622 



A [w] = - a^n k f 1 + ^J sin x 

 where x = kg(x - c t) and from (25) 

 [t] = [w]+ - g 3^"'^g sin X • 



= [w] + Z 3^nk (K/k) sin x 

 - 8 g 



(27) 



(28) 



where [ ]+ and [ ]_ represent averages taken just 

 above and below the discontinuity in density. 



These matching conditions to be applied as z = 

 involve the non-linear forcing provided by the wave 

 groups. The field equations are, however, linear 

 to this order. Above the pycnocline, when d > z > 0, 

 we have Laplace ' s equation for the averaged velocity 



[*] 



(29) 



while in the averaged internal wave equation (21) 

 for z < 0, the non-linear terms are smaller by at 

 least (ng/n)^ << 1 than those in the matching con- 

 ditions, since they involve two horizontal deriva- 

 tives (or one x and one t derivative) of averaged 

 second order quantities. Accordingly, to sufficient 

 accuracy. 



fe'^i"]-^^!^ 



[w] = 



z < 



(30) 



since n/N >> 1 and K/k = 1. The vertical component 

 of the group velocity of the radiated waves is Cg 

 cos i|j sin ijj where Cg is the group of the inter- 

 facial waves, so that the vertical energy flux is 



n2, ^h 



(9/128) a'+n^Nk (1 



(35) 



Although this representation of the density dis- 

 tribution by a discontinuity at the pycnocline, 

 followed by a uniform stratification below, is a 

 gross simplification of typical oceanic conditions, 

 it is of interest to examine the order of magnitude 

 of the vertical energy flux that might be generated 

 in this way. If the interfacial wave amplitude is 

 10 m at a frequency of 5 c.p.h. , having groups 1 km 

 in length and if N = 2 c.p.h., the downwards energy 

 flux is about 2 erg/cm sec, which is of the same 

 order as the 5 erg/cm^ sec. estimated by Garrett 

 and Munk (1972) for the rate of energy dissipation 

 from internal waves by sheer instability. This 

 correspondence is sufficiently close to encourage 

 a more detailed study with N(z) arbitrary, the re- 

 sults of which will be presented elsewhere. 



.ACKNOWLEDGMENT 



This work was supported by the Fluid Dynamics 

 Branch of the Office of Naval Research under con- 

 tract NR 062-245. 



Since the length of the wave groups is large 

 compared with the wavelength of the interfacial 



waves, k << k and it is consistent to assume that 

 kgd << 1, even though kd >> 1. Furthermore n/N << 

 1 while NqN = 0(1). Under these conditions the 

 solutions for the mean pycnocline displacement and 

 the low frequency internal waves radiated downwards 

 are found to be 



[?] 



-r-rr- COS k„(x - c t) 

 bd g g 



(31) 



[w] -^ 



where 



a^nk (2 + ^) cos (k x + k z - n t) , (32) 

 g k g g g 



(33) 



is the vertical wave-number of the radiated field. 

 The horizontal velocity component in the internal 

 wave motion below the pycnocline 



[u] = [w] tan i|j , 



where cos i|; = ng/N and the energy density (twice 

 the kinetic energy density) is 



E = p (^[u]2 + "M^j , 



^ pa'<kg2n2N2 / ^ 

 128n2 V k 



g 



^9£aNifN£ 

 128c 2 



(34) 



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