Finite-Time Transport in Aperiodic Dynamical Systems
In this dissertation, we consider the problem of quantifying finite-time transport in aperiodic dynamical systems. To address this problem, we concentrate primarily on a class of aperiodic systems that we call "transitory." These systems exhibit time-dependent behavior only on a compact time interval, and we present a new method for quantifying transport between isolated coherent structures of such systems, in the globally Liouville case. Both 2D and 3D examples are given. Moreover, our treatment of the 3D case represents the first quantitative analysis of transport between Lagrangian coherent structures in fully 3D aperiodic flows. We also present a numerical method that facilitates the application of our transport formulas to systems defined by discrete velocity data.
In each case we consider, transport is quantified by computing the areas (or volumes in dimensions greater than two) of lobes bounded by codimension-one objects that are past or future invariant, and our method is Lagrangian, in the sense that it relies only on knowing certain key trajectories. These trajectories form a codimension-two set at the intersections of lobe boundary components. Thus, our transport computations require little Lagrangian information relative to various other methods involving finite-time Lyapunov exponents (FTLEs) or distinguished hyperbolic trajectories. To show this, we compare our method to one that identifies coherent structures as regions bounded by ridges of the FTLE field, and additionally present a new computational method for efficiently extracting such ridges.