The response of the global climate–carbon-cycle system to anthropogenic perturbations happens differently at different timescales. The unravelling of the memory structure underlying this timescale dependence is a major challenge in climate research. Recently the widely applied

The global carbon cycle plays a key role in determining the sensitivity of Earth's climate to anthropogenic emissions from fossil-fuel burning, cement production, and land-use change. The increase in atmospheric CO

To gain insight into this combined dynamics of carbon cycle and climate, one must in particular study climate–carbon-cycle feedbacks. Such feedbacks arise from the already mentioned reaction of the global carbon cycle to a change in atmospheric CO

Depending on the speed at which the various feedbacks unfold, climate change may develop differently. Generally, the dynamics of the coupled climate–carbon-cycle system arising in response to perturbations depends on the spectrum of internal timescales of the various processes involved in the response. For instance, the speed at which global climate is warming in reaction to anthropogenic emissions depends largely on the rate at which the oceans can take up heat, and this rate – actually an inverse timescale – is determined by the temporal characteristics of the internal ocean dynamics, like the rate of mixing between upper and deep ocean and the speed at which heat is re-distributed by ocean currents. Similar remarks apply to the uptake and re-distribution of CO

Concerning the analysis of feedbacks, a large step forward was the seminal work by

Quantitative results on the size of global climate–carbon-cycle feedbacks were particularly obtained as part of the Coupled Climate–Carbon Cycle Model Intercomparison (C

The size of these feedbacks depends on the considered timescale

In the present study we employ this generalized framework to study the role of feedbacks and their timescale dependence for airborne fraction. Airborne fraction is classically defined as the fraction of emitted CO

To gain further insight into the timescale dependence of the airborne fraction and in particular how this timescale dependence emerges from the underlying feedbacks, in the present study we analyse by means of the generalized

Most of our present study relies on a single theoretical result of the generalized

Overall, our analysis of the simulation results from the considered set of CMIP5 models will show that one can understand the dynamics of the airborne fraction from the behaviour of the climate–carbon-cycle feedbacks and that it is possible to pinpoint the particular feedback that dominates the observed model spread in the airborne fraction at different timescales. Moreover, it will become clear that by applying the generalized

The outline of the paper is as follows. In the next section we introduce the generalized

To prepare for our investigation of the timescale dependence of the airborne fraction and the underlying feedbacks, we introduce here the generalized

To introduce this framework, we start from the carbon balance equation that couples the different subsystems of the global carbon cycle:

As discussed above, atmospheric CO

In the absence of feedbacks the atmospheric change in carbon content

Note that our reference system is different from that used in

The total feedback function

Concerning the timescale dependence it is important to note that in Eq. (

Such independent behaviour is also the reason for the identical structure of the Laplace-transformed formulas of the generalized

Note also that the timescale dependence of feedbacks cannot be obtained from the original

Our main topic in this study is the timescale dependence of the airborne fraction. As explained in the following, by the generalized framework this timescale dependence can be fully traced back to that of the feedback functions. In its standard definition

This follows by deriving Eq. (

To relate the generalized airborne fraction to the feedbacks, Eq. (

To follow our subsequent investigation of airborne fraction it is important to note that the generalized

The present section prepares for the main investigation of our study (next section). This involves two issues. The first was already shortly addressed at the end of the previous section, namely that we have to demonstrate the predictive power of Eq. (

The second preparatory issue tackled in this section concerns the technical aspects of the calculation of the generalized sensitivities from simulation data. As explained in

To derive the generalized sensitivities we employ our recently developed response function identification (RFI) method

To address complication (ii), we employ following

To demonstrate the predictive power of the generalized framework, all these technical issues must be tackled before we can invoke Eq. (

As explained above, to demonstrate that indeed the timescale-resolved airborne fraction

From given time series for atmospheric carbon content

We thank the reviewer Ian Enting for making us aware that

By comparing now Eq. (

By these considerations, the true airborne fraction

For our demonstration of predictive power we performed impulse-type emission-driven experiments with MPI-ESM and obtained

The second step towards the demonstration of the predictive power of the generalized framework is to calculate the predicted airborne fraction by application of Eq. (

To predict airborne fraction via the total feedback function

To determine the generalized sensitivities from the simulation data we once more invoke our RFI method

Actually, as already explained in the introduction to this section, to obtain linear response functions reliably by the RFI method, additional preparatory effort is needed concerning selection of a pre-processing technique and checks assuring that the underlying linearity assumption is valid for the simulation data used. For those purposes we performed additional rad and bgc simulations with MPI-ESM for a variety of different CO

The final steps to obtain the generalized airborne fraction as predicted by Eq. (

Please note that the way of deriving here the predicted generalized airborne fraction for MPI-ESM is exactly how we derive it also for the other CMIP5 models in the next section, except that the additional preparatory analysis and checks cannot be performed because of the lack of the necessary additional simulations. Accordingly, we will assume that the size of the linear regime obtained for MPI-ESM applies also to these other models and will pre-process also their data by the technique identified to be best for MPI-ESM (see summary of linear regime and best pre-processing technique for each sensitivity in Table

So far in this section, the generalized airborne fraction

Note that for our analysis we calculated

The discrepancy between the two estimates of the airborne fraction observed at timescales shorter than 5 years is expected from two types of error that might have affected the results. The first type affects the predicted airborne fraction and arises from the ill-posedness of the deconvolution problem that must be solved to derive the generalized sensitivities employed in Eq. (

Overall, the close agreement between the two estimates of the airborne fraction demonstrates the predictive power of the generalized

Quality of agreement between the true generalized airborne fraction (Eq.

In the present section we extend our analysis of the timescale dependence of airborne fraction to the set of CMIP5 models listed in Table

CMIP5 data considered in this study. For a description of the experiments please see Table

The whole investigation is based on the calculation of the generalized airborne fraction by means of Eq. (

In this subsection we present our results for the generalized sensitivities of the considered CMIP5 models. The robustness of the recovered sensitivities depends on the quality of the data

We start by discussing the identified generalized sensitivities. In Fig.

Generalized sensitivities (see definition in Eqs.

There is a close agreement between the obtained generalized ocean sensitivities

Figure

In contrast to

The magnitude of

Finally, Fig.

We now turn to the analysis of the plausibility of the recovered generalized sensitivities by means of the prediction of the original

The results of these calculations are shown in Fig.

The results for MPI-ESM-LR can be considered a reference for the achievable agreement between data-derived and predicted sensitivities because for MPI-ESM-LR the generalized sensitivities were obtained in a quality-controlled way by means of additional simulations (see Appendix

Such systematic deviations in

Overall, we consider the results of this comparison as sufficiently convincing to add confidence to the validity of the recovered generalized sensitivities (Fig.

The

Before the main question of this study on the role of feedbacks for airborne fraction can finally be addressed in the next section, another preparatory step is necessary. Key to investigate this question will be Eq. (

To check additivity, we plot in Fig.

Check of the additivity of the biogeochemical and radiative carbon responses in CMIP5 models that underlies the generalized

In this section we tackle the main question of our study, namely how the climate–carbon-cycle feedbacks shape the timescale dependence of the generalized airborne fraction. From here on we take for granted that by the methods presented in the previous sections Eq. (

In Fig.

Airborne fraction and climate–carbon-cycle feedbacks in CMIP5 models as derived by the generalized framework (Eqs.

How the airborne fraction changes in the timescale domain is determined by the climate–carbon-cycle feedbacks. As seen in Fig.

In the mean over all models, the land biogeochemical feedback is at all timescales longer than the ocean biogeochemical feedback: at a timescale of 10 years, it is 1.4 times larger, and at a timescale of 100 years, it is 1.8 times larger. The picture is qualitatively similar for the radiative feedback: at a timescale of 10 years, the land feedback is, despite its small value of 0.03, orders of magnitude larger than the almost absent ocean feedback, and at a timescale of 100 years, the land feedback is 7.4 times larger than its ocean counterpart. Aggregating land and ocean, the mean of the biogeochemical feedback is 22 times larger than the radiative feedback at a timescale of 10 years and 5.6 times larger at a timescale of 100 years. These results are in particular at short timescales in contrast to previous estimates

By Fig.

Analysis of model spread of feedback functions and their influence on the airborne fraction.

An even clearer view about the impact of the different feedbacks on the airborne fraction may be gained by artificially changing the values of these feedbacks to study hypothetical situations and then evaluating the resulting change in the airborne fraction. For instance, one can illustrate how strongly the model spread in the airborne fraction depends on the spread in the land biogeochemical feedback by recalculating the statistics of the airborne fraction taking

The dynamics of the global carbon cycle can be understood in terms of feedbacks arising via the land and ocean carbon cycle when atmospheric CO

Here, we employed this generalized framework to study the timescale dependence of the climate–carbon-cycle feedbacks and the associated airborne fraction for an ensemble of CMIP5 models. In Sect.

Based on experience with MPI-ESM, we quantified in Sect.

Considering global carbon, in the model mean the biogeochemical feedback was found to be 22 times larger than the radiative feedback at a 10-year timescale and 5.6 times larger at a 100-year timescale. This result suggests that at least over shorter timescales the difference between these feedbacks may be even greater than previously thought

The influence of the model spread of the different feedbacks on the airborne fraction was also investigated. It was found that the spread in the airborne fraction arises mostly from the spread in the land biogeochemical feedback, especially for timescales below 30 years. By considering the hypothetical case where this particular feedback would be equal to the model mean, we found that the spread in the airborne fraction would decrease by 82 % at a 10-year timescale and by 61 % at a 100-year timescale, which demonstrates even more clearly that it is indeed the land biogeochemical feedback that causes the spread in airborne fraction between the different models.

While the generalized framework was shown here to reasonably describe the linear dynamics of the global carbon cycle in MPI-ESM, the results obtained for the other CMIP5 models depend on two basic assumptions. The first is that the generalized sensitivities in the CMIP5 models are recoverable with sufficient quality by the same numerical approaches that were appropriate to recover the sensitivities in MPI-ESM. This involves the assumption that for all other considered CMIP5 models the linear perturbation regime is of similar extent to that found for MPI-ESM for the different response variables invoked to recover the sensitivities. This might not be the case – and is probably not for

With these cautionary remarks in mind, our conclusion that the spread in the airborne fraction arises mostly from the spread in the land biogeochemical feedback corroborates the recent finding by

Estimates of the timescale-resolved airborne fraction, by means of Eq. (

As explained in Sect.

Besides investigating the timescale dependence of airborne fraction, our study also demonstrated for MPI-ESM the predictive power of the generalized framework (see Sect.

Furthermore, the generalized framework may be invoked to investigate the contribution of the different feedbacks to committed climate change, where one is interested in understanding the behaviour of the system once atmospheric CO

One aspect emphasized throughout this study is that the generalized framework is valid only for weak perturbations. In fact, we have found in application to the MPI-ESM that the linear regime extends only up to about 100 ppm atmospheric CO

Finally it may be noted that our study is an example for the application of linear response theory – known from statistical mechanics

This appendix complements the results from

Since the land carbon sensitivities

Select a technique to pre-transform the data to account for known non-linearities in the response. Accounting for these non-linearities allows for recovering the generalized sensitivity from experiments with higher perturbation strengths and thus higher signal-to-noise ratio, which improves the quality of the results.

Determine the maximum forcing strength for which no strong non-linearities are present in the response. This gives the best trade-off between signal-to-noise ratio and non-linearity for a particular pre-transformed response data, thereby further improving the quality of the recovery.

Calculate the linear regime of the response, i.e. the range of forcing strengths for which the generalized sensitivity can be used to predict the response of the system. By analysing this linear regime for all generalized sensitivities we will be able to determine the overall linear regime for which the generalized

The final result of this analysis is summarized in Table

C

To perform the analysis described in the three steps above we employ a simple procedure introduced in

Similarly to

Since Eq. (

In the second technique, we consider the logarithm of

This log-transform technique is inspired by the fact that non-linearities in the ocean carbon uptake come mainly from the dissolution of CO

For both techniques

Using the pre-transformation techniques described above, in the following we recover the generalized sensitivity

We start by deriving

Generalized sensitivity

To try to improve the quality of the recovery,

But despite the overall reduction in the prediction error, Fig.

To obtain evidence that the recovered

Figure

Despite these discrepancies, overall Fig.

The conclusions from this subsection therefore suggest that the best pre-transformation technique to derive

In this subsection we recover

We start by recovering

Generalized sensitivity

The resulting

In summary, since the response can be considered linear over the whole 1 % rad experiment, we chose to derive

Following Sect.

Because CO

We start by recovering

Generalized sensitivity

Generalized sensitivity

To assess whether the log-transform technique (Eqs.

Therefore in Figs.

To obtain evidence that the recovered

Note that linear response functions characterize only the ensemble average of the response

As expected from the known logarithmic relationship between radiative forcing and CO

Techniques identified to derive the generalized sensitivities in Sects.

As in Sect.

In summary, the conclusions from this subsection suggest that the best approach to derive

To recover

To derive

With the results presented in the preceding subsections we can now summarize the best techniques identified to recover the generalized sensitivities for our study. A general summary of the identified techniques is given in Table

Because the linear regime for the biogeochemical response of land and ocean carbon is restricted to forcing strengths even smaller than that for temperature responses (

With this subsection we complete the recovery of all generalized sensitivities for the MPI-ESM. The approaches selected here are employed to recover the generalized sensitivities for all CMIP5 models in the main text.

In this appendix we explain in detail how we derived the airborne fraction

Response function

In the first step, we recovered

The resulting impulse response after taking the ensemble average and the fit by the recovered

To make sure that the impulse response is within the linear regime and therefore that the recovery of

Hence, overall these results suggest that the recovery of

Further evidence of the reliability of our numerics is obtained by examining the agreement of the resulting airborne fraction (Fig.

This appendix complements Appendix

After obtaining

To understand how the constraint can be enforced, one has to consider

Knowing how to discretely account for the desired constraint the spectrum

The response function

Show

Show

When calculating the

In the present appendix we demonstrate exemplarily for MPI-ESM that from the generalized airborne fraction

To compute the true standard airborne fraction

To predict now the standard airborne fraction

The results are compared in Fig.

Such large variability in

Prediction of standard airborne fraction

In this appendix we show how from

Using a single global temperature,

Taking a single global temperature,

Since the Laplace-transformed formulation of the generalized framework is completely analogous to that of the original

The scripts and data used to produce the results in this paper can be found at

GLTM led the study, performed the analysis, and wrote the first draft. All authors contributed to the conception and scientific content of the study. The final manuscript was jointly prepared by GLTM and CHR.

The contact author has declared that none of the authors has any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

We would like to thank Matteo Puglini, Peter Landschützer, Ian Enting, and Vivek Arora for their helpful comments on the manuscript and Ludwig Lierhammer for preparing the CMIP5 data used in the study. Guilherme L. Torres Mendonça was supported by the Max Planck Institute for Meteorology and by research grant no. 2023/04579-5 from the São Paulo Research Foundation (FAPESP).

Guilherme L. Torres Mendonça was supported by the Max Planck Institute for Meteorology and by research grant no. 2023/04579-5 from the São Paulo Research Foundation (FAPESP). The article processing charges for this open-access publication were covered by the Max Planck Society.

This paper was edited by Anja Rammig and reviewed by Ian Enting and Vivek Arora.