Analytical Solutions of Seismic-Generated Internal Tsunamis

Chi-Min Liu,Gaetano Giunta

doi:10.1155/2021/1113733

Abstract

Internal tsunamis induced by an instantaneous seabed deformation are theoretically analyzed in this paper. Based on Hammack’s mathematical methods and findings, an in-depth and extensive study is performed to examine tsunami waveforms, especially at the initial stage. Waveforms of surface and internal tsunamis induced by three fundamental seabed uplifts, the rectangle-, cosine-, and sine-shape deformations, are solved to constitute an important base for analyzing waves generated by arbitrary seabed movements. A closed-form solution for the rectangle-shape deformation and analytic solutions for cosine and sine cases are obtained. The effects of spatial parameters on waveforms and the contributions of two frequency modes are investigated for the Boussinesq limit. The derived wave solutions not only improve the understanding of the formation of internal tsunamis but also provide an exact initial waveform for simulating wave propagation by various wave models.

Highlights

Tsunamis are one of the most catastrophic natural disasters for human beings
In the Boussinesq limit, we focus on deriving the closed-form solutions of initial tsunami waveforms. e initial waveform means the wave elevation at the time when the seabed deformation is just completed (t 0+)
For internal tsunamis shown in the left panel, initial waveforms are closer to the seabed deformation for either larger h1/h or smaller h/B

Summary

Introduction

Tsunamis are one of the most catastrophic natural disasters for human beings. In this century, several destructive tsunamis, which include the 2004 Indian Ocean, 2010 Chile, and 2011 Japan Tohoku tsunamis, cause heavy casualties and destruction of infrastructures and buildings. E linear potential theory was adopted to examine tsunamis induced by both of the exponential- and cosine-type rectangle-shape seabed uplifts. In Hammack’s derivation, the term 1/coshkh (k is the wavenumber and h the water depth) appearing in wave solutions demonstrates that the difference between the initial waveform and seafloor deformation will increase as water depth becomes larger even the seafloor movement is instantaneous. Derakhti et al [2] examined the tsunami generations by three seafloor deformations of different shapes and simulated the wave propagation using the fully nonlinear smooth particle hydrodynamic model. Hammack [25] examined barotropic (surface) and baroclinic (internal) tsunamis induced by an instantaneous rectangle-shape seabed uplift in a two-fluid system.

Mathematical Modeling

The Boussinesq Limit

Conclusions