物理守恒律保真格式构造与数值预报斜压原始方程传统谱模式改进研究

THE FORMULATION OF FIDELITY SCHEMES OF PHYSICAL CONSERVATION LAWS AND IMPROVEMENTS ON A TRADITIONAL SCHEME OF BAROCLINIC PRIMITIVE EQUATIONS FOR NUMERICAL WEATHER PREDICTION

文中构造并证明了一般二次和三次物理守恒律时间差分保真格式两个构造定理，以往一些主要时间离散守恒格式构造方案可作为两个定理特例给出。它们不仅可为解决更加广泛类别的时间离散保真格式构造基本问题提供适用数学基础，而且也为结合已有瞬时空间离散守恒格式，解决更加广泛类别的时－空离散意义下保真格式构造基本问题提供适用的数学基础。此外，文中两个定理还可解决两大类问题的线性和非线性计算不稳定性问题。斜压原始方程传统半隐式全球谱－垂直有限差分模式目前是世界上许多国家的业务预报和大气环流模式。本工作利用文中新构定理，构造并且实现了斜压原始方程全球谱－垂直有限差分模式半隐式高阶全能量守恒方案。以往该项基本问题无论在理论还是实践上长期以来一直都未能得到解决。该项全能量守恒半隐式全球谱模式方案适用于实测资料的长时间数值预报积分。使用ＦＧＧＥ夏季资料进行的１３个个例３０ｄ数值积分实验表明：新型全能量半隐式保真方案可以有效地改进传统预报方案中关于能量质量守恒性质的系统性偏差。值得注意的是，实验统计分析还显示：在本文实验条件下，传统方案中由于时间离散过程中原物理守恒律性质破坏导致的系统误差（简称Ｚ类误差），对于实验总体均方根系统误差的贡献?

In this paper,two formulation theorems of time difference fidelity schemes for general quadratic and cubic physical conservation laws are respectively constructed and proved, with earlier major conserving time discretized schemes given as special cases. These two theorems can provide new mathematical basis for solving basic formulation problems of more types of conservative time discrete fidelity schemes, and even for formulating conservative temporal spatial discrete fidelity schemes by combining existing instantly conserving space discretized schenes. Besides, the two theorems can also solve two large categories of problems about linear and nonlinear computational instability. The traditional global spectral vertical finite difference semi implcit model for baroclinic primitive equations is currently used in many countries in the world for operational weather forecast and numerical simulation of general circulation. The present work, however, based on Theorem 2 formulated in this paper, develops and realizes a high order total energy conserving semi implicit time difference fidelity scheme for global spectral vertical finite difference model of baroclinic primitive equations. Prior to this, such a basic formulatin problem remain unsolved for long, whether in terms of theory or practice. The total energy conserving semi implicit scheme formulated here is applicable to real data long term numerical integrations. The experiment of thirteen FGGE data 30 day numerical integrations indicates that,the new type of total energy conserving semi implicit fidelity scheme can surely modify the systematicalalal deviation of energy and mass conserving of the traditional scheme. It should be particularly noted that, under the experimental conditions of the present work, the systematic errors induced by the violation of physical laws of conservation in the time discretized process regarding the traditional scheme designs (called type Z errors for short) can contribute up to one third of the total systematic RMS error at the end of second week of the integration and exceeds one half of the total amount four weeks afterwards. In contrast, by realizing a total energy conserving semi implicit fidelity scheme and thereby eliminating corresponding type Z errors, roughly an average of one fourth of the RMS errors in the traditional forecast cases can be reduced at the end of second week of the integrations, and averagely more than one third reduced at integral time of four weeks afterwards. In addition, experiment results also reveal that, in a sense, the effects of type Z errors are no less great than that of the real topographic forcing of the model. The prospects of the new type of total energy conserving fidelity schemes are very encouraging.