Pretty much all scientific histories of the problem of turbulence start in the same place: with the sketches of wild water flows made by Leonardo da Vinci in the 15th century. What Leonardo was up to was rather profound. In the words of art historian Martin Kemp, Leonardo regarded nature “as weaving an infinite variety of elusive patterns on the basic warp and woof of mathematical perfection.”


Leonardo was trying to grasp those patterns. So when he drew an analogy between the braided vortices in water flowing around a flat plate in a stream, and the braids of a woman’s hair, he wasn’t just saying that one looks like the other—he was positing a deep connection between the two, a correspondence of form in the manner that Neoplatonic philosophers of his age deemed to exist throughout the natural world. He saw fluid flow as a static, almost crystalline entity: His sketches have a solidity to them, seeming almost to weave water into ropes and coils.

Yet what mattered was not the superficial and transient manifestations of these forms but their underlying essence. Leonardo didn’t imagine that the artist should be painting “what he sees,” but rather what he discerns within what he sees. It behooves the artist to invent: painting is “a subtle invention with which philosophy and subtle speculation considers the natures of all forms.” Not a bad definition of science either.

Still, it would take centuries for science to develop Leonardo’s ideas about turbulent flow. It’s not hard to see why. When you look at a turbulent flow—cream being stirred into coffee or a jet of exhaled air traced out in the smoke of a cigarette—you can see that it is full of structure, a profound sort of organization made up of eddies and whirls of all sizes that coalesce for an instant before dissolving again. That’s rather different to what we imply in the colloquial use of the word to describe, say, a life, a history, or a society. There we tend to mean the thing in question is chaotic and random, a jumble within which it is difficult to identify any cause and effect. But pure randomness is not so hard to describe mathematically: It means that every event or movement in one place or at one time is independent of those at others. On average, randomness blurs into dull uniformity. Uniformity that is the veil we see when we hypothesized our inductive narratives about the world we see.

A turbulent flow is at some scales different: It does have order and coherence, but an order in constant flux. Flows of fluids—liquids and gases—generally become turbulent once they start flowing fast enough. When they flow slowly, all of the fluid moves in parallel, rather like ranks of marching soldiers. But as the speed increases, the ranks break up and swirls and eddies begin to form.

water - train of vortices

This transition to turbulence doesn’t happen at the same flow speed for all fluids—more viscous ones can be “kept in line” at higher speeds than runny ones. For flow down a channel or pipe, a quantity, which I posted earlier, called the Reynolds number determines when turbulence appears. Roughly speaking, this encodes the ratio of the flow speed to the viscosity of the fluid. Turbulence develops at high values of the Reynolds number.

Many of the flows we encounter in nature—in rivers and atmospheric air currents like the jet streams—have high Reynolds numbers.

Turbulence provides a perfect example of why a problem is not solved simply by writing down a mathematical equation to predict it.

Such equations exist for all fluid flows, whether laminar or turbulent: They are called the Navier-Stokes equations, and they amount largely to an expression of Isaac Newton’s second law of motion applied to fluids. These equations are the bedrock of the modern investigation of flow in the science of fluid dynamics.

The problem is that, except in a few particularly simple cases, the equations can’t be solved. Yet it’s those solutions, not the equations themselves, that describe the world. What makes the solutions so complicated is that, crudely speaking, each part of the flow depends on what all the other parts are doing. When the flow is turbulent, this interdependence is extreme and the flow becomes chaotic, in the technical sense that the smallest disturbances at one time can lead to completely different patterns of behavior at a later moment.

The constant appearance and disappearance of pockets of organization in a disorderly whole has a beautiful, mesmerizing quality and the perception of order and predictability.

In the 1940s Kolmogorov calculated how much energy is bound up in the eddies of different sizes, showing that there is a rather simple mathematical relationship called a power law that relates the energy to the scale: Each time you halve the size of eddies, the amount of energy contained in all the eddies of that size decreases by some constant factor. This idea of turbulence as a so-called spectrum of different energies at different size scales is one that was already being developed by Heisenberg’s work on the subject. It’s a very fruitful and elegant way of looking at the problem, but one in which the actual physical appearance of turbulent flow is subsumed into something much more recondite. Kolmogorov’s analysis can supply a statistical description of the buffeting, swirling masses of gases in the atmospheres of planets—but what we see, and sometimes what concerns us most, is the individual vortices of a tropical cyclone on Earth or the Great Red Spot on Jupiter.


The photographs of complex flow forms, of turbulent plumes and interfering waves and rippled erosion features in sand, in Schwenk’s 1963 book Sensitive Chaos, offered a reminder that this was how flow manifests itself to human experience, not as an energy spectrum or hierarchical cascade. Such images seem to insist on a spontaneous natural creativity that is a far cry from the deterministic mechanics of a Newtonian universe or our unyielding yearn to systematically smooth out the jaggedness universe.