Research on Time-Space Fractional Model for Gravity Waves in Baroclinic Atmosphere

The research of gravity solitarywavesmovement is of great significance to the study of ocean and atmosphere. Baroclinic atmosphere is a complex atmosphere, and it is closer to the real atmosphere. Thus, the study of gravity waves in complex atmosphere motion is becoming increasingly essential. Deriving fractional partial differential equation models to describe various waves in the atmosphere and ocean can open up a new window for us to understand the fluid movement more deeply. Generally, the time fractional equations are obtained to reflect the nonlinear waves and few space-time fractional equations are involved. In this paper, using multiscale analysis and perturbation method, from the basic dynamic multivariable equations under the baroclinic atmosphere, the integer order mKdV equation is derived to describe the gravity solitary waves which occur in the baroclinic atmosphere.Next, employing the semi-inverse and variationalmethod, we get a newmodel under the Riemann-Liouville derivative definition, i.e., space-time fractional mKdV (STFmKdV) equation. Furthermore, the symmetry analysis and the nonlinear selfadjointness of STFmKdV equation are carried out and the conservation laws are analyzed. Finally, adopting the exp(−Φ(ξ))method, we obtain five different solutions of STFmKdV equation by considering the different cases of the parameters (η, σ). Particularly, we study the formation and evolution of gravity solitary waves by considering the fractional derivatives of nonlinear terms.


Introduction
With the intercross and penetration of different knowledge, Rossby solitary waves have been applied to many fields successfully, such as physical oceanography, atmospheric science, hydraulic engineering, and communication engineering.Particularly, Rossby solitary waves have important theoretical significance and research value in marine atmospheric science.They have largely determined the impact of the oceans on the atmosphere and other climate change.On the time scale, the energy of Rossby waves determines the ocean energy spectrum, which makes the energy spread from east to west to form and maintain a strong ocean boundary flow, such as Kuroshio, Gulfstream, and East Australian flow.Great achievements have been made in this regard.
As we know, Rossby solitary waves in the westerly shear flow were first found by Long [1].He found that the amplitude of the Rossby waves satisfied KdV equation by the -plane approximation   +   +   = 0. ( With the development of Rossby waves theory, Wadati [2] derived the modified KdV equation In view of the barotropic fluid and stratified fluid model, the KdV model and the mKdV model are also generated to describe the generation and evolution of Rossby solitary waves by Redekopp [3].Apart from the KdV model and the mKdV model, for other initial disturbance, employing a different time and space stretching transform, Boussinesq equation was derived by Meng and Lv [4].Afterwards, the Rossby parameters  along with the changes of latitude were discussed by Luo [5] and generalized -plane approximation was obtained.In recent years, in the theoretical study of Rossby waves, many new wave equations were obtained to describe the generation and evolution of various types of fluctuations in the ocean [6,7], such as NLS equation, ILW-Burgers equation, and ZK-Burgers equation.In the past, predecessors studied wave equations in the barotropic atmospheric environment by using the -plane approximation.But we know that the basic dynamic equations for describing the baroclinic atmospheric movement are more in line with the actual situation and are very complicated. And the baroclinic problem in real atmosphere is inevitable [8].In this paper, starting with the basic equations adopting the Bousinesq approximation [9] and under the baroclinic atmospheric environment, using multiscale analysis and turbulence method, we get a new model (mKdV) to describe the Rossby solitary waves.The advantages of basic equations are as follows: (1) The equations are multivariate, and the physical meaning of each variable is clear; (2) The baroclinic atmosphere problem is considered to help us understand the generation and evolution of isolated waves in the ocean.
In recent years, the study of integer partial differential equations has yielded many brilliant achievements [10][11][12][13][14][15].Simultaneously, it has been found that fractional order partial differential equations also play an important role in many fields [16][17][18][19][20][21][22].The fractional differentiation calculus [23,24] was first developed by Liouville primary.Liouville expands the function into an exponential form and defines the order derivatives of this expanded form term by term.Afterwards, Riemann proposed a different definition that can be implemented to a power series with a negative power term.Finally, Ross and Oldham [25,26] unified the two definitions, so that the application of fractional differential was further developed.Subsequently, a version of the Euler-Lagrange equations for problems of calculus of variation with fractional derivatives was formulated by Riewe in 1990s [27,28].Recently, Agrawal [29,30] studied the fractional Euler-Lagrange equation deeper and a series of new methods have been put forward in his research, which provide a new idea for us to study fractional partial differential equations [31,32].The fact has shown that the new fractional equation is more suitable than the integer order equation due to the precise description of the nonlinear phenomena [33,34].At the same time, in the field of oceanography, the fractional partial differential equation can better describe the generation and evolution of solitary waves, which is more favorable for us to study the theory of fluctuation.
Similar to the study of integer order differential equation, the conservation laws of the fractional differential equation are an important branch.As we know, if the fractional differential equation is an Euler-Lagrange equation, then conservation laws can be found using Noethem's theorem by variational Lie point symmetries of this equation [35][36][37].Lie symmetry analysis was proposed by Sophus Lie.The main idea of this method is that the infinitesimal transformation keeps the solution set of the partial differential equation unchanged.The Lie symmetry analysis offers an efficient and powerful tool for the study of conservation laws of fractional partial differential equation [38][39][40][41][42].For this reason, the researchers are very interested in studying the symmetry analysis of fractional differential equations.As far as we know, in the past, the symmetric method was only used for time fraction partial differential equations (TFPDE), but has not been used to analyze space and time fraction partial differential equations (STFPDE) [43,44].In this paper, the Lie symmetry analysis was used to study the conservation law of the STFmKdV equation [45,46].
This paper is organized as follows: In Section 2, using multiscale analysis and turbulence method, from the basic dynamic multivariable equations under the baroclinic atmosphere [59,60], the integer order mKdV equation is derived.In Section 3, we use the semi-inverse method to derive the Lagrangian form of the mKdV equation [61,62] firstly.Then the Lagrangian space and time operator of the mKdV equation have been transformed into the fractional domain of the left Riemann-Liouville fractional differential operator.Finally, applying the variational method, we derive the STFmKdV from this Euler-Lagrangian equation.In Section 4, we first study the symmetry analysis of the fractional equation to obtain the corresponding infinitesimal generator of the equation.Then we discuss the nonlinear self-adjointness of the STFmKdV equation and finally get the conservation vectors of the equation.In Section 5, based on the STFmKdV equation, employing the exp(−Φ()) method, and considering the different cases of the parameters(, ), we obtain five different solutions of the equation.

Derivation of the mKdV Equation
Using the sum of disturbance pressure gradient force and buoyancy force expressing the vertical pressure gradient force and gravity force, and adopting the Bousinesq approximation [60], the dimensionless basic dynamic equations of atmospheric motion are as follows [59]:   = 0. ( where   , V  are the level of the air speed,   is the vertical velocity,   is the atmospheric pressure,  is the temperature field, and  is the Coriolis parameter. 0 is the potential of environmental flow field and   is the density of environmental flow field; they are both the height functions.
Because the second term of the fourth formula in the left side is lesser, we get the following approximation: Let the parameter  =  2 / 2 ( ≪ 1),  2 = / 0 .By varying, (3) transforms to We introduce the multiscale variables (omitting the sign at the top right corner of the variables) so long time and space scales are defined as Further, according to the small parameter ,   , V  ,   ,   ,   in ( 5) can be expended: where , , Θ are the function of , . is the speed of the basic flow,  is the air pressure, and Θ is the temperature field.
Obviously, the zonal flow is in the following forms: and the boundary conditions of (3) are Substituting ( 7) and ( 8) into (3), we get each order form for  as follows: Next, we write the first-, second-, and third-order approximation for  as the following form: where By eliminating  0 ,V 0 , 0 , 0 in (13), we can obtain the equation for  0 where Clearly, ( 15) is a variable separable equation.Assume its solution is and under a certain definite solution condition, we can get p0 .Further, all the solutions of ( 15) can be obtained: Further, we know that Similarly, we cannot get the equation for (, ).So, we eliminate 13) and ( 19), and we can obtain the equation of  1 where Substituting (19) into (20) and observing both ends of (20), we can get the solutions for the variables as follows: Next, we have a further discussion, Simplify ( 13) and ( 23) to the following form: where We know that the homogeneous part in (24) is the same as (15).Substituting ( 18) and ( 22) into ( 19) and ( 23), and according to (13)× 0 /-(20)× 2 /, we can get Integrating ( 27) over the domain [0,  0 ) leads to We know that the two ends of (28) are identically zero, so we can write the simple form as the following form: where

Formulation of the STFmKdV Equation
First we introduce the definition of Riemann-Liouville fractional derivatives and Caputo fractional derivatives [41,63].
Definition 1 (Riemann-Liouville fractional derivative [41,63]).() is a function defined in the [, ], for any nonnegative real , satisfying  − 1 ≤  < , and this formula is called the -order left Riemann-Liouville fractional derivative.And and this formula is the -order right Riemann-Liouville fractional derivative.
Remark 2. We note that if the function () is -order continuous derivable on the interval of [, ], when  tends to , the left fractional derivative is the traditional -order derivative.In addition, if the function () is -order continuous derivable on the interval of [, ], when  tends to , the right fractional derivative is the traditional ( − 1)-order derivative multiplied by (−1)  .Definition 3 (Caputo fractional derivative [41,63]).() is a function defined in the [, ], for any nonnegative real , satisfying  − 1 ≤  < , and this formula is the -order left Caputo fractional derivative.
and this formula is the -order right Caputo fractional derivative.
Lemma 4 (see [61,62]).Riemann-Liouville fractional derivative and Caputo fractional derivative have the following relationship: In this section, based on the generalized mKdV equation obtained in Section 2, we use semi-inverse method and variational method [63] to construct the generalized STFmKdV equation under the Riemann-Liouville derivative definition.Let  0,3 = 1, so that (29) transforms to the following form: where  1 ,  2 are arbitrary constants, (, ) denotes the amplitude of the Rossby waves,  ∈  is the space variable in the propagation of the field, and  ∈ (= [0,  0 ]) is the time variable.The main steps are arranged as follows [62].
First of all, we introduce (, ) as a potential function.Let (, ) =   (, ), so that (36) can be written as The Lagrangian form of ( 36) can be defined using the semiinverse method. is considered as a fixed function.The functional of the potential equation ( 37) can be represented by where  1 ,  2 ,  3 , and  4 are Lagrangian multipliers to be determined later.Integrating (38) by parts and taking Secondly, by applying the variation of this functional with respect to (, ), and integrating each term by parts, optimizing the variation () = 0, we have Equation ( 40) is equivalent to (37), so we can get Mathematical Problems in Engineering

7
A functional relation ( 39) is given to produce the direct Lagrange form of the mKdV equation.

𝐿 (𝑢
Thirdly, according to the Lagrangian form of the integer order, we can obtain the Lagrangian form of the space-time fractional order similarly.

𝐹 (𝐷
where 0 ≤ ,  < 1.The fractional derivative    (, ) or    (, ) in terms of the left Riemann-Liouville fractional derivative is defined by Then, the STFmKdV-Burgers equation takes the following form: The variation of (45) with respect to (, ) leads to Adopting the fractional integration rule and the right Riemann-Liouville fractional derivative,   * () is written as Optimizing the variation of the functional, i.e.,   * () = 0, the Euler-Lagrange form for the STFmKdV equation leads to Substituting ( 43) into (48), we have Substituting    (, ) = (, ) into ( 49), we have the STFmKdV equation In this paper, in order to make the content more complete, then we will study the conservation laws and the solutions of the STFmKdV equation.

Lie Symmetry Analysis and Conservation Laws of the STFmKdV Equation
In this section, we employ Lie symmetry analysis to discuss the conservation laws [64,65] of STFmKdV equaiton which does not contain dissipation item.The details are as follows.

Lie Symmetry Analysis. The STFPDE of a function 𝑓(𝑥, 𝑡)
with two independent variables considered in the Riemann-Liouville sense is defined as the following form: where , ,  are infinitesimal functions and  is a small continuous parameter.The associated Lie algebra is spanned by The prolonged generator can be defined On the basis of the infinitesimal invariance criterion, one can get where The operators  , ,  , ,  , are fractional extended symmetry operators defined as follows [64]: where the symbols   ,   represent the total derivative operators defined by Second, the conserved vectors of the STFmKdV equation are investigated as follows.Applying the second prolongation  (,,2) , we get  , +  , + 2 1     +  1  2  , +  2  , = 0 (59) Substituting ( 57) and ( 58) into (59), and equating the coefficients of alike partial derivatives of , we can obtain the determining equations Solving these equations, the infinitesimals can be derived in the following form: where  1 ,  2 ,  3 are arbitrary constants.Thus, the corresponding infinitesimal generator can be written 4.2.Nonlinear Self-Adjointness.The concept of nonlinear self adjoint is proposed in the application of new conservation theorem to the conservation laws of equations.This concept is extended to the space-time fractional partial differential equations.The lagrangian form of nondissipative STFmKdV equation is given by where V(, ) is a new dependent variable.The adjoint equation of the STFmKdV equation is defined by where / is the Euler-Lagrange operator.According to ( 63) and ( 64), the adjoint equation of STFmKdV equation can be obtained as follows: Here, (   ) * , (   ) * are the adjoint operators of    ,    .For the Riemann-Liouville fractional differential operators, the corresponding adjoint operators [41] have the following form: where  −  ,

𝑚−𝛽 𝑅
are the right fractional integral operators of order  − ,  − , defined by For nonlinear self-adjointness, let us assume V = (, , ), where (, , ) ̸ = 0. Substituting V = (, , ) into (65) By using the method of undetermined coefficients, we obtain the following cases [64,65]: Here,  0 ,  1 ,  2 are arbitrary constants.These solutions of V(, ) are introduced in the Lagrangian form for the construction of conserved vectors in the next subsection.

Conservation Laws.
It is well known that  = (  ,   ) is called the conserved vector that satisfies the conservation equation Noether's theorem is widely adopted to construct the conserved vectors.Using this method, the conserved vectors are obtained by using the Noether operators for the Lagrangian form.According to [40], the fractional Noether operator for the variable ,  has been given.
Before constructing the conserved vectors, we should consider the Lie characteristic function for the Lie symmetry infinitesimal generator  = (  /  ) + (  /  ) + (/).The Lie characteristic function is defined by Then, the fractional Noether operator for  of the STFmKdV-Burgers equation is Similarly, the fractional Noether operator for  of the STFmKdV-Burgers equation is Substituting the Lagrangian form ( 65) and ( 71) into (72) with case 1 in (69), let  0 = 1; the -component of the conserved vectors can be obtained: The -component of the conserved vectors is In case 2 in (69), let  1 = 1,  2 = 0; the -component of the conserved vectors can be obtained as follows: The -component of the conserved vectors is

The Exact Solutions of the STFmKdV Equation
In this part, we deal with the exact solutions [66][67][68] of (50).Using exp(−Φ()) method, the main steps of this method to solve the space time fractional partial equation can be given as follows.
Firstly, we introduce the fractional complex transform: where ] is the wave speed.Substitute (80) into the STFmKdV equation as follows: Equation (81) transforms to an ordinary differential equation Integrating (82) with respect to  and setting the integration constant to zero, we have the following equation: Secondly, balancing the highest order derivative term and the highest order nonlinear term in (83), we get the balancing number  = 1.Thus the solution of (83) takes the form where  0 ,  1 are constants to be determined later.And Φ() satisfies the following auxiliary ordinary differential equation: Substitute ( 84) and ( 81) into (83) and collect all terms with the same degree of exp(−Φ())  together.Equating each coefficient of the same degree of exp(−Φ())  to zero, a set of algebraic equations for  0 ,  1 , , , ] can be obtained.
Solving the algebraic equation system, we can get the solutions.Let  = ±√; the solutions of (86) are expressed by Different solutions of the auxiliary equation ( 85) have been given in [68], so we get the solutions of (81).Different cases are discussed as follows.
Case 1 (hyperbolic function solutions).When Case 2 (trigonometric function solutions).When  2 −4 < 0,  ̸ = 0,  (92) We take the solution of Case 1 as an example to discuss the influence of fractional derivatives  and  on the wave, respectively.
Figure 1 shows that when the space and time derivative are determined, the amplitude of the wave decreases as time increases.
In Figures 2 and 3, we investigate how the parameters ,  affect the nonlinear solitary waves; the corresponding physical interpretations can be given as follows: (1) From Figure 2, we can see that the amplitude of solitary wave increases with increase of  value, while the width of the wave decreases.
(2) Figure 3 shows that when the values of  increase, the amplitude of solitary waves has an increasing tread; however, the amplitude of the wave declines more and more quickly.

Conclusions
In this paper, using multiscale analysis and turbulence method, from the basic dynamic multivariable equations under the baroclinic atmosphere, the integer order mKdV equation is derived.In Section 3, we use the semi-inverse method and variational method to derive the STFmKdV equation under the Riemann-Liouville definition.In Section 4, we extend the symmetry analysis of the fractional equation to obtain the corresponding infinitesimal generator of the equation.Then we discuss the nonlinear selfadjointness of the STFmKdV equation and finally get the conservation vectors of the equation.In Section 5, based on the STFmKdV equation, employing the exp(−Φ()) method, and considering the different cases of the parameters (, ), we obtain five different solutions of the equation.(No. 2017M610436).The authors sincerely thank Xin Chen, Min Guo, Qiulan Zhao, and Jianxin Liu for their contribution to the research.

( 30 )
Remark.According to the above study, we obtain a new model in baroclinic atmosphere.Based on the nonlinear term  2   , the new model is called the generalized mKdV equation.Compared to the model which is obtained in the Mathematical Problems in Engineering barotropic atmosphere, the new mKdV equation is more likely to describe the movement of solitary waves.

Figure 1 : 1 𝑥
Figure 1: The effect of the variation of the dissipation coefficient on the amplitude if the space and time derivative are determined.

Note 5 .
In this paper, we only study the space-time fractional order equation under the Riemann-Liouville derivative definition.In future studies, we can also consider the fractional equation under the Caputo derivative definition.