Experiments on Convective Flows in Geophysical Fluid Systems 



T(x,z) = A • cos (27Tnx/L) • (1 - e"^/") (23) 



to start the convection. The scale height H is so chosen that the perturbation 

 is strongest where the fluid is most unstable. The relations between the time 

 elapsed in cooling and the amplitude of the perturbations on the one hand, and 

 the growth rate and the final form of the convection cells on the other, have not 

 yet been worked out. 



The results for Case 1 are displayed in Fig. 4(a) and 4(b) and Fig. 5, for 

 L = 3 cm, D = 1.5 cm, 3T/3ZJ ^^^ = 1.5°C/cm, and water as the working fluid 

 {v = 1.0 X 10"2 cm^/sec and k = 1.4 x 10"-^ cm^/sec). A time of 16 seconds 

 elapsed before a perturbation of n = 1, A = .001, and H = 1.5 cm was applied. 

 It was foimd that the isotherms are a much more sensitive indicator of the con- 

 vective motions than the streamlines; this is not surprising if we consider that 

 the diffusion coefficient of friction is --7 times that of heat, so that it will take 

 --7 times longer for all thermal fluctuations to die out than for the velocities. 

 Similarly, the temperature field is concentrated into narrower regions, for the 

 small thermal diffusion is ineffective in smearing it out. 



Figures 4(a) and 4(b) represent the time development of the temperature 

 pattern when conduction gives way to convection. The times elapsed between 

 frames are listed in the figure captions. The last two frames for temperature 

 and streamline in Fig. 4(b) are taken at t = 454 seconds. All six frames of 

 isotherm development in this figure had the same (visible) streamline pattern 

 associated with them (shown in the bottom frame). At the onset, a heavy blob 

 of cold fluid forms which penetrates to the bottom and is reflected by the rigid 

 surface. When the reflected upward-moving thermals join the top layer again, 

 a strong "finger" of cold fluid forms which again descends to the bottom and is 

 reflected; however, at this time, two weaker fingers develop also at the side- 

 walls, and the streamline pattern shows that at this time the two-cell pattern 

 breaks into a four-cell pattern. After this time there are three descending and 

 two ascending columns. The foregoing pattern is repeated many times, with the 

 "finger" growing weaker after each cycle until the pattern shown in the final 

 frame eventually emerges. It was also observed that the period of the oscilla- 

 tions increased steadily; this can be understood if we assume the oscillations 

 to be some form of internal gravity waves whose frequency depends on the 

 average vertical temperature gradient. The total kinetic energy and absolute 

 vorticity have converged to four significant figures, yet small but nevertheless 

 visible changes occurred in the isotherms. Though a truly steady state in this 

 problem can never be achieved as the mean temperature of the system decreases 

 linearly, the location of this temperature becomes a constant and the horizontally 

 averaged temperature as a function of depth also becomes a constant. Thus a 

 "quasi- steady" state is possible in the system, but one has to iterate a very 

 long time to damp out the thermal fluctuations. 



Figure 5 shows the vertical variation of the horizontally averaged tempera- 

 ture. In a significant portion of the flow the temperature gradient is reversed: 

 this can be traced to the impinging cold stream on the bottom and its consequent 

 spreading. Because of the relatively weak nature of the upward-moving com- 

 pared to the descending columns, a fluid particle spends a greater time in the 

 former regions and achieves its highest temperature only on the upward 

 passage. 



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