地形边界变化条件下的浅水方程求解及其动力学特征

Due to the natural variation and the anthropogenic factors, the altitude of the topographic boundary maybe changed, which may influence the circulation of atmosphere, and then change the local and global climate. The research project based on the fact of changing terrain, discusses the dynamics process of the effect upon the atmosphere. The topography is divided into two parts, the natural terrain and the disturbance term, the latter one is considered as a small quantity. Small parameter expansion is applied to study, using the method of treatment the boundary condition of the wave dynamics and the skill of the boundary condition of differential equation, we discuss the shallow water wave equation. Through the mathematical derivation, this research obtains the vorticity equation involving the change of the topographic boundary, and can get the KdVequation for the evolution of the amplitude of the vorticity(mKdV equation stands for the stratified fluid). We introduce a new method to solve the KdV equation, and will make this method better to adapt to the KdV equation which contains the topographic boundary. The research emphasis is the mKdV equation, which has been less studied so far. From the analytical solution and the numerical solution of the KdV equation, the influence of the topographic boundary upon the circulation of atmosphere, the blocking high and the cold vortex can be shown. The results can not only help to understand the atmosphere dynamical process, but also can offer scientific basis for the climate model.

由于自然变化和人为因素影响，地形高度会发生变化，进而导致大气环流、局地气候乃至全球气候的变化。本项目针对地形高度变化的事实，深入探讨地形对大气影响的动力过程。将地形变化分为两部分：固有地形项和随时间变化的扰动项；视后者为一小项，对此做小参数展开，运用波动力学边界条件处理方法和微分方程处理技巧，研究地形变化条件下的浅水方程。通过数学推导，由浅水方程导得含地形变化效应的涡度方程，并得到涡度振幅演变满足的KdV方程，对于分层流体则是mKdV方程。我们将引进新的求解KdV方程的方法并加以改进，使之更适合于含地形附加项的KdV方程。研究重点将放在mKdV方程上，该方程至今少有研究。通过求KdV方程的解析解和数值计算，揭示地形变化对大气环流以及阻高、冷涡的影响。研究成果有助于深化了解大气动力过程，可为现有气候模式的陆面过程提供动力学依据。

项目从2015年元月开展以来，在项目组各位成员的共同努力下，执行项目计划，取得了一定的科研成果。主要就浅水方程与涡度方程对地形高度变化的响应，地形效应对Rossby 波传播影响的物理机制，大气突变检测方法，微分方程解的适定性，以及动力系统强迫项的调制做了一定研究。得到含有地形缓慢变化项的浅水方程和涡度方程，据此分析了流体相对高度、固有地形项、随时间变化地形项三者的定量化关系，该结论推广了流体相对位置函数守恒的结果，定量化了它们之间的变化关系，具体为：在地形高度变化下，相对位置函数不再守恒，相对位置函数就要调整，以适应地形的变化；通过对位涡方程研究发现：δ效应对Rossby 波传播起到稳定的作用，特别在中高纬度其作用尤为明显；给出了一种全新方法来对场时间序列突变做出检测。.对于分层流体中浅水方程，由于问题的复杂性和非线性所致，研究进展不顺利。.发表各类学术论文十篇，其中SCI检索论文七篇，其他论文三篇，编辑在审论文一篇，均为第一资助。