When the German physicist Arnold Sommerfeld assigned his most brilliant student a subject for his doctoral thesis in 1923, he admitted that “I would not have proposed a topic of this difficulty to any of my other pupils.” Those others included such geniuses as Wolfgang Pauli and Hans Bethe, yet for Sommerfeld the only one who was up to the challenge of this subject was Werner Heisenberg.

Heisenberg went on to be a key founder of quantum theory and was awarded the 1932 Nobel Prize in physics. He developed one of the first mathematical descriptions of this new and revolutionary discipline, discovered the uncertainty principle, and together with Niels Bohr engineered the “Copenhagen interpretation” of quantum theory, to which many physicists still adhere today.

The subject of Heisenberg’s doctoral dissertation, however, wasn’t quantum physics. It was harder than that. The 59-page calculation that he submitted to the faculty of the University of Munich in 1923 was titled “On the stability and turbulence of fluid flow.”

Sommerfeld had been contacted by the Isar Company of Munich, which was contracted to prevent the Isar River from flooding by building up its banks. The company wanted to know at what point the river flow changed from being smooth (the technical term is “laminar”) to being turbulent, beset with eddies. That question requires some understanding of what turbulence is. Heisenberg’s work on the problem was impressive—he solved the mathematical equations of flow at the point of the laminar-to-turbulent change—and it stimulated ideas for decades afterward. But he didn’t really crack it—he couldn’t construct a comprehensive theory of turbulence.

Heisenberg was not given to modesty, but it seems he had no illusions about his achievements here. One popular story goes that he once said, “When I meet God, I am going to ask him two questions. Why relativity? And why turbulence? I really believe he will have an answer for the first.”

It is probably an apocryphal tale. The same remark has been attributed to at least one other person: The British mathematician and expert on fluid flow, Horace Lamb, is said to have hoped that God might enlighten him on quantum electrodynamics and turbulence, saying that “about the former I am rather optimistic.”

STARRY, STARRY MATH: Vincent van Gogh’s Starry Night offers a gripping way to visualize a hidden regularity of turbulence, a constant mathematical relationship between points, later discovered by Soviet physicist Andrei Kolmogorov.

You get the point: turbulence, a ubiquitous and eminently practical problem in the real world, is frighteningly hard to understand. Nearly a century after Heisenberg, scientists are still trying to figure it out. And it’s still a cutting-edge problem: Russian mathematician Yakov Sinai won the 2014 Abel Prize for mathematics—often seen as the Nobel of math—partly for his work on turbulence and chaotic flow.

Yet I propose that to fully articulate and understand turbulence we need to add the intuitive, contemplative perspective of art to the detailed analysis of science. There is a long-standing dialogue between art and science on this elusive problem. It is no coincidence the science of turbulence has often been forced to fall back on qualitative, descriptive accounts, while art that celebrates turbulence sometimes resembles a quasi-scientific gathering of data and idealization of form: a search for underlying patterns and regularities.

The interplay of the two perspectives can enhance both. Intuition of turbulent flow can serve the mathematician and the engineer, while careful observation and even experiment can benefit the artist. Scientists tend to view turbulence as a form of “complexity,” a semi-technical term which just tells us there is a lot going on and that everything depends on everything else—and that a reductionist approach therefore has limits. But rather than regarding turbulence as a phenomenon awaiting a complete mathematical description, we should see it as one of those concepts, like life, love, language, and beauty, that overlaps with science yet is not wholly contained within it. Turbulence has to be experienced to be grasped.





Into the Storm

Pretty much all scientific histories of the problem of turbulence start in the same place: with the sketches of wild water flows made by Leonardo da Vinci in the 15th century. What Leonardo was up to was rather profound. In the words of art historian Martin Kemp, Leonardo regarded nature “as weaving an infinite variety of elusive patterns on the basic warp and woof of mathematical perfection.”

Leonardo was trying to grasp those patterns. So when he drew an analogy between the braided vortices in water flowing around a flat plate in a stream, and the braids of a woman’s hair, he wasn’t just saying that one looks like the other—he was positing a deep connection between the two, a correspondence of form in the manner that Neoplatonic philosophers of his age deemed to exist throughout the natural world. He saw fluid flow as a static, almost crystalline entity: His sketches have a solidity to them, seeming almost to weave water into ropes and coils.

Yet what mattered was not the superficial and transient manifestations of these forms but their underlying essence. Leonardo didn’t imagine that the artist should be painting “what he sees,” but rather what he discerns within what he sees. It behooves the artist to invent: painting is “a subtle inventione with which philosophy and subtle speculation considers the natures of all forms.” That’s not a bad definition of science either, when you think about it.

We should see turbulence as one of those concepts, like life, love, language, and beauty, that overlaps with science.

Still, it would take centuries for science to develop Leonardo’s ideas about turbulent flow. It’s not hard to see why—and I mean that literally. When you look at a turbulent flow—cream being stirred into coffee, say, or a jet of exhaled air traced out in the smoke of a cigarette—you can see that it is full of structure, a profound sort of organization made up of eddies and whirls of all sizes that coalesce for an instant before dissolving again. That’s rather different to what we imply in the colloquial use of the word to describe, say, a life, a history, or a society. There we tend to mean the thing in question is chaotic and random, a jumble within which it is difficult to identify any cause and effect. But pure randomness is not so hard to describe mathematically: It means that every event or movement in one place or at one time is independent of those at others. On average, randomness blurs into dull uniformity.

A turbulent flow is different: It does have order and coherence, but an order in constant flux. Flows of fluids—liquids and gases—generally become turbulent once they start flowing fast enough. When they flow slowly, all of the fluid moves in parallel, rather like ranks of marching soldiers. But as the speed increases, the ranks break up. You could say that the “soldiers”—little parcels of fluid—begin to bump into one another or move sideways, and so swirls and eddies begin to form.

This transition to turbulence doesn’t happen at the same flow speed for all fluids—more viscous ones can be “kept in line” at higher speeds than runny ones. For flow down a channel or pipe, a quantity called the Reynolds number determines when turbulence appears. Roughly speaking, this encodes the ratio of the flow speed to the viscosity of the fluid. Turbulence develops at high values of the Reynolds number. The quantity is named after Osborne Reynolds, an Anglo-Irish engineer whose pioneering work on fluid flow in the 19th century provided the foundation for Heisenberg’s work.

Copyright Goh Shigetomi

Many of the flows we encounter in nature—in rivers and atmospheric air currents like the jet streams—have high Reynolds numbers. The eddies and knots of air turbulence can make for a bumpy ride when an aircraft passes through them.

Turbulence provides a perfect example of why a problem is not solved simply by writing down a mathematical equation to describe it. Such equations exist for all fluid flows, whether laminar or turbulent: They are called the Navier-Stokes equations, and they amount largely to an expression of Isaac Newton’s second law of motion (force equals mass times acceleration) applied to fluids. These equations are the bedrock of the modern investigation of flow in the science of fluid dynamics.

The problem is that, except in a few particularly simple cases, the equations can’t be solved. Yet it’s those solutions, not the equations themselves, that describe the world. What makes the solutions so complicated is that, crudely speaking, each part of the flow depends on what all the other parts are doing. When the flow is turbulent, this interdependence is extreme and the flow becomes chaotic, in the technical sense that the smallest disturbances at one time can lead to completely different patterns of behavior at a later moment.





A New Confluence

The constant appearance and disappearance of pockets of organization in a disorderly whole has a beautiful, mesmerizing quality. For this reason, turbulence has proved as irresistible to artists as it is intransigent to scientists.

Leonardo’s representations of fluid flow found few takers in the West. But a comparable tradition of seeking fundamental forms in the changing flux was already well developed in Eastern Asia. In the late 17th century, the Chinese painter Shitao drew an analogy between water waves and mountain ranges—a comparison that is explicitly rendered by Shitao’s friend Wang Gai in The Mustard-Seed Garden Manual of Painting. Here the serried ranks of waves could almost be the limestone peaks of Guilin, while the frothy tendrils of breaking wave-crests recall the pitted and punctured pieces of rock with which Chinese intellectuals loved to adorn their gardens.

For Chinese artists, the forms of turbulent flow were defined by the ebb and flow of a natural energy called qi, which supplies the creative spontaneity of Taoist philosophy. The artist captured this energy not with slow, meticulous attention to detail but with a free movement of the wrist that imparted qi to the watery ink on the brush and to the trace it left on silk. The wrist, Shitao wrote, should be “flowing deep down like water.” It is this insistence on dynamic change that makes Chinese art a profound meditation on turbulence.

One can’t help noticing how its traditional schema for depicting flow resemble the attempts of modern fluid dynamicists to capture the essentials of complex flow in so-called streamlines, which, to a rough approximation, trace the trajectories of particles borne along in the fluid. Are these resemblances more than superficial and coincidental? I think so: They express a recognition that turbulent flows contain orderly patterns and forms, and that these have to be visualized in order to be appreciated.

However, for scientists in the 20th century, this “deep structure” of turbulence became increasingly an abstract, mathematical notion. One of the key advances in the science of turbulence came from the Soviet mathematical physicist Andrei Kolmogorov, under whose guidance Sinai began his work in the 1950s. By this time, turbulence was regarded as a hierarchy of eddies of all different sizes, down which energy cascades from the largest to the smallest until ultimately being frittered away as heat in the friction of molecules rubbing viscously against one another. This picture of turbulence was famously captured by the English mathematician Lewis Fry Richardson, another pioneer of turbulence theory, in a 1922 poem indebted to Jonathan Swift:

Big whirls have little whirls

That feed on their velocity,

And little whirls have lesser whirls

And so on to viscosity.

In the 1940s Kolmogorov calculated how much energy is bound up in the eddies of different sizes, showing that there is a rather simple mathematical relationship called a power law that relates the energy to the scale: Each time you halve the size of eddies, the amount of energy contained in all the eddies of that size decreases by some constant factor. This idea of turbulence as a so-called spectrum of different energies at different size scales is one that was already being developed by Heisenberg’s work on the subject. It’s a very fruitful and elegant way of looking at the problem, but one in which the actual physical appearance of turbulent flow is subsumed into something much more recondite. Kolmogorov’s analysis can supply a statistical description of the buffeting, swirling masses of gases in the atmospheres of planets—but what we see, and sometimes what concerns us most, is the individual vortices of a tropical cyclone on Earth or the Great Red Spot on Jupiter.

But there were, at the same time, stranger currents at play. While Heisenberg was juggling with equations, an Austrian forest warden named Viktor Schauberger was grappling toward a more intuitive understanding of turbulent flow. Schauberger’s interest in the subject arose in the 1920s from his wish to improve log flumes so that they didn’t get jammed as they carried timber through the forest. This led him to develop an idiosyncratic theory of turbulent vortices which mutated into something akin to a theory of everything: a view of how energy pervades the universe, which alleged to yield Einstein’s E=mc2 as a special case. It is said that Schauberger was forced by the Nazis to work on secret weapons related to his “implosion theory” of vortices, and even that he was taken for an audience with Hitler. After the war Schauberger was brought to the United States, where he was convinced that all his ideas were being stolen for military use.

Turbulence provides a perfect example of why a problem is not solved simply by writing down a mathematical equation to describe it.

Inevitably this is the stuff of conspiracy theory—Schauberger is said to have designed top-secret flying saucers powered by turbulent vortices. The spirit of his approach can be discerned also in the ideas of the German anthroposophist Theodor Schwenk in the 1950s and ’60s. Schwenk claimed that his work was “based on scientific observations of water and air but above all on the spiritual science of Rudolf Steiner,” and he believed that the flow forms of water, and in particular the organization of vortices, reflects the wisdom of a teleological, creative nature. These “flow forms,” he said, are elements of a “cosmic alphabet, the word of the universe, which uses the element of movement in order to bring forth nature and man.”

Schauberger and Schwenk were not doing science; it is not unduly harsh to say that, in the way they clothed their ideas in arcane theory disconnected from the scientific mainstream, they were practicing pseudo-science. Their appropriation by New Age thinkers today reflects this. But we shouldn’t be too dismissive of them on that account. One way to look at their work is as an attempt to restore the holistic, contemplative attitude exemplified by Leonardo to a field that seemed to be retreating into abstruse mathematics.

The gorgeous photographs of complex flow forms, of turbulent plumes and interfering waves and rippled erosion features in sand, in Schwenk’s 1963 book Sensitive Chaos, offered a reminder that this was how flow manifests itself to human experience, not as an energy spectrum or hierarchical cascade. Such images seem to insist on a spontaneous natural creativity that is a far cry from the deterministic mechanics of a Newtonian universe. Schwenk himself suggested that representations of vortices and waves in primitive art, such as the stone carvings on the Bronze Age burial chamber at Newgrange, Ireland, were intuitions of the fecund cosmic language of flow forms.

Courtesy of the artist and Purdy Hicks Gallery, London





Flow on Film

However sniffy scientists might be about Schauberger and Schwenk, their ideas have captivated artists and designers, and continue to do so. The contemporary British artist Susan Derges, who has made several works concerned with waves and flow in water, says that she was inspired by their ideas. Growing up beside the Basingstoke Canal in southern England, Derges spent a lot of time exploring the towpath walks. “I was intrigued by the mixture of orderly patterning and interference set up by barges and bird life moving through the water,” she says. She began to explore how waves and interference patterns give rise to orderly, stable patterns: “It was a way of revealing a sense of mysterious but ordered processes behind the visible world.”

When she moved to Dartmoor in southwest England in the 1990s, Derges encountered the torrent rivers coming down from the high moor. “I found it fascinating that a huge amount of energy, momentum and complex, chaotic movement could give rise to stable vortices and flow forms that remained in areas of the river’s course,” she says. “It seemed to suggest a metaphor for how one might consider all apparently constant and solid appearances as being sustained by a more fluid energetic underlying process.”

In a series of works in the 1990s, Derges captured these turbulent structures in the River Taw on Dartmoor by placing large sheets of photographic paper, protected with a waterproof covering, just beneath the water surface at night and exposing them with a single bright flash of light. In her inspiration, motives, and techniques, there is little distance between what Derges did and what an experimental scientist might do: Such “shadowgraphs” are commonly used by fluid dynamicists to capture and study flow structures. But for Derges this “data gathering” becomes an artistic moment.

Like Derges, American artist Athena Tacha was inspired by Leonardo’s sketches of vortices, a debt that she made particularly explicit in her 1977 sculpture maquette Eddies/Interchanges (Homage to Leonardo). Much of Tacha’s work over the past several decades is an enquiry into the deep structures of turbulent flow, which she often reduces to their abstract essence and transforms into something more permanent and rigid. Because her work includes large-scale public commissions, these architectural sculptures allow people to literally get inside the forms and experience them as if they were a particle borne along in the flow—for example, in the brickwork-trellis maze of Mariathne (1985-6) and the stepped crescent forms of the terraced courtyard space Green Acres (1985-7). If you want a visceral sense of the real tantalizing confusion of a turbulent maelstrom, no scientific description will improve on Tacha’s photographic series such as Chaos (1998).

Copyright Athena Tacha (1985-6)

“I think I respond to turbulence because I am generally interested in fluid forms that evoke the state of ‘chaos’ in nature—which I consider a different kind of order, with constant irregularities and changes, but ultimately extremely organized,” Tacha says. Kolmogorov and his scientific successors would find little to object to in that claim.

Nothing, perhaps, better captures the sense of a flow frozen into an instant than Tacha’s sculpture Wave, which allows the viewer to experience the terrifying beauty of Japanese artist Hokusai’s Great Wave (c.1831-33) without fear of being pulled under. If this work hints at the connection to an East Asian appreciation of flow, that context is unmistakable in the work of contemporary Japanese artist Goh Shigetomi. Shigetomi has found a way to disperse black sumi ink into natural streams so that it can imprint an image of the flow on paper. He modestly equivocates about his status as an artist at all, since, as he puts it, the water itself “spontaneously draws lines.” Only the right ink and the right paper (Japanese rice paper) will work, and it took years of experimentation to refine the technique.

The results are unearthly, and Shigetomi expresses them in almost magical terms, reminiscent of Schwenk: “ ‘New-born’ water is full of infinite live force.” He believes that “the water remembers every single thing which has happened on and around the earth,” and that one can see “the fragments of the memories in flows and movements of water as certain patterns.”

Can these claims be in any sense true from a scientific viewpoint? Not obviously; they seem closer to a form of thaumaturgy, of divination from natural symbols. (Shigetomi literally believes that a “spirit of water” is sending him messages.) But the complexity of the inky traceries, when seen firsthand, are richer and more subtle than anything I have seen in a strictly scientific photograph. They seem to conjure up much more than a cold physical trace of the technical process of their production. I find it hard not to see these “water figures” as an extension of Shitao’s instruction that the painter must find a spontaneous, unforced way of applying ink to paper, a way that captures the dynamic force of qi. Shigetomi explains that it takes a finely developed sensibility to make these “experiments” work—one cultivated by 38 years of standing in rivers, waiting for the right moment. Derges says the same: “I had to be very aware of the tide and the wave patterns… One would watch and wait for the seventh wave and one needed split second timing.” These artists have developed the same patient, observant sensitivity to flow that characterizes both the meditations of the Chinese Tang Dynasty water poets Li Bai and Du Fu and the sketches of Leonardo.

But can this attitude of contemplative observation, rather than careful testing and measurement, serve the scientist too? Certainly it can. In 1934 the French mathematician Jean Leray proved that the Navier-Stokes equations have so-called “weak” solutions, meaning that there are solutions that satisfy the equations on average but not in detail at every point in space: flow patterns that “fit,” you could say, so long as you don’t examine them with a microscope. And Leray is said to have found much of his inspiration for this mathematical tour de force not by poring over his desk into the small hours but by leaning over the Pont-Neuf in Paris and watching, hour after hour, the eddies of the Seine surging around the piles.





Order and Chaos

There is, however, a more dramatic example of how these intuitions of the form of turbulence can cross boundaries between art and science. One of the most striking and certainly famous artistic depictions of turbulence is Vincent van Gogh’s Starry Night (1889). It is a fantastical vision, of course—the night sky is not really alive with these swirling stellar masses, at least not in a way that the eye can see. But spiral galaxies and stellar nebulae were known in van Gogh’s day, having been revealed in particular by the telescopic studies of William Herschel 100 years earlier. It is tempting to conclude that van Gogh’s notion of a turbulent heavens was simply a metaphor for his tumultuous inner world—but whether or not this is so, the artist seems to have had a startlingly accurate sense of what turbulence is about.

Kolmogorov’s work showed how to relate the velocity of the flow at one point to that at some other point a certain distance away: something that varies from place to place but which has a constant mathematical relationship on average. In 2006, researchers in Mexico, led by physicist Jose Luis Aragon of the National Autonomous University in Mexico City, showed that this same relationship deduced by Kolmogorov also describes the probabilities of differences in brightness, as a function of distance, between points in Starry Night. The same is true of some of van Gogh’s other “swirly” works, such as Road with Cypress and Star (1890) and Wheat Field with Crows (1890). These paintings offer a way to visualize an otherwise recondite and hidden regularity of turbulence: They show us what Kolmogorov turbulence “looks like.”

“I think I respond to turbulence because I am generally interested in fluid forms that evoke the state of ‘chaos’

in nature.”

These works were created when van Gogh was mentally unstable: The artist is known to have experienced psychotic episodes in which he had hallucinations, minor fits, and lapses of consciousness, perhaps indicating epilepsy. “We think that van Gogh had a unique ability to depict turbulence in periods of prolonged psychotic agitation,” says Aragon. Any psychological explanation is sure to be tendentious, but the connection does seem to be more than just chance—other, superficially similar paintings such as Edvard Munch’s The Scream don’t have this mathematical property connecting the brush strokes, for example.

Of course, it would be absurd to suggest that van Gogh had somehow intuited Kolmogorov’s result before the Russian mathematician deduced it. But the incident does imply that a sensitive and receptive artist can penetrate to the core of a complex phenomenon, even if the result falls short of a scientific account. That, I think, is what Derges is implying when she suggests that the most revealing images of turbulent flow patterns “need to be situated in between something that has been closely observed and something that has been emotionally experienced.”

Some scientists agree. Last fall, physical oceanographer Larry Pratt of the Woods Hole Oceanographic Institution in Massachusetts and performing artist Liz Roncka led a workshop near the Massachusetts Institute of Technology (MIT) in Cambridge in which the participants, mostly mathematicians and scientists, were encouraged to dance their interpretation of turbulence. As Genevieve Wanucha, science writer for the “Oceans at MIT” program, reported, Pratt “was able to improvise complex movements that responded fluidly to the motion of his partner’s body, inspired by obvious intuition about turbulence.” Wanucha explains that Pratt uses dance “as a teaching tool to elegantly and immediately represent to the human mind how eddies transport heat, nutrients, phytoplankton, or spilled oil down beneath the ocean surface.” He believes the approach will help young scientists working on ocean flows to “gain a more intuitive understanding” of their work.

An intuitive understanding has been an essential part of any great scientist’s mental toolkit. It is what has motivated researchers to make physical models and draw pictures, immerse themselves in virtual sensory environments that display their data, and create “haptic interfaces” that let them feel their way to understanding. I daresay that dance and other somatic experiences could also be valuable guides to scientists. This interplay of art and science should be especially fruitful when applied to a question like turbulence that is so hard to grasp, so elusive and ephemeral yet also governed and permeated by an underlying regularity. How, one wonders, might Heisenberg have fared if he had set aside his calculations and picked up a sketchbook?





Philip Ball is the author of Serving the Reich and many books on science and art.