Physical mechanisms of the rogue wave phenomenon

Physica D: Nonlinear Phenomena, 2000

The mechanism of the freak wave formation related to the spatial-temporal focusing is studied within the framework of the Korteweg-de Vries equation. A method to find the wave trains whose evolution leads to the freak wave formation is proposed. It is based on the solution of the Korteweg-de Vries equation with an initial condition corresponding to the expected freak wave. All solutions of this Cauchy problem by the reversal of abscissa represent the possible forms of wave trains which evolve into the freak wave. It is found that freak waves are almost linear waves, and their characteristic Ursell parameter is small. The freak wave formation is possible also from the random wave field and the numerical simulation describes the details of this phenomenon. It is shown that freak waves can be generated not only for specific conditions, but also for relative wide classes of the wave trains. This mechanism explains the rare and short-lived character of the freak wave.

2015

We investigate the generation mechanisms and evolution dynamics of rogue (or freak) waves using direct phase-resolved nonlinear wave-field simulations. The focus is on the understanding of the effects of nonlinear broadband wave-wave interactions on rogue wave development and characteristics. Large-scale long-time simulations of nonlinear long-crested wave-fields with various wave spectrum parameters are obtained. Based on these simulations, we find that nonlinear self-focusing of wave groups and nonlinear wave group interactions play a critical role in the formation and development of rogue waves in deep seas. Occurrence of rogue waves is closely correlated to kurtosis of the wave-field, with larger kurtosis associated with rogue waves of higher height. Moreover, occurrence of rogue waves (especially of large height) is usually correlated with broadband wave spectra. Background The occurrence of rogue/freak waves in deep seas has been observed (e.g. Haver 2000). Due to exceptionall...

2014

The aims of this project are to study the wave dynamics, formation mechanisms and statistical properties of extreme and rogue waves on the water surface. Bimodal/crossing seas can generate extreme and rogue wave events and are relatively common in the open ocean. A series of experiments was carried out in the Marintek Ocean Basin in Trondheim under conditions close to those found in the open sea. Rogue waves, i.e. events with crests larger than or equal to five times the standard deviation or wave heights larger than or equal to twice the significant wave height, were observed in each test. The mean height of the highest waves increased with increasing record length and increasing kurtosis. For the bimodal/crossing conditions there was little difference between the observed values of kurtosis and an empirical estimate, particularly if the effect of bound waves are included.

Physical Review E, 2013

We show experimentally that a stable wave propagating into a region characterized by an opposite current may become modulationally unstable. Experiments have been performed in two independent wave tank facilities; both of them are equipped with a wavemaker and a pump for generating a current propagating in the opposite direction with respect to the waves. The experimental results support a recent conjecture based on a current-modified nonlinear Schrödinger equation which establishes that rogue waves can be triggered by a nonhomogeneous current characterized by a negative horizontal velocity gradient. PACS number(s): 05.45.Yv, Ocean waves are characterized by a statistically small steepness and often (but not always; see, for example, [1]) a weakly nonlinear approach is sufficient to capture some of the intriguing aspects hidden in the fully nonlinear primitive equations. This weakly nonlinear approach is also shared by other fields of physics such as nonlinear optics [2] and plasma physics where small parameters can be individuated and asymptotic expansions can be used to simplify the original equations. If the considered physical process is not only weakly nonlinear but also narrow banded then the lion's share is played by the nonlinear Schrödinger equation (NLS). Being an exactly integrable equation via the inverse scattering transform [4], bizarre analytical solutions have been found in the past: besides traveling waves, breathers or multibreather solutions have been found and observed in hydrodynamics , nonlinear optics , and plasma experiments. Starting from , such solutions have been considered as prototypes of rogue waves. The early stages of the so-called Akhmediev breather solution [6] describes the exponential growth of slightly perturbed plane waves, i.e., it corresponds to the classical modulational instability process . For water waves in infinite water depth, the instability is active when εN 1/ √ 2, where ε = k 0 a 0 is the initial steepness of the plane wave, k 0 the wave number, a 0 its amplitude, and N = ω 0 / the number of waves under the modulation; ω 0 is the angular frequency corresponding to the carrier wave number k 0 and the angular frequency of the modulation. The whole picture is by now pretty well understood and relies on the fact that the medium in which waves propagate is homogeneous. In terms of the NLS equation this means that the coefficients of the dispersive and nonlinear terms do not depend on the spatial coordinates. Much more complicated and intriguing is the case in which the medium changes its properties along the direction of propagation of the waves. This situation is much more difficult to treat in terms of simplified models because it turns out that in general the resulting modified NLS does not share the property of integrability as the standard NLS, and analytical breather solutions can be found only in special cases (see some examples in ).

2009

Freak waves, or rogue waves, are one of the fascinating manifestations of the strength of nature. These devastating walls of water appear from nowhere, are short-lived and extremely rare. Despite the large amount of research activities on this subject, neither the minimum ingredients required for their generation nor the mechanisms explaining their formation have been given. Today, it is possible

The European Physical Journal Special Topics, 2010

Most of the processes resulting in the formation of unexpectedly high surface waves in deep water (such as dispersive and geometrical focusing, interactions with currents and internal waves, reflection from caustic areas, etc.) are active also in shallow areas. Only the mechanism of modulational instability is not active in finite depth conditions. Instead, wave amplification along certain coastal profiles and the drastic dependence of the run-up height on the incident wave shape may substantially contribute to the formation of rogue waves in the nearshore. A unique source of long-living rogue waves (that has no analogues in the deep ocean) is the nonlinear interaction of obliquely propagating solitary shallowwater waves and an equivalent mechanism of Mach reflection of waves from the coast. The characteristic features of these processes are (i) extreme amplification of the steepness of the wave fronts, (ii) change in the orientation of the largest wave crests compared with that of the counterparts and (iii) rapid displacement of the location of the extreme wave humps along the crests of the interacting waves. The presence of coasts raises a number of related questions such as the possibility of conversion of rogue waves into sneaker waves with extremely high run-up. Also, the reaction of bottom sediments and the entire coastal zone to the rogue waves may be drastic.

Since the 1990s, the modulational instability has commonly been used to explain the occurrence of rogue waves that appear from nowhere in the open ocean. However, the importance of this instability in the context of ocean waves is not well established. This mechanism has been successfully studied in laboratory experiments and in mathematical studies, but there is no consensus on what actually takes place in the ocean. In this work, we question the oceanic relevance of this paradigm. In particular, we analyze several sets of field data in various European locations with various tools, and find that the main generation mechanism for rogue waves is the constructive interference of elementary waves enhanced by second-order bound nonlinearities and not the modulational instability. This implies that rogue waves are likely to be rare occurrences of weakly nonlinear random seas. According to the most commonly used definition, rogue waves are unusually large-amplitude waves that appear from nowhere in the open ocean. Evidence that such extremes can occur in nature is provided, among others, by the Draupner and Andrea events, which have been extensively studied over the last decade 1–6. Several physical mechanisms have been proposed to explain the occurrence of such waves 7 , including the two competing hypotheses of nonlinear focusing due to third-order quasi-resonant wave-wave interactions 8 , and purely disper-sive focusing of second-order non-resonant or bound harmonic waves, which do not satisfy the linear dispersion relation 9,10. In particular, recent studies propose third-order quasi-resonant interactions and associated modulational instabilities 11,12 inherent to the Nonlinear Schrödinger (NLS) equation as mechanisms for rogue wave formation 3,8,13–15. Such nonlinear effects cause the statistics of weakly nonlinear gravity waves to significantly differ from the Gaussian structure of linear seas, especially in long-crested or unidirectional (1D) seas 8,10,16–19. The late-stage evolution of modulation instability leads to breathers that can cause large waves 13–15 , especially in 1D waves. Indeed, in this case energy is 'trapped' as in a long wave-guide. For small wave steepness and negligible dissipa-tion, quasi-resonant interactions are effective in reshaping the wave spectrum, inducing large breathers via non-linear focusing before wave breaking occurs 16,17,20,21. Consequently, breathers can be observed experimentally in 1D wave fields only at sufficiently small values of wave steepness 20–22. However, wave breaking is inevitable when the steepness becomes larger: 'breathers do not breathe' 23 and their amplification is smaller than that predicted by the NLS equation, in accord with theoretical studies 24 of the compact Zakharov equation 25,26 and numerical studies of the Euler equations 27,28. Typical oceanic wind seas are short-crested, or multidirectional wave fields. Hence, we expect that nonlinear focusing due to modulational effects is diminished since energy can spread directionally 16,18,29. Thus, modulation instabilities may play an insignificant role in the wave growth especially in finite water depth where they are further attenuated 30. Tayfun 31 presented an analysis of oceanic measurements from the North Sea. His results indicate that large waves (measured as a function of time at a given point) result from the constructive interference (focusing) of elementary waves with random amplitudes and phases enhanced by second-order non-resonant or bound non-linearities. Further, the surface statistics follow the Tayfun 32 distribution 32 in agreement with observations 9,10,31,33. This is confirmed by a recent data quality control and statistical analysis of single-point measurements from fixed sensors mounted on offshore platforms, the majority of which were recorded in the North Sea 34. The analysis of

We discuss a method for the determination of the shape of the ocean wave power spectrum that is based on the physics of the modulational instability for the nonlinear Schrödinger and the Zakharov equations. We find that the form of the spectrum includes an enhanced spectral peak and modulational channels that extend to both high and low frequency. Furthermore, this fundamental shape of the spectrum is found to also be contained in the kinetic equation commonly used for wind-wave models provided that the full Boltzmann four-wave interactions are included. We discuss a number of numerical simulations that demonstrate the modulational form of the power spectrum. We furthermore discuss how the enhanced spectral peak governs the formation of rogue wave packets. We provide ways to compute the properties of the rogue waves directly from the nonlinear spectrum of analyzed time series data or from wave forecasts and hindcasts.

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2013

Rogue waves observed in the ocean and elsewhere are often modelled by certain solutions of the nonlinear Schrodinger equation, describing the modulational instability of a plane wave and the subsequent development of multi-phase nonlinear wavetrains. In this paper, we describe how integrability and application of the inverse scattering transform can be used to construct a class of explicit asymptotic solutions that describe this process. We discuss the universal mechanism of the onset of multiphase nonlinear waves (rogue waves) through the sequence of successive multi-breather wavetrains. Some applications to ocean waves and laboratory experiments are presented.

Physics of Fluids, 2013