helum catchment. As indicated above, typical melt conditions of TDD = 5 ∘C d occur at 1500–2000 m a.s.l. usually around mid-February, while at 3500–4000 m a.s.l. similar degree days are obtained for mid-May. The resulting DDFs (see Fig. 10b) show a significant difference, both under clear-sky conditions and under overcast conditions, because of the different input in solar radiation caused by the alteration in solar angle between February and May. Figure 10b shows an additional term DDFA on top of the solar radiation component that represents the increase in incoming solar radiation due to the clearness altitude factor, which takes into account the increase in the clearness index with altitude (see Sect. 3.2.1). Averaging the factors proposed by different solar radiation models (see Fig. 4) results in an additional component DDFA of 0.4 and 1.4 mm ∘C−1 d−1 under clear-sky conditions and of 0.5 and 1.6 mm ∘C−1 d−1 under overcast conditions at 1500–2000 and 3500–4000 m a.s.l., respectively.
While snow albedo is assumed constant at 0.5 in Fig. 10b, taking into consideration the decrease in albedo as the snow ages (see Table 1, e.g. A=0.74 in February and A=0.42 in May) results in a more pronounced difference with altitude, i.e. a DDF of 0.3 compared to 10.5 mm ∘C−1 d−1 under clear-sky conditions and of 2.7 versus 7.3 mm ∘C−1 d−1 under overcast conditions for the two altitudes, respectively.
The increase in DDF with increasing altitude is in line with previous studies (e.g. Hock, 2003; Kayastha and Kayastha, 2020). For example, in the Nepalese Himalayan region, seasonal-average DDF increases from 7.7 to 11.6 mm d−1 ∘C−1 with respect to altitude ranging 4900 to 5300 m a.s.l. (Kayastha et al., 2000), whereas Kayastha and Kayastha (2020) found that the model-calibrated range of the DDF in the central Himalayan basin varies between 7.0–9.0 mm d−1 ∘C−1 over an approximate altitude range of 4000–8000 m a.s.l. As Kayastha et al. (2000) pointed out, higher values of the DDF usually occur at very low temperatures since at higher altitudes the major driving factor for melt is the energy input by solar radiation.
As already discussed in the sections before, snow albedo is a critical parameter for the DDF since according to Eq. (5) the albedo directly controls the net solar radiation flux into the snowpack. While albedo of fresh snow is well above 0.9, hence reflecting most of the incoming shortwave radiation, it drops rapidly when larger grains form due to snow metamorphism. Figure 10c demonstrates the effect of ageing snow after a new snow event, when a simple exponential decay model as given in Eq. (19) is used and typical melting conditions TDD = 5 ∘C d are assumed. Since directly after a new snow event (Day = 0) the fresh snow albedo is high (A=0.95), the overall DDF is generally small. Under clear-sky conditions, if longwave radiation cooling is larger than net shortwave radiation flux, even a negative DDF value, i.e. no melt, may occur. If there is no new snow event in between, albedo will decrease following the exponential decay model to 0.52 after 10 d, resulting in a DDF of 5.8 mm ∘C−1 d−1 under clear-sky conditions and 4.4 mm ∘C−1 d−1 under overcast conditions. The increase in the DDF with exponential decay in albedo is in agreement with the findings of MacDougall et al. (2011), who found that the DDF is sensitive to albedo with values of > 4.0 mm d−1 ∘C−1 at an albedo of 0.6. As described qualitatively in the literature (e.g. Hock, 2003), under all sky conditions the DDF continuously increases with decreasing albedo, with the increase, however, being more pronounced under clear-sky than under overcast conditions.
Since the solar angle rises from its minimum at the winter solstice in December to its maximum on 21 June, the solar radiation component DDFS increases during the snowmelt season, and thus the DDF is expected to increase. Figure 10d shows the influence of season on the DDF at the Brunnenkopfhütte test site during the melt period, assuming average degree days of 1, 4, and 7 ∘C d in March, April, and May, respectively (see Table 1). Under clear-sky conditions, the total DDF increases from a negative value of −3.6 mm ∘C−1 d−1 in March to 6.6 mm ∘C−1 d−1 in May. Under overcast conditions, however, the DDF is virtually stable, ranging from 4.4 to 4.5 mm ∘C−1 d−1 in the same period. The stability of the DDF under overcast conditions found in our study is in agreement with the study of Kayastha et al. (2000), where the DDFs observed during July–August are small compared to June because of prevailing cloud cover due to monsoon activity, which reduces the incoming shortwave radiation.
An evaluation of the individual DDF components shows that under clear-sky conditions the high impact of solar radiation in combination with low degree days at the onset of the snowmelt season is counterweighted by a strong negative longwave radiation component that decreases as the season progresses. Under overcast conditions, DDFL is neutral or slightly positive, while the DDFS component decreases because degree days rise faster than the input from solar radiation, which implies that sky conditions (i.e. overcast, and clear sky) are more decisive for an estimate of the DDF than the day of the year.
The effect of cloud cover is further amplified by the decrease in albedo while the melt season progresses, which becomes more significant under clear-sky conditions. In the present example (with the average monthly albedo as specified in Table 1) only 30 % of incoming solar radiation is contributing to melt in March, while it is about 60 % in May, enhancing the marked increase in the DDF under clear-sky conditions.
In general, the precipitation heat component alone has only a minor effect on the DDF. However, in conjunction with certain weather conditions, like breaking in of warm and moist air, rain over snow events may lead to sudden melt and severe flooding.
Figure 10e shows the different DDF components resulting from a hypothetical rain over snow events assuming an air temperature of 15 ∘C, a precipitation of 70 mm d−1, a daily average wind speed of 10 m s−1, a relative humidity of 100 %, and overcast conditions. Although the amount of precipitation is substantial and the rain temperature is comparatively high, the contribution of DDFP is still modest. However, air temperature, relative humidity, and in particular wind speed associated with such events increase the sensible and latent heat components significantly. Thus, the resulting overall DDF is much higher than under usual melt conditions, which may lead to a considerable melt that adds to the runoff already caused by the heavy rain.
In addition to the analysis of the influence of individual factors on the DDF, the dataset from the Brunnenkopfhütte test site is used to compare energy-flux-based estimates with field-derived DDFs in order to demonstrate how naturally varying meteorological conditions during the melt season, and in particular the cold content of the snowpack, affect the accuracy of DDF estimates.
For this purpose, daily melt was estimated from the daily difference of observed snow water equivalent during melt periods (see Fig. 3). Energy-flux-based melt was calculated by the formulas given in Sect. 3 using observed daily data from the Brunnenkopfhütte automatic snow and weather station (e.g. air temperature, wind speed; see Sect. 2.1) where applicable.
The daily degree-day sum is calculated from hourly air temperature data as proposed by Braithwaite and Hughes (2022). In general, operational degree-day models typically use constant degree-day factors for a certain time period (e.g. 10 d period). In this backdrop, both energy-flux-based and data-derived daily melt values were accumulated on a 10 d basis and divided by the degree days of the respective period. The 10 d averaging procedure also smooths daily noise in the observed data, in particular inaccuracies in the determination of daily melt and unrealistic DDF values because of daily temperature averages just above 0 ∘C.
The comparison between field-derived and estimated (energy-flux-based) DDFs (see Fig. 11) yields a fair agreement with bias = 0.14 mm ∘C−1 d−1 between estimated and field-derived values, and root mean square error (