Let us now use Equations (9.32) to calculate stresses and velocities in an idealized glacier consisting of a slab of ice of infinite horizontal extent resting on a bed with a uniform slope. By appropriate choice of the coordinate system, the problem is thus reduced to two dimensions, or plane strain. The ice is assumed to be isotropic and incompressible. We will consider first the case of a perfectly plastic rheology. Then a more realistic nonlinear flow law is used. Our discussion is based on papers of Nye (1951, 1957), which are classics in glaciology.

Although glaciers consisting of such slabs are uncommon, to say the least, there are several reasons for undertaking this calculation. First, it provides an opportunity to apply some of the material discussed in the previous chapter. Secondly, the stress distributions are representative of those which we expect to find in glaciers, and are commonly used approximations when the required assumptions can be justified by the geometry of a problem. Thirdly, the calculations demonstrate the limitations of analytical methods in situations in which boundary conditions are complex. For calculations involving glaciers with realistic shapes, numerical models are required for all but the simplest situations. Finally, the effect of longitudinal stresses on velocity profiles is elucidated.

Solutions for stresses and velocities in plane strain

The coordinate system to be used for the calculation is shown in Figure 10.1: x is parallel to the glacier surface in the direction of flow and z is directed downward normal to the surface.