Universality and imaginary potentials in advection–diffusion equations in closed flows

Abstract

This article addresses the scaling and spectral properties of the advection–diffusion equation in closed two-dimensional steady flows. We show that homogenization dynamics in simple model flows is equivalent to a Schrödinger eigenvalue problem in the presence of an imaginary potential. Several properties follow from this formulation: spectral invariance, eigenfunction localization, and a universal scaling of the dominant eigenvalue with respect to the Péclet number $\mbox{\it Pe}$. The latter property means that, in the high-$\mbox{\it Pe}$ range (in practice $\mbox{\it Pe},\geq 10^2 \hbox{--} 10^3$), the scaling exponent controlling the behaviour of the dominant eigenvalue with the Péclet number depends on the local behaviour of the potential near the critical points (local maxima/minima). A kinematic interpretation of this result is also addressed.