We study the motion of non-diffusive, passive particles within steady, three-dimensional
vortex breakdown bubbles in a closed cylindrical container with a rotating
bottom. The velocity fields are obtained by solving numerically the three-dimensional
Navier–Stokes equations. We clarify the relationship between the manifold structure
of axisymmetric (ideal) vortex breakdown bubbles and those of the three-dimensional
real-life (laboratory) flow fields, which exhibit chaotic particle paths. We show that the
upstream and downstream fixed hyperbolic points in the former are transformed into
spiral-out and spiral-in saddles, respectively, in the latter. Material elements passing
repeatedly through the two saddle foci undergo intense stretching and folding, leading
to the growth of infinitely many Smale horseshoes and sensitive dependence on initial
conditions via the mechanism discovered by šil'nikov (1965). Chaotic šil'nikov orbits
spiral upward (from the spiral-in to the spiral-out saddle) around the axis and then
downward near the surface, wrapping around the toroidal region in the interior of the
bubble. Poincaré maps reveal that the dynamics of this region is rich and consistent
with what we would generically anticipate for a mildly perturbed, volume-preserving,
three-dimensional dynamical system (MacKay 1994; Mezić & Wiggins 1994a). Nested
KAM-tori, cantori, and periodic islands are found embedded within stochastic regions.
We calculate residence times of upstream-originating non-diffusive particles and
show that when mapped to initial release locations the resulting maps exhibit fractal
properties. We argue that there exists a Cantor set of initial conditions that leads
to arbitrarily long residence times within the breakdown region. We also show that
the emptying of the bubble does not take place in a continuous manner but rather
in a sequence of discrete bursting events during which clusters of particles exit the
bubble at once. A remarkable finding in this regard is that the rate at which an initial
population of particles exits the breakdown region is described by the devil's staircase
distribution, a fractal curve that has been already shown to describe a number of
other chaotic physical systems.