Standardising the “Gregory method” for calculating equilibrium climate sensitivity

<strong class="journal-contentHeaderColor">Abstract.</strong> This comment addresses assertions made by Sherwood and Forest (2024) [SF24] regarding the narrowing of the range of equilibrium climate sensitivity (ECS), particularly at the low end. SF24 challenged a previous study by Lewis (2022) [L22] that found a narrower and substantially lower ECS level. This comment clarifies that, contrary to SF24's claims, L22 did not rule out a high ECS level based on historical evidence, and did identify and correct errors in Sherwood et al. (2020). Those errors included use of an invalid likelihood estimation method that, ironically, substantially underestimated likelihood at high ECS levels for their historical evidence. This comment also discusses the role of priors in Bayesian ECS estimation and explains why the subjective Bayesian approach favoured by SF24 risks producing unreliable inference for uncertain parameters such as ECS. Finally, the importance of considering structural uncertainties in climate models, particularly concerning tropical warming patterns, is extended beyond the points raised by SF24. Such uncertainties could affect ECS estimation not only from historical period evidence but also from climate process understanding, paleoclimate data and emergent constraints, but seem more likely to suggest existing ECS estimates are too high than too low.

Estimates of the Earth's equilibrium climate sensitivity (ECS) from twentieth century observations predict a lower ECS than estimates from climate models, paleoclimate data, and interannual variability. Here we show that estimates of ECS from the twentieth century observations are sensitive to the assumed efficacy of aerosol and ozone forcing (efficacy for a forcer is the amount of warming per unit global average forcing divided by the warming per unit forcing from CO2). Previous estimates of ECS based on the twentieth century observations have assumed that the efficacy is unity, which in our study yields an ECS of 2.3 K (5%–95% confidence range of 1.6–4.1 K), near the bottom of the Intergovernmental Panel on Climate Change's likely range of 1.5–4.5 K. Increasing the aerosol and ozone efficacy to 1.33 increases the ECS to 3.0 K (1.9–6.8 K), a value in excellent agreement with other estimates. Forcing efficacy therefore provides a way to bridge the gap between the different estimates of ECS.

We compare equilibrium climate sensitivity (ECS) estimates from pairs of long (≥800‐year) control and abruptly quadrupled CO2 simulations with shorter (150‐ and 300‐year) coupled atmosphere‐ocean simulations and slab ocean models (SOMs). Consistent with previous work, ECS estimates from shorter coupled simulations based on annual averages for years 1–150 underestimate those from SOM (−8% ± 13%) and long (−14% ± 8%) simulations. Analysis of only years 21–150 improved agreement with SOM (−2% ± 14%) and long (−8% ± 10%) estimates. Use of pentadal averages for years 51–150 results in improved agreement with long simulations (−4% ± 11%). While ECS estimates from current generation U.S. models based on SOM and coupled annual averages of years 1–150 range from 2.6°C to 5.3°C, estimates based longer simulations of the same models range from 3.2°C to 7.0°C. Such variations between methods argues for caution in comparison and interpretation of ECS estimates across models.

Cloud feedback is a dominant source of uncertainty in climate model estimates of equilibrium climate sensitivity (ECS). Cloud controlling factor analysis can observationally constrain cloud feedback. For the first time, we use separate rather than unified frameworks to assess high- and low-cloud feedbacks and constrain the net cloud feedback and subsequently, the ECS. We find a robustly positive cloud feedback (i.e., a negative feedback is &lt;0.5% probable), indicating that clouds amplify global warming. We assess the individual and combined impacts of our cloud feedback constraints on ECS using three approaches. Two indicate an upward ECS shift with reduced uncertainty, preserving ECS–feedback correlations but using cloud feedback as a single line of evidence. The third, a Bayesian framework combining multiple lines of evidence, also suggests a higher ECS but with a smaller increase and broader confidence range.

&amp;lt;p&amp;gt;There is much current debate about the way in which the earth's climate and temperature are responding to anthropogenic and natural forcing. Current forecasts are dependent on the accuracy of how Equilibrium Climate Sensitivity (ECS) is evaluated. In the recent wide-ranging ECS sensitivity review and assessments of Sherwood et al (2020, &amp;lt;em&amp;gt;Rev. Geophys.&amp;lt;/em&amp;gt; &amp;lt;strong&amp;gt;58&amp;lt;/strong&amp;gt;, e2019RG000678), they report a baseline ECS 5%&amp;amp;#8211;95% with a range of 2.3 &amp;amp;#176;C to 4.7 &amp;amp;#176;C. While this is an important and much needed quantification &amp;amp;#8211; this wide ECS uncertainty range, based largely on uncertainty around total radiative forcing (TRF), is also problematic for our geophysical based community. The ECS value as viewed by policy and global strategy negotiators is currently only understood to a 70% (= 2.4 / 3.5) physically resolved level of resolution. An open question (to discuss in this talk) is it possible to resolve this physical measurement more accurately and what are the current main issues that need to be accommodated? As part of this discussion we present the work of Young, Allen and Bruun (2021): &amp;lt;em&amp;gt;a re-evaluation of the Earth's surface temperature response to radiative forcing &amp;lt;/em&amp;gt;(2021, &amp;lt;em&amp;gt;ERL&amp;lt;/em&amp;gt;, &amp;lt;strong&amp;gt;16&amp;lt;/strong&amp;gt;, 054068). In that paper we have re-assessed the current evidence at the globally averaged level by adopting a generic 'data-based mechanistic' modelling strategy that incorporates statistically efficient parameter estimation. This identifies a low order, differential equation model that explains how the global average surface temperature variation responds to the influences of total radiative forcing. The model response includes a novel, stochastic oscillatory component with a period of about 55 years (range 51.6&amp;amp;#8211;60 years) that appears to be associated with heat energy interchange between the atmosphere and the ocean. These 'quasi-cycle' oscillations, which account for the observed pauses in global temperature increase around 1880, 1940 and 2001, appear to be related to ocean dynamic responses, particularly the Atlantic multidecadal oscillation. The model explains 90% of the variance in the global average surface temperature anomaly and yields estimates of the equilibrium climate sensitivity (ECS) (2.29 &amp;amp;#176;C with 5%&amp;amp;#8211;95% range 2.11 &amp;amp;#176;C to 2.49 &amp;amp;#176;C) and the transient climate response (TCR) (1.56 &amp;amp;#176;C with 5%&amp;amp;#8211;95% range 1.43 &amp;amp;#176;C to 1.68 &amp;amp;#176;C), both of which are smaller than most previous estimates. When a high level of uncertainty in the TRF is taken into account, the ECS and TCR estimates are unchanged but the ranges are increased to 1.43 &amp;amp;#176;C to 3.14 &amp;amp;#176;C and 0.99 &amp;amp;#176;C to 2.16 &amp;amp;#176;C, respectively. This then gives the 70% physical resolution limit in ECS mentioned above. Current work is in progress to test this ECS re-evaluation approach using the CMIP6 models. We will discuss some on-going findings of these model signal assessments which include a specific focus on resolution of Atlantic and Pacific Ocean pentedecadal modes. &amp;amp;#160;&amp;lt;/p&amp;gt;&amp;lt;p&amp;gt;Peter C Young, P Geoffrey Allen and John T Bruun (2021). A re-evaluation of the Earth's surface temperature response to radiative forcing, &amp;lt;em&amp;gt;Environ. Res. Lett.&amp;lt;/em&amp;gt; &amp;lt;strong&amp;gt;16&amp;lt;/strong&amp;gt; 054068, https://iopscience.iop.org/article/10.1088/1748-9326/abfa50.&amp;lt;/p&amp;gt;

Equilibrium climate sensitivity (ECS) and hydrological sensitivity describe the global mean surface temperature and precipitation responses to a doubling of atmospheric CO2. Despite their connection via the Earth’s energy budget, the physical linkage between these two metrics remains controversial. Here, using a global climate model with a perturbed mean hydrological cycle, we show that ECS and hydrological sensitivity per unit warming are anti-correlated owing to the low-cloud response to surface warming. When the amount of low clouds decreases, ECS is enhanced through reductions in the reflection of shortwave radiation. In contrast, hydrological sensitivity is suppressed through weakening of atmospheric longwave cooling, necessitating weakened condensational heating by precipitation. These compensating cloud effects are also robustly found in a multi-model ensemble, and further constrained using satellite observations. Our estimates, combined with an existing constraint to clear-sky shortwave absorption, suggest that hydrological sensitivity could be lower by 30% than raw estimates from global climate models. The connections between global mean temperature and precipitation responses to CO2 doubling (equilibrium climate and hydrological sensitivity) are driven through low-cloud responses to surface warming, according to MIROC5 perturbation experiments.

We examine the response of the Community Earth System Model Versions 1 and 2 (CESM1 and CESM2) to abrupt quadrupling of atmospheric CO2 concentrations (4xCO2) and to 1% annually increasing CO2 concentrations (1%CO2). Different estimates of equilibrium climate sensitivity (ECS) for CESM1 and CESM2 are presented. All estimates show that the sensitivity of CESM2 has increased by 1.5 K or more over that of CESM1. At the same time the transient climate response (TCR) of CESM1 and CESM2 derived from 1%CO2 experiments has not changed significantly—2.1 K in CESM1 and 2.0 K in CESM2. Increased initial forcing as well as stronger shortwave radiation feedbacks are responsible for the increase in ECS seen in CESM2. A decomposition of regional radiation feedbacks and their contribution to global feedbacks shows that the Southern Ocean plays a key role in the overall behavior of 4xCO2 experiments, accounting for about 50% of the total shortwave feedback in both CESM1 and CESM2. The Southern Ocean is also responsible for around half of the increase in shortwave feedback between CESM1 and CESM2, with a comparable contribution arising over tropical ocean. Experiments using a thermodynamic slab‐ocean model (SOM) yield estimates of ECS that are in remarkable agreement with those from fully coupled Earth system model (ESM) experiments for the same level of CO2 increase. Finally, we show that the similarity of TCR in CESM1 and CESM2 masks significant regional differences in warming that occur in the 1%CO2 experiments for each model.

Equilibrium climate sensitivity (ECS) refers to the total global warming caused by an instantaneous doubling of CO2 from the preindustrial level. It is mainly estimated through the linear fit between the changes in global-mean surface temperature and top-of-atmosphere net radiative flux, due to the high costs of millennial-length simulations for reaching a stable climate. However, the accuracy can be influenced by the response’s nonlinearity and the internal noise, especially when using a limited-length simulation. Here, we propose a new method that derives a new series using an exponential-interval sampling (EIS) method for the original simulation to reduce the noise and estimate the ECS more accurately. Utilizing the millennial-length simulations of LongRunMIP, we prove that the EIS method can effectively reduce the influence of internal variability, and the estimated ECS based on the first 150 years of simulation is closer to the final ECS in the millennial-length simulations than previous estimations with the deviation rate decreased by around 1/3. The ECS in CMIP6 models estimated by the EIS method ranges from 1.93 to 6.78 K, and suggests that the multimodel mean ECS derived from the original series with previous methods could be underestimated.

Equilibrium climate sensitivity (ECS), defined as the response of the global mean surface temperature response to a sustained doubling of atmospheric CO2 at equilibrium, is a key metric for quantifying the Earth’s climate sensitivity to greenhouse gas emissions. Accurate ECS estimates are therefore fundamental for reliable simulations of the long-term carbon cycle. cGENIE, as an Earth system model of intermediate complexity that integrates ocean circulation, atmospheric energy balance, and global biogeochemical cycling, is widely used to investigate cross-sphere carbon cycle evolution and long-term climate feedback mechanisms. However, previous cGENIE studies have assumed a fixed climate sensitivity (with a default radiative forcing of 4 W m-2 per CO2 doubling), which often led to inaccurate surface temperature estimates compared with proxy reconstructions, limiting the model’s ability to capture state-dependent climate feedbacks. Here we use fully coupled models (e.g., HadCM3 and CESM) to derive the relationship between atmospheric CO2 concentrations and ECS throughout the Phanerozoic. These simulations are considered to closely match proxy reconstructions of temperatures. We then incorporate state-dependent climate sensitivity into cGENIE to enhance its representation of climate feedbacks across varying CO2 levels. Our results show that temperature simulations using the unmodified cGENIE model exhibit substantial discrepancies for periods of rapid cooling and warming, such as the Late Ordovician and the PETM. However, incorporating state-dependent climate sensitivity substantially reduces the discrepancy between simulated and proxy-reconstructed surface temperatures. These findings highlight the importance of accounting for state-dependent climate sensitivity in Earth system models, both for accurately reconstructing past climate extremes and for improving projections of future climate change.

On a minimal model for estimating climate sensitivity

Regression‐based “fingerprinting” methods in climate science employ total least squares (TLS) or orthogonal regression to remedy attenuation bias arising from measurement error due to reliance on climate model‐generated explanatory variables. Proving the consistency of multivariate TLS requires assuming noise variances are equal across all variables in the model. This assumption has been challenged empirically in the climate context but little is known about TLS biases when the assumption is violated. Monte Carlo analysis is used herein to examine coefficient biases when the noise variances are not equal. The analysis allows the explanatory variables to be negatively correlated which is typical in climate applications. Ordinary least squares (OLS) exhibits the expected attenuation bias which vanishes as the noise variances on the explanatory variables disappear. In some cases, TLS corrects attenuation bias but more typically imparts large and generally positive biases. OLS performs well when the true value of whereas TLS performs quite poorly. This implies that TLS is not well suited for tests of the null. When TLS tends to exhibit opposite biases to OLS. Diagnostic information specific to each data sample should be consulted before using TLS to avoid spurious inferences and replacing OLS attenuation bias with other, potentially larger biases.

Abstract. It has been shown that a theoretically derived relation between annual global mean temperature variability and climate sensitivity held in the CMIP5 climate model ensemble (Cox et al., 2018a, hereafter CHW18). This so-called emergent relationship was then used with observations to constrain the value of equilibrium climate sensitivity (ECS) to about 3 °C. Since this study was published, CMIP6, a newer ensemble of climate models has become available. Schlund et al. (2020) showed that many of the emergent constraints found in CMIP5 were much weaker in the newer ensemble, including that of CHW18. As the constraint in CHW18 was based on a relationship derived from reasonable physical principles, it is of interest to find out why it is weaker in CMIP6. Here, we look in detail at the assumptions made in deriving the emergent relationship in CHW18 and test them for CMIP5 and CMIP6 models. We show one assumption, that of low correlation and variation between ECS and the internal variability parameter, a parameter that captures chaotic internal variability and sub-annual (fast) feedbacks, that while true for CMIP5 is not true for CMIP6. When accounted for, an emergent relationship appears once again in both CMIP ensembles, implying the theoretical basis is still applicable while the original assumption in CHW18 is not. Unfortunately, however, we are unable to provide an emergent constraint in CMIP6 as observational estimates of the internal variability parameter are too uncertain.

We analyzed a dataset from an experiment of an earth system model of intermediate complexity, focusing on the change in transient climate response to cumulative carbon emissions (TCRE) after atmospheric CO2 concentration was stabilized in the Representative Concentration Pathway (RCP) 4.5. We estimated the TCRE in 2005 at 0.3–2.4 K/TtC for an unconstrained case and 1.1–1.7 K/TtC when constrained with historical and present-day observational data, the latter result being consistent with other studies. The range of TCRE increased when the increase of CO2 concentration was moderated and then stabilized. This is because the larger (smaller) TCRE members yield even greater (less) TCRE. An additional experiment to assess the equilibrium state revealed significant changes in temperature and cumulative carbon emissions after 2300. We also found that variation of land carbon uptake is significant to the total allowable carbon emissions and subsequent change of the TCRE. Additionally, in our experiment, we revealed that equilibrium climate sensitivity (ECS), one of the 12 parameters perturbed in the ensemble experiment, has a strong positive relationship with the TCRE at the beginning of the stabilization and its subsequent change. We confirmed that for participant models in the Coupled Model Intercomparison Project Phase 5, ECS has a strong positive relationship with TCRE. For models using similar experimental settings, there is a positive relationship with TCRE for the start of the period of stabilization in CO2 concentration, and rate of change after stabilization. The results of this study are influential regarding the total allowable carbon emissions calculated from the TCRE and the temperature increase set as the mitigation target.

Although ordinary least squares (OLS) regression achieves great success in clean image interpolation, its effectiveness is questionable in the scenario of web images which are usually compressed beforehand. The inherent flaw of OLS is that it is asymmetric, the perturbation is only confined on the right side of the linear system. It is not reasonable for web images. Considering the drawback of OLS, in this paper, we propose an efficient web image interpolation algorithm based on total least squares (TLS) regression. In the proposed method, small perturbations are allowed in both side of the system, which are optimized by TLS in a patch-based manner. Furthermore, we develop a weighted version of TLS to consider contribution diversity of different samples and patches in model estimation, which can efficiently remove the influence of outliers in regression. Experimental results on benchmark test images demonstrate the efficiency of our method.

EvolutionVolume 53, Issue 5 p. 1624-1627 CommentFree Access DON'T FORGET THE BIOLOGY: A REPLY TO GREEN W. G. Eberhard, W. G. Eberhard weberhar@cariari.ucr.ac.cr Escuela de Biología, Universidad de Costa Rica, Ciudad Universitaria, Costa Rica Smithsonian Tropical Research InstituteSearch for more papers by this authorB. A. Huber, B. A. Huber Department of Entomology, American Museum of Natural History, New York, New York, 10024Search for more papers by this authorR. L. Rodriguez, R. L. Rodriguez Department of Entomology, Hawarth Hall, University of Kansas, Lawrence, Kansas, 66045Search for more papers by this author W. G. Eberhard, W. G. Eberhard weberhar@cariari.ucr.ac.cr Escuela de Biología, Universidad de Costa Rica, Ciudad Universitaria, Costa Rica Smithsonian Tropical Research InstituteSearch for more papers by this authorB. A. Huber, B. A. Huber Department of Entomology, American Museum of Natural History, New York, New York, 10024Search for more papers by this authorR. L. Rodriguez, R. L. Rodriguez Department of Entomology, Hawarth Hall, University of Kansas, Lawrence, Kansas, 66045Search for more papers by this author First published: 31 May 2017 https://doi.org/10.1111/j.1558-5646.1999.tb05428.xCitations: 29AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Citing Literature Volume53, Issue5October 1999Pages 1624-1627 ReferencesRelatedInformation