### Abstract

An ensemble of weakly interacting capillary waves on a free surface of deep ideal fluid is described statistically by methods of weak turbulence. The stationary kinetic equations for capillary waves have an exact Kolmogorov solution which gives for the spatial spectrum of elevations asymptotics I_{k} = C(P^{1/2}/σ^{3/4})k^{-19/4}. The Kolmogorov constant C is found analytically together with the interval of locality in K-space. Direct numerical simulation of the dynamical equations in the approximation of small surface angles confirms the presence of almost isotropic Kolmogorov spectrum in the large k region. Besides, at small amplitudes of the pumping, an essentially new phenomenon is found: 'frozen' turbulence, in which, despite the big number of interacting waves (of the order of 100) there is no energy flux toward high k. This phenomenon is connected with the finiteness of the region (or, in other words, discreteness of the spectrum in Fourier space). This is believed to be universal for different sorts of nonlinear systems.

Original language | English (US) |
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Pages (from-to) | 98-116 |

Number of pages | 19 |

Journal | Physica D: Nonlinear Phenomena |

Volume | 135 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2000 |

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### ASJC Scopus subject areas

- Applied Mathematics
- Statistical and Nonlinear Physics

### Cite this

*Physica D: Nonlinear Phenomena*,

*135*(1), 98-116. https://doi.org/10.1016/S0167-2789(99)00069-X

**Turbulence of capillary waves - theory and numerical simulation.** / Pushkarev, A. N.; Zakharov, Vladimir E.

Research output: Contribution to journal › Article

*Physica D: Nonlinear Phenomena*, vol. 135, no. 1, pp. 98-116. https://doi.org/10.1016/S0167-2789(99)00069-X

}

TY - JOUR

T1 - Turbulence of capillary waves - theory and numerical simulation

AU - Pushkarev, A. N.

AU - Zakharov, Vladimir E

PY - 2000/1/1

Y1 - 2000/1/1

N2 - An ensemble of weakly interacting capillary waves on a free surface of deep ideal fluid is described statistically by methods of weak turbulence. The stationary kinetic equations for capillary waves have an exact Kolmogorov solution which gives for the spatial spectrum of elevations asymptotics Ik = C(P1/2/σ3/4)k-19/4. The Kolmogorov constant C is found analytically together with the interval of locality in K-space. Direct numerical simulation of the dynamical equations in the approximation of small surface angles confirms the presence of almost isotropic Kolmogorov spectrum in the large k region. Besides, at small amplitudes of the pumping, an essentially new phenomenon is found: 'frozen' turbulence, in which, despite the big number of interacting waves (of the order of 100) there is no energy flux toward high k. This phenomenon is connected with the finiteness of the region (or, in other words, discreteness of the spectrum in Fourier space). This is believed to be universal for different sorts of nonlinear systems.

AB - An ensemble of weakly interacting capillary waves on a free surface of deep ideal fluid is described statistically by methods of weak turbulence. The stationary kinetic equations for capillary waves have an exact Kolmogorov solution which gives for the spatial spectrum of elevations asymptotics Ik = C(P1/2/σ3/4)k-19/4. The Kolmogorov constant C is found analytically together with the interval of locality in K-space. Direct numerical simulation of the dynamical equations in the approximation of small surface angles confirms the presence of almost isotropic Kolmogorov spectrum in the large k region. Besides, at small amplitudes of the pumping, an essentially new phenomenon is found: 'frozen' turbulence, in which, despite the big number of interacting waves (of the order of 100) there is no energy flux toward high k. This phenomenon is connected with the finiteness of the region (or, in other words, discreteness of the spectrum in Fourier space). This is believed to be universal for different sorts of nonlinear systems.

UR - http://www.scopus.com/inward/record.url?scp=0033640486&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0033640486&partnerID=8YFLogxK

U2 - 10.1016/S0167-2789(99)00069-X

DO - 10.1016/S0167-2789(99)00069-X

M3 - Article

AN - SCOPUS:0033640486

VL - 135

SP - 98

EP - 116

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

SN - 0167-2789

IS - 1

ER -