Chaotic advection in a Rayleigh-Bénard flow

We consider the problem of transport of a passive tracer in the time-dependent flow corresponding to a Rayleigh number scrR slightly above the ${\mathit{scrR}}_{\mathit{t}}$ at the onset of the even oscillatory instability for Rayleigh-Bénard convection rolls. By modeling the flow with a stream function, we show how to construct and identify invariant structures in the flow that act as a ‘‘template’’ for the motion of fluid particles, in the absence of molecular diffusivity. This approach and symmetry considerations allow us to write explicit formulas that describe the tracer transport for finite times. In the limit of small amplitude of the oscillation, i.e., when (scrR-${\mathit{scrR}}_{\mathit{t}}$ ${)}^{1/2}$ is small, we show that the amount of fluid transported across a roll boundary grows linearly with the amplitude, in agreement with the experimental and numerical findings of Solomon and Gollub [Phys. Rev. A 38, 6280 (1988)]. The presence of molecular diffusivity introduces a (long) time scale into the problem. We discuss the applicability of the theory in this situation, by introducing a simple rule for determining when the effects of diffusivity are negligible, and perform numerical simulations of the flow in this case to provide an example.