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Volume 20 Issue 4

Jul.  2003

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Article >  ADVANCES IN ATMOSPHERIC SCIENCES  > 2003 >  20(4): 623-630

Generalized Nonlinear Subcritical Symmetric Instability

  • 1.

    Department of Atmospheric Sciences, Nanjing Institute of Meteorology, Nanjing 210044,Department of Atmospheric Sciences, Nanjing Institute of Meteorology, Nanjing 210044


doi: 10.1007/BF02915505

  • Starting from nonlinear equations on the f-plane containing frictional dissipation under the Boussinesqapproximation, a new kind of generalized energy is proposed as the Lyapunov function, and averages aretaken as any functions of (x, z) instead of the commonly-used means of bilinear functions of (x, z), therebyresulting in a new criterion of generalized nonlinear symmetric stability. It shows that not only mustthe dissipative coefficient be greater than a certain critical value but the initial disturbance amplitudemust be synchronously smaller than another marginal value as well. It follows that the latter imposesa crucial constraint on the former, thus leading to the fact that when the amplitude is bigger comparedto another critical value, generalized nonlinear subcritical symmetrical instability 05- occur. The newcriterion contributes greatly to the improvement of the previous results of its kind.
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    [2] Ren Shuzhan, 1994: Note on the Symmetric Stability of Quasi-Homogeneous and Incompressible Rotating Ocean, ADVANCES IN ATMOSPHERIC SCIENCES, 11, 74-78.  doi: 10.1007/BF02656996
    [3] Ren Shuzhan, 1994: Symmetric Stability of Rotation and Boussinesq Fluid in Bounded Domain by Using Normal Mode Method, ADVANCES IN ATMOSPHERIC SCIENCES, 11, 291-295.  doi: 10.1007/BF02658148
    [4] 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
    [5] Zhaoxia PU, Joshua HACKER, 2009: Ensemble-based Kalman Filters in Strongly Nonlinear Dynamics, ADVANCES IN ATMOSPHERIC SCIENCES, 26, 373-380.  doi: 10.1007/s00376-009-0373-9
    [6] Li Chongyin, Liao Qinghai, 1996: Behaviour of Coupled Modes in a Simple Nonlinear Air-Sea Interaction Model, ADVANCES IN ATMOSPHERIC SCIENCES, 13, 183-195.  doi: 10.1007/BF02656861
    [7] Li Yang, Mu Mu, 1996: Baroclinic Instability in the Generalized Phillips’ Model Part I: Two-layer Model, ADVANCES IN ATMOSPHERIC SCIENCES, 13, 33-42.  doi: 10.1007/BF02657026
    [8] Mu Mu, Xiang Jie, 1998: On the Evolution of Finite-amplitude Disturbance to the Barotropic and Baroclinic Quasigeostrophic Flows, ADVANCES IN ATMOSPHERIC SCIENCES, 15, 113-123.  doi: 10.1007/s00376-998-0023-7
    [9] Liu Yongming, 1999: Nonlinear Stability of Zonally Symmetric Quasi-geostrophic Flow, ADVANCES IN ATMOSPHERIC SCIENCES, 16, 107-118.  doi: 10.1007/s00376-999-0007-2
    [10] He Jianzhong, 1994: Nonlinear Ultra-Long Wave and Its Stability, ADVANCES IN ATMOSPHERIC SCIENCES, 11, 91-100.  doi: 10.1007/BF02656998
    [11] Li Yang, Mu Mu, Wu Yonghui, 2000: A Study on the Nonlinear Stability of Fronts in the Ocean on a Sloping Continental Shelf, ADVANCES IN ATMOSPHERIC SCIENCES, 17, 275-284.  doi: 10.1007/s00376-000-0009-6
    [12] Mu Mu, Guo Huan, Wang Jiafeng, LiYong, 2000: The Impact of Nonlinear Stability and Instability on the Validity of the Tangent Linear Model, ADVANCES IN ATMOSPHERIC SCIENCES, 17, 375-390.  doi: 10.1007/s00376-000-0030-9
    [13] Lin Wantao, Ji Zhongzhen, Wang Bin, 2001: Computational Stability of the Explicit Difference Schemes of the Forced Dissipative Nonlinear Evolution Equations, ADVANCES IN ATMOSPHERIC SCIENCES, 18, 413-417.  doi: 10.1007/BF02919320
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    [20] Liu Yongming, Mu Mu, 1992: A Problem Related to Nonlinear Stability Criteria for Multi-layer Quasi-geostrophic Flow, ADVANCES IN ATMOSPHERIC SCIENCES, 9, 337-345.  doi: 10.1007/BF02656943
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    • Manuscript History

    Manuscript received: 10 July 2003
    Manuscript revised: 10 July 2003
    通讯作者: 陈斌, bchen63@163.com
    • 1. 

      沈阳化工大学材料科学与工程学院 沈阳 110142

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    Generalized Nonlinear Subcritical Symmetric Instability

    • 1. Department of Atmospheric Sciences, Nanjing Institute of Meteorology, Nanjing 210044,Department of Atmospheric Sciences, Nanjing Institute of Meteorology, Nanjing 210044
    • Manuscript received: 2003-07-10
    • Manuscript revised: 2003-07-10

    Abstract: Starting from nonlinear equations on the f-plane containing frictional dissipation under the Boussinesqapproximation, a new kind of generalized energy is proposed as the Lyapunov function, and averages aretaken as any functions of (x, z) instead of the commonly-used means of bilinear functions of (x, z), therebyresulting in a new criterion of generalized nonlinear symmetric stability. It shows that not only mustthe dissipative coefficient be greater than a certain critical value but the initial disturbance amplitudemust be synchronously smaller than another marginal value as well. It follows that the latter imposesa crucial constraint on the former, thus leading to the fact that when the amplitude is bigger comparedto another critical value, generalized nonlinear subcritical symmetrical instability 05- occur. The newcriterion contributes greatly to the improvement of the previous results of its kind.

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