Lecture 1: surface water waves and tsunamis

Lecture 2: interactions between nonlinear solitary water waves

Abstract: Because of the enormous earthquake in Sumatra, and the devastating tsunami which followed, there is a heightened interest in the prediction and the study of characteristics of tsunamis and nonlinear ocean waves in general.

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. We will further formulate the full nonlinear fluid dynamical equations as a Hamiltonian system (Zakharov 1968), we will describe 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)(Craig, Guyenne, Nicholls & Sulem 2004) gives a family of model equations taking this into account.

My second lecture will describe a series of recent results in PDE, numerical results, and experimental results on the nonlinear interactions of solitary surface water waves. In contrast with the case of the KdV equations (and certain other integrable PDE), the Euler equations for a free surface do not admit clean (`elastic') interactions between solitary wave solutions. This has been a classical concern of oceanographers for several decades, but only recently have there been sufficiently accurate and thorough numerical simulations which quantify the degree to which solitary waves lose energy during interactions (Cooker, Wiedman & Bale 1997) (Craig, Guyenne, Hammack, Henderson & Sulem 2004). It is remarkable that this degree of `inelasticity' is remarkably small. I'll describe this work, as well as recent results on the initial value problem which are very relevant to this phenomenon (Schneider & Wayne 2000) (Wright 2004).