- Research
- Open Access
Combined ZK-mZK equation for Rossby solitary waves with complete Coriolis force and its conservation laws as well as exact solutions
- Bao Jun Zhao^{1, 2},
- Ru Yun Wang^{1}Email author,
- Wen Jin Sun^{3} and
- Hong Wei Yang^{4}
https://doi.org/10.1186/s13662-018-1492-3
© The Author(s) 2018
- Received: 13 November 2017
- Accepted: 15 January 2018
- Published: 25 January 2018
Abstract
In the present paper, a new partial differential equation has been obtained to describe the Rossby solitary waves with complete Coriolis force by employing multi-scale analysis and perturbation method, we call it combined ZK-mZK equation. The equation can reflect the propagation of Rossby waves on the plane and is more appropriate for the real ocean and atmosphere than the \((1+1)\) dimensional models (such as KdV and mKdV), which can only represent the propagation of Rossby solitary waves in a line. Furthermore, by adopting the multiplier method, we construct conservation laws of the combined ZK-mZK equation, which is meaningful for researching the global stability of solutions. Finally, we deduce the exact solutions of the combined ZK-mZK equation via the semi-inverse variational principle. By applying these exact solutions, some propagation features of Rossby solitary waves are analyzed.
Keywords
- combined ZK-mZK equation
- Rossby solitary waves
- complete Coriolis force
- semi-inverse variational principle
- conservation laws
PACS Codes
- 02.30.Jr
- 43.75.Fg
- 92.10.Hm
1 Introduction
The Rossby waves on behalf of huge vortices which exist in the ocean and atmosphere play an increasingly significant role in transporting energy, which can determine the weather and climate change in the earth to a large extent [1–3]. In the last few decades , the research on constructing mathematical models to study the generation and evolution of Rossby waves has attracted a lot of attention, and a series of mathematical models, such as KdV [4, 5], mKdV [6, 7], ZK [8, 9], modified Kawahara equation [10], and so on [11, 12], have been obtained. Meanwhile, some natural phenomena related to Rossby waves were explained with the help of mathematical models [13]. We notice that the former research has the following two disadvantages:
(1) The motion equations describing ocean and atmosphere, including momentum equation, continuity equation, and so on, are very complicated. For the sake of simplicity, we can find that \((1+1)\) dimensional nonlinear partial differential equations are used to describe the evolution of nonlinear Rossby waves. However, real oceanic and atmospheric motions are not just in one direction. Providing higher dimensional theories for the nonlinear Rossby waves is necessary. So, in this paper, we discuss a new \((2+1)\) dimensional model [14].
(2) All the time, the complete Coriolis force has been an important research hot spot in dealing with the atmosphere and ocean. On the one hand, the horizontal component of the complete Coriolis force shows the imbalance of the motion, which is an important factor that causes the development of weather system. On the other hand, the theoretical research on the nonlinear Rossby waves in the atmosphere has been a very important subject in meteorology. So, it is necessary to analyze the effect of intact Coriolis force on atmospheric dynamics. However, in order to calculate and study it conveniently, many researchers ignore the complete Coriolis force. As we know, the TA is suitable for quantitative studies. In the past, Kasahara [15] pointed out that the errors made by ignoring the vertical acceleration may be much larger than those made by ignoring the horizontal component of complete Coriolis force. However, it is controversial from the dynamical perspective [16, 17]. White and Bromley [18] indicated the necessity to retain the horizontal component of Coriolis force by scale analysis of the zonal momentum equations. It is necessary to include the cosine Coriolis terms for fully understanding the atmospheric motion, which is named ‘non-traditional approximation’ (NTA). Based on the NTA, the near-inertial waves were considered by Gerkema and Shrira [19] from primitive equations. Using the variational method, a conserved potential vorticity equation with complete Coriolis force was obtained by Dellar and Salmon [20]. According to the Hamilton least-action principle [21], Dellar also derived a generalized beta plane equation. So, the NTA is more important for tropical atmosphere, such as the stability theory [22–24], the dispersion relation [25, 26], the Madden-Julian Oscillation(MJO) [27, 28], and so on [29–31].
Conservation laws [32, 33] are a power tool for studying the global stability of solutions and numerical integration for PDEs emerging in nonlinear science. Numerous powerful methods have been used to seek the conservation laws: Laplace direct technique [34], characteristic form given by Stuedel [35], q-homotopy analysis transform method (q-HATM) [36], multiplier approach [37, 38]. In this thesis, based on the modified Camassa-Holm equation [39] and ZK-BBM equations [40] for each multiplier, and the method of Ibragimov (nonlocal conservation method) [41–43], using the multiplier approach, conservation laws and the corresponding conserved quantities are discussed.
The investigation of the soliton solutions [44–46] for PDEs has become highly active in all areas of applied research. Many efficient techniques, such as homogeneous balance technique [47], symmetry theory [48], Jacobi elliptic function method [49], homotopy analysis transform method [50], homotopy perturbation transform method [51], Darboux transformation [52–54], Bilinear method [55, 56], and so on [57], have been proposed to seek solitary waves solutions. In addition, some numerical methods, i.e., modified binomial and Monte Carlo methods [58], high accurate NRK, and MWENO [59–61], have also been used to solve partial differential equations. In the paper, we use the semi-inverse variational principle [62, 63] to obtain the soliton solutions of PDEs.
In this article, by using multi-scale analysis and perturbation method, a \((2+1)\) dimensional combined ZK-mZK model for Rossby solitary waves is obtained. According to the new model, we have a discussion. The construction of this paper is as follows. In Section 2, applying the quasi-geostrophic potential vorticity equation, we obtain a new combined ZK-mZK equation. The property of conservation laws and the corresponding conserved quantities of the new equation are discussed in Section 3. Later, the soliton solutions of the combined ZK-mZK equation are established in Section 4. According to the soliton solution, we research the solitary wave profiles for various timescales and the collisions between two waves with different velocities and various timescales. Finally, some conclusions are drawn.
2 The derivation of combined ZK-mZK equation
Remark
Eq. (35) is a new \((2+1)\) dimensional model. The coefficient \(a_{0}\) is in connection with topographic effect G. The second term \(a_{0}A_{X}\) represents the effect of complete Coriolis force and topography on the evolution of Rossby solitary waves. Based on \(a_{1}AA_{X}\) and \(a_{2}A^{3}A_{X}\), we know that the new equation is the combination of ZK equation and mZK equation. So, the new model Eq. (35) is called combined ZK-mZK equation. Compared with other models, the combined ZK-mZK model can describe Rossby solitary waves more accurately.
3 The derivation of conservation laws
4 The soliton solutions
In order to seek the soliton solution [65, 66] of a partial differential equation, the semi-inverse variational principle is used. Its steps are as follows:
Step 4. Substituting Eq. (37) into Eq. (59), letting \(\frac{\partial I}{\partial p}=0\) and \(\frac{\partial I}{\partial q}=0\), and solving them to determine the constants p and q, respectively, we determine the solitary wave solutions of Eq. (37).
5 Conclusions and discussions
In this paper, using the quasi-geostrophic potential vorticity equation, we get a new \((2+1)\) dimensional combined ZK-mZK model for Rossby solitary waves by applying the multi-scale analysis and perturbation method. Based on the new model, we study conservation laws and the soliton solutions of the new equation. By theory and image analysis, the following conclusions can be obtained:
(1) A combined ZK-mZK equation is a new \((2+1)\) dimensional model which includes the cosine Coriolis terms. Compared with \((1+1)\) dimensional models, it can describe the actual situation of ocean and atmosphere. Besides, in the actual ocean and atmosphere, the cosine Coriolis is not ignored. It is necessary to include the cosine Coriolis terms for fully understanding the atmospheric motion. So, the combined ZK-mZK equation is more suitable to describe Rossby solitary waves.
(2) According to the combined ZK-mZK equation, using the multiplier approach, we obtain conservation laws and the corresponding conserved quantities. In addition, the semi-inverse variational principle is a robust and efficient method which gives the variational principles for nonlinear problems. By using the semi-inverse variational principle, we obtain soliton solution for the new equation. According to the soliton solution, some propagation features of Rossby solitary waves are discussed.
Declarations
Acknowledgements
This work was supported by the National Natural Science Foundation of China (Nos. 41576023, 11571207), China Postdoctoral Science Foundation funded project (No. 2017M610436), CAS Interdisciplinary Innovation Team ‘Ocean Mesoscale Dynamical Processes and Ecological Effect’, Key project of Nanjing Vocational Institute of Transport Technology (No. JZ1701), the Science and Technology Plan Projects of Shandong Province Universities and Colleges (No. J15LI54).
Authors’ contributions
The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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