the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
A radiativeconvective model computing precipitations with the maximum entropy production hypothesis
Quentin Pikeroen
Didier Paillard
Karine Watrin
Abstract. What do we need to compute the pertinent variables for climate? Some highly detailed models exist, called Earth System Models, where all the relevant components of climate are present: the atmosphere, the ocean, the vegetation and the ice sheets. As many as possible phenomena are represented, and for accuracy, there are two ways of doing it. The first is to solve dynamics equations with a grid size as small as possible. This method induces high economic and computational costs. The second method is to compute the subgrid processes with smart parameterizations adapted to the grid size. This method induces a massive amount of parameterizations. Some simpler models exist, e.g. 1D radiativeconvective model, but like the other models, they use parameterizations. For example, to compute the material energy fluxes provoked by temperature gradients, one may use a Fourier law, saying that energy fluxes are locally proportional to temperature gradients. While this law has a welldefined parameter value at the microscopic scale, the parameter needs to be better defined for the climate scale. More than that, the button for this parameter can be turned to make the model closer to observations. This process is called tuning and exists in all accurate climate models. This article uses a new method to compute temperatures and energy fluxes, where tuning is impossible. We hope this method is more physical and universal as we have less range to tell the model to give the desired result beforehand. Therefore, it could be used for climates where few are known, such as paleoclimate or climates of other planets. The method used is based on a thermodynamic hypothesis, the maximum entropy production. For simplicity, we restrict the model to be 1D vertical for a tropical atmosphere. With conservation laws, the problem is an optimization problem under constraints. It is solved with an algorithm making a gradient descent from an initial condition. The result is the maximum of the objective function, the entropy production, where the constraints are satisfied. As constraints, energy conservation and mass conservation already give a suitable temperature profile. This article adds a new constraint on the water cycle. The water vapour is allowed to disappear, leading to precipitations, but it is not allowed to be created. The result for this new set of equations (or constraints) shows precipitations nil everywhere and positive at the top of the troposphere. It is like "cumulonimbus" precipitations. It seems coherent to what happens in the tropics, where the Intertropical Convergence Zone leads to deep convection. Moreover, the computed order of magnitude is correct. Fundamentally, although the water cycle is often described as a complex and multidisciplinary problem, the correct order of magnitude of precipitations can be computed with almost only the knowledge of radiative transfer.
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Quentin Pikeroen et al.
Status: open (until 18 Dec 2023)

RC1: 'Comment on egusphere20232208', Anonymous Referee #1, 24 Nov 2023
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This article presents a 1D radiativeconvective model based on the thermodynamic hypothesis of maximum entropy production. I was pleased to receive the review assignment, but found it challenging. The text was disjointed and unclear, with numerous unfamiliar citations in the earlier sections that required further investigation. After several attempts, I managed to complete my reading. In my understanding, this model serves as a tool for simulating the earth system from scratch. This innovative work holds potential significance for scientific exploration; however, its practical applicability is questionable.
The article suggests that energy conservation and mass conservation have been confirmed in previous studies while introducing a new constraint on the water cycle. Are these two conservation results validated by prior publications? The updated model is tested in tropical regions. It is claimed that the results align with what happens in the tropics – does this refer to average conditions or specific locations within this region? Can it be applied universally across all tropical areas? Furthermore, authors propose that “it could be used for climates where few are known, such as paleoclimate or climates of other planets”. I am skeptical about whether successful testing in Earth’s tropics can be directly applicable to another planet?
Regarding its limited sensitivity to CO_{2} increase, this outcome is expected due to insufficient consideration of physical processes within the model. Once again, I question the utility of this model given these abovementioned concerns. Based on these considerations, I regretfully cannot recommend accepting this manuscript at present.
Citation: https://doi.org/10.5194/egusphere20232208RC1 
RC2: 'Comment on egusphere20232208', Anonymous Referee #2, 24 Nov 2023
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General
The authors have extended a simple 1d climate model to include precipitation. The model is based on a relatively complex radiation scheme in combination with an optimization of entropy production as a closure representing all other processes defining the temperature profile. The precipitation results from an additional constraint imposed to the optimization. The approach follows previous work utilizing the same model and relays on the maximum entropy production (MEP) conjecture.
Although not proven to work for the climate system, MEP has been successfully applied to various problems. Previous work with the same model (but without the precipitation constraint) seem to demonstrate the applicability of this approach (Herbert et al. 2013 & Labarre et al. 2019, as cited by the authors). Thus, in general, the idea followed by the authors seems well founded. However, I have to admit that to me the results presented are not significant enough to warrant publication in the present form. So far, it is hard for me to extract the gain of new knowledge provided by this study/model. I have detailed some points below.
Major
1) Results: It appears that most of the results described in sections 5.1 and 5.2 have been already (and more thoroughly) discussed in Herbert et al. (2013) and Labarre et al. (1019). The authors need to make clearer what is new here.
2) Section 5.4, sensitivity: In my view, the results of the sensitivity study (2xCO2) question the approach used by the authors (either MEP in general or the imposed constraints). I appreciate the honesty in showing these results. However, I think the deficit of the model in reproducing the climate sensitivity and, in particular, the temperature profile is too serious to postpone the investigation of the reasons (and a potential fix) to further investigations (as the authors do e.g. in L251). From the results I would simply conclude that the method as it is now does not work sufficiently well.
3) Discussion and conclusions: I do not see sufficient information on significant new results in the discussion and conclusions. From the discussion, I mainly learn that there can be many (more or less obvious) reasons why the method may not work (and the key result, P of reasonable order, is a "surprise" (L288)). The conclusions appear to question the key result (L310311) and give a quite general outlook only. Overall, this is a bit disappointing.
Minor
1) The abstract needs to be shorten significantly. Most of it might move to the introduction.
2) A figure like Fig. 1 in Labarre et al. (2019) may help the reader to understand the setup of the model.
3) From Fig 1b) it seems that all models show about the same energy flux at the surface, which, in PRECIP, seems to be approx. the latent heat flux consistent with the evaporation/precipitation (as one may expect). That means, all models would predict approx. a similar P by using this quantity and I am wondering whether this would be a fairer comparison instead of using a quantity (water vapour flux convergence) which is not a part of ENERGY & CONV.
4) Fig 1b: adding a moist adiabat may be helpfull. In addition: Do the lower most values represent the surface temperatures (i=0)? If not they may be added.
5) L239/240: I think the fact that the surface radiation budget controls the evaporation (and thus the water cycle) is well known. I do not completely understand why this needs to be emphasized as "of prime theoretical importance"
6) I might be wrong but it seems for me that it might be possible to use a q not equal to qs (e.g. q=rh*qs, rh=const.) also in the precipitation case (and still have P=FW(i+1)FW(i)). If so, I'm wondering about the sensitivity of the results to rh.
7) L307: "[...] leads to a stable atmosphere: the potential temperature decrease with altitude." I guess a typo, as stable means pot. temp. increases with z.
Citation: https://doi.org/10.5194/egusphere20232208RC2 
RC3: 'Comment on egusphere20232208', Anonymous Referee #3, 29 Nov 2023
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General comments
Pikeroen et. al. proposed a radiativeconvective model with no tunable parameter needed based on Maximum Entropy Production (MEP) theory. MEP is an excellent theory, and it has been successfully demonstrated to simulate different physical processes in previous studies. Previous applications of MEP theory demonstrated that the number of parameters can be significantly reduced because the information is efficiently used. Therefore, MEPbased model can potentially reduce the parametric uncertainty in Earth system model simulations/projections, which stem from hundreds of uncertain parameters. However, the derivation of the MEP model in this study involves a lot of assumptions, constraining the application of the proposed model in an atmospheric model to simulate realistic process. Although the authors mentioned the limitations in the discussion section, I cannot see the potential solution to resolve the limitation to extend the proposed model to 2D domain with varying humidity profile and surface conditions. This study will be a significant contribution to atmospheric modeling if they can reduce their assumptions and demonstrate the application in more realistic conditions. Otherwise, I cannot see the motivation of developing the MEPbased radiativeconvective model. Overall, I cannot recommend the publication of this study in current form.
Specific comments
The model is very simplified with many processes ignored, and the assumptions will not be valid for realistic application. For example, Line 97 assumes energy budget is a function of the temperature, with relative humidity and surface albedo being constant. However, energy budget is significantly affected by the surface conditions and the relative humidity is not only affected by temperature.
I cannot understand why and should be taken as zero. Specifically, the shortwave radiation is from the sun, which represents positive flux from space to atmosphere. In addition, there are nonzero energy flux transferred from surface to subsurface of the earth, i.e., ground heat flux. One may argue the ground heat flux are averaged to be zero in longterm, but it is not always zero in land surface energy balance. It will be helpful for the authors to clarify their assumptions.
Line 135 assumes specific humidity to be the saturated specific humidity. This assumption is not consistent with Line 96. Or is used at Line 96? Overall, I don’t think it is reasonable to assume the air is saturated when computing the specific energy.
Line 163 assumes the evaporation of the layer balances the total precipitation from layer . This assumption ignores the vapor movement in horizontal direction. Therefore, the proposed model may only work for very large scales that horizontal vapor movement is negligible than the evaporation for the precipitation. If so, the author should clarify this limitation.
Line 248 – Line 249: could this substantial underestimation caused by the assumptions of the model, for example, fixed relative humidity profile and albedo? I think the climate sensitivity is a very important metric to evaluate a model. If the climate sensitivity cannot be well captured due to the model assumptions, then those assumptions are not valid.
Line 254: where is the plot for the IPSLCM6ALR?
Citation: https://doi.org/10.5194/egusphere20232208RC3
Quentin Pikeroen et al.
Model code and software
MEP Quentin Pikeroen, Didier Paillard, Karine Watrin https://doi.org/10.5281/zenodo.7995540
Quentin Pikeroen et al.
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