Averaging and spectral properties for the 2D advection-diffusion equation in the semi-classical limit for vanishing diffusivity

We consider the two-dimensional advection-diffusion equation on a bounded domain subject to either Dirichlet or von Neumann boundary conditions and study both time-independent and time-periodic cases involving Liouville integrable Hamiltonians that satisfy conditions conducive to applying the averaging principle. Transformation to action-angle coordinates permits averaging in time and angle, leading to an underlying eigenvalue equation that allows for separation of the angle and action coordinates. The result is a one-dimensional second-order equation involving an anti-symmetric imaginary potential. For radial flows on a disk or an annulus, we rigorously apply existing complex-plane WKBJ methods to study the spectral properties in the semi-classical limit for vanishing diffusivity. In this limit, the spectrum is found to be a complicated set consisting of lines related to Stokes graphs. Eigenvalues in the neighborhood of these graphs exhibit nonlinear scaling with respect to diffusivity leading to convection-enhanced rates of dissipation (relaxation, mixing) for initial data which are mean-free in the angle coordinate. These branches coexist with a diffusive branch of eigenvalues that scale linearly with diffusivity and contain the principal eigenvalue (no dissipation enhancement).