# G

where c0 = g/a>0 is the phase velocity complying with the frequency of the maximum in the wind wave spectrum.

Now system (3.5.4a), (3.5.16)-(3.5.18), (3.5.22) and (3.5.23) is closed: it includes only one differential equation (3.5.4a) and five diagnostic relationships to determine six unknown functions: ,.//w, Ew, u^, z0, c0 and h. We find what can be obtained from the system without recourse to integration over time. Keeping this in mind we select any one of the six unknowns (say, u„,) as an independent variable and find the remaining ones as functions of that. But first we introduce the definition

describing the ratio between energy and momentum losses in the surface atmospheric layer, and rewrite (3.5.17) as

from whence

In combination with (3.5.22) and (3.5.23) this equality gives

Substituting (3.5.18) into (3.5.16) and taking into account (3.5.22) we obtain gz0 pwP{c0\3 fr't

where ç = Kuh/ujt. is a dimensionless parameter connected to the resistance coefficient Cu by the relationship £ = kC~ 1/2.

Considering that gz0/ul = (gh/ul)(z0/h), and (h/z0) = e{ (see (3.5.18)) we obtain gh pwßfc0y i (3527)

uf p 3 \uhJ 1 — e Finally, in accordance with (3.5.20), (3.5.22) and (3.5.23), g-^w/pw = ßfco u\ 3 \u„

We only have to find the function C appearing in (3.5.25). To do this we substitute (3.5.18) into (3.5.24), as a result of which we obtain

It follows from (3.5.31) that the function C/uh is limited at all values of its argument £ in the range [0, oo] and obeys the inequality 0.5 < C/uh < 1.0, where the lower limit (C/uh = 0.5) complies with £ = 0; the upper limit (C/uh = 1.0) complies with £ => 1. From this and from conditions of positive-ness of z0, h and c0 follows boundedness of y: under variations of £ in the range [£0, oo] the constant y obeys the inequality 0 < y < (C/uh)i=ioy where ^ 0 describes an initial momentum flux to waves. The equality y = (C/uh)i=io meets physically realized situations, and in (3.5.25)—(3.5.28) it provides a continuous transfer to initial values of c0, z0, h and at i => 0. Thus, £0 is a solution of the algebraic equation (C/uh)io = y.

Formulae (3.5.25)—(3.5.31) allow us to find all the basic parameters of the surface atmospheric layer as functions of c0/uFor real values of the resistance coefficient (Cu ~ 10"3, ^ ~ 10) the component e_i in (3.5.26), (3.5.27) and (3.5.31) is much less than one. If it is ignored, then it follows from (3.5.25)—(3.5.31) that changes in c0/uh make up no more than 30% of maximum value, and that for developing waves (£ => oo) the relation c0/uh approaches its limiting value

'4/3 at y = 0, - = Kl-y)= 1 at 7 = 0.25, h [2/3 at y = 0.5, inherent in maximum possible waves at fixed wind velocity. Based on (3.5.25) and (3.5.30) we have for such waves u2h 9

At the same time, according to observational data for the conditions of developed waves (see Benilov et al. (1978)), c0/uh « 1.14 and ga^/uj « 5.2 x 10"2. As can be seen, at y = 0.25 the theoretical and experimental estimates are in agreement with each other. The same can be said about the roughness parameter of the sea surface (Figure 3.3).

Calculated, by Equation (3.5.27), values of gh/uf demonstrate that at co/u* ^ uh = 10 m/s and y = 0.25, the thickness of the surface atmospheric layer becomes roughly equal to 50 m, and at c0/u^ « 30 (developed waves), Figure 3.3 Dependence of the non-dimensional sea surface roughness gz0/ul on c0/uSf, according to Benilov et al. (1978). Various symbols show the estimates of different authors.

gzo/v-l

Figure 3.3 Dependence of the non-dimensional sea surface roughness gz0/ul on c0/uSf, according to Benilov et al. (1978). Various symbols show the estimates of different authors.

other things being equal, it is 200 m. Thus it turns out that in the process of wind wave development accompanied by momentum and energy transfer to waves the surface atmospheric layer is rebuilt up to heights of the order of hundreds of metres.

In conclusion, we adduce an asymptotic solution describing the evolution of wave parameters at large t. Previously, with the relationships (3.5.22), (3.5.25) and (3.5.31) and definitions r and we reduce Equation (3.5.4a) to the form df \uhJ pw ß uh

At large t (in other words, at large c) the expression for (c0/uh) takes the form i-y-

uu 3

Its substitution into (3.5.32) and elimination of the small terms on the right-hand side yield