624 



FIGURE 1. The development of Kelvin Helmholtz In- 

 stability (K-H.I) in a stratified shear flow [from 

 Thorpe (1971)]. 



Shear affects internal gravity waves in several 

 ways. Perhaps the most important concern the wave 

 speed. Bell (1974) has shown that for any wave 

 mode, the phase speed, c, is a decreasing function 

 of wavenumber, k, which, for waves moving faster 

 than the mean flow at any level, tends to k"lN„_^ 



+ U„, as k increases indefinitely, where N^.^^ 

 is of the Brunt -Vaisala frequency, N, and U[t,ax ^^^ 

 maximum mean flow. (A similar result holds for 

 waves travelling more slowly than the mean flow. ) 

 This result reduces to the well-known property, 

 a < Njj, J,, of internal waves in the absence of shear 

 [Groen (1948) ] where a = ck is the wave frequency 

 relative to the mean flow. It implies that even 

 in a shear flow the wave frequency is less than 

 '^max pJ^ovided the waves are viewed in frame of 

 reference which moves forward at the speed, Uj^^j^. 

 Banks et al. showed further that, at least for 

 simple mean flow profiles, the speed of waves of 



a given mode and wavenumber tends to U 



(from 



above) as Ri decreases. We see a consequence of 

 this result later. 



The vertical structure of internal waves is also 

 changed by shear. Figure 2 shows how the distri- 

 bution of the amplitude of a small amplitude wave 

 of given k varies with z as the shear increases 

 for (a) plane Couette flow of a fluid with constant 

 N and (b) hyperbolic tangent profiles of mean speed 

 and density. The profiles are distorted as Ri 

 decreases with the largest amplitudes displaced 

 towards the level at which the mean speed in the 

 direction of wave propagation is greatest. We shall 

 find it convenient to distinguish these cases by 

 the sign of x = £.-9. ^ H where Q_ is the mean flow' 

 vorticity and c_ the phase speed of the waves in a 

 frame of reference in which the depth averaged mean 

 flow is zero. Positive Uq in Figure 2 corresponds 

 to X > 0, and conversely. 



The shape of waves in a fluid with density and 

 velocity distributed as tanh z (corresponding to 

 Figure 2b) is shown in Figure 3 for (a) backward 

 relative motion in the upper layer, X ^ Oi (b) no 

 shear, (c) forward motion in the upper layer, x ^ 0- 

 The waves in (b) and (c) have narrower crests than 

 troughs, whilst the waves in (a) have wide crests 

 and narrow troughs . 



This second-order effect is not unexpected. It 

 may easily be shown [Thorpe (1974, Appendix C)] 

 that interfacial waves (see Figure 4) which move 

 forward with the speed of the upper layer (the 

 limit, as we have seen, towards which the phase 

 speed of the internal waves tends as Ri decreases) 

 have exactly the same shape as have surface waves 

 on a fluid of depth equal to the lower layer. Con- 

 versely those moving at the speed of the lower layer 

 have the shape of surface waves on a fluid of depth 

 equal to the upper layer, but inverted. This is 

 just the trend shown in Figure 3. The limiting 

 form of the surface wave is one with a sharp apex 

 of 120°. Such an angle can exist in a two- layer 

 flow only in the cases we have considered where 

 the wave speed is the same as the flov; in one of 

 the two layers. Otherwise there is a relative flow 

 around the apex in the upper (or lower) fluid 

 leading to a singularity of infinite flow in the 

 irrotational fluid. In general, some other limiting 

 profile must appear, although it is likely to tend 

 in a continuous way towards the limiting sharp apex 

 profile. Recent work on breaking surface waves 

 [Cokelet (1977) ] cannot be applied even in the 

 special case for the analogy is valid only for 

 steady waves. 



Experiments, however, [Thorpe (1968)] demonstrate 

 how internal waves break in a shear flow. Figure 

 5 shows wave breaking for x ^ 0- A jet of fluid 

 moves forward (that is faster than the waves advance) 

 from the wave crest above the level of the mean 



