stability of and Waves in Stratified Flows 



(50) 



(c-ur 



in which, it must be emphasized, all accents indicate differentiation 

 with respect to y, not z. The rest is strictly similar to the proof 

 for Theorem 1 , except the range of integration is between z = and 

 z = y^ (or between y = y^ and y = 0) , and we want to show w ^ at 

 y = 0, Note also that U"< now guarantees ("pU')' < 0, 



Since the non-existence of singular neutral modes implies the 

 non-existence of unstable modes, we have also 



Theorem 3, If P a^rid U are continuous and analytic , with p' 

 negative and U' positive, and if either ("pU')' and (ln"p)" are both 

 positive throughout, or U" and (In "p) " are negative throughout, 

 the flow is stable. 



This theorem is the natural generalization of Rayleigh's theorem for 

 inviscid homogeneous fluids in shear flow. Previous attempts at this 

 generalization [ Synge 1933, Yih 1957, Drazin 1958] have produced 

 the result that there must be stability if (in dimensional terms) 



does not change sign. This criterion is not useful because it involves 

 not only c^ but also c. , 



VI. SUFFICIENT CONDITIONS FOR INSTABILITY 



Sufficient conditions for instability have seldom been given in 

 studies of hydrodynamic stability. In giving some such conditions, 

 we shall also be able to explain why the a can be multi-valued for 

 the same N, at neutral stability. 



We assume that p and U are analytic, that p' :£ 0, and that 

 at a point where p' = 0, U" is also zero. The value of U at that 

 point will be denoted by U^ , for we shall consider the possibility of 

 having c equal to U _at that point. We demand that at any other 

 point where U = U^. , p' = = U" must be satisfied. If U is mono- 

 tonic, of course there is only one point at which U = U • 



_ Under the assumptions made, p" must be zero at y , since 



p' is never positive, and near y 



231 



