Glacier dynamics

Harding Icefield , Kenai National Wildlife Refuge , Alaska

As glacier dynamics refers to the movement behavior of glaciers , ice caps and ice sheets and their physical description. The weight of the glacier is responsible for the observed movements . For one thing, it deforms the ice, which behaves like a very viscous liquid. This process is called internal deformation. On the other hand, the glacier as a whole can move on its base, which is known as basal gliding. The base itself can also be deformed by the heavy weight and the movement of the ice.

The speed at which glaciers move ranges from a few meters to a few kilometers per year. The movement of the glaciers is influenced by a number of factors, including the slope, the ice size, the nature of the bedrock and the temperature. Even within a glacier, the speed is not homogeneous. It generally increases in the accumulation area (nutrient area) of the glacier, but decreases again in the ablation area (depletion area), and the upper layers of the ice also move faster than the ice near the rock bed.

Mathematically, the flow of glaciers can be described using methods of continuum mechanics by using a flow law to relate the rate of deformation of the ice to the stress . Historically, the study of glacier dynamics developed in the eighteenth and nineteenth centuries through observation of alpine glaciers. In the twentieth century, the great Arctic and Antarctic ice sheets became the focus of research, not least because of their central role in the earth's climate system, for example due to their reflection, i.e. the albedo effect or their influence on sea level.

Research history

Johann von Charpentier, author of the Essai sur les glaciers et sur le terrain erratique du bassin du Rhône and pioneer of glacier research

"Hôtel des Neuchâtelois", under the rock at this point on the Unteraar Glacier, the first glaciological research station, built by Louis Agassiz, was built.

Glaciers were treated in the work of Geography by the ancient Greek historian Strabo , and in modern geographical works for the first time by Sebastian Münster in his Cosmographia . Glaciers were also mentioned in medieval documents, for example to designate borders. In Tyrol, for example, this happened for the first time in a deed of donation from 1260. However, a scientific discourse on the movements of the glaciers did not develop until the 18th century. In the 17th century, the native population of the Swiss Alps still believed that the glaciers grew from the bottom up the mountain. This view was also held by several contemporary scientists such as Johann Gottfried Gregorius in his special encyclopedia Description of the most famous mountains in alphabetical order from 1715.

In 1787, Bernhard Friedrich Kuhn was one of the first to try to explain the movements of the glaciers physically. He assumed that debris warmed by the sun would get under the glacier, melt ice and reduce the stability of the glacier. As soon as so much water has melted that the ice no longer has contact with the rock bed, the glacier then begins to move downhill as a whole. Even if his theory of glacier movements is counted among the weaker parts of his work, he nevertheless made a remarkable contribution to glaciology , because he established a connection between moraines and changes in the mass balance and postulated that the movements of alpine glaciers and arctic ice caps are the same Mechanism. Other early theories of glacier dynamics attempted to explain the downward movement of the glaciers by melting and re-freezing of water. For example, in his work Essai sur les glaciers et sur le terrain erratique du bassin du Rhône , published in 1841 , Johann von Charpentier explained the glacier movements caused by the snowmelt on the glacier surface . The melted water penetrates the interior of the glacier and, when it freezes again at night, causes cracks and deformations that lead to movement of the glacier. Other naturalists such as Johann Jakob Scheuchzer or Ignaz Venetz advocated similar theories. However, it was recognized early on that the temperatures inside the glacier are normally too low to cause deformation through such processes.

In contrast, as early as 1751 Johann Georg Altmann attributed the movements of the glaciers to gravity . This leads to the ice of the glacier being pushed down into the valley. However, according to his ideas, the glaciers as a whole moved, and he had no idea of ​​the flow behavior of the ice itself as a viscous liquid. Other of his ideas also seem rather strange for today's ideas. For example, he assumed that under the surface of the glacier a sea of ​​liquid water extends down to the valley floor, of which the glaciers are only the top layer. In 1779, Horace-Bénédict de Saussure , in the first volume of his Voyages dans les Alpes, put the theory of motion through gravity on a scientifically somewhat more solid basis from today's perspective. Based on the observation that there are often cavities and draining glacial streams at the foot of the glacier, he explained that the ice melts on the rock bed and thus allows the glacier to move, which is moving downhill due to the weight of the ice pressing from above. Even his theory does not take into account the viscous properties of the ice; only James David Forbes correctly recognized this as one of the causes of the movement of the glaciers. His observations at the Mer de Glace contradicted Saussure's theory, as the temperatures were too low to melt ice appreciably. Instead, in his work published in 1842, he assumed a viscous deformation of the ice as the cause of the glacier movements. Even if this theory was finally accepted, it was not initially contradicted: John Tyndall considered it impossible for ice cream to have viscous properties. In this case, a glacier should be able - according to Tyndall - to flow over steep edges instead of breaking. His explanation for the movement of glaciers was the continuous formation and subsequent reclosing of small cracks. These cracks formed as soon as sunlight melted the ice in various places in the glacier and the lower volume of the water compared to the ice could not completely fill the cavity. Accordingly, he saw the air bubbles in glacier ice as the remains of these cracks.

The first systematic measurements of glacier movements also began in the time of Forbes and Tyndall. Louis Agassiz showed that a glacier flows faster in the middle than on its lateral edges. He also found that the speed at the beginning and end of a glacier is lower than in the areas in between. Harry Fielding Reid finally showed in 1896 that the flow lines of a glacier do not run parallel to the rock bed, but rather are inclined downwards in the accumulation area (submergence) and upwards in the ablation area ( emergence ). This can be seen as experimental confirmation of Forbes' theory of the glacier as a viscous liquid.

Great advances were made in the fifties of the twentieth century. Thanks to the work of Glen and Nye , a general law of flow for ice could be formulated for the first time ( Glensches law of flow , see below). In addition, Weertman formulated his theory of the basal sliding of a glacier as a whole over the underlying rock bed in 1957. The description of basal gliding was further refined in the decades that followed. In particular, the role of meltwater on the rock bed and the fact that the rock bed itself is deformable due to the pressure of the overlying glacier were given greater consideration in the models.

The relevance of the Arctic and Antarctic ice sheets to the global climate system and the variation in sea level has led to the fact that the ice sheets have come more into the focus of research in the last few decades, whereas the early work on glacier dynamics dealt almost exclusively with the alpine glaciers. After seismic measurements of the ice thickness had already been carried out during Alfred Wegener's last Greenland expedition, detailed studies of the Antarctic ice sheet did not begin until the Norwegian-British-Swedish Antarctic Expedition from 1949 to 1952.

In addition to developing better experimental methods such as For example, remote sensing , the introduction of numerical simulation represents a radical change in scientific work on glacier dynamics. The first numerical models were developed in the late 1960s. The first three-dimensional model of a glacier was applied to the Barnes Ice Cap on Baffin Island in 1976 ; previously only two-dimensional simplifications were used. In 1977, thermodynamics could be included in the models for the first time. In the meantime, the models are able to reproduce temperature, flow velocity as well as rock bed and surface topography at least on the order of magnitude. Thanks to computer simulations, it is now possible to simulate the influence of individual parameters on the flow behavior as a whole, without having to resort to complicated laboratory measurements. Even if the models have become increasingly powerful in recent times thanks to increasingly powerful computers, caution is advised when interpreting their predictions. For example, the current increase in ice flow in the polar ice sheets has not been predicted in any model. These recent dramatic changes in glaciers and their effects on the global climate system are currently the focus of research.

Crystal structure and deformation

Crystal structure of ice

Ice crystals

Ice I _h : crystal structure

As ice in general, the solid is physical state of the water referred to, which can occur in various manifestations. In glaciology , a further distinction is made between fresh snow and various forms of firn and (glacier) ice, which has a closed pore space and in which air bubbles trapped by the ice no longer have any contact with the outside atmosphere. For the structure in the crystals , this distinction is initially irrelevant: The water molecule consists of an oxygen atom that has two hydrogen atoms attached to it. In the solid state of aggregation, two additional hydrogen atoms bind to the oxygen atom via hydrogen bonds , so that each molecule has four neighbors connected via hydrogen bonds (two starting from the oxygen atom and one from each hydrogen atom).

A molecule with four nearest neighbors can crystallize in different ways. While several crystal structures of ice can be realized under laboratory conditions (currently nine stable as well as several metastable and amorphous structures are known), only the hexagonal form ice I _h occurs in nature , in which six water molecules join together to form rings that form in arrange individual layers. Each molecule belongs to two rings. At 0.276 nm, the distance between two neighboring ring layers is considerably greater than the dislocations within the ring (0.092 nm). The direction perpendicular to the ring layers is called the optical or c-axis, the area defined by the ring layers is called the basal plane.

Deformation of monocrystalline ice

Due to the layered structure of a single ice crystal, its deformation usually takes place parallel to its basal plane, the tension required for deformation along other directions is approximately 100 times higher. Here the ice is first elastically deformed, then it begins to deform permanently as long as the tension persists. Laboratory experiments show that even small stresses cause deformation. This is due to defects within the crystal structure - so-called dislocations , which can move much more easily within the crystal than atoms in a perfect crystal lattice.

Deformation of polycrystalline ice

Ice creep curve

Glacier ice does not consist of a single, large ice crystal, but is composed of many individual individual crystals ( grains ). One cubic meter of glacier ice contains 10 ⁶ to 10 ⁹ individual grains. In contrast to monocrystalline ice consisting of only one crystal, such ice is called polycrystalline. It deforms more slowly than monocrystalline because the orientation of the individual crystals is random and does not allow uniform sliding along the basal plane. Processes that lead to deformation are instead movement of the individual crystals relative to one another, movement of lattice defects within a crystal and dynamic recrystallization, the formation of new crystals that are advantageously oriented for the deformation.

If a constant pressure is exerted, an initial elastic deformation is followed by a phase in which the deformation rate decreases ( primary creep ) until a minimum, the secondary creep rate , is reached. The decrease is caused by disturbances of crystals of different orientations blocking each other. Dynamic recrystallization ultimately leads to crystal structures that are easier to deform and consequently to an increase in the deformation rate ( tertiary creep ).

Deformation due to internal deformation

Glen's law of flow

Forces acting on glacier ice (generally gravitation) cause the ice to deform due to the mechanisms mentioned above. For stresses common in glaciers, the deformation rate can be described as a function of the stress with the factor and the exponent according to ${\displaystyle {\dot {\epsilon }}}$ ${\displaystyle \tau }$ ${\displaystyle A}$ ${\displaystyle n}$

${\displaystyle {\dot {\epsilon }}=A\tau ^{n}.}$

This relationship is called Glensch's law of flow. Glen's law of flow has essentially been found empirically on the basis of various laboratory and field data, whereby the values ​​of and depending on the data set can be very different. The value of varies between about 2 and 3.9, whereby a value of 3 is generally assumed for glacier ice. While the value of can be assumed to be constant for practical applications in glaciology, the value of the rate factor is not a constant, but depends on temperature, pressure and the concentration of impurities in the ice such as sand. With regard to the temperature and the gas constant , an Arrhenius dependence shows : ${\displaystyle A}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle A}$ ${\displaystyle T}$ ${\displaystyle R}$ ${\displaystyle A}$

${\displaystyle A=A_{0}\cdot \exp \left({\frac {-Q}{RT}}\right).}$

The activation energy is about 60 kJ / mol for temperatures below −10 ° C. This means that the deformation rate at −10 ° C is about five times higher than at −25 ° C. If temperatures rise above −10 ° C, the polycrystalline glacier ice deforms even faster, although pure monocrystalline ice does not show this behavior. The increased deformation rate between −10 ° C and 0 ° C can be described by an activation energy of 152 kJ / mol. ${\displaystyle Q}$

The temperature-independent variable is not a constant, but depends on the pressure , which in turn can be described by an exponential equation: ${\displaystyle A_{0}}$ ${\displaystyle P}$

${\displaystyle A_{0}=A_{0}'\cdot \exp \left({\frac {-PV}{RT}}\right)}$

with the activation volume . However, the pressure effect, even for pressures like those on the underside of ice sheets , is very small and far less relevant than the temperature dependence. In addition, the rate of deformation can depend on the crystal size and water content. In a similar way to water, chemical impurities in ice increase its deformability by forming salty solutions with a lower melting point than pure water between the grain boundaries, which make it easier to slide along the grain boundaries. The effect of insoluble impurities, on the other hand, is less clear, as small particles within the crystal structure increase the frequency of lattice defects, which makes the ice more deformable, but on the other hand also makes it more difficult for the ice to slide. However, a measurement of the deformability with different sand contents showed a significant increase with an increasing amount of sand. Overall, the effect of pollution on the deformability of glacier ice has not yet been researched and is difficult to assess, as a connection to other variables such as pressure and temperature is assumed. However, the effects of contamination in the ice should play a major role, especially on the bed of a glacier, as this is where the particle content is highest. ${\displaystyle V}$

Generalized flow law

Usually the shear forces in a glacier act in different directions. Therefore, in the general case, both the deformation rate and the shear stress are treated as tensorial quantities. The stress tensor has the form ${\displaystyle {\dot {\epsilon }}}$ ${\displaystyle \sigma }$

${\displaystyle \sigma ={\begin{pmatrix}\sigma _{xx}&\sigma _{xy}&\sigma _{xz}\\\sigma _{yx}&\sigma _{yy}&\sigma _{yz}\\\sigma _{zx}&\sigma _{zy}&\sigma _{zz}\end{pmatrix}}}$

Since the flow of the glacier is independent of the hydrostatic pressure , only the stress deviator is considered, where the hydrostatic pressure is subtracted from the stress tensor: ${\displaystyle \tau }$

${\displaystyle \tau _{ij}=\sigma _{ij}-{\frac {1}{3}}(\sigma _{xx}+\sigma _{yy}+\sigma _{zz}).\qquad (i,j=x,y,z)}$

Here and in the following, the index or any entry of a tensor stands. The rate of deformation is determined by the speed gradient and is also a tensor variable: ${\displaystyle i}$ ${\displaystyle j}$ ${\displaystyle {\dot {\epsilon }}_{ij}}$

${\displaystyle {\dot {\epsilon }}_{ij}={\frac {1}{2}}\left({\frac {\partial v_{i}}{\partial j}}+{\frac {\partial v_{j}}{\partial i}}\right)}$.

Its diagonal elements describe an expansion or compression along an axis. The off-diagonal elements correspond to shears (the element, for example, a shear of the plane in the direction ). ${\displaystyle {\dot {\epsilon }}_{ii}={\frac {\partial v_{i}}{\partial i}}}$ ${\displaystyle {\dot {\epsilon }}_{xy}}$ ${\displaystyle y}$ ${\displaystyle x}$

A general flow law should mathematically relate stress and deformation rate. A basic assumption here is that the deformation rate and stress deviator are proportional to each other:

${\displaystyle {\text{(a)}}\quad {\dot {\epsilon }}_{ij}=F\tau _{ij}.}$

This is a function of the temperature, the pressure and the applied voltage. Since the flow law must be independent of the chosen coordinate system, it can not be a function of a single element of the stress deviator (dependent on the coordinate system) and the invariants of the two tensors and are of particular interest. Since the stress deviator is track -free , it follows from the assumed linear dependence (equation (a)) that what is equivalent to the assumption of incompressibility for ice. ${\displaystyle F}$ ${\displaystyle F}$ ${\displaystyle {\dot {\epsilon }}}$ ${\displaystyle \tau }$ ${\displaystyle {\dot {\epsilon }}_{xx}+{\dot {\epsilon }}_{yy}+{\dot {\epsilon }}_{zz}=0}$

The second invariant of the deformation rate (or the deviatoric shear stress) is called the effective deformation rate (effective shear stress) and is defined as

${\displaystyle {\text{(b)}}\quad 2{\dot {\epsilon }}={\dot {\epsilon }}_{xx}^{2}+{\dot {\epsilon }}_{yy}^{2}+{\dot {\epsilon }}_{zz}^{2}+2({\dot {\epsilon }}_{xy}^{2}+{\dot {\epsilon }}_{xz}^{2}+{\dot {\epsilon }}_{yz}^{2})}$

respectively

${\displaystyle {\text{(c)}}\quad 2\tau =\tau _{xx}^{2}+\tau _{yy}^{2}+\tau _{zz}^{2}+2(\tau _{xy}^{2}+\tau _{xz}^{2}+\tau _{yz}^{2}).}$

For these two quantities, the relationship of the form corresponding to the experimental observations becomes

${\displaystyle {\text{(d)}}\quad {\dot {\epsilon }}=A\tau ^{n}}$

assumed, which in the case of simple shear (e.g. with all entries except and equal to 0) reduces to Glen's flow law. It is therefore plausible to assume that it can also describe the voltage dependence of in general . This is again a function of pressure and temperature and a parameter to be determined experimentally. ${\displaystyle \tau _{xy}}$ ${\displaystyle \epsilon _{xy}}$ ${\displaystyle F}$ ${\displaystyle A}$ ${\displaystyle n}$

From the equations - it follows that ${\displaystyle (a)}$ ${\displaystyle (d)}$

${\displaystyle {\dot {\epsilon }}=F\tau \qquad {\text{mit}}\quad F=A\tau ^{n-1}.}$

If you insert into the equation , you get the generalized flow law for ice : ${\displaystyle F}$ ${\displaystyle (a)}$

${\displaystyle {\dot {\epsilon }}_{ij}=A\tau ^{n-1}\tau _{ij}.}$

The deformation of the plane in the direction depends not only on the corresponding input of the stress tensor, but also on the shear forces acting in all other directions, which are contained in the effective shear stress . If the shear stress tensor has only one entry, i.e. the force only acts on one surface in one direction, the generalized flow law is equivalent to Glen's flow law. ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle \tau _{xy}}$ ${\displaystyle \tau }$

More complex relationships between strain rate and stress are also given in recent literature. Based on the observation that different flow behavior occurs depending on the cause of a deformation, David L. Goldsby and David Kohlstedt (2001) designed a model in which the total deformation rate is composed of the sum of all contributions of the various deformation mechanisms for polycrystalline ice. Relationships that deviate even further from the form of the general flow law were also discussed. Nevertheless, the generalized flow law in the form given above is used in most glacier dynamic models.

Basal sliding and rock bed deformation

Section through a glacier. Due to melting processes, the ice layer directly above the rock bed is more transparent than the layers above.

Glaciers as a whole can move downhill due to gravity, which is known as basal sliding . The speed of the basal glide depends less on the magnitude of the gravitational force, but more on the conditions on the rock bed, which depend on the temperature of the glacier. If the temperature there is higher than the pressure melting point, a thin film of water can form through melting, which enables the glacier to slide. Otherwise, the sliding happens very slowly and is therefore irrelevant for most cold glaciers , i.e. glaciers whose temperature is below the pressure melting point. A measure of the sliding ability of a glacier rock bed is the drag factor , which connects the shear stress caused by the movement with the sliding speed : ${\displaystyle \psi }$ ${\displaystyle \tau _{b}}$ ${\displaystyle u_{b}}$

${\displaystyle \tau _{b}=\psi u_{b}}$.

The higher the drag factor , the more difficult it is to slide. Its numerical value varies greatly, even for glaciers with melt on the rock bed. For this reason, it is not possible to state generally how relevant basic sliding is for the movement of the glacier as a whole. In glaciers with temperatures above the pressure melting point, it is responsible for about 50% of the total movement, but sometimes also for considerably more.

Basal gliding over a solid rock bed

The fact that the ice can move over bumps on the rock bed is mainly due to two mechanisms that were already described by Deeley and Pfarr in 1914 and mathematically treated by Weertman in 1957 in a first theory of basal sliding. The basic assumption of his theory is an ice body that moves over a thin film of water over a non-deformable rock bed. If an unevenness of the rock bed opposes the river, on the one hand the force of the glacier pressing on the obstacle from above creates a pressure gradient between the two sides of the obstacle. The higher pressure on the side of the obstacle facing the mountain ensures that the pressure melting point on this side is lowered. Since the temperature of the glacier on the rock bed corresponds to the pressure melting point, the ice here is colder than on the valley side. This temperature difference creates a flow of heat that melts the ice on the mountain side. In the liquid state, the obstacle can be overcome, and the water then freezes again. The efficiency of this mechanism depends on the heat flow through the obstacle, which becomes smaller the larger the obstacle. Hence, it is negligible for large obstacles. On the other hand, obstacles opposing the flow create a higher tension, which results in a higher flow velocity. The larger the obstacle, the greater this effect, so it is only relevant for large obstacles. The combination of both effects finally enables movement over both large and small obstacles.

After Weertman first formulated this theory, other theories on basal sliding were developed without any fundamental changes. The mechanisms initially postulated by Weertman have now also been confirmed experimentally. A modification results from the presence of larger water inclusions on the rock bed. The water pressure within these inclusions can become so great that the ice no longer only moves over a thin film of water over the rock bed, but is sometimes completely raised, which reduces the contact area between the glacier and the rock bed. This drastically reduces the frictional resistance and increases the flow rate. A completely correct description of this case has not yet been developed. Furthermore, particles trapped in the ice near the rock bed can increase the frictional resistance, which also has a noticeable effect on the flow velocity.

Basal gliding over a deformable rock bed

In the above considerations it was assumed that the glacier moves over a completely rigid rock bed. However, the heavy weight of the glacier ice can cause the rock bed to deform itself and move the underlying sediment with it. Experimental confirmation of this is provided by the current retreat of the glaciers , which normally does not leave behind a rigid bed of rock, but rather rock debris that was created by the deformation of the rock. Investigations using boreholes have only been carried out on a few glaciers, but they have confirmed the important role played by rock bed deformation, which in some glaciers is the main cause of the basal glacier movements. Rock bed deformations show strong spatial and temporal fluctuations, caused by spatial changes in the geometry and the material of the underlying rock bed and changes in the water content of the glacier. For this reason, too, it is very difficult to describe the exact flow behavior of a glacier with a deformable bed. However, since the deformation of the rock bed significantly influences the flow behavior, it is now attempted to parameterize it by all modern glacier models.

The flow of glaciers

Measured ice flows and speeds at representative locations of some glaciers

glacier	Type	Surface speed (m / a)	Flow velocity in the middle of the glacier (m² / a)	Ice thickness (m)
Storglaciar	Valley glacier	15th	3000	200
Worthington Glacier	Valley glacier	75	15000	200
Columbia Glacier (1977)	Mountain glacier (flowing into the sea)	730	70000	950
Columbia Glacier (1995)	Mountain glacier (flowing into the sea)	2900	230000	800
Pine Island Glacier	Ice stream (with ice shelf )	1500	270000	1800

The flow speed of glaciers varies between a few meters and a few kilometers per year. Basically, two components contribute to the flow velocity: a constant part , which is caused by basal sliding and rock bed deformation, and one which is due to internal deformation and varies depending on the mass of the ice on top, i.e. with the depth of the glacier: ${\displaystyle u}$ ${\displaystyle u_{\mathrm {b} }}$ ${\displaystyle z}$

${\displaystyle u=u_{\mathrm {b} }+\int _{\mathrm {B} }^{z}{\frac {\partial u}{\partial z}}\mathrm {d} z.}$

The proportion of basal sliding is determined by the water content and nature of the rock bed, the internal deformation by the applied tensions and the geometry of the glacier. Both processes are also dependent on the temperature. If you multiply the flow velocity by the cross-sectional area of ​​the glacier, you get the amount of ice that flows through this area per unit of time, the ice flow in the unit . ${\displaystyle \mathrm {m^{3}/s} }$

Based on the specific mass balance , i.e. the mass balance at different points on a glacier, general predictions can be made about the mean equilibrium velocities and the direction of the flow lines. In order to get the exact flow velocities as a function of the depth, the velocities must be calculated using the general law of flow. Except for idealized cases, an analytical solution is no longer possible, so that one has to rely on numerical models.

Measurement of glacier speeds

A glaciologist at work

Surface speeds of glaciers used to be measured by triangulation . For this purpose, poles were distributed over the glacier and the distance between them was determined. The displacements of the rods and thus the speed result from measurements at different times. Even if technical improvements such as automatic angle measurement and laser distance measurement have simplified such measurements, they always require a reference point. In addition, the workload of the regular field measurements required for this is relatively high and unfavorable weather conditions can make measurements at certain times impossible. A fundamental improvement came with the introduction of the Global Positioning System (GPS) and the use of GPS receivers on the surveying poles. If there is a reference station near the glacier, a measurement accuracy of up to one centimeter can be achieved. In addition, data can be measured continuously, not just during a few field measurement excursions. However, measuring methods with rods only ever provide data for a spatially very limited area. In the meantime, terrestrial laser scanners with a range of several kilometers are mainly used to determine surface speed. The glacier speed is determined with them by shifting characteristic structures in successive scans ( feature tracking ).

Using remote sensing and surface speeds of larger and previously inaccessible areas can be measured. Here, data or images from aircraft or satellite measurements are used to determine the speed. If you want to measure the speeds of glaciers that are too small to be resolved using satellite images, remote sensing data can also be obtained with drones . Here too, the speed can be determined by means of feature tracking based on the displacement of prominent fixed points. For this purpose, glacier crevasses , for example , which were recorded in different, chronologically consecutive images of the same area, are used. Alternatively it is possible to obtain the speed by means of microwave interferometry by measuring the phase shift of a microwave signal reflected from the glacier.

Measuring speeds within the glacier are more difficult. One way to do this is to observe the deformation of ice boreholes, which can provide information about the deformation rate and flow velocity.

Equilibrium speed

Equilibrium speed of the Antarctic glaciers

Assuming a balanced mass balance, an average speed, the so-called equilibrium speed , can be calculated for each point of a glacier . This takes advantage of the fact that, due to the conservation of the mass, ice within the glacier can neither arise out of nowhere nor be lost. The change in the total mass of the glacier above a cross-section must therefore correspond to the difference between the accumulation (or ablation ) above this cross-section and the ice flow through it: ${\displaystyle M}$ ${\displaystyle Y}$ ${\displaystyle Q}$

${\displaystyle {\frac {\mathrm {d} M}{\mathrm {d} t}}=\rho \int _{A}{\dot {b}}\mathrm {d} A-\int _{Y}Q\mathrm {d} y}$

The area integral over in the accumulation term corresponds to the area above , is the ice density, the local change in mass. The flow of ice through the area can be mathematically described as ${\displaystyle A}$ ${\displaystyle Y}$ ${\displaystyle \rho }$ ${\displaystyle {\dot {b}}}$ ${\displaystyle Q}$ ${\displaystyle Y}$

${\displaystyle Q=\int _{\mathrm {b} }^{S}\rho (z)u(z)\mathrm {d} z={\overline {\rho u}}Hy}$( is the height of the rock bed, the height of the surface). ${\displaystyle b}$ ${\displaystyle S}$

${\displaystyle H}$stands for the height and the width of the glacier cross-section . If the change in mass is negligibly small compared to the other terms of this equation (i.e. in the case of a glacier with an approximately balanced mass balance and negligible short-term variability due to snowfall or snowmelt), the following applies and therefore for any cross section of the glacier ${\displaystyle y}$ ${\displaystyle Y}$ ${\displaystyle {\frac {\mathrm {d} M}{\mathrm {d} t}}=0}$ ${\displaystyle Y}$

${\displaystyle \rho \int _{A}{\dot {b}}\mathrm {d} A=\rho {\dot {b}}A={\overline {\rho }}HU_{\mathrm {g} }y}$,

with as the area of ​​the glacier above . In words, this equation says that all of the ice that is accumulated above the observed cross-section must also flow through it. The speed at which this happens is called the equilibrium speed . Since it is an averaged quantity, knowledge of the equilibrium speed does not provide any information about the actual speed distribution within the glacier. For this purpose, a more detailed examination of the accelerating and braking forces acting on the glacier is necessary. If the mass balance of a glacier is not balanced, the actual speed will of course also deviate from the equilibrium speed. Clear differences between the measured speed and the equilibrium speed are therefore a sign that the mass balance of the glacier is not balanced. ${\displaystyle A}$ ${\displaystyle Y}$ ${\displaystyle U_{\mathrm {g} }}$

In the case of valley glaciers, the equilibrium speed is positive in the accumulation area, but negative in the ablation area, as the ice flow decreases with increasing depth. Glaciers in arctic regions that end in the sea and the large ice sheets can have a large ice flow even at the edge. They don't lose their ice by melting, but by calving icebergs. In these cases the equilibrium speed is still relatively high on the coast, in the case of Antarctica the areas with the highest equilibrium speed are even directly on the coast, as there is no ablation area on the Antarctic mainland due to the extremely low temperatures.

Vertical velocities: emergence and submergence

Flow lines and mass balance (b) of a typical valley glacier (above) and ice sheet. In the case of the valley glacier, one can clearly see that the flow lines are directed downwards in the accumulation area and upwards in the ablation area.

The equilibrium speed describes the mean horizontal speed of a glacier. If the glacier is actually in equilibrium or does not deviate significantly from it, it is also possible to make statements about the vertical speed of the ice. So that the height of the glacier does not change at any point, the vertical speed must correspond exactly to the net accumulation - i.e. the difference between accumulation and ablation:

${\displaystyle {\dot {b}}=-U_{\mathrm {v} }+U_{\mathrm {h} }\tan(\alpha )}$.

${\displaystyle U_{\mathrm {v} }}$describes the vertical speed here. The term takes into account the fact that the glacier also has a horizontal speed, so that the vertical speed does not exactly correspond to the net accumulation as soon as the glacier is inclined at an angle . ${\displaystyle U_{\mathrm {h} }\tan(\alpha )}$ ${\displaystyle \alpha }$

In the accumulation area is positive, therefore negative. The flow lines point into the glacier and the ice tends to flow downwards. This behavior is called submergence . In the ablation area the situation is exactly the opposite, the flow lines point upwards and the ice flows again towards the glacier surface (emergence) . Only at the level of the equilibrium line does the ice flow parallel to the glacier surface. For valley glaciers, significant deviations from this flow behavior indicate that the glacier is not in equilibrium. Then the vertical speed no longer corresponds to the net accumulation and the glacier loses or gains in substance. This law does not necessarily have to apply to ice sheets, since they also lose ice through other mechanisms such as the calving of icebergs. ${\displaystyle {\dot {b}}}$ ${\displaystyle U_{\mathrm {v} }}$

Forces acting on the glacier

The driving force of the glacier movements is the weight of the ice itself. This can lead to glaciers moving in two different ways:

• A constant height glacier is located on a sloping bed of rock. The force acting on the glacier then corresponds to the downhill force . ${\displaystyle H}$ ${\displaystyle \tau _{\mathrm {g} }=\rho gH\sin(\alpha )}$
• A glacier is located on a flat bed of rock, but has a variable height so that ice flows from the higher to the shallower part of the glacier. In this case applies to the acting force , where the angle symbolizes the inclination of the glacier surface. ${\displaystyle \tau _{\mathrm {g} }=\rho gH\tan(\alpha )}$ ${\displaystyle \alpha }$

A combination of the two situations described here is also possible. Since the surface inclination of glaciers is seldom more than 20 °, the small-angle approximation applies and the acting force can in this case be determined solely by the surface inclination . The force is always related to an area of ​​the glacier, so it has the unit of tension (force per area). In glaciology it is therefore also referred to as tension, even if it does not only affect one surface but the entire glacier. ${\displaystyle \sin(\alpha )\approx \tan(\alpha )\approx \alpha }$

Opposite to gravity are the friction on the rock bed , possible frictional forces on the side of the glacier and compression and expansion of the ice along the direction of flow ( ). Since accelerations within a glacier are minimal, an equilibrium can normally be assumed in which the braking forces compensate for the acceleration caused by gravity: ${\displaystyle \tau _{\mathrm {b} }}$ ${\displaystyle \tau _{s}}$ ${\displaystyle \tau _{k}}$

${\displaystyle \tau _{\mathrm {g} }=\tau _{\mathrm {b} }+\tau _{s}+\tau _{k}}$.

The friction on the rock bed is generally the most important of these three forces and accounts for 50 to 90% of the total braking forces in valley glaciers. In the case of ice shelves and ice flows flowing into the sea , the friction on the “rock bed” is, however, negligibly small and its contribution to the braking forces is almost zero. This is the reason why these glaciers are more dynamic. In addition, in this case local changes in stress in one part of the glacier also have a strong impact on other regions, whereas in glaciers with a rock bed they can remain local due to the friction. ${\displaystyle \tau _{\mathrm {b} }}$

Speed ​​profiles

Simple shear

In order to be able to calculate not only the averaged equilibrium speed but also a value for the speed at a single point on the glacier, one has to translate the acting stresses into speeds using the generalized flow law. In the simplest case, a simple shear ( parallel flow ), only forces act on the z-plane of the glacier (friction on the sides is neglected) and the glacier always flows parallel to the rock bed. This means that the only relevant entry is the stress tensor. With the law of flow and a stress that increases linearly with the height above the bed of rock, this results in the velocity after integration ${\displaystyle \tau _{\mathrm {xz} }}$ ${\displaystyle \epsilon _{\mathrm {xz} }=A\tau _{\mathrm {xz} }^{n}}$ ${\displaystyle \tau _{\mathrm {xz} }=\rho gh\alpha }$

${\displaystyle u(z)=u_{\mathrm {b} }+{\frac {2A}{n+1}}(\rho gh\alpha )^{n}(H)^{n+1}}$.

It describes the movement on the rock bed by basal sliding. Using this simple model, it can be seen that the speed increases significantly with the height above the rock bed and that the ice directly above the rock bed consequently only moves very slowly and can therefore be very old. This is one reason why ice cores can represent a climate archive that extends well into the past. ${\displaystyle u_{\mathrm {b} }}$

The speed equation makes it clear that the speed strongly depends on the parameters (height of the glacier above the rock bed) and (surface inclination) , if one assumes approximately three for the experimentally found value. Therefore, even small changes in the geometry have a strong impact on the speed. The parameter A is also strongly dependent on factors such as temperature, water content and other influences. ${\displaystyle H}$ ${\displaystyle \alpha }$ ${\displaystyle n}$

The properties of the ice near the rock bed differ greatly from the rest of the ice, which can lead to deviations from the expected flow velocities in both directions. A high proportion of soluble impurities leads to an increased flow rate, as is observed, for example, on the Agassiz ice cap . In other places, such as on Devon Island , an exceptionally low speed is observed on the rock bed. In the case of glaciers that still have very old ice, the flow behavior in the deeper layers can increase significantly within the sequence of layers as soon as one encounters a layer with ice from the last ice age. In Greenland and Canadian glaciers, speeds were measured that deviated from the expected value by a factor of about three. This is probably due to a smaller average crystal size in these glaciers caused by impurities, which makes it easier for the individual layers of ice crystals to slide on top of one another.

Real flow behavior

Strictly speaking, the simple relationship shown in the previous section only applies to horizontally infinitely extended glaciers with constant surface inclination and height. Changes in incline or in height above the bed of rock lead to gradients in which affect the speed. In the case of valley glaciers, the lateral boundary of the glacier plays a major role, which leads to additional friction. Accumulation and ablation lead to submergence and emergence, which also cannot be described as simple shear. In this case the flow law takes on a more complicated form, since not all entries of the stress tensor except can be neglected. ${\displaystyle \tau _{\mathrm {xz} }}$ ${\displaystyle \tau _{\mathrm {xz} }}$

With increasing complexity, analytical descriptions reach their limits, so that numerical simulations are required for the solution. Given the given boundary conditions such as the rock bed, the strength of the basal slide and the mass balance, the speeds at every point on the glacier can be determined. Examples of systems with high variability in flow velocities are ice streams , areas of ice sheets that have a significantly higher flow velocity than their surroundings. The cause is often topographical differences on the rock bed: Ice flows are mostly located in subglacial “valleys”, which increases the ice thickness and thus the tension locally. In addition, the higher weight ensures stronger basal gliding. Even with tidal glaciers that end in the sea, the speed is not homogeneous, but increases the closer you get to the end of the glacier, where icebergs form due to calving .

The flow velocities can vary significantly not only spatially but also in time. While the speed of most glaciers varies seasonally, so-called surges show an extreme periodic variability. Periods of ten to 100 years of relative calm alternate with brief (one to 15 year) phases of up to 1000 times higher flow rates.

Effects of climatic changes

Terminus of Kong Oscar Glacier in Greenland

Both mountain glaciers and the polar ice sheets show a significantly negative mass balance in the course of global warming ; a retreat of glaciers has been observed worldwide since the middle of the 19th century . The changed mass balance has an impact on the flow behavior of the glacier. Conversely, changes in flow behavior can in turn influence the mass balance by determining the rate of decline through higher or lower ice flow. The glacier dynamics thus have a massive influence on the expansion of the polar ice caps, indirectly also on the Earth's radiation balance, since the albedo of the ice sheets is greater than that of land or liquid water. The incomplete understanding of the future flow behavior of the polar ice sheets is also a factor of uncertainty for the determination of the global sea ​​level rise .

In the case of mountain glaciers, a higher temperature leads to an increase in the equilibrium line and a decrease in the area of ​​accumulation, which also has an impact on the flow velocity. If meltwater or rainwater penetrates from the glacier surface to the glacier bed, the ice flow can be additionally accelerated by increased basal gliding.The flow speed in the lower end of the Rhône glacier, for example, tripled from 2005 to 2009, while the ice thickness decreased. A glacial rim lake formed and the terminus of the glacier began to float on the lake, leading to increased instability of the glacier. The formation of new glacial edge lakes is a frequently observed phenomenon due to the ongoing glacier retreat. If the equilibrium line of the mountain glaciers shifts so much that it lies above the highest point of the glacier and if it loses more ice than it gains through accumulation in the entire area, this leads to complete melting in the long term. The Alpine glaciers, for example, will largely disappear within the next few decades, while the Himalayan glaciers are also losing mass, but the strong climatic and topographical variability of the Himalayas makes predictions difficult: in the Karakoram , for example, a slight increase in ice mass was observed. The effects of climatic changes on mountain glaciers and their flow behavior are therefore strongly dependent on the respective geometry and local conditions, they can have no influence at all or lead to the complete disappearance of a glacier.

The Greenland Ice Sheet lost large amounts of ice annually between 1992 and 2012. Not only has the melt on the glacier surface increased significantly, but the loss of ice due to outflow into the ocean, which contributes to the mass balance in a comparable order, has also increased. Aircraft-based laser altimeter measurements showed that the Greenland glaciers thinned by up to ten meters per year in the 1990s, with flow speeds almost doubling in some cases. For the whole of Greenland, most studies show a massive increase in ice loss since 2003, while the changes previously were rather small. Estimates obtained using various methods range from losses of between 80 and 360 gigatons of ice per year. Especially glaciers that end in the sea lose massive amounts of ice, as the warmer temperatures and melting glaciers drive various mechanisms that increase the flow velocity and thus indirectly also the loss of ice: A glacier tongue that is shortened by ice loss is accompanied by smaller braking forces. The larger amount of meltwater also drives the circulation of seawater in the fjords into which the glaciers flow. This leads to a lower formation of sea ice and thus faster flow speeds of the glaciers, especially in winter. In contrast, glaciers ending in the country show hardly any changes in the flow speed, the changes in the Greenland Ice Sheet therefore vary greatly from region to region: While in the south-east the ice loss in many glaciers has more than doubled at times and the flow speeds have increased massively in the north and north-east, too There are fewer changes in the southwest, where most glaciers do not end in the sea but further inland. Satellite measurements from the first decade of the 21st century show continuous increases in flow velocity only in the northeast, while the southwest shows variable velocities after a sharp increase at the beginning of the decade and the mean values ​​from 2010 differ only slightly from those from 2005. Overall, there does not seem to be a consistent trend for Greenland as a whole, but rather high regional and temporal variability, which also makes it difficult to estimate future sea level rise.

In the Antarctic Ice Sheet , temperatures are well below freezing all year round, meaning that ice losses on the surface are negligible. The mass of the Antarctic ice sheet is therefore dependent on precipitation and the loss of ice to the oceans. Both variables can only be determined imprecisely, which results in great uncertainties in the overall balance. In general, large differences are observed between West and East Antarctica: While the ice mass in the East hardly changed between 1980 and 2004, West Antarctica and the Antarctic Peninsula lost significant amounts of ice. In the process, the ice sheet thinned and the flow speeds accelerated, as a result of which a larger amount of ice escapes into the ocean than before. On the Antarctic Peninsula, the breaking of several ice shelves also caused an increase in the flow rate. The flow velocities are too high to allow a balanced mass balance. In many areas the flow velocity increased in the 1990s and 2000s, in others it has remained constant since 1992 but consistently above the equilibrium velocity, ie they have been losing more ice for several decades than they have gained through accumulation. A combination of 24 satellite estimates of the ice mass showed that a total of 2725 ± 1400 gigatons of ice was lost between 1992 and 2017. This corresponds to a sea level rise of 7.6 ± 3.9 millimeters. This study also confirms that ice losses occurred mainly in West Antarctica and the Antarctic Peninsula, while East Antarctica is in equilibrium within the forecast accuracy.

literature

Textbooks

• Kurt M. Cuffey, William SB Paterson: The Physics of glaciers. 4th edition. Butterword-Heinemann, Amsterdam et al. 2010, ISBN 978-0-12-369461-4 .
• Ralf Greve, Heinz Blatter: Dynamics of Ice Sheets and Glaciers . Springer, Berlin / Heidelberg 2009, ISBN 978-3-642-03415-2 .
• Roland A. Souchez, Reginald D. Lorrain: Ice composition and glacier dynamics (= Springer Series in Physical Environment. Volume 8). Springer, Berlin et al. 1991, ISBN 3-540-52521-1 .
• Cornelis J. van der Veen: Fundamentals of Glacier Dynamics. AA Balkema, Rotterdam et al. 1999, ISBN 90-5410-471-6 .

Reviews and book chapters

• Andrew Fowler: Mathematical Geoscience (= Interdisciplinary Applied Mathematics. Volume 36). Springer, London et al. 2010, ISBN 978-0-85729-699-3 , Chapter 10 (Glaciers and Ice Sheets) and 11 (Jökulhlaups) .
• Hester Jiskoot: Dynamics of Glacier. In: Vijay P. Singh, Pratap Singh, Umesh K. Haritashya (Eds.): Encyclopedia of Snow, Ice and Glaciers. Springer, Dordrecht 2011, ISBN 978-90-481-2641-5 , pp. 415-428.
• Christian Schoof, Ian Hewitt: Ice-Sheet Dynamics . In: Annual Review of Fluid Mechanics . tape 45 , no. 1 , 2013, p. 217-239 , doi : 10.1146 / annurev-fluid-011212-140632 .

Individual evidence

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