In this study we employed the RCM HIRHAM5 , which was developed at the Danish Meteorological Institute. It is a hydrostatic RCM which combines the dynamical core of the HIRLAM7 numerical forecasting model and physics schemes from the ECHAM5 general circulation model . Model simulations have been successfully validated over Greenland using AWS and ice core data e.g..

While the original HIRHAM5, as described in , used unchanged ECHAM physics, an updated model version, which includes a dynamic surface scheme that explicitly calculates the surface mass budget on the surface of glaciers and ice sheets, is used in this study. This new scheme takes melting of snow and bare ice into account and resolves the retention and refreezing of liquid water in the snow pack . In addition, the five-layer surface scheme in ECHAM has been expanded to 25 layers.

The turbulent energy fluxes were calculated from AWS measurements using a one-level eddy flux model which uses Monin–Obukhov similarity theory and implements different roughness lengths for the vertical profiles of wind, temperature, and water vapour . The model is described in detail in . Uncertainties of this model for example pertain to the aerodynamic roughness length for momentum z0. The majority of z0 values recorded over melting glacier surfaces vary over 2 orders of magnitude (between 1 and 10 mm), but over fresh snow or smooth ice surfaces the roughness length is generally around 0.1 mm . An order of magnitude increase in z0 can more than double the estimated turbulent fluxes , so the chosen roughness length parametrization can greatly affect the performance of the model. Generally, a constant value of z0 is prescribed for snow and/or ice surfaces , which is an oversimplification as the roughness may vary significantly over the ablation season e.g..

However, since measurements of the evolution of z0 over the entire measurement period are not available, a constant roughness length of 1 mm was chosen in the calculation of the non-radiative fluxes. Sensitivity tests were conducted to estimate how large an error this choice of roughness length could lead to at the used AWS sites. A roughness length of 0.1 mm would decrease the calculated turbulent fluxes by 16–22 %, while using a roughness length of 10 mm would increase the calculated fluxes by 10–19 %, depending on the station. Since the contribution of the turbulent fluxes to the total energy balance is generally low, this translates into an increase or a decrease in the total energy balance at the stations by a maximum of 7 %.

The surface air pressure at the station is also needed to calculate the turbulent fluxes, but it is not measured at the AWS sites. Instead it is estimated at the relevant elevation h using synoptic observations from meteorological stations operated by the Icelandic Met Office and the following relationship: P(h)=P(h0)1-0.0065(h-h0)T(h0)5.25, where P(h0) and T(h0) are the air pressure and air temperature, respectively, observed at an elevation h0 e.g. This method has previously been applied successfully at various locations on Vatnajökull and Langjökull e.g..

AWS data from the period 2001–2014 for three Brúarjökull stations and two Tungnaárjökull stations are considered, as well as SMB point measurements from 1995 to 2014. All stations were operated during the summer months, but since 2006 the lowest Brúarjökull station has been operated year round. Comparisons are made between daily averages from the HIRHAM5 model and the in situ observations collected at the AWSs. HIRHAM5 daily means are calculated from 6-hourly outputs, while the AWS daily means are calculated from observations at 10 min intervals.

Comparisons between station values and model values are made by bilinearly interpolating the model output to the measurement position using the four closest model grid points and using only glacier-surface type grid cells.

In order to remove the effect of seasonally varying magnitudes of the energy balance components, the percentage errors listed in Tables – are calculated as the root mean square error (RMSE) divided by the observations.

HIRHAM5 uses an elevation model over Iceland which has been interpolated onto the 5.5 km model grid. Since errors in the elevation of the glacier surface can introduce significant biases in temperature and pressure which are not caused by physical model errors , any elevation bias in the model has to be taken into account before validating the results. The elevation bias was calculated as the difference between the model elevation and GPS observations at each site (Table ).

The temperature was corrected for the elevation bias in order to compare the model results to the AWS measurements at AWS locations. This was done using a constant lapse rate of 6.5 K km-1, which resulted in temperature corrections on the order of 0.1–0.3 K. Pressure is corrected using Eq. () decreasing the bias down to 0.1 to 0.5 hPa. Thus, although the HIRHAM5 elevation is consistently overestimated, the resulting differences are not large enough to introduce significant biases in temperature and surface pressure.

As the sensible and latent heat fluxes are computed using the surface pressure psl, air temperature at 2 m, T2m, relative humidity r2m, and wind speed u, these model variables were evaluated at all five stations at the measurement height. How well these variables are simulated should indicate the model's ability to simulate the turbulent fluxes.

The comparison of modelled and observed mean daily values during the summer months from the period 2001–2014 is shown in Fig.  and Table . The surface pressure, psl, which was not observed at the stations but estimated using Eq. (), is generally forecast with only a small error. At each station there is a high positive correlation (r>0.9) between modelled and estimated pressure (Eq. ), for the entire time series and for each individual year.

The model also captures the 2 m temperatures, T2m, satisfactorily. The largest deviation from the observations is found at the BAB station, which underestimates the temperature by 0.8 K on average. The temperature is also underestimated at the four other stations, but by at most 0.6 K. The model simulates the variation in temperature well; for example, it captures the temperature dampening over a melting glacier surface. This is expressed in the high correlation values for all five stations (r∼0.9). The measured relative humidity, r2m, at all five stations is generally high, with only 1–3 % of the data points at each station falling below 70 %, and the minimum daily value between 42 and 58 %. The model simulates a lower mean humidity than the measured at all five stations, with 8–20 % of the points at each stations having values lower than 70 % and minimum daily values between 18 and 30 %. Since the exchange coefficient for moisture is a function of the atmospheric temperature profile, the underestimation of the relative humidity could be due to a too low temperature gradient between the atmosphere and the surface. This is consistent with the underestimation found in the 2 m temperature. The correlation of between 0.68 and 0.7 indicates that the model simulates the humidity fluctuations satisfactorily.

The lowest wind speed level in HIRHAM5 is at 10 m and the AWS wind speeds are measured at between 2 and 4 m, depending on the year, the HIRHAM5 wind speed is extrapolated to the measurement height using a logarithmic profile with a roughness length of 1 mm. At all five locations, HIRHAM5 simulates winds that are too weak on average. This could be due to the uncertainty arising from the interpolation of the model winds from second-lowest level (30 m) to the lowest level (10 m) under stable conditions, as the wind speed can change significantly over the 20 m interval.

As shown above, HIRHAM5 underestimates the temperature at all five stations, with the largest underestimation at the BAB station. As a result, a similar underestimation of incoming longwave radiation is obtained at all five stations, with the largest difference occurring at the BAB station . The average percentage difference is approximately 8 % for all five locations (see Table ), and falls well within the 10 % uncertainty of the AWS observations. However, Fig. a also shows that 25–30 % of the simulated days have errors larger than 10 %.

The incoming LW radiation is mainly emitted from clouds and atmospheric greenhouse gases, and therefore a source of the underestimation could be that the model underrates cloud formation and/or simulates clouds that are too optically thin in the LW region of the spectrum. An underestimation of the temperature in the atmosphere could also be causing the underestimation.

Figure b shows the comparison of the modelled and measured outgoing LW radiation. There is a small overestimation at the TAC station, and a small underestimation of the other four stations, but in general the model reproduces the daily values well (r∼0.76). The average percentage deviation between the modelled and measured values is only around 3 %, combined with between 0.5 and 2 % of the HIRHAM5 data points having deviations larger than 10 %.

Due to an underestimation of the incoming LW radiation, and only small negative or positive biases in the outgoing LW, the net LW (incoming–outgoing) radiation has a mean negative bias at all AWS locations (-7.9 W m-2).

Figure and Table show the comparisons of the modelled and measured components of the shortwave (SW) radiation as well as the surface albedo. On average, the incoming SW radiation is underestimated at all five stations. This underestimation is also present in the means at all five stations for most years, except in 2002, 2004, 2005, and 2014 at the BAB station. This suggests that there are errors in either the modelling of the clouds, e.g. due to an overestimation of the cloud fraction, the amount of cloud formation, or the optical thickness of the clouds in the shortwave region, and/or because of errors in the clear-sky fluxes.

The albedo comparison is shown in Fig. b. The modelled albedo at the two AB stations has the largest deviation from the observations; this is partly due to the modelled snow cover, which either does not completely disappear or disappears later in the year than the AWS data show. At the BAB station, the ice layer is generally exposed in the model (except in 2001 and 2011–2013), although the snow cover always persists longer than in reality. One exception occurs in 2001, where the modelled albedo never drops down to the ice value, whereas observations show albedo values as low as 0.03. This one year therefore highly contributes to the average overestimation of the albedo. This very low albedo value could be due to a layer of dust or tephra beneath the station, so it may not represent the ice albedo. However, very low ice albedo values down to 0.05 are not uncommon in the ablation zone of Vatnajökull e.g.. Comparisons with the mass balance measurements (discussed in Sect. ) show that the winter balance is overestimated during approximately half of the measured years, which contributes to delay the albedo drop in the model.

At the TAB station, a too-thick modelled snow cover in winter is also the cause of some of the discrepancy. Comparisons with mass balance measurements (Sect. ) show that the winter balance is always overestimated at this station. An overestimation of the snow thickness at the beginning of summer, combined with an underestimation in the radiation and turbulent fluxes, leads to persistent snow cover at the end of summer. As a result, the ice surface is never exposed in the model during any of the modelled years, and the albedo never drops much below 0.4 (the minimum snow albedo), even though the AWS data shows that the ice surface was exposed during all but two years, i.e. 2008 and 2010. During these two years, the simulated albedo fits well with observations.

Another issue which affects both stations is that the MODIS albedo at these points is not as low as the measured albedo. The MODIS ice albedo at these stations is 0.10 (BAB) and 0.16 (TAB), whereas the observations show the albedo can drop as low as 0.01 at both stations. The albedo drops below the MODIS value every year at the BAB, and during 2001–2005 and 2011 at the TAB stations. This is presumably due to the heterogeneity of the albedo in the ablation zone, which means that a low in situ albedo value at a point cannot be captured at the current HIRHAM5 resolution.

At the ELA station, the mean albedo value is underestimated (Table ). Close to the equilibrium line, the albedo is highly variable both temporally and spatially; for example, there is a large difference in albedo depending on whether the previous year's summer surface was exposed or not. In general, the model overestimates the albedo during years where the summer surface was exposed, and underestimates the albedo during years where it was not. In addition, the winter mass balance at this station is always underestimated (Sect. ), meaning that the thickness of snow layer in spring is underestimated and the effect of the underlying ice layer will therefore be overestimated, leading to the underestimation in albedo.

The smallest difference between modelled and observed albedo is found at the two AC stations. The BAC station generally provides the best fit with the observations, while the model tends to underestimate the albedo at the TAC station. An exception to this is found in 2010 and 2011, where the albedo was overestimated by the model at both stations due to ash deposition from the Eyjafjallajökull and Grímsvötn eruptions e.g..

A general reason for the model overestimating the albedo is that it does not take the albedo changes due to dust storms or volcanic dust deposition into account. For instance, the very low albedo values obtained at the TAC station (Fig. b) are due to tephra deposition on the glacier during the 2010 eruption of Eyjafjallajökull e.g.. Even though dust events do not cause as large changes in albedo as a volcanic eruption, they can still significantly lower the albedo e.g. . As previously mentioned, the albedo in HIRHAM5 often reaches its yearly minimum value later in the summer than the observed. Such discrepancy could be explained by dust events, advancing or delaying the drop in surface albedo. investigated 10 dust events which occurred at the BELA station in 2012, and found a lowering in the albedo during all events and showed that the dust storms have a significant effect on the resulting energy balance.

The error in the outgoing shortwave radiation is caused by errors in the albedo and the incoming SW. At the BAB station, the incoming radiation is slightly underestimated but the albedo is overestimated; hence, the outgoing SW is overestimated. The values at the four other stations are all underestimated, due to larger underestimations of the incoming SW radiation and lower albedo errors.

As both the incoming and outgoing SW radiation are underestimated at most stations, the net SW shows a negative bias of ∼-6 to -12 W m-2 at the AC and ELA stations, and of -22 and -28 W m-2 at the two AB stations. The resulting average model error at all five stations is -15.5 W m-2.

As HIRHAM5 underestimates meteorological variables at all stations, similar underestimation is obtained for the turbulent fluxes (Table  and Fig. ). The two AC stations have the largest differences and also the lowest correlation (0.45 and 0.49) between the AWS estimate and the HIRHAM simulation. The other three stations also have significantly lower values in the HIRHAM5 model than in the AWS model, but with higher correlation coefficients (0.69–0.73).

It is important to bear in mind that this comparison is a model–model comparison, so while the eddy flux model may give a good estimate of the turbulent fluxes, model errors still affect the results, e.g. due to the use of a constant roughness length.

After the simulated components of the energy balance were evaluated against AWS observations, the total energy balance was estimated (see Table ). The energy balance (E) is found using E=LWnet+SWnet+Hs+l, where LWnet is the net LW radiation, SWnet is the net SW radiation, and Hs+l are the turbulent fluxes. Overall, the melt energy is underestimated, owing to all elements of the energy balance generally being underestimated. This is in large part due to the underestimation of the modelled incoming radiation. We attribute this to an error in the modelling of the clouds, but since both the incoming SW and LW radiation are underestimated, inaccurate cloud representation cannot be the the only source of the error. Errors in the interaction of clouds and radiation, e.g. error in the optical thickness of the clouds, or in the clear-sky fluxes, could partly explain these discrepancies. The underestimation of the incoming LW radiation could also be due to errors in the vertical atmospheric temperature gradient.

Since the simulated outgoing LW radiation generally only has a small negative bias, the deviation in net LW radiation is governed by the incoming radiation. Errors in the simulated albedo mean that both the in- and outgoing SW radiation greatly contribute to the deviation in net SW radiation. These errors can be partly attributed to ash and dust deposition during volcanic eruptions and dust storms, which are not taken into account in HIRHAM5. In addition, errors in the simulated albedo also stem from snow cover that disappears too slowly compared to AWS records in the ablation zone. As a result, modelled albedo drops too slowly compared to the measured albedo. The underestimation of the net SW and LW radiation and the turbulent fluxes leads to underestimated melt energy. This contributes to the overestimation of the modelled snow thickness.

In order to estimate how much the different components contribute to the energy difference on a year-to-year basis, the mean difference between modelled and observed energy components during each summer (April–October) is shown for each station (Fig. ).

At the BAC station, the contribution of the long- and shortwave radiation and turbulent fluxes to the energy difference is consistent for the entire period, with the error of each component being almost equal, varying between -25 and 0 W m-2. At the TAC station, the error due to the three components is also of the same order of magnitude, except in 2010 and 2011 where the error in the net SW radiation is much larger than that in the other components. This is due to a large drop in the albedo as a result of the Eyjafjallajökull (2010) and Grímsvötn (2011) eruptions. The mean difference between observations and the simulations of the SW radiation for non-eruption years is -3 W m-2, whereas the radiation difference in 2010 is -106 W m-2. Assuming the larger deviation from the mean in 2010 is only due to the volcanic eruption, the increase in available energy due to the eruption is 103 W m-2. If it is further assumed that the surface was always at melting point, the increase in melt due to the 2010 Eyjafjallajökull eruption over the 128-day measuring period would be ∼ 3.1 m w.e. at this station.

At the ELA site, the contribution from the modelled turbulent fluxes to the energy balance deviation generally varies between ±10 W m-2, except in 2013 where the bias is around -25 W m-2. Modelled longwave radiation is consistently underestimated by -10 W m-2. The deviation in the shortwave radiation is more variable, as expected from the results of the albedo comparison. Depending on whether bare ice was exposed or not, the albedo is generally either over- or underestimated. For example, at BELA, the ice surface was reached in, for example, 2007 and 2012, resulting in an overestimation of the albedo. In, for example, 2002 and 2009, however, the albedo was high the entire summer as no ice was exposed, resulting in an underestimation of the predicted albedo.

At the TAB station, both the net LW radiation and the turbulent fluxes agree well with observations for the entire period. The net SW radiation, however, is always underestimated, especially in the period 2001–2003 and 2011. These years, the measured albedo at the station goes below 0.1, while the HIRHAM5 albedo stays around 0.4. As previously discussed, this albedo bias, and hence underestimated net SW radiation, occurs because of an overestimation of the snow cover at the station due to an overestimation of the winter accumulation and possibly also the proximity of the equilibrium line. An underestimation of the incoming SW radiation, which we attribute to an error in cloud cover amount of clear-sky fluxes, also contributes to this error.

At the BAB station, the longwave radiation bias is relatively constant with values close to 0 W m-2 for much of the measurement period. The absolute deviation due to the turbulent fluxes is less than 10 W m-2 for most of the period, although with slightly larger deviations from the period 2007–2010. The SW radiation is always underestimated at this station, mostly due to the previously discussed overestimation of the albedo.

Spatial maps of the (uncorrected) average winter, summer, and net SMB from the 1980–1981 glaciological year until 2013–2014 are shown in Fig. . The approximate location of the average ELA is marked in the figure. The model captures the position of the ELA fairly well, but at, for example, Brúarjökull, where the average ELA is at 1200 m, the position of the average ELA is at a too high elevation. The average deviation between observation and model over the observation period at each measurement location is also shown in Fig. , in order to give an indication of the average error of the model at different parts of the ice cap. The winter balance (Fig. e) is generally overestimated at low elevations and underestimated at high elevations, except for at Öræfajökull, where there is a large overestimation of the winter balance, as discussed in the previous section. As can be seen in Fig. e, there is generally a low SMB bias at high elevations and a high SMB bias at low elevations during the summer. This is consistent with the comparisons with AWS stations, as we found that the bias in the energy available for melt was smaller at high elevation than at low elevation (see Table ). This was partly due to a larger albedo bias for stations in the ablation zone than for stations in the accumulation zone.

In addition to the spatial maps, the winter, summer, and net mass balances of Vatnajökull were calculated for the entire simulation period, and the results were compared with an estimate of the specific balance from 1995 to 2014, created by interpolation of the mass balance measurements e.g.; see Fig. . The model prediction of the mean specific summer mass balance generally fits well with the interpolated observations, with an overall difference of only 0.06 m w.e. The largest deviations are obtained in 1995, where ablation is overestimated in the simulation, and in 1997, 2005, and 2010–2012, where ablation is underestimated, most likely due to ash depositions on the glacier following the 1996 Gjálp eruption, the 2004 and 2011 Grímsvötn eruptions or the 2010 Eyjafjallajökull eruption, which are not taken into account in the model.

Excluding the years where the albedo was affected by volcanic eruptions, the average difference becomes smaller but the model also predicts slightly too much ablation, as the difference becomes -0.02 m w.e.

There is a shift in the summer and annual mass balance calculated by the model and the in situ MB measurements around 1996, with a generally more negative mass balance after 1996 than before. This is consistent with the increase of the annual mean temperature of Iceland in the mid-1990s, which resulted in a mean annual temperature ∼1 K higher in the decade after than the decade prior to 1995. This is likely linked with atmospheric and ocean circulation changes around Iceland, as there was a rapid increase in ocean temperatures off the southern coast in 1996 .

The specific winter mass balance is overestimated in HIRHAM5 for the entire measurement period with an average of 0.54 m w.e. Due to this difference, and only the small negative mean difference in summer mass balance, the annual mass balance of Vatnajökull is overestimated every year with an average difference of 0.50 m w.e.

However, this is mostly due to the large overestimation of the winter accumulation on Öræfajökull; comparison with the mass balance measurements showed that the model overestimated the winter accumulation by 100–200 % compared with the observations. In an attempt to estimate how much this error affects the results, a simple correction was added to the Öræfajökull points by reducing the simulated winter SMB by 50 %. The correction was added to four model grid points around Öræfajökull, due to the high (> 10 m yr-1) annual specific mass balance in these points (see Fig. a). The resulting modelled winter and annual specific balance are shown in Fig. . The winter balance is still overestimated, but the difference between modelled and interpolated values has been reduced to only 0.1 m w.e. In addition, the average difference between the HIRHAM5 and interpolated annual SMB drops to only 0.08 m w.e.

In order to quantify the changes in the model performance resulting from the new albedo scheme used in this study, which utilizes an albedo map based on MODIS data , the results are compared to those of a previous run using a constant ice albedo of 0.3. The average difference in albedo and mass balance over the period 2001–2014 in each grid point are shown in Fig. , as well as the position of the AWS stations.

There is little to no difference between the two runs in the accumulation zone, due to the year-round snow cover. In the ablation zone, however, using the MODIS ice albedo map has a large effect on the simulated albedo. The largest differences are found on the southern outlet glacier Skeiðarárjökull, which is unfortunately a glacier where no mass balance or AWS measurements have been conducted. The BAB and BELA stations are located in areas that are affected by the ice albedo, either because ice is exposed (BAB) or because the underlying surface contributes to the albedo (BELA). The TAB station is located in the ablation area, but the ice surface is never exposed in the model due to an overestimation of the winter accumulation. The albedo estimate at this station was therefore not improved by using the MODIS albedo.

When the model is run with the constant ice albedo of 0.3, the amount of ablation will be lower and thus the specific summer balance will be higher. Compared to the simulation using the MODIS map (Fig. ), the constant ice albedo simulation results in an increase in the specific summer SMB by an average of 0.37 m w.e., or 18 %, per year for the period 1995–2014. The increase in the summer SMB ranges from 14 cm (in 2014) to 85 cm (in 2001) and the percentage increase varies between 8 % (in 2011) and 39 % (in 1995). As the winter balance is not dependant on the ice albedo, there are no changes in the specific winter SMB between the two simulations.