A numerical method for subglacial cavitation posed as a variational inequality

Subglacial cavitation is a phenomenon that occurs at the base of an ice sheet or a glacier where the ice detaches from the bedrock at high water pressures. The process is recognised as an essential mechanism in glacial sliding. A mathematical description of subglacial cavitation involves a free boundary equation and a Stokes equation with contact boundary conditions. These contact boundary conditions model the process of detachment from the bed at each instant in time. 

In this talk we show that the problem can be written as a variational inequality and present a novel approach to solving the equations with finite element methods that exploit the structure of the variational inequality. In particular, we present a formulation involving Lagrange multipliers, which allows us to solve the discrete contact conditions exactly. Thanks to this latter property, the Stokes equations can be solved together with the free boundary equations in a robust and stable manner. A similar method should also prove useful for improving grounding-line calculations.

With this numerical method, we compute a friction law (the relation between sliding velocity and shear stress) for ice flowing over a periodic bed.  We recover existing results for the case when the cavities are in a steady state for a given effective pressure. We extend these results to consider time-varying cavitation driven by changes in subglacial water pressure.