10J h0 cm.

Figure 3.2 Dependence of the sea surface roughness z0 on the average height h0 of wind waves, according to Kitaigorodskii (1970). The various symbols indicate estimates by different authors.

the logarithmic character of dependence Cu on z0 the dispersion of its values turns out to be less.

Let us return to the hydrodynamic classification of the underlying surfaces presented in the preceding section, in accordance with which the hydro-dynamically smooth surface complies with

and the hydrodynamically rough surface with

From this it follows that in the first case the resistance coefficient decreases with increasing wind velocity, and in the second it remains constant. Data from field and laboratory measurements over the water surface do not confirm this conclusion, the reason for which is usually associated with the dependence of Cu on the stage of wind wave development or, in other words, on the wave age c0/u„., where c0 = g/a>0 is the characteristic phase velocity of energetically significant wind waves; co0 is the frequency of maximum in the spectrum of wind waves. However, we note that Cu = (u^/uj2 = ((OoU^/g^icooU./g)'2 and that in accordance with field measurement data co0u./g ~ 1, that is, Cu ~ («o"*/0)2 = (co/u*) 2- In other words, the dependence of the resistance coefficient on c0/u% is predetermined by its expression, and in this sense does not contribute any additional information about the character of the changing hydrodynamic properties of the sea surface.

A clearer understanding of the question can be obtained from wave disturbances uw of the mean wind velocity defined as where u is the actual wind velocity and uT is the wind velocity over the underlying surface undisturbed by waves at the same friction velocity.

If there are no waves, then at the air-water interface the continuity condition for the momentum flux pu\ = pwu^w should be fulfilled, where subscript w means belonging to water. In this case the characteristic scale of turbulent fluctuations of velocity in the near-surface water layer is of the order of = (p/pw)1/2uThese velocity fluctuations will create vertical displacements of the air-water interface having the scale t]T ~ =

g~1(p/Pw)ul- But the thickness <5V of the viscous sublayer is equal to 5v/u^; therefore at typical values of the friction velocity (u^ « 0.3 m/s) and coefficient of molecular viscosity (v « 0.15 x 10_6m2/s) the vertical displacements of the interface will be less than Sv. Hence, in accordance with the hydrodynamic classification of underlying surfaces the surface in question can be considered to be smooth. Taking this into account we assume that the component uT in (3.4.5) follows the logarithmic law for the mean velocity profile over the smooth surface, that is, where z0v = 0.1 v/u^ is the roughness parameter of the smooth surface. Combining (3.4.1), (3.4.5) and (3.4.6) we obtain

k zq from which it can be seen that z0 describes not the mean wind velocity but, rather, its wave component, and because the latter is a function of Cq/u^, then this, in fact, defines the dependence of the roughness parameter of the sea surface on the stage of development of wind waves. A. Yu. Benilov was the first to pay attention to this fact (personal communication). The results of gradient measurements of the mean wind velocity presented by him demonstrate that the dependence of the dimensionless (normalized by u^) wave component uw at height 10 m on c0/u% in the range of c0/u# values from 5 to 90 can be approximated by the polynomial of second degree:

Consequently, maximum negative values of uw/u% and, hence, maximum values of the roughness parameter z0 conform to the initial stages of development of wind waves (cq/m^ < 30). At c0/M* ^ 30-40, that is, for developed waves, the function uw/uie is close to zero and the sea surface resembles a smooth Wall in its hydrodynamic properties (z0 % z0v). In the regime of attenuation of the swell when c0/uw > 30-40, values uw/u% become positive. From the viewpoint of the hydrodynamic classification of underlying surfaces, this is equivalent to the fact that in the regime of attenuation of the swell the sea surface turns out to be smoother than the immovable smooth wall.

Thus the process of development of wind waves is accompanied by significant reorganization of the regime of hydrodynamic resistance of the sea surface. In addition the most distinctive changes occur in the early stages of development of wind waves (at c0/u% < 10-20). This is the first important distinction of the marine surface atmospheric layer from the surface atmospheric layer over an immovable surface. Another distinction is the existence of an additional momentum flux near the air-water interface that is generated by wave disturbances of velocity and pressure. In the stationary case, as follows from the condition of constancy with the height of the friction stress, the appearance of the wave momentum flux has to be accompanied by vertical redistribution of the turbulent momentum flux. Accordingly, the vertical gradients of the mean wind velocity have to change as well: they decrease as compared with those predicted by the logarithmic law in the regime of wave generation, and increase in the regime of attenuation of the swell. But because the thickness of the wave sublayer at all possible values of c0/u* does not exceed 10 m (it equals approximately 0.1/lo, where is the wavelength corresponding to a maximum in the wave spectrum), then at heights z > 10 m the interpretation of gradient measurements of mean wind velocity in terms of and z0 remains justified.

There is an alternative to the method mentioned above describing the logarithmic profile of the mean wind velocity over a smooth surface. In the alternative version the mean velocity uT of a smooth flow at a certain fixed height z coincides with the observed one at the same height, that is, and the friction velocity does not coincide ^ u^). In taking into account that within the limits of the logarithmic boundary layer the square of the friction velocity is proportional to the momentum flux towards the underlying surface, such a description makes it possible to compare momentum fluxes towards the sea surface and the smooth wall.

Indeed, we define the resistance coefficients of the sea surface (Cu = M*/Ma) and of the smooth wall (C„x = u^/ul) and find an expression for their relative deviation <SCU/CU = (Cu — CuT)/Cu. This expression connecting <SCU/CU with the relative difference of momentum fluxes at the sea surface and smooth wall has the form

It is evident now that in the presence of swell when <5Cu/Cu < 0 and, hence, when ulT/ul > 1, the wind obtains an additional momentum from waves. On the other hand, for developing waves when (5CU/CU > 0 and u\Tju\ < 1 the momentum is transmitted from wind to waves (this determines the differences of the sea surface from a smooth wall). Finally, in the intermediate case, when the sea surface is close to the hydrodynamically smooth wall (¿Cu/Cu « 0), waves can be considered as developed and, therefore, weakly interacting with the wind. In this case the complete momentum flux to the sea surface is spent not on developing waves but, rather, on forming a drift current by means of tangential stress and wave breakdown.

In conclusion, we list five of the most popular methods used to estimate the roughness parameter z0 of the sea surface. The first is based on the Rossby assumption about the proportionality of z0 and the average wave height. The latter, in the regime of developed waves, is a function of the friction velocity u^ and gravity g. Taking both these circumstances into account and using dimensional considerations we obtain the well-known Charnock formula z0 = mul/g, where m is a numerical constant varying, according to different authors, from 0.011 to 0.08.

Such a significant spread in the values of m indicates that the roughness parameter should depend not only on the wave height but on some other wave characteristic, say the wavelength. In this case, as was postulated by Hsu (1974), the Charnock formula can be rewritten in the form z0 = m^Ho/LQXul/g), where H0 and L0 are the height and length of energetically uT = In —, uT(za) = u(za), k z0v

significant waves with maximum frequency co0 in the wind wave spectrum; mj is another numerical constant close to 1.0. We note that the combination H0/L0 is nothing but the wave steepness, and that in deep water the phase velocity cQ and wavelength L0 are connected by the relation c0 « (gL0/2n)1/2. Substitution of this relation into the expression mentioned above yields z0 = (mJlnjHoico/Ux)'2. Under conditions of developed waves when c0/m* ss 30 the Hsu formula reduces to the form z0 ~ H0, implying that the roughness parameter is proportional to the height of energetically significant waves.

If, following Kitaigorodskii (1970), we replace H0 by the root-mean-square wave height adefined by the equality on = (2 jo S(co) dco)1/2, where S(co) is the spectral density (co is the frequency of the waves), and take into account the mobility of the wave spectral components moving in deep water with phase velocities c = g/co, then the condition of proportionality of z0 and takes the form and thus ignore the contribution of low-frequency waves with co < co0 to an and z0. Here, /? is a universal constant varying, according to different authors, from 0.4 x 10"2 to 1.0 x 10~2. After substitution of (3.4.10) into the expression for z0 and using the relation co0 = (j?/2)1/4(éf/o-^)1/2 connecting co0 and an, we obtain from which it can be seen that in the initial stage of wave development (at co/u* « 1)» when the flow above the waves is similar to the flow above the immovable roughness elements, the parameter z0 is proportional to the root-mean-square wave height ar In the second stage, complying with developing waves (at c0/u^ ss 0(1)), z0 turns out to depend both on root-mean-square height a^ and on age c0/M*- Finally, at the stage of developed waves (at c0/w* » 1) the roughness parameter z0 ceases to depend on wave age c0/w* and is determined only by u^ and g, as in the Charnock formula.

To confirm the ambiguous dependence of z0 on Cq/u^ at the stage of

We approximate S(co) in the form

developing waves we picture the dimensionless roughness parameter gz0/ul in the form gz0/ul = f(c0/uj. (3.4.12)

We take into account the dispersion relation c0 = g/co0 and approximate the right-hand side of Equation (3.4.12) by the power dependence /(c0/u.,.) ~ (cOoU^/gy. Then, from (3.4.12), at y = 0 we have the Charnock formula gz0/ul = m; at b = — 1 we have the Toba and Koga formula z0co0/ut = n, here n = 0.025 is a numerical constant. Verification of these formulae by data from field and laboratory measurements have shown (see Toba, 1991) that on the phase plane (gz0/u2, a)0u*/9) experimental dots are grouped within the triangle formed by lines gz0/ul = m, gzju\ = n{oj0u^lg)~1 and co0u^lg = 0.05. The latter conforms to the boundary value of the wave age co0ujg separating the regime of mixed waves (wind waves and swell) from the regime of developing waves. In other words, experimental data demonstrate that at the stage of developing waves the dimensionless roughness parameter £jfz0/w2 is determined not only by the wave age a>0u^/g but also by the fetch, for example. As a palliative aimed at practical use, Toba (1991) proposed the approximation formula gz0/u2 = 0.020(coou^./g)~112 occupying, on the phase plane (gz0/ul, co0u^/g), an intermediate position between the Charnock formula and the Toba and Koga formula.

The fourth method basically reduces to a combination of the Hsu and Kitaigorodskii approaches. The expression for the roughness parameter has the form z0 ~ (c0/w*)~2[io S(co) exp( — iKg/aou^) dco]1/2, where the proportionality factor equals 2.80, in accordance with Geernaert et al. (1986). As shown in the same work, the proportionality constant in (3.4.11) is equal to 0.028.

The fifth method, proposed by Donelan (1982), is conceptually close to that proposed by Kitaigorodskii. But the low-frequency components of wind waves are not excluded and are considered as new roughness elements characterized by their own roughness parameters. The latter for low-frequency and high-frequency spectral ranges is given, respectively, in the form zo; = m2-(J2ro° S(co) d«)1/2, zGs = m2-(j2^0 S(co0) dw)1/2, where m2 is a numerical constant equal to 0.0125. Thus it is suggested that all spectral components of wind waves are fixed. This restriction is weakened in the following manner. Resistance coefficients for low- and high-frequency spectral ranges are calculated with the help of known values of z01 and zos. Then the first mentioned coefficient is multiplied by

|m10 — c,/cos <p\(u10 — c,/cos (p)'(UiQ cos (p), the second one by (u10 — c,l2uj"02), that is, it is assumed that low-frequency waves are propagated at an angle (p to the wind, and high-frequency waves are propagated along the wind. Phase velocities of low-frequency (c,) and high-frequency (cs) waves are found by the empirical formulae c, = O.83c0 and cs = 0.415co. The wind velocity u10 at a height of 10 m is assumed to be preset. It is clear that a number of assumptions forming the basis of this and other methods of evaluation of the roughness parameter, and particularly the assumption of additive presentation of the roughness parameter that is intended to account for contributions of different spectral intervals into the formation of the resulting resistance, stand in need of an additional substantiation.

As is well-known, the turbulent velocity fluctuations in the near-surface layer of water are less than the wave disturbances, complying with the energy-carrying components of the wave spectrum. Because of this, it is considered that waves are potential and that their generation is due not to the tangential wind stress but, rather, to the force of normal pressure. On the other hand, motions in the near-surface air layer are turbulent in character and wave disturbances of velocity are quite obvious. These disturbances transform the vertical structure of the surface atmospheric layer, and, therefore, the momentum and energy transfer to waves. Thus, if the problem of simulating the evolution of the wind in the surface atmospheric layer or of waves in the upper layer of the ocean is posed, then its solution can be obtained only within the framework of a coupled wind-wave interaction model.

We consider one such model proposed by Benilov et al. (1978). Assume that wind and wave fields are horizontally homogeneous and non-stationary and that some more simplifying assumptions are fulfilled, namely: (i) the upper ocean layer disturbed by waves can be contracted into to a plane, that is, its effect can be taken into account by assigning proper boundary conditions at the air-water interface; (ii) breakdown of waves generating the conversion of momentum and energy of waves into the momentum and energy of the drift current, and into the energy of small-scale turbulence in water is absent; (iii) wind momentum and energy fluxes supporting the drift current are also absent; (iv) in the surface atmospheric layer one can select the logarithmic layer and an underlying layer between the roughness level and the free ocean surface, within the limits of which the mean wind velocity and viscous stress are equal to zero; (v) because of non-commensurability of vertical scales of the above-mentioned layers the contribution of the latter to the integral (over the entire surface atmospheric layer) turbulent energy dissipation is negligible.

Let us write down the evolution equations for the momentum:

for the kinetic energy of mean motion

dt 2 8z p and for the turbulent energy db d q x du s

dt dz p p 8z p in the surface atmospheric layer. Here b is the kinetic energy of turbulent fluctuations of velocity normalized to the air density (p); s and q are the dissipation rate and diffusion flux, respectively, of the turbulent energy; other symbols are the same.

Momentum and turbulent energy fluxes at the air-water interface which, according to the problem conditions, are spent only on wave generation, can be determined from the evolution equations for the momentum M.„ and energy £w of the wave field. On the basis of assumptions (i)-(iv) these equations take the form

dt dt

These are used as the dynamic boundary conditions at the air-water interface. One more condition at this surface is the continuity of the mean velocity in the absence of drift current. It is written in the form u = 0 at z = 0. (3.5.5)

The following conditions are given at the upper boundary h of the surface atmospheric layer:

The first means choosing a fixed mean wind velocity, the others refer to vanishing of turbulent fluxes of momentum and energy, and also of the turbulent energy itself. We note the approximate character of the last three conditions: they assume that interaction between the surface layer and the upper layers of the atmosphere is lacking.

We turn to the derivation of integral relations for the surface atmospheric layer, not specifying concrete initial conditions for the moment. For this purpose we integrate (3.5.1)—(3.5.3) over z from 0 through h and take advantage of conditions (3.5.4)-(3.5.6). As a result we obtain d Ch dh dJIJp

"u2

ul dh

where the terms on the left-hand side describe the rate of change in the integral (within the limits of the surface atmospheric layer) momentum, kinetic energy of mean motion and kinetic energy of fluctuating motion; the first terms on the right-hand sides of (3.5.7) and (3.5.8) are changes in momentum and kinetic energy of mean motion due to variations in thickness of the surface layer; the second terms on the right-hand sides of (3.5.7) and (3.5.9) are changes in the momentum and kinetic energy of fluctuating motion due to their transport to waves; the second term on the right-hand side in (3.5.8) and the first term on the right-hand side in (3.5.9) are mutual conversions of the kinetic energy of mean and fluctuating motions; and, finally, the third term on the right-hand side in (3.5.9) is the integral (within the limits of the surface atmospheric layer) turbulent energy dissipation.

Let us now recall that the lifetime of turbulent formations in the surface atmospheric layer is much less than the relaxation time of the surface atmospheric layer (which allows us to ignore the term on the left-hand side of Equation (3.5.9)), and take into account the constancy of uh in time. Then, based on (3.5.7)—(3.5.9) we obtain

d |
%h | ||

df |
— |
0 | |

d |
"1 |
*h | |

dr |
2 |
Ew P Next, according to assumptions (iv) and (v) the integral (within the limits of the surface atmospheric layer) turbulent energy dissipation may be replaced by the integral dissipation within the limits of the logarithmic layer. But in the logarithmic layer the local turbulent energy dissipation e/p is determined only by the friction velocity u* and the height z, so from dimensional considerations e/p « u\/z, and, therefore, dz o P or, if we use the logarithmic distribution of the mean velocity in the layer where y is a non-dimensional numerical factor representing the product of the von Karman constant and the proportionality constant from (3.5.12). We also take into account that u= x/p with x/p = Because of this, after substitution of (3.5.13) into (3.5.11) we finally obtain d dt The system (3.5.10) and (3.5.14), together with initial conditions u = un, Ew = 0 at t = 0, (3.5.15) has the two first integrals: the law of conservation of momentum, p (uh — u)dz = A Jo and the law of conservation of energy The system (3.5.16) and (3.5.17) is not closed. To close it according to assumption (iv) we assume that u z and then we take advantage of the momentum and energy definitions and of the Phillips expression for the frequency spectrum S(a>) of wind waves ai w \ wq, where, as before, a> is the wave frequency; oj0 is the frequency of the spectral maximum; is the root-mean-square wave height; pw is density of sea water and ¡i is a numerical universal constant. Substitution of (3.5.21) into (3.5.19) and (3.5.20) yields |

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