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 h distinguishable tsunami waves 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 [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 [6, 13] 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 [3, 4]. It is striking that this degree of ‘inelasticity’ is remarkably small. I will describe this work, as well as recent results on the initial value problem which are very relevant to this phenomenon [14, 18].     

Numerical simulation of landslide impulsive waves by incompressible smoothed particle hydrodynamics

An incompressible‐smoothed particle hydrodynamics (I‐SPH) formulation is presented to simulate impulsive waves generated by landslides. The governing equations, Navier–Stokes equations, are solved in a Lagrangian form using a two‐step fractional method. Landslides in this paper are simulated by a submerged mass sliding along an inclined plane. During sliding, both rigid and deformable landslides mass are considered. The present numerical method is examined for a rigid wedge sliding into water along an inclined plane. In addition solitary wave generated by a heavy box falling inside water, known as Scott Russell wave generator, which is an example for simulating falling rock avalanche into artificial and natural reservoirs, is simulated and compared with experimental results. The numerical model is also validated for gravel mass sliding along an inclined plane. The sliding mass approximately behaves like a non‐Newtonian fluid. A rheological model, implemented as a combination of the Bingham and the general Cross models, is utilized for simulation of the landslide behaviour. In order to match the experimental data with the computed wave profiles generated by deformable landslides, parameters of the rheological model are adjusted and the numerical model results effectively match the experimental results. The results prove the efficiency and applicability of the I‐SPH method for simulation of these kinds of complex free surface problems. Copyright © 2007 John Wiley & Sons, Ltd.     

Computation of wave fields in the ocean around an Isaland

A linear wave equation correct to first order in bed slope is used to calculate the wave field in the sea around an idealized island. This is of circular cylindrical shape and is situated on a paraboloidal shoal in an ocean of constant depth (Figure 1). The sides of the island are assumed fully reflecting. The incident waves are plane and periodic. Wave periods up to 30 min are investigated, and the Coriolis force is neglected. The solution of the wave equation is represented by a finite Fourier series, and a large number of very accurate numerical computations are carried through. The results appear partly in figures showing amplitude and phase angle curves (in some cases extending to the water area of constant depth outside the shoal), partly in figures showing amplitude vs wave period in fixed points. Comparison with solutions to the linearized long-wave equation is made, and the validity range of the corresponding shallow water theory is given. The influence of the shoal is studied by investigating the wave field around an island in an ocean of constant depth. New criteria are given for the applicability of a geometrical optics approach (i. e. refraction). Complete numerical refraction solutions for points at the shoreline (corresponding to many wave orthogonals ending at the point) for shallows water waves, as for the general case, demonstrate the inadequacy of this approach for long-period waves (seismic seawaves: tsunamis). All non-linear effects, including dissipation, are excluded.