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.

Earth system models (ESMs) are widely used to make projections of the future behavior of Earth's climate in the context of anthropogenic climate change. Setting aside uncertainties stemming from the design and implementation of the model, there, nevertheless, remain substantial uncertainties with such projections. Two important ones arise from uncertainties in (i) the initial conditions and (ii) the values of parameters within the model. Here, we systematically investigate the latter: the consequences of parametric uncertainty, as might be explored by perturbed parameter ensembles. Utilizing a low-dimensional system with key characteristics of a climate model, we examine two types of parametric uncertainty through a large ensemble approach. The first, microparametric uncertainty, is akin to microinitial condition uncertainty and explores a situation where one knows the relevant parameter values well but not perfectly. The second, macroparametric uncertainty, explores the situation where there may be substantial uncertainty in parameter values. We also investigate how they interact with each other and with microinitial condition uncertainty. In general, we find that microparametric uncertainty can lead to a much broader range of states than in initial condition ensembles, with the resulting standard deviations being over 2.5-3.5 times higher for slow-and fast-mixing variables alike. Additionally, we show that the scale of the effect may be even larger with macroparametric uncertainty. Finally, we discuss the implications for ensemble design and interpretation and particularly how these results indicate the need for more complex ensemble designs when making projections of climate change within ESMs. SIGNIFICANCE STATEMENT: This study presents a systematic discussion of the sources and consequences of parametric uncertainty in climate models. Two types of parametric uncertainty are found depending on the source and magnitude of uncertainty: a micro-and macrouncertainty, with the latter being subdivided into two types. While parametric ensembles are generally found to quantify a broader range of plausible states and behavior than initial condition ensembles, the results are varied and dependent on multiple factors, such as the number of parameters perturbed and the output variable of interest. Together, these results shed some light on how to better design informative climate model ensembles, particularly given the computational power demands of running such models.

Much effort goes into studying the causes of systematic errors in Earth System Models (ESMs). Reducing them is often seen as a high priority. Indeed, the development of Digital Twin approaches in climate research is founded on the idea that a sufficiently good model would be able to provide reliable and robust, conditional predictions of climate change (predictions conditioned on scenarios of future greenhouse gas emissions). Here, “reliable” encapsulates the idea that the predictions are suitable for use by society in anticipating and planning for future climate change, and “robust” encapsulates the idea that they are unlikely to change as the models are improved and developed.Such an approach, however, begs the question, when is a model sufficiently realistic to be able to provide reliable, detailed predictions? A physical processes view of current ESMs suggests that they are not close to this level of realism while a nonlinear dynamical systems perspective raises questions over whether it will ever be possible to achieve such reliability for the types of regionally-specific, extrapolatory, climate change predictions that we may think society seeks.Given this context, multi-model and perturbed-physics ensembles are often seen as a means to quantify uncertainty in conditional, climate change predictions (commonly referred to as “projections” in the scientific community). In the IPCC atlas (https://interactive-atlas.ipcc.ch/) the most easily accessible output is the multi-model median with the 10th, 25th, 75th and 90th percentiles of the multi-model distribution also prominent. This presentation in terms of probabilities implies that the probabilities themselves have meaning to the users of the data - most users are likely to take them as probabilities of different outcomes in reality. Unfortunately multi-model ensembles cannot be interpreted that way because we have no metric for the shape of model space nor any idea of how to explore it, so the ensemble members cannot be taken as independent samples of possible models. Perturbed-parameter ensembles work in a more defined space of possible model-versions but the shape of that space is also undefined and as a result the ensemble-based probabilities are again arbitrary.When seeking the best possible information for society, multi-model and perturbed physics ensembles would benefit from targeting diversity: the greatest possible range of responses given a particular model structure. Model emulators could be used to systematise this process. Such an approach would provide more reliable information. It changes the question, however, from “when is a model sufficiently realistic” to “how unrealistic does a model have to be to be uninformative about extrapolatory future climatic behaviour?”In this presentation I will discuss and elaborate on these issues. References:Stainforth, D., “What we do with what we’ve got”, Chapter 21 in “Predicting Our Climate Future: What we know, what we don’t know and what we can’t know”, Oxford University Press, 2023.Stainforth, D.A. et al., Confidence, uncertainty and decision-support relevance in climate predictions, Phil.Trans.Roy.Soc., 2007.Stainforth, D.A. et al., Issues in the interpretation of climate model ensembles to inform decisions, Phil.Trans.Roy.Soc., 2007.

When it comes to communicating climate change, both our understanding of what we don’t know and the uncertainties in the science are themselves core elements of our knowledge. That’s to say, what we know about uncertainty is part of what we know. Failing to communicate uncertainty and the limits of our understanding is failing to communicate the full picture of climate change.In 2023, after many years of writing, my book, “Predicting Our Climate Future: What we know, what we don’t know, and what we can’t know”, came out. The book is targeted at a public audience and addresses the many exciting, deep, conceptual and practical challenges that we face in climate change science and climate change social science. It aims to show that there are fundamental questions here that are simply fascinating in themselves: intrinsically interesting irrespective of the social relevance of the research.In doing this it has to shine a spotlight on the many things that we don’t know - particularly our limited ability to describe the climate of the future at local scales, and the consequences of climate change for the societies in which we live. Some might be concerned that doing this could undermine trust in climate science and work against our ability to tackle climate change. In practice the opposite is true. Acknowledging and presenting the limits of our knowledge upfront, increases the credibility of climate change information. It also provides a handle for people and diverse disciplines to actively engage with climate science and to bring their values and attitudes to risk into the debate.Of course it is also important to be clear about what we do know: what really isn’t open to debate and why. Here I will discuss how I approach this balancing act between communicating the exciting aspects of what we don’t know while being clear about what we do. I will also discuss my experience of presenting these issues to public, academic and business audiences. Further materials:Stainforth, D., “Predicting Our Climate Future: What we know, what we don’t know and what we can’t know”, Oxford University Press, 2023.(https://global.oup.com/academic/product/predicting-our-climate-future-9780198812937)Stainforth, D.A. The big idea: can we predict the climate of the future?, The Guardian, 30th Sept 2023(https://www.theguardian.com/books/2023/oct/02/the-big-idea-can-we-predict-the-climate-of-the-future)Podcast: Instant Genius - Can we predict the climate of the future?Podcast: Challenging Climate - Models and uncertaintyPodcast: Empty Space Inbetween - In conversation with David Stainforth

The consequences of climate change are often conceptualised in terms of the changing risks of natural disasters, or as reductions in future economic output. When understood in these terms, it is all too easy to believe that one might “get lucky”—that the floods won’t affect those of us who don’t live by the waterfront, and that the heatwaves won’t affect the salaries or job security of those of us who go to work in air conditioned offices. The consequences of climate change, serious though they may be, seem far away. Contrary to this perspective, we argue that changing risk profiles, even marginal or distant changes, are likely to strain the underlying fabric of societies, and thus have profound consequences for everyone. Even for individuals who are relatively insulated from the direct physical consequences of climate change, it may well be that there is little chance of “getting lucky.” This has important implications for how we perceive and assess the benefits of climate action. We therefore call for greater efforts to understand the system-wide social consequences of increasing disaster risks.

We summarise the contributions to the Topical Collection on quality of climate information for adaptation decision support. Based on these contributions, we draw some further lessons for the development of high-quality climate information and services, bridging between a “credibility-first” paradigm (exemplified by top-down information provision from systematic downscaling or impact projections) and a “salience-first” paradigm (exemplified by user-led tailored information products or storylines) by looking to identify their respective strengths and use cases. We emphasise that a more nuanced collective understanding of the dimensions of information quality in climate information and services would be beneficial to users and providers and ultimately support more confident and effective climate adaptation decisions and policy-making.

Modern research in climate science relies heavily on Earth System Models (ESMs), which are highly complex and nonlinear mathematical representations of the planet Earth. Because ESMs are too complicated to be tractable analytically, one must resort to computers in order to extract useful information from them. Numerically solving these models is, however, a non-trivial task, with two key practical consequences. First, being complex and nonlinear, ESMs require the use of several numerical schemes and involve different computational strategies to accommodate for the multiplicity of scales and components present in such models. These numerical choices vary from model to model and are also dependent on the user needs, as well as the computing resources available to them. Second, being chaotic means that its initial value problem is sensitive to the finest details in the model (e.g. initial condition), and so any future state can only be characterised as a distribution, requiring an ensemble of simulations instead of a single one. Hence, different numerical approximations could in principle lead to differences in the resulting distributions, particularly in shape and extreme values. This is important, as such numerical scheme dependence can cause ambiguity in our interpretation of future climate within a model. Despite that, this issue has been largely neglected in both climate and mathematical literature. In this presentation, we will briefly review and discuss the use of numerical schemes in ESMs and individual components. Using a conceptual but low-dimensional representation of the climate system [1,2], we will then present a systematic study of how different numerical methods can indeed change the resulting climate distribution. In addition to the uncertainties from initial condition, parameter and model formulation, our results suggest a fourth level of “structural” uncertainty in climate modelling – in the numerical (or computational) implementation. This has implications to the design and interpretation of climate ensembles, suggesting that the computational formulation must be accounted as a source of uncertainty when producing climate distributions, particularly for high-end stakeholders such as decision makers. References: [1] de Melo Viríssimo, F. and Stainforth, D.: A low-dimensional dynamical systems approach to climate ensemble design and interpretation, EGU General Assembly 2023, Vienna, Austria, 24–28 Apr 2023, EGU23-14755, https://doi.org/10.5194/egusphere-egu23-14755, 2023. [2] de Melo Viríssimo, F., Stainforth, D. A. and Bröcker, J.: The evolution of a non-autonomous chaotic system under non-periodic forcing: a climate change example, Chaos, 34, 2024, http://doi.org/10.1063/5.0180870, 2024

We first review the way in which Hasselmann’s paradigm, introduced in 1976 and recently honored with the Nobel Prize, can, like many key innovations in complexity science, be understood on several different levels. It can be seen as a way to add variability into the pioneering energy balance models (EBMs) of Budyko and Sellers. On a more abstract level, however, it used the original stochastic mathematical model of Brownian motion to provide a conceptual superstructure to link slow climate variability to fast weather fluctuations, in a context broader than EBMs, and led Hasselmann to posit a need for negative feedback in climate modeling. Hasselmann’s paradigm has still much to offer us, but naturally, since the 1970s, a number of newer developments have built on his pioneering ideas. One important one has been the development of a rigorous mathematical hierarchy that embeds Hasselmann-type models in the more comprehensive Mori–Zwanzig generalized Langevin equation (GLE) framework. Another has been the interest in stochastic EBMs with a memory that has slower decay and, thus, longer range than the exponential form seen in his EBMs. In this paper, we argue that the Mori–Kubo overdamped GLE, as widely used in statistical mechanics, suggests the form of a relatively simple stochastic EBM with memory for the global temperature anomaly. We also explore how this EBM relates to Lovejoy et al.’s fractional energy balance equation.

In this article, we approach the problem of measuring and interpreting the mid-term climate of a non-autonomous chaotic dynamical system in the context of climate modeling. To do so, we use a low-dimensional, conceptual model for the Earth system with different timescales of variability and subjected to non-periodic external forcing. We introduce the concepts of an evolution set and its distribution, which are dependent on the starting state of the system, and explore their links to different types of initial condition uncertainty and the rate of external forcing. We define the convergence time as the time that it takes for the evolution distribution of one of the dependent variables to lose memory of its initial condition. We suspect a connection between convergence times and the classical concept of mixing times, but the precise nature of this connection needs to be explored. These results have implications for the design of influential climate and Earth system model ensembles and raise a number of issues of mathematical interest.

<p>Climate change is a complex, multidisciplinary problem which relates our physical understanding of the consequences of greenhouse gas emissions with economic and socio-political actions to mitigate and adapt to those consequences. An important role that the mathematics of climate change can play involves utilising and developing understanding of nonlinear systems in such a way as to guide the design of ensembles of Global Climate and Earth System Models (ESMs), as well as integrated assessment and economic models. To this end it is informative to view these computer models as high-dimensional nonlinear systems and ask what we can learn about ensemble design from somewhat related, low-dimensional nonlinear systems.</p> <p> </p> <p>This talk will discuss what it means to make a prediction of climate change within a computer model as well as how we can design ensembles to reflect our uncertainty in the real-world, physical climate system. The Lorenz ’84/Stommel ’61 (L84-S61) system will be introduced as a valuable tool for studying issues of ensemble design and will be used to illustrate key sources of uncertainty and sensitivity.</p> <p> </p> <p>First amongst these senstitivities is initial value sensitivity of the sort explored in a variety of single model large ensembles (see session CL4.10/NH11/OS4) - these are known as micro-initial-condition ensembles. However, the results of such ensembles can themselves be dependent on large scale features of the starting conditions - so-called macro-initial-condition uncertainty. Lastly, the sensitivity of ensemble results to model structure and parameter value selection is crucial. How can we identify how close to the target system a model has to be to make useful probabilistic forecasts at different lead times? This question raises the prospect that climate predictions could be vulnerable to the “hawkmoth effect” - the potential for probabilistic forecasts based on initial condition ensembles to be highly sensitive to the finest details of model formulation.</p> <p> </p> <p>Here the different types of initial value and model parameter sensitivities will be illustrated with the L84-S61 system. Based on these, a series of design questions will be raised - questions which suitably-designed ensembles of low-dimensional systems could help us understand and answer, and which could be extremely valuable in improving the design of ensembles of GCMs and ESMs.</p>

<p>Earth System Models (ESMs) are complex, highly nonlinear, multi-component systems described by large number of differential equations. They are used to study the evolution of climate and its dynamics, and to make projection of future climate at both regional and global levels – which underpins climate change impact assessments such as the IPCC report. These projections are subject to several sorts of uncertainty due to high internal variability in the system dynamics, which are usually quantified via ensembles of simulations.</p> <p>Due to their multi component nature of such ESMs, the emerging dynamics also contain different temporal scales, meaning that climate ensembles come in a variety of shapes and sizes. However, our ability to run such ensembles is usually constrained by the computational resources available, as they are very expensive to run. Hence, choices on the ensemble design must be made, which conciliate the computational capability with the sort of information one is looking for.</p> <p>One alternative to gain information is to use low-dimensional climate-like systems, which consists of simplified, coupled versions of atmosphere, ocean, and other components, and hence capture some of the different time scales present in ESMs. This approach allows one to run very large ensembles, and hence to explore all sorts of model uncertainty with only modest computational usage.</p> <p>In this talk, we discuss this approach in detail, and illustrate its applicability with a few results. Particular attention will be given to the issues of micro and macro initial condition uncertainty, and parametric uncertainty – including external, anthropogenic-like forcing. The ability of large ensembles to constrain decadal to centennial projections will be also explored.</p>

Limiting global warming to 1.5$^\circ$C will very likely require, or to 2$^\circ$C may require, large-scale removal of carbon dioxide (CO$_2$) from the atmosphere. Many CO$_2$ removal strategies (CDRSs), or negative emissions technologies, have been proposed, which vary widely in both price per ton of CO$_2$ removed and storage timescale of this removed CO$_2$, as well as mechanism, maturity, scalability, and other factors. It has not yet been assessed whether the benefits, in terms of climate change-related damages avoided, of CDRSs' deployment exceed their costs at current reported prices and storage timescales, nor what cost is required for a CDRS with a given storage timescale to provide net benefits, nor how these depend on socioeconomic assumptions. For a long-storage-timescale CDRS, these questions reduce to whether its price is lower than the social cost of carbon, but for CDRSs with shorter storage timescales, they may also depend on its storage timescale. We show that for CDRSs with reported storage timescales from decades to centuries, the benefits of their deployment outweigh their reported costs under middle-of-the-road socioeconomic assumptions. For some, their benefits still outweigh their costs under optimistic socioeconomic assumptions. These CDRSs' associated benefit-cost ratios vary by more than an order of magnitude, and are strongly influenced by both price and storage timescale. The price threshold where a CDRS yields net benefits depends strongly on storage timescale, particularly for storage timescales $\leq$50 years. Our results provide a framework to assess and compare different CDRSs quantitatively for future CDRSs research, development, and policy.

<p>Low-order coupled models of the atmosphere and ocean can illuminate the role of weather-climate interactions in long-term climate prediction. An important example is models that bring together the interplay between the Atlantic meridional overturning circulation (AMOC) with mid-latitude quasi-geostrophic dynamics of the atmosphere, as in the coupled model introduced by Van Veen et al (2001). In such models, the AMOC can transition from its present thermally driven to a much weaker salinity-driven state, through a tipping point. We show using these coupled models that, for scenarios with intermediate forcing between a strong and weak circulation, the long-term evolution shows extreme sensitivity to initial conditions, due to the appearance of riddled basins of attraction. The literature on dynamical systems has extensively examined such dynamics when two distinct basins of attraction are riddled, that is any small part of one attractor’s basin also includes a piece of the other. Moreover, in the presence of feedback from the atmosphere to the ocean, initial atmospheric conditions are amplified to the extent that long-term prediction in these models is inhibited by the finite precision at which the atmospheric state is known. We propose to describe the various facets of this phenomenon and consider the lessons for understanding and predicting long-term climate (in our case, thermohaline circulation), given initial state uncertainty. Furthermore, the resulting challenges of long-term prediction are not necessarily ameliorated by the real-world asymmetries in the model. When the relevant symmetries that yield riddled basins are broken through perturbations to the vector fields, the asymptotic dynamics become perfectly predictable given the initial conditions; however, long-term uncertainties in the transient state (strong vs weak circulation) persist for centuries, owing to ocean timescales.</p> <p> </p>

<p>Twenty years ago in 2003 the climateprediction.net project was launched. It gave members of the public the opportunity to engage in climate modelling and climate prediction by downloading a comprehensive climate model and running it on their PCs. Participants contributed their results to a large perturbed-parameter ensemble and thus supported an exploration of uncertainty in climate projections. What the project did not do was give the participants much opportunity for participating in the experimental design or data analysis.</p> <p> </p> <p>Nowadays the questions regarding uncertainty in model-based predictions remain. Unlike twenty years ago, however,  there are many more individuals in our societies who have skills in computing, statistics, physics, geophysics etc. and who have an interest in research but are not part of the research community and don’t want a career in academia. Here I will present a potential project to engage such individuals in exploring and quantifying uncertainty in real-world extrapolatory forecasts of the climate system - that’s to say of climate change. Key to this would be the use of a range of simple, low-dimensional stochastic models founded on the Hasselmann model. Participants would be asked to both code and run ensembles of various versions of the model to explore physical science uncertainties in feedback processes, ocean heat uptake, the scale and type of the stochastic forcing, and even the structure of the model. They would participate in a collection of standardised experiments - common across multiple individuals - to allow for verification of results but they would also be encouraged to run their own experiments and to propose extensions to the main project in collaborative teams.</p> <p> </p> <p>Such a project would provide a route to enable skilled and interested individuals throughout society to participate in climate research and also to contribute to the wider communication and understanding of the climate prediction and uncertainty quantification problems. This proposal is for a citizen science project that takes scientific engagement to a new level - a project that enables those in society who want to contribute as active researchers to do so but on a voluntary basis without the pressures and demands of a typical academic career.</p>

The mathematical stochastic energy balance models (SEBMs) pioneered by Hasselmann and Mitchell have long been known to climate scientists to be important aids to gaining both qualitative insight and quantitative information about global mean temperatures. SEBMs are now much more widely visible, after the award of the 2021 Physics Nobel Prize to Hasselmann, Manabe and Parisi. The earliest univariate SEBMs were, however, built around the simplest linear and Markovian stochastic process, enabling Hasselmann and his successors to exploit their equivalence to the Langevin equation of 1908. Multivariate SEBMs have now been extensively studied but this presentation focuses on the continuing value of univariate SEBMs, especially when coupled to economic models, or when used to study longer-ranged memory than the exponential type seen in Hasselmann's Markovian case.I will highlight how we and others are now going beyond the first SEBMs to incorporate more general temporal dependence, motivated by increasing evidence of non-Markovian, and in particular long-ranged, memory in the climate system. This effort has brought new and interesting challenges, both in mathematical methods and physical interpretation. I will highlight our recent paper [Calel et al, Nature Communications, 2021] on using a Markovian Hasselmann-type EBM to study the economic impacts of climate change and variability and our other ongoing work on generalisations (in particular fractional ones) of Hasselmann SEBMs.This presentation updates our preprints [Watkins et al, arXiv; Watkins et al, in preparation for submission to Chaos] to show how the overdamped generalised Langevin equation can be mapped onto an SEBM that generalises Lovejoy et al's FEBE and I will give a progress report on this work. I will also briefly discuss the relation of such non-Markovian SEBMs to fluctuation-dissipation relations.

The challenges of climate prediction are varied and complex. On the one hand they include conceptual and mathematical questions relating to the consequences of model error and the information content of observations and models. On the other, they involve practical issues of model and ensemble design, and the statistical processing of data. A route to understanding the complexity of these challenges is to study them using low-dimensional nonlinear systems that encapsulate the key characteristics of climate and climate change. Doing so facilitates the fast generation of very large ensembles with a variety of designs and target goals. These idealised ensembles can provide a solid foundation for improving the design of ESM/GCM ensembles, making them better suited to evaluating the risks associated with climate change and to providing end-user support through climate services. The ODESSS project - Optimizing the Design of Ensembles to Support Science and Society - is using low-dimensional nonlinear systems to provide solid foundations for the design of climate change ensembles with climate models. In this presentation I will introduce the project and the concepts behind it. First I will discuss the essential characteristics required of a low dimensional nonlinear system to be able to capture the process of climate prediction. Results will then be presented from the coupled Lorentz ’84 - Stommel ’61 system; a low-dimensional nonlinear system which has these characteristics. These results will be used to illustrate the dangers of confounding natural variability with the consequences of initial condition uncertainty[1], and to demonstrate why risk assessments require much larger initial condition ensembles than are currently available with today’s ESMs/GCMs. The difference between micro and macro initial condition ensembles [2,3] will then be introduced, along with an explanation of how this leads to a requirement for ensembles of ensembles: the former exploring macro-initial-condition-uncertainty, the latter micro-initial-conditional-uncertainty. The importance of this distinction will be illustrated with both new results from the Lorentz ‘84 - Stommel ‘61 system, and also a GCM[3]. I will highlight the challenges in designing these ensembles of ensembles to be most informative. These challenges relate closely to the problems of initialization and the optimal use of observations. Finally the subject of model error, multi-model and perturbed-physics ensembles will be discussed. The impact of model error on climate predictions can only be studied effectively if climate change can be accurately quantified within each model. To begin to explore the consequences of model error for climate predictions therefore requires ensembles of ensembles of ensembles: perturbed-physics or multi-model ensembles which themselves consist of both macro and micro initial condition ensembles. Some approaches will be presented for how low-dimensional systems can be used to optimise the design of such multi-layered ensembles with ESMs/GCMs where computational constraints are more restrictive. [1] Daron and Stainforth, On predicting climate under climate change. ERL, 2013. [2] Stainforth et al., Confidence, uncertainty and decision-support relevance in climate predictions. Phil. Trans Roy. Soc., 2007. [3] Hawkins et al., Irreducible uncertainty in near-term climate projections. Climatic Change, 2015.

The risks of climate change are enormous, threatening the lives and livelihoods of millions to billions of people. The economic consequences of many of the complex risks associated with climate change cannot, however, currently be quantified. Here we argue that these unquantified, poorly understood and often deeply uncertain risks can and should be included in economic evaluations and decision-making processes. We present an overview of these unquantified risks and an ontology of them founded on the reasons behind their lack of robust evaluation. These consist of risks missing owing to delays in sharing knowledge and expertise across disciplines, spatial and temporal variations of climate impacts, feedbacks and interactions between risks, deep uncertainty in our knowledge, and currently unidentified risks. We highlight collaboration needs within and between the natural and social science communities to address these gaps. We also provide an approach for integrating assessments or speculations of these risks in a way that accounts for interdependencies, avoids double counting and makes assumptions clear. Multiple paths exist for engaging with these missing risks, with both model-based quantification and non-model-based qualitative assessments playing crucial roles. A wide range of climate impacts are understudied or challenging to quantify, and are missing from current evaluations of the climate risks to lives and livelihoods. Strong interdisciplinary collaboration and deeper engagement with uncertainty is needed to properly inform policymakers and the public about climate risks.

Probability distribution functions (PDFs) are widely used in projections of future climate, projections of the impacts of future climate, and by climate services aiming to provide information to support practical climate change adaptation. Furthermore they are often used as a means of connecting these different activities and linking the variety of disciplines involved in climate science and climate social science. Here we present an assessment of when such probability distributions misrepresent our uncertainty and a discussion of how we might recognise when such misrepresentations occur [1]. We go on to provide a collection of alternatives to probability distributions for use in such situations. We start by categorising the ways that probability distributions can misrepresent the state of our knowledge about future climate. Such misrepresentation is of importance because it may adversely affect practical societal decisions, particularly in regard to adaptation activities, as well as misdirecting other research efforts. We follow this with a discussion of how we might identify such misrepresentations. Doing so would help us communicate climate information better and consequently provided better reasoned and more robust scientific conclusions and societal decisions. Such assessments are an important component in the evaluation of climate information provided by climate services: what aspects of the information can be described as actionable. We consider two perspectives on these issues. On one, available theory and evidence in climate science essentially excludes using probability distributions to represent our uncertainty. On the other, which represents a significant strand of current practice, probability distributions can legitimately be provided by relying on appropriate expert judgement and the recognition of associated risks. We discuss the reasoning behind each perspective, framed in terms of the analysis of climate models and expert judgement. Finally we explore alternatives to the use of probability distributions. We describe two formal alternatives, namely imprecise probabilities and possibilistic distribution functions, as well as some informal possibilistic alternatives. We suggest that the possibilistic alternatives are preferable. [1] Katzav, Thompson, Risbey, Stainforth, Bradley and Frisch, On the appropriate and inappropriate uses of probability distributions in climate projections and some alternatives, Climatic Change, 2021.

The stochastic energy balance models (SEBMs) pioneered by Hasselmann and Mitchell [1] have long been known to climate scientists to be important aids to gaining both qualitative insight and quantitative information about global mean temperatures. SEBMs are now much more widely visible, after the award of last year’s Nobel Prize to Hasselmann, shared with Manabe and Parisi [1]. The earliest univariate SEBMs were, however, built around the simplest linear and Markovian stochastic process, and researchers have very intentionally exploited their equivalence to the Langevin equation of 1908. Although multivariate SEBMs have now been extensively studied [1,2] and provide one important route to incorporating non-Markovian memory effects into climate dynamics, my presentation will discuss the continuing value of univariate SEBMs, especially when coupled to other models. I will also highlight how we and others (e.g. [4,5]) are going beyond the first SEBMs to incorporate more general models of temporal dependence, motivated by evidence of non-Markovian, and in particular long-ranged, memory in the climate system. This effort has brought new and interesting challenges, both in mathematical methods and physical interpretation. I will highlight our recent paper [3] on using a Hasselmann-type EBM to study the economic impacts of climate change and variability and our other ongoing work [6, and its updated version, 7] on generalised (and in particular fractional) Hasselmann univariate SEBMs. I will compare our model [6,7] with Lovejoy and co-workers' FEBE [5], and discuss what the requirements are in order for such non-Markovian SEBMs to exhibit fluctuation-dissipation relations, which have been debated in the SEBM field since the early work of Leith in the 1970s. [1] Scientific background on the Nobel prize in physics 2021, Nobel Committee, Royal Swedish Academy of Sciences. [2] Franzke and O’Kane, eds. Nonlinear and Stochastic Climate Dynamics, CUP, 2017. [3] Calel et al, Nature Communications, 2020. [4] Rypdal et al, Climate, 2018. [5] Lovejoy et al, QJRMS, 2021. [6] Watkins et al, On Generalized Langevin Dynamics and the Modelling of Global Mean Temperature, 2021, https://link.springer.com/chapter/10.1007%2F978-3-030-67318-5_29 [7] Watkins et al, arXiv: https://arxiv.org/abs/2007.06464v2.