Physicists Make Progress On The Bumpy Ride To Understanding Turbulence

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Illustration of the Jovian moon Io, seen against the backdrop of Jupiter’s Great Red Spot. The latter is a vast, cyclonic storm – wider than the entire Earth – that has raged for centuries. Io, a highly volcanic world, is the innermost Galilean moon of Jupiter.

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Turbulence is a big problem — not just for airline passengers, but for physicists and mathematicians. The phenomenon of random, chaotic motion in a fluid is still really hard to describe mathematically. Physicists are still wrangling with problems like exactly how to predict the details of chaotic motion to how to predict when, and how quickly, a fluid flow will go from smooth and orderly to turbulent.

One of the lingering mysteries of turbulence is why, in fluids that only flow in two dimensions (picture the surface of a bubble), the many vortices tend to merge into larger ones. If you want a real-world example, check out Jupiter’s Great Red Spot; it’s a single large vortex that formed from the turbulent eddies of the gas giant’s upper atmosphere. A pair of new studies sheds a little light on the phenomenon.

Order emerges from chaos

The new studies offer some validation for an idea proposed by physicist and chemist Lars Onsager back in 1949. He was studying liquid helium, which has a cool property called superfluidity. Liquid helium and other superfluids have zero viscosity, or internal friction; in other words, the molecules in a superfluid don’t stick together.

In practice, this means that very viscous liquids (like honey, cake batter, or oil) seem thick and resist disturbances from stirring or sloshing. If you swirl a spoon around in a jar of honey, you can stir it, but you’re not going to be able to sit there and watch eddies and swirls in the honey after you take the spoon out. Because there’s so much friction between the molecules, the energy from the stirring dissipates quickly, and the honey settles back into sticky, smooth stillness. On the other hand, if you stir a jar of water, the water will keep swirling around for a while, because there’s less friction between its molecules — but there’s SOME friction, so eventually, the energy will dissipate and the water will settle down. But in a superfluid, like liquid helium, there’s NO friction between the molecules, so if you stir it, it will just keep on swirling, because there’s no friction to dissipate the energy as the molecules slide past each other.

What Onsager proposed is that if you take a turbulent two-dimensional superfluid and add energy to it, all the seemingly random eddies and vortices will eventually merge into larger, more stable vortices. Order gradually emerges from the apparent chaos of turbulence — which is weird for physics, because randomness and disorder (also called entropy) usually increase when you add energy to a system. Think about it: energy usually translates to motion, and the more things bounce around and interact, the more difficult it gets to predict the motion of a particular particle.

In any turbulent fluid, the movement of particles moves energy around the system, which in turn impacts the movement of other particles. In a 3D fluid, the tendency is to move energy on smaller and smaller scales, so that the vortices get smaller and less stable over time. But for some reason, in a 2D fluid, the reverse happens: “Energy flows toward the

largest length scales available, resulting in system scale, persistent vortex flows. This behavior has been observed in systems ranging in scale from soap films to Jupiter’s atmosphere,” wrote Monash University physicist Shaun Johnstone and his colleagues.

Onsager’s proposal applies to normal fluids in two-dimensions, as we’ve seen with the Great Red Spot and cyclones in our own atmosphere, not to mention the pretty swirls on the surface of soap bubbles, but the math works out most neatly with 2D superfluids. That includes things like the flow of neutrons on the surface of neutron stars or electrons in semiconductors.

Rubidium? I’ve never even met him!

To test Onsager’s theory, two separate teams of physicists decided to do what any serious scientist confronted with a problem might do: shoot something with a laser (well, sort of).

Both teams produced what’s called a Bose-Einstein condensate, which essentially means cooling a gas so much that it condenses into a superfluid. That happens at almost absolute zero (the temperature at which there’s so little energy that even the tinest atomic motions cease), which means you probably shouldn’t try it at home, because it almost certainly won’t work (if it does, please email me). The coolest thing about Bose-Einstein condensates isn’t their temperature, though; it’s the fact that certain quantum effects, which usually only show up at a microscopic scale, become visible at a normal, macroscopic scale. For instance, turbulent vortices whose centers would measure just hundred-millionths of a centimeter across instead can be measured in micrometers, or millionths of a centimeter. That’s still tiny, but it’s big enough for scientists to image directly and see what’s going on.

By dragging a grid of laser beams through a cloud of rubidium atoms, Shaun Johnstone and his team kick-started turbulence in their two-dimensional film of rubidium. At first, the turbulence looked pretty wild: some vortices’ nearest neighbors were spinning in the opposite direction, but just as many were hanging out by themselves or close to a vortex rotating the opposite way. Over time, though, vortices rotating in the same direction started to form clusters. Order was beginning to emerge from the chaos, and at the same time, measurements showed that on average, the energy in each vortex was increasing.

Why? As the pairs of opposing vortexes “annihilated” each other, Johnstone and his colleagues suggest, they released energy which then flowed into clusters of vortices that rotated in the same direction.

Meanwhile at the University of Queensland, physicist Tyler Neely and his colleagues watched what happened to two large clusters of vortices, each flowing in the opposite direction. In physics-speak, the fluid was “out of equilibrium,” which means that energy wasn’t evenly spread out through the fluid, but was instead still flowing around and making particles move, powering the vortices. But even in a system that was totally out of equilibrium, the vortex clusters were surprisingly stable over time.

And that, in short, validates what Onsager proposed back in 1949.

The million-dollar question (well, one of them)

Physicists now understand at least one really specific aspect of turbulence a little better, but turbulence in general is still a pretty vexing problem for people who like to be able to describe the world very precisely with numbers and equations. As long as you just want a general, larger-scale idea of how a fluid is going to flow, modern physics has a pretty decent handle on the turbulence problem. But it’s still nearly impossible to predict the exact path and energy of any given molecule swirling and eddying its way around the turbulent surface of a soap bubble.

That’s because eddies in a turbulent fluid split up the initial kinetic energy in the flow – passing it around and concentrating it unevenly, and it’s complex to predict exactly how that’s going to happen. The standard equations that describe how fluids move, called the Navier-Stokes equations, work very reliably to predict how molecules will move along in a smooth, orderly laminar flow of fluid. But they don’t do nearly as good a job at predicting all the chaos of turbulent flows, because of the complexity of how all those vorticies and eddies interact and exchange energy. Too much turbulence basically breaks the equation — or does it? Physicists and mathematicians still aren’t sure.

The Clay Mathematics Institute will award one of its million-dollar Millennium Prizes to anyone who can either prove that the equations will always work, no matter how turbulent the system gets or how many iterations they’re put through, or that they’ll eventually fail and start spitting out crazy results like molecules moving at infinite velocity. They’re not even asking for a perfect mathematical definition of turbulence, just proof of whether the Navier-Stokes equations can or can’t actually describe the fluid.

“After nearly 200 years of experiments, it’s clear the equations work: The flows predicted by Navier-Stokes consistently match flows observed in experiments. If you’re a physicist working in a lab, that correspondence might be enough,” wrote Quanta’s Kevin Hartnett in 2013. “But mathematicians want to know more than that — they want to be able to check if one can follow the equations all the way through, to see exactly how a flow changes moment by moment (for any initial configuration of a fluid) and even to pinpoint the onset of turbulence.”

And that’s yet another challenge. At the moment, we can’t predict the exact moment an orderly, smooth, laminar flow becomes a turbulent flow, or how quickly it will happen. All we have are what Ars Technica‘s Lee Phillips called “rules of thumb” from experiments and real-world observations.

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