Introduction

Antarctic ice shelves typically lie in confined embayments. Suppose that, within its embayment, an ice shelf is held together by weak links to the ice streams that supply it and to the ice rises where it is anchored to bedrock. Then the ice shelf might frag-ment if these weak links were all broken. This is an assumption. However, shear rupture alongside the floating tongue of Byrd Glacier Reference Hughes(Hughes 1979) and shear/fatigue rupture around Crary Ice Rise Reference Barrett(Barrett 1975) have locally fractured the entire thickness of the Ross Ice Shelf. This tendency to disintegrate, if it is real, can be counteracted if retreating grounding lines create new ice-shelf ice as fast as the old ice shelf disintegrates.

An ice shelf is carved away by a calving bay if its calving front retreats faster than its grounding line. Reference Fastook and SchmidtFastook and Schmidt (1982) have examined calving rates, including those expected when climatic warming places the ice shelf in an ablation zone. This paper examines the role of ice thinning on grounding-line retreat rates caused by both creep and melting, with applications to the Ross Ice Shelf.

Thinning by Creep

Two assumptions govern the relationship between the stress tensor τ_ij and the strain-rate tensor έ_ij (tensor subscripts i, j, k refer to the rectilinear axes x, y, z in succession, according to standard tensor notation). First, a given strain-rate component is proportional to the corresponding deviator stress component , where is the Kronecker delta, and σ_kk is the first invariant of stress. Second, the proportionality constant is assumed to be a function only of the second invariant of deviator stress τ², where . The precise functional relationship in the empirical flow law of ice requires that ice be incompressible, έ_kk = 0, and is Reference Glen(Glen 1958):

(1)

where A is a hardness coefficient and n is a viscoplastic exponent.

Consider a horizontal ice shelf with x directed along a surface flow line, y transverse to the flow line, and z vertically upward. Principal surface strain-rates έ₁ and έ₂ are frequently different from strain-rates έ_xx and έ_yy, but έ₃ = έ_zz everywhere. In terms of principal stresses, the first assumption permits a ratio R such that:

(2)

where subscripts 1, 2, 3 denote principal components of stress and strain-rate. Solving (2) for σ₂ gives:

(3)

Substituting (3) into the expression for τ in terms of principal stresses gives:

(4)

Substituting (3) into the expression for in terms of principal stresses gives:

(5)

Substituting Equations (4) and (5) into (1) gives the flow law for a horizontal ice shelf in terms of R and its principal stresses σ₁ and σ₃:

(6)

Terms containing R in (6) can be collected to form a constant R' defined as:

(7)

The principal strain-rates for an ice shelf are then:

(8a)

(8b)

(8c)

where Equation (8c) expresses the first invariant of strain-rate, for conservation of volume (έ_kk=0).

Since hydrostatic pressure increases linearly with depth for an ice shelf having thickness hi and a constant density ρ_I:

(9)

where g is gravity acceleration and z = 0 at the base of the ice shelf. Substituting (9) into (8a) and solving for σ, gives:

(10)

The base of the ice shelf is below sea-level at depth h_w in water of density ρ_W. Balancing hydrostatic pressure in a given ice column by the hydrostatic pressure of water in the column if the ice melted:

(11)

Substituting (10) for σ₁, integrating, and solving for έ₁gives:

(12)

where (7) is substituted for R', buoyancy requires that h_W = (ρ_I/ρ_W)h_I, and the average value of A is:

(13)

Principal strain-rates έ_k are obtained from strain-rates έ_ij using the Mohr circle construction:

(14a)

(14b)

(14c)

where Φ is the angle between coordinates x, y and 1, 2.

Most Antarctic ice shelves occupy embayments, so that ice entering the ice shelf crosses a groundingline perimeter that exceeds the calving perimeter crossed by ice leaving the ice shelf as icebergs. Consequently, a typical flowband experiences bending converging flow from the grounding line to the calving front. Figure 1 shows bending converging flow on the Ross Ice Shelf for a flowband from the Siple Coast. The centerline of the flowband is at radius r from its rotation axis r. The sides of the flowband are arcs about rotation axes r' and r", which are displaced because flow converges. In the absence of rigid body rotation, the flowband experiences a backshear -Δs across its width Δr as it rotates through angle θ. For local velocities u, v along local rectilinear axes x, y, the simple shear strain γ_xy and strain-rate γ_xy., in the plane of the ice shelf are:

(15a)

(15b)

where έ_xy is the pure shear strain-rate, ώ_xy is the rotation rate, dv/dx = 0, and If axes x, y conform to flow referred to cylindrical axes 8, r, where the flowband width decreases from y¹ to y” and mean flowband velocity increases from u’ to u” in arc length x’ - x” taken along the flowband centerline, the strain-rate components are:

(16a)

(16b)

(16c)

Certain implicit assumptions must now be examined. First, (11) strictly applies when hydrostatic forces dominate all other forces. Reference ThomasThomas (1973[b]) treats this point in detail and my (23) shows what other forces may be involved. Second, Reference ThomasThomas (1973[a]) showed that τ in (1) is approximately doubled when ρ1 is constant in (12) instead of varying with depth in a realistic manner. I have assumed that this effect can be absorbed.into A instead of τ, so that A is increased to 2^(n-1)/n_A A and έ₁ computed from (12) can be compared with observed values. Third, έ_xx, έ_yy, and έ_xy, given Equations (16), are the dominant surface strain-rate components for bending converging flow In a fiowband containing no ice rises and free from lateral shear alongside grounding lines. (12) would be Invalid unless these boundary conditions were satisfied.

The derivation of (12) uses the approach presented by Reference HughesHughes (1972), but it can also be obtained from the equations derived by Reference ThomasThomas (1973[b]). Reference Stuiver, Denton, Hughes, Fastook, Denton and HughesStuiver and others (1981) used the Reference HughesHughes (1972) approach, taking R = έ₁/έ₂ instead of R = έ₁/έ₂ where έ₁ is the largest principal extending strain-rate in both treatments. Reference Stuiver, Denton, Hughes, Fastook, Denton and HughesStuiver and others (1981) used a version of (12) to examine the stability of the Ross Ice Shelf toward the Siple Coast, using data from the Ross Ice Shelf Survey (RISS) published by Reference Dorrer, Hofmann and SeufertDorrer and others (1969). Their conclusions will now be checked using the much better data set of the Ross Ice Shelf Geophysical and Glaciological Survey (RIGGS) published by Reference ThomasThomas (1976[a]) and Reference Bentley and JezekBentley and Jezek (in press), and the cylindrical coordinates used in (16).

Figure 1 shows measured surface velocities on the Ross Ice Shelf in relation to a flowband bounded by ice streams B and D from West Antarctica. Bending converging flow is evident; the flowband has a uniformly low velocity acros the Siple Coast grounding line and has no grounding lines along its sides. For flowband transects KL, MN, and OP, the approximate flowband widths are 160, 200, and 310 km, and the approximately uniform velocities are 850, 450, and <50 m a⁻¹, respectively. Provided that the assumptions examined in the previous discussion are valid, Table I lists data used in solving Equations (16), the strain-rate components obtained from Equations (16) and (14), and the mean ice-hardness coefficient obtained from (12) for the principal surface strain-rates and average ice thicknesses ni in areas KLNM and MNPO obtained from data by Reference Bentley, Clough, Jezek and ShabtaieBentley and others (1979).

Fig.1. Flow of the Ross Ice Shelf. Identified are the calving front (hachured line), grounding lines (dotted lines), other shore-lines (solid lines), Crary Ice Rise (C.I.R), Roosevelt Island (R.I.), core holes at J9 and Little America V (crosses), RIGGS stations (dots), RIGGS ice velocities (vectors attached to dots), a Siple Coast flowband (between arcs centered on r' and r"), flowband transects KL, MN, and OP (radiating from r), and West Antarctic ice streams A, B, C, and D (arrows). Bending flow causes back-shear that prevents ice velocity from increasing along the bending radius. The RIGGS ice velocity map was kindly provided by C.R. Bentley, and appears in Reference Bentley and JezekBentley and Jezek (in press). See also Reference ThomasThomas (1976[a]).

Table I. Strain-rates and Apparent Ice Hardness for the Flowband in Figure 1.

The higher ice-hardness coefficient averaged over area MNPO compared to area KLNM can be explained in two ways. In the first explanation, ice in area MNPO is harder because it is colder, being closer in time and space to its primary source area on the West Antarctic polar plateau. Taking Ᾱ = Ᾱ_o exp(Q/n R T), where Q = 67 kJ mol⁻¹ is the activation energy for creep, n = 3, R is the ideal gas constant, T is absolute temperature, and Ᾱ_o is a constant, average values T ₁ and Ᾱ₁ In area MNPO are related to average values T ₂ and Ᾱ₂ in area KLNM as follows:

(17)

Using the values of Ᾱ in Table I and T ₁ = 257 K at the J9 core hole Reference Clausen, Dansgaard, Nielsen and Clough(Clough and Hansen 1979), (17) gives T ₂ « 263 K. The different Ᾱ values would then reflect a temperature difference of 6 K between ice in areas MNPO and KLNM. However, the average temperature at the Little America V core hole is 258 K, which is only 1 deg warmer than the average ice temperature at the J9 core hole. Given the positions of these two core holes in relation to areas MNPO and KLNM, as seen in Figure 1, it seems unlikely that the different Ᾱ values can be explained by invoking harder ice toward the Siple Coast grounding line resulting from colder ice.

In the second explanation, ice in area MNPO only appears to be harder than ice in area KLNM because Ᾱ was computed from an assumption that is valid in area KLNM but not in area MNPO. This assumption is that the general regime of bending converging flow is not disrupted by grounding points that create small ice rises and ice rumples in area MNPO. Reference ThomasThomas (1976[b]) and Reference Thomas and BentleyThomas and Bentley (1978) report evidence for numerous small grounding points toward the Si pie Coast, landward of transect MN. No evidence for grounding points has been reported seaward of transect MN for the flowband in Figure 1. If grounding points exist in area MNPO, the back-stress term involving compressive stress τ_C in (23) would keep έ_zz from attaining the high value it would otherwise have if grounding points were absent. Since έ_zz = έ₃, = -(1 + R) έ₁, by (8c), the value of A computed from (12) using the measured values of έ₁ and R given in Table I will appear to be larger than it actually is. Since ice at the J9 and Little America V core holes have quite comparably temperature profiles, we can assume that the actual Ᾱ value for area MNPO is close to the Ᾱ value computed for area KLNM. Combining the Ᾱ value computed in area KLNM with the values of R computed for areas KLNM and MNPO allows (12) to give an estimate of έ₁ if no grounding points exist in area MNPO. With no grounding points, (8c) then gives

έ_zz=-2.33 × 10⁻³ a⁻¹ for R = -0.719 and

έ_zz = -2.38 × 10⁻³ a⁻¹ for R = -0.454. Taking h_I ≃ 500 m as the mean ice thickness in area MNPO Reference Bentley, Clough, Jezek and Shabtaie(Bentley and others 1979), the ice-thinning rate due to creep is u_z = h_I έ_zz ≃ -1.18 m a⁻¹.

The above creep-thinning rate compares to a thinning rate of -1.37 m a⁻¹ that Reference Stuiver, Denton, Hughes, Fastook, Denton and HughesStuiver and others (1981) computed near the Siple Coast using RISS data Reference Dorrer, Hofmann and Seufert(Dorrer and others 1969) for a West Antarctic flow-band between ice stream A and Roosevelt Island. Reference Stuiver, Denton, Hughes, Fastook, Denton and HughesStuiver and others (1981) concluded that creep thinning by this amount would exceed the combined effects of surface snow-fall, basal freezing, and isostatic rebound that would act to create new grounding points. Consequently, in absence of an appreciable thickness gradient it appears that the Ross Ice Shelf may be becoming unpinned toward the Siple Coast, rather than repinned as Reference ThomasThomas (1976[b]) and Reference Thomas and BentleyThomas and Bentley (1978) concluded from their analyses of RIGGS data. The present study, also using RIGGS data, tends to confirm the conclusion of Reference Stuiver, Denton, Hughes, Fastook, Denton and HughesStuiver and others (1981). Repinning presupposes an unpinned past when the above high creep thinning rates would make repinning unlikely.

Sea-water beneath the Ross Ice Shelf wedges out at a rate of about 50 m per 100 km toward the Siple Coast Reference Greischar and Bentley(Greischar and Bentley 1980). So a creep thinning rate of -1.2 m a⁻¹ in absence of thickness advection would cause the Siple Coast grounding line to retreat at a rate of 1.4 km a⁻¹ after unpinning was complete, provided that surface accumulation, basal freezing, and isostatic rebound rates are all much less than the creep thinning rate. As seen in Figure 1, velocity across the Siple Coast was too low to measure, so thickness advection is probably unimportant. Recent surface-accumulation rates near the Siple Coast are less than 80 mm a⁻¹ Reference Clausen, Dansgaard, Nielsen and Clough(Clausen and others 1979), basal freezing between the Siple Coast and the J9 core-hole site has averaged only 10 mm a⁻¹ over the past 600 a Reference Zotikov, Zagorodnov and Raikovsky(Zotikov and others 1979), and the present rate of isostatic uplift has been computed to be about 25 mm a⁻¹ near the Siple Coast Reference Greischar and Bentley(Greischar and Bentley 1980). These rates total less than 10% of the unpinned rate of creep thinning.

Thinning by Melting

Retreat of the Ross Ice Shelf grounding lines, both around ice rises and up ice streams, occurs when sea-level rises or when ice thins. Retreat since the last ice-age maximum was mostly a result of rising sea-level, but future retreat, if it occurs, would presumably be a consequence of climatic warming that thinned the ice shelf by melting. The surface of the ice shelf coincides with the flotation line of ice, which lies below the surface where the ice shelf is grounded. This grounded ice will float when its surface lowers to coincide with its flotation line, a condition that is met when hydrostatic pressure at the base of the ice column equals the hydrostatic pressure of water at that depth below sea-level. Analytically:

(18)

where h_I is ice thickness at the grounding line, Δh_I is the thickness change at grounded distance Δ_x from the grounding line, h_W is water depth at the grounding line, Δh_W is the change in depth at Δ_x, u_z is the ice-thinning rate, v_z is the rate of rising sea-level, and Δt is the time needed for the grounding line to retreat distance Δx. If v_x is the grounding-line retreat rate:

(19a)

(19b)

where α is the surface slope and β is the bed slope in distance Δx, both slopes positive upward. Substituting Equations (19) into (18) and substituting h_W = (ρ_I/ρ_W)h_I, obtained when Δt = 0, gives the ice-thinning rate:

(20)

Another expression for the ice-thinning rate is:

(21)

where a is the net surface and basal melting rate, έ_zz is the creep-thinning rate of ice, and u_x is tne average ice velocity across the grounding line, all measured at the grounding line. Equating Equations (20) and (21) and solving for the grounding-line retreat rate gives:

(22)

Except for the v_z term, (22) was derived by Reference ThomasThomas (1977). Observationally, u_x can be very large for ice streams but is almost nil for ice rises, α is much smaller for ice streams than for ice rises, and β is generally negative for ice streams and is always positive for ice rises.

Consider a flowband that has width w when it crosses the ice-shelf grounding line, has a length s from the grounding line to the calving front, and encounters an ice rise of radius r on the ice shelf. The creep-thinning rate at the grounding line is then:

(23)

where έ_zz = έ₃ is related to έ₁ by (8c), R' expresses the degree of convergence or divergence of the flowband according to (7), the first term is the longitudinal tensile stress along the flowband derived from (12) for pure shear, the second term is the lateral shear stress alongside the flowband that exists If the average flowband velocity exceeds the average ice-shelf velocity on either side, and the third term is the longitudinal compressive stress between the ice rise and the grounding line. All terms act at the grounding line. Reference ThomasThomas (1973[a]) has τ_cdecreasing inversely with distance up-stream from ice rises.

Comparing Equations (22) and (23) highlights various interactions between changing boundary conditions for ice shelves. Most flowbands on an ice shelf begin as ice streams. If an ice stream punches through the ice shelf, τ_S → 0 and the grounding line may retreat rapidly up the ice stream. If the grounding line retreats faster than the calving front, s could increase and thereby slow retreat of the grounding line. If the flowband thins during this time, the resulting reduction of r could accelerate retreat of the grounding line. If the calving front retreats past the ice rise, τ_c = 0 and retreat of the grounding line may be accelerated even more.

The most important applications of Equations (22) and (23) are for retreat of ice-shelf grounding lines across ice streams and around ice rises. For ice streams we expect that u_x >> 0 during a surge, έ_zz > 0 under all conditions, ȧ < 0 if katabatic winds and tidal pumping ablate surface and basal ice, and β < 0 unless a bedrock sill exists. For ice rises we expect that u_x ≃ 0 at all times, έ_zz < 0 on the up-stream side and έ_zz > 0 on the down-stream side and tend to be offset by basal melting and freezing, A > 0 from surface snow-fall and a < 0 from basal tidal pumping, and a > 0 at all times.

Tidal pumping might ablate basal ice along the grounding line in two ways. Basal ice could be eroded by a slurry of sand entrained during tidal flushing, and also might be melted by frictional heat generated in water flushed back and forth in the tidal zone. Tidal pumping converts gravitational potential energy E_P into a fraction f of kinetic energy E_K and a fraction (1 - f) of thermal energy E_T, which are released in the tidal zone. In broad terms, E_P released during a tidal cycle is the tidal gravity force exerted over the mean elevation change δz that allows length δy of grounding line to sweep across distance δx:

(24)

where the tidal gravity force is the tidal hydrostatic pressure change pj g 6z acting over area δx δy swept by the grounding line during one tidal cycle. The average velocity of water flushed in and out of the tidal zone during the rise and fall of tide is:

(25)

where mW is the mass of water moved back and forth in the tidal zone. The rate of basal melting in the tidal zone is:

(26)

where 365 tidal cycles occur per year (one cycle per day). Basal water wedges out at about 50 m per 100 km toward the Siple Coast grounding line of the Ross Ice Shelf Reference Greischar and Bentley(Greischar and Bentley 1980), where Reference Williams and RobinsonWilliams and Robinson (1979) measured δz = 2 m, so 2 δz/δx = 5 x l0⁻⁴ gives δx * 8 km. Assuming that f = 0.5, (25) gives u_w = 6 m s⁻¹ and (26) gives a = 20 mm a⁻¹. Sand could be entrained for these uy values Reference Cacchione and Southard(Cacchione and Southard 1974), and could then erode basal ice in the tidal zone; how much erosion is unclear.

Widespread surface melting may affect the local melting rates of basal ice. Reference RobinRobin (1979) noted that ocean currents at the freezing point lose heat as they descend under ice shelves and sweep along ice-shelf grounding lines. This heat loss melts basal ice because the pressure-melting point decreases with increasing hydrostatic pressure. Similarly, surface melt water on an ice shelf must lose heat as it descends into crevasses, and the lost heat melts crevasse walls once they become temperate. The descending surface melt water could displace the more dense sea-water that fills crevasses to sea-level when surface crevasses join bottom crevasses. It is proposed that heat lost during descent into crevasses could melt basal ice when the crevasses fracture the entire ice thickness. If volume Vy of water at the surface melts volume Vj of ice at the base, the volume ratio is:

(27)

where C_p = 4.18 × 10³ J kg⁻¹ K⁻¹ is the specific heat of water, Q_L = 334.4 × 10³ J kg⁻¹ is the latent heat of water, and (∂T_M/∂P) = -8.9 × 10⁻³ °C bar⁻¹ is the change in the pressure-melting point T_M of ice with hydrostatic pressure P = ρI g h_I. If melt water from a surface ablation rate of 0.1 m a⁻¹ on the 5.2 × 10⁵ km² area of the Ross Ice Shelf reaches the base through grounding-line crevasses that fracture ice averaging 500 m thick, (27) predicts that 0.26 km³ a⁻¹ of basal ice would be melted. If basal melting were confined to a band 100 m wide along the crevasses, a basal melting rate of about 0.8 m a⁻¹ would occur, should this mechanism operate, along the 2 400 km of Ross Ice Shelf grounding lines. This eightfold increase when the surface melting rate is transformed into a basal melting rate along grounding lines could be the major effect of CO₂-induced climatic warming on ice shelves.

Conclusions

Creep-thinning rates for ice shelves lying in confined embayments are reduced if the general regime of bending converging flow is disrupted by ice rises. An analysis of RIGGS data leads to the conclusion that the Ross Ice Shelf is becoming unpinned instead of repinned toward the Siple Coast. Unpinning should facilitate grounding-line retreat. Ice-melting rates along grounding lines were examined for tidal pumping and descent of surface melt water into crevasses. At the present time, only tidal pumping would facilitate unpinning and grounding-line retreat. However, should future climate warming place the Ross Ice Shelf in an ablation zone, surface melt water entering tidal flexure crevasses could become the dominant process facilitating unpinning and grounding-line retreat. Basal melting by warm ocean currents has been discussed by Reference RobinRobin (1979), and was not examined.