Singular eigenvalue limit of advection-diffusion operators and properties of the strange eigenfunctions in globally chaotic flows

Abstract

Enforcing the results developed by Gorodetskyi et al. [O. Gorodetskyi, M. Giona, P. Anderson, Phys. Fluids 24, 073603 (2012)] on the application of the mapping matrix formalism to simulate advective-diffusive transport, we investigate the structure and the properties of strange eigenfunctions and of the associated eigenvalues up to values of the Péclet number Pe ~ 𝒪(10⁸). Attention is focused on the possible occurrence of a singular limit for the second eigenvalue, ν₂, of the advection-diffusion propagator as the Péclet number, Pe, tends to infinity, and on the structure of the corresponding eigenfunction. Prototypical time-periodic flows on the two-torus are considered, which give rise to toral twist maps with different hyperbolic character, encompassing Anosov, pseudo-Anosov, and smooth nonuniformly hyperbolic systems possessing a hyperbolic set of full measure. We show that for uniformly hyperbolic systems, a singular limit of the dominant decay exponent occurs, log|ν₂| → constant≠0 for Pe → ∞, whereas log |ν₂| → 0 according to a power-law in smooth non-uniformly hyperbolic systems that are not uniformly hyperbolic. The mere presence of a nonempty set of nonhyperbolic points (even if of zero Lebesgue measure) is thus found to mark the watershed between regular vs. singular behavior of ν₂ with Pe as Pe → ∞.