Introduction

In Antarctica today there are three types of ice sheet. There is a “terrestrial ice sheet” in East Antarctica resting on land which, for the most part, is above sea-level and which would be well above sea-level if the ice sheet were removed and isostatic recovery were to take place. In contrast, most of the land beneath the West Antarctic ice sheet is well below sea-level and would remain so even after removal of the ice; the overlying ice sheet here is known as a “marine ice sheet”. Finally, there are the ice shelves, which form where ice flows onto the sea, which occupy the great embayments in the Antarctic coastline, and which serve to smooth the outline of the continent. The East Antarctic ice sheet is the most stable of these; removal of the others would have only a peripheral effect on it and would lead to a high probability of ice-sheet growth, since moist air would be allowed access to a major portion of the ice sheet which today is separated from the sea. Although an ice shelf is to some extent nourished by snow falling on its surface, the large Antarctic ice shelves probably could not exist without a supply of ice from the grounded ice sheets; local snow accumulation alone would not be sufficient to balance loss by bottom melting. Despite this, ice shelves are not merely peripheral features; without their buttressing effect the West Antarctic ice sheet could not exist.

The marine ice sheet in West Antarctica (Fig. 1) flows west into the Ross Ice Shelf, east into the Ronne Ice Shelf, and north into several small ice shelves that fringe the coast and are pinned by small ice rises. The bedrock beneath the grounding line between the ice shelves and West Antarctica is generally rather flat, without any sills or steps to provide a stable grounding-line position. This means that the position of the grounding line is very susceptible to changes in either ice thickness or sea-level. The ice thickness is controlled to some extent by the damming effect of the ice shelf, and removal of the ice shelf would allow the grounding line to retreat. Indeed, Reference MercerMercer (1968) suggested that the present-day West Antarctic ice sheet would collapse almost completely if climatic conditions were not suited to the existence of ice shelves. He proposed that this may have happened during the Sangamon Interglacial, 120000 years ago, when sea-level was approximately 5 m higher than it is today. Because ice shelves exist at sea-level they are particularly susceptible to changes in climate, and Reference MercerMercer (1978) warns of the possibility that the “greenhouse effect” from man-made carbon dioxide may cause destruction of the ice shelves and collapse of the West Antarctic ice sheet. Questions raised by this possibility will be discussed in a later section. In addition to helping us to predict ice-sheet response to possible changes in climate, an understanding of marine-ice-sheet dynamics may provide valuable insights into past glacial epochs since it appears likely that large portions of the Pleistocene ice sheets rested on rock which was below sea-level (Reference Hughes, Hughes, Denton and GrosswaldHughes and others, 1977).

Fig. 1. West Antarctica, showing the areas of bedrock that are above sea-level. The arrows represent ice streams that drain the ice sheet. Most of the ice that flows into the Ross Ice Shelf from West Antarctica drains through ice streams B, D, and E. Pine Island Glacier (P.I.G.) and Thwaites Glacier (T.G.) carry most of the ice that drains northward. These glaciers appear to be relatively unprotected by bounded or partially grounded ice shelves. A vertical section taken along the broken line through ice stream B and the Ross Ice Shelf is shown in Figure 4.

The aim of this paper is to examine the various factors that affect the stability of a marine ice sheet, and to explore possible feedback effects which may influence the rate of growth or decay of the ice sheet. The fundamental requirement for a marine ice sheet is that the ice sheet becomes afloat at its edge, thus, the state of equilibrium of the ice sheet is dependent on conditions in this region. Attempts have been made elsewhere to derive equations for the thickness and migration-rate of ice at the grounding line (Reference WeertmanWeertman, 1974; Reference ThomasThomas, 1977; Reference Thomas and BentleyThomas and Bentley, 1978[b]), and I shall not repeat them here. Instead I shall try to provide a description of the physical significance of these equations and the relative importance of their component parts. I shall also discuss the state of equilibrium of the West Antarctic ice sheet and its likely response to rapid ablation of the ice shelves.

The grounding line

Ice that flows across the grounding line of a marine ice sheet either calves to become icebergs or forms a floating ice shelf that is joined to the ice sheet. Ice shelves tend to form in embayments or where there are sufficient shoals to pin the ice shelf. For an ice sheet that is in equilibrium, ice movement across the grounding line is balanced by accumulation over the ice sheet. Excessive snowfall causes the ice sheet to thicken and the grounding line to advance. Then, if advance takes place over a shoaling seabed, creep-thinning rates decrease so that advance may continue until the grounding line reaches the edge of the continental shelf. Conversely, if the ice at the grounding line is growing thinner, the ice sheet retreats and, for retreat over a bed that slopes downwards towards the centre of the ice sheet, the creep-thinning rate increases and the ice sheet becomes more and more unstable. Under these conditions the ice sheet will collapse completely unless there is a sufficiently high bedrock sill over which the grounding line can achieve equilibrium.

For an ice sheet that flows into a bounded ice shelf or into an ice shelf that contains grounded ice rises, shear between the ice shelf and its sides and compressive stresses up-stream of the ice rises result in a back-pressure at the grounding line which can be sufficient to reduce creep-thinning rates almost to zero (Fig. 2). The grounding line is then able to achieve a new equilibrium position, with the ice shelf acting as a valve to control drainage from the ice sheet. This appears to be the situation that exists today in West Antarctica.

Fig. 2. A bounded ice shelf with ice rises. Velocity profiles are shown for three sections across the ice shelf. The stippling indicates the presence of a back-pressure exerted on the ice shelf by its margins and ice rises. The intensity of the stippling gives an indication of the relative magnitude of the back-pressure. Three bottom profiles are shown: the solid line is for the bounded ice shelf with ice rises and the others are for bounded and unbounded ice shelves without ice rises. The position of the grounding line is strongly dependent on the type of ice shelf that forms.

Most of the drainage from West Antarctica is through large ice streams (Rose, in press) which move principally by sliding along bedrock troughs. Between the ice streams there are promontories of slow-moving ice which is probably frozen to the bed. The equilibrium state of a marine ice sheet is determined mostly by what is happening to the ice streams, and these may not advance or retreat in unison. I shall consider the stability of only one of the ice streams and I shall assume that ice-stream movement is entirely by basal sliding so that all the ice in a vertical column is moving at the same speed. The factors affecting ice thickness at the grounding line are then:

The sum of these three terms gives the net thickening-rate of ice T at the grounding line:

(1)

If T is positive the ice is growing thicker and, since hydrostatic equilibrium must be maintained at the grounding line, there will be a tendency for the ice sheet to advance. If T is negative then the grounding line retreats. Equilibrium is represented by T = 0.

State of Equilibrium

At any time it is unlikely that the thickening-rate T is identically equal to zero. Equilibrium is more probably represented by minor fluctuations in T that balance out over a long period, and the position of the grounding line is likely to oscillate in response to these fluctuations. Such minor variations in T may be caused by random changes in the local accumulation-rate A or in the advection term D, which itself responds to variations in accumulation-rate over the ice sheet. Grounding-line migrations of this type are reversible so long as the changes in A and D are reversible. Thus, we may regard the equilibrium position of the grounding line to be somewhere within a zone that accommodates these minor fluctuations.

Sustained changes in any of the terms on the right-hand side of Equations (1) will result in a permanent shift of the grounding line to a new equilibrium position. For instance, a permanent increase in A or D may force the grounding line to advance to a new position, probably in deeper water, where the creep-thinning rate (— C) increases sufficiently to set T equal to zero. Conversely, a reduction in A or D causes the grounding line to retreat into shallower water. The rate of grounding-line migration and whether migration is stable or unstable depend on the seabed topography, the type of ice shelf that forms to seaward of the grounding line, ice movement across the grounding line, crustal warping, and sea-level changes.

Seabed Topography

To highlight the effects of seabed topography I shall describe the behavior of an ice stream that either flows into an unbounded ice shelf or calves at the grounding line. If the seabed slopes down to seaward then grounding-line migration is limited and is determined by the slope of the seabed (Fig. 3A), if the seabed is steep then the migration will be very small. However, a marine ice sheet rests on large areas of continental shelf which, prior to loading, are comparatively level. After being depressed by the ice sheet the seabed generally slopes up to seaward, and advance of the grounding line is associated with a decrease in creep-thinning rate (—C) so that T becomes more and more positive. In this situation, advance continues to a point where the seabed slopes down into sufficiently deep water. In the extreme case this may be the edge of the continental shelf (Fig. 3B). A retreating grounding line encounters a deeper bed and becomes more and more unstable until a sufficiently high sill is reached.

Fig. 3. Grounding-line migration. In Figure 3A the sea bed slopes down to seaward and a change in ice thickness or sea depth causes only a limited shift in the grounding-line position. If the sea bed slopes up to seaward (Fig. 3B) and the ice sheet is not protected by a bounded or partially grounded ice shelf then there are only two stable modes. These are represented by the “advance” and “retreat”" profiles in Figure 3B. With a suitably bounded or pinned ice shelf, the ice sheet may adopt a stable profile that lies between these two extremes, such as that represented by the solid line. The grounding line may then migrate in response to changes in the ice shelf as well as to changes in ice thickness and sea depth.

The Stability of the West Antarctic Ice Sheet

The collapse of an enlarged West Antarctic ice sheet to achieve its present grounding line with the Ross Ice Shelf has been considered by Reference Thomas and BentleyThomas and Bentley (1978[b]). Our analysis indicated that, during the late Wisconsin, the West Antarctic ice sheet may have extended across the Ross Ice Shelf to reach a line of seabed sills that parallel the edge of the continental shelf. If it did then its collapse was probably initialed by eustatic rise in sea-level, but the ice sheet may have been in such precarious equilibrium that retreat was triggered by depression of the seabed. If no ice shelf formed during retreat then most of the West Antarctic ice sheet probably would have collapsed by about 9000 B.P. Our results support the suggestion of Reference MercerMercer (1968) that the present-day Ross and Ronne Ice Shelves are buttressing the ice sheet, and also indicate that removal of either ice shelf would result in the collapse of the West Antarctic ice sheet within a few decades. Reference MercerMercer (1978) has warned that such a collapse may be one consequence of the currently increasing levels of carbon dioxide in the atmosphere. However, even if climatic warming does become sufficient to prohibit the existence of “steady-state” ice shelves, it is probable that, during collapse of the West Antarctic ice sheet, ice shelves would continue to play some part in slowing retreat-rates. The effect of climatic warming on the ice-sheet-ice-shelf system may be summarized as follows:

• 1. Ice-shelf thinning due to increased melt-rates.

• 2. As bounded ice shelves thin, they exert less back-pressure at their up-stream grounding lines so that there is an increase in flow-rates from the ice sheet into the thinning ice shelves.

• 3. There is a tendency for ice-shelf thinning-rates due to melting to be balanced by increased drainage of thicker ice from the collapsing ice sheet.

In this way ice shelves will continue to exist in a state of quasi-equilibrium with the high melting- and drainage-rates, and so long as these ice shelves transmit a back-pressure to the grounding line they will slow down retreat-rates. Thus, collapse of the marine ice sheet will continue, but at a significantly slower rate than would be the case for complete removal of the ice shelf. Actual retreat-rates would depend critically on how many ice rises remain within the modified ice shelf, and on whether increased flow-rates are associated with progressive softening of ice along the sides of the ice shelf. Preliminary calculations (Reference Thomas, Thomas, Sanderson and RoseThomas and others, 1979) suggest that, even if the “CO₂ greenhouse” effect caused ablation-rates to increase by several metres per year, the collapse of the West Antarctic ice sheet would take more than 200 years. This estimate represents a minimum since it takes no account of the time taken, first for the climate to respond to increased levels of CO₂, and second for the ocean to respond to a changing climate. The ocean response is of particular importance because most of the ablation is likely to take place by bottom melting.

There is considerable disagreement concerning the current state of equilibrium of the West Antarctic ice sheet. Reference HughesHughes (1975) suggested that the ice sheet may be collapsing, and measurements of surface strain-rate that were made near the ice-sheet summit have been interpreted as indicating thinning of perhaps a few centimetres per year (Reference WhillansWhillans, 1973, Reference Whillans1977; Reference HughesHughes, 1973; Reference ThomasThomas, 1976). There is poor agreement between the various estimates of thinning-rate because of lack of information on the behavior of deep ice, and errors are difficult to assess since the result is rather sensitive to errors in the strain-rate measured perpendicular to the flow direction. However, if thinning is taking place it may represent a delayed response of the summit to Holocene retreat of the ice-sheet margins (Reference ThomasThomas, 1976). If this is the case then the West Antarctic ice sheet could be slumping to cover a larger area, but with a lower summit elevation, in an attempt to achieve a new equilibrium profile. This model is consistent with the observation of current thickening in the south-east corner of the Ross Ice Shelf (Reference Thomas and BentleyThomas and Bentley, 1978[a]). Reference WhillansWhillans (1976) was led to the conclusion that evidence from the spacing of internal layers revealed by radio echo-sounding implies that the summit of the ice sheet has been in equilibrium for about 30000 years. This is, however, suspect since the evidence is also consistent with uniform thinning near the summit. A thinning which is approximately uniform is precisely the delayed response we might expect to find as a result of earlier retreat of the ice-sheet margins. Clearly, more data are required before this question can be answered. Perhaps the least ambiguous results could be obtained from a strain network covering the entire summit region. Such an experiment could best be made using techniques that are currently available involving survey equipment mounted in either an aeroplane or a satellite.

Figure 4 shows a section through ice stream B and across the Ross Ice Shelf. Observations suggest that the ice shelf between the grounding line and the Crary Ice Rise is growing thicker by one metre every three years (Reference Thomas and BentleyThomas and Bentley, 1978[a]). If these high thickening-rates were sustained then this entire area would run aground within the next 300 years. The large areas of comparatively flat grounded ice that would form must ultimately offer increased resistance to flow of ice into the ice shelf, so that ice velocities would decrease and ice would pile up in the lower reaches of ice stream B. This process may explain the apparent anomaly of ice stream C, which, from radio echo-sounding evidence (Rose, 1979), should be a very active ice stream. Comparatively simple field experiments in the lower reaches of the West Antarctic ice streams should indicate whether widespread advance is taking place.

Fig. 4. A section through the West Antarctic ice sheet and across the Ross Ice Shelf (see Fig. 1). The bedrock topography is similar to that shown in Figure 3B, and the grounding line is maintained in its present position by back-pressure that is exerted on the Ross Ice Shelf by its sides and by the Crary Ice Rise. Only a very small thickening would be necessar’y for theice sheet to advance to cover Crary Ice Rise. Thereafter, Jilrther advance would require grounding of the ice shelf on theshoals of Pennell Bank.

Along parts of the northern coast of the West Antarctic ice sheet, ice streams appear to flow directly into the sea with little or no ice shelf to buttress them (Fig. 1). It is in this region that partial collapse of the ice sheet may be taking place. If the ice streams do not flow over a sufficiently high bedrock sill, then the grounding lines may retreat a significant distance into the heart of West Antarctica. The Thwaites and Pine Island glaciers are the most likely exits for such a collapse (Reference HughesHughes, 1977); their catchment areas lie over large areas of bedrock that are more than 1000 m below sea-level. Unfortunately, there is no information on bedrock topography near their grounding lines. However, Equations (1) can be solved to give the maximum depth of a bedrock sill that could support an equilibrium ice sheet, in terms of total up-stream accumulation, the width of the grounding line, and the ice flow-law and sliding-law parameters (equation (A13) from Reference Thomas and BentleyThomas and Bentley, 1978[b]). Using available measurements of snow accumulation-rate (Reference Bull and QuamBull, 1971), estimates of grounding-line width from satellite photographs, and the flow-law and sliding-law parameters from Reference Thomas and BentleyThomas and Bentley (1978[b]) I have estimated the equilibrium sill depth below sea-level b ^⋆ for each of these glaciers. For the Thwaites Glacier, b ^⋆ lies between 400 and 550 m, with an associated ice velocity across the grounding line of approximately three kilometres per year. For Pine Island Glacier I have calculated two values of b ^⋆ corresponding to the presence or absence of a small bounded ice shelf which may exist, depending on the precise position of the grounding line. If the ice shelf exists, b ^⋆ lies between 520 and 610 m, and if it does not exist b ^* lies between 410 and 550 m. Associated ice velocities at the grounding line are approximately two and three kilometres per year respectively. If the actual sill depth turns out to be greater than b ^⋆ for either of these glaciers, then the glacier is unstable and there is a high risk of major ice-sheet collapse within its catchment area. This would affect neighbouring glaciers, and it could trigger collapse of a large part of the West Antarctic ice sheet.

The range of values that I have given for b ^⋆ correspond to a wide range of basal-sliding laws and, although there are bound to be large errors in the assumed values of other parameters, we should note that, approximately.

The value of b ^⋆ is most influenced by the creep behavior of ice at the grounding line and, if the assumption that this can be described by ice-shelf dynamics is correct, these values of the equilibrium sill depth should provide a good first approximation. I would suggest that measurement of actual sill depth should take a high priority in future radio echo-sounding programmes. The result would indicate whether additional field work is warranted in order to detect whether or not the ice sheet is collapsing through these glaciers.