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E L S E V I E R Applied Ocean Researcli 28 (2006) 235-245www.elsevier.com/locate/apor
Distribution of crest heights in sea states with abnormal waves
p . P e t r o v a , Z . C h e r n e v a , C . G u e d e s S o a r e s *
Unit of Marine Tecimoiogy and Engineering, Teclmical University' of Lisbon, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisbon, Portugal Received 27 June 2006; accepted 21 November 2006
Available online 20 February 2007
Abstract
T l i e short-term statistical representation o f wave crests i n sea states w i t h abnormal waves is discussed on the basis o f c o m p a i i s o n w i t h the crest statistics i n l o w e r sea states. The analyzed set o f field data was collected d u r i n g a storm i n the N o r t h Sea. Some t y p i c a l properties o f the reported abnormal waves and the associated sea states that have been described i n Guedes Soares et al. [Guedes Soares C, Cherneva Z , A n t a o E . Characteristics o f a b n o r m a l waves i n N o r t h Sea storm sea states. A p p l Ocean Res 2 0 0 3 ; 2 5 : 3 3 7 - 4 4 ] are reviewed and additional relations are derived here.
To describe the observed crest height distributions, the m o d e l o f FoiTistall [ F o n i s t a l l G. Wave crest distributions: Observations and second-order theory. J Phys Oceanogr 2 0 0 0 ; 3 0 : 1 9 3 1 ^ 3 ] has been applied. N o n l i n e a r effects to second second-order are taken i n t o account b y means o f the mean wave steepness and the U r s e l l number. A l t h o u g h the m o d e l is f o u n d to be generally adequate f o r sea states w i t h less severe conditions, the largest crests i n the highest sea states, and especially those w i t h abnormal waves present, ai'e permanently underestimated. I t c o u l d be concluded that the abnormal waves do not c o n f o r m to the u n d e r l y i n g p o p u l a t i o n o f waves i n the associated sea states,
© 2007 Pubhshed b y Elsevier L t d
Keywords: Abnormal waves; Crest height distribution; Short-teiTn distribution; Freak waves; Rogue waves
1. Introduction
The proper short-term statistical model for t h e largest waves in a storm that can be further applied for long-term predictions of extreme events is of importance for t h e safe operation of offshore stractures and vessels. The model has been pursued beyond the Gaussian representation, since it has to capture the enhancement of wave crests i n the nonlinear wave fields, especially those with abnormal waves included.
Still there is no consensus on what an abnormal or freak wave is. Various quantitative criteria are c u i T e n t l y used to distinguish the freak event from the largest wave in a storm. Initially, abnormal waves have been defined as being waves that do not conform with the linear theory. Given a sea state of 20 rrün duration and Rayleigh distributed wave heights, Dean [3] proposed that an abnormal wave is a single wave of height exceeding two times the significant wave height, namely Hmax/Hi/3 > 2. This ratio is known as the amplification or abnormality index, A I . However, the linear theory predictions
* Conesponding author. Tel.: +351 01 841 7607; fax: +351 01 847 4015. E-mail address: guedess@mai.ist.utl.pt (C. Guedes Soaies).
are usually found to underestimate the real probability of occuixence of extreme and freak waves.
Haver [9], on the other hand, assumed second order theory for the underlying population of waves in moderately nonlinear-sea with stationary and homogeneous conditions and defined a freak wave as being an outlier in that population. He proposed a definition based on the ratio between the maximum crest amplitude and significant wave height, which is denoted as the crest amplification index, C I = C„,^^/Hip, given that Cmax is the maximum crest height. The ratio is expected to exceed a factor R, set at 1.2.
Some authors use the combination between C I 0 > 1.3 and A I > 2 [24], or between the abnormality index and some global wave parameters, such as Hmax > 2.15 7 ^ 1 / 3 and Cmax >
0.6 //max [2]. The subscript D in the designation of the crest amplification index, C I B , shows that the value refers to the maximum down-crossing crest height.
Although cuiTently in use, these criteria need further clarification. On one hand, they should change when a high order approximation for the suiface profile is assumed, since the nonlinearity increases the wave steepness, maximum crest and wave height, as compared to the linear theory. Furthennore,
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236 P. Petrova et al./Applied Ocean Research 28 (2006) 235-245 the freak wave definitions are related to the record duration, in
such a way that longer time series increase the probability of encountering an abnormal wave [3,1].
The largest waves in a storm and the associated sea states have been studied so far from different points of view. Various parameters were proposed for the description of the individual extreme wave properties. Myrhaug and Kjeldsen [13] studied the crest front steepness and asymmetry of the large waves, motivated by the cases of capsized fishing vessels in the coastal zone of Norway. Stansberg [18] investigated the shape of the largest crests with respect to. the effect of wave loading on offshore stnictures. Stansell et al. [21] proposed definitions for calculation of the steepness of different parts of the wave and the horizontal asymmetry and used a Generalized Pareto Distribution to predict the steepness, conditional on the extreme wave height. New definitions of individual coefficients of steepness and asymmetry have been recently given by Guedes Soares et al. [7].
In the context of offshore sttuctures, the interest in wave asymmetry is reflected in the studies of the probability distributions of crest heights. Crest heights are preferred to wave heights, since they are more sensitive to nonlinear effects [22] and numerous response problems are more related to steep crests, than to wave heights [10].
Under the Gaussian assumption for the free surface elevation, the crest heights are modelled by the Rayleigh distribution [12], However, measured and laboratory simulated data show that the observed crests are usually higher than the linear ones, as demonstrated by Tayfun [22,23], Haring et al. [8], and Fonistall [5], among others.
Recently, two new iiTegular wave models, which take into account the 3D structure of the wave field, have been developed. The model of Prevosto et al. [16] represents a nonlinear-Rayleigh transformation based on a narrowband second order Stokes expansion for the free suiface elevation. The wave crest distribution is given by a unique expression for both 2D and 3D cases and so can be adapted to all intermediate situations. The model proposed by Foixistall [5] is a perturbed Weibull model. The advantage of this model is the simple formulation. Although it does not consider the effect of variations in the directional spreading on the crest height statistics, it is found to describe weU realistic 3D wind sea [17]. Therefore, the model of Fonistall has been used in this study to describe the observed crest height distributions.
The largest waves in severe sea states need nonlinear theories, in order to be described. Moreover, while the linear wave theory has afi-eady been well validated against experimental data, the nonlinear theories need further verification. The present investigation is based on full-scale data collected during one of the severest storms in the North Sea i n the period of 1994-1997. The aim is to fit the model of Fonistall to the observed set of nonlinear crests looking for the differences in the quality of the fit of the sea states with abnormal waves found and those with large storm waves.
Section 2 discusses the characteristics of the abnormal waves found in the storm. Section 3 focuses on the sea state statistics and the derived relations between theiu. The second order
model of Forristall is described at the beginning of Section 4 and the results of fitting to the sets of observed crest heights are presented in the second part of Section 4. The derived conclusions are reported in Section 5.
2. Characteristics of the abnormal waves in the data sets The analyzed wave data was coUected at the fixed offshore platform North Alwyn in the northern part of the North Sea during the November storm in 1997. The platform is located at 130 m water depth. During the period of the storm, from November 16th to November 22nd, a total of 421 series of 20 min suiface elevation measurements performed at 5 Hz sampling frequency were recorded by means of wave laser altimeters mounted on the platform. Thus, each time series consists of 6000 samples of water surface elevations. The process is assumed to be stationary over the 20 min intervals of measurements.
The records have been corrected for the linear changes in the mean water level. Subsequently, the mean suiface elevation has been subti-acted from each measurement, yielding a new process in units of meters, which has a zero mean. The waves have been determined by down-crossing and up-crossing definitions. A total of 54 245 waves have been found.
The two available definitions for the significant wave height have been used. The spectral one gives the sigiuficant wave height as four times the standard deviation of the elevation process, //,„o = ^^/m, where mo is equal to the variance of the process. According to the statistical formulation, the significant wave height, H 1 / 3 , is the mean height of the largest one-third of the waves. Analysis of full-scale data showed that the latter definition yields approximately 5% smaller values [4]. The conclusion has been validated for the North Alwyn data set by Guedes Soares et al. [6]. The spectral definition was pointed out as the better choice in the distribution model of wave crests [5]. This formulation was adopted in all calculations, for which results are presented and discussed hereafter.
The adopted criteria for the classification of a wave as abnormal one are the crest amplification index, CI, larger than 1.3 and abnormality index, A I , larger than 2, which are used separately or together. The combination of both indices exceeded refers the wave to the class of genuine freak waves, according to Tomita and Kawamura [24].
The set of storm data used herein has recently been studied by Guedes Soares et al. [6] and 23 cases of abnormal waves have been identified in the total of 421 available records, since the ratio C I 0 = C n , a x D / ^ / l / 3 o exceeded 1.3. A f l waves were found to have maximum down-crossing wave height, HnmxD>
larger than 10 m and to be strongly asymmetric, with vertical asymmetry larger than 0.65. Furthermore, eight of the abnormal waves have been classified as genuine freak waves, since they f u l f i l l both criteria, C I D > 1-3 and A l e > 2. Moreover, they also have A I j / > 2. It was discussed that the values of the two calculated parameters, A I D and A l y , can sometimes be significantly different.
It must be recaUed that the significant wave height used for the calculations in the present study is based on the spectrum,
p. Petrova et al./Applied Ocean Research 28 (2006) 235-245 237 Table 1
Chaiacteristics of the abnomial waves and genuine beak waves identified
Data K3 Y4 S Ur T^niax D ^max D
^max D CID A I D 18110110 0.40 0.88 0.0519 0.0016 16.44 0.69 1.56 2.27 2.16 20110151 0.48 1.39 0.0522 0.0027 18.17 0.73 1.53 2.11 2.11 20110531 0.30 0.87 0.0472 0.0027 16.97 0.67 1.43 2.12 2.07 20110731 0.24 0.89 0.0477 0,0015 13.51 0.65 1.32 2.04 2.07 19111611 0.43 0.87 0.0456 0.0022 15.19 0.70 1.44 2.07 2.22 20110131 0.29 0.50 0.0461 0.0027 17.63 0.68 1.51 2.23 2.05 19110911 0.45 1.58 0.0571 0.0026 17.99 0.71 1.40 1.98 1.90 19111031 0.44 0.68 0.0579 0.0030 20.29 0.71 1.50 2.12 1.93 19111831 0.21 0.48 0.0516 0.0029 18.09 0.68 1.40 2.07 1.97 20112051 0.26 0.50 0.0494 0.0009 11.70 0.66 1.36 2.06 1.99 16111053 0.47 0.76 0.0515 0.0009 10.78 0.72 1.34 1.85 2.09 17111930 0.25 0.41 0.0492 0.0013 13.17 0.68 1.38 2.04 1.96 17112130 0.39 0.90 0.0522 0.0012 13.32 0.66 1.36 2.05 1.75 18111950 0.29 0.42 0.0585 0.0036 19.92 0.70 1.36 1.95 1.79 18112210 0.58 1.14 0.0456 0.0070 18.33 0.72 1.22 1.69 1.75 19110751 0.33 0.27 0.0543 0.0037 19.44 0.68 1.35 1.98 1.84 19110951 0.39 0.38 0.0528 0.0024 15.62 0.71 1.34 1.88 1.73 19111211 0.27 0.59 0.0492 0.0028 16.43 0.66 1.28 1.96 1.68 19112011 0.20 0.48 0.0515 0.0020 16.01 0.63 1.31 2.08 1.92 20110031 0.26 0.23 0.0492 0.0018 13.46 0.71 1.31 1.86 1.79 20110311 0.36 0.95 0.0525 0.0020 15.00 0.75 1.44 1.91 2.27 20111011 0.30 0.26 0.0481 0.0016 13.47 0.67 1.32 1.98 1.69 20111131 0.31 0.58 0.0467 0.0019 14.09 0.72 1.43 1.98 2.24
H,„o, and, subsequently, the newly calculated abnormality ratios have different values f r o m those reported in Guedes Soares et al. [6]. However, the results allow identification of the same records as having abnormal waves. Table 1 presents some statistics of the large asymmetric waves found and the corresponding sea states which are discussed in Section 3. The values of mean steepness, S, calculated from the mean period. Tot = mo/nii, with /HQ.I being the zero and first spectral moments, and the Ursell number, Ur, are shown in the table along with the values of Clo and A I , based on H,„o.
Table 1 shows that only the first six cases satisfy the condition that both abnormality indices are larger than 2, while the crest amplification index is larger than 1.3, compared to eight cases based on Hip reported i n [6]. Moreover, only the first four records have abnormality indices larger than 2 for all possible definitions, based on both the spectral and statistical formulations of the significant wave height. These waves w i l l be regarded as genuine freak waves hereafter.
3. Analysis of the sea state statistics
Some of the parameters describing the nonlinear properties of the s t o i T n sea states are discussed in this section. The interest is focused on the values of these parameters typically observed in the sea states with an abnormal wave present, as shown in Table 1 above. Some relations have aheady been derived by Guedes Soares et al. [6], where the set of the 23 abnormal waves along with the New Year wave f r o m the Draupner field was considered. The relations demonstrating the typical conditions contributing to the occuiTence of abnormal waves are reanalyzed here and some new dependencies have been proposed.
The observed sea states are characterized by values of the significant wave height, Hi„o, between 3 and 1 1 m and peak periods, Tp, between 8 and 14 s. The scatter diagram for the two parameters is given in Fig. 1(a). The designation used is subsequently applied to all graphs in the paper: the background population of wave crests is represented by f u l l circles; the sea states producing abnormal waves are shown with h g h t triangles and the four sea states with genuine freak waves observed are shown with fuU squares. The coefficient of correlation between H,„o and Tp is high, R{H,nQ, Tp) = 0.85. It is seen that the majority of wave records, approximately 94%, have H,„o > 5 m, which con'esponds to severe storm conditions.
The scatter diagram between H,„o and the mean wave period, 7bi, is shown in Fig. 1(b). The values of Tbi range between 7 and 13 s. Comparison between Fig. 1(a) and (b) shows that the pah (Hino, Tbi) is more correlated than the pair (TT,„o, Tp), with
coefficient of c o i T e l a t i o n RiH,„o, Tbi) = 0.92. ForristaU [5]
reports that his model fits the crest data better when the mean steepness is based on the mean wave period, Tbi. rather than on the peak period, Tp.
The significant wave height was found to be highly
c o i T e l a t e d with the maximum crest height in the set of storm data from four different locations, analyzed by Guedes Soares et al. [7]. The same tendency can be observed in the data gathered at the North Alwyn platform (Fig. 2(a)). On the other hand. Fig. 2(b) represents the dependence of the maximum wave height on the significant wave height showing a higher c o i T e l a t i o n of 0.86, as compared with 0.78 in Fig. 2(a). The regression lines and the 95% confidence bounds are also demonstrated in the figures. The crests of the abnormal waves fall either on the upper confidence bound, or outside of the confidence interval. Generally, the heights of the
238
(a) 14
12.5
h-'" 11
P. Petrova et al/Applied Ocean Researcii 28 (2006) 235-245 ( b ) i 3 i
12
12
R = 0.92
Fig. 1. Relation between //,„o and (a) the specüal peak period, Tp; (b) the average period, Tq\.
12 (a)1 12
•
R = 0.78 '^mOFig. 2. Correlation between the significant wave height, ff,„o, and (a) the maximum down-crossing wave crests, C|„ax D; (b) tlie maximum down-crossing wave heigllts,
T/maxD-abnormal waves follow the same tendency. Excluding the sets of abnormal waves, the calculated coefficients of correlation for the rest of the data increase to 0.84 for the maximum crest height and 0.90 for the maximum wave height. It is evident from the figures that the highest sea state in the storm does not correspond to the highest wave observed. This is better demonstrated in Fig. 3.
Fig. 3(a) depicts the time history of the significant wave height, H„,Q, plotted along with the time history of the crest amplification index, C I Q (Fig. 3(b)) and the abnoriuahty index, A I D (Fig. 3(c)). The abscissa in each figure shows the consecutive number of records during the storm.
The conclusion that can be derived is that the majority of abnormal waves pertain to the records with numbers from 100 to 300, or expressed by means of time, they are r e f e i T e d
to the interval from 33 to 100 h after the initiation of the storm. This interval contains the most energetic sea states, with significant wave heights larger than the mean significant wave height of the storm in 84% of the cases. The highest sea state is reached after 62 h (record 186) and is represented
by significant wave height /f,„o = 10.78 m, maximum down-crossing wave height H m a x D = 14.53 m and associated crest height CmaxD = 9.23 m. The wave is not classified as being abnormal, considering the values of the abnoriuahty ratios, C I D = 0.86, A I D = 1-35. However, when the significant wave height is derived from the highest one-third of the waves, the severest sea state is reached after 58 h storm duration (record 175) which has significant wave height H\pD = 10.64 m; largest wave height / / m a x D = 19.92 m and maximum crest
C m a x D = 13.88 m. This wave was classified as an abnormal one.
On the other hand, the largest wave in the storm, designated with a f u l l circle in Fig. 3, is observed in the record with number 217, almost 10 h after the most energetic sea state. The coiresponding parameters are: H,„o = 9.57 m, H^axD = 20.29 m and CmaxD = 14.33 m. The largest wave is also a genuine freak wave, when the abnormality indices are calculated from the statistical definition of the significant wave height, Hxp. The values of the calculated ratios are as follows:
p. Pen-ova et a!./Applied Ocean Research 28 (2006) 235-245 239
50 100 150 200 250
Record Number
300 350 400
Fig. 3. Time histories of (a) H„,O; (b) C I Q and (c) A I Q .
As a measure of nonlinearity in the wave field, higher order statistical moments or cumulants, or their normaUzed equivalents are usually applied. To second and third order of approximation, the relevant statistics could be the coefficient of skewness, = and the coefficient of kurtosis, or excess of kurtosis, — 3, where /x,- denotes the ;'th central moment, particularly, 112 = o" is the variance of the suiface elevation process. The coefficient of skewness reflects quantitatively the increased frequency of occurrence of high wave crests, as compared to the hnear theory predictions, while the coefficient of kurtosis, y4, reflects the frequency of encountering large crest-to-trough excursions. Thus, with respect to the freak wave phenomena, the coefficient 74 is regarded as being an adequate measure for the nonlinearity of the wave field and, consequently, for the prediction of abnormal wave occurrence, [6,7,15].
In Fig. 4, the crest amplification index, C I D , and the abnormality index, A I D , are plotted against the coefficient of skewness (Fig. 4(a) and (c)) and the coefficient of kurtosis (Fig. 4(b) and (d)). h is seen that the largest waves, which have also been classified as genuine freak waves, do not appear to be drawn either from the highest skewness, or from the highest kurtosis. However, they are more concentrated in the high range of kurtosis, than skewness. This conclusion is confirmed by the calculated coefficients of con-elation, showing higher correlation between the critical ratios and 5/4, than with yi. The plot in Fig. 4(d) shows similar results to those in Fig. 8 in [6], since it uses the same set of data from the North Alwyn storm, although the values from the Draupner field have not been included here. In the case of both data sets used, strong correlation with 7- = 0.72 has been found and a linear dependence was derived, which led to the conclusion of using
74 as an indicator of third order nonlinearity and of possible freak wave occunence.
The same tendency for the set of abnormal waves pointed out in Fig. 2 is observed in Fig. 4, namely, these waves appear as distinct clouds, with respect to the population of typically observed values of the investigated parameters.
Another parameter, used as a measure of nonlinearity, which is closely related to the increased crest extremes, is the wave steepness, S. The steepness of the considered set of abnormal waves and its relation with different characteristics of nonlinearity has already been discussed in [6]. The sigiuficant wave steepness and the individual steepness of the maximum waves have been found to be concentrated in a very narrow range of values. Relations between the largest wave crests and the proposed coefficients of down-crossing and up-crossing steepness, denoted by So and Su, respectively, which also define the individual waves, have been derived by Guedes Soares et al. [7] based on various full-scale data. High correlation has been reported.
The effect of sea state steepness on the wave crest amplitudes was found to be stronger than the effect on the wave heights [22]. Fig. 5 represents a scatter diagram between the maximum down-crossing crests, C m a x D , and the mean steepness, S, calculated from Eq. (1).
As can be seen in Fig. 5, the dependence of the maximum crest height on the sea state steepness does not show a clear trend. The highest measured wave crests do not always conespond to the largest mean steepness, though there is a positive correlation, i?(5, C m a x o ) = 0.54. The cases of genuine abnonnal waves are concentrated in the midrange of the steepness. Close values of the steepness are found to conespond to difference in the measured crests that could be significant. Furthermore, measured waves with similar steepness values pertain to different classes of waves: asymmetric, genuine freak waves or waves which are not abnormal according to any criteria. A possible reason could be the fact that the estimated steepness is concenttated in a considerably narrow range, which does not allow us to see any
240 (a) 1.8 1.6 i.4 1.2 0.4
P. Petrova et al./Applied Ocean Researcii 28 (2006) 235-245
( b ) 1 R = 0.47 0.2 0.4 Y3 (C) 2.6 2.2 ? 1 (d) 2.6 5 1.8
Fig. 4. Comparisons between (a) C I Q and ^ 3 ; (b) C I ^ and y4; (c) A l f l and yy, (d) A I ^ and 74.
16 13 10 1 0.035 0.045 - 0,055 S 0,065
Fig, 5, Maximuin crest lieiglit, C^ax D. versus tlie mean wave steepness.
clear dependence. The corresponding 95% confidence limits for ah crest data are also shown in the figure. Almost half of the abnormal events fall outside of the interval.
The model of Forristall [5] introduces the Urseh number (Eq. (2)), as a parameter reflecting the dynamical nonlinearity
of the wave field. Having in mind that the Noilh Alwyn platform is located at 130 m water depth, which is aheady deep water conditions for the majority of observed waves, the Ursell number is expected to be very small. The calculated values, of order 10"^ are plotted in Fig. 6 against the values of CmaxD' The estimated correlation coefficient is R{Ui; Cmaxo) = 0.71. The set of abnormal waves shows values of the Ursell parameter that are concentrated between 0.001 and 0.004 (Table 1). Also in this relation, the points corresponding to the abnormal waves are outside or on the upper limit of the confidence interval. The calculated coefficient of correlation excluding the abnormal waves is 0.77.
4. Probability distribution of wave crests
A commonly used model for describing wave amplitudes is the Rayleigh distribution. Contrary to the wave height extremes, which are influenced weakly by the nonhnear effects and could be well described even by the linear wave theory, the largest crest heights are usually underestimated, especially in sea states with an abnormal wave event [14,19]. Consequently, the second order model is chosen to describe the waves. It is found to approximate generally well laboratory simulated data and full-scale field measurements [5,11]. However, certain conditions
p. Petrova et al /Applied Ocean Research 28 (2006) 235-245 241 16 13 10 D X Ü
•
R = 0.71 Ur x 1 0Fig. 6. Maximum crest height, C„^„[), versus the Ursell number.
could produce maxima in random wave trains exceeding significantly the second order predictions, even for moderately steep wave conditions [20].
In the following, the second-order model of Fonistall [5] is first briefly described and the results from its application to the crest data from the Noi1h Alwyn storm are subsequently shown.
4.1. Forristall model
The model is based on a large number of 2D and 3D simulations of a second order elevation process. The time series are modelled thi'ough the standard JONSWAP spectrum. Two integral characteristics are used for the parameterization of the model: the mean steepness, S,
271 H,nQ
( 1 )
measures the geometiical nonlinearity of the wave field, due to nonlinear wave-wave interactions and the Ursell number, Ur, measures the dynanucal nonlinearity, due to the finite water depth effects, given as
Ur KQ^CI
( 2 )
In addition to the nomenclature defined before, in Eqs. (1) and (2) /coi is the deep water wave number, coiTesponding to TQI and d is the local water depth.
The probability of a wave crest, C, to exceed a prescribed value, c, P{C > c), is given by a two-parameter Weibull law of the form
P(C > c) = e x p (3)
The shape parameter, ^6, and the scale parameter, a, are polynomial functions of S and Ur. The coiTesponding polynomials for the 2D and 3D cases are given by Eqs. (4) and
(5), for long-crested waves, and by Eqs. (6) and (7), for short-crested waves. « 2 = 0.3536 - f 0.28925 + 0.1060f/r ,02 = 2 - 2.15975 -f- 0.0968C/r2 a!3 = 0.3536 -f- 0.25685 + 0.0800C/r ft = 2 - 1.79125 - 0.5302C/r + O.lUUr^. (4) (5) (6) (7) It could be deduced from Eqs. (4) to (7) that in deep water, where Ur —> 0, and for infinitesimally small wave steepness, 5 —> 0, the expressions reduce to the Rayleigh equivalents, namely to a = 0.3536 and /J = 2. In this case, Eq. (3) is simply the Rayleigh disttibution applicable to wave crest heights, given in the form
P(C > c) = exp (8) The model, although accounting for the directional stmcture of the wave field, does not consider the variations in the directional spreading [17]. However, it reflects the opposite effect of the directional and unidirectional spreading on the wave crest statistics in deep and shallow water. In particular, the directionally spread waves in shallow water are higher than the unidirectional waves, as opposed to deep water, where the unidirectional waves have larger crest heights.
4.2. Application of the model to full-scale data
The model is fitted to the set of waves observed in different sea states. The form of the distributions and the parameter values, associated with the cases of abnormal waves are outhned and commented in the following.
The shape and scale parameters o f the proposed Fonistall's distiibution are calculated from the polynomial functions given in (4)-(7). Some statistics of the samples of observed parameters are demonstrated in Table 2, where CIL and CIU denote the lower and upper confidence limits of the set of values. It is seen that the long-crested and short-crested seas produce sinoilar values of « 2 and ora, on one hand, and of ^2 and Pi, on the other hand. Moreover, the parameters of the model distribution are close to the Rayleigh equivalents (a = 0.3536, p = 2). The values of the standard deviations and the confidence bounds, given for the level of significance a = 0.05, show that the variability in the observed values of the unidirectional and directional parameters is small.
Table 2
Statistics of the calculated Weibull parameters
Pai'ameter «2 1^2 • /i3
Mean 0.3681 1.8936 0.3664 1.9107
St. dev. 0.0012 0.0089 0.0011 0.0076
C I L 0.3680 1.8927 0,3663 1.9100
C I U 0.3682 1.8945 0.3665 1.9114
Fig. 7 shows the crest height distributions compared with the distiibution of Fonistall for sea states with large waves that have not been classified as abnormal. The crests are normalized by the significant wave height Hi„o. The
242 (a) 10° s 10-1 10-0 Data Rayleigh Forristall 2D Forristall 3D SQ., = 0.03945
\
U^-l = 0.00222P. Petrova etal. /Applied Ocean Research 28 (2006) 235-245
(b) 100 10-' o Data Rayleigh Forristaii 2D Forristaii 3D SQ., = 0.04944 Uo, = 0,00227 10-(c) 10» 0,5 1 Crest Height, GIH^a
0,5 1 Crest Height, C/H^Q (d) 10° 1,5 ( f ) 100 0,5 1 Crest Height, C/H^o
0,5 1 Crest Height, C/H,„o
0,5 1 Crest Height, C / H „ o
0,5 1 Crest Height, C / H „ o
Fig, 7. Exceedance probability of crest heights i n sea states with no abnormal waves identified.
experimental distribution is given by full circles. The solid line corresponds to the Rayleigh distribution; the dash-dot line shows the 2D and the dotted line — the 3D model version. For the present data set the differences between the 2D and 3D disuibution resuhs are found negligible.
The model is seen to describe well the observations over almost the enhre data range. There are cases corresponding to a
perfect fit to the data (Fig. 7(a) and (b)). However, deviations are observed for the small probability levels (Fig. 7(c) and (d)), where the observed crests appear at higher probabilities of exceedance than the predictions of the linear or second order models.
In some cases (Fig. 7(e) and (f)) the fitting shows a pattern where the nonualized crest observations follow the ForristaU's
p. Petrova et al. /Applied Ocean Research 28 (2006) 215-245 243 10-1 10-2 5 10-1 10-0.5 1 Crest Height, C/H 0.5 1 1.5 Crest Height, C/H„,(, O Data Rayleigh Forristaii 2D Forristaii 3D \\ So, =0.05190 Uo, =0.00162 .
\
0 Data Rayleigh Forristaii 2D Forristaii 3D So, =0.05216 U„, = 0.00272 0.5 1 1.5Crest Height, C/H,.„o Crest Height, CIH„
• Data Rayleigh Forristaii 2D
\
Forristaii 3D So, =0.04715 Uo, = 0.00265 Forristaii 3D So, =0.04715 Uo, = 0.00265 V \ . V - '\\
o u 0.5 1.5Crest Height, CIH„
0.5 1 Crest Height, C / H „
Fig. 8. Exceedance probability of the crest heights in sea states with abnormal waves identified.
model, except for the largest sample value, which shows agreement with the Rayleigh distribution given by Eq. (8).
Examples of the approximations to sea states with abnormal waves are demonstrated in Fig. 8. Fig. 8(a) and (b) correspond to abnormal waves with C I D > 1-3 only while Fig. 8(c)-(f) represent the four sea states with genuine abnormal waves. Fig. 8(b) shows the sample distribution in the sea state with the largest wave in the storm. It has also the largest calculated mean
steepness, S. Fig. 8(c) shows the genuine freak wave, as defined herein, which has the largest down-crossing abnoiTuality ratio AIoHniO, calculated from H,i,o and AIDHI/3< calculated from
Hip i n the set of 23 largest waves in Table 1.
For all abnormal sea states the model fits well the observadons until approximately 0.5-0.67T„,o. Fig. 8(c) even shows a perfect fit over the entire range of crest heights, except for the genuine freak wave. However, generally, the observed
244 P. Petrova etal. /Applied Ocean Researcii 28 (2006) 235-245
distributions exhibit long tails and the deviations from the model in the extreme tail are large. The geimine freak events behave as outliers with respect to the background population of wave crests.
5. Conclusions
The short-term statistics of the wave crests in sea states with abnormal waves are discussed. They are compared with the statistics observed in storm sea states with the largest waves being extreme, but not abnormal, as none of the critical ratios have been exceeded.
Some relations between the parameters describing the nonlinear characteristics in the wave field have been studied. It has been found that the crest and height abnormality indices are more dependent on the coefficient of kurtosis than on the coefficient of skewness.
Higher correlation is found between the largest crests and the Ursell number, than between the largest crests and the mean steepness. The estimated values of both parameters for the cases of abnormal waves are concentrated in a very naiTow range, as compared to the total set of data. The steepness of the considered abnormal waves has aheady been discussed in Guedes Soares et al. [6].
The scatter diagrams demonstrated that the set of abnoriual waves usually forms a separate cloud with respect to the background set of values and is normally outside the intervals corresponding to the 95% confidence levels.
The observed samples of nonlinear crest height data have been approximated by the model of Fonistall. The second order nonlinear effects are taken into account by introducing the mean wave steepness and Ursell number. The range of values for the two parameters is found to be very naiTow. On the other hand, the scale and shape parameters of the distiibution, estimated from the polynomial functions for the short-crested and the long-crested waves, show small variance. Moreover they are very close to the corresponding values in the Rayleigh distribution.
The population of storm waves are found to be generally well represented by the theoretical model. However, the sea states with identified abnormal waves differ, showing large discrepancy from the linear and second-order models in the extreme tails of the distributions. The crests associated with the abnormal waves appear as outliers in the underlying populations of nonlinear crest events.
The results from the fittings in the sea states with abnormal waves usually show long tail distributions. The model is found to describe well the observations until approximately 0.5-0.6//„,o and to show large deviation for crests larger than 0.6Hi„o. The freak events occur as outliers in the population of wave crests, which further justifies the name they have received: abnormal waves, i.e. waves that do not conform to the set of waves in the population.
The fittings to the wave crests in sea states with waves that could not be defined as abnormal ones show three types of pattern: sea states where the model is seen to be in perfect agreement with the data; sea states where a discrepancy in
the extreme tail of the distiibution is observed, namely, the observed crests show higher probabiUty of exceedance than the predicted by the linear or second order wave model, and sea states where the second order model fits well the data but the largest value falls on the Rayleigh cuive.
Acknowledgements
The data used in this work was obtained duiing the project "Rogue Waves — Forecast and Impact on Manne Structures (MAXWAVE)", which was partially financed by the European Commission, under the contract EVK3-CT2000-00026.
This work was financed by the Portuguese Foundation for Science and Technology (FCT) under the pluiiaiuiual funding to the Unit of Marine Technology and Engineering.
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