1. Introduction
Equilibrium climate sensitivity (ECS; the global-average equilibrium surface temperature change due to CO2 doubling) is one of our primary measures of the severity of anthropogenic climate change. Because global warming is the result of complex interactions between many aspects of the climate system, estimates of ECS are typically made using sophisticated general circulation models (GCMs). Discrepancies between ECS predictions from different GCMs are often taken as a proxy for global warming uncertainty (Flato et al. 2013), even though intermodel differences do not account for biases shared across models and therefore actually constitute a lower bound on the true uncertainty.
The net feedback λ can be broken down into a sum of individual feedback mechanisms λi. The exact decomposition into individual feedbacks varies somewhat between studies, but the most common partitioning differentiates between Planck (Pl), water vapor (WV), lapse rate (LR), albedo (Alb), and cloud (Cld) terms. The feedback term λPl represents the rapid increase in longwave emission with temperature as encapsulated by Planck’s law; it is negative in sign (acting to stabilize the climate) and has the largest amplitude of all feedbacks. Because water vapor increases with warming and acts as an additional greenhouse gas, λWV is positive. The feedback term λLR is negative because air higher in the troposphere warms more than air near the surface as the planet warms, allowing the planet to radiate to space more efficiently as surface temperature increases. The feedback term λAlb tends to be small and positive and is related to decreases in future snow and/or ice coverage and changes in solar reflectance due to vegetation changes. Clouds have complex effects on TOA radiation. In the shortwave (SW), clouds act primarily to cool the planet by reflecting solar radiation back to space. In the longwave (LW), clouds warm the planet by trapping infrared radiation. SW cloud radiative effects depend primarily on incoming solar radiation (which varies with latitude and season), cloud fraction, and cloud albedo. LW cloud radiative effects depend primarily on cloud fraction and cloud-top altitude. Cloud feedback is positive in most models because cloud amount tends to decrease and cloud-top altitude tends to increase as the planet warms (Zelinka et al. 2012, 2013). These positive feedback effects are slightly offset by robust increases in high-latitude cloud optical depth (Gordon and Klein 2014). Although these behaviors are captured by most GCMs, the magnitude of global warming–induced changes in cloud properties differs between models in subtle but important ways, which lead to large intermodel spread in the magnitude of cloud feedback.
Several methods have been proposed for identifying the processes responsible for intermodel spread in ECS. One approach is to simply compare intermodel variance in each feedback mechanism λi. Since λi has units of watts per meter squared per kelvin, this is fundamentally a comparison of TOA radiative perturbations normalized by the total amount of warming in each model. Variance across models (var) is found to be much larger for λCld than for other feedback mechanisms. Contributions from var(λWV) and var(λLR) are also sizable but largely cancel when combined (e.g., Bony et al. 2006; Soden and Held 2006; Huybers 2010). Using λ as a proxy for ECS is an incomplete solution because it neglects F and because ECS is not proportional to λ. Another approach that has been used to identify sources of ECS spread is to compute intermodel spread in ECS [using Eq. (2)] with all but one term replaced by multimodel mean values. Using this method, Webb et al. (2013) showed intermodel spread in net feedback to be twice as important to ECS as forcing, with cloud feedback again being the dominant source of spread. This approach is also somewhat unsatisfactory because the variances associated with individual processes do not sum to the total variance in ECS. On a related note, Dufresne and Bony (2008, hereafter DB08) point out that the spread associated with a process of interest could equally be well defined as the decrease in var(ECS) due to replacing that process with its multimodel mean while allowing all other processes to vary across models. This new definition would provide a different estimate of the importance of that term to var(ECS), leading DB08 to conclude that approaches involving replacing some terms with multimodel values yield nonunique solutions. To avoid this arbitrariness, DB08 propose a third approach based on decomposing each model’s ECS into temperature perturbations associated with each feedback and forcing mechanism. A major goal of this paper is to show that the DB08 approach is equally arbitrary.
Another aim of this study is to clarify the role of covariances between feedbacks or between forcing and feedback terms in understanding intermodel spread in ECS. Previous studies have identified correlations between λWV and λLR [as noted above and discussed in Cess (1975); Held and Soden 2000; Soden and Held 2006], between λCld and λAlb (Huybers 2010; Mauritsen et al. 2013), between λCld and λWV+LR (Huybers 2010; Mauritsen et al. 2013), and between F and λSW_Cld (Andrews et al. 2012; Webb et al. 2013; Ringer et al. 2014). These previous studies point out in a general way that feedback and forcing terms are not independent. Our study provides a way to quantify the relative importance of spread in individual forcing or feedback terms versus covariance between those terms.
We begin in section 2 by describing how we obtain estimates of ECS, forcing, and feedback terms. In section 3 we describe the DB08 approach to partitioning ECS into a sum of terms and highlight some problems with their methodology. An alternative approach that avoids these problems is presented in section 4. In section 5 we provide a formal method for breaking the variance of a sum into a sum of covariances and apply this decomposition to both partitioning approaches. Resulting insights into the source of ECS spread are discussed in section 6, and conclusions are drawn in section 7.
2. Data
Following Eq. (1), we compute forcing F and net feedback λ for each model as the y intercept and slope (respectively) of the linear regression of ΔN onto ΔT. Once F and λ are computed, ECS is calculated following Eq. (2). This approach, which was pioneered by Gregory et al. (2004), is based on the realization that while ΔN and ΔT respond nonlinearly in time to an abrupt CO2 increase, their relationship tends to be approximately linear. The climate-change signal Δ is computed as the difference between yearly samples from 150 years of abrupt quadrupling of atmospheric CO2 experiment (abrupt4×CO2) data and contemporaneous 21-yr running-mean values from piControl runs. The choice to compute anomalies relative to a running-mean climatology rather than a single static climatology is meant to account for any climate drift that may be present, under the assumption that the drift would be similar in the perturbed (abrupt4×CO2) and preindustrial control (piControl) runs. We have tested the impact of computing anomalies relative to the average over the entire piControl simulation and found averaging length to make little difference.
While linearity is a good first-order approximation, the slope of ΔN versus ΔT actually tends to decrease over time as the strength of individual feedback mechanisms are modulated by the background climate state (Armour et al. 2013; Rose et al. 2014). For some models this decrease is pronounced and induces error in our F and λ calculations (e.g., Andrews et al. 2012). We use the Gregory et al. (2004) approach despite this potential error because very few modeling groups provided sufficient data to compute feedbacks and forcings using other methods. We also note that the current study is focused entirely on intermodel spread, and Fig. 2a of Chung and Soden (2015b) indicates that spread in effective forcing is not strongly affected by the method used to derive it. Because our estimates of F and λ are calculated using the first 150 years of output after quadrupling CO2, they probably do not represent true equilibrium values and are better thought of as estimates of the effective values over this period. In particular, the effective climate sensitivities calculated here are probably underestimates of the true equilibrium climate sensitivity (Williams et al. 2008; Winton et al. 2010; Andrews et al. 2015).
For each mechanism i, annually averaged ΔRi is regressed on corresponding global- and annual-average ΔT values (as done for net λ and F) to get λi (from the slope) and the tropospheric adjustment of i to CO2 forcing Fi (from the y intercept). These tropospheric adjustments act to modify the radiative forcing of CO2.
Error in the kernel method can be quantified by comparing the change in clear-sky TOA radiation computed as 
Calculated values for all models are presented in Table 1 (model data available online at https://pcmdi9.llnl.gov/projects/cmip5/; data accessed September 2015). As recommended by Vial et al. (2013, hereafter Vial13), we include the residual Re between net λ and the sum of its components as an explicit term in our table. Because Shell08 and Soden08 kernels both provide very similar numbers and more models pass the clear-sky linearity test for Soden08 kernels, we use Soden08 values exclusively for the rest of this paper.
Feedbacks and forcing used in this study. Models with names in boldface pass a clear-sky linearity test. Values in parentheses are computed relative to constant RH as advocated by HS12. All values are computed using the Soden08 kernels. (Expansions of acronyms are available at http://www.ametsoc.org/PubsAcronymList.)
Feedbacks using constant RH
Traditionally, λWV is taken to be the radiative impact of water vapor specific humidity qυ increase, λLR is the radiative change associated with vertically nonuniform atmospheric temperature increase, and λPl is the radiative effect of surface warming. Held and Shell (2012, hereafter HS12) note, however, that qυ typically changes in such a way that relative humidity (RH) remains roughly constant. As a result, water vapor increases as atmospheric temperature increases (leading to the correlation between WV and LR feedbacks noted above). HS12 suggest defining the water vapor feedback based on change in qυ associated with changes in relative humidity. To have all λi sum to λ, the radiative impact associated with qυ change at constant RH must then be partitioned such that the component responsible for warming each atmospheric column by its surface ΔT is added to λPl and the impact of qυ change at constant RH associated with nonuniform atmospheric warming is added to λLR. This repartitioning of feedbacks has the advantage (shown below) of reducing covariance between λWV and λLR. As explained in HS12, this new set of feedback definitions also avoids issues with computing feedbacks relative to a physically unrealizable (supersaturated) base state. We consider the impact of this feedback repartitioning in section 6.
3. DB08 method
In addition to being simpler than the derivation in DB08, our derivation clearly shows that the DB08 partitioning is mathematically nonunique. For example, λPl/λPl on the first line in Eq. (5) could be replaced with λ/λ or any other expression of the form x/x to produce alternative ECS partitionings. Fundamentally, this nonuniqueness stems from the fact that moving the summation from the denominator to the numerator of Eq. (2) is ill posed. In general, any linear combination of terms that sum to ECS for each model is a mathematically acceptable partitioning. Some partitionings are more physically defensible than others, however. The DB08 partitioning is physically attractive because TPl is the temperature response needed to balance 
4. A better partitioning
The above discussion leaves us with a few obvious questions: Is there a better way to decompose ECS spread? Do the details of partitioning strategy matter? In this section we describe a new partitioning strategy that avoids the issues noted above. By comparing this new approach against that from DB08, we are able to show in section 6 that partitioning strategy does have a large impact on our understanding of which terms are important for ECS spread.
Nonlinearity in the system is contained entirely in the last two terms of Eq. (7); we combine these terms into a single quantity that we call Tnonlin. We show in section 6 that var(Tnonlin) is negligibly small.
The first term in Eq. (7) is large and is not related to any particular feedback mechanism. This makes linearization inappropriate for determining which processes are important to the multimodel mean magnitude of ECS. Because our paper focuses entirely on ECS spread, this term is not important since it is constant across models and therefore makes no contribution to var(ECS).
The second term in Eq. (7) is the contribution to ECS spread from spread in effective forcings TF. The third term in Eq. (7) is the ECS contribution from λi: 
Since we have linearized the contribution of each mechanism to var(ECS), the new partitioning is only an approximate partitioning of ECS. It is therefore incumbent on us to prove that linearization does not significantly distort our understanding of the main processes contributing to ECS spread. This issue is addressed in detail in the appendix. To summarize, the λi with largest variance is not necessarily the largest contributor to spread in 
5. The variance of a sum
Plotting the Ti covariance matrix is a nice way to visualize the contributions of forcing and feedback processes to ECS spread. This is done in Fig. 1. Variances for each process appear on the main diagonal, while covariance terms are on the off diagonals. Because the covariance matrix is symmetric, each covariance should be included twice. For ease of display, we instead double the value of covariances above the main diagonal and omit the corresponding covariances below the diagonal. Covariance terms can be positive or negative while variances are always nonnegative. For ease of display, each matrix in Fig. 1 has been normalized by var(ECS) (=0.475 K2) such that the sum of all its entries is 1.
Fractional contributions of partitioned terms to the total variance of ECS. Colors indicate magnitude, which is also given by numbers in each cell.
Citation: Journal of Climate 29, 2; 10.1175/JCLI-D-15-0352.1
6. Results
First, note that covariance and variance terms in Fig. 1 are of comparable magnitude regardless of partitioning approach; this implies that ignoring covariance terms is inappropriate.
Next, note that the DB08 and new partitioning methods yield extremely different covariance matrices (Figs. 1a and 1c, respectively). Variance and covariance tend to be much more homogeneously distributed in DB08 because Ti ∝ λi/λ for i ∉ {Pl, F}. Dividing by λ damps contributions from the largest-variance contributors (which drive sympathetic variations in λ) and adds variance to terms that have little on their own. Using λi rather than 
It is also worth pointing out that the rows and columns in Fig. 1c related to feedbacks are identical to the covariance matrix for variance in net feedback (subject to rescaling) because 
One troubling aspect of Fig. 1a is that its variance terms seem different than presented by DB08 for CMIP3 models and by Vial13 for CMIP5 models. In particular, both DB08 and Vial13 found λCld to be the dominant source of intermodel spread, yet λSW_Cld and λLW_Cld in the current study both have magnitude smaller than F and λWV. While some differences are expected as a result of using different models and computing variance instead of standard deviation, the main reason for inconsistency with previous studies is that DB08 and Vial13 presented only the combined LW + SW cloud feedbacks and WV + LR feedbacks rather than individual components. Looking at Fig. 1a, we see that LW and SW Cld contribute 0.1 and 0.3 fractional units to the total spread (respectively) and their covariance terms add an additional 0.1 unit, so the variance contributed by net feedback sums to 0.5 units. Combining WV and LR in a similar way yields a spread of only 0.3 units because WV and LR variations compensate each other strongly. Thus after combining terms λCld again stands out as a dominant source of intermodel spread in ECS. This can also be seen in Fig. 2, which shows feedbacks after combining SW with LW Cld and WV with LR. So is it better to consider these terms separately or in combination? Combining terms simplifies interpretation but reduces information content, so the answer to this question probably depends on the application of interest.
As in Fig. 1, but using standard feedback definitions only and combining λSW_Cld with λLW_Cld and λLR with λWV.
Citation: Journal of Climate 29, 2; 10.1175/JCLI-D-15-0352.1
Even after accounting for combined terms, Fig. 1a indicates greater importance for var(F) than suggested by Vial13. This can be understood by examining the forcing terms in Eqs. (25) and (26) of Vial13. These equations are analogous to our Eq. (5), which is an intermediate stage where forcing terms have not been isolated yet. Isolating forcing variability into a single term [as done in our Eq. (6) and DB08’s Eq. (15)] results in the forcing term being normalized by λ rather than λPl. Since λPl ≈ 3λ, this difference greatly (and inappropriately) reduces their estimates of intermodel variance in F.
Figures 1b,d show the partitioning of intermodel spread using the HS12 definitions for WV, LR, and Pl. As expected, the HS12 approach reduces compensation between WV and LR and reduces intermodel variance in these terms as a result. Even with HS12 definitions, however, covariance terms remain nonnegligible. Switching to HS12 definitions has a large effect on all terms when the DB08 partitioning is used because λPl is used as a scaling factor for all ΔTi. In the new ECS partitioning, feedbacks other than Pl, WV, and LR are unchanged (as expected). Sources of intermodel spread are more consistent between DB08 and our new method when using these new feedback definitions. Reduced ambiguity regarding the source of ECS variability is another good reason to use the HS12 definitions.
Because HS12 definitions are based in sound physics and reduce discrepancies between partitioning methods, we adopt this approach for our analysis of the sources of ECS spread. Note that the nonlinear term in Fig. 1d is negligibly small, a necessary condition for our new approach to be effective. Having a small nonlinear term related to the net feedback is not sufficient proof that individual feedbacks can be linearized, but our analysis in the appendix shows this assumption to be appropriate for our dataset. The residual term TRe, which measures error in the radiative kernels, is also acceptably small.
By far the largest term in Fig. 1d is var(λSW_Cld). This is consistent with previous papers (e.g., Cess et al. 1990, 1997; Webb et al. 2013), which all show that SW cloud feedback is the dominant source of uncertainty in climate sensitivity. The second-largest term in Fig. 1d is covariance between λSW_Cld and F; Ringer et al. (2014) also noted that anticorrelation between forcing and cloud feedback acts to strongly damp intermodel spread in ECS. Other important terms are var(λLW_Cld), var(F), cov(λSW_Cld, λAlb), cov(λLW_Cld, F), cov(λLW_Cld, λAlb), and cov(λSW_Cld, λAlb). Huybers (2010) and Mauritsen et al. (2013) have previously noted compensation between λSW_Cld and traditionally defined λWV+LR. Our use of RH-based feedback definitions allows us to clarify that it is the lapse rate rather than moisture changes that are responsible for this relationship. Our separation of SW and LW Cld feedbacks makes clear that it is primarily LW rather than SW cloud feedbacks that are responsible for this behavior. Huybers (2010) and Mauritsen et al. (2013) also note negative cov(λSW_Cld, λAlb).
It is worth asking whether these covariances are robust or spurious. The best way to answer this question is to come up with an unassailable physical explanation for the relationship. A plausible explanation for the fairly strong negative correlation between LR and LW cloud feedbacks is that models with larger upper-tropospheric warming (and thus a more negative LR feedback) experience a more pronounced deepening of the troposphere (O’Gorman and Singh 2013), which allows high clouds to rise higher, thereby producing a larger LW cloud feedback (Zelinka and Hartmann 2010). Physical explanations for other covariances are still needed.
In the absence of a physical explanation, it is worth noting that cov(x, y) = [var(x)]½[var(y)]½corr(x, y). This means that a large covariance can result either from a robust underlying relationship (i.e., a large-magnitude correlation) or from a large variance amplifying a very weak correlation. This is explored in Fig. 3 by scatterplotting pairs of quantities with substantial covariance. In Fig. 3 we see that there is very little correlation between TSW_Cld and TLW_Cld and between TSW_Cld and TLR, even though these terms had sizable covariance. Relationships with large covariance but small correlation indicate sources of ECS variance due to noise in the ensemble; these relations will probably change from ensemble to ensemble and are not worth exploring. The opposite may also occur; highly correlated quantities may explain a very small fraction of var(ECS). This represents relationships that are real but not very important to climate sensitivity (and therefore of less interest). Such relationships receive small weighting in our covariance matrices. The other correlations in Fig. 3 are of moderate magnitude and warrant further exploration.
Scatterplots of contributions to ECS shown in Fig. 1d to have substantial covariance. All plots are based on HS12 feedback definitions.
Citation: Journal of Climate 29, 2; 10.1175/JCLI-D-15-0352.1
Another way to test whether results are robust is to repeat the analysis using different datasets. We have done this in a limited way by performing our analysis using a 10% cutoff for clear-sky kernel error (which yields 11 models instead of the 20 we use elsewhere) and using all available models regardless of clear-sky kernel error. Results look very similar in both cases (not shown), except cov(λSW_Cld, λAlb) becomes negligible when all models are included and cov(λLW_Cld, λLR) becomes small when a 10% clear-sky linearity test is adopted. This suggests that these covariances may be less robust, but further testing is needed to be definitive.
7. Conclusions
In this study we derive and analyze effective forcing and feedback information for a larger set of CMIP5 models than provided by earlier studies (Vial13; Andrews et al. 2012; Webb et al. 2013). We also explore an alternative way of defining water vapor feedback proposed by Held and Shell (2012). By recasting λWV as the TOA radiative response to changes in relative rather than specific humidity, WV and LR feedbacks are decoupled. We show that this redefinition makes analyzing feedback mechanisms easier, so we hope that the HS12 technique is more widely adopted in the future.
Our main goal is to revisit the question of what causes disagreement in ECS between climate models. We note that the variance of a sum can be written as the sum of all possible combinations of covariances of summands. This insight leads us to show that covariance terms are as important to ECS spread as the more commonly analyzed variance terms. We also note that writing ECS as a sum of terms involving feedback processes is mathematically problematic because there is no good way to move 
Our results confirm previous work showing that λSW_Cld is by far the dominant source of feedback spread. We also find that var(λLW_Cld), var(F), cov(λSW_Cld, F), cov(λLW_Cld, F), cov(λLW_Cld, λLR), and cov(λSW_Cld, λAlb) also contribute substantially to ECS spread. Understanding the physical mechanisms behind these sources of spread and improving their representation in climate models is the most direct path toward reducing uncertainty regarding the magnitude of global warming for a given increase in CO2. The main contribution of this article is to point out that interactions between processes are themselves responsible for a great deal of ECS uncertainty and should be studied more closely. Since this point has not been widely appreciated in the past, there is probably great opportunity for advances in this area.
Acknowledgments
We would like to acknowledge the modeling groups [the Program for Climate Model Diagnosis and Intercomparison (PCMDI) and the World Climate Research Programme’s Working Group on Coupled Modelling] for their roles in making available the CMIP5 multimodel dataset. Thanks also go to Thomas Reichler for making his WMO tropopause code publicly available. Support for these datasets is provided by the U.S. Department of Energy (DOE) Office of Science. This work was supported by the Office of Science (BER) at Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. All authors were supported by BER’s Regional and Global Climate Modeling (RGCM) Program.
APPENDIX
Appropriateness of Linearizing 1/λ
(a) Perturbations normalized by 
Citation: Journal of Climate 29, 2; 10.1175/JCLI-D-15-0352.1
Mathematically, the reason why our counterexample behaved contrary to our linearity-based expectations is that x′ and y′ are of similar magnitude to 
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