A mathematical model for calculation of the run-up of tsunamis

Abstract

To understand the engineering implications of possible wave run-up resulting from tsunamis, a formulation of the run-up process capable of giving quantitative answers is required. In this thesis, a new mathematical run-up model suitable for computer evaluation is proposed and tested. The two-dimensional model uses a flow constrained so that the horizontal velocity is uniform in depth. However, unlike the usual shallow water theory, the terms representing the kinetic energy of the vertical motion are retained. It is shown that this formulation allows a solitary-like wave to propagate as well as giving a more accurate indication of wave breaking. An 'artificial viscosity' term is used to allow the formation of hydraulic shocks. The effects of bottom friction are also included. The model is derived for a linear beach slope, in Lagrangian coordinates. A finite element formulation of the problem is derived that is suitable for digital computer evaluation. Calculations with the model agree satisfactorily with experimental results for the fun-up of solitary waves and bores. The model is used to obtain run-up data on tsunami-like waves, which show the danger of large run-up from low initial steepness waves on shallow slopes. However, the data also show that bottom friction values can significantly attenuate run-up, especially on shallow slopes. Waves generated by a dipole-like displacement of the simulated ocean floor show that the run-up is usually larger when the upwards displacement is nearest the beach than when the downwards displacement is nearest the beach.