变分同化、变分最优分析及流体动力稳定性中若干问题的研究
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摘要
本文针对大气科学和流体动力学中几个重要的理论与应用领域:资料变分同化、变分最优分析及扰动能量有限时间快速增长的问题进行了研究。论文的主要工作分为三个部分:第一部分对于特定模式,理论上分析了变分同化与正则化的关系,给出了误差估计;第二部分,提出了最优分析与正则化相结合的方法,称为广义变分最优分析方法,并用来进行了二维、三维风场的最优变分分析,;第三部分,运用数值计算方法,对线性位涡方程在有限时间内扰动能量的发展问题进行了研究,这个问题对灾害性天气预报有重要的意义。
在第一部分中,利用变分同化方法,在整体观测资料下,对一类简单的预报模式的初值、参数及模式进行修正,讨论了变分同化方法的误差估计问题。文中不仅研究了观测误差和模式误差对于变分同化方法的影响,还考虑了参数误差以及边界扰动情形对于变分同化方法的影响。对变分同化方法反演得到的初值和预报值的收敛性以及收敛精度进行了估计,从理论上分析了变分同化方法的有效性。对于局部观测资料下的变分同化方法,由于问题的不适定性,利用变分同化结合数学物理反问题中的Tikhonov正则化方法,对所讨论的解加上适当的约束条件,引入稳定性泛函和正则化参数,对通常的变分同化方法进行改进,使得原来不适定的问题变得适定。从理论上对结合正则化的变分同化方法反演得到的初值和预报值的收敛性以及收敛精度进行了估计。结果表明,在适当的正则化参数下,利用结合正则化方法的变分同化方法是有效的。
在第二部分中,首先分析了Sasaki提出的变分最优分析方法(VOAM)对于带有高频噪音的观测资料的缺陷,然后通过变分最优分析结合正则化方法,提出了广义变分最优化方法(GVOAM)分析求解二维流场和三维风场,理论分析和数值试验表明该方法不仅能滤去高频噪音,而且可以提高计算的精度,达到与真实流场十分逼近的预报结果。
在第三部分中,讨论了有限时间段内扰动能量快速发展的问题,由于强烈发展的天气尺度和次天气尺度系统本身会造成灾害,本问题研究对灾害性天气预报具有重要的指导意义。本文对三种水平切变条件下的正压准地转线性位涡方程在有限时间内的扰动能量发展问题进行了研究。借助于数值计算手段,分析了有限时间内扰动能量的发展情况。通过分析表明,有限时间内扰动能量的发展情况主要取决于初始扰动和水平切变条件,扰动能量在开始的一段时间内急剧增长,然后随着时间的发展,扰动能量呈现振荡衰减的现象,最终趋向于零。
In this thesis, some theoretical problems are studied in the domains of variational data assimilation, variational optimization analysis and the fast growth of disturbance energy within fininte time interval in atmosphere science and hydrodynamics. Three main parts are included in this work. Firstly, the relation between the variational data assimilation and regularization is analyzed and the error estimates of variational data assimilation methods for some special models are theoretically researched. Secondly, an improved variational data assimilation method is presented, which combines the optimization analysis with the regularization method and is named as the deneralized variational optimization analysis method (GVOAM), and then used in the analysis of 2-D and 3-D wind fields; Thirdly, the growth of disturbance energy of a linear model within a finite period of time is theoretically and numerically investigated with the presented method. The obtained results are of significance in the disaster weather forecasting.
In the first part of this thesis, on condition that the global observed data have been obtained, the initial conditions and the parameters for a diffusion equation are optimally modified by using the variational data assimilation method, and the estimates of prediction errors are given. The impacts of the parameter and boundary errors as well as of the observational and model errors, on variational assimilation analysis are investigated for the diffusion equation. Some mathematical methods are used to determine the convergence and the convergence rates of the assimilated initial values and the prediction solutions, and the variational assimilation method is theoretically proved to be an effective method. For given local observed data, in order to overcome the difficulties caused by the ill-posedness of inverse problems, the Tikhonov regularization method is integrated into the variational data assimilation method, resuling in an improved variational data assimilation method. The convergence and convergence rates of the assimilated initial conditions and the prediction solutions of the improved method are analyzed in the same way. By introducing the appropriate regularization parameters, the improved variational data assimilation method is proved to be rather effective and efficient.
In the second part, the variational optimization analysis method (VOAM) for 2-D flow field and 3-D wind field suggested by Sasaki are first reviewed. It is known that the VOAM can be used efficiently in most cases. However, in the cases where there are high- frequency noises in 2onserved data, it appears to be inefficient. In the present thesisr, based on Sasaki's VOAM, a generalized variational optimization method (GVOAM) is proposed with the aid of regularization ideas, which can deal well with flow field containing high-frequency noises. Some numerical tests for 2-D flow field and 3-D wind field show that observed data can be both variationally optimized and filtered, and that the flow fields,close to real ones can be resulted, much better than those predicted by the VOAM, which indicated that the GVOAM is an efficient method.
In the third part, the fast growth of disturbance energy within a fininte time interval is discussed, for the disaster weather is frequently induced by the strongly-developed synoptic-scale and the subsynoptic-scale systems, and therefore this study is of importance in the disaster weather forecasting. The evolution of disturbance energy for the linear potential vorticity equation within a finite time interval is studied under three different conditions for horizontal shear conditions. The equation is discretized with an implicit difference scheme and the disturbance energy is computed for three different horizontal shear conditions respectively. Through computing, the disturbance energies are all found to be sharply increasing at the beginning. Though the disturbance energy brings about some oscillatory growth phenomena in the following time, it decreases with the time finally.
在第一部分中,利用变分同化方法,在整体观测资料下,对一类简单的预报模式的初值、参数及模式进行修正,讨论了变分同化方法的误差估计问题。文中不仅研究了观测误差和模式误差对于变分同化方法的影响,还考虑了参数误差以及边界扰动情形对于变分同化方法的影响。对变分同化方法反演得到的初值和预报值的收敛性以及收敛精度进行了估计,从理论上分析了变分同化方法的有效性。对于局部观测资料下的变分同化方法,由于问题的不适定性,利用变分同化结合数学物理反问题中的Tikhonov正则化方法,对所讨论的解加上适当的约束条件,引入稳定性泛函和正则化参数,对通常的变分同化方法进行改进,使得原来不适定的问题变得适定。从理论上对结合正则化的变分同化方法反演得到的初值和预报值的收敛性以及收敛精度进行了估计。结果表明,在适当的正则化参数下,利用结合正则化方法的变分同化方法是有效的。
在第二部分中,首先分析了Sasaki提出的变分最优分析方法(VOAM)对于带有高频噪音的观测资料的缺陷,然后通过变分最优分析结合正则化方法,提出了广义变分最优化方法(GVOAM)分析求解二维流场和三维风场,理论分析和数值试验表明该方法不仅能滤去高频噪音,而且可以提高计算的精度,达到与真实流场十分逼近的预报结果。
在第三部分中,讨论了有限时间段内扰动能量快速发展的问题,由于强烈发展的天气尺度和次天气尺度系统本身会造成灾害,本问题研究对灾害性天气预报具有重要的指导意义。本文对三种水平切变条件下的正压准地转线性位涡方程在有限时间内的扰动能量发展问题进行了研究。借助于数值计算手段,分析了有限时间内扰动能量的发展情况。通过分析表明,有限时间内扰动能量的发展情况主要取决于初始扰动和水平切变条件,扰动能量在开始的一段时间内急剧增长,然后随着时间的发展,扰动能量呈现振荡衰减的现象,最终趋向于零。
In this thesis, some theoretical problems are studied in the domains of variational data assimilation, variational optimization analysis and the fast growth of disturbance energy within fininte time interval in atmosphere science and hydrodynamics. Three main parts are included in this work. Firstly, the relation between the variational data assimilation and regularization is analyzed and the error estimates of variational data assimilation methods for some special models are theoretically researched. Secondly, an improved variational data assimilation method is presented, which combines the optimization analysis with the regularization method and is named as the deneralized variational optimization analysis method (GVOAM), and then used in the analysis of 2-D and 3-D wind fields; Thirdly, the growth of disturbance energy of a linear model within a finite period of time is theoretically and numerically investigated with the presented method. The obtained results are of significance in the disaster weather forecasting.
In the first part of this thesis, on condition that the global observed data have been obtained, the initial conditions and the parameters for a diffusion equation are optimally modified by using the variational data assimilation method, and the estimates of prediction errors are given. The impacts of the parameter and boundary errors as well as of the observational and model errors, on variational assimilation analysis are investigated for the diffusion equation. Some mathematical methods are used to determine the convergence and the convergence rates of the assimilated initial values and the prediction solutions, and the variational assimilation method is theoretically proved to be an effective method. For given local observed data, in order to overcome the difficulties caused by the ill-posedness of inverse problems, the Tikhonov regularization method is integrated into the variational data assimilation method, resuling in an improved variational data assimilation method. The convergence and convergence rates of the assimilated initial conditions and the prediction solutions of the improved method are analyzed in the same way. By introducing the appropriate regularization parameters, the improved variational data assimilation method is proved to be rather effective and efficient.
In the second part, the variational optimization analysis method (VOAM) for 2-D flow field and 3-D wind field suggested by Sasaki are first reviewed. It is known that the VOAM can be used efficiently in most cases. However, in the cases where there are high- frequency noises in 2onserved data, it appears to be inefficient. In the present thesisr, based on Sasaki's VOAM, a generalized variational optimization method (GVOAM) is proposed with the aid of regularization ideas, which can deal well with flow field containing high-frequency noises. Some numerical tests for 2-D flow field and 3-D wind field show that observed data can be both variationally optimized and filtered, and that the flow fields,close to real ones can be resulted, much better than those predicted by the VOAM, which indicated that the GVOAM is an efficient method.
In the third part, the fast growth of disturbance energy within a fininte time interval is discussed, for the disaster weather is frequently induced by the strongly-developed synoptic-scale and the subsynoptic-scale systems, and therefore this study is of importance in the disaster weather forecasting. The evolution of disturbance energy for the linear potential vorticity equation within a finite time interval is studied under three different conditions for horizontal shear conditions. The equation is discretized with an implicit difference scheme and the disturbance energy is computed for three different horizontal shear conditions respectively. Through computing, the disturbance energies are all found to be sharply increasing at the beginning. Though the disturbance energy brings about some oscillatory growth phenomena in the following time, it decreases with the time finally.
引文
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[2].黄荣辉,吴津生.卫星资料在数值天气预测中的应用与四维同化分析方法的若干进展.卫星气象文集,北京:气象出版社,国家卫星气象中心,1987:23-26.
[3].Cressman G P.An operational objective analysis scheme.Mon.Wea.Rev.,1959,87:367-374.
[4].张玉玲,吴辉碇,王晓林.数值天气预报.北京:科学出版社,1986:1-472.
[5].曾忠一.气象资料同化.渤海堂出版,台北市,1997:593.
[6].郑祚芳,沈桐立,张秀丽.气象卫星探测资料在数值天气预报中的应用.气象,2001,27(9):3-8.
[7].Sasaki Y K.Proposed inclusion of time variation terms,observational and theoretical,in numerical variational objective analysis.J.Meteor.Soc.,Japan,1969,47:115-124.
[8].Sasaki Y K.Some basic formalisms in numerical variational analysis.Mon.Wea.Rev.,1970,98:875-883.
[9].LeDimet F,Talagrand O.Variational algorithms for analysis and assimilation of meteorological observations:Theoretical aspects.Tellus,1986,38A:97-110.
[10].Talagrant O,Curtier P.Variational assimilation of meteorological observations with the adjoint vorticity equation,Part Ⅰ:Theory.Quart.J.Roy.Meter.Soc.,1987,113(478):1311-1328.
[11].Courtier P,Derber J,Errico R,Louis J F,Vukicevic.Important literature on the use of adjoint,variational methods and the Kalman filter in meteorology.Tellus,1993,45A:342-357.
[12].黄思训,伍荣生.大气科学中的数学物理问题.北京:气象出版社,2001:411-473.
[13].Navon I M.Practical and theoretical aspects of adjoint parameter estimation and identifiablity in meteorology and oceanography.Dyn.Atmos.Oceans,1997,27:55-79.
[14].Penenko V V,Obratsov N N.A variational initialization method for the fields of the meteorological elements(English translation).Soviet Meteorology and Hydrology, 1976,11:1-11.
[15] . Penenko V V. Energy balanced discrete models of atmospheric proceed dynamics. (English translation). Soviet Meteorology and Hydrology, 1977,10:3-20.
[16] . Lewis J M, Derber J C. The use of adjoint equation to solve a variational adjustment problem with advective constraints. Tellus, 1985,37A: 309-322.
[17] . Derker J C. Variational four-dimensionalanalysis using quasi-geostrophic constraints. Mon.Wea.Rev., 1987,115: 998-1008.
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