619 



linear superposition of high modes may make little 

 sense. A linear mode can itself be considered the 

 superposition of two disturbance trains, one propa- 

 gating downwards and the other upwards with reflec- 

 tions either at the bottom or at a region where the 

 buoyancy (or stability) frequency N drops below the 

 wave frequency. McComas ' calculations indicate that 

 the non-linear interaction time of such components 

 at the spectral densities found in the deep ocean, 

 is remarkably short, only a few wave periods in 

 many wave cases. Accordingly, a train of waves 

 generated, say, near the thermocline will in actu- 

 ality have little opportunity to travel to the 

 bottom, reflect upwards, and combine with a down- 

 wards travelling wave to produce a 'mode' as usually 

 conceived. More realistic would be the view of dis- 

 turbances generated in the more active thermocline 

 region, radiated downwards but being 'scrambled' by 

 wave-wave interactions into a more diffuse spectral 

 background. 



This contribution is concerned with some aspects 

 of the energy flux downwards from high frequency, 

 low mode internal waves at the thermocline. If the 

 internal wave frequency is greater than the stability 

 frequency N^j below the thermocline, the waves are 

 of course trapped to the thermocline region. How- 

 ever, as their frequency decreases below Nfj, they 

 become 'leaky' and their energy radiates rapidly 

 downwards as the simple analysis of the next section 

 will demonstrate. Yet, if Brekhofskikh et al.(1975) 

 measurements are at all typical, most of the energy 

 of the low mode internal waves in the thermocline 

 region is at frequencies considerably above N^; 

 indeed, in view of the efficiency with which such 

 low frequency energy is propagated downwards , we 

 would not expect to find much energy at these fre- 

 quencies in the main thermocline. However, one 

 possible link is suggested by the work of Mclntyre 

 (1973) who showed that groups of internal waves in 

 a fluid of constant frequency N, confined between 

 horizontal boundaries, produce second order 'mean' 

 motions, modulated as are the wave groups. There 

 is no reason to believe that these second order dis- 

 turbances are confined only to the particularly 

 simple case that he considered, and indeed in Sec- 

 tion 3 it is shown that they are not. High frequency 

 internal waves , occurring in groups and trapped 

 within the main thermocline, produce second order 

 low frequency disturbances; if the group frequency 

 is less than N^j, their energy is radiated downwards 

 at the group frequency. 



The results presented here are preliminary but 

 intended to provoke consideration of this mechanism 

 as a source of oceanic internal waves. The simplest 

 case of a sharp thermocline overlying a deep, uni- 

 formly stratified region is described in some detail. 

 The more realistic (and complicated) case with a 

 general distribution of N(z) can be considered by 

 asymptotic methods and these results will be de- 

 scribed elsewhere. 



2. RADIATION DOWNWARDS — A "LEAKY MODE" 



Consider the following experiment: a laboratory 

 tank (Figure 1) is stratified with a layer of uni- 

 form density lying over a density jump 6p below 

 which the fluid is uniformly stratified, with N^ = 

 (-p~ g 3p/3z) = constant. A wave-maker at the end 

 of the tank generates a periodic disturbance with 

 (real) frequency n. What are the characteristics 

 of the motion induced? 



It is, I think intuitively evident that if n > N 

 an interfacial wave mode will propagate. The struc- 

 ture of the mode below the pycnocline will be in- 

 fluenced by the stratification but at these high 

 frequencies, no internal waves can be supported in 

 the lower layer and the interfacial wave will propa- 

 gate without loss. If, however, n < N, internal 

 waves induced in the lower region by the interfacial 

 disturbance can carry energy downwards so that the 

 interfacial wave will attenuate. The question is: 

 how rapidly does this occur? 



A linear analyses suffices. Suppose the pycno- 

 cline displacement is represented by the real part 

 of ^ = a exp i(kx - nt) , where n is real and k may 

 be real or complex. Above the pycnocline at z = 0, 

 the motion is irrotational with u = Vi|) and V^(j) = 0. 

 In the uniformly stratified region below, the vert- 

 ical velocity component, w, obeys the internal wave 

 equation 



3t^ 



V^w + N^V^^^. 

 h 



(1) 



where V^^ is the horizontal Laplacian operator, 

 8 /3x in this two-dimensional problem. At the 

 upper free surface at z = d, w = to sufficient 

 accuracy; at the pycnocline the vertical displace- 

 ment and the pressure must both be continuous and 

 as z ^ - «>, the disturbance must either die away 

 or represent internal waves with an energy flux 

 downwards . 



In the upper region, the solution for (j) is 

 readily found to be 



ina cosh k(z 

 He 



d) 



sinh kd 



expi(kx - nt) 



(2) 



while in the lower layer, if 



w = - ina exp [kz + i(kx - nt) ] , (3) 



(which satisfies the condition of continuity of w 

 at z = 0) , then substitution into (1) requires that 



(K/k)' 



(N/n)- 



(4) 



Note that since n is real, K/k is either purely real 

 (if n - N) or purely imaginary (if n < N) . 



The dispersional relation is obtained from the 

 condition that the pressure be continuous at z = ?. 

 In the upper region of density p, 



3t ' C 

 pa[g + (n^/k) coth kd] exp i(kx -nt) , (5) 



P(zl 



.J 



^Wave 

 Absorbers 



FIGURE 1. Tank stratified with a layer of uniform 

 density over a density jump below which the fluid is 

 uniformly stratified. 



