625 



Uo<0 



FIGURE 2. The amplitude of the displace- 

 ment of lines of constant density in 

 internal waves of the first mode with wave 

 number k = ir/H calculated from linear theory 

 (i.e., from the Taylor-Goldstein equation) 

 at various Richardson numbers (as labelled) 

 in 



(a) Couette flow, U = Uo (2z/H - 1), with 

 constant density gradient. Uq is posi- 

 tive for the left hand set of curves 

 and negative for the right hand set. 



(b) Hyperbolic tangent profiles, U = 

 Ugtanh y and density p = Pod " 

 Atanh y) where y = 20z/H - 15. 

 Uq is positive for the first three 

 curves at the left, zero for Hi = " 

 and negative for the three curves on 

 the right. The value of Ri marked on 

 these curves is the minimum mean flow 

 value at z = 3H/4. 



interface where we saw in Figure 2 that the dis- 

 placement was concentrated, and, in Figure 3, where 

 the curvature was greatest. The fluid particles 

 move forward (at speed Cp) more rapidly than the 

 wave advances and this leads to a layered structure 

 with a region of slightly denser fluid overlying 

 less dense fluid with the potential consequence of 

 gravitational instability. Similar 'forward' 

 breaking occurs at the wave troughs when X * 0. 

 The experiments demonstrate clearly the difference 

 between K-H.I of the mean flow (seen in Figure 5j) 

 and the convective instability of the waves . In 

 the former the wave-like disturbances grow, extract- 

 ing energy from the mean flow, whilst in the latter 

 the waves do not grow in amplitude and lose energy 

 as a consequence of instability. 



The condition for convective instability to 

 occur (Cp = c) has been used in a calculation to 

 produce the stability diagrams of Figure 6. These 

 are appropriate only to a particular wavelength 

 and show the wave slope at which instability will 

 occur for a given Ri. The Couette flow (Figure 6a) 

 is stable in the absence of waves for all Ri > 0, 

 but the hyperbolic tangent profile (Figure 6b) is 

 unstable at Ri = 0.25 and the dashed lines show 

 the value Ri = 0.25 at the interface marking the 

 boundary at which K-H.I will occur in a quasi steady 

 flow. These diagrams demonstrate how shear greatly 

 reduces the wave slope at which convective instabil- 

 ity sets in, a partial consequence of the trend of 

 the phase speed toward V^^x ^^'^ hence a reduction 

 of the wave particle speed necessary to promote net 

 speeds, Cp, which exceed the phase speed. The non- 

 linear terms are also very important however, the 

 finite amplitude change in the phase speed being 

 as important as other non-linear effects. 



We may press the analogy between interfacial 

 internal waves in a shear flow and surface waves 

 further. The shape of surface gravity waves 

 (narrower crests than troughs) and their habit of 

 breaking forwards at the crests seems universal, 

 in that it is independent of water depth, being 

 observed and (where theory is available) predicted 

 for both shallow and deep water waves. The internal 

 waves observed in the experiments have similar prop- 

 erties, accepting that the profile is inverted with 

 respect to the surface waves if x * » even though 

 they are not strictly interfacial waves or moving 

 at the speed of one of the layers. This suggests 

 that the shape and breaking, by convective overturn, 

 of long first mode internal waves on a relatively 

 narrow interface between two uniform layers follow 

 the pattern observed in the experiments, independent 

 of the depths of the layers, provided that the 

 Richardson number of the mean flow in the interfacial 

 region is small. 



Figure 6b is not symmetrical, a consequence of 

 the asymmetry introduced by having unequal layer 

 thicknesses above and below the interface. Trans- 

 lated to a situation in which wind is driving a 

 flow above a shallow thermocline, the diagram 

 implies that internal waves travelling with the 

 wind (x > 0) will break at a greater amplitude (or 

 later if the shear flow is increasing) then waves 

 of the same length travelling against the wind. 

 This result also follows from our analogy with 

 surface waves since, for a given wavelength, surface 

 waves of limiting (120° apex) amplitude in deep 

 water (corresponding to the forward moving, X > 0' 

 internal gravity waves) are higher than waves in 

 shallow water (which correspond to the backward 

 moving waves) . Waves moving across the flow will 



