Krishnamurti 



(ii) There is a change in the flow pattern from two-din:ien- 



sional rolls to a three-dimensional flow which is periodic 

 in space and steady in time. The change occurs at a 

 Rayleigh number coinciding with R,. to within the error 

 in determining Rjj^. 



(iii) There is hysteresis in the heat flux as well as in the flow 

 pattern as R is increased from below or decreased from 

 above, indicating that the transition is caused by a finite 

 amplitude instability. 



The third transition is indicated by curve III in Fig, 10, 

 Above this curve, the flow is time dependent with a slow tilting of the 

 cell in the vertical and a faster oscillation which has the nature of 

 hot or cold spots advected with the mean flow. Transition to disorder 

 is seen to result from an increased number and frequency of such 

 oscillations . 



Higher transitions observed by Malkus [ 1954] and confirmed 

 by Willis and Deardorff [ 1967a] have not been discussed. 



The small amplitude nonlinear theories have been quite suc- 

 cessful in a smiall neighborhood of the critical point R^. The obser- 

 vation that transition to turbulence occurs near Re for small Prandtl 

 number in non- rotating convection, and for T > Tc for rotating 

 convection, indicates the possibility of gaining further understanding 

 of transition to turbulence through the nonlinear theories. 



The research reported here was supported by the Office of 

 Naval Research Contract N-OOOi 4-68- A-0 159 and by grant number 

 GK- 18136 from the National Science Foundation, 



REFERENCES 



Busse, F. H. , Dissertation, University of Munich. (Translation 

 from the German by S. H. Davis, the Rand Corporation, 

 Santa Monica, California, 1966), 1962. 



Busse, F. H. , "On Stability of Two-Dimenslonal Convection In Layer 

 Heated from Below, "J. Math, and Physics, 46 , 140, 1968. 



Chandrasekhar , S. , Hydrodynamlc and Hydromagnetlc Stability , 

 Oxford, 1961. 



Chen, M. M. and Whitehead, J. A. , "Evolution of Two -Dimensional 

 Periodic Rayleigh Convection Cells of Arbltary Wave- 

 Numbers," J. Fluid Mech. , 3j.» 1» i968. 



Davis, S. H. , "Convection In a Box: Linear Theory, " J. Fluid 

 Mech. , 30, 465, 1967. 



308 



