In order to understand the complex dynamics of the climate system beyond the radiation balance and to disentangle internal variability from external forcing, one needs to model the time evolution of the climate system's components and their interactions. These include the hydrosphere, the cryosphere, the lithosphere, the biosphere and – that which affects society most on a daily basis – the atmosphere. The GFD perspective describes the evolution of these components based on the principles of mechanics and thermodynamics. These principles can be expressed as partial differential equations in an appropriate form of the Navier–Stokes equations, adapted to the relevant processes at these vast scales, such as stratification and the Coriolis effect.

In order to describe the evolution of an immensely complex system such as the atmosphere and to gain insight into its dynamics, the relevant degrees of freedom and conserved quantities need to be identified. Brian Hoskins recounted how the concepts of (quasi-geostrophic) potential vorticity enabled the development of the first numerical weather prediction models and how isentropic potential vorticity maps helped gain an understanding of weather developments . A century before the introduction of these concepts, severe weather phenomena such as hurricanes had already motivated early meteorologists to analyse their rotating behaviour on weather maps and to understand their dynamics. Kerry Emanuel described how the scientific progress around hurricanes took a step backwards, highlighting a persistent misunderstanding around the processes driving these dynamics. Despite progress made in the 1940s and 1950s , later research ignored the importance of surface enthalpy fluxes and instead focused on latent heat release. This led to a failure of numerical simulations of hurricanes and the persistence of errors in textbook descriptions even until today. Finally, improved observations played a key role in confirming the dominant drivers.

Raymond Pierrehumbert pointed out in his talk that as simple as the rules governing an atmosphere with water vapour might be – i.e. Newtonian mechanics, thermodynamics and radiative transfer – nontrivial properties emerge in the resulting climate system. Again, misconceptions such as the thermostat hypothesis remind us to stay vigilant and encourage us to try reproduce earlier work. This hypothesis, which claimed that the temperature in the tropics could not exceed 302 K due to shading by high convective clouds, was debunked with some effort . In his more recent work, showed how GFD ideas can be revisited on the new testing ground provided by the rich variety of exoplanetary atmospheres, including tidally locked planets and atmospheres with supercritical water.

On our own planet, again, Clara Deser explained how climate projections driven by GFD simulations help understand regional climate change effects. On these scales, internal variability has an enormous impact on the observed local trends in atmospheric variables. In her seminar, she presented a solution to robustly disentangle the effects due to the external forcing (anthropogenic, volcanic, etc.), the model differences and the system's internal stochastic variability: the latter can be identified as the variability observed in large initial-condition ensembles of historical climate simulations .

The disentangling of internal and forced variability was also the central theme of Michael Mann's story of the Atlantic Multidecadal Oscillation. This concept emerged based on the multidecadal (50–70 years) spectral peak in global surface temperature records and rapidly gained traction in scientific works but was subsequently shown to be an artefact caused by the response of the climate system to both anthropogenic and natural forcings rather than an internal mode of variability. Strong evidence for this was provided by the continued presence of this spectral peak in the globally averaged ensemble mean temperatures of CMIP5 (Coupled Model Intercomparison Project Phase 5, ) climate control simulations of the last millennium. Indeed, one would expect the internal variability to be averaged over these ensembles, meaning that any remaining variability can only be attributed to external forcings .

The atmosphere and the climate system as a whole, i.e. the coupled atmosphere, hydrosphere, biosphere, lithosphere and cryosphere, are chaotic dynamical systems, given the sensitive dependence of their evolution on initial conditions . David Ruelle discussed in his seminar the interdisciplinary origins of chaos theory. We learned that, at the beginning of the 20th century, the mathematician Henri Poincaré had already explicitly considered the fact that the weather is sensitive to initial conditions. He stated that meteorologists are not able to predict the exact location of a cyclone because of imperfect and insufficient initial conditions, which cause the weather to be perceived as something random. However, the chaotic nature of the weather did not become generally established until the work of Ed Lorenz and the works that followed, all strongly connected to the spread of numerical computers in the middle of the 20th century. As David Ruelle reported, this was followed by an explosion of interdisciplinary work on chaos theory between 1970 and 2000 involving meteorology, hydrodynamics, chemical kinetics and the astronomy of the solar system . After that, interdisciplinarity declined, and the work nowadays is mainly done in individual disciplines.

In contrast to the GFD perspective, which focuses on equations describing the temporal evolution of dynamic and thermodynamic variables, the dynamical system perspective considers the solution of these equations in the phase space, the set of all possible configurations of the system. In the phase space, one studies trajectories corresponding to different initial conditions, the stability of fixed points, periodic and non-periodic orbits, the geometry of the attractor, and the properties of the invariant measure. Despite the high level of mathematical abstraction, these concepts are useful because they provide unique insights into the system’s dynamics. It becomes clear, for example, that the geometrical properties of the attractor are strongly related to the evolution of trajectories and thus to the dynamics of the system. Similar states or configurations of the climate system are close to each other in the phase space and evolve correspondingly with the local geometry of the attractor in that region, even if they are far away from each other in time. However, as we found out in James Yorke’s seminar, the dynamics of a chaotic system can be very complex due to phenomena such as hetero-chaos . Because of this, very similar initial conditions can lead to completely different evolutions of the weather and thus to serious predictability issues.

These predictability issues are well-known in numerical weather prediction, where it has become clear over time that purely deterministic predictions are insufficient and misleading. Tim Palmer convinced us in his seminar that the predictability of a nonlinear system and the related uncertainty always depend on the initial condition. Thus, further developments in probabilistic prediction, based on enough ensemble members, are crucially important for the future of weather prediction. Also important for the future is exploiting the possibilities offered by artificial intelligence, as well as using stochasticity for improved parameterisations and data assimilation and for making simulations computationally less expensive.

Phenomena such as climate change, tipping points and internal variability cannot be explained based on the classical concept of an attractor with an invariant measure. In his seminar, Michael Ghil talked about pullback attractors of non-autonomous dynamical systems, i.e. with time-dependent forcing . Already, the presence of noise in the form of internal variability makes the notion of a pullback attractor necessary. In the case of constant external forcing and stochastic natural forcing (internal variability), the corresponding attractor is a time-invariant pullback attractor, referred to as a random attractor. The effect of changing radiative forcing turns the attractor into a time-dependent pullback attractor. Denisse Sciamarella showed us how the study of the attractor's topology, with the help of branched manifolds, helps us to understand chaos and the effect of noise. Due to noise-induced chaos, the attractor can be described by different homology groups at different time steps .

The climate system is a complex system featuring a large number of degrees of freedom with components that interact in a nonlinear way . These concepts are strictly related to those relying on the description of turbulent flows, such as the existence of a hierarchy of exponents to fully characterise the nature of the system i.e. the so-called multifractal view;, the observation of a global- vs. local-scale invariance and the role of extreme events as singularities .

Shaun Lovejoy applied the above concepts to describe different dynamical regimes in the climate system: the weather, the macroweather and the climate . Indeed, weather and climate models are based on thermodynamics and continuum mechanics and are successful because they retain only some relevant “macroscopic” variables. However, the role of the “details” see cannot be ignored, requiring the development of a higher-level description of unknown aspects of the climate system. This led to several (high-level) macroweather and climate models based on energy balance and scale invariance, which profoundly affect our forecasting capabilities at monthly and seasonal scales, as well as improved multidecadal climate projections.

Bérengère Dubrulle went deeper into the details by focusing on a ubiquitous aspect of fluids: turbulence . Turbulent flows are characterised by a self-similar energy spectrum, the signature of fluid movements at all scales. This organisation has been described for more than 70 years by the phenomenology of the Kolmogorov–Richardson cascade : the energy injected on a large scale by the work of the force that moves the fluid (e.g. a turbine) is transferred to smaller and smaller scales with a constant dissipation rate, up to the Kolmogorov scale, where it is transformed into heat and dissipated by viscosity. Such cascade phenomenology is at the basis of most turbulent models. Dubrulle further discussed how progress in numerical simulations and laboratory experiments gradually changed such a simple vision (starting from Landau's objection in the 1950s), leading to a new picture where quasi-singularities living beyond the Kolmogorov scale play a central role . This has important impacts on resolution requirements of numerical simulations and calls for new models of turbulence .

Ken Golden focused instead on a key component of Earth's climate system: the polar sea ice. Indeed, sea ice exhibits complexity on length scales ranging from tenths of millimetres to tens of kilometres. A principal challenge in modelling sea ice and its role in climate is how to use information about the small-scale structure to find the effective or homogenised properties on larger scales relevant to coarse-grained climate models. In other words, and strictly connected to previous concepts introduced by Shaun Lovejoy and Bérengère Dubrulle, how can we predict macroscopic behaviour from microscopic laws? Similar questions arise in statistical mechanics, materials science, and many other areas of science and engineering . In his talk, Golden gave an overview of recent results, inspired by composite materials and statistical physics theories, on modelling effective behaviour in the sea ice system over a broad range of scales .

He also discussed how physical sea ice processes influence microbial communities in the ice and upper ocean and vice versa. This work is helping to advance how sea ice is represented in climate models and to improve projections of the fate of the Earth's sea ice packs and the ecosystems they support.

When modelling the climate system, a stochastic behaviour emerges due to the sensitive dependence on the initial conditions and the practical impossibility of resolving the processes down to viscosity. This makes the stochastic perspective indispensable for useful weather predictions, allowing specifically for the estimation of their uncertainty and a proper interpretation of climate model outputs. At the same time, stochastic methods are useful for different aspects of weather and climate modelling, such as parameterisations (i.e. the representation of the impact of the processes not resolved by the numerical model as function of processes that are resolved), to model certain behaviours and processes in the climate system and even to reduce computational costs. Due to its focus on many-particle systems and the expertise in connecting micro- and macroscales, statistical physics offers inspiration and practical tools for understanding the climate system.

As a format break from the other seminars, the authors interviewed Klaus Hasselmann, who won the Nobel Prize in Physics in 2021 together with Syukuro Manabe and Giorgio Parisi. Klaus Hasselmann pioneered the development of stochastic climate models and reduced-order models in order to detect climate change signals . During the discussion, Klaus Hasselmann stressed the importance of his background in physics for his scientific career and how he opened up new research fields and jumped to new endeavours when he got excited by new ideas without ever losing curiosity and a sense of fun while doing research. His work on stochastic climate models led to the development of stochastic climate theory and stochastic parameterisations, now in use in operational models.

Cécile Penland used Klaus Hasselmann's model reduction approach to develop linear inverse models . Linear inverse models are linear dynamical models driven by stochastic noise. In her presentation, she described her scientific path and how she started using stochastic methods. She emphasised the concept of probability as a physical, conserved quantity and pointed out the necessity of using this concept more often in climate science. However, she also stressed that stochastic methods have become more widespread nowadays.

Roberto Benzi introduced the concept of stochastic resonance to explain past climate variability together with Giorgio Parisi . In his presentation, he outlined the many difficulties and misconceptions he faced when first introducing the concept in the fields of both climate dynamics and nonlinear dynamical systems. In the meantime, stochastic resonance has been used in a large number of physical systems.

In his seminar, Giovanni Jona-Lasinio helped us understand the state of non-equilibrium based on macroscopic fluctuations theory. He argued that non-equilibrium is characterised by a variety of phenomena; thus, searching for a unique theory, such as the one describing classical thermodynamics, is prone to fail, and one has to restrict themselves to subclasses of problems. He pointed out that one of the major difficulties is defining adequate thermodynamic functionals in states far from equilibrium, and he showed how large deviation rates are useful in this regard. Large deviation theory is currently applied in climate studies .

Paleoclimate broadens our perspective on the climate system due to the very long timescales considered. These long timescales are essential to understanding the dynamics of the whole system, including the effect of the slow components, such as the hydrosphere and the biosphere.

Pascale Barconnot emphasised in her talk that knowledge about past climate states brings us closer to understanding future changes and helps us evaluate climate models. Based on paleoclimate modelling, climate models can be tested under different conditions than what we experience today. One can test the effect of using different sets of parameters, applying different forcings or conducting simulations based on different experimental protocols. All this is important in order to be confident that models can indeed simulate the climate of the future. The paleoclimate perspective reveals that the various evolution periods in the history of the climate system do not just have a distinct mean state but are instead the result of very different system dynamics, with different internal variability and feedbacks.

Valérie Trouet pointed out, based on tree ring studies, the strong interconnection between different climate system components, discussing the remote effects of the atmospheric dynamics in the upper troposphere on the vegetation growth and wildfires at the surface. Furthermore, she emphasised the devastating effects of fire suppression on the frequency and extension of Californian wildfires. Disturbances are often part of the natural dynamics of the system, and altering their natural occurrence frequency can lead to the opposite of the intended outcome.

On the one hand, humans are an essential part of the climate system. Due to the emission of greenhouse gases, land use and other human activities disturbing the natural balance, they directly affect the state of the system. However, at the same time, the state of the climate influences human activities and societies. On the other hand, the human dimension represents a huge challenge for modelling the future development of the climate system due to the difficulty – or even impossibility – of describing human behaviour and the evolution of societies based on mathematical equations.

Building upon Hasselmann's seminal ideas about climate response and attribution, Gabriele Hegerl gave a retrospective survey of how ideas such as optimal fingerprinting and the detection and attribution of climate change were brought up by her and her colleagues, mentioning Ben Santer and Klaus Hasselmann among others. The rationale for identifying causes of climate change was limited not only by the emergence of an anthropogenic signal but also by the issue of separating forcings and feedbacks in order to address the feasibility of an orthogonal approach to the attribution of changes in specific forcings, e.g. greenhouse gases or aerosols. Spatial patterns of the expected forced response (after Hasselmann) were, in this respect, compared to observations in order to isolate the role of internal variability. New perspectives are currently being drawn in the field by starting to apply the ideas of attribution to the investigation of the impacts of climate change.

In a thought-provoking talk, Hans von Storch stressed the evolution of the study of climate through the 20th century and the first part of the 21st century, starting from the concept of climate determinism, aimed at justifying the European colonisation of Africa and Asia, and moving towards a proper, rigorous, method-based science, relying on CUDOS principles, and towards the issues of post-normal sciences. Von Storch suggested that the scientific process has been challenged by the increasing demand for science-motivated policies. At the same time, policymakers have been progressively asked to take decisions aimed at addressing climate change adaptation and mitigation. In this context, the climate scientist has been required to take a political stance, and motivating public opinion has been, at times, given higher priority than producing additional scientific results. Von Storch then gave tips on how climate scientists may navigate these troubled waters and what are, in his opinion, the prospects ahead.

The contributions by Susan Solomon and Eugenia Kalnay went deeper into how much the interaction between policymakers and climate scientists is relevant in order to address the challenges posed by the changing climate. Susan Solomon gave an insightful retrospective on the path to the Montreal Agreement, which led to a worldwide ban on emissions responsible for widening the ozone hole over Antarctica. This is regarded to be a pioneering step in the virtuous interaction between policymakers and scientists, with scientific evidence motivating the choice of the community of nations. This step has paved the way to the establishment of the United-Nations-led IPCC, the recurrent redaction of the Assessment Reports and the organisation of the Conference of Parties (COP).

Eugenia Kalnay's contribution focused on stressing the importance and urgency of integrating the human system into Earth system modelling. Not only is this related to an innovative approach to the redaction of socio-economic development pathways serving as boundary conditions in Earth system model exercises, but the human system, according to Kalnay, also has to be viewed as a module and should be integrated into the Earth system model via coupling to the other components, such as the atmosphere, hydrosphere and cryosphere. The task is becoming more urgent as most of the world's population is increasingly reliant on fossil fuel combustion, urbanisation and the continuous increase in gross domestic product. Unlike the usual approach in the climate model communities, the Earth system–human system coupling has to be bidirectional and target the specific subdomains. According to Kalnay, the main challenge embedded in this viewpoint is addressing socio-economic policies, especially those that involve the modification of the current social inequalities.