Passive scalars, three-dimensional volume-preserving maps, and chaos

The dynamics of a medium-sized particle (passive scalar) suspended in a general time-periodic incompressible fluid flow can be described by three-dimensional volume-preserving maps. In this paper, these maps are studied in limiting cases in which some of the variables change very little in each iteration and others change quite a lot. The former are called slow variables or actions, the latter fast variables or angles. The maps are classified by their number of actions. For maps with only one action we find strong evidence for the existence of invariant surfaces that survive the nonlinear perturbation in a KAM-like way. On the other hand, for the two-action case the motion is confined to invariant lines that break for arbitrary small size of the nonlinearity. Instead, we find that adiabatic invariant surfaces emerge and typically intersect the resonance sheet of the fast motion. At these intersections surfaces are locally broken and transitions from one to another can occur. We call this process, which is analogous to Arnold diffusion, singularity-induced diffusion. It is characteristic of two-action maps. In one-action maps, this diffusion is blocked by KAM-like surfaces.