The Pliocene Model Intercomparison Project (PlioMIP; ) has presented boundary conditions in order for climate models to simulate the mPWP. This was not a true “time slice” experiment such as the LGM simulations, which represented the climatic average over an interval of 19–23 ka. The much longer mid-Pliocene interval contained multiple ice age cycles, and the mPWP experiments were designed to represent a typical or average interglacial within this period. There were two experiments conducted in PlioMIP. Experiment 1  used atmosphere-only climate models, with the sea surface temperature (SST) boundary condition prescribed from a reconstruction which is discussed further in the next section. For these simulations, we expect the SAT anomaly to be tightly constrained by the imposed boundary conditions (especially over the ocean) and therefore to bear little relationship with the model's sensitivity. The model results bear this out, and thus we do not consider these simulations further. There were 10 models that performed Experiment 2, in which coupled atmosphere–ocean general circulation models were forced with a suite of boundary conditions including a land–sea mask, topography, ice sheet, vegetation, and greenhouse gas concentration (see  for details). For these models, we expect their mPWP simulations (and in particular their SAT response) to be related to their climate sensitivities, since the greenhouse gas boundary condition forms a large part of the total forcing. In order to relate past to future, however, we can only use models for which both the mPWP simulation results and an estimate of the model's sensitivity are available. The GENISIS model is mentioned in , but results are not available in the PlioMIP database, so this condition reduces the ensemble to the 9 models which are listed in Table . The ensemble size, while smaller than might be hoped for given that more than 20 models contributed to the Climate Model Intercomparison Project, CMIP5, is of very similar size to that available for the LGM, where there are eight models in PMIP2 and 9 models in PMIP3 satisfying equivalent criteria. For most models, the values of equilibrium climate sensitivity are taken from the estimates published in Table 1 of . The relevant sensitivity value for the FGOALS model was not included in that paper but has since been published elsewhere .

Raised atmospheric CO2 is one of the more significant changes in boundary conditions for the mPWP, with other forcings contributing less than half as much again , so it seems a priori reasonable to hope for a correlation in the climate model ensemble between their equilibrium sensitivities and their SAT changes at the mPWP. However, the other boundary condition changes are not negligible, and if the models respond very differently to these (or non-linearly to combinations of forcings) then a correlation between global SAT anomaly at mPWP and equilibrium sensitivity may not be observed.

As a first investigation, we tested for a correlation between global SAT anomaly in the mPWP simulations vs. S. As the left plot of Fig.  shows, there is a weak correlation between these two variables of 0.59, but this does not reach the 95 % significance threshold of 0.67. As in  and , we anticipate that the relationship between S and palaeoclimate changes is likely to be stronger if we focus on the tropics for the palaeosimulations, since this will reduce the influence of ice sheet and vegetation changes. This is borne out by the right-hand panels of Fig. , which show the correlations for both pointwise (on a 10∘ grid) and zonally averaged palaeosimulations vs. S. The model ensemble exhibits a strong correlation between mPWP tropical SAT anomaly and S. Integrating over the entire tropical region 30∘ S–30∘ N, the correlation between tropical mPWP SAT anomaly and climate sensitivity is 0.73.

While the small ensemble gives us cause for concern (compare  with  and ) we proceed under the assumption that it is informative regarding the real climate system. In order to test the potential for constraining the climate system using information from the mPWP, we need an estimate of typical tropical temperatures during this period. As our reconstruction of mPWP temperatures we use the PRISM3 SST anomaly field which was presented by . This is based on the PRISM3D dataset, firstly processed into warm peak averages (to represent typical interglacial conditions within the “time slab” of interest) for both February and August, then converted to anomalies relative to modern conditions, and finally smoothed and interpolated in both time and space into complete SST anomaly fields under the assumption that SST patterns were similar to the present. More sophisticated methods could in principle be used for the SST reconstruction, such as were presented by  and , but this is outside the scope of this paper. We use the average of these data fields for our analysis (equivalently, the annual average of the monthly fields that were generated for PlioMIP Experiment 1).

We have also directly investigated the data points, in the form of the annual mean temperature anomaly estimates which were provided by . The locations of these data can be seen in Fig. . The simple mean of the anomalies of the 17 data points which lie within the tropical ocean region of all models is rather low at 0.15 ∘C, which is far outside the full ensemble range (taking model values at the same grid points) of 0.9–2.4 ∘C. The spread of data values is also many times greater than the models, at 2.3 ∘C compared to 0.15–0.6 ∘C across the models (all values 1 standard deviation). In fact, a large majority of the data points lie outside the full range of the model ensemble, in many cases by a substantial margin. Although it is likely that the models do underestimate spatial variation to some extent, it seems reasonable to conclude that much of the model–data discrepancy here is due to uncertainties in the analysis of the data points. Furthermore, we do not expect models to be able to reliably simulate spatial anomaly patterns skilfully at the mPWP, since they fail to do this for other time periods of palaeoclimatic interest where sufficient data have been assembled to test this rigorously . We therefore do not think it is meaningful to constrain the models in this case by a small number of irregularly sampled points and prefer to focus on averages over larger spatial scales where we can reasonably expect the models to have some skill .

To calculate an estimate for equilibrium climate sensitivity, we combine the model estimates for climate sensitivity and the warming at the mPWP, together with the PRISM3 estimate of tropical ocean temperature change, using the approach described in . For consistency with the data, we use sea surface temperature from the climate models, which are of course very close to SAT at the same locations. The interpolated PRISM3 data indicate a warming of 0.8 ∘C  for the SST integrated over 30∘ S to 30∘ N. The calculation of climate sensitivity involves sampling from the uncertain temperature distribution and, for each sample, generating a prediction of the associated sensitivity taking account of the uncertainty in the linear relationship. The PRISM3 reconstruction does not include an estimate of uncertainty in the reconstruction. Initially we take a value of 0.4 ∘C (at 1 standard deviation), based on the hope both that the signal was at least as large as than the noise and that it might come close to matching the value of 0.7 ∘C  (at 2 standard deviations) which was obtained for a recent reconstruction of the LGM tropics . It is of course essential to test the sensitivity of our result to this assumed uncertainty, and we discuss this further below. Figure  shows the result. The regression model generates an estimate for the equilibrium climate sensitivity of 1.9–3.7 ∘C. Only the models with weaker tropical warming are consistent with the data; as these tend to be low sensitivity models, the resulting estimate for S is at the low end of (and extending to values outside) the full range of models that contributed to PlioMIP.

Proxy-based reconstructions of past climates are, of course, uncertain. As mentioned above, however, the size of the uncertainty in the PlioMIP Experiment 1 SST field has not been objectively estimated, and our initial value of 0.4 ∘C  is simply an assumption based in part on previous work focussing on the LGM. It would be reasonable to assume that the Pliocene temperature estimate is in fact more uncertain, so we tested the sensitivity of our result to this. The dashed and dot-dashed blue and black lines in Fig.  show the effect on the estimate of replacing the original estimate of 0.4 ∘C with values of 0.1 and 1 ∘C (all at 1 standard deviation) respectively. It is apparent that reducing the uncertainty even to an extremely low value has relatively little effect on the resulting sensitivity estimate (which only narrows marginally to 2.1–3.6 ∘C), as in this case the spread around the regression line makes a dominant contribution to the total uncertainty. However, none of the models are consistent with this temperature estimate, as all warm by more than 0.8 ∘C in the region, many by a substantial margin. If we increase the SST uncertainty estimate substantially to 1 ∘C, then the uncertainty of the overall result does increase more noticeably to 1.3–4.2 ∘C. At this point, even the models with the strongest warming are consistent with the data, and thus the estimated sensitivity range covers the full range of model values with an extension also to lower values. Note that, at this level of uncertainty, the data would no longer give us confidence even that the mPWP was warmer than the pre-industrial period, at least in the tropics. It would be very useful to have more complete understanding of the uncertainties of temperature reconstructions for the mPWP.

A major issue in simulating the mPWP is that the atmospheric CO2 level corresponding to interglacial peaks is not precisely known. Furthermore, there is hypothesised to be additional forcing due to methane which cannot be directly inferred from proxy data but which has instead been assumed to be proportional to the CO2 forcing. This was implemented within PlioMIP via an increased CO2 concentration. That is, the imposed CO2 forcing was selected to represent not only CO2 but the additional effect of methane. Therefore, we have tested the sensitivity of our result to uncertainty in total GHG forcing. Our approach is rather simplistic and makes the assumption that, for each model, the tropical temperature anomaly will change in direct proportion to the net CO2 forcing (relative to the pre-industrial control). While we do not expect this approximation to be precise, it at least allows us to perform an initial investigation into the sensitivity of our results to changes in the boundary conditions. The PlioMIP protocol imposes a value of 405 ppm CO2, but a value as low as 350 ppm is possible, being at the low end of the average range considered consistent with the data proxies for CO2 (given as “∼ 360–380 ppmv” in ). When we modify the model results accordingly, we obtain the results shown in Fig. . By scaling the modelled results downwards, many more of them are brought into line with the tropical SST estimate derived from the PRISM3 dataset, and the resulting sensitivity estimate increases to 2.0–4.0 ∘C. It seems that the value of 350 ppm is more consistent with the ensemble as a whole than PlioMIP's own estimate of 405 ppm, though of course this cannot be taken to imply that the true value was actually this low.

The model results are dependent on the experimental protocols, both for the mPWP simulation and the calculation of S. For the calculation of S, it is now commonplace to use a regression from an instantaneous 4 × CO2 scenario, with this being used in the IPCC Fifth Assessment Report (AR5) for their model sensitivity values. However, it is increasingly recognised that this regression-based estimate can significantly underestimate the true equilibrium sensitivity. One of the more extreme examples of this is the GISS model, with the sensitivity reported as 2.1 ∘C in the IPCC AR5 but actually estimated as 2.7–2.9 ∘C by the PlioMIP contributors, based on a long simulation . For most other models that have done this comparison, the discrepancy is somewhat smaller . For the Pliocene experiments, the computational cost of long integrations may mean that some model simulations are not fully equilibrated, which could lead to some errors in their estimates of past and present climates. Internal variability is an additional concern if the climatology is generated from a short time series. While global temperature is unlikely to be seriously affected by this factor, regional variabilities can be larger. For PlioMIP the intention is that all simulations should be run for at least 500 years, which should produce a reasonably well equilibrated climate, apart from in the deep ocean. However, there may be significant drifts in some regions beyond that time.

A notable point that is apparent in Fig.  is that the regression lines do not pass through the origin but instead indicate that zero tropical warming at the mPWP corresponds to an equilibrium sensitivity of about 2 ∘C. This may seem a little odd, although it could be argued that, even if the response in the tropics was zero, we would still expect a positive response at higher latitudes and thus also in the global average. Additionally, CO2 is not the only forcing in the mPWP experiments (ice sheets and sea level have changed, and vegetation can also change in some if not all models), which does complicate things somewhat. In the LGM analysis, found that the regression line derived from the PMIP2 ensemble naturally passed close to the origin, so the issue was not an apparent concern there. The purple colour in Fig.  shows the results if we do not include a constant term in the regression. The sensitivity estimate is both lower in its mean value and much more uncertain, with the 5–95 % range reaching from -0.4 to 3.5 ∘C. The increase in uncertainty is due to a combination of there being larger residuals (implying a larger plausible range around the regression line) and also the increased slope of the regression line, which means that uncertainty in the true SST anomaly translates into increased uncertainty in the related sensitivity. However, it is worth noting that, while the constraint is much weaker, high sensitivity values are still excluded by this approach.

As mentioned previously, the mPWP data collation is based on averaging the warm peaks within the mPWP interval. However, different locations may encounter peak warmth at different times, and thus the warmest peaks may not represent any historical equilibrium climate state. Moreover, the boundary conditions for the different warm peaks would also have been somewhat different in reality, with differences in orbital forcing potentially leading to regional variation exceeding 1 ∘C although the variation is lower at larger spatial scales. The comparison between data collected over a wide range of times and a model snapshot with a specific set of boundary conditions is thus challenging, though averaging over broad spatial scales should help to isolate any warming signal most clearly. The next iteration of PlioMIP  plans to address this issue by focussing on a single interglacial for which sufficient proxy data can be obtained.

Robustness of results is a major concern which we have discussed above and summarise here. highlighted the risk of mining for correlations that are not robust, and there are some examples of plausible correlations in the CMIP3 ensemble which disappeared in CMIP5. Thus we focus on relationships that may be reasonably argued to represent our uncertainties in a realistic manner. In particular, it does not seem at all unreasonable to expect that a greater equilibrium response to increased CO2 in the modern era would also imply a greater response to forcing in the past, and vice versa, this being a simple expression of the principle of uniformitarianism. Of course in reality the sensitivity depends on the underlying climate state and the nature of the forcing , so the past is not expected to be a perfect analogue of the future but rather a useful guide. We regard the main result presented here to be a reasonable hypothesis worthy of further investigation, rather than a confident prediction.