Long-Wave Penetration through a Laterally Periodic Continental Shelf

Jeffrey Knowles,Harry Yeh

doi:10.3390/jmse8040241

Abstract

The transformation of long waves—such as tsunamis and storm surges—evolving over a continental shelf is investigated. We approach this problem numerically using a pseudo-spectral method for a higher-order Euler formulation. Solitary waves and undular bores are considered as models for the long waves. The bathymetry possesses a periodic ridge-valley configuration in the alongshore direction which facilitates a means by which we may observe the effects of refraction, diffraction, focusing, and shoaling. In this scenario, the effects of wave focusing and shoaling enhance the wave amplitude and phase speed in the shallower regions of the domain. The combination of these effects leads to a wave pattern that is atypical of the usual behavior seen in linear shallow-water theory. A reciprocating behavior in the amplitude on the ridge and valley for the wave propagation causes wave radiation behind the leading waves, hence, the amplitude approaches a smaller asymptotic value than the equivalent case with no lateral variation. For an undular bore propagating in one dimension over a smooth step, we find that the water surface resolves into five different mean water levels. The physical mechanisms for this phenomenon are provided.

Highlights

There have been many studies to quantify the evolution of localized long waves during shoaling onto a continental shelf
The purpose of our study is to explore the behaviors of a long wave while it climbs over a continental shelf with a lateral periodic variability
The bathymetry is translated to the right of the origin by an amount of l0 = 480 so that the initial wave form is not disturbed by the presence of the continental shelf

Summary

Introduction

There have been many studies to quantify the evolution of localized long waves during shoaling onto a continental shelf. Synolakis & Skjelbreia [2] proposed a “two-zone” shoaling model by performing laboratory experiments which show that at the commencement of the shoaling process (near the toe of the beach) that the rate of shoaling follows Green’s law which is applicable for small-amplitude shallow-water waves propagating over slowly-varying water depths. Numerical results on solitary-wave shoaling generated by a fully-nonlinear wave model are presented by Grilli et al [4] and Guyenne & Nicholls [5], which reveal the various rates of shoaling that may be expected for different beach slopes and wave amplitudes. A comprehensive numerical study of solitary-wave runup by Knowles & Yeh [6] reveals that the two-zone shoaling model is a special case among numerous shoaling behaviors depending on the length and slope of the beach, as well as the initial incident-wave amplitude. The rate of wave shoaling can be slower than what is predicted by Green’s law

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