stability of and Waves in Stratified Flows 



It Is then clear that the system consisting of (32) and (35a, b) 

 gives, for a non-trivial solution, a relationship 



F (cv,N,c) = 0. (36) 



Since c is complex, (36) has a real part and an imaginary part. 

 When Cj is set to zero and c^ eliminated from the two component 

 equations, a relationship 



F2(a,N)=0» (37) 



if one such exists, gives the neutral- stability curve. It is possible, 

 however, that c is real for all values of a and N, in which case 

 c. = in the entire N ~ a plane, and then of course there is no 

 neutral- stability curve because one coinponent equation of (36) is 

 c. = 0, and the other is simply (36) itself, with the c therein real. 



In this section, we shall assume p and U to be continuous, 

 analytic, and monotonic. Furthermore, we assume p'< throughout. 

 We now recall the following known results: 



(i) If J(y) is not less than 1/4 for the entire fluid domain, 

 then the flow is stable [ Miles 1961] , 



(ii) If c. ^ then c^ must be equal to U at some point in 

 the flow, as a consequence of the semi-circle theorem 

 of Howard [ 1961] , and 



(iii) If an eigenfunction exists for (c-, a , N ), then near 

 that point c is a continuous function of a and N, 

 [ Miles 1963 and Lin 1945] . 



Under the assumptions we have made on p and U, and in 

 view of the known results just cited, we conclude that the non-existence 

 of any singular neutral mode, which is a mode with a real c equal to 

 U at some point in the flow, implies the non-existence of unstable 

 modes. The reason is as follows. In the N - a plane there is always 

 a region of stability. For we can imagine g and hence J(y) to in- 

 crease indefinitely, until J(y) is everywhere greater than 1/4, which 

 is attainable since P is nowhere zero. Thus there is a region of 

 large N for which the flow is stable. If unstable modes exist there 

 must then be a stability boundary dividing the region of stability from 

 the region of instability, and hence a neutral-stability curve. As we 

 approach that curve from the region of instability, c^ being within 

 the range of U so long as Cj ^ and continuous in or and N so 

 long as c is an eigenvalue, according to (iii) above, in the limit, 

 when C| = 0, c,. must be within the range of U, i.e. , the limiting 

 mode must be a singular neutral mode. Hence the non-existence of a 

 singular neutral mode implies the non-existence of unstable modes. 



227 



