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A Bussard ramjet is a relativistic spacecraft, fueled by fusion energy of cosmic matter that is collected during the flight. We derive the equation of motion of such a spaceship for a given mass density in space and the fusion mass defect. Two ramjet engine scenarios, where the thrust for propulsion is generated by emission of photons or acceleration of matter, are outlined. As long as not all collected matter is transformed into fusion energy, mass engines are superior to photon engines. If the collected matter is stopped by the spacecraft before fusion it may not reach relativistic terminal velocities. For an ideal ramjet, where no matter is stopped for the generation of energy for propulsion, endless acceleration and relativistic velocities may be obtained such that crossing the universe in a human lifespan would be possible. A journey along one space coordinate and the smallest possible radii of curves were evaluated. The results are compared to the plots in the novel “Tau Zero” by Poul Anderson.

Inclinometry with embedded probes is analyzed with a Stokes model of a solid body floating in a fluid with much smaller viscosity for a two-dimensional flow field. The assumption that such a probe behaves like a Lagrangian unit vector is only justified for probes embedded in a Newtonian fluid with lengths at least four times their width. A fluid with Glen-type rheology results in a slightly smaller rotation rate of the probe compared to Newtonian fluids.

Polythermal conditions are ubiquitous among glaciers, from small valley glaciers to ice sheets. Conventional temperature-based ‘cold-ice’ models of such ice masses cannot account for that portion of the internal energy which is latent heat of liquid water within temperate ice, so such schemes are not energy-conserving when temperate ice is present. Temperature and liquid water fraction are, however, functions of a single enthalpy variable: a small enthalpy change in cold ice is a change in temperature, while a small enthalpy change in temperate ice is a change in liquid water fraction. The unified enthalpy formulation described here models the mass and energy balance for the threedimensional ice fluid, for the surface runoff layer and for the subglacial hydrology layer, together in a single energy-conserving theoretical framework. It is implemented in the Parallel Ice Sheet Model. Results for the Greenland ice sheet are compared with those from a cold-ice scheme. This paper is intended to be an accessible foundation for enthalpy formulations in glaciology.

We present measurements with inclinometer/magnetometer probes inserted in glacier ice, and discuss the underlying mathematical theory to extract velocity-gradient components from these data. Assumptions concerning the ice flow field must be made to reduce the number of unknowns and to close the system of equations obtained from the theory. With the incompressibility assumption and the first-order plane-strain approximation, the evolution equation for the tilt of the main sensor axis can be solved exactly and the obtained function can be fitted to measured data to obtain optimal shear and normal strain components. Daily variations superposed on the tilt evolution reflect variations in sliding and perhaps partial elastic recovery of deformations.

Observations of glacier flow and explanations of its origin started as early as the 18th century. Several mechanisms were suggested before gravity-driven viscous flow became the accepted theory of glacier flow in the 1950s, the early years of the Journal of Glaciology. Since the viscosity of ice is strongly temperature-dependent, the topic of glacier and ice-sheet dynamics became essentially a fluid-dynamical problem. The availability of growing computing power turned the field of glacier mechanics and thermodynamics into a field of numerical modelling with increasing sophistication.

The retreat of Laika Glacier (4.4 km2), part of a small ice cap situated on Coburg Island, Canadian Arctic Archipelago, is analyzed using field data, satellite remote sensing and mass-balance modelling. We present a methodology for merging various data types and numerical models and investigate the temporal and spatial changes of a remote glacier during the past five decades. A glacier mass-balance and surface-evolution model is run for the period 1959–2006, forced with in situ weather observations and climate re-analysis data (ERA-40, NARR). The model is calibrated using the ice-volume change observed between 1959 and 1971, and measured seasonal mass balances. Calculated glacier surface elevation is validated against ICESat GLAS altimeter data and ASTER-derived elevation. Landsatderived glacier outlines are used to validate calculated ice extent. The piedmont tongue of Laika Glacier has retreated considerably and is in a state of disintegration. The modelled glacier mass balance between 1959 and 2006 was −0.41 m w.e. a−1, on average. Model results indicate a significant trend towards higher mass-balance gradients. A complete wastage of Laika Glacier by 2100 is predicted by model runs based on climate scenarios.

The present paper revisits the future surface-climate experiments on the Greenland ice sheet proposed by the Sea-level Response to Ice Sheet Evolution (SeaRISE; Bindschadler et al., 2013) study. The projections of the different SeaRISE participants show dispersion, which has not been examined in detail to date. A series of sensitivity experiments are conducted and analyzed using the ice-sheet model for integrated Earth-system studies (IcIES) by replacing one or more formulations of the model parameters with those adopted in other model(s). The results show that large potential sources of the dispersion among the projections of the different SeaRISE participants are differences in the initialization methods and in the surface mass balance methods, and both aspects have almost equal impact on the results. The treatment of ice-sheet margins in the simulation has a secondary impact on the dispersion. We conclude that spinning up the model using fixed topography through the spin-up period while the temperature is allowed to evolve according to the surface temperature history is the preferred representation, at least for the experiment configuration examined in the present paper. A benchmark model experimental setup that most of the numerical models can perform is proposed for future intercomparison projects, in order to evaluate the uncertainties relating to pure ice-sheet model flow characteristics., The Cryosphere, 10 (1), ISSN:1994-0416, ISSN:1994-0424

The mechanisms controlling the spatial distribution and temporal fluctuations of the thermal structure in polythermal glaciers have, to date, been poorly investigated and are not fully understood. We have investigated the sensitivity of the cold surface layer thickness to different forcing parameters and the causes for an observed thinning of the cold surface layer on Storglaciären, northern Sweden, between 1989 and 2001 using a one-dimensional thermomechanical model and measurements of ice surface temperature, vertical velocity and net mass balance. Similarities between the spatial patterns of the cold surface layer, net mass balance and emergence velocity together with modelled high sensitivity to variations in emergence velocities suggest that the net ablation and vertical ice advection are the dominant forcing parameters. Results from transient model experiments suggest that the cold surface layer reaches a new equilibrium after a perturbation in the forcing within a few decades. No significant change in ice flow or mass balance has been observed at Storglaciären in recent decades. Instead, an increase of 1°C in winter air temperature since the mid-1980s is probably the cause of the observed thinning of the cold surface layer. Increased winter temperatures at the ice surface result in a reduced formation rate of cold ice at the base of the cold surface layer and lead to a larger imbalance between net loss of ice at the surface and freezing of temperate ice at the cold–temperate transition surface. Model results indicate that the cold surface layer is more sensitive to changes in ice surface temperature in areas with lower emergence velocity, which explains the observed complex thinning pattern of the cold surface layer.

In three-dimensional numerical ice-sheet models that use finite-difference schemes, the position of ice margins is poorly represented with a regular quadratic grid. As a result, in a centered difference scheme, the surface gradient term and the flux divergence term computed for the gridpoints next to the ice margin may be inaccurate. In this paper, an improved scheme is presented that computes the horizontal gradients at the ice-sheet margin using an asymmetric (upstream) second-order difference scheme in order to avoid using information from the zero-thickness gridpoints. The model is applied to an idealized synthetic geometry to obtain a steady-state ice-sheet topography. The improved model shows a realistically smooth thickness distribution near the margin. Thermomechanical coupling is found to enhance the error near the margin. The error in simulated thicknesses with the centered-difference method was significantly reduced with the new upstream scheme.

A newly designed eight-arm caliper–inclinometer probe for borehole logging is presented. The caliper sounding gives eight points of cross-sections of the borehole at 1 cm intervals along its axis. Regression circles and ellipses are calculated and used as obvious hypotheses to be tested in order to obtain information on strain rates in the horizontal planes and isotropic or anisotropic borehole closure. The scratching of identifiable marks into the wall of the borehole enables material displacement to be tracked, in particular in the vertical direction, thus providing information for determining vertical strain rates along the borehole. Measurements in two boreholes drilled on high-altitude glaciers in the Swiss Alps are used to demonstrate both the potential and the limitations of the probe and of the mathematical methods presented.

Spatial and temporal variations of the flow of Hansbreen, a tidewater glacier in southern Spitsbergen, Svalbard, are investigated. During summer 1999, surface flow velocities were measured in the ablation zone of Hansbreen with a temporal resolution of 1—2 hours. Short events with strongly increased surface velocities and a typical duration of 1—2 days were observed. These “speed-up events” are related to periods of strongly increased water input to the glacier, due to rainfall or enhanced surface melt. A close relation is found between the surface velocities and water pressure recorded in a moulin. However, there are indications from a short time lag between velocity and water-pressure peak as well as from observed vertical surface uplifts that basal motion is related to basal water storage rather than directly to basal water pressure. The observed short-term velocity variations and associated processes on Hansbreen are very similar to those observed on land-based valley glaciers and suggest that the relevant mechanisms and physical processes that control the flow and its temporal variations are similar. In contrast to the flow of land-based glaciers, sliding velocities on Hansbreen are observed to be high all year round and velocities increase towards the calving front.

An indirect methodology for determining the distribution of mass balance at high spatial resolution using remote sensing and ice-flow modelling is presented. The method, based on the mass-continuity equation, requires two datasets collected over the desired monitoring interval: (i) the spatial pattern of glacier surface-elevation change, and (ii) the mass-flux divergence field. At Haut Glacier d’Arolla, Valais, Switzerland, the mass-balance distribution between September 1992 and September 1993 is calculated at 20 m resolution from the difference between the pattern of surface-elevation change derived from analytical photogrammetry and the mass-flux divergence field determined from three-dimensional, numerical flow modelling constrained by surface-velocity measurements. The resultant pattern of mass balance is almost totally negative, showing a strong dependence on elevation, but with large localized departures. The computed distribution of mass balance compares well (R2 = 0.91) with mass-balance measurements made at stakes installed along the glacier centre line over the same period. Despite the highly optimized nature of the flow-modelling effort employed in this study, the good agreement indicates the potential this method has as a strategy for deriving high spatial and temporal-resolution estimates of mass balance.

Measurements of summer and winter mass balances have been carried out over the past 53 years on Storglaciären, northern Sweden. Repeated surveys of the glacier have resulted in several maps of surface topography as well as a map of the bed topography A new time-dependent ice flow model allows us to compare the observed surface evolution of the glacier with that computed by the model using measured mass-balance maps as input. The computed volume change compares well with the measured change: the model replicates the distribution of surface elevation to within ±10 m over 30 years of integration. On the model side, these deviations can be attributed to the low-resolution discretization of the model domain as well as to the limited accuracy of the ice rheology and omitted basal sliding. On the other hand, the uncertainties of the topography and mass-balance maps match the model uncertainties. In this sense, the experiments are a validation of both model and observations.

During the 1992 summer field season we installed arrays of “plough-meters” and water-pressure transducers beneath Trapridge Glacier. Yukon Territory, Canada, to study hydromechanical coupling at the ice–bed interface. Diurnal signals recorded with two of these ploughmeters appear to correlate with fluctuations in sub-glacial water pressure. These diurnal variations can be explained by changes in basal resistance to sliding as mechanical conditions at the bed vary temporally in response to changes in the subglacial hydrological system. We propose that a lubricating water film, associated with high water pressures, promotes glacier sliding, whereas low pressures cause increased basal drag resulting in “sticky” areas. Using a theoretical model, we analyze the sliding motion of glacier ice over a flat surface having variable basal drag and show that a consistent explanation can be developed. Results from our model calculations provide strong support for the existence of time-varying sticky spots which are associated with fluctuations in subglacial water pressure.

Detailed ground-penetrating radar (GPR) surveys at 50 and 200 MHz on Hansbreen, a polythermal glacier in southern Svalbard, are presented and interpreted. Comparison of the variations in character of the radar reflections with borehole thermometry and water levels in moulins suggests that GPR can be used to study the hydrothermal properties of the glacier. The high resolution of the GPR data shows that the hydrothermal structure of the glacier is highly variable both along the centre line and on transverse profiles. Water contents for many places and depths within the glacier were calculated by estimating radar-wave velocities to point reflectors. We find typical water contents of 1-2% for the temperate ice, but wetter ice associated with surface crevassing and moulins (typically 4% water content). There is evidence that wet ice sometimes overlays drier ice. The hydrothermal structure is thus shown to be very complex. Temperature gradients in the cold ice indicate freezing rates of temperate ice below cold ice of 0.1-0.5 ma-1, while isolated point reflectors within the cold ice indicate large water-filled bodies that are probably related to the regular drainage structure of the glacier.

Numerical methods are used to examine the interaction between the spatial distribution of the basal shear traction and the corresponding basal velocity for an inclined slab geometry. In our improved treatment, we reject the common assumption that basal velocity is a simple function of local variables in favour of a non-local treatment that includes normal deviatoric stress and takes basal velocity to be an integrated response to spatially varying influences. Computationally, one must either iterate the basal velocity with a friction parameterization that relates basal shear traction to basal velocity or, alternatively, prescribe the basal shear traction that results from bed decoupling and substrate déformation.The average of basal shear traction over the entire bed of the ice mass is invariant under changes in sliding distribution and thus constitutes a useful reference; any local relative reduction of traction leads to basal movement, either sliding over the bed or moving with a deforming subglacial layer. The local stress réduction is accompanied by a concentration of traction up-and down-glacier of the moving base. Growth, decay and possible migration of basal stress concentrations may be closely related to short-lived sliding events and to surges.

The set of force equations and stress strain-rate relations for ice masses can be solved with the method of lines and shooting the stress-free conditions at the free surface. Single- and multiple-shooting schemes with fixed point or Newton iterations for solving the stress equations including the deviatoric stress gradients are described and their performances arc discussed. The single-shooting Newton iteration proved to be the fastest seheme for typical valley glaciers, although its horizontal grid limitation is restrictive. Grid resolution can be improved substantially with a multiple-shooting scheme but computation time and storage requirements increase substantially. The Newton iteration allows the handling of mixed basal boundary conditions, partly basal velocity and partly basal shear traction being prescribed. A stick slip free gravity flow illustrates the performance of the scheme.

A three-dimensional, finite-difference model based on a first-order solution of the ice-flow equations is applied to Haut Glacier d’Arolla, Switzerland. The numerical model successfully converges at horizontal resolutions down to 70 m, so a number of detailed comparisons with field data can be made. Modelled surface velocities with no basal sliding component are compared with surface velocities observed on the glacier over four different time periods. The best fit is achieved with over-winter surface velocities (R2 = 0.75) using a rate factor, A, in Glen’s flow law of 0.063 a−1 bar−3. Surface zones of maximum computed effective stress display a high level of coincidence with observed crevassing, the orientation of which is successfully predicted by the direction of the tensile component of the computed principal surface stress. Comparison of the relative magnitude and direction of computed principal stresses with principal strains measured at the ice surface also correspond closely. In an attempt to simulate the observed annual velocity distribution within a cross-section of the glacier tongue, we incorporate two basal-motion patterns into the model. By treating net annual ice motion as a time-weighted composite of three separate flow situations: normal sliding, enhanced sliding and no sliding, we are able to reproduce the key features of the observed cross-sectional ice and basal slip velocity distributions. These experiments indicate there may be substantial decoupling taking place along an elongated narrow zone at the bed of Haut Glacier d’Arolla and that this decoupling interacts in a complex manner with the englacial stress and strain field.

A new and efficient algorithm for computing the three-dimensional stress and velocity fields in grounded glaciers includes the role of deviatoric stress gradients. A consistent approximation of first order in the aspect of ratio of the ice mass gives a set of eight field equations for the five stress and three velocity components and the corresponding boundary conditions. A coordinate transformation mapping the local ice thickness on to unity and approximating the derivatives in the horizontal direction by centered finite-differences yields five ordinary differential and three algebraic equations. This allows use of the method of lines, starting the integration with prescribed stress and velocity components at the base, and a simple iteration procedure converges rapidly.The algorithm can be used for a wide rangе of stress-strain-rate relations, as long as strain only depends on deviatoric and shear stresses and on temperature. Sensitivity tests using synthetic and realistic ice geometries show the relevance of normal deviatoric stresses in the solutions for the velocity components even for ice sheets. Stress and velocity fields may deviate substantially from the widely used shallow-ice approximation.

The initiation and evolution of ice sheets are investigated using a two-dimensional thermo-mechanical ice-sheet model. The importance of the amount of snow accumulation on the ice sheet initiation is summarised in the following three points: (1) an ice sheet can grow from an initial area of less than 50 km diameter with a positive but large-enough accumulation rate; (2) the pattern of multiple steady-state solution branches critically depends on the surface mass balance; and (3) snow accumulation strongly controls the growth rate of the ice mass, which is crucial for ice sheets evolving in the limited time available during Milankovitch cycles.

Englacial temperature measurements in Arctic valley glaciers suggest in the ablation zone the existence of a basal layer of temperate ice lying below the bulk of cold ice. For such a polythermal glacier, a mathematical model is presented that calculates the temperature in the cold part and the position of the cold-temperate transition surface (CTS). The model is based on the continuum hypothesis for ice and the ice-water mixture, and on the conservation laws for moisture and energy. Temperate ice is treated as a binary mixture of ice and water at the melting point of pure ice. Boundary and transition conditions are formulated for the free surface, the base and the intraglacial cold-temperate transition surface. The model is reduced to two dimensions (plane flow) and the shallow-ice approximation is invoked. The limit of very small moisture diffusivity is analysed by using a stationary model with further reduction to one dimension (parallel-sided slab), thus providing the means of a consistent formulation of the transition conditions for moisture and heat flux through the CTS at the limit of negligibly small moisture diffusion.The application of the model to the steady-state Laika Glacier, using present-day conditions, results in a wholly cold glacier with a cold sole, in sharp contrast to observations. The present polythermal state of this glacier is suspected to be a remnant of the varying climatic conditions and glacier geometry during the past few centuries. Steady-state solutions representing a polythermal structure can indeed be found within a range of prescribed conditions which are judged to be realistic for Laika Glacier at the last maximum extent of the glacier.

The dynamics of grounded tidewater glaciers is investigated with a time-dependent numerical flow model, which solves the full equations for the stress and velocity fields and includes a water-pressure-dependent sliding law. The calving criterion implemented in the model shifts the calving front at each time-step to the position where the frontal ice thickness exceeds flotation height by a prescribed value. With this model, the linear relation between calving rate and water depth proposed on empirical grounds is qualitatively reproduced for the situation of a slowly retreating or advancing terminus, but not for situations of rapid changes. Length changes of tidewater glaciers, i.e. especially rapid changes, are dominantly controlled by the bed topography and are to a minor degree a direct reaction to a mass-balance change. Thus, accurate information on the near-terminus bed topography is required for reliable prediction of the terminus changes due to climate changes. The results also confirm the suggested cycles of slow advance and rapid retreat through a basal depression. Rapid changes in terminus positions preferably occur in places where the bed slopes upwards in the ice-flow direction.

A numerical glacier-flow model (finite-element method) is used to suggest the processes that control the flow behind the calving front of a tidewater glacier. The model is developed for grounded calving glaciers and includes an effective-pressure-dependent sliding law The sliding law is implemented by adding a soft basal layer with a variable viscosity The model is applied on Hansbreen, a tidewater calving glacier in Svalbard. Comparison between modeled surface velocities and observed velocity data shows good agreement. We conclude that the flow of a grounded calving glacier can be modeled with an effective-pressure-dependent sliding law

Modelling temperature distribution in non-temperate mountain glaciers presents problems not normally encountered when modelling ice sheets or ice shelves. These problems are mainly concerned with numerical instabilities caused by the high, nonuniform gradients of various input parameters (geometry, mass balance, surface temperature, and flow velocity). Steady-state solutions must be used to check and complete data sets, before using models of greater complexity to calculate temperature fields in a more realistic way. Test runs with a computer model, which allows for true two-dimensional solutions and realistic velocity fields, are described for two examples from the Swiss Alps. These steady-state calculations illustrate, in a semi-quantitative way, that advection of cold ice by glacier flow strongly influences the temperature distribution in both an existing large valley glacier with a cold accumulation zone (Grenzgletscher), and a large piedmont glacier of the last ice age, around 18 ka BP (Rheingletscher). Non-steady-state models are being prepared and tested for future calculations.

From 1974 to 1981, a total of 32 bore holes was drilled on White Glacier and vertical ice-temperature profiles measured. The data obtained allowed the construction of three longitudinal and four transverse profiles of the two-dimensional temperature distributions. Thus, an extensive layer of temperate or near-temperate ice was discovered close to the bedrock in the lowest part of the glacier tongue. It was also found that the temperature distribution cannot be in a steady state, since there is a temperature minimum 100–150 m below the glacier surface in the accumulation area. A simple numerical model calculation shows that this minimum can be mostly explained by the general climatic warming since 1880. The 10 m temperatures show diffuse relations to climatic conditions and balance zones. A simple scheme for extrapolating “surface temperatures” is discussed.

During the North Water Project of the late F. Müller, glaciological studies were carried out on Laika ice cap. In addition to the main climatological investigations, surveying, mapping, mass-balance studies, and englacial temperature measurements were carried out. The mass-balance distribution is strongly determined by the orography. Strong westerly winds erode and transport snow from exposed surfaces, whereas prevailing easterly winds, during precipitation, deposit snow on lee slopes. The balance is negative under the present climate. The history of the glacier-tongue geometry is reconstructed using geomorphological observations and photogrammetric mapping for 1959 and 1971. Englacial temperature measurements revealed a finite layer of temperate basal ice in the ablation zone. The temperature distribution in the accumulation area around the summit of the ice cap is not stationary.

The existence of solutions of the non-sliding shallow-ice-sheet equation on a flat horizontal bed with a mass balance linearly depending on altitude is proven for fixed margins. Free-margin solutions for the same mass balance do not exist. Fixed-margin solutions show unbounded shear stress and nonzero mass flux at the margin. Steady-state solutions with realistic margins, vanishing ice flux and vanishing shear stress are found numerically for ice sheets with Weertman-type sliding.

Ice-penetrating radar and modelling data are presented suggesting the presence of a zone of temperate ice, water ponding or saturated sediment beneath the tongue of Taylor Glacier, Dry Valleys, Antarctica. The proposed subglacial zone lies 3–6 km up-glacier of the terminus and is 400– 1000m across. The zone coincides with an extensive topographic overdeepening to 80m below sea level. High values of residual bed reflective power across this zone compared to other regions and the margins of the glacier require a high dielectric contrast between the ice and the bed and are strongly indicative of the presence of basal water or saturated sediment. Analysis of the hydraulic equipotential surface also indicates strong convergence into this zone of subglacial water flow paths. However, thermodynamic modelling reveals that basal temperatures in this region could not exceed –7˚C relative to the pressure-melting point. Such a result is at odds with the radar observations unless the subglacial water is a hypersaline brine.

We propose a novel Eulerian algorithm to compute the changes of a glacier geometry for given mass balances. The surface of a glacier is obtained by solving a transport equation for the volume of fluid (VOF). The surface mass balance is taken into account by adding an interfacial term in the transport equation. An unstructured mesh with standard stabilized finite elements is used to solve the non-linear Stokes problem. The VOF function is computed on a structured grid with a high resolution. The algorithm is stable for Courant numbers larger than unity and conserves mass to a high accuracy. To demonstrate the potential of the algorithm, we apply it to reconstructed Late-glacial states of a small valley glacier, Vadret Muragl, in the Swiss Alps.

We present a numerical method to estimate the ice thickness distribution within a two-dimensional, non-sliding mountain glacier, given a transient surface geometry and a mass-balance distribution, which are relatively easy to obtain for a large number of glaciers. The inverse approach is based on the shallow ice approximation (SIA) of ice flow and requires neither filtering of the surface topography with a lower slope limit nor approximation of constant basal shear stress. We first address this problem for a steady-state surface geometry. Next, we use an apparent surface mass-balance description that makes the transient evolution quasi-stationary. Then, we employ a more elaborated fixed-point method in which the bedrock solution is iteratively obtained by adding the difference between the computed and known surface geometries at the end of the considered time interval. In a sensitivity study, we show that the procedure is much more susceptible to small perturbations in surface geometry than massbalance. Finally, we present preliminary results for bed elevations in three space dimensions. (Some figures may appear in colour only in the online journal)

For more than a century Alpine glaciers have been retreating dramatically, and they are expected to shrink even more quickly over the coming decades. This study addresses the future evolution of Grosser Aletschgletscher, Switzerland, the largest glacier in the European Alps. A three-dimensional combined surface mass-balance and glacier dynamics model was applied. The ice flow was described with the full Stokes equations. The glacier surface evolution was obtained by solving a transport equation for the volume of fluid. Daily surface melt and accumulation were calculated on the basis of climate data. The combined model was validated against several types of measurements made throughout the 20th century. For future climate change, scenarios based on regional climate models in the ENSEMBLES project were used. According to the median climatic evolution, Aletschgletscher was expected to lose 90% of its ice volume by the end of 2100. Even when the model was driven using current climate conditions (the past two decades) the glacier tongue experienced a considerable retreat of 6 km, indicating its strong disequilibrium with the present climate. By including a model for the evolution of supraglacial debris and its effect in reducing glacier melt, we show that this factor can significantly slow future glacier retreat.

A three-dimensional thermomechanically coupled ice-sheet model including calculation of the first-order stress gradients is developed to perform numerical studies on the effects of stress components neglected in the zeroth-order shallow-ice approximation. Steady-state solutions are obtained with both a shallow-ice and a first-order ice-sheet model for several idealistic, radially symmetric ice-sheet configurations. The results show that the effects of the normal deviatoric-stress gradients on ice thickness are generally small, but the influence on basal temperatures and age profiles at an ice divide is significant.