621 



u = (3(l>/3x) 



na(x + zcot9)cosx 



so that the horizontal energy flux in the upper 

 layer 



Aw = A 



3x 



(uO 



at z = 



Finally, in the lower layer, 



(20) 



Ejj = I up az = 



4n 



exp <- -j^- xtanB |> , (16) 





32 

 3t2 



+ N^ 



3^w 

 3x2 



= Q 



(21) 



the horizontal divergence of which 



where the non-linear terms 



3 Ex 

 3x 



Ijna (x)b sinScosS 



(17) 



provides for the radiative flux in the lower layer. 



This simple example illustrates the way that 

 energy can be radiated downwards by the low fre- 

 quency perturbations produced by groups of high 

 frequency waves , but they have a deeper theoretical 

 interest. Gaster (1977) has pointed out that if 

 the dispersion relation for waves involves complex 

 wave-numbers or frequencies, the usual kinematic 

 definition of group velocity may not be correct, 

 and a simple calculation shows that the solution 

 is an example of this failure. Here the wave- 

 numbers are complex as the energy leaks into the 

 lower layer, but the energy flux is not at the rate 

 represented by the local energy density, v?-!i\^/1 

 cos 6 , times the ordinary group velocity V]^a) = 

 c tan 6 = (b/n) sin 9. The correct interpretation 

 of these situations will be considered elsewhere. 



3x3z3t 

 32 



(u • Vu) 



(u • Vb + — (u • Vw) ) 

 3x2 - 3t 



(22) 



Variations in energy density of the primary waves 

 will propagate with the group velocity, c ; let us 

 therefore average these equations at points fixed 

 with respect to the wave groups but over random 

 phases of the waves themselves , a process repre- 

 sented by brackets [ ] . The averaged interfacial 



conditions are then, to order E 



(ak)' 



A 



3u 



3t 



3x 



[f:] = 



3z3t 



yu 



A [w] = A 

 both at z = , and 



T- (U5) 

 3x 



(23) 



(24) 



3. ENERGY RADIATION DOWNWARDS FROM GROUPS OF 

 INTERFACIAL WAVES 



To illustrate the way in which groups of internal 

 waves produce 'mean,' second order disturbances locked 

 to the wave group, let us consider the same basic 

 stratification as in the previous section, with 

 fluid of depth d and constant density lying over 

 a buoyancy jump b below which the stability fre- 

 quency N is constant. Suppose that interfacial 

 waves with frequency n > N are maintained by high 

 frequency forcing f from the upper layer, perhaps 

 by the surface wave-wave interactions described by 

 Watson, West, and Cohen (1976) . If the internal 

 wave amplitude is characterised by a and the wave- 

 number by k, then, to order £2 = (ak)2, the condition 

 or continuity of pressure across the interface can 

 be expressed as 



'lit -i* 



f = -A 



3^u 

 3zi;t 



+ u 



Vu 



(18) 



at z = 0, where A( ) = ( )+ 

 across the density jumps. Since 



( )_, the difference 



i; = w. 



v; 



? — I 

 ^ 3z 'o 



3? 

 ox 



[C] = [w] - 1^ [uC] , 



(25) 



also at z = 0. The averaged field equation for the 

 lower layer follows similarly from (21) and (22). 

 The linear fluctuating internal wave motion is 

 as given in the previous section when n > N; through 

 the non-linear terms on the right of (23) -(25), this 

 forces a second order mean disturbance [5]/ [v/] , 

 etc., that moves with the velocity of the wave groups. 

 The pycnocline disturbance can be represented as 



? = '2a{c<is(k'x - n't) + cos(k"x 



•t)} 



The form of the forcing functions is simplest when 

 the pycnocline depth is such that kd >> 1, and it 

 is found that (23) reduces to 



3u 

 3x 



b -^ [d 

 3x 



a^nkk < 1 



(c + \c) Sin k (x - c t) 



g g g 



{1 + 0(ak, -a- /n2) } , 



g 



= WO 



3^ <"0^' 



(19) 



to order E^, where the suffixes, C and 0, represent 

 quantities measured at z = C,0( then the condition 

 that ^ be continuous across the interface assumes 

 the form 



a^k n2 ( ^ + M sin k (x - c t) , (26) 

 g \c 2 / g g 



where kg = k' - k", ng = n' - n", and Cg represent 

 the wave-number, frequency, and velocity of the 

 groups. Similarly, from (24) 



