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

• Derivation of an averaged equation with anti-symmetric imaginary potential.
• Characterization of sublinear scaling of eigenvalues with respect to diffusivity.
• Demonstration of validity of averaging for spectra.

We consider the two-dimensional advection–diffusion equation (ADE) 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 problem, we apply complex-plane WKB methods to study the spectrum in the semi-classical limit for 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 emanating from the imaginary axis are subject to various sublinear power laws with respect to diffusivity, leading to convection-enhanced rates of dissipation of the corresponding modes. The solution of the ADE converges in the limit of vanishing diffusivity to the solution of the effective diffusion equation on convective time scales that are sublinear with respect to the diffusive time scales.