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

The K\"uppers-Lortz (KL) instability in Rayleigh-B\'enard convection rotated about a vertical axis was studied experimentally using optical-shadowgraph imaging in the rotating frame for dimensionless rotation rates 6<Ω<20. Two cylindrical convection cells with radius-to-height ratios Γ=40 and 23 were used. The cells contained CO₂ at 33.1 bar and 16.6 bar with Prandtl numbers σ=0.93 and σ=0.83, respectively. Numerical solutions of the Boussinesq equations with parameter values corresponding to the experiments were obtained for comparison. For Γ=40 and 8<Ω<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 Ω>10.5, spontaneous formation of KL domain walls away from the sidewall was observed. For 8<Ω<12, there were differences between the two cells very close to onset, but for ε0.02 the systems were qualitatively similar. For Ω12 there was no qualitative difference in the behavior of the two cells at any {ε}. The average size of a domain containing rolls of approximately the same orientation decreased with increasing {Ω}, 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 {ε} and {Ω}. 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 ωₐ in the frame rotating at angular velocity {Ω}. Quantitative experimental measurements of ωₐ, of a correlation length {ξ}, and of a domain-switching angle Θₛ as functions of ε≡ΔT/ΔTc-1 and {Ω} are presented. For 13Ω18, Θₛ was independent of {Ω} and close to 58^∘. We computed the angle of maximum growth rate ΘKL of KL perturbations, and found it to be 43^∘, distinctly different from Θₛ. The results for ωₐ(ε,Ω) over the range 13Ω20 can be collapsed onto a single curve ${{ ̃ \~{}{}}{{ω}}}ₐ({ε}){≡}{{ω}}ₐ({ε},{Ω})/{{ω}}ᵣ({Ω})by applying an Ω-dependent factor1/{{ω}}ᵣ.$ Similar collapse can be accomplished for ${ ̃ \~{}{}}{{ξ}}({ε})={ξ}({ε},{Ω})/{{ξ}}ᵣ({Ω}).$ An analysis of ${{ ̃ \~{}{}}{{ω}}}ₐ({ε})$ and ${ ̃ \~{}{}}{{ξ}}({ε})in terms of various functional forms is presented. It is difficult to reconcile the ε dependence of{{ ̃ \~{}{}}{{ω}}}ₐ$ and ${ ̃ \~{}{}}{{ξ}}at small ε with the theoretically expected proportionality to ε and{{ε}}^{{-}1/2},$ respectively.