During the history of Earth, geological evidences support the formation of persistent sea ice within the tropical regions, referred to as snowball Earth states e.g.. Because ice is highly reflective, the positive surface albedo feedback strengthens and exceeds the sum of other feedbacks while the Earth cools down. The evolution of the strength of climate feedbacks as a function of temperature is referred to as temperature-dependency. Whilst climate models agree that the surface albedo feedback increases in cold climates e.g., other climate feedbacks such as clouds are less understood in the context of cooling e.g.. Temperature-dependency in climate feedbacks has often been studied in warming climates e.g., but rarely in cold climates e.g..

proposed the Snowball Earth modelling intercomparisong project, or SnowMIP, as a protocol to compare modelling efforts within the field. SnowMIP was made of three atmosphere-ocean models, where two of them are older versions of the models applied in the current study (ECHAM5/MPIOM and CAM). SnowMIP paved the way to identifying the importance of circulation and clouds in snowball Earth initiation across models. Following SnowMIP, the importance of snow-albedo feedbacks and their relation to sea-ice albedo feedback was identified, as well as the interaction of ocean and atmosphere circulations . Due to the complexity of running a model into snowball Earth state, and the increasing cost of running higher-end models, there is to this day only a small number of published snowball Earth simulations and even less feedback analyses. To our knowledge, there is also no study that relates the snowball Earth transition and climate sensitivity, which is the main aim of the present study.

The community definition of equilibrium climate sensitivity (ECS) is the long term global mean surface warming in response to a doubling of CO2 over pre-industrial (PI) levels, excluding temperature changes induced by slow changes in ice sheets . ECS is among the most important geophysical properties for determining future global warming , and has long been the subject of investigation . Currently, multiple lines of evidence are used in combination to constrain ECS, including a process based approach wherein forcing and individual feedbacks are constrained, use of the instrumental temperature record warming, and by interpreting estimates of temperature and forcing in various paleoclimates . The result is a substantially reduced uncertainty relative to earlier assessments .

In this study, we quantify cloud feedbacks as the Earth transitions towards a snowball Earth state from PI conditions under low CO2 concentration. In particular, we focus on the contribution of cloud feedbacks over tropical oceans, as well as the interaction between cloud and sea-ice albedo feedbacks. Our motivation to investigate climate feedbacks in the snowball Earth transition stems from the fact that a number of relatively high climate sensitivity models of the paleoclimate modelling intercomparison project phase 4 (PMIP4) fail to simulate the Last Glacial Maximum (LGM), a cold period with large ice sheets 21 000 years ago e.g., despite being able to simulate warm paleoclimates (e.g. Pliocene, 3.2 million years ago , Eocene, 50 million years ago ). These climate models simulate temperatures for the LGM substantially cooler than indicated by reconstructions, and show hints of runaway behaviour e.g.. Since they have strong positive cloud feedbacks they consequently exhibit high climate sensitivity. It could therefore be possible to use the models that do simulate stable LGM states by evaluating the relationship between simulated LGM temperatures and the ECS of those models as to provide an upper limit constraint on climate sensitivity independent from previous estimates by calculating the temperature at which the Earth transits towards an unstable snowball Earth state. The constraint is essentially based on the fact that during the recent glacial cycles we have evidently not entered the snowball state. The upper bound on ECS is either a large value or not provided for the different lines of evidence used in the latest report of the International Panel on Climate Change (IPCC), where there was no estimate of an upper bound in the combined assessment. Hence, our estimate may become a valuable addition to the next assessment.

To approach this problem with other climate models, we close our study with a short and easy-to-replicate experimental protocol of a modern snowball Earth simulations. The results of this experiment are relevant for both climate sensitivity, as shown in this paper, as well as understanding the challenges around setting up the LGM simulation, a notoriously difficult climate to simulate.

We use the Max Planck Institute for Meteorology Earth system model version 1.2, MPI-ESM1.2  to simulate snowball Earth transitions from PI and LGM initial conditions. The sea-ice and snow-albedo parametrisation in MPI-ESM1.2 is similar to other snowball Earth studies using an older version of this model , except for the addition of meltponds which are described by . We perform abrupt and sustained changes of atmospheric CO2 concentrations or solar constant starting from the equilibrium PI or LGM states. The CO2 concentrations are chosen as one half of the previous concentration, as CO2 has a quasi-logarithmic forcing behaviour , and as to facilitate comparisons with other studies which uses doubling and halving, such as the ones of the non-linear modelling intercomparison project (nonlinMIP, ). Continents, non-CO2 greenhouse gases and orbital configurations are kept unchanged. The initial surface temperature from the PI state is 14.4 °C, whereas it is 10.4 °C from the LGM state. A dynamic vegetation model is used, but vegetation dies-off rapidly due to rapid cooling and low CO2. PI simulations use the coarse resolution MPI-ESM1.2-CR (T31 spectral truncation, 31 atmospheric levels), which is faster and numerically more stable to extreme forcing. LGM boundary conditions are only available in the low resolution MPI-ESM1.2-LR (T63, 47 atmospheric levels). Radiative differences between the two models are small, and the climate sensitivity of MPI-ESM1.2-CR is slightly higher than the LR version. While we do not expect temperature-dependency to vary much between the two model resolutions, individual feedbacks are likely to slightly differ so we also initiate a few simulations from PI with MPI-ESM1.2-LR. Details of all simulations are summarised in Table .

The growth of thick sea ice leads to numerical instabilities in the ocean model component when exceeding 12 m. We do not artificially limit sea-ice growth, as was done in other studies , because this method generates latent heat at the base of sea ice  which changes the required CO2 forcing for snowball Earth initiation . This results in all our simulations reaching numerical instability at some point when approaching a complete snowball Earth state due to the sea ice being too thick in narrow basins, such as the Baltic Sea. This does not affect our results, however, as we evaluate the transition in temperatures much higher than at the numerical instability. Only the 50 ppm simulation was manually stopped as it is expected to reach an equilibrium in a cold non-snowball state (see Table  for details on run length).

We perform experiments with abrupt decreases of CO2 concentrations ranging from 1/8 to 1/2048 of PI CO2 concentration using the MPI-ESM1.2-CR model to simulate snowball Earth transition and break down climate feedbacks (Sect. ). The strengthening surface albedo feedback has often been considered the main driver in the snowball Earth instability, yet positive cloud feedbacks exceed the surface albedo feedback during the initial 10 °C of cooling, relative to PI (Fig. ). This strengthening arises from positive tropical cloud feedbacks (Fig. ) due to an increase in shallow cloud coverage over the tropical oceans (Appendix Fig. ). Cloud feedbacks are increasingly more positive as the initial forcing is increasingly stronger, which amplify tropical cooling. Below -20 °C relative to PI, cloud feedbacks switch sign and are globally negative and weaker than the surface albedo feedback, although there still exists strong positive cloud feedbacks ahead of the advancing sea-ice edge in both hemispheres (Fig. ). Indeed, because the sea-ice surface is cold, clouds are preferentially over open water which is warmer and a source of moisture e.g., where they facilitate further cooling of the ocean. All in all, cloud feedbacks substantially contribute to early cooling in response to reduced CO2. During the transition, the positive surface albedo feedback starts to exceeds the combined cloud, temperature and water vapor feedbacks at temperatures between -15 and -20 °C below PI (Fig. ). This leads to a global instability and is concomitant with the acceleration of the equatorward sea-ice edge into the sub-tropics, resulting in strong global cooling towards the snowball state.

The diagnostics presented above are suggestive that cloud feedback may be an important climate driver towards the snowball transition. To demonstrate that this is indeed the case, we isolate the importance of each feedback using the feedback locking technique (Sect. ) on various simulations, where each feedback mechanism is disabled one after the other (Fig. B). When the cloud feedback is locked, the instability is reached at a similar global mean temperature as in the non-locked simulations, between -15 and -20 °C relative to PI (Fig. A). However, cloud feedback locking drastically increases the forcing required to initiate the snowball Earth transition, and considerably slow down the transition to snowball Earth. We find that simulations at CO2 concentration of 1/64 times PI levels and lower are required to trigger the transition with locked cloud feedbacks, which is a factor ten less than the concentration required with interactive cloud feedbacks. The surface albedo-locked simulation does not reach a snowball instability as it enters quasi-equilibrium below the transition temperature of around -15 °C relative to PI. Similarly, the simulation reaches a quasi-equilibrium below -10 °C relative to PI when the water vapor feedback is locked (Appendix Fig. ). The water vapor feedback actually does not strengthen with cooling so is not itself the cause of the snowball instability, however its large positive feedback is important for allowing the surface albedo to trigger the instability. Thus, the role of cloud feedback is different from that of the surface albedo feedback: (1) positive cloud feedbacks facilitate an early tropical cooling, and this effect is increased with stronger negative forcing, (2) positive cloud feedbacks ahead of the advancing sea-ice edge and its effect on tropical oceans accelerate the transition to snowball Earth and decrease the temperature of instability, as summarized in Fig. ; (3) cloud feedbacks substantially increase the threshold CO2 level required for snowball Earth initiation and (4) the advance of sea ice into the southern subtropics results in the surface albedo feedback exceeding the sum of the negative feedbacks. Whereas the surface albedo feedback is responsible for the transition and the main driver to the complete glaciation, our results emphasise an important contribution of cloud feedbacks. Cloud feedbacks are also the main cause of inter model spread in climate sensitivity e.g., a fact which we shall exploit in Sect. .

The temperature at which the climate transits towards a snowball Earth state is broadly similar across the different CO2 forcing experiments (Fig. ), when taking into account that it depends systematically on the degree of disequilibrium (Fig. ). This is independent of whether the experiment is initialized from a PI or an LGM state, as shown in Appendix , which demonstrates the importance of background temperature control over the climate feedbacks which drive the transition. In particular, sea-ice formation is primarily temperature-dependent, and the amplitude of the sea-ice albedo feedback is practically independent of the strength of the negative forcing (Fig. ). We can expect such temperature-dependency to be similar across climate models, which would contribute to them having a similar temperature of entering the snowball transition. Similarly, the winds push sea ice equatorwards (Appendix Fig. ), in connection with temperature contrast between sea ice and open ocean. This is concomitant with positive cloud feedbacks ahead of sea-ice edge, which is expected to behave in similarly across models since they broadly represent the same general circulation. A simple geometric argument for a transition temperature close to 0 °C is that it happens when the ice edge enters the sub-tropics at about 30° latitude, leaving about half the Earth still ice free and hence above the freezing temperature, and the other half at temperatures below. When abruptly decreasing the CO2 concentration to 50 ppm (around 1/6 of PI CO2), we find hints of instability near the global mean temperature of 0 °C (near -15 °C relative to PI), but the model is stable for 1848 years until it crashes due to thick sea ice (Sect. ). This indicates that it is unlikely that the transition temperature happens at a temperature substantially warmer than 0 °C.

Nevertheless, the transition temperatures of each phase show a slight shift to lower values under stronger negative forcing. Therefore, the climate system deviates from pure temperature-dependent behaviour as the strength of the radiative cooling and the speed of transition to snowball Earth increases. These time-dependent effects have often been described in the literature following extreme cooling rates , and we speculate that this is due to the large heat capacity of the ocean, as well as differences in ocean mixed layer depths and direct CO2 forcing across different regions: in certain circumstances, this can result in local temperatures dropping faster than global temperatures (Appendix Fig. A), and hence out of sync with the sea-ice extent controlling the surface albedo feedback. Ocean convective mixing is also increasingly more efficient at evacuating heat for larger surface cooling, indicated by an increasing peak of the strength of the Atlantic meridional overturning circulation (AMOC) in Appendix Fig. B. The associated ocean currents on the contrary drag sea ice polewards and can slow down the sea-ice edge advance . All in all, we suggest slow, low forcing simulations are preferable when analysing snowball Earth transitions, as (1) fast transitions to snowball Earth are hardly realistic, as geological snowball states may form over millions of years e.g. and (2) fast transitions involve temporal effects which would depart from temperature-dependency. All together, these considerations support the transition temperature close to 0 °C found in the experiments with weak forcing.

While we expect the temperature-dependency to behave similarly across models, it is currently not known whether the temperature threshold of 0 °C that we find in MPI-ESM1.2-CR is indeed universal. The CESM model family has performed several snowball Earth simulations , and while the model CESM2 also transits towards a snowball Earth around 0 °C under similar CO2 forcing , models such as CCSM3 and CCSM4 can have stable waterbelt states with abrupt but smaller transitions, which anyway are not likely consistent with the evidence for recent LGM states. To further test this, we perform an abrupt 1/128×CO2 simulation with the model CESM1.2. At 1/128×CO2, CESM1.2 shows a transition temperature at around -24 °C below pre-industrial, or approximately -10 °C absolute temperature, which is similar as MPI-ESM1.2 at the same forcing level (Fig. ). Due to the time-dependent effects, which we discuss in this study, the transition temperature is displaced under strong forcing. Therefore, under a low forcing, we expect the transition temperature to be closer to 0 °C for CESM1.2, as was the case for MPI-ESM1.2.

There is an ongoing discussion on whether climate models with large cloud feedbacks and consequently high climate sensitivity are compatible with the LGM . Models with high climate sensitivity simulate LGM temperatures around the snowball Earth transition temperature of around 0 °C and are usually unstable. This in fact is to be expected if the transition temperature is controlled by the temperature-dependent behaviour of the climate feedbacks, where the threshold before transiting to a snowball Earth state would be similar across models. In the case of tropical shallow cloud feedbacks, despite considerably increasing the sensitivity of the climate to enter snowball Earth, they do not seem to affect the snowball Earth transition temperature itself (Fig. ).

It is therefore possible to estimate an upper limit of climate sensitivity from the models with stable LGM states with an emergent constraint framework (Sect. ), using as snowball temperature threshold a global mean temperature close to 0 °C. In this way, we find that it is implausible that climate sensitivity exceeds 6.2 °C (3.9 %–8.4 %, 5 %–95 % confidence interval, Fig. ). This upper limit is close to the sensitivity of CESM2 (5.6 °C), which simulates an LGM temperature anomaly of -11.3 °C , just above the global mean temperature of 0 °C. Additional support for this upper limit is obtained from a slightly more sensitive version of the model, which is unstable when run under LGM boundary conditions .

On account of the uncertainty on whether the transition temperature would be close to 0 °C across models, we test the sensitivity of our constraint by decreasing the transition temperature to -5 °C (Fig. ). In this case, we find that the upper bound on ECS cannot exceed 7.6 °C (4.7 %–10.5 %, 5 %–95 % confidence interval). This is a relatively modest change of 1.4 °C in the upper bound, considering the large change in transition temperature of -5 °C. The uncertainty on the transition therefore has a small influence on our upper bound ECS estimate, and the value obtained here is in agreement with estimates from other lines of evidence, or is in many cases a stronger constraint .

Finally, we further test the sensitivity of the upper bound estimates on climate sensitivity using the sea-surface temperature threshold (estimated at close to -7.5 °C using the same approach as in Fig. , in Appendix ), which allows us to include the CESM-family ensemble described in Sect. , and leave CESM2 out of the ensemble. When the CESM-family ensemble is included, and whether CESM2 is or is not in the ensemble, the relationship is robust and displays similar characteristics (Appendix ). The inclusion of CESM2 strengthens the relationship and increases the constraint on the upper bound on climate sensitivity, but also increases the median estimates to 5.5 °C (4.4 %–6.6 %, 5 %–95 % confidence interval), in comparison to 4.9 °C (3.2 %–6.7 %, 5 %–95 % confidence interval) when CESM2 is excluded. As such, the results are not overly sensitive to the inclusion of CESM2, but overall confidence in the constraint would be significantly strengthened if more institutes would perform the LGM simulations.

The described approach to provide an upper bound on climate sensitivity could have a great potential, in particular as independent constraints from paleoclimates are found to be increasingly useful . Nevertheless, to support the constraint, it would be beneficial that modelling centres publish their LGM simulations, even if they fail to stabilise, as this may be used to narrow down the uncertainty on the instability threshold.

It is also important to know whether many models have a threshold gravitating around 0 °C, similarly to that found for MPI-ESM1.2-CR, MPI-ESM1.2-LR and CESM1.2. To tackle this latter problem, we suggest the following experimental design, which is short and easy to replicate:

Run a PI state, with the CO2 concentration set to 1/128 times the PI value, until the model reaches at least the initial instability leading to the snowball Earth state. The chosen CO2 concentration should be a fair trade off between a relatively short simulation length and the time-dependency effects discussed in this study. If time allows, complement with less strong reductions to investigate the time-dependency.

The results could support the novel constraint on climate sensitivity presented in this study, but also help understanding the challenges that some modeling groups experience in LGM simulations, a paleoclimate notoriously difficult to model. We advocate that multi-model comparisons of snowball Earth transitions, as well as single-model ensembles with varying climate sensitivity, would better support the relationship between simulated LGM temperatures and the upper bound on climate sensitivity. The main strength of this constraint is that it is independent of paleoclimate reconstructions and simply relies on the fact that Earth did not undergo a transition to a snowball state 21 000 years ago, or during any earlier glacial cycles during the Pleistocene, providing a valuable additional line of evidence to those already used in the community .