A Numerical Solution for the Coexisting Field of Surface and Internal Solitary Waves

Taro Kakinuma,Kei Yamashita

doi:10.34257/gjreevol20is3pg1

Abstract

The numerical solutions for the coexisting fields of surface and internal solitary waves have been obtained, where the set of nonlinear equations based on the variational principle for steady waves are solved using the Newton- Raphson method. The relative phase velocity of surface-mode solitary waves is smaller in the coexisting fields of surface and internal solitary waves than in the cases without the coexistence of internal waves. The relative phase velocity of internal-mode solitary waves is also smaller in the coexisting fields of surface and internal solitary waves than in the cases without surface waves. The interfacial position of an internal mode internal solitary wave in a coexisting field of surface and internal waves can exceed the critical level determined in the corresponding case without a surface wave. The wave height ratio between internal-mode surface and internal solitary waves is smaller than the corresponding linear shallow water wave solution, and the difference increases, as the relative wave height of internal-mode internal solitary waves is increased.

Highlights

Surface and internal waves coexist in the ocean with stratification development
We focus on solitary waves, such that the number of terms for the expanded velocity potential expressed by Eq (1) is three for both upper and lower layers, i.e., N1 = N2 = N = 3, based on the accuracy verification8) for the surface and internal solitary waves obtained using the fundamental equations
Numerical solutions for surface/internal solitary waves are obtained using the method introduced by Yamashita and Kakinuma8), where the Newton-Raphson method is applied to solve the fundamental equations for steady waves in a coexisting fields of surface and internal waves

Summary

Introduction

Surface and internal waves coexist in the ocean with stratification development. The behaviors of waves in such coexisting fields of surface and internal waves show more complicated characteristics than those which exist individually. The traveling time for a distant tsunami is delayed due to the influence of density stratification in the ocean, according to the theoretical analyses for linear waves1), 2). Fructus and Grue3) used a pressure field for two-layer fluids sandwiched by two fixed horizontal plates, to obtain the surface waves caused by large-amplitude internal waves. A coexisting field of surface and internal waves can be established even in nearshore zones, where surface long waves have great influence on sediment motion and coastal structures as an external force, and internal waves may greatly affect the coastal

Fundamental Equations

Surface-Mode Solitary Waves

Internal-Mode Solitary Waves