We used a 1-D ice-flow model, IcIES-2 (Saito et al., 2020). This model computes the temporal evolutions of the age and temperature profiles of ice columns.

The evolution of the age of the ice is computed using the vertical advection equation: 1∂A∂t=-w∂A∂z+1, where A is the age of the ice, defined as the duration since deposition, and w is the vertical velocity of the ice (a positive value indicates upward velocity). Here, ζ is a normalized coordinate defined as ζ=zH, where z is the height above bedrock, and H is the ice thickness (thus ζ=1 and 0 correspond to the ice surface and base, respectively). The first and second terms on the right-hand side of Eq. (1) represent the vertical advection and aging owing to the lapse of time, respectively.

The vertical velocity of the ice is assumed to be represented as follows: 2wζ=-Ms+Mb-∂H∂tωζ-Mb, where the terms Ms and Mb represent surface (positive indicates ice gain) and basal (negative indicates ice melt) mass balance caused by accumulation and basal melting, respectively, and ∂H∂t is the change in ice thickness over time. The normalized vertical velocity profile, ω, is given as a function of the normalized coordinate derived from Parrenin and Hindmarsh (2007) and Lliboutry (1979): 3ωζ=1-p+2p+11-ζ+1p+11-ζp+2, where ω is 1 at the surface and 0 at the base. Hence, in the case of steady state, ∂H∂t=0, the vertical velocity of the ice at the surface and base equates to -Ms and Mb, respectively. The shape of ω with different p parameters is demonstrated in Fig. 2, indicating that a larger p value yields a larger downward ice velocity. Compared with Fischer et al. (2013), who used a different formulation of the vertical velocity profile with an m parameter (similar role to p of this study) of m=0.5 (Fig. 2, dashed lines), p=3 from Eq. (3) gives a different vertical temperature profile, with a smaller vertical velocity, particularly near the base of the ice.

The temperature of the ice is computed using the following vertical advection and diffusion equation: 4∂T∂t=-w∂T∂z+1ρIcP∂∂zκ∂T∂z, where T is the temperature of the ice [K], κ is the thermal conductivity, ρI is the ice density, and cp is the heat capacity of the ice. The density of ice is set as a constant (910 kg m-2); i.e., we ignore the effects of lower density in the firn column. The strain heating term is neglected in the present study, given that deformation of the ice would be minor near Antarctic domes because of very low horizontal shear. The thermal conductivity and specific heat capacity of the ice are functions of temperature (Greve and Blatter, 2009, following Ritz, 1987): 5κ=9.828e-0.0057TWm-1K-1,6cp=(146.3+7.253T)Jkg-1K-1. Boundary conditions at the surface and base of the ice are required to close the equations. At the ice surface, the age is set as 0, assuming no surface melt, and the temperature is set to the surface temperature at the given time. The basal boundary conditions for temperature depend on the basal condition: 7∂T∂z|b=-Gκif no melting,8Tb=Tpmif melting, where G is the geothermal heat flux (GHF) at the ice–bedrock boundary, and Tpm is the pressure melting point of the ice, which is given as a function of depth using a Clausius–Clapeyron gradient (8.7 × 10-4 K m-1). The basal melting rate at the ice–bedrock interface is determined by the conservation of heat: 9MbρIL=G-κ∂T∂z, where L is the latent heat of the ice (335 kJ kg-1), and ∂T∂z|b is the temperature gradient at the ice–bedrock interface. This model assumes basal melting only occurs at ice–bedrock interfaces, and the temperature gradient at the ice–bedrock interface is calculated using a one-sided difference discretization. The calculated basal melting rate Mb influences the velocity field according to Eq. (2). The model in the present study forecasts temperature in the bedrock, and thus the GHF at the ice–bedrock interface has temporal variations. The bedrock is 3000 m thick, divided vertically into 17 equal layers; constant physical parameters are used for the bedrock (density = 2700.0 kg m-3, heat capacity = 1000.0 J kg-1 K-1, and heat conductivity = 3.0 W m-1 K-1) used in Parizek and Alley (2004).

We adopted different vertical resolution setups in computations of the temperature and age of the ice. The ice profile was discretized with 101 even vertical layers for thermodynamics; it was discretized with 2661 unevenly spaced vertical layers (finer near the base to resolve the thin layers of old ice) for age calculations, which was optimized following Saito et al. (2020). In the typical ice column thickness of 3000 m near DF, the vertical resolution was set to approximately 20 m near the surface and 20 cm near the bedrock, which is sufficient to resolve paleoclimate information (glacial–interglacial annual layer variations) of ∼ 1 ka. We used the rational function-based constrained interpolation profile (RCIP) scheme in the advection equation for the numerical scheme, as in Saito et al. (2020). One significant advantage of this scheme is the avoidance of numerical diffusion and ability to reasonably preserve the time derivative of age, which is critical to the resolution of old ice. We have tested the sensitivity to the vertical resolution of temperature calculation and found that using fine vertical resolution leads to the formation of a temperature inversion layer in the bottom of the ice, which can be a significant error in estimating basal temperature gradient and basal melting. Therefore, we set the number of vertical layers of the model for thermodynamics as 100 (each approximately 30 m thick) to prevent the representation of temperature inversion layers. The time steps of the calculation of temperature and age were set to 20 years.

This section applies the 1-D model to DF under a realistic climate history for model calibration and parameter constraint. Parrenin et al. (2007) determined the p value as ∼ 3.7 for DF, but the chronology of ice older than 335 ka was not established at that time; therefore, we revisited DF to determine the p value covering the entire DF ice-core age–depth dataset. The glaciological boundary conditions at DF are summarized in Table 1: we used an ice thickness of 3028 m, a present-day SMB of 30 i.w.e. mm a-1 (equivalent to 27.3 freshwater mm a-1, based on Kameda et al., 2008, and Fujita et al., 2011), and a mean ice surface temperature at present of -55.5 ∘C. We determined the boundary condition of ice surface temperature by calibrating the temperature profile to be consistent with measured temperature profiles of the top 500 m of the ice, within uncertainty ranges of the observations. The observed present-day 10 m depth annual mean snow temperature is -57.3 ∘C (Kameda et al., 1997), which was also used in Parrenin et al. (2007). We note that the annual mean surface air temperature (SAT) based on meteorological observations was -54.4 ∘C during the period 1995–1997 (Yamanouchi et al., 2003).

The model was forced by a realistic history of SAT and SMB. We used local SAT anomalies at DF for the past 715 kyr (Uemura et al., 2018) and the benthic record of marine oxygen isotope data (Lisiecki and Raymo, 2005) to construct a continuous time series of SAT anomalies during the last 2 Ma. We applied a simple translation of δ18O to scale the temperature change at DF by the amplitude of glacial–interglacial cycles: 10ΔTs=α(β-δ18O), where δ18O is the benthic marine oxygen isotope value [‰]; we set α=4.5 and β=3.23 to scale the amplitude of the glacial cycles, which generated a time series of temperature change over the last 2 Myr, as shown in Fig. 3a. We used past SMB as a function of temperature anomaly compared with the present day following Huybrechts and Oerlemans (1990), which is based on saturation vapor pressure: 11a(t)=a(ref)⋅exp⁡22.47T0Tfref-T0TftTf(ref)Tft2, where a(t) and a(ref) represent past and present SMB rates, respectively; T0=273.16 K is the triple point of water, and Tf is the atmospheric temperature above the inversion layer as a function of surface temperature (Tf [K] = 0.67Ts [K] + 88.9). From this function, an increase in surface air temperature of 1 ∘C increases SMB by approximately 7 % (Fig. 3b). At the Last Glacial Maximum (LGM, approximately 20 ka), when SAT was 8 ∘C cooler, the SMB was approximately 60 % of that of the present day, which is consistent with reconstructions based on the isotopic content of the ice (Parrenin et al., 2016). This relationship between SAT and precipitation changes used herein was within uncertainties estimated from observations and climate model simulations, following a summary by IPCC AR6 (chap. 9.4.2.3; Fox-Kemper et al., 2021), which used the studies of Bracegirdle et al. (2020) and Frieler et al. (2015). Although this relationship is not based on SMB, but rather on precipitation, herein we assume the precipitation change ratio is the same as that of the SMB. The other boundary conditions (ice thickness and GHF) were set as constants in the present study.

We used a result of transient simulation obtained by a 3-D ice-sheet model IcIES, which computes dynamics and thermodynamics of ice sheets using the shallow-ice approximation to simulate past ice thickness history. The experimental design was similar to that of Saito and Abe-Ouchi (2004, 2010) with some changes; the domain of the 3-D model was the whole Antarctic continent, and the horizontal resolution was set to 32 km. The spatial distribution of the GHF was from Martos et al. (2017). The model was initialized using the present-day condition, and it was forced by the same temperature and SMB changes as those of the 1-D model forcing for the past 2 Myr (Fig. 3a). The migrations of the grounding lines were not forecasted; instead the positions of grounding lines were fixed to the present day. We note that the advancement of grounding lines during glacial periods has a minor impact on the ice thickness, in particular around the DF region, compared with the changes in climate forcing (Saito et al., 2010). We extracted the history of changes in the ice thickness at DF and EDC, which showed that the ice thickness was reduced by ∼ 200 m during glacial periods, mainly because of reduced SMB (Fig. 3c).

Using this set of boundary conditions, we conducted simulations with different GHFs (50–70 mW m-2) to calibrate the model with observed values at the DF ice core. We used the depth–age profile of the DF ice core, which was constructed by orbital tuning of a gas record above ∼ 2500 m and matching the AICC2012 chronology below that depth (Kawamura et al., 2017). We also used the measured depth–temperature profiles from the JARE54 surveys conducted during the 2012–2013 Antarctic summer (Buizert et al., 2021). The model was initialized with the conditions of 2 Myr, where the initial age and temperature were set to 0 years and -10 ∘C, respectively, for the entire ice column. All experiments were integrated for 2 Myr to reach the present day; therefore, the age of any ice older than 2 Myr did not appear in the experiments. These simplified initial conditions generated unrealistic temperature fields in the early stage of the simulation, but realistic glacial cycle forcing prevailed over the entire ice column within approximately 100 kyr. Therefore, we mainly analyzed the results of the last 1.5 Myr, which is sufficient to discuss old ice in this study. Furthermore, we also applied this model to the conditions at EDC to check whether the model could simulate the observed temperature and age profiles at this location (Table 1).

We also conducted three sensitivity experiments to investigate the impacts of the p parameters, uncertainty in the amplitude of past temperature changes, and inclusion of past ice thickness changes, respectively. We found that p=3 gave one good age profile when compared with the ice-core data; hence, we set p=3 as the reference in Sect. 3. The uncertainty in the past temperature change was based on a study that proposed that the temperature change at the LGM in interior regions of the East Antarctic ice sheet was less than previously estimated (Buizert et al., 2021). We conducted a set of experiments where SAT anomalies were set to 0 %, 25 %, 50 %, and 75 % of the standard experiments, while keeping changes in SMB the same.

In Fig. 4, the simulated temperature profiles at 0 ka (end of the simulations) with different GHFs under the same p value (p=3) are compared with observations (Fig. 4a). The close-up of the bottom 120 m of the ice column is shown in Fig. 4b; the basal temperature was well below melting point with a GHF of 54 and 56 mW m-2, and at the melting point, the GHF >58 mW m-2. Compared with the observed temperature profile (Fig. 4, black lines), the simulated temperature near the ice base was colder by approximately 1 ∘C. In all simulations, the simulated temperature profiles were generally colder than observed temperature profiles, especially in the middle of the ice columns (Fig. 4a). The generally colder temperature of the ice may have several explanations. One is related to the pressure melting point of the ice. We used a pressure melting point of ice that depended only on local pressure, but there is also a dependence on the impurities and air content of the ice (e.g., Parrenin et al., 2017; Passalacqua et al., 2017). A second explanation is related to the uncertainty in vertical velocity of the ice parameterized with p because a larger vertical advection contributes to a colder ice temperature.

The time series of simulated basal ice melting rates over the last 500 kyr show that there have been significant temporal changes in these rates over time (Fig. 5a). With a GHF of 54 mW m-2, the temperature at the ice base has been below the melting point through the last 500 kyr. In contrast, in the case of a GHF of 56 mW m-2, the basal melting rate is zero at 0 ka, while the maximum basal melting rate of 1 mm a-1 occurs at the later stages of interglacial periods (e.g., 100 ka). This temporal variation in basal melting rate is caused by glacial cycle forcing in SAT and SMB, and minimum basal melting tends to occur at the end of glacial periods as it lags SAT. This result is broadly consistent with previous studies (Saito and Abe-Ouchi, 2004; Van Liefferinge et al., 2018) in that colder ice, which accumulated during glacial maximums, advects towards the ice base owing to an increased SMB during interglacials. A larger GHF (≥60 mW m-2) results in basal melting occurring most of the time, with an increase in the basal melting rate of approximately 1 mm a-1 for every 5 mW m-2 increase in GHF.

The simulated age profiles of the present day are compared with the ice-core-based profiles in Fig. 6a. With a small GHF (54 mW m-2) where basal melting does not occur, the ice age at the ice–bedrock interface is >1.5 Ma. In contrast, if basal melting occurs, the ice age at the ice–bedrock interface can be much younger: for example, 761 or 620 kyr for a GHF of 60 or 62 mW m-2, respectively. The result obtained with a GHF of 60 mW m-2 exhibits the closest fit to the data in terms of the age of ice at the base of the ice column. In this article, we define the “resolution of age” (kyr m-1) as the inverse of annual layer thickness as an indicator of sufficient resolution for the chemical and isotopic contents of the ice (Lilien et al., 2021). In Fig. 6b, the resolution of old ice is compared with the actual DF ice core. The model results largely reproduced the glacial–interglacial contrasts in annual layer thickness caused by the temporal variations of SMB at this locality. The observed resolution of age was approximately 0.5–1 kyr m-1 near the base of the ice core, and the model results using a GHF of 60 mW m-2 reproduced similar values. Furthermore, in a scenario with no significant basal melting, the annual layer thickness of 1.5 Myr ice is approximately 0.1 mm because 1.5 Myr ice appears directly above the bedrock (Fig. 6b, dark blue lines). In accordance with the results described above, a larger GHF tends to result in a higher basal melting rate and younger age of ice at the base of the column. One critical point is that an excessive GHF (i.e., an increase of the order of 2 mW m-2) can have a considerable effect on the age of the ice and the likelihood of old ice.

We also applied this model using the conditions at EDC to enable performance checks at an additional location. We used the parameters listed in Table 1 and conducted sensitivity experiments with different GHFs. For the vertical velocity profile, we used p=2.3 following Parrenin et al. (2007). The model generally resulted in colder temperatures compared with observations, similar to that found at DF (Fig. 7). We note that the pressure melting point of the ice depended only on local pressure in Fig. 7, but several studies have considered the pressure melting point of the ice as a function of the pressure and air content of the ice, which has shown that the basal temperature is at the pressure melting point (Buizert et al., 2021). Modeling using a GHF of 56 mW m-2 gave a basal ice age of approximately 800 kyr (Fig. 8a), which is similar to the value (802 kyr) presented in Veres et al. (2013), and the resolution of age closely fits the chronology estimated from the ice-core analysis (Fig. 8b). One important result is that the threshold of the GHF that allows basal melting is 4 mW m-2 lower at EDC compared with DF. This lower GHF can be attributed to the combination of larger ice thickness, smaller SMB, and higher SAT at the present day. The estimated GHF at EDC is smaller than that given by Parrenin et al. (2017), who estimated it to be 60 mW m-2. This difference can be attributed to the difference in the history of basal melting or the application of past climate history derived from DF to EDC. The results from the application of our model to EDC suggest that it may be applicable to different glaciological conditions, particularly different ice thicknesses and SMBs.

Next, we evaluated the sensitivity of the temperature and age profiles to different vertical velocity profiles, temperature amplitudes, and ice thickness changes over glacial cycles. In Fig. 9, results using different p values under an identical GHF (60 mW m-2) are compared. A larger p value induced a lower basal melting rate because of a larger vertical velocity and downward advection of cold ice from the surface, although this only had a minor impact on the temperature profile. The simulated age profiles indicate that a larger p value induces a younger age of ice at mid-depths within the ice column (Fig. 9b), which is also a result of a larger vertical velocity. The age of the ice at the base of the column was approximately 800 kyr in all five of the variable p-value simulations, partly because of the compensating effects of greater advection and less basal melting.

The results using DF conditions with different amplitudes of temperature change but constant GHF and p parameters (GHF = 60 mW m-2 and p=3) are summarized in Fig. 10. Here, we changed the α value in Eq. (10) (1 is the control case). In the smallest-amplitude experiment (α=0), the temperature was set to the interglacial level and did not change in time. Note that the SMB variation was the same in all sensitivity experiments. The control experiments exhibited colder ice temperatures near the middle of the ice column compared with observations, and this cold bias was reduced when a smaller temperature amplitude over the glacial cycles was used (Fig. 10a), broadly consistent with Buizert et al. (2021). A smaller amplitude of the glacial cycle resulted in a younger age of ice at the bottom of the ice column (Fig. 10b) because of larger basal melting rates (Fig. 10c). This is because the mean temperature over the glacial cycles increases if the temperature amplitude of glacial–interglacial cycles is reduced. The results using a fixed surface temperature (dTs = 0.0) corresponded to the same present-day SAT for the last 2 Myr, which induced basal melting of ∼ 1.5 mm a-1 during most of this time. A slight fluctuation in basal melting still occurred owing to time-dependent SMB.

The results without ice thickness changes did not impact temperature profiles in the present day (Fig. 11a) but impacted the history of basal melting (Fig. 11c). The mean basal melting rates at constant GHF can be reduced if ice thickness changes are included because the reduced ice thickness during glacial periods decreases the pressure melting point. Moreover, the inclusion of ice thickness changes affects the phase of basal melting rates because it reflects the reduction in ice thickness and pressure melting point at the base of the ice during glacials. The minimum in basal melting during the last glacial cycle occurs at the end of the LGM in the control experiment; in contrast, it occurs in the present day in the scenario with no ice thickness change. The absence of ice thickness changes results in larger mean basal melting rates and a younger age of ice at the base of the ice column (Fig. 11b). These results suggest that the basal melting rate in the past can be larger than the present-day rate.

On the basis of the results described in this section, we conclude that using a combination of p=3 and GHF = 60 mW m-2 gives reasonable temperature and age profiles. Therefore, we decided to use these values as calibrated parameters for the DF region; this was performed for the following reasons. Later in the article, we investigate the possibility of old ice in the DF region using different parameters of ice thickness and GHF because glaciological surveys have suggested that there are spatial variations in these parameters (e.g., Carson et al., 2014). Hence, obtaining precise tuning at one specific DF location is unnecessary. In this study, we calibrated the GHF under a vertical velocity profile of p=3, but calibrating the model with the combination of an uncertain GHF and vertical velocity profile is possible. According to the age profile, the results with p=3 may not necessarily be the best because the simulated age profile tends to underestimate the age of ice, particularly 500 m above the bedrock. Therefore, we do not state that a GHF of 60 mW m-2 is a single best estimate for the DF location compared with previous estimates (Burton-Johnson et al., 2020; Talalay et al., 2021) because there were assumptions made in the vertical velocity profiles and experimental design of this study. Furthermore, the calibrated GHF has some dependence on the uncertainty in temperature and ice thickness changes over the glacial cycles.

This section investigates the impact of the three parameters, ice thickness, SMB, and GHF, which may have spatial variations in the DF region. We investigated a range of ice thicknesses between 2000 and 3200 m, based on an ice thickness map of the area around DF (Fig. 1). We used a present-day SMB range of 25–35 i.w.e. mm a-1. There is large uncertainty in GHF; we adopted a range of 50–70 mW m-2. The list of experiments is given in Table 2. Other aspects of the experimental design are the same as in Sect. 3.

In Fig. 12a, the relative effects of ice thickness and GHF on basal temperature are compared using a constant SMB (30 mm a-1). As in Sect. 3, we used an ice thickness of 3028 m, which is comparable to that at DF, and a GHF for basal melting of 60 mW m-2. On the basis of the gradient of contours in Fig. 12a, an increase in ice thickness of 100 m has a comparable impact on the basal temperature as does an increase in GHF of 2 mW m-2. In Fig. 12b, the relative effects of ice thickness and SMB are compared using a constant GHF (60 mW m-2). A larger SMB results in a colder temperature; a 10 % change in GHF leads to a ∼ 4 ∘C change in the basal temperature, while a 10 % change in SMB leads to a ∼ 1 ∘C change. These results are generally consistent with those of Fischer et al. (2013) and suggest that the spatial distribution of SMB (∼ 20 % for the DF area) has a minor impact on the basal temperature compared with that of the ice thickness.

We further investigated the impact of different ice thicknesses on age profiles using the climatic conditions at DF (SMB = 30 i.w.e. mm a-1) and a calibrated GHF (60 mW m-2). Figure 13a shows the simulated age of the ice at 50 and 100 m above the ice–bedrock interface, which were used as indicator depths for potential coring sites by Fischer et al. (2013). The results indicate that the rate of aging of ice decreases with ice thickness between 2800 and 3200 m owing to the occurrence of basal melting. Note that the age of 2 Myr is the limit of the experiments, and the results indicate that the old ice exists 50 m above the bedrock if the ice thickness is thicker than ∼ 2100 m. Figure 13b shows the age resolution of the 1.5 Myr ice, indicating that a larger ice thickness tends to show a finer age resolution. The vertical age profiles and resolution of ice ages at five selected ice thicknesses with constant GHF are shown in Fig. 14. According to Fig. 14b, the expected age resolution of 1.5 Myr ice is approximately 10 kyr m-1 with an ice thickness of 2800 m and 20 kyr m-1 with a smaller ice thickness of 2200 m.

In this section, we apply the 1-D model to interpret the internal layers of the ice near DF under idealized boundary conditions. Here, we used the dataset from 17 December 2017 obtained by ground surveys during JARE59 (2017–2018), which comprises a 54 km long transect from DF to NDF (Fig. 1). The horizontal axis of Fig. 15 indicates the distance from DF, and the vertical axis indicates the depth from the surface. The gray shading indicates the reflectivity, which is an indicator of contours representing ice of the same age. The bedrock elevation, shown by brown lines, was detected based on the maximum reflectivity from the base (Tsutaki et al., 2022). The bedrock elevation was calibrated to match the observed bedrock elevation at DF. We calculated the 1-D age and temperature profiles of the ice at approximately 400 m intervals along the transect. We assumed that the vertical profile of vertical velocity could be determined locally and that there were no horizontal interactions in temperature and age in this simulation. The present-day SMB was linearly interpolated between DF (30 i.w.e. mm a-1) and NDF (25.5 i.w.e. mm a-1). Note that the estimated SMB at NDF is 13 % smaller than that at DF based on shallow ice cores (Oyabu et al., 2023). Because only very limited information on the spatial distribution of GHF is available, we set a uniform value of 60 mW m-2 following the discussion in Sect. 3. As described in Sect. 3, the initial age of the ice was set to 0, the temperature was set to -10 ∘C, and the model was integrated over the last 2 Myr of forcing until it reached the present day (Fig. 3).

In Fig. 15, the computed vertical profiles of the age are overlaid on a radargram using seven selected ages (colored lines), and the simulated basal temperature is indicated by shading in the bottom panel. The colored bar below the radargram indicates the simulated present-day basal temperature. The simulated distribution of ice age captured large-scale features in the black–white contour lines derived from the radargram signal (grayscale color in Fig. 15). The simulated age contours of 21 kyr (approximately 500 m depth) and 128 kyr (approximately 1500 m depth) can be traced from DF, although the deepest horizon corresponding to an age older than 300 kyr is hard to see in this image. Where ice is relatively thick (e.g., 20–25 km from DF), the simulated age of the ice at the ice–bedrock interface is younger than 700 kyr, while ice older than 1.5 Myr occurs where the ice is relatively thin. On the basis of the results shown in Fig. 13b, we note that thin ice gives a poorer age resolution for the old ice. A comparison of the simulated ice age and radargram signal gives an opportunity to examine the validity of the model results. For example, between 5 and 35 km from DF, the computed 128 kyr contour deviates to shallower levels (by 150 m) compared with the traced horizon for the age obtained from the radar measurements, suggesting that the model overestimates the age of the ice near the bedrock in such locations.