A feeling for flow

Here is a slightly different version of my article for Nautilus on turbulence – in particular, with several more images, as this piece seemed to cry out for them.

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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 particular subject was Werner Heisenberg.

Heisenberg went on to be a key founder of quantum theory, for which work he 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 he engineered the “Copenhagen Interpretation” of what quantum theory means, 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 later 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 actually 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 afterwards. 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 says that he was once asked what he would ask God. (Whether Heisenberg possessed a sufficiently unblemished character to get him a divine audience is a matter that still divides historians, but the story implies he had no misgivings about that.) “When I meet God”, he is said to have replied, “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.” You get the point: turbulence, an ubiquitous and eminently practical problem in the real world, is frighteningly hard to understand.

It’s still, almost a century after Heisenberg, a cutting-edge problem in science, exemplified by the award of the 2014 Abel Prize for mathematics – often seen as the “Nobel of maths” – to the Russian mathematician Yakov Sinai, in part for his work on turbulence and chaotic flow.

Yet I propose that, to fully articulate and turbulence, we need the perspectives of both intuitive description and detailed analysis – both art and science. It is no coincidence that the science of turbulence has often been forced to fall back on qualitative accounts, while art that celebrates turbulence sometimes resembles a quasi-scientific gathering of data and idealization of form. 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 that there is a lot going on and that everything depends on everything else – and that a reductionist approach therefore has limits. 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 but is not wholly contained within it. Turbulence has to be experienced to be grasped.

Into the storm

It’s not hard to see why turbulence has been so hard for science to understand – and I mean that literally. When you look at a turbulent flow – cream in stirred 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, a society. Here we tend to mean that the thing on 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. This 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.


Turbulence: complicated, chaotic, but not random

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: this is laminar flow. But as the speed increases, the ranks break up: you could say that the “soldiers” 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 very runny ones. For flow down a channel or pipe, a quantity called the Reynolds number determines when turbulence appears: roughly speaking, this is 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 nineteenth century provided the foundation for Heisenberg’s work.

Many of the flows we encounter in nature have high Reynolds numbers, for example in rivers and atmospheric air currents like the jet streams. 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 = 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 inter-dependence is extreme and the flow therefore becomes chaotic, in the technical sense that the smallest disturbances at one time can lead to completely different patterns of behaviour at a later moment.

Observation and invention

Pretty much all scientific histories of the problem of turbulence start in the same place: with the sketches of turbulent flow made by Leonardo da Vinci in the fifteenth century. For the most part these commentaries do not really know what to do with Leonardo’s efforts, other than to commend him for his careful observation before leaping ahead to the more recognizably scientific work on turbulence by Reynolds. Artists, meanwhile, sidelined Leonardo’s schematic representations in favour of a more impressionistic or ostensibly realistic play of light and movement in chaotic waters – not until the Art Nouveau movement do we see something like Leonardo’s arabesque sketches return.


Leonardo’s drawing of turbulence in an artificial waterfall.

But 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.” He 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. For the artist as much as the scientist, what mattered was not the superficial and transient manifestations of these forms but their underlying essence. This is why Leonardo didn’t imagine that the artist should be painting “what he sees”, but rather, what he discerns within what he sees. It therefore 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, when you think about it.



Sketches of complex flows in water by Leonardo da Vinci (top), in which he saw analogies with braided hair (bottom).

As something approaching a Neoplatonist himself, Leonardo saw this implicit order in fluid flow as a static, almost crystalline entity: his sketches have a solidity to them, seeming almost to weave water into ropes and coils. There can be a similar frozen tangibility to the depictions of turbulent flow in East Asian art, some of which predate Leonardo by several centuries. The early Qing Dynasty painter Shitao in the late seventeenth century 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, working within a context that idealized the artistic contemplation of the Yangtze and the other great waterways, these flow forms are mostly those one can find in rivers and streams. On the island nation of Japan, beset by tsunamis, it is instead the ocean’s waves that supply the archetypes, most famously in the prints of Hokusai.

For Chinese artists, the forms of turbulent flow were defined not by a static but by a dynamic principle: 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 thus 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.

A new confluence?

One can’t help noticing how several of these images in East Asian art resemble the attempts of modern fluid dynamicists to capture the essentials of complex flow in so-called streamlines, which, to a rough approximation, trace out the trajectories of particles borne along in the flow.



Images of water from the seventeenth-century The Mustard-Seed Garden Manual of Painting (top), and streamlines in modern computer simulations of turbulent flow (bottom).

Are these resemblances more than superficial and coincidental? I think so: they express a recognition both 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 twentieth century this “deep structure” of turbulence became increasingly an abstract, scientific notion. One of the key advances in the science or turbulence came from the Soviet mathematical physicist Andrei Kolmogorov, under whose guidance Yakov Sinai began his work in the 1950s. By this time turbulence as 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. 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 atmosphere of Earth or Jupiter – but what we see, and sometimes what concerns us most, is the individual vortices of a tropical cyclone or the Great Red Spot.



Out of turbulence: a cyclone and Jupiter’s Great Red Spot.

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 towards 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 a 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.

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.”


“Flow forms” at a Californian biodynamic vineyard inspired by Theodor Schwenk’s work.

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 images of vortices and waves in primitive art, such as the stone carvings on the Bronze Age burial chamber at Newgrange in Ireland, were intuitions of the fecund cosmic language of flow forms.


Are these swirling forms engraved in rock at the Bronze Age chamber at Newgrange prehistoric intuitions of flow patterns?

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 tow path 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.”


Waves meeting and mingling in Theodor Schwenk’s Sensitive Chaos (1965)

When she moved to Dartmoor 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 in southwest England 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 even techniques, there is very little distance between what Derges did here and what an experimental scientist might do: such “shadowgraphs” of flow structures are commonly used as data in fluid dynamics. But for Derges this ‘data gathering’ becomes an artistic moment.


Susan Derges, image from River Taw Series (1997-9).

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, like Leonardo, she often reduces to their abstract essence and transforms into something more permanent and rigid. Because much of her work involves 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 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).


Athena Tacha, Eddies/Interchanges (Homage to Leonardo) (1977).


Athena Tacha, Marianthe (1985-6; brickwork and cedar), Fort Myers, Florida (now destroyed).


Athena Tacha, Green Acres (1985-7), Trenton, New Jersey.


Athena Tacha, Chaos (1998; work in progress).

“I think I respond to turbulence because I am generally interested in fluid forms that evoke the state of ‘chaos’ in nature”, says Tacha – “which I consider a different kind of order, with constant irregularities and changes, but ultimately extremely organized.” 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 Hokusai’s Great Wave 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 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: as he puts it, the water “spontaneously draws lines”. Only the right ink and the right (Japanese) paper will work, and it took years of experimentation to refine the technique.



Hokusai, The Great Wave (c.1830) (top), and Athena Tacha, Wave (2004-5; lead sheet and silicone sealant) (bottom).

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.”



Flow forms captured in ink on paper by Goh Shigetomi.

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 at first hand, are richer and more subtle than anything I have seen in a ‘strictly scientific’ photograph – there is only one printmaker in all of Japan that can reproduce the images with sufficient fidelity. They seem to conjure up much more than a cold physical trace of the technical process of their production.

Shigetomi denies any connection to the traditions of East Asian art, finding more in common with Leonardo, whose drawings he has examined in the notebooks of the collection housed at the royal Windsor Castle in England. But I find it hard not to see these “water figures” as in some sense 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 in his case 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 had to develop 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 mathematician tour-de-force not by poring over his desk into the small hours but by leaning over the Pont-Neuf in Paris and watching, for hour after hour, the eddies of the Seine surging around the piles.

A sense of order and chaos

There is, however, a still 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 one of the most 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 a hundred 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.


Vincent van Gogh, Starry Night (1889).

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 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). In other words, these paintings offer a way to visualize an otherwise recondite and hidden regularity of turbulence: they show us what Kolmogorov turbulence “looks like”.

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 the team leader Jose Luis 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. And here, even what might seem to be a flawed or mystical view of the natural world can offer guidance towards useful insights. How, for example, did Leonardo manage to produce sketches of the aerial topography of mountains laced by river networks that look almost identical to modern satellite images? He was surely guided by his Neoplatonic conviction of a correspondence between the microcosm of the human body and the macrocosm of the wider world: when he spoke of rivers as being the “blood of the earth”, it wasn’t just a visual pun on the resemblance to vein networks.



Leonardo’s sketch of the topography of northern Italy (top), and a modern satellite image of mountainous terrain carved by rivers (bottom).

As scientists strive to make sense of ever more complex phenomena such as turbulence, then, perhaps it is worthwhile listening to what artists think about them. As Derges puts it, “I feel there will probably always be a movement back and forth between the controlled and chaotic environments of simulated and real fluid events in order to be able to make images that communicate something of the mystery of what lies behind the visible.” The most revealing images of flow patterns, she says, “need to be situated in between something that has been closely observed and something that has been emotionally experienced.”

That something which is “emotionally experienced” should find any place in science might horrify some scientists. It needn’t. We now know that emotional experience plays a significant role in cognition: it can be a part of what allows us to grasp the essence of what happens. There are researchers who already accept the value of this. Last fall, for example, physical oceanographer Larry Pratt of the Woods Hole Oceanographic Institution in Massachusetts and performing artist Liz Roncka led a workshop near 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.” His hope is that such an 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. It seems unlikely that Heisenberg’s quest will ever be completed until we cultivate a feeling for flow.