Zero-Prandtl-number convection with slow rotation

Abstract

We present the results of our investigations of the primary instability and the flow patterns near onset in zero-Prandtl-number Rayleigh-Bénard convection with uniform rotation about a vertical axis. The investigations are carried out using direct numerical simulations of the hydrodynamic equations with stress-free horizontal boundaries in rectangular boxes of size (2π/kx) × (2π/ky) × 1 for different values of the ratio η = kx/ky. The primary instability is found to depend on η and Ta. Wavy rolls are observed at the primary instability for smaller values of η (\documentclass[12pt]{minimal}\begin{document}$1/\sqrt{3} \le \eta \le 2$\end{document}1/3≤η≤2 except at η = 1) and for smaller values of Ta. We observed Küppers-Lortz (KL) type patterns at the primary instability for \documentclass[12pt]{minimal}\begin{document}$\eta = 1/\sqrt{3}$\end{document}η=1/3 and Ta ≥ 40. The fluid patterns are found to exhibit the phenomenon of bursting, as observed in experiments [K. M. S. Bajaj, G. Ahlers, and W. Pesch, “Rayleigh-Bénard convection with rotation at small Prandtl numbers,” Phys. Rev. E 65, 056309 (2002)]. Periodic wavy rolls are observed at onset for smaller values of Ta, while KL-type patterns are observed for Ta ≥ 100 for \documentclass[12pt]{minimal}\begin{document}$\eta =\sqrt{3}$\end{document}η=3. In case of η = 2, wavy rolls are observed for smaller values of Ta and KL-type patterns are observed for 25 ≤ Ta ≤ 575. Quasi-periodically varying patterns are observed in the oscillatory regime (Ta > 575). The behavior is quite different at η = 1. A time dependent competition between two sets of mutually perpendicular rolls is observed at onset for all values of Ta in this case. Fluid patterns are found to burst periodically as well as chaotically in time. It involved a homoclinic bifurcation. We have also made a couple of low-dimensional models to investigate bifurcations for η = 1, which is used to investigate the sequence of bifurcations.