the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 09 Sep 2024
Impulse response functions as a framework for quantifying ocean-based carbon dioxide removal
Abstract. Limiting global warming to 2 °C by the end of the century requires dramatically reducing CO2 emissions, and also implementing carbon dioxide removal (CDR) technologies. A promising avenue is marine CDR through ocean alkalinity enhancement (OAE). However, quantifying carbon removal achieved by OAE deployments is challenging because it requires determining air-to-sea CO2 transfer over large spatiotemporal scales–and there is the possibility that ocean circulation will remove alkalinity from the surface ocean before complete equilibration. This challenge makes it difficult to establish robust accounting frameworks suitable for an effective carbon market. Here, we propose using impulse response functions (IRFs) to address such challenges. We perform model simulations of a short-duration alkalinity release (the “impulse”), compute the resultant air-sea CO2 flux as a function of time, and generate a characteristic carbon uptake curve for the given location (the IRF). Applying the IRF method requires a linear and time-invariant system. We attempt to meet these conditions by using small alkalinity forcing values and creating an IRF ensemble accounting for seasonal variability. The IRF ensemble is then used to predict carbon uptake for an arbitrary-duration alkalinity release at the same location. We test whether the IRF approach provides a reasonable approximation by performing OAE simulations in a global ocean model at locations that span a variety of dynamical and biogeochemical regimes. We find that the IRF prediction can typically reconstruct the carbon uptake in continuous-release simulations within several percent error. Our simulations elucidate the influences of oceanic variability and deployment duration on carbon uptake efficiency. We discuss the strengths and possible shortcomings of the IRF approach as a basis for quantification and uncertainty assessment of OAE, facilitating its potential for adoption as a component of the carbon removal market’s standard approach to Monitoring, Reporting, and Verification (MRV).
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CC1: 'Comment on egusphere-2024-2697', Benoit Pasquier, 16 Sep 2024
I just stumbled on this article and noticed a few issues with the maths that I thought I should point out to the authors in case it is useful:
- Eq. (1) misdefines the Dirac delta function δ, of which the value at 0 is not 1. The Dirac δ does not have a finite value at 0 and is not technically a function. The Dirac δ is instead a "generalised" function, or more specifically a "distribution". It cannot be defined the way Eq. (1) is written. While the first line of Eq. (1) is correct (the Dirac δ is 0 everywhere except at 0), the second line is incorrect and should be replaced with its integral over the entire real line being equal to 1: ∫ δ(t) dt = 1.
- Eq. (2) uses the discrete summation symbol ∑, which seems to suggest that t is an integer, which does not seem correct to me. Why not use the integral notation, x(t) = ∫ δ(t') x(t') dt', instead?
- Eq. (3) could work without the intermediate equality (the one with the summation symbol ∑)
- Eq, (3) and throughout the paper, parentheses must be placed around the functions being convoluted, as in (x ∗ h)(t) instead of x(t) ∗ h(t), the latter being easily confused for simple multiplication otherwise.
This looks like a timely and worthy article otherwise!
Benoît PasquierCitation: https://doi.org/10.5194/egusphere-2024-2697-CC1 -
AC1: 'Reply on CC1', Elizabeth Yankovsky, 16 Sep 2024
Hello Benoît, thank you very much for your comments. We will make these corrections to the manuscript.
Citation: https://doi.org/10.5194/egusphere-2024-2697-AC1 -
CC2: 'Reply on AC1', Benoit Pasquier, 18 Sep 2024
No worries!
Apologies, there is a typo in my comment for Eq. (2): it should be x(t) = ∫ δ(t - t') x(t') dt' instead of x(t) = ∫ δ(t') x(t') dt'.
Good luck with the review!
Citation: https://doi.org/10.5194/egusphere-2024-2697-CC2
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CC2: 'Reply on AC1', Benoit Pasquier, 18 Sep 2024
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AC1: 'Reply on CC1', Elizabeth Yankovsky, 16 Sep 2024
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RC1: 'Comment on egusphere-2024-2697', Anonymous Referee #1, 02 Oct 2024
Review for Yankovsky et al.
The authors have explored nuances of a possible path forward that the community has been collectively been considering for model-based MRV for marine CDR: impulse response functions. IRFs can be used to estimate efficiency losses from incomplete air-sea gas exchange following an intervention. The authors present some mathematics and schematics explaining how this might be able to be done. They do this using a model that captures some seasonal and interannual variability in ways that alternative approaches to deriving IRFs (e.g., from transport matrices) cannot.
My main criticism for this manuscript is that it addresses a “cat” in the room rather than the “elephants.” Specifically, the questions that are addressed by the model are:
- Can an IRF be effectively discretized using 4 distributed seasonally-specific IRFs? (This question is adequately addressed.)
- To what degree does the IRF break down due to interannual variability as the length of time between its definition and use grows? (This question is not well addressed or posed, but some of the results speak to the issue.)
The authors do not address the main problems with IRFs, which swirl around the question of whether the model is adequately representing the true Earth system (are the resolution; parameterizations of biological, atmospheric, and terrestrial feedbacks; parameterizations for mixing, parameterizations of gas exchange; initialization; and forcing adequate to resolve the signal of interest?). Normally that would be okay, and a nice paper can be written that does a targeted analysis of a limited question, except that the authors present their analysis as an assessment of the viability of IRFs for MRV generally. This makes the central argument of the paper feel a bit like a “strawman” argument. When a subset of these other issues are raised, they are mostly dismissed using the logic that (Paraphrased to make a point... this is not a quote) “We don’t have to worry about some challenges to the IRF framework because they only become relevant if we do mCDR in a way that might affect the Earth system.” If this is a fair argument, then the paper is making itself irrelevant by arguing that these approaches to mCDR are not viable at a meaningful scale. It would be better if these issues were more quickly brought up and listed as issues that are not addressed at present rather than presented as issues that can be dismissed.
Another limitation of the paper is that it seems to rely on access to IRFs that are specific to a both a location and a time of release. Several recent studies have blanketed the global surface ocean with IRF estimates and the great Zhou et al. study referenced indeed provides seasonally varying global IRFs. However, it seems unlikely that most people using IRFs will have estimates that are specific to the same year as the release, as is assumed in this study. It would therefore be helpful if the authors could use their analysis to propose a more quantitative approach for assessing uncertainty in IRFs when they are used in different years from when they are determined (or better yet, from another year in another model entirely). The author’s results speak to interannual variability, but the presentation feels anecdotal and doesn’t provide actionable recommendations for quantifying this uncertainty at a general location.
I also would criticize the presentation on two accounts: the complexity and the organization.
Regarding complexity: The main ideas of the paper are reasonably simple, but they are presented in unnecessarily complicated ways. Admittedly, for a formal MRV approach it is important to explicitly present every step in a calculation and drill down into the details to make certain that the calculations are being done in ways that are both practical and defensible. This paper therefore has appropriate ambitions despite the simplicity of the underlying calculations provided the authors can make the underlying math both exactingly correct and highly accessible. I worry that at present the manuscript seems to do neither, and manages to make the simple math behind the proposed idea appear complicated. The presented math also seems to have errors, as noted by an earlier public comment. A solution to both would be to keep the summation notation rather than switching to an integral notation and just accepting that any OAE intervention can be approximated as a sequence of discrete releases rather than an infinite number of infinitesimal releases. This seems likely to be how the IRFs will be implemented in practice in any event, and it seems strange to worry about discretizing the release much finer than the 4x/year IRF functions that will need to be interpolated. Indeed, it is not clear that further discretization of the releases beyond 4x 3-month-long releases could even potentially result in any disagreement whatsoever from the instance with 4x 3-month-long releases because the authors have only modeled releases with no temporal variability between the beginning and end of the release (except in one schematic, which implies a great deal of short-timescale variability). It is possible that linear interpolation of response functions results in a non-linear effective response function, but that math wasn't fully explored.
Regarding organization: The treatment of the various issues that are not addressed (aside from dismissal as mentioned above) and the introduction of the natural “thermostat” hypothesis are mentioned at an unhappy medium level of detail: just enough to convince the readers that these topics will be addressed by the manuscript, but not enough to address them. This makes the presentation somewhat confusing and longer than it needs to be. Similarly, there is some repetition of ideas (e.g., in captions and text) and scattering of methods text throughout the document that makes the paper longer than it needs to be.
In summary, this paper has a worthy if modest aim and good "ingredients" (that is, analysis and simulations). However, it is sufficiently miss-marketed and occasionally overstated that I believe it needs to be rewritten to be shorter, more focused, simpler, and more straightforward in its aims.
Line by line comments are transcribed as going through the paper for the first time... some questions raised in these comments are answered later in the manuscript, but are kept in these comments because they were questions or objections that should likely have been dealt with before that point in the paper.
1-5: why is the abstract focused on OAE when the title is not and the issues raised are not specific to OAE vs. e.g., DOC?
10-15: this has a stronger statement than warranted, as the approach only tests the fidelity of an IRF for a continuous release within the same model, and the real world should be considered a different “model” entirely. The statement is mostly okay, but should be qualified and moderated.
30: the sentence beginning with “A thermostat…” is possibly missing multiple words or is just incorrect.
39: Jumping around timescales here is problematic, as the premature subduction of TA is irrelevant on the timescales that are relevant for the Earth system feedbacks that were the focus of the beginning of the paragraph. Consider dropping the discussion of the natural thermostat to save length and to focus on how mCDR methods tend to create pCO2 deficits in the surface ocean relative to unmodified conditions, how mCDR doesn’t happen until the deficit is eliminated by air-sea exchange, and how subduction slows this equilibration (to timescales that are too slow for climate mitigation strategies).
53: a strategy can consider multiple scales.
57: This implies that computation is our main limitation for modeling. I would argue it is process parameterization and understanding.
58: finish the point by explaining why we don’t do counterfactual experiments
65: There is a disconnect in the community currently with some researchers deliberately avoiding the use of the word “efficiency” to refer to nu with others continuing to use the term to refer to nu. The argument against this term is that there is not a 1:1 equivalence between DIC and TA so the DeltaDIC excess relative to DeltaTA does not fit within the “wasted work” paradigm typically reserved for the term (in)efficiency. Several recent publications have instead taken to using efficiency to refer to the fraction of the expected DIC increase from thermodynamic equilibria that has been achieved, e.g., https://iopscience.iop.org/article/10.1088/1748-9326/ad7477/meta, https://essopenarchive.org/doi/full/10.22541/essoar.170957083.34212619, https://agupubs.onlinelibrary.wiley.com/doi/full/10.1029/2022EF002816
It would be helpful if the authors would adopt this practice or remark on why they do not.
The first schematic left me confused. Figure 1C, why is the Alk input varying over time if the alk pulse was instantaneous at t=1 in the subplot in A (the answer is implied in the main text, but opaque from the figure and its caption)? What is the color representing in A? Why does t sometimes have a prime?
79-88: Why are these conditions important? I can guess, but it should be stated.
106: it is not clear how this is addressed in Figure 2.
125-130: don’t IRFs typically include some degree of subduction and re-emergence?
Figure 3: how different would this figure and figure 4 be if the atmospheric pCO2 were not fixed?
175: if we don’t expect an impact on pCO2atm then why are we doing mCDR? (partially addressed a few lines later, but this should be addressed immediately)
215: this argument cuts both ways. Having a small impact means that a small carbon cycle change from a small and local perturbation to the ecosystem function could result in a significant fractional loss in the expected impact.
256: there are a lot of methods mixed in these results with a fair bit of repetition. It would be best to bring them together and remove them from the results.
293: that appears to be a 50% increase in nu if I am reading it correctly?
309: On the contrary, to my eye, there appears to be more seasonal variability in 4 realizations than in 14 interannual ensemble members. Please explore this point quantitatively rather than visually. Visually, the point might be more easily seen without the 3 sigma envelope. The two sources of variability appear quantitatively dissimilar at this point in the manuscript... they only appear similar in the context of the later figures that the reader has not yet encountered. It might be better to do a comparison across regions initially.
310: presumably, all variants converge over infinite time, though it is interesting that the seasonal variations seem to converge more slowly
313: this has started to address one of many concerns for the IRF method. This claim is too strong.
324: This seems problematic unless the authors feel that in practice it is likely that IRFs will be computed from the same year as the release. If such simulations are available, then why bother with IRFs at all? Wouldn’t it be better to define an ensemble of IRFs for these locations and then test them against an ensemble of releases in various years? The strength of IRFs is that they can be "precomputed" and used later.
Figure 12... nice figure!
345: most of this belongs (and is repeated in) in the caption
346: This seems to significantly undercut the utility of these results.
360: generally this section is well written, but the phrasing of this initial statement is too strong
375: or coastal processes, which may be significant for the many proposed coastal mCDR approaches.
376: lead to or prevent
Citation: https://doi.org/10.5194/egusphere-2024-2697-RC1 -
AC2: 'Reply on RC1', Elizabeth Yankovsky, 18 Dec 2024
Below we quote the original text by Reviewer 1 and reply point by point. The reviewer's text is italicized, our response is not.
The authors have explored nuances of a possible path forward that the community has been collectively been considering for model-based MRV for marine CDR: impulse response functions. IRFs can be used to estimate efficiency losses from incomplete air-sea gas exchange following an intervention. The authors present some mathematics and schematics explaining how this might be able to be done. They do this using a model that captures some seasonal and interannual variability in ways that alternative approaches to deriving IRFs (e.g., from transport matrices) cannot. My main criticism for this manuscript is that it addresses a “cat” in the room rather than the “elephants.” Specifically, the questions that are addressed by the model are:
- Can an IRF be effectively discretized using 4 distributed seasonally-specific IRFs? (This question is adequately addressed.)
- To what degree does the IRF break down due to interannual variability as the length of time between its definition and use grows? (This question is not well addressed or posed, but some of the results speak to the issue.)
The authors do not address the main problems with IRFs, which swirl around the question of whether the model is adequately representing the true Earth system (are the resolution; parameterizations of biological, atmospheric, and terrestrial feedbacks; parameterizations for mixing, parameterizations of gas exchange; initialization; and forcing adequate to resolve the signal of interest?). Normally that would be okay, and a nice paper can be written that does a targeted analysis of a limited question, except that the authors present their analysis as an assessment of the viability of IRFs for MRV generally. This makes the central argument of the paper feel a bit like a “strawman” argument. When a subset of these other issues are raised, they are mostly dismissed using the logic that (Paraphrased to make a point... this is not a quote) “We don’t have to worry about some challenges to the IRF framework because they only become relevant if we do mCDR in a way that might affect the Earth system.” If this is a fair argument, then the paper is making itself irrelevant by arguing that these approaches to mCDR are not viable at a meaningful scale. It would be better if these issues were more quickly brought up and listed as issues that are not addressed at present rather than presented as issues that can be dismissed.
Another limitation of the paper is that it seems to rely on access to IRFs that are specific to a both a location and a time of release. Several recent studies have blanketed the global surface ocean with IRF estimates and the great Zhou et al. study referenced indeed provides seasonally varying global IRFs. However, it seems unlikely that most people using IRFs will have estimates that are specific to the same year as the release, as is assumed in this study. It would therefore be helpful if the authors could use their analysis to propose a more quantitative approach for assessing uncertainty in IRFs when they are used in different years from when they are determined (or better yet, from another year in another model entirely). The author’s results speak to interannual variability, but the presentation feels anecdotal and doesn’t provide actionable recommendations for quantifying this uncertainty at a general location.
We first thank the reviewer for the detailed assessment of the manuscript and the line-by-line suggestions; we will implement many of these ideas into our revisions. We entirely agree with the need for model validation and will do better with discussing the potential biases the 1-degree model may introduce into the IRF approach as well as model limitations. Numerous literature has been published that employs NCAR’s CESM 1 degree model, including Zhou et al. 2024. We thus do not believe that our manuscript is a place to do a detailed validation of the model but agree that there is a need to address the model’s performance and limitations in the context of the IRF methodology. We have partially done this by citing appropriate literature and speaking about the gaps in resolving mesoscale/submesoscale turbulence (which may present challenges to implementing IRFs). We are performing additional investigations into these questions using higher-resolution regional models, and will speak more about this in the manuscript. The resolution dependence of uptake efficiency and variability is a deeper question that requires additional research efforts.
Regarding the dismissal of some challenges as not being relevant until mCDR is done on a larger scale: the scaling of the mCDR industry must pass through a period where deployments are small in scale, but still require robust verification to support transactions of carbon removal. We believe IRFs may have a role to play here. The main place where this is done is in assuming a “non-interactive” atmosphere, i.e. the pCO2 in the atmosphere doesn’t change for the duration of the OAE simulation. This is a simplification but we have cited literature (Tyka 2024) that addresses this by performing simulations with and without an interactive atmosphere; The IRF curves presented here, as well as the results by Zhou et al. 2024, are concerned with the intrinsic OAE efficiency, i.e. relative to a direct atmospheric removal (such as DAC) of the same magnitude. It has been shown (Tyka 2024) that the use of a prescribed atmosphere yields an efficiency metric which only measures this relative efficiency and that is what the present paper is concerned with. On the other hand, the absolute efficiency, meaning the reduction of atmospheric CO2 following some intervention, is obtained when using responsive atmospheres or earth system models. However, this conflates the intrinsic efficiency with the effects of any negative or positive emissions on the rebalancing of CO2 inventories between various interconnected reservoirs (e.g. Ocean, Atmosphere and Terrestrial biosphere). As these are common to all negative emission technologies, they are not addressed here. If there are other specific places where we have inadequately spoken about simplifications we will be happy to explain in more detail.
Regarding the last paragraph of the above text: we have presented quantification of interannual variability for numerous locations in which we tested the IRF methodology. Our 5-year releases involve releasing alkalinity for 5 years and using an IRF that was developed for only the first year (we quantified how well these perform and in most regions they do well). Please see Section 5.3. However, we will add additional discussion on the longer-term variability and some thoughts on how to apply the IRFs incorporating knowledge of the interannual variability (please also see Reviewer 3’s comments).
I also would criticize the presentation on two accounts: the complexity and the organization. Regarding complexity: The main ideas of the paper are reasonably simple, but they are presented in unnecessarily complicated ways. Admittedly, for a formal MRV approach it is important to explicitly present every step in a calculation and drill down into the details to make certain that the calculations are being done in ways that are both practical and defensible. This paper therefore has appropriate ambitions despite the simplicity of the underlying calculations provided the authors can make the underlying math both exactingly correct and highly accessible. I worry that at present the manuscript seems to do neither, and manages to make the simple math behind the proposed idea appear complicated. The presented math also seems to have errors, as noted by an earlier public comment. A solution to both would be to keep the summation notation rather than switching to an integral notation and just accepting that any OAE intervention can be approximated as a sequence of discrete releases rather than an infinite number of infinitesimal releases. This seems likely to be how the IRFs will be implemented in practice in any event, and it seems strange to worry about discretizing the release much finer than the 4x/year IRF functions that will need to be interpolated. Indeed, it is not clear that further discretization of the releases beyond 4x 3-month-long releases could even potentially result in any disagreement whatsoever from the instance with 4x 3-month-long releases because the authors have only modeled releases with no temporal variability between the beginning and end of the release (except in one schematic, which implies a great deal of short-timescale variability). It is possible that linear interpolation of response functions results in a non-linear effective response function, but that math wasn't fully explored.
We will correct the errors pointed out in the review comments (we thank the reviewers for catching those), and will add text on the temporal discretization. We will add a few sentences to provide intuition for the mathematical calculations (there are also many resources online that discuss the mathematics of IRFs and convolutions for those readers that are interested).
Regarding organization: The treatment of the various issues that are not addressed (aside from dismissal as mentioned above) and the introduction of the natural “thermostat” hypothesis are mentioned at an unhappy medium level of detail: just enough to convince the readers that these topics will be addressed by the manuscript, but not enough to address them. This makes the presentation somewhat confusing and longer than it needs to be. Similarly, there is some repetition of ideas (e.g., in captions and text) and scattering of methods text throughout the document that makes the paper longer than it needs to be. In summary, this paper has a worthy if modest aim and good "ingredients" (that is, analysis and simulations). However, it is sufficiently miss-marketed and occasionally overstated that I believe it needs to be rewritten to be shorter, more focused, simpler, and more straightforward in its aims.
We appreciate the feedback and agree that the paper can be streamlined and more focused; we will revise the manuscript accordingly. Please see the more specific responses below. In general we somewhat disagree about “mis-marketing” and view the IRF approach as a promising tool that requires additional research in models with increased turbulent variability. It does raise many challenging problems of how to handle the rich variability across a variety of temporal and spatial scales that profoundly affects carbon uptake (only part of which are captured in the model presented here). We will better caveat the limitations of using the 1-degree CESM in our revisions. However, once the challenging problem of encapsulating different forms of variability into an IRF library is accomplished, the IRF approach can greatly facilitate the MRV problem.
Line by line comments are transcribed as going through the paper for the first time... some questions raised in these comments are answered later in the manuscript, but are kept in these comments because they were questions or objections that should likely have been dealt with before that point in the paper.
We appreciate these comments and address them below; we also note that the ordering of some of the information presented in the paper is a matter of personal preference/style, and different readers will have different questions arise at various points in the manuscript. Our aim is to have all of the information present and logically ordered, and we will make some modifications in the revisions to improve upon this.
1-5: why is the abstract focused on OAE when the title is not and the issues raised are not specific to OAE vs. e.g., DOC?
Please see lines 59-61: “Here, we develop the idea of using impulse response functions (IRFs) as a statistical MRV tool for mCDR. We use OAE as a testbed for the IRF approach, but note this methodology should be suitable for other mCDR intervention strategies, such as direct ocean removal.” Similar to the Zhou et al. 2024 study, we are performing simulations of OAE specifically (which is why OAE is in the abstract) but we have noted early in the introduction that the IRF methodology does apply to DOC. The title is already quite long so we have not included “ocean alkalinity enhancement” there.
10-15: this has a stronger statement than warranted, as the approach only tests the fidelity of an IRF for a continuous release within the same model, and the real world should be considered a different “model” entirely. The statement is mostly okay, but should be qualified and moderated.
Noted, we will do this in the revisions.
30: the sentence beginning with “A thermostat…” is possibly missing multiple words or is just incorrect.
We believe this sentence is grammatically correct, it is saying that a natural thermostat operates in the climate system. However, we will rephrase this in the revisions for better clarity.
39: Jumping around timescales here is problematic, as the premature subduction of TA is irrelevant on the timescales that are relevant for the Earth system feedbacks that were the focus of the beginning of the paragraph. Consider dropping the discussion of the natural thermostat to save length and to focus on how mCDR methods tend to create pCO2 deficits in the surface ocean relative to unmodified conditions, how mCDR doesn’t happen until the deficit is eliminated by air-sea exchange, and how subduction slows this equilibration (to timescales that are too slow for climate mitigation strategies).
The motivation for the natural thermostat text is to emphasize that OAE is attempting to accelerate a natural phenomenon; we think this is an important point given the reticence many feel towards geoengineering. However, we appreciate this comment and will think about shortening/reorganizing that paragraph. Also we don’t agree that premature subduction of TA is irrelevant on the timescales that are relevant for Earth system feedbacks. If TA is injected in the Labrador Sea, we don’t see it back at the surface for a few thousand years, and the Earth system feedbacks can be quite a bit faster than that.
53: a strategy can consider multiple scales.
The message here is that it’s hard to observe a small signal over the entire ocean, we will edit this sentence to make that clearer.
57: This implies that computation is our main limitation for modeling. I would argue it is process parameterization and understanding.
Absolutely agreed that process understanding and parameterization is the primary challenge (that will always be a challenge regardless of how high-resolution ocean models become). The sentence reads “the inherent difficulties in representing the ocean component of the climate and its interactions with the atmosphere, land, and biosphere using finite computational resources”. “Representing the ocean component and its interactions” encapsulates parameterizations and process understanding (necessary for physics-based parameterizations). Having worked a lot on ocean parameterization development, “representation” is often used as a synonym for “parameterization” which is why I chose that language here.
58: finish the point by explaining why we don’t do counterfactual experiments
Here we’re speaking about observations, so if we’re performing an OAE deployment then by definition we can’t have a “counterfactual” for the same time and place. For modeling studies we do counterfactual experiments (stated later).
65: There is a disconnect in the community currently with some researchers deliberately avoiding the use of the word “efficiency” to refer to nu with others continuing to use the term to refer to nu. The argument against this term is that there is not a 1:1 equivalence between DIC and TA so the DeltaDIC excess relative to DeltaTA does not fit within the “wasted work” paradigm typically reserved for the term (in)efficiency. Several recent publications have instead taken to using efficiency to refer to the fraction of the expected DIC increase from thermodynamic equilibria that has been achieved, e.g., https://iopscience.iop.org/article/10.1088/1748-9326/ad7477/meta, https://essopenarchive.org/doi/full/10.22541/essoar.170957083.34212619, https://agupubs.onlinelibrary.wiley.com/doi/full/10.1029/2022EF002816
It would be helpful if the authors would adopt this practice or remark on why they do not.
We understand your point, we used the conventions presented in Zhou et al. 2024 but have noticed that this convention sometimes leads to confusion. We will bring up this issue in the revision. The problem is that calculating the fraction of the expected DIC increase requires knowing what the expected DIC increase actually is, but that depends on the parameterization of the model itself. We are open to suggestions for a better word/term for DIC/Alk, which is a more solidly defined metric in our opinion.
The first schematic left me confused. Figure 1C, why is the Alk input varying over time if the alk pulse was instantaneous at t=1 in the subplot in A (the answer is implied in the main text, but opaque from the figure and its caption)? What is the color representing in A? Why does t sometimes have a prime?
In subplot A, we probe the system with the impulse, and in panel C we use the IRF to predict the effect of an arbitrary time series of alkalinity forcing. That is what is meant by “an arbitrary OAE deployment”. We will edit the figure caption to make this clearer and will define the t’ (t’ is just a dummy variable for time). Also, panel A shows the alkalinity plume spreading (see figure caption). We will explicitly state that the color represents that to make it clear.
79-88: Why are these conditions important? I can guess, but it should be stated.
These are the mathematical requirements for invoking the impulse response function convolution integral; we will state this more explicitly here.
106: it is not clear how this is addressed in Figure 2.
As stated in lines 106-108, writing the uptake curve in the form of equation 4 allows one to bypass performing additional model simulations to obtain the uptake curve. Instead, one can simply perform the convolution pictured in Figure 2. Figure 2 shows that the convolution provides an alternative to perform additional model integrations.
125-130: don’t IRFs typically include some degree of subduction and re-emergence?
Of course; as we say in those lines we first consider purely the “chemistry” problem and then add in ocean dynamics (which of course include subduction and re-emergence).
Figure 3: how different would this figure and figure 4 be if the atmospheric pCO2 were not fixed?
Please see the recent paper by Tyka (2024): https://egusphere.copernicus.org/preprints/2024/egusphere-2024-2150/
The effect on the uptake curves is addressed there and does not affect the validity of the results presented here. See the discussion citing Tyka (2024) in lines 391-397.
175: if we don’t expect an impact on pCO2atm then why are we doing mCDR? (partially addressed a few lines later, but this should be addressed immediately)
We address this question in a couple spots in the manuscript, including citing Tyka (2024) in lines 391-397 but we will bring it up sooner in the revised manuscript. The assumption isn’t that OAE yields no impact on atmospheric pCO2, but rather that for these small deployments we can accurately capture the first-order carbon uptake curve by assuming a non-interactive atmosphere (to make the modeling more affordable). This assumption was also utilized in Zhou et al. 2024.
215: this argument cuts both ways. Having a small impact means that a small carbon cycle change from a small and local perturbation to the ecosystem function could result in a significant fractional loss in the expected impact.
Agreed; but this part of the text is just describing how we envision modeling OAE deployments and the need for higher-resolution modeling to capture the short-term plume evolution. Is there a specific modification you’d like us to make here? The point we make here is that the "efficiency" is measured relative to a direct air removal. The earth system feedback is common to all negative emission technologies and we're intentionally factoring those out, looking only at the relative, intrinsic efficiency of OAE.
256: there are a lot of methods mixed in these results with a fair bit of repetition. It would be best to bring them together and remove them from the results.
Noted, will streamline this text.
293: that appears to be a 50% increase in nu if I am reading it correctly?
Sorry for the confusion; the goal was to say it’s 1.5x larger (150%), but yes, it’s a 50% increase. Will edit this.
309: On the contrary, to my eye, there appears to be more seasonal variability in 4 realizations than in 14 interannual ensemble members. Please explore this point quantitatively rather than visually. Visually, the point might be more easily seen without the 3 sigma envelope. The two sources of variability appear quantitatively dissimilar at this point in the manuscript... they only appear similar in the context of the later figures that the reader has not yet encountered. It might be better to do a comparison across regions initially.
Noted, we will edit this part of the manuscript to avoid confusion in comparing seasonal and interannual variability magnitude (and be more quantitative in comparisons).
310: presumably, all variants converge over infinite time, though it is interesting that the seasonal variations seem to converge more slowly
Agreed, though here we make the point that there’s convergence on the time scale of several years. Also note that this is not the case for all locations (such as the Labrador Sea) where variance increases through the end of the simulation.
313: this has started to address one of many concerns for the IRF method. This claim is too strong.
The IRF method hinges upon two requirements - linearity and time invariance. We have shown that linearity is not a significant concern, but the time invariance poses a challenge. It is a fair statement to say that if time invariance is accounted for then the IRF method is valid; our findings and figures support that claim. Please elaborate on which part of this is too strong.
324: This seems problematic unless the authors feel that in practice it is likely that IRFs will be computed from the same year as the release. If such simulations are available, then why bother with IRFs at all? Wouldn’t it be better to define an ensemble of IRFs for these locations and then test them against an ensemble of releases in various years? The strength of IRFs is that they can be "precomputed" and used later.
That’s a great point (one we’ve also asked ourselves), and we addressed this when we considered the multiyear release experiments as well as the ensembles looking at interannual variability. In the multiyear release experiments, the IRF is ONLY obtained for year 1, that same IRF is used for the entire 5 years of the alkalinity release. In other words, we’re using an IRF from a different year than alkalinity is being deployed in for 4 years; though this prediction obviously doesn’t perform as well as using the IRF from the same year as the release, it still creates a remarkably accurate prediction as our results show. Looking at Figure 10, as well as Figure 12, we see that the IRF from year one reconstructs the uptake curves to maximum a few percent error for the 5 year releases. We argue that this is due to the standard deviation between ensemble members decreasing with time at most locations. Note that this is not always the case. In the Labrador Sea for instance, the standard deviation increases over the entire time period considered here, and the IRF from year 1 does not do very well in reproducing the 5 year continuous release results. However, this is a fairly anomalous case and we find that in most locations we can account for the variability (time variance) sufficiently just by using a seasonally varying IRF from one year.
Figure 12... nice figure!
Thank you!
345: most of this belongs (and is repeated in) in the caption
Noted, will revise.
346: This seems to significantly undercut the utility of these results.
Which aspect of this undercuts the utility? We are honest in saying that the aim isn’t really to have an IRF for the same year as the release (there’s less utility in that), but to use the IRF to predict the evolution of subsequent years. We do this with the 5-year release case. The IRF from year 1 is used to predict the uptake of all the subsequent years, and does so successfully in most regions.
360: generally this section is well written, but the phrasing of this initial statement is too strong
Noted, will revise.
375: or coastal processes, which may be significant for the many proposed coastal mCDR approaches.
Noted, will add.
376: lead to or prevent
Noted, will add.
Citation: https://doi.org/10.5194/egusphere-2024-2697-AC2
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RC2: 'Comment on egusphere-2024-2697', Anonymous Referee #2, 22 Oct 2024
This manuscript explores the use of impulse response functions (IRFs) to quantify carbon dioxide removal (CDR) through ocean alkalinity enhancement (OAE), a promising marine CDR method. The authors test the IRF approach across various oceanic conditions, finding that it can reliably approximate carbon uptake with minimal error.
Major Concerns:
1. Simplified Assumptions in IRFs
The authors used IRFs to predict OAE-CDR efficiency from a chemical perspective, relying on assumptions of linearity and time invariance. These IRF predictions were then compared to the outcomes of a coarse Earth System Model version 2 (ESMV2) simulation. However, this comparison is effectively circular (at least to me): the IRFs do not account for water mass exchanges or mesoscale processes, and the coarse ESM also omits these physical dynamics. As a result, the match between the two datasets is unsurprising, as both primarily reflect chemical hydrodynamics without fully considering the physical processes. It's important to emphasize that without capturing fine-scale physical variability—such as water currents, downwelling, and other dynamic features—the comparison may be less robust than it appears.
2. Figure 12c and the Overlap with ηmax
The close match in Figure 12c is also expected. After five years of TA release, most of the ocean surface has reached or close to equilibrium (based on Figures 9–11) through sea-air exchange, either quickly or slowly. Essentially, this comparison in Figure 12 is like comparing ηmax after considering seasonal and interannual dynamics, which could be determined by simpler methods using datasets like GLODAPv2 (as shown in Figure 3a). Thus, I question the added value of using IRFs to predict CDR efficiency over longer timescales (years) and across much larger spatial coverage (~100 km), when not accounting the physical processes that prevent reaching ηmax.
3. Limited Application for Mesoscale Processes
Given the limitations of IRFs, I also question the claim that IRFs can guide future regional modeling efforts aimed at resolving mesoscale turbulence or submesoscale dynamics (as suggested in lines 400 of the text). Since the IRFs were not designed to capture these more complex physical processes, their applicability to regional models aiming to resolve such variability seems quite constrained.Additionally, Equation (5) appears incorrect. As currently written, η(t) would never approach ηmax but reach to 0 as time increases , which is problematic for the model.
Citation: https://doi.org/10.5194/egusphere-2024-2697-RC2 -
AC3: 'Reply on RC2', Elizabeth Yankovsky, 18 Dec 2024
This manuscript explores the use of impulse response functions (IRFs) to quantify carbon dioxide removal (CDR) through ocean alkalinity enhancement (OAE), a promising marine CDR method. The authors test the IRF approach across various oceanic conditions, finding that it can reliably approximate carbon uptake with minimal error.
Major Concerns:
- Simplified Assumptions in IRFs
The authors used IRFs to predict OAE-CDR efficiency from a chemical perspective, relying on assumptions of linearity and time invariance. These IRF predictions were then compared to the outcomes of a coarse Earth System Model version 2 (ESMV2) simulation. However, this comparison is effectively circular (at least to me): the IRFs do not account for water mass exchanges or mesoscale processes, and the coarse ESM also omits these physical dynamics. As a result, the match between the two datasets is unsurprising, as both primarily reflect chemical hydrodynamics without fully considering the physical processes. It's important to emphasize that without capturing fine-scale physical variability—such as water currents, downwelling, and other dynamic features—the comparison may be less robust than it appears.
There is a misinterpretation of the IRF assumptions in this comment. We do not only treat OAE from a chemical perspective, that was just the first step for deriving the maximum efficiency (Section 3.1) and considering chemistry-induced nonlinearities. Please see the two sections directly after that, Sections 3.2 and 3.3. The flow physics and variability contribute significantly to introducing time variance, which makes the IRF methodology tricky to apply. This is why we create a seasonally varying IRF library and also perform ensembles to consider the interannual variability. We are using the CESM 1 degree model, which does simulate ocean biogeochemistry (through MARBL) and has fidelity in reproducing oceanic variability and current systems. The large scale ocean circulation is captured in this model (albeit mesocale and submesoscale dynamics are not resolved, see our response to your point 3). This model does account for physical dynamics, just not the fine-scale variability. We entirely agree that it is important to extend this work to higher resolution models, and we are in the process of doing so. We used the OAE Atlas of Zhou et al. 2024 as a starting point for our IRF study. As discussed in Zhou et al., there are many physical differences in mixing, seasonality, and subduction/advection of alkalinity present in this model, and we are excited by the success of IRFs here. It will indeed be interesting to examine how the finer-scale physical variability will affect the problem. We want to emphasize that at each location the IRF encapsulates the physical dynamics at that location, i.e. the IRFs themselves deviate from ηmax based on the physical subduction/mixing/circulation present at that location. We are comparing IRFs derived from the CESM model for 1998 to continuous releases of alkalinity in that same model extending for several years post-1998 (that’s the exciting matchup that we discuss in the later figures). We will modify the language in the revisions to avoid confusion on this point and better caveat the lack of mesoscale/submesoscale variability.
- Figure 12c and the Overlap with ηmax
The close match in Figure 12c is also expected. After five years of TA release, most of the ocean surface has reached or close to equilibrium (based on Figures 9–11) through sea-air exchange, either quickly or slowly. Essentially, this comparison in Figure 12 is like comparing ηmax after considering seasonal and interannual dynamics, which could be determined by simpler methods using datasets like GLODAPv2 (as shown in Figure 3a). Thus, I question the added value of using IRFs to predict CDR efficiency over longer timescales (years) and across much larger spatial coverage (~100 km), when not accounting the physical processes that prevent reaching ηmax.
Please see the prior comment; it’s not a matchup with the chemistry-derived ηmax but rather a close matchup of the seasonal IRFs derived for each polygon with the 5-year continuous alkalinity releases that we’re showing in Figure 12. The IRFs do encapsulate the physical dynamics specific to that region. Please see Figure 7 for example. The IRF curves are lower than ηmax in cases of seasonal variability and subduction of water masses. Many of the locations we consider do not achieve ηmax and that’s not the comparison that we’re performing here. The GLODAP data in section 2.1 again is just to compute ηmax and test the nonlinearity of large alkalinity perturbations. We are attempting to encapsulate flow and seasonal variability into our IRF predictions; in some locations where ηmax is achieved rapidly this isn’t as important, but in most places we do need to construct an IRF that accounts for flow variability. Since this point was missed we will strive to make this clearer in the revisions.
- Limited Application for Mesoscale Processes
Given the limitations of IRFs, I also question the claim that IRFs can guide future regional modeling efforts aimed at resolving mesoscale turbulence or submesoscale dynamics (as suggested in lines 400 of the text). Since the IRFs were not designed to capture these more complex physical processes, their applicability to regional models aiming to resolve such variability seems quite constrained.
This is an open question and one that we agree needs to be explored. We are presently performing regional simulations to test this hypothesis and this will be the subject of future work. However, if we consider the OAE Atlas of Zhou et al. 2024, the IRF methodology provides a route by which those results may be extended. The fact that the model doesn’t capture mesoscale processes does not invalidate all work stemming from that model. Instead, we should view this as a starting point which can be built upon and refined in future work. Since we constructed the IRFs from that model, indeed they will not predict how mesoscale and submesoscale dynamics will affect uptake. However, we can study that in higher resolution models and apply what we learn to refine how we use IRFs and how we interpret results from the OAE atlas (i.e. does mesoscale/submesoscale turbulence decrease efficiency? does it increase variability? do we need additional ensembles to encapsulate turbulent variability into our IRFs? etc.). We can not do everything in one study; nonetheless the IRFs are shown to be a promising strategy to be explored further in higher resolution models.
Additionally, Equation (5) appears incorrect. As currently written, η(t) would never approach ηmax but reach to 0 as time increases , which is problematic for the model.
Thanks for catching this, there should be a minus sign in that equation (as in Zhou et al. 2024) and this will be corrected.
Citation: https://doi.org/10.5194/egusphere-2024-2697-AC3
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AC3: 'Reply on RC2', Elizabeth Yankovsky, 18 Dec 2024
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RC3: 'Comment on egusphere-2024-2697', Anonymous Referee #3, 01 Nov 2024
Review of Yankovsky et al.
General comments
The authors developed the use of impulse response functions (IRFs) for predicting the carbon uptake from ocean alkalinity enhancement (OAE) interventions and tested the accuracy of their approach against simulations of continuous-release OAE scenarios using a Community Earth System Model. They found that IRFs sufficiently meet the requirements of linearity and time invariance that they can predict the carbon uptake from continuous alkalinity release scenarios within several percent error and suggest that IRFs may be a viable approach for Monitoring, Reporting and Verification (MRV) of OAE interventions. A key advantage of the IRF approach is that the pre-computed functions can greatly simplify the estimation of carbon uptake from OAE without the need for full biogeochemical simulations. Although the manuscript is generally worthy and well-written, there are some major issues that the authors need to address before publication.
Major issues:
Lack of independent model validation:
As noted by another reviewer, the IRFs were developed and tested using the same ESM and thus have many of the same biases in their representation of physical and biogeochemical processes. The resulting agreement between the IRFs and model simulations should therefore be interpreted cautiously. The authors should address this issue and explicitly discuss the physical/biogeochemical processes and feedbacks represented and not represented in the models and potential sources of biases in the IRFs.
Period of validity of IRFs:
Can the authors speak on the performance of IRFs older than 5 years and provide recommendations on their reliable application? When should IRFs be recomputed?
Confusing mathematical notation:
As noted by other reviews and a public comment, the mathematical notation is confusing and there are some errors. The switch between δ and h(t) is confusing, and it is not clear what h(t) is referring to. The manuscript refers to both h(t) and y(t) = η(t) as “impulse response functions.”
Equation 5 should be η(t) = ηmax – exp(-t/τ) as in Zhou et al. (2024).
What does the prime notation in t’ refer to?
Minor comments:
Calculation of ηmax (Lines 148-151): Please provide a complete description of how ηmax is calculated (i.e., that ΔDIC is the difference between the initial DIC and the final DIC after the alkalinity perturbation and complete equilibration with atmospheric p(CO2)). What are the assumptions made about atmospheric p(CO2) in the calculations and the values used?
Fig. 4: Is this computed at a constant temperature and salinity or with the in situ values from the GLODAP dataset?
Fig. 6 and Lines 272-274: What is the reason for the 5 year curve in the lower right panel for η(t) being different from the others, and why aren’t there visible kinks on the other curves at the 1 year and 1 month mark?
Line 136: It may be helpful to the reader to include the range of equilibration timescales in the ocean.
Fig. 7: The color scheme makes it difficult to distinguish the curves for different months.
Citation: https://doi.org/10.5194/egusphere-2024-2697-RC3 -
AC4: 'Reply on RC3', Elizabeth Yankovsky, 18 Dec 2024
The authors developed the use of impulse response functions (IRFs) for predicting the carbon uptake from ocean alkalinity enhancement (OAE) interventions and tested the accuracy of their approach against simulations of continuous-release OAE scenarios using a Community Earth System Model. They found that IRFs sufficiently meet the requirements of linearity and time invariance that they can predict the carbon uptake from continuous alkalinity release scenarios within several percent error and suggest that IRFs may be a viable approach for Monitoring, Reporting and Verification (MRV) of OAE interventions. A key advantage of the IRF approach is that the pre-computed functions can greatly simplify the estimation of carbon uptake from OAE without the need for full biogeochemical simulations. Although the manuscript is generally worthy and well-written, there are some major issues that the authors need to address before publication.
We appreciate the feedback and positive assessment; please see our responses below. We will edit the manuscript to caveat the use of the 1-degree CESM better and address in more detail the question of how long an IRF is valid for.
Major issues:
Lack of independent model validation: As noted by another reviewer, the IRFs were developed and tested using the same ESM and thus have many of the same biases in their representation of physical and biogeochemical processes. The resulting agreement between the IRFs and model simulations should therefore be interpreted cautiously. The authors should address this issue and explicitly discuss the physical/biogeochemical processes and feedbacks represented and not represented in the models and potential sources of biases in the IRFs.
Agreed; we will dedicate additional discussion to potential model biases and clearly stating what the model is missing. We have done some of this by mentioning that the model does not resolve mesoscale/submesoscale activity and therefore the variability in the model is less than in the real ocean. We also state that CESM uses the biogeochemistry model MARBL (those interested can look into references). We extensively cite the Zhou et al. 2024 study which has now been published and utilizes the NCAR CESM 1 degree model. Our work uses the same model as a starting point for testing the IRF approach in order to extend the OAE Atlas to arbitrary alkalinity release durations and magnitudes. As numerous scientific literature exists that makes use of this model; it is beyond the scope of our work to “validate” this model. However, we entirely agree we should address the potential biases it may introduce to the IRF approach and will elaborate more on this.
Period of validity of IRFs:
Can the authors speak on the performance of IRFs older than 5 years and provide recommendations on their reliable application? When should IRFs be recomputed?
This is an interesting question and one that we can only speculate on, given that our simulations run for a maximum of 20 years. In some regions, such as the Labrador Sea, we find that interannual variations are very substantial and the IRF approach leads to relatively high errors even with a 5-year old IRF. Other regions in the ocean exhibit smaller interannual variability on the timescale of years-decades. The frequency with which IRFs need to be recomputed will vary depending on the variability of the geographic region under consideration and the model that’s used to compute the IRFs. The relatively laminar 1-degree CESM exhibits less turbulence and flow variability than a higher-resolution (mesoscale or submesoscale permitting) model. Also, we found that the standard deviation in ensemble members generally decreases in most regions (not the case everywhere, such as the Labrador Sea). Based on the CESM results, it appears that using an IRF library from one year and then performing an alkalinity release for 5-10 years afterwards will lead to a reasonable prediction in most regions. One must of course consider the background oceanic and atmospheric variability that ultimately sets the OAE efficiency at a given release location. We are now exploring in greater detail the extent to which IRFs vary across different initial condition scenarios, and model resolutions. We will address this point of the period of validity of IRFs in the revised discussion.
Confusing mathematical notation:
As noted by other reviews and a public comment, the mathematical notation is confusing and there are some errors. The switch between δ and h(t) is confusing, and it is not clear what h(t) is referring to. The manuscript refers to both h(t) and y(t) = η(t) as “impulse response functions.”
Equation 5 should be η(t) = ηmax – exp(-t/τ) as in Zhou et al. (2024).
What does the prime notation in t’ refer to?
Noted, thank you for catching these errors. We will improve upon and clarify the mathematical notation in the revisions. The t’ is a dummy variable for time (can just be thought of as time); we’ll clarify in the revisions.
Minor comments:
Calculation of ηmax (Lines 148-151): Please provide a complete description of how ηmax is calculated (i.e., that ΔDIC is the difference between the initial DIC and the final DIC after the alkalinity perturbation and complete equilibration with atmospheric pCO2. What are the assumptions made about atmospheric pCO2 in the calculations and the values used?
Thanks for the comment, we’ll expand upon this more immediately. We state later in the manuscript that atmospheric pCO2 is assumed constant. The calculation is analogous to that presented in Zhou et al. 2024.
Fig. 4: Is this computed at a constant temperature and salinity or with the in situ values from the GLODAP dataset?
In situ values from GLODAP, we will clarify in the revisions.
Fig. 6 and Lines 272-274: What is the reason for the 5 year curve in the lower right panel for η(t) being different from the others, and why aren’t there visible kinks on the other curves at the 1 year and 1 month mark?
The data are discretized by 1 month intervals (so there isn’t a kink for the 1-month deployments since that’s just one data point). If we zoom into the 1-year release curves there are indeed small/negligible kinks at the 1 year mark. They are small because the DIC is increasing rapidly at 1 year, whereas at 5 years there is more of a plateauing in the uptake curve, making the change in delta ALK more obvious.
Line 136: It may be helpful to the reader to include the range of equilibration timescales in the ocean.
Thanks for the comment, we’ll add this in the revisions.
Fig. 7: The color scheme makes it difficult to distinguish the curves for different months.
Distinguishing month by month isn’t the main goal of this figure; we just aim to show the general seasonal trends. However, we will modify this to a “spectral” colormap to increase the amount of colors present and improve readability.
Citation: https://doi.org/10.5194/egusphere-2024-2697-AC4
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AC4: 'Reply on RC3', Elizabeth Yankovsky, 18 Dec 2024
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