On Generalized Donating Regions: Classifying Lagrangian Fluxing Particles through a Fixed Curve in the Plane

For a fixed simple curve in a nonautonomous flow, the fluxing index of a passively-advected Lagrangian particle is the total number of times it goes across the curve within a given time interval. Such indices naturally induce flux sets, equivalence classes of the particles at the initial time. This line of research mainly concerns donating regions, the explicit construction of the flux sets from a sufficiently continuous velocity field. In the author’s previous paper [Q. Zhang, SIAM Review, 55(3), 2013], the flux sets with indices ±1 were constructed from characteristic curves of the flow field. This work generalizes the notion of donating regions and shows that flux sets are index-by-index equivalent to the generalized donating regions for any finite integer index, provided that the two backward streaklines seeded at the two endpoints of the simple curve neither intersect nor self-intersect. All Lagrangian particles marked by their initial positions are thus classified by their fluxing indices.