178 research outputs found
On stationarity of Lagrangian observations of passive tracer velocity in a compressible environment
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
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
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 and the Hurst
exponent , 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 (),
persistent fractional Brownian motions () 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
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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