Abstract

Using the method of multiple scales and perturbation method, a set of coupled models describing the envelope Rossby solitary waves in ()-dimensional condition are obtained, also can be called coupled NLS (CNLS) equations. Following this, based on trial function method, the solutions of the NLS equation are deduced. Moreover, the modulation instability of coupled envelope Rossby waves is studied. We can find that the stable feature of coupled envelope Rossby waves is decided by the value of . Finally, learning from the concept of chirp in the optical soliton communication field, we study the chirp effect caused by nonlinearity and dispersion in the propagation of Rossby waves.

1. Introduction

Wave phenomenon exists widely in nature. As a special and important branch of waves, Rossby solitary waves have important theoretical significance and research value. Meanwhile, with the intercross and penetration of different knowledge, the Rossby solitary waves theory has applied to many other fields successfully, such as physical oceanography, atmospheric physics, hydraulic engineering, communication engineering, and thermal power engineering. In the case of application of solitary waves in engineering, the application of optical soliton in communication engineering is the most representative. Meanwhile, the Rossby waves theory was widely used in the study of mesoscale eddies and the interaction of large and medium scale motions. Rossby solitary waves in the westerly shear flow were first found by Long [1]. Afterward, Benny [2] amplified this research and found that velocity and amplitude of Rossby waves were proportional and depicted the importance of nonlinearity. In view of the barotropic fluid and stratified fluid model, the KdV and mKdV equation are also generated to describe the generation and evolution of Rossby solitary waves by Redekopp [3]. Compared to KdV equation, the mKdV equation is more suitable to express the condition with stronger perturbation. With the development of solitary waves, a variety of equation models for describing the Rossby solitary waves such as ILW-Burgers equation and ZK-Burgers equation were discussed by Yang et al. [4, 5]. Moreover, the generation and evolution of solitary waves in different topography condition and different fluid depths were discussed. Recently, the Rossby parameters along with the changes of latitude were discussed by Luo [6] and plane approximation was obtained. These Rossby solitary waves are called classical solitary waves which related to the KdV-type equations. Later, many researchers have studied the Rossby wave equation in many aspects [7, 8], such as integrable system [9, 10], the integrable coupling of equations [11], and Hamiltonian structures [12]. Unlike the KdV-type equation, the nonlinear Schrödinger equations were used to study the evolution of envelope classical Rossby solitary waves. From the end of the 70’s to the 80’s, driven by the study of atmospheric blocking dynamics [13], nonlinear Rossby wave theory had been developing rapidly and gradually formed Rossby solitary waves theory and dipole waves theory. In addition, beside the above two theories, envelope Rossby solitary waves also dropped in the research scope of the theme. The envelope Rossby solitons in the barotropic shear and uniform flows were first investigated by Benney [14] and Yamagata [15]. Afterward, Luo [16] tried to use this envelope Rossby solitons to explain atmospheric blocking phenomenon. Later, dissipative NLS equation in rotational stratified fluids and its solution were obtained by Shi et al. [17]. As we all know, using the nonlinear Schrödinger equation describing the Rossby solitary waves, we can introduce the concept of chirp in optical soliton communication [18], to study the influence of dispersion and nonlinearity on solitary waves propagation process. In the optical soliton communication field, the concept of chirp [19, 20] is the phenomenon that the central wavelength shifts when the pulse is transmitted. It is helpful to analyze the propagation characteristics and the formation mechanism of solitary waves.

In the domain of solitary wave models, it is necessary to obtain the exact solutions of solitary wave models by all kinds of solving methods and analyze the feature of solitary waves in propagation process based on the exact solutions. Many methods to solve the equations are proposed, such as traveling wave method [21], Darboux transformation method [22–24], Hirota method [25, 26], homogeneous balance method [27], Jacobi elliptic function method [28], Symmetry method [29, 30], Rational solutions [31–33]; meanwhile the features of equations are also discussed [34–36]. In this paper, we plan to adopt the trial function method and derive the exact solution of model. The difference between the two-dimensional and three-dimensional model will be given and some features of three-dimensional NLS equation will be discussed.

We note that the above researches commonly considered two-dimensional model or single ()-dimensional model to reflect the evolution of envelope Rossby solitary waves. There are two disadvantages:(1)The two-dimensional model can only be applied to describe the evolution of envelope Rossby solitary waves in a line.(2)The velocity of the KdV-type soliton is larger than the real observation.

While, as we know, the ()-dimensional model can be applied to reflect the evolution of envelope Rossby solitary waves in a plane, which is more suitable for the real ocean and atmosphere, in this paper, by using the method of multiple scales and perturbation method, starting from the barotropic atmospheric vorticity equation, we will derive the coupled ()-dimensional nonlinear Schrödinger equations for envelope Rossby solitary waves in Section 2. Not only is the model ()-dimensional and more suitable to describe the feature of two envelope Rossby solitary waves in a plane, particularly, but also it is a coupled model and can show the interaction process between two waves. Then, based on trial function method, we will deduce the solution of the CNLS equations group and the envelope solitary waves characteristics in Section 3. Thirdly, we study the modulation instability of Rossby waves trains in ()-dimensional condition in Section 4. Finally, the concept of chirp in optical soliton communication is introduced, and the chirp effect caused by dispersion and nonlinearity is also discussed in Section 5.

2. Derivation of the (2 + 1)-Dimensional CNLS Equations

The tropical atmospheric motion is quasihorizontal and quasiconvergent. The governing equation is the quasigeostrophic barotropic vorticity equation [37].and the boundary condition iswhere and are the local Cartesian coordinates pointing east and north. In this , is Rossby parameter. is characteristic velocity, is characteristic length, and is the width of the beta-channel. is small parameter, on behalf of nonlinear strength. is the Jacobian of .and is Laplace operatorIn general, it is difficult to get analytic solution of (1). But, because of the nonlinear term with a small parameter , we use the multiple scales method and perturbation method to obtain the nonlinear solution of it. As a preliminary study, we will consider two waves.

Let us introduce the slow time and space variablesso long time and space scales are defined asSubstituting (6) into (1), we obtainFurtherand the boundary condition isThe stream function is expanded according to the small parameter and, substituting (9) into (8), we get the multiple order questions of the stream function .

First, one introduces an operatorso thatAssumeand, substituting (13) into , we haveand furtherand the boundary condition iswhereFrom (15), we can get the solution of :Formula (15) under the boundary condition poses a standard Sturm-Liouville problem. The effects of zonal flows on linear equation trapped waves were treated in detail by many researchers. The analytic solution of (15) can be obtained when takes some specific functions. Under normal circumstance one can only seek numerical solution, so we consider the higher order questionand, further, we getwhereand the special solution corresponding to the second, third, and forth item of (21) right side iswhere , , and satisfied and boundary conditionWe assume the special solution corresponding to the first item of the right side of (20) iswhere satisfied The following operators are introduced:Substituting (28) into (27) and because is infinitesimal, we can ignore the first item Obviously the solution for may be expressed in the following form:So we obtained the solution of (20):So as to obtain the solution of , we continue to consider the question of . Substituting (19) and (30) into and collecting the secular-producing items proportional to , we haveFor the sake of obtaining the evolution equation of , we consider another class of nonhomogeneous solutions, assumingintroducing (33) into (32), whereWe assume two special nonhomogeneous solutions of (33) areand, substituting (35) into (34), we getmultiplying (35) by , and integrating on from to . The two sides of (35) are equal to zero, so we get the solution conditions:We get the evolution equations group of wave amplitude, that is, the coupled equations group. To simplify, we introduce the following transformation:and then (37) can be written aswhere In (38), coefficients , , , and are dispersion coefficients, , are Landau coefficients, , are interaction coefficients, and from their expressions it can be seen that their values are related to the base flow function . The (38) is called CNLS equations group.