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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.

Surveys in Geophysics, 32 (4-5), ISSN:0169-3298, ISSN:1573-0956

Three methods based on the three-dimensional shallow ice approximation of glacier flow are devised that infer a glacier’s subglacial topography from the observation of its time-evolving surface and mass balance. The quasi-stationary inverse method relying on the apparent surface mass-balance description of the glacier’s evolution is first exposed. Second, the transient inverse method that iteratively updates the bedrock topography with the surface topography discrepancy is formulated. Third, a shape optimization algorithm is presented. The aim of the paper is to collect these methods, analyze their differences, and identify what brings the sophistication of shape optimization for reconstructing subglacial topographies. The three methods are compared to the ice thickness estimation method (ITEM) on direct measurements on Gries glacier, Swiss Alps. The paper concludes with a detailed discussion on the sensitivity of the shape optimization method to the model parameters., Computational Geosciences, 18 (6), ISSN:1420-0597, ISSN:1573-1499

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.

A climate forecast provided by a General Circulation Model (GCM), a glacier mass balance model and a glacier flow model is applied to a sample of 11 small glaciers. Another sample of six glaciers and six large, heavily glacierized areas in the arctic were modelled using only the climate forecast and the glacier mass balance model. The climate forecast of two different GCM's with identical experimental setup takes into account a gradual increase in atmospheric content of CO 2 and other greenhouse gases to a doubled CO 2 equivalent in 2050 corresponding to the IPCC Scenario IS92a. The differences between the two GCM's are significant for the future development of the glacierization of arctic areas. The glacier mass balance model consists of a temperature index melt model, which uses potential direct clear sky radiation to obtain a better spatial and temporal resolution, combined with imposed snow precipitation. Static mass balance sensitivities span −0.2 to −1.5 mwe a −1 °C, the higher and lower sensitivities apply for glaciers in more maritime and continental climates, respectively. All of the modelled glaciers and glacierized regions show a strong decrease in the net mass balance. Temperature changes dominate the effect of snow precipitation on the net mass balance. A temperature increase results in substantial increase in melt especially through the extension of the melt season in spring and fall. The mass balance projections for the sample of 11 glaciers are then used in a glacier flow model to simulate the dynamic reaction of the glacier to the changing climatic conditions. An iteration procedure accounts for possible feedback of changes in the glacier area and the ice surface elevation to the projected mass balance. For the modelled glaciers, an average volume loss of 60% until 2050 is predicted. Without the above iteration and assuming a constant glacier area, the volume change is overestimated by about 20%. A comparison with the climate predictions of the two different GCMs shows only a difference of about 10% in the mass loss obtained by flow modelling of four glaciers.

Journal of Inverse and Ill-posed Problems, 22 (6), ISSN:0928-0219, ISSN:1569-3945

To predict the evolution of glaciers in an enhanced greenhouse climate, results from a global climate model, a glacier melt/accumulation model, and a glacier flow model were combined. The method was applied to Storglaciaren, a small well-studied glacier in northern Sweden. The difference between the present climate and a 2 × CO2 climate around the year 2050 was extracted from a model experiment with the ECHAM4-T106 high resolution climate model for time slices at present and in 2050, using prescribed boundary conditions of sea surface temperature and sea-ice distribution, which are derived from a lower resolution transient run of the ECHAM4-T42/OPIC-coupled atmosphere ocean model between present and 2050. The local climatic conditions on the glacier for 2050 were obtained by adding the modelled local climate changes to the observed local present-day climate. The combination of the comprehensive models presented offers a tool to test and calibrate simplified models which are applicable to a much larger sample of glaciers. For the region of Storglaciaren, the GCM projected temperature is found to increase most strongly during the winter months, but also shows a warming during the transition from spring to summer, and again between summer and fall, thus extending the melt season by three to four weeks. Precipitation, on the other hand, decreases by approximately 5% during May to September while there is a stronger increase of approximately 14% for the rest of the year. The consequent increase in winter accumulation on Storglaciaren is more than compensated by the increase in ablation during the melt season. The glacier flow model predicts a 300 m retreat of the glacier terminus by the middle of the next century, and a loss of 30% of the present ice mass.

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 shape optimization algorithm is presented that estimates the ice thickness distribution within a three-dimensional, shallow glacier, given a transient surface geometry and a mass-balance distribution. The approach is based on the minimization of the surface topography misfit in the shallow ice approximation by means of a primal-dual procedure. The method?s essential novelty is that it uses surface topography and mass-balance data only within the context of a time-dependent problem with evolving surface topography. Moreover, the algorithm is capable of computing some of the model parameters concurrently with the ice thickness distribution. The method is validated on synthetic and real-world data, where the choice of its Tikhonov regularization parameter by means of an L-curve criterion is discussed. HighlightsThe bedrock topography of glaciers is reconstructed with an optimal control technique.Bedrock topography and model parameters can be computed concurrently.A Tikhonov regularization based on the topography?s area is used.The whole computations underlying the algorithm are detailed.

The enthalpy method for the thermodynamics of polythermal glaciers and ice sheets is tested and verified by a one-dimensional problem (parallel-sided slab). The enthalpy method alone does not include explicitly the transition conditions at the cold-temperate transition surface (CTS) that separates the upper cold from the lower temperate layer. However, these conditions are important for correctly determining the position of the CTS. For the numerical solution of the polythermal slab problem, we consider a two-layer front-tracking scheme as well as three different one-layer schemes (conventional one-layer scheme, one-layer melting CTS scheme, one-layer freezing CTS scheme). Computed steady-state temperature and water-content profiles are verified with exact solutions, and transient solutions computed by the one-layer schemes are compared with those of the two-layer scheme, considered to be a reliable reference. While the conventional one-layer scheme (that does not include the transition conditions at the CTS) can produce correct solutions for melting conditions at the CTS, it is more reliable to enforce the transition conditions explicitly. For freezing conditions, it is imperative to enforce them because the conventional one-layer scheme cannot handle the associated discontinuities. The suggested numerical schemes are suitable for implementation in three-dimensional glacier and ice-sheet models., Comment: 16 pages, 8 figures

The growth and reduction of Northern Hemisphere ice sheets over the past million years is dominated by an approximately 100,000-year periodicity and a sawtooth pattern (gradual growth and fast termination). Milankovitch theory proposes that summer insolation at high northern latitudes drives the glacial cycles, and statistical tests have demonstrated that the glacial cycles are indeed linked to eccentricity, obliquity and precession cycles. Yet insolation alone cannot explain the strong 100,000-year cycle, suggesting that internal climatic feedbacks may also be at work. Earlier conceptual models, for example, showed that glacial terminations are associated with the build-up of Northern Hemisphere 'excess ice', but the physical mechanisms underpinning the 100,000-year cycle remain unclear. Here we show, using comprehensive climate and ice-sheet models, that insolation and internal feedbacks between the climate, the ice sheets and the lithosphere-asthenosphere system explain the 100,000-year periodicity. The responses of equilibrium states of ice sheets to summer insolation show hysteresis, with the shape and position of the hysteresis loop playing a key part in determining the periodicities of glacial cycles. The hysteresis loop of the North American ice sheet is such that after inception of the ice sheet, its mass balance remains mostly positive through several precession cycles, whose amplitudes decrease towards an eccentricity minimum. The larger the ice sheet grows and extends towards lower latitudes, the smaller is the insolation required to make the mass balance negative. Therefore, once a large ice sheet is established, a moderate increase in insolation is sufficient to trigger a negative mass balance, leading to an almost complete retreat of the ice sheet within several thousand years. This fast retreat is governed mainly by rapid ablation due to the lowered surface elevation resulting from delayed isostatic rebound, which is the lithosphere-asthenosphere response. Carbon dioxide is involved, but is not determinative, in the evolution of the 100,000-year glacial cycles.

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 response of the Greenland ice sheet geometry to glacial/interglacial climatic transition is studied by using a numerical ice sheet dynamical model. The possible contribution due to the increase in melting, snow accumulation, ice temperature and accumulation of harder ice during the Holocene is examined. The results imply that: (1) within 1000 years after the termination of the Ice Age, the present ice sheet became thicker than that during the Ice Age, mainly due to the increase in snow accumulation, (2) at present, about 10,000 years after the transition, the ice sheet is still reacting to the change in ice temperature and the advection of harder Holocene ice, effects which are partly compensating each other, (3) the central part of the ice sheet may be slightly by about 4 mm per year because of the somewhat quicker response of the ice sheet to temperature change than to advection of the harder Holocene ice, (4) the influence due to the repetition of the ice age cycles is negligible. Since the rate of change at present is below the detection thresholf, observation of the elevation change must be explained by the short-term climatic change.

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.

[1] A mathematical model for polythermal glaciers and ice sheets is presented. The enthalpy balance equation is solved in cold and temperate ice together using an enthalpy gradient method. To obtain a relationship between enthalpy, temperature, and water content, we apply a brine pocket parameterization scheme known from sea ice modeling. The proposed enthalpy formulation offers two advantages: (1) the discontinuity at the cold-temperate transition surface is avoided, and (2) no treatment of the transition as an internal free boundary is required. Fourier's law and Fick-type diffusion are assumed for sensible heat flux in cold ice and latent heat flux in temperate ice, respectively. The method is tested on Storglaciaren, northern Sweden. Numerical simulations are carried out with a commercial finite element code. A sensitivity study reveals a wide range of applicability and defines the limits of the method. Realistic temperature and moisture fields are obtained over a large range of parameters.

With the constitutive equations given in Sects. 4.3 and 4.4, we are now able to formulate the mechanical and thermodynamical field equations for the flow of ice in an ice sheet. Figure 5.1 shows the typical geometry (cross section) of a grounded ice sheet with attached floating ice shelf (the latter will be treated in Chap. 6), as well as its interactions with the atmosphere (snowfall, melting), the lithosphere (geothermal heat flux, isostasy) and the ocean (melting, calving). Also, a Cartesian coordinate system is introduced, where x and y lie in the horizontal plane, and z is positive upward. These coordinates are naturally associated with the set of basis vectors {e x , e y , e z }. The free surface (ice-atmosphere interface) is given by the function z = h(x, y, t), the ice base by z = b(x, y, t) and the lithosphere surface by z = z l(x, y, t).

In agreement with the scope of the series Advances in Geophysical and Environmental Mechanics and Mathematics (AGEM2), it is our intention that this book serves the purposes

As mentioned in the introduction (Chapter 1), the size of land ice masses spans several orders of magnitude, from large ice sheets of a few thousand kilometres in diameter down to small glaciers of a few hundreds of metres in length. Consequently, the scaling given for ice sheets in Chapter 5 [Eqs. (5.5) and (5.102)] is not valid for smaller ice caps and glaciers, and needs to be modified. However, the Froude number (5.7) and Coriolis-force-to-pressure-gradient ratio (5.10) are always extremely small compared to unity, and therefore the Stokes flow problem formulated in Sect. 5.1 is applicable to land ice masses of all shapes and sizes. On the other hand, the applicability of the approximations defined in Sects. 5.2 to 5.4 is limited by the size of the ice masses.

Continuum mechanics is concerned with the motion and deformation of continuous bodies (for instance, a glacier). A body consists of an infinite number of material elements, called particles. For any time t, each particle is identified by a position vector x (relative to a prescribed origin O) in the physical space e, and the continuous set of position vectors for all particles of the body is called a configuration κ of the body. If t is the actual time, the corresponding configuration is called the present configuration κ t . In addition, we define a reference configuration κ r which refers to a fixed (or initial) time t 0.

The frozen part of the terrestrial climate system is referred to as the cryosphere. The cryosphere consists of several subsystems, namely ice sheets, ice shelves, ice caps, glaciers, sea ice, lake ice, river ice, ground ice and snow. Ice sheets are ice masses of continental size (area greater than 50,000 km2) which rest on solid land, whereas ice shelves consist of floating ice nourished by the inflow from an adjacent ice sheet, typically stabilised by large bays. Extended land-based masses of ice covering less than 50,000 km2 are termed ice caps, and smaller ice masses constrained by topographical features (for instance a mountain valley) are called glaciers. Sea ice floats on the ocean; however, in contrast to an ice shelf it forms directly by freezing sea water.

The phase of H2O ice which exists at pressure and temperature conditions encountered in ice sheets and glaciers is called ice Ih. It forms hexagonal crystals, that is, the water molecules are arranged in layers of hexagonal rings (Fig. 4.1). The plane of such a layer is called the basal plane, which actually consists of two planes shifted slightly (by 0.0923 nm) against each other. The direction perpendicular to the basal planes is the optic axis or c-axis, and the distance between two adjacent basal planes is 0.276 nm.

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.

An individual observer located at one position in space collecting information from the retarded light cone is confronted with aberration and Doppler shift, but also with a shift in intensity and pressure of the observed radiation. The latter are derived using the photon hypothesis and are applied to the cosmic microwave background. The interaction of matter with the cosmic background radiation tends to bring the matter to rest in a preferred frame in which the radiation is isotropic.

[1] The European Ice Sheet Modelling Initiative (EISMINT) intercomparison experiments with thermomechanical coupling are repeated with an ice sheet model that applies the first-order approximation for computing the flow field. The experiments impose radially symmetric boundary conditions. Most of the previous results have shown the loss of implied radial symmetry, i.e., the formation of distinct, regularly spaced spokes of cold ice in the simulated basal temperatures. A similar result is also observed in the presented first-order model results. The computed velocity components scatter widely along the marginal grid points. This indicates that the spokes may be triggered by the poor representation of the margin with a regular grid, where the steep gradients in the surface enhance the numerical errors. Additionally, the generally applied second-order discretization scheme tends to decouple even and odd numbered grid points, thus leading to wavy solutions with a wavelength of two grid cells. These patterns strongly suggest that the loss in radial symmetry is a numerical artifact.

[1] Storglaciaren, northern Sweden, is temperate in most parts except for a cold surface layer in the ablation zone. One of four possible sources for liquid water in temperate ice is melting due to strain heating. Velocity fields are calculated with an ice flow model, so that calculated and observed surface velocities agree. Meltwater accumulation is computed by integrating strain heating along trajectories starting at the surface in the accumulation area and ending at the cold-temperate transition surface in the ablation zone. The distribution of moisture content due to strain heating alone is mapped in a longitudinal section of Storglaciaren. Values reach more than 10 g of water per kilogram ice-water mixture in the lowest parts of the temperate domain. For this moisture content the rate factor is more than 3 times higher than for water-free ice, and therefore water production by strain heating is important for the modeling of temperate and polythermal glaciers.

[1] The volume fraction of liquid water in temperate glacier ice is important not only for the flow of glaciers and the analysis and processing of ground penetrating radar data from glaciers but also for the stability of the thermal layering in polythermal glaciers. However, little is known about the spatial variations of water content in glaciers. We use relative backscatter strength of ground-penetrating radar signals to estimate the spatial distribution of water content close to the cold-temperate transition on Storglaciaren, northern Sweden, in an area close to the equilibrium line. The values of relative backscatter strength are calibrated using determinations of absolute water content from temperature measurements across the cold-temperate transition and the thermodynamic boundary condition at the freezing front. The results show a water content of 0.80%, 0.75%, and 0.58% at three calibration points and a mean water content of 0.8% with a standard deviation of ±0.26% for the extrapolated water content. The extrapolated water content shows a distinct pattern, with lower water content on one side of the glacier center line and higher water content on the other side, with higher water content on the northern side. We hypothesize that the different water contents result from the fact that the ice on either side of the center line originates from different cirques, thus implying spatial variations in the entrapment of water in the firn-ice transition process in the different cirques.

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