If you’re a frequent flyer, the word “turbulence” will have you reaching for your seatbelt. It’s a term that, for many, begins and ends with closed food trays and shaken bellies. In the world of fluid dynamics, however, it’s a slippery phenomenon that’s observed in everything from cigarette smoke to rivers flowing over rocks. Yet despite this ubiquity, it is exceptionally hard to pin down in precise mathematical terms.

Last week, a paper published in *Science* attempted to elucidate the mysteries of turbulence, offering crucial evidence for how energy moves and dissipates around turbulent fluids. The researchers’ findings could help a whole range of scientific fields, from the works of climatologists to aeronautical engineers. But what is turbulence, and why has it proven difficult to explain?

## What is turbulence?

When you pour milk into your morning coffee, you’ll notice that what starts as an orderly flow quickly becomes seemingly chaotic swirls. The same goes for the smoke stemming from the tip of a cigarette – it rises upwards in a straight line but then curls into branching twists, slowing down the spread of the gas so that it lingers in front of your face.

These are what’s known as turbulent flows, in contrast to the laminar flow of those first straight moments of cigarette smoke. With laminar flow, the particles of a fluid (liquid or gas) travel in parallel layers, following a smooth path; not interfering with each other. With turbulent flow, excessive kinetic energy means these particles do interfere with each other, leading to whirlpools and varying velocities in different parts of the fluid.

A significant factor here is viscosity. If a fluid has a low viscosity (say, milk compared to honey), it will have less of a so-called damping effect. This essentially means it will be easier for that flow to become turbulent, rather than maintaining a streamlined, laminar flow. In either case, a dimensionless quantity called the Reynolds number can help to predict when a flow will change from being laminar to turbulent – but even with this it is incredibly hard to model fluid flow.

Another key piece of mathematics, called the Navier-Stokes equations, describes the flow of fluids, but these are exceptionally hard to solve in practice, and so engineers and scientists tend to use simplified theoretical models when trying to design things that may involve turbulence, such as aeroplanes and pipes.

### What causes turbulence on planes?

When in the air, planes will often encounter clear-air turbulence (CAT), which is the most common form of turbulence for pilots to deal with.

CAT is most often found around jet streams, which are fast-moving flows of air that can travel at speeds of up to 250mph. Depending on which way the plane is travelling, and whether this is in the same direction as the jet stream, pilots will either want to avoid the incoming wind, or use its energy as a tailwind to increase speeds.

If you imagine the jet stream as an invisible, airborne river, it will make turbulence easier to visualise. Like water, the flow of a river will break into eddies and swirls that move at different velocities. These swirls of air, travelling at different speeds, will put force on the aeroplane – making it judder and meander.

The good news is that, while turbulence can feel dramatic, it’s very unlikely to be dangerous to a plane and its occupants. Commercial aeroplanes are designed to withstand a remarkable amount of stress, so you don’t need to worry about the wing being ripped off. The best thing you can do is strap in.

### How can we create a turbulence forecast?

In terms of flight, there are certain predictions that can be made about where turbulence will occur, such as altitude and whether the path goes over a mountain range. Areas where air of different speeds meet, like the Intertropical Convergence Zone above Africa, are also prone to turbulence. Aside from these general turbulence forecasts, it’s difficult to reliably predict exactly where turbulence will occur.

In terms of the science, modelling the movements of turbulent flow is one of physics’ great mysteries. In fact, it has proven so difficult to understand that the Clay Mathematics Institute has said solving the Navier-Stokes equations is one of seven most important open problems in mathematics, known as Millennium Prize Problems. The first person to solve each of these problems will be given $1,000,000.

In the paper published this month in Science, the team from the Technical University of Madrid claims it has been able to simulate for the first time how turbulence spreads a fluid’s kinetic energy across eddies and swirls of smaller and smaller scale. The researchers call this a “turbulence cascade”, which references a theory by Russian mathematical physicist Andrei Kolmogorov in the 1940s. This theory states that the large vortexes of a turbulent flow break down into increasingly smaller ones, in a fractal fashion.

“In this model, the transfer of kinetic energy occurs rather like a baton being passed around runners in a relay race,” one of the authors of the paper, aeronautical engineer José Cardesa, told* Nature*. “But one in which the runners get progressively smaller and more numerous.”

While it might not quite solve the Navier-Stokes equations, the latest research could be an important part to a puzzle that has eluded scientists and mathematicians for centuries. If a simpler, fuller model can be found, it would have a big impact on everything from developing energy sources from wave energy, to modelling aircraft. And that means better ways to forecast clear-air turbulence – the type that knocks your tiny bottle of wine onto the floor.

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