Regelation at subfreezing temperatures

At the moment, glaciologists are almost unanimous in the view that glacier sliding can occur only where the basal ice is at the melting temperature, and not where it is sub-freezing (Reference PatersonPaterson, 1981, p. 112), because pressure melting and regelation are necessary for slip past the smallest-scale elements of the bed roughness (Reference NyeNye, 1970, p. 390). The direct field observations generally cited as confirmation, however, namely those of Reference GoldthwaitGoldthwait (1960) and Reference HiltyHilty (1960, p. A3) carried out in a tunnel under an ice cliff in Greenland (basal ice temperature −11° C) and those of Reference Holdsworth and BullHoldsworth and Bull (1970, p, 206) in one under Meserve Glacier in Antarctica (−18° C), at best show only that the slip is very small compared to what it would be at the melting temperature. Moreover, indirect evidence indicates that sliding at sub-freezing temperatures might be possible, albeit slow. The migration of included solid particles in sub-freezing ice under a temperature gradient (Reference Hoekstra and MillerHoekstra and Miller, 1967; Reference Römkens and MillerRömkens and Miller, 1973) points strongly to the presence of a liquid or liquid-like layer surrounding such particles, similar to the layer on sub-freezing free ice surfaces proposed by Reference FaradayFaraday (1850), argued by Reference WeylWeyl (1951) and Reference FletcherFletcher (1962), and demonstrated by Reference Nakaya and MatsumotoNakaya and Matsumoto (1953), Reference JellinekJellinek (1967), and others. The regelation of ice well below the pressure-melting temperature behind wires pulled transversely through it, as demonstrated by Reference Telford and TurnerTelford and Turner (1963) and confirmed by Reference GilpinGilpin (1980), suggests even more compellingly that sliding at sub-freezing temperatures might be possible.

Reference Telford and TurnerTelford and Turner (1963) used 0.45 mm diameter steel wires under a driving stress (as defined by Reference Drake and ShreveDrake and Shreve, 1973, p. 55) of about 3 MPa (30 bar). Speeds ranged from 160 mm a⁻¹ at −3.5° C to 800 mm a⁻¹ at −0.7° C. A 50% increase in driving stress caused a three-fold increase in speed which, in the light of the high driving stress, suggests that plastic flow of the ice may have been significant. Other evidence on this point, however, is at best equivocal. The apparent activation energy calculated from the temperature dependence was 350 kJ mol⁻¹ (recalculated from the data presented), which is about twice the mean activation energy found by Reference Mellor and TestaMellor and Testa (1969, p. 135) for secondary creep of polycrystalline ice in the same temperature range. Also, passage of the wire through a very thin gold foil embedded in the ice observably displaced it over a width about 40% greater than the diameter of the wire. Thus, Telford and Turner’s quantitative results probably were affected by plastic flow. Nevertheless, their pioneering experiments demonstrated the existence of regelation at sub-freezing temperatures.

Reference GilpinGilpin (1980) used 0.0127 to 0.059 mm diameter tungsten wire under driving stresses from 0.27 to 0.68 MPa (3 to 7 bar), 0.076 mm diameter Chromel (nominally 90% Ni, 10% Cr) wire under 0.28 MPa (3 bar), and 0.127 to 0.381 mm Constantan (nominally 55% Cu, 45% Ni) wire under 0.04 to 0.20 MPa (0.4 to 2 bar). Speeds ranged from 1.3 mm a⁻¹ for 0.0127 mm diameter tungsten wire under 0.68 MPa at −35° C to 40 mm a⁻¹ for 0.381 mm diameter Constantan wire under 0.04 MPa at −0.006° C (Reference GilpinGilpin, 1980, p. 443, fig. 8, using p. 442, eq. 18). For comparison, the lowest speeds observed by Reference Drake and ShreveDrake and Shreve (1973, p. 56–58) at 0° C (blocks immersed in melt water and crushed ice) were around 160 mm a⁻¹. At constant temperatures the speeds increased linearly as the driving stress increased (Reference GilpinGilpin, 1980, p. 443, fig. 7); hence, plastic flow of the ice probably was insignificant. Thus, the experiments of Gilpin not only confirmed the existence of regelation at sub-freezing temperatures but also supplied detailed quantitative information on its rate.

Essentials of Gilpin’s model

Gilpin undertook his experiments to test a model of the liquid-like layer adjacent to foreign solids in sub-freezing ice that he had proposed the previous year. The essential idea of the model (Reference GilpinGilpin, 1979, p. 238) is that the foreign solids attract adjacent water, which lowers its chemical potential by an amount assumed inversely proportional to some power α of the distance γ from the interface. The attraction causes an excess of pressure near the interface which in a thin, planar layer of uniform thickness is given by

(1a)

(derived from Reference GilpinGilpin, 1979, p. 239, eq. 5), where p = P_w(h) is the pressure at the outer surface of the layer, ρw and ρi are the densities of water and ice, C is the rate of decrease of melting temperature with pressure (from the Clapeyron equation), and b and α are experimentally determined constants. The gradient in the chemical potential makes equilibrium between water and ice at the outer surface dependent upon the layer thickness; thus, the pressure-melting temperature θ and pressure ρ are related by

(1b)

(derived from Reference GilpinGilpin, 1979, p. 239, eq. 11), where θ_PM is the melting temperature at pressure θ_PM of ice in contact with an infinitely thick water layer (such as 0° C at 1 atm, or 0.1013 MPa, for example); that is, θ_PM is the nominal melting temperature at pressure P_PM.

From his regelation experiments Reference GilpinGilpin (1980, p. 445, 446) found α = 2.4 and b = 20 nm^2.4° C for the three metals used, in agreement with the results of nuclear magnetic resonance (n.m.r.) measurements made by Reference Barer, Barer, Kvlividze, Kurzayev, Sobolev and ChurayevBarer and others (1977) of the amount of unfrozen water in ice containing dispersed fine silica (Aero-sil) particles. Below about −15° C the points based on his measurements deviated from the fitted relationship, whereas the n.m.r. data did not, a discrepancy he attributed (Reference GilpinGilpin, 1980, p. 445) to excess water viscosity due to the extreme thinness of the liquid layer at the lower temperatures. It appears, however, that the discrepancy may actually be due to the formula he used to extrapolate the viscosity to the lower temperatures, which was a second-degree polynomial fitted to the viscosity data for water between −10 and +10° C (Reference GilpinGilpin, 1980, p. 444). From −10 to −24° C it gives values which are increasingly lower than the measured values which Reference HallettHallett (1963, p. 1049) determined using 0.2 mm diameter glass capillary tubes, the difference reaching about 24% at −24° C. An alternative formula,

fits Hallett’s measurements from 0 to −24° C with a root-mean-square error of 0.5% and a maximum error of about 1.1%. When extrapolated to higher temperatures, it gives values increasingly too low, but the deviation is only 1.6% at +10° C and 3.5% at +20° C. When extrapolated to −35° C, it gives a value of 12.9 mPa s (whereas Gilpin’s formula gives only 5.9 mPa s). Inasmuch as this value eliminates the discrepancy between the regelation experiments and the n.m.r. measurements at the lower temperatures, it appears permissible to apply Gilpin’s model and constants to ice in contact with silica and hence, with considerable confidence, with rock.

Modification of Nye’s theory

Clearly, Reference WeertmanWeertman’s (1957, Reference Weertman1964) model of glacier sliding applies even at sub-freezing temperatures, inasmuch as all its basic assumptions are satisfied (at least down to −35° C): a thin basal water layer, pressure melting and regelation, and plastic deformation of the ice. To estimate the magnitude of the sliding speed it suffices to modify the theory of Reference NyeNye (1969, Reference Nye1970), which approximates the basal ice as a Newtonian viscous fluid, rather than that of Reference KambKamb (1970), which takes into some account its actual power-law rheology, or that of Reference LliboutryLliboutry (1968), which includes subglacial cavitation. Hence, except as stated, the derivation that follows makes the same assumptions and uses the same notation as Nye’s theory.

From Equation (1b) Nye’s Fourier-transformed relationship between boundary temperature and pressure (1969, p. 449, unnumbered equation preceding eq. 2)

(2)

where h_m ^−α is the value of h^−α averaged over the glacier bed, the overbars denote Fourier transforms, and k, the argument of the Fourier-transformed functions, is the wave number. The quantity h_m is the thickness the water layer would have if the bed were planar. Thus, the unperturbed thickness is always finite, whereas in Nye’s theory (Reference NyeNye, 1969, p. 447) it is zero, which, amongst other things, obviates the difficulty with negative liquid-layer thicknesses pointed out and discussed by Reference Nye, Whalley, Whalley, Jones and GoldNye (1973[a]). This is a fundamental difference: Nye’s theory cannot be recovered simply by setting b to zero in the modified equations.

In terms of the bed-perturbation parameter ε,

(3)

Mathematically, ε has to be small enough to ensure the convergence of the series. Physically, however, it approximates the slopes of the bed roughness. Thus, for application δ₁, δ₂, …, have to be small relative to 1/ε.

Nye’s relationship from regelation physics (Reference NyeNye, 1969, p. 452, eq. 25) then becomes, letting K be the mean thermal conductivity of ice and rock,

(4)

on z = 0, the mean elevation of the bed.

The new feature is the second term in the right-hand side of Equation (4). It forces explicit consideration of the flow in the liquid layer, which in Nye’s theory was unnecessary. Let q_w be the mass rate of flow in the liquid layer. Then conservation of mass requires

(5)

and, disregarding the comparatively small net shear across the layer, slow viscous flow requires

(6a)

(Reference BatchelorBatchelor, 1967, p. 220), where w_n(x) is the outward component of the basal ice velocity perpendicular to the liquid layer (Reference NyeNye, 1969, p. 448) and η_w is the viscosity of the water in the layer. Substitution of Equation (1a) into Equation (6a) and of the perturbation expansions for p and h^−α into the result leads to

(6b)

to first order in ε. The second term in the square brackets represents a pressure gradient that causes flow toward regions of smaller liquid-layer thickness. Fourier transformation of Equations (5) and (6b) then gives

(7a)

and

(7b)

from which, as w _n(k) = εW(k) (Reference NyeNye, 1969, p. 450),

(8)

on z = O.

Nye’s relationship for the perturbed ice flow (1969, p. 452, eq. 24) remains unchanged, namely,

(9)

on z = 0, where n_j is the viscosity of the basal ice, assumed Newtonian but with value dependent upon the basal effective shear stress, U is the speed of sliding, and z = εf(x) is the profile of the bed.

Simultaneous solution of Equations (4), (8), and (9) gives finally

(10a)

(10b)

(10c)

in which

(11)

With p ₁(k) given by Equation (10a), Nye’s formula for the drag (1970, p. 385, eq. 5) becomes

(12a)

where S_b(k) is the spectral power density of the glacier bed profile. Because k₀ and k₁ are both positive, the denominator is positive throughout the range of integration, by Descartes’s rule of signs. Taking

(12b)

(Reference NyeNye, 1970, p. 386–387), where 2∏/k is to be chosen much larger than the distance over which the running mean is taken in obtaining the datum from which the bed profile is measured (Reference NyeNye, 1970, p. 387), the drag is

(13)

Calculation of sliding speeds

Table I gives the data and their sources used in computing k₀ and k_l for various temperatures ∆θ = θ – θ_PM + C(p – p_PM) relative to the nominal pressure-melting point of pure water. Treating the temperature in this way eliminates from explicit consideration the average pressure due to the weight of the overlying ice. For ∆θ in the range from −10⁻⁸ to −20° C the term involving k₁ ² in the denominator of the integrand in Equation (13) is negligible compare to the sum of the other two terms. Omitting this term, performing the integration, and dropping terms negligible because K and k₁ are small compared to k₀, leads to

(14)

Figure 1shows the sliding speeds computed from this formula for a = 0.022 and drag and basal effective stress both 100 kPa (1 bar).

Fig. 1. Speed of glacier sliding and thickness of liquid layer as functions of temperature of basal ice for various average solute concentrations in liquid layer. Temperature is relative to nominal pressure-melting point of pure water. Dashed portions of curves based on extrapolation of Reference GilpinGilpin’s (1980, p. 445) data toward the pressure-melting point. Speeds calculated for drag and basal effective stress both 100 kPa (1 bar) and bed roughness spectrum as measured by Reference NyeNye (1970, p. 386–387).

Table I. Basic Data used in the Calculations

The value a = 0.022 was computed by Reference NyeNye (1970, p. 388–389) from a profile measured near the terminus of Austerdalsbreen, Norway, in 1963 using the formula

(15)

where r = 0.016 is the root-mean-square deviation of the profile from its simple centered running mean over distances of the order of 100 m (Reference NyeNye, 1970, p. 385, 389). It may not be accurate for wavelengths 2Π/k less than a few millimeters (Reference NyeNye, 1970, p. 387).

Equation (14) is not valid above about −10^−8oC, because k₀ is not large compared to k₁ at the higher temperatures. This is because, as the thickness of the liquid layer exceeds about 10 μm, the flow of heat through the basal ice and the glacier bed, rather than the flow of water through the liquid layer, becomes the rate-limiting factor in the regelation process. This transition thickness is an order of magnitude greater than expected from Reference NyeNye’s theory (1973[b], p. 190), even though the important wavelengths, which are about one metre, and hence the geometry of heat and ice flow, are comparable in the two theories. The discrepancy is due to the fact that in the modified theory the gradient in pressure from stoss to lee of obstacles caused by shearing of the ice over the bed is partially offset by an opposing gradient caused by the smaller thickness of the liquid layer on the stoss side.

As already noted, it is necessary that δ1, δ2,…, in Equation (3) be small relative to 1/ε. For drag and temperature externally fixed, as would normally be the case, this requirement sets a non-zero lower limit on ε in addition to the upper limit ε « 1. Equations (14) and (15) show that halving ε, and therefore r, for example, will quarter a and quadruple U; Equation (10c) shows that this in turn will quadruple δ₁, Thus, as ∊ decreases, 1/ε increases more slowly than δ₁; hence, the condition δ₁ « 1/ε cannot hold for sufficiently small ε.

To make this more quantitative requires an estimate of δ₁, which requires inverse transformation of Equation (10c). This in turn cannot be done without more information about f(x) or f(k) than is given by the spectral power density, Equation (12b). A conservative over-estimate can be made, however, by replacing with its maximum the function of k multiplying f(k) in Equation (10c) and performing the inverse transformation.

Before doing so it is neessary to put Equation (10c) in terms of f’(k) = −ikf(k), the transform of f’ (x) = df/dx, because | f’(x) | is of order 1, in asmuch as ε approximates the slopes of the bed roughness whereas | f(x) | has no particular order of magnitude, inasmuch as it is a dimensional quantity.

Making this change, dropping negligible terms involving k₁, solving for the maximum, which occurs for | k |³ = | k₀ |^3/2, performing the inverse transformation, eliminating ηi^Uk ₀ and a by means of Equations (14) and (15), substituting | ∆θ | for bh_m ^−α, and seting | δ₁ | max « 1/ε leads finally to

(16)

as the conservative minimum permissible range of ε.

For <τ_xy> = 100 kPa (1 bar) the lower limit in Equation (16) varies from 0.01 to 0.1 as | ∆θ | decreases from approximately 0.1 to 0.002° C.

Modification of present theory

Because the thickness of the liquid layer is very nearly uniform, the theory is easily modified to take into approximate account the presence of dissolved solutes.

Assuming that the solute is not influenced by the presence of the foreign solid, as seems likely for the dissolved air and various ions most likely to be present, Equation (1a) remains unchanged and the right-hand side of Equation (1b) has an additional term -Mc, where M is the rate of decrease of melting temperature with solute concentration c, assumed dilute. Hence, the right-hand side of Equation (2) also has an additional term -M(c-c_m), where C_m is the value of c averaged over the glacier bed and is considered to be known. In terms of the small bed-perturbation parameter ε,

(17)

Thus, the right-hand side of Equation (4) has an additional term -Mc_m γ ₁ (k)/C. Equations (8) and (9) have no additional terms, however, because their derivation does not involve Equation (1b). In addition, h_m in Equations (2), (4), and (8) becomes h_m*, as a reminder that it is to be computed from the modified Equation (1b) using the average solute concentration c_m.

The advection of the solute by the flow in the liquid layer is balanced by diffusion in the opposite direction, so that

(18)

approximately, assuming partitioning of the solute entirely to the liquid and letting D be the diffusivity of the solute. Substitution of the perturbation expansions, Equation (3) and (17), followed by Fourier transformation and substitution of ε W(k) for w _n(k), as before, then gives

(19)

Simultaneous solution of the modified equations gives the same expressions as before for p ₁ (k) and W(k), Equations (10a) and (10b), and hence for the drag, Equations (12), (13), and (14), provided k₀ ³ is replaced by

(20)

For δ ₁(k) the expression is as in Equation (10c) with h_m and k₀ replaced by h_m* and k_0* and a term

added inside the square brackets. Finally, from Equations (10b) and (19)

(21)

Effect on sliding speeds

Table I gives the additional data and their sources used in computing k_0*; and Figure 1shows the sliding speeds computed from Equations (14) and (20) for various average concentrations of NaCl in the liquid layer as functions of the temperature below the nominal pressure melting point of pure water. In the range of validity of Equation (14) (K and k₁ samll compared to k₀), the presence of the solute increases the speed of sliding because it increases the thickness of the liquid layer, and hence the flow of water, without significantly affecting the temperatures of melting and freezing, and hence the flow of heat, which is exactly the opposite of its effect in Nye’s theory (as modified by Reference HalletHallet, 1976[b], p. 213).

With solute present it is necessary that both δ₁ and γ₁ be small relative to 1/ε. Applying the procedure used to derive Equation (16) to the modified Equations (10c) and (21) yields

(22a)

as the conservative minimum permissible range of ε, where

(22b)

(22c)

and

(22d)

which is the temperature measured downward from the nominal pressure-melting point of a solution of concentration c_m.

For <τ_xy> = 100 kPa (1 bar) and c_m = 1 p.p.m. the lower limit in Equation (22a) varies from 0.01 to 0.1 | ∆θ_* |as decreases from approximately 0.1 to 0.002° C. The corresponding range for c_m = 10⁴ p.p.m. is approximately 0.1 to 0.005° C.

Sliding at sub-freezing temperatures

Figure 1 indicates that the basal sliding speeds of typical subfreezing glaciers will be extremely low, and it shows why the direct field observations failed to detect sliding. The effect of such low speeds on glacier motion will be completely negligible. The total distance of sliding during the lifetimes of large glaciers, on the other hand, can be of consequence. In 10⁵ a, for instance, it ranges from roughly 35 m at −20° C to roughly 350 m at −5° C for drag and basal effective stress both 100 kPa (1 bar) and would be five times as great at 200 kPa (2 bar), inasmuch as the sliding speed increases as the stress raised to the power 7/3. Assuming that rocky debris was incorporated into the basal ice when the glacier initially began accumulating, the formation of bedrock striations, though not of grooves, seems highly plausible. In addition, the formation of subglacial carbonate or other coatings on suitable bedrock, as described by Reference HalletHallet (1975; Reference Hallet1976[a]) for temperate glaciers, is not precluded. Thus, neither striations nor coatings necessarily indicate temperate ice. If coatings form, the largest rock fragments in them, if any are present, should be very much smaller than those observed by Reference HalletHallet (1979, p. 325–326). If observable, they might permit rough estimation of the liquid-layer thickness and hence of the basal ice temperature.

The maximum contribution to the drag comes from the logarithmic wave-number interval d(ln k) located at k = k₀/2^1/3 = 0.749k₀. Imposing low- and high-frequency cut-offs on tne bed spectrum in Equation (13), as done by Reference NyeNye (1970, p. 391), shows that 90% of the drag comes from that part of the spectrum between 9k₀ and k₀/9, which at temperatures below about −1° C corresponds to wavelengths of 0.1 to 10 mm for basal effective stress 100 kPa (1 bar) and solute concentrations from 0 to 10⁴ p.p.m. These wavelengths vary inversely as the basal effective stress, raised to the power 2/3, in accordance with Equation (11) and the formulae of note (5) of Table I. In any case, they are small enough that the value of a used in computing the sliding speeds plotted in Figure 1may be inaccurate.

Sliding with net melting

Equation (1b) implies that the liquid layer can never be at the nominal pressure-melting temperature, even in a temperate glacier, because it can never be infinitely thick. Instead, it will be at some lower temperature corresponding to its actual thickness, the ambient pressure, and the solute concentration. How this temperature is governed depends upon whether more basal ice melts into the liquid layer than re-freezes, that is, whether there is net melting. With no net melting the temperature is governed by the temperature of the ice above and the rock below, and the thickness adjusts accordingly. With net melting, on the other hand, the thickness is governed by the rate of escape of the excess water, and it is the temperature that adjusts accordingly. Thus, the proper distinction is not between temperate and cold, which in this context are meaningless, but between net melting and no net melting.

With net melting the liquid-layer thickness and, concomitantly, the temperature and the sliding speed increase until the rate of escape balances the rate of melting. The ease of escape is governed by the proximity of subglacial water passages (like those considered by Reference LliboutryLliboutry, 1968, p. 54–56; Reference RöhlisbergerRöthilisberger, 1972, p. 200–201; Reference ShreveShreve, 1972, p. 211–213; Reference WeertmanWeertman, 1972, p. 291–308; and Reference NyeNye, 1973[b], p. 191–192) and by the pressures in them, and is likely to be highly variable from case to case. No conflict such as discussed by Reference NyeNye (1973[b]) arises between the thickness requirement for run-off water flow and that for regelation water flow, although the thickness may become so great that k₁ is not negligible or the perturbation method of solving the equations becomes inapplicable. The theory thus applies whether or not there is net melting.