Linear vs nonlinear transport during chaotic advection in fluid flows

Abstract

The goal of this study is explicit demarcation of the region of validity of a linear canonical representation for chaotic advection of Lagrangian fluid parcels in “chaotic seas” in two-dimensional (2D) and three-dimensional (3D) time-periodic fluid flows governed by Hamiltonian mechanics. The concept of lobe dynamics admits exact and unique geometric demarcation of this region and, inherently, distinction of the portions of chaotic seas with essentially linear vs nonlinear Lagrangian transport. This, furthermore, admits explicit establishment of a topological equivalence between the (embedded) Hamiltonian structure of the Lagrangian dynamics in 2D (3D) flows and their canonical form. The linear transport region in physical space encompasses four adjacent subregions that each corresponds to one of the four quadrants in canonical space and may exchange material with their environment in two essentially nonlinear ways. First, exchange between quadrants within the linear transport region and, second, exchange with the exterior of this region. Both forms of exchange can be linked to specific subsets of material elements defined by interacting lobes and combined give rise to circulation through the quadrants of the linear transport region that systematically exchanges the material with the exterior.