Dynamics of extreme waves

We investigate the nonlinear physics that leads to the formation of extreme and rogue waves. Research is conducted using phase resolving models and laboratory experiment. Emphasis is given to generic random directional sea states in infinite and finite water depth. Effect on wave statistics are considered and deviations from standard theoretical distributions for ocean waves are studied.

Wave turbulence and intermittency in directional wave fields

Records of the water surface elevation in a directional wave tank are used to explore the effect of different spectral wave conditions on the properties of wave turbulence and intermittency. Random directional wave fields were generated mechanically (i.e. with the movement of a plunger and no wind forcing) by imposing a desired input directional wave spectrum at the wave-maker. After generation, the wave field propagated freely along the basin. Nonlinear wave interactions were the only driving factor on wave dynamics.

Figure 1: Exponent of the structure ζp function as a function of the order p as calculated in the middle of the basin for unidirectional waves and directionally spread waves. The dashed line denotes the line ζp =p/2, corresponding to a non-intermittent behaviour. N stands for the wave directional spreading coefficient (N = 1000 for unidirectional waves and N = 10 for directional wave fields).


Figure 2: Structure function exponent as a function of the directional spreading parameter N. On the left axis, teal circles denotes ζ4 normalised by 2 (i.e. reference value for p = 4 in absence of intermittency). On the right axis, orange squares denotes ζ6 normalised by 3 (i.e. reference value for p = 6 in absence of intermittency). The error bar is computed as two times the standard deviation computed for the six probes in the array.

Results indicate that the surface elevation displays an intermittent behaviour with deviation from classical wave turbulence predictions (Fig. 1). Nevertheless, experiments demonstrate that intermittency strength reduces with the increase of the level of wave directionality (Fig. 2). This behaviour is consistent with the fact that the available energy for each spectral component is reduced. In this regard, it is worth remarking the analogy between statistical properties of the surface elevation and intermittency strength. However, although the surface elevation transitions from a strongly to a quasi-Gaussian process, intermittency still persists without completely vanishing.

Fadaeiazar, E., Alberello, A., Onorato, M., Leontini, J., Frascoli, F., Waseda, T., Toffoli, A., 2018. Wave turbulence and intermittency in directional wave fields. Wave Motion, 83, 94–101, doi:10.1016/j.wavemoti.2018.09.002

A preprint can be found here.

Wind Generated Rogue Waves in an Annular Wave Flume

We investigate experimentally the statistical properties of a wind-generated wave field and the spontaneous formation of rogue waves in an annular flume. Unlike many experiments on rogue waves where waves are mechanically generated, here the wave field is forced naturally by wind as it is in the ocean. What is unique about the present experiment is that the annular geometry of the tank makes waves propagating circularly in an unlimited-fetch condition. Within this peculiar framework, we discuss the temporal evolution of the statistical properties of the surface elevation. We show that rogue waves and heavy-tail statistics may develop naturally during the growth of the waves just before the wave height reaches a stationary condition. Our results shed new light on the formation of rogue waves in a natural environment.

Experimental setup (not in scale, panel a); example of wind speed (panel b); and example of water surface elevation (normalized by four times the standard deviation of the 10-minute record), including a rogue wave with wave height 2.7 times higher than the significant wave height (panel c).

Temporal evolution of the kurtosis k of the wave envelope (main panel) and PDF of the normalized wave intensity P=hPi (inset) at the time of maximum kurtosis (2100 s) and at full development (5700 s). The wave intensity is defined as the square modulus of the wave enveloped divided by its mean. The PDF for a Gaussian random process, i.e., exp(−P/<P>), is shown as a reference.



Rogue waves in opposing currents: an experimental study on deterministic and stochastic wave trains

Interaction with an opposing current amplifies wave modulation and accelerates nonlinear wave focusing in regular wavepackets. This results in large-amplitude waves, usually known as rogue waves, even if the wave conditions are less prone to extremes. Laboratory experiments in three independent facilities are presented here to assess the role of opposing currents in changing the statistical properties of unidirectional and directional mechanically generated random wavefields. The results demonstrate in a consistent and robust manner that opposing currents induce a sharp and rapid transition from weakly to strongly non-Gaussian properties. This is associated with a substantial increase in the probability of occurrence of rogue waves for unidirectional and directional sea states, for which the occurrence of extreme and rogue waves is normally the least expected.

Toffoli, A., Waseda, T., Houtani, H., Cavaleri, L., Greaves, D. and Onorato, M., 2015. Rogue waves in opposing currents: an experimental study on deterministic and stochastic wave trains. Journal of Fluid Mechanics769, pp.277-297.

Experimental evidence of the modulation of a plane wave to oblique perturbations and generation of rogue waves in finite water depth

We carried out a laboratory experiment in a large directional wave basin to discuss the instability of a plane wave to oblique side band perturbations in finite water depth. Experimental observations, with the support of numerical simulations, confirm that a carrier wave becomes modulationally unstable even for relative water depths k0h < 1.36 (with k the wavenumber of the plane wave and h the water depth), when it is perturbed by appropriate oblique disturbances. Results corroborate that the underlying mechanism is still a plausible explanation for the generation of rogue waves in finite water depth.

Spectral evolution along the basin for relative water depth kh = 1.24: test with collinear perturbations (upper panels); test with oblique perturbations (lower panels).

Toffoli, A., Fernandez, L., Monbaliu, J., Benoit, M., Gagnaire-Renou, E., Lefevre, J.M., Cavaleri, L., Proment, D., Pakozdi, C., Stansberg, C.T. and Waseda, T., 2013. Experimental evidence of the modulation of a plane wave to oblique perturbations and generation of rogue waves in finite water depth. Physics of Fluids25(9), p.091701.

Evolution of weakly nonlinear random directional waves: laboratory experiments and numerical simulations

Nonlinear modulational instability of wavepackets is one of the mechanisms responsible for the formation of large-amplitude water waves. Here, mechanically generated waves in a three-dimensional basin and numerical simulations of nonlinear waves have been compared in order to assess the ability of numerical models to describe the evolution of weakly nonlinear waves and predict the probability of occurrence of extreme waves within a variety of random directional wave fields. Numerical simulations have been performed following two different approaches: numerical integration of a modified nonlinearSchrödinger equation and numerical integration of the potential Euler equations based on a higher-order spectral method. Whereas the first makes a narrow-banded approximation (both in frequency and direction), the latter is free from bandwidth constraints. Both models assume weakly nonlinear waves. On the whole, it has been found that the statistical properties of numerically simulated wave fields are in good quantitative agreement with laboratory observations. Moreover, this study shows that the modified nonlinear Schrödinger equation can also provide consistent results outside its narrow-banded domain of validity.

A. TOFFOLI, O. GRAMSTAD, K. TRULSEN, J. MONBALIU, E. BITNER-GREGERSEN, and M. ONORATO, “Journal of Fluid Mechanics,” Cambridge University Press, Journal of Fluid Mechanics, 664 (2010), pp. 313-336.

Onorato, M., Waseda, T., Toffoli, A., Cavaleri, L., Gramstad, O., Janssen, P.A.E.M., Kinoshita, T., Monbaliu, J., Mori, N., Osborne, A.R. and Serio, M., 2009. Statistical properties of directional ocean waves: the role of the modulational instability in the formation of extreme events. Physical review letters102(11), p.114502.


Experimental basin used for the mechanical wave generation