## 1. Introduction

The spread of the equilibrium or transient surface temperature response to a CO_{2} doubling as predicted by atmosphere–ocean coupled models is still large (Meehl et al. 2007), and an open question is to identify the primary sources of this spread. Global warming estimates depend on radiative forcing, feedback processes that may either amplify or dampen the climate response and, in the transient case, ocean heat uptake. For individual models, it has been suggested that atmospheric processes were the most critical factors for estimating global temperature changes in transient simulations (e.g., Williams et al. 2001; Meehl et al. 2004; Collins et al. 2007). Here our purpose is to investigate whether these results extend to multimodel ensembles, and how much the various feedbacks and the ocean heat uptake contribute to the multimodel mean and spread of global warming estimates. The main radiative feedbacks are associated with changes in water vapor (WV), temperature lapse rate (LR), clouds, and surface albedo. The associated feedback parameters have been diagnosed for some multimodel ensembles (e.g., Colman 2003; Soden and Held 2006; Webb et al. 2006), but they have not been translated into temperature changes. This makes it difficult to compare the temperature change associated with each feedback with that from other processes, such as the ocean heat uptake.

In this paper we show that it is possible to decompose, and thus to compare, the contributions of the different climate feedbacks, and eventually of the ocean heat uptake, to the global temperature response to a specified forcing. After a brief presentation of the feedback analysis framework (section 2), the decomposition methodology is presented (section 3) and, after gathering the required data (feedback parameters, radiative forcing, and ocean heat uptake; section 4), this methodology is applied to an ensemble of models that participated in the World Climate Research Programme’s (WCRP’s) third phase of the Coupled Model Intercomparison Project (CMIP3) in support of the Intergovernmental Panel on Climate Change (IPCC) Fourth Assessment Report (AR4; section 5). There is very little in this paper that is entirely new. Rather, we propose a new presentation of existing results that allows us to quantify in a more straightforward way the relative contribution of different processes to intermodel differences in global mean temperature changes.

## 2. The feedback analysis framework

*F*

_{t}

^{o}= 0 of the global mean net flux at the top of the atmosphere (TOA) and a time average value

*T*

_{s}

^{o}of the global mean surface temperature. Let us impose on the climate system a radiative forcing, such as a change in the greenhouse gas concentration or in the TOA incoming solar radiation. In the absence of surface temperature change, this forcing translates into a radiative flux perturbation Δ

*Q*at the TOA, called radiative forcing. In response to this disequilibrium, the surface temperature changes. It appears that at any time, the anomalies Δ

_{t}*T*and Δ

_{s}*F*of the surface temperature and the TOA flux from their unperturbed initial steady state are approximately related through the following equation:where

_{t}*λ*is the “climate feedback parameter,” and the fluxes are positive downward. This relationship holds both for transient and equilibrium conditions. If the temperature changes until a new equilibrium is reached, the TOA net flux reaches its steady-state value (Δ

*F*= 0) and the equilibrium temperature change isThe total feedback parameter is commonly split as the sum of five terms,which are the Planck (

_{t}*P*), water vapor (

*w*), lapse-rate (

*L*), cloud (

*c*), and surface albedo (

*α*) feedback parameters. In this approach it is assumed that everything is linear [see, e.g., the appendix of Bony et al. (2006) for more details on this approach and for a discussion of the approximations].

*T*,

_{s}*that would be obtained if the temperature change was horizontally and vertically uniform and was only modifying the infrared emission through a change in the Planck function (e.g., Hansen et al. 1984),Because the total feedback parameter may be decomposed as*

_{P}*λ*=

*λ*

_{P}+ Σ

_{x≠P}

*λ*

_{x}[cf. Eq. (3)], at equilibrium (Δ

*F*= 0) Eq. (1) readswhere

_{t}*g*

_{x}= −(

*λ*

_{x}/

*λ*

_{P}) is called the feedback gain for the variable

**. If the total feedback gainis positive (negative), the temperature change Δ**

*x**T*is larger (smaller) than the temperature change Δ

_{s}*T*,

_{s}*associated with the Planck response.*

_{P}## 3. Relative contribution of each feedback to the global temperature change

### a. Equilibrium temperature change

*x*is active in addition to the Planck response, the equilibrium temperature change resulting from this feedback is simply and uniquely defined from Eq. (5) as the difference

*δ*

_{1}

*T*,

_{s}*between the temperature change with and without this feedback*

_{x}*x*,When several feedbacks are active, various approaches may be used. The first approach is to quantify, as previously, the effect of each feedback as the difference between the temperature change with and without this feedback

*x*[Eq. (7)]. A second possibility is to quantify this effect as the difference

*δ*

_{2}

*T*,

_{s}*between the temperature change when all the feedbacks are active and when all the feedbacks but*

_{x}*x*are active,In this definition, the effect of a feedback loop

*x*on the temperature change depends both on its gain

*g*and on the gain

_{x}*g*of all feedbacks (e.g., Hansen et al. 1984; Hall and Manabe 1999); thus, it cannot be defined independently of the rest of the system. The temperature change obtained with these two definitions may be very different.

*T*,

_{s}*associated with each feedback*

_{x}*x*plus the temperature change Δ

*T*

_{s,P}associated with the Planck response equals the total temperature change Δ

*T*,From Eq. (5), it follows thatThis expression can also be directly obtained by noting that Δ

_{s}*T*[Eq. (5)] cannot be directly decomposed into additive contributions associated with each feedback, whereas the difference Δ

_{s}*T*− Δ

_{s}*T*,

_{s}*can. This new definition leads to partial temperature changes that have some interesting properties. If the feedback parameter*

_{P}*λ*of a feedback

_{x}*x*is multiplied by a factor

*α*and the total gain

*g*is unchanged (in this case, other feedback parameters have also to be modified), the temperature change Δ

*T*,

_{s}*associated with this feedback*

_{x}*x*is multiplied by

*α.*If the feedback parameters of two feedbacks

*x*and

*y*are both multiplied by a factor

*α*, the ratio (Δ

*T*

_{s,x}/Δ

*T*

_{s,y}) is not modified. If the feedback parameters of all the feedbacks are multiplied by a same factor

*α*, the ratio (Δ

*T*

_{s,x}/Σ

_{y≠P}Δ

*T*

_{s,y}), that is, the relative fraction of the temperature change resulting from each feedback

*x*is not modified. Therefore, this definition of the partial temperature change allows us to compare and to add the contribution of the various feedbacks to the temperature response.

*x*[Eq. (10)] are of different natures because of the very specific role of the Planck response (the “basic” response on which the others are feedbacks). Equation (10) may also be written as follows:In this equation, the left-hand side is the change of the TOA flux resulting from the partial temperature change Δ

*T*

_{s,}

*if the temperature change was uniform and only affected the thermal emission. The right-hand side is the change of the TOA flux Δ*

_{x}*F*,

_{t}*resulting from the total temperature change Δ*

_{x}*T*through the feedback

_{s}*x.*The temperature change Δ

*T*,

_{s}*associated with a feedback*

_{x}*x*is the temperature change that would be necessary to produce the same perturbation Δ

*F*,

_{t}*of the TOA flux through thermal emission. This illustrates how the Planck response compensates for the flux disequilibrium associated with each feedback.*

_{x}### b. Transient temperature change

*F*of the net flux at the TOA is equal to the ocean heat uptake and is related to the surface temperature change Δ

_{t}*T*bywhere

_{s}*κ*is the ocean heat uptake efficiency (<0). This assumption is common and useful despite its limited validity. For instance, it is valid neither when the climate tends toward equilibrium (Δ

*T*increases slowly, whereas Δ

_{s}*F*decreases to zero) nor immediately after applying an abrupt forcing (Δ

_{t}*T*

_{s}≈ 0, whereas Δ

*F*≈ Δ

_{t}*Q*). Using Eqs. (1) and (12), the transient temperature change [also called the transient climate response (TCR)] can be expressed asAlthough the ocean heat uptake is not a feedback, the only difference between the expression of the equilibrium [Eq. (2)] and the transient [Eq. (13)] temperature changes is that in the later one, the ocean heat uptake efficiency

_{t}*κ*is added to the total feedback parameter

*λ*. Using the same approach as that for the equilibrium temperature, we thus require the total temperature change Δ

*T*to be the sum of the temperature change resulting from the Planck response, climate feedbacks, and ocean heat uptake. We obtain the same equation as for the equilibrium temperature, except that the ocean uptake efficiency has to be added to the sum over

_{s}*x*in Eqs. (6) and (9). The contribution Δ

*T*,

_{s}*of a feedback*

_{x}*x*to the global temperature change is then given by Eq. (10), where the gain

*g*is replaced by

*g*′ =

*g*+

*g*

_{o}with

*g*

_{o}= −(

*κ*/

*λp*) and the contribution of the ocean heat uptake is given by Δ

*T*

_{s,o}= (

*g*

_{o}/1 −

*g*′)Δ

*T*

_{s,P}.

*g*′ differs from

*g*, and the transient temperature change Δ

*T*

^{t}

_{s,x}associated with a feedback

*x*differs from that at equilibrium Δ

*T*

^{e}

_{s,x}. The transient temperature change Δ

*T*

^{t}

_{s}also differs from that at equilibrium Δ

*T*

^{e}

_{s}, and a direct consequence of Eq. (10) is that the contribution of a feedback

*x*to the global temperature change is the same in both equilibrium and transient conditions,

## 4. CMIP3/AR4 atmosphere–ocean GCMs

We now apply the above decomposition to the global surface temperature response to a CO_{2} doubling predicted by an ensemble of 12 coupled atmosphere–ocean GCMs (AOGCMs) participating in the CMIP3/AR4 (Meehl et al. 2005; Randall et al. 2007a). For this purpose, for each model we need the global mean values of the radiative forcing, the climate feedback parameters, and the ocean heat uptake efficiency.

### a. 2 × CO_{2} radiative forcing

In this study we use the radiative forcing for a CO_{2} doubling reported by Forster and Taylor (2006) and Randall et al. (2007b). These forcings have been computed after stratospheric adjustment in all-sky conditions and are averaged over the globe for a year (Table 1). For the 12 GCMs considered here, the multimodel average of the net radiative forcing (3.71W m^{−2}) is very close to previous Myhre et al. (1998) results, and the relative intermodel standard deviation is about 6% (Table 2).

In another intercomparison study, for 16 GCMs Collins et al. (2006) obtained an intermodel spread of the net radiative forcing as large as 15% (Randall et al. 2007a). These forcing have been computed at 200 hPa for a unique atmospheric profile (midlatitude summer climatological conditions) in clear-sky conditions, without any stratospheric adjustment. When compared with Forster and Taylor (2006) results, the relative values of the intermodel standard deviation of the longwave (LW) forcing are similar in both studies (8%, see Table 1). This is not the case in the shortwave (SW) domain, and the difference is even larger for the net radiative forcing. In the results of Collins et al. (2006), as reported by Randall et al. (2007a), the standard deviation of the net forcing is larger than the quadratic sum of the standard deviation of the SW and LW forcings, which indicates that the SW and LW intermodel differences are positively correlated. The opposite is found in Forster and Taylor (2006), which indicates that the error in the SW and LW domains are anticorrelated, and that stratospheric adjustment can explain part of it. We believe that the intermodel spread of the forcing reported by Forster and Taylor (2006) is the most relevant for our study because the global warming estimates are derived from global simulations, including clouds and a stratospheric temperature response.

*T*are proportional to Δ

_{s}*T*,

_{s}*[Eq. (10)], and therefore to the forcing Δ*

_{P}*Q*[Eq. (4)]. Part of intermodel differences in these contributions may thus arise from intermodel differences in the radiative forcing. To quantify this part, for each model we compute Δ

_{t}*T*

_{s,P}for a reference forcing value Δ

*Q*

^{r}

_{t}(set to the multimodel mean forcing estimate, viz., 3.71W m

^{−2}; see Table 1) and we add a term that represents the impact of the discrepancy

*δQ*on Δ

_{t}*T*between the actual forcing of each model and the reference value,

_{s}### b. Feedback parameters

As reviewed by different authors (e.g., Soden et al. 2004; Stephens 2005; Bony et al. 2006), several approaches have been followed to decompose the total feedback parameter into its several components (water vapor, clouds, surface albedo, etc.), with each method having its own strengths and weaknesses. Soden and Held (2006) computed these feedback parameters for 12 CMIP3/AR4 models (Table 1), using the Special Report on Emissions Scenarios (SRES) A1B simulations, and their results are fairly consistent with previous results obtained by Colman (2003) with older GCMs (cf. Bony et al. 2006). The multimodel mean and standard deviation of the total feedback parameters (*λ*^{−2}, *σ*_{λ} = 0.3 W m^{−2}, Table 3) are consistent with the values obtained by Forster and Taylor (2006) for a larger set of CMIP3/AR4 models and for different ensembles of runs. When analyzing the 1% yr^{−1} increase of CO_{2} simulations performed by 20 AOGCMs, they found a multimodel mean value of the total feedback parameter *λ*^{−2} and a standard deviation *σ*_{λ} = 0.3 W m^{−2}. When considering another set of experiments, namely, doubled-CO_{2} equilibrium runs from 11 atmospheric GCMs coupled to slab oceans, they found a mean value *λ*^{−2} and a standard deviation *σ*_{λ} = 0.3 W m^{−2}.

### c. Ocean heat uptake efficiency

We computed the ocean heat uptake efficiency *κ* using Eq. (12). For each model, the TOA flux *F _{t}* and the surface air temperature

*T*were averaged over the 20-yr period centered at the time of CO

_{s}_{2}doubling, that is, year 70 for the 1% yr

^{−1}simulation. The differences with the corresponding period of the control simulation were performed and the values of

*κ*reported in Table 1.

### d. Representativity of the ensemble of models considered

Using the values reported in Table 1, the equilibrium and transient temperature changes are computed for each of the 12 models as Δ*T*^{e}_{s} = −Δ*Q*_{t}/*λ* and Δ*T*^{t}_{s} = −Δ*Q*_{t}/(*λ* + *κ*), respectively. This leads to a multimodel mean ±1 standard deviation of the equilibrium temperature change of 3.1° ± 0.7°C. These numbers are comparable with those of the AR4 equilibrium climate sensitivity estimates derived from 18 atmospheric GCMs coupled to slab oceans (3.3° ± 0.7°C; see Meehl et al. 2007). For the transient temperature change, we obtain 2.0° ± 0.3°C, which is closed to the AR4 values reported on the basis of 19 coupled atmosphere–ocean GCMs (1.8° ± 0.3°C; see Meehl et al. 2007). As far as global temperature change is concerned, the subset of 12 models considered here is therefore representative of the larger set of CMIP3/AR4 models.

## 5. Results

### a. Decomposition of equilibrium temperature changes

The multimodel mean of the equilibrium temperature change and the contributions associated with the Planck response [Eq. (4)] and each feedback [Eq. (10)], computed for a reference radiative forcing, are shown in Fig. 1a and reported in Table 3. On average, for the set of models considered here, the Planck response represents about a third of the total temperature response (1.2° versus 3.1°C), while climate feedbacks account for two-thirds of it. The increase of water vapor with warming enhances the absorption of longwave radiation and enhances the warming by 1.7°C. Lapse-rate changes are associated with a negative feedback, resulting from the moist-adiabatic structure of the tropical atmosphere. Because of the strong anticorrelation between these two feedbacks, it is convenient to consider the sum of both of them (WV + LR; Soden and Held 2006). This combined feedback increases the temperature by 0.9°C, which is slightly less than the Planck response. The cloud feedback’s contribution to the warming is, on average, slightly weaker than that of the WV + LR feedback, and the surface albedo feedback’s contribution is the smallest.

However, Fig. 2 shows that for each feedback there are some intermodel differences, especially for the cloud feedback contribution, and that the amplitude of the equilibrium temperature change is primarily driven by the cloud feedback component. This appears also clearly when considering the intermodel standard deviation of the temperature change resulting from each feedback normalized by the intermodel standard deviation of the total temperature change (Fig. 1b). The standard deviation resulting from cloud feedback represents nearly 70% the standard deviation of the total temperature change. The temperature spread resulting from the radiative forcing is comparable to the spread resulting from the WV + LR feedback and the spread resulting from the surface albedo feedback is the smallest.

### b. Decomposition of transient temperature changes

The transient temperature changes (or TCR) from individual GCMs, as well as the contribution of the various feeedbacks, are displayed in Fig. 3. The multimodel mean and standard deviation are displayed in Fig. 4 and reported in Table 3. The temperature damping resulting from the ocean heat uptake is about −0.4°C, and its absolute value is comparable to the multimodel contributions of the WV + LR (0.6°C) and cloud (0.4°C) feedback. The mean transient temperature change is nearly 2/3 of that at equilibrium; therefore, the transient temperature changes associated with each feedback scale with it [cf. Eq. (14)]. The intermodel standard deviation of the temperature change resulting from cloud feedback represents nearly 90% of the standard deviation of the total temperature change (Fig. 4b). Similarly for the equilibrium case, cloud feedbacks thus constitute the main source of spread of the transient temperature response among GCMs. The WV + LR feedback, the ocean heat uptake, and the radiative forcing constitute secondary and roughly comparable sources of spread, and the surface albedo feedback constitutes the smallest one.

The intermodel standard deviation of the global temperature change may also be normalized with the multimodel mean global temperature change. This relative standard deviation is comparable in both equilibrium and transient conditions; the spread in equilibrium is slightly larger (23% versus 16%). The same holds for the relative standard deviation of the temperature change associated with each feedback. Therefore, the contribution of the various feedbacks to the total spread is, in relative terms, as important in the transient case as in the equilibrium case.

## 6. Summary and conclusions

In this note we propose a simple decomposition of the equilibrium and transient global temperature responses to an external forcing into a sum of contributions associated with the Planck response, the different climate feedbacks, and, eventually, the ocean heat uptake. This allows us to quantify how the various processes contribute to the multimodel mean and intermodel spread of the global temperature change. This is illustrated (Figs. 1 –4) using published results for the feedback parameters and the radiative forcings (Soden and Held 2006; Forster and Taylor 2006; Randall et al. 2007b), and by diagnosing the ocean heat uptake efficiency from model outputs. In transient simulations, the absolute values of the contributions of the WV + LR feedback, the cloud feedbacks, and the ocean heat uptake to the global temperature response appears to be comparable (Fig. 4a). However, for the ensemble of models considered here, the spread of the transient temperature change resulting from intermodel differences appears to be primarily due to cloud feedback. The spread resulting from WV + LR feedback, ocean heat uptake, or radiative forcing appears to be of the same order of magnitude and roughly one-third of the spread resulting from the cloud feedback (Fig. 4b). Note that the radiative forcing associated with non-CO_{2} greenhouse gases and aerosols is more uncertain than that associated with CO_{2} (Forster et al. 2007). Therefore, the intermodel spread of radiative forcing estimates might be larger either for twentieth-century simulations or for climate change simulations, based on emission scenarios that include changes in aerosol concentrations, than in this study. This difference is mitigated, however, by the fact that the relative contribution of aerosols versus greenhouse gases is likely to decrease in the future (Dufresne et al. 2005).

Our analysis shows that the contribution of each feedback and of the radiative forcing to intermodel differences in temperature change is roughly similar, in a normalized sense, in equilibrium and transient simulations (Figs. 1b and 4b). In particular, cloud feedbacks appear to be the main source of spread in both cases. Intermodel differences in cloud feedbacks have been shown to arise primarily from the response of low-level clouds (Bony and Dufresne 2005; Webb et al. 2006; Wyant et al. 2006). Understanding and evaluating the physical processes that control these cloud responses thus appears to be of primary importance for better assessing the relative credibility of climate projections from the different models.

## Acknowledgments

We thank Jean-Yves Grandpeix for frequent and useful discussions on feedback. We are grateful to Jonathan Gregory and an anonymous reviewer for useful comments on this study, and we acknowledge the modeling groups for making their simulations available for analysis, the Program for Climate Model Diagnosis and Intercomparison (PCMDI) for collecting and archiving the CMIP3 model output, and the WCRP’s Working Group on Coupled Modelling (WGCM) for organizing the model data analysis activity. The WCRP CMIP3 multimodel dataset is supported by the Office of Science, U.S. Department of Energy. This research was supported by CNRS, byINSU-EVE French Program under the project MissTerre, and by the European Commission under the project ENSEMBLES (Contract GOCE-CT-2003-505539).

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The 2 × CO_{2} radiative forcing Δ*Q _{t}*, total feedback parameter

*λ*, and ocean heat uptake efficiency

*κ*estimates of the 12 CMIP3/AR4 models used in this paper, and their multimodel mean and standard deviation.

Multimodel mean and intermodel standard deviation of the LW, SW, and net radiative forcing (W m^{−2}) for a CO_{2} doubling computed by GCMs in two intercomparison studies, with two different numerical setups (see text). In parenthesis, the standard deviation is computed relative to the mean.

Multimodel mean and intermodel standard deviation of total feedback parameter *λ* and its components *λ _{x}* (W m

^{−2}K

^{−1}), the ocean heat uptake efficiency

*κ*(W m

^{−2}K

^{−1}), the 2 × CO

_{2}radiative forcing Δ

*Q*(W m

_{t}^{−2}), and their associated equilibrium and transient temperature changes (°C). The multimodel mean and standard deviation of the equilibrium (Δ

*) and transient (Δ*

^{e}_{s}*T*) temperature changes (°C) are also given.

^{t}_{s}