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It may appear random to the casual observer, but turbulence nonetheless follows a cascading pattern that can be studied and, at least partially, explained using mathematical equations.

“Imagine you are standing on a bridge, and you watch the river flow. You will see swirls on the surface, and these swirls are not random. They arrange themselves in specific patterns, and these kinds of patterns can be predicted by physical laws,” said Yongxiang Huang, lead authorof the study that published Tuesdayin the scientific journal Physics of Fluids. Huang is a researcher at State Key Laboratory of Marine Environmental Science & College of Ocean and Earth Sciences at Xiamen University in southeastern China.

“The Starry Night” is an oil-on-canvas painting that, the study noted, depicts a view just before sunrise from the east-facing window of the artist’s asylum room at Saint-Rémy-de-Provence in southern France. Van Gogh had admitted himself to an asylum there after mutilatinghis left ear.

Using a digital image of the painting, Huang and his colleagues examined the scale of its 14 main whirling shapes to understand whether they aligned with physical theories that describe the transfer of energy from large- to small-scale eddies as they collide and interact with one another.

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In the 1940s, Soviet mathematicianAndrey Kolmogorovdescribed a mathematical relationship between the fluctuations in a flow’s speed and the rate at which its energy dissipates.

Huang and the team also found that the paint, at the smallest scale, mixes around with some background swirls and whirls in a fashion predicted by turbulence theory, following a statistical pattern known as Batchelor’s scaling. Batchelor’s scaling mathematically represents how small particles, such as drifting algae in the ocean or pieces of dust in the wind, are passively mixed around by turbulent flow.

“This is cool. Indeed this is the type of statistics you would expect from algae blooms being swept around by ocean currents, or dust and particulates in the air,” said James Beattie, a postdoctoral researcher in the department of astrophysical sciences at Princeton University in New Jersey, in an email. Beattie wasn’t involved in this study but has conducted similar research on the artwork.

“In my paper, I only ever really looked at the large (swirls in the painting), so I didn’t see this second relation,” he said, referring to the Batchelor’s scaling.

Of course, Huang said, van Gogh would not have been aware of such equations but likely he spent a lot of time observing turbulence in nature.

“I think this physical relationship must be embedded in his mind so that’s why when he made this famous ‘Starry Night’ painting, it mimics the real flow,” Huang said.

Beattie agreed: “It’s an amazing coincidence that Van Gogh’s beautiful painting shares many of the same statistics as turbulence,” he said.

“This makes some sense — the models have been constructed to try to capture the statistics of eddies and swirls on multiple scales, each swirl communicating with other swirls through the turbulent cascade. In some sense, Van Gogh painted something that represents this phenomenon, so why shouldn’t there be some convergence between the theoretical models and the statistics of Van Gogh’s swirls?”

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“Unlike ‘The Starry Night,’ this painting lacks well-defined swirling patterns, but the clouds are rich of structures with different scales, resembling those frequently seen in the sky,” the study noted of Constable’s artwork.

On display at theMuseum of Modern Artin New York, “The Starry Night” is an enormously popular work of art that has been recreated inLego bricks,dronesanddominoes.

Huang said that scientists had long struggled to describe turbulent flow in fluid dynamics in a way that would allow them to predict the phenomenon and that a complete explanation remains a prevailing mystery of physics. A thorough understanding would help with weather forecasting, flight turbulence and many other processes, he said.

“Even after more than 100 years (of) study, we even don’t know how to define this complex phenomenon,” Huang said. “It’s extremely important, but it’s extremely difficult.”

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However, Beattie said that he was a huge fan of the work of art and that it reflected universality and the beauty of turbulence.

“I deeply love the fact that I can take my understanding of the turbulence in the plasma between galaxies and apply it to the turbulence between stars, between Earth and the Sun or in our own lakes, oceans and atmosphere,” he said.

“What I take away from studies like this is that (van Gogh) captured some of this universality in the beautiful (‘Starry Night’),” Beattie added. “And I think people know this. They know that something wonderful has been embedded in this painting and we are drawn to it.”

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