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https://www.quantamagazine.org/december-3-2014-learning-to-move-20131209/ December 9, 2013

Could Knots Unravel Mysteries of Fluid Flow?By Natalie Wolchover

Atrefoil vortex knot in water generated by a hydrofoil.

Spaghetti-thin shoelaces, sturdy hawsers, silk cravats — all are routinely tied in knots. So too,physicists believe, are water, air and the liquid iron churning in Earth’s outer core. Knots twist andturn in the particle pathways of turbulent fluids, as stable in some cases as a sailor’s handiwork. Fordecades, scientists have suspected the rules governing these knots could offer clues for untanglingturbulence — one of the last great unknowns of classical physics — but any order exhibited by theknots was lost in the surrounding chaos.

Now, with deft new tools at their fingertips, physicists are beginning to master the art of tying knotsin fluids and other flowable entities, such as electromagnetic fields, enabling controlled study oftheir behavior. “Now that we have these knots, we can measure the shape of them in 3-D; we canlook at the flow field around them,” said William Irvine, a physicist at the University of Chicago. “Wecan really figure out what the rules of the game are.”

Knots and linked loops exist in turbulent fluids like Earth’s outer core because they arise when arotation coincides with a flow. (As the fluid rotates, the particle pathways, or “streamlines,” get

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dragged around and entangled in an effect similar to tying a shoelace.) Investigating knotted fluidsboth on paper and in the lab could provide a much richer picture of how these tangles, once formed,affect the future evolution of the fluids. The researchers say this new means of probing fluid flowcould eventually advance the scientific understanding of the plasma rising off the surface of the sun,thermonuclear fusion, Earth’s interior and atmosphere, and other systems embroiled in turbulence.

“This is all a realization of this dream of understanding fluids in terms of the knots and links of thestreamlines,” said Randy Kamien, a professor of physics and astronomy at the University ofPennsylvania.

Illustrations of knots and links, including a trefoil knot, top left, in an 1869 paper by Lord Kelvin onhis knotted vortex theory of atoms. Related Video: Knot Possible?

The dream began in the 1860s with an ingenious knot theory of nature. Lord Kelvin proposed thatatoms were knotted vortexes swirling in the ether, an invisible, fluidlike medium believed at the timeto fill space. One element would be the simplest knot, called a trefoil, another a figure-eight knot,and so on — and none could transform into another. Though incorrect, Kelvin’s idea spawned thebranch of mathematics known as knot theory and ultimately led to the realization that knots do morethan passively form in fluids; they can have a pivotal, though as yet poorly understood, influence onturbulent fluid dynamics. In seminal work published in 1969, Keith Moffatt, then a young CambridgeUniversity lecturer, proved that the measure of the total knottedness and linkage in ideal fluids —ones, like liquid helium, that lack viscosity — stays constant over time. In viscous fluids, thismeasure, called “helicity,” fluctuates, and knots can transform or unravel. But scientists still don’tknow when and why helicity dissipates.

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William Irvine of the University of Chicago.

“There is a vast literature about what happens to knottedness in fluids, but it has been really hard todo experiments for a long time,” Irvine said. “It wasn’t until recently that we got these great toolsfor making and measuring things in 3-D, which is essential for knots.”

Earlier this year, Irvine’s team used water displacing objects called hydrofoils, created through 3-Dprinting, to fashion a trefoil knot out of a water vortex — the first vortex knot ever created in the lab.Using lasers, Kamien’s group constructed a knotlike structure in liquid crystals, the self-aligningfluids found in LCD television screens. And a third group — led by Mark Dennis, a theoreticalphysicist at the University of Bristol in the United Kingdom — tied knots in filaments of darknessswirling inside laser beams.

Alongside the experimental advances, researchers have also formulated new mathematicaldescriptions of knotted fluids and fields that can be analyzed on paper rather than in the lab.

Electromagnetic fields — entities that fill space and oscillate at different frequencies, some of whichour eyes perceive as light — are mathematical solutions to a set of laws known as Maxwell’sequations. As reported in October in Physical Review Letters, Irvine and his colleagues HrideshKedia, Iwo Bialynicki-Birula and Daniel Peralta-Salas discovered a large class of solutions in whichthe contours of the electromagnetic fields, called “field lines,” twist and turn in knots.

A static, knotted electromagnetic field was derived in the 1990s, but “the new work is much moregeneral,” said Moffatt, now a professor emeritus of mathematical physics at Cambridge. “Theyprovide a technique for finding a really huge variety of knots.”

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A Knotty Picture

To visualize a knotted electromagnetic field, imagine three-dimensional space partitioned into doughnut-shaped tori of a continuous range of sizes, nested like Russian dolls. Collections of field lines (shown above inorange and blue) form the surface of each torus. In the trefoil knot field, each line wraps twice around theperimeter of the torus and three times through the center. As a result, any two lines on the same torus areknotted, and any two lines from different tori are knotted too. “It gives you a way of filling space with knots,”Irvine said. Then, as the field propagates through space and time, “these knots travel with the light.”(Illustration: Irvine Lab)

Irvine and coauthors will show in forthcoming work that there are corresponding knotted solutionsto Euler’s equations, which govern ideal fluids. Because they have zero viscosity, these fluids flowperfectly smoothly, much like the light fields studied by the researchers. “It illustrates that we canbe talking about very different physical systems with the same sorts of solutions,” Dennis noted. Thisequivalence means that if physicists discover the principles behind knots in Earth’s core, the samerules should apply to the tangled vortexes near an airplane wing.

The knotted light fields that Irvine and his colleagues derived on paper may be realizableexperimentally, he said, within a tightly focused and polarized laser beam. By shining the knottedbeam onto another material, such as plasma, it should also be possible to “transfer the knottednessonto that thing,” he said, enabling controlled study of knots in a range of settings.

At present, almost nothing is experimentally proven about how knots in fluids and fields evolve overtime despite decades of speculation and extensive computer simulations.

“Suppose William [Irvine] made two trefoil knots in a fluid and shot them at each other,” Kamiensaid. “What do they do? How do they interact? That’s completely beyond the scope of what weunderstand.” The answers to these seemingly simple questions, he added, are central to “how fluidswork.”

For starters, when do knots unravel and when do they not? Moffatt proved that helicity staysconstant in zero-viscosity fluids — a law of nature analogous to the conservation of energy infrictionless systems. But just as friction saps energy from a car, particle collisions suck helicity out ofviscous fluids like water and plasma. “We know helicity is not exactly conserved, but how is it notexactly conserved?” Kamien asked. “Nobody really knows.”

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Asupercomputer model of flow patterns in Earth’s liquid iron outer core. The fluid’s complex topologyis believed to play a central role in the “dynamo effect,” which generates the planet’s magnetic field.

The most pressing question is what happens when knotted or linked vortices in a viscous fluid crossand separate — a common process called reconnection. Some researchers hypothesize that link orknot helicity is converted into “twist helicity,” or faster swirling of the vortices, keeping the totalhelicity constant. However, preliminary work by Moffatt and Yoshifumi Kimura, a professor of fluiddynamics at Nagoya University in Japan, suggests that helicity dissipates during reconnection. “It’san open question,” Moffatt said.

Reconnection is central to many turbulent processes, such as feedback between large and smalleddies in Earth’s atmosphere, the heating of the solar corona and the generation of Earth’s magneticfield. In thermonuclear fusion — a solar process in which atoms fuse together, releasing massiveamounts of energy — a turbulent plasma constantly undergoes reconnection as it relaxes to itsminimum energy state. Understanding whether helicity remains constant during this process will

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help researchers correctly model and replicate fusion in the laboratory. “That’s why it’s an importantissue to try to understand,” Moffatt said. “The long-term hope for mankind is to produce energy fromfusion.”

Quantities that are “conserved,” or stay constant in time, “give you powerful ways to look atcomplicated problems,” Irvine explained. “Understanding a new conserved quantity, helicity, couldhave a huge impact on how we understand flows. It’s one of those holy grails.”

Once the rules of knottedness are established, some scientists say it might be possible to harnessthem through clever system design to control turbulence. The findings might suggest, for example, abetter shape for airplane wings. “Could you braid the turbulence, and would that make it possiblefor planes to fly closer together?” Kamien asked. “Turbulence appears to be random. But is theresome way to keep it from being random?”

This article was republished on ScientificAmerican.com.