### Abstract

In this paper we highlight that extreme events such as freak waves are a transient phenomenon in keeping with the old fisherman tale that these extreme events seem to appear out of nowhere. Janssen (J. Phys. Oceanogr., vol. 33, 2003, pp. 863-884) obtained an evolution equation for the ensemble average of the excess kurtosis, which is a measure for the deviation from normality and an indicator for nonlinear focusing resulting in extreme events. In the limit of a narrow-band wave train, whose dynamics is governed by the two-dimensional nonlinear Schrödinger (NLS) equation, the excess kurtosis is under certain conditions seen to grow to a maximum after which it decays to zero for large times. This follows from a numerical solution of the problem and also from an analytical solution presented by Fedele (J. Fluid Mech., vol. 782, 2015, pp. 25-36). The analytical solution is not explicit because it involves an integral from initial time to actual time. We therefore study a number of properties of the integral expression in order to better understand some interesting features of the time-dependent excess kurtosis and the generation of extreme events.

Language | English |
---|---|

Pages | 790-818 |

Number of pages | 29 |

Journal | Journal of Fluid Mechanics |

Volume | 859 |

DOIs | |

State | Published - 25 Jan 2019 |

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### Keywords

- surface gravity waves
- waves/free-surface flows

### Cite this

*Journal of Fluid Mechanics*,

*859*, 790-818. DOI: 10.1017/jfm.2018.844

}

*Journal of Fluid Mechanics*, vol. 859, pp. 790-818. DOI: 10.1017/jfm.2018.844

**Asymptotics for the long-time evolution of kurtosis of narrow-band ocean waves.** / Janssen, Peter A.E.M. (Corresponding author); Janssen, Augustus J.E.M.

Research output: Contribution to journal › Article › Academic › peer-review

TY - JOUR

T1 - Asymptotics for the long-time evolution of kurtosis of narrow-band ocean waves

AU - Janssen,Peter A.E.M.

AU - Janssen,Augustus J.E.M.

PY - 2019/1/25

Y1 - 2019/1/25

N2 - In this paper we highlight that extreme events such as freak waves are a transient phenomenon in keeping with the old fisherman tale that these extreme events seem to appear out of nowhere. Janssen (J. Phys. Oceanogr., vol. 33, 2003, pp. 863-884) obtained an evolution equation for the ensemble average of the excess kurtosis, which is a measure for the deviation from normality and an indicator for nonlinear focusing resulting in extreme events. In the limit of a narrow-band wave train, whose dynamics is governed by the two-dimensional nonlinear Schrödinger (NLS) equation, the excess kurtosis is under certain conditions seen to grow to a maximum after which it decays to zero for large times. This follows from a numerical solution of the problem and also from an analytical solution presented by Fedele (J. Fluid Mech., vol. 782, 2015, pp. 25-36). The analytical solution is not explicit because it involves an integral from initial time to actual time. We therefore study a number of properties of the integral expression in order to better understand some interesting features of the time-dependent excess kurtosis and the generation of extreme events.

AB - In this paper we highlight that extreme events such as freak waves are a transient phenomenon in keeping with the old fisherman tale that these extreme events seem to appear out of nowhere. Janssen (J. Phys. Oceanogr., vol. 33, 2003, pp. 863-884) obtained an evolution equation for the ensemble average of the excess kurtosis, which is a measure for the deviation from normality and an indicator for nonlinear focusing resulting in extreme events. In the limit of a narrow-band wave train, whose dynamics is governed by the two-dimensional nonlinear Schrödinger (NLS) equation, the excess kurtosis is under certain conditions seen to grow to a maximum after which it decays to zero for large times. This follows from a numerical solution of the problem and also from an analytical solution presented by Fedele (J. Fluid Mech., vol. 782, 2015, pp. 25-36). The analytical solution is not explicit because it involves an integral from initial time to actual time. We therefore study a number of properties of the integral expression in order to better understand some interesting features of the time-dependent excess kurtosis and the generation of extreme events.

KW - surface gravity waves

KW - waves/free-surface flows

UR - http://www.scopus.com/inward/record.url?scp=85057374937&partnerID=8YFLogxK

U2 - 10.1017/jfm.2018.844

DO - 10.1017/jfm.2018.844

M3 - Article

VL - 859

SP - 790

EP - 818

JO - Journal of Fluid Mechanics

T2 - Journal of Fluid Mechanics

JF - Journal of Fluid Mechanics

SN - 0022-1120

ER -