Properties of Surface and Internal Solitary Waves

Abstract

Numerical solutions of surface and internal solitary waves are obtained through a new method, where advection equations on physical quantities including surface/interface displacements and velocity potential are solved to find convergent solutions by applying the Newton-Raphson method. The nonlinear wave equations derived using a variational principle are adopted as the fundamental equations in the present study. Surface and internal solitary waves obtained through the proposed method are compared with the corresponding theoretical solutions, as well as numerical solutions of the Euler equations, to verify the accuracy of solutions through the proposed method especially for internal solitary waves of large amplitude progressing at a large celerity with a flattened wave profile. Properties of surface and internal solitary waves are discussed considering vertical distribution of horizontal and vertical velocity, as well as kinetic and potential energy. Numerical simulation using a time-dependent model has also been performed to represent propagation of surface and internal solitary waves with permanent waveforms.

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

“…(2), (3), (5), and (7), are transformed to finite difference equations, which are solved using an implicit scheme 9) . In the present study, numerical solutions for surface/internal solitary waves are obtained using the method introduced by Yamashita and Kakinuma 8) , where the Newton-Raphson method is applied to solve the fundamental equations for steady waves in a coexisting fields of surface and internal waves. We substitute the advection equation ∂F/∂t = −C ∂F/∂x into the time derivative terms of Eqs.…”

“…e-mail: kyamashita@irides.tohoku.ac.jp (4) The lower and 2 nd layer 1 1 1 2 2 2 2 2 2 2 2 2 2 2 , 2 1 1 2 2 2 2 , 2 1 1 2 2 = − − + −  −  + + +   − + − + + In this study, 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., N 1 = N 2 = N = 3, based on the accuracy verification 8) for the surface and internal solitary waves obtained using the fundamental equations.…”

A Numerical Calculation for Internal Waves Over Topography

“…In order to represent both the nonlinearity and dispersion of internal waves propagating from deep to shallow, or from shallow to deep, water, the numbers of terms for the expanded velocity potentials introduced in Eq. (1), are N1 = 3 and N2 = 5 for the upper and lower layers, respectively, referring to the accuracy of the numerical results obtained by Yamashita & Kakinuma (2015), using the same equations for the internal solitary waves.…”

Numerical Simulation for Tsunami-Height Reduction Using a Very Large Floating Structure

“…A solitary wave solution, obtained using the numerical method developed by Yamashita and Kakinuma (2014), is given as an incident tsunami, where the above-described governing equations, are reduced only for stable water waves, without a thin-plate. The initial location of a wave peak, is x = 2.0 km at t = 0.0 s, while the offshore end of a VLFS, is located at x = 4.0 km.…”

“…The offshore end of the offshore-side VLFS, is located at x = 4.0 km. A solitary wave solution, obtained using the numerical method developed by Yamashita and Kakinuma (2014), is given as an incident tsunami. The initial tsunami height a is 5.0 m, and the wave peak is initially located at x = 2.0 km when t = 0.0 s. The tsunami first attacks the offshore-side VLFS.…”