draining cycles in the subglacial-lake problem. We introduce two computational benchmarks where the known lake volume change is used to measure the accuracy of the numerical method. 

Abstract Magmatic systems are composed of melt accumulations and crystal mush that evolve with melt transport, contributing to igneous processes, volcano dynamics, and eruption triggering. Geophysical studies of active volcanoes have revealed details of shallow-level melt reservoirs, but little is known about fine-scale melt distribution at deeper levels dominated by crystal mush. Here, we present new seismic reflection images from Axial Seamount, northeastern Pacific Ocean, revealing a 3–5-km-wide conduit of vertically stacked melt lenses, with near-regular spacing of 300–450 m extending into the inferred mush zone of the mid-to-lower crust. This column of lenses underlies the shallowest melt-rich portion of the upper-crustal magma reservoir, where three dike intrusion and eruption events initiated. The pipe-like zone is similar in geometry and depth extent to the volcano inflation source modeled from geodetic records, and we infer that melt ascent by porous flow focused within the melt lens conduit led to the inflation-triggered eruptions. The multiple near-horizontal lenses are interpreted as melt-rich layers formed via mush compaction, an interpretation supported by one-dimensional numerical models of porous flow in a viscoelastic matrix.