Long-Time Transport and Mixing in Cellular Flow Due to Resonant Processes and Chaotic Advection

Abstract

We present a quantitative theory of resonance-induced mixing in near-integrable, volume-preserving, 3-D, non-autonomous flows. As an example of such a flow we use a simplified model proposed by Solomon and Mezic (2003). When the nominal autonomous flow is subject to time-periodic perturbation, mixing occurs due to resonance processes. We discuss two different resonance phenomena. We show that scattering on resonance is the primary reason for long-time mixing. It is the accumulation of “jumps” in the value of adiabatic invariant that occur during the process of crossing resonance that ultimately leads to chaotic advection and mixing. For multiple crossings, the value of these “jumps” can be considered a random variable. We show that the second moment-of-distribution of the adiabatic invariants over a long time, for a large number of tracers starting from similar initial conditions, follows a linear trend; hence the process can be described by a one-dimensional diffusion equation in the space of adiabatic invariants. Volume fraction of the mixing, as well as the rate of mixing are computed as functions of frequency of perturbation. We describe the transport properties using the evolution of the probability distribution function in the space of adiabatic invariants.