A low-dimensional dynamical systems approach for ensemble design and interpretation in climate science

de Melo Viríssimo, Francisco; Stainforth, David A.

doi:10.5194/egusphere-2026-4138

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https://doi.org/10.5194/egusphere-2026-4138

Preprints

24 Jul 2026

| 24 Jul 2026

A low-dimensional dynamical systems approach for ensemble design and interpretation in climate science

Francisco de Melo Viríssimo and David A. Stainforth

Abstract. The design and use of conceptual models have been a longstanding and successful practice of mathematicians and scientists to study and gain insights about complex phenomena in climate and beyond. Here, we demonstrate how low-dimensional dynamical systems can also be useful in the design and interpretation of climate model ensembles. We argue that, provided they possess a small number of key characteristics, such systems can serve as computationally inexpensive laboratories for investigating questions of ensemble methodology that would be difficult to explore systematically in comprehensive Earth System Models. To this end, we identify the characteristics required of informative low-dimensional systems, formalise a framework for their use, and illustrate the approach through a series of examples.

Received: 17 Jul 2026 – Discussion started: 24 Jul 2026

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Francisco de Melo Viríssimo and David A. Stainforth

Status: final response (author comments only)

RC1:
'Comment on egusphere-2026-4138', Anonymous Referee #1, 25 Aug 2026

The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2026/egusphere-2026-4138/egusphere-2026-4138-RC1-supplement.pdf

Citation: https://doi.org/10.5194/egusphere-2026-4138-RC1
- AC1:
  'Reply on RC1', Francisco de Melo Viríssimo, 20 Oct 2026
  Response to Reviewer #1
  We thank Reviewer #1 for taking the time to read our manuscript and provide detailed comments. We are grateful for the opportunity to clarify aspects of the manuscript that were not sufficiently clear in the original submission. We believe that the comments have usefully highlighted areas where the framing and positioning of the manuscript's original contributions, and its connection to the wider literature, can be strengthened. These revisions will help make the paper more accessible and relevant to both ESD readers and the broader climate science community.
  
  In response to the review, we will revise the manuscript to:
  
  1. further clarify the scope and purpose of the proposed framework;
  
  2. better distinguish our contribution from existing uses of low-dimensional systems in weather predictability and forecast research;
  
  3. expand discussion of relevant literature on short/mid-range ensemble prediction and dynamical systems;
  
  4. strengthen discussion of climate projection problems, particularly around external forcing, and non-autonomous systems;
  
  5. broaden discussion of related methodological approaches, including machine learning-based emulators;
  
  6. make the principal contributions of the framework more explicit throughout the manuscript.
  
  Below, we provide a point-by-point response to the reviewer's comments and outline the changes that will be incorporated into the revised manuscript in response to them.
  
  --------
  
  Part 1 - Overview:
  
  The authors present an argument that low dimensional conceptual models have not played a substantial role in the design and understanding of “how different experimental designs affect the information that can be extracted from systems possessing a number of key characteristics shared with climate models”.
  
  We thank the reviewer for engaging carefully with the manuscript.
  We believe there may be a misunderstanding regarding the central aim of the paper. Our intention is not to argue that low-dimensional dynamical systems have not contributed to ensemble research, nor that they have not previously been used to investigate questions relevant to prediction, predictability, or even uncertainty.
  Rather, the purpose of the paper is to formalise and articulate a particular role for low-dimensional dynamical systems as methodological laboratories for systematically investigating questions of ensemble design and interpretation as they may apply to large, complex models (e.g. Earth System Models). The paper further focuses on the use of such large models in understanding decadal to centennial climate change. The manuscript argues that certain questions can be systematically and efficiently explored in appropriately chosen low-dimensional systems before committing substantial computational resources to Earth System Model experiments.
  We recognise that this distinction was not made sufficiently clearly in the original submission and will revise the abstract and introduction to state this purpose more explicitly.
  
  To this end they propose a conceptual framework to guide ensemble design and interpretation in suitable low dimensional chaotic systems providing a brief illustration and review of their work using the five ODE Lorenz84-Stommel61 as a simple example to illustrate the basic properties of initialization and parameter uncertainty.
  
  We agree that the manuscript proposes a conceptual framework and illustrates it using examples derived largely from studies of the L84-S61 system.
  However, the intention of these examples is not simply to review previous work, but to demonstrate how the proposed framework can be applied in practice to investigate questions related to ensemble design and interpretation in the context of climate projections. The reason these examples feature prominently is that they were originally developed precisely to explore the methodological role for low-dimensional dynamical systems that the present manuscript seeks to formalise.
  We recognise that this broader purpose can be made clearer. In the revised manuscript we will strengthen the discussion of how the proposed framework can be applied beyond the L84-S61 system and clarify the extent to which its underlying principles are transferable to other low-dimensional dynamical systems possessing the characteristics identified in Section 2.
  Regarding the second point, we respectfully disagree with the characterisation of these examples as illustrations of only the "basic properties" of initialisation and parameter uncertainty. The studies discussed were not intended merely to demonstrate the existence of such uncertainties, which are of course well established. Rather, they were designed to investigate how different ensemble-design choices affect the information that can be inferred about system behaviour under conditions relevant to climate-model experimentation.
  More specifically, the examples explore questions concerning the interpretation of ensemble distributions, the consequences of different uncertainty representations, and the extent to which methodological choices influence scientific conclusions. These questions lie at the core of ensemble design and interpretation and motivate the framework proposed in the manuscript.
  We recognise, however, that these broader methodological insights were not emphasised sufficiently clearly in the original submission and will revise the manuscript to make them more explicit.
  
  --------
  Part 2 – Specific comments:
  
  The article is inadequate on several levels.
  
  We appreciate the reviewer's detailed assessment and address each of the specific comments below. We believe the revisions proposed in response will strengthen the manuscript and clarify its intended contribution.
  
  The title does not represent the article as neither theoretical nor computational methods are presented rather a descriptive program is offered.
  
  We thank the reviewer for this comment.
  We interpret this concern as relating to the nature of the contribution presented in the manuscript. The manuscript does not introduce a new numerical algorithm or mathematical theorem. Rather, it formalises a methodological framework for investigating questions of ensemble design and interpretation using appropriately selected low-dimensional dynamical systems.
  We consider this to constitute a methodological contribution, but we recognise that the current title and abstract may not sufficiently communicate the intended nature of the contribution to all readers. We will therefore revisit both and consider revisions that better reflect the conceptual and methodological focus of the manuscript.
  
  The basic premise that somehow the climate community is unaware of the key characteristics required to design appropriate ensemble is unsustainable. The four key characteristics cited are basic concepts one should expect any climate modeller to be fully aware of. The general thesis that somehow ensemble methodology has been neglected belies the deep connections between the dynamical systems, geophysical fluids and weather, seasonal and decadal prediction communities and the many examples illustrated using low order models.
  
  We agree with the reviewer that the characteristics discussed in the manuscript, including multiscale behaviour, multiple interacting components, nonlinearity, chaos and non-autonomous forcing, are well established within atmospheric and oceanic physics and climate science.
  Importantly, however, the manuscript does not intend to present these characteristics as novel discoveries, nor does it suggest that they are unknown within the climate modelling community. Likewise, we do not claim that ensemble methodology has been neglected.
  Rather, our argument is that these characteristics provide useful criteria for evaluating whether a low-dimensional system is suitable for investigating particular questions related to ensemble design and interpretation. The novelty lies not in the characteristics themselves, but in their use as a framework for identifying systems capable of supporting methodological investigations relevant to climate model ensembles.
  We recognise that this distinction was not sufficiently emphasised and will clarify it throughout the manuscript.
  We also agree that low-dimensional dynamical systems have played an important role in studies of predictability, ensemble forecasting and uncertainty quantification in atmospheric/weather prediction. A revised version of the manuscript will acknowledge these contributions more explicitly and discuss their relationship to the framework proposed here.
  
  The manuscript fails to adequately review the literature of ensemble prediction. Specifically, low dimensional models have been extensively employed to understand the general mathematical properties of error growth, projection onto various choices of dynamical vectors e.g., LVs, BVs, CLVs, FTNMs, SVs etc. Further, the emergence of local attractor states and characterization of local dimension underpins the understanding on which operational ensemble prediction systems across weather to decadal timescales are based. However, the article makes no mention of that body of work and the authors own cited work spans a highly limited set of chaotic low order models.
  
  We thank the reviewer for highlighting this body of work.
  We agree that the manuscript would benefit from a broader discussion of literature relating to ensemble prediction, predictability, error growth, dynamical vectors, attractor geometry and local dynamical properties.
  Many of the studies identified by the reviewer emerged primarily within the context of weather prediction and forecast systems, whereas our primary focus is on questions of ensemble design and interpretation in Earth System Model applications and multi-decadal climate projections. Nevertheless, we recognise that there are important conceptual links between these areas and that these links are currently not developed in the manuscript.
  In the revised version we will expand the literature review accordingly and clarify both the connections and distinctions between existing predictability-focused uses of low-dimensional systems and the framework proposed here.
  We note that this revision will also address a related recommendation made by Reviewer #2 regarding links between weather forecasting and climate ensemble methodologies.
  
  There are now a growing number of machine learning approaches to developing stable climate emulators that can effectively learn pde’s of the slow manifold enabling cost effective generation of huge ensembles. Examples include neural operator [1] and transformer [2] architectures, or alternate approaches capable of simulating the climate response for any given bespoke emission pathway [3]. None are mentioned. No mention is made of the many applications of data driven and optimization methods for dimension reduction in high dimensional non-autonomous flows to specifically separate physical modes on the slow manifold from the fast decaying modes e.g., [4,5].
  
  We thank the reviewer for drawing our attention to these developments and references.
  We agree that recent progress in machine learning-based climate emulation represents an important methodological development that should be acknowledged.
  At the same time, we view these approaches as addressing a different objective from that considered here. Climate emulators are generally designed to reproduce aspects of climate model behaviour at reduced computational cost, thereby enabling large numbers of simulations. By contrast, our focus is on the use of simplified dynamical systems as conceptual laboratories for investigating methodological questions related to ensemble design and interpretation.
  We will add discussion of these emerging approaches and clarify the relationship between emulator-based and low-dimensional system-based strategies for supporting climate model experimentation.
  
  As distinct from initial condition uncertainty, climate is primarily a boundary value problem. Once errors from uncertainties in the initial conditions saturate and phase information is lost, it is uncertainties in boundary conditions and a shift in focus to the systems responses to forcing specifications that becomes the focus i.e., prediction – to – projection. This is barely discussed with no representative example given.
  
  We agree that externally forced responses and boundary condition uncertainty play a central role in climate projections and that this deserves greater emphasis in the manuscript.
  Our intention was not to suggest otherwise. Indeed, the discussion of non-autonomous systems and Example 1 were intended in part to illustrate circumstances in which initial condition uncertainty can remain relevant on climatically significant timescales, while recognising that projected responses to forcing become increasingly important as timescales increase.
  More specifically, Example 1 was included to demonstrate that the influence of initial-condition uncertainty need not become negligible immediately after the loss of phase information in the atmosphere. The results indicate that different forms of macro-scale initial condition uncertainty, including those arising from spin-up procedures or observational initialisations, can continue to affect projected system behaviour over multi-decadal timescales. This finding is consistent with earlier studies e.g. Hawkins et al. (2016), and has recently received further support from work such as Deser et al. (2025). These results suggest that initial condition uncertainty may remain relevant to climate projections for considerably longer than is often assumed, highlighting the importance of careful ensemble design when seeking to characterise and interpret projected climate behaviour. They also help identify which ensemble design choices are likely to be most effective in capturing these uncertainties, while illustrating the risk of underestimating uncertainty when macro initial condition variability is neglected or insufficiently sampled.
  Nevertheless, we acknowledge that the balance of discussion in the manuscript currently places greater emphasis on initial condition and parametric uncertainty than on forced responses. In response, we will expand discussion of projection problems, non-autonomous forcing and the transition from prediction-focused to projection-focused questions.
  
  Mostly the article is a review of the authors own work – much of which is restricted to the L84-S61 system of 5 ODEs - but there is a general paucity of new insights or methodological advances.
  
  We respectfully disagree with the characterisation of the manuscript as primarily a review of our own work.
  The manuscript cites more than 60 references spanning climate science, uncertainty quantification, dynamical systems, ensemble methodologies and conceptual modelling, of which only a minority (13) involve one or more of the present authors.
  We recognise, however, that the illustrative examples draw heavily on our own previous studies. This reflects the fact that these studies were specifically developed to investigate aspects of the framework that the present manuscript seeks to formalise, as explained extensively above.
  To address the reviewer's concern, we will broaden discussion of related examples from the wider literature and make clearer how the framework can be applied beyond the specific L84-S61 system.
  With regard to the comment concerning new insights and methodological advances, we will also revise the introduction and conclusions to better articulate the contributions of the manuscript. These include:
  Formalising a framework in which low-dimensional dynamical systems can be used as methodological laboratories for ensemble design and interpretation rather than solely as simplified representations of the climate system;
  
  Identifying a set of characteristics that can be used as practical criteria for evaluating the suitability of low-dimensional systems for methodology-focused ensemble research;
  
  Demonstrating how questions concerning initial condition uncertainty, parametric uncertainty, ensemble structure and experiment design can be investigated within a common conceptual framework;
  
  Providing a rationale for using computationally inexpensive dynamical systems to explore methodological questions prior to undertaking costly Earth System Model experiments.
  
  We will make these contributions more explicit throughout the revised manuscript.
  
  To conclude, we thank Reviewer #1 again for their thoughtful and detailed comments. While we do not agree with all aspects of the reviewer's interpretation of the manuscript, we believe the comments have identified several areas where the scope, positioning, motivation and justification of the work can be clarified and strengthened. We are grateful for the opportunity to improve the paper and believe the proposed revisions will make its contribution clearer and more useful to the broader climate modelling community.
  
  References:
  Deser, C., Kim, W. M., Wills, R. C. J., Simpson, I. R., Yeager, S., Danabasoglu, G., Rodgers, K., and Rosenbloom, N.: Effects of Macro vs. Micro Initialization and Ocean Initial-Condition Memory on the Evolution of Ensemble Spread in the CESM2 Large Ensemble, Climate Dynamics, 63, 1–21, https://doi.org/10.1007/s00382-024-07553-z, 2025.
  Hawkins, E., Smith, R. S., Gregory, J. M., and Stainforth, D. A.: Irreducible uncertainty in near-term climate projections, Climate Dynamics, 46, 3807–3819, https://doi.org/10.1007/s00382-015-2806-8, 2016.
  
  Citation: https://doi.org/10.5194/egusphere-2026-4138-AC1
RC2:
'Comment on egusphere-2026-4138', Anonymous Referee #2, 20 Sep 2026
This manuscript presents a new conceptual framework for using low-dimensional dynamical systems as computational tools for the design and interpretation of climate model ensembles. The authors argue that low-dimensional systems possessing a small set of key characteristics shared with climate models (complexity, multiscale behaviour, multiple components, nonlinearity/chaos, and non-autonomy) can provide valuable insights into ensemble methodology that would otherwise be difficult to obtain using computationally expensive ESMs. The framework is illustrated through a series of examples drawn from previous studies of initial condition uncertainty, parametric uncertainty, and ensemble experimentation tools.
Overall, I found this to be a thoughtful, well-written, and timely contribution. Climate science increasingly relies on ensembles, while at the same time the costs of comprehensive climate model experiments continue to grow. The manuscript addresses an important methodological gap by proposing a clear rationale for investigating ensemble design questions in simplified systems before committing substantial computational resources to ESM experiments. The paper is accessible, logically structured, and succeeds in articulating a compelling vision for an expanded role of conceptual models in climate science.
The principal strength of the paper is its original perspective. Low-dimensional dynamical systems have long been used to understand climate dynamics, predictability, and nonlinear behaviour, but the authors convincingly argue for an additional role: understanding ensemble methodologies themselves rather than the climate system directly. This distinction is clearly articulated and provides a useful conceptual contribution that is likely to resonate with both dynamical systems researchers and climate modellers.
A second strength is the clear identification of the key characteristics that make climate-model ensembles challenging to design and interpret. The discussion of complexity, multiscale dynamics, coupled components, chaos, and non-autonomous forcing provides a useful organizing framework for assessing whether a low-dimensional system is suitable for a particular methodological question.
Rather than remaining purely conceptual, the manuscript demonstrates how the proposed framework has already been applied in practice to questions of initialization, uncertainty characterization, and ensemble construction. These examples help convince the reader that the approach is not merely philosophical but can provide practical guidance for climate-model experimentation.
I have only a few suggestions that could further strengthen the paper:
Clarify the limits of transferability. While the manuscript appropriately emphasizes that low-dimensional systems should not be interpreted as quantitative surrogates for ESMs, the discussion could benefit from a slightly more explicit account of how researchers might assess whether a lesson learned in a conceptual model is likely to transfer to a high-dimensional climate model. A few concrete criteria or examples would help readers apply the framework.

Strengthen the connection to contemporary large-ensemble practice. The paper discusses ensemble design broadly, but a brief discussion of how the proposed framework could specifically inform modern large-ensemble initiatives, perturbed-parameter ensembles, or km-scale modelling efforts would increase its relevance for a wider climate-modelling audience.

Links with weather forecasting. The authors may wish to briefly discuss whether ensemble design techniques developed in probabilistic weather forecasting can inform climate ensemble design. Although weather and climate predictions are affected by different uncertainty sources and timescales, both face the challenge of constructing informative ensembles. A short discussion of which methods or principles may transfer, and how low-dimensional dynamical systems could help evaluate their applicability to climate projections, would broaden the paper's methodological relevance.

In summary, this paper makes a valuable methodological contribution by formalizing a framework through which low-dimensional dynamical systems can be used to investigate ensemble design and interpretation. The ideas are thoughtful, well-motivated, and clearly communicated. I believe the manuscript will be of interest to researchers working in climate modelling, uncertainty quantification, ensemble prediction, and nonlinear dynamical systems.
Citation: https://doi.org/10.5194/egusphere-2026-4138-RC2
- AC2:
  'Reply on RC2', Francisco de Melo Viríssimo, 20 Oct 2026
  Response to Reviewer #2
  
  We thank Reviewer #2 for taking the time to read our manuscript and for providing such a careful and constructive review. We are pleased that the reviewer found our contribution useful, timely and clearly communicated, and that the proposed approach for using low-dimensional dynamical systems to investigate ensemble methodologies was well received. We believe that the suggestions provided will strengthen the manuscript further and make it more relevant to ESD readers and the broader climate science community.
  
  Below, we provide a point-by-point response to the reviewer's comments and outline the revisions that will be incorporated into the revised manuscript.
  
  ----------
  Part 1 – Overview
  
  This manuscript presents a new conceptual framework for using low-dimensional dynamical systems as computational tools for the design and interpretation of climate model ensembles. The authors argue that low-dimensional systems possessing a small set of key characteristics shared with climate models (complexity, multiscale behaviour, multiple components, nonlinearity/chaos, and non-autonomy) can provide valuable insights into ensemble methodology that would otherwise be difficult to obtain using computationally expensive ESMs. The framework is illustrated through a series of examples drawn from previous studies of initial condition uncertainty, parametric uncertainty, and ensemble experimentation tools.
  
  Thank you for this comment. We are pleased that the purpose and scope of the manuscript were communicated clearly. This summary captures accurately the central argument we sought to make regarding the role of suitably chosen low-dimensional dynamical systems in supporting climate ensemble design and interpretation research.
  
  Overall, I found this to be a thoughtful, well-written, and timely contribution. Climate science increasingly relies on ensembles, while at the same time the costs of comprehensive climate model experiments continue to grow. The manuscript addresses an important methodological gap by proposing a clear rationale for investigating ensemble design questions in simplified systems before committing substantial computational resources to ESM experiments. The paper is accessible, logically structured, and succeeds in articulating a compelling vision for an expanded role of conceptual models in climate science.
  
  Thank you for these encouraging comments. A central motivation for the manuscript is the increasing reliance on large and computationally expensive ensemble experiments across climate science. We are therefore pleased that the reviewer considers the proposed framework a potentially useful way of investigating methodological questions before committing substantial Earth System Model resources.
  
  The principal strength of the paper is its original perspective. Low-dimensional dynamical systems have long been used to understand climate dynamics, predictability, and nonlinear behaviour, but the authors convincingly argue for an additional role: understanding ensemble methodologies themselves rather than the climate system directly. This distinction is clearly articulated and provides a useful conceptual contribution that is likely to resonate with both dynamical systems researchers and climate modellers.
  
  Thank you for this observation. We are particularly pleased that the reviewer recognised the distinction between using low-dimensional dynamical systems to investigate the behaviour of the climate system and using them to investigate ensemble methodologies themselves. This distinction is central to the manuscript and will be emphasised further in the revised version, also in response to Reviewer #1’s feedback.
  
  A second strength is the clear identification of the key characteristics that make climate-model ensembles challenging to design and interpret. The discussion of complexity, multiscale dynamics, coupled components, chaos, and non-autonomous forcing provides a useful organizing framework for assessing whether a low-dimensional system is suitable for a particular methodological question.
  
  Thank you. We are pleased that the reviewer found this organisational framework strong and useful. While the characteristics themselves are well established within climate science, one of our aims was to make explicit their role as practical criteria for assessing whether a low-dimensional system is suitable for investigating particular ensemble design and interpretation questions. We will further clarify this distinction in the revised manuscript.
  
  Rather than remaining purely conceptual, the manuscript demonstrates how the proposed framework has already been applied in practice to questions of initialization, uncertainty characterization, and ensemble construction. These examples help convince the reader that the approach is not merely philosophical but can provide practical guidance for climate-model experimentation.
  
  Thank you. One of our concerns when preparing the manuscript was that the framework might be perceived as purely conceptual. We are therefore pleased that the examples were viewed as demonstrating practical applications of the approach and helping to illustrate its relevance to climate model experimentation. But we also believe that this aspect can be strengthened further. So in line with comments from Reviewer #1, we will expand the discussion of the examples and make their methodological implications, particularly for ensemble design and interpretation, more explicit in the revised manuscript.
  ----------
  Part 2 – Specific Comments
  
  I have only a few suggestions that could further strengthen the paper:
  
  Clarify the limits of transferability. While the manuscript appropriately emphasizes that low-dimensional systems should not be interpreted as quantitative surrogates for ESMs, the discussion could benefit from a slightly more explicit account of how researchers might assess whether a lesson learned in a conceptual model is likely to transfer to a high-dimensional climate model. A few concrete criteria or examples would help readers apply the framework.
  
  Thank you for this suggestion. We agree that the question of transferability is important and deserves further discussion.
  In the revised manuscript, we will expand the discussion of transferability and clarify the circumstances under which insights obtained from low-dimensional systems might reasonably be expected to inform ESM experimentation. In particular, we will emphasise that transferability should not be assumed automatically and will depend on the extent to which the mechanisms explored in the low-dimensional system reflect features relevant to the climate model application of interest. We will relate that to the examples provided to help readers apply this reasoning in practice.
  For example, findings concerning the consequences of different approaches to sampling macro-scale initial condition uncertainty may be transferable because they depend primarily on the coexistence of multiple distinctive timescales and interacting components, rather than on the detailed representation of any specific climate process. In such cases, the methodological question being investigated may depend more strongly on the structure of the system and the representation of uncertainty than on the precise level of detail of the physical processes included in the model.
  
  Strengthen the connection to contemporary large-ensemble practice. The paper discusses ensemble design broadly, but a brief discussion of how the proposed framework could specifically inform modern large-ensemble initiatives, perturbed-parameter ensembles, or km-scale modelling efforts would increase its relevance for a wider climate-modelling audience.
  
  Thank you for this suggestion. We agree that strengthening the connection with contemporary ensemble practices would improve the manuscript's relevance for a broader climate modelling audience.
  While we intentionally kept the original manuscript concise, we recognise the value of explicitly discussing how the proposed framework may inform questions arising in contemporary large ensemble initiatives, perturbed-parameter ensembles, high-resolution modelling efforts, and related experimental programmes.
  In the revised manuscript, we will add a dedicated discussion examining these connections and illustrating how low-dimensional dynamical systems may be used to explore questions of experiment design, uncertainty sampling, and ensemble interpretation in such contexts.
  
  Links with weather forecasting. The authors may wish to briefly discuss whether ensemble design techniques developed in probabilistic weather forecasting can inform climate ensemble design. Although weather and climate predictions are affected by different uncertainty sources and timescales, both face the challenge of constructing informative ensembles. A short discussion of which methods or principles may transfer, and how low-dimensional dynamical systems could help evaluate their applicability to climate projections, would broaden the paper’s methodological relevance.
  
  Thank you for this suggestion, also brought up by Reviewer #1. Although weather forecasting and climate projection differ widely in many aspects, including in their dominant sources of uncertainty and relevant timescales, both face common challenges concerning ensemble construction, uncertainty representation, quantification and interpretation. We therefore agree that a discussion of the conceptual connections between ensemble methodologies developed within weather forecasting and those used in climate science would add value to the manuscript.
  In the revised version, we will include a discussion of relevant ensemble design approaches developed within the weather prediction community, including those mentioned by Reviewer #1, and consider how low-dimensional dynamical systems may be used to explore their applicability, as well as strengths and limitations within climate projection contexts.
  
  In summary, this paper makes a valuable methodological contribution by formalizing a framework through which low-dimensional dynamical systems can be used to investigate ensemble design and interpretation. The ideas are thoughtful, well-motivated, and clearly communicated. I believe the manuscript will be of interest to researchers working in climate modelling, uncertainty quantification, ensemble prediction, and nonlinear dynamical systems.
  
  We thank Reviewer #2 again for their careful reading of the manuscript, for their constructive suggestions, and for the encouraging assessment of the work. We are pleased that the central motivation and intended contribution of the paper were clearly communicated, and we believe that the suggested revisions will strengthen the manuscript further – especially after complemented by those provided by Reviewer #1. We are grateful for the reviewer's time and thoughtful engagement with the work.
  
  Citation: https://doi.org/10.5194/egusphere-2026-4138-AC2

Francisco de Melo Viríssimo and David A. Stainforth

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Short summary

This study shows that conceptual mathematical representations of complex climate models can help improve the way climate simulations are designed and understood. We identify the essential features these simplified systems need, propose a clear approach for using them, and demonstrate it through practical examples. Because they require far less computing power than full climate models, they provide an efficient way to test methods and gain insights that would otherwise be difficult to obtain.


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