THE SPECTRAL WAVE CLIMATE IN THE BARENTS SEA

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1 Proceedings of OMAE 2002: 21 st International Conference on OFFSHORE MECHANICS AND ARCTIC ENGINEERING June 2002, Oslo, NORWAY THE SPECTRAL WAVE CLIMATE IN THE BARENTS SEA Leonid Loatoukhin, St. Petersburg State University (De. Oceanology). St. Petersburg, Russia, State Oceanograhic Institute, St. Petersburg, Russia Valentine Rozhkov, St. Petersburg State University (De. Oceanology). St. Petersburg, Russia, State Oceanograhic Institute, St. Petersburg, Russia Alexander Boukhanovsky, Institute for High Performance Comuting and Data Bases, St.Petersburg, Russia Alexander Degtyarev, Institute for High Performance Comuting and Data Bases, St.Petersburg, Russia Kirill Sas'kov, Institute for High Performance Comuting and Data Bases, St.Petersburg, Russia Gerassimos Athanassoulis, National Technical University of Athens, Greece Christos Stefanakos National Technical University of Athens, Greece Harald E. Krogstad, Norwegian University of Science and Technology, Trondheim, Norway ABSTRACT Based on hindcast Barents Sea wave fields, an aroach to a directional sectral wave climate descrition is roosed. A set of classes of climatic wave sectra are identified and the robability of occurrence of sectra in each class is determined. The satial and temoral variability of the climatic wave sectral arameters for the different classes are considered to be nonrandom functions of underlying random arguments. INTRODUCTION Linear wind waves, as a stochastic hydrodynamic rocess, are characterized by a directional sectrum S ζ (, θ, r, t), deending on the frequency, the wave direction θ, the satial location r, and the time t. This rocess is by itself multi-cycled, as wind waves and swell may exist simultaneously. Synotic, annual and year-to-year variability roduce, in turn, olymodulation of the associated wave field æ(r, t). By an ensemble of climatic wave sectra we mean a collection {S ζ (, θ, r, t)} for r R, t T, where R is a satial region, and T a time san of the order of decades. Wave measurements (e.g., ocean weather shis, buoys, latforms, etc) allow one to gather hundreds [Ochi, 1978] or even thousands [Buckley, 1988, 1993] of wave sectra in searate oints. In the simlest case, climatic sectra are S [Buckley, 1993], or in defined as mean frequency sectra, ( ) a more general case, as quantile sectra S () occurring with a certain robability. Here S () may belong to different wave conditions (e.g., swell and wind waves) [Buckley, 1988, 1993]. In a series of aers [Boukhanovsky, Loatoukhin 1998; Loatoukhin et al. 1999] ensembles of climatic wave sectra were arametrized and aroximated based on a set of arameters, Ξ. The arametrization allowed one to write wind wave sectra, S ww, and swell sectra, S sw, as non-random functions S = S(, θ,ξ ) (1) deendent on a set of random arguments, Ξ. In general, the cardinality and meaning of the arameters in Ξ deends on the urose of investigation [see, e.g., Barstow et al., 2001]. The feasibility of an aroach like Eqn. (1) obviously deends on to which accuracy the sectrum S (,è) may be secified by the arameters Ξ taken from their multidimensional distribution F Ξ (ξ). 1 Coyright #### by ASME

2 An investigation based on the 6000 sectra measured in the Black Sea [Boukhanovsky et al., 2000; Loatoukhin et al., 1999], allowed one to select classes of frequency sectra S(ù). The selected classes were wind waves, swell, combined wind waves and swell with and without searation in frequency, and the robability of belonging to each classes and the transition from one class to another were estimated. Reanalysis of wind fields [Kalnay et al., 1996] has allowed for numerically hindcasting wave fields and hence creating ensembles of comuted climatic directional wave sectra, S(,è). Below we are going to resent an analysis of sectra from a hindcast study using the NCEP/NCAR re-analyzed wind fields from 1983 to 1999 and the WaveWatch model [Tolman, 1999]. The simulation was set u using a grid, covering the North Atlantic, Greenland, and the Norwegian and Barents seas. The full directional sectrum S(,è) was calculated at each grid oints with 24 values in direction θ and 25 values in frequency. The time ste was 6 hours so that in any oint r, a collection of more than sectra with more than 13 millions numerical values had to be considered. The main objectives of sectral wave climate investigations are in broad terms to select classes of wave sectra and estimate their robability; to roose a arameterization which allows one to justify a choice of difference between the various classes (i.e., the selection of discriminant variables); to aroximate the ensemble {S(,è)} in terms of its robabilistic characteristics; to elaborate a stochastic model of the sectral wave climate; to estimate climatic wave sectra for extreme waves (design waves). Mainly because of size limitations, the resentation here will focus on the first three oints. The results of a discriminant analysis using the selected variables, behavior of satialtemoral variability of classes and their borders, stochastic models of the climatic ensemble and estimates of quantile sectra (including extremes), are the toics of future investigations. CLIMATIC DIRECTIONAL WAVE SPECTRA In the resent study, arameters of the sectrum related to wave height, sectral shae, the frequency of the sectral eak,, and the main wave direction, θ, are selected as arameters in Ξ. The single field model sectrum may be written S ( /, θ θ, Ξ r ), where Ξ signifies the rest of the r arameters. More general sectra S(,è) are obtained as S N (, θ) = m00 = 1 γ S, θ θ, Ξ r (2) where m 00, the 0 th moment of the sectrum, is equal to the total variance of wave field, N is the number of wave fields (eaks in the sectrum), and γ are weight factors for each system so that, N γ = = 1. 1 A simle frequency model sectrum which covers both wind sea and swell is the Gamma-sectrum, n S ( n) n Γ,, = (3) n n ex 1 n 1 Note that S (,, ) 0 Γ n d = 1, and that the sectral eak occurs for =. The high frequency tail of the sectrum is asymtotically roortional to n. The eakedness of the sectrum is also governed by the size of n. Famous sectra of this shae are the Pierson-Moskowitz sectrum for n=5, the Davidan sectrum for n=5.5, and the Davidan/Massel swell sectrum for n=6. Similarly, a well known directional distribution, and the one used below, is Q m (, θ,m) = C cos ( θ θ ), θ θ < π 2 θ, (4) 0 m / where C m is a normalizing arameter such that 2 π Q0 ( θ, θ, m) dθ = 1, and the m arameter determines the width 0 of the angular distribution. In general, the arameter m in (4) deends on frequency. However, we have assumed that m is a constant in the resent study. In accordance with numerous exeriments [Longuet- Higgins, 1963, Massel, 1996, etc.], m 2 for moderate and rough seas, and m 4 for high seas. This coincides with directional distribution function recommended by ISSC at the time of the study [4 th International Shi Structures Congress, 1970]. The single field model sectrum used below is therefore S (, θ, θ, n, m) S (,, n) Q ( θ, θ, = Γ 0 m) This means that the building blocks of S are defined in terms of four arameters, θ, n, and m. The full sectrum shown in (2) is comletely defined by secifying the overall energy in the waves, m 00, the weights γ i, and the arameters for each system, { i, θ i, n i, m i }. Below we shall also refer to the marginal sectra, S() and Q(θ), defined from The integration of the exression in Eqn. (2) over direction and frequency, resectively. The arameters m 00, i, and θ i are found directly from the sectra, whereas n i, m i, γ i are obtained by a Monte-Carlo method in a nonlinear otimization. The values of m 00, i and θ i, i = 1, L, N, define the total energy of sectrum, and the location an d number of the individual wave fields (essentially the number of eaks in the sectrum). Let us consider the roblem of minimizing the functional J ( S ) = 2π 0 0 (, θ) [ S(, θ) S (, θ) ] sim 2 ddθ (5) 2 Coyright #### by ASME

3 with resect to n i, m i (the arameters of for the sectral shae) and γ i (the distribution of energy between the wave systems). Here S sim (,θ) is the data from the hindcast, S is the sectrum in Eqn. (2), and (,θ) is a weight function. As a first aroximation, we have set (,θ) 1, since the analyzed sectra are obtained by numerical modeling, where the reliability should be fairly uniform over the and θ region. In general, analytic minimization of the functional in (5) generates a system of 3N-1 non-linear algebraic equations. Since the value of N deends on the wave conditions and varies from sectrum to sectrum, the dimensionality of the feasible domain is not fixed, and for simlicity, a discrete technique was emloyed here. The search (Monte Carlo) was limited to discrete values of m i and n i, also taking into account that 0 γ i 1. This consisted in a search for the minimum of (5) among an ensemble of all ossible feasible realizations {n i, m i, γ i }. The algorithm has linear convergence in N, and is therefore rather slow. Since the number of unknowns is not rohibitively large, this is not very critical in these days of fast comuters. The discrete solution may be ultimately be imroved by some of the classical methods, e.g. Conjugate Gradient or Newton-Ralhson.. The raw data from the simulations, S sim (,θ), will have a lot of local eaks, and as a result, a reliable and consistent classification is hard to obtain. Therefore, the rimary classification for and θ is based on the marginal sectra S() and Q(ù). In this way, three main classes and some subclasses are obtained. Examles of inut one, two and multieaked sectra and their aroximation by the model sectra are shown in the Fig.1. In more detail, the selected classes are the following: One -eaked sectra (I). One wave system revails either wind waves (I-1) or swell (I-2). In relation (2), N=1, γ 1 =1, and only one extreme (, θ ) occurs in the marginal S() and Q(è). The searation between wind waves and swell is based on nondimensional steeness defined as 2 2 gτ π g δ = = 2 h m s If δ>300, then sectrum belongs to a swell, otherwise to wind waves. Two-eaked sectra (II). Two wave systems exist simultaneously. In relation (2) N = 2, γ 1 = γ, γ 2 =1 γ. We can recognize three subclasses (deending on the number of ima in the marginal sectra): II-1. Mixed sectra with searation both in frequency and direction. In this case, there are two ronounced ima ( 1,θ 1 ), and ( 2,θ 2 ), both in the frequency sectrum and marginal angular distribution. II-2. Mixed sectra with searation only by direction. In this case, there is one eak in the frequency sectrum and two eaks in the angular distribution. As a result in the two-dimensional sectra, there will be eaks at the same frequency ( 1,θ 1 ) and ( 1,θ 2 ). II-3. Mixed waves with searation only by frequency. The angular distribution has only one eak and the frequency sectrum is broad with a not so ronounced second eak. Multieaked sectra (III). Comlicated wave fields with two or more swell fields. In this case, the angular distribution has more than two ronounced eaks. The estimated robability for belonging to each class of sectra in different months and during the year is resented in Table Table 1 The robability of classes of climatic wave sectra in the western art of the Barents Sea. Probability, % Ñlasse Subclass January Aril July October Annual s I I I II II II II III III We observe that wind waves revail in only 32% of the time during summer, whereas this increases to 48% in the autumn. Similarly, a single swell system has a robability of 24% in the summer, droing to only 17% in January. Doubleeaked sectra have a robability of occurrence equal to 36% during July. During the year, 26% of a total of 34%, the twoeaked sectra are searated only by direction, i.e. the frequency sectra have only one eak. Hence the classification using two-dimensional sectra has a definite advantage over classifications based only on frequency sectra. The statistics of sectral arameters for the I and II classes are shown in Table 2. 3 Coyright #### by ASME

4 Fig. 1. Classes of climatic wave sectra. 1 Hindcasting, 2 - Aroximation 4 Coyright #### by ASME

5 Table 2. Probability characteristics of arameters of tyical climatic sectra. Western art of the Barents Sea. Ww: means wind waves; S: swell. For n and m the 80% robability intervals are resented. Class Wave height Parameters of wave systems Wave H S, (m) g, % system T, (s) q,( 0 ) n m Mean Std. Mean Std. Mean Std. Mean Std. January I 1 3,2 1,7 Ww 7,7 2, I 2 2,9 1,4 S 10,9 2, II 1 2,1 1,2 0,67 0,14 Ww 6,6 1, S 11,3 2, II 2 2,5 1,2 0,70 0,17 Ww 7,5 1, S 10,4 2, II 3 2,3 1,5 0,51 0,14 Ww 4,6 0, S 9,1 3, July I 1 1,5 0,6 Ww 5,7 1, I 2 0,9 0,5 S 6,2 1, II 1 1,1 0,6 0,63 0,19 Ww 5,0 0, S 8,0 1, II 2 0,9 0,5 0,61 0,19 Ww 5,3 1, S 6,8 1, II 3 0,7 0,3 0,76 0,20 Ww 4,5 1, S 7,9 1, The sectra of class I-1 (wind waves) belong to the most heavy waves conditions. Two-eaked sectra are, as a rule, tyical for lower waves, although excetions exist. For all twoeaked sectra (class II), the mean values and the standard deviations of wind wave inut ã are stable, 50-70% in mean with the ó = 14-20%. The general directions of wind wave and swell are quite different for this art of Barents Sea. Wind waves are mainly from west-south-west, and swell is from east in the winter, and from both west-south-west and east during the summer. Very narrow sectra of swell with arameters n 8 and m 18 are sometimes observed for mixed seas (class II). Since in these cases, the directions of wind wave and the swell are areciably different and the swell intensity is quite small, it is conceivable that the swell is due to distant storms in southeast-east art of the Barents Sea. Such swell is definitely different from swell formed within the numerical region where n=5 6 and m=6 16. As seen from the Table 2 and Fig. 1, only the classes I and II could be clearly searated. The distributions of the arameters γ,, θ overla heavily for subclasses II 1, II 2, II 3. Therefore, it is necessary to introduce a set of discriminate variables (DVs) as distinctive indices for the searation. In the first stage, consider the following DVs: The fraction of wind waves in the sea, γ; The difference in frequency between the eaks, = (1) (2) ; The difference between the directions of the eaks, θ= θ (1) θ (2) ; Let us consider the vector of correlated values z ={γ,, θ}, and the corresonding 1 samle Fisher discriminants (SFD) fk [Fisher, 1958]. All SFDs are linear combinations of the DV's: 3 f k = ek zk = e1kγ + e2 k + e3k θ = 1 The coefficients e k in (6) should to be such that the values f k are uncorrelated, and that the distinction between the mean values of z for each class is imized. For this to be case, it is shown in Johnson and Wichern [1992], that e k ={e 1k,e 2k,e 3k } are the eigenvectors of the matrix W 1 B, where W is a samle within classes covariance matrix, and B is a samle between classes covariance matrix. Geometrical interretation of this basis from W 1 B is as follows. If we consider the DV's z as a oint in multidimensional Euclidean sace, each sectrum is an object in the same sace. Therefore, distinctive classes may be reresented as accumulation of oints in some regions of the sace. The vectors e k are an otimal basis for the searation of these grous of oints by means of linear boundaries, and the values of the SFDs are just the coordinates of oints corresonding to this basis. For examle, searation of subclasses II 1, II 2, II 3 for July in the lane of the two first SFDs is shown in Fig. 2. (6) 5 Coyright #### by ASME

6 ACKNOWLEDGMENTS This work has been carried out within the INTAS Project «Estimation of extreme metocean events». Hydrodynamic modeling was carried out on suercomuters (Parsytec CCe/20 and self-made cluster Paritet, see htt:// of Institute for High Performance Comuting, St.Petersburg. REFERENCES Barstow S.F., Athanassoulis G.A., Cavaleri L., Mork G., Belibassakis K.A., Stefanakos Ch.N., Georistathis Th.P., Sclavo M., Bertotti L., Krogstad H.E. (2001) EUROWAVES: a user-friendly tool for the evaluation of wave conditions at any Euroean oint condition. Final Technical Reort. Fig. 2. Classes of two eaked sectra in discriminant lane. Here number is the number of subclass of II. As shown in the Fig. 2, only subclass II 3 is clearly searated. We have not such clear situation for subclasses II 1 and II 2. In this aer we have not considered the roblem of comarison of modeling sectra with measurement results, but this would be one of ossible continuations of the research. A second one could be is exansion of determinant values vector (including of form arameters n i, m i ) and imrovement of searation the rocedure (by means of nonlinear methods). CONCLUSIONS The aer has studied the wave climate in the Barents Sea based on an ensemble of directional wave sectra obtaining by hindcasting. Stable generic classes of sectra have been indentified in the two-dimensional sectrum ensemble. Most of the classes corresond to secific wave conditions, and the robably of occurrence for each class has been identified. The sectra are aroximated by sums of single field model sectra based on four arameters, namely, the eak frequency, the main direction, and the arameters n and m relating to the shae of the frequency sectrum and the width of the directional distribution. In this way, the sectra may be obtained as deterministic functions of a set of random arameters. The robabilistic characteristics of the arameters for each class of wave sectra have been comuted. Finally, it is demonstrated how methods of linear discriminative analysis may be used for identification of sectra with adjusted climatic classes. Boukhanovsky A.V., Loatoukhin L.J., Rozhkov V.A. (1998) Wave climate sectra and wave energy resources in some Russian seas./ World Meteorological Organization/ Marine Meteorology and Related oceanograhic Activities. WMO/TD-No 938. Provision and engineering/oerational alication of ocean wave data. UNESCO Paris, Setember. A conference cosonsored by WMO, Meteo France and CNES, Boukhanovsky A.V., Loatoukhin L.J., Rozhkov V.A., Divinsky B.V., Kos an R.D. (2000). A classification of wind waves in the Black sea using instrumental data. Oceanology N2, (Russian translation) Buckley W.H. (1988) Extreme and climatic wave sectra for use in structural design of shis. Naval Engineers Journal, Set Buckley W.H. (1993) Design Wave Climates for the World Wide Oerations of Shis.. Part 1: Establishments of Design Wave Climate. Int. Maritime Organisation (IMO), Selected Publications. Fisher R.A. (1958), Statistical Methods for Research Workers, 13 th ed., Prentice-Hall, Englewood Cliffs, N.J., 1958 Johnson R.A., Wichern D.W. (1992) Alied multivariate statistical analysis. Prentice Hall International, Inc., London, 642. Kalnay E., Kanamitsu M., Kistler R., Collins W., Deaven D., Gandin L., Iredell M., Saha S., White G., Woollen J., Zhu Y., Leetmaa A., Reynolds R., Chelliah M., Ebisuzaki W., Higgins W., Janowiak J., Mo K. C., Roelewski C., Wang J., Roy Jenne, Dennis Joseh, (1996) The NCEP/NCAR 40-Year Reanalysis Project. Bulletin of 6 Coyright #### by ASME

7 the American Meteorological Society. March, 77, N3, Longuet-Higgins M.S., Cartwright D.E., Smith N.D. (1963) Observations of the directional sectrum of sea waves using the motions of a floating buoy. //Proc. Conf. Ocean Wave Sectra. Easton, Loatoukhin L.J., Boukhanovsky A.V., Rozhkov V.A., Divinsky B.V. (1999). Climatic wave sectra of the Black sea. Proc. Int. MEDCOAST Conf. Wind and wave climate of the Mediterranean and the Black sea, Antalya, Turkey, Massel S.R. (1996) Ocen Surface Waves: their hysics and rediction. World Scientific. London, 1996, 491. Ochi M.K. (1978). Wave statistics for the design of shis and ocean structures. Trans. Soc. Naval. Architects and Mar. Eng. 86, Tolman H. (1999). User manual and system documentation of WAVEWTCH-III, version 1.18/ U.S, De. of Commerce, NOAA, Technical Note, th International Shis Structures Congress. (1970) Reort Committee 1, Environmental Conditions. Tokyo 7 Coyright #### by ASME