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A slip law for glaciers on deformable beds

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Science  03 Apr 2020:
Vol. 368, Issue 6486, pp. 76-78
DOI: 10.1126/science.aaz1183

Slipping on till

How do glaciers flow over the ground underlying them? We know that friction, ice stream velocity, and water pressure at the ice bed all matter, but we still do not know how to represent the process over both hard beds (which are solid rock) and soft ones (composed of unconsolidated erosion products called till). Zoet and Iverson present experimental results describing how glacial ice moves over watersaturated till (see the Perspective by Minchew and Joughin). These observations should help to solve the long-standing problem of constructing a generalized slip law that combines the processes of hard-bedded sliding and bed deformation.

Science, this issue p. 76; see also p. 29

Abstract

Slip of marine-terminating ice streams over beds of deformable till is responsible for most of the contribution of the West Antarctic Ice Sheet to sea level rise. Flow models of the ice sheet and till-bedded glaciers elsewhere require a law that relates slip resistance, slip velocity, and water pressure at the bed. We present results of experiments in which pressurized ice at its melting temperature is slid over a water-saturated till bed. Steady-state slip resistance increases with slip velocity owing to sliding of ice across the bed, but above a threshold velocity, till shears at its rate-independent Coulomb strength. These results motivate a generalized slip law for glacier-flow models that combines processes of hard-bedded sliding and bed deformation.

The potential collapse of the West Antarctic Ice Sheet is the single largest source of uncertainty in estimations of future sea level rise (1, 2), and this uncertainty results, in part, from imperfectly modeled ice sheet processes (35). Among these processes is the rapid slip of marine-terminating ice streams over their beds, which transfers most of the ice sheet’s discharge to the ocean (6). This rapid slip usually occurs where ice at its pressure-melting temperature rests on unlithified, water-saturated till—a mixture of mud, sand, and larger particles that can shear beneath the ice if it is weakened sufficiently by high pore-water pressure (7). Therefore, it is of central importance in glacier-flow models to accurately relate slip resistance to slip velocity and water pressure where ice rests on deformable till (i.e., soft bed) (35)—a condition met not only in West Antarctica but beneath many fast-flowing, marine-terminating glaciers elsewhere that contribute substantially to sea level rise (8).

The slip relation most widely applied in glacier-flow models is derived from considering ice resting on an irregular rock surface (i.e., hard bed). In this scenario, movement of ice past rock bumps by regelation and by creeping flow causes slip resistance to increase with slip velocity—the equivalent of viscous drag at the bed (9). Although viscous drag is usually assumed in glacier-flow models even where glaciers rest on deformable till, this is at odds with experiments with till that indicate its shear resistance is either independent of its deformation rate or highly insensitive to it (10). This observation, together with till shear resistance that increases linearly with effective stress (total-bed normal stress less pore-water pressure), has helped motivate use of Coulomb slip laws in some glacier-flow models (11, 12). In modeling studies in which effects of both viscous and Coulomb slip laws are compared, ice-loss and sea level–rise projections are acutely sensitive to the form of the slip law, with viscous and Coulomb relations yielding distinct glacier responses (35).

This study is motivated by the hypothesis that physical processes at the interface between ice and deformable till contribute to the form of the slip relation, such that experiments with ice on hard beds or with till alone provide an incomplete description of slip behavior on soft beds. We present results of the first experiments that address coupled ice-bed mechanics where ice, under pressure and at its melting temperature, moves steadily over water-saturated till. These results provide a slip relation for soft-bedded glaciers that merges processes of hard-bedded sliding and till deformation.

Experiments are conducted with a ring-shear device designed to study glacier slip over hard (13) or soft (14) beds. This device rotates a horizontal ring of ice (0.9 m in outside diameter, 0.2 m wide, and ~0.2 m thick) at a steady velocity and under a prescribed effective stress of the order expected in natural glaciers across a till bed, while the resultant shear stress is measured (fig. S1). The ice is kept at its pressure-melting temperature by dilute ethylene glycol, regulated to 0.01°C, which circulates around the chamber that holds the ice. Calibrated thermistors measure ice temperature to the nearest 0.01°C. The bed is a 60-mm-thick layer of water-saturated, sandy till. Vertical columns of beads positioned within the till and ice allow the vertical distribution of horizontal displacement in both materials to be measured at the ends of experiments. In one kind of experiment, all grains larger than one-tenth of the till thickness (>6 mm) are removed from the till. In a second kind of experiment, designed to increase bed surface roughness, 11 isolated oblate clasts (axial lengths of ca. 40 mm by 30 mm by 30 mm) are embedded partway in the till surface (see supplementary materials for more methodological details).

Most shear deformation of the till occurs in a zone close to the ice-bed interface. Bead columns indicate that deformation in experiments without clasts occurs only in a narrow zone (7.5 mm) immediately adjacent to the sole of the ice ring (Fig. 1). Approximately 35% of ice motion is caused by till deformation, whereas markers in the ice reveal no discernible strain in the nonboundary ice, indicating that ~65% of total motion results from slip at the ice-till interface. Till deformation in the experiment with clasts results in a concave-up deformation profile and is deeper, extending to a depth of ~25 mm (Fig. 1). Again, no discernible strain is observed in the nonboundary ice, indicating that ~45% of the total motion is accommodated by till deformation and ~55% results from slip at the ice-till interface (Fig. 1). The deformed bed thicknesses are ~125 and ~85% of the maximum grain diameter in the experiments without and with clasts, respectively. When clasts are present, ~25% of their radius protrudes above the interface, and they produce prow-like ridges at their downstream edges, indicating that clasts were dragged by the sliding ice through the yielding till surface (so-called plowing; see fig. S2).

Fig. 1 Till deformation.

Horizontal displacement of beads inserted in initially vertical columns in the till bed from the experiments without (black) and with clasts (red) at the bed surface. Error bars indicate ±1 standard deviation of measurements from four bead columns.

In both types of experiments, steady-state resistance to slip increases with slip velocity (rate strengthening) until a transition speed, ut, at which the shear stress equals the Coulomb shear strength of the till, the stress at which it deforms permanently, as measured independently in tests with a direct-shear device (14). Thus, at higher slip velocities, shear stress is limited by the shear strength of the till, and shear stress is independent of slip velocity. The value of ut is lower with clasts at the bed surface (Fig. 2).

Fig. 2 Slip resistance.

Steady-state shear stress supported by the till bed divided by its Coulomb shear strength for the two kinds of experiments. Slip velocity is abruptly increased, and, subsequently, the shear stress requires hours to days to attain a steady-state condition when stress fluctuates around a mean value that remains approximately steady with time. The stress values reported are the mean values, and the error bars indicate ±1 standard deviation of the fluctuations.

The motion at the ice-till interface and shear of till near the bed surface, together with the rate-strengthening and Coulomb drag responses, indicate that processes of both sliding and till deformation are needed to describe slip on a soft bed. This observation agrees with most measurements beneath modern glaciers, which indicate that basal motion occurs primarily at the surfaces of till beds or by till shear to only shallow depths (7). Thus, to develop accurate slip laws for ice streams and other glaciers on till beds, both types of movement must be considered.

Our results indicate that the transition to bed deformation occurs if the drag from ice flow exerted on particles at the bed surface causes them to plow, mobilizing the bed at its failure strength (Fig. 2). At slip velocities below ut, ice viscously deforms and regelates around static particles at the bed surface, giving rise to a rate-strengthening slip relation. At slip velocities above ut, these processes cause sufficient drag on large particles—those closest to the controlling obstacle size of sliding theory (9)—to cause them to plow at a stress limited by the till’s Coulomb strength, thereby shearing the bed. This hypothesis is supported by plowing structures at the ice-bed interface, deformation depths that are comparable to the largest particle diameters at the bed surface, and the dependence of ut on the largest particle sizes at the bed surface. Smaller particles require higher slip velocities to cause drag sufficient to deform the bed by plowing (Fig. 2).

We propose a traditional rate-strengthening sliding rule [i.e., (9)] for slip velocities below ut and, for higher slip velocities, Coulomb resistance similar to that measured in deformation experiments with till alone (10). A form of slip law that describes both processes and corresponds to our experimental observations isτb=min[Ntan(ϕ),(Cub)1/m](1)where N is the effective stress; ϕ is the friction angle of the till, such that tan ϕ is a friction coefficient; ub is the slip velocity; m is a sliding exponent; and C is a constant that depends on ut and the bed roughness, as controlled by the sizes of particles at the bed surface. The magnitude of C can be estimated from the velocity, ut, at which plowing deforms the bed: C=(Ntan(ϕ))utm. Equation 1 has the same form as sliding rules of some numerical models (4) but has a different physical basis, as ut is here explicitly estimated from plowing mechanics and then used to parametrize C, unlike previous implementations that used a viscous slip law with an assumed C to estimate ut. Low speeds and high effective stresses result in viscous Weertman-style sliding; high sliding speeds and low effective stresses—conditions commonly present at the bases of ice streams—promote shallow deformation of the bed at its Coulomb strength.

The value of ut can be estimated independently. The shear stress needed to cause a partially buried clast to plow, τp, is independent of sliding velocity, whereas the stress that sliding ice exerts on a static clast, τc, increases with sliding velocity. If τp = τc, plowing will occur. By modeling the upper surfaces of clasts partly buried in the bed as sinusoids, the transition speed, ut, can be estimated by setting the shear stress exerted by sliding ice (15) equal to that required for clasts of radius R to plow (16)ut=(1η(Ra)2k03+4C1(Ra)2k0)(NFN)(2+NFk)(2)where η is the effective ice viscosity, k0=2π4R, C1 is a regelation parameter dependent on thermal properties of the ice and rock, a is the fraction of a clast radius that protrudes from the bed surface, NF is a bearing-capacity factor for the till related to its strength, and k accounts for the till-strength reduction resulting from the ice-pressure shadow in the lee of clasts (see supplementary materials). With an estimate of ut from Eq. 2, τb of Eq. 1 can be estimated for different sliding speeds and clast sizes (fig. S3). For a single clast size, τb has a sharp inflection at ut, at which the slip behavior changes. More realistically, when slip is considered over a range of clast sizes (e.g., R = 15 to 30 mm), clasts begin to plow over a range of velocity (40 to 80 m/year) (fig. S3), so the average value of τb transitions smoothly to Coulomb behavior (fig. S3).

The viscous and Coulomb behavior of Eq. 1 can be approximated with a sliding rule of the formτb=Ntan(ϕ)(ubub+ut)1/p(3)where p is the slip exponent. In experiments both with and without clasts at the bed surface, p ~ 5 (Fig. 3), indicating insensitivity of this parameter to the detailed geometry of the bed surface. The form of this function approximates the Coulomb behavior expected at high slip velocity (ub > ut) and low effective pressure, which causes ut in Eq. 2 to approach zero. This sliding relationship is not only consistent with our experimental results (Fig. 3) but, through Eq. 3, is based unambiguously on physical processes at the ice-bed interface and the properties of tills. Estimating the clast radius R for calculating ut in Eq. 3 requires noting that plowing clasts larger than a threshold size will occupy a sufficient fraction of the bed surface to cause pervasive deformation of the bed at its Coulomb strength. This threshold value of R could thus be estimated from a till grain-size distribution and by noting that plowing clasts deform adjacent till across distances comparable to clast diameters (17). The form of Eq. 3 is similar to that estimated for sliding over a hard bed with cavity formation (18), so a slip law of this form may be generally applicable without knowledge of the bed type.

Fig. 3 Generalized model.

Measured steady-state shear stresses compared with the approximate model of Eqs. 2 and 3, with R = 15 and 3 mm for till beds with and without clasts, respectively, at the bed surface. Wall drag was removed according to (14). Other parameter values for the model are listed in table S1.

The proposed slip law agrees with some observations of glacier surface velocity and affects their interpretations. Surface velocities and numerical modeling of flow at Hofsjökull ice cap, Iceland, indicate that, at velocities below a few tens of meters per year, basal drag increases with velocity but that faster-moving ice obeys a Coulomb slip relation (19), in agreement with our observations. Observations of glaciers in Greenland (with velocities >1000 m/year), which indicate that basal drag is highly insensitive to velocity, also agree with our observations but were interpreted to reflect no effect of basal friction on glacier flow (20). For low effective pressures commonly observed beneath ice streams, the transition velocity will be low, causing the frictional, velocity-independent slip resistance that characterizes Coulomb behavior to dominate.

These results demonstrate that classical mechanisms of glacier sliding over rigid beds and till deformation mechanics collectively act to shape the basal-slip relation where glaciers rest on deformable sediment. By distilling these processes that are usually considered separately, Eqs. 2 and 3 provide a slip relation for sediment-floored parts of ice sheets and other glaciers that is empirically justifiable and grounded, with minimal parameterization, on relevant processes.

Supplementary Materials

science.sciencemag.org/content/368/6486/76/suppl/DC1

Materials and Methods

Figs. S1 to S3

Table S1

References (2125)

References and Notes

Acknowledgments: Funding: Construction and use of the device for the ice-till experiments were made possible by the U.S. National Science Foundation (grants OPP-0618747 and EAR-1023586 to N.R.I.). Author contributions: L.K.Z. ran the experiments and analyzed the data, L.K.Z. and N.R.I. wrote the paper, and N.R.I. built the apparatus. Competing interests: The authors have no competing interests. Data and materials availability: The data from these experiments are archived at the University of Wisconsin–Madison and can be obtained at https://uwmadison.box.com/s/a3t60exvoeoo5378aab88y8ohp44gsra.

Correction (7 April 2020): Owing to an inadvertent typesetting error, the variable ut in Eq. 2 was incorrectly shown as part of the numerator. This error has been corrected, and Eq. 2 now displays properly in the PDF.

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