The ORCHIDEE model is the terrestrial component of the IPSL ESM. It simulates the water, carbon and energy exchanges between the land surfaces and the atmosphere. ORCHIDEE includes the Explicit Snow module developed to represent the snowpack over continental surfaces . It has recently been adapted for glaciers and ice sheets in order to simulate their surface mass balance . This module represents the snowpack as a one-dimensional system with 12 layers of snow overlying 8 layers of ice. It simulates surface and internal snow physical processes, including snow densification, heat diffusion, snow aging, surface melting, sublimation, liquid water percolation and refreezing. Snow albedo is parameterized with the formulation proposed by depending on snow age, with a decaying exponential function from a fresh snow albedo value to a minimum albedo value (see Sect. 3, Eqs. S5–S8 in the Supplement). The snow age is computed through a dependency on snow temperature and is reset to zero depending on a fixed threshold of precipitation. Low temperatures slow the snow aging while temperatures near to the melting point increase the aging. The snow temperature is distributed across layers as a function of the snow thermal properties, with the snow thermal conductivity and heat capacity both parameterized as functions of density see Eqs. S1–S4 in the Supplement. The snow mass evolves as a function of the liquid and solid precipitation rate, sublimation and the runoff of liquid water that cannot be retained in the snowpack. ORCHIDEE uses a bucket scheme regarding liquid water retention and percolation. The maximum liquid water holding capacity increases for densities above 200 kgm-3 , meaning that a denser snow can retain more liquid water. The ice layers beneath the snowpack are considered as an infinite reservoir of ice mass, which is only used to compute the contribution of ice melting to runoff in case of complete snow removal during the ablation season. The snow and ice modules have been extensively described in . Both modules were implemented in ORCHIDEE version 2 . Here, we rely on the Explicit Snow scheme from updated in ORCHIDEE version 4. Changes between both ORCHIDEE versions do not affect the representation of snow processes over ice sheets.

Prior to the new developments presented in this article, we have addressed several limitations identified in the previous version of Explicit Snow. Previously, refreezing was computed before percolation of liquid water and was therefore limited to the meltwater produced and retained within a given layer. We have revised this approach to allow refreezing to occur after percolation and to depend on the amount of energy available to refreeze, following the methodology implemented in CROCUS . These modifications enable the snowpack to refreeze as much liquid water as its cold content allows, rather than being restricted by the maximum water retention capacity of individual layers. Additionally, we increased the maximum snow density in the model from 750 to 917 kgm-3 to represent the transition from snow to ice densities. Finally, we modified the maximum snow mass threshold from 3000 to 10 000 kgm-2. This adjustment ensures the formation of a deeper snowpack to better represent percolation, refreezing and runoff.

We use the ORCHIDEE model in offline mode with a 30 min time step. The atmospheric forcing is provided by the polar-oriented regional atmospheric model MAR , coupled to the SISVAT (Soil Ice Snow Vegetation Atmosphere Transfer) land surface scheme , that includes a multi-layered snow model based on the CROCUS model . MAR has been extensively evaluated to represent surface mass balance, near-surface temperature and solid precipitation in comparison with in situ observations over Antarctica and with observations and model reconstructions over Greenland . Moreover, the intercomparison exercise GrSMBMIP higlights the good comparison of MAR with in situ SMB measurements over Greenland . MAR has also been shown to outperform the ERA5 reanalysis in representing the summer near-surface temperatures .

The ORCHIDEE model is forced every 3 h in terms of solid and liquid precipitation, surface pressure, wind speed at 10 m, temperature and specific humidity at 2 m, downward shortwave and longwave radiation. At each time step, the model uses incoming radiative fluxes in the short and longwave bands from MAR, and computes the upward shortwave and longwave radiation and the surface latent and sensible heat fluxes. These turbulent fluxes are obtained following the Monin–Obukhov turbulence theory using the wind speed at 10 m, a drag coefficient and the atmosphere/surface gradient of temperature and specific humidity at 2 m . The model also calculates the surface mass balance by accounting for solid and liquid precipitation from the forcings, as well as modeled sublimation and runoff over snow and ice.

We set up two configurations, one for GrIS and one for the AIS using the same stereographic grid as in the atmospheric forcing. For Greenland, we use MAR version 3.13 with a horizontal resolution of 15 km, covering 1980–2019. For these offline ORCHIDEE experiments over GrIS, we rely on 2 sets of albedo parameters. We first use the assimilated albedo parameters from MODIS satellite albedo products , referred hereafter to as ASIM12L. However, showed that the ASIM12L albedo parameters led to a significant underestimation of runoff over GrIS. Therefore, we also use the albedo parameters set obtained from a manual tuning approach and designed to improve the agreement between the runoff simulated by MAR and ORCHIDEE. These albedo parameters will be referred to as OPT12L and are used only in Sect. . Since the MARv3.13 forcings were not available for Antarctica at the time this study was initiated, we use the MAR version 3.12, spanning 1980–2019 with a 35 km horizontal resolution . Both AIS and GrIS MAR simulations were forced every 6 h at their lateral boundaries by the ERA5 reanalysis . As albedo is more spatially uniform and temporally stable over the AIS than over the GrIS, we manually adjusted the albedo parameters in ORCHIDEE over the AIS, to better represent the surface conditions at Dome C. All AIS simulations use the same albedo parameters. The values for these three sets of albedo parameters can be found in Table S1 in the Supplement.

Following this, to investigate the effect of the developments proposed in this study, we define three different ORCHIDEE configurations (Table ). The reference simulation, presented as OR OLD, uses the same configuration introduced in . This experiment shows an underestimation of density at the surface and an excessive compaction at depth (Fig. c and f) and starts the simulation from a snow-free state over the ice sheet. The OR OLD + INIT configuration uses the new initialization procedure (Sect. ), while OR NEW further includes updated model physics, with the new surface density parameterization (Sect. ) and the calibrated viscosity parameters for polar regions (Sect. ).

To improve the representation of snow densification, we use observations of snow temperature, accumulation and density in Greenland and Antarctica. From the SUMup collaborative database , we use 3749 observations of snow temperature collected at different locations in Greenland, between 1930 and 2023, and 905 in various sites over Antarctica between 1957 to 2021 (Fig. a and d). The observations of snow accumulation are sourced, for Antarctica, from the GLACIOCLIM-SAMBA dataset, detailed in and updated by , with 2981 observations over Antarctica, recorded between 1930 and 2009, and from the SUMup dataset for Greenland with 358 accumulation observations, collected between 1952 and 2023. Over Greenland and Antarctica, we only retain data for which the duration of observations is specified (ranging from a few days to several years). To investigate surface snow density, we use density profiles from the SUMup database and additional density sources listed in Table , containing measurements within the top 20 cm (560 observations in Greenland and 139 in Antarctica).

The representation of the vertical densification is assessed using 126 profiles extracted from the SUMup database (79 in Greenland and 49 in Antarctica) and two additional profiles located at Dome C in Antarctica (Fig. b and e). Among these observations, 102 reach depths of at least 10 m (79 in Greenland and 23 in Antarctica). For our study, we select two sites characterized by dry snow conditions: Summit (72.58° N, 38.50° W; 3254 ma.s.l.) and Dome C (75.06° S, 123.21° E; 3233 ma.s.l.). These locations present different climatic conditions. Dome C is characterized by an extremely cold and arid climate with mean annual surface temperatures of -55 °C and accumulation rates close to 30 kgm-2yr-1 28.6 kgm-2yr-1 observed between 2006 and 2013; . Summit exhibits a less cold environment, with a mean annual surface temperature observed of -31.4 °C observed between 1998 and 2017 and significantly higher accumulation rates of approximately 242 kgm-2yr-1 recorded at Summit Camp between 2017 and 2020 . At Summit, five density profiles are selected, reaching at least 10 m depth, collected between 2006 and 2017 . At Dome C, we use two density profiles collected in the 2012–2013 austral summer . Figures c and f represent the observed density profiles at Summit and Dome C, respectively. To filter out density measurements fluctuations, we have applied a third-degree polynomial interpolation to the raw density profiles (Fig. c and f).

The observations described above are used for distinct purposes (see Table ). The snow temperature, accumulation, and surface density observations are used to develop and calibrate the initialization procedure. The surface density observations are employed to design and calibrate the parameterization of surface snow densification. Finally for the snow compaction scheme, the vertical density profiles at Summit and Dome C are used for calibration, while the complete set of density profiles serves as an independent validation dataset.

To compare point observations and model gridded outputs, the observations are averaged on the ORCHIDEE stereographic grid (15 km resolution in Greenland, 35 km in Antarctica). After averaging temperature observations onto the model grid, we obtain 180 points in Greenland and 770 in Antarctica (Fig. a and d). We also compute the grid averaged accumulation, weighted by the corresponding duration of observations. This way, we obtain 76 grid-cell averages over Greenland, and 589 for Antarctica. For surface densities, we obtain 149 grid cell averages for Greenland and 79 for Antarctica (Fig. b and e).

ORCHIDEE was previously initialized without snow over ice sheet, relying on model accumulation through a 5-year spin-up period . Here, we introduce a new method for initializing the snowpack over the Antarctic and Greenland ice sheets. This procedure enables the formation of a 10 m water equivalent (w.e.) snowpack in low accumulation areas, such as the interior of AIS where several decades of simulation would be necessary to form a thick snowpack. Our initialization method expresses the initial density profile as a function of latitude and surface elevation only, making it easily adaptable to any snowpack model. It also initializes the surface temperature of the snowpack as well as the snowpack depth, which is considered uniform for Antarctica and variable for Greenland between the interior of the ice sheet and areas where net ablation is expected (see Appendix ). Attention must be paid, as this initialization was developed using observations from recent decades, making it potentially not appropriate for very different climatic contexts from the present-day one. This method is presented in detail in Appendix , and summarized hereafter. Here, we use the densification formulation proposed by and further adapted by : 5dρdz=C⋅g⋅γ(ln⁡(A‾))⋅exp⁡Eg-EcRTs‾⋅ρ(z)×(ρice-ρ(z))⋅dz where A‾ is the local mean annual accumulation (in kgm-2yr-1), Ts‾ is the local mean annual surface temperature (in K), g the gravitational acceleration (9.81 ms-2), ρice the ice density (917 kgm-3), dz the snow layer thickness (in m), R the gas constant (8.31 JK-1mol-1), Eg and Ec the activation energies of grain growth (4.24 × 10⁴ Jmol-1) and compaction (6 × 10⁴ Jmol-1), respectively. C is an unitless constant with different values above (0.03) and below (0.07) the critical density ρ = 550 kgm-3, reflecting a faster densification rate near the surface.

The γ factor represents a correction to the accumulation term introduced by . Their study revealed that the ratio of modeled to observed depths at which snow reaches critical densities of 550 and 830 kgm-3 is strongly correlated with mean annual accumulation across Antarctica. To account for this dependency, we use the same corrective term except that we applied a scaling factor of 0.9 for densities greater than 550 kgm-3 to better match the observed density profile at a depth corresponding to 830 kgm-3 in Antarctica: 6γ(x,y)=max⁡(1.435-0.151⋅ln⁡(A‾(x,y)),γmin⁡) for ρ≤550kgm-3max⁡(0.9⋅[2.366-0.293⋅ln⁡(A‾(x,y))],γmin⁡) for ρ>550kgm-3 where γmin⁡ is the minimal correction equal to 0.25 as in . This γ term is generally calibrated to account for surface climate biases .

After integration of Eq. () (Appendix ), the explicit formulation of the density profile is expressed as a function of the mean surface temperature Ts‾, the logarithm of the mean annual accumulation rate ln⁡(A‾), and the surface density ρs. We estimate these key variables as a function of surface elevation and latitude, based on physical considerations and empirical relationships derived from observations and modeling outputs, and detailed in Appendix . This method provides a final parameterization of the density profile that depends only on latitude and elevation. Other studies have used densification equations to initialize snow models, such as CRYOWRF, which relies on the Firn Densification Model to calculate the initial density profile . The strength of our method is its independence from additional models, enabling standalone initialisation for any snowpack model. Over Greenland, the parameterizations yield root mean square errors (RMSE) representing 11 % of the mean accumulation logarithm (RMSE = 0.60 ln⁡ (kgm-2yr-1)) and 11 % of the mean surface density (36.4 kgm-3). For the surface temperature, the RMSE is equal to 3.1 °C in Greenland (compared to an observed mean temperature of -20.6 °C), while over Antarctica, the RMSE reaches 4.6 °C (compared to an observed mean temperature of -32.8 °C). The corresponding errors over the AIS represent 12.2 % of the mean accumulation logarithm (0.56 ln⁡ (kgm-2yr-1)) and 8 % of the mean surface density (29.4 kgm-3). When evaluated over both ice sheets, the parameterized surface mean temperature correlates strongly with the observed 10 m snow temperature, with a difference of only 0.4 °C and a correlation coefficient of 0.89. (Fig. a). This initialization method can generate a polar snowpack of prescribed depth over the AIS and GrIS. As ablation is negligible in Antarctica, we consider a spatially uniform snowpack thickness set to 10 m. Conversely, in Greenland, the snow depth varies spatially from the edges to the interior of the ice sheet where it is also fixed at 10 m (see Appendix ). The estimated mean annual surface temperature of the snowpack is also used to initialize snow and ice temperatures of all snow and ice layers at the beginning of the simulation.

The initialized snowpack density is compared to observations under dry snow conditions, in order to isolate the densification driven by the snow compaction (Fig. b and c). In ablation zones, where melt is significant, additional densification occurs from the refreezing of meltwater retained within the snowpack. This process is not accounted for in the initialization procedure. Overall, the initialized snowpack density agrees well with observations under dry snow conditions. The RMSE represents 7 % of the mean snow density at 5 m depth (RMSE = 32.4 kgm-3). At 10 m depth, the comparison with observations performs even better, with reduced RMSE (26.3 kgm-3, 5.1 % of the mean snow density at this depth) and bias (4.3 kgm-3; 0.9 %).

Over Greenland and Antarctica, surface snow densities measured in the uppermost centimeters of the snowpack are significantly higher than the range of the Pahaut (1976)'s formulation, ranging from 200 to 400 kgm-3 . In this way, applied to GrIS and AIS this formulation leads to a significant underestimation of surface snow density when compared to observations (Sect. ).

To represent high surface densities in polar regions, we choose to compensate the limited physical understanding of the processes involved, by introducing a new surface density parameterization, common to both ice sheets and based only on wind speed. In the following, surface snow density will be considered as the mean density of the upper 20 cm of the snowpack. The new parameterization is applied to the modeled fresh snow density in ORCHIDEE, i.e. only at time steps when snowfall occurs, in order to determine the deposited snow density. Similar approaches have been employed in Arctic and Antarctic studies . Moreover, using both Greenland and Antarctic surface snow observations, we find that the observed surface density correlates more strongly with wind speed than with air temperature, supporting our choice to use a wind parameterization for surface density (Fig. ).

If surface density is expressed as a linear function of instantaneous wind speed, then the temporal average of surface density can be related to wind speed weighted by precipitation. In order to calibrate the formulation, we compare gridded-averaged surface snow density observations from Greenland and Antarctica with the precipitation weighted 10 m wind speed climatological average (1980–2019) simulated by MAR (Fig.).

In Antarctica, a linear relationship appears between wind speed and surface snow density. In contrast, Greenland shows a more heterogeneous pattern, with observations clustering around 320 kgm-3, in agreement with . Here, we propose a new expression for surface snow density ρs, defined as a function of the wind speed at 10 m, 7ρs(ws10)=ρsMIN+(ρsMAX-ρsMIN)×1+tanhws10-ws10CUTws10INT/2 with ρsMIN = 260 kgm-3 the minimum surface snow density reached at low wind speeds (< 5 ms-1), ρsMAX = 450 kgm-3 the maximum surface snow density reached at high wind speeds (> 15 ms-1), ws10CUT = 8.5 ms-1 the wind speed at the inflection point of the hyperbolic tangent, wssINT = 4.0 ms-1 the width of the interval over which the hyperbolic transition from minimum to maximum density takes place. For Greenland, we imposed a maximum density of 400 kgm-3 to ensure a realistic range of surface densities in this region, as observations generally do not exceed this value . For Antarctica, the maximum density remains equal to ρs MAX.

Applied to both ice sheets, the formulation yields a RMSE of 40.3 kgm-3 (11.3 % compared to the averaged surface density) and a bias of 1.9 kgm-3 (i.e. 0.5 %), indicating a satisfactory agreement with observations. Although the parameterization does not fully capture the observed spatial heterogeneity of surface snow density, particularly over Greenland, it nevertheless reproduces the observed mean surface density and thereby corrects the strong negative bias resulting from Eq. ().

History Matching (HM) is a method for calibrating parameters , implemented within the ORCHIDEE Data Assimilation Systems ORCHIDAS. This procedure is based on Gaussian process emulation and is designed to constrain the free parameters of complex models. It has been applied to calibrate atmospheric, ocean and Earth system models , as well as ice sheet models and land surface models . Originally developed to calibrate model parameters at the process level, HM operates in ORCHIDAS using offline ORCHIDEE simulations in a single-column model framework. This approach has previously been applied to the ORCHIDEE model to simulate net ecosystem exchange and latent heat fluxes . This calibration approach differs from the traditional data assimilation method, which primarily focuses on identifying the optimal parameters that best match observations by minimizing a cost function. In contrast, HM aims to identify regions of the parameter space that are “not plausible” given the constraints predefined through observations, i.e. leading to model outputs inconsistent with observations.

The calibration starts by sampling the prior parameter space using a Latin Hypercube design , to efficiently explore the space and maximize the information gained from each parameter combination. Following the recommendation of , we use a sample size equal to N=10⋅P, where P is the number of parameters being calibrated. One-dimensional ORCHIDEE simulations are performed for each sampled set of parameters. Iteratively, Gaussian process emulators are then constructed to capture the relationship between model parameters and scalar metrics, specifically designed to characterize the physical processes under study. The construction of the Gaussian process emulators is done following the methodology presented in and the R-scripts from . Each metric is associated with its own emulator, which is then used to predict metric values across the full parameter space. This enables the identification of parameter regions that are inconsistent with observations using an implausibility criterion I, used to reduce the parameter space. The implausibility is defined as follows: 8I(x)=|m-E[G(x)]|σobs2+σOR2+Var[G(x)] with x being a sample of the parameter space, m the reference observed metric, E[G(x)] the value predicted by the emulator G(x). The implausibility accounts for the aggregated uncertainty, computed as the square root of the sum of squared contributions from three sources: the observational (σobs), model (σOR), and emulator (Var[G(x)]) uncertainties.

This criterion is computed independently for each metric, and used to define the subregion of the parameter space that is not inconsistent with the reference observational metrics, also called the NROY space for “Not Ruled Out Yet”. The NROY space corresponds to the region of the parameter space where the implausiblity is smaller than a defined threshold. For the first iterations (waves) of the algorithm, it is common to set the implausibility cut-off to 3, following the 3σ rule stating that 99.7 % of the probability density for any unimodal distribution is comprised within 3 standard deviations of the mean. As the NROY space is progressively reduced during successive waves, this threshold is lowered to 2 for the final wave. With the progressive reduction in emulator uncertainty, decreasing the implausibility cut-off is a standard step in the calibration procedure .

In this study, we use the HM method, described in Sect. , to calibrate three parameters of the snow viscosity defined by : η0, aη and bη. We calibrate ORCHIDEE using 1D simulations over two dry snow locations, one in Greenland (Summit, 5 density profiles) and one in Antarctica (Dome C, 2 density profiles), presented in Sect.  (Fig. c and f). For each density profile, we apply a third degree polynomial interpolation (called “fit” herefafter) to smooth the observations. In this way, in the 1D simulations, the surface snow density (ρs) is prescribed using the average of the fitted surface snow density (Table ). In addition, to characterize the dry snow densification profiles, we define two metrics for each location: density at 5 m (ρ5m) and at 10 m (ρ10m). Each metric is obtained from the fit of the observed density profiles, and is associated with an uncertainty estimate that accounts for both observational and model-related uncertainty.

Observational uncertainties are estimated by combining two sources of uncertainty. We distinguish within-profile uncertainty, which captures the vertical variability for a given profile around the considered depth, and between-profiles uncertainty, which reflects variability across different profiles at the same site, related to either spatial or temporal variability. Here, we present the observational uncertainties for the metrics (ρ5m and ρ10m) and the observed surface density (ρs). First, within-profile variability is quantified as the Root Mean Square (RMS) of the mean deviation (taken as σn) across a vertical window around the target depth (4 to 6 m for ρ5m, 9 to 11 m for ρ10m, 0 to 1 m for ρs), e.g. for ρ5m: 9σwithin-profile5m=1l∑i=1lσ(ρobsi4m<z<6m)ni2 where l is the number of observed density profiles available, ni the number of observations of a given profile i within the vertical window around the target depth, ρsmoothedi and ρobsi the observed density of the given profile i within the same vertical window.

Second, between-profiles variability is estimated as the standard deviation of the density metrics across all profiles for each location, divided by the square root of the number of profiles, e.g. for ρ5m: 10σbetween-profiles5m=σ(ρsmoothed5m1,…,l)l where ρsmoothed5m1,…,l is the vector of the l density metrics (here ρ5m) resulting from the polynomial interpolation of the corresponding observed density profile.

These two components (σwithin-profile and σbetween-profiles) are then combined using the root sum square to consider independently both uncertainties, yielding the observational uncertainty for each metric (σobs5m, σobs10m) and for surface density (σρsobs). In this way, at 5 m, mean densities from fitted profiles are equal to 409.1 and 367.3 kgm-3 at Summit and Dome C, associated with observational uncertainties equal to ± 9.3 and ± 5.7 kgm-3. At 10 m, we obtain, respectively 497.5 and 415.6 kgm-3, with observational uncertainties of ± 6.8 and ± 3.4 kgm-3. The observed surface densities are equal to 311.4 ± 10.2 kgm-3 at Summit and 316.4 ± 9.2 kgm-3 at Dome C (Table ).

In addition to the observational uncertainty, we need to account for modeling uncertainties. We evaluate the sensitivity of the metrics ρ5m and ρ10m to biases in the modeled snow temperature, accumulation and surface snow density using the steady-state formulation of (Eq. , Sect. ). These biases are evaluated by considering the 40 years integrated accumulation and the final year of snow temperature simulated by ORCHIDEE in comparison with the in situ observations presented in Sect. . We obtain temperature biases of +2.0 °C at Dome C and +2.6 °C at Summit, and accumulation biases of 12.4 kgm-2yr-1 at Dome C and -54 kgm-2yr-1 at Summit. We compute new steady state density profiles after adding the absolute value of these biases to the observed temperature and accumulation, using both positive and negative perturbations. The difference between the two perturbed steady state profiles is assumed to represent the model sensitivity to temperature and accumulation biases. The uncertainty is then estimated for ρ5m and ρ10m, as half this difference (δaccu model and δT model). The same method is applied for ρs by injecting observed surface snow density uncertainty σobsρs, defined previously, in the steady state densification equation for each location separately.

Then, the final model uncertainty for each metric is obtained by Root Sum Square aggregation of the uncertainties presented previously, e.g. for ρ5m: 11σρ5m=σobsρ5m2+δaccu model5m2+δTmodel5m2+δρsmodel5m2

This yields total uncertainties of ± 15.3 and ± 19.2 kgm-3 for ρ5m and ρ10m at Summit, and ± 15.0 and ± 14.0 kgm-3, respectively, at Dome C (Table ).

The HM method is applied to calibrate the viscosity parameters of Eqs. () and (), with the objective of identifying parameter sets consistent with snow densification at both Summit and Dome C. Since we calibrate three viscosity parameters, the sample size is equal to 3×10 sets of parameters, sampled at each wave of the HM algorithm, following (see also Sect. ). For each parameter set and each location, we run 1D ORCHIDEE simulations (1 at Summit and 1 at Dome C) and compute the corresponding metrics. It results in 60 simulated ρ5m and ρ10m values for each wave (3 parameters × 10 simulations × 2 locations), used to construct the Gaussian process emulators. In total, we perform 5 waves, to progressively reduce the parameter space and obtain parameter values providing consistent densification metrics ρ5m and ρ10m.

During the first wave, the simulated densities at 5 and 10 m depths exhibit a wide spread, ranging from 300 to 900 kgm-3, highlighting the strong sensitivity of modeled densification to the viscosity parameters (Fig. a and b). After five waves, the parameter space is reduced by 99.6 %, and the resulting viscosity parameters are able to capture observed densification rates. At wave 5, the simulated ρ5m and ρ10m remain within the bounds of the metrics uncertainties at both Summit and Dome C.

The HM method also leads to a substantial reduction of the viscosity parameter ranges between the first and fifth wave (Fig. c and e; Table ). The sensitivity of snow viscosity to temperature, governed by aη, is increased from 0.081 to values converging between 0.138 and 0.17 K-1, highlighting the stronger control of temperature on polar densification compared to the former formulation. This suggests that snow viscosity was underestimated in the original parameterization , due to an insufficient influence of the temperature. In contrast, the sensitivity of the viscosity to density, controlled by parameter bη, remains close to the values proposed by , with a final range between 0.018 and 0.02 m-3kg-1 (0.018 m-3kg-1initially). By comparison, the parameter η0, which sets the reference viscosity, exhibits limited influence on the metrics, as its variability does not significantly affect model performance within the explored range. This suggests that either the selected metrics are not sensitive enough to constrain η0 or that polar snow densification is primarily driven by the temperature and density sensitivities (aη and bη), with η0 playing only a minor role. Equifinality may also occur, with the calibrated sensitivity to temperature and density compensating for variations in the reference viscosity. The analysis of the final parameter space reveals a strong correlation between aη and bη (Fig. ). These parameter refinements are reflected in the reduction of the aggregated RMSE, i.e. the RMSE computed combining the four density metrics (at 5 and 10 m depths over Summit and Dome C): 121n∑i=0n[(ρi5m-ρi5mobs)2+(ρi10m-ρi10mobs)2] where n is the number of locations studied (Summit and Dome C). The aggregated RMSE decreases substantially from an initial range of 50 to 500 kgm-3 to a reduced range between 6.85 and 9.87 kgm-3 after five calibration waves (Fig. c–e). We select the parameter set that minimizes the aggregated RMSE (6.85 kgm-3) across both locations (Table ).

We compare the new formulation of the surface snow density computed by ORCHIDEE with observations of surface snow densities (Fig. ). This comparison is made for the former formulation and the new one proposed in this study (Eq. ). Surface snow densities are computed as the average density of the top 20 cm of the snowpack over a 40 years period (1980–2019). As previously mentioned, the parameterization strongly underestimates surface snow density for both AIS and GrIS, with a negative bias of -188.3 kgm-3, representing 53 % of the average observed ρs. The new formulation shows an overall good representation of surface density with a bias of only 5.8 kgm-3. Regional analysis reveals a positive bias over Greenland (20.6 kgm-3, i.e. 6.2 % with reference to the observed average ρs) and a negative bias over Antarctica (-23.8 kgm-3, 6.3 %). However, surface snow densities can be affected by melt-refreeze processes, particularly over Greenland. These processes increase snow density through the accumulation of refrozen meltwater. Furthermore, this additional densification is represented within the snow scheme, but is not explicitly included in the surface snow density parameterization. This can constitute a potential bias for our evaluation.

We account for this effect by excluding simulated surface snow densities where the 40 years average refreezing rate in the top 20 cm of the snowpack exceeds 0.1 kgm-2d-1. These locations are represented by transparent dots in Fig. a.

Excluding the Greenland locations affected by refreeze processes improves the computed surface density, when considering both ice sheets, with a negative bias of -3.5 kgm-3 and a reduced RMSE equal to 34.8 kgm-3 (1 % and 9.8 % compared to the observed averaged surface snow density, respectively). This highlights the need for future studies to investigate the ORCHIDEE representation of refrozen areas in Greenland. Moreover, the new parameterization generally reproduces the large-scale spatial gradients in surface snow density over both ice sheets (Fig. b and c). It captures the higher densities along the wind exposed coastal margins, especially over Antarctica.

In addition, we conduct an inter-model comparison by implementing in ORCHIDEE the surface snow parameterizations from MAR over Antarctica , from the IMAU-Firn densification model over both ice sheets IMAU-FDM; and SNOWPACK over Antarctica (see Eqs. S9–S12 in the Supplement). Both MAR and SNOWPACK formulations of surface snow density are based only on 10 m wind speed and were developed for Antarctica. The IMAU-FDM formulation calibrated for Antarctic conditions relies on the 10 m wind speed and near surface temperature (T2m), while the one developed for Greenland depends only of the near surface temperature.

We test these formulations with offline ORCHIDEE simulations over Greenland and Antarctica for the 1980–2019 period. Model outputs are evaluated considering the top 20 cm for surface snow densities (Fig. ). Both the IMAU-FDM over Antarctica (IMAU ANT) and MAR parameterizations perform well over Antarctica, with biases of only -1.3 and 1.5 kgm-3, respectively, representing differences of only 0.3 % and 0.4 % with the averaged observed surface density (Table ). However, their performance degrades over Greenland, where biases increase to 60.8 kgm-3 (18.4 %; IMAU ANT) and 65.3 kgm-3 (19.8 %; MAR). Overall, this behavior highlights the challenge of representing both ice sheets with a single parameterization, given their distinct surface conditions. The SNOWPACK and IMAU GR parameterizations show a similar pattern. The IMAU GR performs best over Greenland, with a reduced bias (11.9 kgm-3, 3.6 %) and a relatively small RMSE (47.4 kgm-3, 14.3 %) but its performance degrades over Antarctica, with a strong underestimation of surface density (bias: -92.7 kgm-3). Similarly, the SNOWPACK parameterization shows good performance over Greenland, with a relatively small bias and RMSE (-20.3 and 45.2 kgm-3, respectively). Nevertheless, it underestimates the surface density over Antarctica with an important bias (-76 kgm-3), likely because, for consistency in comparing surface density parameterizations, only the surface densification term was implemented (see Eq. S12 in the Supplement), without the additional wind-dependent enhancement of the strain rate . Our proposed formulation reproduces reasonably the surface density in both regions, with biases of only 6.2 % and 7.2 % (20.6 kgm-3 and -23.8 kgm-3) for Greenland and Antarctica, respectively, and a relatively small RMSE (47.3 kgm-3 in Greenland; 44.1 kgm-3, in Antarctica). Combined over both ice sheets, the proposed formulation gives a reduced bias (5.8 kgm-3, 1.6 %) and RMSE (44.1 kgm-3, 12.4 %).

To assess whether the densification improvements obtained in 1D simulations extend to the ice sheet scale, we now compare modeled and observed snow densities at 5 and 10 m in 2D configurations for both ice sheets (Fig. ). The agreement between ORCHIDEE outputs and observations is generally good. At 5 m, we note a negative bias of -4.2 kgm-3 (i.e. 1.1 % of the ρ5m average) and a RMSE of 43.2 kgm-3 (9.5 % with reference to the average ρ5m). A modest underestimation of density can be noted over Antarctica (Fig. a). A similar performance is obtained at 10 m with differences of 1.6 % and 8.5 % for the bias and the RMSE, respectively. These statistics demonstrate that the model reproduces the vertical structure of observed density profiles with reasonable accuracy. To further illustrate the model performance in reproducing the ρ5m and ρ10m observed metrics, we perform an evaluation over all available locations (see Figs. S1, S2 and S4 in the Supplement). Here, we selected ten dry snow observation sites (five in Greenland and five in Antarctica), chosen to cover the different regions of both ice sheets. For each location, the previous ORCHIDEE configuration is compared with the updated version of the model, including the new developments (Fig. b). Density profiles correspond to the annual average for the year 2019, computed from monthly outputs. Two systematic biases appear consistently across all sites when we consider the previous representation of density in ORCHIDEE (OR OLD). First, the model underestimates surface snow density by about 150 kgm-3. Second, it overestimates the compaction, leading to an increased density within the snowpack of around 100 to 200 kgm-3 compared to the observations. In the updated version of ORCHIDEE (OR NEW), these biases are considerably reduced and the vertical structure of snow density is significantly improved, despite a slight underestimation in some specific Antarctic locations such as Mount Johns (Fig. b).

To investigate the effect of the initialization procedure as well as the developments proposed in this study, we choose to analyze the monthly evolution of the modeled snow density at Summit under three ORCHIDEE configurations: OR OLD, OR OLD + INIT and OR NEW described at Sect.  and Table . In OR OLD, it takes approximately 30 years for the model to accumulate 10 m of snow at Summit (Fig. a). The initialized simulation OR OLD + INIT begins with a 10 m snowpack, with an initialized steady-state density consistent with observed density profiles at Summit (Fig. b and h). Rapid thinning of the snowpack is observed in the first few years due to the excessive densification rate that causes OR OLD to deviate quickly from the initial density profile.

We further analyze the convergence time of the model towards its equilibrium state by considering anomalies between monthly snow density and the average profile from the final simulation year (Fig. d–f). In OR OLD without initialization, it takes more than 35 years to converge towards the final density profile (Fig. d), while OR OLD + INIT converges slightly faster, within roughly 30 years (Fig. e). However, OR OLD + INIT clearly diverges from the initial profile during the first decade, as indicated by the pronounced negative anomaly during this period. By the end of the simulation, the resulting density profile is significantly denser by more than 150 kgm-3 compared to the initialized profile, particularly in the deeper layers.

Finally, when comparing monthly modeled density profiles with observed density profile at Summit (taken as the polynomial fit on the 05-2017 profile), both OR OLD and OR OLD + INIT produce too low densities near the surface (Fig. g and h), confirming known biases in surface densification presented in Sect. . Below the surface, the OR OLD densities are systematically too high compared to observations, regardless of the initialization (Fig. g and h). This confirms the excessive densification found by using the Anderson densification parameterization in polar regions. In contrast, the OR NEW configuration exhibits a more realistic vertical structure, with densities ranging between 300 kgm-3 and 600 kgm-3 over the top 17 m, closely matching the observed density profile (Fig. c and i). The bias remains below 30 kgm-3 through the snow column. Furthermore, OR NEW reaches equilibrium within approximately five years, substantially reducing the spin-up time (Fig. f). The anomaly with respect to the initial profile remains low during the entire simulation, confirming the consistency of the initialization with the long-term behavior of the model.

In the ORCHIDEE snow scheme, snow thermal conductivity and snow heat capacity are expressed as functions of density (see Sect.  and Eqs. S1–S4 in the Supplement). As a result, improvements in the representation of snow density should translate into a better representation of heat diffusion within the snowpack, and thus of the vertical temperature profile. In the following, the simulated temperature at 10 m (T10m), yearly averaged over the year 2019, are compared to observations using both the previous configuration (OR OLD) and the updated version (OR NEW) including all developments. In the original configuration, T10m is persistently overestimated across both ice sheets, with a bias of 8.6 °C in Greenland and 3.1 °C in Antarctica (Fig. a and c). The updated version improves the performance over Greenland and Antarctica, reducing the bias to 2.3 and 1.4 °C, respectively (Fig. b and d).

To assess the influence of our developments on modeled melt–refreezing processes in Greenland, we focus our analysis on the southeastern coast of GrIS, a region characterized by high snow accumulation and intense ablation (Fig. a). This area is particularly relevant for investigating refreezing dynamics, with MAR modeled mean melt rates exceeding 4.5 myr-1w.e. over 2010–2019 and recurrent episodes of strong melting and subsequent refreezing (Fig. b and c). Because, as presented in Sect. , the albedo parameters from lead to an underestimation of melt-refreeze processes , we perform the same simulations OR NEW and OR OLD + INIT with the OPT12L albedo parameters, to verify that the influence of the developments is consistent with both sets of albedo parameters. Overall, these sets of parameters lead to similar conclusions. Moreover, since our analysis focuses on melt-refreeze processes, we will use the OPT12L configuration in the following to facilitate the interpretation (Fig. ). First we find that even with the OPT12L albedo parameters, ORCHIDEE underestimates both melt and refreezing fluxes in comparison to MAR (Fig. b and c). The mean seasonal melt modeled by MAR, averaged over the region presented in Fig. a, ranges approximately between 6.5 and almost 10 myr-1w.e. between 2010 and 2019, while ORCHIDEE values are lower by 2 to 3 myr-1w.e.. Refreezing is also underestimated, with differences between OR OLD + INIT and MAR of about 0.5 to 1 myr-1w.e. (Fig. c). Nevertheless, the updated version (OR NEW) simulates enhanced refreezing relative to the initial configuration (OR OLD + INIT).

To better understand these differences, we analyze the 2012 melt season, which featured exceptional melt events across GrIS (Fig. b). Although simulated melt rates are similar between OR OLD + INIT and OR NEW, refreezing differs notably early in the melt season (May–June 2012, see Fig. S8 in the Supplement). We further investigate the behavior of the model during the 2012 season at a representative site (Fig. a). We compare modeled snow density, temperature, liquid water, and refreezing concentration (computed, respectively as the liquid and refreezing content divided by the layer thickness). As stated previously, OR NEW corrects the underestimation of surface density and the overestimation of compaction at depth (Fig. d), directly influencing the snowpack temperature and liquid water content (Fig. e and f). Because thermal conductivity is increasing with density, the revised density profile increases snow conductivity at the surface in OR NEW. This results in a higher cold content during winter and spring, with temperature differences ranging from -1.5 to -4.5 °C between the surface and around 6 m depth (Fig. e). As refreezing is controlled by the available cold content, OR NEW promotes a more important refreezing of percolating meltwater (Fig. g). In particular, at the onset of the melt season, when the snowpack is still dry, the intense melt event of 3–14 May 2012 induces a pronounced refreezing peak, reaching 8.2 kgm-2d-1 in OR NEW, nearly twice the rate in OR OLD + INIT (3.8 kgm-2d-1), while the average melt amount during this melt event is similar in both simulations (12 kgm-2d-1w.e. in OR NEW in comparison with 13.1 kgm-2d-1w.e. in OR OLD + INIT). Throughout the melt season, refreezing rates near the surface remain consistently higher in OR NEW, with differences on the order of 8 to 10 kgm-3d-1 (Fig. g). This increased refreezing in OR NEW directly lowers the liquid water content compared to OR OLD + INIT, as observed in May 2012 (Fig. f) with differences of -6 to -9 kgm-3 between 4 and 10 m depth.

In conclusion, the new developments promote higher refreezing rates by modifying the snow density and viscosity profiles. In OR OLD + INIT, the snowpack remains almost too warm to refreeze additional meltwater. In contrast, the colder snowpack in OR NEW allows for greater refreezing potential, which is consistent with the need to increase the melt and refreeze content in line with MAR.

As shown previously, compared with MAR, ORCHIDEE underestimates refreezing within the snowpack, due to an underestimation of meltwater production. This has an impact on densification in the Greenland lower percolation zone, defined generally as the low elevation area of the ice sheet where annual snow melt exceeds accumulation. Indeed, the resulting meltwater significantly influences water percolation and thus refreezing within the snowpack, which in turn affects densification.

In the lower percolation zone, ORCHIDEE underestimates snow density at both 5 and 10 m (Fig. a), with a larger bias at 5 m (-56.8 kgm-3) than at 10 m (-21.3 kgm-3). At 10 m, the influence of refreezing on densification diminishes, as the water does not percolate significantly at this depth, and the compaction becomes the dominant densification process. Since the compaction scheme was calibrated using dry snow conditions, where refreezing is negligible, the model performs more accurately at 10 m. These results likely indicate that accurate representation of density in the percolation zones may depend on the ability of the model to simulate melt and refreeze processes.

Further qualitative analysis is done by comparing observed and modeled density profiles from ORCHIDEE in the percolation zone (Fig. d). The profiles 2–5 show some successful reproduction of densification trends in this area. Exception is made for the first profile, which represents a modeled snowpack without clear evidence of refreezing (with surface densities around 600 kgm-3), whereas density observations suggest that refreezing occurs, leading to a densification of the snowpack. Other available observed profiles can be found in Figs. S1–S3 in the Supplement and show similar results.