Articles | Volume 12, issue 1
Research article 08 Jan 2018
Research article | 08 Jan 2018
Influence of temperature fluctuations on equilibrium
ice sheet volume
Troels Bøgeholm Mikkelsen et al.
No articles found.
Johannes Lohmann, Daniele Castellana, Peter D. Ditlevsen, and Henk A. Dijkstra
Earth Syst. Dynam., 12, 819–835,Short summary
Tipping of one climate subsystem could trigger a cascade of subsequent tipping points and even global-scale climate tipping. Sequential shifts of atmosphere, sea ice and ocean have been recorded in proxy archives of past climate change. Based on this we propose a conceptual model for abrupt climate changes of the last glacial. Here, rate-induced tipping enables tipping cascades in systems with relatively weak coupling. An early warning signal is proposed that may detect such a tipping.
Johannes Lohmann and Peter D. Ditlevsen
Clim. Past, 15, 1771–1792,Short summary
Greenland ice core records show that the climate of the last glacial period was frequently interrupted by rapid warming events, followed by cooling episodes of vastly different duration. We fit a generic waveform to the noisy ice core record in order to extract a robust climate signal and empirically study what controls the amplitude and duration of the warmings and coolings. We find that cooling transitions are more predictable than warmings and are influenced by different climate forcings.
Johannes Lohmann and Peter D. Ditlevsen
Clim. Past, 14, 609–617,Short summary
The climate of the last glacial period was frequently interrupted by rapid warming events, the cause of which is still unknown. One open question is whether the occurrence of events is random or externally controlled. We studied the temporal characteristics of warm and cold phases using statistical null models and find that they are well described as random processes modulated by two different external climate factors. This may help distinguish physical mechanisms for rapid climate change.
A. Messerli, N. B. Karlsson, and A. Grinsted
The Cryosphere Discuss.,
Preprint withdrawnShort summary
We use 2014 Landsat-8 imagery in combination with a newly developed feature tracking toolbox, ImGRAFT to produce velocity fields and flux for five major Greenland marine terminating glaciers: Helheim, Kangerdlugssuaq, Nioghalvfjerdsbræ, Petermann and Jakobshavn Isbræ. A major finding of the paper documents the increased velocities observed at Jakobshavn Isbræ. We measure velocities over 50m/day which exceed that of the previously recorded maximum.
I. Daruka and P. D. Ditlevsen
Clim. Past Discuss.,
Revised manuscript not accepted
H. C. Steen-Larsen, S. J. Johnsen, V. Masson-Delmotte, B. Stenni, C. Risi, H. Sodemann, D. Balslev-Clausen, T. Blunier, D. Dahl-Jensen, M. D. Ellehøj, S. Falourd, A. Grindsted, V. Gkinis, J. Jouzel, T. Popp, S. Sheldon, S. B. Simonsen, J. Sjolte, J. P. Steffensen, P. Sperlich, A. E. Sveinbjörnsdóttir, B. M. Vinther, and J. W. C. White
Atmos. Chem. Phys., 13, 4815–4828,
The Cryosphere, 7, 141–151,
Related subject area
Ice SheetsRemapping of Greenland ice sheet surface mass balance anomalies for large ensemble sea-level change projectionsBrief communication: On calculating the sea-level contribution in marine ice-sheet modelsA simple stress-based cliff-calving lawScaling of instability timescales of Antarctic outlet glaciers based on one-dimensional similitude analysisA statistical fracture model for Antarctic ice shelves and glaciersModelled fracture and calving on the Totten Ice ShelfDesign and results of the ice sheet model initialisation experiments initMIP-Greenland: an ISMIP6 intercomparisonIncreased West Antarctic and unchanged East Antarctic ice discharge over the last 7 yearsGPS-derived estimates of surface mass balance and ocean-induced basal melt for Pine Island Glacier ice shelf, AntarcticaAnalysis of ice shelf flexure and its InSAR representation in the grounding zone of the southern McMurdo Ice ShelfBoundary layer models for calving marine outlet glaciersLiquid water content in ice estimated through a full-depth ground radar profile and borehole measurements in western GreenlandDynamic influence of pinning points on marine ice-sheet stability: a numerical study in Dronning Maud Land, East AntarcticaPersistence and variability of ice-stream grounding lines on retrograde bed slopesSimilitude of ice dynamics against scaling of geometry and physical parametersAn ice-sheet-wide framework for englacial attenuation from ice-penetrating radar dataInversion of geothermal heat flux in a thermomechanically coupled nonlinear Stokes ice sheet modelThe influence of a model subglacial lake on ice dynamics and internal layeringSheet, stream, and shelf flow as progressive ice-bed uncoupling: Byrd Glacier, Antarctica and Jakobshavn Isbrae, GreenlandSeaRISE experiments revisited: potential sources of spread in multi-model projections of the Greenland ice sheetElevation change of the Greenland Ice Sheet due to surface mass balance and firn processes, 1960–2014Ice sheet mass loss caused by dust and black carbon accumulationTemporal variations in the flow of a large Antarctic ice stream controlled by tidally induced changes in the subglacial water systemEvolution of ice-shelf channels in Antarctic ice shelvesOceanic and atmospheric forcing of Larsen C Ice-Shelf thinningHow do icebergs affect the Greenland ice sheet under pre-industrial conditions? – a model study with a fully coupled ice-sheet–climate modelSeismic wave propagation in anisotropic ice – Part 1: Elasticity tensor and derived quantities from ice-core propertiesSeismic wave propagation in anisotropic ice – Part 2: Effects of crystal anisotropy in geophysical dataSimulating the Greenland ice sheet under present-day and palaeo constraints including a new discharge parameterizationElevation and elevation change of Greenland and Antarctica derived from CryoSat-2The importance of insolation changes for paleo ice sheet modelingParameterization of basal friction near grounding lines in a one-dimensional ice sheet modelA range correction for ICESat and its potential impact on ice-sheet mass balance studiesBrief Communication: Further summer speedup of Jakobshavn IsbræCreep deformation and buttressing capacity of damaged ice shelves: theory and application to Larsen C ice shelfScatter of mass changes estimates at basin scale for Greenland and AntarcticaInfluence of ice-sheet geometry and supraglacial lakes on seasonal ice-flow variabilityHindcasting to measure ice sheet model sensitivity to initial statesSurface undulations of Antarctic ice streams tightly controlled by bedrock topographyManufactured solutions and the verification of three-dimensional Stokes ice-sheet modelsGreenland ice sheet contribution to sea-level rise from a new-generation ice-sheet modelRadar diagnosis of the subglacial conditions in Dronning Maud Land, East AntarcticaIce velocity changes in the Ross and Ronne sectors observed using satellite radar data from 1997 and 2009A simple inverse method for the distribution of basal sliding coefficients under ice sheets, applied to AntarcticaThin-layer effects in glaciological seismic amplitude-versus-angle (AVA) analysis: implications for characterising a subglacial till unit, Russell Glacier, West GreenlandBorehole temperatures reveal details of 20th century warming at Bruce Plateau, Antarctic PeninsulaKinematic first-order calving law implies potential for abrupt ice-shelf retreatWarming of waters in an East Greenland fjord prior to glacier retreat: mechanisms and connection to large-scale atmospheric conditionsThe sea level fingerprint of recent ice mass fluxesA comparison of basal reflectivity and ice velocity in East Antarctica
Heiko Goelzer, Brice P. Y. Noël, Tamsin L. Edwards, Xavier Fettweis, Jonathan M. Gregory, William H. Lipscomb, Roderik S. W. van de Wal, and Michiel R. van den Broeke
The Cryosphere, 14, 1747–1762,Short summary
Future sea-level change projections with process-based ice sheet models are typically driven with surface mass balance forcing derived from climate models. In this work we address the problems arising from a mismatch of the modelled ice sheet geometry with the one used by the climate model. The proposed remapping method reproduces the original forcing data closely when applied to the original geometry and produces a physically meaningful forcing when applied to different modelled geometries.
Heiko Goelzer, Violaine Coulon, Frank Pattyn, Bas de Boer, and Roderik van de Wal
The Cryosphere, 14, 833–840,Short summary
In our ice-sheet modelling experience and from exchange with colleagues in different groups, we found that it is not always clear how to calculate the sea-level contribution from a marine ice-sheet model. This goes hand in hand with a lack of documentation and transparency in the published literature on how the sea-level contribution is estimated in different models. With this brief communication, we hope to stimulate awareness and discussion in the community to improve on this situation.
Tanja Schlemm and Anders Levermann
The Cryosphere, 13, 2475–2488,Short summary
We provide a simple stress-based parameterization for cliff calving of ice sheets. According to the resulting increasing dependence of the calving rate on ice thickness, the parameterization might lead to a runaway ice loss in large parts of Greenland and Antarctica.
Anders Levermann and Johannes Feldmann
The Cryosphere, 13, 1621–1633,Short summary
Using scaling analysis we propose that the currently observed marine ice-sheet instability in the Amundsen Sea sector might be faster than all other potential instabilities in Antarctica.
Veronika Emetc, Paul Tregoning, Mathieu Morlighem, Chris Borstad, and Malcolm Sambridge
The Cryosphere, 12, 3187–3213,Short summary
The paper includes a model that can be used to predict zones of fracture formation in both floating and grounded ice in Antarctica. We used observations and a statistics-based model to predict fractures in most ice shelves in Antarctica as an alternative to the damage-based approach. We can predict the location of observed fractures with an average success rate of 84% for grounded ice and 61% for floating ice and mean overestimation error of 26% and 20%, respectively.
Sue Cook, Jan Åström, Thomas Zwinger, Benjamin Keith Galton-Fenzi, Jamin Stevens Greenbaum, and Richard Coleman
The Cryosphere, 12, 2401–2411,Short summary
The growth of fractures on Antarctic ice shelves is important because it controls the amount of ice lost as icebergs. We use a model constructed of multiple interconnected blocks to predict the locations where fractures will form on the Totten Ice Shelf in East Antarctica. The results show that iceberg calving is controlled not only by fractures forming near the front of the ice shelf but also by fractures which formed many kilometres upstream.
Heiko Goelzer, Sophie Nowicki, Tamsin Edwards, Matthew Beckley, Ayako Abe-Ouchi, Andy Aschwanden, Reinhard Calov, Olivier Gagliardini, Fabien Gillet-Chaulet, Nicholas R. Golledge, Jonathan Gregory, Ralf Greve, Angelika Humbert, Philippe Huybrechts, Joseph H. Kennedy, Eric Larour, William H. Lipscomb, Sébastien Le clec'h, Victoria Lee, Mathieu Morlighem, Frank Pattyn, Antony J. Payne, Christian Rodehacke, Martin Rückamp, Fuyuki Saito, Nicole Schlegel, Helene Seroussi, Andrew Shepherd, Sainan Sun, Roderik van de Wal, and Florian A. Ziemen
The Cryosphere, 12, 1433–1460,Short summary
We have compared a wide spectrum of different initialisation techniques used in the ice sheet modelling community to define the modelled present-day Greenland ice sheet state as a starting point for physically based future-sea-level-change projections. Compared to earlier community-wide comparisons, we find better agreement across different models, which implies overall improvement of our understanding of what is needed to produce such initial states.
Alex S. Gardner, Geir Moholdt, Ted Scambos, Mark Fahnstock, Stefan Ligtenberg, Michiel van den Broeke, and Johan Nilsson
The Cryosphere, 12, 521–547,Short summary
We map present-day Antarctic surface velocities from Landsat imagery and compare to earlier estimates from radar. Flow accelerations across the grounding lines of West Antarctica's Amundsen Sea Embayment, Getz Ice Shelf and the western Antarctic Peninsula, account for 89 % of the observed increase in ice discharge. In contrast, glaciers draining the East Antarctic have been remarkably stable. Our work suggests that patterns of mass loss are part of a longer-term phase of enhanced flow.
David E. Shean, Knut Christianson, Kristine M. Larson, Stefan R. M. Ligtenberg, Ian R. Joughin, Ben E. Smith, C. Max Stevens, Mitchell Bushuk, and David M. Holland
The Cryosphere, 11, 2655–2674,Short summary
We used long-term GPS data and interferometric reflectometry (GPS-IR) to measure velocity, strain rate and surface elevation for the PIG ice shelf – a site of significant mass loss in recent decades. We combined these observations with high-res DEMs and firn model output to constrain surface mass balance and basal melt rates. We document notable spatial variability in basal melt rates but limited temporal variability from 2012 to 2014 despite significant changes in sub-shelf ocean heat content.
Wolfgang Rack, Matt A. King, Oliver J. Marsh, Christian T. Wild, and Dana Floricioiu
The Cryosphere, 11, 2481–2490,Short summary
Predicting changes of the Antarctic Ice Sheet involves fully understanding ice dynamics at the transition between grounded and floating ice. We map tidal bending of ice by satellite using InSAR, and we use precise GPS measurements with assumptions of tidal elastic bending to better interpret the satellite signal. It allows us to better define the grounding-line position and to refine the shape of tidal flexure profiles.
Christian Schoof, Andrew D. Davis, and Tiberiu V. Popa
The Cryosphere, 11, 2283–2303,Short summary
We show mathematically and computationally how discharge of ice from ocean-terminating glaciers is controlled by a combination of different forces acting on ice near the grounding line of a glacier and how that combination of forces is affected by the process of iceberg formation, which limits the length of floating ice tongues extending in front of the glacier. We show that a deeper fjord may lead to a longer ice tongue providing greater drag on the glacier, slowing the rate of ice discharge.
Joel Brown, Joel Harper, and Neil Humphrey
The Cryosphere, 11, 669–679,Short summary
We use ground-penetrating radar surveys in conjunction with borehole depth and temperature data to estimate the liquid water content (wetness) of glacial ice in the ablation zone of an outlet glacier on the western side of the Greenland Ice Sheet. Our results show that the wetness of a warm basal ice layer is approximately 2.9 % to 4.6 % in our study region. This high level of wetness requires special attention when modelling ice dynamics or estimating ice thickness in the region.
Lionel Favier, Frank Pattyn, Sophie Berger, and Reinhard Drews
The Cryosphere, 10, 2623–2635,Short summary
We demonstrate the short-term unstable retreat of an East Antarctic outlet glacier triggered by imposed sub-ice-shelf melt, compliant with current values, using a state-of-the-art ice-sheet model. We show that pinning points – topographic highs in contact with the ice-shelf base – have a major impact on ice-sheet stability and timing of grounding-line retreat. The study therefore calls for improving our knowledge of sub-ice-shelf bathymetry in order to reduce uncertainties in future ice loss.
Alexander A. Robel, Christian Schoof, and Eli Tziperman
The Cryosphere, 10, 1883–1896,Short summary
Portions of the Antarctic Ice Sheet edge that rest on upward-sloping beds have the potential to collapse irreversibly and raise global sea level. Using a numerical model, we show that changes in the slipperiness of sediments beneath fast-flowing ice streams can cause them to persist on upward-sloping beds for hundreds to thousands of years before reversing direction. This type of behavior is important to consider as a possibility when interpreting observations of ongoing ice sheet change.
Johannes Feldmann and Anders Levermann
The Cryosphere, 10, 1753–1769,
T. M. Jordan, J. L. Bamber, C. N. Williams, J. D. Paden, M. J. Siegert, P. Huybrechts, O. Gagliardini, and F. Gillet-Chaulet
The Cryosphere, 10, 1547–1570,Short summary
Ice penetrating radar enables determination of the basal properties of ice sheets. Existing algorithms assume stationarity in the attenuation rate, which is not justifiable at an ice sheet scale. We introduce the first ice-sheet-wide algorithm for radar attenuation that incorporates spatial variability, using the temperature field from a numerical model as an initial guess. The study is a step toward ice-sheet-wide data products for basal properties and evaluation of model temperature fields.
Hongyu Zhu, Noemi Petra, Georg Stadler, Tobin Isaac, Thomas J. R. Hughes, and Omar Ghattas
The Cryosphere, 10, 1477–1494,Short summary
We study how well the basal geothermal heat flux can be inferred from surface velocity observations using a thermomechanically coupled nonlinear Stokes ice sheet model. The prospects and limitations of this inversion is studied in two and three dimensional model problems. We also argue that a one-way coupled approach for the adjoint equations motivated by staggered solvers for forward multiphysics problems can lead to an incorrect gradient and premature termination of the optimization iteration.
Eythor Gudlaugsson, Angelika Humbert, Thomas Kleiner, Jack Kohler, and Karin Andreassen
The Cryosphere, 10, 751–760,Short summary
This paper explores the influence of a subglacial lake on ice dynamics and internal layers by means of numerical modelling as well as simulating the effect of a subglacial drainage event on isochrones. We provide an explanation for characteristic dip and ridge features found at the edges of many subglacial lakes and conclude that draining lakes can result in travelling waves at depth within isochrones, thus indicating the possibility of detecting past drainage events with ice penetrating radar.
T. Hughes, A. Sargent, J. Fastook, K. Purdon, J. Li, J.-B. Yan, and S. Gogineni
The Cryosphere, 10, 193–225,Short summary
The Antarctic and Greenland ice sheets are drained primarily by fast ice streams that end as ice shelves if they become afloat. Smooth transitions from slow sheet flow to fast stream flow to confined shelf flow are obtained and applied to Byrd Glacier in Antarctica after two upstream subglacial lakes suddenly drained in 2006, and to Jakobshavn Isbrae in Greenland after a confined ice shelf suddenly disintegrated in 2002. Byrd Glacier quickly stabilized, but Jakobshavn Isbrae remains unstable.
F. Saito, A. Abe-Ouchi, K. Takahashi, and H. Blatter
The Cryosphere, 10, 43–63,Short summary
This article, as the title denotes, is a follow-up study of an ice-sheet intercomparison project SeaRISE, which focuses on the response of the Greenland ice sheet to future global warming. The projections of the different SeaRISE prticipants show diversion, which has not been examined in detail to date. This study detects the main sources of the diversion by a number of sensitivity experiments and shows the importance of initialization methods as well as climate forcing methods.
P. Kuipers Munneke, S. R. M. Ligtenberg, B. P. Y. Noël, I. M. Howat, J. E. Box, E. Mosley-Thompson, J. R. McConnell, K. Steffen, J. T. Harper, S. B. Das, and M. R. van den Broeke
The Cryosphere, 9, 2009–2025,Short summary
The snow layer on top of the Greenland Ice Sheet is changing: it is thickening in the high and cold interior due to increased snowfall, while it is thinning around the margins. The marginal thinning is caused by compaction, and by more melt. This knowledge is important: there are satellites that measure volume change of the ice sheet. It can be caused by increased ice discharge, or by compaction of the snow layer. Here, we quantify the latter, so that we can translate volume to mass change.
T. Goelles, C. E. Bøggild, and R. Greve
The Cryosphere, 9, 1845–1856,Short summary
Soot (black carbon) and dust particles darken the surface of ice sheets and glaciers as they accumulate. This causes more ice to melt, which releases more particles from within the ice. This positive feedback mechanism is studied with a new two-dimensional model, mimicking the conditions of Greenland, under different climate warming scenarios. In the warmest scenario, the additional ice sheet mass loss until the year 3000 is up to 7%.
S. H. R. Rosier, G. H. Gudmundsson, and J. A. M. Green
The Cryosphere, 9, 1649–1661,Short summary
We use a full-Stokes model to investigate the long period modulation of Rutford Ice Stream flow by the ocean tide. We find that using a nonlinear sliding law cannot fully explain the measurements and an additional mechanism, whereby tidally induced subglacial pressure variations are transmitted upstream from the grounding line, is also required to match the large amplitude and decay length scale of the observations.
The Cryosphere, 9, 1169–1181,Short summary
Floating ice shelves extend the continental ice of Antarctica seawards and mediate ice-ocean interactions. Many ice shelves are incised with channels where basal melting is enhanced. With data and modeling it is shown how the channel geometry depends on basal melting and along-flow advection (also for channels which are not freely floating), and how channel formation imprints the general flow pattern. This opens up the opportunity to map the channel formation from surface velocities only.
P. R. Holland, A. Brisbourne, H. F. J. Corr, D. McGrath, K. Purdon, J. Paden, H. A. Fricker, F. S. Paolo, and A. H. Fleming
The Cryosphere, 9, 1005–1024,Short summary
Antarctic Peninsula ice shelves have collapsed in recent decades. The surface of Larsen C Ice Shelf is lowering, but the cause of this has not been understood. This study uses eight radar surveys to show that the lowering is caused by both ice loss and a loss of air from the ice shelf's snowpack. At least two different processes are causing the lowering. The stability of Larsen C may be at risk from an ungrounding of Bawden Ice Rise or ice-front retreat past a 'compressive arch' in strain rates.
M. Bügelmayer, D. M. Roche, and H. Renssen
The Cryosphere, 9, 821–835,
A. Diez and O. Eisen
The Cryosphere, 9, 367–384,
A. Diez, O. Eisen, C. Hofstede, A. Lambrecht, C. Mayer, H. Miller, D. Steinhage, T. Binder, and I. Weikusat
The Cryosphere, 9, 385–398,
R. Calov, A. Robinson, M. Perrette, and A. Ganopolski
The Cryosphere, 9, 179–196,Short summary
Ice discharge into the ocean from outlet glaciers is an important component of mass loss of the Greenland ice sheet. Here, we present a simple parameterization of ice discharge for coarse resolution ice sheet models, suitable for large ensembles or long-term palaeo simulations. This parameterization reproduces in a good approximation the present-day ice discharge compared with estimates, and the simulation of the present-day ice sheet elevation is considerably improved.
V. Helm, A. Humbert, and H. Miller
The Cryosphere, 8, 1539–1559,
A. Robinson and H. Goelzer
The Cryosphere, 8, 1419–1428,
G. R. Leguy, X. S. Asay-Davis, and W. H. Lipscomb
The Cryosphere, 8, 1239–1259,
A. A. Borsa, G. Moholdt, H. A. Fricker, and K. M. Brunt
The Cryosphere, 8, 345–357,
I. Joughin, B. E. Smith, D. E. Shean, and D. Floricioiu
The Cryosphere, 8, 209–214,
C. P. Borstad, E. Rignot, J. Mouginot, and M. P. Schodlok
The Cryosphere, 7, 1931–1947,
V. R. Barletta, L. S. Sørensen, and R. Forsberg
The Cryosphere, 7, 1411–1432,
I. Joughin, S. B. Das, G. E. Flowers, M. D. Behn, R. B. Alley, M. A. King, B. E. Smith, J. L. Bamber, M. R. van den Broeke, and J. H. van Angelen
The Cryosphere, 7, 1185–1192,
A. Aschwanden, G. Aðalgeirsdóttir, and C. Khroulev
The Cryosphere, 7, 1083–1093,
J. De Rydt, G. H. Gudmundsson, H. F. J. Corr, and P. Christoffersen
The Cryosphere, 7, 407–417,
W. Leng, L. Ju, M. Gunzburger, and S. Price
The Cryosphere, 7, 19–29,
F. Gillet-Chaulet, O. Gagliardini, H. Seddik, M. Nodet, G. Durand, C. Ritz, T. Zwinger, R. Greve, and D. G. Vaughan
The Cryosphere, 6, 1561–1576,
S. Fujita, P. Holmlund, K. Matsuoka, H. Enomoto, K. Fukui, F. Nakazawa, S. Sugiyama, and S. Surdyk
The Cryosphere, 6, 1203–1219,
B. Scheuchl, J. Mouginot, and E. Rignot
The Cryosphere, 6, 1019–1030,
D. Pollard and R. M. DeConto
The Cryosphere, 6, 953–971,
A. D. Booth, R. A. Clark, B. Kulessa, T. Murray, J. Carter, S. Doyle, and A. Hubbard
The Cryosphere, 6, 909–922,
V. Zagorodnov, O. Nagornov, T. A. Scambos, A. Muto, E. Mosley-Thompson, E. C. Pettit, and S. Tyuflin
The Cryosphere, 6, 675–686,
A. Levermann, T. Albrecht, R. Winkelmann, M. A. Martin, M. Haseloff, and I. Joughin
The Cryosphere, 6, 273–286,
P. Christoffersen, R. I. Mugford, K. J. Heywood, I. Joughin, J. A. Dowdeswell, J. P. M. Syvitski, A. Luckman, and T. J. Benham
The Cryosphere, 5, 701–714,
J. Bamber and R. Riva
The Cryosphere, 4, 621–627,
R. W. Jacobel, K. E. Lapo, J. R. Stamp, B. W. Youngblood, B. C. Welch, and J. L. Bamber
The Cryosphere, 4, 447–452,
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The atmospheric temperature increase poses a real risk of ice sheets collapsing. We show that this risk might have been underestimated since variations in temperature will move the ice sheets to the tipping point of destabilization. We show this by using a simple computer model of a large ice sheet and investigate what happens if the temperature varies from year to year. The total volume of the ice sheet decreases because a cold year followed by an equally warm year do not cancel out.
The atmospheric temperature increase poses a real risk of ice sheets collapsing. We show that...