HadCM3 is a coupled atmosphere–ocean general circulation model (Gordon et al., 2000; Valdes et al., 2017). It accurately reproduces the heat budget of the present-day climate (Gordon et al., 2000) and has been used for future climate projections in the Intergovernmental Panel on Climate Change Fourth Assessment Report (IPCC AR4) (e.g. Solomon et al., 2007) and paleoclimate reconstructions such as PMIP2 (Braconnot et al., 2007) and PlioMIP (Haywood and Valdes, 2003; Dolan et al., 2011, 2015; Haywood et al., 2013a). The atmosphere module of HadCM3 has a resolution of 2.5∘ latitude by 3.75∘ longitude. The ocean is modelled at a horizontal resolution of 1.25∘ by 1.25∘, with 20 vertical layers. In the model set-up by Berends et al. (2018), the climate matrix consists of two GCM snapshots of, respectively, the pre-industrial period (PI) and the Last Glacial Maximum (LGM), produced by Singarayer and Valdes (2010) with HadCM3. Here, we include several additional snapshots focussing specifically on the Pliocene.

To simulate the evolution of the ice sheets we use ANICE, a coupled 3-D ice-sheet–shelf model (Bintanja and Van de Wal, 2008; de Boer et al., 2013, 2014, 2017). It combines the shallow shelf approximation (SSA; Morland, 1987) for floating ice shelves with the shallow ice approximation (SIA; Morland and Johnson, 1980) for grounded ice to solve the ice flow. A Mohr–Coulomb plastic law for basal sliding is included, with basal stresses included in the SSA equations. The basal stress is calculated as a function of a till stress, which in turn depends on the local bedrock elevation (Winkelmann et al., 2011; de Boer et al., 2013). For grounded ice, the velocities resulting from both approximations are summed, resulting in a smooth transition zone between slow-flowing land ice and fast-flowing floating ice. This approach allows the grounding line to respond to changes in shelf buttressing, resulting in proper glacial–interglacial differences in Antarctic ice volume (de Boer et al., 2013; Berends et al., 2018) by the advance of grounding lines in the Filcher–Ronne and Ross basins toward the continental shelf. The surface mass balance is parameterised using an insolation–temperature scheme using monthly temperatures and precipitation, refreezing of water and a correction for orographic forcing of precipitation; a more detailed model description is provided by de Boer et al. (2013) and references therein. The horizontal resolution of ANICE for this application is 20 km for Greenland and 40 km for the other three regions (North America, Eurasia and Antarctica). The highly parameterised climate forcing and resulting computational efficiency of ANICE allow for transient simulations of multiple glacial cycles to be carried out within 10–100 h on single-core systems, making ensemble simulations feasible. Melt underneath the ice shelves is calculated using a linear relation to ocean temperature change (Pollard and DeConto, 2009; Martin et al., 2011), a parameterisation of subshelf cavity circulation based on the shortest linear distance to the open ocean (Pollard and DeConto, 2009) and the glacial–interglacial variance parameterisation by Pollard and DeConto (2009). A more detailed explanation is provided by de Boer et al. (2013), who tuned this approach to produce realistic present-day Antarctic shelves and grounding lines. A simple threshold thickness of 200 m is used to describe ice calving, whereby any shelf ice below this thickness is removed.

Using the definition by Pollard (2010), a climate matrix is a collection of pre-calculated output data from several steady-state GCM simulations, called “snapshots”, that differ from each other in one or more key parameters, such as prescribed atmospheric pCO2, orbital configuration or ice-sheet configuration, each creating a separate dimension of the matrix. When performing a simulation with an ice-sheet model, at every point in time during the simulation the prescribed climate forcing is determined by combining the climate states constituting the matrix according to the position of the model state within the matrix. This constitutes a middle ground between methods of offline forcing, such as a glacial index method, and fully coupled ice-sheet–climate models. When the ice-sheet model is in a state corresponding to one of the GCM snapshots, the climate from this snapshot will be prescribed, containing the effects of the altitude–temperature and albedo–temperature feedbacks of the ice sheets, the effect of ice sheet geometry on large-scale atmospheric circulation and precipitation, and possibly also the effects of changed freshwater fluxed on ocean circulation, depending on the GCM. When the ice-sheet model is in a state lying in between different GCM snapshots, the prescribed climate is a spatially variable linear interpolation of these snapshots. This means that non-linearities in different feedback processes, such as the effects of ice-sheet geometry on atmospheric circulation and precipitation, are difficult to properly account for.

In this study, we use the model set-up developed by Berends et al. (2018), who created a matrix with the HadCM3 climate states of the pre-industrial and the Last Glacial Maximum from Singarayer and Valdes (2010) and used it to force the ANICE ice-sheet model. In this set-up, temperature fields from the two climate states are combined based on a prescribed value for pCO2 and on the internally modelled ice sheets, with the feedback of the ice sheets on the climate based via the effect on absorbed insolation through changes in surface albedo. This interpolation is carried out separately for all four ice sheets. The altitude–temperature feedback is parameterised by a constant lapse rate derived from the GCM snapshots. Precipitation fields are combined based on changes in surface elevation, reflecting the orographic forcing of precipitation and resulting plateau desert caused by the presence of a large ice sheet. Berends et al. (2018) demonstrated the viability of this method by simulating the evolution of the North American, Eurasian, Greenland and Antarctic ice sheets throughout the entire last glacial cycle, showing that model results agree well with available data in terms of ice-sheet extent, sea-level contribution, ice-sheet surface temperature and contribution to benthic δ18O.

In this study, we extended the PI-LGM climate matrix by adding several climate states from the study by Dolan et al. (2015). The four different ice sheet configurations they used are shown in Fig. 2. The “PRISM” ice sheets are based on the PRISM3 reconstruction (Dowsett et al., 2010), which is a time-slab representation of average peak warm conditions during the MPWP. The “Small” ice sheets are present-day conditions. The “Medium” and “Large” ice sheets were based on the ICE-5G reconstruction of the last deglaciation (Peltier, 2004) at 8 and 11 kyr ago, respectively. Each of these configurations was used as boundary conditions for two simulations with HadCM3: one with 280 ppmv and one with 220 ppmv pCO2, both with 3.3 Myr orbital parameters. This adds up to eight different snapshots, plus one additional “Plio_Control” simulation with the PRISM3 ice sheet configuration and orbital parameters, and 405 ppmv pCO2. These simulations allow the climate matrix to separate effects on climate by pCO2 and ice-sheet extent and provide valuable information on climates that are both warmer and colder than the present day. Although the new climate matrix is relatively sparse for warmer-than-present worlds, containing only one snapshot (Plio_Control) for pCO2 > 280 ppmv and only three snapshots for smaller-than-present ice sheets (Plio_Control, PRISM_280 and PRISM_220), we believe it is still suitable for simulating the warm Pliocene. The matrix used by Berends et al. (2018) to simulate the last glacial cycle only contained two GCM snapshots in total and still produced satisfactory results.

This extended matrix therefore allows for a more accurate simulation of both the warm Late Pliocene and the cold MIS M2 glaciation. For the North American and Eurasian modules of ANICE, we added the simulations of the Medium and Large ice sheets (both the 280 and 220 ppmv pCO2 versions) by Dolan et al. (2015), since those provide extra information on the effect on climate of intermediate-sized ice sheets, as well as the Plio_Control simulation for its information on pCO2 levels above 280 ppmv. In the case of North America and Eurasia, we did not use the Small and PRISM states, as there is no ice on either continent, meaning these simulations contain no additional constraints for these ice-sheet models. For the Greenland and Antarctica models, we chose to only add the simulations of the two PRISM states and the Plio_Control simulation, because they provide new information on the effect on climate of smaller-than-present-day ice sheets. The Medium and Large simulations were left out of the matrix because the ICE-5G ice sheets (Peltier, 2004) that were used to force those HadCM3 simulations have the exact same horizontal extent as the ICE-5G LGM ice sheets. Not only does this make it difficult to distinguish between these states in the interpolation routines, it also means the effect on local climate, other than through the altitude–temperature feedback, is likely to have been small.

The inverse forward modelling approach used to determine pCO2 based on the difference between modelled and observed benthic δ18O is very similar to that described by de Boer et al. (2013). Their method calculates how the climate at high latitudes, described by a single, spatially uniform temperature offset ΔTNH, should have evolved, such that its effect on deep ocean temperature and land ice volume reproduces the observed benthic δ18O signal. This is achieved by comparing the modelled benthic δ18O value δ18Omod at every time step to the observed value δ18Oobs. If it is too positive, then either the ocean is not cold enough or there is not enough land ice. Global mean surface temperatures are then lowered in the next time step, leading to both a cooling in the deep ocean and an increase in ice growth. This relation is quantified by de Boer et al. (2013) by the following equation: 1ΔTNH=ΔTNH‾+20δ18Omod-δ18Oobst+0.1kyr. Here, ΔTNH‾ is the mean surface temperature anomaly between 40 and 80∘ latitude at sea level over the preceding 2 kyr. The modelled benthic δ18O is calculated using ice volume, ice-sheet δ18O and deep-water temperatures relative to the present day (PD) for every 100 years. The spatially variable isotope content of the individual ice sheets is tracked through time, with the surface isotope balance based on the observed present-day relation between precipitation rates and isotope content according to Zwally and Giovinetto (1997). Benthic δ18O is assumed to be linearly dependent on the global mean deep-water temperature anomaly, which is calculated by temporally smoothing the global mean surface temperature anomaly. The optimum values of 2 kyr for the length of the CO2 averaging window, 3 kyr for the deep-water temperature averaging window and 20 for the scaling parameter, were determined by de Boer et al. (2013), producing a value of ΔTNH=-15 K at LGM.

Since the climate matrix used in our model determines the regional climate based on the modelled ice sheet and the scalar atmospheric CO2 concentration, we adapted this approach for our model set-up by using the difference between modelled and observed δ18O to calculate a value for pCO2, which is subsequently forwarded to the climate matrix. That way, our model reconstructs how pCO2 should have evolved in order to change global climate in such a way that the resulting changes in deep ocean temperature and land ice volume reproduce the observed benthic δ18O signal. The algorithm then becomes 2pCO2=pCO2‾+120δ18Omod-δ18Oobst+0.1kyr. As the constrained quantity is the change in pCO2, the scaling factor changes to 120 ppmv/‰ δ18O change in order to produce a glacial–interglacial contrast of 90 ppmv pCO2. Based on the results of preliminary experiments, the length of the CO2 averaging time window was increased to 8.5 kyr, in line with the higher values given by Stap et al. (2016). For forcing the inverse routine, the LR04 stack was used. Although a few different globally distributed stacks are available (e.g. Imbrie et al., 1984; Lisiecki and Raymo, 2005; Zachos et al., 2001, 2008; Cramer et al., 2009), the differences among them are small for the Late Pliocene and the Pleistocene. In order to maintain consistency with earlier reconstructions based on inverse modelling methods (de Boer et al., 2013; Stap et al., 2016; van de Wal et al., 2011), we decided to use the LR04 stack.

A conceptual visualisation of the inverse-method-forced matrix model is shown in Fig. 3.

Several recent studies have investigated the evolution of bedrock topography in the geological past. Although ANICE does include a regional solid Earth model to calculate vertical bedrock movement in response to changes in ice distribution, other processes such as erosion and plate tectonics are currently not accounted for within the model and require external forcing. For the last glacial cycle benchmark simulation, such effects are assumed to be negligibly small, but this assumption might no longer be valid when going millions of years back in time.

In our study, we use the palaeotopography reconstruction of the Barents Sea area by Butt et al. (2002), shown in Fig. 4, based on a reversal of the erosion of sediments by the Pleistocene ice sheets. In order to investigate the effect on atmospheric and oceanic circulation, Hill (2015) used HadCM3 to perform simulations both with this topography reconstruction and with present-day topography, both at 405 and 220 ppmv pCO2. By subtracting the calculated climate fields (temperature and precipitation) for the palaeotopography simulation from the present-day topography simulation, and adding the resulting “fingerprint” to the climate fields generated by our climate matrix, we take into account the effect of this change in topography (and the accompanying change in the land–ocean mask) on the global climate.

Preliminary experiments showed that forcing the ice-sheet model with the palaeotopography reconstruction without applying this climate “fingerprint” resulted in the persistent presence of a small (∼5 m sea-level-equivalent) ice sheet over the newly exposed Barents Land. The climate fingerprint obtained from the simulations by Hill (2015) changes the local climate from an oceanic to a continental climate, with colder, dryer winters and warmer, wetter summers, resulting in more summer melt and an overall more negative mass balance, implying less ice.

Although other areas of the world might have been eroded by ice sheets (i.e. the Canadian Archipelago, Dowsett et al., 2016; Antarctica, Wilson and Luyendyk, 2009), no GCM simulations investigating the effect on global climate of reversing those changes are currently available. We have therefore chosen not to apply any of these other topography reconstructions to our model.

In order to assess the performance of the model when calculating pCO2 with the inverse routine instead of prescribing it directly from an ice-core record, we first performed a simulation of the last four glacial cycles, similar to the work by Berends et al. (2018). The model was calibrated by tuning the ablation parameter for the four individual ice sheets such that their volumes at LGM match the ICE-5G reconstruction. We then performed a sensitivity analysis similar to the experiment described by Berends et al. (2018), investigating the sensitivity of the modelled sea-level drop and benthic δ18O to the uncertainty in the prescribed forcing (range based on the uncertainty reported by Lisiecki and Raymo, 2005), the ablation tuning parameter (range based on the allowed values found by Berends et al., 2018) and SIA/SSA enhancement factors (increasing the ice velocities calculated for isotropic ice to more closely match those calculated for anisotropic ice according to the approach by Ma et al. 2010; ranges based on the allowed values reported by Ma et al., 2010), as well as several new model parameters involved in the inverse forcing method: the averaging time for the modelled pCO2 (range based on the values reported by de Boer et al., 2013 and Stap et al., 2016), the ratio between surface temperature anomaly and deep-sea water temperature anomaly and the tuning parameter relating pCO2 to the difference between observed and modelled δ18O (ranges based on the values reported by de Boer et al., 2010, and Bintanja and van de Wal, 2008), resulting in 17 individual simulations. The values that were used for all these parameters are listed in Table 1. The 17 ensemble members thus yield an estimate of the uncertainty related to both model parameters and forcing.

The simulated pCO2 record is compared to the European Project for Ice Coring in Antarctica (EPICA) Dome C ice-core record (Lüthi et al., 2008) in Fig. 5. The ranges of modelled values for pCO2 and sea-level drop at LGM for all investigated model parameters are listed in Table 2. Based on these uncertainties, the model shows that LGM pCO2 is 188–197 ppmv and that the sea-level-equivalent volume of the four continental ice sheets at LGM was 83–100 m, agreeing well with the values of 185 ppmv pCO2 from the EPICA ice core and 100 m sea-level-equivalent ice volume from the ICE-5G reconstruction (Peltier, 2004). The modelled pCO2 values match the EPICA record better than the values simulated by Stap et al. (2016), as demonstrated by the linear correlation and root mean squared error (RMSE) between the EPICA Dome C record and the reconstructions; R2=0.46 and RMSE = 23.7 ppmv for Stap et al. (2016) and R2=0.71 and RMSE = 15.2 ppmv for our simulation. The reconstruction by van de Wal et al. (2011) performs very similarly to ours (R2=0.72, RMSE = 14.7 ppmv), but since it was partly derived from the EPICA record, the comparison is not independent and therefore cannot be compared to our results.

Benthic oxygen isotope abundance and its contributions from ice volume and deep-sea water temperature are shown in Fig. 6 and compared to reconstructions by Lisiecki and Raymo (2005) and by Shakun et al. (2015), who made a proxy-based decomposition of the respective contributions to the benthic δ18O from land ice and deep-sea temperature. The simulated benthic δ18O shows a near-perfect match with the LR04 stack (Lisiecki and Raymo, 2005) that was used to force the model, as is to be expected when using the inverse forward modelling approach. The observed rapid drop in δ18Osw at the inception, between 120 and 110 kyr BP, is reproduced well, as is the drop in Tdw. Deep-water temperature between 60 ka and LGM appears to be too high, more so than for the CO2-forced model version by Berends et al. (2018).

Surface temperature anomalies over Greenland and Antarctica compared to ice-core records (EPICA Dome C; Jouzel et al., 2007; GISP2; Alley, 2000; NGRIP; Kindler et al., 2014) throughout the last glacial cycle are shown in Fig. 7. The performance of the new model version in terms of ice surface temperature is comparable to that of the model by Berends et al. (2018), as illustrated by the linear correlation coefficients and root mean square error between the modelled temperatures and the ice-core records: R2 = 0.87 and RMSE = 0.86 K for Antarctica in this model versus R2 = 0.84 and RMSE = 0.91 K in the old version. For Greenland, the new model produces a value of R2=0.74 and RMSE = 2.2 K versus R2=0.65 and RMSE = 2.6 K for the old model.

The mismatch during the inception of the glacial cycle between isotope-derived Antarctic surface temperature and ice-core CO2 on the one hand and benthic δ18O and sea level on the other hand, reported by Bintanja and van de Wal (2008), van de Wal et al. (2011), de Boer et al. (2014), Niu et al. (2017) and Berends et al. (2018), is much better in the simulations here. The linear correlation coefficient R2 between modelled and reconstructed Antarctic surface temperatures between 120 and 80 kyr ago increased from a value of 0.49 for Berends et al. (2018) to a value of 0.74 for this study. For Greenland, this value increased from 0 to 0.36. The CO2-forced model (Berends et al., 2018) produced Antarctic surface temperatures that were in good agreement with the isotope-based proxy record but failed to reproduce the strong sea-level drop. The δ18O-forced model from this study reproduces benthic δ18O and its different contributions and shows a too-strong decrease in pCO2 but is in overall agreement with proxy records of CO2, sea level and temperature, indicating that there is an added value of using the climate matrix method as applied here.

The comparisons between model results and (proxy) data for the simulations of the last four glacial cycles indicate that the model accurately reproduces pCO2, ice volume and general geometry (not shown), and surface temperatures. We therefore proceeded to apply the new model set-up to the Late Pliocene. We chose to start our transient simulations 3.65 Myr ago, capturing the warm period between 3.6 and 3.4 Ma. The simulations were run until 2.75 Myr ago, since the density of available pCO2 proxy data is much higher after MIS M2, allowing for a more detailed comparison of modelled pCO2 to proxy-based pCO2 reconstructions. The model was initialised with the same PRISM3 ice sheets (Dowsett et al., 2010) that were also used to force the PRISM and Plio_Control HadCM3 experiments by Dolan et al. (2015). Due to the nature of the inverse coupling method, initialising the model with present-day ice sheets quickly converges to the same result. Topography was set to the present day plus the Barents Sea erosion reversal from Butt et al. (2002) and a glacial isostatic adjustment (GIA) correction accounting for the difference in ice loading over Greenland and Antarctica according to the PRISM3 reconstruction (Dowsett et al., 2010). Insolation and benthic δ18O were prescribed according to Laskar et al. (2004) and Lisiecki and Raymo (2005), respectively. In order to estimate the uncertainty in the modelled ice volume, we performed the same sensitivity analysis as for the last glacial cycle, with the same parameter values shown in Table 1. The resulting simulated pCO2 record is shown in Fig. 8 and compared to other model reconstructions (van de Wal et al., 2011; Stap et al., 2016) and to proxy-based data derived from alkenones (Seki et al., 2010; Badger et al., 2013; Zhang et al., 2013) and δ11B ratios (Seki et al., 2010; Bartoli et al., 2011; Martínez-Botí et al., 2015; Stap et al., 2016).

The ranges of modelled values pCO2 and sea-level change at MIS M2 and at KM5c (3.205 Myr ago) for all investigated model parameters are listed in Table 2. KM5c is used because it has been identified as a time slice representing the mid-Pliocene Warm Period (Haywood et al., 2013b). These simulations show that during MIS M2, pCO2 is 233–249 ppmv and that the sea-level-equivalent volume of the four continental ice sheets was 10–25 m bigger than the present day, with the uncertainty based on the spread in the results from the ensemble of simulations. The uncertainty in modelled pCO2 becomes much larger for warmer-than-present climates, as shown by the modelled ranges for KM5c. The sensitivity to the benthic δ18O forcing is especially high, resulting in modelled pCO2 values of 303–384 ppmv. The reason for this is that the climatological forcing resulting from the climate matrix is less constrained for warmer-than-present-day climates. Whereas the matrix contains six snapshots describing climates with more ice and/or lower pCO2, there is only one ice sheet configuration smaller than the present day (PRISM) and only one snapshot with a pCO2 higher than 280 ppmv (the Plio_Control simulation, with 405 ppmv).

The resulting modelled sea-level contributions over time are shown in Fig. 9. The modelled ice sheets over the Northern Hemisphere and Southern Hemisphere at MIS M2 and KM5c are shown in Figs. 10 and 11, respectively. In North America, MIS M2 is clearly visible as a strong peak in ice volume, which immediately disappears when pCO2 rises again. Most of the ice forms over northeastern Canada, with a smaller ice sheet developing over the northern Cordillera. In Eurasia, only small ice caps form on Svalbard and Nova Zembla (no longer islands but now small mountain areas bordering the newly exposed Barents Land), with no sizeable ice sheets forming even at the peak of MIS M2. Greenland is mostly ice-free until MIS M2, when it rapidly develops an ice sheet slightly larger than the present day. After MIS M2, the ice sheet disappears, advancing and retreating several times during the following period. Similar behaviour is observed on West Antarctica, while East Antarctica remains stable throughout the simulation. While both Greenland and Antarctica continue to show substantial variability throughout the remainder of the simulations, North America does not glaciate again until the onset of the Pleistocene glacial cycles, 2.8 Myr ago.

Global mean sea level is compared to two different reconstructions in Fig. 12. Our model results generally lie between the δ18O-based reconstruction by Miller et al. (2011) and the reconstruction based on geological backstripping from New Zealand by Miller et al. (2012). During warm periods, our model generally shows lower sea levels and less variability than Miller et al. (2011, 2012). During cold periods, our model generally shows less sea-level drop than the δ18O-based reconstruction by Miller et al. (2011) but more than the reconstruction based on geological backstripping. The δ18O-based reconstruction by Miller et al. (2011) is based on a linear relation between benthic δ18O and sea level, which is an oversimplification of separating the contributions to the benthic δ18O signal (e.g. Bintanja et al., 2005; de Boer et al., 2013). Miller et al. (2012) noted that reconstructing absolute values for local relative sea level based on geological backstripping is difficult due to the required corrections for GIA and dynamic topography. However, the relatively short duration of MIS M2 means that the reconstructed drop in sea level, of about 20 m relative to the background level, is likely to be accurate. Our model produces a value of about 24 m, in good agreement with this value.

The evolution of the West Antarctic ice sheet agrees partially with information derived from the AND-1B sediment core, recovered from beneath the northwest part of the Ross Ice Shelf by the ANDRILL programme (Naish et al., 2009; McKay et al., 2012). Information derived from this core by de Schepper et al. (2014) is compared to model results in Fig. 13. AND-1B shows ice-free conditions in the Ross Sea up to 3.4 Myr ago, followed by glacial deposits up to 3.24 Myr ago. Our model results show ice-free conditions up to 3.32 Myr ago, just prior to MIS M2. The ice-free conditions shown in our model results around KM5c cannot be validated by AND-1B due to a lack of data. Between 3.14 and 3.04 Myr ago, AND-1B again contains glacial deposits when our model results indicate ice-free conditions. The glacial conditions between 3.04 and 2.95 Myr ago and the subsequent ice-free conditions between 2.95 and 2.90 Myr ago indicated by AND-1B match with our model results. However, since the AND-1B sediment core contains several sizeable data gaps due to geological unconformities, the possibility that observed ∼40 kyr cycles in ice-rafted debris concentration have been incorrectly matched with 40 kyr cycles in the δ18O age model cannot be precluded. We therefore conclude that the AND-1B sediment core record cannot be used to confirm or refute our model results.