## Abstract

Uncertainties in projections of future climate change have not lessened substantially in past decades. Both models and observations yield broad probability distributions for long-term increases in global mean temperature expected from the doubling of atmospheric carbon dioxide, with small but finite probabilities of very large increases. We show that the shape of these probability distributions is an inevitable and general consequence of the nature of the climate system, and we derive a simple analytic form for the shape that fits recent published distributions very well. We show that the breadth of the distribution and, in particular, the probability of large temperature increases are relatively insensitive to decreases in uncertainties associated with the underlying climate processes.

The envelope of uncertainty in climate projections has not narrowed appreciably over the past 30 years, despite tremendous increases in computing power, in observations, and in the number of scientists studying the problem (*1*). This suggests that efforts to reduce uncertainty in climate projections have been impeded either by fundamental gaps in our understanding of the climate system or by some feature (which itself might be well understood) of the system's underlying nature. The resolution of this dilemma has important implications for climate research and policy.

We investigate a standard metric of climate change: Climate sensitivity is defined as the equilibrium change in global and annual mean surface air temperature, Δ*T*, due to an increment in downward radiative flux, Δ*R*_{f}, that would result from sustained doubling of atmospheric CO_{2} over its preindustrial value (2 × CO_{2}). It is a particularly relevant metric for current discussions of industrial emissions scenarios leading to the stabilization of CO_{2} levels above preindustrial values (*2*). Studies based on observations, energy balance models, temperature reconstructions, and global climate models (GCMs) (*3*–*13*) have found that the probability density distribution of Δ*T* is peaked in the range 2.0°C ≤ Δ*T* ≤ 4.5°C, with a long tail of small but finite probabilities of very large temperature increases. It is important to ask what determines this shape and, in particular, the high Δ*T* tail, and to what extent we can decrease the distribution width.

Climate consists of a set of highly coupled, tightly interacting physical processes. Understanding these physical processes is a massive task that will always be subject to uncertainty. How do the uncertainties in the physical processes translate into an uncertainty in climate sensitivity? Explanations for the range of predictions of Δ*T*, summarized in (*14*), have focused on (i) uncertainties in our understanding of the individual physical processes (in particular, those associated with clouds), (ii) complex interactions among the individual processes, and (iii) the chaotic, turbulent nature of the climate system, which may give rise to thresholds, bifurcations, and other discontinuities, and which remains poorly understood on a theoretical level. We show here that the explanation is far more fundamental than any of these.

We use the framework of feedback analysis (*15*) to examine the relationship between the uncertainties in the individual physical processes and the ensuing shape of the probability distribution of Δ*T*. Because we are considering an equilibrium temperature rise, we consider only time-independent processes.

Let Δ*T* = λΔ*R*_{f}, where λ is a constant. In the absence of feedback processes, climate models show λ ≡ λ_{0} = 0.30 to 0.31 [K/(W/m^{2})] (where λ_{0} is the reference climate sensitivity) (*16*), giving an equilibrium increase Δ*T*_{0} ≈ 1.2°C in response to sustained 2 × CO_{2}. Because of atmospheric processes, however, the climate sensitivity has a value Δ*T* ≠ Δ*T*_{0}. Conceptually, the forcing Δ*R*_{f} produces a temperature change Δ*T*, which induces changes in the underlying processes. These changes modify the effective forcing, which, in turn, modifies Δ*T*. We assume that the total change in forcing resulting from these changes is a constant *C* times Δ*T*. Thus, Δ*T* = λ_{0} (Δ*R*_{f} + *C*Δ*T*), or $$mathtex$$\[\ \frac{{\Delta}T}{{\Delta}R_{f}}{\equiv}{\lambda}=\frac{{\lambda}_{0}}{1-f}\ \]$$mathtex$$(1) Here, the total feedback factor *f* ≡ λ_{0} *C* (*15*). Clearly, the gain *G* ≡ Δ*T*/Δ*T*_{0} >1 if *f* > 0, which appears to be the case for the climate system. The range 2°C ≤ Δ*T* ≤ 4.5°C corresponds to 1.7 ≤ *G* ≤ 3.7 and 0.41 ≤ *f* ≤ 0.73. Under our definitions, the feedback factors for individual processes are linearly additive, but the temperature changes, or gains, from individual processes are not [see the supporting online material (SOM)].

The uncertainties in measurements and in model parameterizations can be represented as uncertainties in *f*. Let the average value of *f* be *f̄* and let its SD be σ_{f}, the sum of uncertainties from all the component feedback processes. σ_{f} can be interpreted in three ways: uncertainty in understanding physical processes, uncertainty in observations used to evaluate *f*, and, lastly, inherent variability in the strengths of the major feedbacks. If σ_{f} is fairly small, we see from Eq. 1 that the uncertainty in the gain, δ*G*, is $$mathtex$$\[\ {\delta}G{\approx}\frac{1}{(1-f)^{2}}{\sigma}_{f}{\equiv}({\bar{G}})^{2}{\sigma}_{f}\ \]$$mathtex$$(2) Thus for *Ḡ* ≈ 3 (corresponding to $$mathtex$$\(\overline{{\Delta}T}\)$$mathtex$$ ≈ 3.6°C), uncertainties in feedbacks are magnified by almost an order of magnitude in their effect on the uncertainties in the gain. A second point is that even if σ_{f} is not large, δ*G* will be large if *f̄* approaches 1: Uncertainty is inherent in a system where the net feedbacks are substantially positive.

Finally, Eq. 2 shows that it is the sum of all the uncertainties in the feedbacks that determines δ*G*; the uncertainties in the large positive feedbacks are not more important than the others. For example, a compilation of values of the feedback factors extracted from several GCMs (*17*) finds considerable intermodel scatter in the albedo feedback, and although the average magnitude of this feedback is not high, this scatter has an important impact on the uncertainty in the total climate sensitivity.

We now derive the shape of the distribution *h*_{T}(Δ*T*): the probability density that the climate sensitivity is Δ*T*. The important features of this distribution are the location of its peak and the shape and extent of the distribution at large Δ*T*. We focus on the relationships between these features and the parameters of the feedback distribution, *f̄* and σ_{f}.

Figure 1 is a schematic picture of the relationships linking *h*_{T}(Δ*T*) to *h*_{f}(*f*), the probability distribution of *f*. The reason for the long tail of typical climate sensitivity distributions is immediately evident [see also (*3*)]. Uncertainties in climate processes, and hence feedbacks, have a very asymmetric projection onto the climate sensitivity. As the peak in the *h*_{f}(*f*) distribution moves toward *f* = 1, the probability of large Δ*T* also grows. The basic shape of *h*_{T} (Δ*T*) is not an artifact of the analyses or choice of model parameters. It is an inevitable consequence of a system in which the net feedbacks are substantially positive.

Formally, *h*_{T}(Δ*T*) is related to *h*_{f}(*f*) by the relationship $$mathtex$$\(h_{T}({\Delta}T)=h_{f}(f({\Delta}T))(df{/}d{\Delta}T)={\Delta}T_{0}{/}({\Delta}T)^{2}h_{f}(1-\frac{{\Delta}T_{0}}{{\Delta}T})\)$$mathtex$$. As is commonplace, we assume the errors in the feedback factors are normally distributed: $$mathtex$$\(h_{f}(f)=(\frac{1}{{\sigma}_{f}\sqrt{2{\pi}}})exp[-\frac{1}{2}(\frac{(f-{\bar{f}})}{{\sigma}_{f}})^{2}]\)$$mathtex$$. Although the general features of our results do not depend on this assumption, it facilitates our analysis. Then, $$mathtex$$\[\ h_{T}({\Delta}T)=\left(\frac{1}{{\sigma}_{f}\sqrt{2{\pi}}}\right)\frac{{\Delta}T_{0}}{{\Delta}T^{2}}\ \]$$mathtex$$(3) $$mathtex$$\[\ {\times}exp\left[-\frac{1}{2}\left(\frac{(1-{\bar{f}}-\frac{{\Delta}T_{0}}{{\Delta}T})}{{\sigma}_{f}}\right)^{2}\right]\ \]$$mathtex$$ Equation 3 shows how uncertainties in feedbacks lead to uncertainty in the response of a system of linear feedbacks. It can be shown that it is algebraically equivalent to a Bayesian derivation of a “posterior” distribution *h*(Δ*T*) based on a uniform previous distribution on feedbacks (SOM). As noted above, several studies have described climate sensitivity distributions similar in form to that indicated in Fig. 1 (*4*–*13*), but the particular power of Eq. 3 is that it provides a simple interpretation of the shape of these distributions. It is also a function that maps uncertainties in feedback processes onto uncertainties in climate sensitivity and therefore permits an analysis of its parametric dependencies. Figure 2 shows *h*_{T}(*T*) and *p*_{cum(ΔTc),} the cumulative probability that the climate sensitivity Δ*T* will exceed a given threshold, Δ*T*_{c,} for a range of values of *f̄* and σ_{f}. From Eq. 3, it can be shown that, for all σ_{f}, half the area under the curve occurs for Δ*T* < Δ*T*_{0.5.} Decreasing either σ_{f} or *f̄* concentrates the distribution around Δ*T* = Δ*T*_{0.5.}

The cumulative probability distributions show that decreasing σ_{f} or *f̄* steadily reduces the cumulative probability of large climate changes (e.g., Δ*T* ≥ 8°C). However, the probability that Δ*T* lies in the interval immediately outside the range of the Intergovernmental Panel on Climate Change (IPCC) (say, 4.5°C ≤ Δ*T* ≤ 8°C) is very insensitive to σ_{f} and *f̄* and changes little with Δ*T*_{c}. The cumulative probability distributions (Fig. 2, C and D) are driven by the extreme tail of the *h*_{T}(Δ*T*) distribution, which is a consequence of our choice of a Gaussian for *h*_{f}(*f*). Even if an *h*_{f}(*f*) without an extreme tail is assumed, the probability distributions in the interval beyond the IPCC range remain insensitive to changes in σ_{f} (SOM).

Thus, foreseeable improvements in the understanding of physical processes, and in the estimation of their effects from observations, will not yield large reductions in the envelope of climate sensitivity. This relative insensitivity of the probability distributions to σ_{f} is also a likely reason why uncertainty in climate sensitivity estimates has not diminished substantially in the past three decades.

We next compare *h*_{T} (Δ*T*) from Eq. 3 with selected published distributions of climate sensitivity. Figure 3 shows a distribution, determined from the multi-ensemble climateprediction.net experiment, for different representations of several cloud processes (*9*, *10*). Independent estimates of feedback parameters for two different suites of GCMs (*17*, *18*) determine values for (*f̄*, σ_{f}) of (0.70, 0.14) and (0.62, 0.13), respectively (*19*). We calculate the implied climate sensitivity distributions from Eq. 3. These match the numerically derived distributions of (*10*) quite well. We obtained a closer match by solving Eq. 3 for the *f̄* and σ_{f} that effectively characterize the feedback processes within the model used by (*9*, *10*) and used these parameters to generate the distribution.

Fits to several other published distributions (obtained by a wide variety of techniques), shown in Fig. 4, are also quite successful, with values 0.15 ≤ *f̄* ≤ 0.86 and 0.10 ≤ σ_{f} ≤ 0.35. Some differences are seen, especially in the tail of the distributions. This is to be expected because some studies explicitly analyze the nonGaussian distribution of uncertainties in the physics and various a priori assumptions. Nonetheless, all of these published distributions are, to a good approximation, consistent with propagation of physical-process uncertainties in a simple system of linear feedbacks.

The shape of *h*_{T} (Δ*T*), including its tail, is crucially dependent on the magnitude of *f̄*, which we have assumed is independent of Δ*T*. Is there any chance that, as warming continues, the probability of extreme values of Δ*T* will actually diminish? This might result if the feedback factors are functions of temperature. We have performed a preliminary analysis of the changes in the *h*_{T} (Δ*T*) distribution that would result from adding nonlinear terms in the Stefan-Boltzmann and water vapor feedbacks (SOM). This analysis shows that these second-order effects are equivalent to decreasing *f̄* by about 0.01 to 0.02, the small effect of which can be gauged from the curves in Fig. 2. To remove the skewness completely would require changes in feedback strength that are about 25 times as great (SOM). Several nonlinear interactions of this strength might, however, contribute an additional σ_{f} ∼0.1, which would change the particulars of a given probability distribution but not its fundamental characteristics. The identification and quantification of these nonlinear interactions are enormously harder tasks than the analysis of the linear feedbacks. It may be that a practical limit to the predictability of climate sensitivity should be anticipated.

It is tempting to speculate on what we can learn about the extreme tail of *h*_{T} (Δ*T*) from paleoclimate data. For instance, Eq. 3 can be extended to evaluate how uncertainties in reconstructions of past temperatures and net radiative forcing propagate to uncertainties in feedback strengths. Moreover, the data that we have on extreme climates [for example, the Eocene warmth and Proterozoic “snowball Earth” (*20*, *21*)] suggest that the climate system may have been acutely sensitive to radiative forcing during some intervals of Earth's history. Our results imply that dramatic changes in physical processes are not necessary for dramatic changes in climate sensitivity, provided that those changes in processes can all align in the same direction toward increased sensitivity. These are events of low but not zero probability.

Despite the enormous complexity of the climate system, the probability distribution of equilibrium climate sensitivity is well characterized by Eq. 3, which reflects the straight-forward, compounding effect of essentially linear feedbacks and depends on only the two parameters *f̄* and σ_{f}. We have shown that the uncertainty in the climate sensitivity in 2 × CO_{2} studies is a direct and general result of the fact that the sum of the underlying climate feedbacks is substantially positive. Our derivation of *h*_{T} (Δ*T*) did not depend on nonlinear, chaotic behavior of the climate system and was independent of details in cloud and other feedbacks. Equation 3 appears to explain the range of climate sensitivities reported in previous studies, which are well synthesized by the IPCC (*1*). Furthermore, reducing the uncertainty in individual climate processes has little effect in reducing the uncertainty in climate sensitivity. We do not therefore expect the range presented in the next IPCC report to be greatly different from that in the 2007 report. On the basis of the values of *f̄* and σ_{f} compiled from our analysis of a large number of published results, it is evident that the climate system is operating in a regime in which small uncertainties in feedbacks are highly amplified in the resulting climate sensitivity. We are constrained by the inevitable: the more likely a large warming is for a given forcing (i.e., the greater the positive feedbacks), the greater the uncertainty will be in the magnitude of that warming.

**Supporting Online Material**

www.sciencemag.org/cgi/content/full/318/5850/629/DC1

SOM Text

Fig. S1

References