Wave Turbulence: Computational and Theoretical Investigations of a Story Far From Over

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Wave Turbulence is about understanding the long time statistical behavior of solutions of weakly nonlinear field equations describing a sea of random waves. The weak nonlinearity allows one to obtain a natural closure of the BBGKY hierarchy of the Fourier space moment equations which includes a kinetic equation which describes how the wavenumber density is redistributed throughout the spectrum by resonances. These equations have especially relevant finite flux solutions (Kolmogorov-Zakharov or KZ) which capture how conserved densities such as energy flow from sources to sinks. What makes Wave Turbulence an open problem is that these KZ solutions are almost never uniformly valid at all scales and so one is left with the challenge of finding out what happens in the regions of wavenumber space where they break down. The work in this proposal continues the authors' efforts to fill in these missing gaps and to make the theory complete. This is not an easy task because once breakdown occurs, the new states may contain strongly nonlinear coherent structures. In particular, we are working on the problem of condensation and whitecap formation. Wave turbulence is about understanding the statistics of systems of waves. Imagine the sea surface after a storm. There are waves of all wavelengths, travelling in all directions. It is absolutely remarkable that one can, over a large range of scales, say what the longtime energy spectrum looks like. The energy spectrum is a graph showing how much energy there is in waves of different lengths and different directions. Even though the initial graph may reflect how the sea is first excited, the sea, because of resonances between waves of different wavelengths and directions, relaxes to a statistically steady universal state called the Kolmogorov-Zakharov or KZ spectrum. But not quite. If the storm is strong enough, at the smaller scales, the statistics of the sea surface is not described by the KZ spectrum. In fact, an observer will see that the more stormy the sea is, the higher the density of whitecaps (locally breaking waves which combine air and water into a frothy emulsion). The challenge is to find out what replaces the KZ spectrum at these small scales. We are developing a new theory of this behavior. In connection with this work, we are also interested in a phenomenon which has been known by mariners for centuries but which only now is being studied, the sudden emergence of freak waves in an otherwise relatively placid sea