Theoretical foundations of emergent constraints: relationships between climate sensitivity and global temperature variability in conceptual models

Abstract

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 ...

What could we learn about climate sensitivity from variability in the surface temperature record?

“…Ψ and τ are closely related and co-vary very similarly over a wide range of sensitivity when other model parameters are held fixed. Henceforth in this section we focus solely on Ψ as it is more precisely defined and has been recently discussed in some detail (Williamson et al, 2019). However very similar results are also obtained when equivalent experiments are performed using τ .…”

“…Formally, the value of the observations is fully summarised by the likelihood function P (O|Θ), but we primarily present our results as posterior pdfs in order to provide an easily interpreted output which can be directly compared to previously published results. We therefore use a uniform prior in S as this is typically the implicit assumption in emergent constraint analyses (Williamson et al, 2019). This choice results in the posterior being visually equivalent to the likelihood even though their interpretation is somewhat different.…”

What could we learn about climate sensitivity from variability in the surface temperature record?

“…Rypdal and Rypdal, 2014). Therefore, we use here a slightly more complex twolayer model based on Winton et al (2010) and Held et al (2010). This model has been shown to reasonably replicate the transient behaviour of the CMIP5 ensemble of complex climate models (Geoffroy et al, 2013a, b).…”

“…C m and C d are the heat capacities of the mixed-layer and deep ocean respectively, and γ represents the ocean heat transfer parameter. The parameter was introduced by Winton et al (2010) to represent the deep-ocean heat uptake efficacy, and while it is not important for our analysis, we include it for consistency with the broader literature. In a slight modification to Winton et al (2010), we add a noise term δ t to the first equation to represent the internal variability in the system as was originally introduced in a single-layer energy balance climate model by Hasselmann (1976).…”

“…Formally, the value of the observations is fully summarised by the likelihood function P (O| ), but we primarily present our results as posterior probability density functions (pdf's) in order to provide an easily interpreted output which can be directly compared to previously published results. We therefore use a uniform prior in S as this is typically the implicit assumption in emergent constraint analyses (Williamson et al, 2019). This choice results in the posterior being visually equivalent to the likelihood even though their interpretation is somewhat different.…”