Chaotic advection in three-dimensional stationary vortex-breakdown bubbles: Shil'nikov's chaos and the devil's staircase (Q2755422)

The authors study the motion of non-diffusive passive particles within steady three-dimensional vortex breakdown bubbles in a closed cylindrical container with rotating bottom. The velocity fields are obtained by solving numerically three-dimensional Navier-Stokes equations. The authors clarify a relationship between the manifold structure of axisymmetric (ideal) vortex breakdown bubbles and those of three-dimensional real-life (laboratory) flow fields, which exhibit chaotic particle paths. It is shown that upstream and downstream fixed hyperbolic points in the former fluid fields can be transformed into spiral-out and spiral-in saddles, respectively, in the latter fluid fields. Material elements passing repeatedly through the two saddle foci undergo intense stretching and folding which leads to the growth of infinitely many Smale horseshoes and to sensitive dependence on initial conditions via the mechanism discovered by \textit{L. P. Shil'nikov} [Sov. Math., Dokl. 6, 163--166 (1965); translation from Dokl. Akad. Nauk SSSR, 160, 558--561 (1965; Zbl 0136.08202)]. Chaotic Shil'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 [\textit{R. S. MacKay}, J. Nonlinear Sci. 4, No. 4, 329--354 (1994; Zbl 0805.58037); \textit{I. Mezic} and \textit{S. Wiggins}, J. Nonlinear Sci. 4, No. 2, 157--194 (1994; Zbl 0796.76021)]. Nested KAM-tori, cantori, and periodic islands are found embedded within stochastic regions. The authors calculate the residence times of upstream-originating non-diffusive particles and show that, when mapped to initial release locations, the resulting maps exhibit fractal properties. The authors argue that there exists a Cantor set or initial conditions that leads to arbitrarily long residence times within the breakdown region. It is also shown 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.