The Comninou–Dundurs effect and position stability of subglacial lakes

Abstract

The Comninou–Dundurs effect (C-D effect) has yet to be considered in slip processes at the base of ice sheets and glaciers. This effect exists whenever slip occurs between two solids of differing mechanical properties and at least one of the solids is compressible. It gives rise to an additional normal traction stress across a non-uniformly slipping interface, a stress which otherwise would not exist if the two solids had identical properties. the C-D effect should exist to some degree, therefore, wherever ice slips over glacial till. It will be enhanced where large slip gradients exist, such as at boundaries of subglacial lakes. In this paper it is shown that, because of the C-D effect, subglacial lakes that lie at a basal till/ice interface may migrate in the direction of increasing ice thickness.

1. Introduction

The purpose of this paper is to investigate what change the Comninou–Dundurs effect (C-D effect) may produce on subglacial lakes and on sliding conditions at the base of an ice sheet or glacier in regions where the basal ice rests on basal till. the C-D effect equations (Dundurs, 1967, 1969; Comninou, 1977,1978; Comninou and Dundurs, 1980) essentially were developed for smeared-out glide- and climb-edge dislocations on an interface between elastic solids of differing elastic constants. the C-D effect may be important to some basal processes of ice sheets and glaciers. Appendix A gives a simple physical explanation of its origin. (In another geophysical area, a generalization of the C-D effect under dynamic conditions provides an explanation for the existence of fast-moving localized packets of smeared glide-edge dislocations (slip pulses) that move on earthquake faults, and an explanation for the apparent lack of frictional heating on the San Andreas Fault, CA, U.S.A. (reviewed in Ben-Zion, 2001; Ranjith and Rice, 2001).)

Subglacial lakes (water lenses) whose horizontal dimensions are large compared with their depth can be considered to be cracks. They are subjected to basal shear loading if they are situated anywhere except under an ice dome. If the average slope of the upper ice surface is not zero, a shear traction stress is exerted at the base of an ice sheet. No shear traction stress, however, can be placed across an effective water-lens crack where the water depth is appreciable. the shear traction is concentrated, therefore, in the regions at the crack tips where the water depth is very small. In this paper the subglacial lake is treated as a crack loaded in shear. (For simplicity, only two-dimensional mode II shear cracks in plane strain are considered in this paper.)

In the case of an elastic solid, the solution of a shear crack that lies parallel to the x axis can be generated from the stress–strain-displacement field of a glide-edge dislocation of Burgers vector bx in the x direction (Weertman, 1996). (The dislocations dealt with in this paper are not dislocations of atomic dimensions but mathematically defined dislocations in an elastic continuum that can have any macroscopic scale (see Weertman, 1996).) the traction shear stress σxy for a dislocation situated at the origin is equal to

(1)

where G is the shear modulus and σ =1–ν, where ν is Poisson’s ratio. the shear stress arising from a distribution Bx(x) of glide-edge dislocations smeared out on the slip plane is therefore

(2)

The shear crack solution found for Equation (2) is (Weertman, 1996)

(3)

and

(4)

where σ_A is the applied shear stress and Dx(x) is the net displacement in the horizontal direction (x direction) at x across the crack faces. Equations (3) and (4) are valid for –a ≤ x ≤ a, where a is the half-length of the crack. (Bx = Dx = 0 for | x |> a.)

No normal (tensile or compressive) traction stress σyy is produced at the slip plane (or crack plane) by a discrete glide-edge dislocation or a smeared dislocation distribution Bx(x) of glide-edge dislocations. to have a normal stress on the crack plane produced by dislocations requires a distribution By(x) of climb-edge dislocations whose Burgers vectors are in the vertical direction (y direction). Equations (1–4) apply for climb-edge dislocations if σxy is replaced with σyy, bx with by, Bx(x) with By(x), and Dx(x) with Dy(x).

2. Comninou–Dundurs Equations

Equations (1–4) must be modified if a shear crack (or a tensile crack) exists on an interface separating elastic half-spaces A and B (with solid A above and solid B below) whose elastic constants differ. A glide-edge dislocation of Burgers vector bx situated at the origin at the interface produces the following long-range (inverse distance dependence) and short-range (exists only where the dislocation exists) traction stresses across the interface

(5a)

(5b)

A climb-edge dislocation of Burgers vector by produces the traction stresses

(6b)

(6b)

In these equations δ(x) is the Dirac delta function δ (→ ∞ when x → 0, and δ = 60 when x ≠ 0) and β is the Dundurs parameter defined as (Dundurs, 1967, 1969)

(7)

The effective shear modulus and the effective alpha term are (Weertman,1996)

(8)

In the equations above, σ_A =1–ν_A and σ_B =1–ν_B, where ν_A and ν_B are Poisson’s ratio of solid A and solid B, and G_A and G_B are the shear moduli of solids A and B. (See appendix C inWeertman,1996 for a derivation of these equations.)

Just as Equation (2) above arose from Equation (1), Equations (5) and (6) for the interface traction stress of discrete dislocations lead to the following equations for the traction stress across the interface for a smeared-out distribution Bx(x) of glide-edge dislocations and a smeared-out distribution By(x) of climb-edge dislocations.

(9)

Equations (9) are the Comninou–Dundurs equations. They give the shear traction stress σxy and the normal traction stress σyy across an interface (with solid A above and solid B below) that arise from a smeared-out distribution Bx(x) of glide-edge dislocations and a smeared-out distribution By(x) of climb-edge dislocations. the inverse Comninou–Dundurs equations

are (after a sign correction is made inWeertman,1996)

(10)

The non-traction stress σxx at the interface is given by

(11a)

where the upper sign is used above the interface and the lower sign below it. (This equation can be obtained from the p. 452 equations in appendix C ofWeertman,1996. See also p. 50–51 of the same work.) Thus

(11b)

where σxx+ is the value of σxx above the interface, and σxx– is the value below it. Similarly, the rotation pseudo-stress σω on either side of the interface is given by

(12a)

and

(12b)

The rotation pseudo-stress is defined by where ω is the elastic rotation.

Note in Equation (9) that when β ≠ 0 a distribution of glide-edge dislocations Bx(x) gives rise not only to a shear traction stress on the slip plane but also to a normal traction stress. the normal traction stress exists only where the dislocations are present. Figure 1 (for β > 0) indicates that glide-edge dislocations of opposite sign produce normal stress of opposite sign. the Dundurs parameter β is equal to zero if If the two solids are incompressible then and β = 0. Note that if the upper half-space is incompressible and the lower is not, the Dundurs parameter β is always positive. (If G_A = G_B, and then

Fig. 1 Dislocations of opposite sign on interface between half-spaces A and B when β >0, showing regions of compressive (negative) normal stress σyy where Bx>0 and of tensile (positive) normal stress σyy where Bx<0. (Curvature of interface is ignored.)

2.1. Solution of Comninou–Dundurs equations for shear crack

Let a positive shear stress σ_A (such as is applied at the base of an ice sheet if the ice thickness is larger on the left side, or

negative x side, of the crack than on the right side, or positive x side) be applied to a crack. the solution of the Comninou– Dundurs equations gives the displacements at the crack faces for an interface mode II shear crack, with no shear or tensile traction stress acting on the crack faces. the solution is

(13a)

(13b)

if interpenetration of the crack faces is allowed. Here Dx is the net displacement across the crack faces in the horizontal direction (x) and Dy is the net displacement in the vertical direction (y). (See p.393–395 inWeertman,1996 and note that these pages gives the solution for the mode I rather than the mode II crack. the solution of the mode II crack is the same as for the mode I crack if the displacements Dx and Dy are interchanged and one sign is reversed.) In Equation (13), a is the half-length of the crack (–a ≤ x ≤ a); σ_A is the applied shear stress, γ₀ – cosh ηπ and η = (1/π)arctanh β . (For and γ₀ ≈1.02.) the crack center is at x = y = 0. Note in Equation (13) the weird Williams oscillations (Williams, 1959) of ever-decreasing spacing as a crack tip is approached. Oscillations cease where a – | x |>

2a exp(–π/2η). for a – |x|> 5.37610^–11a. the Williams oscillations die out at very small distances from a crack tip and can be ignored. In the Equation (13) solution the crack faces are traction-free because the short-range traction stresses arising from the C-D effect are canceled by the long-range stresses of glide- and climb-edge dislocation distributions. (The climb- and glide-edge dislocation distributions are found from Equation (13) using the relationships Bx = –dDx/dx and By = –dDy/dx:)

Figure 2a and b show plots of normalized displacements and andnormalized dislocation distributions and vs normalized distance x/a from the center of the crack found from Equation (13) for the case of and a positive applied shear stress σ_A. the curves are cut off before the anomalous Williams oscillations appear. (The displacement Dx and edge-glide dislocation density distribution Bx is similar to that of a Griffith–Inglis mode II crack. the displacement Dy and edge-climb dislocation density distribution By has the type of symmetry of a Zener–Stroh–Koehler mode I crack (Weertman,1996) but differs from it because the net Burgers vector of climb dislocations is zero rather than of finite amount. the crack is mixed-mode I and II in character.) Note that on the right side Dy is negative. If interpenetration of the crack faces is not allowed, at the right side the crack faces make contact over an extended area. the more complicated solution of Equation (9) is required in this situation. If the Dundurs constant β = 0 then Dy = 0 and Dx has the value of a Griffith–Inglis mode II crack.

Fig. 2 (a) Normalized plot of displacement Dx and Dy vs x/a from Equation (8) for the case of (b)Normalized plot of dislocation density components Bx = –dDx/δx and By = –dDy/dx vs x/a.

It is seen in Figure 2b that the glide-edge dislocation distribution Bx is very large near the tips of the crack. the existence of these dislocations gives rise to large compressive stress to the right of the righthand tip, and to large tensile stress to the left of the lefthand tip. the stress from these dislocations opens the left tip and closes the right tip. Figure 3 shows schematically the glide-edge dislocations that might exist near the tips of a water-lens shear crack at the base of an ice sheet subjected to a positive shear stress σ_A.

Fig. 3 Anomalous C-D effect pressure regions at ice–till contact with thick, lake-like water lens. the contact zone is subjected to an overall shear stress.

3. Application to Base of An Ice Mass

The Comninou–Dundurs equations apply at an interface between elastic solids. for non-elastic solids it is reasonable to expect a similar (generalized) C-D effect to exist because the C-D effect arises from the difficulty, shown schematically in Figure 6 in Appendix A, of fitting together two half-spaces of different material whose interface in any region is subjected to non-uniform slip, and at least one of the half-spaces contains a compressible solid.

At the base of an ice sheet or a glacier the interface is between basal ice (solid A or ice of Fig. 1 and 3), which deforms by high (homologous) temperature creep, and the bed (solid B or till of Fig. 1 and 3), which may be elastic if it is bedrock and non-elastic if it is made of glacial till. High-temperature creep deformation of ice is incompressible (provided that no voids appear at grain boundaries during creep and trapped air bubbles have negligible volume). If the ice sheet or glacier bed is elastic bedrock the β parameter can be expected to be insignificant. Note that the shear modulus of rock is at least an order of magnitude greater than that of cold ice. Thus, using Equation (7) for a rough measure of an effective Dundurs parameter, β is small and can be ignored. the C-D effect, therefore, should be unimportant when basal ice rests on bedrock (or on bedrock veneered with only a thin till layer).

Suppose an ice mass rests on a thick till layer. Tulaczyk and others (2000) showed that till, taken from under Whillans Ice Stream (formerly Ice Stream B), Antarctica, is compressible. Thus a generalized Dundurs parameter β is positive. It is prudent, therefore, to consider the C-D effect when basal ice rests on till. Consider a water lens within basal till, shown schematically in Figure 3, whose horizontal dimension is large compared with its thickness. Assume the water thickness is so large that a shear stress cannot be transmitted across the lens because bed irregularities do not penetrate the water layer and make contact with the overlying ice. However, an overall shear stress exists across the base of the ice mass as indicated in Figure 3. (A shear stress of sense given in this figure is produced at the base of an ice sheet if the ice thickness decreases from left to right.) the water lens acts a mode II-type shear crack.

(If the horizontal dimension of the water lens is very much greater than the thickness of an ice sheet the upper ice surface will be horizontal. In this situation no shear stress will be transmitted to the base of the ice sheet over most of the area of the ice lens. In the regions at the edges of the ice lens, however, a shear stress will exist because the upper ice surface no longer is horizontal there. the edge regions of the water lens act as edge shear cracks.)

If the elastic solids of the half-spaces are replaced with non-elastic solids the qualitative behavior of the crack will not change. the mode II shear crack faces should still try to open up on one side (the left side of Fig. 2b and 3) and try to interpenetrate on the opposite side (the right side of Fig. 2b and 3). If the right side of a water lens closes and the left side opens, the crack may shift its position towards the left. This is shown schematically in Figure 4a and b. (If only the left side shifts, the water lens becomes longer in the horizontal direction, as indicated schematically in Figure 4c.)

Fig. 4 (a) Same as Figure 3. (b) Water lens moved to the left through opening of left-side low-pressure region and closing of right-side high-pressure region. (c) Water-lens left boundary moved to the left through opening of left-side low-pressure region but with right boundary stationary. (Ice thickness increases towards the left.)

In Tulaczyk and others (2000, fig. 4) are shown stress– strain curves of till obtained in triaxial tests. These stress– strain curves approximate well those of an elastic perfectly plastic solid. Tulaczyk and others (2000) find that the shear stress at failure is proportional to an effective normal stress. (Alley (2000) discusses the uncertainties in our understanding of the rheological properties of till present beneath ice sheets.) the deformation of ice itself can be modeled as that of an elastic perfectly plastic solid. As is well known, early (and reasonable) estimates of ice sheet and glacier profiles were made using this ice rheology.

Assume now, for the purpose of investigating the influence of the C-D effect at an ice-sheet base, that reasonable qualitative results can be obtained by considering both ice and till to be elastic perfectly plastic solids. Assume ice to be incompressible and till to be compressible. the Dundurs parameter β is positive. for simplicity, assume the yield stress σ₀ of ice and in situ basal till to be the same. If the yield stress of till is very much greater than the effective one for ice, for all practical purposes, basal ice rests on bedrock and the C-D effect almost entirely disappears. But if the yield stress of till is very much greater than that of ice, the generally accepted concept of deformable soft till glacier beds, based on field observations, is untenable.

Figure 5a and b show, by means of trajectories across which the shear stress is a maximum, the stress solution near the lefthand tip (Fig. 5a) and the righthand tip (Fig. 5b). the crack tip is at the point where the fan-sector trajectories converge. the crack face is indicated in the figure. the stress solution consists of fan-type sectors and block-type sectors. In each sector the effective stress magnitude is at the yield stress, except in the sector with dashed trajectories. Plastic deformation takes place in the fan sectors. Equations for the stress field in each sector are given in Appendix B. Because of the asymmetry in Figure 5a and b, plastic deformation leads to the closing of the crack faces at the righthand tip and to the opening at the lefthand tip.

Fig. 5 (a) Stress-field maximum shear stress trajectories in basal ice and till at left side of water lens when there is a C-D effect. Rheology of ice and till is approximated as an elastic perfectly plastic solid of yield stress σ₀. (σ_D is expressed in units of σ₀.) (b) Same as (a) but for right side of water lens.

It is difficult to make a good estimate of just how fast a water lens might migrate. A more refined analysis than the one presented here is required. Avery rough and unreliable estimate can be made using Figure 2a. This figure shows that the vertical elastic displacements are about an order of magnitude smaller than the horizontal elastic displacements for a crack in an elastic solid. Hence one might expect for a crack in a solid that deforms by creep that the vertical crack face velocity V_V is an order of magnitude smaller than the horizontal velocity V_H. the horizontal velocity can be expected to be smaller than the horizontal velocity at the upper ice surface. If the depth of the lake is D, its right side (right side of Fig. 3) would close in a time of about 10D/V_H. (The left side can advance by opening the material beyond the lefthand crack tip.) If the half-length a of the water lens is smaller than the ice thickness, the drift velocity could be of the order of or smaller than aV_H/10D. (If the water-lens half-length is much larger than the ice thickness, the water lens becomes an edge crack of effective length equal to ice thickness.) the drift velocity could be large (ma^–1 to km a^–1) until the water lens reaches a region of low upper-ice-surface horizontal velocity.

4. Discussion

Many subglacial lakes with appreciable water volume are now known to exist in Antarctica (Siegert and others, 1996; Dowdeswell and Siegert, 1999; Siegert, 2000a,b). (The first indication of a subglacial lake under an Antarctic ice sheet was found by Robin and others (1970).) It is interesting to note that most of the subglacial lakes catalogued by Siegert and others, as they point out, are close to an ice divide. of course, the greater ice thickness at ice divides favors basal meltwater production and lake creation there, but the heat produced by frictional sliding also favors increased basal meltwater production and possible lake formation away from ice divides. the observation that the location of a subglacial lake is biased to be near an ice divide suggests that some mechanism exists that drives a basal water lens to move towards increasing ice thickness. (Of course, the bottom topography at any one location may have large bowl-shaped elevation variation and a large geothermal heat flux that favor a stationary subglacial lake.The analysis of this paper is not applicable to such lakes.)

This paper suggests a mechanism by which crack-shaped basal water lenses situated between basal ice and basal till can shift their positions over time. the mechanism is based on the Comninou–Dundurs equations that are applicable to dislocation distributions on interfaces including those of the special cases of interface cracks. Although the Comninou– Dundurs equations apply to dislocations at interfaces between elastic solids, they should indicate reasonably well the qualitative behavior of interface cracks between non-elastic solids. Thick basal water lens and subglacial lakes essentially are interface cracks subjected to mode II shear crack loading. Because of the C-D effect a mode II shear crack takes on an additional mode I character. At one end of the crack (that nearest an ice divide) a tensile mode I component is added that opens the crack and causes that end to move towards the ice divide. At the other end of the crack (that furthest from an ice divide) a compressive mode I component is added that closes the crack and causes that end also to move towards the ice divide. Hence a water lens could move over time towards an ice divide if not prevented from doing so by large bottom topography variations and by lack of a thick layer of basal till. If subglacial lakes are able to shift their positions over time, it is possible that larger lakes devour smaller ones. (Moreover, the expected high-pressure zone just beyond the downstream side edge of a crack-like lake might impede water flow.) Under Pleistocene ice sheets, subglacial lakes may grow so large and so unstable that they dump and contribute to Heinrich events.

The position and history of subglacial lakes obviously depends on various factors. These include water flux though them, their thermal history, basal melting and freezing of water, and pressure gradients within them. the C-D effect discussed in this paper may be almost as important a factor for some types of subglacial lakes.