Basal sliding Current formalization

Weertman (1957, 1964) considered basal sliding in terms of two discrete processes: enhanced deformation and regelation. Enhanced deformation

Local stress fields are established within basal ice as it passes over a rough bed. Locally enhanced stoss-face stresses induce enhanced local basal drag (tb), forcing the ice to deform around bedrock hummocks particularly rapidly. As velocity scales with strain rate times distance, the larger the hummock of a given shape the greater the ice deformation rate Ud, yielding (for a linear flow law) a relation of the form

where a is the hummock height (or size in general for a given shape). Regelation

Regelation involves melting and refreezing cycles that result from the pressure dependence of the melting point of ice. The process therefore operates most effectively at the pressure melting point. Enhanced stoss-face stresses locally lower the freezing point of ice, inducing melting. The meltwater thereby produced flows along the local pressure gradient to the lee side of the bump, where the local pressure is reduced, the freezing temperature is raised and the water film refreezes. This refreezing releases latent heat, which is conducted back through the rock to the stoss face, where it fuels further melting. Due to this requirement for heat

Figure 67.2 Results of the modelled response of Tsanfleuron Glacier to a 75 m rise in ELA. The current measured glacier surface profile is given as a solid line and the modelled steady-state profile is given as a dashed line: (a) multilayer rheology model and (b) single-layer rheology model. (After Hubbard et al. (2003) with the permission of the International Glaciological Society.)

Distance along flowline (m)

Figure 67.2 Results of the modelled response of Tsanfleuron Glacier to a 75 m rise in ELA. The current measured glacier surface profile is given as a solid line and the modelled steady-state profile is given as a dashed line: (a) multilayer rheology model and (b) single-layer rheology model. (After Hubbard et al. (2003) with the permission of the International Glaciological Society.)

flow, regelation sliding velocity Ur is greater around smaller bumps, yielding Role of bedrock roughness

As sliding by enhanced ice deformation is predominantly controlled by stresses around large bedrock irregularities and regela-tion occurs most effectively around small bedrock irregularities, an intermediate, or controlling, hummock size may be identified that presents the greatest resistance to basal sliding. Other things being equal, it is the predominance of roughness elements of this size that determines differences in sliding velocity over different bedrock substrates. Making use of this fact to minimize the summed velocities from enhanced deformation and regelation allows total sliding speed Ut to be expressed in terms of the basal drag tb and roughness r, defined as the ratio of hummock size to hummock spacing, yielding (for n = 3)

Further, the value of r may be weighted towards the controlling obstacle size and the analysis may be refined to cater, for example, for hummocks of different sizes and spacing by considering the bed as a spectrum of roughness waveforms (e.g. Nye, 1970). Role of basal meltwater

The presence of water-filled cavities at the ice-bedrock interface enhances sliding speed through separating ice from its bed, increasing the shear stress on the locations remaining in contact, and exerting a net down-glacier force on the overlying ice. Although these processes have not been incorporated formally into sliding theory, the most applicable relation expresses sliding speed Ut as an inverse function of effective pressure N (defined as ice pressure minus basal water pressure)

The values of p and q may be determined empirically. Field data

Studies focusing on periods of enhanced motion generally indicate a positive correlation between rates of measured or inferred basal motion and basal water pressures (Willis, 1995). For example, in their pioneering study Iken & Bindschadler (1986) reported simultaneous records of surface velocity and borehole water pressure (a surrogate for basal water pressure) from Findelengletscher, Switzerland. When plotted against each other these two variables revealed a positive non-linear correlation (Fig.

180 140 100 60 20

Borehole water level (m below surface)

Figure 67.3 Measured relationship between horizontal velocity of a surface stake and water level in a borehole at Findelengletscher, Switzerland. (After Iken & Bindschadler (1986) with the permission of the International Glaciological Society.)

67.3). Jansson (1995) compared these with similar data from Storglaciaren, Sweden, both of which matched

although the multiplier terms in the original equations were an order of magnitude different. Although this is similar to Equation (7) above, simultaneous field data that also allow the basal drag term to be calibrated are rare. One exception was a laboratory-based study by Budd et al. (1979) which yielded results that were consistent with Equation (7), indicating values for p and q of 3 and 1 respectively. Basal water pressure, however, could not be varied in these tests.

Despite the general correspondence between basal water pressure and sliding velocity, the precise details of the relationship are unclear. For example, peak velocities may actually coincide with periods of rising subglacial water pressure (e.g. see Sugiyama, this volume, Chapter 68). This raises the possibility that sliding velocity varies in a complex manner in response to a combination of controls, including basal water pressure, rate of (positive) change of pressure, and perhaps even the extent of ice-bed separation.

One means of evaluating rates of basal sliding in detail is to instrument the glacier bed via access provided by boreholes drilled through the ice. Engelhardt et al. (1978) used a borehole video camera to record differential ice-bed velocity as a recognizable object, such as a pebble, moved across the field of view. More recently, dragspools have been used to record high-resolution time series of sliding (see Fischer & Hubbard, this volume, Chapter 76). Blake et al. (1994) first used these instruments, recording a basal sliding component of <70% of the total surface speed of Trapridge Glacier, Canada. One drawback with the use of dragspools, however, is that the anchor needs to be inserted into subglacial material, which can itself deform.

Finally, large basal cavities formed naturally beneath thin ice or artificially by hydropower companies provide researchers with direct access to the basal interface. A small number of experiments carried out in such environments indicate that sliding can occur at subfreezing temperatures (Cuffey et al., 2000a) and that sliding may involve significant slip of the ice over bedrock (Cohen et al., 2000; Hubbard, 2002).

67.4.2 Subglacial sediment deformation

Basal motion of ice masses may also be achieved through the deformation of unconsolidated subglacial sediments, the recognition of the importance of which began with the work of Boulton and co-workers only some 20 years ago. Today, subglacial sediment deformation is widely considered to be responsible for the fast flow of many ice masses, key examples being the well-documented ice streams of the Siple Coast, Antarctica. Current formalization

Two general forms ofconstitutive relation have been advanced for subglacial sediments. The most widely used model is a viscous approximation with an inverse dependence on effective pressure, proposed by Boulton & Hindmarsh (1987)

Here, the yield strength (t0) is defined by a Coulomb failure criterion

where C0 is sediment cohesion and 0 is the angle of internal friction.

More recently, several researchers have proposed a highly nonlinear rheology for subglacial sediments, arguing that their deformation is closer to plastic failure than flow. Hooke et al. (1997), for example, characterized this rheology as e ~ ek

for values of t > t0. The constant k has a value of between 10 and 60.

A less well understood process is that of ploughing, whereby clasts protrude from the basal ice into the underlying sediment. Although the effect is not yet incorporated into models of icemass motion, Hooyer & Iverson (2002b) considered this effect in terms of the basal stress supported by ploughing clasts.

Figure 67.4 Displacement of initially vertical segmented rods emplaced for ca. 5 days in subglacial sediments beneath Breidamerkurjokull, Iceland. (After Boulton & Hindmarsh (1987) with the permission of the American Geophysical Union.) Field data

The first direct records of sediment deformation were reported by Boulton & Jones (1979) and Boulton & Hindmarsh (1987). This study involved installing vertical arrays of deformation pegs through a hole excavated in the floor of a subglacial tunnel at Breiôamerkurjokull, Iceland. The pegs were left in place for about 5 days, during which time local ice surface velocity and sediment pore-water pressure were measured, and then re-excavated. Over the period, the glacier surface moved ca. 0.42 m, of which ca. 90% was achieved by sediment deformation. The shape of the excavated deformation array revealed that this motion was focused within the uppermost 0.5 m of the sediment layer, and that no motion occurred below this depth (Fig. 67.4). Data from this experiment indicated that the empirical constants a and b in Equation (9) above have values close to 1.

Examination of the deforming sediment layer at Breiôamerkurjokull indicated that the upper 'A' horizon was characterized by a high porosity (generally >55%), whereas the underlying 'B' horizon, which had not deformed, had a porosity of <45%. This substrate structure is compatible with observations of the structure of unconsolidated sediments remaining in other glaciated terrains, providing general support for the viscous model (e.g. Hart, 1995b).

Further direct evidence of the nature of the deformation of subglacial sediments has come from borehole-based instrumentation, including sediment samplers, penetrometers, tilt meters and plough meters (described in Fischer & Hubbard, this volume, Chapter 76). Such studies generally indicate a positive relationship between sediment deformation rate and subglacial water pressure (e.g. Hooke et al., 1997). However, the details of the relationship may be complex. For example, research by Iverson et al. (1995) at Storglaciaren, Sweden, indicated that the glacier base may decouple from the bed at high subglacial water pressures. Under such conditions, sediment deformation may actually decrease. Further, subglacial sediments retrieved from Ice Stream B were tested in the laboratory and found by Kamb (1991) to be very weak; indeed, it is likely that the basal sediments of such ice streams exert very little restraining influence at all (e.g. Tulaczyk et al., 2000a) and that they are only restrained by occasional basal protrusions (termed sticky spots). Sediment beneath Black Rapids Glacier, Alaska, also failed rather than deformed viscously, but in this case at a depth of some metres rather than near the layer's upper surface (Truffer et al., 2000). Together, these studies, along with laboratory-based shear testing by Iverson et al. (1997), support a highly non-linear stress-strain relationship for subglacial sediments.

Many valley glaciers may be neither wholly bedrock-based nor wholly sediment-based. In such cases it may not be possible to discriminate at the glacier-wide scale between motion by sliding and motion by deformation. On the other hand, research at such glaciers may provide important information on controls common to both components of basal motion. Borehole-based research at Haut Glacier d'Arolla, for example, points to the key role played by the structure of the basal drainage system in controlling the glacier's overall dynamics. Initially, Harbor et al. (1997) noted that a major subglacial channel that develops in a similar location each melt season acts as a corridor of low basal traction (a slippery zone). More recently, results from three-dimensional velocity measurements at the glacier indicate that these events generate extrusion flow in the ice located immediately above this preferential drainage axis (Willis et al., 2003). In contrast, deformation profiles either side of the axis are characterized by standard power-functions (Fig. 67.5).

These variations at temperate valley glaciers also have a temporal dimension. Intra-annual velocities vary markedly, with dramatic speed-ups being recorded through the summer melt-season relative to the winter (Willis, 1995). This summer speed-up is caused by the onset of basal motion as large volumes of meltwater reach the glacier bed. Conversely, winter flow fields reflect motion by ice deformation alone. Short-lived early melt-season speed-ups also have been noted at several temperate (Gudmundsson et al., 2000, Gudmundsson, 2002, Mair et al., 2001) and polythermal (Bingham et al., 2003) valley glaciers. These spring events reflect the initial delivery of surface-generated meltwater to an underdeveloped basal drainage system. Finally, intradiurnal velocity variations also have been recorded at a few glaciers during the summer melt season, being faster during the afternoon and evening than overnight and during the morning (e.g. Gudmundsson et al., 2000; Sugiyama, this volume, Chapter 68).

Although it is generally accepted that subglacial sediment deformation is only effective where the interface is at the pressure melting point (where liquid water weakens the sediments), researchers who have accessed subfreezing sedimentary beds also report enhanced and localized motion. Echelmeyer & Wang (1987), for example, recorded enhanced deformation and discrete shearing within sediment-rich basal ice that was at a mean temperature of <-4°C. More recently, Fitzsimons et al. (1999) reported similar enhanced motion and shearing at the base of Suess Glacier, Antarctica.

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