Analysis of Energy Characteristics in the Process of Freak Wave Generation

China Ocean Eng, Vol. 28, No.2, pp. 193 - 205◎2014 Chinese Ocean Engineering SocietyDOI 10.1007/s13344-014-0015-6，ISSN 0890-5487Analysis of Energy Characteristics in the Process of Freak Wave Generation*HU Jin-peng (胡金鹏).b and ZHANG Yun-qiu (张运秋):d1a School of Civil Engineering and Transportation, South China University of Technology, Guangzhou 510641, Chinab Naval Architecture and Ocean Engineering R & D Center of Guangdong Province,South China University of Technology, Guangzhou 510641, China“Guangzhou Institute of Energy Conversion, Chinese Academy of Sciences, Guangzhou 510640, Chinad Key Laboratory of Renewable Energy and Gas Hydrate, Chinese Academy of Sciences, Guangzhou 5 10640, China(Received 4 June 2012; received revised form 9 November 2012; acepted 28 February 2013)ABSTRACTThe energy characteristics in the evolution of the wave train are investigated to understand the inherent cause of the freakwave generation. The Morlet wavelet spectrum method is employed to analyze the numerical, laboratory and field evolution dataof this generation process. Their energy distributions and variations are discussed with consideration of corresponding surfaceelevations. Through comparing the energy charateristics of three cases, it is shown that the freak wave generation depends notonly on the continuous transfer of wave train energy to a certain region where finally the maximum energy occurs, but also on thedistinct shift of the converged energy to high-frequency components in a very short time. And the typical energy characteristics offreak waves are also given.Key words: freak wave; generation process; wavelet spectrum; energy characteristics1. IntroductionFreak waves (Klinting and Sand, 1987), usually much higher than the normal waves in the ocean,have been paid much attention to for their disastrous destruction of ocean engineering structures(Kharif and Pelinovsky, 2003; Zhao et al, 2009, 2010). They occur occasionally and disappearinstantaneously in some unexpected positions. Hence the field data of freak waves are very scarce andprecious. Many researchers have used the numerical and laboratory simulations to study theirgeneration mechanisms and characteristics. However, the energy characteristics of the freak wavegeneration process are still unclear, and require intensive study.Energy characteristics are traditionally analyzed with the Fourier energy spectrum method. Thismethod displays a time-mean energy distribution in frequency and is suitable for analyzing a stationaryrandom process. However, it is insufficient for analyzing a non-stationary process such as a wave traincontaining a freak wave (Chien et al, 2002). A proper method is the wavelet spectrum method. Thewavelet spectrum method is another relatively new energy analysis method that has been applied toocean engineering for nearly twenty years (Massel, 2001; Lin and Liu, 2004). Because it can clearlydisplay the local energy properties of a signal in frequency and time domain, many oceanographersThe work was financially supported by the National Natural Science Foundation of China (Grant Nos. 10902039 and41106031).1 Corresponding author. E-mail: zhangyq6@ms.gec.ac.cn中国煤化工YHCNMH G194HU Jin-peng and ZHANG Yun-qiu/ China Ocean Eng, 28(2), 2014, 193 - 205have used this method to analyze the energy structures of freak waves.Now the“New Year wave", measured in the Central North Sea on January 1, 1995, is still themost famous freak wave. Jakobsen et al. (2001) used the wavelet transformation to analyze this waverecord and noticed that, compared with the Mexican hat wavelet, the time-mean Morlet wavelet energyspectrum fits well with the Fourier energy spectrum. Liu and Mori (2000, 2001) and Mori et al. (2002)analyzed several periods of continuous wave records from the Sea of Japan during 1986-1990. Theirresults indicated that freak waves can be readily identified from the Morlet wavelet spectrum. At theinstant a freak wave occurs, its energy density is strong and instantly surges and seemingly carries overto the high-frequency components. They also regarded wavelet transform analysis as an ideal approachto discern the localized characteristics of freak waves. The freak wave event is also one of the frequenthazards around the Taiwan coast. Chien et al. (2002) used the Morlet wavelet spectrum to analyzecoastal freak waves from Cheng-Kung wave stations. The wavelet spectrum of a freak wave series,whose crest is much larger than its trough, shows its energy components are very complex, and itsfrequency band covers a wider range. Zhang (2008) analyzed 25 sets of freak wave records from 10years of Hualien buoy data. He found that at the moment when a freak wave occurred, the Morletwavelet spectrum was wide and the total energy was the largest. However, the moment when thelargest peak energy density emerges, does not necessarily indicate a freak wave. Wu et al. (2010) usedthe wavelet scalogram of the above records to investigate the energy characteristics during theoccurrence of freak waves. The wave energy and phase analysis illuminated that the component waveswould lead to constructive superposition due to similar phases. Lee et al. (2011) also studied these 25sets of data by using the Morlet spectrum, revealing that freak waves display the maximum waveenergies and the instantaneous phase spectrum concentrations. Different from these measured datainvestigations, Kwon et al. (2005) investigated the laboratory freak wave and its impact force throughMorlet transform. Their coherence analysis revealed that some high-frequency components werehighly correlated with the impact force. The study demonstrated that the wavelet transform can be analternative tool in the analysis of the strongly nonlinear freak wave and its impact. Huang et al. (2009)used the Morlet spectrum to study the time-frequency characteristics of laboratory freak wavegeneration process. They noticed that the energy distribution was distinct before and after the freakwave and that high-frequency components existed at the occurrence of freak waves. They alsosuggested that the development stage of a freak wave can be discerned via its energy distribution.Most of those previous investigations focus on the instantaneous time-frequency characteristics offield freak wave records. The measured wave data primarily comes from some single-pointobservations, so it is hard to acquire the continuous energy information of freak waves employed inthis study through analyzing the field data. This energy evolution analysis was only carried out by fewresearchers by using laboratory data (Huang et al, 2009). In the present study, for better understandingthe inherent cause of freak wave generation, we use the numerical, laboratory and field data to studythe energy characteristics of its generation process. Firstly, the wavelet spectrum method is presentedin brief. Secondly, the numerical, laboratory, and measured evolution records of freak wave generationare collcted. And then the time-frequency evolution characteristics of the energy during thegeneration of freak waves are discussed. Finally, the main conclusions are drawn.中国煤化工MHCNMH GHU Jin-peng and ZHANG Yun-qiu 1 China Ocean Eng.. 28(2), 2014, 193 - 205952. Overview of Wavelet Spectrum Method2.1 Wavelet SpectrumThe wavelet is a wave with a very short duration. It must meet the following admissibilitycondition:C,=J°!1y(0)-do< +∞,(1)0where 0 is the angular frequency and vj(@) is the Fourier transformation of the time functionψ(t). And here y(t) is the mother wavelet. Through scaling and shifting, the following family ofanalyzing wavelets is obtained:for a>0,b∈R,(2)Ja"where a and b are the scaling parameter and shifting parameter, respectively. In Fourier space, thewavelet ψ。 。,(1) is transformed into.(0)= F {w..()}= JL y.()exp( - ion)dt =√avy(aw)exp(-iob). (3)The continuous wavelet transform of any finite energy signal f(t) is defined as:Wf(a,b)=-J f()y"(4)where' indicates a complex conjugate.The squared modulus of the wavelet transform presents the signal energy distribution in thetime-scale plane. Thus, |Wf(a,b)" is the scaling spectrum of the signal (Tang, 2006). For theconvenience of analysis, the wavelet spectrum |Wf(f,)" is used in the present study considering thescaling parameter a (= f/f) and the shifting parameter b (=t), where the value of f is in therange of signal frequency, and f。 is the center frequency of the mother wavelet.2.2 Morlet WaveletMany wavelet functions can be applied to wavelet transforms (Tang, 2006). In this study, thecomplex Morlet wavelet is selected as the mother wavelet:√2πσexp2σ2|exp(io.t),(5)where σ is the bandwidth parameter, and 0。is the center frequency of the mother wavelet. In theFourier space, it is transformed intoσ(∞- o)y(@)= exp|(6)2When o, ≥5/σ，this complex wavelet is satisfied with the admissibility condition Eq. (1) andapproximately becomes an analytic function.中国煤化工YHCNMH G196HU Jin-peng and ZHANG Yun-qiu/ China Ocean Eng, 28(2), 2014, 193 - 2053. Evolution Records of Freak Wave Generation3.1 Numerical Evolution RecordsThe numerical evolution records of freak wave generation are taken from the studies of Zhang etal. (2009). In their simulations, the random waves described by JONSWAP were employed as initialconditions, with the Phillips parameter a = 0.0162, the enhancement coefficient γ= 7，and thedominant frequency fo =0.1 Hz. The four-order nonlinear Schrodinger equation proposed by Lo andMei (1985) serves as the governing equation, which controls the evolution of the complex envelope indeep water. The dimensionless wave train evolves within 2π period, the sideband instabilitymechanism is applied to the simulation of freak wave generation, and the conservation law is verifiedin this evolution. The up-crossing zero method is used to estimate the suface elevation, and theappearance of freak waves is determined by the definition of Klinting and Sand (1987). Fig. 1 presentsthe evolution of a dimensionless wave train in a coordinate system moving at its group velocity for thescale factor of the governing equation that equals 0.4. In order to clearly display surface elevationvariations, the interval of two neighboring observation points is gradually reduced from the initialposition to the freak wave in Fig. 1. The initial wave train irregularly fluctuates. As the fetch increases,this wave train gradually splits into two large wave groups. One is short with a larger wave height,whereas the other is long. As the sideband instability develops, in a very short fetch from 1.0 to 1.122,the peaks of the wave groups continue to grow, and the surface elevations on both sides drop,eventually leading to the freak wave generation in the short wave group. Although the surfaceelevations before this generation do not meet the definition of freak waves (Klinting and Sand, 1987),they are similarly considered as extreme waves in the ocean engineering.“ofuwWMMim“ofmJhwW]ww一站，p=0.下n=1.082: ofumMWwin “ofusmWMwn=1.092ofmJmWmw1 n7=1.102ofuJmwM一1=0.8n=1.112” ofu八ofuulwm~上1 n7=1.1227=1.0”ofr lmwwiofulwwM0.0.5.0.52.070.0 0.5 1.0 1.5 2.0/πζFig. 1. Numerical dimensionless evolution records of the freak wave (η = 1.122 ) generation under the condition of randomwaves described by the JONSWAP spectrum.中国煤化工YHCNMH GHU Jin-peng and ZHANG Yun-qiu 1 China Ocean Eng.. 28(2), 2014, 193 - 205973.2 Laboratory Evolution RecordsThe laboratory evolution data of freak wave generation are obtained from the physicalexperiments conducted in the State Key Laboratory of Coastal and Offshore Engineering, DalianUniversity of Technology. The experimental tank is 56 m long, 3 m wide, and 1 m deep. Twenty-onewave gauges, which are concentrated near the wave focusing position, are set in this tank to capturelarge waves, and the detailed layout is like Fig. 1 in Huang et al. (2009). Random waves characterizedby the Pierson-Moscowitz (P-M) spectrum are used in this experiment with the following initialconditions: 0.05 m significant wave height; 0.89 Hz dominant frequency; 0.0236 s sampling interval;about 120 s sampling length; and 100 component waves. One third of these components with the sameinitial phases are arranged to simulate the formation of freak waves.As shown in Fig. 2, the surface elevation of about 100 s is selected for the computationconvenience. The first wave gauge records a longer wave group from 60 s to 80 s at the initial randomwave train propagation. This group gradually splits into several smaller wave groups and a short wavegroup with a higher wave height. This higher wave develops into a freak wave at about 92 s, asrecorded in #20 gauge. Upon reaching #21 gauge, this wave is still higher, but no longer meets thedefinition of a freak wave. In addition, as the wave train develops, a higher wave group is formedbetween 20 s and 40 s, with a highly symmetric surface elevation.105101首商#192021s2(001(S)Fig. 2. Laboratory evolution records of the freak wave (#20) generation under the condition of random wavesdescribed by the P-M spectrum.中国煤化工YHCNMH G198HU Jin-peng and ZHANG Yun-qiu/ China Ocean Eng, 28(2), 2014, 193 - 2053.3 Observed Evolution RecordsThe in situ freak wave record is the“New Year wave”" from the Draupner oil platform in the NorthSea (see the surface elevation recorded from 15:20 in Fig. 3). The largest wave is 25.6 m high with acrest of 18.5 m, and lasts about 12 s, whereas the significant wave height is only 11.9 m at that time.Field wave measurements are often made at a single point. Although many stations may be set in somesea areas, it is still impossible to obtain the evolution records of a wave train if the observed points arenot perfectly fixed on the direction of wave propagation. In addition, the uncertainty of freak waveoccurrences makes their temporal and spatial evolution records very diffcult to be obtained. Thus,surface elevations recorded 1 h before and after the freak wave observation from this plaform are alsogiven for comparison. These wave data are a single-point evolution process gathered by the laser wavegauge set on the platform. The sampling frequency is 2.1333 Hz, and the observation interval is about20 min.1014:20宜-1015:2016:2010-004006001(S)Fig. 3. Observed evolution records of the freak wave (15:20) generation from the Draupner oil platform in the North Seaon January l, 1995.4. Wavelet Spectrum Analysis of the Freak Wave Generation ProcessWe assume that the bandwidth parameter of the Morlet wavelet function is a unit, and adjust thecenter frequency of the mother wavelet to analyze evolution records on freak wave generations. Thiscenter frequency has an important effect on the wavelet spectrum. When this frequency is high, thetime-domain resolution of the wavelet spectrum will increase, and its corresponding frequency-domainresolution will decrease. In contrast, if the center frequency is low, the time-domain resolution willdecrease, and the frequency-domain resolution will increase. Thus, the lower center frequency isemployed to explore the high-frequency characteristics of freak waves when the Morlet wavelet issatisfied with the admissibility condition.4.1 Wavelet Spectrum Analysis of Numerical Evolution RecordsFig. 4 presents the wavelet spectra of the numerical evolution process on the freak wave generation中国煤化工MHCNMH GHU. Jin-peng and ZHANG Yun-qiu 1 China Ocean Eng.. 28(2), 2014, 193 - 205199in Fig. 1. The vertical and horizontal axes denote frequency and time, respectively. The curves are thecontours of Morlet wavelet spectra, with the color variation from blue to red exhibiting energy densitygrowth. At the initial position, the random wave energy is mainly distributed from 0.06 Hz to 0.2 Hz.With the increase of dimensionless fetch, the wave train energy gradually converges on the forming wavegroups (see Fig. 1) under the effect of sideband instability. Thus, the group energy densities around thedominant frequency continue to grow, and the corresponding wavelet spectrum contours becomeintensive. At η = 1.0, the outside equivalent density curves of the left short group begin to extend to thehigh frequency. At η = 1.05 , this phenomenon disappears. From η= 1.082 to η=1.122 , this energyshifting characteristic of the left short wave group reoccurs, and is gradually strengthened, whereas itscentral energy densities do not remain increasing all along. As the freak wave occurs, in a very shorttime, the outside contours of its energy density evidently extend to high frequencies, and the spectrumbandwidth increases to its maximum. However, we note that energy densities of the freak wave,although very high at that time, are not the maximum, because energy densities of the short groupbefore its occurrence and those of the right long group are even higher at some points.).4-0.η=0n= 1.05御.0L0.0)4-0.4n=0.27= 1.082.李.41-=0.41= 1.092.)A_η=0.6η= 1.102..00.0lη= 0.8η= .112是n=1.01=1.122 t001.02.00.5.52.53.03.:Fig. 4. Wavelet spectrum variation of the numerical freak wave generation process under the condition of random waves.中国煤化工MYHCNMH G200HU Jin-peng and ZHANG Yun-qiu/ China Ocean Eng, 28(2), 2014, 193 - 205Fig. 5 presents the energy variations of the above process against time. This wavelet energy is theintegration of the wavelet spectrum on frequency. The wave train energy is scattered at the initialposition, gradually converged on wave groups in the evolution, and highly concentrated on the locationof the freak wave generation. Although the energy densities of the right long wave group reach themaximum in Fig. 4, its corresponding energy is still lower than that of the freak wave. Thisphenomenon occurs because the wave group components are mainly distributed around the dominantfrequency, and their amplitudes are larger. In addition, the corresponding short wave group energy isrelatively large before the freak wave generation. Hence, such a short wave group is also verydestructive.In particular, wave trains at η=1.0 and η=1.122 have the similar wavelet spectra and energycurves, but the former is not recognized to contain a freak wave because of the wave height estimationwith the up-rossing zero method employed. If a down-crossing zero method is applied, this large wave isalso considered as a freak wave (Klinting and Sand, 1987). Furthermore, the energy transferring processto the high-frequency components before and after η=1.0 in Fig. 4 has not been observed. This meansthe distance between these two sampling points is too large to capture this property.η=0η= 1.050.00.0L2.2.0η=0.21= 1.0822.0 |η=0.41= 1.092 10.0 Iη=0.61= 1.102 -0.0620pn=0.8 .1= .112η=1.0”= 1.12211.0.02Fig. 5. Wave energy variation of the numerical freak wave generation process under the condition of random waves.中国煤化工MYHCNMH GHU. Jin-peng and ZHANG Yun-qiu 1 China Ocean Eng.. 28(2), 2014, 193 - 205014.2 Wavelet Spectrum Analysis of Laboratory Evolution RecordsFig. 6 shows the wavelet spectra of laboratory evolution records on the freak wave generation.The energy of wave train 1 near the wave maker mainly ranges from 0.7 Hz to 1.8 Hz, and its densitiesare much lower during the observation period. Moreover, its contours have no noticeable extension tohigher frequencies but two moments of low densities around the dominant frequency. As the fetchincreases, the energy densities of wave trains gradually grow at the moments of forming some large shortwave groups (see Fig. 2). When these wave groups arrive at a certain point, e.g.. wave train 18, theirenergy densities around the dominant frequencies reach the maximum and then drop. At wave gauge #20,as a freak wave appears, the central energy densities of this large wave are higher, and the outside oneshave noticeable instantaneous extensions to the high frequency. Hence, the wavelet spectrum bandwidthepisodically increases to the peak of the total evolution process. Although the large short wave groupbetween 20 s and 30 s (Fig. 2) also has the higher energy densities at around 1 Hz, its energy densitycurves do not exhibit the above characteristics of the energy transfer to high-frequency components. Inaddition, it should be pointed out that here the energy shifting characteristic also does not have achanging process on the freak wave generation. The reason for this may be that the sampling space istoo large.4-#10.至25◎080 066 0◎00●c◎司 《一16-o◎❽o目B.@o60ooCaeFcoo。❹o0。目。四0。。C#19ξoc●◎0@.6o。050◎。0心.❻0oo20 -h2630 o66a◎8x◎0◎◎e。0oo#21全2三f.coo ◎@◎④◎ooco0a。@o.02工02(10501001(s)0.5.01.2.3.03.4.0Fig. 6. Wavelet spectrum variation of laboratory freak wave generation process under the condition of random waves.中国煤化工MYHCNMH G202HU Jin-peng and ZHANG Yun-qiu/ China Ocean Eng, 28(2), 2014, 193 - 205Fig. 7 displays the energy variations of the above process. The wave train energy at the first wavegauge is low and scattered except for few high wave groups. In the evolution of the wave train, theenergy gradually focuses on these wave groups to bring out the growth of their energies. As a freakwave occurs, the energy rapidly increases, and reaches the maximum. The energy of the large shortwave group between 20 s and 30 s is higher, but it persists longer. In addition, the energy of large shortwave groups before or after the freak wave formation quite approaches that of the freak wave, andthus, their potential damages should be paid high attention to.wNM nuN Mwan.swo _#101016uhl18Aaaassn19120218(0Fig. 7. Wave energy variation of laboratory freak wave generation process under the condition of random waves.4.3 Wavelet Spectra of Observed Evolution Records on Freak Wave GenerationFor the convenience of computations, the wave records from the Draupner oil plaform are selectedto be about 1000 s. Their wavelet spectra are given in Fig. 8. The wave train energy mainly ranges from0.06 Hz to 0.15 Hz. At most of the time without the freak wave, energy densities are lower, and do notshift to high frequencies. Although the contour lines occasionally extend to high frequencies, theircorresponding densities around the dominant frequency are all very low. Besides, the wavelet spectra still中国煤化工YHCNMH GHU Jin-peng and ZHANG Yun-qiu 1 China Ocean Eng.. 28(2), 2014, 193 - 20503have some equivalent curves with high energy densities for the wave trains recorded from the initial time14:20 and 16:20. Looking back to the measured surface elevations in Fig. 3, these high densitiescorrespond to the occurrence of large short wave groups. In the wavelet spectrum of the wave trainmeasured from 15:20, there also exists a distribution of high wave energy densities. This high density partjust corresponds to the generation of“New Year wave". The energy densities of the freak wave are veryhigh in the central frequencies, but not the largest ones among the three wavelet spectra. Moreover, theoutside equivalent density curves instantly surge and extend to the high-frequency components, andsimultaneously, the spectrum bandwidth rapidly reaches its maximum.Fig. 9 presents the energy variations of the above process. Without the occurence of freak wavesand wave groups, the wave train energy is low. In the large wave groups, their energies are high, andoften last long. As the freak wave appears, its wave energy is high and much larger than that at othermoments. However, this phenomenon occurs only for a very short time.).3-14:2.0L)3-15:20。A16:20pDCCEO6回C‘G10001(5)30405(6(0 100Fig. 8. Wavelet spectrum variation of observed freak wave generation process from the Draupner platform.10014:20' 15:20出。5001(s)Fig. 9. Wave energy variation of observed freak wave generation process from the Draupner platform.中国煤化工MHCNMH G204HU Jin-peng and ZHANG Yun-qiu/ China Ocean Eng, 28(2), 2014, 193 - 2055. ConclusionsThis work studied the time-frequency energy characteristics of the freak wave generation process.The numerical, laboratory, and observed evolution records on the freak wave generation are analyzedby using the Morlet wavelet spectrum method. The wavelet energy spectra of the numerical evolutionrecords explicitly display the process of the dominan-frequency energy densities highly converging onthe freak wave and the corresponding energy distribution shifting to high-frequency components. Theenergy variations versus time present the continuous transfer of the wave train energy to the focusingpoint. The wavelet energy spectra and the energy variations of the laboratory evolution records exhibitthe energy focusing properties are similar to those of the numerical process, but their correspondingenergy shifting to the high-frequency components only occurs at the freak wave moment. The waveletenergy spectra of the observed single-point evolution records remarkably present the properties of thehigh energy densities around the dominant frequency and the instantaneous energy shift andconvergence for the typical freak wave. Thus it can be concluded that the generation of a freak wavedepends not only on the continuous transfer of wave train energy to a certain focusing region wherefinally the maximum energy occurs, but also on the distinct shift of the highly converged energy tohigh-frequency components in a very short time and fetch. Moreover, the well. defined freak wave hasthe following time-frequency characteristics:(1) Its wavelet spectrum energy densities around the dominant frequency are very high but notnecessary to be the largest one throughout the evolution.because some outside closed curves of its high energy density in the wavelet spectrum extend to(3) Its wavelet energy is highly concentrated and reaches its maximum in the evolution of thewave train.(4) The above energy characteristics persist only for a very short time.Furthermore, the wavelet spectrum analysis also indicates that the large short wave group also hasthe high energy densities, sometimes higher than those of freak waves. However, the group does notexhibit the prominent characteristics of the energy transfer to high- frequency components. Hence, it isnot enough to define a freak wave (Klinting and Sand, 1987) based only on the high densities of theAcknowledgements- The authors thank Statoil for prmission to use the observed freak wave records from the Noth Sea,and also thank Ph. D. PEI Yu-guo for providing the laboratory freak wave data.ReferencesChien, H, Kao, C. C. and Chuang, L. Z. H, 2002. On the characterstics of observed coastal freak waves, Coast.Eng, 44(4): 301-319.Huang,, Y. X., Pei, Y. G and Zhang, N. C., 2009. Time-frequency characteristics analysis of generating process offreak waves based on wavelet transform, Journal of Hydrod)ynamics, 24(6): 754- -760.Jakobsen, J. B., Haver, S. and Odegard, J. E., 2001. Study of freak waves by use of wavelet transform, Proc. 1IthISOPE Conf, Stavanger, Norway, 17-22.中国煤化工YHCNMH GHU. 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