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Some Remarks on the Establishment of the Kolmogorov-Zakharov Cascade in 3-Wave Turbulence
Colm Connaughton
LANL
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Some Remarks on the Establishment of the Kolmogorov-Zakharov Cascade in 3-Wave Turbulence Colm Connaughton LANL |
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ABSTRACT Assuming that wave turbulence is adequately described by the classical 3-wave kinetic equation. One can ask the following question : given zero initial density of waves and spectrally compact forcing, how long does it take to establish the energy cascade? We already know of two scenarios, referred to as the infinite capacity and finite capacity cases respectively. In the former case, the cascade requires an infinite amount of time to set up. In the latter case, a phenomenon similar to the dissipative anomaly in hydrodynamic turbulence, ensures that the cascade sets up in finite time. In this talk I will discuss these scenarios and outline how the 3-wave kinetic equation is structurally similar to the mean field rate equation for some particular fragmentation-coalescence process. The onset of dissipation in finite time for wave systems is analogous to the so-called "gelation transition"in aggregating particle systems. Pushing this analogy further, one can ask whether the phenomenon of "instantaneous gelation", well known in the world of particle aggregation, has an analogue for wave systems. This would correspond to the establishment of a cascade in zero time. I will finish with some circumstantial evidence for this scenario. |