  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.
