The climate model HadSM3 (Williams et al., 2001) is an atmosphere–slab ocean general circulation model (GCM). It has a resolution of 2.5∘ in latitude and 3.75∘ in longitude, with 19 levels in the vertical (Gordon et al., 2000). The MOSES-1 scheme is chosen as the land surface scheme (Cox et al., 1999) with a tundra-like albedo on the Antarctic continent where no ice is present and an albedo for snow where ice is present. Sea surface temperatures are reconstructed based on a best estimate from Evans et al. (2017) for the late Eocene in order to calibrate the corrective heat fluxes from the slab ocean model. These corrective heat fluxes represent the seasonal deep-water exchange and horizontal heat transport that is present in the real ocean. The oceanic heat fluxes are exchanged between the atmosphere and the slab ocean model in the mixed layer, which is 50 m thick in our simulations. In this way, realistic sea surface temperatures are simulated for the different climate model simulations.

The model is chosen because of its good performance over the Antarctic ice sheet for the present day and the Last Glacial Maximum (Connolley and Bracegirdle, 2007; Maris et al., 2012). Moreover, HadSM3 is a computationally efficient climate model that allows for performing a large number of experiments (Valdes et al., 2017). The paleogeographic reconstruction for the simulations is based on the method presented in Baatsen et al. (2016) and makes use of the GPlates software (Müller et al., 2018). The paleogeographic reconstruction represents the continental configuration at 39 Ma as representative for the late Eocene. As a result, the Antarctic continent has a slightly different position compared to the present-day. The bedrock topography for Antarctica is derived from the Wilson et al. (2012) maximum bedrock elevation reconstruction (Fig. 1). Simulated temperatures for a warm orbital configuration (maximum austral summer insolation) and a cold orbital configuration (minimum austral summer insolation) are shown in Fig. 2a and b, respectively.

The Antarctic ice sheet model AISMPALEO is a three-dimensional thermomechanical ice sheet and ice shelf model (Huybrechts and de Wolde, 1999) used for simulating ice sheet dynamics during periods in the past (Goelzer et al., 2016a, b) and the future (Seroussi et al., 2020; Van Breedam et al., 2020). The Shallow Ice Approximation is used to calculate the grounded ice flow, which is a result of internal deformation and basal sliding where the pressure melting point is reached. The model comprises a component taking into account the solid Earth response due to ice loading. This component consists of a rigid elastic plate on top of a viscous asthenosphere to allow for deviations from local isostatic loading. The surface mass balance is computed using the positive degree day (PDD) method (Janssens and Huybrechts, 2000), where the yearly sum of daily average temperatures above 0 ∘C is used to determine the melt potential. The standard deviation of the mean daily temperature is 4.2 ∘C, representing random weather fluctuations and the daily cycle. The difference in snow and ice albedo in the ice sheet model is taken into account by using a PDD factor for snow melting of 0.003 m ice equivalent (i.e.) ∘C-1 d-1 and a PDD factor for ice melting of 0.008 m i.e. ∘C-1 d-1. Monthly mean temperature and precipitation are used from HadSM3 to drive the PDD model. The rain limit is chosen at 1 ∘C and determines whether precipitation falls as snow or as rain. Meltwater retention allows runoff to be retarded and/or to eventually refreeze in the snowpack. Ice shelf formation is included and calculated using the shallow shelf approximation. Ice shelves start to form when the grounding line reaches the coast and the influx of ice from the continent exceeds the ablation (surface ablation and basal melting). A constant basal melt rate of 1 m yr-1 is used in all the simulations. The ice sheet model is run at a resolution of 40 km to allow for long integrations.

The Gaussian process (GP) principal component analysis (PCA) emulator (Wilkinson, 2010; Bounceur et al., 2015; Lord et al., 2017) used in this study is a statistical representation of the climate model HadSM3 (the simulator). The emulator is calibrated on a relatively small number of climate model runs and aims to predict the climate for any combination of climatic forcing of the original climate model runs. To allow for reliable predictions, the initial climate model runs need to fill the entire multi-dimensional input space. In our case, the multi-dimensional input space consists of the orbital parameters (eccentricity e, obliquity ε and longitude of perihelion ϖ), the carbon dioxide concentration forcing, and the ice sheet size and extent defined by the ice sheet parameter. The theoretical background has already been discussed in detail in previous papers (Araya-Melo et al., 2015; Bounceur et al., 2015; Lord et al., 2017), and here the focus is on the implementation of the dynamic ice sheet component.

It is recommended to have at least 10 experiments per input parameter (Loeppky et al., 2009). Therefore, the recommended minimum number of experiments with our five input parameters (three orbital parameters, CO2 concentration and ice sheet parameter) would be 50. Since the atmosphere–slab ocean model is time efficient, we have chosen to run 100 climate model runs with five variable forcing parameters. The ice sheets have a very distinct climatic imprint compared to the orbital parameters and the CO2 level, which all result in smooth climatic fields. Because of the large difference in albedo between ice and tundra at the edge of the ice sheet, the climatic imprint of a certain ice sheet geometry has a sharp boundary. The number of ice sheet geometries taken into the model design of the emulator might therefore have a large impact on the performance of the emulator. To test the impact of the ice sheet parameter, four different emulators are constructed based on a different number of predefined ice sheet geometries or based on a different spread of the ice sheet geometries (Figs. 3, A2, A3 and A4). The different emulators are named according to the number of ice sheet geometries in the model design: 8, 12, and 20 for EMULATOR_8, EMULATOR_12a and EMULATOR_12b, and EMULATOR_20, respectively.

Except for the number of predefined ice sheet geometries, the spread of the different prescribed geometries also varies between the different emulators, depending on whether the ice sheet parameter is defined by ice area or by ice volume. EMULATOR_8, EMULATOR_12a and EMULATOR_20 have a good spread between the different ice sheet geometries in terms of ice volume and ice area. EMULATOR_12b is well defined for ice volume but poorly defined by ice sheet area as there are several experiments with the same ice sheet area but different ice sheet geometry (Fig. 4 and Table A1 in Appendix A for the experimental parameter values). In this way, the influence of the spread in ice sheet volume and ice area on the emulated climate is investigated. The spacing of the different ice sheet geometries is expected to be crucial for medium-sized ice sheets because they constitute a transition zone towards a fully glaciated continent. EMULATOR_20 has the smallest spacing of ice sheet geometries around the crucial medium-sized ice sheets, separated at the minimum distance that corresponds to the resolution of the climate model. The maximum ice sheet geometry in the model design for EMULATOR_12 is smaller than the maximum ice sheet geometry in the model design for EMULATOR_20 and EMULATOR_8. The objective of designing EMULATOR_12 is to evaluate to what extent the emulator can still be used in an extrapolation regime beyond the largest ice sheet geometry.

A model design with 100 GCM experiments is constructed where each experiment has a different combination of orbital parameters, CO2 concentration values and ice sheet geometry (see Table A1 in Appendix A). Insolation values are well approximated as a linear combination of the eccentricity and longitude of perihelion (Loutre, 1993), and therefore the terms esin⁡ϖ and ecos⁡ϖ in combination with the obliquity ε are used for the orbital parameter variation in the model design. The range of orbital parameters is taken from Laskar et al. (2004) for the period 40 to 33 Ma. The eccentricity has a maximum value during the period between 40 and 33 Ma of 0.063, and the obliquity is sampled in the range of 22–24.5∘. The CO2 interval ranges from 550 to 1150 ppmv, roughly equivalent to 2×CO2 to 4×CO2​​​​​​​. The ice sheet parameter consists of 8, 12 or 20 predefined ice sheet geometries. They are constructed based on preliminary steady-state ice sheet model runs for a range of different climatic forcings (for EMULATOR_12a and EMULATOR_12b) or from ice sheet geometry snap shots during the build-up of a continental scale ice sheet (for EMULATOR_8 and EMULATOR_20). The final ice sheet geometries are chosen to provide a range from an almost ice-free Antarctic continent up to a fully glaciated continent. Tundra is present between the ice sheet margin and the coast. The parameter combinations are constructed using a Latin hypercube design where the minimum Euclidean distance between two parameter combinations is maximized (Fig. 5). With this model design, 100 GCM runs are performed until the climate (atmosphere and slab ocean) is in steady state with the forcing (40 years).

The design matrix of input data D (n×p) has 100 rows (number of experiments) and five (esin⁡ϖ, ecos⁡ϖ, ε, CO2, ice volume or ice area) or six columns (esin⁡ϖ, ecos⁡ϖ, ε, CO2, ice volume, ice area). Each simulation performed by the climate model is characterized by a row of matrix D, called the input vector xi. The climatic output (temperature and precipitation) where HadSM3 is a function of the input vector is called f(xi) is from all 100 experiments saved in the matrix Y. Each column of Y contains the output for one experiment. Here, the matrix Y only contains climatic output data on the ice sheet model grid (201 × 201 grid points) because our interest is the climate evolution over Antarctica.

The climatic output is modelled as a Gaussian process, defined by a mean function m(x)​​​​​​​ and a covariance function V(x,x′). The prior mean function is defined by a linear combination of a set of linear regression functions (Eq. 1), where β is a vector of regression coefficients corresponding to the mean function and h(x) is a vector of known regression functions of the inputs. The covariance function consists of the correlation function and the scaling value σ2 (Eq. 2). The squared exponential correlation function is chosen with the inclusion of the so-called nugget ν and correlation length δ hyperparameters (Eq. 3). I is an operator that equals 1 when x=x′ and equals 0 in all other cases. di is the distance between x and x′​​​​​​​. The nugget was originally meant to deal with sampling variability of the simulator output, but even when this sampling variability is small, it is effective to prevent numerical instability and to compensate for inadequate correlation priors (Andrianakis and Challenor, 2012; Araya-Melo et al., 2015). The correlation length δ can be understood as a measure of how quickly the model output is changing as a function of the input (Wilkinson, 2010). The larger the distance between two input vectors, the quicker the correlation goes to zero. 1m(x)=h(x)Tβ2Vx,x′=σ2cx,x′3cx,x′=exp⁡-∑i=1pdiδi2+νI The formalism followed here is Bayesian, and the prior mean and prior covariance functions (Eqs. 1–3) are updated (Eqs. 4–5) based on the climate model output data. All values of β are a priori equally probable, and we assume a vague conjugate prior (β, σ2) that is proportional to σ2. The posterior distribution of the model data (temperature and precipitation) is a Student's t distribution with n-q degrees of freedom (which is close to Gaussian) with a mean m∗(x) and covariance V∗(x,x′) as follows: 4m∗(x)=h(x)Tβ^+t(x)A-1y-Hβ^,5V∗x,x′=y-Hβ^TA-1(y-Hβ^)n-q-2×cx,x′+νI-t(x)TA-1tx′+P(x)HTA-1H-1Px′T, where β^=(HTA-1H)-1HTA-1y. y is the model output, defined by the normal distribution N∼(Hβ,σ2A), with Aij=c(xi,xj), and H the design point regression matrix. t(xi)=c(x,xi), and P(x)=h(x)T-t(x)TA-1H. A PCA is performed on the climatic output, and between 17 and 20 components are kept before calibrating the emulator (see model code for principal components, length scales and nugget values). The Gaussian process model with the posterior means and variances is applied to each principal component. Once the PCA emulator is calibrated (by optimizing the nugget and length scales), the scores of each principal component can be estimated for arbitrary input values, with an associated covariance that effectively measures the prediction uncertainty.

As described above, four different emulators are constructed (EMULATOR_8, EMULATOR_12a, EMULATOR_12b and EMULATOR_20), and each emulator is calibrated with either ice volume, ice area, or with both ice volume and ice area as the ice sheet parameter. This gives a total of 4×3=12 distinct calibrated emulators to simulate the ice sheet evolution. Calibration of an emulator is achieved by adjusting the length scales δ, the nugget ν (uncertainty band) and the number of principal components (PCs) in order to minimize the root-mean-square error between the simulated and predicted climatic fields in leave-one-out experiments. It is assumed that the sampling error introduced by model variability is almost negligible because we are using a slab ocean climate model version where the internal climate variability is small and the climate states quickly converge to a mean value. Therefore, the adopted nugget is chosen to be 0.001 (small non-zero uncertainty band around the data) for the temperature and 0.01 for the precipitation emulation. The emulator calibration is done for precipitation and temperature data for each month. During the calibration process, the number of PCs is varied between 5 and 25. It is chosen to keep 20 PCs to explain the observed variation in temperature and 17 PCs to explain the variation in precipitation, since these numbers gave the best emulator performance (as explained below).

R's Nelder–Mead optimization function (Nelder and Mead, 1965) is used to maximize the likelihood of the emulator (Kennedy and O'Hagan, 2000; Lord et al., 2017). We obtain a low length scale for the ice sheet parameter (between 0.02 and 0.5). When looking at the summer temperature for all GCM runs, there is a smooth, almost linear dependency between the ice sheet parameter and the simulated summer temperatures (Fig. 6), and therefore we increased the length scale for the ice sheet parameter. The optimization function is used to get the correlation lengths for the orbital parameters and the CO2 forcing right and the correlation length for the ice sheet parameter was manually chosen to be 1.2 for all emulators.

The emulator performance is determined by the variance of the emulator and the reliability of the emulator. The variance is a measure of the uncertainty of the mean predictions of the emulator. The reliability of the emulator determines how well the emulator is calibrated (how well the emulator estimates its own uncertainty). Ideally, the emulator is well calibrated and the uncertainty is low. The calibration is investigated by leave-one-out experiments where the simulated temperature is predicted based on the calibrated emulator while leaving out one experiment at a time. The results are visualized in Fig. 7 where the number of grid points that is predicted within 1 (grey) to 4 (red) standard deviations from the simulated temperature is given for each of the GCM runs for EMULATOR_20. Overall, all emulators perform well, since > 68 % of the grid points are predicted within 1 standard deviation (Table 1). The calibration based on ice volume and ice area separately shows a very similar reliability and uncertainty. Even though the variance for the emulators calibrated on both ice area and on ice volume is lower, it has a lower reliability because it struggles to capture the output dependency on both variables simultaneously.

The spatial difference between the simulated and the predicted climatic fields is visualized in Fig. 8. January temperatures are shown for experiment xaemdk, which performs poorly, and for experiment xaemdv, which performs quite well. These experiments are run with the second smallest and smallest ice sheet geometry, respectively. The predicted temperatures are warm-biased up to 8 ∘C for the tundra region and cold-biased up to 10 ∘C for the ice-covered region for experiment xaemdk. The bias is much smaller for experiment xaemdv with errors of less than 2 ∘C over most of the Antarctic continent. Other experiments that perform poorly, such as xaemdg and xaembb, have a very high eccentricity, high obliquity, and a summer during aphelion or perihelion. They have the most extreme insolation values and lay at the edge of the experimental design, which may explain why the emulator does a poorer job of predicting the simulated temperatures.

Due to computational limitations, the coupling procedure between ice sheets and climate on a multi-million year timescale is always asynchronous. However, the emulator–ice sheet coupling leaves the possibility for a very short coupling time because once the emulator is calibrated, it can be run in stand-alone modus (without the need to use the simulator). In these simulations, the ice sheet model is initialized from an ice-free state. After a predefined time (1000 years in the standard experiments), the ice sheet model passes the ice sheet parameter (the actual ice volume, ice area, or both ice volume and ice area) to the emulator and the emulator provides temperatures and precipitation as a function of the orbital parameters, CO2 forcing and the ice sheet parameter. In our simulations, temperature and precipitation are interpolated to the ice sheet model grid using a bilinear interpolation scheme, in order to have smooth climatic fields. In the standard experiments, a constant lapse rate correction of 5 ∘C km-1 is applied between HadSM3 and AISMPALEO. The climatic information is lapse rate corrected for the nearest input ice sheet geometry in terms of ice volume or ice area.

The ice sheet parameter represents the shape and area of the ice sheets. In previous studies (Araya-Melo et al., 2015; Lord et al., 2017), it has been defined by indexing the different ice sheet geometries that have been used to generate the simulation outputs. However, there are several other options as to how to define the ice sheet parameter. Here, it is proposed to define the ice sheet parameter by quantifying the ice sheet volume, the ice sheet area or a vector combining both aspects. This is done for the four different experiment designs. The ice sheet area and ice sheet volume are good parameters to define the ice sheet's influence on climate because the first parameter affects the local albedo and the latter has an influence on the elevation and hence on the temperatures through adiabatic cooling. In case the ice sheet parameter is defined by both the ice sheet volume and the ice sheet area, both variables are calculated after each iteration of the ice sheet model.

Figure 9a shows the ice sheet evolution for the four emulators calibrated on ice volume. EMULATOR_12a, EMULATOR_12b and EMULATOR_20 show the transition towards a continental scale ice sheet in a very narrow CO2 interval of 845 to 875 ppmv. On the other hand, EMULATOR_8 does not seem to show any sensitivity to the CO2 forcing during the 3 Myr simulation (and also not on a longer timescale). The reason is that the prescribed ice sheets are separated too much in the initial climate model runs. Because of the large difference in albedo between ice and tundra, the prescribed ice sheets create a sharp boundary at the ice sheet margin that is visible in the temperature field. If insufficient prescribed ice sheet geometries are used, the ice sheet does not grow enough to see the temperature regime obtained when HadSM3 is run with the next input ice sheet geometry. Consequently, the emulated temperatures remain too high at the ice sheet margin. It appears that the threshold on the number of needed ice sheet geometries is somewhere between 8 and 12. Using 20 input ice sheet geometries does not change the model performance much.

Another option is to calibrate the emulator based on the ice area, which is more directly linked to the albedo. The glaciation threshold for EMULATOR_12a and EMULATOR_20 shows a very similar sensitivity to CO2 of about 860 ppmv (Fig. 9b). EMULATOR_8 grows immediately to a medium-sized ice sheet and cannot grow further towards a continental-scale ice sheet for the reasons already quoted. EMULATOR_12b was poorly defined in terms of ice area (several ice sheet geometries had a similar area but different geometry), and the ice sheet grows immediately towards a continental scale. Therefore, in addition to having sufficient ice sheet geometries, a good spacing for the ice sheet parameter is another requirement to use an emulator for coupled ice sheet–climate simulations.

The simulated temperatures during the first 100 kyr of the simulations are visualized during a strong insolation maximum after 47 kyr and a strong insolation minimum after 60 kyr (Fig. 10) for all four emulators calibrated on ice volume. The corresponding ice sheet sizes are given in Fig. A5 (Appendix). The temperature patterns are very similar, especially above the tundra regions. The main difference in simulated temperatures between the different emulators is caused by the size of the ice sheet, which in turn is determined by the number and spacing of ice sheets used for the calibration.

When the coupling is based on both the ice volume and the ice area, the transition to a fully glaciated climate gives very distinct results for each of the emulators (Fig. 9c). EMULATOR_12a shows the transition to a continental-scale glaciation for a similar CO2 threshold than for the emulators tuned on ice volume or ice area of around 890 ppmv. EMULATOR_12b shows the transition to a fully glaciated continent immediately when the simulations start because of the poor definition on ice area. Remarkably, the transition to a fully glaciated continent for EMULATOR_20 happens for a much lower CO2 threshold of 765 ppmv. However, the reliability of EMULATOR_20 calibrated both on ice area and ice volume is lower than the reliability of the emulator on either ice volume or ice area (see Sect. 2.3), even though the variance is smaller when additional information on the ice sheet parameter is added. The poor emulation originates from calibrating the emulator based on six variables, while only five input forcing parameters are actually reasonably independent. The additional information on the ice sheet parameter is strongly correlated in most cases (though not everywhere) because the spread between ice volume and ice area is not equal. This is visualized in Fig. 11 where three different schematic ice sheet geometries are shown with their respective ice volume and ice area (normalized). For the second ice sheet geometry, the ice area increases by 0.8 units, while the ice volume increases by 0.2 units. In contrast, the next ice sheet geometry is defined by an ice volume increase of 0.8 units and an ice area increase of 0.2 units. In EMULATOR_20, the smallest prescribed ice sheet geometries have a significant increase in ice area, while the ice volume increase is relatively small. On the other hand, the largest prescribed ice sheet geometries have a much larger increase in ice volume than ice area (Fig. 4). For EMULATOR_12a, the relative increase in ice area and ice volume is more equal, and therefore the reliability and the performance of the emulator is better. EMULATOR_8 grows to a fully glaciated continent for a CO2 threshold around 810 ppmv. As mentioned earlier, the lack of sufficient ice sheet geometries is also visible here with complete growth and decline close to the glaciation threshold.

The difference in ice sheet area and ice sheet volume evolution during the build-up of an ice sheet is further visualized in Fig. 12. When the ice sheet starts growing, both the ice sheet area and the ice volume increase. However, the timing between ice sheet volume growth and ice sheet area growth are not fully synchronous. When the climate is cool during an austral summer insolation minimum, the area where the mass balance is positive increases, and first the area where ice is accumulating increases. It takes more time for the ice volume to adjust, and by the time the ice volume reaches a maximum, the ice area starts decreasing again during an austral summer insolation maximum. This can be seen in Fig. 12c, where the first snapshot of the ice sheet geometry shows a relatively large area with ice, but ice volume at that moment is negligible. The ice area will start decreasing towards an insolation maximum, while the ice volume is still growing in response to the large area with a positive mass balance. The delay of the ice volume response compared to ice area is even more clear when an ice sheet is growing from bare bedrock to a continental-scale ice sheet(Fig. 12a and b).

These remarks suggest a possible improvement that would consist of a better experiment design with ice sheets spanning a 2D space more optimally. On the other hand, the difference in relative magnitude of ice volume and ice area during the build-up of an ice sheet also suggests that it is not easy to create an optimal set of variables for the model design.

The coupling time in the first set of experiments is 1000 years. The sensitivity of the ice sheet evolution to the coupling time is tested by applying five different coupling times, ranging from 10 to 2000 years. The smallest time step is of the same order of magnitude as the time step used in the ice sheet model. In this way, it can be regarded as an example of a direct coupling between the climatic component and the ice sheet model. Fig. 13 shows the ice sheet evolution during one precession cycle for the five coupling times. The climatic information for the largest time step of 2000 years is updated 11 times during this interval, and the ice sheet evolution clearly responds stepwise. Another observation is that the higher the coupling time step, the more delayed the ice sheet response to the forcing and the lower the amplitude of the ice sheet volume. Decreasing the coupling time step results in a smoother ice volume evolution. The differences between a coupling time of 500, 250 and 10 years become very small, suggesting that the solution converges. To make a compromise between model efficiency and model accuracy, we opted for performing the multi-million-year sensitivity simulations with a coupling time step of 500 years as a lower limit and 2000 years as an upper limit.

The coupling time is doubled and halved to respectively 2000 and 500 years to test its influence on the ice sheet evolution for EMULATOR_12a and EMULATOR_12b calibrated with ice volume as the ice sheet parameter (Fig. 14). Halving the coupling time step increases the computational time with about 40 %, while doubling the coupling time decreases the computational time with the same percentage. When the coupling time decreases, the ice sheet volume has a larger amplitude and is slightly more sensitive to changes in insolation.

For EMULATOR_12a, the glaciation threshold is more sensitive to CO2 changes when the coupling time is decreased. The difference in glaciation threshold between a coupling time step of 500 and 1000 years is negligible, but the difference with a coupling time step of 2000 years is about 30 ppmv. The continental-scale glaciation for EMULATOR_12b with a coupling time step of 1000 years occurs for lower CO2 values than for a coupling time step of 500 and 2000 years due to the complex interaction between ice sheet response and forcing. The shorter the coupling time, the more sensitive the ice sheet is to the forcing. For a coupling time step of 1000 years, the ice sheet grows more than for a coupling time step of 2000 years, but does not decline as much as for a coupling time step of 500 years and therefore grows quicker to the fully glaciated state.

Comparing the glaciation threshold with respect to the CO2 forcing between the different coupling time steps, there is a decreasing difference between EMULATOR_12a and EMULATOR_12b for a decreasing coupling time step. The difference in glaciation threshold is about 40 ppmv for a coupling time step of 2000 years, 30 ppmv for a coupling time step of 1000 years and 10 ppmv for a coupling time step of 500 years, suggesting that the solution converges for decreasing coupling time steps.

The lapse rate is the change in temperature with elevation, and its value is highly dependent on the moisture in the air. Since the climate model is relatively coarse compared to the ice sheet model, we need to apply a lapse rate correction to the elevation difference between the climate model and ice sheet model. The lapse rate is, in reality, both temporarily and spatially variable, and here we test different model choices. One experiment includes temporal variations in the lapse rate correction, and another experiment includes both spatial and temporal variations. The temporally variable lapse rate is calculated as the average near-surface lapse rate for all grid points on the Antarctic continent for each month (Figs. 15 and A1). The spatially variable lapse rate is included by calculating the local near-surface lapse rate (dT/dZ) simulated by HadSM3 for the four adjacent ice-covered grid points for each month. The average monthly lapse rate varies roughly between the wet adiabatic lapse rate during summer (December–January) and the dry adiabatic lapse rate in winter (June–July). These values are higher than the constant lapse rate that was applied in the standard experiments, and therefore the lapse-rate-corrected temperatures are lower for a growing ice sheet. The resulting ice sheet evolution shows a more stepwise change towards full glaciation (Fig. 16). The larger temperature difference between the climate model and the ice sheet model makes it harder for the ice sheet to grow further until a threshold is reached and a large area is cold enough for snow accumulation.