Introduction

Under global warming, increased glacier melting may contribute to a rise in global sea level over the coming century (Reference Church and HoughtonChurch and others, 2001). Despite much effort (Reference Oerlemans and FortuinOerlemans and Fortuin, 1992; Reference Zuo and OerlemansZuo and Oerlemans, 1997; Reference Gregory and OerlemansGregory and Oerlemans, 1998; Reference Van de Wal and WildVan de Wal and Wild, 2001; Reference Braithwaite and RaperBraithwaite and Raper, 2002; de Woul and Hock, 2005; Reference Raper and BraithwaiteRaper and Braithwaite, 2006), there is still no general agreement on how to calculate changes in glacier melting for projections of future sea level. Most people would probably agree, however, that the ultimate aim is to develop a glacier melt model that can (1) receive input from a Global Climate Model (GCM), and (2) provide output to a model of glacier dynamics and/or geometry.

Mountain glaciers and ice caps are usually much smaller than GCM grid resolution, and probably the best way to treat them is to aggregate glacier areas into larger units. For example, Reference Oerlemans and FortuinOerlemans and Fortuin (1992) regressed mass-balance sensitivity (calculated for 12 glaciers with an energy-balance model) onto precipitation and applied the (non-linear) equation to 100 glacier regions of varying size, from Ellesmere Island (80 000km²) to Mount Kenya (1 km²). Reference Zuo and OerlemansZuo and Oerlemans (1997), Reference Gregory and OerlemansGregory and Oerlemans (1998) and Reference Van de Wal and WildVan de Wal and Wild (2001) apply GCM results to these same 100 regions. This involves some clumsy interpolation because of the different sizes of the regions, and there is also the question of variations within the larger regions. Reference Braithwaite and RaperBraithwaite and Raper (2002) and RReference Raper and Braithwaiteaper and Braithwaite (2006) propose a gridded approach based on G. Cogley’s estimate of glacier areas within 1˚ grid squares (GGHYDRO Release 2.2, ftp://ftp.trentu.ca/pub/gghydro), which covers all glacierized areas outside Greenland and Antarctica. We now extend this approach by applying the degree-day model to 0.5˚ gridded data within some selected regions. We choose to work at 0.5˚ because a climatology (Reference New, Hulme and JonesNew and others, 1999) is available at this scale and because we can aggregate results again to fit a coarser grid if the analysis reveals fairly small variations at the 0.5˚ gridscale.

Equilibrium-Line Altitude (ELA)

The ELA is the altitude on a glacier at which the mass balance (specific balance) is zero with annual ablation equal to annual accumulation. ELA is routinely measured in glacier mass-balance studies and is generally found to fluctuate quite widely from year to year on the same glacier.

Aside from its role in mass-balance study, the ELA is related to the annual snowline, sometimes called firn line, which has been discussed since the 18th century. Reference KurowskiKurowski (1891) suggested that the firn line is approximately equal to the arithmetic mean altitude of the glacier (1) when the glacier is in balance and (2) when mass balance is a linear function of altitude. For a glacier with an approximately symmetrical distribution of area with altitude, the mean elevation is approximately equal to the median elevation, i.e. the altitude with accumulation-area ratio AAR = 0.5. The median glacier altitude, mistakenly referred to as ‘mean’ elevation, was therefore recommended for inclusion in the World Glacier Inventory (Reference Müller, Caflisch and MüllerMüller and others, 1977), and data are now available for many thousands of glaciers in different parts of the world.

If we take a minimum of 5 years of record as roughly ‘representative’ of the ELA for the glacier in question, we have data for 144 glaciers, based on mass-balance data from Reference BraithwaiteBraithwaite (2002) and Reference Dyurgerov, Meier and ArmstrongDyurgerov and others (2002). The choice of 5 years of record is a judgment call, as a shorter series of ELA data may be unduly influenced by a single anomalous year, while restricting ourselves to longer series will greatly reduce the number of glaciers considered. Plotting these ELA values against median glacier elevation (Fig. 1) shows that the two concepts are highly correlated across a wide range of glacial conditions from the Arctic to the highlands of central Asia. The regression equation between observed ELA and median elevation in Figure 1 is very close to 1 : 1. The median elevation should therefore express the broad geographical variations in ELA that we need for our study. For detailed studies of single glaciers, or small regions, however, it may be worth working out various ‘corrections’ to apply to the median elevation. For example, Reference Braithwaite and MüllerBraithwaite and Müller (1980) found a range of AARs for steady-state ELA, with higher values for continental glaciers and lower values for maritime glaciers. However, most glaciers are ‘fat near the middle’, so a large difference in AAR only changes ELA by a few tens of metres.

Fig. 1. Correlation between observed ELA and median glacier elevation. The observed ELA refers to glaciers where ELA has been measured for at least 5 years as part of a mass-balance measurement programme.

For this study we wanted several regions that are well defined and well covered in the World Glacier Inventory (World Glacier Monitoring System, University of Zürich, Switzerland, http://nsidc.org/NOAA/wgms_inventory). The five major regions of European glacierization from Svalbard to the Caucasus were obvious candidates, and we added Axel Heiberg Island and New Zealand to represent extreme cases. We extracted data from the World Glacier Inventory for the chosen regions (Table 1). ELAs for individual glaciers were averaged for the same 0.5˚ latitude/longitude grid squares as used in the 0.5˚ climatology of Reference New, Hulme and JonesNew and others (1999). The ranges of estimated ELAs in Table 1 refer to the grid square averages of median glacier elevation.

Table 1. Summary data for the seven glacier regions extracted from the World Glacier Inventory. ELA is estimated from median glacier elevation or the average of maximum and minimum elevation

Degree-Day Model

Various forms of the degree-day model have been applied to mountain glaciers and ice caps (Reference HockHock, 2003) and to the Greenland ice sheet (Reference Marshall and KnightMarshall, 2006). The ‘physics’ of the model in terms of surface energy balance is discussed by Reference BraithwaiteBraithwaite (1981, Reference Braithwaite1995) and Reference Braithwaite and OlesenBraithwaite and Olesen (1990).

The present paper is part of the same study as Reference Braithwaite, Raper and ChutkoBraithwaite and others (2006), which is a follow-up to earlier work by Braithwaite and others (2003). In the earlier work, the degree-day model is tuned by fitting the model to the observed distribution of mass balance vs altitude. However, such data are only available for relatively few glaciers (e.g. 61 glaciers (Reference Braithwaite, Zhang and RaperBraithwaite and others, 2002)). Reference Braithwaite, Raper and ChutkoBraithwaite and others (2006) therefore relocate the model to the glacier ELA where accumulation is equal to the ablation calculated with the degree-day model. If we further assume that the ELA is approximated by the median glacier altitude (Fig. 1), we can apply the degree-day model to thousands of glaciers for which data are available.

The temperature at the ELA is estimated by extrapolating temperature from the gridded climatology of Reference New, Hulme and JonesNew and others (1999) using a vertical lapse rate. We apply an empirical ‘cooling effect’ to account for the observed temperature difference between a glacier weather station and a station on ice-free land at the same altitude (Reference BraithwaiteBraithwaite, 1980). The sum of positive temperature and the probability of freezing temperature at the ELA are calculated from monthly mean temperature assuming that temperature is normally distributed within the month (Reference BraithwaiteBraithwaite, 1985). The melting of snow and ice is calculated from the annual sum of positive temperature using different degree-day factors (DDFs) for ice and snow (Reference BraithwaiteBraithwaite, 1995), and snow accumulation at the ELA is set identical to the calculated snowmelt. More generally, snow accumulation is the annual sum of monthly precipitation multiplied by monthly probability of freezing taken from the degree-day model. We invert this relationship to calculate precipitation at the ELA from accumulation, assuming that the precipitation at the ELA follows the same annual distribution as the precipitation in the gridded climatology. Calculated in this way, the annual precipitation at the ELA (‘glacier precipitation’) is generally greater than the precipitation in the gridded climatology (‘grid precipitation’) and this probably reflects a real increase of precipitation with altitude (Reference Braithwaite, Zhang and RaperBraithwaite and others, 2002).

The above model involves several potential errors and it would be foolish to apply the degree-day model uncritically. Reference Braithwaite, Raper and ChutkoBraithwaite and others (2006) attempt to verify the model by comparing calculated accumulation with observed winter balance, although accumulation and winter balance are not identical concepts,. They found that uncertainties in vertical lapse rate (VLR) and DDF for snow need not have a large effect on calculations of annual accumulation if low, medium and high VLRs are paired with low, medium and high DDFs. For further work, we adopt the medium estimates of VLR and DDF from Reference Braithwaite, Raper and ChutkoBraithwaite and others (2006): 6.5 K km^–1 for VLR and 4mmd^–1 K^–1 for DDF. This DDF is well within the range of observed values (Reference HockHock, 2003). Reference Braithwaite, Raper and ChutkoBraithwaite and others (2006) find generally good agreement for the 180 glaciers where winter balance data are available, and the agreement improves when both are averaged for the five large regions (Arctic, North America, Iceland, Europe and former Soviet Union/Asia) from which most mass-balance data are available.

Results for the Seven Regions

The degree-day model was applied to the ELA and climate data for each 0.5˚ grid square in each of the seven regions. The main variables calculated by the model are listed in Table 2. The ELAs for the seven regions are shown in Figure 2 where the error bars refer to standard deviation. There is obviously considerable variation within each region as well as between regions. For example, several of the regions show broad geographical trends and effects of aspect that deserve further examination in a future study. Axel Heiberg Island and Svalbard both represent ‘Arctic’ regions, but ELAs are notably higher in Axel Heiberg Island. There is also a clear trend of rising ELA as one goes south from Svalbard to the Caucasus, no doubt reflecting the trend of rising temperature in the large-scale atmosphere.

Fig. 2. Estimated ELA for seven glacial regions. ELA is estimated from median glacier elevation in the World Glacier Inventory, or from maximum and minimum glacier elevation. Error bars represent standard deviation.

Table 2. Main variables calculated by degree-day model

The calculated accumulation at the ELA (Fig. 3) shows wide variations between regions as well as within regions. In addition to a medium value of DDF for snow (4mmd^–1 K^–1), parallel calculations for low and high DDF are made for a sensitivity study. Model accumulation obviously increases with assumed DDF, but this does not change the basic pattern of variability: Axel Heiberg Island has the lowest accumulation and New Zealand has the highest. As low accumulation at the ELA implies low melt, we can interpret the pattern in Figure 3 as a transition from cold-dry to warm-wet. Axel Heiberg Island is obviously drier than Svalbard, and its ELAs are consequently higher. There is surprisingly little difference in accumulation between northern Scandinavia, southern Norway, the Alps and the Caucasus, which probably explains why the ELA is so much higher for the Caucasus, with higher equivalent-altitude temperatures. The high accumulation for New Zealand glaciers no doubt accounts for the relatively low ELA in Figure 2.

Fig. 3. Accumulation at the ELA calculated by the degree-day model for seven glacial regions. Low, medium and high values of DDF for snow are 3.5, 4 and 4.5 mmd^–1 K^–1. Error bars represent standard deviation.

After the degree-day model calculates glacier precipitation, the effect of a 1 K temperature change is simulated by increasing temperature and recalculating accumulation (for the same glacier precipitation) and ablation at the same altitude as before, i.e. at what was the ELA where mass balance is now negative. Mass-balance sensitivity for two different temperature changes is considered, 1 K for the whole year and 1 K for the summer months June–August, but we only show results for the former (Fig. 4) as the two concepts are well correlated. The mass-balance sensitivity for a 10% precipitation increase is also well correlated with the other two sensitivities. As the increased melt at the former ELA is in the form of ice melt, the mass-balance sensitivity for temperature depends upon the DDF for melting ice. Aside from our preferred ‘medium’ estimate of 7mmd^–1 K^–1, we make parallel calculations for low and high DDFs for melting ice (6 and 8mmd^–1 K^–1) which are well within the range of observed values (Reference HockHock, 2003). Mass-balance sensitivity obviously increases with assumed DDF, but this does not change the basic pattern in Figure 4. The cold-dry environments (Axel Heiberg Island and Svalbard) have low mass-balance sensitivity, while New Zealand (warm-wet) has extremely high mass-balance sensitivity. Aside from New Zealand, the other results are in good agreement with previous estimates from Reference Oerlemans and FortuinOerlemans and Fortuin (1992) and Reference Braithwaite, Zhang and RaperBraithwaite and others (2002) (for 12 and 61 glaciers respectively). The high mass-balance sensitivity for New Zealand agrees with the high sensitivity found for Icelandic glaciers by de Woul and Hock (2005), so we should all adjust our ideas in that direction.

Fig. 4. Temperature sensitivity of mass balance calculated by the degree-day model for seven glacial regions. Sensitivity refers to a 1˚C temperature change throughout the whole year. Low, medium and high values of DDF for ice are 6, 7 and 8mmd^–1 K^–1. Error bars represent standard deviation.

The degree-day model has the facility to calculate balance gradients in the upper ablation and lower accumulations areas respectively, i.e. from 100m below the ELA to 100m above it. The calculations are approximate because they assume that precipitation is constant in the 200m altitude band around the ELA. Results (Fig. 5) confirm that balance gradients in the upper ablation area are significantly higher than in the lower accumulation area, with a general increase in balance gradients from cold-dry to warm-wet environments. The results agree with ranges given by Reference SchyttSchytt (1967) who ascribed higher ablation gradients to maritime glaciers and lower values to continental glaciers. A claimed difference between balance gradients in ablation and accumulation areas is the basis of the ‘balance ratio’ method of determining the ELA of former glaciers (Reference Furbish and AndrewsFurbish and Andrews, 1984), but our choice of DDFs gives a ratio (Fig. 5) somewhat less than their preferred value of 2.

Fig. 5. Balance gradients near the ELA calculated by the degree-day model for seven glacial regions. Upper ablation area is 100m altitude band below ELA, and lower accumulation area is 100m altitude above ELA. Medium DDFs for snow and ice are used, i.e. 4 and 7 mmd^–1 K^–1. Error bars represent standard deviation.

Discussion and Conclusions

The average conditions in the seven regions are summarized in Table 3. The annual temperature range is often taken as an index of maritime/continental climate (outside the tropics), and with the exception of the Caucasus the trend is for decreasing temperature range as we move down the list. The apparently anomalous temperature range for the Caucasus covers a wide variation within the region. The well-known association between summer mean temperature and annual accumulation (Reference Ohmura, Kasser and FunkOhmura and others, 1992) is obvious in Table 3, with a trend from cold-dry to warm-wet, and there is a nearly parallel increase in duration of melt season and mass-balance sensitivity. The mass-balance sensitivity is a combination of changes in annual ablation and accumulation (not shown), with both increasing down Table 3. For example, the high average mass-balance sensitivity for New Zealand (–1.87ma^–1 K^–1) is made up of sensitivities of –0.40 and –1.47ma^–1 K^–1 for accumulation and ablation respectively.

Table 3. Summary of glaciological characteristics for the seven regions as calculated by the degree-day model using medium estimates of vertical lapse rate and DDFs. Numbers refer to averages over all grid squares in each region

We have chosen to present our results as averages for the different regions (Figs 3–5) and to interpret them in terms of a transition from cold-dry to warm-wet conditions. We could have presented our results in terms of correlations with grid precipitation, but there is not enough space to give the relevant scatter diagrams in this short paper. The following correlation coefficients must suffice: annual accumulation and grid precipitation r = 0.77; mass-balance sensitivity and grid precipitation r = –0.87; balance gradient in upper ablation area and grid precipitation r = 0.87; balance gradient in lower accumulation area and grid precipitation r = 0.87. All sample sizes are 367 and all four correlations are significant at <1% probability. With such high correlations it should be possible to extrapolate glaciological conditions from the seven regions to the rest of the world (with known grid precipitation) using the appropriate regression equation as Raper and Braithwaite (2006) did for balance gradient.

There are other areas, notably the Gulf of Alaska (Reference Arendt, Echelmeyer, Harrison, Lingle and ValentineArendt and others, 2002) and Patagonia (Reference Rignot, Rivera and CasassaRignot and others, 2003), where high mass-balance sensitivity similar to New Zealand and Iceland might prevail over large areas and could contribute substantial rises in global sea level. In principle, we could extend our modelling approach to cover the entire world, but there are large areas that are still not covered by glacier inventory, especially in Canada and the USA as well as ‘local glaciers’ in Greenland and Antarctica. The glacier inventory work in both Canada and the USA probably ‘bogged down’ in the face of the overwhelming amount of detailed data that had to be included. However, if the relevant agencies were aware that we only need a single glacier elevation (e.g. median elevation) and a total area, they may be motivated to complete a ‘useful minimum’ inventory. Alternatively, we will investigate ways of applying the degree-day model to regions where glacier inventory data are lacking. We thereby hope to apply the degree-day model to all mountain glaciers and ice caps in the world.