Convection under rotation for Prandtl numbers near 1: Kuppers-Lortz instability

The Kuppers-Lortz (KL) instability in Rayleigh-Benard convection rotated about a vertical axis was studied experimentally using optical-shadowgraph imaging in the rotating frame for dimensionless rotation rates 6<Omega<20. Two cylindrical convection cells with radius-to-height ratios Gamma = 40 and 23 were used. The cells contained CO2 at 33.1 bar and 16.6 bar with Prandtl numbers sigma=0.93 and sigma=0.83, respectively. Numerical solutions of the Boussinesq equations with parameter values corresponding to the experiments were obtained for comparison. For Gamma = 40 and 8 < Omega < 10.5, the initial pattern above onset was time dependent. Its dynamics revealed a mixture of sidewall-nucleated domain-wall motion characteristic of the KL instability and of dislocation-defect motion. For Omega>10.5, spontaneous formation of KL domain walls away from the sidewall was observed. For 8<Omega<12, then were differences between the two cells very close to onset, but for epsilon greater than or similar to 0.02 the systems were qualitatively similar. For Omega greater than or similar to 12 there was no qualitative difference in the behavior of the two cells at any epsilon. The average size of a domain containing rolls of approximately the same orientation decreased with increasing Omega, and the time dependence speeded up and became dominated by domain-wall propagation. The numerical solutions were qualitatively similar, although there was a tendency for the domains to be larger at the same epsilon and Omega. The replacement of domains of one orientation by those with another led to a rotation in Fourier space which was characterized by a rotation frequency omega(a) in the frame rotating at angular velocity Omega. Quantitative experimental measurements of omega(a), of a correlation length xi, and of a domain-switching angle Theta(s) as functions of epsilon=Delta T/Delta T-c-1 and Omega are presented. For 13 less than or similar to Omega less than or similar to 18, Theta(s) was independent of Omega and close to 58 degrees. We computed the angle of maximum growth rate Theta(KL) Of KL perturbations, and found it to be 43 degrees, distinctly different from Theta(s). The results for omega(a)(epsilon,Omega) over the range 13 less than or similar to Omega less than or similar to 20 can be collapsed onto a single curve <(omega)over tilde>(a)(epsilon)=omega(a)(epsilon,Omega)/omega(r)(Omega) by applying an Omega-dependent factor l/omega(r). Similar collapse can be accomplished for <(xi)over tilde>(epsilon)=xi(epsilon,Omega)/xi(r)(Omega). An analysis of <(omega)over tilde>(a)(epsilon) and <(xi)over tilde>(epsilon) in terms of various functional forms is presented. It is difficult to reconcile the epsilon dependence of <(omega)over tilde>(a) and <(xi)over tilde> at small epsilon with the theoretically expected proportionality to epsilon and epsilon(-1/2), respectively. [S1063-651X(98)12911-2].