Studies of the Approaches for Computing Stream Function and Velocity Potential in a Limited Domain in the Past 70 Years and Their Recent Developments
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摘要: 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。Abstract: 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.
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图 1
$\bigtriangledown $ χ、vχ和χ气象学梯度方向示意图Figure 1. Schematic diagram of the directions of
$\bigtriangledown $ χ, vχ and the meteorological gradient of χ
图 2 （a）2018年11月30日22:00（北京时，下同）新疆伊犁河谷地区距地面约0.5 km高度处的风场分布（灰框显示D1、D2、D3区域范围，阴影代表风速值，单位：m s−1）以及（b）地形高度（单位：m）；（c）2020年6月25日20:00至26日02:00京津冀地区极大风分布（阴影代表蒲福风级）以及（d）地形高度（单位：m）
Figure 2. Distributions of (a) wind fields at approximately 0.5 km over the surface and (b) terrain height (units: m) in Ili Valley in Xinjiang at 2200 BJT (Beijing time) on November 30, 2018; the gray lines denote the boundaries of D1, D2, and D3, and the shaded areas denote velocity (units: m s−1). Distributions of (a) maximum winds and (d) terrain height (units: m) over the Beijing–Tianjin–Hebei Region from 2000 BJT June 25 to 0200 BJT June 26, 2020
图 3 调和—余弦试验（a、b）RealD1、（c、d）RealD2和（e、f）RealD3中的原始风场（黑线）和三种方法重建的vs（左列）和vn（右列）沿边界的分布（单位：m s−1）。绿、蓝和红线分别代表调和—余弦法、调和—正弦余弦混合法和调和-余弦扰动法
Figure 3. Distributions of the original (black lines) and reconstructed vs (left column) and vn (right column) along the boundary of harmonic cosine expansion method experiments (a, b) RealD1, (c, d) RealD2, and (e, f) RealD3. The green, blue, and red lines denote the constructed wind components by the harmonic cosine expansion method, combined sine and cosine corrected method, and perturbation corrected method, respectively
图 4 2020年6月24日20:00至25日19:00的原始风场与SOR法（左列）、混合法（右列）重建风场的SCC在整层的分布：（a，b）Res001；（c，d）Res0125；（e，f）Res05；（g，h）Res25。横坐标代表时刻，粗线代表SCC≥0.98，（c）中阴影填充色代表SCC≤0.8
Figure 4. Distributions of SCC (spatial correlation coefficient) at all vertical levels between the original winds and those reconstructed by the SOR method (left column) and the hybrid method (right column) in (a, b) Res001, (c, d) Res0125, (e, f) Res05, and (g, h) Res25. The x-axis denotes the temporal evolution from 2000 BJT 24 to 1900 BJT 25 June 2020. The thick lines denote the regions with SCC≥0.98, and the gray-shaded circles in (c) denote the regions with SCC≤0.8
表 1 近70年有限区域(ψ, χ)算法分类及基本信息
Table 1. Basic information of the five categories of algorithms for computing (ψ, χ) in limited domains
代表算法 优点 不足 后续应用 直接法 Phillip（1958） 计算快 物理假设不合理、精度低 迭代法 迭代法 Sangster（1960） 计算快 精度低、不收敛 诊断分析
数值模式Endlich（1967） 精度较低、不收敛 SOR法 复杂流场精度低、难以客观选择收敛系数 谱展开法 双傅氏变换法 计算快 只能求解规则区域 有限区域谱模式 调和—正（余）弦法 迭代不收敛、
只能求解规则区域变分法 正则化方法 物理意义清楚、计算精度较高 难以客观选择正则化参数 沿岸流等不规则区域求解 格林函数法 Xu et al.（2011） 物理意义清楚、精度高、迭代收敛 复杂流场精度较低 不规则区域、复杂流场求解 混合法 精度高、适合复杂流场、迭代收敛 无上述缺点 
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表 2 调和—余弦试验中不同方法计算结果在可解性条件公式（6-1）和（6-2）左端项的归一化值和精度指标（加粗表示达到精度指标）
Table 2. Normalized values at both sides of the solvability conditions (Eqs. 6-1 and 6-2) and accuracy indices in the four experiments. Bold fonts denote those values where both SCC (spatial correlation coefficient) and RRD (relative root-mean-square difference) satisfy the accuracy adequacy criterion given at the end of section 2.2
调和—余弦试验组 方法 ∫Svsds ∫Svnds SCC RRD Ana 调和—余弦法 0.868 1.043 0.9905 0.1044 调和—正弦余弦混合法 0.885 0.883 0.9998 0.1045 调和—余弦扰动法 0.868 0.843 0.9998 0.1396 RealD1 调和—余弦法 1.527 1.120 0.7875 0.4806 调和—正弦余弦混合法 1.537 0.955 0.7836 0.4800 调和—余弦扰动法 0.922 0.912 0.9882 0.1389 RealD2 调和—余弦法 0.576 1.149 0.8713 0.2976 调和—正弦余弦混合法 0.471 1.260 0.8661 0.2986 调和—余弦扰动法 0.775 0.918 0.9873 0.1357 RealD3 调和—余弦法 0.725 0.983 0.8715 0.5215 调和—正弦余弦混合法 0.683 0.777 0.8753 0.5483 调和—余弦扰动法 1.029 1.005 0.9863 0.1660
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表 3 分辨率试验组基本信息
Table 3. Details of resolution experiments
分辨率试验 水平分辨率 纬度范围 经度范围 垂直分辨率 垂直范围 Res001 1 km 40.2°～
41°N116.2°～117°E 0.5 km 0.5～10 km Res0125 0.125° 35°～45°N 110°～120°E 50 hPa 1000～50 hPa Res025 0.25° 25°～45°N 100°～120°E Res05 0.5° 15°～55°N 90°～130°E Res10 1.0° 5°S～75°N 70°～150°E Res25 2.5° 75°S～75°N 40°～160°E
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表 4 六组分辨率试验在2020年6月25日18:00 900 hPa（或1 km）高度上的SCC、RRD（加粗表示达到精度指标）和CPU时间
Table 4. Accuracy indices and CPU times in the six resolution experiments at 900 hPa (or 1 km above the surface) at 1800 BJT on June 25, 2020
分辨率试验 方法 SCC RRD CPU时间/s Res001 SOR法 0.6995 0.5140 0.135 混合法 0.9995 0.0190 0.533 Res0125 SOR法 0.7464 0.9794 0.135 混合法 0.9992 0.0379 0.533 Res025 SOR法 0.9238 0.2757 0.135 混合法 0.9991 0.0267 0.533 Res05 SOR法 0.9530 0.1655 0.135 混合法 0.9972 0.0432 0.533 Res10 SOR法 0.9096 0.2847 0.135 混合法 0.9939 0.0639 0.533 Res25 SOR法 0.9904 0.1351 0.113 混合法 0.9930 0.1206 0.395
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