178 research outputs found

    On stationarity of Lagrangian observations of passive tracer velocity in a compressible environment

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    We study the transport of a passive tracer particle in a steady strongly mixing flow with a nonzero mean velocity. We show that there exists a probability measure under which the particle Lagrangian velocity process is stationary. This measure is absolutely continuous with respect to the underlying probability measure for the Eulerian flow.Comment: Published at http://dx.doi.org/10.1214/105051604000000945 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

    Quantifying the dissipation enhancement of cellular flows

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    We study the dissipation enhancement by cellular flows. Previous work by Iyer, Xu, and Zlato\v{s} produces a family of cellular flows that can enhance dissipation by an arbitrarily large amount. We improve this result by providing quantitative bounds on the dissipation enhancement in terms of the flow amplitude, cell size and diffusivity. Explicitly we show that the mixing time is bounded by the exit time from one cell when the flow amplitude is large enough, and by the reciprocal of the effective diffusivity when the flow amplitude is small. This agrees with the optimal heuristics. We also prove a general result relating the dissipation time of incompressible flows to the mixing time. The main idea behind the proof is to study the dynamics probabilistically and construct a successful coupling.Comment: 21 pages, 2 figure

    Phase Diagram for Turbulent Transport: Sampling Drift, Eddy Diffusivity and Variational Principles

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    We study the long-time, large scale transport in a three-parameter family of isotropic, incompressible velocity fields with power-law spectra. Scaling law for transport is characterized by the scaling exponent qq and the Hurst exponent HH, as functions of the parameters. The parameter space is divided into regimes of scaling laws of different {\em functional forms} of the scaling exponent and the Hurst exponent. We present the full three-dimensional phase diagram. The limiting process is one of three kinds: Brownian motion (H=1/2H=1/2), persistent fractional Brownian motions (1/2<H<11/2<H<1) and regular (or smooth) motion (H=1). We discover that a critical wave number divides the infrared cutoffs into three categories, critical, subcritical and supercritical; they give rise to different scaling laws and phase diagrams. We introduce the notions of sampling drift and eddy diffusivity, and formulate variational principles to estimate the eddy diffusivity. We show that fractional Brownian motions result from a dominant sampling drift

    Quenched large deviations for diffusions in a random Gaussian shear flow drift

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    We prove a full large deviations principle in large time, for a diffusion process with random drift V, which is a centered Gaussian shear flow random field. The large deviations principle is established in a ``quenched'' setting, i.e. is valid almost surely in the randomness of V.Comment: 29 page
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