We examine the spectral structure of the two-dimensional advection–diffusion operator in flows with mixed phase space at very large Péclet number. Using Fourier discretization combined with symmetry reduction and Krylov–Arnoldi methods, we compute on the order of one hundred leading eigenpairs reliably in the asymptotic, weak-diffusion regime. While the principal eigenvalue is asymptotically diffusive and localized on the largest regular region, the broader spectrum exhibits a rich organization controlled by local phase-space geometry. In particular, exponential mixing in chaotic regions rapidly suppresses correlations, whereas algebraic mixing in integrable regions generates long-lived coherent structures that dominate the slow and intermediate parts of the spectrum. We identify three distinct classes of eigenmodes: advective modes associated with transport on invariant tori, diffusive modes and, within the diffusive branch, tunneling modes arising from weak coupling between dynamically separated regular regions. Drawing on a semiclassical analogy, we assign quantum-number-like labels to these families and predict the appearance, scaling, and ordering of their sub-spectra directly from the Hamiltonian phase-space structure. The coexistence of these families implies that no uniform control of the spectral gap exists across the full spectrum: although the slowest mode is diffusive, arbitrarily small gaps arise between competing families at higher mode numbers. As a result, finite-time advection–diffusion dynamics is generically governed by persistent modal competition rather than single-mode dominance, even at asymptotically large Péclet number.
Amey et al. (Mon,) studied this question.