Christian Schoof

Professor

Research Classification

Ice and Snow
Transformation and Evolution of the Earth Surface
Fluid Mechanics
Hydraulic
Asymptotic and Classical Applied Analysis
Differential Equation

Research Interests

Glaciology
ice sheet dynamics
glacier hydrology
applied mathematics

Relevant Degree Programs

 

Research Methodology

mathematical modelling
field instrumentation

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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).

Graduate Student Supervision

Doctoral Student Supervision (Jan 2008 - May 2019)
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 (2010 - 2018)
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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