Phillips 



The first of the interaction equations, Eq. (17a), vanishes identically, since 

 cos 0j = also, and the other two, Eqs. (17b) and (17c), become 



20-2 = -kj cos Q sin 6 a^aj , (24a) 



(24b) 



Clearly, 



a^ + a^ - const. , (25) 



SO that from Eq. (21) a^ = const. The energy density of the shearing motion is 

 unaltered in time; though it plays a central role in the interaction, it does not 

 partake of the energy exchange. 



The complete solution to the interaction equations is very simple in this 

 case. 



cij =-(-^kj cos 6 sin d a Ah.. 



1 V 



Ty k J cos 6 sin 8 aj 



so that if a^ = and a3 = a at t = , 



ttj = -A sin at , (26a) 



a^ = A COS at , (26b) 



where 



a = 2 COS 9 sin 6 aj k^ . (27) 



The energy density of the component with wavenumber kj then varies as sin^at 

 and that of the k3 component as cos^ at; the energy oscillates completely from 

 one component to the other with a frequency that is proportional to the vorticity 

 in the steady shearing motion. 



This kind of behavior is remarkable in several respects. The steady shear- 

 ing motion takes part in the interaction but does not share in the energy ex- 

 change; it acts as a kind of catalyst for the interaction. As far as I know, this 

 is the first kind of interaction in which this effect has been demonstrated. It 

 has some interesting physical consequences also. Suppose, in the ocean ther- 

 mocline, there is some distribution of horizontal velocity with an internal wave 

 with wavenumber V^ propagating upward through this region. This wave will 

 select from the steady shearing motion the Fourier component with wavenumber 

 kj = 2k 2 sin with which to interact (resonant interactions with other compo- 

 nents being not possible). As a result of the interaction between this wave and 

 the steady motion, energy begins to drain from the first wave component and 

 generates another with the same frequency but moving in the downward direction 



542 



