### 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.

Original language | English (US) |
---|---|

Pages (from-to) | 64-85 |

Number of pages | 22 |

Journal | Physica D: Nonlinear Phenomena |

Volume | 184 |

Issue number | 1-4 |

DOIs | |

State | Published - Oct 1 2003 |

### Fingerprint

### Keywords

- Kolmogorov-Zakharov spectrum
- Non-stationary
- Wave turbulence

### ASJC Scopus subject areas

- Applied Mathematics
- Statistical and Nonlinear Physics

### Cite this

*Physica D: Nonlinear Phenomena*,

*184*(1-4), 64-85. https://doi.org/10.1016/S0167-2789(03)00213-6

**Non-stationary spectra of local wave turbulence.** / Connaughton, Colm; Newell, Alan C; Pomeau, Yves.

Research output: Contribution to journal › Article

*Physica D: Nonlinear Phenomena*, vol. 184, no. 1-4, pp. 64-85. https://doi.org/10.1016/S0167-2789(03)00213-6

}

TY - JOUR

T1 - Non-stationary spectra of local wave turbulence

AU - Connaughton, Colm

AU - Newell, Alan C

AU - Pomeau, Yves

PY - 2003/10/1

Y1 - 2003/10/1

N2 - 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.

AB - 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.

KW - Kolmogorov-Zakharov spectrum

KW - Non-stationary

KW - Wave turbulence

UR - http://www.scopus.com/inward/record.url?scp=0141642222&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0141642222&partnerID=8YFLogxK

U2 - 10.1016/S0167-2789(03)00213-6

DO - 10.1016/S0167-2789(03)00213-6

M3 - Article

AN - SCOPUS:0141642222

VL - 184

SP - 64

EP - 85

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

SN - 0167-2789

IS - 1-4

ER -