  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.
