## 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.