Three-Dimensional Coupled NLS Equations for Envelope Gravity Solitary Waves in Baroclinic Atmosphere and Modulational Instability

Baojun Zhao,Hongwei Yang,Ruyun Wang

doi:10.1155/2018/4276591

Abstract

Envelope gravity solitary waves are an important research hot spot in the field of solitary wave. And the weakly nonlinear model equations system is a part of the research of envelope gravity solitary waves. Because of the lack of technology and theory, previous studies tried hard to reduce the variable numbers and constructed the two-dimensional model in barotropic atmosphere and could only describe the propagation feature in a direction. But for the propagation of envelope gravity solitary waves in real ocean ridges and atmospheric mountains, the three-dimensional model is more appropriate. Meanwhile, the baroclinic problem of atmosphere is also an inevitable topic. In the paper, the three-dimensional coupled nonlinear Schrödinger (CNLS) equations are presented to describe the evolution of envelope gravity solitary waves in baroclinic atmosphere, which are derived from the basic dynamic equations by employing perturbation and multiscale methods. The model overcomes two disadvantages: (1) baroclinic problem and (2) propagation path problem. Then, based on trial function method, we deduce the solution of the CNLS equations. Finally, modulational instability of wave trains is also discussed.

Highlights

According to media reported, “6.1 the Oriental Star cruise ship capsized event” was caused by the violence rainstorm attack which was brought by a sudden squall lines process
Through the analysis of previous research on the gravity wave we can get the following two points: (1) In order to simplify the calculation, the former researches studied the propagation of gravity solitary waves in two-dimension space and barotropic atmosphere [37]
The paper will be organized as follows: (1) using multiscale analysis and turbulence method, from the basic dynamic equations of multivariable in baroclinic environment, we derived out the gravity wave model in Section 2, which is a coupled nonlinear Schrodinger equation (CNLS)

Summary

Introduction

“6.1 the Oriental Star cruise ship capsized event” was caused by the violence rainstorm attack which was brought by a sudden squall lines process. From the original equation of two layers of f-plane, under the shearing basic flow, the famous KdV equation was derived out by Li [6], and he pointed out that the nonlinear characteristic of gravity waves was the generation mechanism of squall lines. Two-dimensional dissipative nonlinear Schrodinger equation [9] was obtained to reflect the evolution of envelope solitary Rossby waves. (1) In order to simplify the calculation, the former researches studied the propagation of gravity solitary waves in two-dimension space and barotropic atmosphere [37]. The paper will be organized as follows: (1) using multiscale analysis and turbulence method, from the basic dynamic equations of multivariable in baroclinic environment, we derived out the gravity wave model, which is a coupled nonlinear Schrodinger equation (CNLS). By observing the structures of the solution, the evolution characteristics of gravity solitary waves was obtained in Section 4. (3) modulational instability of a uniform three-dimensional gravity waves trains was discussed

Derivation of the Three-Dimensional CNLS Equations Group

The Solutions of the Three-Dimensional CNLS Equations Group

Energy Variation Characteristics of Coupled Envelope Gravity Waves

Modulation Instability of a Uniform Gravity Waves Trains in 3-Dimension Space

Conclusions