dorsal/arxiv
View SchemaNon-stationary Spectra of Local Wave Turbulence
| Authors | Colm Connaughton, Alan C. Newell, Yves Pomeau |
|---|---|
| Categories | |
| ArXiv ID | physics/0304076 |
| URL | https://arxiv.org/abs/physics/0304076 |
| DOI | 10.1016/S0167-2789(03)00213-6 |
Abstract
The evolution of the Kolmogorov-Zakharov (K-Z) spectrum of weak turbulence is studied in the limit of strongly local interactions where the usual kinetic equation, describing the time evolution of the spectral wave-action density, can be approximated by a PDE. If the wave action is initially compactly supported in frequency space, it is then redistributed by resonant interactions producing the usual direct and inverse cascades, leading to the formation of the K-Z spectra. The emphasis here is on the direct cascade. The evolution proceeds by the formation of a self-similar front which propagates to the right leaving a quasi-stationary state in its wake. This front is sharp in the sense that the solution remains compactly supported until it reaches infinity. If the energy spectrum has infinite capacity, the front takes infinite time to reach infinite frequency and leaves the K-Z spectrum in its wake. On the other hand, if the energy spectrum has finite capacity, the front reaches infinity within a finite time, t*, and the wake is steeper than the K-Z spectrum. For this case, the K-Z spectrum is set up from the right after the front reaches infinity. The slope of the solution in the wake can be related to the speed of propagation of the front. It is shown that the anomalous slope in the finite capacity case corresponds to the unique front speed which ensures that the front tip contains a finite amount of energy as the connection to infinity is made. We also introduce, for the first time, the notion of entropy production in wave turbulence and show how it evolves as the system approaches the stationary K-Z spectrum.
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"abstract": "The evolution of the Kolmogorov-Zakharov (K-Z) spectrum of weak turbulence is\nstudied in the limit of strongly local interactions where the usual kinetic\nequation, describing the time evolution of the spectral wave-action density,\ncan be approximated by a PDE. If the wave action is initially compactly\nsupported in frequency space, it is then redistributed by resonant interactions\nproducing the usual direct and inverse cascades, leading to the formation of\nthe K-Z spectra. The emphasis here is on the direct cascade. The evolution\nproceeds by the formation of a self-similar front which propagates to the right\nleaving a quasi-stationary state in its wake. This front is sharp in the sense\nthat the solution remains compactly supported until it reaches infinity. If the\nenergy spectrum has infinite capacity, the front takes infinite time to reach\ninfinite frequency and leaves the K-Z spectrum in its wake. On the other hand,\nif the energy spectrum has finite capacity, the front reaches infinity within a\nfinite time, t*, and the wake is steeper than the K-Z spectrum. For this case,\nthe K-Z spectrum is set up from the right after the front reaches infinity. The\nslope of the solution in the wake can be related to the speed of propagation of\nthe front. It is shown that the anomalous slope in the finite capacity case\ncorresponds to the unique front speed which ensures that the front tip contains\na finite amount of energy as the connection to infinity is made. We also\nintroduce, for the first time, the notion of entropy production in wave\nturbulence and show how it evolves as the system approaches the stationary K-Z\nspectrum.",
"arxiv_id": "physics/0304076",
"authors": [
"Colm Connaughton",
"Alan C. Newell",
"Yves Pomeau"
],
"categories": [
"physics.flu-dyn"
],
"doi": "10.1016/S0167-2789(03)00213-6",
"title": "Non-stationary Spectra of Local Wave Turbulence",
"url": "https://arxiv.org/abs/physics/0304076"
},
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