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Dynamic modification of the shallow water equation on the slowly changing underlying surface condition

Da Chao-Jiu Feng Ai-Xia Gong Zhi-Qiang Song Jian

Dynamic modification of the shallow water equation on the slowly changing underlying surface condition

Da Chao-Jiu, Feng Ai-Xia, Gong Zhi-Qiang, Song Jian
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  • In this paper, the shallow water equation is discussed, when the underlying surface is slowly changing. From the continuity equation and the equation of motion controlling the movement of the atmosphere, the slowly changing of the underlying surface is superimposed on the topography function as a small quantity, using the upper and lower boundary conditions of the atmosphere, the modified shallow water equation is obtained. In the modified shallow water equation, the slowly changing local horizontal divergence modifies the equilibrium between the local horizontal divergence and the local change of the thickness of the atmosphere. On the basis of this, the vorticity equation is obtained, which contains the slowly changing underlying surface.
    • Funds: Project supported by the National Basic Research Program of China (Grant No. 2012CB955902), the National Natural Science Foundation of China (Grant Nos. 40930952, 41175067), the Key Technologies R & D Program of China (Grant No. 2009BAC51B04), the Meteorological Special Project of China (Grant Nos. GYHY201106016, GYHY201206009), the Natural Science Foundation of Inner Mongolia, China (Grant No. 2011MS0112) and the Fundamental Research Funds for the Central Universities, China (Grant No. zyz2011079).
    [1]

    Tehart W J, Shenlelon G W, Giffiths D C 1990 The IPCC Impacts Assessment (Canberra: Australian Government Publishing Service)

    [2]

    Perry J S 1993 Understanding Our Own Planet: An Overview of Major International Scientific Activities (Paris: ICSU Secretariat)

    [3]

    IPCC 1998 The Regional Impacts of Climate Change: An Assessment of Vulnerability (Cambridge: Cambridge University Press)

    [4]

    IPCC 2001 Climate Change 2001: Impacts, Adaptation & Vulnerability (Cambridge: Cambridge University Press)

    [5]

    Zhang Z Q, Sun C Q 1999 Chin. Sci. Bull. 44 464 (in Chinese) [张志强, 孙成权 1999 科学通报 44 464]

    [6]

    Sha W Y, Shao X M, Huang M 2002 Scientia Sinica Terrae 32 317 (in Chinese) [沙万英, 邵雪梅, 黄玫 2002 中国科学(D辑) 32 317]

    [7]

    Liu S Y, Ding Y J, Li J, Shangguan D H, Zhang Y 2006 Quaternary Sciences 26 762 (in Chinese) [刘时银, 丁永建, 李晶, 上官冬辉, 张勇 2006 第四纪研究 26 762]

    [8]

    Zhang S X, Zhang L, Sun S P, Feng G L 2011 Plateau Meteor. 30 659 (in Chinese) [张世轩, 张璐, 孙树鹏, 封国林 2011 高原气象 30 659]

    [9]

    Sun S P, Zhang L, Hou W, Feng G L 2011 Acta Phys. Sin. 60 029201 (in Chinese) [孙树鹏, 张璐, 侯威, 封国林 2011 物理学报 60 029201]

    [10]

    Zong X P, Li M H, Hu J G, Feng G L 2010 Acta Phys. Sin. 59 8272 (in Chinese) [宗序平, 李明辉, 胡经国, 封国林 2010 物理学报 59 8272]

    [11]

    Zhao M S, Neilson R P, Yan X D 2002 Acta Geog. Sin. 57 28 (in Chinese) [赵茂盛, Ronald P. Neilson, 延晓东 2002 地理学报 57 28]

    [12]

    Zhou G S, Zhang X S 1995 Acta Phytoecologica Sinica 20 9 (in Chinese) [周广胜, 张新时 1995 植物生态学报 20 9]

    [13]

    Zhou G S, Wang Y H 1999 Journal of Natural Resources 14 318 (in Chinese) [周广胜, 王玉辉 1999 自然资源学报 14 318]

    [14]

    Li X B, Chen Y H 2002 Advances in Earth Science 17 254 (in Chinese) [李晓兵, 陈云浩 2002 地球科学进展 17 254]

    [15]

    Wang C H, Dong W J, Wei Z G 2001 Acta Geog. Sin. 56 523 (in Chinese) [王澄海, 董文杰, 韦志刚 2001 地理学报 56 523]

    [16]

    Lü X M, Wu S H, Yang Q Y 2003 Progress in Geography 22 260 (in Chinese) [吕新苗, 吴绍洪, 杨勤业 2003 地理科学进展 22 260]

    [17]

    Yang G S, Shi Y F 1995 Acta Geog. Sin. 50 302 (in Chinese) [杨桂山, 施雅风 1995 地理学报 50 302]

    [18]

    Douglas B C 1991 J. Geo. Phys. Research 96 6981

    [19]

    Pirazzoli PA 1989 Global and Planetary Change 1 241

    [20]

    Yang G S, Zhu J W 1993 Advances in Earth Science 8 70 (in Chinese) [杨桂山, 朱季文 1993 地球科学进展 8 70]

    [21]

    Li J L, Zhang D F, Yang X P, Tong Y Q, Zhang R S 2005 J. Catastrophology 20 49 (in Chinese) [李加林, 张殿发, 杨晓平, 童亿勤, 张忍顺 2005 灾害学 20 49]

    [22]

    Shi Y F 1996 Journal of Natural Disasters 5 106 (in Chinese) [施雅风 1996 自然灾害学报 5 106]

    [23]

    Ji Z X, Jiang Z X, Zhu J W 1993 Acta Geog. Sin. 48 516 (in Chinese) [季子修, 蒋自巽, 朱季文 1993 地理学报 48 516]

    [24]

    Hou W, Zhang D Q, Zhou Y, Yang P 2011 Acta Phys. Sin. 60 109202 (in Chinese) [侯威, 章大全, 周云, 杨萍 2011 物理学报 60 109202]

    [25]

    Feng G L, Yang J, Wan S Q, Hou W, Zhi R 2009 Acta Meteor. Sin. 67 61 (in Chinese) [封国林, 杨杰, 万仕全, 侯威, 支蓉 2009 气象学报 67 61]

    [26]

    Gong Z Q, Wang X J, Zhi R, Feng G L 2009 Acta Phys. Sin. 58 4342 (in Chinese) [龚志强, 王晓娟, 支蓉, 封国林 2009 物理学报 58 4342]

    [27]

    Wan S Q, Wang L, Feng G L, He W P Acta Phys. Sin. 58 5083 (in Chinese) [万仕全, 王令, 封国林, 何文平 2009 物理学报 58 5083]

    [28]

    Xiong K G, Feng G L, Wang Q G, Hu J G 2009 Acta Phys. Sin. 58 8107 (in Chinese) [熊开国, 封国林, 王启光, 胡经国 2009 物理学报 58 8107]

    [29]

    Zhang D Q, Yang J, Wang Q G, Feng G L 2009 Acta Phys. Sin. 58 4354 (in Chinese) [章大全, 杨杰, 王启光, 封国林 2009 物理学报 58 4354]

    [30]

    Da C J, Chou J F 2008 Acta Phys. Sin. 57 2595 (in Chinese) [达朝究, 丑纪范 2008 物理学报 57 2595]

    [31]

    Da C J, Sun S P, Song J, Yang L G 2011 Plateau Meteor. 30 349

    [32]

    Dong W J, Chou J M, Feng G L 2007 Adv. Atmos. Sci. 24 336

    [33]

    Wang B, Weng H Y 1981 Geophysical Fluid Dynamics (Beijing: China Ocean Press) pp7-141 (in Chinese) [王斌, 翁衡毅 1981 地球物理流体动力学导论 (北京: 海洋出版社)第60–141页]

    [34]

    Joseph Pedlosky 1986 Geophysical Fluid Dynamics (New York: Springer Verlag) pp7-9

    [35]

    Da C J 2005 M. S. Dissertation (Hohhot: Inner Mongolia University) (in Chinese) [达朝究 2005 硕士学位论文 (呼和浩特: 内蒙古大学)]

    [36]

    Guo B R B 1990 the Introduction of Linear and Nonlinear Wave Theory (Beijing: China Meteorological Press) (in Chinese) [郭秉荣 1990 线性波与非线性波导论 (北京: 气象出版社)]

  • [1]

    Tehart W J, Shenlelon G W, Giffiths D C 1990 The IPCC Impacts Assessment (Canberra: Australian Government Publishing Service)

    [2]

    Perry J S 1993 Understanding Our Own Planet: An Overview of Major International Scientific Activities (Paris: ICSU Secretariat)

    [3]

    IPCC 1998 The Regional Impacts of Climate Change: An Assessment of Vulnerability (Cambridge: Cambridge University Press)

    [4]

    IPCC 2001 Climate Change 2001: Impacts, Adaptation & Vulnerability (Cambridge: Cambridge University Press)

    [5]

    Zhang Z Q, Sun C Q 1999 Chin. Sci. Bull. 44 464 (in Chinese) [张志强, 孙成权 1999 科学通报 44 464]

    [6]

    Sha W Y, Shao X M, Huang M 2002 Scientia Sinica Terrae 32 317 (in Chinese) [沙万英, 邵雪梅, 黄玫 2002 中国科学(D辑) 32 317]

    [7]

    Liu S Y, Ding Y J, Li J, Shangguan D H, Zhang Y 2006 Quaternary Sciences 26 762 (in Chinese) [刘时银, 丁永建, 李晶, 上官冬辉, 张勇 2006 第四纪研究 26 762]

    [8]

    Zhang S X, Zhang L, Sun S P, Feng G L 2011 Plateau Meteor. 30 659 (in Chinese) [张世轩, 张璐, 孙树鹏, 封国林 2011 高原气象 30 659]

    [9]

    Sun S P, Zhang L, Hou W, Feng G L 2011 Acta Phys. Sin. 60 029201 (in Chinese) [孙树鹏, 张璐, 侯威, 封国林 2011 物理学报 60 029201]

    [10]

    Zong X P, Li M H, Hu J G, Feng G L 2010 Acta Phys. Sin. 59 8272 (in Chinese) [宗序平, 李明辉, 胡经国, 封国林 2010 物理学报 59 8272]

    [11]

    Zhao M S, Neilson R P, Yan X D 2002 Acta Geog. Sin. 57 28 (in Chinese) [赵茂盛, Ronald P. Neilson, 延晓东 2002 地理学报 57 28]

    [12]

    Zhou G S, Zhang X S 1995 Acta Phytoecologica Sinica 20 9 (in Chinese) [周广胜, 张新时 1995 植物生态学报 20 9]

    [13]

    Zhou G S, Wang Y H 1999 Journal of Natural Resources 14 318 (in Chinese) [周广胜, 王玉辉 1999 自然资源学报 14 318]

    [14]

    Li X B, Chen Y H 2002 Advances in Earth Science 17 254 (in Chinese) [李晓兵, 陈云浩 2002 地球科学进展 17 254]

    [15]

    Wang C H, Dong W J, Wei Z G 2001 Acta Geog. Sin. 56 523 (in Chinese) [王澄海, 董文杰, 韦志刚 2001 地理学报 56 523]

    [16]

    Lü X M, Wu S H, Yang Q Y 2003 Progress in Geography 22 260 (in Chinese) [吕新苗, 吴绍洪, 杨勤业 2003 地理科学进展 22 260]

    [17]

    Yang G S, Shi Y F 1995 Acta Geog. Sin. 50 302 (in Chinese) [杨桂山, 施雅风 1995 地理学报 50 302]

    [18]

    Douglas B C 1991 J. Geo. Phys. Research 96 6981

    [19]

    Pirazzoli PA 1989 Global and Planetary Change 1 241

    [20]

    Yang G S, Zhu J W 1993 Advances in Earth Science 8 70 (in Chinese) [杨桂山, 朱季文 1993 地球科学进展 8 70]

    [21]

    Li J L, Zhang D F, Yang X P, Tong Y Q, Zhang R S 2005 J. Catastrophology 20 49 (in Chinese) [李加林, 张殿发, 杨晓平, 童亿勤, 张忍顺 2005 灾害学 20 49]

    [22]

    Shi Y F 1996 Journal of Natural Disasters 5 106 (in Chinese) [施雅风 1996 自然灾害学报 5 106]

    [23]

    Ji Z X, Jiang Z X, Zhu J W 1993 Acta Geog. Sin. 48 516 (in Chinese) [季子修, 蒋自巽, 朱季文 1993 地理学报 48 516]

    [24]

    Hou W, Zhang D Q, Zhou Y, Yang P 2011 Acta Phys. Sin. 60 109202 (in Chinese) [侯威, 章大全, 周云, 杨萍 2011 物理学报 60 109202]

    [25]

    Feng G L, Yang J, Wan S Q, Hou W, Zhi R 2009 Acta Meteor. Sin. 67 61 (in Chinese) [封国林, 杨杰, 万仕全, 侯威, 支蓉 2009 气象学报 67 61]

    [26]

    Gong Z Q, Wang X J, Zhi R, Feng G L 2009 Acta Phys. Sin. 58 4342 (in Chinese) [龚志强, 王晓娟, 支蓉, 封国林 2009 物理学报 58 4342]

    [27]

    Wan S Q, Wang L, Feng G L, He W P Acta Phys. Sin. 58 5083 (in Chinese) [万仕全, 王令, 封国林, 何文平 2009 物理学报 58 5083]

    [28]

    Xiong K G, Feng G L, Wang Q G, Hu J G 2009 Acta Phys. Sin. 58 8107 (in Chinese) [熊开国, 封国林, 王启光, 胡经国 2009 物理学报 58 8107]

    [29]

    Zhang D Q, Yang J, Wang Q G, Feng G L 2009 Acta Phys. Sin. 58 4354 (in Chinese) [章大全, 杨杰, 王启光, 封国林 2009 物理学报 58 4354]

    [30]

    Da C J, Chou J F 2008 Acta Phys. Sin. 57 2595 (in Chinese) [达朝究, 丑纪范 2008 物理学报 57 2595]

    [31]

    Da C J, Sun S P, Song J, Yang L G 2011 Plateau Meteor. 30 349

    [32]

    Dong W J, Chou J M, Feng G L 2007 Adv. Atmos. Sci. 24 336

    [33]

    Wang B, Weng H Y 1981 Geophysical Fluid Dynamics (Beijing: China Ocean Press) pp7-141 (in Chinese) [王斌, 翁衡毅 1981 地球物理流体动力学导论 (北京: 海洋出版社)第60–141页]

    [34]

    Joseph Pedlosky 1986 Geophysical Fluid Dynamics (New York: Springer Verlag) pp7-9

    [35]

    Da C J 2005 M. S. Dissertation (Hohhot: Inner Mongolia University) (in Chinese) [达朝究 2005 硕士学位论文 (呼和浩特: 内蒙古大学)]

    [36]

    Guo B R B 1990 the Introduction of Linear and Nonlinear Wave Theory (Beijing: China Meteorological Press) (in Chinese) [郭秉荣 1990 线性波与非线性波导论 (北京: 气象出版社)]

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  • Received Date:  04 July 2012
  • Accepted Date:  07 September 2012
  • Published Online:  05 February 2013

Dynamic modification of the shallow water equation on the slowly changing underlying surface condition

  • 1. School of Mathematics and Computer Science Institute, Northwest University for Nationalities, Lanzhou 730030, China;
  • 2. College of Atmospheric Sciences, Lanzhou University, Lanzhou 730000, China;
  • 3. Laboratory for Climate Studies, National Climate Center, China Meteorological Administration, Beijing 100081, China;
  • 4. College of Science, Inner Mongolia University of Technology, Hohhot 010062, China
Fund Project:  Project supported by the National Basic Research Program of China (Grant No. 2012CB955902), the National Natural Science Foundation of China (Grant Nos. 40930952, 41175067), the Key Technologies R & D Program of China (Grant No. 2009BAC51B04), the Meteorological Special Project of China (Grant Nos. GYHY201106016, GYHY201206009), the Natural Science Foundation of Inner Mongolia, China (Grant No. 2011MS0112) and the Fundamental Research Funds for the Central Universities, China (Grant No. zyz2011079).

Abstract: In this paper, the shallow water equation is discussed, when the underlying surface is slowly changing. From the continuity equation and the equation of motion controlling the movement of the atmosphere, the slowly changing of the underlying surface is superimposed on the topography function as a small quantity, using the upper and lower boundary conditions of the atmosphere, the modified shallow water equation is obtained. In the modified shallow water equation, the slowly changing local horizontal divergence modifies the equilibrium between the local horizontal divergence and the local change of the thickness of the atmosphere. On the basis of this, the vorticity equation is obtained, which contains the slowly changing underlying surface.

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