The effects of wind-driven waves and ocean spray on the drag coefficient and near-surface wind profiles over the ocean

Transcription

1 Acta Oceanol. Sin., 216, Vol. 35, No. 11, P DOI: 1.17/s The effects of wind-driven waves and ocean spray on the drag coefficient and near-surface wind profiles over the ocean ZHANG Ting 1, 2, SONG Jinbao 1 *, LI Shuang 1, YANG Liangui 2 1 Ocean College, Zhejiang University, Hangzhou 31, China 2 School of Mathematical Science, Inner Mongolia University, Hohhot 1, China Received 15 November 215; accepted 5 February 216 The Chinese Society of Oceanography and Springer-Verlag Berlin Heidelberg 216 Abstract By introducing a wave-induced component and a spray-induced component to the total stress, a mathematical model based on the Ekman theory is proposed to detail the influence of wind-driven waves and ocean spray on the momentum transport in a marine atmosphere boundary layer (MABL). An analytic solution of the modified Ekman model can be obtained. The effect of the wave-induced stress is evaluated by a wind wave spectrum and a wave growth rate. It is found that the wave-induced stress and spray stress have a small impact compared with the turbulent stress on the drag coefficient and the wind profiles for low-to-medium wind speed. The spray contribution to the surface stress should be much more taken into account than the winddriven waves when the wind speed reaches above 25 m/s through the action of a spray stress. As a result, the drag coefficient starts to decrease with increasing wind speed for high wind speed. The effects of the winddriven waves and spray droplets on the near-surface wind profiles are illustrated for different wave ages, which indicates that the production of the spray droplets leads the wind velocity to increase in the MABL. The solutions are also compared with the existed field observational data. Illustrative examples and the comparisons between field observations and the theoretical solutions demonstrate that the spray stress has more significant effect on the marine atmosphere boundary layer in the condition of the high wind speed compared with wave-induced stress. Key words: wind-driven waves, Ekman theory, marine atmosphere boundary layer, spray droplets Citation: Zhang Ting, Song Jinbao, Li Shuang, Yang Liangui The effects of wind-driven waves and ocean spray on the drag coefficient and near-surface wind profiles over the ocean. Acta Oceanologica Sinica, 35(11): 79 85, doi: 1.17/s Introduction Accurately modeling the inclusion of the sea surface wave-induced stress and sea spray has been recognized a crucial requirement to understand the air-sea exchanges of momentum, moisture and heat. On the other hand, the exchanges of momentum result in the storm surges, the waves and the currents in the ocean. A lot of research demonstrated that surface waves play an important part on the dynamics and kinematics of the MABL (Janssen, 1989; Belcher and Hunt, 1993; Sullivan et al., 24; Song, 29; Song et al., 215) for weak and medium winds over the ocean. Nevertheless, as the wind speed approaching a hurricane strength, ocean spray droplets proliferate. The process that the sea sprays thrown into the air extracts the momentum from the near-surface wind and slows the speed of the near-surface wind has been pointed (Munk, 1955; Wu, 1973; Andreas, 22, 24). However, quite a few of recent research and field observations (Kudryavtsev and Makin, 211; Tang et al., 213; Rastigejev and Suslov, 214) have demonstrated that the sea spray droplets suppress the turbulence to accelerate the air flow near the sea surface. The research to explicitly study the effect of the wind-driven waves and spray droplets on the MABL is not enough. The sea spray droplets are generated on the sea surface by the wind tearing off the wave breaking crests mainly. During wave breaking, air is entrained into the water, producing bubbles that burst upon emerging on the sea surface, ejecting droplets into the air. Mechanical mixing of spray with the near-surface airflow influences the momentum exchange between the atmosphere and the upper sea layer (Bortkovskii, 1973; Borisenkov, 1974; Ling and Kao, 1976). The rate at which the spray droplets of any given size are produced on the sea surface is essential for many applications. An approximate sea spray generation function is introduced, which quantifies the rate of the sea spray droplets produced on the sea surface (Fairall et al., 1994; Andreas, 1998, 22). Moreover, the theory of the motion of suspended particles in a turbulent flow of incompressible fluid was developed by Barenblatt (1955). This approach was adopted by Kudryavtsev and Makin (211) to develop a model of the MABL in present of the spray. In this study, considering the Coriolis force in the momentum conservation equation, the momentum flux is not constant in the MABL (Song et al., 215). The modified Ekman mod- Foundation item: The National Natural Science Foundations of China under contract Nos and ; the National High Technology Research and Development Program (863 Program) of China under contract No. 213AA12283; the Strategic Priority Research Program of the Chinese Academy of Sciences under contract No. XDA *Corresponding author, songjbju.edu.cn

2 8 ZHANG Ting et al. Acta Oceanol. Sin., 216, Vol. 35, No. 11, P el is presented by dividing the total stress into the turbulent stress, the wave-induced stress and the spray stress. The theory solution can be obtained for the model. The effects of the waveinduced stress and the spray droplets on the drag coefficient and the steady near-surface wind profiles in the MABL are illustrated for different wind speeds by choosing a wind wave spectrum. The solutions are also compared with observations from existed field observational data (Liu et al., 212). 2 The modified Ekman model The modified Ekman equation (Makin, 28; Song et al., 215) is written t if ½ a (U U g ) = ; (1) where t = tx + i ty is the complex total stress; f is the Coriolis parameter; ½ a is the density of the air ( ½ a is assumed a constant here); U = u + iv is the complex horizontal wind velocity with u pointing to the x direction and v pointing to the y direction; and U g = u g + iv g is the complex geostrophic wind velocity. Taking into account the effects of the wind-driven waves and the spray droplets, the total stress can be written as t (z) = (z) + w (z) + sp (z) ; (2) where (z) is the complex turbulent stress; w (z) is the wave-induced stress; and sp (z) is the spray stress. Because the Coriolis force is included in Eq. (1), the total stress is not a constant with a height in an atmosphere surface layer. As above, we denote a friction velocity u * right on the ocean surface (Janssen, 1991): ½ a u 2 = t () = () + w () + sp () : (3) As usual, in the atmospheric surface layer, the turbulent stress (Andreas, 24; Song et al., 215) can be written as (z) ½ a K M = ½ a (z + z ; where K M (z) is the vertical eddy viscosity coefficient; κ=.4, is von Karman s constant; v = (j ()j =½ a ) 1=2, is the local surface friction velocity; and z is the sea surface aerodynamic roughness length. The wave-induced stress w (z) (Makin et al., 1995; Semedo et al., 29; Song et al., 215) is defined as w (z) = ½ w Z Z! k K E (k; µ) e 2kz dkdµ; (5) where ½ w is the density of sea water; ω is the angular frequency of the waves; K = k x + ik y is k = k2 x + ky 2 1=2! = p the horizontal complex wavenumber and ; gk, is given by the dispersion relation of gravity waves, g is the gravity acceleration; E (k; µ) is the directional wavenumber spectrum of surface waves and θ is the direction of a wave vector; and β is the rate of growth of waves. According to the model of Kudryavtsev and Makin (211), we assume that the droplets are injected into the airflow with a velocity equal to the wind velocity once generated. In other words, the droplets did not extract the momentum from the airflow to be (4) accelerated as Andreas (24) suggested. After being injected at the altitude of breaking crests, the droplets are falling and transferring the momentum from upper layers of higher velocity to the airflow in the spray generation layer. In this situation, the spray stress should be regarded as negative, written as µ z sp (z) = sp () exp ; (6) H s =7 H s = :63p u (:91 2 c p ) 3=2 where, is the significant wave g height for the wind-driven sea (Toba, 1972; Liu et al., 212); c p is the phase speed of the spectral peak; U 1 is the wind speed at 1 m height above the sea surface; and sp () (Andreas, 24; Innocentini and Gonçalves, 21) is the spray stress right on the sea surface defined as sp () = 4 3 ½ wu sp Zr H r L r 3 df dr dr; (7) u sp = u µ ln zsp where r is the droplet radius; is the droplet horizontal velocity, in which z sp = :63 H s represents all the z droplets above the sea surface (Andreas, 1992). df The sea spray generation function satisfies the relation: dr 8 df >< a 1 r 1 (3 < r < 75 ) ; dr = Re1:5 a 2 r 3 (75 < r < 2 ) ; (8) >: a 3 r 8 (2 < r < 5 ) ; where a 1 = 7: ; a 2 = 44:1; and a 3 = 1: The result for the integral of Eq. (7) calculated from r L = 3 to r H = 5 is 8: Re 1:5 m/s; Re is windsea Reynods number (Innocentini and Gonçalves, 21), defined as Re = u2 ¾ p º ; (9) where ¾ p = 2 :13g, is the peak angular frequency for the 28u fully developed sea spectrum; and º = 1: m 2 /s, is the kinematic viscosity of air. The boundary conditions are U = z = ; (1a) U = U g z! 1: (1b) Supposing V = U U g, Eq. (1) and the boundary conditions Eq. (1) can be rewritten as where (z + z if V + T w + T sp = ; (11) T w (z) Z Z w = 2½ w=½ a V = U g z = ; (12a) V = z! 1; (12b)!K E (k; µ) e 2kz dkdµ; (13)

3 ZHANG Ting et al. Acta Oceanol. Sin., 216, Vol. 35, No. 11, P T sp (z) µ µ sp z z = sp () =½ a exp : (14) H s =7 H s =7 The solution of Eqs (11) (14) can be determined theoretically. In this study, we are interested in the impact of the wind-driven waves and the spray droplets on the MABL. Rewriting Eq. (11), we have (z + z V(z) 2 if V(z) = T w (z) + T sp (z) : (15) (z) = ½ a (z + z ½ = ½ a K ( ) [I ( ) K ( ) I ( ) K ( )] Z z K ( ) [T w (z) + T sp (z)] dz+ [K ( ) I ( ) I ( ) K ( )] [T w (z ) + T sp (z )] dz p if (z + z ) K ¾ ( ) K ( ) U g ; (2) The general solution of Eq. (15) (Song et al., 215) is V(z) = A I ( ) + B K ( ) + à (z) ; (16) where A and B are constants to be determined; I and K are the first and second kinds of the modified Bessel functions with s if (z + z ) = 2 ; and à (z) is a special solution of Eq. (15), that is 8 à (z) = 2 < : I ( ) z K ( ) [T w (z ) + T sp (z )] dz K ( ) 9 = I ( ) [T w (z ) + T sp (z )] dz ; ; z (17) s if (z = 2 + z ) in which. From the boundary condition Eq. (12), we can conclude A=, and B = 2 ½ I ( ) K ( ) K ( ) [T w (z) + T sp (z)] dz 9 = I ( ) [T w (z) + T sp (z) dz] ; U g K ( ) ; (18) r if z where = 2. Thus, the solution of Eqs (11) (14) reduces to where I and K are the derivatives of I and K respectively. The first two terms on the right of Eq. (2) are produced by the interaction of three factors, the wave, the spray and the turbulence, which modify the turbulent stress. Using Eq. (2), we have () = ½ a z = ½ a z= K ( ) 8 +1 < Z K : ( ) [T w (z) + T sp (z)] dz p o if z K ( ) U g F (z ; v ) : (21) According to the definition of the local friction velocity v * and Eq. (3), the relation between the friction velocity u * and the local friction velocity v * reduces to u 2 = 1 ½ a [F (z ; v ) + w () sp ()] ; (22) v 2 = jf (z ; v )j = ½ a u 2 w () + sp () ; (23) where jf (z ; v )j denotes the modulus of complex number F (z ; v ). 3 Illustrative examples Before calculating the wind profile of Eq. (19), taking ½ a = 1:2 kg/m 3, ½ w = 1 25 kg/m 3, f=1-4 s -1 for the model Eq. (1). As the ocean spray droplets produced under a medium high wind speed are considered, the roughness length z of the sea surface is determined as Charnock s (Andreas, 24) relation z = :18 5 u2 g : (24) U(z) = V(z) + U g = B K ( ) + à + U g = 2 ½ I ( ) I ( ) K ( ) K ( ) Z z K ( ) [T w (z) + T sp (z)] dz+ [K ( ) I ( ) I ( ) K ( )] [T w (z ) + ¾ T sp (z )] dz + 1 K ( ) U g : K ( ) (19) According to the expression of the wave-induced stress Eq. (5), we use the spectrum for long and short wind-driven waves (Elfouhaily et al., 1997; Polnikov, 213) described by µ 2 E (µ; k) = k 3 (B l + B h ) cos 2 µ; (25) where B 1 is the low-frequency spectrum, that is B l = 1 p c p c F p; (26) Using the solution Eq. (19), from the definition of the turbulent stress Eq. (4), we have where c =!=k is the phase speed of waves; F p = L pm J p e 1 p 1 ³q k kp 1 ; (27)

4 82 ZHANG Ting et al. Acta Oceanol. Sin., 216, Vol. 35, No. 11, P = ³ L pm = e 5 kp 2 4 k ; (28) J p = ; (29) = e ³q k kp 1 2= 2¾ 2 ; ¾ = : ; (3) k p = g U ; (31) ( 1:7 :84 < 1 < 1; 1:7 + 6 ln 1 1 < 1 < 5; In addition, B h is the high-frequency spectrum, that is: F m = e 1 4 m = 1 2 B h = 1 2 m ³ k (32) :23 F m (33) c km 1 2; k m = 2g=:23 2 (34) ( 1 + ln (u =c m ) u < c m ; ln (u =c m ) u > c m ; (35) where a wave age Ω is defined as c p =U 1. Seas are said to be fully developed, mature and young when the inverse wave age Ω 1 has values close to.84, 1.5 and 2.. For the wind wave spectrum, the growth rate β (Belcher and Hunt, 1993; Song et al., 215) is specified as = c! ½ ³ a u 2 ; (36) ½ w c where c β is the wave growth rate coefficient for both high-frequency spectrum and low-frequency spectrum (Stewart, 1974), defined as µ cu c = 1:5 1 ln : (37) kz Assuming the wind is directed to the x axis, we have by using Eq. (5) and Eq. (13) p wx (z) = 2½ w g Z k 1=2 cos µe (k; µ) e 2kz dkdµ; (38) p T wx (z)= 4½ w =½ a g Z k 3=2 cos µe (k; µ) e 2kz dkdµ; (39) wy (z) = T wy (z) = : (4) 3.1 The drag coefficient with spray and wind-driven waves Commonly, the traditional formulation connecting the momentum flux on the surface to the mean wind speed at some reference level (usually at 1 m) U 1 via the drag coefficient C d (Kudryavtsev and Makin, 211; Innocentini and Gonçalves, 21; Liu et al., 212) is adopted as ½ a u 2 = ½ a C d U 2 1: (41) Rearranging terms of Eq. (41), we have µ 2 u C d = : (42) U 1 For a given 1 m wind speed and a wind wave spectrum denoted by Eq. (25), geostrophic wind velocity U g can be calculated using Eq. (19) and F (z ; v ) can be calculated by substituting Eqs (14) and (39) into Eq. (21). At last, the value of frictional velocity u * can be obtained from Eqs (22) and (23). Furthermore, the value of C d can be calculated from Eq. (42). The dependence of the 1 m drag coefficient C d on the 1 m wind speed U 1 for different wave ages is shown in Fig. 1. We can see that C d reaches a maximum but then decreases when the 1 m wind speed exceeds 25 m/s (see the dashed curves and solid curves in Fig. 1). That means the spray droplets take effects on the momentum for medium-to-high wind speed, which is consistent with the results of Andreas (24) and Kudryavtsev and Makin (211). The reduction of the drag coefficient under the high wind speed is due to that the sea spray droplets generated by wave breaking and wind tearing wave crests prevent the sea surface from being dragged by the wind directly. It also noticed that the younger surface waves lead to the values of the drag coefficient bigger for the high wind speed by comparing the solid lines in the figures, which is agreement with the results of Liu et al. (212). Correspondingly, we can find that the larger the drag coefficient, the higher wind speed at which the drag coefficient begin to decrease. Fig. 1. The estimate of the 1 m drag coefficient for different inverse wave ages with spray and the wind-driven waves (solid line), only with the spray (dashed line), and only with the wind-driven waves (dotted line). a. Ω -1 =.84, b. Ω -1 =1.5, and c. Ω -1 =2..

5 ZHANG Ting et al. Acta Oceanol. Sin., 216, Vol. 35, No. 11, P Table 1. The values of u *, v *, wx(), sp() for different wind speeds 1, 2, 3, 4 and 5 m/s U 1 /m s u * /m s v * /m s τ wx ()/Pa τ sp ()/Pa The little difference between the dashed line and the solid line states that the wind-induced stress acts to reduce the value of C d for medium and high wind speeds with the spray present. Generally, the wave-induced stress is positive, denoting the momentum is supplied from the atmosphere to the sea surface. Nevertheless, the addition of wave-induced can also influence the turbulent stress shown in Eq. (2) to reduce the momentum flux on the water surface, which results in the reduction of C d in present of spray. The values of u *, v *, wx() and sp() are shown in Table 1 for different wind speeds. The vertical profiles of the wave-induced stress wx (z) and the spray stress sp (z) along the wind direction for fully developed seas are shown in Fig. 2 to investigate the air-sea exchange in the MABL further. Figure 2a shows the stress for the low wind condition with 1 m wind speed U 1 =1 m/s, which illustrates that the wave-induced stress and spray stress are almost equivalent, while the values of both the wave-induced stress and the spray stress are much smaller than the turbulent stress for the low wind speed through calculation. The corresponding vertical profiles of the stress for the medium wind speed with U 1 =2 m/s are shown in Fig. 2b, where the wave-induced stress is so small compared with the spray stress that can be almost neglected for the total stress. It noted from Figs 2a and b that the wave-induced stress grows slowly with the increasing wind speed, whereas the rate of growth of the waveinduced stress is so much smaller than the spray stress that the values of wave-induced stress are totally ignored from the total stress for the high wind speed, such as the profiles of the wave-induced stress in Fig. 2c. The behavior of the drag coefficient in Fig. 1a is related to the stress shown in Fig. 2. The three overlapped lines under low-to-medium wind speeds shown in Fig. 1a illustrate that the drag coefficient is mainly determined by the turbulent stress rather than both the wave-induced stress and the spray stress. What is more, the spray droplets make the drag coefficient C d decrease to a great extent, while the effect of the winddriven waves acts to slightly reduce the values of C d under medium-to-high wind speeds. This process is agreement with the theoretical results and field and experimental observations proposed by Liu et al. (212). 3.2 The wind profiles with spray and wind-driven waves Considering the values of the wave-induced stress are small for various wind speeds as mentioned above, we assume the effect of the wave-induced stress on the near-surface wind speed can be neglected for the high wind speed. Thus, in this section, we account for the effect of the spray droplets on the near-surface wind speed for various wave ages under different high wind speeds. The vertical profiles of magnitude of the wind velocity are calculated from Eq. (19) for various 1 m wind speeds in this section. The solutions presented by Eq. (19) are obtained for the 1 m wind speed 3, 5 m/s with different wave ages are shown in Fig. 3. Figure 3 shows the wind profiles can be modified by the ocean spray for the high wind speeds. The small difference between the three colorful solid curves in Fig. 3 demonstrates that the wind speed is also related to the wave states. The acceleration of the wind is mainly due to the fact that the sea spray droplets transport the momentum from an up layer to the airflow near the sea surface. Besides, the spray droplets suspended in the air also suppress the turbulence in the MABL, which is agreement with the results of the model proposed by Rastigejev and Suslov (214). It is also noted that the greater wind speed is, the more ocean spray droplets are generated, resulting in the more obvious effect on rising the near-surface wind speed. 4 Comparison with observational data Figure 4 shows the comparisons of the relation between the drag coefficient and the 1 m wind speed for different wave ages proposed in Section 2 with the field observations. The sources of the field observational data come from the measurements by Banner (1999), Jarosz et al. (27) and Powell et al. (23). The experimental values of Powell et al. (23) were obtained from the wind profiles. Powell et al. (23) measured the wind profiles in several layers above the sea surface by releasing GPS wind drop sondes in 15 storms from 1997 to The values of estimated from the wind profiles measured in an above 2 16 m surface layer are considered most valid and shown in Fig. 4. Moreover, the bottom-up determination method of the air-sea momentum exchange were used by Jarosz et al. (27) to measure the sea surface stress under the hurricane wind condition. Fig. 2. The values of u *, v *, wx(), sp() for different wind speeds 1, 2, 3, 4 and 5 m/s.

6 84 ZHANG Ting et al. Acta Oceanol. Sin., 216, Vol. 35, No. 11, P Fig. 3. wind profiles with spray (solid lines) and without spray (dashed lines) for inverse wave age =.84 (blue lines), 1.5 (red lines) and 2. (green lines) for 1 m wind speed is 3 m/s (a) and 5 m/s (b). Fig. 4. Comparison between the drag coefficient and wind speed relations calculated by Eq. (42) for inverse wave age.84 (blue line), 1.5 (red line) and 2. (green line). The blue marks are the field measurements from Banner (1999) by the airplane; the magenta marks are field measurements from Jarosz et al. (27) based on resistance coefficient.1; and the light blue marks are observations from Powell et al. (23) measured by GPS dropsondes in the vicinity of the hurricane eye walls. Their estimates were based on the observations on the outer continental shelf in the northeastern Gulf of Mexico as Hurricane Ivan passed directly over on 15 September 24. The drag coefficients under the wind speeds ranging from 2 m/s to 48 m/s were obtained by them (shown in Fig. 4). Most of the field observations we used in present study are obtained under moderate to high wind speed. In this situation, the wind waves and the spray droplets are more dominant than swell to influence the MABL. Figure 4 shows that the results presented by Eq. (42) can cover the range of the existing field observations well, and can generalize the scatter of current observations to some extent. The reduction of the drag coefficient at the wind speed exceeding 25 m/s shows in both the field observations and the solutions of our model for different wave ages. The maximum values of the drag coefficient for wind generated sea reach about.2 3 resulting from Eq. (42), which is very consist with the field measurements by Jarosz et al. (27) and Powell et al. (23). 5 Discussion and conclusions We have introduced an atmospheric boundary layer model based on the Ekman theory to investigate the effect of the winddriven waves and the sea spray on the marine atmosphere boundary layer from low to strong wind. An analytic solution is obtained for the modified Ekman model when the surface total stress is broken up into the turbulent stress, the wave-induced stress and the spray stress. The solutions of the drag coefficient and wind profiles near the sea surface depend on the directional spectrum of waves, the wave age, wave growth rate, the significant wave height, the sea spray generation function, the geostrophic wind velocity, the Coriolis parameter and the densities of water and air. As illustrative examples, the ocean states are specified by the directional spectrum for long and short wind-driven waves (Elfouhaily et al., 1997; Polnikov, 213). The wave growth rate defined by the formulation of Belcher and Hunt (1993). The ocean spray generation function presented by Andreas (22) is adopted to develop the expression of the spray stress (Andreas, 24; Innocentini and Gonçalves, 21), which has a considerable impact on the MABL when the wind speed reaches above 25 m/s. At first, we present the relations between the 1 m drag coefficient and 1 m wind speed with different wave ages. The results are also compared with those obtained by neglecting the waveinduced stress and the spray stress. The vertical profiles of the wave-induced stress and the spray stress are illustrated to show the comparison of the wave-induced stress and the spray stress for different wind speeds for fully developed wind generated sea. The wind profiles calculated for both medium high and high wind conditions with different wave ages are also illustrated to restate the effect of the wind-driven waves and the spray droplets on the near-surface wind speed. The comparisons between the field observations and the theoretical predictions demonstrate that the momentum in the MABL is significantly influenced by the wind-driven waves and the sea spray droplets from low to high wind speed. The winddriven waves slightly modify the surface friction velocity and the drag coefficient, whereas the impact of spray on the surface friction velocity and drag coefficient becomes more and more significant with the increasing wind speed. Although we only illustrate the results for the directional spectrum for long and short wind-driven waves, this model can be extended to other kinds of wind wave spectrums. It is noted that we only use the spectrum of the win-generated sea to discuss the effect of surface waves on the drag coefficient and the near-surface wind profiles without investigating the effect of swell on the MABL. It is also noted that we neglect many other

7 ZHANG Ting et al. Acta Oceanol. Sin., 216, Vol. 35, No. 11, P phenomena over the ocean, such as atmospheric stability and atmospheric rolls. These effects should be further considered to accurately correct the model. References Andreas E L Sea spray and the turbulent air-sea heat fluxes. J Geophys Res, 97: Andreas E L A new sea spray generation function for wind speeds up to 32 m/s. J Phys Oceanogr, 28: Andreas E L. 22. A review of sea spray generation function for the open ocean. In: Perrie W A, ed. Atmosphere-Ocean Interactions, Vol. 1. Southampton, UK: WIT Press, 1 46 Andreas E L. 24. Spray stress revisited. J Phys Oceanogr, 34: Banner M L, Chen Wei, Walsh Edward J, et al The southern ocean waves experiment. Part Ⅰ: overview and mean results. J Phys Oceanogr, 29: Barenblatt G I On the motion of suspended particles in a turbulent flow taking up a half-space or a plane open channel of finite depth. Prikl Mat Meh, 19: Belcher S E, Hunt J C R Turbulent shear flow over slowly moving waves. J Fluid Mech, 251: Borisenkov E P Some mechanisms of atmosphere-ocean interaction under stormy weather conditions. Problems Arctic and Antarctic, 1: Bortkovskii R S On the mechanism of interaction between the ocean and the atmosphere during a storm. Fluid Mech Sov Res, 2: Elfouhaily T, Chapron B, Katsaros K, et al A unified directional spectrum for long and short wind-driven waves. J Geophys Res, 12: Fairall C W, Kepert J D, Hollannd G J The effect of sea spray on surface energy transports over the ocean. Global Atmos Ocean Syst, 2: Innocentini V, Gonçalves I A. 21. The impact of spume droplets and wave stress parameterizations on simulated near-surface maritime wind and temperature. J Phys Oceanogr, 4: Janssen P A E M Wave-induced stress and the drag of air flow over sea waves. J Phys Oceanogr, 19: Janssen P A E M Quasi-linear theory of wind-wave generation applied to wave forecasting. J Phys Oceanogr, 21: Jarosz E, Mitchell D A, Wang D W, et al. 27. Bottom-up determination of air-sea momentum exchange under a major tropical cyclone. Science, 315: Kudryavtsev V N, Makin V K Impact of ocean spray on the dynamics of the marine atmospheric boundary layer. Boundary- Layer Meteorology, 14: Ling S C, Kao T W Parameterization of the moisture and heat transfer process over the ocean under whitecap sea states. J Phys Oceanogr, 6: Liu Bin, Guan Changlong, Xie Lian The wave state and sea spray related parameterization of wind stress applicable from low to extreme winds. J Geophys Res, 117: CJ22 Makin V K. 28. On the possible impact of a following-swell on the atmospheric boundary layer. Boundary-Layer Meteorology, 129: Makin V K, Kudryavtsev V N, Mastenbroek C Drag of the sea surface. Boundary-Layer Meteorology, 73: Munk W H Wind stress on water: an hypothesis. Quarterly Journal of the Royal Meteorological Society, 81: Polnikov V G Extended verification of the model of dynamic near-surface layer of the atmosphere. Izvestiya, Atmospheric and Oceanic Physics, 49: Powell M D, Vickery P J, Reinhold T A. 23. Reduced drag coefficient for high wind speeds in tropical cyclones. Nature, 422: Rastigejev Y, Suslov S A E-ε model of spray-laden near-sea atmospheric layer in high wind conditions. J Phys Oceanogr, 44: Semedo A, Saetra Ø, Rutgersson A, et al. 29. Wave-induced wind in the marine boundary layer. Journal of the Atmospheric Sciences, 66: Song Jinbao. 29. The effects of random surface waves on the steady Ekman current solutions. Deep-Sea Res: Ⅰ, 56(5): Song Jinbao, Fan Wei, Li Shuang, et al Impact of surface waves on the steady near-surface wind profiles over the ocean. Boundary-Layer Meteorology, 155: Stewart R W The air-sea momentum exchange. Boundary-Layer Meteorology, 6: Sullivan P P, McWilliams J C, Melville W K. 24. The oceanic boundary layer driven by wave breaking with stochastic variability: Part1. Direct numerical simulations. Journal of Fluid Mechanics, 57: Tang Jie, Li Weibao, Chen Shumin, et al Impacts of sea spray on the boundary layer structure of Typhoon Imbudo. Acta Oceanologica Sinica, 32(11): Toba Y Local balance in the air-sea boundary processes: I. On the growth process of wind waves. Journal of the Oceanographical Society of Japan, 28: Wu J Spray in the atmospheric surface layer: laboratory study. J Geophys Res, 78: