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Volume 10 Issue 4

Oct.  1993

Article Contents
Article >  ADVANCES IN ATMOSPHERIC SCIENCES  > 1993 >  10(4): 393-406

Group Velocity of Anisotropic Waves-Part I: Mathematical Expression

  • 1.

    Department of Earth, Atmospheric and Planetary Sciences, MIT, MA 02139, USA


doi: 10.1007/BF02656964

  • The group velocity used in meteorology in the last 30 years was derived in terms of conservation of wave energy or crests in wave propagation. The conservation principle is a necessary but not a sufficient condition for deriving the mathematical form of group velocity, because it cannot specify a unique direction in which wave energy or crests propagate. The derived mathematical expression is available only for isotropic waves. But for anisotropic waves, the traditional group velocity may have no a definite direction, because it varies with rotation of coordinates. For these reasons, it cannot be considered as a general expression of group velocity. A ray defined by using this group velocity may not be the trajectory of a reference point in an anisotropic wave train. The more general and precise expression of group velocity which is applicable for both isotropic and anisotropic waves and is independent of coordinates will be derived following the displacement of not only a wave envelope phase but also a wave reference point on the phase.
  • [1] Yong L. McHall, 1993: Group Velocity of Anisotropic Waves-Part II: Conservative Properties, ADVANCES IN ATMOSPHERIC SCIENCES, 10, 407-414.  doi: 10.1007/BF02656965
    [2] LI Nan, WEI Ming, TANG Xiaowen, PAN Yujie, 2007: An Improved Velocity Volume Processing Method, ADVANCES IN ATMOSPHERIC SCIENCES, 24, 893-906.  doi: 10.1007/s00376-007-0893-0
    [3] Fu Baopu, 1987: VARIATION IN WIND VELOCITY OVER WATER, ADVANCES IN ATMOSPHERIC SCIENCES, 4, 93-104.  doi: 10.1007/BF02656665
    [4] ZHAI Guoqing, ZHOU Lingli, WANG Zhi, 2007: Analysis of a Group of Weak Small-Scale Vortexes in the Planetary Boundary Layer in the Mei-yu Front, ADVANCES IN ATMOSPHERIC SCIENCES, 24, 399-408.  doi: 10.1007/s00376-007-0399-9
    [5] Zhao Ming, 1987: ON THE PARAMETERIZATION OF THE VERTICAL VELOCITY AT THE TOP OF PLANETARY BOUNDARY LAYER, ADVANCES IN ATMOSPHERIC SCIENCES, 4, 233-239.  doi: 10.1007/BF02677070
    [6] LIU Guimei, WANG Hui, SUN Song, HAN Boping, 2003: Numerical Study on the Velocity Structure around Tidal Fronts in the Yellow Sea, ADVANCES IN ATMOSPHERIC SCIENCES, 20, 453-460.  doi: 10.1007/BF02690803
    [7] LI Gang, HE Guangxin, Xiaolei ZOU*, and Peter Sawin RAY, 2014: A Velocity Dealiasing Scheme for C-band Weather Radar Systems, ADVANCES IN ATMOSPHERIC SCIENCES, 31, 17-26.  doi: 10.1007/s00376-013-2251-8
    [8] Chen Panqin, 1987: MEASUREMENTS OF THE DRY DEPOSITION VELOCITY FOR SUSPENDED PARTICLES OVER THE SUBURBS OF BEIJING, ADVANCES IN ATMOSPHERIC SCIENCES, 4, 323-331.  doi: 10.1007/BF02663602
    [9] SUN Jianning, JIANG Weimei, CHEN Ziyun, YUAN Renmin, 2005: A Laboratory Study of the Turbulent Velocity Characteristics in the Convective Boundary Layer, ADVANCES IN ATMOSPHERIC SCIENCES, 22, 770-780.  doi: 10.1007/BF02918721
    [10] Botao ZHOU, Ying SHI, Ying XU, 2016: CMIP5 Simulated Change in the Intensity of the Hadley and Walker Circulations from the Perspective of Velocity Potential, ADVANCES IN ATMOSPHERIC SCIENCES, 33, 808-818.  doi: 10.1007/s00376-016-5216-x
    [11] LI Pingyang, JIANG Weimei, SUN Jianning, YUAN Renmin, 2003: A Laboratory Modeling of the Velocity Field in the Convective Boundary Layer with the Particle Image Velocimetry Technique, ADVANCES IN ATMOSPHERIC SCIENCES, 20, 631-637.  doi: 10.1007/BF02915506
    [12] Jeong-Eun LEE, Sung-Hwa JUNG, Hong-Mok PARK, Soohyun KWON, Pay-Liam LIN, GyuWon LEE, 2015: Classification of Precipitation Types Using Fall Velocity-Diameter Relationships from 2D-Video Distrometer Measurements, ADVANCES IN ATMOSPHERIC SCIENCES, 32, 1277-1290.  doi: 10.1007/s00376-015-4234-4
    [13] Wansuo DUAN, Xixi WEN, 2019: Errors in Current Velocity in the Low-latitude North Pacific: Results from the Regional Ocean Modeling System, ADVANCES IN ATMOSPHERIC SCIENCES, 36, 397-416.  doi: 10.1007/s00376-018-8140-4
    [14] Jie CAO, Qin XU, Haishan CHEN, Shuping MA, 2022: Hybrid Methods for Computing the Streamfunction and Velocity Potential for Complex Flow Fields over Mesoscale Domains, ADVANCES IN ATMOSPHERIC SCIENCES, 39, 1417-1431.  doi: 10.1007/s00376-021-1280-y
    [15] 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
    [16] G. S. GOLITSYN, 2009: Tropical Cyclones and Polar Lows: Velocity, Size, and Energy Scales, and Relation to the 26oC Cyclone Origin Criteria, ADVANCES IN ATMOSPHERIC SCIENCES, 26, 585-598.  doi: 10.1007/s00376-009-0585-z
    [17] CAO Jie, Qin XU, 2011: Computing Streamfunction and Velocity Potential in a Limited Domain of Arbitrary Shape. Part II: Numerical Methods and Test Experiments, ADVANCES IN ATMOSPHERIC SCIENCES, 28, 1445-1458.  doi: 10.1007/s00376-011-0186-5
    [18] Qin XU, CAO Jie, GAO Shouting, 2011: Computing Streamfunction and Velocity Potential in a Limited Domain of Arbitrary Shape. Part I: Theory and Integral Formulae, ADVANCES IN ATMOSPHERIC SCIENCES, 28, 1433-1444.  doi: 10.1007/s00376-011-0185-6
    [19] ZHANG Zhanhai, ZHOU Mingyu, Sharon ZHONG, Donald H. LENSCHOW, Qing WANG, 2009: Parameterizations of the Daytime Friction Velocity, Temperature Scale, and Upslope Flow over Gently Inclined Terrain in Calm Synoptic Conditions, ADVANCES IN ATMOSPHERIC SCIENCES, 26, 577-584.  doi: 10.1007/s00376-009-0577-z
    [20] Ma Yimin, 1992: Preliminary Study on Vertical Velocity Caused by Katabatic Wind in Antarctica and Its Influence on Atmospheric Circulation, ADVANCES IN ATMOSPHERIC SCIENCES, 9, 247-250.  doi: 10.1007/BF02657515
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    • Manuscript History

    Manuscript received: 10 October 1993
    Manuscript revised: 10 October 1993
    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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    Group Velocity of Anisotropic Waves-Part I: Mathematical Expression

    • 1. Department of Earth, Atmospheric and Planetary Sciences, MIT, MA 02139, USA
    • Manuscript received: 1993-10-10
    • Manuscript revised: 1993-10-10

    Abstract: The group velocity used in meteorology in the last 30 years was derived in terms of conservation of wave energy or crests in wave propagation. The conservation principle is a necessary but not a sufficient condition for deriving the mathematical form of group velocity, because it cannot specify a unique direction in which wave energy or crests propagate. The derived mathematical expression is available only for isotropic waves. But for anisotropic waves, the traditional group velocity may have no a definite direction, because it varies with rotation of coordinates. For these reasons, it cannot be considered as a general expression of group velocity. A ray defined by using this group velocity may not be the trajectory of a reference point in an anisotropic wave train. The more general and precise expression of group velocity which is applicable for both isotropic and anisotropic waves and is independent of coordinates will be derived following the displacement of not only a wave envelope phase but also a wave reference point on the phase.

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