Chaotic dynamics in two-dimensional Rayleigh-Bénard convection

Abstract

We investigate the origin of various convective patterns using bifurcation diagrams that are constructed using direct numerical simulations. We perform two-dimensional pseudospectral simulations for a Prandtl number 6.8 fluid that is confined in a box with aspect ratio Γ=22–√. Steady convective rolls are born from the conduction state through a pitchfork bifurcation at r=1, where r is the reduced Rayleigh number. These fixed points bifurcate successively to time-periodic and quasiperiodic rolls through Hopf and Neimark-Sacker bifurcations at r≃80 and r≃500 respectively. The system becomes chaotic at r≃750 through a quasiperiodic route to chaos. The size of the chaotic attractor increases at r≃840 through an "attractor-merging crisis" which also results in travelling chaotic rolls. We also observe coexistence of stable fixed points and a chaotic attractor for 846≤r≤849 as a result of a subcritical Hopf bifurcation. Subsequently the chaotic attractor disappears through a "boundary crisis" and only stable fixed points remain. Later these fixed points become periodic and chaotic through another set of bifurcations which ultimately leads to turbulence.