4 - Barriers to Chaotic Advection

Summary

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.