Notations and Numerical Values for Blue Temperate Ice

Related to Phase Changes

Introduction

The theories of glacier sliding and glacial erosion have been developed considering temperate ice as an impermeable and dry material. The time has come to adopt a more realistic model.

The permeability of bubble-free (“blue”) temperate glacier ice is a well-documented and accepted fact (Reference LliboutryLliboutry, 1971). Reference Raymond and HarrisonRaymond and Harrison (1975) have examined the water veins at three-grain intersections just after ice cores were retrieved from depths up to 60 m. They inferred the following percolation rates, when the pressure gradient in a vertical vein is the same as in stagnant glacier ice: 19 cm a⁻¹ in fine-grained ice (grain-size ∼ 2 mm), 0.8 cm a⁻¹ in coarse-grained blue ice (grain-size ∼ 2cm) and 0.05 cm a⁻¹ in coarse-grained white ice.

Note that the percentage of blue ice increases with depth. When coring to the bottom of glacier d’Argentière, down to 107 m, only 10–20% of blue ice was found in the cores, the remnant being white. Below 107 m, the amount of blue ice increased and reached 90–95% between 180 m and the bottom at 237 m (Reference Hantz and LliboutryHantz and Lliboutry, 1983).

From chemical analyses, Reference Berner, Stauffer and OeschgerBerner and others (1977) estimated that 2–4 cm of water percolate per year through Griegsgletscher. At depth, it dissolves air under pressure. In this way, after several centuries, white ice might become bubble-free.

Reference HutterHutter (1982) and Reference Blatter and HutterBlatter and Hutter (1991), in their study of polythermal glaciers, considered a Fourier-type diffusion of moisture in temperate ice. Since neither the corresponding diffusivity coefficient nor the boundary conditions for water fluxes could be estimated, they decided to neglect in their numerical applications any diffusion of moisture through ice. Reference FowlerFowler (1984) suggested that recrystallization and flow of ice might strongly enhance the flow of water through the capillary network but he did not offer a predictive model. Therefore, Reference ShoemakerShoemaker (1990) considered the two extreme cases only: impermeability, or a permeability sufficiently large to allow perfect drainage of any internally produced meltwater.

As for wetness, temperate ice contains much more of the liquid phase than the infilling of the capillary network. The water content has been measured, from surface to bottom, in the accumulation zone of Mer de Glace (Reference Vallon, Petit and FabreVallon and others, 1976), and in the ablation zone of glacier d’Argentière (Reference MontmollinMontmollin and others, unpublished). It varies at random at the decimetric scale, between 0 and 2%, without any correlation with the grain-size or the fabric. This important result may be found in Reference Lliboutry and DuvalLliboutry and Duval (1985) or in Reference LliboutryLliboutry (1987b, p. 122–24), where a tentative explanation is given. It is grounded on the fact that most of the water is found in water lenses at grain boundaries, of similar shape (their convexity being governed by the interfacial energies). The lenses enlarge with time, because of the viscous dissipation of heat, until they reach a vein and empty. (As we shall see, the water pressure is always less in the veins.) Therefore, the random distribution of the water content that resulted from the different salt contents of successive snowfalls is perpetuated.

Contrary to previous statements, in stressed ice the very local temperature at an ice—water interface is the melting temperature that corresponds to the hydrostatic pressure in the water, which is equal to the normal stress in the ice that acts on the interface (Reference LliboutryLliboutry, 1964). As shown later, capillary forces (that cause unequal normal pressures against both sides of an interface) and salt content (that lowers the temperature) have negligible effects. Therefore, melting and refreezing at adjacent water inclusions should occur until lenses exist only at the grain boundaries where the normal pressure is the highest (the local temperature the lowest). Thus, with several assumptions that are discussed in the first half of this paper, we may consider that the temperature of wet glacier ice is determined by the local stress, more precisely by the minimum normal stress (when tensile stresses are taken as positive). They are not determined, as in standard theory, by the temperatures or heat fluxes at the distant boundaries of the body and by Fourier’s law. Of course, there are heat fluxes governed by Fourier’s law but they may be drawn directly from the stress field.

I have put forward this theory of “locally stress-controlled temperatures” at a symposium held in 1985 at Interlaken (Reference LliboutryLliboutry, 1986). As a stress field, I used the asymptotic solution given by Reference NyeNye (1969) for the sliding on a hard bed of a Newtonian viscous body, without ice-bedrock separation (cavitation, in the glaciological sense). The surprising result was an extremely large freezing trend at the glacier sole and internal melting in its vicinity. In my book (Reference LliboutryLliboutry, 1987b), I presented this theory on p. 160–64, using this time the third-power law of viscosity but always the asymptotic solution. Later, I improved the stress field by considering a bed profile with finite amplitudes. This can significantly change the results, as shown in the second half of this paper. More importantly, I realized that the water which is produced a short distance above the bed more or less equals the amount of water that the freezing trend at the glacier sole requires. My results have been given in two manuscripts, submitted in December 1989 to the Journal of Glaciology, fused into a single one in June 1990, and finally rejected in March 1991 because it was too long.

Meanwhile, one of those attending the Interlaken symposium picked up the idea of locally stress-controlled temperatures and submitted to the same journal a paper on this topic in May 1989 (Reference ShoemakerShoemaker, 1990). He did not consider the varying stress field due to sliding on a bumpy bed. He made a calculation only in the asymptotic case where deviatoric stresses in the ice would derive from shrinking due to the loss of internal meltwater. This assumption is untenable, because the overlying ice can contract and cope with this loss of volume. Moreover, the many delicate points of the theory have been overlooked. They will be successively addressed in this paper (which is not merely a shortened version of the rejected paper):

• 1. When water appears continuously within the ice, can it drain off?

• 2. Which time-scale must be adopted in order to have a local thermodynamic equilibrium, although a temperate glacier as a whole is not in thermodynamic equilibrium?

• 3. Can the local stress, at the space scale of a crystal, or even of a water lens, be equated with the macro-scopic stress that is considered in continuum mechanics?

• 4. Can we use as a model for the bed a series of smooth bumps, ignoring the minute asperities?

• 5. Which is the best analytical expression of the stress field over a wavy bed, with non-linear rheology and finite amplitudes?

• 6. How to cope with the mathematical singularities that this analytical expression yields?

• 7. How to choose the bed parameters in order to obtain realistic freezing trends?

It will be shown that, with locally stress-controlled temperatures, the water film that, in standard sliding theory, flows at the interface from the stoss sides of the bumps to their lee sides does not exist. Water migrates by the capillary network, within a bottom ice layer of thickness h_w. Nevertheless, this change does not lead to dramatic changes in the sliding laws.

Since, as an average, ice is continuously formed at the sole of a temperate glacier, while water appears above and drains off, a permanent layer of accreted regelation ice, with mean thickness h_i exists at the sole. The values of h_w and h_i will be estimated. They are essential for any study of the salt content or isotopic composition of bottom ice. The accreted layer may appear as a thin silty layer, which is actually observed in many mountain temperate glaciers. It has to be considered in any modelling of glacial abrasion, a topic that will be dealt with in another paper.

Water Pressure in Lenses and Water Veins

The microscopic scale, i.e. the scale of water inclusions and ice grains, will be considered first. To ground assertions made in the introduction, first recall some of the laws of capillarity and thermodynamics.

At an ice-water interface, the interfacial energy 7 causes the normal pressures in the ice (p_n) and in the water (p_w) to be different. Denoting r₁ , r₂ , the principal radii of curvature of the interface (both positive in the case of a water lens), we have:

(1)

For a given p_n, p_w is lower in veins than in lenses because of the opposite concavities of their walls. For a lens with diameter δ, according to Walford (quoted by Reference Nye and MaeNye and Mae (1972); η = r ₁ = r ₂ = δ/(2 sin 16°)). With λ = 0.034 J m⁻², for δ = 1 mm, it follows p_w = p_n + 37.4 Pa. For veins, according to Reference Raymond and HarrisonRaymond and Harrison (1975), r ₁ ∼ 8.5 × 10⁻⁵ m, and 1/r ₂ = 0, whence p_w = p_n − 400 Pa. Therefore, in unstressed ice, when a lens on a grain facet becomes large enough to reach a vein at the periphery of the facet, it empties into the capillary network.

In stressed ice, the water-pressure difference between lenses and veins is still larger, because the lenses are found at grain boundaries more or less perpendicular to the maximum compressive stress. With r ₁ r ₂ r ₃ denoting this time, the radii of curvature of the three cylindrical facets of a vein, and p ₁, p ₂, p ₃ the corresponding values of p _n:

(2)

Since the p _i values are unequal, the curvatures are unequal. Nevertheless, the last term remains of the same order as before, −400 Pa. The walls of a vein are more or less in three directions at 120° from each other. Taking the z axis in the direction of the vein, a simple calculation shows that in this case:

(3)

Since this value is less than the maximum compressive stress, whereas at lens walls it is equal, the water pressure in the veins is lower than in a water lens by more than 437 Pa. In veins perpendicular to the lens, it is lower than in veins parallel to it.

The water discharge in a vein is k _v, denoting a constant, s _v, the cross-sectional area of the vein, and g_z , the component of gravity in the direction of the vein:

(4)

The first term is smaller than the second one by two orders of magnitude and may be neglected, as long as we deal with the vicinity of an uneven bed.

Temperatures at the Microscopic Scale

The Kelvin temperature Τ in the local equilibrium state depends on p _n p _w, and on the entropy difference between water and ice per unit mass, that is L/T, with L denoting the latent heat of melting. It also depends on the concentration of dissolved salts in the inclusions (μ, in equivalents per kg) and of dissolved air (υp _w, in moles per kg). Substituting the value of Equation (1) of p _w, the formula reads:

(5)

with the values of C _m, C _s and C _r that are indicated at the beginning of this paper.

The total curvature of a vein wall is of order −10⁴ m⁻¹, and the one of a lens wall of order 10³m⁻¹. Thus, the last term in Equation (5) is of order 10⁻⁴ K. Salt concentration in wet glacier ice is of order 10⁻⁴ equiv./kg at most (Reference Souchez, Lorrain and LemmensSouchez and others, 1973) and, since the water content of bottom ice is about 2%, μ is less than 10⁻². The corresponding lowering of the melting temperature is less than 0.01 K. In deforming ice, near the glacier sole, p_n may differ from one wall to another, according to its orientation, by up to 1 MPa, providing temperature differences up to 0.1 K. In bottom ice, the temperature is lowered a little because of the salt content, but its variations depend almost exclusively on the variations of p _n. For modelling heat transfers in deforming ice, only the stress-dependence of temperature has to be considered.

The assumption that ice is deforming, i.e. that the deviatoric stresses are conspicuous, is important. The author had the opportunity to examine stagnant temperate ice at the bottom of Mer de Glace, thanks to a tunnel dug at the tip of its tongue by the hydro-electric agency EDF, in the 1970s. (This tip disappeared in the succeeding years, impeding more detailed and quantitative observations.) The cross-section of the bed there had a V-shape, with large boulders in the furrow, amongst which ran the subglacial stream. All the ice around the tunnel was blue, excepting ribbons of white ice with very large air bubbles that rose from the ceiling, parallel to the almost vertical rockwall on one side. Near the boulders, crystals were unusually large (5–10 cm) and some water pockets (~10 cm ³) were observed. Clearly, this ice had suffered post-kinematic recrystallization over a long time, with the interfacial energies as the main driving factor.

For the general case, the mean crystal area on thin slices of bottom (blue) ice is 0.4–1.3 cm² (Reference MontmollinMontmollin and others, unpublished). To assume local thermo-dynamic equilibrium, at the centimetric scale of a grain, means that temperature and stress are uniform within a grain. Then, σ₁ < σ₂ < σ₃, denoting the three principal stresses (compressive, and thus negative):

• (a) Water lenses are more or less perpendicular to σ₁;

• (b) The local Celsius temperature is:

(6)

When flow over an uneven bed causes the stress to change, the temperature must follow the change in σ ₁ It may happen that the stress change causes lenses to disappear and others with a different orientation to appear. On the other hand, the heat flux caused by ∇T (at the macroscopic scale) may supply heat or cold to the crystal. There is also some viscous dissipation of heat (it will be found to be negligible in front of the former heat transfer). All these causes induce melting or refreezing at the lens walls, hence very local volume changes and the corresponding stresses.

Ice diffusivity is large enough (1.16mm²s⁻¹) to ensure that a steady temperature field within the crystal is reached in a few minutes. Then, the temperature gradients between melting lenses and freezing ones is several hundredths of a degree over several millimetres. The corresponding heat fluxes are of order 10 J m⁻² s ₋₁. They correspond to melting or freezing at the lens walls at rates of order 0.1 mm h⁻¹. It should take some hours for a new set of lenses to be formed. Since the temperate ice viscosity, in transient creep, is less than 10⁷ Pas, and the elastic shear modulus of ice is 3.45 × 10⁹ Pa, the time constant of stress relaxation is of order 10⁻²s at most and very local stresses due to volume changes remain negligible.

The slowest governing factor for local temperatures is the slow change of the macroscopic stress field. Large stress changes may demand several hours before the local thermodynamic equilibrium is restored. Note that, in laboratory experiments with non-recrystallizing stressed ice, Reference NyeNye and Mae (1972) found about 1 h.

Such large stress changes occur when bottom ice slides and flows from the stoss side to the lee side of a bump. Even with fast sliding, e.g. 2 cm h⁻¹, local thermal equilibrium is always reached, unless the wavelength is centimetric or smaller. For this reason, and because of the grain-size, Equation (6) is not valid when dealing with minute bumps. For minute asperities, the classical sliding theory must be used instead.

Microscopic Stresses and Macroscopic Stress

At the grain scale, strain rates and stresses in deforming glacier ice are not uniform, because individual crystals are strongly anisotropic. The standard homogenization procedure, which has been developed to deal with the plastic deformation of cubic metallic crystals, assumes that the microscopic strain rate is uniform and equals the macroscopic one, whereas microscopic and macroscopic stresses differ. (With this assumption, equilibrium conditions for stress are not obeyed.) For creeping ice, the opposite assumption, the same stress but different strains in adjacent grains, should be much closer to the truth, for the following reasons (Reference LliboutryLliboutry, 1987b, p. 455-57):

• 1. In deforming temperate glacier ice, a multi-maxima fabric is always found. With this fabric, grains are never in an unfavourable orientation for shear by gliding on basal planes. Also, they are not in the most favourable one.

• 2 Given the third-power viscous law, a small local perturbation of stress can ensure a large perturbation of the strain rate.

• 3 Compatibility of strains in two adjacent crystals is readily obtained in two ways: (a) thanks to grain-boundary migration, as shown by Reference Means and JesselMeans and Jessel (1986), and more intuitively explained in my quoted book. Coincidence-site lattices (Reference HigashiHigashi, 1978), that stabilize the multi-maxima fabric, also make boundary migration easy, (b) Thanks to micro-slip on the grain boundaries, or something similar, linked with the modification of bundles of dislocations that are gathered at these boundaries. This process should account for most of the recoverable transient creep on unloading (Reference LliboutryLliboutry, 1987b, p. 428–31).

The reason why polycrystalline ice is much more viscous than single isolated crystals is because of piling up of dislocations at grain boundaries. In single crystals they escape freely at the surface. It is not, as often asserted, due to the existence of misoriented hard grains with very high local stresses that would govern the macroscopic strain rate.

Therefore, in Equation (4) giving the water flux within the ice, and in Equation (6) relating the temperature to the principal stress σ ₁, macroscopic stresses may be used. The only requirement is that the macroscopic stress field is almost uniform at the space scale of a grain. Again, we cannot use these formulas to deal with sliding on very small obstacles. In the following, sliding on a sine curve will be tackled. Its wavelength λ must be large in front of the centimetric size of the grains. (As an application, λ = 0.24 m will be used.)

Ice Accretion and Internal Melting Assuming Nye’s Stress Field

Consider a hard bed whose micro-relief is independent of the transverse coordinate y, and whose longitudinal profile is the sine curve:

(7)

The stress field is assumed to be periodic in x. At the scale of the micro-relief, the bed is assumed to be perfectly smooth (no shear stress against it). Which values of α and ω allow the best modelling of a real bed will be considered later. At a large scale (not large enough to require gravity to be considered in the stress equations), the average value of τ _xz is τ_b, and the average value of τ_xz is τ_b. For given ice properties, when there is no ice-bed separation, the average forward velocity (the sliding velocity), say U, is then determined.

Further simplifications are to assume ice to be Newtonian viscous and to consider the asymptotic case αω ≪ 1. This approximation is so crude that no adjustment to field data will be attempted, but it will allow the introduction of the main new concepts. Ignoring phase changes, the asymptotic solution for this plane problem in continuum mechanics, given independently by Reference NyeNye (1969) and Reference KambKamb (1970), is:

(8)

The effective shear stress is:

(9)

The principal stresses are σ₁ = σ₀ − τ, σ ₂ = σ ₀, σ ₃ = σ ₀ + τ. Thus, with the new theory, the temperature field is (in Celsius):

(10)

The vertical heat flux follows:

(11)

In particular, the heat flux entering the ice at the ice-bed interface (z = 0), say ϕ_i , is never negative, as is (1 + sin ωx).

Geothermal heat reaches the interface. There is also a heat flux through the bedrock due to the oscillations of Τ at the interface, and governed by ∇² T = 0 in the bedrock; its average is zero. The mean ice-accretion rate at the interface is thus (() denoting an average for all values of x):

(12)

A is the “melting-refreezing parameter” that is defined at the beginning of the paper. Since A = 21.4 cm² a₋₁ bar⁻¹, τ_b ∼ 1 bar and ϕ_g/(ρL) ∼ 1 cm a⁻¹, there is a trend in ice accretion when a 40 cm. Of course, enough water must be available for that; otherwise ice would become cold and the theory would not hold.

Within the ice, the volume rate of ice melting per unit volume is:

(13)

It is independent of x because, for a Newtonian viscous body, ∇²σ₀ = 0, and in the considered case τ is x-independent. Above z = 2/ω, m becomes negative, indicating a freezing rate; a small one because of the exponential factor. The total melting rate in a vertical column of cross-section unity, up to level z, is

(14)

Ice rises above the bed with the vertical velocity given by Equation (12), but it melts continuously. Since the water content cannot be larger than about 2%, the meltwater is continuously expelled. Therefore, the vertical ice velocity is:

(15)

In the general case, this velocity is positive for z = 0 and it changes sign at some value h _i of z that is less than 1/ω. Thus, a permanent layer of regelation ice is found at the bottom of the glacier, Above h_i small rock debris embedded in the glacier moves down with the ice, but it cannot reach the bed. (Only large debris fragments may touch the bed and scratch it.)

Water Fluxes Assuming Nye’S Stress Field

From Equation (14), the total water produced within the ice per unit time and unit area is equivalent to a volume of ice 2Aτ_b /α. It is larger than the one ice accretion demands by ϕ_g /L, if the flow in the water veins is in the right direction. This point will now be examined.

According to Equation (4), the water fluxes in the x-and z-directions are:

(16)

where k is a positive permeability coefficient that depends on the grain-size and on the cross-sectional area of the veins. The latter may vary, hence k. Any vertical water flux is possible but its direction is determined by the stress field. Since q _z 0 at z = 3/ω = h_w , no water flux crosses this level.

In the layer 0 < z < h_w, water (lows up over stoss sides (sinwx < 0). Part of it allows the freezing that occurs in the layer 2/ω < z < 3/ω. Water flows down over the lee sides (sin ωχ > 0), draining off meltwater that appears below z < 2/ω. As a balance, the outflow would allow (without geothermal flux) a mean accretion rate:

(17)

It is larger than required by

Over z = h _w slight freezing occurs at the expense of the surface water. The global balance of water fluxes is the same as if some surface water crossed the barrier at 2 = h_w and reached the bed, at a rate 0.2 Aτ_b/α.

Best Analytical Expression of the Stress Field Over a Sine Profile Having any Amplitude, with Non-Linear Rheology

To begin with, phase changes will be ignored.

With a realistic rheology, namely third-power law isotropic viscosity, the velocity and stress fields over a sine profile cannot be expressed with known functions, even in the asymptotic case aw→0 Nevertheless, approximate analytical expressions of both fields, including adjustable parameters, may be written. (They are called “trial functions”.) Next, the two complementary variational theorems allow calculation of the best value of the adjustable parameters. The method leads to an upper and a lower bound for very close to each other when the trial functions have been cleverly chosen. The method has been described in Reference LliboutryLliboutry (1987b, p. 35659). In the asymptotic case, the result is:

(18)

The accuracy does not guarantee that the trial fields, with adjusted parameters, are accurate. Therefore, Reference MeysonnierMeysonnier (1983) computed the solution by the finite-element method for different non-zero values of αω. He found that the analytical fields are accurate provided that the overall shear stress τ_b is taken into account, as did Reference KambKamb (1970). Meysonnier parameterized his results as the following law:

(19)

Therefore, my trial stress field, with the above-mentioned correction, is the best analytical expression that is available to date, and will be used henceforth. Whether Equation (19) has to be modified to take phase changes into account will be examined later. This stress field is drawn from the Airy stress function:

(20)

where c is a numerical coefficient that decreases slowly from 0.81447 when αω increases from zero. It is calculated and given in Appendix I.

The components of stress are:

(21)

To simplify their expressions, the following notations and reduced variables will be used.

(22)

It is found:

(23)

Note that the shear stress on the bed, at the scale of the micro-relief, is not 7¾. Subscripts μ referring to the exact values on the sine profile, it is:

(24)

with tan β = —αω sin X. The leading term is found to be:

(25)

which is a good approximation, without trend, of the boundary condition τ_bμ = 0.

The trouble with this improved stress field is that θ has singularities at the points of the (X − Ζ − αω) space where it vanishes. It happens when θ_xx and θ_xz are both zero. These points are sought in Appendix II. In the vicinity of these points, and thus Thus, ∂θ/∂X jumps from a negative to a positive value when X crosses the value X = X ₀. In the vicinity of a singularity |∇0| becomes extremely large. These spikes must be smoothed out, because they have no physical meaning. A mathematical derivative is defined by giving to the variables x or z infinitesimal increments, and increments smaller than the grain-size cannot be considered.

Water Production and Water Fluxes

The Celsius temperature within the ice is:

(26)

Contour lines of the reduced temperature field (S ₀-θ) are displayed in Figure 1, in the asymptotic case αω = 0, and when αω = 0.5. Since θ is always positive, the area in the X–Z plane on the lee side of the bump where Τ − C_mσ_b is positive is smaller than the area on the stoss side where it is negative, and it has steeper gradients. Therefore, more heat comes from the interface on the lee side than goes to the stoss side. For finite amplitudes, the contour lines become skew, and X = ±90° are no longer axes of symmetry.

The melting rate of ice per unit volume within the ice is:

(27)

The result is given in Appendix III. The area where it is negative (i.e. where freezing occurs) is shaded in Figure 1. Since ∇T vanishes at infinity upwards, the global balance of ice melting within the ice is (n denoting the unit normal to the bed, pointing upwards):

(28)

The analytical expression of R_i is given in Appendix III.

Fig. 1

Reduced temperature field in the ice, when it is locally stress-controlled, (S ₀-θ ). The Celsius temperature is The areas where there is freezing instead of melting are shaded. The barrier for capillary flow, at Ζ = Z _w, is above the drawn area.

The components of the water flux within the ice are found to be:

(29)

(As a check, for Newtonian viscous ice, y = 1, and Equation (14) are found.) The water motion in the ice is similar to the one described when ice was assumed to be Newtonian viscous. Nevertheless, the level z = h _w that water does not cross is higher. It is:

(30)

For a = 0, h_w = 4.936/ω, and for αω = 0.5, h_w = 9.000/ω, when for Newtonian viscosity and αω = 0 it was h _w = 3/ω.

Ice-Accretion Rate

The ice-accretion rate (melting rate when negative) reads:

(31)

The term in R _b corresponds to the heat fluxes in the bedrock, caused by the non-uniform temperature at the ice—bed interface. The ratio of the thermal conductivities in the bedrock and in the ice, K _b/K _i is introduced because A has been defined with K _i whereas the heat fluxes in the bed are proportional to K _b.

R _i as given by Equation (28), has been computed for 24 values of X, 15° apart from each other, and for several values of αω. Some spikes appear near the zeros of Θ. As stated in Appendix II, when Ζ = Z_u , the zeros of θ are:

• (1) For

• (2) For

• (3) For

As previously explained, these spikes without physical meaning must be smoothed out. The smoothing may be done in several ways. This arbitrary choice introduces a significant inaccuracy in the mean value of R _i only when this mean is very small. The figures given in Table 1 correspond to the following rule: “when θ_μ < (maxθ_μ)/8, and R _i(X − 15°), R _i(X), R _i(X + 15°), not ordered, then R _i(X) is replaced by the arithmetic mean of the three values”.

Table 1.

Dimensionless accretion rate at the interface, ignoring geothermal heat and assuming K_b = K_i. It may be written R_i+ R_b = R₀ + R₁ sin Χ + …

The temperature at the interface reads:

(32)

Temperatures in the bedrock have been approximated by a truncated series that obeys ∇ ² T = 0 and tends towards

(33)

The b _is have been calculated by fitting this expression

to 24 values of Τ_μ , corresponding to steps in X of 15°, by a least-squares procedure. The method and the results are given in Appendix IV. Next, the reduced value R _b of the heat flux entering the bedrock from the interface is easy to calculate. With the notations:

(34)

it is found:

(35)

In the asymptotic case αω = 0, R _i and R _b can be rigorously calculated. Then:

(36)

The mean value of R _i is (R _i) = 4c ³/π Since

(37)

the temperature field in the bedrock is exactly given by an infinite Fourier series. It yields:

(38)

This series does not converge when sin X = 0, but this singularity must be smoothed out, as explained before.

Values of R _i and R _b at different abscissae are displayed in Figure 2. In the asymptotic case αω = 0, R _i is never negative. When finite amplitudes are considered, negative values appear in the hollows of the micro-relief, with a minimum at about X = −167°, almost opposite to the maximum at about X = 138°. This unexpected large oscillation is not a mathematical artifact due to the vicnity of a singularity, because in this case a similar oscillation would be found near 0°.

Fig. 2

Components of the reduced freezing rate at the interface (melting when negative). R_i is due to the cold coming from the ice (ignoring the viscous dissipation of heat, which is negligible). R_b is due to the cold coming from the bed, ignoring the geothermal flux.

As for R _b, in the asymptotic case it is negative over a smaller extent than in the standard theory. With finite micro-relief amplitudes, an oscillation appears at the crests of the bumps, with a maximum at about X = 0°, and a minimum at 30°.

Mean values of R _i, R _b and of their sum are given in Table 1. 〈R _b〉 is always very small but 〈R _i〉 hence 〈R _i + R _b〉 = R ₀ diminishes strongly for increasing values of αω. A rough parameterization is:

(39)

The fact that 〈r _b〉 is not zero modifies Equation (12). Let us state it again. The mean thickness of the regelation layer is given by the following balance between accretion rates and internal melting, averaged for all X:

(40)

The second term on the righthand side equals 〈r _i〉 and both disappear. (The equation says then that the heat flux entering the ice at the bed is found again at z = h _i.) With reduced variables (in particular Z _i=cwh _i):

(41)

Since Z _i is a positive root of

(42)

It is sufficiently accurate for our needs to use the following approximation:

(43)

Sliding Velocity, Taking the Melting-Refreezing Process Into Account

The fact that on the stoss sides most of the melting occurs within the ice and not against the bed does not impede this melting process to allow in part the surmounting of bumps by sliding ice. Instead of ice disappearing against the stoss face, it disappears mainly within the ice, which contrasts vertically because meltwater cannot accumulate. The only difference with standard sliding theory is that the water migration from stoss sides to lee ones is not by the mythical, never-observed, water film of micro-metric thickness between ice and rock, but thanks to well-known ice permeability.

In Nye’s asymptotic solution for impermeable, dry and Newtonian viscous ice sliding on a sine profile, a stream line runs across the crests of all the bumps. The thickness of the regelation ice below is a_m(1 − cos wx). The velocity and stress fields are the same as though there were no phase changes, and ice was sliding without friction on the sine profile z=a _m+a _v cos wx with a _v=a–a _m In this case, we have η denoting the viscosity, and assuming K _b = K _i (Reference LliboutryLliboutry, 1987b, p. 146-49):

(44.1)

(from the stress-strain-rate field)

(44.2)

(from global equilibrium)

(44.3)

(from geometry)

(44.4)

(from the standard temperature field). By eliminating σ _zμ a _v and a _m are obtained, and a _w + a _m = a yields:

(45)

The sliding velocity may be split into two parts, corresponding to viscous flow and to melting-refreezing, respectively.

With the new theory, another reasoning holds. The stress field for incompressible ice sliding over z = a cos wx is not modified when internal melting occurs, provided that the contact at the bed is kept. And, this can be ensured by an extra velocity U _m. Thus, U given by Equation (19), without changing a is, in fact, U _v. It remains to calculate U _m.

The accretion rate is no longer proportional to sin wx, as resulted from Equation (44.2) and Equation (44.4). It is given by Equation (31), and may be written as a Fourier series:

(46)

The trend, that tilts all the stream lines, may be ignored. Only the leading term in sin ωχ will be kept. This truncation of the series corresponds to modifying a little the sine-profile model.

Since we do not solve the problem by next writing Fourier integrals in ω, nothing compels us to model a series of bumps by an exact sine curve.

The stress field, and ice fluidity, provide a vertical velocity at the bed U _v αω sin wx Ice contraction due to internal melting provides an extra vertical velocity r. The oscillation of the sum equals the actual oscillation of the vertical velocity at the bed, that is U αω sin wx. Thus,

(47)

Values of R ₁ are given in Table 1.

Realistic Values of the Parameters and Numerical Results

With non-linear rheology, sliding over any micro-relief cannot be solved by summing solutions for each sine component of the micro-relief profile. Therefore, to obtain a realistic sliding law on any bed from theory is very difficult (Reference LliboutryLliboutry, 1987a). It is even worse when cavitation must be accounted for, because the behaviour of a sine profile is a singular one, quite different from the behaviour of a series of bumps of unequal heights. Nevertheless, the main prediction of this paper is the existence of an ice-accretion trend 〈r〉, and of a bottom layer of regelation ice with thickness h _i. To estimate their values, the micro-relief model with a sine profile should be sufficient, if the values of U and τ_b, that are related by Equation (47), are realistic. Since there are two parameters in the micro-relief model, one of them has to be arbitrarily fixed.

Since, for given values of τ_b and αω, 〈r〉 is inversely proportional to α, we must consider the smallest bumps existing on the bed for which the theory of locally stress-controlled temperatures applies. When the bumps are too small, the stress field as given by continuum mechanics varies significantly over distances of the same order as the grain-size (~ 1 cm), and this theory is no longer valid. We return to the standard temperature field, which is controlled by the temperatures or heat fluxes at the boundaries of the glacier. Therefore, a wavelength λ=2π/w=0.24 m has been adopted.

The realistic values τ_b = 1 bar and K _b = K _i, have been assumed. A and Β are given at the beginning of this paper. Since water forms continuously in bottom ice, the water content should reach its maximum value, about 2%. Therefore, I have adopted B = 1 bar⁻³ a⁻¹, a larger value than the recommended one for the bulk of the glacier: B = 0.44 bar⁻³ a⁻¹ (Reference LliboutryLliboutry, 1987b, p. 451). Last, the value ϕ_g/(ρL) = 1 cm a⁻¹ has been used. It corresponds to a geothermal flux of 2 HFU, a standard value for young mountains such as the Alps. Nevertheless, this flux may be locally quite different. It may vanish when meltwater cools the bedrock below the glacier. (This is not mere speculation. In the car tunnel beneath Mont Blanc, cold water pours out, and the ancillary tunnel planned for injecting cool air is actually used to evacuate this water.) The geothermal flux may be larger in deep glacier valleys, because of orographic effects (cf. Reference Forster and SmithForster and Smith, 1989).

The results obtained using these values are given in Table 2.

Table 2.

Results with

For high sliding velocities, as for αω = 0.1, there should be cavitation that modifies the stress field in the ice. Nevertheless, the expression that holds without cavitation should remain roughly valid, if we consider z = α cos X as being the lower boundary of the ice, which includes the ceilings of the cavities. The temperatures Τ _μ that govern the heat fluxes in the bedrock are constant along the floor of the cavitities, and this floor is no longer z = α cos X. Therefore, 〈R _i〉 is somewhat changed but it is a very small term in front of 〈R _i〉For very slow sliding velocities, as for αω = 0.7, there is no accretion trend, and accordingly Equation (42) and Equation (43) yield no solution for h _i. However, strain rates might not be large enough to cause dynamic recrystallization. Then, as reported above, crystals enlarge considerably and λ = 24 cm is too small a value for the theory to be valid. Therefore, in this case 〈r〉 = −1 cm a⁻¹ rather than −0.66 cm a⁻¹.

It appears that the internal-melting-ice-accretion effect is important for any sliding velocity greater than 1 ma⁻¹. Between two crests 24 cm apart, at moderate sliding velocities, the layer of accreted ice thickens by about 2 mm. Internal melting impedes this bottom-regelation ice layer to grow infinitely: its thickness h _i stands at 3–4 cm.

Of course, these figures are only rough estimations, because the adopted wavelength, λ = 0.24 m is somewhat arbitrary. The possibility that the best modelling value differs by a factor of 2 cannot be dismissed. Field observations and sample analysis which allow the determination of h _i and h _w might constrain the model better.

Conclusion

Although the consideration of non-linear viscosity and finite amplitudes led to cumbersome hand calculations, it was necessary, because there might have been an objection in that the results yielded by locally stress-controlled temperatures are totally different in the asymptotic case aw–0 It has been stated that this is not the case, unless sliding velocities are very small (totally negligible when calculating surface velocities).

The main consequence of considering ice permeability and wetness is the prediction of a bottom layer, some decimetres thick, in which water flows from the stoss sides of decimetric bumps to the lee sides. The main victim of the theory is the popularized concept of a continuous water film between ice and bedrock. Lee cavities (that remain possible) cannot be considered as “thickenings of the water film”. It is within this decimetric bottom layer that ions from the bedrock can diffuse into the ice, or migrate to lee faces and provide calcite deposits.

This result is at variance with a previous speculation by Reference RobinRobin (1976), the first person to consider the effect of ice permeability on sliding, for two reasons:

• (1) There is internal melting even over the lee sides of the bumps. It is not the mean normal pressure in the ice that governs the local ice temperatures but the pressure against the water lenses, which is more or less equal to the maximum compressive stress, and this fact leads to ∇² T > 0 near the bed, even over the lee sides.

• (2) Anyway, because of capillary effects, lenses may freeze without the water in the veins being frozen. Therefore, water can always reach the lee sides of the bumps.

On the other hand, there should be a thinner bottom layer, some centimetres thick, which consists of regelation ice (with possibly small debris picked up from the bed). Since this regelation ice recrystallizes continuously, it cannot be distinguished by its fabric from old ice derived from the firn. The fact that bottom ice is bubble-free is irrelevant, as shown by field evidence. Isotopic studies, as those done by Reference Jouzel and SouchezJouzel and Souchez (1982) would be the only way of checking this prediction.

To check directly the theory of locally stress-controlled temperatures seems almost impossible in the laboratory, because ice creeping steadily during many hours and recrystallizing must be observed. Experiments such as those by Reference Nye and MaeNye and Mae (1972) are not pertinent to this problem. Steady shear of a large ice volume has been obtained by using the huge viscosimeter Penelope (Reference BrepsonBrepson, 1979; Reference MeysonnierMeysonnier, 1983), but Penelope only worked well at some tenths of a degree below the melting point, using dry ice, and it also took far too much time to open the chamber and retrieve an ice sample.

In situ observations on the walls of a tunnel dug along the bed seem to be more promising. However, they could be obstructed by the fact that temperate-ice properties are of little interest either to industry or climatology. Therefore, to obtain a research grant for the study of such an unusual material becomes quite a problem.

To my knowledge, the only solid material existing in Nature entirely at its melting point, other than ice, is the parent rock from which magma flows towards a magma chamber, a fashionable topic today. Although in melting rocks the melting point is not well defined, this paper might interest volcanologists. Unfortunately, they still ignore the fact that glacier ice is a metamorphic rock, and would be reluctant to call meltwater a magma!