the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 15 May 2025
On defining climate by means of an ensemble
Abstract. We study the suitability of an initial condition ensemble to form the conceptual basis of defining climate. We point out that the most important criterion for this is the uniqueness of the probability measure on which the definition relies. We first naively propose, in harmony with earlier work, to represent such a probability measure by the distribution of ensemble members that has, loosely speaking, converged to the natural probability measure of the so-called snapshot or pullback attractor of the dynamics; this attractor is time dependent in the presence of external forcing. Then we refine the proposal by taking a probability measure that is conditional on the (possibly time-evolving) state of modes characterized by time scales of convergence that are longer than the horizon of a particular study. We discuss the applicability of such a definition in the Earth system and its realistic models, and conclude that the practically relevant probability measure may, hopefully, become accessible by a few decades of convergence after initialization; for this, initialization may perhaps need to rely on the observed state of the slower-converging modes. However, the absence of sufficient separation of time scales of convergence between modes or regime transitions in variables corresponding to slower-converging modes might preclude uniqueness, perhaps in certain subsystems. In uniqueness holds, time evolution of slower-converging modes may induce unforced climate changes, leading to the need for targeted investigations to determine the forced response. We propose an initialization scheme for studying all these issues in Earth system models.
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RC1: 'Comment on egusphere-2025-2030', Michael Ghil, 15 Jul 2025
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The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2025-2030/egusphere-2025-2030-RC1-supplement.pdf
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AC1: 'Reply on RC1', Gabor Drotos, 02 Aug 2025
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# Authors' first response on RC1
We are grateful to Michael Ghil for evaluating our manuscript and formulating helpful thoughts for improvement so that final publication will be worthwhile. Following the peer review policy of the journal, we provide a brief response below, then actually proceed with further steps later on.
Indeed, as suggested by the Reviewer, readers may benefit from mathematics literature about nonlinear dynamical systems, so that we will include references to the works mentioned by him. Although our considerations do not rely on the precise formulation of the mathematical statements underlying them (hence we cannot actually 'define' climate), we agree that acknowledging the existence of this background and directing the interested reader there is important.
Regarding the practical issues mentioned by the Reviewer, we believe that dedicating an entire section of the main text to the naive approach was confusing, so that we will relegate this part to an appendix. With further streamlining, we hope it will become clearer that the novelty is handling a wide range of time scales that describe convergence, through providing a conditional definition of climate.
We also agree that the title should be improved. Notwithstanding, we feel that the word 'describing' does not reflect our goal well. We would really like to point out by the title that the concept of climate had not been defined sufficiently well in the past in our view, and that we outline a possible solution to the issue on the grounds of the theory of dynamical systems; without, however, precisely formulating an actual definition in the present work. We will propose a new title by the time we are ready with revising the manuscript at the latest, and we will strictly follow any further instructions received at that time even if they perhaps happen to disagree with our preference. We reiterate that we ourselves are not satisfied with the current form of the title either.
We hereby thank once more for the Reviewer's kind work.
Citation: https://doi.org/10.5194/egusphere-2025-2030-AC1
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AC1: 'Reply on RC1', Gabor Drotos, 02 Aug 2025
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RC2: 'Comment on egusphere-2025-2030', Stéphane Vannitsem, 19 Jul 2025
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See the attached document.
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AC2: 'Reply on RC2', Gabor Drotos, 02 Aug 2025
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# Authors' first response on RC2
We are grateful to Stéphane Vannitsem for evaluating our manuscript and writing a supporting and constructive review. Following the peer review policy of the journal, we provide a brief response below, which clarifies some points in this case along with outlining corresponding modifications of the text, then actually proceed with further steps later on.
We first prefer to clarify that we used PlaSim only for an illustration of the naive approach whereby approximate convergence is ensured to the actual pullback attractor of the entire system, i.e., in terms of all of the modes defined by the relevant spectrum. We now see, also based on the review of Michael Ghil, that dedicating a full section to this approach is confusing. We will relegate this section to an appendix in the revised version of the manuscript.
As for our considerations about a conditional definition, they are in fact not affected by "how" or "why" modes of different time scales arise (e.g. interaction between system components). Our idea is to perform an analysis numerically and at the same time "blindly" with respect to the underlying background but, most importantly, knowing that the time scales of convergence are governed by the spectrum mentioned above, in the way described in the manuscript. In particular, an appropriate gap will provide an opportunity for a relatively clear definition, while we acknowledge in the manuscript that the lack thereof will pose difficulties, as suggested by the Reviewer as well. Besides rearranging the content, we will also work on streamlining the formulation so that the above message will hopefully become clearer than in the current version.
We thank the Reviewer for pointing out that the analyses of Lovejoy and coworkers deserve more attention. We will elaborate on the issue with more care in the revised version.
Regarding the title, we see that improving it is important; however, we are not fully happy with the word 'describing'. By the time we formulate our final response, we will select our new proposal.
We will take care of all of the remaining as well and thank once more for the Reviewer's kind work.
Citation: https://doi.org/10.5194/egusphere-2025-2030-AC2
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AC2: 'Reply on RC2', Gabor Drotos, 02 Aug 2025
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CC1: 'Comment on egusphere-2025-2030', Jochen Broecker, 05 Aug 2025
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This paper aims to contribute to the debate on how to define ``climate’’. There is agreement that the collection of physical quantities relevant for meteorological phenomena on our planet should be regarded as the state of a very large and complex dynamical system (stochastic or deterministic), with the governing equations reflecting the relevant fundamental laws of physics. There is also agreement that while the term ``weather'' essentially refers to a single state of this dynamical system (or in other words a single value for each meteorological variable), the term ``climate’’ should refer to a range of values, or a distribution of states. This distribution should be time dependent so as to account for climate change (i.e. if the climate system is non-autonomous due to natural or anthropogenic interventions), and for the fact that we have observations of the actual weather which drastically reduces the plausible range of states.
In my opinion, the debate should focus on identifying mathematical concepts that are suitable for serving as ``climate’’, and this is what the manuscript is trying to do. If I am correct though with my assessment as to what the community currently agrees on, the debate needs to face the devil in the details. The presented manuscript however struggles with this. There are a number of rather relevant issues where the paper is extremely vague and imprecise. In a sense, this renders the paper hard to criticise since I am never 100% sure what the authors actually mean. Yet for the same reason, it is difficult to say what the actual original contribution of the paper is. The assertions of the paper are not sufficiently precise to tell how they actually differ from or improve upon what has already been said elsewhere.
As a particular example, it is not clear to me what the paper adds to the discussion compared to Flandoli et al (2022). The authors state that ``Flandoli et al. (2022) did not restrict the form of forcing either, but they based their analysis on a slow time dependence compared with the convergence of time averages to ensemble averages, which is generically not the case and is certainly not so during the ongoing global warming’’. Firstly, I don’t see how the definition of climate provided by the authors resolves this issue. Secondly, the analysis in Flandoli et al is not, in my opinion, fundamentally based on some time scale separation. In fact, that paper asserts the existence of (potentially many) candidate climates without any reference to time scale separation, whence the definition of the climate becomes a question of uniqueness. Flandoli et al. then propose one possible mechanism to resolve this (which one may argue indeed involves separation of timescales). The presented manuscript likewise identifies uniqueness as the essential question but I struggle to understand what alternatives the authors propose.
Citation: https://doi.org/10.5194/egusphere-2025-2030-CC1 -
CC2: 'Reply on CC1', Tamás Bódai, 09 Aug 2025
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Dear Jochen,
Thank you for your comment and feedback on what aspects of the paper should be improved.
Among them you address a quite fundamental requirement: that the paper be scientific as in posing testable hypotheses. Classic examples of pseudo-science are Marxism and psychoanalysis. We name two conditions to hold so that the temporal evolution of some ensemble-wise statistics can represent "climate change".
(1) Time scale separation;
(2) Avoiding the initialisation of the ensemble in the course of a (fast) regime transition of the slow system.
Condition (2) is hopefully made concrete enough by the example in the Appendix. Condition (1) is indeed a bit more subtle to precisely formulate because of the various ways that time scales can be defined. Yet, i think that we clearly state that for (1), time scales are defined as the reciprocals of the eigenvalues of the Frobenius-Perron (transfer) operator.
Then, about testabilty, our paper culminates in a proposal for a ensemble experiment initialisation scheme by which we can verify if "the temporal evolution of some ensemble-wise statistics can represent "climate change"". That is the case, we suggest, if pairs of subensembles converge, i.e., in the case of uniquenes, independence of ensemble initialisation. As such an experiment has never been carried out, clearly, there is room for doubt whether the protocol is mathematically well-posed. (Any scrutiny of it and feedback would be most welcome.) As for verifying convergence of ensemble-wise statistics, on the other hand, seems to me a rather trivial task.
We also point out that the concept of climate change in this sense delineates from any idea of forced climate change (forced response) because such a climate change would occur in an autonomous dynamical system due to the drift of the slow system. Please correct me if i'm wrong, but this also seems a novel insight, and an important one too.
I believe that Gabor will provide his reflections shortly including the "connection with Flandoli et al.".
Citation: https://doi.org/10.5194/egusphere-2025-2030-CC2 -
CC3: 'Reply on CC2 by the other author', Gabor Drotos, 09 Aug 2025
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Dear Tamás and Jochen,
Let me first reflect on some of Tamás' remarks.
"Yet, i think that we clearly state that for (1), time scales are defined as the reciprocals of the eigenvalues of the Frobenius-Perron (transfer) operator."
In my view, what is told here is not actually sufficient or true. First of all, this approach as described in Appendix A works for autonomous dynamical systems only (or those that can be converted to autonomous ones; i.e., periodically forced ones). Instead of explicitly addressing the nonautonomous case, we practically state in the manuscript that the nonautonomous case works more or less similarly, and direct the reader to Froyland et al. (2010) for the mathematics of that problem. Second, each eigenvalue in fact defines two different time scales through its real and imaginary part even in the autonomous case. Only this recognition makes the description complete and correct; "the reciprocal" of an eigenvalue is a complex number in a generic case and hence cannot itself be identified with a time scale. (We thank Mickaël Chekroun for having warned us to pay attention to the complex nature of these eigenvalues.)
What I am writing here about the nonautonomous case means that I do acknowledge that the mathematics underlying the problem is not discussed in full detail in our manuscript. But I believe it is not actually required to proceed, as explained in AC3 of the discussion webpage.
"As for verifying convergence of ensemble-wise statistics, on the other hand, seems to me a rather trivial task."
This may be true, but note the multitude of possible quantifiers of how different two "probability distributions" are. Certainly, we mention two families in Footnote 12 but recommend an actual choice to be made separately in each practical problem. On the one hand, I think this is not a shortcoming in terms of preciseness; cf. discussions of similar issues in the manuscript. On the other hand, I guess that results will be insensitive to the choice.
Now let me discuss Flandoli et al. (2022) here so as not to render the comment tree too complicated and as anticipated by Tamás. The authors write in their Section 1.1.1 that"the time average corresponds for instance to the average over a period of a few years, defining values of temperature that we associate to the idea of “climate”. \par In the climate example, however, the process Xt(ω) is sometimes not stationary, for instance in the past and present century."
Based on this, they introduce a time scale separation parameter epsilon, based on which they further introduce a "macroscopic view" in Section 1.1.2. Throughout the paper, they investigate the 0 limit of epsilon, ensuring clear separation between microscopic and macroscopic time scales. What is most important is that, if I understand correctly, they _assume_ that climate changes are characterised by the macroscopic time scale. This, unfortunately, does not apply e.g. to the ongoing climate change of Earth. But please let me know in case I am wrong about the mentioned assumption.
Kind regards,
GáborCitation: https://doi.org/10.5194/egusphere-2025-2030-CC3
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CC3: 'Reply on CC2 by the other author', Gabor Drotos, 09 Aug 2025
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AC3: 'Reply on CC1 on behalf of all co-authors', Gabor Drotos, 09 Aug 2025
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Dear Jochen,
What follows below has been accepted by both of us. After earlier discussion with Tamás, I wrote it in the hope of being pragmatic.
We thank you for pointing out the importance of the topic.
We believe what we actually present with regard to the underlying mathematics (in Section 4 and especially Appendix A) is sufficient to proceed and to conclude about a sufficiently precise operational definition in Section 5.1 and about an algorithm for its application in practice in Section 5.2.1. We will see if we can provide clearer formulations in the latter sections so as to avoid any confusion about preciseness and novelty, and we will definitely reorganise earlier sections with the same purpose (as already anticipated in relation to the two referee comments published so far). First and foremost, we will keep in mind that the reader must not be distracted by unnecessary or vague discussions by the time a paper reaches its conclusions.
Regarding Flandoli et al. (2022), I will provide a separate comment a bit later.
Kind regards,
GáborCitation: https://doi.org/10.5194/egusphere-2025-2030-AC3
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CC2: 'Reply on CC1', Tamás Bódai, 09 Aug 2025
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