Integrated model for a wave boundary layer

In the paper a new version of semi-phenomenological model is constructed, which allows to calculate the friction velocity u* via the spectrum of waves S and the wind at the standard horizon W. The model is based on the balance equation for the momentum flux, averaged over the wave-field ensemble, which takes place in the wave-zone located between troughs and crests of waves. Derivation of the balance equation is presented, and the following main features of the model are formulated. First, the total momentum flux includes only two physically different types of components: the"wave"part TAUw associated with the energy transfer to waves, and the"tangential"part TAUt that does not provide such transfer. Second, component TAUw is split into two constituents having different mathematical representation: (a) for the low-frequency (energy-containing) part of the wave spectrum, the analytical expression of momentum flux TAUw is given directly via the local wind at the standard horizon, W; (b) for the high-frequency part of the wave spectrum, flux TAUw is determined by friction velocity u*. Third, the tangential component of the momentum flux TAUt is parameterized by using the similarity theory, assuming that the wave-zone is an analogue of the traditional friction layer, and in this zone the constant eddy viscosity is realized, inherent to the wave state. The constructed model was verified on the basis of simultaneous measurements of two-dimensional wave spectrum S and friction velocity u*, done for a series of fixed values of W. It is shown that the mean value of the relative error for the drag coefficient, obtained with the proposed model, is 15-20%, only.


Introduction
This work is devoted to constructing a conceptually new model of the dynamic boundary layer of the atmosphere (or the wave boundary layer, WBL, in terms of Chalikov (1995) ) and is a natural extension of our previous paper (Polnikov 2009). Herewith, it is believed that the main purpose of the WBL-model is to establish the mathematical relation between characteristics of the boundary layer of the atmosphere (in particular, the friction velocity ) with the twodimensional spatial spectrum of wind waves * u ( , ) x y S k k (or the frequency-angular analogue, S ω θ ), and the mean wind at standard horizon W. The mathematical basis of this model is the balance equation for the momentum flux of the form 1 . (1) The physical content of the left-hand side of (1) must follow from the fundamental equations of hydrodynamics.
The task of constructing a physical model of the WBL can be solved in different ways. In Polnikov (2009) a classification of approaches to solving this problem is given where semiphenomenological models (Janssen 1991;Zaslavskii 1995;Chalikov 1995;Makin and Kudryavtsev 1999) were mentioned (among numerous others) as the most acceptable in practical terms. More recent alternatives (Kudryavtsev and Makin 2001;Makin and Kudryavtsev 2002) also belong to this class 2 . In the terms mentioned, the advantages of such models are their physical justification based on the equations of hydrodynamics and the lack of overly cumbersome analytical or numerical details. In practical models the latter are replaced by a number of quite clear and simple phenomenological constructions. Based on the analysis mentioned, the models by Zaslavskii (1995) and Makin and Kudryavtsev (1999) were indicated as the most promising ones for further development. The disadvantages of these models were mentioned in Polnikov (2009) as well. Taking in mind these findings, we will try to advance further in the problem.
The physical basis of all these WBL-models is one or another version of the analytical representation of the left-hand side of balance equation (1) by a set of summands which generally has the form (Makin and Kudryavtsev 1999) 2 * w t u ν τ τ τ τ + + = = . (2) In the left-hand side of (2) it is used to distinguish three components of the total stress τ : the wave component w τ , viscous one ν τ , and turbulent one t τ . The first of them is usually associated with the transfer of energy from the wind to waves, whilst the last two create the tangential stress at the interface, not affecting the energy of waves. The problem is to find an adequate analytical representation for each of these components 3 .
In semi-phenomenological approach it is widely accepted (see the same references) to describe the wave part of momentum flux w τ via the known function that corresponds to the rate of energy pumping waves by the wind. It is traditionally represented as (Komen et al. 1994;Polnikov 2009) Without dwelling on the technical point of choosing function ( , , , ) w W β ω θ θ , one can assume that the mathematical representation of flux w τ is known, and its estimation does not cause any principal difficulties. 3 It should be especially noted that the equation of the form (2) is traditionally ascribed to a narrow atmospheric layer located directly at the air-water interface (see the mentioned references). In this case, it is implicitly assumed that the flux to a random and non-stationary surface is essentially similar to the flux towards a solid surface. The stochasticity of a wave field, implying the presence of abrupt changes in shape of the air-water interface, and its non-stationarity due to multiple vertical wave motions, are not taken into account in such approach.
Regarding to the procedure of calculating values t τ and ν τ , in contrast to estimation of w τ , there is not a unity of approach. Therefore, in each of the above-mentioned versions for WBLmodels, calculations of these quantities vary considerably.
For example, in the model (Makin and Kudryavtsev 1999), the calculation of t τ is based on the well-known theoretical relation for the turbulent part of the momentum flux (Monin and Yaglom 1971) where the dimensional coefficient K has a meaning of the turbulent viscosity in the WBL. Using representation (2) (considered beyond the space of influence of ν τ ), by constructing an additional specific model for the turbulent viscosity in the WBL of the atmosphere, Makin and Kudryavtsev (1999) have obtained Result (6) allows to complete the solution by using a known expression for the vertical profile of the wave flux ( ) w z τ . In the referred paper, the latter is adopted as the exponentially decaying function: (see details in the references). Herewith, the essential point of the model by Makin and Kudryavtsev (1999) is the assertion that just the turbulent component of friction velocity, , is to be taken in the integrand of (4) (instead of the full friction velocity, ), in the course of calculating w τ by formula (4). However, the results of the model verification, obtained in (Polnikov 2009), show that this approach to constructing the WBL-model does not provide an adequate range of variability for dependence , and requires some modification. * ( , ) u S W The sequential works by the same authors (Kudryavtsev and Makin 2001;Makin and Kudryavtsev 2002) serve as an example of such, rather significant modification of the approach mentioned. Since they have not been discussed previously in (Polnikov 2009), it is worthwhile to bring shortly the following important points. Thus, Kudryavtsev and Makin (2001) rejected the idea of using the differential relation (5), and began to assess the value of t τ by accepting the idea of matching the linear profile of the average wind speed with the standard logarithmic velocity profile at the boundary of the molecular viscous sublayer (where profile depends . It means that they consider equation (2)  S ω θ and magnitude of , it was shown that for the model discussed, the mean relative error of the total flux (or the drag coefficient C d ), given by the ratio is of the order of 20-25% in a wide range of values for . All the said testifies to the significant advantage of the new model (Kudryavtsev and Makin 2001) compared with the earlier one (Makin and Kudryavtsev 1999). * u However, peering and detailed consideration of the concept of constructing the integrated model (Kudryavtsev and Makin, 2001) raises several questions.
First. The introduction of term fs τ is interesting itself from the standpoint of physics, though it does somewhat artificially complicate the task of constructing the WBL-model. Indeed, due to linearity in the spectrum, the main contribution of the air-flow separation stress fs τ to the total stress τ , in fact, can be accounted for in the traditional representation for w τ by choosing proper parameters for the empirical increment ( , , , ) w W β ω θ θ . Appropriateness of the said is provided by the fact that inevitable breaking events are automatically taken into account in the empirical parameterization of ( , , , w W ) β ω θ θ (Drennan et al. 1999;Komen at al. 1994;Chalikov and Rainchik 2010; among others).
In addition, after averaging the balance equation over the wave-field ensemble (see details in Section 2), a possible contribution of fs τ to the tangential stress becomes uncertain, and this circumstance should be taken into account in the assessment of t τ . Thus, the introduction into consideration the air-flow separation, as well as the introduction of breaking events, complicates the basing a validity of using the traditional (molecular) viscous sublayer for the determination of tangential stress t τ , realized in (Kudryavtsev and Makin 2001).
Second. The validity of using the traditional viscous sublayer in the situation with a random, highly non-stationary, and spatial-inhomogeneous surface experiencing the breaking, invokes a serious doubt in the method of assessing t τ used in the model said. Breaking and randomness of the interface are clearly not in accordance with the traditional approach based on existence of molecular viscous sublayer, applicable to a firm and fixed surface. It seems that in view of stochasticity of the interface, the traditional concept of the viscous sublayer should be refused and properly replaced.
Third. The final representation of the balance equation in form (7), resulting from introduction a set of postulates, hypotheses, and fitting constants of the model ( fs C , fs k t C , c ν ), is very cumbersome and difficult to treat it due to its irrational kind with respect to .
Therefore, there arises a natural need in constructing a simpler, but equally physically meaningful WBL-model. * u Thus, all the said above is the basis for attempts to construct a new semi-phenomenological WBL-model in a frame of less complicated and physically reasonable assumptions and fitting parameters. This paper is aimed to solve this problem.
The structure of the paper is the following. In Section 2 we mention shortly the main points of the balance equation derivation with the aim to introduce the new concept of "the wave-zone".
That allows us to get new treating the terms of this equation. In Sections 3 and 4 parameterizations for two constituents of wave-part stress w τ are specified. Section 5 is devoted to constructing parameterization for t τ by means of the similarity theory. The method and results of the model verification are given in Section 6, and the final conclusive remarks are presented in Section 7.

Derivation of the Balance Equation. Introducing the Concept of "the Wave-zone"
In order to establish the physical content of the left-hand side of equation (1) applied to the problem with a random wavy interface, and taking into account the comments made in footnote 3, we briefly consider the balance equation derivation for a wavy surface (the main calculations in this section are a courtesy of VN Kudryavtsev). We show that the kind of representation of the left-hand side of (1) is largely determined not by the detailed account of the dynamic equations at the wavy interface ( , ) t η x but is done by the rules of averaging the balance equation.
To obtain equation (1), the Navier-Stokes equations are used as initial ones, written (for generality) for the three-dimensional, unsteady turbulent flow of air with the mean speed profile over the wavy interface: Here, for index ( 1, 2) α = it is sufficient to take value 1, whilst the index i takes the values 1,2,3, corresponding to x, y, z components of the velocity field u; the pressure field p is given in the normalization on the density of air; i α τ is the viscous-stress tensor; the remaining notations are taken from the monograph (Monin and Yaglom 1971). The task is to obtain equations for the momentum fluxes. For this purpose a procedure of integrating equation (9) over the vertical variable from the interface ( , ) t η x to horizon z located far from the wavy surface is used. In such a case, the formula of differentiation of the integral by parameter (Leibniz's formula) is applied (Monin and Yaglom 1971): After integrating (9) and using (10) with ( , ) a t η = x and b z = , it follows (Makin et al. 1995) where the appearing index β is analogous to index α. Now, it is necessary to perform the spatial and temporal averaging (11) (for 1 α = ) ), supposing for simplicity that , =0, means the turbulent fluctuations of the air velocity). Herewith, it should be noted that the averaging is to be done both on the turbulent scales and the scales of the wave-field variability.
The first type of averaging leaves the integral forms in (11) practically unchanged, while the "free" terms in (11) yield the momentum fluxes that we are looking for.
For a steady flow over a solid and horizontal surface, the integral terms in (11) vanish. For a moving and horizontally inhomogeneous surface, these terms can also be cleared but only if one passes to the scales of horizontally homogeneous and steady-state description of the wave field, i.e. after the wave-ensemble averaging. Incidentally, it is the averaging that allows us to pass to the spectral representation for waves, used in the WBL-model representation. As a result of these actions, the final equation of the form (1) appears, which can be written in the form where the small angle brackets <..> denotes averaging over the turbulent scales, and large ones does the wave-ensemble averaging. Note that balance equation (12) is not ascribed to the certain wave profile, as well as the wave spectra are not ascribed to the certain water level. In this case, generally speaking, it should be formulated a special treatment of the terms of equation (12), due to non-stationarity and inhomogeneity of interface ( , ) x t η .
Traditionally, the first term in the left-hand side of (12) is interpreted as the momentum flux associated with the work of pressure forces, while the second term to the viscous flux component. It means that the meaning of w τ is prescribed for the first term, and the meaning of ( t ν τ τ + ) does for the last one (see eq. (2)). Herewith, equation (12) is often interpreted as the balance equation written at the interface (see footnote 3 and the treatment of viscous terms in Kudryavtsev and Makin (2001)), implying that the average over the wave scales is realized by sliding along the current border between the media ( Figure 1a).
Though, such kind averaging does implicitly assume the horizontal and temporal invariance of the vertical structure of the WBL (i.e. uniformity in the mean-wind profile for any value of z, measured from the current position of the boundary), regardless to the wave phase. It is caused by wishing to use directly the traditional interpretation of the terms, mentioned above. Fig. 1. а) a part of wave record ( , ) x t η , b) an ensemble of two hundreds parts of the same wave record ( , ) x t η .
The time scales are given in conventional units.
However, it is easy to imagine that in this, "monitor" coordinate system, the conditions of stationarity and horizontal homogeneity of the mean motion are violated (to say nothing about breaking), despite of the necessity of existing these conditions for removing the integral terms in (11).
In fact, the dynamics of an air flow over a wavy surface (and the vertical structure of the WBL) varies significantly on the ridges, troughs, windward and leeward sides of the wave profile, i.e. the horizontal and temporal invariance of the vertical structure of WBL is violated.
This "speculative", but fairly obvious conclusion is confirmed by the numerical solution of the Navier-Stokes equations (see, for example, Li et al. 2000;Sullivan et al. 2000;Chalikov and Rainchik 2010). Therefore, the result of averaging equation (11) for the case of a random wave surface, which is required not only to eliminate the non-stationarity and horizontal inhomogeneity of the vertical structure of WBL, but also for the transition to a spectral representation of waves, gives rise to search for different interpretation of (12) .
To achieve this goal, it is more natural to treat the averaging equation (11) as the averaging performed over the statistical ensemble of wave surfaces, conventionally depicted in Fig. 1b.
Clearly, in this case, that equation (11) is averaged over the entire "wave-zone" located between the certain levels of troughs and crests of waves. This zone occupies the space from -H to H in vertical coordinate measured from the mean surface level, and the value of H is of the order of the standard deviation D given by the ratio Besides to the said, in the course of ensemble averaging, each term in the left-hand side of (12) undergoes to such changes which can not be identified and treated in advance, i.e. their contributions to w τ и t τ can not be distinguished. Therefore, we can assume that in the left-hand side of the final balance equation there are only two types of terms: the generalized wave stress w τ (traditionally called "the form drag") which corresponds to the entire transfer of energy from wind to waves; and the generalized tangential stress (called "the skin drag") which is not associated with the wave energy. Thus, the resulting balance equation becomes somewhat "simpler" and takes the form * * ( , , , ) ( , , , ) In (14) contributions of the individual terms of the left-hand side of equation (12) to magnitudes of the wave and tangential stresses are already not detailed.
It should be noted that the writing momentum balance equation in form (14) is not new and, as mentioned above, is widely used (Zaslavskii 1995;Makin and Kudryavtsev 1999;Polnikov 2009; among others). It only states the fact that contributions of the terms in the left-hand side of equation (12) into the left-hand side of (14) are not a priori distinguishable. Equations (12) and (14) are not ascribed to a certain vertical coordinate; they are valid throughout the wave-zone, as well as the wave spectrum is not tied to a specific level of the wavy interface.
Next, as usual, we assume that wave stress w τ in (14) can be determined via the pumping function (4) (14) (see details below).
The value tot τ , standing in the right-hand side of (14), has a meaning of the total momentum flux from the wind to wavy surface, averaged over the wave ensemble. It is quite natural to assume that this momentum flux tot τ has the value actually measured at some horizon in the WBL, located highly from the largest wave crests; that is, as usual, where is the friction velocity.  (2010)). It is this scrutiny of features for w τ allows us to give its analytical representation in the form of (4), where the pumping function IN is used, mathematical representation of which has widely accepted theoretical and empirical justification.

IN
, , The most detailed representation of , useful for further, can be written in the form (Komen et al. 1994;Polnikov 2009) ( ) % are usually parameterized on the basis of various experimental measurements or numerical calculations, due to what its analytical representation is not uniquely defined (Komen et al. 1994;Donelan et al., 2006;Chalikov and Rainchik, 2010). 5 In a narrow domain of frequencies ω (or wave numbers k), covering the energy-containing interval of gravity waves within the limits (the so called low-frequency domain) if the parameterization of via u * is used. There are also present parameterizations of via W, among which it is appropriate to mention one of the last (Donelan et al. 2006): where ] w ε θ θ is the empirical function of the wave component steepness, ( ) k ε , the direction of its propagation, θ , and the local wind direction, w θ .
In a higher and broader frequency domain, which includes the capillary waves ranging up to 100 rad/s (where the most significant part of the wave momentum flux is accumulated), the contact measurement of ( * / , , W u c ω ) β θ θ % becomes impossible due to evident technical reasons.
This deficit of information can be compensated by another means, at certain extent. Namely, remote sensing and numerical estimates indicate the fact that, in the mentioned frequency domain, β % has a shape of a broad asymmetric "dome" with the maximum at frequencies of about 3rad/s (corresponding to wave numbers of the order of 1m -1 ). Function β % drops to zero both at the low frequencies corresponding to condition W/ c ω <1 (i.e. when the waves overtake 5 There are more detailed representation of β % , using dependences on wave age А and wave component steepness Donelan et al., 2006). But they are not discussed here. the wind) and at high frequencies corresponding to the region of the phase-velocity minimum (Elfouhaily et al. 1997;Kudryavtsev et al. 2003) (Fig. 2). Additionally, regarding to estimates of w τ based on formula (4), it should be noted that the integral in the right-hand side of (4) is slowly convergent (Polnikov 2009;Zaslavskii 1995;Makin and Kudryavtsev 1999;among others). It is for this reason the limits of integration in (4) should cover the domain from the maximum frequency of the wave spectrum p ω , having the order of (0.5-1)rad/s, up to frequencies of about 100 rad/s (k ≅ 1000rad/m), including the domain of capillary waves existence where β % is close to zero. The using this extended domain ensures convergence of the integral in (4). However, the wave spectrum in such domain is very difficult to obtain both by contact measurements and by regular numerical calculations. For this reason, a sufficiently accurate assessment of flux w τ requires using a special approach. It consists in sharing the wave spectrum into two fundamentally different parts: the low-frequency spectrum of gravity waves (LFS), actually measured (or calculated by some numerical model), and the L S high-frequency spectrum H S L H S (HFS), a modeling representation of which can be found by using a variety of remote measurements and their theoretical interpretation (Elfouhaily et al. 1997;Kudryavtsev et al. 2003). Thus, the farther progress in constructing the WBL-model consists in using representation of the wave spectrum in the form of two terms, for example, in the form proposed in (Elfouhaily et al. 1997 ): cut A ω is the known cutoff factor.
Since we are interested in integrated estimations of the type of ratio (4) where γ is the water surface tension normalized on the gravity acceleration g, and d is the local depth of the water layer.
Thus, the explicit shearing the wave-part stress into two summands is introduced in (21) (Elfouhaily et al. 1997;Kudryavtsev et al. 2003). Assuming that the both parts of the total (gravity-capillary) wave spectrum can principally be represented in a quantitative form, one can state that the magnitude of the wave-part stress is known as a function the following arguments: wave spectrum S, wind speed W, friction velocity , wind direction * u W θ , peak-wave-component direction p θ , wave age , and mean wave-field steepness A ε .
Introducing the dimensionless variables, ˆw L τ and ˆw H τ , we can write the ratio 2 * * [ ( , , , , , , ( , , ) the summands of which, ˆw L τ and ˆw H τ , are potentially known. Herewith, we especially reserve the wind speed W as the argument, to keep the possibility of getting dependence on W as the solution of equation (14) with respect to unknown function .
Besides the said, to simplify the procedure of assessing the high-frequency component ˆw H τ , it is proposed to tabulate the numerical representation of the latter, calculated on the basis of known shape-function for the HFS (Elfouhaily et al. 1997;Kudryavtsev et al. 2003).

Numerical Specification of ˆw H
τ .
With the aim to specify function , let us calculated it numerically as a function of friction velocity and wave age A, defining A via the wave number of dominant waves, k p .
For this purpose, we use the modeling representation of HFS spectrum (Kudryavtsev et al., 2003) where the matter is disclosed with a maximum specificity. Since, according to the reference, the analytical presentation of * , w u ( , , H S k θ θˆw H is extremely cumbersome, to save the space, here we confine ourselves to graphic illustrations of τ , made by the method described in (Kudryavtsev et al., 2003). This approach allows us to obtain the quantitative representation of function * ) w u ( , detailed examination showed that this effect is due to the above-described behavior of the growth increment β % (see Figure 2). Indeed, while increasing k p , the lower limit of integration domain H Ω in (21) moves up in the wave number scale, and the corresponding value of β % begins to decline, providing less weighting factor under the second integral in (21), what leads to the effect under consideration.
The observed numerical effect does somewhat complicate the problem of solving equation (14), because the magnitude of ˆw H τ , as it would seem, is to be determined mainly by the shape of the HFS, i.e. depends on the friction velocity , only. However, Fig. 4  In contrast to flux w τ , the value of tangential flux t τ , transferring the momentum to the underlying surface as a whole, is practically not determined in the light of equation (12) derivation. To this regard, it is appropriate to note that in ocean-circulation models, ignoring the existence of waves at the interface, the value of t τ is associated with the full wind stress tot τ , as the source of drift currents. In fact, besides from the drift currents, the wind generates waves.
Therefore, there is a large and physically justified difference between tangential stress t The author has recently processed the data of numerical experiments by Chalikov and Rainchik (2010), devoted to the joint simulation of the wind and wave fields in the case of air flow over the wavy surface. Kindly provided data of the simulated wind field W(x,z), obtained by the mentioned authors in the curvilinear coordinates following to the wavy interface, have been converted to the Cartesian coordinates, and the profile of mean wind W(z) was built in the wavezone, i.e. in the area between troughs and crests of waves where water and air present alternately.
Details of these calculations can be found online at the site ArXiv.org (Polnikov 2010). It turned out that in the wave-zone, the mean wind profile W(z) has a linear dependence on z (Fig. 5).
Herewith, linear profile W(z) is spread from -D to 3D in the height measured relative to the mean water level assumed to be zero. Here, D is the standard deviation of the wavy surface, given by (13)  Results are shown in the dimensionless units; zero value of z corresponds to the mean surface level η . 7 Though, starting from height 3D over the mean surface level η , the mean wind profile W(z) becomes close to the logarithmic one, in full accordance with the experimental observations (Drennan et al. 1999) (see Fig 5b).
Herewith, upper the wave-zone, the wave-part flux w τ has the exponential decay law close to the traditional representation as ( ) exp( ) w z k p z τ ∝ − (Polnikov, 2010).
In the frame of the proposed approach, we believe that the wave-age parameter A, being associated only with the ratio of the dominant waves phase-velocity to the wind speed, is less crucial for evaluation of than the mean steepness of waves w K ε . Therefore, at present, preliminary stage of the WBL-model specification, the account of dependence (А) may be excluded, and the sought function can be written in the form: S k θ should be measured simultaneously. According to the literature, such kind of measurements are rather common (see references in Makin and Kudryavtsev 1999;Kudryavtsev and Makin 2001;Babanin and Makin 2008). Nevertheless, they are hardly available for us, and this is the main technical difficulty of performing the verification procedure. Results of simultaneous measurements ( , ) S ω θ and , being at our disposal, are very limited and based completely on the data described in (Babanin and Makin 2008), kindly provided by Babanin.
Full information about the measured parameters of the "wave-boundary layer" system is described in the reference; so, it is further provided in an extremely compressed form (Table 1) sufficient to analysis of verification results.

Specification of the Model
The model is given by equation (14) (27) corresponds to the generalized parameterization of empirical , given by formula (18) and proposed by Yan(1987). This form is widely used by the author (see references in Polnikov 2009). In the present case, approximation (27-28) is tentative but not the final one.
(...) β % postulate that the transfer of momentum from the atmosphere to the gravity waves is determined by the value of mean local wind W. From a technical standpoint, this postulate is needed to ensure that in the balance equation (14), written for friction velocity , wind W is also present.
This approach provides the sought dependence . From physical point of view, this approach is fully justified, since wind W is the primary source of both the wave energy and the statistical structure of the interface.  Note. The column for wave steepness ε is shaded for better identification. Sign (!) in this column does mean "a mismatch" between values of ε and k p . Nevertheless, it should be noted that the results reported here are not final yet, and still require a large series of comparisons based on a more extensive database of measurements, before the optimal choice of the used fitting functions and parameters of the model will be finally determined. Such work is planned in the nearest future.
In addition to the said, and in order to analyze the quality of experimental results presented in Tab. 1, one should pay attention to the following. Even in our selected series, sometimes there is a clear discrepancy in values between too small wave steepness ε and very high values of the peak wave number k p (see note to Tab. 1). It is in these cases, there are the most significant errors of the discussed model. The foregoing demonstrates the need of careful controlling the measurements accuracy and selection of empirical data involved to verification of such models.

Conclusion
Summarizing, we formulate the main regulations of the proposed approach to solving the problem of relation between friction velocity , as the main parameter of the atmospheric layer, and the main characteristics of the wind-wave system: the local wind at the standard horizon W, and the two-dimensional spectrum of waves * u ( S , ) ω θ .
First. The derivation of the initial balance equations for the momentum flux at the wavy interface (Section 2) shows the need of rethinking interpretation of the components of balance equation (1), as far as the latter is to be averaged in the wave-zone covering the area between through and crests of waves (Fig. 1b). To solve this issue, it seems to be required not so experimental efforts but rather the detailed theoretical analysis of features of the wind-flow dynamics near a wavy surface, small-scale details of which can be obtained mainly by numerical simulation (for example, by analogy with simulations in (Li et al. 2000;Sullivan et al. 2000;Chalikov and Rainchik 2010;among others).
Second. It is proposed to share the total momentum flux from wind to wavy surface tot τ into two principally different components, only: the wave part w τ responsible for the energy transfer to waves, and the tangential part t of the interface dynamics. It is simply a convenient technical tool allowing achieving the task solution.
Fifth. Regarding to tangential stress t τ , there is not commonly used and theoretically justified analytical representation for it in terms of the system parameters. Therefore, to describe t τ , in the present model it is proposed using the similarity theory. The main idea is based on the proposition that the wave-zone, located between troughs and crests of waves, is an analogue of the traditional friction layer. This idea is supported by the results of the author's analysis (Polnikov 2010) of the numerical simulations performed in (Chalikov and Rainchik 2010).
According to this results, the mean wind profile is linear in vertical coordinate z what is typical for a vicious layer. Thus, by using formula (5), function ( ) W z t τ can be parameterized simply via the mean-wind gradient ∂ and a constant turbulent viscosity K , as functions of the system parameters (formulas (5), (21)-( 24)). As a result, the balance equation (14) becomes closed what provides the problem solution. and suitable in accuracy, to be used for the WBL-model verification. Problems and ways of solving these issues are considered in detail in (Polnikov 2009).