Christian Schoof


Research Classification

Research Interests

Ice and Snow
Transformation and Evolution of the Earth Surface
Fluid mechanics
Asymptotic and Classical Applied Analysis
Differential Equation
applied mathematics
glacier hydrology
ice sheet dynamics

Relevant Thesis-Based Degree Programs

Affiliations to Research Centres, Institutes & Clusters


Research Methodology

mathematical modelling
field instrumentation


Doctoral students
Any time / year round

Glacier surging, glacier and ice sheet hydrology, modelling ice sheet dynamics

I am open to hosting Visiting International Research Students (non-degree, up to 12 months).

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Graduate Student Supervision

Doctoral Student Supervision

Dissertations completed in 2010 or later are listed below. Please note that there is a 6-12 month delay to add the latest dissertations.

Inverse modelling of subglacial hydraulic processes (2023)

Numerous studies have documented that water at the ice-bed interface can affect the ice flow dynamics of both mountain glaciers and ice sheets. Water at the bed is routed through a complex network of conduits that form a subglacial drainage system, which evolves over the melt season in response to the changes in the meltwater supply. Despite its significance, our understanding of the subglacial hydraulic processes remains limited due to the inaccessibility of the glacier bed. This thesis aims to address this knowledge gap by using a dense record of basal pressure time series from a mountain glacier in the St. Elias Mountains, Yukon Territory called South Glacier and answering: (i) what insights can we gain by using inverse modelling to reconstruct the seasonal evolution of the subglacial drainage system, and (ii) whether we can identify and improve our understanding of physical processes that are driving abrupt changes in the behaviour of a pair of basal pressure time series, that we refer to as `switching events'.We developed an inverse model for estimating the permeability of the subglacial drainage and its changes over the melt season. Our results indicate that the inverse model can effectively reconstruct the permeability field and its seasonal changes, and, therefore, provide valuable information about the evolution of the subglacial drainage system under the South Glacier. More importantly, this method can be applied to other glaciers with sufficiently dense borehole data. It is worth pointing out, that a thorough sensitivity study on synthetic data highlighted the importance of considering factors such as heterogeneous englacial water storage, offsets in hydraulic potential measurements, and the delays in the estimates of meltwater supply to the glacier bed when interpreting the inversion results. In the second part of this thesis, we thoroughly examined two potential mechanisms that could be driving switching events, and found that they were unable to explain the behaviour in observed data. However, the created catalogue of switching events has proven to be a valuable resource in evaluating the validity of hypotheses related to the underlying physical processes behind the switching events.

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Modelling the propagation of crevasses and calving in ice shelves (2023)

Ice shelves are floating extensions of glaciers and ice sheets that terminate in the ocean, and their stability and mass balance play a crucial role in controlling the sea level. One of the major processes affecting their stability is the calving of icebergs, which is a complex and poorly understood phenomenon. In order to understand and predict the mass balance of ice shelves, it is essential to investigate the physical processes that control iceberg calving. Despite the challenging nature of this problem, due to the lack of observational data and the mathematical and numerical difficulties involved in modeling crevasses penetration, this topic remains of great scientific and practical importance.This Ph.D. thesis focuses on the investigation of the fundamental parameters that influence crackgrowth and, as a result, the rate of iceberg calving. The study considers the effect of hydrologicalparameters, such as water table height, as well as geometry aspect ratio, basal and surface crevasse positions, and ice shelf extensional forcing. The research begins with the assumption of ice as an elastic medium with abrupt crevasse penetration, and progresses to a more advanced visco-elastic model that accounts for gradual changes in parameters. The model is compared to several recent related models and the advantages and disadvantages of each approach are discussed. The findings of this study reveal that the purely instantaneous stress-based calving laws that have become popular in large-scale ice sheet mechanics are too simplistic to accurately describe the complex process of iceberg calving. Instead, the study proposes a more comprehensive approach that takes into account the interplay between multiple parameters and their gradual changes over time.In conclusion, this Ph.D. thesis provides a valuable contribution to the field of stability of ice shelves by investigating the physical processes that control iceberg calving. The results of this research will inform and support the efforts to better understand and predict the behavior of crevasses on ice shelves, and their indirect impact on global sea level.

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Subglacial drainage characterization from eight years of continuous borehole data on a small glacier in the Yukon Territory, Canada (2019)

The subglacial drainage system is one of the main controls on basal sliding but remains only partially understood, constituting one of the most significant sources of uncertainty in glacier dynamics models. Increasing the accuracy of such models is of great importance to correctly forecast the availability of water in glaciated basins and the global sea level rise. While current glacial hydrology models are successful in reproducing the general seasonal change in surface speed and the structure of the subglacial drainage system, they fail to reproduce significant features observed in boreholes. Here we use an eight-year dataset of borehole observations on a small, alpine polythermal valley glacier in the Yukon Territory, to assess which missing physical processes in current glacier hydrology models can explain borehole observations. Our primary tool to analyze the borehole dataset and make inferences about the structure and evolution of the subglacial drainage system is a custom methodology to cluster water pressure time series according to their similarities. We find that the standard picture of a distributed drainage system that progressively channelizes throughout the melt season explains many features of the dataset. However, our observations underline the importance of hydraulically disconnected parts of the bed. Different regions of the bed are generally either hydraulically well-connected or disconnected, and the transition between the two states is abrupt in time (minutes to a few hours) and space (
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Modelling the migration of ice stream margins (2015)

The Siple Coast ice streams are long, narrow bands of ice within the Antarctic ice sheet. They move significantly faster than the surrounding ice ridges, and therefore discharge significantly more ice. Observations suggest that their fast flow is due to sliding along a water-saturated bed, while the bed of the neighbouring ridges generally appears to be frozen. The ice stream velocities and widths vary on decadal to centennial time scales, and these variations include the migration of the ice stream margins, where the fast flow slows down to the speed of the surrounding ice. In this thesis I show that conventional thin film models, which are used to calculate the evolution of ice sheets on continental scales, are only able to reproduce the inwards migration of ice stream margins and the subsequent shutdown of an ice stream. These processes are the result of an insufficient heat dissipation and freezing at the bed. Conversely, I find that the widening of ice streams into regions where the bed is frozen can only be modelled by taking small-scale heat transfer processes in the ice stream margin into account. Previous research has shown that ice stream widening results from an interplay of heating through lateral shearing in the ice stream margin and inflow of cold ice from the adjacent ridges. However, the relative importance of the different effects on the migration speed has not yet been quantified. To account for these processes, I derive a new boundary layer model for ice stream margins. The numerical solution of this model provides the margin migration speed as a function of large-scale ice stream properties such as ice stream width, ice thickness, and geothermal heat flux. The influence of different basal boundary conditions and temperate ice properties on the margin migration velocity is also investigated. To derive a parameterization of ice stream widening that can be used in continental-scale models, I consider asymptotic solutions with high heat production rates and high advection velocities, a limit that likely applies in real ice stream margins.

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Continuum limits of granular systems (2014)

Despite a century of study, the macroscopic behaviour of quasistatic granular materials remains poorly understood. In particular, we lack a fundamental system of continuum equations, comparable to the Navier-Stokes equations for a Newtonian fluid. In this thesis, we derive continuum models for two-dimensional granular materials directly from the grain scale, using tools of discrete calculus, which we develop.To make this objective precise, we pose the canonical isostatic problem: a marginally stable granular material in the plane has 4 components of the stress tensor σ, but only 3 continuum equations in Newton’s laws ∇ ‧σ = 0 and σ = σT. At isostaticity, there is a missing stress-geometry equation, arising from Newton’s laws at the grain scale, which is not present in their conventional continuum form.We first show that a discrete potential ψ can be defined such that the stress tensor is written as σ = ∇ × ∇ × ψ, where the derivatives are given an exact meaning at the grain scale, and converge to their continuum counterpart in an appropriate limit. The introduction of ψ allows us to understand how force and torque balance couple neighbouring grains, and thus to understand where the stress-geometry equation is hidden.Using this formulation, we derive the missing stress-geometry equation ∆(F^ : ∇∇ψ) = 0, introducing a fabric tensor F^ which characterizes the geometry. We show that the equation imposes granularity in a literal sense, and that on a homo- geneous fabric, the equation reduces to a particular form of anisotropic elasticity.We then discuss the deformation of rigid granular materials, and derive the mean-field phase diagram for quasistatic flow. We find that isostatic states are fluid states, existing between solid and gaseous phases. The appearance of iso- staticity is linked to the saturation of steric exclusion and Coulomb inequalities.Finally, we present a model for the fluctuations of contact forces using tools of statistical mechanics. We find that force chains, the filamentary networks of con- tact forces ubiquitously observed in experiments, arise from an entropic instability which favours localization of contact forces.

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Master's Student Supervision

Theses completed in 2010 or later are listed below. Please note that there is a 6-12 month delay to add the latest theses.

Pattern-forming instabilities in the coupling of ice sheets and subglacial drainage systems (2018)

Sharp spatial changes discovered in the basal conditions of an ice sheet do not always have an obvious source. By modelling instabilities in the coupling of an ice sheet and subglacial drainage system, we describe physical feedback mechanisms that force the formation of sharp spatial structures in basal conditions and ice flow. This model predicts the spontaneous formation of periodic subglacial `sticky spot'-lake pairs, that correspond in shape to previous empirical and modelled descriptions of similar structures. The instability that forms this structure is driven by a feedback whereby periodic humps in ice thickness redirect subglacial water to slippery spots that lie immediately downstream of the ice humps: the slippery regions increase ice flux into the ice humps, making them grow.Scaling a one-dimensional model ice sheet coupled to a basal drainage system, we find conditions for the instability with linear stability analysis. Solutions in the full nonlinear model are simulated numerically, using operator splitting and finite difference methods. The instability requires a bed permeability weakly dependent on water pressure changes, negligible bed slopes, and a water velocity much greater than ice velocity. The `sticky spot'-lake pairs are predicted to form with periodic spacing and migrate upstream.

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