## The coupling between a glacier and its bed

In Chapter 4 we found that the rate of deformation of ice, ee, could be related to the applied stress, oe, by: ee = (ae/B)n (Equation (4.5)). The rigorous basis for this flow law will not be developed until Chapter 9, but some indications of the complexities involved in applying it have already been mentioned. Despite these complexities, calculations using it are reasonably accurate. Computed deformation profiles are an example. This is, in large part, because ice is a crystalline solid with relatively uniform properties. The principal causes of inaccuracy in such calculations are a consequence of impurities in the ice, including water, of anisotropy associated with the development of preferred orientations of crystals, and of incomplete knowledge of the temperature and boundary conditions.

As mentioned briefly in Chapter 5 (Figure 5.5), glaciers also move over their beds readily when the basal temperature is at the pressure melting point. However, the rate at which this movement occurs is far more difficult to analyze. This is again, in part, because the boundary conditions, principally the water pressure and the morphology of the bed, are not known. However, a more fundamental problem is the fact that granular rock debris is usually present, either in the ice or between the ice and the bed, or both. There is considerable uncertainty surrounding the processes involved in the deformation of such material and the appropriate constitutive relations describing its deformation and that of ice containing it. Furthermore, unlike the situation with pure ice, the properties of the rock debris vary, not only from glacier to glacier, but also from point to point beneath a single glacier.

Figure 7.1. Bed geometry used in Weertman's (1964) analysis of basal sliding.

Figure 7.1. Bed geometry used in Weertman's (1964) analysis of basal sliding.

Although it may be impossible to know the boundary conditions well enough to predict accurately the rate of movement of a glacier over its bed, it is nevertheless important to understand the processes in order to place limits on the rate. Thus, a significant effort has been made to analyze these processes, using judicious assumptions where necessary as a substitute for detailed data. This analysis has led to relations between the rate of movement and measurable quantities such as water pressure and driving stress that can be tested with field data.

We start this discussion by looking at the movement of clean ice over an irregular hard rigid bed - the traditional sliding problem. Some of the principal shortcomings of the analysis are then discussed. Finally we take up the problem of deformation of the granular materials over which many glaciers move.

### Sliding

The basic processes by which ice moves past an obstacle on a rigid bed, regelation and plastic flow, were first discussed by Deeley and Parr (1914) and later quantified by Weertman (1957a, 1964). Regelation involves melting of ice in the region of high pressure on the upglacier or stoss side of the obstacle and refreezing of that water in the region of lower pressure on its lee face. Plastic flow is simply deformation of ice in a three-dimensional flow field around the obstacle. In his analysis, Weertman used a simplified model of the bed geometry, sometimes called the tombstone model, consisting of uniformly spaced rectangular blocks on a flat surface (Figure 7.1). This model has been roundly criticized as being

Higher pressure, Colder

Lower pressure, Warmer

Figure 7.2. Pressure and temperature on stoss and lee sides of a rectangular bump on a glacier bed.

unrealistic, and inappropriately defended by arguing that fudge factors can be inserted to make it applicable to real situations. The real value of the model is that the physical principles involved in the sliding process are illustrated without resorting to sophisticated mathematical techniques.

Consider the bed shown in Figure 7.1. For simplicity we will use cubical obstacles with sides of length, I, instead of Weertman's rectangular ones. The mean spacing between obstacles is L, and we therefore define r = UL as the roughness of the bed. The mean drag on the bed is t . As the ice is separated from the bed by a thin film of water, we assume that faces parallel to the flow cannot support a shear stress. Thus the entire drag over an area L2 must be supported by one obstacle. The total force on the obstacle is then F = tL2, so the stress difference across the obstacle, as - al, is:

rL"

where the subscripts, s and l, refer to stoss and lee, respectively.

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