## Reconstruction of icesurface profiles and calculation of basal shear stress

From studies of modern glaciers, the basal shear stress is known to be related to the surface profile and thickness of glaciers, expressed by the equation:

where tb is basal shear stress, r and g the ice density and gravitational constant, respectively, a the surface slope of the glacier, and h the glacier thickness (Paterson, 1994). The value of tb (the basal shear stress) can be calculated from measurements of ice thickness h and surface slope a. Most calculated values of basal shear stress vary between 50 and 100 kPa, suggesting that glacier ice behaves like a perfectly plastic material with a yield stress of 100 kPa.

For evaluation of the effects of sides of valley glaciers and ice streams on stresses and velocities, the shape factor F was introduced, given by the half-width/ice thickness on the centre line. Thus, in a valley glacier the valley walls support part of the weight of the glacier, inducing a basal shear stress on the centre line less than its value for a wide channel. A factor in the range of 0.5-0.9 may therefore be inserted in the denominator on the right-hand side of the equation (4.8).

Measurements along flow-lines of modern ice sheets show that most of their profiles can

Table 4.5 Values of basal shear stress (tb) and the corresponding k value be expressed by different parabola and ellipse equations. Glaciologically this is explained by the plastic properties of ice being the dominating factor determining the shape of glaciers and ice sheets (Paterson, 1994). For perfect plastic ice sheets the profile can be expressed by the equation:

where t0 is shear stress and L — x the distance from the ice margin along a flow line.

In constructing ice-sheet profiles, a simple approach in the case of a perfectly plastic ice sheet can be used:

where H is the altitude of the glacier surface at distance L from the margin, both values in metres. The constant k refers to basal shear stress and may be as high as 4.7 (basal shear stress of 100 kPa). Calculated basal shear stress in ice sheets, however, varies between 0 and 100 kPa, with an average around 50kPa, giving k = 3.4. Values of basal shear stress (kPa) and the corresponding k values are given in Table 4.5.

The shear stress can be calculated when the glacier thickness and surface gradient are known and when the glacier profile is mapped by means of lateral moraines and/or trimlines. One property of the parabola equations is that the altitudinal difference between glacier profiles becomes smaller with increasing distance from respective glacier termini.

Small irregularities (less than lm high) in the glacier sole effectively reduce basal sliding. If the glacier is at the pressure melting point, a thin water film is commonly present between the glacier sole and the substratum. If the water film is thick enough to reduce the friction between the ice and the ground, the basal shear stress will be reduced significantly. This will in turn lead to low-gradient glacier profiles and ice thickness may be halved at the transition between cold/polar and temperate ice. An important requirement for such a reduction of the basal shear stress is that the water is not draining away.

In polar/cold glaciers, all or most of the glacier movement occurs as internal deformation. When the temperature in the ice mass is below the pressure melting point, the ice does not deform so easily as it does when it is on the pressure melting point. The terrain relief is important for glaciers with low-gradient surface profiles. The overall gradient of the landscape is, however, of minor importance for the surface profile. Deformation of underlying sediments reduces the basal shear stress and cause low-gradient ice-sheet and glacier profiles. An important factor for subglacial sediment deformation is high sediment pore-water pressure. Rapid retreat of fjord glaciers may be explained by break-up and calving when the glacier front starts to float. A rapid retreat of fjord glaciers normally causes steep and dynamically unstable glaciers. The precipitation distribution over an ice sheet can influence surface profiles. Of peculiar importance are variations in the relationship between accumulation and ablation (the nivometric coefficient) on different parts of the glacier.

Glacier profiles based on parabolic equations require that the glacier is in equilibrium. However, such a balance between ablation and accumulation will not normally occur on large ice masses. The ice thickness will remain more or less constant if the volume of ice flowing through the glacier is of similar magnitude to the net balance. If this relationship is not maintained, local accumulation and ablation effects can influence the shape of the glacier profile significantly.

Table 4.5 Values of basal shear stress (tb) and the corresponding k value

 lb k 5 I.) 10 1.5 20 2.1 30 2.6 40 3.0 50 3.4 60 3.7 70 4.0 SO 4.3 90 4.5 100 4.7

reconstruction of ice-surface profiles and calculation of basal shear stress 115

As the continental shelf off Norway and large areas of mainland Scandinavia covered by the Late Weichselian ice sheet were underlain by deformable sediments, we must take into consideration the recent information on subglacial processes in reconstructing the thickness of the ice sheet. As a result, glacier flow in the present shelf area could have been a product of deformation of basal till and sediments as well as deformation of ice. The ice thickness may therefore have been influenced by the yield stress of the subglacial sediments.

Modelling of the Laurentide and Scandinavian ice sheets that incorporates the effects of possible substrate deformation indicates a central ice-dome surface about 400-800 m lower than those based on the assumption of a rigid bed, perfectly plasticity and a yield stress of 100 kPa (Boulton et al, 1985; Fisher et al, 1985; Clark et al, 1996). Low-gradient ice-sheet profiles have been reconstructed in the southern Baltic region, over the present continental shelf area in the North Sea, along the fjord areas of southern Norway (Nesje and Sejrup, 1988), and in NW Scotland (e.g. Ballantyne et al, 1998). These gently sloping ice-sheet profiles can best be explained by subglacial sediments deforming at low yield strength, or perhaps by water-lubricated sliding, or a combination of the two. The evidence for thin, low-gradient ice margins in the area covered by the Scandinavian ice sheet is in accordance with the reconstruction of former lobes and outlet glaciers of the Late Wisconsin Laurentide ice sheet, which indicate similar thin ice margins (e.g. Klassen and Fisher, 1988).

Licciardi et al (1998) presented a series of numerical reconstructions of the Laurentide ice sheet during the last deglaciation from 18,000 to 7000 yr bp, evaluating the sensitivity of the ice-sheet geometry to subglacial sediment deformation. Their reconstructions assumed that the Laurentide ice sheet flowed over extensive areas of water-saturated, deforming sediments. Their reconstructions suggest a relatively thin and multidomed ice sheet.

In addition, several factors like the land-sea distribution, precipitation pattern, equilibrium line altitude and ice-sheet frontal fluctuations, and isostatic movements, influenced the geometry of the Late Weichselian Scandinavian ice sheets (Nesje et al, 1988; Nesje and Dahl, 1990).

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