47,066 research outputs found
Modulation instability and capillary wave turbulence
Formation of turbulence of capillary waves is studied in laboratory
experiments. The spectra show multiple exponentially decreasing harmonics of
the parametrically excited wave which nonlinearly broaden with the increase in
forcing. Spectral broadening leads to the development of the spectral continuum
which scales as , in agreement with the weak turbulence
theory (WTT) prediction. Modulation instability of capillary waves is shown to
be responsible for the transition from discrete to broadband spectrum. The
instability leads to spectral broadening of the harmonics, randomization of
their phases, it isolates the wave field from the wall, eventually allows the
transition from 4- to 3-wave interactions as the dominant nonlinear process,
thus creating the prerequisites assumed in WTT.Comment: 6 pages, 5 figure
Capillary rogue waves
We report the first observation of extreme wave events (rogue waves) in
parametrically driven capillary waves. Rogue waves are observed above a certain
threshold in forcing. Above this threshold, frequency spectra broaden and
develop exponential tails. For the first time we present evidence of strong
four-wave coupling in non-linear waves (high tricoherence), which points to
modulation instability as the main mechanism in rogue waves. The generation of
rogue waves is identified as the onset of a distinct tail in the probability
density function of the wave heights. Their probability is higher than expected
from the measured wave background.Comment: 4 pages, 5 figure
Poisson process approximation: From Palm theory to Stein's method
This exposition explains the basic ideas of Stein's method for Poisson random
variable approximation and Poisson process approximation from the point of view
of the immigration-death process and Palm theory. The latter approach also
enables us to define local dependence of point processes [Chen and Xia (2004)]
and use it to study Poisson process approximation for locally dependent point
processes and for dependent superposition of point processes.Comment: Published at http://dx.doi.org/10.1214/074921706000001076 in the IMS
Lecture Notes Monograph Series
(http://www.imstat.org/publications/lecnotes.htm) by the Institute of
Mathematical Statistics (http://www.imstat.org
Turbulence damping as a measure of the flow dimensionality
The dimensionality of turbulence in fluid layers determines their properties.
We study electromagnetically driven flows in finite depth fluid layers and show
that eddy viscosity, which appears as a result of three-dimensional motions,
leads to increased bottom damping. The anomaly coefficient, which characterizes
the deviation of damping from the one derived using a quasi-two-dimensional
model, can be used as a measure of the flow dimensionality. Experiments in
turbulent layers show that when the anomaly coefficient becomes high, the
turbulent inverse energy cascade is suppressed. In the opposite limit
turbulence can self-organize into a coherent flow.Comment: 4 pages, 4 figure
Stein's method, Palm theory and Poisson process approximation
The framework of Stein's method for Poisson process approximation is
presented from the point of view of Palm theory, which is used to construct
Stein identities and define local dependence. A general result (Theorem
\refimportantproposition) in Poisson process approximation is proved by taking
the local approach.
It is obtained without reference to any particular metric, thereby allowing
wider applicability. A Wasserstein pseudometric is introduced for measuring the
accuracy of point process approximation. The pseudometric provides a
generalization of many metrics used so far, including the total variation
distance for random variables and the Wasserstein metric for processes as in
Barbour and Brown [Stochastic Process. Appl. 43 (1992) 9-31]. Also, through the
pseudometric, approximation for certain point processes on a given carrier
space is carried out by lifting it to one on a larger space, extending an idea
of Arratia, Goldstein and Gordon [Statist. Sci. 5 (1990)
403-434]. The error bound in the general result is similar in form to that
for Poisson approximation. As it yields the Stein factor 1/\lambda as in
Poisson approximation, it provides good approximation, particularly in cases
where \lambda is large. The general result is applied to a number of problems
including Poisson process modeling of rare words in a DNA sequence.Comment: Published by the Institute of Mathematical Statistics
(http://www.imstat.org) in the Annals of Probability
(http://www.imstat.org/aop/) at http://dx.doi.org/10.1214/00911790400000002
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