Dynamics of zero-Prandtl number convection near the onset

Abstract

In this paper we present various convective states of zero-Prandtl number Rayleigh-B\'enard convection using direct numerical simulations (DNS) and a 27-mode low-dimensional model containing the energetic modes of DNS. The origin of these convective states have been explained using bifurcation analysis. The system is chaotic at the onset itself with three coexisting chaotic attractors that are born at two codimension-2 bifurcation points. One of the bifurcation points with a single zero eigenvalue and a complex pair $(0,\pm i \omega)$ generates chaotic attractors and associated periodic, quasiperiodic, and phase-locked states that are related to the wavy rolls observed in experiments and simulations. The frequency of the wavy rolls are in general agreement with $\omega$ of the above eigenvalue of the stability matrix. The other bifurcation point with a double zero eigenvalue produces the other set of chaotic attractors and ordered states such as squares, asymmetric squares, oscillating asymmetric squares, relaxation oscillations with intermediate squares, some of which are common to the 13-mode model of Pal et al.(2009).