We consider the two-dimensional advection-diffusion equation on a bounded domain subject to Dirichlet or von Neumann boundary conditions involving a Liouville integrable Hamiltonian. Transformation to action-angle coordinates permits averaging in time and angle, resulting in an equation that allows for separation of variables. The Fourier transform in the angle coordinate transforms the equation into an effective diffusive equation and a countable family of non-self-adjoint Schrödinger equations. For the corresponding Liouville-Sturm problems, complex-plane WKB methods were applied to study the spectrum in the semi-classical limit of vanishing diffusivity. The spectral limit graph is found to consist of analytic curves (branches) related to Stokes graphs forming a tree-structure. Eigenvalues in the neighborhood of branches are subject to various sublinear power laws with respect to diffusivity that can be inferred from the behavior of the frequency function near the boundary and critical points. A consequence is convection-enhanced rates of dissipation of the corresponding modes. The solution of ADE converges in the limit of vanishing diffusivity to the solution of an effective diffusion equation on convective timescales that are sublinear with respect to the diffusive ones.