Changing forms and sudden smooth transitions of tsunami waves
Update: 2014-08-13
Description
In some tsunami waves travelling over the ocean, such as the one approaching the eastern coast of Japan in 2011, the sea surface of the ocean is depressed by a small meter-scale displacement over a multi-kilometer horizontal length scale, lying in front of a positive elevation of comparable magnitude and length, which together constitute a "down-up" or ``breather'' wave. Shallow water theory shows that the latter travels faster than the former and, according to the extended Korteweg-de Vries model presented here, the waves undergo a transition. Firstly, the two parts coincide at a given position and time producing a maximum elevation, whose amplitude depends on the shape of the approaching wave. Typically this amplitude is larger than the initial displacement magnitude by a factor which can be as large as two, which may explain anomalous elevations of tsunamis at particular positions along their trajectories. It is physically significant that for these small amplitude waves, no wave breaking occurs and there is no excess dissipation. Secondly, following the transition, the elevation wave moves ahead of the depression wave and the distance between them increases either linearly or logarithmically with time.The implications for how these ``down-up'' tsunami waves reach beaches are considered. This is joint work with Chow & Hunt.
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