389 research outputs found
Do Finite-Size Lyapunov Exponents Detect Coherent Structures?
Ridges of the Finite-Size Lyapunov Exponent (FSLE) field have been used as
indicators of hyperbolic Lagrangian Coherent Structures (LCSs). A rigorous
mathematical link between the FSLE and LCSs, however, has been missing. Here we
prove that an FSLE ridge satisfying certain conditions does signal a nearby
ridge of some Finite-Time Lyapunov Exponent (FTLE) field, which in turn
indicates a hyperbolic LCS under further conditions. Other FSLE ridges
violating our conditions, however, are seen to be false positives for LCSs. We
also find further limitations of the FSLE in Lagrangian coherence detection,
including ill-posedness, artificial jump-discontinuities, and sensitivity with
respect to the computational time step.Comment: 22 pages, 7 figures, v3: corrects the z-axis labels of Fig. 2 (left)
that appears in the version published in Chao
Spatial Patterns in Chemically and Biologically Reacting Flows
We present here a number of processes, inspired by concepts in Nonlinear
Dynamics such as chaotic advection and excitability, that can be useful to
understand generic behaviors in chemical or biological systems in fluid flows.
Emphasis is put on the description of observed plankton patchiness in the sea.
The linearly decaying tracer, and excitable kinetics in a chaotic flow are
mainly the models described. Finally, some warnings are given about the
difficulties in modeling discrete individuals (such as planktonic organisms) in
terms of continuous concentration fields.Comment: 41 pages, 10 figures; To appear in the Proceedings of the 2001 ISSAOS
School on 'Chaos in Geophysical Flows
Lagrangian coherent structures in n-dimensional systems
Numerical simulations and experimental observations reveal that unsteady fluid systems can be divided into regions of qualitatively different dynamics. The key to understanding transport and stirring is to identify the dynamic boundaries between these almost-invariant regions. Recently, ridges in finite-time Lyapunov exponent fields have been used to define such hyperbolic, almost material, Lagrangian coherent structures in two-dimensional systems. The objective of this paper is to develop and apply a similar theory in higher dimensional spaces. While the separatrix nature of these structures is their most important property, a necessary condition is their almost material nature. This property is addressed in this paper. These results are applied to a model of Rayleigh-Bénard convection based on a three-dimensional extension of the model of Solomon and Gollub
Reduced-order Description of Transient Instabilities and Computation of Finite-Time Lyapunov Exponents
High-dimensional chaotic dynamical systems can exhibit strongly transient
features. These are often associated with instabilities that have finite-time
duration. Because of the finite-time character of these transient events, their
detection through infinite-time methods, e.g. long term averages, Lyapunov
exponents or information about the statistical steady-state, is not possible.
Here we utilize a recently developed framework, the Optimally Time-Dependent
(OTD) modes, to extract a time-dependent subspace that spans the modes
associated with transient features associated with finite-time instabilities.
As the main result, we prove that the OTD modes, under appropriate conditions,
converge exponentially fast to the eigendirections of the Cauchy--Green tensor
associated with the most intense finite-time instabilities. Based on this
observation, we develop a reduced-order method for the computation of
finite-time Lyapunov exponents (FTLE) and vectors. In high-dimensional systems,
the computational cost of the reduced-order method is orders of magnitude lower
than the full FTLE computation. We demonstrate the validity of the theoretical
findings on two numerical examples
Evidence for a k^{-5/3} spectrum from the EOLE Lagrangian balloons in the low stratosphere
The EOLE Experiment is revisited to study turbulent processes in the lower
stratosphere circulation from a Lagrangian viewpoint and resolve a discrepancy
on the slope of the atmospheric energy spectrum between the work of Morel and
Larcheveque (1974) and recent studies using aircraft data. Relative dispersion
of balloon pairs is studied by calculating the Finite Scale Lyapunov Exponent,
an exit time-based technique which is particularly efficient in cases where
processes with different spatial scales are interfering. Our main result is to
reconciliate the EOLE dataset with recent studies supporting a k^{-5/3} energy
spectrum in the range 100-1000 km. Our results also show exponential separation
at smaller scale, with characteristic time of order 1 day, and agree with the
standard diffusion of about 10^7 m^2/s at large scales. A still open question
is the origin of a k^{-5/3} spectrum in the mesoscale range, between 100 and
1000 km.Comment: 19 pages, 1 table + 5 (pdf) figure
Chaotic advection of reacting substances: Plankton dynamics on a meandering jet
We study the spatial patterns formed by interacting populations or reacting
chemicals under the influence of chaotic flows. In particular, we have
considered a three-component model of plankton dynamics advected by a
meandering jet. We report general results, stressing the existence of a
smooth-filamental transition in the concentration patterns depending on the
relative strength of the stirring by the chaotic flow and the relaxation
properties of planktonic dynamical system. Patterns obtained in open and closed
flows are compared.Comment: 5 pages, 3 figues, latex compiled with modegs.cl
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