Random focusing of tsunami waves

Abstract

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

Constructing analytically tractable ensembles of stochastic covariances with an application to financial data

“…Non-stationarities are also seen when wave packets travel through disordered systems. Even if the disorder is static, the correlations between the wave intensities measured at different positions versus time will change, when the direction or the composition of the wave packet is altered [3][4][5]. Finance provides another important example for this type of non-stationarity.…”

Branched flow and caustics in nonlinear waves

“…Probably the most sensational manifestation are giant ocean freak waves of more than 20 m height, which have been confirmed to exist by measurements for the first time only little more than 20 years ago [1]. Heavy tailed intensity distributions and thus rogue waves, however, can be found in a variety of different contexts including optics [1] and microwave transmissions [2,3], as well as the propagation of sound [4] and tsunami waves [5].…”

“…Breathers have been extensively studied both theoretically and experimentally in optics and water waves [12][13][14][15][16][17]. Likewise have branched flows been studied in a variety of systems ranging from transport of electrons in the two-dimensional electron gas scattered by the disorder potential of impurities [18][19][20], via the scattering of wind driven ocean waves by ocean currents [21,22] to the focusing of tsunamis by small variations of the ocean floor topography [5]. Not only do branched flows in general lead to heavy-tailed intensity distributions [10,23], but experiments on microwave transmissions through disordered arrays of scatterers have shown that in combination with fluctuating sources they can lead to breather-like isolated rogue events [2].…”

Predictability of extreme intensity pulses in optically injected semiconductor lasers

“…In hydrodynamics, when the height of a wave is larger by a factor of 3 than average, this wave is considered extreme. For example, regarding the wave height of tsunamis, maxima several times higher than the average wave height have been reported [6]. In optics, however, fluctuations of much higher amplitudes compared to the average are often observed [25].…”

“…Extreme events are ubiquitous in complex systems [1][2][3][4][5][6] and a lot of efforts are nowadays focused on developing reliable analysis techniques for their detection and prediction [7][8][9][10][11][12][13]. In many scientific fields, if an extreme event occurs in an unexpected way, it often has disastrous consequences.…”