Tsunamis: Non-Breaking and Breaking Solitary Wave Run-Up
Creators: name: Li, Ying
Publisher: name: California Institute of Technology
Year: 2000

description: This study considers the run-up of non-breaking and breaking solitary waves on a smooth sloping beach. A non-linear theory and a numerical model solving the non-linear shallow water equations (NLSW) were developed to model this physical process. Various experiments to obtain wave amplitude time-histories, water particle velocities, wave free surface profiles, and maximum run-up were conducted and the results were compared with the analytical and numerical models.

A higher order theoretical solution to the non-linear shallow water equations, which describes the non-breaking wave characteristics on the beach, was sought and is presented in this study. The solution was obtained analytically by using the Carrier and Greenspan (1958) hodograph transformation. It was found that the non-linear theory agreed well with experimental results. The maximum run-up predicted by the non-linear theory is larger than that predicted by Synolakis (1986) at the order of the offshore relative wave height for a given slope. This correction for non-breaking waves on beach decreases as the beach slope steepens, and increases as the relative incident solitary wave height increases.

A unique run-up gage that consists of a laser and a photodiode camera was developed in connection with this study to measure the time-history of the tip of the run-up tongue of a non-breaking solitary wave as it progresses up the slope. The results obtained with this run-up gage agree well with other measurements, and this technique provides a simple and reliable way of measuring run-up time-histories. The run-up of breaking solitary waves was studied experimentally and numerically since no fully theoretical approach is possible. The wave characteristics such as wave shape and shoaling characteristics, and, for plunging breakers, the shape of the jet produced are presented. The experimental results show that wave breaking is such a complicated process that even sophisticated numerical models cannot adequately model its details.

Two different plunging wave breaking and resultant run-up were found from the experiments. The point where the tip of the incident jet produced by the plunging breaking wave  impinges determines the characteristics of the resulting splash-up. If the jet impinges on a dry slope, no splash-up occurs and the plunging breaker simply collapses. If the impingement point is located on the free surface, splash-up including a reflected jet is formed, which further increases the turbulence and energy dissipation associated with wave breaking. It is hypothesized that both clockwise and counterclockwise vortices may be generated by the impinging plunging jet and the reflected jet associated with the splash-up when the jet impinges on the front face of a breaking wave or on the still water surface in front of the wave.

If only the run-up process and maximum run-up are of interest, the wave and the water flow produced after breaking can be simplified as a propagating bore, which is analogous to a shock wave in gas dynamics. A numerical model using this bore structure to treat the process of wave breaking and propagation was developed. The non-linear shallow water equations were solved using the weighted essentially nonoscillatory (WENO) shock capturing scheme employed in gas dynamics. Wave breaking and propagation is handled automatically by this scheme and no ad-hoc term is required. A computational domain mapping technique proposed by Zhang (1996) is used in the numerical scheme to model the shoreline movement. This numerical scheme is found to provide a somewhat simple and reasonably good prediction of various aspects of the run-up process. The numerical results agree well with the experiments corresponding to the run-up on a relatively steep slope (1:2.08) as well as
on a more gentle slope (1:19.85).

A simple empirical estimate of maximum run-up based on energy conservation considerations is also presented where the energy dissipation associated with wave breaking was estimated using the results from the numerical model. This approach appears to be useful and the maximum run-up predicted agrees reasonably well with the experimental results.

The splash-up of a solitary wave on a vertical wall positioned at different locations on a gentle slope was also investigated in this study to understand the degree of protection from tsunamis afforded by seawalls. It was found that the effect of breaking wave kinematics offshore of the vertical wall on the splash-up is of critical importance to the maximum splash-up. The maximum slope of the front face of the wave upon impingement of the wave on the wall, which represents the maximum water particle acceleration, was important in defining the maximum sheet splash-up as well as the trend for splash-up composed of drops and spray.
description: © 2000 Ying Li All Rights Reserved. W. M. Keck Laboratory of Hydraulics and Water Resources. California Institute of 
Technology.

This report is essentially the thesis of the same title submitted by the author on May 2000 to the California Institute of Technology, in partial fulfillment of the
requirements for the degree of Doctor of Philosophy in Mechanical Engineering.

This material is based upon work supported by the National Science Foundation under Award CMS-9523414. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the author and do not necessarily reflect the views of the National Science Foundation.