Formation of Extraordinarily High Waves in Space and Time

Proceedings of the …, 2008

It is well established that the modulational instability enhances the probability of occurrence for extreme events if waves are long crested, narrow banded and sufficiently steep. As a result, substantial deviations from commonly used second-order theory-based distributions can be expected. However, the coexistence of directional wave components can suppress the effects related to the modulational instability. To get better insight into the effect of wave directionality and its implication for design work, numerical simulations based on the truncated potential Euler equations were used. The analysis has been concentrated primarily on the wave crest distribution.

2018

Modulational or Benjamin-Feir instability is a well known phenomenon of Stokes' periodic waves on the water surface. In this dissertation, we study this phenomenon for periodic traveling wave solutions of various shallow water wave models. We study the spectral stability or instability with respect to long wave length perturbations of small amplitude periodic traveling waves of shallow water wave models like Benjamin-Bona-Mahony and Camassa-Holm equations. We propose a bi-directional shallow water model which generalizes Whitham equation to contain the nonlinearities of nonlinear shallow water equations. The analysis yields a modulational instability index for each model which is solely determined by the wavenumber of underlying periodic traveling wave. For a fixed wavenumber, the sign of the index determines modulational instability. We also includes the effects of surface tension in full-dispersion shallow water models and study its effects on modulational instability.

Comptes Rendus de l Académie des Sciences - Series IIB - Mechanics

The influence of wind on extreme wave events in shallow water is investigated numerically. A series of numerical simulations using a pressure distribution over the steep crests given by the modified Jeffreys' sheltering theory shows that wind blowing over a strongly modulated wave group due to the dispersive focusing of a chirped long wave packet increases the time duration and maximal amplitude of the extreme wave event. These results are coherent with those obtained within the framework of deep water. However, steep wave events are less unstable to wind perturbation in shallow water than in deep water.

Physics of Fluids, 2007

The long-term evolution of a nonlinear wave train in deep water with varied initial wave steepness between 0.1 and 0.3 was experimentally investigated in a super wave flume ͑300 m long, 5 m wide, and 5.2 m deep͒. The initial wave train was the combination of one carrier wave and a pair of imposed sideband components which is the most unstable mode, referred to as sideband instability theory. Sixty-six wave gauges were installed downstream along the wave flume to simultaneously measure the evolution of a wave train. Increasing modulation of the wave train was observed due to sideband instability until a critical value which either wave breaking is initiated or maximum modulation is reached. The near recurrence of the initial state of the wave train is exhibited for the nonbreaking case. An effective frequency downshift of the wave spectrum accompanied with wave breaking is observed as the initial wave steepness is larger than 0.11. At postbreaking, the wave train reveals periodic modulation and demodulation, meanwhile, the related wave spectrum shows a down and upshift, respectively. Present results support the hypothesis that the frequency downshift induced by wave breaking is not permanent.

2010

Progression of nonlinear wave groups to breaking was studied numerically and experimentally. Evolution of such wave group parameters as distance to breaking and modulation depth was described. The wave modulation depth R is a height ratio of the highest Hh and the lowest Hl waves in the group: R = Hh-Hl (Babanin et al. 2009). This parameter, together with distance to breaking, was studied by means of the fully-nonlinear Chalikov-Sheinin (CS) model (Chalikov and Sheinin, 2005). Subsequent experiment demonstrated a good qualitative agreement with the numerical results. In the present study, both in numerical simulations and laboratory experiments, a wave group was initially generated as a superposition of two waves with primary and secondary wave steepnesses and close wave numbers, and allowed to evolve. It was shown that the modulation depth decreases as a function of the primary wave steepness, that distance to breaking also decreases with primary wave steepness, but grows as a func...

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

2019

Motivated by the relatively new phenomenon observed in Oslofjord where ships moving with subcritical speeds across substantial depth changes generated long waves with heights up to 1.4m, the linear generation mechanism for these upstream waves is investigated using the linear shallow water equations. The simulations are performed both with one and two horizontal dimensions where the average depth at the location where the bottom variation happens is twice the change in depth, ∆h/h̄ = 0.5. Analytical calculations on the amplitude of the generated free waves as the source moves over a step in bottom topography shows good agreement with the numerical results. For ships moving from deep to shallow water, the maximum elevation of the generated waves grow accroding to Fr, where n is in the range 3.6− 4.6. The simulations with two horizontal dimensions are compared to a dispersive model to give a measure of how well the linear shallow water model captures this phenomenon.

Http Dx Doi Org 10 1175 2010jpo4405 1, 2010

Progression of nonlinear wave groups to breaking was studied numerically and experimentally. Evolution of such wave group parameters as a function of distance to breaking and modulation depth-the height ratio of the highest and the lowest waves in the group-was described. Numerical model results demonstrated good agreement with experimental results in describing the behavior of the distance to breaking and modulation depth as functions of initial wave steepness. It was shown that energy loss appears to be a function of the modulation depth at the breaking onset. Energy loss grows with modulation depth up to a certain threshold of the latter. It was also shown that breaking probability for wave groups with modulation depth below 2.2 is very low.

Journal of Physical Oceanography, 2018

Wave statistical properties and occurrence of extreme and rogue waves in crossing sea states are investigated. Compared to previous studies a more extensive set of crossing sea states are investigated, both with respect to spectral shape of the individual wave systems and with respect to the crossing angle and separation in peak frequency of the two wave systems. It is shown that, because of the effects described by Piterbarg, for a linear sea state the expected maximum crest elevation over a given surface area depends on the crossing angle so that the expected maximum crest elevation is largest when two wave systems propagate with a crossing angle close to 90°. It is further shown by nonlinear phase-resolving numerical simulations that nonlinear effects have an opposite effect, such that maximum sea surface kurtosis is expected for relatively large and small crossing angles, with a minimum around 90°, and that the expected maximum crest height is almost independent of the crossing ...

Coastal Engineering Proceedings, 2014

Nonlinear four-wave interactions amplify wave heights of deep-water generating extreme wave such as a freak wave. However, it is not clear the behavior of generated freak waves in deep-water shoaling to shallow water regions. In this study, a series of physical experiments and numerical simulations with several bathymetry configurations were conducted for unidirectional random waves from deep to shallow water regions. The maximum wave heights increase with an increase in kurtosis by third-order nonlinear interactions in deep water regions. The dependence of the kurtosis on the freak wave occurrence is weakened due to second-order nonlinear interactions associated with wave shoaling on the slope. Moreover, it is possible to understand the behavior of the high-order nonlinearity and the freak wave occurrence in shallow water regions if appropriate correction of the insufficient nonlinearity of more than O(ε 2 ) to the standard Boussinesq equation are considered analytically.

Water

We present a study of the physical characteristics of traveling waves at shallow and intermediate water depths. The main subject of study is to the influence of nonlinearity on the dispersion properties of waves, their limiting heights and steepness, the shape of solitary waves, etc. A fully nonlinear Serre–Green–Naghdi-type model, a classical weakly nonlinear Boussinesq model and fifth-order Stokes wave solutions were chosen as models for comparison. The analysis showed significant, if not critical, differences in the effect of nonlinearity on the properties of traveling waves for these models. A comparison with experiments was carried out on the basis of the results of a joint Russian–Taiwanese experiment, which was carried out in 2015 at the Tainan Hydraulic Laboratory, and on available experimental data. A comparison with the experimental results confirms the applicability of a completely nonlinear model for calculating traveling waves over the entire range of applicability of t...

Applied Mathematical Modelling, 2007

Modulational, Benjamin-Feir, instability is studied for the down-stream evolution of surface gravity waves. An explicit solution, the soliton on finite background, of the NLS equation in physical space is used to study various phenomena in detail. It is shown that for sufficiently long modulation lengths, at a unique position where the largest waves appear, phase singularities are present in the time signal. These singularities are related to wave dislocations and lead to a discrimination between successive 'extreme' waves and much smaller intermittent waves. Energy flow in opposite directions through successive dislocations at which waves merge and split, causes the large amplitude difference. The envelope of the time signal at that point is shown to have a simple phase plane representation, and will be described by a symmetry breaking unfolding of the steady state solutions of NLS. The results are used together with the maximal temporal amplitude MTA, to design a strategy for the generation of extreme (freak, rogue) waves in hydrodynamic laboratories.

Physical Review Letters, 2006

Here we consider a simple weakly nonlinear model that describes the interaction of two-wave systems in deep water with two different directions of propagation. Under the hypothesis that both sea systems are narrow banded, we derive from the Zakharov equation two coupled nonlinear Schrödinger equations. Given a single unstable plane wave, here we show that the introduction of a second plane wave, propagating in a different direction, can result in an increase of the instability growth rates and enlargement of the instability region. We discuss these results in the context of the formation of rogue waves.

Physical Review Letters, 2009

We discuss two independent, large scale experiments performed in two wave basins of different dimensions in which the statistics of the surface wave elevation are addressed. Both facilities are equipped with a wave maker capable of generating waves with prescribed frequency and directional properties. The experimental results show that the probability of the formation of large amplitude waves strongly depends on the directional properties of the waves. Sea states characterized by long-crested and steep waves are more likely to be populated by freak waves with respect to those characterized by a large directional spreading.

Physics of Fluids, 2000

We propose a new approach for modeling weakly nonlinear waves, based on enhancing truncated amplitude equations with exact linear dispersion. Our example is based on the nonlinear Schrödinger ͑NLS͒ equation for deep-water waves. The enhanced NLS equation reproduces exactly the conditions for nonlinear four-wave resonance ͑the ''figure 8'' of Phillips͒ even for bandwidths greater than unity. Sideband instability for uniform Stokes waves is limited to finite bandwidths only, and agrees well with exact results of McLean; therefore, sideband instability cannot produce energy leakage to high-wave-number modes for the enhanced equation, as reported previously for the NLS equation. The new equation is extractable from the Zakharov integral equation, and can be regarded as an intermediate between the latter and the NLS equation. Being solvable numerically at no additional cost in comparison with the NLS equation, the new model is physically and numerically attractive for investigation of wave evolution.