Soliton Collision in Random Seas
SSoliton Collision in Random Seas
Hendrik Fischer a , Marten Hollm a , Leo Dostal a a Institute of Mechanics and Ocean Engineering, Hamburg University of Technology,21073 Hamburg, Germany
Abstract
Although extreme or freak waves are repeatedly measured in the open ocean,their origin is largely unknown. The interaction of different water waves isseen as one reason for their emergence. One way to consider nonlinear wavesin deep water is to look at solutions of the nonlinear Schr¨odinger equation,which plays an important role in the determination of extreme waves. Onespecific solution is the soliton solution. Therefore the question arises, hownonlinear waves behave as they interact or collide. Using a relaxation pseudospectral scheme for the computation of solutions of the nonlinear Schr¨odingerequation, the behavior of colliding solitons is studied. Thereby, differentwave amplitudes and angles of collision are considered. In addition to this,the influence of an initial perturbation by random waves is studied, which isgenerated using a Pierson-Moskowitz spectrum.
Keywords: surface gravity waves, random wave interactions, rogue waves,nonlinear Schr¨odinger equation, disturbed solitons
1. Introduction
The phenomenon of extreme waves was first observed in connection withocean waves and has been studied since the seventies of the last century. Theamplitude of these waves exceeds the amplitude of the surrounding wavesby a factor of more than two [11]. Furthermore the behavior of extremewaves can be characterized by their sudden, unpredictable appearance anddisappearance without any trace. These waves can be accompanied by deeptroughs in front and behind the crest of the wave [9].
Email addresses: he.fischer @tuhh.de (Hendrik Fischer), [email protected] (Marten Hollm), dostal @tuhh.de (Leo Dostal)
Preprint submitted to August 28, 2020 a r X i v : . [ n li n . PS ] A ug hereby, extreme waves are a real danger for ships, offshore structuresand all persons present [11]. The appearance of an extreme wave can alsohave fatal consequences for ships as well. According to [9], 22 super tankerssank between 1969 and 1994 with a total of 525 fatalities.Although it has been a well-documented phenomenon, which has beenintensively studied in recent years, the causes of such waves are widely un-known. The lack of understanding makes it extremely difficult to predict suchwaves. In [11] physical mechanisms such as dispersion, refraction, chaotic be-havior, Benjamin-Feir instability and soliton wave interaction are mentionedas possible causes of extreme wave development. More recently, the presenceof random wind forcing was investigated [6]. There it was also shown thatsolitons may persist under wind forcing.The characteristic steepness of extreme waves suggests that nonlinearitieshave an important influence on the development of extreme waves. The sameresult can be concluded from the satellite observations in the wake of theMaxWave project [11]. These show that extreme waves occur more often thanpredicted by the linear wave theory, which was confirmed by experiments atMarintek [10]. This implies the necessity to consider nonlinear wave theory.In most problems in offshore engineering, it is sufficient to consider thenonlinear Euler equations instead of the Navier-Stokes equations [3]. Butthe numerical computation of these equations is still very expensive due tothe unknown water surface required as a boundary condition. Therefore, afurther problem reduction is of particular interest. Zakharov [17] has shownthat weakly nonlinear solutions of the one dimensional Euler equations canbe reduced to solutions of the nonlinear Schr¨odinger equation (NLS), whichdescribe complex wave envelopes.One solution of the NLS is the soliton solution. Based on the assumptionin [11] that the collision of solitons could be a possible factor in the develop-ment of extreme waves, the collision of solitary waves is a point of research.Thereby, Fedele et. al. [7] have found that smooth solitary waves appear tointeract elastically, but no results are determined for the non-smooth case ofwaves which are disturbed by an irregular sea.A study on emergence of breather rogue waves in random seas was pre-sented in [14]. Thereby, the Peregrine breather, which is another solution ofthe NLS, has been considered [12]. Furthermore, it has been shown that theemergence of the Peregrine breather dynamics can be also attributed to amore general context of higher-order soliton interaction [13].In this work soliton wave interactions in irregular sea states are inves-2igated with focus on the effect of these disturbances. In this respect, thecreation of a disturbed soliton in a realistic random sea is achieved by theapplication of the well known model of random sea waves described in [4, 5]in conjunction with the soliton solution. This model is basically built on thesuperposition of harmonic waves resulting from the linear wave theory.The work is structured as follows: In section 2 the nonlinear Schr¨odingerequation and the soliton solution are introduced, the modeling of the in-teraction of these solitons is described and the model for the generation ofrandom ocean waves is presented. Building on this, section 3 evaluates thecollision of solitons in regular seas. These results provide the starting pointfor the subsequent analysis of the collision of disturbed solitons in randomseas. Finally, this work ends with a conclusion in section 4.
2. Nonlinear water waves in an irregular sea
As noted above, the time-consuming numerical computation of the Eu-ler equations motivates a further problem reduction. It can be shown thatweakly nonlinear solutions of these equations can be reduced to a complexwave envelope which satisfies the nonlinear Schr¨odinger equation. Such a re-duction can be achieved by means of the method of multiple scales presentedfor example in [6]. An initial disturbance finally leads to nonlinear waves inirregular seas.
The NLS is derived using the multiple scale method, taking into accountterms up to the order O ( ε ) with the wave steepness ε (cid:28)
1. The derivationis shown in [6] in detail and leads to the equation for the case of deep wateri ψ τ = αψ ξξ + β | ψ | ψ, (1)where α = ω k and β = ω k . Additionally, ψ ( ξ, τ ) ∈ C describes the waveenvelope, ξ = ε ( x − c g t ) is the scaled spatial coordinate including the deepwater group velocity c g = ω k and τ = ε t is the scaled time. Furthermore, k is the wave number and ω is the frequency of the carrier wave.According to [15], the evaluation of the free water surface follows in firstorder from the NLS by η ( x, t ) = Re [ ψ ( x, t ) exp(i( kx − ω t ))] (2)3nd in second order by η ( x, t ) = Re (cid:20) ψ ( x, t ) exp(i( kx − ω t )) + 12 k [ ψ ( x, t )] exp(2i( kx − ω t )) (cid:21) . (3)Additionally, the wave period T and the wavelength λ are calculated by T = 2 πω , λ = g π T , (4)where g is acceleration due to gravity.For all further examinations, ω and k are chosen as ω = 1 rad/s and k = 1 /g . Therefore, the coefficients α and β are also determined. With respect to the soliton interaction, it is insufficient to consider astationary solution of the NLS. A solution extended by a motion must betaken into account. According to [16, 2] this soliton solution can be describedby ψ ( ξ, τ ) = a sech (cid:34) a (cid:114) β α ( ξ − ξ − vτ ) (cid:35) exp(i( cξ − wτ )) , (5) c = − v α , w = − αc + 12 βa with the amplitude a , spatial shift ξ and velocity v of the soliton. Theparameters α and β are set by the NLS (1) itself. Thereby, v is the velocityof the soliton in the ( ξ, τ ) coordinate system, which can be transformed intothe ( x, t ) coordinate system by v x,t = c g + εv .The soliton solution in Fig. 1 is shown from two different perspectives andis characterized by the parameters a = 1 m, ξ = 500 m and v = − igure 1: Analytical soliton solution with amplitude a = 1 m, spatial shift ξ = 500 mand velocity v = − A well-known model of random sea waves is given by the superpositionof harmonic waves with wave numbers κ ( ω ) and wave frequencies ω corre-sponding to a one-sided spectral density S ( ω ), cf. [4, 5]. Common sea spec-tral densities S ( ω ) are hereby the JONSWAP spectrum for shallow-waterwaves and the Pierson-Moskowitz spectrum for deep-water waves. In the onedimensional case, beside the amplitude of the waves, the direction of prop-agation γ has to be taken into account too. In order to obtain an initiallyirregular or random wave surface, a random phase shift ε ( ω ) is added, whichis uniformly distributed in [0 , π ). With this, an irregular wave surface canbe written as Z ( x, t ) = (cid:90) ∞ (cid:90) π − π cos ( ωt − κ ( ω )( x cos( γ )) + ε ( ω )) (cid:112) S ( ω ) D ( γ )d γ d ω. (6)Here, the integral is not a Riemann integral but a summation rule over thefrequencies ω and the directions of propagation γ . The propagation function D ( γ ) leads to a scattering of the directions of propagation of the single har-monic waves. The function D ( γ ) has to be normalized in the domain [ − π, π ],i. e. (cid:90) π − π D ( γ )d γ = 1 . (7)5 propagation function D ( γ ), which is often used, is D ( γ )d γ = 2 γ R cos (cid:18) πγ R ( γ − γ ) (cid:19) with | γ − γ | ≤ γ R . (8)Here, γ is the main propagation direction and γ R has to be chosen such thatEq. (7) holds. Under the additional assumption of a long-crested sea state,this can be further reduced to Z ( x, t ) = (cid:90) ∞ cos ( ωt − κ ( ω ) x cos( γ ) + ε ( ω )) (cid:112) S ( ω )d ω, (9)since all superpositioned harmonic waves have the same direction of propa-gation γ .In the following calculations the Pierson-Moskowitz spectrum is usedwhich can be described by the significant wave height H s and the modalfrequency ω m by S J ( ω ) = 0 . H s ω m ω exp (cid:26) − . (cid:16) ω m ω (cid:17) (cid:27) . (10)If not explicitly mentioned, in all further investigations the significant waveheight H s = 0 . ω m = 0 .
25 rad/s and direction of prop-agation γ = 0 are selected.Using this model, the initial condition of the numerical simulation, de-scribed by the analytical solution (5), is adjusted to simulate a realistic ir-regular sea state. Let η d ( ξ,
0) be the initial disturbed free sea surface and η ( ξ,
0) the sea surface obtained from the NLS eq. (2) or (3), respectively.Here, we use the scaled space-coordinate ξ . In order to achieve η d ( ξ,
0) = η ( ξ,
0) + Z ( ξ, , (11)we use the following disturbance for ψ ( ξ, (cid:101) ψ ( ξ,
0) = (cid:18) Z ( ξ, | ψ ( ξ, | (cid:19) ψ ( ξ, . (12)By doing so, the amplitude of the disturbed wave elevation can be achievedby | (cid:101) ψ ( ξ, | = || ψ ( ξ, | + Z ( ξ, | , (13)6. e. by adding the amplitude of the undisturbed sea surface and the distur-bance Z ( ξ,
0) and taking the absolute value. Figure 2a shows such a randomsea state calculated by means of Eq. (9) and Fig. 2b the application to thesoliton solution using Eq. (12).
Figure 2: (a) Generated random ocean sea and (b) the corresponding perturbed solitonsolution for τ = 0. For the study of soliton interaction, the initial conditions are selectedaccording to Eq. (5). But this equation only describes one soliton. Con-sequently, a way must be found to combine solitons. For this, the generalapproach described in [16, 8] results in the required superposition of singlesolitons. Thus, the initial condition for N combined waves is given by ψ ( ξ ) = N (cid:88) i =1 ψ i ( ξ, , (14)whereby ψ i ( ξ,
0) is determined by Eq. (5). Furthermore, each wave ψ i ( ξ, v i , shift in space ξ i and amplitude a i .It is important to ensure that the solitons do not already interact witheach other in the initial condition in order to guarantee that the entire inter-action behavior takes place in the simulation period. This can be done by asuitable choice of the spatial shift ξ i for each wave.In order to obtain an irregular or random sea state scenario, the initialcondition can be modified using Eq. (12). This leads to the initial condition (cid:102) ψ ( ξ ) = (cid:18) Z ( ξ, | ψ ( ξ ) | (cid:19) ψ ( ξ ) (15)7hich is used for the following calculations. Figure 3 shows such an initialcondition for the regular and irregular case calculated using Eq. (14) andEq. (15), respectively. Figure 3: Combined initial condition for two solitary waves for the regular and irregularsea state.
3. Results
Based on the theory presented in section 2, this section evaluates thesimulations of the soliton interaction. The following simulation results aregenerated numerically by a relaxation pseudo spectral scheme described in[6]. The main part of this analysis evaluates the maximum of the waveenvelope amplitude | ψ | during the interaction, i.e.M = max ξ,τ | ψ ( ξ, τ ) | . (16)In the following simulations different scenarios are considered, which differin wave amplitude and velocity. With regard to the velocities, the magnitudeand the direction are of interest. The focus is primary on the interaction oftwo solitons, since according to [1] the simultaneous collision of several waterwaves can be assumed as a very unlikely event.8n order to be able to make a general statement about the interactionbehavior in irregular seas, the regular case is considered first, i. e. the casewithout any disturbances. Based on the corresponding results, the scenariowith the greatest potential in terms of amplitude development will be an-alyzed in the irregular case. Due to the stochasticity in the calculation ofrandom ocean waves in subsection 2.3, an empirical evaluation is of particularrelevance for the irregular sea state. In order to analyze various scenarios, the initial conditions from Eq. (14)are adjusted accordingly. In this context the choice of the parameters v i , ξ i and a i , which specify the velocities, the shifts in space and the amplitudesof the particular waves, is of significant importance.Different soliton interaction scenarios are presented in Figs. 4, 5 and6, where each solution is shown from two different perspectives. Figure 4shows an interaction scenario of solitons with equal wave envelope amplitudeand equal velocity but with different directions of movement. In Fig. 5,however, the amplitudes are adjusted such that a scenario with differentwave amplitudes is considered. In contrast to the previous figures, Fig. 6provides an insight into an interaction scenario of solitons with the sameamplitude and the same direction of movement, but with different velocities. Figure 4: Interaction behavior of solitons with amplitudes a = a = 1 m and velocities v = − v = − . igure 5: Interaction behavior of solitons with amplitudes a = 2 a = 1 m and velocities v = − v = − . a = a = 1 m and velocities v = 3 v = − . In each scenario considered, a significant rise in the amplitude of the waveenvelope can be observed around the time of collision, but its developmentand magnitude depends on the chosen initial condition. As a result, thepresented simulations have indicated that the amplitudes and the directionsof movement of the solitons have a considerable influence on the interac-tion behavior. Before and after the collision, however, the solitons show anunchanged course. 10dditional simulations presented in Fig. 7 have revealed that when a sce-nario of two waves moving towards each other is considered, the magnitudesof the velocities are a negligible factor with respect to the evolution of themaximal wave envelope amplitude. However, if the solitons have the samedirection of motion, a change in the magnitude of the velocities will resultin different peaks of the wave envelope amplitude at the time of collision. Ingeneral, the peaks in these scenarios are limited by the peak of the envelopeamplitude of the scenario presented in Fig. 4.While computing the result in Fig. 7a, both magnitudes of the velocitiesare varied equally. In contrast to this, Fig. 7b was generated by fixing thevelocity of the inner soliton to v = − m/s, whereas the velocity v of theouter wave was varied. Figure 7: Maximal wave envelope amplitude during soliton collision in relation to magni-tude of the velocity. Figure 7a presents the results of solitons with opposite and Fig. 7bwith the same direction of movement.
With regard to the topic of extreme waves, it can be assumed that sce-narios of the kind shown in Fig. 4 offer the greatest potential for the creationof extreme waves. Scenarios of the type shown in Fig. 6 can also producecomparable peaks of the wave envelope amplitude. In this case, however,the development of the peaks depends very much on the appropriate choiceof the initial condition, such that the scenarios mentioned in Fig. 4 can beassumed to be more suitable.
The aim of this section is to make general statements about the interac-tion behavior in an irregular or random sea state. For this purpose the initial11ondition of the regular case described in Eq. (14) is modified according toEq. (15). The result of this procedure is illustrated in Fig. 3.The development of disturbed solitons shown in Fig. 8 illustrates thefundamental difference between the development of solitons in irregular andregular waves. The constant wave envelope amplitude outside the collisionperiod, which is characteristic for the soliton, is replaced by an oscillatingbehavior in the irregular case. Identical parameters for the initial conditionwere chosen for all simulations. This implies that the differences in oscillatorybehavior are generated by the stochastic perturbation according to Eq. (12).This justifies the importance of an empirical analysis.Furthermore, the impact of disturbances with varying significant waveheights on the soliton course is examined in Fig. 9 and Fig. 10. The majorfocus is on the effects caused by an increase of the significant wave height.Although no empirical analysis has been performed in this context, the resultsindicate an enhancement of the maximum of the wave envelope amplitudewith rising significant wave height. Moreover, a minor loss of the generalshape of the soliton wave is observed in this scenario.
Figure 8: Interaction behavior of disturbed solitons with amplitudes a = a = 1 m andvelocities v = − v = − . igure 9: Interaction behavior of disturbed solitons with amplitudes a = a = 1 m, ve-locities v = − v = − H s = 0 . H s = 0 . H s = 1 . H s = 2 . = 2 . = 2 . = 2 . = 2 . igure 10: Zoom into the result for H s = 2 . In order to carry out the analysis of the irregular case, the results ofthe regular case are considered first. Here it was verified that the scenarioof interaction of two solitons with the same amplitude and same velocitybut with opposite direction of movement offers the greatest potential for thedevelopment of greater wave envelope amplitudes. This information leadsthe focus on this scenario in the following analysis.For further reduction of the analysis, it is first shown that, despite theoscillating wave behavior, the time that elapses before the interaction has anegligible influence on the actual interaction and the resulting wave envelopeelevation.Since an irregular sea state is considered here, this is shown empiricallyin Fig. 11, which presents the maximum wave envelope amplitude during theinteraction as a function of the time elapsing before the interaction. The fig-ure indicates that the average wave elevation has an approximately constantvalue and furthermore that the standard deviation and the extrema of themaximum wave envelope amplitude are bounded nearly equally for all times.This leads to the conclusion that the time elapsing before the interaction isnegligible with respect to the expected wave envelope amplitude.14 igure 11: Maximal wave envelope amplitude during interaction for varying time passingbefore the interaction. For each time 100 simulations were performed.
However, the visualization in Fig. 11 only allows statements to be madeabout the average and limits of the wave envelope amplitude in irregular seastates, but no statements can be deduced about a stochastic distribution ofthese.Nevertheless, due to the previous statements, this empirical analysis canbe reduced to only one interaction scenario. For this purpose, the scenariopresented in Fig. 8 is selected. The main focus is again on the maximumwave envelope amplitude. Therefore, the histogram shown in Fig. 12 rep-resents the absolute elevation of the maximal wave envelope amplitude thatoccurred in the interaction. Thus, this provides insight into the correspond-ing stochastic distribution. Together with the results from Fig. 13, whichverifies the number of simulations as sufficient, the validity of the histogramis proven. Furthermore, the distribution of the maximal wave envelope canbe identified as a normal distribution. It should be emphasized here that themean value approximately corresponds to the maximal amplitude of the reg-ular interaction. In addition, Fig. 13b reveals that an increased amplitudeoccurs in over half of all simulations.15 igure 12: Histogram of the maximal wave envelope amplitude during interaction for atotal of 2000 simulations.Figure 13: (a) Average maximal wave envelope amplitude and (b) percentage with max-imum wave envelope amplitude above the amplitude of the regular case with respect tothe number of simulations.
Besides the analysis of the interaction itself, a further point of investiga-tion is the impact of the interaction on the disturbed solitons. At first thedegree to which the interaction affects the movement of the disturbed solitonis studied. 16his is accomplished by comparing the positions of two disturbed solitonsat the end of the simulation period, whereby one of the waves has undergonean interaction and the other one has taken an isolated course. From anyother point of view, the waves are chosen identically. The wave position ischaracterized by the location of the maximum wave envelope amplitude ofthe disturbed soliton. In the following, the scenario shown in Fig. 8 is used,where in the interaction case the disturbed solitons with negative velocityis analyzed. For the isolated case the other wave with positive velocity iseliminated.In the regular case in Fig. 4, no interaction effect can be determined anddue to the symmetric course, the final position is ξ = 750 m. In contrast,the empirical results for the irregular case presented in Fig. 14 indicatethat the interaction is most likely to cause a spatial shift in the direction ofmovement of the disturbed soliton. However, opposite shifts are also possible,but rather unlikely. Moreover, the average position of the disturbed solitonsin the isolated case is almost identical to the regular case.But, according to the scale of the spatial simulation area, all shifts of thismagnitude must be interpreted as minor. Figure 14: (a) Histogram of disturbed soliton positions for the isolated and interactionexposed case and (b) the average disturbed soliton position depending on the number ofsimulations, for a total of 1000 simulations.
Finally, beyond the impact of the interaction on the spatial displacementof the disturbed soliton wave, the impact of the interaction on the amplitudeof the wave envelope is investigated. For this purpose, the averaged amplitudeof the disturbed soliton before and after the interaction is compared. The17verage amplitude before the interaction is calculated in the time interval[0 s ,
500 s] and after the interaction in the time interval [1000 s , Figure 15: (a) Histogram of the change in amplitude of the disturbed soliton envelopeinduced by the interaction and (b) the average amplitude change depending on the numberof simulations, for a total of 2000 simulations.
4. Conclusions
The analysis of the soliton collision in a regular sea state confirms thescenario of solitons moving towards each other at the same speed and with thesame amplitude as the scenario with the best potential for the developmentof an extreme wave. Thus, the empirical analysis of the collision of disturbedsolitons in a random sea is based on exactly this scenario. For this, wepresented an approach which allows a regular sea to be transformed into arandom sea.Overall, it can be summarized that the amount of time before the dis-turbed solitons collide in a random sea makes a negligible contribution to thedevelopment of an extreme wave. Also the examined impact of the collisionon changes in the amplitude and spatial displacement of the disturbed solitoncan be considered insignificant. 18evertheless, the application of a realistic random sea state enhances thedevelopment of larger wave envelope amplitudes during the soliton collisioncompared to the regular sea state. Although the maximal amplitude for theregular and irregular state are almost identical on average, the distribution inthe irregular case can be approximately identified as a normal distribution.Applied to the topic of extreme waves, this suggests that a more realis-tic random sea state can promote the development of these waves and themanifestation of their high amplitude.
Acknoledgements
We thank F. Fedele for fruitful discussions about soliton propagation.
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