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Spatial evolution of the kurtosis of steep unidirectional random waves

Published online by Cambridge University Press:  02 December 2020

Tianning Tang Open the ORCID record for Tianning Tang [Opens in a new window] ,
Wentao Xu ,
Dylan Barratt Open the ORCID record for Dylan Barratt [Opens in a new window] ,
H. B. Bingham ,
Y. Li Open the ORCID record for Y. Li [Opens in a new window] ,
P. H. Taylor ,
T. S. van den Bremer Open the ORCID record for T. S. van den Bremer [Opens in a new window]  and
T. A. A. Adcock Open the ORCID record for T. A. A. Adcock [Opens in a new window]
Tianning Tang
Affiliation:
Department of Engineering Science, University of Oxford, OxfordOX1 3PJ, UK
Wentao Xu
Affiliation:
School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai200240, PR China
Dylan Barratt
Affiliation:
Department of Engineering Science, University of Oxford, OxfordOX1 3PJ, UK
H. B. Bingham
Affiliation:
Department of Mechanical Engineering, Technical University of Denmark, LyngbyDK-2800, Denmark
Y. Li*
Affiliation:
School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai200240, PR China
P. H. Taylor
Affiliation:
Department of Engineering Science, University of Oxford, OxfordOX1 3PJ, UK Faculty of Engineering and Mathematical Sciences, University of Western Australia, Crawley, WA6009, Australia
T. S. van den Bremer
Affiliation:
Department of Engineering Science, University of Oxford, OxfordOX1 3PJ, UK
T. A. A. Adcock
Affiliation:
Department of Engineering Science, University of Oxford, OxfordOX1 3PJ, UK
*
†Email address for correspondence: [email protected]
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Abstract

We study the evolution of unidirectional water waves from a randomly forced input condition with uncorrelated Fourier components. We examine the kurtosis of the linearised free surface as a convenient proxy for the probability of a rogue wave. We repeat the laboratory experiments of Onorato et al. (Phys. Rev. E, vol. 70, 2004, 067302), both experimentally and numerically, and extend the parameter space in our numerical simulations. We consider numerical simulations based on the modified nonlinear Schrödinger equation and the fully nonlinear water wave equations, which are in good agreement. For low steepness, existing analytical models based on the nonlinear Schrödinger equation (NLS) are found to be accurate. For cases which are steep or have very narrow bandwidths, these analytical models over-predict the rate at which excess kurtosis develops. In these steep cases, the kurtosis in both our experiments and numerical simulations peaks before returning to an equilibrium level. Such transient maxima are not predicted by NLS-based analytical models. Above a certain threshold of steepness, the steady-state value of kurtosis is primarily dependent on the spectral bandwidth. We also examine how the average shape of extreme events is modified by nonlinearity over the evolution distance, showing significant asymmetry during the initial evolution, which is greatly reduced once the spectrum has reached equilibrium. The locations of the maxima in asymmetry coincide approximately with the locations of the maxima in kurtosis.

JFM classification

Waves/Free-surface Flows: Surface gravity waves Geophysical and Geological Flows: Ocean processes Instability: Nonlinear instability
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 908 , 10 February 2021 , A3
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

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