WIND INPUT FUNCTION. Ray-Qing Lin 1 and Lihwa Lin 2 1

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1 WIND INPUT FUNCTION Ray-Qing Lin 1 and Lihwa Lin 2 1 David Taylo Model Basin, Division of Seakeeping, 9500 MaAth Blvd, West Bethesda, NSWCCD, MD U.S. Amy Enginee Reseah and Development Cente 3909 Halls Fey Road, Viksbg, MS INTRODUCTION Wind is one of majo soes fo geneating wate sfae gavity waves. To pedit waves, an aate inpt fntion mst be fist developed. The stdy of the mehanis of -wave geneation stated moe than half enty ago (Phillips, 1957; Miles, 1957; Cavalei and Malanotte-Rizzoli, 1981). Howeve, the wave pedition based on lassial theoies tends to ndeestimate the wave enegy gowth (Hasselmann et al. 1973). In the advaned oean wave model WAM (WAMDI gop, 1988) and a eently developed oastal wave model (Lin and Hang, 1996a, 1996b; Lin and Peie, 1997a, 1997b, 1999), the inpt fntion is fomlated based pon the state-of-the-at tehnology. These eent models ae geneally eliable in the pedition of wave height in the deep wate bt not always aate fo wave dietion and in oastal egions. It appeas that the oean wave geneation by s is still not flly ndestood. Fo instane, oeanogaphi sientists have notied that geneated waves do not always popagate along the down dietion and, sometimes, the angle between and waves an be vey lage (Wang et al. 2000). This angle dfeene between and wave popagation dietions an have a signiant impat to the sfae wave dynamis, espeially in the oastal egion whee the wave popagation is moe dependent on the bottom topogaphy. In the pesent stdy, a new fomlation of the inpt fntion at the oean model sale is poposed and investigated based on the physis and wave data olleted in the intemediate depth wate. In WAM, the physial paametes applied in the inpt fntion have inlded wave feqeny,, wave popagation dietion, θ, speed,, dietion, θ, and wave enegy density, E (, θ ). A linea elationship is assmed among the speed, wave popagation angle, and wave feqeny (WAMDI gop, 1988). Howeve, this linea elationship is not always the ase bease the wave geneation is a vey omplex physial poess. The impovement of the inpt fntion in wave pedition models shold be based on the -foing wave poess to detemine the elationship among basi physial paametes. Fthemoe, two moe physial paametes shold be also onsideed: wave phase veloity,, and wave age. The se of the wave phase veloity is in the light that the wave gowth shall be a fntion of the dfeene of speed and wave phase veloity,, instead of the speed alone. The effet of wave age is ndestood fom the fat that the low feqeny wave enegy density in a naow dietion band is dfilt to gow while a yong wave spetm in the high feqeny ange and wide dietion band gows moe easily. In the pesent stdy, the wave age is epesented by new inpt fomla is poposed to be the fntion of the wave age along with the othes. Finally, the mehanis abot the angle between and wave fo newly geneated waves mst be onsideed (Phillips, 1957). The inpt fntion is also investigated fo the ase of a lage angle between and wave popagation dietions and possible ases assoiated with the ondition besides the Phillips mehanis. To aately pedit the wave geneation, it is essential to onside the nonlinea wave-wave inteation, wave beaking, and bottom fition, espeially in the shallow wate. These effets wee disssed in detail in sepaated papes (Lin and Peie, 1977a and b; Lin and Lin, 2004). To illstate the new inpt fomla, a seies of dietional wave enegy speta meased offshoe of Cape Fea, NC, available fom 2000 to 2003 wee seleted fo the following thee ases: (1) modeate, with steady speed and dietion, blowing seawad fo the new wave geneation, (2) modeate, with and a

2 steady speed and dietion, blowing in the same dietion as the old wave popagation, and (3) pesistent stong with a onstant angle to the mean wave dietion. 2. FIELD DATA Oean waves sally appea in iegla foms and popagate in mltiple dietions. To stdy the physial phenomena of oean waves, it is meaningfl to se data olleted in the field. In the pesent stdy, and wave measements offshoe of Cape Fea, NC, wee seleted fo the investigation of inpt fntion. Wave data wee olleted as a seies of dietional wave enegy speta fom five wave gages maintained by the Wilmington Habo Monitoing Pogam ( The wave data olletion was ondted in a thee-yea peiod fom Septembe 2000 to Jne Fige 1 shows the loation of the five wave gages elative to Cape Fea. Table 1 pesents the loation, wate depth, and data olletion peiods of the five wave gages. Offshoe sea sfae data ae available fom two meteoologial stations maintained by the National Data Boy Cente ( Fige 2 shows the loation map of the two meteoologial stations and Table 2 pesents the loation, wate depth, and data olletion peiod of the two stations. The two meteoologial stations also meased wave enegy speta in Howeve, bease these wave speta meased fom the two meteoologial stations do not ontain the wave dietion infomation, they wee not sed in the inpt fntion investigation. Fige 1. Loation map of dietional wave gages

3 Fige 2. Loation of two meteoologial stations Boy (sqae) and Platfom FPSN7 (tiangle); also shown ae dietional wave gages (geen iles) and wate depth ontos in metes Table 1 Dietional Wave Gage Infomation Station Coodinates Nominal depth (m) Data olletion peiod Oak Island 33 o N, 78 o W 7 Septembe 2000 May 2003 Bald Head Island 33 o N, 78 o W 5.8 Septembe 2000 Jne 2003 Mond Cest 33 o N, 78 o W 7 Jly 2001 Jne 2002 Mond Offshoe 33 o N, 78 o W 12.8 Agst 2001 Jly 2002 Mile o N, 78 o W 12.8 Septembe 2000 Jne 2003 Table 2 Meteoologial Station Infomation Station Coodinates Nominal depth (m) Data olletion peiod FPSN7 33 o N, 77 o W 14 Novembe 1984 pesent Boy o N, 79 o W 38 Jne 1978 pesent

4 3. WIND WAVE GENERATION The inpt fntion, denoted as S in, is a soe tem in the wave ation onsevation eqation in eently developed New Coastal Wave Model (Lin and Hang, 1996a, 1996b; Lin and Peie, 1997a, 1997b, 1999) as A [( + t gx + ) A] [( + x + v) A] [ g θ A] [ g A] + + = S y θ gy in + S dp + S nl, (1) whee A is the wave ation density, defined as the atio of spetal enegy density to intinsi feqeny, ; gx, gy, gθ, and g ae gop veloities elative to x, y (easten and nothen axes of the wate sfae), θ (angle between noth and eah dietion) and, espetively; and v ae the ent veloity omponents in the x and y dietions, espetively. S ds is a sink tem fo the enegy dissipation and S nl is a fntional tem fo nonlinea wave-wave inteations. The nmeial sheme fo solving the left hand side eqation was desied in Lin and Hang (1996a, 1996b). The theoy and method of allating S nl wee desibed in Lin and Peie (1997a, 1997b, 1999). The impotane and a genei fomlation of S ds will be disssed in a sepaate pape by athos (Lin and Lin, 2004). The pesent pape foses on the investigation and fomlation of S. in 3.1 Phillips s Mehanis As Hasselmann et al. (1973) pointed ot that the wave geneation pedited by Phillips s (1957) and Miles s mehanis (1957) is geneally one ode of magnitde smalle than the obseved infomation. Howeve, Phillips s mehanis pesents an inteesting ondition that waves do not popagate in the down dietion the wave phase veloity is smalle than the speed. The angle between and wave popagation dietions is detemined fom os 1 ϕ = 0,, < whee = k (3) is the wave phase veloity and k is the wave nmbe. If the wave phase veloity is eqal to o geate than the speed, aoding to Phillips s mehanis, new waves will popagate in the down dietion, bt the wave phase veloity is smalle than speed, then thee is a angle, ϕ, between the wave popagation and dietions. As an example, Fige 3 shows dietional wave enegy speta obseved fom Mile 11, Mond Offshoe, Mond Cest, and Bald Head Island between 09:38 and 11:35 GMT in Novembe 29 th, 2001, nde a athe weak sotheastely ( speed is 3m/se and dietion is fom 140 o, elative to the noth). The nit of the dietional wave enegy density is shown in m 2 se/adian. These meased wave speta lealy show that the wave gowth is in the down dietion. In the ase of a stong ove the sea sfae, the wave phase veloity is smalle than the speed, waves an gow at an obliqe angle to the dietion. To illstate the ase, Fige 4 displays two dietional speta meased fom Mond Cest and Mile 11 at 20:15 and 20:35 GMT, espetively, in Novembe 16 th, 2001, nde a stong notheastely ( speed is 13m/se and dietion is 20 o ). (2)

5 These two speta show that longe waves with a pimay peak at abot 0.11 Hz have popagated towad the oastline (shoe nomal at 180 o ) as eslts of efation. Fo the pimay peak of the spetm fom Mond Cest, = 10.6 m/se and ϕ = 35.1 o an be obtained fom Eqations (3) and (2), espetively, and the pedited wave popagation dietion is 55.1 o. Fo the pimay peak of the spetm fom Mile 11, =12.2 m/se andϕ =20.2 o, and the pedited wave popagation dietion is 40.2 o. In these two speta, the wave gowth appeas in the highe feqeny ange as a seonday peak at 0.2 Hz w and 120 o. Fo this seonday peak at 0.2 Hz, = 7.7 m/se and ϕ = 53.1 o wee detemined fom Eqations (3) and (2), espetively, fo both Mond Cest and Mile 11, and the pedited wave popagation dietion is 73.1 o. Spisingly, obseved wave dietions vay moe fom the dietion than pedited by Phillips. Thee mst be othe mehanis involved, whih ase the wave popagation away fom the down dietion. Fige 3. Meased dietional wave speta fom (a) Bald Head Island, (b) Mond Cest, () Mond Offshoe, and (d) Mile 11 between 09:38 and 11:35 GMT, Novembe 29 th, 2001 ( speed is 3m/se and dietion is fom 140 o, shown as dash lines)

6 Fige 4. Meased dietional wave speta fom (a) Mond Cest at 20:15 GMT and (b) Mile 11 at 20:35 GMT, Novembe 16 th, 2001 ( speed is 13m/se and dietion is 20 o, shown as dash lines) 3.2 The Nonlinea Effet of Wind Inpt to Wave Geneation Oeanogaphes and oean eseah sientists have eently notied that geneated waves indeed may not popagate in the down dietion and, sometimes, the angle of the dfeene between and wave dietions an be geate than 60 o (Wang, et al. 2000). This is patilaly of inteest fo the wave geneation, espeially in the intemediate depth wate. What ae the physis, besides Phillips s mehanis, that an ase the -wave popagation to divet fom the dietion? It is likely de to the nonlinea wave-wave inteation that is epesented by the fntional tem S nl in Eqation (1). Fist, we examined the five-wave inteation mehanis bease many sientists have sggested, as have Lin and Peie (1997b), the mehanis plays an impotant ole in the spetal wave enegy balane poess. To investigate this, it is neessay to examine the elationship among the speed, spetal peak feqeny, and the angle between and wave popagation dietions. Fo flly developed sea states in a finite depth, h, the peak feqeny,, an be appoximated by (Gabe and Madsen, 1988) paek peak = g tanh k os( θ peak h θ wave ) whee k paek is the wave nmbe oesponding to the peak feqeny. Fige 5 shows an example of this elationship fo the demonstation: (1) the peak feqeny of geneated waves ineases as the speed deeases, and (2) the peak feqeny of geneated waves ineases as the angle between and wave popagation dietions ineases. If the mehanis is de to five-wave inteations, the dominated mehanis mst tansfe the enegy fom lowe to highe feqenies. At Mile 11, whee h=12.8 m, and the peak feqeny is 0.2 Hz, so kh = 2.1. This ondition oesponds to the intemediate wate egime and, as a eslt, the fo-wave inteations shold dominate. If the peak feqeny is 0.11 Hz, then kh=0.8. It also oesponds to the intemediate wate egime that fo-wave inteations dominate. Fthemoe, Figes 4a and 4b show that wave enegy densities in both lowe and highe feqeny tails ae not symmetial along the dietion. Theefoe, five-wave inteations annot be the majo mehanis. In this ase, the fo wave inteations shold dominate and, as a onseqene, most wave enegy mst tansfe fom highe to lowe feqenies along the down dietion.

7 Fige 5. Relation of peak feqeny, speed, and angle between and wave popagation dietions What is the mehanis esponsible fo waves to popagate away fom the down dietion besides the Phillips mehanis? Based on the nonlinea mehanis by Lin and Peie (1997), the long wave inteating with loal waves is the dominant mehanis in the oastal egion. The long wave an be a swell, an edge wave, o a bottom topogaphy wave. In Fige 4, the long wave is a swell fom the open oean that has absobed the enegy of loal waves by fo-wave inteations. Theefoe, the swell gained wave enegy fom loal waves bt it etained its oiginal popagation dietion. To fthe demonstate the ase, the and wave infomation olleted fom Mond Offshoe, Bald Head Island, and Oak Island wee sed in the investigation. Fige 6a displays the dietional wave spetm olleted fom Mond Offshoe at 19:00 GMT in Novembe 16 th, 2001, ding a stong notheastely ( speed is 13 m/se and dietion is 20 o ). The dietional spetm shows a swell with the spetal peak at 0.12 Hz and popagation dietion fom the SSE (160 o ). Fige 6b displays the dietional wave spetm olleted at 19:38 GMT of the same day fom Bald Head Island. It indiates that the swell is assoiated with the peak at 0.12 Hz and popagation dietion is moe fom the soth (170 o ). Fige 6 displays the dietional spetm at 19:55 GMT of the same day fom Oak Island. This spetm also shows the swell is moe fom the soth (170 o ). It is evident in Fige 6a, 6b, and 6 that the swell gadally tns its dietion fom the SSE towad the noth fom the open oean to oastal egion as a eslt of wave efation. The swell does not follow the loal dietion. On the othe hand, the spetal tail o highe feqeny enegy densities have expeiened less wave efation and ae affeted moe by the loal. Bease mh geate enegy ontent is assoiated with the swell than in the spetal tail, as shown in Fige 6, the swell mst have absobed the enegy of highe feqeny waves thogh the fo-wave inteations (Lin and Peie, 1997b).

8 Fige 6. Meased dietional wave speta fom (a) Mond Offshoe at 19:00 GMT, (b) Bald Head Island at 19:38 GMT, and () Oak Island at 19:55, Novembe 16 th, 2001 ( is 13m/se fom 20 o, as dash lines) 4. The NEW WIND INPUT FUNCTION Setion 3 demonstates that geneated waves an be stongly inflened by an initial spetm thogh nonlinea fo-wave inteations, tansfeing the enegy fom highe to lowe feqenies. In the ases whee the loal dietion is dfeent fom a swell o longe waves in intemediate depth wate, the gowth of waves appeas in the swell absobing the enegy fom highe feqeny waves. Althogh the wave geneation has been stdied moe than half enty, it is still not flly ndestood. Existing inpt fntions applied in eent wave models ae still based on eithe empiial o semi-empiial fomlas (Hasselmann et al. 1973; WAMDI gop, 1988). The lak of sffiient and dietional wave data in the past has limited the investigation of the inpt fntion. In the pesent stdy, the fomlation of the inpt fntion is detemined by how its physial poess was intepeted by the athos and is expessed as

9 S in whee a1 * = F1( ) F2 ( ) EPM ( ) Φ( θ ) + g a2 F1( ) 2 ) g ( F F F1( 2 F ( ) = 0, osϕ, ) = 1, os( θ θ ) (, θ ), 3 ( ) v v v v < < E (, θ ), (4),, F ( log ) = 0, [( 10 3 ) 1 ], v v <, a g * 1 0 and E PM ) = exp( 0.74 ). * 2 4 ( 5 4 g / Hee, E PM ( ) is the fntional fom of Pieson-Moskowitz spetm, is the Phillips s onstant, and 8 4 Φ ( θ ) = os ( θ θ ) fo θ θ < π / 2. 3π is a nomalized dietional speading. Fntion F 1 is de to the stess effet, F 2 is de to Phillips mehanis with ϕ being defined in Eqation (2), and F 3 is de to the wave age effet. Fo long waves (old waves), the phase veloity is geneally lage and F 3 < 1. In the ondition that 0 =, then, F 3 =0. Fo shot waves (yong waves), the phase veloity is geneally small and F 3 > 1. A least sqaes otine is sed to estimate oeffiients a 1 and a 2 in the following two steps: Step 1: Compte = N 2 ( S data Sin ) i i= 1, (5) whee S data is the wave enegy fom the obsevational data, and sbsipt i is the index of feqeny and dietion bands. Step 2: Solve = 0, and = 0. (6) a a 1 2 By sing the data olleted in Mile 11 and Mond Offshoe in the intemediate depth wate, and theefoe negleting the dissipation fom bottom fition, in the ase of new wave geneation nde modeate seawad s, and eliminating stong wave beaking, a 1 = and a 2 = wee obtained fom Eqations (5) and (6). Fige 7, as an example of applying Eqation (4) fo a depth of 18 m, shows the wave enegy gowth in the feqeny spetm fo the fist 5 hos nde the steady of 5 m/se ( dietion is 90 o ) ove a alm sea. Fige 8 shows the dietional wave spetm at the end of the 5-ho simlation. The eslts in both Figes 7 and 8 do not onside nonlinea wave-wave inteations, wave beaking, and bottom fition effets. Theefoe, highe feqeny waves at the end of the 5-ho simlation shold aleady beak and longe waves shold appea as a eslt of wave-wave inteations tansfeing wave enegies fom high to low feqenies.

10 Fige 7. Wave enegy gowth in the feqeny spetm fo the fist 5 hos nde the steady ondition ( speed is 5 m/se, fom 90 o ) ove a alm sea in wate depth of 18 m Fige 8. Dietional wave spetm (in the nits of m 2 se/adian) fom a 5-ho simlation nde the steady ondition ( speed is 5 m/se and dietion fom 90 o ) ove a alm sea in wate depth of 18 m

11 5. SUMMARY The new inpt soe in the New Coastal Wave Model (Lin and Hang, 1996a and b; Lin and Peie, 1997a and b, 1999) is fomlated as a fntion of the dfeene between speed and wave phase veloity, wave age, Phillips mehanis, as well as those taditional paametes sed in WAM, sh as wave feqeny,, wave popagation dietion, θ, dietion, θ, and wave enegy density, E (, θ ). To aately estimate the inpt fntion, one mst onside the dfeene between speed and wave phase veloity, instead of speed alone. It is also essential to inlde the wave age effet bease long waves sh as a swell ae moe dfilt to gow while yong waves an gow easie. In addition, to inlde the Phillips mehanis and the effet of nonlinea wave-wave inteations (Lin and Peie, 1997a and b), the new inpt is able to pedit the wave popagation dietion divegent fom the dietion. It has been notied eently that the wave gowth in the oean is not always in the same dietion as the, and sometimes the dfeene of dietion between the two an be lage. The divesion of the wave popagation fom the dietion may signiantly impat the wave geneation. In the ase of a stong ove yong waves with the wave phase veloity smalle than the speed, the oesponding wave popagation dietion will be dfeent fom the dietion, espeially in intemediate depth wate. Phillips sggested (1957) that the angle between and wave popagation dietions is os 1. In the ase of a mild whee the speed is less than the wave phase veloity of the spetm, the wave popagation will be in the down dietion. Besides the Phillips mehanis, the wave popagation dietion is also affeted by the initial spetm and a new nonlinea fo-wave inteation mehanis sggested by Lin and Peie (1997b). Fo instane, when a swell popagates fom the open oean to oastal egions, it an absob the enegy fom loal waves thogh nonlinea fo-wave inteations and the mean wave popagation will be stongly assoiated with the oiginal swell dietion. Based on the ndestanding of physial poesses of the wave gowth at sea, a new inpt fntion fomla is expessed in Eqation (4), whih is a fntion of speed, wave phase veloity, wave age, and othe basi paametes (,θ, θ, and E (, θ ) ) with empiial onstants a 1 = and a 2 = being estimated fom a least sqae method based on the obsevational data. To moe aately estimate the wave, one mst ople the nonlinea wave-wave inteation tem (Lin and Peie, 1997a, 1999) and wave beaking fntion (Lin and Lin, 2004). Aknowledgements The athos wold like to thank M. Mihael Davis fo poviding many sefl omments and sggestions to this stdy. We also thank D. John Bakyomb fo sppoting the nmeial modeling pojet nde the Offie of Naval Reseah In Laboatoy Independent Reseah Pogam. Finally, we wold like to thank M. Cal Mille fo poviding the field expeiment data sets and M. Clf Baon assisting in the management of the data. The U.S. Amy Cops of Enginees (USACE) Soth Atlanti Division s Wilmington Distit maintains the Coastal Monitoing Pogam. Refeenes Cavalei, L. and Malanotte-Rizzoli, P Wind Wave Pedition in Shallow Wate: Theoy and Appliations. J. Geophys. Res., 86, No. C11, 10,961-10,973. Gabe, H.C. and Madsen, O.S A Finite-Depth Wind-Wave Model. Pat I: Model Desiption, J. Phys. Oeanog., 18, Hasselmann, K., Banett, T.P., Bows, E., Calson, H., Catwight, D.E., Enke, K., Ewing, J.A., Gienapp, H., Hasselmann, D.E., Ksemann, P., Meebg, A., Mlle, P., Olbes, D.J., Rihte, K., Sell, W. and Walden, H., Measements of Wind-Wave Gowth and Swell Deay ding the Joint Noth Sea Wave Pojet (JONSWAP), Dtsh. Hydog. Z. Sppl., 12, A8.

12 Lin, R.-Q. and Hang, N.E. 1996a. The Goddad Wave Model, Pat I. The Nmeis, J. Phys. Oeanog., 26, Lin, R.-Q. and Hang, N.E. 1996b. The Goddad Wave Model, Pat II. Kinematis, J. Phys. Oeanog., 26, Lin, L. and Lin, R.-Q Wave Beaking Fntion. 8 th Intenational Wokshop on Wave Hindasting and Foeasting, Noth Shoe, Oah, Hawaii (Aepted). Lin, R.-Q. and Peie, W. 1997a. A New Coast Wave Model Pat III. Nonlinea Wave-wave Inteations, J. Phys. Oeanog., 27, Lin, R.-Q. and Peie, W. 1997b. A New Coastal Wave Model Pat V. Five-wave Inteations, J. Phys. Oeanog., 28, Lin, R.-Q. and Peie, W Wave-wave Inteations in Finite Wate Depth (A New Coastal Wave Model, Pat IV), J. Geophys. Res., 104, No. C5, Miles, J. W., On the Geneation of Sfae Waves by Shea Flows. J. Flid Meh. 3, Phillips, O. M., On the Geneation of Waves by Tblent Wind. J. Flid Meh. 2, Pat 5, WAMDI gop, The WAM Model - A Thid Geneation Oean Wave Pedition Model. J. Phys. Oeanog., 18, Wang, W.D., Hwang, P.A., Kaihat, J.M., and Roges, W.E Bimodal Dietional Distibtion of Oean Waves in Mixed Seas. The 6 th Intenational Wokshop On Wave Hindasting and Foeasting, Monteey, Calonia