According to everyone's experience, predicting the weather reliably over more than 8 d seems an impossible task for our best weather agencies. At the same time, politicians and citizens are asking scientists for climate projections several decades into the future to guide economic and environmental policies, especially regarding the maximum admissible emissions of

In this review we will investigate this question, focusing on the topic of predictions of transitions between metastable states of the atmospheric or oceanic circulations. Two relevant examples are the switching between zonal and blocked atmospheric circulation at mid-latitudes and the alternation of El Niño and La Niña phases in the Pacific Ocean. The main issue is whether present climate models, which necessarily have a finite resolution and a smaller number of degrees of freedom than the actual terrestrial system, are able to reproduce such spontaneous or forced transitions. To do so, we will draw an analogy between climate observations and results obtained in our group on a laboratory-scale, turbulent, von Kármán flow in which spontaneous transitions between different states of the circulation take place. We will detail the analogy, investigate the nature of the transitions and the number of degrees of freedom that characterize the latter, and discuss the effect of reducing the number of degrees of freedom in such systems. We will also discuss the role of fluctuations and their origin and stress the importance of describing very small scales to capture fluctuations of correct intensity and scale.

The present review paper is based on the lecture delivered by Bérengère Dubrulle on the occasion of her reception of the Lewis Fry Richardson Medal 2021. The story around this lecture
started back in the year 2000, when Bérengère became interested in climate change and started discussions with colleagues at the Laboratoire des Sciences du Climat et de l'Environnement (LSCE). She was intrigued by a strange behaviour of the temperature curves discussed in the IPCC reports: they all exhibited a constant, quasi-linear increase with time, linearly following the rise of

She became all the more puzzled as the climate community was starting to acknowledge the possibility of occurrence of tipping points

On the other hand, the climate science community has achieved considerable success in predicting the increase in Earth average temperatures in the last few decades. This success shows that present climate models, though still incomplete and perfectible, capture the correct evolution of atmospheric or oceanic circulation, at least in terms of large-scale features and most probable events. Recent versions of climate models also include non-linear effects on oceanic circulation

In the atmosphere, characterizing the geometry and the dynamics of the polar jet is still an open problem, as it can be in an almost zonally symmetric state with strong zonal currents associated with trains of extratropical cyclones or in broken, flower-like, so-called “blocked” states which induce hot or cold waves depending on the season and the geography

In fact, non-linear phenomena arise in all the “spheres”. In the ocean, while ENSO is reproduced by most models, correctly reproducing the magnitude and frequency of its occurrence is still challenging, and the fate of the thermohaline circulation remains to be determined, whereas the biosphere is a mine of non-linear interactions between living species directly breathing the non-linear atmospheric chemistry and reacting non-linearly (losing leaves, migrating, hibernating, etc.) to changes in their physical environment.

Considering all that, it appeared clear to Bérengère that the richness of non-linear interactions in climate needed to be further understood, and she had a feeling that laboratory experiments of turbulence could help guide intuition regarding the role of such non-linearities in climate bifurcations, as already demonstrated earlier by

This review is the story of the long journey she undertook with four main collaborators into theoretical, laboratory, and numerical explorations to try and understand these mysterious apparent contradictions and examine how many modes it takes to capture transitions in climate models.

Simulating climate is an arduous task, for we need to describe the interaction of fluid envelopes (atmosphere, ocean) with the lithosphere, cryosphere, and biosphere under solar forcing. This makes the climate a non-equilibrium non-linear and complex system. “Complex” here means that there are several interacting scales at which the energy is accumulated and distributed in space, time, and towards other scales through energetic processes that are visible to humans as meteorological and oceanic phenomena, e.g. cyclones, thunderstorms, marine currents, or iceberg break-off. How many degrees of freedom are needed to take into account this complexity is still an open question. The number we choose is therefore fixed by necessity (i.e. by computing capabilities) rather than by reason. Let us take for example the case of the fluid envelopes.

Their basic physics obey the non-linear, partial differential equations proposed by Navier and Stokes 200 years ago. They describe the dynamics of a velocity field

In recent years there have been developments in understanding that this computational nightmare can be partially solved by applying neural networks or, more generally, machine learning approaches (see e.g.

We are then led by necessity to simulate far fewer degrees of freedom, typically a few thousands in the atmosphere or ocean for recent climate models. How reasonable is this drastic reduction of the number of degrees of freedom? It now depends on the flow physics: the self-similar energy spectrum is an indication that some scales or modes may play a more prominent role than others. So, maybe,
the theoretical

Climate models therefore use a simpler idea and currently select modes based on length scales: the idea is that the largest scales are very energetic and do not directly feel the viscosity, which becomes active only at scales of the order of

Replacing air and water with tar and honey does not look too appealing. Moreover, there are a lot of more complex processes like beating/backscatter/intermittency

Picture and diagram of the experimental set-up. The black arrows indicate the direction of turbine rotation. Symmetry: the system is symmetric by any rotation

Our experimental set-up is summarized in Fig.

The torque or velocity applied by each of the impellers to the turbulent flow. These are global measurements of the state of the system.

Local velocity maps obtained using stereoscopic particle image velocimetry (s-PIV). These measurements give us access to the three instantaneous components of the velocity field in a vertical plane crossing the centre of the experiment with an acquisition rate of about

Averaging a large number of instantaneous velocity fields during statistically steady regimes allows us to observe the mean flow established in the geometry. It takes the form of a large-scale circulation carrying the vertical angular momentum from one impeller to the other. This circulation is analogous to the atmospheric or thermohaline circulation, transporting heat from the Equator to the pole, under the action of either solar radiative forcing and/or surface forcing by winds. The analogy is summarized in Table 1.

Analogy between the Earth system fluid envelopes and the von Kármán turbulent swirling flow stirred by impellers (Fig.

We have used our von Kármán set-up in two complementary forcing modes. In the first mode, the rotation rates of the impellers are kept fixed. In this case one can set the mean rotation rate of the impellers,

One can control the degree of perturbation in the von Kármán flow by controlling the level of velocity fluctuations for a given set of forcing conditions and viscosity. This may be achieved by changing the curvature of the impeller blades, which will be used as an analogue of changes in

We can also change the number of degrees of freedom in the von Kármán flow by considering more or less viscous fluids, going from

In the von Kármán system, vertical angular momentum of opposite signs is injected by the top and bottom impellers. The resulting imbalance produces a large-scale meridional circulation analogous to the large-scale oceanic and atmospheric circulations between the Equator and the poles. The topology of the large-scale circulation is strongly influenced by the symmetries of the experimental set-up.
Arbitrary rotations around the cylinder axis obviously leave the experimental set-up invariant. Its symmetry group thus contains the symmetry group of the oriented circle, the special orthogonal group SO(2).
This symmetry usually carries over, in the time-averaged sense, to the large scales of the flow.
Another basic symmetry of the system is the

However, it is well known that the distinct distribution of continental masses between the two hemispheres or the insulation distribution on seasonal timescales breaks the mirror symmetry of the Earth with respect to the Equator. The

The different large-scale flow configurations observed in the von Kármán flow in the rotation-rate-controlled mode when

Different states observed in our system.

When

However, the situation in that case can also be considered to be analogue to the atmospheric situation in a single hemisphere: it consists of two toroidal recirculation cells arranged on either side of a transition plane, separated by a highly turbulent shear layer – a configuration somewhat equivalent to the Hadley circulation of the atmosphere: positive vertical angular momentum fluid is confined to the lower half of the cylinder, negative vertical angular momentum fluid is confined to the upper half, and vertical angular momentum transport between the two is only mediated by the turbulent fluctuations, which have a visual appearance strikingly similar to that of synoptic-scale atmospheric disturbances.
This situation prevails for small values of the impeller rotation rate imbalance

Given the properties of the large-scale circulation, we are therefore able to produce the equivalent of a “seasonal cycle” in our experiment by modulating the forcing asymmetry as a function of time

Seasonal cycle in the von Kármán experiment, imposed by modulating the vertical angular momentum imbalance

We can now apply a perturbation to our system by imposing higher-velocity fluctuations (“increasing

Seasonal cycle in the von Kármán experiment for a case with high fluctuations. The response of the mean vertical angular momentum

The symmetric T state is then particulary difficult to reach, as it becomes marginally stable: starting from a symmetric state at

Up to now, we focused on circulation changes induced by externally imposed changes in impeller rotation rate imbalance

Spontaneous jumps between two circulation configurations observed by measuring the impeller rotation rates for different, fixed, values of the torque imbalance

These observations show that fast spontaneous transitions between long-lived states may arise in a complex system with many degrees of freedom and large fluctuations even when the external forcing does not vary as a function of time. Similar transitions could then occur in the atmospheric and oceanic circulation on Earth should the level of perturbations – our greenhouse gas emissions – become sufficiently high. Current models employ forcing functions for solar dynamics and

Attractors for different values of the torque imbalance

Despite their apparent complexity, the spontaneous transitions can actually be characterized by low-dimensional objects called attractors. This is illustrated in Fig.

The existence of a low-dimensional stochastic attractor in the experiment illustrates the fact that some degrees of freedom are probably not necessary to capture the bifurcations. It leaves the hope that maybe the essential features of the bifurcation will remain even if we decrease the number of degrees of freedom. So, let us see what happens when we reduce the number of degrees of freedom by increasing the viscosity, which is what is done in climate models when turbulent viscosity is introduced.

We have therefore conducted additional studies in the von Kármán cell filled with glycerol. This amounts to reducing the degrees of freedom to

Different states observed in our reduced system – using glycerol – with fewer degrees of freedom.

One can also produce a forced seasonal cycle in this configuration by varying

Seasonal cycle in the von Kármán experiment for a reduced number of degrees of freedom – using glycerol.

The instantaneous global vertical angular momentum

In a sense, this is quite an achievement: with a much smaller number of modes than the real system, we are able to forecast the dynamics of natural systems at a climatic level. Is this really so? Given that the number of degrees of freedom is still larger than the dimensions of the stochastic attractor, we may think that we should also be able to capture the bifurcations and the dynamics when switching to more curved blades and imposing a torque imbalance.

This hope however was shattered by our experiments: at high viscosity, the multiple-branch region of Fig.

Disappearance of the bifurcation when the Reynolds number or the fluctuations are decreased.

We have seen in the previous section that the small-scale velocity fluctuations are essential to get the rich transition dynamics between metastable states. The usual mental image invoked in such cases is that of an “energy” landscape (Fig.

How does one get fluctuations of sufficient amplitude to trigger these transitions? Let us take a closer look at fluctuations in our von Kármán flow. They are shown in Fig.

What is the physical process that generates strong velocity gradients in a turbulent flow? Obviously not the viscous dissipation, which has the opposite effect of smearing out such gradients. In fact, only the non-linear term of the Navier–Stokes equations

Can we quantify the connection between strong velocity fluctuations and large energy transfer events in a more rigorous way? To do so, we can introduce a local diagnostic quantity that will prove useful in the understanding of the local dynamics of the energy transfer. It is the “local scaling exponent”, defined as

We see that Hölder continuity uses the velocity increment

However, these two hypotheses are violated. First, it is clear that unsigned moments converge much more rapidly and easily than signed moments, as the latter are prone to cancellation effects and sensitive to large fluctuations of positive or negative values. Due to these phenomena, signed moments are sensitive to possible sub-leading correction to scaling and are avoided by taking the absolute value. In that sense, the local scaling exponent we define is more robust but maybe less sensitive to subtle scaling effects like oscillatory scaling behaviour. In the von Kármán flow, we have checked that indeed

While we cannot consider the

The corresponding mathematical description can be built using the large deviation theory

This is a Legendre transform. In that description, the value of

We have computed the multifractal spectrum in the von Kármán flow using both experimental measurements and numerical simulations. The result is shown in Fig.

If this simple model were valid in the von Kármán flow, our measured value of

At all the places where

Using the local energy transfers, we are actually able to observe a smaller local scaling exponent very close to

Indeed, at the smallest scales of the flow, viscous effects curb the growth of the local energy transfer. The typical scale below which viscous effects become effective can be estimated from the dimensional analysis arguments of

The sink term reads

On Earth, strong velocity gradients develop in regions associated with the planetary boundary layer and may result in large-scale extreme weather events: tornadoes, hurricanes, or supercells. Besides viscosity, there are other types of regularizing mechanisms for such phenomena. The energy can indeed be dissipated directly on solid surfaces with sometimes dramatic consequences for human beings, animals, and vegetation. Such a process is however akin to a “large-scale friction” that is not present in our experiment. Extreme weather phenomena are still difficult to forecast. To capture them, is it enough to refine our everyday weather forecasts at a resolution of

We have seen in Sect.

We seem to have understood why climate models work in the first place: the large-scale topology and externally forced transitions do not depend very much on the value of the viscosity and may in fact be described with tools from statistical mechanics

A key result from this experiment is that the large scales of even highly turbulent flows are not restricted to having purely relaxational dynamics, decaying monotonically to an “energy landscape pit” from which escape is impossible, the turbulent fluctuations “randomizing away” any escape attempt. The forced rotation rate experiments (Sect.

The fact that this time evolution of the large-scale topology of the flow can be described by a low-dimensional attractor, corresponding to a few degrees of freedom, is very encouraging and justifies the search for procedures for cutting down the number of degrees of freedom.
However, the destruction of the coherent large-scale dynamics by the large increase in the viscosity in the glycerol experiments shows that a too trivial procedure might preclude a sufficient representation of potentially vitally important large-scale Earth system processes.
Indeed, our study suggests that a good practice to approach complexity is to use both simple conceptual models such as the one presented in

We have seen that these fluctuations are generated at very small scales by a concentration of local energy transfer ending up in point-like quasi singularities, with large fluctuations over small scales. These small scales are really small, even smaller than the Kolmogorov scale

Code is available upon request to berengere.dubrulle@cea.fr.

Data are available upon request to berengere.dubrulle@cea.fr.

BD, FD, DF, BSM and LM conceived and designed the analysis. LM and BSM collected the data. DF contributed analysis tools. BD, BSM and DF performed the analysis. BD (main text) and BSM (figure editing) wrote the paper. BD, FD, DF, LM and BSM reviewed the paper.

The contact author has declared that neither they nor their co-authors have any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This article is part of the special issue “Centennial issue on nonlinear geophysics: accomplishments of the past, challenges of the future”. It is not associated with a conference.

Even though Bérengère Dubrulle received the medal as an individual, most of this work has been done through and thanks to collaboration with students, post-docs, and colleagues, some of them co-authors of this paper. Didier Paillard co-supervised with Bérengère on topics related to the subjects and fuelled our thinking with many interesting references and ideas. She learned a lot about climate and dynamical systems from discussion with Pascal Yiou, Robert Vautard, Didier Roche, and other members of their team.

Thanks to Jean-Philippe Laval, Vishwanath Shukla, Florian Nguyen, Hugues Falller, Caroline Nore, Jean-Luc Guermond and Loïc Cappanera for providing the numerical data and analysis. François Daviaud, Arnaud Chiffaudel, Adam Cheminet, Jean-Marc Foucaut, Christophe Cuvier, Yashar Ostovan, Vincent Padilla, Cécile Wiertel, Pantxo Diribarne, Pierre-Philippe Cortet, Éric Herbert, Davide Faranda, Ewe-Wei Saw, Valentina Valori, Romain Monchaux, Brice Saint-Michel, Simon Thalabard, Denis Kuzzay, Paul Debue, and Damien Geneste performed the particle velocimetry measurements and analysis. The bifurcation measurements in water and glycerol owe much to the PhD work of Louis Marié, Florent Ravelet, and Brice Saint-Michel. This work has been supported by Agence Nationale de la Recherche (ANR) EXPLOIT, grant agreement no. ANR-16-CE06-0006-01.

This research has been supported by the Agence Nationale de la Recherche (grant no. ANR-16-CE06-0006).

This paper was edited by Daniel Schertzer and reviewed by two anonymous referees.