The European Physical Journal Special Topics

, Volume 226, Issue 10, pp 2247–2262 | Cite as

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

Regular Article
Part of the following topical collections:
  1. Aspects of Statistical Mechanics and Dynamical Complexity

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 → ∞.

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© EDP Sciences and Springer 2017

Authors and Affiliations

  1. 1.Dipartimento di Ingegneria Chimica DICMA Facoltà di Ingegneria, La Sapienza Università di Roma via Eudossiana 18RomaItaly
  2. 2.Materials Technology, Eindhoven University of TechnologyEindhovenThe Netherlands

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