Internal Wave Interactions 



b = -gp'/Po ■ 



An elementary infinitesimal disturbance satisfying this equation with the non- 

 linear terms neglected is of the form 



u = a cos X , (3a) 



b = N^^-^ sin X, (3b) 



n 



where x= k-x-nt+e and visa unit vector vertically upward. The vector a 

 specifies the maximum particle velocity in the wave motion and, in virtue of the 

 incompressibility condition v • u = 0, is always normal to k: 



a • k = . 



Note that the particle trajectories are in the vertical plane, so that [a.k.v] = 0. 

 The frequency of the wave motion is given by 



n = N cos ^ , (4) 



where d is the angle between the wavenumber vector and the horizontal in the 

 same vertical plane. Note also that 



a. 'V - a. cos e ^ 1^1 max • (5) 



These wave components are capable of resonant interaction; energy can be 

 exchanged among sets of waves whose wavenumber s kj.kj.kj and frequencies 

 nj,n2,n3 Satisfy the conditions 



ki + kj = kj , (6a) 



Hj + Hj = nj . (6b) 



The last of these conditions is equivalent to 



cos 6'i + cos ^2 ~ ^°^ ^3 • ' ' 



It has been shown (2) that provided the rates of shear in the internal wave modes 

 are small compared with the Brunt-Vaisala frequency, the energy exchange is 

 limited to this set of wavenumbers; the other components generated by the inter- 

 action (for example, the component with wavenumber kj - kj) are bounded in 

 amplitude and remain small. They play no part in the resonant interaction and 

 in the continuing energy exchange from one mode to another, at least to this 

 order. 



We consider, then, a set of three interacting internal wave components: 



3 



u = V a^. cos X^ + u , (8a) 



r= 1 



537 



