Self-Conveating Flows 



Close agreement between theoretical and experimental results was 

 obtained when we used 3K/2k = 6 or K/k = 4. We have included 

 in the same figure the experimental results from Series III which 

 had a markedly different value of P; these too were found to agree 

 very closely with the theoretical prediction based on D = 0. 2 and 

 K/k = 4, indicating that the dependence of these latter two parameters 

 on p is probably very weak. We have used these values of D and 

 K/k for all subsequent comparisons of experimental and theoretical 

 results. The coefficient of virtual potential energy cannot be much 

 different from unity, according to its definition in Eq. (38). The 

 total kinetic energy of a rising vortex-pair was thus found to be 

 about four times larger than the kinetic energy associated with its 

 linear convection alone, an indication of the intensity of motion 

 inside the vortex-pair, which contributes to its total kinetic energy. 

 This finding lends important support to the strong circulation 

 assumption. 



A comparison between the predicted and actual maximum 

 heights of rise of a vortex-pair convected in a linearly density- 

 stratified medium is shown as Fig. 12. The figure includes a 

 prediction based on the simplified asymptotic solution for small B 

 (or S) , Eq. (59), and a prediction obtained from the exact solution 

 of Eq. (49) for Zf^Qx* Generally there is good agreement between 

 the experimentally measured maximum height of vortex-pairs and 

 the exact theoretical solution. 



Finally, in Fig. 13, we compare the measured trajectories 

 (height versus time) of vortex-pairs with their theoretically pre- 

 dicted trajectories. The vortex-pairs included in the Figure all had 

 different starting conditions and they were moving through media 

 with different density-stratifications. However, their trajectories, 

 as depicted in the figure are shown to depend only on the values of 

 the two lumped parameters G and S which combine their starting 

 conditions with the properties of the surrounding medium. 



The fact that trajectories of different vortex-pairs are 

 grouped according to their G and S values confirms the validity 

 of the scaling laws and scaling parameters used herein, while the 

 agreement obtained between the experimental and theoretical tra- 

 jectories lends further support to the validity of the simplified 

 theory presented here for the motion of vortex-pairs in stratified 

 media. 



VI. SUMMARY AND CONCLUSIONS 



A theory for the motion of two-dimensional turbulent vortex- 

 pairs in homogeneous media has been developed based on separate 

 velocity scaling of the internal and external flow fields involved in 

 the motion. These two flow fields are depicted to be separated by a 

 thin region of high shear, which also forms the boundary of the 



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