Abstract. Climate sensitivity to CO2 remains the key uncertainty in projections of future climate change. Transient climate response (TCR) is the metric of temperature sensitivity that is most relevant to warming in the next few decades and contributes the biggest uncertainty to estimates of the carbon budgets consistent with the Paris targets. Equilibrium climate sensitivity (ECS) is vital for understanding longer-term climate change and stabilisation targets. In the IPCC 5th Assessment Report (AR5), the stated “likely” ranges (16 %–84 % confidence) of TCR (1.0–2.5 K) and ECS (1.5–4.5 K) were broadly consistent with the ensemble of CMIP5 Earth system models (ESMs) available at the time. However, many of the latest CMIP6 ESMs have larger climate sensitivities, with 5 of 34 models having TCR values above 2.5 K and an ensemble mean TCR of 2.0±0.4 K. Even starker, 12 of 34 models have an ECS value above 4.5 K. On the face of it, these latest ESM results suggest that the IPCC likely ranges may need revising upwards, which would cast further doubt on the feasibility of the Paris targets. Here we show that rather than increasing the uncertainty in climate sensitivity, the CMIP6 models help to constrain the likely range of TCR to 1.3–2.1 K, with a central estimate of 1.68 K. We reach this conclusion through an emergent constraint approach which relates the value of TCR linearly to the global warming from 1975 onwards. This is a period when the signal-to-noise ratio of the net radiative forcing increases strongly, so that uncertainties in aerosol forcing become progressively less problematic. We find a consistent emergent constraint on TCR when we apply the same method to CMIP5 models. Our constraints on TCR are in good agreement with other recent studies which analysed CMIP ensembles. The relationship between ECS and the post-1975 warming trend is less direct and also non-linear. However, we are able to derive a likely range of ECS of 1.9–3.4 K from the CMIP6 models by assuming an underlying emergent relationship based on a two-box energy balance model. Despite some methodological differences; this is consistent with a previously published ECS constraint derived from warming trends in CMIP5 models to 2005. Our results seem to be part of a growing consensus amongst studies that have applied the emergent constraint approach to different model ensembles and to different aspects of the record of global warming.

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Despite major advances in climate science over the last 30 years, persistent uncertainties in projections of future climate change remain. Climate projections are produced with increasingly complex models that attempt to represent key processes in the Earth system, including atmospheric and oceanic circulations, convection, clouds, snow, sea ice, vegetation, and interactions with the carbon cycle. Uncertainties in the representation of these processes feed through into a range of projections from the many state-of-the-art climate models now being developed and used worldwide. For example, despite major improvements in climate models, the range of equilibrium global warming due to doubling carbon dioxide still spans a range of more than 3. Here a promising way to make use of the ensemble of climate models to reduce the uncertainties in the sensitivities of the real climate system is reviewed. The emergent constraint approach uses the model ensemble to identify a relationship between an uncertain aspect of the future climate and an observable variation or trend in the contemporary climate. This review summarizes previous published work on emergent constraints and discusses the promise and potential dangers of the approach. Most importantly, it argues that emergent constraints should be based on well-founded physical principles such as the fluctuation-dissipation theorem. This review will stimulate physicists to contribute to the rapidly developing field of emergent constraints on climate projections, bringing to it much needed rigor and physical insights.

Background: The emergent constraint approach has received interest recently as a way of utilizing multimodel General Circulation Model (GCM) ensembles to identify relationships between observable variations of climate and future projections of climate change. These relationships, in combination with observations of the real climate system, can be used to infer an emergent constraint on the strength of that future projection in the real system. However, there is as yet no theoretical framework to guide the search for emergent constraints. As a result, there are significant risks that indiscriminate data-mining of the multidimensional outputs from GCMs could lead to spurious correlations and less than robust constraints on future changes. To mitigate against this risk, Cox et al 1 (hereafter CHW18) proposed a theory-motivated emergent constraint, using the one-box Hasselmann model to identify a linear relationship between equilibrium climate sensitivity (ECS) and a metric of global temperature variability involving both temperature standard deviation and autocorrelation (Ψ). A number of doubts have been raised about this approach, some concerning the application of the one-box model to understand relationships in complex GCMs which are known to have more than the single characteristic timescale.Objectives: To study whether the linear Ψ-ECS proportionality in CHW18 is an artefact of the one-box model. More precisely we ask 'Does the linear Ψ-ECS relationship feature in the more complex and realistic two-box and diffusion models?'. Methods:We solve the two-box and diffusion models to find relationships between ECS and Ψ. These models are forced continually with white noise parameterizing internal variability. The resulting analytical relations are essentially fluctuation-dissipation theorems.Results: We show that the linear Ψ-ECS proportionality in the one-box model is not generally true in the two-box and diffusion models. However, the linear proportionality is a very good approximation for parameter ranges applicable to the current state-of-the-art CMIP5 climate models. This is not obviousdue to structural differences between the conceptual models, their predictions also differ. For example, the two-box and diffusion, unlike the one-box model, can reproduce the long term transient behaviour of the CMIP5 abrupt4xCO2 and 1pcCO2 simulations. Each of the conceptual models also predict different power spectra with only the diffusion model's pink 1/f spectrum being compatible with observations and GCMs. We also show that the theoretically predicted Ψ-ECS relationship exists in the piControl as well as historical CMIP5 experiments and that the differing gradients of the proportionality are inversely related to the effective forcing in that experiment. Conclusions:We argue that emergent constraints should ideally be derived by such theory-driven hypothesis testing, in part to protect against spurious correlations from blind data-mining but mainly to aid understanding. In this approach, an underlying model is proposed, the ...

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