Climate Bifurcations in a Schwarzschild Equation Model of the Arctic Atmosphere
 ^{1}Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark
 ^{2}Dept. of Mathematics & Statistics, University of Guelph, Guelph, Canada
 ^{3}Faculty of Science, Ontario Tech University, Oshawa, Canada
 ^{1}Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark
 ^{2}Dept. of Mathematics & Statistics, University of Guelph, Guelph, Canada
 ^{3}Faculty of Science, Ontario Tech University, Oshawa, Canada
Abstract. A column model of the Arctic atmosphereocean system is developed including the nonlinear responses of surface albedo and water vapor to temperature. The atmosphere is treated as a gray gas and the flux of longwave radiation is governed by the twostream Schwarzschild equations. Representative carbon pathways (RCPs) are used to model carbon dioxide concentrations into the future. The resulting ninedimensional twopoint boundary value problem is solved under various RCPs and the solutions analyzed. The model predicts that under the highest carbon pathway, the Arctic climate will undergo an irreversible bifurcation to a warm steady state, which would correspond to an annually icefree situation. Under the lowest carbon pathway, corresponding to very aggressive carbon emission reductions, the model exhibits only a mild increase in Arctic temperatures. Under the two moderate carbon pathways, temperatures increase more substantially, and the system enters a region of bistability where external perturbations could possibly cause an irreversible switch to a warm, icefree state.
Kolja L. Kypke et al.
Status: closed

RC1: 'Comment on npg20222', Anonymous Referee #1, 17 Feb 2022
This paper presents a a column model for the Artic atmosphereocean system where surface albedo and water vapor effects are considered. The most relevant result is the existence of a bistable regime for certain values of CO2 concentration where the system could switch to a warm, icefree state. Although the model is a simplification of the climate system in the area, it contains enough ingredients to make qualitative predictions, and the authors really do a complete and careful study of parametrizations, functional forms of different processes, and calibration of the model. The results are interesting and the authors identify the role and importance of different physical mechanisms (at difference of some previous works) in a possible bifurcation to an undesired icefree state. The paper is ready for publication in my opinion. Maybe it is sometimes difficult to follow since there are many details, in particular in the appendixes, and the authors could try to improve the readibility of the paper in these sections.

RC2: 'Comment on npg20222', Marek Stastna, 31 Mar 2022
This manuscript presents a single column model of the Arctic atmosphere. The authors nondimensionalize this model and present a bifurcation analysis. They then draw fairly standard conclusions (Arctic change is nonlinear with possibly irreversible changes). The manuscript is well written, and the subject matter fits NPG well. The model is highly over simplified, and I would consider it unlikely the climate modelling community will pay it much attention. The mathematical analysis is fairly standard, and the numerical techniques used, in Matlab, should not trouble the majority of the readership. The model is built up in large part following Pierrehumbert’s book, and is presented in a similar spirit. Conclusions are drawn on published climate system scenarios, and this is a very nice touch. I have detailed suggestions below, but all of these are consistent with a manuscript that deserves to be published once appropriate changes are made.
Where I am somewhat troubled is by the “tail wagging the dog” aspect of the entire exercise. The point here is to use standard ODE theory, and modelling assumptions are made to suit this. The authors are reasonably transparent about this in their comments throughout the paper. Where they could be clearer is in acknowledging what does not fit their assumptions; namely the Arctic Ocean. In essence, the fundamental problem I have with the model is that it is not a simplification of a “more complete” model. There is no way for someone to say, “Ok I like these results now I will include a more realistic representation of the zonally asymmetric ocean” (or some other aspect of the climate system that is near and dear to their heart). I don’t think this would get me to suggest that the paper should not be published, but I think the authors could put some thought into this.
Detailed comments:
The introduction is one sided on box/column models. These models do have advantages, but some criticisms as well. Both points of view should be outlined.
The authors state that the Arctic atmosphere is, to a good approximation, zonally symmetric. The Arctic Ocean, however, is not. This needs to be clear.
The stratosphere is never mentioned. It is clear it is not part of the model, but it likely merits a short discussion at the very least.
The model is described in words; a diagram is essential.
The model appears to lack any moisture in the main troposphere. This merits discussion.
The strength of the model is the detailed exposition that explains it, and I want to add a positive comment as a note of appreciation for the authors’ efforts.
The table of quantities, B2, is very useful. Adding references to it earlier in the main text would be even more useful. If there is room, adding a column for equations the quantity appears in would be helpful.
Overall the figures are very nice, but could be improved.
The figures would be much improved by using plot(quantity1,quantity2,’linewidth’,2) and in some cases the “grid on” command in Matlab.
Panel labels, as opposed to titles are likely preferable, and if this is the case the captions should be expanded.
Figure 3 could use a legend for the bottom panel.
I would fill in the region between the curves in Figure 4 and increase the font side of the text in the figure.
So much good material is in the Appendices. Can the authors strongly push the readers to actually read the Appendices?

AC1: 'Comment on npg20222', Allan Willms, 20 Apr 2022
The authors thank the reviewers for their time and thoughtful comments on the manuscript.
We agree with the first reviewer that there are many details in the paper. They are included because we feel it important that the model be fully explained and so that the results could be replicated. We attempted to strike an appropriate balance between material in the text itself, and what should be put in the appendices. We feel strongly that the appendix material is not simply an addendum, but is essential to the paper, which concurs with the second reviewer's request that we promote reading of the appendices. The first reviewer expressed a desire that the appendices be more readable, but at the same time, the second reviewer commended the detailed explanation. We will therefore make a few more references within the paper itself, directing the reader to the appendices for further information, and we will edit the appendices to help enhance the clarity of the main points from each subsection therein. In particular we will edit Appendix B so that all of the parameter values will be presented in two tables rather than spread out over five.
The main concerns of the second reviewer were 1) that we were not sufficiently clear about the aspects of the Arctic climate that do not fit our assumptions, particularly the Arctic Ocean, and 2) that our model is not a simplification of a "more complete" model. The first concern is a valid criticism, and we will edit the paper to emphasize the zonal symmetry being assumed and how this symmetry is not present in the Arctic Ocean. We will also emphasize that the model is primarily an atmospheric model by editing the abstract to say "Arctic atmosphere" rather than "Arctic atmosphereocean system". However, we do not agree with the reviewer's second assertion. Zonal symmetry of the Earth's atmosphere is a reasonable and wellutilized approximation for a simple annually averaged model of climate. Our current model with a cylindrical atmosphere can be thought of as taking a limit as one restricts the Earth's atmosphere to a vanishingly small region centered at the North Pole. In this limit, the zonally averaged atmosphere becomes a onedimensional column with downward flow and the PDEs governing the atmospheric fluid flow become ODEs. Alternatively, one can view the model as a meridional and zonal average over a cylinder centred at the North Pole. Although the authors had this view of the model from the beginning, and it is expressed in the sixth paragraph of the introduction, we recognize that the manuscript may not adequately convey this point of view, hence we will edit to emphasize how our model arises from this simplification/limiting process, particularly by adding a paragraph near the beginning of Section 2. Further, once the model is understood as representing a small region around the North Pole, the use of a single scalar to represent ocean heat transport becomes a reasonable approximation, regardless of the fact that the ocean is not zonally symmetric. For calibration of our model we used values of ocean and atmospheric heat transport (F_O and F_A)
measured at 70N latitude. Although this clearly does not correspond to a small region around the North Pole, measured values of these heat transports are not readily available further north. Further, and partly to alleviate the concern around what values of F_O and F_A are used, we did a bifurcation analysis varying these two parameters, showing our conclusions are generic regardless of the precise values used (old Figure 5). We also used the solar insolation value, Q, for the portion of the Earth north of 70N latitude. This value came to 185 W/m^2. This value does not change much by restricting to a region closer to the pole; the limiting value is 173.8 W/m^2.Below we respond to the detailed comments of the second reviewer.
1) In the past, 1D column/box models have been used to describe the globallyaveraged climate of the Earth. These provide little detail and generally arise from the gross approximation of the entire atmosphere as a uniform slab or altitudevarying column. Here instead, our 1D model results from the assumption of a zonally symmetric atmosphere, making the polar axis invariant, and the limiting process as one considers a small region centered at the North Pole. Thus we believe our 1D model is relevant there. Furthermore, the Earth's climate is changing most rapidly in the high Arctic, so a polar model can be informative. To our knowledge, a 1D polar model has not been studied before.
2) It is true that the Arctic Ocean is not zonally symmetric, if one is considering the entire Arctic. However, if one is considering a small region around the North Pole, as explained above, this problem is minimized, and a single number can represent ocean heat transport.
3) The stratosphere is not part of the model, as was recognized by the reviewer. Since the air density is very small in the stratosphere there will be minimal absorption of radiation in the stratosphere, however the effect is not zero. As part of our modelling efforts, we did investigate a simplified stratosphere model attached to the present model, however the resulting quantitative changes to the radiation terms were considered not sufficiently large to warrant the additional complication of modelling the stratosphere. (Actually, the manuscript contained a notation in equations (A17) and (A24) that was a holdover from our stratosphere modelling that did not get edited out; these equations refer to the downward longwave radiation at the troposphere being the constant I_^{TP} and its nondimensional version K_. These will be removed and replaced with zero.)
4) The reviewer requested that we add a schematic figure of the model at its introduction. We will do so at the beginning of Section 2.
5) The reviewer indicates that the model lacks moisture in the main troposphere. This is a misreading by the reviewer. The entire atmosphere has moisture content governed by the ClausiusClapeyron equation and a linear decay of the relative humidity with altitude. These things are discussed in sections 2.1.3 and A.3.1. The absorption of longwave radiation due to moisture in the air is the third term in the expression for kappa given in Equation (8). Perhaps the reviewer's oversight was due to the fact that just the symbol kappa appears in Equation (11) and onward. We will add text to emphasize the role of watervapor feedback contained in the factor kappa. This feedback is certainly essential to our model.
6) We thank the reviewer for the commendation; we put considerable effort into explaining the model sufficiently so that it could be replicated.
7) We will refer to Table B2 earlier in the text, and will add equation numbers to the table as requested. In addition, to aid readability, we will combine the values from (old) Tables B4, B6, and B7 into Table B2.
8) We will make adjustments to the figures, including wider lines, panel labels, expanded captions, and legends, as requested.
Other changes to be made to the manuscript:
We will replace the variable notation M_{max} with M_{tot} since it refers to a total amount, not a maximum. Replace several references to F_{Amax} with F_A^{tot}, as they should have been. Replace constant F_{A0} with F_{A1} as it is more consistent with the fact that it is the value of F_A at 1. Fix a few other minor typos.
Status: closed

RC1: 'Comment on npg20222', Anonymous Referee #1, 17 Feb 2022
This paper presents a a column model for the Artic atmosphereocean system where surface albedo and water vapor effects are considered. The most relevant result is the existence of a bistable regime for certain values of CO2 concentration where the system could switch to a warm, icefree state. Although the model is a simplification of the climate system in the area, it contains enough ingredients to make qualitative predictions, and the authors really do a complete and careful study of parametrizations, functional forms of different processes, and calibration of the model. The results are interesting and the authors identify the role and importance of different physical mechanisms (at difference of some previous works) in a possible bifurcation to an undesired icefree state. The paper is ready for publication in my opinion. Maybe it is sometimes difficult to follow since there are many details, in particular in the appendixes, and the authors could try to improve the readibility of the paper in these sections.

RC2: 'Comment on npg20222', Marek Stastna, 31 Mar 2022
This manuscript presents a single column model of the Arctic atmosphere. The authors nondimensionalize this model and present a bifurcation analysis. They then draw fairly standard conclusions (Arctic change is nonlinear with possibly irreversible changes). The manuscript is well written, and the subject matter fits NPG well. The model is highly over simplified, and I would consider it unlikely the climate modelling community will pay it much attention. The mathematical analysis is fairly standard, and the numerical techniques used, in Matlab, should not trouble the majority of the readership. The model is built up in large part following Pierrehumbert’s book, and is presented in a similar spirit. Conclusions are drawn on published climate system scenarios, and this is a very nice touch. I have detailed suggestions below, but all of these are consistent with a manuscript that deserves to be published once appropriate changes are made.
Where I am somewhat troubled is by the “tail wagging the dog” aspect of the entire exercise. The point here is to use standard ODE theory, and modelling assumptions are made to suit this. The authors are reasonably transparent about this in their comments throughout the paper. Where they could be clearer is in acknowledging what does not fit their assumptions; namely the Arctic Ocean. In essence, the fundamental problem I have with the model is that it is not a simplification of a “more complete” model. There is no way for someone to say, “Ok I like these results now I will include a more realistic representation of the zonally asymmetric ocean” (or some other aspect of the climate system that is near and dear to their heart). I don’t think this would get me to suggest that the paper should not be published, but I think the authors could put some thought into this.
Detailed comments:
The introduction is one sided on box/column models. These models do have advantages, but some criticisms as well. Both points of view should be outlined.
The authors state that the Arctic atmosphere is, to a good approximation, zonally symmetric. The Arctic Ocean, however, is not. This needs to be clear.
The stratosphere is never mentioned. It is clear it is not part of the model, but it likely merits a short discussion at the very least.
The model is described in words; a diagram is essential.
The model appears to lack any moisture in the main troposphere. This merits discussion.
The strength of the model is the detailed exposition that explains it, and I want to add a positive comment as a note of appreciation for the authors’ efforts.
The table of quantities, B2, is very useful. Adding references to it earlier in the main text would be even more useful. If there is room, adding a column for equations the quantity appears in would be helpful.
Overall the figures are very nice, but could be improved.
The figures would be much improved by using plot(quantity1,quantity2,’linewidth’,2) and in some cases the “grid on” command in Matlab.
Panel labels, as opposed to titles are likely preferable, and if this is the case the captions should be expanded.
Figure 3 could use a legend for the bottom panel.
I would fill in the region between the curves in Figure 4 and increase the font side of the text in the figure.
So much good material is in the Appendices. Can the authors strongly push the readers to actually read the Appendices?

AC1: 'Comment on npg20222', Allan Willms, 20 Apr 2022
The authors thank the reviewers for their time and thoughtful comments on the manuscript.
We agree with the first reviewer that there are many details in the paper. They are included because we feel it important that the model be fully explained and so that the results could be replicated. We attempted to strike an appropriate balance between material in the text itself, and what should be put in the appendices. We feel strongly that the appendix material is not simply an addendum, but is essential to the paper, which concurs with the second reviewer's request that we promote reading of the appendices. The first reviewer expressed a desire that the appendices be more readable, but at the same time, the second reviewer commended the detailed explanation. We will therefore make a few more references within the paper itself, directing the reader to the appendices for further information, and we will edit the appendices to help enhance the clarity of the main points from each subsection therein. In particular we will edit Appendix B so that all of the parameter values will be presented in two tables rather than spread out over five.
The main concerns of the second reviewer were 1) that we were not sufficiently clear about the aspects of the Arctic climate that do not fit our assumptions, particularly the Arctic Ocean, and 2) that our model is not a simplification of a "more complete" model. The first concern is a valid criticism, and we will edit the paper to emphasize the zonal symmetry being assumed and how this symmetry is not present in the Arctic Ocean. We will also emphasize that the model is primarily an atmospheric model by editing the abstract to say "Arctic atmosphere" rather than "Arctic atmosphereocean system". However, we do not agree with the reviewer's second assertion. Zonal symmetry of the Earth's atmosphere is a reasonable and wellutilized approximation for a simple annually averaged model of climate. Our current model with a cylindrical atmosphere can be thought of as taking a limit as one restricts the Earth's atmosphere to a vanishingly small region centered at the North Pole. In this limit, the zonally averaged atmosphere becomes a onedimensional column with downward flow and the PDEs governing the atmospheric fluid flow become ODEs. Alternatively, one can view the model as a meridional and zonal average over a cylinder centred at the North Pole. Although the authors had this view of the model from the beginning, and it is expressed in the sixth paragraph of the introduction, we recognize that the manuscript may not adequately convey this point of view, hence we will edit to emphasize how our model arises from this simplification/limiting process, particularly by adding a paragraph near the beginning of Section 2. Further, once the model is understood as representing a small region around the North Pole, the use of a single scalar to represent ocean heat transport becomes a reasonable approximation, regardless of the fact that the ocean is not zonally symmetric. For calibration of our model we used values of ocean and atmospheric heat transport (F_O and F_A)
measured at 70N latitude. Although this clearly does not correspond to a small region around the North Pole, measured values of these heat transports are not readily available further north. Further, and partly to alleviate the concern around what values of F_O and F_A are used, we did a bifurcation analysis varying these two parameters, showing our conclusions are generic regardless of the precise values used (old Figure 5). We also used the solar insolation value, Q, for the portion of the Earth north of 70N latitude. This value came to 185 W/m^2. This value does not change much by restricting to a region closer to the pole; the limiting value is 173.8 W/m^2.Below we respond to the detailed comments of the second reviewer.
1) In the past, 1D column/box models have been used to describe the globallyaveraged climate of the Earth. These provide little detail and generally arise from the gross approximation of the entire atmosphere as a uniform slab or altitudevarying column. Here instead, our 1D model results from the assumption of a zonally symmetric atmosphere, making the polar axis invariant, and the limiting process as one considers a small region centered at the North Pole. Thus we believe our 1D model is relevant there. Furthermore, the Earth's climate is changing most rapidly in the high Arctic, so a polar model can be informative. To our knowledge, a 1D polar model has not been studied before.
2) It is true that the Arctic Ocean is not zonally symmetric, if one is considering the entire Arctic. However, if one is considering a small region around the North Pole, as explained above, this problem is minimized, and a single number can represent ocean heat transport.
3) The stratosphere is not part of the model, as was recognized by the reviewer. Since the air density is very small in the stratosphere there will be minimal absorption of radiation in the stratosphere, however the effect is not zero. As part of our modelling efforts, we did investigate a simplified stratosphere model attached to the present model, however the resulting quantitative changes to the radiation terms were considered not sufficiently large to warrant the additional complication of modelling the stratosphere. (Actually, the manuscript contained a notation in equations (A17) and (A24) that was a holdover from our stratosphere modelling that did not get edited out; these equations refer to the downward longwave radiation at the troposphere being the constant I_^{TP} and its nondimensional version K_. These will be removed and replaced with zero.)
4) The reviewer requested that we add a schematic figure of the model at its introduction. We will do so at the beginning of Section 2.
5) The reviewer indicates that the model lacks moisture in the main troposphere. This is a misreading by the reviewer. The entire atmosphere has moisture content governed by the ClausiusClapeyron equation and a linear decay of the relative humidity with altitude. These things are discussed in sections 2.1.3 and A.3.1. The absorption of longwave radiation due to moisture in the air is the third term in the expression for kappa given in Equation (8). Perhaps the reviewer's oversight was due to the fact that just the symbol kappa appears in Equation (11) and onward. We will add text to emphasize the role of watervapor feedback contained in the factor kappa. This feedback is certainly essential to our model.
6) We thank the reviewer for the commendation; we put considerable effort into explaining the model sufficiently so that it could be replicated.
7) We will refer to Table B2 earlier in the text, and will add equation numbers to the table as requested. In addition, to aid readability, we will combine the values from (old) Tables B4, B6, and B7 into Table B2.
8) We will make adjustments to the figures, including wider lines, panel labels, expanded captions, and legends, as requested.
Other changes to be made to the manuscript:
We will replace the variable notation M_{max} with M_{tot} since it refers to a total amount, not a maximum. Replace several references to F_{Amax} with F_A^{tot}, as they should have been. Replace constant F_{A0} with F_{A1} as it is more consistent with the fact that it is the value of F_A at 1. Fix a few other minor typos.
Kolja L. Kypke et al.
Kolja L. Kypke et al.
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