Dynamics of Internal Envelope Solitons in a Rotating Fluid of a Variable Depth

Yury Stepanyants

doi:10.3390/fluids4010056

Abstract

We consider the dynamics of internal envelope solitons in a two-layer rotating fluid with a linearly varying bottom. It is shown that the most probable frequency of a carrier wave which constitutes the solitary wave is the frequency where the growth rate of modulation instability is maximal. An envelope solitary wave of this frequency can be described by the conventional nonlinear Schrödinger equation. A soliton solution to this equation is presented for the time-like version of the nonlinear Schrödinger equation. When such an envelope soliton enters a coastal zone where the bottom gradually linearly increases, then it experiences an adiabatical transformation. This leads to an increase in soliton amplitude, velocity, and period of a carrier wave, whereas its duration decreases. It is shown that the soliton becomes taller and narrower. At some distance it looks like a breather, a narrow non-stationary solitary wave. The dependences of the soliton parameters on the distance when it moves towards the shoaling are found from the conservation laws and analysed graphically. Estimates for the real ocean are presented.

Highlights

The effect of the Earth’ rotation on the dynamics of nonlinear waves in the oceans was extensively studied in the last decades
The process of solitary wave decay is more complicated in reality and leads eventually to the formation of envelope solitons described by the nonlinear Shrödinger (NLS) equation or its modifications [19–24]
In the real ocean, when a Korteweg–de Vries (KdV) soliton approaches a coastal zone, it can experience a terminal decay in the domain where the depth is constant, so that it can be transformed into an NLS envelop soliton, and the envelop soliton can enter into the inhomogeneous domain where an oceanic depth gradually decreases

Summary

Introduction

The effect of the Earth’ rotation on the dynamics of nonlinear waves in the oceans was extensively studied in the last decades (see, for example, References [1–8] and references therein). In the real ocean, when a KdV soliton approaches a coastal zone, it can experience a terminal decay in the domain where the depth is constant, so that it can be transformed into an NLS envelop soliton, and the envelop soliton can enter into the inhomogeneous domain where an oceanic depth gradually decreases. It is a matter of interest to study the adiabatic dynamics of an NLS envelop soliton when it approaches a shoaling zone. To this end, we consider below different models of NLS-type equations for water waves in a rotating ocean, their solutions in the form of envelop solitons, and the dynamics of such solitons in the ocean with a gradually decreasing depth

The Variable Coefficients Ostrovsky Equation

The Variable Coefficients NLS Equation

Generalised Variable Coefficients NLS Equation

What Is the Most Probable Frequency of Envelope Soliton?

Estimations for the Real Oceanic Conditions

Discussion and Conclusions