#eddies

(CNN) — The dappled starlight and swirling clouds of Vincent van Gogh’s “The Starry Night” are thought to reflect the artist’s tumultuous state of mind when he painted the work in 1889.

Now, a new analysis by physicists based in China and France suggests the artist had a deep, intuitive understanding of the mathematical structure of turbulent flow.

As a common natural phenomenon observed in fluids — moving water, ocean currents, blood flow, billowing storm clouds and plumes of smoke — turbulent flow is chaotic, as larger swirls or eddies, form and break down into smaller ones.

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 author of the study that published Tuesday in 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 mutilating his 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.

‘The Starry Night’ and turbulence theories

The atmospheric motion of the painted sky cannot be directly measured, so Huang and his colleagues precisely measured the brushstrokes and compared the size of the brushstrokes to the mathematical scales expected from turbulence theories.

To gauge physical movement, they used the relative brightness or luminance of the varying paint colors.

They discovered that the sizes of the 14 whirls or eddies in “The Starry Night,” and their relative distance and intensity, follow a physical law that governs fluid dynamics known as Kolmogorov’s Theory of Turbulence.

In the 1940s, Soviet mathematician Andrey Kolmogorov (1903–1987) described 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.

‘An amazing coincidence’

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?”

The study team performed the same analysis and detected the same phenomenon in two other images:

— a painting, “Chain Pier, Brighton,” created by British artist John Constable in 1826-7;

— a photograph of Jupiter’s Great Red Spot, taken by NASA’s Voyager 1 spacecraft on 5 March 1979.

“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 the Museum of Modern Art in New York, “The Starry Night” is an enormously popular work of art that has been recreated in Lego bricks, drones and dominoes.

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.”

"The fact that “The Starry Night” matched statistical models of turbulence even though the artwork doesn’t actually move could suggest that the statistical methods and tools are less precise than scientists may have thought," Beattie said.

"The painting can’t be precisely measured because it’s “actually not turbulence. … (I)t has no kinetic energy,” he said.

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.