Statistics of caustics in an underwater sound channel

Gribova, E. Z.

doi:10.1134/s1063771007060115

“…In such systems, the consecutive effect of weak scattering events that are correlated in space leads to strong focusing in branch-like patterns, connected to the formation of random caustics, which are regions where the flow is focused. Examples of this phenomenon include the random focusing of electrons in semiconductors [8,[21][22][23][24] and microwaves propagating in cavities with weak scatterers [9], as shown in Fig.1.2, and it is considered to explain the twinkling of starlight due to its propagation through the slightly inhomogeneous atmosphere [25][26][27][28][29], as well as the focusing of sound waves in the ocean due to water density variations [30][31][32][33][34][35][36]. All those phenomena can be modeled using Schrödinger-type equations, which can then be studied using Hamiltonian rays.…”

Section: 0mentioning

confidence: 99%

“…(1) i and the diffusion coefficients D (2) ij are generally given by [55] 36) where once again the summation convention applies. In our case, the coefficients are…”

Section: 2mentioning

confidence: 99%

Random focusing of tsunami waves

Degueldre

¹

,

Metzger

²

,

Geisel

³

et al. 2015

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Branched flow is a universal phenomenon of random focusing that occurs in wave or particle flows that propagate in weakly scattering, correlated random media. The consecutive effect of small random forces leads to regions of strong focusing which have the appearance of branches and originate from the formation of random caustics. This phenomenon has been experimentally and theoretically studied in various systems, ranging from experimental observations in electronic microdevices on the micrometer scale to theoretical predictions for the propagation of sound waves in the ocean, on the scale of thousands of kilometers. Reconstructions of the tsunami of March 2011 exhibited strong fluctuations in the tsunami height, associated with a filamentation of the flow, reminiscent of the structures observed for example in electron flows in semiconductor microstructures. This raises the question, to what extent are the same mechanisms at play in these very different physical systems and what impact do they have for tsunami predictions. Developing a theory of random caustics and branching in tsunami waves is the main purpose of this thesis.We will start by showing that tsunamis indeed exhibit strong focusing even when propagating over a weakly scattering region of the ocean floor. We will therefore develop the stochastic theory for the characteristic length scale on which random caustics appear in the propagations of tsunamis described by ray equations. We then confirm that the focusing regions of tsunami waves follow the scaling predicted by stochastic ray dynamics with respect to the parameters of the bathymetry. We thus show that tsunamis are indeed subject to the phenomenon of branched flow. We will furthermore demonstrate that, due to the fact that already tiny bathymetry fluctuations can be a source of branched flow, bathymetry has a severe impact on the predictability of tsunami heights. Small uncertainties in the knowledge of the ocean's bathymetry can lead to drastically wrong predictions.Because the ocean floor bathymetry is known to exhibit anisotropies and to be correlated on several length scales due to the various geological processes contributing to its formation, we later extend the general theory of branched flows to systems where the random medium is correlated on more than one single length-scale, both for tsunami waves and Hamiltonian rays, as it is also relevant to many other systems. We calculate how such correlations affect the typical length scale of branching. Our theory is then applicable to a large variety of correlation functions, either anisotropic or isotropic with multiple correlation lengths. We conclude with a proposal for an experiment which scales a tsunami event down to the size of a tank in a laboratory to study the focusing effect of bathymetry structures. Such a tool could be useful in tsunami studies and forecasting and it would allow us to experimentally verify our theoretical and numerical results on random focusing of tsunami waves.3

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“…Examples of this phenomenon include the random focusing of electrons in semiconductors [8,[21][22][23][24] and microwaves propagating in cavities with weak scatterers [9], as shown in Fig. 1.2, and it is considered to explain the twinkling of starlight due to its propagation through the slightly inhomogeneous atmosphere [25][26][27][28][29], as well as the focusing of sound waves in the ocean due to water density variations [30][31][32][33][34][35][36]. All those phenomena can be modeled using Schrödinger-type equations, which can then be studied using Hamiltonian rays.…”

Section: 0mentioning

confidence: 99%

Random Focusing of Tsunami Waves

Degueldre¹

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Branched flow is a universal phenomenon of random focusing that occurs in wave or particle flows that propagate in weakly scattering, correlated random media. The consecutive effect of small random forces leads to regions of strong focusing which have the appearance of branches and originate from the formation of random caustics. This phenomenon has been experimentally and theoretically studied in various systems, ranging from experimental observations in electronic microdevices on the micrometer scale to theoretical predictions for the propagation of sound waves in the ocean, on the scale of thousands of kilometers. Reconstructions of the tsunami of March 2011 exhibited strong fluctuations in the tsunami height, associated with a filamentation of the flow, reminiscent of the structures observed for example in electron flows in semiconductor microstructures. This raises the question, to what extent are the same mechanisms at play in these very different physical systems and what impact do they have for tsunami predictions. Developing a theory of random caustics and branching in tsunami waves is the main purpose of this thesis.We will start by showing that tsunamis indeed exhibit strong focusing even when propagating over a weakly scattering region of the ocean floor. We will therefore develop the stochastic theory for the characteristic length scale on which random caustics appear in the propagations of tsunamis described by ray equations. We then confirm that the focusing regions of tsunami waves follow the scaling predicted by stochastic ray dynamics with respect to the parameters of the bathymetry. We thus show that tsunamis are indeed subject to the phenomenon of branched flow. We will furthermore demonstrate that, due to the fact that already tiny bathymetry fluctuations can be a source of branched flow, bathymetry has a severe impact on the predictability of tsunami heights. Small uncertainties in the knowledge of the ocean's bathymetry can lead to drastically wrong predictions.Because the ocean floor bathymetry is known to exhibit anisotropies and to be correlated on several length scales due to the various geological processes contributing to its formation, we later extend the general theory of branched flows to systems where the random medium is correlated on more than one single length-scale, both for tsunami waves and Hamiltonian rays, as it is also relevant to many other systems. We calculate how such correlations affect the typical length scale of branching. Our theory is then applicable to a large variety of correlation functions, either anisotropic or isotropic with multiple correlation lengths. We conclude with a proposal for an experiment which scales a tsunami event down to the size of a tank in a laboratory to study the focusing effect of bathymetry structures. Such a tool could be useful in tsunami studies and forecasting and it would allow us to experimentally verify our theoretical and numerical results on random focusing of tsunami waves.

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“…In this way, even minute perturbations can quickly lead to drastic effects -a clear hint that the equations describing this phenomenon are nonlinear. There are many examples of this random focusing: The twinkling of starlight which propagates through the slightly inhomogeneous atmosphere [23][24][25][26][27] or light in an optical fibre focused by small perturbations [28], freak waves in the ocean which can even appear in calm seas and without an additional focusing mechanism such as an underwater island but just by small fluctuations in the wave velocity [29][30][31][32][33][34][35][36][37], the focusing of sound waves in the ocean because of the fluc- tuating water density [38][39][40][41][42][43][44], and the random focusing of electrons in semiconductor devices because of a random disorder potential created by donor atoms [45][46][47], which can drastically alter the transport properties of such devices. Random caustics have also been described as a mechanism for the activation of rainshowers [48], and have recently been observed in microwave cavities [49].…”

mentioning

confidence: 99%

Branched Flow and Caustics in Two-Dimensional Random Potentials and Magnetic Fields

Johannes¹

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Branched flow is a universal phenomenon of two-dimensional wave or particle flows which propagate through a weak random potential. Its origin is the formation of caustics, which are locations where the flow is focused by the cumulative effect of weak random forces acting along the flowpath. Branched flow has been observed on length scales spanning at least twelve orders of magnitude and in a variety of systems. For example, it has been studied in semiconductor microdevices, has been argued to be the mechanism underlying the formation of giant freak sea waves and has been predicted for the propagation of sound through the ocean on scales of several thousands of kilometers. A thorough understanding of the mechanism dictating how a random potential can cause such drastic effects such as branching is therefore important in many areas of physics, and is interesting to experimentalists and theoreticians alike.In this thesis, we contribute to the theory of branched flow in the following ways. First, we consider the statistics of caustics along particle trajectories in a random potential with an additional deterministic focusing mechanism, a constant magnetic field. By extending existing theories and with detailed numerical simulations we can study the interplay between random focusing by the disorder potential and deterministic focusing by the magnetic field.We then apply our theory to data from a magnetic focusing experiment in a semiconductor microstructure. We can reproduce the results of the experiments numerically and show them to be a result of random and deterministic focusing. Our results have important consequences for the conductance properties of semiconductor microstructures.In the second part of the thesis, we consider the statistics of branches transverse to the flow, since this, although not as directly analytically and numerically accessible, is a quantity which can be measured more easily in an experiment. For the first time, we obtain statistics of the number of branches valid for all distances from a source, analytically and numerically. Also for the first time, we analyze the effect of different correlation functions and find an analytic expression for the universal curve describing the number of branches, which is valid for a wide range of correlation functions and parameters of the random potential. KurzfassungBranched flow ("verästelter Fluss") ist ein universelles Phänomen zweidimensionaler Teilchen-und Wellenflüsse, die durch ein schwaches, korreliertes Zufallspotential propagieren. Ihm liegt die Entstehung von Kaustiken zugrunde, an denen die Flussdichte besonders hoch ist. Die Flussverästelung ist auf Längenskalen, die sich auf mehr als zwölf Größenordnungen erstrecken, und in einer Vielzahl verschiedener Systeme beobachtet worden. Sie wurde unter anderem in Halbleiter-Mikrostrukturen untersucht, als Ursache für Riesenwellen (Monsterwellen) beschrieben und für die Propagation von Schallwellen durch den Ozean auf Skalen von tausenden von Kilometern vorhergesagt. Eine genaues Verständnis des Mec...

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Statistics of caustics in an underwater sound channel

Cited by 3 publications

References 3 publications

Random focusing of tsunami waves

Random focusing of tsunami waves

Random Focusing of Tsunami Waves

Branched Flow and Caustics in Two-Dimensional Random Potentials and Magnetic Fields

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