Introduction

In this paper we estimate the sensitivity of glacier mass balance to temperature changes using a degree-day model. The model is applied to five Swiss glaciers (Fig. 1) for which detailed mass-balance data are available. We did this work as a pilot study for a new assessment of world sea-level rise from increased melting of glaciers, following up earlier work by Reference MeierMeier (1984), Reference Oerlemans and FortuinOerlemans and Fortuin (1992) and Reference Kuhn, Warrick, Barrow and WigleyKuhn (1993). Alpine glaciers are too small to have any direct effect on world sea level, but in the present paper we describe a simple methodology and test it on some well-documented glaciers; we will apply it later to larger but less known ice masses.

Fig. 1. Locations of Swiss glaciers with long mass-balance records, together with nearby high-elevation weather stations.

The present study uses the mass balances of five Swiss glaciers that have been measured over a series of years by scientists from the Swiss Federal Institute of Technology (ETH) in Zürich, i.e. Griesgletscher (1961–95), Limmern-gletscher (1947–85), Plattalvagletscher (1947–85), Rhonegletscher (1979–83) and Silvrettagletscher (1959–92). These data refer to the direct glaciological method whereby the mass balance is measured as a function of altitude by a combination of ablation stakes and accumulation pits. Mass-balance data from two other Swiss glaciers are not used because they do not give the mass-balance vs altitude relation: Aletschgletscher where the hydrological and “index” stake methods are used, and Claridenfirn where the “index” stake method is used. Data for the five glaciers are summarized in the publications of the Permanent Service on the Fluctuations of Glaciers (Reference KasserKasser, 1967, Reference Kasser1973; Reference MüllerMüller, 1977) and the World Glacier Monitoring Service (Reference HaeberliHaeberli, 1985; Reference Haeberli and MüllerHaeberli and Müller, 1988; Reference Haeberli and HoelzleHaeberli and Hoelzle, 1993). The most recent balance–altitude data for Griesgletscher are given by Reference Funk, Morelli and StahelFunk and others (1997). The most detailed mass-balance study, including separate measurements of winter and summer balances, is for Rhonegletscher (Reference FunkFunk, 1985), while only annual balances are available for the other glaciers.

For glaciological purposes, the best climate stations are at relatively high elevations, where essentially the same climatic conditions are sampled as those at the glaciers. For present purposes, the stations at Gütsch ob Andermatt and Weissfluhjoch (Table 1; Fig. 1) are relatively close to the five glaciers, and their data are used in the present study. The data for these and other Swiss stations are given in annual publications (Annalen der Schweizerischen Meteorologischen Zentralanstalt) from the Swiss Meteorological Office in Zürich. Climate observations have also been made for more than a century at Säntis at 2500 m a.s.l. just to the northeast of the study area (Fig. 1).

The simplest, and possibly most obvious, way to link mass-balance data to climate is by correlation of mass-balance measurements with climate data. For example, Reference LiestølLiestøl (1967), Reference MartinMartin (1975), Reference BraithwaiteBraithwaite (1977), Reference Laumann, Tvede and HuttunenLaumann and Tvede (1989), Reference Chen and FunkChen and Funk (1990) and Reference Müller-Lemans, Funk, Aellen and KappenbergerMüller-Lemans and others (1995) have all calculated multiple regression models for mean specific mass-balance data in terms of summer mean temperature and annual precipitation at some suitable nearby climate station. According to this approach, the sensitivity of the mass balance to temperature change would be represented by the temperature coefficient in the regression equation. Although we prefer to calculate the mass-balance sensitivity with the degree-day model, we first calculate multiple regression models so that we can compare the two approaches.

Regression Models

The first problem of the regression approach is that it is difficult to prescribe the length of summer for temperature averages, or the choice of hydrological year for the annual precipitation. This point is illustrated by correlating the mean specific mass balance of each glacier with mean temperatures for different periods, i.e. T ₇ for July temperatures only, T _7–8 for the July–August average, T _6–8 for June–August, T _6–9 for June–September, T _5–9 for May–September and T _5–10 for May–October.

There are only 3 years of data for Rhonegletscher, so no regression models are calculated for that glacier. The highest correlations for the other four glaciers (Table 2) appear for T _6–8, suggesting that the June–August period (92 days) is the best choice for the summer period for the whole glacier. This seems a little too short, however. For example, the equilibrium-line altitude (ELA) is generally regarded as the most representative altitude on the glacier (Reference Ohmura, Kasser and FunkOhmura and others, 1992), and extrapolation of climate-station temperatures to the ELAs of the four glaciers suggests a longer melting season at this altitude: from 112 days at Griesgletscher to 147 days at Silvrettagletscher. There is also a hint in Table 2 of a secondary maximum correlation coefficient for mass balance and temperature for May–September (153 days), but this seems much too long a period to represent the melting season at the ELA.

The mean specific mass balances were also correlated with annual precipitation for different choices of hydrological year: P _8–7 for the total precipitation from August in one calendar year to July in the following year, P _9–8 for September–August, P _10–9 for October–September and P _11–10 for November–October. The correlations between mass balance and annual precipitation (Table 3) are generally weaker than those for temperature. There is a very slight maximum for correlations between mass balance and P _9–8, but other choices of hydrological year cannot be precluded.

The patterns of correlations in Tables 2 and 3 suggest the following multiple regression model:

where is the mean specific balance of the glacier, α is the intercept in the multiple regression equation and β and γ are regression coefficients for summer temperature (June–August) and annual precipitation (September–August), respectively. The intercept and regression coefficients in Equation (1) are readily determined for a matrix of mass-balance, temperature and precipitation data using a commercial personal computer data package.

Mass Balance and Degree-Day Model

The model used here is based on that developed by Reference BraithwaiteBraithwaite (1977, Reference Braithwaite1980, Reference Braithwaite1985) and used by Reference Braithwaite and ThomsenBraithwaite and Thomsen (1989). The earlier model attempted to take account of the refreezing of meltwater and rainfall in subpolar glaciers, which is assumed to be negligible for the temperate glaciers considered in this paper.

The mass balance of the glacier is characterized by observed mass-balance values at regular altitude intervals on the glacier, i.e. 100 m intervals. The observed balance at the jth altitude is b_j , and the corresponding mean specific balance for the whole glacier is defined by:

where s_j is the area of the jth altitude band and S is the total area of the glacier. The corresponding modelled balances are b_j ^* and , respectively. The modelled balance at jth altitude is given by:

where c_j ^* and a_j ^* are the annual accumulation and annual ablation at the jth altitude in the model.

The annual ablation a ^* is generally made up of a sum of ice ablation a _i ^* and snow ablation a _s ^*, and is calculated by the degree-day model (Reference Braithwaite, Olesen and OerlemansBraithwaite and Olesen, 1989):

where PDD_i and PDD_s are the annual positive degree-day sums (sum of positive air temperatures measured at screen height above the glacier) at the altitude in question for the periods of ice melt and snowmelt, respectively. The parameters k _i and k _s are the degree-day factors for ice melt and snowmelt. (To avoid a cluttered notation, the subscript j is implicit for PDD variables.)

The degree-day model has already been used to calculate the sea-level rise from the increased melting of the Greenland ice sheet (Reference Huybrechts, Letréguilly and ReehHuybrechts and others, 1991) and can be applied to other glacier areas if the appropriate k values are known. There is ample evidence from both fieldwork and modelling studies that snowmelt generally has a lower degree-day factor than ice melt, but there is otherwise no evidence of any single set of universal factors that can be applied to all situations. A summary of positive degree-day factors from the literature is given in Table 4, together with values for ice ablation on Griesgletscher (at 2510 and 2560 m a.s.l.) measured by the authors in July–August 1996 and August 1997.

The reasons for the different degree-day factors in Table 4 are not immediately obvious. The three highest values in Table 4 for Spitsbergen and Greenland reflect rather cold low-ablation situations where higher degree-day factors may be expected from energy-balance considerations (Reference BraithwaiteBraithwaite, 1995b). The greater altitude of Griesgletscher compared with the Norwegian and Greenland glaciers in Table 4 would suggest that Griesgletscher ought to have lower turbulent fluxes, and therefore a lower degree-day factor for ice (Reference BraithwaiteBraithwaite, 1995a), in agreement with, for example, Reference KasserKasser (1959), but this is contradicted by the field data.

The modelled balance is then given by:

The degree-day sum for the snowmelt period PDD_s is C_j ^*/k _s, so the degree-day sum for the ice-melt period is given by:

and

where PDD is the annual positive degree-day total equal to PDD_i + PDD_s. Equation (6a) applies to the ablation area, and Equation (6b) to the accumulation area.

Snow accumulation at any particular altitude is estimated for each month from monthly mean temperature and monthly precipitation by assuming that precipitation is split between rain and snow according to the probability that air temperatures within the month lie above or below 0°C. This probability is estimated from monthly mean temperature (Fig. 2) by assuming that temperatures within the month are distributed according to the normal distribution with standard deviation σ = 4°C (Reference BraithwaiteBraithwaite, 1985). The estimation of accumulation here implicitly assumes that (1) precipitation falls as snow if air temperature is below 0°C, and (2) the probability of precipitation occurring is independent of temperature. Neither of these assumptions is exactly correct, but they may compensate (e.g. snow often falls when surface temperatures are above the freezing point, but precipitation in the Alps also generally occurs in the colder part of a summer month).

Fig. 2. Probability of freezing vs monthly mean temperature assuming that temperatures are normally distributed within the month, with σ = 4°C.

Rain in the model is assumed to run off the glacier and not to contribute to the mass balance, and the annual snow accumulation C _j ^* in the model is found by summing the calculated accumulation for each month.

The degree-day total is calculated for each month from the monthly mean temperature according to the Reference BraithwaiteBraithwaite (1985) model which assumes that temperatures are normally distributed within the month. Annual degree-day total is then found by summing the monthly totals, and the mass balance is obtained from Equations (5), (6a) and (6b) using the appropriate degree-day factors. For example, degree-day factors of 8 and 4.5 mm d⁻¹ °C⁻¹ are used for ice and snow ablation, respectively, in Figure 3. The former is close to that found for ice in Greenland (Reference Braithwaite, Olesen and OerlemansBraithwaite and Olesen, 1989) and on Griesgletscher, while the latter is reported for seasonal snow in Switzerland (Reference ZinggZingg, 1951; Reference De QuervainDe Quervain, 1979). Whenever other values of the degree-day factor for ice were used in the present study, the same ratio of snow to ice factors, i.e. 4.5/8.0 = 0.6, was maintained.

Fig. 3. Monthly ablation for snow and ice vs monthly mean temperatures assuming that temperatures are normally distributed within the month, with σ = 4°C. Degree-day factors for ice and snow are taken to be 8.0 and 4.5 mm d⁻¹ °C⁻¹, respectively.

The monthly mean air temperatures for the accumulation and ablation calculations are estimated at each altitude by extrapolating from a suitable weather station below the glacier using a standard lapse rate (e.g. 0.006–0.008°C m⁻¹).

Model Tuning

As the annual precipitation for each glacier is not well known, precipitation is treated as a tuning variable in the model and varied to make the model fit the observed data. The time distribution of precipitation over the glacier is assumed to be the same as at the climate station, although partition between rain and snow varies according to temperature. For Griesgletscher it is sufficient to use the same precipitation for all altitudes, although the vertical lapse of temperature ensures an increase of accumulation with altitude.

At the beginning of the study, which started with Griesgletscher, the temperature lapse rate and degree-day factor for ice melt were also regarded as uncertain, and model calculations were made for a range of values. In all, 96 different runs of the model were made for three values of lapse rate (0.006–0.008°C m⁻¹), two values of degree-day factor (8 or 9 mm d⁻¹ °C⁻¹ and 16 values of precipitation (1.02.5 m w.e. a⁻¹ in intervals of 0.1 m w.e. a⁻¹). The accuracy of any model run is expressed by the error between mean specific values of modelled balance and observed balance .

For many of the model runs, there is little resemblance between the modelled and observed mass balance, but a number of runs appear to give a reasonable fit between model and data. For example, the model error in Figure 4 is small for a number of different combinations of the model variables. This shows that tuned models are not unique. In the present approach, increased precipitation can usually be offset by increasing the degree-day factor to melt more snow, or by reducing the lapse rate to give higher temperature (and ablation) at the altitude in question. Presumably, more sophisticated models like the energy-balance model of Reference Oerlemans and FortuinOerlemans and Fortuin (1992) will also suffer from this problem of non-uniqueness.

Fig. 4. Model errors and assumptions for 96 runs of the degree-day model for Griesgletscher.

From a visual examination of the balance–altitude relations for the different model runs, the results of model run 26 appear to give the best overall agreement between model and data for Griesgletscher. The parameters for this model are annual precipitation = 1.9 m w.e. a⁻¹, temperature lapse rate = 0.007°C m⁻¹ and degree-day factor for ice = 8.0 mm d⁻¹ °C⁻¹, which are all plausible values. However, it is clear from Figure 5 that the model cannot exactly reproduce the observed data, which show a distinct “kink” between about 2950 and 3150 m a.s.l. Presumably this is caused by a local precipitation anomaly. The results of model 92 are also shown in Figure 5, to illustrate the fact that some models can fit the data very well in one altitude range (near the snout in the present case) whilst deviating greatly elsewhere (in the accumulation area).

Fig. 5. Observed and modelled mass balances vs altitude for Griesgletscher.

The tuning is repeated in a similar way for the other four Swiss glaciers: Limmerngletscher, Plattalvagletscher, Rhonegletscher and Silvrettagletscher. We became confident enough to prescribe the lapse rate and degree-day factor for ice, with values of 0.007°C m⁻¹ and 8.0 mm d⁻¹ °C⁻¹, respectively, rather than treat them as parameters. For the first three glaciers it was also found necessary to vary precipitation with altitude to fit the balance–altitude models to data. For Limmerngletscher the modelling indicated particularly heavy accumulation on the snout, while Rhonegletscher needs a strong precipitation increase with altitude to reproduce its vertical mass-balance curve. In the case of Rhonegletscher, the modelled accumulation of 2.09 m w.e. a⁻¹ can be compared with the measured winter balance of 1.89 m w.e. a⁻¹ for the 4 years of record. In the nature of things, annual accumulation (whole year) must be somewhat higher than winter balance (net balance for part of the year), so the modelled and observed data are in reasonable agreement.

The modelled and observed averages for mean specific balance are given in Table 5 for the five glaciers. The periods of record are incomplete in a couple of cases because balance–altitude data could not be found for all years. In all cases, however, the model uses temperature data for the same period as covered by the mass-balance data.) The tuning process naturally involves reducing the error between model and data, but over-tuning was avoided. For example, model run 26 was judged to give the best overall agreement with observed altitude distribution of mass balance for Griesgletscher, although it did not give the lowest error for mean specific balance.

Effect of Temperature Change

Once the mass-balance model has been satisfactorily tuned, the effect of temperature change is simulated by recalculating the mass balance with the same parameters as before, but with a changed temperature. In the present paper we are only concerned with a uniform change of 1°C throughout the year. This is purely an illustration, and different temperature changes can easily be applied to different months when plausible climate-change scenarios become available for the Swiss Alps.

The effect of a 1°C temperature change on the mass balance of Griesgletscher is illustrated in Figure 6 for model run 26. It is noteworthy that the largest change, involving an increased ablation of > 1 m w.e.a⁻¹, occurs at the snout, with a progressively smaller increase with greater altitude on Griesgletscher, in agreement with Reference Oerlemans and HoogendoornOerlemans and Hoogendoorn (1989), Reference Jóhannesson, Sigurdsson, Laumann and KennettJόhannesson and others (1993) and Reference Laumann and ReehLaumann and Reeh (1993). There are probably two reasons for this: (1) variations in ablation increase with the magnitude of the ablation itself, and (2) snow ablation, with a lower degree-day factor than ice ablation, becomes more common with increasing altitude. The first point is illustrated by the ablation curves in Figure 3 where the ablation–temperature gradients increase as ablation increases.

Fig. 6. Modelled mass balance for Griesgletscher for present climate and for a +1°C temperature rise.

The overall change in the mass balance of Griesgletscher as a result of the increased temperature, i.e. the sensitivity of the mean specific mass balance to temperature change, is −0.69 m w.e. a⁻¹ °C⁻¹ for model run 26. This represents an area average of the balance changes at different altitudes. The same procedure can also be applied to the other model runs that give a reasonable fit to the data (e.g. the 11 cases in Figure 4 where the model error is less than ±0.1 m w.e. a⁻¹, and a range of sensitivities results. The sensitivity is strongly correlated with the annual precipitation (Fig. 7), in agreement with Reference Oerlemans and FortuinOerlemans and Fortuin (1992) who proposed an explicit functional relation between mass-balance sensitivity and precipitation. A more obvious explanation is that where there is high precipitation there is also high ablation and therefore greater negative sensitivity.

Fig. 7. Sensitivity vs assumed annual precipitation for Griesgletscher.

The range of sensitivities in Figure 7 casts a new light on the problem of uniqueness for tuned models. In this case, we have a number of model runs with a similarly small error, as shown in Figure 4, and it may seem unimportant which one we choose, but they do have different sensitivities.

The mass-balance sensitivities calculated for the five glaciers by the degree-day model are considerably larger than those for the regression model (Table 6). This may be partly a statistical artefact, i.e. regression models “smooth” the processes that they purport to describe, but it may also be because the degree-day model is physically more realistic. For example, in the degree-day model, higher temperature will explicitly reduce the model accumulation (by reducing the proportion of precipitation falling as snow) even if the precipitation remains the same, while this effect can only be implicit, at best, in the temperature coefficient in the multiple regression model. Also, the regression model cannot distinguish between ice-and snowmelting (Reference Vincent and VallonVincent and Vallon, 1997).

More fundamentally, the regression model explicitly assumes a linear relation between mean specific balance and temperature, while the degree-day model implies no such relation. The mass-balance sensitivity quoted here refers to the mass-balance change for the first increase of 1°C because this is the convention in discussions of glaciers and sea level (e.g. Reference MeierMeier, 1984; Reference Oerlemans and FortuinOerlemans and Fortuin, 1992; Reference Kuhn, Warrick, Barrow and WigleyKuhn, 1993), although most workers agree that the relation between mass balance and mean temperature must be somewhat non-linear.

Discussion

The temperature-change experiments (above) were carried out holding precipitation constant, but it was found that model accumulation fell by 5–8% with a temperature rise of 1°C. This reflects the reduced proportion of snowfall at higher temperatures in the model. As future climate changes could involve changes in precipitation as well as temperature, a further experiment was made for Griesgletscher (model run 26) where precipitation was increased or decreased by 20% to compare with the effect of raising or lowering temperatures by 1°C. The results (Table 7) show that increased precipitation would partly offset the effect of higher temperatures, but even a 20% increase in precipitation cannot compensate for the increased ablation due to a 1°C temperature rise. By contrast, decreased precipitation would substantially reinforce the effects of increased temperature.

A small experiment was performed by rerunning the model for Griesgletscher (model run 26) and varying the ratio of degree-day factors k _s/k _i from the standard value of 0.6 for this study to a value of 1.0. The result is generally to increase ablation by > 0.5 m w.e. a⁻¹ at the glacier snout, and to straighten the balance-altitude curve somewhat. Insofar as it would be more difficult to fit these straighter curves to the observed data, the use of differing degree-day factors for ice and snow in this and other studies (Reference Huybrechts, Letréguilly and ReehHuybrechts and others, 1991; Reference Jóhannesson, Sigurdsson, Laumann and KennettJόhannesson and others, 1993; Reference Laumann and ReehLaumann and Reeh, 1993; Reference Vincent and VallonVincent and Vallon, 1997) seems justified. This means that the degree-day approach is more realistic than the power-law model of Reference Krenke and KhodakovKrenke and Khodakov (1966), recently resurrected by Reference Davidovich and AnanichevaDavidovich and Ananicheva (1996) and Reference PfefferPfeffer and others (1997), because it distinguishes between ice and snow. The latter model is obviously based upon a very extensive dataset from the former Soviet Union, and we urge our Russian colleagues to re-analyze their data in terms of the differing ablation properties of ice and snow.

The relation between temperature sensitivity and altitude is shown for all five glaciers in Figure 8, supporting the earlier finding for Griesgletscher. It is interesting that the five glaciers have roughly similar sensitivities for the same altitudes. The strong relation between sensitivity and altitude suggests that climate-change experiments should be performed for individual altitude bands rather than for whole glaciers. For example, as the tongue of a glacier, with the highest sensitivity, is melted away, the average sensitivity of the remaining part of the glacier must be reduced.

Fig. 8. Temperature sensitivity vs altitude for five Swiss glaciers.

The sensitivities found for the five glaciers, i.e. −0.7 to −0.9 m w.e. a⁻¹ °C⁻¹ in round figures, probably represent an intermediate range in global terms, with lower values for subpolar and high-altitude glaciers and higher values for very maritime glaciers. From an energy-balance study of 12 glaciers from different regions, Reference Oerlemans and FortuinOerlemans and Fortuin (1992) suggested a global sensitivity of −0.4 m w.e. a⁻¹ °C⁻¹, which would give a world sea-level rise of 0.6 mm a⁻¹ °C⁻¹. As the latter figure is only half that given in the 1990 Intergovernmental Panel on Glimate Ghange report (Reference Warrick, Oerlemans, Houghton, Jenkins and EphraumsWarrick and Oerlemans, 1990), it is important to confirm or refute this lower figure by further work.

Conclusions and Outlook

The degree-day model can be tuned to fit the mass balance of five Swiss glaciers by varying precipitation in the model and assuming a temperature lapse rate of 0.007°C m⁻¹ and a degree-day factor for melting ice of 8.0 mm d⁻¹ °C⁻¹. A 1°C temperature rise in the model gives an increased ablation of >1 m w.e.a⁻¹ at the glacier snouts, with a progressively smaller increase with greater altitude. A 20% precipitation increase could partly offset the increased ablation caused by the 1°C temperature rise but could not compensate for it.

The area-averaged sensitivities for the five Swiss glaciers are −0.7 to −0.9 m w.e. a⁻¹ °C⁻¹, probably representing an intermediate range in global terms, i.e. higher than sensitivities for subpolar glaciers and lower than for maritime glaciers.