A Kind of New Coupled Model for Rossby Waves in Two Layers Fluid

In this article, a new model for Rossby waves in two layers fluid is studied. Form the dimensionless baroclinic quasi-geostrophic vortex equations include exogenous and dissipative, the new (2 + 1)-dimensional coupled ZK-mZK equations are established by multiscale analysis and perturbation method. Based on the semi-inverse and Agrawal’s method, the time-fractional coupled ZK-mZK equations are derived. Then, Lie symmetries and conservation laws of time-fractional equations are analyzed. Finally, the exact and numerical solutions of the time-fractional coupled ZK-mZK equations are obtained by the Jacobi elliptic function expansion method and alternative variational iteration method. The relative errors between solutions show that the alternative variational iteration method gives a high-precision numerical solution. Further, propagation of Rossby waves in two layers fluid is affected by time, order of fractional derivative and coefficient of coupling term.


I. INTRODUCTION
Many large-scale fluids in nature and engineering, such as the atmosphere and oceans, are stratified. The study of stratified flow plays an important role in national economic construction, people's life and environmental protection [1]- [3]. The existence of solitary waves in hydrodynamics has been known over a century. If there are waves in stratified fluid, they will interact with each other. Therefore, it is of practical significance to establish the coupling model in the stratified fluid. Compared with the single equation of low dimension, the coupled equations of high dimension are more practical in the actual marine-atmosphere system [4]- [6]. The coupled equations models are also used in many scientific fields, such as fluid dynamics, aerodynamics and mass transport, but it mostly describes processes with weak perturbations.
As a matter of fact, the problems we studied often do not entirely meet ideal situation. Therefore, the effect of friction dissipation and viscous dissipation should be considered. The application of dissipative effect in the atmospheric dynamics The associate editor coordinating the review of this manuscript and approving it for publication was Lei Wang. and oceanic dynamics is more obvious. So, the shallow water equation is still too rough as the starting equation in the treatment of such problems [7], [8]. In this article, we only consider the large scale shallow water equations with dissipation in the horizontal direction [9]. In the future, we will study the shallow water equation with dissipation in both the horizontal and vertical directions.
Because of the complexity of nature world, many of the systems we studied are non-conservative [10]. In 1996, the fractional derivative term was introduced into the functional by Riewe [11], [12] to obtain the non-conservative term needed in the differential equation. In the years since, fractional calculus has developed on the basis of nonlinear science [13]- [15]. Nowadays, it is more important to establish fractional derivative model for daily production and life. Conservation laws mean the mathematical formula in which the total amount of a physical quantity remains constant during the evolution of a physical system [16]- [18]. The Lie symmetries and conservation laws of nonlinear partial differential equations play important roles in the study of nonlinear physical phenomena. VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ The solution of partial differential equation is of great practical significance [19], [20]. Therefore, many different methods are used to calculate the exact and numerical solutions of the equation [21]- [23]. In order to reflect the dissipative effect more intuitively, the Jacobi elliptic function expansion method which can produce periodic solution can be chosen to derive the exact solution [24], [25]. Compared with exact solutions, numerical solutions have fewer restrictions on the equation and are allowed complex structures, but they have certain requirements on the selection of initial values. In 1998, Chinese mathematician He proposed the variational iterative method (VIM) for solving fractional differential equations. This method has been applied to the solution of strongly nonlinear equations. Some ameliorative measures are put forward to improve the convergence speed and extend the convergence interval of the solution of VIM series. Variational iterative method has been widely adopted by many scholars after years of research [26], [27].
The structure of the paper is as follows: In Section 2, we consider the Rossby waves in two layers fluid with dissipation and exogenous. Form the dimensionless baroclinic quasi-geostrophic vortex equations include exogenous and dissipative, the new coupled ZK-mZK equations are yielded by using the method of multiscale expansion and disturbance analysis. As the the Agrawal method, the semiinverse method, and the fractional variational principle used, the time-fractional coupled ZK-mZK equations are obtained in section 3. In section 4, according to the Lie group theory, the generators are obtained, and then the conservation laws of the equations can be established. In Section 5, the exact and numerical solutions of fractional equations are separately given by using the Jacobi elliptic function expansion method and alternative variational iteration method. Especially, the relative errors between the exact solution and numerical solution are analyzed. In addition, we analyze and discuss the effect of time, order of fractional derivative, and coefficient of coupling term. Finally, the full text is summarized.

II. DERIVATION OF THE COUPLED ZK-mZK EQUATIONS
Stratified flow has multiple layers of fluid. To facilitate the study, we first simplified the model to a two-layer fluid. A two-layer fluid can be divided into upper and lower layers. If there are waves in two layers fluid, they will interact with each other. In this article, we study the dimensionless barotropic quasi-geostrophic vortex equations including exogenous and dissipative in two layers fluid as following where ψ A and ψ B are the stream functions of upper layer and lower layer fluid respectively, F 1 and F 2 represent the weak coupling coefficients between two layers fluid respectively, f is the Coriolis parameter, s = N 2 /f , N is a physical quantity that measures the level of layer stability, and stands for Brunt-Vaisala frequency, ρ s = ρ s (z) for density. The boundary condition is z = 0. Then (1) can be expanded to Next, in order to obtain the coupled ZK-mZK equations, we separate the stream functions ψ A and ψ B into the basic stream functions and the disturbance stream functions. The form of the stream functions are as follows Consider that the coupling between the two layers fluid is weak, and the rotation effect of the Earth is smaller. Therefore we set transformations as where ε is a small parameter. Taking the x-directional longwave approximation, we also introduce the stretched variables as Then, we expend the perturbation stream functions into the following form Substitute (3), (4), (5), and (6) into (2). Equations for the coefficient on the parameter ε 2 are obtained as follows Integrating with respect to X and y, we get a set of equations that are related only to y, Form the expression in (8), we assume that functions φ A 1 and φ B 1 have the form of separate variables and can be rewritten as the following special form Equations (8) are equal to We obtained the following equations for the coefficient on the parameter ε 3 ε 3 : Substituting (9) and (10) into (11) and integrating with respect to X , we get Form the expression in (12), we take the form of separate variables of φ A 2 and φ B 2 as the following form where m i (i = 1, 2, . . . , 8) can be determined later. Substituting (13) into (14), we can obtain Collecting the coefficient of the parameter ε 4 , we have ε 4 : Substituting (9), (10), and (13) into (15), we have a set of identities. In the general process of dealing with such identities, φ A 3 and φ B 3 would be set to zero. But this approach may lead to inconsistencies. To avoid this kind of situation, simple and ambiguous methods are usually used in previous research discussions, e.g., integrating of variable y from y 1 to y 2 . But in this article, we do not take φ A 3 = φ B 3 = 0 nor to integrate of variable y from y 1 to y 2 . In order to obtain the coupled ZK-mZK equations for A 1 and A 2 in this article, we only give out one potential alternative of φ A 3 and φ B 3 as where and R i (i = 1, 2, . . . , 11), S i (i = 1, 2, . . . , 11) are in Appendix A. By calculation, the terms containing y can be eliminated, and the equations with X and T can be obtained. We get that A 1 and A 2 are suitable for the relationship of the VOLUME 8, 2020 following coupled ZK-mZK system: where constants α ij (i = 1, 2, . . . , 6, j = 1, 2) are arbitraries in R i (i = 1, 2, . . . , 11) and S i (i = 1, 2, . . . , 11).
Equations (17) are the two-dimensional coupled ZK-mZK (Zakharov-Kuznetsov-modified Zakharov-Kuznetsov) equations. Compared with the general mZK equation, coupled ZK-mZK equations are a system of equations with weak coupling terms α 13 (A 1 A 2 ) X , α 23 (A 1 A 2 ) X . Compared with the general ZK system, coupled ZK-mZK equations have both coupling terms α 13 (A 1 A 2 ) X , α 23 (A 1 A 2 ) X and high-order nonlinear terms α 14 (A 3 1 ) X , α 24 (A 3 2 ) X . It can be seen from the coefficient relationship of the equations that the coupling constants have effects on the coefficients of terms of A 1X , but have no effect on the coefficients of terms of (A 3 1 ) X , (A 3 2 ) X . And we can see that the coupling terms are up to the square terms, and the highest degree terms are not coupled.

III. THE TIME-FRACTIONAL COUPLED ZK-mZK EQUATIONS
In order to build a more suitable model to describe the non-conservative system, we extend the integral derivatives equations (17) to the field of fractional derivatives. Take integral derivatives equations as a special case of fractional derivatives equations. In this section, we will derive the timefractional coupled ZK-mZK equations by the semi-inverse method, the variational method and Agrawal method.
In the first place, we introduce two potential functions Substitute potential functions into (17), and the potential equations of the coupled ZK-mZK equations can be written as Next, in order to obtain the Lagrangian equations of the coupled ZK-mZK equations, we have written the functional of (18) as where = R×R×T * , and m i , n i (i = 1, . . . , 9) are Lagrange coefficients which will be determined later. For the function in the first parenthesis in (19), integration by parts of U , and for the function in the second parenthesis, integration by parts of V , and taking We derive the first order variational equations of functional equation (20) by using the variational method. Variational equations can be expressed as By the semi-inverse method, it is obvious that (18) and (21) are equal. So, the Lagrange coefficients can be obtained, that is m 1 = m 2 = m 4 = m 6 = m 7 = n 1 = n 2 = n 4 = n 6 = n 7 = 1/2, m 3 = n 3 = 1/3, m 5 = n 5 = 1/4. Substituting the Lagrange coefficients into (21), we get Lagrangian forms as At the moment, we have the Lagrangian equations for integral derivatives. Then, we use Agrawal's method to extend the Lagrangian equations for integral derivatives to the Lagrangian equations for fractional derivatives. The Lagrangian forms of the time-fractional coupled ZK-mZK equations are given as where D α T is the operator of fractional derivative, T is independent variable, α is the order of the derivative and α can be a fraction. Similar to the integral derivatives equations, the functional of the time-fractional coupled ZK-mZK equations is Using the variational method of Agrawal's method, we have For the function in the first parenthesis in (25), derive the first order variational equations of U , and for the function in the second parenthesis, derive the first order variational equations of V . The Euler-Lagrange equations for the time-fractional coupled ZK-mZK equations can be expressed as The last step is to substitute the expressions for F 1 , F 2 given by (23) into (26), and we get , and the final equations are Equations (28) are the time-fractional coupled ZK-mZK equations. When α = 1, they are equal to (17). This new set of fractional derivative equations is more suitable to describe the non-conservative system. It provides more possibilities for future research.

IV. CONSERVATION LAWS OF THE TIME-FRACTIONAL COUPLED ZK-mZK EQUATIONS A. LIE SYMMETRY ANALYSIS
We assume that (28) are invariant under a one parameter Lie group of point transformations in the following form are the prolongations of infinitesimal functions which are defined as VOLUME 8, 2020 where D T , D X , and D Y are the total derivative operators as follows Apply the generalized Leibnitz rule as follows where C n = (−1) n−1 α (n−α) (1−α) (n+1) , and α > 0. The chain rule for compound function which is defined as For the chain rule (33), when f (T ) = 1, we can get where i = 1, 2, and So, when i = 1, j = 2 or i = 2, j = 1, (34) can be written as The infinitesimal generator V can be defined as Under the infinitesimal transformations, the invariance of the system (28) lead to the invariance condition as follows According to (37) and (38), we can gain the following invariance criterion Substituting (30), (31), and (36) into (39), by uniting the similar terms of A and its derivative, we can get a bunch of equations about the coefficients. When α 11 = α 12 , by solving the equations, a set of Lie algebra of point symmetries will be obtained as follows Hence, a series of Lie algebra of point symmetries can be written as

B. CONSERVATION LAWS
We obtain Lie symmetry generator above. According to it, we will discuss conservation laws of the time-fractional coupled ZK-mZK equations in this section. We know that the conserved vectors satisfy conservation equation in the following form where C T , C X , and C Y are conserved vectors. A formal Lagrangian for the time-fractional coupled ZK-mZK equations can be presented as follows where θ 1 = θ 1 (X , Y , T ) and θ 2 = θ 2 (X , Y , T ) are new dependent variables. Accorded to the formal Lagrangian, an action integral is defined as the following form So, we can get the adjoint equations of (28) as Euler-Lagrange equations where δ δA 1 and δ δA 2 are the Euler-Lagrange operators. They are defined as δ δA 1 = where (D α T ) * is the adjoint operators of the Riemann-Liouville fractional differential operator D α T , which is given by where I n−α p is the right-sided fractional integral operator of order n − α. And C T D α p is the right-sided fractional differential operator of order α. So, the adjoint equations (45) can be rewritten as On the basis of last section, we get infinitesimal symmetry of (28). In order to get the conservation laws, we assume that the Lie characteristic functions W i (i = 1, 2) are as follows Applying on the V j (j = 1, 2, 3, 4) of the symmetry (41), we have By using the Riemann-Liouville fractional derivative, the component of conserved vectors of (28) are defined as , ) + · · · + · · · , where m = 1, 2, 3, 4 and i, j = X , Y . When n = [α] + 1, J is the integral as follows (50) VOLUME 8, 2020 When m = 4, we can have the following components of conserved vectors Equations (41) are Lie algebra of point symmetries of the time-fractional coupled ZK-mZK equations and (51) are conservation laws of the time-fractional coupled ZK-mZK equations. Lie symmetries and conservation laws are one of the properties of fractional partial differential equations. The forms of Lie symmetries and conservation laws play an important role in the analysis of the stability of equations and the solution of some special structures. However, in the process of literature review, it is found that the conservation laws of fractional order equations are rarely mentioned.

V. SOLUTIONS OF THE TIME-FRACTIONAL COUPLED ZK-mZK EQUATIONS
In this section, two methods will be used to solve the exact and numerical solutions of (28). Firstly, we calculate the exact solution by applying the Jacobi elliptic function expansion method. Next, we will derive the asymptotic numerical solution of the system by referring to AVIM. Then, the relative errors between the numerical solution obtained by the AVIM method and the exact solution obtained by the Jacobi elliptic function expansion method at different points are compared. In the end, one set of numerical solutions is taken as an example to analyze the influence of time, order of fractional derivative and coefficient of coupling term.

A. EXACT SOLUTIONS OF THE TIME-FRACTIONAL COUPLED ZK-mZK EQUATIONS
We apply the Jacobi elliptic function expansion method to calculating the exact solutions. In the first place, we use the wave variable as follows: where k, l, and n are undetermined positive parameters, and n is the velocity of propagation. So, (28) simplify to polynomials of A 1 , A 2 , and its total derivatives In order to cancel out higher order nonlinear term A 2 1 A 1 , A 2 2 A 2 and higher derivative terms A 1 , A 2 , the functions are assumed to have solutions of the form where Z (ξ ) satisfies the Jacobi elliptic equation: and a, b, c, a 0 , a 1 , b 0 , b 1 are constants which can be determined later. Substituting (54) along with (55) into (53) and collecting all the coefficients of Z i (ξ )(i = 0, 1, 2, . . .), then setting these coefficients to zero, we yield a set of algebraic equations, which can be solved to find the values of k, l, n, a, b, c, a 0 , a 1 , b 0 In this case, (28) have the solutions where ξ = k(X + Y − (α 11 + 2α 12 a 0 + α 13 ). In the particular case where m → 1, sn(ξ ) → tanh(ξ ), the solutions are: where ξ = k(X + Y − (α 11 + 2α 12 a 0 + α 13 In this case, (28) have the solutions ). In the particular case where m → 1, cn(ξ ) → sech(ξ ), the solutions are: where ξ = k(X + Y − (α 11 + 2α 12 a 0 + α 13 In this case, (28) have the solutions where ξ = k(X + Y − (α 11 + 2α 12 a 0 + α 13 ). In the particular case where m → 1, dn(ξ ) → sech(ξ ), the solutions are: where ξ = k(X + Y − (α 11 + 2α 12 a 0 + α 13 ). In order to give a better understanding of the properties of Rossby waves, we take the appropriate parameters of the above exact solution and draw some three-dimensional diagrams. In the process of plotting, we find that (57) and (58) have the same tendency under the appropriate parameters. So we only plot Case 3.
It can be seen from the Fig. 1-4 that the fractional derivative may cause drastic changes in the form of the solution, or it may be more in line with the actual situation. In the subsequent process of numerical solution calculation,   the deformation of solutions (58) with a more stable trend is adopted.

B. SOLUTIONS OF THE TIME-FRACTIONAL COUPLED ZK-mZK EQUATIONS BY AVIM
In this part, we will apply the alternative variational iteration method to determine the solution of the time-fractional coupled ZK-mZK equations. Considering (28) and applying the alternative variational iteration method, we construct the following correction functionals where λ 1 and λ 2 are the general Lagrange multiplier whose optimal value are found via variational theory, And A 1 (X , Y , 0), A 2 (X , Y , 0) are initial approximations. Make the above correction functionals stationary, and use the initial conditions: we yield the Lagrange multiplier as the following forms λ 1 = −1, and λ 2 = −1.
Therefore (59) can be rewritten as In our work, based on the alternative variational iteration method, we can obtain the variational iteration solution: u(x, t) = ∞ k=0 v k (x, t), using the following iteration formula for m − 1 < α ≤ m: which converges to the solution if 0 < γ < 1 exists such that v k+1 ≤ γ v k , ∀k ∈ N ∪ 0. The selection of initial value has a crucial influence on the solution of numerical solution. So, we choose the initial value of the numerical solution based on the exact solution. It can be seen from Fig. 1-4 that the form of (58) is more stable. In these circumstances, we start with initial conditions By the above formulae (62), we can obtain a few first terms being calculated: where q 1 = α 15 m + α 16 n, q 2 = α 25 m + α 26 n, p 1 = α 12 c 1 + α 13 c 2 , p 2 = α 23 c 1 + α 22 c 2 . Due to the complexity of the structure of the numerical solutions, we give the form of A 12 and A 22 in Appendix B.
The higher-order approximation can be calculated to the appropriate order using the Maple or Mathematica package, where the infinite approximation leads to the exact solutions.

C. RESULTS AND DISCUSSION
In order to judge the practicability, veracity and reliability of the methods proposed in this study, the relative errors are discussed between the numerical solution obtained by the alternative variational iteration method and the exact solution obtained by the Jacobi elliptic function expansion method.
Take the first equation of (58) as the exact solution, and calculate A 14 as the numerical solution. We take α = 0.3, α = 0.7, and α = 1, respectively, in Tab. 1-3 at different points of X and T . When α = 1, it means that time derivative is in the order of integers. From the tables, it is obvious to see that the numerical results of the the time-fractional coupled ZK-mZK equations obtained by the alternative variational iteration method are satisfactory. This is because we  have solved both the exact solution and the numerical solution, and the exact solution is the basis of the numerical solution. This is a good way to ensure the stability of the solution.
The quasi-geostationary vortex equations are of great practical significance, and the solution of the model established in this article is instructive to the practical production and life in the future. Therefore, we select appropriate values for different parameters to study and discuss their variation trends.
In Fig. 5, three curves with different trends can be clearly seen. When α = 1, A 1 > 0. When α = 0.5, A 1 on the left half of the X -axis can take a negative value, but can not take a negative value when X > 1. And when α = 0.3, A 1 can get negative values on both sides of the X-axis. That is to say, the order of fractional derivative will affect the number and height of peaks and troughs produced by the interaction. Although the fractional order model and integer order model have obvious differences in some places, all of them can describe natural phenomena. Of course, which model is better depends on the question being studied. In Fig. 6, the number and height of peaks and troughs also varies significantly with the time of interaction. Within a certain range, the number of peaks and troughs increases as the time of interaction increases. That is to say, in the short term, an already unsta-  ble wave becomes more unstable as the time of interaction increases. Fig. 7 and Fig. 8 are the effect of the coefficients of the coupling terms of (28) on the interaction. The corresponding two sets of curves in Fig. 7 are obtained by taking symmetric values from the coupled ZK-mZK equations, that is, the interaction problems between the two fluid are completely symmetric. At this time, as the coupling term coefficient increases, the wave width will increase and the wave height will decrease. This is probably due to the fact that the wave widens and cancels out with another wave. The corresponding two sets of curves in Fig. 8 are obtained by the coupled ZK-mZK equations without symmetry, that is, the interaction problems between the two fluid are not symmetric. At this point, the coupling term coefficient of the large increase first, the coupling term coefficient of the small increase. This may be due to the large coupling term coefficient and the large influence in the interaction process. The relative positions of its peaks and troughs are not very different.

VI. CONCLUSION
In this article, the Rossby waves in two layers fluid with dissipation and exogenous is studied. Form the dimensionless baroclinic quasi-geostrophic vortex equations include exogenous and dissipative, a new (2 + 1)-dimensional coupled ZK-mZK equations is established. In the weakly coupled system, the coupling terms are generally linear, but in this article, the coupling terms we get are nonlinear terms. We can see that the coupling terms are up to the square terms, and the highest degree terms are not coupled. Then, we deduced the timefractional coupled ZK-mZK equations and analyze its conservation laws, which provides a basis for the theoretical study of the Rossby waves in two layers fluid. In order to study the properties of solitary waves, we derived the exact solutions and the numerical solutions of the time-fractional coupled ZK-mZK equations. By comparing the relative errors of the numerical solutions and the exact solutions, the accuracy of the alternative variational iteration method can be judged. From the numerical results above, it can be seen that the alternative variational iteration method gives a high-precision numerical solution of the time-fractional coupled ZK-mZK equations. This suggests that the stability of the solution is greatly improved when the stable exact solution is taken as the initial value of the numerical solution. It can be seen form trends of variation in figures that the wave width, the wave height, and the number of peaks are changed when time, order of fractional derivative, and coefficient of coupling term changed. Especially, as the coupling term coefficient increases, the wave width will increase and the wave height will decrease, which has the instruction to the future actual production life.