Analytical Approximations of Dispersion Relations for Internal Gravity Waves Equation with Shear Flows

Vitaly Bulatov,Yury Vladimirov

doi:10.3390/sym12111865

Abstract

The problem of internal gravity waves fields in a stratified medium of finite depth is considered for model distributions of background shear currents. For the analytical solution of the problem, a constant distribution of the buoyancy frequency and various linear dependences of the background shear current on depth were used. The dispersion dependences are obtained, which are expressed in terms of the modified Bessel function of the imaginary index. Under the Miles–Howard stability condition and large Richardson numbers, the Debye asymptotics of the modified Bessel function of the imaginary index were used to construct analytical solutions. The dispersion equation is solved using the proposed analytical approximation. The properties of the dispersion equation are studied and the main analytical characteristics of the dispersion curves are investigated depending on the parameters of background shear flows.

Highlights

Among the large variety of observed wave processes of different physical nature in the ocean and the Earth’s atmosphere, the interaction between generated waves and hydrodynamic flows is of particular interest [1,2,3,4,5,6]
The problem of internal gravity waves (IGW) field in a stratified medium of finite depth was solved for model distributions of background shear currents
The qualitative properties of the dispersion equation were studied, and the basic analytic properties of dispersion curves were investigated depending on the characteristics of model background shear currents

Summary

Introduction

Among the large variety of observed wave processes of different physical nature in the ocean and the Earth’s atmosphere, the interaction between generated waves and hydrodynamic flows is of particular interest [1,2,3,4,5,6]. An important characteristic of natural stratified media (ocean, atmosphere) is the presence of background flows with vertical displacement of velocity which rather weakly depend on time and horizontal coordinates [7] In the ocean, such currents can manifest in the area of seasonal thermocline and have a noticeable effect on the IGW dynamics. Using the analytical properties of the dispersion relations, one can construct the asymptotics of the far IGW fields in a stratified ocean with shear flows. The goal in this paper is to study the dispersion relations for constructing the solutions which describe the far fields of IGW in a stratified medium of finite depth for model distributions of shear flows

Statement of the Problem

Basic Properties of Dispersion Curves

Analytical Approximation of Dispersion Relations

Dispersion

Conclusions