Numerical Computation of Tsunami Response for Island Systems

Abstract

A numerical procedure has been developed to determine the tsunami response of an island system and the surrounding underwater topography. The main algorithm in based on the Eulerian equations of motion and continuity which correspond to the classical, linear, long-wave equation in the absence of friction and rotation. It performs an integration of these equations in terms of an explicit scheme based on centered differences. The computations are carried out on a Cartesian grid network and employ a condition of no normal flow at the island shoreline representation as well as an approximate radiation condition for the scattered portion of the wave field at the outer boundaries. The responses at the island shorelines are determined by introducing a time sequence input with a stipulated spectrum covering the tsunami range (4?90 min). Computations are continued for a duration sufficient to establish a statistical equilibrium within the system. The shoreline time series of water elevations are then Fourier-analyzed to obtain the spectrum for each point. Each spectrum is normalized by the spectrum of a time series taken of the forcing function, propagating on a grid of constant, deep water depth at a central position in the model. The normalized spectra or energy ratios are averaged for each island and for the entire set of shoreline points to produce the responses. The results are presented as graphs of averaged normalized spectra and as contours of (energy ratio)1/2 on period vs island perimeter plots. An application of the technique is given for the Hawaiian Islands with a northern (Alaskan) wave approach angle. The results indicate energetic response is possible at ten periods from 12.5 to 73.1 min.