This content has been downloaded from IOPscience. Please scroll down to see the full text.

Download details:

IP Address: 54.39.106.173

This content was downloaded on 01/06/2020 at 14:05

Please note that terms and conditions apply.

You may also be interested in:

Sea level change as an indicator of cryosphere instabilities

Kurt Lambeck

Snow: a reliable indicator for global warming in the future?

H-W Jacobi

The application of remote sensing image sea ice monitoring method in Bohai Bay based on C4.5

decision tree algorithm

Wei Ye and Wei Song

Tests of piles in melted and frozen soils in creep mode: relaxation conditions

Petr Korolev and Mikhail Korolev

CryoClim: A new system and service for climate monitoring of the cryosphere

R Solberg, M A Killie, L M Andreassen et al.

Northern Eurasia Earth Science Partnership Initiative

Pavel Groisman and Amber J Soja

Spatial and temporal patterns of greenness on the Yamal Peninsula, Russia: interactions

ofecological and social factors affecting the Arctic normalized difference vegetationindex

D A Walker, M O Leibman, H E Epstein et al.

Chemistry of the atmosphere and the climate

K Ya Kondrat'ev

Late Cenozoic Paleoceanography of the Central Arctic Ocean

Matt O'Regan

IOP Concise Physics

Introduction to the Physics of the Cryosphere

Melody Sandells and Daniela Flocco

Chapter 1

The cryosphere

1.1 Definition of the cryosphereThe cryosphere encompasses all regions that experience water in ice form for someportion of the year. This includes the Arctic, Antarctic, large parts of North America,Eurasia and some parts of the Southern Hemisphere such as the Andes of Chile,Bolivia and Argentina. Frozen soil, permafrost, snow, glaciers, ice sheets and shelves,and river, lake and sea ice are all features of the Earth’s surface in the cryosphere, withsnow and sea ice being among the most dynamic (Foster et al 2005). Some of theterms used to describe parts of the cryosphere have very specific definitions. Perma-frost is ground that has been at a temperature of 0 °C or below for two or more years.Glaciers are flowing bodies of ice covering 0.1 km2 or more of a particular topo-graphic feature and may be covered in rock or soil debris, as shown in figure 1.1. Icemasses that cover a mountain or mountain range, or land surface up to an area of50 000 km2 are known as ice fields if they are constrained by the topography and icecaps if they are not. A particular feature of ice caps is a dome-like shape, where the iceflows away from the highest point. Ice sheets are bodies of ice covering more than50 000 km2 of the land surface; ice shelves are attached to land as they are formed as aresult of land ice flowing into the sea, but are floating on the water. Icebergs aresections of freshwater ice that have broken away from the land ice. Sea ice is formedfrom frozen sea water and therefore differs from land ice because it contains salt.

Figure 1.2, reproduced with kind permission from Key et al (2007), shows theglobal distribution of the cryosphere; it exists at all latitudes for some portion of theyear. To explore the cryosphere in more detail, visit the webpages of the NationalSnow and Ice Data Center (NSIDC), who have developed an interactive atlas of thecryosphere. The areal extents of cryospheric components, adapted from Vaughanet al (2013), are given in table 1.1. Snow and sea ice have large seasonal variabilityand are thus more sensitive to climate change.

The variability in snow and sea ice areas and concentrations is illustrated infigure 1.3 for the Northern and Southern Hemispheres, and compares coverage

doi:10.1088/978-1-6270-5303-7ch1 1-1 ª Morgan & Claypool Publishers 2014

Figure 1.2. Global distribution of cryospheric components (from Key et al 2007, reproduced with kindpermission).

Figure 1.1. Ice-cored terminal moraine of Hilda Glacier, Banff National Park. Photograph reproduced withkind permission from Sarah Boon.

Introduction to the Physics of the Cryosphere

1-2

Figure 1.3. Seasonal changes in sea ice and snow cover for the Northern Hemisphere (top) and SouthernHemisphere (bottom). Images are shown for 1 March 2013 (a), (c) and for 1 September 2013 (b), (d ).

Table 1.1. Areal extent of cryospheric components.

Cryospheric element Areal extent (106 km2)

Seasonal frozen ground 48.7Permafrost 13.3–17.7Seasonal snow 1.9–45.2Glaciers 0.7Greenland ice sheet 1.8Antarctic ice sheet 12.3Northern Hemisphere lake and river ice 1.6Antarctic ice shelves 1.6Antarctic sea ice 2.9–18.9Arctic sea ice 6.2–14.1

Introduction to the Physics of the Cryosphere

1-3

in March with the September distribution for 2013. These images were formed fromthe NASA Near-Real-Time SSM/I-SSMIS EASE-Grid Daily Global Ice Con-centration and Snow Extent data set, derived from the measurement of naturallyemitted radiation at microwave frequencies (Nolin et al 1998). In March, sea ice andsnow extents are largest in the Arctic but small in the Antarctic. In September, theextents are reversed. A video, derived from a similar data set, shows how Arctic seaice has changed since 1978. An equivalent video of snow mass changes since 1992has been provided by the GlobSnow project.

Changes in the weather and longer-term changes in the climate result in evolutionof the ice and/or melt. In turn, these changes affect ecology, the economy and humanactivities. There are a multitude of reasons why it is important to observe the state ofthe cryosphere, monitor changes and understand its behaviour.

1.2 Importance of cryospheric knowledgeCryospheric knowledge has many different impacts—social, economic and ecological—although the monetary value of the cryosphere is not easily quantified. From asocietal point of view, ice and snow are important sources of water, and Barnett et al(2005) estimated that more than a sixth of the global population requires melt fromsnow and glaciers for their water supply. Efficient management of this resourcerequires knowledge of the timing and volume of the supply. Water from melt can beused for many things: drinking, manufacturing, agriculture, hydropower andthermoelectric power. Carroll et al (2003) calculated that the contribution of snowto US gross domestic product was US$1.7 trillion dollars annually.

Some effort has been put into quantifying the economic impact of the changingcryosphere (the physical reasons for the changes are described in the followingsection). The changes are not necessarily negative from a financial point of view:with a smaller sea ice extent, more transport options become available. On Friday 27September 2013, a freighter navigated the Northwest Passage for the first time,ironically on the day the first section of the Intergovernmental Panel on ClimateChange (IPCC) 5th Assessment Report was released, describing how climate changeeffects are more certainly than ever attributable to the actions of mankind. Thusclimate change offers an apparent economic benefit for the shipping industry.Conversely, warmer conditions also lead to permafrost melt, which damages iceroads and other infrastructure, affecting the stability of exploration platforms. Otherimpacts from a changing cryosphere include coastal erosion, sea level rise, tourismand insurance risks (Key et al 2007).

Kivalina, Alaska, is a small village in a vulnerable coastal position. Loss of sea ice,which has traditionally protected the village against coastal erosion, means that it maybe underwater by 2025. It is estimated that the cost of relocating Kivalina would beUS$54m (Anisimov et al 2007), although the value reported in the media was an orderof magnitude greater.

Introduction to the Physics of the Cryosphere

1-4

Cold regions are hazardous and a good understanding of the physical behaviourof the environment can help to monitor and mitigate the risks. For example, snowavalanche risk is assessed by an examination of the meteorological data and diggingsnow pits. Evidence of poorly bonded layers indicate that it might not be wise to ski.One of the most memorable presentations I ever saw was from an avalancheresearch group. They have an observation bunker on one side of a valley and set up avideo system to capture avalanche events on the other side of the valley in order tounderstand mass flow mechanisms. The video started from the initiation of theavalanche with explosive charges through to full flow of the avalanche down thevalley. The language and tone of the commentators made clear the precise momentat which they realized that the avalanche was not going to stop at the bottom of thevalley. It took 4 hours to dig the shocked researchers out of the bunker, which wasunder 5 m of snow. As the snow was very dense, they broke all their metal shovelsand ended up using chainsaws to cut through the snow. Other sudden changes in thecryosphere can also occur through glacial collapse, glacial lake outbursts andlandslides. Ice can scour the sea floor and break pipes and cables if they are notburied sufficiently deep, and these are expensive to repair.

There are many ecosystems that depend on the state of the cryosphere. Theimpact of climate change on polar bears is often documented in the media: loss ofsea ice and hunting opportunities mean that these creatures either starve, drown orspend more time on land, with greater risk to human life. Figure 1.4 shows a polar

Figure 1.4. A polar bear peeking around the corner at the Churchill Northern Studies Centre, Churchill,Canada. Photograph reproduced with kind permission from the exceptionally brave Arvids Silis. Warning:objects shown in your camera may be closer than they appear. Many others have not been as lucky as Arvids.

Introduction to the Physics of the Cryosphere

1-5

bear making a surprise appearance at the Churchill Northern Studies Centre,Churchill, Canada.

For land mammals, snow cover provides insulation from cold air temperaturesand also camouflage. Small mammals dig tunnels beneath the snow. However,without snow covering the land, white Arctic hares and foxes would be a shiningbeacon to predators. Trees are also dependent on the cryosphere: in the AmericanNorthwest, vegetation growth is limited by the availability of soil moisture so treestend to follow topographic features and grow in areas where snow drifts form, asshown in figure 1.5. This creates a feedback effect, as the vegetation traps blowingsnow leading to more ground accumulation and a greater supply of water for thetrees. In the Swedish Arctic, it is said that the trees ‘walk’ across the landscape,due to preferential growth and decay linked to snow patterns (Pauker et al 1996).The mechanisms for treeline movement are complicated though, with Rundqvistet al (2011) identifying vegetation differences consistent with climate change,whereas Van Bogaert et al (2011) found only a small link with climate change.

1.3 Cryosphere and climateOne of the most important impacts of the cryosphere on climate is on the albedoof the Earth’s surface: this is the proportion of incoming sunlight reflected at thesurface. This is given by equation (1.1):

α = ↑↓

SS

(1.1)

Figure 1.5. Tree growth in Reynolds Creek Experimental Watershed, Idaho, USA, follows snow drift patterns.

Introduction to the Physics of the Cryosphere

1-6

where α is the albedo, S↑ is the upwelling (reflected) solar radiation and S↓ is thedownwelling (incoming) solar radiation. From this, the amount of solar radiationabsorbed by the surface, Snet is

α= − ↓S S(1 ) . (1.2)net

Approximate values for the albedo of common natural features of the Earth’ssurface are given in table 1.2. These values are typical, but there is quite a bit ofvariation due to factors such as roughness (or conversely how flat and smooth asurface is), chemical constituents or impurities, species of plant, water content of thesoil and, for snow, the size of the snow crystals. Figure 1.6 illustrates a change inalbedo for soil: a section of the soil has been recently plowed, bringing moisture to

Table 1.2. Approximate albedo values for somenatural materials.

Material Albedo

Soil 0.2Fresh snow 0.85Vegetation (trees) 0.15Sea water 0.06Sea ice 0.6

Figure 1.6. The soil in the recently ploughed section appears darker because it has a higher moisture content.Photograph reproduced with kind permission from Ian Davenport.

Introduction to the Physics of the Cryosphere

1-7

the surface. The wetter soil has a lower albedo, so reflects less sunlight and appearsdarker to the eye.

The impact of melt on the energy input to the planet can be illustrated with thealbedo values in table 1.2. For a given energy input of 800Wm−2 from the Sun, theamount absorbed by soil would be 640Wm−2, whereas if the soil was covered withfresh snow, the amount of solar radiation absorbed would only be 120Wm−2.Similarly, the difference is large for sea and sea ice. The kicker is that under awarming climate, the cryosphere shrinks, resulting in lower albedos and more energyabsorbed by the planet. This is known as a feedback loop, where warming causesphysical changes that lead to more warming, and is one of the reasons why thecryosphere is so important.

Another important equation governing cryospheric behaviour in a warmingclimate is the Clausius–Clapeyron equation: this relates the amount of moisture theair can hold to its temperature

=e

T

Le

RT

d

d(1.3)v v

2

where ev is the saturation vapour pressure, T is the temperature, R is the universalgas constant and L is the specific latent heat (also known as enthalpy) of evaporationif phase changes between liquid and gas are considered, or the specific heat ofsublimation if phase change takes place between the solid and gas phases.

Neglecting the small temperature dependence of the latent heat, this equation canbe integrated to form

= − −⎡⎣ ⎤⎦ ⎡⎣ ⎤⎦eLR

Tln (1.4)e

e

T

Tv

v0vf

10

f

where ev0 and ev

f are the initial and final values of the vapour pressure, and T 0 and T f

are the initial and final values of the temperature, giving a final equation

= −⎧⎨⎩

⎡⎣⎢

⎤⎦⎥

⎫⎬⎭e eLR T T

exp1 1

. (1.5)vf

v0

0 f

From equation (1.5), it can be seen that for higher temperatures, the vapourpressure is higher i.e. the air will hold more moisture. It is this property that governsprecipitation around the globe: water is picked up from warmer areas, movedaround by wind (in turn linked to pressure differences) and deposited in colderregions.

Thus the impacts of climate change on the cryosphere are complicated. Broadly, awarming climate causes more melt, with an additional warming feedback throughthe albedo changes, but may also bring more precipitation. This could be snow orrain, depending on the temperature. Snow is good news for the cryosphere: moresolar radiation is reflected and less heat is absorbed. Rain is very bad news for thecryosphere as its high energy content causes rapid melt or thaw of frozen surfaces,and potential flooding. Clouds themselves shield the surface from solar radiation butemit and contribute to the thermal radiation absorbed by the surface. Whether this

Introduction to the Physics of the Cryosphere

1-8

warms or cools the surface depends on the energy balance, and we use conservationequations to understand changes in the cryosphere.

1.4 Conservation equationsThe physics of the cryosphere, used to predict the behaviour of cryospheric com-ponents, relies on two main principles: conservation of mass and conservation ofenergy. Water is not destroyed but is converted from one form to another, betweenice, liquid and vapour depending on the energy balance. Similarly, energy is con-verted from one form to another and may result in a local warming or cooling of thecryosphere. The basic equations are presented here: these will crop up again indifferent forms throughout the book, though they are essentially the same but withdifferent assumptions. Not all terms may be included (for example, the energybalance equation excludes the kinetic energy of precipitation because it is small).Here, they will be applied to a generic ‘block of cryosphere’. This could be a block ofglacial ice, a section of snowpack, or a cuboid of sea ice.

The generic conservation equation in integral form for a volume dV is:

∫ ∫ ∫∑γ σΨ = − ⋅ +J St

V Vdd

d d d (1.6)V k S V

k

or, alternatively, the change in quantity Ψ is equal to the fluxes across the boundariesplus sources and/or sinks. In equation (1.6), t is time, V is the volume, and γk is thepartial density of constituent k, where k is air, water vapour, liquid water or ice. Ψ isthe quantity being conserved, J is the flux across the boundary enclosing boundary S,and σ is the source/sink density.

The treatment of the first term depends on whether the grid is fixed (Eulerian) ormoving with the material (Lagrangian) as

= ∂∂

+ ∂∂

∂∂t t

zt z

dd

. (1.7)

For a Lagrangian grid whose boundaries move but which contains largely thesame material (apart from fluxes across the boundaries and sources/sinks), thesecond term in equation (1.7) drops out and the derivative with respect to time inthe conservation equation (1.6) can be replaced with the partial derivative.

The partial density γk of a material is a key concept of mixture theory—wheredifferent phases of the same material are present. Snow is a good example as it is amixture of ice, liquid water, vapour and air. γk is the mass of that material per unitvolume of the whole mixture. Therefore the partial density of the ice phase of snow issubstantially lower than the intrinsic density of ice (mass of a constituent per volumeof that constituent), which is 916.7 kg m−3 at 0 °C.

Thus to keep track of the ice mass, the equation might look like:

∫ ∫ ∫∑γ δ∂∂

= − ⋅ + −′ ′ ′( )U S

tV M Vd d d 1 (1.8)

V S k Vk k k ki i

Introduction to the Physics of the Cryosphere

1-9

where i denotes ice, Ui is the ice mass flux into the system, Mk′k is the mass rate ofchange from phase k′ to k. δ is the Kronecker delta function so there is no con-tribution if k′ = k. One form of equation (1.8) could be:

= ++ − −

change in ice mass snowfall condensation from vapourrefreeze melt sublimation

although another flux term would be needed if it is raining, other terms may beneglected if they are not significant, as shown in figure 1.7. If the amount of liquidwater is important for a particular application (e.g. for surface runoff or infiltration)then a separate mass conservation equation for the liquid component might beconsidered. The mass balance equation can also be applied to the entire cryosphericblock, making no distinction between ice, liquid and vapour phases. It is, after all, allwater. In this case, there is no third term in the equation, and it is only flow throughthe surface and through the base of the material that contribute to mass changes.

In principle, mass flux input is easy to determine: go out and measure it, or usenumerical weather prediction outputs. For now, we will pretend that these give youperfect values. Determining phase change rates, however, is more difficult and that iswhere the energy balance equations come in.

A simple energy balance equation for a cryospheric block on Earth might looklike this:

= ↓ − ↑ + ↓ − ↑ + + + +Q S S L L H LE M G (1.9)

where Q is the change in energy, which will result in either a phase change or atemperature change. S and L are solar and thermal radiation, respectively. The upand down arrows denote the direction of radiative flux. Downward solar andthermal radiation can be measured with radiometers (a pyranometer for solar and apyrgeometer for thermal). In the case of solar radiation, also known as shortwaveradiation, the upward radiation is purely due to reflection of the downwellingradiation. The proportion of solar radiation reflected is given by the albedo, asdescribed earlier in equation (1.1). Thermal radiation, also termed longwave due to

Figure 1.7. Inclusion of mass balance terms depends on the application, but it is important to recognize theassumptions that are made in the development of any model. Twitter feed reproduced with kind permissionfrom Mark Brandon aka @icey_mark.

Introduction to the Physics of the Cryosphere

1-10

its longer wavelength, has a small reflected component, governed by the thermalemissivity ε. For ice, thermal emissivity is around 0.99, so only about 1% ofdownwelling thermal radiation is reflected at the surface and this is often assumed tobe negligible in models. The surface also emits thermal radiation, so the bulk of theupwelling thermal radiation will be emission, governed by the Stefan–Boltzmannequation:

ε σ↑ =L T (1.10)4

where T is the surface temperature and σ is the Stefan–Boltzmann constant(5.67 × 10−8Wm−2 T−4).

Returning to the energy balance equation (1.9): H denotes sensible heat transfer:this is convection by another name. LE is latent heat transfer, i.e. evaporation/sublimation/condensation processes. Both H and LE are dependent on the wind,temperature and humidity profile of the atmosphere above the surface. There areways of measuring these variables (Martin and Lejeune 1998), but this can beparticularly tricky in cold regions as the amount of vapour in the air is smallercompared to warmer regions (see the earlier Clausius–Clapeyron equation (1.5)) andthe wind profile can be a lot smoother. There are also ways of modelling H and LE(Morris 1989): this is rather complicated and, depending on the application, maynot be important, so has not been described here.

There are two terms left in the energy balance equation. M is advected heat: theheat transferred from rain or snow precipitation at a different temperature Tp to thesurface Ti, where c is the specific heat capacity and ρΔz is the precipitation massadded.

ρ= Δ −( )M c z T T . (1.11)p i

If the precipitation is at the same temperature as the ice component then M = 0.G is the ground heat flux, the conduction of heat from the surface underneath theblock. This can be measured, or modelled with a simple heat diffusion equation inone dimension:

= ∂∂

G KTz

. (1.12)

The change in energy can result in a change in temperature for a unit volume:

γ= ΔQ c T , (1.13)

and/or a phase change:

γ=Q L . (1.14)k

Some cryospheric components move rapidly, e.g. icebergs or melt water, so aconservation of momentum would be needed to determine the rate of flow. Anexample of this is given in chapter 5.

Introduction to the Physics of the Cryosphere

1-11

1.5 Properties of iceThe physical properties of ice at the microscale govern the macroscale behaviour ofice bodies. Even though the cryosphere is composed of different elements, they oftenreact in the same way to changes in local environmental conditions. Water is aunique material and governs the world around us, and its behaviour is all due to itsmolecular structure.

Ice is the crystalline form of water, H2O, formed at temperatures below 0 °C; it iscolourless, transparent and its crystal lattice is of the hexagonal system. Themolecular crystalline structure of ice has four hydrogen atoms linked to each oxygenatom: two by hydrogen bonds (the water molecule) and two by polar covalent links,as illustrated in figure 1.8. The bond angles are near tetrahedral.

Water is fairly unique in that it is one of the only materials to exist in vapour,liquid and solid forms at a temperature and pressure that we commonly experience.A rise in pressure results in a drop in freezing point, termed freezing-point depression.This can be seen in the phase diagram in figure 1.9. In the sea this phenomenon isgenerally negligible: an external pressure of 100 atm (around 1000m depth), woulddecrease the ice melting point by 1 °C. However, at higher pressures the freezingpoint depression may allow pressure melting to occur, as shown in figure 1.10 and inthe time lapse video demonstrating the process of regelation. It is this property thatresults in melting at the base of ice bodies, known as basal melting, and rapid flow ofice bodies such as glaciers, as described in chapter 4. The presence of impurities suchas salt, particularly important for sea ice, can also lower the freezing point, asdescribed in chapter 5.

At a temperature of 0 °C the density of ice is 916.7 kg m−3, indicating that itfloats on water. In fact, there is enough interstitial space in the crystal lattice for awater molecule, although this is not thought to occur commonly. As temperaturedecreases, the density of the ice lattice increases to 934.0 kg m−3 at −180 °C.

Hydrogen Hydrogen

Oxygen

Hexagonal lattice

Figure 1.8. Molecular structure of ice. Adapted from http://commons.wikimedia.org/wiki/File:H2O_%28water_molecule%29_white.png and http://en.wikipedia.org/wiki/File:Liquid-water-and-ice.png.

Introduction to the Physics of the Cryosphere

1-12

Being less dense than liquid water, ice expands as it forms and can be responsiblefor some of the worst weathering of natural and man-made materials. Changesbetween the liquid and ice phase can have a large impact on the structure of thelandscape because of the change in the volume occupied by the water when it

Figure 1.10. Regelation of ice. A wire or thread with suspended weights placed across a block of ice will slowlycut through the ice by pressure melting and the ice will resolidify once the pressure has been removed, i.e.above the wire, to leave a solid block. (a) Time at 0 h and (b) time at approximately 2 h.

374T (°C)

0.01

611.73

22.106

p(Pa)

Triplepoint

Gas

Liquid

Solid

Figure 1.9. Simple illustration of ice phase diagram. Solid, liquid and gas phases coexist at the triple point.A more detailed ice phase diagram is given in chapter 5. Adapted from http://commons.wikimedia.org/wiki/File:WaterPhaseDiagram.svg.

Introduction to the Physics of the Cryosphere

1-13

changes phase, leading to distinct features unique to the cryosphere. The followingchapter on permafrost describes how freeze–thaw processes lead to the formationof pingos, palsas, retrogressive thaw slumps and the sorting of stones into circles,stripes and polygons.

ReferencesAnisimov O A, Vaughan D G, Callaghan T, Furgal C, Marchant H, Prowse T D, Vilhjálmsson H

and Walsh J E 2007 Polar regions (arctic and antarctic) Climate Change 2007: Impacts,Adaptation and Vulnerability. Contribution of Working Group II to the Fourth AssessmentReport of the Intergovernmental Panel on Climate Change ch. 15 pp 653–85

Barnett T P, Adam J C and Lettenmaier D P 2005 Potential impacts of a warming climate onwater availability in snow-dominated regions Nature 438 303–9

Carroll T, Cline D, Berkowitz E and Savage D 2003 Snow economics and the NOHRSC SnowInformation System (SNOW-INFO) for the United States EGS-AGU-EUG Joint Assemblyp 1605

Foster J L, Sun C, Walker J P, Kelly R, Chang A, Dong J and Powell H 2005 Quantifying theuncertainty in passive microwave snow water equivalent observations Remote Sens. Environ.94 187–203

Key J, Drinkwater M and Ukita J 2007 IGOS Cryosphere Theme Report WMO/TD 100 http://igos-cryosphere.org/docs/cryos_theme_report.pdf

Martin E and Lejeune Y 1998 Turbulent fluxes above the snow surface Ann. Glaciol. 26 179–83Morris E M 1989 Turbulent transfer over snow and ice J. Hydrol. 105 205–23Nolin A W, Armstrong R and Maslanik J 1998 Near-Real-Time SSM/I-SSMIS EASE-Grid

Daily Global Ice Concentration and Snow Extent version 4 (Boulder, CO: NASA DAAC atthe National Snow and Ice Data Center)

Pauker S J and Seastedt T R 1996 Effects of mobile tree islands on soil carbon storage in tundraecosystems Ecology 77 2563

Rundqvist S, Hedenås H, Sandström A, Emanuelsson U, Eriksson H, Jonasson C and CallaghanT V 2011 Tree and shrub expansion over the past 34 years at the tree-line near Abisko,Sweden AMBIO 40 683–92

Van Bogaert R, Haneca K, Hoogesteger J, Jonasson C, De Dapper M and Callaghan T V 2011A century of tree line changes in sub-Arctic Sweden shows local and regional variability andonly a minor influence of 20th century climate warming J. Biogeography 38 907–21

Vaughan D G et al 2013 Observations: cryosphere Climate Change 2013: The Physical ScienceBasis. Contribution of Working Group I to the Fifth Assessment Report of the Intergovern-mental Panel on Climate Change ed T F Stocker et al (Cambridge: University Press)

Introduction to the Physics of the Cryosphere

1-14