Ice streams are narrow bands of rapidly sliding ice within an otherwise
slowly flowing continental ice sheet. Unlike the rest of the ice sheet,
which flows as a typical viscous gravity current, ice streams experience
weak friction at their base and behave more like viscous 'free films' or
membranes. The reason for the weak friction is the presence of liquid
water at high pressure at the base of the ice; the water is in turn
generated as a result of dissipation of heat by the flow of the ice
stream. I will explain briefly how this positive feedback can explain the
observed (or inferred, as the time scales are rather long) oscillatory
behaviour of ice streams as a relaxation oscillation. A key parameter in
simple models for such ice stream 'surges' is the width of an ice stream.
Relatively little is understood about what controls how the width of an
ice stream evolves in time. I will focus on this problem for most of the
talk, showing how intense heat dissipation in the margins of an ice stream
combined with large heat fluxes associated with a switch in thermal
boundary conditions may control the rate at which the margin of an ice
stream migrates. The relevant mathematics involves a somewhat non-standard
contact problem, in which a scalar parameter must be chosen to control the
location of the contact region. I will demonstrate how the problem can be
solved using the Wiener-Hopf method, and show recent extensions of this
work to more realistic physics using a finite element discretization.