球面效应和地转水平分量对非线性水波的影响研究

In the research on large-scale water waves, one usaully adopts so-called "traditional approximation",i.e.the horizontal component of the Earth's rotation is not taken into account. In recent years, it is shown that the influence of the horizontal component of the Earth's rotation on internal waves, especially on near-inertial waves, can not be neglected. In non-traditional case, many phenomena are found by internal wave theory which never appear in traditional case. However, the research work on water waves with non-traditional hypothesis has almost been confined to linear range so far. In the present project, we start from the primary equations which control geophysical fluid, and focus on the influence of spherical surface effect, rotation effect(including the horizontal component of the Earth's rotation), topography effect on nonlinear water waves.By taking use of nonlinear mathods such as asymptotic analysis to establish a more perfect mathematical model for potetial vorticity, and further to reveal the mechanism of the influence of the every dynamic element mentioned above on nonlinear water waves. The potential research findings would simulate the evolutionary process of real fluid better, and also enrich the theory system of geophysical fluid dynamics greatly.

在大尺度水波研究中，通常采用所谓"传统近似"，即未考虑地转向量水平分量对水波的影响。近年来的研究发现，地转向量水平分量对内波、尤其是对近惯性内波及大尺度波动的影响不容忽视。在非传统近似下，即引入地转向量水平分量，通过对内波理论研究已经发现了很多在传统近似下无法发现的现象。目前，在非传统近似下的研究大多仅局限在线性理论范畴。本项目从控制地球流体的原始非线性方程组出发，同时考虑球面效应、旋转效应（包括地转水平分量）、地形效应对非线性波动的影响。利用渐近分析等非线性手段，建立更加完善的位涡方程模型，并通过对模型的进一步研究，揭示地转向量水平分量对非线性波动的影响机理。其成果可以更精确地模拟真实流体的演化过程，极大地丰富了地球流体力学的理论体系。

非线性波动是自然界普遍存在的物理现象。地球物理流体-大气及海洋的运动就是典型的非线性波动现象，研究大气及海洋中的非线性波动动力学问题对于理解及预测天气及气候变化有着指导性作用，是前沿性学科问题。本项目开展了关于大气中大尺度运动的波动力学物理机制研究，通过大气运动的基本方程组或准地转位涡守恒方程，试图探究球面效应及地转水平分量在非线性Rossby孤立波的生成、发展及衰退过程中的作用，同时也将考虑地形、热外源及耗散等物理因子的影响。推导得到了描述非线性Rossby波振幅演变所满足的一系列偏微分方程，如KdV方程、mKdV方程、Schrodinger方程、Zakharov-Kuznetsov等，从定性角度及定量角度揭示多物理因素作用下的非线性Rossby波波动的物理机制，对于流体力学的学科发展具有重要的学术价值。