Kinematic First-order Calving Law Implies Potential for Abrupt Ice-shelf Retreat

Recently observed large-scale disintegration of Antarctic ice shelves has moved their fronts closer towards grounded ice. In response, ice-sheet discharge into the ocean has accelerated, contributing to global sea-level rise and emphasizing the importance of calving-front dynamics. The position of the ice front strongly influences the stress field within the entire sheet-shelf-system and thereby the mass flow across the grounding line. While theories for an advance of the ice-front are readily available, no general rule exists for its retreat, making it difficult to incorporate the retreat in predictive models. Here we extract the first-order large-scale kinematic contribution to calving which is consistent with large-scale observation. We emphasize that the proposed equation does not constitute a comprehensive calving law but represents the first-order kinematic contribution which can and should be complemented by higher order contributions as well as the influence of potentially heterogeneous material properties of the ice. When applied as a calving law, the equation naturally incorporates the stabilizing effect of pinning points and inhibits ice shelf growth outside of embayments. It depends only on local ice properties which are, however, determined by the full topography of the ice shelf. In numerical simulations the parameterization reproduces multiple stable fronts as observed for the Larsen A and B Ice Shelves including abrupt transitions between them which may be caused by localized ice weaknesses. We also find multiple stable states of the Ross Ice Shelf at the gateway of the West Antarctic Ice Sheet with back stresses onto the sheet reduced by up to 90 % compared to the present state.


Introduction
Recent observations have shown rapid acceleration of local ice streams after the collapse of ice shelves fringing the Antarctic Peninsula, such as Larsen A and B (De Angelis and Skvarca, 2003;Scambos et al., 2004;Rignot et al., 2004;Rott et al., 2007;Pritchard and Vaughan, 2007).Lateral drag exerted by an ice-shelf's embayment on Figures the flow yields back stresses that restrain grounded ice as long as the ice shelf is intact (Dupont andAlley, 2005, 2006).Ice-shelf disintegration eliminates this buttressing effect and may even lead to abrupt retreat of the grounding line that separates grounded ice sheet from floating ice shelf (Weertman, 1974;Schoof, 2007;Pollard and Deconto, 2009).The dynamics at the calving front of an ice shelf is thus of major importance for the stability of the Antarctic Ice Sheet and thereby global sea level (Cazenave et al., 2009;Thomas et al., 2004;Bamber et al., 2009).Small-scale physics of calving events is complicated and a number of different approaches to derive rates of large-and smallscale calving events have been proposed (Bassis, 2011;Amundson and Truffer, 2010;Grosfeld and Sandh äger, 2004;Kenneally and Hughes, 2002;Pelto and Warren, 1991;Reeh, 1968).Different modes of calving will depend to a different extent on material properties of the ice and on the local flow field.As an example, considerable effort has been made to incorporate fracture dynamics into diagnostic ice shelf models (Rist et al., 1999;Jansen et al., 2010;Hulbe et al., 2010;Humbert et al., 2009;Saheicha et al., 2006;Luckman et al., 2011;Albrecht and Levermann, 2011).Furthermore ice fractures will alter the stress field and thereby also influence the flow field and calving indirectly (Larour et al., 2004a,b;Vieli et al., 2006Vieli et al., , 2007;;Khazendar et al., 2009;Humbert and Steinhage, 2011).
Here we take a simplifying approach and seek the first-order kinematic contribution to iceberg calving ignoring higher order effects as well as potential interactions of material properties with the kinematic field.We build on earlier observations which found that the integrated large-scale calving rate shows a dependence on the local ice-flow spreading rate (Alley et al., 2008) and that ice fronts tend to follow lines of zero across-flow spreading rates (Doake et al., 1998;Doake, 2001).Taking a macroscopic viewpoint, we propose a simple kinematic first-order calving law applicable to three-dimensional ice shelves, which is consistent with these observations (Fig. 1) and unifies previous approaches in the sense that it extracts the first order dependence on the spreading rate tensor.Introduction

Conclusions References
Tables Figures

Back Close
Full 2 First-order kinematic contribution to calving Direct observations in Antarctica show that local calving rates increase with along-flow ice-shelf spreading-rates, ˙ , i.e. the local spatial derivative of the horizontal velocity field along the ice-flow direction.Across different ice shelves the calving rate is proportional to the product of this along-flow spreading-rate and the width of the shelf's embayment (Alley et al., 2008).Dynamically this width strongly influences the spreading rate perpendicular to the main flow ˙ ⊥ (Fig. 2).Observations furthermore suggest that spreading perpendicular to the main flow controls formation and propagation of intersecting crevasses in the ice and thereby determines potential calving locations and influences calving rates (Kenneally and Hughes, 2006).The reason is that negative ˙ ⊥ represents a convergence of ice at a certain location, which, if not too strong in magnitude, tends to close crevasses and inhibit large-scale ice-shelf calving.Positive ˙ ⊥ , on the other hand, denotes dilatation of ice, which tends to enhance ice weaknesses and thereby promotes calving.Such small scale considerations are consistent with the large-scale observation (Doake et al., 1998) that ice-shelf fronts tend to follow lines of zero ˙ ⊥ (Doake, 2001).
All of this suggests that the relevant dynamic quantity for large-scale ice-shelf calving is the overall dilatation of ice flow as represented by the determinant of the horizontal spreading-rate matrix.Mathematically the determinant is one of two rotationallyinvariant characteristics of the horizontal spreading-rate matrix and thus particularly representative of isotropic effects in ice dynamics.The determinant is computed as the product of the spreading rates in the two principal directions of horizontal flow on ice shelves, ˙ + • ˙ − .In most areas along the calving front, these so-called eigen-directions will coincide with directions along and transversal to the flow, i.e. ˙ + ≈ ˙ and ˙ − ≈ ˙ ⊥ .We thus propose that in regions of divergent flow, where ˙ ± > 0, the rate of large-scale calving, C, is Here K

Conclusions References
Tables Figures

Back Close
Full for calving.The macroscopic viewpoint of Eq. ( 1) comprises a number of small-scale physical processes of different calving mechanisms as well as intermittent occurrences of tabular iceberg release averaged over a sufficient period of time (see reference (Hughes, 2002) for a detailed discussion).Consistent with Eq. ( 1), we find that the product ˙ + • ˙ − as derived from satellite observations (Joughin, 2002;Joughin and Padman, 2003) of the ice flow field of the ice shelves Larsen C and Filchner is indeed proportional to the calving rate estimated by the terminal velocity at the ice front (Fig. 1 and Appendix for details on the data).
Equation ( 1) can be considered the first-order kinematic contribution to calving under three basic assumptions: (a) stable calving fronts exist, (b) material properties of ice near the calving front can be assumed to be isotropic to leading order and (c) of all available kinematic properties it is the vertically averaged spreading rate tensor on which calving depends to leading order.In other words, if a calving law is sought that assumes isotropic material properties of the ice and one asks for the dependence of this law on the horizontal spreading rate field, then such a law can be expanded in terms of the the eigenvalues of the spreading rate tensor as follows: with parameters K ± i which may depend on the material properties of the ice.In this formal expansion a number of terms need to be dismissed on the ground of large-scale physical reasoning.Since the largest eigenvalue ˙ + generally increases within the ice shelf when approaching the grounding line upstream (compare for example Fig. 3c),  K + 1 has to vanish or otherwise ice shelves would generally not be stable at all.On the other hand, a non-vanishing K − 1 would mean that calving is determined mainly by ice divergence across the main flow direction.While we argue that this is indeed a relevant quantity, observations (Fig. 1 and reference Alley et al., 2008) suggest that there has to be a leading role of ˙ + .The same arguments make it necessary that both K + 2 and K − 2 vanish.Thus the first-order term of such a law has to be the one associated with K Introduction

Conclusions References
Tables Figures

Back Close
Full It has to be noted that not all calving will fulfill the three assumptions above.For example, some calving processes might mainly depend on the vertical shear within the ice.In cases where the assumptions are met, Eq. ( 1) can be applied both in analytic calculations and numerical simulations of ice-shelf dynamics.Here it was implemented into the Potsdam Parallel Ice Sheet Model (PISM-PIK) (Winkelmann et al., 2011;Martin et al., 2011) for the Ross Ice Shelf (Please find more details on the experimental setup in the Appendix.)For low values of the proportionality constant, K ± 2 , calving is not sufficient to ensure a stable calving front (Fig. 3a, b).Above a certain threshold, K * ≈ 1×10 9 ma, the calving front is dynamically stable.The lack of bottom drag induces a maximum spreading rate along the ice-flow direction downstream of, but close to, the grounding line (Fig. 3c).Thus, as argued above, a calving rate which depends only on ˙ + would increase towards the grounding line so that no stable ice-shelf front would develop.Stability arises through the dependence on the overall dilatation of the velocity field, which is negative for most of the shelf area (Fig. 3d) and prohibits calving there.
Spreading rate fields and thereby areas of potential calving according to Eq. ( 1) are strongly influenced by the geometry of the shelf ice.The proposed kinematic calving law thereby incorporates the effect of pinning points, such as Roosevelt Ice Rise (RIS) amidst the Ross Ice Shelf.These pinning points produce convergence of ice flow and thereby stabilize the calving front.In contrast to these regions of convergence, ice flow strongly expands towards the mouth of the shelf's embayment.Here, calving rates strongly increase and hinder ice-shelf growth outside the bay.
For the Larsen A and B Ice Shelves, application of Eq. ( 1) in numerical simulations with PISM-PIK yields a stable calving front that compares well with the observed state before 1999 (Fig. 4a).These fronts are obtained with the same proportionality factor K ± 2 = 5 × 10 9 ma as used for the Ross Ice Shelf.Eigencalving thus allows to reproduce calving fronts with varying ice thickness as also found in simulations of the whole Antarctic Ice Sheet (Martin et al., 2011).Introduction

Conclusions References
Tables Figures

Back Close
Full  dar et al., 2007;Vieli et al., 2006).Instantaneous introduction of ice-free rifts in the vicinity of these locations for the minimal duration of one time step strongly alters the stress field within the ice shelf (Fig. 4).Consequently spreading rates and potential calving rates are changed and the ice shelf undergoes an abrupt transition towards a second stable state (Fig. 5 and animation in Supplement).
Further perturbation or different initialization of the ice shelf area reveals a third stable front.These states are dynamically robust with respect to the proportionality constant K ± 2 ≥ K * and compare reasonably well with observed stages of Larsen B disintegration between 1998 and 2002 (Fig. 5).The corresponding calving fronts tend to follow lines of zero-ice-flow-compression at ˙ − = 0, as had been suggested by earlier reconstructions of ˙ − from observed ice velocity data (Doake et al., 1998;Doake, 2001).These contours of ˙ − = 0 change with ice-shelf geometry which makes multiple dynamically stable fronts possible.These ice fronts can, however, not be inferred simply from a static velocity field, but evolve under the dynamical interaction of ice flow field and calving rate.
Rapid retreat as governed by Eq. ( 1) occurs within several months after the perturbation depending on the value of K ± 2 (compare animation in Supplement).The analysis of satellite data show that the 2002-Larsen-B disintegration was not caused by localized calving near the ice front but by emergence of surface ponds which weakened the ice shelf and caused a sudden collapse (MacAyeal et al., 2003;Glasser and Scambos, 2008).Our simulations thus do not provide an explanation of this event but show that the emerging stable calving fronts are consistent with the calving, which is governed by the first-order kinematic law (Eq.1) and may explain temporal stability of these fronts.Introduction

Conclusions References
Tables Figures

Back Close
Full Simulations of the Ross Ice Shelf reproduce the presently observed calving front and suggests two additional stable front positions (Fig. 3c).These are associated with reductions in ice-shelf areas by 37 % and 60 %.Back stress onto the West Antarctic Ice Sheet in these states is reduced by up to 90 % (Fig. 6).

Discussions and conclusions
Reliable projection of the evolution of the Antarctic Ice Sheet under future climate change need to take calving-front dynamics into account.We present a calving law that reproduces presently observed ice fronts of ice shelves with strongly varying ice thickness, e.g. the Larsen Ice Shelf and the Ross Ice Shelf.The calving law has been applied in simulations of the entire Antarctic Ice Sheet (Martin et al., 2011) where it produces a realistic ice front for all major ice shelves.The calving law of Eq. ( 1) represents the first order kinematic contribution of calving under the assumption that calving depends kinematically to first-order on the horizontal strain rate field and that ice properties are to first-order isotropic.Not all calving has to fulfill these assumptions.While isotropic material properties can be captured in the proportionality constant K ± 2 , ice can be anisotropic near the ice front.Furthermore, some calving processes might mainly depend on the vertical shear within the ice as opposed to the horizontal strain rate field.We propose eigencalving as a first-order law for large-scale calving comprising a number of small-scale processes.For a comprehensive description of calving front dynamics our approach may be complemented by additional processes that are not captured by Eq. ( 1).Introduction

Conclusions References
Tables Figures

Back Close
Full (RADARSAT Antarctic Mapping Mission).Strain rates were computed as the spatial derivative of the velocity field (neglecting projection errors) with a centered scheme (along the boundaries inward).In order to reduce effect of undulations of length scale 5-8 km that are due to the measuring and processing technique, we long-pass filtered the velocity field with a two-dimensional sliding window of 21 × 21 km size.In order to avoid data from grounded ice only data points with a velocity magnitude larger than a threshold of 10 m yr −1 within this window were taken into account.Independent data points were computed in 30 km sections along the ice shelf front in which average velocities and horizontal spreading rate eigenvalues were computed.Figure 1 shows the relation between ice velocity at the ice front (assumed to be a measure for the calving rate for a steady calving front) and the product of the spreading rate eigenvalues, i.e. the determinant of the spreading rate matrix for Larsen B and Filchner Ice Shelf.
While the results for Ronne Ice Shelf do not contradict our calving law, the range of calving rates that is observed in this ice shelf is very narrow and not sufficient to observe any trend which is also reflected in a very narrow range of spreading rate eigenvalues.Introduction

Conclusions References
Tables Figures

Back Close
Full derneath the ice sheets.Verification of the code with exact analytical solutions were used to test its numerical accuracy (Bueler et al., 2007).Ice softness, A(T * ), is determined following Paterson and Budd (1982).PISM generally applies a combination of the Shallow Ice Approximation (SIA) and the Shallow Shelf Approximation (SSA) (Morland, 1987;Weis et al., 1999) to simulate grounded ice.In all simulations presented here ice is floating and thus the velocity field is computed by application of SSA only.Numerical solutions of the SSA-matrix inversion were obtained using the Portable Extensible Toolkit for Scientific computation (PETSc) library (Balay et al., 2008).Compared to PISM, the Potsdam Parallel Ice Sheet Model (PISM-PIK) as used here, applies two major modifications to the original version of PISM.First, a subgrid scale representation of the calving front was implemented in order to be able to apply a continuous calving rate (compare Albrecht et al., 2011, for a detailed description).Secondly, we introduced physical boundary conditions at the calving front to ensure that the non-local velocity calculation yields a reasonable and physically based distribution of the velocity throughout the ice shelf which is crucial for this study.For a full description of PISM-PIK please confer Winkelmann et al. (2011).The models performance under present day boundary conditions is provided in Martin et al. (2011).Introduction

Conclusions References
Tables Figures

Back Close
Full

Experimental setup of ice-shelf simulations
In the experimental setups used in this study an ice flow from a region of grounded ice is prescribed by Dirichlet boundary conditions (fixed ice thickness and velocity) and the dynamically evolving ice-shelf is computed according to SSA.Simulations have all been integrated into equilibrium for several thousand of years.Grounded ice was not taken into account (margins fixed) and in order to ensure that no grounding occurs during the simulation the bathymetry within the confining bay is set to 2000 m.

C1 Ross Ice Shelf (RIS)
The Ross Ice Shelf setup has been derived from data used for the ice-shelf modelintercomparison, EISMINT II-Ross (MacAyeal et al., 1996).Ice thickness, surface temperature and accumulation rate of the main part of the Ross ice shelf were applied on a 6.82 km resolution grid (Fig. 3).The computational domain in total measures 1000×1000 km 2 , approximately between 150 • W and 160 • E, as well as between 75 • S and 85 • S. Inflow velocities at the grounding line have been prescribed from RIGGS data (Ross Ice shelf Geophysical and Glaciological Survey, Thomas et al., 1984;Bentley, 1984) west, where the 3894 m high volcano Mount Erebus can be found, and the Edward-VII-Peninsula (EP) with Cape Colbeck (CC) in the east.These areas had been roughly extrapolated in the original model intercomparison, but they are crucial in our studies in finding a more realistic steady state of the calving front position and are thus more realistically represented in our simulations.

C2 Larsen Ice Shelves A and B
Our computational setup incorporates part A and B of the Larsen Ice Shelf which has undergone some dramatical retreats in the last two decades (Fig. 5).These comparably small ice shelves are fringing the Antarctic Peninsula between 64.5 • S-66 • S and 63 • W-59 • W. Surface elevation and velocity data have been raised in the Modified Antarctic Mapping Mission (MAMM), by the Byrd Center group (Liu et al., 1999;Jezek et al., 2003).We apply them on a 2 km grid as provided by Dave Covey from the University of Alaska, Fairbanks (UAF).The computational domain is 200 × 170 km 2 and has been re-gridded to 1 km resolution.Gaps in the data were filled as average over existing neighboring values up to the ice front in the year 1999 for Larsen B (Ice front of Larsen A at position of about beginning 1995).The surface temperature (Comiso, 2000) and accumulation data (Van de Berg et al., 2006) has been taken from presentday Antarctica SeaRISE Data Set on a 5 km grid (Le Brocq et al., 2010), regridded to 1 km resolution.Ice shelf thickness is calculated from surface elevation, assuming hydrostatic equilibrium.The grounding line was expected to be located where the surface gradient exceeds a critical value (also for ice rises).Observed velocities and calculated ice thicknesses were set as a Dirichlet condition on the supposed grounding line in inlet regions simulating the ice stream inflow through the mountains.Six important tributaries can be located along the western grounding line, from south to north: Leppard Glacier (LG), Flask Glacier (FG), Crane Glacier (CG), Evans Glacier (EG) and Hektoria Glacier (HG).Drygalsky Glacier (JP).Larsen C is much larger and not taken into account in these studies.There, ice is cut off at the coastline south of Cape Framnes (CF).

Appendix D Animation of simulation of Larsen A and B Ice Shelves
The animation in this Supplement shows the smaller eigenvalue of the spreading rate matrix ˙ − for the Larsen A and B Ice Shelves.Time is plotted as a running number in years of integration.The animation begins without any calving starting from a steady state with front position comparable to the observed situation of the year 1997.Consequently, the shelves grow out of the embayment where it builds up a region with strongly positive ˙ − .After 100 years eigencalving (following Eq. ( 1) with K Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | 4 Multiple stable fronts and abrupt transitions between them Inverse computation of ice softness prior to the major disintegration of Larsen B Ice Shelf in 2002 suggested regions of strongly weakened ice rheology along the fringing pinning points of Robertson Island (RI), Seal Nunataks Islands (SN) in the North East and Jason Peninsula (JP) and Cape Disappointment (CD) in the South West (Khazen- Discussion Paper | Discussion Paper | Discussion Paper | for the computation of the spreading rates were derived by SAR speckle tracking algorithms and are provided on a regular grid of grid length 1 km in Polar Stereographic projection with 71 • S as the latitude of true scale(Joughin, 2002).Data are based on observations of the years 2006-2008 for Larsen C (JAXA ALOS L-band SAR) and 1997-2000 for Filchner-Ronne(Joughin and Padman, 2003) , we apply a derived version of the open-source Parallel Ice Sheet Model version stable 0.2 (PISM, see user's manual (PISM-authors) and model description paper(Bueler and Brown, 2009)).Starting from PISM stable 0.4 most features of the Potsdam Parallel Ice Sheet Model (PISM-PIK,Winkelmann et al., 2011) have been integrated in the openly available PISM code.It is a thermo-mechanically coupled model developed at the University of Alaska, Fairbanks, USA.The thickness, temperature, velocity and age of the ice can be simulated, as well as the deformation of Earth un- . Broad ice streams drain into the shelf over the Siple Coast (SC) in the south-east draining the West Antarctic Ice Sheet (WAIS), smaller ones urge through the Transantarctic Mountains (TAM) in the west, e.g. the Byrd Glacier (BG), feeding the Ross Ice Shelf with ice from East Antarctica.Properties in the grounded part are held constant during the simulation and friction is prescribed at the margin.The Roosevelt (RIR) and Crary Ice Rises (CIR) are prescribed as elevations of the bedrock.RIGGS velocity data, acquired at a few hundred locations in the years 1973-1978, has been used to validate the SSA-velocity calculations of the model.Compared to the original data the setup has been slightly modified in the vicinity of the outer pinning points according to BEDMAP (Lythe et al., 2001) data, such as Ross Island (RI) in the Introduction Discussion Paper | Discussion Paper | Discussion Paper | (DG) is feeding Larsen A north of the Seal Nunataks Islands (SNI) with the prominent Robertson Island (RI) that separate Larsen A and B Ice Shelves; Larsen B and C are physically divided by Jason Peninsula 2710 Discussion Paper | Discussion Paper | Discussion Paper |

Fig. 1 .Fig. 2 .Fig. 3 .Fig. 4 .Fig. 5 .Fig. 6 .
Fig. 1.Concept of eigen-calving -(a) Schematic illustrating proposed kinematic calving law: the calving rate is proportional to the spreading rates in both eigen-directions of the flow which generally coincide with directions along (green arrows) and perpendicular to (red arrows) the flow field.In confined region of the ice shelf, e.g. in the vicinity of the grounding line, convergence of ice flow perpendicular to the main flow direction yields closure of crevasses, inhibits large-scale calving and stabilizes the ice shelf.Near the mouth of the embayment, the flow field dilatation occurs in both eigen-directions and large-scale calving impedes ice-shelf growth onto the open ocean.(b)The observed calving rate determined as the ice flow at the calving front increases with the product of the two eigenvalues which is proposed here as a first-order kinematic calving law in Eq. (1).Details on the data can be found in Appendix.