74 research outputs found
Pattern formation in Hamiltonian systems with continuous spectra; a normal-form single-wave model
Pattern formation in biological, chemical and physical problems has received
considerable attention, with much attention paid to dissipative systems. For
example, the Ginzburg--Landau equation is a normal form that describes pattern
formation due to the appearance of a single mode of instability in a wide
variety of dissipative problems. In a similar vein, a certain "single-wave
model" arises in many physical contexts that share common pattern forming
behavior. These systems have Hamiltonian structure, and the single-wave model
is a kind of Hamiltonian mean-field theory describing the patterns that form in
phase space. The single-wave model was originally derived in the context of
nonlinear plasma theory, where it describes the behavior near threshold and
subsequent nonlinear evolution of unstable plasma waves. However, the
single-wave model also arises in fluid mechanics, specifically shear-flow and
vortex dynamics, galactic dynamics, the XY and Potts models of condensed matter
physics, and other Hamiltonian theories characterized by mean field
interaction. We demonstrate, by a suitable asymptotic analysis, how the
single-wave model emerges from a large class of nonlinear advection-transport
theories. An essential ingredient for the reduction is that the Hamiltonian
system has a continuous spectrum in the linear stability problem, arising not
from an infinite spatial domain but from singular resonances along curves in
phase space whereat wavespeeds match material speeds (wave-particle resonances
in the plasma problem, or critical levels in fluid problems). The dynamics of
the continuous spectrum is manifest as the phenomenon of Landau damping when
the system is ... Such dynamical phenomena have been rediscovered in different
contexts, which is unsurprising in view of the normal-form character of the
single-wave model
Braids of entangled particle trajectories
In many applications, the two-dimensional trajectories of fluid particles are
available, but little is known about the underlying flow. Oceanic floats are a
clear example. To extract quantitative information from such data, one can
measure single-particle dispersion coefficients, but this only uses one
trajectory at a time, so much of the information on relative motion is lost. In
some circumstances the trajectories happen to remain close long enough to
measure finite-time Lyapunov exponents, but this is rare. We propose to use
tools from braid theory and the topology of surface mappings to approximate the
topological entropy of the underlying flow. The procedure uses all the
trajectory data and is inherently global. The topological entropy is a measure
of the entanglement of the trajectories, and converges to zero if they are not
entangled in a complex manner (for instance, if the trajectories are all in a
large vortex). We illustrate the techniques on some simple dynamical systems
and on float data from the Labrador sea.Comment: 24 pages, 21 figures. PDFLaTeX with RevTeX4 macros. Matlab code
included with source. Fixed an inconsistent convention problem. Final versio
Rotation shields chaotic mixing regions from no-slip walls
We report on the decay of a passive scalar in chaotic mixing protocols where
the wall of the vessel is rotated, or a net drift of fluid elements near the
wall is induced at each period. As a result the fluid domain is divided into a
central isolated chaotic region and a peripheral regular region. Scalar
patterns obtained in experiments and simulations converge to a strange
eigenmode and follow an exponential decay. This contrasts with previous
experiments [Gouillart et al., Phys. Rev. Lett. 99, 114501 (2007)] with a
chaotic region spanning the whole domain, where fixed walls constrained mixing
to follow a slower algebraic decay. Using a linear analysis of the flow close
to the wall, as well as numerical simulations of Lagrangian trajectories, we
study the influence of the rotation velocity of the wall on the size of the
chaotic region, the approach to its bounding separatrix, and the decay rate of
the scalar.Comment: 4 pages, 12 figures, RevTeX 4 styl
Slow decay of concentration variance due to no-slip walls in chaotic mixing
Chaotic mixing in a closed vessel is studied experimentally and numerically
in different 2-D flow configurations. For a purely hyperbolic phase space, it
is well-known that concentration fluctuations converge to an eigenmode of the
advection-diffusion operator and decay exponentially with time. We illustrate
how the unstable manifold of hyperbolic periodic points dominates the resulting
persistent pattern. We show for different physical viscous flows that, in the
case of a fully chaotic Poincare section, parabolic periodic points at the
walls lead to slower (algebraic) decay. A persistent pattern, the backbone of
which is the unstable manifold of parabolic points, can be observed. However,
slow stretching at the wall forbids the rapid propagation of stretched
filaments throughout the whole domain, and hence delays the formation of an
eigenmode until it is no longer experimentally observable. Inspired by the
baker's map, we introduce a 1-D model with a parabolic point that gives a good
account of the slow decay observed in experiments. We derive a universal decay
law for such systems parametrized by the rate at which a particle approaches
the no-slip wall.Comment: 17 pages, 12 figure
Moving walls accelerate mixing
Mixing in viscous fluids is challenging, but chaotic advection in principle
allows efficient mixing. In the best possible scenario,the decay rate of the
concentration profile of a passive scalar should be exponential in time. In
practice, several authors have found that the no-slip boundary condition at the
walls of a vessel can slow down mixing considerably, turning an exponential
decay into a power law. This slowdown affects the whole mixing region, and not
just the vicinity of the wall. The reason is that when the chaotic mixing
region extends to the wall, a separatrix connects to it. The approach to the
wall along that separatrix is polynomial in time and dominates the long-time
decay. However, if the walls are moved or rotated, closed orbits appear,
separated from the central mixing region by a hyperbolic fixed point with a
homoclinic orbit. The long-time approach to the fixed point is exponential, so
an overall exponential decay is recovered, albeit with a thin unmixed region
near the wall.Comment: 17 pages, 13 figures. PDFLaTeX with RevTeX 4-1 styl
2010 program of study : swirling and swimming in turbulence
Swirling and Swimming in Turbulence was the theme at the 2010 GFD Program. Professors
Glenn Flierl (M.I.T.), Antonello Provenzale (ISAC-CNR, Turin) and Jean-Luc Thiffeault
(University of Wisconsin) were the principal lecturers. Together they navigated an elegant
path through topics ranging from mixing protocols and efficiencies to ecological strategies,
schooling and genetic development. The first ten chapters of this volume document these
lectures, each prepared by pairs of this summer’s GFD fellows. Following on are the written
reports of the fellows’ own research projects.Funding was provided by the Office of Naval Research under Contract No. N000-14-09-10844 and the National Science Foundation through Grant No. OCE 082463
Topological entropy and secondary folding
A convenient measure of a map or flow's chaotic action is the topological
entropy. In many cases, the entropy has a homological origin: it is forced by
the topology of the space. For example, in simple toral maps, the topological
entropy is exactly equal to the growth induced by the map on the fundamental
group of the torus. However, in many situations the numerically-computed
topological entropy is greater than the bound implied by this action. We
associate this gap between the bound and the true entropy with 'secondary
folding': material lines undergo folding which is not homologically forced. We
examine this phenomenon both for physical rod-stirring devices and toral linked
twist maps, and show rigorously that for the latter secondary folds occur.Comment: 13 pages, 8 figures. pdfLaTeX with RevTeX4 macro
Topological Entropy of Braids on the Torus
A fast method is presented for computing the topological entropy of braids on
the torus. This work is motivated by the need to analyze large braids when
studying two-dimensional flows via the braiding of a large number of particle
trajectories. Our approach is a generalization of Moussafir's technique for
braids on the sphere. Previous methods for computing topological entropies
include the Bestvina--Handel train-track algorithm and matrix representations
of the braid group. However, the Bestvina--Handel algorithm quickly becomes
computationally intractable for large braid words, and matrix methods give only
lower bounds, which are often poor for large braids. Our method is
computationally fast and appears to give exponential convergence towards the
exact entropy. As an illustration we apply our approach to the braiding of both
periodic and aperiodic trajectories in the sine flow. The efficiency of the
method allows us to explore how much extra information about flow entropy is
encoded in the braid as the number of trajectories becomes large.Comment: 19 pages, 44 figures. SIAM journal styl
Walls Inhibit Chaotic Mixing
We report on experiments of chaotic mixing in a closed vessel, in which a
highly viscous fluid is stirred by a moving rod. We analyze quantitatively how
the concentration field of a low-diffusivity dye relaxes towards homogeneity,
and we observe a slow algebraic decay of the inhomogeneity, at odds with the
exponential decay predicted by most previous studies. Visual observations
reveal the dominant role of the vessel wall, which strongly influences the
concentration field in the entire domain and causes the anomalous scaling. A
simplified 1D model supports our experimental results. Quantitative analysis of
the concentration pattern leads to scalings for the distributions and the
variance of the concentration field consistent with experimental and numerical
results.Comment: 4 pages, 3 figure
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