SEMI-PHENOMENOLOGICAL MODEL FOR A WIND-DRIFT CURRENT. Vladislav G. Polnikov

Transcription

1 UD SEMI-PHENOMENOLOGICAL MODEL FOR A WIND-DRIFT CURRENT Vladislav G. Polnikov A.M. Obukhov Insiue of Amospheric Physics of RAS, Moscow, Russia; polnikov@mail.ru Absrac Some daa of he drif curren, U d, measured on a wavy surface of waer in a laboraory and he field, are briefly described. Empirical formulas for U d are given, and heir incompleeness is noed, regarding o absence of he drif curren dependence on surface-wave parameers. Wih he purpose of heoreical jusificaion of empirical formulas, a semi-phenomenological model of he phenomenon is consruced. I is basing on he known heoreical and empirical daa abou he hree-layer srucure of he air-waer inerface: 1) he air boundary layer, 2) he wave-zone, and 3) he waer upper layer. I is shown ha a presence of linear drif-curren profile in he wave-zone, U(z), makes i possible o obain a general formula for he drif curren on he wavy waer surface, U d. This model is based on he balance equaion for he momenum-flux and curren-gradien, aking place in he wave-zone. The proposed approach allows us o give an inerpreaion of he empirical resuls, and indicae he direcion of heir furher deailed specificaion. eywords: air-waer inerface, wind waves, drif curren, momenum flux, verical profile of curren, urbulen viscosiy.

2 1. Inroducion Presence of air curren (wind) above a calm surface of waer leads very quickly o an appearance of waves and he drif curren, U d, on a surface of waer. The sable wind profile, W(z), is formed in he air boundary layer and he drif velociy profile, U d (z), do in he waer upper layer, depending on he sae of surface waves. All hese phenomena are caused by he appearance of a urbulen momenum flux from he wind o he inerface. This flux is ofen referred o as he wind sress, τ. For manifesaion of hese phenomena, i is required o ge he regime of sufficienly high Reynolds number, Re WL /, (where, L is he spaial scale of he a wind-variabiliy, a is he kinemaic viscosiy of air), i.e. he wind speed should excess a cerain hreshold value. According o experimens in he anks, i is 2-3 m /s (Monin & Yaglom, 1971; Phillips, 1977; Wu, 1975; Longo e al., 2012a). Due o he simulaneous combinaion of shear flows, wavy and urbulen moions, aking place in he viciniy of moving air-waer boundary (hereinafer referred o as he inerface), he hydrodynamics of he inerface sysem is very complicaed (Monin & Yaglom 1977; Phillips 1978). In Secion 3, i will be shown ha he enire sysem of inerface can be raher clearly shared ino hree consiuen pars: he air boundary layer (ABL), where he air is permanenly presen; he wave-zone (WZ), where he air and waer are alernaely presen; and he upper waer layer (WUL), where waer is always presen 1. In he descripion dynamics of he inerface sysem, informaion abou he wind-wave sae is very significan, due o he fac ha windwaves are he mediaor of all he movemens near inerface. However, here we will consider he parameers of ABL and WZ as given ones, and confine ourselves o he ask of consrucing a heoreical descripion of wind-drif curren U d over he WUL, as a funcion of he wind speed, W, and he wave parameers nominaed below. This is he purpose of his work. A presen ime, here is no heoreical model ha allows giving he descripion of wind-drif curren, menioned above. The soluion of his problem assumes exensive using all he available experimenal daa abou he relaion of drif-curren U d wih boh he wind and waves parameers and he srucure of inerface. This kind of informaion on he srucure of inerface was esablished only in recen years (see below), wha opens up an opporuniy for solving he ask. Despie he fac ha values of U d are small in comparison wih he wind speed a he sandard horizon z, W z, or wih he celeriy of wind waves, he heoreical descripion of winddrif is of boh physical and pracical ineres. The firs of hem is deermined by he necessiy o 1 We mean ha he layer is a horizonally exended region of space, for which a leas one of he verical boundaries is no well defined (ABL, WUL), whils he zone is a layer, boh verical boundaries of which are defined.

3 clarify he naure of phenomenon, and he laer is imporan for solving he problems of navigaion and marine aciviies safey, and for managing environmenal problems (esimaing he rae and region of impuriy disribuion on he waer surface) (Wu, 1975; Longo e al., 2012a; Churchil & Csanady, 1983; Babanin, 1988; Malinovskii e al, 2007; udryavsev e al., 2008; among ohers). Experimenal sudies of he wind-drif are very numerous (see, for example, references in he papers menioned above), wha is explained by availabiliy of he objec of research and simpliciy of measuremen for i. However, his simpliciy, in fac, is only apparen, since accurae measuremens on wavy (oscillaing) inerface are far from o be easy. Indeed, when measuring drif velociies in he field (for example, Churchil & Csanady, 1983; Babanin, 1988; Malinovskii e al, 2007; udryavsev e al., 2008), here are appearing a lo sources for errors: nonsaionariy of wind speed W, unconrolled background currens U b in he measuremen area; unconrolled exen of sraificaion for air in he ABL and waer in he WUL, and so on. A he same ime, he advanage of field-measuremens is a very wide range of realizaion for he windwave condiions. An alernaive mehod for sudying he regulariies of drif currens formaion is based on laboraory (ank) measuremens. Bu here are drawbacks in his approach. Indeed, in he ank measuremens (see he examples in Wu, 1975; Longo e al., 2012a), he causes of errors are an influence of he verical and laeral boundaries of he ank, he presence of reverse currens, he small scale of wind- and wave- fech, and so on. All hese limiaions affec on he kind of wind profile, and, consequenly, on esablishing he corresponding dependencies of U d on parameers of he ABL. Small dimensions of anks are decreasing he possibiliy of esablishing dependences U d on he wave sae, in a wide range of is parameers. However, he ank measuremens have an imporan advanage, as far as all he parameers of wind-wave condiions are compleely conrolled during such an experimen. In his regard, i is he ideal way o carry ou experimens in anks wih ransverse dimensions of he order of a meer and a lengh of several ens of meers (Huang & Long, 1980). The lised and oher sources of he errors impose significan limiaions on he accuracy of resuls measured. Herewih, he experimens hemselves require a careful preparaion boh for he measuremens and heir analysis, which is described in deail in he cied references. Alhough esimaes of he accuracy of measuring drif velociy U d are no always presened, an analysis of scaering he published daa allows us o assume ha he measuremen errors for U d are abou 10% (see, for example, figures in Wu, 1975; Longo e al., 2012a; Churchil & Csanady, 1983; Babanin, 1988; Malinovskii e al, 2007; udryavsev e al., 2008). This errors value is also

4 confirmed by direc he esimaes of measuremen errors in (Babanin, 1988; Malinovskii e al, 2007). Before represening he empirical formulas for drif velociy U d and informaion on he inerface srucure, needed for consrucing he model, we noe ha in he absence of background curren U b, he velociy vecor of he surface curren on a wavy surface, U S, includes, in fac, wo erms: U S = α 1 U S + α 2 U d. (1) Here α 1 and α 2 are he proporionaliy coefficiens close o uniy (Malinovskii e al, 2007), and U S is he Sokes-drif vecor, which is differen for each of specral componens of surface waves and direced along he propagaion-vecor for each of hese componens (Sokes, 1847). As is known, he Sokes drif (ranspor of he liquid paricles along he direcion of wave propagaion) is creaed by he uncloseness of he orbis for nonlinear wave moions, i.e. i is due only o he wave processes and is no direcly relaed o he wind. Therefore, he Sokes drif is addiive one o he wind-drif, U d, whose vecor coincides wih he wind-sress vecor, τ. Noe, however, ha, in he field experimens, he direcion of wind-sress vecor τ, and he vecor of wind-drif, U d, may no coincide wih he direcion of local wind W (Babanin, 1988; Malinovskii e al, 2007; udryavsev e al., 2008). This effec is due o he influence of Coriolis forces (realized on he spaial scales of hundreds kilomeers), associaed wih he roaion of he Earh. Furher in his paper, his effec will no be considered, wha means adoping he approximaion of he "spaial localiy" for he air-sea ineracion processes on he wavy waer surface. The expression for he magniude of curren U S, caused by a nonlinear graviaional wave wih ampliude a, frequency ω, and wave number k, was obained by Sokes sill in 1847, and i is well known (Phillips, 1977; Sokes, 1847): U ( / k )( ka ) 2 S = ( a )( ka ). (2) Here wo forms for represening U S are given, showing he relaionship beween he Sokes drif, he slope of wave, ε = ka, and he horizonal phase velociy of he graviaional wave, c ph = ω / k (he firs equaliy), and wih he modulus for verical velociy of fluid paricles, u 3 = ωa (he second equaliy). As shown in (Wu, 1975; Longo e al., 2012a; Churchil & Csanady, 1983), he value of U S has approximaely 10-15% of wind-drif U d, wha indicaes he necessiy o ake i ino accoun in measuremens and pracical problems, especially in he presence of inense long waves having a high phase velociy (for example, in sorms). However, furher, in he view of addiiviy for he erms in (1), and in he conex of he problems ha we solve, we do no need o ake ino accoun he Sokes drif in his paper.

5 2. Empirical daa and analysis In our problem, here are imporan experimenal daa concerning boh he empirical formulas for he drif velociy on waer surface, U d, and he measured feaures of mass-ransfer velociy profiles in all hree pars of he inerface: in he air-layer, in he wave-zone, and in he waer-layer. Leaving descripion of he inerface srucure and he wind- and curren-profiles for he subsequen subsecion, firs, we discuss he empirical formulas for wind-drif velociy U d. Herewih, we noe ha all he resuls under consideraion are referring o he long-erm averaged drif-velociy on he waer surface, he level of which is no specified. 2a. Formulas for U d Le's sar from he measuremens of wind-drif velociy, performed in laboraory anks under sricly conrolled condiions (see references in Wu, 1975; Longo e al., 2012a). In he well-known work by Wu (1975), which became he classical one, a simple linear relaion was found U d = α d u *a, (3) where u *a is he fricion velociy in he ABL, and α d The value of u*a was esablished in Wu (1975) wih a sandard mehod, by measuring he wind velociy values W(z) a a number of horizons z locaed far from he mean inerface-line, on he basis of he well-known formula for he logarihmic law (Monin & Yaglom, 1971; Phillips, 1977): W( z ) ( u / )ln( z / z ). (4) *a 0 I is imporan o noe ha in paper Wu (1975), here was no esablished a direc dependence of U d on wave parameers: for example, he average wave-heigh, H 2<a> (<a> is he variance of free surface elevaions η()), he average wave slope, ε = k p <a>, and he age of waves, A, defined by he relaion A= c ph (ω p )/W 10. (5) In (5), c ph (ω) is he phase velociy as funcion of frequency, aken a he peak frequency of wind-wave specrum, ω p ; and W 10 is he wind speed a he sandard horizon, z=10 m. Herewih, in Wu (1975) i was clearly shown ha he drif velociy, U d, decreases wih he wave-fech increasing, wha is in a good agreemen wih he known variabiliy of he fricion velociy (see survey of daa in (Polnikov e al., 2003) and deailed paramerizaion in (Polnikov, 2013)). Formulas for U d, similar o (3), were obained in all oher experimenal works boh in anks (Longo e al., 2012a, and references herein), and in field condiions (Babanin, 1988;

6 Malinovskii e al, 2007; udryavsev e al., 2008). These formulas differ only by coefficien α d on he righ-hand side of (3). For example, in he recen paper by Longo a al. (2012a), where he mos modern laboraory measuring equipmen was used, he value of coefficien α d 0.4 was esablished. Herewih, in his case, he srange growh of fricion velociy u*a wih he fech was fixed, accompanied by he proper growh of U d ; alhough, as i should be, he seepness of waves fell down wih he fech (see Tables 1 and 6 in Longo (2012)). In order o explain hese effecs, i was noed in Longo a al. (2012a) ha all he differences in he laboraory measuremens of U d, which obained by differen auhors, are simply relaed o he geomery of he anks, where he experimens are performed. In he view of he above remarks abou he drawbacks of ank experimens, here one can agree wih he his inerpreaion given by he auhors of he paper menioned, and proceed o he resuls of field experimens. In he field measuremens (for example, Tsahalis, 1979; Babanin, 1988; Malinovskii e al, 2007; udryavsev e al., 2008), he same formula (3) was esablished for he drif velociy, hough he values of α d are varying wihin a wide range: from 0.24 in (Babanin, 1988) o 1.5 (Tsahalis, 1979). Such a wide scaering is due o boh he naural variabiliy of he wind-wave condiions and he echnical difficulies of performing accurae measuremens in he field experimens. The las reason, apparenly, explains he lack of informaion abou he direc dependences of U d on he surface-wave parameers, which could be realized under he field condiions. However, i should be noed ha boh a significan scaering values of coefficien α d and no monoonic dependence of raio U d / u*a on W (Wu,1975) can mean an exisence of cerain (hough "hidden") dependence of α d on wave parameers, which is no esablished experimenally ye. I can be assumed ha all he dependences of drif velociy on he wave sae are "hidden" in he direc proporionaliy beween U d and fricion velociy u *a, whils he laer, as is well known, depends explicily on he above-menioned wave parameers: H, ε and A (see references in Polnikov e al. (2003), Polnikov (2013)). However, he absence of direc empirical dependences of U d on wave parameers, in our opinion, requires is jusificaion basing on specialized experimens for heir deerminaion. 2b. The srucure of inerface and wind-curren-profiles Here, firs of all, i is necessary o represen he facs confirming boh he exisence of a hree-layer inerface srucure, menioned in he inroducion, and he physical expediency of accouning such a srucure in furher heoreical consrucions.

7 a) b) Fig. 1. (a) The elemen of wave record () (he ime scale is given in convenional unis). (b) The ensemble of wo hundred segmens of wave record () of he same realizaion. The solid lines show: in (a) he condiional mean waer-surface level; in (b) he condiional boundaries of he wave-zone. Firs, in order o visualize he fac of exising he wave-zone, (WZ), which should have is own physical properies, we represen here figures 1(a, b) aken from (Polnikov, 2010, 2011). In Fig. 1a, he single ime realizaion of he free-surface elevaion of waer, η(), is shown, on he background of which any experimen is usually performed. Figure 1b shows he ensemble of wo hundred such realizaions of he same lengh, aken a he same poin. Such an ensemble corresponds o he ime scale of he order of hundreds of periods for he dominan wave (corresponding o he peak of wave specrum). On his ime-scale all average values of he sysem under consideraion are described, including he wave specrum. As can be seen from Fig. 1b, on such a scale, in he viciniy of he mean waer level, he coninuous zone is formed (WZ), in which he air and waer are alernaely presen. I is naural o assume ha his zone is an independen elemen of he inerface sysem, he physical characerisics of which, on he menioned ime scales, should be described in a unified manner. 0,6 0,5 Z 1 z 0,4 0,1 0,3 0,01 0,2 0,1 W(z) 0, ,1 0,2 0,3 0,4 0,5 0,6-0,1 W(z) 0,0001 0,001 0,101 0,201 0,301 0,401 0,501 0,601 (a) linear coordinaes (b) semi-logarihmic coordinaes Fig. 2. Calculaed profile of he mean wind W(z) over a wavy waer surface (from Polnikov, 2011). The resuls are given in dimensionless quaniies, and zero value of z corresponds o he mean waer level.

8 Second, as i was demonsraed for he firs ime in Polnikov(2011), basing on he analysis of numerical simulaions execued in (Chalikov & Rinechik, 2011), he profile of average windvelociy, W(z), differs in he wave-zone significanly from he radiional logarihmic one. I varies linearly wih he heigh from level z -h o level z of he order of 3h, relaive o he mean waer level (Fig. 2) (where h = <a> is he variance of waer-surface deviaions). Third, hese calculaions are fully suppored by he Longo's measuremens (Longo e al, 2012b) (Fig. 3). I can be clearly seen from Fig. 3 ha, in a wide range of horizons above he mean waer level, and in a somewha smaller region below, he linear profiles are really observed, boh for wind speed W(z) and for curren velociy U(z) (in our noaions). Ouside his region, which, in fac, deermines he WZ, he wind profiles, W(z) in he ABL, and curren U(z) in he WUL, ge he form close o he wall-urbulence logarihmic profiles of form (4), having heir own parameers u * and z 0 in each of hem (see he deails in Longo e al, 2012b). Regarding he physical expediency of inroducing he WZ, as he separae elemen of inerface, i is due o he fac ha dependencies of he mean wind and curren on z, W(z) and U(z), have he linear profiles. From he hydrodynamics poin of view, such velociy profiles are he genuine characerisic for regions of viscous flows suppored by viscosiy coefficien independen of z. Due o he urbulen naure of moions in WZ, his quaniy,, should be considered as he urbulen viscosiy, he effecive formula for which is he objec of heoreical consrucions. Fig. 3. General scheme for he mean flows disribuion in he inerface sysem (from Longo e al, 2012b). The speed scale for he ABL is shown a he op (U, m/s), for he WUL does a he boom ( u, m/s).

9 Thus, summing up he facs presened in his subsecion, we can sae he following conclusions: 1) he wavy inerface sysem has he hree-layer srucure, including: ABL, WZ, and WUL; 2) from he hydrodynamic poin of view, he WZ is an analog of fricion layer, having linear profiles for mean velociy (boh wind and curren), locaed beween he ABL and WUL; 3) in ABL and WUL, heir own wall-urbulence profiles of form (4) are realized for mean wind and curren, wih parameers deermined by he presence of wind waves on he free surface. 3. The model for a wind-drif curren 3a. Basic grounds On he basis of empirical daa menioned in he previous secion, a physical model, allowing jusifying he observed relaion of form (2), can be consruced be he following mean. Le us sar from he fac ha in he ABL here is he wind-induced downward momenumflux (or wind sress) a, going from he wind in he ABL o WZ, and furher ino he WUL. This flux could be wrien in he form w w u, (6) 2 a 1 3 *a where, on he righ-hand side, here is he fricion velociy in he air, u *a (formula (6) is, acually, he definiion of u *a erms of is modulus and in normalizaion o he air densiy, o he second power ). Here and furher, he wind sress, a, is wrien a ; whils he wind velociy componens, w 1, w 3, correspond o wind-vecor represenaion as W = (w 1, w 2, w 3 ). Angular brackes, as usual, mean he averaging over a saisical ensemble. I is physically clear (for he jusificaion, see, for example, Janssen, 1991; Polnikov, 2011) ha only he cerain par,, (called as he skin drag ) of he oal flux, creaing a surface curren, i.e. he wind drif. The oher par, is expended on he growh of wave energy. Thus, one can wrie a, is spen on w, ( form drag ) of he oal flux a w. (7) In works of he auhor (Polnikov, 2011, 2013), where quaniy is called as he angenial componen of he oal flux, he model was proposed for calculaing magniudes of each erms in (7), as funcions of wind speed W 10 and previously menioned wave parameers. This model is no used here, bu i is imporan o noe ha value can be furher considered as he known one. According o esimaes of paper (Polnikov, 2013), i ranges from 40 o 60% of oal flux a, and his proporion depends on he wave sae.

10 Thus, flux a is coming from he ABL o he WZ, where i is shared ino wo menioned pars. In urn, a par of he energy acquired by he waves in he WZ is carried by waves away, due o heir progressive feaure, and some par of he wave energy dissipaes immediaely wih he rae E wd, ransmiing he momenum fluxes o boh he WZ and he WUL. In our undersanding (Polnikov, 2013), he main mechanism for he wind-wave dissipaion is precisely he ineracion of waves wih he urbulence in WZ and WUL. Herewih, he naure of his urbulence is no imporan, since i is clear ha he urbulence is generaed by all possible mechanisms: breaking of cress; shear insabiliy of mean currens and orbial wave moions; pulsaions of he air pressure; formaion of droples and bubbles in WZ and WUL, ec. (for a discussion of alernaive mechanisms of wave energy dissipaion, see Babanin, 2009). A par of he dissipaing energy flux, E wd1, is los in he WZ, and he remainder par, E wd2, does in he whole WUL, which exends up o he deph of abou a half of he dominan wave lengh (due o he exponenial feaure of decaying ampliude for wave-specrum componens: a( k,z ) exp( kz )) (Phillips, 1977). Here, we will assume ha he second par of he dissipaive flux, E wd2 is compleely ransmied o he urbulence of he WUL. The firs par, E wd1, by he maer of physics, generaes he urbulen viscosiy in he WZ, and, possibly, in some small fracion of i, ransmis some momenum o he drif curren (for example, due o breaking processes). Consequenly, he final value of he momenum flux, ransmied o currens in he inerface sysem,, should be slighly higher han he value menioned above. However, since oday here is no clear assessmen of an addiional inflow of he momenum from he dissipaing waves o he drif curren, we assume ha he angenial flux of he horizonal momenum, realized a he upper boundary of he WZ, has he following quaniy 0.5 a = u. (8) *a Furher, we assume ha i is momenum (8) does form he wind drif boh in he WZ and in he WUL (hough, he exac digi in (8) does no any principal role). Noe, however, ha in order o deermine he drif velociy in he waer, including WZ, i should be used he angenial momenum flux, w, normalized o he densiy of waer, w. Therefore, by virue of he coninuiy of angenial par of momenum flux hrough he inerface, he value of flux in waer, w, is given by he raio / ro 05. u, (9) 2 w a w *a,

11 where ro a / w 10-3 is he raio of air and waer densiies. Formula (9) also deermines he fricion velociy in waer, u *w, by he raios u ro ro 05. u. (10) 2 2 *w w *a Now, le us clarify he geomery of verical disribuion of he drif curren. On he basis of he observaion resuls, presened in Longo e al. (2012b) and shown in Fig 3, i can be saed ha he drif curren localized in he WZ has a linear profile. A he upper boundary of he WZ, he sough surface drif, U d0, akes place (furher, he lower index "0" will be omied, for simpliciy of designaion). I is his value is considered as he observaion resul. And a he lower boundary of he WZ, he drif velociy should be deermined by he fricion velociy in waer, u *w by formula (10). Velociy u*w, given is also used in he formula of logarihmic profile of form (4) for he WUL, saring from he lower boundary of he WZ. According o (10), u ( ro / ) u = 12 2 / *w *a u *a, i is always valid he inequaliy: u *w << u *a, and, consequenly, i is valid ha u*w << U d, wha is very imporan for our aims. Then, o close he momenum flux, we use he sandard mehod of he urbulence heory (e.g., Monin & Yaglom, 1971; Phillips, 1977; Wu, 1975; Longo e al., 2012a), according o which he consan verical momenum flux, w, in he layer of viscous flow (he analogue of which is he WZ, as was shown above) is balanced by he verical gradien of mean velociy, i.e. Here, w U d( z ). (11) z is an unknown urbulen viscosiy, aking place in he WZ, he value of which is provided by he whole complicaed hydrodynamics of his zone. To complee he consrucion of model, i remains o add ha: a) he value of gradien in (11) is also consan, according o he experimenal daa. is consan in he heigh; b) he verical velociy 3b. Semi-phenomenological approximaion of he model According o he said above abou profile U(z) in he WZ, wih a high degree of reliabiliy, he esimae of he velociy gradien in (11) is given by U z d (U u ) / ( c h ) U / h d 0 *w zw d 2, (12) where h = <a> is he average wave ampliude a he surface-poin under consideraion; and, in he final expression of (12), index "0" is omied, wih he aim of simpliciy. In he firs, exac

12 equaliy, he dimensionless coefficien, c zw, is formally inroduced, he value of which follows from he measuremens. In he final, approximae equaliy, he heigh of WZ is aken from he resuls of (Longo e al., 2012b) in he explici form. The urbulen viscosiy funcion is easy o be parameerized, as is cusomary used (Monin & Yaglom, 1971; Phillips, 1977; Polnikov, 2011,2013), in he frame of dimensional consideraions. Here we sae ha: 1) he average wave heigh, h, is he characerisic size; and 2) he value of drif curren a he upper boundary of he WZ, U d, is he characerisic velociy. In his case, we obain he expression f ( h,,a,...) U h, (13) d in which here is he unknown dimensionless funcion, f ( h,,a,...), depending only on he wave parameers. As a resul, he balance (11) acquires he form of an equaion for deermining drif velociy U d on he upper boundary of he WZ: ro 0. 5u ( S,W ) ( f U h ) (U / 2h ) f ( h,,a,...)u / 2. (14) 2 2 *a 10 d d d Here, in he lef-hand side, he known dependence of fricion velociy, u *a( S,W 10 ), on wave specrum S and wind speed W 10 (for example, see Polnikov e al, 2003, Polnikov, 2013) is noed for compleeness of descripion, and, in he righ-hand side, a poenial dependence of unknown funcion f on he wave parameers is noed as f (h,ε, A,.), which will be symbolically denoed laer as f (.). From equaion (14), one can immediaely found he soluion for he sough drif velociy on he wavy surface of waer in he form 12 / U ro / f (.) u. (15) d *a From he above, i is clearly seen ha resul (15) coincides wih he observaions: in he range of values U ( ) u ), he values of f (.) should be varying wihin he range: f (.) d a* , including a possible dependence of f (.) on he wave parameers. Such an esimae of values for coefficien f (.), inroduced in formula (13), is quie plausible, since in he pracice of sudying boundary phenomena here is a large number of dimensionless quaniies of such order. For example, he Phillips parameer, α Ph, sanding in he formula for inensiy of he ail for a sauraed wind-wave specrum (Phillips, 1977): S( ) g 2 5 Ph, (16) has he magniude of he order of α Ph Anoher examples is he Charnock s parameer, α Ch, defined by he formula

13 z / ( u / g ), (17) 2 Ch 0 *a which has he same order (Phillips, 1977). As can be seen from (17), α Ch is he dimensionless characerisic for he roughness heigh, z 0, used in he formula of logarihmic layer (4), wrien in he erms of fricion velociy u *a and graviy acceleraion g. Moreover, i is well known ha boh dimensionless values in (16), (17) are he funcions of wave parameers and wave-formaion facors, and heir magniudes can vary wihin a wide range (Phillips, 1977). The foregoing gives he sufficien grounds for he saing ha formula (15) is he heoreical jusificaion for empirical formula of form (3), wha complees, in general, he soluion of he problem posed. 4. Discussion of he resuls Firs of all, i is ineresing o consider an alernaive solving he poin of paramerizaion for he urbulen viscosiy funcion,. Le us ake he fricion velociy in waer, u *w, as he scale for velociy (insead of U d ), changing correspondingly unknown funcion f (.) by he oher unknown one, f w (.). Then, i is easy o obain from (10) and (13) ha he funcional represenaion of U d ges he form: U d ~ ro 1/2 u *a /f w (.). By changing he noaion of unknown, f w (.), by oher he unknown, f 1/2 (.), one can sae ha he laer represenaion of Ud does quie correspond o raio (15). Therefore, furher discussion of he resul obained will be performed on he basis of formula (15). I is also ineresing o analyze an applicabiliy of raio (15) in he limiing case of complee absence of waves. In his case, here is no WZ, he verical gradien of drif velociy and he inroducion of viscosiy funcion become o be no needed, and funcion f (.) degeneraes ino a uniy. Then, from (15) i follows ha velociy U d is simply equal o he fricion velociy in waer, u *w, which is deermined by formula (10) wihou coefficien 0.5. In oher words, he funcional form of formula (15) is preserved, wha allows us o analyze i furher. According o observaions (see Secion 2), he drif velociy a he surface of waer, U d, depends only on he fricion velociy in he air. The explici dependences of U d on he wave parameers are no esablished in he experimens ye (ill now). This can be sipulaed by eiher he physics of processes considered (wha, as shown above, is fully reaed heoreically on he basis of formula (15) under he condiion of invariabiliy of f (.) ) or he lack of proper experimenal daa. In his regard, model (15) gives he possibiliy o make a more deailed descripion of he drif curren properies.

14 In paricular, le's ry o answer he quesion: wha dependences of he drif curren on he parameers of wind-waves: heigh h, seepness ε, age of waves A, and oher characerisics of he wave specrum (in dimensionless combinaions of parameers), U d (h, ε, A ), could be expeced on he basis of raio (15). According o he adoped model, all he sough dependencies are deermined by he paramerizaions for he drif-curren gradien, U d / z, and for he urbulen viscosiy,. The esimaion of magniude for he verical gradien of drif velociy in he WZ, U d / z, is compleely based on he measuremens, and i does no allow any funcional changes. In his case, he explici inverse dependence of U d / z on he wave heigh is compensaed (in he adoped model) by he explici linear dependence of he urbulen viscosiy on h. Therefore, in his heory, he only degree of freedom, which allows he possibiliy for addiional dependence of on he wave parameers, remains he specificaion of paramerizaion for dimensionless funcion f (.). Thus, if he dependence of f (.) on wave parameers is heoreically possible, hen i appears a possibiliy for dependence of U d on he wave parameers, U d (h,ε, A, ). Since funcion f (.) is responsible for he inensiy of urbulen viscosiy, i is necessary o look for hose physical processes (and parameers) of wind waves which can change he degree of mixing in he WZ, and, herefore, affec he magniude of urbulen viscosiy coefficien in WZ, i.e. on he value of f (.). The mos likely processes of his kind can be: a) micro- and macrobreaking of wave cress (in erminology of paper by Longo e al, 2012a); and b) he shear insabiliy of orbial wave moions. The inensiy of he firs process, unequivocally, mus grow wih he increase of wave seepness (Babanin, 2009); and he second does wih he growh of wave ampliude (including he growing wave age) due o he increase of local Reynolds number for orbial wave moions. An increase of he breaking frequency is also possible while sharpening cress of waves, appearing wih he age growh, due o he addiional horizonal impac of wind on he cress presening on he background of deep roughs (Babanin, 2009). Now i should be noed ha an increase of inensiy for he verical moions in he WZ corresponds, obviously, o a decrease in he effecive viscosiy, i.e. o he decrease of value for f (.). Thus, on he basis of he said above, i can be assumed ha funcion f (.) will decrease wih increasing boh he seepness of waves, and, possibly, heir age. As a resul, according o formula (15), we can expec an increase of drif velociy U d wih increasing seepness and wave age for a fixed value of he fricion velociy. I is impossible o predic heoreically he funcional form of dependences U d (ε) and U d (А), because of he saisical naure of insabiliy forming hem. Besides, i should be remembered ha wih growing wave age A, boh wave seepness ε and depending on i fricion

15 velociy u *a end o decrease (Drennan e al., 2003; Polnikov e al, 2003). Therefore, wih increasing A, here are appearing he mulidirecional rends, which can compensae each oher o a large exen, canceling ou poenial dependence U d (А). Neverheless, in he fuure, i seems quie reasonable o search for poenially possible direc empirical dependences of drif velociy boh on seepness, U d (ε), and on wave age, U d (А), in addiion o he explici proporionaliy, Ud ~ u *a, already esablished. 5. Conclusions The resuls of he work allow us o draw he following conclusions In order o describe he drif velociy on a wavy waer surface, he concep of he hreelayer srucure for he wind-wave-curren sysem is inroduced. The wave-zone (WZ) wih dimensions of he order of wo-hree variance of wave record η(), in which he air and waer are presen alernaely, has is own dynamics of processes in erms of average values. I is in his zone, he measured drif curren is formed, which has a shear profile The experimenally esablished linear profile of he average drif curren, U(z), in he wave zone (Fig. 3) allows us o consider his zone as he analog of fricion layer, in which a consan verical flux of momenum is conserved, and consan coefficien of urbulen viscosiy is realized The balance beween he momenum flux and he verical gradien of mean velociy, applied in he WZ, makes he basis of he drif curren model (formula 11) In he simples case of independence of dimensionless funcion f (.) (in formula (13) for urbulen viscosiy ) of wave parameers, he model gives he linear proporionaliy beween drif velociy on he waer surface, U d, and fricion velociy in he air, u *a, (formula 15) In a more general case, formula (15) predics a possible increase of drif velociy U d wih an increase of mean wave slope ε and heir age A. This effec is physically sipulaed by a decrease of urbulen viscosiy (via funcion f (.)) wih an increase of inensiy of wave breaking, leading o increasing inensiy of verical movemen dynamics The absence of direc empirical dependences of U d on wave parameers ε and A (and ohers) requires is convincing empirical confirmaion or heir search on he basis of careful special experimens. 6. Acknowledgemens The projec was suppored by he RFBR gran No

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