Impact of Swell on the Marine Atmospheric Boundary Layer

Transcription

1 934 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 34 Impact of Sell on the Marine Atmospheric Boundary Layer V. N. KUDRYAVTSEV Marine Hydrophysical Institute, Sevastopol, Ukraine, and Nansen Environmental and Remote Sensing Center, Bergen, Noray V. K. MAKIN Royal Netherlands Meteorological Institute (KNMI), De Bilt, Netherlands (Manuscript received 8 February 003, in final form 19 September 003) ABSTRACT A model that describes the impact of sell on the marine atmospheric boundary layer is proposed. The model is based on the to-layer approximation of the boundary layer: the near-surface inner region and the outer region above. The sell-induced momentum and energy fluxes are confined ithin the inner region. Sell loses energy to the atmosphere and enhances the turbulent kinetic energy in the inner region. The transfer of momentum results in acceleration or deceleration of the airflo near the surface. Folloing-ind sell accelerates the flo, hich for a very lo ind results in a sell-driven ind. Opposite-ind sell decelerates the airflo, hich for a steep sell could cause the reverse airflo. The sea drag in the case of opposite-ind sell is considerably enhanced as compared ith the folloing-ind sell case. Cross-ind sell causes the rotation of the ind velocity vector ith height, hich leads to the deviation of the turbulent stress vector from the ind velocity vector. The impact of sell becomes more pronounced hen the ind speed decreases or hen the sell phase velocity increases. In fact, it is the ave age of the sell that characterizes the sell impact because the dimensionless ave-induced fluxes of energy and momentum are proportional to the ave age parameter. Both fluxes are also proportional to the sell slope so that the sell impact is stronger for a steeper sell. The model reproduces qualitatively and quantitatively the main experimental findings for the ocean sell: the impact of sell on the sea drag is very pronounced for opposite-ind sell, is less pronounced for cross-ind sell, and is only at lo ind speeds. 1. Introduction Experimental estimates of the sea drag coefficient demonstrate a large scatter ithin individual experiments and beteen different experiments. It is suggested that one of the reasons for such a large scatter could be the unsteady ind conditions and the variable sea state (Drennan et al. 1999). In the case of a developing ind sea, several studies have shon that the drag coefficient is dependent on ave age (e.g., Donelan et al. 1993; Maat et al. 1991; Smith et al. 199). In terms of the Charnock parameter this dependence reads z 0 g/ u * (u * /C p ) n, here z 0 is the roughness scale, u * is the friction velocity, g is the acceleration of gravity, C p is the phase velocity at the peak of the ave spectra, and n is a positive exponent. Despite differences in the experimental estimate of n, measurements revealed the undoubted fact that the younger ind sea supports the larger sea drag. Makin and Kudryavtsev (00) argued that the airflo separation from breaking dominant Corresponding author address: Dr. V. K. Makin, KNMI, P.O. Box 01, 3730 AE De Bilt, Netherlands. makin@knmi.nl ind-generated aves is responsible for the observed enhancement of the sea drag in the young sea. A young sea usually corresponds to the fetch- and/ or duration-limited conditions. That is not typical for the open ocean. Mixed ind sea sell or pure sell conditions are the most remarkable features of the open ocean. Yelland and Taylor (1996) found that in the open ocean at lo ind speeds, the drag coefficient increases ith a decrease in the ind speed. This rather rapid increase cannot be explained by the aerodynamically smooth conditions of the sea surface. The authors mentioned that during the ocean cruise, the ave field at lo inds as alays dominated by sell, and that it is very likely that the impact of sell on the atmosphere could result in a strong increase in the drag coefficient at lo inds. Donelan et al. (1997) found a strong increase in the drag coefficient at lo inds in the presence of sell traveling across or opposite to the ind. This finding as confirmed by Drennan et al. (1999), ho also observed the enhanced sea drag under the same conditions. They concluded that much of the scatter in the drag coefficient could be attributed to the presence of sell. Guo-Larsen et al. (003), using measurements from the Baltic Sea, have shon that the cross sell 004 American Meteorological Society

2 APRIL 004 KUDRYAVTSEV AND MAKIN 935 enhances the sea drag as compared ith the folloingsell case and that the magnitude of the drag coefficient is increased ith increasing the angle of sell propagation to the ind. Hoever, no quantitative relations describing the impact of sell on the sea surface drag ere proposed, nor ere the physical mechanisms responsible for the impact revealed. Another related phenomenon is the occasional observations of the upard momentum transfer (e.g., Davidson and Frank 1973; Grachev and Fairall 001), hich is manifested in a ave-driven ind. Sell traveling faster than the ind transfers momentum to the atmosphere and accelerates the airflo close to the sea surface (Harris 1966; Smedman et al. 1999). It should be clearly realized hoever that the reported impacts of sell on the atmospheric boundary layer can be attributed only ith some certainty to aves. A possibility exists that mesoscale turbulent or convective motions in the marine atmospheric boundary layer, completely independent of sell, are entirely or predominantly responsible for the observed peculiarities. For example, it could be realized (Grachev and Fairall 001, their Fig. 7) that the reported upard momentum transport assigned by authors to sell is mainly due to motions ith frequency around Hz. This points at mesoscale motions rather than at sell signature. Lo inds and the presence of sell are typical of tropical ocean regions, here the atmosphere accumulates heat and moisture and then transfers them to higher latitudes. In that respect the understanding of sell impact on the exchange processes at the air sea interface is one of the key issues in climate research. In the present paper a model that describes the impact of sell on the marine atmospheric boundary layer (MABL) and, in particular, on the sea drag is proposed. The physical mechanism responsible for that impact is explained and quantitative estimates of the impact of sell on the momentum flux and the ind profile in the MABL at various ind sell conditions are given. It is shon that under certain conditions the impact of sell on the MABL can be dramatic.. Model a. Governing equations A neutrally stratified marine atmospheric boundary layer in the presence of sell is considered. Sell is characterized by the amplitude A, avenumber K, and frequency connected by the dispersion relation for aves on deep ater. In the Cartesian coordinate system moving ith the phase velocity of sell, C /K, the x 1 axis aligned ith the sell propagation direction, the x axis parallel to the sell crest, and the x 3 axis directed upard, the sea surface displacement is ikx z(x, t) Ae 1. (1) Hereinafter only the real part of this or any other quantity is to be considered. The ind and ave fields are assumed to be stationary and spatially homogeneous, and the ind direction is arbitrary ith respect to the sell direction. Governing equations describing the vertical structure of the onedimensional averaged over the horizontal coordinates MABL above aves are the momentum and the turbulent kinetic energy (TKE) conservation equations (e.g., Makin and Mastenbroek 1996). In the present study the equations are refined by reriting them in a more convenient and idely used ave-folloing orthogonal coordinate system, hich is defined in the appendix. Derivations of the equations are also given there. The momentum and TKE conservation equations in the ave-folloing i -coordinate system [see (A14) and (A17)] are () () 0 and () U ũ t i F ij Diss 0, (3) j here 3 is the vertical ave-folloing coordinate; 1, ; and i, j 1,, 3. In addition, () is the ave-induced momentum flux defined by (A13), () is the turbulent shear stress, 0 is the total shear stress in the MABL far aay from the sea surface here the impact of aves on the stress is negligible, F t is the vertical flux of the TKE, Diss is the viscous dissipation of the TKE, U is the mean ind velocity in the moving coordinate system related to u in the fixed coordinate system as U 1 u 1 C, and U u. The momentum conservation equation () shos that the sum of the ave-induced and turbulent stress is constant over height. At the sea surface 0 (here 0 is the roughness scale of the avy surface) the ave-induced stress (form drag) is s1 p sz 1, (4) here s1 1( 0 ), z 1 z/x 1 is the sea surface slope, and p s is the ave-induced pressure at the sea surface. The component of the form drag parallel to the ave crest is 0: s ( 0 ) 0. The TKE production due to the ork of the aveinduced turbulent stress ij against the shear in the ind velocity [third term on the left-hand side of (3)] can be expressed through the energy balance of the ave-induced motion (A15), here this term describes the sink of the ave-induced energy into the TKE [last term in (A15)]. With (A15) the TKE balance equation (3) is ritten as U t (F F ) ( ) Diss 0, (5) here F is the vertical flux of the ave-induced energy. When the ave-induced momentum flux is comparable ith the turbulent stress, that is,, the effect of aves on the MABL could be significant. In this case the vertical flux of the TKE is much loer than the

3 936 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 34 vertical flux of the ave-induced energy. It is easy to sho that in this case the ratio F t /F is of order F t /F u * /C K 1, here u * is the friction velocity. Neglecting F t in comparison ith F and accounting for the momentum conservation equation () the TKE balance equation finally reads U F 0 Diss 0. (6) According to (A16), the energy flux of the ave-induced motion at the sea surface is F s p s s ũs 1s (7) Cp z ũ, (8) s 1 s 1s here s and ũ s are the vertical and horizontal components of the ave surface orbital velocity, and s1 is the ave-induced variation of the shear stress at the surface. This energy flux provides the groth or attenuation of the ave energy due to the ork of normal and tangential stresses. Their relative contributions to the total ave groth rate depend on the ind and ave conditions and ill be analyzed later. To complete the problem a turbulence closure scheme has to be chosen. In the present study the simplest mixing length closure scheme is used. Assuming that the mixing length is proportional to the distance from the sea surface in the ave-folloing coordinate system, the turbulent shear stress and the TKE dissipation in Eqs. () and (6) are ritten as b 3/ Diss, (9) du k, and (10) d 1/ k b, (11) here b is the TKE and k is the turbulent eddy viscosity coefficient. With (9) (11) the governing equations of the problem are du 1/ b 0 () d and (1) U b 3/ F 0 0. (13) Using (1) the TKE balance equation (13) transforms into F () 1/ b 0[ 0 ()] b. (14) If the vertical profiles of () and F () are knon and the ind speed at the reference level h far aay from the sea surface here the ave-induced fluxes are vanished is specified, the solution of Eqs. (1) and (14) gives a full description of the MABL: the drag coefficient, the ind velocity, the TKE profile, and so on. The solution of this problem for pure ind seas [the last term on the right-hand side of (14) can be then ignored] is given, for example, by Makin and Kudryavtsev (1999). In the presence of sell the last term on the right-hand side of (14) plays a significant role. It is easy to check that hen the effects of aves on the atmospheric boundary layer are important (i.e., s1 0 ), the ratio of the second term to the first term on the right-hand side of (14) is of order C/ U l, here U l is the ind speed at the height of the inner region defined in section b. Thus in the sell conditions (C/ U l k 1) the divergence of the ave energy [the last term on the right-hand side of (14)] is the dominant source of the TKE production. b. To-layer approximation A to-layer approximation of the ave boundary layer is used for its description. According to the rapid distortion theory of turbulence applied to modeling the airflo above aves (e.g., Tonsend 197; Belcher and Hunt 1993; Belcher 1999; Kudryavtsev et al. 001), numerical simulations of the airflo above aves based on Reynolds equations ith a second-order turbulent stress closure scheme (e.g., Mastenbroek et al. 1996), direct numerical simulations (e.g., Sullivan et al. 000), the ave-induced stress (), and the corresponding vertical flux of the ave-induced energy F () are confined to a thin near-surface inner region (IR) characterized by the vertical scale l. They rapidly vanish outside the IR at l in the outer region (OR). The scale of the IR can be defined as the vertical spreading height of the oscillation in the turbulent boundary layer caused by a avy surface: l k l T a, here k l is the effective eddy viscosity inside the IR, T a 1/(K U l ) is the scale of the oscillation period, and U l U 1 (l) u 1 (l) C is the ind velocity in a frame of reference moving ith the ave phase speed at l. At arbitrary ind angle ith respect to the ave propagation direction the eddy viscosity for the ave-induced turbulent stress variation is k (1 cos )b 1/. (Meirink et al. 003). If e define k l as the eddy viscosity at l [i.e., k l k(l)] then the scale of the IR reads 1/ Kl (1 cos )b / U l. (15) At 0 and b 1/ u *, (15) corresponds to the definition of the IR scale proposed by Belcher and Hunt (1993) and Cohen and Belcher (1999). The scales of the IR in the folloing-, cross-, and opposite-ind directions for the logarithmic ind velocity profile u() u * / ln(/ 0 ) are shon in Fig. 1 as a function of the inverse ave age parameter u /C, here u is the ind speed at level /K. At the opposite- and cross-ind directions the dimensionless IR scale Kl is very small; its magnitude varies beteen 10 1 and 10. At the folloing-ind direction the IR scale is also small for fast- and slo-

4 APRIL 004 KUDRYAVTSEV AND MAKIN 937 FIG. 1. Inner region depth Kl as a function of the inverse ave age parameter u /C for the sell-ind angle 0 (solid line ith open circles), 90 (solid line ith crosses), and 180 (solid line ith pluses). The height of the critical layer K cr is shon by the solid line. (ith respect to the ind speed) propagating surface aves. When the phase velocity of the surface ave is close to the ind speed, the IR scale Kl increases to about 1. The height of the critical level K cr K 0 exp(c/u * ) the level here the ind speed is equal to the ave phase velocity in the case of a folloingind ave is also shon in Fig. 1. In the range of the inverse ave age parameter u /C 1.5, the height of the critical layer is ell inside the IR. As as discussed by Belcher and Hunt (1993), Mastenbroek et al. (1996), and Kudryavtsev et al. (001), this fact means that the dynamics of the critical layer is significantly affected by the turbulent stress and the nonseparated sheltering mechanism (Belcher and Hunt 1993) is the most plausible one in the description of the ave generation by the ind. The concept of the IR allos for a significant simplification of the description of the ave boundary layer. According to Fig. 1, in most cases (except at u /C 1.5 for the folloing-ind ave), the IR is a thin layer. Since () and F () are confined to the IR, a to- layer approximation of the ave boundary layer can be adopted. The to layers are the IR ( l) and the OR ( l). Inside the IR the ave-induced stress is constant over height and vanishes outside: 1 () s1, at 0 l and (16) 1 () 0, at l, (17) here s1 is the ave-induced stress at the sea surface (the form drag) defined by (4) (notice that component s 0). Correspondingly the profile of the vertical flux of the ave-induced energy has the folloing shape: ln(l/) F () F s, at 0 l, and (18) ln(l/ ) 0 F () 0, at l, (19) F s here is the energy flux at the sea surface defined by (8). The logarithmic shape of the F profile (18) follos from the ave-induced energy conservation equation (A15). The assumption that 1 is constant in- side the IR dictates the logarithmic shape of the F () profile. With (16) (19) the profile of the ind velocity and the TKE inside the IR ( 0 l) follos from (1) and (14): 0 s 1/ u () ln and (0) b 0 [ ] 1/ F s 0 0 s U l b ( ) *, (1) here * 0 1/ s is the friction velocity inside the IR and U l u l(l) C is the ind velocity at z l. In the OR l, the ind profile patches in to the ind velocity at the top of the IR and has a form 0 u () ln u (l), () u* l here u * 0 1/ is the friction velocity in the OR. The resistance la relating the stress to the given ind velocity u h at the reference level h results from (0) and (): 1/ 0 uh sb ln(l/ 0). (3) 1/ u* ln(h/l) u*b ln(l/ ) In the present study the aerodynamic roughness scale is specified through the modified Charnock relationship: 0 a1/ a * * /g, (4) here a and a 0.01 are constants, and is the molecular viscosity of the air. The parameterization of the roughness scale (4) is based on the IR friction velocity. The reason for this is that the momentum flux for very short ind aves parameterized by the second term on the right-hand side of (4) and the viscous surface stress parameterized by the first term on the right-hand side of (4), are both functions of the friction velocity in the IR rather than in the OR. c. Parameterization of the form drag and ave energy flux at the surface The form drag s and the ave energy flux at the sea surface are the governing parameters of the mod- F s 0

5 938 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 34 el. Unfortunately, the experimental estimates of these fluxes for the ocean sell are lacking. Hoever, useful information related to this study can be extracted from a laboratory experiment by Donelan (1999) ho directly measured the groth and attenuation rates of aves propagating along and opposite to the ind. Waves propagating along the ind ere generated by this ind. In order to generate aves propagating against the ind, a avemaker (paddle) as used. Donelan found that the groth/attenuation rate of the ave energy E could be parameterized by 1 E a u / u / c 1 1, (5) E t C C here u / is the ind velocity at height x 3 /K and c is an empirical constant, hich is equal to 0.8 for aves traveling along the ind and 0.11 for aves traveling opposite the ind. Measurements ere performed in the range of the dimensionless inverse ave age parameter u //C from 3 to 7 for the ind-generated aves propagating along the ind, and from 1.5 to 4.5 for the paddle-generated aves propagating against the ind. The question arises, hoever, hether the parameterization (5) could be used for the ocean sell, hich is characterized by much smaller values of the inverse ave age parameter. Taking for the ocean sell the typical value of the phase velocity C 15 m s 1 and the ind speed u / 3ms 1 results, according to (5), in 10 4 for opposite-ind sell and for folloing-ind sell, hich means that the spatial scale of sell attenuation L () 1 C g (C g is the sell group velocity) is L 50 km and L 130 km, respectively. This estimate shos that the parameterization (5) being applied to the ocean sell (extrapolated to loer values of the inverse ave age parameter) gives too large of an attenuation rate. It is a knon fact that sell may propagate large distances (thousands of kilometers) across the ocean (e.g., Snodgrass et al. 1966). It is concluded that the parameterization (5) cannot be directly applied to the ocean sell. It is valid only in the range of measurements. The latter fact ill be used further in the derivation of the analytical expression of sell attenuation. Expressions describing the groth/attenuation rate of a surface ave traveling at an arbitrary angle ith respect to the ind direction are based on the simplified model of the airflo above aves by Kudryavtsev et al. (001), and its extension to account for the arbitrary ind direction by Meirink et al. (003). The details of the derivation are given in the appendix. The attenuation/groth rate of the ave due to the ork of the surface stress against the ave orbital velocity described by the second term on the right-hand side of (8) is given by (A4). Correspondingly the energy flux at the sea surface due to the ork of the shear stress is expressed through as 1 3 F s ũs s1 C (AK). (6) The energy flux due to the action of pressure correlated ith the ave slope is described by the first term on the right-hand side of (8). The parameterization of the groth rate parameter p is given by (A9) in the appendix. In terms of p the energy flux at the sea surface due to the action of pressure is 1 3 F sp Cp sz1 pc (AK). (7) The total energy flux is 1 3 F s F s F sp ( p)c (AK). (8) Correspondingly, the form drag supported by sell [(4)] is 1 s1 p sz1 pc (AK). (9) Figures and 3 sho the groth rate parameters p and obtained according to (A4) and (A9), and by numerical calculations based on the model of Meirink et al. (003). Results are presented in terms of the groth rate coefficient defined as C [,p] [,p] a u* C. (30) In calculations a ave ith an infinitesimal slope as assumed, so that its influence on the mean structure of the MABL is negligible. In this case the friction velocities in the IR and in the OR are equal (i.e., * u * ), and the TKE b inside the IR is b u *. Figure displays and their sum C C[,p] Cp C as a function of the angle beteen the ind and the ave direction for three values of the inverse ave age parameter defined in the 0 angle direction; namely, u /C 0.3, 1.5, and 8. Sell is traditionally defined as aves that do not receive energy from the ind. For aves traveling along the ind direction that are propagating faster than the ind; that is, u cos/c 1.. At opposite- and cross-ind directions all aves can be regarded as sell, as they attenuate on the ind. In oceans, hen the ind rapidly turns, the value of u /C can be rather high (1.5). Case u /C 8 represents a pure ind- sea case hen aves actively receive energy from the ind at the folloing- and close to folloing-ind directions. At the opposite-ind direction this case represents sell generated in laboratory conditions (Donelan 1999). For the pure sell case, u /C 0.3, both mechanisms (ork of the stress and pressure) are equally important in the transfer of energy from sell to the ind. The angular distribution is very asymmetric: moving from 0 to 90 the absolute magnitude of the attenuation rate decreases. Moving farther to 180 it rapidly increases. The rate of attenuation is times

6 APRIL 004 KUDRYAVTSEV AND MAKIN 939 FIG.. Groth rate coefficient defined by (30) as a function of the sell-ind angle: (left), (middle), and (right) their sum C C Cp C Cp. The inverse ave age parameter u /C (top) 0.3, (middle) 1.5, and (bottom) 8. Solid lines are (A4) and (A9) and their sum; circles are model results of Mierink et al. (003). as large for the opposite-ind direction than for the folloing-ind direction. Case u /C 1.5 represents a mixed ind sea sell sea. The pressure term is dominant in the folloing-ind direction, but ith an increase of the angle it rapidly falls to zero and becomes negative. In the cross- and opposite-ind directions both terms are important in sell attenuation. At high values of u /C the pressure mechanism dominates and pro- vides the groth of surface aves propagating at an angle to the ind. At an angle they attenuate. Notice that at large u /C the angular distribution of C is very close to cos cos, as as mentioned by Meirink et al. (003). The distribution of C and their sum C [,p] as a function of the inverse ave age parameter u /C for the folloing-ind sell ( 0), cross-ind sell ( 90), and opposite-ind sell ( 180) is shon in Fig. 3. The pressure term dominates the stress term for the ind-sea aves. For sell u /C 1. the terms are equal. For cross- and opposite-ind sells both mechanisms are equally important. The magnitude of the attenuation rate is maximal for the opposite-ind sell. In the hole range of the inverse ave age parameter and the angle, the expressions obtained approximate the model results of Meirink et al. (003) ith reasonable accuracy. By comparing obtained values of C ith measurements (e.g., Plant 198; Donelan 1999) it becomes clear that they are several times as small as the measured values in the range of data validity, that is, for sloly propagating aves u /C. This is a knon feature of the ind-over-aves theoretical models based on the Reynolds equations (e.g., Tonsend 197; Belcher and Hunt 1993; Mastenbroek et al. 1996). Recently Kudryavtsev and Makin (00) shoed that this discrepancy can be partly explained by the variation along the ave profile of the aerodynamic surface roughness (mainly due to modulation of small-scale breaking aves), hich leads to an increase of the modeled ave groth/attenuation rate. An accurate account of this effect is out of the scope of the present study. To assess the impact of sell on the sea drag, the right magnitude of the sell attenuation rate is needed. It is assumed that (A4) and (A9) correctly reproduce the spectral and angular behavior of and p in the hole range of ind and ave parameters considered. Then a proportionality factor f is introduced, hich allos for fitting the magnitude of the modeled groth rate r p to the measured values in the range of data validity, that is, at u / /C. The groth rate according to parameterization (5) and the modeled value f ith f 5 are shon in Fig. 4. Corrected in this ay, the modeled groth rate is no consistent ith measurements for both folloing- and oppositeind aves. Last, it is assumed that expressions (A4) and (A9) multiplied by 5 are also valid in the range of u / /C not covered by measurements, that is, at u / /C. These corrected expressions (their sum) are used further in the study. Considering again the ocean sell ith C 15 m s 1 and 3ms 1 the u /

7 940 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 34 FIG. 3. Groth rate coefficient defined by (30) as a function of the inverse ave age parameter u /C: (top) and (bottom) their sum C C [,p] C Cp. Results are shon for different sell-ind angles: (left) 0, (middle) 90, and (right) 180. Solid lines are (A4) and (A9) and their sum; open circles are modeled Cp; crosses are modeled C ; triangles are modeled C. FIG. 4. (a) Groth rate for the ind aves. (b) Attenuation rate for the paddle-generated aves traveling opposite the ind. Open circles are parameterization (5) of Donelan (1999); solid lines are defined as the sum of (A4) and (A9) multiplied by a factor of 5. Range of the inverse ave age parameter u / /C corresponds to measurements by Donelan (1999).

8 APRIL 004 KUDRYAVTSEV AND MAKIN 941 corrected values of the attenuation rate are for opposite-ind sell and for folloing-ind sell. Corresponding spatial scales of sell attenuation are L 600 and 1000 km. This scale is 10 times that predicted by (5). 3. Model results In this section the impact of ocean sell on the sea drag is studied using the model developed above. The equations describing the impact of sell on the MABL are the resistance la (3), the TKE balance equation (1), the momentum balance equation (), and equations for the ave-induced momentum flux (9), the ave energy flux at the sea surface (8), the scale of the IR, and for the ind velocity profile in the IR (0) and OR (). The input parameters to the model are the ind speed and the angle beteen the ind and sell propagation direction, the phase velocity, and the slope of the sell. Equations are solved by iterations starting from a no-sell MABL state and resulting in a ne MABL state adjusted to the presence of sell. As the sell attenuation rate (A4) and p (A9) and, consequently, the ave-induced momentum flux s and the ave energy flux F s are functions of local parameters of the MABL, namely, the friction velocity in the IR, sell and the MABL form a coupled system ith a feedback beteen the sell and the atmosphere. We are interested in the ocean sell characterized by a small value of the inverse ave age parameter u / C 1. hen the pronounced impact of sell on the boundary layer is expected. The characteristic slope of sell in the ocean is AK 0.1 and the phase velocity is 10 C 30 m s 1. Results are presented in terms of the drag coefficient at 10-m height, u* 10 u 10 CD, (31) and the drag coefficient at the sea surface defined as * CD, (3) sur here u 10 is the ind speed at 10-m height. In real conditions the reference level of 10 m is alays located in the OR so that u * is the friction velocity in the outer region, hile * is the surface friction velocity located inside the IR. The surface friction velocity is an important parameter for a number of applications (e.g., generation of capillary gravity aves, and for gas, heat, and moisture transfers, etc.), as all of these processes are governed by the turbulent momentum flux at the sea surface. When the impact of sell is pronounced and the form drag is comparable to the turbulent stress, the surface friction velocity defined through the surface turbulent stress differs from the friction velocity taken far aay from the surface. Figure 5 displays CD 10 as a function of the ind speed u 10 FIG. 5. Modeled and measured drag coefficient CD 10 at 10-m height as a function of the ind speed u 10 and for different sell-ind angles: folloing-ind sell (solid line ith open circles: modeled 0; open circles: data), cross-ind sell (solid line ith pluses: modeled 90; pluses: data), and opposite-ind sell (solid line ith crosses: modeled 180; crosses: data). No-sell case (solid line: modeled; dots: data). The sell slope AK 0.06, and the phase velocity C 15ms 1, sell parameters typical for the ocean experiments of Donelan et al. (1997) and Drennan et al. (1999). Data are taken from Donelan et al. (1997, their Table 1). for folloing-, cross-, and opposite-ind sell. The sell slope and the phase velocity are specified in the model as AK 0.06 and C 15 m s 1, values that are typical for the experiment in the open ocean presented in Donelan et al. (1997). The experimental data of Donelan et al. (1997) (taken from their Table 1) are also shon in Fig. 5. According to the model, the impact of sell is quite pronounced for a lo ind speed, u 10 6ms 1. It increases ith as the angle beteen the sell and the ind increases, and is maximum for oppositeind sell. These features are consistent ith Donelan et al. s (1997) data. The model magnitude of the drag coefficient at lo inds and opposite-ind sell in general agrees ith experimental estimates by Donelan et al. (1997) though the scatter in the data is large. This qualitative and quantitative agreement ith observations is encouraging, and belo the impact of sell on the MABL is analyzed in more details. In Fig. 6 the drag coefficient CD 10 and the surface drag coefficient CD sur for folloing-, cross-, and opposite-ind sell characterized by the phase velocity C 15 m s 1 and the slope AK 0 (no sell), AK 0.05, and AK 0.1 are shon as a function of the ind speed. The obvious result is that the impact of sell drastically depends on the sell slope AK, increasing as the slope is increased. This directly follos from (6),

9 94 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 34 FIG. 6. (top) Modeled drag coefficient CD 10 at 10-m height, and (bottom) the surface drag coefficient CD sur as a function of the ind speed u 10 and for different sell-ind angles. (left) Folloing-ind sell 0, (middle) cross-ind sell 90, and (right) opposite-ind sell 180. No-sell case is solid lines; the sell slope AK 0.05 is dotted lines; the sell slope AK 0.1 is dashed lines. The sell phase velocity is C 15ms 1. (7), and (9) because energy and momentum transfers are proportional to the slope squared. Another anticipated result is that the impact is most pronounced at lo inds as the parameter u 10 /C (or u * /C) goes don and the aveinduced fluxes dominate the turbulent fluxes. As as already mentioned above, the impact of sell on CD 10 is strongest in the opposite-ind direction. For folloing-ind sell the magnitude of the surface turbulent stress (surface drag coefficient CD sur )at lo inds considerably exceeds the stress in the nosell case and is larger than CD 10. As sell attenuates and transforms energy and momentum to the atmosphere the surface turbulent stress inside the IR should be increased compared to the no-sell case. This directly follos from the condition of the constant-flux layer (): as 1 (0) 0, the surface turbulent stress 1 (0) should be increased to compensate for the negative ave-induced flux and keep 0 constant ith height. In fact their sum 0 appears to be slightly larger than the initial turbulent flux in the no-sell case. This leads to a slight increase of the turbulent stress in the OR and thus CD 10 as compared to the no-sell case. For crossand especially for opposite-ind sell the situation is quite different. For cross-ind sell, CD sur and CD 10 are comparable. This is due to the generation of the stress component being aligned ith the direction of sell propagation. For opposite-ind sell CD sur is only slightly larger than in the no-sell case hile CD 10 is increased considerably. This is explained by the folloing. The surface turbulent stress is increased as compared to the no-sell case because sell generates additional turbulence in the IR. The momentum is transferred from sell to the ind so that (0) 0. The surface turbulent stress is also negative, 1(0) 0, as the ind is in the opposite direction. Both stresses combine to increase considerably the absolute value of the total stress 0 and thus the turbulent stress in the OR and the CD 10. There are to peculiarities observed: one is the minimum of CD 10 for the folloing-ind steep sell, and the other is the minimum of CD sur for the opposite-ind steep sell, both at a very lo ind. To find out the cause of these peculiarities let us consider the dependence of the TKE inside the IR and the ind velocity profile in the MABL on the ind and sell parameters. Sell transfers its energy to the atmosphere, providing the additional source of the TKE. According to (1) it enhances the TKE inside the IR as compared ith the no-sell case. In the no-sell case the TKE b * u, so that in the presence of sell b is alays larger * than. That is illustrated in Fig. 7, here b/ u 10 is * plotted as a function of the ind speed and the slope (cf. ith values of CD sur * / u 10 plotted in Fig. 6). The ind profile is shon in Fig. 8. As sell loses momentum to the atmosphere ( 1 0), the ind ve- locity gradient 1

10 APRIL 004 KUDRYAVTSEV AND MAKIN 943 FIG. 7. Turbulent kinetic energy b/ u 10 in the IR [(1)] as a function of the ind speed u 10 and for different sell-ind angles: (a) folloing-ind sell 0, (b) cross-ind sell 90, and (c) opposite-ind sell 180. No-sell case is solid lines; the sell slope AK 0.05 is dotted lines; the sell slope AK 0.1 is dashed lines. The sell phase velocity is C 15ms 1. du 0 1/ d ln b deviates from one in a no-sell case. In the folloingind direction and a very lo ind speed the airflo inside the IR is accelerated in comparison ith a nosell case. This effect generates a sell-driven ind, hich is quite pronounced in Fig. 8a in the form of the jetlike shape of the ind profile. The maximum of the ind velocity is located at the IR height. In this case the shear stress belo the IR height is positive, hile that above is negative (the momentum flux is directed upard). With an increase of the ind speed the bump in the ind profile disappears, and the shear stress in the OR also becomes positive. A change of the sign of the shear stress in the OR explains the origin of the minimum of CD 10 in Fig. 6. Figure 8b illustrates the ind velocity profile for opposite-ind sell. As the x 1 axis is directed along the sell direction the ind velocity at the reference level is negative. In this case sell transfers the momentum to the IR and decelerates the airflo reducing the absolute ind speed. At a very lo ind this effect could causes a reverse flo. A change of the sign of the shear stress in the IR explains the existence of the CD sur minimum in Fig. 6. At all ind speeds the airflo in the IR is decelerated as compared ith the no-sell condition, hich results in the enhanced drag coefficient at the reference level CD 10 (see also explanation above in terms of the stress). Cross-ind sell (Figs. 8c,d) loses its momentum and generates the x 1 component of the ind velocity (Fig. 8c). This leads to the rotation of the ind vector ith height. In this ay cross-ind sell causes the difference beteen the ind and the shear stress vectors. This effect is illustrated in Fig. 9 for the turbulent stress in the OR and in the IR. As before, the effect is proportional to the sell slope and eakens as the ind speed increases. In general the deviation of the shear stress in the OR and in the IR has the opposite sign ith respect to the ind direction at the reference level. When the angle beteen the ind direction and the sell is 0 180, the shear stress in the OR deviates to the left from the ind vector, hile the surface stress deviates to the right. The situation is opposite hen Generation of ave-driven ind as observed by Harris (1966) in a ave tank experiment. Harris revealed that the paddle ave generates the mean air motion in the initially still air above. His measurements of the ind velocity caused by the paddle ave ith period 0.8 s and amplitude 5.1 cm (ave height 10. cm) at three fixed levels are shon in Fig. 10. The experiment evidently shos the existence of ave-driven ind. Though the measurements are on three levels only, one can see that the ave-driven ind speed increases toard the ater surface. The model simulation as performed for the same ave parameters and given 0 ambient ind speed at the reference level of 6 m [that is equal to the roof height in the experiment by Harris (1966)]. The model simulation shon in Fig. 10 exhibits a jetlike ind velocity profile ith a maximum at the IR height. Above the IR the ind speed decreases ith height as as observed in the experiment. The maximum of the ind speed profile is not resolved by the measurements, as they ere done in the fixed coordinate system above the ave crest (i.e., above the IR height). The model simulation is performed in the ave-folloing coordinate system. Hoever, above the ave crest at the level of measurements the ind speeds in the ave-folloing and fixed coordinate systems are not distinguishable. The model reproduces the effects of generation of ave-driven ind in the laboratory conditions, though the observed ind speed is somehat higher than the modeled. The dependence of CD 10 and CD sur on the sell phase velocity in the range of 10 C 30 m s 1 for folloing-, cross-, and opposite-ind sell ith the slope AK 0.05 is shon in Fig. 11. When the phase velocity increases, the impact of sell becomes more and more pronounced. This follos directly from (8) and (9), describing the ave-induced energy and momentum flux. Being normalized on the ind speed (or the friction

11 944 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 34 FIG. 8. Vertical profile of the ind velocity normalized on the ind speed u 10 in the presence of sell ith the slope AK 0.1 and phase velocity C 15ms 1 : (a) folloing-ind sell; (b) opposite-ind sell; (c) cross-ind sell, x 1 component (along the sell direction) of the ind velocity; and (d) cross-ind sell, x component (across the sell direction) of the ind velocity. Solid lines: u ms 1 ; dashed: u m s 1 ; dotted: u 10.0 m s 1 ; dashed dotted: u m s 1. velocity), the dimensionless fluxes are proportional to the ave age parameter C/u 10 (or C/u * ). Decreasing u 10, fixing C, or increasing C ith the ind speed fixed leads to the increase of the ave-induced fluxes and thus the sell impact on the drag. 4. Conclusions Sell propagating over the ocean interacts ith the atmosphere and loses its momentum and energy to the ind. The exchange of energy is realized through to mechanisms: through the correlation of the surface pressure ith the rate of change of the ave surface (vertical orbital velocity), and through the ork of the surface turbulent stress against the ave orbital velocity. For ocean sell characterized by a small absolute value of the inverse ave age parameter u 10 cos()/c 1., both mechanisms are equally important. The attenuation rate is maximal for opposite-ind sell, and in this case sell attenuates times as fast as the folloing-ind sell. A model that accounts for the impact of sell on the marine atmospheric boundary layer is described. The model is based on the to-layer approximation of the MABL: the near-surface inner region and the outer region above. The ave-induced momentum and energy fluxes are confined ithin the IR, so that the IR is directly affected by sell. Sell transfers energy to the atmosphere and enhances the turbulent kinetic energy in the IR. The transfer of momentum results in acceleration or deceleration of the airflo near the surface, hich leads to the fact that the surface turbulent stress differs from the turbulent stress (sea drag) far above the sea surface. Folloing-ind sell accelerates the flo, hich for a very lo ind results in the jetlike profile of the ind velocity (sell-driven ind) ith its maximum at the IR height. In this case the surface stress is strongly enhanced, hile the sea drag is less affected. Opposite-ind sell decelerates the airflo, hich for a steep sell could cause the reverse airflo. The sea drag in this case is considerably enhanced as compared ith the folloing-ind sell case. Cross-ind sell

12 APRIL 004 KUDRYAVTSEV AND MAKIN 945 FIG. 9. Angle beteen the turbulent shear stress vector and the ind velocity at 10-m-height vector as a function of the sell-ind angle : (a) u , (b) u , (c) u 10.0, and (d) u m s 1. The sell phase velocity C 15 m s 1. Solid lines ith pluses and crosses sho the turbulent stress at 10-m height and the sell slopes AK 0.1 and AK 0.05, respectively. Solid lines and dashed lines sho the turbulent stress at the surface and the sell slopes AK 0.1 and AK 0.05, respectively. causes the rotation of the ind velocity vector ith height, hich leads to the deviation of the turbulent stress vector from the ind velocity vector. All of these peculiarities in the ind profile and the stress are quite pronounced at very lo inds and disappear as the ind speed increases or as the sell phase velocity decreases. In fact, it is the inverse ave age parameter u 10 /C (or u * /C) that characterizes the impact: the smaller u 10 /C, the stronger is the impact of the sell on the MABL. The impact of sell directly depends on the sell slope being more pronounced for a steeper sell. This is because both the ave-induced momentum and the energy fluxes increase rapidly ith an increase of the slope. Given the values of the ind speed, the sell phase velocity, and the slope typical for the ocean experiments of Donelan et al. (1997) and Drennan et al. (1999), the model is able to reproduce qualitatively and quantitatively the main experimental findings: the impact of sell on the sea drag is ell pronounced for oppositeind sell, less pronounced for cross-ind sell, and occurs only at lo ind speeds u 10 6ms 1. At lo inds (and zero ambient ind) the model predicts the existence of ell-pronounced ave-driven ind, observed, for example, by Harris (1966) in the laboratory. We remind the reader that the model possesses a tuning constant f 5, hich as chosen to fit the model ind attenuation/groth rate to the laboratory data of Donelan (1999). Since only these laboratory data are currently available (at least for the attenuation rate), future corrections of the magnitude of the attenuation rate of sell in real conditions could result in the revision of the effects of sell on the atmospheric boundary layer. FIG. 10. Vertical distribution of ave-driven ind: solid line is modeled; open circles are data from Harris (1966).

13 946 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 34 FIG. 11. (top) Drag coefficient CD 10 at 10-m height and (bottom) the surface drag coefficient CD sur as a function of the ind speed u 10 and for different sell-ind angles: (left) folloingind sell 0, (middle) cross-ind sell 90, and (right) opposite-ind sell 180. No-sell case is solid lines; the sell phase velocity C 10 m s 1 is dashed dotted lines; C 0ms 1 is dashed lines; C 30ms 1 is dotted lines. The sell slope is AK Acknoledgments. The authors acknoledge the support by the Netherlands Organization for Scientific Research (NWO) through the Dutch Russian Scientific Co-operation Programme 001, Project , and by the Office of Naval Research (ONR), Grant N (PR No. 03PR ). VNK acknoledges the support of the Noregian Space Centre under Contract JOP , and INTAS cooperation via Project INTAS APPENDIX Interaction of Turbulent Airflo ith a Surface Wave a. Governing equations A ave-folloing orthogonal coordinate system ( 1,, 3 ) defined as Kx x Ae sinkx 1, (A1) x, and (A) Kx x Ae coskx 1 (A3) is used in the present study (e.g., Sullivan et al. 000). As it follos from (A3), the surface 3 0 coincides ith the avy surface (1): z A coskx 1. The Navier Stokes equations ui p (uiu) j f i (A4) t x x and the continuity equation u j 0 x j j i (A5) ritten above in the Cartesian coordinate system, in the ave-folloing -coordinate system, take the form 1 ui j,ip (ui j) f i and (A6) J t J j j 0, j j (A7) here u i is the ind velocity component in the Cartesian coordinate system x i, p is the pressure normalized on the air density, f i is the component of the molecular viscosity force, j,i j /x i, J 1,1 3,3 1,3 3,1 is the Jacobian of the coordinate system transformation, and i is the component of the contravariant flux velocity defined by i uj i,j /J. (A8) To derive (A6) and (A7), the identity

14 APRIL 004 KUDRYAVTSEV AND MAKIN 947 as used. j,i 0 J j (A9) b. Momentum and energy conservation equations Any airflo quantity y can be split into a sum of the mean (averaged over 1 and : y), the ave-correlated (defined as a deviation of y averaged over at given 1 y from its mean value y : ỹ y y), and the random (turbulent) (y y y) components. For example, the decomposition of the air pressure is p p p p. (A10) Averaging of (A6) over at a given 1 results in the folloing equation: 1 ui (ui j) J t j 1 (j,ij p) ( ij), (A11) j here the action of the molecular viscosity force on the mean and the ave-induced motion is neglected; ij u i j is the tensor of the turbulent stress based on the multiplication of the Cartesian and contravariant velocity components. The one-dimensional momentum conservation equation for the horizontal component is found by averaging (A11) over 1 and reads ( ) 0, (A1) 3 here 1or, 3 is the component of the turbulent shear stress, and is the ave-induced mo- mentum flux defined as ũ 3 p 3,/J. (A13) Equation (A1) provides the condition of the constantflux layer: the sum of the ave-induced stress and the shear stress is constant over the height and is equal to the vertical flux of the horizontal momentum 0 far aay from the sea surface; that is, 0. (A14) The conservation equation for the ave-induced energy can be derived by multiplying (A11) by u i and averaging the obtained equation over the horizontal space U ũ i F ij 0, (A15) 3 3 j here F is the energy flux of the ave motion: F p ũ 3 ũj j3, (A16) and U u C. The expression for F is ritten ith second-order accuracy in the ave slope. To derive j (A15), (A1) is used, hich, being multiplied by U, gives the energy conservation equation for the mean flo (its explicit form is not needed here). Equation (A15) shos that the energy of the ave-induced motion is provided by the balance of the divergence of the ave energy flux, the ork of the ave-induced momentum flux against the mean ind velocity gradient, and the energy loss due to the correlation of the turbulent stress and the gradient of the ave-induced ind velocity. The energy balance equation for the TKE is obtained by multiplying (A6) by u i, averaging it (hich results in the conservation equation for the total energy), and subtracting (A15). The TKE conservation equation reads U ũ t i F ij Diss 0, (A17) 3 3 j here F t is the vertical flux of the TKE, Diss is the TKE viscous dissipation, and the second and the third terms describe the production of the TKE by the mean and the ave-induced motion. Notice that the latter term describes the energy sink of the ave motion into the turbulence [the last term in (A15)]. Using (A15) the TKE balance equation can be ritten as U t (F F ) ( ) Diss 0. (A18) 3 3 c. Equations for a small ave-induced disturbance in the airflo To obtain expressions for the surface energy and momentum flux, equations describing a small linear aveinduced variation in the airflo are needed. To the first order in ave slope (AK K 1) the linearized (A11) ritten in terms of the amplitude ŷ (hich is a complex variable) of the harmonic oscillation ỹ ŷ exp(ikx 1 ) are du1 ˆ 13 iku1û1 ˆ ikpˆ and (A19) 3 d3 3 pˆ iku1û3, (A0) 3 here terms containing the normal component of the Reynolds stress are omitted [as it as argued by Kudryavtsev et at. (001), their role is not important]. In AK order, the amplitude of the contravariant vertical velocity ˆ 3 in (A19) is ˆ 3 û 3 3,1 U 1. Correspondingly, in AK order the continuity equation [(A7)] reads ˆ ikû 3 1 3,1,3 U 1, (A1) 3 here 3,1,3 3 /x 1 x 3. Equations (A19), (A0), and (A1) describe the linear undulation of the turbulent airflo over the surface ave.

15 948 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 34 d. Attenuation/groth rate of surface aves First, the attenuation/groth of aves due to the ork of the surface stress against the ave orbital velocity described by the second term in (8) is considered. The amplitude of the ave-induced variation in the shear stress inside the IR ˆ 1 is given by Meirink et al. [003, their (3)]. In present terms this equation reads (neglecting terms of the second order) û 1/ ˆ (1 cos )b 1 s1 û 1/ (1 cos )b 1. (A) ln(l/ 0) The second equality results from an estimate û 1 / ln û 1 /ln(l/ 0 ), here û 1 is the drop of the ind velocity variation amplitude over the IR: û 1 û 1 (l) C. The horizontal velocity variation in the OR is given by Kudryavtsev et al. (001) as K û () U () exp(k) Ue d 1, U 1 () u 1 () C, and the velocity drop in (A) at small Kl can be expressed as u* û1 Ul C cos ln(kl). (A3) Accounting for (A) and (A3) the ave energy attenuation/groth rate is ũs s1 E [ ] a ln(kl) C * (1 cos ) cos 1, ln(l/ ) u C 0 l (A4) here E ga / is the sell energy and u l */ (b 1/ ) ln(l/ 0 ) is the ind speed at l [see (0)]. The energy transfer due to the action of the ave slope correlated pressure is described by the first term in (8). The imaginary part of the amplitude of the surface pressure pˆ s, hich is responsible for the energy transfer, is found from (A0) integrated over height: 3 z 0 Im(pˆ ) K Re(û )U d ŵ U, (A5) s 1 l K here U K U 1 (K 1 ) and ŵ l is the real (elevation correlated) part of the vertical velocity at l. To obtain the estimate of Im(pˆ s) the folloing conditions ere accounted for: inside the OR, Re(û 3 ) attenuates exponentially [i.e., Re(û 3 ) ŵ l exp(k); see Kudryavtsev et al. 001, their (57)]; the pressure is approximately constant over the IR; and at small Kl the integral K # l U 1 exp(k)d can be approximated by U K. Inside the IR the profile of the real part of the vertical velocity is approximately linear over the height: Re(û) (/l) l. Assuming that the dynamics of the IR results from the balance beteen the advective term and the Reynolds shear stress terms, and that the pressure term does not play a dominant role, the real part of the vertical velocity on the top of the IR ŵ l follos from (A19) and (A1): ˆ s1 ŵ l, (A6) Ul */ cos here ˆ s1 is the real quantity defined by (A) ith (A3). For fast-propagating surface aves (U l C) the estimate of the vertical velocity is 1 1/ ŵl (1 cos ) ln (l/ 0)b. For aves propagating ith the ind speed (U l 0), 1/ ŵl (1 cos )b, and for sloly propagating aves [U l u 1 (l)] the ver- tical velocity is 1 1/ ŵl (1 cos ) ln (l/ 0)b. Thus the real part of the vertical velocity attains the maximal magnitude in the vicinity of U l 0. This is consistent ith the model calculations of Mastenbroek et al. (1996), Sullivan et al. (000), and Kudryavtsev et al. (001). For practical applications (A6) ith (A) is ritten as 1/ 1 ŵl (1 cos )b ln (l/ 0)(Kl, ), (A7) here (Kl, ) is a correction function, hich allos for all three above-mentioned asymptotes: (Kl, ) 1 c p(kl/) ln(l/ 0) sign(u l)kl ln(kl)/ cos/[(1 cos )]. (A8) Here c p is a tuning constant of order 1 chosen as c p by fitting the analytical expression to the model calculations. Last, the dimensionless attenuation/groth rate due to the action of pressure on the ave slope is found from (A5) and (A7): acp sz1 a UK * p (1 cos )(Kl, ). (A9) E û C REFERENCES Belcher, S. W., 1999: Wave groth by non-separated sheltering. Eur. J. Mech., 18B, , and J. C. R. Hunt, 1993: Turbulent shear flo over sloly moving aves. J. Fluid Mech., 51, Cohen, J. E., and S. E. Belcher, 1999: Turbulent shear flo over fastmoving aves. J. Fluid Mech., 386, Davidson, K. L., and A. K. Frank, 1973: Wave-related fluctuations in the airflo above natural aves. J. Phys. Oceanogr., 3, Donelan, M. A., 1999: Wind-induced groth and attenuation of lab- l