# 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.

# Authors' final response on RC3

We thank the Reviewer for evaluating the manuscript and formulating his main concern about novelty in a very clear way.

The latter allows us to give a simple explanation of our stance about the aim and novelty of our work. In particular: no publication ever (to our knowledge) has provided a framework (i.e., a description of the dynamics of probability densities) according to which the "well-known" definition of climate, quoted by the Reviewer from the manuscript, could be constructed. Unless such a construction is available, it is not at all obvious that such a definition can be meaningful from a mathematical or (geo-) physical point of view: what kind of time scales one refers to in the definition, what do they characterise, why is that characterisation relevant? The principal aim and achievement of our manuscript is filling this gap.

In terms of the stochastic description of the Reviewer, we agree that it is obvious how climate should be defined; however, the question is whether one has any a priori reason to assume that such a description can be realistic. In turn, we provide an a posteriori reason through an ab initio approach (according to the best of the scientific community's knowledge in terms of modelling the system in question).

We empasise that we do not claim that such a description *is* actually realistic, or, to be more precise, the desired separation of time scales exists. Our claim is that posing the question of this existence is meaningful; i.e., a well-defined climate *may* exist, although only if certain conditions are met.

While we do not actually answer whether these conditions are met, we make an "educated guess" that they do so, based on existing simulations of AOGCMs and comprehensive Earth system models. More importantly, we present a scheme for intialising a set of ensemble simulations to a posteriori assess the issue in any dynamical model of practical interest.

We reiterate from AC1 and AC3 that we do not need all of the mathematical details for our exposition, especially that the above-mentioned practical assessment (which seems to be the only viable one) does not rely on them. We prefer to add here that the targeted audience is the geoscientific community, writing papers like Hawkins et al. (2016), Deser et al. (2020; 2025), etc. cited in our manuscript, suggesting that they may probably be more comfortable with argumentations like those of our manuscript in its present form.

The claim of RC1 is nevertheless right that our "guess", as we see, for "what climate may be" is (more or less) the modus operandi just in the mentioned community. In order to make our aim and message clearer to them as well as to any interested party, we have chosen the new title to be

    "Can we define climate by means of an ensemble? A tale of time scales of convergence"

and also decided to formulate a new abstract which reads as follows:

    "It is hardly questioned today that climate can be described in theory by an ensemble of trajectories differing in their initial conditions, which is then translated to numerical ensembles in climate models. It is also widely accepted that any evolution observed within a few decades after initialization is not relevant to climate. Evolution at a later stage, instead, is then used to characterize climate and its change, under the implicit assumption that slower processes do not considerably contribute to differences between ensemble members, letting internal variability of climate be identified with these differences. However, a justification for this practice is as yet lacking. In particular, a definition of climate in support of this practice is outstanding, including the identification of the kind of time scales at play through providing an argumentation for their relevance. Our study aims at filling this gap. After pointing out that the most important criterion for a definition of climate is the uniqueness of the probability measure on which the definition relies, we first recall the naive proposal 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. We then consider the time scales of convergence and refine the proposal by taking a probability measure that is conditional on the (possibly time-evolving) state of modes characterized by convergence time scales longer than the horizon of a particular study. We design an ensemble simulation initialization scheme for studying convergence time scales and uniqueness of ensembles in Earth system models."

Along with the rearrangements anticipated in the previous ACs, we hope that these reformulations will facilitate a clear communication of the novelties in our work. Among others, we keep our promise to relegate the detailed discussion of the naive proposal to an appendix. We hereby thank once more for the Reviewer's helpful evaluation.

# 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.

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,

Gábor

# 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.

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,

Gábor

# Authors' final response on RC3

We thank the Reviewer for evaluating the manuscript and formulating his main concern about novelty in a very clear way.

The latter allows us to give a simple explanation of our stance about the aim and novelty of our work. In particular: no publication ever (to our knowledge) has provided a framework (i.e., a description of the dynamics of probability densities) according to which the "well-known" definition of climate, quoted by the Reviewer from the manuscript, could be constructed. Unless such a construction is available, it is not at all obvious that such a definition can be meaningful from a mathematical or (geo-) physical point of view: what kind of time scales one refers to in the definition, what do they characterise, why is that characterisation relevant? The principal aim and achievement of our manuscript is filling this gap.

In terms of the stochastic description of the Reviewer, we agree that it is obvious how climate should be defined; however, the question is whether one has any a priori reason to assume that such a description can be realistic. In turn, we provide an a posteriori reason through an ab initio approach (according to the best of the scientific community's knowledge in terms of modelling the system in question).

We empasise that we do not claim that such a description *is* actually realistic, or, to be more precise, the desired separation of time scales exists. Our claim is that posing the question of this existence is meaningful; i.e., a well-defined climate *may* exist, although only if certain conditions are met.

While we do not actually answer whether these conditions are met, we make an "educated guess" that they do so, based on existing simulations of AOGCMs and comprehensive Earth system models. More importantly, we present a scheme for intialising a set of ensemble simulations to a posteriori assess the issue in any dynamical model of practical interest.

We reiterate from AC1 and AC3 that we do not need all of the mathematical details for our exposition, especially that the above-mentioned practical assessment (which seems to be the only viable one) does not rely on them. We prefer to add here that the targeted audience is the geoscientific community, writing papers like Hawkins et al. (2016), Deser et al. (2020; 2025), etc. cited in our manuscript, suggesting that they may probably be more comfortable with argumentations like those of our manuscript in its present form.

The claim of RC1 is nevertheless right that our "guess", as we see, for "what climate may be" is (more or less) the modus operandi just in the mentioned community. In order to make our aim and message clearer to them as well as to any interested party, we have chosen the new title to be

    "Can we define climate by means of an ensemble? A tale of time scales of convergence"

and also decided to formulate a new abstract which reads as follows:

    "It is hardly questioned today that climate can be described in theory by an ensemble of trajectories differing in their initial conditions, which is then translated to numerical ensembles in climate models. It is also widely accepted that any evolution observed within a few decades after initialization is not relevant to climate. Evolution at a later stage, instead, is then used to characterize climate and its change, under the implicit assumption that slower processes do not considerably contribute to differences between ensemble members, letting internal variability of climate be identified with these differences. However, a justification for this practice is as yet lacking. In particular, a definition of climate in support of this practice is outstanding, including the identification of the kind of time scales at play through providing an argumentation for their relevance. Our study aims at filling this gap. After pointing out that the most important criterion for a definition of climate is the uniqueness of the probability measure on which the definition relies, we first recall the naive proposal 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. We then consider the time scales of convergence and refine the proposal by taking a probability measure that is conditional on the (possibly time-evolving) state of modes characterized by convergence time scales longer than the horizon of a particular study. We design an ensemble simulation initialization scheme for studying convergence time scales and uniqueness of ensembles in Earth system models."

Along with the rearrangements anticipated in the previous ACs, we hope that these reformulations will facilitate a clear communication of the novelties in our work. Among others, we keep our promise to relegate the detailed discussion of the naive proposal to an appendix. We hereby thank once more for the Reviewer's helpful evaluation.