Internal Wave Interactions 



with equal inclination; this continues until all the energy is extracted from the 

 first wave and has drained off to the second one. This, in turn, interacts with the 

 shear, producing the first wave again traveling upward. As a result of this con- 

 tinual interchange between components whose propagation directions are inclined 

 upward and downward, the internal waves are confined to a horizontal channel, in 

 which the energy density in the upward and downward moving components are, on 

 the average, equal. The kind of channeling is quite distinct from the well-known 

 effect of the thermocline itself in restricting internal gravity waves to regions 

 where n < N, and may have consequences that are equally important. If a local 

 source generates internal gravity waves in the presence of steady motion, with 

 an irregular velocity distribution, the resulting disturbance is limited to a range 

 of depths surrounding the source, as shown in the following figure, even though 



SOURCE 



the stratification may be uniform far beyond. This range can be estimated sim- 

 ply from Eq. (27). The interaction time is t = 1/(1/2) cos 6 sin 6 a^ki, and the 

 group velocity is c = (n/kj) sin6i in the direction normal to kj, or (n/kj) 

 cos 6 sin d in the vertical. The distance that the disturbance propagates in the 

 interaction time is therefore 



Z = c_ COS d T 



2n, 



a, k, k, 



4N cos 6 sin 

 a, k,^ 



(28) 



The maximum depth of the disturbance zone is found when 6 = 45°, and the dis- 

 turbance frequency equals N/'/T; in this case z == 2N/ajki^. When n-»0, e-^-n/i, 

 and when n -•N , 6* -» o; in either case z ^ and the disturbance is concentrated at 

 the level of the source. The low-frequency limit is of particular interest in the 

 oceanic and atmospheric case. It implies that low-frequency internal wave dis- 

 turbances distribute themselves horizontally but very little vertically — a result 

 that is consistent with the commonly observed zones of horizontal motions in 

 the atmosphere, whose vertical extent is very limited. 



Another interesting corollary of these results is that if a^ = o, then 

 ttj = const., a3 = const. Two internal waves of the same frequency pass 

 through one another without interaction in a uniformly stratified ambient fluid 

 at rest. The time seems to have come when controlled experiments on these 

 effects would be of great value. 



543 



