2.1. The degree-day calculation

The degree-day calculation is illustrated in Figure. 2. The annual temperature cycle is approximated by a cosine function

(1)

where, TMA is the long-term mean annual temperature, AMP is the amplitude of the long-term average annual temperature cycle, TD is the daily mean temperature, t is time, A = 1 year and ϕ is the phase angle.

Fig. 2.

Annual temperature variation used to calculate the number of positive degree-days per year. TD represents the long-term average daily temperature, TMA is the long-term mean annual temperature, AMP is the amplitude of the long-term average annual cycle. See text for further information.

Equation (1) gives a very good representation of the 30 year temperature normals for the climate stations near the glaciers with root-mean-square deviations between 0.4 and 0.6 Κ. The climate stations used as temperature stations are Førde, Bjørkehaug and Skåbu for Ålfotbreen, Nigardsbreen and Hellstugubreen, respectively (see Fig. 1). The input to the model is 30 year average annual temperature cycles in terms of mean monthly temperatures, from which TMA, TMA and ϕ are determined by means of least-squares fitting. In order to account for temperature deviations from the long-term annual cycle, a stochastic term has been introduced. This is important for the degree-day calculation, since random temperature fluctuations in the spring or in the fall (may be even in the winter) will cause positive temperatures, even though the long-term average temperature in these seasons is way below the freezing point. Moreover, although the daily temperature cycle contains a large deterministic component, it can also be accounted for by the stochastic term. In the model, the combined effect of these two terms on the calculation of positive degree-days is approximated by a statistic, which is normally distributed, centred on the curve given by Equation (1) and with a standard deviation σ (see Fig. 2). This approach follows that suggested by Reference Braithwaite,Braithwaite (1985).

The positive degree-days in 1 year can therefore be calculated as

(2)

where TD is given by Equation (1) and A = 1 year.In order to calculate the positive degree-days at different elevations, TD must be determined as a function of elevation. This is done by adjusting temperatures observed at the climate stations to higher elevations, using a constant lapse rate (see section 3).

The degree-day calculation by means of Equation (1) is tested using data from the climate station Bjorkehaug in Jostedalen (see Fig. 1) for the period 1964–82. During this period, temperature was recorded three times a day, viz. at 0700, 1300 and 1900 h. In addition maximum and minimum temperatures were observed, i.e. a total of five observations d⁻¹. By means of these data, positive degree-days were summed for each year and the average annual positive degree-days in the period were calculated as 2035 Κ d.

Next, mean temperatures for each day in the 19 year period (1964–82) were calculated based on the five temperature observations per day. For each calendar day of the year, the average of the resulting 19 mean daily temperatures could then be determined. These average mean daily temperatures were taken to represent the long-term annual temperature cycle. The average annual standard deviation of the observed temperature deviations from this long-term annual temperature cycle is σ = 4.4 ± 0–8 Κ and contains a contribution from the daily temperature cycle (five observations per day) as well as a contribution from random temperature deviations from the long-term average annual cycle (19 years of observation). This standard deviation, therefore, represents the standard deviation σ used in Equation (1).

In Figure. 3, positive degree-days (PDD) calculated by means of Equation (1), with TD determined by using the Bjørkehaug 1964–82 long-term annual temperature cycle, are plotted against σ. PDD = 2035 Κ d corresponds to σ = 4.6 K, in good agreement with the σ value of 4.4 ± 0.8 Κ calculated from the detailed data. This confirms that the statistical approach can be used as a good approximation for calculating positive degree-days, as previously demonstrated by Reference Braithwaite,Braithwaite (1985). Also, it is worth mentioning that the σ values found above are in very good agreement with the value of 4.5 Κ suggested by Reference Reeh,Reeh (1991) for degree-day calculations in Greenland.

Fig. 3.

Positive degree-days calculated by means of the model for the period 1964–82 at Bjørkehaug, using different standard deviations. The dotted line corresponds to the positive degree-days (2035) calculated by using daily observations in the same period. The corresponding standard deviation is 4.6 K.

2.2. Snow and ice melt

The positive degree-days as calculated by means of Equation (1) are used to melt snow and ice as follows: snow, if present, is melted. The meltwater percolates into the snow cover where it may refreeze as superimposed ice. Run-off does not begin until the cold storage of the winter snow has been eliminated. In the model this process is not followed in detail. Instead, run-off is assumed not to start until the amount of refrozen melt (superimposed ice) exceeds a certain fraction (P _max) of the winter snow accumulation. The value of P_max should be chosen considering the freezing potential of the snow layer at the end of the winter season. For the Greenland study (Reference Reeh,Reeh, 1991), a value of P _max = 0.6 appeared to be appropriate. For glaciers in southern Norway, the freezing potential of the winter-snow cover is much lower and can be neglected (Laumann, unpublished). Accordingly, in this study, P _max is set equal to zero.

If the positive degree-days exceed the number needed for melting the snow, the remaining degree-days are used to melt glacier ice, but with a different degree-day factor (see section 3).

2.3. Precipitation

In the original version of the melt model, which was designed for Greenland ice-sheet conditions, it was assumed that the total annual precipitation fell in the form of snow. This may be a reasonable assumption for the Greenland ice sheet but certainly not for the precipitation on the glaciers in Norway. Therefore, modifications to the model have had to be introduced to allow calculation of the fraction of the annual precipitation that falls as snow at different elevations.

In general, the available climatic data will be monthly mean values of temperature and precipitation measured at one or more climate stations near the glacier. The input data needed for the mass-balance calculation, however, are the annual temperature cycle and the total annual snow precipitation as a function of elevation. Temperature and precipitation are therefore adjusted to higher elevations by means of constant elevation gradients (see section 3). Experience from Norwegian glaciers shows that precipitation will fall as snow, even for a small positive temperature (TS = 1°C). Therefore, the annual snow accumulation is determined as the precipitation falling in the season when the temperature is below TS. The calculation is performed by using monthly mean values of temperature and precipitation at various elevations. To account for the error that would be introduced if the total precipitation in a month with a mean temperature slightly above or below TS was counted as either rain or snow, temperatures are assumed to be normally distributed around the mean monthly temperature with standard deviation σ _M The fraction of the total monthly precipitation that falls as snow is then determined as the fraction of the month with temperatures below TS.

This method of calculating the snow accumulation is tested by means of the above-mentioned data from Bjørkehaug. The model-calculated snow accumulation at different elevations is compared with the snow accumulation determined by means of daily observations of temperature and precipitation in the period 1964–81, (see Fig. 4). In the test, the elevation gradient of the precipitation is neglected, so that the amount of snow falling at a given elevation is only dependent on the temperature. The standard deviation σ _M is determined as the standard deviation of the daily mean temperatures from the mean monthly temperatures in the period 1964–82. This standard deviation (3.6 K) should not be mistaken for the standard deviation a used in the degree-day calculation (see section 2.1).

Fig. 4.

Snow precipitation as a function of elevation as calculated by means of the model (dashed and dotted curves) compared with calculations using daily observations (solid line) at Bjørkehaug for the period 1964–82. The elevation gradient of the precipitation is set to zero. The dotted line results when mean monthly temperatures are used for estimating the snow precipitation. The dashed line is the result when monthly mean temperatures for days with precipitation are used.

The full curve in Figure. 4 represents the snow accumulation calculated by using daily observations of temperature and precipitation, whereas the dotted curve represents the snow accumulation calculated by means of the model described above using the statistical approach. Although the dotted curve gives a reasonable estimate of the snow accumulation, it obviously overestimates the snow accumulation in the lower reaches and underestimates the snow accumulation in the upper reaches. The explanation is that, in the winter, the temperature is generally higher on a day with precipitation than it is on a “dry” day. In the summer, on the contrary, the temperature is generally lower on a day with precipitation than it is on a “dry” day. This means that, in the winter season, a lower proportion of the precipitation will fall as snow in the lower reaches of the glacier than estimated when using mean monthly temperatures, whereas the use of mean monthly temperatures gives essentially correct estimates for the upper reaches of the glacier. In the summer season, conditions are reversed: a higher proportion of the precipitation will fall as snow in the upper reaches of the glacier than estimated when using mean monthly temperatures, whereas the use of mean monthly temperatures gives essentially correct estimates for the lower reaches of the glacier.

If the calculation is based on mean monthly temperatures determined for days with precipitation only (dashed curve in Fig. 4), a very close fit to the snow-accumulation curve as determined by means of the detailed data (full curve in Fig. 4) is obtained.

Unfortunately, temperature statistics for days with precipitation are not generally available. A simple approximate way of accounting for the fact that the temperature on a day with precipitation is different from the temperature on a “dry” day would be to reduce the amplitude (AMP) of the fitted annual temperature cycle given by Equation (1) by 1–2 K. This would have the desired effect of lowering summer temperatures and increasing winter temperatures. (For the degree-day calculation, the full temperature amplitude should, of course, be used.) However, in view of the uncertainties of the other model parameters (i.e. degree-day factors, temperature-elevation gradients, precipitation-elevation gradients), mean monthly temperatures will be used in this work to calculate the fraction of the total precipitation which falls as snow.

3.1 Ålfotbreen

Of all the glaciers in Norway, Ålfotbreen is the one that is located in the most maritime environment. The precipitation in this region is very large (more than 3000 mm year⁻¹ at sea level and more than 5000 mm year⁻¹ in the coastal mountains), and the precipitation varies strongly from place to place dependent on local conditions. In this study, the precipitation at sea level near Ålfotbreen is therefore determined as the average of the observations at three stations located around the glacier, i.e. Svelgen, Grøndalen and Àlfoten (see Fig. 1). The temperature data are taken from the climate station at Førde.

The determination of the model parameters is impeded by the fact that almost identical results can be obtained with more than one choice of model parameters. Consequently, it is important to restrain as many of the parameters as possible by means of observed quantities before attempting to fit the remaining parameters.

Beside the mass-balance, the temperature at the front of Ålfotbreen has been measured in the period 1965–81. For this period, the average temperature lapse rate in the summer season June-September from Førde at an elevation of 11 m to the glacier front at about 900 m elevation was −0.73 ± 0.02 K/100 m. The summer lapse rate ranges between −0.78 K/100 m for June and –0.64 K/100 m for September. Using an average temperature lapse rate of −0.7 K/100 m, the 1961–90 average annual temperature cycle at Førde, and the observed winter balance on the glacier as input, the remaining unknown parameters are the degree-day factors for snow and ice melt. The best-fit values of these parameters are determined by comparing model-calculated — and ob-served elevation profiles of average summer—and net balances for the period 1961–90 (see Fig 5a). The best-fit values of the degree-day factors are 4.5 mm of H₂O/PDD and 6 mm of H₂O/PDD for snow and ice melt, respectively.

Fig. 5.

Calculated mass balance (dashed and solid lines) for the period 1961–90 and observed mass balance (large dots) for the period 1963–90 as a function of elevation for (a) Ålfotbreen, (b) Nigardsbreen and (c) Hellstugubreen. Input data for the calculations are the observed winter balances (dotted lines) and mean monthly temperatures at (a) Førde, (b) Bjørkehaug and (c) Skåbu (for location see Fig. 1 ). For other parameter values, see Table 1. Legend.---summer balance; — net balance;.... winter balance.

Measurements done on Ålfotbreen by Norges Vass-drags-og Elektrisitetsvesen (NVE) in 1965 gave degree-day factors of 5.3 mm of H₂O/PDD at the equilibrium line and 7.5 mm of H₂O/PDD on the glacier tongue. These values and other degree-day factors obtained by measurements on Norwegian glaciers will be discussed in section 3.4.

The last step in the determination of the model parameters is to introduce the observed precipitation at climate stations near Ålfotbreen instead of the observed winter balance and use the method described in section 2.3 to calculate the snow precipitation as a function of elevation. A good fit to the observed winter, summer and net balances at elevations above 1000 m is obtained by using a precipitation gradient of 7% increase per 100 m. The results are presented in Figure 6a. The difference between observed and calculated winter balances above 1350 m elevation and below 1000 m elevation can probably be explained as being caused by wind transport of snow from the upper to the lower reaches of the glacier. This process is not included in the model. Nevertheless, the overall fit between model results and observations is satisfactory.

Fig. 6.

Calculated mass balance (dashed and solid lines) for the period 1961–90 and observed mass balance (large dots) for the period 1963–90 as a function of elevation for (a) Ålfotbreen, (b) Nigardsbreen and (c) Hellstugubreen. Input data for the calculations are mean monthly temperatures at (a) Førde, (b) Bjørkehaug and (c) Skåbu, and the mean monthly precipitation at (a) Ålfoten II, Grøndalen and Svelgen (average), (b) Bjørkehaug and (c) Bygdin (for location see Fig. 1). For other parameter values, see Table 1. Legend.---summer balance; — net balance;.... winter balance.

3.2. Nigardsbreen

For the mass-balance modelling on Nigardsbreen, temperature and precipitation as observed at the nearby climate station Bjørkehaug are used. The temperature lapse rate in the area can be estimated from temperature observations at Bjørkehaug (324 m a.s.l.) and Steinmann (1630 m a.s.l.) for the period 1965–81. The average lapse rate for the summer months June, July and August was −0.75 ± 0.01 K/100 m ranging between −0.78 K/100 m for June and –0.72 K/100 m for August. Using the average temperature lapse rate of −0.75 K/100 m, the 1961–90 average annual temperature cycle for Bjørkehaug and the observed winter balance on Nigardsbreen as input, a good fit to the observed summer and net balance curves is obtained with degree-day factors of 4 mm of H₂O/PDD for snow melt and 5.5 mm of H₂O/PDD for ice melt (see Fig. 5b). These values are slightly lower than the degree-day factors found for Ålfotbreen. The explanation is probably that higher wind speeds and humidity in the maritime environment at Ålfotbreen result in a larger melting effect for the same temperature.

In Figure 6b, observed and model-calculated winter, summer and net balances are compared. In these calculations, the observed winter balance on the glacier is replaced by the snow accumulation determined from the observed precipitation at Bjørkehaug by means of the calculation method described in section 2.3. The good fit in Figure 6b is obtained by using a precipitation gradient of 8% increase per 100 m. The fit would have been even better if the monthly mean temperatures used to calculate the fraction of the precipitation falling as snow were replaced by monthly mean temperatures for periods with precipitation only (see discussion in section 2.3 and Figure. 4).

3.3. Hellstugubreen

For the mass-balance calculations for Hellstugubreen, Skåbu (865 m a.s.l.) is used as the temperature station and Bygdin (1055 m a.s.l.) is used as the precipitation station. Temperature-lapse rates have not been observed near Hellstugubreen, for which reason the lapse rate of −0.75K/100m found for Nigardsbreen is also used for Hellstugubreen. The result of the calculation with application of the 1961–90 average annual temperature cycle for Skåbu and the observed winter balance on Hellstugubreen is shown in Figure 5c. A good fit to the observed summer and net balances is obtained with degree-day factors of 3.5 and 5.5 mm H₂O/PDD for snow and ice melt, respectively. The degree-day factor for snow melt is slightly lower than that found for Nigardsbreen, whereas the degree-day factor for ice melt is the same as that found for Nigardsbreen.

In Figure 6c, observed and model-calculated winter, summer and net balances are compared. In these calculations, the observed winter balance on the glacier was replaced by the snow accumulation determined from the observed precipitation at Bygdin by means of the calculation method described in section 2.3. The good fit in Figure 6c is obtained by using a precipitation gradient of 7% increase per 100 m. Also, for Hellstugubreen, the fit would have been even better if the monthly mean temperatures used to calculate the fraction of the precipitation which falls as snow were replaced by monthly mean temperatures determined for periods with precipitation only (see discussion in section 2.3 and Figure. 4.)

3.4 Discussion of model parameters

An overview of the derived parameters is given in Table 1. In general, the parameters show surprisingly little variation from one glacier to another. The temperature gradient is slightly higher (less negative) for Ålfotbreen than for the other two glaciers, probably reflecting the more humid conditions near the Atlantic coast giving rise to more frequent overcast skies and more days with precipitation than in the more continental environment of the other two glaciers. The derived precipitation gradients are very similar and the small variation from 7% increase per 100 m to 8% increase per 100 m most likely reflects the influence of local topography, i.e. that precipitation at a station on the lee side of a mountain range, for example, Bjørkehaug at Nigardsbreen, is reduced compared to the regional average.

The degree-day factors for snow and ice melt decrease from west to east. The snow-melt factor ranges from 4.5 mm H₂O/PDD in the west to 3.5 mm H₂O/PDD in the east, whereas the ice-melt factor ranges from 6 mm H₂O/ PDD in the west to 5.5 mm H₂O/PDD in the east. The trends in the melt factors can very likely be explained as being due to higher wind speeds and humidity in the maritime environment in the west compared to the calmer and drier conditions in the east, causing higher melt rates for the same temperature in the west than in the east. This explanation is supported by Table 1 of Reference Roland, and Haakensen,Roland and Haakensen (1985), which shows that the fraction of the ablation which is due to convection/condensation decreases from 0.5–0.7 on glaciers in western Norway to 0.3–0.4 on glaciers farther to the east.

The derived values can be compared to degree-day factors obtained by measurements on different glaciers in Norway. As mentioned in section 3.1, measurements on Ålfotbreen by Norges Vassdrags-og Elektrisitetsvesen (NVE) in 1965 gave degree-day factors of 5.3 mm H₂O/PDD at the equilibrium line and 7.5 mm H₂O/PDD on the glacier tongue. The corresponding numbers found by Reference Pytte, and LiestølPytte and Liestel (1966) on Storsteinsfjellbreen i Nordland (68° 15′Ν, 17°54′E) are 5.6 mm H₂O/PDD and 7.5 mm H₂O/PDD, respectively. In a measuring program from 1967 to 1974 on Fillefjell (61°10′Ν, 08°07′E), Reference Furmyr, and Tollan,Furmyr and Tollan (1975) found an average degree-day factor for snow melt of 3.9 mm H₂O/PDD. Finally, Liestel (1967) calculated an average value for melting on Storbreen (61°34′N, 08°08′E) in the period 1949–65 of 5.5 mm H₂O/PDD.

It appears that the degree-day factors for snow melt derived in this study are in good agreement with the observations. However, the derived degree-day factors for ice melt are slightly lower than most of the values previously reported for Norwegian glaciers. The degree-day factors also deviate slightly from those found by observations on the ice margin in southwest Greenland (Reference Braithwaite,, Olesen, and Oerlemans,Braithwaite and Olesen, 1989). These authors found degree-day factors of 2.5 mm H₂O/PDD and 7–8 mm H₂O/PDD for snow and ice melt, respectively.

Part of the explanation for the low ice-melt factors derived in this study compared to the “observed” values could be that the latter are based on temperatures measured on, or very close to the locations where the melt rates were observed. This is in contrast to the present study in which the temperatures used for deriving melt-rate factors are determined by means of observations at meteorological stations at some distance from the glaciers combined with measured or estimated temperature-elevation gradients. A small change in the temperature gradient will, for example cause a significant change in the melt-rate factors. As an illustration, scattered observations indicate that the temperature on the glacier tongue of Nigardsbreen at the same elevation as the nearby Bjørkehaug meteorological station is 1–2 Κ lower than the temperature measured at Bjørkehaug. If this temperature difference had been taken into account, a higher melt-rate factor (about 7 mm H₂O/PDD) would have been found. The reason why the derived snow-melt factor shows a better agreement with the observations is that the derivation of this factor, to a large extent, is based on the temperature conditions at the high elevated area of Nigardsbreen, which is close to the local Steinmann temperature station. The melt-rate factors derived in this study should therefore be considered as fitted to the temperature observations at the available meteorological stations.