Mass balance (glaciology)

Mass balance of the Silvretta Glacier from 1960 to 2020. The cumulative annual balance can no longer be shown in the diagram from 2003; the value for 2020 is −25.6 meters of water equivalent

In glaciology, the mass balance is the difference between mass inflow ( accumulation ) and mass loss ( ablation ) of an ice body. The total mass gain or loss of a glacier , an ice cap or an ice sheet over a hydrological cycle - usually one year - is called the total mass balance. The specific mass balance is the change in mass over a period of time in relation to a point on the glacier. Mostly the total mass balance is achieved through integration measured specific mass balance data distributed over the glacier area. By dividing the total mass balance by the area of the glacier, the mean specific mass balance is obtained , which enables the behavior of different glaciers to be compared. This is the mainly published size, it is usually given in millimeters or meters of water equivalent per year and can be understood as the "average change in ice thickness". It is often referred to as the annual mass balance for short . If the mass balance is positive over several years, a glacier will advance, if it is negative it will retreat. If a glacier is in equilibrium with the climate , its mass balance is balanced.

Most of the accumulation occurs through snowfall , influenced by wind movements and avalanches . The greatest loss of mass in most glaciers is caused by the melting of snow , firn or ice on the surface. But other processes can also be important: in the case of ice shelves and tidal glaciers , calving plays a major role, steep, hanging glaciers lose a lot of mass due to avalanches, and in dry regions the sublimation of blown snow is a factor that should not be neglected.

There are various methods of determining the mass balance of a glacier. The oldest and still fundamental method today is the so-called glaciological method . This measures the change in the surface level at various points distributed over the glacier. From this, the specific mass balance at this point is determined by estimating the near-surface firn or ice density . Knowledge of the total mass of a glacier is not required to determine the mass balance, and it is often not exactly known at all.

Historical development

The oldest known efforts to determine a mass balance began as early as 1874 on the Rhone Glacier . The research at that time was carried out by the so-called “Glacier College”, which was initiated in 1869 by the Swiss Alpine Club (SAC) and the Swiss Natural Research Society (SNG, today SCNAT). The aim of the research at that time was to understand the historical development of the glacier and the relationship between changes in the glacier surface and glacier advances. The data collected at that time do not meet today's standards, mainly because the density of the firn in the glacier's nutrient zone was not determined. For the period from 1884 to the end of the series of measurements at that time in 1909, certain assumptions and extrapolations made it possible to compare the data with today's data. The mean specific mass balance of this period was −130 millimeters of water equivalent .

The Storglaciären is the glacier with the longest series of measurements on the mass balance

Continuous measurements of the specific mass balance at two locations on the Claridenfirn have been carried out since 1914. The Swedish glaciologist Hans Ahlmann (1889–1974) made pioneering contributions to mass balance measurements in today's sense, which include the entire glacier, in the 1920s and 1930s. He initially carried out these measurements every year for a different glacier, later the importance of multi-year, directly comparable data on a glacier was recognized. For the Storglaciar in northern Sweden, mass balance data have been recorded in uninterrupted succession since 1945, the longest series in the world. This was later followed by the Taku Glacier in southeast Alaska , the Storbreen in Norway and a growing number of glaciers in the Alps .

It was soon recognized that it was necessary to largely standardize the procedure for determining the mass balance in order to be able to compare and aggregate the data of different researchers. An early suggestion for this came from Mark Meier in 1962 . After some discussion, a consensus emerged under the leadership of the International Association of Scientific Hydrology (IASH, today IAHS), the key points of which were published in 1969 in the Journal of Glaciology . This publication became the de facto standard with a few amendments published a little later. In the meantime, various inconsistencies have arisen in the interpretation of some terms of this standard, and there was also the need to better cover the mass balance determination of the ice sheets, so that the International Association of Cryospheric Sciences (IACS) published a document in 2011 with the aim of continuing the standardization.

Basics

Contribution of the glacier surface to the mass balance

In most glaciers, the processes that are decisive for the mass balance take place in the area of the glacier surface that is most accessible for measurements. The main ones are snowfall , avalanches , melting, re-freezing of water, sublimation and resublimation as well as wind displacement . Another important factor is the loss of mass due to calving in glaciers ending in water. While the majority of the mass loss in valley glaciers is due to runoff in the channel, in Greenland, for example, the calving of outlet glaciers into the sea is responsible for almost 50% of the ice loss.

In the case of polar glaciers in particular, however, the processes inside the glacier cannot be completely neglected. For example, while the melt water in the depletion area of valley glaciers can flow off practically unhindered, in the nutrient area of polar ice fields it is assumed that 60% of the melt water will freeze again. Volcanism or geothermal springs can lead to noticeable ablation at the bottom of the glacier , which is the case, for example, in the northern Greenland ice sheet .

Budget year, summer and winter balance

Idealized seasonal cycle of the surface balance at one point on the glacier

The period between two annual minima of the glacier mass is one of the definitions for the balance sheet or budget year of a glacier. For mid-latitude glaciers, the budget year therefore begins in autumn, at the end of the ablation period. The glacier surface at the beginning of a budget year can be reconstructed afterwards in some places on the dirty intermediate layer. A second special point in time is at the end of the accumulation period, for most glaciers in spring, when the ice thickness is at its maximum. The data determined between these points in time are called the winter and summer balance. With this definition, which is based on the sequence of shifts (Stratigraphic System) , the budget years are not always the same length due to inconsistent weather conditions, which impairs the comparability of the data. Also, the minimum and maximum do not occur at the same time in all places, especially with large glaciers.

Another definition therefore sets a fixed calendar date for the beginning of the budget year and a distinction between winter and summer balances (fixed-date system) . For glaciers in the mid-latitudes of the northern hemisphere, the budget year usually begins on October 1st , based on the hydrological year , and March 1st is the limit between winter and summer. If it is not possible - for example due to the weather - to actually carry out the measurements at the respective date, an attempt is made to extrapolate the data of the actual appointment, for example using the data from weather stations located in the vicinity . If the cycle of the fixed-date system is adhered to approximately, but such an extrapolation is dispensed with and therefore unequal long budget years are accepted, this is referred to as the floating-date system . If several of these approaches are combined in order to obtain the data matching several definitions, one calls the combined system . Viewed over a longer period of time, the data of all systems does not differ significantly.

It should be noted, however, that on the basis of a measurement of the surface change taking place twice a year, as is at least necessary to differentiate between summer and winter balance, in none of the definitions actually the complete accumulation and ablation can be measured - for example, since snowfall in the summer months is possible. Such a distinction between summer and winter balance offers the only practical possibility to estimate the influence of the different climatic factors. There are glaciers for which there is no such seasonal cycle and no such distinction between winter and summer balance is possible. For example, glaciers in monsoonal climates have an active phase during which both the bulk of accumulation and ablation take place.

terminology

The specific mass balance is the local change in mass of a glacier in relation to an area and can be specified in kilograms per square meter (symbol ). ${\ displaystyle {b}}$

{\ displaystyle b \; {\ begin {pmatrix} {\ frac {kg} {m ^ {2}}} \ end {pmatrix}}}

Similar to precipitation , which is given as water depth in relation to an area, the information is often given in the form of a change in the thickness of the ice. Since the density of the glacier ice is not uniform, the density of the water ( ) is usually used as a proxy and the specific mass balance is expressed in meters of water equivalent. ${\ displaystyle \ rho _ {w}}$

{\ displaystyle b_ {we} = {\ frac {b} {\ rho _ {w}}} \; {\ begin {pmatrix} m \ end {pmatrix}}}

In order to express the time reference explicitly, the data are also presented in the form of the specific mass balance rate ( ). The specific mass balance is obtained by integrating the mass balance rate over time. ${\ displaystyle {\ dot {b}}}$

{\ displaystyle b = \ int _ {t} {\ dot {b}} \ dt}

Most of the data in the mass balances relate implicitly to the period of one year. In particular, if the winter ( ) and summer ( ) balance are determined separately, the annual balance is also referred to as the net balance. ${\ displaystyle {b_ {w}}}$ ${\ displaystyle {b_ {s}}}$

{\ displaystyle b_ {n} = b_ {w} + b_ {s}}

When using the glaciological method, calculations are usually carried out the other way around at points with a negative net balance, i.e. the net balance is measured as a change from the previous year and the summer balance is determined from the difference to the winter balance.

The total mass balance ( ) results from the integration of the specific mass balances over the area of the glacier ( ). By dividing the total mass balance by the area of the glacier, the mean specific balance is obtained ( ). ${\ displaystyle B_ {n}}$ ${\ displaystyle A}$ ${\ displaystyle {\ bar {b}} _ {n}}$

{\ displaystyle B_ {n} = \ int _ {A} b_ {n} \ dA}

{\ displaystyle {\ bar {b}} _ {n} = {\ frac {B_ {n}} {A}}}

Height dependence and equilibrium line

Schematic representation of the height dependence of the specific mass balance

The specific mass balance differs significantly at different points on the glacier. For most glaciers there is a clear separation between a higher accumulation zone in which the annual specific net balance is positive everywhere, and a deeper ablation zone in which it is negative. The dividing line at which the mass balance is exactly balanced (that is true) is the equilibrium line ( Equilibrium Line altitute called ELA). For most glaciers, the equilibrium line is close to the firn line at the end of summer. An exception are polar glaciers, in which ice is formed in the lower part of the nutrient area by re-freezing meltwater, so-called superimposed ice . ${\ displaystyle b_ {n} = 0}$

Another parameter of a glacier derived from the mass balance is the ratio between the nutrient area and the total area ( Accumulation Area Ratio , AAR). This ratio is small in warm years or years with little snow. In the case of valley glaciers, it is assumed that these are in equilibrium with the climate at a ratio between 55% and 65%. In the case of Pasterze , the ratio in four budget years in the period from 2005 to 2010 was between 45% and 49%, there was an outlier in 2008 with only 16%.

The so-called mass balance gradient expresses the rate of change of the specific mass balance in relation to the altitude. A high mass balance gradient indicates that the glacier is sensitive to the climate. The mass balance gradient in the area of the equilibrium line is also referred to as the activity index.

But there are also glaciers in which the nutrient and feeding area cannot be clearly separated: In the case of glaciers in the Antarctic, the nutrient area can extend over the entire glacier; they lose their mass almost exclusively through calving. Avalanches, coastal fog or shadowing can also result in deeper "islands" with a positive mass balance.

Methods

There are several methods of determining the mass balance of a glacier. The oldest and still fundamental today is the so-called direct glaciological method , in which the changes on the glacier surface are measured on site. All other methods are called "indirect". However, this is usually only emphasized if the mass balance of a glacier is also estimated on the basis of the directly determined past data using data that is easier to collect or less data in the following years. There are also other methods, in particular the geodetic method , in which the glacier does not have to be entered for measurement. However, none of the methods is suitable for all glaciers and provides sufficiently accurate results for every glacier. In order to better estimate the accuracy of the result, it is therefore advisable to combine several methods.

Measurement of ablation at Sperry Glacier in Glacier National Park

Direct glaciological method

With the direct glaciological method, the surface changes are determined at the most representative measurement points possible and the specific mass balance is determined from this. On the basis of the data obtained by this measuring network, the specific mass balances for the entire glacier area are estimated by interpolation and the mean specific mass balance is calculated from this. Measuring points are needed in both the nutrient and the feeding area.

To measure the ablation, rods, also known as the ablation level, have to be drilled deep enough into the ice that they do not fall out at the end of the ablation period - a drilling depth of ten meters near the end of the glacier may not be enough for this. The next time you visit the glacier, the change in altitude is measured. Assuming an ice density of 900 kilograms per cubic meter, the change in mass is calculated from this. If it is to be expected that the ablation will also extend to the area above the firn border, poles must also be set there and, moreover, the density profile in the vicinity of the pole must be determined in advance to be on the safe side.

Digging a shaft to measure the firn density on the Taku Glacier

Poles are also set to measure accumulation. In the case of large amounts of snow, it may be impossible to prevent them from disappearing into the snow - there are various strategies for finding such poles again, for example attaching a transmitter or a strong magnet. At the end of the accumulation period, the height of the fallen snow must be determined. In the case of glaciers in mid-latitudes, it is usually not difficult to determine the layer before the start of the accumulation period - it is “dirty” due to the dust collected during the ablation period and is harder than the surrounding layers due to frozen meltwater. Marking on the pole can also be helpful; in very difficult cases, dark-colored sawdust can also be scattered around the pole. To determine the density of the accumulated snow, a shaft is usually dug near the pole and the snow profile on the wall of the shaft is analyzed. A drill core can also be removed to determine the density, but there is a risk that the snow will be compacted when it is removed, which can lead to an overestimation of the density.

The exact position of the rods is determined while measuring the surface change. The fact that the bars moved with the ice is usually not taken into account. The accuracy of the mass balance determined in this way can be difficult to assess, especially in the case of glaciers with extensive areas that are difficult to access, such as crevasse zones . The glaciological method requires a comparatively high expenditure of time and personnel.

Indirect methods based on the glaciological method

Annual mass balance and AAR for Vernagtferner for the years 1965 to 2010. The points from the last three years are highlighted. The coefficient of determination (R²) of the regression line is 0.94 in this case, which is a good approximation .

Past measurements have shown that the height profile of the specific mass balances of many glaciers is very similar over several years and essentially only shifts depending on the weather in the respective year. This makes it possible to limit oneself to a few measuring points (index stakes) that are as representative as possible in subsequent years and still be able to estimate the mass balance of the entire glacier with sufficient accuracy. In many glaciers there is also a correlation between the mean specific mass balance and the height of the equilibrium line (ELA) or the ratio of the area of the nutrient area to the total area (AAR). Thus, the specific mass balance can be approximately calculated on the basis of a formula from ELA or AAR determined from the past data obtained by means of the direct glaciological method. What is attractive about it is that ELA and AAR can be determined on the basis of aerial photographs taken at the end of the ablation period, so that no measurements are required on site. The procedure does not work, however, if the snow limit is not identical to the equilibrium line due to meltwater freezing again . You should not miss the last possible point in time for a useful exposure, because early snowfall can make it impossible to determine the equilibrium line.

Geodetic method

In the geodetic method, the change in volume is determined by comparing the elevation model of the glacier at two specific points in time, often over a period of several years. The change in mass is calculated from the change in volume, assuming the density. It should be noted that a change in the thickness of the ice at one point can be caused by a loss or gain in mass or simply by the flow of the ice. The change in volume of an ice column at a point on the glacier is made up of a contribution that can be assigned to the mass balance and another contribution caused by the movement of the ice:

{\ displaystyle \ Delta V = \ Delta V_ {bil} + \ Delta V_ {dyn}}

The contribution of the glacier dynamics can certainly exceed that of the change in mass. This means that, for example, at points at which an increase in volume is measured, the ablation can nevertheless be greater than the accumulation, that is to say a negative specific mass balance is present.

Emergence and submergence make the essential contribution to these vertical movements on the glacier surface . These are usually directed downwards in the nutrient area (submergence) and upwards in the consumption area (emergence). These movements are essential to ensure that a glacier that is in equilibrium with the climate maintains its shape by compensating for the increases and decreases in volume caused by accumulation and ablation. For the glacier as a whole, the vertical movements cancel each other out as long as its overall density does not change.

As long as these vertical movements are not known precisely enough, the geodetic method cannot be used to determine the mass balance for parts of the glacier, nor can accumulation and ablation be quantified separately. Exact topographical maps are the basis for determining the change in volume and, in the last few decades, digital elevation models that are obtained from aerial or satellite images, laser scanning and radar interferometry are also used. Difficulties with this method can be caused by the lack of contrast, especially in the snow-rich accumulation area. The estimation of the density of the ice and in particular of the snow can be very imprecise, and it can also be necessary to calculate corrections for the deeper layers of the glacier that settle. The geodetic method is particularly suitable as a supplement to the glaciological method, in particular to uncover systematic errors.

Hydrological method

From a hydrological point of view, the total mass balance of a glacier can be determined by subtracting the losses due to runoff and evaporation from the sum of the precipitation in the catchment area of the glacier . Furthermore, the changes in the water not stored in the form of glacier ice also play a role, be it groundwater or water within the glacier, the amount of which increases sharply, especially at the beginning of the ablation period. The actual measurement density required for precipitation measurement in mountain regions can hardly be achieved in practice. A sufficiently accurate measurement of the amount of water runoff is also extremely time-consuming. Therefore, the mass balance determination using the hydrological method is not particularly accurate - the error rate is often in the order of 100% - which is why it is normally only used in combination with other methods. In contrast to the glaciological method, however, changes in mass inside and at the bottom of the glacier are also recorded.

Model-based methods

In this approach, numerical models are used, similar to the methods for weather forecasting , which simulate the behavior of a glacier in interaction with weather and climate, which is relevant for the mass balance. The modeling approaches focus primarily on ablation. Relatively simple degree-day approaches are used as well as more detailed energy balance models that also take into account solar radiation, albedo or wind , for example . The choice of the procedure depends not least on what data is available. The temporal and spatial distribution of precipitation can usually only be roughly mapped. Such models must first be calibrated using data from nearby weather stations and otherwise determined mass balance data from the past . Glacier movements that are unrelated to the climate, such as avalanches or surges, are a problem.

Other methods

The flow of the glacier is also included in various ways. For example, the ice flow is determined by a glacier cross-section (flux gate) . This can be of particular interest in the case of calving glaciers or outlet glaciers . This data is often combined with data obtained elsewhere. The approach of combining the different flow velocities of the glacier surface with the data obtained using the geodetic method (flux divergence) goes even further in order to be able to derive a spatial distribution of the mass balance, which is not possible with the geodetic method alone. So far, the accuracy of the data is not sufficient, as the models of glacier dynamics are currently only inadequately able to depict vertical ice movements.

Also gravimetric methods were used large glaciated areas already for determining the mass balances. At present, only the Gravity Recovery And Climate Experiment (GRACE) can provide usable data for this . It is controversial whether this procedure can also be used for smaller-scale mass balance determinations.

Goals and Results

The aim of determining the mass balance of glaciers has always been to be able to better understand and predict the behavior of glaciers, especially with regard to disasters caused by glaciers such as glacial lake eruptions . Furthermore, the development of the mass balance of a glacier is usually a reaction to a changed climate, which occurs practically without delay. Therefore, an important motivation for the detailed determination of mass balances is to better understand the relationships between the climate and the resulting changes in the glacier, the glacier dynamics . On the one hand, this enables well-founded conclusions to be drawn about the climate at that time from historical glacier behavior, but on the other hand, it also enables a more precise mapping of the behavior of the glaciers in climate models . The hydrological aspect is also important , on the one hand at the regional level with regard to the future drinking water supply , and on the other hand globally with the forecast of the expected rise in sea level . Whether the ice sheets of Greenland and Antarctica or the other glaciers and ice caps on earth will make the greater contribution to sea level rise in the first half of the 21st century is a matter of dispute.

Glaciers and ice caps

Direct measurements of the mass balance have so far been carried out on around 300 glaciers worldwide and roughly cover the period since the second half of the 20th century. Of this, the data from around 250 glaciers was collected by the World Glacier Monitoring Service (WGMS) as a contribution to the Global Terrestrial Network for Glaciers (GTN-G) and made available in a standardized format. For the period between 1980 and 2010, however, the data were only collected for 37 glaciers without gaps. These glaciers, known as “reference glaciers”, do not represent a representative selection of glaciers worldwide. The total amount of all glaciers with mass balance data certainly provides a clearly distorted picture. Most of them are in the Alps or Scandinavia , some are in North America and the high mountains of Central Asia. In contrast, the glaciers in northern Asia and South America are completely underrepresented ; the ice sheets of Greenland and Antarctica have to be considered separately anyway. This selection of glaciers is also unbalanced from other perspectives: on the one hand, small glaciers are overrepresented, and the accessibility of the glaciers also logically plays a role, as well as whether the weather at all makes it possible to take measurements on site frequently enough. The extent to which it is still possible to draw conclusions about glaciers worldwide on the basis of this data is controversial. There is agreement that series of measurements should be started in previously underrepresented regions. Another strategy is the attempt to derive mass balances from cumulative changes in length of the glaciers. This is attractive because changes in length are much easier to determine and there is much more historical data. At least the order of magnitude of the mass balance can be estimated in this way.

For the 37 glaciers with seamless, directly determined mass balance data between 1980 and 2010, the average annual mean specific mass balance in the first decade of the 21st century was −0.75 meters of water equivalent . The mass loss has thus doubled since the 1970s. In the 1980s, a third of these glaciers still had a positive mass balance; in the first decade of the 21st century it was only a fifth, which suggests that the glacier retreat is covering more and more areas completely. In the case of some glaciers, it has been observed that there is an increase in the mass balance gradient. This is caused by increased ablation in the nutrient area and an opposite, somewhat smaller increase in accumulation in the nutrient area - the slightly higher temperatures obviously lead to more precipitation at higher altitudes. This makes the glaciers more sensitive to further temperature changes.

Schematic cross-section and specific mass balance ( ) of a typical valley glacier (above) and an ice sheet

{\ displaystyle b}

Greenland and Antarctic ice sheets

The mass balances of the two ice sheets are of great interest because their behavior is crucial for sea level rise . If they were to melt completely, this would mean an increase of around 65 to 70 meters.

With the exception of the deeper, coastal areas of the Greenland Ice Sheet, there are no significant losses of mass due to melting in the polar ice sheets. The specific mass balance is therefore shaped by continentality , since the precipitation is mainly concentrated in the areas that are a few hundred kilometers from the sea. This means that the specific mass balance decreases with distance from the coast. In the Antarctic , the annual balance on the coast is typically between 300 and 600 millimeters of water equivalent, at the South Pole it is less than 100 millimeters. The ice sheets lose their mass mainly through calving , in the Antarctic this accounts for 90% and in Greenland 50% of the mass loss. In Antarctica, subglacial melting at the bottom of the ice shelves is another factor.

At the end of the 1990s, the mass balance of the ice sheets was almost unknown. Even at the beginning of the 21st century, the measurement uncertainties did not allow a statement to be made as to whether the ice masses in Greenland and Antarctica were increasing or decreasing. Three different, largely independent procedures are currently in use:

Mass balance method (Mass Budget Method) : Here, the accumulation and ablation is determined on the surface, in addition, the ice flow is determined on the edges of the ice sheet. The surface balance is determined using simulation models that are calibrated or verified using directly obtained measurement data. In order to determine the runoff at the edges, the flow velocity and ice thickness of the ice streams and outlet glaciers are measured with the help of satellites.
Geodetic method (Altimetry Method) : The changes in the height of the surface are determined by means of laser scanning and radar interferometry by satellites such as ERS I / II , Geosat or ICESat , from which the changes in volume and mass are derived.
Gravimetric method (Gravity Method) : Since April 2002, the two satellites of the GRACE project have been measuring the earth's gravitational field and its changes over time. In order to draw conclusions about the changes in mass, various other effects such as tides have to be factored out.

Corrections due to the postglacial land elevation must be taken into account in the gravimetric method and, to a lesser extent, in the geodetic method. It should also be noted that the ice for sea level rise is effective as soon as it swims. For this purpose, the line from which the ice of the ice shelf or the glacier tongue begins to swim on the sea must be determined, the so-called grounding line . With the gravimetric method, the floating ice is not part of the current ice mass anyway. With the other methods, you have to estimate the course of the grounding line and also take into account if it shifts in the direction of the coastline due to the thinning ice.

Mass balance 1992–2011
region	Balance ( Gt / year)
Greenland Ice Sheet	−142 ± 49
Antarctic Peninsula	0−20 ± 14
East Antarctic Ice Sheet	0-14 ± 43
West Antarctic Ice Sheet	0−65 ± 26
Total Antarctic Ice Sheet	0−71 ± 53
Total ice sheets	−213 ± 72

All procedures have their weaknesses. By combining the methods, an attempt is made to obtain a more accurate result. A study from 2012 tried to summarize the data from previous measurements and evaluate them according to the latest findings. It is emphasized here that long series of measurements are important so that temporary fluctuations do not impair the informative value of the results. For the period between 1992 and 2011, an average mass balance of approximately −213 gigatons per year was determined. By far the largest part was accounted for by the Greenland Ice Sheet with around −142 gigatons per year, the Antarctic Peninsula and West Antarctica also had a negative mass balance, while that of East Antarctica showed a positive trend. 360 gigatons correspond to a sea level rise of one millimeter, so according to this study, the ice sheets have caused a total sea level rise of 11.2 millimeters since 1992. The Greenland ice sheet becomes thinner mainly at its edges, which is also due to increased melting processes on the surface. The positive mass balance in East Antarctica could be due to the increased precipitation due to the rise in temperature, but it could also be a natural fluctuation. Basically, a change in glacier dynamics can be observed in the two ice sheets, the flow velocities in the edge areas and outlet glaciers have increased, which means that more ice is released into the oceans.

literature

Kurt M. Cuffey, WSB Paterson: The Physics of Glaciers. Fourth Edition Butterworth-Heinemnn, Burlington 2010, ISBN 0-12-369461-2
Georg Kaser , Andrew Fountain, Peter Jansson: A manual for monitoring the mass balance of mountain glaciers - with particular attention to low latitude characteristics. International Commission on Snow and Ice (ICSI), 2002 ( online ; PDF; 3.1 MB)
Roger LeB. Hooke: Principles of Glacier Mechanics. Second edition. Cambridge University Press, Cambridge 2005, ISBN 0-521-83609-3
Wilfried Haeberli: Glacier Mass Balance. In: Vijay P. Singh, Pratap Singh, Umesh K. Haritashya (Eds.): Encyclopedia of Snow, Ice and Glaciers. Springer , Dordrecht 2011, pp. 399-408, ISBN 978-90-481-2641-5
Eric Rignot : Ice Sheet Mass Balance. In: Vijay P. Singh, Pratap Singh, Umesh K. Haritashya (Eds.): Encyclopedia of Snow, Ice and Glaciers. Springer, Dordrecht 2011, pp. 608-612, ISBN 978-90-481-2641-5
JG Cogley et al .: Glossary of Glacier Mass Balance and Related Terms. IHP-VII Technical Documents in Hydrology No. 86, IACS Contribution No. 2, UNESCO-IHP, Paris 2011 ( online ; PDF; 2.7 MB)
G. Østrem, M. Brugman: Glacier mass-balance measurements: a manual for field and office work. National Hydrological Research Institute (NHRI), Saaskaton 1991
World Glacier Monitoring Service (WGMS): Fluctuations of Glaciers 2005–2010 (Vol. X). Zurich 2012 ( online ; PDF; 4.8 MB)

Individual evidence

↑ ^a ^b ^c ^d Wilfried Haeberli: Glacier Mass Balance. See literature
↑ Kaser et al .: A manual for monitoring the mass balance of mountain glaciers. Page 21f; see literature
↑ ^a ^b J. G. Cogley et al .: Glossary of Glacier Mass Balance and Related Terms. P. 6, see literature
↑ Peter Kasser: 100 Years of the Glacier Commission, their creation and history. In: Swiss Academy of Natural Sciences: Glaciers in constant change: Jubilee symposium of the Swiss Glacier Commission. Verbier 1993, p. 11 ( Google books )
↑ Jiyang Chen, Martin Funk: Mass balance of Rhone Glacier during 1882 / 83–1986 / 87. In: Journal of Glaciology. Volume 36, 1990, pp. 199–209 ( online ( Memento from February 9, 2016 in the Internet Archive ); PDF; 1.2 MB)
↑ ^a ^b Roger J. Braithwaite: After six decades of monitoring glacier mass balance we still need data but it should be richer data. In: Annals of Glaciology. Volume 50, 2009, pp. 191–197 ( online ( Memento from March 1, 2014 in the Internet Archive ); PDF; 235 kB)
↑ ^a ^b J. G. Cogley et al .: Glossary of Glacier Mass Balance and Related Terms. P. 2f, see literature
^ Mark. F. Meier: Proposed definitions for glacier mass budget terms. In: Journal of Glaciology. Volume 4, 1962, pp. 252–263 ( online ( memento of February 18, 2013 in the Internet Archive ); PDF; 8.4 MB)
↑ Anonymous: Mass-Balance Terms. In: Journal of Glaciology. Volume 8, 1969, pp. 3–7 ( online ( Memento from February 18, 2013 in the Internet Archive ); PDF; 3.7 MB)
↑ Graham Cogley: Mass balance terms revisited. In: Journal of Glaciology. Volume 46, 2010, pp. 997–1001 ( online ; PDF; 81 kB)
↑ ^a ^b J. G. Cogley et al .: Glossary of Glacier Mass Balance and Related Terms. See literature
↑ ^a ^b ^c ^d ^e ^f Cuffey, Paterson: The Physics of Glaciers. Fourth Edition. Pp. 96-109, see literature
^ ^A ^b Hooke: Principles of Glacier Mechanics. Pp. 17-41, see literature
↑ Cuffey, Paterson: The Physics of Glaciers. Fourth Edition pp. 116–121, see literature
↑ JG Cogley et al .: Glossary of Glacier Mass Balance and Related Terms. P. 10, see literature
↑ Cuffey, Paterson: The Physics of Glaciers. Fourth Edition pp. 91–96, see literature
↑ ^a ^b Kaser et al .: A manual for monitoring the mass balance of mountain glaciers. Pp. 9-14, see literature
↑ JG Cogley et al .: Glossary of Glacier Mass Balance and Related Terms. P. 86f, see literature
↑ Jostein Bakke, Atle Nesja: Equilibrium-Line Altitute. 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. 268-277
^ World Glacier Monitoring Service (WGMS): Fluctuations of Glaciers 2005–2010 (Vol. X). P. 182, see literature
↑ ^a ^b ^c ^d ^e ^f ^g Cuffey, Paterson: The Physics of Glaciers. Fourth Edition pp. 127–131, see literature
↑ Kaser et al .: A manual for monitoring the mass balance of mountain glaciers. Pp. 34-39, see literature
↑ Østrem, Brugman: Glacier mass-balance measurements: a manual for field and office work. P. 34 ff., See literature
↑ Kaser et al .: A manual for monitoring the mass balance of mountain glaciers. P. 42 ff., See literature
↑ World Glacier Monitoring Service (WGMS): Glacier Mass Balance Bulletin No. 11 (2008-2009). Zurich 2011, p. 14 ( online ( memento of November 2, 2012 in the Internet Archive ); PDF; 9.6 MB)
↑ ^a ^b ^c ^d ^e ^f ^g Kaser et al .: A manual for monitoring the mass balance of mountain glaciers. Pp. 21-26, see literature
^ A. Fischer: Comparison of direct and geodetic mass balances on a multi-annual time scale. In: The Cryosphere . , Volume 5, 2011, pp. 107–124 ( online ; PDF; 3.3 MB)
^ Hooke: Principles of Glacier Mechanics. P. 91f, see literature
↑ Kathrin Marowsky: The validation of the glacier model Surges using the example of Vernagtferner and Northern and Southern Schneeferner. Diploma thesis, Munich 2010 ( online ; PDF; 14.8 MB)
↑ Stefan Reisenhofer: Modeling the mass and energy balance of a glacier, using the Pasterze as an example. Diploma thesis, University of Vienna, Vienna 2009 ( online ; PDF; 3.1 MB)
↑ Kaser et al .: A manual for monitoring the mass balance of mountain glaciers. Pp. 15-20, see literature
↑ Østrem, Brugman: Glacier mass-balance measurements: a manual for field and office work. P. 1 ff., See literature
↑ Mark. F. Meier et al .: Glaciers dominate eustatic sea-level rise in the 21st century. In: Science. Volume 317, 2007, pp. 1064-1067 ( online ; PDF; 171 kB)
↑ ^a ^b Eric Rignot et al .: Acceleration of the contribution of the Greenland and Antarctic ice sheets to sea level rise. In: Geophysical Research Letters. Volume 38, 2011, pp. L05503 – L05508 ( online ( memento from October 20, 2013 in the Internet Archive ))
↑ World Glacier Monitoring Service (WGMS): Glacier Mass Balance Bulletin No. 11 (2008-2009). Zurich 2011, p. 85 ( online ( memento of November 2, 2012 in the Internet Archive ); PDF; 9.6 MB)
↑ M. Zemp, M. Hoelzle, W. Haeberli: Six Decades of glacier mass-balance observations: a review of the worldwide monitoring network. In: Annals of Glaciology. Volume 50, 2009, pp. 101–111 ( online ( Memento from May 2, 2013 in the Internet Archive ); PDF; 330 kB)
↑ M. Hoelzle et al .: Secular glacier mass balances derived from cumulative glacier length changes. In: Global and Planetary Change. Volume 36, 2003, pp. 295–306 ( online ( Memento from March 4, 2016 in the Internet Archive ); PDF; 577 kB)
^ World Glacier Monitoring Service (WGMS): Fluctuations of Glaciers 2005–2010 (Vol. X). Page 71, see literature
↑ Mark B. Dyurgerov, Mark F. Meier: Glaciers and the Changing Earth System: A 2004 Snapshot. Institute of Arctic and Alpine Research, University of Colorado, Bolder 2005, ISSN 0069-6145 , pp. 7, 22 ff. ( Online ; PDF; 2.5 MB)
↑ ^a ^b ^c Eric Rignot : Ice Sheet Mass Balance. See literature
↑ Cuffey, Paterson: The Physics of Glaciers. Fourth Edition pp. 575-578, see literature
↑ ^a ^b ^c ^d Andrew Shepherd et al .: A Reconciled Estimate of Ice-Sheet Mass Balance. In: Science. Volume 338, 2012, pp. 1183–1189 ( online ; PDF; 786 kB)

Web links

Commons : Glacier mass balance - collection of images, videos and audio files

[Haeberli2011-1] Wilfried Haeberli: Glacier Mass Balance. See literature

[2] Kaser et al .: A manual for monitoring the mass balance of mountain glaciers. Page 21f; see literature

[Cogley2011_6-3] J. G. Cogley et al .: Glossary of Glacier Mass Balance and Related Terms. P. 6, see literature

[4] Peter Kasser: 100 Years of the Glacier Commission, their creation and history. In: Swiss Academy of Natural Sciences: Glaciers in constant change: Jubilee symposium of the Swiss Glacier Commission. Verbier 1993, p. 11 ( Google books )

[ChenFunk-5] Jiyang Chen, Martin Funk: Mass balance of Rhone Glacier during 1882 / 83–1986 / 87. In: Journal of Glaciology. Volume 36, 1990, pp. 199–209 ( online ( Memento from February 9, 2016 in the Internet Archive ); PDF; 1.2 MB)

[Braithwaite2009-6] Roger J. Braithwaite: After six decades of monitoring glacier mass balance we still need data but it should be richer data. In: Annals of Glaciology. Volume 50, 2009, pp. 191–197 ( online ( Memento from March 1, 2014 in the Internet Archive ); PDF; 235 kB)

[Cogley2011_2-7] J. G. Cogley et al .: Glossary of Glacier Mass Balance and Related Terms. P. 2f, see literature

[Meier1962-8] Mark. F. Meier: Proposed definitions for glacier mass budget terms. In: Journal of Glaciology. Volume 4, 1962, pp. 252–263 ( online ( memento of February 18, 2013 in the Internet Archive ); PDF; 8.4 MB)

[9] Anonymous: Mass-Balance Terms. In: Journal of Glaciology. Volume 8, 1969, pp. 3–7 ( online ( Memento from February 18, 2013 in the Internet Archive ); PDF; 3.7 MB)

[Cogley2010-10] Graham Cogley: Mass balance terms revisited. In: Journal of Glaciology. Volume 46, 2010, pp. 997–1001 ( online ; PDF; 81 kB)

[Cogley2011-11] J. G. Cogley et al .: Glossary of Glacier Mass Balance and Related Terms. See literature

[CuffeyPaterson2010-4.2-12] ↑ ^a ^b ^c ^d ^e ^f Cuffey, Paterson: The Physics of Glaciers. Fourth Edition. Pp. 96-109, see literature

[Hooke3-13] A ^b Hooke: Principles of Glacier Mechanics. Pp. 17-41, see literature

[CuffeyPaterson2010-4.5-14] Cuffey, Paterson: The Physics of Glaciers. Fourth Edition pp. 116–121, see literature

[Cogley2011_10-15] JG Cogley et al .: Glossary of Glacier Mass Balance and Related Terms. P. 10, see literature

[CuffeyPaterson2010-4.1-16] Cuffey, Paterson: The Physics of Glaciers. Fourth Edition pp. 91–96, see literature

[Kaser1-17] Kaser et al .: A manual for monitoring the mass balance of mountain glaciers. Pp. 9-14, see literature

[Cogley2011_86-18] JG Cogley et al .: Glossary of Glacier Mass Balance and Related Terms. P. 86f, see literature

[BakkeNestje2011-19] Jostein Bakke, Atle Nesja: Equilibrium-Line Altitute. 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. 268-277

[FoG10_182-20] World Glacier Monitoring Service (WGMS): Fluctuations of Glaciers 2005–2010 (Vol. X). P. 182, see literature

[CuffeyPaterson2010-4.7-21] ↑ ^a ^b ^c ^d ^e ^f ^g Cuffey, Paterson: The Physics of Glaciers. Fourth Edition pp. 127–131, see literature

[Kaser6-22] Kaser et al .: A manual for monitoring the mass balance of mountain glaciers. Pp. 34-39, see literature

[23] Østrem, Brugman: Glacier mass-balance measurements: a manual for field and office work. P. 34 ff., See literature

[Kaser7-24] Kaser et al .: A manual for monitoring the mass balance of mountain glaciers. P. 42 ff., See literature

[GMBB11_14-25] World Glacier Monitoring Service (WGMS): Glacier Mass Balance Bulletin No. 11 (2008-2009). Zurich 2011, p. 14 ( online ( memento of November 2, 2012 in the Internet Archive ); PDF; 9.6 MB)

[Kaser3-26] ↑ ^a ^b ^c ^d ^e ^f ^g Kaser et al .: A manual for monitoring the mass balance of mountain glaciers. Pp. 21-26, see literature

[Fischer2011-27] A. Fischer: Comparison of direct and geodetic mass balances on a multi-annual time scale. In: The Cryosphere . , Volume 5, 2011, pp. 107–124 ( online ; PDF; 3.3 MB)

[Hooke5.8-28] Hooke: Principles of Glacier Mechanics. P. 91f, see literature

[Marowsky2010-29] Kathrin Marowsky: The validation of the glacier model Surges using the example of Vernagtferner and Northern and Southern Schneeferner. Diploma thesis, Munich 2010 ( online ; PDF; 14.8 MB)

[Reisenhofer2009-30] Stefan Reisenhofer: Modeling the mass and energy balance of a glacier, using the Pasterze as an example. Diploma thesis, University of Vienna, Vienna 2009 ( online ; PDF; 3.1 MB)

[Kaser2-31] Kaser et al .: A manual for monitoring the mass balance of mountain glaciers. Pp. 15-20, see literature

[32] Østrem, Brugman: Glacier mass-balance measurements: a manual for field and office work. P. 1 ff., See literature

[Meier2007-33] Mark. F. Meier et al .: Glaciers dominate eustatic sea-level rise in the 21st century. In: Science. Volume 317, 2007, pp. 1064-1067 ( online ; PDF; 171 kB)

[Rignot2011b-34] Eric Rignot et al .: Acceleration of the contribution of the Greenland and Antarctic ice sheets to sea level rise. In: Geophysical Research Letters. Volume 38, 2011, pp. L05503 – L05508 ( online ( memento from October 20, 2013 in the Internet Archive ))

[GMBB11_85-35] World Glacier Monitoring Service (WGMS): Glacier Mass Balance Bulletin No. 11 (2008-2009). Zurich 2011, p. 85 ( online ( memento of November 2, 2012 in the Internet Archive ); PDF; 9.6 MB)

[Zemp2009-36] M. Zemp, M. Hoelzle, W. Haeberli: Six Decades of glacier mass-balance observations: a review of the worldwide monitoring network. In: Annals of Glaciology. Volume 50, 2009, pp. 101–111 ( online ( Memento from May 2, 2013 in the Internet Archive ); PDF; 330 kB)

[Hoelzle2003-37] M. Hoelzle et al .: Secular glacier mass balances derived from cumulative glacier length changes. In: Global and Planetary Change. Volume 36, 2003, pp. 295–306 ( online ( Memento from March 4, 2016 in the Internet Archive ); PDF; 577 kB)

[FoG10_71-38] World Glacier Monitoring Service (WGMS): Fluctuations of Glaciers 2005–2010 (Vol. X). Page 71, see literature

[Dyurgerov2005-39] Mark B. Dyurgerov, Mark F. Meier: Glaciers and the Changing Earth System: A 2004 Snapshot. Institute of Arctic and Alpine Research, University of Colorado, Bolder 2005, ISSN 0069-6145 , pp. 7, 22 ff. ( Online ; PDF; 2.5 MB)

[Rignot2011-40] Eric Rignot : Ice Sheet Mass Balance. See literature

[CuffeyPaterson2010-14.1-41] Cuffey, Paterson: The Physics of Glaciers. Fourth Edition pp. 575-578, see literature

[Shepherd2012-42] Andrew Shepherd et al .: A Reconciled Estimate of Ice-Sheet Mass Balance. In: Science. Volume 338, 2012, pp. 1183–1189 ( online ; PDF; 786 kB)