How sensitive is climate sensitivity?



[1] Estimates of climate sensitivity are typically characterized by highly asymmetric probability density functions (pdfs). The reasons are well known, but the situation leaves open an uncomfortably large possibility that climate sensitivity might exceed 4.5°C. In the contexts of (1) global-mean observations of the Earth's energy budget and (2) a global-mean feedback analysis, we explore what changes in the pdfs of the observations or feedbacks used to estimate climate sensitivity would be needed to remove the asymmetry, or to substantially reduce it, and demonstrate that such changes would be implausibly large. The nonlinearity of climate feedbacks is calculated from a range of studies and is shown also to have very little impact on the asymmetry. The intrinsic relationship between uncertainties in the observed climate forcing and the climate's radiative response to that forcing (i.e., the feedbacks) is emphasized. We also demonstrate that because the pdf of climate forcing is approximately symmetric, there is a strong expectation that the pdf of climate feedbacks should be symmetric as well.

1. Introduction

[2] Climate sensitivity (≡ T), the equilibrium response of global-mean, annual-mean, near-surface air temperature to a doubling of carbon dioxide above preindustrial concentrations, is a conceptually convenient metric for comparing different methods of estimating climate change. However, both the observations from which T is estimated and the climate simulations from which T is derived are uncertain, so that we cannot establish a single value but only its probability density function (pdf), image Both observations and simulations yield highly skewed pdfs, with finite probabilities of large sensitivities [e.g., Knutti and Hegerl, 2008].

[3] Because the large asymmetry of image has been questioned [e.g., Hannart et al., 2009; Zaliapin and Ghil, 2010; Solomon et al., 2011, section 3.2], it is appropriate to revisit the underlying assumptions on which its derivation rests. First, image must be consistent with observations, so we analyze what modifications of those observations would lead to a significantly more symmetric pdf. Secondly, we examine the effect of relaxing assumptions underlying the simple model of Roe and Baker [2007, hereafter RB07], who derived an asymmetric image from the pdf of the total feedback factor f.

2. Estimates of Climate Sensitivity From Observations

[4] A linearization of Earth's annual-mean, global-mean energy budget is H = Rλ−1T, where H is ocean storage, R is radiative forcing, and λ−1T is the climate response in terms of the global-mean, annual-mean, near-surface air temperature change, T, and the climate sensitivity parameter, λ [e.g., Gregory et al., 2002]. Let R be the forcing due to a doubling of CO2 over pre-industrial values (≃3.7 Wm−2). Computation of image can be made purely from observations of the modern state via the relationship:

equation image

since H is zero in equilibrium. Simplifying notation, let Fobs = RobsHobs. Pdfs of these quantities are related by:

equation image

where hFobs and hTobs are the pdfs of the observations. Both are found to be nearly normal distributions [e.g., Forster et al., 2007, Figure 2.20], given by

equation image

and image = ϕ(Fobs, equation imageobs, σF). Various estimates of Fobs and Tobs have been made. We use values from Armour and Roe [2011, hereafter AR11] of equation imageobs ± σF = 0.90 ± 0.55 Wm−2, and equation imageobs ± σT = 0.76 ± 0.11°C, which are the same as Forster et al. [2007] and Trenberth et al. [2007], but updated with new ocean storage observations [Hobs ± σH = 0.74 ± 0.08 Wm−2, Lyman et al., 2010; Purkey and Johnson, 2010] (see auxiliary material). We assume independent errors.

[5] Thus, from equation (2) and the aforementioned uncertainties, the skewed nature of image estimated from global-mean observations (Figure 1) is an inevitable result of the fractional uncertainty in Fobs being much larger than the fractional uncertainty in Tobs [e.g., Gregory et al., 2002]: the skewed tail towards high climate sensitivity is because the observations allow for the possibility that the relatively-well constrained observed warming might have occurred with little or no net climate forcing. Allen et al. [2006] present several other estimates for various time periods: in all cases, observations and reconstructions are more constrained for temperature than forcing.

Figure 1.

Pdfs of T computed from perturbed physics ensembles [Sanderson et al., 2008], model-estimated climate feedbacks (RB07), and modern instrumental observations (AR11). A histogram of T from IPCC AR4 [Solomon et al., 2007] models is also shown. The highest T in the ensemble is 12.0°C, and the (T05, T50, T95) quantiles for the pdfs based on feedbacks and observations are, in °C, (2.1, 3.4, 8.6) and (1.5, 3.0, 12.1), respectively. All pdfs are normalized between 0 and ∞.

3. Can Observation-Based image Be Unskewed?

[6] How different would the aforementioned assumptions have to be in order to significantly reduce the asymmetry of image As a metric for the symmetry of the sensitivity pdfs, we define

equation image

where Tx is xth quantile. S is the interquantile skewness [e.g., Hinkley, 1975], mapped onto the range 0 to ∞. The more common moment skewness (i.e., E[(xμ)/σx]3), is infinite for the form of equations (2) and (3). The use of 90% bounds is prevalent in other studies of T. A symmetric distribution has S = 1, whereas for image based on observations, S = 6.0. We now focus on image because it matters much more than hTobs. Let image now be represented by the so-called ‘skew normal’ distribution:

equation image

For αF = 0 this is the normal distribution given by equation (3); for αF ≠ 0 the skewness of image has the same sign as that of αF.

[7] The parameters necessary to achieve S ≈ 1 are given in Table 1, and the corresponding pdfs are shown in Figures 2a and 2c. It is obvious that to remove the skewness completely would require a drastically different image (e.g., a 100-fold reduction in σF). We thus conclude that, without exceedingly large reductions in forcing uncertainty, or compelling arguments why image has to be highly asymmetric, some skewness is inevitable in image For the rest of the paper, we ask whether that skewness might perhaps be, if not completely removed (i.e., S = 1), then moderated substantially, and pick S = 2 as our measure. Table 1 shows this requires an approximate halving of σF, a five-fold increase in equation imageobs, or an αF ≃ 2.0. The accompanying distributions are shown in Figures 2b and 2d. Table 1 gives guidance to the search for lower S by means of new observations.

Figure 2.

The effect of (top) altered pdfs of radiative forcing observations on (bottom) the asymmetry of image The thick grey curve shows current uncertainties (AR11, αF = 0, S = 6.0) for comparison. (a and c) S ≃ 1. (b and d) S = 2. The pdfs are normalized between 0 and ∞.

Table 1. Variations in pdf of Forcing, image and the Impact on the Asymmetry, S, of imagea
equation imageσFαFS
  • a

    The first line are the standard combination of parameters for image in equation (5), and subsequent lines show the changes in parameters necessary to obtain the given value of the asymmetry parameter, S. In each case only a single parameter has been altered (shown underlined).


4. Estimates of Climate Sensitivity From Models

[8] Climate sensitivity may also be estimated by diagnosing feedbacks within climate models. Let f be the linear sum of individual climate feedbacks, f ≡ Σifi. There then is a one-to-one correspondence between values of this total feedback factor, f, and T [e.g., Roe, 2009]. Thus the pdf of T can be calculated from hf, the pdf of f. To derive estimates of image RB07 further assumed: 1) hf is Gaussian:

equation image

and 2) feedbacks are independent of temperature, which led to the relationship between sensitivity T and f:

equation image

where λ0 = 0.3, T0 = λ0R ≈ 1.2°C. Assumptions (6) and (7) yield an asymmetric image For current best estimates σf = 0.13, equation image = 0.65 the resulting pdf has S = 4.0.

[9] Given these assumptions, the skewed nature of image is an inevitable result of the asymmetric amplification by the feedback response on the high side of the mode of hf. This amplification serves to underscore the magnitude of the challenge of refining model-based estimates of the high side of image It requires a high degree of confidence in the shape of the high side of hf and, moreover, how that shape changes with mean climate state.

[10] In previous work [RB07; Roe and Baker, 2011, hereafter RB11], we have shown that a model based on equations (6) and (7) is supported by its ability to reproduce the multi-thousand member ensemble results of; by observational studies that find an approximately Gaussian distribution to the total feedback factor [e.g., Allen et al., 2006]; and by the fact that for a system of many feedbacks, the Central Limit Theorem would suggest that the distribution of hf would converge on a Gaussian.

[11] Despite these successes of the model, assumptions (6) and (7) have been questioned. Hannart et al. [2009, hereafter HDN09] take issue with the RB07 result that it is hard to reduce the likelihood that T is higher than the IPCC ‘likely range’ (i.e., >4.5°C) by reducing uncertainty in climate parameters, or equivalently in observations [Allen et al., 2006]. They point out that equation (6) allows the possibility that f ≥ 1, which they argue is an indictment of the model. The applicability of equation (7) has also been questioned by HDN09, and Zaliapin and Ghil [2010]. It is therefore appropriate to examine the effect of relaxing assumptions (6) and (7) on the symmetry parameter S.

4.1. Can Model-Based image Be Unskewed?

[12] We consider the following set of analyses, taken one at a time:

[13] - Vary equation image, σf, keeping relationships (6) and (7). We extend the arguments of RB07 here.

[14] - Let the pdf of feedbacks be asymmetric: hf = Ψsn(f, equation image, σf, αf): in order to decrease the asymmetry in image αf must be negative.

[15] - Let the feedbacks be nonlinear: f(T) = f0 − 20T, where f0 is independent of temperature, and the constant a must be positive to reduce the asymmetry of image

[16] In our opinion, Table 2 shows that it is virtually impossible to achieve S ≈ 1 by any single parameter change in the RB07 model: it requires either a 104-fold reduction in σf, or a highly skewed hf with αf ≤ − 5. The lowest value of S achievable for non-negative equation image is 1.2. Table 2 also shows single parameter variations in the model that result in S ≈ 2.0. The corresponding hfs and image are shown in Figure 3, as well as RB07's model for comparison.

Figure 3.

(a) The effect on image (y-axis) of varying the parameters controlling the shape of hf (x-axis). Parameters correspond to those given in Table 2 for S = 2. Solid line shows RB07's model. (b) The effect of feedback nonlinearity parameter, a, on image The grey lines show the fT relationships. See auxiliary material for calculations of a from previous model studies. The pdfs of image are normalized between 0 and ∞.

Table 2. Variation of Feedback Model Parameters and the Impact on S
σfequation imageαfaS

4.2. Nonlinear Feedbacks

[17] Allowing for nonlinearities (see RB11, and auxiliary material), equation (7) is replaced by

equation image

The auxiliary material derives the value of a from a large number of published studies. We find a ≤ 0.06, from which S ≥ 2.8. To achieve S ≈ 1 requires a to be 20 times greater (Table 2). Figure 3b shows the image implied by equation (8) after adjusting f0 so all curves pass through f = 0.65, T = 3.5°C, the best linear estimate for today's climate (see auxiliary material). For a = 0.11, S = 2 and the high sensitivity tail (T ≳ 8°C) is eliminated, while at lower values of a, image is virtually identical to the linear model.

5. Why Are Observation-Based and Model-Based Estimates of image So Similar?

[18] A striking feature of Figure 1 is that observation-based and model-based estimates of climate sensitivity are very similar. If they differed wildly, it might perhaps imply that there was important unused information, or that there were troubling biases among different methods. Another reason for their similarity is also worth emphasizing. From equation (1) and the fact λ = λ0/(1 − Σifi), we can write

equation image

λ0 is known, H and T are well constrained in the current climate (i.e., Section 2), and estimating λ is the goal. Term (i) on left-hand side of equation (9) reflects the principal source of uncertainty in global-mean energetics (the radiative forcing of aerosols), and term (ii) on the right-hand side reflects the uncertainty in climate feedbacks. Equation (9) shows that these two approaches are equivalent to each other. Therefore, because hR is broad (relative to hH and hT) and nearly symmetric [e.g., Forster et al., 2007] hf should be too. Moreover, to the extent that ensembles of models (1) adequately simulate T and H and (2) faithfully represent the observed forcing uncertainties, the modeled hf must behave similarly. As noted previously [e.g., Knutti, 2008], the AR4 ensemble undersamples observed hR, implying an undersampling of hf, and consequently of image (Figure 1).

6. Discussion

[19] We have used the frameworks of global-mean energy budget observations and global-mean feedback analysis to explore how asymmetry in image might be reduced. While we have only varied the parameters one at a time, we've shown that the asymmetry cannot be eliminated by any realistic change to the parameters of either the observed uncertainty distribution or RB07's model (see auxiliary material for multiple parameter changes). We have also shown that estimates of image based on global energetics and estimates based on feedbacks are intrinsically linked. Therefore HDN09, for example, overreach in asserting that the analysis of RB07 is “a mathematical artifact with no connection whatsoever to climate”. Moreover it is critical for future climate projections to appreciate that uncertainties in forcing are not independent of uncertainties in λ, though this is sometimes overlooked [e.g., Ramanathan and Feng, 2008; Hare and Meinhausen, 2006].

[20] Our results add to a body of work demonstrating that, if further substantial progress is to be made on constraining the fat tail of image it will likely only come from combining multiple estimates of image or going beyond the global mean to spatio-temporal comparisons of models and observations. Bayesian approaches that combine multiple estimates can in principle lead to narrower and less skewed distributions [Annan and Hargreaves, 2006], though there are some formidable challenges to objectively establishing the independence and relative quality of the different estimates [e.g., Lemoine, 2010; Henriksson et al., 2010]. Hegerl and Knutti [2008] review the many studies that constrain image by minimizing disagreement between observations and models. Such estimates of image as well as those based on model-diagnosed feedbacks are typically somewhat narrower than that permitted by modern observations (Figure 1), and are narrower still when including apparent correlations among feedbacks [Huybers, 2010]. Confidence in these model-based estimates depends on whether models fully sample the observed uncertainties, and whether they adequately represent the relationship between other aspects of the climate system and the global-scale energetics with sufficient skill [e.g., Knutti et al., 2010]. Finally, a practical measure of the acceptance of any of these narrower estimates is whether they become formally used as constraints to narrow uncertainties in current climate forcing (AR11).

[21] Ominous consequences have been thought to follow from the skewness of image [e.g., Weitzman, 2009]. The argument has been made that we should focus our efforts on decreasing the probabilities of high T by making more accurate observations. Our results provide clear targets in terms of improved observations or more certainty among models. However, this focus is to some extent misplaced. Firstly, because, as shown by RB07 and the present analysis, it would take large decreases in observed or modeled uncertainties to have much of an impact. Moreover, a reduction of uncertainty in Fobs or f moves the mode of image to higher values (e.g., Figures 1 and 2). So, as noted by RB07, while the probabilities become more focussed, in other words the range—however measured—gets less, the cumulative likelihood beyond 4.5°C remains stubbornly persistent. Secondly, and more fundamentally, T is only a metric of a hypothetical global mean temperature rise that might occur thousands of years into the future. Very high temperature responses, if they develop, are associated with the very longest time scales [e.g., Baker and Roe, 2009]. On the other hand, in this century we face the very real threat of climate changes that will have very damaging impacts on life and society [e.g., Intergovernmental Panel on Climate Change, 2007]. While understanding the basic relationship between radiative forcing, climate feedbacks and climate sensitivity is important, arguments about the details of the pdf shape are not.


[22] The authors thank M. Baker for formative guidance, two reviewers for constructive comments that improved the paper, and Noah Diffenbaugh, the editor.

[23] The Editor thanks the two anonymous reviewers for their assistance in evaluating this paper.