Hyperbolic and elliptic transport barriers in three-dimensional unsteady flows

We develop a general theory of transport barriers for three-dimensional unsteady flows with arbitrary time-dependence. The barriers are obtained as two-dimensional Lagrangian Coherent Structures (LCSs) that create locally maximal deformation. Along hyperbolic LCSs, this deformation is induced by locally maximal normal repulsion or attraction. Along shear LCSs, the deformation is created by locally maximal tangential shear. Hyperbolic LCSs, therefore, play the role of generalized stable and unstable manifolds, while closed shear LCSs (elliptic LCSs) act as generalized KAM tori or KAM-type cylinders. All these barriers can be computed from our theory as explicitly parametrized surfaces. We illustrate our results by visualizing two-dimensional hyperbolic and elliptic barriers in steady and unsteady versions of the ABC flow.