A Lagrangian perspective on nonautonomous advection-diffusion processes in the low-diffusivity limit

Abstract

We study mass preserving transport of passive tracers in the low-diffusivity limit using Lagrangian coordinates. Over finite-time intervals, the solution-operator of the nonautonomous diffusion equation is approximated by that of a time-averaged diffusion equation. We show that leading order asymptotics that hold for functions [Krol, 1991] extend to the dominant nontrivial singular value. This answers questions raised in [Karrasch & Keller, 2020]. The generator of the time-averaged diffusion/heat semigroup is a Laplace operator associated to a weighted manifold structure on the material manifold. We show how geometrical properties of this weighted manifold directly lead to physical transport quantities of the nonautonomous equation in the low-diffusivity limit.