1. Introduction
Equilibrium climate sensitivity (ECS) is a measure of the increase in equilibrium global mean surface temperature (GMST) generated by doubling atmospheric CO2 (Sherwood et al. 2020). It is calculated:
\(\ ECS=\ \Delta F_{2\times CO_{2}}\lambda^{-1}\) (1)
where \(\Delta F_{2\times CO2}\)(units: W m-2) is the change in radiative forcing brought on by a doubling of atmospheric CO2 and \(\lambda\) (units: W m-2K-1) is the net feedback factor, defined as the sum of all climate system feedbacks that either amplify or reduce the effect of radiative forcing on temperature (Knutti & Hegerl, 2008). While the exact value of ECS is unknown, general circulation models (GCMs) have exhibited a persistent spread in this quantity for decades due to various structural and parametric differences (Knutti et al., 2017; Zelinka et al., 2020). For example, the most recent generation of GCMs, associated with Phase 6 of the Coupled Model Intercomparison Project (CMIP6: Eyring et al. 2016), has ECS values ranging from 1.8 to 5.6K (Zelinka et al., 2020). In recent years, the Intergovernmental Panel on Climate Change (IPCC) narrowed its likely range (66%) for ECS from 1.5-4.5K to 2.5-4K (IPCC, 2021). The IPCC based its assessment on a synthesis of many prior studies (Sherwood et al. 2020), considering diverse methodological perspectives such as process understanding, the instrumental record, paleoclimates, and emergent constraints, to arrive at a holistic estimate of ECS. Even so, the current uncertainty in ECS remains undesirably large because it implies a large spread in possible future warming values (IPCC, 2021), leading to drastically different climate outcomes (IPCC, 2022). It has also been estimated that reducing the uncertainty in ECS could save trillions of dollars (Hope, 2015).
One method used to estimate ECS, emergent constraints, is designed to reduce uncertainty in climate change metrics which are otherwise difficult or intractable to measure. An emergent constraint requires two quantities: a modeled, but observable, climate quantity, and some uncertain metric of future climate change. The relationship between these quantities must be causal and statistically robust (Hall et al., 2019; Brient, 2020; Williamson et al., 2021). This feature is sometimes referred to as the “emergent relationship.”
Over the past decade, numerous studies have proposed emergent constraints on ECS (Caldwell et al., 2018; Williamson et al., 2021). One prior study, Cox et al. (2018a), suggested that model spread in ECS could be reduced through an emergent constraint based on historical global mean surface temperature (GMST) variability. They defined a metric of interannual temperature variability (\(\psi\)), which was found to be closely tied to ECS:
\(\psi=\sigma_{T}\ (-\ln\alpha_{1T})^{-0.5}\) (2)
where \(\sigma_{T}\) (units: K) is the standard deviation of global temperature and \(\alpha_{1T}\) is the one-year-lag autocorrelation of GMST (Cox et al., 2018a). The metric is then computed for every overlapping 54-year window in the GMST record, after detrending, and averaged to produce the observational constraint (Cox et al., 2018a). A similar metric was proposed by Nijsse et al. (2019). Using piControl experiments, the study found that a metric of decadal temperature trend variability (\(\sigma_{b}\)), defined as the standard deviation of the decadal trend in GMST, was correlated strongly with ECS over the CMIP5 model ensemble, although it did not propose an emergent constraint. We refer to \(T\) as the timeseries of GMST (units: K) and \(t\) as time (units: yr). For any window of time spanning ten years, the decadal trend in GMST (\(b\)) is defined as
\(b\ =\ \sigma_{\text{Tt}}{\sigma_{t}}^{-1}\) (3)
where for an arbitrary decadal window, \(\sigma_{\text{Tt}}\) is the covariance of \(T\) and \(t\) while \(\sigma_{t}\) is the standard deviation of \(t\) (Nijsse et al., 2019). Estimates of \(b\) are computed for every non-overlapping decadal window over the entire record, and then the standard deviation of the resulting slopes is computed, \(\sigma_{b}\).
For an emergent constraint to be useful, it must be credible, meaning that it has a solid theoretical foundation and is bolstered by empirical evidence (Klein & Hall, 2015; Hall et al., 2019; Williamson et al., 2021). Theoretical support for the proposed emergent relationships is founded on the fluctuation-dissipation theorem (FDT), a dynamical relationship pertaining to systems that obey detailed balance (Leith, 1975). FDT states that for such a system, transient fluctuations in one of its thermodynamic state variables will be directly proportional to the long-term response of that same variable to external forcing (Kubo, 1957). In the climate system, this theoretically implies a direct relationship between GMST variability and ECS. Empirical support for the proposed relationship comes from the CMIP5 ensemble (Taylor et al., 2012), where both metrics of GMST variability (\(\text{ψ\ }\text{and}\ \sigma_{b})\) were shown to exhibit a strong linear relationship with ECS (Figure 1; Supplemental Table 1).
Despite sources of theoretical and empirical support, the proposed constraints have some important caveats to consider. For one, FDT relies on an assumption of thermal equilibrium, which is violated by external forcing in the climate system (Brown et al., 2018; Po-Chedley et al., 2018; Rypdal et al. 2018). For another, the instrumental record of GMST is limited to approximately 100 years, arguably too short to properly characterize its variability (Annan et al., 2020). Finally, the empirical foundation of the constraints is placed into question by a distinct lack of support for their relationships in the CMIP6 ensemble (Figure 1; Schlund et al., 2020). This type of “out-of-sample” testing is typically used to assess statistical robustness in an emergent constraint (Hall et al., 2019), so the lack of a meaningful relationship in CMIP6 raises concerns. Therefore, more investigation into the relationship between ECS and temperature variability is needed. In this study, we adapt the methods of Cox et al. (2018a) and Nijsse et al. (2019) to examine whether their proposed emergent relationships are consistent with a longer, relatively independent record of temperature variability: the past millennium (850-1999). By focusing on this period, we hope to answer some of the outstanding theoretical and empirical questions revolving around this potentially valuable emergent constraint. After doing so, we will conclude by contextualizing our results within the broader effort to constrain the value of ECS.