The role of normal pressure

Another effect that is overlooked in the sliding theories discussed above is that of normal stresses. Budd et al. (1979) carried out some laboratory experiments in which ice blocks upon which a normal load, N, had been placed, were dragged across rough rock surfaces. Temperature control was achieved by immersing the ice and rock surfaces in an ice-water bath. They found that S a t 3/N. The cubic dependence on t might suggest that plastic flow was the dominant sliding process, and this may very well have been the case as it was possible for melt water formed at the interface to escape to the surrounding bath. Thus, heat released by refreezing of this melt water in the lee of the obstacle might not have been available. In that case, the only heat available for melting would be frictional heat and heat conducted from the bath to the interface. (The interface would be colder than the bath owing to depression of the melting point.)

More puzzling is the inverse dependence on N; this is what one expects in a purely frictional system, such as would be provided by a rock being dragged across a bedrock surface. To the extent that the ice-rock interface in the experiments was perfectly lubricated by a melt film, however, we would presume that no tractions could have been supported parallel to the surface. In this case, the sliding speed should have been independent of the normal pressure. However, some erosion of the rock surface occurred during the experiments, and the erosion rate was proportional to N2/3. Rock particles entrained in the basal ice and in contact with the bed would have increased the drag. It seems doubtful, however, that the small amount of rock debris involved could account for a significant reduction in sliding speed.

Budd et al. suggested that in studies of real glaciers, N should be replaced by the effective normal pressure, Ne, or the normal pressure minus the water pressure, a factor first vigorously emphasized by Lliboutry (1968 and earlier). The importance of water pressure on sliding speed is now widely recognized (see, for example, Figure 7.8), but, as we will discuss next, some of the mechanisms involved are not frictional as first suggested.

Figure 7.8. Diurnal variations in surface speed on Storglaciaren, Sweden, measured with the use of a computer-controlled laser distance meter. The distance from a point off the glacier to a stake on the glacier was determined every 10 min. The dashed line shows corresponding water pressures measured in nearby boreholes. Only the major peaks in speed are clearly related to water pressure peaks. (Modified from Hanson et al., 1998. Reproduced with permission of the author and the International Glaciological Society.)

Figure 7.8. Diurnal variations in surface speed on Storglaciaren, Sweden, measured with the use of a computer-controlled laser distance meter. The distance from a point off the glacier to a stake on the glacier was determined every 10 min. The dashed line shows corresponding water pressures measured in nearby boreholes. Only the major peaks in speed are clearly related to water pressure peaks. (Modified from Hanson et al., 1998. Reproduced with permission of the author and the International Glaciological Society.)

Cavities and the effect of water pressure

Elevated water pressures increase the sliding speed in two ways: (1) by increasing the degree of separation of ice from the bed, thereby increasing the shear traction on parts of the bed still in contact with the ice; and (2) by exerting a net downglacier force on ice that bridges cavities. In addition, they weaken any deforming subglacial till over which the glacier is moving, thus increasing ud (Figure 5.5). Here, we consider the first two of these. The third will be addressed in connection with our discussion of subglacial till deformation.

The degree of separation of ice from the bed in the lee of obstacles is increased when water pressures remain elevated for periods of a few days or weeks. Let us briefly examine the conditions required for such separation in an idealized situation. The pressure at the bed is:

where a o is the ice overburden pressure and Po(x, y) is a fluctuating contribution that is positive on the stoss sides of bumps and negative on lee sides (Iken and Bindschadler, 1986). The basal drag due to this effect can be expressed in terms of Po, thus:

Ar J dx

Here, the x-axis is directed downglacier and z is normal to the mean bed and positive upward, so a positive dz/dx is a slope opposing flow, while negative values indicate downglacier-sloping surfaces. Ar is an area of the bed large enough to be representative of average conditions.

If we consider a sinusoidal bed of amplitude a and wavelength X, Po will vary sinusoidally; Iken and Bindschadler (1986) find that its maximum amplitude is:

where (Po2} is the root mean square of Po. The minimum pressure will occur at inflection points on the downglacier faces of the undulations, and is:

XT b

Note that Pmin decreases as rb increases, and hence as the sliding speed increases. If the water pressure exceeds this minimum value, separation occurs and cavities will grow to a size determined by the degree to which the water pressure exceeds Pmjn. Roughness elements on the bed that are bridged by such cavities no longer exert any drag on the ice. The task of balancing the driving force, pgha, is thus shifted to places where the ice is still in contact with the bed. S increases, and this results in the necessary increase in shear traction on these surfaces. (Note that S is the independent variable in this situation, while Tb is dependent. This is a subtle but important distinction.)

The second mechanism by which elevated water pressures lead to acceleration of a glacier is a type of hydraulic jacking. If the subglacial drainage system is reasonably well connected to cavities on the lee sides of bumps, increasing water pressures in the drainage system result in increased water pressures in the cavities. (Pressures in the water film on the stoss sides of the bumps are always in excess of the overburden pressure, and thus are not affected appreciably by changes in the (lower) pressure in the cavities.) The water in the cavity thus pushes upglacier against the bedrock and downglacier against ice. The result is a downglacier force that is added to the downglacier component of the body force. Drag forces on the bed must then increase to balance this

Figure 7.9. Flow around a subglacial cavity in the lee of a sinusoidal bump based on a numerical model using the finite element method. (a) The water pressure in the cavity, 2.36 MPa, is too low, so the cavity is shrinking. (b) The water pressure in the cavity is too large, so the cavity is expanding.

Light-faced numbers show pressure at bed. Bold numbers show change in pressure following a 0.07 MPa change in water pressure in the cavity. The basal drag was 0.103 MPa in both experiments, and the mean pressure at the bed was 2.72 MPa. The cavity would be stable at a pressure of 2.41 MPa. (After Rothlisberger and Iken, 1981, Figure 3. Reproduced with permission of the authors and the International Glaciological Society.)

additional downglacier force. Again, an increase in sliding speed results in this increase in drag.

Over time spans of the order of hours, cavity sizes cannot change appreciably because such changes require ice flow. Thus, on bedrock beds, diurnal changes in speed resulting from input of meltwater or from storms (Figure 7.8) must be a result, principally, of hydraulic jacking. Of course, cavity size increases during hydraulic jacking as a result of the increased flow rate, so if high speeds are sustained the degree of separation will increase sufficiently to result in a significant further increase in speed.

The effect of changing water pressures in a lee-side cavity is nicely illustrated by a numerical modeling study conducted by Rothlisberger and Iken (1981) (Figure 7.9). When the pressure in the model cavity was 0.05 MPa lower than the 2.41 MPa necessary to support it, velocity vectors were toward the cavity (Figure 7.9a), tending to close it. An increase in pressure of only 0.07 MPa was sufficient to start enlarging the

Figure 7.10. Diagram illustrating calculation of Pcr|t.

cavity at a rate of about 10 mm d-1 (Figure 7.9b). Note, in particular, the substantial reductions in normal pressure, indicated by the bold numbers in Figure 7.9b; the decrease of over 1.2 MPa atthe crest ofthe bump could easily have resulted in freezing there in a real situation, as suggested by Robin (Figure 7.6b).

Iken (1981) notes that for a given adverse bed slope, ft, measured with respect to the average slope of the bed, there is a critical pressure, PCTit, above which the glacier may accelerate without bound. Such acceleration would occur if (Figure 7.10):

Here, Pw is the pressure in the cavity and k sin ft is the projected area of the cavity face, normal to the back slope of the bump, against which Pw acts. Thus, this is the force trying to push the glacier up the back slope of the bump. The right-hand side of the equation is the component of the body force acting parallel to the back slope of the bump and in the upglacier direction. Using the expressions for ao and Tb shown in Figure 7.10 and the trigonometric identity sin (ft — a) = sin ft cos a - cos ft sin a, we obtain:

tan ft

On an actual glacier bed consisting of a variety of sizes and shapes of obstacles, Pcrit would not be exceeded everywhere simultaneously. Thus, for most situations, Pcrit « ao is probably more realistic.

Working on Findelengletscher (Findelen glacier), Iken and Bind-schadler (1986) have collected an outstanding set of field data on the

Figure 7.11. Speed of a stake on the surface of Findelengletscher as a function of water pressure, here represented by the water level in a borehole. Were the water level to rise to 18 m below the surface, the glacier would float. (After Iken and Bindschadler, 1986, Figure 6. Reproduced with permission of the authors and the International Glaciological Society.)

w ro

"O

1 20

10 L"

Speed in fall, 1982

180 140 100 60 20 Water level in borehole, m below glacier surface relation between water pressure and surface speed (Figure 7.11). Here, the expected exponential increase in speed with increased water pressure, with water pressure asymptotically approaching the limit ao, is clearly demonstrated.

Iken and Bindschadler suppose that the character of the bed in front of Findelengletscher is similar to that beneath the glacier, and thus are able to calculate sliding speeds using Kamb's (1970) theory. For wavelengths and roughnesses that they believe to be appropriate, the theory gives sliding speeds that are too large, compared with the surface speed, to be realistic. They attribute the discrepancy largely to failure of the theory to take rock-to-rock friction into consideration.

Jansson (1995) has studied the relation between effective pressure, Ne, and surface speed, us, on Findelengletscher and Storglaciaren, using Iken and Bindschadler's (1986) data (Figure 7.11) for Findelengletscher, and finds that a relation of the form us = CN-0'4 (7.16)

fits the data well (Figure 7.12). Note that tb does not vary significantly within either of the two data sets, so its effect is incorporated into the

Vs.

Findelengletscher -

- ^ --Us = 37^ TVg-0'40 ■

S= 263TVg-061

^—-______ ▲

rrc—r «s=30

~~~~ s = 24We-°49

Effective normal pressure, MPa

Figure 7.12. Relation between surface speed, us, sliding speed, 5, and effective pressure, Ne, on Findelengletscher and Storglaciären. Dashed lines show sliding speed estimated by subtracting internal deformation from us. (After Jansson, 1995. Reproduced with permission of the author and the International Glaciological Society.)

constant factor, C, which is more than an order of magnitude higher on Findelengletscher (Figure 7.12). Even after subtracting the contribution of internal deformation, estimated with the use of Equation (5.16), Jansson found that Findelengletscher still seemed to be sliding more than ten times as fast as Storglaciaren under comparable effective pressures. Such a difference in sliding speed would have resulted in a basal drag beneath Findelengletscher that was two to three times that beneath Storglaciaren, but driving stresses (albeit uncorrected for longitudinal stress gradients; see Figure 12.7) at the sites of the measurements were nearly equal on the two glaciers.

More recent data from Findelengletscher (Iken and Truffer, 1997) serve only to further emphasize our lack of understanding of the effect of water pressure on sliding. By 1985, three years after the measurements shown in Figures 7.11 and 7.12, the surface speed had decreased 25% for comparable water pressures. By 1994 there had been an additional 35% decrease. There have not been any changes in the geometry of the glacier, and hence in driving stress, that could explain this deceleration. Iken and Truffer suggest that the basal water system was better connected in 1982, so that high water pressures reached more subglacial cavities. Thus, in effect, there may have been more subglacial hydraulic jacks urging the glacier forward in earlier years.

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