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Chin. Phys. B, 2017, Vol. 26(11): 114701    DOI: 10.1088/1674-1056/26/11/114701
ELECTROMAGNETISM, OPTICS, ACOUSTICS, HEAT TRANSFER, CLASSICAL MECHANICS, AND FLUID DYNAMICS Prev   Next  

Establishment of infinite dimensional Hamiltonian system of multilayer quasi-geostrophic flow & study on its linear stability

Si-xun Huang(黄思训)1,2, Yu Wang(王宇)1, Jie Xiang(项杰)1
1. Institute of Meteorology and Oceanography, National University of Defense Technology, Nanjing 211101, China;
2. State Key Laboratory of Satellite Ocean Environment Dynamics, Second Institute of Oceanography, State Oceanic Administration, Hangzhou 310012, China
Abstract  A multilayer flow is a stratified fluid composed of a finite number of layers with densities homogeneous within one layer but different from each other. It is an intermediate system between the single-layer barotropic model and the continuously stratified baroclinic model. Since this system can simulate the baroclinic effect simply, it is widely used to study the large-scale dynamic process in atmosphere and ocean. The present paper is concerned with the linear stability of the multilayer quasi-geostrophic flow, and the associated linear stability criteria are established. Firstly, the nonlinear model is turned into the form of a Hamiltonian system, and a basic flow is defined. But it cannot be an extreme point of the Hamiltonian function since the system is an infinite-dimensional one. Therefore, it is necessary to reconstruct a new Hamiltonian function so that the basic flow becomes an extreme point of it. Secondly, the linearized equations of disturbances in the multilayer quasi-geostrophic flow are derived by introducing infinitesimal disturbances superposed on the basic flows. Finally, the properties of the linearized system are discussed, and the linear stability criteria in the sense of Liapunov are derived under two different conditions with respect to certain norms.
Keywords:  infinite dimensional Hamiltonian system      multilayer quasi-geostrophic flow      linear stability  
Received:  24 March 2017      Revised:  20 July 2017      Accepted manuscript online: 
PACS:  47.10.Df (Hamiltonian formulations)  
  68.65.Ac (Multilayers)  
  47.15.Fe (Stability of laminar flows)  
Fund: Project supported by the National Natural Science Foundation of China (Grant Nos. 41575026, 41275113, and 41475021).
Corresponding Authors:  Yu Wang     E-mail:  wangyu20140901@163.com

Cite this article: 

Si-xun Huang(黄思训), Yu Wang(王宇), Jie Xiang(项杰) Establishment of infinite dimensional Hamiltonian system of multilayer quasi-geostrophic flow & study on its linear stability 2017 Chin. Phys. B 26 114701

[1] Yih C S 1982 Fluid Mechanics(Beijing:Higher Education Press)(in Chinese)
[2] Georgescu A 1985 Hydrodynamic Stability Theory(Berlin:Spring Science)
[3] Huang S X and Wu R S 2001 Methods of Mathematical Physics in Atmospheric Science(3rd Edn.)(Beijing:China Meteorological Press)(in Chinese)
[4] Sumaira A, Saleem A and Adeel A 2017 Chin. Phys. B 26 014704
[5] Kuo H L 1949 J. Metro. 6 105
[6] Pedlosky J 1987 Geophysical Fluid Dynamics(Berlin:Springer)
[7] Lu W S 1992 The Principle of Dynamic Stability(Beijing:China Meteorological Press)(in Chinese)
[8] Oliver T S and Ulrich R 2011 J. Fluid Mech. 688 569
[9] Wang S C and Huang S X 2016 Appl. Math. Mech. Engl. Ed. 37 181
[10] Arnold V I 1965 Dokl. Akad. Nauk. SSSR 162 773
[11] Arnold V I 1966 Am. Math. Soc. Transl. 79 267
[12] Mu M 1991 Sci. China:Chem. 12 1516
[13] Mu M, Zeng Q C, Shepherd T G, et al. 1994 J. Fluid Mech. 264 165
[14] Zhang G, Xiang J and Li D H 2002 Appl. Math. Mech. Engl. Ed. 23 79
[15] Zhang G and Xiang J 2002 Appl. Math. Mech. 23 1195(in Chinese)
[16] Swaters G E 2000 Introduction to Hamiltonian Fluid Dynamics and Stability Theory (New York:Chapman& Hall/CRC)
[17] Tang Q and Chen C M 2007 Appl. Math. Mech. 28 1071
[18] Liu C, Liu S X, Mei F X and Guo Y X 2008 Acta Phys. Sin. 57 6709(in Chinese)
[19] Chen X W, Li Y M and Mei F X 2014 Appl. Math. Mech. 35 1392
[20] Jin S X and Zhang Y 2014 Chin. Phys. B. 23 054501
[21] Wang Y, Mei F X, Xiao J and Guo Y X 2017 Acta Phys. Sin. 66 054501(in Chinese)
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