Vincent van Gogh's most famous painting is The Starry Night (1889), created (along with several other masterpieces) during the artist's stay at an asylum in Arles following his breakdown in December 1888. Where some have seen the swirling vortices of the night sky depicted in Starry Night as a reflection of van Gogh's own inner turmoil, physicists often see a masterful depiction of atmospheric turbulence. According to a new paper published in the journal Physics of Fluids, the illusion of movement in van Gogh's blue sky is also due to the scale of the paint strokes—a second kind of "hidden turbulence" at the microscale that diffuses throughout the entire canvas.

“It reveals a deep and intuitive understanding of natural phenomena,” said co-author Yongxiang Huang of Xiamen University in China. “Van Gogh’s precise representation of turbulence might be from studying the movement of clouds and the atmosphere or an innate sense of how to capture the dynamism of the sky.”

As previously reported, in a 2014 TED-Ed talk, Natalya St. Clair, a research associate at the Concord Consortium and coauthor of The Art of Mental Calculation, used Starry Night to illuminate the concept of turbulence in a flowing fluid. In particular, she talked about how van Gogh's technique allowed him (and other Impressionist painters) to represent the movement of light across water or in the twinkling of stars. We see this as a kind of shimmering effect, because the eye is more sensitive to changes in the intensity of light (a property called luminance) than to changes in color.

In physics, turbulence relates to strong, sudden movements within air or water, usually marked by eddies and vortices. Physicists have struggled for centuries to mathematically describe turbulence. It's still one of the great remaining challenges in the field. But a Russian physicist named Andrei Kolmogorov made considerable progress in the 1940s when he predicted there would be a mathematical connection (now known as Kolmogorov scaling) between how a flow's speed fluctuates over time and the rate at which it loses energy as friction.

That is, some turbulent flows exhibit energy cascades, whereby large eddies transfer some of their energy to smaller eddies. The smaller eddies, in turn, transfer some of their energy to even smaller eddies, and so forth, producing a self-similar pattern at many spatial size scales. Experimental evidence since then showed that Kolmogorov wasn't that far off with his prediction.

In 2019, two Australian graduate students mathematically analyzed the painting and concluded it shares the same turbulent features as molecular clouds (where literal stars are born), based on a 2004 Hubble image of turbulent eddies of dusty clouds moving around a supergiant star. They examined digital photographs of several van Gogh paintings and measured the brightness varied between any two pixels, calculating the probability that two pixels at a given distance would have the same luminance. They found evidence of something remarkably close to Kolmogorov scaling, not just in Starry Night, but also in two other paintings from the same period in van Gogh's life: Wheatfield with Crows and Road with Cypress and Star (both painted in 1890).

Brush strokes at the microscale

Huang is a marine scientist who collaborated with physicists to take a closer look at the turbulent patterns lurking in van Gogh's masterpiece. They focused on studying the spatial scales of the 14 primary whirling vortices in the painting, using the relative brightness of the paint colors as an analog for kinetic energy. Specifically, they precisely measured the typical brushstroke size and then compared those scales to what is predicted by fluid dynamics.

Their findings confirmed the 2019 conclusion that the overall painting closely aligns with Kolmogorov's law. The team also found that, at the microscale, the paint strokes align with a different phenomenon known as Batchelor's scaling, named after Australian mathematician George Batchelor, who specialized in fluid dynamics. It's similar to Kolmogorov's law, except instead of describing the smallest scales of turbulence before viscosity becomes dominant in a system, Batchelor scaling describes the smallest-length scales of fluctuations before diffusion becomes dominant. Per the authors, it's quite rare to find both these kinds of scaling in one atmospheric system.

This is yet more evidence that van Gogh had an exquisitely fine-tuned intuitive sense of turbulence, and he captured it beautifully in Starry Night. There may also be implications for fluid dynamics. “Turbulence is believed to be one of the intrinsic properties of high Reynolds flows dominated by inertia, but recently, turbulence-like phenomena have been reported for different types of flow systems at a wide range of spatial scales, with low Reynolds numbers where viscosity is more dominant,” Huang said. “It seems it is time to propose a new definition of turbulence to embrace more situations.”

Physics of Fluids, 2024. DOI: 10.1063/5.0213627  (About DOIs).