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.