ABRUPT CHANGES IN ICE SHELVES AND ICE STREAMS: MODEL STUDIES
Abstract
Ice sheets are among the most important components of the Earth system because of their ability to force changes in climate and sea level. Ice streams are efficient pathways of mass flux from the interior of ice sheets. Thus an understanding of ice-stream dynamics is integral to an understanding of ice sheets and their interplay with sea level and climate.
Here a 1-d model of the coupled mass and momentum balance of ice streams
and shelves is developed. Longitudinal deviatoric stress is included in the force-balance component model. The mass-balance component model is time-dependent and thus allows simulation of the dynamic consequences of changes in boundary conditions or parameters. An improved, computationally efficient algorithm of the discretization of the mass-balance equation is outlined. All model parameters are non-dimensional.
The model is applied to two problems. In the first study we address the sensitivity of ice-stream/ice-shelf systems to changes in ice-shelf buttressing. We find that for reasonable parameter values such systems are markedly sensitive to a loss of buttressing. Response includes net grounding-line retreat on the order of 10% of the length scale for the system and a roughly 30% loss in the volume of ice above flotation. In the second study we examine the conditions under which ice flowing over a sill will tend to create a reversed ice/air surface slope. Here we find that such slope reversals occur within the range of reasonable parameter values, and thus should be expected. Hence, ice shelf grounding on a sill can trap water and drive subsequent thickening, eventually tending toward outburst flooding.
Table of Contents
List of Tables………………………………………………………………………………………………………………………………………. v
List of Figures……………………………………………………………………………………………………………………………………. vi
Acknowledgments…………………………………………………………………………………………………………………………… vii
Preface………………………………………………………………………………………………………………………………………………… ix
Chapter 1. Introduction……………………………………………………………………………………………………………………. 1
1.1 Sea level, marine-ice-sheet instability and buttressing…………………………………………. 1
1.2 Outburst floods and slope-reversal…………………………………………………………………………… 8
1.3 Summary and preview………………………………………………………………………………………………….. 9
Chapter 2. Derivation of the momentum balance equation for ice streams and iceshelves 10
2.1 Introduction………………………………………………………………………………………………………………….. 10
2.2 x-directed stress equilibrium equation………………………………………………………………….. 10
2.3 Vertical integration……………………………………………………………………………………………………… 11
2.4 Lateral integration………………………………………………………………………………………………………. 13
2.5 Pressure and the vertical stress equilibrium………………………………………………………… 14
2.6 Scaling……………………………………………………………………………………………………………………………. 18
2.7 Constitutive relations…………………………………………………………………………………………………. 21
2.8 Boundary conditions………………………………………………………………………………………………….. 23
2.9 Nondimensionalizing………………………………………………………………………………………………….. 26
Chapter 3. When buttressing matters: a sensitivity study…………………………………………………….. 30
3.1 Introduction………………………………………………………………………………………………………………….. 30
3.2 Model Description……………………………………………………………………………………………………….. 30
3.2.1 governing equations and boundary conditions…………………………………………………………….. 32
3.3 Experiments and Results…………………………………………………………………………………………… 34
3.4 Conclusions…………………………………………………………………………………………………………………… 37
Chapter 4. Conditions for the reversal of ice-stream surface slope…………………………………….. 45
4.1 Introduction………………………………………………………………………………………………………………….. 45
4.2 Model Description……………………………………………………………………………………………………….. 46
4.2.1 governing equations…………………………………………………………………………………………………………….. 47
4.2.2 numerical model & experiments……………………………………………………………………………………….. 50
4.3 Results & Discussion…………………………………………………………………………………………………… 50
4.4 Conclusions…………………………………………………………………………………………………………………… 52
Appendix A. Notation……………………………………………………………………………………………………………………… 63
Appendix B. Numerical model of ice-stream/ice-shelf evolution equation………………………. 66
B.1 Introduction…………………………………………………………………………………………………………………. 66
B.2 Spatial Discretization…………………………………………………………………………………………………. 67
B.3 Time Discretization…………………………………………………………………………………………………….. 71
B.4 Summary……………………………………………………………………………………………………………………….. 72
Bibliography…………………………………………………………………………………………………………………………………….. 74
Chapter 1 Introduction
Ice sheets are agents of sea-level, climate and geomorphological dynamics. Ice sheets are reservoirs of water and thus sea level: if the present-day Greenland and Antarctic ice sheets were to melt, sea level would rise approximately 75 meters [Warrick et al.,
1995, p. 372]. During the last glacial maximum (∼ 25 ka) sea level was approximately 120 meters lower than present, suggesting that ice sheets held ∼200 meters of sea-level equivalent at that time. Thus, the mass balance of ice sheets plays a key part in the evolution of sea level. Through feedbacks with the atmosphere and ocean, ice sheets also influence climate dynamics. One of these is the ice/albedo feedback, which taken to its extreme runs away to the hypothesized snowball-earth [e.g., Hoffman et al., 1998]. Climate also feels the effects of ice sheets through the influence of injections of ice-berg armadas and meltwater pulses on ocean circulation [e.g., Broecker, 1994; Bond et al.,
1999]. The geomorphic power of ice sheets and glaciers is evidenced in vast till sheets [Alley, 1991] and overdeepenings [Alley et al., 2003b].
1.1 Sea level, marine-ice-sheet instability and buttressing
In 1968, Mercer called into question the stability of the West Antarctic Ice Sheet (WAIS, see figure 1.1) on the basis of geologic evidence for a sea-level high-stand during the previous interglacial (∼120 ka) that was approximately 5 meters higher than present [Mercer, 1968]. The suggestion was that the WAIS, which presently holds the equivalent of ∼ 5.5 meters of sea level ([e.g., Drewry, 1983]), has collapsed relatively recently in geologic terms. Scherer et al. [1998] used diatom and10Be evidence to confirm the notion of recent WAIS collapse, albeit only pinning the timing of this collapse to the late Pleistocene. The implication of a recent WAIS collapse is that if this ice sheet has a natural tendency toward collapse, then it is plausible that this ice sheet will collapse again, either naturally or in response to anthropogenic forcing.
The work of Weertman [1961, 1974] andThomas and Bentley [1978] bolstered this notion of WAIS collapse by identifying and simulating a physically plausible mechanism for such a collapse. The main idea is that a marine ice sheet (one whose bed is generally well below sea-level), such as the WAIS, is fundamentally unstable due to a positive feedback between the flux and thickness at the grounding line, where ice from the interior goes afloat as it flows into the surrounding ice shelves. Clearly then, the viability of this “marine-ice-sheet instability” mechanism hinges on correctly modeling the nature of the feedback between grounding-line thickness and flux. This flux is dominated by ice streams, which drain the interior of the ice sheet. Therefore, the crux of the instability lies in resolving the physical interaction between ice streams and ice shelves.
The stress condition at the grounding line is one facet of this interaction. Of particular import is the concept of ice-shelf buttressing, wherein the ice shelf buffers the grounded ice from the stretching force that would apply if the ice front were at the grounding line. The stretching force derives from the difference between the depthintegrated static pressures within the ice and the ocean, respectively. The stretching force yields a positive depth-integrated longitudinal (along-flow) deviatoric stress. Figure (1.2)
Fig. 1.1. Location map for the West Antarctic Ice Sheet, modified from Alley and Bindschadler [2001] .
shows the origin of this stretching force at an ice front, the map-view boundary between the ice shelf and the ocean. Buttressing reduces the stretching force required, and is accomplished by the ice shelf through drag along its lateral boundaries and through drag in localized areas of grounding.
Van der Veen [1986] attempted to model the interplay between grounded and floating (shelf) ice by specifying a transition zone for the stress regime, going from the longitudinal-deviatoric-dominated regime near the grounding line to what is presumed to be the vertical-shear-dominated regime upglacier.Van der Veen [1986] assumed that in this transition zone the longitudinal deviatoric stress decays linearly to zero as one moves upglacier. In addition, he derived the longitudinal deviatoric stress applied at the grounding line from the assumption of an appended ice shelf at equilibrium. The main advantage of this model was that it was simple to implement within the context of coarser-resolution models of inland ice flow. Unfortunately, the ad-hoc treatment of the longitudinal deviatoric stress and the assumption of equilibrium ice shelves are unrealistic and thus unreliable when assessing the interaction of ice streams and ice shelves. Subsequent work byVan der Veen and Whillans [1996] aimed at examining the evolution of ice streams neglects longitudinal deviatoric stresses altogether. Their work, and similar investigations Hindmarsh and Le Meur, 2001; Le Meur and Hindmarsh, 2001; Hindmarsh and Le Meur, 2002], thus do not elucidate the role of buttressing, which manifests itself through its effect on the longitudinal deviatoric stress at the grounding line. Interestingly, some have cited observations of little or no tensile strain-rate at the grounding line of particular ice streams to indicate that ice-shelf buttressing cannot be important [e.g., Whillans et al., 2001]. However, it is this very lack of tensile strain-rates
Fig. 1.2. Illustration of the origin of stretching in ice shelves [from Hughes, 1998]. Here the large left and right triangles represent the pressure as a function of elevation within the ice and seawater, respectively. The areas of these triangles are the depthintegrated static pressures of the ice and water. The area of the small triangle on the right represents the static integrated pressure difference across the ice front. This static pressure difference requires a compensating stretching force and hence leads to a longitudinal strain rate.
that shows these grounding lines are buttressed. Without buttressing, these sites would have the tensile strain-rate of ice fronts and freely-floating ice shelves.
From the observed thickness and strain-rate fields, Thomas and MacAyeal [1982] calculated what they referred to as the ’back force’ within the Ross Ice Shelf (RIS). The back force is essentially the difference between the depth-integrated longitudinal deviatoric stress expected if the ice shelf were freely-spreading, and that calculated from the observed strain-rate field. Higher values of back force thus indicate greater buttressing. As shown in figure (1.3), the downstream ends of the ice streams draining into the RIS are clearly buttressed. The work ofMacAyeal et al. [1987, 1989] showed the importance of buttressing in the vicinity of the Crary Ice Rise. They used a two-dimensional, in planview, force-balance model in conjunction with strain-rate measurements to diagnose the forces acting within the study area. From their work it is clear that the strain-rate at the grounding line upstream of the ice rise would be much greater without the drag provided by the ice rise. However, their work was diagnostic in nature, and thus cannot reveal the prognostic consequences, stable or otherwise, of changes in the drag experienced by ice shelves and the resulting buttressing. The same can be said of the more-recent diagnostic work ofRignot et al. [2002] andSchmeltz et al. [2002] addressing the Pine Island embayment of West Antarctica.
Subsequent work by MacAyeal [1992] employed a prognostic model of the whole West Antarctic Ice Sheet, and included thermodynamics, crude sediment dynamics, and both inland and stream/shelf flow regimes. The main goal of this work was to simulate the evolution of WAIS over glacial cycles. The results showed irregular variations in ice volume, with episodes of complete collapse. The resolution of this study was by necessity
Fig. 1.3. Contour plot showing lines of equal back force as calculated by Thomas and MacAyeal [1982] . This back force is measured in MN per unit width of the ice shelf. Effectively, the larger the back force, the more buttressed the ice shelf is. In these units, the total spreading force in the absence of this back force would range from ≈ 300 near the grounding lines to ≈ 50 near the ice front; thus, the back force offsets almost all of the spreading stress in places and must be included to allow accurate calculations almost everywhere.
coarse in space and time. This is unfortunate given the short spatial (order of one ice thickness) and temporal scales within which ice stream and ice shelf dynamics operate
[Bindschadler and Vornberger, 1998; Rignot, 1998; Joughin et al., 1999; Shepherd et al., 2001, 2003]. Given this coarse resolution as well as the complexity of interaction between the thermodynamic, sediment and force-balance components of the model, assessment of the results is difficult and the question of the near-term (millennial or shorter) stability of WAIS remains open.
1.2 Outburst floods and slope-reversal
Glaciers and ice sheets are some of the post powerful agents for geomorphic change [e.g., Hallet et al., 1996; Cutler et al., 2002]. Subglacial water is one of the key ingredients for this geomorphic dynamism. Storage of subglacial water in sufficient quantity can lead, upon release, to flooding events. Present-day examples of such events, sometimes referred to as jokulhlaups, can be exceptionally large and destructive in comparison to ’ordinary’ floods, [e.g., Johannesson, 2002], although most recent outbursts have been smaller. Alley et al. [2003a] hypothesize that larger ice masses, either extant (e.g., Greenland and West Antarctic ice sheets) or otherwise (e.g., Laurentide and Fennoscandian ice sheets), are able to produce even larger outburst flood events.
One of the key features of the outburst flood hypothesis of Alley et al. [2003a] is the presence of an overdeepening with closed bathymetric contours, which allows for the trapping and storage of subglacial water. Such overdeepened basins are in evidence today as fjords and sounds as well as still-occupied glacial valleys and basins, and are characterized by long, deep channels terminated by shallow sills.
Clearly, an additional ingredient for creating large outburst floods is the ability to provide a seal for the storage and pressurization of subglacial water. One way to seal the subglacial water system is to have the ice freeze onto the subglacial substrate around the reservoir. A second mechanism is the creation of a reversal in hydrologic gradient by reversing the ice-air surface slope over the sill. Alley et al. [2003a] suggested that the localized basal friction experienced as the ice grounds and flows over the sill should produce such surface-slope reversals. Modeling such slope reversals requires the inclusion of longitudinal deviatoric stresses in order to communicate the localized momentum loss through basal drag to the upglacier ice.
1.3 Summary and preview
The consideration of ice-shelf buttressing and surface-slope reversal requires proper treatment of longitudinal-deviatoric stress. The work presented here involves the development and application of a model that includes this stress and can capture the resulting dynamic consequences. Ultimately, the utility of the model lies in its efficiency, wrought through its conceptual simplicity, which allows for a relatively inexpensive examination of the importance of the various parameters inherent to the ice shelves and ice streams.