基于三次插值函数算法的时间积分方案与二阶时空余差数值模式——以原始大气运动方程与理想全球模拟个例为例  

INTRODUCTION TO AN INTEGRATION SCHEME OF FITTING CUBIC INTERPOLATION FUNCTIONS AND 2–ORDER TIME AND SPACE DIFFERENTIAL REMAINDER NUMERICAL MODEL ——An Ideal Global Simulation Case With Primitive Atmospheric Equations

作　　者：辜旭赞[1] 张兵[1] 王明欢[1] 

机构地区：[1]中国气象局武汉暴雨研究所,湖北武汉430074

出　　处：《热带气象学报》2011年第5期669-678,共10页Journal of Tropical Meteorology

基　　金：国家自然科学基金面上项目(40575053)--双三次数值模式研究资助

摘　　要：从大气运动原始方程和欧拉算符出发,用泰勒级数展开,给出二阶时空微商余项预报方程。进而讨论用三次插值函数——双三次曲面拟合求上游点的准拉格朗日时间积分方案与相应的二阶时空余差数值模式——"双三次模式"。则双三次模式是通过实现各个大气物理量场的二阶可导,从而可对预报方程做空间非线性("三次")时间离散积分,成为"双三次曲面拟合——时间步积分——双三次曲面拟合——……"一种新算法数值模式。讨论双三次数值模式的数学基础:三次插值函数及其数值分析极性定律用于数值模式。指出:双三次模式和谱模式都具有数学"收敛性";而Coons双三次曲面具有对变量场拟合二阶可导"最优性";和Hermite双三次曲面片具有对网格变量场二阶可导运算"等价性"。又指出:有限差分模式的中央差近似斜率和曲率,分别是三次样条斜率和曲率作"三点平滑"。双三次模式适合采用原始大气运动方程,适合采用准拉格朗日时间积分方案,并给出一个理想全球模拟个例。因大气运动本质上是非线性的,理论上可按变量场双三次曲面曲率判断,以采用符合物理诠释的局域或单点平滑,以保持模式时间积分稳定性。且未来容易实现全球多重/时变套网格双三次数值模式。In this paper,from the Navier-Stokes primitive equations and Eulerian operator,forecasting equations are deduced with 2–order time and space differential remainder by Taylor series expansion,and incorporated to the Bicubic Numerical Model（BiNM for short）,which is with a quasi-Lagrangian integration scheme of fitting cubic spline/bicubic surface to all physical variables in atmospheric equation sets on spherical discrete meshes.Their first-order and second-order derivatives as well as their upstream points were determined,and discrete time integration was performed in cubic space for the governing equations,i.e.with a new algorithm of ＂fitting bicubic surface-time step integration-fitting bicubic surface-……＂.Then,BiNM＇s mathematical foundation of numerical analysis was discussed for the cubic spline and its mathematical polar characters.It was pointed out that,as a spectrum model,BiNM shows mathematical ＂convergence＂ of the cubic spline and the bicubic surface contracting to the original function as well as its first–order and second-order derivatives,with the ＂optimality＂ of the second-order derivative of the cubic spline being optimal approximation to that of the original function.It was indicated that Hermite bicubic patches are equivalent in performing operation to the secondary derivative ＂mesh＂ variables.It was identified that the slope and curvature of the centred difference are respectively three-point smooth of that of the cubic spline.Using a global BiNM with latitude-longitude grids,and keeping the non-static and total field compressible,adiabatic and non-frictional,and running the so called ＂shallow atmosphere＂ equations in the spherical coordinate,and along with a quasi-Lagrangian time integration scheme,an ideal global simulation case was shown by adopting the re-analysis data of NCEP for getting an initial model atmosphere.Lastly,we had to say that,because atmospheric motion can be essentially non-linear,future BiNM could smooth reasonable local area or simple point

关 键 词：数值预报与模拟 二阶时空余差数值模式 三次插值函数 Navier-Stokes大气运动方程组 准拉格朗日时间积分 欧拉向前差分时间积分 全球模拟个例 

分 类 号：P435[天文地球—大气科学及气象学]