the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Solving a North-type energy balance model using boundary integral methods
Abstract. Simplified climate models such as energy balance models (EBMs) are useful conceptual tools, in part because their reduced complexity often allows for studies using analytical methods. In this paper, we solve a North-type EBM using a boundary integral method (BIM). The North-type EBM is a diffusive one-dimensional EBM with a non-linear albedo feedback mechanism. We discuss this approach in light of existing analytical techniques for this type of equation. Subsequently, we test the proposed method by solving multiple North-type EBMs with a zonally symmetric continent featuring an altered ice-albedo feedback dynamic. We demonstrate that the introduction of a continent results in new equilibrium states characterized by multiple ice edges and ice belts. Furthermore, we show that the BIM serves as an efficient framework for handling unconventional ice distributions and model configurations for North-type EBMs.
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Status: open (until 14 Aug 2024)
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RC1: 'Comment on npg-2024-11', Anonymous Referee #1, 22 May 2024
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The manuscript introduces the boundary integral method (BIM) to find equilibrium solutions, and their corresponding stability properties, in North-type energy balance models (EBMs). I fully support the use of EBMs to understand and characterize the main features of climate states under different conditions. The manuscript is well written and it is surely appropriate for NPG. I have some moderate suggestions to compare the results of this paper with previously obtained ones in a similar framework.
Main comments
- An interesting result is the "striped" pattern obtained in Figure 3. This pattern has been also reported in 1D EBMs including vegetation (e.g., Nevison et al., 1999; Adams et al., 2003; Alberti et al., 2015). I would suggest the authors to compare their results with those papers in terms of active feedbacks, especially those related to the ice-albedo feedback.
- The authors consider a step-wise function for the albedo with a latitude dependence. However, they do not include in the feedback the extension of the continent (e.g., Wood et al., 2008; Rombouts and Ghil, 2015; Alberti et al., 2018). I would recommend to comment on this point and how the BIM can be generalized towards including additional contributions in the ice-albedo feedback.
- I found the results on the larger number of equilibria very interesting. I would suggest to include more discussion on the bifurcation diagrams reported in Figure 2. What about inspecting how the extension of the stable and unstable regimes depend on ε? This would suggest and provide more information on the residence time in a specific state for the model that can be useful for a broader discussion on the role of ice-ocean-land distribution on the climate states.
Minor comments
- Line 38: the values are slightly different from those usually reported in literature (Budyko, 1969; Rombouts and Ghil, 2015). Are these values corrected for current levels of greenhouse effect?
- Line 60: is in the definition of β a missing Ts or Eq. (3) is just written in dimensionless temperature (T → T/Ts)? Please clarify.
- Eq. (8): please clarify that this is valid in a linear approximation around an equilibrium solution.
- Eqs. (11)-(12): should these conditions valid for each selection of the ice-albedo feedback? Are there for boundary conditions at the poles, right?
- Line 144: why choosing l = π/4? Please also clarify that in the ice-albedo feedback there is no ice-ocean-land feedback.
- Figure 3: I would recommend to increase the quality of the figure since it seems that θ labels are not fully resolved.
Suggested references
- Adams, B., et al. (2003) J. Theor. Biol., 223, 505.
- Adams, B., and Carr, J. (2003) Nonlinearity, 16, 1339.
- Alberti, T., et al. (2015) Phys. Rev. E, 92, 052717.
- Alberti, T., et al. (2017) ApJ, 844, 19.
- Alberti, T., et al. (2018) J. Phys. Comm., 2, 065018.
- Nevison, C., et al. (1999) Tellus B, 53, 288.
- Rombouts, J., and Ghil, M. (2015) NPG, 22, 275.
- Wood, A.J., et al. (2008) Rev. Geophys., 46, RG1001.
Citation: https://doi.org/10.5194/npg-2024-11-RC1 -
CC1: 'Comment on npg-2024-11', Francesco Berrilli, 16 Jul 2024
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Report for the manuscript NPG-2024-11 "Solving a North-type energy balance model using boundary integral methods" authors Aksel Samuelsberg and Per Kristen Jakobsen
The work is well organized and written, however some variations or further explanations are required in order to be accepted for publication.
Here is the list not in order of relevance:
p.2 eq.1: please specify the physical meaning of parameters A, B and D. Even if a bibliographical reference is reported, the reader will be helped to understand the equation;
p.3 eq.3-eq.8: there are several points to clarify:
1) please explain the dependence (delta,T) of the variables T(delta) and h(delta,T);
2) detail the steps that allow eq.6+eq.7 to eq.8 (for example by inserting the first equations reported in Appendix A and inserting them before eq.8). This makes it easier for the reader.
3) please number all equations;
4) In eq.8 we seem to be missing a term in sin(xi). In particular I find: sin(xi)T(xi)= integral to the right of eq.8. Did I do something wrong?
p.2, p.8, p.9 the authors use the term solar constant to define the control parameter Q of the incident radiation. The term generally accepted today in scientific literature is Total Solar Irradiance (TSI). This removes the ambiguity of the adjective "constant" as we know that the TSI varies on different time scales due to different physical processes that occur on our star. I suggest introducing the term TSI instead of solar constant.
Citation: https://doi.org/10.5194/npg-2024-11-CC1 -
RC2: 'Comment on npg-2024-11', Anonymous Referee #2, 25 Jul 2024
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General
In the present manuscript the authors apply the boundary integral method (BIM) to find equilibrium solutions of a North-type one dimensional energy balance model. The proposed method is described in detail and applied using some case studies concerning the effect of a continent. The study demonstrates that BIM is a valuable method and may be applicable to conceptual models of similar type, which can still contribute significantly to our process understanding. The results of the showcase experiments indicate the existence a variety of equilibrium solutions depending on the location of the continent.
The manuscript is well written. The methodology is sound and sufficiently explained together with an appropriate application. Thus, in general, I can recommend publication. However, the authors may like to add some more discussion of the results (see Specific).
Specific
While I am happy with the methodology, there is (unfortunately) very little discussion of the (interesting) results. One may wonder why the authors perform (sort of) a sensitivity study with different continental set ups only to illustrate that the method is able to find equilibrium solutions. For this, one example would have been sufficient. In my view, a more thorough discussion of the results would add significant to the value of this paper.
Minor
1) I may have overlooked it, but the authors may state which root searching algorithm they have used.
2) It seems that starting from section 2 the variable T is non-dimensional (T/Ts)) which may be confusing given the dimensional T in Eq. (1). The authors may use different symbols for dimensional and non-dimensional variables, respectively.
3) All equations should be numbered.
Citation: https://doi.org/10.5194/npg-2024-11-RC2
Model code and software
Code for "Solving a North-type energy balance model using boundary integral methods" A. Samuelsberg and P. K. Jakobsen https://doi.org/10.5281/zenodo.11083624
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