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Volume 6 Issue 2

Apr.  1989

Article Contents

Variational Principle of Instability of Atmospheric Motions


doi: 10.1007/BF02658013

  • Problems of instability of rotating atmospheric motions are investigated by using nonlinear governing equations and the variational principle. The method suggested in this paper is universal for obtaining criteria of instability in all models with all possible basic flows. For example, the model can be barotropic or baroclinic, layer or continuous, quasi-geostrophic or primitive equations; the basic flow can be zonal or nonzonal, steady or unsteady.Although the basic flows possess a great deal of variety, they all are the stationary points in the functional space determined by an appropriate invariant functional. The basic flow is an unsteady one if the conservation of angular momentum is included in the associated functional.The second variation, linear or nonlinear, gives the criteria of instability. Especially, the general criteria of instability for unsteady basic flow, orographically disturbed flow as well as nongeostrophic flow are first obtained by the method described in this paper.It is also shown that the difference between the criteria of instability obtained by the linear theory and our variational principle clearly indicates the importance of using nonlinear governing equations.In the appendix the theory is extended to cases such as in a β-plane where the fluid does not possess finite total energy, hence the variational principle can not be directly applied. However, a generalized Liapounoff norm can still be obtained on the basis of variational consideration.
  • [1] Zeng Qingcun, 1991: Comments on Zeng’s Paper “Variational Principle of Instability of Atmospheric Motions”, ADVANCES IN ATMOSPHERIC SCIENCES, 8, 125-127.  doi: 10.1007/BF02657371
    [2] LIU Shikuo, LIU Shida, FU Zuntao, XIN Guojun, LIANG Fuming, 2004: The Structure and Bifurcation of Atmospheric Motions, ADVANCES IN ATMOSPHERIC SCIENCES, 21, 557-561.  doi: 10.1007/BF02915723
    [3] Kang Hongwen, Gu Xiangqian, Liu Chongjian, Wang Pengyun, 2002: Adaptive Grid Technique Based on the Variational Principle and Its Weight Functions, ADVANCES IN ATMOSPHERIC SCIENCES, 19, 705-718.  doi: 10.1007/s00376-002-0010-3
    [4] Liu Shikuo, Huang Wei, Rong Pingping, 1992: Effects of Turbulent Dispersion of Atmospheric Balance Motions of Planetary Boundary Layer, ADVANCES IN ATMOSPHERIC SCIENCES, 9, 147-156.  doi: 10.1007/BF02657505
    [5] Zeng Xinmin, Zhao Ming, Su Bingkai, 2000: A Numerical Study on Effects of Land-Surface Heterogeneity from “Combined Approach” on Atmospheric Process Part I: Principle and Method, ADVANCES IN ATMOSPHERIC SCIENCES, 17, 103-120.  doi: 10.1007/s00376-000-0047-0
    [6] Su Jeong LEE, Myoung-Hwan AHN, Yeonjin LEE, 2016: Application of an Artificial Neural Network for a Direct Estimation of Atmospheric Instability from a Next-Generation Imager, ADVANCES IN ATMOSPHERIC SCIENCES, 33, 221-232.  doi: 10.1007/s00376-015-5084-9
    [7] Chaoqun MA, Tijian WANG, Zengliang ZANG, Zhijin LI, 2018: Comparisons of Three-Dimensional Variational Data Assimilation and Model Output Statistics in Improving Atmospheric Chemistry Forecasts, ADVANCES IN ATMOSPHERIC SCIENCES, 35, 813-825.  doi: 10.1007/s00376-017-7179-y
    [8] Li Yang, Mu Mu, 1996: On the Nonlinear Stability of Three-Dimensional Quasigeostrophic Motions in Spherical Geometry, ADVANCES IN ATMOSPHERIC SCIENCES, 13, 203-216.  doi: 10.1007/BF02656863
    [9] Mu Mu, Zeng Qingcun, 1991: Criteria for the Nonlinear Stability of Three-Dimensional Quasi-Geostrophic Motions, ADVANCES IN ATMOSPHERIC SCIENCES, 8, 1-10.  doi: 10.1007/BF02657360
    [10] LU Weisong, SHAO Haiyan, 2003: Generalized Nonlinear Subcritical Symmetric Instability, ADVANCES IN ATMOSPHERIC SCIENCES, 20, 623-630.  doi: 10.1007/BF02915505
    [11] Yong. L. McHall, 1992: Nonlinear Planetary Wave Instability and Blocking, ADVANCES IN ATMOSPHERIC SCIENCES, 9, 173-190.  doi: 10.1007/BF02657508
    [12] CHEN Jinbei, HU Yinqiao, ZHANG Lei, 2007: Principle of Cross Coupling Between Vertical Heat Turbulent Transport and Vertical Velocity and Determination of Cross Coupling Coefficient, ADVANCES IN ATMOSPHERIC SCIENCES, 24, 89-100.  doi: 10.1007/s00376-007-0089-7
    [13] Ji Zhongzhen, Lin Wantao, Yang Xiaozhong, 2001: Problems of Nonlinear Computational Instability in Evolution Equations, ADVANCES IN ATMOSPHERIC SCIENCES, 18, 397-403.  doi: 10.1007/BF02919318
    [14] Gao Shouting, 2000: The Instability of the Vortex Sheet along the Shear Line, ADVANCES IN ATMOSPHERIC SCIENCES, 17, 525-537.  doi: 10.1007/s00376-000-0016-7
    [15] Shen Xinyong, Ni Yunqi, Ding Yihui, 2002: On Problem of Nonlinear Symmetric Instability in Zonal Shear Flow, ADVANCES IN ATMOSPHERIC SCIENCES, 19, 350-364.  doi: 10.1007/s00376-002-0027-7
    [16] Sun Shuqing, 1985: GRAVITY WAVES ON THE AXIS OF LOW LEVEL JET AND THEIR INSTABILITY, ADVANCES IN ATMOSPHERIC SCIENCES, 2, 112-123.  doi: 10.1007/BF03179743
    [17] Li Hongji, Xu Hong, Wang Ronghua, 1988: A HIGH-RESOLUTION ANALYSIS METHOD OF INSTABILITY ENERGY, ADVANCES IN ATMOSPHERIC SCIENCES, 5, 75-86.  doi: 10.1007/BF02657348
    [18] WANG Yunfeng, WANG Bin, 2003: The Variational Assimilation Experiment of GPS Bending Angle, ADVANCES IN ATMOSPHERIC SCIENCES, 20, 479-486.  doi: 10.1007/BF02690806
    [19] Wang Yunfeng, Wu Rongsheng, Wang Yuan, Pan Yinong, 1999: Application of Variational Algorithms in Semi-Lagrangian Framework, ADVANCES IN ATMOSPHERIC SCIENCES, 16, 419-430.  doi: 10.1007/s00376-999-0020-5
    [20] ZHANG Ming, ZHAO Yanling, HUANG Hong, LIANG Danqing, 2007: The Generalized Energy Equation and Instability in the Two-layer Barotropic Vortex, ADVANCES IN ATMOSPHERIC SCIENCES, 24, 147-151.  doi: 10.1007/s00376-007-0147-1

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Manuscript History

Manuscript received: 10 April 1989
Manuscript revised: 10 April 1989
通讯作者: 陈斌, bchen63@163.com
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Variational Principle of Instability of Atmospheric Motions

  • 1. Laboratory of Numerical Modelling for Atmospheric Sciences and GFD, Institute of Atmospheric Physics, Academia Sinica, Beijing, China

Abstract: Problems of instability of rotating atmospheric motions are investigated by using nonlinear governing equations and the variational principle. The method suggested in this paper is universal for obtaining criteria of instability in all models with all possible basic flows. For example, the model can be barotropic or baroclinic, layer or continuous, quasi-geostrophic or primitive equations; the basic flow can be zonal or nonzonal, steady or unsteady.Although the basic flows possess a great deal of variety, they all are the stationary points in the functional space determined by an appropriate invariant functional. The basic flow is an unsteady one if the conservation of angular momentum is included in the associated functional.The second variation, linear or nonlinear, gives the criteria of instability. Especially, the general criteria of instability for unsteady basic flow, orographically disturbed flow as well as nongeostrophic flow are first obtained by the method described in this paper.It is also shown that the difference between the criteria of instability obtained by the linear theory and our variational principle clearly indicates the importance of using nonlinear governing equations.In the appendix the theory is extended to cases such as in a β-plane where the fluid does not possess finite total energy, hence the variational principle can not be directly applied. However, a generalized Liapounoff norm can still be obtained on the basis of variational consideration.

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