3. The Offshore-to-Nearshore (OtN) Wave Transformation Technique

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2 backgrund database cmbined with user-friendly and sphisticated mdelling and analysis sftware tls, with the aim t make high quality wave infrmatin available t a larger grup f users than is the case tday and at lwer cst. In the absence f lng term wave data cllected at a site f interest, the calculatin f wave statistics at a castal area requires varius data sets t be assembled, including temprally and spatially lng term representative directinal wave data ffshre f the site, as well as bathymetric and castline data. Further, a suitable wave mdel is required, which is capable f mdelling the transfer f the ffshre cnditins t the site, incrprating the relevant shallw water wave phenmena. EUROWAVE simplifies the mdelling f wave cnditins in castal waters by integrating the fllwing under a single sftware package fr all Eurpean waters: High quality lng-term wave data ffshre f all Eurpean casts derived frm the integratin f wave mdel data frm peratinal and hindcast runs f the WAM mdel at the Eurpean Centre fr Medium- Range Weather Frecasts (ECMWF), tgether with high precisin satellite altimeter data and buy data frm 13 Eurpean cuntries. Eurpean-wide bathymetric data frm U Navy s Digital Bathymetric Data Base (DBDB-V) Eurpean-wide castline data frm the GMT High Reslutin Castline Database (GMT-HRCD). Tw shallw water wave mdels: WAN (imulating Waves Nearshre), a state-f-the-art third generatin wave mdel [1,], and VENICE, a traditinal backward ray-tracing mdel [5]. phisticated ffshre and nearshre wave statistics tlbxes. A gegraphic mdule allwing the user t easily zm in t the area f interest n gegraphic maps, displaying bathymetry and castline, tgether with tls t assist the user in setting up the wave mdel grid. The surce f EUROWAVE wave data, fr the Mediterranean ea, is ECMWF wave analysis data, cvering the 8-year perid n a 0.5 grid, based n the T13 meterlgical mdel up t 1998 and t T319 afterwards. The analysis results frm blending the WAM data and assimilatin f ER-1 and, subsequently, ER- altimeter wave data int the mdel, started frm August In EUROWAVE, the surce wave data have been further cmpared t satellite data frm different altimeters (TOPEX/POEIDON), and buy data (where available), fr validatin and calibratin purpses. The EUROWAVE satellite-calibrated data have als been used in the present study t describe the ffshre wave cnditins in the examined gegraphical area. Mre infrmatin abut the prject, including an nline dem, is available frm the prject Web ite ( Additinal details can be fund in the final reprt f EUROWAVE prject [6]. 3. The Offshre-t-Nearshre (OtN) Wave Transfrmatin Technique T illustrate the Offshre-t-Nearshre (OtN) sea-state transfrmatin technique, enabling us t derive nearshre wave infrmatin frm ffshre wave data, cnsider as a typical example the gegraphical area shwn in Fig. 1 (arniks bay near Athens). This area has a size f apprximately 1km 1km, and is partially bunded by land. The uthern and the Western sides are pen-sea bundaries. The depth almst all alng the Western side f the seaward bundary is mre than 70m, and ranges frm 70m t 0m (land) alng the uthern side. Offshre wave cnditins, represented either by the directinal spectrum ( ω, θ ) r by apprpriate spectral parameters (significant wave height, mean wave perid, mean wave directin), are cnsidered knwn all alng the seaward bundary (taken frm a glbal frecasting mdel). The ffshre wave cnditins represent the main frcing f the examined system, and are schematically presented by using a cntinuus distributin (arrws) alng the seaward bundary in Fig. 1. Given the ffshre bundary cnditins, the phaseaveraged mdel (WAN) is applied t calculate the wave cnditins inside the cmputatinal dmain D g, with a sufficient spatial reslutin. Based n this calculatin, the directinal wave spectrum ( ω, θ ) in the lcal nearshre area D l is determined; see Fig. 1. Having calculated the directinal spectrum at each pint f the lcal area, the crrespnding spectral mments can be easily btained by integratin. Varius ther physical quantities f interest, as the significant wave height and the wave pwer, are easily btained frm the spectral mments. The basic equatin used in WAN mdel is the radiative transfer equatin expressing actin balance [7,8] F N + cxn + cyn + cωn + cθn =, (1) t x y ω θ ω = dentes the wave actin =, expressed as a functin f ω =, where k is the wavenumber mdulus, g is the gravitatinal acceleratin h = h x,y is the water depth) and the wave in which N N ( ωθ, ;x,y,t) density N ( ω, θ;x,y,t ) / ω frequency ω ( kg tanh( kh) and ( ) directin θ. Actin density is parametrically dependent n the hrizntal (Cartesian) crdinates x and y, and n time t. In Eq. (1), c,c,c,c x y ω θ are the prpagatin velcity cmpnents in the physical and the Furierparameter spaces, respectively. Let it be nted that the abve frmulatin can take int accunt the interactin between waves and an ambient prespecified current (f

3 Nearshre area the pulse-based mdel f Hasselman, see, e.g., [7]. Fr bttm frictin the drag law mdel f [9] is used. The prcess f depth-induced wave breaking is mdeled with the dissipatin f a bre applied t the breaking waves in a randm field, using the spectral versin f the bre mdel [10]. As cncerns nn-linear interactins, the cmputatin f quadruplet wave-wave interactins is carried ut using the discrete interactin apprximatin (DIA) f [11]. Finally, fr mdeling the triad wave-wave interactins, the Lumped Triad Apprximatin (LTA) derived by [1] is used. The numerical integratin f the actin balance equatin (1) and () has been implemented in WAN by using finite difference schemes in all five dimensins (time, gegraphic space, and Furier space). In the gegraphical space the discrete scheme is based n a first rder upwind difference scheme. In the Furier space a mixed upwind/central difference scheme is used. Mre details abut the WAN mdel frmulatin, implementatin and validatin can be fund in the wrks [1,]. Figure 1. The OtN wave transfrmatin technique, using the spectral wave mdel WAN. hrizntal velcity U ), in which case the abslute frequency ω is replaced by the relative frequency σ = ω ku, where k is the wavenumber vectr ( k = k ). The left-hand side term f Eq. (1) represents respectively the lcal rate f change f actin density in time, the prpagatin f actin in x and y gegraphical directins, the frequency shifting due t depth variatin and currents, and the depth- and current-induced refractin. Thus, the abve frmulatin uses linear wave thery fr wave prpagatin and des accunt fr diffractin effects. The frcing term F = F ( ωθ, ;x,y,t) is the energy surce term. This term is decmpsed t a number f separate surce terms, as fllws: ( ω θ) ( ω θ) ( ω θ) ( ω θ) F, = F, + F, + F,, () in ds nl where F in (, ) wave energy by wind, F (, ) ω θ represents the effects f generatin f ds ω θ the dissipatin f wave energy due t whitecapping (deep-water wave breaking), wave-bttm interactins and depth-induced wave F ω, θ represents breaking in very shallw water, and ( ) the wave energy transfer due t cnservative nnlinear fur- and three-wave interactins. Wind generatin is described in WAN by the sum f linear [5] and expnential grwth terms [7]. The dissipatin term is cmpsed by the cntributins f whitecapping, bttm frictin and depth-induced breaking. The whitecapping dissipatin is mdeled using nl 4. Definitin f Wave Energy Ptential Using the definitin f the directinal wave spectrum ( ω, θ ) r ( f,θ ), where f = ω / π is the (linear) frequency, we are able t express the stchastic (shrttime) average value f the wave pwer flux P thrugh a W vertical sectin (Σ), at a specific angle f rientatin (α), extending ver the whle water clumn, and having ne meter f hrizntal width. T fix ideas let us cnsider the wave system characterised by the spectrum ( f,θ ), and the wave pwer flux P W thrugh a vertical sectin f ne meter width, at an angle f rientatin a; see Fig.. In this case [13], under the assumptin f deep water, the average wave pwer flux per meter f sectin width is defined as fllws + θ= θ ρ g 1 PW = f, cs csa+ sin sina dfd f ( θ )( θ θ ) θ, (3) f = 0 θ= θ where θ Θ π ± = ±, and g=9.81 m/sec. In the case when the directinal spreading f the spectrum is weak, and the spectrum can be well apprximated by a factrizatin f the frm ( θ ) = ( ) ( θ Θ) f, f D, (4) where ( f ) dentes the ne-dimensinal frequency spectrum, then Eq. (3) reduces t the fllwing simpler frm W ( 64 ) ( ) P = ρg π G Θ;a H T. (5) 10

4 ( f, θ ) Θ Σ 5. A Case tudy: Nearshre Wave Ptential in the Area f uth-eastern Part f Evia Island, in the Central Aegean ea 1m α mean wave directin Θ Figure. Definitin f wave pwer flux per meter f hrizntal width thrugh a vertical sectin Σ, in cnjunctin with a directinal wave spectrum ( f,θ ). In Eq. (5) the term G ( Θ ;a) is defined as fllws + θ= θ ( ) = ( )( + ) G Θ ;a D θ Θ csθ csa sinθ sina dθ, (6) θ= θ and the significant wave height H (in m) and the mean energy perid T 10 (in sec) are defined thrugh the m,m 0 1 mments f the frequency spectrum ( f;h s,t) by the usual expressins, 0 f = 0 ( ) H = 16m = 16 f df, m 16 1 T ( f ) df. f 1 10 = = m0 H f = 0 If all wave pwer is assumed t be cncentrated in the mean wave directin, i.e. D ( θ Θ) δ ( θ Θ), and the sectin is aligned t the lcal wavefrnt, Θ = α, then the functin G ( Θ ;a) is maximized, taking the value G ( Θ ;a) =1. This prvides us with a maximum value f the wave pwer flux per meter f wavefrnt, which using standard values fr the density f seawater reduces t the frmula (7) The examined gegraphical area is the nearshre area f uth-eastern part f Evia Island, in the Central Aegean ea, which is enclsed by the bx 4 10 E, 4 40 E N, N. This area is shwn in the map f Fig. 3 (frm EUROWAVE tlbx), tgether with the nearest ffshre data pints O1, O and O3 (shwn by bullets). In this area, 5 representative nearshre target pints have been selected, shwn by using white circles in Fig. 3, in rder t cver the whle nearshre regin in the examined gegraphical area. The detailed analysis, presented in [14], has shwn that it is the nearshre area arund TP5 (Nrth side f Cape Kafireas), where the available wave ptential is maximized. Thus, we shall fcus the presentatin f ur results n the latter nearshre/castal area. 6. Wave Data Offshre the Examined Area The ffshre wave parameters and the wave ptential PW, 10, calculated using (8) at the Offshre pint O, as representative f the ffshre wave climate and ptential in the examined area, are presented in Fig. 4. Wave data are available in the frm f time-series cvering the 8- year perid , regularly sampled at 6-hur intervals. The ttal sample size f the ffshre data is The annual wave statistics cncerning the distributins f the significant wave height H, the mean wave directin Θ, the exceedance prbability f the wave pwer flux PW, 10, and the mean mnthly values f P are shwn in Fig. 4. W, 10 P 05. H T, in kw/m. (8) W, The abve frmula is valid in deep water. In the case f intermediate water depth the maximum value f the wave pwer flux is less (and the calculatin is mre tedius). ince shallw water effects lead t a reductin f the value f maximum wave pwer flux, we have based ur results n Eq. (8), as an indicatin f the maximum available wave ptential (wave pwer flux per meter f wavelength) in the nearshre area. Figure 3. Map f uth-eastern part f Evia Island in Central Aegean ea, and dmain f wave calculatin

5 Figure 4. Annual distributin f wave parameters and wave ptential at the ffshre pint O (available frm EUROWAVE database) Figure 5. Annual distributin f wave parameters and wave ptential at the nearshre area TP5 (calculated)

6 7. Wave Data at the Nearshre Target Pints The EUROWAVE methdlgy fr btaining nearshre wave infrmatin at specific target pints, frm the available ffshre wave data has been applied. The kernel part f this methdlgy is the frmulatin f an effective Offshre-t-Nearshre (OtN) wave-parameter transfrmatin. The latter transfrmatin is nn-linear and, thus, it is numerically perfrmed. It is decmpsed in the fllwing main parts: First, a representative time series f ffshre spectra is set up frm the wave parameters at the ffshre pints cvering the seaward bundary f the gegraphical area surrunding the target pints. Then, the OtN wavespectrum (r sea-state) transfrmatin is perfrmed by using WAN, fr each ne f the ffshre spectra, which essentially specify the bundary cnditins f the examined gegraphical area. Finally, the btained time series f nearshre spectra have been analysed, and the varius spectral mments have been calculated, btaining thus the time-series f nearshre wave parameters, including the wave ptential, at all target pints. The main results f the present study fr the nearshre target area TP5, cncerning the distributins f the same (as in the ffshre case) parameters are presented in Fig Cnclusins In the present wrk, the wave climate and wave ptential in the nearshre regin f the uth-eastern part f Evia Island, has been studied, using the EUROWAVE methdlgy. The verall mean value f the maximum available wave ptential in this castal area has been calculated t be PW, 10 ( TP 5) = 38. kw / m, which significantly differs frm the crrespnding ffshre value ( PW, 10 ( O ) = 7. kw / m). Hwever, this result may be prved satisfactry fr explitatin purpses. In the same nearshre/castal area, the mnths during the winter perid (Octber t March) present the highest values f maximum available ffshre wave pwer (f the rder f 6 kw/m). During the spring mnths (April, May, June) the ffshre wave ptential presents the lwest values (f the rder f kw/m). Acknwledgement This study has been perfrmed at the hip and Marine Hydrdynamics ectin f the Department f Naval Architecture and Marine Engineering f the Natinal Technical University f Athens (NTUA), and has been partially supprted by f the Center f Renewable Energy urces (CRE), Greece. References [1] N. Bij, R.C. Ris, L.H. Hlthuijsen, A thirdgeneratin wave mdel fr castal regins. 1. Mdel descriptin and validatin, J. Gephys. Res., 104(C4), 1999, [] R.C. Ris, L.H. Hlthuijsen, N. Bij, A thirdgeneratin wave mdel fr castal regins.. Verificatin, J. Gephys. Res., 104(C4), 1999, [3] Y. Gda, Randm eas and Design f Maritime tructures (ingapre: Wrld cientific, 000). [4] G.A. Athanassulis,.F. Barstw, L. Cavaleri, Eurwaves, an easy apprach fr lcal wave climatlgy, Prc. Castal Engineering 99, Lemns, Greece, 1999 (in T. Graczyk, T. Jastrzebski, C.A. Brebbia, (Eds.), Marine Technlgy III, WIT Press, 000, ). [5] L. Cavaleri, P. Malantte-Rizzzli, Wind wave predictin in shallw water: Thery and applicatins, J. Gephys. Res., 86(11), 1981, [6].F. Barstw, G.A. Athanassulis, L. Cavaleri et al., EUROWAVE: A user friendly tl fr the evaluatin f wave cnditins at any Eurpean lcatin (Final Technical Reprt f EUROWAVE Prject MA3-CT , 001). [7] G.J. Kmen, L. Cavaleri, M. Dnelan, K. Hasselman,. Hasselman, P.A.E.M. Janssen, Dynamics and Mdelling f Ocean Waves (Cambridge: Cambridge University Press, 1994). [8]. Massel, Ocean urface Waves: Their Physics and Predictin (ingapre: Wrld cientific, 1996). [9] J.I. Cllins, Predictin f shallw water spectra. J. Gephys. Res., 77(15), 197, [10] Y. Eldebersky, J.A. Battjes, Parameterizatin f triad interactins in wave energy mdels. Prc. Castal Dynamics Cnference 95, Gdansk, Pland, [11]. Hasselmann, K. Hasselmann, J.H. Allender, T.P. Barnett, Cmputatins and parametrizatins f the linear energy transfer in a gravity wave spectrum, II, Parametrizatin f the nnlinear transfer fr applicatin in wave mdels. J. Phys. Oceangr., 15, 1985, [1] Y. Eldeberky, Nnlinear transfrmatin f wave spectra in the nearshre zne, (Delft: Ph.D. thesis, Delft University f Technlgy, 1996). [13] G.A. Athanassulis, Predictin f design wave spectra and wave energy ptential in Greek eas (NTUA, Athens: Final Reprt f Research Prject funded by the Greek General ecretariat f Research & Technlgy, 1990). [14] G.A. Athanassulis, K.A. Belibassakis, Ch.N. tefanaks, tudy f the wave climate and wave ptential in the nearshre area f uth Eastern Evia Island, enclsed by the bx [4 ο 10 Ε, 4 ο 40 Ε] x [38 ο 05 Ν, 38 ο 30 Ν] (NTUA, Athens: Final Reprt f Research Prject funded by the Greek Centre f Renewable Energy urces, 00).