Drag coefficient reduction at very high wind speeds

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 111,, doi: /2005jc003114, 2006 Drag coefficient reduction at very high wind seeds John A. T. Bye 1 and Alastair D. Jenkins 2 Received 21 June 2005; revised 25 November 2005; acceted 10 December 2005; ublished 31 March [1] The correct reresentation of the 10-m drag coefficient for momentum (K 10 )at extreme wind seeds is very imortant for modeling the develoment of troical deressions and may also be relevant to the understanding of other intense marine meteorological henomena. We resent a unified boundary layer model for (K 10 ), which takes account of both the wave field and sray roduction, and asymtotes to the growing wind wave state in the absence of sray. The theoretical develoment is based on an air-sea system with shear layers in both fluids and contains three constants that must be determined emirically. This is done using data from observations, and the resulting behavior is interreted in terms of sray. A feature of the results is the rediction of a broad maximum in K 10. For a sray velocity of 9 m s 1, it is found that a maximum of K occurs for a 10-m wind seed, u m s 1, in agreement with recent GPS sonde data in troical cyclones. Thus K 10 is caed at its maximum value for all higher wind seeds exected. A hysically based model, where sray drolets are injected horizontally into the airflow and maintained in susension by air turbulence, gives qualitatively similar results. The effect of sray is also shown to flatten the sea surface by transferring energy to longer wavelengths. Citation: Bye, J. A. T., and A. D. Jenkins (2006), Drag coefficient reduction at very high wind seeds, J. Geohys. Res., 111,, doi: /2005jc Introduction [2] It is of imortance to be able to accurately arameterize air-sea exchange rocesses at extreme wind seeds in order to understand the mechanisms which control the evolution of troical cyclones [Emanuel, 2003]. There are also indications that raid increases in wind seed may tend to deress the height of surface waves and thus erhas reduce the drag coefficient by the flattening of sea surface roughness elements [Jenkins, 2002]. Here we consider momentum exchange, and resent a seamless formulation which redicts the drag coefficient over the comlete range of wind seeds. The results are calibrated against the data set of Powell et al. [2003], obtained by Global Positioning System drowindsonde (GPS sonde) releases in troical cyclones. The theoretical develoment is based on an air-sea system with shear layer in both fluids, and contains three constants that must be determined emirically. This is erformed using the roerties of the fully develoed growing wind wave sea, and two field data sets collected in storm systems, and the resulting behavior is interreted in terms of sray. [3] The basis of the analysis is to aly a general exression for the drag coefficient (K 10 ), that has been derived from the inertial couling relations [Bye, 1995], which take account of the wave field [Bye et al., 2001], to 1 School of Earth Sciences, University of Melbourne, Melbourne, Victoria, Australia. 2 Bjerknes Centre for Climate Research, Bergen, Norway. Coyright 2006 by the American Geohysical Union /06/2005JC003114$09.00 the wave boundary layer [Bye, 1988] in the situation occurring under very high wind seeds, when sray lays a significant role in the air-sea momentum transfer. The analysis shows how the roduction of sray may lay an essential role in the frictional regime which revails in storm systems. The inertial couling relation may be regarded as a arameterization of the dynamical effect of ocean waves within the couled system containing the atmosheric and oceanic near-surface turbulent boundary layers [Jenkins, 1989, 1992]. [4] We outline the derivation of the general exression for the 10-m drag coefficient and the Charnock constant [Charnock, 1955] in section 2, and then (section 3) introduce a simle formulation, which characterizes the sea state in storm systems, and gives rise to a maximum in the 10-m drag coefficient. In section 4, the inertially couled boundary layer analysis is interreted in terms of sray roduction, which is thought to be of great imortance in very high wind conditions; see, for examle, Lighthill [1999]. In articular, in section 4.6, a hysical model in which sray drolets are injected horizontally into the airflow and are maintained in susension by turbulence is introduced, which gives qualitatively similar redictions for the variation of the 10-m drag coefficient with wind seed. 2. General Exressions for the 10-m Drag Coefficient (K 10 ) and the Charnock Constant (A) [5] In the wave boundary layer [Bye, 1988], u 10 ¼ u 1 u * =k ln ð zb =z 10 Þ; ð1þ 1of9

2 where u 10 is the wind velocity at 10 m, z 10 = 10 m, and u 1 (which will be called the surface wind) is the wind velocity at the height z B = 1/(2 k 0 ), where k 0 is the eak wave number of the wave sectrum, u * is the friction velocity and k = 0.4 is von Kármán s constant. On introducing the inertial couling relationshis [Bye, 1995; Bye and Wolff, 2004], u * ¼ K 1=2 I ðu 1 u 2 =eþ ð2þ eu L ¼ 1 ð 2 eu 1 þ u 2 Þ ð3þ in which the reference velocity has been set equal to zero for convenience, is the inertial drag coefficient, and e = (r 1 /r 2 ) 1/2, where r 1 and r 2 are the densities of air and water, resectively, and u 2 (which will be called the surface current) is the current velocity at the deth z B, at which the article velocities in the wave motion become negligible, and eu L is the wave-induced velocity in water (the sectrally integrated surface Stokes velocity (the surface Stokes drift velocity)), and u L is the wave-induced velocity in air (the sectrally weighted hase velocity), and also the relation [Bye and Wolff, 2001] eu L ¼ rðu 2 Þ; ð4þ where r is the ratio of the Stokes shear to the Eulerian shear in the water. We obtain the drag law in which u * 2 ¼ K R u 1 2 K R ¼ =R 2 ; ð5aþ ð5bþ where R = 1 2 (1 + 2r)/(1 + r), and K R is the intrinsic drag coefficient for the couled system. For R = 1, in which the Eulerian shear in the water is negligible in comarison with the Stokes shear, K R =. In the situation in which the Eulerian shear ooses the Stokes shear (r < 0), a frictional drag occurs in which R > 1, and K R <, which indicates the formation of a sli surface at the air-sea interface. On now substituting for u 1 in (1), we obtain 1= ffiffiffiffiffiffiffi ffiffiffiffiffiffi K 10 ¼ 1= K R ð1=kþln ½ 1= ð 2z10 k 0 ÞŠ; ð6þ where K 10 = u * 2 /u 2 10 is the 10-m drag coefficient. Next, with the introduction of the relation c 0 =u 1 ¼ B; where B is the ratio of the hase seed of the eak wave, c 0 = (g/k 0 ) 1/2, to the surface wind, u 1, g being the acceleration due to gravity, equation (6) yields the 10-m drag relation 1= ffiffiffiffiffiffiffi ffiffiffiffiffiffi h i K 10 ¼ 1= K R ð1=kþln B 2 u 2 * = ð 2z 10gK R Þ ð7þ ð8þ and (5) yields the exression for the wave age, c 0 =u * ¼ B= ffiffiffiffiffiffi K R: ð9þ Finally, on defining the Charnock constant, a ¼ z 0 g=u 2 * ; ð10þ where the air-sea roughness length (z 0 ) satisfies the relation ð1=kþlnðz 10 =z 0 Þ ¼ 1= ffiffiffiffiffiffiffi K 10; ð11þ we obtain, from (8), the exression a ¼ 1 ffiffiffiffiffiffi 2 B2 =K R ex k= K R : ð12þ Equations (8) and (12) are general exressions for K 10 and a, resectively, in terms of the wave boundary layer arameters K R and B. [6] It is the urose of this aer to aly these relations to model the form of the 10-m drag coefficient at the very high wind seeds, which occur in hurricanes, where sray may have an imortant influence. The hurricane is the most intense examle of a cyclonic storm system in which the effects of rotation are clearly of imortance. At the outset, however, we retreat to the simler environment characterized by the growing wind wave sea, in which rotation lays a negligible role. 3. Characterization of Sea States by the Frictional Regime, Which Occurs in the Wave Boundary Layer [7] The inertial couling formulation introduced in section 2 incororates the frictional regime of the wave boundary layer through the arameter, r in (4), or equivalently, the arameter R in (5). We consider first the situation for the growing wind wave sea Fully Develoed Growing Wind Wave Sea [8] The wave field in the growing wind wave sea is generated imulsively by an ideal steady rectilinear wind. The fully develoed growing wind wave sea occurs when the wave field is indeendent of fetch. In this situation, it was shown by Bye and Wolff [2001], by evaluating both the sectrally integrated surface Stokes velocity (the Stokes drift) and the sectrally weighted hase velocity of the wave sectrum that the Stokes shear dominates the Eulerian shear, r! ±1 (R = 1), such that the intrinsic drag coefficient (K R ) is the inertial drag coefficient ( ). The roerties of the fully develoed growing wind wave sea, in which (1) the Charnock constant a = [Wu, 1980] and (2) the inverse wave age u * /c 0 = A, where A = [Toba, 1973], can be used to estimate and B. On substituting the conditions 1 and 2 in (12), with R =1, we obtain = , and on substituting for in (9) with R =1,B = 1.3. We will use these estimates of and B below when considering the wind sea in a storm system. An extended discussion of the alication of the 2of9

3 Table 1. Storm System Data Sets u 10,ms 1 u *,ms 1 K 10,10 3 R MBL a MBL a MBL a MBL a JASIN b a MBL x-y, mean boundary layer wind seed grou (m s 1 ). Estimates of u * and K 10 have been extracted from Figures 3a and 3c, resectively, of Powell et al. [2003]. b JASIN (Joint Air-Sea Interaction) exeriment, mean wind seed (m s 1 ). Estimates of u * and K 10 have been extracted from Figure 1 of Nicholls [1985]. inertial couling relations to the fully develoed growing wind wave sea is given by Bye and Wolff [2004], in which it is shown that should remain aroximately constant in more general wave conditions. The arameter B would be exected to be aroximately constant because of the fetch-indeendent conditions which occur in the storm systems Frictional Balance in a Storm System [9] In a storm system, rotation lays an imortant role. The frictional balance can be addressed through a model of the couled Ekman layers of the ocean and the atmoshere. A suitable model has been develoed by Bye [2002], in which the velocity and shear stress at the edge of the wave boundary layer in the ocean and the atmoshere are matched with an outer layer of constant density and viscosity using the inertial couling relation (2). This model is of similar form to the steady state two-layer lanetary boundary layer (PBL), which has been found to rovide a good reresentation of the PBL velocity structure over land [Garratt and Hess, 2003]. [10] In the model, the eddy viscosities in the constant viscosity layers in the atmoshere and ocean are reresented by the similarity exressions: n 1 ¼ Cku * 2 =f ; ð13aþ n 2 ¼ Ckw * 2 =f ; f > 0; ð13bþ the Northern Hemishere) of the surface geostrohic velocity in the atmoshere (u g ) are K g ¼ u * 2 =u 2 g ¼ ðr þ 1Þ 2 = r 2 þ 1 ð15aþ m ¼ tan 1 ð1=rþ ð15bþ resectively. Thus the wave field in the storm system is controlled by a different frictional regime to the fully develoed growing wind wave sea. This regime is characterized by an angle of turning (m), which is determined by the frictional arameter (r). [12] We will consider two data sets that have been obtained in storm systems, which enable r (or R) tobe determined. The first data set was obtained in moderate conditions in the Joint Air-Sea Interaction (JASIN) exeriment in the Atlantic Ocean northwest of Scotland [Nicholls, 1985]. The second data set was obtained in very high wind seeds in the troical Atlantic and Pacific Oceans during the assage of 15 hurricanes [Powell et al., 2003]. These data are summarized in Table 1 in four ranges of u 10 for the hurricane data, and for the mean conditions of the JASIN exeriment, and the corresonding values of R have been obtained by the numerical solution of (8), using g = 9.8 m s 2, k = 0.4, = , and B = 1.3. [13] Figure 1 indicates that the data can be fitted by a linear regression in which 1 ð1=rþ ¼ au * ; ð16þ where a = m 1 s, although there is a considerable scatter, which arises from the sensitivity of R to the mean observed value of u * for each u 10 range. The substitution of (5a) in (16) yields R ¼ R 0 þ u 1 =q 0 ð17aþ, R ¼ R 0 1 u * ffiffiffiffiffi ; ð17bþ q 0 where w * = eu *, and f =2W sinf is the Coriolis arameter, in which W is the angular seed of rotation of the Earth, f is the latitude, C is a similarity constant, and the matching of the two layers in the atmoshere occurs at z B = Cu * /f. A key result was that n o r ¼ 1 þ ½Ck= ð2 ÞŠ ; ð14þ which demonstrates that, since C > 0, a steady state equilibrium is only ossible for 1 < r < 1 (R >1)[Bye, 2002]. Equation (14) links the frictional roerties in the inner wave boundary layer and the outer constant viscosity layer of the Ekman layer, and shows that r is determined by the constant (C). [11] It was also found that for a zero reference velocity in the ocean, the geostrohic drag coefficient and the angle of rotation of the surface shear stress to the left-hand side (in Figure 1. Inverse frictional arameter (1/R) as a function of u * for the data sets resented in Table 1. 3of9

4 R m = 1.19 (r m = 3.58). Other roerties at the maximum in K 10 are the following: friction velocity u * m ¼ q 0 ð2 =kþ= 1 þ 2 ffiffiffiffiffi ; ð22þ k 10-m velocity ðu 10 Þ m ¼ q ffiffiffiffiffi 0 2 ln 2B 2 q 2 0 = z ð 10gk 2 Þ k 1 þ k= 2 ffiffiffiffiffi ; ð23þ Figure 2. Drag coefficient (K 10 ) obtained from equation (8) as a function of u 10 for q 0 = 100 m s 1, q 0 = 300 m s 1, and q 0!1shown by shaded curves. The solid curve shows K 10 comuted from the jet ejection model for drolets (equation (49)). where R 0 = 1, and q 0 = 1/(a ffiffiffiffiffi ) is a scale velocity, from which we have K R ¼ = ð1 þ u 1 =q 0 Þ 2 ð18aþ h. ffiffiffiffiffi i 2: K R ¼ 1 u * ð18bþ At very large surface wind velocities, K R! 0, and q 0 ffiffiffiffiffi u * ¼ q 0 ; ð19þ where q 0 is the sole velocity which determines u *, and hence u * tends to a constant. For a = m 1 s, we have q m s 1. The key roerty of this frictional regime can be deduced by differentiating (8) with resect to u *, which yields 1 2 K 10 3=2 dk 10 =du * ¼ 1= ffiffiffiffiffi 2= ðkrþ dr=du* 2= ku * : ð20þ [14] Equation (20) indicates that for a constant R, K 10 increases monotonically with u 10. This is the traditional form for the drag coefficient relationshi. For the linear deendence of R on u 1, reresented by (17a) (17b), however, we find from (20) that a maximum in drag coefficient with resect to u * (or u 10 ) occurs for R = R m, where R m ¼ 1 þ 2 ffiffiffiffiffi ; ð21þ k which indicates that the maximum drag coefficient occurs for an intrinsic drag coefficient (K R ) which is indeendent of the scale velocity (q 0 ), and on evaluating (21) we obtain 10-m drag coefficient *,( " ðk 10 Þ m ¼ q 0 ðu 10 Þ m 1 þ k= 2 ffiffiffiffiffi #) + 2 : ð24þ [15] The 10-m drag laws resulting from the alication of (8) for a series of scale velocities (q 0 ) are illustrated in Figure 2. For q 0! 1, the monotonic behavior of the growing wind wave sea occurs, whereas for q 0 = 300 m s 1 (which aroximately reresents the observations shown in Table 1) a maximum drag coefficient, (K 10 ) m, of occurs at (u 10 ) m =42ms 1 with (u * ) m = 1.88 m s 1.It is also aarent that the drag coefficient has a broad maximum with resect to u 10. For q 0 = 100 m s 1, the maximum occurs at a much lower wind seed, u 10, and the gradual aroach to the high surface wind seed limit (19), which occurs for u * = 3.87 m s 1, at which K 10! 0 and u 10!1, is clearly shown. [16] The linear model thus reroduces both the osition and shae of the maximum in the drag coefficient. The imortant question is what is its hysical basis? From the oint of view of the frictional regime, the constant q 0 model imlies an atmosheric Ekman layer in which the similarity constant (C) decreases with u 10, giving rise to a frictional arameter (R) and an angle of turning (m) which both increase, reaching resectively, R = 1.3 (r = 2.7, C = 0.021) and m =21 for the highest wind seeds shown in Table 1, at which the intrinsic drag coefficient K R has decreased to The hysical mechanism reresented by this evolution is the rogressive formation of a sli surface at the sea surface. In section 4, we argue that this is due to sray roduction. 4. Sray Model 4.1. Nature of Sray [17] The resence of sray at the sea surface indicates that the momentum imarted by the wind is artitioned between wave generation and sray roduction; see Andreas [2004]. The hysical rocesses occurring in the growing wind wave sea, where the Stokes shear dominates over the Eulerian shear, makes no allowance for the existence of sray. The frictional loss occurring in the storm system, however, is fundamentally due to sray roduction, which is essentially the waste roduct of the wave generation mechanism. [18] We will now interret (17), as a sray model, assuming that the calibration, q 0 = 300 m s 1 is alicable. The consequences of this calibration for various asects of the air-sea dynamics will be investigated. 4of9

5 friction velocity increases, there is a rogressive increase in the return flow of momentum from the ocean to the atmoshere because of the oceanic (Eulerian) shear in comarison with that from the atmoshere to the ocean because of the atmosheric shear. This two-way momentum exchange across the air-sea interface is reresented by the two terms on the right-hand side of (2), the first of which arises from the atmosheric shear, and the second from the oceanic shear. Using (3) and (4), the ratio of the two shears, u 2 =u ð 1 Þ ¼ 1= ð2r þ 1Þ: ð27þ Figure 3. Ratio u 2 /(eu 1 ) as a function of u * for q 0 = 300 m s Flattening of the Sea State [19] A characteristic of the sea state in hurricane winds is that the waves aear to be flattened by the wind. This effect can be quantified using the sray model. We adot the Toba wave sectrum for the growing wind wave sea, truncated at the eak wave number (k 0 ), for which [22] For the growing wind wave sea, u 2 /(eu 1 ) = 0, whereas with the inclusion of sray roduction, u 2 /(eu 1 ) increases with u *, and at r = r m, u 2 /(eu 1 ) = 0.16 (Figure 3). The increase over the range in u 10 from about 30 to 60 m s 1 gives rise to an almost constant K 10 over this range through corresonding changes in z B and u 1 /u * Sray Velocity [23] We look now at the energetics of sray formation, making use of the following exression for the rate of working on the wave field: E ¼ 1 3 g 0u * c 0 3 =g 2 ; ð25þ W ¼ r 1 u * 2 u L ; ð28þ where E = hz 2 i is the root mean square wave height, and g 0 is Toba s constant. On substituting for u *, we obtain E ¼ 1 3 g c0 4 ffiffiffiffiffi, g 2 B ; ð26þ where g = g 0 /R. Hence the reduction in wave energy, due to sray, can be interreted in terms of a reduced Toba constant (g). In the limit of large surface wind velocities, g! 0, indicating a totally flattened sea state, and at (K 10 ) m, g/g 0 = 0.84, indicating a mild flattening in which the wave height is reduced by about 8%. The eak wave seed, c 0!1for large surface wind velocities, and at (K 10 ) m, c 0 increases by about 20% because of the sray effect. Thus the roduction of sray tends to increase the wave seed of the eak wave, i.e., to transfer energy to longer wavelengths. The level of redicted flattening is in general agreement with that obtained by indeendent reasoning by Jenkins [2002] The Similarity Profile at Extreme Wind Seeds [20] The key result of section 3 is that the drag coefficient asses through a maximum, (K 10 ) m, with wind seed, and then is almost constant over a wide range of higher seeds, see Figure 2. Hence for the uroses of hurricane dynamics, where (K 10 ) m occurs at about 40 m s 1, the drag coefficient is caed at its maximum value over the full range of extreme wind seeds that are likely to occur. [21] The hysical rocesses which bring about this aarent similarity regime for extreme wind seeds are a dilation of the wave boundary layer, in which its thickness (z B ) and nondimensional velocity scale (u 1 /u * ) both increase, but without a significant change in K 10 ; see (1). The dynamical rocess which is occurring, is that as the where u L is the velocity at which the transfer of momentum to the wave field is centered [Bye and Wolff, 2001]. On substituting for u L, using (3) and (4), we obtain W ¼ 1 2 r 1u * 3 ð2r 1Þ= ffiffiffiffiffi : ð29þ The rate of working (W) can be usefully artitioned into the two comonents, W ¼ W 0 þ W S ; ð30þ where W 0 = 1 r 2 1 u * 3 ffiffiffiffiffi / is the rate of working on the growing wind wave field, and W S ¼ r 1 u * 2 is the rate of working which generates the sray, where. ffiffiffiffiffi ¼ u * ðr 1Þ ð31þ ð32þ is the sray velocity. At the maximum of the 10-m drag coefficient, (K 10 ) m, ðw S =W 0 Þ m ¼ 4 ffiffiffiffiffi ; ð33þ k and the sray velocity, () m = 2(u * ) m /k. Hence, on evaluating (33), we find that just over one quarter of the rate of working is used for sray roduction, and three quarters are used for wave growth [(W S /W 0 ) m = 0.39]. This artitioning of the rate of working highlights that the changes occurring in the wave field, described in section 4.2, 5of9

6 Figure 4. Ejection of fluid from a breaking-wave crest [after Jenkins, 1994]. The major axis of the overturning loo is aroximately 8g 1/3 2/3, where is the flux of fluid in the jet. The vertical and horizontal axes are labeled in terms of the length scale g 1/3 2/3. The relative seed of the fluid in the jet and the main body of water at the imact oint is 6.9(g ) 1/3. Figure # Cambridge University Press 1994, rerinted with ermission. are due to sray roduction. For q 0 = 300 m s 1, the sray velocity, () m = 9.4 m s 1, and for W 0 = W s, the friction velocity (u * ) is 3.9 m s 1, which is very similar to that of 4.2 m s 1, redicted by Andreas and Emanuel [2001] for the condition that the sray stress and the interfacial stress are equal, strongly suorting the choice of q 0 = 300 m s 1 in the sray model Proerty Transfer Across the Sea Surface [24] The imlications of the artitioning of the rate of working into a wave (W 0 ) and a sray (W S ) comonent are aosite. The wave comonent (W 0 ) has no significance for roerty transfers across the sea surface; these are encomassed by the sray comonent (W S ). In the event that rocesses other than sray roduction are unimortant at extreme wind seeds, as roosed by Emanuel [2003], heat and momentum transfer should be governed by the same hysics. Thus, on exressing the surface shear stress (t S = r 1 u * 2 ) in terms of the sray velocity, we have t S ¼ r 1 C S 2 ; ð34þ where C S is a drag coefficient aroriate to the sray roduction, and the net uward heat flux is F ¼ r 1 C C S T ð S T W Þ; ð35þ where the drag coefficients (C S ) in (34) and (35) are identical, T S is the surface water temerature, T W is the wet bulb temerature of the descending sray articles, and C is the secific heat of water at constant ressure [Emanuel, 2003]. Equation (35) is of the same form as that alicable for heat exchange due to rainfall, in which is relaced by the reciitation velocity (P) [see, e.g., Bye, 1996], excet that, while P is a vertical velocity, is a horizontal velocity. Allowance for evaorative heat exchange can also be made, and it is found that the drag coefficient for enthaly transfer at the temeratures occurring in hurricanes is similar to that for heat [Emanuel, 2003]. [25] In summary, at extreme wind seeds in which roerty transfers across the sea surface are dominated by sray roduction, the drag coefficients (C S ) for momentum and heat transfer, relative to the sray velocity (), and hence also the drag coefficients (K 10 ) relative to u 10, are identical, and since the momentum drag coefficient (K 10 )is caed, as discussed in section 4.3, that for heat transfer is also caed Volume Flux, Vertical Distribution of Sray Drolets, and Effect on Mean Flow Profile [26] In the above analysis, the sray dynamics are reresented through two rocesses: (1) the horizontal velocity (32) of the sray articles at formation and (2) the roughness relation (12), which modifies the wind rofile because of the resence of the sray articles. It is instructive to consider these two rocesses using a hysically based model Sray Production [27] For a wind-sea state given by (25), we may assume that the momentum flux r 1 u 2 * from the atmoshere acts to increase the wave momentum, and that the greater art of the wave momentum thereby generated is dissiated more or less immediately by wave breaking. The breaking of surface waves, though it is a comlicated, time-deendent rocess, is, when sufficiently vigorous, usually characterized by the ejection of water in a forward directed jet at the crest. One of the simler arameterizations of wave breaking which reroduces this feature is the stationary otentialflow model of Jenkins [1994], in which the jet is attached to a modified Stokes 120 corner flow, and where there is a unique relation between the geometrical length scale of the breaking structure and the flux of fluid in the jet (see Figure 4). In the frame of reference moving with the wave crest, the jet imacts the forward surface of the wave with a velocity, v J, which deends on the size of the breaking-crest structure, and which in ractice will be a fraction of the wave hase seed c. On contact with the forward face of the wave, the dissiation of the kinetic energy may go toward reducing the wave energy, but may also contribute to increasing the surface interfacial energy by the formation of drolets [Andreas, 2002]. [28] In this rocess, a roortion (b J ) of the surface shear stress (r 1 u * 2 ) would be used in sray generation, and give rise to a mass flux er unit area, G ¼ b J r 1 u * 2 =v J ; where, on comaring with the relations in section 4.4, b J ¼ W s =W ¼ b J v J : ð36þ ð37þ ð38þ Sray Vertical Distribution [29] To estimate the vertical distribution of sray drolets, we assume that they diffuse randomly with a (turbulent) diffusion coefficient ku * z, but descend under gravity at a 6of9

7 terminal velocity w t. To determine the terminal velocity, we need to secify a tyical drolet radius r s : in fact, a tyical radius for the largest drolets, since the mass of a drolet is roortional to the cube of its radius. We assume that r s is determined by a balance between the airflow tending to tear the drolet aart (reresented by r 1 u 1 2 = r 1 u * 2 / ) and the forces of surface tension (T) holding it together. By dimensional analysis, we have. 2 r s a r T r 1 u * ; ð39þ where a r is a constant. To comute w t we note that a tyical value for r s would be 87.5 mm (for a r = 1.0, T Nm 1, , r kg m 3, and u * 1 m s 1 ), and drolets of this radius fall in the atmoshere in a regime intermediate between Stokes flow and fully turbulent flow [e.g., Beard, 1976]. Beard derived a relatively comlicated exression for the deendence of w t on r s, but this may be simlified by insection of his Figure 6, which gives the following aroximate relation: w t f s r s ; ð40þ with f s = s 1, for drolets of radius between aroximately 0.01 mm and 1 mm. The terminal velocity for larger drolets increases more slowly with increasing radius, as a result of the drolet shae becoming flattened, and tends to a constant value of aroximately 9 m s 1 for the largest drolets. [30] If sray drolets susended in the air contain a mass r s of water er unit volume, in a steady state with no net vertical sray flux we will have ku * zdr ð s =dzþþw t r s ¼ 0: ð41þ Solutions to this equation are of the form wt= ku ðr s =r s0 Þ ¼ ðz=z 0 Þ ð * Þ ; ð42þ where r s0 is the surface value of r s, which, from (36), must satisfy, under steady state conditions, w t r s0 ¼ b J r 1 u * 2. v J : ð43þ [31] It should, however, be noted that the integral of the solution in (42) diverges as z!1if u * (u * ) c, where (u * ) c = w t /k, so a steady state vertical distribution of sray drolets will not be attainable in this case. On evaluating, we obtain, (u * ) c = 1.2 m s 1, which interestingly is similar to the friction velocity likely to be encountered in very high winds, see Table 1, and consistent with the anecdotal statement that in hurricane conditions, the air is too thick to breathe and the water is too thin to swim in [Kraus and Businger, 1994,. 58]. Nevertheless, we assume that the drolets do become distributed according to (42) in a sufficiently dee layer for our uroses. This resonse arises from the classical form of the diffusivity, D=ku * z, used in (41). For a constant D, the solution of (41) is r s =r s0 ¼ exðw t z=dþ; the integral of which converges unconditionally for all D. This model was used by Lighthill [1999] in an elegant study of the sray distribution brought about by wind gusts, in which he showed that D = (1/6) Z 2 /T, where T is the time of flight for the coherent vertical dislacement of a small article of air because of a random gust which gives it a vertical dislacement of equal robability over the range Z to +Z Effect of Susended Sray Drolets on the Mean Flow Profile [32] The dynamical effect of sray drolets has been estimated by Makin [2005], using the theory of Barenblatt [1953, 1979] for the effect of susended articles in a turbulent flow. Barenblatt s theory alies only in the case where u * (u * ) c, and the redicted effect of the drolet susension on the mean flow deends only on the terminal velocity and not on the drolet concentration. In this section we emloy a different theory: a modification of the Monin- Obukhov theory for stratified boundary layers. We assume that kz=u * ðdu=dzþ ¼ f1 ðz=lþ; ð44þ where the Monin-Obukhov length L is given by L ¼u * 3 r 1 = ðkgf b Þ ¼ u * 3 r 1 = ðkgw t r s Þ; ð45þ where F b is the vertical turbulent buoyancy flux, in the steady state equal to w t r s, and the universal function f 1 (z/l) is, according to Businger et al. [1971]: f 1 ðz=lþ ¼1 þ 6z=L; for 0 < z < L: ð46þ [33] The value of f 1 (z/l) for z > L from exerimental measurements aears to be rather uncertain, but in the calculations we resent below, L is always much greater than the reference height of 10 m. [34] From (42) (45) we obtain wt= ku L ¼ u * v J = ðgkb J Þ ð z=z0 Þ ð * Þ ð47þ du=dz ¼ u * = ðkzþ þ 6g wt= ku ð bj =v J Þðz=z 0 Þ ð * Þ ; 0 < z < L: ð48þ [35] Now the boundary condition at the surface (z = z 0 ) should not be u = 0, but u =(r s0 /r 1 )v J, to account for the sray being injected horizontally into the water column [Kudryavtsev, 2005, also Effect of sea dros on atmosheric boundary layer at high wind conditions, rerint, 2005]. Integrating uward from z = z 0, we obtain u ¼ u * =k ln ð z=z0 Þþðr s0 =r 1 Þv J þ 6gðb J =v J Þ h i 1z0 1wt= ku 1 w t = ku * ðz=z 0 Þ ð Þ 1 : ð49þ 7of9

8 [36] Equation (49) suggests that if u * (u * ) c, the effect of the sray formation on the velocity rofile is very small: However, for u * (u * ) c it becomes significant. The black curve in Figure 2 shows the value of the 10-m drag coefficient, K 10 =(u * /u 10 ) 2, comuted from (49), with the following arameters: k = 0.4, b J = 0.15 (which was estimated from (37) for R 1.1), a = 0.018, T = Nm 1, r 1 = 1.2 ffiffiffiffiffi kg m 3, r 2 = 1000 kg m 3, a r = 1.0, and v J = 0.5u * /( ), which follows directly from (38). Note that the dearture from the growing wind wave sea relation (q 0!1) becomes significant for u m s 1 (u * 1.2 m s 1 ). We see that there are still some discreancies between the value of the drag coefficient comuted by this method and by (8): notably that the reduction in drag coefficient begins at a higher wind seed. The reason for this effect may be that we have assumed that the drolets have only one radius, and that this radius decreases relatively raidly with increasing wind stress (r s / u * 2 ). In reality, the drolets have a comlex size distribution [Andreas, 2002, 2004], which may, by modifying the vertical distribution of drolet mass in (42), tend to reduce the negative sloe of the drag coefficient curve in Figure Conclusion [37] We have resented a unified boundary layer model for redicting the drag coefficient (K 10 ) for momentum exchange at the sea surface, which takes account of wave growth and also sray roduction. It is found that K 10 asses through a broad maximum rimarily because of the return flow of momentum from the ocean to the atmoshere, which increases with friction velocity (u * ). The hysical rocesses, which become evident in this extreme wind seed similarity range are the flattening of the sea surface with the transfer of energy to longer wavelengths, together with the roduction of sray. On the assumtion that heat transfer across the sea surface at extreme wind seeds is mainly due to sray roduction [Emanuel, 2003], it is argued that the drag coefficient for heat should be similar to that for momentum, and also caed at extreme wind seeds. [38] The analysis uses a simle exression (17) to model sray roduction, which has the effect that the sea surface becomes asymtotically flat for wind seeds well beyond those exected in nature. Equation (17) is essentially a linear exansion about the classical growing wind wave state, which takes account of sray roduction, and is aroriate for an oen ocean environment. We also consider in section 4.6 a hysically based model for the drag reduction, with exlicit assumtions for the sray drolet size and the horizontal velocity of injection of sray drolets into the air column [Kudryavtsev, 2005, also Effect of sea dros on atmosheric boundary layer at high wind conditions, rerint, 2005], which, when calibrated using the arameters of the inertially couled boundary layer model, gives the same qualitative behavior for the wind velocity deendence of the drag coefficient. An esecially interesting finding is that the classical exression for diffusivity in (41) suggests that a critical friction velocity, (u * ) c is alicable for drolets of a secified terminal velocity, above which their effect on the dynamics (and thermodynamics) of the lanetary boundary layer becomes very significant. [39] The analysis suggests that the growing wind wave sea can be regarded as an oen-ended sea state, which evolves into a mature sea state of intensity set by the synotic situation, and with frictional roerties determined by the atmosheric Ekman layer, through the similarity constant C (and hence r). [40] A similar exansion to (17) can be made about the wave state alicable in wave tanks by a suitable choice of R 0 and q 0. An analysis of the laboratory exeriments at high wind seeds, however, is beyond the scoe of this aer. [41] Acknowledgments. This work was begun while J.A.T.B. was a Visiting Fellow at the Bjerknes Centre for Climate Research in Setember 2003 and was comleted during a Fellowshi at the Hanse Institute for Advanced Study in Delmenhorst, Germany, in July and August A.D.J. is suorted by the Research Council of Norway under Project /700. This is Publication A 115 of the Bjerknes Centre for Climate Research. Helful comments by the Editor and two referees are gratefully acknowledged. References Andreas, E. L. (2002), A review of the sea sray generation function for the oen ocean, in Atmoshere-Ocean Interactions, vol. 1, edited by W. Perrie,. 1 46, WIT Press, Southamton, U. K. Andreas, E. L. (2004), Sray stress revisited, J. Phys. Oceanogr., 34, Andreas, E. L., and K. A. Emanuel (2001), Effects of sea sray on troical cyclone intensity, J. Atmos. Sci., 58, Barenblatt, G. I. (1953), On the motion of susended articles in a turbulent flow, Prikl. Mat. Mekh., 17, Barenblatt, G. I. (1979), Similarity, Self-Similarity, and Intermediate Asymtotics, 218., Sringer, New York. Beard, K. V. (1976), Terminal velocity and shae of clouds and reciitation dros aloft, J. Atmos. Sci., 33, Businger, J. A., J. C. Wyngaard, Y. Izumi, and E. F. Bradley (1971), Fluxrofile relationshis in the atmosheric surface layer, J. Atmos. Sci., 28, Bye, J. A. T. (1988), The couling of wave drift and wind velocity rofiles, J. Mar. Res., 46, Bye, J. A. T. (1995), Inertial couling of fluids with large density contrast, Phys. Lett. A, 202, Bye, J. A. T. (1996), Couling ocean-atmoshere models, Earth Sci. Rev., 40, Bye, J. A. T. (2002), Inertially couled Ekman layers, Dyn. Atmos. Oceans, 35, Bye, J. A. T., and J.-O. Wolff (2001), Momentum transfer at the oceanatmoshere interface: The wave basis for the inertial couling aroach, Ocean Dyn., 52, Bye, J. A. T., and J.-O. Wolff (2004), Prediction of the drag law for air-sea momentum exchange, Ocean Dyn., 54, Bye, J. A. T., V. K. Makin, A. D. Jenkins, and N. E. Huang (2001), Couling mechanisms, in Wind Stress Over the Ocean, edited by I. S. F. Jones and Y. Toba, , Cambridge Univ. Press, New York. Charnock, H. (1955), Wind stress on a water surface, Q. J. R. Meteorol. Soc., 81, Emanuel, K. (2003), A similarity hyothesis for air-sea exchange at extreme wind seeds, J. Atmos. Sci., 60, Garratt, J. R., and G. D. Hess (2003), Neutrally stratified boundary layer, in Encycloedia of Atmosheric Sciences, edited by J. R. Holton, J. A. Curry, and J. A. Pyle, , Elsevier, New York. Jenkins, A. D. (1989), The use of a wave rediction model for driving a near-surface current model, Dtsch. Hydrogr. Z., 42, Jenkins, A. D. (1992), A quasi-linear eddy-viscosity model for the flux of energy and momentum to wind waves, using conservation-law equations in a curvilinear coordinate system, J. Phys. Oceanogr., 22, Jenkins, A. D. (1994), A stationary otential-flow aroximation for a breaking-wave crest, J. Fluid Mech., 280, Jenkins, A.D. (2002), Do strong winds blow waves flat?, in Ocean Wave Measurement and Analysis: Proceedings of the Fourth International Symosium, WAVES 2001: Setember 2 6, 2001, San Francisco, Cali- 8of9

9 fornia, edited by B. L. Edge and J. M. Hemsley, , Am. Soc. of Civ. Eng., Reston, Va. Kraus, E. B., and J. A. Businger (1994), Atmoshere-Ocean Interaction, 362., Oxford Univ. Press, New York. Kudryavtsev, V. N. (2005), On the marine atmosheric boundary layer at very strong winds, aer resented at General Assembly, Eur. Geosci. Union, Vienna, Austria, Aril. Lighthill, J. (1999), Ocean sray and the thermodynamics of troical cyclones, J. Eng. Math., 35, Makin, V. K. (2005), A note on the drag of the sea surface at hurricane winds, Boundary Layer Meteorol., 115, Nicholls, S. (1985), Aircraft observations of the Ekman layer during the Joint Air-Sea Interaction Exeriment, Q. J. R. Meteorol. Soc., 111, Powell, M. D., P. J. Vickery, and T. A. Reinhold (2003), Reduced drag coefficient for high wind seeds in troical cyclones, Nature, 422, Toba, Y. (1973), Local balance in the air-sea boundary rocess III. On the sectrum of wind waves, J. Oceanogr. Soc. Jn., 29, Wu, J. (1980), Wind-stress coefficients over sea surface near neutral conditions A revisit, J. Phys. Oceanogr., 10, J. A. T. Bye, School of Earth Sciences, University of Melbourne, Melbourne, Victoria 3010, Australia. (jbye@unimelb.edu.au) A. D. Jenkins, Bjerknes Centre for Climate Research, Geohysical Institute, Allégaten 70, N-5007 Bergen, Norway. (alastair.jenkins@ bjerknes.uib.no) 9of9