1. Introduction
The junction of an ice sheet and an ice shelf occurs at the point, as the ice flows to the sea, at which the ice mass first is afloat and consequently loses contact with the bed. Reference HughesHughes (1972) has suggested that the position of the junction of the West Antarctic ice sheet and of the Ross Ice Shelf is unstable. He concluded that at the present time this junction is retreating into the interior of West Antarctica at the velocity of about 70 m a−1.
The problem of junction stability that Hughes’ work has emphasized is important. Large shifts in the position of the ice sheet-ice shelf junction produce relatively large changes in the thickness of an ice sheet. In this paper we attempt to obtain, for the two dimensional problem, the basic equations that determine the position of the region in which an ice sheet turns into a floating ice shelf; we also examine the conditions that must be satisfied if the ice sheet is even to exist.
2. Theory
Consider a symmetric, two-dimensional ice sheet. Let ice shelves be attached to both sides of this ice sheet as shown in Figure 1. Let D(x) represent the depth of the bedrock below sea-level, where x is the distance measured in the horizontal direction from the center of the ice sheet. (We consider x always to be a positive quantity whether it is measured to the right or to the left of the center of the ice sheet.) Assume that if no ice sheet were present D(x) would be a linearly increasing function of distance x out to the edges of the continent. Thus
for 0 ≤ x ≤ Lo and, for all practical purposes,
for x > L 0. Here x = L 0 is the position of the edges of the continent, D 0 is a constant, and β is the slope of the bed rock away from the center of the continent. (We consider only values of β in the range β ≥ 0.)
Fig. 1. Cross-section of ice sheet with attached ice shelves.
Isostatic sinking occurs under an ice sheet. If h(x) is the thickness of the ice sheet, D(x) after isostatic sinking is given by
in the region 0 ≤ x ≤ L where x = L gives the position of the edges of the ice sheet. Of necessity the position L of the junction between the ice sheet and an ice shelf must satisfy the relationship L ≤ L 0. The quantities p, pw and pr in Equations (2) are the average densities of ice, sea-water, and bedrock, respectively. In the region under the floating ice shelf (x ≥ L) the depth D(x) is given by Equations (1).
A rise or fall in the sea-level results in an identical increase or decrease in the value of D 0 in Equations (1) and (2). The change in the value of the ice sheet-ice shelf junction position L with change in sea-level can be found only from an analysis of the mechanics of flow of ice sheets and ice shelves.
Assume that the ice sheet and ice shelves of Figure 1 are in a steady-state condition. From a mass balance argument it can be seen that
where a(x) is the accumulation rate at x and U(x) is the average value of the horizontal ice velocity component. If there is melting or freezing at the bottom surface this mass loss or gain must be included in the term a(x).
Differentiation of Equations (3) produces the equation
where is the longitudinal strain-rate and γ = —dh/dx = α + α *,, where α is the slope of the upper ice surface (considered to be a positive quantity if the upper surface decreases in elevation as x increases) and α* is the slope of the bottom ice surface (considered to be a positive quantity if the lower ice surface increases in elevation with increasing value of x).
Assume that, in the case of the ice sheet, the value of the velocity is given approximately by the equation (see chapter 9 of Patterson, 1969)
where B, τ0 and n are constants, and τ is the basal shear stress. The value of n and the expression B/τ0 n depend on factors such as whether or not sliding occurs, and the temperature distribution within the ice sheet. Reasonable values of B and n are n and n ≈ 3 and B ≈ 50 m a–1 if τ0 = 1 bar.
Equations (5) is derived (Reference PatersonPaterson, 1969, chapter 9) under the assumption that the dominant stress producing flow in an ice sheet is the shear stress that acts across any surface that is approximately parallel to the top or bottom surface. When large longitudinal stresses are present this equation is not valid.
In the ice shelves U is given by
where the longitudinal creep rate is
In Equations (6) and (7), U L is the horizontal velocity at x = L, Δp = pw — p, m ≈ n ≈ 3, and A is a constant whose value depends on the temperature and density profile of an ice shelf. A reasonable value of A is ≈ 2 × 10-3 a-1 for τ0 = 1 bar. Equations (7) is found from the analysis of the creep rate of an unconfined, two-dimensional ice shelf (Reference WeertmanWeertman, 1957).
3. Perfectly Plastic Approximation
Considerable insight is often gained in problems in glacier mechanics by making the assumption that ice is a perfectly plastic solid. Such is the case with the ice sheet-ice shelf junction problem.
The perfect plasticity approximation is made in our previous equations by taking the limit of n = m = ∞. If n is set equal to ∞ in Equations (7) for an ice shelf with a positive accumulation, that equation predicts that the ice shelf must have the thickness h 1p (independent of x) equal to
(If the thickness of an ice shelf were greater than h 1p it would strain at an infinite rate until its thickness were reduced to the value h 1p. If the thickness were smaller than h 1p the ice shelf would not creep but its thickness would increase in time because of the positive accumulation rate until it reached the value h 1p.) For Δρ ≈ 0.1 ρw and τ0 = 1 bar the value of h 1p is approximately 400 m.
By similar reasoning it can be seen from Equations (5) that the basal shear stress of the ice sheet must take on the value τ = τ0. The ice sheet profile thus is found by integrating the equation
where
If the ice sheet bed was flat before the ice sheet was placed on it (β = 0) the solution of Equations (9) and (10) is
where H, given by
is the thickness of the ice sheet at the center (x = 0) and h 2 is the thickness of the ice sheet at its edges (x = L). Because the ice sheet is afloat at its edges the thickness h 2 is equal to
When [In line Eq.-16] the ice-sheet profile is given by the equation
where
The value of H in Equations (14) is found by setting h = h 2 at x = L.
3.1. Condition at ice sheet—ice shelf junction
At x = L, the position of the junction of the ice sheet and an ice shelf, the thickness of the ice shelf, h = h 2, is such that the ice sheet is afloat. The ice shelf also is afloat. Thus the thickness of the ice shelf, h = h 1p, must satisfy the condition
Suppose h 1p were appreciably smaller than h 2, a situation illustrated in Figure 2. The ice sheet is chopped off essentially at x = L. The effect of this truncation at the edge of the ice sheet is to cause a large longitudinal tensile stress to be set up in the ice sheet near its edge. The magnitude of this tensile stress must be of the same order as that of the stress existing in an ice shelf of thickness h 2. The longitudinal strain-rate in the ice sheet near its edge must thus be of the order given by Equations (7). But since h2 ≫ h1p this strain-rate is infinite if ice is a perfectly plastic solid. Therefore it is not possible to have h2 ≫ h1p and have the ice sheet and the ice shelves in a steady-state condition.
Fig. 2. Junction between ice sheet and ice shelf. No restriction on thickness of ice sheet at x = L. Perfect plasticity approximation.
We conclude that
is a necessary condition at the junction of an ice sheet and an ice shelf.
An appreciable longitudinal tensile stress exists within the ice near the edge of an ice sheet even if Equations (17) is satisfied. The tensile stress that exists within an ice shelf cannot die out abruptly at the ice sheet—ice shelf junction. The basal shear stress near the edge of the ice sheet must be reduced in value in order that the effective stress that produces plastic deformation does not exceed the limit τ0. (The effective stress that produces plastic deformation is equal to the sum of the squares of the deviator stress components divided by [In line Eq.-19] (Reference PatersonPaterson, 1969).) Therefore when Equations (9) is used to determine the ice-sheet profile a reduced value of the term τ0 should be employed near the edge of the ice sheet.
Figure 3 shows the cross-section of the junction region between the ice sheet and an ice shelf when h2 ≈ h1p . No sharp change in ice thickness occurs across this zone. The slope of the upper ice surface is reduced in the transition region at the edge of the ice sheet because of the need to reduce the basal shear stress when a longitudinal tensional stress is present.
Fig. 3. Junction between ice sheet and ice shelf. A transition region exists in which the longitudinal tensile stress and the shear stress are comparable in magnitude. Perfect plasticity approximation.
The junction between the ice sheet and the ice shelf occurs at that value of x = L for which
When Equation (1a) is inserted in Equations (18) the following equation is found for L:
If the value of L predicted by Equations (19) is larger than Lo (see Fig. 1) the edge of the ice sheet occurs at the value x = L 0 instead of x = L. In this situation the edge of the ice sheet cannot be afloat because h 1 p > (pw/p) D(Lo . Under these circumstances the equivalent of ice falls must exist at the edges of the ice sheet. The ice shelves form below these ice falls.
A bed that was flat before an ice sheet was placed on it (β = 0) becomes entirely covered by ice out to x = L 0 if [In line Eq.-22] no ice sheet can exist at all. In the former case the ice sheet would turn catastrophically into a floating ice shelf if the sea-level were to rise until h 1p becomes less than ( pw / p ) D 0.
4. Realistic Flow Law
How are the results of the last section modified if ice is assumed to obey the more realistic flow laws of Section 2 that use finite values of n and m (m ≈ n ≈ 3) rather than the infinite values of the perfectly plastic approximation? The thickness of the ice shelves no longer is a constant. Instead of the unique thickness given by Equations (8) the thickness h 1 of the ice shelf at the junction is given by the following equation (found by combining Equations (4) and (7)):
where h 1p is given by Equations (8), y 1, and a 1, are the values of y and a for the ice shelf at the junction, and <a> is the average accumulation rate given by
Because the exponent 1/(m+2) in Equations (20) is approximately equal to it, very large changes in <a>, a 1 and y 1 are required to change the value of h 1 by more than a factor of 2 to 3. Thus the results in Section 3 are expected to remain valid as a first approximation.
In the transition region (see Figure 3) at the edge of an ice sheet longitudinal as well as shear stresses are expected to be important. The deformation and flow in the transition region thus are determined by the triaxial creep equations of Nye (see chapter 6 of Reference PatersonPaterson, 1969). The following equation, that takes the triaxial stresses into account and replaces Equations (7), gives a rough approximation of the longitudinal strain-rate έ in the transition zone
where c 1 and c 2 are constants of order of magnitude of 1.
Similarly Equations (5) for the average ice velocity U is replaced by the equation
as a rough approximation in the transition region.
Combining Equations (4), (22) and (23) gives
when n = m and the approximation Uγ ≫ is valid. From Equations (2) it is possible to show that
Combining Equations (3) and (23) and setting n = 3 gives
Combining Equations (24), (25) and (26) and assuming that α ≫ β gives for the upper surface slope α the equation
Combining Equations (24) and (27) gives the following equation for the ice thickness h = h 2 in the transition region
For <a> = 0.1 m a−1, L = 1000 km, h 1p = 400 = 2 X 10−3, B = 50 m a−1, Δp/p = 0.1, and p/p = 1/3 the thickness h 2 = 543 m and the slope α = 0.019. These results justify a posteriori the assumptions thatU γ/h a and α ≫ β for reasonable values of a, L and β.
The distance x = L to the edge of an ice sheet can be found from Equations (19) if h 2 is substituted for h 1p in that equation. The following is found for L:
where
(The term g = 22.7 m7/9 for the values of <a>, etc., mentioned before.)
The solution for Equations (29) can be found graphically by plotting the function βL+D 0 versus L and the function gL 2/Q versus L. Such a plot is given in Figure 4. Depending upon the values of β and D 0. there may be one, two, or no real, positive values of L that satisfy Equations (29).
When β = 0 and D 0 > 0 there is one real, positive value of L that satisfies Equations (29) (see Figure 4). It is given by the equation
When β = 0 a steady-state ice sheet whose half-width L is given by Equations (31) clearly is in unstable equilibrium. It only need be noted that Equations (31) predicts that L decreases in value if the accumulation rate <a> increases or if the sea-level drops. A non-steady-state ice sheet whose edge occurs at a value of L which is larger than that given by Equations (31) will grow in size until it reaches the edge of the continental shelf at x = L 0. If the half-width is smaller than that given by Equations (31) this ice sheet will shrink with time until it disappears. If L given by Equations (31) is larger than L 0 clearly no ice sheet can exist. These results are very similar to those found for an ice sheet that obeys the equations of a perfectly plastic solid for the case in which β = 0.
Fig. 4. Plots of the functions gL 2/9 and (βL+ D 0) versus L. Curves plotted for g = 22.7 m7/9, 200 m and (for one curve) 250 m, and β = 2 x 10−3, 6 x 10−4, 4 x 10–4 and 0. Values of L for steady-slate ice sheets occur where the curves of gL 2/0 and (βL +D 0) cross.
If the curve of the function βL+D 0 intersects the curve of the function gL 2/9 twice (see Figure 4) the intersection corresponding to the smaller value of L represents the solution of a steady-state ice sheet in unstable equilibrium. The other intersection represents an ice sheet in stable equilibrium. For example, if D 0 is small compared with h 2 the solution of Equations (3) for the larger value of L is
This equation predicts that L increases if <a> increases or if the sea-level drops. Equations (32) gives the half-width of an ice sheet that is in stable equilibrium. This result too is analogous to that found for the perfectly plastic approximation when β ≠ 0.
If the value of L for the steady-state ice sheet in unstable equilibrium is such that βL ≪ D0 the solution of Equations (29) is
From Figure 4 it can be seen that if D 0 and β are made sufficiently large in value no solution for L exists. By considering the special case in which the curves of the functions βL+D 0 and gL 2/9 are tangent to each other it is simple to show that no solution exists when the following inequality is satisfied
For the value of g given previously (g = 22.7 m7/9) if the term β 32/7 D 0 is larger than 28 m no ice sheet can exist.
If the ice sheet is small the assumption that U γ L/h ≫ a breaks down. In this situation an ice sheet can exist if h 1, the thickness of the ice shelves given by Equations (20), is larger than D 0. An approximate half-width for such a small ice sheet can be found by replacing in Equations (19) the termh 1p with the termh 1. In the approximation m = 3 and <a> [In line Eq.-35] a the half-width L is given by
Thus if
an ice sheet can exist.
5. Discussion
The results of this paper show that an ice sheet that rests on a flat bed (flat before the ice sheet was placed on it) situated below sea-level is inherently unstable. Depending upon the depth of the bed below sea-level the ice sheet either will shrink in size until it disappears or it will grow until its edge is at the border of the continental shelf. A stable ice sheet can occur if the bed slopes away from the center of the ice sheet. The generalization of our results to other bed shapes is rather obvious.
We have not attempted to analyse the three-dimensional situation in which an ice sheet flows into a confined ice shelf. The results of Reference ThomasThomas (1973[a], Reference Thomas1973[b]) and Reference BuddBudd (1966) on the deformation of confined ice shelves might be used to extend our theory to this situation.
We hope that the theory of this paper can be applied to the question Hughes raised: Is the West Antarctic ice sheet disintegrating? The bedrock under a large fraction of this ice sheet is below sea-level now and would still be below sea-level if the ice sheet were removed and complete isostatic rebound occurred. From the cross-section of the West Antarctic ice sheet through Byrd Station and the Ross Ice Shelf shown in Figure 6 in the paper of Reference HughesHughes (1972) or in Figure 4 of Reference Robin, Robin, Swithinbank and SmithRobin and others (1970) the following estimate can be made: [In line Eq.-36] and [In line Eq.-37]. If these values of β and D 0 are inserted into Equations (29) and if g is assumed to equal 22.7 m7/9 it is found (see Fig. 4) that [In line Eq.-38] for the stable steady-state ice sheet. This value is approximately the half-width of the present West Antarctic ice sheet along this flow line. However we emphasize strongly that one should not conclude from this calculation, which is only an approximate one, that the present ice sheet is indeed in a stable steady-state condition. If the estimated values of β and D 0 were changed to β ≈ 6 x 10–4 and, o ≈ 250 m, values well within reasonable error limits, no steady-state ice sheet could exist (see Fig. 4) and the present West Antarctic ice sheet must shrink until it disappears. In addition any complete treatment of this problem must also take into account the fact that fast-moving ice streams exist near the edge of the West Antarctic ice sheet where it flows into the Ross Ice Shelf. It appears within possibility that Hughes’ suggestion that the West Antarctic ice sheet is disintegrating is correct. Certainly this suggestion cannot be ruled out as physically unlikely on the basis of our theory and calculations.
Acknowledgements
This paper was written as a result of my participation at the Cambridge Workshop on Temperature and Isotopic Profiles in Polar Ice Sheets held at Scott Polar Research Institute, Cambridge, England, in April 1973. I wish to thank the Royal Society of London for a travel grant that made my participation possible and Dr Gordon Robin for the invitation to attend.
