Max Planck Institute for Dynamics and Self-Organization -- Department for Nonlinear Dynamics and Network Dynamics Group
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Random focusing of tsunami waves

Tsunamis are probably among the deadliest, frequently recurring natural disasters. Typically, submarine earthquakes and landslides excite these extremely long wavelength gravity waves that travel the oceans with speeds of several hundred km/h. Due to the unpredictable nature of the earthquakes that cause them, long term forecasts of tsunamis seem to be far out of reach. However, despite their speed, due to the large distances they might travel over the ocean, there can be hours between their excitation and the time when they hit the shore. Following the devastating tsunami catastrophes in recent times, great efforts are made to use this time for tsunami warning systems. A prerequisite, however, is a thorough scientific understanding not only of the excitation of tsunamis but also of their propagation in the ocean.

 NOAA reconstruction 

Figure 1: Reconstruction of the tsunami on the 11th of March 2011 in the Pacific Ocean by the US National Oceanic and Atmospheric Administration (NOAA)  [nctr.pmel.noaa.gov]. The colors code the maximum wave height, which shows a strong filamentation of the wave flow.

An important aspect is that tsunamis exhibit surprisingly strong height fluctuations: run-up measurements and detailed simulations of past tsunami events have shown pronounced fluctuations in the crest height of tsunamis (see Fig. 1). While in-detail numerical modeling of the wave propagation in the measured depth-profile of the ocean floor captures many aspects of these fluctuations, only detailed studies of the focusing mechanisms that cause them, will allow to scrutinize and improve the assumptions made in these models. It has been known, e.g, that the presence of large underwater islands [1] or the shape of the tsunami source [2] can affect the waves heights. We have recently shown that the consecutive effect of even tiny fluctuations in the profile of the ocean floor (the bathymetry) can cause unexpectedly strong fluctuations in the wave height of tsunamis with maxima several times higher than the average wave height [3], with severe implications for tsunami predictions, as will be discussed below.

 PIC 

Figure 2: Measurement of the electron flow emitted from a narrow, nano-scaled opening, a quantum point contact, into the two dimensional electron gas of a semiconductor heterostructure [4]. The flow shows pronounced branches (on length scales much shorter than the mean free path of approximately 7μm), caused by consecutive scattering by the very weak disorder potential generated by impurities. (Taken from [4].)

Electron waves that are scattered by weak impurities in semiconductor crystals show a pronounced branching of the flow (see Fig. 2), which is closely connected to the occurrence of random caustics [56]. Recently, we contributed substantially [678910] to the theoretical and experimental understanding of this very general mechanism leading to the formation of rogue waves. Visually Fig. 2 exhibits strong similarities to the filamentation of the tsunami wave shown above. Therefore the questions posed itself, whether tsunamis on length scales of thousands of kilometers are subject to the same focusing mechanisms as electrons on the micrometer scale and which impact this might have for their predictability. Tsunami waves are shallow water waves because their wavelength – ranging from ten to several hundreds of kilometers – is much larger than the ocean depths. While propagating over the deep ocean, tsunamis are very well described by the linear shallow water wave equations because their amplitude is on the order of a meter whereas the ocean depth is several kilometres. However, the structure of these equations and their ray approximation are quite different from the Schrödinger-equation and its semi-classics, so that the existing statistical theory of branched flows could not simply be used to analyse tsunami waves scattered by small structures in the bathymetry.

 Tsunami focussing 

Figure 3: (a) Simulated tsunami event showing pronounced branching in the bathymetry of Indian Ocean taken from the GEBCO database. An island-free region was chosen in which the bathymetry B(x, y) only fluctuates with a standard deviation of approx. 6.9% of the average ocean depth H0 4km, illustrated in panel (b).(c+d) Statistical analysis of the focusing of tsunami waves in computer generated bathymetries, showing the average distance to the strongest fluctuations of the waves (c) and the mean distance to the first caustic (d) calculated in the ray approximation . [Adapted from [3], where details of the simulations can be found.]

First, we demonstrate that tsunamis are indeed affected even by very weak fluctuations in the profile of the ocean floor. We used the GEBCO database [www.gebco.net] and chose a region of the Indian Ocean (see Fig. 3) where the bathymetry fluctuations β(x, y) have a standard deviation Sqrt{<\beta^2>} of less than 7% of the average ocean depth. In this bathymetry we simulated a tsunami excited by a well localized source in the south-western corner. The time integrated wave intensity (i.e. the square of the wave amplitude), which is a good measure of the wave energy, is shown in Fig. 3 and exhibits clear branching patterns. In a second step, we theoretically and numerically studied the focusing of tsunami waves and their corresponding rays in random bathymetries. We found that the strongest focusing occurs at distances df that scale with the characteristic quantities of the disordered bathymetry according to

Scaling law (1)

where c is the correlation length of the fluctuations. This scaling behaviour is in excellent agreement with the numerics as shown in Fig. 3c+d for waves (c) and rays (d).

 Predictability of tsunamis 

Figure 4: (a)+(c) The same bathymetry region and the branched intensity of the simulated tsunami wave shown in Fig. 3. (b)+(d) Adding a small additional disorder well below the uncertainties in the measured bathymetry (see text) strongly changes the branching pattern. (e) This can be most clearly seen in the cuts along the green in red curve in panels (c) and (d). The wave intensity shows peaks up to 6 times the average at completely unrelated positions.

From Fig. 3c we can read off that already fluctuations as small as 4% standard deviation lead to focussing of the tsunami waves in a distance of approximately 1000km from the source. Such variations are smaller than the uncertainties in the available bathymetry data but might be highly relevant for actual predictions. We tested this by adding random fluctuation of this strength to the measured bathymetry data form Fig. 3. An example is shown in Fig. 4: Panel (a) and (c) show the original bathymetry data and simulation result from Fig. 3 in comparison to the bathymetry (b) and integrated intensity (d) of the the simulation with added fluctuations. Panels (c) and (d) show that the focusing directions are indeed strongly affected by the additional disorder. Cuts through the intensity along the green and red arc, reveal this even more clearly (panel e): pronounced peaks can be seen at completely unrelated positions.

Our work shows that the understanding of the branched flow of tsunamis is crucial for reliable wave height predictions. It is imperative to analyse in detail not only the magnitude of the uncertainties in the bathymetry data but their spatial correlations as well, and to systematically study the effect of additional random, correlated height fluctuations on the predicted wave heights.

[1]    M. V. Berry. Focused tsunami waves. Proc. R. Soc. A 463:3055 (2007).

[2]    U. Kanoglu et al. Focusing of long waves with finite crest over constant depth. Proc. R. Soc. A 469:20130015 (2013).

[3]     H. Degueldre, J.J. Metzger, T. Geisel, and R. Fleischmann, Random focusing of tsunami waves, Nature Physics, Nat Phys 12, 259–262 (2016).

[4]     M. A. Topinka, B. J. LeRoy, R. M. Westervelt, S. E. J. Shaw, R. Fleischmann, E. J. Heller, K. D. Maranowski, and A. C. Gossard. Coherent branched flow in a two-dimensional electron gas. Nature, 410:183, March 2001.

[5]    Lev Kaplan. Statistics of branched flow in a weak correlated random potential. Phys. Rev. Lett., 89, 184103 (2002).

[6]     Jakob J. Metzger, Ragnar Fleischmann, and Theo Geisel. Universal statistics of branched flows. Phys. Rev. Lett. 105, 020601 (2010).

[7]    D. Maryenko et al., How branching can change the conductance of ballistic semiconductor devices, Phys. Rev. B 85, 195329 (2012).

[8]    S. Barkhofen et al., Experimental Observation of a Fundamental Length Scale of Waves in Random Media, Phys. Rev. Lett. 111, 183902 (2013).

[9]    J. J. Metzger, R. Fleischmann, and T. Geisel, Intensity Fluctuations of Waves in Random Media: What Is the Semiclassical Limit?, Phys. Rev. Lett. 111, 013901 (2013).

[10]    J. J. Metzger, R. Fleischmann, and T. Geisel, Statistics of Extreme Waves in Random Media, Phys. Rev. Lett. 112, 203903 (2014).

[11]    D.T. Sandwell, S.T. Gille, and W.H.F. Smith, Bathymetry from space: Oceanography, geophysics, and climate, Geoscience Professional Services, Bethesda, Maryland (2002), www.igpp.ucsd.edu/bathymetry_workshop


Contact:  Ragnar Fleischmann 

Members working within this Project:

 Ragnar Fleischmann 
 Theo Geisel 

Former Members:

 Jakob J. Metzger 
 Henri Degueldre 

Selected Publications:

H. Degueldre, J.J. Metzger, E. Schultheis, and R. Fleischmann (2017).
Channeling of Branched Flow in Weakly Scattering Anisotropic Media
Phys. Rev. Lett. 118:024301 . download file

H. Degueldre, J.J. Metzger, T. Geisel, and R. Fleischmann (2016).
Random Focusing of Tsunami Waves
Nature Physics 12:259–262.