Internal Wave Interactions 



ki • a^ = (kj- kj) 0-2 = '^a '*'2 



(since k^. • a^ - o), together with its permutations. The details will not be given 

 here; suffice it to say that the right side vanishes identically and that 



aj2 + a^ + a^ - const. (21) 



These interactions provide the means of energy exchange in a field of weak 

 interacting gravity waves and play a role in determining the spectral properties 

 of an ensemble of such waves. In this paper, however, we are concerned with a 

 more restricted problem: the interaction between these waves and steady, hori- 

 zontal shearing motions, with a periodic variation in the vertical. 



INTERACTIONS WITH STEADY, HORIZONTAL SHEAR 



A very slowly varying, horizontal shearing motion can be considered as the 

 limit of an internal wave whose frequency, Oj , say, approaches zero as the 

 wavenumber approaches the vertical. The frequency condition for resonance 

 then reduces to 



nj = ng , (22a) 



cos ^2 = «^os ^3 . (22b) 



Consequently, the internal waves capable of resonant interaction with the steady 

 motion have wavenumbers that form the two equal sides of an isosceles triangle 

 as indicated in the following figure. Clearly, 



-d^= e^= e , say , . / 



kj -v = kj , / 



kj * V = kj sin ^j - ~^^2 ^^" ^ ' \ 



k^ • V - kj sin 6^ - \^^ sin d , ^ \ 



2 sin 



The motion then represents the interaction of two internal waves of the same 

 frequency but of opposite inclination to the vertical with a steady, horizontal 

 motion whose variation in the vertical is specified by kj. Under these condi- 

 tions, the interaction coefficients simplify considerably: 



G23 = , (23a) 



Nk,' cos^g (23b) 



541 



