1. Introduction
During coming decades and centuries, climate change is expected in response to anthropogenic emissions into the atmosphere, especially of carbon dioxide. Projections of global climate change, for instance as assessed by Meehl et al. (2007) for the Intergovernmental Panel on Climate Change, are based principally on the results of three-dimensional atmosphere–ocean general circulation models (AOGCMs), which simulate relevant dynamical and physical processes at a horizontal resolution typically of 2°–3°. Despite the complexity of the system, model results indicate that its global-mean behavior can be described in rather simple terms as linear responses and feedbacks. The resulting conceptual framework is useful for comparison of the models with one another and the real world.
In recent years, increasing attention has been paid to the possible responses of terrestrial ecosystems and ocean biogeochemistry to increasing atmospheric CO2 concentration and the consequent climate change. Perturbations to the carbon cycle will change the storage of carbon on land and in the ocean, constituting a feedback on the atmospheric CO2 concentration. Because of their potential importance for future climate change, representations of the relevant processes are being incorporated into AOGCMs. Simulated carbon cycle changes exhibit a large spread, indicating systematic uncertainty in the models (Friedlingstein et al. 2006; Meehl et al. 2007; Plattner et al. 2008). As with the physical climate system, model comparison may identify the sources of uncertainty and help to reduce it. Simple global-mean metrics are useful for this purpose.
It appears to us that there are important analogies between the responses of the carbon cycle and the climate to forcing, but the ways in which they are described in recent literature are rather different, and this obscures the similarity and makes comparison difficult. The first aim of the present paper is to present and relate various conceptual frameworks. We show how the magnitude of carbon cycle and climate feedbacks can be compared directly by putting them into the same terms. (For convenience we have summarized the notation in Table 1 and the main results in Table 2.) We then illustrate the use of these metrics in studying past and future changes, and we discuss shortcomings and improvements.
2. Climate change and feedback
In this section, we review the formalism used to quantify climate feedback, which we will compare in subsequent sections with carbon cycle feedback; Roe (2009) provides a useful and more detailed account of feedback formalism. Changes in composition of the atmosphere from anthropogenic emissions of greenhouse gases and aerosols primarily have their climatic effect through perturbing the heat balance of the climate system. Their influence is measured in terms of their radiative forcing F (W m−2). To be stable to radiative perturbations, the climate system must tend to resist F with an opposing radiative response H, which is the increase resulting from climate change in the global-mean rate of heat loss to space by the system. The net heat flux into the system is thus N = F − H. On multiannual time scales, the net downward radiative flux at the top of the atmosphere and the net heat flux into the ocean are practically equal definitions of N, because nearly all of the heat capacity resides in the ocean. Climate change evolves while N ≠ 0 on a time scale determined by this thermal inertia.
A new steady state in equilibrium with the forcing is attained when N = 0 ⇒ H = F. If the climate system behaved like a blackbody, the system would warm up by enough to compensate for the forcing, according to the Stefan–Boltzmann law. The anthropogenic forcings of relevance to coming centuries would produce global-mean temperature changes of a few kelvins. Because this is small compared with the global-mean temperature of about 255 K required to balance solar irradiance, the blackbody response can be linearized as HBB = λBBT, with λBB = 4σSB 2553 = 3.8 W m−2 K−1, where T is the change in global-mean temperature with respect to the unperturbed climate, and σSB is the Stefan–Boltzmann constant (Hansen et al. 1984).














In Eq. (2) the terms all have the same form. Each is an independent radiative response of the climate system [although actually they are not entirely independent; e.g., see Randall et al. (2007) and Soden et al. (2008)]. Their net effect H tends to resist the imposed F, and they increase together with T until H balances F. An analogy to this “resistance” picture is to consider the forcing F as a weight hanging on a spring whose consequent extension is T. Then λ is the spring constant (tension per unit extension). The blackbody response alone is like a stiff spring with a large spring constant, giving a small extension to support a given weight. Positive feedbacks mean the spring is weaker and extends more.
In Eq. (4) the blackbody response has a special status, which is somewhat arbitrary and artificial, because it is hard to isolate and quantify this effect within the complexity of the climate response to forcing, and it has limited practical relevance. Another drawback with the gain interpretation of heat balance is that the terms are not additive in their effect on T, because their sum appears in the denominator of G. If the processes are added one by one to Eq. (4), and T is evaluated each time, with T0 being the response resulting from all of the processes already considered, a G = T/T0 can be assigned to each process; however, owing to the nonlinear combination, these results depend on the order in which the processes are included.




3. Coupling of carbon cycle and climate
The carbon cycle is so called because it involves a flux of carbon through various stores, with the total mass of carbon in the unperturbed system remaining fixed. On centennial time scales, the storage of carbon in rocks and sediments changes little, and the relevant stores are in the atmosphere, the terrestrial biota, the organic carbon in the soil, and in the ocean as dissolved and particulate organic and inorganic compounds and marine biota. Anthropogenic emissions of carbon dioxide from fossil fuels add initially to the carbon content of the atmosphere, but this carbon is subsequently repartitioned among the stores, so the effect is to increase the total mass of carbon in the system. For simplicity we neglect anthropogenic emissions of fossil methane in this argument.


In the Coupled Climate–Carbon Cycle Model Intercomparison Project (C4MIP), Friedlingstein et al. (2006) carried out an analysis of the changes in the carbon cycle simulated by a set of earth system models in response to the SRES A2 scenario of anthropogenic CO2 emissions (from fossil fuels and land use change) during the twenty-first century (omitting non-CO2 anthropogenic emissions). Some of the models were AOGCMs coupled to models of the terrestrial biosphere and marine biogeochemistry, and others were earth system models of intermediate complexity (EMICs). The formalism used by Friedlingstein et al. for their analysis of the results implies various simplifying assumptions, which we now describe, that are not made in the models themselves. Through most of this work we follow these same assumptions, while in section 6 we examine their validity.
Following Friedlingstein et al. (2003), Friedlingstein et al. (2006) assume that the simulated changes in land and ocean stores can be approximated as a linear combination of effects resulting from the change in atmospheric CO2 concentration, and effects resulting from climate change. Thus, CL = CLβ + CLγ and CO = COβ + COγ, where the final β and γ subscripts (chosen for consistency with later notation) denote these two classes of effect. We call these classes the concentration (i.e., biogeochemical) and climate effects of CO2. The dominant contribution to CLβ is carbon fertilization (a positive term, resulting from the stimulation of photosynthesis by elevated CO2 concentration). The term CLγ represents the effect of climate change, mainly through temperature and precipitation change, on photosynthesis, plant respiration, soil respiration, and the abundance and distribution of vegetation (terms of both signs, whose sum is found to be negative during the twenty-first century by all C4MIP models). The term COβ is principally the increased dissolution of carbon dioxide in the ocean to maintain equilibrium with the atmospheric concentration (a positive term). The term COγ is predominantly the reduction of vertical transport in the ocean resulting from increased stability and reduced solubility in warmer water (both negative); in current models, change in marine biological productivity is relatively unimportant.








Anthropogenic land use change, especially deforestation, also affects the partition of carbon among the stores. Its immediate effect is to transfer carbon from CL to C or vice versa; thus, in Eq. (7) we can count land use change in CE, even though it is not fossil carbon, if we conserve carbon by making an equal and opposite change to CL. The latter cannot be shown in Eq. (8) because land use change is imposed, rather than being a response, so it cannot appear in either Cβ or Cγ. However, the response of the climate and carbon systems to the perturbation in C from land use change will be similar to their response to other C perturbations. It is therefore acceptable to include net emissions from land use change in CE only in Eqs. (8) and (9). However, land use management modifies the nature of the response of the terrestrial carbon system to atmospheric CO2 change and climate change, so it may make β and γ variable in Eq. (9).
4. Climate change as carbon cycle feedback










Using β and uγ we can compare the magnitudes of the concentration–carbon and climate–carbon feedbacks on the carbon system, because they are both dimensionless (whereas γ is not). Their values in the C4MIP models of Friedlingstein et al. (2006) are shown in Table 3. On average, the concentration–carbon feedback is about 4 times stronger, and hence dominant. The uncertainty in β is nearly twice as large as that in uγ. The land carbon models of the Lawrence Livermore National Laboratory (LLNL) and the University of Maryland (UMD) have unusual land contributions to β; LLNL has a particularly large one and UMD a particularly small one, both being outside ±2 standard deviations of the C4MIP mean (Denman et al. 2007, their Table 7.4). Excluding these two models reduces the uncertainty in β by 25% to a standard deviation of 0.21, which still exceeds that of uγ.
The ratio u = CE/C of the emitted carbon to the increase in atmospheric carbon content can be decomposed, according to Eq. (14), into contributions from the atmosphere, the concentration–carbon response, and the climate–carbon response. In Fig. 1 we show these contributions to u, further divided between land and ocean.










5. Carbon cycle as climate feedback






The values of rβ and rγ for the C4MIP models are shown in Table 3. In Fig. 2 we compare these carbon cycle feedbacks with atmosphere and surface climate feedbacks (Soden and Held 2006) and ocean heat uptake efficiency (Gregory and Forster 2008). These values for the noncarbon feedbacks represent a wider range of recent GCMs than the C4MIP set of models. Translating the carbon cycle feedbacks into the same terms as climate feedbacks allows us to compare them numerically with each other and with the noncarbon feedbacks.
The concentration–carbon response rβ is negative, because β > 0. The concentration–carbon response, by sequestering a large fraction of the emitted carbon, is an important feedback that opposes warming, being on average 60% of the magnitude of the blackbody response. The concentration–carbon feedback is 4 times larger in magnitude than the climate–carbon feedback rγ, which is positive because γ < 0, and it is of about the same average magnitude as cloud feedback.


The concentration–carbon feedback magnifies the uncertainty σ(ρ) associated with the noncarbon feedbacks (the first term) by 1 +
The combination of carbon cycle feedback parameters rβ + rγ = 1.9 ± 0.6 W m−2 K−1 is of comparable size and uncertainty to the noncarbon climate resistance ρ = 2.1 ± 0.8 W m−2 K−1, after accounting for combination of uncertainties (Table 3). Huntingford et al. (2009) and Booth et al. (2008, manuscript submitted to Nature), both of whom follow a different approach of varying model parameters, also conclude that for projecting climate change in response to CO2 emissions the uncertainties in the carbon system are of comparable magnitude to those of the climate system.




The formulas for gCC and GCC are identical with those of Eqs. (17) and (18). In section 4 we derived them, following Friedlingstein et al. (2006), to quantify the amplification resulting from the climate system of the increase in C following a CO2 emission. Here they quantify the amplification resulting from the carbon system of the increase in T following a CO2 emission. The extra CO2 and extra warming are obviously related, as Friedlingstein et al. point out in their discussion. Both approaches are quantifying the climate–carbon interaction. There is symmetry about whether climate is regarded as a feedback on carbon, or vice versa.
However, we repeat our remark of section 2 that the gain formalism involves an arbitrary choice of basic response. We therefore think it is better to admit that T and C cannot meaningfully be decomposed into additive contributions resulting from different causes, and instead quantify the contributions to the parameters ρ and u, which measure resistance to change.
6. Experimental design and results
For each model in C4MIP, Friedlingstein et al. (2006) carried out two experiments with the same emissions scenario: one experiment (coupled) in which both the climate system and the carbon system respond to increasing atmospheric CO2; the other (uncoupled) in which only the carbon system responds to CO2, while the climate system experiences the unperturbed control value of CO2. In the former case, both the concentration–carbon and the climate–carbon responses operate. In the latter case, there is no greenhouse radiative forcing from CO2.








When we introduced β and γ to describe the response of the carbon cycle to forcing (section 3), we pointed out that this description implies that the concentration–carbon and climate–carbon feedbacks can be combined linearly. This can be tested with neither the experimental design of C4MIP nor that of Hibbard et al. (2007). To do so requires an additional experiment, in which CO2 has its radiative effect and forces climate change, but the carbon system experiences the unperturbed control CO2 and the concentration–carbon response is suppressed, which is the reverse of the uncoupled arrangement. With three experiments to distinguish, we need clearer terminology. We will use the term fully coupled for the coupled experiment of Friedlingstein et al. (2006), biogeochemically coupled (indicating that CO2 is biogeochemically active) for their uncoupled, and radiatively coupled for the additional experiment described here. Our naming convention thus indicates which aspects of the system are coupled in the model.
We have carried out all three experiments with the HadCM3LC model (Cox et al. 2001) using a scenario of CO2 increasing at 1% yr−1, which is a standard idealized experiment for AOGCM studies. Temperature change T(t) in the fully coupled experiment is similar to the sum of the biogeochemically and radiatively coupled experiments (Fig. 3a), with most of T arising from radiative forcing; that is, climate change is largely suppressed in the biogeochemically coupled experiment. However, there is some small climate change in this experiment, which is also noted by Cox et al. (2000). Climate change can be caused by modification to surface properties, such as albedo, arising from changes in the characteristics and distribution of vegetation in response to higher CO2 concentration (Matthews 2007). This is not significant in our biogeochemically coupled experiment. The warming is caused by the physiological response of stomatal closure to elevated CO2 concentrations (Sellers et al. 1996), which effectively gives rise to a small radiative forcing from CO2 (Doutriaux-Boucher et al. 2009; Dong et al. 2009).
Carbon uptake Cu(t) in the fully coupled experiment is less than the sum of the other two experiments (Fig. 3b). The difference increases with time; at the end of the experiments, the fully coupled experiment is two-thirds that of the sum. Matthews (2007) followed a similar experimental design, considering only the terrestrial carbon cycle, and in contrast found that the carbon uptake in the fully coupled experiment was greater than the sum of the other two. In either case, the nonlinear combination of the concentration–carbon and climate–carbon responses is an obstacle to their reliable diagnosis, so we advocate that fully, radiatively, and biogeochemically coupled experiments should be carried out with other models.
Friedlingstein et al. (2006) note that the climate–carbon gain factor gCC ∝ −γ/(1 + β) [Eq. (18)] increases with time during their experiments (their Fig. 2b) in all models, by various amounts. They attribute this to the inconstancy of γ exhibited by their Figs. 2e,f, which show that the carbon released from land and ocean resulting from climate change rises more rapidly than linearly with T, that is, γ becomes increasingly negative. In addition, Friedlingstein et al.’s Figs. 2c,d show a weak tendency for the carbon taken up on account of increasing atmospheric CO2 concentration to rise less rapidly than linearly with C; that is, β decreases.
The inconstancy of β and γ indicates that the linear formulas Cβ(C) = βC and Cγ(T) = γT are inadequate, but it does not necessarily contradict the assumption that Cβ is a function of C alone and Cγ of T alone. This can be tested only by trying different scenarios, such that given values of C and T are approached along various trajectories. We have carried out biogeochemically coupled experiments with both HadCM3LC and the IPSL CM4 LOOP (Cadule et al. 2008, manuscript submitted to Proc. Natl. Acad. Sci. USA), with CO2 increasing at 0.5% and 2% yr−1 as well as at 1% yr−1. Comparison of these experiments shows that Cβ is not a unique function of C (Fig. 4); in each model, for a given C, uptake is greater with a smaller rate of increase of CO2. This is because the processes of land and ocean uptake are not instantaneous, as would have to be the case for carbon uptake Cu to be a function of state. The results also confirm that Cβ is not proportional to C; for a given increment in C, uptake becomes smaller as C increases, that is, β decreases, as in the C4MIP experiments.
By contrast, Cγ is more nearly a unique and linear function of T in radiatively coupled experiments (Fig. 5; these experiments were done only with HadCM3LC), although for a given T, there is a relatively weak tendency for carbon release to be greater with a smaller rate of increase of CO2. Our result that Cγ ∝ T suggests that a constant γ is a reasonable approximation for these scenarios.




As noted above, there is some climate change in the biogeochemically coupled experiment (Fig. 3a) resulting from the physiological response of vegetation to CO2 and the change in vegetation cover, rather than from CO2 radiative forcing. In the biogeochemically coupled experiment, the carbon uptake is therefore actually C′u = β′C + γT ′. This adds −γT ′ to γ̂, tending to make it less negative because γ < 0. On the other hand, the estimate of β by the C4MIP method β̂ = C′u/C = β′ + γT ′/C will be less positive than β′.
If the other C4MIP models behave qualitatively like HadCM3LC, there are compensating errors in the equivalent climate feedbacks terms rβ and rγ (section 5 and Table 3) that reduce the magnitude of both, and rγ includes a (negative) contribution from their interaction. These complications in the diagnosis of carbon cycle feedbacks from C4MIP results arise because Cγ can be estimated only as a difference if the radiatively coupled experiment is not carried out. We think that they underline the need to run all three experiments. It is essential that the analysis techniques are not simplified to the point of giving misleading results for the sensitivities of carbon–cycle components.
7. Metrics of the climate carbon system








However, unforced interannual variability can lead to large variations in the incremental airborne fraction, causing it to fluctuate around the cumulative airborne fraction (Fig. 7.4b of Denman et al. 2007). Furthermore, as we have seen (section 6), the assumptions of the C4MIP analysis do not hold exactly. Consequently, the incremental and cumulative airborne fractions are neither constant (cf. Denman et al. 2007) nor equal. As β decreases, A and Ai increase [Eq. (15)]. However, we find that in our fully coupled experiments with HadCM3LC and IPSL-CM4-LOOP, when following different rates of CO2 increase, the incremental airborne fraction is a more scenario-independent function of CO2 (Fig. 6) than of time, though this holds better for HadCM3LC than IPSL-CM4-LOOP. Furthermore, it holds better for Ai than A, which may indicate that Ai is actually more nearly a function of the instantaneous state than A (cf. Boer and Arora 2009). In both models, the airborne fraction is greater at any given CO2 concentration for larger rates of atmospheric CO2 increase, consistent with our earlier observation (section 6) that carbon uptake is less for larger rates of increase.
For a prescribed C(t), there is systematic uncertainty in Ai, and hence in RE(t). Although the decomposition of the airborne fraction into components from the concentration–carbon and climate–carbon feedbacks is quantitatively only as accurate as the assumptions of the C4MIP analysis, it is very likely that our earlier conclusion still holds for the relative sizes of their uncertainties (section 4). Hence, the concentration–carbon feedback dominates the uncertainty in determining the emissions consistent with a given concentration scenario.
If the policy objective is stated in terms of a climate change scenario instead of a concentration scenario, we will want to know the relationship between T and carbon emissions, including carbon cycle uncertainties. The re-lationship of T to carbon concentrations is often measured by the equilibrium climate sensitivity F2×/λ. Having represented the carbon cycle as contributions to climate feedback λ, we might think of defining an analogous quantity F2×/(λ + rβ + rγ). We might suppose this to be the eventual steady-state warming resulting from the emission of an amount CE2× of carbon equal to the initial carbon content of the atmosphere, because F2× = F(CE2×). Actually, on the long time scale of approach to a steady state, β and γ are certainly not constant. The eventual steady-state T will probably be zero, because C will return to zero over millennia, resulting from slow processes of marine carbon storage (Denman et al. 2007, their Table 7.3). Possibly there is irreversibility in the system, such that C would approach some nonzero value, but the processes determining this would not be correctly reflected by β and γ evaluated from projections under SRES scenarios. If the terrestrial carbon cycle alone were considered, disregarding slow processes of carbon storage on land as well as carbon uptake by the ocean, there would be a nonzero steady-state T and C.


Matthews et al. (2009) propose a related metric to TCRE, which they call the climate–carbon response. In effect, this quantity is TCRE per gigaton of carbon emitted, TCRE/CE2× = ϕA/ρ (K GtC−1). It avoids the arbitrary choice of a particular cumulative emission and suggests greater generality. To avoid confusion, we note that the climate–carbon response depends on both the climate–carbon and the concentration–carbon feedback, not just on the former.
TCRE and the climate–carbon response of Matthews et al. (2009) are not constant in general, because none of the factors ϕ, A, and ρ are really constant. However, Matthews et al. point out that in the C4MIP experiments, while emissions are sustained and CO2 is rising, ϕA/ρ is approximately constant, because ϕ declines and A increases as C rises, leading to a remarkably accurate and fortuitous cancellation of these independent variations in time. Moreover, ϕA/ρ is fairly independent of the emissions scenario, because of compensating variations in A and ρ; as we found above, A is larger for greater rates of emission, mainly related to terrestrial carbon uptake, but so is ρ (W m−2 K−1), because ocean heat uptake is less efficient for a faster increase in radiative forcing (Gregory and Forster 2008). Because ϕA/ρ in C4MIP models is therefore nearly time and scenario independent, T is proportional to cumulative carbon emissions (as noted by Van Vuuren et al. 2008; Matthews et al. 2009). However, once emissions cease, CE becomes constant, but C and T slowly fall thereafter (Meehl et al. 2007; Solomon et al. 2009), so A and T per gigaton of carbon emitted must eventually tend to zero.
8. Summary and conclusions
Perturbations to the carbon cycle could have a substantial influence on future climate change and atmospheric CO2 concentration, through the concentration–carbon feedback, resulting from the uptake of carbon by land and ocean as a biogeochemical response to the CO2 concentration, and the climate–carbon feedback, resulting from the effect of climate change on carbon fluxes.
In this paper we have demonstrated the similarities between the formalisms used to describe climate change and carbon cycle change. In both cases, the response to forcing can be expressed in two ways, which we call resistance and gain. In the resistance form, a sum of the terms that are individually proportional to the change (in temperature T or CO2 concentration C) has the net effect of opposing the forcing. We prefer this form, which is nowadays more commonly used in analysis of climate feedback in response to imposed radiative forcing, because it is more suitable for comparing the terms and their uncertainties. For the carbon cycle there is an analogous decomposition, which measures its resistance to the imposed emissions. To make the decomposition, we follow the C4MIP intercomparison of coupled climate–carbon cycle models in assuming that the concentration–carbon response is proportional to C, the climate–carbon response is proportional to T, and that they can be combined linearly.
In the resistance form, with these assumptions, we show that the effect of the carbon cycle can be regarded as two extra climate feedback terms. This translation allows us to compare these feedbacks with others, and shows that in present models the climate–carbon feedback is positive for warming and is of similar size to the cloud feedback. The concentration–carbon feedback is negative, 4 times larger than the climate–carbon feedback, and more uncertain. The net carbon cycle feedback has a comparable uncertainty to the noncarbon climate response in modeling the climate response to a scenario of CO2 emissions. In an analogous way, climate feedback can be translated into a feedback on the carbon cycle. The concentration–carbon response is the dominant source of uncertainty in the allowable CO2 emissions, which are consistent with a given CO2 concentration scenario.
In the gain form for climate change or carbon cycle change, one of the terms is regarded as the basic response, and the others are treated as feedbacks that add to the imposed forcing and hence amplify the response. The designation of the basic term is somewhat arbitrary and the consequent amplifications (feedback factors) are not additive. The attention that has been given to the uncertainty of the climate–carbon gain may have distracted attention from the uncertainty in the concentration–carbon response.
We have revisited the design of the experiments and quantities used to measure the carbon cycle response by C4MIP. Using additional experiments with two models, we show that, contrary to the assumptions made above, the concentration–carbon and climate–carbon responses do not combine linearly, and the former in particular is dependent on scenario and time. Consequently, our feedback formalism applies only approximately, though we think it remains useful as an interpretative tool and indicative of the sources of uncertainty. We note that the transient climate response and the climate resistance, used to measure the noncarbon response of the climate system to radiative forcing (Gregory and Forster 2008), are likewise useful despite their not being truly constant. The inconstancy of the climate resistance, and the inaccuracy of the linear approximation for CO2 radiative forcing employed in this analysis (following Scheffer et al. 2006; Friedlingstein et al. 2006), mean that the translation between carbon cycle and noncarbon feedbacks is not exact, but this does not alter our qualitative conclusions.


We would make the following recommendations to improve the quantification of the carbon cycle feedbacks in models:
We propose that three experiments should be carried out for each scenario, which we call fully coupled (C4MIP coupled), biogeochemically coupled (C4MIP uncoupled, in which the CO2 increase has no radiative forcing), and radiatively coupled (in which the CO2 increase has no biogeochemical effect). The radiatively coupled experiment is additional to the C4MIP design. It allows the concentration–carbon and climate–carbon effects to be separated cleanly. Insofar as the response to radiative forcing is independent of the nature of the forcing agent, this experiment can be regarded also as a simulation of the climate and carbon cycle response to non-CO2 greenhouse gas forcing, which has relevance to the analysis of multigas mitigation scenarios.
We support the recommendation of Hibbard et al. (2007), adopted in this work and by Plattner et al. (2008), to use scenarios of prescribed concentration rather than emission, because it simplifies the analysis of carbon uptake.
We advocate that carbon cycle metrics should be evaluated using the scenario of prescribed atmospheric CO2 concentration increasing at 1% yr−1. This idealized scenario implies a similar magnitude of emissions to typical detailed socioeconomic scenarios for the twenty-first century, and has the advantage of simplicity. It has been a standard for several years for studies of climate response in AOGCMs and is likely to remain in use, whereas the policy-related scenarios are updated more frequently. Adopting the 1% scenario for carbon cycle studies will facilitate comparison of the climate and carbon feedback terms.
We suggest that the airborne fraction of CO2 emissions could be used to quantify the net effect upon the carbon cycle of all the climate and carbon feedbacks. As a precisely defined metric for model intercomparison, corresponding to the transient climate response (TCR), we propose the airborne fraction defined as the increase of atmospheric CO2 content relative to the initial state at the time of 2 × CO2, divided by the cumulative CO2 emissions up to that time, in the 1% scenario.
Acknowledgments
We are grateful for discussions with and comments from Peter Cox, Ron Stouffer, Karl Taylor, George Boer, Mark Webb, Mark Ringer, Keith Williams, Olivier Boucher, and Ben Booth, and to Damon Matthews, Gian-Kasper Plattner, and the third referee for their encouraging and constructive reviews. Jonathan Gregory was supported by the National Centre for Atmospheric Science, Climate Division. Work at the Hadley Centre was supported by the Joint DECC, Defra, and MoD Integrated Climate Programme, DECC/Defra (GA01101), MoD (CBC/2B/0417_Annex C5). Work at IPSL was supported by the ENSEMBLES project funded by the European Commission (GOCE-CT-2003-505539). Computer time was provided by CEA and CNRS.
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APPENDIX A
Non-CO2 Radiative Forcing
In this paper, we have confined our attention to CO2 forcing, for simplicity and because it is the dominant forcing in scenarios for the twenty-first century. Here, we consider non-CO2 forcing.






APPENDIX B
Relationship of C and T under CO2 forcing



Comparison of concentration–carbon and climate–carbon responses expressed as contributions to the ratio CE/C of the emitted carbon to the increase in atmospheric carbon content. This ratio is the reciprocal of the airborne fraction. Positive terms indicate positive storage of carbon when C increases. The land and ocean contributions are as calculated in this paper from the C4MIP models of Friedlingstein et al. (2006). The bars indicate approximate 5%–95% confidence intervals (mean ± 1.65 × SD). There is no uncertainty in the atmospheric contribution, which is unity by definition.
Citation: Journal of Climate 22, 19; 10.1175/2009JCLI2949.1

Comparison of concentration–carbon and climate–carbon responses expressed as contributions to the ratio CE/C of the emitted carbon to the increase in atmospheric carbon content. This ratio is the reciprocal of the airborne fraction. Positive terms indicate positive storage of carbon when C increases. The land and ocean contributions are as calculated in this paper from the C4MIP models of Friedlingstein et al. (2006). The bars indicate approximate 5%–95% confidence intervals (mean ± 1.65 × SD). There is no uncertainty in the atmospheric contribution, which is unity by definition.
Citation: Journal of Climate 22, 19; 10.1175/2009JCLI2949.1
Comparison of concentration–carbon and climate–carbon responses expressed as contributions to the ratio CE/C of the emitted carbon to the increase in atmospheric carbon content. This ratio is the reciprocal of the airborne fraction. Positive terms indicate positive storage of carbon when C increases. The land and ocean contributions are as calculated in this paper from the C4MIP models of Friedlingstein et al. (2006). The bars indicate approximate 5%–95% confidence intervals (mean ± 1.65 × SD). There is no uncertainty in the atmospheric contribution, which is unity by definition.
Citation: Journal of Climate 22, 19; 10.1175/2009JCLI2949.1

(top) A comparison of components of climate feedback and ocean heat uptake, and (bottom) a comparison of the combined noncarbon response (the climate resistance, the sum of the terms in the top part) with the carbon cycle feedbacks for forcing resulting from CO2 emissions. The blackbody, cloud, surface albedo, and WV + LR (water vapor and lapse rate) terms are from Soden and Held (2006). The heat uptake term is the ocean heat uptake efficiency κ from Gregory and Forster (2008) and the climate resistance is their ρ evaluated from CMIP3 AOGCMs, with its uncertainty amplified by the concentration–carbon feedback as described in the text. The carbon cycle contributions are as calculated in this paper from the C4MIP models of Friedlingstein et al. (2006). We show climate feedback parameters as −λi, so that positive terms tend to increase climate warming for a positive forcing; the blackbody response is shown as a negative climate feedback −λBB, and the ocean heat uptake efficiency and climate resistance are likewise negative terms. The bars indicate approximate 5%–95% confidence intervals (mean ± 1.65 × SD).
Citation: Journal of Climate 22, 19; 10.1175/2009JCLI2949.1

(top) A comparison of components of climate feedback and ocean heat uptake, and (bottom) a comparison of the combined noncarbon response (the climate resistance, the sum of the terms in the top part) with the carbon cycle feedbacks for forcing resulting from CO2 emissions. The blackbody, cloud, surface albedo, and WV + LR (water vapor and lapse rate) terms are from Soden and Held (2006). The heat uptake term is the ocean heat uptake efficiency κ from Gregory and Forster (2008) and the climate resistance is their ρ evaluated from CMIP3 AOGCMs, with its uncertainty amplified by the concentration–carbon feedback as described in the text. The carbon cycle contributions are as calculated in this paper from the C4MIP models of Friedlingstein et al. (2006). We show climate feedback parameters as −λi, so that positive terms tend to increase climate warming for a positive forcing; the blackbody response is shown as a negative climate feedback −λBB, and the ocean heat uptake efficiency and climate resistance are likewise negative terms. The bars indicate approximate 5%–95% confidence intervals (mean ± 1.65 × SD).
Citation: Journal of Climate 22, 19; 10.1175/2009JCLI2949.1
(top) A comparison of components of climate feedback and ocean heat uptake, and (bottom) a comparison of the combined noncarbon response (the climate resistance, the sum of the terms in the top part) with the carbon cycle feedbacks for forcing resulting from CO2 emissions. The blackbody, cloud, surface albedo, and WV + LR (water vapor and lapse rate) terms are from Soden and Held (2006). The heat uptake term is the ocean heat uptake efficiency κ from Gregory and Forster (2008) and the climate resistance is their ρ evaluated from CMIP3 AOGCMs, with its uncertainty amplified by the concentration–carbon feedback as described in the text. The carbon cycle contributions are as calculated in this paper from the C4MIP models of Friedlingstein et al. (2006). We show climate feedback parameters as −λi, so that positive terms tend to increase climate warming for a positive forcing; the blackbody response is shown as a negative climate feedback −λBB, and the ocean heat uptake efficiency and climate resistance are likewise negative terms. The bars indicate approximate 5%–95% confidence intervals (mean ± 1.65 × SD).
Citation: Journal of Climate 22, 19; 10.1175/2009JCLI2949.1

(a) Global-mean, annual-mean surface air temperature change T, and (b) carbon uptake by land and ocean Cu, as a function of time relative to the control climate in fully, biogeochemically, and radiatively coupled experiments with the HadCM3LC climate–carbon model under a scenario of atmospheric CO2 increasing at 1% yr−1.
Citation: Journal of Climate 22, 19; 10.1175/2009JCLI2949.1

(a) Global-mean, annual-mean surface air temperature change T, and (b) carbon uptake by land and ocean Cu, as a function of time relative to the control climate in fully, biogeochemically, and radiatively coupled experiments with the HadCM3LC climate–carbon model under a scenario of atmospheric CO2 increasing at 1% yr−1.
Citation: Journal of Climate 22, 19; 10.1175/2009JCLI2949.1
(a) Global-mean, annual-mean surface air temperature change T, and (b) carbon uptake by land and ocean Cu, as a function of time relative to the control climate in fully, biogeochemically, and radiatively coupled experiments with the HadCM3LC climate–carbon model under a scenario of atmospheric CO2 increasing at 1% yr−1.
Citation: Journal of Climate 22, 19; 10.1175/2009JCLI2949.1

Carbon uptake by land and ocean relative to the control climate in annual means from biogeochemically coupled experiments with (a) HadCM3LC and (b) IPSL-CM4-LOOP, plotted against the atmospheric CO2 concentration (1 ppm = 2.1 GtC). Results are shown for CO2 increasing at three different rates of 0.5%, 1%, and 2% yr−1.
Citation: Journal of Climate 22, 19; 10.1175/2009JCLI2949.1

Carbon uptake by land and ocean relative to the control climate in annual means from biogeochemically coupled experiments with (a) HadCM3LC and (b) IPSL-CM4-LOOP, plotted against the atmospheric CO2 concentration (1 ppm = 2.1 GtC). Results are shown for CO2 increasing at three different rates of 0.5%, 1%, and 2% yr−1.
Citation: Journal of Climate 22, 19; 10.1175/2009JCLI2949.1
Carbon uptake by land and ocean relative to the control climate in annual means from biogeochemically coupled experiments with (a) HadCM3LC and (b) IPSL-CM4-LOOP, plotted against the atmospheric CO2 concentration (1 ppm = 2.1 GtC). Results are shown for CO2 increasing at three different rates of 0.5%, 1%, and 2% yr−1.
Citation: Journal of Climate 22, 19; 10.1175/2009JCLI2949.1

Carbon uptake (actually release when negative) by land and ocean relative to the control climate in decadal means from radiatively coupled experiments with HadCM3LC, plotted against global-mean surface air temperature change. Results are shown from experiments with CO2, increasing at three different rates of 0.5%, 1%, and 2% yr−1.
Citation: Journal of Climate 22, 19; 10.1175/2009JCLI2949.1

Carbon uptake (actually release when negative) by land and ocean relative to the control climate in decadal means from radiatively coupled experiments with HadCM3LC, plotted against global-mean surface air temperature change. Results are shown from experiments with CO2, increasing at three different rates of 0.5%, 1%, and 2% yr−1.
Citation: Journal of Climate 22, 19; 10.1175/2009JCLI2949.1
Carbon uptake (actually release when negative) by land and ocean relative to the control climate in decadal means from radiatively coupled experiments with HadCM3LC, plotted against global-mean surface air temperature change. Results are shown from experiments with CO2, increasing at three different rates of 0.5%, 1%, and 2% yr−1.
Citation: Journal of Climate 22, 19; 10.1175/2009JCLI2949.1

(a),(c) Incremental and (b),(d) cumulative airborne fraction as a function of atmospheric CO2 concentration in decadal means from fully coupled experiments with (a),(b) HadCM3LC and (c),(d) IPSL-CM4-LOOP. Results are shown from experiments with CO2 increasing at three different rates of 0.5%, 1%, and 2% yr−1.
Citation: Journal of Climate 22, 19; 10.1175/2009JCLI2949.1

(a),(c) Incremental and (b),(d) cumulative airborne fraction as a function of atmospheric CO2 concentration in decadal means from fully coupled experiments with (a),(b) HadCM3LC and (c),(d) IPSL-CM4-LOOP. Results are shown from experiments with CO2 increasing at three different rates of 0.5%, 1%, and 2% yr−1.
Citation: Journal of Climate 22, 19; 10.1175/2009JCLI2949.1
(a),(c) Incremental and (b),(d) cumulative airborne fraction as a function of atmospheric CO2 concentration in decadal means from fully coupled experiments with (a),(b) HadCM3LC and (c),(d) IPSL-CM4-LOOP. Results are shown from experiments with CO2 increasing at three different rates of 0.5%, 1%, and 2% yr−1.
Citation: Journal of Climate 22, 19; 10.1175/2009JCLI2949.1
Notation and units for main quantities discussed. Note that C is often given in ppm and β in GtC ppm−1: 1 GtC ≡ 0.47 ppm of C. Following electronics literature, the terms gain factor and feedback factor (both dimensionless) are sometimes used the other way round (Roe and Baker 2007); however, feedback parameter (W m−2 K−1) is unambiguous.


Summary of main results for climate and carbon response and feedbacks. References are to equation numbers in the text. These formulas apply to time-dependent change in which constant ocean heat uptake efficiency κ is a good approximation.


The upper section of the table shows parameters quantifying climate and carbon responses in the earth system models of C4MIP (Friedlingstein et al. 2006), computed from the values in their Table 3, using formulas from our Table 2. Details of all the models are given by Friedlingstein et al. (2006). The parameters are defined in Table 1, except for λβγ, which is defined in appendix A; ρ, rβ, rγ, and λβγ are in W m−2 K−1; β, uγ, A, and gCC are dimensionless; γ is in GtC K−1. SD is the standard deviation across models, and CV is the coefficient of variation, that is, the ratio of the SD to the magnitude of the mean, expressed as a percentage. Following Eq. (14), β can be interpreted as the concentration–carbon response parameter, that is, uβ ≡ β, corresponding to the climate–carbon response parameter uγ. The lower section of the table shows the climate and carbon contributions to the climate response to CO2 emissions, evaluated by combining the climate resistance of CMIP3 AOGCMs with the carbon cycle parameters of the C4MIP models.

