Chapter 9 Almost-Invariant and Finite-Time Coherent Sets: Directionality, Duration, and Diffusion

Regions in the phase space of a dynamical system that resist mixing over a finite-time duration are known as almost-invariant sets (for autonomous dynamics) or coherent sets (for nonautonomous or time-dependent dynamics). These regions provide valuable information for transport and mixing processes; almost-invariant sets mitigate transport between their interior and the rest of phase space, and coherent sets are good transporters of 'mass' precisely because they move about with minimal dispersion (e.g. oceanic eddies are good transporters of water that is warmer/cooler/saltier than the surrounding water). The most efficient approach to date for the identification of almost-invariant and coherent sets is via transfer operators. In this chapter we describe a unified setting for optimal almost-invariant and coherent set constructions and introduce a new coherent set construction that is suited to tracking coherent sets over several finite-time intervals. Under this unified treatment we are able to clearly explain the fundamental differences in the aims of the techniques and describe the differences and similarities in the mathematical and numerical constructions. We explore the role of diffusion, the influence of the finite-time duration, and discuss the relationship of time directionality with hyperbolic dynamics. All of these issues are elucidated in detailed case studies of two well-known systems.