Chapter 18

Ice sheet dynamics

Ice sheets are essentially large viscous drops, supplied by a surface accumulation of ice supplied by the compaction of snow. A model for their motion is derived and simplified using the lubrication approximation. Various features of the resulting equations are then outlined. In particular, if the viscosity is independent of temperature (in reality, it depends significantly on temperature), the model reduces to a nonlinear diffusion equation for the ice thickness H. This equation is degenerate at the margins, giving the usual waiting time behaviour and finite speed of advance, with a singular slope there. There is a non-zero steady state (because of the accumulation), and this is linearly stable (the stability analysis requires use of the method of strained coordinates, because of the singularity at the margin). Incidentally, this singularity was what caused Nye's 1960 analysis of waves on glaciers to be in error at the snout, see Fowler and Larson (1980) for a correction to this.

Non-isothermal flow can be analysed to some extent using the ideas of large activation energy asymptotics, which are appropriate. The flow is dominated by a shear layer at the base, and the resulting plug flow can be obtained. If this description is coupled to a hydraulically based sliding law for the ice over deformable till, it is possible to derive a rational theory for the occurrence of large scale ice sheet surges, such as are thought to have occurred in the Laurentide ice sheet in the last glaciation, causing Heinrich events.