  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).
