Review of Torres-Mendonca et al, bg-2023-101, by I. G. Enting

This paper (hereafter denoted TM23) is suitable for publication.  The authors may wish to consider the following comments. 

References cited below refer to those listed in the bibliography  of TM23 or additional  references listed below.

Some of my points are, for convenience, illustrated by citing my  own publications. This not in itself a suggestion that

these papers should be cited, nor that they are the most appropriate references. 

OVERVIEW

This paper considers feedbacks in the carbon cycle, expressing the gain  1/(1-f) from feedbacks  in terms of the total  feedback, f.

A generic result for linear feedbacks is that the combined loop feedback from multiple feedbacks is the sum of the feedbacks so

that linear feedback contributions can be partitioned or aggregated without restriction.

The analysis is in terms of the alpha, beta, gamma description (Friedlingstein et al, 2003), where carbon cycle feedback, f, is partitioned

into a concentration feedback, beta, and a radiative feedback alpha*gamma. Each of these is often further partitioned into

Land and Ocean contributions (and may be further partitioned, cf Enting and Clisby 2019 appendix).

The description by (Friedlingstein et al, 2003) is history-dependent. A scenario-independent form requires generalising each of alpha, beta, gamma  to functions which can be  combined by convolutions over time. 

These combinations take a simpler form as Laplace transforms.  (In this regard, as in the study by Enting and Clisby (2019), Laplace transform

expressions can be regarded as a compact expression for calculations that are actually performed in the time domain, analogous to the

way that vector expressions for electromagnetic fields define expressions that are ultimately calculated using specific components).

The paper uses results from the CMIP intercomparison to estimate  the generalised alpha, beta, gamma (in each case

partitioned into Land and Ocean) and combines them into a generalised airborne fraction.

 COMMENTS

Line 71. This notes characterising climate change in terms of a single global temperature change. This is the basis for "pattern scaling" which is

often used in impact assessments (Mitchell 2003, which should be cited). However this is not what is done later (see comment on eqn 4,5).

At some point between lines 155 and 175, it could be worth combining all the terms into one equation and write down the original 

alpha, beta, gamma  formalism for comparison Delta C [1+ beta +alpha * gamma ] = integrated emissions.

Equations 2 and 3.

These have several issues created by working with separate land and ocean

temperatures:

* it contradicts the initial statement about using a single global  temperature

* the numbers are going to differ from other work that has a single

global alpha and absorbs the land-ocean differences into gamma

* working with a single alpha simplifies the inclusion of other radiative forcing (CH4, volcanoes etc) which would be needed for possible

future work. Even within the context of the present work, including these terms helps make the important point that the carbon-climate

coupling amplifies these forcings by the same  factor that applies to amplification of the CO2 response determined by the beta term

(Gregory 2009, eqns 8a,b of Freidlingstein 2003).

The results shown in panels c and f of figure 2, suggest that for most, but not all, models the results are consistent with the assumptions

of pattern scaling, with the land response about twice the ocean response on all timescales. It would be interesting to analyse this

in more detail than what can be gleaned from low resolution plots-- eg a plot of the ratios of the two responses as a function of timescale.

Eqn 6. Introducing CO2 concentrations (rather than CO2 content as originally done by Friedlingstein et al 2003) (and thus introducing

the factor k) seems an unnecessary complication which hinders comparisons with other work. (Of course k is given by the mass 

of the atmosphere scaled by ratio of molecular weights).

Line 188 seems poor wording of the situation. You can work in the time domain (and in many cases do so), but you have to

incorporate history and not just single times (which is what Oeschger and Heimann pointed out in 1983).

Circa line 295. At around this point it could be helpful to have one or two sentences summarising the key aspects

of the RFI technique (and maybe a longer summary in the relevant appendix).

Eqn 15, (and associated definitions). What this means is that chi is the CO2 impulse response function that is widely used in the definition of

GWP and was the subject of an extensive intercomparisons by Joos et al (2013). This should be noted and Joos et al cited at this point.

Eqn 16, goes back at least to Enting (1990), as a Laplace transform.

As a relation for growth at single time-scale it is implicit in the results of Oeschger et al 1980.

The overall response for CO2 is given by 1/p/(1+beta(p) +alpha(p) * gamma(p) )  Enting (2009) suggested that models  could give similar fits to

20th century changes (p approx 0.02)  whether or not the gamma feedback  was included, simply by changing beta (specific examples were given).

While the CMIP data presented here do not give details of how the various models were calibrated, the results suggest that a similar inter-model

flexibility appears here.  

Panels d and e of figure 2 suggest that on timescales of about 50 years, the models with smaller positive beta have smaller negative gamma.

This interpretation is supported by figure 6a, where the difference  between the true AF spread and that from eqn (22) (which assumes that

the spreads are independent), suggests that the spreads of beta and gamma are not independent.

* POTENTIAL FUTURE EXTENSIONS

An indication of the quality of the work is the extent that it suggests potential future studies. Some possibilities  (realising that such work

may already be in progress) are:

1: Calculate the airborne fraction, dC/dt/E, as a function of time for various cases of RCP-SSP (Representative

Concentration Profiles - Shared Socio-economic Pathways). Non-linearity will limit the accuracy, but the least affected will

be the ones that are of most interest as likely to have the greatest change from near-constant airborne fraction.

2: Using the formalism to analyse the  CMIP historical runs.

These two extensions would require extending eqn (4,5) to include non-CO2 radiative forcing.

3: A more speculative possibility is the extension to response functions describing radiocarbon (C14). As noted by Enting and Clisby (2021)

and Enting (2022), expressions for the responses to exponential forcing were described by Oeschger et al (1980).

Oeschger et al also described corresponding responses for  C14 perturbations and how these relate to total carbon (see also Enting 1990).

Since such responses to exponential forcing  are the Laplace transforms of the impulse response

functions (eg see eqn 2.11 of Enting 2022), the Oeschger et al results suggest the possibility of defining generalised

sensitivities for C14. This would be of most interest in the analysis of historical trends.

MINOR POINTS

 Line 182: the PgC/ppm CO$_2$ should be in upright font, since these are not symbols representing mathematical variables.

Similarly, throughout, the labels A, O, L  should be in upright font.

In the figure captions, it may be helpful to the reader if the symbol (and maybe number of defining equation)

was included after the text description.

Note that actual (as opposed to CMIP model) human CO2 emissions  include cement production and not just fuel use and also land use

change and forestry.

ADDITIONAL REFERENCES

\bibitem[Enting(1990)]{enting90}

\newblock Ambiguities in the calibration of carbon cycle models.

\newblock \emph{Inverse Problems}, 6:\penalty0 L39--L46, 1990.

\bibitem[Enting((on line, 2009))]{enting09}

\newblock Seeking carbon-consistency in the climate-science-to-policy

  interface.

\newblock \emph{Biogeochemistry}, pages DOI 10.1007/s10533--009--9351--7, (on

  line, 2009).

\bibitem[Mitchell(2003)]{mitchell03}

\newblock Pattern scaling: An examination of the accuracy of the technique for

  describing future climates.

\newblock \emph{Climatic Change}, 60\penalty0 (3):\penalty0 217--242, 2003.

\newblock ISSN 0165-0009.

\newblock \doi{10.1023/A:1026035305597}.

\bibitem[Oeschger et~al.(1980)Oeschger, Siegenthaler, and Heimann]{oeschger80}

\newblock The carbon cycle and its perturbations by man.

\newblock In W.~Bach, J.~Pankrath, and J.~Williams, editors, \emph{Interactions

  of Energy and Climate}, pages 107--127. Reidel, Dordrecht, 1980.

Authors present a new framework for representing the carbon feedbacks in the climate system that takes into account the history/memory of the system by using a convolution function based on Volterra series. This is indeed a new development that is welcome. The paper is written extremely well and should be published. I only have minor comments to improve the readability/clarity of the paper. I note the background of the first author in math. This may not be the case for a lot of carbon cycle folks, including myself. Hence a lot of math related questions in the following minor comments and my request to simplify/clarify things for a more general audience.



I also apologize for taking such a long time to review. This is a long paper. Unfortunately, I still haven’t made my way through the entire appendix, but I don’t want to hold this process on any longer.



Minor comments

1. Recall that the carbon feedbacks framework can use results from any two of the three runs (COU, RAD, and BGC). Please note this in your manuscript and clarify that this manuscript uses the RAD and BGC runs.
2. Lines 28 and 29. Please changes “reaction” to “response”.
3. Lines 6-80. This sentence is too long. Please also reword “the negative biogeochemical feedback is in terms of radiative forcing more than four times stronger than the positive radiative feedback” to make it more clear.
4. Equation (1) – I think E(s)ds should be changed to E(t)dt for easy interpretation.
5. Line 255. Why is there is square in CAF(t)^2?
6. Line 267. I am not a math expert but I didn’t follow what the plus(+) sign in “lim(t->0+)” means.
7. Line 304 needs rewording – “Accordingly, when studying in the next section also these other CMIP5 models, we …”
8. Line 319. Does “definition (12)” actually mean “equation (12)”?
9. Why does the Laplace transform of equation (12) yields a p in the denominator in equation (13), and the Laplace transform of equation (15) doesn’t (in equation 16).
10. Lines 379-381 are somewhat difficult to follow. Can you simply say a delta CO2 of how many ppm is considered a linear regime?
11. The results in Figure 1 correspond to which scenario?
12. Equation (17) has a lot of meaning. It implies that airborne fraction in the frequency domain is not a function of emissions but rather a function of the feedbacks. Is this correct interpretation? If yes, please bring out this message more clearly.
13. Equation (18) has a lot going on. In particular, what does the term between the last two “=” means physically?
14. I felt, Section 3.3 needs more description of the experiments/simulations.
15. Line 405. “In addition, because the two curves were obtained from very different simulations …”. Sorry, what simulations are being referred to here.
16. In the context of the airborne fraction, A, is it correct interpretation that A(t) depends on the emissions scenario while A(p) does not. If yes, again this is profound and should be brought out more clearly.
17. Please make it clear that you have assumed T*=0, i.e. the temperature change in the BGC simulation is ignored.
18. Lines 511-512. “In contrast, for all models the predicted beta(O)(t) is for times larger than 15 years systematically too high, and …”. This sentence is unclear.
19. Note that typically we want the perturbation to be larger to enhance the response. In the usual carbon feedbacks analysis the feedback metrics are highly variable when c’ and especially T’ are small. Only when c’ and T’ have increased sufficiently then the feedback metrics settle down.
    
    
    
    In contrast, your analysis requires c’ < 95 ppm to keep things in the linear regime.
    
    
    
    Can these two statements be reconciled?
20. First sentence of section 4.2 – “Before in the next section finally the main question of this study on the role of feedbacks …” needs rewording.
21. In Figure 5b what is the y-axis unit for “Feedback function”?
22. Lines 594-595, “These results are in particular at short time scales in contrast with previous estimates (Gregory et al., 2009; Arora et al., 2013) using Friedlingstein’s framework, which suggested that the biogeochemical feedback is about 4 times larger than the radiative feedback”.
    
    
    
    Note that the 4 times number was in the context of C units (Pg C). Hence my question in bullet 21 (what are the y-axis units in Figure 5b).
23.  Lines 676-679. This last sentence of this paragraph is unclear. Please consider rewording.
24. Line 813. I do not follow how X_beta_ln(O)(t) = X_beta(O)(t).
25. What are the units of prediction error in Figure A1a on y-axis?