Random Focusing of Tsunami Waves (2015)

Branched flow is a universal phenomenon of random focusing that occurs in wave or particle&#13;\nflows that propagate in weakly scattering, correlated random media. The consecutive effect of&#13;\nsmall random forces leads to regions of strong focusing which have the appearance of branches&#13;\nand originate from the formation of random caustics. This phenomenon has been experimentally&#13;\nand theoretically studied in various systems, ranging from experimental observations in electronic&#13;\nmicrodevices on the micrometer scale to theoretical predictions for the propagation of sound&#13;\nwaves in the ocean, on the scale of thousands of kilometers.&#13;\nReconstructions of the tsunami of March 2011 exhibited strong fluctuations in the tsunami&#13;\nheight, associated with a filamentation of the flow, reminiscent of the structures observed for&#13;\nexample in electron flows in semiconductor microstructures. This raises the question, to what&#13;\nextent are the same mechanisms at play in these very different physical systems and what impact&#13;\ndo they have for tsunami predictions. Developing a theory of random caustics and branching in&#13;\ntsunami waves is the main purpose of this thesis.&#13;\nWe will start by showing that tsunamis indeed exhibit strong focusing even when propagating&#13;\nover a weakly scattering region of the ocean floor. We will therefore develop the stochastic theory&#13;\nfor the characteristic length scale on which random caustics appear in the propagations of&#13;\ntsunamis described by ray equations. We then confirm that the focusing regions of tsunami waves&#13;\nfollow the scaling predicted by stochastic ray dynamics with respect to the parameters of the&#13;\nbathymetry. We thus show that tsunamis are indeed subject to the phenomenon of branched flow.&#13;\nWe will furthermore demonstrate that, due to the fact that already tiny bathymetry fluctuations&#13;\ncan be a source of branched flow, bathymetry has a severe impact on the predictability of&#13;\ntsunami heights. Small uncertainties in the knowledge of the ocean’s bathymetry can lead to&#13;\ndrastically wrong predictions.&#13;\nBecause the ocean floor bathymetry is known to exhibit anisotropies and to be correlated on&#13;\nseveral length scales due to the various geological processes contributing to its formation, we later&#13;\nextend the general theory of branched flows to systems where the random medium is correlated&#13;\non more than one single length-scale, both for tsunami waves and Hamiltonian rays, as it is also&#13;\nrelevant to many other systems. We calculate how such correlations affect the typical length scale&#13;\nof branching. Our theory is then applicable to a large variety of correlation functions, either&#13;\nanisotropic or isotropic with multiple correlation lengths.&#13;\nWe conclude with a proposal for an experiment which scales a tsunami event down to the size of&#13;\na tank in a laboratory to study the focusing effect of bathymetry structures. Such a tool could be&#13;\nuseful in tsunami studies and forecasting and it would allow us to experimentally verify our&#13;\ntheoretical and numerical results on random focusing of tsunami waves.