BIFURCATIONS TO PERIODIC, QUASIPERIODIC, AND CHAOTIC REGIMES IN ROTATING AND CONVECTING FLUIDS *

We have performed a comparative experimental study of the bifurcations in circular Couette flow and Rayleigh-Btnard convection, two systems which long served as classical prototypes for experimental and theoretical investigations of hydrodynamic stability.”* In circular Couette flow in its simplest form a fluid is contained between concentric cylinders with the inner cylinder rotating, and in Rayleigh-Bknard convection the fluid is contained between horizontal thermally conducting plates heated from below. The bifurcation parameter for the Couette flow system can be taken as the Reynolds number R , which describes the distance away from equilibrium and is proportional to the angular velocity of the inner cylinder. Similarly, for the Rayleigh-Btrnard problem the distance away from equilibrium is given by a dimensionless number R , the Rayleigh number, which is proportional to the difference between the temperatures of the two horizontal plates. At small R the flow in the Couette cell is purely azimuthal, but when the Reynolds number exceeds a critical value R , the azimuthal circular Couette flow is no longer stable and there is a bifurcation to a flow with a horizontal toroidal vortex pattern superimposed on the azimuthal flow. This bifurcation was predicted and observed by Taylor in 1923 in work that stands as a classic study of hydrodynamic ~ t a b i l i t y . ~ In 191 6 Rayleigh showed, in the pioneering theoretical paper on the convective instability, that above R, the pure conduction state is unstable to horizontal disturbances, and the system bifurcates to a new state consisting of parallel convection As R is increased further both the circular Couette flow and Rayleigh-Btnard systems bifurcate from the time-independent vortex patterns to time-dependent flows. These secondary bifurfactions have been studied theoretically and experimentally for the past few years; however, there has been little detailed quantita-