INTERNAL WAVE INTERACTIONS 



O. M. Phillips 



The Johns Hopkins University 



Baltimore, Maryland 



ABSTRACT 



The second-order resonant interaction equations are derived for three 

 internal wave modes in a fluid with constant Brunt -Vaisala frequency, 

 N. These specify the energy interchanges among the modes, the redis- 

 tribution of energy from one wavenumber to another. Integrals are 

 shown to exist, representing the partition of energy among the modes 

 and the conservation of energy among all three modes. 



One particularly interesting type of interaction is that among two inter- 

 nal wave modes and a zero frequency horizontal drifting motion in the 

 stratified fluid. Simple solutions are derived, and it is shown that the 

 energy densities of the two wave nnodes are proportional to cos^(crt) 

 and sin^(crt), respectively, where ct is proportional to the vorticity in 

 the steady horizontal motion. The energy exchange depends on, but 

 does not involve, the horizontal motion; the latter acts as a catalyst for 

 the interactions. From these results, it is inferred that an irregular 

 vertical profile of horizontal velocity results in a horizontal channeling 

 of low frequency internal wave nnotion, restricting their vertical spread 

 even when the Brunt-Vaisala frequency is constant and much greater 

 than the wave frequency. 



INTRODUCTION 



The interactions among modes in wave motions have been studied consider- 

 ably in recent years, and it is becoming evident that they are involved in a sur- 

 prising number of physical phenomena, some of which have long been observed 

 but not understood. Probably the most striking such example is the instability 

 of the classical Stokes wave discovered by Benjamin and Feir (1) and described 

 in the paper before the preceding one. The effect is well known; it is common 

 knowledge among wave -tank people that if a plunger generates a smooth train of 

 finite amplitude waves in a long tank, then as the waves propagate down the tank, 

 they gradually degenerate into chaos, and that the steeper the initial wave, the 

 sooner this happens. This has been regarded as a nuisance, probably "some- 

 thing to do with the plunger," and it was not until Benjamin and Feir's important 

 work that this has been recognized as a genuine hydrodynamic instability of the 

 wave train itself. Although their analysis is set up in terms of small perturba- 

 tions to the Stokes wave train, with wavenumbers parallel and almost equal to 

 that of the primary wave, it has since been shown (2) that the Benjamin-Feir 



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