Glacier melt: a review of processes and their modelling€¦ · Glacier ice and snow cover exert a...

Progress in Physical Geography 29 3 (2005) pp 362ndash391

copy 2005 Edward Arnold (Publishers) Ltd 1011910309133305pp453ra

I Introduction1 BackgroundGlacier ice and snow cover exert a majorcontrol on the dynamics of the Earth withrespect to both climate and hydrology From aclimatologic point of view snow and ice playan important role interacting with the atmos-phere over a range of spatial and temporalscales involving complex and sensitivefeedback mechanisms From a hydrologicalperspective glaciers represent importantwater resources contributing significantly tostreamflow Glaciers exert a considerableinfluence on catchment hydrology especiallyin mountain areas by temporarily storing

water as snow and ice on many timescales(Jansson et al 2003) Typical characteristicsof glacier runoff involve marked melt-induceddiurnal cyclicity and a concentration of annual flow during the melt season (Figure 1)An increasing demand for fresh water hasstimulated the need to predict melt-derivedstreamflow as a basis for efficient waterresource management with respect to issuessuch as water supply management of hydro-electric facilities and flood forecasting Thesuccess of modelling glacier-derived runoffstrongly depends on the formulation of themelt processs Melt modelling has tradition-ally been motivated by runoff forecasts but in

Glacier melt a review of processes and their modelling

Regine HockInstitute for Atmospheric and Climate Science Swiss Federal Institute of Technology CH-8057 Zuumlrich Switzerland and Department of Physical Geography and Quaternary Geology Stockholm University SE-106 91 Stockholm Sweden

Abstract Modelling ice and snow melt is of large practical and scientific interest including issuessuch as water resource management avalanche forecasting glacier dynamics hydrology andhydrochemistry as well as the response of glaciers to climate change During the last few decadesa large variety of melt models have been developed ranging from simple temperature-index tosophisticated energy-balance models There is a recent trend towards modelling with both hightemporal and spatial resolution the latter accomplished by fully distributed models This reviewdiscusses the relevant processes at the surface-atmosphere interface and their representation inmelt models Despite considerable advances in distributed melt modelling there is still a need torefine and develop models with high spatial and temporal resolution based on moderate input datarequirements While modelling of incoming radiation in mountain terrain is relatively accuratemodelling of turbulent fluxes and spatial and temporal variability in albedo constitute majoruncertainties in current energy-balance melt models and thus need further research

Key words distributed models energy balance glacier melt modelling temperature-index models

Regine Hock 363

recent years interest has risen in particular inspatially distributed estimates of snow and icemelt for many other purposes These includeavalanche forecasts assessment of the contri-bution of melting ice to sea-level rise as wellas studies in glacier dynamics hydrochemistryand erosion

Meltwater production in particular fromsnow cover has received extensive examina-tion in terms of both measurements andmodelling There is a complete hierarchy ofmelt models relating ablation to meteorologi-cal conditions varying greatly in complexityand scope These range from models basedon the detailed evaluation of the surfaceenergy fluxes (energy-balance models) tomodels using air temperature as the sole indexof melt energy (temperature-index models)Much work has been done at the point scaleHowever promoted by increased availabilityof digital terrain models and computationalpower increasing efforts have been devotedto areal melt modelling using fully distributedmodels In addition there is a trend towardshigh temporal resolution modelling (eg withhourly time steps) The latter is essential

for predicting peak flows in glacierized orsnow-covered basins High spatial resolutionis needed to account for the large spatialheteorogeneity with respect to ice and snowmelt typically encountered in steeply sidedterrain as a result of the effects of surroundingtopography

Previous reviews have addressed individualaspects of melt and its modelling Lang (1986)and Roumlthlisberger and Lang (1987) present anoverview of glacier melt and dischargeprocesses An exhaustive review of radiationand turbulent heat transfer at a snow surfaceis given by Male and Granger (1981) Male(1980) and Dozier (1987) summarize theprocesses of snow melt and hydrologyKirnbauer et al (1994) focus on distributedsnow models Recent trends in temperature-index melt modelling are discussed in Hock(2003) This review considers recent advancesin the simulation of melt focusing on glaciersand distributed modelling The relevantprocesses involved in energy exchange at the glacier surface are discussed withrespect to their representation in meltmodels Emphasis is on models suitable for

Figure 1 Hydrograph of hourly discharge at Vernagtferner Austria 1990 displaying seasonal and diurnal variations typical of glacier regimesSource data from Commission for Glaciology of the Bavarian Academy of ScienceMunich

364 Glacier melt a review of processes and their modelling

operational purposes hence on methodsrequiring moderate data input

2 Historical overviewThe relationship between glacier behaviourand climate has long been a central issue inglaciology This applies in particular tomountainous regions where people haveexperienced damage to farmlands and villageseither by direct advance of a glacier or byglacier outburst floods (eg Finsterwalder1897 Hoinkes 1969) Early studies attemptedto investigate the causes of glacier fluctu-ations Walcher (1773) was one of the firstto propose that glacier fluctuations arecaused by variations in climatic conditionsFinsterwalder and Schunk (1887) suggested aclose relation between air temperature andablation Hess (1904) recognized radiation asthe most important source of energy for meltAringngstroumlm (1933) stressed the importance of temperature radiation and wind as agentsfor melting

Pioneering work concerning the details ofenergy exchange between the atmosphereand a glacier surface was performed in theNordic countries In the 1920s Ahlmannrelated ablation measurements to simultane-ous meteorological observations and hederived the first empirical formula for thecomputation of ablation from known valuesof incident radiation air temperature andwind velocity (Ahlmann 1935 1948) Sverdruprsquosstudy in West Spitzbergen in 1934 providedthe foundation for most glacier and snowcover energy transfer studies to follow(Sverdrup 1935 1936) He computed acomplete energy balance although primarilyfocusing on the turbulent heat fluxes He wasthe first to apply gradient flux techniques toice and snow In the 1940s Walleacuten (1949)conducted a detailed glaciometeorologicalexperiment during six successive summers on Karingrsaglacier in northern Sweden Heconcluded that studies of glacier variationsshould deal with changes in the total volume of glaciers and not only with frontvariations

In the Alps early comprehensive investiga-tions to quantify the relationships betweenglacier and climate fluctuations and toassess the water balance of glacierized catch-ments were started in the 1930s (Luumltschg-Loetscher 1944) Hoeck (1952) focused onsnow melt and systematically investigated the climatic and topographic variables deter-mining the energy exchange of a snow coverDetailed investigations of the radiation budgetover glacier surfaces were initiated in 1938 atSonnblick (3100 m asl) in the Austrian Alpsand further developed in the 1950s (Saubererand Dirmhirn 1952) A comprehensive studyof water ice and energy budgets was startedon several glaciers in Oetztal Austria in 1948and greatly expanded during the InternationalHydrological Decade in the 1950s (Hoinkesand Untersteiner 1952 Hoinkes 1955)during which several long-term mass balanceseries were initiated (Hoinkes and Steinacker1975 Reinwarth and Escher-Vetter 1999)

Although restricted to snow surfaces ageneral and very thorough discussion of snowcover energy exchange and melt processesfrom both a theoretical and practical view-point was given by the Corps of Engineers(1956) and Kuzmin (1961) based on exhaus-tive studies in the USA and the former SovietUnion respectively Their work containedgeneralized snow melt equations based ontheoretical and empirical considerations andformed the basis of many of the snow and icemelt models which were to follow

In the 1960s the first computer simulationmodels of accumulation and ablation processeswere developed (Anderson 1972 Crawford1973 WMO 1986) This early stage of massbalance modelling pertained to snow coversand generally aimed at providing the melt-water input for watershed models Sincethen a large variety of snow models rangingfrom simple temperature-index models (egAnderson 1973 Braun and Aellen 1990) tosophisticated energy-balance models includ-ing simulation of the internal state of thesnow cover have been developed (eg Brunet al 1989) Modelling attempts focusing on

glaciers and ice sheets began in connectionwith the growing concern about potentiallyenhanced greenhouse warming and theeffects on global sea level (eg Braithwaiteand Olesen 1990a Oerlemans and Fortuin1992) as well as an increased interest in tapping glacial water for hydroelectricpurposes (eg Braithwaite and Thomsen1989 van de Wal and Russel 1994) Themost recent development concerns the incor-poration of remote sensing data into meltmodels providing a particularly useful tool inbasins inaccessible for detailed groundsurveys (eg Seidel and Martinec 1993Reeh et al 2002) In addition much effort isfocused on enhancing the spatial and tempo-ral resolution of melt models by moving frompoint-scale to distributed modelling and fromfor example daily time steps to hourly timesteps (Burlando et al 2002)

3 Characteristics of snow and ice relevant to meltGlacier melt is determined by the energybalance at the glacier-atmosphere interfacewhich is controlled by the meteorologicalconditions above the glacier and the physicalproperties of the glacier itself Glacier-atmosphere interactions are complex Theatmosphere supplies energy for melt whileatmospheric conditions are modified by the presence of snow and ice due to thespecific properties of snow and ice and theirhigh temporal variability In general snow and ice are characterized by (Male 1980Kuhn 1984)bull fixed surface temperatures during melting

(0C)bull penetration of shortwave radiationbull high and largely variable albedobull high thermal emissivitybull variable surface roughnessBecause the surface temperature of a meltingsnow or ice surface cannot exceed 0Cstrong temperature gradients can develop in the air immediately above the surfaceConsequently during the melt season the airis generally stably stratified thus suppressing

turbulence Gradients may reach more than5 K m1 within the first 2 m above the surface(Holmgren 1971 Oerlemans and Grisogono2002) Temperature stratification combinedwith typical glacier surface slopes inducesgravity flows (Ohata 1989 van den Broeke1997) On small valley glaciers this glacierwind typically reaches a maximum between05 and 3 m above the surface Due to thefact that surface temperature cannot increasebeyond 0C turbulent fluxes at some pointbecome independent of radiation (Holmgren1971 Greuell et al 1997)

The vapour pressure of a melting surface is611 hPa This relatively low value favoursvapour pressure gradients towards the sur-face and leads to condensation Since thelatent heat of evaporation (2501 106 J kg1

at 0C) is 75 times larger than the latent heatof fusion required for melting of snow and ice(0334 106 J kg1) condensation can be animportant energy source (eg Sverdrup1935 de Quervain 1951 Orvig 1954) Withreversed vapour gradients evaporationoccurs significantly reducing the energyavailable for melt due to the high energyconsumption involved in evaporation Thusthe process of evaporation is considered toplay an important role in maintaining low-latitude glaciers and the present ice sheets(Ohmura et al 1994)

Shortwave radiation penetrates ice andsnow to a depth of about 10 m and 1 mrespectively depending on their physicalproperties (Warren 1982 Oke 1987) Onlyabout 1ndash2 of global radiation (shortwaveincoming radiation) penetrates into a snowcover (Ohmura 1981 Konzelmann andOhmura 1995) and due to the exponentialdecline of transmitted radiation most of theenergy is absorbed in the first few mm belowthe surface However the process is impor-tant for heating the snow cover during pre-melt periods and for internal meltingwhich may even occur when the surface isfrozen due to net outgoing longwave radia-tion (Holmgren 1971) La Chapelle (1961)observed measurable amounts of snow

Regine Hock 365


melting as deep as 20 cm below the summersurface On Peyto Glacier 20 of the dailysnow melt took place internally as a result of penetration of shortwave radiation (Foumlhn1973) Winther et al (1996) attributedsubsurface melt layers exceeding 05 m inthickness in blue ice areas in Antarctica to thisprocess On glacier ice internal melting isimportant for the formation of a low-densitylsquoweathering crustrsquo in the top layer of the ice(Muumleller and Keeler 1969 Munro 1990)

Snow is generally characterized by a higheralbedo than ice varying roughly between 07and 09 compared to 03 to 05 for ice(Paterson 1994) In the infrared part of thespectrum both snow and ice behave asalmost perfect black-bodies (Kondratyev1969) with emissivities of about 098ndash099for snow and 097 for ice (Muumlller 1985) Thethermal conductivity of typical snow layers isless than one tenth of that of ice (Table 1)rendering snow a particularly good insulatorNevertheless snow temperatures near thesurface can drop rapidly during periods of nomelt occurring in particular at high altitudeson clear nights Combined with high albedoand high thermal emissivity snow representsa radiative sink during such periods Itsthermal insulating properties prevent efficientcompensation of these radiative losses Snowand ice melt at 0C However melting willnot necessarily occur at air temperatures of0C since melt is determined by the surfaceenergy balance which in turn only indirectly isaffected by air temperature (Kuhn 1987)

II Energy-balance melt modelsA physically based approach to compute meltinvolves the assessment of the energy fluxesto and from the surface At a surface temper-ature of 0C any surplus of energy at thesurface-air interface is assumed to be usedimmediately for melting The energy balancein terms of its components is expressed as

(1)

where QN is net radiation QH is the sensibleheat flux QL is the latent heat flux (QH andQL are referred to as turbulent heat fluxes)QG is the ground heat flux ie the change inheat of a vertical column from the surface tothe depth at which vertical heat transfer isnegligible QR is the sensible heat flux suppliedby rain and QM is the energy consumed bymelt As commonly defined in glaciology apositive sign indicates an energy gain to thesurface a negative sign an energy loss Meltrates M are then computed from the avail-able energy by

(2)

where w denotes the density of water and Lfthe latent heat of fusion Energy-balancemodels fall into two categories point studiesand distributed models The former assessthe energy budget at one location usually thesite of a climate station The latter involveestimating the budget over an area usually on a square grid

Examples of point studies on glaciers aregiven in Table 2 complementing similar sum-maries by for example Ohmura et al (1992)and Willis et al (2002) Complete energybudget measurements are seldom availableand if so only over short periods of time due to the enormous equipment and mainte-nance requirements Hence methods ofcomputing the energy budget componentsfrom standard meteorological observationshave been developed and applied in moststudies (see Table 2 for references) Despitesimplifying assumptions inherent to thesemethods they have provided reliable

MQ

L= M

w f

Q Q Q Q Q QN H L G R M+ + + + + =0

Regine Hock 366

Table 1 Some properties of snow andice at 0C (from Oke 1987 Paterson1994)

Typical Specific Thermal density heat capacity conductivity kg m3 J kg1 K1 W m1 K1

Fresh 50ndash150 2009 008snow

Old snow 200ndash500 2009 042

Ice 900 2097 21

Regine Hock 367T

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estimates of ablation In general most of theenergy used for melt is supplied by radiationfollowed by the sensible heat flux and only aminor fraction is derived from latent heat(Table 2) The importance of net radiationrelative to the turbulent fluxes tends toincrease with altitude as a result of reducedturbulent fluxes due to the vertical lapse ratesof air temperature and vapour pressure(Roumlthlisberger and Lang 1987)

Direct comparisons of different studiesshould be treated with caution as most stud-ies extend only over timescales of days orweeks rather than for the entire ablation sea-son The relative importance of the differentcomponents of the energy balance dependsstrongly on weather conditions and theirrelative contributions may change during themelt season On Devon Island Ice CapHolmgren (1971) found relative contributionsof net radiation sensible heat flux and thelatent heat flux of 70 20 and 8 on clear-skydays with light winds On overcast days withstrong winds the percentages changed to 4446 and 10 respectively In addition differ-ent accuracy in instrumentation and methodsof computation restrict such direct compari-son Distributed grid-based energy-balancestudies over ice and snow are comparativelyscarce (Table 3) The main challenge for

distributed studies is the extrapolation ofinput data and energy budget components to the entire grid

1 Net radiationNet all-wave radiation of a surface is thedifference between the incoming and out-going energies absorbed or emitted by thesurface (Kondratyev 1965) Traditionallyradiation is classified as shortwave or long-wave The former covers the wavelengthrange of approximately 015ndash4 m and pre-dominantly originates directly from the sunwhereas the longwave radiation refers to thespectrum of 4ndash120 m and is mainly thermalradiation of terrestrial and atmospheric originIn mountainous regions the radiative fluxes in particular the direct sun radiation varyconsiderably in space and time as a result ofthe effects of slope aspect and effectivehorizon These effects include reduction ofincoming radiation by obstruction of the skyas well as reflection and emission of thesurrounding slopes Thus the radiationbalance may be written as (Kondratyev 1965)

(3)

where I is direct solar radiation Ds is diffusesky radiation Dt is reflected radiation fromthe surrounding terrain (I Ds Dt is

Q I D D L L LN s t s t)+= + + minus + + uarrdarr darr( )(1

Regine Hock 368

Table 3 Grid-based energy-balance melt models applied to mountain glaciers andsnow-covered mountain areas

Location Time resolution Grid spacing Reference

GlaciersRhonegletscher (187 km2) day 100 m Funk 1985Vernagtferner (91 km2) half-hour 100 m Escher-Vetter 1985bHaut Glacier drsquo Arolla (63 km2) hour 20 m Arnold et al 1996Storglaciaumlren (31 km2) hour 30 m Hock and Noetzli 1997Moteratschgletscher (172 km2) hour 25 m Klok and Oerlemans 2002

SnowLaumlngental Austria (9 km2) hour 25 m Bloumlschl et al 1991Tedorigawa basin Japan (247 km2) day 540 469 m Ujihashi et al 1994Mount Iwate Japan (11 km2) hour 125 m Ohta 1994Davos (16 km2) day 25 m Pluumlss 1997

Regine Hock 369

referred to as global radiation) is albedo Ldarr

s is longwave sky radiation Ldarrt is longwave

radiation from the surrounding terrain and Luarr is the emitted longwave radiationMeasurements of net radiation on glaciers are seldom available and it is thereforenecessary to parameterize the individualcomponents

2 Global radiationUpon entering the atmosphere solar radia-tion is partioned into direct and diffusecomponents This is mainly due to scatteringby air molecules scattering and absorption byliquid and solid particles and selective absorp-tion by water vapour and ozone all processeshaving different wavelength dependenciesBesides atmospheric conditions and cloudssite-specific characteristics such as slopeangle and aspect are also crucial for the computation of shortwave radiation incomplex topography Potential clear-skydirect solar radiation on an inclined surface Iccan be expressed by (eg Iqbal 1983)

(4)

where I0 is the solar constant (~1368 W m2)R is the sun-Earth distance (with subscript mrefering to the mean) a is the atmosphericclear-sky transmissivity P is atmosphericpressure P0 is mean atmospheric pressure atsea level Z is the local zenith angle and isthe angle of incidence between the slope-normal and the solar beam A widely usedsolution for the incidence angle is given byGarnier and Ohmura (1968)

(5)

where is the slope angle Z is the zenithangle and sun and slope are the solar azimuthand the slope azimuth angles respectivelyThe zenith angle can be approximated as a function of latitude solar declination andhour angle (eg Iqbal 1983) Atmospheric

attenuation of shortwave radiation can bedescribed by Bouguerrsquos law (also calledLambertrsquos or Beerrsquos law) and is proportionalto the atmospheric pathlength and the initialradiation flux In atmospheric models thisprocess is often parameterized by using differ-ent transmission coefficients for the mostattenuative molecules and aerosols (egDozier 1980) Transmissivities tend to behighest in winter and lowest in summer andtend to increase with latitude due to thelower atmospheric water vapour and dustcontent both in winter and at high latitudesClear-sky transmissivities vary between 06and 09 (Oke 1987) The amount of diffuseradiation depends largely on atmosphericconditions The fractions of diffuse radiationto global radiation ranged from 36 to 51 inspring and 24 to 41 in summer at eightstations between latitudes of 40 and 60(Kondratyev 1969) At the ETH camp inGreenland 40 of global radiation was dif-fuse on average during the summer months(Konzelmann and Ohmura 1995) On clear-sky days this portion ranged from 13 to 17Holmgren (1971) reported 16 of globalradiation as diffuse on Devon Ice Cap andOhmura (1981) found 15 to 21 on AxelHeiberg Island

In complex topography diffuse radiationoriginates from two sources ndash the sky and thesurrounding topography ndash and consists ofthree components1 radiation that is initially scattered out of

the beam by molecules and aerosols (skyradiation)

2 backscattered radiation or the globalradiation that is reflected by the snowsurface and subsequently redirected down-ward by scattering and reflection in theatmosphere

3 radiation reflected from adjacent slopesConsequently surrounding topography affectsthe amount of diffuse radiation in twoopposing ways sky radiation is reduced aspart of the sky is obscured while diffuseradiation is enhanced by reflection from adja-cent slopes The backscattered component

cos cos cos sin sincos( )

= +minusZ Z

sun slope

I IR P

P zc

maR

0=⎛⎝⎜

⎞⎠⎟0

2

cos cos



strongly depends on albedo and it is greatlyincreased by the presence of snow due toenhanced multiple reflections between theground and the atmosphere

Clear-sky diffuse radiation is significantlyanisotropic (Kondratyev 1969) The intensityis greatest in the direction of the sun and nearthe horizon because of the greater opticalthickness of the atmosphere at large zenithangles This effect is considered in the radia-tion model by Dozier (1980) However mostmelt models assume isotropy This simplifica-tion is acceptable in most applications becausediffuse radiation is most anisotropic for clearskies when diffuse radiation is relatively lowand is nearly isotropic under overcast condi-tions (Garnier and Ohmura 1968) whendiffuse radiation tends to be high

Many theoretical and empirical formulaeto calculate diffuse radiation have been pro-posed (eg List 1966 Wesley and Lipschutz1976) Diffuse sky irradiance onto an inclinedsurface is given by (Kondratyev 1965)

(6)

where D(h) is the radiance from the direc-tion determined by the angular height h rela-tive to the horizontal plane and the azimuthangle h() is the lowest angular height ofthe point in the sky unobstructed by topog-raphy in the azimuth and is the anglebetween the vector normal to the slope andan arbitrary direction Most melt modelsrefrain from integrating diffuse radiation over azimuth and zenith angles and prefer asimpler so-called view-factor relationship toaccount for the effects of topography Theview-factor Vf is related to the fraction of thehemisphere unobstructed by surroundingslopes and can be approximated by

(7)

where H is the average horizon angle (Marksand Dozier 1992) A widely used simpli-fication is given by Kondratyev (1969) andapplied for example by Arnold et al (1996)

(8)

with the slope of the surface Strictlyspeaking this approximation only refers to a sloping surface without surrounding obst-ructions Funk (1985) has shown that onRhonegletscher results differed by roughly1310 from those obtained by numerical inte-gration of the horizon angle Diffuse radiationon the inclined surface including the effects oftopography can be approximated by

(9)

where D0 is diffuse sky radiation of an unobstructed sky G is global radiation and mis the mean albedo of the surroundings Thefirst term refers to sky radiation the secondterm to terrain radiation

For point calculations a large variety ofempirical functions have been developed to relate global radiation to climatic andtopographic variables without explicitly dif-ferentiating between direct and diffuse radia-tion Based on measurements in the AlpsSauberer (1955) proposed a parameterizationfor global radiation Wagner (1980a) presentsscattergrams for daily sums of global radiationin the Alps depending on the month andassuming mean daily cloudiness Kasten(1983) computed global radiation as a func-tion of top of atmosphere radiation optical airmass and turbidity deriving the latter twofrom zenith angle elevation air temperatureand humidity Based on Kasten (1983)Konzelmann et al (1994) derived a para-meterization for global radiation using airtemperature vapour pressure albedo cloudamount and elevation at the ETH camp onWest Greenland

On a larger scale numerous distributedradiation models have been developed tocompute global radiation for each gridelement of an elevation model Munro andYoung (1982) and Varley and Beven (1996)proposed global radiation models for steepterrain treating direct and diffuse radiationseparately Dozier (1980) developed a highlysophisticated spectral model for clear-skysolar radiation incorporating separate trans-mission functions for various absorption

D D V G V= + minus( )0 1f m f

Vf = 2cos ( )2

V Hf = cos ( )2

D D h h hhs d d=

2

intint ( ) cos cos( )

0

2

Regine Hock 370

and scattering processes as a function ofwavelength and explicitely accounting for the interactions with the snow surface andsurrounding terrain Models of this type are complicated and require estimates ofvarious atmospheric attenuation parametersand of the ozone and water vapour distribu-tion with altitude as input which are oftenunknown

In some distributed melt models this prob-lem is circumvented by including measuredglobal radiation into model formulations thusscaling calculations to measurements (egArnold et al 1996) Spatial variations due totopographic effects are considerably morepronounced for the direct than the diffusecomponent Hence in distributed modellingmeasured global radiation needs to be splitinto direct and diffuse components prior toextrapolation Hock and Holmgren (2005)accomplished this by adapting an empiricalrelationship between the ratio of diffuse radi-ation to global radiation and the ratio of globalradiation to top of atmosphere radiation assuggested by Collares-Pereira and Rabl(1979) Following Ohta (1994) direct solarradiation thus obtained Is divided by potentialclear-sky direct radiation at the weatherstation Ics was then used to compute directradiation for each grid cell I

(10)

where Ic is potential clear-sky directradiation for the grid cell to be calculated (seeequation 4) The ratio IsIcs which is assumedconstant in space accounts for the deviationsfrom clear-sky conditions and tends todecrease with increasing cloudiness Escher-Vetter (2000) and Hock and Noetzli (1997)extrapolated global radiation without explicitseparation into the direct and diffuse compo-nents by multiplying the ratio of measuredglobal radiation to Ics by Ic to obtain globalradiation for each grid cell Both methods canonly be applied if the grid cell to be computedand the measuring site are not topographi-cally shaded

3 AlbedoWarren (1982) gives a comprehensive reviewof snow albedo Albedo generally defined asthe averaged reflectivity over the spectrumfrom 035 to 28 m varies considerably onglaciers both in space and time Ranging from01 for dirty ice to more than 09 for freshsnow it controls the spatial and temporal dis-tribution of meltwater production to a largeextent thus rendering albedo a key para-meter in glacier melt simulations Summersnowfall events can reduce melt and runoffconsiderably because of abruptly enhancedalbedo Snow albedo displays considerableshort-term fluctuations Fresh-snow albedomay drop by 03 within a few days due tometamorphism Temporal variations of icealbedo are small compared to those of snow(Cutler and Munro 1996 Brock et al 2000aJonsell et al 2003) However small-scalespatial albedo variations may occur on glacierice and cause large spatial differences in abla-tion (van de Wal et al 1992 Konzelmann andBraithwaite 1995) Glacier albedo often isconsiderably modified by deposits of sedimentor rock debris

Albedo is determined by factors related tothe surface itself such as grain size watercontent impurity content surface roughnesscrystal orientation and structure and byfactors related to the incident shortwaveradiation such as the wavelength or whetherthe sunlight is diffuse or direct Water ininterstices between grains indirectly affectsalbedo by increasing grain size which in turnreduces albedo Surface albedo increases withincreasing cloudiness and atmospheric watercontent Clouds preferentially absorb near-infrared radiation thus increasing the fractionof visible light for which albedo is higher(Marshall and Warren 1987) This effect isenhanced by multiple reflection between thecloud base and the glacier surface Snowalbedos have been found to increase by3ndash15 when moving from clear-sky to over-cast conditions (Holmgren 1971 Greuell andOerlemans 1987) Jonsell et al (2003) reportshort-term albedo variations over snow by

III

Is

scc=

Regine Hock 371

gt01 due to cloud fluctuations while theresponse of albedo over ice to varying cloudi-ness was considerably smaller Albedo ishighest at low angles of incidence as a resultof the Mie scattering properties of ice andsnow grains dominating other effects such asthe spectral change in global radiation

Modelling albedo is complicated as it isexceedingly difficult to quantitatively relatealbedo variations to their causes Opposingfactors may influence the local conditions indifferent ways On glaciers snow and icealbedo must be treated separately due to theirsubstantial difference and different temporalvariability Radiative transfer modellingstudies indicate that snow albedo is primarilyexplained by the snow grain size Modelsaccounting for the effects of grain size as well as atmospheric controls have beenproposed by Warren and Wiscombe (1980)Choudhury and Chang (1981) Marshall andWarren (1987) and Marks and Dozier (1992)However due to large data requirementsthey are generally not applicable for opera-tional purposes and surrogate methods areused instead A common surrogate method isthe so-called aging curve approach whichcalculates the decreasing snow albedo as afunction of time after the last significantsnowfall The formulation of the Corps ofEngineers (1956) has been widely used

(11)

where 0 is the minimum snow albedo nd thenumber of days since the last significantsnowfall and b and k are coefficients Kuzmin(1961) presented nomogramms for the decayof snow albedo for different snow depths A large variety of snow albedo parameteriza-tions have been proposed incorporating oneor more variables such as snow depth snowdensity melt rate sun altitude air tempera-ture and accumulated daily maximum airtemperatures since the last snowfall (egBrock et al 2000a Willis et al 2002 ndash seesummary in Brock et al 2000a) Althoughthese generally generate satisfactory albedosimulations for the site developed and

calibrated they need further independenttesting to promote confidence in their generalapplicability

In contrast to snow albedo very few stud-ies have focused on ice albedo (eg Cutlerand Munro 1996 Brock et al 2000a) Icealbedo is often treated as a temporal andspatial constant (Konzelmann and Braithwaite1995 Hock and Noetzli 1997) albedo jump-ing from a fixed or variable snow value to a lower fixed ice value as soon as all snow has melted Oerlemans (1992) proposed anenergy-balance model including modelling ofglacier albedo His albedo parameterization isbased on the assumption of a downglacierdecrease in albedo owing to an increase inconcentrations of dust and debris He definesa lsquobackground albedo profilersquo as a function ofthe elevation relative to the equilibrium line atthe end of the ablation season and variousempirically determined constants Snowalbedo is superimposed on this profile as afunction of snow depth Arnold et al (1996)slightly modified this approach by consideringfresh snow separately

4 Longwave radiationLongwave incoming radiation is emitted mostlyby atmospheric water vapour carbon dioxideand ozone Variations are largely due to varia-tions in cloudiness and in the amount and tem-perature of the water vapour Due to the latterlongwave irradiance tends to decrease withaltitude Although downward radiation is emit-ted from all levels of the atmosphere thelargest portion reaching the surface originatesfrom the lowest layers Irradiance can be mod-elled using radiative transfer equations requir-ing input of temperature and water vapourprofiles and the distribution of the concentra-tions of CO2 and O3 A review of models isgiven by Ellingston et al (1991) In melt modelslongwave irradiance is usually estimated fromempirical relationships based on standardmeteorological measurements exploiting thefact that longwave irradiance correlates wellwith air temperature and vapour pressure atscreen level usually at 2 m above the surface

= + minus0 be n kd


(Kondratyev 1969) Over melting snow and icesurfaces however the use of screen-leveltemperature may underestimate longwaveirradiance due to the proximity of the meltingsurface which restricts temperature increaseof the lowest air layers (van den Broeke 1996)Generally the equations for longwave irradi-ance Ldarr take the form

(12)

where c is the full-spectrum clear-skyemissivity is the Stefan-Boltzmannconstant (567 108 W m2 K4) Ta is airtemperature (K) and F(n) is a cloud factordescribing the increase in radiation due toclouds as a function of cloud amount nEffective or apparent emissivity e defined bythe product of c and F(n) ranges fromapproximately 07 under clear-sky conditionsto close to unity under overcast conditions A variety of parameterizations have beensuggested to relate clear-sky emissivity toscreen-level measurements of air tempera-ture and humidity Widely quoted expres-sions are those of Aringngstroumlm (1916)

(13)

and Brunt (1932)

(14)

where A B C a 051 and b 0066 are empirically determined coefficients isabsolute humidity and ea is vapour pressure[hPa] The coefficients obtained by differentauthors are very variable as is to be expectedsince they vary with the time of year andlocation These equations do not work onsubdaily timescales Wagner (1980b) foundgood results on Hintereisferner using Bruntrsquosand Aringngstroumlmrsquos formulae although Bruntrsquosformula tended to yield 7ndash8 lower valuesthan Aringngstroumlmrsquos expression A more physicallybased approach has been developed byBrutsaert (1975) with the advantage that it does not require calibration to local condi-tions It is based on the integration of theSchwarzschildrsquos transfer equations for simpleatmospheric profiles of temperature [K] and

vapour pressure [hPa] but neglects greenhousegases other than water vapour

(15)

Over glaciers Braithwaite and Olesen (1990b)and Arnold et al (1996) applied the expres-sion for effective emissivity e suggested byOhmura (1981) which includes the effect of clouds

(16)

where Ta is air temperature [K] thus account-ing for the increase in absolute humidity withtemperature and n is the cloud amount Thecoefficient k is a function of cloud type forwhich Ohmura (1981) listed eight valuescorresponding to different cloud typesKonzelmann et al (1994) derived parameteri-zations for hourly and daily longwave irradi-ance at the ETH camp in Greenland Thehourly formulation for effective emissivity is

(17)

Clear-sky emissivity c as obtained frommeasured longwave radiation under clear-skyconditions (c LdarrTa

4) was related tomeasured eT to fit a modified version of Brutsaertrsquos (1975) equation yieldingc 023 0484(eaTa)18 with ea in Pa Byconsidering greenhouse gases other thanwater vapour (c is different from zero (023)in a completely dry atmosphere) this for-mulation was superior to the original one by Brutsaert (1975) The emissivity of acompletely overcast sky oc 0952 and thecoefficient p 4 were obtained empiricallyfrom observations of Ldarr Ta ea and cloudamount Although the high power (p 4)resulted from the site-specific cloud climatol-ogy where high cloudiness was primarilycaused by low clouds and low cloudiness byhigh clouds the expression has been used atother sites in Greenland (eg Greuell andKonzelmann 1994 Zuo and Oerlemans1996) and in the Alps (Pluumlss 1997)Coefficients were adjusted to the conditionson Pasterzegletscher by Greuell et al (1997)

ep

ocpn n= minus +( )c 1

e aT kn= times +minus ( )8 733 10 13 0 788

c = ( ) 1 24 1 7e Ta a

c = +a b ea

c-= minus sdotA B 10 C

L = cdarr T F na4 ( )

Regine Hock 373

and used on Moteratschgletscher byOerlemans (2000) At a low-elevation site inGreenland van den Broeke (1996) obtainedconsiderable underestimation of longwaveirradiance using equation 17 The same wastrue when the formula derived in polarregions by Koenig-Langlo and Augstein(1994) was applied Van den Broeke (1996)attributes the underestimation to a strongersurface inversion than at the Greenland ETHcamp for which the coefficients were deter-mined In general satisfactory results can beachieved with such parameterization how-ever it is necessary to adjust coefficients for different sites

In mountainous areas surrounding topog-raphy can cause significant spatial variationsin longwave irradiance This is neglected inmany distributed energy-balance modelswhich assume longwave irradiance is spatiallyuniform Longwave sky irradiance is reducedby obstructed sky due to surrounding terrainConversely the surface receives additionalradiation from the surrounding terrain andthe air between the terrain and the receivingsurface The importance of terrain contribu-tions to the incoming longwave radiation has been pointed out by Olyphant (1986)Pluumlss and Ohmura (1997) found variationsbetween 160 and 240 W m2 in longwaveincoming radiation Ldarr on a clear day in a high alpine area solely due to topographicinfluence Marks and Dozier (1979) consid-ered topographic modification of longwaveirradiance by

(18)

where a and s are emissivities of the air andthe surrounding surface respectively Ta andTs are the temperature of the air and the sur-face respectively and Vf is the thermal viewfactor related to the unobscured fraction ofthe hemisphere (see equations 7 and 8) Thefirst term represents sky irradiance and the second term refers to terrain irradianceThe effect of the air between the emittingsurface and the receiving surface was neg-lected Based on calculations with a spectral

radiation transfer model (LOWTRAN7)Pluumlss and Ohmura (1997) revealed thatneglecting this effect may lead to significantunderestimation of irradiance especially if theair temperature exceeds the snow surfacetemperature which is typical of melting sur-faces They derive a simple parameterizationfor longwave irradiance in completely snow-covered mountainous terrain accounting forthis effect Despite the large anisotropy inlongwave irradiance under clear-sky conditions(Kondratyev 1969) Pluumlss and Ohmura (1997)showed that the assumption of isotropyproduced only small errors in all investigatedcases Under cloudy conditions longwaveirradiance is nearly isotropic

Longwave outgoing radiation Luarr referringto the radiation emitted by and reflected fromthe surface can be calculated from

(19)

with s the emissivity of the snow cover In many computations the snow and icesurfaces are assumed to be at melting pointCombined with an assumed emissivity ofunity the corresponding longwave emissionamounts to 3156 W m2

5 Turbulent heat fluxesThe turbulent fluxes of sensible and latentheat are driven by the temperature and mois-ture gradients between the air and the surfaceand by turbulence in the lower atmosphere asthe mechanism of vertical air exchange Thesefluxes are generally small when averaged overperiods of weeks or months and compared tothe net radiation flux (Table 2 Willis et al2002) but they can exceed the radiationfluxes over short time intervals of hours anddays and in mid-latitude maritime environ-ments also over longer periods (eg Hogg et al 1982 Marcus et al 1984) Highest meltrates often coincide with high values of theturbulent fluxes (Hay and Fitzharris 1988)The latent heat flux is of major importance forthe short-term variations of melt rates ontemperate glaciers (Lang 1981) Sublimation is

L = s suarr + minus darr T Ls4 1( )

L = ( a a4

f s s4

fdarr + minus) ( )( ) T V T V1


important at high altitudes and high latitudessuch as the blue-ice areas of the Antarctic ice sheet where all ablation occurs by sublima-tion (Bintanja 1999) A review of turbulent heat transfer over snow and ice is given byMorris (1989)

The turbulent heat fluxes can be measureddirectly by eddy-correlation techniquesThese require sophisticated instrumentationwith continuous maintenance which renderthem unsuitable for operational purposesConsequently such studies are rare andrestricted to short periods of time (egMunro 1989 Forrer and Rotach 1997 vander Avoird and Duynkerke 1999) Thereforethe turbulent heat fluxes are often describedby gradient-flux relations These are based onthe theoretical work of Prandtl (1934) andLettau (1939) and they were first introducedto snow and ice by Sverdrup (1936) In thesurface layer the relations are based on theassumption of constant fluxes with heightand horizontal homogeneous conditionsAccordingly the turbulent energy fluxes ofsensible heat QH and of latent heat QE areproportional to the time-averaged gradientsof potential temperature and specifichumidity q in the surface boundary layer andcan be expressed by

(20)

(21)

where a is the air density cP is the specificheat capacity of air Lv is the latent heat ofevaporation z is the height above the surfaceKH and KE are the eddy diffusivities for heatand vapour exchange respectively KH and KLspecify the effectiveness of the transferprocess and depend on wind speed surfaceroughness and atmospheric stability

The profile method involves measure-ments of q and wind speed u at preferablymore than two levels within the first fewmetres above the surface (eg de laCasiniegravere 1974 Forrer and Rotach 1997)

The method has the disadvantage of largesensitivity to instrumental errors especially if only two levels of measurements areemployed Because detailed profile measure-ments are seldom available the so-called bulk aerodynamic method has frequentlybeen applied for practical purposes (egBraithwaite et al 1998 Oerlemans 2000) It exploits the fact that the surface conditionsof a melting surface are well defined(T 0C e 611 hPa) thus allowing forthe computation of the sensible QH and thelatent heat flux QE from only one level ofmeasurements Integrating equations 20 and21 the bulk aerodynamic expressions read

(22)

(23)

where u is mean wind speed z and s aremean potential temperatures and qz and qs aremean specific humidities at height z and thesurface respectively In practice the specifichumidity term (qz qs) is often replaced by(0622p)(ez e0) with p the atmosphericpressure and e the mean vapour pressureThe exchange coefficients for heat CH andvapour pressure CE are given by

(24)

(25)

where k 04 is the von Kaacutermaacuten constant is the integrated form of the Monin-Obukhovstability functions as an atmospheric stabi-lity correction the subscripts M H and Lrefering to momentum heat and watervapour respectively Under neutral condi-tions the functions assume a value of zeroThe roughness length for wind z0 defined asthe height above the surface where u 0 isrelated in a complex way to the roughness of

Ck

z z z L z z

z LE

EM

E

=minus

minus[ln( ) ( )] [ln( )

( )]

2

0 0

Ck

z z z L z z

z LT

HM

H

=minus

minus[ln( ) ( )] [ln( )

( )]

2

0 0

Q L C u q qz sE a v E= minus( )

Q c C u z sH a P H= minus( )

Q L KqzE a v E= part

part

Q c KzH a P H= part

part

Regine Hock 375

the surface The roughness lengths fortemperature z0T and vapour pressure z0e arescaling parameters lacking a well-definedphysical meaning The stability functions depend on the stability parameter zL whereL denotes the Monin-Obukhov-length whichin the stable boundary layer can be interpre-tated as the height at which the rate of tur-bulent energy production by shear stressbalances the energy consumption by buoy-ancy forces (Obukhov 1946) The quantityzL in turn depends on QH among otherfactors thus an iterative loop is required tocompute the turbulent fluxes (eg Munro1989) The exchange coefficient is often alsoreferred to as the transfer or drag coefficientHowever direct comparison of numericalvalues between studies is often difficultbecause the number and type of para-meters and coefficients lumped into thesenumerical values vary among authors Forexample Moore (1983) and Price and Dunne (1976) lump the wind speed into theexchange coefficient expressions (equations24 and 25)

A variety of empirical expressions havebeen proposed to define the form of thestability functions (Houmlgstroumlm 1988) A fre-quently used approximation is zL where 5 for stable stratification (Dyer 1974)Forrer and Rotach (1997) showed that forlarge stability (zL 04) the linear form ofthe stability functions was not appropriate ata site on the Greenland ice sheet but thenonlinear expressions suggested by Beljaarsand Holtslag (1991) yielded better resultsDue to its simplicity a commonly usedstability criterion is the Richardson numberwhich in its bulk form is defined by

(26)

where g is the acceleration of gravity and Tzand Ts are absolute temperatures at height zand the surface respectively For stable strat-ification (Rb 0) which prevails over meltingglaciers one way to relate the Richardson

number to the stability function is given byWebb (1970) where the exchange coefficientC for the property x to be transported reads

(27)

The magnitude of correction either by the Monin-Obukhov stability or by theRichardson number increases considerably as the wind speed decreases

Although the bulk aerodynamic approachis very convenient due to its simplicity muchuncertainty about its application to glaciersurfaces remains One problem concerns thespecification of roughness lengths They canbe derived from detailed measurements ofwind temperature and humidity profilesHowever these are seldom available and theroughness lengths are often estimated frompublished data (eg Greuell and Oerlemans1989 Konzelmann and Braithwaite 1995)This poses a problem because the roughnesslengths for wind reported over snow and icevary by several orders of magnitude (sum-maries in Moore 1983 Braithwaite 1995a)ranging from 0004 mm (Inoue 1989) to70 mm (Jackson and Carroll 1978) oversnow and from 0003 mm (Antarctic blue icearea Bintanja and van den Broeke 1995) to120 mm (van den Broeke 1996) over glacierice Generally z0-values of a few mm tend tobe assumed in glacier applications Changes inz0 and z0T by one order of magnitude canresult in differences in the turbulent heatfluxes by a factor of two (Munro 1989 Hockand Holmgren 1996) demonstrating thesignificance of accurate roughness lengthdetermination

The relationship between the roughnesslengths of wind heat and vapour pressure isanother matter of discussion The principle ofsimilarity is often invoked (eg Streten andWendler 1968 Hay and Fitzharris 1988Brun et al 1989 Munro 1989 Zuo andOerlemans 1996) although there is evidencethat the surface roughnesses for heat andvapour pressure are smaller than z0 by one or

Ck

z z z zR

xx b= minus

2

0 0

21 5 2ln( ) ln( )

( )

Rg

TT T z z

uz

z s

zb =

minus minus( )( )02


two orders of magnitude (Sverdrup 1935Holmgren 1971 Ambach 1986 Beljaars and Holtslag 1991 Smeets et al 1998)Conversely Morris et al (1994) concludedthe opposite from energy-balance modellingnamely that z0T and z0E were considerablylarger than the aerodynamic roughnesslengths Data analysis from Greenland gavevalues for zT of about 10 to 100 times largerthan z0 (Calanca 2001) Braithwaite (1995a)suggests that the assumption of unequalroughness lengths is not strictly necessarybecause an lsquoeffective roughness lengthrsquosatisfying z0 z0T z0E can be chosen so that the exchange coefficient gives the samevalue when applying equations 24 and 25The roughness length of vapour pressure isgenerally assumed to be equal to the one for heat However very few studies existdue to the inherent difficulty of accuratelymeasuring vapour pressure profiles over gla-ciers In addition roughness lengths will varyin space and time (Pluumlss and Mazzoni 1994Greuell and Konzelmann 1994 Calanca2001) Holmgren (1971) observed an increasein roughness lengths with decreasing windspeed Anderson (1976) a decrease with timeduring the snow melt season

Alternative approaches to estimatingroughness lengths have been suggested byLettau (1969) who calculated them from theheight and cross-sectional area of surfaceforms and by Andreas (1987) who used sur-face renewal theories and expressed z0Tz0and z0Ez0 as functions of the roughnessReynolds number In a study on Greenlandvan den Broeke (1996) obtained z0 08 mmfrom wind profiles but at the same locationz0 120 mm from the microtopographicalsurvey according to Lettau (1969) clearlyshowing the difficulties involved in obtainingaccurate estimates of roughness lengths Thelatter value yielded more realistic results forthe energy balance Herzfeld et al (2000)designed a sensor recording variations inmicrotopography at 02 m 01 m resolutionwhen pulled across the ice surface The dataare analysed using geostatistical methods

More studies are needed to evaluate themethod in terms of suitability for use in turbu-lent flux calculations Because of the difficul-ties involved in specifying transfer coefficientsor roughness lengths some authors treatthem as residuals in the energy-balanceequation (eg La Chapelle 1961 Braithwaiteet al 1998 Zuo and Oerlemans 1996Oerlemans 2000) Sufficient accuracy canonly be obtained over longer periods of timeA unique attempt to incorparate spatial andtemporal variations in z0 into a distributedenergy-balance model has been made by onHaut Glacier drsquoArolla (Brock et al 2000b)Over snow z0 was parameterized as a func-tion of accumulated daily maximum tempera-tures since the last snowfall to account forincreasing snow roughness during snowmeltHowever no relationship could explain theobserved variation over ice when relating z0derived from the formula of Lettau (1969) tovarious meteorological variables The para-meterization over snow needs further testingat other sites

Many energy-balance models applied tosnow and ice do not include a formal stabilitycorrection in the bulk approach (eg Foumlhn1973 Hogg et al 1982 Konzelmann andBraithwaite 1995 Oerlemans 2000) thusassuming logarithmic wind profiles despiteprevailing stable stratification Braithwaite(1995a) points out that the uncertainty in sur-face roughness may cause larger errors thanneglecting stability Some authors identify atendency of energy-balance modelling tounderestimate glacier melt and attribute thiserror to an underestimation of the turbulentfluxes (eg Harding et al 1989 Konzelmannand Braithwaite 1995 van den Broeke1996) Brun et al (1989) concluded thatunder light wind conditions energy gain atthe surface by heat conduction through theair and by vapour diffusion due to vapourgradients in the air may be higher than byturbulent transfer In order to obtain largerfluxes they modified equations 22 and 23 intheir snow model by replacing the wind speedu by the empirical relation a bsdotu where

Regine Hock 377

a and b are experimentally determined Kingand Anderson (1994) and Martin and Lejeune(1998) showed that turbulent heat fluxes aresensitive to orography and concluded thatexchange coefficients in complex topographymust be increased Martin and Lejeune(1998) proposed an empirical parameteriza-tion which reduces the Richardson numberunder stable atmospheric conditions thusallowing for turbulent heat transfer even withstrong stability However they emphasizethat parameterization of the turbulent fluxesis highly dependent on the site and that auniversal or simple formula can not be deter-mined De la Casiniegravere (1974) and Halberstamand Schieldge (1981) observed temperatureprofile anomalies that would lead to seriouserrors in an estimate of the turbulent heatfluxes if the data at standard height (2 m)were used Due to radiative heating of the airabove the surface the temperature maximaoccurred within the first couple of decimetresabove the surface thus violating the assump-tion of constant fluxes with height Munroand Davies (1972) gave an upper limit of 1 mfor the surface boundary layer thicknessContrary to theory Grainger and Lister(1966) showed that the logarithmic windprofile is valid over a wide range of stabilityand on Greenland Forrer and Rotach (1997)found no tendency for the stability functionfor heat to increase with increasing atmos-pheric stability although they lack an explana-tion in terms of boundary layer theory

Experimental and theoretical evidence ismounting that Monin-Obukhov theory onwhich both profile and bulk methods arebased is not applicable over a sloping glaciersurface due to violation of assumptions suchas homogeneous infinite flat terrain and con-stant flux with height (Holmgren 1971Denby and Greuell 2000) The turbulentscaling laws used in the theory are altered bythe wind-speed maximum By observationsand second-order modelling Denby andGreuell (2000) showed that profile methodswill severely understimate turbulent fluxeswhen a wind-speed maximum is present but

found that the bulk method was appropriateat least in the region below the wind-speedmaximum despite large scatter in the data

The discussion above reveals that largeuncertainty remains in the determination ofturbulent fluxes over glaciers in terms of thesuitability of bulk and profile methods thedetermination of exchange coefficients orroughness lengths and their spatial andtemporal variability and the application ofstability corrections Further research isneeded to explore and develop alternativenew methods that are more suitable oversloping glacier surfaces subject to glacierwinds More eddy correlation measurementsare needed on valley glaciers In glaciologysuch studies have been conducted over icesheets while to date only one reported studyconcerns a valley glacier (Smeets et al 1998)

6 Ice heat fluxBefore surface melting can occur the tem-perature of the icesnow surface must beraised to 0C The energy necessary to heat acold snow or ice mass to 0C defines the coldcontent given by

(28)

where is the snow or ice density cP is thespecific heat of snow or ice T is the tempera-ture at depth z (C) and Z is the maximumdepth of subfreezing temperatures The coldcontent of both winter snow cover and thesurface ice layers can be an important reten-tion component significantly contributing tothe delay between surface melt and melt-derived streamflow (eg Kattelmann andYang-Daqing 1992) Closely connected tothe cold content is the formation of super-imposed ice (Koumlnig et al 2002) and internalaccumulation (Trabant and Mayo 1985Schneider and Jansson 2004) The formerforms when water percolating through thesnow layer refreezes at the impermeable cold ice surface while the latter refers towater refreezing in the firn area below thelast summer surface These processes can be

C z c T z zz

( ) ( )= minusint P d0


Regine Hock 379

crucial for glacier mass balance especially inpolar regions and for correct calculations ofequilibrium line altitude The energy balanceis also affected as the albedo and surfaceroughness are modified

a Ice Near-surface glacier ice is warmedup primarily by heat conduction and byabsorption of penetrating shortwave-radiation For temperate glacier ice the heatflux is zero except during occasional noctur-nal freezing For cold glacier ice or glacierswith a perennial cold surface layer heat fluxinto the ice can be a substantial energy sinkAt a site in North Greenland 11 of the totalenergy surplus at the surface was used toheat up the ice and thus was not available formelt (Konzelmann and Braithwaite 1995)Heat flux across the ice surface is given by

(29)

with Tt the rate of change of ice tem-perature This flux can be estimated fromtemperature-depth profiles down to the baseof the seasonally affected layer usually 10ndash15 mbelow the surface (Paterson 1994) The heatflux is then computed from the change in cold content with time Konzelmann andBraithwaite (1995) propose a method tocompute the ice heat flux when temperaturemeasurements are shallower than the depthof seasonal temperature fluctuations how-ever the temperature changes with time are assumed constant Then the heat fluxthrough the glacier surface is calculated as theintercept in a regression equation of the heatflux at depth z versus depth where theenglacial heat flux is calculated from ice tem-perature gradients and thermal conductivityGreuell and Oerlemans (1987) model thetemperature profile inside the glacier down toa depth of ~25 m considering conductionand advection of snowice normal to thesurface and energy release or consumption by phase changes Results showed that abla-tion is considerably overestimated at higherelevations if glacier temperatures were kept

at 0C thus emphasizing the need to considerthe occurrence of subfreezing glaciertemperatures in models

b Snow Quantitative assessment of theenergy transfer within a cold snowpack ismore difficult due to a complex interplay ofhydrology and heat transfer (Colbeck 1972)The mechanisms of heat transfer in snow aregoverned by penetration of shortwave radia-tion internal movement of waterwatervapour and phase changes The most efficientprocess for snowpack warming is the releaseof latent heat by refreezing of percolatingmelt or rain water Physical-based snowmodels accounting explicitly for heat and mass transfer within a snow coverincluding simulation of the evolution oftemperature density water content meta-morphism and snow stratigraphy have beensuggested by several authors (eg Anderson1976) those few which have been convertedinto fully developed computer programs andwidely tested include CROCUS (Brun et al1989) SNTHERM (Jordan 1991) andDAISY (Bader and Weilenmann 1992)

More conceptual approaches have beendeveloped in order to circumvent excessivedata requirements Van de Wal and Russel(1994) developed an algorithm where energydeficits from previous time steps were com-pensated before allowing melt This approachwas extended by adjusting iteratively thesurface temperature until the trial ablation for the time step becomes zero in casenegative melt was computed (Escher-Vetter1985b Braithwaite et al 1998 Hock andHolmgren 2005) An iterative procedure isnecessary because surface temperature alsoaffects outgoing longwave radiation the tur-bulent heat fluxes and the heat flux suppliedby rain A common way to consider the coldcontent of snow in conceptual melt models isto use the concept of lsquonegative meltrsquo (egBraun and Aellen 1990) The amount ofrefreezing water is computed from air tem-perature and a factor of refreezing in casecomputed melt turns negative

Q cTt

zG p

Z= part

partint d0

7 Other heat fluxesThe sensible heat flux of rain is generallyunimportant in the overall surface energy bal-ance of a glacier and is thus neglected inmany models A rainfall event of 10 mm at10C on a melting surface would produce aheat flux of 24 W m 2 averaged over a dayhence negligible compared to other heatfluxes Precipitation may be a significantshort-term heat source only when precipita-tion is heavy prolonged and warm asencountered in maritime areas exposed tostorms originating over warm oceans Oneday on Ivory Glacier New Zealand the heatflux by rain contributed 37 of the daily abla-tion of 62 mm (Hay and Fitzharris 1988)The heat flux by rain QR is given by

(30)

where w is the density of water cw is thespecific heat of water (42 kJ kg1 K1) R isthe rainfall rate and Tr and Ts are the temper-atures of rain and the surface respectivelyAlthough energetically of minor importancerain may affect melt indirectly by increasingthe liquid water content of a glacier surfaceand thus reducing its albedo

On alpine glaciers advection of warm air from adjacent valley slopes and morainesmay contribute substantially to glacier melt(Wendler 1975) This advective heat flux isnot considered in the vertical flux equationused in the one-dimensional energy-balancemodels generally applied in melt modellingFor alpine tundra environments modellingstudies have demonstrated that local advec-tion of heat from snow-free patches consider-ably enhances snow melt rates along theleading edges of snow-covered patches(Olyphant and Isard 1988 Marsh andPomeroy 1997 Essery 1999)

Moore (1991) investigated the possibility ofsensible heat being advected by supraglacialrunoff Through numerical modelling he con-cluded that such advection will be negligibleon a macroscale under most conditions butmay cause microscale variations in ice melt

III Temperature-index melt modelsTemperature-index models or degree-daymodels assume an empirical relationshipbetween melt and air temperature based on astrong and frequently observed correlationbetween these quantities A detailed reviewof the method and recent advances in distrib-uted temperature-index modelling is given byHock (2003) Since air temperature is usuallythe most readily available data quantity and isreasonably easy to extrapolate and forecasttemperature-index models have been themost widely used method of ice and snowmelt computations Applications are wide-spread and include the prediction of melt for operational flood forecasting and hydro-logical modelling (WMO 1986) glacier massbalance modelling (eg Laumann and Reeh1993 Oerlemans et al 1998) and assessmentof the response of snow and ice to pre-dicted climate change (eg Braithwaite and Zhang 1999) Most operational runoffmodels eg HBV model (Bergstroumlm 1976)SRM model (Martinec and Rango 1986)UBC model (Quick and Pipes 1977)HYMET model (Tangborn 1984) and evenversions of the physically based SHE model(Boslashggild et al 1999) apply temperature-index methods for melt modelling

Although the concept involves a simplifica-tion of complex processes that are moreproperly described by the surface energybalance temperature-index models oftenmatch the performance of energy-balancemodels on a catchment scale (WMO 1986Rango and Martinec 1995) The success ofthe temperature-index method is generallyattributed to the high correlation of tem-perature with various components of theenergy-balance equation Longwave incom-ing radiation and the turbulent heat fluxesdepend strongly on temperature and temper-ature in turn is affected by global radiationalthough not in a simple way (Kuhn 1993Ohmura 2001)

The classical degree-day model relates iceor snow melt M [mm] during a period of n time intervals t to the sum of positive

Q c R T TR = minusw w r s( )


air temperatures of each time interval Tduring the same period

(31)

The factor of proportionality is the degree-dayfactor DDF expressed in mm d1 K1 for t expressed in days and temperature in C(Braithwaite 1995b Hock 2003)

Degree-day factors vary considerably inspace and time because they implicitelyaccount for all terms of the energy budgetwhich vary in relative importance Highrelative contributions of sensible heat flux to the heat of melt favour low degree-dayfactors (Ambach 1988) On average reportedvalues for different sites based on melt meas-urements range from 25 to 116 mm d1 K1

and 66 to 200 mm d1 K1 for snow and ice respectively (see summary in Hock2003) Degree-day factors for ice generallyexceed those for snow due to lower albedo ofice compared to that of snow In additionenormous small-scale spatial variabilityoccurs partially due to topographic effectsOn Storglaciaumlren (3 km2) hourly degree-dayfactors computed from melt determined from a distributed energy-balance model on a30 m resolution grid ranged from 0 to 16 mmd1 K1 for the same hour of the day (Hock1999) Seasonal variations in the DDF can beexpected due to the seasonal variation inclear-sky direct radiation and in the case ofsnow due to snow metamorphism whichgenerally decreases snow albedo thusincreasing the DDF as the melt seasonprogresses (Kuusisto 1980) Measurements(Singh and Kumar 1996) and modellingstudies (Hock 1999) indicate large diurnalvariations in degree-day factors (0 to gt15 mmd1 K1) caused by diurnal radiation fluctua-tions implying that constant melt factors areinadequate for subdaily (eg hourly) meltcomputations

Many temperature-index based runoffmodels consider seasonal variations in meltfactors For instance the UBC runoff model

(Quick and Pipes 1977) uses a monthlyvariable melt factor while the HBV-ETHmodel (Braun et al 1993) determines themelt factor from sinusoidal interpolationbetween a minimum value on 21 Decemberand a maximum value on 21 June Schreideret al (1997) and Arendt and Sharp (1999)varied the degree-day factor according toalbedo

Because degree-day factors are influencedby all components of the energy balancemany attempts have been made tostrengthen the physical foundation of themethod by incorporating more variables suchas wind speed vapour pressure or radiation(eg Willis et al 1993) There is a gradualtransition from simple degree-day approachesto energy-balance type expressions byincreasing the number of input variables in melt computations The widely quotedcombination method by Anderson (1973)applies a simple degree-day approach duringdry periods and a simplified empirical energy-balance formulation during rainy periods TheUBC runoff model (Quick and Pipes 1977)and the HYMET runoff model (Tangborn1984) include the daily temperature range inaddition to air temperature itself as climaticinput to their melt routines Various studieshave added a radiation term often in form of shortwave or net radiation balance(Martinec 1989 Kustas and Rango 1994Kane and Gieck 1997) thus achieving betterresults on a daily or hourly basis compared to the classical degree-day method

To account for spatial variability in meltrates while employing spatially constantdegree-day factors melt-runoff models oftendivide the basin into elevation bands toconsider a decrease in melt with increasingelevation or also into aspect classes toaccount for enhanced melt on south-facingslopes compared to north-facing slopes(Braun et al 1994) In recent years suchapproaches have been considerably advancedby explicitly varying the degree-day factor foreach slopeaspect class (Brubaker et al 1996Dunn and Colohan 1999) or for each grid

M DDF T ti

n

i

n

= sdot∆=

+

=sum sum

1 1

Regine Hock 381

element of a digital elevation model in a fully distributed manner Daly et al (2000)increase the melt factor for each grid elementas a function of an accumulated temperatureindex while Cazorzi and Fontana (1996) andHock (1999) incorporate clear-sky shortwaveradiation to account for the spatial heteoro-geneity of radiation and melt conditions incomplex terrain

IV Discussion and concluding remarksDue to the availability of air temperaturedata temperature-index methods will proba-bly remain the most widely used approach tocompute melt for many purposes Thesemethods lump the surface energy exchangeinto one or a few parameters with the disad-vantage that melt factors vary from site tosite so model calibration is requiredConversely energy-balance melt modelsmore properly describe the physical processesat the glacier surface but require much moredata which are often not available

There is a trend to replace simplerschemes with more sophisticated onesHowever a fundamental question concernswhether such an increase in sophistication iswarranted In Figure 2 hourly discharge sim-ulations for Storglaciaumlren Sweden based ondifferent melt models are intercomparedClearly the classical degree-day model cap-tures the seasonal pattern in discharge butnot the daily discharge fluctuations (Figure2a) Model performance is significantlyenhanced by incorporating potential directsolar radiation as an index of local and dailyvariability in the energy available for melt(Figure 2b) Although this model assumesclear-sky conditions a third approach includ-ing measured global radiation to incorporatecloud effects does not yield any improvementin performance (Figure 2c Hock 1999) Thisis probably due to increases in other energyfluxes under cloudy conditions such as long-wave irradiance Both energy-balanceapproaches (Figure 2d and e) yield goodresults The models differ in their levels ofsophistication Model (d) assumes a melting

surface constant snow albedo and spatiallyinvariant longwave irradiance while model(e) parameterizes these quantities thusaccounting for spatial variability (Hock1998) In general model performanceimproved with increasing model complexity

For runoff modelling using spatially lumpedmodels and typical time steps of one daymodelling intercomparison projects (egWMO 1986) revealed that model complexitycould not be related to the quality of thesimulation results Simple models providedcomparable results to more sophisticatedmodels which is generally attributed to thedifficulties of assigning proper model para-meters and meteorological input data to eachcatchment element The example in Figure 2indicates that the appropriate level of modelsophistication is related to the temporal andspatial scale of interest and the objectives ofthe study For catchment-scale studiesemploying daily time steps simple tempera-ture-index methods are often sufficientHowever if higher resolution in space andtime is required more sophisticated energy-balance models are preferable Physicallybased models are more suited to quantifyingthe response of melt and discharge to futureclimate changes since parameters in simplermodels may not be the same under a differentclimate In highly glacierized basins a higher(eg hourly) temporal resolution is neces-sary for accurate runoff modelling especiallywith respect to peak flow estimates becauseof the enormous melt-induced diurnal dis-charge fluctuations This aspect of temporalresolution is crucial in attempts to predict thechanges associated with a warming climateas diurnal melt-cycles are expected to beamplified significantly increasing peak flows(Hock et al 2005) The spatial aspect isimportant in complex topography in massbalance studies as well as in studies related to avalanche forecasting erosion solifluctionsolute transport or vegetation patterns

Generally speaking significant advances indistributed melt modelling have been made inrecent years based on both temperature-index


Regine Hock 383

Figure 2 Simulated and measured hourly discharge (m3 s1) of StorglaciaumlrenSweden using different temperature-index and energy-balance models for the meltmodelling and a linear reservoir model for water routing Melt calculations are basedon the classical degree-day method (a) a modified temperature-index model includingpotential direct solar radiation (b) a modified temperature-index model includingpotential direct solar radiation and global radiation (c) a simple energy-balance model(d) and a more sophisticated energy-balance model (e) Model performance is given interms of the efficiency criterion R2 (Nash and Sutcliffe 1970) Also given are hourlymeasurements of global radiation (Wm2) wind speed (m s1) air temperature (C)and precipitation (mm h1)Source Models (andashc) from Hock (1999) model (d) from Hock and Noetzli (1997)model (e) from Hock (1998)

and energy-balance methods Such app-roaches require further testing and refine-ment and would benefit from more extensiveexploitation of increasingly available ofremote sensing data for both model inputand verification The need to extrapolatemeteorological input data places major limita-tions on the quality of model simulations(Charbonneau et al 1981) Often windspeed and relative humidity are assumed spa-tially constant and air temperature is variedwith altitude according to a simple lapse rateThe inclusion of wind models may aid inbetter interpolating meteorological inputdata although the interfacing may be difficultdue to differences in spatial resolution

Energy-balance models yield sufficientlyaccurate estimates of the spatial and toporalpatterns of incoming radiation in mountainterrain However the determination of tur-bulent fluxes and surface albedo are currentlyidentified as the most prominent uncertain-ties Increasing evidence suggests that exist-ing boundary layer theory does not apply overthe typically inclined glacier surface which issubject to gravity winds Much uncertaintyconcerns the determination of exchangecoefficients and the consideration of stabilityeffects Evidence seems to mount that theturbulent heat fluxes over valley glaciers are larger than existing theory predictsFurther research needs to be directedtowards the development of realistic con-cepts in the glacier environment based oncareful re-evaluation of existing data sets and in particular more eddy correlationstudies on valley glaciers Determining accu-rate spatial distributions of turbulent fluxes is an even larger challenge that needs to be tackled Besides the problems in dataextrapolation accounting for the spatial and temporal variations in exchange coeffi-cients is a yet rather unexplored issueDespite a large variety of proposed albedoparameterizations future research needs tofocus on quantifying physical controls onglacier albedo and the development of models capturing the full range of spatial and

temporal albedo variability In additionenergy gain due to advected heat may not benegligible on valley glaciers and needs furtherinvestigation

Few studies have attempted to link thevariations in ablation and energy partitioningat the point scale to larger-scale atmosphericconditions (eg Hoinkes 1968 Cline 1997Hannah et al 1999) Such relationships maybe exploited for forecasting the timing of melt-induced runoff from mountainous regions aslarge-scale air mass characteristics are morepredictable than local wind temperature andrelative humidity patterns Further researchwill need to focus on the links between thedifferent energy fluxes and the synopticweather pattern and investigate their poten-tial for operational use in melt forecasting

AcknowledgementsBjoumlrn Holmgren contributed through criticaldiscussions Heinz Blatter Andrew Fountainand Christian Pluess are acknowledged forvaluable comments on the manuscriptCareful reviews were provided by Ian Willisand an anonymous reviewer

ReferencesAhlmann HW 1935 Scientific results of the

Norwegian-Swedish Spitzbergen expedition 1934Part 5 The Fourteenth of July Glacier GeografiskaAnnaler 17 167ndash218

mdash 1948 Glaciological Research on the north Atlanticcoasts Research Series No 1 London GeographicalSociety 83 pp

Ambach W 1963 Untersuchungen zum Energieumsatzin der Ablationszone des groumlnlaumlndischen InlandeisesMeddelser om Groslashnland 174(4) 311 pp

mdash 1986 Nomographs for the determination of meltwaterfrom snow and ice surfaces Berichte des naturwis-senschaftlich-medizinischen Vereins in Innsbruck 73 7ndash15

mdash 1988 Interpretation of the positive-degree-daysfactor by heat balance characteristics ndash WestGreenland Nordic Hydrology 19 217ndash24

Ambach W and Hoinkes HC 1963 The heatbalance of an Alpine snowfield (Kesselwandferner3240 m Oetztal Alps 1958) Wallingford IAHSPublication 61 24ndash36

Anderson EA 1972 Techniques for predicting snowcover runoff In The role of snow and ice in hydrologyProceedings of the Banff Symposium 1972Wallingford IAHS Publication 107 840ndash63


mdash 1973 National weather service river forecast systemsnow accumulation and ablation model NOAATechnical Memorandum NWS HYDRO-17 SilverSpring MD US Department of Commerce 217 pp

mdash 1976 A point energy and mass balance model of a snowcover NOAA Technical Report NWS 19Washington DC NOAA 150 pp

Andreas EL 1987 A theory for the scalar roughnessand the scalar transfer coefficient over snow and seaice Boundary-Layer Meteorology 38 159ndash84

Aringngstroumlm A 1916 Uumlber die Gegenstrahlung derAtmosphaumlre Meteoroische Zeitschrift 33 529ndash38

mdash 1933 On the dependence of ablation on air tempera-ture radiation and wind Geografiska Annaler 15264ndash71

Arendt A and Sharp M 1999 Energy balance meas-urements on a Canadian high arctic glacier and theirimplications for mass balance modelling In TranterM Armstrong R Brun E Jones G Sharp Mand Williams M editors Interactions between thecryosphere climate and greenhouse gases Proceedingsof the IUGG Symposium Birmingham 1999Wallingford IAHS Publication 256 165ndash72

Arnold NS Willis IC Sharp MJ RichardsKS and Lawson WJ 1996 A distributed surfaceenergy-balance model for a small valley glacier IDevelopment and testing for Haut Glacier drsquoArollaValais Switzerland Journal of Glaciology 42 77ndash89

Bader H and Weilenmann P 1992 Modelingtemperature distribution energy and mass flow in a(phase-changing) snowpack I Model and casestudies Cold Regions Science and Technology 20157ndash81

Beljaars ACM and Holtslag AAM 1991 Fluxparameterization over land surfaces for atmosphericmodels Journal of Applied Meteorology 30 327ndash41

Bergstroumlm S 1976 Development and application of aconceptual runoff model for Scandinavian catch-ments Department of Water Resources EngineeringLund Institute of TechnologyUniversity of LundBulletin Series A 52 134 pp

Bintanja R 1999 On the glaciological meteorologicaland climatological significance of Antarctic blue iceareas Reviews of Geophysics 37 337ndash59

Bintanja R and van den Broeke MR 1995Momentum and scalar transfer coefficients overaerodynamically smooth Antarctic surfacesBoundary-Layer Meteorology 74 89ndash111

Bloumlschl G Kirnbauer B and Gutknecht D 1991Distributed snowmelt simuations in an Alpinecatchment 1 Model evaluation on the basis of snow cover patterns Water Resources Research 273171ndash79

Boslashggild CE Knudby CJ Knudsen MB andStarzer W 1999 Snowmelt and runoff modelling ofan arctic hydrological basin in east GreenlandHydrological Processes 13 1989ndash2002

Braithwaite RJ 1995a Aerodynamic stability andturbulent sensible-heat flux over a melting ice sur-face the Greenland ice sheet Journal of Glaciology41 562ndash71

mdash 1995b Positive degree-day factors for ablation on theGreenland ice sheet studied by energy-balancemodelling Journal of Glaciology 41 153ndash60

Braithwaite RJ and Olesen OB 1990a Responseof the energy balance on the margin of the Greenlandice sheet to temperature changes Journal ofGlaciology 36 217ndash21

mdash 1990b A simple energy-balance model to calculateice ablation at the margin of the Greenland ice sheetJournal of Glaciology 36 222ndash28

Braithwaite RJ and Thomsen OO 1989Simulation of run-off from the Greenland ice sheet for planning hydro-electric power IlulissatJakobshavn West Greenland Annals of Glaciology13 12ndash15

Braithwaite RJ and Zhang Y 1999 Modellingchanges in glacier mass balance that may occur as aresult of climate changes Geografiska Annaler81A 489ndash496

Braithwaite RJ Konzelmann T Marty C andOlesen OB 1998 Reconnaisance study of glacierenergy balance in north Greenland 1993ndash94 Journalof Glaciology 44 239ndash247

Braun LN and Aellen M 1990 Modelling dischargeof glacierized basins assisted by direct measurementsof glacier mass balance In Lang H and Musy A edi-tors Hydrology of mountainous regions I Proceedingsof two Lausanne Symposia 1990 Wallingford IAHSPublication 193 99ndash106

Braun LN Aellen M Funk M Hock RRohrer MB Steinegger U Kappenberger Gand Muumlller-Lemans H 1994 Measurement andsimulation of high alpine water balance componentsin the Linth-Limmern head watershed (NortheasternSwitzerland) Zeitschrift f uumlr Gletscherkunde undGlazialgeologie 30 161ndash85

Braun LN Grabs W and Rana B 1993Application of a conceptual precipitation-runoffmodel in the Langtang Khola basin Nepal HimalayaIn Young GJ editor Snow and glacier hydrologyProceedings of the Kathmandu Symposium 1992Wallingford IAHS Publication 218 221ndash37

Brock B Willis IC and Sharp MJ 2000aMeasurement and parameterization of albedo varia-tions at Haut Glacier drsquoArolla Switzerland Journalof Glaciology 155 675ndash88

Brock B Willis IC Sharp MJ and ArnoldNS 2000b Modelling seasonal and spatial varia-tions in the surface energy balance of Haut GlacierdrsquoArolla Switzerland Annals of Glaciology 31 53ndash62

Brubaker K Rango A and Kustas W 1996Incorporating radiation inputs into the snowmeltrunoff model Hydrological Processes 10 1329ndash43

Regine Hock 385

Brun E Martin E Simon V Gendre C andColeou C 1989 An energy and mass model ofsnow cover suitable for operational avalancheforecasting Journal of Glaciology 35 333ndash42

Brunt D 1932 Notes on radiation in the atmosphere IQuartely Journal of the Royal Meteorological Society58 389ndash420

Brutsaert W 1975 On a derivable formula for long-wave radiation from clear skies Water ResourcesResearch 11 742ndash44

Burlando P Pellicciotti F and Strasser U 2002Modelling mountainous water systems Betweenlearning and speculating Looking for challengesNordic Hydrology 33 47ndash74

Calanca P 2001 A note on the roughness length fortemperature over melting snow and ice QuarterlyJournal of the Royal Meteorological Society 127255ndash60

Cazorzi F and Fontana GD 1996 Snowmelt mod-elling by combining air temperature and a distributedradiation index Journal of Hydrology 181 169ndash87

Charbonneau R Lardeau JP and Obled C 1981Problems of modelling a high mountainous drainagebasin with predominant snow yields HydrologicalSciences Bulletin 26 345ndash61

Choudhury BJ and Chang ATC 1981 On theangular variation of solar reflectance of snow Journalof Geophysical Research 86 465ndash72

Cline DW 1997 Snow surface energy exchanges and snowmelt at a continental midlatitude Alpinesite Water Resources Research 33 689ndash701

Colbeck SC 1972 A theory of water percolation insnow Journal of Glaciology 11 369ndash85

Collares-Pereira M and Rabl A 1979 The averagedistribution of solar radiation correlations betweendiffuse and hemispherical and between daily andhourly insolation values Solar Energy 22 155ndash164

Corps of Engineers 1956 Summary report of the snowinvestigations snow hydrology Portland OR USArmy Engineer Division 437pp

Crawford N 1973 Computer simulation techniques forforecasting snowmelt runoff In The role of snow andice in hydrology Proceedings of the Banff Symposium1972 Wallingford IAHS Publication 107 1062ndash72

Cutler P and Munro DS 1996 Visible and near-infrared reflectivity during the ablation period onPeyto Glacier Alberta Canada Journal of Glaciology42 333ndash40

Daly SF Davis R Ochs E and Pangburn T2000 An approach to spatially distributed snowmodelling of the Sacramento and San Joaquin basinsCalifornia Hydrological Processes 14 3257ndash71

de la Casinier AC 1974 Heat exchange over amelting snow surface Journal of Glaciology 1355ndash72

de Quervain M 1951 Zur Verdunstung derSchneedecke Archiv fuumlr Meteorologie Geophysik undBioklimatologie B3 47ndash64

Denby B and Greuell W 2000 The use of bulk andprofile methods for determining surface heat fluxes inthe presence of glacier winds Journal of Glaciology46 445ndash452

Dozier J 1980 A clear-sky spectral solar radiationmodel for snow-covered mountainous terrain WaterResources Research 16 709ndash18

mdash 1987 Recent research in snow hydrology Reviews of Geophysics 25 153ndash161

Dunn SM and Colohan RJE 1999 Developingthe snow component of a distributed hydrologicalmodel a step-wise approach based on multi-objectiveanalysis Journal of Hydrology 223 1ndash16

Dyer AJ 1974 A review of flux-profile relationshipsBoundary-Layer Meteorology 7 363ndash72

Ellingson RG Ellis J and Fels S 1991 The inter-comparison of radiation codes used in climate modelslong-wave results Journal of Geophysical Research96 8929ndash53

Escher-Vetter H 1985a Energy balance calculationsfor the ablation period 1982 at Vernagtferner OetztalAlps Annals of Glaciology 6 158ndash60

mdash 1985b Energy balance calculations from five yearsmeteorological records at Vernagtferner OetztalAlps Zeitschrift fuumlr Gletscherkunde und Glazialgeologie21 397ndash402

mdash 2000 Modelling meltwater producton with a distrib-uted energy balance method and runoff using a linearreservoir approach ndash results from VernagtfernerOetztal Alps for the ablation seasons 1992 to 1995Zeitschrift fuumlr Gletscherkunde und Glazialgeologie 36119ndash50

Essery R 1999 Parameterization of heterogeneoussnowmelt Theoretical Applied Climatology 6225ndash30

Finsterwalder S 1897 Der VernagtfernerWissenschaftliche Ergaumlnzungshefte zur Zeitschrift desDeutschen und Oesterreichischen Alpenvereins 1(1)1ndash112

Finsterwalder S and Schunk H 1887 DerSuldenferner Zeitschrift des Deutschen undOesterreichischen Alpenvereins 18 72ndash89

Foumlhn PMB 1973 Short-term snow melt and ablationderived from heat- and mass-balance measurementsJournal of Glaciology 12 275ndash289

Forrer J and Rotach M 1997 On the turbulencestructure in the stable boundary layer over theGreenland ice sheet Boundary-Layer Meteorology 85111ndash36

Funk M 1985 Raumlumliche Verteilung der Massenbilanzauf dem Rhonegletscher und ihre Beziehung zu Klimaelementen Zuumlrcher Geographische Schriften 24 Department of Geography ETHZuumlrich 183 pp

Garnier B and Ohmura A 1968 A method of calculating the direct shortwave radiation income on slopes Journal of Applied Meteorology 7 796ndash800


Grainger ME and Lister H 1966 Wind speedstability and eddy viscosity over melting ice surfacesJournal of Glaciology 6 101ndash27

Greuell W and Konzelmann T 1994 Numericalmodelling of the energy balance and the englacialtemperature of the Greenalnd ice sheet Calculationsfor the ETH-camp location (West Greenland 1155 masl) Global Planetary Change 9 91ndash114

Greuell W and Oerlemans J 1987 Sensitivitystudies with a mass balance model includingtemperature profile calculations inside the glacierZeitschrift fuumlr Gletscherkunde und Glazialgeologie 22101ndash24

mdash 1989 Energy balance calculations on and nearHintereisferner (Austria) and an estimate of theeffect of greenhouse warming on ablation InOerlemans J editor Glacier fluctuations and climaticchange Glaciology and Quaternary GeologyDordrechtKluwer 305ndash23

Greuell W Knap WH and Smeets PC 1997Elevational changes in meteorological variables alonga mid-latitude glacier during summer Journal ofGeophysical Research 102(D22) 25941ndash54

Halberstam I and Schieldge JP 1981 Anomalousbehaviour of the atmospheric surface layer over amelting snow pack Journal of Applied Meteorology20 255ndash65

Hannah M Gurnell AM and McGregor GR1999 Indentifying links between large-scale atmos-pheric circulation and local glacier ablation climates inthe French Pyrenees In Tranter M Armstrong RBrun E Jones G Sharp M and Williams Meditors Interactions between the cryosphere climateand greenhouse gases Proceedings of the IUGGSymposium Birmingham 1999 Wallingford IAHSPublication 256 155ndash63

Harding RJ Entrasser N Escher-Vetter HJenkins A Kaser M Kuhn M Morris EMand Tanzer G 1989 Energy and mass balancestudies in the firn area of the Hintereisferner InOerlemans J editor Glacier fluctuations and climaticchange Glaciology and Quaternary GeologyDordrecht Kluwer 325ndash41

Hay JE and Fitzharris BB 1988 A comparison ofthe energy-balance and bulk-aerodynamicapproaches for estimating glacier melt Journal ofGlaciology 34 145ndash53

Herzfeld U Mayer H Feller W and Mimler M2000 Geostatistical analysis of glacier-roughnessdata Annals of Glaciology 30 235ndash42

Hess H 1904 Die Gletscher Braunschweig Druck undVerlag von Friedrich Vieweg und Sohn 426 pp

Hock R 1998 Modelling of glacier melt and dischargeZuumlrcher Geographische Schriften 70 Department ofGeography ETH Zuumlrich 140pp

mdash 1999 A distributed temperature-index ice- andsnowmelt model including potential direct solarradiation Journal of Glaciology 45 101ndash11

mdash 2003 Temperature index melt modelling in mountainareas Journal of Hydrology 282(1-4) 104ndash15 Doi101016S0022-1694(03)00257-9

Hock R and Holmgren B 1996 Some aspects ofenergy balance and ablation of StorglaciaumlrenSweden Geografiska Annaler 78A 121ndash31

mdash 2005 A distributed energy balance model for com-plex topography and its application to StorglaciaumlrenSweden Journal of Glaciology in press

Hock R and Noetzli C 1997 Areal melt anddischarge modelling of Storglaciaumlren Sweden Annalsof Glaciology 24 211ndash17

Hock R Jansson P and Braun L 2005 Modellingthe response of mountain glacier discharge to climatewarming In Huber UM Reasoner MA andBugmann H editors Global change and mountainregions ndash a state of knowledge overview Advances inGlobal Change Series Dordrecht Springer 293ndash52

Hoeck E 1952 Der Einfluss der Strahlung und derTemperatur auf den Schmelzprozess der SchneedeckeBeitraumlge zur Geologie der Schweiz ndash Geotechnische Serie ndash Hydrologie Bern 36 pp

Hogg IGG Paren JG and Timmis RJ 1982Summer heat and ice balances on Hodges glaciersouth Georgia Falkland dependencies Journal ofGlaciology 28 221ndash38

Houmlgstroumlm U 1988 Non-dimensional wind and tem-perature profiles in the atmospheric surface layer a re-evaluation Boundary-Layer Meteorology 4255ndash78

Hoinkes HC 1955 Measurements of ablation andheat balance on alpine glaciers Journal of Glaciology2 497ndash501

mdash 1968 Glacier variation and weather Journal ofGlaciology 7 3ndash19

mdash 1969 Surges of the Vernagtferner in the Oetztal Alpssince 1599 Canadian Journal of Earth Science 6853ndash61

Hoinkes HC and Steinacker H 1975Hydrometeorological implications of the massbalance of Hintereisferner 1952ndash53 to 1968ndash69 InProceedings of the Snow and Ice Symposium Moscow1971 Wallingford IAHS Publication 104 144ndash49

Hoinkes HC and Untersteiner N 1952Waumlrmeumsatz und Ablation auf Alpengletschern IVernagtferner (Oetztaler Alpen) August 1950Geografiska Annaler 34(1-2) 99ndash158

Holmgren B 1971 Climate and energy exchange on asub-polar ice cap in summer Arctic Institute of NorthAmerica Devon Island Expedition 1961-1963 UppsalaMeteorologiska Institutionen Uppsala UniversitetMeddelande 107 Part AndashE

Inoue J 1989 Surface drag over the snow surface ofthe Antarctic plateau 1 factors controlling surfacedrag over the katabatic wind region Journal ofGeophysical Research 94(D2) 2207ndash2217

Iqbal M 1983 An introduction to solar radiationLondon Academic Press 390 pp

Regine Hock 387


Jackson BS and Carroll JJ 1978 Aerodynamicroughness as a function of wind direction overasymmetric surface elements Boundary-LayerMeteorology 14 323ndash30

Jansson P Hock R and Schneider T 2003 Theconcept of glacier storage ndash a review Journal ofHydrology 282 115ndash29 101016S0022-1694(03)00258-0

Jonsell U Hock R and Holmgren B 2003 Spatialand temporal variations in albedo on stoglaciaumlrenJournal of Glaciology 49 59ndash68

Jordan R 1991 A one-dimensional temperaturemodel for a snow cover technical documenta-tion for SNTHERM 89 CRREL Special Report91ndash16

Kane DL and Gieck RE 1997 Snowmelt modelingat small Alaskan arctic watershed Journal ofHydrological Engineering 2 204ndash10

Kasten F 1983 Parameterisierung der Globalstrahlungdurch Bedeckungsgrad und Truumlbungsfaktor Annalender Meteorologie 20 49ndash50

Kattelmann R and Yang-Daqing Y 1992 Factors delaying spring runoff in the upper UrumqiRiver basin China Annals of Glaciology 16 225ndash30

King JC and Anderson PS 1994 Heat and watervapour fluxes and scalar roughness lengths over anAntarctic ice shelf Boundary-Layer Meteorology 69101ndash21

Kirnbauer R Bloumlschl G and Gutknecht D 1994Entering the era of distributed snow models NordicHydrology 25 1ndash24

Klok EJ and Oerlemans J 2002 Model study of the spatial distribution of the energy and mass balanceof Morteratschgletscher Switzerland Journal ofGlaciology 48 505ndash18

Koenig-Langlo G and Augstein F 1994Parameterization of the downward long-waveradiation at the Earthrsquos surface in polar regionsMeteorologische Zeitschrift 3 343ndash47

Kondratyev KY 1965 Radiative heat exchange in theatmosphere Oxford Pergamon Press 411 pp

mdash 1969 Radiation in the atmosphere New YorkAcademic Press 912 pp

Koumlnig M Wadham J Winther JG Kohler Jand Nuttall AM 2002 Detection of superim-posed ice on the glaciers Kongsvegen and midreLovenbreen Svalbard using SAR satellite imageryAnnals of Glaciology 34 335ndash42

Konya K Matsumoto T and Naruse R 2004Surface heat balance and spatially distributedablation modelling at Koryto Glacier Kamchatkapeninsula Russia Geografiska Annaler 86A 337ndash48

Konzelmann T and Braithwaite RJ 1995Variations of ablation albedo and energy balance atthe margin of the Greenland ice sheet KronprinsChristian Land eastern north Greenland Journal ofGlaciology 41 174ndash82

Konzelmann T and Ohmura A 1995 Radiative fluxesand their impact on the energy balance of theGreenland ice sheet Journal of Glaciology 41 490ndash502

Konzelmann T van de Wal RSW Greuell WBintanja R Henneken EAC and Abe-Ouchi A 1994 Parameterization of global andlongwave incoming radiation for the Greenland icesheet Global Planetary Change 9 143ndash64

Kuhn M 1984 Physikalische Grundlagen des Energie-und Massenhaushalts der Schneedecke In BrechtelH editor Schneehydrologische Forschung inMitteleuropa volume 7 DVWK (Deutscher Verbandfuumlr Wasserwirtschaft und Kulturbau) 5ndash56

mdash 1987 Micro-meteorological conditions for snow meltJournal of Glaciology 33 263ndash72

mdash 1993 Methods of assessing the effects of climaticchanges on snow and glacier hydrology In YoungGJ editor Snow and glacier hydrology Proceedingsof the Kathmandu Symposium 1992 WallingfordIAHS Publication 218 135-44

Kustas WP and Rango A 1994 A simple energybudget algorithm for the snowmelt runoff modelWater Resources Research 30 1515ndash27

Kuusisto E 1980 On the values and variability ofdegree-day melting factors in Finland NordicHydrology 11 235ndash42

Kuzmin PP 1961 Melting of snow cover Israel Programfor Scientific Translation 290 pp

La Chapelle E 1959 Annual mass and energyexchange on the Blue Glacier Journal of GeophysicalResearch 64 443ndash49

mdash 1961 Energy exchange measurements on the BlueGlacier Washington IAHS Publication 54 302ndash10

Lang H 1981 Is evaporation an important componentin high alpine hydrology Nordic Hydrology 12 217ndash24

mdash 1986 Forecasting meltwater runoff from snow-covered areas and from glacier basins In KraijenhoffDA and Moll JR editor River flow modelling andforecasting Dordrecht D Reidel 99ndash127

Laumann T and Reeh N 1993 Sensitivity to climate change of the mass balance of glaciers in southern Norway Journal of Glaciology 39656ndash65

Lettau H 1939 Atmosphaumlrische Turbulenz LeipzigAkademische Verlagsgesellschaft

mdash 1969 Note on aerodynamic roughness-parameter esti-mation on the basis of roughness-element descriptionJournal of Applied Meteorology 8 828ndash32

List RJ 1966 Smithsonian Meteorological Tables (sixthrevised edition) Washington DC SmithsonianInstitution

Luumltschg-Loetscher O 1944 Zum Wasserhaushalt desSchweizer Hochgebirges Beitraumlge zur Geologie derSchweiz ndash Geotechnische Serie ndash Hydrologie 1(1) Bern101 pp

Male DH 1980 The seasonal snow cover In ColbeckS editor Dynamics of snow and ice masses New York Academic Press 305ndash91

Male DH and Granger RJ 1981 Snow surface and energy exchange Water Resources Research 17609ndash27

Marcus MG Moore RD and Owens IF 1984Short-term estimates of surface energy transfers andablation on the lower Franz Josef Glacier SouthWestland New Zealand New Zealand Journal ofGeology and Geophysics 28 559ndash67

Marks D and Dozier J 1979 A clear-sky longwaveradiation model for remote alpine areas Archiv fuumlrMeterologie Geophysik und Bioklimatologie 27159ndash78

mdash 1992 Climate and energy exchange at the snowsurface in the alpine region of the Sierra Nevada 2Snow cover energy balance Water ResourcesResearch 28 3042ndash54

Marsh P and Pomeroy JW 1997 Sensible heat fluxand local advection over a heterogeneous landscapeat an Arctic tundra site during snowmelt Annals ofGlaciology 25 132ndash36

Marshall SE and Warren SG 1987 Parameteri-zation of snow albedo for climate models InGoodison BE Barry RG and Dozier J editorsLarge scale effects of seasonal snow cover Proceedingsof the Vancouver Symposium 1987 WallingfordIAHS Publication 166 43ndash50

Martin E and Lejeune Y 1998 Turbulent fluxesabove the snow surface Annals of Glaciology 26179ndash83

Martin S 1975 Wind regimes and heat exchange on glacier de Saint-Sorlin Journal of Glaciology 1491ndash105

Martinec J 1989 Hour-to-hour snowmelt rates andlysimeter outflow during an entire ablation period InColbeck SC editor Glacier and snow covervariations Proceedings of the Baltimore SymposiumMaryland 1989 Wallingford IAHS Publication 183 19ndash28

Martinec J and Rango A 1986 Parameter values forsnowmelt runoff modelling Journal of Hydrology 84197ndash219

Moore R 1983 On the use of bulk aerodynamicformulae over melting snow Nordic Hydrology 14193ndash206

Moore RD 1991 A numerical simulation ofsupraglacial heat advection and its influence on icemelt Journal of Glaciology 37 296ndash300

Morris EM 1989 Turbulent transfer over snow andice Journal of Hydrology 105 205ndash23

Morris EM Anderson PS Bader HWeilenmann P and Blight C 1994 Modellingmass and energy exchange over polar snow using the DAISY model In Jones HG Davies TDOhmura A and Morris EM editors Snow and ice covers interactions with the atmosphere and ecosystems Proceedings of the YokohamaSymposium 1993 Wallingford IAHS Publication223 53ndash60

Muumlller F and Keeler CM 1969 Errors in short-termablation measurements on melting ice Journal ofGlaciology 8 91ndash105

Muumlller H 1985 Review paper on the radiation budgetin the Alps Journal of Climatology 5 445ndash62

Munro DS 1989 Surface roughness and bulk heattransfer on a glacier comparison with eddy correla-tion Journal of Glaciology 35 343ndash48

mdash 1990 Comparison of melt energy computations andablatometer measurements on melting ice and snowArctic and Alpine Research 22 153ndash62

Munro DS and Davies JA 1972 An experimentalstudy of the glacier boundary layer over melting iceJournal of Glaciology 18 425ndash36

Munro DS and Young GJ 1982 An operational netshortwave radiation model for glacier basins WaterResources Research 18 220ndash30

Nash JE and Sutcliffe JV 1970 River flowforecasting through conceptual models Part I ndash adiscussion of principles Journal of Hydrology 10282ndash90

Obukhov AM 1946 Turbulentnostrsquo v temeraturno-neodnorodnoy atmosfere (Turbulence in the ther-mally inhomogeneous atmosphere) Trudy InstitutaTeoreticheskoyi Geofiziki Akademiya Nauk SSSR 195ndash115

Oerlemans J 1992 Climate sensitivity of glaciers in southern Norway application of an energy-balance model to Nigardsbreen Hellstugubreen and Alfotbreen Journal of Glaciology 38 223ndash32

mdash 2000 Analysis of a 3-year meteorological recordfrom the ablation zone of MoteratschgletscherSwitzerland energy and mass balance Journal ofGlaciology 46 571ndash79

Oerlemans J and Fortuin JP 1992 Sensitivity ofglaciers and small ice caps to greenhouse warmingScience 258 115ndash17

Oerlemans J and Grisogono B 2002 Glacier windsand parameterisation of the related surface heatfluxes Tellus 54A 440ndash52

Oerlemans J Anderson B Hubbard AHuybrechts P Johannesson T Knap WH andSchmeits M 1998 Modelling the response of glaciersto climate warming Climate Dynamics 14 267ndash74

Ohata T 1989 The effect of glacier wind on localclimate turbulent heat fluxes and ablation Zeitschriftfuumlr Gletscherkunde und Glazialgeologie 25 49ndash68

Ohmura A 1981 Climate and energy balance on arctictundra Axel Heiberg Island Canadian ArcticArchipelago spring and summer 1969 1970 and 1972Zuumlrcher Geographische Schriften 3 Department ofGeography ETH Zuumlrich 448 pp

mdash 2001 Physical basis for the temperature-based melt-index method Journal of Applied Meteorology40 753ndash61

Ohmura A Kasser P and Funk M 1992 Climateat the equilibrium line of glaciers Journal of Glaciology38 397ndash411

Regine Hock 389

Ohmura A Konzelmann T Rotach M Forrer JWild M Abe-Ouchi A and Toritani H 1994Energy balance for the Greenland ice sheet by observa-tion and model computation In Jones HG DaviesTD Ohmura A and Morris EM editors Snow and ice covers interactions with the atmosphere andecosystems Proceedings of the Yokohama Symposium1993 Wallingford IAHS Publication 223 163-74

Ohta T 1994 A distributed snowmelt prediction modelin mountain areas based on an energy balancemethod Annals of Glaciology 19 107ndash13

Oke TR 1987 Boundary layer climates (second edition)London Routledge 435 pp

Olyphant G 1986 Longwave radiation in mountainousareas and its influence on the energy balance of alpinesnowfields Water Resources Research 22 62ndash66

Olyphant GA and Isard SA 1988 The role ofadvection in the energy balance of late-lying snow-fields Niwot Ridge Front Range Colorado WaterResources Research 24 1962ndash68

Orvig S 1954 Glacio-meteorological observations onicecaps in Baffin Island Geografiska Annaler 36197ndash318

Paterson WSB 1994 The physics of glaciers (thirdedition) Oxford Pergamon Press 480 pp

Pluumlss C 1997 The energy balance over an alpinesnowcover Zuumlrcher Geographische Schriften 65Department of Geography ETH Zuumlrich 115 pp

Pluumlss C and Mazzoni R 1994 The role of turbulentheat fluxes in the energy balance of high alpine snowcover Nordic Hydrology 25 25ndash38

Pluumlss C and Ohmura A 1997 Longwave radiationon snow-covered mountainous surfaces Journal ofApplied Meteorology 36 818ndash24

Prandtl L 1934 The mechanics of viscous fluids(Aerodynamic theory) Berlin Durand (ed) vol IIIdivision G

Price AG and Dunne T 1976 Energy balancecomputations of snowmelt in a subarctic area WaterResources Research 12 686ndash94

Quick MC and Pipes A 1977 UBC watershedmodel Hydrological Sciences Bulletin 221 153ndash61

Rango A and Martinec J 1995 Revisiting thedegree-day method for snowmelt computationsWater Resources Bulletin 31 657ndash69

Reeh N Mohr JJ Krabill WB Thomas ROerter H Gundestrup N and Boggild CE2002 Glacier specific ablation rate derived by remotesensing measurements Geophysical Research Letters16 1010292002GLO15307

Reinwarth O and Escher-Vetter H 1999 Massbalance of Vernagtferner Austria from 196465 to199697 results for three sections and the entireglacier Geografiska Annaler 81A 743ndash51

Roumlthlisberger H and Lang H 1987 Glacialhydrology In Gurnell AM and Clark MJ editorsGlacio-fluvial sediment transfer An Alpine perspectiveNew York Wiley 207ndash84

Sauberer F 1955 Zur Abschaumltzung der Globalstrahlungin verschiedenen Houmlhenstufen Wetter und Leben 722ndash29

Sauberer F and Dirmhirn I 1952 DerStrahlungshaushalt horizontaler Gletscherflaumlchen auf dem Hohen Sonnblick Geografiska Annaler 34261ndash90

Schreider T and Jansson P 2004 Internal accumula-tion in firm and its significance for the mass balance ofStorglaciaumlren Sweden Journal of Glaciology 5025ndash34

Schreider SY Whetton PH Jakeman AJand Pittock AB 1997 Runoff modelling for snow-affected catchments in the Australian alpineregion eastern Victoria Journal of Hydrology 2001ndash23

Seidel K and Martinec J 1993 Operational snowcover mapping by satellites and real time runoffforecasts In Young GJ editor Snow and glacierhydrology Proceedings of the Kathmandu Symposium1992 Wallingford IAHS Publication 218 123-32

Singh P and Kumar N 1996 Determination ofsnowmelt factor in the Himalayan region HydrologicalSciences Journal 41 301ndash10

Smeets P Duynkerke PG and Vugts HF 1998Turbulence characteristics of the stable boundarylayer over a mid-latitude glacier Part i a combinationof katabatic and large-scale forcing Boundary-LayerMeteorology 87 117ndash45

Streten NA and Wendler G 1968 The midsummerheat balance of an Alaskan maritime glacier Journalof Glaciology 7 431ndash40

Sverdrup HU 1935 Scientific results of theNorwegian-Swedish Spitzbergen Expedition in 1934Part IV The ablation on Isachsenrsquos plateau and on the Fourteenth of July Glacier in relation to radiationand meteorological conditions Geografiska Annaler17 145ndash66

mdash 1936 The eddy conductivity of the air over a smoothsnow field Geofysiske Publikasjoner 11 5ndash69

Tangborn WV 1984 Prediction of glacier derivedrunoff for hydro-electric development GeografiskaAnnaler 66A 257ndash65

Trabant DC and Mayo LR 1985 Estimation and effects of internal accumulation on five differentglaciers in Alaska Annals of Glaciology 6 113ndash17

Ujihashi Y Takase N Ishida H and Hibobe E1994 Distributed snow cover model for a mountain-ous basin In Jones HG Davies TD Ohmura Aand Morris EM editors Snow and ice covers interac-tions with the atmosphere and ecosystems Proceedingsof the Yokohama Symposium 1993 WallingfordIAHS Publication 223 153ndash62

van de Wal RSW and Russell AJ 1994 Acomparison of energy balance calculations meas-ured ablation and meltwater runoff near SoslashndreStroslashmfjord West Greenland Global PlanetaryChange 9 29ndash38


Regine Hock 391

van de Wal RSW Oerlemanns J and HageJC 1992 A study of ablation variations on thetongue of Hintereisferner Austrian Alps Journal ofGlaciology 38 319ndash24

van den Broeke M 1996 Characteristics of the lowerablation zone of the west Greenland ice sheet forenergy-balance modelling Annals of Glaciology 23160ndash66

mdash 1997 Structure and diurnal variation of the atmos-pheric boundary layer over a mid-latitude glacier insummer Boundary-Layer Meteorology 83 183ndash205

van der Avoird E and Duynkerke PG 1999Turbulence in a katabatic flow Does it resembleturbulence in stable boundary layers over flat sur-faces Boundary-Layer Meteorology 92 39ndash66

Varley MJ and Beven KJ 1996 Modelling solarradiation in steeply sloping terrain Journal ofClimatology 16 93ndash104

Wagner HP 1980a Strahlungshaushaltsuntersuchungenan einem Ostalpengletscher waumlhrend derHauptablationsperiode Teil I KurzwelligeStrahlungsbilanz Archiv fuumlr Meteorologie Geophysikund Bioklimatologie B(27) 297ndash324

mdash 1980b Strahlungshaushaltsuntersuchungen an einemOstalpengletscher waumlhrend der HauptablationsperiodeTeil II Langwellige Strahlung und StrahlungsbilanzArchiv fuumlr Meteorologie Geophysik und BioklimatologieB(28) 41ndash62

Wagnon P Ribstein P Francou B and Pouyaud B1999 Annual cycle of the energy balance of ZongoGlacier Cordillera Real Bolivia Journal of GeophysicalResearch 104 3907ndash23

Walcher J 1773 Nachrichten von den Eisbergen inTyrol Wien 99 pp

Walleacuten CC 1949 The shrinkage of the Karingrsa Glacierand its probable meteorological causes GeografiskaAnnaler 1-2 275ndash91

Warren SG 1982 Optical properties of snow Reviewsof Geophysics and Space Physics 20 67ndash89

Warren SG and Wiscombe WJ 1980 A model forthe spectral abledo of snow II Snow containingatmospheric aerosols Journal of Atmopheric Science37 2734ndash45

Webb EK 1970 Profile relationships the log-linear range and extension to strong stability Quartely Journal of the Royal Meteorological Society 9667ndash90

Wendler G 1975 A note on the advection of warm airtowards a glacier A contribution to the InternationalHydrological Decade Zeitschrift fuumlr Gletscherkundeund Glazialgeologie 10 199ndash205

Wesley ML and Lipschutz RC 1976 A methodfor estimation hourly averages of diffuse and directsolar radiation under a layer of scattered clouds SolarEnergy 18 467ndash73

Willis IC Arnold NS and Brock BW 2002Effect of snowpack removal on energy balance meltand runoff in a small supraglacial catchmentHydrological Processes 16 2721ndash49

Willis IC Sharp MJ and Richards KS 1993Studies of the water balance of MidtdalsbreenHardangerjoumlkulen Norway I The calculation ofsurface water inputs from basic meteorological dataZeitschrift fuumlr Gletscherkunde und Glazialgeologie2728 97ndash115

Winther J Elvehoy H Boslashggild C Sand K andListon G 1996 Melting runoff and the formationof frozen lakes in a mixed snow and blue-ice field in Dronning Maud Land Antarctica Journal ofGlaciology 42 271ndash78

World Meterology Organization (WMO) 1986Intercomparison of models for snowmelt runoff Operational Hydrology Report 23 (WMO no 646)

Zuo Z and Oerlemans J 1996 Modelling albedo and specific balance of the Greenland ice sheetcalculations for the Soslashndre Stroslashmfjord transectJournal of Glaciology 42 305ndash33

Regine Hock 363

recent years interest has risen in particular inspatially distributed estimates of snow and icemelt for many other purposes These includeavalanche forecasts assessment of the contri-bution of melting ice to sea-level rise as wellas studies in glacier dynamics hydrochemistryand erosion

Meltwater production in particular fromsnow cover has received extensive examina-tion in terms of both measurements andmodelling There is a complete hierarchy ofmelt models relating ablation to meteorologi-cal conditions varying greatly in complexityand scope These range from models basedon the detailed evaluation of the surfaceenergy fluxes (energy-balance models) tomodels using air temperature as the sole indexof melt energy (temperature-index models)Much work has been done at the point scaleHowever promoted by increased availabilityof digital terrain models and computationalpower increasing efforts have been devotedto areal melt modelling using fully distributedmodels In addition there is a trend towardshigh temporal resolution modelling (eg withhourly time steps) The latter is essential

for predicting peak flows in glacierized orsnow-covered basins High spatial resolutionis needed to account for the large spatialheteorogeneity with respect to ice and snowmelt typically encountered in steeply sidedterrain as a result of the effects of surroundingtopography

Previous reviews have addressed individualaspects of melt and its modelling Lang (1986)and Roumlthlisberger and Lang (1987) present anoverview of glacier melt and dischargeprocesses An exhaustive review of radiationand turbulent heat transfer at a snow surfaceis given by Male and Granger (1981) Male(1980) and Dozier (1987) summarize theprocesses of snow melt and hydrologyKirnbauer et al (1994) focus on distributedsnow models Recent trends in temperature-index melt modelling are discussed in Hock(2003) This review considers recent advancesin the simulation of melt focusing on glaciersand distributed modelling The relevantprocesses involved in energy exchange at the glacier surface are discussed withrespect to their representation in meltmodels Emphasis is on models suitable for

Figure 1 Hydrograph of hourly discharge at Vernagtferner Austria 1990 displaying seasonal and diurnal variations typical of glacier regimesSource data from Commission for Glaciology of the Bavarian Academy of ScienceMunich















Regine Hock 365





(1)


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able

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elt

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lle 1

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tsch

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ce2

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181

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0)R

oumlthl

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rger

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g 1

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Ale

tsch

glac

ier

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g 1

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m s

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Wor

thin

gton

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cier

16

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127

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Stre

ten

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Wen

dler

196

8A

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a ic

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er 2

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m14

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y 19

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(44)

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Foumlhn

197

3H

odge

s G

laci

er

111

197

3ndash4

419

7447

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6)

3 (

3)

86 (

97)

Hog

g et

al

198

2So

uth

Geo

rgia

460

mSt

Sor

lin G

laci

er 2

700

m11

d in

sum

mer

32 (5

7)24

(43)

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(7)

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(93

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artin

197

5H

inte

reis

fern

er ic

e10

d in

198

619

1 (9

0)22

(10)

4

(2)

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9 (

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Gre

uell

and

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lem

ans

198

7Iv

ory

Gla

cier

53

d in

Jan

ndashFeb

197

273

76 (5

2)44

(30)

23 (1

6)

14

7 (

100)

Hay

and

Fitz

harr

is 1

988

New

Zea

land

150

0m

Stor

glac

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n 13

70m

ice

197

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81

994

73 (6

6)33

(30)

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)

3 (

3)

122

(97

)H

ock

and

Hol

mgr

en 1

996

Pate

rze

glac

ier

2205

m i

ce24

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99

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180

(74)

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van

den

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199

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ka

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Rai

n su

pplie

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2

(2

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O

nly

whe

n m

eltin

g oc

curr

ed






(3)



Regine Hock 368





Regine Hock 369





(4)


(5)








= +minusZ Z

sun slope

I IR P

P zc

maR

0=⎛⎝⎜

⎞⎠⎟0

2

cos cos






(6)


(7)


(8)


(9)





Vf = 2cos ( )2

V Hf = cos ( )2

D D h h hhs d d=

2


0

2

Regine Hock 370



(10)




III

Is

scc=

Regine Hock 371



(11)





= + minus0 be n kd



(12)


(13)

and Brunt (1932)

(14)



(15)


(16)


(17)



ep



c = ( ) 1 24 1 7e Ta a

c = +a b ea



Regine Hock 373



(18)




(19)




L = ( a a4

f s s4





(20)

(21)




(22)

(23)


(24)

(25)


Ck

z z z L z z

z LE

EM

E

=minus

minus[ln( ) ( )] [ln( )

( )]

2

0 0

Ck

z z z L z z

z LT

HM

H

=minus

minus[ln( ) ( )] [ln( )

( )]

2

0 0




part

Q c KzH a P H= part

part

Regine Hock 375



(26)



(27)




Ck

z z z zR

xx b= minus

2

0 0

21 5 2ln( ) ln( )

( )

Rg

TT T z z

uz

z s

zb =

minus minus( )( )02






Regine Hock 377






(28)


C z c T z zz



Regine Hock 379



(29)





Q cTt

zG p

Z= part

partint d0


(30)










(31)








M DDF T ti

n

i

n

= sdot∆=

+

=sum sum

1 1

Regine Hock 381








Regine Hock 383











































Regine Hock 385
































































Regine Hock 387



































































Regine Hock 389

































Regine Hock 391



































Regine Hock 365





(1)


(2)



MQ

L= M

w f

Q Q Q Q Q QN H L G R M+ + + + + =0

Regine Hock 366





Ice 900 2097 21

Regine Hock 367T

able

2Po

int

ener

gy-b

alan

ce s

tudi

es o

n A

lpin

e va

lley

glac

iers

Net

rad

iatio

n Q

N s

ensi

ble

heat

flux

QH

lat

ent

heat

flux

QL i

ce h

eat

flux

QG

and

the

ener

gy fo

r m

elt

QM

(her

e de

fined

as

nega

tive)

are

giv

en in

W m

2

Val

ues

in b

rack

ets

are

in

of t

otal

ene

rgy

sour

ce o

r si

nk T

he e

nerg

y ba

lanc

e do

es n

ot n

eces

saril

y ba

lanc

e if

QM

is o

btai

ned

from

abl

atio

n m

easu

rem

ents

inst

ead

of fr

om c

losi

ng t

he e

nerg

y ba

lanc

e

Loc

atio

n m

as

l

Perio

d (d

da

ys)

QN

QH

QL

QG

QM

Ref

eren

cesu

rfac

e ty

pe

Ver

nagt

fern

er 2

970

m i

ce45

d in

Aug

Se

p 19

50ndash5

314

3 (8

4)23

(14)

4 (2

)0

(0)

17

0 (

100)

Hoi

nkes

195

5K

esse

lwan

dfer

ner

20 d

in 1

958

43 (6

7)21

(33)

1

(2)

64

(98

)A

mba

ch a

nd H

oink

es 1

963

3240

m s

now

Blu

e G

laci

er 2

050

m12

7ndash

208

195

885

(63)

50 (3

7)

3 (

2)

132

(98

)L

a C

hape

lle 1

959

Ale

tsch

glac

ier

2220

m i

ce2

ndash27

819

6512

9 (7

1)38

(21)

14 (8

)

181

(10

0)R

oumlthl

isbe

rger

and

Lan

g 1

987

Ale

tsch

glac

ier

3ndash1

98

1973

44 (9

2)4

(8)

3

(6)

45

(94

)R

oumlthl

isbe

rger

and

Lan

g 1

987

3366

m s

now

Wor

thin

gton

Gla

cier

16

7ndash

18

1967

127

(51)

68 (2

9)47

(20)

22

4 (

100)

Stre

ten

and

Wen

dler

196

8A

lask

a ic

ePe

ytog

laci

er 2

510

m14

d in

Jul

y 19

7080

(44)

87 (4

8)15

(8)

18

1 (

100)

Foumlhn

197

3H

odge

s G

laci

er

111

197

3ndash4

419

7447

(54)

42 (4

6)

3 (

3)

86 (

97)

Hog

g et

al

198

2So

uth

Geo

rgia

460

mSt

Sor

lin G

laci

er 2

700

m11

d in

sum

mer

32 (5

7)24

(43)

4

(7)

53

(93

)M

artin

197

5H

inte

reis

fern

er ic

e10

d in

198

619

1 (9

0)22

(10)

4

(2)

20

9 (

98)

Gre

uell

and

Oer

lem

ans

198

7Iv

ory

Gla

cier

53

d in

Jan

ndashFeb

197

273

76 (5

2)44

(30)

23 (1

6)

14

7 (

100)

Hay

and

Fitz

harr

is 1

988

New

Zea

land

150

0m

Stor

glac

iaumlre

n 13

70m

ice

197

ndash27

81

994

73 (6

6)33

(30)

5 (5

)

3 (

3)

122

(97

)H

ock

and

Hol

mgr

en 1

996

Pate

rze

glac

ier

2205

m i

ce24

6ndash

99

1994

180

(74)

51 (2

1)11

(5)

24

2 (

100)

van

den

Bro

eke

199

7Z

ongo

Gla

cier

515

0m

9

1996

ndash81

997

17 (6

5)6

(23)

17

(65

)3

(12)

9

(35

)W

agno

n et

al

199

9ic

esn

owM

orte

rats

chgl

etsc

her

1

101

995ndash

309

199

815

2 (8

0)31

(16)

8 (4

)

191

(10

0)O

erle

man

s 2

000

2100

m i

ces

now

Kor

yto

Gla

cier

10

8ndash

89

2000

43 (3

3)59

(44)

31 (3

3)

133

(10

0)K

onya

et a

l 2

004

Kam

chat

ka

840

m s

now

Rai

n su

pplie

d 4

W m

2

(2

)

O

nly

whe

n m

eltin

g oc

curr

ed






(3)



Regine Hock 368





Regine Hock 369





(4)


(5)








= +minusZ Z

sun slope

I IR P

P zc

maR

0=⎛⎝⎜

⎞⎠⎟0

2

cos cos






(6)


(7)


(8)


(9)





Vf = 2cos ( )2

V Hf = cos ( )2

D D h h hhs d d=

2


0

2

Regine Hock 370



(10)




III

Is

scc=

Regine Hock 371



(11)





= + minus0 be n kd



(12)


(13)

and Brunt (1932)

(14)



(15)


(16)


(17)



ep



c = ( ) 1 24 1 7e Ta a

c = +a b ea



Regine Hock 373



(18)




(19)




L = ( a a4

f s s4





(20)

(21)




(22)

(23)


(24)

(25)


Ck

z z z L z z

z LE

EM

E

=minus

minus[ln( ) ( )] [ln( )

( )]

2

0 0

Ck

z z z L z z

z LT

HM

H

=minus

minus[ln( ) ( )] [ln( )

( )]

2

0 0




part

Q c KzH a P H= part

part

Regine Hock 375



(26)



(27)




Ck

z z z zR

xx b= minus

2

0 0

21 5 2ln( ) ln( )

( )

Rg

TT T z z

uz

z s

zb =

minus minus( )( )02






Regine Hock 377






(28)


C z c T z zz



Regine Hock 379



(29)





Q cTt

zG p

Z= part

partint d0


(30)










(31)








M DDF T ti

n

i

n

= sdot∆=

+

=sum sum

1 1

Regine Hock 381








Regine Hock 383











































Regine Hock 385
































































Regine Hock 387



































































Regine Hock 389

































Regine Hock 391




























Regine Hock 365





(1)


(2)



MQ

L= M

w f

Q Q Q Q Q QN H L G R M+ + + + + =0

Regine Hock 366





Ice 900 2097 21

Regine Hock 367T

able

2Po

int

ener

gy-b

alan

ce s

tudi

es o

n A

lpin

e va

lley

glac

iers

Net

rad

iatio

n Q

N s

ensi

ble

heat

flux

QH

lat

ent

heat

flux

QL i

ce h

eat

flux

QG

and

the

ener

gy fo

r m

elt

QM

(her

e de

fined

as

nega

tive)

are

giv

en in

W m

2

Val

ues

in b

rack

ets

are

in

of t

otal

ene

rgy

sour

ce o

r si

nk T

he e

nerg

y ba

lanc

e do

es n

ot n

eces

saril

y ba

lanc

e if

QM

is o

btai

ned

from

abl

atio

n m

easu

rem

ents

inst

ead

of fr

om c

losi

ng t

he e

nerg

y ba

lanc

e

Loc

atio

n m

as

l

Perio

d (d

da

ys)

QN

QH

QL

QG

QM

Ref

eren

cesu

rfac

e ty

pe

Ver

nagt

fern

er 2

970

m i

ce45

d in

Aug

Se

p 19

50ndash5

314

3 (8

4)23

(14)

4 (2

)0

(0)

17

0 (

100)

Hoi

nkes

195

5K

esse

lwan

dfer

ner

20 d

in 1

958

43 (6

7)21

(33)

1

(2)

64

(98

)A

mba

ch a

nd H

oink

es 1

963

3240

m s

now

Blu

e G

laci

er 2

050

m12

7ndash

208

195

885

(63)

50 (3

7)

3 (

2)

132

(98

)L

a C

hape

lle 1

959

Ale

tsch

glac

ier

2220

m i

ce2

ndash27

819

6512

9 (7

1)38

(21)

14 (8

)

181

(10

0)R

oumlthl

isbe

rger

and

Lan

g 1

987

Ale

tsch

glac

ier

3ndash1

98

1973

44 (9

2)4

(8)

3

(6)

45

(94

)R

oumlthl

isbe

rger

and

Lan

g 1

987

3366

m s

now

Wor

thin

gton

Gla

cier

16

7ndash

18

1967

127

(51)

68 (2

9)47

(20)

22

4 (

100)

Stre

ten

and

Wen

dler

196

8A

lask

a ic

ePe

ytog

laci

er 2

510

m14

d in

Jul

y 19

7080

(44)

87 (4

8)15

(8)

18

1 (

100)

Foumlhn

197

3H

odge

s G

laci

er

111

197

3ndash4

419

7447

(54)

42 (4

6)

3 (

3)

86 (

97)

Hog

g et

al

198

2So

uth

Geo

rgia

460

mSt

Sor

lin G

laci

er 2

700

m11

d in

sum

mer

32 (5

7)24

(43)

4

(7)

53

(93

)M

artin

197

5H

inte

reis

fern

er ic

e10

d in

198

619

1 (9

0)22

(10)

4

(2)

20

9 (

98)

Gre

uell

and

Oer

lem

ans

198

7Iv

ory

Gla

cier

53

d in

Jan

ndashFeb

197

273

76 (5

2)44

(30)

23 (1

6)

14

7 (

100)

Hay

and

Fitz

harr

is 1

988

New

Zea

land

150

0m

Stor

glac

iaumlre

n 13

70m

ice

197

ndash27

81

994

73 (6

6)33

(30)

5 (5

)

3 (

3)

122

(97

)H

ock

and

Hol

mgr

en 1

996

Pate

rze

glac

ier

2205

m i

ce24

6ndash

99

1994

180

(74)

51 (2

1)11

(5)

24

2 (

100)

van

den

Bro

eke

199

7Z

ongo

Gla

cier

515

0m

9

1996

ndash81

997

17 (6

5)6

(23)

17

(65

)3

(12)

9

(35

)W

agno

n et

al

199

9ic

esn

owM

orte

rats

chgl

etsc

her

1

101

995ndash

309

199

815

2 (8

0)31

(16)

8 (4

)

191

(10

0)O

erle

man

s 2

000

2100

m i

ces

now

Kor

yto

Gla

cier

10

8ndash

89

2000

43 (3

3)59

(44)

31 (3

3)

133

(10

0)K

onya

et a

l 2

004

Kam

chat

ka

840

m s

now

Rai

n su

pplie

d 4

W m

2

(2

)

O

nly

whe

n m

eltin

g oc

curr

ed






(3)



Regine Hock 368





Regine Hock 369





(4)


(5)








= +minusZ Z

sun slope

I IR P

P zc

maR

0=⎛⎝⎜

⎞⎠⎟0

2

cos cos






(6)


(7)


(8)


(9)





Vf = 2cos ( )2

V Hf = cos ( )2

D D h h hhs d d=

2


0

2

Regine Hock 370



(10)




III

Is

scc=

Regine Hock 371



(11)





= + minus0 be n kd



(12)


(13)

and Brunt (1932)

(14)



(15)


(16)


(17)



ep



c = ( ) 1 24 1 7e Ta a

c = +a b ea



Regine Hock 373



(18)




(19)




L = ( a a4

f s s4





(20)

(21)




(22)

(23)


(24)

(25)


Ck

z z z L z z

z LE

EM

E

=minus

minus[ln( ) ( )] [ln( )

( )]

2

0 0

Ck

z z z L z z

z LT

HM

H

=minus

minus[ln( ) ( )] [ln( )

( )]

2

0 0




part

Q c KzH a P H= part

part

Regine Hock 375



(26)



(27)




Ck

z z z zR

xx b= minus

2

0 0

21 5 2ln( ) ln( )

( )

Rg

TT T z z

uz

z s

zb =

minus minus( )( )02






Regine Hock 377






(28)


C z c T z zz



Regine Hock 379



(29)





Q cTt

zG p

Z= part

partint d0


(30)










(31)








M DDF T ti

n

i

n

= sdot∆=

+

=sum sum

1 1

Regine Hock 381








Regine Hock 383











































Regine Hock 385
































































Regine Hock 387



































































Regine Hock 389

































Regine Hock 391

























(1)


(2)



MQ

L= M

w f

Q Q Q Q Q QN H L G R M+ + + + + =0

Regine Hock 366





Ice 900 2097 21

Regine Hock 367T

able

2Po

int

ener

gy-b

alan

ce s

tudi

es o

n A

lpin

e va

lley

glac

iers

Net

rad

iatio

n Q

N s

ensi

ble

heat

flux

QH

lat

ent

heat

flux

QL i

ce h

eat

flux

QG

and

the

ener

gy fo

r m

elt

QM

(her

e de

fined

as

nega

tive)

are

giv

en in

W m

2

Val

ues

in b

rack

ets

are

in

of t

otal

ene

rgy

sour

ce o

r si

nk T

he e

nerg

y ba

lanc

e do

es n

ot n

eces

saril

y ba

lanc

e if

QM

is o

btai

ned

from

abl

atio

n m

easu

rem

ents

inst

ead

of fr

om c

losi

ng t

he e

nerg

y ba

lanc

e

Loc

atio

n m

as

l

Perio

d (d

da

ys)

QN

QH

QL

QG

QM

Ref

eren

cesu

rfac

e ty

pe

Ver

nagt

fern

er 2

970

m i

ce45

d in

Aug

Se

p 19

50ndash5

314

3 (8

4)23

(14)

4 (2

)0

(0)

17

0 (

100)

Hoi

nkes

195

5K

esse

lwan

dfer

ner

20 d

in 1

958

43 (6

7)21

(33)

1

(2)

64

(98

)A

mba

ch a

nd H

oink

es 1

963

3240

m s

now

Blu

e G

laci

er 2

050

m12

7ndash

208

195

885

(63)

50 (3

7)

3 (

2)

132

(98

)L

a C

hape

lle 1

959

Ale

tsch

glac

ier

2220

m i

ce2

ndash27

819

6512

9 (7

1)38

(21)

14 (8

)

181

(10

0)R

oumlthl

isbe

rger

and

Lan

g 1

987

Ale

tsch

glac

ier

3ndash1

98

1973

44 (9

2)4

(8)

3

(6)

45

(94

)R

oumlthl

isbe

rger

and

Lan

g 1

987

3366

m s

now

Wor

thin

gton

Gla

cier

16

7ndash

18

1967

127

(51)

68 (2

9)47

(20)

22

4 (

100)

Stre

ten

and

Wen

dler

196

8A

lask

a ic

ePe

ytog

laci

er 2

510

m14

d in

Jul

y 19

7080

(44)

87 (4

8)15

(8)

18

1 (

100)

Foumlhn

197

3H

odge

s G

laci

er

111

197

3ndash4

419

7447

(54)

42 (4

6)

3 (

3)

86 (

97)

Hog

g et

al

198

2So

uth

Geo

rgia

460

mSt

Sor

lin G

laci

er 2

700

m11

d in

sum

mer

32 (5

7)24

(43)

4

(7)

53

(93

)M

artin

197

5H

inte

reis

fern

er ic

e10

d in

198

619

1 (9

0)22

(10)

4

(2)

20

9 (

98)

Gre

uell

and

Oer

lem

ans

198

7Iv

ory

Gla

cier

53

d in

Jan

ndashFeb

197

273

76 (5

2)44

(30)

23 (1

6)

14

7 (

100)

Hay

and

Fitz

harr

is 1

988

New

Zea

land

150

0m

Stor

glac

iaumlre

n 13

70m

ice

197

ndash27

81

994

73 (6

6)33

(30)

5 (5

)

3 (

3)

122

(97

)H

ock

and

Hol

mgr

en 1

996

Pate

rze

glac

ier

2205

m i

ce24

6ndash

99

1994

180

(74)

51 (2

1)11

(5)

24

2 (

100)

van

den

Bro

eke

199

7Z

ongo

Gla

cier

515

0m

9

1996

ndash81

997

17 (6

5)6

(23)

17

(65

)3

(12)

9

(35

)W

agno

n et

al

199

9ic

esn

owM

orte

rats

chgl

etsc

her

1

101

995ndash

309

199

815

2 (8

0)31

(16)

8 (4

)

191

(10

0)O

erle

man

s 2

000

2100

m i

ces

now

Kor

yto

Gla

cier

10

8ndash

89

2000

43 (3

3)59

(44)

31 (3

3)

133

(10

0)K

onya

et a

l 2

004

Kam

chat

ka

840

m s

now

Rai

n su

pplie

d 4

W m

2

(2

)

O

nly

whe

n m

eltin

g oc

curr

ed






(3)



Regine Hock 368





Regine Hock 369





(4)


(5)








= +minusZ Z

sun slope

I IR P

P zc

maR

0=⎛⎝⎜

⎞⎠⎟0

2

cos cos






(6)


(7)


(8)


(9)





Vf = 2cos ( )2

V Hf = cos ( )2

D D h h hhs d d=

2


0

2

Regine Hock 370



(10)




III

Is

scc=

Regine Hock 371



(11)





= + minus0 be n kd



(12)


(13)

and Brunt (1932)

(14)



(15)


(16)


(17)



ep



c = ( ) 1 24 1 7e Ta a

c = +a b ea



Regine Hock 373



(18)




(19)




L = ( a a4

f s s4





(20)

(21)




(22)

(23)


(24)

(25)


Ck

z z z L z z

z LE

EM

E

=minus

minus[ln( ) ( )] [ln( )

( )]

2

0 0

Ck

z z z L z z

z LT

HM

H

=minus

minus[ln( ) ( )] [ln( )

( )]

2

0 0




part

Q c KzH a P H= part

part

Regine Hock 375



(26)



(27)




Ck

z z z zR

xx b= minus

2

0 0

21 5 2ln( ) ln( )

( )

Rg

TT T z z

uz

z s

zb =

minus minus( )( )02






Regine Hock 377






(28)


C z c T z zz



Regine Hock 379



(29)





Q cTt

zG p

Z= part

partint d0


(30)










(31)








M DDF T ti

n

i

n

= sdot∆=

+

=sum sum

1 1

Regine Hock 381








Regine Hock 383











































Regine Hock 385
































































Regine Hock 387



































































Regine Hock 389

































Regine Hock 391





















Regine Hock 367T

able

2Po

int

ener

gy-b

alan

ce s

tudi

es o

n A

lpin

e va

lley

glac

iers

Net

rad

iatio

n Q

N s

ensi

ble

heat

flux

QH

lat

ent

heat

flux

QL i

ce h

eat

flux

QG

and

the

ener

gy fo

r m

elt

QM

(her

e de

fined

as

nega

tive)

are

giv

en in

W m

2

Val

ues

in b

rack

ets

are

in

of t

otal

ene

rgy

sour

ce o

r si

nk T

he e

nerg

y ba

lanc

e do

es n

ot n

eces

saril

y ba

lanc

e if

QM

is o

btai

ned

from

abl

atio

n m

easu

rem

ents

inst

ead

of fr

om c

losi

ng t

he e

nerg

y ba

lanc

e

Loc

atio

n m

as

l

Perio

d (d

da

ys)

QN

QH

QL

QG

QM

Ref

eren

cesu

rfac

e ty

pe

Ver

nagt

fern

er 2

970

m i

ce45

d in

Aug

Se

p 19

50ndash5

314

3 (8

4)23

(14)

4 (2

)0

(0)

17

0 (

100)

Hoi

nkes

195

5K

esse

lwan

dfer

ner

20 d

in 1

958

43 (6

7)21

(33)

1

(2)

64

(98

)A

mba

ch a

nd H

oink

es 1

963

3240

m s

now

Blu

e G

laci

er 2

050

m12

7ndash

208

195

885

(63)

50 (3

7)

3 (

2)

132

(98

)L

a C

hape

lle 1

959

Ale

tsch

glac

ier

2220

m i

ce2

ndash27

819

6512

9 (7

1)38

(21)

14 (8

)

181

(10

0)R

oumlthl

isbe

rger

and

Lan

g 1

987

Ale

tsch

glac

ier

3ndash1

98

1973

44 (9

2)4

(8)

3

(6)

45

(94

)R

oumlthl

isbe

rger

and

Lan

g 1

987

3366

m s

now

Wor

thin

gton

Gla

cier

16

7ndash

18

1967

127

(51)

68 (2

9)47

(20)

22

4 (

100)

Stre

ten

and

Wen

dler

196

8A

lask

a ic

ePe

ytog

laci

er 2

510

m14

d in

Jul

y 19

7080

(44)

87 (4

8)15

(8)

18

1 (

100)

Foumlhn

197

3H

odge

s G

laci

er

111

197

3ndash4

419

7447

(54)

42 (4

6)

3 (

3)

86 (

97)

Hog

g et

al

198

2So

uth

Geo

rgia

460

mSt

Sor

lin G

laci

er 2

700

m11

d in

sum

mer

32 (5

7)24

(43)

4

(7)

53

(93

)M

artin

197

5H

inte

reis

fern

er ic

e10

d in

198

619

1 (9

0)22

(10)

4

(2)

20

9 (

98)

Gre

uell

and

Oer

lem

ans

198

7Iv

ory

Gla

cier

53

d in

Jan

ndashFeb

197

273

76 (5

2)44

(30)

23 (1

6)

14

7 (

100)

Hay

and

Fitz

harr

is 1

988

New

Zea

land

150

0m

Stor

glac

iaumlre

n 13

70m

ice

197

ndash27

81

994

73 (6

6)33

(30)

5 (5

)

3 (

3)

122

(97

)H

ock

and

Hol

mgr

en 1

996

Pate

rze

glac

ier

2205

m i

ce24

6ndash

99

1994

180

(74)

51 (2

1)11

(5)

24

2 (

100)

van

den

Bro

eke

199

7Z

ongo

Gla

cier

515

0m

9

1996

ndash81

997

17 (6

5)6

(23)

17

(65

)3

(12)

9

(35

)W

agno

n et

al

199

9ic

esn

owM

orte

rats

chgl

etsc

her

1

101

995ndash

309

199

815

2 (8

0)31

(16)

8 (4

)

191

(10

0)O

erle

man

s 2

000

2100

m i

ces

now

Kor

yto

Gla

cier

10

8ndash

89

2000

43 (3

3)59

(44)

31 (3

3)

133

(10

0)K

onya

et a

l 2

004

Kam

chat

ka

840

m s

now

Rai

n su

pplie

d 4

W m

2

(2

)

O

nly

whe

n m

eltin

g oc

curr

ed






(3)



Regine Hock 368





Regine Hock 369





(4)


(5)








= +minusZ Z

sun slope

I IR P

P zc

maR

0=⎛⎝⎜

⎞⎠⎟0

2

cos cos






(6)


(7)


(8)


(9)





Vf = 2cos ( )2

V Hf = cos ( )2

D D h h hhs d d=

2


0

2

Regine Hock 370



(10)




III

Is

scc=

Regine Hock 371



(11)





= + minus0 be n kd



(12)


(13)

and Brunt (1932)

(14)



(15)


(16)


(17)



ep



c = ( ) 1 24 1 7e Ta a

c = +a b ea



Regine Hock 373



(18)




(19)




L = ( a a4

f s s4





(20)

(21)




(22)

(23)


(24)

(25)


Ck

z z z L z z

z LE

EM

E

=minus

minus[ln( ) ( )] [ln( )

( )]

2

0 0

Ck

z z z L z z

z LT

HM

H

=minus

minus[ln( ) ( )] [ln( )

( )]

2

0 0




part

Q c KzH a P H= part

part

Regine Hock 375



(26)



(27)




Ck

z z z zR

xx b= minus

2

0 0

21 5 2ln( ) ln( )

( )

Rg

TT T z z

uz

z s

zb =

minus minus( )( )02






Regine Hock 377






(28)


C z c T z zz



Regine Hock 379



(29)





Q cTt

zG p

Z= part

partint d0


(30)










(31)








M DDF T ti

n

i

n

= sdot∆=

+

=sum sum

1 1

Regine Hock 381








Regine Hock 383











































Regine Hock 385
































































Regine Hock 387



































































Regine Hock 389

































Regine Hock 391


























(3)



Regine Hock 368





Regine Hock 369





(4)


(5)








= +minusZ Z

sun slope

I IR P

P zc

maR

0=⎛⎝⎜

⎞⎠⎟0

2

cos cos






(6)


(7)


(8)


(9)





Vf = 2cos ( )2

V Hf = cos ( )2

D D h h hhs d d=

2


0

2

Regine Hock 370



(10)




III

Is

scc=

Regine Hock 371



(11)





= + minus0 be n kd



(12)


(13)

and Brunt (1932)

(14)



(15)


(16)


(17)



ep



c = ( ) 1 24 1 7e Ta a

c = +a b ea



Regine Hock 373



(18)




(19)




L = ( a a4

f s s4





(20)

(21)




(22)

(23)


(24)

(25)


Ck

z z z L z z

z LE

EM

E

=minus

minus[ln( ) ( )] [ln( )

( )]

2

0 0

Ck

z z z L z z

z LT

HM

H

=minus

minus[ln( ) ( )] [ln( )

( )]

2

0 0




part

Q c KzH a P H= part

part

Regine Hock 375



(26)



(27)




Ck

z z z zR

xx b= minus

2

0 0

21 5 2ln( ) ln( )

( )

Rg

TT T z z

uz

z s

zb =

minus minus( )( )02






Regine Hock 377






(28)


C z c T z zz



Regine Hock 379



(29)





Q cTt

zG p

Z= part

partint d0


(30)










(31)








M DDF T ti

n

i

n

= sdot∆=

+

=sum sum

1 1

Regine Hock 381








Regine Hock 383











































Regine Hock 385
































































Regine Hock 387



































































Regine Hock 389

































Regine Hock 391





















Regine Hock 369





(4)


(5)








= +minusZ Z

sun slope

I IR P

P zc

maR

0=⎛⎝⎜

⎞⎠⎟0

2

cos cos






(6)


(7)


(8)


(9)





Vf = 2cos ( )2

V Hf = cos ( )2

D D h h hhs d d=

2


0

2

Regine Hock 370



(10)




III

Is

scc=

Regine Hock 371



(11)





= + minus0 be n kd



(12)


(13)

and Brunt (1932)

(14)



(15)


(16)


(17)



ep



c = ( ) 1 24 1 7e Ta a

c = +a b ea



Regine Hock 373



(18)




(19)




L = ( a a4

f s s4





(20)

(21)




(22)

(23)


(24)

(25)


Ck

z z z L z z

z LE

EM

E

=minus

minus[ln( ) ( )] [ln( )

( )]

2

0 0

Ck

z z z L z z

z LT

HM

H

=minus

minus[ln( ) ( )] [ln( )

( )]

2

0 0




part

Q c KzH a P H= part

part

Regine Hock 375



(26)



(27)




Ck

z z z zR

xx b= minus

2

0 0

21 5 2ln( ) ln( )

( )

Rg

TT T z z

uz

z s

zb =

minus minus( )( )02






Regine Hock 377






(28)


C z c T z zz



Regine Hock 379



(29)





Q cTt

zG p

Z= part

partint d0


(30)










(31)








M DDF T ti

n

i

n

= sdot∆=

+

=sum sum

1 1

Regine Hock 381








Regine Hock 383











































Regine Hock 385
































































Regine Hock 387



































































Regine Hock 389

































Regine Hock 391

























(6)


(7)


(8)


(9)





Vf = 2cos ( )2

V Hf = cos ( )2

D D h h hhs d d=

2


0

2

Regine Hock 370



(10)




III

Is

scc=

Regine Hock 371



(11)





= + minus0 be n kd



(12)


(13)

and Brunt (1932)

(14)



(15)


(16)


(17)



ep



c = ( ) 1 24 1 7e Ta a

c = +a b ea



Regine Hock 373



(18)




(19)




L = ( a a4

f s s4





(20)

(21)




(22)

(23)


(24)

(25)


Ck

z z z L z z

z LE

EM

E

=minus

minus[ln( ) ( )] [ln( )

( )]

2

0 0

Ck

z z z L z z

z LT

HM

H

=minus

minus[ln( ) ( )] [ln( )

( )]

2

0 0




part

Q c KzH a P H= part

part

Regine Hock 375



(26)



(27)




Ck

z z z zR

xx b= minus

2

0 0

21 5 2ln( ) ln( )

( )

Rg

TT T z z

uz

z s

zb =

minus minus( )( )02






Regine Hock 377






(28)


C z c T z zz



Regine Hock 379



(29)





Q cTt

zG p

Z= part

partint d0


(30)










(31)








M DDF T ti

n

i

n

= sdot∆=

+

=sum sum

1 1

Regine Hock 381








Regine Hock 383











































Regine Hock 385
































































Regine Hock 387



































































Regine Hock 389

































Regine Hock 391























(10)




III

Is

scc=

Regine Hock 371



(11)





= + minus0 be n kd



(12)


(13)

and Brunt (1932)

(14)



(15)


(16)


(17)



ep



c = ( ) 1 24 1 7e Ta a

c = +a b ea



Regine Hock 373



(18)




(19)




L = ( a a4

f s s4





(20)

(21)




(22)

(23)


(24)

(25)


Ck

z z z L z z

z LE

EM

E

=minus

minus[ln( ) ( )] [ln( )

( )]

2

0 0

Ck

z z z L z z

z LT

HM

H

=minus

minus[ln( ) ( )] [ln( )

( )]

2

0 0




part

Q c KzH a P H= part

part

Regine Hock 375



(26)



(27)




Ck

z z z zR

xx b= minus

2

0 0

21 5 2ln( ) ln( )

( )

Rg

TT T z z

uz

z s

zb =

minus minus( )( )02






Regine Hock 377






(28)


C z c T z zz



Regine Hock 379



(29)





Q cTt

zG p

Z= part

partint d0


(30)










(31)








M DDF T ti

n

i

n

= sdot∆=

+

=sum sum

1 1

Regine Hock 381








Regine Hock 383











































Regine Hock 385
































































Regine Hock 387



































































Regine Hock 389

































Regine Hock 391























(11)





= + minus0 be n kd



(12)


(13)

and Brunt (1932)

(14)



(15)


(16)


(17)



ep



c = ( ) 1 24 1 7e Ta a

c = +a b ea



Regine Hock 373



(18)




(19)




L = ( a a4

f s s4





(20)

(21)




(22)

(23)


(24)

(25)


Ck

z z z L z z

z LE

EM

E

=minus

minus[ln( ) ( )] [ln( )

( )]

2

0 0

Ck

z z z L z z

z LT

HM

H

=minus

minus[ln( ) ( )] [ln( )

( )]

2

0 0




part

Q c KzH a P H= part

part

Regine Hock 375



(26)



(27)




Ck

z z z zR

xx b= minus

2

0 0

21 5 2ln( ) ln( )

( )

Rg

TT T z z

uz

z s

zb =

minus minus( )( )02






Regine Hock 377






(28)


C z c T z zz



Regine Hock 379



(29)





Q cTt

zG p

Z= part

partint d0


(30)










(31)








M DDF T ti

n

i

n

= sdot∆=

+

=sum sum

1 1

Regine Hock 381








Regine Hock 383











































Regine Hock 385
































































Regine Hock 387



































































Regine Hock 389

































Regine Hock 391






















(12)


(13)

and Brunt (1932)

(14)



(15)


(16)


(17)



ep



c = ( ) 1 24 1 7e Ta a

c = +a b ea



Regine Hock 373



(18)




(19)




L = ( a a4

f s s4





(20)

(21)




(22)

(23)


(24)

(25)


Ck

z z z L z z

z LE

EM

E

=minus

minus[ln( ) ( )] [ln( )

( )]

2

0 0

Ck

z z z L z z

z LT

HM

H

=minus

minus[ln( ) ( )] [ln( )

( )]

2

0 0




part

Q c KzH a P H= part

part

Regine Hock 375



(26)



(27)




Ck

z z z zR

xx b= minus

2

0 0

21 5 2ln( ) ln( )

( )

Rg

TT T z z

uz

z s

zb =

minus minus( )( )02






Regine Hock 377






(28)


C z c T z zz



Regine Hock 379



(29)





Q cTt

zG p

Z= part

partint d0


(30)










(31)








M DDF T ti

n

i

n

= sdot∆=

+

=sum sum

1 1

Regine Hock 381








Regine Hock 383











































Regine Hock 385
































































Regine Hock 387



































































Regine Hock 389

































Regine Hock 391























(18)




(19)




L = ( a a4

f s s4





(20)

(21)




(22)

(23)


(24)

(25)


Ck

z z z L z z

z LE

EM

E

=minus

minus[ln( ) ( )] [ln( )

( )]

2

0 0

Ck

z z z L z z

z LT

HM

H

=minus

minus[ln( ) ( )] [ln( )

( )]

2

0 0




part

Q c KzH a P H= part

part

Regine Hock 375



(26)



(27)




Ck

z z z zR

xx b= minus

2

0 0

21 5 2ln( ) ln( )

( )

Rg

TT T z z

uz

z s

zb =

minus minus( )( )02






Regine Hock 377






(28)


C z c T z zz



Regine Hock 379



(29)





Q cTt

zG p

Z= part

partint d0


(30)










(31)








M DDF T ti

n

i

n

= sdot∆=

+

=sum sum

1 1

Regine Hock 381








Regine Hock 383











































Regine Hock 385
































































Regine Hock 387



































































Regine Hock 389

































Regine Hock 391























(20)

(21)




(22)

(23)


(24)

(25)


Ck

z z z L z z

z LE

EM

E

=minus

minus[ln( ) ( )] [ln( )

( )]

2

0 0

Ck

z z z L z z

z LT

HM

H

=minus

minus[ln( ) ( )] [ln( )

( )]

2

0 0




part

Q c KzH a P H= part

part

Regine Hock 375



(26)



(27)




Ck

z z z zR

xx b= minus

2

0 0

21 5 2ln( ) ln( )

( )

Rg

TT T z z

uz

z s

zb =

minus minus( )( )02






Regine Hock 377






(28)


C z c T z zz



Regine Hock 379



(29)





Q cTt

zG p

Z= part

partint d0


(30)










(31)








M DDF T ti

n

i

n

= sdot∆=

+

=sum sum

1 1

Regine Hock 381








Regine Hock 383











































Regine Hock 385
































































Regine Hock 387



































































Regine Hock 389

































Regine Hock 391























(26)



(27)




Ck

z z z zR

xx b= minus

2

0 0

21 5 2ln( ) ln( )

( )

Rg

TT T z z

uz

z s

zb =

minus minus( )( )02






Regine Hock 377






(28)


C z c T z zz



Regine Hock 379



(29)





Q cTt

zG p

Z= part

partint d0


(30)










(31)








M DDF T ti

n

i

n

= sdot∆=

+

=sum sum

1 1

Regine Hock 381








Regine Hock 383











































Regine Hock 385
































































Regine Hock 387



































































Regine Hock 389

































Regine Hock 391

























Regine Hock 377






(28)


C z c T z zz



Regine Hock 379



(29)





Q cTt

zG p

Z= part

partint d0


(30)










(31)








M DDF T ti

n

i

n

= sdot∆=

+

=sum sum

1 1

Regine Hock 381








Regine Hock 383











































Regine Hock 385
































































Regine Hock 387



































































Regine Hock 389

































Regine Hock 391


























(28)


C z c T z zz



Regine Hock 379



(29)





Q cTt

zG p

Z= part

partint d0


(30)










(31)








M DDF T ti

n

i

n

= sdot∆=

+

=sum sum

1 1

Regine Hock 381








Regine Hock 383











































Regine Hock 385
































































Regine Hock 387



































































Regine Hock 389

































Regine Hock 391





















Regine Hock 379



(29)





Q cTt

zG p

Z= part

partint d0


(30)










(31)








M DDF T ti

n

i

n

= sdot∆=

+

=sum sum

1 1

Regine Hock 381








Regine Hock 383











































Regine Hock 385
































































Regine Hock 387



































































Regine Hock 389

































Regine Hock 391






















(30)










(31)








M DDF T ti

n

i

n

= sdot∆=

+

=sum sum

1 1

Regine Hock 381








Regine Hock 383











































Regine Hock 385
































































Regine Hock 387



































































Regine Hock 389

































Regine Hock 391






















(31)








M DDF T ti

n

i

n

= sdot∆=

+

=sum sum

1 1

Regine Hock 381








Regine Hock 383











































Regine Hock 385
































































Regine Hock 387



































































Regine Hock 389

































Regine Hock 391




























Regine Hock 383











































Regine Hock 385
































































Regine Hock 387



































































Regine Hock 389

































Regine Hock 391





















Regine Hock 383











































Regine Hock 385
































































Regine Hock 387



































































Regine Hock 389

































Regine Hock 391






























































Regine Hock 385
































































Regine Hock 387



































































Regine Hock 389

































Regine Hock 391
















































Regine Hock 385
































































Regine Hock 387



































































Regine Hock 389

































Regine Hock 391




















































































Regine Hock 387



































































Regine Hock 389

































Regine Hock 391


















































Regine Hock 387



































































Regine Hock 389

































Regine Hock 391























































































Regine Hock 389

































Regine Hock 391




















































Regine Hock 389

































Regine Hock 391





















































Regine Hock 391





















Regine Hock 391





















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