Self-Conveoting Flows 



I 



Kp. Yi:^=^ Li^^+^^)didi (37) 



and 



k(pj - Pe)zR^ = J ^P - PeHz + ;) de d; (38) 



In most practical instances where one is dealing with a mass 

 of fluid convected through a honmogeneous or stratified medium such 

 as the ocean or the atmosphere, the difference between the densities 

 of the convected and surrounding masses is very small; that is 

 Ap/pg « 1 , being usually of the order of 10" , and therefore Pj/p© 

 can be taken as 1. This assumption, frequently referred to as the 

 Boussinesq approximation, see Phillips [ 1966] , will be used through- 

 out the analysis presented herein. 



Using the Boussinesq approximation and the identity dz = w dt, 

 the three conservation statements, Eqs. (33), (34) and (36), may be 

 reduced to the following form in the case of a planar motion: 



R = Pz, (39) 



dpi , 2Ap _ ^ 



"dT T~" ^' 



or (40) 



d /Apw2/ApN ^ 

 dz Vpg / zVpg / 



.2 



^^^^0^%)*'^^(7f^-)^ = <' <^'> 



where a = - (1 /pe)(dpe/dz) and Ap = (pj - Pg). 



Finally, explicit general solutions of (40) and (41) may be 

 found. They are: 



^= n^) _^']z-'+ ^¥ (42) 



Pe L V Pe ^0 3 -• 3 



(%) " I ' (1 + 2D) ^ ^ (i +D/2 " 1 + 2d)J ^ 



345 



