Internal Waves 



O. M. Phillips 



Johns Hopkins University 



Baltimore, Maryland 



ABSTRACT 



It has become evident in the past few years that 

 the wave-number, frequency spectrum of deep ocean 

 oscillations has a remarkably consistent form close 

 to that which would be expected for statistical 

 equilibrium among the modes under wave-wave resonant 

 interactions. The energy sources that maintain deep 

 oceanic internal waves are, however, not well under- 

 stood. 



In the vicinity of the thermocline, the energy 

 density (per unit mass) of internal wave activity 

 is generally much greater than in the ocean depths. 

 Relatively high frequency internal waves, generated 

 in a variety of ways, are to a first approximation, 

 trapped in this region. Disturbances whose fre- 

 quencies are less than N(j, the deep stability fre- 

 quency, do however radiate downwards effectively. 

 Also, groups of high frequency, low mode waves 

 generate second order mean perturbations to the 

 thermocline structure, and if the group frequency 

 is less than N^j, again energy radiates down. The 

 flux of energy into the deep ocean is illustrated 

 first in a simple model in which a sharp pycnocline 

 lies over uniformly weakly stratified water. The 

 more general problem involving an arbitrary strati- 

 fication is formulated and some preliminary asymp- 

 totic solutions are presented. 



1. INTRODUCTION 



During the last 10 years or so, a variety of new 

 and ingeneous oceanographic observations has been 

 made on the structure of internal waves fluctua- 

 tions in the ocean. Twelve years ago, in the first 

 edition of The Dynamics of the Upper Ocean, I was 

 forced to write that in view of the difficulty and 

 expense involved in the systematic study of oceanic 

 internal waves, "those (measurements) that do exist 

 are correspondingly rare and valuable." The present 

 situation is gratifyingly different. Deep oceanic 



observations of internal waves are no longer rare , 

 but they remain valuable; Cairns (1975) , Katz (1975) , 

 Gould, Simmons, and Wunsch (1974), and a number of 

 others have provided different kinds of observations 

 from which a consistent pattern is emerging. It ap- 

 pears that the deep oceanic internal wave spectrum 

 has a remarkably universal form close to that speci- 

 fied by the Garrett-Munk (1975) spectrum, though why 

 this is so cannot yet, I think, be asserted with con- 

 fidence. McComas' (1975) calculations on resonant 

 wave-wave interactions indicate that the Garrett- 

 Munk spectrum is close to what one would expect in 

 a state of statistical equilibrium under the balance 

 of these interactions. On the other hand, there are 

 indications, such as the occurrence of sporadic, 

 isolated patches of turbulence in the stably strati- 

 fied regions of the ocean which suggest that local 

 instabilities may be limiting the wave spectral 

 density. 



Soviet investigations, such as those of 

 Brekhovskikh et al. (1975) have concentrated on the 

 low mode structure in the thermocline region whose 

 energy density (per unit mass) exceeds, usually by 

 an order of magnitude, that of the deep oceanic in- 

 ternal waves. The characteristic frequencies are 

 also about an order of magnitude higher. The cal- 

 culations of Watson, West, and Cohen (1975) among 

 others indicate that the lowest modes are generated 

 quite rapidly by interactions among surface wave 

 components; a number of studies along these lines 

 are described in the useful review by Thorpe (1975) 

 and by the present author (1977) . The upper ocean 

 is certainly the site of considerable dynamical 

 activity, but how much of it is radiated downwards 

 ■to provide a source for those motions encountered 

 in the deeper, less strongly stratified region be- 

 low? According to the usual linear analysis, the 

 low mode, relatively high frequency waves are trapped 

 to the strongly stratified thermocline region; only 

 the low frequency high modes have structure that can 

 penetrate great depth . 



Yet the description of deep oceanic motions as a 



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