Meteorological forcing of the coupled model consists of an hourly time series, describing air temperature, atmospheric pressure, wind speed, relative humidity, fractional cloud cover, and precipitation (Fig. ). The series was primarily assembled using data from the CM AWS, located on the Signalkuppe summit at 4560 m a.s.l. (Fig. a). The station records air temperature, barometric pressure, wind speed and direction, and global radiation (Table ). The measured parameters are available as hourly instantaneous values.

The consistent availability (>95 %) of long-term data (since mid-2002) makes the series a valuable dataset for high-alpine research. Still, extensive evaluation and processing of the data are essential due to the extreme measuring conditions . In particular, freezing of the anemometer – as well as snow and ice accumulation interfering with the pyranometer – are relatively common occurrences. Therefore, we used the hourly data from eight other high-altitude AWSs located in the region (Fig. b, id 2–9 in Table ) to perform quality checks, to fill gaps in the CM time series, and to provide data for parameters not measured at CM. Our selection of which stations to include was determined by the availability of different parameters measured at each site, by the effort to provide an unbiased geographic coverage in all directions from the CM AWS, and by the station tendency towards concurrent failures in challenging conditions, such as winter storms.

All AWS series were pre-processed to remove clock errors, detected with cross-correlation analysis. Then each series was entirely reconstructed from the data of the best correlated others using quantile mapping , to provide a reference for robust outlier detection. High-resolution reanalysis series of the parameters measured at CM were also collected as an additional basis for comparison COSMO-REA2 and COSMO-REA6:. Due to the large volume of data involved, an automated pre-filtering routine was implemented, based on objective criteria (absolute values, rates of change, comparison with reconstructed series and reanalysis) to mark single values as potential outliers, which were then manually checked. The CM AWS was always processed last in order to be compared to the highest-quality data. After quality check, all gaps in the hourly CM series (Table ) were filled with the corresponding reconstructed values. The close match of the reconstructed series (Table ) despite the large elevation differences involved (Table ) confirms the benefit of the extensive processing described.

The CM AWS does not record series of relative humidity (RH) or cloudiness, which are required as model input. Thus RH was entirely derived from the other quality-checked time series; cloudiness was reconstructed from the incoming SW radiation at CM when available and otherwise computed from the incoming LW radiation and observed cloud cover, respectively at Stockhorn and Plateau Rosa. Lapse rates of temperature and pressure – used to extend the AWS record to the gridded model domain – were computed from the difference between CM and the other stations. Despite the large elevation differences (Table ), a robust computation of lapse rates was possible thanks to the availability of multiple stations within a close distance; the spread of the individual rates was usually 1 order of magnitude smaller than their absolute values. Moreover, the close proximity of the CM AWS to the domain minimizes the amount of extrapolation needed, as does the narrow elevation range of the domain itself. The other variables (wind speed, cloud cover, and RH) were held constant over the model domain, as no measurements were available to estimate their spatial variability. Further details on the series processing can be found in ; Fig.  shows the final time series of all meteorological parameters assigned to CM and used as model input.

To determine some of the model calibration parameters (Table ), we also re-analyzed data from two additional weather and energy-balance stations (10 and 11 in Table ), temporarily installed above 4000 m a.s.l. within the ALPCLIM project . The Seserjoch station operated from September 1998 to October 2000 for the detailed SEB investigations of ; the Colle del Lys station was active between 1996 and 2000.

For model validation (Sect. 4.1) we used archived firn temperature profiles (Fig. a), measured between 2003 and 2018. Besides the CG saddle, we included two profiles from the Grenzgletscher slopes and four from Seserjoch. In total, we considered 25 temperature profiles from 18 boreholes, where some locations have been measured more than once. Detailed description of the profiles is reported in and . Earlier information on the deep temperatures was also used as reference during model setup (Sect. 3.4).

Moreover, we compiled point measurements of annual accumulation (Fig. a), derived from layer thicknesses in ground-penetrating radar (GPR) profiles and from the stake network of , as well as from archived ice core measurements (densities and annual dating) acquired at CG between 1982 and 2019. In total, 14 core profiles were available; only one (core Zumsteinkern, from 1991) is located on the high-accumulation, south-facing Zumsteinspitze slope (Fig. a). Description of the individual cores can be found in and .

Finally, we hand-drilled a 5.5 m core (unifr-2019, Fig. a) near the CG saddle point on 25 June 2019, analyzing density and stratigraphy in the field. Core description is presented in Appendix A.

The model grid was based on a 20 m digital elevation model (DEM). Due to the Swiss–Italian border crossing the simulation domain, the DEM was produced by merging the stereo-photogrammetric SwissAlti3D dataset (acquired in 2015 with a vertical accuracy of 1–3 m) on the Swiss side and the ICE lidar digital terrain model (acquired in 2011 with a vertical accuracy of 0.3–0.6 m) on the Italian side . Elevation mismatch along the border (rms error of 3.2 m) was corrected via smoothing, to avoid unwanted biases in slope and aspect.

The model computes incoming SW radiation as 2SWin(x,y)=TOAshaded(x,y)trgtwtatcl,3tcl(n)=1-an-bn2. Here, TOAshaded(x,y) is the unattenuated top-of-atmosphere radiation (W m-2), corrected for topographic shading and angle of incidence on the surface. Coefficients trg, tw, ta, and tcl are the gaseous, water vapor, aerosol, and cloud transmissivities; n is the fractional cloud cover; and a and b are calibration parameters. Values for a and b (Table ) were derived from as calibrated in high-alpine terrain, unlike the EBFM defaults which were tuned from measurements in the Arctic . The parameter values of increase the dependence of tcl on the cloud fraction, allowing a stronger decrease in the incoming SW flux under overcast conditions. Such a higher dependence was confirmed by an analysis of the distribution of the incoming radiation flux measured at the CM AWS, supporting our parameter choices (Table ).

Reflected SW radiation is controlled by a broadband isotropic surface albedo αsnow (Eq. ). Albedo evolution is modeled after as an exponentially decaying function of time t since the last significant snowfall (Eq. ), bounded by constant values for fresh snow (αfresh) and firn (αfirn). The precipitation threshold to reset albedo is 0.1 mm w.e. h-1 . The timescale of albedo decay is a function of snow surface temperature (Eq. ) to account for slower metamorphism in cold conditions : 4SWout=αsnowSWin,5αsnow(t)=αfirn+(αfresh-αfirn)e-tt*,6t*(Ts)=twet*Ts=0∘Ctdry*+K|max⁡(Ts,Tmax,t*)|Ts<0∘C, where t* is the computed albedo decay timescale (d), Ts snow surface temperature (K), twet* (tdry*) timescale (d) for a melting (dry) surface at 0 ∘C, K a calibration parameter (d ∘C-1), and Tmax,t* a temperature cut-off value (K) for decay slow-down (Table ).

Incoming LW radiation is computed with the Stefan–Boltzmann law for grey-body radiation (Eq. ); sky emissivity is modeled after as a function of cloud cover, air temperature, and humidity (Eqs.  and ): 7LWin=eσT4,8e=ecs(1-n2)+ecln2,9ecs=0.23+cVPT0.125, where σ is the Stefan–Boltzmann constant (W m-2 K-4), ecs clear-sky emissivity, c a calibration parameter (K0.125 Pa-0.125), VP vapor pressure (Pa), T air temperature (K), e sky emissivity, and ecl cloud emissivity.

We calibrated several parameters of the radiation routines (Table ) to reflect the local conditions of the high-alpine CG site. We selected the parameters to be tuned based on their relevance for our site, the availability of local measurements, and the simplicity of comparison within the model result. Within the albedo routine we optimized parameters αfresh, twet*, and K (Eqs.  and ), which refer to situations often observed at CG (respectively a fresh snow surface, a melting surface, and a sub-freezing surface). By contrast, we kept the default values for αfirn and tdry*, as they correspond to conditions which are almost never encountered at our site (respectively a bare firn surface and a non-melting surface at 0 ∘C). We also used the original formulations of trg, tw, and ta (Eq. ): this is because the modeled incoming SW flux (Eqs.  and ) is derived from measured radiation at CM, through the series of reconstructed cloud cover (Sect. 2.1). Thus, the effect of these transmissivity coefficients is already taken into account in the cloud cover series. Furthermore, we optimized parameters c and ecl in the LW radiation module (Eqs.  and ), since locally measured values were available from the Seserjoch station.

To perform the tuning, we applied the EBFM radiation routines individually (outside the full model runs), driving them with the meteorological and energy-balance measurements of the Seserjoch and Colle del Lys stations at 10 min resolution. Then, we adjusted the parameter values to find the best match (in terms of bias and RMSE) between the simulated and measured series of albedo and incoming LW radiation.

In the EBFM, turbulent heat exchange is modeled with the glacier katabatic wind parametrization of . This was developed with a focus on large valley glaciers; notably, it computes heat fluxes which are independent of the ambient wind field, since wind speeds are estimated from the katabatic flow model . This was deemed inadequate for a high-alpine, wind-exposed saddle: thus we re-implemented the EBFM computation of turbulent heat exchange, following the bulk aerodynamic equations of . These were chosen due to their operational simplicity, allowing calculation of turbulent fluxes from a single measurement level of wind speeds. The fluxes are computed as 10QSH=ρacp,aChV1(T1-Ts),11QLH=ρaLs,vChV1(q1-qs), where ρa is air density (kg m-3), cp,a specific heat of dry air (J kg-1 K-1), V wind speed (m s-1), Ls,v latent heat of sublimation or vaporization (J kg-1, chosen depending on the modeled surface temperature), and q specific humidity (kg kg-1), and the s and 1 subscripts refer respectively to the snow surface and the measurement level (2 m). Exchange coefficient Ch is defined as 12Ch=Chnfh, where Chn is the value under neutral conditions and fh a correction for atmospheric stability, expressed in terms of the bulk Richardson number RiB. 13Chn=k2log⁡z1z0-214fh=(1+10RiB)-1RiB≥0(stable)1-10RiB1+10Chn-RiBfz-1RiB<0(unstable)15RiB=gz1V12T1-TsT1+q1-qsq1+ϵ(1-ϵ)-116fz=14z0z10.5 Here, k is the von Kármán constant, z1 the measurement level (m), z0 the surface roughness length (m), g the gravity acceleration (m s-2), and ϵ the ratio of molecular weights for water and dry air.

At CG, surface roughness length is a poorly constrained parameter, due to frequent scouring by extreme winds which alter the snow surface. In our simulation, we used the value computed by from measurements of wind profiles at Seserjoch. In Appendix B we examine the sensitivity of our simulation to this parameter (Fig. a and b).

We adapted the model precipitation routine to reproduce the extreme spatial gradient of snow accumulation distinctive of the site (Sect. 1). Since the EBFM does not include a blowing snow routine, the simple model of linear precipitation rates with altitude was replaced by a gridded precipitation time series, already corrected for snow lost to wind scouring. This was computed with a three-phase anomaly method inspired from , by combining a fixed climatological grid, an annual anomaly series, and a temporal down-scaling coefficient. Specifically, for grid cell (x,y), at simulation time step t, in year i, precipitation was expressed as 17P(x,y,t)=C(x,y)A(i)D(t), where C is the long-term annual accumulation climatology (m w.e. yr-1), A the domain-wide annual anomaly, and D the down-scaling coefficient. The main assumption of the method is that spatial patterns of relative accumulation do not change over time. The climatological grid C was assembled by interpolating point values of long-term net accumulation, estimated from the snow mass of dated firn cores and from the mean layer thickness in GPR profiles (Fig. a). To extend data coverage and reduce the occurrence of extrapolation, stake measurements from were also used in the western and southern domain regions. Because single-year stake measurements are in principle not representative of the long-term means, we re-scaled their values with a conversion factor. We computed this as the mean ratio between the stake and the firn core/GPR point values, taken at the locations of overlap.

For annual anomalies A, the multiplicative snow mass anomaly of the KCC deep core was found to be moderately anti-correlated with wind speed measured at the CM AWS (Fig. b). Linear fit over nine annual data points yielded the formulation 18A=1.46-0.21V‾, where A is core snow mass anomaly and V‾ the median wind speed (m s-1) observed at CM over the corresponding year. The computed time series of annual anomaly was then applied to the whole domain. Equation () represents a minimal model of the inter-annual variability of wind scouring at CG, accounting for increased erosion rates at higher wind speeds.

Finally, the down-scaling coefficients D were computed by normalizing to a unit sum each year of hourly precipitation, averaged over the three closest rain gauges (Passo Monte Moro, Bocchetta delle Pisse, and Rifugio Zamboni: Fig. b, Table ). Equation () produces an hourly series of gridded precipitation (already corrected for wind erosion) which was used to force the EBFM. The model uses local air temperature to compute the fraction of precipitation falling as snow: this increases linearly from 0 % to 100 % within a 2 ∘C interval, symmetric around a threshold Ts/r=0.6 ∘C.

The EBFM represents the sub-surface with a Lagrangian discretization: model layers move freely along the depth axis, following the addition or removal of mass at the surface. A new layer is created whenever snowfall and riming push the topmost layer thickness beyond a fixed threshold. This approach prevents numerical diffusion and at the same time accounts for heat advection towards depth .

Layer temperature evolves according to processes of heat conduction and water refreezing : 19ρfcp(Tf)∂Tf∂t=∂∂zκ(ρf)∂Tf∂z+FLm, where ρf and Tf are layer density (kg m-3) and temperature (K), cp firn heat capacity (J kg-1 K-1), z depth (m), κ effective conductivity (W m-1 K-1), F refreezing rate (kg m-3 s-1), and Lm latent heat of melting (J kg-1). Parametrizations for cp and κ are taken respectively from and : 20cp(Tf)=152.2+7.122Tf,21κ(ρf)=0.138-1.01×10−3ρf+3.23×10−6ρf2. Layer density is governed by gravitational settling and water refreezing : 22∂ρf∂t=Kg(ρf,Tf)+F,23Kg(ρf,T)=baccg(ρice-ρf)exp⁡-EcRTf+EgRTavg⋅CLig(bacc). Here, Kg is gravitational densification (kg m-3 s-1) as in , bacc accumulation rate (mm yr-1), ρice ice density (kg m-3), R universal gas constant (J mol-1 K-1), Ec (60 kJ mol-1), and Eg (42.4 kJ mol-1) activation energies of creep by respectively lattice diffusion and grain growth, and Tavg is year-averaged firn temperature (K). CLig is a correction based on the accumulation rate, accounting for different densification regimes above and below the critical density value of 550 kg m-3 .

In the EBFM the only source of water at depth is infiltration from the surface (after melt, rainfall, and moisture condensation), since sub-surface melting is not simulated. The model features a parametrized routine to account for preferential percolation : as long as the near-surface layers are not impermeable, liquid water is instantly routed from the surface to a prescribed sub-surface distribution, defined by its shape along the vertical axis (constant, linear, or Gaussian) and maximum depth reached (zlim). Here we selected the Gaussian profile, and we tuned zlim to a value of 4 m, to match the 20 m firn temperature of the SP location (Fig. a) as measured by . The chosen reference depth allows the minimization of the impact of the annual temperature cycle and of any recent temperature trends.

Water in the sub-surface can subsequently refreeze until reaching an upper bound on layer density (the value of glacier ice) and temperature (the melting point). Excess water is partly retained by capillary and adhesive forces (irreducible water content) and partly routed to deeper layers until it is depleted; if an impermeable layer is reached, the resulting slush water drains gradually according to the linear reservoir model . The maximum irreducible water content θmi of a layer (kg kg-1) is computed from its porosity np following : 24θmi=0.0143exp⁡(3.3np). Appendix B presents a sensitivity analysis of the EBFM to parameters zlim and θmi, among others; further details on the sub-surface model can be found in and .

We initialized the model grid to steady-state conditions by looping eight times over the 2004–2011 weather input. The spin-up duration (64 years) enables a complete adjustment of the whole grid to the mean surface forcing (up to 20 m depth), thus avoiding transitory periods at the beginning of the actual simulation. The selected sub-period excludes the extreme melt year of 2003 and the increasing temperature warming of the 2010s. Then we performed two main model runs, with 20 m/1 h and 100 m/3 h spatiotemporal resolution. We introduced the coarser version to decrease the large computational volume of sub-surface investigations (Fig. ) and to examine the impact of different spatiotemporal resolutions. On the depth axis, the model grid included 250 layers up to 10 cm thick, for an effective modeling depth of about 20 m due to layer compaction.

Comparison of modeled firn temperatures to measured borehole profiles (Table , Figs.  and ) shows that most model deviations are similar in magnitude to the spatial variability of firn temperatures. Indeed, profiles CG08-1/08 and CG08-2/08 (acquired in the flat region, on the same day, and within a radius of 20 m: Fig. e and f) report measured temperature differences in excess of 2 ∘C at all depths. Model residuals show a clear spatial pattern: simulated profiles tend to be too cold on shaded north-facing slopes and too warm in the flat or south-facing regions of CG and Seserjoch. As such, model bias is moderately correlated with mean annual accumulation and potential incoming solar radiation (correlation coefficients of respectively 0.42 and 0.64). Moreover, model residuals appear to be maintained over time for boreholes with repeated measurements (e.g., CG05-1 and CG13-1).

The time series of modeled firn temperatures (Fig. ) show large differences over rather small distances at the CG saddle, depending on surface aspect. Still, relative annual deviations are consistent across locations. The annual cycle on average reaches an amplitude of 40 ∘C at the snow surface and is fully damped at a depth of 20 m.

The model simulates the frequent summer occurrence of infiltration and refreezing, reflected in sudden near-surface warming events (inset in Fig. ). Magnitude of these events shows a high variability, with especially large heat amounts simulated in the summers of 2003, 2008, 2015, and 2017. With more meltwater refreezing, the sun-exposed ZS slope commonly shows sustained near-melting temperatures in the topmost 4 m; conversely, these conditions almost never appear at the shaded SK location. A slight positive temperature anomaly can be seen at depth after the extreme melt year of 2015, persisting over the following years despite non-record melt amounts.

The spatial distribution of 20 m firn temperatures (Fig. a) shows strong spatial gradients, reflecting surface elevation and aspect. Temperatures range from -17 ∘C at the shaded NE cliff to near-temperate conditions below 4300 m on the Grenzgletscher slope (Fig. a).

Modeled temperature trends are relatively uniform over the saddle area above 4400 m a.s.l., where the overall 2003–2018 warming at 20 m amounts to 0.64–0.75 ∘C (Fig. b). The steep NE cliff constitutes an exception: there, the simulation indicates a very slight and non-uniform tendency towards decreasing temperatures. At lower elevations, temperature trends have a much higher spatial variability, ranging from 0 ∘C yr-1 in the near-temperate area to 0.13 ∘C yr-1 (or 2.5 times the atmospheric warming rate) on the west-facing slopes, about 200 m away.

Modeled mean annual melt amounts have an extreme spatial variability (Fig. ), broadly reflecting surface elevation, slope, and aspect. Values increase from less than 1 cm w.e. yr-1 on the steepest slopes of the Signalkuppe to 17 cm w.e. yr-1 at the saddle point and about 23 cm w.e. yr-1 on the Zumsteinspitze slope. Even higher melt amounts, exceeding 30 cm w.e. yr-1, are simulated for the lower-elevation Grenzgletscher slopes (towards the western border of the domain) and for Seserjoch. Grid average is 21 cm w.e. yr-1. Within the overall SEB, melt represents a relatively minor component (Fig. ): the largest mean monthly contribution, in August, is well below 10 % of the total energy turnover, and in every month sublimation is a more effective energy sink than melt. Still, refreezing at depth transfers heat deep into the snowpack (Fig. ), compared to the slower processes of diffusion and advection which proceed from the surface. In the NE domain region – where wind scouring is strongest – annual melt amounts correspond to a significant fraction of net accumulation (Fig. a). With regard to temporal patterns, the entire surface was found to always refreeze at night over the modeled period. Also, no melt is simulated between November and March, with only minor amounts in April and October (up to 1 % of the annual totals at ZS; Fig. ).

Despite the large spatial heterogeneity, and an inter-annual variability exceeding 50 %, a common trend of melt increase could be detected in the annual time series (Fig. ): the fitted slope ranges from (3±2) to (4±3) mm w.e. yr-2 across the saddle. While the trend is somewhat masked by the 2003 extreme melt year at the very beginning, it becomes statistically significant over the rest of the period (p<0.05 for 2004–2018).

The EBFM shows a marked tendency towards small melt amounts: frequency of modeled melt events decays exponentially with their magnitude (Fig. a), and a significant fraction of total melt amounts is contributed by micro-melt events under 4 mm w.e. in a single day (Fig. b).

Investigation of the weather conditions leading to melt occurrence (Fig. ) reveals the relationships between weather variables and surface melt. Air temperature provides a critical control over melt rates, unlike cloud cover, which appears to have almost no effect (Fig. a). The majority of melt amounts happens at slightly positive air temperatures (Fig. b), which correspond to clear-sky conditions (cloud cover <0.1), in more than 50 % of the cases. Still, significant melt is simulated between −5 and 0 ∘C: at SP, SK, and ZS melt at sub-freezing air temperatures accounts for respectively 22 %, 34 %, and 17 % of the total simulated amounts. Non-zero (though minimal) melt amounts are modeled down to air temperatures of −7 ∘C, under clear skies, and moderate to high humidity. Conversely, in very dry conditions (Fig. c and d) sublimation losses hinder melt even at slightly positive temperatures. Finally, wind speed appears to have a minor effect on mean melt rates. Enhanced turbulent heat losses can be seen slightly decreasing the likelihood of melt under high winds, at air temperatures between -5 and 0 ∘C (Fig. e and f).

The EBFM shows considerable potential at simulating cold firn. In addition to reproducing individual borehole profiles, the model confirms broader patterns such as the strong firn temperature gradient towards the Grenzgletscher slopes and the depth of zero annual temperature oscillation, at about 20 m . This last observation indicates a realistic simulation of heat conduction within the firn pack.

The aspect-dependent spatial pattern of temperature residuals (Fig. ) could be affected by several factors. The locations of the largest temperature under-estimation coincide with a positive bias of modeled density by 5 %–30 % compared to measured core profiles below 2 m depth. This density bias leads to a higher thermal conductivity (Eq. ), resulting in colder firn temperatures as shown in Appendix B. The density bias could be due to a lack of local calibration for the accumulation-dependent densification model, developed over Antarctic firn . Another possibility is a deep density increase caused by the percolation routine. Specifically, after each melt event water is distributed between the surface and the depth of zlim, and there it refreezes, increasing density over that entire vertical extent – even in the case of repeated melt–freeze cycles (which would melt a same ice surface, not increasing density). We expect this effect to be especially significant at locations where melt amounts represent a large fraction of accumulation: as such, it would be amplified by the accumulation model, which computes precipitation amounts already corrected for losses from wind scouring. Indeed, accumulation at CG results from summer precipitation events (Sect. 1), but modeled precipitation is distributed more evenly throughout the year, as it is based on weather station measurements from lower elevations (Sect. 3.3). Thus, we expect an under-estimation of the strong seasonal gradient that favors summer accumulation, so that modeled melt and refreeze can temporarily approach (or even locally exceed) the low accumulated snow amounts in summer, hence exacerbating the density bias. Simple sensitivity tests support this interpretation, by showing that the simulation of temperature profiles at low-accumulation borehole CG05-1 (Figs.  and ) improves substantially with an artificial increase in precipitation amounts. Future development of the percolation model could mitigate the density bias (and thus improve the firn temperature simulation), for example introducing a dependence of percolation depth on the meltwater supply: this would prevent very small meltwater amounts from percolating to unrealistically large depths and escaping repeated melt–freeze cycles.

A deficit in summer precipitation totals in the model (with a corresponding winter excess) could also introduce a systematic deviation in the advected heat, due to a different deposition temperature of snow between summer and winter. The resulting firn temperature bias would be roughly proportional to mean accumulation (Appendix B): thus, a more extreme accumulation seasonality (with no winter precipitation) could potentially amplify the spatial pattern of modeled temperature residuals (Fig. ). Still, complete removal of winter precipitation would be problematic for the albedo decay routine (Eq. ). A more realistic model of wind scouring should account for the timing of snow erosion, which at CG can happen several months after deposition e.g.,.

Some boreholes with positive model bias (Fig. ) are simulated with too strong near-surface temperature gradients, leading to sharp positive deviations in the profile near the surface (e.g., Fig. f, k, p, q). Such a behavior suggests too deep refreezing is occurring in their simulation, which could again be linked to the parametrized preferential infiltration routine (Sect. 3.4). Indeed, below a depth of 0–4 m (where refreezing is occurring) the simulation appears to be unbiased at most locations. The vertical distribution of refreezing is also affected by residual saturation θmi, controlling water storage after melt events and potentially affecting firn temperatures. In practice, we show in Appendix B that sensitivity to this parameter is very low in our setup and limited to high-melt locations.

The spatial pattern of model residuals (Fig. ) could in principle be affected by the lack of SW radiation reflected from the surrounding terrain in the modeled SEB. This process could induce a net energy transfer from the more sun-exposed cells towards the more shaded ones. We quantified its magnitude applying a simple Lambert reflection model e.g., to the modeled series of SW radiation reflected by each grid cell. We found that radiation interception can indeed introduce a mean energy flux of up to about 3 W m-2 in the SEB, but the net effect (taking into account mutual radiation exchange and the high surface albedo) is always smaller than ±0.2 W m-2. As such, redistribution of reflected radiation cannot be considered a major contributor to the aspect-dependent model temperature residuals. Another effect which is not captured by our SEB formulation is the penetration of SW radiation within the snowpack. This process – enabling sub-surface melt – challenges the EBFM assumption of water originating entirely at the snow surface and was found to attenuate a cold bias in firn temperatures within the model of at CdD.

The relative spatial distribution of modeled firn temperatures (Fig. a) is in good agreement with the interpolated result shown by . Features such as the regular temperature gradient across the CG saddle and the near-temperate south-facing slope below 4300 m a.s.l. are reproduced well. The high spatial variability of firn temperature trends (Fig. b) is consistent with the borehole observations of and the model results of at CdD. Below 4400 m a.s.l., firn warming rates are generally high on west-facing slopes and low on south-facing ones: this could be related to a prevalent role of solar radiation over air temperature in the firn thermal regime at south-facing locations. As warming progresses, it is possible that such pronounced spatial patterns also migrate towards the higher saddle region, where warming rates are at present more uniform. In the context of increasing firn temperatures and melt amounts, the large spatial variability of warming trends will likely have a growing importance for the localization of future ice core drilling campaigns.

Among past modeling efforts at CG, both and formulated independent predictions for firn temperature evolution by 2020. As the target time frame for verification is reached, a major limitation of their modeled scenarios is found in the expected magnitude of atmospheric warming: while they assumed linear air temperature increases by respectively 0.4 and 0.45 ∘C between 2000 and 2020, the fitted trend over the CM AWS annual means (series in Fig. ) amounts to (0.05±0.03) ∘C yr-1, corresponding to a much stronger warming of 1 ∘C over the period. For the CG saddle region, the two studies predict firn temperature increases at 18 m by 0.42 and 1.06 ∘C respectively between 2000 and 2020. By comparison, the EBFM simulates 0.70 ∘C of warming between 2003 and 2018 at SP, which can be uniformly rescaled over 2000–2020 to 0.88 ∘C. Such a firn warming is consistent with the results of the two studies but lower when compared to the measured air temperature increase. A contributing factor could be the EBFM spin-up: temperatures were initialized with repeated model runs over 2004–2011; thus the initial grid at all depths is in equilibrium with the mean forcing over that period. By contrast, an adjustment time of 2–4 years is to be expected at the considered depth . Therefore at the beginning of the simulation the EBFM may slightly over-estimate deep firn temperatures (Fig. a/b), resulting in a lower trend for 2003–2018.

At the CdD site, estimated firn warming by 2030 and 2050 using a thermo-mechanical coupled model, forced by three climate projections within the A1B emission scenario . They found a dependence of the warming rate on advection velocities and percolation amounts. Their reported firn warming at 20 m depth is in the range 0.0–1.8 ∘C over 2010–2030 (depending on location and climate scenario), similar to our results modeled for 2000–2020 over the CG domain (Fig. b).

A key parameter used for model setup is the percolation depth zlim, which defines the maximum depth reached by infiltrated meltwater through preferential flow. In cold firn such a parameter is crucial: simulated firn temperatures (Fig. ) indicate that all meltwater can be expected to refreeze not far from the initial location (controlled by the parametrized vertical distribution: Sect. 3.4). Thus zlim effectively determines not only the initial meltwater distribution, but also the depth of latent heat release. The effect of this parameter on firn temperatures is indeed large at all depths (Appendix B). The calibrated value of 4 m, together with the Gaussian vertical distribution, corresponds to a mean preferential percolation depth of 1.06 m. Unfortunately, in situ quantitative measurements of infiltration depths are scarce and mostly indirect. Evidence from winter snowpacks and glacier accumulation areas shows that percolation and refreezing are strongly dependent on the meltwater supply amounts and rates , the temperature of the firn matrix , and its stratigraphy , notably affected by the previous history of infiltration and refreezing. An often observed consequence is the increase in percolation depths over the melting season . At CG, observed slight melting but no meltwater percolation during a warm spell in summer 1981. attempted to directly track meltwater refreezing by continuously logging a temperature profile at Seserjoch in 1999, but the setup failed before the onset of summer melt. reported evidence for increasing infiltration depths, indicating (at Seserjoch and on the Grenzgletscher slopes) a transition to a percolation regime spanning several annual firn layers. On the cold firn of CdD, tracked sub-surface temperatures over the summer of 2012, measuring percolation depths of up to 4–5 m: these would match the value of zlim in our simulation. However, the stratigraphy of core unifr-2019 (Fig. ) revealed dry firn layers interspersed with thin ice crusts, suggesting small amounts of infiltration and refreezing, except for a thick, ice-rich layer at 4.5 m depth. Therefore in our simulation we consider zlim more as a tuning parameter than a realistic percolation depth.

Stratigraphy of the firn core points to a shortcoming of the subsurface model: meltwater distributed in cold firn along the parametrized vertical profile usually refreezes in place, producing a diluted density increase in the simulation. Instead, distinct ice layers are known to form at depth after preferential percolation, affecting the mechanical, hydrological, and thermodynamical properties of the snowpack . In the present EBFM formulation, such ice layers would not appear even with a very fine model grid. Refreezing after parametrized infiltration also distributes heat instantly over a fixed, large vertical extent, resulting in unrealistic, frequent warm pulses at depth after each melt event – no matter how small (inset in Fig. ). These observations provide motivation for future work on a physically based percolation routine in the EBFM, accounting for the time evolution of zlim and its dependence on snow density and stratigraphy.

Modeled melt amounts can be compared to the amount of refrozen ice observed in core unifr-2019 (Fig. ). According to the computed climatology (Fig. a), the core location has a mean long-term accumulation of about 50 cm w.e. yr-1: thus the 5.5 m core should span an estimated period of about 5–6 years (with a fairly large uncertainty due to the inter-annual accumulation variability). The model predicts about 85–100 cm w.e. of melt over such a time span (Fig. ). Instead, the core was found to contain only 31 cm of ice layers. These correspond to just 14 cm w.e. of refrozen ice after subtraction of the mean density of the ice-free sections (such a correction is fairly uncertain due to the high density variability of the profile). Some observations can be brought forward to put the apparent discrepancy into context.

First, refrozen ice amounts recorded in the core are affected by repeated cycles of melt–refreeze. Indeed, the very small amounts of meltwater produced during less intense (but rather frequent: Fig. ) melt events can be expected to refreeze in the very first snow centimeters. Then any subsequent melt occurring before the next snowfall would affect the same ice surface, contributing to total melt amounts but without significant increases in ice layer thickness. Such surface crusts of relatively impermeable ice have already been observed at CG e.g.,. Since melt mostly happens in clustered patterns (almost only in summer and within a specific set of weather conditions: Figs.  and ), contribution of repeated melt–refreeze cycles could potentially be very large.

Because ice-equivalent thicknesses of daily melt amounts are often of the same order as the size of single crystals (Fig. a), it is suggested that detection of some refrozen forms would require a resolution not achieved during our field analysis. This is supported by the observation in the core of sections of icy firn, as opposed to well-defined ice layers (see Appendix A). Investigations of such minimal melt processes are very limited in the literature. Still, performed hot-box experiments on the formation of thin refreeze layers in Antarctic snow, finding that a profile resolution of 1 mm was necessary to capture small-scale melt processes. Moreover, in their experiments the wetted and refrozen snow next to melt layers did not show any type of melt feature detectable in firn core stratigraphy , possibly enabling some refrozen layers to remain undetected. The possibility of an overlooked vertical ice gland embedded in our core also cannot be ruled out (measured core sections were not broken up after analysis). In fact, the presence of undetected ice would be consistent with the high density variability encountered in the core profile (Fig. ). Still, this observation could also be linked to wind compaction and its interplay with wind erosion exposing older, denser snow.

proposed an estimation of refreezing amounts at CG from the measured density anomalies over ideal dry densification profiles. At the Sattelkern and Zumsteinkern cores (respectively close to SP and to ZS) the reported refreezing rates have confidence intervals of 1–13 and 3–33 cm ice yr-1 . The EBFM predicts approximately 19 and 25 cm ice yr-1 of melt. On the shaded Signalkuppe flank, values span the range 0–15 cm ice yr-1 from multiple cores; the EBFM result over the same region is between 5 and 12 cm ice yr-1 of melt. Considering the very strong inter-annual variability of melt amounts (Fig. ), the results are largely compatible.

The same density anomaly approach had been applied at CdD by to estimate melt amounts of the 2011 summer season. Based on 14 firn density profiles (measured between 4230 and 4310 m a.s.l.) the authors computed melt amounts in the range 1–18 cm w.e., significantly correlated with potential incoming solar radiation. At both CG and CdD the density anomaly method shows considerable potential and consistent results but does not account for the occurrence of repeated melt–refreeze cycles. Moreover, at CG the method involves sizable uncertainties reflecting a high variability of surface density .

The significant melt amounts modeled at sub-freezing 2 m air temperatures suggest reconsideration of degree-day models for simulating melt at high-alpine (and possibly high-latitude) locations. This is consistent with the findings of at Seserjoch, who observed several surface melt events but no days with positive mean temperatures over the whole 1999 summer. Indeed, laboratory experiments by revealed the occurrence of melt already at −4 ∘C with 475 W m-2 of incoming SW radiation. At the CM AWS, values in excess of 1000 W m-2 are a common summer occurrence (195 h per year on average in the 2003–2018 series). This supports the plausibility of melt under even colder conditions (Fig. ). From a theoretical perspective, analytically explored a standard SEB equation (neglecting sub-surface heat conduction), in relation to common weather situations on an alpine glacier. The conclusion was that melt onset can likely happen at air temperatures between −10 and +10 ∘C: our results appear to corroborate such a range (Fig. ).