Using the inverse spectral theory of the nonlinear Schrodinger (NLS) equation
we correlate the development of rogue waves in oceanic sea states characterized
by the JONSWAP spectrum with the proximity to homoclinic solutions of the NLS
equation. We find in numerical simulations of the NLS equation that rogue waves
develop for JONSWAP initial data that is ``near'' NLS homoclinic data, while
rogue waves do not occur for JONSWAP data that is ``far'' from NLS homoclinic
data. We show the nonlinear spectral decomposition provides a simple criterium
for predicting the occurrence and strength of rogue waves (PACS: 92.10.Hm,
47.20.Ky, 47.35+i).Comment: 7 pages, 6 figures submitted to Physics of Fluids, October 25, 2004
  Revised version submitted to Physics of Fluids, December 12, 200

Islas, Alvaro

Schober, Constance

English

arXiv

Using the inverse spectral theory of the nonlinear Schrodinger (NLS) equation we correlate the development of rogue waves in oceanic sea states characterized by the Joint North Sea Wave Project (JONSWAP) spectrum with the proximity to homoclinic solutions of the NLS equation. We find in numerical simulations of the NLS equation that rogue waves develop for JONSWAP initial data that are  near  NLS homoclinic data, while rogue waves do not occur for JONSWAP data that are  far  from NLS homoclinic data. We show the nonlinear spectral decomposition provides a simple criterium for predicting the occurrence and strength of rogue waves

Islas, A. L.

Schober, C. M.

University of Central Florida (UCF): STARS (Showcase of Text, Archives, Research & Scholarship)

University of Central Florida STARS Faculty Bibliography 2000s Faculty Bibliography 1-1-2005 Predicting rogue waves in random oceanic sea states A. L. Islas University of Central Florida C. M. Schober University of Central Florida Find similar works at: https://stars.library.ucf.edu/facultybib2000 University of Central Florida Libraries http://library.ucf.edu This Article is brought to you for free and open access by the Faculty Bibliography at STARS. It has been accepted for inclusion in Faculty Bibliography 2000s by an authorized administrator of STARS. For more information, please contact STARS@ucf.edu. Recommended Citation Islas, A. L. and Schober, C. M., "Predicting rogue waves in random oceanic sea states" (2005). Faculty Bibliography 2000s. 5297. https://stars.library.ucf.edu/facultybib2000/5297 Predicting rogue waves in random oceanic sea statesA. L. Islas and C. M. Schobera!Department of Mathematics, University of Central Florida, Orlando, Florida 32816sReceived 27 October 2004; accepted 18 January 2005; published online 16 February 2005dUsing the inverse spectral theory of the nonlinear Schrödinger sNLSd equation we correlate thedevelopment of rogue waves in oceanic sea states characterized by the Joint North Sea Wave ProjectsJONSWAPd spectrum with the proximity to homoclinic solutions of the NLS equation. We find innumerical simulations of the NLS equation that rogue waves develop for JONSWAP initial data thatare “near” NLS homoclinic data, while rogue waves do not occur for JONSWAP data that are “far”from NLS homoclinic data. We show the nonlinear spectral decomposition provides a simplecriterium for predicting the occurrence and strength of rogue waves. © 2005 American Institute ofPhysics. fDOI: 10.1063/1.1872093gIntroduction. Rogue waves are rare, large amplitudewaves whose heights exceed 2.2 times the significant waveheight of the background sea. One of the proposed mecha-nisms for the development of rogue waves in deep water isnonlinear focusing due to the Benjamin–Feir sBFdinstability.1,2 The BF instability is a modulational instabilityin which a uniform train of surface gravity waves is unstableto a weak amplitude perturbation. The BF instability is de-scribed approximately by the focusing nonlinear SchrödingersNLSd equation3iut + uxx + 2uuu2u = 0, s1dand in the simplest setting homoclinic orbits of the unstableStokes solution of the NLS equation have been used formodeling rogue waves.4,5 Homoclinic solutions of the NLSequation, obtained when two or more unstable modes arepresent, can be phase modulated to provide striking ex-amples of wave amplification where the amplification is dueto both the BF instability and the additional phasemodulation.5The NLS equation is the leading order equation in ahierarchy of envelope equations and is derived from the fullwater wave equations under the assumption of a narrow Osedbanded spectrum. This bandwidth constraint limits the appli-cability of the NLS equation in two dimensions as it resultsin energy leakage to high wave number modes.6 The broaderbandwidth NLS sBBNLSd equation, obtained by assumingthe bandwidth is Os˛ed and by retaining higher order termsin the asymptotic expansion for the surface wave displace-ment, has been successful in reducing the energy leakage.6An alternate approach is to “enhance” the NLS equation withexact linear dispersion, whereby the equation has improvedbandwidth resolution and stability properties.7 All thesehigher order equations, whether narrow or broader band-width or enhanced, may be viewed as perturbations of theNLS equation. Homoclinic orbits of the Stokes wave havebeen shown to persist for the BBNLS equation.5,8 This per-sistence result suggests homoclinic solutions of the NLSequation may be significant in modeling rogue waves forrandom oceanic states.Onorato et al.9 examined the generation of extremewaves for typical random oceanic sea states characterized bythe Joint North Sea Wave Project sJONSWAPd power spec-trum. In numerical simulations of the NLS equation it wasfound that rogue waves occur more often for large values ofthe Phillips parameter a and the enhancement coefficient gin the JONSWAP spectrum. Even so, they observed thatlarge values of a and g do not guarantee the development ofextreme waves.In this Letter we clarify the dependence of rogue waveevents on the phases in the “random phase” reconstruction ofthe surface elevation fsee Eq. s2dg. We find that the phaseinformation is as important as the amplitude and peakednessof the wave sgoverned by a and gd when determining theoccurrence of rogue waves. Random oceanic sea states char-acterized by JONSWAP data are not small perturbations ofStokes wave solutions. As a consequence, it is difficult toinvestigate the generation of rogue waves in more realisticsea states using a linear stability analysis sas in theBenjamin–Feir instabilityd. Our approach is based on theNLS equation and its inverse spectral theory, used to exam-ine a nonlinear mode decomposition of JONSWAP type ini-tial data. Such analysis allows us to determine the nonlinearmode content of the data and the proximity smeasured interms of a parameter dd to instabilities and homoclinic solu-tions of the NLS equation.Our main results are s1d JONSWAP data can be quitenear data for homoclinic orbits of complicated N-phase so-lutions. For fixed values of a and g in the JONSWAP spec-trum, as the phases in the initial data are randomly varied,the proximity d to homoclinic data varies. s2d In hundreds ofsimulations of the NLS, where the parameters and the phasesin the JONSWAP initial data are varied, we find that roguewaves develop for JONSWAP data that are “near” NLS ho-moclinic data, while rogue waves do not occur for JON-SWAP data that are “far” from NLS homoclinic data. Con-sequently, we find that the nonlinear spectral decompositionprovides a simple criterium, in terms of the proximity toadAuthor to whom correspondence should be addressed. Telephone: 407-823-0147. Fax: 407-823-6253. Electronic mail: cschober@mail.ucf.eduPHYSICS OF FLUIDS 17, 031701 s2005d1070-6631/2005/17~3!/031701/4/$22.50 © 2005 American Institute of Physics17, 031701-1homoclinic solutions, for predicting the occurrence andstrength of rogue waves. This is the first time homoclinicsolutions have been correlated with rogue waves for realisticoceanic conditions.Random oceanic sea states. To examine the generationof rogue waves in a random sea state, we note that the sur-face elevation h is related to u, the solution of the NLSequation, by h=Rehiueikxj /˛2k. Using the Hilbert transformof h and its associated analytical signal, the initial conditionfor u can be modeled as the random wave processusx,0d = on=1NCn expfiskn−1x − fndg , s2dwhere Cn is the amplitude of the nth component with wavenumber kn= sn−1dk, k=2p /L, and random phase fn, uni-formly distributed on the interval s0,2pd. The spectral am-plitudes, Cn=−i˛2Sn /L, are obtained from the JONSWAPspectrum,9Ssfd = af 5 expF− 54S f0f D4Gg r, r = expF− 12S f − f0s0f0 D2G .s3dHere f0 is the dominant frequency, determined by the windspeed at a specified height above the sea surface, s0=0.07s0.9d for f ł f0sf . f0d and fn=n /L is the wave fre-quency. The parameter g is the peak-shape parameter; as g isincreased, the spectrum becomes narrower about the domi-nant peak. For g.1 the wave spectra continues to evolvethrough nonlinear wave-wave interactions even for very longtimes and distances. It is in this sense that JONSWAP spectradescribe developing sea states rather than a fully developedsea. The scale parameter a is related to the amplitude andenergy content of the wavefield. Based on an “Ursell num-ber,” the ratio of the nonlinear and dispersive terms of theNLS equation s1d in dimensional form, the NLS equation isconsidered to be applicable for 2,g,8.9 Typical values ofalpha are 0.008,a,0.02.We examine a nonlinear spectral decomposition of theJONSWAP initial data, which takes into account the phaseinformation fn. This decomposition is based upon the in-verse scattering theory of the NLS equation, a procedure forsolving the initial value problem analogous to Fourier meth-ods for linear problems. We find that we are able to predictthe occurrence of rogue waves in terms of the proximity d todistinguished points of the discrete spectrum. We briefly re-call elements of the nonlinear spectral theory of the NLSequation.Floquet spectral theory. The integrability of the NLSequation s1d is related to the following pair of linear systemssthe so-called Lax paird:Lsxdf = SD + il − uu* D − il DSf1f2 D = 0, Lstdf = 0, s4dwhere D denotes the derivative with respect to x, l is thespectral parameter and f is the eigenfunction.3 These sys-tems have a common nontrivial solution fsx , t ;ld, providedthe potential usx , td satisfies the NLS equation. Lstd is notspecified explicitly as it is not implemented in our analysis.The first step in solving the NLS using the inverse scat-tering theory is to determine the spectrum ssud= hlPC uLsxdf=0, ufu bounded " xj of the associated linearoperator Lsxd, which is analogous to calculating the Fouriercoefficients in Fourier theory. For periodic boundary condi-tions, usx+L , td=usx , td, the spectrum of u is expressed interms of the transfer matrix Msx+L ;u ,ld across a period,where Msx ;u ,ld is a fundamental solution matrix of the Laxpair s4d. Introducing the Floquet discriminant Dsu ,ld=TrfMsx+L ;u ,ldg, one obtains3ssud = hl P CuDsu,ld P R,− 2 ł Dsu,ld ł 2j . s5dThe distinguished points of the periodic/antiperiodic spec-trum, where Dsl ,ud= ±2, are: sad simple points hl js udD /dlÞ0j and sbd double points hl jd udD /dl=0,d2D /dl2Þ0j. TheFloquet discriminant functional Dsu ,ld is invariant under theNLS flow and encodes the infinite family of constants ofmotion of the NLS sparametrized by the l jsd.The Floquet spectrum s5d of a generic NLS potentialconsists of the entire real axis plus additional curves scalledbandsd of continuous spectrum which terminate at the simplepoints l js. N-phase solutions are those with a finite number ofbands of continuous spectrum. Double points arise when twosimple points have coalesced and their location is important.Using the direct spectral transform, any initial conditionor solution of the NLS can be represented in terms of a set ofnonlinear modes. The spatial structure and dynamical stabil-ity of these modes is determined by the order and location ofthe corresponding l j as follows:10 sad Simple points corre-spond to stable active degrees of freedom. sbd Double pointslabel all additional potentially active degrees of freedom.Real double points correspond to stable inactive szero ampli-tuded modes. Complex double points are associated with allthe unstable active modes and label the corresponding ho-moclinic orbits.Figure 1 shows the spectrum of a typical unstableN-phase solution. There are N bands of spectrum determinedby the 2N simple points l js. The 2M complex double pointsl jd indicate that the solution is unstable and that there is aFIG. 1. Spectrum of an unstable N-phase solution.031701-2 A. L. Islas and C. M. Schober Phys. Fluids 17, 031701 ~2005!homoclinic orbit. The simple periodic eigenvalues are la-beled by circles and the double points are labeled by crosses.An example of a spectrum for a nearby semistable N-phasesolution where the complex double point is split Osed isgiven in Fig. 2sad.Explicit formulas for the N-phase solutions,Qsu1 , . . . ,uNd, are obtained in terms of the simple spectrum.The phases evolve according to u j =k jx+V jt+u j0, k j=2pnj /L, where k j and V j are determined by l js ssince thespectrum is invariant k j and V j are constantsd. For a givenN-phase solution, the isospectral set sall NLS solutions withthe same spectrumd comprises an N-dimensional torus char-acterized by the phases u j. If the spectrum contains complexdouble points, then the N-phase solution may be unstable.The instabilities correspond to orbits homoclinic to theN-phase torus.JONSWAP data and the proximity to homoclinic solu-tions of the NLS. In the numerical simulations the NLSequation is integrated using a pseudospectral scheme with256 Fourier modes in space and a fourth-order Runge–Kuttadiscretization in time sDt=10−3d. The nonlinear mode con-tent of the data is numerically computed using the directspectral transform described above, i.e., the system of ordi-nary differential equations s4d is numerically solved to obtainthe discriminant D. The zeros of D±2 are then determinedwith a root solver based on Muller’s method.10 The spectrumis computed with an accuracy of Os10−6d, whereas the spec-tral quantities we are interested in are in the rangeOs10−2d–Os10−1d.Complex double points typically split under perturbationinto two simple points, l±, thus opening a gap in the band ofspectrum fsee Fig. 2sadg. We denote the distance betweenthese two simple points by dsl+ ,l−d= ul+−l−u and refer to itas the splitting distance. We use d to measure the proximityin the spectral plane to homoclinic data, i.e., to complexdouble points and their corresponding instabilities. Since theNLS spectrum is symmetric with respect to the real axis andreal double points correspond to inactive modes, in subse-quent plots only the spectrum in the upper half complex lplane will be displayed.We begin by determining the spectrum ofJONSWAP initial data given by s2d for various combinationsof a=0.008,0.012,0.016,0.02, and g=1,2 ,4 ,6 ,8. For eachsuch pair sg ,ad, we performed fifty simulations, each with adifferent set of randomly generated phases. As expected, thebasic spectral configuration and the number of excited modesdepended on the energy and the enhancement coefficient aand g. However, the extent of the dependence of the spec-trum upon the phases in the initial data was surprising.As a typical example of the results, Figs. 2sad and 3sadFIG. 2. sad Nonlinear spectrum and sbd evolution of Umax for JONSWAPdata sg=4 and a=0.016d that are near homoclinic data. Dashed curve cor-responds to 2.2Hs.FIG. 3. sad Nonlinear spectrum and sbd evolution of Umax for JONSWAPdata sg=4 and a=0.016d that are far from homoclinic data. Dashed curvecorresponds to 2.2Hs.031701-3 Predicting rogue waves in random oceanic sea states Phys. Fluids 17, 031701 ~2005!show the numerically computed nonlinear spectrum ofJONSWAP initial data when g=4 and a=0.016 for two dif-ferent realizations of the random phases. We find thatJONSWAP data correspond to “semistable” N-phase solu-tions, i.e., we interpret the data as perturbations of N-phasesolutions with one or more unstable modes fcompare Fig.2sad with the spectrum of an unstable N-phase solution inFig. 1g. In Fig. 2sad the splitting distance dsl+ ,l−d<0.07,while in Fig. 3sad dsl+ ,l−d<0.2. Thus the JONSWAP datacan be quite near homoclinic data as in Fig. 2sad or far fromhomoclinic data as in Fig. 3sad, depending on the values ofthe phases fn in the initial data. For all the examined valuesof a and g we find that, when a and g are fixed, as thephases in the JONSWAP data vary, the spectral distance d oftypical JONSWAP data from homoclinic data varies.Most importantly, irrespective of the values of theJONSWAP parameters a and g, in simulations of the NLSequation s1d we find that extreme waves develop forJONSWAP initial data that are near NLS homoclinic data,whereas the JONSWAP data that are far from NLS ho-moclinic data typically do not generate extreme waves. Fig-ures 2sbd and 3sbd show the corresponding evolution of themaximum surface elevation, Umax, obtained with the NLSequation. Umax is given by the solid curve and as a reference,2.2HS sthe threshold for a rogue waved is given by the dashedcurve. HS is the significant wave height and is calculated asfour times the standard deviation of the wave amplitude. Fig-ure 2sbd shows that when the nonlinear spectrum is nearhomoclinic data, Umax exceeds 2.2HS sa rogue wave developsat about t=40d. Figure 3sbd shows that when the nonlinearspectrum is far from homoclinic data, Umax is significantlybelow 2.2HS and a rogue wave does not develop. In this way,we correlate the occurrence of rogue waves characterized byJONSWAP spectrum with the proximity to homoclinic solu-tions of the NLS equation.The results of hundreds of simulations of the NLS equa-tion consistently show that proximity to homoclinic data is acrucial indicator of rogue wave events. For example, Fig. 4shows the synthesis of 200 random simulations of the NLSequation for JONSWAP initial data for different sg ,ad pairsswith g=2,4 ,6 ,8, and a=0.012,0.016d. For each such pairsg ,ad, we performed 25 simulations, each with a differentset of randomly generated phases. Each circle represents thestrength of the maximum wave sUmax/HSd attained duringone simulation as a function of the splitting distancedsl+ ,l−d. The results for the particular pair sg=4,a=0.012d is represented with an asterisk. A horizontal line atUmax/HS=2.2 indicates the reference strength for rogue waveformation. We identify two critical values d1=0.08 and d2=0.22 that clearly show that sad if d,d1 snear homoclinicdatad rogue waves will occur; sbd if d1,d,d2, the likeli-hood of obtaining rogue waves decreases as d increases and,scd if d.d2 the likelihood of a rogue wave occurring is ex-tremely small.This behavior is robust. As a and g are varied, thestrength of the maximum wave and the occurrence of roguewaves are well predicted by the proximity to homoclinicsolutions. The individual plots of the strength vs d for par-ticular pairs sg ,ad are qualitatively the same as in Fig. 4 ascan be seen by the highlighted case sg=4, a=0.012d. Theseresults give strong evidence of the relevance of homoclinicsolutions of the NLS equation in investigating rogue wavephenomena for more realistic oceanic conditions and identi-fies the nonlinear spectral decomposition as a simple diag-nostic tool for predicting the occurrence and strength ofrogue waves. Finally we remark that the nonlinear spectralanalysis can be implemented for other general data snon-JONSWAPd in order to predict the occurrence of roguewaves.This work was partially supported by NSF Grant No.NSF-DMS0204714.1Rogue Waves 2000, edited by M. Olagnon and G. Athanassoulis sIfremer,France, 2001d, Vol. 32.2C. Kharif and E. Pelinovsky, “Physical mechanisms of the rogue wavephenomenon,” Eur. J. Mech. A/Solids 22, 603 s2003d.3M. Ablowitz and H. Segur, Solitons and the Inverse Scattering TransformsSIAM, Philadelphia, 1981d.4A. Osborne, M. Onorato, and M. Serio, “The nonlinear dynamics of roguewaves and holes in deep-water gravity wave trains,” Phys. Lett. A 275,386 s2000d.5A. Calini and C. Schober, “Homoclinic chaos increases the likelihood ofrogue waves,” Phys. Lett. A 298, 335 s2002d.6K. Trulsen and K. Dysthe, “A modified nonlinear Schrödinger equation forbroader bandwidth gravity waves on deep water,” Wave Motion 24, 281s1996d.7K. Trulsen, I. Kliakhandler, K. Dysthe, and M. Velarde, “On weakly non-linear modulations of waves on deep water,” Phys. Fluids 12, 2432s2000d.8M. Ablowitz, J. Hammack, D. Henderson, and C. Schober, “Long timedynamics of the modulational instability of deep water waves,” Physica D152–153, 416 s2001d.9M. Onorato, A. Osborne, M. Serio, and S. Bertone, “Freak wave in ran-dom oceanic sea states,” Phys. Rev. Lett. 86, 5831 s2001d.10N. Ercolani, M. G. Forest, and D. McLaughlin, “Geometry of the modu-lational instability III. Homoclinic orbits,” Physica D 43, 349 s1990d.FIG. 4. Strength of Umax/Hs vs the splitting distance dsl+ ,l−d.031701-4 A. L. Islas and C. M. Schober Phys. Fluids 17, 031701 ~2005!

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Predicting rogue waves in random oceanic sea states

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