Numerical approximation of viscous contact problems applied to glacial sliding

Abstract

Viscous contact problems describe the time evolution of fluid flows in contact with a surface from which they can detach and reattach. These problems are of particular importance in glaciology, where they arise in the study of grounding lines and subglacial cavities. In this work, we propose a novel numerical method for solving viscous contact problems based on a mixed formulation with Lagrange multipliers of a variational inequality involving the Stokes equations. The advection equation for evolving the geometry of the domain occupied by the fluid is then solved via a specially-built upwinding scheme, leading to a robust and accurate algorithm for viscous contact problems. We first verify the method by comparing the numerical results to analytical results obtained by a linearised method. Then we use this numerical scheme to reconstruct friction laws for glacial sliding with cavitation. Finally, we compute the evolution of cavities from a steady state under oscillating water pressures. The results depend strongly on the location of the initial steady state along the friction law. In particular, we find that if the steady state is located on the downsloping or rate-weakening part of the friction law, then the cavity evolves towards the upsloping section, indicating that the downsloping part is unstable.

The evolution of isolated cavities and hydraulic connection at the glacier bed. Part 1: steady states and friction laws

“…The cavity roof h * C remains very close to the bed b * in the cavities initially, which are therefore easier to discern in the normal stress distribution −σ * nn (panel a2). The pattern of normal stress shown here is common to the steady state cavity solutions computed elsewhere (Fowler, 1986;Schoof, 2005;Gagliardini et al, 2007;Stubblefield et al, 2021;de Diego et al, 2021de Diego et al, , 2022: compressive normal stress is continuous at the upstream end of the cavity, with larger values immediately outside the cavity than inside acting to contain the water in the cavity, and normal stress has a positive singularity at the downstream cavity end. I show in appendix A4 that this stress pattern necessarily follows from the inequalities ( 7) and ( 9)…”

“…Computation of steady state friction τ b (the dynamic case being even more complicated, see e.g. de Diego et al (2022) and also Gilbert et al (2022)) therefore requires not only knowledge of u b and N , but also of the prior history of the bed and of hydraulic connections that have been made. This suggests that at least one additional state variable may need to be included in the formulation of steady-state basal friction laws, possibly the cavitation ratio of Thøgersen et al (2019) defining the fraction of the bed that has become cavitated.…”

“…In process-scale models, hydraulic connectedness typically occurs through the bed itself: the bed is highly permeable, offering ready access to water sourced from an ambient drainage system at some given water pressure. That water will force its way between ice and bed as soon as compressive normal stress in the ice drops to the level of the water pressure, causing a cavity to form (Schoof, 2005;Gagliardini et al, 2007;Helanow et al, 2020Helanow et al, , 2021Stubblefield et al, 2021;de Diego et al, 2021de Diego et al, , 2022 These assumptions are at odds with a growing set of observations (Hodge, 1979;Murray and Clarke, 1995;Andrews et al, 2014;Lefeuvre et al, 2015;Rada and Schoof, 2018) indicating that hydraulic connections at the glacier bed are often patchy, and evolve in time: while the bed itself may be somewhat permeable, that permeability is too low to allow significant water transport on the time scales over which the drainage system evolves. On these time scales, water must then flow predominantly along the ice-bed interface, and the topology of the conduit network present there (consisting of subglacial cavities and other forms of void space like R-channels) may not provide a connection to all parts of the bed.…”

A Consistent Framework for Coupling Basal Friction With Subglacial Hydrology on Hard‐Bedded Glaciers

“…Under a steady‐state situation, cavity geometry is at equilibrium with the sliding velocity u b and the effective pressure $$N={p}_{i}-{p}_{w}$$ (where $${p}_{i}$$ and $${p}_{w}$$ denote the ice and water pressure, respectively) such that basal shear stress $${\tau }_{b}$$ is only a function of $${u}_{b}$$ and N (Diego et al., 2022; Gagliardini et al., 2007; Schoof, 2005). This is however no longer the case in a transient situation, where cavity geometry does not necessarily have the time to fully adjust to changing sliding velocities and effective pressures.…”

“…This is however no longer the case in a transient situation, where cavity geometry does not necessarily have the time to fully adjust to changing sliding velocities and effective pressures. In this case, the friction law is expected to be of the form $${\tau }_{b}=f\left({u}_{b},N,\theta \right)$$ (Diego et al., 2022; Iken, 1981), where θ is a variable describing the cavity geometry. Although calculations of force balance at the sliding interface suggest that a transient sliding law should incorporate an instantaneous dependency on the effective pressure (Iken, 1981; Schoof, 2005); here, we neglect this aspect (as also done in Thøgersen et al.…”