Lagrangian transport through surfaces in volume-preserving flows

Advective transport of scalar quantities through surfaces is of fundamental importance in many scientific applications. From the Eulerian perspective of the surface it can be quantified by the well-known integral of the flux density. The recent development of highly accurate semi-Lagrangian methods for solving scalar conservation laws and of Lagrangian approaches to coherent structures in turbulent (geophysical) fluid flows necessitate a new approach to transport from the (Lagrangian) material perspective. We present a Lagrangian framework for calculating transport of conserved quantities through a given surface in $n$-dimensional, fully aperiodic, volume-preserving flows. Our approach does not involve any dynamical assumptions on the surface or its boundary.