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.
- Preprint
(723 KB) - Metadata XML
- BibTeX
- EndNote
Status: open (until 24 Jul 2024)
-
RC1: 'Comment on npg-2024-11', Anonymous Referee #1, 22 May 2024
reply
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
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
Viewed
HTML | XML | Total | BibTeX | EndNote | |
---|---|---|---|---|---|
264 | 30 | 10 | 304 | 8 | 7 |
- HTML: 264
- PDF: 30
- XML: 10
- Total: 304
- BibTeX: 8
- EndNote: 7
Viewed (geographical distribution)
Country | # | Views | % |
---|
Total: | 0 |
HTML: | 0 |
PDF: | 0 |
XML: | 0 |
- 1