Chaotic Mixing in a Torus Map

Abstract

The advection and diffusion of a passive scalar is investigated for a map of the 2-torus. The map is chaotic, and the limit of almost-uniform stretching is considered. This allows an analytic understanding of the transition from a phase of constant scalar variance (for short times) to exponential decay (for long times). This transition is embodied in a short superexponential phase of decay. The asymptotic state in the exponential phase is an eigenfunction of the advection-diffusion operator, in which most of the scalar variance is concentrated at small scales, even though a large-scale mode sets the decay rate. The duration of the superexponential phase is proportional to the logarithm of the exponential decay rate; if the decay is slow enough then there is no superexponential phase at all.

arXiv comment12 pages, 4 figures. RevTeX4 and psfrag macros. Final version

References in corpus (6)

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• The Decay of Passive Scalars Under the Action of Single Scale Smooth Velocity Fields in Bounded 2D Domains : From non self similar pdf's to self similar eigenmodes
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• Using multiscale norms to quantify mixing and transport
• Dissipation Enhancement by Mixing
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• Stirring up trouble: Multi-scale mixing measures for steady scalar sources
• Rotation shields chaotic mixing regions from no-slip walls
• Moving walls accelerate mixing
• Open-flow mixing: Experimental evidence for strange eigenmodes
• Diffusion of passive scalar in a finite-scale random flow
• Self-similar turbulent dynamo
• Scalar Decay in Chaotic Mixing
• The Strange Eigenmode in Lagrangian Coordinates
• Eigenfunctions of the Perron-Frobenius operator and the finite-time Lyapunov exponents in uniformly hyperbolic area-preserving maps
• Scalar decay in a three-dimensional chaotic flow
• Scarring in classical chaotic dynamics with noise
• A Computational Study of Mixing Microchannel Flows
• Spectral gap of sparse bistochastic matrices with exchangeable rows with application to shuffle-and-fold maps