Wave Crest Distributions: Observations and Second-Order Theory

Transcription

1 1AUGUST 000 FORRISTALL 1931 Wave Cret Ditribution: Obervation and Second-Order Theory GEORGE Z. FORRISTALL Shell E&P Technology, Houton, Texa (Manucript received February 1999, in final form 14 September 1999) ABSTRACT Many empirical and heuritic ditribution function for wave cret height have been propoed, but their prediction differ coniderably. Part of the lack of agreement i due to the difficulty of making meaurement that accurately record the true height of the wave cret. Surface following buoy effectively cancel out the econd-order nonlinearity by making a Lagrangian meaurement. Preure tranducer filter the nonlinear component of the ignal in complicated way. Wave taff have varying degree of enitivity to pray. The location of the intrument alo play an important role. There i clear evidence from meaurement in the North Sea that puriou cret due to pray are a problem downwind even from mounting upport that appear tranparent. Much of the theoretical nonlinearity can be captured by calculation correct to econd order. Explicit calculation of the interaction of each pair of component in a directional pectrum i traightforward although computationally intenive. Thi technique ha the advantage that the effect of wave teepne, water depth, and directional preading are included with no approximation other than the truncation of the expanion at econd order. Comparion with meaurement that are believed to be of the bet quality how good agreement with thee econd-order calculation. Simulation for a et of JONSWAP pectra then lead to parametric cret ditribution, which can be ued eaily in application. 1. Introduction Qualitatively, the harpening of the cret of urface wave i the mot obviou manifetation of nonlinearity in the ocean. Yet a detailed quantitative decription of thi phenomena accurate enough for engineering ue remain eluive. The problem i to calculate the tatitical ditribution of cret height given the directional pectrum of the wave and the water depth. By cret height, we mean the highet point on a wave trace between the time it croe above mean water level and the time it croe below mean water level, the zerocroing cret height. The alternate definition of a cret a a local maxima i not ueful for engineering purpoe ince wave record can how many mall maxima between zero croing. To firt order, the water urface can be repreented a Gauian noie with a reaonably narrow frequency band. The cret height then have the ame ditribution a the envelope of the noie, which ha the Rayleigh ditribution [ ] c H P( ) exp 8, (1) Correponding author addre: Dr. George Z. Forritall, Shell E&P Technology, Box 481, Houton, TX gzforritall@hellu.com 1/ where c i the cret height, H 4m 0 i the ignificant wave height, and m 0 i the variance of the wave pectrum. We ue thi definition of the ignificant wave height throughout thi paper intead of it original definition a the average of the highet ⅓ of the zero croing wave ince m 0 i the more fundamental meaure of the energy in the pectrum, predicted by all modern wave forecating program. Forritall (1978) and many other have hown that the H 1/3 definition uually give value about 5% lower than the definition from the variance. Real wave how a mall but eaily noticed departure from a Gauian urface. The cret are higher and harper than expected from a ummation of inuoidal wave with random phae, and the trough are hallower and flatter. It i eay to tell by inpection whether a wave record i right ide up. The hape of regular progreive wave can now be calculated to a very high degree of accuracy, but no complete theory for the tatitic of random wave exit. Longuet-Higgin (1963) ued a Gram-Charlier erie to decribe the probability denity function of the urface elevation. He calculated the moment that determine the propertie of the ditribution from a weakly nonlinear wave theory. The major problem with the reult i that the denity function i ometime negative. Srokoz (1998) ha recently overcome thi problem by forming a Pearon ditribution with a pecified poitive value of kewne and zero kurtoi. Thee ditribution 000 American Meteorological Society

2 193 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 30 have the poible flaw of predicting bound on the maximum and minimum value of the urface elevation. In addition, neither the theory of Longuet-Higgin or that of Srokoz predict the ditribution of zero-croing cret height, which i of mot interet in application. A theory for the econd-order interaction of wave in a random directional ea ha been available for ome time. It i not clear how to derive a cret height ditribution directly from the theory, but it i reaonably imple to imulate time erie of wave that include thee interaction. Sample ditribution correct to econd order can thu be found through analyi of long imulated record. Seriou exploration of thi approach ha recently begun. Jha (1997) ha performed uch imulation for unidirectional wave and compared them to laboratory and field meaurement. Prevoto (1998) extended the work to directionally pread wave. He calculated the kewne of the urface elevation and the mot probable cret height for many combination of pectral hape and water depth. We ue eentially the ame technique but extend the work by comparing the reult to everal et of field meaurement. In addition, we fit Weibull ditribution to imulation with a wide variety of wave teepne and Urell number to produce parametric verion of wave cret ditribution correct to econd order, which can be ued very eaily in application. We begin with a review of ome of the method that have previouly been propoed for calculating wave cret height for engineering deign. The next ection dicue the principle of operation of ome intrument commonly ued to meaure wave and the problem they may have in accurately meauring cret. Then we preent the equation for the econd-order interaction between wave component in a random directional ea in intermediate water depth. The reult of uing thoe equation to imulate wave cret are compared to meaurement and to previou method. Simulation for a et of JONSWAP pectra then lead to parametric cret ditribution. We cloe with concluion and recommendation for future work.. Previou etimate of extreme cret For deign purpoe, a cret height i often etimated by taking the height and period of the deign wave and applying a high-order regular wave theory uch a Stoke fifth order. Since uch regular wave are often ued a input to calculate force on a tructure, thi method ha the advantage that the cret height ued to et the deck elevation i conitent with the wave ued in the force calculation. It ha the diadvantage of neglecting the random and directionally pread nature of the real ea. A popular empirical cret height ditribution wa preented by Haring et al. (1976) and i given a 1 m d d 0 P( ) exp c [ ] () Thi equation wa derived by empirical fitting to 376 hour of torm wave record including meaurement with Baylor wave taff in the Gulf of Mexico and Waverider buoy in the North Sea and the Gulf of Alaka. The ditribution i a function of the variance in the pectrum, m 0, divided by water depth, d. It give a higher probability of high cret for hallow water, a would be expected theoretically from the increaed nonlinearity of hoaling wave. It doe not have any dependence on wave teepne and it reduce to the Rayleigh ditribution in very deep water, which cannot be trictly correct, ince deep water wave are alo nonlinear. Equation () i, however, ignificantly different from the Rayleigh ditribution for large wave anywhere on the continental helf. It i reaonable to uppoe that the nonlinear wave urface could be approximated by an amplitude-modulated Stoke wave. In deep water, the nonlinear cret amplitude from thi model i 1 r a ka, (3) where a i the linear wave amplitude and k i the wavenumber. D. L. Kriebel (1998 peronal communication) ugget uing the wavenumber aociated with the frequency of the highet wave, which he etimate a 0.95/ f p where f p i the peak frequency of the pectrum. Thi choice eem reaonable ince we are trying to etimate the ditribution of the highet cret. Tayfun (1980) and Huang et al. (1986) produced cret height ditribution from the Stoke model. There i ome diagreement between thee author on the exact form of the reulting ditribution. Equation (4) below i taken from the review by Tucker (1991): 8 1/ P(c ) exp [(1 R/H ) 1] R, (4) where R kh i the wave teepne. Kriebel and Dawon (1991) produced a verion of thi ditribution that doe not involve taking the a quare root, which i given by [ ] [ ] 3 P(c ) exp 8 exp 8R. (5) 3 H H If the wave teepne become large, the probability denity function for Eq. (5) can become negative, and Kriebel and Dawon (1993) ued the ame aumption to derive the lightly different formula, P(c ) exp 8 1 R. (6) H H [ 1 ]

3 1AUGUST 000 FORRISTALL 1933 Kriebel and Dawon (1993) alo extended their ditribution to intermediate water depth uing the depthdependent term from the Stoke econd-order expanion. The reulting ditribution i the ame a Eq. (6) with R replaced by R *, an effective teepne given by R * kh f (kd), (7) where cohkd( cohkd) 1 f (kd). (8) 3 inh kd inhkd It eem reaonable to ue R * in a hallow water verion of Eq. (4) a well. 3. Intrumentation Verification of cret height ditribution ha been hampered by the difficulty of accurately meauring wave cret in extreme ea tate. Meaurement from one type of enor often diagree with thoe from another type, and there i no agreement on which i correct. The baic problem i the lack of any abolute tandard againt which the accuracy of the enor can be judged. The note below illutrate the problem for ome of the enor in common ue. They are not meant to be exhautive. a. Buoy Buoy are the mot popular intrument for collecting information on wave climate. Intrument comparion, mot notably thoe in WADIC (Allender et al. 1989), have demontrated that the popular model can accurately meaure integral propertie of the wave field. Cret meaured by buoy are, however, generally maller than thoe meaured by other intrument. Thi underetimation of cret height i often thought to be due to the buoy partially ubmerging in a cret or liding ideway away from the highet point on a high cret. Thee mechanim may play a role, but even a perfect urface following buoy will underetimate wave cret. A buoy that act a a particle on the urface will move forward in the direction of wave propagation in the cret and backward in the trough. It will therefore pend more time than a wave taff at a fixed location in the cret, and le time in the trough. The orbital motion of the buoy ditort the hape of the wave profile, but doe not by itelf make the cret meaurement lower. However, almot all buoy meaure wave elevation through double integration of the vertical acceleration. The abolute value of the till water elevation i thu not known, and cret height are meaured from the mean of the elevation meaurement. Since the buoy pend extra time in the cret, the mean water level will be lightly higher than the true till water level, and the cret height above mean water level will be lightly too mall. The motion of the buoy i a finite amplitude effect. Jame (1986), Srokoz and Longuet-Higgin (1986), and Longuet-Higgin (1986) have all conidered apect of thi problem and how that to econd order, the vertical diplacement of the mean urface i equal to the amount that the cret i raied. Therefore, the Lagrangian motion of the buoy cancel out the econd order nonlinearity of the wave cret. The detail of the buoy motion are greatly complicated by it mooring line and the random nature of the real ea, and have not yet been worked out completely. Neverthele, the main feature of the argument mut till be important, o accelerometer buoy cannot be conidered a a real choice for meauring the ditribution of wave cret. b. Preure tranducer Preure tranducer are ueful for meauring wave at hallow water ite or on platform where the enor can be mounted relatively cloe to the ea urface. The ignal mut be corrected for the attenuation of the preure fluctuation, which increae with increaing depth and increaing wavenumber. Thee correction are reaonably accurate for at leat the low frequency part of the wave pectrum, but it i hard to ee how the reult could be ueful for etimating the cret ditribution. If the correction are made with firt-order theory, a i uual, the reulting urface will be Gauian. If the difficult problem of making higher-order correction i faced, the reult will only be an expreion of the theory ued rather than an independent check of it. c. The Baylor wave taff The Baylor wave taff conit of a pair of tainle teel wire rope eparated by inulator about 0 cm long. The tranducer meaure the natural frequency of the inductive loop made by the two wire and the ea urface, from which the length of the loop i found. The intrument i robut and relatively immune to fouling. It ha been particularly popular for wave meaurement from platform in the Gulf of Mexico. Tet have hown that it ha a linearity better than 1% and that it will record change of elevation at leat a fat a 300 m 1. Our experience from calibrating Baylor taff i that quite a firm hort i neceary before the enor repond. It thu eem unlikely that it would be affected by pray. d. EMI laer The EMI laer i a puled range finder operating in the near-infrared region. Narrow pule of light are produced by a laer diode, and the radiation from the target i ued to top a time interval meaurement. The time of travel i converted to an analog voltage proportional to the ditance to the reflector. The repone of the optical unit i 10 to 15 Hz, but the output i uually filtered by a -Hz Butterworth filter to eliminate high frequency

4 1934 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 30 noie. We are aware of ome tet that howed that the intrument reponded to artificial pray, but we do not have detail of the tet. e. Marex radar The Marex wave radar i a ranging device derived from a radar altimeter. The radar operate in the microwave (J) band with a beamwidth of 6. It mut be poitioned on a tructure o that idelobe of the beam do not reflect from member of the tructure. We do not know of any tet of it repone to pray. 4. Second-order wave profile A econd-order expanion of the ea urface can capture the effect of wave teepne, water depth, and directional preading with no approximation other than the truncation of the expanion at econd order. Higherorder interaction and other effect will of coure influence the ditribution of real wave cret. In particular, wave breaking could be important. The effect of wave breaking on wave height in deep water wa conidered by Tayfun (1981), and Thornton and Guza (1983) conidered the hallow water cae. The point of our invetigation i, however, to ee how well a traightforward application of econd-order theory can match obervation. The econd-order wave interaction for infinite water depth were calculated by Longuet-Higgin (1963), and the calculation were extended to intermediate water depth by Sharma and Dean (1979). We reproduce the latter reult for completene. Let the firt-order water urface be given by N n n1 a co(k x t ), (9) n n n where t i time; x i the poition vector in the plane; n, n, and k n are, repectively, the radian frequency, phae, and vector wavenumber of Fourier wave component n; and a n i it amplitude. The frequencie and wavenumber are related by the linear diperion equation n g k n tanh( k n d), (10) where g i the acceleration of gravity and d i the water depth. The econd-order correction to the wave urface given by Sharma and Dean (1979) i then where i 1 aa{k co( ) N N () j i j 4 i1 j1 K co(i j)}, (11) 1/ K [Dij (k i kj RR)](R i j ir) j (Ri R j) (1) 1/ K [Dij (k i kj RR)](R i j ir) j D D (Ri R j) (13) (Ri R j){r j(ki R i) R i(kj R j)} ij (Ri R j) kij ij tanhk d (Ri R j)(k i kj RR) i j (14) (R R ) k tanhk d i j ij ij (Ri R j)(k i kj RR) i j ij (Ri R j) kij ij tanhk d (Ri R j){r i(kj R j) R j(ki R i)} (Ri R j) kij tanhkij d (15) kij ki k j (16) kij ki k j (17) Ri k i tanh( k n d) i/g and (18) i kix it i. (19) For infinite water depth, Eq. (7) reduce to Eq. (3.7) of Longuet-Higgin (1963), except that the latter equation i miing a factor of ½. a. Form of the interaction kernel The poitive interaction term given by Eq. (15) occur at the um of the frequencie of the interacting wave component. They produce the harpening of the cret and flattening of the trough that we aociate with econd-order Stoke wave. The negative interaction term given by Eq. (14) occur at the difference of the frequencie of the firt-order wave component. Thee interaction give the etdown of the water level under wave group. The qualitative behavior of the interaction can be undertood by conidering the imple cae of two component wave with nearly the ame frequency. For thi cae the poitive interaction term in Eq. (11) ha a frequency approximately twice that of the linear component wave. It will thu have poitive peak in phae with both the cret and trough of the linear component. The frequency of the negative interaction term i the difference of the frequencie of the component, which i the beat frequency or envelope of the linear wave group. Since the interaction kernel given by Eq. (1) i negative, thi low frequency econd-order wave will be negative under high wave group. Figure 1 how an example of the reult of umming all of the econd-order interaction term in a JONSWAP

5 1AUGUST 000 FORRISTALL 1935 FIG. 1. An example of firt- and econd-order wave for a JON- SWAP pectrum with a peak period of 1 and a ignificant wave height of 11. m in 40-m water depth. The olid line how the firtorder imulation, the hort dahed line how the um of the poitive econd-order term, and the long dahed line how the um of the negative econd-order term. pectrum. The example pectrum had a peak period of 1 and a ignificant wave height of 11. m. The water depth wa 40 m. A expected, the um of the poitive interaction term i poitive in both the trough and cret of the large wave, while the um of the negative term i generally mall except for the etdown under the group of large wave in the middle of the record. The kewne kernel i defined a (K K )/4 and i a meaure of the trength of the econd-order interaction. It i a function of the frequencie of the two interacting wave a well a their angular eparation and the water depth. The trength of the interaction i much greater in hallow water, matching the obervation that the wave profile i more kewed in hallow water. Figure how an example of the interaction kernel for relatively hallow water. The depth i 1 m and f The ratio of the frequencie of the two wave i hown on the x axi and the difference in the direction of travel of the wave i on the y axi, with 0 in the center of the cale. In water thi hallow the peak of the interaction doe not occur for colinear wave, but for wave eparated by a mall angle. The kernel ha a depreion for wave that are nearly in the ame direction, which i due to the large value of the negative interaction term for wave with nearly the ame direction and frequency in hallow water. Thi depreion i particularly deep and harp near f 1 / f 1. It doe not appear in deep water. Figure how that wave etdown effect will be much more important for unidirectional wave produced in a wave tank than they are for natural wave in the ocean, which are alway pread directionally to ome extent. b. Numerical implementation The calculation indicated by Eq. (11) (19) are eay to implement, but repreent a fair amount of computation ince there are very many interaction to be calculated, particularly in the three-dimenional cae. A FIG.. Second-order interaction kernel for wave in hallow water.

6 1936 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 30 typical imulation would have 4096 time tep at 4 Hz. A firt-order imulation of a given wave pectrum i produced from Eq. (9) by chooing the phae n from a uniform random ditribution and calculating the wave height uing an invere fat Fourier tranform to perform the ummation in eq. (9). The econd-order interaction are then calculated for each pair of component wave in the firt-order imulation. The ummation in Eq. (11) give time erie of the poitive and negative interaction a hown in Fig. 1. Adding the time erie of the poitive and negative interaction to the firt-order imulation then give the econd-order imulation. The interaction do not have to be calculated for all of the 048 frequencie in the Fourier tranform of the time erie, partly becaue the energy in the firt-order pectrum i very mall at the higher frequencie, but alo becaue the energy in the meaured pectrum at thee high frequencie appear to be motly due to nonlinear interaction. It i poible to ue an iterative cheme to produce a firt-order pectrum from the meaured pectrum, but it i much impler to truncate the econd-order calculation at a mall multiple of the peak frequency of the pectrum. Thi implification ha little effect on the cret height ditribution. The calculation decribed in thi paper were typically truncated for f 1 f greater than four or five time the peak frequency. Repeating a imulation with preciely the ame pectral variance at each Fourier line doe not include all of the natural variability of wave. The variance are actually random variable with a Chi-quared ditribution. If a directional pectrum i imulated, the addition of everal wave component with different direction and the ame frequency will automatically produce thi ditribution in the frequency pectrum. For conitency, we therefore multiply the input pectral line in two-dimenional imulation by a Chi-quared random variate. We generally want to perform many imulation uing the ame pectrum in order to produce table tatitic for rare cret height. Since the kewne kernel i the ame for all of the repetition, it i efficient to calculate it once for each pectrum and tore it. The calculation alo exploit the fact that the kernel i ymmetric in the two frequencie and depend only on the difference in the two angle. We ue an angular reolution of 1 for the three-dimenional imulation. Repeating a imulation many time with different random number give enough ample to tabilize the tatitic at low probability level. In a typical example, we would ue 00 repetition of a 4096 time tep imulation at 4 Hz for a pectrum with a peak period of 8. Running thi example everal time howed that the tandard deviation of the cret height normalized by the ignificant wave height wa at a probability level of and at a probability level of The length of the individual imulation doe not have any meaurable effect on the tatitic, at leat within the limit ued in thi tudy. To tet thi aertion, we did repetition of a unidirectional imulation with 4096 time tep and 5000 repetition of a imulation of the ame pectrum with 819 time tep. At a probability level of 0.001, the 4096 point imulation gave a normalized cret height of 1.56 and the 819 point imulation gave a normalized cret height of Different ample rate and number of repetition were ued in ome of the cae tudie dicued below. In general, we ued the ame ample rate a the meaurement. A dicued by Tayfun (1993) and hown in the Camille meaurement below, the ample rate can have ome effect on the tatitic if there are too few ample point per fundamental wave period. Since D imulation are much more economical than 3D imulation, we often ued many more repetition in the D imulation. 5. Comparion with meaurement a. Storm at Tern in the North Sea The Tern platform i located in the northern North Sea between the Shetland Iland and Norway. It i about 150 km from the nearet horeline and in 167-m water depth. It i a fairly tandard eight legged teel oil production platform that wa intalled in It wa equipped with a tructural monitoring ytem including train gauge, two wave height enor, and an electromagnetic current meter. Jonathan et al. (1994) and Jonathan and Taylor (1995) give decription of the meaurement ytem and the oceanographic condition in the torm conidered here. Figure 3 how an outline plan of the platform with the enor location marked. The rectangle in the figure how the outline of the tructural member of the platform at mean ea level (MSL), at the 41-m depth of the Marh McBirney electromagnetic current meter, and at the mud line. A Marex wave radar wa mounted under the outheat corner of the platform deck and an EMI laer wave enor wa mounted under the deck on the outhwet corner. Both of thee enor meaure the ditance from the intrument to the intantaneou water urface. Three torm at Tern have been tudied in detail. One of the torm occurred in January 199 and two in January The torm on 4 January 1993 i referred to a 93a and the one on 17 January 1993 i referred to a 93b. Condition in all three torm were extreme, with peak ignificant wave height above 1 m. Storm 93a and 93b are particularly ueful for wave proce tudie ince the condition remained tationary, nearly within the limit of ampling variability, for 8 or 9 hour. Storm 9 built quickly to a peak H of 13.8 m and then declined. Figure 4 and 5 how the probability ditribution of the wave cret height for torm 93a and 9 repectively. The cret height are normalized by the ignifi-

7 1AUGUST 000 FORRISTALL 1937 FIG. 3. Outline plan of the Tern platform with location of the wave height enor and the mean wave direction for the torm. cant wave height during the hour the cret wa meaured. If the wave were linear o that the urface elevation had a Gauian ditribution, the cret height between zero croing would have a Rayleigh ditribution. Thi theoretical ditribution i hown by the olid line in the figure. A expected, the ample ditribution from the meaurement how an exce of high cret height above the Rayleigh curve. The ample ditribution alo how a ignificant diagreement between the reult from the two wave enor. In torm 93a, the cret meaured by the Marex radar are about 10% higher than the cret expected from linear theory, but the highet cret from the EMI laer meaurement are 0% 30% higher than linear theory. In torm 9, however, the ituation i revered. The meaurement from the EMI laer are about 10% above the linear curve while thoe from the Marex radar are much higher. It eem quite likely that the difference between the torm can be explained by the difference in wave direction, a hown in Fig. 3. In torm 93a, the wave were propagating to the north, o that their cret paed the Marex radar before encountering any tructural element on the platform. On the other hand, the leg on the outhwet corner of the platform i up- FIG. 4. Probability ditribution of normalized cret height meaured at Tern during the torm on 4 Jan The cret height are normalized by the ignificant wave height during each hour of the meaurement. Nine hour of meaurement with an average ignificant wave height of about 1 m were combined to produce the oberved ditribution. FIG. 5. Probability ditribution of normalized cret height meaured at Tern during the torm on 1 Jan 199. The cret height are normalized by the ignificant wave height during each hour of the meaurement. Eight hour of meaurement with an average ignificant wave height of about 11 m were combined to produce the oberved ditribution.

8 1938 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 30 FIG. 6. Ditribution of cret height in torm 93a at Tern on 4 Jan The ordinate how the cret height normalized by the height predicted at that probability level by the Rayleigh ditribution. The meaurement are from the Marex wave radar. wave from the EMI enor, and it i quite likely that pray caued by a wave cret hitting that leg would ometime pa under the EMI laer. The repone of wave enor in the preence of pray i not known with any certainty, but it i reaonable to uppoe that pray from tructural interference could caue a laer gauge to record higher cret than thoe in the ambient wave. Thi hypothei i upported by the fact that in torm 9 the wave were propagating to the eat o that the EMI laer wa on the windward ide of the platform while the Marex radar wa in the lee of tructural member. Apparently, both of the enor meaured cret higher than the ambient wave due to pray from tructural member, although there i no obviou evidence of pray in the meaured time erie. The location of a wave enor with repect to the platform it i mounted on thu may be at leat a important a the repone characteritic of the enor itelf. Given the likely platform interference, we compare the econd-order imulation with the Marex meaurement during torm 93a and the EMI meaurement during torm 9. Figure 6 how the comparion for 93a. The ordinate in thi figure give the ratio of the cret height to the height predicted by the Rayleigh ditribution at the probability level of the abcia. It thu how that the meaured wave cret were about 10% higher than expected from linear theory and that the ratio increae lightly at lower probability level. The meaurement are for 9 hour of the torm, which included about 3000 cret. The imulation were baed on pectra calculated from the meaurement for each hour. For the two-dimenional imulation, 00 repetition for each of the nine pectra were made. The imulation were made at a 5.1-Hz ample rate, the ame a the meaurement, and each imulation wa 4096 point long. There were therefore about 44 hour of imulation for each hour of meaurement, o the tatitic of the imulation are FIG. 7. Ditribution of cret height during torm 9 at Tern on 1 Jan 199. For an explanation of the curve, ee Fig. 6. much more table than thoe of the meaurement. The three-dimenional imulation were baed on directional pectra calculated from the wave radar and the electromagnetic current meter. Since the three-dimenional imulation demand much more computer time, only 30 repetition were made for each pectrum, giving 6.67 hour of imulation for each hour of meaurement. The tatitic of the imulation are very imilar to thoe of the meaurement, about 10% higher than linear theory and increaing lightly at lower probability level. The three-dimenional imulation are about % lower than the two-dimenional imulation. The difference i caued by the lightly lower value of the econd-order interaction kernel for wave that are not colinear. The two-dimenional imulation appear to be lightly higher than the meaurement while the threedimenional imulation appear to be lightly lower. Figure 7 how the meaured and imulated tatitic for torm 9. Both the meaurement and the imulation are for 8 hour at the peak of the torm when the ignificant wave height ranged from 7.85 to m. The D and 3D imulation are very imilar to thoe for torm 93a, but the data point in the top 10% of the cret are conitently higher than the imulation. There i no obviou difference in the pectra of the two torm that would explain the difference in their cret tatitic. It i poible that the repone of the EMI and Marex enor i lightly different, even when they have good expoure to the ambient wave. b. Hurricane Opal Thi torm entered the Bay of Campeche in the Gulf of Mexico on 30 September 1995 a a tropical depreion. It intenified rapidly a it accelerated north northwetward on 3 and 4 October on it way to a landfall in the panhandle of Florida. It maximum intenity wa oberved early on 4 October a the eye paed 7.3N,

9 1AUGUST 000 FORRISTALL 1939 FIG. 8. Ditribution of cret height during Hurricane Opal at Bullwinkle. For an explanation of the curve, ee Fig W with a 916-mb central preure and a very mall eye 10 n mi in diameter. Wave from Hurricane Opal were meaured at the Bullwinkle oil production platform located at 7.9N, 90.9W in 410 m of water. A comprehenive intrumentation ytem wa intalled on Bullwinkle in order to meaure the environment and the repone of the tructure. Swanon and Baxter (1989) give a good decription of thi ytem. A Baylor wave taff wa ued to meaure wave elevation and three Marh McBirney electromagnetic current meter meaured the wave particle velocitie. Since the eye of the tight torm paed 15 mile to the eat of Bullwinkle, the wave there were never extremely high. We ued the 14 hour of meaurement from 0500 to 1800 UTC 4 October 1995 in our analyi, and the ignificant wave height during that time ranged from 4.78 to 6.14 m. The meaurement were recorded at 4 Hz. Figure 8 how the oberved and imulated ditribution of cret height for Hurricane Opal. The imulation are about.5% lower than the imulation for the Tern torm ince the wave were not a teep. The three-dimenional imulation produce cret about 1% maller than the two-dimenional imulation, but they are till % larger than the bulk of the obervation. c. Hurricane Camille Camille, whoe path croed the north-central Gulf of Mexico, wa one of the mot intene and detructive torm to trike the United State thi century. Wave were recorded by a Baylor wave taff at tation 1 of the ODGP (Hamilton 1976), which wa on the South Pa 6A oil production platform located at 90450N, W. Camille paed almot directly over the tation, and the large wave that were meaured had a great influence on etting the tandard criteria for platform deign in the gulf. The meaurement were recorded on an analog tape recorder and, unfortunately, FIG. 9. Ditribution of cret height during Hurricane Camille at SP6 for CDT 17 Aug For an explanation of the curve, ee Fig. 6. were only digitized at a 1-Hz ampling rate. The wave taff broke at 1630 CDT 17 Augut 1969, which hindcating tudie how to be very cloe to the peak of the torm. Meaurement and imulation of the cret height from 100 to 1630 CDT are hown in Fig. 9. The ignificant wave height grew from 9.96 to m during thi period. Only two-dimenional imulation are hown ince directional pectra were not meaured during the torm. The imulation made at a 1-econd time tep agree reaonably well with the meaurement, although there appear to be coniderable catter in the meaurement which caue the ditribution to deviate from a mooth curve. We alo made imulation with a ¼-econd time tep, and they are ignificantly different, about.5% 3% higher. The difference i not due to the energy content of the ea at frequencie above 1 Hz, which i very mall, but to the fact that 1-Hz ampling frequency i likely to mi the very peak of the wave. Thi effect wa invetigated in ome detail by Tayfun (1993), who found that the higher wave would be underetimated by approximately ( /6)(/T ) where i the ampling interval and T i the average wave period. For the 10- average wave period in thi part of Camille, the formula give an error of 1.6%, cloe to the difference in the imulation. The relatively long ampling interval in the digitized record of the Camille wave clearly caue an underetimate of the true cret height. d. Shallow water wave in Lake Ontario The Canada Centre for Inland Water maintain a reearch tower at the wetern end of Lake Ontario near Toronto. Many intereting tudie of wave procee have been conducted there, including the definitive meaurement of directional pectra by Donelan et al. (1985). The tower i in 1 m of water. An eaterly a

10 1940 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 30 FIG. 10. Ditribution of cret height at the CCIW tower on 10 Jan FIG. 11. Comparion of cret height ditribution at Tern for D 167 m, T 13.6, H 1.0 m. torm on 10 January 1977 produced wave that reached a ignificant height of 3 m at the tower. During that torm, 14 wave taff on the tower and extenion from it were operational, and we have examined meaurement from them during one hour at the peak of the torm. The meaurement were recorded at 5 Hz. Depite that the tructure appear to be very tranparent, a few unuually high cret were oberved in the lee of the platform but not by the wave taff on the windward ide of the platform. Therefore in Fig. 10 we only included data from the 6 wave taff with the clearet upwind expoure. The two-dimenional imulation agree very well with the meaurement, while the threedimenional imulation appear to be a bit high, at leat at low probabilitie. In hallow water, wave with a narrow directional preading can be more nonlinear than unidirectional wave becaue the interaction kernel reache it maximum for wave component lightly eparated in direction, a hown in Fig.. Figure 10 alo how the reult of a two-dimenional imulation made with the water depth increaed to 1000 m, labelled Deep Water. The cret in thi imulation are about 3% lower than thoe for the true water depth of 1 m, howing the increaed nonlinearity of the wave due to the hallow water at the tower. well with thi imulation. The reult from Eq. (4), labelled Tayfun in the figure, are a few percent lower. For the teepne of 0.68, there i a noticeable difference between thee two verion of the ditribution derived from modulated Stoke wave. The ditribution of Haring et al. (1976) i even lower becaue the water depth at Tern i deep enough that the nonlinear adjutment in Eq. () i too mall. We alo teted Eq. () for the 410-m water depth at Bullwinkle, and the reulting ditribution (not hown) wa very cloe to the Rayleigh ditribution. Cret height calculated from Stoke fifth-order wave are hown a mall circle in Fig. 11. The wave height at everal probabilitie of exceedence were calculated from the Rayleigh ditribution and the cret height of a Stoke fifth wave with that height and a period of 13.6 wa found. Thee cret are lightly lower than thoe from the econd-order imulation and agree very cloely with the ditribution from modulated Stoke wave given by Eq. (4). The tep in thi cal- 6. Comparion with previou etimate Figure 11 and 1 compare our two-dimenional, econd-order wave imulation with previou method of etimating cret height. Figure 11 i for condition during the peak of torm 93a at Tern, for D 167 m, T p 14.3, H 1.0 m. Uing T 0.95T p, we have T 13.6 for ue in Eq. (6). The teepne, R 0.68, and the effective hallow water teepne R * are virtually identical for thi relatively deep water. Our two-dimenional econd-order imulation i hown a the olid line in the figure. The Kriebel and Dawon (1993) ditribution from Eq. (6) agree quite FIG. 1. Comparion of cret height ditribution at the CCIW tower for D 1 m, T 7.94, H 3.0 m.

11 1AUGUST 000 FORRISTALL 1941 culation of cret height from regular wave are tandard practice, but they are omewhat inconitent ince actual trough to cret wave height are lower than given by the Rayleigh ditribution. If an empirical ditribution uch a the one due to Forritall (1978) were ued to etimate the wave height, the Stoke cret height for thi example would be about the ame a given directly by a Rayleigh ditribution of cret height. Figure 1 i for condition during the peak of the torm in Lake Ontario, for D 1 m, T p 8.36, T 7.94, and H 3.0 m. The teepne i R 0.51, and the effective hallow water teepne R * We ued the hallow water teepne in Eq. (6) and (4). The Kriebel and Dawon (1993) hallow water ditribution i unrealitically higher than any of the other, probably becaue the very high value of effective teepne invalidate the mathematical aumption that were made in the derivation of the ditribution by reverion of erie. Equation (4), labeled Tayfun, give a more reaonable reult, but it too i coniderably higher than the imulation (and the data). The cret height from individual Stoke fifth-order wave again agree with Eq. (4), and the Haring et al. (1976) ditribution give about the ame reult in thi cae, but all are coniderably larger than the imulation. The Stoke cret height may be too high becaue the regular wave method effectively concentrate all of the energy of the pectrum at one frequency, and the poitive interaction term i larget for elf interaction. The method alo doe not include the negative interaction which produce a et down under high wave group. The Haring et al. ditribution i too low for very deep water and too high for very hallow water. Thee difference hould not be too urpriing ince the water depth in thee example are outide the range of water depth in the data from which thi empirical ditribution wa developed. The higher wave height during thi torm were omewhat higher than the normally recommended limit of applicability of Stoke econd-order theory. However, for a wave height of 6 m and a period of 7.94, Stoke wave theory give a cret height of 3.9 m while a 15thorder Chappelear numerical wave expanion give a cret height of 4.08 m. The difference i noticeable, but mall compared to the dicrepancie in Fig. 1, and in the oppoite direction. FIG. 13. Normalized cret height ditribution for a mean JON- SWAP pectrum with a peak period of 10 in a water depth of 0 m. Solid line how unidirectional imulation, mall circle how directionally pread imulation and dahed line how the Weibull fit to the pread imulation. Simulation for four value of teepne are hown with S p 0.01, 0.03, 0.05, and 0.07 tarting from the bottom et of curve. 7. Parametric cret height ditribution from imulation The econd-order imulation appear to match a variety of meaurement relatively well. Performing the imulation i, however, rather time conuming epecially for directionally pread wave. For application, it would be ueful to have imple functional form that match the reult of the imulation. In order to produce uch parametric ditribution, we imulated cret for a variety of wave teepne and Urell number. The pectra were contructed uing the equation of Goda (1985), which give pectra imilar to the JONSWAP form but with a pecified peak period and ignificant wave height. The pectra had peak period of 8, 10, and 1. The teepne baed on the peak period T p, H S p, (0) g T p varied from 0.01 to 0.10 in tep of Mot of the imulation were made uing the tandard JONSWAP peak enhancement factor 3.3, but ome imulation were alo made with 1.0 and The run were repeated for water depth of 10, 0, and 40 m and for infinite water depth. Combination of teepne and water depth that gave phyically impoible wave height were not imulated. The three-dimenional imulation ued a co preading function with the parameter taken from the fetch-limited meaurement of Ewan (1998). Each cae of the two-dimenional imulation included repetition of a 104- time erie at 4 Hz for tatitical tability. The three-dimenional imulation require much more computer time, o only 50 repetition were done for each cae. In addition, only about half of the two-dimenional cae were included in the directionally pread imulation. Figure 13 how ome example reult from the imulation in a water depth of 0 m. The imulation are for a tandard JONSWAP pectrum with a peak period of 10. Four et of imulation are hown with T p 0.01, 0.03, 0.05, and 0.07, tarting from the lowet et of curve. The directionally pread imulation how more ampling variability than the unidirectional imulation ince they include fewer data point. For teep wave in thi water depth, the directionally pread cret

12 194 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 30 are higher at low probability level than the unidirectional cret. A mentioned before, thi effect i due to the maller low frequency etdown term in the directionally pread wave. The parameterization of the imulation involved two tep of fitting. Firt, the imulation for each cae were fit to a Weibull ditribution of the form: [ ] P(c ) exp. (1) H Then imple expreion for the Weibull parameter and were found a function of the water depth and wave pectrum. Thee expreion are baed on parameter that characterize the degree of nonlinearity of the wave, that i, the wave teepne and Urell number. We found that baing the wave teepne on the mean wave period, rather than the peak period, produced good fit for pectra with the ame peak period but different peak enhancement factor. The teepne parameter ued in the fit i thu H S 1, () g T 1 where T 1 i the mean wave period calculated from the ratio of the firt two moment of the wave pectrum, m 0 /m 1. The tandard parameter for characterizing the effect of water depth on the nonlinearity of wave i the Urell number. The Urell number baed on the ignificant wave height and mean period i H Ur 3, (3) kd where k 1 i the wavenumber for a frequency of 1/T 1. Propoal have been made for nonlinearity parameter which combine the effect of teepne and water depth, but we found that the fit were better when the two were included eparately. The fit are forced to match the Rayleigh ditribution with 1/ 8 and at zero teepne and Urell number. The fit to the two-dimenional imulation are then S Ur (4).1597S Ur (5) and the fit to the three dimenional imulation are S Ur (6) S Ur 0.84Ur. (7) Example of thee fit to the three-dimenional imulation are hown a the dahed line in Fig. 13. From the bottom ditribution up, S , , , and The Urell number are , 0.13, 0.04, and One of the more intereting feature of the imulation FIG. 14. The contour line how the ratio between the cret height of directionally pread and unidirectional wave at a probability level of 1/1000 a a function of teepne and Urell number. i the influence of directional preading on the cret height. Figure 14 how the ratio between the cret height at the 1/11000 probability level for pread wave and the cret height for unidirectional wave a determined from the fit in Eq. (4) (7). The ratio are hown a contour line a a function of wave teepne, S 1, and Urell number. For deep water wave with Ur 0, the unidirectional cret are lightly bigger, but in hallow water the cret in directionally pread wave are alway larger. A mentioned before, thi behavior i due to the much maller etdown term in the interaction kernel for pread wave, a hown for example in Fig.. 8. Concluion Second-order imulation of wave cret agree well with meaurement of high wave cret made in both deep and hallow water. Three-dimenional imulation that account for the directional preading of wave produce cret that are about % lower than two-dimenional imulation in deep water. Shallow water make the imulated wave more nonlinear and the cret higher a expected. Directionally pread cret can be higher than unidirectional wave when the water i hallow enough. All of thee feature appear to agree with meaurement, but we cannot be too definite about the accuracy of the imulation becaue there i till doubt about the accuracy of the meaurement. The meaured height of wave cret i apparently influenced by both the type of enor ued and the location of the enor on a platform. The two influence have been difficult to ort out becaue there have been

13 1AUGUST 000 FORRISTALL 1943 few comparion of different intrument placed near each other. To remedy thi ituation, the Wave Cret Senor Intercomparion Study (WACSIS) wa conducted on the Meetpot Noordwijk reearch platform in the North Sea through the winter of 1997/98 (van Unen et al. 1998). In that project, wave were continuouly recorded uing a Baylor wave taff, Marex radar, Saab radar, and EMI laer, and video recording of the wave were made at Hz during daylight hour. The meaurement in thi dataet hould provide a good tet of econd-order theory. The econd-order imulation appear to have a greater range of applicability than previou method, which have been propoed for etimating cret height ditribution. Simulation of peudo-jonswap pectra indicate that the cret height ditribution increae almot linearly with wave teepne. A ytematic invetigation of imulation for variou input pectra and water depth ha led to parametric ditribution that match the obervation and are accurate enough for engineering ue. Acknowledgment. Mark Donelan provided the data from Lake Ontario and Mike Vogel ent u the data from Hurricane Opal. The Tern data were provided by Peter Troman and Paul Taylor, with whom I have had many timulating dicuion on the ubject of wave tatitic and dynamic. Dave Kriebel wa very helpful in dicuing the ue of hi ditribution function with me. REFERENCES Allender, J., and Coauthor, 1989: The WADIC project: A comprehenive field evaluation of directional wave intrumentation. Ocean Eng., 16, Donelan, M. A., J. Hamilton, and W. H. Hui, 1985: Directional pectra of wind-generated wave. Philo. Tran. Roy. Soc. London, 315A, Ewan, K. C., 1998: Obervation of the directional pectrum of fetchlimited wave. J. Phy. Oceanogr., 8, Forritall, G. Z., 1978: On the tatitical ditribution of wave height in a torm. J. Geophy. Re., 83, Goda, Y., 1985: Random Sea and Deign of Maritime Structure. Univerity of Tokyo Pre, 33 pp. Hamilton, R. C., and E. G. Ward, 1976: Ocean Data Gathering Program: Quality and reduction of data. J. Petrol. Technol., 8, Haring, R. E., A. R. Oborne, and L. P. Spencer, 1976: Extreme wave parameter baed on continental helf torm wave record. Proc. 15th Int. Conf. on Coatal Engineering, Honolulu, HI, Huang, N. E., L. F. Bliven, S. R. Long, and C. C. Tung, 1986: An analytical model for oceanic whitecap coverage. J. Phy. Oceanogr., 16, Jame, I. D., 1986: A note on the theoretical comparion of wave taff and wave rider buoy in teep gravity wave. Ocean Eng., 13, Jha, A. K., 1997: Nonlinear tochatic model for load and repone of offhore tructure and veel. Ph.D. diertation, Stanford Univerity, 70 pp. Jonathan, P., and P. H. Taylor, 1995: On irregular, nonlinear wave in a pread ea. J. Offhore Mech. Arctic Eng., 119, ,, and P. S. Troman, 1994: Storm wave in the northern North Sea. Proc. Seventh Int. Conf. on the Behavior of Offhore Structure (BOSS), Boton, MA, ASCE, Kriebel, D. L., and T. H. Dawon, 1991: Nonlinear effect on wave group in random ea. J. Offhore Mech. Arctic Eng., 113, , and, 1993: Nonlinearity in wave cret tatitic. Proc. Second Int. Symp. on Ocean Wave Meaurement and Analyi, New Orlean, LA, ASCE, Longuet-Higgin, M. S., 1963: The effect of non-linearitie on tatitical ditrbution in the theory of ea wave. J. Fluid Mech., 17, , 1986: Eulerian and Lagrangian apect of urface wave. J. Fluid Mech., 173, Prevoto, M., 1998: Effect of directional preading and pectral bandwith on the nonlinearity of the irregular wave. Proc. Eighth Int. Offhore and Polar Engineering Conference, Montreal, PQ, Canada, International Society of Offhore and Polar Engineer, Sharma, J. N., and R. G. Dean, 1979: Development and evaluation of a procedure for imulating a random directional econd order ea urface and aociated wave force. Ocean Engineering Rep. 0, Univerity of Delaware, 11 pp. [Available from Ocean Engineering Laboratory, Univerity of Delaware, Newark, DE ] Srokoz, M. A., 1998: A new tatitical ditribution for the urface elevation of weakly nonlinear water wave. J. Phy. Oceanogr., 8, , and M. S. Longuet-Higgin, 1986: On the kewne of ea urface elevation. J. Fluid Mech., 164, Swanon, R. C., and G. D. Baxter, 1989: The Bullwinkle platform intrumentation ytem. Proc. 1t Annual Offhore Technology Conf., Houton, TX, Mar. Technol. Soc., Tayfun, M. A., 1980: Narrow-band nonlinear ea wave. J. Geophy. Re., 85, , 1981: Breaking-limited wave height. J. Waterway, Port, Coatal Ocean Eng., 107, , 1993: Sampling-rate error in tatitic of wave height and period. J. Waterway, Port, Coatal Ocean Eng., 119, Thornton, E. B., and R. T. Guza, 1983: Tranformation of wave height ditribution. J. Geophy. Re., 88, Tucker, M. J., 1991: Wave in Ocean Engineering: Meaurement, Analyi, Interpretation. Elli Horwood, 487 pp. van Unen, R. F., A. A. vand Beuzekom, G. Z. Forritall, J.-P. Mathien, and J. Starke, 1998: Waci Wave cret enor intercomparion tudy at the Meetpot Noordwijk meaurement platform. Ocean 98, Nice, France, IEEE,