14 th Australasian Fluid Mechanics Conference Adelaide University, Adelaide, Australia 10-14 December 001 Wind wave evolution in finite deth water I.R. Young 1 and A.V. Babanin 1 Faculty of Engineering, Comuter & Mathematical Sciences Adelaide University, Adelaide, South Australia, 5005 AUSTRALIA Deartment of Civil & Environmental Engineering Adelaide University, Adelaide, South Australia, 5005 AUSTRALIA Abstract This aer resents recent research aimed at develoing a detailed redictive caability for wind generated ocean waves in shallow coastal waters. The detailed evolution of the directional sectrum, and measurements of the source terms resonsible for this evolution are resented. Introduction Wind generated ocean waves are imortant for a number of reasons. From an engineering viewoint, they reresent the major loading on coastal and offshore structures. From the environmental context, they have a governing role in the stability of coastlines and the flooding of near-coastal regions during storm condition. Wind generated waves also largely govern the aerodynamic roughness of the ocean surface and hence lay a major role in determining the fluxes of heat, energy, momentum and ossibly carbon dioxide across the ocean surface. Thus accurate rediction of ocean wave conditions may also be a key variable in redicting long-term climate. The study of gravity waves has been a fertile area of fluid mechanics for many centuries. Our abilities to redict the generation and evolution of ocean waves on a global or regional scale only started to develo in the 1950s (Gelci et al, 1956 [4]). The sohistication of such models has increased as both comuting ower grew and our understanding of the, often comlex, nonlinear rocesses governing wind-wave evolution increased. This develoment has culminated in the introduction of 3 rd Generation sectral models, caable of redicting wave fields on either a global of regional scale (WAMDI, 1988 [10]). Desite the significant develoments which have occurred in this area, the resulting models have largely been confined to areas of dee water, where the influences of finite deth can be ignored. Although dee-water regions include the vast majority of the world s oceans, it is in the relatively small shallow coastal regions where the bulk of human activity is concentrated. Until recently, both the form of wind generated waves in shallow coastal regions and the mechanisms resonsible for their evolution have been oorly understood. This aer describes a number of integrated field research rojects aimed at addressing this lack of knowledge. Governing Equations For the simlifying case of a constant water deth, the evolution of the directional wave sectrum can be described by the energy balance equation F + C g F = S tot (1) t where F is the directional wave sectrum, t is time, C g is the rocesses which transfer energy to, from or between comonents of the wave sectrum F. To first order, S tot can be reresented as the sum of a number of searate rocesses S tot = S in + S wc + S bf + S 4nl + S 3nl () where each of the terms reresents a secific hysical rocess S in inut from the wind S wc dissiation due to white-ca breaking at the surface S bf dissiation due to friction with the bottom S 4 nl 4-wave resonant non-linear interactions between comonents in the sectrum S 3 nl 3-wave or triad interactions between comonents of the sectrum Exerimental Descrition Little detailed knowledge was available on either the form of the wave sectrum, F or of the individual source terms in () for finite deth conditions. As a result, a detailed set of field exeriments were conducted at Lake George, near Canberra, Australia. Lake George is aroximately 0 km in length by 10 km wide. It has a remarkably flat bottom, with a deth that varies seasonally. Two major sets of exeriments were conducted at Lake George. The first of these involved measuring the evolution of the wave sectrum as a function of fetch (distance from shore), under conditions were the wind was constant and blew aroximately erendicular to the shoreline (often termed Fetch Limited Conditions). Effectively, such data yields the evolution of the sectrum, as shown on the left side of (1). The second set of exeriments involved detailed measurements of the individual source terms in (), that is, the right side of (1). Analysis of the data should close the balance in (1). Non-dimensional growth curves Dimensional analysis indicates that the evolution of the total wave energy and of a reresentative wave frequency can be exressed in terms of a small number of non-dimensional arameters. Based on the very extensive Lake George data set, Young and Verhagen (1996a) [13] have shown that the nondimensional energy, ε and non-dimensional eak frequency,ν can be exressed in terms of the non-dimensional fetch, χ and non-dimensional water deth, δ. Examles of these relationshis are shown in Figures 1 and. grou velocity and S tot is a source term which reresents all 79
In these relationshis the non-dimensional variables are defined 4 as ε = g E / U10, fu10 / ν = g and δ = gd / U10, where g is gravitational acceleration, E is the variance or total energy of the water surface elevation, U10 is the wind seed, f is the frequency of the sectral eak and d is the water deth. Equations (3) describes the evolution of the total energy and sectral eak frequency as a function of fetch, wind seed and water deth, but yield no information about the detailed evolution of the sectrum. Sectral evolution Young and Verhagen (1996b) [14] comiled an extensive data set of the evolution of the one-dimensional frequency sectrum F( f ), an examle of which is shown in Figure 3. Figure 1. Scatter lot of the non-dimensional energy, ε against non-dimensional fetch, χ. Only data with values of nondimensional deth, δ between 0.3 and 0.4 are shown. Equation (3) is shown for these two extremes of δ by the dashed lines and the dee-water asymtote by the solid line. Equations Figure 3. An examle of the develoment of the finite deth wave sectrum with fetch. The stations where the measurements were made are labeled S1 to S5 (increasing fetch from 1.3km to 6.7km). The wind seed was 10.8 m/s. Figure. Scatter lot of the non-dimensional frequency,ν against non-dimensional fetch, χ. Only data with values of nondimensional deth, δ between 0.3 and 0.4 are shown. Equation (6) is shown for these two extremes of δ by the dashed lines and the dee-water asymtote by the solid line. Young and Verhagen (1996a) [13] reresented these data by a relationshis of the form 1.74 3 B1 ε = 3.64 10 tanh A1 tanh (3) tanh A1 where and 0.75 A1 = 0.493δ (4) 3 0.57 B1 = 3.13 10 χ (5) ν = 0.133 tanh A tanh B tanh A 1.01 0.37 (6) A = 0.331δ (7) 4 0.73 B = 5.15 10 χ (8) Young and Verhagen (1996b) [14] arameterized these results in terms of the well-known JONSWAP sectral form (Hasselmann et al, 1973 [5]) ( π) 4 5 5 F( f ) = αg f ex 4 ( f f ) ex σ f γ Φ (9) where Φ is a Jacobean. Young and Verhagen (1996b) could find no systematic trends within their data for the values of γ and σ and adoted their mean values γ =.70 and σ = 0.1. They found thatα could be reresented in terms of the non-dimensional wave number κ = U 10 k / g, where k is the wave number of the sectral eak, as 0.4 f f 4 α = 0.0091κ (10) Equation (9) aears to indicate a sectrum with a high frequency face roortional to f 5, but inclusion of the Jacobean means that the magnitude of the exonent decreases as 3 the water deth decreases, aroaching f in very shallow 80
water. The data of Young and Verhagen (1996b) [14] clearly showed this evolution. Directional sreading As art of the Lake George exeriments Young et al (1996) [15] deloyed a high-resolution directional array, caable of determining the directional roerties of the waves. In keeing with revious dee-water reresentations, Young et al (1996) [15] reresented their data in the form s θ F( f, θ ) = A( f ) F( f )cos (11) whereθ is the direction of wave roagation and the exonent s determines the width of the directional sreading (high vales of s corresond to narrow sreading). A( f ) is a normalizing function. Figure 4 shows values of s as a function of the normalized frequency f / f and non-dimensional deth arameter k d. A summary of finite-deth evolution The data set described above resents a consistent icture of the evolution of the finite deth wave sectrum with fetch. As the fetch increases, the total energy of the sectrum increases and the eak frequency decreases. At long fetch, this evolution ceases as a result of interaction with the bottom. As a result, this asymtotic limit to growth decreases as the water deth decreases. In comarison to dee-water sectra, the onedimensional sectral form for finite deth waves has a high frequency face which decays more slowly. The directional sreading of finite deth sectra is similar to that of their deewater counter arts, although the sreading is ossibly a little broader. Young et al (1996) [15] seculated that the aarently broader directional sreading in finite deth situations may be due to the behaviour of S 4 nl in such situations. In finite deth situations S 4nl is caable of couling sectral comonents more widely searated in frequency and direction sace, thus ossibly leading to a more isotroic sectral form. The detailed interaction of the various source terms in finite deth situations is, however, still to be fully understood. The formulations resented above rovide a reasonably accurate reresentation of the evolution of the directional wave sectrum with fetch, that is df / dt, or the right hand side of (1). Determination of source terms Following the exerimental camaign aimed at determining the sectral evolution, a searate rogram was mounted to determine the source terms on the right hand side of (1). An integrated set of measurements in the atmosheric and sub-surface boundary layers as well as on the surface itself were conducted. Figure 4. Values of the directional exonent s as a function of the normalized frequency f / f. Values have been artitioned based on the relative deth arameter, kd. 1.0 < k d < 1.5 o,1.5 < k d <.0 +, k d >.0 *. These results have been arameterized as.7 f 11 for f < f f s =.4 f 11 for f f f (1) Further analysis by Young et al (1995) [1] and more recently by Wang and Hwang (001) [11] has revealed that the directional sreading may be more comlex than indicated by (1). These analyses indicate that the uni-modal for reresented by (1) is reasonable to first order, but that the detailed sreading may, in fact, be bi-modal with slightly more energy roagating at angles to the wind, than in the wind direction. The exerimental site develoed for these exeriments included an observational latform, with a shelter to accommodate electronics and equiment as well as researchers during observations. An anemometer mast, housing 3 wind robes at 10 m and 5.65 m elevations over the water surface, was erected 10 m from the latform. Another anemometer mast, housing 5 wind robes at four heights closer to the surface, was set 6 m off the bridge to ensure undisturbed airflow for these lower anemometers. The bridge was used for the majority of the wave measurements. Two shelters constructed onshore rovided basic storage and accommodation for researchers during their stay at the site. The location had vehicular access, a simle offshore bathymetry and water of aroximately 0.5 m deth within 50 m of the shoreline, ideal for the objectives of the shallow water study. The latform, located aroximately 50 m offshore was shore-connected with an elevated walkway and was thus accessible in any weather conditions. Comuter facilities allowed for reliminary data analysis to be erformed on site, though the main data rocessing was conducted subsequently. The full exerimental facility is shown in Figures 5 and 6. To measure white-caing dissiation, four different but integrated instrument systems were emloyed. Since individual waves naturally change their heights whilst roagating within irregular wind wave grous, direct in situ estimates of energy lost by breaking were obtained in terms of the integrated grou energy, measured with an array of caacitance wave robes. The array also allowed measurement of directional wave sectra. 81
Simultaneously, measurements in the atmosheric boundary layer were also conducted. The anemometer masts had six cu anemometers logarithmically saced from 0.5 m to 10 m and two wind direction vanes. The data logging was synchronized with the wave and breaking recordings. In addition to the wind rofile measurements, a sonic anemometer was also deloyed on one mast to rovide direct estimates of the momentum fluxes. A wave following ressure system, develoed and oerated by Mark Donelan from the University of Miami, USA, was deloyed during AUSWEX to exlore wave-induced stresses and ressure fluctuations in the atmoshere. Figure 5. A view of the observation latform from the shore. The elevated walkway can be seen in the foreground, with the accommodation module visible on the latform. It was initially roosed to measure the bottom shear stress with a bottom mounted shear late. Preliminary tests, however, indicated that the mud bottom of the lake was not suitable for this instrument. Hence, a high recision traversing system was used to measure vertical velocity rofiles in the bottom boundary layer with one of the ADVs, the measurement volume of which was reduced to 1 mm for this urose. To determine bottom friction estimates, recise laboratory exeriments were conducted. These measurements were designed to determine accurate values of the bottom roughness and friction arameters for the cohesive Lake George mud to be used in further estimates of the bottom dissiation. Full analysis of the data will include a modelling hase, in which the EXACT-NL model of Hasselmann will be used. The model will be forced with the source terms measured at the exerimental site and sectral evolution comared with the comrehensive data reviously obtained at the Lake George site (Young et al., 1996). Figure 6. View of the observation latform taken from an offshore location. The two anemometer masts can be seen to the left. The major measurement bridge, where most of the measurements were conducted joins the latform at the right to the anemometer masts. A wave gauge is seen in the foreground. Three additional mobile wave robes were ositioned to measure satial decay of breaking wave grous. An observer also marked data records electronically, in resonse to visual observations of breaking. Facilities for quick in situ calibrations of the wave robes were available. Three Acoustic Doler Current Meters (ADV) were emloyed for simultaneous measurement of dissiation rates through the water column in the vicinity of the array. During the exeriment (termed AUSWEX), these measurements of the turbulence were sulemented by Dobeam measurements of wave-number velocity sectra. The Dobeam was rovided by Kendall Melville of the Scris Institution of Oceanograhy, USA for the eriod of the exeriment. An underwater hydrohone, the outut of which is related to both the strength and dimension of breakers, was located on the bottom beneath the array. Video recording of the surface sot around the array was used to identify breaking events and to suly information on the satial dimensions of whitecaing. The logging of all systems, including the video, was synchronized. Data from the Lake George site were obtained for three years, including the intensive AUSWEX eriod. A database, containing hundreds of hours of wave, wind, air turbulence, sub-surface currents and turbulence, under-water sound, humidity, air and water temeratures and other records, as well as hotograhic images of the surface, was reared, documented and archived on seven CDs. About 90 hours of relevant video and hydrohone sound records, with synchronizing time code, were stored on two sets of 40 Suer-VHS video taes. The joint AUSWEX exeriment was carried out from 14 August to 18 Setember, 1999. The data acquired comrise numerous synchronized records by the wave array, anemometer masts, sonic anemometer, three acoustic doler velocimeters (ADV) located at different levels in the water column, video camera, hydrohone, humidity and temerature robes, as well as by the Dobeam, which was traversed to different deths, and by the wave follower. These data comrise four CDs and about 45 hours of video and hydrohone sound records. A number of laboratory tests were conducted to clarify fine details of the hysical rocesses and the comlex behaviour of measurement devices. Atmosheric Inut The atmosheric inut to the wave field can be measured directly from the atmosheric ressure - water surface sloe correlation, measured at the water surface. Hence, measurements need to be made of the extremely small wave induced air flow ressure fluctuations, very close to the moving water surface. In the resent exeriments, this was achieved using an electrically driven servo-feedback wave follower. This system held a disc shaed ressure robe (calibrated to measure static ressure), a fixed distance above the water surface. The system was extensively tested and was found to have a "flat" transfer function for frequencies lower than 5 Hz. 8
Desite the excellent frequency resonse of the wave follower, the ressure signal needs to be carefully corrected for surious fluctuations introduced by the motion of the ressure robe and associated tubing and for the frequency resonse of the tubing (amlitude and hase). Based on these data, the atmosheric inut source term was reresented in a arametric form as: ρa Sin ( f ) = gγ ( f ) ff( f ) (13) ρ where, γ ( f ) is a sectral growth increment function a w U ( f ) a( 1) c λ ( f )/ γ = (14) ρ and ρ are the densities of air and water resectively, c is w the wave hase seed and U λ / is the wind velocity, measured at a height equal to one-half wavelength above the mean water surface. Equations (13) and (14) can be used to directly calculate the wind inut based on the wave sectrum and the local mean wind information. The coefficient a, was calculated from the Lake George data as 0.14. Donelan (1999) [3], however, based on laboratory data, obtained a value 0.8, twice as large as the resent data set. Hence, it aears that this coefficient is not a universal constant, but a function of other arameters. enhancement associated with the breaking waves. The bottom anel shows a running average of the bottom ressure. The enhancement associated with the breaking events in Figure 7 aears to be associated with atmosheric flow searation which occurs above breaking waves. The searation "bubble" significantly changes the hase difference between the ressure and the water surface elevation, thus enhancing inut. A full understanding of the detailed deendence of a on the breaking activity is resently being investigated. White-ca dissiation Of all the source functions resonsible for wave evolution, the white-caing dissiation term is the most oorly understood. Hence, initial analysis of the extensive Lake George data set focused on the study of the roerties of wave breaking. The major goals were: 1. to find reliable techniques for the detection of breakers in the records by instrumental means and without the use of subjective criteria and. to study breaking statistics as a function of sectral roerties The concentration on sectral measures of breaking statistics was aimed at imlementation in sectral wave rediction models. The acoustic signature of the breakers, as recorded by the hydrohone has been used as an objective measure of breaking. Sectrograms below (Figure 8) clearly show breaking events as local maxima which san a frequency bandwidth in excess of 4 KHz. The breaking events shown in the sectrograms have been confirmed by visual insection of video records. Further investigation indicated that a is a function of the wave breaking activity. Figure 7 shows an examle of a tyical wave record, and the associated atmosheric inut. The shar-crested waves in the nd, 7 th and 1 th seconds in the to lot were identified as downwind roagating breakers by means of the synchronized video records. These breakers are also detected by the bottom ressure sensor (bottom lot), which was subsequently used to identify breaking events. The middle sublot shows local running average values of the momentum flux from the wind to the waves. There is clear enhancement of the inut, associated with the breakers. Figure 8. One minute sectrogram of the wave acoustic data. The signatures of breaking waves can clearly be seen as the local maxima (dark regions). An eisode of 14 breaking waves which occurred under U 10 =19.8 m/s wind at f = 0.37 Hz eak frequency and H s = 0.46 m significant wave height. Banner et al. (000) [1] have investigated breaking in dee water and found the robability of breaking is closely associated with the hydrodynamic deformation of wave grous. Using the Lake George data set, Babanin et al. (001) [] have extended this concet to finite deth conditions. Figure 7. A tyical water surface elevation record (to). Breaking waves occur at the nd, 7 th and 1 th seconds. The middle anel shows the atmosheric inut to the waves, with clear They have reresented the robability of breaking, b T as: b T H s ( ξ ) ( )( ν) = 6.16 0.55 1 + 1 + 1 + d 1.91 (15) 83
where ξ is the significant steeness of the sectral eak and H 1.3 f = 4 F( f ) df 0.7 f ξ = H (16) 1/ /. us u0 = is the ratio of the wind induced shear current to the maximum orbital velocity, which has been aroximated by U10 / εc = 0.01 ( ). The Lake George data, as resented by Babanin et al. (001) [] is remarkably well reresented by (15), as shown in Figure 9. suitable for the Lake George data set to model the dissiation rofile: const z Hs / 3 1 Dz ( ) = z z Hs /3, U10 < 7.5m/s z z Hs / 3, U10 7.5m/s (17) The above relationshi, together with measurements of the turbulent velocity sectrum at various deths within the water column rovides the basis for estimating the dissiation for secific cases. Extension of such results to a relationshi which can be used generally, requires an understanding of the functional deendence of these measured dissiation rates. These tyes of deendencies are resently being investigated. Examles of measured dissiation rates are resented later in this aer. Bottom Friction Direct measurements of the bottom friction dissiation in the field are very demanding and have not been attemted as art of the exeriment. Rather, a detailed set of laboratory exeriments have been conducted, in an effort to more fully understand the bottom friction source term under a sectrum of wind generated waves. The bottom friction source term, Sbf ( f ) is often reresented in terms of the bottom shear stress τ 0 as: Figure 9. The robability of breaking as a function of the arameters reresented in (15). Data are resented from both dee water and finite deth situations. (after Babanin et al., 001 []) An understanding of the breaking robability and its deendence on other hysical arameters is imortant, and rovides an essential first ste in determining the dissiation source term. It does not, however, rovide a direct measure of energy loss during breaking events. Such estimates can be made from measurements of the total energy dissiation in the water column. At Lake George, two different devices, the standard SonTek ADV and the Dobeam (Veron and Melville, 1999 [9]), were used to obtain local dissiation rates by measuring Kolmogorov turbulence sectra. The ADV works only in the frequency domain, whereas the Dobeam rovides both frequency and wavenumber sectra of the velocity fluctuations. Once the local dissiation rates are obtained at a number of water deths, they need to be integrated over the whole water deth to rovide an estimate of the total dissiation er unit of wavy surface. Therefore, knowledge of the deth rofile of the dissiation is required, which is imossible to measure recisely using devices such as the ADV or Dobeam, articularly very close to the surface. The study by Terray et al. (1996) [8] rovided a arameterization of the dissiation rofile as a z function of the water deth z. The z deendence was confirmed at Lake George, however, the arameterization was revised in terms of the dissiation behaviour near the surface and the wind influence on the transition from the linear wall layer to the quadratic function. Clearly, this form of deendence cannot be extended to z = 0. The following, iecewise relationshi was found S ( f ) = τ u ( f ) (18) bf where u b is the wave-induced bottom velocity. The bottom shear stress is further reresented in terms of a quadratic friction law τ0 = 0.5ρ w f w U b (19) where f w is a bottom friction factor andu b the maximum value of the bottom orbital velocity. Mirfenderesk and Young (001) [7] investigated the functional deendence of f w using direct measurements of the bottom shear stress obtained with a mechanical shear late. These measurements were conducted in a laboratory wave flume, with a number of difference bed roughness values. There results are broadly consistent with earlier results and can be summarized in the friction factor diagram in Figure 10. Bottom material from the Lake George exerimental site has been tested in the laboratory to determine tyical values of the bed roughness k s. This information, together with the results in Figure 10, rovide sufficient information to determine the energy dissiation due to bottom friction. 0 b 84
Figure 10. The friction factor results of Mirfenderesk and Young (001) [7]. The figure shows the friction factor f w as a function of the Reynolds Number R e. The family of lines shown are from Jonsson (1966) [6] and are for constant values of A / k, where A b is the bottom wave orbital excursion and k s is the bed roughness. The numbers on the figure reresent values of A / k obtained from the laboratory exerimental data. b s The Energy Balance As the water deth at Lake George was quite shallow ( d < 0.5m), and the fetch was a number of kilometers, it is reasonable to assume the sectrum was in equilibrium and had reached a fully develoed state. Hence, from () we obtain Stot df = 0 = Sindf + Swcdf + Sbf df (0) That is, the total inut should balance the total dissiation. Note that in (0), the nonlinear terms have been excluded as they are, by definition, conservative, with the integral always equalling zero. As each of the terms in (0) can be calculated from the Lake George data, this relationshi can be tested. The resulting balance is shown in Figure 11. The agreement is quite remarkable, considering the extremely comlex nature of the exeriments. Conclusions A detailed understanding of the many comlex rocesses which are active in the generation and evolution of surface gravity waves will robably never be comletely understood. Nevertheless, many activities require a level of understanding where redictions of wave conditions can be made with reasonable accuracy. The exerimental rojects described in this aer reresent a body of more than 10 years of research, aimed at develoing such an understanding. The data sets obtained during this extended exerimental camaign are extensive and unique. The analysis of these data will involve many more years of work. Although the full analysis still awaits, rogress has been considerable. b s Figure 11. The total dissiation, (white-ca lus bottom friction) on the vertical axis, as a function of the total atmosheric inut on the horizontal axis. As the waves are in equilibrium, these arameters should balance. The solid line shows this equilibrium. Data oints are from Lake George. A detailed understanding of the develoment of the directional sectrum under fetch-limited finite-deth conditions now exists. This understanding rovides both a redictive caability (although it is largely emirical) and a "test bed" for the evaluation of theory. Initial results aimed at determining the source terms resonsible for the observed sectral evolution have been encouraging. The atmosheric inut term has been arameterized to first order in terms of the environmental arameters. As well as a deendence on the seed of the wind relative to wave hase seed, wave breaking aears to also lay a significant role. Atmosheric inut is considerably enhanced above breaking waves, due to the atmosheric flow searation which occurs in such cases. A detailed reresentation of the magnitude of this enhancement is the subject of further analysis of the data. A detailed study of the robability of breaking has shown that the modulation of waves within roagating grous lays a major role in determining whether breaking occurs. The robability of breaking has been arameterized in terms of a eak sectral steeness, shear in the water column and the relative water deth. The resulting relationshi is caably of redicting breaking under a wide range of conditions. Detailed measurements of the energy loss during breaking have been obtained from measurements of the turbulent velocity sectrum at various deths within the water column. These results look very encouraging and aear to confirm that in the severely deth-limited conditions of Lake George inut is indeed balanced by dissiation. A detailed arameterization of the energy loss during breaking in terms of hysical arameters is the subject of further analysis. Acknowledgements The Lake George roject has been sonsored by both the Australian Research Council (ARC) and the US Office of Naval Research (ONR). This ongoing suort is gratefully acknowledged. 85
References [1] Banner, M.L., A.V. Babanin, and I.R. Young, Breaking Probability for Dominant Waves on the Sea Surface, J. Phys. Oceanogr., 30, 000, 3145-3160. [] Babanin, A.V., I.R. Young, and M.L. Banner, Breaking Probabilities for Dominant Surface Waves on Water of Finite Constant Deth, J. Geohys. Res., 106, 001, 11659-11676. [3] Donelan, M.A., Wind-induced growth and attenuation of laboratory waves, in Wind-over-Wave Coulings. Persective and Prosects, S.G.Sajadi, N.H.Thomas and J.C.R.Hunt, Eds., Clarendon Press, Oxford, 1999, 183-194. [4] Gelci, R., J. Cazale, and J. Vassal, Utilisation des diagrammes de roagation a la rovision energitique de la houle, Bull. Inf. Comite Cen. Oceanogr. Etudes Cotes, 8, 1956, 169-187. [5] Hasselmann, K. et al., Measurements of wind-wave growth and swell decay during the Joint North Sea Wave Project (JONSWAP), Dtsch. Hydrogh. Z., Sul. A, 8, 1, 1973, 95. [6] Jonsson, I.G., Wave boundary layer and friction factors, Coastal Engineering Conf. 1966. [7] Mirfenderesk, H. and I.R. Young, Direct Measurements of the Bottom Friction Factor beneath Surface Gravity waves, 001, Coastal Eng. (in ress). [8] Terray, E.A., M.A. Donelan, Y.C. Agrawal, W.M. Drennan, K.K. Kahma, A.J. Williams III, P.A. Hwang and S.A. Kitaigorodskii, Estimates of kinetic energy dissiation under breaking waves, J. Phys. Oceanogr., 6, 1996. [9] Veron, F. and W.K. Melville, Pulse-to-ulse coherent Doler measurements of waves and turbulence, J. Atmos. Ocean Technology, 16, 1999, 1580-1597. [10] The WAMDI Grou (S. Hasselmann, K. Hasselmann, E. Bauer, P.A.E.M. Janssen, G.J. Komen, L. Bertotti, P. Lionello, A. Guallaume, V.C. Cardone, J.A. Greenwood, M. Reistad, L. Zambresky, and J.A. Ewing), The WAM model a third generation ocean wave rediction model, J. Phys. Oceanogr., 18, 1988, 1775-1810. [11] Wang, D.W., and P.A. Hwang, Evolution of the bimodal directional distribution of ocean waves, J. Phys. Oceanogr., 31, 001, 100-11. [1] Young, I.R., L.A. Verhagen, and M.L. Banner, A note on the bimodal directional sreading of fetch-limited wind waves., J. Geohys. Res., 100, 1995, 733-778. [13] Young, I.R., and L.A. Verhagen, The growth of fetch limited waves in water of finite deth, art I, Total energy and eak frequency, Coastal Eng., 9, 1996a, 47-78. [14] Young, I.R., and L.A. Verhagen, The growth of fetch limited waves in water of finite deth, art II, Sectral evolution, Coastal Eng., 9, 1996b, 79-100. [15] Young, I.R., L.A. Verhagen, and S.K. Khatri, The growth of fetch limited waves in water of finite deth, art III, Directional sectra, Coastal Eng., 9, 1996, 101-11. 86