WHOI PO

Mixing in near-integrable flows with a clear separation of time or spacial scales is notoriously difficult to achieve. In many flows the intrinsic symmetries create invariant surfaces that act as barriers to chaotic advection and mixing. Thus, a key to efficient mixing is to add to the original (symmetric) flow a certain kind of perturbation that destroys those symmetries. In the present talk we discuss a quantitative long-time theory of mixing due to the presence of resonances in 3-D near-integrable Stokes flows. The resonance phenomena, such as scattering on resonance, capture into resonance, and separatrix crossings, involving different components of the original flow and the perturbation may destroy the invariant surfaces, paving a way to the large-scale mixing in a big fraction of the fluid flow. We explain the extend and the rate of mixing in terms of the evolution of the adiabatic invariants of the system. We show that when the leading phenomenon is scattering on resonances or separatrix crossings, the resulting mixing can be described in terms of a single 1-D diffusion-type equation, with parameters of the diffusion equation defined by the averaged statistics of a single passage through resonance or separatrix.