ESD Ideas: A Global Warming Scaling Law

Verbitsky, Mikhail; Mann, Michael

doi:https://doi.org/10.5194/esd-2021-87

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https://doi.org/10.5194/esd-2021-87

Preprints

03 Nov 2021

| 03 Nov 2021

Status: this preprint was under review for the journal ESD but the revision was not accepted.

ESD Ideas: A Global Warming Scaling Law

Mikhail Verbitsky and Michael Mann

Abstract. In this study, we highlight a component of global warming variability, a scaling law that is based purely on fundamental physical properties of the climate system. We suggest that three similarity parameters define the system response to external forcing, and an argument of physical similarity with observed climate responses in the past can be made when all three parameters are identical for the current and historical climates. We determined that the scaling law of global warming is the (𝜆 + 1 + m) – power of time, where 𝜆 is prescribed by external forcing and m is defined by climate system internal dynamics. When the climate system develops in the direction of intensified positive feedbacks, the power m changes from m = −1 (negative feedbacks dominate) to m ≥ 1 (positive feedbacks dominate). We also establish that a “hothouse” climate with dominant positive feedbacks will be preceded by a climate having a property of incomplete similarity in feedbacks similarity parameters. It implies that the same future scenario may be produced by climate feedbacks of different magnitudes as long as their positive-to-negative ratio is the same.

Received: 01 Nov 2021 – Discussion started: 03 Nov 2021

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Mikhail Verbitsky and Michael Mann

Status: closed

CC1:
'Comment on esd-2021-87', Richard Rosen, 03 Nov 2021

Frankly, as a physicist, I don't know how to evaluate this paper since the main references for the methodology are so old that I do not have access to them. Also, there is no physics in the paper to support the main hypothesis (equation #1). So if this paper is seriously considered for publication it should have enough physics in it to demonstrate the hypotheses. Right now it seems rather speculative.

Citation: https://doi.org/10.5194/esd-2021-87-CC1
- AC1:
  'Reply on CC1', Mikhail Verbitsky, 03 Nov 2021
  
  Dear Dr. Rosen,
  Thank you for your comment. I think the book of G. I. Barenblatt (Barenblatt, G. I.: Scaling. Cambridge University Press, Cambridge, 2003) provides a very good introduction into dimensional analysis that may be helpful to you.
  https://www.cambridge.org/core/books/scaling/E08325F4C8A14AAD4742E39FE5D0A6B3
  I think Introduction and Chapters 1-4 will be sufficient.
  You are raising a very good question regarding the physical content of the equation (1), and I do not want to be brief in answering your question. I would like to start with the quote from the book I just recommended: “Applied mathematics is the art of constructing mathematical models in nature, engineering, and society” (page xi). Why art? Because every model is a simplification or idealization of a phenomenon (otherwise a model would be as difficult for interpretation as the nature itself), and therefore it is a decision of a researcher what factors of a phenomenon are of critical importance and what factors can be neglected, a decision that like any artistic action cannot be formalized but is based on researchers’ experience and best judgement. We call equation (1) a hypothesis, a hypothesis that suggests that the internal dynamics of the climate systems is largely governed by magnitudes of its positive and negative feedbacks. This is the physical content of the equation (1). Then, using dimensional requirements, we proceed to the scaling law. Thus the idea of the paper is as following: If the climate system is indeed governed by the magnitudes of its feedbacks, it should follow these laws. To challenge our scaling law, one needs to present an argument why our hypothesis is not valid.
  Thank you again for your question. I am sure that our conversation will help other readers as well.
  Respectfully,
  Mikhail Verbitsky
  
  Citation: https://doi.org/10.5194/esd-2021-87-AC1
  - CC2: 'Reply on AC1', Richard Rosen, 03 Nov 2021
    
    Of course, one can summarize the fact that the very complex earth system is a product of it positive and negative feedbacks, but that is not saying anything that has much content. One problem is that those feedbacks are very complicated and change over time in non-linear ways. So you still have to demonstrate your case with real physics. Applied mathematics says nothing about the earth system directly, though mathematics is always a useful tool for doing physics.
    
    Citation: https://doi.org/10.5194/esd-2021-87-CC2
    
    AC2: 'Reply on CC2', Mikhail Verbitsky, 12 Nov 2021
    
    We are aware about complexity of the climate system.
    Our ambition though is to highlight a simple scaling law. It is the same ambition that motivates generations of researches to complement 3D general circulation models with models of intermediate complexity, dynamical models, energy-balance models, etc. In pursuit of simplicity, or generality, we, again, propose that most important governing parameters are magnitudes of system’s feedbacks (you seem to agree with that) regardless of these feedbacks’ physical nature.
    Yet, specific examples of the feedbacks are not absent. In paragraph 2.2 we talk about Plank feedback as a dominant negative feedback of the recent history and in paragraph 2.3 we refer to Steffen et al (2018) for multiple examples of potential positive feedbacks.
    Your comment prompts us to be more explicit about this in the revised version of the paper - thank you.
    
    Citation: https://doi.org/10.5194/esd-2021-87-AC2
    
    CC3: 'Reply on AC2', Richard Rosen, 12 Nov 2021
    
    Good - just to be clear, please be much more explicit about what a scaling law is, and how it can be determined quantitatively by studying the positive and negative feedbacks in a complex system. I am quite sure that this is a totally new concept for most climate scientists and general readers of climate science research such as the readers of ESD. It is totally new to me. Again, you have to show that certain mathematical properties of some physical systems apply to the physical properties of the climate system in particular, and you have to show how you know this.
    
    Citation: https://doi.org/10.5194/esd-2021-87-CC3
    
    CC4: 'Reply on CC3', Richard Rosen, 12 Nov 2021
    
    In fact, it might speed your publication process if you were to try out such an explanation on me by replying to my last comment.
    
    Citation: https://doi.org/10.5194/esd-2021-87-CC4
    
    EC1: 'Handling Editor's Reply on CC4', Gabriele Messori, 12 Nov 2021
    
    Dear Richard,
    I very much welcome community interactions and comments on preprints in ESD. However, I would ask you to limit the discussion to the scientific contents of the article and refrain from commenting on the editorial evaluation and processing of the paper.
    Best Regards,
    Gabriele Messori
    
    Citation: https://doi.org/10.5194/esd-2021-87-EC1
    
    CC5: 'Reply on EC1', Richard Rosen, 12 Nov 2021
    
    Sure, No problem. I was just trying to be helpful.
    
    Citation: https://doi.org/10.5194/esd-2021-87-CC5
RC1:
'Comment on esd-2021-87', Anonymous Referee #1, 06 Dec 2021

Review of "A global warming scaling law" by M. Y. Verbitsky and M. E. Mann for consideration in the 'ESD Ideas' section of Earth System Dynamics.

In this manuscript the authors suggest that the evolution of the global temperature is governed by two time scales of positive and negative feedbacks, external forcing strength and evolution, as well as time. Forming non-dimensional groups, time is normalised and an equation for the temperature evolution (2) is derived. Then different cases are discussed where first positive feedbacks dominate, leading to stable climate states, and when positive feedbacks dominate leading to climate instability.

This manuscript is problematic in countless aspects, and I really don't know where to begin, and to end. Although I don't regularly use it, I am familiar with Buckinham's pi-theorem, and feel confident enough to know that if you provide it with a poor hypothesis, then the result will not be more insightful or fundamental. It does make sure the formulation is independent of the units used, but it is not magic.

A guiding principle in the climate sciences is conservation of energy, at least Arrhenius (1896) used it, but probably also earlier studies. Arrhenius realised that studying the energy balance at the top of the atmosphere was a useful starting point and identified forcing from CO2, negative temperature feedback and positive water vapour and surface albedo feedbacks. Not using this as a starting point requires justification.

Typically, it is found that a two-layer formulation of the climate system with a shallow atmosphere+ocean mixed-layer reservoir coupled to a deep ocean provides a good starting point (Hansen et al. 1985), and this type of model is used with slight modification in countless places, and it is well-justified (e.g. Gregory and Forster 2008). Does the here discovered theory in some way predict aspects of climate change that the simple two-layer model does not?

Usually, we would think of feedbacks as dependent on temperature only to first order (e.g. Sherwood et al. 2015). There are modifications to this, for example some feedbacks may change during a transient as the system equilibrates (e.g. Held et al. 2010, Geoffroy et al. 2013), or as the temperature changes (e.g. Bloch-Johnson et al. 2015). But these are higher order effects and the starting point is that the feedback scales with temperature. In their formulation the authors assume that there are negative feedbacks with one time scale and positive feedbacks with another time scale without any justification.

The paper by Steffen et al. (2018) appears to be used as a kind of confirmation of the theory. However, it is well known, and represented by the above mentioned theory, that if the feedback parameter becomes positive one enters an instability and a run-away climate, for instance snow-ball Earth (e.g. Budyko 1969). However, there is no evidence that the hothouse hypothesis of Steffen et al. 2018 is correct as assessed by IPCC AR6.

The other piece of evidence provided is the near linear temperature increase over a select period of time. But that is not proof.

Overall, though, what I find most problematic with this manuscript is that there is no attempt made to connect with relevant studies. A proposed idea can be radically different from whatever is already there, an attempt to reinvent the wheel, if you want. But authors need to do their homework and explain why they did what they did and how that is different and potentially better than existing approaches. It is therefore not possible for me to recommend publication.

---

Arrhenius, 1896, https://www.rsc.org/images/Arrhenius1896_tcm18-173546.pdf

Bloch-Johnson et al. 2015, http://dx.doi.org/10.1002/2015GL064240

Hansen et al. 1985, https://doi.org/10.1126/science.229.4716.857

Held et al. 2010, https://doi.org/10.1175/2009JCLI3466.1

Geoffroy et al. 2013, https://doi.org/10.1175/JCLI-D-12-00196.1

Gregory and Forster, 2008, https://doi.org/10.1029/2008JD010405

Sherwood et al. 2015, http://dx.doi.org/10.1175/BAMS-D-13-00167.1

Citation: https://doi.org/10.5194/esd-2021-87-RC1
- AC3: 'Reply on RC1', Mikhail Verbitsky, 07 Dec 2021
  
  Dear Referee 1, Thank you for your review. The following is our response to your comments.
  Comment: In this manuscript the authors suggest that the evolution of the global temperature is governed by two time scales of positive and negative feedbacks, external forcing strength and evolution, as well as time. Forming non-dimensional groups, time is normalised and an equation for the temperature evolution (2) is derived. Then different cases are discussed where first positive feedbacks dominate, leading to stable climate states, and when positive feedbacks dominate leading to climate instability.
  Answer: With the exception of highlighted misprint (should be “negative”) you correctly captured most of the paper content. Unfortunately, in your brief overview you didn’t mention the third case – a climate having property of incomplete similarity. This is the critical part of our study for a number of reasons including a perspective of novelty that you raised below.
  Action: No action is required
  Comment: This manuscript is problematic in countless aspects, and I really don't know where to begin, and to end. Although I don't regularly use it, I am familiar with Buckinham's pi-theorem, and feel confident enough to know that if you provide it with a poor hypothesis, then the result will not be more insightful or fundamental. It does make sure the formulation is independent of the units used, but it is not magic.
  Answer: We definitely agree that a physical hypothesis is a cornerstone of any study.
  Action: In the revised manuscript we have sought to better clarify what the underlying physical hypotheses are in the current study, which is really more of a “think piece” (that’s all that is really possible within the tight 2-page length constraint of ESD Ideas).
  Comment: A guiding principle in the climate sciences is conservation of energy, at least Arrhenius (1896) used it, but probably also earlier studies. Arrhenius realised that studying the energy balance at the top of the atmosphere was a useful starting point and identified forcing from CO2, negative temperature feedback and positive water vapour and surface albedo feedbacks. Not using this as a starting point requires justification.
  Answer: Obviously, conservation of energy is a paramount principle, but starting with the hypothesis (1) does not mean that this principle somehow is not observed. Frankly, we internally debated how to better introduce the hypothesis (1). One of the options we considered was to review an energy-balance model (it is currently equation (8)) and to use it as an “inventory” of the governing parameters. We decided against this approach because it may create an impression that our analysis is based purely on the energy-balance model like (8). This is not the case because equation (8) does not have a property of incomplete similarity. We wanted a more general approach and decided in favor of equation (1). We can see now that complete avoiding mentioning of the energy conservation may create confusion.
  Action: We will articulate our approach better in the revised version of the paper.
  Comment: Typically, it is found that a two-layer formulation of the climate system with a shallow atmosphere+ocean mixed-layer reservoir coupled to a deep ocean provides a good starting point (Hansen et al. 1985), and this type of model is used with slight modification in countless places, and it is well-justified (e.g. Gregory and Forster 2008). Does the here discovered theory in some way predict aspects of climate change that the simple two-layer model does not?
  Answer: The purpose of our study wasn’t to introduce the entire hierarchy of models of increasing complexity (particularly within the two-page constraint of ESD Ideas). Instead, it is a thought piece that investigates different low-dimensional descriptions of the climate system. The zero-dimensional EBM, for example, is the simplest. From there one can of course build all the way up fully coupled three dimensional ocean-atmosphere models with interactive carbon cycle and cryosphere, etc.
  Yes, it does. Because of its simplicity, our approach is insightful. We suggest that the “hot-house” climate may be preceded by a climate having a property of incomplete similarity. To our knowledge, this is a new proposition.
  Action: No action is required
  Comment: Usually, we would think of feedbacks as dependent on temperature only to first order (e.g. Sherwood et al. 2015). There are modifications to this, for example some feedbacks may change during a transient as the system equilibrates (e.g. Held et al. 2010, Geoffroy et al. 2013), or as the temperature changes (e.g. Bloch-Johnson et al. 2015). But these are higher order effects and the starting point is that the feedback scales with temperature. In their formulation the authors assume that there are negative feedbacks with one time scale and positive feedbacks with another time scale without any justification.
  Answer: Yes, we consider two cases where positive and negative feedback scales are different. One case we relate to the recent climate history based on the simple zero-dimensional EBM experiments described by Mann et al (2014) and another case of the “hot-house” climate is hypothetical.
  Action: No action is required
  Comment: The paper by Steffen et al. (2018) appears to be used as a kind of confirmation of the theory. However, it is well known, and represented by the above mentioned theory, that if the feedback parameter becomes positive one enters an instability and a run-away climate, for instance snow-ball Earth (e.g. Budyko 1969). However, there is no evidence that the hothouse hypothesis of Steffen et al. 2018 is correct as assessed by IPCC AR6.
  Answer: We do not consider paper of Steffen et al. (2018) as a proof of our findings but simply as an example of possible regime where positive feedbacks may dominate over some range of variation (e.g., over the range where positive methane feedbacks play a dominant role - a process that ultimately saturates at some level of warming since the available carbon reservoir isn’t infinite).
  Action: No action is required
  Comment: The other piece of evidence provided is the near linear temperature increase over a select period of time. But that is not proof.
  Answer: We do not prove our scaling law but discover its parameters, a power degree. For a climate with dominant negative feedback, we use historical records to determine m = -1 and for a climate with dominant positive feedback we use equation (8) to find m >= 1. Additional studies will be needed to prove (or challenge) our scaling law.
  Action: No action is required
  Comment: Overall, though, what I find most problematic with this manuscript is that there is no attempt made to connect with relevant studies. A proposed idea can be radically different from whatever is already there, an attempt to reinvent the wheel, if you want. But authors need to do their homework and explain why they did what they did and how that is different and potentially better than existing approaches. It is therefore not possible for me to recommend publication.
  Action: Our presentation is very brief because of the 2-page ESD Ideas format, but we will revise the manuscript, within these length constraints, to clarify some of the points raised and responded to in this review.
  Mikhail Verbitsky and Michael E. Mann
  
  Citation: https://doi.org/10.5194/esd-2021-87-AC3
- AC4: 'Additional Reply on RC1', Mikhail Verbitsky, 09 Dec 2021
  
  Dear Referee 1,
  We would like to share with you an additional thought that has been motivated by your comment.
  We do not think we “attempt to reinvent the wheel” but instead we hope we are closing the gap.
  Since 1969, the Budyko's model (e.g., our equation (8) and similar) became a most popular simple tool of climate studies. The next level of sophistication, as you quite rightly noted, was "shallow atmosphere + ocean mixed-layer reservoir coupled to a deep ocean (Hansen et al. 1985)", followed by intermediate complexity models and 3D models.
  But Budyko's model doesn't have a property of incomplete similarity and the next-level atmosphere-mixed-layer model is already too complex for incomplete similarity to be easily recognized (it would require a specially designed studies that, to our knowledge, nobody did). Thus, a possibility of the incomplete-similarity regime remained unnoticed. Our dimensional analysis, in fact, closes this gap. It is more general than zero-dimensional EBM but obviously simple enough to reveal a possibility of incomplete similarity.
  Mikhail Verbitsky and Michael E. Mann
  
  Citation: https://doi.org/10.5194/esd-2021-87-AC4
- AC5:
  'Reply on RC1-suggested changes to the paper', Mikhail Verbitsky, 21 Dec 2021
  
  The comment was uploaded in the form of a supplement: https://esd.copernicus.org/preprints/esd-2021-87/esd-2021-87-AC5-supplement.pdf
  
  Citation: https://doi.org/10.5194/esd-2021-87-AC5
  - CC6: 'Reply on AC5', Richard Rosen, 22 Dec 2021
    
    It seems to me that your last sentence in your new reply is analytically true independent of the right equations describing the energy balance for the earth. Namely, the "climate’s behavior may be largely governed by magnitudes of its positive and negative feedbacks". Am I missing something? Once one takes into account the incoming radiation from the sun that heats the earth in appropriate detail, with clouds, oceans, biomass, land, etc., then of course the climate's behavior is due to the magnitudes of all the feedbacks. Thus, I don't understand what your new formulations contributes to understanding the value of your original article.
    
    Citation: https://doi.org/10.5194/esd-2021-87-CC6
    
    AC6: 'Reply on CC6', Mikhail Verbitsky, 23 Dec 2021
    
    Since the magnitudes of positive and negative feedbacks are defined not by a single similarity parameter but by groups (conglomerates) of similarity parameters, the hypothesis that “climate’s behavior may be largely governed by magnitudes of its positive and negative feedbacks” speaks to the reduction of the number of effective parameters and allows us to move from several similarity parameters (e.g., eight in the case of the system (1) - (3)) to just three similarity parameters and, in some circumstances, to a single conglomerate similarity parameter, the V-number.
    
    Citation: https://doi.org/10.5194/esd-2021-87-AC6
    
    CC7: 'Reply on AC6', Richard Rosen, 23 Dec 2021
    
    What you say may be true of your artificial model of the climate (your set of equations), but I still don't see the connection between your termology and model and the real physical world. The real physical world is far more complex and uncertain than you small set of equations. There are many feedback loops, positive and negative, that we still do not understand. So I do not understand how your model and mathematical terminology help advance the scientific analysis of climate change relative to the coupled atmospheric models that have existed for many years.
    
    Citation: https://doi.org/10.5194/esd-2021-87-CC7
    
    AC7: 'Reply on CC7', Mikhail Verbitsky, 23 Dec 2021
    
    Dr. Rosen, We thank you for engaging but these exchanges now seem to have exhausted their usefulness, so we will, respectfully, leave this where it currently stands.
    Mikhail Verbitsky and Michael E. Mann
    
    Citation: https://doi.org/10.5194/esd-2021-87-AC7
RC2:
'Comment on esd-2021-87', Anonymous Referee #2, 17 Jan 2022

This paper presents an interesting idea to derive a scaling law for global warming based on the physical fundamental physical properties of the climate system and the Buckingham-pi theorem for dimensional analysis. The scaling laws at two extreme conditions of positive or negative feedbacks dominating (complete similarity) are first derived, and then the case of incomplete similarity is discussed. I find this work novel and interesting, and a very good fit for an ESD Idea paper. I have a number of questions and suggestions, mostly about the dimensional analysis part, which I hope the authors address.

- Line 57 and eq 3 (similarly, line 73 and eq 6): it is not clear to me how you can go from \phi(t/\tau_n, \lembda) to (t/\tau_n)^m in eq. (3). In Buckingham-Pi, the function \phi of some variables can be written as the product of each variable to an unknown power, but it is unclear to me why in eq. 3 there is no \lambda^q (q being another unknown power) in eq. 3 multiplied by the rest of the terms. This is my main comment/question about the method. Please clarify.

- Line 46: the dimension of \epsilon. I guess it is correctly written as sec^{-\lambda-1), however, it does not appear clearly as far as I see. Please take a look and clarify if possible

- line 53: you may want to change this to ".... global temperature response T/(\epsilon t^{\lambda+1}) is a function of ....."

- Also you may want to clarify around line 50 that here there are 6 variables in (1), there are two fundamental dimensions (time and temperature), so Buckingham-pi states that there would be 4 dimensionless (pi) groups, as presented on lines 52-53.

- I find the word "dimensionless" just sounding better than "adimensional" but it is up to you which one to use.

- line 99: m should be in the math mode

- line 22: the use of Buckingham-pi theorem in climate science has been rather limited, and the cited paper by Golitsyn is certainly a great example. You may want to also mention some of the recent papers using this approach to study the climate system, e .g.

+Chavas, D.R. and Emanuel, K., 2014. Equilibrium tropical cyclone size in an idealized state of axisymmetric radiative–convective equilibrium. JAS

+Yang, D. and Ingersoll, A.P., 2014. A theory of the MJO horizontal scale. GRL

+Nabizadeh, E., et al. 2019. Size of the atmospheric blocking events: Scaling law and response to climate change. GRL

- line 16: is the word "similarity" in ".... in feedbacks similarity parameters ...." needed?

Citation: https://doi.org/10.5194/esd-2021-87-RC2
- AC8: 'Reply on RC2', Mikhail Verbitsky, 17 Jan 2022
  
  Dear Referee 2,
  Thank you very much for the encouraging and insightful review. The following is our response to your comments.
  Comment: This paper presents an interesting idea to derive a scaling law for global warming based on the physical fundamental physical properties of the climate system and the Buckingham-pi theorem for dimensional analysis. The scaling laws at two extreme conditions of positive or negative feedbacks dominating (complete similarity) are first derived, and then the case of incomplete similarity is discussed. I find this work novel and interesting, and a very good fit for an ESD Idea paper.
  Answer: We appreciate your evaluation.
  Comment: I have a number of questions and suggestions, mostly about the dimensional analysis part, which I hope the authors address.
  - Line 57 and eq 3 (similarly, line 73 and eq 6): it is not clear to me how you can go from \phi(t/\tau_n, \lembda) to (t/\tau_n)^m in eq. (3). In Buckingham-Pi, the function \phi of some variables can be written as the product of each variable to an unknown power, but it is unclear to me why in eq. 3 there is no \lambda^q (q being another unknown power) in eq. 3 multiplied by the rest of the terms. This is my main comment/question about the method. Please clarify.
  Answer: Your observation is correct. Indeed, 𝜆^q may appear in equations (3) and (6). But since 𝜆 and 𝜆^q are constants, 𝜆^q has been absorbed by the experimental constant k.
  Action: We will clarify this reasoning in the revised version of the paper.
  Comment: - Line 46: the dimension of \epsilon. I guess it is correctly written as sec^{-\lambda-1), however, it does not appear clearly as far as I see. Please take a look and clarify if possible
  Answer: The dimension of ε is [^oC sec^(-𝜆-1)]. It looks like the pdf file of the preprint introduced some distortion in line 46 that may cause confusion.
  Action: We will make sure that it is clear in the final version of the paper.
  Comment: - Also you may want to clarify around line 50 that here there are 6 variables in (1), there are two fundamental dimensions (time and temperature), so Buckingham-pi states that there would be 4 dimensionless (pi) groups, as presented on lines 52-53.
  Answer: Agreed
  Action: This clarification will be included in the final version of the paper.
  Comment: - line 22: the use of Buckingham-pi theorem in climate science has been rather limited, and the cited paper by Golitsyn is certainly a great example. You may want to also mention some of the recent papers using this approach to study the climate system, e .g.
  +Chavas, D.R. and Emanuel, K., 2014. Equilibrium tropical cyclone size in an idealized state of axisymmetric radiative–convective equilibrium. JAS
  +Yang, D. and Ingersoll, A.P., 2014. A theory of the MJO horizontal scale. GRL
  +Nabizadeh, E., et al. 2019. Size of the atmospheric blocking events: Scaling law and response to climate change. GRL
  Answer: Thank you for your suggestion. We agree that it will be helpful to our readers.
  Action: Per editor’s approval (number of references in ESD Ideas papers is limited to 12) we will be glad to include all your recommended references.
  Minor comments:
  - line 53: you may want to change this to ".... global temperature response T/(\epsilon t^{\lambda+1}) is a function of ....."
  - I find the word "dimensionless" just sounding better than "adimensional" but it is up to you which one to use.
  - line 99: m should be in the math mode
  - line 16: is the word "similarity" in ".... in feedbacks similarity parameters ...." needed?
  Action: All minor comments will be taken care of.
  Mikhail Verbitsky and Michael E. Mann
  
  Citation: https://doi.org/10.5194/esd-2021-87-AC8
- AC9: 'Reply on RC2- pdf version', Mikhail Verbitsky, 17 Jan 2022
  
  We also publish a pdf version of our response so that mathematical symbols are not distorted
  
  Citation: https://doi.org/10.5194/esd-2021-87-AC9
RC3:
'Comment on esd-2021-87', Anonymous Referee #3, 25 Mar 2022

The authors present an approach to analyze the time evolution of global warming based on century-old dimensionality arguments that are commonly used in the field of fluid dynamics. They include more recent insights into the origins of incomplete similarity to hypothesize a characteristic scaling of T ~ t² for an intermediate pre-hothouse climate regime, before the positive feedbacks start to dominate.

First of all, it could be clarified why other dimensional physical parameters that play an important role in the climate system, such as the active ocean layer depth/mass that responds nonlinearly to global warming, albedo etc. do not feature. Alternatively, explain how they are absorbed into the limited set of parameters shown here. It is possible that they can all be reduced to the proposed time scales, but the way it's presented feels like an extreme simplification.

Secondly, there's a significant difference with fluid dynamics that is not highlighted in this study. In contrast to fluid dynamics, in climate there is no multitude of experiments that provide the required observations time series to determine the scaling parameters. Of course, this does not mean that no information can be distilled from the limited historical climate record - clearly, as many studies have shown, hypotheses can be presented and analyses based on (ensemble) climate methods can confirm or refute these hypotheses, using the historical climate record. However, the proposed scaling behaviour is only feebly, if at all supported by the historical data. I realise that this is due to the nature of the problem - we have to make do with the observations we have. As a result, however, I find the evidence for the theory too tenuous to publish as-is.

A straightforward way to fix this, is that the authors test their hypothesis through another means. Indeed, the simplicity of the assumptions implies that they can use fairly simple general circulation models, or even simpler zero-dimensional energy balance models, to generate temperature time series for the different scenarios. This will allow them to check whether the scaling behaviour holds. Unless this can be provided, I do not recommend publication of this manuscript, as interesting as the underlying ideas are.

Citation: https://doi.org/10.5194/esd-2021-87-RC3
- AC10: 'Reply on RC3', Mikhail Verbitsky, 26 Mar 2022
  
  Dear Referee 3,
  Thank you very much for your review. The following is our response to your comments.
  Comment: The authors present an approach to analyze the time evolution of global warming based on century-old dimensionality arguments that are commonly used in the field of fluid dynamics. They include more recent insights into the origins of incomplete similarity to hypothesize a characteristic scaling of T ~ t^2 for an intermediate pre-hothouse climate regime, before the positive feedbacks start to dominate.
  Answer: Your short description of the paper is correct
  Comment: First of all, it could be clarified why other dimensional physical parameters that play an important role in the climate system, such as the active ocean layer depth/mass that responds nonlinearly to global warming, albedo etc. do not feature. Alternatively, explain how they are absorbed into the limited set of parameters shown here. It is possible that they can all be reduced to the proposed time scales, but the way it's presented feels like an extreme simplification.
  Answer: We agree that our readers will benefit from more detailed description of our reasoning.
  Action: Additional substantiation of our main hypothesis will be provided (please also review our response AC5 to Referee 1 https://editor.copernicus.org/index.php?_mdl=msover_md&_jrl=430&_lcm=oc108lcm109w&_acm=get_comm_sup_file&_ms=98841&c=216702&salt=18451585431668037296 )
  Comment: Secondly, there's a significant difference with fluid dynamics that is not highlighted in this study. In contrast to fluid dynamics, in climate there is no multitude of experiments that provide the required observations time series to determine the scaling parameters. Of course, this does not mean that no information can be distilled from the limited historical climate record - clearly, as many studies have shown, hypotheses can be presented and analyses based on (ensemble) climate methods can confirm or refute these hypotheses, using the historical climate record. However, the proposed scaling behaviour is only feebly, if at all supported by the historical data. I realise that this is due to the nature of the problem - we have to make do with the observations we have. As a result, however, I find the evidence for the theory too tenuous to publish as-is.
  A straightforward way to fix this, is that the authors test their hypothesis through another means. Indeed, the simplicity of the assumptions implies that they can use fairly simple general circulation models, or even simpler zero-dimensional energy balance models, to generate temperature time series for the different scenarios. This will allow them to check whether the scaling behaviour holds. Unless this can be provided, I do not recommend publication of this manuscript, as interesting as the underlying ideas are.
  Answer: Thank you for recognizing our ideas as interesting. Your suggestion to strengthen our arguments is well understood.
  Action: Though dedicated experiments with GCM models are outside of the scope of this ideas-format paper, additional analytics with an energy-balance model will, indeed, be provided as well as a compilation of appropriate already published studies.
  Mikhail Verbitsky and Michael E. Mann
  
  Citation: https://doi.org/10.5194/esd-2021-87-AC10

Status: closed

CC1:
'Comment on esd-2021-87', Richard Rosen, 03 Nov 2021

Frankly, as a physicist, I don't know how to evaluate this paper since the main references for the methodology are so old that I do not have access to them. Also, there is no physics in the paper to support the main hypothesis (equation #1). So if this paper is seriously considered for publication it should have enough physics in it to demonstrate the hypotheses. Right now it seems rather speculative.

Citation: https://doi.org/10.5194/esd-2021-87-CC1
- AC1:
  'Reply on CC1', Mikhail Verbitsky, 03 Nov 2021
  
  Dear Dr. Rosen,
  Thank you for your comment. I think the book of G. I. Barenblatt (Barenblatt, G. I.: Scaling. Cambridge University Press, Cambridge, 2003) provides a very good introduction into dimensional analysis that may be helpful to you.
  https://www.cambridge.org/core/books/scaling/E08325F4C8A14AAD4742E39FE5D0A6B3
  I think Introduction and Chapters 1-4 will be sufficient.
  You are raising a very good question regarding the physical content of the equation (1), and I do not want to be brief in answering your question. I would like to start with the quote from the book I just recommended: “Applied mathematics is the art of constructing mathematical models in nature, engineering, and society” (page xi). Why art? Because every model is a simplification or idealization of a phenomenon (otherwise a model would be as difficult for interpretation as the nature itself), and therefore it is a decision of a researcher what factors of a phenomenon are of critical importance and what factors can be neglected, a decision that like any artistic action cannot be formalized but is based on researchers’ experience and best judgement. We call equation (1) a hypothesis, a hypothesis that suggests that the internal dynamics of the climate systems is largely governed by magnitudes of its positive and negative feedbacks. This is the physical content of the equation (1). Then, using dimensional requirements, we proceed to the scaling law. Thus the idea of the paper is as following: If the climate system is indeed governed by the magnitudes of its feedbacks, it should follow these laws. To challenge our scaling law, one needs to present an argument why our hypothesis is not valid.
  Thank you again for your question. I am sure that our conversation will help other readers as well.
  Respectfully,
  Mikhail Verbitsky
  
  Citation: https://doi.org/10.5194/esd-2021-87-AC1
  - CC2: 'Reply on AC1', Richard Rosen, 03 Nov 2021
    
    Of course, one can summarize the fact that the very complex earth system is a product of it positive and negative feedbacks, but that is not saying anything that has much content. One problem is that those feedbacks are very complicated and change over time in non-linear ways. So you still have to demonstrate your case with real physics. Applied mathematics says nothing about the earth system directly, though mathematics is always a useful tool for doing physics.
    
    Citation: https://doi.org/10.5194/esd-2021-87-CC2
    
    AC2: 'Reply on CC2', Mikhail Verbitsky, 12 Nov 2021
    
    We are aware about complexity of the climate system.
    Our ambition though is to highlight a simple scaling law. It is the same ambition that motivates generations of researches to complement 3D general circulation models with models of intermediate complexity, dynamical models, energy-balance models, etc. In pursuit of simplicity, or generality, we, again, propose that most important governing parameters are magnitudes of system’s feedbacks (you seem to agree with that) regardless of these feedbacks’ physical nature.
    Yet, specific examples of the feedbacks are not absent. In paragraph 2.2 we talk about Plank feedback as a dominant negative feedback of the recent history and in paragraph 2.3 we refer to Steffen et al (2018) for multiple examples of potential positive feedbacks.
    Your comment prompts us to be more explicit about this in the revised version of the paper - thank you.
    
    Citation: https://doi.org/10.5194/esd-2021-87-AC2
    
    CC3: 'Reply on AC2', Richard Rosen, 12 Nov 2021
    
    Good - just to be clear, please be much more explicit about what a scaling law is, and how it can be determined quantitatively by studying the positive and negative feedbacks in a complex system. I am quite sure that this is a totally new concept for most climate scientists and general readers of climate science research such as the readers of ESD. It is totally new to me. Again, you have to show that certain mathematical properties of some physical systems apply to the physical properties of the climate system in particular, and you have to show how you know this.
    
    Citation: https://doi.org/10.5194/esd-2021-87-CC3
    
    CC4: 'Reply on CC3', Richard Rosen, 12 Nov 2021
    
    In fact, it might speed your publication process if you were to try out such an explanation on me by replying to my last comment.
    
    Citation: https://doi.org/10.5194/esd-2021-87-CC4
    
    EC1: 'Handling Editor's Reply on CC4', Gabriele Messori, 12 Nov 2021
    
    Dear Richard,
    I very much welcome community interactions and comments on preprints in ESD. However, I would ask you to limit the discussion to the scientific contents of the article and refrain from commenting on the editorial evaluation and processing of the paper.
    Best Regards,
    Gabriele Messori
    
    Citation: https://doi.org/10.5194/esd-2021-87-EC1
    
    CC5: 'Reply on EC1', Richard Rosen, 12 Nov 2021
    
    Sure, No problem. I was just trying to be helpful.
    
    Citation: https://doi.org/10.5194/esd-2021-87-CC5
RC1:
'Comment on esd-2021-87', Anonymous Referee #1, 06 Dec 2021

Review of "A global warming scaling law" by M. Y. Verbitsky and M. E. Mann for consideration in the 'ESD Ideas' section of Earth System Dynamics.

In this manuscript the authors suggest that the evolution of the global temperature is governed by two time scales of positive and negative feedbacks, external forcing strength and evolution, as well as time. Forming non-dimensional groups, time is normalised and an equation for the temperature evolution (2) is derived. Then different cases are discussed where first positive feedbacks dominate, leading to stable climate states, and when positive feedbacks dominate leading to climate instability.

This manuscript is problematic in countless aspects, and I really don't know where to begin, and to end. Although I don't regularly use it, I am familiar with Buckinham's pi-theorem, and feel confident enough to know that if you provide it with a poor hypothesis, then the result will not be more insightful or fundamental. It does make sure the formulation is independent of the units used, but it is not magic.

A guiding principle in the climate sciences is conservation of energy, at least Arrhenius (1896) used it, but probably also earlier studies. Arrhenius realised that studying the energy balance at the top of the atmosphere was a useful starting point and identified forcing from CO2, negative temperature feedback and positive water vapour and surface albedo feedbacks. Not using this as a starting point requires justification.

Typically, it is found that a two-layer formulation of the climate system with a shallow atmosphere+ocean mixed-layer reservoir coupled to a deep ocean provides a good starting point (Hansen et al. 1985), and this type of model is used with slight modification in countless places, and it is well-justified (e.g. Gregory and Forster 2008). Does the here discovered theory in some way predict aspects of climate change that the simple two-layer model does not?

Usually, we would think of feedbacks as dependent on temperature only to first order (e.g. Sherwood et al. 2015). There are modifications to this, for example some feedbacks may change during a transient as the system equilibrates (e.g. Held et al. 2010, Geoffroy et al. 2013), or as the temperature changes (e.g. Bloch-Johnson et al. 2015). But these are higher order effects and the starting point is that the feedback scales with temperature. In their formulation the authors assume that there are negative feedbacks with one time scale and positive feedbacks with another time scale without any justification.

The paper by Steffen et al. (2018) appears to be used as a kind of confirmation of the theory. However, it is well known, and represented by the above mentioned theory, that if the feedback parameter becomes positive one enters an instability and a run-away climate, for instance snow-ball Earth (e.g. Budyko 1969). However, there is no evidence that the hothouse hypothesis of Steffen et al. 2018 is correct as assessed by IPCC AR6.

The other piece of evidence provided is the near linear temperature increase over a select period of time. But that is not proof.

Overall, though, what I find most problematic with this manuscript is that there is no attempt made to connect with relevant studies. A proposed idea can be radically different from whatever is already there, an attempt to reinvent the wheel, if you want. But authors need to do their homework and explain why they did what they did and how that is different and potentially better than existing approaches. It is therefore not possible for me to recommend publication.

---

Arrhenius, 1896, https://www.rsc.org/images/Arrhenius1896_tcm18-173546.pdf

Bloch-Johnson et al. 2015, http://dx.doi.org/10.1002/2015GL064240

Hansen et al. 1985, https://doi.org/10.1126/science.229.4716.857

Held et al. 2010, https://doi.org/10.1175/2009JCLI3466.1

Geoffroy et al. 2013, https://doi.org/10.1175/JCLI-D-12-00196.1

Gregory and Forster, 2008, https://doi.org/10.1029/2008JD010405

Sherwood et al. 2015, http://dx.doi.org/10.1175/BAMS-D-13-00167.1

Citation: https://doi.org/10.5194/esd-2021-87-RC1
- AC3: 'Reply on RC1', Mikhail Verbitsky, 07 Dec 2021
  
  Dear Referee 1, Thank you for your review. The following is our response to your comments.
  Comment: In this manuscript the authors suggest that the evolution of the global temperature is governed by two time scales of positive and negative feedbacks, external forcing strength and evolution, as well as time. Forming non-dimensional groups, time is normalised and an equation for the temperature evolution (2) is derived. Then different cases are discussed where first positive feedbacks dominate, leading to stable climate states, and when positive feedbacks dominate leading to climate instability.
  Answer: With the exception of highlighted misprint (should be “negative”) you correctly captured most of the paper content. Unfortunately, in your brief overview you didn’t mention the third case – a climate having property of incomplete similarity. This is the critical part of our study for a number of reasons including a perspective of novelty that you raised below.
  Action: No action is required
  Comment: This manuscript is problematic in countless aspects, and I really don't know where to begin, and to end. Although I don't regularly use it, I am familiar with Buckinham's pi-theorem, and feel confident enough to know that if you provide it with a poor hypothesis, then the result will not be more insightful or fundamental. It does make sure the formulation is independent of the units used, but it is not magic.
  Answer: We definitely agree that a physical hypothesis is a cornerstone of any study.
  Action: In the revised manuscript we have sought to better clarify what the underlying physical hypotheses are in the current study, which is really more of a “think piece” (that’s all that is really possible within the tight 2-page length constraint of ESD Ideas).
  Comment: A guiding principle in the climate sciences is conservation of energy, at least Arrhenius (1896) used it, but probably also earlier studies. Arrhenius realised that studying the energy balance at the top of the atmosphere was a useful starting point and identified forcing from CO2, negative temperature feedback and positive water vapour and surface albedo feedbacks. Not using this as a starting point requires justification.
  Answer: Obviously, conservation of energy is a paramount principle, but starting with the hypothesis (1) does not mean that this principle somehow is not observed. Frankly, we internally debated how to better introduce the hypothesis (1). One of the options we considered was to review an energy-balance model (it is currently equation (8)) and to use it as an “inventory” of the governing parameters. We decided against this approach because it may create an impression that our analysis is based purely on the energy-balance model like (8). This is not the case because equation (8) does not have a property of incomplete similarity. We wanted a more general approach and decided in favor of equation (1). We can see now that complete avoiding mentioning of the energy conservation may create confusion.
  Action: We will articulate our approach better in the revised version of the paper.
  Comment: Typically, it is found that a two-layer formulation of the climate system with a shallow atmosphere+ocean mixed-layer reservoir coupled to a deep ocean provides a good starting point (Hansen et al. 1985), and this type of model is used with slight modification in countless places, and it is well-justified (e.g. Gregory and Forster 2008). Does the here discovered theory in some way predict aspects of climate change that the simple two-layer model does not?
  Answer: The purpose of our study wasn’t to introduce the entire hierarchy of models of increasing complexity (particularly within the two-page constraint of ESD Ideas). Instead, it is a thought piece that investigates different low-dimensional descriptions of the climate system. The zero-dimensional EBM, for example, is the simplest. From there one can of course build all the way up fully coupled three dimensional ocean-atmosphere models with interactive carbon cycle and cryosphere, etc.
  Yes, it does. Because of its simplicity, our approach is insightful. We suggest that the “hot-house” climate may be preceded by a climate having a property of incomplete similarity. To our knowledge, this is a new proposition.
  Action: No action is required
  Comment: Usually, we would think of feedbacks as dependent on temperature only to first order (e.g. Sherwood et al. 2015). There are modifications to this, for example some feedbacks may change during a transient as the system equilibrates (e.g. Held et al. 2010, Geoffroy et al. 2013), or as the temperature changes (e.g. Bloch-Johnson et al. 2015). But these are higher order effects and the starting point is that the feedback scales with temperature. In their formulation the authors assume that there are negative feedbacks with one time scale and positive feedbacks with another time scale without any justification.
  Answer: Yes, we consider two cases where positive and negative feedback scales are different. One case we relate to the recent climate history based on the simple zero-dimensional EBM experiments described by Mann et al (2014) and another case of the “hot-house” climate is hypothetical.
  Action: No action is required
  Comment: The paper by Steffen et al. (2018) appears to be used as a kind of confirmation of the theory. However, it is well known, and represented by the above mentioned theory, that if the feedback parameter becomes positive one enters an instability and a run-away climate, for instance snow-ball Earth (e.g. Budyko 1969). However, there is no evidence that the hothouse hypothesis of Steffen et al. 2018 is correct as assessed by IPCC AR6.
  Answer: We do not consider paper of Steffen et al. (2018) as a proof of our findings but simply as an example of possible regime where positive feedbacks may dominate over some range of variation (e.g., over the range where positive methane feedbacks play a dominant role - a process that ultimately saturates at some level of warming since the available carbon reservoir isn’t infinite).
  Action: No action is required
  Comment: The other piece of evidence provided is the near linear temperature increase over a select period of time. But that is not proof.
  Answer: We do not prove our scaling law but discover its parameters, a power degree. For a climate with dominant negative feedback, we use historical records to determine m = -1 and for a climate with dominant positive feedback we use equation (8) to find m >= 1. Additional studies will be needed to prove (or challenge) our scaling law.
  Action: No action is required
  Comment: Overall, though, what I find most problematic with this manuscript is that there is no attempt made to connect with relevant studies. A proposed idea can be radically different from whatever is already there, an attempt to reinvent the wheel, if you want. But authors need to do their homework and explain why they did what they did and how that is different and potentially better than existing approaches. It is therefore not possible for me to recommend publication.
  Action: Our presentation is very brief because of the 2-page ESD Ideas format, but we will revise the manuscript, within these length constraints, to clarify some of the points raised and responded to in this review.
  Mikhail Verbitsky and Michael E. Mann
  
  Citation: https://doi.org/10.5194/esd-2021-87-AC3
- AC4: 'Additional Reply on RC1', Mikhail Verbitsky, 09 Dec 2021
  
  Dear Referee 1,
  We would like to share with you an additional thought that has been motivated by your comment.
  We do not think we “attempt to reinvent the wheel” but instead we hope we are closing the gap.
  Since 1969, the Budyko's model (e.g., our equation (8) and similar) became a most popular simple tool of climate studies. The next level of sophistication, as you quite rightly noted, was "shallow atmosphere + ocean mixed-layer reservoir coupled to a deep ocean (Hansen et al. 1985)", followed by intermediate complexity models and 3D models.
  But Budyko's model doesn't have a property of incomplete similarity and the next-level atmosphere-mixed-layer model is already too complex for incomplete similarity to be easily recognized (it would require a specially designed studies that, to our knowledge, nobody did). Thus, a possibility of the incomplete-similarity regime remained unnoticed. Our dimensional analysis, in fact, closes this gap. It is more general than zero-dimensional EBM but obviously simple enough to reveal a possibility of incomplete similarity.
  Mikhail Verbitsky and Michael E. Mann
  
  Citation: https://doi.org/10.5194/esd-2021-87-AC4
- AC5:
  'Reply on RC1-suggested changes to the paper', Mikhail Verbitsky, 21 Dec 2021
  
  The comment was uploaded in the form of a supplement: https://esd.copernicus.org/preprints/esd-2021-87/esd-2021-87-AC5-supplement.pdf
  
  Citation: https://doi.org/10.5194/esd-2021-87-AC5
  - CC6: 'Reply on AC5', Richard Rosen, 22 Dec 2021
    
    It seems to me that your last sentence in your new reply is analytically true independent of the right equations describing the energy balance for the earth. Namely, the "climate’s behavior may be largely governed by magnitudes of its positive and negative feedbacks". Am I missing something? Once one takes into account the incoming radiation from the sun that heats the earth in appropriate detail, with clouds, oceans, biomass, land, etc., then of course the climate's behavior is due to the magnitudes of all the feedbacks. Thus, I don't understand what your new formulations contributes to understanding the value of your original article.
    
    Citation: https://doi.org/10.5194/esd-2021-87-CC6
    
    AC6: 'Reply on CC6', Mikhail Verbitsky, 23 Dec 2021
    
    Since the magnitudes of positive and negative feedbacks are defined not by a single similarity parameter but by groups (conglomerates) of similarity parameters, the hypothesis that “climate’s behavior may be largely governed by magnitudes of its positive and negative feedbacks” speaks to the reduction of the number of effective parameters and allows us to move from several similarity parameters (e.g., eight in the case of the system (1) - (3)) to just three similarity parameters and, in some circumstances, to a single conglomerate similarity parameter, the V-number.
    
    Citation: https://doi.org/10.5194/esd-2021-87-AC6
    
    CC7: 'Reply on AC6', Richard Rosen, 23 Dec 2021
    
    What you say may be true of your artificial model of the climate (your set of equations), but I still don't see the connection between your termology and model and the real physical world. The real physical world is far more complex and uncertain than you small set of equations. There are many feedback loops, positive and negative, that we still do not understand. So I do not understand how your model and mathematical terminology help advance the scientific analysis of climate change relative to the coupled atmospheric models that have existed for many years.
    
    Citation: https://doi.org/10.5194/esd-2021-87-CC7
    
    AC7: 'Reply on CC7', Mikhail Verbitsky, 23 Dec 2021
    
    Dr. Rosen, We thank you for engaging but these exchanges now seem to have exhausted their usefulness, so we will, respectfully, leave this where it currently stands.
    Mikhail Verbitsky and Michael E. Mann
    
    Citation: https://doi.org/10.5194/esd-2021-87-AC7
RC2:
'Comment on esd-2021-87', Anonymous Referee #2, 17 Jan 2022

This paper presents an interesting idea to derive a scaling law for global warming based on the physical fundamental physical properties of the climate system and the Buckingham-pi theorem for dimensional analysis. The scaling laws at two extreme conditions of positive or negative feedbacks dominating (complete similarity) are first derived, and then the case of incomplete similarity is discussed. I find this work novel and interesting, and a very good fit for an ESD Idea paper. I have a number of questions and suggestions, mostly about the dimensional analysis part, which I hope the authors address.

- Line 57 and eq 3 (similarly, line 73 and eq 6): it is not clear to me how you can go from \phi(t/\tau_n, \lembda) to (t/\tau_n)^m in eq. (3). In Buckingham-Pi, the function \phi of some variables can be written as the product of each variable to an unknown power, but it is unclear to me why in eq. 3 there is no \lambda^q (q being another unknown power) in eq. 3 multiplied by the rest of the terms. This is my main comment/question about the method. Please clarify.

- Line 46: the dimension of \epsilon. I guess it is correctly written as sec^{-\lambda-1), however, it does not appear clearly as far as I see. Please take a look and clarify if possible

- line 53: you may want to change this to ".... global temperature response T/(\epsilon t^{\lambda+1}) is a function of ....."

- Also you may want to clarify around line 50 that here there are 6 variables in (1), there are two fundamental dimensions (time and temperature), so Buckingham-pi states that there would be 4 dimensionless (pi) groups, as presented on lines 52-53.

- I find the word "dimensionless" just sounding better than "adimensional" but it is up to you which one to use.

- line 99: m should be in the math mode

- line 22: the use of Buckingham-pi theorem in climate science has been rather limited, and the cited paper by Golitsyn is certainly a great example. You may want to also mention some of the recent papers using this approach to study the climate system, e .g.

+Chavas, D.R. and Emanuel, K., 2014. Equilibrium tropical cyclone size in an idealized state of axisymmetric radiative–convective equilibrium. JAS

+Yang, D. and Ingersoll, A.P., 2014. A theory of the MJO horizontal scale. GRL

+Nabizadeh, E., et al. 2019. Size of the atmospheric blocking events: Scaling law and response to climate change. GRL

- line 16: is the word "similarity" in ".... in feedbacks similarity parameters ...." needed?

Citation: https://doi.org/10.5194/esd-2021-87-RC2
- AC8: 'Reply on RC2', Mikhail Verbitsky, 17 Jan 2022
  
  Dear Referee 2,
  Thank you very much for the encouraging and insightful review. The following is our response to your comments.
  Comment: This paper presents an interesting idea to derive a scaling law for global warming based on the physical fundamental physical properties of the climate system and the Buckingham-pi theorem for dimensional analysis. The scaling laws at two extreme conditions of positive or negative feedbacks dominating (complete similarity) are first derived, and then the case of incomplete similarity is discussed. I find this work novel and interesting, and a very good fit for an ESD Idea paper.
  Answer: We appreciate your evaluation.
  Comment: I have a number of questions and suggestions, mostly about the dimensional analysis part, which I hope the authors address.
  - Line 57 and eq 3 (similarly, line 73 and eq 6): it is not clear to me how you can go from \phi(t/\tau_n, \lembda) to (t/\tau_n)^m in eq. (3). In Buckingham-Pi, the function \phi of some variables can be written as the product of each variable to an unknown power, but it is unclear to me why in eq. 3 there is no \lambda^q (q being another unknown power) in eq. 3 multiplied by the rest of the terms. This is my main comment/question about the method. Please clarify.
  Answer: Your observation is correct. Indeed, 𝜆^q may appear in equations (3) and (6). But since 𝜆 and 𝜆^q are constants, 𝜆^q has been absorbed by the experimental constant k.
  Action: We will clarify this reasoning in the revised version of the paper.
  Comment: - Line 46: the dimension of \epsilon. I guess it is correctly written as sec^{-\lambda-1), however, it does not appear clearly as far as I see. Please take a look and clarify if possible
  Answer: The dimension of ε is [^oC sec^(-𝜆-1)]. It looks like the pdf file of the preprint introduced some distortion in line 46 that may cause confusion.
  Action: We will make sure that it is clear in the final version of the paper.
  Comment: - Also you may want to clarify around line 50 that here there are 6 variables in (1), there are two fundamental dimensions (time and temperature), so Buckingham-pi states that there would be 4 dimensionless (pi) groups, as presented on lines 52-53.
  Answer: Agreed
  Action: This clarification will be included in the final version of the paper.
  Comment: - line 22: the use of Buckingham-pi theorem in climate science has been rather limited, and the cited paper by Golitsyn is certainly a great example. You may want to also mention some of the recent papers using this approach to study the climate system, e .g.
  +Chavas, D.R. and Emanuel, K., 2014. Equilibrium tropical cyclone size in an idealized state of axisymmetric radiative–convective equilibrium. JAS
  +Yang, D. and Ingersoll, A.P., 2014. A theory of the MJO horizontal scale. GRL
  +Nabizadeh, E., et al. 2019. Size of the atmospheric blocking events: Scaling law and response to climate change. GRL
  Answer: Thank you for your suggestion. We agree that it will be helpful to our readers.
  Action: Per editor’s approval (number of references in ESD Ideas papers is limited to 12) we will be glad to include all your recommended references.
  Minor comments:
  - line 53: you may want to change this to ".... global temperature response T/(\epsilon t^{\lambda+1}) is a function of ....."
  - I find the word "dimensionless" just sounding better than "adimensional" but it is up to you which one to use.
  - line 99: m should be in the math mode
  - line 16: is the word "similarity" in ".... in feedbacks similarity parameters ...." needed?
  Action: All minor comments will be taken care of.
  Mikhail Verbitsky and Michael E. Mann
  
  Citation: https://doi.org/10.5194/esd-2021-87-AC8
- AC9: 'Reply on RC2- pdf version', Mikhail Verbitsky, 17 Jan 2022
  
  We also publish a pdf version of our response so that mathematical symbols are not distorted
  
  Citation: https://doi.org/10.5194/esd-2021-87-AC9
RC3:
'Comment on esd-2021-87', Anonymous Referee #3, 25 Mar 2022

The authors present an approach to analyze the time evolution of global warming based on century-old dimensionality arguments that are commonly used in the field of fluid dynamics. They include more recent insights into the origins of incomplete similarity to hypothesize a characteristic scaling of T ~ t² for an intermediate pre-hothouse climate regime, before the positive feedbacks start to dominate.

First of all, it could be clarified why other dimensional physical parameters that play an important role in the climate system, such as the active ocean layer depth/mass that responds nonlinearly to global warming, albedo etc. do not feature. Alternatively, explain how they are absorbed into the limited set of parameters shown here. It is possible that they can all be reduced to the proposed time scales, but the way it's presented feels like an extreme simplification.

Secondly, there's a significant difference with fluid dynamics that is not highlighted in this study. In contrast to fluid dynamics, in climate there is no multitude of experiments that provide the required observations time series to determine the scaling parameters. Of course, this does not mean that no information can be distilled from the limited historical climate record - clearly, as many studies have shown, hypotheses can be presented and analyses based on (ensemble) climate methods can confirm or refute these hypotheses, using the historical climate record. However, the proposed scaling behaviour is only feebly, if at all supported by the historical data. I realise that this is due to the nature of the problem - we have to make do with the observations we have. As a result, however, I find the evidence for the theory too tenuous to publish as-is.

A straightforward way to fix this, is that the authors test their hypothesis through another means. Indeed, the simplicity of the assumptions implies that they can use fairly simple general circulation models, or even simpler zero-dimensional energy balance models, to generate temperature time series for the different scenarios. This will allow them to check whether the scaling behaviour holds. Unless this can be provided, I do not recommend publication of this manuscript, as interesting as the underlying ideas are.

Citation: https://doi.org/10.5194/esd-2021-87-RC3
- AC10: 'Reply on RC3', Mikhail Verbitsky, 26 Mar 2022
  
  Dear Referee 3,
  Thank you very much for your review. The following is our response to your comments.
  Comment: The authors present an approach to analyze the time evolution of global warming based on century-old dimensionality arguments that are commonly used in the field of fluid dynamics. They include more recent insights into the origins of incomplete similarity to hypothesize a characteristic scaling of T ~ t^2 for an intermediate pre-hothouse climate regime, before the positive feedbacks start to dominate.
  Answer: Your short description of the paper is correct
  Comment: First of all, it could be clarified why other dimensional physical parameters that play an important role in the climate system, such as the active ocean layer depth/mass that responds nonlinearly to global warming, albedo etc. do not feature. Alternatively, explain how they are absorbed into the limited set of parameters shown here. It is possible that they can all be reduced to the proposed time scales, but the way it's presented feels like an extreme simplification.
  Answer: We agree that our readers will benefit from more detailed description of our reasoning.
  Action: Additional substantiation of our main hypothesis will be provided (please also review our response AC5 to Referee 1 https://editor.copernicus.org/index.php?_mdl=msover_md&_jrl=430&_lcm=oc108lcm109w&_acm=get_comm_sup_file&_ms=98841&c=216702&salt=18451585431668037296 )
  Comment: Secondly, there's a significant difference with fluid dynamics that is not highlighted in this study. In contrast to fluid dynamics, in climate there is no multitude of experiments that provide the required observations time series to determine the scaling parameters. Of course, this does not mean that no information can be distilled from the limited historical climate record - clearly, as many studies have shown, hypotheses can be presented and analyses based on (ensemble) climate methods can confirm or refute these hypotheses, using the historical climate record. However, the proposed scaling behaviour is only feebly, if at all supported by the historical data. I realise that this is due to the nature of the problem - we have to make do with the observations we have. As a result, however, I find the evidence for the theory too tenuous to publish as-is.
  A straightforward way to fix this, is that the authors test their hypothesis through another means. Indeed, the simplicity of the assumptions implies that they can use fairly simple general circulation models, or even simpler zero-dimensional energy balance models, to generate temperature time series for the different scenarios. This will allow them to check whether the scaling behaviour holds. Unless this can be provided, I do not recommend publication of this manuscript, as interesting as the underlying ideas are.
  Answer: Thank you for recognizing our ideas as interesting. Your suggestion to strengthen our arguments is well understood.
  Action: Though dedicated experiments with GCM models are outside of the scope of this ideas-format paper, additional analytics with an energy-balance model will, indeed, be provided as well as a compilation of appropriate already published studies.
  Mikhail Verbitsky and Michael E. Mann
  
  Citation: https://doi.org/10.5194/esd-2021-87-AC10

Mikhail Verbitsky and Michael Mann

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Cumulative views and downloads (calculated since 03 Nov 2021)

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Nov 2021	493	130	9	632
Dec 2021	221	54	9	284
Jan 2022	130	39	6	175
Feb 2022	67	20	0	87
Mar 2022	90	16	5	111
Apr 2022	59	19	0	78
May 2022	43	14	1	58
Jun 2022	16	3	1	20
Jul 2022	20	5	0	25
Aug 2022	12	6	0	18
Sep 2022	5	7	0	12
Oct 2022	20	4	1	25
Nov 2022	29	4	1	34
Dec 2022	15	7	0	22
Jan 2023	13	20	0	33
Feb 2023	17	6	0	23
Mar 2023	25	4	0	29
Apr 2023	9	3	0	12
May 2023	13	4	3	20
Jun 2023	12	16	2	30
Jul 2023	10	23	0	33
Aug 2023	11	24	0	35
Sep 2023	19	20	1	40
Oct 2023	17	11	2	30
Nov 2023	11	1	0	12
Dec 2023	22	2	1	25
Jan 2024	10	8	0	18
Feb 2024	13	28	5	46
Mar 2024	14	26	2	42
Apr 2024	19	2	6	27

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