Layer Of Temperate Ice Research Articles

Icequakes and Large Shear Wave Velocity Drop in the Kuoqionggangri Glacier of Tibetan Plateau Observed with Fiber Optic Seismometer Array

We developed a kind of fiber optic seismometer array for a high mountain glacier and first tested it on the Kuoqionggangri Glacier in the Tibetan Plateau. The array clearly recorded substantial passive seismic source signals of various icequakes, including shallow, deep and hybrid events. These fracturing activities indicate that crevasses and/or fractures developed in the glacier. We further obtained the glacial thickness of about 40 m by analyzing the active seismic source after hitting the glacier surface with a hammer based on the seismic scattering method. Most importantly, we observed a low shear wave velocity layer with a large velocity drop of ~28% and thickness of about ~7 m in the lower glacier. It is inferred that the low-velocity layer may represent a temperate ice layer. Our experiment provides a kind of feasible seismic observation to study icequakes and the englacial structure of Tibetan glaciers, offering new insights for evaluating glacier change in the Tibetan Plateau.

Complex motion of Greenland Ice Sheet outlet glaciers with basal temperate ice.

Uncertainty associated with ice sheet motion plagues sea level rise predictions. Much of this uncertainty arises from imperfect representations of physical processes including basal slip and internal ice deformation, with ice sheet models largely incapable of reproducing borehole-based observations. Here, we model isolated three-dimensional domains from fast-moving (Sermeq Kujalleq/Store Glacier) and slow-moving (Isunnguata Sermia) ice sheet settings in Greenland. By incorporating realistic geostatistically simulated topography, we show that a spatially highly variable layer of temperate ice (much softer ice at the pressure-melting point) forms naturally in both settings, alongside ice motion patterns which diverge substantially from those obtained using smoothly varying BedMachine topography. Temperate ice is vertically extensive (>100meters) in deep troughs but thins notably (<5meters) over bedrock highs, with basal slip rates reaching >90 or<5% of surface velocity dependent on topography and temperate layer thickness. Developing parameterizations of the net effect of this complex motion can improve the realism of predictive ice sheet models.

Internal drainage system of Aldegondabreen, Spitsbergen, according to speleological studies

The systems of internal drainage of glaciers have been studied mainly by indirect methods. In order to reveal the structure of the internal drainage network inside Aldegondabreen, moulins and glacial caves were investigated by speleological methods in 2001–2021, which was accompanied by a semi-instrumental topographic survey in the cavities. This allowed us to see the change in the glacial cavities over time. There are three types of moulins in Aldegondadreen: active, dead and healed ones. We visited active and dead moulins. The depth of the entrance pits in the moulins varies from 52 to 65 m (moulin group No 1), from 70 to 75 (moulin group No 2) and from 45 to 60 m (moulin group No 3). The depth of moulins is equal to the thickness of the cold ice layer. Using the structure of the moulins, we show that the water from moulin group No 1 flows to the right marginal part of the glacier tongue. The water from moulin groups No 2 and No 3 flows to the left margin part of the glacier tongue, which is confirmed by the mapping of healed moulins locations. We find that the number of active and dead moulins has been decreasing since 2001, while the number of healed moulins has increased. We attribute this to a decrease in the thickness of the temperate ice layer at the base of the glacier due to climate change. Many moulins have narrow meanders at the lower part of the entrance pits, which usually finish by siphons. None of the moulins reaches the glacier bed, their lower parts are usually located in clean transparent ice. The lifetime of the moulins usually does not exceed 6 years. Our study of the caves on the glacier tongue revealed that they can be englacial or subglacial, and they originate along sub-horizontal thrusts located in the ice. We assume that the moulins reach the slip planes along thrusts close to the glacier bed. The water from the moulins flows along these slip planes as a film in early summer and turns into channels in mid- or late summer. The presence of thrusts in the ice depths can explain the development of internal drainage systems in glaciers (regardless of their size), outbursts of glacial lakes, surges and the formation of eskers. Clastic material for eskers formation can penetrate into a cave channel from the contact areas of the thrusts with uplifts on the bed. The results obtained can help in the interpretation of the available geophysical data for this glacier.

UAV and GPR Data Integration in Glacier Geometry Reconstruction: A Case Study from Irenebreen, Svalbard

Although measurements of thickness and internal structure of glaciers are substantial for the understanding of their evolution and response to climate change, detailed data about polythermal glaciers, are scarce. Here, we present the first ground-penetrating radar (GPR) measurement data of Irenebreen, and high-resolution DEM and orthomosaic, obtained from unmanned aerial vehicle (UAV) photogrammetry. A combination of GPR and UAV data allowed for the reconstruction of the glacier geometry including thermal structure. We compare different methods of GPR signal propagation speed determination and argue that a common midpoint method (CMP) should be used if possible. Our observations reveal that Irenebreen is a polythermal glacier with a basal temperate ice layer, the volume of which volume reaches only 12% of the total glacier volume. We also observe the intense GPR signal scattering in two small zones in the ablation area and suggest that intense water percolation occurs in these places creating local areas of temperate ice. This finding emphasizes the possible formation of localised temperate ice zones in polythermal glaciers due to the coincidence of several factors. Our study demonstrates that a combination of UAV photogrammetry and GPR can be successfully applied and should be used for the high-resolution reconstruction of 3D geometries of small glaciers.

Distribution of cold and temperate ice in glaciers on the Nordenskiold Land, Spitsbergen, from ground-based radio-echo sounding

Data of ground-based radio-echo sounding of 16 glaciers located on the Nordenskiold Land, Spitsbergen, carried out in springs of 1999, 2007 and 2010–2013, allowed defining five glaciers as of the cold thermal type while other eleven ones were polythermal glaciers. In the last ones (polythermal) the average thickness of the upper layer of cold ice and the bottom layer of temperate ice was equal to 11-66 m and 15-96 m, respectively. The ratio of these thicknesses varies from 0.32 to 2.28, and the volume fraction of temperate ice in the total volume of the glaciers varies from 1 to 74% and changes from 0 to 50% in the ablation zone up to 80% in the accumulation zone. Thickness of cold ice was determined by measured delay time of radar reflections from cold-temperate surface (CTS) while thickness of temperate ice was derived as a difference between the total thickness of the glacier and the thickness of its cold ice. For interpretation of radar reflections from CTS we used the noticeable distinction in character of the radar reflections from the upper and lower thicknesses of glacier: absence of internal reflections (excluding reflections from buried crevasses and glacier wells) from upper cold ice layer and a great number of reflections of hyperbolic form from the lower layer related to strong scattering of radio waves by water inclusions in the temperate ice. According to the measurements, relative power of the radar reflections from CTS is by 5,5–14,2 dB smaller than those from the bedrock, that can be considered as an indicator of smaller water content at CTS; so, the repeated measurements of their relative power can be used for estimation of temporal changes in the water content at these boundaries. In layers of the temperate ice, the series of vertical hyperbolic reflections penetrating the cold ice down to CTS and further to the bedrock were detected. Such reflections are related to buried crevasses and/or the glacier wells and can serve as sources of the water permeating during the melt periods from the glacier surface down to CTS and bedrock and, thus, influencing on the ice viscosity and fluidity as well as on velocity of the bottom sliding in the polythermal glaciers. Repeated measurements of relative power of reflections from buried crevasses and wells can also be used to study processes of freezing them through and emptying during the period before start of the surface melting. Relation between volume of temperate ice and area of 16 studied glaciers was used to estimate the probability of existence of polythermal glaciers with a temperate ice core in all 202 glaciers in the Nordenskiold Land. 72 glaciers with areas exceeding 1.79 km2 may be referred to the polythermal type. The probable total volume of temperate ice in these glaciers amounts roughly to 10 km3, and with the 95% confidence it is within the interval from 8 to 33 km3. Almost 80% of the whole temperate ice may be concentrated in only five glaciers with area more than 17 km2, that makes up 2.5% of the total number of glaciers and about 30% of their total area. Data presented in this paper demonstrate more sophisticated pattern of the cold and temperate ice distribution within the glaciers than it was earlier known that should be taken into consideration when modeling and forecasting dynamics of the polythermal glaciers and investigating internal processes of the temperate ice formation in such glaciers.

Sliding of temperate basal ice on a rough, hard bed: creep mechanisms, pressure melting, and implications for ice streaming

Abstract. Basal ice motion is crucial to ice dynamics of ice sheets. The classic Weertman model for basal sliding over bedrock obstacles proposes that sliding velocity is controlled by pressure melting and/or ductile flow, whichever is the fastest; it further assumes that pressure melting is limited by heat flow through the obstacle and ductile flow is controlled by standard power-law creep. These last two assumptions, however, are not applicable if a substantial basal layer of temperate (T ∼ Tmelt) ice is present. In that case, frictional melting can produce excess basal meltwater and efficient water flow, leading to near-thermal equilibrium. High-temperature ice creep experiments have shown a sharp weakening of a factor 5–10 close to Tmelt, suggesting standard power-law creep does not operate due to a switch to melt-assisted creep with a possible component of grain boundary melting. Pressure melting is controlled by meltwater production, heat advection by flowing meltwater to the next obstacle and heat conduction through ice/rock over half the obstacle height. No heat flow through the obstacle is required. Ice streaming over a rough, hard bed, as possibly in the Northeast Greenland Ice Stream, may be explained by enhanced basal motion in a thick temperate ice layer.

Hydrothermal structure of a polythermal glacier in Spitsbergen by measurements and numerical modeling

Thickness of the upper cold ice layer in the ablation area of the polythermal glacier Grønfjordbreen (Spitsbergen) was estimated by means of numerical modeling. The results were compared with data of radio-echo sounding of the same glacier obtained in 1979 and 2012. Numerical experiments with changing water content in the lower layer of temperate ice and surface snow cover thickness made possible to compare calculated and modeled cold ice thicknesses and to estimate their changes for 33‑year period caused by regional climate change. According to data of radio-echo sounding, thickness of the cold ice layer decreased, on average, by 34 m. Numerical modeling shown similar results: the cold ice layer became thinner by 31 m and 39 m at altitudes 100–300 a.s.l. under the snow cover thickness of 1 m and 2 m. We explain this by rising of annual mean air temperature by 0,6 °С as compared to data of the nearest meteorological station Barentsburg in the same period. We believe that changes in cold ice layer thickness in polythermal glaciers can be used for estimation of changes in such regional climatic parameter as mean air temperature at different altitudes of the glacier surface in the ablation area.

Thermal structure of Svalbard glaciers and implications for thermal switch models of glacier surging

AbstractSwitches between cold‐ and warm‐based conditions have long been invoked to explain surges of High Arctic glaciers. Here we compile existing and new data on the thermal regime of six glaciers in Svalbard to test the applicability of thermal switch models. Two of the large glaciers of our sample are water terminating while one is land terminating. All three have a well‐known surge history. They have a thick basal layer of temperate ice, superimposed by cold ice. A cold terminus forms during quiescence but is mechanically removed by calving on tidewater glaciers. The other three glaciers are relatively small and are either entirely cold or have a diminishing warm core. All three bear evidence of former warm‐based thermal regimes and, in two cases, surge‐like behavior during the Little Ice Age. In Svalbard, therefore, three types of glaciers have switched from slow to fast flow: (1) small glaciers that underwent thermal cycles during and following the Little Ice Age (switches between cold‐ and warm‐based conditions), (2) large terrestrial glaciers which remain warm based throughout the entire surge cycle but develop cold termini during quiescence, and (3) large tidewater glaciers that remain warm based throughout the surge cycle. Our results demonstrate that thermal switching cannot explain the surges of large glaciers in Svalbard. We apply the concept of enthalpy cycling to the spectrum of surge and surge‐like behavior displayed by these glaciers and demonstrate that all Svalbard surge‐type glaciers can be understood within a single conceptual framework.

50 MHz helicopter-borne radar data for determination of glacier thermal regime in the central Chilean Andes

AbstractDespite their importance as freshwater reservoirs for downstream river systems, few glaciers in central Chile have been comprehensively surveyed. This study presents ground-penetrating radar (GPR) and field-based observations for characterizing the englacial and basal conditions of Glaciar Olivares Alfa (33°110 S, 70°130 W), central Chilean Andes. Using a 50 MHz radar mounted onto a helicopter platform, data were collected covering large portions of the glacier accumulation and ablation zones. The radar data revealed boundaries of a temperate-ice layer at the base of the eastern body of Glaciar Olivares Alfa which appears to be covered by colder ice that extends throughout large parts of the glacier. The thickness of the temperate ice layer is highly variable across the glacier, being on average 40% of the total ice thickness. Radar data analyses reveal regions of cold ice at the bottom/base of the glacier and also patterns of highly saturated sediments beneath the glacier. Using GPR data, this study represents the most exhaustive analysis of glacier ice structure performed in the central Chilean Andes. The results will enable improved estimations of the glacier’s mass balance and ice dynamics, helping us to understand its further development and its impact on water availability.

Vertical distribution of water within the polythermal Storglaciären, Sweden

Knowledge of water content and its distribution in polythermal glaciers is required to model their flow and thermal state. However, observations of water content variations with depth in polythermal glaciers are scarce. Water content can be estimated from radio wave speed because they depend on one another. We obtained continuous profiles of radio wave speed variations with depth from zero‐offset radar profiles collected in boreholes approximately 80 m deep in the upper ablation area of Storglaciären, northern Sweden. These profiles show that the microcrystalline water system in the temperate ice is relatively homogeneous. The overall hydrothermal structure at this location is composed of a 20 m thick upper layer of cold, water‐free ice, underlain by a temperate ice layer whose average water content is 0.6% ± 0.3%. These results are corroborated by surface radar and thermistor measurements, which show that the depth of the cold temperate transition is 21 m and the calculated water content at that transition is 0.6% ± 0.1%. These findings imply that the whole temperate ice layer is from 3 to 4 times softer than the cold ice and, consequently, that realistic ice flow models of polythermal glaciers should include the effect of water content on viscosity.

The near−surface ice thermal structure of the Waldemarbreen, Svalbard

The near−surface ice thermal structure of the Waldemarbreen, a 2.5−square km glacier located at 78 N1 2E in Spitsbergen, Svalbard, is described here. Traditional glacio− logical mass balance measurements by stake readings and snow surveying have been con− ducted annually since 1996. The near−surface ice temperature was investigated with auto− matic borehole thermistors in the ablation and accumulation areas in 2007-2008. The mean annual surface ice temperatures (September-June) of the ablation area were determined to be −4.7C at 1 m depth and −2.5C at 9 m. For the accumulation area, they were −3.0 Ca t 2 m, and −2.3C at 10 m depth between September and August. On the Waldemarbreen, at 10 m depth, the mean annual near−surface ice temperature was 4.0C above the mean annual air temperature in the accumulation area. The Waldemarbreen may thus be classified as a polythermal type with cold ice which is below the pressure melting point and a temperate ice layer in the bottom sections of the glacier and with a temperate surface layer only during summer seasons. At a depth of 10 m, temperatures are of the order of −2C to −3C. Key wor ds: Arctic, Spitsbergen, ice temperature, thermal regime, glacier.

Calving icebergs indicate a thick layer of temperate ice at the base of Jakobshavn Isbræ, Greenland

The puzzling fact that Jakobshavn Isbræ, West Greenland, is flowing very fast but without any significant seasonal velocity changes, despite big amounts of surface-derived meltwater entering the ice stream (Echelmeyer and Harrison, 1990), has been explained by a combination of different types of measurements. It is now well established from seismic measurements and radio-echo sounding that Jakobshavn Isbræ flows through a deeply eroded subglacial trench that, even 50 km inland of the grounding line, extends as far as 1500 m below sea level (Clarke and Echelmeyer, 1996; Legarsky and Huang, 2006). Temperature measurements in boreholes down to 65% of the 2500 m thick ice stream at site B, some 50 km upstream of the calving front (Fig. 1b; Iken and others, 1993), were used to infer the presence of a substantial layer of temperate ice, the thickness of which was estimated to be at least 300 m by modeling and matching internal layering structures (Funk and others, 1994; Lüthi and others, 2002). The presence of a thick layer of temperate ice under very high driving stress allows for high ice-deformation rates, which contribute substantially to the observed fast flow velocities. Basal motion, while certainly important, seems to be barely influenced by the seasonal meltwater input (Echelmeyer and Harrison, 1990).

Temporal changes in the radiophysical properties of a polythermal glacier in Spitsbergen

AbstractIn order to study the seasonal and inter-seasonal variations in radio-wave velocity (RWV), radiophysical investigations were made at Hansbreen, a polythermal glacier in Spitsbergen, in July– August 2003 and April 2004. These investigations included repeated radar profiling (20 and 25 MHz) along a transverse profile, repeated common-midpoint measurements, continuous radar measurements during 8 days at a fixed site, meteorological observations, and continuous ice surface velocity monitoring by differential GPS. Seasonal and inter-seasonal RWV changes in the temperate ice layer are attributed, respectively, to rapid water redistribution within it during the summer, and to variations in water content from 2.1% in summer to 0.4% in spring. The reflection properties of the temperate ice layer correlate well with the air temperature, with a nearly semi-diurnal time lag. The temporal variability of the reflection properties of the internal horizon suggests enlargement of water inclusions or water drainage from the horizon. Repeated profiling shows a stable spatial pattern in bed reflection power interpreted as changes in water content controlled by bedrock topography. The spatial variations of internal reflection energy along the repeated profile correlate with the thickness of the cold ice layer and the occurrence of drainage and crevasse systems.

Mechanisms of fast flow in Jakobshavn Isbræ, West Greenland: Part III. Measurements of ice deformation, temperature and cross-borehole conductivity in boreholes to the bedrock

AbstractAt a site on the ice sheet adjacent to the Jakobshavn ice stream in West Greenland, ice deformation rates and temperatures have been measured in boreholes to the bedrock at 830 m depth. Enhanced deformation rates were recorded just below the Holocene–Wisconsin transition at 680 m depth. A 31 m layer of temperate ice and the temperature minimum of −22°C at 520 m depth were detected. The good agreement of these data with results of a two-dimensional thermomechanically coupled flow model implies that the model input is adequate. Discrepancies between modelled and measured temperature profiles on a flowline at the ice-stream centre have been attributed to effects not accounted for by the model. We have suggested that the convergent three-dimensional flow leads to a vertical extension of the basal ice entering the stream. A thick basal layer of temperate and Wisconsin ice would explain the fast flow of this ice stream. As a test of this hypothesis, the new core-borehole conductivity (CBC) method has been used to compare conductivity sequences from the ice stream to those of the adjacent ice sheet. The correlation thus inferred suggests that the lowest 270 m of the ice sheet correspond to the lowermost 1700 m of the stream, and, consequently, that the lower part of the ice stream has experienced a very large vertical extension.

Application of a Polythermal Three-Dimensional Ice Sheet Model to the Greenland Ice Sheet: Response to Steady-State and Transient Climate Scenarios

Abstract Steady-state and transient climate-change computations are performed with the author’s three-dimensional polythermal ice sheet model Simulation Code for Polythermal Ice Sheets for the Greenland Ice Sheet. The distinctive feature of this model is the detailed consideration of the basal temperate ice layer, in which the water content and its impact on the ice viscosity are computed; its transition surface to the cold ice region is accounted for by continuum-mechanical jump conditions on this interface. The simulations presented include steady states subject to a range of physical parameters and two different climates (present and glacial conditions), as well as three types of transient scenarios, namely (i) sinusoidal Milankovic-period forcing, (ii) paleoclimatic forcing from the Greenland Ice Core Project core reconstruction, and (iii) future greenhouse warming forcing.

Polythermal modelling of steady states of the Antarctic ice sheet in comparison with the real world

An approach to simulate the present Antarctic ice sheet with reaped to its thermomechanical behaviour and the resulting features is made with the three-dimensional polythermal ice-sheet model designed by Greve and Hutter. It treats zones of cold and temperate ice as different materials with their own properties and dynamics. This is important because an underlying layer of temperate ice can influence the ice sheet as a whole, e.g. the cold ice may slide upon the less viscous binary ice water mixture. Measurements indicate that the geothermal heat flux below the Antarctic ice sheet appears to be remarkably higher than the standard value of 42 m W m−2 that is usually applied for Precambrian shields in ice-sheet modelling. Since the extent of temperate ice at the base is highly dependent on this heat input from the lithosphere, an adequate choice is crucial for realistic simulations. We shall present a series of steady-state results with varied geothermal heat flux and demonstrate that the real ice-sheet topography can be reproduced fairly well with a value in the range 50–60 m W m−2. Thus, the physical parameters of ice (especially the enhancement factor in Glen’s flow law) as used by Greve (1995) for polythermal Greenland ice-sheet simulations can be adopted without any change. The remaining disagreements may he explained by the neglected influence of the ice shelves, the rather coarse horizontal resolution (100 km), the steady-state assumption and possible shortcomings in the parameterization of the surface mass balance.

Polythermal modelling of steady states of the Antarctic ice sheet in comparison with the real world

An approach to simulate the present Antarctic ice sheet with reaped to its thermomechanical behaviour and the resulting features is made with the three-dimensional polythermal ice-sheet model designed by Greve and Hutter. It treats zones of cold and temperate ice as different materials with their own properties and dynamics. This is important because an underlying layer of temperate ice can influence the ice sheet as a whole, e.g. the cold ice may slide upon the less viscous binary ice water mixture.Measurements indicate that the geothermal heat flux below the Antarctic ice sheet appears to be remarkably higher than the standard value of 42 m W m−2 that is usually applied for Precambrian shields in ice-sheet modelling. Since the extent of temperate ice at the base is highly dependent on this heat input from the lithosphere, an adequate choice is crucial for realistic simulations. We shall present a series of steady-state results with varied geothermal heat flux and demonstrate that the real ice-sheet topography can be reproduced fairly well with a value in the range 50–60 m W m−2. Thus, the physical parameters of ice (especially the enhancement factor in Glen’s flow law) as used by Greve (1995) for polythermal Greenland ice-sheet simulations can be adopted without any change. The remaining disagreements may he explained by the neglected influence of the ice shelves, the rather coarse horizontal resolution (100 km), the steady-state assumption and possible shortcomings in the parameterization of the surface mass balance.

Mechanisms of fast flow in Jakobshavns Isbræ, West Greenland: Part II. Modeling of englacial temperatures

AbstractA model for the calculation of two-dimensional temperature fields is described and applied along the central flowline of Jakobshavns Isbræ, West Greenland, and along a flowline through the adjacent ice sheet. The model calculates the velocity-depth distribution based on Glen’s flow law and subject to the condition that the calculated velocities agree with the measured surface velocity and the estimated sliding velocity. The model allows for two-dimensional conduction and advection, for deformational energy dissipation and for the development of a basal layer of temperate ice. The results of modeling are compared to the englacial temperatures measured in boreholes reaching a depth of 1550 m which corresponds to 60% of the total depth at the center line. While there is a good agreement of the measured and modeled minimum temperatures, the shape of the temperature—depth profiles is quite different. We attribute this difference in shape to a characteristic three-dimensional ice deformation taking place in the convergent sub-surface channel of the actual ice stream. The model does not account for this three-dimensional effect. Adjustment of the modeled central temperature profile, so that its shape matches that of the measured profile, leads to an increase of thickness of the temperate basal layer by about 30%. Hence, the predicted temperate basal layer in the ice stream is likely to be about 300 m thick while the two-dimensional model suggests about 230 m. Such a thickening of the temperate basal layer by three-dimensional ice deformation may be an important mechanism of fast ice-stream flow.

Mechanisms of fast flow in Jakobshavns Isbræ, West Greenland: Part II. Modeling of englacial temperatures

AbstractA model for the calculation of two-dimensional temperature fields is described and applied along the central flowline of Jakobshavns Isbræ, West Greenland, and along a flowline through the adjacent ice sheet. The model calculates the velocity-depth distribution based on Glen’s flow law and subject to the condition that the calculated velocities agree with the measured surface velocity and the estimated sliding velocity. The model allows for two-dimensional conduction and advection, for deformational energy dissipation and for the development of a basal layer of temperate ice. The results of modeling are compared to the englacial temperatures measured in boreholes reaching a depth of 1550 m which corresponds to 60% of the total depth at the center line. While there is a good agreement of the measured and modeled minimum temperatures, the shape of the temperature—depth profiles is quite different. We attribute this difference in shape to a characteristic three-dimensional ice deformation taking place in the convergent sub-surface channel of the actual ice stream. The model does not account for this three-dimensional effect. Adjustment of the modeled central temperature profile, so that its shape matches that of the measured profile, leads to an increase of thickness of the temperate basal layer by about 30%. Hence, the predicted temperate basal layer in the ice stream is likely to be about 300 m thick while the two-dimensional model suggests about 230 m. Such a thickening of the temperate basal layer by three-dimensional ice deformation may be an important mechanism of fast ice-stream flow.

Polythermal conditions in arctic glaciers

AbstractEnglacial 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.