The dynamical core is the MIT general circulation model (MITgcm) , which solves the Navier–Stokes equations for the atmosphere and the ocean on the same cubed-sphere (CS) grid . In particular, we deploy MITgcm (version c68s) in a coupled setup including atmosphere, ocean, thermodynamic sea ice, and land (denoted as Coupled MITgcm Setup). We use the so-called CS32 configuration, where each face of the cube has 32×32 grid cells, corresponding to an average horizontal resolution of 2.8°. This or similar configurations have already been used for studying idealized configurations such as the coupled aquaplanet , deep-time climates , and the present-day climate .

Physical parameterizations for the atmosphere are based on the 5-layer SPEEDY model (Simplified Parameterizations, privitivE-Equation DYnamics) . SPEEDY is a model of intermediate complexity for the atmosphere, and includes simplified representations of convection, large-scale condensation, vertical diffusion, surface fluxes of momentum and energy. The radiative scheme uses two spectral bands for the shortwave radiation and four for the longwave radiation. Cloud cover and thickness are defined diagnostically from the values of relative and absolute humidity. The cloud albedo depends on latitude, as done in , to reduce net solar radiation at high latitudes and therefore to have better agreement with observational data . Five pressure levels are represented, from 1000 hPa to 0 Pa, where the bottom level represents the planetary boundary layer, the upper one the stratosphere and the remaining three the free troposphere. SPEEDY has been evaluated against the NCEP-NCAR and ERA5 reanalysis and, despite its simplified parameterizations, has been assessed to provide a realistic description of the atmosphere, with the advantage of requiring fewer computer resources than state-of-the-art Atmospheric GCMs. A simple 2-layer land model  is coupled to SPEEDY.

The physics packages used in the ocean component are the K-profile parameterization scheme  to account for vertical mixing in the water column and the Gent and McWilliams scheme   to capture residual-mean circulation and mixing by mesoscale eddies. The Winton model  is used to represent sea ice thermodynamics (THSICE), while sea ice dynamics is neglected. In our setup, there are 25 vertical nonuniform levels in the ocean, with thickness ranging between 20 m near the surface and 1300 m at the bottom.

The Coupled MITgcm Setup requires the following boundary conditions: bare-surface albedos, vegetation fraction, bathymetry, topography, runoff routing map, salinity, and ocean temperature for all ocean levels (the latter two files are used to initialize the coupled model and then updated by the dynamical core as the integration proceeds). Orbital parameters can be set by specifying obliquity, precession, and eccentricity. In addition, the duration of the day and the solar incoming radiative influx can be modified. The whole system runs at a speed of about 200 years per day using 25 cores, which is equivalent to 300 CPU hours for 100 simulated years.

Our model can be applied to the present-day and deep-time climates using different paleogeographies. Although the present day observed continental configuration is well known, we employ reconstructions of paleogeography for deep-time climates. Several options are available . Although in previous studies MITgcm made use of the PANALESIS reconstructions , the model can be initialized using any boundary conditions (i.e. alternative paleogeographical reconstructions, idealized land-ocean configurations, or aquaplanet) and ocean depths, which is useful for exoplanet or conceptual studies.

In both present-day and deep-time applications, the procedure for adapting the high resolution geographical maps to the MITgcm grid is the same. Using the present-day ETOPO global relief model  or a paleogeographic reconstruction, the input geography is given in a latitude/longitude coordinate system with arc-sec horizontal resolution. Since simulations are performed with the cubed-sphere CS32 grid, interpolation and smoothing are used to upscale the resolution to 2.8° . Then, isolated oceanic points or lakes are removed, irrespective of their size, to avoid numerical instability when there are enclosed areas of water. Moreover, very shallow waters are also prone to instability, especially when sea ice develops; all oceanic points shallower than -20 m are set to this depth.

BIOME4 is a vegetation model that predicts the global steady state of the vegetation distribution corresponding to long-term averages of monthly mean 2-m air temperature (denoted as surface air temperature in the following or SAT), sunshine and precipitation . Additional inputs are soil depth and texture, which are used to determine water holding capacity and percolation rates . Moreover, the atmospheric CO2 content needs to be specified. All these quantities are obtained from steady-state simulations performed using MITgcm in the coupled configuration described in Sect. . Bare-surface albedos and the vegetation map obtained as outputs of BIOME4 are then reused in the next iteration of the coupling system. Water holding capacity and percolation rates are kept constant in our simulations and set to the present-day average value.

BIOME4 follows the principle that ecosystems can be divided into a set of biomes characterised by the performance of plant functional types (PFTs), i.e. key parameters used to classify plant species having a similar response to the environment . The model selects among a set of 12 PFTs the subset that can be present in a grid cell on the basis of physiological and climatic constraints, like minimal temperature and water supply. Using a coupled carbon and water flux model, BIOME4 calculates the net primary productivity (NPP) of each PFT and the corresponding seasonal maximum leaf area index (LAI) that maximises NPP. At this point, competition among PFTs is simulated by selecting the PFT with the optimal NPP as the dominant plant type. Opposing effects due to light competition and wildfires are included through semi-empirical rules. The final output is the vegetation distribution in terms of the dominant and secondary PFTs, total LAI and NPP, which can be classified into biome types, for a total of 27 biomes (28 including land ice). Land-ice points are determined by the ice sheet extent computed by MITgcmIS (see next section). Therefore, the grid points that are identified as land ice overwrite the biome number given by BIOME4.

Main differences between BIOME4 and earlier versions (developed by ) are the inclusion of new PFTs to represent vegetation types in the polar regions, and the calculation of photosynthetic pathways (for C3 and C4 plants) that depend on the PFT. BIOME4 and its earlier versions have been used to investigate climate-biosphere interactions in the past , and coupled to MITgcm in for the Permian-Triassic Boundary. BIOME4 has also been used to assess the impact of current climate changes on the distribution of vegetation types .

We have developed a Python code, called MITgcmIS, that describes the evolution of ice sheets at the global scale on the same cubed-sphere grid used by MITgcm. A global-scale ice sheet model is required when the climate state allows for the presence of ice sheets at low latitudes, for example during glaciation periods, waterbelt states or snowball states . Since we are interested in simulating the main processes occurring at spatial resolutions of around 2° or coarser, we neglect basal melting and other fine-scale processes, as calving and ice streams, since they cannot be resolved at such coarse resolutions. Note that there is already an MITgcm module, called STREAMICE, which implements in Fortran these small-scale processes at the km-scale , however this package is not used in our coupled setup.

In MITgcmIS, we use the shallow-ice approximation to model the ice-sheet movement and the Positive Degree Day (PDD) method to compute the surface mass balance. In particular, we choose the PDD approach as described in instead of the Surface Energy Balance (SEB) method, since the latter requires quantities at the km-scale to estimate the energy budget, such as layer structure, surface roughness, and stability of the surface terrain to obtain latent and sensible heat fluxes . Thus, the SEB method is generally used in regional climate models, which are able to reach the required accuracy in the representation of the climatic fields (especially clouds), in general provided by reanalyses .

Although the PDD approach succeeds in representing the surface mass balance in coarse simulations, it can underestimate the melting in past periods of high insolation . Moreover, since this method does not account for energy exchanges between the ice sheet and the other components of the climate system, it cannot ensure a closed energy budget. Despite its limitations, we believe that the PDD method is the best choice when the representation of the atmosphere and snow processes are at the global scale on a coarse grid and simplified as in the SPEEDY and land modules (Sect. 2.1).

In our study, we need to consider different continental configurations corresponding to the Earth's evolution, under a range of ice sheet loading. Hence, for each new configuration, we need to recalculate the runoff map. The Coupled MITgcm Setup needs as an input a file with three arrays, specifying for each land point Li the corresponding precipitation storage area Ai and the ocean point Oi where it is drained (outlet point).

For present-day as well as for past (palæo-) topographies, we discriminate between land and ocean points by defining a contour line at 0 m in elevation. If any area with negative values are fully enclosed within area with positive values, they are considered as “lakes” (unless elevation reaches values below -2000 m). In such cases, we re-assign the elevation (without affecting the ocean volume) in order to remove the depression and reroute the flow direction. For this purpose, as well as for cleaning local pits, depressions and flat terrains are corrected using the fill_pits, fill_depressions, and resolve_flats functions from pysheds . This ensures a continuous Digital Elevation Model (DEM) with no single pixel or stagnant areas where water would not flow. Finally, we clip the DEM to all elevations above 0 m and retrieve the closest outlet point.

Every point in the MITgcm grid is hence defined as “continental” or “oceanic” depending on whether or not it is located inside the land (positive elevation) or not. Using the corrected topography, we generate a flow direction by applying an eight-direction (D8) flow routing algorithm from pysheds. This method assumes that water from each cell in the DEM will flow to one of its eight neighboring cells, the one that results in the steepest descent. The D8 algorithm is computationally efficient and widely used for hydrological modelling.

The slope from a cell c to each of its eight neighbors i is calculated as: 13si=Zc-Zidi where Zc is the elevation at the center cell, Zi is the elevation of the ith neighbor, and di is the distance to that neighbor. For cardinal directions (N, E, S, W), di=1, and for diagonal directions (NE, SE, SW, NW), di=2.

Then, the flow direction is determined by selecting the neighbor with the maximum positive slope: 14flow_dir=arg⁡max⁡isi,where si>0.

The resulting direction is encoded using a directional mapping: (N, NE, E, SE, S, SW, W, NW]=[64,128,1,2,4,8,16,32]. Each cell is assigned one of these values in the resulting flow direction map, indicating the direction water would flow from that cell based on the steepest slope. We then trace the flow path for every continental point using the flow direction until we reach the ocean, and define the nearest oceanic point as its outlet. Each initial continental point ultimately is being assigned one oceanic outlet, while initial oceanic points are their own outlet. Note that this approach mimicking the drainage system is different from grid schemes used in CMIP6 models .

Moreover, since the MITgcm has no proper online ice sheet model, excess water that would accumulate to form ice sheets is instead evacuated via runoff. More precisely, in the MITgcm code, snow precipitation Psnow exceeding the tolerated limit (usually set to 10 m) is automatically redirected into the ocean via the runoff. This creates an artificial excess of runoff in our asynchronous coupling, where Psnow is now used in the surface mass balance accumulation term, and hence a correction is necessary in the first steps of the coupling procedure, until a steady state is reached and the ice sheet is stabilised. Thus, we introduce the following correction in the precipitation storage area Ai at each land point: 15Ai′=Ai1-PsnowiPtoti+Psnowi where Ptoti is the total precipitation at the land point i and Psnowi≤Ptoti. This correction has an effect only on land points where the ice sheet is developing.

Offline coupling between the Coupled MITgcm Setup and BIOME4 has been already successfully applied in . Here, we will document the comprehensive framework that includes BIOME4, the new ice sheet module MITgcmIS and the runoff map calculation for different boundary conditions (present-day, paleo or idealised configurations), as schematically illustrated in Fig. .

A first simulation is run until the Coupled MITgcm Setup has reached a steady state, defined by having a surface energy balance Fs<0.2Wm-2 (usually several thousands of simulated years are required). Afterwards, additional 30 year are run with monthly frequency for the variables required by BIOME4, and with daily frequency for the variables required by MITgcmIS.

At this point, the offline coupling workflow can start. Before running the ice sheet model, the following corrections are required. For representing an advancing ice flow over shallow ocean, we mimic this process as follows: if the sea ice thickness is equal to the ocean depth (up to a depth of -20 m), the ocean point becomes a land point and hence the ice sheet can develop on it. Then, the corrected topography file is given as an input to the ice-sheet model, together with daily MITgcm outputs per grid cell for SAT and snow precipitation, which are used to calculate ablation and accumulation rates, respectively. MITgcmIS is run for 40–100×103 years, during which the isostatic adjustment is calculated and the lapse rate from the previous convergence step is used, until the ice sheet reaches a steady state . Sea level correction and salt compensation (Sect. ) are included at this stage, giving rise to new topography (including ice sheet height), mask and bathymetry files. Afterwards, pysheds is applied using the mask and the topography, closing lakes and small passages if necessary, and giving a new runoff map as output.

Finally, the vegetation model is run based on the new files. The outputs needed from MITgcm and the ice-sheet model (namely, precipitation, SAT and sunshine) are converted on a latitude/longitude grid and then given to the BIOME4 model. The equilibrium biome distribution is converted in new files for vegetation fraction and surface albedo using the values reported in . Due to the coordinate change, some land and ocean points can be inverted on the cubed-sphere grid. Hence, a vegetation fraction equal to 0 and an albedo value equal to the default water value of 0.07 are assigned to new ocean points on the CS grid, while the value of the closest land point is assigned to new land points.

Before running a second iteration, salinity and sea temperature at all ocean levels are averaged over the last 30 yr to generate new input files for the Coupled MITgcm Setup. Then, these files, along with the new vegetation fraction, albedo, topography, bathymetry, runoff and mask files, are given back to the Coupled MITgcm Setup to run the whole coupling process at least twice, so that the GCM has time to adjust to the new input files. Convergence is considered achieved when less than 10 % of the land points experience a change in biome distribution and total ice sheet volume. Moreover, the Coupled MITgcm simulation needs to have a surface energy imbalance Fs<0.2Wm-2, corresponding to extremely low drifts in both the global ocean temperature and the surface air temperature. The whole procedure forms the BIG-MITgcm simulation tool, which thus describes the climatic steady state, including those components with a slow response, like deep-ocean dynamics, vegetation and ice sheets.

To assess the performance of our climate framework BIG-MITgcm, we run three simulations. The first one, denoted as run1, is the pre-industrial simulation at 280 ppm starting from an input map of land elevation in the absence of ice. This simulation will be assessed against two CMIP6 models, as there is a lack of observational data for this period. The second simulation (run2), which corresponds to the 1979–2009 period with an average CO2 concentration of 360 ppm , is started from the run1 steady state by applying a constant increase of CO2 of 1 ppmyr-1 for 80 year and keeping the ice sheet fixed. This run is evaluated against reanalysis and observational data. Finally, a third simulation (run3) is run using the Coupled MITgcm Setup at 280 ppm starting from ETOPO2 (thus, including present-day ice sheets) and the vegetation cover obtained in run1. Comparing run1 and run3 helps evaluate the ability of BIG-MITgcm to grow plausible ice sheets.

More specifically, for assessing the pre-industrial run at 280 ppm, we use the output data from the IPSL-CM6A-LR model  and the NorESM2-LM model using the piControl dataset . We chose these two CMIP models because they include dynamical vegetation (see Table ). For the second run at 360 ppm, we compare our data with the following datasets: ERA5  for the atmosphere, OSRA5  for ocean diagnostics, MODIS for the vegetation, BedMachine for the surface elevation of Greenland and Antarctica , RAPID observations for the Atlantic Meridional Overturning Circulation (AMOC) profile , and the Sea Ice index for the sea ice extent .

Five iterations of the procedure illustrated in Fig.  were necessary to reach convergence in run1. The last 30 years of the fifth iteration are used for diagnostics. To be consistent in comparisons, all diagnostics are converted to the same longitude-latitude coordinate system with a spatial resolution of 2°×2°.

The outputs were treated using Matlab version 2023b and the figures were made using python. The MITgcm took approximately 1500 simulated years per week to reach equilibrium using 25 processors for each iteration. BIOME4 and pysheds run in less than 5 min on a desktop computer. MITgcmIS needs around 1 h of CPU time for 40 thousand years due to the daily stepping of Eqs. ()–() for all ice-covered points.

To start the first run of our simulations, it is necessary to provide initial conditions that are representative of the present-day climate. The three-dimensional distributions of sea potential temperature and salinity are derived from the Levitus World Ocean Atlas . Orbital forcing is prescribed at present-day values, with a solar constant of 1365.4 Wm-2 and obliquity of 23.45°. Topography (including ice sheets and glaciers), bathymetry and the corresponding files are taken from the ETOPO2 dataset with a resolution of 2 arcminutes. Annual mean values of bare-surface albedos (in the absence of snow or ice) and fraction of land-surface covered by vegetation are the same as those used in  and derived from the ERA dataset. Interpolation and smoothing are applied to convert these maps to the resolution of the MITgcm CS32 grid .

In order to assess the ability of MITgcmIS to correctly generate the ice sheets, we need to provide an input map of land elevation in the absence of ice. This map was generated using the BedMachine dataset that is part of the MEaSUREs program of NASA , with the addition of an isostatic correction as in for Antarctica and Greenland. The BedMachine dataset provides a density-corrected satellite-based DEM of the ice sheet surface, as well as a data-constrained estimate of bedrock elevation and a mask for identifying the different parts of the ice sheets. There are two separate products for Antarctica  and Greenland , respectively. The surface elevation of the ice sheets are used to assess the performance of MITgcmIS, while bedrock elevations with isostatic adjustment are used as initial boundary conditions, as shown in Fig. . Note that, regarding paleoclimate simulations, PANALESIS or other reconstructions directly provide unloaded bedrock elevations.

Finally, it is important to also describe the tuning procedure. In order to obtain pre-industrial conditions at 280 ppm, once all the albedo values for vegetation cover, snow and ice have been fixed and set to values within observational ranges, we tune the relative humidity threshold for the formation of low clouds (a parameter denoted as RHCL2 in SPEEDY) so that the average global SAT becomes approximately equal to the observed value of 13.7 °C . The adjusted value, RHCL2=0.7727, is applied to all simulations. Moreover, the coefficient a in Glen's law, which in our simulations is assumed to be constant (see Eq. ) and governs the ice sheet formation, is determined as follows.

Evaluating the surface mass balance produced for Antarctica in our setup is necessary to calibrate the Glen's law parameter using the total ice sheet volume. The reason is that the volume is strongly sensitive to both the net surface mass balance and the Glen's law coefficient; without a good estimate of the surface mass balance, the correct volume can be achieved for the wrong reasons. The surface mass balance of Antarctica is estimated by the Coupled MITgcm Setup started from the present-day ice sheets (ETOPO2), and turns out to be approximately 1500 Gtyr-1. This value can be compared with the ensemble mean obtained from a comparison of Regional Climate Models (RCMs) in . In that study, the ensemble mean over the grounded ice sheet is estimated at 2073±306 Gtyr-1. Although the value obtained in our simulation is lower than this ensemble mean, as well as lower than the values obtained by all individual models in that comparison, it is important to consider that the spatial interpolation and smoothing applied to obtain 2.8° resolution in our simulations  implies a different representation of the Antarctic continent compared to models with finer resolutions (25–50 km). Additionally, there are known limitations in the representation of snow processes in the land module, as discussed in Sect. . Therefore, even if our value is lower than those obtained from RCMs, it remains within the same order of magnitude. Using this value for the surface mass balance, we apply our ice sheet model MITgcmIS (which always starts from the bedrock and isostatic adjusted topography) with a range of a values. This yields different Antarctic volumes (Table ), while maintaining a similar surface elevation profile. We selected a=0.6×10-15 Pa-3yr-1 as a compromise, as it produces a volume within 10 % of the observed value and surface elevations consistent with observations.

To evaluate if our coupling setup correctly reproduces the modern Earth climate, we examined several diagnostics of the dynamical behavior of atmosphere, ocean, vegetation and cryosphere.

We now demonstrate how the asynchronous procedure can be applied to a very different continental configuration. As an example, we consider the Permian-Triassic bedrock paleogeography as reconstructed by PANALESIS . Three alternative climatic steady states have been identified for this paleogeography by , with mean surface air temperatures (SAT) higher than present-day values. However, the so-called cold state, with an average SAT of around 17 °C and a meridional overturning circulation characterised by a negative cell (i.e. flowing from north to south at the surface), corresponds to climate conditions where a small ice cap can develop in the Northern Hemisphere. Therefore, we applied our procedure to investigate this possibility.

We find that a small ice cap can indeed develop in the Northern Hemisphere in the cold state, as shown in the resulting topography in Fig. A. The associated runoff routing map is also shown in Fig. B. Global annual values of the main climatic variables, listed in Table , fall within the same range as those reported by , with slightly lower SAT and higher sea ice extent, indicating that this climatic state can be represented by the same attractor. The ice sheet extends in the north polar region with a volume of 7.8×106 km3, corresponding to approximately -20 m of sea-level change relative to a climate state without ice sheets, and -65 m relative to the present-day value. Interestingly, this value falls within the range of eustatic variations in the Early Triassic reconstructed by global stratigraphic data .

Regarding the vegetation cover, the main differences in NPP distribution relative to  occur at high latitudes, as shown in Fig. , due to the equatorward migration of temperate forests and the disappearance of tundra in regions covered by the ice sheet.