620 



to the first order in the wave amplitude. In the 

 lower region, where the density is p + 6p - pN z/g, 

 the horizontal pressure gradient 





to the lowest order, 



= i(p + 6p)an^(K/k) exp i(kx - nt) 

 at z = from (3) , so that 



PO = (p + i5p) an^ (K/k^) exp i(kx - nt) 

 At z = C, below the pycnocline, 



p = - (p + 6p)a(g - n^K/k-) exp i(kx - nt) (6) 



From (5) and (6) it follows that 



2 _ (5p/p) gk bk 



n' 



coth kd + (K/k) coth kd + (K/k) 



(7) 



to the Boussinesq approximation, when 5p/p << 1, and 

 where b is the contrast in buoyancy across the pycno- 

 cline. 



For high frequency oscillations, when n 2 N, 

 equation (4) shows that K/k is real and less than 

 unity; from (7) k is real and the waves propagate 

 without attenuation. The additional restoring 

 forces provided by th'e stratification below do how- 

 ever increase the wave frequency for given k and b 

 above the value for an unstratified lower layer by 

 the ratio 



[coth kd + 1] / [coth kd + (1 - N^/n^)'^] , 



The case when n < N is algebraically simplest when 

 |kd| ^ ". In view of the upper boundary conditions, 

 the real part of k > 0, while from (4) 



-=±i(l 



n2; 



= ±i tan 



C8) 



The ratio of the spatial attenuation rate to the 

 wave-number is simply tan 9 = (N /n - 1)^; when n 

 is significantly less than N the attenuation dis- 

 tance is short as the energy leaks downwards very 

 effectively. 



Expressions for the motion in the upper and 

 lower regions can be written down simply. In the 

 lower layer energy flows along the characteristics 

 5 = X cos 9 + z sin 8 = const. , and the distribu- 

 tion of vertical velocity is 



xna exp 



inaj (5 



bcos^e 



exp 



• ( "' r 



nt 



(12) 



where ai(5) is the amplitude of the interfacial 

 wave at the point where the characteristic inter- 

 sects the pycnocline. The horizontal component 

 of the velocity field in the lower layer is u = 

 - w tan 9, since the motion here consists of alter- 

 nate layers sliding relative to one another along 

 the characteristic surface inclined at an angle 9 

 to the vertical. The pressure fluctuation can be 

 found most simply from the horizontal momentum 

 equation: 



. (• n^5 1 

 p = - ai (£;)bsin9 (sin9 + icos9)exp 1 L up^qo — ^^} • 



(13) 



where n = N cos 9. From (7) 



The vertical energy flux is therefore 



2 

 E^ = -55na]^(C)b sin9cos9 , 



k = (n^/b) (1 



i tan 



(9) 



Since the interfacial waves attenuate in the posi- 

 tive x-direction as energy leaks downwards , the 

 positive sign in (9) is relevant and the vertical 

 wave -number 



K = iktan 9 , 



= (n^/b) ( - tan^e + i tan 



(10) 



The motion of the pycnocline is therefore repre- 

 sented by 



a exp 



exp 1 



nt 



and the total energy flux, directed downwards along 

 the characteristics 5 = const is 



E = Sjnaj (g) bsin 9 



(14) 



In the upper region, the fluctuations in pressure 

 are found from (2) : 



P = - 37= 



exp [-kz + i (kx - nt) ] , 



when kd >> 1, whose real part, in view of (9) and 

 (12) reduces to 



p = - a{x + zcot9)b(cos^9cosx + cos9sin9sinx) , (15) 



a(x) exp 1 [— nt 



(11) 



where 



where 



X = ^(x 



ztane) - nt 



o(x) = a exp 



tan 9 



The real part of the horizontal velocity field is 

 likewise 



