the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 30 Jul 2024
Advances in conceptual modelling of the variable nature of Dansgaard-Oeschger events
Abstract. This study introduces a novel dynamical systems model designed to capture the highly non-periodic nature of Dansgaard-Oeschger (DO) events. Such events are difficult to model adequately due to their variable durations — some lasting around a century, while others span multiple millennia — and the occurrence of short precursor events that precede the longer DO events despite similar boundary climate conditions. Utilizing a simplified two-equation framework derived from the Stommel model, our approach integrates an internal control parameter which acts as a feedback parameter on the Antarctic Bottom Water (AABW) formation. Through both analytical and numerical methods, we establish a suitable parameter domain within which the newly adjusted models can accurately replicate the paleoclimatic records of DO events as described by summary statistics derived from ice-core data. The analysis also shows that without the novel control parameter, the model does not have a suitable parameter domain in which it can reproduce the wide range of event characteristics seen in the ice-core record. The study provides new insights into the underlying mechanisms driving these highly significant climate phenomena and the necessary timescale in which they are forced, by allowing the new model's parameters to vary through time. This allows our model to achieve unprecedented precision in capturing a realistic sequence of DO events with timing characteristics matching those of the observational record. This refined model not only enhances our understanding of the DO cycles but also demonstrates the potential of simple dynamical systems to simulate complex climate interactions.
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RC1: 'Review of Melcher et al', Anonymous Referee #1, 15 Aug 2024
Melcher and colleagues present an extended dynamical systems model designed to simulate Dangaard-Oeschger events. The aim of the model extension is to capture the irregular periodicity of observed DO-events which traditional modelling approaches have been unable to capture so far. The extended model is based on a Stommel-type box model formulated by Vettoretti et al (2020) and includes a novel control parameter that feeds back on AABW production. By varying the control parameters in time, the model is able to realistically capture the highly variable nature of stadial and interstadial duration. The authors thoroughly assess the behaviour of the new model and make a convincing case for the inclusion of the new control parameter.
Overall, the manuscript is well written, the figures are well prepared and the results are very relevant for the wider DO- and AMOC community. In the current form of the manuscript it difficult to get through the technical details in some places, and some points need clarification or additional context. Below I list my detailed comments (which are mostly minor).General Comments
---------------------------------------------
Derivation of the model:
I appreciate the effort the authors put into introducing the model in small steps and thoroughly illustrating the baseline model from Vettoretti et al (2022). At the same time, the order of things did not always work for me and confused me in parts. Especially the presentation of the equations was confusing. In three different places the text referred to a different set of equations as the model's equations (eq1&2, eq3&4, eq8-10). I see how the three sets are related but while reading I stumbled a lot and things only became clear at the very end. I am not sure about the best sequence of things. Perhaps it could work to keep the first part of 2.2 free of equations and discuss only in terms of processes. Regarding the presentation of the equations, for me the logical sequence would have been to start with the general form of equations (currently eq3&4), introduce the Vettoretti 2022 model (eq8&9) and then introduce the updated model including alpha (eq8&10). I am sure there are multiple valid ways of restructuring Section 2.2 and its subsections. I would however ask the authors to revisit the section and improve its clarity.Control parameters:
For non-experts in dynamical systems theory it would be helpful to always state which exactly are the control parameters in each context (instead of just referring to "the control parameters"). And also to always clarify whether a control parameter is constant throughout a simulation or being varied. In case of Fig.3: Would the system remain constant in time (except for noise) in the cases of a and c? What makes the system move along the delta_b manifold in case of constant control parameters?Precursor Events:
How many of these precursor events are present in the actual ice-core record? Can you elaborate a bit more on why it is important to be able to replicate them? I am also somewhat confused by the discussion of the different precursor types in Section 4. Can the Type I event be distinguished at all from a regular event? Is the existence of the Type 1 events purely theoretical? Do you see all three types of precursor events in different realisations? Fig9 only shows the one example.Alpha:
In some places it is unclear, whether the reproduced non-periodicity comes from introducing alpha or from *varying* alpha? From my understanding, the non-periodicity comes from varying alpha. This is especially important in the comparison with more complex GCMs (e.g., l419-423). Most GCMs that have produced self-sustained DO oscillations have done so with constant forcing. I would therefore think, that the self-sustained oscillations are more comparable with a constant alpha. The complex models are able to simulate the bipolar sea saw, therefore I think that some feedbacks comparable to alpha are already present in the models (whether they are realistic is another question of course). Some clarification and more detailed discussion of what the processes would represent alpha in the real ocean would be helpful here.Detailed Comments
---------------------------------------------l40-53 How are the two stochastic approaches different? Perhaps it si more useful to group into studies that are based on the Stommel model and those that are not? Which approaches do include the bipolar sea saw and which do not? Perhaps you could mention the FitzHugh-Nagumo model here as well already? Also make it clearer that the baseline for your new model is the model in Vettoretti et al 2022 and explain how the Vettoretti box model relates to previous box models.
l54 include also other comprehensive models that oscillate for completeness (e.g. MPI-ESM Klockmann et al (2020,https://doi.org/10.1029/2020GL090361), HadCM3 Armstrong et al (2023,https://doi.org/10.1007/s00382-022-06564-y) or Malmierca-Vallet et al (2023,https://doi.org/10.5194/cp-19-915-2023) for an overview).
l104 if you keep the order of section 2.2. make it explicit that eq 1 and 2 are not the final form, but a more general form of the equations, and that the equations refer to the baseline model without alpha (see general comment on model derivation)
l109/110 the dependence of q on delta_b is not obvious from eq1
Fig3 caption: "as lambda changes" what does "lambda" refer to here? The Eigenvalues have not been introduced yet
l157-157 is that also what you do or is it in contrast to what you do?
l163 what makes the southern bouyancy flux increase?
l166 is the salt-advection feedback acting on the northern or the southern box? Not equal to the typical AMOC-North Atlantic salt advection feedback?
l162-177 be careful with words. transport/flow/circulation seem to be used for q and AMOC simultaneously? What makes the system move along the manifold? I am probably lacking some basic dynamical systems theory here (see general comment on control parameters).
l182 what about the heat loss removing buoyancy in northern box/real ocean?
l193 I find this information confusing here without context.
l194 link eq3/4 back to eq1/2 (see general comment on model derivation)
l212 are there haline and thermal modes in this model?
eq7 what are B_c and b_c?l228-230 mention this earlier?
l242 which control parameters do refer to here?
l274ff state stability criteria up front for convenience, otherwise the reader needs to guess what the criterion is (negative Re(lambda1,2)? ad if yes, what does that mean?)
Fig5 Is the case with alpha = 0 in c equivalent to the cases presented in Fig3a-c?
l282 the change from b to d seems quite drastic, not so much from d to f
l296 which control parameter do you refer to here?
2.2.3 refer to left hand side of Fig.5 throughout the section to illustrate
l295ff interpretation of b0/value of b0?
l314 How do you set the initial conditions? Simply as a set of values for delta b and B?
l323 how do the 100 members differ? only in the noise realisation?
l327/l344 the age is impossible to determine from the crosses in Fig6. The caption suggests the crosses should be numbered, but they are not.
Fig6(especially a/b) How do mu_E=0 and a variable mu_P go together? If there is no event, should mu_p not always be either 0 or 1?
l335 "with decreasing gamma, mu_P' moves to lower values" is only true for gamma between 1.5 and 3. for gamma < 1.5 mu_P' increases with decreasing gamma. Or am I reading the plot wrong?
l338 what exactly do you mean by "extends the arches downward"? Do you mean that each line of constant gamma reaches lower values of mu_E'?
l342/343 again not sure what you mean by "increasing sigma moves the entire arch downward"?
l382-385 does this refer to the sequence in time?
l386 Are 91 and 21ka b2k the right timings? To me 97 and 24ka b2k would seem more appropriate? How relevant is this misfit?
l389/390 It might be worth including this member also with a separate colour to a and b.
l425-427 but would you have expected anything else given that the target are summary statistics and not the actual timeseries?
Editorial Comments
---------------------------------------------
l33 first mentioning of NGRIP here but the acronym is only introduced in l87l209 and Eq6 say the same thing
l269 "close to 1" instead of "intermediate"?
l271-273 difficult to read, please reformulate more clearly
l278 It should be "delta_b" in the equation and not "b"
l323/324 move sentence with Runge-Kutta to l316, after "[...]run for 5371 steps."?
l375 Is "linear interpolation" correct? The caption of Fig7 says cubic spline fit
l410 "improve" instead of "determine"?
l412 What is "origo"?
l421 "might be able" instead of "might be possible"
l456 "vertical jumps" instead of "vertical jump"
Citation: https://doi.org/10.5194/egusphere-2024-2156-RC1 -
AC1: 'Reply on RC1', Jonathan Ortved Melcher, 22 Oct 2024
The authors would like to thank reviewer one for their time and effort, in formulating a precise, and detailed review, please find line-by-line comments attached in Reply_RC1/RC1.pdf
All line and section numbers refer to the line numbers in the enclosed annotated manuscript (Reply_RC1/annotated_manuscript.pdf), with suggested changes have been highlighted. We are aware that we have not been encouraged to submit a revised version yet but find it more fruitful in this case to demonstrate how we will change the manuscript in response to the reviews.
-
AC1: 'Reply on RC1', Jonathan Ortved Melcher, 22 Oct 2024
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RC2: 'Comment on egusphere-2024-2156', Anonymous Referee #2, 18 Sep 2024
The paper by Melcher et al “Advances in conceptual modelling of the variable nature of DO events” introduces a novel model that realistically reproduces the patterns of DO events in the NGRIP record using modulating ocean parameters. The paper is well written and can be published after a minor revision.
My comments are as follows:
The current title of the paper is more suitable for a review than for a novel research item, and it can be improved as follows: “A novel model of the dynamics of Dansgaard-Oeschger events in the NGRIP record”.
Multiple acronyms used in the paper would be convenient to introduce in a small table in the beginning.
Writing style could be improved. In lines 28-29, replace “supports the essential understanding of the underlying processes required to predict” with “is essential for prediction”. In line 45, “taking another approach, other studies have approached” must ne improved. In line 103, “the relationship between the model and the time series of the model” can be omitted (obviously, the time series produced by the model is its trajectory).
Type I & II events are denoted with or without capital letters, and sometimes with a dash. They, as well as interstadials, are mentioned with a reference but not well introduced for general readership.
In Fig.1, panel b, the y-axis for P(t) has not units – is it in kiloyears? If so, aren’t values within 0.8 kyr too small for stadials in a window of 20 kyr?
In Eqs. 1,2, NOISE could be replaced with some notation, and a short description of the noise would be useful (distribution, scaling properties?)
“Box model”, “earth system models” do not need to be capitalised.
The fixed points are repeatedly called “fixpoints” (including the legend of Figure 3).
In line 258, remove either “equations” or “Eqs”
Add dot at the end of Eq.10. Remove comma after “where” in line 264.
In line 310, remove “achievable”. This is discussed in the section and does not need to be mentioned in the section title.
In line 359, “because” should be with a small letter.
In line 370, e.g. should be between commas.
Add the missing dot at the end of the caption of Figure 7.
In line 453, brackets are not necessary.
In several places (Eqs. 7,9,11), the text line after the equation is indented – either remove the blank line after the equasion or add command \noindent
Citation: https://doi.org/10.5194/egusphere-2024-2156-RC2 -
AC2: 'Reply on RC2', Jonathan Ortved Melcher, 22 Oct 2024
The authors would like to thank the reviewer for their comprehensive and insightful comments. We have considered all suggestions and have incorporated the majority into the revised manuscript. Please find line-by-line comments attached in Reply_RC2/RC2.pdf
All line and section numbers refer to the line numbers in the enclosed annotated manuscript (Reply_RC2/annotated_manuscript.pdf), with suggested changes have been highlighted. We are aware that we have not been encouraged to submit a revised version yet but find it more fruitful in this case to demonstrate how we will change the manuscript in response to the reviews.
-
AC2: 'Reply on RC2', Jonathan Ortved Melcher, 22 Oct 2024
Status: closed
-
RC1: 'Review of Melcher et al', Anonymous Referee #1, 15 Aug 2024
Melcher and colleagues present an extended dynamical systems model designed to simulate Dangaard-Oeschger events. The aim of the model extension is to capture the irregular periodicity of observed DO-events which traditional modelling approaches have been unable to capture so far. The extended model is based on a Stommel-type box model formulated by Vettoretti et al (2020) and includes a novel control parameter that feeds back on AABW production. By varying the control parameters in time, the model is able to realistically capture the highly variable nature of stadial and interstadial duration. The authors thoroughly assess the behaviour of the new model and make a convincing case for the inclusion of the new control parameter.
Overall, the manuscript is well written, the figures are well prepared and the results are very relevant for the wider DO- and AMOC community. In the current form of the manuscript it difficult to get through the technical details in some places, and some points need clarification or additional context. Below I list my detailed comments (which are mostly minor).General Comments
---------------------------------------------
Derivation of the model:
I appreciate the effort the authors put into introducing the model in small steps and thoroughly illustrating the baseline model from Vettoretti et al (2022). At the same time, the order of things did not always work for me and confused me in parts. Especially the presentation of the equations was confusing. In three different places the text referred to a different set of equations as the model's equations (eq1&2, eq3&4, eq8-10). I see how the three sets are related but while reading I stumbled a lot and things only became clear at the very end. I am not sure about the best sequence of things. Perhaps it could work to keep the first part of 2.2 free of equations and discuss only in terms of processes. Regarding the presentation of the equations, for me the logical sequence would have been to start with the general form of equations (currently eq3&4), introduce the Vettoretti 2022 model (eq8&9) and then introduce the updated model including alpha (eq8&10). I am sure there are multiple valid ways of restructuring Section 2.2 and its subsections. I would however ask the authors to revisit the section and improve its clarity.Control parameters:
For non-experts in dynamical systems theory it would be helpful to always state which exactly are the control parameters in each context (instead of just referring to "the control parameters"). And also to always clarify whether a control parameter is constant throughout a simulation or being varied. In case of Fig.3: Would the system remain constant in time (except for noise) in the cases of a and c? What makes the system move along the delta_b manifold in case of constant control parameters?Precursor Events:
How many of these precursor events are present in the actual ice-core record? Can you elaborate a bit more on why it is important to be able to replicate them? I am also somewhat confused by the discussion of the different precursor types in Section 4. Can the Type I event be distinguished at all from a regular event? Is the existence of the Type 1 events purely theoretical? Do you see all three types of precursor events in different realisations? Fig9 only shows the one example.Alpha:
In some places it is unclear, whether the reproduced non-periodicity comes from introducing alpha or from *varying* alpha? From my understanding, the non-periodicity comes from varying alpha. This is especially important in the comparison with more complex GCMs (e.g., l419-423). Most GCMs that have produced self-sustained DO oscillations have done so with constant forcing. I would therefore think, that the self-sustained oscillations are more comparable with a constant alpha. The complex models are able to simulate the bipolar sea saw, therefore I think that some feedbacks comparable to alpha are already present in the models (whether they are realistic is another question of course). Some clarification and more detailed discussion of what the processes would represent alpha in the real ocean would be helpful here.Detailed Comments
---------------------------------------------l40-53 How are the two stochastic approaches different? Perhaps it si more useful to group into studies that are based on the Stommel model and those that are not? Which approaches do include the bipolar sea saw and which do not? Perhaps you could mention the FitzHugh-Nagumo model here as well already? Also make it clearer that the baseline for your new model is the model in Vettoretti et al 2022 and explain how the Vettoretti box model relates to previous box models.
l54 include also other comprehensive models that oscillate for completeness (e.g. MPI-ESM Klockmann et al (2020,https://doi.org/10.1029/2020GL090361), HadCM3 Armstrong et al (2023,https://doi.org/10.1007/s00382-022-06564-y) or Malmierca-Vallet et al (2023,https://doi.org/10.5194/cp-19-915-2023) for an overview).
l104 if you keep the order of section 2.2. make it explicit that eq 1 and 2 are not the final form, but a more general form of the equations, and that the equations refer to the baseline model without alpha (see general comment on model derivation)
l109/110 the dependence of q on delta_b is not obvious from eq1
Fig3 caption: "as lambda changes" what does "lambda" refer to here? The Eigenvalues have not been introduced yet
l157-157 is that also what you do or is it in contrast to what you do?
l163 what makes the southern bouyancy flux increase?
l166 is the salt-advection feedback acting on the northern or the southern box? Not equal to the typical AMOC-North Atlantic salt advection feedback?
l162-177 be careful with words. transport/flow/circulation seem to be used for q and AMOC simultaneously? What makes the system move along the manifold? I am probably lacking some basic dynamical systems theory here (see general comment on control parameters).
l182 what about the heat loss removing buoyancy in northern box/real ocean?
l193 I find this information confusing here without context.
l194 link eq3/4 back to eq1/2 (see general comment on model derivation)
l212 are there haline and thermal modes in this model?
eq7 what are B_c and b_c?l228-230 mention this earlier?
l242 which control parameters do refer to here?
l274ff state stability criteria up front for convenience, otherwise the reader needs to guess what the criterion is (negative Re(lambda1,2)? ad if yes, what does that mean?)
Fig5 Is the case with alpha = 0 in c equivalent to the cases presented in Fig3a-c?
l282 the change from b to d seems quite drastic, not so much from d to f
l296 which control parameter do you refer to here?
2.2.3 refer to left hand side of Fig.5 throughout the section to illustrate
l295ff interpretation of b0/value of b0?
l314 How do you set the initial conditions? Simply as a set of values for delta b and B?
l323 how do the 100 members differ? only in the noise realisation?
l327/l344 the age is impossible to determine from the crosses in Fig6. The caption suggests the crosses should be numbered, but they are not.
Fig6(especially a/b) How do mu_E=0 and a variable mu_P go together? If there is no event, should mu_p not always be either 0 or 1?
l335 "with decreasing gamma, mu_P' moves to lower values" is only true for gamma between 1.5 and 3. for gamma < 1.5 mu_P' increases with decreasing gamma. Or am I reading the plot wrong?
l338 what exactly do you mean by "extends the arches downward"? Do you mean that each line of constant gamma reaches lower values of mu_E'?
l342/343 again not sure what you mean by "increasing sigma moves the entire arch downward"?
l382-385 does this refer to the sequence in time?
l386 Are 91 and 21ka b2k the right timings? To me 97 and 24ka b2k would seem more appropriate? How relevant is this misfit?
l389/390 It might be worth including this member also with a separate colour to a and b.
l425-427 but would you have expected anything else given that the target are summary statistics and not the actual timeseries?
Editorial Comments
---------------------------------------------
l33 first mentioning of NGRIP here but the acronym is only introduced in l87l209 and Eq6 say the same thing
l269 "close to 1" instead of "intermediate"?
l271-273 difficult to read, please reformulate more clearly
l278 It should be "delta_b" in the equation and not "b"
l323/324 move sentence with Runge-Kutta to l316, after "[...]run for 5371 steps."?
l375 Is "linear interpolation" correct? The caption of Fig7 says cubic spline fit
l410 "improve" instead of "determine"?
l412 What is "origo"?
l421 "might be able" instead of "might be possible"
l456 "vertical jumps" instead of "vertical jump"
Citation: https://doi.org/10.5194/egusphere-2024-2156-RC1 -
AC1: 'Reply on RC1', Jonathan Ortved Melcher, 22 Oct 2024
The authors would like to thank reviewer one for their time and effort, in formulating a precise, and detailed review, please find line-by-line comments attached in Reply_RC1/RC1.pdf
All line and section numbers refer to the line numbers in the enclosed annotated manuscript (Reply_RC1/annotated_manuscript.pdf), with suggested changes have been highlighted. We are aware that we have not been encouraged to submit a revised version yet but find it more fruitful in this case to demonstrate how we will change the manuscript in response to the reviews.
-
AC1: 'Reply on RC1', Jonathan Ortved Melcher, 22 Oct 2024
-
RC2: 'Comment on egusphere-2024-2156', Anonymous Referee #2, 18 Sep 2024
The paper by Melcher et al “Advances in conceptual modelling of the variable nature of DO events” introduces a novel model that realistically reproduces the patterns of DO events in the NGRIP record using modulating ocean parameters. The paper is well written and can be published after a minor revision.
My comments are as follows:
The current title of the paper is more suitable for a review than for a novel research item, and it can be improved as follows: “A novel model of the dynamics of Dansgaard-Oeschger events in the NGRIP record”.
Multiple acronyms used in the paper would be convenient to introduce in a small table in the beginning.
Writing style could be improved. In lines 28-29, replace “supports the essential understanding of the underlying processes required to predict” with “is essential for prediction”. In line 45, “taking another approach, other studies have approached” must ne improved. In line 103, “the relationship between the model and the time series of the model” can be omitted (obviously, the time series produced by the model is its trajectory).
Type I & II events are denoted with or without capital letters, and sometimes with a dash. They, as well as interstadials, are mentioned with a reference but not well introduced for general readership.
In Fig.1, panel b, the y-axis for P(t) has not units – is it in kiloyears? If so, aren’t values within 0.8 kyr too small for stadials in a window of 20 kyr?
In Eqs. 1,2, NOISE could be replaced with some notation, and a short description of the noise would be useful (distribution, scaling properties?)
“Box model”, “earth system models” do not need to be capitalised.
The fixed points are repeatedly called “fixpoints” (including the legend of Figure 3).
In line 258, remove either “equations” or “Eqs”
Add dot at the end of Eq.10. Remove comma after “where” in line 264.
In line 310, remove “achievable”. This is discussed in the section and does not need to be mentioned in the section title.
In line 359, “because” should be with a small letter.
In line 370, e.g. should be between commas.
Add the missing dot at the end of the caption of Figure 7.
In line 453, brackets are not necessary.
In several places (Eqs. 7,9,11), the text line after the equation is indented – either remove the blank line after the equasion or add command \noindent
Citation: https://doi.org/10.5194/egusphere-2024-2156-RC2 -
AC2: 'Reply on RC2', Jonathan Ortved Melcher, 22 Oct 2024
The authors would like to thank the reviewer for their comprehensive and insightful comments. We have considered all suggestions and have incorporated the majority into the revised manuscript. Please find line-by-line comments attached in Reply_RC2/RC2.pdf
All line and section numbers refer to the line numbers in the enclosed annotated manuscript (Reply_RC2/annotated_manuscript.pdf), with suggested changes have been highlighted. We are aware that we have not been encouraged to submit a revised version yet but find it more fruitful in this case to demonstrate how we will change the manuscript in response to the reviews.
-
AC2: 'Reply on RC2', Jonathan Ortved Melcher, 22 Oct 2024
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