Abstract
The complete form of the vertical vorticity tendency equation (the complete-form vorticity equation) is derived from the Ertel potential vorticity equation to contain thermodynamic factors. In this study, a new complete-form vorticity equation, which has the same form as the original complete-form vorticity equation, is deduced from the absolute vorticity vector equation combined with the continuity equation and the expression of three-dimensional (3D) entropy gradient. By comparing the complete-form vorticity equation with the classical vertical vorticity equation, it is found that regardless of whether or not the isentropic surface is tilting, the two vorticity equations are in essence the same. The “baroclinic term” of the complete-form vorticity equation is exactly equal to the solenoidal term of the classical one, and there is a significant amount of cancellation between the two baroclinic items (the “slantwise term” and the horizontal vorticity change term) in the complete-form vorticity equation. In operational weather analysis, the tilt of the isentropic surface can be diagnosed according to the density of the isotherm on the upper-level isobaric map. For synoptic-scale motion, the vertical vorticity produced by the tilt of the isentropic surface is due to the contribution of atmospheric baroclinicity, which is measured by the solenoid. The 3D solenoid is parallel to the isentropic surface, so the more tilted the isentropic surface, the bigger the projection of the 3D solenoid in the vertical direction. The baroclinic contribution can be interpreted based on the PV thinking theory, but the relationship between the vorticity field and the potential vorticity field is not immediate.
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References
Chen Zhongming, 2006: Comparison of the different forms of vertical vorticity tendency equation. J. Nanjing Univ. (Natl. Sci.), 1, 535–542. (in Chinese)
Ertel, H., 1942a: Ein neuer hydrodynamischer Wirbelsatz. Meteor. Z., 1, 277–281.
Ertel, H., 1942b: Ein neuer hydrodynamischer Erhaltungssatz. Naturwissenschaften, 1, 543–544.
Holton, J. R., 2004: An Introduction to Dynamic Meteorology, Fourth Edition. Elsevier Academic Press, San Diego, 228–230 pp.
Hoskins, B., 1997: A potential vorticity view of synoptic development. Meteor. Appl., 1, 325–334.
Hoskins, B. J., M. A. Pedder, and D. W. Jones, 2003: The omega equation and potential vorticity. Quart. J. Roy. Meteor. Soc., 1, 3277–3303.
Hu Bowei, 2003: Some opinions about the “potential vorticity thinking” and its application. J. Nanjing Inst. Meteor., 1, 111–115. (in Chinese)
Jaw, J.-J., 1946: The formation of the semipermanent centers of action in relation to the horizontal solenoidal field. J. Meteor., 1, 103–114.
Jiang Yongqiang, Chen Zhongyi, Zhou Zugang, et al., 2004: Slantwise vorticity development and meso-β scale low vortex. J. PLA Univ. Sci. Technol., 1, 81–87. (in Chinese)
Li Na, Ran Lingkun, Zhou Yushu, et al., 2013: Diagnosis of the frontogenesis and slantwise vorticity development caused by the deformation in the Beijing “7.21” torrential rainfall event. Acta Meteor. Sinica, 1, 593–605. (in Chinese)
Li Ying and Duan Xu, 1999: Slantwise vorticity development and severe precipitation in winter over Yunnan. J. Nanjing Inst. Meteor., 1, 705–710. (in Chinese)
Liu Chongjian, Liu Ying, and Xu Hui, 2007: Entropy flow and the evolution of the atmospheric systems. Chinese J. Atmos. Sci., 1, 1251–1256. (in Chinese)
Liu Shida and Liu Shikuo, 2011: Dynamics of Atmospheric Eddy. China Meteorological Press, Beijing, 41–42 pp. (in Chinese)
Lü Meizhong and Peng Yongqing, 1990: Tutorial of the Dynamical Meteorology. China Meteorological Press, Beijing, 131–132 pp. (in Chinese)
Lü Meizhong, Hou Zhiming, and Zhou Yi, 2004: Dynamic Meteorology. China Meteorological Press, Beijing, 22–23 pp. (in Chinese)
Ma Leiming, Qin Zenghao, Duan Yihong, et al., 2002: Case study on the impact of atmospheric baroclinicity to the initial development of Jianghuai cyclone. Acta Oceanol. Sinica, 24(S1), 95–104. (in Chinese)
Wang Ying, Wang Yuan, Zhang Lixiang, et al., 2007: The development of slantwise vorticity near a weakened tropical cyclone. J. Trop. Meteor., 1, 47–52. (in Chinese)
Wang Ziqian, Zhu Weijun, and Duan Anmin, 2010: A case study of snowstorm in Tibetan Plateau induced by Bay of Bengal storm: Based on the theory of slantwise vorticity development. Plateau Meteor., 1, 703–711. (in Chinese)
Wu Guoxiong, 2001: Comparison between the completeform vorticity equation and the traditional vorticity equation. Acta Meteor. Sinica, 1, 385–392. (in Chinese)
Wu, G. X., and H. Z. Liu, 1998: Vertical vorticity development owing to down-sliding at slantwise isentropic surface. Dyn. Atmos. Oceans, 1, 715–743.
Wu Guoxiong and Liu Huanzhu, 1999: Complete form of vertical vorticity tendency equation and slantwise vorticity development. Acta Meteor. Sinica, 1, 1–14. (in Chinese)
Wu Guoxiong, Zheng Yongjun, and Liu Yimin, 2013: Dynamical and thermal problems in vortex development and movement. Part II: Generalized slantwise vorticity development. Acta Meteor. Sinica, 1, 15–25.
Wu Rongsheng, 1990: Atmospheric Dynamics. China Meteorological Press, Beijing, 85–86 pp. (in Chinese)
Zhou Xiaogang, Liu Shijun, Wang Xiuming, et al., 2011: An investigation into the expression of potential vorticity in the common meteorological coordinate systems. Acta Phys. Sinica, 1, 059201. (in Chinese)
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Supported by the National Natural Science Foundation of China (41475042 and 41175043) and China Meteorological Administration Special Public Welfare Research Fund (GYHY201406002).
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Wang, X., Zhou, X., Tao, Z. et al. Discussion on the complete-form vorticity equation and slantwise vorticity development. J Meteorol Res 30, 67–75 (2016). https://doi.org/10.1007/s13351-016-5040-3
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DOI: https://doi.org/10.1007/s13351-016-5040-3