Self-Conveoting Flows 



motions in homogeneous flows the internal velocity scale grows 

 steadily relative to the translational (external) velocity, the ratio 

 approaching a value considerably larger than unity, while at the 

 same time, the virtual momentum coefficient associated with the 

 vortex motion approaches the value zero. The data also suggest 

 that vorticity ingested from around one half of the vortex pair is 

 almost annihilated through mixing with vorticity ingested from the 

 opposite side* 



IV. SIMPLIFIED THEORY (VORTEX PAIR MOTION IN STRATIFIED 

 MEDIA) 



Convected masses in nature are often rising or falling in a 

 medium of varying density, as in the case of a chimney plume pro- 

 jected upwards into a stable atmosphere. The latter may be 

 ch arac terized by a characteristic time (the Vaissala period), 

 1 /Va-g , where a = - (l/pe)(dpe/dz) and Pg is the potential density 

 of the atmosphere. The same definition can be used to characterize 

 any density stratified media. 



The motion of the convecting mass may also be characterized 

 at any instant by the time, R/W, It is almost apparent that when 

 the latter time is long in comparison to the Vaissala period that the 

 effect of stratification will dominate, and conversely. That is, 



r,. .sri 4-1 J : RVag , , stratification 



Stratification decreasing — rrr^ increasing „ , 



,r • 1 ^ f W J' Dominates 



Vanishes ^ ^ 



Quite clearly, too, as the motion proceeds in time, the ratio R/W 

 increases continuously, so that stratification must eventually 

 dominate. When this happens , the vertical motion of vortex-pairs 

 may become oscillatory, and is accompanied by the collapse and 

 horizontal spreading of the convected mass, as illustrated in Fig. 8. 

 This behavior is, of course, not consistent with similarity either 

 complete or of the kind assumed in the preceding section. 



It is sometimes desirable to be able to estimate the tra- 

 jectory of a vortex pair while it is rising in a stratified media and 

 particularly to predict the maximum height of rise and the time 

 required to reach the maximum. For this purpose, we adopt here 

 a simplified theory based essentially on the assumption of strong 

 circulation and annihilation of ingested vorticity. In fact, the parti- 

 cular assumptions adopted would apply if the velocity ratio, ^, had 

 already closely approached its limiting value. These assumptions 

 are: (i) the motion is determined by conservation of volume, mass, 

 and energy (neglecting vorticity and momentum); (ii) complete 

 similarity (dR/dz = (3 , a constant). For further justification of these 



341 



