7.1. Model Sensitivity to Variations in z0v and the Stability Correction
 The turbulent fluxes play a significant role in the surface energy balance, but also have a large uncertainty, as they need to be calculated from other quantities. In literature, many different methods have been employed to calculate the turbulent fluxes, mainly depending on the measurements available. As stable stratification is common over melting glaciers, turbulence is expected to be reduced and stability corrections should be applied. However, Konzelmann and Braithwaite  and Hock and Holmgren  found that the measured surface melt was underestimated when stability corrections were included in the calculation of the turbulent fluxes. For this and other reasons, stability corrections have been neglected in several energy balance studies on glaciers [Hock and Holmgren, 1996; Oerlemans and Klok, 2002; Konya et al., 2004]. Another uncertainty is associated with the specification of roughness lengths, as these are both temporally and spatially highly variable [Brock et al., 2006]. Often, the available measurements are not sufficient to determine estimates of the roughness lengths. In several studies, values for the roughness lengths are therefore taken from published studies [e.g., Klok et al., 2005; Arnold et al., 2006] or used as tuning parameters to match measured melt or sublimation [Wagnon et al., 1999; Oerlemans and Klok, 2002; Hock and Holmgren, 2005]. Previous studies showed that when the roughness lengths for momentum, heat and moisture are assumed to be the same, an order-of-magnitude increase in these roughness lengths leads to a doubling of the turbulent fluxes [Munro, 1989; Braithwaite, 1995]. Similarly, in a study where only z0v was increased by an order of magnitude, a 25% error in the turbulent fluxes was reported [Denby and Greuell, 2000]. Munro  noticed that this error becomes smaller when z0T is not prescribed, but calculated following Andreas . In Andreas' model, an increase in z0v results in a decrease in z0T, which partly compensates for the change in z0v. Klok et al.  applied this model and found an increase in the turbulent fluxes of 19% for a tenfold increase in z0v, while omitting the stability functions lead to a much larger increase (43%). Our energy balance model is similar to their model, the main difference is that in our model Ts is calculated and not taken from measurements. In line with these previous studies, we determined the sensitivity of our energy balance model to the value of z0v and the stability correction applied.
 The values for z0v for ice and snow were varied separately by two orders of magnitude within the ranges most commonly observed on glaciers [Brock et al., 2006]. The roughness length is expected to fluctuate within this range. Hence the runs with very high and low values of z0v give extreme values that will not likely be attained. We determined the effect of a varying z0v by using one single value for z0v for snow and ice surfaces, so z0v,i = z0v,s = z0v. We report the sensitivity with respect to the reference value z0v = 1 mm instead of values used in this study, to provide more general results for comparison with other models. The effect of a separate roughness length for snow was investigated by keeping z0v,i constant at the reference value while z0v,s was varied. The entire measurement period was included in the calculations; results are listed in Table 5. Model results for the z0v values used in this study are also reported to allow for a comparison with the reference value.
Table 5. Sensitivity of the Mean Turbulent Fluxes Δ (Hsen + Hlat), Total Surface Melt ΔM and Modeled Surface Temperature (Expressed as the Mean Deviation ΔTs From the Observed Surface Temperature) to the Values of z0v,i, z0v,s and Stability Correctionsa
|z0v (mm)||Δ (Hsen+ Hlat) (%)||ΔM (%)||ΔTs (°C)|
|No Stability Correction|
|Full Stability Correction|
 Taking z0v = 1 mm as the reference case, we see that an increase (decrease) in z0v of one order of magnitude results in a 11% increase (16% decrease) in the mean turbulent fluxes and an increased (decreased) ice melt of 5% (7%). Using a separate value for z0v over snow has little effect on the surface melt, since the majority of the melt occurs when the surface is ice. Variation of z0v has more impact on the turbulent fluxes for ice surfaces, while over snow the variation in ΔTs is larger between the different runs. These results can be explained by the model design. Given a value for z0v, the model determines in every time-step the set of values for Ts, z0T, z0q and LMO which, by the associated values for Lout, G and the turbulent fluxes, results in a closure of the energy balance. When the optimal value for Ts is higher than the melting point temperature, Ts is set back to 0°C and the excess energy is used for melting. For a melting surface, Lout and G have determined values and only the turbulent fluxes and the melt energy are variable. The ice surface is often at the melting point temperature while the temperature of the snow surface can generally vary over a wider range. Hence changing z0v for an ice surface mainly affects the turbulent fluxes and the melt energy, for a snow surface Lout and G will also change and the relative change in the turbulent fluxes and the melt energy will be smaller. The model used here is less sensitive to the choice of z0v than models used in several other studies [Braithwaite, 1995; Hock and Holmgren, 1996; Brock et al., 2006], because a change in z0v does not directly lead to a change in turbulent fluxes and melt, but to a different solution of the energy balance which generally damps the effect. By solving the energy balance, the effect of small errors in measurements and model parameters is reduced and the time evolution of the surface energy fluxes is consistent. However, when the errors are larger because of measurement problems or an incorrect representation of the surface energy balance, the model may still be able to obtain values for the energy fluxes, but the residual in the energy balance will be larger and ΔTs will increase.
 In this study we use separate, but constant values for z0v,i and z0v,s. In reality, these values will vary on hourly to seasonal time scales. Assuming that values for z0v fluctuate within the ranges presented in Table 5, the errors in individual values of the turbulent fluxes are estimated to be 20%. The numbers reported in Tables 3 and 4 represent melting conditions, when the sensitivity of the turbulent fluxes to z0v is largest. Recalculating the numbers in Table 3 using z0v = z0v = 5 mm, which we regard as an upper limit for z0v on Midtdalsbreen, the mean value of the turbulent fluxes increases by 9 W m−2 (18%). A 9 W m−2 increase in the turbulent fluxes is also found for clear-sky and overcast conditions separately, with the difference that Hsen is mainly affected for clear skies (+8 W m−2), while for overcast skies the increase in Hsen is only 2 W m−2 larger than the increase in Hlat. The larger turbulent fluxes result in a 4% increase in their contribution to melt (Table 4), a similar decrease in the contribution by Snet and less than 1% change in the other fluxes. These numbers also apply to clear-sky and overcast conditions separately, although for clear skies the contribution by Hsen changes by 4%, for overcast skies the contributions by Hsen and Hlat both change by 2%.
 Furthermore, we investigated the effect of the stability correction by running the model without applying a stability correction and with the full (not limited) stability correction. The results reveal that neglecting atmospheric stability corrections or using the full stability correction for stable cases, has a larger impact on the turbulent fluxes and surface melt than an order-of-magnitude change in the value of z0v (Table 5). This is contrary to the findings of Braithwaite  who found a larger effect for changes is z0v. In that study, equal values were chosen for the roughness lengths of momentum and heat, which explains the large sensitivity to a change in z0v. The values found here for ice surfaces are of the same magnitude as the values found by Klok et al. , which is not surprising considering the similarity of the models used. Omitting the stability corrections does not significantly deteriorate the model performance, although the overall value for ΔTs is larger. On the other hand, using the full stability correction leads to a systematic underestimation of Ts and a larger RMSE (1.5°C, not shown).
 From these results, it is difficult to say whether applying a stability correction is necessary. If the correction is omitted, the amount of ice melt increases by 10%, lessening the agreement with the melt measured at the ablation stakes. Omitting the stability correction for overcast conditions induces changes in the turbulent fluxes during melt comparable to changing z0v to 5 mm. For clear-sky conditions the effect on the turbulent fluxes is larger (+17 W m−2), as the air is generally stably stratified under clear skies. Applying the full stability correction hardly changes the fluxes under overcast skies, for clear-sky conditions the turbulent fluxes decrease by 14 W m−2. For clear-skies, the contribution of the turbulent fluxes to melt (Table 4) increases (decreases) by 8% (7%) when the stability correction is omitted (not limited). For overcast skies, the respective contributions only change by +4% and −2%. Interestingly, changing z0v or the stability correction significantly affects the values reported for the turbulent fluxes in Tables 3 and 4, but the absolute value of Lin (and hence Lnet) does not change by more than 1 W m−2 and the contribution to melt varies less than 1% for all cloud conditions.
 The results indicate that one should be careful with stability corrections when the turbulent fluxes are calculated from measurements made at a level above the wind speed maximum. We can conclude that when using this type of model, the errors introduced by the uncertainty in the turbulent fluxes are of the same order of magnitude as uncertainties in the radiation measurements and other model parameters, for example the value assumed for the ice density. Therefore, when attempting to further improve the model, all these factors need to be addressed.
7.2. Interannual Variability in the Total Melt
 The interannual variability in the total melt is a combined result of changes in surface properties, which are largest at the moment the snow cover disappears, and variations in meteorological conditions. To estimate the separate effects of these two processes on the total annual melt, we performed additional model runs with prescribed albedo and snow depth series. When snow is present, the snow depth is set to 1 m. On a given date, snow depth is set to zero and the albedo is switched from snow to ice albedo. Hence the moment of ice appearance is controlled and not determined by the meteorological conditions. To remove the effect of small variations in albedo that could complicate the comparison, we used two constant albedo values representing snow and ice conditions. Together with measured Sin, these values prescribe Snet in the model. Albedo values of 0.70 for melting snow and 0.31 for ice produce annual melt values less than 0.2 m w.e. larger than the melt calculated with the measured albedo series, except in 2002 where the melt is overestimated by 0.4 m w.e. The differences originate when a constant albedo is not a good approximation, for example during summer snowfall events. Nevertheless, the resemblance is good enough for this analysis.
 The model was run with five different albedo series: always ice (albedo = 0.31), always snow (albedo = 0.70), snow disappears on 11 June as in 2002 (the earliest date), snow disappears on 18 July as in 2005 (the latest date) and snow disappears on the observed date. The modeled melt for these five runs is shown in Figure 10. Melt is defined as the amount of water produced at the surface, still including the water that later possibly refreezes in the snowpack. For the hypothetical all-year ice surface, total melt is almost twice as large as for an all-year snow cover. This result demonstrates the dominating influence of the albedo on melt. In May and June, melt is most sensitive to albedo changes, in both months melt increases by approximately 0.8 m w.e. when the surface is ice instead of snow (not shown). Varying the date of snow disappearance by 37 days according to the spread in the observations induces a mean change in melt of 0.9 m w.e. with little variability between the years. The four simulations with identical albedo series for all years demonstrate the influence of the meteorological conditions during the melt season on interannual variability, amounting to a maximum difference in melt between 2001 and 2002 of 0.8 m w.e. We see that even when the albedo effect of the thin snowpack is removed, 2002 still has the largest amount of melt, a result of the high summer temperatures (Figure 3). The simulations also show that the melt in 2002 and 2005 would have been comparable when the snow cover had disappeared around the same time, provided that the meteorological conditions for the data gap in 2005 have been simulated adequately. Even with the albedo effect removed, 2001 remains the year with the minimum total melt. We can conclude that for these five years, the interannual variability in total melt induced by meteorological conditions during the melt season is of the same magnitude as the albedo effect of an early or late disappearance of the snowpack. Under identical meteorological conditions, the date of ice appearance depends solely on the total winter accumulation. The experiment shows that variations in winter mass balance at the AWS location not only affect the net mass balance directly, but also have a significant effect on the total melt, and hence net mass balance, via a positive albedo feedback mechanism. Unfortunately, this finding cannot simply be generalized to other years or other locations, which may have different variability in snow accumulation and meteorological conditions.
Figure 10. Modelled annual melt for five albedo series, where the snow albedo is 0.70 and the ice albedo 0.31. The labels refer to runs with always ice, always snow, snow disappearance on 11 June, snow disappearance on 18 July, and snow disappearance on the observed date.
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7.3. Comparison With Morteratschgletscher
 In this section, we compare the seasonal cycles of meteorological variables and surface energy fluxes on Midtdalsbreen with measurements from an IMAU–AWS (2100 m a.s.l.) on Morteratschgletscher, Switzerland [Oerlemans and Klok, 2002]. This comparison of two isolated records from two different regions should not be regarded as a comparison of the climates in the European Alps and Norway. Irrespective of the question of representability, the two glaciers differ in many aspects: Morteratschgletscher is situated at a lower latitude, in a drier environment and is surrounded by higher mountains than Midtdalsbreen. The comparison of two long AWS records from these locations gives insight in differences in meteorological variables and the surface energy balance originating from the characteristics of the glaciers and their environment.
 The two AWSs have a similar design, except that on Morteratschgletscher the measurement level is at 3.5 m above the ice surface. This data set spans the period 8 July 1998 to 14 May 2007. We selected a subset of the data corresponding to the period used for Midtdalsbreen (2 October 2000 to 7 September 2006) and applied the same energy balance model (section 4.1) to the data. Only the initial subsurface temperature profile and the snow density (300 kg m−3) were changed. The method to derive cloudiness described in section 3.4 did not give robust results for Morteratschgletscher, cloud fractions tended to be overestimated in summer because of higher minimum values for Lin in this season. Although a detailed cloud analysis could not be done in this case, the mean cloud fraction calculated for Morteratschgletscher (0.47) is considerably lower than on Midtdalsbreen (0.61, Table 2). At the AWS site on Morteratschgletscher the maximum snow depth ranges between 0.3 and 2 m, the surface already becomes snow-free in May. During the period considered here, the measured ice ablation amounts to 40.8 m, almost twice as much as on Midtdalsbreen (24.0 m).
 The seasonal cycles of meteorological variables on the two glaciers (Figures 11a and 11b) show clear differences. Throughout the year, air temperatures are higher at the AWS site on Morteratschgletscher, despite the 650 m higher altitude of this station. The temperature difference is largest in spring and autumn, probably because the seasonal cycle in solar irradiance is larger on Midtdalsbreen and the snow cover disappears later. In winter, surface temperatures are much lower on Morteratschgletscher because of more frequent clear-sky conditions, values are more comparable to surface temperatures on Midtdalsbreen under clear skies (Figure 5a). Wind speeds are higher on Midtdalsbreen, especially in winter. The wind climate on Midtdalsbreen is dominated by the large-scale circulation (section 5.3), while katabatic winds prevail on Morteratschgletscher [Oerlemans and Grisogono, 2002]. The absolute values and the seasonal cycles of specific humidity are very similar on the two glaciers, although relative humidity values are much lower on Morteratschgletscher (annual mean 64.9%). Hence climatic conditions on Midtdalsbreen are generally colder, cloudier, more humid and windier than on Morteratschgletscher.
Figure 11. Monthly average values of air and surface temperature (Ta and Ts in °C), wind speed (v in m s−1) and specific humidity (q in g kg−1) for (a) Midtdalsbreen and (b) Morteratschgletscher. Ts calculated with the energy balance model is shown here. Monthly average values of the surface energy fluxes for (c) Midtdalsbreen and (d) Morteratschgletscher.
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 The seasonal cycles of the surface energy fluxes (Figures 11c and 11d) show that the melt season starts earlier on Morteratschgletscher. The melt energy already reaches a maximum in June, while on Midtdalsbreen the melt energy is maximum in July. From July to September, absolute values for the melt energy are comparable at the two AWS locations. However, the relative contribution of the surface energy fluxes to the melt energy is different for the two glaciers. Snet is larger and more dominant on Morteratschgletscher because of an earlier disappearance of the snowpack and a lower ice albedo. From May to July, Sin is larger on Midtdalsbreen, even though the incoming solar radiation at the top of the atmosphere is smaller and the cloud fraction is larger. At the AWS location on Morteratschgletscher, Sin is significantly reduced because of shading by the surrounding topography [Klok and Oerlemans, 2002], but multiple reflection of solar radiation by the snow cover may also contribute to a higher Sin on Midtdalsbreen. Despite the lower air temperatures, Lnet is more positive on Midtdalsbreen throughout the year; a result of the frequent clouds on Midtdalsbreen which increase Lin. The seasonal cycle of Hsen is comparable at the two locations; the effect of lower air temperatures on Midtdalsbreen is compensated by the higher wind speeds. In June, Hsen is considerably larger on Morteratschgletscher due to the higher air temperatures. Hlat is more negative in spring and more positive in summer on Midtdalsbreen. As the annual cycle in specific humidity is comparable at the two locations, this is also a result of the windier conditions on Midtdalsbreen.
 The number of half-hourly periods with melt is similar at the two AWS sites (35% of the year for Midtdalsbreen and 36% for Morteratschgletscher), but the mean melt energy during these periods is 36% larger on Morteratschgletscher (202 W m−2). The mean value of Snet during melt is 68 W m−2 (60%) larger on Morteratschgletscher and contributes on average 90% of the melt energy. The differences between the other fluxes are smaller; mean values for Lnet and Hlat during melt are 11 W m−2 and 7 W m−2 smaller than on Midtdalsbreen, Hsen is 5 W m−2 larger. Hence the larger ablation on Morteratschgletscher primarily results from an earlier start of the melt season, a thinner snowpack and a lower ice albedo.