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

1College of Oceanography, Hohai University, Nanjing 210098, China 2College of Roads and Bridges, Nanjing Vocational Institute of Transport Technology, Nanjing 211188, China 3College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, Shandong 266590, China 4Key Laboratory of Meteorological Disaster of Ministry of Education, Nanjing University of Information Science and Technology, Nanjing 210044, China


Introduction
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.Unfortunately, the forecast for the severe weather such as squall lines is very difficult.As we know, the nonlinearity concentration of the gravity waves makes the energy assemble together and forms disastrous weather phenomena, such as squall lines and rainstorm.So constructing the theoretical model of gravity waves suits the real atmosphere condition and analyzing the formation mechanism of disastrous weather phenomena based on the theoretical model has significant scientific meaning and application value.
The first discovery of solitary waves was attributed to Russell [1]; then, the study of solitary wave continued to deepen.There was vast research literature devoted to the solitary waves.Solitary waves in the westerly shear flow were first found by Long [2].Afterward, Benny [3] amplified the conclusions and got a conclusion that velocity and amplitude of solitary waves were related.Recently, a variety of equation models which described the solitary waves, such as ILW-Burgers equation and ZK-Burgers equation, were discussed by Yang et al. [4,5].Meanwhile the generation and evolution of solitary waves in different topography condition and different fluid depths were discussed.In recent years, in the solitary waves community, gravity solitary wave as a rising star had been paid more and more attention by many scientists.In the 60s, Long [2] proposed that the amplitude of atmospheric gravity wave satisfied KdV equation and got the solutions of amplitude.Then a lot of researchers obtained nonlinear KdV equations of gravity waves from the basic dynamic equation.From the original equation of two layers of -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.S. K. Liu and S. D. Liu [7] based on nonlinear atmospheric motion equation deduced out nonlinear gravity wave solutions and found that gravitational wave amplitude with propagation speed was proportional.Later, Boussinesq equation was derived in Luo [8] to describe the algebraic gravity solitary wave in atmosphere.Additionally, two-dimensional dissipative nonlinear Schrödinger equation [9] was also obtained to reflect the evolution of envelope solitary Rossby waves.
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].But for the envelope of gravity waves in the real atmosphere, it was not sufficient to consider only two dimensions.And the initial equation was a simple model in barotropic atmosphere where  = / 0 ,  0 =  * 0 / 0 ,  * 0 =  *  0 , and  * = (( −   )/).These variables were simplified.Therefore, the threedimensional model in baroclinic atmosphere is more in line with the actual atmospheric conditions.
(2) Previous studies have focused on the solution of gravity waves and the dynamic characteristics of the wave equation mainly based on the numerical calculation from the original equation.As we all know, many of the variables in the original equation are simplified and dimensional; therefore, numerical simulation directly based on the original equation is usually imprecise and difficult.So it is necessary for us to find the appropriate model and solve it to study the evolution of the waves.
In this paper, overcoming the limitations of calculations and using the appropriate method we obtain a new model.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 Schrödinger equation (CNLS).Not only is the model three-dimensional and more suited to describe the feature of two envelope gravity solitary waves in a plane, specially, but also it is a coupled model and can show the interaction process between two waves.(2) Based on the CNLS model, using the trial function method to solve the equations, the analytical solution was obtained in Section 3. By observing the structures of the solution, the evolution characteristics of gravity solitary waves was obtained in Section 4. (3) Finally, modulational instability of a uniform three-dimensional gravity waves trains was also discussed.

Derivation of the Three-Dimensional CNLS Equations Group
Using the sum of disturbance pressure gradient force and buoyancy force, express the vertical pressure gradient force and gravity force, adopt Boussinesq approximation, and the basic dynamic equations of atmospheric motion are as follows: in the above equations,  0 is the temperature of environmental flow field and  0 is the density;  =  0 /.Each variable above will be dimensionless.Let where  is the height of the homogeneous atmosphere,  is the characteristics pressure of the ground,  , is the horizontal pressure changes, and   is the pressure changes in the vertical direction.Substituting (3) into (2), get the dimensionless equations as follows: supposing  ∼  in the above.Because the second term of the left side of the fourth formula is lesser, get the following approximate: introducing parameter  =  2 / 2 ( ≪ 1), where  2 = / 0 .By varying, (4) transforms to the following: Introduction of multiscale variables (omit the sign at the top right corner of the variables) so long time and space scales are defined as , V  ,   ,   ,   in (6) were expanded according to the small parameter : where , , , Θ are the functions of , , where  is the speed of the basic flow,  is the air pressure, and Θ is the temperature field.Substituting ( 8) and ( 9) into ( 6), the zero-order approximation of  can be obtained: It is shown that basic flow is geostrophic equilibrium and static balance.For a basic flow we can further assume that in the first formula in (10), derivation of , and the second formula in (10), derivation of , so that Further, take the first-order approximation of  and introduce the new variables After eliminating other variables in (14), we can get the equation about  0 , as follows: where Clearly, ( 15) is a variable separable equation; assume its solution is under a certain definite solution condition, we can get p0 ; further, all the solutions of ( 14) can be obtained: Taking the second-order approximation of  and introducing the new variables we have let Substitute ( 18) into ( 21) After eliminating other variables in (20), the equation about  1 can be obtained as follows: where Similarly, ( 23) is also a variable separable equation; with comparison on both ends of ( 23), assume its basic solution is further, we can get all solutions of (23): Similar to the above, introducing the new variables, the third-order approximation of  transforms to Let Substitute ( 18) and ( 26) into ( 29) and collecting the secular-producing terms proportional to  (  −  ) , we have After eliminating other variables in (28), the equation about  2 can be obtained as follows: +  3, ( 1 ) +  4, ( 1 ) . ( Introduce the following variables: There are conditions for solution of (31), where

The Solutions of the Three-Dimensional CNLS Equations Group
In the past, Hui [38] gave a solitary wave solution of (2 + 1) dimensional nonlinear Schrödinger equation with the help of a space coordinate transformation.In this chapter, this method will be applied.Based on trial function method, we will discuss the solution of the CNLS equations group.The equations group is If we further introduce (35) reduces to where By rescaling the spatial coordinate and both amplitudes, we can reduce the number of nontrivial coefficients of the CNLS system; the simplified system is where the new unknowns and coefficients are The steadily oscillating solutions of this are solutions and their spatially periodical generalizations.Solitons and cnoidal waves are obtained by assuming where ,  and , V are assumed to be real.Note that, in spite of the cross-modulation, the frequencies in the two modes are generally different.Substitute ( 41) into (39), a set of two coupled ODES equations' group can obtained and can be called SCNLS equations group By rescaling the spatial coordinate and both amplitudes, it is possible to reduce the number of nontrivial coefficients of the SCNLS system from six to three.The simplified system is where the new unknowns and coefficients are To compute the solitary and cnoidal waves, it is always sufficient to solve this system of two ODES.However, to collide solitary waves it is necessary to shift the steadily oscillating solitary waves into a frame of reference such that they move at some translational velocity .
If  1 (, ),  2 (, ) are solutions to the CNLS equations group the another solution is where the velocity  is which implies that the constants  1 and  2 must be proportional as indicated set  =  and define where (, ) is a solution to Substitute (48) into the ODE system (42) and collecting terms in  3 and imposing the condition that the coefficients must be equal 2, the 2 × 2 matrix system is obtained ) .
The solution is The limiting solitary waves for both branches are So the solution of CNLS equations group is

Energy Variation Characteristics of Coupled Envelope Gravity Waves
In this section we will discuss two waves energy variation characteristics in the process of interaction; the first equation of ( 33) is multiplied by  * 1 ; the conjugate equation is multiplied by  1 , subtracting the two types.In the same way, the second equation of (33) Equations (55 Obviously  1 and  2 are energy of two waves.This suggests that the energy of waves is conserved, when the coefficients satisfy or Equation ( 33) has infinite conservation law.Except the two kinds of energy conversation  1 and  2 above, (33) Equations ( 58) to (61) are the momentum of two waves, so we can confirm the conservation of momentum.

Modulation Instability of a Uniform Gravity Waves Trains in 3-Dimension Space
We will discuss the modulation instability for threedimensional gravity wave trains in this section.Based on (53) we can obtain the solution of a uniform wave train.Initial solution of the equation can take the following form: according to the studies of Yuen and Lake [39], the perturbed wave trains may be expressed as follows: where, in fact,  0 denotes the amplitude of a uniform disturbance stream function,  ± is the amplitude of a sideband disturbance and is small enough compared with  0 to permit linearization,  is eigenvalue to be determined, and its conjugate is  * .Informed by Craik [40], ±, ± denotes the prescribed zonal and meridional wave number perturbations in the  and  directions.Here, we define (±, ±) = (±, ±), in which  ≪ 1 and  ≪ 1 are required.Linearization with respect to  0 satisfied the following eigenvalue equation: Mathematical Problems in Engineering If  0 =  0 and  2 =  2 / 4 are defined, we can get if  and  satisfy the uniform gravity wave train is unstable.When  = 0, the result resembles the instability diagram of Benjamin-Feir instability [41].The case of  ̸ = 0 has not been studied yet.To obtain more detailed results, for gravity solitary waves, we only study the case where the fundamental number is 1; that is, where  0 is the latitude.We will discuss the influence of  and latitude value on the instability regions.Taking  = 6,  1 = 2 and  0 = 0.5.In this case the instability growth rates 35 Im  (35 × Im ) of a uniform gravity wave train at 30 ∘ N, 45 ∘ N, and 60 ∘ N for  = 0, 0.08, 0.24 are shown in Figures 1, 2, and 3.
From Figure 1 we know that the instability region is || ≤ 0.38 for  = 0.08, || ≤ 0.37 for  = 0.08, and || ≤ 0.30 for  = 0.24 at the latitude 30 ∘ N.This shows that the instability region of  will become narrower if  increase in the small value limit.Meanwhile, at latitudes 45 ∘ N and 60 ∘ N, there are the same properties as those at latitude 30 ∘ N, as shown in Figures 2 and 3.
In addition, as shown in the figures we can also note that the instability of  will become wider if the latitude increases for the same .It is worth noting that the higher the latitude, the more unstable the uniform gravity wave trains.This conclusion is valid for all small values of .In this paper, the effects of external forcing on the modulational instability of gravity wave trains are not discussed.This problem remains to be further studied.

Conclusions
In this paper, overcoming the limitations of calculations and using the appropriate method, we obtained a new model.Using the 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 Schrödinger equation (CNLS).Not only is the model three-dimensional and more suited to describe the feature of two envelope gravity solitary waves in a plane, specially, but also it is a coupled model and can show the interaction process between two waves.Based on the CNLS model, using the trial function method to solve the equations, the analytical solution was obtained.By observing the structures of the solution, the evolution characteristics of gravity solitary waves were obtained.Finally, modulational instability of a uniform three-dimensional gravity waves trains were also discussed.This shows that the instability region of  will become narrower if  increase in the small value limit.Meanwhile, at latitudes 45 ∘ N and 60 ∘ N, there are the same properties as those at latitude 30 ∘ N. In this paper, the effects of external forcing on the modulational instability of gravity wave trains are not discussed.This problem remains to be further studied.