Chaos - James Gleick [59]
For the sake of a clean description, Kolmogorov imagined that these eddies fill the whole space of the fluid, making the fluid everywhere the same. This assumption, the assumption of homogeneity, turns out not to be true, and even Poincaré knew it forty years earlier, having seen at the rough surface of a river that the eddies always mix with regions of smooth flow. The vorticity is localized. Energy actually dissipates only in part of the space. At each scale, as you look closer at a turbulent eddy, new regions of calm come into view. Thus the assumption of homogeneity gives way to the assumption of intermittency. The intermittent picture, when idealized somewhat, looks highly fractal, with intermixed regions of roughness and smoothness on scales running down from the large to the small. This picture, too, turns out to fall somewhat short of the reality.
Closely related, but quite distinct, was the question of what happens when turbulence begins. How does a flow cross the boundary from smooth to turbulent? Before turbulence becomes fully developed, what intermediate stages might exist? For these questions, a slightly stronger theory existed. This orthodox paradigm came from Lev D. Landau, the great Russian scientist whose text on fluid dynamics remains a standard. The Landau picture is a piling up of competing rhythms. When more energy comes into a system, he conjectured, new frequencies begin one at a time, each incompatible with the last, as if a violin string responds to harder bowing by vibrating with a second, dissonant tone, and then a third, and a fourth, until the sound becomes an incomprehensible cacophony.
Any liquid or gas is a collection of individual bits, so many that they may as well be infinite. If each piece moved independently, then the fluid would have infinitely many possibilities, infinitely many “degrees of freedom” in the jargon, and the equations describing the motion would have to deal with infinitely many variables. But each particle does not move independently—its motion depends very much on the motion of its neighbors—and in a smooth flow, the degrees of freedom can be few. Potentially complex movements remain coupled together. Nearby bits remain nearby or drift apart in a smooth, linear way that produces neat lines in wind-tunnel pictures. The particles in a column of cigarette smoke rise as one, for a while.
Then confusion appears, a menagerie of mysterious wild motions. Sometimes these motions received names: the oscillatory, the skewed varicose, the cross-roll, the knot, the zigzag. In Landau’s view, these unstable new motions simply accumulated, one on top of another, creating rhythms with overlapping speeds and sizes. Conceptually, this orthodox idea of turbulence seemed to fit the facts, and if the theory was mathematically useless—which it was—well, so be it. Landau’s paradigm was a way of retaining dignity while throwing up the hands.
Water courses through a pipe, or around a cylinder, making a faint smooth hiss. In your mind, you turn up the pressure. A back-and–forth rhythm begins. Like a wave, it knocks slowly against the pipe. Turn the knob again. From somewhere, a second frequency enters, out of synchronization with the first. The rhythms overlap, compete, jar against one another. Already they create such a complicated motion, waves banging against the walls, interfering with one another, that you almost cannot follow it. Now turn up the knob again. A third frequency enters, then a fourth, a fifth, a sixth, all incommensurate. The flow has become extremely complicated. Perhaps this is turbulence.