Lewis fry Richardson Medal Lecture-How many modes are needed to predict climate bifurcations?: Lessons from an experiment

According to everyone’s experience, predicting the weather reliably over more than 8 days 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 CO2. To what extent is this request scientifically admissible? In this lecture we will investigate this question, focusing on the topic of predictions of transitions between metastable states 5 of the atmospheric or oceanic circulations. Two relevant exemples are the switching between zonal and blocked atmospheric circulation at midlatitudes and the alternance of El Niño and La Niña phases in the Pacific ocean. The main issue is whether present climate models, that 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 10 transitions between different states of the circulation take place. We will detail the analogy, and investigate the nature of the transitions, the number of degrees of freedom that characterizes 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.

shows that each phase of this cycle is associated with a different oceanic circulation (Sarnthein et al., 1994). (b) Recording of the rapid climate variability of the last ice age obtained by the δ 18 O isotope profile for the Greenland core. The x-axis represents the time spent (in ka). Our present time is located on the origin of this axis. Traditionally, the temperature is reconstructed from a relationship between temperature and δ 18 O . However, many biases such as the seasonality of precipitation or the temperature of regions of evaporation of water masses lead to an underestimated reconstructed temperature. Thus the temperature difference between the current period and the last glacial maximum at 21,000 years is estimated at 20 • C while a Dansgaard-Oeschger (numbered) event is estimated between 10 • C and 16 • C. [Picture courtesy Didier Paillard].
orators joined us during the travel, bringing new scientific ideas, helping with data acquisition and processing, and assisting in shaping the following summary of our findings.
2 How many degrees of freedom are theoretically and practically involved in climate simulations?
Simulating climate is an arduous task, for we need to describe the interaction of fluid envelopes (atmosphere, ocean) with the lithosphere, cryosphere, biosphere under solar forcing. This makes the climate a non-equilibrium non-linear and complex 60 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 for humans as meteorological and oceanic phenomena, e.g. cyclones, thunderstorms, marine currents, 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. 65 Their basic physics obey the non-linear, partial differential equations proposed by Navier and Stokes two hundred years ago.
They describe the dynamics of a velocity field u(x, t) under the action of a pressure field, viscous dissipation and stirred by volume forces. From the work of Kolmogorov on turbulence (Kolmogorov, 1941), hereafter referred to as the K41 theory of turbulence, we know that the balance between forcing and dissipation results in a self-similar organization of the fluid. Kinetic energy is injected at scale L f and is transferred at a constant rate by the energy flux Π I down to the scale η where the energy 70 flux becomes compensated by the viscous flux, with η = (ν 3 / ) 1/4 and ν is the viscosity. At smaller scales, the energy flux is transported by the viscous processes to the smallest hydrodynamic scale, where it is dissipated into heat. A turbulent flow then displays vortices of all sizes in between L f and η, and its energy spectrum scales like E(k) ∼ 2/3 k −5/3 . This means that if we want to capture the flow physics (e.g; location of vortices and their dynamics in the flow, energy dissipation, interplay between vortices at different scales), we need to discretize the Navier-Stokes equations on a grid that is (L f /η) × (L f /η) × (L f /η), 75 containing a priori N ∼ (L f /η) 3 degrees of freedom (as we shall discuss later, the actual number may even be larger!) . If we put numbers corresponding to the atmosphere (L = 10 3 km, η = 10 mm), we get an astronomical number: N ∼ 10 24 , larger than the Avogadro number. Reading-in or writing out this volume of data at each time-step to advance the flow would take 73 billion years of CPU time at the pace of the fastest massively parallel computers! We are then led by necessity to simulate much fewer degrees of freedom, typically a few thousands in the atmosphere or ocean for recent climate models.

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How reasonable is this drastic reduction of the number of degrees of freedom? It now depends on the flow physics: the selfsimilar energy spectrum is an indication that some scales or modes may play a more prominent role than others. So, maybe, the theoretical N ∼ (L f /η) 3 figure overestimates the actual number of modes that is needed to represent the flow dynamics, and we could circumvent the computational obstacle by a clever selection of grid points or modes. Climate models 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 85 becomes active only at scales of the order of η. The large scales are however indirectly influenced by the viscosity through their interaction with the small scales, that play the role of a "turbulent viscosity" with respect to the large scales. Hence, current climate models only keep the largest modes, and add a "turbulent viscosity" to take into account the influence of the small scales. This turbulent viscosity is significantly higher than the classical Newtonian viscosity of the fluids under study: in the atmosphere, turbulent viscosities are equivalent to the Newtonian viscosity of tar, while in the ocean, they are equal to that of 90 honey.
This procedure may sound foolish and bound to failure. Yet, numerical simulations based on these assumptions provide a fairly realistic picture of the past and present state of circulation in the ocean and in the atmosphere XXX(Flato et al.,

2013)XXX.
There is therefore a rationale in this procedure that we should try to understand and improve. To do so, we considered a laboratory model experiment that retains some essential properties of natural flows (forcing, dissipation, symmetries, 95 wide range of spatial and temporal scales) but with simple boundary conditions, and few interacting components, in order to limit the number of relevant observables to be analyzed. To some extent, this may be similar to the simplification obtained by considering a perfect monoatomic gas to describe a real gas.
3 The von Kármán flow, an analogue for oceanic or atmospheric large scale circulation Our experimental setup is summarized in Fig. 2: we consider a transparent cylindrical tank of aspect ratio 1.8, filled with water.

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The tank is closed at each end by two coaxial and counter-rotating impellers with curved blades. Such a device provides a very  efficient, inertial stirring of the liquid inside the tank, which easily reaches speeds of the order of 1 m/s for an impeller rotation frequency of 10 Hz in a 10 cm radius tank. The flow produced in the device (called von Kármán flow) is highly turbulent, characterized by a Reynolds number in water around 10 6 -about 1000 times lower than in the atmosphere. We characterize the flow using two complementary types of measurements: 105 1. The torque or velocity applied by each of the impellers onto the turbulent flow. These are global measurements of the state of the system.
2. 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 center of the experiment with an acquisition rate of about 15 Hz. This rate is insufficient to resolve in time all turbulent scales but offers reliable 110 statistics on the velocity field, obtained by averaging several thousands of instantaneous maps.

Impeller geometry
Observables Mean and imbalance Mean and imbalance Global mean T angular momentum flux impeller rotation rate Global Angular momentum I = ru θ dV Fluids Air, Water Water

Reduced system equivalent Tar, Honey Glycerol
A priori degrees of freedom N ∼ 10 24 N ∼ 10 13 Reduced system degrees of freedom Nr ∼ 10 4 Nr ∼ 10 4 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 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 115 Table 1.
We have used our von Kármán set-ups 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, (f 1 + f 2 )/2, but also the normalized rotation rate difference of the impellers, θ = (f 1 − f 2 )/(f 1 + f 2 ). Varying the first of these parameters, one can adjust the difference of fluid specific angular momentum between the top and bottom boundaries. Varying the second of these parameters 120 allows one to adjust continuously the degree of R π -symmetry of the flow forcing. Such a forcing mechanism is analogous to convective Rayleigh Benard systems in which ∆T , the difference between the top and bottom temperature, is fixed. However, in natural systems, the forcing is rather done by imposing the heat flux that traverses the system (due to solar radiation). We have thus also performed experiments in a second forcing mode, in which we controlled the torque applied to the von Kármán flow by the impellers. In this mode, the mean torque (c 1 + c 2 )/2 is a measure of the angular momentum flux from top to 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 CO 2 concentration in natural flows.
On the other hand, we can change the number of degrees of freedom in the von Kármán flow by considering more or less viscous fluids, going from N ∼ 10 13 for a water filled experiment at Re ∼ 10 6 down to N ∼ 10 4 for a glycerol filled experiment at Re ∼ 10 2 .This is an experimental equivalent of the reduction of the number of degrees of freedom that is explicitly done 135 when using turbulent viscosities in present numerical simulations of atmospheric or oceanic circulation. Eventually, the only major difference between natural flows and the von Kármán flow is the symmetry of the geometry, which is spherical rather than cylindrical. From the point of view of a theoretical physicist, though, this difference only affects the detailed shape of the circulation. The topology of the large-scale circulation is strongly influenced by the symmetries of the experimental set-up. Arbitrary 145 rotations around the cylinder axis obviously leave the experimental set-up invariant. Its symmetry group thus contains the symmetry group of the oriented circle, 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 R π symmetry, which exchanges the two impellers by rotation of 180 degrees around any axis of the middle plane which intersects the cylinder axis (the "hourglass reversal" symmetry, see Fig. 2). When the two impellers rotate at exactly the same frequency f 1 = f 2 , corresponding to a relative speed difference 150 θ = (f 1 − f 2 )/(f 1 + f 2 ) = 0, the system is strictly symmetric with respect to R π and its symmetry group is O(2). When θ = 0, R π is broken, the experiment symmetry group reduces to SO(2).

Large scale circulation properties
Except on "Aqua-planets" (Ferreira et al., 2011), the symmetry group of the continents is trivial. If one neglects the continental influences, the symmetry group of the atmosphere is SO(2) × Z 2 (rotations around the polar axis and mirror-symmetry with respect to the Equator), while the symmetry group of each hemisphere taken separately is SO(2).

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These symmetry considerations guide us in our search for analogies between the von Kármán system and the Earth system: in some respects, the symmetry group of the R π -asymmetric von Kármán flow is the same as the symmetry group of the atmosphere in a single hemisphere, and insight gained in the study of angular momentum transport in the von Kármán flow may be expected to carry over to the transport of heat from the Equator to one pole in the natural atmosphere.
However, it is well known that the distinct distribution of continental masses between the two hemispheres, or the insulation 160 distribution on seasonal time scales, break the mirror-symmetry of the Earth with respect to the Equator. The SO(2) × Z 2 symmetry of the Earth system is thus in fact quite imperfect, and in some respects closer to the SO(2) symmetry of the von Kármán flow in R π -asymmetric conditions. It can thus be hoped that some aspects of the von Kármán flow could also bear resemblance to phenomena involving the Earth system as a whole, such as the seasonal motion of the ITCZ at the boreal summer/austral summer transitions, the oceanic thermohaline circulation or the glacial/interglacial climate transitions. the sign of θ. Negative I correspond to b − states ("austral summer", henceforth "A" states), whereas positive I correspond to b + states ("boreal summer", henceforth "B" states). The B state obtained for a value of θ is transformed by R π into the A state obtained for −θ.
When θ becomes close to 0, the configuration obtained is closer to R π -symmetry: it consists of two toroidal recirculation cells arranged on either side of the median plane, rotating in opposite azimuthal directions (Fig. 3a). The two halves of the 175 von Kármán flow are almost R π -symmetric images of one another and the total mean angular momentum is close to zero: In a way, the situation can also be considered to resemble the global Earth system, in which the two hemispheres are approximately image of one another by reflection across the Equatorial plane, separated by an interface that shifts gradually as the symmetry of the forcing is varied (in analogy to the seasonal cycle of insulation, which brings the latitude of maximal solar energy input into the Earth system alternatively in the two hemispheres). This situation is thus reminiscent of 180 a "spring" or "autumn" transition situation (henceforth "T" state), or of an interglacial situation.
But 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 angular momentum fluid is confined to the lower half of the cylinder, negative angular momentum fluid is confined to the upper half, and angular 185 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 θ, for which the mid-plane shear layer (analogue of the midlatitudes "storm track") shifts continuously in "latitude" as θ varies, moving gradually away from the fastest-rotating impeller.  Figure 3. Different states observed in our system: (a) Two-cell symmetric state ("spring" or "autumn" transition, or "T", state ) (b) One-cell asymmetric state b+ ( "austral summer", or "A", state). (c) One-cell asymmetric state b− ( "boreal summer", or "B", state). The normalized azimuthal flowũ θ = 2u θ /R(f1 + f2) is shown in color (from negative blue to positive red), while the radial and vertical speeds (ur, uz) are represented by arrows. The resolution of the fields has been degraded for better visibility. The positions x and z have been rescaled by the radius of the cylinder R.

Seasonal cycle in water experiment
Given the properties of the large-scale circulation, we are therefore able to drive the equivalent of a "seasonal cycle" in our experiment by modulating the forcing asymmetry as a function of time θ(t). This modulation reflects the yearly modulation of the heat flux imbalance between the poles and the Equator as the Earth revolves around the Sun. An example of the reaction of the mean angular momentum I to such a seasonal cycle in the von Kármán water experiment is shown in Figure 4a. On average, the flow responds continuously -yet in a non-linear fashion -to the forcing imbalance θ: the shift between the A and B 195 states is progressive, as observed on Earth. Indeed boreal summer and austral summer circulations have different characteristics on our planet and they are often studied separately (Vrac et al., 2014). For a given value of θ, the instantaneous value of I is however fluctuating, as illustrated in Figure 4b, and similarly to the fluctuations of the daily temperatures on Earth.

Transitions induced by changes in the velocity fluctuations intensity
We can now apply a perturbation to our system by imposing higher velocity fluctuations (increasing "CO 2 "), using blades 200 that are more curved. In that case, we observe an interesting bifurcation: when the level of fluctuations is sufficiently high, the seasonal cycle becomes discontinuous and the transition between the A and B states becomes brutal, with a jump in the global mean angular momentum I , as seen in Figure 5. In addition, this discontinuous seasonal cycle shows hysteresis: once we have switched to one of the summer branches, if we decrease again θ, the flow jumps directly to the other summer branch, without going through the T state (Ravelet et al., 2004;Ravelet, 2005). Once the flow is in one of the two summer states, its 205 configuration remains the same permanently, unless the parameters of the forcing are changed. The transition from one of the summer states to the T state seems forbidden, and we have never been able to observe it.
The symmetric T state is then particulary difficult to reach, as it becomes marginally stable: starting from a symmetric state at θ ∼ 0, the flow switches to either of the summer states after a time that diverges as θ −6 (Ravelet, 2005).

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Up to now, we focused on circulation changes induced by externally-imposed changes of impeller rotation rate imbalance θ: these are forced transitions, similar to the ones caused by changes of solar radiation on Earth, resulting in seasonal cycles or glacial-interglacial transitions. But the von Kármán flow is actually even more interesting. Using another type of forcing, which is actually closer to the forcing conditions of the natural flows on Earth, we also observed interesting spontaneous transitions of the circulation (Marié, 2003;Ravelet, 2005;Saint-Michel et al., 2013). In this situation we set the torque imbalance applied new type of dynamics only happens when the level of fluctuations is sufficiently high. It is illustrated in Figure 6, where we observe spontaneous transitions between a summer state and the T state. The transitions can be quasi-periodic (case (a)) or very rare (case (b)), depending on the value of γ. They can be viewed as a laboratory equivalent of "weather regimes" (Vautard, 1990), of El Niño events, or of the Heinrich events of Figure 1.

The low-dimensional attractor
Despite their apparent complexity, the spontaneous transitions can actually be characterized by low-dimensional objects called attractors. This is illustrated in Figure 7, where we show the joint probability density function (PDF) of having simultaneously

Circulation properties in glycerol
We have therefore conducted additional studies in a von Kármán cell filled with glycerol. This amounts to reducing the degrees 255 of freedom to ∼ 10 4 , a number two orders of magnitude larger than the dimension of the stochastic attractor. In this reduced system, we observe again three different states, one symmetric with respect to the R π symmetry, and two states that are exchanged by applying the R π symmetry. They are shown in Figure 8. They have the same global topology than the equivalent states in the full system, even though they differ in details: for instance, the global rotation of the fluid is significantly more localized in the summer states, and only extends in the vicinity of the rotating impeller.

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One can also drive a forced seasonal cycle in this configuration by varying θ and examining the associated variations of the angular momentum I , as shown in Figure 9-a. One sees that the seasonal cycle I (θ) is continuous from winter to summer, similarly to the case of the full system with low fluctuations. The seasonal cycle also looks almost linear with θ and shows a smaller contrast of the mean global angular momentum I between summer and winter.
The instantaneous global angular momentum I(t) (displayed in Figure 9b) shows very little fluctuations as a function of 265 time regardless of the impeller frequency imbalance θ: the fluctuations corresponding to the varying daily temperatures are suppressed. In this reduced system, we are thus able to capture forced circulation transitions, but not the short term fluctuations of the observable: in other words, we are able to conduct "climate" experiments (testing the switching between glacial and interglacial state), but not weather experiments (reproducing the daily temperature records) or amplitude of transition between summer and winter.

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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 got shattered by our experiments: at low viscosity, the multiple-branches region of Figure 5 disappears 275 (Ravelet, 2005), along with the bifurcations and their associated dynamics, as shown in Figure 10. The contrast with the flow in water -the full system -is striking: not only does the stochasticity disappear (a reasonable observation if we believe that stochasticity is induced by the small scales), but so does the attractor itself! This means that our way of separating the large and small scales is not suitable for our system, as it has suppressed some of the degrees of freedom that are essential for the large scales dynamics. Instead, one has to pick up carefully chosen large, intermediate and small scales, to represent both the 280 attractor, and the transitions between its branches. The question of how to do it is however still open and left for future work.

Fluctuations and small-scale properties
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. 11) in which the summer and spring states are represented by potential wells of different depths. Once the system is e.g. in the spring state 285 (at position x − ), it can only jump towards the other potential well (at location x + ) following fluctuations exceeding the energy barrier ∆U 0 . This classical problem of stochastic processes results in an exponential distribution of escape times from the potential well, that we have observed experimentally (Ravelet, 2005). This kind of scenario has also recently been observed in stochastically forced PLASIM model of climate (Margazoglou et al., 2021). by H. Faller and collaborators, using the SFEMANS code (Cappanera et al., 2020). Large velocity fluctuations are primarily observed at two locations: one being close to the 'corner' of the experiment (i.e. close to the impellers and the outer wall), and the second being at mid-height. The first location obviously corresponds to the fluctuations induced by the moving boundary conditions which are then advected by the mean velocity of the spring state of Figure 3a. The second location is associated to 295 a strong shear layer resulting from the counter rotation of the two main toroidal cell of the spring state. We then observe, in agreement with our physical intuition, that larger velocity gradients produce stronger velocity fluctuations.
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, ionly the non-linear term of the Navier-Stokes equations (u · ∇)u can produce finer length scales through triadic mode interactions. At a given scale, the intensity of the 300 effect of the non-linear term can be estimated through the "local energy transfer" term defined as: Π = (δ r u) 3 , where δ r u = u(x + r) − u(x) is the velocity increment over a distance r, and x indicates that we average the distance r over a ball of size centered on x . Indeed, this term provides the contribution of non-linear interactions to the energy budget, and provides the quantity of energy that is transferred through scales (Dubrulle, 2019). In other words, the larger Π , the higher the energy cascade towards smaller scales, and the stronger the final gradient (and velocity fluctuations) at this location. The 305 quantity Π is shown in Figure 12b): as expected, areas of large velocity fluctuations correspond to areas of large local energy transfers Π .

Scaling exponents and intermittency
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 310 energy transfer. It is the "local scaling exponent", defined as h = ln(δ r u)/ ln(r), or equivalently, δ r u ∼ r h . From dimensional analysis, it is then possible to show that the local energy transfer scales like Π ∼ 3h−1 (Dubrulle, 2019). So, when h > 1/3, Π decreases with scale and tends to zero, while for h < 1/3 , Π increases with decreasing scales, and is able to provide larger and larger fluctuations. The case h = 1/3 is special, and corresponds to a case where the local energy flux is constant. This is the situation that was assumed in the original self-similar theory of turbulence proposed by Kolmogorov. This theory leads to 315 an energy spectrum scaling as k −5/3 , yet we see from Figure 12b that the local energy transfer term is highly variable in space, contrasting with the assumptions of homogeneity and isotropy made by Kolmogorov. This means that we have to forget about global homogeneity or scale invariance for turbulent flows, and find a way to describe them using local scaling laws.

The multifractal spectrum
The corresponding mathematical description can be built using the large deviation theory (Eyink, 2007(Eyink, -2008) that allows us to 320 describe the probability of finding an exponent h at a given place or time in the flow as where C(h) is the rate function of h, also called multifractal spectrum. Formally, 3 − C(h) can be interpreted as the dimension of the set where the velocity increment at a distance scales like h . This observation allows to single out the exponents where C(h) > 3 (possibly C(h) → ∞), corresponding to exponents with a very low probability of occurrence. We have computed this 325 multifractal spectrum in the von Kármán flow, using both experimental measurements and numerical simulations. The result is shown in Figure 12b. We see that the minimum of C(h), corresponding to the exponent with highest probability, is h = 0.35, very close but slightly larger than the 1/3 Kolmogorov exponent. At all the places where h < 1/3 (and we see that there are quite a few), the local energy transfer increases with decreasing scales. This means that there are many places in a turbulent flow where large fluctuations can build up. The smallest exponent we were able to measure is h ≈ 0.17. If we extrapolate our 330 measurements to the value where C(h) = 3, we expect that the smallest exponent we may be able to find in the flow with a non-zero probability is h min −0.2. For such a negative value of h, the local energy transfer increases sharply towards the smaller scales like Π ∼ −1.6 . Does it increase without limit? Is there any process to stop it?

The regularizing scale
Indeed, at the smallest scales of the flow, viscous effects curb the growth of the local energy transfer. The typical scale below 335 which viscous effects become effective can be estimated from the dimensional analysis arguments of (Paladin and Vulpiani, 1987). Their argument is that the local energy budget of a turbulent flows at a given scale includes a sink term due to viscous dissipation besides the local energy transfer we mentioned in the previous sections (for a complete expression of the energy budget, see Dubrulle (2019)).
The sink term reads Π ν = ν(δ r u) 2 and scales like Π ν ∼ ν 2h−2 at locations where δ r u ∼ r h . The viscous term therefore 340 balances the local energy transfer term at a scale η h ∼ ν 1/(1+h) . This scale thus appears as a fluctuating cut-off scale, which depends on the scaling exponent and therefore on x. This is a generalization of the Kolmogorov scale η K ∼ ν 3/4 . Below η h , the sink term due to dissipation becomes dominant in the energy budget, the local energy transfer vanishes and the flow becomes regular, whereas this regularization occurs everywhere at η K in the K41 model.
On Earth, regions where strong velocity gradients develop are associated with extreme weather events: tornadoes, hurricanes 345 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, despite the good representation of the "everyday"' weather. Yet resolution is now refined to a level (∼ 1 km) where convective flows can be represented (Hohenegger et al., 2015). In the next 350 section we will try to discuss whether such refinement is enough if we want to capture bifurcations.

The computational nightmare continues!
We have seen in Section 4 that we need to include the velocity fluctuations of the von Kármán turbulent flow down to their smallest scales to accurately reflect the large-scale properties and dynamics of its seasonal cycle. For any given local exponent h < 1/3, the local energy transfer increases with decreasing scales down to = η h . Therefore, we have to make sure that our 355 modelling resolution δ (the size of our grid) is below η h for any h present in the flow. Otherwise, we may underestimate the magnitude of the fluctuations by a factor (δη h ) h . A stringent criterion to make sure all small scales are properly taken into account would be to choose δ equal to η h=hmin ∼ Re 5/4 ∼ η K Re −1/2 . This choice actually increases the number of degrees of freedom required to accurately model turbulent flows by a factor Re 3/2 compared to our initial estimate based on the Newtonian viscosity of the fluid and the K41 turbulent model. We then reach N ∼ 10 23 degrees of freedom (the Avogadro number) in our 360 water experiment, or N ∼ 10 40 for the atmosphere!

Conclusion
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 be in fact described with tools from statistical mechanics (Thalabard et al., 2015), involving only a few modes that can easily be captured by present climate models.

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However, when it comes to understanding whether present climate model have enough degrees of freedom to capture global bifurcations, we have reached contrasted conclusions using our laboratory experiment.
A key result from this experiment is that the large scales of even highly turbulent flows are not restricted to have 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 (section 4.3) show that this is 370 definitely not the case: when brought to the symmetric forcing θ = 0 situation, the flow remembers with seemingly infinite memory how it has been prepared, and remains in its "A" or "B" state indefinitely (the "T" state is peculiar in that its decay time diverges when θ → 0, but the transition seems a priori possible). In this respect, though the level of turbulent fluctuations is very high, the large scales of the flow are as deterministic as a light switch. This behaviour is in fact well known in aerodynamics, where it is encountered in the "stalling" transition of airfoils (Sarraf et al., 2005). The fixed flux experiments (section 4.4) 375 are even more spectacular, in that they show that the large scales of the flow can have non-trivial temporal dynamics, with an elaborate scenery of fast transitions between long-lived states, with residence times in the states either sharply distributed around a finite value, or exponentially distributed with a long characteristic time of decay.
The fact that this time-evolution of the large scale topology of the flow can be described by a low dimensional attractor, corresponding to few degrees of freedom, is very encouraging, and justifies the search for procedures for cutting down the 380 number of degrees of freedom. However, the destruction of the coherent large-scale dynamics by the large increase of 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.
We have seen that these fluctuations are generated at very small scales, by a concentration of local energy transfer ending up into point-like quasi singularities, with large fluctuations over small scales. These small scales are really small, even smaller 385 than the Kolmogorov scale η K . Resolving these small scales using direct numerical simulations comes at a colossal computational cost, out of reach of current (and future) computing facilities. So the problem of small scale parametrization is in fact even more acute than we thought. From what we have learned, one thing we must take care of is to devise a model that is able to reproduce and include the large velocity fluctuations observed at very small scale in experimental turbulent flows without explicitly resolving them. This is clearly the next challenge in climate modeling.