大气动力学方程组的定性理论及其应用

基于完整的湿大气动力学方程组，利用无穷维动力系统的新理论和新方法，系统讨论了强迫耗散的非线性大气系统的定性理论及其应用。将完整的强迫耗散非线性湿大气动力学方程组化为Hilbert空间中一个等价的算子方程，研究了算子的性质及其物理意义，在此基础上得到湿大气系统全局吸引子的存在性定理，揭示出系统向外源的非线性适应特征，并把结果推广到有地形动力作用和非定常外源强迫的情形。同时探讨了大气方程组惯性流形的存在，大气多平衡态产生的根源以及强迫、耗散和非线性对系统解的渐近行为的影响。在理论结果的基础上，提出强迫耗散的非线性动力系统中存在三类时间边界层、方程组简化准则、分解算法的算子约束原则以及支撑吸引子基底的少数自由度的构造方法，探讨了理论在非线性发展方程差分格式的设计和计算稳定性分析、多平衡态的数值分析、数值模式延伸预报的改进、短期气候预测以及一类中尺度系统分析与预测中的应用，指出描述长期过程动力学模式的必备条件，给出初值与模式相协调的合理解释。最后，对今后的研究方向作了展望。

The Qualitative Theory of the Dynamical Equations of Atmospheric Motion and Its Applications

Based on the complete dynamical equations of the moist atmospheric motion, the qualitative theory of nonlinear atmosphere with dissipation and external forcing and its applications are systematically discussed by new theories and methods on the infinite dimensional dynamical system. The complete nonlinear dynamical equations of the moist atmospheric motion with dissipation and external forcing are rewritten as an equivalent operator equation in Hilbert space, and the properties of operators and their physical senses are studied. On the basis of the above, the existence of global attractor of the moist atmospheric system is proved, and the characteristic of nonlinear adjustment to external forcing is revealed. Furthermore, the results mentioned above are extended to the cases with orographic dynamical effect and non-stationary external forcing. Meanwhile, the existence of the inertial manifold of the atmospheric equations is investigated. On the basis of theoretical results, this paper presents three classes of time boundary layers existed in the forced dissipative nonlinear system, the principle of simplification of atmospheric equations, the principle of operator restriction of splitting algorithm and the constructive method of a few freedoms to support the base of attractor. Besides, the applications of the theoretical results to the design of difference schemes, the computational stability analysis of nonlinear evolutionary equations, the numerical analysis of atmospheric multiple equilibria, the improvement of extended forecast of numerical models and the short-term climate forecast, the analysis and prediction of one class of mesoscale system are discussed; the necessary conditions possessed in the dynamic models describing the long-time process are pointed out; and a rational explanation of the harmonization of initial values with model is given. Finally, the studied trends in future are prospected.