stability of and Slaves in Stratified Flows 



in which the asterisk denotes the complex conjugate, and the t, now 

 in terms of d/V, is dimensionless , as is f« Considering the singu- 

 lar neutral case, for which Cj =0, it is easy to see from (40) and 

 (41) that f'f* is real for y > yc and equal to - iir for y < y^.. Hence 

 (f'f*). suffers a jump at y^,. Since f'f* is zero at both rigid 

 boundaries, it cannot afford this jump, [if a ^ 0, this jump cor- 

 responds to a jump in the Reynolds stress. But we do not have to 

 consider the jump in t, and can consider merely the jump in (f'f*). .] 

 Consequently a singular neutral mode with J(y ) equal to i/4 is im- 

 possible. And we can henceforth concentrate on the case J(y ) < l/4. 



For J(y ) < 1/4 Miles [ 196l] gave the solutions of (32): 



Vy) = (y - Yc) w* 



(44) 



in which 



w^= 1 +A(y - yc)/(l ± V) + ..., (45) 



with A given by (40) [but with y = (1 ± v)/2 therein] and 



V = (1 - 4Jc) 



Jc= J(yc)- 



(46) 



We can use (44) and (45) with all terms therein considered dimension- 

 less. Miles [ 1961 , pp. 506-507] showed that for J < 1 /4 the solu- 

 tion, if one exists, must be either f+ or f_. We can demonstrate 

 our point by considering f+ as the solution. The demonstration for 

 the other case is strictly similar. 



The study of the eigenvalue problem defined by (32) and (35a, b) 

 naturally leads to a study of the zeros of f. Since f is given by 

 (44), it leads to the study of the zeros of w+. This in turn leads us 

 to consider the differential equation for w (from which the subscripts 

 are removed for convenience). Denoting w^ or w_ by w, we can 

 easily obtain that equation: 



(pz^/wV +z^^ 



-J,pz-2 +VP-Z-' +-£34; -a' 



^P'^' ''^T^ "^ P - 



Np' 



w = 0, (47) 



(c - U)^J 



with z = Y - y^f and y = (1 ± v)/2. 



We are now in a position to present 



Theorem 1. If p and U are continuous and analytic, with p' < 

 and U' > 0, and if CpU')' and (In "p) " are positive throughout, then 

 singular neutral naodes are impossible. 



229 



