Modeling of the Turbulence in the Water Column under Breaking Wind Waves

Transcription

1 Journal of Oceanography, Vol. 59, pp. 1 to 41, Modeling of the Turbulence in the Water Colun under Breaking Wind Waves HONG ZHANG* and ENG-SOON CHAN Tropical Marine Science Institute, National University of Singapore, Singapore 119 (Received 1 Noveber 1; in revised for Deceber ; accepted Deceber ) Past studies have shown that there is a wave-enhanced, near-surface ixed-layer in which the dissipation rate is greater than that derived fro the law of the wall. In this study, turbulence in water coluns under wind breaking waves is investigated nuerically and analytically. Iproved estiations of dissipation rate are paraeteried as surface source of turbulent kinetic energy (TKE) for a ore accurate odelling of vertical profile of velocity and TKE in the water colun. The siulation results have been copared with the experiental results obtained by Cheung and Street (1988) and Kitaigorodskii et al. (198), with good agreeent. The results show that the nuerical full odel can well siulate the near-surface wave-enhanced layer and suggest that the vertical diffusive coefficients are highly epirical and related to the TKE diffusion, the shear production and the dissipation. Analytical solutions of TKE are also derived for near surface layer and in deep water respectively. Near the surface layer, the dissipation rate is assued to be balanced by the TKE diffusion to obtain the analytical solution; however, the balance between the dissipation and the shear production is applied at the deep layer. The analytical results in various layers are copared with that of the full nuerical odel, which confirs that the wave-enhanced layer near the surface is a diffusion-doinated region. The influence of the wave energy factor is also exained, which increases the surface TKE flux with the wave developent. Under this region, the water behavior transits to satisfy the classic law of the wall. Below the transition depth, the shear production doinantly balances the dissipation. Keywords: Turbulent kinetic energy, wave breaking, wave-enhanced layer, dissipation, surface roughness. 1. Introduction The surface layer of the ocean is a ixed-layer controlled ainly by the theral and oentu fluxes. A better understanding of the turbulence under breaking waves is essential in deterining the echaniss of heat and oentu exchanges between atosphere and ocean. However, it is typically difficult to easure the velocity and dissipation rates of the turbulence near the ocean surface, especially when there are extree waveinduced otions. The absence of stable observation platfors and the coplexity of extracting the essential coponents describing ixing also ake easureents difficult. In the past a great deal of effort has been directed towards deterining the near-surface vertical distribution of the velocity. By tracking drifters and drogues, using * Corresponding author. E-ail: tsh@nus.edu.sg Copyright The Oceanographic Society of Japan. acoustic travel tie and copass sighting techniues, it has been found that velocity profiles were logarithic at depths up to the order of 1 below the surface water (Churchill and Csanady, 198). In laboratory studies, it has been noted that waves affected the ean flow, and velocity profiles reained essentially logarithic with depth (Cheung and Street, 1988). Using linear statistical techniues, the root-ean-suare (rs) turbulent velocities obtained through field easureent exhibit a strong dependence on the wave energy (Kitaigorodskii et al., 198). They found that a dissipation rate in the upper layer is two orders of agnitude larger than the expected value for a constant stress layer, and this intense turbulence is generated by waves. When there are breaking waves, wave-turbulence interaction could be responsible for the enhanced nearsurface dissipation of the turbulent kinetic energy (TKE) (Anis and Mou, 1995). The dissipation rate is closely related to wind stress production and exhibits an exponential decay with depth. Derivation of vertical diffusive 1

2 coefficients through acoustic easureents of bubbles has been attepted using an upward-directed, high-freuency sonar (Thorpe, 1984). However, the data were too scattered and only the trend of vertical diffusion coefficient of TKE near the surface could be derived. The experient can only be used to deonstrate the techniue. The observation data were also inadeuate to yield the diffusion coefficient as a function of water depth and wind speed. To date, epirical paraeters are still needed to predict the diffusion due to turbulent ixing. In the earlier studies, the surface layer dissipation rates were derived based on the classical law of the wall. In easureents, however, uch higher dissipation rates were found near the ocean surface (Kitaigorodskii et al., 198). The upper layer, up to a depth of 1ζ rs ( rs is the rs wave aplitude), is a region of intense turbulence generated by wave breaking. Siilarly, dissipation rates near the surface of a large lake were found to be one or two orders of agnitude greater than those estiated through the law of the wall (Terray et al., 1996). The scaling of the dissipation rate as a function of wind speed, wave characteristics and depth has been proposed by Terray et al. (1996). Near the surface, within one wave height, the dissipation rate is great and reains constant. Beyond this region it decays with depth as. In the deepest layer, the dissipation rate is controlled by the law of the wall. These estiations can be applied to new odels to siulate any physical, cheical and biological processes related to the turbulent intensity of ixing in the very near surface layer. In ore recent years, nuerical odeling has been extensively applied to analye the dynaics of turbulent ixing. Typical turbulent ixed layer odels ay be classified into two basic types according to their forulation: differential and bulk odels. Models such as those developed by Mellor and Yaada (1974, 198), and Leos (1991) are odels in the sense that the euations for oentu, heat, salt and TKE are used in their priitive for and are not integrated over the ixed layer. The ixed layer in these odels is defined as a region where the local TKE is sufficient to provide a certain level of vertical ixing. Craig and Banner (1994) and Craig (1996) eployed an iproved level 1/ closure odel to predict near-surface turbulence, in which the dissipation rate decays as ε = /Bl, where B is a constant of proportionality, is turbulent velocity and l is ixing length. The influence of wave breaking was odeled by including turbulent energy input at the surface. The waveenhanced layer was found. In the latter paper, wind effects are considered as a boundary input. In bulk odels, e.g. those of Garwood (1977) and Niiler (1975), the ixed layer was assued to be a well ixed layer, which is unifor in teperature and salinity. The governing euations for the bulk odels were obtained by integrating the priitive euations over the depth of the ixed layer. Martin (1985) tested the ability of the ixed layer odels to siulate the seasonal evolution of the ixed layer at weather-ship stations Noveber and Papa, in the eastern North Pacific. The odels were sensitive to external paraeters such as surface heat flux, sea-water turbidity, and abient diffusivity below the ixed layer. Most recently, Burchard (1) siulated dynaics in the wave-enhanced layer by using a k ε odel, and showed that the easured near-surface dissipation rate under-breaking waves can be siulated by considering a shear-dependent closure for the second oents. In our study, we have adoptedan enhanced estiate of the dissipation rates based on the proposal by Terray et al. (1996). Recent advanced field easureent techniues have been successfully applied to the sea surface layer easureent. Based on extensive field data, Terray et al. (1996) proposed the wave-dependent scaling of the dissipation rate which was controlled by various characteristics of the wave field, leading to a better description of the vertical turbulent structure near the ocean surface. The proposed scaling included the turbulent kinetic energy. With the iproved estiation of dissipation rate as TKE source ter to describe the TKE fluxes, the velocity and TKE were odeled and verified with easureent data fro Cheung and Street (1988) and Kitaigorodskii et al. (198). The effects of diffusion and shear production were analyed analytically in different depth ones, respectively. The results showed that the diffusion was ore iportant at the surface, but that shear production was doinant in deep water.. Description of the Model.1 Governing euations In this study, we have developed a one-diensional, steady-state odel of wave-enhanced turbulence in the ocean surface, siilar to the odel described by Craig and Banner (1994) and Craig (1996), but with a ore accurate and realistic forula of the dissipation rate. As illustrated in Fig. 1, the sea-bed is located at = H and the surface is located at =. Wave breaking is assued to be doinant and therefore to play an iportant role in enhancing turbulence in the upper ocean layer. Neglecting the stratification, the governing euations for the odel include oentu euations and a TKE euation as follows: Du Dt Dv Dt 1 fv = + fu = ρ P x K u Fx + + () 1 1 P y K v Fy + + ( ) ρ H. Zhang and E.-S. Chan

3 The first ter in (6) represents the vertical diffusion of TKE and the second ter is the energy generated by velocity shear. Following Mellor and Yaada (198), the eddy viscosity K and the vertical TKE diffusive coefficient K can be expressed as: K = S l (7) K = S l, (8) where l is the ixing length (turbulent length scale); S and S are epirical oentu and turbulent coefficients, respectively. Fig. 1. Structure of the water layers, Z b =.6h s, Z t =.κch s /u. D Dt K u v = + K F + + ε. () In (1) and (), ρ and ρ are reference and in-situ densities, is the vertical coordinate, f is the Coriolis paraeter, K is the eddy viscosity, and F x, F y are horiontal diffusion ters. In (), K is the vertical diffusion coefficient for TKE; ε is the dissipation due to turbulent otion; is the turbulent velocity scale, defined as the suare root of twice the TKE ( /). Neglecting advection, horiontal diffusion and Coriolis force, the oentu and TKE euations in the steady case are given by K u 4 = ( ) K v 5 K = ( ) + + u v K ε. 6 = ( ). Mixing length and surface roughness The ixing length l can be odeled siply as the ixed layer depth (Skyllingstad and Denbo, 1995), or in a ore coplex for using a differential euation siilar to (6). In the present study, l is assued to follow the law of the wall near the water surface, i.e., l = κ( ), in which κ is the von Karan constant, assued to be.4; is the roughness length for the surface. This reflects the loss of oentu to sea surface. Here, an epirical odel of Donelan et al. (199) is applied to copute the roughness length according to the wind speed and the wave status, 9. 5 U1 U = g C p ( ) in which U 1 is the wind velocity at 1 eters height above the water surface and C p is wave velocity corresponding to the spectral peak freuency. The wind velocity profile has a logarithic height dependence U 1 ua 1 = ( 1) κ ln. The friction velocity of air u a and water u w can be written as: τ s = ρa C U ( ) 1 1, 11 u a τ s ρ a = / ( 1) u w τ s ρ w = / ( 1) where τ s is surface stress; ρ a and ρ w are the density of air and water respectively; and C 1 is the drag coefficient, C 1 = ( U 1 )1 (Wu, 198). The wave energy factor α is described as in Terray et al. (1996): Modeling of the Turbulence in the Water Colun under Breaking Wind Waves

4 c α =, u cp c ~. 5cp, for < u cp c ~ 15u, for > u ( 14) where c p is the wave phase velocity and c p /u is defined as the wave age. Euation (9) indicates that the surface roughness is inversely associated with wave age, since U 1 /C p ~ u w /C p ~ 1/α which represents the wave age and characteries the wind stress and the wave developent stages. The surface roughness would influence the dissipation under wind wave breaking.. Dissipation rate In general, the surface layer dissipation estiates agree satisfactorily with the structure of a classical law of the wall. This has been widely applied in any TKE odeling exercises, such as Noh and Ki (1999), Craig and Banner (1994) and Mellor and Yaada (198). The dissipation due to turbulent otion is scaled as ε = /Bl (Batchelor, 195), which does not take account of the influence of wind and wave. However, for wind-driven waves, field easureents in the upper layer of the ocean showed that dissipation rate ε() under wind wave breaking is uch higher than predicted by the scaling of u /κ (Soloviev et al., 1988). Kitaigorodskii et al. (198) found that ε() is two order greater than the value given by the law of the wall. Melville (198) suggested that dissipation decayed as n, with n ranging Terray et al. (1996) also proposed a scale for the dissipation rate based on wind and wave paraeters and the water depth, as follows:. uch s ε =,, = 6. h ( 15) b b b s. uch s. κchs ε =, t b, t = ( 16) u u ε =, H ( ) κ ( ) t, 17 where c is the effective phase velocity. This estiate is based on the assuption that the water is deep and the region is directly stirred by the wave breaking. This threelayer structure is illustrated in Fig. 1. Near the surface, within one wave height depth, the dissipation rate is high and is aintained constant. At sufficient depth the dissipation rate declines asyptotically to values given by traditional wall law. A coparison between the dissipation rate obtained fro the iproved prediction using (15) (17) and the wall law is shown in Fig. 6, which illustrates that Es. (15) (17) can describe a dissipation rate closer to the easureents reported by Kitaigorodskii et al. (198). We have therefore applied the to our nuerical and analytical odel for analying the velocity and TKE in the water colun under wind-wave breaking..4 Boundary conditions The shear flow near the air/water interface is analogous to the turbulent flow over a rough surface. For this sort of flow it is often assued that there is a region adjacent to the surface where the stress can be considered constant. Churchill and Csanady (198) stated that a sublayer exists with linear velocity, very close to the surface. Assuing that the wind direction is the sae as that of horiontal velocity u, the following conditions (Craig and Banner, 1994) are satisfied at the surface layer ( = ), K K K u u = ( ) 18 v = ( 19) α u. = ( ) If the oentu euation (4) and boundary condition (18) are satisfied, one can conclude that boundary condition (18) is valid for the water colun in the entire range H < <. In the work reported by Ly (1986), the TKE at = is assued to satisfy the boundary condition, / = 4.66u, which is a function of wind friction velocity only and is not related to the wave paraeters. However, in Craig (1996) s analytical analysis, TKE is derived as a function of wave age and the diffusive coefficients at the surface, = β u, ( 1) / 4 1 / β = α B + B, ( ) S S which is applied in the present analysis. At the sea-bed = H, a ero-flux TKE is assued, i.e. 4 H. Zhang and E.-S. Chan

5 and the no-slip condition is iposed, = ( ) u =, v =. (4).5 Nuerical ethod In order to solve this proble, we ust resort to a nuerical coputation schee, as follows. Neglecting the Coriolis force, the right-hand sides of (4) and (5) eual ero. Substituting the following two euations into (4)and (5), u u ls = ( 5 ) v =, ( 6 ) the proble then reduces to solving (5), and rewriting (6) as κ s Q Q Q l l + l, 4 1 / u s l Q + ( ) ε = ( 7) = ( 8) which can be viewed as the discretiation of points along the water colun. The vertical doain is discretied as = [] > [1] > [i] > [N + 1] = H. Euations (5) and (7) can be discretie as follows: i 1 i li 1 li 4 u 1 / si l ( Qi) κ s + ui l Q Q + l i 1 Q Q + Q l l l l i 1 i i+ 1 i 1 i ( i i+ 1) ε = ( 9) 1 ui u = 1 /, i 1 li κsi l ( Qi) i ( ) ( ) which can be solved by applying Newton s ethod. An initial guess is ade for the unknown values of the (Q 1, Q,..., Q i,..., Q N ). All the other dependent variables are coputed on the basis of this guess. They are then eventually updated by the Newtonian Iteration Schee until they converge with an absolute error of less than 1 8 at all grids. The boundary point of Q can be calculated by (1). Q N+1 will be discussed in the next section. The nuerical results are shown in Section 4.. Analytical Analysis.1 Shear production balancing dissipation in deep layer To analye the effects of the shear generation of TKE, we siplify (6) using only ters describing the balance between the shear production and dissipation of the turbulent kinetic energy in deep layer, which can be written as κls u v + Substituting (5) and (6) into (1) leads to ε =. ( 1 ) u b =, b ( ). κs h cl s u =, t b ( ). κs h cl s u, H t. 4 S = ( ) Substituting () (4) into (5) and integrating it with the boundary condition of u = at = H yields. hc s u t + 6. hc s. hc s u = + ln +, b κ H + b t 5 ( ) u t + hc s hc s u = + + t b ( ) κ ln.., 6 H + u u = H t κ ln H +,. t b ( 7) In the deeper depth ( H t ), is strictly a constant and u varies logarithically. This layer is defined as the shear layer. The distribution of obtained using the above analytical ethod is shown by dashed lines in Fig.. When the results are copared with the full nuerical odel, it is found that they are in good agreeents in Modeling of the Turbulence in the Water Colun under Breaking Wind Waves 5

6 9. chsu = κs 1 b b ln ln b αu + βu κs ln, b b b. ( 4) t b Fig.. Coparison of profiles /u (Cheung and Street, 1988), obtained by nuerical full odel and the analytical solutions, of case 4 in Table 1. deeper water, since the shear generation of TKE is balanced doinantly by dissipation. This shear layer is unaffected by the presence of the wave-enhanced dissipation near the ocean surface. Therefore, at the sea botto, N+1 = u /S is used as a boundary condition. However, the analytical solutions in this section for upper layers are eaningless and can be ignored, since in the near surface upper layers the influence of diffusion is significant.. Diffusion balancing dissipation at near surface layer The effect of diffusion is ore significant in TKE production at the surface. To eliinate the shear production, the TKE euation (6) can be rewritten as a balance of the downward TKE diffusion and its dissipation, K ε. 8 = ( ) By integrating (8) with the boundary condition (1) at =, can be solved for analytically as follows: αu 9. chsu = u + β ln ln, κs κs b 9 ( ) b Close to the surface, Es. (9) and (4) hold. But we did not derive the analytical solution through (8) in deeper water, since E. (8) could not be valid there. No analytic solution of u has so far been derived due to its coplexity. Figure shows a coparison of the profile of obtained by analytical solutions and the full nuerical odel. The solid line is the result given by the analytical solution fro the balancing between the diffusion and the dissipation, which is in good agreeent with that of the nuerical solutions (dashed-dotted line, in Fig. ) near the surface. This very near surface layer ( b ) is defined as the wave-enhanced layer following Craig and Banner (1994). In this wave-enhanced layer, the TKE fro wave breaking is the ain source to balance the dissipation. A coparison of the results fro the full odel with the analytical solutions based on the balance between the shear production and dissipation is given by a plot of the TKE results obtained by nuerical and analytical ethods given in Fig., which clearly illustrates TKE transition fro the wave-enhanced layer to the shear layers. In the diffusive near-surface layer, a balance between the surface flux of TKE and dissipation is the doinant effect while further beneath this layer, dissipation would satisfy the classic law of the wall, which shows that the shear generation of TKE is balanced by dissipation 4. Coparison with Experiental Data 4.1 Calculation of S and S The Level 1/ odel described by Mellor and Yaada (198) gives: G = l l Gh = Ps + Pb ε u ( 41) 1 4 θ β g ( ) ( ) ( ) = B1 S G + S G, 4 h h 6 H. Zhang and E.-S. Chan

7 Table 1. Characteristic paraeters for different cases of wind-generated waves. u (/s) u S (/s) () u (/s) ũ o (/s) η () f D (H) case case case case The paraeters are denoted as: u - the wind speed, u S - the Eulerian surface drift, ũ o - the rs surface orbital velocity, - the depth at which the turbulent shear stress vanishes, η - the wave-decay depth, f D - the doinant freuency of the wave. where θ is teperature or salinity, P s is the shear production of turbulent energy, and P b is the buoyant production. For the experient of wind generated wave breaking (Cheung and Street, 1988), it was assued that there is no stratification, the θ flux at the surface boundary layer is ero, i.e. θ/ =, which leads to G h =. Then S and S h can be written as, S h A1 Ps + Pb = A B ε 1 ( ) greater than.5 s 1 (Thorpe, 199), it was sufficient to cause breaking waves. Therefore, the lowest three speeds did not create fully turbulent boundary layers, and hence only the runs with speeds of s 1 are considered in our coparison. Table 1 lists the characteristic paraeters of the runs. In coparison between the nuerical and experiental data, u and have been converted fro the nondiensional paraeters u + and +, + = ( 46) u u u u S 6A S = A1 1 C1 B 1 1 Ps + Pb, ε ( 45) + u ν = ( 47) w where (A 1, B 1, A, B, C 1 ) = (.9, 16.6,.74, 1.1,.8). In the following analysis, s is set eual to s h. Their experient ignores the buoyancy effects. At the surface layer, the diffusion ter is doinant,while the shear production is relatively sall copared to dissipation. As P s /ε << 1, S and S h eual.7 and.74, respectively, as applied to the analytical solution in Section. However, in the nuerical solution discussed in Subsection.4, S and S h are solved nuerically using (44) and (45), which are depth dependent. 4. Experients of Cheung and Street (1988) and Kitaigorodskii et al. (198) In our study, we copared the nuerical solutions with the experiental results of Cheung and Street (1988) and Kitaigorodskii et al. (198). Cheung and Street (1988) conducted a series of experients in a laboratory channel to obtain ean and turbulent velocity data in the water layer beneath wind-generated water waves. Wind-generated wave experients were run at seven different wind speeds fro s 1. When the wind speed was where ν w is the kineatic viscosity of water. The phase velocity c p can be estiated as η f D fro the data in Table 1. The wave energy factor α and botto roughness can be calculated by (14) and (9) respectively. The significant wave height h s can be calculated by using the given rs surface orbital velocity as follows, h s u πf o D =. ( 48) Table shows the wave-age-dependent c p and, which are used in the analytic and nuerical analysis. We used the full nuerical odel to calculate and u, and copared the values with the experiental data. The results are shown in Figs. and 4, which show that the easureents of and u are in good agreeent with both the analytical and nuerical solutions. The easureent results of turbulent velocity coponents beneath nature, wind-generated waves on Lake Modeling of the Turbulence in the Water Colun under Breaking Wind Waves 7

8 Table. The wave age dependent c p and for different cases of wind-generated waves. u (/s) c p (/s) α () case case case case Table. Characteristic paraeters for different cases of field experients. U 1 (/s) ζ (c) u (c/s) c p (/s) case case Fig.. Profile coparison of easured /u (Cheung and Street, 1988) with nuerical and analytical solutions of cases 1 4 in Table 1. Ontario published by Kitaigorodskii et al. (198) are also copared with the nuerical solution. The wave paraeters are listed in Table. Table 4 lists the easured and ε at different depths. First, the dissipation rate calculated by using ε = /Bl and (15) (17) are copared and shown in Fig. 5. As the depths of the easured data are out of the wave-enhanced layer, the analytical solution is not valid for the coparison of. Therefore, it is only copared with the full nuerical odel. A satisfactory agreeent is also found, as shown in Fig The effects of wave energy factor α For wind generated waves, the wave energy factor α is a good easure of the sea state. The experiental values of α found by Cheung and Street (1988) are listed in Table. If the wave energy factor increases, it eans that the surface flux of the TKE is increasing. Fro (9) and (1), the following relationships exist, Fig. 4. Profile coparison of easured u (Cheung and Street, 1988) with nuerical and analytical solutions of cases 1 4 in Table ~ α ( 49) / ~ α 1. ( 5) u Figure 7 shows the profile of /u for different values of α. It is found that in the wave-enhanced layer the wave energy factor α has a significant influence on the odel results; however, deep in the water, its effect on and u vanishes. 5. Conclusions The classic wall law is widely used in estiating the dissipation and odeling of TKE. However, recent experiental results show that a wave enhanced turbulent layer exists in a wind-force auatic surface in which the dissipation rate is higher than the prediction given by the law of the wall. In the present odel, the dissipation is 8 H. Zhang and E.-S. Chan

9 Table 4. The easured and ε in Kitaigorodskii et al. (198). Depth (c) (c/s) ε (c /s ) case case Fig. 6. Coparison of easured /u (Kitaigorodskii et al., 198) with the results predicted by the nuerical full odel of cases 5 and 6 in Table. Fig. 5. Coparison of easured dissipation rate (Kitaigorodskii et al., 198) with results by ε = /Bl and the estiation of Terray et al. (1996) which is used to odify our odel. described by a three-layer structure, and scaling with both wind forcing and wave paraeters (Terray et al., 1996), which can predict the dissipation rate ore accurately and realistically. The odified odel results of velocity and TKE are in reasonably good agreeent with those of laboratory (Cheung and Street, 1988) and field (Kitaigorodskii et al., 198) easureents. Near the surface the odel is siplified by balancing the diffusion of TKE with the dissipation rate. The analytical solution can be derived for the near-surface layer. In deep water, the shear production is the only ter to balance the dissipation rate. The analytical solution can also be found for both u and. The good agreeents obtained in various layers confir that the wave-enhanced layer near the surface is a diffusion-doinated region Fig. 7. The profiles of /u and u of case 4 in Table 1 for different α. Modeling of the Turbulence in the Water Colun under Breaking Wind Waves 9

10 (Craig, 1996). Under this region, the water behavior transits to satisfy the classical law of the wall. Below the transition depth, the shear production predoinantly balances the dissipation. Therefore, coparison aong easureents, analytical solutions and full odel solutions illustrates that the present odel appears to work well in reproducing our present knowledge, siulating turbulence in the ocean surface layer under wind-wave breaking, in which the wind-wave effects have been taken into account in TKE euation. It has also been found that the wave energy factor has significant influence on the surface layer. During the developent phase of the wave field, with increasing wave age, the surface roughness reduces and ore wind energy ight go into the wave field and is lost by wave breaking, this does not affect the ixing in deeper water. This odel has produced predictions that can be applied to benchark further experiental results. It can also be extended in the future to odel the transport of cheical and physical process under wind-wave breaking. When applying this wave-enhanced ixing layer schee to the general circulation odel, however, the teporal and horiontal spatial effects are neglected. The influence of variable tie and geoetry on the wave-enhanced layer needs to be investigated and evaluated in future studies. Notation C 1 drag coefficient for U 1 c effective wave phase velocity [LT 1 ] c p wave phase velocity [LT 1 ] f D doinant wave freuency [T 1 ] H water depth [L] h s significant wave height [L] K eddy viscosity [L T 1 ] K vertical TKE diffusive coefficient [L T 1 ] l ixing length [L] turbulent velocity scale [LT 1 ] + nondiensionied turbulent velocity scale S epirical oentu coefficient S epirical TKE coefficient t tie [T] U 1 wind velocity at 1 height above the sea surface [LT 1 ] u, vhoriontal velocity coponents [LT 1 ] u S Eulerian surface drift [LT 1 ] ũ o rs surface orbital velocity [LT 1 ] u wind velocity in experient [LT 1 ] u friction velocity at the surface [LT 1 ] u a friction velocity of air [LT 1 ] u b friction velocity of sea bed [LT 1 ] u + nondiensionied horiontal velocity vertical coordinate [L] surface roughness length [L] b botto roughness length [L] η wave decay depth [L] nondiensinied vertical coordinate + α wave energy factor ε TKE dissipation rate [L T ] κ von Karan constant δ viscous sub layer thickness [L] depth at which the turbulent shear stress vanishes [L] ν w dynaic viscosity of water [L T 1 ] ρ a density of air [ML ] τ s surface stress [ML 1 T ] References Anis, A. and J. N. Mou (1995): Surface wave-turbulence interaction: Scaling ε() near the sea surface. J. Phys. Oceanogr., 5, Batchelor, G. K. (195): The Theory of Hoogenous Turbulence. Cabridge University Press. Burchard, H. (1): Siulation the wave-enhanced layer under breaking surface waves with two-euation turbulence odels. J. Phys. Oceanogr., 1, Cheung, T. K. and R. L. Street (1988): The turbulent layer in water at an air-water interface. J. Fluid Mech., 194, Churchill, J. H. and G. T. Csanady (198): Near-surface easureents of Quasi-Lagrangian velocities in open water. J. Phys. Oceanogr., 1, Craig, P. D. (1996): Velocity profiles and surface roughness under breaking waves. J. Geophys. Res., 11, Craig, P. D. and M. L. Banner (1994): Modeling wave-enhanced turbulence in the ocean surface layer. J. Phys. Oceanogr., 4, Donelan, W. A., F. W. Dobson and S. D. Sith (199): On the depandence of sea surface rougness on wave developent. J. Phys. Oceanogr.,, Garwood, R. W. (1977): An oceanic ixed-layer odel capable of siulating cyclic sates. J. Phys. Oceanogr., 7, Kitaigorodskii, S. A., M. A. Donelan, J. L. Luley and E. A. Terray (198): Wave-turbulence interactions in upper ocean. Part II: Statistical characteristics of wave and turbulent coponents of the rando velocity field in the arine surface layer. J. Phys. Oceanogr., 1, Leos, C. M. (1991): Wave Breaking. Springer-Verlag. Ly, L. N. (1986): Modeling the interaction between the atospheric and oceanic boundary layers, Including a surface wave layer. J. Phys. Oceanogr., 16, Martin, P. J. (1985): Siulation of the ixed layer at OWS Noveber and Papa with several odels. J. Phys. Oceanogr., 9, Mellor, G. L. and T. Yaada (1974): A hierarchy of turbulence closure odels for planetary boundary layers. J. Atos. Sci., 1, Mellor, G. L. and T. Yaada (198): Developent of a turbulence closure odel for geophysical fluid probles. Rev. Geophys.,, Melville, W. K. (1996): The role of surface-wave breaking in air-sea interaction. Ann. Rev. Fluid Mech., 8, Niiler, P. P. (1975): Deepening of the wind ixed layer. J. Mar. Res.,, H. Zhang and E.-S. Chan

11 Noh, Y. and H. J. Ki (1999): Siulations of teperature and turbulence structure of the oceanic boundary layer with the iproved near-surface process. J. Geophys. Res., 14, Skyllingstad, E. D. and D. W. Denbo (1995): An ocean largeeddy siulation of Languir circulations and convection in the surface ixed layer. J. Geophys. Res., 1, Soloviev, A. V., N. V. Vershinsky and V. A. Beverchnii (1988): Sall-scale turbulence easureents in the thin surface layer of the ocean. Deep-Sea Res., 5, Terray, E. A., M. A. Donelan, Y. C. Agrawal, W. M. Drennan, K. K. Kaha, A. J. Willias, III, P. A. Hwang and S. A. Kitaigorodskii (1996): Estiates of kinetic energy dissipation under breaking waves. J. Phys. Oceanogr., 6, Thorpe, S. A. (1984): On the deterination of k v in the nearsurface ocean fro acoustic easureents of bubbles. J. Phys. Oceanogr., 14, Thorpe, S. A. (199): Bubble clouds and the dynaics of the upper ocean. Quarterly Journal od the Meteorological, 118, 1. Wu, J. (198): Wind-stress coefficients over sea surface fro breee to hurricane. J. Geophys. Res., C87, Modeling of the Turbulence in the Water Colun under Breaking Wind Waves 41