|dc.description.abstracteng||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.||de