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

Peter A.E.M. Janssen (Corresponding author), Augustus J.E.M. Janssen

Research output: Contribution to journalArticleAcademicpeer-review

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.

LanguageEnglish
Pages790-818
Number of pages29
JournalJournal of Fluid Mechanics
Volume859
DOIs
StatePublished - 25 Jan 2019

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kurtosis
Water waves
narrowband
oceans
Nonlinear equations
normality
Fluids
nonlinear equations
deviation
fluids
decay

Keywords

  • surface gravity waves
  • waves/free-surface flows

Cite this

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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 Schr{\"o}dinger (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.",
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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.

In: Journal of Fluid Mechanics, Vol. 859, 25.01.2019, p. 790-818.

Research output: Contribution to journalArticleAcademicpeer-review

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