Max Planck Institute for Dynamics and Self-Organization -- Department for Nonlinear Dynamics and Network Dynamics Group
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Counting Catastrophes

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Date: 03.08.2010

When a current is influenced by a spatially varying random force, “catastrophes” occur: At different locations within the flow, lines featuring a drastically enhanced flux density emerge. An example for this type of behavior are monster waves in the ocean that even in relatively calm seas can pile up to a height of more than 25 meters. Scientists from the Max Planck Institute for Dynamics and Self-Organization (MPIDS) in Göttingen (Germany) have now developed a theory that can predict the number of such catastrophes within a flow. Surprisingly, this law is universal, i.e.: it is independent of the physical system.

So-called monster waves in the ocean are driven by tiny fluctuations within the ocean current. Again and again, this irregularity creates gigantic waves. The same principle can be found in the microscopic world: In thin semiconductor layers the electron flow bundles into flow lines of high density. This is caused by tiny impurities within the semiconductor. But regardless of whether water or electrons are flowing, scientists call such catastrophically enhanced flux densities caustics. How many of such caustics arise within a flow? And how strongly can these flow lines branch out?

"Until now, there was no satisfying answer to these questions“, explains Dr. Jakob Metzger from the MPIDS. Only one thing was for sure: Far away from the random force initiating the caustics the number of flow lines with high density increases rapidly. So rapidly, that the lines smudge out into a single diffuse flow. For the first time, the researchers from Göttingen now succeeded in mathematically describing the area close to the random force.

In their calculations the scientists concentrated on two-dimensional flows. “Since both the surface of the ocean and semiconductor layers are in principle two-dimensional, this is not a grave impairment”, explains Metzger. The scientists succeeded in deriving mathematical expressions describing the number of caustics in two limits: very close to the source and far away. By combining both expressions a law arises that is valid universally. In a second step, the scientists simulated different physical systems numerically in order to test their law.

All in all, the researchers found that the number of caustics in every system develops in the same way – independently of the exact type of applied random force. “At a certain distance from the source several caustics suddenly appear at the same time”, Metzger describes the findings he made in collaboration with Dr. Ragnar Fleischmann from MPIDS and Prof. Theo Geisel, director at MPIDS. Further along the flow, the caustics branch out and new ones appear so that their number increases continuously.

"The new law allows to predict caustics more easily“, says Ragnar Fleischmann. Now, the scientists intend to extend their results to further systems. One area of application, for example, is sound propagation in the ocean.

Original publication:

Jakob J. Metzger, Ragnar Fleischmann, and Theo Geisel:
Universal statistics of branched flows.
Physical Review Letters 105, 020601 (2010).