Chaotic Advection in a Bounded 3-Dimensional Potential Flow

3-dimensional potential, or Darcy flows, are central to understanding and designing laminar transport in porous media; however, chaotic advection in 3-dimensional, volume-preserving flows is still not well understood.\footnote{Wiggins, J. Fluid Mech. {\bf 654} (2010). } We show results of advecting passive scalars in a transient 3-dimensional potential flow that consists of a steady dipole flow and periodic reorientation. Even for the most symmetric reorientation protocol, neither of the two invarients of the motion are conserved; however, one invarient is closely shadowed by a surface of revolution constructed from particle paths of the steady flow, creating in practice an adiabatic surface. A consequence is that chaotic regions cover 3-dimensional space, though tubular regular regions are still transport barriers. This appears to be a new mechanism generating 3-dimensional chaotic orbits. These results contast with the experimental and theoretical results for chaotic scalar transport in 2-dimensional Darcy flows.\footnote{Metcalfe et al, Phil. Trans. R. Soc. {\bf A368} (2010a,b).}$^,$\footnote{Lester et al, Phys. Rev. {\bf E80} (2009), {\bf E81} (2010).}