Breaking Internal Waves in Shear Flow 



S . S . Thorpe 



Institute of Oceanographic Sciences, 



Wormley , United Kingdom 



ABSTRACT 



During and following periods of strong winds, the 

 Richardson number (the square of the ratio of the 

 Brunt-Vaisala frequency to the shear) in the 

 thermocline is of order unity, and the shear becomes 

 an important factor in determining the properties 

 of internal gravity waves. These properties are 

 discussed and the shape and breaking of waves in a 

 shear flow is investigated in laboratory experiments. 

 These experiments show that the waves may break at 

 their crests or their troughs depending on the sign 

 of a certain vector scalar product. An analogy 

 between surface waves and interfacial waves is 

 invoked to account for this behaviour. Breaking 

 is observed to occur by particles of fluid moving 

 forward more rapidly than the wave crest advances, 

 leading to gravitational instability. The effect 

 of breaking in the ocean will not only enhance 

 diffusion rates, but it will modify the directional 

 spectrum of the internal waves. 



Although many acoustic backscatter observations 

 from ships reveal clearly the presence of internal 

 waves in the ocean seasonal thermocline, very few 

 have been published which appear to show signs of 

 their breaking. This is surprising in view of the 

 clear and not infrequent evidence of 'breaking 

 events ' in the equivalent acoustic or Doppler radar 

 measurements in the atmosphere . Our knowledge of 

 internal wave breaking in the ocean still rests 

 almost entirely on the direct observations by divers 

 using dye in the Mediterranean thermocline [Woods 

 (1968)]. The present towed, moored, or dropped 

 instruments give inadequate information on the 

 nature or structure of the intermittent mixing events 

 in the ocean to be certain of their cause, or even 

 of the scales of motion which contribute most to 

 diffusion across density surfaces in spite of its 

 great importance to the prediction of the thesrmo- 

 cline structure of the upper ocean. 



It is against this background of poorly known 

 dynamical structures that this paper is presented. 



One aim is to describe the patterns which accompany 

 wave breaking, for without a knowledge of such 

 patterns it is difficult to design the appropriate 

 experiment to detect wave breaking or, conversely, 

 to correctly identify the processes involved once 

 observations are available. 



It would be naive to ignore the effect of wind 

 in a description of breaking waves on the surface 

 of the sea in deep water [see, for example, Phillips 

 and Banner (1974) ] . (Wave breaking on a beach is 

 a different matter) . It is similarly inappropriate 

 to ignore the effect of mean shear on internal waves 

 in the seasonal thermocline, since the Richardson 

 number there is low, especially during, and follow- 

 ing, storms [Halpern (1974)]. Internal gravity 

 waves can exist and propagate in a shear flow just 

 as they can when a mean flow is absent. These waves 

 belong to a group which Banks, Drazin, and Zaturska 

 (1976) have classified as 'modified' (-by shear) 

 ' internal gravity waves ' . They may sometimes coexist 

 with a set of wavelike disturbances which grow in 

 amplitude (the 'unstable wave solutions' of the 

 Taylor-Goldstein equation) and which may eventually 

 lead to turbulence (Figure 1) . It is known however 

 that (for steady mean flows) the latter solution cor- 

 responding to Kelvin-Helmholtz instability (K-H.I) 

 only exists if the Richardson niimber, Ri, in the flow 

 is somewhere less than a quarter [Miles (1961) , 

 Howard (1961) ] and even then in some flows an un- 

 stable solution may not exist. One way in which 

 internal gravity waves may break is by themselves 

 causing or augmenting a mean shear to induce regions 

 of such low Ri that small-scale disturbances may 

 grow as K-H.I and generate turbulence. It appears 

 that Woods' (1968) billows were generated in this 

 way, and similar structures in Loch Ness [Thorpe, 

 Hall, Taylor, and Allen (1976)] may have a like cause. 

 It is however known that internal waves may break in 

 quite a different way, by what has been termed 'con- 

 vective instability' [Orlanski and Bryan (1969) ] . 

 This form of instability becomes much more likely in 

 the presence of a mean shear. 



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