In this chapter, we will discuss barriers to purely advective transport in velocity fields that may have complex spatial features but a simple (recurrent) temporal structure: steady, periodic or quasiperiodic. Such velocity fields can be integrated for all times on bounded domains and hence their trajectories can be interrogated over infinite time intervals. While such exact recurrence is atypical in nature, mixing processes with precisely repeating stirring protocols are abundant in technological applications. Here, we survey classic results on temporally recurrentvelocity fields partly for motivation, partly for historical completeness and partly because their predictions in distinguished (recurrent) frames coincide with the predictions of Lagrangian coherent structure (LCS) methods to be discussed in the next chapter. For this reason, recurrent velocity fields are ideal benchmarks for LCS techniques because their transport barriers can be unambiguously identified. There are also a number of technological mixing processes in which the velocity field is engineered to be spatially recurrent, and hence the techniques discussed here apply directly to them.

Here, we take our first step to discover barriers to transport outside the idealized setting of temporally recurrent (steady, periodic or quasiperiodic) velocity fields. While we can no longer hope for even approximately recurring material surfaces in this general setting, we can certainly look for material surfaces that remain coherent. We perceive a material surface to be coherent if it preserves the spatial integrity without developing smaller scales. Those smaller scales would manifest themselves as protrusions from either side of the material surface without a break-up of that surface. In other words, using the terminology of the Introduction, we seek advective transport barriers in nonrecurrent flows as Lagrangian coherent structures (LCS). We will refer to this instantaneous limit of LCSs as objective Eulerian coherent structures (OECSs). These Eulerian structures act as LCSs over infinitesimally short time scales and hence their time-evolution is not material. Despite being nonmaterial, OECSs have advantages and important applications in unsteady flow analysis, as we will discuss separately.

While the transport of concentration fields arising in nature and technology is often predominantly advective, it invariably has at least a small diffusive component as well. The inclusion of diffusivity in transport studies increases their complexity significantly, as we will see.At the same time, introducing the diffusivity creates an opportunity to settle on a broadly agreeable definition for a transport barrier. Indeed, diffusive transport through a material surface is a uniquely defined, fundamental physical quantity, whose extremizing surfaces can be defined without reliance on any special notion of coherence. In the limit of zero diffusivity, the results we describe in this chapter also give a unique, physical definition of purely advective LCSs as material surfaces that will block transport most efficiently under the addition of the slightest diffusion or uncertainty to the velocity field.