Dynamical systems offers a geometric point of view in which to study Lagrangian particle dynamics. While the geometric theory of mixing has a solid foundation for two-dimensional flows with regular time-dependence, such techniques fall short in applications to aperiodic velocity fields. We discuss recent results that enable one to study mixing in finite-time numerical or exprimental velocity fields with general time dependence. We also discuss applications to oceanic and atmospheric problems, including mixing in a double-gyre ocean model near the arctic ozone hole.

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