Antarctic ice sheet mass loss estimates using Modified Antarctic Mapping Mission surface flow observations



[1] The long residence time of ice and the relatively gentle slopes of the Antarctica Ice Sheet make basal sliding a unique positive feedback mechanism in enhancing ice discharge along preferred routes. The highly organized ice stream channels extending to the interior from the lower reach of the outlets are a manifestation of the role of basal granular material in enhancing the ice flow. In this study, constraining the model-simulated year 2000 ice flow fields with surface velocities obtained from InSAR measurements permits retrieval of the basal sliding parameters. Forward integrations of the ice model driven by atmospheric and oceanic parameters from coupled general circulation models under different emission scenarios provide a range of estimates of total ice mass loss during the 21st century. The total mass loss rate has a small intermodel and interscenario spread, rising from approximately −160 km3/yr at present to approximately −220 km3/yr by 2100. The accelerated mass loss rate of the Antarctica Ice Sheet in a warming climate is due primarily to a dynamic response in the form of an increase in ice flow speed. Ice shelves contribute to this feedback through a reduced buttressing effect due to more frequent systematic, tabular calving events. For example, by 2100 the Ross Ice Shelf is projected to shed ~40 km3 during each systematic tabular calving. After the frontal section's attrition, the remaining shelf will rebound. Consequently, the submerged cross-sectional area will reduce, as will the buttressing stress. Longitudinal differential warming of ocean temperature contributes to tabular calving. Because of the prevalence of fringe ice shelves, oceanic effects likely will play a very important role in the future mass balance of the Antarctica Ice Sheet, under a possible future warming climate.

1 Introduction

[2] Currently, the Earth's climate is in an interglacial period that possibly will continue for another 50 kyr [Berger and Loutre, 2002]. The relative abundance of glaciers, compared with two of the last three interglacial periods, suggests that sea level rise (SLR) is possible from the current cryosphere (L. Thompson, personal communication, 2005). As the largest potential contributor to SLR, the total mass balance of the Antarctic Ice Sheet (AIS, Figure 1) is critical for an understanding of the global hydrological cycle and its fragile polar ecosystem consequences. The AIS, especially the West Antarctica Ice Sheet (WAIS), has been studied extensively [Mercer, 1978; Oppenheimer, 1998; Vaughan, 2006; Van den Broeke et al., 2006; Rignot et al., 2008; Bamber et al., 2009; Joughin and Alley, 2011]. Much of the grounded ice in West Antarctica lies on a bed descending inland and extending well below sea level (Figures 1b and 1c). This bathymetry makes the ice sheet subject to marine-ice sheet runaway instability [Joughin and Alley, 2011; Schoof, 2007]. When the ice warms as a consequence of climate change, the primary stabilizing factor of the WAIS is buttressing ice shelves [Thomas, 1973]. Because a significant portion of the WAIS inland ice has a weak bed (partially a consequence of basal melting, which produces higher pore pressure), the gravitational driving stresses are not locally balanced. Ice shelves have very flat (upper/sub-aerial) surface elevations and require little resistive stress to maintain balance, except near the calving front. Thus, compared with the background flow (the ice flow speeds up immediately after leaving the grounding line), there is little along-flow-direction acceleration [Martin et al., 2011]. Along the lateral direction, hydrostatic pressure from the submerged portion of ice shelves provides the primary resistive stress for neighboring coastal land ice, to balance the gravitational driving stress from uneven surface topography. Lateral resistance along the sides of embayments works only for a narrow zone less than 10 km wide (limited compared with the Ross Ice Shelf width). Resistance in the direction perpendicular to the confining coasts is also apparent, as ice flow decreases and reaches local minima at ice rises. Warming from below the marine-based ice sheet and ice shelves could release this potentially fragile stability and lead to an accelerated grounding line retreat and enhanced sliding of the WAIS. Warming factors include sudden increases of geothermal activity from very large, sustained volcanic eruptions. Although they have low probabilities, they are not disregarded because of their potentially high impact over a short time period [Blankenship et al., 1993]. More important is the gradual but widespread oceanic and atmospheric warming driven by anthropogenic greenhouse effects, anticipated to be salient during the 21st century [IPCC AR4, 2007].

Figure 1.

(a) The Antarctica land-ice-ocean mask based on SeaRISE 5 km resolution digital elevation, ice thickness, and bedrock elevation data. In the color shading, white is ice, yellow (brown) is bare ground (L), and blue is ocean. The ice shelves are cross-hatched areas; land ice with base under sea level (marine based) is hatched. West Antarctica has more complex ice-water-bedrock configurations than the rest of Antarctica. WAIS is defined as the ice sector confined by the Transantarctica Mountains and 40°W longitude. The Peninsula has a limited ice volume (<3.3 × 104 Gt) compared with land-based ice of WAIS (~2.8 × 106 Gt). (b) and (c) The West Antarctica land-ice-ocean mask. In the color shading, white is ice, yellow (brown) is bedrock, and blue is ocean. The left panel is a cross-section along the F-R shelf/Amundsen direction, as indicated in the inset (red dashed line). The right panel is along the Siple coast direction, as the red line in the inset. In a future warming climate, ocean waters likely are entering the WAIS through the Siple coast pathway. The extensive troughs (if ice is removed) can extend to depths of more than 2 km. Color shading in the insets is surface elevation over the AIS. For reference of the following discussion, “W” is Mount Waesche, “AP” is Antarctic Peninsula, and “L-A” is the Lambert Glacier-Amery Ice Shelf system. Some ice shelves, glaciers, and seas also are labeled: Wilkins ice shelf (WIS), Bindschadler ice stream (BIN), Ross ice shelf (RIS), Filchner-Ronne ice shelf (FRIS), Amery ice shelf (AmIS), West ice shelf (WeIS) and Shackleton ice shelf (ShIS).

[3] Ice velocity is fundamental to measuring the ice transport from the inte is fundamental to measuring the ice transport from the interior toward the oceans, and the ice flow patterns indicate the locations of the preferred channels of ice transport. In addition to surficial/basal mass balance, ice velocity divergence and convergence indicate how the ice mass evolves with time. This study investigates ice volume change using surface velocity measurements from the radar interferometry or seven patches of the Canadian Space Agency's Radarsat-1 SAR sensor data from fall 2000 via the Modified Antarctic Mapping Mission (MAMM, Jezek [2003]; Compared with the very recently available MEaSURES [, Joughin et al., 2011], MAMM has the advantage of control points distributed about the entire continent, as well as having a single measurement campaign from which to compile the data. This minimized the uncertainty involved in cross-comparisons and integration over an extended time period. Although glacier flows were mapped independent of weather conditions, such as cloud cover, MAMM does not provide complete topographic coverage near the South Pole and, more importantly, vertical velocities are not yet suitable for estimating surface elevation changes (detailed below). The study is based on a thermo-mechanically coupled ice evolution scheme designed and implemented as one component of a scalable and extensible geofluid modeling system referred to as SEGMENT-Ice [Ren et al., 2011a]. SEGMENT-Ice provides prognostic fields of the driving and resistive forces and describes the flow fields and the dynamic evolution of thickness profiles of the medium. The inner ice domain follows ice rheology, whereas a granular layer is permitted between the ice and bedrock. Ren et al., 2011b and Ren and Leslie [2011] provided the granular law formulation, justification of the granular layer, discussion of consistent choices of ice constitutive law and Weertman basal sliding [Weertman, 1974], new parameterizations of surface melting and runoff, and ocean-ice interaction at the calving front. In SEGMENT-Ice, the dependence of dry granular viscosity on clasts size, shape, bulk density, dry repose angle, and effective confining pressure is parameterized according to Jop et al. [2006]. Because the till underlying the ice is saturated, the soil moisture effects on granular viscosity follow the approach of Ren et al. [2008]. SEGMENT-Ice has a vertical sublayer division within the sliding medium. Not only the viscosity itself but the granular layer (a term more generic than the till layer in Alley et al. [1987a, 1987b]) thickness also plays a critical role in determining overlying ice flow speeds. Retrieval of basal granular properties and layer thickness is feasible because of the tight relationship between flow fields and basal conditions.

[4] The tributary feature of ice sheet flow is a manifestation of the importance the basal processes. This study retrieves basal granular thickness and the mechanical properties of the granular material. The vast area of the AIS prevents a detailed survey of its basal conditions. The proposed data assimilation scheme, which retrieves mechanical properties and thicknesses of basal granular material, is original and possibly has wider application in the ice research community. A physically sound forward model is critical for a successful retrieval. In this sense, the state-of-the-art tabular calving scheme proposed here contributes to the successful retrievals of the important basal parameters and to reliable projections of the AIS's sensitivity to the transient climate warming in the 21st century.

[5] Section 2 describes the ice model, especially the tabular calving process, and the data sources. Section 3 presents the verification of the SEGMENT-Ice present-day simulations and the mass loss projections for the 21st century. Section 4 draws conclusions based on the results of section 3 and indicates the primary role of granular basal sliding processes. Atmospheric warming and the associated increase in precipitation all contribute to future mass loss of the AIS, in addition to increased oceanic erosion of the grounding line and ice shelves.

2 Methodology

2.1 The Tabular Calving Scheme

[6] An ice sheet is composed of fast-moving, channelized ice streams that drain thick, slower-moving inland ice. Due to the present relatively cold ocean temperatures around the AIS, inland ice streams discharge into the ice shelves, forming the floating extensions. Not activated in previous applications is the ice shelf mechanics component of SEGMENT-Ice. The ice shelf parameterization in SEGMENT-Ice is derived from a physical understanding of an ice shelf's life cycle, which is an advancing-thinning-attrition cycle (Figure 2). Ice is brittle at higher strain rates, especially under tension, with a melting point diffusivity ~10−15 m2/s, much lower than that for elemental metals. For inland ice, crevasses/cracks collocate with locations of concentrated strain rates, for example, after ice flows from relatively gentle bedrock into a much deeper bedrock slope (Figure 2), which is the case for the Pine Island Glacier of the WAIS. At the flanks of ice stream, crevasses caused by transverse strain also are prevalent. When ice crosses the grounding line and floats, its contribution to sea level change is negligible. Nevertheless, the ice shelves are an integral component of the ice sheet's thermo-dynamic system. For example, the discharging rate of inland ice is influenced by the buttressing restraint provided by ice shelves [Thomas and MacAyeal, 1982; Dupont and Alley, 2005].

Figure 2.

Shelf ice calving in SEGMENT-Ice. (a) The upper panels are schematic diagrams of ice profiles, showing the different flow regimes. Note that, due to granular material, the basal velocity is not exactly zero. The acceleration of the ice (to the right) causes the ice to be torn. Inside the ice shelf, the spreading tendency is restrained mainly by longitudinal stretching (along-flow stress). Ice shelves thin by creep thinning. The negative vertical strain rate (compression) causes a horizontal divergent (positive) strain rate. The ice shelf's velocity increases toward the calving front as determined by the spatial integral of the horizontal strain rate. In the diagram, white bulk arrows are stress (hydrostatic pressure) on the right side of the calving front exerted by the ocean, decreasing to zero at sea level. The red bold arrows are static stresses exerted on the left side of the front, decreasing linearly to zero at ice upper/sub-aerial surface. At the shelf bottom, hydrostatic stresses from both sides are almost equal. The red curve is the net horizontal stress, which reaches maximum at sea level. Hence, the vertical variation of horizontal shear and the vertical profile of the velocity have a turning point at sea level. The diagram of the ice shelf is partially adapted from T. Hughes via R. Bindschadler (personal communication, 2011). The vertical profile of horizontal ice velocity field determined that there will be a “mushroom”-shaped spread section that are not in hydrostatic balance with the ocean water, although the bulk of the shelf section is. There is a limit to the length of this section before it breaks off from the main body of the shelf (bk in the diagram). (b) A cross-section of the Amery ice shelf (5 km DEMs are used in the plotting, as indicated in the inset white dashed line). (c) A further zoom-in of the red line confined region in Figure 2b with 1 km resolution ice thickness and surface elevation maps to illustrate the forward slanting of the calving front. Color shading in the insets are ice thicknesses.

[7] Ice shelves are critical components of the AIS. Following is a flow-dependent ice shelf calving scheme that is unique to SEGMENT-Ice. Figure 2 illustrates the calving physics implemented in SEGMENT-Ice. Although the shelf as a whole is in near hydrostatic balance with the ocean waters, the flow structure inside the ice shelf determines that it is a dynamic scenario of advancing-thinning-breaking, from groundling line toward the calving front (Figure 2b). Figure 2a is a conceptualization of Amery Ice Shelf (Figure 2b). Compared with land ice, where shear resistance counters most of the surface elevation caused by gravitational driving, ice shelves thin by creep thinning. Except at the very bottom, the cryostatic pressure inside the ice is always higher than the hydrostatic pressure at the same level from the ocean water. The differences (net horizontal stress) reach a maximum at sea level (the thin red curve in Figure 2a). Hence, the vertical variation of horizontal shear and the vertical profile of the velocity have a turning point at sea level. The negative vertical strain rate (compression) causes a horizontal divergent (positive) strain rate. Inside the ice shelf, the spreading tendency is restrained mainly by along-flow stress. The ice shelf's velocity increases toward the calving front as determined by the spatial integral of the horizontal strain rate. Ice shelves spread under their own weight, and the imbalance pushes most at the calving front and around the grounding line. According to Reeh [1968], as a consequence of this horizontal compression, the geometry is that of an anvil-shaped outreach at the calving front [Thomas, 1973; Hughes, 1992, 2002]. The forward slanting of the Amery Ice Shelf at the calving front (Figure 2c) is a manifestation of the vertical shear in ice flow [Sanderson and Doake, 1979].

[8] As shown in Figure 2, although the bulk of the shelf is in hydrostatic balance with the ocean waters, the “mushroom”-shaped, girder-like spread section is usually not. The limiting length of this portion, before it breaks off from the main body of the shelf (bk in the diagram), is limited by the tensile strength of ice (~2 Mpa at present for marginal regions of the Ross Ice Shelf), the ice thickness (H), and the ice creeping speed and vertical shear, ΔU. In general, the tensile strength of ice can vary over a wide range depending on temperature, strain rate, and grain size (as natural ice is polycrystalline, e.g., Currier and Schulson [1982]). This length is regularly approached and causes the systematic calving of an ice shelf. There are random components in the calving processes, such as hydro-fracturing [Scambos et al., 2000; Doake and Vaughan, 1991], which produce ice shelf lobes. Based on elastic mechanics, we propose the following:

display math(1)

[9] where ρw = 1028 kg m− 3 is the density of sea water, ρi is the density of ice, g is gravitational acceleration, CT is a factor taking tidal and sea wind swelling into consideration, and DT is a dimensionless factor measuring the ratio of ice flow shear to surface ice velocity. DT is a function of ice temperature. The density of ice varies with loading pressure, according to Thomas [1973]. Ice density is sensitive to overloading because the ice has air bubbles encapsulated during the transition from snow to firn and further into glacial ice. In the Appendix, a more rigorous derivation is provided of equation (1), based on a cantilever beam approximation.

2.2 Grounding Line Dynamics

[10] In SEGMENT-Ice, ocean-ice interactions are parameterized so freezing point depression by soluble substances, salinity dependence of ocean water thermal properties, and ocean current-dependent sensible heat fluxes are included [Ren and Leslie, 2011]. SEGMENT-Ice has a chemical potential submodel to estimate the effects of ocean water salinity changes on the grounding line retreat of water terminating glaciers and the erosion of ice shelves. SEGMENT-Ice follows a molar Gibbs free energy bundle in considering phase changes. Melting/refreezing is determined by the chemical potential of H2O in both states, across the interface. SEGMENT-Ice estimates ice temperature variations and calculates the fraction of melted ice. When the terminal heat source becomes a heat sink, freezing occurs, and ice can extend beyond the initial interface, simulating the advance/retreat of the ice shelf grounding line. SEGMENT-Ice is designed such that when the newly formed ice is less than the dimension of the grid mess, it records the fraction, which melts first when heat flux reverses. If the newly formed ice fills an entire grid, SEGMENT-Ice adjusts its “phase-mask” array to indicate the new water/ice interface. The reverse (melting) process is analogous.

2.3 Data

[11] SEGMENT-Ice uses a terrain following coordinate system, the sigma coordinate system, σ (grid lines in Figure 1b), defined as σ = (h − r)/H, where h is the distance from the ice surface to the Earth's center, H is the local ice thickness, and r is the independent variable in the radial direction in the spherical coordinate system. A vertical integration of the incompressible continuity equation, with surface mass balance rate and basal melt rate as boundary conditions, gives the following:

display math(2)

where t is time, R is the Earth's radius, θ is longitude, φ is latitude, u and v are the horizontal velocity components, and w is the vertical velocity component and is expanded using the continuity equation, assuming incompressible ice. The subscripts “b” and “s” mean respectively evaluated at the bottom and upper ice surfaces. Equation (2) diagnoses the temporal evolution of the surface elevation and also is the ice thickness because bedrock is assumed unchanged over the time scale of several hundred years. The surface elevation varies as a function of velocity fields and boundary sources. The surface mass balance rate, b, also includes basal melt rate. Over the AIS, it primarily is the net snowfall (precipitation minus sublimation and wind redistribution) minus basal melting, in ice thickness equivalent. The change in the ice thickness multiplied by the grid area gives the volume ice loss for that grid. The total ice loss is calculated by summation over the entire simulation domain.

[12] In equation (2), the first term on the right-hand side is the vertical velocity (w) evaluated at the upper ice surface. In MAMM measurements, vertical velocities are based on resolving the along- and across-track velocities (the u and v components before they are projected onto the South Polar stereographic map, with a 70°S secant plane) with a slope derived from the surface elevation (K. Jezek, personal communication, 2011). That is, math formula. If the three velocity components from MAMM are used in equation (2), the dynamic component of total ice mass balance would always be exactly zero. Therefore, dynamic elevation changes cannot be estimated directly from vertical velocities provided by MAMM. The vertical velocity components also are not useful in calibrating ice dynamic models because of the way vertical velocity is estimated in MAMM measurements, as it assumes an ice sheet in mass equilibrium and ice flows perpendicular to material surface contours. The first assumption clearly is incorrect because of significant climate warming since the 1970s [Hansen et al., 2006]. The second assumption is a simplification of the gravitational driving stress equation of motion. In reality, ice sheet flow is 3-D and, for the AIS, has sophisticated basal conditions, including the incorporation of flow effects of basal lakes and sedimentation.

[13] Here, the available horizontal velocities from MAMM improve the modeled 3-D ice flow fields through improved parameter settings for sensitive physical parameters in SEGMENT-Ice. With improved model parameters, SEGMENT-Ice is driven by monthly meteorological parameters obtained from coupled general circulation models (CGCMs) to make projections of future total mass loss of the AIS.

[14] A high-resolution digital elevation map (DEM) and ice thickness data are key inputs for SEGMENT-Ice in investigating gravitational driven ice creeping. The SeaRISE project ( provides surface DEM at 5 km horizontal resolution on a South Polar Stereographic projection. Surface topography and bedrock topography were obtained from: The surface DEM is used to calculate strain, stress, and surface slope. The latter in turn is used to estimate the meltwater redistribution. However, surface melt is very limited over the AIS. For places with basal melt (usually coexisting with basal granular material), the basal slope is used to compute meltwater redistribution. Geothermal heat flux, because it operates steadily for extended periods, is an important control on the ice sheet temperature profile. The geothermal heat flux is obtained from the SeaRISE website and originally from Shapiro and Ritzwoller [2004]. In addition to this static data set, monthly atmospheric and oceanic parameters also are inputs.

[15] This study investigates both atmospheric and oceanic driving of the AIS mass changes, especially those from ice shelves. The key controls on melting, such as the volume and properties of the subtropical water intrusion, the pattern of along-shore winds, and the associated changes in weather patterns, likely are intertwined. Therefore, atmospheric and oceanic parameters projected by coupled climate models are necessary drivers of SEGMENT-Ice. Refining the horizontal model resolution improves regional simulations of precipitation [Genthon, 1994; Ohmura et al., 1996]. Consequently, the climate models used are required to have a relatively fine horizontal resolution. In this study, the NCAR CCSM3 has a resolution of ~1.4° (Collins et al. [2005]), the MIROC-hires a resolution of 1.125° (K-1 Model developers [2004]), and the ECHAM5/MPI-OM has a resolution of 1.875° (Jungclaus et al. [2005]). They provide all input variables needed by SEGMENT-Ice. For ice melting, SEGMENT-Ice uses an energy balance scheme, rather than the positive degree days schemes, which are popular in paleo-climate studies. The 20th century simulations (20C3M runs) are used to spin up SEGMENT-Ice. The steady ice flow field from the spin-up also is the “initial guess” state vector in the optimization scheme that retrieves the basal sliding parameters. Estimates of the warming effects on future mass loss from the AIS employ the SRES A1B scenario, which assumes a balanced energy source in a future world of rapid economic growth, reflecting recent trends in the driving forces of emissions. In addition to atmospheric parameters, to investigate the ice ocean interactions at the ice shelves, the ocean flow speed (math formula), potential temperature (T), salinity (S), and density (ρ) also are needed [Ren and Leslie, 2011]. The CCSM3, MIROC-hires, and ECHAM5/MPI-OM ocean model monthly output at 0, 10, 20, 30, 50, 75, 100, 500, and 1000 m depths all are interpolated to SEGMENT-Ice grids.

[16] The ice dynamics/thermodynamics model is run on 5 km resolution. The monthly atmospheric forcings are obtained from climate models and are of much coarser resolution. Fortunately, because the surface of the AIS is very flat, the coarser atmospheric forcings still are representative of the higher resolution. The surface topography is primarily determined by precipitation climatology, whereas the bottom geography is determined primarily by geology. That the surface topography of the AIS is much smoother than the bedrock topography is evidence of the uniformity of the atmospheric forcings (compared with bedrock topography).

[17] In the spin-up run of SEGMENT-Ice, the subglacial particle properties of the AIS are specified according to studies using boreholes and seismic methods [Bentley, 1991; Englehardt et al., 1990]. The rocks under the WAIS are mostly volcanic, and the basalt clasts are of sizes ~10 cm. Loose, ice-cemented volcanic debris also is widespread around Mt. Waesche (77°S, 130°W) and the northern Antarctica Peninsula and its constituent blocks. In assigning granular particle sizes, geothermal patterns also are referenced, as repeated phase changes at the interface of ice/rock arguably are the most efficient means of erosion and reducing the granular particle sizes [Anderson, 2006].

[18] In addition to mechanical properties such as particle size, porosity, bulk density, cohesion, and repose angle, the thickness of the granular material is critical in determining the magnitude of the ice flow and the erosion rate. It is assumed that clasts are generated primarily by abration of bedrock, which clearly is the case as glacial creeks always are turbid. The granular mechanical properties and present thickness are inversely retrieved using SEGMENT-Ice, constrained by the goodness of fit between model simulated and observed surface ice velocities over the entire AIS. The adjoint-based retrieval scheme is described by Ren [2004]. Ice flow is sensitive to the granular basal parameters and the granular material thickness. Based on this sensitivity relationship, repeated runs of the SEGMENT-Ice model are performed with “present-day” ice geometry and ice temperature profiles but with automatically varying granular layer thickness to best fit the observed surface ice velocities. The metric for goodness of fit between the modeled and observed velocity fields is defined as follows:

display math(3)

where u, v are horizontal components of the full vector velocity. The subscripts “obs” and “model” are respectively the observed and modeled velocities. The observed ice velocities are obtained from the Radarsat-1 SAR sensor via Modified Antarctic Mapping Mission (MAMM, Jezek [2008]). Because the seven regional composites of MAMM do not cover the entire AIS, there is a “polar hole,” so the summation in the metric is only over regions with MAMM observations.

[19] The formal expression of the retrieval scheme is presented in the Appendix (subsection 2). The form of the cost function is trivial. However, to minimize it with a numerical ice model's solution is not. Many forward and adjoint (backward) model runs are required. In each optimization iteration, updated control variables (granular material thickness and mechanical properties) through a forward run of SEGMENT-Ice produce updated surface velocities in equation (3). The value of cost function gradually is minimized. The optimization procedure is terminated if two adjacent cost-function values are close to one thousandth of their average value or a maximum iteration number is reached. Limited by the size of this paper, the technical aspects are not expanded upon here. Interested readers are referred to Ren [2004] for details of the dynamic data assimilation (parameter retrieval) procedure.

[20] Upon convergence, the overall agreement between modeled and observed velocities is high, with a correlation coefficient of 0.92 for velocity direction and 0.9 for velocity magnitude. Figure 3 is the distribution of the estimated granular material thickness over Antarctica. The initial guess sliding material thickness is 0.1 m for regions with basal sliding. The cost function decreased by 40% in this retrieval experiment. The overlain vector fields are the surface ice flow fields, at convergence of the retrieval, representative the flow field at the “present” time (i.e., year 2000 ± 5 years). The concentrated tributary patterns of granular material thickness distribution are similar to the measured surface flow field. For example, under the WAIS, the five branches of meandering ice streams over the Siple coast correspond well with granular material accumulation. Their formation may be a positive feedback between ice flow and granular material production (Ren and Leslie [2011]). Granular basal slip is a significant component of ice motion [Boulton and Hindmarsh, 1987], especially for the WAIS [Alley et al., 1987a] and the peripheral region of the eastern AIS. Examining the spatial distribution of clast size reveals another feature that helps stabilize the WAIS. Because the bedrock slope tilts inland, the finer clasts are washed toward the inland sectors, and the coarser granular materials are left at the periphery. The slowly increasing clast size downstream, assisted by the smaller loading from the overlying ice, resulted in a decrease of granular viscosity downstream. Notice that the ice flow is driven by surface slope and flows to the periphery. This configuration is a stable configuration [Alley et al., 1987b]. Alley et al., 1987b did not incorporate the granular size effects because they used a simplified 2-D numerical model that did not include recent developments in granular rheology that have occurred since the Alley et al. [1987b] study was published. Many idealized assumptions on ice-till layer-bedrock geometry now are either unnecessary or inappropriate in their 2-D numerical model. However, the general features of flutes patterns of till layer spatial distribution remain a common feature. The mechanisms illustrated by their model also agree with SEGMENT-Ice. Although the 3-D SEGMENT-Ice model and the data assimilation scheme are computationally more demanding, a global view over the entire AIS is possible with SEGMENT-Ice.

Figure 3.

SEGMENT-Ice retrieved granular material depth under the ice (color shading, in meters), using MAMM surface velocity observations. Model simulated surface velocities (upon convergence of retrieval procedure) are overlain. The top portion of Mt. Waesche (“W”) and the ridge of Whitmoor Mountains (“WM”) have shallower debris accumulations. Note that the granular layer is thickest under the WAIS. Under the lakes of Eastern Antarctica, the granular thickness also is relatively thick.

[21] For the interior ice, because of its long existence, gravitational training of the c3 axis orientation of the unit cells also is a factor in explaining the meandering branches of the ice stream extending inland. As mentioned, the ice surface topography over inland east Antarctica is nearly flat. The dividing ridges in the surface topography are only ~100 m above surrounding areas within several hundred kilometers. This implies that the flow caused by gravitational driving is very small (<5 m/yr) and takes a long time finally to be shed into surrounding oceans. However, it provides sufficient time for the gravitational training of the originally random c3 axis orientation of the ice crystals. This effect is parameterized as a function of the surface elevation slope and can be referred to as “slope enhancement” of ice flow. In SEGMENT-Ice, it is parameterized in a manner analogous to Wang and Warner's [1999] ice aging enhancement scheme. This scheme and the granular parameterization are critical for the successful simulation of the extended channeled stream flow over the AIS.

3 Results

[22] The simulated present 3-D ice flow field, using the retrieved granular properties, current ice geometry, and current climate conditions (Figure 4a), provides more information than the InSAR measurements (Figure 4d). For example, from MAMM, the Ross Ice Shelf has a large downstream flow speed. The model simulation (Figure 4c) indicates that for most of the flat section of the ice shelf, the ice flow has little flow-direction accelerations. Only near the edge does the ice flow accelerate toward the ocean, and the unique vertical velocity profile in Figure 2 then becomes clear. Moreover, right south of the Byrd Ice Stream, there is a stagnant region with ice flow speed less than 50 m/yr. The small ice velocities for the central area primarily are a result of the meeting of ice streams from Siple coast with the Byrd Ice Stream. The small dynamic mass balance for the interior part of the ice shelf explains the fact that the ocean melting there mostly balances the precipitation. Precisely because of this dynamic effect, the ice shelf usually is called a buttressing ice shelf, which also prevents direct ocean-ice interaction for inland ice. With the massive rate of basal ablation of the ice shelf by underlain ocean water [Bindschadler, 2006], it is uncertain whether or not ice shelves can maintain a near balance under a warming climate.

Figure 4.

SEGMENT-Ice simulated present ice flow fields (meter/a). (a) Surface level. Overlain are geothermal heat fluxes (milliwatt, mW/m2). (b) MAMM measured surface velocities. (c) Vertical u-velocity component profiles at the cross mark (inland ice), circle mark (shelf ice close to transitional zone) and circle-with-vertical-bar mark (shelf ice close to ice/ocean front) of Figure 4a. At the cross marks, ice extends from 114 m elevation to 2100 m. This velocity shape is characteristic of shear-thinning fluid. Velocity profiles within the top 450 m are shown for the two locations on Ross Ice Shelf. (d) A zoomed-in view of the Ross Ice Shelf (blue rectangle in Figure 4a) surface ice speeds. There is a stagnant area on the Ross Ice Shelf (<100 m/yr) of thicker ice at the central part off the steep Trans Antarctic Mountain coast. Outside this stagnant region, the ice speeds accelerate toward the ice/ocean front and can reach 2000 m/yr at the calving front.

[23] Bedrock topography has a clear impact on the ice flow patterns. For the Thwaites and Pine Island Glaciers, simulated maximum speeds are close to observations, at ~2300 m/year for Thwaites Glacier and ~4000 m/year at the core of Pine Island Glacier. However, for the Drygalsky ice tongue, the transectional direction covers only four 5 km grids, a resolution not fine enough to represent the ice geometry. Consequently, the modeled maximum speeds are only half the observed speeds (400 m/yr versus ~790 m/yr). However, the point-to-point flow directions have an RMS error less than 0.7 degrees. The model simulated surface flow field at the Lambert Glacier-Amery Ice Shelf system was compared with InSAR mosaic composites from Yu et al. [2010]. Agreements are close for both flow speed and direction (correlations are both ~0.9). Note that flow features and ice geometry features on sub-km scales are likely variable. For example, due to its northern location, the Antarctic Peninsula (AP) around Deception Island experiences significant summer melting. SEGMENT-Landslide indicates that this area is prone to avalanches. Smaller (200 m) scale flow features from InSAR measurements possibly are caused by random variability in atmospheric forcing, so only flow features at >5 km are climatologically informative.

[24] From the Gravity Recovery and Climate Experiment (GRACE) measurements, the mass loss rate for land-based ice in the Antarctica is about 193 km3/year during 2003–2009 (Figure 5b). To examine the credibility of SEGMENT-Ice mass loss rate estimates, the National Centers for Environmental Prediction/National Center for Atmospheric Research (NCEP/NCAR) reanalysis data were merged with atmospheric forcing provided by CGCMs from 1948 to 2009. Coupled ocean-atmospheric climate models have difficulty in reproducing the phases of observed interannual and decadal climate variations. Thus, they cannot be used as climate forcing for SEGMENT-Ice model validation against observations on interannual to decadal scales. More realistic climate forcing is provided by the NCEP/NCAR reanalysis [Kalnay et al., 1996]. The reanalyses are used by climate researchers as surrogates for real observations on large spatial scales.

Figure 5.

(a) SEGMENT-Ice simulated geographic distributions of rates of surface elevation changes over Antarctica between 2003 and 2009 (m/yr equivalent water thickness change). (b) GRACE ice mass change rate over the same period. The post-glacial rebound (PGR) is removed using the method of Ivins and James [2005]. A decorrelation filter and a 300 km Gaussian smoothing have been applied to the raw data. Note that model (in Figure 5a) gives more details than observations (in Figure 5b). West Antarctica facing Amundsen Sea has systematic mass loss (>0.6 m/yr in the model, and the smoothed GRACE observations show >0.1 m/yr reduction in surface elevation) during the 5 year period. As a whole, the WAIS is losing mass, but the ridges are gaining mass slightly during the 5 year observation period, a feature captured by the model but is not well differentiated by the coarse resolution of GRACE.

[25] Figure 5a shows the surface elevation changes of the AIS between 2003 and 2009. Currently, the mass loss in Antarctica is dominated by ice flow acceleration in parts of West Antarctica. In this respect, SEGMENT-Ice simulated mass loss rates are close to GRACE measurements. SEGMENT-Ice also reveals details that GRACE, limited by its horizontal resolution, cannot identify. For example, the model simulations show clearly that it is the peripheral sectors of WAIS that have lost mass most significantly due to ice dynamics. In addition to changes in surface mass balance, these sectors also have suffered enhanced submarine melting and a consequent grounding line retreat and acceleration. In contrast, the Whitmoor Mountains gained mass because of increased snow precipitation. Shelf dynamics likely play a role because the peripheral regions that show significant (>10 cm/yr) mass losses are mostly near major ice shelves: for example, the Amery and West Ice shelves and the fringing shelves around Dronning Maud Land in East Antarctica. Examining the present ice-water geometry near the calving front of the Ross Ice Shelf indicates that, for the next systematic tabular calving to occur, the ice surface elevation needs to be lowered by ~20 m. This indicates that the backward stress it provides to the inland ice is lower than its “climatological” value. Without considering ocean and atmospheric warming, buttressing gradually will be restored in the next two decades. Despite the very different resolutions (the model has a 5 km resolution), the overall mass loss rate of ~180 km3/yr is close to GRACE observations.

[26] From equation (2), the total ice mass loss is composed of that from ice flow divergence (dynamic mass loss) and from boundary mass input/loss (e.g., surface mass balance). During the reference period (1900–1920) for ice mass loss from climate warming, it is assumed that dynamic mass loss almost equals surface mass balance, with the slight difference being the interglacial residual trend. Thus, the following best represents the climate warming effect on the AIS total mass balance. Surface elevation changes of the AIS between 2000 and 2060 are examined further under the CCSM3 A1B scenario. In general, the peripheral areas are losing ice from climate warming. For the Antarctica Peninsula, a −1 m/yr lowering of the surface elevation is sustainable (Figure 6). This area loses more mass primarily because of higher surface air temperatures and summer season surface melt. Precipitation over this area also is highest, but the increase of precipitation does not compensate for the mass loss due to the extra melting. Because of the higher temperatures, the ice flow speeds also are higher. This is the sector of most rapid change, at present, over the widest area and with the greatest impact on total ice mass loss. Some interior regions have significant surface elevation increases, such as 0.2 m/yr upstream of Ross Ice Shelf. Many small-scale elevation increase/decrease pairs located in the interior of the ice sheet may not be persistent as they may change sign. They likely are due to the random fluctuations in snowfall. There are signals that are caused by dynamic flow response to a warming climate. For example, near the Foundation ice stream, there is significant mass gain in the 21st century related to the convergence of ice flow. The meandering banded patterns of mass loss, which are more significant near the coast but reach several hundred kilometers inland, also are persistent features of the dynamic response of ice flow to a warming climate.

Figure 6.

SEGMENT-Ice simulated geographic distributions of rates of surface elevation changes between 2000 and 2060 (m/yr), with atmospheric parameters provided by CCSM3 A1B scenario run. Peripheral areas and AP lose mass more significantly. Upper streams of the Foundation Glacier gain mass because of increased snow precipitation. The inland alternative, tributary patterns of mass loss and gain arise from dynamic response to climate warming (corresponding to the ice flow convergence and divergence patterns).

[27] The marine-based ice sector of West Antarctica, confined by Marie Byrd Land, Siple coast and the Whitmoor Mountains, discharge ice primarily to the Ross Ice Shelf. As the climate warms, increased snowfall partially compensates the effect of flow divergence for the marine-based ice. The increases in ocean temperature are ~0.2 K for the surrounding oceans (figure not shown), and the increased erosion of the ice shelves from oceans is ~1 m/yr. The elevation increases caused by net snow accumulation are less than 0.1 m/yr. Horizontal spreading within an ice shelf is laterally accelerating. This would be a thinning effect for local ice thickness if the shelf has uniform thickness. Interestingly, the thickness profile already has adjusted to this flow pattern and is thinner oceanward. Thus, the ice shelf resembles a stream-tube. The thickness change of ice from convergence/divergence is minimal. Consequently, the restoring of the buttressing stress is slowed by oceanic erosion of the shelf thickness, and the discharging of marine-based ice is enhanced. The marine-based (actually land ice with base elevations below sea level) sector of ice mass loss also has a SLR contribution. Here we discuss another mechanism from oceanic warming that causes reduction in the buttressing effect. Due to the small vertical shear in the flow profile, the calving front is slanted forward (the semimushroom-shaped structure, right-tip end of Figure 2c). This means that the oceanward section is not fully supported by the buoyancy, and extra weight is put on the section toward the grounding line, causing this section to be submerged more than is necessary to support its own weight. This trend continues until a critical point is reached and a tabular calving event occurs. At this point, the calved iceberg lowers the weight center, and the remaining ice shelf rebounds upward (less submerged in the water). With this cycle, the buttressing effect fluctuates. Ocean water has a poleward temperature gradient. For the fringing ice shelves of Antarctica, the ocean temperature at the calving front is warmer than around the grounding line. If this gradient is larger than 0.0007 ∇ H°C/m, where H is the shelf ice thickness, ocean melt also contributes to tabular calving and to a reduced buttressing effect for the inland ice.

[28] The above results are only for CCSM3 under the SRES A1B scenario. Atmospheric and oceanic forcing parameters were used from two other CGCMs: MIROC-hires and ECHAM/MPI-AM. These are independently developed climate models. With 1948–2009 replaced with reanalysis atmospheric forcing, total mass loss rates are different between the CGCMs (Figure 7). Figure 7 shows the mass change compared with the control period (1900 ± 10 years). Although the atmospheric forcing during 1948–2009 reanalysis period are identical, different CGCMs have their respective control period. This explains why there are large spreads in the curves. Only the CCSM mass loss rates are close to GRACE measurements. The lower MIROC-hires loss rate is due to its warm global air temperature bias of about 2° above observations. Thus, reanalysed temperature field, compared to the MIROC-hires 1900–1920 reference period, is too low, and there is less ice melting around the AP, resulting in a slower mass decrease than reality. Systematic biases in ECHAM/MPI-AM temperature and precipitation are lower. Outside the period 1948–2009, mass loss rates are similar among the CGCMs. Using the 1900–1920 reference periods of each climate model, there are large intermodel and interscenario similarities in mass loss rates, with a magnitude of spread less than 10% of the absolute decrease, compared with the 1900 level, up to year 2100. Compared with other mountainous regions, the AIS surface is flat, with the steepest slope less than 14° in the 5 km SeaRISE topography data set. This poses less of a challenge for atmospheric components of the CGCMs, and all three models have similar weather pattern evolutions over the entire AIS. Region-specific biases thus are not of great concern. The high intermodel and interscenario consistency adds confidence in the quantification of future sea level change contributions from the AIS.

Figure 7.

Total ice volume changes obtained by forcing SEGMENT-Ice with atmospheric parameters from three CGCMs, under the IPCC A1B, B1, and A2 scenarios. The 1948–2008 period uses NCEP/NCAR reanalysis data (identical across models and scenarios). Clearly, the intermodel differences are larger than the interscenario differences. However, the rates of decrease are similar between the models.

[29] As ice temperature increases, the viscosity decreases, and ice flow increases. Ice flow divergences also increase, resulting in a more significant mass loss. The sensitivity of the AIS ice flow to a warming climate likely results from three positive feedback mechanisms described by Ren and Leslie [2011], most notably the positive feedback between granular basal slip and the ice flow. The coastal sectors are increasingly coupled with the interior regions along the preferred channels of ice streams. These ice streams form because of the slow turnover time of the ice that changed its rheological property under gravity on slopped surfaces. The ice along steeper slopes creeps progressively faster, forming an ice stream. Downstream toward the ocean, there is granular material formation and accumulation. A warming climate enhances ice deformation, and positive feedbacks are triggered among ice flow, granular material accumulation, and reduced resistance to further deformation. These processes provide an explanation for the accelerated rate of mass loss in the 21st century. At the flanks of ice streams, large horizontal transverse stress creates crevasses. These crevasses, when advected downstream to ice shelves, act as seeding cracks that enhance tabular calving.

[30] Figure 8 is the SEGMENT-Ice simulated calving speed of ice shelves and the possible systematic cracks in inland ice around the year 2050, with atmospheric/oceanic forcing parameters from CCSM3 A1B scenario run. In general, large ice shelves (the Filchner-Ronne and Ross Ice Shelves) and ice shelves at cold locations (Dronning Maud Land, Amery Ice Shelf) calve slowly, for example, at ~10–25 years on average over the Ross Ice Shelf. Shallower (thinner) ice shelves in relatively warmer environments calve faster, but the resulting icebergs are smaller (e.g., the fringe ice shelves along the Amundsen Sea coast). The model output is the average width of the ice calved by a systematic tabular type of calving. For the Ross Ice Shelf, the average width of such a calving event is ~10–22 km. Each systematic calving will discharge ~40 km3 ice to the ocean. In contrast, smaller ice shelves along the coast of Amundsen Sea sector of the WAIS calve more frequently but are of smaller size. Limited by the 5 km DEM data resolution, the peninsular region is not well represented. The vertical temperature profile within ice shelves is almost linear, rising downward from the surface air temperature to the pressure melting temperature at the bottom (the ice/ocean interface). Increased air and ocean temperatures both increase the vertical shear (ΔU in Figure 2). This effect reduces the calving period by about 5 years by the year 2050, compared with the present calving period for the Ross Ice Shelf. Before each tabular calving, the sprout girder is not in hydrostatic balance with surrounding waters, which means that the connected “main body” is submerged more than required to support its own weight. After systematic attrition, the main body of a shelf will rebound, thereby reducing the buttressing effect. More frequent tabular calving in a warming future climate also is a positive feedback that contributes to an accelerated inland ice discharge rate.

Figure 8.

(a) SEGMENT-Ice simulated calving iceberg width, in meters and (b) calving time scale (rapidity or frequency, days) around 2050, with atmospheric forcing parameters from CCSM A1B scenario run. The inland ice areas that may crack also are labeled in Figure 8b. In Figure 8b, for clarity, places with calving periods longer than 50,000 days are highlighted (small squares). The largest tabular icebergs are from the Ross Ice Shelf and F-R Ice Shelf. The Amery Ice Shelf and Dronning Maud Land coast calve less frequently because of ice creeping slower than in other peripheral areas. The AP and WAIS facing Amundsen Sea sectors calve more frequently, but the iceberg sizes are smaller than those from the Ross Ice Shelf and those from the Amery glaciers.

[31] As the climate warms, surface ice flows faster. The strain rate gradient is enhanced, and new cracks will be produced. Figure 8b shows regions that SEGMENT-Ice indicates are prone to tensile cracking. Notably, the Pine Island Glacier at (100°W and 75°S) is prone to cracking for the inland section because surface slopes are steep and the above-mentioned positive mechanism operates long enough such that the ice flow is concentrated and fast (~3 km/yr near the central line). The northern slopes of Mt. Sidley/Mt. Waesche also are prone to cracking. In addition to the proposed mechanisms [Jacobs et al., 1992; Nicholls et al., 2009; Jenkins and Doake, 1991; Horgan et al., 2011; Schoof, 2007], warming caused by enhanced snow precipitation contributes to the cracking of these glaciers. The large long wave radiation loss of the ice surface over Antarctica is compensated for primarily by sensible heat flux from the atmosphere. From Figure 9, the atmospheric temperature throughout the snow-producing layer (which is up to 600 mb during the Austral summer) has temperatures higher than the surface ice temperatures. This is true all year for Antarctica in the present climate. Snow precipitation occurs when warm air masses intrude. Snow precipitation thus is a heating process for local ice surfaces. According to the CCSM A1B scenario, heating caused by increased snow precipitation is a heating source comparable with atmospheric warming and oceanic interaction. However, for inland ice over the AP, it accounts for over 5% of the additional heating, from the increased precipitation over the region. Atmospheric warming affects ice flow through a mechanism illustrated in Ren et al. [2011b], in which the effects of increased surface flow are amplified by frictional heating and propagate to greater depths than possible simply by diffusion.

Figure 9.

Precipitation is a heating source for the AIS ice surface. Daily averaged vertical profiles of air temperature (black line) and specific humidity (red dashed line) are shown for two locations in Antarctica (filled marks). Skin temperature (black dashed lines) and precipitation rate (red dashed horizontal line) are labeled. Precipitation is the right axis (shared with specific humidity) in mm/day. Insets in Figures 9a and 9b show the locations of interest using a filled mark. (a, c, and e) For a location on the Siple coast. (b, d, and f) For a location near the Amery Ice Shelf. Days of year are labeled. Snow is generated and falls through a layer of air with temperature greater than the skin temperature (which holds for all Antarctica and all seasons). This is not surprising because Antarctica is a cold source for the atmosphere. Precipitation occurs as warm incursions. It is a link in the chain of the energy transport cycle in which transient eddies transport energy from the midlatitudes to the polar regions, to balance the radiative net energy loss (primarily a long wave radiation loss of energy into space). As snowflakes fall through a layer of air warmer than the surface air temperature, snowfall is a heat gain for the ice surface. NCEP/NCAR Reanalyses atmospheric fields.

4 Conclusions

[32] The AIS ice flow field consists of slow moving sheet flow, channelized stream flow, and free creeping of the fringe ice shelves (Figure 2). Although the physics of ice flow and transitions between these three flow regimes are treated in detail in SEGMENT-Ice, many basal parameters are still uncertain. Over the vast AIS, there currently are very few observational resources (for example, except for sporadic boreholes, little is known of the basal conditions), and relatively little research effort has been dedicated to investigating the rheological properties of natural Antarctic ice and basal sediments despite their demonstrated relevance to ice flow concentration. Here, an inverse method is proposed to retrieve highly sensitive but uncertain basal granular material properties, using remotely sensed surface flow fields to constrain the ice dynamics model. The retrieval method is shown to perform satisfactorily, and this study has led to the following major conclusions.

[33] For inland ice, ice surface topography is primarily a result of persistent snow precipitation and surface ablation patterns, with bottom topography playing a very limited role. When a surface is uneven, there is gravity-driven creeping. If the ice flow cannot find a path downstream to the ocean, there will be ice accumulation downstream. As convergence continues, the original slope angle will get progressively smaller, and eventually the ice flow will cease. Only those streams that can be directed to the ocean can further develop into larger ice streams. Ice streams, like valley troughs, develop from advection and diffusion [Dietrich and Perron, 2004]. Reduced ice viscosity enhances these processes. Granular material produced by bedrock is not necessarily present for frozen ice in contact with bedrock. If liquid water is present, such as in areas with strong geothermal activity, basal ice temperatures approach the pressure melting point, or if there are crevasses channeling surface meltwater to the bedrock, the rate of granular material production increases dramatically. Saturated granular material has a viscosity 6–7 orders of magnitude smaller than frozen ice and thus enhances the flow of the overlain ice. A strong subglacial hydrological system can very effectively transfer the produced granular material downstream. With liquid water washing away the granular material, further erosion can continue and steepen both the bedrock slope and the ice surface slope through hydrostatic adjustment. Steepened surfaces enhance ice flow and produce more severe erosion, forming a positive feedback loop.

[34] Finding a pathway to the ocean determines the longevity and scale of an ice stream. It explains ice stream branching, with smaller tributaries in the interior of the AIS merging with larger glacier ice streams as they approach the coast. Lateral diffusion is decisive in forming the graded structure. For example, the tributaries of Lambert Glacier extend inland through higher ground with higher basal friction. Some tributaries emerge as narrow, fast streams at the flanks of the Gamburtsev Mounts, as shown in the elevation change map (Figure 5a). The narrow mountain flank streams merge with the slow, meandering wider tributary indicating mass accumulation (red color, Figure 5a) and are a dynamic response to a warming climate.

[35] In this study, the sensitive and uncertain basal granular material properties are retrieved from InSAR observed surface ice velocities (e.g., using a best fit of model simulated surface velocity and observed surface velocity as retrieval criteria). With improved granular rheological parameterizations, SEGMENT-Ice is driven by atmospheric parameters provided by CGCMs to project future mass shed from the AIS. Compared with reanalyses, there are apparent mean biases among these climate models. The goal here aims at getting the climate warming response from the AIS. It is assumed that the ice sheet is in a near mass balance during the pre-industrial control period. Significant anthropogenic climate warming occurred in the past century. As is known, there is significant thermal inertia of the AIS, and it never is in an exact balance with the prevailing climate at any specific moment. We made such an assumption “surface mass balance and ice dynamics are in near balance in their (CGCMs) respective control period” so that future effects from a warmer climate can be estimated. If one argues that there is a residual term (e.g., the two terms do not cancel each other during the control period), then we assumed the residual term remains constant in the following years. Our purpose of examining the mass loss rates is minimally affected by the large intermodal differences in atmospheric/oceanic forcings. There is a high level of intermodel and interscenario consistency, with all indicating that the mass loss rate increases with time and is expected to reach ~220 km3/yr by 2100. Although they have no direct sea level change contribution, ice shelves are integral components of the AIS. Periodic calving, as a normal ablation process, releases tabular icebergs to maintain a dynamic balance of the AIS. In a warming future climate, increased air and ocean temperatures thin the ice shelves by surface erosions, but also they increase the vertical shear near the ice shelf front and cause more frequent tabular calving. Direct erosional ice shelf thinning and ice sheet rebounding, after calving, both signify reduced buttressing effects that lead to further increases in the inland ice mass loss rate.

Appendix A

[36] The first section derives the tabular calving scheme proposed in this study, and the second section contributes to the dynamic data assimilation (parameter retrieval) scheme relevant for the retrieval efforts involved in this research.

A1 Cantilever Beam Approximation for Ice-Shelf Attrition

[37] As shown in Figure A1, it is assumed that the material is stiff and the deflection in the z direction is sufficiently small that linear deformation theory for elastic material is valid. It further is assumed that the cross-section is rectangular to simplify the derivation by removing variations in the y direction and work only in the x-z plane (the convention for moment and torque are all in a right-handed coordinate system, as in Figure A1). The beam material has an elastic modulus E and density ρ. Other geometrical parameters are as labeled in the figure.


Figure A1. Cantilever beam assumption for an ice shelf. H is the ice thickness of ice at the hanging side (connected with the main shelf). Assume that the y-z cross-section is rectangular-shaped. Ice thickness h(x) is assumed to be a linear function of x. T denotes external loading, such as tides or random hits by other icebergs. Small deflections (in the z direction) also are assumed.

[38] As the cross-section is assumed to be rectangular, the area moment of inertia is math formula at point O. The momentum around O exerted by the weight of the beam and an external loading (representative of tides and other random factors), T, located at the tip end of the shelf is expressed as follows:

display math(A1)

where g is gravitational acceleration (9.8 m/s2), and M is the moment in the positive y direction. In a static state, the resistance moment should be in the −y direction and of the same magnitude. As a result of moment drive, there is potential energy stored around the cross-section passing through O:

display math(A2)

where r is the curvature of the beam at O. At the yielding condition, math formula is the tensile strength of ice. In equation (A2), the factor of 2 appears because mass conservation is assumed, so that the cross-sectional area experiencing compression and the area experiencing expansion are the same. The relation between strain and Young's modulus are applied, producing the factor of 2. The stored potential energy and the moment should have the same value. That is,

display math(A3)

[39] Equation (A3) is the master equation for obtaining the limiting length of ice shelf before attrition. For example, if we assume the linear ice thickness profile,

display math(A4)

where ρw and ρi are respectively the density of water and ice, then substituting into equations (A1) and (A2) and using equation (A3) gives the following:

display math(A5)

where math formula. For the case without external loading, bk = [fcH/(2ρig(1 − ρi/ρw))]0.5. This is similar to equation (1). In equation (1), we assumed a more realistic ice profile in reference to ice flow vertical shear and made the further assumption of equation (A5), assuming T is small compared with the integral part of equation (A1) (the first term on the right-hand side).

[40] Ice shelf calving in essence is a fatigue process of visco-plastic ice. In SEGMENT-Ice, the von Mises yielding criteria (a critical point for ice to deform plastically) is applied to identify initial seminal crevasses for inland ice. In principal deviatoric stress form,

display math(A6)

[41] Because different parts of the ice shelf calving front have different ice thicknesses and speeds, it is usually unlikely a through cut (e.g., a crack through the entire depth of shelf and all across the ice front arch) is finished at once. The normal scenario is that one sector breaks first, and the crack propagates laterally to the neighboring area. Tides and other random oscillating factors play a role in the following development speed of the crack because a crack that does not cyclically open and close does not grow rapidly. Tide amplitude, a measure of its energy, around Antarctica generally is small compared with ice thickness except for the portion facing South America. Only those tides with the resonance frequencies of the ice shelves have significant effects on the crack tearing rates. In SEGMENT-Ice, the crack tearing rate is expressed as

display math(A7)

[42] Where a is the crack length starting from 0, N is time steps, c1 is the Paris coefficient [Timoshenko and Gere, 1963], C0 is a negative number indicating the exponential damping of the tidal tearing when it is out of synchronization with the natural frequency of the ice shelf, υ is the tidal frequency, υ0 is resonance frequency of the ice shelf under consideration, and ΔK is the range of stress intensity change (proportional to the square of the amplitude of the tide). The resonance frequencies are determined for all peripheral ice shelves, using a Newton-Raphson iteration method applied to the deflection equation of the 3-D ice shelf configuration, with distributed ice density and temperature.

A2 A General Statement of Sensitivity Study and Data Assimilation Using Adjoint

[43] Parameter sensitivity studies and data assimilation may seem to be different subjects. However, using a variational optimization approach, an adjoint model can be developed for both (Errico, 1997). In the following, a general variational framework is described, in which adjoint sensitivity and variational data assimilation are special cases.

[44] Consider a physical model that is represented in vector form by the following system of coupled nonlinear differential equations:

display math(A8)

where U(t) is an Ns × 1 state vector (Ns is the number of state variables, augmented to include all spatial locations for a distributed physical system), and F is an Ns × 1 nonlinear vector operator that is a function of the states themselves and the Np × 1 vector of time invariant spatially distributed model parameters α (e.g., granular material's mechanical properties). Forcings such as observed meteorological inputs are included in F.

[45] Next, define a scalar “measurement importance functional” [Backus and Gilbert, 1970; Marchuk, 1981] by the following:

display math(A9)

where ϕ is a nonlinear function of the state variables and model parameters, x is the spatial variable, and t is time. In a sensitivity study, J may represent a quantity of interest (e.g., diurnally averaged evaporation), whereas for data assimilation it would represent the cost function, which generally is a least square performance metric that indicates the difference between model predictions and measurements (e.g., equation (3)). In either case, interest is in obtaining sensitivity derivatives [Errico, 1997]. For sensitivity studies, these derivatives give the sensitivity of the model response to various model parameters, providing insight into the relative sensitivities to different parameters, as well as which physical processes are most important in the system. In data assimilation, generally the derivatives are needed of the least squares cost function with respect to target parameters (also called control variables) as input to an optimization scheme.

[46] With the above general task in mind, first adjoint the model to the “measurement importance functional” using a vector of Lagrange multipliers [Yu and O'brien, 1991] λT:

display math(A10)

where T represents transpose operator, and L is the Lagrangian which transforms a constrained minimization problem into an unconstrained problem.

[47] Taking the first-order variation of L with respect to U and α gives the following:

display math(A11)

where it is assumed that the assimilation period is [t0, t1]. Denoting the values of the Lagrange multiplier at the initial and final times as λ0 and λf and assuming λf = 0 [Thacker, 1988], equation (A11) can be simplified as

display math(A12)

[48] Minimization of L requires that


[49] Equation (A13a) is called the adjoint system (of equation (A8)) as the Lagrange multipliers are called the adjoint variables. The adjoint model is integrated backward in time as a terminal value problem. Note that the homogeneous part of the linear differential equation (the first two terms on the left-hand side of equation (A13a)) is totally determined by the Jacobian of the forward model with respect to U (i.e., math formula), while the forcing math formula is dependent on the gradient of the model response. The homogeneous adjoint model (the so-called adjoint system) is constructed separately and used in conjunction with different measurement importance functionals.

[50] Our ultimate goal is for math formula to approach zero. However, once λ is known by the integrating of the adjoint model, (A13b) and (A13c) does provide the gradient of L with respect to parameter α and initial condition of the control variable U(t0) during the minimization process. This is one of the primary benefits of the adjoint method in that once the adjoint model is developed, all of the derivatives are obtained efficiently through a single forward model run and a single adjoint model run. This is possible because the adjoint model is developed from the forward model and propagates the sensitivity information backward in time over the model integration

[51] Closely related to the adjoint model is the tangent linear model (TLM, defined in equation (A15) below). For the convenience, the forward system is written in discrete integration form (evolution form) rather than in differential form as in equation (A8). The conception of a small perturbation is found in Marchuk et al., 1996, among many other sources. M is designated for the discrete system of F, and x is designated for model control variable U.

[52] Selecting a control vector c ∈ Rn and letting math formula, the orbit can be computed using x(k) = M(x(k − 1), α) = … = M(k)(c + δc, α). The actual evolution of the perturbation is given by

display math(A14)

[53] As the computation of Mk(x,α) is difficult in general, this quantity is usually approximated using a first-order Taylor series expansion:

display math

Here JM is the Jacobian of M with respect to x.

[54] Similarly, math formula is an approximation to x(1) to first-order accuracy. Thus, we have math formula. By denoting math formula, math formula is thus an approximation to x(2).

[55] Continuing this argument, inductively defining

display math(A15)

results in math formula as an approximation to x(k + 1).

[56] Note that equation (A15) is a non-autonomous linear dynamical system where the one-step transition matrix math formula is evaluated along the base state. Equation (A15) is called the tangent linear system (TLM). For convenience, introducing the following notation math formula, equation (A15) can be rewritten as δx(k + 1) = δx(k)JM(k) = JM(k)JM(k − 1)JM(k − 2) … JM(1)JM(0)δc. Further defining, for i ≤ j,JM(i : j) = JM(j)JM(j − 1) … JM(i), the equation (A15) can be reduced to δx(k + 1) = JM(0 : k)δc, or δx(k) = JM(0 : k − 1)δc.

[57] The iterative scheme that defines the TLM as in equation (A15) can also be written in a matrix-vector form:

display math(A16)

where F is an N by N block partitioned matrix given by

display math(A17)

[58] δx = (δx(1), δx(2), δx(3),  δx(N))T and b is a block-partitioned vector given by b = (JM(0)δc, 0, 0, …  0)T. Interestingly, the homogeneous adjoint model is formally a transpose of the TLM model. For detailed derivations and explanations of the relationship between TLM and adjoint model, see Appendix 2 of Ren [2004].

[59] From a review of the 4DVAR data assimilation literature (e.g., LeDimet and Talagrand, 1986; Courtier and Talagrand [1987]; Talagrand and Courtier [1987]), it can be concluded that the adjoint method brings a model trajectory as close as possible to the data by varying control variables. The closeness of a model trajectory to a period of data is quantified by a cost function. An iterative algorithm is used to minimize the cost function: Starting from a first guess, the values of the cost function and its gradient with respect to the control variables are used to improve the estimation of the vector of the control variables. Under a series of forcings composed of the model-data misfits, a backward integration of the adjoint model provides the gradient vector of the cost function with respect to control variables.

[60] For our retrieval of basal granular property, parameter retrieval works as follows: one varies a small amount of the initial estimates till thickness and reruns the ice dynamics forward model (SEGMENT-Ice), which provides updated surface ice velocities. These ice velocities are different from the MAMM observations. Adjoint-based optimization guarantees the direction the till thickness is modified minimizes the differences of the modeled and observed ice velocity fields. When the simulated surface flow field is satisfactory (close enough to observations), the retrieval scheme is terminated, and the till thickness is retrieved. Other mechanical parameters are retrieved the same way and simultaneously.


[61] This study was supported by the Australian Sustainable Development Institute (ASDI), Curtin University, Perth, Western Australia. We thank Professors T. Hughes, R. Thomas, H. Conway, and D. MacAyeal for insightful comments on calving mechanisms and general climate patterns around Antarctica Ice Sheet. We thank Dr. A. Hu from NCAR for helpful discussions of oceanic warming of the Southern Ocean in the upcoming century.