the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 12 Sep 2023
Variational Techniques for a One-Dimensional Energy Balance Model
Abstract. A one-dimensional climate energy balance model (1D-EBM) is a simplified climate model that describes the evolution of Earth's temperature based on the planet's energy budget. In this study, we examine a 1D-EBM that incorporates a bifurcation parameter representing the impact of carbon dioxide on the energy balance. Firstly, independent of the value of the additive parameter, we demonstrate the existence of a steady-state solution by solving the associated variational problem, which involves minimizing a potential functional. Again using variational techniques, we can give sufficient conditions to prove the existence of at least three-steady state solutions. Secondly, we establish the uniqueness of the solution for the variational problem by examining the differentiability of the value function, which represents the minimum value of the potential functional across all temperature profiles. Lastly, we explore how this characterization provides valuable insights into the structure of the bifurcation diagram. Specifically, we demonstrate a one-to-one correspondence between the derivative of the value function and the mean value of the minimizer for the variational problem. Furthermore, we show the applicability of our findings to more general reaction-diffusion spatially heterogeneous models.
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Notice on discussion status
The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
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Preprint
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Supplement
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The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
- Preprint
(1399 KB) - Metadata XML
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Supplement
(346 KB) - BibTeX
- EndNote
- Final revised paper
Journal article(s) based on this preprint
Interactive discussion
Status: closed
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RC1: 'Comment on egusphere-2023-1994', Anonymous Referee #1, 06 Nov 2023
This paper presents a rigorous analysis of the stable solution of a 1D energy balance model with a parameter that explicitly represents the radiative effects of CO2 concentration. The existence of a stable solution is proven, and conditions to prove the existence of more stable solutions are provided. The authors also provide a generalization of their results to other reaction-diffusion equations.
The paper is well structured, and the results are carefully presented.
Overall, the paper presents a rigorous and detailed discussion of the results. However, a discussion of the implications of these results is missing. I could only find the following sentence expressing the link between what is discussed in the paper and the climate system:
"Our physical interpretation of these results is that either the climate will fluctuate around a single equilibrium state, other states are exponentially less likely, and the global mean temperature will change smoothly with changes in CO2; or there are several equally likely states, some of which must differ in their global mean temperature."
Given the broad audience of the NPG journal, I think that the paper would be strengthened if the discussion about the implications of these results for climate science were presented in more detail. Also, it would be nice to provide an interpretation of some results in terms of the physics of the system at stake in the result section. This can make the paper more friendly to the audience of the journal and increase its impact within the geoscientific community. So, my recommendation is that the paper should be revised to strengthen the discussion of the physical interpretation of the results and their implications for climate science.
Below, I provide a few minor, specific comments.
L50 Could the authors provide more insight on the physical interpretation of parameter \nu ?
L56 Using u as the zonally averaging temperature may be confusing. Why not use T* or any other variant more consistent with the notation used in the 0D model?
u_{xx} is used in Equation 4, and \Delta u is used later to represent the same quantity. Also, u' is used instead of u_x in this same equation.
Eq. 5. Could the authors clarify the notation for the second term of the right hand side of this equation?
L70 It would be nice to point out that parameter \kappa represents heat transport by the atmospheric dynamics, whose variability is known to be related to temperature gradients (an effect that is not explicitly accounted for in the simplified model).
L125, where q is assumed to be independent of latitude. It would be nice to include some discussion about how realistic this assumption is (e.g., https://agupubs.onlinelibrary.wiley.com/doi/full/10.1002/2017JD027221 ).
At the end of page 6, "is parametrized by a smooth, monotonically increasing function ". Is this statement correct? Shouldn't the albedo be a monotonically decreasing function of the temperature?
L140 The covariance of \mu is defined using \Delta which is an operator. Could the authors clarify this aspect?
L147, could the authors clarify the meaning of the equation that is just before the sentence starting with "A rigorus..."?
L156 Could the authors provide more detail on how these simulations have been performed to obtain the results shown in Figure 2?
Citation: https://doi.org/10.5194/egusphere-2023-1994-RC1 - AC1: 'Reply on RC1', Gianmarco Del Sarto, 21 Jan 2024
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RC2: 'Comment on egusphere-2023-1994', Anonymous Referee #2, 27 Nov 2023
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2023/egusphere-2023-1994/egusphere-2023-1994-RC2-supplement.pdf
- AC2: 'Reply on RC2', Gianmarco Del Sarto, 21 Jan 2024
Interactive discussion
Status: closed
-
RC1: 'Comment on egusphere-2023-1994', Anonymous Referee #1, 06 Nov 2023
This paper presents a rigorous analysis of the stable solution of a 1D energy balance model with a parameter that explicitly represents the radiative effects of CO2 concentration. The existence of a stable solution is proven, and conditions to prove the existence of more stable solutions are provided. The authors also provide a generalization of their results to other reaction-diffusion equations.
The paper is well structured, and the results are carefully presented.
Overall, the paper presents a rigorous and detailed discussion of the results. However, a discussion of the implications of these results is missing. I could only find the following sentence expressing the link between what is discussed in the paper and the climate system:
"Our physical interpretation of these results is that either the climate will fluctuate around a single equilibrium state, other states are exponentially less likely, and the global mean temperature will change smoothly with changes in CO2; or there are several equally likely states, some of which must differ in their global mean temperature."
Given the broad audience of the NPG journal, I think that the paper would be strengthened if the discussion about the implications of these results for climate science were presented in more detail. Also, it would be nice to provide an interpretation of some results in terms of the physics of the system at stake in the result section. This can make the paper more friendly to the audience of the journal and increase its impact within the geoscientific community. So, my recommendation is that the paper should be revised to strengthen the discussion of the physical interpretation of the results and their implications for climate science.
Below, I provide a few minor, specific comments.
L50 Could the authors provide more insight on the physical interpretation of parameter \nu ?
L56 Using u as the zonally averaging temperature may be confusing. Why not use T* or any other variant more consistent with the notation used in the 0D model?
u_{xx} is used in Equation 4, and \Delta u is used later to represent the same quantity. Also, u' is used instead of u_x in this same equation.
Eq. 5. Could the authors clarify the notation for the second term of the right hand side of this equation?
L70 It would be nice to point out that parameter \kappa represents heat transport by the atmospheric dynamics, whose variability is known to be related to temperature gradients (an effect that is not explicitly accounted for in the simplified model).
L125, where q is assumed to be independent of latitude. It would be nice to include some discussion about how realistic this assumption is (e.g., https://agupubs.onlinelibrary.wiley.com/doi/full/10.1002/2017JD027221 ).
At the end of page 6, "is parametrized by a smooth, monotonically increasing function ". Is this statement correct? Shouldn't the albedo be a monotonically decreasing function of the temperature?
L140 The covariance of \mu is defined using \Delta which is an operator. Could the authors clarify this aspect?
L147, could the authors clarify the meaning of the equation that is just before the sentence starting with "A rigorus..."?
L156 Could the authors provide more detail on how these simulations have been performed to obtain the results shown in Figure 2?
Citation: https://doi.org/10.5194/egusphere-2023-1994-RC1 - AC1: 'Reply on RC1', Gianmarco Del Sarto, 21 Jan 2024
-
RC2: 'Comment on egusphere-2023-1994', Anonymous Referee #2, 27 Nov 2023
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2023/egusphere-2023-1994/egusphere-2023-1994-RC2-supplement.pdf
- AC2: 'Reply on RC2', Gianmarco Del Sarto, 21 Jan 2024
Peer review completion
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Gianmarco Del Sarto
Jochen Bröcker
Franco Flandoli
Tobias Kuna
The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
- Preprint
(1399 KB) - Metadata XML
-
Supplement
(346 KB) - BibTeX
- EndNote
- Final revised paper