2.2.1. The records

Eleven bore holes connected with the subglacial drainage system for different periods of time (in a pilot study in 1980, five holes out of eight connected). Near the margin, drilling was less successful than in the central part of the glacier. In the connected holes the water level dropped quickly to a depth of approximately 70 m below the surface. In at least two cases this happened before the drill had reached the glacier bed; when the holes were re-drilled later, they could be made a few metres deeper. Some of the holes, or their connections with the subglacial drainage system, soon became obstructed in spite of efforts to flush them. The water-level records indicate whether there was an efficient connection between a hole and the subglacial drainage system; if the connection was poor, the water level would rise when the water input was turned on, or drop when the water input was turned off. The pressure records for such holes do not give a reliable indication of the pressure in the subglacial drainage system.

The pressure records of four holes which had good connections with the subglacial drainage system during most of the time are shown in Fig. 2b, together with the velocities of two poles, the discharge of the terminal stream, and air temperatures. Figure 2a gives additional velocity data. The variations of water level in these holes were in general quite similar and obviously correlated with the velocity variations. On a more detailed scale, the individual records show frequent, irregular deviations from the general trend. In June, diurnal variations in water levels became distinct, with peaks in the late evening. There were two cases when a systematic phase shift occurred between records of bore holes at different locations with respect to distance down-glacier. the large peaks on 30 May and 12 June, These events will be considered in a later paragraph. Although the four holes were located in different zones with respect to slope and longitudinal compression of the glacier (Fig 3), the water level was very roughly, that is within approximately 10 m, at the same depth below surface in all of them. This applies also to holes 8 and 9 during the brief period when they had connections.

Fig. 2a. Velocity records for poles near the centre line of the glacier. Broken lines indicate mean velocities over periods when no short-interval measurements were made.

b. Top: velocity of two poles of profile c. Broken lines indicate mean velocities over periods when no short-interval measurements were made.

Upper middle: depth of the water level below surface in four bore holes.

Lower middle: discharge of the terminal stream (by courtesy of Grande Dixence, S.A.). Broken line: a dam had broken and discharge data refer to only part of the outlet stream.

Bottom: thick line: air temperature near the glacier terminus (by courtesy of Grande Dixence, S.A.).

Thin line: temperature of the free atmosphere at the 700 mbar level near Payerne (by courtesy of Schweizerische Meteorologische Anstalt).

Fig. 3. Longitudinal sections through the part of glacier studied. (a) Vertical section along a straight line through bore holes 1, 7, and 10. Poles or bore holes which were located at a significant distance from this section are indicated with broken tines. (b) Section following approximately the Talweg of the glacier and passing through bore hole 4.

In the two semi-marginal holes, 1 and 5, where the ice is only 110 and 85 m thick, respectively, the water levels tended to be somewhat deeper. That is, the piezometric surface, which approximately paralleled the ice surface in the central part of the glacier, appeared to drop slightly towards the margin; transverse profiles over the ice surface were almost horizontal in the study area. The record of hole 5, which was situated on a relatively steep slope (and thus may not be representative for semi-marginal positions in general) is shown in Figure 4. The record of the other hole, 1, was only very brief. On the whole, the conclusion that water levels in semi-marginal positions were lower, is not absolutely certain, because data are too few.

Fig. 4. Water-level record in semi-marginal bore hole 5 (the record in hole 4 is shown by a thin line for comparison). At hole 5, the glacier is only 85 m thick, and the pressure gauge was at a depth of 78 m. The record is shown by a broken line when the water level was below the gauge.

Water-level records for holes 7 and 10 are quite different from the others; the daily peaks occurred earlier and the water levels were usually higher (see Fig 5). A small local melt-water stream was flowing into hole 7. Additional water input through one or two hoses is indicated in Figure 5 with “+” signs; it caused a significant rise in water level in hole 7. This means that the hole had a poor connection with the subglacial drainage system. The maximum daily water level is therefore markedly shifted towards the time of maximum discharge into the hole. In early June, the snow depth was 0.5 m in this area. According to Reference BurkimsherColbeck and Davidson (1973), the maximum melt-water volume flux from such a snow cover should be expected in the early afternoon. No stream was flowing into hole 10 from the surface but it is conceivable that the hole communicated with a system of nearby crevasses. The records would then indicate impeded drainage of that system and a considerable storage capacity at shallow depth (approximately 20 m below surface). Storage at that depth caused a time delay of daily maximum water levels. In this hole the water level was not much affected by water input through hoses.

Fig. 5. Water-level record in bore holes 7 and 10 with particular drainage conditions. (The record of bore hole 11 is shown for comparison.)

+ water is led into hole 7.

‡ water input enlarged (flow through two hoses).

− water input through one hose stopped.

= water input through both hoses stopped.

When the water-level record was greatly disturbed by artificial input through the hoses, it is shown with a broken line.

In summary, the majority of bore holes had similar water-level records with daily maximum levels in the late evening. These records correlated with the velocity variations of the glacier. In the two semi-marginal holes, the water levels were often a few metres deeper. Two holes had quite different records; the water levels tended to be higher and reached the daily maximum several hours earlier. The water level of one of these holes (7) was definitely affected by a local melt-water stream flowing into the hole.

2.2.2. Preliminary conclusions on the drainage system

In the study area, small melt-water streams were flowing into many crevasses and presumably reached the glacier bed. It is noteworthy that the existence of such streams did, in general, not cause deviations from the uniform depth of water levels in different bore holes. Furthermore, the very marked Footnote * diurnal variations in discharge of these streams did not affect the variations of subglacial water pressure. The recorded pressure maxima occurred up to 6 h later than the maxima of melt-water flow from the thin local snow cover. The synchronous response of pressure variations in different bore holes excludes a delay due to water storage as a possible reason. We believe that the water-pressure variations corresponded to the variations of melt-water flow originating in the large area up-glacier where the snow cover was still 2–3 m thick. Which are the conditions to render possible the dominating influence of water flow from areas up-stream and the suppression of effects of the numerous local streams?

First, it is necessary to assume that the discharge from the upper glacier contributed the largest proportion of the total discharge (precisely, of the total discharge in that section of a conduit where the largest part of the total hydraulic head is consumed). Secondly, one has to assume that the loss of hydraulic head along the local tributaries was small, even during times of peak discharge. Either they joined a larger stream with melt water from the upper glacier after a quite short distance (this presupposes a large number of such streams) or the gradient of hydraulic head along the tributaries was small, or both. (The records of hole 7 demonstrate what happens if this condition is not fulfilled.)

Subglacial streams may either flow in semi-cylindrical channels melted into the ice or flow through existing subglacial cavities (Reference LliboutryLliboutry, 1968), or seep through sediments. Which of these flow modes appears to correspond best to the observed facts?

Seepage through till can be ruled out as a major mechanism of subglacial flow in this area because of the very low seepage velocity in till (Reference Lambe and WhitmanLambe and Whitman, 1979, tables 3.2 and 19.1). More permeable sediments, namely remnants of a former outwash plain, are not likely to be present in the study area but may exist down-stream of it, below a riegel (personal communiccation from C. Schluechter).

If water flows through subglacial cavities, a slight increase of water pressure causes an immediate expansion of the cavities and also opens connections to neighbouring strings of cavities, which prevents a large build-up of water pressure. In contrast, a compact channel adjusts much more slowly to a change in pressure.

The characteristics outlined above suggest that the small local melt-water streams largely flowed through subglacial cavities. That would explain why the strong variations of discharge were accompanied by only small variations in water pressure. Furthermore, extensive bed separation would also provide an explanation for the sensitive reaction of the velocity of the glacier to the variations of water pressure. The subglacial water pressure can affect the sliding velocity only if it acts on a large proportion of the glacier bed (and not just in the vicinity of a few channels).

2.4.1. Drainage of a small lake into the glacier

In the night and early morning of 6/7 June approximately 3000 m3 of water drained from a small lake through the moraine near theodolite-station A. The water was then flowing along the glacier margin and into a moulin just up-stream of profile B.

Features in the data were:

• (a) On 7 June, between 10.00 and 16.30 h, the horizontal velocity increased strongly at all poles of profile D but at no other survey marker. (The velocity of pole D3 is shown in Figure 2a.)

• (b) During the same period (i.e. between two surveys) pole Cl moved upward by 14 mm in addition to the normal vertical movement and later downward by the same amount (Fig 7). None of the other poles showed a significant deviation from the regular vertical movement.

• (c) The records of subglacial water pressure were not affected. (Holes 3, 4, 6, 7, and 11 were functioning at that time.)

Fig. 7. Vertical displacement of poles of profiles C and D. In the night of 6/7 June, a small lake had drained into the glacier; only at pole CI the vertical displacement seems to have been affected.

These observations suggest that the water could not enter the main cavity network before it had reached approximately profile D. It seems to have flowed in the area beneath pole Cl causing there a temporary uplift that was possibly localized because of the small ice depth. Apparently, the conduits in the marginal zone were not connected with the main cavity network. This is plausible if indeed the water level was parallel to the glacier surface, which implies a relatively small subglacial water pressure near the margin. The scarcity of connecting bore holes near the margin can perhaps be explained in this way. At profile D and just below, the glacier is crevassed and appears to descend over a step or knob on the south side of the valley. Possibly, the water could enter the cavity network near this irregularity in the glacier bed.

The local velocity increase should be accompanied by longitudinal straining of the ice and consequently by a downward movement of the ice at or up-stream of profile D. This was, however, not observed. A very speculative explanation is that the downward movement was compensated by an upward movement due to subglacial water storage.

2.4.2. Waves of high water pressure travelling down the glacier

Two events of waves of high water pressure were observed in the spring of 1982: on 30 May and on 12 June. A similar event had been observed in 1980. That event had been preceded by a period of very strong melt while a similar relationship to such periods was not obvious in 1982.

The wave of 30 May 1982

The situation before the event: 25 and 26 May were sunny and warm; the following days were cooler and more cloudy. At profile c the snow melt varied between 4 cm of snow/d from 24 to 25 May, 5.5 cm/d from 26 to 28 May, and 3 cm/d from 29 to 31 May. The mean snow depth was 0.6 m on 29 May. The velocity of the glacier and the discharge of the terminal stream increased slightly and gradually from 26 to 29 May (Fig 2b).

The event: in the evening of 29 May the water level in the bore holes began to rise very rapidly; at first in the upper bore holes but some hours later in the lower ones. In two holes, 3 and 11, the water rose to a higher level than corresponded to the ice-overburden pressure (no water flowed into these holes from the surface). The water level remained high for a few hours and then dropped gradually. The typically sharp rise and gradual decrease in water levels and of seismic activity is depicted in Figure 8. The times of rising and of maximum water levels in various bore holes are listed in Table I together with data on seismic activity. The scatter of wave velocities, also given in this table, is in part due to uncertainty in interpretation (the leading edges of pressure curves, for instance, were disturbed by the ordinary daily fluctuations of water pressure). To some extent, however, this is real. After the event the water pressure fell to particularly low values and the connections of some bore holes with the subglacial drainage system became obstructed.

Fig. 8. Fig. 8. Water level in two bore holes and seismic activity at two seismographs during the passing of a wave of high water pressure.

Various water-filled crevasses and a small supraglacial lake drained in the night 29/30 May. A lake of turbid water formed temporarily at the southern margin between profiles C and D; its level had a maximum before 08.00 h on 30 May. The discharge of the terminal stream increased strongly on 30 May. A dam broke, and therefore only part of the total discharge was recorded, and the time of maximum discharge may have been later than shown in Figure 2b.

In the night of 29/30 May the horizontal velocity nearly doubled (during a shorter, not observed period it was probably much larger). At profile A, and only there, the velocity was already quite high in the afternoon of 29 May (Fig 2a), The velocity data do not demonstrate the propagation of the wave beyond that, because no measurements had been made overnight.

In the records of vertical displacement, however, the passing of the wave can be recognized quite distinctly (Fig 9). The descending part of these graphs resembles the decrease of a draining reservoir and is thus suggestive of temporary water storage during the passage of the wave of high water pressure. This question is considered in detail in Appendix III. The analysis in Appendix III suggests that temporary water storage, equivalent to a layer of 100 mm thickness, had occurred. That analysis is based on the assumption that the (few) measurements of longitudinal straining of the ice at the glacier surface are not very different from the longitudinal straining averaged over the ice thickness. In a recent paper by Balise and Raymond (1985) it has, however, been shown that this assumption is wrong, if the perturbation of sliding velocity is restricted to a section of the glacier bed which has a length in the order of the ice thickness. The validity of the result in Appendix III is therefore doubtful.

Fig. 9. Vertical displacement of several poles during the passing of the wave of high water pressure.

The measurements show that the maximum upward displacement was completed 4–5 h after the maximum of water pressure, at a time when the water level had dropped by about 15 m. (The last statement refers to hole 11 and profile D. The maximum water pressure was recorded at hole 11 at 05.30 h. If the wave of high water pressure travelled with a velocity of, say, 125 m/h down-glacier, it would have arrived at pole D2 at 05.50 h and at pole D3 at 06.10 h. The maximum upward displacement of poles D2 and D3 was measured at approximately 10.00 and 11.00 h, respectively.)

The wave of 16 June 1980

In that year not much melt had occurred before 12 June. The snow cover was still 2 m thick at profile B. Between 12 and 14 June an unusually high temperature with a clear sky and strong winds caused very high melt-water production which stopped abruptly on 15 June. One bore hole, 80 m east-north-east of hole 11 (Fig 1), was functioning in this pilot study. The water-level record shows two peaks, on 15 and 16 June (Fig 10). The maximum level (on 16 June) was 26 m below the surface, which corresponds to a basal water pressure of 0.9 bar below the ice-overburden pressure. Glacier movement was measured at profiles B and D. The data suggest the propagation of a wave (Fig 10). The velocity of propagation can, very approximately, be determined from the records of vertical displacement. The result is a velocity of ≈ 67 m/h which is comparable to that of 12 June 1982.

Table I. Velocities of Waves of High Water Pressure on 29/30 May 1982 and on 12 June 1982

(The distances given in the table are the projections of the distances between two holes on the valley axis.)

Fig. 10. Fig. 10. Wave of high water pressure on 16 June 1980: horizontal velocity and vertical displacement of two poles B3* and D3 * near the 1982 poles B3 and D3. water level in a bore hole between these poles, discharge at the terminus, and air temperature at the terminus and of the free atmosphere at 700 mbar (by courtesy of Grande Dixence, S.A. and Schweizerische Meteorologische Anstalt). The dotted line is an extrapolation of the vertical displacement of pole D3*, indicating the probable time of maximum upward displacement.

2.5.1. Dye tests

Uranin, rhodamin B extra, and tinopal ABP solutions were injected into bore holes and later in the summer into moulins, approximately at the time of daily maximum discharge (Reference MoeriMoeri, unpublished). Usually the maximum dye concentration was detected at the terminus 1–4 h after injection, corresponding to mean velocities in the range 0.12–0.48 m/s if the streams were flowing along straight lines. These velocities are within the range usually found in such experiments but on its lower side (Reference StenborgStenborg, 1969; Reference Ambach, Ambach, Behrens, Bergmann and MoserAmbach and others, 1972; Reference Behrens, Behrens, Bergmann, Moser, Ambach and JochumBehrens and others, 1975; Reference Lang, Lang, Leibundgut and FestelLang and others, 1981; Reference BoultonBurkimsher, 1983). In June, when the experiments commenced, and in the autumn the velocities were smaller than in summer. For comparison, the mean velocity in straight cylindrical channels is given in Table II using various assumptions. (β = 8°corresponds to a straight channel under Findelengletscher; β = 3° corresponds to a meandering channel.)

Table II. Mean Velocity ū_t of Turbulent Flow Through Straight Cylindrical Channels for Different Values of Manning-Roughness, Discharge, and Channel Slope. The Pressure Gradient is Taken to be Zero

A characteristic result was the presence of multiple peaks of dye concentration in samples taken at the snout. Large secondary maxima tended to occur in the afternoons of the following days, during times of rising or high discharge and subglacial water pressure. Such secondary peaks were not significantly broader than the maximum peak. As an example, the result of a test in bore hole 1 is shown in Figure 11. Besides these large secondary maxima, several irregular pulses of increased concentration were detected; they became more frequent and sharper in the autumn.

Fig. 11. Tracer experiment in bore hole I, 14 June 1982. Input of dye at 17.10 h (from Moeri, unpublished).

The multiple peaks indicate a complex subglacial drainage system, with various branches and/or storage spaces. The particular finding that secondary peaks were often sharp and tended to occur in the afternoons of the following days cannot be understood by simply assuming flow through various channnels of different length, but an additional assumption is needed:

• (a) One possibility is to assume that some of the subglacial conduits had pressure-controlled valves, so that water could flow through them at times of high water pressure but was trapped during times of low water pressure.

• (b) Another possibility is to assume intermittent rinsing of storage spaces by the transfer of local surface melt water at times of high melt rate. (The simpler concept of storage spaces without flow through them would result in a different timing of the concentration peaks, which should then occur at times of low water pressure, when subglacial cavities are shrinking and releasing stored water.)

2.5.2. Conductivity measurements

Reference CollinsCollins (1979) recorded the electrical conductivity of the terminal stream of Findelengletscher as well as of the neighbouring Gornergletscher. On the assumption that the two components of a terminal stream, “subglacial flow” and “englacial flow”, had a certain constant solute concentration, he derived the proportion of subglacial flow in relation to the total discharge. (Collins defined “subglacial flow” as flow in intense contact with the glacier bed, and “englacial flow” as flow with insignificant contact with the bed. The latter definition includes large streams at the glacier bed.) His analysis revealed three principal differences in the subglacial drainage systems of Findelengletscher and Gornergletscher:

• (a) Diurnal variations of subglacial flow were approximately in phase with those of the total discharge at Findelengletscher; they were out of phase by nearly 12 h at Gornergletscher.

• (b) The mean solute concentration was four times larger in the terminal stream of Findelengletscher than in that of Gornergletscher.

• (c) In the event of an unusually large run-off, the subglacial flow of Findelengletscher increased quite suddenly and took over the greater part of the total discharge (this refers to Collins’ (1979) figure 4; 11 August 1977). Nothing similar has been observed at Gornergletscher.

These differences can be explained if one assumes numerous small conduits under Findelengletscher and fewer but larger conduits under Gornergletscher. In particular, if the conduits under Findelengletscher consist in part of subglacial cavities, they can adjust quickly to a sudden increase in run-off and water pressure by expansion of bed separation.

In the case of Gornergletscher, Collins assumed that the subglacial flow was retained in times when discharge and pressure in the major drainage channels were large. At low water pressure, however, water from subglacial storage was released into these channels.

In summary, Collins’ results on the drainage of Findelengletscher agree well with the results of the present study. The different type of drainage system of Gornergletscher, namely the smaller frequency of subglacial conduits, may be responsible for the absence of diurnal velocity variations in that glacier (Reference IkenIken, 1978).

3.1.1. The relation in a simple model

We consider a perfectly lubricated bed (no tangential stress) undulating sinusoidally in the flow direction. The pressure which the sliding ice mass exerts on the bed is not distributed evenly but is larger than the ice-overburden pressure on the up-stream faces of undulations and smaller on the down-stream faces. The pressure at the glacier bed can in general be expressed as

where σ ₀ is the ice-overburden pressure and P ₀(x,y) is a fluctuating contribution which is related to the basal shear stress, τ_b, by

Here A _r is an area large enough to be statistically representative of the glacier bed; the x-axis points downstream in the direction of the mean bed slope, and the z-axis points upward, in the case of the simple sinusoidal bed, considered now, P ₀(x) fluctuates sinusoidally with the amplitude

(2)

where <P₀ ²>^1/2 is the root-mean-square value of P ₀, λ is the wavelength, and a is the amplitude of the bed undulations.

The minimum pressure occurs at the inflection points of the down-stream faces of the bed undulations and is termed the separation pressure, p _s.

(3)

The separation pressure has been introduced and given in quantitative terras by Reference LliboutryLliboutry (1968), Reference NyeNye (1969), Reference KambKamb (1970), Reference MorlandMorland (1976), and others. If the subglacial water pressure exceeds the value p _s, the ice starts to separate from the bed, water-filled cavities form, and the sliding velocity increases. If the subglacial water pressure exceeds a critical value, namely

(4)

the glacier is pushed upward along the steepest tangent of up-stream faces of bed undulations with accelerating velocity. β_m is the angle between the steepest tangent of the up-stream faces and the direction of the mean bed slope (Reference IkenIken, 1981).

Fig. 12. Sketch of the functional relationship between the subglacial water pressure and the sliding velocity on a sinusoidal bed supporting no tangential stress.

p_s separation pressure.

p_c critical pressure.

σ₀ ice-overburden pressure.

v_s sliding velocity at the separation pressure.

— perfect connections between cavities (low-pressure zones) and subglacial drainage system.

--- degree of interconnection of cavity network decreases as the pressure in the subglacial drainage system decreases. The water pressure in the disconnected cavities remains above the pressure in the subglacial drainage system.

.... pressure in disconnected cavities has adjusted to the minimum pressure in the drainage system by seepage. During the subsequent increase in water pressure in the drainage system, the pressure in the disconnected cavities remains too low. At high water pressure, connections are re-established.

In the case of a sinusoidal bed,

and

(5)

In Figure 12 the functional relationship between subglacial water pressure and sliding velocity is sketched for the case of steady cavities and a constant basal shear stress. The sketch illustrates the two characteristic features:

• (a) The sliding velocity is constant at water pressures below the separation pressure.

• (b) The sliding velocity becomes infinite when the subglacial water pressure attains the critical value pc.

3.1.2. Comparison with the empirical relation

Comparing the above sketch with Figure 6 suggests that the range of measured water pressures was considerably above the presumed “separation pressure” of Findelengletscher. One may speculatively infer a value for the separation pressure at the bed of Findelengletscher by extrapolation of the curve in Figure 6 until it has a horizontal tangent. The corresponding depth of the water level is in the range 140–240(m below the surface. As an illustration, we will consider two cases in detail:

• (a) The separation pressure corresponded to the depth of the water level of 180 m below the surface, which is equivalent to a water pressure of zero bar (the glacier is 180 m thick).

• (b) The separation pressure corresponded to the depth of the water level of 140 m below the surface. This represents an upper limit to the possible separation pressures obtained by extrapolation of the curve in Figure 6.

• Case. (a). Inserting a water pressure of zero bar for p _s in Equation (3) and σ₀ = 15.9 bar, and τ_b ≈ 1 bar, yields a bed roughness a/λ ≈ 0.02. This is a relatively low value and compares well with the appearance of the now exposed bed of Findelengletscher 2 km down-glacier from the study area (Fig 13). On the other hand, if we insert a/λ = 0.02, σ₀ = 15.9 bar, and τ_b ≈ 1 bar into Equation (5), we obtain a critical pressure p _c = 7.9 bar, corresponding to a depth of the water level of 99 m below surface. Obviously, no sliding instability was observed anywhere near this value at Findelengletscher (Fig 6). The result therefore indicates that the idealized model is not adequate for an interpretation of the data. Possible reasons for the discrepancy will be discussed later.

• Case (b). An analogous analysis leads to a/λ = 0.027 and p _c = 9.9 bar (which is also far too low). In this context, it is also interesting to compare the sliding velocity predicted by the model with the measured velocities.

Fig. 13. The former glacier bed 2 km down-stream of the study area. Bedrock undulations with wavelengths of several metres or less have small amplitudes.

The sliding velocity at the separation pressure can be calculated from Reference KambKamb’s (1970) equations (92) and (96). Numerical values have to be specified for the following parameters:

• N and n, parameters of the flow law of ice ;

• Г, a constant in Kamb’s theory expressing thermal properties of ice and rock (Г is only influential if regelation is significant);

• λ = 2π/h, the wave length; and

• ζ = a/λ the roughness of the bed.

The roughness has been inferred above: ζ = 0.02 (case (a)). Other numerical values adopted are:

• n = 3.

• N = 1.32 bar a^1/3 (a value inferred from bore-hole deformation at Athabaska Glacier (Reference RaymondRaymond, 1973)).

• Г = 120 bar a m⁻² (a value from the range suggested by Kamb).

For two choices of λ, 3 m and 2.05 m, the sliding velocity has been calculated; the results are 13.4 mm/h and 9.2 mm/h, respectively.

With λ = 2.05 m, a sliding velocity compatible with the measured surface velocity is found, while for λ = 3 m the calculated sliding velocity is certainly too large. On the other hand, the visual appearance of the bed undulations (Fig 13) would suggest an even greater wavelength.

The value chosen for N is a relatively large one; in fact, Reference Raymond and ColbeckRaymond (1980) suggested a smaller value in a reinterpretation of the Athabaska Glacier data. A smaller value of N would result in even higher values of calculated sliding velocities. On the whole, the idealized model tends to predict too large a sliding velocity.

3.1.3. Possible actual conditions preventing instability of the glacier

Based on the low values of separation pressure and bed roughness inferred from the measurements, the idealized models predict instability at a far too low water pressure and too high sliding velocities in general. Possible reasons for this discrepancy are discussed below.

(4) Friction between glacier sole and bed

“Perfect lubrication”, i.e. no tangential stress at the ice/rock interface, has been assumed in the idealized model. In reality, rock particles, partly embedded in the basal ice or jammed at the bedrock, cause friction. The friction effectively reduces the basal shear stress available for sliding over the large-scale undulations, and determining sliding velocity, separation pressure, and critical pressure. If the reduction of the basal shear stress by friction were the same at the separation pressure and at the critical pressure, then the relation

(6)

which is valid for a perfectly lubricated sinusoidal bed, would still hold in the case of friction. This implies that the method of estimating the separation pressure from Figure 6 and then deducing the critical pressure would still give the same unrealistically low critical pressure as derived above. However, an analysis based on Reference HalletHallet’s (1981) theory shows that the reduction of basal shear stress by friction is larger in the case of extensive bed separation than without separation, for a given debris content in the basal ice. This will be described in a separate paper. The consequence for the present discussion is that the critical pressure is no longer given by Equation (6) but it is larger. Although this is the desired effect, it is noteworthy that the observations do not only indicate a higher critical pressure than predicted by the idealized model but a coincidence of critical pressure and ice-overburden pressure.

In summary, a variety of natural conditions may be responsible for the absence of instability and for the reduction of sliding velocity. We presume that debris in the basal ice, and locally below it, are the main factors.

3.2.1. Calculation of steady-state water pressure in straight cylindrical channels in the ice near the glacier bed

Assumptions and results of calculation: these calculations are based on a theory by Rothlisberger (1972). At a given discharge, channel closure due to ice pressure and channel expansion due to the melting of ice at the channel wall balance for a certain water pressure. This water pressure is calculated. Important simplifying assumptions are:

• (a) No heat is transported by the water.

• (b) The stress in the ice is isotropic and homogeneous.

The governing equation is

(7)

σ₀ is the ice-overburden pressure, p _w the water pressure, ρ_w the density of water, β the bed slope, D (= 3.63 × 10¹⁰ (N m⁻²)^11/8 m^−3/8) a constant related to material properties of water and ice, K (= 10 m^1/3 s⁻¹), the roughness coefficient of the conduit wall, Q the discharge, and n and A the flow-law parameters if the flow law is written

The x-axis is horizontal and curvilinear following the longitudinal centre line. β(x) and σ⁰(x) are calculated from the glacier geometry. Near the terminus the discharge was taken to be 1 m³ s⁻¹ as is suggested by the discharge record (Fig 2b). At × = 0 (3 km up-stream of the terminus) a discharge of 0.7 m³ s⁻¹, increasing linearly towards the terminus, has been assumed. The calculations have been carried out for four different sets of flow-law parameters.

In Figure 14 the results are depicted as hydraulic grade lines which represent the heights of water levels in piezo-metric pipes connected with the drainage channel. Case 1 refers to the flow-law parameters derived by Reference NyeNye (1953) from measurements of ice-tunnel closure. In this case, the hydraulic grade line is below the bed. In case 2, the height of the hydraulic grade line is comparable to that of the recorded water levels. The flow-law parameter A is, however, much smaller than any value given in the literature (e.g. Reference PatersonPaterson, 1981, p. 38; Reference LliboutryLliboutry, 1983, p. 220). Case 3 refers to the flow-law parameters derived by Reference GlenGlen (1955) with n = 4.2. The hydraulic grade line is only slightly higher than in case 1. In case 4, the parameter n also equals 4.2, while the parameter A is smaller by a factor of 10 (which corresponds to an even greater reduction in viscosity than was assumed in case 2 relative to case 1). The water-equivalent line is also shown in Figure 14. (The water-equivalent surface is obtained if the ice is replaced, column by column, by an equivalent mass of water.)

Fig. 14. Fig. 14. Longitudinal section through Findelengletscher and hydraulic grade lines of the main stream calculated with different flow-law parameters.

1: A = 5.8 × 10⁷ N m⁻² s^1/n, n = 3.

2: A = 1 × 10⁷ N m⁻² s^1/n, n = 3.

3: A = 0.96 × 10⁷ N m⁻² s^1/n, n = 4.2.

4: A = 1 × 10⁶ N m⁻² s^1/n, n = 4.2.

WE: water-equivalent line.

3.2.2. Comparison of calculated and measured water pressure

The calculated examples suggest that the hydraulic grade line of a large cylindrical channel is much lower than the recorded water levels. Can such a channel exist in connection with the cavity network to which the recorded water levels refer? Given this case, two observed features cannot be understood:

• (1) In general, the water level was at approximately the same shallow depth in all bore holes. (An interesting exception is the water level in hole 4, which at the end of June was about 20 m deeper than elsewhere with a trend towards a progressively greater depth.)

• (2) The time of daily maximum water pressure recorded in the bore holes did not coincide with the time of maximum local melt-water input into the cavity network. The time lag was 6 or 8 h in early June, at least twice the unsystematic scatter of peak times of individual bore-hole levels (Figs 5 and 2b).

The first condition excludes the existence of tributaries which transferred substantial quantities of local surface melt water from the cavity network to the considered large drainage channel with a low grade line. Because these tributaries would also have relatively low grade lines (Fig 15), they would therefore cause a considerable drop in the water level in the adjacent parts of the cavity network, in contrast to the observed uniform, shallow depth of the water levels.

Fig. 15. Transverse section of the glacier at profile C with grade lines of tributaries flowing towards a major drainage channel near the valley axis. The discharge of the tributaries increases from 1 l/s at the margin to 49 l/s at the confluence. Two hypothetical conditions of the main stream are considered:

A: the grade line of the main stream is at the glacier bed.

B: the grade line of the main stream is 70 m below the surface.

The label 1 refers to flow-law parameters A = 5.8 × 10⁷ N m⁻² s^1/3 and n = 3.

The label 2 refers to A = 10⁷ N m⁻² s^1/3, n = 3.

WE is the water-equivalent line which has here a shallow local depression at a distance of 200 m from the margin.

The second observed fact has a similar implication; the gradient of the hydraulic head along the outlet(s) from the cavity network (i.e. the connection(s) between the cavity network and the main channel) cannot have been large, otherwise the considerable daily variations in discharge of local surface melt water would have been accompanied by considerable variations in water pressure at the head of the connection(s). The storage capacity of the cavity network can, to some extent, dampen and delay water pressure variations. This delay is likely to be different at different bore holes. The relative synchroneity of the recorded water-level variations therefore suggests that the storage capacity was not responsible for the absence of a water-pressure peak at the time of maximum local melt-water input.

It is concluded that the height of the grade line of a major channel connected with the cavity network was not very different from the recorded water levels, in contrast to the result of the calculations. Which of the assumptions made in the calculation were not adequate?

In the snow-melt season, the discharge generally increases and is not steady as assumed. Calculations by Reference SpringSpring (unpublished, fig 3.28), based on a theory of conduit flow of a fluid through its solid phase (Reference Spring and HutterSpring and Hutter, 1981), show, however, that the new steady state is almost reached within a few days after a change in discharge. Periods of this length with a constant daily mean discharge did occur; in such cases calculations based on Equation (7) should give adequate results.

In order to obtain agreement between the calculated grade line and the water levels measured at Gornergletscher, Rethlisberger (1972) assumed flow-law parameters corresponding to a relatively small viscosity for the basal ice:

The high content of impurities and unfrozen water in the basal ice, as well as its structure, may reduce its viscosity significantly, though probably less than required here (Reference DuvalDuval, 1977). A further reduction in the apparent viscosity may be attributed to the non-isotropic stress state (Reference GlenGlen, 1958) and to the occurrence of transient rather than secondary creep. (A very small apparent viscosity has been inferred from strain-rate measurements in ice tunnels at the foot of ice falls (Reference HaefeliHaefeli, 1951; Reference GlenGlen, 1956).) Reference LliboutryLliboutry (1983) has discussed several of these aspects including the effect of a non-homogeneous stress distribution along a channel at the glacier bed. The latter situation may also give rise to a higher energy dissipation which is required to prevent channel closure.

Altogether, it seems justified to assume a flow-law parameter A smaller than 5.8 × 10⁷ Nm⁻² s^1/n (with n = 3).

In the case of Findelengletscher, many streams originate at various crevasses through which surface melt water reaches the bed. Initially, these streams largely follow the existing cavity network and take a meandering path. We believe that this type of conduit was, to a large extent, preserved during the observation period. We will return to this point in section 4 and consider here only the consequences for the present discussion; the mere assumption of a multitude of conduits with a smaller discharge does not introduce a significant change in the calculated steady-state water pressure. This follows from the small exponent of Q in Equation (7); (σ₀ − p _w) is proportional to Q ^1/4n. However, the conduits considered here meander which implies a smaller conduit slope, 6, as well as a greater conduit roughness (a smaller value of K) than originally assumed. Furthermore, these conduits do not generally have circular cross-sections, which may be expressed as a further reduction of A.

In summary, a combination of changes of the numerical values in Equation (7) seems to be appropriate. It is doubtful, however, whether these changes are sufficient to produce the required result. Since Equation (7) really applies to englacial channels, one should expect more substantial modifications if the channels are subglacial and if basal sliding is substantial. This aspect will be considered in more detail in section 4.