The chaotic fluid motion known as turbulence is notoriously difficult to describe precisely and to understand theoretically. In Physical Review Letters researchers report the first detection of a type of turbulence posited theoretically several decades ago, in which fluid flow can be modeled as a collection of the individual wave motions known as solitons. Such clean observations of predicted phenomena are rare in the study of turbulence, so experts see it as an important step. But some theoretical puzzles remain.
A soliton in water is a single “hump” that propagates without changing shape and can appear when the water depth is not much greater than the wave amplitude (see Focus Landmark from 2013). Solitons are solutions of the Korteweg-deVries (KdV) equation, a nonlinear equation that governs wave motion in this situation.
In general, turbulent flow consists of waves and eddies co-existing on many length scales, forming a constantly changing pattern that can’t be represented as a sum of simpler motions. Over years ago, Vladimir Zakharov, working at what is now the Budker Institute for Nuclear Physics in Novosibirsk, Russia, conceived of a novel kind of turbulence that could appear in situations ruled by the KdV equation. He described collective fluid motion that looked like typical turbulence but was in fact the aggregate result of numerous solitons of different sizes . But so far no one has observed soliton turbulence.
Al Osborne, who runs the Nonlinear Waves Research Corporation in Arlington, Virginia, and his colleagues have been working with a wave measuring team led by Don Resio of the University of North Florida in Jacksonville. Resio’s team had measured wind-driven wave action during a 2002 storm in Currituck Sound, a shallow stretch of ocean water between North Carolina’s Outer Banks and the mainland. The dataset is a string of sea surface height measurements, recorded about 5 times per second over a period of hours. The water depth was meters, with wave heights averaging about half a meter.
Researchers describe turbulence using a plot called a power spectrum that shows the wave energy in the fluid motion at each wave frequency. Analysis of the Currituck Sound dataset yielded a surprisingly simple power spectrum. For low frequencies, from up to Hz, the power diminishes with the reciprocal of the frequency. Above Hz the power rises to a sharp peak and then declines again at higher frequencies. The low-frequency spectrum agreed with an earlier prediction by Zakharov for the behavior of so-called weak wave turbulence in shallow water, under conditions far from those in which solitons are expected to appear .
Nevertheless, Osborne and his colleagues looked for evidence of solitons using a mathematical technique called finite gap theory, which he and other colleagues had previously used to reveal isolated solitons in the Adriatic Sea . For the Currituck Sound data, the researchers decomposed the low-frequency turbulence into a set of distinct solitons. In a typical 20-minute section of the data, they identified as many as 120 solitons, with a few reaching heights of 0.1 meter or more. The duration of the average soliton peak was about 10 seconds.
The researchers say this is the first observation of soliton turbulence. Currituck Sound has just the right depth that there was a clear demarcation between waves controlled by the KdV equation and those at higher frequencies that follow a deep water equation—a happenstance that was “a gift from the gods,” Osborne says. He believes that storm winds generated conventional water waves mostly near the peak of the power spectrum (0.5 Hz) and that interactions redistributed this energy into disturbances at higher and lower frequencies.
The fact that the low-frequency spectrum matches Zakharov’s prediction for weak wave turbulence seems to be simply a coincidence, says Gennady El of Loughborough University in the UK. El is currently working on a theoretical explanation for what he calls Osborne’s “dramatic” results. Efim Pelinovsky of the Institute of Applied Physics of the Russian Academy of Sciences in Nizhny Novgorod agrees that the theoretical situation is unclear, but he says recent numerical work by himself and others is consistent with the new observations and with Osborne’s interpretation. If the wave motion instigating turbulence has a narrow power spectrum, Pelinovsky says, then his simulations show that interactions among solitons downshifts them to lower frequencies.
David Lindley is a freelance writer in Alexandria, Virginia, and author of Uncertainty: Einstein, Heisenberg, Bohr and the Struggle for the Soul of Science (Doubleday, 2007).
- V. E. Zakharov, J. Exp. Theor. Phys. 33, 538 (1971).
- V. E. Zakharov, J. Exp. Theor. Phys. 11, 10 (1967).
- A. R. Osborne, E. Segre, G. Boffetta, and L. Cavaleri, “Soliton Basis States in Shallow-Water Ocean Surface Waves,” Phys. Rev. Lett. 67, 592 (1991).