IMAU-FDM is a 1D semi-empirical model that simulates density, temperature, and meltwater content in the firn layer. The model is originally developed by and has undergone multiple updates . We use IMAU-FDM v1.2A for the Antarctic domain and IMAU-FDM v1.2G for the Greenland domain, which only differ from each other in their ice-sheet dependent calibration. Both versions have been evaluated extensively against firn density and temperature observations .

The firn and ice column are divided into 200–3000 layers, traced in a Lagrangian framework. Snowfall is added to the topmost layer of the column, burying other layers and thereby representing downward advection. No layers are removed during a simulation. The layers have a maximum vertical resolution of 0.15 m. Layer routines are built in that split the uppermost layer if it exceeds the 0.15 m threshold and merges the two uppermost layers if the thickness of the uppermost layer is smaller than 0.05 m. When the firn column has less than 200 layers, 100 very thin layers of pure ice are added to the bottom of the column.

The fresh snow density is an upper boundary condition with different parameterizations for the AIS and GrIS domain. In IMAU-FDM v1.2A (AIS), the fresh snow density ρ0 (kgm-3) depends on instantaneous surface temperature Ts (K) and 10 m wind speed ff10m (ms-1) (Eq. ) : 1ρ0=82.97+0.769Ts+11.67ff10m In IMAU-FDM v1.2G (GrIS), ρ0 is based on the surface temperature averaged over the year prior to the snowfall Ts‾ (K) (Eq. ) : 2ρ0=362.1+2.78(Ts‾-273.15) The densification of snow and the underlying firn in each layer is parameterized by a semi-empirical equation (Eq. ) based on : 3dρdt=Db˙gMO(ρi-ρ)e-EcRT+EgRTb with b˙ the average accumulation rate (kgm-2yr-1) over the spin-up period, g the gravitational acceleration (9.81 ms-2), ρi the density of bubble-free ice (917 kgm-3), ρ the instantaneous layer density (kgm-3), Ec the activation energy for creep (60 000 Jmol-1), Eg the activation energy for grain growth (42 400 Jmol-1), R the universal gas constant (8.3145 Jmol-1K-1), T the instantaneous layer temperature (K), and Tb the instantaneous temperature of the bottom firn layer. D is a constant and MO is a correction term that improves model alignment with observations of the 550 and 830 kgm-3 density level depths . The correction term differs for both domains . In this equation, e-EcRT represents the temperature-dependency of ice deformation; eEgRTb is a correction term arising from the estimated grain size (Ec>Eg). Ice grains grow faster in warmer conditions, while large grains deform slower than smaller grains, hence reducing the densification. As the current grain size is the result of the grain growth since deposition, its impact on densification must be approximated by the typical temperature conditions. Therefore, the mean firn temperature, estimated by the temperature of the lowermost layer (Tb), is used in Eq. ().

The physical description for water percolation and heat transport is identical in both model versions. Liquid water is added to the top of the column. Its transport through the layers is calculated using the “bucket scheme”. In this scheme, each layer has an irreducible water content which is the maximum amount of water that can be retained by capillary forces. The irreducible water content decreases with increasing density . Water refreezes when in a firn layer cold content is available, reducing the pore space of the firn.

In IMAU-FDM, ice layers form through subsequent events of wetting and refreezing, whereby the density is increased and the pore space is reduced until the ice density of 917 kgm-3 is reached. Ice layers primarily form near the surface where sufficient meltwater and cold content allow for repeated cycles of pore space filling and refreezing. Therefore, in the current model setup, a single melt event cannot completely fill and refreeze the pore space of a firn layer. The bucket scheme does not allow standing water over ice layers. Instead, water percolates to the next layer when no pore space is available. The water percolation formulation implies that, in one model time step, water can percolate through the whole firn layer until it reaches the firn-ice interface. Any leftover water is assumed to run off.

Heat conduction through the firn layer is described by the 1D heat transfer equation, with conductivity being a function of density . Refreezing acts as a heat source within the firn. Heat transfer at the upper boundary is governed by the prescribed surface temperature. A constant heat flux through the bottom layer is applied as the lower boundary condition.

IMAU-FDM is forced at the surface with surface temperature, 10 m wind speed, snowmelt, snowfall, sublimation, snowdrift erosion, and rainfall. These climate forcings originate from the regional atmospheric climate model RACMO2.3p2 , dynamically downscaling ERA5 reanalysis data . The AIS data covers the period 1979–2023 with a horizontal resolution of 27km×27km and is extended from . The GrIS data spans 1 September 1939–1 December 2023 with a horizontal resolution of 5.5km×5.5km and is a renewed and extended version of the simulation presented by .

We force the model with four climate forcing time steps (dtforce) at the surface: 3 h, 6 h, 1 d and 1-month, referred to as dtforce=3 h, 6 h, 1 d, and 1 m respectively. We chose these values because (1) they are typically used to force firn models, and (2) they provide examples with and without a diurnal cycle in the input data (Fig. ). In addition, dtforce=3h is the smallest available climate forcing time step from RACMO2.3p2 and is commonly used to force IMAU-FDM. We averaged the 3-hourly surface mass fluxes, surface temperature, and 10 m wind speed data to get the input data for dtforce=6h, 1 d, and 1 m. IMAU-FDM interpolates the forcing data onto a 15 min model time step in which we assume a constant value for the whole forcing period for the surface mass fluxes, surface temperature and 10 m wind speed (Fig. ). To summarize, numerical time integration is done using a model time step of 15 min, but the climate forcing time step can be 3 h, 6 h, 1 d, or 1-month.

In IMAU-FDM, the firn layer is initialized by spinning up the model over a repeated reference period until it reaches a steady state where the firn air content (FAC) and surface height no longer change. FAC is the vertically integrated pore space in a column expressed in meters, representing the change in depth that would result from compressing the firn layer to the density of glacier ice (here assumed to be 910 kgm-3). The full firn layer, up to the pore close-off density of 830 kgm-3, is refreshed during the initialization. To avoid shocks after the initialization, this reference period should not exhibit large climate trends. The AIS spin-up period covers 1979–2020 . After 2020, the AIS has gained mass due to higher than average precipitation , and therefore 2020–2023 is excluded from the spin-up period. The GrIS SMB has been decreasing since 1990 . Therefore, we chose 1 September 1939–31 December 1970 as the reference period. The climate forcing time step was identical for spin-up and the rest of the simulation.

We select the APIS and the southern GrIS, two areas with a broad range of surface climate conditions, from low to high accumulation and with and without melt (Fig. ). Accumulation represents the net result of snowfall, sublimation and drifting snow processes (deposition/erosion).

In the APIS, the main climate gradient is from the wet and warm west to the dry and cold east. The ice shelves are the lowest-lying areas (Fig. a) and experience most snowmelt (up to 500 kgm-2yr-1) (Fig. b), whereas the snowmelt is limited on the higher-elevated grounded ice. In contrast, the grounded ice has higher accumulation, peaking in the north-west of the APIS. Accumulation is lower on the ice shelves (Fig. c).

In the southern GrIS, the lowest-lying marginal areas (Fig. d) experience most snowmelt (>750 kgm-2yr-1) (Fig. e). Accumulation is highest in the east (>2000 kgm-2yr-1) (Fig. f). Snowmelt rates exceeding 500 kgm-2yr-1 combined with high accumulation, as in the southeastern GrIS, typically results in firn aquifers, or, in drier climate zones like the southwest, in ice slabs .

Figure  shows the FAC3h (Fig. a and e), ΔFAC6h (Fig. b and f), ΔFAC1d (Fig. c and g), and ΔFAC1m (Fig. d and h) for the APIS and southern GrIS. Figure b, c, d, f, g, and h show that FAC3h is generally lower than FAC with larger dtforce in both the APIS and southern GrIS. The larger the dtforce becomes, the higher the ΔFAC.

On the APIS, FAC is smallest (0–5 m for FAC3h) in the lower-lying areas, corresponding to the ice shelves (Fig. a). ΔFAC is largest on the ice shelves (15 % for ΔFAC1d, increasing up to 44 % for ΔFAC1m). ΔFAC6h is relatively small (3 %). Interestingly, an area with a negative ΔFAC is situated along the high-elevation southern spine of the APIS (red hues in Fig. b–d). In a relative sense, these differences are small, of the order of ∼0.5 m on typical FAC3h values of 15–20 m. In both an absolute and a relative sense, ΔFAC is largest for the locations with low FAC3h.

In the southern GrIS, FAC increases from the ice margin where it equals zero in the ablation zone, to higher elevations, with the steepest spatial gradients on the east coast (Fig. e). In the southern GrIS, in the ablation zone, most clearly visible on the western margin, ΔFAC is small (Fig. f–h). Here, firn pore space is depleted across all simulations, regardless of dtforce. At high elevations (>2000 m), ΔFAC is also relatively small, with average differences <1 % in ΔFAC6h up to 3 % in ΔFAC1m. At intermediate elevations, a band with larger ΔFAC is apparent, increasing with greater dtforce (Fig. f, g, and h). In the east and west, ΔFAC in this band is quite similar in an absolute sense: on average, ΔFAC6h=0.3m, ΔFAC1d=2.1m, and ΔFAC1m=4.0m. However, relative differences are larger in the west, where absolute FAC3h values are much smaller (typically <20 m) than in the east (typically >20 m).

We discuss the processes that contribute to ΔFAC in IMAU-FDM. Figure  gives an overview of the dominant processes in a firn layer that cause the forcing time step dependency of the FAC and that are described in depth later in this section. The processes described below are computed every 15 min model time step during both the spin-up and the simulation periods.

FAC is governed by a balance between the processes that create pore space (snowfall) and that deplete it (densification, snowmelt, snowdrift erosion, and sublimation). Figure  shows the relative importance of several processes determining FAC3h and FAC1d for two example locations depicted in Fig. . Figure a represents a location in the APIS and Fig. b a location in the southern GrIS. Both locations experience similar melt and ΔFAC1d, but the location on the GrIS has more snowfall. For clarity, a positive excursion of the elevation change difference in Fig. c and d means that the elevation increases more for dtforce=1d.

Snowfall is the major source of firn air in both locations (Fig. a and b). The contribution of snowfall to FAC for the APIS is slightly higher for dtforce=3h than for dtforce=1d (Fig. c), whereas it is almost the same for the GrIS (Fig. d). Larger differences arise for processes that reduce FAC. In both locations, more FAC3h is removed by snowmelt than FAC1d (Fig. c and d). For the dry location (Fig. a), densification is relatively slow, and FAC depletion due to melt is largest. For the wet location (Fig. b), the densification dominates reduction of the FAC. At both locations, melt removes more FAC3h and has slower firn densification at dtforce=3h (Fig. c and d).

Summarizing, ΔFAC is mainly controlled by melt and densification rate, with a minor contribution from snowfall. Now, we further zoom in on those processes to understand the ΔFAC they cause.

Figure a shows that ΔFAC on the APIS increases with snowmelt. Accordingly, ice shelves have largest positive ΔFAC values (Fig. b–d), reflecting their relatively high snowmelt rates (Fig. b). Consequently, the chosen dtforce is crucial in assessing whether the firn air is depleted or could still buffer meltwater. Ice shelves are more vulnerable to hydrofracturing when the firn air is depleted. Therefore, neglecting the daily temperature and melt cycle in firn simulations could lead to underestimated ice shelf vulnerability.

Figure b presents a FAC timeseries of a single grid cell on the Larsen C ice shelf (see Fig.  for location). The overall variations in FAC are similar across all simulations, with ΔFAC primarily established during the spin-up. The equilibrium state of the firn column has higher FAC for larger dtforce, where reduced FAC removal during melt events is balanced by enhanced densification.

The wider range of climatic conditions in terms of snowmelt and accumulation in the southern GrIS (Fig. a) compared to the APIS (Fig. a), complicates the isolation of processes governing ΔFAC. To facilitate the discussion, we divide the accumulation-snowmelt space into four domains (Fig. a).

The largest increases in ΔFAC1d occur at snowmelt rates of ∼500 kgm-2yr-1 and near a Melt-Over-Accumulation (MOA) ratio of 1 (Fig. a). Climate zones with snowmelt below 500 kgm-2yr-1 (Fig. a, climate zone I), corresponding to the interior of the southern GrIS (Fig. e), behave similarly to the APIS: higher snowmelt rates increase ΔFAC1d.

Firn aquifers can form when snowmelt exceeds 500 kgm-2yr-1 . In climate zone II (Fig. a and b), which are the points with annual snowmelt between 500–800 kgm-2yr-1, firn aquifers formed under dtforce=3h during spin-up, but did not for dtforce=1d. The firn has a higher heat content throughout the entire column for dtforce=3h, due to the deeper refreezing of meltwater. The higher heat content limits the refreezing capacity. We hypothesize that more liquid water remains in the firn layer and the aquifer can refill during the next melt season. In contrast, the cold content for dtforce=1d still facilitates refreezing of all liquid water, and therefore, aquifers did not form during spin-up.

In climate zone III, snowmelt exceeds 800 kgm-2yr-1 and the MOA is below 1. Firn aquifers form under both dtforce=3h and 1 d (Fig. b). Here, ΔFAC1d increases with snowmelt as less pore space is removed for larger dtforce (Fig. , process 1), but this process is increasingly counterbalanced by enhanced firn densification at sites with higher accumulation (Fig. a). FAC3h is lower in this climate zone, so less pore space is available for water storage, resulting in less liquid water content (LWC) for dtforce=3h whereas the liquid water is still retained for dtforce=1d (Fig. b). More excess water for dtforce=3h leads to more runoff. This example illustrates that the runoff limit also depends on dtforce (Fig. c). In the southeastern GrIS, the runoff limit shifts more than 10 km inland for dtforce=3h compared to dtforce=1d, and up to 35 km compared to dtforce=1m. This result shows that dtforce plays a role in estimates of future GrIS mass balance projections.

When the MOA exceeds 1 (Fig. a climate zone IV), ΔFAC equals 0, as the firn is depleted in both simulations.

Unlike the example of the APIS (Fig. b), long-term FAC variability in Fig. d depends on dtforce (Fig. d). For dtforce=3h, firn aquifers are present at the start of the simulation, whereas for dtforce=1d they develop around 2005 (climate zone II) (Fig. e). In this year, the change in FAC1d diverges from FAC3h, and eventually, FAC1d even drops below FAC3h. This results from the densification equation (Eq. ): liquid water in the aquifers sets the instantaneous temperature (T in Eq. ) to 273.15 K. The bottom temperature (Tb in Eq. ) is higher for dtforce=3h (Sect. ), because of the deeper refreezing, slowing down densification. The faster densification under dtforce=1d reduces FAC1d. Hence, for locations where aquifers form during the simulation, the time series of FAC (and surface elevation) become different and their trends possibly of opposite sign. This has important implications for the correct interpretation of satellite altimetry.

Figure b shows an example from the APIS where the FAC differences due to dtforce are established during the spin-up period (1979–2020), during which the climate is assumed to be in an approximate steady state. The simulation continues for only three more years. Therefore, the ΔFAC for this location is mainly established during the spin-up period. Figure d shows a GrIS aquifer location, where ΔFAC develops both during and after the spin-up, which ends in 1970. The changing climate after 1970 causes FAC3h and FAC1d variations to diverge further. In this example, the ΔFAC divergence is associated with the formation of firn aquifers.

Figure  shows the spatial distribution of ΔFAC1m during the last year of the simulation without the contribution from the spin-up period. The magnitude of ΔFAC established after the spin-up is larger on the GrIS than on the APIS, because the GrIS simulation covers a larger period with a changing climate. In addition, on the GrIS, the changing climate leads to formation of firn aquifers, causing the FAC to diverge after the spin-up under different dtforce (Fig. d and e). The mechanisms described in Sect.  lead to the ΔFAC differences established both during and after the spin-up.

As simulated FAC depends on dtforce, then so does firn depth and hence surface height change, which is in turn used to correct altimetry data. For the same APIS sample location as shown in Fig. b, surface height change under dtforce=1d and 1 m differs from dtforce=3h by 0.2 and 0.4 m, respectively (Fig. a). ΔFAC already emerges during spin-up. At a GrIS location in transition to an aquifer, as shown in Fig. d and e, ΔFAC is primarily established after the spin-up and shows a stronger divergence of surface height changes, at 8 %–30 % of the FAC (Fig. b). Altimetry corrections based on IMAU-FDM simulations with dtforce=1d and 1 m would give higher surface height increases for the two examples. Consequently, during periods of ice thinning, subtracting the modeled firn depth change from the altimetry signal leads to an underestimation of ice loss, while during periods of ice thickening it leads to an underestimation of ice gain.