Viscous contact problems in glaciology
Keywords: non Newtonian flow
finite element
numerical analysis
friction law
variational inequality
glaciology
marine ice sheet
grounding line
ice
subglacial cavity

Viscous contact problems describe the time evolution of fluid flows in contact with a surface from which they can detach and reattach. They can be modelled by coupling the Stokes equations with contact boundary conditions and a free boundary equation that evolves the geometry of the domain occupied by the fluid. These problems are of particular importance in glaciology, where they arise in the study of grounding lines and subglacial cavities. This work investigates the numerical approximation of viscous contact problems with applications to these two examples in glaciology.
We commence by formulating the viscous contact problems that model subglacial cavitation and marine ice sheets in Chapter 1. We state the different variational inequalities that arise in these problems and are equivalent to the Stokes equations with contact boundary conditions.
We then propose a novel framework for building numerical schemes for these problems in Chapter 2. This framework considers a family of discrete variational inequalities and establishes certain conditions that should be satisfied when approximating the free boundary equations. We then describe the numerical scheme that is used for the remainder of this work and compare it with different schemes that fit the framework introduced beforehand.
Chapter 3 is dedicated to the numerical analysis of one of the Stokes variational inequalities formulated in this work. We give rigorous proofs on the conditions under which it is well-posed and its finite element approximation converges. By developing theoretical tools based on existing work in the numerical analysis of variational inequalities and p-Stokes systems, our analysis deals with three substantial difficulties arising in this system: the presence of rigid modes in the space of admissible velocity fields, the nonlinear rheology used in glaciology, and the friction boundary condition enforced at the base of glaciers.
Chapter 4 presents a numerical investigation of subglacial cavitation and its application to glacial sliding under steady and unsteady conditions. We reconstruct steady friction laws by calculating several steady cavity shapes. These steady results are validated by comparing them to a linearised analytical method. We then perturb some of these steady states with oscillating water pressures that reveal an interplay between the frequency of the perturbations and the resulting sliding speed and cavity volume. Moreover, we find that if the steady state is located on the downsloping or rate-weakening part of the friction law, the cavity evolves towards the upsloping section, indicating that the downsloping part is unstable.
Finally, we explore steady marine ice sheet configurations in Chapter 4. We do so by computing steady states to the parallel slab marine ice sheet problem, which we propose in this chapter. In this problem, a slab of ice of uniform thickness flows down an inclined bedrock into the ocean. We enforce influx conditions that allow us to explore a spectrum of flow regimes, ranging from sliding to shear-dominated flow. We find that the flux-thickness relationship at the grounding line takes the form of power laws with exponents n+1 and n+2 in these two limits, respectively, where n is a non-dimensional parameter in Glen's law, the power law rheology used commonly used in glaciology. We derive analytical approximations in these limits which resemble our numerical findings closely, with visible deviations in some cases. Our numerical results allow us to understand the shortcomings of these commonly used analytical methods. Moreover, in the context of the parallel slab problem, we find that the flux-thickness relationships are strictly monotonically increasing and that the bedrock plays a prominent role in the sliding dominated regime.