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有限区域风速场求解流函数和速度势场的有效方案
An Effective Method to Solve the Streamfunction and Velocity Potential from a Wind Field in a Limited Area
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摘要: 流函数和速度势是表示风场的一种变量, 在数值天气预报模式和分析、同化方案中经常使用, 通常可以用风速分量场求解Poisson方程得到。对于有限区域系统, 往往采用差分方法, 但由于存在边界问题, 用计算所得到的流函数和速度势场重建风速场, 在边界附近经常出现明显的偏差。基于差分方法、利用有限区域风速场求解流函数和速度势场的基本方法和特点的分析, 在Arakawa A网格分布的有限区域, 设计了一种用差分方法求解流函数和速度势场的有效方案。在该有效方案中, 通过将有限区域向外扩展二圈, 风速场线性外推, 改进计算边界风速值和边界定解条件的效果; 尽可能使用协调、一致的差分格式, 提高求解精度; 最后利用一种增量订正迭代方法, 迭代2~3次就可以获得令人满意的结果。实例试验的对比、检验显示, 用该方案计算求得的流函数和速度势场重建风速场, 具有非常高的精度。Abstract: The streamfunction and the velocity potential are variables commonly used in the scheme design for atmospheric data assimilation and initial field analysis of numeric weather prediction.They can be obtained in partitioning a wind field by solving the Poisson equations of the vorticity and the divergence of the horizontal components of the wind field. Difference method is usually adopted in a limited area; nevertheless, an obvious departure exits between the original wind field and the reconstructed one from the sum of the streamfunction and velocity potential components in the vicinity of the boundary of the limited area.An effective scheme is designed to solve by difference method the streamfunction and velocity potential of a wind field in an Arakawa-A grid limited area, based on analyzing detailedly the approach and characteristics in the process of the solution. The key techniques in the scheme include the following. Firstly, the solution domain is expanded by two rings by extrapolating linearly the wind field to improve a calculation of boundary values. Secondly, consistent difference schemes are introduced in the solution procedure to enhance the solution precision. And finally only two to three iterations are imposed on incremental corrections to get a satisfactorily accurate result. Experiments are carried out with real wind data and their results indicate that the streamfunction and velocity potential of a wind field can be acquired by the scheme and the reconstructed wind field is reproduced with a very high precision.
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图 1 2003年7月20日12:00的水平风速场 (单位: m·s-1)
(a)1000 hPa u, (b)1000 hPa v, (c)500 hPa u, (d)500 hPa v, (e)150 hPa u, (f)150 hPa v
Fig. 1 Components of wind speed at 12:00 on July 20, 2003 (unit:m·s-1)
(a) u at 1000 hPa, (b) v at 1000 hPa, (c) u at 500 hPa, (d) v at 500 hPa, (e) u at 150 hPa, (f)v at 150 hPa
图 2 2003年7月20日12:00的流场 (单位:m·s-1)
(a)1000 hPa, (b)500 hPa, (c)150hPa
Fig. 2 Wind fields at 12:00 on July 20, 2003(unit:m·s-1)
(a)1000 hPa, (b)500 hPa, (c)150hPa
图 3 2003年7月20日12:00的流函数和速度势场 (单位: 106m2·s-1)
(a)1000 hPa Ψ, (b)1000 hPa χ, (c)500 hPa Ψ, (d)500 hPa χ, (e)150 hPa Ψ, (f)150 hPa χ
Fig. 3 Streamfunctions (Ψ) and velocity potentials (χ) at 12:00 on July 20, 2003 (unit:106m2·s-1)
Ψ at 1000 hPa, (b)χ at 1000 hPa, (c)Ψ at 500 hPa, (d)χ at 500 hPa, (e) Ψ at 150 hPa, (f)χ at 150 hPa
表 1 计算Poisson方程时采用3种不同差分格式方案迭代前得到的500 hPa风速绝对误差比较 (单位:m·s-1)
Table 1 The comparison of absolute errors between wind speeds and their reconstructed ones from three difference schemes in solving the Poisson equations without any iteration correction at 500 hPa (unit:m·s-1)
表 2 计算Poisson方程时采用3种不同差分格式方案迭代3次后得到的500 hPa风速绝对误差比较 (单位:m·s-1)
Table 2 Same as in Table 1, but with three-iteration corrections
表 3 4种不同计算方案的500 hPa风速绝对误差比较 (单位:m·s-1)
Table 3 Same as in Table 1, but their reconstructed ones from four difference schemes in expanding the solution domain and adding iteration corrections (unit:m·s-1)
表 4 不扩展区域方案迭代10次计算得到的风速绝对误差 (单位:m·s-1)
Table 4 Same as in Table 3, but for ten-iteration corrections without expanding the solution domain (unit:m·s-1)
表 5 扩展区域方案迭代10次计算得到的风速绝对误差 (单位:m·s-1)
Table 5 Same as in Table 4, but for ten-iteration corrections with expanding the solution domain (unit:m·s-1)
表 6 2006年7月10—19日12:00 10 d 500 hPa风速绝对误差及统计结果 (单位:m·s-1)
Table 6 Same as in Table 3, but for ten days from 10 to 19 July 2006 and their average, according to the scheme of expanding the solution domain and adding three-iteration corrections (unit:m·s-1)
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