用有限区域风速场准确求解流函数和速度势场的方法

An Accurate Solution Method of Stream Function and Velocity Potential from the Wind Field in a Limited Area

流函数和速度势是气象业务和研究中常用于表述风速的一组变量。用有限区域风速场,使用有限差分方法求解得到的流函数和速度势场重建初始风速场,由于受区域边界的限制往往有明显的偏差。虽然有许多求解方法的研究,但是,至今仍尚未见到一种真正准确的求解计算方案。首先,介绍用Arakawa A网格和D网格分布的有限区域风速场求解流函数和速度势场的一般有限差分计算方法,探讨用它们的解重建风速场产生误差的原因。然后,针对这些原因,对给定的有限区域,通过线性外推初始风速场,扩展求解计算区域,使用协调、一致的有限差分格式方案,准确计算求解区域的边界有旋风速、散度风速和速度势的定解边界条件,以及恰当选择流函数、速度势、涡度和散度等变量的分布网格,设计了用上述两种网格分布的风速场准确求解流函数、速度势场的方案,并对其正确性加以证明,它们可以推广应用于其他Arakawa网格。用实际资料试验同样显示,方案避免了重建风速场误差的出现,与初始风速场相比,全场风速最大偏差精度达到10-12m/s或以上,在计算机精度造成的计算误差影响范围内。本文的研究很好解决了长期以来用有限区域风速场、使用有限差分方法无法准确求解流函数和速度势场的问题。

The wind field is often represented by stream function and velocity potential in the meteorological commu- nity. For a limited area, the finite-difference approach is used to calculate the stream function and velocity potential from the wind field. But there are often large departures between the original and reconstructed wind fields caused by the unsuitable boundary conditions. The basic finite-difference method for Arakawa A- and D-grids is introduced and the comprehensive discussion about the problems in the previous methods is given. Then a new method is presented, whose unique features in- clude （a） the linear extrapolation of the original wind field and the extension to the computation domain, （b） the consistent finite-difference scheme, （c） the accurate calculations of non-divergent wind, irrotational wind and solu- tion-determining condition of velocity potential at the boundary, and （d） the suitable grid distribution of variables. The correctness of the new method is proved mathematically. The new method can be also used for all other Araka- wa grids. The results of experiments using the real data show the maximum magnitude of departures between the o- riginal and reconstructed wind fields is only 10-12 m/s, which is produced by the round-off error of the computer. The longstanding problem on accurate solution of the stream function and velocity potential from the wind field in a limited area by using the finite-difference approach, is perfectly solved in the present study.