1. Introduction

The mechanism by which a temperate glacier slides over its bed is a topic of considerable interest, and some controversy, at present. To date, theoretical studies have predominated; Reference LliboutryLliboutry (1968) has listed most of the relevant papers. To measure the sliding velocity of a glacier is difficult. The usual approach is to calculate its value from measurements of surface velocity by the formula (Reference NyeNye, 1952):

(1)

Here u _b, is sliding velocity, u _s velocity at the surface, h ice thickness, and A and n are parameters in the flow law of ice. The basal shear stress T is given by

(2)

where p is the density of ice, p the acceleration due to gravity, α the surface slope, and F is a “shape factor” introduced to allow in an approximate way for the effect of the valley walls.

Reference NyeNye (1952) pointed out several reasons why this method of calculating u _b, is not very reliable: the underlying assumption, that the longitudinal strain-rate in the glacier is zero, may be false; the calculated value of τ may not be equal to the actual shear stress; the values of A and n are not precisely known for the type of ice in the glacier in question. Thus an alternative method of calculation may be useful.

The sliding velocity of Athabasca Glacier has been determined at two points where bore holes reached the glacier bed (Reference Savage and PatersonSavage and Paterson, 1963). We have devised a method of estimating, by extrapolation from these measured values, the sliding velocity at twelve other points where ice thickness and other parameters have been measured. This method rests on few assumptions and, in particular, does not use the flow law of ice. In this paper we describe the method and discuss how the resulting values of sliding velocity are related to basal shear stress and other parameters.

2. Athabasca Glacier

Athabasca Glacier (lat 52.2° N., long. 117.2° W) is one of the main outlet glaciers from the Columbia Icefield in the Canadian Rocky Mountains. The glacier descends from the Icefield in a series of three ice falls over a distance of 2 km. The measurements to be discussed were made in the section between the foot of the lowest ice fall and the terminus. The length of this section is 3.8 km, its average width 1.1 km. The width varies only slightly, and the only bend is a slight one about 1 km from the terminus. The surface slope, generally between 2° and 6°, increases to about 20° at the terminus. The maximum ice thickness is about 325 m. This section of the glacier is virtually free of crevasses except near its sides and in two small areas, one immediately above the terminus and one about 1.5 km from it. Ice velocity decreases from about 75 m a⁻¹ just below the lowest ice fall to about 10 m a⁻¹ near the terminus.

3. Measurements

We shall mention briefly how we measured those parameters used in the subsequent analysis. Readers who want further details are referred to previous papers (Reference Paterson and SavagePaterson and Savage, 1963[a]; Reference Savage and PatersonSavage and Paterson, 1963).

Bed slope

The slope of the glacier bed at each seismic station was calculated from the move-out of a reflection trough across the twelve traces of a single reflection record. This is an accurate method of determining the average slope over the distance covered by the geophone spread (about 100 m). Thus the measurements of bed slope at each station are independent of the measurements of ice thickness. The standard error of a measurement of bed slope is estimated to be 20%.

The average slope of the bed between adjacent stations, also needed in the analysis, was determined from the difference in elevation of the bed between the two stations. The elevation of (he bed was calculated from ice thickness and surface elevation.

Figure 1 shows the profile of the glacier along the center-line and the locations of seismic stations and bore holes. (Seismic measurements were made at the bore holes.)

Fig. 1.

Cross-section of the lower part of Athabasca Glacier, along the center-line, with seismic stations. Recorded reflection surfaces are shown as heavy line segments; bore holes as vertical lines. There is no vertical exaggeration in scale.

4. Estimation of sliding velocity

The method consists of calculating the longitudinal strain-rate at the glacier bed. Then, if we start at a point where the sliding velocity u _b is known, we can calculate u _b at the next seismic station, and so on.

We have already shown that, in Athabasca Glacier, the longitudinal strain-rate varies with depth (Paterson, unpublished; Reference Paterson and SavagePaterson and Savage, 1963[b], Savage and Paterson. 1963). Our method consisted of calculating , the strain-rate averaged over the ice thickness. and showing that it differed significantly from , the strain-rate measured at the surface. Reference Raymond and KambRaymond and Kamb (1968) have subsequently confirmed this conclusion by measurements in several bore holes in the vicinity of our hole 322. That we cannot use to extrapolate a measured value of u _b to other points is obvious from the following fact: the sliding velocity at hole 209 is about 23.5 m a ⁻¹ less than that at hole 322; the difference in surface velocity is only 10 m a⁻¹.

We use a coordinate system fixed in space. The origin is below the glacier bed. The x-axis is horizontal, in the direction of flow along the center-line (assumed straight), positive in the direction of flow. The y-axis is vertical, positive upwards. The z-axis is chosen to make the system right-handed. The corresponding velocity components are u, v, w. Strain-rates are denoted by , and time the by τ The glacier surface is

and the glacier bed is

(3)

where h _s, h _b are arbitrary functions.

Because ice is incompressible, we have

Thus

Integration with respect to y, between the limits h _b and h _s, gives

(4)

where v _s, v _b are the vertical components of velocity at the surface and bed and are strain-rates averaged over the ice thickness h = h _s—h _b

If we assume that ice is neither formed nor melted at the glacier bed, there can be no flow normal to the bed. The condition for this is

where D denotes differentiation following the motion. In the present case, from Equation (3), if u _b, v _b, w _b denote the velocity components at the bed,

Substitution in Equation (4) gives

For points on the center-line of the glacier, w_b = 0 and we have

(5)

Except for a change of coordinate system, this is the formula that Reference Savage and PatersonSavage and Paterson (1963) used to calculate .

When is known, we calculate from the measured value of by the formula

(6)

This rests on the assumption of simple bending, namely that varies linearly with depth. This is the simplest assumption that can be made; to assume a more complex relation seems hardly justified. This assumption was made previously in the analysis of bore-hole deformation (Reference Paterson and SavagePaterson and Savage, 1963[b]). It leads to a value of 0.000 3 a⁻¹ for at hole 322. which agrees satisfactorily with the observation of Reference Raymond and KambRaymond and Kamb (1968) that the strain-rate “approaches zero toward the bottom” in this region.

In the calculations we have to assume that does not vary with depth, so that in Equation (5) we can replace by . As is small, this procedure is unlikely to introduce appreciable errors.

The calculations were carried out in a coordinate system in which the x-axis, instead of being horizontal, was parallel to the average bed of the glacier between the two stations. A separate coordinate system was used for each pair of stations. (In the following paragraph. quantities are measured in this coordinate system and ∂h _b/∂x denotes the difference between the slope of the bed at the station and the average bed slope.)

The calculations were carried out as follows. At hole 322, all quantities on the right-hand side of Equation (5) were measured. Thus was calculated from Equation (5) and then from Equation (6). We then use this value to calculate

(7)

Here Δx is the distance between hole 322 and L19 (the next seismic station down-glacier; and Δu _b is the correction to be added to the value of u _b at hole 322 to give a first approximation to u _b at L19. (Use of a coordinate system parallel to the average bed eliminates a term in Δy from Equation (7).) With this approximation, we then calculate at L19 by means of Equations (5)and (6). We then calculate a new value of Δu _b from Equation (7) using the average of the values of at hole 322 and L19. This gives a second approximation to u _b at L19. The process is repeated until there is no further change in the value of u _b at L19. The final value of at L19 is then used in Equation (7) to give a first approximation to u _b at the next station.

I am indebted to Dr J. F. Nye for pointing out an alternative to the numerical solution namely, to solve the differential equation for u _b analytically. Equation (7) is equivalent to

where

and

The solution is

For the special case when the glacier bed is a plane, g(x) = 0 and the solution reduces to

However, for the case of Athabasca Glacier, f(x) and g(x) cannot be represented by simple functions. The numerical solution has to be used.

The calculations were carried, through five intermediate seismic stations, to hole 209. The calculated value of u _b at hole 209 was 9.0 m a⁻¹. We consider that this is satisfactory agreement with the measured value of 6.5±3.5 m a⁻¹.

The sliding velocity was also calculated at four seismic stations up-glacier from hole 322 and three stations below hole 209. (The lowest station, L37, had to be excluded because the vertical velocity was not measured there). There is no check on these values however,

This method of calculating sliding velocity is based on the following assumptions:

1. (1) Ice is incompressible.

2. (2) There is no flow normal to the glacier bed.

3. (3) The longitudinal strain-rate varies linearly with depth.

4. (4) The transverse strain-rate does not vary with depth.

The method is not restricted to a glacier in a steady state nor to any particular flow, Nor do we have to assume that the surface and bed are planes.

This is certainly not a precise method of calculating sliding velocity. The measurements of ice thickness, bed slope, velocity, and strain-rate are subject to experimental inaccuracies. And assumptions (3) and (4) will not be strictly true. Nevertheless we think that the calculated values should at least show the general trend of sliding velocity in this part of the glacier.

Table I lists, for each seismic station, the horizontal components of surface and sliding velocity, ice thickness, surface and bed slopes and basal shear stress. The bed slope given is that measured at each station, not the average slope between stations. The basal shear stress was calculated from Equation (2), with values of shape factor F obtained from Table IV of Reference NyeNye (1965) on the assumption that the glacier has a parabolic cross-section.

Table I.

Surface and sliding velocities, ice thickness, surface and bed slopes, and basal. shear stress at seismic stations on Athabasca Glacier

5 Discussion of results

Theoretical studies of glacier sliding have yielded relations between sliding velocity, basal shear stress, and other parameters. For lack of data, however, these relations have not been adequately tested. Perhaps the most widely used relation is that of Reference WeertmanWeertman (1957, 1964[b]):

(8)

Here u _b, is sliding velocity, τ basal shear stress, S measures the smoothness of the glacier bed (the larger S, the less rough the bed) and B, m and p are constants. The value of m is about 2, that of p about 4. A major difficulty in trying to verify this relation is that u _b depends on a high power of S, a quantity that is very difficult to measure.

Reference MeierMeier (1968) has recently examined the relation between u _b and τ for Nisqually Glacier, He used Equation (1) to calculate u _b, from measurements of u_s, estimates h, and values of τ calculated from Equation (2). The need to determine S was eliminated by confining the analysis lo one point on the glacier bed. A range of velocities was obtained because (he values of u_s h and τ changed by large amounts during the 22 year period of observation. Meier concluded that u _b was not a simple, single-valued function of τ.

The fact that S is unknown presents a major obstacle to using the Athabasca Glacier data to test Equation (8). If we assume, as in some theoretical studies, that S has the same value at all points on the glacier bed, we would expect the data to show a relation between u _b and τ. However, the measured sliding velocity at hole 322 is nearly five limes that at hole 209, though the value of τ is slightly smaller (1.1 bar against 1.4 bar). Again, regression analysis of the data in Table I showed no significant relation between u _b and τ (On the other hand, a significant relation was found between sliding velocity and ice thickness.) Thus one or more of the following statements must be true:

1. 1 The roughness of the glacier bed is not uniform.

2. 2 The basal shear stress cannot be calculated by Equation (2).

3. 3 The data do not fit Equation (8).

The assumption of uniform roughess is unlikely to be correct. As Figure. 1 shows, the glacier bed appears to be much rougher down glacier from station L27 than it is above that point. And sliding velocities below L27 are much smaller than they are above, as Equation (8) predicts. However, the scale of measurement of S must be considered. On Weertman's theory, the sliding velocity is mainly controlled by obstacles of a certain size on the glacier bed. The critical size is a matter of controversy (Reference Kamb and LaChapelleKamb and LaChapelle, 1964: Reference WeertmanWeertman, 1964[a], |b]); however, it is probably in the range 1 mm to 1 m, Presumably S should be measured at this scale. On Weertman's theory, S is assumed to have the same value for obstacles of all sizes; but this is unlikely to be true in a real glacier. Thus the roughness shown in Figure 1, which has a scale of 100 m, should not be used as an indication of the small-scale roughness. To obtain information on (his will not be easy; radar sounding, at least at the wave-lengths used at present (0.7 m to 10 m), will not provide the answer.

There are two possible sources of inaccuracy in values of basal shear stress calculated from Equation (2): inaccuracies in allowing for the effect of the shape of the glacier channel, and neglect of the effect of variations in longitudinal stress along the glacier, Use of the shape factor appears to be the best method available for allowing for the effect of the channel. The method is only approximate, however, and it is difficult to estimate the amount by which the calculated value may differ from the actual shear stress.Footnote *

Reference LliboutryLliboutry (1958) has shown that, when the longitudinal stress varies along the glacier, a correction term should be added to Equation (2). Reference CollinsCollins (1968) has derived the formula rigorously and made clear the assumptions on which a calculation of the correction term must be based. These assumptions are:

1. (1) The slope of the bed is small.

2. (2) The shear strain-rate is small compared with the direct strain-rates where bars denote average values over the ice thickness. (Collins assumed that all strains involving z are zero.)

3. (3) does not vary with depth.

The slope of the bed of Athabasca Glacier, in the area in question, reaches 17°. We know that the third assumption does not hold. The second assumption can be tested at the bore holes, where all the strain-rates have been measured. At hole 322, is –0.012 a⁻¹ and is − 0.011.a⁻¹ At hole 209, is –0.049 a⁻¹ and − 0.003 a⁻¹ Thus the second assumption also breaks down. We are therefore unable to calculate Collins' correction term, Reference LliboutryLliboutry (1968, equation (54)) has also derived a correction term to the equation for the shear stress. However the correction includes a term involving ∂σ_x/∂x integrated over the ice thickness. where σ_x denotes longitudinal stress. It does not appear possible to evaluate this term.

For comparison with results from Athabasca Glacier, Table II lists all published measurements (as distinct from estimates) of sliding velocity known to us. (Reference HaefeliHaefeli (1951) made observations in a tunnel in Glacier du Mt Collon. However, the tunnel reached the glacier bed at a point where the ice descended over a vertical cliff, 60 m high, and was separated from the rock by a gap of 2 to 4 m. We do not consider that the ice velocity can be regarded as a sliding velocity. Reference IheaksloneTheakstone (1967) has measured basal sliding in a natural tunnel near the edge of Østerdalsisen, but he does not state the ice thickness or surface slope.) In some cases the data were insufficient for calculation of a shape factor, so all shear stresses were calculated without it. Thus the values of τ are not very reliable, particularly in the cases of Vesl-Skautbreen, a small steep corrie glacier, and Blue Glacier, where the measurements were made near the side of the glacier in an ice fall. In the Blue Glacier tunnel, the smoothness of the glacier bed was measured and Reference WeertmanWeertman (1964[b]) showed that the sliding velocity was in reasonable agreement with the value calculated from Equation (8).

Regression analyses of the data in Table II gave results similar to those from Athabasca Glacier: no significant relation between sliding velocity and basal shear stress, but a significant relation between sliding velocity and ice thickness. These data thus lend no support to Equation (8). As in the case of the Athabasca Glacier data however, the discrepancy could be accounted for by variations in roughness of the glacier bed, inaccuracies in calculated values of shear stress, or both.

Table 2.

Published measurements of sliding. velocity, ice thickness, surface slope and basal shear stress

For the combined data in Tables I and II, the regression of sliding velocity on ice thickness is

(9)

with h in meters and u _b in m a⁻¹. Figure 2 shows this relation and the observations. This relation is presented merely as an empirical fit to the data. We do not suggest that it implies any causal relation between ice thickness and sliding velocity. Nor do we suggest that an equation of this form has any wide application. In fact, we can see, by considering the situation in an ice fall, that such a relation cannot be true everywhere. The ice is relatively thin in an ice fall; yet velocities are high and sliding probably accounts for a large part of the total motion. Moreover, u _b must be related to other factors as well as h, otherwise the direction of sliding would be undetermined.

Fig. 2.

Relation between sliding velocity and ice thickness. The line represents Equation (9)

6. Acknowledgements

I should like to thank Drs J. F. Nye and J. Weertman for valuable comments on a draft of this paper.