## Subglacial water conduits

In Chapter 8 we applied Equation (12.22) to closure of subglacial water conduits. As noted there, problems arise when one attempts to estimate closure rates of semicircular conduits, owing to drag on the bed. Even more profound difficulties arise in attempting to estimate closure rates of broad low conduits, as stresses in the ice are no longer symmetrically distributed about the conduit.

Here, we look into another problem of interest: the normal stresses on the bed at the boundaries of a semicircular conduit, and in particular, the gradient in these stresses outward from the conduit (Figure 12.6). This problem was first studied by Weertman (1972). If pressures are Figure 12.6. Stresses around a semicircular conduit: orr is extending and ogg is compressive (hence the minus sign on P). The variation in ogg away from the tunnel is shown schematically. For n > 2, ogg is less compressive than P, and conversely.

higher adjacent to the conduit, water in a film at the ice-bed interface will be forced away from the conduit, and conversely.

The significance of this problem lies in its application to water flow beneath polar ice sheets. Several authors have suggested that for conduits to exist beneath such ice masses in the absence of water inputs from the glacier surface, there must be an influx of water from adjacent parts of the bed (Alley, 1989a; Walder, 1982; Weertman and Birchfield, 1983; Ng, 2000a). The problem of the existence of such conduits is fundamental; where they are present, subglacial water pressures are probably appreciably lower than otherwise. Thus, any attempt to explain, for example, the fast flow of ice streams hinges upon an understanding of the nature of the water flow system.

The relevant stress in this problem is ogg. Thus, let us start with the expression for ogg in Equation (12.14a). Note that in so doing, we tacitly assume that the bed is flat and slippery so that shear stresses do not impede movement of ice inward toward the tunnel. The appropriate value for oa is now the difference between the pressure in the ice and that in the water in the conduit, AP. As before, we add a pressure, —P, everywhere to account for the weight of the ice. With a little rearranging, Equation (12.14a) thus becomes:

Note that ogg is negative, or compressive, as P always exceeds the first term on the right.

It may appear from Equation (12.14b) that ogg will not support the weight of the glacier when n > 2, as then ogg ^ —P as r ^to but a ee > —P near the conduit. In other words, a ee is sufficiently compressive to support the glacier far from the conduit but not near or beneath it. However, a(rr = R) is more compressive than a(rr = a) (Equation (12.17a)), and this provides the additional support. In other words, referring to Figure 12.6, the vertical force acting on the surface at radius R balances that on the bed, which is: 2 /0( AP — P)dr + 2 /R(aee —P) dr.

Let us now consider a semicircular conduit at a depth ho on a horizontal bed beneath an ice sheet of uniform thickness and infinite horizontal extent. Taking the derivative of aee with respect to r along the bed yields da ee 2 /n — 2\ 2 /-2,,\

If n > 2, as might be expected, daee /dr is negative. Thus aee decreases, or becomes more negative, or more compressive, away from the tunnel (Figure 12.6). In this case, water in a film will be forced toward the conduit, enhancing discharge in it. However, when one considers coupling of stresses, particularly where there is a shear stress on the bed parallel to the conduit, the situation is not so simple. It appears that in this case water flow may be away from the tunnel (Weertman, 1972, pp. 299-300).

The physical reason for the change in behavior of da ee/dr with n is not obvious. We might expect that if a cavity is introduced at the base of a glacier, compressive stresses adjacent to the cavity would increase in order to support that part of the weight of the glacier that is no longer supported by the bed under the cavity. However, toward the tunnel u, and hence e.r, increase and this requires an increase in a'rr. The way in which the stress field is modified to satisfy this requirement, and hence the way in which the pressure on the bed is redistributed, depends upon n. A more intuitive explanation of this effect is elusive.

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