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Snell’s Law Applied to Tsunamis: Simulations and Observations

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Abstract

At the edge of a continental shelf, where the depth of the ocean column can vary rapidly by a factor of \(\sim \) 20, tsunamis are expected to be severely refracted in application of Snell’s Law. We use beaming techniques applied to numerical simulations under the shallow-water approximation, in the geometry of two real-life provinces of the Pacific Basin featuring sharp bathymetric discontinuities and extended shelves (Southeastern Alaska and Nicaragua). We conclude that tsunamis do indeed undergo refraction under Snell’s Law with equivalent refraction indices as high as 5. We apply the same technique to actual records of the 2011 Tohoku tsunami by an array of seafloor pressure sensors off the coast of Southern California, and similarly observe Snell refraction despite a smaller velocity contrast and the presence of a shelf with extremely irregular bathymetry. Finally, we show numerically that for a source and a linear array of receivers both deployed on a well developed continental shelf bordering a deep basin, the tsunami will develop a head wave in all ways similar to the familiar case of refraction seismology.

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Notes

  1. Snell’s Law takes its name from the Dutch mathematician Snellius (1621), but is known in French-speaking countries as Loi de Descartes, since it was stated in an appendix to the French philosopher’s Discours de la Méthode in 1637. A longstanding controversy regarding the knowledge of Snell’s work by R. Descartes, who was living in the Netherlands at the time, was never fully resolved (Kwan et al., 2002). At any rate, the geometry of refraction was already described in 984 A.D., i.e., 650 years earlier, by A. Ibn Sahl, a Persian scholar living in Baghdad, who was probably himself inspired by early but fragmentary work by Ptolemy, dating back to the second century A.D. (Rashed, 1990). In this paper, we will stick with the traditional name of “Snell’s Law”.

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Acknowledgements

I thank A. B. Rabinovich for pointing out his 1993 monograph. The data used in Sect. 3 were obtained from the IRIS Data Center. Maps were drawn using the GMT software (Wessel & Smith, 1991).

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  1. Department of Earth and Planetary Sciences, Northwestern University, Evanston, IL, 60208, USA

    Emile A. Okal

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Okal, E.A. Snell’s Law Applied to Tsunamis: Simulations and Observations. Pure Appl. Geophys. 178, 4969–4983 (2021). https://doi.org/10.1007/s00024-021-02703-4

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