ESD Ideas: A Global Warming Scaling Law
 ^{1}Gen5 Group, LLC, Newton, MA, USA
 ^{2}UCLouvain, Earth and Life Institute, LouvainlaNeuve, Belgium
 ^{3}The Pennsylvania State University, Department of Meteorology and Atmospheric Science, University Park, PA, USA
 ^{1}Gen5 Group, LLC, Newton, MA, USA
 ^{2}UCLouvain, Earth and Life Institute, LouvainlaNeuve, Belgium
 ^{3}The Pennsylvania State University, Department of Meteorology and Atmospheric Science, University Park, PA, USA
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 positivetonegative ratio is the same.
Mikhail Verbitsky and Michael Mann
Status: closed

CC1: 'Comment on esd202187', 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.

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 14 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

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 nonlinear 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.

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, energybalance 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.

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.

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.

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
 CC5: 'Reply on EC1', Richard Rosen, 12 Nov 2021

EC1: 'Handling Editor's Reply on CC4', Gabriele Messori, 12 Nov 2021

CC4: 'Reply on CC3', Richard Rosen, 12 Nov 2021

CC3: 'Reply on AC2', Richard Rosen, 12 Nov 2021

AC2: 'Reply on CC2', Mikhail Verbitsky, 12 Nov 2021

CC2: 'Reply on AC1', Richard Rosen, 03 Nov 2021

AC1: 'Reply on CC1', Mikhail Verbitsky, 03 Nov 2021

RC1: 'Comment on esd202187', 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 nondimensional 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 pitheorem, 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 twolayer formulation of the climate system with a shallow atmosphere+ocean mixedlayer 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 welljustified (e.g. Gregory and Forster 2008). Does the here discovered theory in some way predict aspects of climate change that the simple twolayer 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. BlochJohnson 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 runaway climate, for instance snowball 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_tcm18173546.pdf
BlochJohnson 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/JCLID1200196.1
Gregory and Forster, 2008, https://doi.org/10.1029/2008JD010405
Sherwood et al. 2015, http://dx.doi.org/10.1175/BAMSD1300167.1

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 nondimensional 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 pitheorem, 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 2page 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 energybalance 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 energybalance 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 twolayer formulation of the climate system with a shallow atmosphere+ocean mixedlayer 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 welljustified (e.g. Gregory and Forster 2008). Does the here discovered theory in some way predict aspects of climate change that the simple twolayer model does not?
Answer: The purpose of our study wasn’t to introduce the entire hierarchy of models of increasing complexity (particularly within the twopage constraint of ESD Ideas). Instead, it is a thought piece that investigates different lowdimensional descriptions of the climate system. The zerodimensional EBM, for example, is the simplest. From there one can of course build all the way up fully coupled three dimensional oceanatmosphere models with interactive carbon cycle and cryosphere, etc.
Yes, it does. Because of its simplicity, our approach is insightful. We suggest that the “hothouse” 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. BlochJohnson 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 zerodimensional EBM experiments described by Mann et al (2014) and another case of the “hothouse” 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 runaway climate, for instance snowball 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 2page 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

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 mixedlayer 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 nextlevel atmospheremixedlayer 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 incompletesimilarity regime remained unnoticed. Our dimensional analysis, in fact, closes this gap. It is more general than zerodimensional EBM but obviously simple enough to reveal a possibility of incomplete similarity.
Mikhail Verbitsky and Michael E. Mann

AC5: 'Reply on RC1suggested changes to the paper', Mikhail Verbitsky, 21 Dec 2021
The comment was uploaded in the form of a supplement: https://esd.copernicus.org/preprints/esd202187/esd202187AC5supplement.pdf

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.

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 Vnumber.

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.
 AC7: 'Reply on CC7', Mikhail Verbitsky, 23 Dec 2021

CC7: 'Reply on AC6', Richard Rosen, 23 Dec 2021

AC6: 'Reply on CC6', Mikhail Verbitsky, 23 Dec 2021

CC6: 'Reply on AC5', Richard Rosen, 22 Dec 2021

AC3: 'Reply on RC1', Mikhail Verbitsky, 07 Dec 2021

RC2: 'Comment on esd202187', 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 Buckinghampi 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 BuckinghamPi, 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^{\lambda1), 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 Buckinghampi states that there would be 4 dimensionless (pi) groups, as presented on lines 5253.
 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 Buckinghampi 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?

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 Buckinghampi 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 BuckinghamPi, 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^{\lambda1), 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 [^{o}C sec^(𝜆1)]. 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 Buckinghampi states that there would be 4 dimensionless (pi) groups, as presented on lines 5253.
Answer: Agreed
Action: This clarification will be included in the final version of the paper.
Comment:  line 22: the use of Buckinghampi 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
 AC9: 'Reply on RC2 pdf version', Mikhail Verbitsky, 17 Jan 2022

AC8: 'Reply on RC2', Mikhail Verbitsky, 17 Jan 2022

RC3: 'Comment on esd202187', Anonymous Referee #3, 25 Mar 2022
The authors present an approach to analyze the time evolution of global warming based on centuryold 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 prehothouse 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 asis.
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 zerodimensional 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.

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 centuryold 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 prehothouse 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 asis.
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 zerodimensional 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 ideasformat paper, additional analytics with an energybalance model will, indeed, be provided as well as a compilation of appropriate already published studies.
Mikhail Verbitsky and Michael E. Mann

AC10: 'Reply on RC3', Mikhail Verbitsky, 26 Mar 2022
Status: closed

CC1: 'Comment on esd202187', 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.

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 14 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

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 nonlinear 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.

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, energybalance 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.

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.

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.

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
 CC5: 'Reply on EC1', Richard Rosen, 12 Nov 2021

EC1: 'Handling Editor's Reply on CC4', Gabriele Messori, 12 Nov 2021

CC4: 'Reply on CC3', Richard Rosen, 12 Nov 2021

CC3: 'Reply on AC2', Richard Rosen, 12 Nov 2021

AC2: 'Reply on CC2', Mikhail Verbitsky, 12 Nov 2021

CC2: 'Reply on AC1', Richard Rosen, 03 Nov 2021

AC1: 'Reply on CC1', Mikhail Verbitsky, 03 Nov 2021

RC1: 'Comment on esd202187', 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 nondimensional 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 pitheorem, 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 twolayer formulation of the climate system with a shallow atmosphere+ocean mixedlayer 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 welljustified (e.g. Gregory and Forster 2008). Does the here discovered theory in some way predict aspects of climate change that the simple twolayer 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. BlochJohnson 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 runaway climate, for instance snowball 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_tcm18173546.pdf
BlochJohnson 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/JCLID1200196.1
Gregory and Forster, 2008, https://doi.org/10.1029/2008JD010405
Sherwood et al. 2015, http://dx.doi.org/10.1175/BAMSD1300167.1

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 nondimensional 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 pitheorem, 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 2page 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 energybalance 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 energybalance 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 twolayer formulation of the climate system with a shallow atmosphere+ocean mixedlayer 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 welljustified (e.g. Gregory and Forster 2008). Does the here discovered theory in some way predict aspects of climate change that the simple twolayer model does not?
Answer: The purpose of our study wasn’t to introduce the entire hierarchy of models of increasing complexity (particularly within the twopage constraint of ESD Ideas). Instead, it is a thought piece that investigates different lowdimensional descriptions of the climate system. The zerodimensional EBM, for example, is the simplest. From there one can of course build all the way up fully coupled three dimensional oceanatmosphere models with interactive carbon cycle and cryosphere, etc.
Yes, it does. Because of its simplicity, our approach is insightful. We suggest that the “hothouse” 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. BlochJohnson 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 zerodimensional EBM experiments described by Mann et al (2014) and another case of the “hothouse” 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 runaway climate, for instance snowball 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 2page 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

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 mixedlayer 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 nextlevel atmospheremixedlayer 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 incompletesimilarity regime remained unnoticed. Our dimensional analysis, in fact, closes this gap. It is more general than zerodimensional EBM but obviously simple enough to reveal a possibility of incomplete similarity.
Mikhail Verbitsky and Michael E. Mann

AC5: 'Reply on RC1suggested changes to the paper', Mikhail Verbitsky, 21 Dec 2021
The comment was uploaded in the form of a supplement: https://esd.copernicus.org/preprints/esd202187/esd202187AC5supplement.pdf

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.

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 Vnumber.

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.
 AC7: 'Reply on CC7', Mikhail Verbitsky, 23 Dec 2021

CC7: 'Reply on AC6', Richard Rosen, 23 Dec 2021

AC6: 'Reply on CC6', Mikhail Verbitsky, 23 Dec 2021

CC6: 'Reply on AC5', Richard Rosen, 22 Dec 2021

AC3: 'Reply on RC1', Mikhail Verbitsky, 07 Dec 2021

RC2: 'Comment on esd202187', 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 Buckinghampi 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 BuckinghamPi, 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^{\lambda1), 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 Buckinghampi states that there would be 4 dimensionless (pi) groups, as presented on lines 5253.
 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 Buckinghampi 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?

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 Buckinghampi 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 BuckinghamPi, 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^{\lambda1), 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 [^{o}C sec^(𝜆1)]. 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 Buckinghampi states that there would be 4 dimensionless (pi) groups, as presented on lines 5253.
Answer: Agreed
Action: This clarification will be included in the final version of the paper.
Comment:  line 22: the use of Buckinghampi 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
 AC9: 'Reply on RC2 pdf version', Mikhail Verbitsky, 17 Jan 2022

AC8: 'Reply on RC2', Mikhail Verbitsky, 17 Jan 2022

RC3: 'Comment on esd202187', Anonymous Referee #3, 25 Mar 2022
The authors present an approach to analyze the time evolution of global warming based on centuryold 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 prehothouse 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 asis.
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 zerodimensional 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.

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 centuryold 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 prehothouse 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 asis.
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 zerodimensional 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 ideasformat paper, additional analytics with an energybalance model will, indeed, be provided as well as a compilation of appropriate already published studies.
Mikhail Verbitsky and Michael E. Mann

AC10: 'Reply on RC3', Mikhail Verbitsky, 26 Mar 2022
Mikhail Verbitsky and Michael Mann
Mikhail Verbitsky and Michael Mann
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