Linear interference of random waves and the appearance of abnormally large waves

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Introductory remarks

Water waves propagate with different velocities and in different directions; as a result of this complex interference, very irregular wave system containing weak and strong peaks is formed. Such large waves (freak waves) can be a serious threat to ships, oil platforms, port facilities and tourist areas on the coast. Mechanisms of abnormally large wave formation are described in the books [Kurkin et Pelinovsky, 2004; Kharif et al, 2009, Dotsenko et Ivanov, 2006] and numerous articles and reviews that will be cited in the thesis as needed. This chapter focuses on only one mechanism of large wave occurrence – the mechanism of dispersive focusing a ssociated with the dispersion of water waves (dependence of the propagation velocity of the spectral components from their frequency). This mechanism is very popular for the freak wave generation in the laboratory, where their reliable reproducibility is necessary [Brown et Jensen, 2001; Johannesen et Swan, 2001; Clauss, 2002; Kurkin et Pelinovsky, 2004; Kharif et al, 2008, 2009, Shemer et al., 2007; Shemer et Dorfman, 2008; Shemer et Sergeeva, 2009].
Particular attention will be paid to the study of kinematics and statistics of large waves in linear random wind wave fields. Paragraph 2.2 is based on basic equations of water waves and a description of dispersive focusing mechanisms is also presented. Examples of single freak wave appearances in the framework of this mechanism are given. The interaction of simultaneously moving swell waves with weak wind waves in the framework of potential theory is considered in paragraph 2.3. It is noted that in the case of variable wind, swell waves can be focused at a distance from the original area – in a storm area, forming an abnormally large wave (or « freak wave »). An investigation of the visibility of freak waves of different shapes from a background of wind waves is made. The formation of « freak waves » at vertical barrier (rock or cliff) is studied in paragraph 2.4.

Mechanism of dispersive focusing of freak wave appearance

Suddenly appearing short-lived abnormal waves on the sea surface (freak waves) attract the attention of specialists because they can be a serious threat to ships, oil platforms, port facilities and tourist areas on the coast.
Numerous observations of freak waves in various areas of the oceans are presented in books [Lavrenov, 1998; Kurkin et Pelinovsky, 2004; Kharif et al, 2009] and in articles [Lavrenov, 1985; Lopatuhin et al., 2003; Divinsky et al., 2004; Badulin et al., 2005; Didenkulova et al, 2006; Liu, 2007; Nikolkina et Didenkulova, 2011, 2012]. Through the mechanisms of freak wave generation in the open sea there are [Kharif et al, 2009]: а) superpositions of a large number of individual spectral components moving with different velocities and in different directions (geometrical and dispersive focusing); b) nonlinear mechanisms of modulation instability; c) interactions of waves with the bottom and currents.
Each of these mechanisms has its own specificity, which manifests in the corresponding probability of freak wave appearance and its lifetime. Each mechanism leads to different waveforms of freak waves and different scenarios of their manifestation. All of these important features have not been sufficiently studied yet.
In this paragraph the scenario of the appearance of freak waves in the sea, based on dispersive focusing of wave packets propagating in the same direction, is considered. When the group wave velocity depends on the frequency, this mechanism “works” for dispersive waves of all physical natures. In this case, the faster waves overtake the slower ones. It is obvious that for a significant focus of wave energy, the convergence of large numbers of quasi-monochromatic packets is necessary.
This mechanism « works » for deterministic and for random waves, leading to a natural or accidental occurrence of abnormally high waves. It can occur in both linear and nonlinear theory of water waves, but of course, nonlinearity leads to its features in the wave field [Pelinovsky et Kharif, 2000; Pelinovsky et al., 2003; Kharif et al, 2001; Pelinovsky et al, 2000; Shemer et al., 2007; Shemer and Dorfman, 2008].
Theoretical (analytical) results on wave packet focusing in water are obtained mainly using linear theory, specifically in the framework of the parabolic equations for wave packet envelopes [Clauss et Bergmann, 1986; Magnusson et al., 1999; Shemer et al., 2002; Pelinovsky et al., 2003; Shemer et Dorfman, 2008].
The parabolic equation for wave packets can be derived for weakly modulated waves in waters of any depth, not necessarily infinitely deep. However dispersion decreases in shallow water, and processes of dispersion convergences occur over very long times (distances), which can exceed the physical dimensions of water reservoir. Therefore, the main application of parabolic equations is associated with the finite but not the small depth. One exact solution of this equation is Gaussian impulse [Clauss et Bergmann, 1986; Magnusson et al, 1999; Pelinovsky et Kharif, 2000], which demonstrates the process of the emergence of abnormally high waves and their disappearance. The description of the process of single freak wave generation will be given in this paragraph.
In this chapter, the following will be assumed:
1. The liquid is assumed to be ideal, incompressible and unstratified.
2. We will consider two-dimensional potential wave motion (both a horizontal and vertical coordinate).
3. The water depth is constant and there is no water filtration through the « solid » bottom.
4. The action of the wind flow is neglected, and the atmospheric pressure is constant.
In this case, the original equations are two-dimensional Euler equations:
Thus, we have a closed system for two functions(x,t) and(x,z,t). It is a linear Laplace equation (2.8) with linear (2.9) and nonlinear (2.10), and (2.12) boundary conditions. The function(x,t) could be neglected with the help of (2.12) and a closed nonlinear boundary value problem for the potential could be obtained. The derivation of this system can be found in many textbooks of hydrodynamics and it is presented here in a shortened form. In any case, this is a complex system, which is why the number of analytical solutions for water waves are seldomly found.
In the case of linear wave motions in a reservoir of finite depth, the Euler equations can be simplified by transferring boundary conditions from the unknown free surface z =(x,t) to the plane z = 0. Additionally, all nonlinear terms in the kinematic and dynamic boundary conditions can be omitted. The basic equation for wave motion is the Laplace equation (2.8) which, of course, remains unchanged.
As a result, we obtain a closed linear boundary value problem for the potential, consisting from the Laplace equation (2.8) with linear boundary conditions (2.9) and (2.15).
Because of homogeneity of the boundary value problem with respect to the transformation of the horizontal coordinate and time, the solution of this boundary value problem can be found by separation of variables
Scenario of a single freak wave appearance in the framework of the mechanism of dispersive focusing is given in the article Pelinovsky E., Shurgalina E., and Chaikovskaya N. The scenario of a single freak wave appearance in deep water – dispersive focusing mechanism framework. Nat. Hazards Earth Syst. Sci., 2011, 11, 127-134.
Let us say a few words about the manifestation of the same effect in a fluid of finite depth. If the waves are long enough, they all propagate with the same velocity ( c(k) gH ) and hence they cannot overtake each other. Thus, in purely shallow water the effect of dispersion focusing is impossible. Freak waves in nonlinear traveling (Riemann) waves are also absent [Didenkulova et Pelinovsky, 2011]. Then other effects must play a role here. In the case of water with small but finite depth, the velocity of spectral component propagation in the approximation of the linearized Korteweg-de Vries equation takes the following form:
and shorter components may overtake each other. This process is considered in the articles [Pelinovsky et al, 2000; Talipova et Pelinovsky, 2009] taking into account nonlinear effects. It is important to emphasize that due to small dispersion the overtaking process takes a long time, thus the lifetime of the freak waves in shallow water in the framework of mechanisms of dispersion focusing is greatly increased. In this sense, it decreases the risk because such a wave can appear at a great distance from the ship or from the shore, thus preparations for the meeting with the danger wave can be done.

Various forms of freak waves in case of swell and wind wave interaction

A large number of photos and eyewitness stories of freak waves in the ocean has been accumulating during the past decade. Available collected data proves the existence of freak waves of various forms [Kurkin et Pelinovsky, 2004, Faulkner, 2000, Kharif et al., 2009]. In literature there are descriptions of abnormally large waves in the form of « white wall », « single tower », « three sisters » (a group of several individual waves).
Sometimes in front of freak waves there are depressions of several meters deep – « hole in the sea ». Oftentimes these waves have sharp fronts and are asymmetric, indicating the nonlinear character of freak waves. Typical records of abnormal waves, including one and two « sisters » are shown on
Figure 2.2 Temporary record of abnormally high wave in the Black Sea (a), received on 22 November 2000 [Divinsky et al., 2004], and a group of abnormally large waves (b) in the Sea of Japan [Mori et al., 2002] – 24 January 1 987.
One of the mechanisms of freak wave formation is a combined effect of geometric (spatial) and dispersive wave focusing in case of imposition of waves moving in different directions (“crossing sea”). One reason is that wave superposition can be the interaction of swell, coming from a storm area, with wind waves in the area of local storm. Usually for wind waves description statistical methods are used, and for swell description at large distances from the storm deterministic methods are used. Water waves are dispersive, and out of the storm area swell is a frequency-modulated packet and longer waves with greater velocity of propagation are ahead of the shorter ones. This fact has already been used in practice to determine the distance to the storm zone by changing swell current frequencies [Snodgrass et al, 1966]. Wind in storm area is not constant, and it leads to wave packet generation with a very complicated law of frequency changing with time, including the generation of packets when short waves propagate ahead of long ones. It is obvious that such packets will focus in the anomalous wave due to dispersion, and then spread out over large distances from the storm area. Thus, at intermediate distances from the storm area, we can expect the appearance of abnormally large swell waves (freak waves), which will interact with a random wind wave field associated with a local storm. The main purpose of this section is to estimate the lifetime of abnormally large swell waves in wind wave fields.
To simplify the problem, we will assume that the anomalous swell wave is already formed at the initial time and stands out against a background of wind waves. It could have a different shape, as was mentioned in the beginning, thus we consider several possible freak wave forms. Spreading of abnormally large swell waves on an unperturbed (smooth) water surface and the interaction of swell with a random wind wave field will be studied.
Fig. 2.7 shows that with increasing the number of waves in the initial group’s wave lifetime increases with the square of the average number of waves; more precisely the exponent in the regression curves vary from 1.7 to 1.8.
Figure 2.7 – Dependence of freak wave lifetimes and the n umber of waves for different critical threshold values of visibility (the critical amplitude of the « visible » waves, m: 1 – level of 0.3, 2 – 0.4, 3 – 0.5. The dashed lines are regression curves).
The value of freak wave lifetimes in the framework of deterministic problems strongly depends their « level of visibility ». It is more important to understand the influence of nonlinearity on the value of the lifetime and the applicability of linear theory to the description of the formation of abnormal waves. In the article [Shemer et al, 2007], there are descriptions of a laboratory experiment and numerical simulations of the dispersive focusing of the wave packet in the framework of the nonlinear theory. It is shown that the freak wave lifetime is about of 1-3 minutes (steepness is 0.2-0.3). The same estimation follows from our results for a single wave. Taking this into account we hope that our estimations of lifetimes of abnormal waves of different shapes (from fig. 2.7) are the same for nonlinear theory.
Swell waves propagate in the background of wind waves caused by the action of local wind. We assume that the wind is weak enough and it generates waves with small amplitudes, thus that abnormal swell waves are visible in the background. The interaction time of wind with waves is large enough (it is determined by the ratio of the water density to the air density, which could takes hours) [Kharif et al. 2008, 2009] and it is much longer than the lifetime of freak waves (a few minutes, as we will see below). Thus we will not take into account winds in the model.
Space-time (x – t) diagrams are a good way to study the wave evolution since they make it easy to separate swell and wind waves moving with different speeds. This method is widely used for the analysis of dispersive wave packets. Space-time diagrams for all considered forms of abnormal swells are shown in Fig. 2.10. The planes of wave fields exceeded the amplitudes of freak waves afr = 0.38 m are presented in the diagrams. The clearly observed bright line starting from the origin of the coordinate system corresponds to the anomalous swell wave transforming in the wind wave field. This line is extended from the moment of time 4000 s at x = 0 (in cases c and d) due to the periodic boundary conditions used in the calculation, yet as previously mentioned, we will not analyze such long times. A significant number of randomly appearing single points and short lines appeared outside of bright line. The slope of these lines and single points differ from the slope of the main line, thus freak waves appear mainly in wind wave field.
It is important to mention that the « natural » freak waves in the wind wave field occur frequently in accordance with the predictions of the linear statistical theory based on the Rayleigh distribution. Space-time diagrams show that the number freak waves depends on the computational domain size – which is still poorly investigated. Lastly, the diagrams are similar qualitatively for different initial set of random phases.
The « right » and « left » diagrams in Fig. 2.10, correspond to two realizations of wind waves, which differ slightly in general. While the localization of abnormal waves in the background is varied, « lines » of freak waves have almost the same intensity. The space-time diagrams can be used for the estimation of freak wave lifetimes. Lifetimes of freak waves are random due to the random nature of wind waves. Even the lifetime of abnormal swell wave is also changing because the bright, almost continuous line time becomes discontinuous after some time (Fig. 2.10).
Using the amplitude criteria in equation (2.32), we found that the anomalous wave « one sister » disappears in about 4-8 minutes, « two sisters » – 30-40 minutes, « three sisters » – 60-70 min, and « four sisters » – more than 2 hours.
These values are higher than the lifetimes of deterministic signals, where the threshold of visibility was selected artificially. However, the tendency of increasing of freak wave lifetimes with the growth of number of individual waves in the wave packet is saved in both cases. The values of freak wave lifetimes are unreal in some sense, because the lifetime depends on the ratio between the swell height and wave background. In our case, the amplitude of the anomalous swell was chosen to be sufficiently large (0.6 m), and a fr / as is about 3.2.
Such waves are rarely observed on the ocean surface. If we consider only wave amplitudes that are very large, three times higher than large amplitude waves, the number of such waves is much less on the space-time diagram (Fig. 2.11). Virtually all « natural » freak waves disappear, and only noticeable blurring anomalously large swell.

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Table of contents :

1 Introduction 
2 Linear interference of random waves and the appearance of abnormally large waves 
2.1 Introductory remarks
2.2 Mechanism of dispersive focusing of freak wave appearance
2.3 Various forms of freak waves in case of swell and wind wave interaction
2.4 Wave interaction with a vertical barrier
2.5 Conclusion
3 Two-soliton interactions in nonlinear models of long water waves 
3.1 Introductory remarks
3.2 Observation of solitons in the coastal zone and the basic equations
3.3 Two-soliton interactions in the framework of the Korteweg – de Vries equation
3.4 Two-soliton interactions in the framework of the modified Korteweg – de Vries equation
3.5 Conclusion
4 Soliton turbulence in the framework of some integrable long-wave models
4.1 Introductory remarks
4.2 Nonlinear dynamics of irregular soliton ensembles in the framework of the Korteweg – de Vries equation
4.3 Unipolar soliton gas in the framework of the modified Korteweg – de Vries equation
4.4 Freak waves in soliton fields in the framework of the modified Korteweg – de Vries equation
4.5 Conclusion
5 Conclusions

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