STABILITY OF AND WAVES IN STRATIFIED FLOWS 



Chia-Shun Yih 



University of Michigan 

 Ann AvhoVy Michigan 



ABSTRACT 



A theorem giving sufficient conditions for stability 

 of stratified flows, which is a natural generalization 

 of Rayleigh's theorem for shear flows of a homogeneous 

 fluid, is given. Sufficient conditions for the existence 

 of singular neutral modes , and consequently of unstable 

 modes, are also presented, and in the development the 

 possibility of multi-valued wave number for neutral 

 stability of the same flow is explained. Finally, neutral 

 waves with a wave velocity outside of the range of the 

 velocity of flow (non-singular modes) are studied, and 

 results concerning the possibility of these waves are 

 given. In addition. Miles' theorem [ 1961] on the stability 

 of stratified flows for which the Richardson number Is 

 nowhere less than 1/4, and Howard's semi- circle theorem 

 [ 1961] are extended to fluids with density discontinuities,. 



I. INTRODUCTION 



The stability of stratified flows of an Invlscid fluid has been 

 studied In a general way, l,e, , without specifying the actual density 

 and velocity distributions, by Synge [ 1933] , Ylh [ 1957] , Drazln 

 [ 1958] , Miles [ 1961 , 1963] , Howard [ 1961] , and others. Of these. 

 Miles has made particularly substantial contributions to the subject. 

 However, many questions still remain open. Among these are the 

 following: 



(I) Miles [ 1961] showed that If the Richardson number Is 

 nowhere less than 1/4, the flow must be stable. This 

 Is a sufficient condition for stability. What can one say 

 regarding the stability of the flow when the Richardson 

 number Is less than l/4 In part or all of the fluid? Are 

 there then some sufficient conditions for stability not 



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