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众所周知, 风场分解与重构最有效的方法就是引入速度势和流函数, 其一般通过求解两个Poisson 方程得到. 由于速度势和流函数在边界上的耦合性质,有限区域风场分解是不唯一的, 这对风场分解带来了很大困难. 本文采用变分伴随结合正则化方法来克服风场分解的不唯一性, 其核心是把速度势和流函数的边值作为控制变量来反演. 目标泛函由两部分组成, 一是衡量重构风场误差大小的观测项; 二是保证风场分解问题适定的正则化项, 其目的在于寻求具有气象意义的稳定正则化解. 数值试验结果表明, 在正确选取正则化参数后, 利用变分伴随结合正则化方法进行有限区域风场分解与重构是有效可行的.As is well known, the efficient method to wind partitioning and reconstruction is to introduce the velocity potential and stream function which are calculated from divergence and vorticity by solving two Poisson's equations. Since velocity potential and stream function are coupled at the boundary of limited domain, the wind partitioning problem is nonunique. To vercome the nonuniqueness of the wind portioning, a new variational adjoint method combined with regularization is proposed in this paper, which is based on the control of velocity potential and stream function boundary values under Dirichlet conditions. The cost function is composed of two parts, one is the observation term to minimize the error of the reconstructed wind field, and the other is the regularization term to guarantee the uniqueness of the reconstruction problem by seeking a stable regularization solution within meteorological content. The results of numerical experiments demonstrate that after choosing an appropriate regularization parameter, the new variational adjoint method combined with regularization is efficient and suitable for wind portioning and reconstruction in a limited domain.
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Keywords:
- variational adjoint method /
- regularization method /
- stream function and velocity potential /
- wind partition and reconstruction
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[2] Ran L K, Gao S T, Li C Y 1995 Chin. J. Atmos. Sci. 19 209 (in Chinese) [冉令坤, 高守亭, 李崇银 1995 大气科学 19 209]
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[4] Ding Y H, Hu G Q 2003 Acta Mete. Sin. 61 129 (in Chinese) [丁一汇, 胡国权 2003 气象学报 61 129]
[5] Gao S T, Sun J H, Cui X P 2008 Chin. J. Atmos. Sci. 32 854 (in Chinese) [高守亭, 孙建华, 崔晓鹏 2008 大气科学 32 854]
[6] Baede A P M, Jarraud M M, Cubasch U 1979 ECMWF Tech. Rep. 15 39
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[8] Phillips N A 1958 Geophysica 6 389
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[10] Shukla J, Saha K R 1974 Mon. Wea. Rev. 102 419
[11] Stephens J, Johnson K 1978 Mon. Wea. Rev. 106 1452
[12] Lynch P 1989 Mon. Wea. Rev. 117 1492
[13] Bijlsma S J, Hafkensheid L M, Lynch P 1986 Mon. Wea. Rev. 114 1547
[14] Chen Q S, Kuo Y H 1992 Mon. Wea. Rev. 120 91
[15] Chen Q S, Kuo Y H 1992 Mon. Wea. Rev. 120 2653
[16] 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]
[17] Zhou Y S, Cao J 2010 Acta Phys. Sin. 59 2898 (in Chinese) [周玉淑, 曹洁 2010 59 2898]
[18] Bishop H C 1996 J. Atmos. Sci. 53 241
[19] Tikhonov A N, Arsenin V Y 1977 Solution of Ill-Posed Problems (New York: Winston and Sons) p224
[20] Huang S X, Wu R S 2005 Mathematical and Physical Problems in Atmospheric Science (Beijing: Meteorological Press) p422 (in Chinese) [黄思训, 伍荣生 2005 大气科学中的数学物理问题 (北京: 气象出版社) 第422页]
[21] Sheng Z, Huang S X 2010 Acta Phys. Sin. 59 1734 (in Chinese) [盛峥, 黄思训 2010 59 1734]
[22] Zhao Y L, Huang S X, Du H D, Zhong J Q 2011 Acta Phys. Sin. 60 079202 (in Chinese) [赵延来, 黄思训, 杜华栋, 仲跻芹 2011 60 079202]
[23] Li K T, Ma Y C 1990 Hilbert Space Methods for Mathematical Equation (Vol. 1) General Function and Sobolev Spaces (Xi'an: Xi'an Jiao- tong University Press) p222 (in Chinese) [李开泰, 马逸尘 1990 数理方程 Hilbert 空间方程方法(上)广义函数和Sobolev空间 (西安: 西安交通大学出版社) 第222页]
[24] Le Dimet F X, Mohamed O 1993 Tellus A 45 449
[25] Li Z J, Chao Y, McWilliams J C 2006 Mon. Wea. Rev. 134 3384
[26] Kirsch A 1996 An Introduction to the Mathematical Theory of Inverse Problems (New York: Springer-Verlag) p48
[27] Liu D C, Nocedal J 1989 Math. Program. 45 503
[28] Engl H W 1987 J. Optim. Theory. Appl. 52 209
[29] Barker D M, Huang W, Guo Y R, Bourgeois A J, Xiao Q N 2004 Mon. Wea. Rev. 132 897
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[1] Krishnamurti T, Ramanathan Y 1982 J. Atmos. Sci. 39 1290
[2] Ran L K, Gao S T, Li C Y 1995 Chin. J. Atmos. Sci. 19 209 (in Chinese) [冉令坤, 高守亭, 李崇银 1995 大气科学 19 209]
[3] Zhang L, Zhang L F, Wu H Y, Li G 2010 Acta Phys. Sin. 59 44 (in Chinese) [张亮, 张立凤, 吴海燕, 李刚 2010 59 44]
[4] Ding Y H, Hu G Q 2003 Acta Mete. Sin. 61 129 (in Chinese) [丁一汇, 胡国权 2003 气象学报 61 129]
[5] Gao S T, Sun J H, Cui X P 2008 Chin. J. Atmos. Sci. 32 854 (in Chinese) [高守亭, 孙建华, 崔晓鹏 2008 大气科学 32 854]
[6] Baede A P M, Jarraud M M, Cubasch U 1979 ECMWF Tech. Rep. 15 39
[7] Bourke W 1974 Mon. Wea. Rev. 102 687
[8] Phillips N A 1958 Geophysica 6 389
[9] Sangster W E 1960 J. Atmos. Sci. 17 166
[10] Shukla J, Saha K R 1974 Mon. Wea. Rev. 102 419
[11] Stephens J, Johnson K 1978 Mon. Wea. Rev. 106 1452
[12] Lynch P 1989 Mon. Wea. Rev. 117 1492
[13] Bijlsma S J, Hafkensheid L M, Lynch P 1986 Mon. Wea. Rev. 114 1547
[14] Chen Q S, Kuo Y H 1992 Mon. Wea. Rev. 120 91
[15] Chen Q S, Kuo Y H 1992 Mon. Wea. Rev. 120 2653
[16] 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]
[17] Zhou Y S, Cao J 2010 Acta Phys. Sin. 59 2898 (in Chinese) [周玉淑, 曹洁 2010 59 2898]
[18] Bishop H C 1996 J. Atmos. Sci. 53 241
[19] Tikhonov A N, Arsenin V Y 1977 Solution of Ill-Posed Problems (New York: Winston and Sons) p224
[20] Huang S X, Wu R S 2005 Mathematical and Physical Problems in Atmospheric Science (Beijing: Meteorological Press) p422 (in Chinese) [黄思训, 伍荣生 2005 大气科学中的数学物理问题 (北京: 气象出版社) 第422页]
[21] Sheng Z, Huang S X 2010 Acta Phys. Sin. 59 1734 (in Chinese) [盛峥, 黄思训 2010 59 1734]
[22] Zhao Y L, Huang S X, Du H D, Zhong J Q 2011 Acta Phys. Sin. 60 079202 (in Chinese) [赵延来, 黄思训, 杜华栋, 仲跻芹 2011 60 079202]
[23] Li K T, Ma Y C 1990 Hilbert Space Methods for Mathematical Equation (Vol. 1) General Function and Sobolev Spaces (Xi'an: Xi'an Jiao- tong University Press) p222 (in Chinese) [李开泰, 马逸尘 1990 数理方程 Hilbert 空间方程方法(上)广义函数和Sobolev空间 (西安: 西安交通大学出版社) 第222页]
[24] Le Dimet F X, Mohamed O 1993 Tellus A 45 449
[25] Li Z J, Chao Y, McWilliams J C 2006 Mon. Wea. Rev. 134 3384
[26] Kirsch A 1996 An Introduction to the Mathematical Theory of Inverse Problems (New York: Springer-Verlag) p48
[27] Liu D C, Nocedal J 1989 Math. Program. 45 503
[28] Engl H W 1987 J. Optim. Theory. Appl. 52 209
[29] Barker D M, Huang W, Guo Y R, Bourgeois A J, Xiao Q N 2004 Mon. Wea. Rev. 132 897
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