538 



1.5 



1.0- 

 08- 



06- 



Z 



X 

 QA- 



02 



0.5 



A.O 



FIGURE 16. Another representation of the 3/2 power law 



for individual waves (all trough-to-trough) . H„ and f , 



N N 

 represent the wave height and the frequency, respectively, 



normalized by values for waves of the maximum energy 



density. [Cited from Tokuda and Toba (1978).] 



face, depends not only on the wind speed but also 

 on the state of the water surface. Various attempts 

 have been made to obtain a functional form of the 

 roughness length incorporating the state of wind 

 waves or the wave breaking. However, in view of 

 the complexity of the expressions, together with 

 the wide scattering of data points, a simple 

 dimensional formula by Charnock (1955) has been 

 cited most frequently, but with various values of 

 a constant of proportionality, although the formula 

 contains only a parameter representing the wind 

 field. 



A dimensional consideration leads to an expression: 



ZQ* = zo*(u** 



n 



(5) 



where zo"* 



zq/v is the dimensionless roughness 



parameter, u** s u* /gv the dimensionless friction 

 velocity representing the overall wind effect, and 

 Cm* - u*aj^/g the single parameter representing the 

 wind-wave field as stated in the previous section, 

 where Oj^ is the frequency at the energy maximum. 

 Charnock 's formula 



zo 



= BU 2/g 



(6) 



is equivalent to 



Since the difference in the local wind drift is 

 caused by the mean wind stress, the self -adjustment 

 is expressed by the condition that the wave current 

 is proportional to u*, namely. 



zo^ 



6u. 



(7) 



which is a form of (5) in which a^* is disregarded. 

 It is shown that another simple form for zg*, using 



„* hereafter: 



a^a^/gu. 



constant 



This is transformed immediately to 

 H* = B'a*-3/2 



(3) 



(4) 



which is equivalent to (2) , where a* = u*a/g. 



The condition of constant steepness may arise 

 from the similarity requirement. The combination 

 of the 3/2-power law relationship and the constant 

 steepness condition leads to the apparent phase 

 speed proportional to the square root of the 

 frequency. These three relationships, which have 

 been shown by the experiments to be satisfied by 

 the individual waves, are self-consistent with one 

 another, and may thus result from the strongly 

 nonlinear effects. 



The 3/2-power law makes it possible that the 

 wind-wave field is represented by a single dimension- 

 less parameter of the frequency at the energy 

 maximum as discussed by Toba (1978a) . One of the 

 consequences of the above paper is that the growth 

 of the wind wave field is expressed by the evolution 

 of the dimensionless single parameter in a form of 

 error function of the parameter itself, in which 

 the value of the parameter approaches a final value 

 as a simple stochastic process, irrespective of its 

 initial conditons, through a rapid self-adjustment 

 of the state. 



6. WIND STRESS OVER WIND WAVES 



The final topic of this paper concerns the expression 

 of wind stress over wind waves. It has been pointed 

 out on many occasions that the roughness length, or 

 equivalently the drag coefficient of the water sur- 



10' 



10^ 



u.'/gb 



lO:* 



10" 



FIGURE 17. Data plots for the relationship (6) in 

 a dimensionless form. Data by Toba (1972) and 

 Kunishi (1963) are from wind-wave tunnel experiments, 

 and data by Kawai et al. (1977) and Mitsuyasu et al. 

 (1971) are from tower-station observations. [Cited 

 from Toba (1978b) . I 



