Improved methods for estimating equilibrium climate sensitivity from transient warming simulations

Abstract

Equilibrium climate sensitivity (ECS) refers to the total global warming caused by an instantaneous doubling of atmospheric CO2 from the pre-industrial level in a climate system. ECS is commonly used to measure how sensitive a climate system is to CO2 forcing; but it is difficult to estimate for the real world and for fully coupled climate models because of the long response time in such a system. Earlier studies used a slab ocean coupled to an atmospheric general circulation model to estimate ECS, but such a setup is not the same as the fully coupled system. More recent studies used a linear fit between changes in global-mean surface air temperature (ΔT) and top-of-atmosphere net radiation (ΔN) to estimate ECS from relatively short simulations. Here we analyze 1000 years of simulation with abrupt quadrupling (4 × CO2) and another 500-year simulation with doubling (2 × CO2) of pre-industrial CO2 using the CESM1 model, and three other multi-millennium (~5000 year) abrupt 4 × CO2 simulations to show that the linear-fit method considerably underestimates ECS due to the flattening of the −dN/dT slope, as noticed previously. We develop and evaluate three other methods, and propose a new method that makes use of the realized warming near the end of the simulations and applies the −dN/dT slope calculated from a best fit of the ΔT and ΔN data series to a simple two-layer model to estimate the unrealized warming. Using synthetic data and the long model simulations, we show that the new method consistently outperforms the linear-fit method with small biases in the estimated ECS using 4 × CO2 simulations with at least 180 years of simulation. The new method was applied to 4 × CO2 experiments from 20 CMIP5 and 19 CMIP6 models, and the resulting ECS estimates are about 10% higher on average and up to 25% higher for models with medium–high ECS (> 3 K) than those reported in the IPCC AR5. Our new estimates suggest an ECS range of about 1.78–5.45 K with a mean of 3.61 K among the CMIP5 models and about 1.85–6.25 K with a mean of 3.60 K for the CMIP6 models. Furthermore, stable ECS estimates require at least 240 (180) years of simulation for using 2 × CO2 (4 × CO2) experiments, and using shorter simulations may underestimate the ECS substantially. Our results also suggest that it is the forced −dN/dT slope after year 40, not the internally-generated −dN/dT slope, that is crucial for an accurate estimate of the ECS, and this forced slope may be fairly stable.

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Acknowledgements

We thank Bo Dong for helping set up some of the CESM1 simulations and making the control run used in this study, and Dr. Cao Li of MPI and Dr. David Paytner of GFDL for providing us their model data. We acknowledge the CMIP5 and CMIP6 modeling groups and NCAR CESM project, the Program for Climate Model Diagnosis and Intercomparison and the WCRP’s Working Group on Coupled Modelling for their roles in making available the WCRP CMIP multi-model datasets. A. Dai acknowledges the funding support from the U.S. National Science Foundation (Grant No. AGS–1353740 and OISE-1743738), the U.S. Department of Energy's Office of Science (Award No. DE–SC0012602), and the U.S. National Oceanic and Atmospheric Administration (Award nos. NA15OAR4310086 and NA18OAR4310425). B. Rose was supported by NSF (Grant no. AGS-1455071).

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Appendix: a note on the estimates of the slopes in noisy data

Appendix: a note on the estimates of the slopes in noisy data

For two correlated noisy time series xi and yi, we can use least-squares fitting to estimate the slopes in the following equations

$$y_{i} = a_{y} + b_{y} x_{i} + \varepsilon_{yi}$$
(5)
$$x_{i} = a_{x} + b_{x} y_{i} + \varepsilon_{xi}$$
(6)

as

$$b_{y} = r_{xy} \frac{{\sigma_{y} }}{{\sigma_{x} }},\quad {\text{and}}\quad b_{x} = r_{xy} \frac{{\sigma_{x} }}{{\sigma_{y} }}$$
(7)

where \(r_{xy} = r\left( {x,y} \right) = \frac{{cov\left( {x,y} \right)}}{{\sigma_{x} \sigma _{y} }}\) is the correlation coefficient between xi and yi, σx and σy are the standard deviation of xi and yi, respectively, and εyi and εxi are the residuals from the fitting and are considered as noise here. Thus, by <  bx if σy < σx, and by ≠ 1/bx if rxy ≠ 1.

For an exact relationship: y = a + b x, we have rxy = 1, σy = b σx, so that by = b, bx =1/b. Adding weakly correlated noise (with zero mean) to x and y to form two new variables: X = x + εx, and Y = y + εy with r(εx, εy) ≈ 0. Then, we have \(\sigma_{X}^{2} = \sigma_{x}^{2} + \sigma_{\varepsilon x}^{2} {\text{ and }} \sigma_{Y}^{2} = \sigma_{y}^{2} + \sigma_{\varepsilon y}^{2}\), and

$$r_{XY} = \frac{{cov\left( {X,Y} \right)}}{{\sigma_{X} \sigma_{Y} }} = \frac{{cov\left( {x,y} \right) + cov\left( {\varepsilon_{x} ,\varepsilon_{y} } \right)}}{{\sigma_{X} \sigma_{Y} }}\approx\frac{{cov\left( {x,y} \right)}}{{\sigma_{X} \sigma_{Y} }} = \frac{{cov\left( {x,y} \right)}}{{\sigma_{x} \sigma_{y} }} \frac{{\sigma_{x} \sigma_{y} }}{{\sigma_{X} \sigma_{Y} }} = r_{xy} \frac{{\sigma_{x} \sigma_{y} }}{{\sigma_{X} \sigma_{Y} }} = \frac{{\sigma_{x} \sigma_{y} }}{{\sigma_{X} \sigma_{Y} }}$$
(8)

Following (7), the slope between X (as the predictor) and Y (as the predictand) is

$$b_{Y} = r_{XY} \frac{{\sigma_{Y} }}{{\sigma_{X} }}\approx\frac{{\sigma_{x} \sigma_{y} }}{{\sigma_{X} \sigma_{Y} }} \frac{{\sigma_{Y} }}{{\sigma_{X} }} = \frac{{\sigma_{x} \sigma_{y} }}{{\sigma_{X}^{2} }} = \frac{{b \sigma _{x}^{2} }}{{\sigma_{x}^{2} + \sigma_{\varepsilon x}^{2} }} = \frac{b}{{1 + {\raise0.7ex\hbox{${\sigma_{\varepsilon x}^{2} }$} \!\mathord{\left/ {\vphantom {{\sigma_{\varepsilon x}^{2} } {\sigma_{x}^{2} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${\sigma_{x}^{2} }$}}}}.$$
(9)

Thus, \(\left| {b_{Y} } \right| < \left| b \right|\), and the difference between the estimated slope \(b_{Y}\) and the true slope b increases with the squared noise-to-signal ratio (\({\raise0.7ex\hbox{${\sigma_{\varepsilon x}^{2} }$} \!\mathord{\left/ {\vphantom {{\sigma_{\varepsilon x}^{2} } {\sigma_{x}^{2} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${\sigma_{x}^{2} }$}}\)). Therefore, one should avoid using the variable with large noise (e.g., N(t)) as the predictor in estimating the slope between two data series. For X = T(t) (i.e., the global-mean temperature change series) and Y =  = N(t) (i.e., the TOA net radiation change series), the estimated slope (−dN/dT) using least squares fitting should underestimate the true slope between the forced T and N changes (the signals) due to the existence of the noise induced by internal variability (Dai and Bloecker 2019). Since ECS = F/(−dN/dT), this underestimation should lead to an overestimation of ECS in Gregory et al. (2004)’s method. However, as stated in the main text of this paper, the use of the data from the first 40 years or so greatly increase the magnitude of the slope (−dN/dT), whose effect dominates over the effect of noise and leads an underestimation of ECS by the Gregory’s method.

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Dai, A., Huang, D., Rose, B.E.J. et al. Improved methods for estimating equilibrium climate sensitivity from transient warming simulations. Clim Dyn 54, 4515–4543 (2020). https://doi.org/10.1007/s00382-020-05242-1

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Keywords

  • Climate sensitivity
  • Equilibrium response
  • Climate feedback
  • Climate models
  • CMIP5
  • Global warming