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The potential vorticity theory and diagnostic techniques are based on the potential vorticity equation and expression in the common meteorological coordinate systems. In this paper, the potential vorticity equation and expression in the isobaric and isoentropic coordinates are gotten via coordinate transformation with the two methods. First, starting from the three-dimensional vector motion equation, the potential vorticity equations and expressions are gotten by the combination of the three-dimensional vorticity equation, continuity equation, and thermodynamic equation. Second, the potential vorticity equations and expressions are directly gotten from the corresponding scalar motion equations in the isobaric and isoentropic coordinates. The results show that potential vorticity expression is different with one method from that with the other in the isobaric coordinate system, and it is the same as each other in the isoentropic coordinate system. It was found, based on further analysis of the physical nature of the coordinates, that the isobaric and isoentropic coordinates are essentially treated as a mathematical coordinate system with the first method despite the coordinate transformation made for the term of pressure gradient force in the vector motion equation. From the procedure for the second method it is clearly seen that the isobaric and isoentropic coordinate systems are the physical coordinate system under the assumption of static equilibrium, which are not simply used as a mathematical coordinate system. As far as the isobaric coordinate is concerned, only the potential vorticity equation obtained from the scalar motion equations is the strict potential vorticity equation. As for the isoentropic coordinate, owing to the potential temperature gradient perpendicular to the isoentropic plane, the potential vorticity equation and expression are the same regardless of the coordinate being viewed as the physical or the mathematical.
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Keywords:
- potential vorticity /
- coordinate transformation /
- isobaric coordinate /
- isoentropic coordinate
[1] Cai Q F, Huang S X, Gao S T, Zhong K, Li Z Q 2008 Acta Phys. Sin. 57 3912 (in Chinese)[蔡其发、黄思训、高守亭、钟 科、李自强 2008 57 3912]
[2] Wu G X 2001 Acta Meteor. Sin. 59 385 (in Chinese)[吴国雄 2001 气象学报 59 385]
[3] Zhou Y S, Ran L K 2010 Acta Phys. Sin. 59 1366 (in Chinese)[周玉淑、冉令坤 2010 59 1366]
[4] Cao J, Gao S T, Zhou Y S 2008 Acta Phys. Sin. 57 2600 (in Chinese)[曹 洁、高守亭、周玉淑 2008 57 2600]
[5] Zhou Y S, Cao J, Gao S T 2008 Acta Phys. Sin. 57 6654 (in Chinese)[周玉淑、曹 洁、高守亭 2008 57 6654]
[6] Tao J J, Li C K 2009 Acta Phys. Sin. 58 4313 (in Chinese)[陶建军、李朝奎 2009 58 4313]
[7] Huang S X, Cai Q F, Xiang J, Zhang M 2007 Acta Phys. Sin. 56 3022 (in Chinese)[黄思训、蔡其发、项 杰、张 铭 2007 56 3022]
[8] Ertel H 1942 Met.Z. 59 271
[9] Hoskins B J, McIntyre M E, Robertson A W 1985 Quart. J. Roy.Meteor. Soc. 111 877
[10] Wu G X, Cai Y P, Tang X J 1995 Acta Meteor. Sin. 53 387 (in Chinese)[吴国雄、蔡雅萍、唐晓菁 1995 气象学报 53 387]
[11] Liu X Q,Wang T M,Wu B J,Hu S C,Chen Q J 1994 Chin. J. Atmos. Sci. 18 570 (in Chinese)[留小强、王田民、吴宝俊、胡圣昌、陈乾金 1994 大气科学 18 570]
[12] Liu H Z, Zhang S Q 1996 J. Appl. Meteor. Sci. 7 275 (in Chinese)[刘还珠、张绍晴 1996 应用气象学报 7 275]
[13] Gao S T, Lei T, Zhou Y S, Dong M 2002 J. Appl. Meteor. Sci. 13 662 (in Chinese)[高守亭、雷 霆、周玉淑、董 敏 2002 应用气象学报 13 662]
[14] Schubert W H, Hausman S A, Garcia M, Ooyama K V, Kuo H C 2001 J. Atmos. Sci. 57 3148
[15] Zhang W, Tao Z Y, Hu Y Y, Wang H Q, Huang W 2006 Acta Sci. Natura. Uni. Pek. 42 61 (in Chinese)[张 伟、陶祖钰、胡永云、王洪庆、黄 炜 2006 北京大学学报(自然科学版) 42 61]
[16] Yu Y B, Yao X P 2003 Acta Meteor. Sin. 61 769 (in Chinese)[于玉斌、姚秀萍 2003 气象学报 61 769]
[17] Yao X P, Peng G, Yu Y B 2009 Plateau Meteor. 28 507 (in Chinese)[姚秀萍、彭 广、于玉斌 2009 高原气象 28 507]
[18] Wang T H, Yang S 2009 Acta Meteor. Sin. 67 522 (in Chinese)[王东海、杨 帅 2009 气象学报 67 522]
[19] Browning K A 1997 Meteor. Appl. 4 317
[20] Browning K A, Roberts N M 1996 Quart J. Roy. Meteor. Soc.122 1845
[21] Davis C A 1992 J Atmos. Sci. 49 1397
[22] Zhang S W, Wang S G 2001 Plateau Meteor. 20 468 (in Chinese)[张述文、王式功 2001 高原气象 20 468]
[23] Zhao B K, Liu Y M, Liang P 2008 Plateau Meteor. 27 158 (in Chinese) [赵兵科、刘屹岷、梁 萍 2008 高原气象 27 158]
[24] Hu Y J, Zhong Z, Min J Z 2008 Chin. J. Atmos. Sci. 32 90 (in Chinese)[胡轶佳、钟 中、闵锦忠 2008 大气科学 32 90]
[25] Liu Y, Liu C J, Xu H, Zhao Y M 2006 Plateau Meteor. 25 651 (in Chinese)[刘 英、柳崇健、徐 辉、赵永明 2006 高原气象 25 651]
[26] Yan H M, Ren J Z, Duan W 2006 J. Nanjing Inst. Meteor. 29 491 (in Chinese) [晏红明、任菊章、段 玮 2006 南京气象学院学报 29 491]
[27] Brennan M J,Lackmann G M, Mahoney K M 2008 Wea. Forecasting 23 168
[28] Hu B W 2003 J. Nanjing Inst. Meteor. 26 111 (in Chinese)[胡伯威 2003 南京气象学院学报 26 111]
[29] Yuan Z J 1999 Chin. J. Atmos. Sci. 23 199 (in Chinese)[袁卓建 1999 大气科学 23 199]
[30] Ding Y H 1989 Diagnosis method of weather dynamics(Beijing: Science Press)p178 (in Chinese)[丁一汇 1989 天气动力学中的诊断分析方法(北京:科学出版社)第178页]
[31] Zhao G X, Cheng L S, Li X S 2007 Plateau Meteor. 26 615 (in Chinese)[赵桂香、程麟生、李新生 2007 高原气象 26 615]
[32] Wu R S 2002 Atmospheric dynamics (Beijing: Higher Education Press ) p44 (in Chinese)[伍荣生 2002 大气动力学(北京:高等教育出版社) 第44页]
[33] Lu M Z, Peng Y Q 1990 Dynamic meteorology tutorial (Beijing: China Meteorological Press) p132 (in Chinese)[吕美仲、彭永清1990 动力气象学教程(北京:气象出版社)第132页]
[34] Yang D S, Liu Y B, Liu S K 1980 Dynamic meteorology (Beijing: China Meteorological Press) p119 (in Chinese)[杨大 升、刘余滨、刘式适 1980 动力气象学(北京:气象出版社)第119页]
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[1] Cai Q F, Huang S X, Gao S T, Zhong K, Li Z Q 2008 Acta Phys. Sin. 57 3912 (in Chinese)[蔡其发、黄思训、高守亭、钟 科、李自强 2008 57 3912]
[2] Wu G X 2001 Acta Meteor. Sin. 59 385 (in Chinese)[吴国雄 2001 气象学报 59 385]
[3] Zhou Y S, Ran L K 2010 Acta Phys. Sin. 59 1366 (in Chinese)[周玉淑、冉令坤 2010 59 1366]
[4] Cao J, Gao S T, Zhou Y S 2008 Acta Phys. Sin. 57 2600 (in Chinese)[曹 洁、高守亭、周玉淑 2008 57 2600]
[5] Zhou Y S, Cao J, Gao S T 2008 Acta Phys. Sin. 57 6654 (in Chinese)[周玉淑、曹 洁、高守亭 2008 57 6654]
[6] Tao J J, Li C K 2009 Acta Phys. Sin. 58 4313 (in Chinese)[陶建军、李朝奎 2009 58 4313]
[7] Huang S X, Cai Q F, Xiang J, Zhang M 2007 Acta Phys. Sin. 56 3022 (in Chinese)[黄思训、蔡其发、项 杰、张 铭 2007 56 3022]
[8] Ertel H 1942 Met.Z. 59 271
[9] Hoskins B J, McIntyre M E, Robertson A W 1985 Quart. J. Roy.Meteor. Soc. 111 877
[10] Wu G X, Cai Y P, Tang X J 1995 Acta Meteor. Sin. 53 387 (in Chinese)[吴国雄、蔡雅萍、唐晓菁 1995 气象学报 53 387]
[11] Liu X Q,Wang T M,Wu B J,Hu S C,Chen Q J 1994 Chin. J. Atmos. Sci. 18 570 (in Chinese)[留小强、王田民、吴宝俊、胡圣昌、陈乾金 1994 大气科学 18 570]
[12] Liu H Z, Zhang S Q 1996 J. Appl. Meteor. Sci. 7 275 (in Chinese)[刘还珠、张绍晴 1996 应用气象学报 7 275]
[13] Gao S T, Lei T, Zhou Y S, Dong M 2002 J. Appl. Meteor. Sci. 13 662 (in Chinese)[高守亭、雷 霆、周玉淑、董 敏 2002 应用气象学报 13 662]
[14] Schubert W H, Hausman S A, Garcia M, Ooyama K V, Kuo H C 2001 J. Atmos. Sci. 57 3148
[15] Zhang W, Tao Z Y, Hu Y Y, Wang H Q, Huang W 2006 Acta Sci. Natura. Uni. Pek. 42 61 (in Chinese)[张 伟、陶祖钰、胡永云、王洪庆、黄 炜 2006 北京大学学报(自然科学版) 42 61]
[16] Yu Y B, Yao X P 2003 Acta Meteor. Sin. 61 769 (in Chinese)[于玉斌、姚秀萍 2003 气象学报 61 769]
[17] Yao X P, Peng G, Yu Y B 2009 Plateau Meteor. 28 507 (in Chinese)[姚秀萍、彭 广、于玉斌 2009 高原气象 28 507]
[18] Wang T H, Yang S 2009 Acta Meteor. Sin. 67 522 (in Chinese)[王东海、杨 帅 2009 气象学报 67 522]
[19] Browning K A 1997 Meteor. Appl. 4 317
[20] Browning K A, Roberts N M 1996 Quart J. Roy. Meteor. Soc.122 1845
[21] Davis C A 1992 J Atmos. Sci. 49 1397
[22] Zhang S W, Wang S G 2001 Plateau Meteor. 20 468 (in Chinese)[张述文、王式功 2001 高原气象 20 468]
[23] Zhao B K, Liu Y M, Liang P 2008 Plateau Meteor. 27 158 (in Chinese) [赵兵科、刘屹岷、梁 萍 2008 高原气象 27 158]
[24] Hu Y J, Zhong Z, Min J Z 2008 Chin. J. Atmos. Sci. 32 90 (in Chinese)[胡轶佳、钟 中、闵锦忠 2008 大气科学 32 90]
[25] Liu Y, Liu C J, Xu H, Zhao Y M 2006 Plateau Meteor. 25 651 (in Chinese)[刘 英、柳崇健、徐 辉、赵永明 2006 高原气象 25 651]
[26] Yan H M, Ren J Z, Duan W 2006 J. Nanjing Inst. Meteor. 29 491 (in Chinese) [晏红明、任菊章、段 玮 2006 南京气象学院学报 29 491]
[27] Brennan M J,Lackmann G M, Mahoney K M 2008 Wea. Forecasting 23 168
[28] Hu B W 2003 J. Nanjing Inst. Meteor. 26 111 (in Chinese)[胡伯威 2003 南京气象学院学报 26 111]
[29] Yuan Z J 1999 Chin. J. Atmos. Sci. 23 199 (in Chinese)[袁卓建 1999 大气科学 23 199]
[30] Ding Y H 1989 Diagnosis method of weather dynamics(Beijing: Science Press)p178 (in Chinese)[丁一汇 1989 天气动力学中的诊断分析方法(北京:科学出版社)第178页]
[31] Zhao G X, Cheng L S, Li X S 2007 Plateau Meteor. 26 615 (in Chinese)[赵桂香、程麟生、李新生 2007 高原气象 26 615]
[32] Wu R S 2002 Atmospheric dynamics (Beijing: Higher Education Press ) p44 (in Chinese)[伍荣生 2002 大气动力学(北京:高等教育出版社) 第44页]
[33] Lu M Z, Peng Y Q 1990 Dynamic meteorology tutorial (Beijing: China Meteorological Press) p132 (in Chinese)[吕美仲、彭永清1990 动力气象学教程(北京:气象出版社)第132页]
[34] Yang D S, Liu Y B, Liu S K 1980 Dynamic meteorology (Beijing: China Meteorological Press) p119 (in Chinese)[杨大 升、刘余滨、刘式适 1980 动力气象学(北京:气象出版社)第119页]
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