IMPROVED SPECTRAL WAVE MODELLING OF WHITE-CAPPING DISSIPATION IN SWELL SEA SYSTEMS

Transcription

1 Proceedings of OMAE 2004: 23rd International Conference on Offshore Mechanics and Arctic Engineering June 2004, Vancouver, Canada OMAE IMPROVED SPECTRAL WAVE MODELLING OF WHITE-CAPPING DISSIPATION IN SWELL SEA SYSTEMS David P. Hurdle Alkyon Hydraulic Consultancy & Research b.v. Marknesse, The Netherlands. Gerbrant Ph. Van Vledder Alkyon Hydraulic Consultancy & Research b.v. Marknesse, The Netherlands. ABSTRACT An iproved ethod for coputing the dissipation by whitecapping in full spectral discrete wind wave odels is presented. In this ethod the rate of dissipation of a certain spectral wave coponent is deterined by the cuulative wave steepness of all spectral coponents with lower frequencies. This ethod is known as the cuulative wave steepness ethod (CSM) and was described by Van Vledder and Hurdle (2002). In this paper soe iproveents of the CSM are presented which are related to directional and scaling effects. The iproved CSM was ipleented in the latest release of the SWAN odel (version of February 2004) and a calibration against a paraetric growth curve was perfored. Further, the benefits of the CSM in swell-sea systes are illustrated and copared to the ethod of Koen et al. (1994). INTRODUCTION Spectral wave odels are an iportant tool for transforing deep-water wave conditions to shallow water for deriving design loads on arine structures and coastal protection systes. In the Netherlands the SWAN odel (Booij et al., 1999) is used to derive wave boundary conditions along the water defenses. A weak point of SWAN, and siilar full spectral odels, is a structural under-prediction of wave period easures in any shallow water situations. Detailed analyses of wave observations and wave odel runs with SWAN indicate that this under-prediction is stronger in locations were relatively long offshore waves coexist with shorter locally generated waves. Analyses of spectral shapes clearly indicate an under-prediction of low-frequency energy and an overprediction of the wind sea part. The under-prediction of lowfrequency wave energy was also noted by Rogers et al. (2003). They analyzed the under-prediction of swell energy in two coastal areas and an inland lake, and they concluded that the Koen et al. (1994) ethod for odeling dissipation by whitecapping is responsible for the enhanced dissipation of swell waves. This is rearkable; since it sees very unlikely that whitecapping should affect swell waves. Most present spectral wave odels, including WAM, SWAN and WAVEWATCH, are third-generation wave prediction odels. The ai of such odels is to reproduce each individual physical process on the basis of first principles in the for of source ters. This ai is rather abitious in view of our lack of understanding of any of these processes. One of the least understood processes is the dissipation by white-capping. A coonly used forulation for white-capping is the one proposed by Koen et al. (1994). A key eleent of this forulation is the use of a ean wave steepness. This steepness is used to scale the aount of white-capping. The forulation works well in idealized growth situations as long as the wave spectra are uni-odal. As indicated in Van Vledder and Hurdle (2002) this white-capping forulation has erroneous properties in double-peaked wave spectra, especially in relation with its effect on ean period easures. The presence of a sall aount of swell energy lowers the ean wave steepness, resulting in a lower rate of dissipation of the whole spectru. As a result the wind sea peak grows too fast, resulting in an over-estiation of high-frequency energy and an underestiation of the ean wave period. Moreover, a sall wind sea super-iposed on a swell syste increases the ean wave steepness, resulting in a higher dissipation rate of the whole spectru. As a result an unrealistic dissipation of the swell takes place, also leading to an under-prediction of the ean wave period. These probles can be avoided by the cuulative wave steepness ethod, introduced by Van Vledder and Hurdle (2002). In this ethod the dissipation rate at a certain spectral coponent is deterined by the cuulative wave steepness of 1 Copyright 2004 by ASME

2 all spectral coponents with lower frequencies. This ethod is known as the cuulative wave steepness ethod (CSM). The ain advantage of the CSM is that is does not use integral wave paraeters. Use of integral paraeters has several unwanted properties. One of the is an unrealistic coupling between the dissipation of various wave systes, as described in the introduction. Another property is the dependence of the integral paraeter values on the energy at higher frequencies. Especially the higher frequency oents are very sensitive on the upper integration liit. In the CSM these probles are avoided because the dissipation of a certain wave coponent only depends on the wave energy at lower frequencies. In this paper new features and results of the CSM ethod are presented. The CSM has been extended in a natural way to include directional effects. It has been ipleented in the SWAN odel and calibrated against an established paraetric growth curve. Finally, soe results of applying the CSM in a swell-sea syste are deonstrated. NOMENCLATURE A C cs C nl4 noralisation coefficient in CSM scaling coefficient of CSM scaling coefficient of source ter for quadruplet wave-wave interactions d water depth () E(ω,θ) energy density spectru ( 2 /rad) F * diensionless fetch (-) g acceleration due to gravity (/s 2 ) H 0 significant wave height () k wave nuber (1/) paraeter controlling directional dependence M agnitude of whitecapping source function N nuber of calibration points p S st (ω,θ) steepness spectru S (ω,θ) source function for whitecapping T 01 ean wave period (s) u * friction velocity (/s) ε error easure ε * diensionless wave energy λ shape paraeter of wave nuber configuration θ direction (rad) ν * diensionless peak frequency ω radian frequency ω in iniu discrete frequency of SWAN axiu discrete frequency of SWAN ω ax THE CUMULATIVE WAVE STEEPNESS METHOD FOR WHITE-CAPPING Based on ideas of Donelan (1999) and Booij (1999, inforal counication), Van Vledder and Hurdle (2002) proposed a possible solution for the erroneous behavior of the standard white-capping dissipation in swell-sea systes. They argue that the dissipation of a spectral coponent should not be based on the average steepness of the whole spectru. Instead they propose to link this dissipation rate to the cuulative wave steepness of all spectral coponents up to the frequency considered. Therefore this ethod is known as the Cuulative Steepness Method (CSM). In the CSM the cuulative (squared) wave steepness is coputed as: 2 ( ) ( ') S ω = keω dω' (1) st ω 0 In which the wave nuber k and the radian frequency ω are related via the linear dispersion relation ω 2 =gktanh(kd) withd the water depth. Equation (1) is the integral of the steepness spectru up to the frequency ω. Using the the cuulative steepness S st, the source ter for whitecapping is given by: ( ωθ, ) = ( ω) p ( ωθ, ) S Ccs Sst E (2) with C cs a tunable coefficient and p a paraeter controlling the proportionality of the dissipation rate on the steepness. In the present paper p=1 is assued. Expression (2) does not include explicit directional effects, but they are included in this ethod by considering the physical process of the straining echanis. In this echanis short waves propagating on top of larger waves are copressed on the forward face of the longer wave and stretched at the backward face. The copressed waves becoe steeper, resulting in enhanced dissipation. It is assued that this process does not act when waves propagate at an angle of 90 with respect to one another. It acts siilarly when the waves have opposing direction coponents of propagation and weaker when the waves are travelling at an oblique angle. A siple way to account for this effect is to introduce a cosine-type dependence of the angular difference in cobination with an absolute operation. This leads to the following extended forulation for coputing the cuulative steepness at a certain frequency- direction bin: ω 2π 2 ( ) = ( ) ( ) S ωθ, k cos θ θ' E ω', θ' dθ' dω' st (3) 0 0 In this expression is a tunable coefficient, which controls the directional dependence. It is expected that this coefficient is order 1 if the straining echanis is doinant. If directional effects do not play a role, all coponents contribute equally to the cuulative steepness, the coefficient takes the value 0 and expression (3) reduces to expression (1). On the other hand, if the whitecapping can be considered to be directionally decoupled (i.e. the dissipation is not influenced at all by waves in other direction sectors, except opposing wave directions) then approaches infinity. 2 Copyright 2004 by ASME

3 Equation (3) has the property that for a given vale of C cs the directional distribution of the steepness depends on the coefficient. For practical reasons it is desirable to isolate directional effects, as controlled by the coefficient, frothe total aount of dissipation, as controlled by the coefficient C cs. This is achieved by noralizing the right-hand side of expression (3) by assuing that the cos (θ) function is a distribution function. Then, the noralization coefficient A is such that: 2π A 0 ( ) cos θ = 1 (4) This leads to the following expression for A : A = ( / 2 1) ( /2 0.5) 1 Γ + π Γ + The expression for the steepness spectru then becoes: ω 2π 2 ( ) = ( ) ( ) S ωθ, A k cos θ θ' E ω', θ' dθ' dω' st 0 0 (6) The source ter for the whitecapping is given by: ( ωθ, ) = ( ωθ, ) p ( ωθ, ) (5) S Ccs Sst E (7) Expression (7) has been ipleented in SWAN with the paraeters C cs and as tunable paraeters and p=1. In this paper this SWAN odel version has been used to calibrate and validate the CSM ethod. PROPERTIES OF THE CSM METHOD FOR WHITECAPPING Soe basic properties of the CSM are illustrated using a uniodal and a bi-odal spectru. The directional integrated cuulative wave steepnesses for these spectra are shown in Figure 1. As can be seen, the cuulative steepness is practically zero for the lowest frequencies of the swell syste. It increases quite rapidly for frequencies above the peak of the wind sea spectru. Especially for the higher frequencies the extra contribution of the swell syste is visible. The corresponding directionally integrated source functions of the CSM are shown in Figure 2, together with a coparison of the Koen et al. (1994) source function. Inspection of this figure clearly shows that the Koen forulation reduces the dissipation of the wind sea in the presence of a swell syste, whereas the CSM slightly increases this dissipation. Further, the Koen ethod increases the dissipation of the swell syste inthepresenceofawindseasyste. The applicability of the noralization coefficient A was tested for a JONSWAP type spectru with a cos-shaped directional distribution, and various values of the coefficient. Tothat end the CSM source function was coputed using a range of values for and a fixed value for C cs. The total dissipation rate was then coputed by nuerical integration of the source function for white-capping according to: 2π ωax 0 ωin ( ωθ, ) M S dωdθ = (8) in which ω in and ω ax are the lowest and highest frequencies in the discrete SWAN spectru. The noralized variation of M as a function of is shown in Figure 3. It can be seen that M is practically constant with varying power. Thisipliesthat the coefficients C cs and can practically be treated as independent variables. The dependence of the directional distribution of the dissipation rate on the coefficient is illustrated in Figure 4 for a ean JONSWAP spectru with a cos 4 (θ) directional distribution. The directional distribution of the whitecapping dissipation is coputed by integration over the frequencies as: ω ax ( ) (, ) S θ = S ω θ dω (9) ω in Figure 4 contains three curves. The solid curve is for =0, the dashed curve is for =2 and the dotted curve is for =1000. The directional resolution of the discrete spectru is 10. The results in this figure show that with increasing power the directional distribution of the white-capping dissipation becoes narrower. This behavior is new and intriguing since up to now the only source ter affecting the directional distribution is the one for the quadruplet wave-wave interactions. CALIBRATION OF CSM The two coefficients of the CSM have been deterined by eans of a calibration against the growth curve of Kaha and Calkoen (1992). To that end the SWAN odel was applied in one-diensional ode to generate deep water growth curves for a fetch of 25 k and two wind speeds of 10 /s and 20 /s. The diensionless fetch laws were taken fro the coposite data set of Kaha and Calkoen (1992). They are given by: ε = F ν * * 0.27 * = 3.08 F* (10) with ε * and ν * the diensionless wave energy and peak frequency, respectively, as noralized by the friction velocity u *. In the calibration of SWAN also the coefficients C nl4 and λ of the source ter for quadruplet wave-wave interactions were optiized. This was necessary since the source ters for whitecapping and quadruplet interactions act together during active wave growth and because they both affect the directional distribution. The calibration of SWAN was treated as a ulti-variate iniization proble of the following error easure: 3 Copyright 2004 by ASME

4 1 ε = N + i= 1 i= 1 N N i i i i ( H0, S H0, KC) ( T01, S T01, KC) 2 2 (11) In this expression N is the nuber of calibration points. It was equal to 20, corresponding to 5 locations along the fetch (5 k, 10 k, 15 k, 20 k and 25 k), 2 wind speeds (10 /s and 20 /s) and the paraeters H 0 and T 01. The coputed values have the subscript S (of SWAN) and the target values have the subscript KC (of Kaha and Calkoen). The results of the calibration yielded the following rounded paraeter values: C cs =0.4, =2,C nl4 =4e7, and λ=0.28. An exaple of a coputed growth curve and the coparison with the paraetric growth curve of Kaha and Calkoen is shown in Figure 5. The agreeent is very satisfactory. The developent of the frequency spectra along the fetch is shown in Figure 6. The spectra are very realistic and they clearly show the overshoot at higher frequencies. WAVE GROWTH IN A SWELL SEA SYSTEM The benefits of the of the CSM over the Koen ethod are deonstrated in a situation of wave growth with and without the presence of a background swell. To that end two sets of wave odel runs with the SWAN odel were ade. In each set two coputations were perfored, one with the Koen ethod for the white-capping dissipation and one with the CSM. The first run coprises an ideal deep water wave growth situation starting with zero waves and a wind speed of 20 /s. In the second set of runs a background swell with a peak period T p =10 s, a significant wave height of H 0 =0.5 and a cos 10 (θ) directional spreading was specified as initial condition. The growth curves of the significant wave height H 0 and ean wave period T 01 are presented in Figures 7 and 8, for the Koen ethod and the CSM, respectively. The corresponding frequency spectra at fetches of 1 k, 5 k, 10 k, 15 k, 20 k and 25 k are shown in the Figures 9 and 10. The results in these figures show that the presence of a background swell in the CSM yields a slight increase in the overall level of the significant wave height H 0 and ean wave period T 01, whereas the Koen ethods yields uch ore growth. Coparison of the frequency spectra, as shown in the Figures 6, 9 and 10, indicates striking differences in behavior between the CSM and the Koen ethod. The Koen ethods yields unrealistically enhanced wave growth, whereas in the CSM the wind-sea spectra of swell situation are slightly lower than for the pure wave growth situation. This is due to the increased level of dissipation by the presence of the background swell. As the waves becoe ore developed the wind-sea and swell start to interact via the CSM and the quadruplet wave-wave interactions. dissipation is presented. The results show that the CSM was successfully ipleented in the SWAN odel and could be calibrated against an established growth curve. The ain benefit of the CSM is its ability to predict realistic wave growth in a situation with a background swell. The CSM forulation deviates considerably fro the wellknown Koen et al. (1994) or the extended Koen forulation, proposed by Holthuijsen et al. (2001). Both of these ethods use ean properties of the wave spectru, which are not used in the CSM. The extended Koen forulation reduces to the standard Koen forulation in the case of uni-odal spectra. However, the CSM uses a copletely different ethod for estiating the dissipation by white-capping, which ay lead to a further questions and understanding of the source ter balance in wave growth situations. Further work is in progress to study the directional behavior of the CSM in cross-seas, occurring in situations with rapidly changing wind directions, and the behavior in situations with double peaked wave spectra, which occur frequently in the Dutch coastal waters. Other points of attention are the perforance of the CSM in fully developed wave conditions, and the nature of the interactions between the wind-sea and swell systes by non-linear quadruplet wave-wave interactions. The dependence of the paraeter p (see Eq. 2) on the growth stage of the waves is of special interest in these studies. REFERENCES Booij, N., R.C. Ris, and L.H. Holthuijsen, 1999: A third generation wave odel for coastal regions. 1. Model description and validation, J. Geophys. Res., 104, Donelan, M.A., 1999: Presentation at WISE eeting, Annapolis, USA. Holthuijsen, L.H., R.C. Ris, N. Booij and E. Cecchi, Swell and Whitecapping, A Nuerical Experient. Proc. 27th Int. Conf. on Coastal Eng., Sydney, Australia. Pp Kaha, K.K., and C.J. Calkoen, 1992: Reconciling discrepancies in the observed growth rates of wind waves. J. Phys. Oceanogr., Vol. 22, Koen, et al., 1994: Dynaics and odelling of ocean waves. Cabridge Univ. Press. Rogers, W.E., P.A. Hwang, and D.W. Wang, 2003: Investigation of wave growth and decay in the SWAN odel: Three Regional-Scale Applications. Journal of Phys. Oceanogr., Vol. 33, Van Vledder, G.Ph., and D.P. Hurdle, 2002: Perforance of forulations for whitecapping in wave prediction odels. Proc. OMAE DISCUSSION AND CONCLUSION In the present paper an iproved version (with respect to Van Vledder and Hurdle, 2002) of the CSM for white-capping 4 Copyright 2004 by ASME

5 Figure 1: Frequency spectra (upper panel) and cuulative wave steepness (lower panel) for a uni-odal spectru and a bi-odal spectru. Figure 3: Noralized variation of white-capping source ter agnitude with the paraeter for a ean JONSWAP with a cos 2 (θ) directional distribution. Figure 2: Frequency spectra (upper panel), Koen whitecapping source ters (iddle panel) and CSM white-capping source ters (lower panel) for a uni-odal and a bi-odal spectru. Figure 4: Directional distribution of CSM white-capping for a ean JONSWAP spectru with a a cos 2 (θ) directional distribution for =0, 1 and Copyright 2004 by ASME

6 . Figure 5: Fetch-liited growth of significant wave height H 0 and ean wave period T 01 for a wind speed of 20 /s, coputed with the calibrated SWAN and paraetric growth curve of Kaha and Calkoen. Figure 8: As for Fig. 7, but now with CSM. Figure 6: Frequency spectra at fetches of 1, 5, 10, 15, 20 and 25 k for the situation described in legend of Figure 5. Figure 9: Frequency spectra at fetches of 1, 5, 10, 15, 20 and 25 k for the situation described in legend of Figure 7, with Koen ethod for white-capping. Figure 7: Fetch liited growth of significant wave height H 0 and ean wave period T 01 for a wind speed of 20 /s and an incoing swell with H 0 =0.5 T p =10 s. Figure 10: Frequency spectra at fetches of 1, 5, 10, 15, 20 and 25 k for the situation described in legend of Figure 7, with CSM for white-capping. 6 Copyright 2004 by ASME