- Research
- Open Access
Analytical research of (\(3+1\))-dimensional Rossby waves with dissipation effect in cylindrical coordinate based on Lie symmetry approach
- Yanwei Ren1,
- Mengshuang Tao2,
- Huanhe Dong2 and
- Hongwei Yang2Email author
https://doi.org/10.1186/s13662-019-1952-4
© The Author(s) 2019
- Received: 15 September 2018
- Accepted: 8 January 2019
- Published: 18 January 2019
Abstract
Rossby waves, one of significant waves in the solitary wave, have important theoretical meaning in the atmosphere and ocean. However, the previous studies on Rossby waves commonly were carried out in the zonal area and could not be applied directly to the spherical earth. In order to overcome the problem, the research on (\(3+1\))-dimensional Rossby waves in the paper is placed into the spherical area, and some new analytical solutions of (\(3+1\))-dimensional Rossby waves are given through the classic Lie group method. Finally, the dissipation effect is analyzed in the sense of the above mentioned new analytical solutions. The new solutions on (\(3+1\))-dimensional Rossby waves have important value for understanding the propagation of Rossby waves in the rotating earth with the influence of dissipation.
Keywords
- Rossby waves
- Lie group
- Cylindrical coordinate
- Dissipation effect
1 Introduction
As is well known, Rossby waves play a central role in the atmosphere and ocean, which depicts an essential phenomenon. The oceans’s response to the atmosphere change and climate change can be determined by Rossby waves. In addition, Rossby waves have significant theoretical meaning and real value in the atmosphere and ocean. In recent years, more and more researchers have focused on the study of Rossby waves [1–4]. Many studies on Rossby waves have been conducted in the zonal area, and many meaningful results have been achieved [5, 6]. However, as everyone knows, the propagation of Rossby waves happens in the earth which is a spherical area [7], so the above mentioned achievements could not be directly applied. It is necessary to discuss the propagation characteristic of (\(3+1\))-dimensional Rossby waves in the spherical area under the influence of dissipation. Here, in order to overcome the problem, our research is carried out in cylindrical coordinate, which better matches with the real condition.
With the development of soliton theory, Rossby waves have been becoming an important research direction in the field of the nonlinear partial differential [8–10]. In recent years, some weakly nonlinear models for the evolution of Rossby waves have been extensively studied [11–13]. More importantly, some methods are found to study the nonlinear models [14–18] and some significant properties are discussed [19–23]. In the past, Rossby waves were often studied in the zonal area. However, Rossby waves are prominently affected via the rotation effect of the earth. Therefore, in order to study some propagation characteristics of Rossby waves, we use the (\(3+1\))-dimensional quasi-geostrophic vorticity equation with dissipation effect in the cylindrical coordinate to describe the dynamic behavior of Rossby waves.
In this paper, the (\(3+1\))-dimensional Rossby waves with dissipation effect in cylindrical coordinate will be discussed through the classic Lie group method. In Sect. 2, we analyze the (\(3+1\))-dimensional Rossby waves with dissipation effect in cylindrical coordinate by using the classic Lie group method. In Sect. 3, the solution of the (\(3+1\))-dimensional Rossby waves with dissipation effect in cylindrical coordinate can be obtained. In addition, some conclusions are placed in Sect. 4.
2 Symmetry analysis for the (\(3+1\))-dimensional Rossby waves with dissipation effect in cylindrical coordinate
3 The new solution of the (\(3+1\))-dimensional Rossby waves with dissipation effect in cylindrical coordinate
4 Discussion and conclusion
In this section, the solution of the (\(3+1\))-dimensional dissipation Rossby waves can be discussed relying on Eq. (30).
Obviously, Rossby waves were established in the zonal area and could not be used directly to the spherical earth in the previous research. However, in this paper, we get the solution of the (\(3+1\))-dimensional dissipation Rossby waves by using the classic Lie group method in cylindrical coordinate, and the new solution overcomes the problem. According to theoretical analysis, we can make the following conclusion: In the spherical earth, the dissipation effect could give rise to a decrease in amplitude \(e^{-\alpha t}\), where α denotes the dissipation coefficient from Eq. (31).
Declarations
Funding
This work was supported by the Natural Science Foundation of Shandong Province of China (No. ZR2018MA017), China Postdoctoral Science Foundation funded project (No. 2017M610436).
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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