550 



To complete the calculation of Section 2 for 

 long-wave growth rate due to the non-uniformity of 

 short-wave growth rate , we require the part of c . 

 that is in phase with C • For the analysis of short 

 waves. Section 2 uses an expression equivalent to 



p k- (c' 



u) {a- + 16') (1 - akC)C' (42) 



where all quantities are real. This assumes that 

 the real and imaginary part of c are modulated by 

 the large scale in exactly the same proportions. 

 Thus by this assumption, 



aky 



(Cc - u ) 

 1 w 



a + ie 



(103) 



/g/k+kT + '"^^^' and C 



Cg + u 



Should these assumptions not be exactly correct, a 

 would have a small imaginary part, which will be 

 ignored. 



4. NUMERICAL RESULTS 



We have carried out the calculations described in 

 3 using the Orr-Sommerf eld solver developed by 

 Gustavsson (1977) . This is an implicit method 

 which uses an Adam' s integration technique. One 

 particularly attractive feature of the program is 

 its variable step size. Thus it is possible with 

 a reasonable number of points to have a fine mesh 

 in the "wall" layer and other regions of high gra- 

 dients and to coarsen the mesh as one moves out 

 into the boundary layer. The programs and results 

 will be more fully described in a subsequent publi- 

 cation. Only one set of calculations will be 

 reported here . 



The shear flow profile and its derivatives are 

 modelled with continuous functions that approximate 

 the mean profile of a turbulent boundary layer. 

 Calculations were done at a friction velocity, uT, 

 of 30 cm/sec; conditions were chosen so that the 

 ratio, ut/uoo, was .05, a typical value for wind- 

 tunnel experiments. Interaction between long waves 

 of 100, 75, 50, 36, 20, and 16.5 cm with short 

 waves of 2, 1, 0.75, and 0.6 cm were investigated. 

 Although many interesting features of the flow can 

 be investigated using this approach (such as the 

 distortion of the mean profile as the large wave 

 passes, and the variation of the wave speed, local 

 growth rate, and amplitude of the short waves along 

 the large waves) , the only systematic investigation 

 we have yet performed concerns the energy input to 

 the large waves due to the modulation of the short- 

 wave Stokes drift. 



The linear temporal growth rate of wind-driven 

 waves Q^ = kc^ is of course a direct output of the 

 calculations. Figure 2 shows fij ~ sec"l as a 

 function of wave number, k - cm-1, for u = 30 cm/ 

 sec. The growth rates we obtained are slightly 

 higher than Miles 's viscous calculations [Miles 

 (1962)] but when we used his shear-flow profile, we 

 obtain close agreement. For u^ = 30 cm/sec, all 

 the waves we investigated were viscously dominated, 

 that is, their critical layers were sufficiently 

 close to the free surface to be essentially merged 

 with the surface viscous layer. Thus, little in- 

 sight to the behavior of these flows can be obtained 



1 1 — I I I I I u lOO 



- I.I 



k ~ cm 



FIGURE 2. Linear temporal growth rate @ u =30 cm/sec; 

 , present calculations; present calcu- 

 lations with miles profile Ui = 5U*. 



from an inviscid model of the behavior of shear 

 flows. The real part of the wave speed, Cj., also 

 shown in Figure 2, differs very little from the 

 free wave speed of gravity-capillary waves, cq. 

 The energy input to the large waves from the 

 small waves is given by (66) . With T = kct, and 

 S' from (54), the dimensional temporal growth rate 

 of the large waves can be written 



^=?0 [«i+^k^l^k'2e2] (104) 



c'-c 



g 



where Q- is the linear growth rate 6k/2. We define 

 the coupling coefficient C as 



-ag'k 



(105) 



where the minus sign is introduced because, contrary 

 to our expectations, a turned out to be negative 

 for the cases we investigated. Thus the growth of 

 the large-wave amplitude is given by 



dC 

 dt 



c-c' 



g 



(106) 



Thus for C positive, an energy input to long waves 

 comes from short waves whose group velocity, c', 

 is slightly less than the long wave phase velocity, 

 c. The theory also predicts that long waves will 

 decay if c' > c. Since waves satisfying this con- 

 dition will be shorter capillary waves which will 

 be more strongly damped by viscosity, we expect a 

 net energy input to the large waves. Of course 

 the theory does not hold at c 

 interactions must be considered. 



Numerical values of the coupling coefficient, C, 

 are shown in Figure 3 as a function of A for various 

 X'. C is certainly 0(1) having a maximum value of 

 3 at X ' =1 cm. It is also a slowly varying function 

 of X . It has its maximum value about A ' = 1 cm 

 which corresponds to the maximum in the linear 

 growth rate for short waves for these conditions. 

 It drops off more rapidly with decreasing wave 



c where non-linear 



