A numerical study of the tsunami response at an island

Abstract

A method is presented for the evaluation of the modification phase of the tsunami phenomenon that occurs during the advancement of the waves from deep water into the more shallow water around islands and seamounts. The numerical experiments are carried out with an orthogonal coordinate system of particular use in the study of the response to long gravity waves near islands of irregular shape. Evaluation of the coordinate system involves a conformal mapping of the island onto an image plane in which the island contour is a unit circle. The mapping relation is evaluated by an iterative procedure that has the advantage of being applicable to an island contour for which the contour relation r(θ), in polar coordinates, is multivalued for a given range of θ. This generality is achieved by employing a parametric representation of the island contour with the arc length on the island contour entering as a parameter. A numerical integration procedure is used to evaluate the wave field produced by the interaction of the scattered and incident wave fields in the near field of an island. Plane monochromatic waves with periods in the tsunami range were selected as incident waves. The wave equation chosen for representation is the classical linearized long wave equation and the boundary condition at the island is that of no radial flow. A radiation condition is used in the far field for the scattered waves. Verification of the numerical procedure is restricted to two axially symmetric islands for which analytical solutions are available in polar coordinates. A comparison is made between the refraction and diffraction analyses for a third axially symmetric island. ...