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 $\propto f^{-2.8}$, 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