Diagnostic model analysis of spatial mass, energy and melt

Northern Researcli Basins Water Balance (Proceedings of a workshop held at Victoria, Canada, March 2004). IAHS Publ. 290, 2004 257

Diagnostic model analysis of spatial mass, energy and melt distribution in a catchment in northeast Greenland

CARL EGEDE B0GGILD The Geological Survey of Denmark and Greenland (GEUS), 0ster Voldgade 10, DK-1350 Copenhagen K, Denmark ceb lg teeus .dk

Abstract Surface climate and resulting snow accumulation and melt distribution is examined for a catchment in northeast Greenland. From satellite observations, areas of bare ice in winter (no snow accumulation) indicate locally high wind speeds, which coincide with areas displaying high melt rates in summer. A network of climate stations was installed during the summer of 1995. The data serve as input to drive a simple spatially-distributed energy balance model. Using a simple model to drive the katabatic wind field on the glacier, a map of the snow distribution is proposed, which compares well with observations from a satellite image. It is difficult to determine whether the extent of the presented spatial variability in climate and mass balance is valid for other parts of the northeast Greenland ice sheet margin. However, this approach may improve modelling of the climate and melting distribution, because it attempts to move beyond a simple, elevation-dependent model toward a fully distributed formulation, which is necessary for describing the mass balance of a low ablation area with significant variability. K e y w o r d s ene rgy ba l ance ; G r e e n l a n d ; ka taba t i c w i n d s ; m e l t ra tes ; N o r t h - A t l a n t i c ; s n o w drift; spat ial m o d e l l i n g ; sur face o b s e r v a t i o n s

INTRODUCTION

Compared to the southwestern parts of Greenland, relatively little is known about the rest of the country. In particular, information about northeastern Greenland is limited (Williams & Ferrigno, 1995). Many basins in northeast Greenland are entirely ice covered, with the ice margin reaching the sea. The Storstrommen catchment area is one such ice filled basin with a glacier-covered area of 3068 W a n d an ice-free area along some of the ice margin of approximately 500 km 2. During 1988-2000, glaciological investigations were carried out in this region in a combined Danish and German effort (Boggild etal, 1993). The first mass-balance information from the Storstrommen glacier at 77°N, covering the period from 1989 to 1992, has documented the elevation gradient of ablation, which is highly irregular with a "noisy" decline in melt rates with altitude (Boggild et al, 1994). For example, during the mass balance year from August 1989 to August 1990, the annual melt rate roughly declined with 0.8 m per 1000 m elevation (the equilibrium line was around 1000 m a.s.l.). Along this gradient the standard deviation was 0.44 m or approximately equal to the mean annual ablation, giving a low signal-to-noise ratio. A mass balance and melt rate reconstruction was carried out using a statistically-derived altitude-dependent term, which treats deviations from the general mass balance gradient (Boggild et al, 1994). This type of

258 Carl Egede Boggild

statistical approach provides a mass balance reconstruction, but offers little insight into the general mass balance gradient of the region. Furthermore, the results obtained are only valid for the Storstrommen glacier; subsequently they cannot be extrapolated directly to other regions of the northeast Greenland (NE-GRL) ice margin. This paper tries to address this problem of spatial mass balance variability by presenting surface observations and modelling results of the mean daily distribution in the spatial climate and mass balance fields over a period of 58 days in the summer of 1995.

Water balance observations of heavily glacierized basins are not available in this part of Greenland. But, since storage is negligible and so is evaporation (very small values for observed latent fluxes—to be presented later), a water balance can be approximated from known ablation and accumulation. In an area of 3500 km", with an annual accumulation (snow and rain) of 255 mm year"1, the total precipitation is some 8.9 x 108 m 3, and the ablation is some 6.5 x 109 m 3. Notice the melting volume is almost an order of magnitude higher than accumulation volume. The excess melting is from ice flowing into the basin from the interior of the ice sheet (beyond the runoff line). In this paper the surface climate and resulting mass balance is examined. From satellite observations, areas of bare ice in winter (no snow accumulation) indicate high wind speeds locally, which coincide with areas displaying high ablation rates in summer. In addition, during field campaigns we observed very consistent katabatic wind flow down along the glacier surface slope. Based on this information, a network of climate stations was installed during the summer of 1995 in representative locations of the glacier. The climate and mass balance data serve as input to drive a simple, spatially distributed energy balance model, which will be described further below. The model treats the bulk katabatic wind-field, which is used separately to examine the snow redistribution in winter as well.

FIELD PROGRAMME AND DATA DESCRIPTION

The Storstrommen glacier basin is situated at the margin of the Greenland ice sheet at 77°45'N (Fig. 1). The gentle slope (around 1%) and the high elevation of the equilibrium line (due to the low snow accumulation rates, around 170 mm water equivalent per year) makes the glacier one of the largest single outlet glacier basins in northeast Greenland, with an ablation area of approximately 2900 km 2.

During a field campaign in the summer of 1994, four automatic weather stations (AWS) were deployed at the glacier to study the spatial ablation variations. However, given the large area and the complex spatial climate, four stations were not sufficient. In 1995 the number of AWS was increased to eight. The difficulty of remote access to NE-GRL demands that AWS equipment packages be light and compact for shipment to the field and quick to install; this instrumentation must be deployed within a two-day operation using a small four-seat helicopter. Between 30 June and 27 August 1994, all stations were in operation with only minor gaps in the time series. A detailed description of the field experiment is given in Boggild (1996).

Based on this data set only two periods covering 14 days can safely be identified as being clearly fine weather periods. The rest of the days are lumped into a data set that will here be termed as a period of "mixed weather". The treatment of the data set

Diagnostic model analysis of spatial mass, energy and melt distribution in northeast Greenland 259

-1(1 -81» -71) - 6 U -SO -411 -.111 -211 - I I I (I

Fig. 1 Storstrommen glacier basin situated in northeast Greenland.

in two separate groups is carried out to gain insight into changes during periods with predominant katabatic winds.

From observations, it is evident that there is a persistent decline in wind velocities from the equilibrium line toward sea level. Approaching the glacier from above and over the less inclined accumulation zone, the glacier winds will increase in strength toward the steeper Storstrommen glacier. But, since no air sounding data are available from the ice margin in NE-GRL, an explanation of the declining velocities would be speculative. It is worth mentioning that a decline in wind velocities nearest to the coast


in Antarctica is interpreted to be an effect of de-coupling of the glacier wind from the surface in the vicinity of the nearly horizontal ice shelf regions (Bintanja, 1998). A similar process may, at least partly, be controlling the wind field at the Storstrommen glacier, since winds terminate in the ice-covered Borgfjorden. Superimposed on the declining trend in the wind field, a local positive deviation occurs in the 35CM50 m elevation band. This elevation band represents a local slope region which, in satellite images from the winter, indicates bare glacier ice on the generally snow-covered glacier. A similar bare ice area occurs in the elevation band around 850-1000 m elevation, just above the uppermost AWS. However, due to logistical limitations, this upper slope region, as well as the area above, could not be examined with AWS.

THE MODEL

The present model is basically a boundary layer surface energy balance model, which reads:

where Qm is the energy available for melt, Qs„ is the net short-wave radiation balance, Qin the net long-wave radiation balance, Qn the sensible heat flux, Qe the latent heat flux, and Qg the conduction to the ground. Melt only occurs when Qm is positive; i.e. when fluxes are toward the ground.

Most energy balance models used in glacier ablation studies are confined to single point measurements and calculations, whereas this model attempts to cover the entire glacier surface at separate grid points in a DEM. The wind field and subsequent turbulent fluxes are assessed by simulating the katabatic wind-field when this flow is dominant. The lack of vertical atmospheric sounding or balloon data, to provide information on the vertical structure of the boundary layer, has restricted the present approach to deal with only vertically averaged properties of wind speed, temperature, and humidity for the katabatic flow. Whereas all other terms in the present model are treated on a seasonal basis, the short-wave radiation balance is calculated on an hourly basis, and geometrically corrected for exposure to the sim at each grid point. The surface albedo field is retrieved from a Landsat TM image and calibrated with surface observations. To assess the global radiation in the entire DEM, observations are extrapolated to all grid points on an hourly basis to account for the grid points' exposure to the sun. The long-wave radiation field is taken directly from measurements. Heat conduction to the glacier is negligible.

The radiation budget

For an inclined surface, the intensity of global radiation at the surface <bs is given by:

Qm=Qsn+Qla+Qh+Qe+Qt d )

O s = / / C o s ( 4 ) (2)

where is defines the angle between the solar direction vector and the vector normal to the inclined surface. 7/ is the radiation normal to a horizontal surface. When combining


the contributions from seasonal, diurnal and local topographic effects (Boggild, 1996, provides a detailed description), the global radiation reaching any arbitrarily oriented glacier surface is determined by (Kondratyev, 1969):

sin(ô)sin(.B)cos(8)sm(f)

where b is the hour angle of the sun, B is the azimuth of the sloping surface, 8 is the solar declination, t is the horizontal position of the sun, and (D is the latitude. A shortcoming of this model is that it does not account for the influence of terrain shading from the surrounding ice-free terrain. Terrain shading has proven crucial on alpine glaciers with steep slopes and a mountainous terrain surrounding the glacier (Esher-Vetter, 1980).

A Landsat TM image from 14 August 1994 (late summer with minimum snow cover) was used in the present study to distribute the albedo field. Also, water vapour reduces the radiative penetration through the atmosphere, making additional atmospheric correction necessary. With suitable ground observations, a simple and efficient way to circumvent the atmospheric bias is to perform a linear regression using at-satellite reflectance values as the independent values against measured ground reflectance values as the dependent values (Hall et al, 1990). Ground truth measurements were carried out in 1994 with a portable SE 590 spectrometer on the Storstrommen glacier. Boggild et al. (1996) have shown that late in the ablation season the albedo field of the glacier surface does not show significant inter-annual variability. After spectral processing of the image, the image was re-sampled to a 1-km2 resolution, which is used in the DEM of the glacier.

The flux of incoming long wave radiation has solely been obtained by interpolating the observed incoming radiation between station 4 and station 5; i.e. the highest and lowest stations, respectively. For the melting surface the outgoing long wave radiation was taken to be constant at 316 W m"2.

The katabatic wind field

We assume that the force of katabatic flow is the gradient force, which in turn, is controlled by both the inversion strength and the slope of the ice surface. In a two dimensional space in the horizontal plane, the Coriolis force has to be accounted for. The resulting wind vector of the katabatic layer comes from a balance between the Coriolis force Fc, the friction force Ff and the gradient force Fg of the dense katabatic layer. With the x-axis aligned along the direction of maximum slope, the following equations can be written (Takahashi et al, 1988):

cos(z's) = [cos(è)sin((p)-sin(è)cos((p)cos(/i)]sin(ô)

[cos((p)cos(è) + sin((p)sin(è)cos(/i)]cos(S)cos(?)- (3)

0 = (CdVu)/H + fv + g(A0/e)sin a

0 = (CdVu)/H-fu

(4)

(5)

Cd is the drag coefficient, V is the absolute wind speed, u and v are the x and y components of the wind, respectively. We will here treat the drag coefficient and the

262 Carl Egede Beggild

inversion depth H as a constant friction coefficient k, as suggested by Broeke (1996). Cd and H are lumped into a common friction coefficient k. Equations (4) and (5) can be solved with respect to the wind velocity "V" and with respect to the deflection angle "cos B" between maximum slope and resulting wind vector (Takahashi et al, 1988):

V = {Fg/k)cosB (6)

cosB = f2/{2Fgk)+[{f2/2Fg) 1/2

(V)

From equations (6) and (7) the direction and intensity of the katabatic flow can be calculated in an entire DEM.

Snowdrift

With the wind field solved, spatial redistribution of the snow in winter can be estimated as snowdrift size relates to wind speed. Based on Takahashi (1985), the snow drift rate Q0 at the Mizuho station in Antarctica has been estimated as a function of the absolute wind speed Vn, following the empirical power law:

Qo=dKm (8)

where d and m are constants. From the above relation, Takahashi et al, (1988) show that drift rates at other regions having a different wind speed F can be obtained by:

Q = Q0(v/v0)m (9)

Along a streamline in the wind direction, a change in inclination will influence the snowdrift rates and subsequently the accumulation pattern. This is expressed by the "divergence" of drifting snow (divQ), which is estimated by (Takahashi et al, 1988):

divQ = {Q2-Ql)/DL (10)

where DL is the horizontal distance between two points. When combining equations (9) and (10) it follows that:

divQ = Q0((vJvoy-(v2/v0T)/DL (11)

Turbulent fluxes

A survey of numerical methods for estimation of the vertical turbulent fluxes (i.e. the fluxes of sensible and latent heat) has been carried out by Boggild (1991). It was found that methods which include the Monin-Obukhov similarity theory, and methods which account for both stability and aerodynamic roughness, are most valid under most normal conditions on glacier surfaces. However, the computational demand and the need for surface roughness data cannot justify the inclusion of such turbulent flux schemes in this type of diagnostic study. Gray & Male (1981) provide a simple parameterization of the sensible heat flux based on only air temperature "Ta" and the


(12)

assuming the surface to be at the melting point (Ts = 0). The latent heat flux (Gray & Male, 1981) uses a similar bulk transfer coefficient De, valid for latent heat transfer. With a vapour pressure of 611 Pa on the melting surface, the latent heat transfer can be obtained when the vapour pressure of the air ea is known (measured):

MODEL RESULTS AND EVALUATIONS

Radiation budget

For the radiation budget, the previously described division into separate weather categories has not been earned out. Instead, hourly values of global radiation have been calculated in discrete hourly time-steps using observed incoming solar radiation. To get the full effect of terrain exposure to solar radiation, all incoming radiation is assumed to be direct radiation, since the diffuse radiation (typically some 10% of the total solar radiation) is much less terrain dependent. By taking average hourly values of the global radiation from stations 4 and 5, and using the data to mn the model in 1391, successive 1-h time-steps (the period of measurement), an incoming short-wave radiation field of the glacier surface is obtained. Figure 2(a) shows the distribution of global radiation on the Storstrommen surface, with the conditions described above. The mean seasonal global radiation ranges within 190 and 195 W m"~; i.e. it varies only some 2-3% on the glacier. This can be contrasted to similar studies on alpine glaciers, where surface exposure almost entirely controls ablation (e.g. Esher-Vetter, 1980; Funic, 1985). This is probably due to the geographical location at 76°N, where the solar elevation angle displays little diurnal variation during a period of up to two months after summer solstice. Figure 2(b) shows the absorbed global radiation with the measured altitude dependence of global radiation (from data) and the spatial distribution of surface albedo retrieved from the Landsat TM image. Perhaps the distinct increase in global radiation with altitude is compensated by a similar increase in the surface albedo, giving little overall spatial variability in the distribution of absorbed global radiation. High absorption rates of global radiation (more than 140 W m"2, see Fig. 2(b)) occur only in a zone nearest to the ice margin. But this is an artefact from the DEM gridding process. The long-wave radiation budget is calculated by assuming a 316 W m"2 emittance from the melting surface and interpolating measured incoming radiation as a function of elevation with the values provided.

Turbulent fluxes

For analyses purposes, the sub-division into different weather types is performed by assuming that during "mixed weather" conditions the wind velocity depends only on elevation, as partly suggested by the data. During "fine weather" the wind field is

(13)

264 Carl Egede Boggtid

'."lobal radiation

190 191 192 193 191 195

(c)

-»xx \ • • • --•» X

\ , - H V X •»

A .

•A v » v

Shortwave radiat ion balance

w/m 2 w/m 2

f i i i i i i — r — i

50 60 70 SO 90 100 .110 120 130 110

ri *-1

r.r

• - 20 ,00

Katabat ic windtïeld (m/s)

Sensible heatfhex ( W . non-katabatic windfield)

[ — — — - , w ^ m 2

(e-' (f)

É«flH>flH

•A: 'M

Sensible lu - . i i f lH ;c ( W . kai.ib.'wic windfield)

w/m 2

Latent heatflux

w / m 2

- 1 . 2 S - 0 . 9 2 - 0 . 5 8 - 0 . 2 5 0 . 0 8 0.42 0 .75

Fig. 2 (a) The spatial distribution of global radiation, (b) The short-wave radiation balance, (c) The wind-field, (d) The sensible heat flux under "mixed" weather, (e) The sensible heat flux under "fine" weather; (f) The latent heat flux under "fine" weather.


assumed controlled by katabatic forces. Figure 2(c) shows the calculated katabatic wind-field using an inversion strength of 5°K, as suggested by Broeke (1996). Along the ice margin some local topographic effects result in unrealistic (up-glacier) wind vectors. Again, this is attributed to the applied 1 km 2 grid size, which along the ice margin results in local artefacts.

Comparing the simulated wind field with observations, it appears that at nearly horizontal surfaces—i.e. at stations 1 and 5 (see Table 1)—the simulated wind-field provides somewhat too low velocities. However, on slopes where the turbulent heat fluxes are important—i.e. near stations 3 and 6—the velocities match observations. Using the "mixed weather" linear increase in wind velocities with altitude, the model results show a maximum of sensible heat to the ice surface, at the intermediate elevations of the glacier (see Fig. 2(d)). Spatial distribution of turbulent fluxes under "fine weather" conditions shows a complex pattern with the highest fluxes occurring at the E—SE facing slope around stake 7. Another maximum around the grounding line, which is estimated to be some 10 km behind the terminus, is associated with the relatively high air temperatures near sea level. In the turbulent flux budget the latent heat flux (Fig. 2(f)) is less than 5 W m"2 (not greater than 1 W m"2) and is therefore negligible at seasonal scale.

Table 1 1995 Climate stations at Storstremmen.

Station Location Logger type Sensors Type

Station 1 NE9206 Campbell Ventilated air temp. Vector 302 BDR 320 Anemometer

Wind direction RIS0 Aanderaa

Station 2 NE927.2 Campbell Ventilated air temp. Vector 302 BDR 320 Anemometer

Wind direction RIS0 Aanderaa

Station 3 NE9207 Campbell Air temp./ humidity Vaisala BDR 320 Anemometer

Pyranometer HMP310FA Vector A102R Kipp and Zonen CM7

Station 4 NE9211 Campbell Ventilated air temp. Vector 302 CR10 Pyranometer

Anemometer Wind direction Pyrgeometer Snow/ice depth

Kipp and Zonen CM7 Aanderaa Aanderaa Eppley PIR Campbell SR50

Station 5 NE2992 Campbell Ventilated air temp. Vector 302 CR10 Pyranometer

Anemometer Wind direction Pyrgeometer Snow/ice depth

Kipp and Zonen CM7 Aanderaa Aanderaa Eppley PIR Campbell SR50

Station 6 NE957.1 Campbell Ventilated air temp. Vector 302 CR10 Anemometer

Wind direction Pyrgeometer Snow/ice depth

Aanderaa Aanderaa Eppley PIR Campbell SR50

Station 7 NE9227 Ryan Inst. RTM Air temp. PT100 2000 Wind-way Lambrecht 1440

Station 8 NE9209 Ryan Inst. RTM Air temp. PT100 2000 Wind-way Lambrecht 1440

266 Cad Egede B0ggUd

Ablation rates

When including all individual fluxes shown in Fig. 2 in the energy balance budget (equation (1)), the average daily melt rate results as shown in Fig. 3 are obtained. The numbers in Fig. 3(a) refer to observed daily mean ablation rates at each AWS in mm day -1. Some discrepancies occur between the model results and measured ablation at the highest and lowest regions of the glacier. In these two regions, the modelled ablation is slightly overestimated; i.e. near sea level (at station 5 at 150 m a.s.l.) observation vs modelled ablation is 27-30 mm day"1, and at stake 8 (at 560 m a. s. 1.) the ratio is 15-19 mm day-1. In the intermediate elevations the model results are in agreement with observations. The diagnostic model captures the low daily ablation rates in the almost horizontal areas (i.e. around stakes 6 and 8) and the high local ablation in the slope region around stake 7. Most of this spatial variability is attributed to the variability in the sensible heat flux (Fig. 2(e)). If no slope-driven wind field was included in this diagnostic model, the measured locally high ablation rates around stake 7 would not have been reproduced. As for the single fluxes in Fig. 2, the daily ablation rates are likely to be unrealistic for the 1 to 3 grid elements nearest to the ice margin.

Snowdrift and accumulation distribution

In order to estimate the snow accumulation pattern on the Storstrommen glacier the following assumptions have been made: (a) Influx of drifting snow equals the out-flux from the glacier. Box (pers. com.) states that mass loss by drifting snow is negligible for the ice sheet, (b) Annual accumulation is assumed to be 170 mm/yr as observed in Danmarkshavn nearby (Boggild et al., 1994). (c) The wind field in winter is similar to the wind field in summer, as observed on the more northerly location at Kronprins Kristian Land (unpublished). Applying these assumptions, the diagnostic spatial field in snow distribution results in Fig. 3(b).

(a) (b)

Fig. 3 (a) Calculated daily melt rates (in mm day" 1); the numbers are observed daily. Mean ablation rates at the respective localities during the 1995 observation period, (b) Modelled distribution of accumulated snow water equivalent (swe, mm).


The model results indicate that the highest snow thickness is often found adjacent to bare snow areas; i.e. the net snow transport is over short distances. Similar results are seen in northern Alaska and are confirmed by both observations and modelling (Liston & Sturm, 1998).

Since observation of snow distribution is lacking by the end of the winter, the snow distribution observed by satellite becomes an important source for verification. The diagnostic wind field shows a complex pattern, which is only partly confirmed by observations. Along the eastern flank of the glacier margin the model produces a field lacking snow accumulation, which observations do not confirm. This artefact is most likely due to the highly convex surface in this region. But along the centre line of the glacier, i.e. the AWS profile where the influence from the adjacent land is least dominant, the model results resemble observations fairly well. The observed area of bare ice around the local slope section at stake 7 is likewise reproduced by the model.

CONCLUSIONS AND DISCUSSION

In this paper, emphasis has been on diagnostic analysis of the spatial mass and energy balance distribution. Such an effort is needed since generally little is known about the mass balance and melt conditions at this part of the ice sheet margin. Furthermore, the most intensively studied glacier Storstrommen displays a highly "noisy" mass balance profile.

The climate observations behind the model study show two striking features; the first is a persistent decline in observed wind velocities down along the profile of observation to sea level. Since radiosonde data is not available, an explanation would be speculative. However, in Antarctica a similar decrease occurs in the vicinity of the ice shelves as an effect of de-coupling of the glacier wind from the surface (Bintanja, 1998). The second striking feature is the observed lapse rate of only 0.16°C 100 m"1, which is well below an adiabatic lapse rate. It is therefore likely that the air masses moving down the glacier are cooled significantly by the surface.

The short-wave radiation budget has been shown to control the spatial ablation distribution on alpine glaciers (e.g. Escher-Vetter, 1980). But, in this region, the combination of low relief of the adjacent land and the gentle slope of the ablation zone all favour little terrain effect. And, with a geographical location on 76°N and the short ablation season around summer-solstice, the relative importance of the incoming shortwave radiation on the distributed energy balance budget is nearly negligible. This makes the wind field and sensible heat flux distribution relatively most important for explaining the observed spatial variability in ablation and mass balance.

The model has proved capable of reproducing the somewhat higher melt rates on the slope in the vicinity of stake 7, as well as the lower melt rates at the adjacent and nearly horizontal areas above and below this slope. The local "noise" as well as the general ablation gradient can be reproduced with the present spatially distributed energy balance model. But some local discrepancies from observations are evident near sea level and near the equilibrium line, as well as closest to the ice margin, although the latter can probably be attributed solely to boundary effects in the DEM.

The increased ablation at the local slope around stake 7 gives confidence in addressing the katabatic (slope dependent) wind field as the primary controlling factor,


because, when neglecting the slope component (Fig. 2(d)), the sensible heat flux shows less variability in the vicinity of stake 7. Moreover, snow is removed by wind from this slope in the winter.

It is difficult to determine to what extent the presented spatial variability in climate and mass balance is valid for other parts of the NE-GRL ice sheet margin. However, the approach offers possibilities for linking surface observation of mass balance and climate, which have traditionally been point observations, to the large scale global circulation models, thus bringing mass balance observations into a larger climate perspective.

Acknowledgements This paper has been published with the permission of GEUS. I would like to express my gratitude to Dr Hans Oerter and to Niels Reeh for giving me precious helicopter time to install and operate the climate stations within the combined AWI and GGU field campaign at the Storstrommen glacier. Dr F. Obleitner and Dr I. Vergainer provided valuable comments on an early version of this paper. The present work has been supported by the Danish Research Councils and by the Commission for Scientific Research in Greenland.

REFERENCES

Bintanja, R. (1998) The contribution of snow drift subl imation to the surface mass balance of Antarctica. Ann. Glaciol. 27 , 2 5 1 - 2 5 9 .

Boggi ld , C. E. (1991) En smeltende snepakkes masse- og energfluxe—belyst med beregnlngsmetoder (Mass- and energy fluxes of a mel t ing snow pack—determined with numerical methods) . M S c Thesis , Universi ty of Copenhagen , Denmark.

Boggild, C. E. (1996) Cl imate and mel t ing of ice sheet margins—assessed with observation and model ing. P h D Thesis , Universi ty of Copenhagen , Denmark .

Boggi ld , C. E., Reeh, N . & Oerter, H. (1993) Ablat ion reconstruction and mass-balance sensitivity to cl imate change-assessed by mode l ing on Stors t rommen, north-east Greenland (ed. by N . Reeh & H. Oerter) Open File Series Gronlands Geologiske Undersogelse 93/5, 6 0 - 6 3 .

Boggild, C. E., Reeh, N . & Oerter, H. (1994) Model ing ablation and mass-balance sensitivity to cl imate change of Stors t rommen, north-east Greenland. Global and Planetary Change 9 , 7 9 - 9 0 .

Boggild, C. E., Oerter, H. & Tukiainen, T. (1996) Increased ablation of Wiscons in ice in eastern north Greenland: Observat ions and model ing. Ann. Glaciol. 2 3 , 1 4 4 - 1 4 8 .

Broeke , M . van den. (1996) The atmospheric boundary layer over ice sheets and glaciers. P h D Thesis , University of Utrecht, The Nether lands .

Esher-Vetter , H. (1980) Der stralungshaushalt des Vernagtferners als basis der energihaushaltberechnung zur bestimmung der schmellzwasserproduktion elnes Alpenglelschers. Wissenshaftl iche Mittei lung, no 39 . Munchen , Germany.

Funk, M. (1985) Raumliche Verteilung der Massenbilanz auf dem Rhonegletscher und Ihre Bezelhung zu Klimaelementen. Zurcher Geographsiche Schriften, no. 24. Zurich, Germany.

Gray, D. M . & D. H. Ma le (1981) Handbook of Snow: Principles, Processes, Management and Use. Pergamon Press Canada.

Hall , D. K., Bindschadler , R. A., Forster, J. L., Chang, A. T. C. & Siddalingaiah, H. (1990) Compar i son of m situ satellite derived reflectances of Forbindels Glacier, Greenland. Int. J. Remote Sens. 11, 4 9 3 - 5 0 4 .

Kondratyev, K. Y. (1969) Radiation of the Atmosphere. Academic Press, N e w York.

Liston, G., & Sturm, M . (1998) A snow-transport model for complex terrain. J. Glaciol. 44 (148), 4 9 8 - 5 1 6 .

Takahashi , S. (1985) Characteristics of drifting snow at Mizuho Station, Antarct ica. Ann. Glaciol. 6, 7 1 - 7 5 . Takahashi , S., Naruse , R., N a k a w o , M . & M a e , S. (1988) A bare ice field in East Queen Maud Land, Antarctica, caused by

horizontal divergence of drifting snow. Ann. Glaciol. 1 1 , 1 5 6 - 1 6 0 .

Wll iams, R. S. Jr., & Ferrigno, J. G. (1995) Satellite Image At las of the World; Greenland. US Geol. Survey Prof. Paper I386-C.

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Diagnostic model analysis of spatial mass, energy and melt