## 1. Introduction

The equilibrium climate sensitivity is the conventional measure of the equilibrium climate response to radiative forcing resulting from greenhouse gases and other anthropogenic and natural causes. It is defined as the steady-state change in global-average surface temperature due to a doubling of the carbon dioxide concentration, and is estimated to lie between 1.5 and 4.5 K (Cubasch et al. 2001), largely on the basis of experiments with general circulation models (GCMs). This wide range was informally obtained from the model results, and does not correspond to any particular probability limits. Despite considerable improvements in many aspects of the simulation of twentieth-century climate by GCMs, the range has remained essentially unchanged during the last two decades, and is the greatest source of uncertainty in climate change projections for the twenty-first century.

*T.*During time-dependent climate change, the imbalance between the imposed radiative forcing

*Q*and the radiative response

*λ*Δ

*T,*

*λ*being a constant, is absorbed by the heat capacity of the system, which resides overwhelmingly in the ocean (Levitus et al. 2001). Hence,

*F*

*t*

*Q*

*t*

*λ*

*T*

*t*

*t*is time and

*F*is the heat flux into the ocean. Equation (1) has often been employed as the basis for energy-balance climate models.

In the unperturbed steady-state climate, *Q* = *F* = 0 and Δ*T* = 0. If *Q* is raised from zero to some positive value, *F* becomes positive, additional heat is stored in the ocean, and Δ*T* rises. If *Q* then remains constant, *F* returns to zero over time, as the climate approaches a new steady state in which Δ*T* = *Q*/*λ.* From its definition, the equilibrium climate sensitivity Δ*T*_{2×} = *Q*_{2×}/*λ,* where *Q*_{2×} is the forcing that results from a doubling of the CO_{2} concentration.

Although it is defined in terms of a steady-state climate, the climate sensitivity can be estimated from any climate state. Provided we know *F,* *Q,* and Δ*T,* we can calculate *λ* from Eq. (1) and, hence, Δ*T*_{2×} (e.g., Cubasch et al. 2001). Some results with coupled atmosphere–ocean GCMs (AOGCMs) suggest that Δ*T*_{2×} (called the “effective” climate sensitivity when calculated from an unsteady climate) might not be constant even on the century timescale (Senior and Mitchell 2000), although AOGCM experiments do not give rise to any expectation that it will change rapidly. If Δ*T*_{2×} is not constant, its usefulness for predicting future climate change is of course limited, and an estimate based on recent climate change is the most appropriate one to use. The utility of the climate sensitivity also depends on the response being independent of the nature of the agent causing the radiative forcing.

## 2. Method

Recent studies aimed at setting constraints on the climate sensitivity have used climate models in which *λ* can be varied and heat uptake by the ocean is modeled simply (Wigley et al. 1997; Andronova and Schlesinger 2001; Forest et al. 2002). The approach is systematically to adjust the parameters and inputs of the model, comparing the simulated results with observed surface temperature changes. The results give a range for Δ*T*_{2×} that is even wider than 1.5°–4.5°C.

^{22}J. The stated uncertainty, of two standard deviations, relates to measurement and sampling uncertainties. Denoting the time average for 1957–94 by an overbar, Eq. (1) becomes

*F*

^{−2}(expressed per unit area of the entire world, not just the ocean surface).

Here Δ*T* is defined with respect to the steady-state climate for zero forcing. Sufficient measurements exist to estimate global-average temperature changes back to 1860, but the climate of that period was not a steady state, not least because anthropogenic greenhouse gases began to increase in the latter part of the eighteenth century. In fact, there has probably never been a steady-state climate, because solar output fluctuations and volcanism produce continual variations in radiative forcing on a shorter timescale than that required for the climate system to reach equilibrium.

*T*

*Q*

*F*

*T*

No observations exist of past changes in radiative forcing, so this quantity must be estimated. We take into account the effects of greenhouse gases (carbon dioxide, methane, nitrous oxide, halocarbons, and tropospheric ozone), anthropogenic sulfate aerosols, solar variation, and volcanic aerosols (Table 1). Greenhouse gas and sulfate aerosol forcing are dominant and of opposite signs. The former is calculated using historical concentrations of the gases and formulas for radiative forcing; there is estimated to be a range of uncertainty of ±10% on the results (Ramaswamy et al. 2001).

The effect of sulfate aerosol is much less precisely known. The patterns of temperature change are sensitive to aerosol forcing. We derive limits for the forcing (Table 1) by comparison of the spatiotemporal patterns of temperature change in observations and experiments with the Hadley Centre AOGCM (HadCM3; Stott et al. 2000). The method (see Allen et al. 2002) assumes that the patterns simulated by the AOGCM are realistic, but does not depend on the model's forcing or climate sensitivity.

Solar output is thought to have increased in the early twentieth century, giving a positive contribution to *Q**Q*^{−2} (Andronova et al. 1999), or −0.4 W m^{−2} (Crowley 2000).

We obtain a ±2*σ* interval for forcing change *Q*^{−2}, by combining the ranges of the various terms (Table 1), assuming the individual ranges to be normal ±2*σ* intervals. We make this assumption on pragmatic grounds, as we lack knowledge of the probability density function of any of the terms, although we note that a tendency toward normality as one adds more terms is consistent with the central limit theorem.

To complete the calculation, information is needed about the average heat flux into the ocean during 1861–1900, in order to calculate *F**F* with a one-dimensional upwelling-diffusion ocean model], as tuned by Cubasch et al. (2001) to reproduce results from a range of AOGCMs. In simulations with anthropogenic forcing alone, starting at the onset of substantial industrial emissions of greenhouse gases in the late eighteenth century, *F*^{−2}. In simulations with natural forcings alone (volcanoes and solar variation; Crowley 2000), starting in the year 1000, *F*^{−2}. It is positive because the climate is recovering from substantial negative forcing in preceding decades. Making the usual assumption that forcings can be combined linearly, these results suggest an estimate for *F*^{−2}, whereas a steady state would have *F**σ* interval and calculating the difference from *F**σ* interval for *F*^{−2}.

## 3. Results

We calculate *λ* from Eq. (3) as a function of *T**F**Q**T*_{2×} using *Q*_{2×} = 3.71 W m^{−2} (Myhre et al. 1998; Fig. 1). We compute the probability distribution of resulting values (Fig. 2), assuming *T**F**Q**Q*_{2×} (Myhre et al. 1998), which is negligible by comparison. The effect of internal (unforced) variability of the climate system on *F**Q**T*_{2×} we obtain a 90% confidence interval, whose lower bound (the 5th percentile) is 1.6 K. The median is 6.1 K, above the canonical range of 1.5–4.5 K; the mode is 2.1 K.

A positive *F**F**T*_{2×} increases to 2.0 K. On the other hand, if the climate system were assumed always to be in steady state, that is, *F**T*_{2×} would be 1.3 K. Use of a low-diffusivity ocean model might underestimate heat uptake, thus giving smaller Δ*T*_{2×}.

The 90% confidence interval for Δ*T*_{2×} extends up to infinity, and beyond to negative values (cf. Fig. 1). Here Δ*T*_{2×} < 0 if *Q**F**T*_{2×} is unphysical, because it implies that the unforced climate system would be unstable to any perturbation generated by internal variability. We infer that *Q**F**Q**F**T*_{2×} is extremely large. Such values can be excluded by paleoclimatic studies, which show that the climate sensitivity of the real world is of roughly the size indicated by GCMs (e.g., Hoffert and Covey 1992), but do not constrain it more tightly.

The dominant uncertainty in the calculation of climate sensitivity is clearly that pertaining to the estimates of radiative forcing, especially the aerosol forcing (cf. Forest et al. 2002; Knutti et al. 2002; Allen et al. 2002). While representing the state of current knowledge, the radiative forcing estimates we have employed are imprecise and undoubtedly incomplete in some respects. Some known negative radiative forcings have been omitted (stratospheric ozone depletion, aerosol from biomass burning, albedo change from land use change; Ramaswamy et al. 2001), whose inclusion would tend to raise the lower bound of Δ*T*_{2×}. Mineral dust and black carbon aerosol, also omitted, could give positive forcing (Ramaswamy et al. 2001). If we make an informal allowance for the possibility of substantial additional positive forcing by raising the upper bound of the sulfate aerosol forcing to zero, following Andronova and Schlesinger (2001), the 5th percentile of the climate sensitivity falls to 1.1 K. Although the HadCM3 simulations from which the sulfate aerosol forcing was derived did not include nonsulfate anthropogenic aerosols, these may have a somewhat similar geographical distribution to that of sulfate aerosols. To the extent that this is so, the sulfate aerosol forcing resulting from the method includes them as well; otherwise, their omission will be reflected in a greater forcing uncertainty.

We consider that the lower bound is an important constraint on climate sensitivity, because it is objectively derived, and independent of GCM results for Δ*T*_{2×}. Although the lower bound does not lead us to reject any of the AOGCMs used by Cubasch et al. (2001) in projections for the twenty-first century, it does exclude substantially smaller values. Improved understanding of physical processes of climate change and refinement of climate models is essential to reducing uncertainty in climate prediction. However, reducing the uncertainty on the inputs to the method described here offers an alternative route to obtaining better constraints on climate sensitivity. For example, with *Q*^{−2} and Δ*T*_{2×} = 2.0 K, if the ranges of uncertainty on *Q**F**T*_{2×} from this method would be 1.7–2.3 K. A range as narrow as that would be a great improvement on the current state of knowledge.

## Acknowledgments

We are grateful to Myles Allen, Tom Crowley, John Mitchell, Tom Wigley, Michael Schlesinger, John Antonov, Syd Levitus, and the anonymous reviewers for discussions, comments, and other help. Work at the Hadley Centre was supported by the U.K. Department of the Environment, Food and Rural Affairs under Contract PECD 7/12/37 and by the Government Meteorological Research contract. Sarah Raper was supported at AWI by HGF Strategiefonds Projekt 2000/13 SEAL.

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Probability distribution for the effective climate sensitivity Δ*T*_{2×}, computed assuming *T**F**Q**T*_{2×}, and the vertical dotted line the median. Although the distribution is shown here only up to 10 K, the probability of larger values was accounted for in deriving the statistics and confidence interval

Citation: Journal of Climate 15, 22; 10.1175/1520-0442(2002)015<3117:AOBEOT>2.0.CO;2

Probability distribution for the effective climate sensitivity Δ*T*_{2×}, computed assuming *T**F**Q**T*_{2×}, and the vertical dotted line the median. Although the distribution is shown here only up to 10 K, the probability of larger values was accounted for in deriving the statistics and confidence interval

Citation: Journal of Climate 15, 22; 10.1175/1520-0442(2002)015<3117:AOBEOT>2.0.CO;2

Probability distribution for the effective climate sensitivity Δ*T*_{2×}, computed assuming *T**F**Q**T*_{2×}, and the vertical dotted line the median. Although the distribution is shown here only up to 10 K, the probability of larger values was accounted for in deriving the statistics and confidence interval

Citation: Journal of Climate 15, 22; 10.1175/1520-0442(2002)015<3117:AOBEOT>2.0.CO;2

Radiative forcing difference *Q*′ (W m^{−2}) between the periods 1957–94 and 1861–1900