Spatial evolution of the kurtosis of steep unidirectional random waves

Tianning Tang, Wentao Xu, Dylan Barratt, Harry B. Bingham, Y. Li*, P. H. Taylor, T. S. van den Bremer, T. A. A. Adcock

*Corresponding author for this work

Research output: Contribution to journalJournal articleResearchpeer-review


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 Schrodinger equation and the fully nonlinear water wave equations, which are in good agreement. For low steepness, existing analytical models based on the nonlinear Schrodinger 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.
Original languageEnglish
Article numberA3
JournalJournal of Fluid Mechanics
Number of pages21
Publication statusPublished - 2020


  • Surface gravity waves
  • Ocean processes
  • Nonlinear instability

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