近70年有限区域流函数速度势算法研究的回顾和新进展

Studies of the Approaches for Computing Stream Function and Velocity Potential in a Limited Domain in the Past 70 Years and Their Recent Developments

流函数和速度势能很好反映流体的涡度和散度特征,一直广泛应用于全球和区域大气和海洋环流分析、污染物扩散和资料同化等研究领域。近年发现,有限区域流函数速度势常用算法计算中小尺度系统复杂流场和复杂下垫面驱动的边界层流场时,精度显著下降。本文全面回顾上世纪五十年代以来的五类常用算法,从数学原理和物理意义两方面简述优缺点,总结其适用范围;指出常用的调和—余弦法在可解性条件方面的科学问题,并设计订正方案,以提高其在求解复杂流场问题中的适用性和计算精度;通过理想函数和实际天气过程复杂流场的多组数值试验,直观定量显示并归纳总结适于不同分辨率资料的算法。本文旨在为流函数速度势及其相关变量在极端天气气候事件机理分析和数值预报等领域的有效应用,提供科学依据。

Stream function and velocity potential can represent the vorticity and divergence of flow fields,respectively,and have been widely used in studies of global and regional atmospheric and oceanic circulations,pollutant diffusion,and data assimilation for a long time.In recent years,accuracy has decreased sharply when the commonly used algorithms are applied to complex flows driven by mesoscale and storm-scale weather systems,particularly in boundary layers over complex terrains and/or heterogeneous underlying surfaces.This paper presents a comprehensive review of the algorithms developed since the 1950s in five categories,emphasizing their strengths and weakness from the perspective of mathematical principles and physical meanings and summarizing their scopes of applications.The previously developed harmonic cosine series expansion spectral approach is revisited and corrected for its ill-positioned solvability conditions to improve its suitability and accuracy in solving complex flow fields.Numerical experiments based on both idealized and real flow fields are performed to illustrate/highlight and summarize the applicability and accuracy of the algorithms in each category for different types of datasets with different spatial resolutions.The objective of this paper is to provide a solid scientific basis for the correct and efficient applications of stream function,velocity potential,and their derived variables in the diagnostic analysis and numerical prediction of extreme weather and climate events.