Numerical Solution of Ostrovsky Equation over Variable Topography Passes through Critical Point Using Pseudospectral Method

Nik Nur Amiza Nik Ismail ,Azwani Alias,Fatimah Noor Harun

doi:10.13189/ms.2021.090523

Abstract

Internal solitary waves have been documented in several parts of the world. This paper intends to look at the effects of the variable topography and rotation on the evolution of the internal waves of depression. Here, the wave is considered to be propagating in a two-layer fluid system with the background topography is assumed to be rapidly and slowly varying. Therefore, the appropriate mathematical model to describe this situation is the variable-coefficient Ostrovsky equation. In particular, the study is interested in the transition of the internal solitary wave of depression when there is a polarity change under the influence of background rotation. The numerical results using the Pseudospectral method show that, over time, the internal solitary wave of elevation transforms into the internal solitary wave of depression as it propagates down a decreasing slope and changes its polarity. However, if the background rotation is considered, the internal solitary waves decompose and form a wave packet and its envelope amplitude decreases slowly due to the decreasing bottom surface. The numerical solutions show that the combination effect of variable topography and rotation when passing through the critical point affected the features and speed of the travelling solitary waves.

Highlights

The classical model for weakly nonlinear waves propagating over a uniform topography, h, yields to the formation of the Korteweg–de Vries (KdV) equation
The KdV is succeeded by the variable-coefficient Korteweg–de Vries equation
The main interest of this study is to investigate the evolution of the internal solitary wave over variable topography when the nonlinearity term slowly and rapidly passes the critical point in the presence of the background rotation, which is an extension from the studies by [17, 24]

Summary

Introduction

The classical model for weakly nonlinear waves propagating over a uniform topography, h, yields to the formation of the Korteweg–de Vries (KdV) equation. The effect of the variable topography, must be taken into consideration when retrieving the mathematical model. The KdV is succeeded by the variable-coefficient Korteweg–de Vries (vKdV) equation. Johnson [1] was the first person who derive the vKdV equation for surface waves, in which Q = c, and were discussed recently by Grimshaw et al [2] for internal waves. In the form of the vKdV equation, Grimshaw [3, 4] carried out a detailed analysis and an effective model of solitary wave propagation over the variable topography. On the assumption that the flow is two-dimensional, with x and z indicating horizontal and vertical coordinates, respectively, the end result is

Methods

Results

Conclusion