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Glacier definitions
Glaciers are large masses of ice which form in favorable climate conditions through accumulation and consolidation of snow which became capable of coherent motion under its own weight. They appear when the snow accumulation rate exceeds the melting rate over a long period of time. Currently, this is the part of cryosphere which contains the absolute majority of ice on Earth.
Depending on average size, placement, and relation between geometric dimensions, ice masses are usually divided into (i) mountain glaciers that are relatively small, placed on mountain slopes, and typically have an elongated shape with commensurate length and thickness (Figure 2, left); (ii) ice caps that are larger, have more a dome-like shape which usually entirely covers the solid topography, and can have many tongues forming radial flows (Figure 2, right); and (iii) ice sheets that are huge ice caps with the planar sizes orders of magnitude bigger than the thickness.
Currently there are two ice sheets on Earth: Greenland and Antarctica. In this thesis, we focus on Greenland, although many of the results obtained could be applied to Antarctica as well. The Greenland Ice Sheet (GrIS) with an area of 1 710 000 km² covers almost entirely the island of Greenland. It contains about 7.3 m sea-level-equivalent of water (Morlighem et al., 2017). Generalizing, GrIS has a dome-like shape, meaning that the ice is much thicker in the inland central areas compared to the margins where it thins over short distance. This shapes the universal radial gravity-driven ice flow from the center towards the ice sheet margins, regardless of the predominantly inverse direction of the bed slope. The ice flow is relatively slow (several tens of meters per year) and uniform over inland areas and increases toward the margin in a non-uniform way, forming distinct outlet glaciers with a velocity that ranges from tens of meters to several kilometers per year. As for rivers, they have drainage basins defining the catchment area where the ice comes from and usually follow the subglacial topographical valleys near the ice-sheet margin.
Two types of outlet glaciers can be distinguished in regards to how they terminate. The land-terminating type (Figure 3-a) corresponds to glaciers that terminate on land and not in contact Seasonal flow variability of Greenlandic glaciers: satellite observations and numerical modeling to study driving processes with water. If a glacier terminates in contact with an ocean, it is called marine-terminating (or tidewater). As ice is less dense than water, some glaciers start floating at the ending section when ice reaches hydrostatic equilibrium. Thus, two subtypes of marine-terminating glaciers can be defined: grounded tidewater glaciers terminating in a vertical ice cliff and floating tidewater glaciers ending with a floating ice shelf (Figure 3-b). At a land-terminating glacier, the speed decreases to zero at the terminus, while at a marine-terminating glacier speed usually stays fast or even increases towards the calving front.
Glaciers are often composed of two specific zones depending on the experienced surface mass regimes. The upper one is the accumulation zone where the annual accumulation rate of snow is greater than the melting rate, meaning that surface mass is gained. From there, ice mass is transferred by the ice flow process toward the ablation zone, where the annual surface ablation rate, mainly consisting of surface melting, exceeds accumulation. The elevation where the two processes of accumulation and ablation compensate each other defines the limit between these two zones or the equilibrium line. The surface mass balance (SMB) of a glacier defines the net difference over a time period between total snow accumulation and total surface ice melting. The total mass balance (MB) is the difference between accumulation and all « ice-removing » processes, i.e. surface ablation, ice discharge (D) into lakes or the ocean, and, not yet commonly included in this concept, basal melting. The latter is a key process of basal mass balance (BMB) which describes ice mass loss and gain processes happening at the bedrock interface, its first estimations only start to appear (e.g. Karlsson et al., 2020). Note that D term is commonly estimated as a flux across the grounding line, thereby floating sections do not account in the mass budget.
By definition a land-terminating glacier does not discharge ice, and so its MB is equal to mass exchanges on the top and bottom surfaces. In contrast, a marine-terminating glacier discharges ice directly into ocean through calving (detachment of ice blocks at the front) and submarine melting. As the discharge rate depends on the ice flow speed, MB of such glaciers is highly influenced by their dynamics. The glacier flow causes the displacement of ice from accumulation to ablation areas and calving front; thereby, the glacier shape, speed and mass balance are closely related and cross-influencing. For a glacier in a steady state, the annual MB is null, and the mean annual speed and geometry are relatively stable.
The glacier geometry is partly shaped by external constraints and partly depends on the mass balance. The upper surface of a glacier corresponds to the free, unrestricted upper face of the ice mass which adapts over short and long terms being a result of the balance between local melting, local accumulation, and mass transfer by the ice flow from one area to others. In contrast, the basal surface under grounded ice is restricted and on a large scale follows the bedrock topography. Its change depends on the evolution of underlying rocks, mainly due to glacier erosion, which is usually considered as a relatively slow process with typical time scales of tens to hundreds of years. The front line (or terminus) is the furthest downstream extension limit of a glacier where the whole incoming ice volume melts or where icebergs calving happens. Sometimes, on the line of a faster flow the elongated tongue is shaped, clearly outstanding downstream from the main ice field (Figure 2, left). When a glacier ends in an ice shelf, the grounding line delimits the location where the ice detaches from the bed. Both the front and grounding lines are highly dynamic borders which can advance and retreat over time. Depending on the mutual displacement of both lines which is semi-independent and can be in opposite directions, the same tidewater glacier can switch between grounded and floating types during its lifetime.
The melt water produced at the surface or base of a glacier is routed outside through a hydrological system which develops on, in and below the ice (Figure 3). The majority of melting happens on the top surface, where water can be captured by snow and firn, filling the surface depressions creating lakes, being routed by supraglacial rivers toward the margin or by vertical crevasses and moulins to the bed. Some water can be generated as well right at the bed thanks to the local heat sources such as geothermal flux or basal friction. Together with surface water, it is routed more or less efficiently in the direction of hydraulic potential which at the scale of the entire glacier follows the surface slope. If the local area of water input or generation has the opportune connection, the water from there will be finally discharged out from the subglacial environment; otherwise it accumulates in cavities under the glacier.
Glacier motion
Generalizing, the main driver of glacier motion is gravity. The uneven distribution of ice mass generates the gradient of pressure exerted by it. This causes the ice to deform and slide at the bed. In this section, the physics basis behind the ice flow processes are explained first for an idealistic block of ice and then developed for the more realistic glacier configurations. Note that in this section we do not consider the details and equations of the mentioned phenomena; however, some of them can be found in Chapter 3.2.
Ice mechanics and rheology
After a certain threshold, under a pool of forces acting in different directions per area unit (stresses), a unit of ice starts to deform (strain) with changes in the shape. The ice strain mainly occurs as a permanent deformation, meaning that it keeps its new configuration after the stress is removed. The opposite elastic deformation can be important to explain the glaciers’ behavior in the specific situations like ice shelf response to the ocean tides, but usually it can be neglected. If the deformation happens in the failure way, a crack appears in the matter block; otherwise, the ductile deformation causes a flow of matter (or creep). The second type of reaction is the key process of the glacier’s motion; in turn, the former is responsible for crevasses appearance, including calving events.
The ability of a material to deform within time is its strain rate, which can be measured as the motion of different parts of material relative to each other. The relation between applied stress and strain rate, especially how the latter varies with respect to changes in the stress or its duration, is defined by rheological properties of the material. For ice, it was experimentally established (Duval et al., 1983; Glen, 1952; Lliboutry and Duval, 1985) that this relation is non-linear and that ice can be considered as a highly-viscous incomprehensible (non-Newtonian) fluid. Its viscosity (resistance to motion) depends non-linearly on temperature and also demonstrates the anisotropy (easier deformation in a preferred direction predefined by crystals orientation).
The force component acting perpendicular to the matter surface (normal stress) causes the compressive and tensile actions (Figure 4-b), while the component acting parallel to the matter surface (shear stress) leads to the shearing between the matter layers (Figure 4-c). A block of ice exerts a persistent force by its weight on the underlying bed. When considered parallel to the gravity, ice pressure force is directly proportional to the ice column thickness. In the same way all overlying ice layers exert pressure force on the underlying layers. Acting on the horizontal bed, pressure is equivalent to the normal stress (Figure 4-d); on the inclined bed (or, equivalent in the turned coordinate system, under a top-surface inclined ice block) it can be decomposed, and each of the components becomes dependent on ice thickness (measured vertically) and ice surface slope. The shear component, acting parallel to the bed in the slope direction, is the gravitational driving stress (Figure 4-e). It can be considered as the pressure force gradient in the bed-face plane which defines the rate and direction of the deformation and sliding on the ice-bed interface. Thereby, it is a key forcing of the glaciers motion, i.e. the coherent displacement of the entire ice mass in the same direction. Pressure forces exerted on other faces can also be important for flow occurrence in specific cases, for instance, on the marine-exposed face of the glacier, where the ice-to-sea difference of ice- and water-generated pressures creates a « pulling action » which causes the ice to flow towards the ocean.
The resistive forces act in the direction opposite to the driving stress. They occur due to the drag on the ice block boundaries and by the matter viscosity. These can be basal drag or lateral drag, depending on the boundary considered, or longitudinal stress gradient which occurs from the spatial variations in pushing and pulling forces along the flow. In the majority of glacier configurations, the former is the most important one. The basal shear stress, commonly called basal friction, results from it as a stress component parallel to the bed and, thus, complementary to the driving stress (Figure 4-e). In the idealistic case of a small, uniformly moving, side-free ice block, where any other resistive stresses are neglected, both will be equal.
The motion resulting from ice matter deformation has a relatively slow rate. The inertial forces of such flow are very small and can be neglected, meaning that the internal velocity field of an ice block in each next moment does not depend on the previous state and is governed only by the matter properties and current conditions on the external boundaries. Such motion satisfies the principles of mass and momentum conservation; thus, its internal fields of velocity and pressure can be derived from the matter properties and boundary conditions alone, using the Stokes equation from fluid mechanics.
Glacier flow
The flow of a glacier, meaning the ice mass displacement in space, is the same process of ice deformation as described above, which happens uniformly across the glacier and is accompanied by some additional processes on the ice-bed interface. The ice temperature across the glacier and the boundary conditions can vary is space and time, therefore, a field of acting stresses is complex.
Depending on the glacier’s shape and bed type, the relative contribution of various stresses changes. To simplify the problem and bring to the fore the principal mechanisms in a specific situation, the approach of force budget is widely used (Van Der Veen and Whillans, 1989). It assumes that despite the complexity of a real glacier, its average driving stress is closely balanced by a limited number of the most important resistive stresses, with the possibility to neglect some of them according to the required accuracy. For instance, for a glacier grounded on a relatively flat topography the most simplistic approximation is that resistive forcings can be approximated by the basal friction alone, which thereby is equal to the driving stress. Another example: under a floating shelf the basal drag on the bottom ice-water interface is zero, thus, one can simplify the resistive stresses to the drag on the shelf’s lateral borders in contact with the bedrocks.
The fact that different glacier faces experience unequal stresses leads to the more pronounced deformations concentrated on specific locations. Usually in the case of a grounded glacier, the majority of processes which drive the glacier motion occurs close to the bottom face, where the major stress – basal friction – acts. To simplify the further considerations, we decompose the internal ice-motion field into two components (Figure 6): internal shear deformation component related to the deformation occurring along the entire ice column, and the basal sliding component related to the cumulative effect of all processes occurring close to the ice-bed interface.
The « internal shear deformation » component refers to the ice creep which occurs along the entire vertical profile of the ice column. Its cumulative effect causes slow displacement of ice layers relative to the basal face with certain deformation speed ud (Figure 6, green line). Deformation speed has a non-uniform vertical profile along the ice column, because the shear rate increases towards the bed, where the weight of the overlying ice is larger. Additionally, it is typical for Greenland that the lower ice layers have a higher temperature, so they are easily deformable (e.g. Maier et al., 2019; Young et al., 2019). Thereby, a faster creep develops on the lower section of the ice column, while it is negligible in the upper ice layers.
Under the term « basal sliding », we consider the cumulative effect of all micro-scale processes taking place near the ice-bed interface and resulting in the uniform – at the scale of the ice column – motion of the entire upper ice column with a certain basal speed ub (Figure 6, blue line). The list of corresponding processes includes enhanced deformation of the basal ice layers around large obstacles, « displacement » of ice by melting and refreezing around small-scale obstacles, sliding on a water film at bedrock the interface, and others (Benn and Evans, 2010). Note that usually the presence of a water film on the ice-bed interface is assumed to exist, so no « true » dry friction directly between materials takes place; instead, the term « basal friction » refers to the general flow retention effect of bedrock’s obstacles. Except the slip on water film, those processes involve the local deformation of ice or its state change. The appearance and dominance of some of them depend on the glaciers’ underlying bed type, which can be hard non-deformable rocks and frozen sediments or soft deformable sediments (till). In the latter case, under a certain shear stress at the till interface, till also starts to deform in the direction of the glacier flow. For simplification, we will also include this process in the « basal sliding » component of the glacier motion, as it provides additional speed to the underlying glacier with a uniform contribution across the ice column.
Note that while the discussion above considers only the basal face, similar processes take place on the lateral sides of a glacier between ice and rocks (e.g. a glacier confined in a valley or fjord), under the analogical lateral friction.
The speed of motion observed at the glacier surface us (surface speed) is a cumulative sum of creep and sliding components. Usually they occur together (Figure 7-c). However, limit cases of glacier motion can be found in nature and described with simplistic approximations. For instance, an unconfined floating ice shelf does not experience any basal drag on the ice-water interface which is needed to generate the shear stress and then a deformation speed component;
its internal velocity field can be approximated as vertically uniform sliding on the water interface (Figure 7-a). A glacier frozen at its base, which practically does not slide but experiences high basal shear stress, provides an opposite limit case; its flow can be approximated by the creep component alone (Figure 7-b).
Figure 7 The simplified internal glacier velocity fields under varying conditions: (a) only sliding motion but no shear deformation, e.g. shelf slipping on the water face without basal drag; (b) only deformation motion but no basal sliding, e.g. glacier frozen to bedrock; (c) both components contribute to the ice surface motion, e.g. temperate glacier lying on the rocks.
As the motion-influencing factors are neither spatially uniform nor temporally stationary, the creep and basal sliding components can strongly vary in time and space, both in absolute magnitude and their relative contribution to the surface speed. This is observed, for example, in borehole measurements in Greenland (Maier et al., 2019; Young et al., 2019). It is commonly expected that the creep is the main motion mechanism of internal parts of the GrIS as they are most probably frozen to the bed (MacGregor et al., 2016), what explains why these regions move relatively slowly. The ice sheet periphery moves much faster (Mouginot et al., 2017), being mainly non-frozen at their base (MacGregor et al., 2016) and, thus, sliding-compatible. The range of speed can be very broad here, from tens of meters per year up to two tens of kilometers per year (Mouginot et al., 2017). As the majority of these velocities are much higher than the estimated creep-related scope (MacGregor et al., 2016), the basal sliding is expected to be dominating in the surface speed for many of these glaciers.
Drivers of velocity change over time
Remarkable variations of surface speed over different time scales have been widely observed across the GrIS on both land- and marine-terminating margins. Thereby, the questions of processes driving those variations are widely investigated.
To change the glacier’s velocity over time, changes in the ice rheology properties and/or boundary conditions are required. Different factors affecting them evolve over different time scales, being thereby more or less important when a specific time scale is examined. Here, we will address the changes in the key factors affecting the ice rheology, geometry of glacier boundaries, or conditions in the bounding environments, including:
• ice temperature which affects the ice viscosity;
• ice thickness and surface slope which define the driving stress;
• grounding line and ice front displacement (mainly on tidewater glaciers) which changes the glacier geometry and thus the distribution of stresses;
• basal friction which usually is the major resistive stress on glaciers.
Ice temperature
The ice temperature is a parameter affecting ice viscosity and deformation rate, which increases with ice temperature. Thus, all other factors being equal, a warmer glacier deforms easier and so moves faster.
The temperature of a glacier depends on heat exchange at the top and bottom interfaces and on internal heat generated by ice deformation. The first point implies that usually temperature is not vertically uniform. Accurate description of the vertical profile, especially for the basal layers where the majority of shear deformation happens, is necessary to obtain the realistic simulations of other motion-related processes and conditions, for instance, the state of the basal environment (Habermann et al., 2017; Seroussi et al., 2013).
At the surface, the ice is at the air temperature. However, the low thermal conductivity of ice leads to the very slow propagation of surface temperature changes within the underlying layers. For instance, the typical greenlandic 50-degree amplitude of annual changes in air temperature does not propagate more than two tens of meters below the surface, while the mean annual air temperature is persistently kept deeper (Cuffey and Paterson, 2010). This means that the GrIS which is up to 3 km thick keeps in its current thermal regime the influence from the previous geological epochs; the deepest ice layers are only affected by air temperature changes on the millennium time scale (Dahl-Jensen et al., 1998; Goelzer et al., 2017).
The heat fluxes coming from the bottom – geothermal, frictional and dissipated by water – are more important for the greenlandic glaciers flow. They warm the basal glacier layers above the average temperature of the overlying ice. The geothermal heat component is usually assumed to evolve over very long time scales. The heat flux coming from the basal friction varies with the sliding speed. Finally, the thawing effect of infiltrating surface water is expected to exist along the margins, following the seasonal melt cycle (Karlsson et al., 2021). Between these three heat sources, friction is estimated to be the major source under the Greenlandic margins, while the geothermal flux is the only one existing under the central parts (Karlsson et al., 2021). However, because across the margins the lowest ice layers are already close to the melting point (Doyle et al., 2018; Harrington et al., 2015; Hills et al., 2017; MacGregor et al., 2016), the possible small seasonal variations of basal ice temperature are commonly ignored. Generalizing, if no outstanding flow acceleration happens, one could consider the long-time absence of flow acceleration provoked by changes in external heat fluxes.
Besides the externaly-induced temperature changes, the process of ice deformation itself is accompanied by heat emission. In case of a newly emerging persistent and intense deformation, this source is able to change the temperature profile of the ice column at a relatively important rate. For instance, it has been modeled that Jakobshavn Isbræ’s margin warmed up by almost 2° due to the twenty-year enhanced strain after the front disintegration and velocity acceleration by about +5 km/yr (Bondzio et al., 2017). Nevertheless, when the deformations have the short and alternating character, like seasonal speedups do, no significant or widespread changes in temperature would occur.
Ice thickness and surface slope
Changes in thickness and surface slope affect directly the driving stress and, thus, the basal stress. All factors being equal, first of all, effective pressure on the bed, a thicker glacier has a higher surface speed; in the same way, a steeper glacier moves faster.
While the bed topography under grounded ice can be considered unchanged over centuries (Goelzer et al., 2017), the upper free surface evolves with time. For the GrIS, thickness variations due to evolution of the SMB or flow dynamics are widely observed within the timeframe of several years (Csatho et al., 2014; Helm et al., 2014). On the same time scale an increase of an average glacier slope can be provoked.
Table of contents :
Introduction
1. Glaciers
1.1 Glacier definitions
1.2 Glacier motion
1.2.1 Ice mechanics and rheology
1.2.2 Glacier flow
1.3 Drivers of velocity change over time
1.3.1 Ice temperature
1.3.2 Ice thickness and surface slope
1.3.3 Grounding line and ice front displacement
1.3.4 Basal friction
1.4 Summary
2. Satellite observations of the surface ice speed
2.1 Study areas
2.2 Velocity database
2.2.1 State of the art methods for deriving glacier surface displacements
2.2.1.1 Feature-tracking approaches and their limitations
2.2.1.2 SAR interferometry (InSAR) approaches
2.2.1.3 Axes of further development
2.2.2 Implementation of the ice velocity retrieval workflow
2.2.2.1 Used sensors
2.2.2.2 Automated velocity-tracking workflow
2.2.2.3 Uncertainties assignment
2.2.2.4 Data fusion and geo-database
2.2.3 Overview of the obtained database
2.2.3.1 Data quantity
2.2.3.2 Average precision and accuracy
2.2.3.3 Comparison with similar databases
2.2.3.4 Seasonal direction deviation
2.2.4 Post-processing of dense ice velocity time-series
2.2.5 Summary
2.3 Seasonal variations in surface speed on selected glacier in 2015-2019
2.3.1 Russell sector
2.3.1.1 Observed seasonal variations of the ice speed
2.3.1.2 Physical drivers of the seasonal dynamics
2.3.2 Petermann
2.3.2.1 Observed seasonal dynamics of the ice speed
2.3.2.2 Physical drivers of the seasonal dynamics
2.3.3 Upernavik Isstrøm
2.3.3.1 Observed seasonal dynamics of the ice speed 8/151
2.3.3.2 Physical drivers of the seasonal dynamics
3. Modelling of seasonal dynamics of glacier basal environment
3.1 Modelling approaches
3.2 Case study of the Russell sector: ice flow seasonal dynamics
Conclusion & perspectives
