Chaotic traveling rolls in Rayleigh-B\'enard convection

In this paper we investigate two-dimensional (2D) Rayleigh-B\'enard convection using direct numerical simulation in Boussinesq fluids of Prandtl number $P = 6.8$ confined between thermally conducting plates. We show through the simulation that in a small range of reduced Rayleigh number $r$ ($770 < r < 890$) the 2D rolls move chaotically in a direction normal to the roll axis. The lateral shift of the rolls may lead to global flow reversal of the convective motion. The chaotic traveling rolls are observed in simulation with {\it free-slip} as well as {\it no-slip} boundary conditions on the velocity field. We show that the traveling rolls and the flow reversal are due to an interplay between the real and imaginary parts of the critical modes.