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Project supported by the National Natural Science Foundation of China (Grant No. 41406018).
In the past few decades, the (1+1)-dimensional nonlinear Schrödinger (NLS) equation had been derived for envelope Rossby solitary waves in a line by employing the perturbation expansion method. But, with the development of theory, we note that the (1+1)-dimensional model cannot reflect the evolution of envelope Rossby solitary waves in a plane. In this paper, by constructing a new (2+1)-dimensional multiscale transform, we derive the (2+1)-dimensional dissipation nonlinear Schrödinger equation (DNLS) to describe envelope Rossby solitary waves under the influence of dissipation which propagate in a plane. Especially, the previous researches about envelope Rossby solitary waves were established in the zonal area and could not be applied directly to the spherical earth, while we adopt the plane polar coordinate and overcome the problem. By theoretical analyses, the conservation laws of (2+1)-dimensional envelope Rossby solitary waves as well as their variation under the influence of dissipation are studied. Finally, the one-soliton and two-soliton solutions of the (2+1)-dimensional NLS equation are obtained with the Hirota method. Based on these solutions, by virtue of the chirp concept from fiber soliton communication, the chirp effect of envelope Rossby solitary waves is discussed, and the related impact factors of the chirp effect are given.
It is generally known that the ocean and atmosphere move endlessly in complex patterns and on diverse scales. To discuss the whole dynamics of this enormous spectrum of motions is almost impossible. So our special interest is the so-called large scale motions which are prominently affected by the rotation effect of the earth. Among these large scale motions, the nonlinear Rossby waves have attracted considerable attention from the oceanic and atmospheric field in recent years. Long first found the Rossby solitary waves in the westerly shear flow in the 1960s, then Benney, Redekopp and other investigators pushed the research forward. In the 1970s∼1980s, driven by blocking dynamics, the nonlinear theory of Rossby solitary waves achieved rapid development and formed the solitary waves theory, dipole waves theory, and envelope solitary waves theory. For the Rossby solitary waves, some weakly nonlinear models for the evolution of Rossby solitary waves have been extensively studied in the past. Among these, the Korteweg–de Vries (KdV) equation,[4–6] modified KdV equation,[7,8] and Boussinesq equation were derived to describe the classical Rossby solitary waves; the Benjamin–Ono equation[10,11] was obtained to show the algebraic Rossby solitary waves with non-uniform horizontal shear of zonal flow in infinite depth fluid, while the Intermediate Long Wave equation[12,13] was used to describe the condition in finite depth fluid. For the envelope Rossby solitary waves, the nonlinear Schrödinger (NLS) equation was first obtained by Benney and Yamagata, respectively. Later, Luo also derived the NLS equation to express the envelope Rossby solitary waves and showed that the atmospheric blocking in the mid-high latitudes could be explained better by employing a dipole envelope Rossby solitary wave. With the development of theory, the higher order NLS equation was derived to describe the dynamic behavior of envelope Rossby solitary waves. However, all the above-mentioned models are (1+1)-dimensional and used to describe the propagation of Rossby solitary waves or envelope Rossby solitary waves in a line, which are not enough to reflect the evolution of Rossby solitary waves in the real ocean and atmosphere. It seems that few previous researches involved the problem. The purpose of this paper is to obtain a new (2+1)-dimensional dissipation nonlinear Schrödinger equation (DNLS), which is more suitable to describe the evolution of envelope Rossby solitary waves in the real ocean and atmosphere. Especially, due to the dissipation character of the real oceanic and atmospheric motion, we will embed the dissipation effect in the study.
On the other hand, in the fiber soliton communication field, the chirp effect is proposed to express frequency modulation in the process of transmission. The interaction of the nonlinear and dispersive effect can cause the excursion of the center wave and lead to chirp effect. Drawing lessons from the chirp concept in fiber soliton communication, Song investigated the chirp effect of an internal wave, while we will discuss the chirp effect during the propagation of envelope Rossby solitary waves.
Besides the above-mentioned problems, we shall use these methods to solve the soliton equations, such as the first integral method, similarity transformation, Jacobi elliptic function expansion method, Bäcklund transformation method, Darboux transformation method, and so on. In the paper, we will derive the one-soliton and two-soliton solutions of the (2+1)-dimensional NLS equation by employing the Hirota method and analyze the chirp effect based on the analytical solutions.
This paper is organized as follows. In Section 2, from the rotational potential vorticity-conserved equation with dissipation, by constructing a (2+1)-dimensional stretching transformation of time and space, a (2+1)-dimensional DNLS equation is derived, which is a generalization of the (1+1)-dimensional NLS equation obtained by the former. In Section 3, by theoretical analyses, the character of the (2+1)-dimensional envelope Rossby solitary waves under the influence of dissipation is studied. Section 4 will be devoted to investigating the one-soliton and two-soliton solutions of the DNLS equation by using the Hirota method and dissipation effect. Based on these analytical solutions, the chirp effect of envelope Rossby solitary waves will be discussed in Section 5. Some conclusions are given in the last section.
The basic equation of motion used in previous studies is the following non-dimensional quasi-geostrophic equation in the form of potential-vorticity:
But we note that a disadvantage exists for Eq. (
where some symbols need to be explained, ψ denotes the non-dimensional stream function, β = β0(L2/U) and β0 = (ω0/R0) cos ϕ0, in which R0 represents the earth’s radius, ω0 is the angular frequency of the earth’s rotation, ϕ0 is the latitude; the first term on the right-hand side of Eq. (
Because of the multiple time and space scales feature of Rossby waves, in order to consider the nonlinear effect, we take the following slow scales:
where ɛ is a small parameter which weighs the weakness of the nonlinearity.
Based on the WKB method, we can expand the stream function as
where the first term on the right-hand side of Eq. (
Here for the sake of achieving a balance between nonlinearity and turbulent dissipation
meanwhile restricting ourselves to discuss the dissipation caused by the perturbation stream and eliminate the dissipation caused by the basic stream, so we take the external source as
Substituting Eqs. (
where for simplicity, we redefine the Laplace operator ∇2 in the plane polar coordinate as
Assume the perturbation does not exist at the boundaries, so we take the boundary conditions as
We look for an amplitude equation, so in the order of ɛ1, assuming ψ1 is periodic in θ and t in the form of
where A is the complex wave amplitude; k represents the polar angle direction wave number of Rossby waves; ω denotes the frequency of Rossby waves; Re stands for the real part.
Substituting Eq. (
The polar axis direction structure on the small scale r of ϕ1 can be determined at the leading order, meanwhile, we can obtain the dispersion relation of linear Rossby waves
and the corresponding eigenfunction
where the eigenvalue m = nπ/r1, n = ±1,2,3, …, and ϕ1 satisfies
In order to obtain the governing equation of complex amplitude A, we need to consider the higher order problem. At the next order, O(ɛ2), by introducing Eq. (
denotes the group velocity of Rossby waves. By employing Eq. (
But we can find that the solution ψ2 cannot be obtained in the first two order equations, so in order to obtain the form of ψ2, it is necessary to consider the next order equation.
To determine ψ2, we then substitute the solution (
where p is a constant. Introducing the transformation Θ = Θ1 − CgT1 and using the time scale T2 = ɛ T1 in Eq. (
Due to ɛ ≪ 1, in view of the fact that the first term of the first equation in Eq. (
It is easy to find that the solution of Eq. (
Substituting Eqs. (
Then multiply by ϕ1 and integrate from 0 to r1 with respect to r, by virtue of the boundary conditions, it turns out that the nonlinear equation for the evolution of complex amplitude A satisfies the following form
where the coefficients are as follows:
Based on Ref. , adopting the following coordinate transformations:
then equation (
The above equation depends on either the time variable T or the space variables Θ and R; iλA stands for the dissipation effect, when λ = 0, i.e., the dissipation effect is absent, the equation (
The theoretical model of envelope Rossby solitary waves has been derived in the above section, now we will discuss the evolution of envelope Rossby solitary waves with the dissipation effect. As we know, many physical characteristics of solitary waves remain invariant during the various processes which happen in the physics world; these invariant physics characteristics are called conservation laws. Conservation laws play an important role in the research on the unsteady problem of waves. We will focus on the change of conservation laws of envelope Rossby solitary waves under the influence of dissipation.
In the following, assuming A,A* → 0 as |R| → +∞ and the values of A,A* at Θ = 0 equal those at Θ = 2π, where the superscript * stands for the complex conjugate.
The complex conjugate equation of Eq. (
Integrating Eq. (
It is easy to find that
where A0 is the initial value of A. Equation (
In the following, performing equation (
Then equation (
Integrating Eq. (
In the same way, by a direct calculation, we have
Based on Eq. (
Finally, let us continue to consider the third conservation quantity. equation (
From Eq. (
By Eq. (
As mentioned above, three conservation quantities of (2+1)-dimensional envelope Rossby solitary waves have been derived. Furthermore, the conservation laws are destroyed due to dissipation, which shows that the dissipation has an important influence for the evolution of (2+1)-dimensional envelope Rossby solitary waves.
In the above section, we have analyzed the variation of conservation laws under the influence of the dissipation effect and revealed the impact of dissipation on the evolution of envelope Rossby solitary waves. This section will be devoted to discussing the chirp effect of envelope Rossby solitary waves and assume that the dissipation is absent. In order to make the reader understand the content more clearly, it is necessary to explain the concept of chirp. The concept of chirp comes from the optical soliton communication area and is also called the frequency modulation effect. During the propagation of the wave, due to the nonlinear and dispersion effect, the excursion phenomenon of the center wave will occur, i.e., the chirp effect will occur. Here drawing lessons from the chirp concept of optical soliton communication, we will study the frequency modulation effect of (2+1)-dimensional envelope Rossby solitary waves. It is very meaningful for recognizing the character of envelope Rossby solitary waves. In order to accomplish this aim, we must explore the analytical solutions of the (2+1)-dimensional NLS equation. As noted previously, many methods have been proposed to solve soliton equations; here we adopt the Hirota method.
Neglecting the dissipation effect (λ = 0), equation (
For Eq.(38), first let us make the following bilinear transform
substituting Eq. (
substituting Eq. (
substituting Eq. (
where p1,q1∈ C. Substituting Eq. (
Letting ξ = 1, then we can get the one-soliton solution of Eq. (
In fact, we can also derive the two-solitons solution of Eq. (
substituting Eq. (
where p1,q1, p2,q2 ∈ C.
After similar procedure and tedious calculation, we have
Then, we can obtain
By virtue of Eq. (
where mi (i = 1,2,3,4), ki (i = 1,2, …,8), li (i = 1,2, …,9) are all constants; for the detailed form see Appendix A. Taking ξ = 1 and substituting Eqs. (
In the following, we will study the chirp effect of the (2+1)-dimensional envelope Rossby solitary waves based on the solution (
according to Eq. (
Observing the time T from 0 to ΔT, where ΔT is an infinitesimal variable, then the following approximate solution of Eq. (
then we can get the phase of the wave as
the chirp caused by dispersion effect is
Next, focusing on the chirp caused by nonlinear, then equation (
In the same way, observing the time from 0 to ΔT, then the approximate solution is
then it is easy to find that the phase of the wave is
So the chirp caused by nonlinear is
By using Eqs. (
By analyzing Eq. (
the whole chirp ΔνS = 0, which means that the dispersion balances to the nonlinear; when
i.e., |ΔνD| > |ΔνN|, then the dispersion effect is more than the nonlinear effect; when
i.e., |ΔνD| < |ΔνN|, then the dispersion effect is less than the nonlinear effect.
In conclusion, the chirp effect depends on the initial amplitude, the critical value is given in Eq. (
In the paper, based on the multiscale and perturbation method, by constructing a new (2+1)-dimensional multiscale transform, the (2+1)-dimensional dissipation NLS equation to describe envelope Rossby solitary waves propagating in a plane under the influence of dissipation is derived. By theoretical analysis, the conservation laws of (2+1)-dimensional envelope Rossby solitary waves is obtained. Furthermore, their variation state under the influence of dissipation reveals the impact of dissipation on the evolution of envelope Rossby solitary waves. By employing the Hirota method, after tedious calculation, the one-soliton, two-solitons solutions of the (2+1)-dimensional NLS equation are given. One-soliton solution is used to discuss the chirp effect and two-solitons solution possesses potential application in the inaction of envelope Rossby solitary waves. Finally, with the help of the one-soliton solution, the chirp effect of (2+1)-dimensional envelope Rossby solitary waves is studied. By analysis, we find the critical value of the chirp effect and the related factors with chirp effect, i.e., the polar angle direction wave number k, the frequency of Rossby waves Ω, the Coriolis force parameter β as well as the modal number m of (2+1)-dimensional envelope Rossby solitary waves.