Numerical stabilisation of grounding line dynamics in Stokes problems

The grounding line marks the boundary between grounded and floating ice, and is a critical region for ice-sheet stability and sea-level projections. The complex ice-flow at the grounding line, where the stress regime moves from vertical shear to horizontal extension over a relatively short distance, is prone to numerical instability in transient full-Stokes simulations. Furthermore, boundary conditions change at the grounding line, switching from a friction law in grounded ice to an ocean pressure force at the ice-ocean interface. Grounding-line full-Stokes problems have been successfully stabilised by the sea spring stabilisation scheme in Elmer/Ice (Durand et al., 2009) which mimicks an implicit time-stepping scheme by predicting the surface elevation and corresponding ocean pressure corrections in the next time step. We extend on this stabilisation approach by introducing the Free-Surface Stabilisation Approximation (FSSA) to the ice-ocean interface. FSSA has been proven successful in allowing larger stable time steps in grounded problems with an evolving ice-atmosphere interface (Löfgren et al., 2022; Löfgren et al., 2024). This stabilisation approach incorporates a boundary condition term into the weak-form of the Stokes equations representing the predicted stress adjustment between the current and next time step. Using a synthetic MISMIP set up (Pattyn et al., 2012), we investigate the applicability of FSSA to the ice-ocean interface.G. Durand, O. Gagliardini, B. de Fleurian, T. Zwinger, and E. Le Meur. Marine ice sheet dynamics: Hysteresis and neutral equilibrium. Journal of Geophysical Research, 114(F3):F03009, 2009. doi: 10.1029/2008JF001170.A. Löfgren, T. Zwinger, P. Råback, C. Helanow, and J. Ahlkrona. Increasing numerical stability of mountain valley glacier simulations: implementation and testing of free-surface stabilization in Elmer/Ice. The Cryosphere, 18(8):3453–3470, 2024. doi: 10.5194/tc-18-3453-2024.A. Löfgren, J. Ahlkrona, and C. Helanow. Increasing stable time-step sizes of the free-surface problem arising in ice-sheet simulations. Journal of Computational Physics: X, 16:100114, 2022. doi: 10.1016/j.jcpx.2022.100114.F. Pattyn, C. Schoof, L. Perichon, R. C. A. Hindmarsh, E. Bueler, B. de Fleurian, G. Durand, O. Gagliardini, R. Gladstone, D. Goldberg, G. H. Gudmundsson, P. Huybrechts, V. Lee, F. M. Nick, A. J. Payne, D. Pollard, O. Rybak, F. Saito, and A. Vieli. Results of the Marine Ice Sheet Model Intercomparison Project, MISMIP. The Cryosphere, 6(3):573–588, 2012. doi: 10.5194/tc-6-573-2012.