the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 11 Dec 2023
Linkage of tropical glaciation to supercontinents: a thermodynamic closure model
Abstract. Precambrian tropical glaciations pose a significant challenge to our understanding of Earth’s climate. A popular explanation invokes runaway ice-albedo feedback leading to “iceball earth”, an extreme state conflicting however with the sedimentary evidence of an open ocean and active hydrological cycle. We point out flawed physics of the runaway scenario, which overlooks potency of the ocean heat transport in deterring the perennial sea ice. Nor is frozen ocean needed for tropical glaciation as the latter requires only that the tropical land be cooled to below the marking temperature of the glacial margin, which is necessarily above the freezing point to counter the yearly accumulation. Since tropical glaciations generally coincide with Precambrian supercontinents, we posit that it is their blockage of the brighter tropical sun that causes the required cooling. To test this hypothesis, we formulate a minimal two-box model, which is nonetheless thermodynamically closed and yields lowering tropical/polar temperatures with increasing tropical land, whose crossings of the glacial marking temperature would divide non/polar/pan-glacial regimes—the last being characterized by tropical glaciation abutting an open ocean. Given the observed chronology of paleogeography, our theory may provide a unified account of the faint-young-sun paradox, Precambrian tropical glaciations and glacio-epochs through Earth’s history.
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RC1: 'Comment on esd-2023-32', Christoph Braun, 29 Feb 2024
First of all I want to clarify that I was not familiar with the concept of maximum entropy production (MEP) before reviewing this manuscript and therefore will not attempt to judge, whether the basic and additional assumptions of this concept are valid here. Thus, I put my focus on whether I can follow the arguments presented here and how to align the conclusions with previous studies.
I first answer some general questions regarding the manuscript. Then I provide a general comment, as well as some specific further remarks and references supporting my comments.
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Does the paper address relevant scientific questions within the scope of ESD?
Yes, the manuscript discusses implications of the postulated basic thermodynamic principle of maximum entropy production on the interactions of continental configuration, ocean heat transport and land/sea ice extent.
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Does the paper present novel concepts, ideas, tools, or data?
I would prefer leaving the judgement of this to the editor and other reviewers. The presented core idea is relating the major cycles of global glaciation on Earth to the continental configuration based on the concept of maximum entropy production using a highly idealized box model.
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Are substantial conclusions reached?
Yes, the main conclusions reached is the claim that the major cycles of global glaciation (none, polar, and pan-glaciation) are determined by the continental configuration only and the sea-ice albedo feedback plays a negligible role for pan-glacial states during Earth’s history.
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Are the scientific methods and assumptions valid and clearly outlined?
I have problems of relating my understanding of the ice-albedo feedback with the applied model
One assumption that from my point of view is key to the importance of the presented results is the treatment of the ice-albedo feedback in the applied model. This is quite unclear to me.
An explicit assumption regarding ocean albedo and how it changes with the formation of sea ice as well as the albedo of snow/ice covered land is not made. However, in line 246 it is mentioned that there is an ice-albedo feedback in the model. How is it incorporated?
Another aspect that from my point of view may have a strong impact on the sea ice extent, but is not considered here, is the potential formation of high latitude sea glaciers that may be ~1km thick and thus cause gravitational spread of sea ice towards the equator (see Fig. 16 in Hoffman et al., 2017). Please clarify, or justify why the inclusion of this effect may not be necessary.
Lines 228 to 230: This content relates to my questions regarding sea ice. I do not understand the following formulation:
“and with the differential temperature fixed as such, MEP implies a maximized OHT. This leads immediately to vanishing perennial ice since any such ice would curb the ocean cooling hence OHT, contravening its maximization”
Specific points, I do not understand, are:
1) To what (value/state) is the differential temperature fixed, and why does MEP then imply maximized OHT?
2) Is atmospheric heat transport also considered here? If not, why?
3) Why do sea ice formation and maximized OHT exclude each other?
Besides that, mostly, the assumptions are clearly outlined. Some of the assumptions require further clarification; see list below.
Line 164: Assumption on orography to produce glacier flow towards sea. In this context a discussion of Walsh et al., 2019 might be beneficial.
line 181: The assumption on cloud albedo seems very strong, given the zonal-mean cloud albedo of present-day climate; cf e.g. Södergren and McDonald, 2022; Fig. 1 c). I suggest to further justify this assumption and include a sensitivity test of the derived results regarding this assumption. E.g., how would results look like for polar/tropical cloud albedo of 0.4/0.2 ?
line 182: land reflectance. I suggest to also include a sensitivity test here.
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Are the results sufficient to support the interpretations and conclusions?
Besides my points lined out above (4.): If accepting the principle of MEP and the assumptions made throughout the manuscript, the presented theory seems to support the interpretations and conclusions.
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Is the description of experiments and calculations sufficiently complete and precise to allow their reproduction by fellow scientists (traceability of results)?
Yes, as far as I see the equations provided throughout the manuscript allow the calculation of the qualitative results shown in the figures.
I have two points, though:
L173: Although it is explained, I do not understand why the factor 2 needs to be taken into account here.
Regarding Fig 3c: Why is there unglaciated polar land for T2 = 0°C ?
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Do the authors give proper credit to related work and clearly indicate their own new/original contribution?
Yes.
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Does the title clearly reflect the contents of the paper?
Yes.
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Does the abstract provide a concise and complete summary?
Yes.
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Is the overall presentation well structured and clear?
The structure of the manuscript is clear. However I have problems in relating sections 2 and 3.
E.g., in section 2 the author refers to MEP based on turbulent wind, whereas in section 3, the author discusses MEP in the context of ocean heat transport.
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Is the language fluent and precise?
In my perception the language used throughout the article hinders to access the content of the article. This is mainly due to some of the used vocabulary and sentence structure. To some extent the language even appears offensive to me (e.g. the usage of the word “flawed”).
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Are mathematical formulae, symbols, abbreviations, and units correctly defined and used?
In Fig. 3 b) the meaning of “I” labelling the right hand side axis is not mentioned in the figure caption.
It is not clear to me, whether A and A* refer to land area or land area fraction.
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Should any parts of the paper (text, formulae, figures, tables) be clarified, reduced, combined, or eliminated?
As mentioned in 10. I would appreciate more guidance on relating sections 2 and 3.
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Are the number and quality of references appropriate?
Yes
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Is the amount and quality of supplementary material appropriate?
There is no supplementary material. I would appreciate if a sensitivity study of relevant parameters would be added here (see point 4 for suggestions).
General comment
From my point of view more complex approaches of modeling climate states of Earth’s past based on dynamic three-dimensional equations serve to focus on single aspects of Earth system dynamics. Yet, I see the point of the author, that these approaches are highly idealized and thus may overlook the interaction of fundamental processes. I do not see a strong conflict between the approach based on MEP presented here and more complex modeling approaches of pan-glacial states on Earth. Thus, I would appreciate and recommend to include a discussion of how the presented main conclusion in this manuscript relates to more complex modeling approaches lined out below. The points below are partly taken from Braun, 2022 chapter 2.3. Cryogenian waterbelt scenarios; see there for further details.
1) Hyde et al., 2000 found tropical land ice with a sea ice edge located at 25° lat and a continuous transition between this climate state and climate states with less ice cover, i.e. without an ice-albedo feedback. They applied “GCM GENESIS 2 coupled to a thermodynamic mixed-layer ocean and thermodynamic sea ice, neglecting ocean and sea-ice dynamics”.
2) Based on simulations with the GCM FOAM, Poulsen et al. (2001) “found iceball states with mixed-layer ocean but not with a fully coupled ocean”, i.e. in their model “ocean heat transport counteracts the expansion of sea ice towards the equator”.
3) Based on a “coupled ocean–atmosphere model of intermediate complexity (CLIMBER-2)”, “Donnadieu et al. (2004b) considered the major stabilizing effect on the simulated waterbelt climate to be meridional atmospheric heat transport via the Hadley circulation.”
4) Rose (2015) found that “ocean heat convergence at the ice edge driven by a feedback between ice extent, wind stress and ocean circulation was found to stabilize the waterbelt state”.
Further remarks
l30: Please explain abbreviation pCO2 l55: Please clarify the statement regarding the unbalanced initial state.
L78: What does external conditions explicitly refer to?
L105: Please explain the expression “linear in the corresponding blackbody 105 radiance”.
L162 to 165: I find it hard to follow and understand the argument about the marking temperature (here and throughout the text)
l225: High obliquity hypothesis: The box model assumes a specific zonal distribution of radiative forcing to polar/tropical boxes. The high obliquity hypothesis is based on the recognition that zonal distribution of radiative forcing changes. Thus, the distribution of radiative forcing to polar and tropical regions in the box model would need to be adjusted for application to a high obliquity state. I suggest to discuss this or apply the box model with modified radiative forcing to further investigate this statement.
L314, 315: See my point regarding cloud albedo in present-day climate above.
References
Hoffman, Paul F., Dorian S. Abbot, Yosef Ashkenazy, Douglas I. Benn, Jochen J. Brocks, Phoebe A. Cohen, Grant M. Cox, et al. “Snowball Earth Climate Dynamics and Cryogenian Geology-Geobiology.” Science Advances 3, no. 11 (2017). https://doi.org/10.1126/sciadv.1600983.
Walsh, Amber, Thomas Ball, and David M Schultz. “Extreme Sensitivity in Snowball Earth Formation to Mountains on PaleoProterozoic Supercontinents.” Scientific Reports 9, no. 1 (2019): 1–7. https://doi.org/10.1038/s41598-019-38839-6.
Södergren, A. H., and A. J. McDonald. “Quantifying the Role of Atmospheric and Surface Albedo on Polar Amplification Using Satellite Observations and CMIP6 Model Output.” Journal of Geophysical Research: Atmospheres 127, no. 12 (2022): e2021JD035058. https://doi.org/10.1029/2021JD035058.
Braun, dissertation 2022: https://publikationen.bibliothek.kit.edu/1000150229 Hyde, William T, Thomas J Crowley, Steven K Baum, and W Richard Peltier. “Neoproterozoic ‘Snowball Earth’ Simulations with a Coupled Climate/Ice-Sheet Model.” Nature 405, no. 6785 (2000): 425–29. https://doi.org/10.1038/35013005.
Poulsen, Christopher J., Raymond T. Pierrehumbert, and Robert L. Jacob. “Impact of Ocean Dynamics on the Simulation of the Neoproterozoic ‘Snowball Earth.’” Geophysical Research Letters 28, no. 8 (2001): 1575–78. https://doi.org/10.1029/2000GL012058.
Donnadieu, Yannick, Yves Goddéris, Gilles Ramstein, Anne Nédélec, and Joseph Meert. “A ‘Snowball Earth’Climate Triggered by Continental Break-up through Changes in Runoff.” Nature 428, no. 6980 (2004): 303–6. Rose, Brian E. J. “Stable ‘Waterbelt’ Climates Controlled by Tropical Ocean Heat Transport: A Nonlinear Coupled Climate Mechanism of Relevance to Snowball Earth.” Journal of Geophysical Research: Atmospheres 120, no. 4 (2015): 1404–23. https://doi.org/10.1002/2014JD022659.
Citation: https://doi.org/10.5194/esd-2023-32-RC1 -
AC1: 'Reply on RC1', Hsien-Wang Ou, 19 Mar 2024
The comment was uploaded in the form of a supplement: https://esd.copernicus.org/preprints/esd-2023-32/esd-2023-32-AC1-supplement.pdf
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RC2: 'Comment on esd-2023-32', Anonymous Referee #2, 01 Mar 2024
Title: Linkage of tropical glaciation to supercontinents: a thermodynamic closure model
Author: Hsien-Wang Ou
In this paper, the author uses a simplified two-box model of the Earth and investigates how the tropical and polar temperature distribution changes with the change in the area of tropical land. The main finding is that, as the tropical land area increases, solar radiation is reflected effectively, and both tropical and polar temperatures tend to decrease towards the ice-forming temperature, although the land area in the polar region decreases. This tendency may explain tropical glaciations due to the appearance of Precambrian supercontinents, as well as some subsequent glacial epochs triggered by continental drifts. This reviewer finds these results basically interesting. However, the model assumes intricate relationships between radiations and temperatures that are not explained explicitly in this paper. Also, some of the model parameters and assumptions are stated so vaguely that readers of this paper cannot verify the validity of the results obtained. This reviewer therefore recommends major revisions of this paper regarding the comments below.
Major comments:
1. Equation (1) and the sensitivity of s = 0.5 °C/(W/m2).
This sensitivity (0.5) is used to estimate the dependence of the global-mean temperature on the tropical land area A in Fig. 3b, and therefore plays a significant role. This value, however, is quite large compared to the well-known theoretical value calculated from the Stefan-Boltzmann law: F = σ T4, from which one can derive dT/dF = 1/(4 σ T3) ≈ 0.2 for T ≈ 280 K. Although the used value is more than twice as large as the well-known theoretical value, the rationale for this is not clearly explained in the text or the cited reference, making the estimate somewhat doubtful. Please provide a valid reason for using this value.2. Equation (2) and the air-sea transfer coefficient of α = 15 (W/m2/°C).
This coefficient implies the reciprocal of the temperature sensitivity, and is used to estimate the tropical and polar temperatures from the deviation of solar radiation from the average. The corresponding sensitivity, 1/α ≈ 0.07, then plays a crucial role in determining the temperatures of both regions. This sensitivity is extremely low compared to the global sensitivity of 0.5, resulting in a somewhat strange result of a decreasing polar temperature with increasing area A, even with a slight increase in the net solar radiation in the polar region (q2 in Fig. 3a). Thus, these two parameters (1/α and s) determine the general temperature dependence shown in Fig. 3b. However, there is no rational explanation for this low sensitivity value in this paper or the cited reference (Ou, 2018, Appendix B). Also, I suspect this transfer coefficient applies to air-sea interaction, not air-land interaction. The author invokes an MEP principle for justification, but then the basic logic for the estimation from that principle should be explained.Minor comments:
1. L104-105: “clouds would self-adjust to stabilize the temperature constrained by intrinsic water properties”.
Here, the surface temperature seems to be related to the cloud amount. If so, it would be easier to understand if the cloud amount was also shown in Fig. 1.2. Figure 2.
I cannot understand what are the lines shown on the top-right side of this figure. Also, there is a protruding structure in the tropical region. It would be better to explain what these mean.3. L171-172: Equations (3) and (4).
In these equations, the reflectance of the sea surface appears to be assumed to be zero. If so, this assumption should be stated in the text as the sea surface reflectance is known to be about 0.1, which is larger than zero.4. L182-183: “the land reflectance is that of a desert set to r = 0.5, ... for a total land area set to 0.3”.
Here, the land reflectance (r) seems to be changed for a desert (0.5) and total land (0.3). However, it is unclear how the reflectance is changed in the model calculations shown in Fig. 3. Please resolve this ambiguity.5. L217-218: “not ... the ice-albedo feedback”.
In this model, the land reflectance is fixed to a value (0.3 or 0.5, see the above comment 4). So this model does not include the ice-albedo effect. If so, one should not deny the ice-albedo feedback by the results obtained from a model that contains no ice-albedo effect.6. Figure 3b: the profile of T1.
There is a bend in the temperature profile (T1) in this figure, and I cannot understand why. Could you explain the reason?7. Equation (9).
I cannot understand the meaning of this equation. Please explain this equation in more detail.Citation: https://doi.org/10.5194/esd-2023-32-RC2 -
AC2: 'Reply on RC2', Hsien-Wang Ou, 19 Mar 2024
The comment was uploaded in the form of a supplement: https://esd.copernicus.org/preprints/esd-2023-32/esd-2023-32-AC2-supplement.pdf
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AC2: 'Reply on RC2', Hsien-Wang Ou, 19 Mar 2024
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We postulate that a tropical supercontinent, by blocking the brighter sun, may cool the tropics to below the marking temperature of the glacial margin, enabling the tropical glaciation. Since this marking temperature is above the freezing point to counter the accumulation, glacial margin is abutting an open ocean, contrary to the snowball-earth hypothesis. The theory provides a unified account of faint-young-sun paradox, Precambrian tropical glaciations and glacio-epochs through Earth’s history.
We postulate that a tropical supercontinent, by blocking the brighter sun, may cool the tropics...