Previous discussions of glacier flow have assumed that there is a rigid glacier bed, and that the flow results from a combination of deformation of the ice and slipping over the bed. However field work (Reference BoultonBoulton, in press) has shown that for some glaciers deformation of the bed contributes most of the forward movement of the ice, and there is evidence indicating that the flow of the Laurentide ice sheet was controlled by the deformable sediments over which it passed.
Consider a glacier resting on a bed of unlithified till. If the glacier is melting at the bottom, the melt water produced penetrates into the till so that the usual mechanisms for glacier sliding are not available. The rheology of the bed is governed by the relative sizes of the particles involved, by the types of material present, and by the amount of water present. If the bed consists of coarse gravel, the melt water percolates freely through the bed, and the bed only becomes unstable if large amounts of water are trapped in it. If, however, the bed consists of fine till containing clay particles, the clay adsorbs the water and tends to immobilize it in the bed, lowering the permeability. If melt water is now produced faster than it can percolate through the bed, the surplus ‘‘free” water is dammed near the surface producing a slurry which behaves like a Newtonian fluid with viscosity (Reference RoscoeRoscoe, 1952)
where w is the volume fraction of free water, and η 0 is the viscosity of water. Since most of the free water remains close to the interface, the bed is highly mobile at the top, but at the limit of effective free-water penetration w becomes zero, the viscosity becomes infinite, and the bed is effectively rigid.
In order to study the contribution of bed deformation to the overall flow-rate of a glacier, we consider the simplest situation of an ice sheet of uniform thickness H resting on a bed whose inclination to the horizontal is α (Fig. 1). We take coordinate axes x in the direction of maximum slope, y perpendicular to the bed, and z in the transverse direction, and choose the origin to be at the interface, so that 0 < y ⩽ H corresponds to the ice layer and y < 0 to the till. The shear stress and velocity are assumed to be continuous across the interface ν = 0.
Fig. 1. Ice–till model showing coordinate axes.
If we assume that water penetration into the till is independent of both x and z, the problem reduces to that of determining the velocity u of the ice/till in the x direction as a function of the single independent variable y.
Treating the ice as a Glen fluid, the velocity u(y) for 0 ⩽ y ⩽ H is given by (Reference NyeNye, 1957)
where u s is the surface velocity and u 0 is the velocity at the interface. The tangential shear at the interface is τ 0 = ρigH sin α.
Within the till, we assume that the free water penetrates to a depth l, and that the amount of free water present varies linearly with depth, so that w is given by
where 0 ⩽ f ⩽ 1 is the volume fraction of free water at the interface. All deformation of the till is confined to the region −l ⩽ y ⩽ 0, and in this region the effective viscosity of the till is
The density of the mixture also varies with depth, being ρ = ρ w w(y) + ρ t(I−w(y)), where ρ w and ρ t are the densities of water and saturated till respectively.
Substituting these values into the Navier-Stokes equations we have, for −l ⩽ y ⩽ 0,
This equation is solved subject to the boundary conditions that the shear at y = 0 is ρigH sin α and that the velocity at y = −l is zero, giving
A typical profile for u(y) is shown in Figure 2. Setting y = 0, we obtain
since we expect l ≪ H. For ice thickness about 30 m and penetration depth about 30 cm, ρigHl/η 0 is of the order of 1015 m/year, so for realistic values of u 0 we must have f 5/2 sin α ≈ 10−15. Hence, for finite slopes, significant deformation of the till occurs when the proportion f of free water at the interface is about 10−6, and when f is 10−5 we find that u 0 ≈ 100 sin α m/year. Since doubling f produces a six-fold increase in interface velocity, values of f greater than 10−5 will produce velocities in the surge range unless sin α is very close to zero; i.e. the bed is essentially horizontal. This implies a very sensitive mechanism for triggering glacier surges. If the normal rate of melt-water production is just below the rate at which water can percolate through the bed, any damming of the bed would lead to the trapping of free water in the bed and a rapid rise in the velocity until the obstruction is removed. Alternatively, a glacier whose rate of melt-water production rose slowly but steadily following a corresponding increase in glacier thickness, would surge when the melting rate passed the critical level. The consequent thinning would lead to a lowering in the temperature at the bed, and hence a reduction in basal melt-water production, and the glacier would return to its quiescent state.
Fig. 2. Typical velocity profile in deforming till and adjacent ice.
Conclusion
If the bed of a glacier consists of unlithified till, then the stability of the bed depends critically on the rate of melt-water production at the bottom of the ice. Small excess amounts of water liquidize the top of the till, making it very mobile, and resulting in high sliding velocities in the till. If the glacier bed has a finite slope this may result in surging.
