The Ice-sheet and Sea-level System Model (ISSM) provides numerical solutions for ice sheet dynamics using finite element and fine mesh adaption. However, considering ISSM is compatible only with central processing units (CPUs), it has limitations in economizing computational time to explore the linkage between climate forcings and ice dynamics. Although several deep learning emulators using graphic processing units (GPUs) have been proposed to accelerate ice sheet modeling, most of them rely on convolutional neural networks (CNNs) designed for regular grids. Since they are not appropriate for the irregular meshes of ISSM, we use a graph convolutional network (GCN) to replicate the adapted mesh structures of the ISSM. When applied to transient simulations of the Pine Island Glacier (PIG), Antarctica, the GCN successfully reproduces ice thickness and velocity with a correlation coefficient of approximately 0.997, outperforming non-graph models, including fully convolutional network (FCN) and multi-layer perceptron (MLP). Compared to the fixed-resolution approach of the FCN, the flexible-resolution structure of the GCN accurately captures detailed ice dynamics in fast-ice regions. By leveraging 60–100 times faster computational time of the GPU-based GCN emulator, we efficiently examine the impacts of basal melting rates on the ice sheet dynamics in the PIG.

The momentum conservation equation for glacier flow can be described through minimization of an action functional. Several software packages for glacier flow modeling use this action principle in the design of numerical solution procedures. We derive here an equivalent dual action principle for the shallow stream approximation and implement this model using the finite element method. The key feature of the dual action is that the flow law and friction law are both inverted, which changes the character of the non-linearities. This altered character makes it possible to implement numerical solvers for the dual form that work even when the ice thickness or strain rate are exactly equal to zero. Solvers for the primal form typically fail on such input data and require regularization of the problem. This robustness makes it possible to implement iceberg calving in a simple way: the modeler sets the ice thickness to zero in the desired area. We provide several demonstrations and a reference implementation.

There follows the open discussion which took place at the IGS symposium on ‘The Edges of Glaciology’, in July 2023. The discussion was curated by Doug Benn. The time of speaking in minutes and seconds into the Panopto recording is given in bold figures. The recording itself is provided as electronic supplementary material. It has been transcribed and edited by Andrew Fowler, with much (and much-needed) assistance from the participants. Footnotes (mostly references) are editorial intrusions.

The time-evolution of glacier basal motion remains poorly constrained, despite its importance in understanding the response of glaciers to climate warming. Athabasca Glacier provides an ideal site for observing changes in basal motion over long timescales. Studies from the 1960s provide an in situ baseline dataset constraining ice deformation and basal motion. We use two complementary numerical flow models to investigate changes along a well-studied transverse profile and throughout a larger study area. A cross-sectional flow model allows us to calculate transverse englacial velocity fields to simulate modern and historical conditions. We subsequently use a 3-D numerical ice flow model, Icepack, to estimate changes in basal friction by inverting known surface velocities. Our results reproduce observed velocities well using standard values for flow parameters. They show that basal motion declined significantly (30–40%) and this constitutes the majority (50–80%) of the observed decrease in surface velocities. At the same time, basal resistive stress has remained nearly constant and now balances a much larger fraction of the driving stress. The decline in basal motion over multiple decades of climate warming could serve as a stabilizing feedback mechanism, slowing ice transport to lower elevations, and therefore moderating future mass loss rates.

Ice-contact lakes modify glacier geometry and dynamics by shifting the majority of mass loss from the ice surface to the terminus. Lake-terminating glaciers are known to experience greater thinning rates and higher velocities than land-terminating glaciers, but the controls on variability in surface elevation change and ice flow between lake-terminating glaciers in different regions remain poorly explored. We combined existing datasets of glacier velocity, surface elevation change and glacial lake area to characterise the evolution of 352 lake-terminating and land-terminating glaciers within three Himalayan sub-regions between 2000 and 2019. These analyses show that the influence of ice-contact lakes propagates up-glacier across only the lowermost 30% of the hypsometric distribution, even where lakes are well established. We find that ice-contact lakes only affect glacier behaviour when the lakes reach an advanced evolutionary stage; most clearly manifested in the Eastern Himalaya by statistically robust differences in glacier-wide surface elevation change between lake-terminating (–0.68 ± 0.05 m a–1) and land-terminating (–0.54 ± 0.04 m a–1) glaciers. These differences are driven by the presence of a greater number of well-developed ice-contact lakes in the Eastern Himalaya compared to in the Western and Central Himalaya, resulting from greater mass loss rates to date.

Upernavik Isstrøm, the largest contributor to sea-level rise in northwest Greenland, has experienced complex and contrasting ice-flow-speed changes across its five outlets over the last two decades. In this study, we present a detailed remote-sensing analysis of the ice dynamics at Upernavik's outlets from 2000 to 2021 to evaluate the details of these changes. Previous research suggested that the presence or absence of floating ice tongues strongly influences Upernavik's ice dynamics. We use several lines of evidence to document the presence of floating ice tongues, and find that, while several outlets experienced ice-tongue formation and/or loss during the study period, these changes do not explain observed fluctuations in ice-flow velocity. Further exploration of ice-dynamic forcings using a flowline model suggests that changes in basal slipperiness near the terminus have a strong impact on upstream ice dynamics and can explain the velocity variations. Our results suggest that speed fluctuations at Upernavik's outlets may be seasonally and interannually controlled by bed conditions near the terminus, and highlight the need for further research on the influence of basal conditions on complex tidewater glacier dynamics.

To better constrain meltwater transport and ice viscosity in temperate glaciers, particularly in ice stream shear margins, we use a custom permeameter to study the untested model relationship between the permeability of temperate ice and its liquid water content. The permeability of lab-made ice of two mean grain diameters (1.8 and 4.2 mm) is measured, and water content is controlled with the ice salinity and measured calorimetrically. Fluorescein dye is added to through-flowing, chilled water to highlight flow pathways through the ice after experiments. As predicted by a simple model, permeability increases with approximately the square of the water content and by about three orders of magnitude across water contents of 0.1–4.4%. However, permeability values are less than those of the model by average factors of 2.6 and 4.1 for the finer and coarser ice, respectively. This discrepancy is likely due to tortuous, truncated or air-clogged veins. The order-of-magnitude agreement between measured and modeled values may indicate that reduced permeability from these factors is nearly compensated by preferential flow in oversized veins that are isolated or arborescent. Both kinds of preferred flow pathways are observed but the latter only in fine-grained ice at water contents > 2%.

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.

We investigate unusual discontinuous glacier motion on Thompson Glacier, Umingmat Nunaat, Arctic Canada, using synthetic aperture radar (SAR) images and ice-flow modeling. A novel intensity-rescaling scheme is developed to reduce errors in high-resolution speckle tracking, resulting in a ~25% improvement in accuracy. Interferometric SAR (InSAR) and speckle tracking using high resolution RADARSAT-2 data indicate velocity discontinuities of up to 1 cm d−1 across deep and longitudinally extensive supraglacial channels on Thompson Glacier. We use a cross-sectional finite-element ice-flow model to determine the conditions under which velocity discontinuities of the observed magnitude and signature are possible. The modeling suggests that discontinuous motion across (long and straight) supraglacial channels can occur without ice fracture and under a wide variety of glacier thermal structures, including in fully temperate glaciers. Despite the wide range of conditions conducive to discontinuous motion, the form we observe requires that the associated channels be deep, longitudinally extensive and located in regions of lateral shearing. We speculate that these combined conditions are rare except on polythermal glaciers, where drainage features such as moulins are comparatively scarce and lower deformation rates allow channels to incise consistently and persist over many years.

Bed topography is a critical parameter for determining the modern-day and future dynamics of ice sheets and their outlet glaciers. This is because the topography controls the state of stress for glaciers. At glacier termini, topography can influence the timing of terminus retreat by controlling access to warm ocean waters and/or by influencing the ability of a glacier terminus to retreat over bed bumps (moraines). Inland from the terminus, the topography can also influence where glacier retreat and thinning can stabilize. In part, this is because of knickpoints in bed topography created through glacial erosion that may influence the extent to which thinning can diffuse inland for an individual glacier and thus, the timing and magnitude of long-term mass loss. Here we provide a review of the current literature on these topics. While much of the reviewed literature assumes that topography is stable on relevant timescales to humans, new research suggests that topography may change much faster than previously thought and this further complicates our ability to project future outlet glacier change.

To investigate the mechanisms driving recent changes in outlet glaciers in Antarctica, we measured the glacier front position, flow velocity and surface elevation of five outlet glaciers flowing into Lützow-Holm Bay in East Antarctica. After a steady advance from 2008 to 2015, all the glaciers synchronously retreated by 0.4–6.0 km between 2016 and 2018. The initiation of the retreat coincided with the breakup of land-fast sea ice in Lützow-Holm Bay in 2016, which resulted in the largest sea-ice loss in the region since 1998. Similar flow variations and surface elevation changes were observed near the grounding line of Shirase, Skallen and Telen glaciers. The slowdown in 2011–15 (by 13%) and the speedup in 2016–18 (by 7%) coincided with the respective increase and decrease in surface elevation. Simultaneous retreat and acceleration after the land-fast sea-ice breakup implies that sea ice has a significant influence on glacier dynamics. Thickening/thinning observed near the grounding line was attributed to a reduced/enhanced stretching flow regime during the deceleration/acceleration period. Our results demonstrate that land-fast sea ice affects not only terminus positions, but also the flow speed and ice thickness of the Antarctic glaciers.

Ice streams are regions of rapid ice sheet flow characterised by a high degree of sliding over a deforming bed. The shallow shelf approximation (SSA) provides a convenient way to obtain closed-form approximations of the velocity and flux in a rapidly sliding ice stream when the basal drag is much less than the driving stress. However, the validity of the SSA approximation breaks down when the magnitude of the basal drag increases. Here we find a more accurate expression for the velocity and flux in this transitional regime before vertical deformation fully dominates, in agreement with numerical results. The closed-form expressions we derive can be incorporated into wider modelling efforts to yield a better characterisation of ice stream dynamics, and inform the use of the SSA in large-scale simulations.

This study provides the first comprehensive reconstruction of the dynamics of Iceberg Glacier, located on western Axel Heiberg Island, and reveals detailed observations of a complete surge for the first time in the Canadian Arctic. Historical aerial photographs, declassified intelligence satellite photographs, optical satellite imagery and synthetic aperture radar data were used to quantify changes in terminus position, ice velocity and glacier thickness since the 1950s. A surge initiated at the terminus in 1981 and terminated in 2003, suggesting a 22-year active phase. High surface velocities, reaching ~2300 m a−1 in 1991, were accompanied by a maximum terminus advance of >7 km and a large transfer of mass down-glacier, causing significant median trunk-wide surface elevation changes attaining >3 ± 1 m a−1. We suggest that the retreat from a pinning point, flotation of the terminus, the removal of sea-ice from the ice front, and an increase in subglacial meltwater availability from relatively high air temperatures in 1981 likely contributed to surge initiation. The ensuing quiescent period has seen a continual decrease in surface flow rates to an average centreline velocity of 11.5 m a−1 in 2020–21, a gradual steepening of the glacier surface and a > 2.5 km terminus retreat.

Mass loss from iceberg calving at marine-terminating glaciers is one of the largest and most poorly constrained contributors to sea-level rise. However, our understanding of the processes controlling ice fracturing and crevasse evolution is incomplete. Here, we use Gabor filter banks to automatically map crevasse density and orientation through time on a ~150 km2 terminus region of Narsap Sermia, an outlet glacier of the southwest Greenland ice sheet. We find that Narsap Sermia is dominated by transverse (flow-perpendicular) crevasses near the ice front and longitudinal (flow-aligned) crevasses across its central region. Measured crevasse orientation varies on sub-annual timescales by more than 45$^\circ$ in response to seasonal velocity changes, and also on multi-annual timescales in response to broader dynamic changes and glacier retreat. Our results show a gradual up-glacier propagation of the zone of flow-transverse crevassing coincident with frontal retreat and acceleration occurring in 2020/21, in addition to sub-annual crevasse changes primarily in transition zones between longitudinal to transverse crevasse orientation. This provides new insight into the dynamics of crevassing at large marine-terminating glaciers and a potential approach for the rapid identification of glacier dynamic change from a single pair of satellite images.

Numerical glacier and ice-sheet models compute evolving ice geometry and velocity fields using various stress-balance approximations and boundary conditions. At high spatial resolution, with horizontal mesh/grid resolutions of a few kilometers or smaller, these models usually require time steps shorter than climate-coupling time scales because they update ice thickness after each velocity solution. High-resolution performance is degraded by the stability restrictions of such explicit time-stepping. This short note, which considers the shallow ice approximation and Stokes models as stress-balance end members, clarifies the scaling of numerical model performance by quantifying simulation cost per model year in terms of mesh resolution and the number of degrees of freedom. The performance of current-generation explicit time-stepping models is assessed, and then compared to the prospective performance of implicit schemes. The main results highlight the key roles played by the algorithmic scaling of stress-balance solvers and coupled, implicit-step solvers.

In this study, we investigate the differences between two transient, three-dimensional, thermomechanically coupled ice-sheet models, namely, a first-order approximation model (FOM) and a ‘full’ Stokes ice-sheet model (FSM) under the same numerical framework. For all numerical experiments, we take the FSM outputs as the reference values and calculate the mean relative errors in the velocity and temperature fields for the FOM over 100 years. Four different boundary conditions (ice slope, geothermal heat flux, basal topography and basal sliding) are tested, and by changing these parameters, we verify the thermomechanical behavior of the FOM and discover that the velocity and temperature biases of the FOM generally increase with increases in the ice slope, geothermal heat flux, undulation amplitude of the ice base, and with the existence of basal sliding. In addition, the model difference between the FOM and FSM may accumulate over time, and the spatial distribution patterns of the relative velocity and temperature errors are in good agreement.

We perform Bayesian inference of the parameters of a time-dependent model of ice flow and calving at Helheim Glacier, East Greenland. We find that, while a time-independent calving parameterization can recover the mean observed terminus position, such a model is unable to recover sub-annual variability, even when forced with seasonally varying climate. To address this, we develop a simple stochastic model relating surface runoff rates and calving threshold. Again inferring model parameters from observations, we find that this parameterization is able to reproduce observations with respect to both mean position and characteristic temporal variability. This result demonstrates the importance of considering potential sub-annual controls on calving rates in numerical models, which may include variable undercutting rates or surface runoff-dependent surface crevasse propagation.

Sea-level projections depend sensitively on the parameterization used for basal slip in glacier flow models. During slip over rock-beds, ice-bed separation increases with slip velocity and basal water pressure. We present a method for using these variables and measured bed topography to estimate the average bed slope in contact with ice, ${\bar m}$. Three-dimensional numerical modeling of slip over small areas of former beds has shown that changes in ${\bar m}$ with increasing slip velocity and water pressure mimic changes in basal drag. Computed values of ${\bar m}$ can thus provide the form of the slip law that relates drag to velocity and water pressure, avoiding computationally expensive numerical modeling. The method is applied to 618 sections from four former glacier beds. Results generally show an increase in ${\bar m}$, and hence inferred basal drag, with slip velocity up to a limiting value, consistent with a regularized Coulomb slip law.

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.