1. Introduction

Athabasca Glacier in the Canadian Rocky Mountains has been the scene of studies of glacier flow (Reference Savage, and Paterson,Savage and Paterson, 1963, Reference Savage, and Paterson,1965; Reference Paterson, and Savage,Paterson and Savage 1963[a], Reference Paterson, and Savage,[b]; Reference Paterson,Paterson, 1964, Reference Paterson,1970; Reference Raymond, and Kamb,Raymond and Kamb, 1968; Reference Raymond,Raymond, unpublished), seismic investigations (Reference Clee,, Clee,, Savage, and Neave,Clee and others, 1969; Reference Neave, and Savage,Neave and Savage, 1970), and measurements of electrical properties of glacier ice (Reference Watt, and Maxwell,Watt and Maxwell, 1960; Reference Keller, and Frischknecht,Keller and Frischknecht, 1960, Reference Keller,, Frischknecht, and Raasch,1961). Interpretation of these studies requires knowledge of whether the ice is at the pressure melting point; difficulties in thermal drilling suggested that it might not be (Reference Savage, and Paterson,Savage and Paterson, 1963). Some temperature measurements have now been made; they are discussed in this paper.

2. Techniques

Ten-meter temperatures were measured, within a few days of completion of drilling, in bore holes made with a hand drill. Electrically powered thermal drills were used for bore holes deeper than 10 m. Except in one case, a multicore cable containing pairs of thermistors at intervals was attached to the power cable and descended into the ice with it. On completion of drilling, the thermistor cable, power cable and drill all remained in the hole. In the exceptional case, the thermistor cable was placed in the hole after the drill had been withdrawn. Temperatures were measured at intervals over periods of up to 3 years. Figure 1 shows locations of bore holes, each designated by its depth in meters. With reference to previous flow studies (Reference Paterson, and Savage,Paterson and Savage, 1963), hole 259 was 100 m down glacier from the 1960 position of hole 322, hole 209 was close to the 1960 position of the old hole 209 and hole 72 was close to the 1960 position of stake L 34. Hole 100 was hole 4a of Reference Raymond,Raymond (unpublished). Table I contains information about the bore holes. Measurements were made only in some holes in April and June 1967; the other cables could not be found beneath the snow because the stakes marking them had broken. Holes 259 and 209 had to be abandoned when the cables broke, in hole 259 this happened before any measurements had been made. Measurements in the other holes were discontinued when sufficient data had been obtained.

Fig. 1.

Map of lower part of Athabasca Glacier with locations of bore holes used in the present study. Each bore hole is designated by its depth in meters. Contour elevations are in meters above sea-level.

TABLE I.

Bore-Hole Information

As the holes contained water when they were drilled, we must discuss whether subsequent temperature measurements are reliable. Because the water level in a bore hole varied and because, occasionally, the drill appeared to penetrate water-filled cavities in the ice, the bore holes seemed to be connected to a “water table” in the glacier. In summer, the water table is near the surface but, when the supply of surface melt water is cut off at the end of summer, the water table probably falls to near the base of the glacier. Previous experiences with cased bore holes provide some evidence for this. After the winter, the casings were almost always blocked by ice near the surface. After this ice had been penetrated, the casing was usually found to be free of ice until near the bottom, where further ice plugs were encountered. The lower plugs probably indicate the position of the water table in winter. Reference Shreve, and Sharp,Shreve and Sharp (1970), who observed the same phenomenon on Blue Glacier, have discussed it in detail. Once the water drains out of a bore hole, ice flow will begin to close it and eventually the thermistor cable will be gripped by the ice. There is some evidence for this: the cables could not be moved in the holes after the first summer and, in addition, some of the cables broke. For example, in April 1967 the cable in hole 209 was intact, by June it had broken somewhere between 154 and 199 m, by August it had broken again between 39 and 62 m, and by the following August it had broken above 39 m. Such behaviour strongly suggests that the cable was firmly fixed in the ice. That the cables in holes 72 and 100 have not broken yet can be attributed to the stronger type of cable used there.

The strongest evidence that the cables were firmly embedded in the ice, not in a water-filled bore hole, is the fact that, although measurements were made in more than one year and at different times of year, at no point did the measured temperature ever change. Also, apart from two exceptions to be discussed later, all temperatures were below the 0°C measured at all depths at the time of drilling. Again, two thermistors, originally at a depth of 9 m in hole 72, gave readings in the range —0.2 to —0.5°C at different times before they melted out 3 years later; thus these thermistors were in ice, not in a water-filled hole.

Another possible source of error is that the water originally in each bore hole perturbs the temperature of the surrounding ice. Reference Bullard,Bullard (1947) has studied a similar problem. Calculations based on his formula show that, if the hole were full of water for 3 months, the temperature disturbance would be reduced to 5% of its initial value 5 months after the water drained.

The mean difference between the readings of the two thermistors at each depth, 0.03 deg for hole 209 and 0.02 deg for the other holes, has been taken as the standard error of the measured temperature.

3. Measured Temperatures

The average temperature at a depth of 10 m in the ablation area is —0.5°C. These measurements will be discussed separately (Paterson, in preparation). Figure 2 shows measured temperatures below 10 m and the pressure melting temperature at each depth, calculated using a gradient of 0.00742 deg bar^–1 equivalent to 6.62 × 10^–4 deg per m of ice (density 0.91 Mg m^–3). Two thermistors (one at 24 m and one at 39 m) indicate temperatures of 0°C although the other thermistors at the same depths do not. The suggested explanation is that the resistances of these two thermistors had changed slightly since calibration. These anomalous values were included in the statistical analysis.

Fig. 2.

Measured temperatures in Athabasca Glacier. The broken lines join the readings of the two thermistors at each depth. The solid line indicates the pressure melting point.

For analysis, the data were divided into two parts; measurements above 70 m (3 bore holes) and measurements below 70 m (hole 209 only). Above 70 m, data from the three bore holes appear consistent with each other and 12 of the 16 points lie to the left of the pressure melting line. The probability of this, or a more extreme, distribution occurring if the temperatures were equal to the pressure melting value, is only 4%. The t-test confirms this result; the temperatures above 70 m are significantly (at the 5% level) below the pressure melting point, although the mean difference is only 0.01 deg. Similar tests show that the temperatures below 70 m do not differ significantly from the pressure melting point. An alternative way of subdividing the data is to separate the measurements in hole 209 from those in the other holes. Statistical tests then show that the temperatures in holes 72 and 100 are significantly below the pressure melting point, whereas those in hole 209 are not, except for the point at 154 m where the discrepancy is larger than can reasonably be attributed to experimental inaccuracies. As holes 209 and 72 are both on the center line of the glacier and only about 650 m apart, one would not expect their thermal regimes to differ. Moreover, as regards distance along the glacier, hole 209 lies between the other two holes. For these reasons, I think that the difference in temperature conditions is probably a function of depth rather than position. However, the data are inadequate to establish this conclusively.

4. Expected Temperatures

For the moment we ignore the very small deviations from the pressure melting point at depths between 17 and 70 m and consider another problem. Larger deviations were expected, for three reasons that we now discuss.

Freezing in bore holes. Reference Savage, and Paterson,Savage and Paterson (1963) have described cases of this, (i) One bore hole was blocked by ice that formed within the casing less than 5 d after completion of drilling, (ii) Each summer a thermal drill had to be used to remove ice plugs formed in the casings during the previous winter. During this operation there was a continuous hazard of ice reforming above the drill. Three bore holes were permanently blocked in this way. (iii) One bore hole was permanently blocked when ice formed above an acid bottle used for measurements of inclination. All these blockages were at depths well below the 10 m cold layer at the surface. This suggested that temperatures at depth were at least one or two degrees below the pressure melting point. As this is not the case, the refreezing must have resulted from contamination by antifreeze as described by Reference Clee,, Clee,, Savage, and Neave,Shreve and Sharp (1970). Briefly, antifreeze placed in the casing leaks through the joints into the surrounding ice. Some melting occurs and the temperature drops. This causes the solution inside the casing to freeze, because the leakage has reduced the concentration of antifreeze there.

Heat conduction to the surface. The heat balance at the surface appears to be such that the 10 m temperature is maintained at about —0.5°C. Thus temperatures would be expected to be below the pressure melting point for some distance below 10 m. To calculate the magnitude of this effect, consider heat conduction in a semi-infinite medium, moving with velocity v (negative and equal to the ablation rate of 3.8 m a^–1), with temperature T₀ (= 0°C) at time t = 0, and the surface maintained at T ₁(= —0.5°C) for t > 0. Depth below 10 m is denoted by y and thermal diffusivity by k.

In the following analysis k is assumed constant. However, as a result of impurities, glacier ice contains pockets of water. Thus any temperature change involves a phase change of some water or ice. It follows that impure ice has a higher specific heat than pure ice; the difference can be large near the melting point. Thermal conductivity also varies with temperature though to a lesser extent. Thus, in ice near the melting point, k is a function of temperature. Calculations, based on formulae given by Reference Schwerdtfeger,Schwerdtfeger (1963) and on a salinity of 10^–6, a reasonable value for glacier ice (Reference Langway,Langway, 1967, p. 46), show that at — 0.5°C and below k has the same value as for pure ice, at — 0.1°C the value is halved, while at —0.01°C the value is only about one hundredth that of pure ice. Because this effect has been ignored the following calculation is only approximate.

The equation of heat transfer is

Reference Carslaw, and Jaeger,Carslaw and Jaeger (1959, p. 388) give the solution

Here

For large values of t, the solution tends to a steady-state solution which, since v is negative, is

Calculations show that, for t ⩾ 10 a, temperatures are within 0.01 deg of the steady-state value. On the assumption that the annual heat balance at the surface has not shown any significant trend in the past 10 years, we adopt the steady-state solution, shown in Figure 3.

Fig. 3.

Steady-state temperatures, calculated from Equation (3), on the assumption that the 10 m temperature remains at –0.5°C throughout the year.

Comparison of Figures 2 and 3 shows that the temperatures measured between 17 and about 30 m are higher than expected. Thus there must be a heat source in the ice.

Reduction of pressure melting point. As ice flows towards the surface, where it is removed by ablation, the hydrostatic pressure on any element of ice is progressively reduced and so its pressure melting temperature rises. Thus the ice in the ablation area should be below the pressure melting point unless there is a heat source within the glacier. It has long been recognized that, for a glacier to be temperate, there must be heat sources and sinks within the ice. If H is the amount of heat required annually to maintain unit volume of ice at depth y at the pressure melting point

Here ρ is the density of ice, c its specific heat, λ is the decrease in pressure melting point per unit decrease in distance below the ice surface, and v(y) is the velocity component normal to the ice surface. Substitution of numerical values gives

For consistency with the observation that the longitudinal strain-rate varies linearly with depth and with the fact that ice is incompressible, we should take v to be a quadratic function of y. However, for present purposes it is sufficiently accurate to use the linear relation

Here v_s is the value of v at the surface (the ablation rate) and h is the ice thickness. The broken lines in Figure 4 show H as a function of depth for holes 322 and 209. We next consider possible sources of this heat.

Fig. 4.

Solid lines: heat produced by ice deformation. Broken lines: heat required to maintain ice at pressure melting point as hydrostatic pressure is reduced by ice flow towards the surface. Data for (a) hole 322 and (b) hole 209 of Reference Savage, and Paterson,Savage and Paterson (1963).

5. Heat Sources within the Glacier

Any heating of the interior of the glacier must be by heat generated within it. At the base, geothermal heat and heat produced by sliding cannot be conducted into the glacier because the temperature gradient (the pressure-melting gradient) is in the wrong direction. Moreover, the fact that the 10 m temperature is negative indicates a net loss of heat at the surface. There are two heat sources in the glacier: heat of strain-work and latent heat released when water within the ice freezes.

Strain-work. The amount of heat can be calculated from the bore hole measurements of Reference Savage, and Paterson,Savage and Paterson (1963). The formula is

Here W is heat per unit volume per unit time, the ⋵ are strain-rates and the σ and τ are stresses. The shear strain-rates are defined by

and similar relations, where u,V are velocity components. On the center line of the glacier, the relation reduces to

Here the x coordinate is measured along the center line, the y coordinate is measured perpendicular to the glacier surface positive downwards and σ_x’ is the longitudinal stress deviator. The values of ⋵x, ⋵xy, τxy were calculated as described in previous papers (Reference Savage, and Paterson,Savage and Paterson, 1963; Reference Paterson, and Savage,Paterson and Savage, 1963[b]; ⋵x was taken to vary linearly with depth as it has been shown to do (Reference Savage, and Paterson,Savage and Paterson, 1963). Values of σx’ were calculated from ⋵x and the flow law of ice, with values of the constants derived from the bore-hole data.

Figure 4 shows W as a function of depth in holes 322 and 209. The value is much smaller in hole 322 than in hole 209 because sliding accounts for about 75% of the glacier movement at hole 322 but for only about 10% at hole 209. Also shown in Figure 4 is H, the heat needed to maintain the ice at the pressure melting point, discussed in the preceding section. In each case, strain-work provides enough heat only in about the lower half of the ice. Above this level, heat must be supplied by the refreezing of water already present within the glacier. Figure 4 also shows that, throughout most of the lower half, strain-work produces more than enough heat to maintain the ice at the pressure melting point. As the temperature gradient in the ice is linear (the pressure melting gradient), this surplus heat cannot be conducted upwards or downwards; it must melt ice where it is generated. The amount melted is very small however; the heat generated at a depth of 190 m in hole 209 would melt about 1.5% of the ice in 100 a.

Water within the glacier. We have shown that a heat source is necessary to maintain temperatures at 17 m and below within 0.01 deg of the pressure melting point, in spite of the fact that the surface energy balance is such as to maintain the 10-meter temperature at —0.5°C. As strain-work cannot supply this heat, it must be produced by continuous freezing of water in the ice. Thus, between 10 and 17 m, there must be an interface below which the ice contains water but above which it does not. Indirect evidence for this comes from the following observation of Raymond (personal communication): “In a given bore hole (in Athabasca Glacier), the near-surface zone, where short-time-scale refreezing occurred, seemed to have a sharp bottom... typically around 12 to 13 m”.

We can estimate the amount of water from the position of the interface in the following way. Let the water content, by volume, below the interface be f, assumed to be constant. Let y denote depth below the initial 10 m level. At time t, the 10 m level will be at y = u_st where u_s is the ablation rate, taken positive. Let T(y, t) denote temperature. We suppose that T(u_st, t) = T₀, a constant. Let Y(t) be the y coordinate of the interface. At the interface, the rate of production of latent heat equals the rate at which heat is conducted upwards. Thus, at y = Y,

Here L is the latent heat of fusion of ice, ρ the density, and K the thermal conductivity. We now make the simplifying assumption that, at all times, the temperature gradient in the ice above the interface is linear. Thus

. If d Y/d t were to increase, the temperature gradient above the interface would be reduced, less heat would be carried away and so the rate of freezing of water, and therefore d Y/d t,would be reduced. We therefore expect a steady state to be set up such that d Y/d t = u_s. If Y’ is the steady state value of Y—u_st we have = T ₀/Y’ and so

With T₀ = —0.5°C, v_s = 3.8 m a^–1, and appropriate values for the other constants, we have

which gives the values in Table II. On the assumption that our observations correspond to such a steady state, the fact that Y’ 7 m indicates that the water content of the ice is about 0.5 or 1%.

TABLE II.

Level of Water in a Glacier for Various Water Contents

6. Origin of Water within the Glacier

We are concerned here solely with the water dispersed in the tiny channels and cavities between the ice grains. In addition, the glacier contains water in crevasses, moulins, stream channels and large isolated cavities. These undoubtedly influence the ice temperature in their vicinity but there are not enough of them to affect more than a small part of the whole glacier.

Water in the glacier might originate in at least five ways: by percolation from the surface, by strain-work, by melting induced by changes in hydrostatic pressure, by being trapped in small pockets when the ice formed from firn, and by a mechanism proposed by Bader. In discussing these mechanisms we do not mean to imply that all the water necessarily originates in the same way.

Percolation of melt water from the surface is apparently excluded, except near moulins and crevasses, by the cold layer that persists at a depth of about 10 m throughout the ablation season. The mechanism of Reference Bader,Bader (1950) can also be excluded. He postulated that, as ice flows towards the surface in the ablation area, the pressure in air bubbles is kept approximately equal to the hydrostatic pressure in the surrounding ice by pressure melting of ice around each bubble. Because the water occupies less volume than the ice occupied, the air can expand. Bader measured the air content of four surface ice samples from Malaspina Glacier and, on the basis of his postulate, calculated the water content. He obtained values up to 28%; these are unreasonably high. Moreover, there is no source of heat adequate to produce such amounts of water.

The heat of strain-work melts some ice. As previously stated, the deformation (mainly shear) near the bottom of hole 209 (⋵xy = 0.2 a^–1, τzy = 1.2 bar) produces enough heat to melt 1.5% of the ice in 100 a. Near the valley walls, the value of the transverse shear is probably about the same as this. To melt this quantity of ice outside the marginal and basal zones of the glacier would require longitudinal stresses of these magnitudes. In Athabasca Glacier, longitudinal strain-rates of 0.2 a^–1 or more are encountered only in the ice falls and the ice travels through them in very much less than 100 a. We conclude that, though the heat of deformation produces some water, it cannot produce concentrations of the order of 1% throughout the ice.

Some water will also be produced by the reverse process to one considered previously. As an element of ice is progressively buried in the accumulation area, the hydrostatic pressure increases, the pressure melting temperature is lowered and some ice will melt. The broken curves in Figure 4 can be taken to indicate the amount of heat involved if the values shown are divided by about 4, because the accumulation rate in Athabasca Glacier is roughly 1 m a^–1 compared with the ablation rate of 3.8 m a^–1. The amount of heat is much smaller than the heat of deformation near the bottom of hole 209. As ice probably takes, at most, a few hundred years to travel through the accumulation area, we conclude that this process cannot produce a water concentration of the order of 1%. This process will also operate in the ablation area in the region immediately below the icefalls where the ice thickens rapidly in the down-glacier direction. In this region, the velocity vectors at depth will be inclined downwards relative to the ice surface. However, the ice travels through this region in a few years. Moreover, the velocity vectors in the near-surface layers will still be inclined upwards relative to the surface, to compensate for ablation; thus no water will be formed in these layers.

Thus we are left with the last explanation namely that most of the water in the ablation area is water trapped during the transformation of firn to ice in the accumulation area. A corollary is that glacier ice is impermeable to water. Reference Joubert,Joubert (1963) has determined the water content of ice at depths between 35 and 60 m in the accumulation area of the Mer de Glace. The measured concentrations (0.15 to 1%) are about the same as those required to explain the temperature measurements in Athabasca Glacier. Joubert’s observation that water content varied periodically with depth, with layers of maximum water content 3 or 4 m apart, appears to support the view that glacier ice is impermeable, otherwise such fluctuations should be smoothed out. On the other hand, Reference Nye, and Frank,Nye and Frank (in press) believe that ice at the melting point is permeable to water.

7. Temperatures Below Pressure Melting Point

It remains to discuss why some temperatures are, on the average, about 0.01 deg below the calculated pressure melting point. Two possible explanations have already been mentioned, others relate to impurities in the ice, curvature of ice-water interfaces, and the effect of air bubbles.

One possibility is that the bore holes had not closed around the thermistor cables but still contained water. In this case the density of water, not ice, should be used in calculating the pressure melting temperature. This would explain the observed discrepancy. However we do not think that the bore holes contained water, for the reasons discussed in section 2.

Another possible explanation is the fact that, throughout half of the ice thickness, strain-work does not provide enough heat to maintain the ice at the pressure melting point as the hydrostatic pressure is reduced. However, we have shown that there must be water in the ice to provide latent heat to maintain the ice close to the pressure melting point in spite of heat loss at the surface. This water should be sufficient to provide the small amount of additional heat needed to compensate for the reduction in hydrostatic pressure.

Reference Langway,Langway (1967, p. 46) has summarized available data on the chemical composition of glacier ice. For glaciers in temperate regions, the total salt content (by weight) was found to be about 1 part in 10⁶ in Austria and about one tenth of this value in Norway. Glacier ice at the melting point consists of crystals of pure ice surrounded by films of brine (Reference Quincke,Quincke, 1905; Reference Renaud,Renaud, 1949). Thus for ice with a water content of 1 per cent and measured salt content of 1 in 10⁶, the effective salt content is 1 in 10⁴. From the linear relation between freezing point and salinity given by Reference Pounder,Pounder (1965, p. 4) we deduce that the above salt content would lower the melting point by about 0.005 deg. Thus the presence of impurities might perhaps explain the observed temperatures.

Under a pressure of 1.013 bar, ice and water are in equilibrium at 273.15 K only if the interface is plane. Reference Williams,Williams (1968, p. 41) gives the following formula for ΔT, the change in equilibrium temperature, when the radius of curvature of the ice-water interface is r:

Here T = 273.15 K, V is the specific volume of water, y is the surface free energy of the ice-water interface, and L is the latent heat of fusion of ice. Substitution of numerical values gives

Thus, for ΔT = 0.01 deg, r = 5 × 10^–3 mm. The relevant radius of curvature is that of the ice-water interface at a water vein lying at a three-gain intersection. As this radius of curvature might perhaps be as small as 5 × 10^–3 mm, this explanation cannot be completely dismissed. However, the estimated water content of 1% could hardly be contained in veins of this size.

The explanation in terms of air bubbles is related to Bader’s hypothesis, already discussed. As ice flows towards the surface, pressure melting will occur round air bubbles with heat supplied by the surrounding ice. Because there is not enough heat to maintain the bubble pressure at the hydrostatic pressure, the ice will be at the melting temperature corresponding to the bubble pressure not to the hydrostatic pressure of the surrounding ice. We suggest that this is the most likely explanation of the measured temperatures. The observed difference, 0.01 deg, corresponds to a pressure difference of 1.5 bar. This is a reasonable value. Reference Bader,Bader (1950) measured bubble pressures of 1 to 2 bar in ice at the surface of Malaspina Glacier. Reference Coachman,, Coachman,, Hemmingsen and Scholander,Coachman and others (1956) measured bubble pressures of 1 to 3 bar in surface samples from Stor Glaciären. That the difference may not extend below 70 m can perhaps be explained by the fact that the rate of change of hydrostatic pressure varies inversely with depth. The greater the depth, the more time there is for the bubbles to reach equilibrium with the hydrostatic pressure, partly by Bader’s mechanism and also by plastic flow of the ice around each bubble. As there may be slight differences in pressure between bubbles at the same depth, there may also be small inhomogeneities in temperature. A non-uniform distribution of impurities could also produce such inhomogeneities.