多级扰动法——静力扣除及其在非静力模式设计中的可应用性

A Multilevel Perturbation Method-Hydrostatic Deduction and Its Applicability to Designing the Nonhydrostatic Model

在尺度分析的基础上,利用大气垂直运动方程中各项量级的分布特点,提出了多级扰动法。通过它可以使该方程中量级最大的垂直气压梯度力项(VPGF)和重力项(G)的相应扰动项的量级随级次n的增加明显减小,而其他项的量级不变;而且,方程中最大的垂直截断误差的量级也随n的增加而减小,直到高级扰动项不再是扰动方程中仅有的最大项时为止。另外,因截断误差的量级和被离散项的量级成正比,在方程中VPGF的截断误差就是该方程中最大的垂直截断误差,可以表示该方程的总的垂直截断误差。故垂直气压梯度力扰动项的量级的减小还可以使该方程中最大的垂直截断误差减小,用多级扰动法可以使最大垂直截断误差控制在可容许的范围内,使垂直气压梯度力扰动项和重力扰动项计算准确,可以较好地描写对流活动。这样,最大垂直截断误差对倾向的影响就不致歪曲或掩盖垂直柯氏力项和曲率项对倾向的贡献。还可以证明:扰动平衡方程的解是静力扰动。多级扰动法实质上是多次利用扰动平衡方程的解,从低级扰动中扣除其所含和解大小相同的静力扰动部分,使高一级扰动项的量级减小,达到其与其他项量级接近,以减小方程中的最大量级差的一种方法。而且,n的变化不影响方程的性质。可以说,多级扰动法使大气基本态垂直廓线有较大的改善,使其与大气实际垂直廓线很接近,通过静力平衡,使基本模式大气不包含垂直声波。

Based upon the results from the scale analysis of the vertical equation of motion for the atmosphere and forecast experience, especially numerical weather forecast experience, the multilevel perturbation method has been proposed. It can be proved that in the vertical perturbation equation the magnitude of the perturbation decreases with n, n=1, 2, ... But, on the other hand, since the magnitude of truncation error is proportional to that of the discretized term, in the vertical equation of motion or the perturbation equation the truncation error of the discretized vertical pressure gradient force （VPGF） is the largest vertical truncation error, which can represent the total vertical truncation error in the equation. Hence the decrease in the magnitude of Vt＇GF with n is favorable for reduction of the total vertical truncation error and use of the method can limit such an error within a permissible range, to make computation of VPGF more correct and describe convective activity better than ever. Therefore the impact of the lar- gest truncation error on the tendency of w would be unable to distort or cover up the contribution of vertical Coriolis force term and that of curvature term to the tendency. Further, it can be proved that the solution of level n perturbation balance equation is a hydrostatic perturbation. The method is, substantially, a tool to be used to deduct the hydrostatic part from level n perturbation, in order to reduce the level n^l perturbation in magnitude. In this way a high level perturbation VPGF may easily come closer to the other terms without perturbation in magnitude. Furthermore, because the sum of VPGF and gravity does not vary with n, the physical properties of the vertical equation of motion would not change. Therefore, the method may be named the multilevel perturbation hydrostatic deduction method. It can diminish the difference between the improved basic state vertical profile and the actual atmospheric vertical profile quite small, and under hydrostatic equilibrium in the vast majority of the model atmosphere there would he almost no vertical sound waves.