Extreme waves in water of arbitrary depth

It is well established that third-order nonlinearity produces a strong deviation from Gaussian statistics in water  of infinite depth, provided the wave field is long crested, narrow banded and sufficiently steep. A reduction of  third order effects is however expected when the wave energy is distributed on a wide range of directions. In water of arbitrary depth, on the other hand, third-order effects tend to be suppressed by finite depth effects if  waves are long crested. Numerical simulations of the truncated potential Euler equations are here used to address the combined effect of directionality and finite depth on the statistical properties of surface gravity waves; only relative water depth kh greater than 0.8 are here considered. Results show that random directional wave fields in intermediate water depths, kh=O(1), weakly deviate from Gaussian statistics independently of the degree of directional spreading of the wave energy.
Toffoli A., Benoit M., Onorato M., Bitner-Gregersen E.M., 2009. The effect of third order nonlinearity on statistical properties of random directional wave fields. Nonlin. Processes Geophys., 16, 131-139.

Extreme waves in a directional wave tank

The occurrence of extreme (or freak or rogue) waves is a serious threat for marine infrastructure and operations. There are many mechanisms which are responsible for the formation of such events. In the absence of ambient currents, large amplitude waves are likely to be related to the instability of wave packets (modulational instability). For a random sea state, this instability produces strong deviation from Normality, provided the wave field sufficiently steep and narrow banded both in frequency and direction. As the wave spectrum broadens, however, a transition from strongly to weakly non-Gaussian statistics is observed. Recently, two independent laboratory experiments in directional wave tanks have confirmed this transition:

• Onorato M., Waseda T., Toffoli A., Cavaleri L. Gramstad O., Janssen P.A.E.M., Kinoshita T., Monbaliu J., Mori N., Osborne A.R., Serio M., Stansberg C.T., Tamura H., Trulsen K., 2009a. On the statistical properties of directional ocean waves: the role of the modulational instability in the formation of extreme events. Physical Review Letters, 102, 114502, doi: 10.1103/PhysRevLett.102.114502.
• Onorato M., Cavaleri L., Fouques S., Gramstad O., Janssen P.A.E.M., Monbaliu J., Osborne A.R., Pakozdi C., Serio., Stansberg C.T., Toffoli A., Trulsen K., 2009b. Statistical properties of mechanically generated surface gravity waves: a laboratory experiment in a 3D wave basin. J. Fluid Mech., 627, 235-257, doi:10.1017/S002211200900603X. 

A brief description of the laboratory experiments described in Onorato et al. (2009b) can be find at this page.

Maximum wave steepness in the ocean

Laboratory and field data were analyzed to study the shape of ocean waves. Results show that waves may reach a maximum front steepness ka = 0.55 (where k is the peak wavenumber and a is half the significant wave height) and a rear steepness of 0.45.
Toffoli, A., A. Babanin, M. Onorato, and T. Waseda (2010), Maximum steepness of oceanic waves: Field and laboratory experiments, Geophys. Res. Lett., 37, L05603, doi:10.1029/2009GL041771. 

Evolution of the directional wave spectrum

The evolution of the directional wave spectrum is discussed using numerical simulations of the potential Euler equations. Due to nonlinear wave interaction, the wave spectrum show a downshift of the spectral peak and the broadening of the directional distribution. For long time evolution, the wave spectrum shows bimodal properties, which first develops in the short wave portion of the spectrum and eventually reaches the spectral peak. No wind forcing or dissipation were accounted for.
Toffoli, A., M. Onorato, E. M. Bitner-Gregersen, and J. Monbaliu (2010), Development of a bimodal structure in ocean wave spectra, J. Geophys. Res., 115, C03006, doi:10.1029/2009JC005495.