Retrogressive slumping could accelerate sea-level rise if ice-sheet retreat generates ice cliffs much taller than observed today. The tallest ice cliffs, which extend roughly 100 m above sea level, calve only after ice-flow processes thin the ice to near flotation. Above some ice-cliff height limit, the stress state in ice will satisfy the material-failure criterion, resulting in faster brittle failure. New terrestrial radar data from Helheim Glacier, Greenland, suggest that taller subaerial cliffs are prone to failure by slumping, unloading submarine ice to allow buoyancy-driven full-thickness calving. Full-Stokes diagnostic modeling shows that the threshold cliff height for slumping is likely slightly above 100 m in many cases, and roughly twice that (145–285 m) in mechanically competent ice under well-drained or low-melt conditions.
Ocean-ending ice in almost all cases forms a near-vertical cliff (Hollin, 1962) from which icebergs break (e.g., James et al., 2014; Murray et al., 2015). Faster submarine mass loss produces an overhang that breaks off to restore the cliff; faster subaerial mass loss creates analogous buoyant submerged ice that breaks and “falls” upward (Wagner et al., 2016). Observations show that sufficient warming removes floating ice shelves (Alley et al., 2015), leaving grounded ice cliffs from which icebergs break off directly. Recent modeling (DeConto and Pollard, 2016) and data (Wise et al., 2017) indicate that this could greatly accelerate ice-sheet mass loss, particularly within wide embayments in which outlet glaciers flow over retrograde beds.
The tallest modern subaerial ice cliffs are ∼100 m (Scambos et al., 2011; Nick et al., 2013), occurring where ice shelves have been lost recently. Observations of these (e.g., Joughin et al., 2012) indicate that a large calving event generally shifts the cliff into ice too thick to float, although the cliff remains too short for rapid spontaneous failure. The loss of buttressing from the calved block is compensated by resistive stresses englacially and/or by increased side or basal drag upglacier. This increases deviatoric stresses and strain rates in and near the grounding zone (transitional region between grounded and floating ice), causing flow thinning until the ice approaches flotation. This likely allows greater tidal bending or other stresses and reduces basal friction, leading to another calving event (Joughin et al., 2012).
Taller cliffs have greater stress imbalance. Above some cliff-height threshold, which may slightly exceed 100 m (Hanson and Hooke, 2003; Bassis and Walker, 2012), the internal deviatoric stresses would exceed the ice strength, causing rapid failure without thinning to flotation. The pressure in a glacier from the weight of ice above exceeds that in adjacent air and water, with the largest imbalance immediately above the water line (Fig. 1A; Fig. DR3A in the GSA Data Repository1). Tension in the ice opposes the resulting horizontal differential pressure and tends to open near-vertical surface crevasses, as well as generates vertical deviatoric compression. The combined stresses promote failure along a surface slanting down to near the water line, favoring slumping (Fig. 1B; Figs. DR3B and DR3C) entirely analogous to slumping of rock cliffs and hillslopes. Removal of the weight of the slump mass produces an excess upward buoyancy force and an offsetting distribution of enhanced englacial deviatoric stresses, promoting basal crevassing and submarine block-rotation calving (Fig. 1B; Figs. DR3D–DR3F) (e.g., Murray et al., 2015), as documented for Jakobshavn Isbræ, Greenland (Walter et al., 2012; Sergeant et al., 2016).
Because the tallest modern ice cliffs are shorter than the estimated threshold height for rapid failure, we do not now expect frequent, spontaneous slumping followed by submarine calving. However, because the tallest cliffs approach the threshold height, we hypothesize that slumping followed by submarine calving would sometimes be observed, likely aided by tidal flexure or other processes. To test this hypothesis, we examined Greenland data, finding that, at times, slumping occurs near the onset of full-thickness calving. We also coupled new modeling with fracture-mechanics calculations to refine estimates of limiting cliff heights.
From 9 to 15 August 2014, we imaged the elevation and line-of-sight (LOS) velocity of Helheim Glacier, East Greenland (Fig. 2), with a real-aperture terrestrial radar interferometer (e.g., Voytenko et al., 2015) (Fig. DR2; see methods in the Data Repository). Here we focus on one major calving event (06:00–07:00 UTC, 12 August 2014; Figs. 1 and 3).
Beginning ∼2 h before the event, acceleration along flow began for the ice behind the highest, northern ∼1.5 km of the ∼6-km-wide calving cliff, extending ∼400 m upglacier to a series of prominent, recently expanding crevasses. Acceleration likely continued until failure, with loss of interferometric coherency beginning ∼10 min before calving, likely due to motion becoming too rapid to be captured by the sampling interval. Total anomalous forward motion of this block relative to the rest of the glacier prior to loss of coherency was <0.5 m. This northern mass was rotating top-upglacier (Figs. 3A and 3C) just before full-thickness calving (Figs. 1B, 3A, and 3C), consistent with a classic slump along a listric (concave upward) surface.
The initial slump was followed immediately by a large, full-thickness calving event across all except the southern ∼1.5 km of the calving front (Fig. 3A), with the top rotating upglacier and temporarily reversing glacier motion (Figs. 3B and 3C). The post-calving subaerial glacier-front height stabilized at ∼90 m (Figs. 1C and 3C). A similar Helheim slumping event, followed almost immediately by full-thickness calving, was filmed on 12 July 2010 (James et al., 2014).
Our data are in agreement with earlier GPS measurements of block rotation reversing glacier motion during calving (Murray et al., 2015). We also find a staged failure progressing along the ice front, starting with slumping of ice above and near the water line. Prior to our slumping event, the calving front was near or possibly at flotation, with unloading by slumping of the highest part of the calving front rapidly producing a large buoyancy-force imbalance (Fig. 1B; Fig. DR3D), allowing buoyancy-driven block rotation without ice-front advancement into deeper water.
Similar slump failures of a subaerial cliff (∼100 m high) have been observed at Jakobshavn Isbræ, followed by large calving events that removed the ice beneath. Time matching of seismic recordings and video footage of one such event showed that a smaller-amplitude precursor signal matches the timing of the slumping event, followed by a characteristic, larger-amplitude, block rotation–sourced signal (Walter et al., 2012). Furthermore, in 38 of the 76 glacial earthquake events that were analyzed across Jakobshavn Isbræ, Helheim Glacier, and Kangerdlugssuaq Glacier in Greenland, similar precursory seismic signals were found, occurring 2–12 min before the main calving and/or block-rotation event (Olsen and Nettles, 2017). Other sources of precursory events are possible, including basal-crevasse propagation prior to block rotation triggering the slumping, but the coincident timing of listric failures with their precursory seismic signals is consistent with our hypothesis that slumping triggers block rotation.
We use (1) a diagnostic plane-strain Stokes model of idealized outlet glacier flow and (2) engineering and glaciological failure criteria (e.g., Schulson, 2001; see methods in the Data Repository) to mechanistically assess ice-cliff failure. Through large-ensemble numerical experiments, we evaluate tensile, shear, and compressive failure for a range of subaerial cliff heights as well as material and boundary parameters.
The ice-front pressure difference generates tensile ice stresses that tend to open nearly vertical surface crevasses behind the cliff. These may reach the bed if meltwater filled (Alley et al., 2005), but if well drained, extend downward a maximum of ∼55%–60% of the subaerial cliff height in the limit of zero tensile ice strength (Fig. DR4A), at which point there is a transition to compression (Nye, 1957). (In crevasse fields such as those observed on Helheim’s surface [Fig. DR2], penetration depths of air-filled crevasses are reduced somewhat [van der Veen, 1998] by stress interactions.)
Failure deeper than surface crevassing is most likely through compressive wing-crack growth (Renshaw and Schulson, 2001), which occurs for lower stresses than shear failure (see methods in the Data Repository). Along-flow tension and coupled vertical compression behind an ice cliff cause shear slip along favorably oriented grain boundaries or other features (half-length c), which produces secondary wing cracks, if the greatest principal compressive strain rate exceeds the ductile-to-brittle transition (Fig. DR4C) so creep does not blunt the cracks and relieve the stress. Subsequent linkage of neighboring wing cracks creates a throughgoing damage zone angling down toward the cliff near the water line (Figs. DR4C and DR4D; Moore et al., 2010). For grain-face cracks of c = 5 cm (see methods in the Data Repository) and a low-end estimate of ice fracture toughness (critical stress intensity factor KIc = 0.05 MPa m1/2; Moore et al., 2010), a connected failure zone forms from upper surface to cliff face for a minimum subaerial cliff height of ∼200 m (145–285 m for doubling and halving c, respectively; Fig. 4). This height range agrees with a depth-integrated analytical estimate for failure of intact ice (Bassis and Walker, 2012).
This range provides an upper limit on stable cliff height, however, because (1) failure mechanisms can interact, reducing total strength; (2) we have not included subcritical crack growth, the resulting deformation, and subsequent stress loading of the remaining, still-competent ice; and (3) preexisting cracks much larger than individual grain boundaries typically exist at tall marine-terminating ice fronts that currently end in grounded or nearly grounded ice cliffs (Fig. DR2). Water pressure–driven fracturing from the surface also may lower the maximum stable cliff height.
DISCUSSION AND IMPLICATIONS
The Helheim Glacier and Jakobshavn Isbræ calving events highlighted here involved crevasse opening producing a near-vertical “head scarp”, then block rotation likely on a listric surface extending downward and forward from the scarp base to the calving cliff just below the water line. Similar rotational slumps are widespread on hillslopes, and are suggested by previous modeling of subaerial ice cliffs (Hanson and Hooke, 2003; Bassis and Walker, 2012).
Near the water line, ice mélange can provide some cliff stability (e.g., Amundson et al., 2010) (see methods in the Data Repository), and the increase in water pressure with depth reduces the glaciostatic-hydrostatic stress imbalance across the calving front, so this imbalance is highest at the top of the mélange or at the water level in the absence of strong mélange. Any seawater seeping upglacier along cracks in the cliff face, or englacial drainage to a water table graded to the water line, would favor cliff instability by lubricating any failure surface and reducing the surface energy of new cracks (Griffith, 1921) and thus the work required for wing-crack formation. A failure surface running almost horizontally upglacier from just below the waterline could thereby connect through a failure surface from wing-crack interactions rising upglacier to the base of a tensile crevasse. Such a pattern matches our observations, and perhaps also those at Jakobshavn Isbræ (Walter et al., 2012), where nearly horizontal first seismic motion at the start of slumping may represent failure of the lower part of a listric surface and where precursor signals indicative of slumping occurred in >50% of the 43 recorded calving events between 2009 and 2013 (Olsen and Nettles, 2017).
Importantly, extensive and pervasive preexisting cracks near calving cliffs are much larger than the millimeter- to centimeter-scale grain size (to >10 m) (Fig. DR2), reducing maximum stable cliff height (Fig. 4; see methods in the Data Repository). The 200-m-high ice front of Eqip Sermia in west Greenland is not vertical but slopes notably (∼48°) (Engell, 2014; Lüthi and Vieli, 2016). Repeated subaerial slumping failure within its narrow fjord (∼4 km) indicates that the ice is too weak to support 200 m cliffs. (Shallow calving-front water depth of only 20–40 m limits buoyant ice loss after slumping, preventing formation of a vertical cliff and thus providing stability against fast calving retreat.) The observed slumping of ∼100 m cliffs, despite some stabilization from side drag and possibly from mélange, similarly suggests that the ice of Helheim Glacier and Jakobshavn Isbræ is much weaker than intact, undamaged ice, and/or we are still omitting key elements of the relevant mechanics.
Higher cliffs are more prone to failure. While stress increases linearly with cliff height, and creep inhibiting brittle failure increases approximately with the third power of the stress (see methods in the Data Repository), the rate of subcritical crack growth promoting failure increases approximately as the 30th power (e.g., Atkinson, 1984; Hallet, 1996) of the stress intensity. Therefore, higher cliffs have relatively less viscous relaxation due to the outpacing rate of crack growth.
Future cliff failure–driven rates of ice-sheet retreat and sea-level rise are likely to depend at least partially on the amount of surface melting, and on the rate at which damage occurs behind a cliff retreating into previously intact ice. Nevertheless, sufficiently high cliffs (∼200 m) are likely to retreat by brittle failure, and thus to cause rapid sea-level rise at rates that surpass those predicted by parameterizations that limit wastage rates to those that have been recently observed.
The work was supported by NASA grants NNX15AH84G (Parizek), NNX16AM01G (Christianson), NNX12AB69G (Christianson and Holland), NNX12AK29G (Dixon), and NNX12AD03A (Walker); U.S. National Science Foundation grants PLR-1443190 (Parizek), ANT-0424589 (Parizek and Alley), AGS-1338832 (Parizek and Alley), and PLR-1443284 (Walker); and New York University Abu Dhabi Research Institute grant G1204 (Holland). The field data-collection effort was organized by Denise Holland. We thank Kira Olsen for fruitful discussions, and both Kira Olsen and Meredith Nettles for freely sharing their scientific results. Finally, we would like to recognize the efforts of the Science Editor, Dennis Brown, and Ravindra Duddu and two anonymous reviewers, whose critical reviews were invaluable.