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Volume 23 Issue 5

Sep.  2006

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Article >  ADVANCES IN ATMOSPHERIC SCIENCES  > 2006 >  23(5): 809-814

On Nonlinear Stability Theorems of 3D Quasi-geostrophic Flow

  • 1.

    Key Laboratory of Geographic Information Science, Ministry of Education, East China Normal University, Shanghai 200062, State Key Laboratory of Numerical Modeling for Atmospherics Sciences and Geophysical Fluid Dynamics, Institute of Atmospheric Physics,,Department of Mathematics, East China Normal University, Shanghai 200062


doi: 10.1007/s00376-006-0809-4

  • Nonlinear stability criteria for quasi-geostrophic zonally symmetric flow are improved by establishing an optimal Poincar′e inequality. The inequality is derived by a variational calculation considering the additional invariant of zonal momentum. When applied to the Eady model in a periodic channel with finite zonal length, the improved nonlinear stability criterion is identical to the linear normal-mode stability criterion provided the channel meridional width is no greater than 0.8605· · · times its channel length (which is the geophysically relevant case).
  • [1] Liu Yongming, Mu Mu, 1994: Arnol’d’s Second Nonlinear Stability Theorem for General Multilayer Quasi-geostrophic Model, ADVANCES IN ATMOSPHERIC SCIENCES, 11, 36-42.  doi: 10.1007/BF02656991
    [2] 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
    [3] Mu Mu, Wu Yonghui, Tang Mozhi, Liu Haiyan, 1999: Nonlinear Stability Analysis of the Zonal Flows at Middle and High Latitudes, ADVANCES IN ATMOSPHERIC SCIENCES, 16, 569-580.  doi: 10.1007/s00376-999-0032-1
    [4] Li Yang, 2000: Baroclinic Instability in the Generalized Phillips’ Model Part II: Three-layer Model, ADVANCES IN ATMOSPHERIC SCIENCES, 17, 413-432.  doi: 10.1007/s00376-000-0033-6
    [5] 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
    [6] 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
    [7] 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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    [9] 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
    [10] Qin XU, Wei GU, GAO Shouting, 2005: Nonlinear Oscillations of Semigeostrophic Eady Waves in the Presence of Diffusivity, ADVANCES IN ATMOSPHERIC SCIENCES, 22, 49-57.  doi: 10.1007/BF02930869
    [11] Ren Shuzhan, 1991: Nonlinear Stability of Plane Rotating Shear Flow under Three-Dimensional Nondivergence Disturbances, ADVANCES IN ATMOSPHERIC SCIENCES, 8, 129-136.  doi: 10.1007/BF02658089
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    • Manuscript History

    Manuscript received: 10 September 2006
    Manuscript revised: 10 September 2006
    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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    On Nonlinear Stability Theorems of 3D Quasi-geostrophic Flow

    • 1. Key Laboratory of Geographic Information Science, Ministry of Education, East China Normal University, Shanghai 200062, State Key Laboratory of Numerical Modeling for Atmospherics Sciences and Geophysical Fluid Dynamics, Institute of Atmospheric Physics,,Department of Mathematics, East China Normal University, Shanghai 200062
    • Manuscript received: 2006-09-10
    • Manuscript revised: 2006-09-10

    Abstract: Nonlinear stability criteria for quasi-geostrophic zonally symmetric flow are improved by establishing an optimal Poincar′e inequality. The inequality is derived by a variational calculation considering the additional invariant of zonal momentum. When applied to the Eady model in a periodic channel with finite zonal length, the improved nonlinear stability criterion is identical to the linear normal-mode stability criterion provided the channel meridional width is no greater than 0.8605· · · times its channel length (which is the geophysically relevant case).

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