Introduction
In recent years, marine ice sheets have been a focus of attention for both the glaciological and paleoclimatological communities. While terrestrial ice sheets rest on bedrock that is largely above contemporary sea-level, marine ice sheets are based on bedrock predominantly below contemporary sea-level. They are drained almost exclusively by fast-flowing ice streams that discharge either directly into the ocean or into an attached ice shelf (Reference HughesHughes, 1977). Ice shelves are in hydrostatic equilibrium with the ocean; the junction between the ice shelf and the ice stream, where the ice stream first becomes afloat, is the grounding line. It has been suggested (Reference HughesHughes, 1973, Reference Hughes1975; Reference WeertmanWeertman, 1974) that marine ice sheets are inherently unstable and cannot exist at steady state. While a terrestrial ice sheet interacts directly only with the atmosphere, a marine ice sheet is also in contact with the ocean and responds to changes in oceanic temperatures, circulation, and sea-level. This is the reason for the significant difference in response times to climatic change, on the order of thousands of years for terrestrial ice sheets and of hundreds of years for marine ice sheets (Reference Bentley, Hansen and TakahashiBentley, 1984). Marine ice sheets may thus well play an important role in shaping our near-future climatic environment and, especially in view of changes in the radiative properties of the atmosphere resulting from the burning of fossil fuels and industrial and thermal pollution, it is a question of some urgency to better our understanding of their behavior (National Research Council, 1985). On longer time scales, marine ice sheets may have been a critical component of the complex internal feed-back mechanisms active in the waxing and waning of the large Pleistocene ice sheets (Reference Denton and HughesDenton and Hughes, 1983). It is therefore important to clarify the dynamical differences between terrestrial and marine ice sheets in trying to explain the past.
Starting with Reference WeertmanWeertman’s (1974) analysis of the stability of the junction of a marine ice sheet and an ice shelf and Reference ThomasThomas’s (1977) zero-dimensional model for the grounding-line retreat rate of a marine ice stream, several studies have addressed the question of marine ice-sheet and ice-stream dynamics. We have developed a much simplified but dynamically consistent model for a two-dimensional coupled marine ice-stream-ice-shelf system. Stresses and velocities are determined explicitly in terms of the ice thickness; ice- thickness profiles are obtained numerically. Numerical results will be published in a forthcoming paper (paper in preparation by Reference Muszynski and BirchfieldI. Muszynski and G. E. Birchfield).
A discussion of potential roles of marine ice sheets in the climate changes of the late Pleistocene follows in the next section. A qualitative picture contrasting dominant physical processes in marine ice streams and terrestrial ice sheets is followed by presentation of the simplified model that explicitly incorporates dominant processes appropriate for marine ice streams (with details of derivation delegated to Appendix A). Discussion of the coupled ice-stream- ice-shelf model follows with conclusions.
Climatological Role of Ice Sheets
Large polar ice masses, whether terrestrial or marine, interact with the mean climatic state in a variety of ways. For the Earth as a whole, a balance exists on long time-scales between the absorption of incoming shortwave solar radiation and the emission of longwave radiation back to space. The amount of solar energy absorbed depends on the planetary albedo. The planetary albedo is a function of the surface albedo of the Earth, the atmospheric concentrations of aerosols, and the characteristics of the cloud cover. The pioneering energy-balance modeling experiments of Reference BudykoBudyko (1969) and Reference SellersSellers (1969) raised the possibility of an ice-covered Earth catastrophe due to the dependence of the surface albedo on temperature and thus on ice-sheet extent, and spurred intensive investigation of global climate sensitivity to this albedo-temperature feed-back mechanism. Extensive discussions of the ice-albedo feed-back and of climate sensitivity can be found in Reference Schneider and DickinsonSchneider and Dickinson (1974), Reference North, North, Cahalan and CoakleyNorth and others (1981), and Reference DickinsonDickinson (1985). Another important feed-back mechanism involves the dependence of the longwave flux on the surface temperature of the ice sheet; the temperature itself is a complicated function of the lapse rate, the surface elevation of the ice sheet, the accumulation rate at the surface of the ice sheet, and the rate of isostatic adjustment of the bedrock under the ice load (Reference OerlemansOerlemans, 1980; Reference BowmanBowman, 1982; Reference PollardPollard, 1982; Reference Birchfield and GrumbineBirchfield and Grumbine, 1985; Reference Muszynski and BirchfieldMuszynski and Birchfield, 1985). The zonal radiation budget is unbalanced, with a net excess of energy at low latitudes and a net deficit at high latitudes. The resulting meridional thermal gradient is the main driving force of the atmospheric and oceanic circulations. A high-latitude cooling due to growth of the ice cover will intensify the Equator to pole temperature gradient and will be counteracted by an increased poleward flux of latent and sensible heat (Reference Schneider and DickinsonSchneider and Dickinson, 1974; Reference Saltzman, Saltzman, Sutera and HansenSaltzman and others, 1982). A complex but potentially very important feed-back concerns the effects of ice-sheet topography and extent on the zonal atmospheric circulation. Disturbances in the planetary wave structure and winds resulting from a temperature-inversion layer over the ice sheet would both result in the Equatorward advection of cold air (Reference Schwerdtfeger and OrvigSchwerdtfeger, 1970; Reference Hartmann, Hansen and TakahashiHartmann, 1984). Not only would these cold air masses enhance the albedo feed-back by helping preserve winter snow cover on land but they would also increase the land-ocean temperature contrast and thereby influence sea-ice extent, sea-surface temperature, and cyclone tracks and precipitation patterns (Reference Ruddiman and McIntyreRuddiman and McIntyre, 1981[a]; Reference Hartmann, Hansen and TakahashiHartmann, 1984; Reference Manabe and BroccoliManabe and Broccoli, 1985[b]). Other feed-back mechanisms are related to the temperature dependence of ice rheology (Reference GlenGlen, 1955): as its temperature rises, ice softens and strain-rates increase. Down-draw of the ice-sheet interior results (Reference Hughes, Denton and HughesHughes, 1981) and the lapse-rate feed-back causes a further rise in temperature at the surface of the ice sheet. Higher air temperatures will also lead to higher surface melt rates. This melt water percolates into crevasses and either refreezes inside the ice column or reaches the bed. The latent heat released upon refreezing of the melt water will soften the ice and compound the effect of the temperature increase (Reference HughesHughes, 1986). An increase in the amount of basal water will result in a higher sliding velocity at the bed through increased bed separation and basal water pressure (e.g. Reference BindschadlerBindschadler, 1983; Reference Weertman and BirchfieldWeertman and Birchfield, 1983).
Marine ice streams and ice shelves are involved in an additional series of feed-back mechanisms that affect the stability of the grounding line and may either enhance or reduce the effects of the above feed-back loops. The rate of advance or retreat of the grounding line depends on the local surface and bedrock slopes, the local mass balance and the rate of sea-level change (Reference Thomas and BentleyThomas and Bentley, 1978). Through non-linear positive feed-back mechanisms, a small perturbation in one of these parameters may result in a run-away process in which the initial small displacement of the grounding line is amplified until the marine ice sheet completely collapses or extends to the edge of the continental shelf (Reference ThomasThomas, 1979[b]). If the bedrock depth increases inland from the grounding line, a rise in sea-level, for example, may lift a part of the ice stream free of its bed and effectively float the marginal zone of the ice stream, causing the grounding line to retreat inland to a region of greater ice thicknesses. Because of the strong dependence of the creep rate at the grounding line on ice thickness (Reference WeertmanWeertman, 1957), ice discharge at the grounding line will increase, resulting in down-draw of the ice-sheet surface in the drainage basin of the ice stream, a decrease in basal shear stress, and thus an acceleration of the collapse of the marine ice sheet (Reference HughesHughes, 1975, Reference Hughes, Denton and Hughes1981). These are all positive feed-back mechanisms that will amplify the initial perturbation. In the absence of an ice shelf, complete collapse of the marine ice sheet may result unless a bedrock sill anchors the grounding line (Reference Crabtree and DoakeCrabtree and Doake, 1982; Reference Thomas, Hansen and TakahashiThomas, 1984; Reference DoakeDoake, 1985) by slowing down ice discharge and causing up-stream thickening of the ice.
The two major mechanisms for stabilizing the grounding line, however, involve the interaction of an ice stream with an ice shelf (e.g. Reference ThomasThomas, 1979[a], [b], 1984; Reference Bentley, Hansen and TakahashiBentley, 1984). If an ice stream has to expend work to push its ice into and through an ice shelf, there will be less gravitational potential energy available for conversion into frictional energy dissipation at the glacier base and the discharge rate at the grounding line will decrease. A distinction is traditionally made between unbounded or freely floating ice shelves and bounded ice shelves which flow in a channel or embayment. The retarding force exerted by these solid boundaries on the ice-shelf flow leads to a decrease in ice-shelf thinning rates and thereby has a damming effect on ice-stream discharge into the ice shelf. Grounding-line thicknesses are extremely sensitive to the dynamic state of the ice shelves into which the ice streams drain. While the grounding-line thickness of an ice stream attached to a freely floating ice shelf would be approximately 400 m (Reference WeertmanWeertman, 1974), thicknesses of 1000 m are not uncommon. The excess compressive stress above that provided by buoyancy alone at an ice-stream grounding line has been termed back-pressure by Reference ThomasThomas (1973). The back-pressure depends on the geometry of the embayment (Reference SandersonSanderson, 1979) and on the length of the ice shelf, and thus on the relative rates of grounding-line movement and iceberg calving at the front of the ice shelf (Reference HughesHughes, 1982; Reference Fastook, Hansen and TakahashiFastook, 1984). A climatic warming, resulting in larger amounts of melt water at the surface of the ice shelf and in a change in calving mechanism from intermittent production of large tabular icebergs to “explosive” production of small icebergs (Reference Thomas, Hansen and TakahashiThomas, 1984), may thus decrease the ice-shelf back-pressure and destabilize the ice stream. Shallow bedrock shoals, on which the ice shelf runs aground, provide another source of back-pressure by impeding the ice-shelf flow, Ice shelves are very vulnerable to increased bottom melting rates which would result in two competing effects (Reference ThomasThomas, 1985; Reference MacAyealMacAyeal, unpublished): increased thinning rates, with the possibility of ice-shelf ungrounding at the deeper pinning points, would increase the ice-stream discharge rate at the grounding line, while the selective removal of the warmest and thus softest ice from the ice column would decrease the depth-averaged ice temperature and thus stiffen the remaining ice, thereby decreasing ice-shelf strain-rates and stabilizing the grounding line. This second effect would be enhanced by the increased advection of cold ice-stream ice. It is not yet clear, however, whether a climatic warming would significantly affect the temperature of the waters circulating in the sub-ice-shelf cavities (Reference MacAyealMacAyeal, 1984; Reference JacobsJacobs, 1985). Ice-shelf pinning or unpinning may also occur as the result of isostatic rebound (Reference ThomasThomas, 1976).
The scale of climatic variability associated with a particular climatic component depends upon its response time to external forcings and to changes in the internal configuration of other climatic sub-systems (Reference SaltzmanSaltzman, 1983). Because the time needed for an ice sheet to adjust its volume to a change in external boundary conditions is very long, terrestrial ice sheets contribute mainly to long-term climatic trends, although some short-term climatic variability may be associated with the fast response time of surface conditions associated with a snow and ice cover, such as the amount of summer melt water. Reference Bentley, Hansen and TakahashiBentley (1984) estimated that the potential instability of the grounding line of a marine ice sheet leads to a decrease in its response time by about an order of magnitude, from thousands to hundreds of years. The reaction of the West Antarctic ice sheet, which is today’s only truly marine ice sheet, to a CO2-induced climatic warming therefore is the cause of great concern among the glaciological and climatological communities; unfortunately, there is as yet no clear consensus as to the likelihood of its collapse or the time-scale over which such a collapse would take place (see, for example, Reference MercerMercer, 1978; National Research Council, 1985).
Climatic variability on time-scales of 10−400 ka is thought to be the consequence of astronomical forcing (e.g. Reference SaltzmanSaltzman, 1983). The astronomical or Milankovitch theory postulates that the small insolation fluctuations due to the quasi-periodic changes in the obliquity of the Earth’s axis and in the eccentricity and longitude of perihelion of its orbit can be amplified sufficiently by internal feed-back mechanisms to cause the Pleistocene succession of glacial and interglacial episodes observed in the paleoclimatic record (e.g. Reference Imbrie and ImbrieImbrie and Imbrie, 1979; Reference ImbrieImbrie, 1985). The main spectral power of orbital variations occurs near periods of 413, 100, and 54 ka for eccentricity; 41 ka for obliquity; and 23 and 19 ka for precession. The spectrum of the Earth’s response, as measured in proxy paleoclimatic records, contains main peaks at periods of 100, 41, 23, and 19 ka, and is clearly correlated with the astronomical forcing for the Northern Hemisphere. As it stands today, the Milankovitch theory has two puzzling features. The first of these concerns the response of the climate system at the eccentricity-related 100 ka period, which is much stronger than the corresponding orbital forcing. This suggests that the spectral response at 100 ka represents a non-linear response of the climate system. As we have seen, ice sheets participate in a very rich spectrum of feed-back loops; they obey a non-linear flow law, have a very slow volume response time, and interact with the lithosphere on long time-scales. They are thus a prime candidate for explanation of the 100 ka paradox. A series of studies has explored this possibility (e.g. Reference Birchfield, Birchfield, Weertman and LundeBirchfield and others, 1981; Reference OerlemansOerlemans, 1982[a]; Reference Le Treut and GhilLe Treut and Ghil, 1983; Reference Peltier, Hyde, Berger, Berger, Imbrie, Hays, Kukla and SaltzmanPeltier and Hyde, 1984; Reference Pollard, Berger, Berger, Imbrie, Hays, Kukla and SaltzmanPollard, 1984; Reference Birchfield and GrumbineBirchfield and Grumbine, 1985). Although they clearly show the correlation between ice-volume changes and Northern Hemisphere insolation fluctuations, and do produce simulated ice-volume curves that resemble those deduced from the oxygen stable-isotope record, their spectral power at 100 ka is still somewhat less than is seen in the proxy record. A common characteristic of these modeling studies is that they consider the interaction between a single, Northern Hemisphere continental ice sheet and the insolation forcing. They are thus inadequate to explore the role of Southern Hemisphere ice sheets, in particular the Antarctic ice sheet, in the Pleistocene ice ages.
The second Milankovitch paradox concerns interhemispheric interactions. While the insolation anomalies associated with the obliquity are symmetric between the hemispheres, those associated with the precession of perihelion are anti-symmetric. Global circulation-model studies indicate that, perhaps because it is largely oceanic, the Southern Hemisphere is hardly affected by the presence of large Northern Hemisphere ice sheets (Reference Manabe and BroccoliManabe and Broccoli, 1985[b]). Detailed studies of ice-extent markers, however, indicate that mountain-glacier fluctuations in both hemispheres were nearly synchronous (Reference Mercer, Hansen and TakahashiMercer, 1984), and foraminiferal evidence suggests that sea-surface temperature changes in the sub-Antarctic slightly pre-date the ice-volume changes associated with the demise of the large Northern Hemisphere ice sheets (Reference Hays, Hays, Imbrie and ShackletonHays and others, 1976). Moreover, glacial terminations are slightly more abrupt in the Southern than in the Northern Hemisphere (Reference Broecker, Berger, Berger, Imbrie, Hays, Kukla and SaltzmanBroecker, 1984). How, then, are deglaciations driven? Part of the answer probably lies in the existence, in the climate system, of one or more fast mechanisms able to transform Northern Hemisphere insolation fluctuations into world-wide temperature changes. In the last few years, attention has been increasingly focused on the role of the deep ocean, more particularly the cross-equatorial heat flux and the effect of high-latitude deep convection on bottom-water production and atmospheric CO2 concentration (Reference Boyle and KeigwinBoyle and Keigwin, 1982; Reference Broecker, Berger, Berger, Imbrie, Hays, Kukla and SaltzmanBroecker, 1984; Reference Broecker, Takahashi, Hansen and TakahashiBroecker and Takahashi, 1984; Reference Ennever, McElroy, Sundquist and BroeckerEnnever and McElroy, 1985; Reference Manabe and BroccoliManabe and Broccoli, 1985[a]; Reference Toggweiler, Sarmiento, Sundquist and BroeckerToggweiler and Sarmiento, 1985; Reference Wenk, Siegenthaler, Sundquist and BroeckerWenk and Siegenthaler, 1985). Further important clues were furnished by Reference Ruddiman and McIntyreRuddiman and McIntyre (1981[b]), who found evidence that the last glacial Northern Hemisphere ice sheets may have substantially decreased in volume before shrinking in area, thereby raising the possibility that down-draw effects in the drainage basins of marine components (Reference Stuiver, Stuiver, Denton, Hughes, Fastook, Denton and HughesStuiver and others, 1981) played a critical role, and by Reference Denton and HughesDenton and Hughes (1983), who asserted that, at the last glacial maximum, a critical part of the large Northern Hemisphere ice sheets may have become marine because of isostatic adjustment under the ice load. They proposed that summer melting of Northern Hemisphere terrestrial ice-sheet margins, caused by a favorable orbital configuration, leads to a change in eustatic sea-level that, in turn, destabilizes all marine ice-sheet components. Increased interior down-draw and grounding-line retreat, migration of ice divides between adjacent terrestrial and marine drainage basins, and albedo and other positive feed-back mechanisms all amplify the initial marginal melting and ultimately lead to total collapse of the ice sheets. To summarize, then, there is considerable evidence that marine ice sheets may be critical cryospheric components on short, as well as long, time-scales of climatic variability.
A Marine Ice-Stream Model
Marine ice-stream physics, a qualitative model
The flow regime of an ice mass is largely determined by conditions at the ice-bedrock interface. A terrestrial ice sheet which is frozen to its bed is subject to two forces: gravity and the drag force exerted by the bedrock on the ice; the resulting normal stress field is therefore essentially isotropic, that is, the normal deviatoric stresses are small compared with the shear stress. The longitudinal strain-rate or stretching may then be neglected compared with the shear strain-rate and movement will be by shearing only. Equating the normal stresses to the glaciostatic pressure, it is then possible to obtain the shear stress by balancing the gravitationally induced driving stress and the basal shear stress acting along the ice-bedrock interface. The velocity, in turn, follows from the rheology by integration of the shear strain-rate.
If the ice is not frozen to the bed, but basal conditions are such that the sliding velocity and its down-stream gradient are small, the response is the same except for the addition of a bottom-sliding component to the velocity. As the sliding velocity increases (due to pressure-melting of the basal ice, for example), its down-stream gradient may become comparable in magnitude to the shear strain-rate and stretching may no longer be negligible compared with shearing. In a surging glacier, for example, the sliding velocity is so large that stretching becomes the dominant deformation process (e.g. Reference McMeeking and JohnsonMcMeeking and Johnson, 1986 and references therein). Corrections to the frozen-bed solution under these circumstances have been obtained by asymptotic methods (Reference HutterHutter, 1981; Reference Shoemaker and MorlandShoemaker and Morland, 1984; Reference McMeeking and JohnsonMcMeeking and Johnson, 1985).
For a marine ice stream with attached freely floating ice shelf, strain heating, inland seepage of sea-water, and subglacial buoyancy effects associated with the marine character of the ice stream all contribute to establishing a strong gradient in basal sliding velocity between the head of the ice stream and its grounding line. This suggests that the dominant deformation process in the marine ice stream is longitudinal stretching. We shall assume that the predominance of stretching over shearing is a distinguishing property of marine ice streams. Since, in the floating ice shelf, shear stresses must be small, the dominant deformation rate must also be a stretching. Reference WeertmanWeertman (1957) and Reference ThomasThomas (1973) have demonstrated that this stretching is induced by the deviatoric normal stress which itself is ultimately due to the density difference between the ice shelf and sea-water. It is obvious that in the region of the grounding line the longitudinal strain-rate produced by these two distinct mechanisms varies continuously from the ice stream to the ice shelf. It is through this coupling that the basal-sliding control of the ice stream affects the flow of the ice shelf and, vice versa, that changes in the dynamic state of the ice shelf, for example through grounding or ungrounding, affect the dynamics of the ice stream. The up-stream limit of this domain of influence of the ice shelf on the ice stream is estimated in the discussion. A further feed-back mechanism that may be important in this coupling is also discussed there.
Marine ice-stream physics, a scaled model
As the dynamics of a laterally unbounded ice shelf are well understood (Reference WeertmanWeertman, 1957; Reference ThomasThomas, 1973), we limit our discussion here primarily to the marine ice-stream domain. We first present the full, unsimplified system of equations and boundary conditions; this is followed by a simple, but fundamental scaling analysis which substantiates the discussion in the previous section and provides the basis for our coupled model. The simplified ice-stream model is then presented, with details relegated to Appendix A. To demonstrate overall dynamical consistency of the coupled model, a scale analysis of the freely floating ice shelf is presented in Appendix B.
Consider a Cartesian coordinate system, with x horizontal along the longitudinal axis and y along the vertical axis, as illustrated in Figure 1. The y-axis denotes the up-stream limit of the ice-stream flow regime and is assumed fixed in space and time. The ice shelf extends from the grounding line, at x = x*. to x = L. For simplicity, we assume that the ice stream rests on a rigid bed and that the bedrock profile is linear with slope ᶲ > 0. The bedrock depth is zero at x = 0. In both ice sheet and ice shelf, h′ is the ice-surface elevation and H the total ice thickness. In the ice sheet, h is the bedrock depth, h = ᶲx, while in the ice shelf it is the depth to the bottom ice surface. Thus, H = h’ + h. The surface slope of the ice is ∂h’/∂x = −α (α > 0), the bottom slope ∂h/∂x = ᶲ. If ρw is the density of sea-water, ρ i the density of ice (both assumed constant), and ∆ρ = ρ w−ρ i, hydrostatic equilibrium in the ice shelf requires that
The bottom ice surface is thus known everywhere. The values of h’, h, and H at the grounding line are h’ *, h*, and H*, and S is sea-level.
Fig. 1. Model geometry. The surface slope is α, the bedrock slope ɸ. The total ice thickness is H and the surface elevation h’. In the ice stream, h = ɸx, in the ice shelf,
Basic equations. We denote normal stresses by σii, shear stresses by σij (i ≠ j), and strain-rates by
with, for the marine ice stream, boundary conditions at y = h′:
at y = −h:
at x = 0:
at x = x*:
and, for the ice shelf at y = h’:
at y = −h:
at x = x*:
at x = L,
(Equation 1) is a vertically integrated statement of mass continuity for an incompressible material; a − b is the net accumulation rate, equal to net surface accumulation minus bottom melting (in meters of ice equivalent/year). Equations (2) are stress equilibrium equations, with g the vertical acceleration due to gravity. Equations (3) define the deviatoric stresses as being deviations from isotropy in the normal stress field, while (Equation 4) defines the effective stress in terms of the deviatoric and shear stresses. Equations (5) constitute a statement of ice rheology and relate the stress and strain-rate fields. We have adopted a non-linear power-law rheology with exponent equal to three; the flow-law parameter A is assumed constant. Equations (6.1)–2), (10.1–2), and (11.1–2) are stress-continuity conditions at the ice-air and ice-water interfaces (atmospheric pressure is assumed to be zero); Equations (6.3), (10.3), and (11.3) are kinematic conditions describing the location of the free surfaces. Equation (7.1) states that there is no penetration of ice into the bedrock, and Equation (7.2) is the conventional sliding law relating basal velocity and basal shear stress. F’ is an empirical function of (x, t), and m a constant exponent, both discussed below. The boundary conditions at the origin (Equations (8))) and at the grounding line (Equations (9)) state that the ice thickness and the normal component of the stress vector are continuous at the junctions of the ice stream with the ice sheet at x = 0, and with the ice shelf at x = x*. Because the ice shelf is afloat everywhere, boundary condition (9.1) is also a flotation condition on the ice stream at the grounding line. (Equation 12) expresses that the mass flux is continuous across the ice-stream-ice-shelf transition, and (Equation 13) is an integrated stress continuity condition at the ice-shelf front.
For small α and ø,
where F = F ϕ cos – cos m 2ϕ and Equation (7.1) has been used to eliminate the vertical velocity v from Equation (7.2).
Fundamental scale analysis. The traditional way to reduce the above equations in the case of a terrestrial ice sheet is to neglect the normal strain-rate equations (Equations (5.1)–2) and to retain only the shear strain-rate equation (Equation (5.3)). As pointed out above, this is not appropriate for a marine ice stream, as it implies isotropy of the normal stress field and dominance of shearing over stretching. We first note that the down-stream aspect ratio, i.e. the ratio of ice thickness, H c, to length, L c, denoted by ω2 of most large ice bodies is small, ω2 ≈ 10−2−10−3. Because of the dominance of gravity in the vertical, typical normal stresses in such bodies are σc ≈ ρi gH c. Furthermore, as seen from Equation (2.1), the ratio of shear stress to normal stress must be of the order of the aspect ratio ω2. For the rapidly flowing ice stream in the neighborhood of the grounding line, we assert that the normal deviatoric stress must be larger than the shear stress. Suppose that this was not true, and that instead they were of the same order of magnitude ω2σc. Estimating the longitudinal strain-rate as the ratio of a typical basal sliding velocity U c, to the length of the ice stream L c, by equation (5.1) we must have
But for typical values U C ≈ 200 m a−1, L C ≈ 100 km, A C ≈ 1 × 10−17 a−1 Pa−3, and σC ≈ 2.5 × 106 Pa,
Provided
the basal sliding velocity clearly dictates large deviatoric stresses, of order
in the ice stream.
In the freely floating ice shelf, the deviatoric stress can be shown to be of order
Strain-rate continuity in the neighborhood of the grounding line then requires
The basal sliding velocity of the ice stream and the dynamics of the ice shelf are thus coupled through the relationship
Finally, with this coupling constraint on U c, the ratio of shear stress to deviatoric stress in the ice stream becomes
so that the condition for dominance of stretching over shearing in the ice stream reduces to the same condition as in the ice shelf,
Marine ice-stream model. The above scale analysis, as applied in Appendix A, asserts that, for a fast-flowing marine ice stream, the original system of (Equations 1)–(5) can be replaced by the following, much simpler system, together with boundary conditions (6’), (7′), (8), and (9).
This system is integrated, using Equations (6)’ and (7′), to give the final form for the marine ice-stream model:
with boundary conditions (8) and (9) unchanged by the scale analysis.
The vertical normal stress in an ice shelf, whether bounded or not, is equal to the glacial overburden (Reference WeertmanWeertman, 1957; Reference ThomasThomas, 1973; Equation (B28)) and thus has the same functional expression as the vertical norma! stress in the marine ice stream (Equation (15.2)), Equation (3.1) then shows that, at the grounding line, the continuity condition on the normal stress (Equation (9.2)) can be replaced by a continuity condition on the deviatoric stress. In a freely floating ice shelf, the deviatoric stress is
(Reference WeertmanWeertman, 1957; Reference ThomasThomas, 1973; Equation (B30)). If the ice shelf is grounded, however, this is no longer true. To allow for this case, we state boundary condition (9.2) simply as
where Σ’ * is a function of the dynamic state of the ice shelf.
Discussion and Conclusions
We have shown that the most important control of the dynamics of a fast-moving ice stream coupled to a freely floating ice shelf is associated with first-order buoyancy effects, that is, longitudinal stretching dominates shearing deformation in both the ice shelf and the ice stream provided the aspect ratio ω2 « ∆ρ/ρ w. This constraint is the basis for our simplified ice-stream model consisting of the three coupled differential (Equations 17), (18), and Reference Gow, Gow, Ueda and Garfield(21) with boundary conditions (8), (9.1), and (9.2’). Given initial and boundary conditions, the model predicts time and spatial changes of ice thickness, longitudinal velocity, and deviatoric stress. Auxiliary diagnostic relations (19) and (20) provide the normal and shear stresses.
The domain up-stream of the grounding line over which this dynamic regime extends can be estimated from the ratio of shear stress to deviatoric stress; the magnitude of the former is estimated here explicitly in terms of the free surface slope α from (Equation 20),
Solving for L c, we have
Since the distance up-stream from the grounding line over which the ratio of shearing to stretching remains small varies as the ¼th power of the ratio of basal sliding velocity to flow-law constant, it is fairly insensitive to changes in U c and A c. For A c ≈ x 10−17 a−1 Pa−3, U c ≈ 200 m a−1 and α ≈ 5 x 10−3, L c ≈ 150 km.
Our knowledge of conditions at the base of ice streams is still limited. It does seem likely, however, that, largely as a result of strain heating and geothermal heat flux, the basal ice of most West Antarctic ice streams is at the pressure melting-point (Reference Gow, Gow, Ueda and GarfieldGow and others, 1968; Reference RoseRose, 1979) and that an extensive basal water layer is present (Reference Weertman and BirchfieldWeertman and Birchfield, 1982). Under these conditions, the effective bed roughness is decreased and separation of the basal ice by cavity formation on the lee side of obstacles and subglacial buoyancy effects is favored, and high sliding velocities are to be expected (e.g. Reference PatersonPaterson, 1981, chapter 7;Reference Bindschadler Bindschadier, 1983). The sliding law u = Fτb m incorporates both flow around bed protuberances by creep deformation and regelation through the basal shear stress term τb m , and subglacial water effects through the function F. Theoretical considerations and modeling studies indicate that 1 ≤ m ≤ 3 (Reference WeertmanWeertman, 1957; Reference FowlerFowler, 1981; Reference Budd, Budd, Jenssen and SmithBudd and others, 1984; Reference Mclnnes and BuddMclnnes and Budd, 1984) and that F is an inverse function of the ice thickness above buoyancy,
(Reference BindschadlerBindschadier, 1983; Reference Budd, Budd, Jenssen and SmithBudd and others, 1984; Reference Mclnnes and BuddMclnnes and Budd, 1984). The role of F in the dynamics of the ice stream is seen by estimating U c in Equation (23) from (Equation 21), U c ≈ F c(ρi gH cα) m
for m ≈ 3. The distance over which the ice-stream flow regime can be maintained increases with basal lubrication (F); the relative importance of shearing versus the sliding-induced stretching increases as A increases, i.e. as the ice becomes softer. In the present model, A should be regarded as a vertically averaged rheological constant. Inclusion of the temperature dependence of A in the model would give rise to a differential softening of the ice in the basal layer compared to that in the near-surface layer (e.g. Reference HutterHutter, 1983, chapter 3) and thus to a thermally induced shearing. Due to the large increase in creep-activation energy at temperatures approaching the melting point (Reference Weertman, Whalley, Whalley, Jones and GoldWeertman, 1973), and to the relatively temperate surface conditions prevalent near the edges of a major polar ice sheet such as in Antarctica, this effect may be relatively insignificant compared to the large ice-shelf-induced stretching.
Using (Equations 18) and (20), the basal shear stress in our model,
is found to be
where τc = ρi gHα is the “classical” basal shear stress that exactly balances the gravitational driving stress and
is a buoyancy correction to τc.
Since σ’ xx contains a second derivative of the surface elevation, τm contains a third derivative. Because of the correction τm, the total basal shear stress τb will now be a function not only of local conditions but also of conditions at some distance up-stream and down-stream (Reference BuddBudd, 1975; Reference McMeeking and JohnsonMcMeeking and Johnson 1985). Further significance of τm becomes clear if we look at the strain-rate,
From (Equation 21), the velocity in terms of τc and τm is
and hence,
It is now apparent that τm introduces an internal feed-back between down-stream changes in longitudinal strain-rate, ice thickness, and basal velocity. This feed-back is positive if
The buoyancy basal shear stress τm thus provides a mechanism whereby the ice-stream profile and sliding velocity adjust to conditions at the grounding line so as to generate a stretching of the same order of magnitude as that imposed at the grounding line by the ocean or the ice shelf into which the ice stream discharges. In the absence of an ice shelf or if the ice shelf is freely floating, the down-stream strain-rate gradient is large and positive, the surface slope is small, and τm is positive. If the back-pressure exerted by the ice shelf increases, the ice-shelf strain-rate at the grounding line decreases and may even become negative, and there will be a region up-stream of the grounding line in which the negative feed-back associated with a negative τm contributes in decreasing the sliding velocity and strain-rate in the ice stream to the grounding-line value. It is important to realize, however, that as the back-pressure changes, both F and τc will also be affected; this discussion is meant only to highlight the qualitative role of τm in adjusting the ice-stream sliding velocity to conditions at the grounding line. The detailed mechanism of this adjustment is explored fully in the numerical model (paper in preparation by Reference Muszynski and BirchfieldI. Muszynski and G.E. Birchfield).
Previously published marine ice-sheet models (Reference ThomasThomas, 1977; Reference Thomas and BentleyThomas and Bentley, 1978; Reference Hughes, Denton and HughesHughes, 1981; Reference Stuiver, Stuiver, Denton, Hughes, Fastook, Denton and HughesStuiver and others, 1981; Reference YoungYoung, 1981; Reference OerlemansOerlemans, I982[b]; Reference Budd, Budd, Jenssen and SmithBudd and others, 1984; Reference Fastook, Hansen and TakahashiFastook, 1984, Reference Fastook1985; Reference LingleLingle, 1984; Reference Mclnnes and BuddMclnnes and Budd, 1984; Reference Lingle and ClarkLingle and Clark, 1985; Reference Veenvan der Veen, unpublished) have used the “classical” sliding law u = Fτb m with F derived either from theoretical considerations of the sliding process or by fitting to field data. The basal shear stress in these models is thus a function of local conditions only, and any change in back-pressure will act on the sliding velocity through the resulting changes in ice-stream thickness and surface slope only. Inclusion of τm not only ensures dynamical self-consistency but provides an explicit feed-back mechanism that links changes in back-pressure directly to changes in the ice-stream strain-rate.
Field measurements show that the surface curvature of ice streams is positive, i, e. concave up (e.g. Reference HughesHughes, 1975, fig. 6; Reference RoseRose, 1979, fig. 1). This was attributed by Reference HughesHughes (1975) to partial uncoupling of the base of the ice stream from its bed in the presence of large amounts of basal melt water. The stress-continuity requirement at the grounding line (Equation (9.2)’) can be rewritten as an expression for the surface curvature on the ice-stream side of the grounding line by substitution of (Equation 18):
The term involving the down-stream gradient of F stresses the importance of subglacilal water-pressure effects in maintaining positive ice-stream surface curvatures.
Acknowledgements
Part of this work was completed while the authors were visiting the Institute of Oceanographic Sciences, Wormley, England; the second author was on a Guggenheim Fellowship at the time of this visit and gratefully acknowledges the support therefrom. Appreciation is also extended to the Scott Polar Research Institute, University of Cambridge, for the use of their library. The figure was drafted by C. Cheverton. The authors are indebted to two anonymous reviewers whose comments resulted in considerable improvements to the manuscript. This research was supported by grant ATM8306251 from the Climate Dynamics Section, U.S. National Science Foundation.
Appendix A Scale Analysis
We use the approach of asymptotic analysis (Reference Morland and JohnsonMorland and Johnson, 1980, Reference Morland and Johnson1982; Reference HutterHutter, 1981, Reference Hutter1983; Reference JohnsonJohnson, 1981; Reference Shoemaker and MorlandShoemaker and Morland, 1984; Reference McMeeking and JohnsonMcMeeking and Johnson, 1985) to develop a greatly simplified but dynamically consistent ice-stream-ice-shelf model based on the scale analysis in the text. The basic system of equations is non- dimensionalized with a set of characteristic scales X c such that the non-dimensional form of a variable X is given by
Let
We scale all stresses with
Since the empirical function F in the basal sliding law is the ratio of the velocity to a power of the basal shear stress, we scale F as the ratio of these two quantities,
The characteristic strain-rate in the ice shelf is
From continuity, the characteristic velocity for the ice stream is then
Substituting for all dimensional variables in terms of characteristic quantities and non-dimensional variables, we obtain the following system of equations and boundary conditions:
with boundary conditions at
at
at
at
As shown in the text, for predominance of stretching over shearing, ω2 « Δρ/ρw, the latter ratio being a constant of our problem. For all ice streams of interest to us, the relationship
Based on the analysis in the text, we now incorporate explicit statements as to the relative magnitudes of normal, shear, and deviatoric stresses and of normal and shear strain-rates into the equations by expanding all dependent variables in n-term asymptotic series of the form
(for example, see Reference NayfehNayfeh, 1973) where the O symbol means that the error, T, committed by truncating the series after m = (n−1) terms is such that
If m 0 = 0, the leading term in the asymptotic expansion for X is
Scaled Equations
We now replace all variables by their asymptotic expansions and group terms according to powers of ω, Since the resulting equations are valid for any value of ω, the coefficients of ω have to vanish separately. By equating the coefficients of successive powers of ω to zero, the original system of equations is separated into a hierarchy of equations representing successively smaller corrections to the lowest-order system obtained by setting ω equal to zero. The zero-order system can be written
with boundary conditions at
at
Equations (A11)–(A15) may be reduced to:
Equations (A16)–(A19), with (A10), constitute the zero-order solution. The first-order system is then
with boundary conditions at
at
The solution of this system is again straightforward:
Adding the zero- and first-order solutions, we can write the non-dimensional equations to O(ω2) as
Finally, applying the transformation
Appendix B A Scaled Ice-Shelf Model
Scale analysis
We first perform the coordinate translation ξ = x, η = y – S. The upper ice surface is now given by
The dependent variables are next expanded in n-term asymptotic series in ω. The expansions for the stresses and velocities are identical to those for the marine ice stream. The ice thicknesses are expanded as
The top and bottom boundary conditions and the integral over the ice thickness in the boundary condition at the front of the ice shelf ((Equation 13)) are then expanded in Taylor series about
Scaled equations
The zero-order system consists of the equations
with boundary conditions at
at
at
The solution to this zero-order system is
The first-order system is given by the equations
with boundary conditions at
at
at
From Equations (B8.2) and (B11.2),
and, from Equation (B12.2),
Equations (B8.2), (B9), (B10.1), and (B10.2) show that
so that the deviatoric stress (Equation (9)) is
The velocity is obtained by integration of Equation (10.1)
The shear stress, finally, follows from Equations (8.1) and (11.1)
The complete solution to order ω2 is thus
In the original x, y coordinate system, the dimensional form of the ice-shelf equations is then, to order ω2,
with, at x = x*, boundary condition (12) on mass-flux continuity. Aside from environmental factors, the dynamical state of the ice shelf is controlled by the dominance of stretching over shearing assured by ω2« Δρ/ρw, by the position of the grounding line, the discharge rate from the inland ice at the grounding line, and of course by the calving physics at the ice-shelf front.
