Blinking rolls: Chaotic advection in a three-dimensional flow with an invariant

We study a simple, nonautonomous, three-dimensional, incompressible flow corresponding to sequentially active two-dimensional rolls with distinct axes. A feature of the model is that an analytical map is obtained. We show that when the roll axes are orthogonal, motion is confined to two-dimensional topological spheres. The dynamics on each surface ranges from nearly regular to largely chaotic. We study the transport and mixing on each surface and their dependence upon parameters in the system.