On the Greenland Ice Sheet, hydrofracture connects the supraglacial and subglacial hydrologic systems, coupling surface runoff dynamics and ice velocity. In recent decades, the growth of low-permeability ice slabs in the wet snow zone has expanded Greenland's runoff zone, but observations suggest that surface-to-bed connections are rare, because meltwater drains through crevasses into the porous firn beneath ice slabs. However, there is little quantitative evidence confirming the absence of surface-to-bed fracture propagation. Here, we use poromechanics to investigate whether water-filled crevasses in ice slabs can propagate vertically through an underlying porous firn layer. Based on numerical simulations, we develop an analytical estimate of the water injection-induced effective stress in the firn given the water level in the crevasse, ice slab thickness, and firn properties. We find that the firn layer substantially reduces the system's vulnerability to hydrofracture because much of the hydrostatic stress is accommodated by a change in pore pressure, rather than being transmitted to the solid skeleton. This result suggests that surface-to-bed hydrofracture will not occur in ice slab regions until all pore space proximal to the initial flaw has been filled with solid ice.

Convolutional neural networks (CNN) trained from high-order ice-flow model realisations have proven to be outstanding emulators in terms of fidelity and computational performance. However, the dependence on an ensemble of realisations of an instructor model renders this strategy difficult to generalise to a variety of ice-flow regimes found in the nature. To overcome this issue, we adopt the approach of physics-informed deep learning, which fuses traditional numerical solutions by finite differences/elements and deep-learning approaches. Here, we train a CNN to minimise the energy associated with high-order ice-flow equations within the time iterations of a glacier evolution model. As a result, our emulator is a promising alternative to traditional solvers thanks to its high computational efficiency (especially on GPU), its high fidelity to the original model, its simplified training (without requiring any data), its capability to handle a variety of ice-flow regimes and memorise previous solutions, and its relatively simple implementation. Embedded into the ‘Instructed Glacier Model’ (IGM) framework, the potential of the emulator is illustrated with three applications including a large-scale high-resolution (2400x4000) forward glacier evolution model, an inverse modelling case for data assimilation, and an ice shelf.

Data assimilation in high-order ice flow modeling is a challenging and computationally costly task, yet crucial to find ice thickness and ice flow parameter distributions that are consistent with ice flow mechanics and mass balance while best matching observations. Failing to find these distributions that are required as initial conditions leads to a disequilibrium between mass balance and ice flow, resulting in nonphysical transient effects in the prognostic model. Here we tackle this problem by inverting an emulator of the Stokes ice flow model based on deep learning. By substituting the ice flow equations using a convolutional neural network emulator, we simplify, make more robust and dramatically speed up the solving of the underlying optimization problem thanks to automatic differentiation, stochastic gradient methods and implementation of graphics processing unit (GPU). We demonstrate this process by simultaneously inferring the ice thickness distribution, ice flow parametrization and ice surface of ten of the largest glaciers in Switzerland. As a result, we obtain a high degree of assimilation while guaranteeing an equilibrium between mass-balance and ice flow mechanics. The code runs very efficiently (optimizing one large-size glacier at 100 m takes < 1 min on a laptop) while it is open-source and publicly available.

Glacier motion responds dynamically to changing meltwater inputs, but the multi-decadal response of basal sliding to climate remains poorly constrained due to its sensitivity across multiple timescales. Observational records of glacier motion provide critical benchmarks to decode processes influencing glacier dynamics, but multi-decadal records that precede satellite observation and modern warming are rare. Here we present a record of motion in the ablation zone of Saskatchewan Glacier that spans seven decades. We combine in situ and remote-sensing observations to inform a first-order glacier flow model used to estimate the relative contributions of sliding and internal deformation on dynamics. We find a significant increase in basal sliding rates between melt-seasons in the 1950s and those in the 1990s and 2010s and explore three process-based explanations for this anomalous behavior: (i) the glacier surface steepened over seven decades, maintaining flow-driving stresses despite sustained thinning; (ii) the formation of a proglacial lake after 1955 may support elevated basal water pressures; and (iii) subglacial topography may cause dynamic responses specific to Saskatchewan Glacier. Although further constraints are necessary to ascertain which processes are of greatest importance for Saskatchewan Glacier's dynamic evolution, this record provides a benchmark for studies of multi-decadal glacier dynamics.

Glacier slip is usually described using steady-state sliding laws that relate drag, slip velocity and effective pressure, but where subglacial conditions vary rapidly transient effects may influence slip dynamics. Here we use results from a set of laboratory experiments to examine the transient response of glacier slip over a hard bed to velocity perturbations. The drag and cavity evolution from lab experiments are used to parameterize a rate-and-state drag model that is applied to observations of surface velocity and ice-bed separation from the Greenland ice sheet. The drag model successfully predicts observed lags between changes in ice-bed separation and sliding speed. These lags result from the time (or displacement) required for cavities to evolve from one steady-state condition to another. In comparing drag estimates resulting from applying rate-and-state and steady-state slip laws to transient data, we find the peaks in drag are out of phase. This suggests that in locations where subglacial conditions vary on timescales shorter than those needed for cavity adjustment transient slip processes control basal drag.

Changes in water pressure at the beds of glaciers greatly modify their sliding rate, affecting rates of ice mass loss and sea level change. However, there is still no agreement about the physics of subglacial sliding or how water affects it. Here, we present a new simplified physical model for the effect of transient subglacial hydrology on basal ice velocity. This model assumes that a fraction of the glacier bed is connected by an active hydrologic system that, when averaged over an appropriate scale, is governed by two parameters with limited spatial variability. The sliding model is reminiscent of Budd's empirical sliding law but with fundamental differences including a dependence on the fractional area of the active hydrologic system. With periodic surface meltwater forcing, the model displays classic diffusion-wave behavior, with a downstream time lag and decay of subglacial water pressure perturbations. Testing the model against Greenland observations suggests that, despite its simplicity, it captures key features of observed proglacial discharges and ice velocities with reasonable physical parameter values. Given these encouraging findings, including this sliding model in predictive ice-sheet models may improve their ability to predict time-evolving velocities and associated sea level change and reduce the related uncertainties.

Ice stream discharge responds to a balance between gravity, basal friction and lateral drag. Appreciable viscous heating occurs in shear margins between ice streams and adjacent slow-moving ice ridges, altering the temperature-dependent viscosity distribution that connects lateral drag to marginal strain rates and ice stream velocity. Warmer ice deforms more easily and accommodates faster flow, whereas cold ice supplied from ice ridges drives advective cooling that counteracts viscous heating. Here, we present a two-dimensional (three velocity component), steady-state model designed to explore the thermal controls on ice stream shear margins. We validate our treatment through comparison with observed velocities for Bindschadler Ice Stream and verify that calculated temperatures are consistent with results from previous studies. Sweeping through a parameter range that encompasses conditions representative of ice streams in Antarctica, we show that modeled steady-state velocity has a modest response to different choices in forcing up until temperate zones develop in the shear margins. When temperate zones are present, velocity is much more sensitive to changes in forcing. We identify key scalings for the emergence of temperate conditions in our idealized treatment that can be used to identify where thermo-mechanical feedbacks influence the evolution of the ice sheet.

The representation of iceberg calving in numerical models is a key source of uncertainty in century-scale sea-level rise projections. Parameters central to model representations of calving, including the tensile strength of glacier ice, remain poorly constrained. Grain-size and sample-size dependence make it difficult to reconcile laboratory and in situ estimates of ice tensile strength. Further, assumptions of various numerical models obscure comparison of the ‘strength’ parameter with a physically observable value. Here, we address the problem of fracture during calving using an analogous natural laboratory: a viscoelastic analysis of observed surface deformation and associated stresses in the 2015 collapse of eastern Skaftá cauldron, Vatnajökull ice cap, Iceland. We find that the ice within the cauldron could have experienced instantaneous elastic stress on the order of several MPa. We observe surface crevasses at the cauldron edges and center, but find that large areas of ice remain intact despite high stress. Our findings suggest a tensile strength of glacier ice on the order of 1 MPa, consistent with laboratory estimates but exceeding previous glacier-specific estimates.

Seismic studies of glaciers yield insights into spatio-temporal processes within and beneath glaciers on scales relevant to flow and deformation of the ice. These methods enable direct monitoring of the bed in ways that complement other geophysical techniques, such as geodetic or ground penetrating radar observations. In this work, we report on the analysis of passive seismic data collected from the interior of the North East Greenland Ice Stream, the Greenland ice sheet's largest outlet glacier. We record thousands of basal earthquakes, many of which repeat with nearly identical waveforms. We also record many long-duration glacial tremor episodes that migrate across the seismic network with slow velocities (e.g. ~4–12 m s−1). Analysis of the basal earthquakes indicates a transition between times of individual event activity and times of tremor activity. We suggest that both processes are produced by shear slip at localized asperities along the bed. The transition between discrete and quasi-continuous slipping modes may be driven by pore-water pressure transients or heterogeneous strain accumulation in the ice due to strength contrasts of the underlying till.

Hydrofracturing can enhance the depth to which crevasses propagate and, in some cases, allow full depth crevasse penetration and iceberg detachment. However, many existing crevasse models either do not fully account for the stress field driving the hydrofracture process and/or treat glacier ice as elastic, neglecting the non-linear viscous rheology. Here, we present a non-local continuum poro-damage mechanics (CPDM) model for hydrofracturing and implement it within a full Stokes finite element formulation. We use the CPDM model to simulate the propagation of water-filled crevasses in idealized grounded glaciers, and compare crevasse depths predicted by this model with those from linear elastic fracture mechanics (LEFM) and zero stress models. We find that the CPDM model is in good agreement with the LEFM model for isolated crevasses and with the zero stress model for closely-spaced crevasses, until the glacier approaches buoyancy. When the glacier approaches buoyancy, we find that the CPDM model does not allow the propagation of water-filled crevasses due to the much smaller size of the tensile stress region concentrated near the crevasse tip. Our study suggests that the combination of non-linear viscous and damage processes in ice near the tip of a water-filled crevasse can alter calving outcomes.

Subglacial rock friction is an important control on the sliding dynamics and erosive potential of hard-bedded glaciers, yet it remains largely unconstrained. To explore the relative influence of basal melt rate, effective stress and ice temperature on frictional resistance, we conducted abrasion experiments in which limestone beds were slid beneath a fixed slab of ice laden with granitic rock fragments. Shear stress scales linearly with melt rate and cryostatic stress, confirming that both viscous drag and effective stress are first-order controls on the contact force in drained conditions. Furthermore, temperature gradients in the ice increase the contribution of viscous drag on basal shear stress. In all experiments, the relationship between melt rate and shear stress is best explained by a model that accounts for the effects of regelation and viscous creep on the bed-normal drag force. We interpret this to mean fluid flow around entrained clasts contributed to basal drag even at subfreezing temperatures. Incorporating premelting dynamics into the Watts/Hallet model for subglacial rock friction, we find that the predicted debris-bed drag decreases by approximately an order of magnitude, with a corresponding ~3.5 × increase in the transition radius. This is lower than we observe for ice slightly below the pressure melting point.

Modelling paleo-glacier networks in mountain ranges on the millennial timescales requires ice flow approximations. Hybrid models calculating ice flow by combining vertical shearing (shallow ice approximation) and longitudinal stretching (shallow shelf approximation) have been applied to model paleo-glacier networks on steep terrain, yet their validity has not yet been assessed quantitatively. Moreover, hybrid models consistently yield higher ice thicknesses than Last Glacial Maximum geomorphological reconstructions in the European Alps. Here, we compare results based on the hybrid Parallel Ice Sheet Model (PISM) and the Stokes model Elmer/Ice on the Rhine Glacier, a catchment of the former European Alpine Icefield. For PISM, we also test two magnitudes of flux limitation in a scheme that reduces shearing velocities. We find that the flux limitation typically used in PISM yields significantly reduced shearing speeds and increases ice thicknesses by up to 500 m, partly explaining previous overestimations. However, reducing the ice flux limitation allows the hybrid model to minimize this mismatch and captures sliding speeds, ice thicknesses, ice extent and basal temperatures in close agreement with those obtained with the Stokes model.

Johannes Weertman published his first glaciological paper in 1957 only 5 years after getting his DSc in metallurgy from the Carnegie Institute of Technology. The paper presented the very first sliding law developed quantitatively from first principles, and involved the unconventional idealization of bed roughness using cubic ‘tombstones’ of rock. Since 1957, there has been a great deal of progress in understanding glacier sliding, but few studies, if any, have preserved the original tombstone geometry that was a hallmark of this first theory. The current study presents a partial reanalysis of the sliding process over a bed with tombstone obstacles using modern numerical methods. The result confirms the enduring applicability of Weertman's model as a pedagogical tool and motivates new questions about (1) folding flow near bedrock obstacles that invert normal ice stratigraphy, (2) the presence and role of stress singularities on sharp edges of bedrock, and (3) the validity of a presumption that regelation flow can be plug-like.

The evolution of glaciers and ice sheets depends on processes in the subglacial environment. Shear seismicity along the ice–bed interface provides a window into these processes. Such seismicity requires a rapid loss of strength that is typically ascribed to rate-weakening friction, i.e., decreasing friction with sliding or sliding rate. Many friction experiments have investigated glacial materials at the temperate conditions typical of fast flowing glacier beds. To our knowledge, however, these studies have all found rate-strengthening friction. Here, we investigate the possibility that rate-weakening rock-on-rock friction between sediments frozen to the bottom of the glacier and the underlying water-saturated sediments or bedrock may be responsible for subglacial shear seismicity along temperate glacier beds. We test this ‘entrainment-seismicity hypothesis’ using targeted laboratory experiments and simple models of glacier sliding, seismicity and sediment entrainment. These models suggest that sediment entrainment may be a necessary but not sufficient condition for the occurrence of basal shear seismicity. We propose that stagnation at the Whillans Ice Stream, West Antarctica may be caused by the growth of a frozen fringe of entrained sediment in the ice stream margins. Our results suggest that basal shear seismicity may indicate geomorphic activity.

The non-linearly viscous ice flow in the vicinity of a glacier terminus, an observation region, depends crucially on the upstream flow as well as on the local surface and bed conditions. The former requires a likely complex solution of the balance laws and boundary conditions for the complete glacier. However, if the profile and downstream surface tangential velocity in the observation region are measured at an observation time t = 0, and a two-dimensional flow approximation is satisfactory, the complete stress and velocity fields satisfying local reduced model equations in the observation region at time t = 0 can be determined by asymptotic expansions in upstream distance from the (moving) terminus. Thus the full strain-rate and stress tensors are determined without prescribing the basal conditions. The terminus velocity is determined in terms of the net accumulation or melt flux and surface velocity at the terminus, with bounds for advance or retreat. The analysis and illustration are presented for a plane flow approximation.

Basal slip accounts for a large part of the flow of ice streams draining ice from Antarctica and Greenland into the ocean. Therefore, an appropriate representation of basal slip in ice flow models is a prerequisite for accurate sea level rise projections. Various friction laws have been proposed to describe basal slip in models. Here, we compare the influence on grounding line (GL) dynamics of four friction laws: the traditional Weertman law and three effective pressure-dependent laws, namely the Schoof, Tsai and Budd laws. It turns out that, even when they are tuned to a common initial reference state, the Weertman, Budd and Schoof laws lead to thoroughly different steady-state positions, although the Schoof and Tsai laws lead to much the same result. In particular, under certain circumstances, it is possible to obtain a steady GL located on a reverse slope area using the Weertman law. Furthermore, the predicted transient evolution of the GL as well as the projected contributions to sea level rise over a 100-year time horizon vary significantly depending on the friction law. We conclude on the importance of choosing an appropriate law for reliable sea level rise projections and emphasise the need for a coupling between ice flow models and physically based subglacial hydrological models.

Understanding grounding-line dynamics is necessary for predictions of long-term ice-sheet stability. However, despite growing observations of the tidal influence on grounding-line migration, this short-timescale migration is poorly understood, with most modeling attempts assuming beam theory to calculate displacements. Here we present an improved model of tidal grounding-line migration that treats migration as an elastic fracture problem, forced by the additional ocean water pressure from the tide. This new model predicts that the grounding line cannot be assumed to be in hydrostatic equilibrium and, furthermore, that migration is inherently asymmetric and nonlinear, with migration distances that are not proportional to the tidal load. Specifically, for constant surface slope, the grounding line migrates upstream approximately ten times further as the tide rises from mean sea level to high tide than it migrates downstream as the tide falls from mean sea level to low tide, and migration distances are substantially larger than simple flotation arguments suggest. Numerical tests also show that the dependence of migration distance on elastic moduli and ice-sheet thickness are inconsistent with predictions of beam theory for a range of realistic values. Finally, applying the new model to observations in Antarctica results in new estimates of bed slopes, though these estimates remain uncertain due to imperfect knowledge of the relevant rheological parameters.

Estimating a glacier’s volume by inferring properties at depth (e.g. bed topography or basal slip) from properties observed at the surface (e.g. area and slope) creates a calculation instability that grows exponentially with the size of the glacier. Random errors from this inversion instability can overwhelm all other sources of error and can corrupt thickness and volume calculations, unless problematic short spatial wavelengths are specifically excluded. Volume/area scaling inherently filters these short wavelengths and automatically eliminates the instability, while numerical inversions can also give stable solutions by filtering the correct wavelengths explicitly, as is frequently done when ‘regularizing’ a model. Each of the scaling and numerical techniques has applications to which it is better suited, and there are trade-offs in resolution and accuracy; but when calculating volume, neither the modeling nor the scaling approach offers a fundamental advantage over the other. Both are significantly limited by the inherently ‘ill-posed’ inversion, and even though both provide stable volume solutions, neither can give unique solutions.

The behavior of marine-terminating ice sheets, such as the West Antarctic ice sheet, is of interest due to the possibility of rapid grounding-line retreat and consequent catastrophic loss of ice. Critical to modeling this behavior is a choice of basal rheology, where the most popular approach is to relate the ice-sheet velocity to a power-law function of basal stress. Recent experiments, however, suggest that near-grounding line tills exhibit Coulomb friction behavior. Here we address how Coulomb conditions modify ice-sheet profiles and stability criteria. The basal rheology necessarily transitions to Coulomb friction near the grounding line, due to low effective stresses, leading to changes in ice-sheet properties within a narrow boundary layer. Ice-sheet profiles ‘taper off’ towards a flatter upper surface, compared with the power-law case, and basal stresses vanish at the grounding line, consistent with observations. In the Coulomb case, the grounding-line ice flux also depends more strongly on flotation ice thickness, which implies that ice sheets are more sensitive to climate perturbations. Furthermore, with Coulomb friction, the ice sheet grounds stably in shallower water than with a power-law rheology. This implies that smaller perturbations are required to push the grounding line into regions of negative bed slope, where it would become unstable. These results have important implications for ice-sheet stability in a warming climate.

The orientations of individual crystals within a polycrystalline aggregate subjected to stress have a strong influence on its bulk strain rate and flow behavior. The ability to include the effect of crystal fabric and recrystallization processes in an ice flow law, especially at the bottom of glaciers and ice sheets where temperature is close to the pressure-melting point, is important because the stratigraphy of the ice body may be affected and the paleoclimate reconstruction hampered. We present herein three newly developed deformation apparatuses offering the possibility, from single experiments, of investigating different finite strain stages and their corresponding c-axis fabric and grain texture patterns in various confined, shear flow configurations (simple shear, pure shear and compression/extension bending). The technical set-ups and major advantages compared to classical methods are explained, and results from experiments are discussed in order to illustrate the functioning and purposes of the methods. In all experiments, significant variations in the microstructural development have been observed that reflect the varying orientations of the anisotropy and its relationship to the stress pattern. In monocrystalline ice-bending experiments, the pre-existing c-axis fabric is shown to have a profound influence on the response to stress and possibly to the type of slip system activated.