Theoretical Research On Some Problems Of Variational Data Assimilation Method, Variational Optimization Analysis And Hydrodynamical Stability

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.