On the dynamical origin of asymptotic It* dispersion of a nondiffusive tracer in incompressible laminar flows

where the velocity field is periodic in one or more of the variables (x,y,z) with the remaining variables (if any) bounded (this setting can be relaxed, as we will explain later). The study of nondiffusing passive scalars has gained more attention in the past few years, and this is probably related to the interest in the notion of chaotic advection and stirring of fluids, which has largely been’concerned with the situation of zero molecular diffusion. As examples of recent work along these lines, Jones and Young’ study dispersion in pipes due to chaotic advection and find t” dispersion. Pasmanter?’ and Ridderinkhof and Zimmerman3 study dispersion in models of shallow tidal flows and find t2 dispersion. Weiss and Knobloch perform a numerical study of dispersion in modulated traveling waves in binary fluid convection and find a dispersion exponent of 1.93. As their results were numerical, they were only able to compute for a IMe length of time. The result in this paper applies in each of these settings and gives conditions under which the dispersion should behave asymptotically (in time) like t2. As such, it may also prove to be’ a useful guide for numerical investigations. We denote the domain of the velocity field generally by A. We assume that the dependence of the veIocity field on time is either periodic or quasiperiodic (time independent velocity fields are also permitted). In light of the nature of the assumed time dependence we rewrite (1) as