Most of us would instinctually agree that Earth is a complex system, without even requesting a precise definition of this concept, and likely also that unraveling the planet’s complexity is a fundamentally important role for science. Less recognized is how our understanding of this complexity greatly benefits from developments in the study of nonlinear geophysical processes as well as of exotic concepts in statistical physics.

Until recently, the Nobel Committee for Physics has been more used to awarding scientists for tracking down the elementary building blocks of the universe. Yet in October 2021, the committee awarded the prize jointly to three scientists who revolutionized nonlinear physics with insights into complex systems. Specifically, Syukuro Manabe and Klaus Hasselmann were awarded “for the physical modelling of Earth’s climate, quantifying variability and reliably predicting global warming,” and Giorgio Parisi was awarded “for the discovery of the interplay of disorder and fluctuations in physical systems from atomic to planetary scales.”

The diverse approaches and work of the three recipients, though groundbreaking, do not detract from the global challenges posed by complex systems [*Nicolis and Nicolis*, 2009, 2012]. And in giving these awards, the committee clearly identified the significance of understanding these systems, highlighting, for example, the importance of turbulence and, more precisely, the ubiquitous and multifaceted phenomenon of intermittency. This recognition affirms the primordial importance of these phenomena and validates the investments and work that have gone into the study of nonlinear physics and complex systems over the past several decades.

### Defining Complex Systems and Intermittency

According to the French Roadmap for Complex Systems, which was developed to coordinate and focus research on complex systems, “a complex system is in general any system comprised of a great number of heterogeneous entities, among which local interactions create multiple levels of collective structure and organization…[that] cannot be easily traced back to the properties of the constituent entities.” Natural examples of complex systems range from biomolecules and cells to social systems and the ecosphere; sophisticated artificial systems, such as the Internet, power grids, and large-scale distributed software systems, also qualify.

Turbulent flows—either oceanic or atmospheric, for example—are classic examples of complex systems, as they comprise a large number of eddies with a wide range of sizes. This multiplicity of eddies causes these flows to be very agitated and to experience continuous mixing and velocity fluctuations in both amplitude and direction. Laminar flows, in contrast, are smooth, and their streamlines are easy to identify. Understanding the transition from laminar to turbulent flow is considered a prime example illustrating the difficulties of characterizing the boundary between order and disorder in nonlinear systems (i.e., systems whose response is not proportional to the perturbation) [*Nicolis*, 1995].

There is a further degree of complexity in turbulent flows called intermittency, which refers to the fact that the agitation in a turbulent flow is far from homogeneous. This trait is commonly experienced by air travelers when planes are violently shaken by pockets of turbulence corresponding to clusters of strong velocity gradients. Intermittency is, in fact, a ubiquitous property of complex systems—including Earth’s climate—in which increasingly high levels of activity are generally concentrated in ever smaller fractions of space.

### Climate Models and Spin Glasses

The scientific study of climate and attempts to uncover its past and project its future behavior rely on both deterministic and stochastic models. These models can be either simple (low-order) mathematical models or detailed (high-order) numerical ones based on physical principles and developed to simulate climatic phenomena that operate over multiple time and space scales within the Earth system.

Syukuro Manabe was recognized by the Nobel committee for designing and developing one of the first consistent, high-resolution, deterministic numerical global climate models. His work combining multiple climate components (e.g., displaying an ocean-continent configuration) operating at different time and space scales into a single model laid the groundwork on which all modern climate models, known as general circulation models, are based [*Manabe and Wetherald*, 1975].

Klaus Hasselmann developed a simple climate system forced by random fluctuations (Gaussian white noise in technical terms) representing small-scale weather components (e.g., daily local temperature) that evolve much faster than climatic components averaged over longer time intervals (typically over a year). The spectral properties of the climatic variables obtained with Hasselmann’s model agreed with those observed over large-scale subranges (i.e., averaged variables over time ranges of interest as obtained experimentally) [*Hasselmann*, 1976], providing a fair return for Joseph Fourier, the father of spectral analysis, who had perceived the greenhouse effect as early as 1824.

Decades of innovation in climate modeling have occurred since the seminal works of Manabe and Hasselmann and their colleagues. However, the inevitably finite memory of computers used to conduct numerical experiments still forces researchers to simplify or truncate the equations modeling the full physical system at hand (as described by the laws of nature). A typical simplification involves resolving the mathematics of numerical computations at time and space scales much larger than those at which physical phenomena operate in reality. For example, the dissipation scale of turbulence is of the order of millimeters, whereas global climate models resolve system behavior over scales of at least a few tens of kilometers. As a result, one cannot access probably the most characteristic dynamical feature of the atmosphere and climate, namely, intermittency.

Giorgio Parisi, in his work, further emphasized the role of fluctuations in complex systems to the point of rethinking statistical physics of nonequilibrium systems. For example, he established that magnetic fluctuations could break the “replica symmetry” of spin glasses. Spin glasses—media (e.g., metal alloys) whose magnetic spins, as well as their coupling, are randomly distributed, unlike classical ferromagnetic media in which all spins are aligned and their coupling is homogeneous—are iconic examples of systems that, like turbulence, cannot reach equilibrium. Both of these types of media generate extremely complex and rather similar energy landscapes. Statistical properties of spin glasses used to be calculated by averaging over many copies or replicas (the “replica trick”), therefore assuming a symmetry among these replicas, but that approach proved to be adequate only for high temperatures. Parisi resolved this problem for all temperatures, and he and others subsequently found that his insights and mathematical formulations regarding spin glasses could also apply to and help explain a multiplicity of other complex systems, including artificial intelligence and, indirectly, turbulence and its intermittency.

### Nonlinear Geophysics Through the Years

At several stimulating conferences in 1983, scientists discussed the growing conviction that the activity of turbulence and other nonlinear systems does not cluster on a single fractal set as previously supposed [e.g., *Frisch et al.*, 1978] but, rather, over a hierarchy of multiple fractal sets corresponding to different levels of activity. Particularly notable that summer was the “Turbulence and Predictability in Geophysical Fluid Dynamics” meeting held in Varenna, Italy, and organized by Michael Ghil, Roberto Benzi, and Parisi. It was at this conference that *Parisi and Frisch* [1985] presented a first version of their multifractal formalism for turbulence—and in fact, this is when the term “multifractal” itself was introduced. Also raised were the first questions about the relationship of this statistical formalism with stochastic cascade models, whose paradigm can be traced back to *Richardson* [1922] and which, by way of iterated random multiplications, generate an increasingly heterogeneous flux of energy to smaller and smaller scales. In particular, it was argued that the multifractal formalism could not yield the extremes of the cascade models [*Schertzer and Lovejoy*, 1984].

Debates on these issues have continued over the years in other conferences and publications, for example, in the “Nonlinear Variability in Geophysics” conference series [*Schertzer and Lovejoy*, 1991], in multiple sessions at AGU and European Geosciences Union meetings, and in the journal *Nonlinear Processes in Geophysics*. The advances that have come out of these debates and Parisi’s work on spin glasses have served to widely extend to more abstract statistical physics the well-known role of the Legendre transform in thermodynamics to map conjugate couples of statistical variables onto each other (e.g., temperature or potential onto entropy or energy and vice versa). In the case of intermittency, the scale dependence of statistical moments maps onto that of probabilities, succeeding to link two apparently different statistical approaches to understanding intermittency and providing new means to measure and simulate it.

Studies of the physics of complex systems, deeply rooted in nonlinear geophysics, have matured since these pioneering works and are still going strong (for overviews, see, e.g., *Schertzer and Tchiguirinskaia* [2020] and *Lovejoy and Schertzer* [2013]), despite a lack of funding for fundamental research. Ongoing work is increasingly advancing our understanding of complex systems involved in Earth’s climate and other natural and artificial processes. For all we’ve learned, though, much remains unknown, motivating further work.

The Nobel committee, in explaining its 2021 award in physics, was probably right to quote Philip Anderson (himself a physics Nobel laureate in 1977), who once wrote that “a real scientific mystery is worth pursuing to the ends of the Earth for its own sake, independently of any obvious practical importance or intellectual glamour.” Intermittency most likely belongs in this category of mystery, regardless of the fact that its practical importance is quite obvious!

### Acknowledgments

D.S. acknowledges stimulating discussions with Ioulia Tchiguirinskaia.

**References**

Frisch, U., P.-L. Sulem, and M. Nelkin (1978), A simple dynamical model of intermittent fully developed turbulence, *J. Fluid Mech*.,* 87*(4), 719–736, https://doi.org/10.1017/S0022112078001846.

Hasselmann, K. (1976), Stochastic climate models, part 1: Theory, *Tellus*, *28*, 473–485, https://doi.org/10.3402/tellusa.v28i6.11316.

Lovejoy, S., and D. Schertzer (2013), *The Weather and Climate: Emergent Laws and Multifractal Cascades*, 475 pp., Cambridge Univ. Press, New York.

Manabe, S., and R. T. Wetherald (1975), The effects of doubling the CO_{2} concentration on the climate of a general circulation model, *J. Atmos. Sci*.,* 32*(1), 3–15, https://doi.org/10.1175/1520-0469(1975)032%3C0003:TEODTC%3E2.0.CO;2.

Nicolis, G. (1995), *Introduction to Nonlinear Science*, 254 pp., Cambridge Univ. Press, New York, https://doi.org/10.1017/CBO9781139170802.

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Nicolis, G., and C. Nicolis (2012), *Foundations of Complex Systems: Emergence, Information and Prediction*, 2nd ed., 384 pp., World Sci., Singapore, https://doi.org/10.1142/8260.

Parisi, G., and U. Frisch (1985), On the singularity structure of fully developed turbulence, in *Turbulence and Predictability in Geophysical Fluid Dynamics and Climate Dynamics*, edited by M. Ghil, R. Benzi, and G. Parisi, pp. 84–88, North Holland, Amsterdam.

Richardson, L. F. (1922), *Weather Prediction by Numerical Process*, Cambridge Univ. Press, New York.

Schertzer, D., and S. Lovejoy (1984), On the dimension of atmospheric motions, in *Turbulence and Chaotic Phenomena in Fluids*, edited by T. Tatsumi, pp. 505–512, Elsevier Sci., Amsterdam.

Schertzer, D. and S. Lovejoy (1991), *Non-linear Variability in Geophysics: Scaling and Fractals*, 318 pp., Kluwer Acad., Dordrecht, Netherlands.

Schertzer, D., and I. Tchiguirinskaia (2020), A century of turbulent cascades and the emergence of multifractal operators, *Earth Space Sci*.,* 7*(3), e2019EA000608, https://doi.org/10.1029/2019EA000608.

### Author Information

Daniel Schertzer (daniel.schertzer@enpc.fr), École des Ponts ParisTech, Marne-la-Vallée, France; also at Imperial College London, U.K.; and Catherine Nicolis (cnicolis@meteo.be), Institut Royal Météorologique de Belgique, Brussels