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Surface water waves and tsunamis

Because of the enormous earthquake in Sumatra on December 26, 2004, and the devastating tsunami which followed, I have chosen the focus of my mini-course lectures at this year’s PASI to be on two topics which involve the dynamics of surface water waves. These topics are of interest to mathematicians interested in wave propagation, and particularly to Chilean scientists, I believe, because of Chile’s presence on the tectonically active Pacific Rim.

My first lecture will describe the equations of fluid dynamics for the free surface above a body of fluid (the ocean surface), and the linearized equations of motion. From this we can predict the travel time of the recent tsunami from its epicenter off of the north Sumatra coast to the coast of nearby Thailand, the easy coasts of Sri Lanka and south India, and to Africa. In fact the signal given by ocean waves generated by the Sumatra earthquake was felt globally; within 48 hours distinguishable tsunamiwaves were measured by wave gages in Antarctica, Chile, Rio di Janeiro, the west coast of Mexico, the east coast of the United States, and at Halifax, Nova Scotia.

To describe ocean waves we will formulate the full nonlinear fluid dynamical equations as a Hamiltonian system (Zakharov 1968 [19]), and we will introduce the Greens function and the Dirichlet-Neumann operator for the fluid domain along with the harmonic analysis of the theory of their regularity. From an asymptotic theory of scaling transformations we will derive the known Boussinesq-like systems and the KdV and KP equations which govern the asymptotic behavior of tsunami waves over an idealized flat bottom. When the bottom is no longer assumed to be perfectly flat,a related theory (Rosales & Papanicolaou 1983 [13]) (Craig, Guyenne, Nicholls & Sulem 2005 [6]) gives a family of model equations taking this into account.

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Surface water waves and tsunamis