The one-dimensional energy balance model is a further development of the models presented by , , and ; here only the main features are described. The energy balance of an infinitesimally thin surface layer (the “skin” layer) is defined as follows: 1M=SW↓+SW↑+LW↓+LW↑+QS+QL+QG, where positive fluxes are defined to be directed towards the surface. SW↓ and SW↑ are the incoming and reflected shortwave radiation, LW↓ and LW↑ are the downward and upward longwave radiation, QS and QL the turbulent sensible and latent heat fluxes and QG is the conductive subsurface heat flux. We neglect latent energy from rain. M is the energy used to melt snow or ice and is non-zero only when the surface has reached the melting point of ice (Ts=273.15K). Throughout this paper, melt and accumulation amounts are expressed in terms of millimetre water equivalent (mm w.e.), which equals kg m-2. In order to calculate QG and allow for densification, meltwater percolation and refreezing, a snow–firn model is used, initialised with 70 layers. The layer thickness varies from 1 cm at the top to 2 m at the bottom (25 m depth). We impose a no-energy flux boundary condition at the lowermost model level. New snow density is parameterised following the expression of , which relates it to the prevailing surface temperature (Ts) and 10 m wind speed (V10m) and imposes a lower limit of new snow density ρs,0. Meltwater percolation is based on the tipping-bucket method e.g., allowing for immediate downward transport (within a single timestep of 10 s) of remaining water if a layer has attained its maximum capillary retention, as modelled using the expressions of . Meltwater refreezing increases the density and temperature of a layer. At the bottom of the firn layer, the meltwater is assumed to run off immediately, i.e. the model does not allow for slush/superimposed ice formation or lateral water movement. Turbulent fluxes are calculated following the “bulk” method, which is based on Monin–Obukhov similarity theory (see e.g. for relevant equations) between a single measurement level (2 m for temperature and humidity, 10 m for wind) and the surface, assuming the latter to be saturated with respect to ice and using the stability functions according to for unstable and for stable conditions.

Subsurface penetration of shortwave radiation is calculated using a spectral model , based on the parameterisation by , which is in turn based on the two-stream radiation model of . The impact on modelled melt and the quantification of the SMAF is discussed in the relevant sections.

The terms in Eq. () are either based on observations or can be expressed as a function of the skin temperature Ts. The SEB is solved iteratively by looking for a value of Ts that closes the SEB to within 0.005 K between iterations: if Ts>273.15K, it is reset to 273.15 K and excess energy M is used for surface melt. To evaluate model performance, the modelled value of Ts is compared to observed Ts calculated from LW↑, using Stefan–Boltzmann's law for a longwave emissivity ϵ=1: 2LW↑=σTs4, where σ=5.67⋅10-8 W m-2 K-4 is the Stefan–Boltzmann constant.

Surface roughness lengths for momentum, heat and moisture are related through the expression of : 3ln⁡z0*z0,m=a1+a2ln⁡Re*+a3ln⁡Re*2, where z0* represents either z0,h or z0,q, the roughness lengths for heat and moisture respectively, and a1, a2 and a3 are coefficients determined by for various regimes of the roughness Reynolds number Re*=u*z0,mν with kinematic viscosity ν and friction velocity u*.

Because the shortwave radiation sensor faces the sky and includes a significant direct component, measured SW↓ suffers from relatively large uncertainties owing to poor sensor cosine response, sensor tilt and/or rime formation . In order to improve the accuracy of observed net shortwave radiation used in the SEB calculations (Sect. ), we calculate SWnet based on SW↑, which is diffuse and hence much less sensitive to these errors. To further decrease the impact of these errors, we use a 24 h moving average albedo, as described in . In Sect. , in which albedo is parameterised to study melt–albedo feedbacks, for consistency we use measured SW↑ in combination with parameterised albedo to estimate SWnet.

In Sect. , the parameterised surface albedo α is described as a base albedo αS, modified by perturbations describing the effect of changing solar zenith angle θ (dαu), the cloud optical thickness τ (dατ) and the concentration of black carbon in the snow (dαc) : 4α=αS+dαu+dατ+dαc. For Antarctica, we neglect the impact of impurities in the snow (dαc=0); dαu and dατ both depend on the base albedo αS, dαu in addition depends on the solar zenith angle (u=cos⁡θ) and dατ on the cloud optical thickness τ: 5dαu=0.53αS(1-αS)(1-0.64x-(1-x)u)1.2,6dατ=0.1τ(αS+dαc)1.3(1+1.5τ)αS, where x=min⁡τ3u,1. The base albedo depends on the snow grain size re (in metres): 7αS=1.48-1.27048re0.07, in which the snow grain size re on time step t is parameterised as 8re(t)=re(t-1)+dre,dry+dre,wetfo+re,0fn+re,rfr. Here, dre,dry and dre,wet describe the metamorphism of dry and wet snow respectively, fo, fn and fr are the fractions of old, new and refrozen snow, and re,0 and re,r are the grain sizes of new and refrozen snow. Dry snow metamorphism is parameterised following : 9dre,drydt=dredt0η(re-re,0)+η1/κ, where re,0 is the new snow grain size, and the coefficients dredt0, η and κ are obtained from a look-up table. This look-up table is compiled based on simulations with the SNICAR model , which calculates the snow metamorphism resulting from temperature gradient metamorphism. dre,wet is a function of the snow grain size re itself and the liquid water content fliq : 10dre,wetdt=Cfliq34πre2, where C is a constant (4.22⋅10-13 m3 s-1).

The fractions fo, fn and fr are derived from the snow/firn model, and the grain sizes of new and refrozen snow are constants; the method for determining their values from a tuning exercise is described in Sect. .

To determine cloud optical thickness τ, an empirical relation between τ and the longwave-equivalent cloud cover Nϵ is used following : 11τ=c1exp⁡(c2Nϵ)-1, with fitting parameters c1 and c2. Nϵ is determined using a method described by , which relates hourly values of downward longwave radiation LW↓ to near-surface air temperature T2m as illustrated in Fig. a. Red lines indicate quadratic fits through the upper and lower 5 percentiles of the data, assumed to represent fully cloudy and clear conditions, respectively. Nϵ is obtained by linearly interpolating between these upper and lower bounds, yielding values between 0 and 1. Hourly values for cloud cover are then used to obtain values for τ (Fig. b). The values used for the fit parameters c1=5.404 and c2=2.207 (both dimensionless) differ somewhat from , who used daily values for the fit.

The SEB model is forced with data from the meteorological observatory at the German research station Neumayer, situated on the Ekström ice shelf . The observatory has been operational since 1981 and was relocated in 1992 and 2009. In 2016, its location was 70∘40′ S, 8∘16′ W (Fig. ). The observatory is one of only four Antarctic stations – and the only one situated on an ice shelf – that is part of the Baseline Surface Radiation Network (BSRN), a global network of stations with high-quality (artificially ventilated) radiation observations, coordinated by the Alfred Wegener Institute (AWI). We use hourly averages of 2 m temperature (T2m) and specific humidity (q2m), 10 m wind speed (V10m), surface pressure (p) and radiation fluxes for the period April 1992–January 2016 (24 years) to force the SEB model; their uncertainty ranges are provided in Table . Approximately 4.1 % of the data points contained at least one missing variable, which mostly come from daily performed visual inspection of the data. To obtain a continuous data set, all missing data were replaced: pressure, relative humidity, wind speed, temperature and longwave radiation were simply linearly interpolated. In the case of shortwave radiation, the missing value was replaced by imitating the average daily cycle of the 2 preceding days. As the measurement station is visited and maintained every day, the impact of rime formation is limited, as is the tilt of the observation mast, resulting in a high-quality meteorological data set.

Accumulation observations are only available from stake measurements, provided by AWI, which were performed weekly for the period April 1992–January 2009. As timing of precipitation is important for correctly simulating the effects of new snow on snow albedo, we combined the stake observations with precipitation predicted by the regional atmospheric climate model RACMO2.3p2 to obtain realistic timing of precipitation in between stake observations, as well as for the post-2009 period. The amount of precipitation modelled by RACMO2 was scaled such that the modelled surface height changes agree with stake measurements; this required a 15.3 % upward adjustment of the modelled precipitation flux.

There are several SEB model parameters for which the exact values or formulations are unknown, e.g. the surface roughness length for momentum z0,m, the density of new snow ρs, the stability functions (required to calculate the turbulent scales) and the effective conductivity, which couples the magnitude of QG to the temperature gradient in the snow. We estimated the impact of observational and model uncertainties on modelled melt by running the model 600 times while randomly varying all hourly observations within the specified measurement uncertainty ranges (Table ) and using multiple expressions for the heat conductivity and stability functions. Model performance is quantified by comparing modelled with observed Ts and assessing the changes in modelled 24-year cumulative melt. Note that in this section, the albedo based on observations is used to obtain SWnet.

The choice of expressions for the stability functions and heat conductivity did not significantly impact the modelled amount of melt (total within 30 mm w.e. or 2.7 %, not shown). The model outcome is more sensitive to the choice of surface roughness length for momentum z0,m and the lower limit of density of new snow ρs,0: when z0,m is varied between 0.5 and 50 mm and ρs,0 between 150 and 500 kg m-3, the cumulative amount of surface melt over the 24-year period varies between 900 and 1220 mm w.e., with higher melt values for smaller values of z0,m and ρs,0. Optimal values in terms of simulated Ts are z0,m=1.65mm and ρs,0=280 kg m-3, resulting in a Ts bias of 0.01 K and an RMSD of 0.79 K (Fig. ). We use these values in the remainder of this study. Figure a and b show modelled 24-year cumulative melt and annual melt (March–February) at Neumayer, combined with uncertainties associated with model parameters. The annual values for year X are obtained by summing monthly values for March of year X until February of year X+1. The total melt amounts to 1154 mm w.e., with a small uncertainty associated with measurement uncertainties (1σ≈3mmw.e., i.e. 0.3 %). The method adopted to estimate this uncertainty has its limitations, as measurement errors are probably autocorrelated: if a measurement at one time is disturbed in some way, it is probably disturbed in a similar way at the next time step. Therefore, this result could be interpreted as a lower bound of the uncertainty range, which is supported by the larger uncertainty estimates (∼15 %) by , who applied a constant systematic error which can be interpreted as an upper bound on the modelled uncertainty range. This also explains why the model outcome is much more sensitive to different values of z0,m, as these runs effectively introduce a systematic error between the true (unknown) value and the chosen value. Furthermore, this approach assumes the true value to be constant, which likely is an oversimplification .

The sensitivity of modelled cumulative melt to z0,m is somewhat unexpected. Following Eq. () both z0,h and z0,q decrease for increasing z0,m; in combination with the bulk method this acts to dampen the effect of z0,m on the magnitude of the turbulent fluxes. Our interpretation of this result is that decreasing z0,m and ρs,0 lead to a decrease in the turbulent fluxes as well as the ground heat flux QG. This reduces the efficiency with which heat is removed from the surface, in turn allowing more energy to be invested in melt. The obtained value of z0,m=1.65mm is high compared to the average value of z0,m=0.1mm found during a field campaign at Neumayer in 1982 but it is not uncommon for snow surfaces .

Measured values of Ts in excess of the melting point in Fig.  only occurred in the first six seasons; from 1998–1999 onwards they were removed by additional post-processing. These measurements mainly reflect uncertainties in the adopted unit value of longwave emissivity and in measured LW↑, e.g. from sensor window heating and the fact that the downward-facing radiation sensor also measures longwave radiation emitted by the relatively warm air between the surface and the sensor.

Annual (March–February) mean values of near-surface meteorological quantities and SEB components are presented in Table , with seasonal cycles of SEB components presented in Fig. b. These show that the summertime SEB is dominated by the radiation fluxes; despite the high albedo of the snow surface, SWnet is the dominant heat source for the skin layer, whereas LWnet extracts energy from the surface, most efficiently so in summer, when the surface is heated by the sun. In summer, QL becomes a significant source of heat loss in the SEB (sublimation), preventing strong negative QS (convection). The seasonal cycle of QG is small, indicating a small net transport of heat away from the surface in summer and towards the surface in winter. The net annually integrated amount is less than zero as a result of the refreezing of meltwater, warming the subsurface snow layers.

Statistically significant and previously unreported trends over the full 24-year period (not shown) are detected in LW↑ (-0.28±0.14 W m-2 yr-1) and QS (+0.21±0.07 W m-2 yr-1). Both of these are a result of wintertime trends. LW↑ is linked directly to Ts, which shows a statistically insignificant negative trend (-0.029±0.026 K yr-1), which in magnitude exceeds the negative trend in T2m (-0.0045±0.02 K yr-1; assuming a normal distribution, the probability that the negative trend in Ts is greater in magnitude than the trend in T2m is 0.76). As a result, the air temperature gradient near the surface has increased, enhancing QS. The negative trend in Ts originates from a decrease in LW↓ (-0.26±0.17 W m-2 yr-1), which is in turn driven by a slight decrease in cloud cover (-0.003±0.001 yr-1). This is suggested independently by the decrease in average winter humidity (-0.004±0.002 g kg-1 yr-1). These findings agree with and , who determined from satellite observations that summer cloud cover has decreased over that part of coastal Antarctica in the period 1979–2011.

Melt occurs at Neumayer from November to February (Fig. ) but is highly variable from year to year. The mean annual amount of melt is 50 mm w.e. with an interannual variability of 42 mm w.e. and a range of 2 mm w.e. in 1999–2000 to 176 mm w.e. in 2012–2013. Most melt occurs in December and January and the surface only sporadically reaches melting point in February. Only in 2007 did melt occur in November, and no melt occurs outside these 4 months. The cumulative melt occurring at Neumayer shows stepwise increases (Fig. a), which represent the peaked melt seasons, in which melt occurs on average on 18±10 d. The uncertainty in the number of melt days due to the chosen values of z0,m and ρs,0 is relatively small compared to the interannual variability in melt totals (Fig. ), implying that this choice does not significantly affect the modelled melt duration, but it does affect the total melt.

To investigate the link between melt and climate, we compare the two summers with the highest (2003–2004 and 2012–2013, on average 145 mm w.e.) and lowest (1999–2000 and 2014–2015, on average 4 mm w.e.) melt amounts. Figure  shows the meteorological and SEB components for these years, averaged over December and January. The largest differences are found in T2m (+2.3 K) and SWnet (+17 W m-2); based on the measurement uncertainties (Table ), these differences are significant. In cold summers, the low T2m corresponds to a stronger temperature inversion (T2m-Ts), more longwave cooling, less sublimation and a larger QS. SW↓ and LW↓ show almost no difference between high and low melt seasons; therefore, the difference in SWnet cannot be caused by a change in cloud cover and is likely caused solely by surface albedo, which suggests an important role for the SMAF. This will be elaborated upon in the next section. Finally, the direction of QG is reversed: in high melt years, the surface is warmed from below, while in low melt years the surface loses heat to the subsurface. More refreezing of meltwater in high melt years warms the near surface snow layers, which in turn leads to a conductive heat flux towards the surface.

Using the subsurface radiation model of , the influence of subsurface penetration of shortwave radiation is estimated. Its inclusion increases the modelled cumulative amount of melt by 13 %, from 1154 to 1326 mm w.e. The absorbed shortwave radiation heats the subsurface layers, but the heat cannot be transported away as effectively as would happen at the surface by turbulent fluxes and longwave radiation. This leads to an increase in total melt.

The findings presented in this section are in good agreement with , who used a similar approach to calculate the SEB at Neumayer but used a lower value for z0,m=0.32mm and a higher snow density that was assumed constant with depth (420 kg m-3 in their study vs. 280 kg m-3 in this study). Compared to melt estimates from the Larsen C ice shelf, obtained through a similar modelling approach by , melt at Neumayer is weak. Owing to its more northerly location, on the Larsen C ice shelf an annual (2009–2011) average melt energy of 2.8 W m-2 is obtained, compared to the 2009–2011 annual average of 0.7 W m-2 obtained at Neumayer. Furthermore, in November and February melt occurs much more frequently on the Larsen C ice shelf.

The albedo parameterisation, and especially the expression for snow grain size (Eq. ), contains several parameters that are not well constrained, such as new snow grain size re,0 and refrozen snow grain size re,r. These parameters were varied within reasonable ranges to optimise the results: new snow grain sizes between 0.04 and 0.3 mm and refrozen snow grain sizes between 0.1 and 10 mm. The best comparison with observed albedo was achieved when using the look-up table for dry snow metamorphism, dre,dry, corresponding to a grain size of 0.055mm.

The first step in optimising the parameterisation was to split the summer season into two parts, the “dry” and the “wet” season. The respective starts of the dry and wet seasons are the first day on which the sun rises more than 15∘ above the horizon and the first day that surface melt occurs. The wet season ends when the sun no longer rises higher than 15∘. For the dry season, we varied the dry snow metamorphism factor and the new snow grain size to best match observed SW↑. This resulted in a new snow grain size of 0.25mm. This value is then used in the second step, in which the refrozen snow grain size re,r is varied to best match the modelled cumulative melt using observed albedo. This was achieved for a refrozen snow grain size of 1.45 mm.

This value for refrozen snow grain size is compatible with the typical largest grains in dry metamorphosed snow of O(1 mm) and which used as a lower limit for refrozen snow grains. and present observations of snow grain sizes on the Antarctic plateau during field campaigns in 2012–2013 and 2013–2014 as well as estimates from satellite observations. On the plateau, summer temperatures are comparable to Neumayer winter temperatures. report summertime snow grain size estimates of approximately 0.11 mm (Fig. 6 in their study, reported as a specific surface area SSA=3ρire, where ρi is the density of ice and re is the snow grain size). In our study, wintertime snow grain sizes approach 0.21 mm. The difference is expected as the plateau is generally much colder than Neumayer. The seasonal cycle of modelled average specific surface area in the upper 7 cm (Fig. ) is comparable to the one presented in , although the wintertime values are probably too low. For the purpose of this study, however, the accurate representation of surface albedo during winter is less relevant as there is no shortwave radiation in winter.

When the adopted albedo values are combined with the observations of SW↑, the model adequately reproduces the incoming shortwave radiation (Fig. , bias=+0.93 W m-2, RMSD=7.3 W m-2), providing confidence in the modelled albedo.

Three experiments with the SEB model were carried out in addition to the original run (R0), which uses the measured albedo:

R1: the average measured albedo (0.84, determined by adding all SW↓ and SW↑ for all measurements when the sun is higher than 15∘ above the horizon and taking the ratio between the two) is prescribed for the entire period.

Figure a and b show time series of modelled cumulative and seasonal surface melt for the four experiments. Experiment R1 underpredicts melt in most seasons, yielding a mean annual amount of surface melt of 39±27 mm w.e. yr-1 (compared to 50±42 mm w.e. yr-1 for experiment R0). More melt was modelled in the 1995–1996 melt season, which was characterised by frequent precipitation events and cloudy conditions, keeping observed albedo higher than the long-term mean. Because the albedo parameterisation (used in experiment R2) has been calibrated to match observed albedo, experiment R2 adequately reproduces the amount of seasonal melt (50±34 mm w.e. yr-1), although melt, e.g. in the 2012 melt season, is underestimated. Run R3 represents the situation in which the SMAF has been switched off, leading to significantly underpredicted melt (21±16 mm w.e. yr-1).

Defining the strength of the SMAF as the ratio between the total seasonal surface melt in experiments R2 and R3, we obtain an average value of 2.6, with a range of 1.3 (1996–1997) to 4.8 (1993–1994; see Fig. c). The effect of subsurface penetration of shortwave radiation on this result is estimated by repeating the above experiments with an inclusion of the radiation penetration model of . This yielded an average SMAF of 2.3, ranging from 1.5 (2005–2006) to 3.2 (2002–2003). The main difference between the two experiments is the reduced interannual variability: including penetration of shortwave radiation does not yield SMAF values larger than 3.5. Shortwave radiation penetration heats the subsurface, causing subsurface melt which is less affected by the SMAF because the radiative flux is smaller in the subsurface. Therefore, the “extreme” years in the sense of SMAF are less distinct in the experiment with shortwave radiation penetration. The effect of shortwave radiation penetration is included in the uncertainties indicated in Fig. c. Combining this with the uncertainties in observed SW↑ and the determination of τ (Fig. b) leads to uncertainties in the determination of the SMAF of typically 15 %, with a range of 4 % (1995–1996) to 32 % (1993–1994).

A weak positive correlation was found between SMAF and SW↓ (R2=0.15, p=0.07): if SW↓ increases, more energy is available at the surface for melting, which is then in turn further intensified by SMAF. Another weak negative correlation was found between SMAF and summer precipitation (R2=0.13, p=0.1): snowfall inhibits SMAF as it effectively “resets” the surface albedo as was also shown by in a dry region.

Only few studies report on the SMAF concerning the darkening of snow rather than disappearance of it. provide relationships between anomalies of seasonal T2m and SWnet (Figs. 5 and 12 of ). They find a negative relationship for accumulation regions, i.e. lower 2 m temperatures are associated with smaller SWnet. No such relationship is found for Neumayer (not shown).