Since the beginning of the Quaternary period (2.58 Ma to present), the Earth's climate has repeatedly switched between cold “glacial” periods and warmer “interglacial” periods, which are collectively known as glacial–interglacial cycles. Glacial periods are characterized by the presence of large ice sheets covering North America and Fennoscandia, while interglacials refer to times when ice sheets are restricted to Greenland and Antarctica, such as the current Holocene period (11.7 ka to present). The glacial–interglacial cycles are widely documented in natural climate archives. For example, the oxygen isotope ratio (δ18O) recorded in the shells of micro-organisms that accumulated on the ocean floor – hereafter referred to as benthic δ18O – serves as a valuable proxy for global ice volume and deep-ocean temperatures spanning millions of years into the past . Additionally, Antarctic ice cores contain valuable information over the past 800 kyr such as atmospheric CO₂ concentrations and surface temperature changes .

Despite extensive research, a comprehensive understanding of what caused the glacial–interglacial cycles has remained elusive, although the importance of changes in the Earth's orbital parameters has been widely acknowledged. Changes in these orbital parameters are thought to determine when the Earth switches between glacial and interglacial climates by redistributing the incoming solar radiation (insolation) the Earth receives across latitudes and seasons. These variations in the orbital parameters encompass changes in the shape of the Earth's orbit (eccentricity) on 100 and 413 kyr cycles, changes in the tilt (obliquity) of the Earth's rotational axis on 41 kyr cycles, and changes in the time of the year when the Earth is closest to the Sun (climatic precession) on 19 and 23 kyr cycles. The most favoured of the orbital hypotheses comes from Milutin Milanković, who proposed that glaciation occurs during times of reduced insolation at the high northern latitudes during the summer months, when obliquity is low and the Northern Hemisphere (NH) summers coincide with the Earth's furthest distance from the Sun (and vice versa for deglaciation). were the first to discover that benthic δ18O records contained cycles with periods of 23, 41 and 100 kyr, in correspondence to the Earth's orbital cycles. Subsequent research has extended the benthic δ18O record further back in time and led to compiled records from across the globe , which together with other proxy records provides compelling evidence for an orbital control on climate change. Nevertheless, despite a distinct orbital “rhythm” to glacial–interglacial cycles, important characteristics of the benthic δ18O record cannot be easily explained by Milanković theory, including a shift in the dominant periodicity observed around 1 Ma.

The Mid-Pleistocene Transition (MPT) from 1.25 to 0.7 Ma represents a transition in the dominant periodicity of glacial–interglacial cycles from 41 kyr in the Early Pleistocene to ∼ 100 kyr in the Late Pleistocene, occurring without significant variations in the orbital cycles . Hypotheses for the MPT are abundant in the literature . Prominent theories include the removal of a subglacial regolith beneath the NH ice sheets , a gradual decline in atmospheric CO₂ concentrations , the merging of the Laurentide and Cordilleran ice sheets over North America , phase locking between Antarctica and the NH ice sheets , or some combination of these mechanisms . Even the ∼ 100 kyr cycles in glacial ice volume following the MPT were not an expected outcome of Milanković theory, as they correspond to changes in the Earth's eccentricity, which has a negligible direct influence on summer insolation . Instead, these ∼ 100 kyr cycles have been explained as the skipping of one (80 kyr) or two (120 kyr) obliquity cycles (averaging to 100 kyr), leading to longer glacial cycles in the Late Pleistocene , where the precise timing of deglaciation is set by the combined forcing of precession and obliquity . Independent modelling studies have also highlighted the importance of precession, and its modulation by eccentricity, for generating the ∼ 100 kyr cycles of the Late Pleistocene . Interestingly, models that have simulated the MPT and/or the ∼ 100 kyr cycles of the Late Pleistocene still struggle to reproduce the dominant 41 kyr cycles seen in the Early Pleistocene .

While modelling experiments of the Early Pleistocene capture 41 kyr cycles in ice volume, in correspondence to the Earth's obliquity, they generate stronger 19 and 23 kyr precession cycles than observed in the benthic δ18O record . This is unsurprising given that climatic precession controls the intensity of summer insolation. Indeed, for various metrics of summer insolation variability, including mid-month insolation (i.e. 21 June), monthly mean insolation or the caloric summer half year, precession has a strong influence at latitudes where NH ice sheets grow and melt. It is worth noting that precession cycles are detectable prior to the MPT , but they become more pronounced across the Quaternary, with much stronger signals observed in the Late Pleistocene compared to the Early Pleistocene . Theories that account for weaker 19 and 23 kyr precession cycles during the Early Pleistocene include a counterbalancing between summer insolation intensity and summer duration , the cancellation of out-of-phase precession cycles between the NH ice sheets and Antarctica , or the influence obliquity has on the poleward flux of moisture and accumulation rates on ice sheets .

Models of varying levels of complexity have been employed to address these questions relating to glacial–interglacial cycles. Global climate models (GCMs) are too computationally expensive for simulations on these timescales, so studies have instead relied on conceptual models , energy balance models , Earth system models of intermediate complexity , climate parameterization based on discrete GCM “snapshots” or benthic δ18O records . Among these, zonally averaged energy balance models (EBMs) provide computationally efficient tools for studying the response of the Earth's climate to changes in the Earth's orbital parameters . EBMs calculate the distribution of surface temperature with latitude by considering the conservation of energy on a sphere subject to heating by solar insolation, cooling by terrestrial radiation and the diffusive redistribution of heat from the Equator to the poles. Since the original works of and , EBMs have long been used to study the Earth's climate sensitivity and have shown success at simulating both present-day and glacial climate states . Moreover, the simplicity of EBMs enables the isolation and identification of important processes and feedbacks . Consequently, EBMs provide valuable tools for exploring the glacial–interglacial cycles of the Quaternary and guiding further investigations with more realistic models.

In this study, we introduce the Zonally Averaged Energy and Moisture BAlance (ZEMBA) climate model to study the response of the Earth's climate to changes in the orbital parameters. The model is designed to place a particular emphasis on physical processes that may influence the relative contributions of obliquity and precession to climate variability during periods such as the Early Pleistocene. The model spans both hemispheres and is forced by the full seasonal cycle in insolation. Unlike previous EBMs used for studies of glacial–interglacial cycles , ZEMBA includes a hydrological cycle to simulate precipitation and snowfall. Before using ZEMBA in experiments relating to glacial–interglacial cycles, it is important to ensure the model can simulate the present-day (or pre-industrial) climate with reasonable accuracy and to constrain the sensitivity of the model. Consequently, the following sections document the model and its climate sensitivity. Firstly, we describe a “control” simulation of ZEMBA for the pre-industrial period and compare the broad-scale features with a GCM and atmospheric reanalysis (Sect. ). Then, we test the sensitivity of the model for a doubling of the atmospheric CO₂ concentration and a 2 % increase in solar insolation (Sect. ). Finally, we evaluate its performance for a simulation of the Last Glacial Maximum (LGM) (Sect. ). Prior to that, we provide a detailed description of the atmospheric (Sect. ), land (Sect. ) and ocean (Sect. ) components of ZEMBA.

ZEMBA is primarily based on the EBM from , which comprises a single atmospheric layer overlying a surface divided into land and ocean. While utilizing the same shortwave and longwave radiation scheme as in , in addition to the same ocean transport model, ZEMBA includes a hydrological cycle to estimate precipitation and snowfall rates. Moreover, instead of parameterizing the surface albedo over land as a function of surface temperature, ZEMBA directly estimates snow coverage through the competition between snow accumulation (from the hydrological cycle) and ablation (from the surface energy balance). Following recent studies , atmospheric heat transport is now proportional to meridional gradients in near-surface moist static energy – instead of classic (or “dry”) EBMs that diffuse heat along temperature gradients. Moreover, a Hadley cell parameterization is included to produce an equatorward flux of moisture in the tropics . The radiative and turbulent heat fluxes are calculated separately over land and ocean, but atmospheric temperatures and humidities are set to the zonal average of land and ocean at the end of each model time step. The subsequent sections will describe in more detail the individual atmospheric, land and ocean components of ZEMBA.

We present equilibrium simulations of the model for the pre-industrial period (ZEMBA-PI), a doubling of atmospheric CO₂ conditions relative to the pre-industrial period (ZEMBA-2×CO2), a 2 % increase in solar insolation (ZEMBA-So+2 %) and the Last Glacial Maximum (ZEMBA-LGM). The model takes approximately 3000 model years to reach an equilibrium due to the inclusion of the ocean model with a large heat capacity. Key model parameters remain the same between experiments (Table ), with the exception of the insolation forcing, atmospheric CO₂ concentration and land ice extent (Table ). The following sections describe output from ZEMBA prior to coupling with an interactive ice sheet model, with fixed land ice fractions and zonal-mean elevations over land taken from ICE-6G_C .

Our simulation of the pre-industrial period demonstrates that ZEMBA can describe the zonally averaged climate of this period with reasonable accuracy. Surface air temperatures are in strong agreement with NorESM2 and ERA5 1940–1970 both annually and seasonally (Fig. a–c), generally falling within 4 °C of those estimated by NorESM2 (Fig. d–f). Most notably, given the simplicity of ZEMBA, the model shows success in emulating precipitation and snowfall rates for the pre-industrial climate (Fig. a), particularly at the polar latitudes. Furthermore, the model captures the poleward enhancement of the surface and planetary albedo due to the presence of snow and sea ice cover (Fig. ), with the seasonal range of sea ice cover in the NH corresponding nicely to both NorESM2 and ERA5, varying between ∼ 0.8 and ∼ 1.4 × 10¹³ m² (Fig. b). The ocean transport model shows success in emulating zonally averaged ocean heat transport, which makes a significant contribution to total heat transport at the lower latitudes, peaking with values of around 1 and 1.5 PW for the SH and NH, respectively (Fig. b). Similarly, the model predicts that atmospheric heat transport dominates in the middle to high latitudes and reaches its maximum levels at around ∼ 45° in each hemisphere. Finally, looking specifically at the partition of atmospheric heat transport into its dry static and latent components, the model captures the relative contributions of each flux to the total atmospheric heat transport, with dry static fluxes peaking at ∼ 15° N/S and latent heat transport peaking at 45° N/S.

Despite these favourable comparisons with state-of-the-art climate models and reanalysis data, there are limitations of ZEMBA in its current state. For example, while the model does reasonably well at capturing the radiative fluxes at the TOA and the surface (Fig. ), one of the largest discrepancies between ZEMBA and NorESM2 or ERA5 1940–1970 resides in the outgoing longwave radiation at the TOA. As the outgoing longwave flux is overestimated around the tropics, the net radiation received at these latitudes is up to ∼ 25 W m⁻² lower than NorESM2 and ERA5 1940–1970 (Fig. c). Given the comparatively large surface area in the tropics, the underestimation of these net radiative fluxes results in a significant reduction in the surplus radiative energy which needs to be transported poleward. Consequently, both the total and the atmospheric heat transport in ZEMBA is much lower (Fig. a–b), with atmospheric heat transport peaking at ∼ 3 PW in each hemisphere, whereas it should be closer to 5 PW . In addition, one of the challenging aspects of a simple model such as ZEMBA is the underestimation of the winter maximum in snow cover over land (Fig. a), resulting in a smaller seasonal amplitude in snow cover. However, we note that the annual-mean rates of snowfall correspond well to NorESM2 and ERA5 1940–1970 over the northern high latitudes (Fig. a). Similarly to NH snow cover, the seasonal range of sea ice cover in the SH is reduced when compared to NorESM2 and ERA5 1940–1970 (Fig. b) and snowfall rates over Antarctica are underestimated by about 50 % (Fig. a).

Given their simplicity, the ability of EBMs like ZEMBA to accurately reproduce the latitudinal pattern of surface temperature is impressive but has been well established since the original works by and . In reference to other studies, ZEMBA is most similar to later iterations of EBMs that contain vertical resolution (i.e. an atmospheric and surface layer), the division of the surface into land and ocean, and a seasonal cycle in insolation . In comparison to the “present-day” simulation from the EBM developed by , which effectively serves as the basis for ZEMBA, most climate variables exhibit qualitative agreement with the pre-industrial output shown in Sect. , although the global mean surface air temperature reported in is 1.09 °C warmer due to a higher 350 ppm CO₂ forcing in their model experiment. Other than the hydrological cycle, the primary contrast between the models lies in their representation of land-based snow cover. In the study, the seasonal cycle in snow cover is parameterized as a function of surface air temperature and appears to be in better agreement with observations than ZEMBA, which explicitly calculates snow coverage over land. Overall, many such features of the Earth's climate are relatively well produced by various EBMs , including ZEMBA, in part because they are “tuned” to match the present-day climate. Of greater importance for EBMs than achieving an exact replication of observations and/or GCM output is the investigation of different climate processes and feedbacks, which necessitates appropriate sensitivities to alterations in external or internal forcings.

The responses of ZEMBA to both a doubling of the atmospheric CO₂ concentration and a 2 % increase in solar insolation are in qualitative and quantitative agreement with one another (Fig. ), including polar amplification of warming, which is stronger in the NH than the SH and concentrated in the winter months. The climate sensitivity of ZEMBA is larger than that of the EBM of and similar such studies . For example, the So+2 % experiment generates much stronger annual-mean warming in the high northern latitudes (>14°) than (<5°). In addition, the ECS of ZEMBA (3.6 °C) is higher than that of the original model of 1.9–2.2 °C . However, more recent EBMs have generated global mean and polar-amplified warming in response to CO₂ forcing which is similar to or even greater in magnitude than that of ZEMBA . Moreover, the ECS of ZEMBA is consistent with the ECS range projected by GCMs that contributed to the Coupled Model Intercomparison Project Phase 5 (CMIP5) and CMIP6 and other works . The strong seasonal asymmetry of surface warming in the polar regions (Fig. b–d) is in accordance with both observations and GCM results . As in , the winter maximum in surface air temperature is related to sea ice loss, causing a greater absorption of shortwave radiation, which is released from the ocean surface into the overlying atmosphere during the winter months. For the 2xCO₂ experiment, the polar amplification in warming – normalized (divided) by the global mean warming – exceeds 4 in the high latitudes of the Arctic and reaches up to 1.5 in the Antarctic (Fig. a). While polar amplification in warming that is strongest in the NH is in agreement with both observations and GCM results, warming in the high northern latitudes that reaches 4 times the global mean resides in the upper boundary of estimates from GCM simulations . When averaged from 60 to 90° N, however, the normalized Arctic warming is 3.2, which is still less than the 3.5 estimated over the historical period from 1979 to 2014 according to the observational dataset HadCRUT5 . Overall, we see that ZEMBA exhibits a climate sensitivity – in terms of both global mean and polar-amplified warming – that appears broadly consistent with other EBMs, GCMs and observations.

There are comparatively few EBMs which incorporate a hydrological cycle . developed an EBM, consisting of a vertically averaged atmosphere overlying an ocean layer, to study the influence of the hydrological cycle on climate. Using an optimization procedure, the modelled precipitation and evaporation, amongst other climate variables, compare very well with contemporary observations . While both ZEMBA and the work of contain a hydrological cycle, ZEMBA includes land cover and a seasonal cycle, which is suitable for investigating the response of climate to variations in the orbital parameters. introduced a moist static energy balance model with a hydrological cycle coupled to a carbon cycle model, suited for studying the long-term relationship between the carbon cycle, hydrological cycle and climate. However, the modelled precipitation does not impact the surface albedo, which is instead simplified as a function of surface temperature, and, like the work of , it lacks the seasonal insolation cycle necessary to study the climate response to Milanković cycles. Previous EBMs used for studies of glacial–interglacial cycles prescribe either a present-day or a spatially uniform distribution of precipitation, which is not perturbed across climates, or instead parameterize precipitation and snowfall as a function of surface air temperatures and ice sheet size . The inclusion of a hydrological cycle enables precipitation and snowfall to be calculated internally in ZEMBA, which can account for the influence of changes in both local air temperatures and poleward moisture transport for precipitation and snowfall rates. It has been suggested that changes in the Earth's obliquity – by altering the meridional gradient in insolation – have a relatively strong influence on ice sheet volume due to changes in poleward moisture transport . The inclusion of a hydrological cycle enables ZEMBA to explore these processes in the context of glacial–interglacial cycles.

It is important that ZEMBA, intended for studies of glacial–interglacial cycles, can simulate climates other than the present-day or pre-industrial periods. When our simulation of the LGM is compared to other reconstructions, the model compares favourably regarding changes in both surface air temperature (Fig. a) and precipitation (Fig. b). Indeed, ZEMBA captures the polar amplification of cooling in both hemispheres due to positive feedbacks relating to snow cover and sea ice expansion, although this cooling appears somewhat underestimated around Antarctica and slightly overestimated in the northern high latitudes. Moreover, the global mean cooling of -4.11 °C is similar to that estimated by the ensemble averages from PMIP3 and PMIP4 and the data assimilation product from . It should be noted that other data assimilation reconstructions suggest much stronger global cooling of between 6.1 and 6.8 °C , which perhaps signals the importance of including feedbacks relating to clouds, dust and/or vegetation for reproducing the LGM cooling. However, the reconstructions from and are based on a single climate model (CESM1.2), which produces one of the coldest LGM climates in PMIP4, whereas the reconstruction incorporates the wide range of climates generated across the PMIP ensemble.

Evaluating changes in meridional heat transport during the LGM is made difficult by the large (and often conflicting) range of total, atmospheric and ocean heat transport generated across PMIP3 and PMIP4 at all latitudes . For the ZEMBA-LGM experiment, changes in total heat transport are most intense in the NH (Fig. c), in response to large steepening of the Equator-to-pole temperature gradient. The upsurge in total heat transport between 15–45° N is somewhat consistent with PMIP3–PMIP4 , although most of this increase in ZEMBA is generated by stronger atmospheric heat transport (Fig. c–d), whereas PMIP3–PMIP4 also shows stronger ocean heat transport at these latitudes. Stronger ocean heat transport in PMIP3–PMIP4 can be attributed to a stronger and sometimes deeper Atlantic Meridional Overturning Circulation (AMOC) , which is inconsistent with proxy-based reconstructions of ocean circulation and is associated with stronger surface winds over the northern North Atlantic . For ZEMBA, on the other hand, there is a large decreases in NH ocean heat transport during the LGM, reaching ca. -0.5 PW at 60° N (Fig. c). The prominent reduction in ocean heat transport, not observed in any of the PMIP3–PMIP4 simulations, is associated with the expansion of sea ice and a surface ocean layer that resides near the freezing point of seawater. Consequently, horizontal heat fluxes due to both advection (representing overturning) and diffusion (representing eddies and gyres) drop to zero at latitudes now covered by sea ice, which perhaps highlights the limitation of the simplified ocean model used in ZEMBA, where meridional heat transport is limited to the surface and bottom ocean layers.

The sensitivity experiments performed for the LGM boundary conditions (Fig. ) recreate the experiment made by using the original EBM. For ZEMBA, the global mean cooling caused by the combined ice sheet and CO₂ forcing is 4.11 °C, which is stronger than the 3.3 °C of cooling noted by and is in better agreement with LGM reconstructions. The stronger cooling in ZEMBA can be partially attributed to differences in the LGM boundary conditions used in this study, which involves a stronger CO₂ reduction and larger LGM ice sheet extents than those in . In our study, the addition of LGM ice sheets generates cooling and drying that are primarily localized to the NH, whereas CO₂ lowering generates widespread cooling and drying with stronger global mean cooling than that induced by ice sheet expansion (Tables  and ). Additionally, by shifting the mid-point of ocean circulation from 5 to 15° S, surface air temperatures are in better agreement with the LGM reconstructions, as first noted by . As for the strength of ocean circulation, found that reducing flow velocities in their ocean model led to temperatures in better agreement with contemporary LGM temperature reconstruction. In this study, however, reductions in ocean overturning instead lead to larger discrepancies with modern LGM constructions. Indeed, it remains uncertain to what degree overturning changed during glacial conditions, with some studies suggesting the mean state of overturning was not “sluggish” across glacial cycles or the LGM . Therefore, for simulations of glacial–interglacial cycles using ZEMBA, it may be important to consider the impact of variations in both the mid-point and the overall strength of ocean circulation.

ZEMBA is intended to be used as a computationally efficient tool for studies of the glacial–interglacial cycles of the Quaternary. The PI and the LGM experiments indicate ZEMBA is able to simulate glacial and interglacial climate states for a given insolation, CO₂ and ice sheet extent. As the model includes (1) both hemispheres, (2) a seasonal cycle and (3) a hydrological cycle, it is able to explore mechanisms invoked to explain the dearth of precession cycles in ice volume observed during the Early Pleistocene, such as (1) out-of-phase precession cycles between the hemispheres , (2) a counterbalancing between summer insolation intensity and summer duration , or (3) obliquity-induced variations in atmospheric moisture transport . In future work, we intend to explore both the equilibrium of ZEMBA and its transient response to changes in the Earth's obliquity and precession, prior to simulations of the glacial–interglacial cycles of the Early Pleistocene via coupling to an ice sheet model.

In this study, a simple climate model (ZEMBA) is introduced to simulate zonally averaged climate fields including surface temperatures and precipitation. ZEMBA is largely built on the zonally averaged energy balance climate model from , comprising an atmospheric layer overlying a surface divided into a land component and a six-layered, zonally averaged ocean transport model. Unlike its predecessor, ZEMBA incorporates a hydrological cycle to estimate snowfall and precipitation with latitude.

Simulations of the pre-industrial period compare favourably with GCMs and reanalysis data, including surface temperatures (and their seasonal cycle), precipitation, surface and TOA radiative fluxes, sea ice, snow cover, and meridional heat transport. However, the underestimation of the net TOA radiation received in the tropics leads to a reduction in atmospheric heat transport, and there is also an underestimation of the seasonal amplitude in snow cover over land.

The responses of ZEMBA to increases in the atmospheric CO₂ concentration or the solar constant are in qualitative agreement with other EBMs, GCMs and observations, such as polar amplification in surface warming, which is strongest over the NH and focused in the winter months. The new additions to ZEMBA appear to increase climate sensitivity in comparison to older EBMs, but its results are still broadly consistent with the global mean and polar-amplified warming projected by climate models of higher complexity.

As the purpose of ZEMBA is for studies of the glacial–interglacial cycles of the Quaternary, it is important that the model can simulate climates other than present-day or pre-industrial climates. A simulation of the LGM indicates ZEMBA is able to capture changes in surface temperature and precipitation in qualitative and quantitative agreement with state-of-the-art climate models and data assimilation products, despite neglecting climate feedbacks relating to dust, vegetation and clouds. In particular, ZEMBA reproduces the polar amplification of cooling in both hemispheres and global mean cooling in accord with reconstructions from more elaborate models.

The overall conclusion from this study is that ZEMBA is suitable for studies of climatic change on large spatial and temporal scales, with a particular emphasis on glacial–interglacial cycles and the response of the climate system to changes in the orbital parameters. In future work, we intend to explore both the equilibrium of ZEMBA and its transient response to changes in the Earth's obliquity and precession, prior to simulations of the glacial–interglacial cycles of the Early Pleistocene via coupling to an ice sheet model.