Chaotic dynamics in two-dimensional Rayleigh-B\'enard 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 $\Gamma = 2\sqrt{2}$. 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 $r \simeq 80$ and $r \simeq 500 $ respectively. The system becomes chaotic at $r \simeq 750$ through a quasiperiodic route to chaos. The size of the chaotic attractor increases at $r \simeq 840$ 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 \le r \le 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.