# Water pressure and glacier quarrying

Quarrying is an important process of glacier erosion. In quarrying, blocks of bedrock must first be loosened, either along preglacial joints or along fractures formed by subglacial processes. Then they must be entrained by the glacier. Because rock fragments that have been loosened but not removed are uncommon on deglaciated bedrock surfaces, Hallet (1996) argues that loosened blocks are readily entrained. He thus concludes that fracture must be the rate-limiting process.

To analyze the stresses causing fracture, Hallet considers an idealized bed consisting of steps as shown in Figure 8.31. The distance between steps is L. The treads slope upglacier at an angle p. The crests of the steps are at the same elevation, so the average bed slope is 0. Ice sliding over the crest of one step separates from the bed, forming a water-filled cavity under pressure Pw. The ice regains contact with the bed a distance S from the crest. The slope of the cavity roof is a. The glacier is supported in part by water in the cavity and in part by a vertical stress, an, on the crest of the step, so summing forces:

From Equation (8.28), noting that the effective pressure (or pressure causing closure of the cavity), Pc, is Pi — Pw, we obtain:

The slopes a and ß are related by:

Now an acts on the crest of a step, promoting fracture, but Pw supports the rock face, resisting fracture. Thus, the total stress causing fracture (the non-hydrostatic or deviatoric stress - see Chapter 2) is:

To determine an we need to know S.

Hallet assumes that the rate of closure of the cavity roof, uc, can be approximated by a relation of the form of Equation (8.3), thus:

uc = kSPc3 (8.32)

where k is a constant involving, among other things, the ice viscosity parameter, B. If uc is considered to be vertical and uniform over the cavity roof and ub is the horizontal sliding speed, then:

Combining Equations (8.29), (8.32), and (8.33) then yields1:

Thus, for a given L, Pc, ub, and p, Equation (8.34) can be solved for S and Equation (8.31) then gives a'n.

Hallet then uses principles of fracture mechanics (Chapter 4), into which we will not delve further here, to estimate the rate of crack growth. In his calculations, deviatoric stresses in the ice are constrained to be less than the tensile strength of ice. By assuming that the crack is initiated at a distance (L - S) from the crest of the step (Figure 8.31), he then calculates the quarrying rate (Figure 8.32) for two rock types, a granite and a marble. The softer marble erodes 1000 times faster than the granite!

As expected, quarrying rates increase with sliding speed. In addition, as ub increases maximum quarrying rates occur at higher effective pressures. To understand the latter, note in Equation (8.31) that as S ^ L, the term in brackets increases without bound. An increase in S can result either from an increase in ub or from a decrease in Pc. This is because both of these changes cause ice in the roof of the cavity to regain contact with the bed further from the crest of the preceding step. If an increase in S is a consequence of a decrease in Pc, a'n does not increase so much because the two effects offset one another in Equation (8.31) (note that S = L when Pc = 0 because the glacier is then afloat). Thus, quarrying

1 This differs from Hallet's analysis because he takes uc to be normal to the cavity roof. Considering the approximations in Equation (8.32) and the simplifications achieved by using Equation (8.33), taking uc to be vertical seems justified.

Figure 8.32. Theoretical rates of quarrying of marble (solid lines) and granite (dashed lines) on a bed like that shown in Figure 8.31. Parameters used in the calculation were L = 10 m, 0 = 11.5o, and k = 5 a-1 MPa-3. (Based on Hallet, 1996, Figure 2. Reproduced with permission of the author and the International Glaciological Society.)

Figure 8.32. Theoretical rates of quarrying of marble (solid lines) and granite (dashed lines) on a bed like that shown in Figure 8.31. Parameters used in the calculation were L = 10 m, 0 = 11.5o, and k = 5 a-1 MPa-3. (Based on Hallet, 1996, Figure 2. Reproduced with permission of the author and the International Glaciological Society.) rates have maxima at some fairly low Pc (Figure 8.32). On the other hand, if the increase in S is a consequence of an increase in ub, then the maxima in a and the quarrying rate can occur at a higher Pc. Also noteworthy in Figure 8.32 is the sensitivity of the quarrying rate to Pc; under 450 m of ice, for example, Pc may vary from 0 to MPa, but significant erosion occurs over <10% of that range.

Because stresses that can be generated in the rock in this way are limited by the tensile strength of ice, the maximum steady a that can be applied to a rock is about 10 MPa, while tensile strengths of strong crystalline rocks without macroscopic flaws generally range from 10 to 20 MPa (Hallet, 1996). Water-pressure fluctuations in cavities in the lee of a bump on a glacier bed provide a mechanism for fracturing this strong rock. They also accelerate propagation of fractures in weaker rocks or those with macroscopic flaws and play a role in the entrainment process (Rothlisberger and Iken, 1981; Iverson, 1989). Let us consider fracture first.

Water-pressure fluctuations are particularly rapid where moulins provide connections from the surface of a glacier to the bed. Water inputs from rain or melt can then cause subglacial cavities to fill and drain faster than they can adjust by flow of the ice. The resulting pressure fluctuations transfer the weight of the glacier first to and then from the tops of bumps. Under 250 m of ice, for example, the pressure could vary from a relatively uniform 2.2 MPa on all faces of a bump to over 12 MPa, say, on the top, and nearly zero on the lee face.

All rocks contain microcracks, fractions of a millimeter to a few millimeters in length, and the stress variations resulting from these water pressure fluctuations can lead to propagation of tensile fractures at the tips of favorably oriented cracks (Griffith, 1924). This can occur even at stresses well below the experimentally determined tensile strength of the rock (Atkinson and Rawlings, 1981; Atkinson, 1984; Segall, 1984). The likelihood of crack growth increases when the water pressure within cracks remains elevated while that in an adjacent cavity drops, or when stress corrosion resulting from repeated pressure changes reduces the strength of the rock (Iverson, 1991). Even higher and more concentrated stress differences can result when a cobble or boulder is dragged over a bump by the ice. Thus, it now seems safe to conclude that even sound crystalline rocks can be fractured subglacially through the action of ice and water, despite the fact that the ice is weaker than the underlying rock.

Boulders with smooth stoss faces and plucked lee faces that are embedded in till, called bullet boulders, provide convincing field evidence for subglacial fracture (see, for example, Sharp, 1982). These boulders must have been transported by the ice and become lodged in the till as the basal ice melted. They would not have had their characteristic shape prior to lodgement, nor could they have been transported to their present location, intact, if they had pre-existing fractures. Thus their shape must have been produced by the overriding glacier after they became lodged.

Once a block of rock is isolated by fractures, bed-parallel forces tending to slide it out of position must exceed frictional forces tending to hold it in place in order for entrainment to occur (Iverson, 1989). Both the bed-parallel and the frictional forces are affected by fluctuations in water pressure. As noted earlier (Figure 7.6), pressure-release freezing may occur on tops of bumps when increases in water pressure in cavities transfer part of the weight of a glacier away from the bumps (Robin, 1976), and similar cold patches can also develop owing to simple flow of the ice from the stoss side of a bump to its crest. Both processes increase the drag exerted by the ice on the block. The latter process is more effective at higher sliding velocities, so increases in subglacial water pressure that cause increases in sliding speed should increase its effectiveness.

Frictional forces resisting dislodgement of loosened blocks are reduced as water pressures rise (Iverson, 1989). This is because the normal pressure that ice exerts on a bedrock surface upglacier from a Figure 8.33. Longitudinal section of Storglaciaren, Sweden, approximately along a flowline showing cirque, overdeepened basins, water-input points (crevasse zones and bergschrund), and inferred locations of quarrying (indicated by AAA). Here w.e.l. = water equivalent line; circles (o) show heights of water in boreholes. (Modified from Hooke, 1991, Figure 2.)