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.

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

For two correlated noisy time series x_i and y_i, 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 x_i and y_i, σ_x and σ_y are the standard deviation of x_i and y_i, respectively, and ε_yi and ε_xi are the residuals from the fitting and are considered as noise here. Thus, b_y <  b_x if σ_y < σ_x, and b_y ≠ 1/b_x if r_xy ≠ 1.

For an exact relationship: y = a + b x, we have r_xy = 1, σ_y = b σ_x, so that b_y = b, b_x =1/b. Adding weakly correlated noise (with zero mean) to x and y to form two new variables: X = x + ε_x, and Y = y + ε_y 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.